Atlas › Modular forms › Weight 4 › Level 93 › 93.4.a › 93.4.a.f

Newform orbit 93.4.a.f

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 3q^{2} + 18q^{3} + 33q^{4} + 12q^{5} + 9q^{6} + 47q^{7} + 6q^{8} + 54q^{9} + 32q^{10} + 9q^{11} + 99q^{12} + 57q^{13} - 135q^{14} + 36q^{15} + 161q^{16} + 111q^{17} + 27q^{18} + 268q^{19} + 72q^{20} + 141q^{21} + 61q^{22} - 48q^{23} + 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} - 15x^{5} - 370x^{4} + 7611x^{3} - 27923x^{2} - 57444x - 17796$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
31	$-1$

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
$\operatorname{tr} a_p$	3	18	12	47	9	57	111	268	−48	0	186	116	−309	408	−369	−72	−597	289	1509	−30	1652	911	1983	786	−1032

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.108 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, 3, 31, 93]: [1, 2, 9, 34] (certified at ell = 2147482951)
T = [(2, 1), (5, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 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(93, 4)[5]   # 93.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 93:4 1000