Atlas › Modular forms › Weight 10 › Level 14 › 14.10.a › 14.10.a.c

Newform orbit 14.10.a.c

Trace form

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

$2q - 32q^{2} - 14q^{3} + 512q^{4} - 2730q^{5} + 224q^{6} + 4802q^{7} - 8192q^{8} + 75982q^{9} + 43680q^{10} + 44940q^{11} - 3584q^{12} + 100282q^{13} - 76832q^{14} + 503160q^{15} + 131072q^{16} - 870408q^{17} - 1215712q^{18} + 508774q^{19} - 698880q^{20} - 33614q^{21} - 719040q^{22} + 79800q^{23} + 57344q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2305})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} + 8204x + 6168084$

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
$\operatorname{tr} a_p$	−32	−14	−2730	4802	44940	100282	−870408	508774	79800	2006328	2188732	−20723576	19016592	4193716	−74542524	−3239748	−133642362	227801686	332930272	−167985720	−44684276	269642776	−183105762	791657748	−4169480

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.294 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]: [1, 3, 7, 20] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 2 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, 10)[2]   # 14.10.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 14:10 1000