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

Newform orbit 202.4.a.e

Trace form

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

$7q - 14q^{2} - 2q^{3} + 28q^{4} - 19q^{5} + 4q^{6} - 14q^{7} - 56q^{8} + 77q^{9} + 38q^{10} - 72q^{11} - 8q^{12} - 21q^{13} + 28q^{14} - 106q^{15} + 112q^{16} - 83q^{17} - 154q^{18} - 103q^{19} - 76q^{20} - 108q^{21} + 144q^{22} - 185q^{23} + 16q^{24} + O(q^{25})$

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

$x^{7} + 21x^{6} - 348x^{5} - 6960x^{4} + 14173x^{3} + 343493x^{2} - 369675x - 72123$

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 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
$\operatorname{tr} a_p$	−14	−2	−19	−14	−72	−21	−83	−103	−185	−654	−315	−212	−486	−1236	−1199	−732	−1076	1978	336	247	930	237	−714	722	1684

$\operatorname{tr} a_n$ 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)[4]   # 202.4.a.e

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

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