# Properties

 Label 4026.2.a.s Level 4026 Weight 2 Character orbit 4026.a Self dual Yes Analytic conductor 32.148 Analytic rank 0 Dimension 4 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4026 = 2 \cdot 3 \cdot 11 \cdot 61$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4026.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$32.1477718538$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.26825.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ q^{2}$$ $$- q^{3}$$ $$+ q^{4}$$ $$+ ( 1 - \beta_{3} ) q^{5}$$ $$- q^{6}$$ $$+ ( - \beta_{1} + \beta_{2} ) q^{7}$$ $$+ q^{8}$$ $$+ q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ q^{2}$$ $$- q^{3}$$ $$+ q^{4}$$ $$+ ( 1 - \beta_{3} ) q^{5}$$ $$- q^{6}$$ $$+ ( - \beta_{1} + \beta_{2} ) q^{7}$$ $$+ q^{8}$$ $$+ q^{9}$$ $$+ ( 1 - \beta_{3} ) q^{10}$$ $$- q^{11}$$ $$- q^{12}$$ $$+ ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{13}$$ $$+ ( - \beta_{1} + \beta_{2} ) q^{14}$$ $$+ ( -1 + \beta_{3} ) q^{15}$$ $$+ q^{16}$$ $$+ ( 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{17}$$ $$+ q^{18}$$ $$+ ( 1 - \beta_{1} ) q^{19}$$ $$+ ( 1 - \beta_{3} ) q^{20}$$ $$+ ( \beta_{1} - \beta_{2} ) q^{21}$$ $$- q^{22}$$ $$+ ( 1 - \beta_{1} ) q^{23}$$ $$- q^{24}$$ $$+ ( 4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{25}$$ $$+ ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{26}$$ $$- q^{27}$$ $$+ ( - \beta_{1} + \beta_{2} ) q^{28}$$ $$+ ( \beta_{1} + \beta_{2} ) q^{29}$$ $$+ ( -1 + \beta_{3} ) q^{30}$$ $$+ ( - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{31}$$ $$+ q^{32}$$ $$+ q^{33}$$ $$+ ( 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{34}$$ $$+ ( 2 - 4 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{35}$$ $$+ q^{36}$$ $$+ ( -3 + 3 \beta_{1} + 2 \beta_{2} ) q^{37}$$ $$+ ( 1 - \beta_{1} ) q^{38}$$ $$+ ( -3 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{39}$$ $$+ ( 1 - \beta_{3} ) q^{40}$$ $$+ ( 2 + \beta_{2} + \beta_{3} ) q^{41}$$ $$+ ( \beta_{1} - \beta_{2} ) q^{42}$$ $$+ ( -5 + \beta_{2} ) q^{43}$$ $$- q^{44}$$ $$+ ( 1 - \beta_{3} ) q^{45}$$ $$+ ( 1 - \beta_{1} ) q^{46}$$ $$+ ( 5 - 2 \beta_{2} - \beta_{3} ) q^{47}$$ $$- q^{48}$$ $$+ ( - \beta_{2} - 2 \beta_{3} ) q^{49}$$ $$+ ( 4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{50}$$ $$+ ( -2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{51}$$ $$+ ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{52}$$ $$+ ( 3 - 2 \beta_{1} + \beta_{2} ) q^{53}$$ $$- q^{54}$$ $$+ ( -1 + \beta_{3} ) q^{55}$$ $$+ ( - \beta_{1} + \beta_{2} ) q^{56}$$ $$+ ( -1 + \beta_{1} ) q^{57}$$ $$+ ( \beta_{1} + \beta_{2} ) q^{58}$$ $$+ ( 4 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{59}$$ $$+ ( -1 + \beta_{3} ) q^{60}$$ $$- q^{61}$$ $$+ ( - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{62}$$ $$+ ( - \beta_{1} + \beta_{2} ) q^{63}$$ $$+ q^{64}$$ $$+ ( 11 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{65}$$ $$+ q^{66}$$ $$+ ( -2 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{67}$$ $$+ ( 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{68}$$ $$+ ( -1 + \beta_{1} ) q^{69}$$ $$+ ( 2 - 4 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{70}$$ $$+ ( -2 + 3 \beta_{1} + \beta_{3} ) q^{71}$$ $$+ q^{72}$$ $$+ ( -2 + 3 \beta_{1} - 3 \beta_{2} ) q^{73}$$ $$+ ( -3 + 3 \beta_{1} + 2 \beta_{2} ) q^{74}$$ $$+ ( -4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{75}$$ $$+ ( 1 - \beta_{1} ) q^{76}$$ $$+ ( \beta_{1} - \beta_{2} ) q^{77}$$ $$+ ( -3 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{78}$$ $$+ ( 3 + 3 \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{79}$$ $$+ ( 1 - \beta_{3} ) q^{80}$$ $$+ q^{81}$$ $$+ ( 2 + \beta_{2} + \beta_{3} ) q^{82}$$ $$+ ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{83}$$ $$+ ( \beta_{1} - \beta_{2} ) q^{84}$$ $$+ ( 5 + 4 \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{85}$$ $$+ ( -5 + \beta_{2} ) q^{86}$$ $$+ ( - \beta_{1} - \beta_{2} ) q^{87}$$ $$- q^{88}$$ $$+ ( 3 + 2 \beta_{1} + 3 \beta_{3} ) q^{89}$$ $$+ ( 1 - \beta_{3} ) q^{90}$$ $$+ ( 3 - 6 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{91}$$ $$+ ( 1 - \beta_{1} ) q^{92}$$ $$+ ( \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{93}$$ $$+ ( 5 - 2 \beta_{2} - \beta_{3} ) q^{94}$$ $$+ ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{95}$$ $$- q^{96}$$ $$+ ( -2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{97}$$ $$+ ( - \beta_{2} - 2 \beta_{3} ) q^{98}$$ $$- q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut 4q^{2}$$ $$\mathstrut -\mathstrut 4q^{3}$$ $$\mathstrut +\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 3q^{5}$$ $$\mathstrut -\mathstrut 4q^{6}$$ $$\mathstrut -\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut 4q^{8}$$ $$\mathstrut +\mathstrut 4q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut 4q^{2}$$ $$\mathstrut -\mathstrut 4q^{3}$$ $$\mathstrut +\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 3q^{5}$$ $$\mathstrut -\mathstrut 4q^{6}$$ $$\mathstrut -\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut 4q^{8}$$ $$\mathstrut +\mathstrut 4q^{9}$$ $$\mathstrut +\mathstrut 3q^{10}$$ $$\mathstrut -\mathstrut 4q^{11}$$ $$\mathstrut -\mathstrut 4q^{12}$$ $$\mathstrut +\mathstrut 13q^{13}$$ $$\mathstrut -\mathstrut 2q^{14}$$ $$\mathstrut -\mathstrut 3q^{15}$$ $$\mathstrut +\mathstrut 4q^{16}$$ $$\mathstrut +\mathstrut 3q^{17}$$ $$\mathstrut +\mathstrut 4q^{18}$$ $$\mathstrut +\mathstrut 2q^{19}$$ $$\mathstrut +\mathstrut 3q^{20}$$ $$\mathstrut +\mathstrut 2q^{21}$$ $$\mathstrut -\mathstrut 4q^{22}$$ $$\mathstrut +\mathstrut 2q^{23}$$ $$\mathstrut -\mathstrut 4q^{24}$$ $$\mathstrut +\mathstrut 13q^{25}$$ $$\mathstrut +\mathstrut 13q^{26}$$ $$\mathstrut -\mathstrut 4q^{27}$$ $$\mathstrut -\mathstrut 2q^{28}$$ $$\mathstrut +\mathstrut 2q^{29}$$ $$\mathstrut -\mathstrut 3q^{30}$$ $$\mathstrut -\mathstrut q^{31}$$ $$\mathstrut +\mathstrut 4q^{32}$$ $$\mathstrut +\mathstrut 4q^{33}$$ $$\mathstrut +\mathstrut 3q^{34}$$ $$\mathstrut +\mathstrut q^{35}$$ $$\mathstrut +\mathstrut 4q^{36}$$ $$\mathstrut -\mathstrut 6q^{37}$$ $$\mathstrut +\mathstrut 2q^{38}$$ $$\mathstrut -\mathstrut 13q^{39}$$ $$\mathstrut +\mathstrut 3q^{40}$$ $$\mathstrut +\mathstrut 9q^{41}$$ $$\mathstrut +\mathstrut 2q^{42}$$ $$\mathstrut -\mathstrut 20q^{43}$$ $$\mathstrut -\mathstrut 4q^{44}$$ $$\mathstrut +\mathstrut 3q^{45}$$ $$\mathstrut +\mathstrut 2q^{46}$$ $$\mathstrut +\mathstrut 19q^{47}$$ $$\mathstrut -\mathstrut 4q^{48}$$ $$\mathstrut -\mathstrut 2q^{49}$$ $$\mathstrut +\mathstrut 13q^{50}$$ $$\mathstrut -\mathstrut 3q^{51}$$ $$\mathstrut +\mathstrut 13q^{52}$$ $$\mathstrut +\mathstrut 8q^{53}$$ $$\mathstrut -\mathstrut 4q^{54}$$ $$\mathstrut -\mathstrut 3q^{55}$$ $$\mathstrut -\mathstrut 2q^{56}$$ $$\mathstrut -\mathstrut 2q^{57}$$ $$\mathstrut +\mathstrut 2q^{58}$$ $$\mathstrut +\mathstrut 13q^{59}$$ $$\mathstrut -\mathstrut 3q^{60}$$ $$\mathstrut -\mathstrut 4q^{61}$$ $$\mathstrut -\mathstrut q^{62}$$ $$\mathstrut -\mathstrut 2q^{63}$$ $$\mathstrut +\mathstrut 4q^{64}$$ $$\mathstrut +\mathstrut 36q^{65}$$ $$\mathstrut +\mathstrut 4q^{66}$$ $$\mathstrut -\mathstrut 3q^{67}$$ $$\mathstrut +\mathstrut 3q^{68}$$ $$\mathstrut -\mathstrut 2q^{69}$$ $$\mathstrut +\mathstrut q^{70}$$ $$\mathstrut -\mathstrut q^{71}$$ $$\mathstrut +\mathstrut 4q^{72}$$ $$\mathstrut -\mathstrut 2q^{73}$$ $$\mathstrut -\mathstrut 6q^{74}$$ $$\mathstrut -\mathstrut 13q^{75}$$ $$\mathstrut +\mathstrut 2q^{76}$$ $$\mathstrut +\mathstrut 2q^{77}$$ $$\mathstrut -\mathstrut 13q^{78}$$ $$\mathstrut +\mathstrut 19q^{79}$$ $$\mathstrut +\mathstrut 3q^{80}$$ $$\mathstrut +\mathstrut 4q^{81}$$ $$\mathstrut +\mathstrut 9q^{82}$$ $$\mathstrut +\mathstrut 12q^{83}$$ $$\mathstrut +\mathstrut 2q^{84}$$ $$\mathstrut +\mathstrut 27q^{85}$$ $$\mathstrut -\mathstrut 20q^{86}$$ $$\mathstrut -\mathstrut 2q^{87}$$ $$\mathstrut -\mathstrut 4q^{88}$$ $$\mathstrut +\mathstrut 19q^{89}$$ $$\mathstrut +\mathstrut 3q^{90}$$ $$\mathstrut +\mathstrut q^{91}$$ $$\mathstrut +\mathstrut 2q^{92}$$ $$\mathstrut +\mathstrut q^{93}$$ $$\mathstrut +\mathstrut 19q^{94}$$ $$\mathstrut +\mathstrut 5q^{95}$$ $$\mathstrut -\mathstrut 4q^{96}$$ $$\mathstrut -\mathstrut q^{97}$$ $$\mathstrut -\mathstrut 2q^{98}$$ $$\mathstrut -\mathstrut 4q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut -\mathstrut$$ $$2$$ $$x^{3}\mathstrut -\mathstrut$$ $$5$$ $$x^{2}\mathstrut +\mathstrut$$ $$5$$ $$x\mathstrut +\mathstrut$$ $$5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 3 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.696894 2.92520 −1.68442 1.45612
1.00000 −1.00000 1.00000 −2.78091 −1.00000 −1.12055 1.00000 1.00000 −2.78091
1.2 1.00000 −1.00000 1.00000 −1.14111 −1.00000 −0.293619 1.00000 1.00000 −1.14111
1.3 1.00000 −1.00000 1.00000 3.40047 −1.00000 3.20612 1.00000 1.00000 3.40047
1.4 1.00000 −1.00000 1.00000 3.52154 −1.00000 −3.79195 1.00000 1.00000 3.52154
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$11$$ $$1$$
$$61$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4026))$$:

 $$T_{5}^{4}$$ $$\mathstrut -\mathstrut 3 T_{5}^{3}$$ $$\mathstrut -\mathstrut 12 T_{5}^{2}$$ $$\mathstrut +\mathstrut 25 T_{5}$$ $$\mathstrut +\mathstrut 38$$ $$T_{7}^{4}$$ $$\mathstrut +\mathstrut 2 T_{7}^{3}$$ $$\mathstrut -\mathstrut 11 T_{7}^{2}$$ $$\mathstrut -\mathstrut 17 T_{7}$$ $$\mathstrut -\mathstrut 4$$ $$T_{13}^{4}$$ $$\mathstrut -\mathstrut 13 T_{13}^{3}$$ $$\mathstrut +\mathstrut 34 T_{13}^{2}$$ $$\mathstrut +\mathstrut 133 T_{13}$$ $$\mathstrut -\mathstrut 514$$
{"ft_lang_label":"__label__en","ft_lang_prob":0.57109684,"math_prob":1.00001,"size":1883,"snap":"2019-26-2019-30","text_gpt3_token_len":790,"char_repetition_ratio":0.32410857,"word_repetition_ratio":0.0,"special_character_ratio":0.4944238,"punctuation_ratio":0.040441178,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000099,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-16T18:20:58Z\",\"WARC-Record-ID\":\"<urn:uuid:d4cca003-94bb-4dd6-8a1d-38b733f75364>\",\"Content-Length\":\"93736\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:483c2229-4a53-40e4-ba1d-70b73be311c1>\",\"WARC-Concurrent-To\":\"<urn:uuid:eeb541c8-a0e6-464c-83dc-29991f1fbcb1>\",\"WARC-IP-Address\":\"35.241.19.59\",\"WARC-Target-URI\":\"http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/4026/2/a/s/\",\"WARC-Payload-Digest\":\"sha1:Y6PUEE4BQG35NJCW2NMM6B6LN32XPEK4\",\"WARC-Block-Digest\":\"sha1:JPX6NXAXY7YTYATHXGQBZVLPDR3OFS4E\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195524685.42_warc_CC-MAIN-20190716180842-20190716202842-00288.warc.gz\"}"}