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

Newform orbit 94.6.a.a

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 16q^{2} + 6q^{3} + 64q^{4} - 68q^{5} - 24q^{6} + 88q^{7} - 256q^{8} + 46q^{9} + 272q^{10} - 510q^{11} + 96q^{12} + 578q^{13} - 352q^{14} + 98q^{15} + 1024q^{16} - 1226q^{17} - 184q^{18} + 658q^{19} - 1088q^{20} - 2660q^{21} + 2040q^{22} - 4968q^{23} - 384q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 62x^{3} - 2577x^{2} - 102766x + 2468941$

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 odd 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
$\operatorname{tr} a_p$	−16	6	−68	88	−510	578	−1226	658	−4968	−15722	−13926	−10026	−11030	−9294	−8836	8774	1288	29918	43248	−38922	−9828	121078	57588	−101578	−39358

$\operatorname{tr} a_n$ 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)[0]   # 94.6.a.a

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

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