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

Newform orbit 615.2.a.e

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} + 10x^{4} - 2555x^{3} - 37356x^{2} + 485365x - 1124961$

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
$\operatorname{tr} a_p$	1	−5	5	0	9	0	7	−4	−2	29	−1	5	−5	−7	−11	16	16	3	−4	35	−13	−8	−16	22	4

$\operatorname{tr} a_n$ 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)[4]   # 615.2.a.e

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

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