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

Newform orbit 230.4.a.g

Trace form

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

$3q - 6q^{2} - q^{3} + 12q^{4} + 15q^{5} + 2q^{6} + 7q^{7} - 24q^{8} + 10q^{9} - 30q^{10} + 27q^{11} - 4q^{12} + 75q^{13} - 14q^{14} - 5q^{15} + 48q^{16} + 127q^{17} - 20q^{18} - 185q^{19} + 60q^{20} - 258q^{21} - 54q^{22} + 69q^{23} + 8q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 20x^{2} - 7653x + 12972$

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
$\operatorname{tr} a_p$	−6	−1	15	7	27	75	127	−185	69	344	397	978	−575	812	−270	510	142	−49	1616	−471	−780	−860	−288	−90	1321

$\operatorname{tr} a_n$ 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)[6]   # 230.4.a.g

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

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