Atlas › Modular forms › Weight 2 › Level 751 › 751.2.a › 751.2.a.a

Newform orbit 751.2.a.a

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q - 5q^{2} - 8q^{3} + 17q^{4} - 16q^{5} - 13q^{6} - 7q^{7} - 12q^{8} - 10q^{10} - 22q^{11} - 16q^{12} - 8q^{13} - 19q^{14} - 7q^{15} + 3q^{16} - 51q^{17} + 2q^{18} - 9q^{19} - 31q^{20} - 19q^{21} + 4q^{22} - 12q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} + 13x^{23} + 29x^{22} - 279x^{21} - 1294x^{20} + 1580x^{19} + 15952x^{18} + 4933x^{17} - 92905x^{16} - 85703x^{15} + 290200x^{14} + 353949x^{13} - 521684x^{12} - 695482x^{11} + 577084x^{10} + 719985x^{9} - 420999x^{8} - 394300x^{7} + 199978x^{6} + 104890x^{5} - 54060x^{4} - 10041x^{3} + 6323x^{2} - 107x - 131$

Atkin–Lehner signs

$p$	$w_p$
751	$+1$

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
$\operatorname{tr} a_p$	−5	−8	−16	−7	−22	−8	−51	−9	−12	−49	−17	6	−99	−1	−12	−15	−18	−29	17	−29	−36	−33	−21	−78	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 751]: [0, 63] (certified at ell = 2147482501)
T = [(2, 1), (3, 1)]; h of degree 62 = 1 x 62 (phi(ord eps) x dim S_k^new); 6 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(751, 2)[0]   # 751.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 751:2 1000