Atlas › Modular forms › Weight 2 › Level 415 › 415.2.a › 415.2.a.e

Newform orbit 415.2.a.e

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q + 18q^{4} + 11q^{5} - 5q^{6} - 3q^{7} + 3q^{8} + 21q^{9} - q^{11} + 2q^{12} + q^{13} + 12q^{14} + 20q^{16} + 28q^{17} - 26q^{18} - 2q^{19} + 18q^{20} + 3q^{21} - 20q^{22} + q^{23} - 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{11} - 248x^{9} + 335x^{8} + 22121x^{7} - 48198x^{6} - 856784x^{5} + 2210696x^{4} + 13158705x^{3} - 31622405x^{2} - 43975783x - 12032336$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
83	$+1$

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	0	11	−3	−1	1	28	−2	1	21	4	−10	39	−13	8	21	5	14	−42	3	3	−20	−11	37	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.195 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, 5, 83, 415]: [0, 1, 8, 44] (certified at ell = 2147483077)
T = [(2, 1), (3, 3)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 4 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(415, 2)[4]   # 415.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 415:2 1000