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

Newspace 61.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(61))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
61.8.a.a	16	degree 16	$16q - 17q^{2} - 82q^{3} + 741q^{4} - 571q^{5} - 1295q^{6} - 355q^{7} + O(q^{8})$	−	$-1$
61.8.a.b	19	degree 19	$19q + 23q^{2} + 134q^{3} + 1317q^{4} + 429q^{5} + q^{6} + 1017q^{7} + O(q^{8})$	+	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
61.8.a.a	$x^{16} + 99x^{15} - 16352x^{14} - 1622284x^{13} + 103699976x^{12} + 9826881320x^{11} - 345358898275x^{10} - 27570864062745x^{9} + 709302038473666x^{8} + 37250125907847464x^{7} - 902839201612347144x^{6} - 21223478014506106512x^{5} + 604278964808845391792x^{4} + 1942719037362214910896x^{3} - 149353388029218659045472x^{2} + 1293346987491953807417728x - 3423161070436268107790592$
61.8.a.b	$x^{19} - 157x^{18} - 19452x^{17} + 3935214x^{16} + 111307486x^{15} - 38942725392x^{14} + 6585808481x^{13} + 201579711896603x^{12} - 2333402261158404x^{11} - 603737436766341432x^{10} + 9304518292983566192x^{9} + 1087717855071623089616x^{8} - 15794644321086717819312x^{7} - 1180482471443915394800496x^{6} + 12181494901509568072952192x^{5} + 736159171497289498719988736x^{4} - 3484954022133273395794671104x^{3} - 223496925835967520364397856512x^{2} + 95933493568065711362985882624x + 21814786220227436998967371978752$

Orbit

Characteristic polynomial of

T

61.8.a.a

x^{16} + 99x^{15} - 16352x^{14} - 1622284x^{13} + 103699976x^{12} + 9826881320x^{11} - 345358898275x^{10} - 27570864062745x^{9} + 709302038473666x^{8} + 37250125907847464x^{7} - 902839201612347144x^{6} - 21223478014506106512x^{5} + 604278964808845391792x^{4} + 1942719037362214910896x^{3} - 149353388029218659045472x^{2} + 1293346987491953807417728x - 3423161070436268107790592

61.8.a.b

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

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(61))$	35
Old forms	0
Cusp forms $S_{8}(\Gamma_0(61))$	35
Eisenstein series $E_{8}(\Gamma_0(61))$	2
Modular forms $M_{8}(\Gamma_0(61))$	37

Decomposition of the old subspace

The old subspace is 0.

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)

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

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

← 60.8.a · 62.8.a →