Atlas › Modular forms › Weight 2 › Level 633 › 633.2.a › 633.2.a.e

Newform orbit 633.2.a.e

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 3q^{2} + 14q^{3} + 23q^{4} - q^{5} + 3q^{6} + 13q^{7} + 6q^{8} + 14q^{9} + 8q^{10} - q^{11} + 23q^{12} + 9q^{13} - 3q^{14} - q^{15} + 33q^{16} - 3q^{17} + 3q^{18} + 12q^{19} - 19q^{20} + 13q^{21} - 6q^{22} + 21q^{23} + 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{14} - 2x^{13} - 78x^{12} + 213x^{11} + 2033x^{10} - 6936x^{9} - 19061x^{8} + 84163x^{7} + 30317x^{6} - 318258x^{5} + 71599x^{4} + 351040x^{3} - 34375x^{2} - 39344x - 24$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
211	$+1$

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	14	−1	13	−1	9	−3	12	21	−8	14	25	−19	11	11	3	−9	38	32	13	−4	25	−7	−23	1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.219 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, 3, 211, 633]: [0, 1, 18, 72] (certified at ell = 2147482681)
T = [(2, 1), (5, 1)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 4 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(633, 2)[4]   # 633.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 633:2 1000