Atlas › Modular forms › Weight 2 › Level 941 › 941.2.a › 941.2.a.a

Newform orbit 941.2.a.a

Trace form

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

$28q - 3q^{2} - 10q^{3} + 9q^{4} - 8q^{5} - 9q^{6} - 21q^{7} - 3q^{8} + 12q^{9} - 11q^{10} - 29q^{11} - 13q^{12} - 8q^{13} - 3q^{14} - 16q^{15} - 21q^{16} + 3q^{17} - 8q^{18} - 48q^{19} - 15q^{20} - 7q^{21} - 12q^{22} - 15q^{23} - 16q^{24} + O(q^{25})$

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

$x^{28} + 13x^{27} + 13x^{26} - 479x^{25} - 1599x^{24} + 6974x^{23} + 35583x^{22} - 48635x^{21} - 402471x^{20} + 123423x^{19} + 2777536x^{18} + 515305x^{17} - 12562191x^{16} - 5317000x^{15} + 38327743x^{14} + 19772983x^{13} - 79102266x^{12} - 40500406x^{11} + 108122766x^{10} + 47416378x^{9} - 92936657x^{8} - 29277715x^{7} + 45240156x^{6} + 7227330x^{5} - 9969926x^{4} + 62850x^{3} + 567256x^{2} - 80295x + 3175$

Atkin–Lehner signs

$p$	$w_p$
941	$+1$

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 odd 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	−10	−8	−21	−29	−8	3	−48	−15	−15	−48	−35	−7	−50	−16	14	−35	−16	−77	−44	−20	−44	−2	5	−8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.3 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, 941]: [0, 79] (certified at ell = 2147473241)
T = [(2, 1), (3, 1)]; h of degree 78 = 1 x 78 (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(941, 2)[0]   # 941.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 941:2 1000