Atlas › Modular forms › Weight 12 › Level 15 › 15.12.a › 15.12.a.b

Newform orbit 15.12.a.b

Trace form

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

$2q - 22q^{2} + 486q^{3} - 636q^{4} - 6250q^{5} - 5346q^{6} - 10864q^{7} - 18744q^{8} + 118098q^{9} + 68750q^{10} - 361792q^{11} - 154548q^{12} - 2133732q^{13} - 3986664q^{14} - 1518750q^{15} - 5326320q^{16} - 7804588q^{17} - 1299078q^{18} - 15562224q^{19} + 1987500q^{20} - 2639952q^{21} + 37395424q^{22} - 37450248q^{23} - 4554792q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{1609})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} + 10886x - 2586004376$

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 odd 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
$\operatorname{tr} a_p$	−22	486	−6250	−10864	−361792	−2133732	−7804588	−15562224	−37450248	−70320668	298584872	236000956	−464942588	−242208600	−4375796920	−2189541388	−5480385856	14557903980	−15918388888	1120561024	−24521574348	−79243055560	9245226696	22117321236	−160363673468

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.335 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]: [2, 5, 7, 24] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 5 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, 12)[1]   # 15.12.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 15:12 1000