Atlas › Modular forms › Weight 2 › Level 753 › 753.2.a › 753.2.a.f

Newform orbit 753.2.a.f

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 6q^{2} + 6q^{3} + 8q^{4} - 4q^{5} - 6q^{6} - 7q^{7} - 15q^{8} + 6q^{9} - 6q^{10} - 9q^{11} + 8q^{12} - 5q^{13} + 11q^{14} - 4q^{15} + 16q^{16} - 30q^{17} - 6q^{18} - 7q^{19} - 3q^{20} - 7q^{21} + 8q^{22} - 9q^{23} - 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} + 10x^{5} + 32x^{4} + 21x^{3} - 74x^{2} - 138x - 67$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
251	$-1$

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
$\operatorname{tr} a_p$	−6	6	−4	−7	−9	−5	−30	−7	−9	−23	9	6	−10	−5	−21	−16	3	−5	8	−11	−8	−5	−22	6	33

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.244 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, 3, 251, 753]: [0, 1, 22, 86] (certified at ell = 2147482501)
T = [(2, 1), (5, 1)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 5 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(753, 2)[5]   # 753.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 753:2 1000