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

Newform orbit 238.4.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 6q^{2} + 7q^{3} + 12q^{4} + 19q^{5} + 14q^{6} + 21q^{7} + 24q^{8} + 54q^{9} + 38q^{10} - 8q^{11} + 28q^{12} + 32q^{13} + 42q^{14} - 28q^{15} + 48q^{16} - 51q^{17} + 108q^{18} + 122q^{19} + 76q^{20} + 49q^{21} - 16q^{22} + 254q^{23} + 56q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{3} - 64x^{2} + 629x + 4726$

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 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
$\operatorname{tr} a_p$	6	7	19	21	−8	32	−51	122	254	138	−5	−334	19	123	162	−213	−538	19	−845	−656	529	−1286	−156	12	297

$\operatorname{tr} a_n$ 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)[5]   # 238.4.a.f

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

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