Atlas › Modular forms › Weight 2 › Level 407 › 407.2.a › 407.2.a.d

Newform orbit 407.2.a.d

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + q^{2} + 8q^{3} + 13q^{4} + 5q^{5} - 10q^{6} + 7q^{7} - 3q^{8} + 20q^{9} + 3q^{10} + 12q^{11} + 17q^{12} + 14q^{13} - 13q^{14} + 4q^{15} + 31q^{16} + 9q^{17} - 17q^{18} - 2q^{19} - 2q^{20} + 7q^{21} + q^{22} + 8q^{23} - q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} - 9x^{11} + 4x^{10} + 164x^{9} - 357x^{8} - 770x^{7} + 2626x^{6} + 120x^{5} - 5037x^{4} + 3217x^{3} + 204x^{2} - 134x - 11$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
37	$+1$

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
$\operatorname{tr} a_p$	1	8	5	7	12	14	9	−2	8	−17	−10	−12	4	9	10	16	10	−2	3	−3	31	0	−14	−22	63

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.188 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, 11, 37, 407]: [0, 2, 3, 40] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 31 = 1 x 31 (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(407, 2)[3]   # 407.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 407:2 1000