Atlas › Modular forms › Weight 6 › Level 93 › 93.6.a › 93.6.a.d

Newform orbit 93.6.a.d

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 9q^{2} + 72q^{3} + 193q^{4} + 58q^{5} + 81q^{6} + 304q^{7} + 684q^{8} + 648q^{9} + 318q^{10} + 834q^{11} + 1737q^{12} + 564q^{13} + 109q^{14} + 522q^{15} + 4745q^{16} - 1760q^{17} + 729q^{18} + 4512q^{19} - 1470q^{20} + 2736q^{21} - 5917q^{22} + 100q^{23} + 6156q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} - 67x^{7} - 20785x^{6} + 1261398x^{5} + 133446413x^{4} - 6744283937x^{3} - 256853159373x^{2} + 7359438520174x + 89950798027792$

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
$\operatorname{tr} a_p$	9	72	58	304	834	564	−1760	4512	100	−1040	−7688	18620	14544	9928	1640	−21432	−62436	−35124	18324	−103908	114344	66702	4854	5620	183410

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 3, 15, 56] (certified at ell = 2147482951)
T = [(2, 1), (5, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 6 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, 6)[3]   # 93.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 93:6 1000