Atlas › Modular forms › Weight 4 › Level 46 › 46.4.a › 46.4.a.c

Newform orbit 46.4.a.c

Trace form

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

$2q - 4q^{2} - q^{3} + 8q^{4} + 10q^{5} + 2q^{6} + 6q^{7} - 16q^{8} + 131q^{9} - 20q^{10} + 72q^{11} - 4q^{12} - 111q^{13} - 12q^{14} - 128q^{15} + 32q^{16} + 124q^{17} - 262q^{18} + 22q^{19} + 40q^{20} - 126q^{21} - 144q^{22} - 46q^{23} + 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{41})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} - 29x + 118$

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
$\operatorname{tr} a_p$	−4	−1	10	6	72	−111	124	22	−46	15	67	18	485	−440	215	240	792	456	240	−705	27	594	394	−486	2152

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.102 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, 2, 7, 20] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 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, 4)[2]   # 46.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 46:4 1000