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

Newform orbit 15.10.a.d

Trace form

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

$2q + 31q^{2} - 162q^{3} + 541q^{4} + 1250q^{5} - 2511q^{6} + 14112q^{7} + 26133q^{8} + 13122q^{9} + 19375q^{10} - 21512q^{11} - 43821q^{12} + 24284q^{13} + 461664q^{14} - 101250q^{15} + 387265q^{16} - 156956q^{17} + 203391q^{18} - 95896q^{19} + 338125q^{20} - 1143072q^{21} - 2901532q^{22} - 735264q^{23} - 2116773q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{241})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} - 14143x + 22554706$

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
$\operatorname{tr} a_p$	31	−162	1250	14112	−21512	24284	−156956	−95896	−735264	−2678212	10782432	21968332	26060372	−7191160	−31580240	3131116	−35494664	341497340	−288195816	210286064	−232663084	−24755040	−372082152	−427639116	1771658884

$\operatorname{tr} a_n$ 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)[3]   # 15.10.a.d

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

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