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

Newform orbit 94.6.a.b

Trace form

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

$4q + 16q^{2} - 22q^{3} + 64q^{4} - 68q^{5} - 88q^{6} - 304q^{7} + 256q^{8} + 46q^{9} - 272q^{10} - 1050q^{11} - 352q^{12} - 1118q^{13} - 1216q^{14} - 2634q^{15} + 1024q^{16} - 1490q^{17} + 184q^{18} - 3034q^{19} - 1088q^{20} - 428q^{21} - 4200q^{22} + 264q^{23} - 1408q^{24} + O(q^{25})$

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

$x^{4} + 90x^{3} - 437x^{2} - 56134x - 252655$

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	−22	−68	−304	−1050	−1118	−1490	−3034	264	−10922	−5242	−7682	22506	14102	8836	−20930	39472	8070	19616	−31822	79916	−36510	−24812	−59530	163058

$\operatorname{tr} a_n$ 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)[1]   # 94.6.a.b

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

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