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

Newform orbit 230.4.a.h

Trace form

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

$4q - 8q^{2} - 4q^{3} + 16q^{4} - 20q^{5} + 8q^{6} - q^{7} - 32q^{8} + 32q^{9} + 40q^{10} - 39q^{11} - 16q^{12} - 20q^{13} + 2q^{14} + 20q^{15} + 64q^{16} - 23q^{17} - 64q^{18} + 53q^{19} - 80q^{20} + 300q^{21} + 78q^{22} - 92q^{23} + 32q^{24} + O(q^{25})$

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

$x^{4} + 7x^{3} - 14513x^{2} + 91533x + 40877636$

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	−4	−20	−1	−39	−20	−23	53	−92	161	388	466	484	894	−265	576	−94	1153	−1472	200	1147	−908	−1048	−1784	−2047

$\operatorname{tr} a_n$ 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)[7]   # 230.4.a.h

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

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