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

Newform orbit 407.2.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 3x^{3} - 2x^{2} - 4x - 1$

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 odd 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	−4	−5	−1	4	−12	−13	−2	−6	1	−8	4	20	−7	−6	8	−6	−22	−5	9	−33	−4	18	−18	−29

$\operatorname{tr} a_n$ 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)[1]   # 407.2.a.b

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

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