Atlas › Modular forms › Weight 2 › Level 629 › 629.2.a › 629.2.a.f

Newform orbit 629.2.a.f

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - 3q^{3} + 4q^{4} - 5q^{5} - 7q^{6} - 5q^{7} + 5q^{9} - 13q^{10} - 6q^{11} - 12q^{12} - q^{13} - 6q^{14} - 19q^{15} - 4q^{16} - 8q^{17} + 14q^{18} - 26q^{19} + 8q^{20} + 10q^{21} - 7q^{22} - 14q^{23} - 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{8} + 69x^{7} + 941x^{6} - 16724x^{5} - 368949x^{4} - 190732x^{3} + 19193125x^{2} + 50421239x - 33653744$

Atkin–Lehner signs

$p$	$w_p$
17	$+1$
37	$+1$

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
$\operatorname{tr} a_p$	0	−3	−5	−5	−6	−1	−8	−26	−14	−9	−14	−8	2	−19	24	18	−5	−29	−41	3	20	−41	1	−21	−29

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.245 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, 17, 37, 629]: [0, 2, 3, 58] (certified at ell = 2147482801)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 49 = 1 x 49 (phi(ord eps) x dim S_k^new); 11 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(629, 2)[5]   # 629.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 629:2 1000