Atlas › Modular forms › Weight 8 › Level 61 › 61.8.a › 61.8.a.b

Newform orbit 61.8.a.b

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 23q^{2} + 134q^{3} + 1317q^{4} + 429q^{5} + q^{6} + 1017q^{7} + 4479q^{8} + 18249q^{9} + 5029q^{10} + 21895q^{11} + 13639q^{12} + 4293q^{13} + 28184q^{14} + 28006q^{15} + 30149q^{16} - 3512q^{17} - 52668q^{18} + 62964q^{19} + 128631q^{20} + 151802q^{21} + 380355q^{22} + 194883q^{23} + 643935q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} - 157x^{18} - 19452x^{17} + 3935214x^{16} + 111307486x^{15} - 38942725392x^{14} + 6585808481x^{13} + 201579711896603x^{12} - 2333402261158404x^{11} - 603737436766341432x^{10} + 9304518292983566192x^{9} + 1087717855071623089616x^{8} - 15794644321086717819312x^{7} - 1180482471443915394800496x^{6} + 12181494901509568072952192x^{5} + 736159171497289498719988736x^{4} - 3484954022133273395794671104x^{3} - 223496925835967520364397856512x^{2} + 95933493568065711362985882624x + 21814786220227436998967371978752$

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
$\operatorname{tr} a_p$	23	134	429	1017	21895	4293	−3512	62964	194883	343584	376400	630086	1584539	−93664	1752508	127548	5826729	−4312639	−10206319	4358512	−15042353	3473903	19386052	10913876	55835432

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.291 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, 37] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 8 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, 8)[1]   # 61.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 61:8 1000