Atlas › Modular forms › Weight 2 › Level 253 › 253.2.a

Newspace 253.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(253))$

Newforms

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

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 $T$
253.2.a.a	$x^{3} + 6x^{2} - x - 5$
253.2.a.b	$x^{3} - 6x^{2} + 9x - 3$
253.2.a.c	$x^{5} + 9x^{4} + 26x^{3} + 28x^{2} + 8x - 1$
253.2.a.d	$x^{6} - 10x^{5} + 25x^{4} + 36x^{3} - 224x^{2} + 265x - 67$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(253))$	17
Old forms	6
Cusp forms $S_{2}(\Gamma_0(253))$	23
Eisenstein series $E_{2}(\Gamma_0(253))$	3
Modular forms $M_{2}(\Gamma_0(253))$	26

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(253)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 11, 23, 253]: [0, 2, 3, 26] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 2 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(253, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 253:2 1000

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