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

Newform orbit 46.8.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 24q^{2} - 28q^{3} + 192q^{4} + 390q^{5} + 224q^{6} - 470q^{7} - 1536q^{8} - 1519q^{9} - 3120q^{10} - 1872q^{11} - 1792q^{12} - 14732q^{13} + 3760q^{14} + 34642q^{15} + 12288q^{16} + 5344q^{17} + 12152q^{18} + 85786q^{19} + 24960q^{20} + 109554q^{21} + 14976q^{22} - 36501q^{23} + 14336q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{3} - 362x^{2} - 252599x + 76043764$

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
$\operatorname{tr} a_p$	−24	−28	390	−470	−1872	−14732	5344	85786	−36501	391908	−107892	−168438	−137384	−1110278	180748	804268	−30336	931612	−189720	4480752	2533364	−4671024	−922204	16643190	−24176888

$\operatorname{tr} a_n$ 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)[1]   # 46.8.a.b

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

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