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

Newform orbit 107.4.a.b

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q + 4q^{2} + 13q^{3} + 80q^{4} + 39q^{5} + 9q^{6} + 68q^{7} + 51q^{8} + 189q^{9} + 42q^{10} + 7q^{11} + 132q^{12} + 313q^{13} - 13q^{14} + 74q^{15} + 484q^{16} + 178q^{17} + 300q^{18} + 187q^{19} + 237q^{20} + 248q^{21} + 253q^{22} + 178q^{23} + 203q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 17x^{15} - 279x^{14} + 5330x^{13} + 30324x^{12} - 673834x^{11} - 1560379x^{10} + 44012893x^{9} + 31646572x^{8} - 1577181469x^{7} + 332845590x^{6} + 29979347727x^{5} - 24358448723x^{4} - 255574842790x^{3} + 283431673162x^{2} + 472313926916x + 9713514088$

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 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
$\operatorname{tr} a_p$	4	13	39	68	7	313	178	187	178	70	−140	913	−420	−173	105	−39	1003	2713	1455	−318	2730	26	1012	914	7342

$\operatorname{tr} a_n$ 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)[1]   # 107.4.a.b

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

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