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

Newform orbit 94.4.a.b

Trace form

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

$2q + 4q^{2} - 9q^{3} + 8q^{4} - 20q^{5} - 18q^{6} - 19q^{7} + 16q^{8} - 11q^{9} - 40q^{10} - 88q^{11} - 36q^{12} - 34q^{13} - 38q^{14} + 70q^{15} + 32q^{16} + 27q^{17} - 22q^{18} - 20q^{19} - 80q^{20} + 128q^{21} - 176q^{22} - 90q^{23} - 72q^{24} + O(q^{25})$

Coefficient field

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

$x^{2} + 29x + 149$

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	−9	−20	−19	−88	−34	27	−20	−90	48	318	193	78	−584	−94	−743	−947	497	−1088	−1635	348	−963	360	−903	879

$\operatorname{tr} a_n$ 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)[1]   # 94.4.a.b

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

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