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

Newform orbit 623.2.a.f

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 3q^{2} + 4q^{3} + 23q^{4} + 6q^{5} - 2q^{6} - 17q^{7} + 9q^{8} + 31q^{9} + 4q^{10} + 5q^{11} + 11q^{12} + 7q^{13} - 3q^{14} + 11q^{15} + 39q^{16} + 20q^{17} + 12q^{18} - 11q^{19} + 26q^{20} - 4q^{21} + 31q^{22} + 10q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{17} - 7x^{16} - 43x^{15} + 357x^{14} + 600x^{13} - 6714x^{12} - 3362x^{11} + 61015x^{10} + 10850x^{9} - 295347x^{8} - 50500x^{7} + 756222x^{6} + 205519x^{5} - 908000x^{4} - 314834x^{3} + 369917x^{2} + 88888x - 29932$

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
$\operatorname{tr} a_p$	3	4	6	−17	5	7	20	−11	10	11	−25	31	−5	28	7	22	−18	0	19	8	55	13	−5	17	75

$\operatorname{tr} a_n$ 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)[5]   # 623.2.a.f

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

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