Atlas › Modular forms › Weight 2 › Level 661 › 661.2.a

Newspace 661.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(661))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
661.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	−	$+1$
661.2.a.b	23	degree 23	$23q - 9q^{2} - 7q^{3} + 15q^{4} - 11q^{5} - 10q^{6} - 6q^{7} + O(q^{8})$	+	$-1$
661.2.a.c	29	degree 29	$29q + 7q^{2} + 5q^{3} + 33q^{4} + 9q^{5} + 8q^{6} + 10q^{7} + O(q^{8})$	−	$+1$

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 $T$
661.2.a.a	$x^{2} - 2x - 1$
661.2.a.b	$x^{23} + 16x^{22} + 68x^{21} - 176x^{20} - 1922x^{19} - 1702x^{18} + 17890x^{17} + 36839x^{16} - 73752x^{15} - 232844x^{14} + 120871x^{13} + 702344x^{12} + 20875x^{11} - 1096428x^{10} - 282538x^{9} + 863795x^{8} + 276425x^{7} - 300407x^{6} - 81301x^{5} + 31666x^{4} + 8628x^{3} - 538x^{2} - 198x - 9$
661.2.a.c	$x^{29} - 12x^{28} - 49x^{27} + 1092x^{26} - 753x^{25} - 39102x^{24} + 94671x^{23} + 721625x^{22} - 2587472x^{21} - 7436949x^{20} + 36277245x^{19} + 42122117x^{18} - 302761304x^{17} - 105115787x^{16} + 1588625986x^{15} - 113354791x^{14} - 5341321031x^{13} + 1437982132x^{12} + 11463346065x^{11} - 3640598996x^{10} - 15281258236x^{9} + 3801570744x^{8} + 11840607958x^{7} - 1050162530x^{6} - 4488524491x^{5} - 492753867x^{4} + 417132388x^{3} + 56508819x^{2} - 3989722x - 145919$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(661))$	54
Old forms	0
Cusp forms $S_{2}(\Gamma_0(661))$	54
Eisenstein series $E_{2}(\Gamma_0(661))$	1
Modular forms $M_{2}(\Gamma_0(661))$	55

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.223 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, 661]: [0, 55] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 54 = 1 x 54 (phi(ord eps) x dim S_k^new); 6 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(661, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 661:2 1000

← 660.2.a · 662.2.a →