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

Newform orbit 226.4.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 18q^{2} + 13q^{3} + 36q^{4} + 41q^{5} + 26q^{6} + 2q^{7} + 72q^{8} + 158q^{9} + 82q^{10} + 92q^{11} + 52q^{12} + 80q^{13} + 4q^{14} + 59q^{15} + 144q^{16} + 234q^{17} + 316q^{18} + 212q^{19} + 164q^{20} + 107q^{21} + 184q^{22} + 158q^{23} + 104q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{9} - 54x^{8} + 397x^{7} + 22153x^{6} - 397638x^{5} - 762974x^{4} + 60074668x^{3} - 372132516x^{2} - 289941656x + 5709790152$

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 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
$\operatorname{tr} a_p$	18	13	41	2	92	80	234	212	158	177	37	68	449	362	−61	−165	−195	−657	131	−1258	−113	−1723	−1080	634	384

$\operatorname{tr} a_n$ 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)[3]   # 226.4.a.d

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

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