Atlas › Modular forms › Weight 2 › Level 655 › 655.2.a

Newspace 655.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(655))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
655.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 2q^{4} - q^{5} + 6q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$	655a1 rank 2
655.2.a.b	8	degree 8	$8q - 5q^{2} - 7q^{3} + 5q^{4} + 8q^{5} + 4q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
655.2.a.c	8	degree 8	$8q - 3q^{2} - q^{3} + 5q^{4} - 8q^{5} - 6q^{6} + O(q^{8})$	+ +	$-1$
655.2.a.d	13	degree 13	$13q + 4q^{2} + 5q^{3} + 10q^{4} + 13q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
655.2.a.e	13	degree 13	$13q + 5q^{2} + 6q^{3} + 15q^{4} - 13q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
655.2.a.a	$x + 5$
655.2.a.b	$x^{8} + 12x^{7} + 45x^{6} + 29x^{5} - 162x^{4} - 305x^{3} - 115x^{2} + 30x + 1$
655.2.a.c	$x^{8} + 4x^{7} - 7x^{6} - 31x^{5} + 26x^{4} + 67x^{3} - 61x^{2} + 5$
655.2.a.d	$x^{13} - 9x^{12} - 8x^{11} + 255x^{10} - 319x^{9} - 2312x^{8} + 4008x^{7} + 9464x^{6} - 14908x^{5} - 20725x^{4} + 14373x^{3} + 24042x^{2} + 9117x + 1085$
655.2.a.e	$x^{13} - 11x^{12} + 12x^{11} + 211x^{10} - 445x^{9} - 1708x^{8} + 3676x^{7} + 7322x^{6} - 12852x^{5} - 16347x^{4} + 19477x^{3} + 15342x^{2} - 10335x - 2843$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(655))$	43
Old forms	22
Cusp forms $S_{2}(\Gamma_0(655))$	65
Eisenstein series $E_{2}(\Gamma_0(655))$	3
Modular forms $M_{2}(\Gamma_0(655))$	68

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(655)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(131))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.219 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, 5, 131, 655]: [0, 1, 12, 68] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 43 = 1 x 43 (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(655, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 655:2 1000

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