Atlas › Modular forms › Weight 4 › Level 230 › 230.4.a › 230.4.a.j

Newform orbit 230.4.a.j

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 8q^{2} + 14q^{3} + 16q^{4} + 20q^{5} + 28q^{6} + 8q^{7} + 32q^{8} + 64q^{9} + 40q^{10} + 21q^{11} + 56q^{12} + 70q^{13} + 16q^{14} + 70q^{15} + 64q^{16} + 56q^{17} + 128q^{18} + 173q^{19} + 80q^{20} - 120q^{21} + 42q^{22} - 92q^{23} + 112q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7}$ , whose minimal polynomial is

$x^{4} - 38x^{3} - 4620x^{2} + 110078x - 611293$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$-1$
23	$+1$

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
$\operatorname{tr} a_p$	8	14	20	8	21	70	56	173	−92	−118	17	−343	139	−50	367	−353	−453	−327	−455	195	−633	−1140	−1199	−2170	−703

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.206 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, 5, 10, 23, 46, 115, 230]: [1, 2, 3, 7, 7, 20, 38, 112] (certified at ell = 2147483053)
T = [(3, 1), (7, 3)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 6 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(230, 4)[9]   # 230.4.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 230:4 1000