Atlas › Modular forms › Weight 4 › Level 109 › 109.4.a › 109.4.a.b

Newform orbit 109.4.a.b

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 4q^{2} + 17q^{3} + 72q^{4} + 40q^{5} + 31q^{6} + 24q^{7} + 51q^{8} + 190q^{9} - 4q^{10} + 212q^{11} + 96q^{12} + 15q^{13} + 47q^{14} + 102q^{15} + 392q^{16} + 166q^{17} + 218q^{18} + 238q^{19} + 405q^{20} + 160q^{21} + 247q^{22} + 293q^{23} + 373q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} - 21x^{14} - 204x^{13} + 6214x^{12} + 5691x^{11} - 681564x^{10} + 1186992x^{9} + 34790152x^{8} - 96195800x^{7} - 858948365x^{6} + 2610615488x^{5} + 9759575116x^{4} - 29172345320x^{3} - 35492139456x^{2} + 116926215616x - 60803961088$

Atkin–Lehner signs

$p$	$w_p$
109	$+1$

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	17	40	24	212	15	166	238	293	212	92	−503	−144	−517	1033	139	3212	−92	454	1444	747	1009	4515	4791	679

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.105 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, 109]: [1, 29] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 27 = 1 x 27 (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(109, 4)[1]   # 109.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 109:4 1000