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

Newform orbit 635.2.a.g

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + q^{2} + 27q^{4} + 18q^{5} + 5q^{6} + 5q^{7} + 34q^{9} + q^{10} + 7q^{11} - 13q^{12} + 15q^{13} + 2q^{14} + 41q^{16} - 9q^{17} - 12q^{18} + 10q^{19} + 27q^{20} + 17q^{21} - 10q^{22} + 4q^{23} - 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{18} - x^{17} - 80x^{16} + 103x^{15} + 2396x^{14} - 3561x^{13} - 34074x^{12} + 52776x^{11} + 247345x^{10} - 348367x^{9} - 992820x^{8} + 1041545x^{7} + 2305894x^{6} - 1197943x^{5} - 2916555x^{4} - 22994x^{3} + 1411228x^{2} + 511278x + 9533$

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
$\operatorname{tr} a_p$	1	0	18	5	7	15	−9	10	4	10	18	51	25	4	−14	−19	4	29	13	−15	10	24	−40	57	12

$\operatorname{tr} a_n$ 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)[6]   # 635.2.a.g

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

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