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

Newform orbit 415.2.a.d

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{7} + 18x^{6} + 48x^{5} - 469x^{4} - 921x^{3} + 5365x^{2} - 4606x + 964$

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 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
$\operatorname{tr} a_p$	−3	−5	−7	−6	−2	−5	−25	6	−9	−2	1	−13	−15	−5	−12	−31	−14	7	10	9	−15	6	−7	−37	−20

$\operatorname{tr} a_n$ 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)[3]   # 415.2.a.d

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

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