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

Newform orbit 633.2.a.d

Trace form

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

$8q + 4q^{2} - 8q^{3} + 8q^{4} + 3q^{5} - 4q^{6} + 7q^{7} + 15q^{8} + 8q^{9} + 9q^{11} - 8q^{12} - 7q^{13} + 5q^{14} - 3q^{15} + 8q^{16} + 3q^{17} + 4q^{18} - 2q^{19} + 7q^{20} - 7q^{21} + 10q^{22} + 29q^{23} - 15q^{24} + O(q^{25})$

Coefficient field

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

$x^{8} - 7x^{7} - 10x^{6} + 128x^{5} - 114x^{4} - 393x^{3} + 770x^{2} - 480x + 101$

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
$\operatorname{tr} a_p$	4	−8	3	7	9	−7	3	−2	29	16	4	−3	9	17	21	22	2	−46	24	27	−19	−9	−8	3	−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)[3]   # 633.2.a.d

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

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