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

Newform orbit 415.2.a.c

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{6} - 11x^{5} - 80x^{4} + 840x^{3} + 1448x^{2} - 13371x - 8779$

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
$\operatorname{tr} a_p$	2	3	−6	0	4	7	21	2	8	10	−13	1	10	−3	14	9	20	−13	16	5	3	2	6	45	22

$\operatorname{tr} a_n$ 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)[2]   # 415.2.a.c

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

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