Atlas › Modular forms › Weight 2 › Level 626 › 626.2.a › 626.2.a.f

Newform orbit 626.2.a.f

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{12} - 110x^{11} + 2155x^{10} + 137382x^{9} - 5288339x^{8} - 17553804x^{7} + 2525816784x^{6} - 14526186982x^{5} - 313368985412x^{4} + 1904004246270x^{3} + 10848335691769x^{2} - 52149770837876x + 50484818112634$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
313	$+1$

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
$\operatorname{tr} a_p$	12	4	6	5	9	5	6	8	−12	19	7	14	13	−3	−18	1	−11	−2	−8	−21	−11	−9	−18	−6	22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.35 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, 2, 313, 626]: [0, 1, 26, 80] (certified at ell = 2147482273)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 27 = 1 x 27 (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(626, 2)[5]   # 626.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 626:2 1000