Atlas › Modular forms › Weight 2 › Level 634 › 634.2.a › 634.2.a.a

Newform orbit 634.2.a.a

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 10x^{3} + 34x^{2} + 45x + 19$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
317	$-1$

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
$\operatorname{tr} a_p$	4	−3	−7	−7	−7	−7	−12	1	−7	−10	−2	0	−11	16	−1	4	−13	−18	17	−4	−5	8	19	−23	−15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.217 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, 2, 317, 634]: [0, 1, 27, 81] (certified at ell = 2147479753)
T = [(3, 1), (5, 1)]; h of degree 26 = 1 x 26 (phi(ord eps) x dim S_k^new); 3 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(634, 2)[0]   # 634.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 634:2 1000