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

Newform orbit 226.4.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 10q^{2} - 11q^{3} + 20q^{4} - 29q^{5} - 22q^{6} - 26q^{7} + 40q^{8} + 14q^{9} - 58q^{10} - 84q^{11} - 44q^{12} - 180q^{13} - 52q^{14} - 61q^{15} + 80q^{16} - 106q^{17} + 28q^{18} - 92q^{19} - 116q^{20} - 313q^{21} - 168q^{22} - 164q^{23} - 88q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + 40x^{4} + 535x^{3} + 2583x^{2} + 1920x - 8724$

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
$\operatorname{tr} a_p$	10	−11	−29	−26	−84	−180	−106	−92	−164	−131	−255	18	−491	116	791	−449	−867	95	1435	−590	−435	2239	−1844	−1176	432

$\operatorname{tr} a_n$ 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)[0]   # 226.4.a.a

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

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