Atlas › Modular forms › Weight 2 › Level 617 › 617.2.a › 617.2.a.b

Newform orbit 617.2.a.b

Trace form

The 28 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$28q + 3q^{2} + 11q^{3} + 29q^{4} + 2q^{5} + 5q^{6} + 39q^{7} + 6q^{8} + 27q^{9} + 10q^{10} + 4q^{11} + 13q^{12} + 27q^{13} - 6q^{14} + 6q^{15} + 23q^{16} + 2q^{17} - 2q^{18} + 23q^{19} - 14q^{20} + 8q^{21} + 2q^{22} + 8q^{23} - 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 28. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{28} - 14x^{27} - 5x^{26} + 903x^{25} - 2852x^{24} - 20884x^{23} + 112859x^{22} + 187826x^{21} - 1925650x^{20} + 163328x^{19} + 17954082x^{18} - 16541212x^{17} - 99776640x^{16} + 141683001x^{15} + 345362706x^{14} - 607296711x^{13} - 768869371x^{12} + 1520905549x^{11} + 1144449363x^{10} - 2314979609x^{9} - 1193978579x^{8} + 2115923035x^{7} + 892532452x^{6} - 1076105881x^{5} - 439542455x^{4} + 238563524x^{3} + 104632508x^{2} - 2405034x - 767942$

Atkin–Lehner signs

$p$	$w_p$
617	$-1$

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	11	2	39	4	27	2	23	8	1	56	28	2	33	21	−3	6	29	20	−1	59	52	−5	−11	40

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.209 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, 617]: [0, 52] (certified at ell = 2147481337)
T = [(2, 1), (3, 1)]; h of degree 51 = 1 x 51 (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(617, 2)[1]   # 617.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 617:2 1000