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

Newform orbit 615.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7} + 5T_{17} + 7T_{19} + 9T_{23}$ , whose minimal polynomial is

$x^{6} - 69x^{5} - 11073x^{4} + 1440957x^{3} - 61388631x^{2} + 1110654234x - 7096340803$

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	6	−6	8	0	12	4	6	−2	4	4	16	−6	18	−18	−2	−4	−6	26	6	2	18	−2	−4	26

$\operatorname{tr} a_n$ 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)[5]   # 615.2.a.f

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

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