Atlas › Modular forms › Weight 2 › Level 635 › 635.2.a › 635.2.a.f

Newform orbit 635.2.a.f

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - 6x^{8} - 22x^{7} + 141x^{6} + 202x^{5} - 1042x^{4} - 1005x^{3} + 2232x^{2} + 984x - 1481$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
127	$-1$

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
$\operatorname{tr} a_p$	3	3	−9	12	0	9	1	6	3	4	2	31	−29	20	6	19	−2	10	33	10	22	9	19	−33	40

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.22 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, 5, 127, 635]: [0, 1, 11, 66] (certified at ell = 2147483269)
T = [(2, 1), (3, 1)]; h of degree 43 = 1 x 43 (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(635, 2)[5]   # 635.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 635:2 1000