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

Newform orbit 202.4.a.d

Trace form

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

$5q - 10q^{2} + 7q^{3} + 20q^{4} + 6q^{5} - 14q^{6} + 14q^{7} - 40q^{8} + 50q^{9} - 12q^{10} + 71q^{11} + 28q^{12} + 18q^{13} - 28q^{14} - 16q^{15} + 80q^{16} + 155q^{17} - 100q^{18} + 30q^{19} + 24q^{20} - 108q^{21} - 142q^{22} + 229q^{23} - 56q^{24} + O(q^{25})$

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

$x^{5} - 13x^{4} - 321x^{3} + 4362x^{2} + 14849x - 225206$

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
$\operatorname{tr} a_p$	−10	7	6	14	71	18	155	30	229	413	121	189	922	827	577	1043	917	−911	59	−37	−36	−2011	1051	−360	−1338

$\operatorname{tr} a_n$ 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)[3]   # 202.4.a.d

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

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