Atlas › Modular forms › Weight 4 › Level 113 › 113.4.a › 113.4.a.a

Newform orbit 113.4.a.a

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q - 4q^{2} - 18q^{3} + 32q^{4} - 24q^{5} - 55q^{6} - 106q^{7} - 45q^{8} + 70q^{9} - 144q^{10} - 58q^{11} - 196q^{12} - 188q^{13} - 85q^{14} - 130q^{15} + 8q^{16} - 204q^{17} - 227q^{18} - 222q^{19} - 456q^{20} - 94q^{21} - 253q^{22} - 414q^{23} - 468q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} + 22x^{11} + 36x^{10} - 1480x^{9} - 4684x^{8} + 28734x^{7} + 71925x^{6} - 239624x^{5} - 281328x^{4} + 855224x^{3} - 77149x^{2} - 461316x + 146871$

Atkin–Lehner signs

$p$	$w_p$
113	$-1$

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
$\operatorname{tr} a_p$	−4	−18	−24	−106	−58	−188	−204	−222	−414	68	−522	146	−10	−1512	−672	446	504	−114	−1256	78	−1298	−4052	−488	−328	−3992

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 113]: [1, 30] (certified at ell = 2147480161)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 4 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(113, 4)[0]   # 113.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 113:4 1000