Atlas › Modular forms › Weight 6 › Level 61 › 61.6.a › 61.6.a.b

Newform orbit 61.6.a.b

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 11q^{2} + 26q^{3} + 261q^{4} + 71q^{5} + 205q^{6} + 109q^{7} + 543q^{8} + 1728q^{9} - 115q^{10} + 1893q^{11} + 295q^{12} + 499q^{13} + 2188q^{14} + 862q^{15} + 10397q^{16} + 1780q^{17} + 17370q^{18} + 4016q^{19} + 5607q^{20} + 2500q^{21} + 7355q^{22} + 11017q^{23} + 2699q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} - 37x^{13} - 2440x^{12} + 79492x^{11} + 2599692x^{10} - 64837100x^{9} - 1517512423x^{8} + 24774504295x^{7} + 485659259986x^{6} - 4462759655616x^{5} - 76215923769224x^{4} + 366066156099648x^{3} + 4744562847617552x^{2} - 13957028681264016x - 70580968683671904$

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 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
$\operatorname{tr} a_p$	11	26	71	109	1893	499	1780	4016	11017	14816	11462	−17702	5801	8194	−16308	−4394	99977	52094	66109	251686	169601	136045	199016	153370	−12398

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.232 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, 27] (certified at ell = 2147482921)
T = [(2, 1), (3, 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(61, 6)[1]   # 61.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 61:6 1000