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

Newform orbit 46.8.a.d

Trace form

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

$4q + 32q^{2} + 96q^{3} + 256q^{4} + 180q^{5} + 768q^{6} + 534q^{7} + 2048q^{8} + 9398q^{9} + 1440q^{10} + 5100q^{11} + 6144q^{12} + 570q^{13} + 4272q^{14} + 26702q^{15} + 16384q^{16} - 2470q^{17} + 75184q^{18} + 40026q^{19} + 11520q^{20} + 23746q^{21} + 40800q^{22} + 48668q^{23} + 49152q^{24} + O(q^{25})$

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

$x^{4} - 276x^{3} - 168383x^{2} + 24068794x + 8344577640$

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 even 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
$\operatorname{tr} a_p$	32	96	180	534	5100	570	−2470	40026	48668	64698	213500	−102088	621698	−1039614	−1077364	−3324790	−75636	3632274	−735076	5085744	−13346	11668016	5973712	6024960	−190638

$\operatorname{tr} a_n$ 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)[3]   # 46.8.a.d

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

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