Atlas › Modular forms › Weight 2 › Level 623 › 623.2.a › 623.2.a.e

Newform orbit 623.2.a.e

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 6x^{15} - 39x^{14} + 272x^{13} + 424x^{12} - 4262x^{11} - 494x^{10} + 28255x^{9} - 7783x^{8} - 91464x^{7} + 23586x^{6} + 146128x^{5} - 3955x^{4} - 95881x^{3} - 25789x^{2} + 3954x + 1268$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
89	$+1$

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
$\operatorname{tr} a_p$	0	6	6	16	−3	15	12	21	0	−3	35	31	15	14	−1	−8	−2	34	−9	−12	47	−23	−7	−16	59

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.221 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, 7, 89, 623]: [0, 1, 8, 62] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 45 = 1 x 45 (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(623, 2)[4]   # 623.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 623:2 1000