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

Newform orbit 237.4.a.d

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 4q^{2} + 42q^{3} + 80q^{4} + 12q^{5} + 12q^{6} + 47q^{7} + 84q^{8} + 126q^{9} + 97q^{10} + 69q^{11} + 240q^{12} + 119q^{13} + 121q^{14} + 36q^{15} + 548q^{16} + 127q^{17} + 36q^{18} + 178q^{19} + 24q^{20} + 141q^{21} + 363q^{22} + 222q^{23} + 252q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{14} - 16x^{13} - 1251x^{12} + 20120x^{11} + 581867x^{10} - 9667599x^{9} - 122064467x^{8} + 2227308818x^{7} + 10656232273x^{6} - 253739295188x^{5} - 151043773545x^{4} + 13438039702717x^{3} - 23094999063722x^{2} - 262419627089376x + 804673493253504$

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	42	12	47	69	119	127	178	222	−12	829	511	−123	1334	164	1162	1311	1285	1788	228	1486	1106	−2402	−3069	2880

$\operatorname{tr} a_n$ 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)[3]   # 237.4.a.d

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

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