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

Newform orbit 46.8.a.c

Trace form

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

$3q + 24q^{2} - 12q^{3} + 192q^{4} - 570q^{5} - 96q^{6} - 1382q^{7} + 1536q^{8} - 3759q^{9} - 4560q^{10} - 11792q^{11} - 768q^{12} - 12q^{13} - 11056q^{14} - 12990q^{15} + 12288q^{16} - 25584q^{17} - 30072q^{18} - 42966q^{19} - 36480q^{20} - 59166q^{21} - 94336q^{22} - 36501q^{23} - 6144q^{24} + O(q^{25})$

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

$x^{3} + 582x^{2} + 53961x - 9376060$

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
$\operatorname{tr} a_p$	24	−12	−570	−1382	−11792	−12	−25584	−42966	−36501	−11196	33724	90570	−452488	8074	160540	1288796	742720	4089196	−7060728	−5044032	−8838444	−3851056	−8398012	−1319178	−1020584

$\operatorname{tr} a_n$ 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)[2]   # 46.8.a.c

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

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