Atlas › Modular forms › Weight 6 › Level 46 › 46.6.a › 46.6.a.b

Newform orbit 46.6.a.b

Trace form

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

$2q + 8q^{2} - 22q^{3} + 32q^{4} - 68q^{5} - 88q^{6} - 312q^{7} + 128q^{8} - 100q^{9} - 272q^{10} - 12q^{11} - 352q^{12} - 1482q^{13} - 1248q^{14} - 452q^{15} + 512q^{16} + 1768q^{17} - 400q^{18} - 1124q^{19} - 1088q^{20} + 4248q^{21} - 48q^{22} + 1058q^{23} - 1408q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 90x - 1847$

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
$\operatorname{tr} a_p$	8	−22	−68	−312	−12	−1482	1768	−1124	1058	−5118	−3110	36	−7234	28816	−19918	−8508	−46944	−61860	−25428	−49974	11610	−11340	−16352	−126600	67744

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.234 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, 11, 32] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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, 6)[1]   # 46.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 46:6 1000