Atlas › Modular forms › Weight 2 › Level 615 › 615.2.a › 615.2.a.c

Newform orbit 615.2.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7} + 5T_{17} + 7T_{19} + 9T_{23}$ , whose minimal polynomial is

$x^{2} + 18x + 13$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	$+1$
41	$+1$

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	−2	−2	0	−5	0	−9	4	0	3	−1	−5	−2	3	−1	−14	4	−1	−2	−9	−17	−8	6	0	14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.337 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, 3, 5, 15, 41, 123, 205, 615]: [0, 1, 1, 4, 4, 16, 22, 88] (certified at ell = 2147482921)
T = [(2, 1), (7, 3), (17, 5), (19, 7), (23, 9)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 7 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(615, 2)[2]   # 615.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 615:2 1000