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

Newform orbit 15.12.a.c

Trace form

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

$2q - 13q^{2} - 486q^{3} - 3111q^{4} - 6250q^{5} + 3159q^{6} + 7784q^{7} + 35139q^{8} + 118098q^{9} + 40625q^{10} + 295568q^{11} + 755973q^{12} + 657492q^{13} + 3176796q^{14} + 1518750q^{15} + 2974065q^{16} + 8579948q^{17} - 767637q^{18} + 17627976q^{19} + 9721875q^{20} - 1891512q^{21} + 25972696q^{22} - 29841072q^{23} - 8538777q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{1801})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} - 7771x - 5771617196$

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
$\operatorname{tr} a_p$	−13	−486	−6250	7784	295568	657492	8579948	17627976	−29841072	−201881948	−71057008	705858484	−327655148	3192552120	2053064720	−2304299452	−1478770576	−8264891460	24212177528	−20218888256	25879583268	22324995440	48014508984	79209683076	−37075227452

$\operatorname{tr} a_n$ 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)[2]   # 15.12.a.c

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

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