Atlas › Modular forms › Weight 12 › Level 14 › 14.12.a › 14.12.a.d

Newform orbit 14.12.a.d

Trace form

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

$2q + 64q^{2} - 350q^{3} + 2048q^{4} + 3738q^{5} - 11200q^{6} + 33614q^{7} + 65536q^{8} + 412894q^{9} + 119616q^{10} + 953700q^{11} - 358400q^{12} - 225722q^{13} + 1075648q^{14} + 1463664q^{15} + 2097152q^{16} + 5116272q^{17} + 13212608q^{18} + 16531942q^{19} + 3827712q^{20} - 5882450q^{21} + 30518400q^{22} - 14101728q^{23} - 11468800q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{352969})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 3388x - 2777868$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
7	$-1$

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
$\operatorname{tr} a_p$	64	−350	3738	33614	953700	−225722	5116272	16531942	−14101728	−70893504	−5164292	107289472	172190088	878848156	−1299140556	7439756316	2219271558	−9739798838	2779087912	21197540088	−14015230628	−21708444104	10851702750	−125407177980	104692220416

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.341 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, 2, 7, 14]: [2, 4, 9, 24] (certified at ell = 2147483647)
T = [(3, 1), (5, 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(14, 12)[3]   # 14.12.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 14:12 1000