Atlas › Modular forms › Weight 2 › Level 553 › 553.2.a

Newspace 553.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(553))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
553.2.a.a	7	degree 7	$7q - 6q^{2} - 3q^{3} + 8q^{4} - 4q^{5} - 2q^{6} + 7q^{7} + O(q^{8})$	− −	$-1$
553.2.a.b	8	degree 8	$8q + 3q^{2} + 3q^{3} + 7q^{4} + 4q^{5} + 6q^{6} - 8q^{7} + O(q^{8})$	+ −	$+1$
553.2.a.c	11	degree 11	$11q - 5q^{2} - 5q^{3} + 11q^{4} - 10q^{5} - 2q^{6} - 11q^{7} + O(q^{8})$	+ +	$-1$
553.2.a.d	13	degree 13	$13q + 7q^{2} + q^{3} + 17q^{4} + 4q^{5} - 6q^{6} + 13q^{7} + O(q^{8})$	− +	$+1$

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 $T$
553.2.a.a	$x^{7} + 9x^{6} + 22x^{5} - 5x^{4} - 50x^{3} + 8x^{2} + 13x - 1$
553.2.a.b	$x^{8} - 6x^{7} - 11x^{6} + 99x^{5} + 15x^{4} - 486x^{3} + 67x^{2} + 624x + 121$
553.2.a.c	$x^{11} + 10x^{10} + 12x^{9} - 129x^{8} - 307x^{7} + 304x^{6} + 1080x^{5} + 31x^{4} - 1064x^{3} - 316x^{2} + 244x + 67$
553.2.a.d	$x^{13} - 8x^{12} - 13x^{11} + 227x^{10} - 223x^{9} - 1651x^{8} + 2263x^{7} + 5053x^{6} - 5638x^{5} - 7243x^{4} + 3116x^{3} + 2369x^{2} - 482x - 123$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(553))$	39
Old forms	12
Cusp forms $S_{2}(\Gamma_0(553))$	51
Eisenstein series $E_{2}(\Gamma_0(553))$	3
Modular forms $M_{2}(\Gamma_0(553))$	54

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(553)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.203 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, 7, 79, 553]: [0, 1, 7, 54] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 39 = 1 x 39 (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(553, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 553:2 1000

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