Atlas › Modular forms › Weight 4 › Level 133 › 133.4.a

Newspace 133.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(133))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
133.4.a.a	1	$\mathbb{Q}$	$q + 4q^{2} + 8q^{3} + 8q^{4} + 6q^{5} + 32q^{6} - 7q^{7} + O(q^{8})$	+ +	$+1$
133.4.a.b	6	degree 6	$6q - 3q^{2} - 3q^{3} + 25q^{4} - 13q^{5} - 38q^{6} - 42q^{7} + O(q^{8})$	+ −	$-1$
133.4.a.c	6	degree 6	$6q + 3q^{2} + 7q^{3} + 29q^{4} - 29q^{5} + 34q^{6} - 42q^{7} + O(q^{8})$	+ +	$+1$
133.4.a.d	7	degree 7	$7q - 8q^{2} - 17q^{3} + 18q^{4} - 33q^{5} + 44q^{6} + 49q^{7} + O(q^{8})$	− +	$-1$
133.4.a.e	8	degree 8	$8q + 2q^{2} + 13q^{3} + 30q^{4} + 37q^{5} - 12q^{6} + 56q^{7} + O(q^{8})$	− −	$+1$

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 $T$
133.4.a.a	$x - 12$
133.4.a.b	$x^{6} + 6x^{5} - 61x^{4} - 68x^{3} + 779x^{2} - 378x - 1196$
133.4.a.c	$x^{6} - 10x^{5} - 127x^{4} + 1138x^{3} + 4369x^{2} - 29974x - 3828$
133.4.a.d	$x^{7} + 25x^{6} + 95x^{5} - 1285x^{4} - 7653x^{3} - 14615x^{2} - 10924x - 2372$
133.4.a.e	$x^{8} - 15x^{7} - 75x^{6} + 1735x^{5} - 593x^{4} - 59221x^{3} + 109120x^{2} + 628964x - 1584848$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(133))$	28
Old forms	10
Cusp forms $S_{4}(\Gamma_0(133))$	38
Eisenstein series $E_{4}(\Gamma_0(133))$	4
Modular forms $M_{4}(\Gamma_0(133))$	42

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(133)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.112 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]: [1, 3, 6, 42] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (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(133, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 133:4 1000

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