Atlas › Modular forms › Weight 4 › Level 107 › 107.4.a › 107.4.a.a

Newform orbit 107.4.a.a

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 2q^{2} - 17q^{3} + 20q^{4} - 31q^{5} - 39q^{6} - 44q^{7} - 21q^{8} + 27q^{9} - 138q^{10} - 59q^{11} - 228q^{12} - 259q^{13} - 125q^{14} - 46q^{15} - 44q^{16} - 162q^{17} + 30q^{18} - 79q^{19} - 323q^{20} - 172q^{21} - 363q^{22} - 236q^{23} - 373q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 19x^{9} + 21x^{8} - 1030x^{7} - 2819x^{6} + 13327x^{5} + 40797x^{4} - 22536x^{3} - 90676x^{2} + 5184x + 44752$

Atkin–Lehner signs

$p$	$w_p$
107	$-1$

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
$\operatorname{tr} a_p$	−2	−17	−31	−44	−59	−259	−162	−79	−236	192	−248	−1043	−20	−867	503	−1075	901	−1247	−451	−14	−2450	−1132	34	−790	−1442

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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, 107]: [1, 28] (certified at ell = 2147482621)
T = [(2, 1), (3, 1)]; h of degree 26 = 1 x 26 (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(107, 4)[0]   # 107.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 107:4 1000