Atlas › Modular forms › Weight 2 › Level 461 › 461.2.a

Newspace 461.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(461))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
461.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 3q^{3} - q^{4} + 4q^{6} - 2q^{7} + O(q^{8})$	+	$-1$
461.2.a.b	3	degree 3	$3q - 2q^{2} - q^{5} - 3q^{7} + O(q^{8})$	+	$-1$
461.2.a.c	7	degree 7	$7q - 3q^{3} + 2q^{4} - 4q^{5} - 8q^{6} - 5q^{7} + O(q^{8})$	+	$-1$
461.2.a.d	26	degree 26	$26q + 3q^{2} + 8q^{3} + 39q^{4} + q^{5} - 2q^{6} + 10q^{7} + O(q^{8})$	−	$+1$

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 $T$
461.2.a.a	$x^{2} + 4x - 1$
461.2.a.b	$x^{3} + 2x^{2} - 8x - 8$
461.2.a.c	$x^{7} + 3x^{6} - 5x^{5} - 9x^{4} + 6x^{3} + 7x^{2} - x - 1$
461.2.a.d	$x^{26} - 11x^{25} - 40x^{24} + 824x^{23} - 580x^{22} - 23713x^{21} + 53252x^{20} + 344769x^{19} - 1123481x^{18} - 2780000x^{17} + 12130383x^{16} + 12327581x^{15} - 77586773x^{14} - 23992183x^{13} + 308131014x^{12} - 22145406x^{11} - 763167150x^{10} + 207951723x^{9} + 1150606318x^{8} - 396559071x^{7} - 1010330981x^{6} + 302360438x^{5} + 489321595x^{4} - 77272839x^{3} - 105860022x^{2} - 3682848x + 1540216$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(461))$	38
Old forms	0
Cusp forms $S_{2}(\Gamma_0(461))$	38
Eisenstein series $E_{2}(\Gamma_0(461))$	1
Modular forms $M_{2}(\Gamma_0(461))$	39

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.195 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, 461]: [0, 39] (certified at ell = 2147478721)
T = [(2, 1), (3, 1)]; h of degree 38 = 1 x 38 (phi(ord eps) x dim S_k^new); 4 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(461, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 461:2 1000

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