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

Newform orbit 237.4.a.c

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q - 6q^{2} - 33q^{3} + 44q^{4} - 12q^{5} + 18q^{6} - 23q^{7} - 66q^{8} + 99q^{9} - 11q^{10} - 69q^{11} - 132q^{12} - 37q^{13} - 51q^{14} + 36q^{15} + 196q^{16} - 111q^{17} - 54q^{18} + 66q^{19} - 414q^{20} + 69q^{21} - 221q^{22} - 318q^{23} + 198q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{11} + 18x^{10} - 629x^{9} - 8880x^{8} + 143787x^{7} + 1360797x^{6} - 13001237x^{5} - 89716956x^{4} + 417376709x^{3} + 2418378732x^{2} - 1420065157x - 9393958125$

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	−33	−12	−23	−69	−37	−111	66	−318	−504	105	−835	−369	−1052	−1032	−1182	−2079	−25	−1332	−240	1526	869	−2724	−1827	1416

$\operatorname{tr} a_n$ 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)[2]   # 237.4.a.c

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

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