Atlas › Modular forms › Weight 4 › Level 94 › 94.4.a › 94.4.a.a

Newform orbit 94.4.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 4q^{2} - q^{3} + 8q^{4} - 10q^{5} + 2q^{6} + 9q^{7} - 16q^{8} - 11q^{9} + 20q^{10} - 58q^{11} - 4q^{12} - 26q^{13} - 18q^{14} - 80q^{15} + 32q^{16} - 117q^{17} + 22q^{18} - 80q^{19} - 40q^{20} - 132q^{21} + 116q^{22} - 200q^{23} + 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{85})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 11x + 9$

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
$\operatorname{tr} a_p$	−4	−1	−10	9	−58	−26	−117	−80	−200	−42	98	357	−112	454	94	−507	53	−443	1098	−235	422	−283	−580	−1983	−9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 13, 38] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (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(94, 4)[0]   # 94.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 94:4 1000