Atlas › Modular forms › Weight 2 › Level 647 › 647.2.a › 647.2.a.c

Newform orbit 647.2.a.c

Trace form

The 38 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$38q + 6q^{2} + 4q^{3} + 46q^{4} + 2q^{5} - 4q^{6} + 18q^{7} + 15q^{8} + 54q^{9} + 17q^{10} + q^{11} + 39q^{13} - 2q^{14} + 2q^{15} + 62q^{16} + 27q^{17} + 14q^{18} + 12q^{19} - 10q^{20} + 7q^{21} + 43q^{22} + 11q^{23} - 22q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 38. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{38} - 10x^{37} - 91x^{36} + 1181x^{35} + 2787x^{34} - 60183x^{33} - 9653x^{32} + 1763836x^{31} - 1633165x^{30} - 33277067x^{29} + 52469613x^{28} + 427658135x^{27} - 860816885x^{26} - 3863183851x^{25} + 8970065066x^{24} + 25001501920x^{23} - 63383764479x^{22} - 117586315312x^{21} + 311317699661x^{20} + 407854106532x^{19} - 1070069477330x^{18} - 1060648301364x^{17} + 2561349545718x^{16} + 2092447503844x^{15} - 4199692805544x^{14} - 3109928506401x^{13} + 4568253385809x^{12} + 3357859294324x^{11} - 3110271085440x^{10} - 2468189785923x^{9} + 1176348942817x^{8} + 1125380062293x^{7} - 169598424907x^{6} - 275695129707x^{5} - 18069844445x^{4} + 26963727625x^{3} + 5623560745x^{2} + 1968270x - 34670547$

Atkin–Lehner signs

$p$	$w_p$
647	$-1$

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
$\operatorname{tr} a_p$	6	4	2	18	1	39	27	12	11	15	10	50	7	65	−16	24	−6	18	52	−24	67	33	−7	−20	88

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.225 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, 647]: [0, 55] (certified at ell = 2147482943)
T = [(2, 1), (3, 1)]; h of degree 54 = 1 x 54 (phi(ord eps) x dim S_k^new); 6 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(647, 2)[2]   # 647.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 647:2 1000