Newform orbit 470.2.a.i

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$; the sum of their $q$-expansions is

$3q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} + 2q^{6} + q^{7} + 3q^{8} + 3q^{9} + 3q^{10} + 2q^{12} - q^{13} + q^{14} + 2q^{15} + 3q^{16} + 2q^{17} + 3q^{18} + 4q^{19} + 3q^{20} - 13q^{21} + 7q^{23} + 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$, whose minimal polynomial is

$x^{3} - 11x^{2} - 2157x - 20104$

Atkin–Lehner signs

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$. The Fricke sign is $w_N = -1$, so the sign of the functional equation is $(-1)^{k/2} w_N = +1$: each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.44 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

newform_orbits(470, 2)[8]   # 470.2.a.i

cargo run --release -p sagebrush-web --example atlas -- 470:2 1000