Atlas › Modular forms › Weight 2 › Level 937 › 937.2.a › 937.2.a.b

Newform orbit 937.2.a.b

Trace form

The 43 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$43q + 12q^{2} + 13q^{3} + 50q^{4} + 7q^{5} + 3q^{6} + 12q^{7} + 33q^{8} + 54q^{9} - 6q^{10} + 21q^{11} + 24q^{12} + 3q^{13} + 12q^{14} + 40q^{15} + 56q^{16} + 19q^{17} + 27q^{18} - q^{19} + 10q^{20} + 2q^{21} + 7q^{22} + 102q^{23} + 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 43. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{43} - 25x^{42} + 150x^{41} + 1278x^{40} - 18196x^{39} + 17199x^{38} + 691114x^{37} - 2772488x^{36} - 11435149x^{35} + 89681667x^{34} + 37083607x^{33} - 1513496120x^{32} + 1833736369x^{31} + 15277024166x^{30} - 36735437159x^{29} - 93693086740x^{28} + 360009734864x^{27} + 313484652418x^{26} - 2212190034325x^{25} - 175670925526x^{24} + 9182521923789x^{23} - 3289939373106x^{22} - 26667319243353x^{21} + 16001592822754x^{20} + 55353319214789x^{19} - 38539921424737x^{18} - 83282186047583x^{17} + 54838132979630x^{16} + 91039865218435x^{15} - 45474828321928x^{14} - 70441308225023x^{13} + 18774411658812x^{12} + 35628305533786x^{11} - 1385765304534x^{10} - 10184911772257x^{9} - 1164756999878x^{8} + 1484706340949x^{7} + 304690084083x^{6} - 100416646223x^{5} - 25924749294x^{4} + 2717032179x^{3} + 740607621x^{2} - 17921372x - 601316$

Atkin–Lehner signs

$p$	$w_p$
937	$-1$

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
$\operatorname{tr} a_p$	12	13	7	12	21	3	19	−1	102	23	10	13	12	7	97	76	32	−25	32	100	−32	18	55	7	−33

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.303 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, 937]: [0, 78] (certified at ell = 2147472289)
T = [(2, 1), (3, 1)]; h of degree 77 = 1 x 77 (phi(ord eps) x dim S_k^new); 8 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(937, 2)[1]   # 937.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 937:2 1000