Atlas › Modular forms › Weight 4 › Level 226 › 226.4.a › 226.4.a.b

Newform orbit 226.4.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 14q^{2} - 7q^{3} + 28q^{4} - 13q^{5} + 14q^{6} - 2q^{7} - 56q^{8} + 40q^{9} + 26q^{10} - 56q^{11} - 28q^{12} + 60q^{13} + 4q^{14} - 57q^{15} + 112q^{16} - 30q^{17} - 80q^{18} - 100q^{19} - 52q^{20} - 101q^{21} + 112q^{22} - 200q^{23} + 56q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{7} + 20x^{6} - 389x^{5} - 7477x^{4} + 24274x^{3} + 534500x^{2} - 233004x - 7137136$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
113	$-1$

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
$\operatorname{tr} a_p$	−14	−7	−13	−2	−56	60	−30	−100	−200	−533	−587	−550	−971	150	−1651	−765	−1539	127	315	−1172	1539	1763	−112	−1810	3776

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.142 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, 113, 226]: [1, 2, 30, 88] (certified at ell = 2147480161)
T = [(3, 1), (5, 1)]; h of degree 28 = 1 x 28 (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(226, 4)[1]   # 226.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 226:4 1000