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

Newform orbit 629.2.a.h

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q - q^{2} + 7q^{3} + 27q^{4} + 4q^{5} + 5q^{6} + 14q^{7} - 9q^{8} + 24q^{9} + 23q^{11} + 8q^{12} + q^{13} - 7q^{14} + 5q^{15} + 51q^{16} - 17q^{17} - 25q^{18} + 21q^{19} + 9q^{20} + 2q^{21} + 6q^{22} - 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{17} - 138x^{16} + 4387x^{15} + 245421x^{14} - 20403912x^{13} + 358822008x^{12} + 9656994280x^{11} - 450185268148x^{10} + 3095115953386x^{9} + 122645099039559x^{8} - 2387004693201682x^{7} - 35831885346591x^{6} + 350738356315365656x^{5} - 2442535238507453881x^{4} - 11816998364687857989x^{3} + 170501139890762548558x^{2} - 260912435279216555791x - 1372742734537900969372$

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 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
$\operatorname{tr} a_p$	−1	7	4	14	23	1	−17	21	0	14	14	17	8	16	−6	13	0	16	45	22	−2	31	−33	−53	−26

$\operatorname{tr} a_n$ 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)[7]   # 629.2.a.h

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

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