Atlas › Modular forms › Weight 8 › Level 46 › 46.8.a › 46.8.a.a

Newform orbit 46.8.a.a

Trace form

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

$2q - 16q^{2} - 28q^{3} + 128q^{4} - 110q^{5} + 224q^{6} + 74q^{7} - 1024q^{8} + 4348q^{9} + 880q^{10} - 3614q^{11} - 1792q^{12} + 11006q^{13} - 592q^{14} + 5110q^{15} + 8192q^{16} - 61190q^{17} - 34784q^{18} + 12932q^{19} - 7040q^{20} - 166446q^{21} + 28912q^{22} + 24334q^{23} + 14336q^{24} + O(q^{25})$

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

$x^{2} + 138x - 3739$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
23	$-1$

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	−28	−110	74	−3614	11006	−61190	12932	24334	49466	−327956	−74018	−331214	879216	163820	349872	−4542988	−2020716	−3362	−3841752	−2128338	2725172	6895674	−4123764	−13462214

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.285 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, 23, 46]: [1, 3, 15, 44] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 4 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(46, 8)[0]   # 46.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 46:8 1000