Atlas › Modular forms › Weight 10 › Level 15 › 15.10.a › 15.10.a.c

Newform orbit 15.10.a.c

Trace form

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

$2q + 19q^{2} + 162q^{3} + 1521q^{4} - 1250q^{5} + 1539q^{6} - 11872q^{7} + 49647q^{8} + 13122q^{9} - 11875q^{10} + 35488q^{11} + 123201q^{12} + 143676q^{13} - 245196q^{14} - 101250q^{15} + 707265q^{16} + 385156q^{17} + 124659q^{18} - 403296q^{19} - 950625q^{20} - 961632q^{21} - 4278368q^{22} + 223704q^{23} + 4021407q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{4729})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} + 11853x + 31547096$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
5	$+1$

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
$\operatorname{tr} a_p$	19	162	−1250	−11872	35488	143676	385156	−403296	223704	−74572	−5027128	5373628	14211332	27748920	95966440	−64305596	187863136	154080060	33592376	−228270976	−33122316	−932406760	207040152	224518164	387134596

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.298 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 engine's checks

dim M^+ at levels [1, 3, 5, 15]: [1, 4, 5, 20] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 3 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(15, 10)[2]   # 15.10.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 15:10 1000