Atlas › Modular forms › Weight 2 › Level 577 › 577.2.a

Newspace 577.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(577))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
577.2.a.a	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{3} - 4q^{4} + 6q^{5} - 2q^{7} + O(q^{8})$	−	$+1$
577.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{6} - q^{7} + O(q^{8})$	−	$+1$
577.2.a.c	3	degree 3	$3q + q^{2} + q^{3} + 3q^{4} + 8q^{6} + 6q^{7} + O(q^{8})$	−	$+1$
577.2.a.d	18	degree 18	$18q + 8q^{2} + 24q^{4} - 2q^{5} - 11q^{6} + 11q^{7} + O(q^{8})$	−	$+1$
577.2.a.e	22	degree 22	$22q - 13q^{2} - 8q^{3} + 21q^{4} - 8q^{5} - 4q^{6} - 20q^{7} + O(q^{8})$	+	$-1$

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 $T$
577.2.a.a	$x^{2} - x - 3$
577.2.a.b	$x^{2} - 5x + 5$
577.2.a.c	$x^{3} - 2x^{2} - 19x + 45$
577.2.a.d	$x^{18} - 8x^{17} - 23x^{16} + 274x^{15} + 165x^{14} - 3683x^{13} - 783x^{12} + 24958x^{11} + 7586x^{10} - 86624x^{9} - 44341x^{8} + 135462x^{7} + 80836x^{6} - 92227x^{5} - 46234x^{4} + 31162x^{3} + 6992x^{2} - 4834x + 537$
577.2.a.e	$x^{22} + 21x^{21} + 151x^{20} + 232x^{19} - 2299x^{18} - 10631x^{17} + 974x^{16} + 90983x^{15} + 121172x^{14} - 314176x^{13} - 731016x^{12} + 427619x^{11} + 1869473x^{10} + 1571x^{9} - 2435409x^{8} - 531550x^{7} + 1617912x^{6} + 445774x^{5} - 485480x^{4} - 131361x^{3} + 57400x^{2} + 12950x - 1487$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(577))$	47
Old forms	0
Cusp forms $S_{2}(\Gamma_0(577))$	47
Eisenstein series $E_{2}(\Gamma_0(577))$	1
Modular forms $M_{2}(\Gamma_0(577))$	48

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.204 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, 577]: [0, 48] (certified at ell = 2147481793)
T = [(2, 1), (3, 1)]; h of degree 47 = 1 x 47 (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(577, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 577:2 1000

← 576.2.a · 578.2.a →