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

Newform orbit 633.2.a.c

Trace form

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

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

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

$x^{8} + 5x^{7} - 17x^{6} - 42x^{5} + 94x^{4} + 56x^{3} - 159x^{2} + 65x - 4$

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 odd 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	−8	−2	−11	−8	4	−9	−7	−25	6	−2	4	−1	−18	−40	−41	−15	16	−22	−22	−10	−12	−27	3	39

$\operatorname{tr} a_n$ 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)[2]   # 633.2.a.c

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

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