Atlas › Modular forms › Weight 4 › Level 61 › 61.4.a › 61.4.a.a

Newform orbit 61.4.a.a

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 7q^{2} - 13q^{3} + 17q^{4} - 16q^{5} - 29q^{6} - 27q^{7} - 81q^{8} - 15q^{9} - 31q^{10} - 180q^{11} - 77q^{12} - 54q^{13} - 134q^{14} - 74q^{15} + 69q^{16} - 50q^{17} + 402q^{18} + 7q^{19} + 87q^{20} + 160q^{21} + 391q^{22} - 77q^{23} + 443q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 20x^{5} + 123x^{4} + 117x^{3} - 1046x^{2} - 2229x - 1088$

Atkin–Lehner signs

$p$	$w_p$
61	$-1$

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
$\operatorname{tr} a_p$	−7	−13	−16	−27	−180	−54	−50	7	−77	−463	−235	−99	−841	190	−72	−329	−1852	366	444	−917	1941	442	−1291	−242	4825

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.095 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, 61]: [1, 17] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 3 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(61, 4)[0]   # 61.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 61:4 1000