Atlas › Modular forms › Weight 4 › Level 238 › 238.4.a › 238.4.a.a

Newform orbit 238.4.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 4q^{2} - 5q^{3} + 8q^{4} + 9q^{5} + 10q^{6} - 14q^{7} - 16q^{8} + 5q^{9} - 18q^{10} - 4q^{11} - 20q^{12} - 28q^{13} + 28q^{14} + 24q^{15} + 32q^{16} + 34q^{17} - 10q^{18} + 12q^{19} + 36q^{20} + 35q^{21} + 8q^{22} + 50q^{23} + 40q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{93})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} - 22x - 251$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
7	$+1$
17	$-1$

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
$\operatorname{tr} a_p$	−4	−5	9	−14	−4	−28	34	12	50	−208	189	−320	−457	−69	−774	−731	296	−339	467	216	821	30	−358	−918	−915

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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, 2, 7, 14, 17, 34, 119, 238]: [1, 2, 3, 8, 6, 16, 38, 112] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 5 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(238, 4)[0]   # 238.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 238:4 1000