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

Newform orbit 237.4.a.b

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 4q^{2} - 27q^{3} + 40q^{4} + 8q^{5} - 12q^{6} + 5q^{7} + 54q^{8} + 81q^{9} - 31q^{10} + 85q^{11} - 120q^{12} + 15q^{13} + 61q^{14} - 24q^{15} + 212q^{16} + 59q^{17} + 36q^{18} + 66q^{19} + 121q^{20} - 15q^{21} + 284q^{22} + 550q^{23} - 162q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{9} - 12x^{8} - 622x^{7} + 5020x^{6} + 112550x^{5} - 280182x^{4} - 5251333x^{3} - 11969298x^{2} + 3374048x + 21072992$

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 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
$\operatorname{tr} a_p$	4	−27	8	5	85	15	59	66	550	202	−19	123	497	870	544	1140	485	−1855	840	−400	−1830	−711	2894	839	−3756

$\operatorname{tr} a_n$ 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)[1]   # 237.4.a.b

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

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