Atlas › Modular forms › Weight 4 › Level 62 › 62.4.a › 62.4.a.d

Newform orbit 62.4.a.d

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 6q^{2} + 4q^{3} + 12q^{4} + 17q^{5} + 8q^{6} + 21q^{7} + 24q^{8} + 3q^{9} + 34q^{10} + 24q^{11} + 16q^{12} - 20q^{13} + 42q^{14} - 50q^{15} + 48q^{16} - 98q^{17} + 6q^{18} - 67q^{19} + 68q^{20} - 226q^{21} + 48q^{22} - 42q^{23} + 32q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 21x^{2} + 96x - 94$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
31	$-1$

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
$\operatorname{tr} a_p$	6	4	17	21	24	−20	−98	−67	−42	−146	93	−180	−141	310	244	538	795	328	1828	55	234	1212	−640	−3048	−1083

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 31, 62]: [1, 2, 9, 26] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(62, 4)[3]   # 62.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 62:4 1000