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

Newform orbit 226.4.a.c

Trace form

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

$7q - 14q^{2} - q^{3} + 28q^{4} + 27q^{5} + 2q^{6} + 26q^{7} - 56q^{8} + 76q^{9} - 54q^{10} + 76q^{11} - 4q^{12} - 44q^{13} - 52q^{14} + 63q^{15} + 112q^{16} - 30q^{17} - 152q^{18} - 62q^{19} + 108q^{20} + 67q^{21} - 152q^{22} + 122q^{23} + 8q^{24} + O(q^{25})$

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

$x^{7} - 26x^{6} - 339x^{5} + 9225x^{4} + 20748x^{3} - 739674x^{2} + 1976472x - 807948$

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
$\operatorname{tr} a_p$	−14	−1	27	26	76	−44	−30	−62	122	377	361	214	1041	506	1457	1071	1567	−885	−723	1688	−2567	−567	−140	1964	−1584

$\operatorname{tr} a_n$ 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)[2]   # 226.4.a.c

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

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