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

Newform orbit 93.4.a.e

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 3q^{2} - 9q^{3} + 5q^{4} + 8q^{5} - 9q^{6} + 19q^{7} + 45q^{8} + 27q^{9} + 67q^{10} + 59q^{11} - 15q^{12} + 25q^{13} + 80q^{14} - 24q^{15} + 41q^{16} + 71q^{17} + 27q^{18} + 52q^{19} + 105q^{20} - 57q^{21} + 9q^{22} + 120q^{23} - 135q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} - 11x^{2} - 112x + 964$

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	−9	8	19	59	25	71	52	120	−70	−93	−198	403	32	675	454	1159	−563	801	1306	282	1025	1393	−1452	148

$\operatorname{tr} a_n$ 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)[4]   # 93.4.a.e

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

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