Frequency Spectrum of a sequence
You're missing a step: pulse shaping. The sequence [.7+.7i, ...] is a sequence of pulse amplitudes. As such, it should not be considered to be signal; in consequence, it does not have a spectrum.

Say your sequence of amplitudes is $A=a_0, a_1, a_2,\ldots$. Now, you need to specify your symbol rate (how many pulses per second you'll transmit). Call this rate $R_p$ and let $T_p=1/R_p$ be the pulse interval. Finally, you need to specify a pulse $p(t)$ that has this property: $$\int_{-\infty}^\infty p(t-kT_p)p(t-lT_p)dt=\begin{cases}1,\textrm{ if $k=l$}\\0,\textrm{ if $k\neq l$}\end{cases}$$for $k$ and $l$ integers. Now you're ready to create your actual signal: $$s(t)=\sum_i a_ip(t-iT_p).$$ This signal:


Has a spectrum; specifically, its bandwidth is the same as the bandwidth of the pulse $p(t)$.
Can be up-converted to a carrier frequency.
Can be converted to an analog signal and physically transmitted over a medium.


For completeness, I'll explain how to recover the amplitudes $a_i$ from $s(t)$ (ignoring noise for simplicity). The receiver can estimate $a_k$ by doing this operation: $$\hat{a_k}=\int_{-\infty}^\infty s(t)p(t-kTp).$$ You can easily verifty that this is the case, using the properties of $p(t)$. Such an operation can be implemented with a matched filter.