Embedding Format (reviewed)

Let $f$ be newform of level $N$, weight $k$ and character $\chi$.

Further, let $a_n$ denote the trace of the Hecke operator $T_n$.

For each embedding $\iota$ of the coefficient field, we may consider the embedded modular form $\iota(f)$. We present these embeddings sorted numerically according to the two values that define their label.

We may present $\iota(f)$ in four formats:

• Complex embeddings: For each embedding $\iota$, we present the real and imaginary values $\iota(a_n)$ with the desired number of significant digits of the stored values. The stored values are the approximations to 53 bits of $\iota(a_n)$ initially computed up to 200 bits of precision.

• Normalized embeddings: the same as above, but now we present $a_n$ analytically normalized, i.e., $\iota(a_n) / n^{(k-1)/2}$

• Satake Parameters where we use $\iota(a_p)$, these are deduced from the angles. See below.

• Satake angles: the arguments of the Satake Parameters $\theta_p$ where $\theta_p \in [-\pi, \pi]$