Atlas › Modular forms › Weight 6 › Level 94 › 94.6.a › 94.6.a.c

Newform orbit 94.6.a.c

Trace form

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

$6q - 24q^{2} + 6q^{3} + 96q^{4} + 32q^{5} - 24q^{6} - 108q^{7} - 384q^{8} + 856q^{9} - 128q^{10} + 458q^{11} + 96q^{12} - 850q^{13} + 432q^{14} - 2642q^{15} + 1536q^{16} + 278q^{17} - 3424q^{18} + 590q^{19} + 512q^{20} + 5324q^{21} - 1832q^{22} + 5372q^{23} - 384q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{6} - 38x^{5} - 16903x^{4} + 496934x^{3} + 90318235x^{2} - 1506714180x - 156065911650$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
47	$-1$

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
$\operatorname{tr} a_p$	−24	6	32	−108	458	−850	278	590	5372	13358	13146	−6054	54150	−17318	13254	33546	67064	−2638	−73128	3898	−117900	50014	52324	189070	105510

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.265 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, 2, 47, 94]: [1, 2, 21, 62] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 5 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(94, 6)[2]   # 94.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 94:6 1000