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

Newform orbit 615.2.a.g

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 2q^{2} + 8q^{3} + 14q^{4} + 8q^{5} + 2q^{6} + 8q^{9} + 2q^{10} - q^{11} + 14q^{12} + 2q^{13} - 3q^{14} + 8q^{15} + 30q^{16} - q^{17} + 2q^{18} + 12q^{19} + 14q^{20} + 2q^{22} + 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7} + 5T_{17} + 7T_{19} + 9T_{23}$ , whose minimal polynomial is

$x^{8} - 135x^{7} + 764x^{6} + 449344x^{5} - 10611252x^{4} - 327133905x^{3} + 8944645390x^{2} + 64301282472x - 1830946225063$

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 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	8	8	0	−1	2	−1	12	6	1	15	−1	8	5	−5	6	−12	29	−4	−7	21	−12	−20	4	−6

$\operatorname{tr} a_n$ 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)[6]   # 615.2.a.g

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

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