Atlas › Modular forms › Weight 2 › Level 133 › 133.2.a

Newspace 133.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(133))$

Newforms

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

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
133.2.a.a	$x^{2} + 12x + 16$
133.2.a.b	$x^{2} + 10x + 12$
133.2.a.c	$x^{2} - 10x + 20$
133.2.a.d	$x^{3} - 11x^{2} + 8x + 16$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(133))$	9
Old forms	2
Cusp forms $S_{2}(\Gamma_0(133))$	11
Eisenstein series $E_{2}(\Gamma_0(133))$	3
Modular forms $M_{2}(\Gamma_0(133))$	14

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(133)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.17 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, 19, 133]: [0, 1, 2, 14] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 9 = 1 x 9 (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(133, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 133:2 1000

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