Atlas › Modular forms › Weight 2 › Level 413 › 413.2.a › 413.2.a.f

Newform orbit 413.2.a.f

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 7q^{3} + 8q^{4} + 3q^{5} + q^{6} + 9q^{7} - 3q^{8} + 8q^{9} + 5q^{10} + 3q^{11} + 5q^{12} + 13q^{13} - 3q^{15} + 2q^{16} + q^{17} + 4q^{18} + 14q^{19} - 2q^{20} + 7q^{21} - 12q^{22} + 7q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - 7x^{8} - 7x^{7} + 122x^{6} - 60x^{5} - 683x^{4} + 439x^{3} + 1392x^{2} - 521x - 457$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
59	$+1$

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

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	0	7	3	9	3	13	1	14	7	−20	16	4	3	19	9	−9	−9	22	22	−21	10	11	14	−8	26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.191 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.

The engine's checks

dim M^+ at levels [1, 7, 59, 413]: [0, 1, 6, 42] (certified at ell = 2147481901)
T = [(2, 1), (3, 1)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 3 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(413, 2)[5]   # 413.2.a.f

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 413:2 1000