Atlas › Modular forms › Weight 4 › Level 132 › 132.4.a

Newspace 132.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(132))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
132.4.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 12q^{5} + 14q^{7} + O(q^{8})$	+ −
132.4.a.b	1	$\mathbb{Q}$	$q - 3q^{3} + 2q^{7} + O(q^{8})$	+ +
132.4.a.c	1	$\mathbb{Q}$	$q - 3q^{3} + 22q^{5} - 20q^{7} + O(q^{8})$	+ −
132.4.a.d	1	$\mathbb{Q}$	$q + 3q^{3} + 10q^{5} + 8q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
132.4.a.a	$x - 1648$
132.4.a.b	$x + 1256$
132.4.a.c	$x - 1274$
132.4.a.d	$x - 806$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(132))$	4
Old forms	62
Cusp forms $S_{4}(\Gamma_0(132))$	66
Eisenstein series $E_{4}(\Gamma_0(132))$	12
Modular forms $M_{4}(\Gamma_0(132))$	78

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(132)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(22))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(66))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132]: [1, 2, 2, 3, 5, 4, 9, 11, 14, 21, 40, 78] (certified at ell = 2147483171)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 4 = 1 x 4 (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(132, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 132:4 1000

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