Atlas › Modular forms › Weight 12 › Level 5 › 5.12.a › 5.12.a.b

Newform orbit 5.12.a.b

Trace form

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

$2q - 20q^{2} - 220q^{3} + 6976q^{4} - 6250q^{5} + 60184q^{6} + 57900q^{7} - 246240q^{8} - 20846q^{9} + 62500q^{10} - 618176q^{11} - 1927040q^{12} + 3414260q^{13} + 1334472q^{14} + 687500q^{15} + 6005632q^{16} + 1317940q^{17} - 12548020q^{18} + 5325320q^{19} - 21800000q^{20} + 3836184q^{21} - 89491840q^{22} + 58943940q^{23} + 122180160q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{151})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 240x - 203644$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$

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
$\operatorname{tr} a_p$	−20	−220	−6250	57900	−618176	3414260	1317940	5325320	58943940	94140380	244543464	21003220	−745743316	629950100	−1402061540	1138320580	7317515560	−1516425676	15734290140	32938471544	−29982848860	−3302823120	13299102420	−12674770860	−3080703740

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.303 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, 5]: [2, 7] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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(5, 12)[1]   # 5.12.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 5:12 1000