Atlas › Modular forms › Weight 4 › Level 237 › 237.4.a › 237.4.a.a

Newform orbit 237.4.a.a

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 6q^{2} + 18q^{3} + 4q^{4} - 8q^{5} - 18q^{6} - 37q^{7} - 36q^{8} + 54q^{9} - 43q^{10} - 85q^{11} + 12q^{12} - 141q^{13} + 9q^{14} - 24q^{15} - 108q^{16} - 43q^{17} - 54q^{18} - 278q^{19} + 79q^{20} - 111q^{21} + 76q^{22} - 198q^{23} - 108q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} + 14x^{5} - 104x^{4} - 1958x^{3} - 1300x^{2} + 52386x + 129443$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
79	$+1$

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
$\operatorname{tr} a_p$	−6	18	−8	−37	−85	−141	−43	−278	−198	−118	−287	−159	−77	−872	−516	−120	−373	−845	−1896	644	−270	−474	1172	1129	−1348

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.153 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 engine's checks

dim M^+ at levels [1, 3, 79, 237]: [1, 2, 21, 82] (certified at ell = 2147483053)
T = [(2, 1), (5, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 7 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(237, 4)[0]   # 237.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 237:4 1000