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

Newform orbit 14.12.a.c

Trace form

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

$2q - 64q^{2} + 350q^{3} + 2048q^{4} + 266q^{5} - 11200q^{6} - 33614q^{7} - 65536q^{8} + 13294q^{9} - 8512q^{10} + 773500q^{11} + 358400q^{12} + 3275622q^{13} + 1075648q^{14} + 6479648q^{15} + 2097152q^{16} + 9376528q^{17} - 425408q^{18} + 1741642q^{19} + 272384q^{20} - 5882450q^{21} - 24752000q^{22} - 11993072q^{23} - 11468800q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{153169})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 616x - 74038932$

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	266	−33614	773500	3275622	9376528	1741642	−11993072	−23355904	142349508	530129728	−206465112	1036645444	−2847603444	−4876085316	−6723217942	−9724278438	8448611288	−26281683512	61139797628	37890465096	−46563770750	80689325220	−9944168416

$\operatorname{tr} a_n$ 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)[2]   # 14.12.a.c

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

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