Atlas › Modular forms › Weight 4 › Level 202 › 202.4.a › 202.4.a.f

Newform orbit 202.4.a.f

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 16q^{2} + 10q^{3} + 32q^{4} + 3q^{5} + 20q^{6} + 46q^{7} + 64q^{8} + 98q^{9} + 6q^{10} + 92q^{11} + 40q^{12} + 113q^{13} + 92q^{14} + 338q^{15} + 128q^{16} + 95q^{17} + 196q^{18} - 59q^{19} + 12q^{20} + 204q^{21} + 184q^{22} + 287q^{23} + 80q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{8} - 13x^{7} - 1138x^{6} + 17660x^{5} + 292897x^{4} - 5896917x^{3} + 17933007x^{2} + 39217743x - 5098998$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
101	$-1$

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
$\operatorname{tr} a_p$	16	10	3	46	92	113	95	−59	287	−56	13	−238	−44	−80	497	530	−940	−976	−1940	631	−412	861	1434	4	−266

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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, 101, 202]: [1, 2, 27, 79] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 25 = 1 x 25 (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(202, 4)[5]   # 202.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 202:4 1000