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

Newform orbit 937.2.a.a

Trace form

The 34 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$34q - 13q^{2} - 15q^{3} + 27q^{4} - 9q^{5} - q^{6} - 18q^{7} - 36q^{8} + 17q^{9} - 6q^{10} - 19q^{11} - 28q^{12} - 15q^{13} - 12q^{14} - 36q^{15} + 17q^{16} - 27q^{17} - 18q^{18} - 5q^{19} - 14q^{20} - 2q^{21} - 11q^{22} - 88q^{23} + 14q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 34. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{34} + 28x^{33} + 286x^{32} + 880x^{31} - 5870x^{30} - 53945x^{29} - 71723x^{28} + 746313x^{27} + 2949163x^{26} - 2524847x^{25} - 33583966x^{24} - 31518380x^{23} + 188809875x^{22} + 399398464x^{21} - 524119607x^{20} - 2092347719x^{19} + 262928545x^{18} + 6262134671x^{17} + 2817202187x^{16} - 11389460505x^{15} - 9343364459x^{14} + 12513387930x^{13} + 14033301360x^{12} - 7949619850x^{11} - 11403425845x^{10} + 2826125922x^{9} + 5064225359x^{8} - 703770747x^{7} - 1246429464x^{6} + 166212277x^{5} + 159149294x^{4} - 29388472x^{3} - 7483257x^{2} + 2216500x - 140788$

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 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
$\operatorname{tr} a_p$	−13	−15	−9	−18	−19	−15	−27	−5	−88	−27	−10	−9	−6	−25	−93	−82	−40	5	−32	−94	−14	−32	−41	−15	−15

$\operatorname{tr} a_n$ 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)[0]   # 937.2.a.a

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

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