Atlas › Modular forms › Weight 4 › Level 48 › 48.4.a

Newspace 48.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(48))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
48.4.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 18q^{5} - 8q^{7} + O(q^{8})$	+
48.4.a.b	1	$\mathbb{Q}$	$q - 3q^{3} + 14q^{5} + 24q^{7} + O(q^{8})$	+
48.4.a.c	1	$\mathbb{Q}$	$q + 3q^{3} + 6q^{5} + 16q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{11} + 7T_{23} + 8T_{29}$ 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 $T$
48.4.a.a	$x + 2084$
48.4.a.b	$x + 1284$
48.4.a.c	$x + 1460$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(48))$	3
Old forms	15
Cusp forms $S_{4}(\Gamma_0(48))$	18
Eisenstein series $E_{4}(\Gamma_0(48))$	12
Modular forms $M_{4}(\Gamma_0(48))$	30

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(48)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 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, 8, 12, 16, 24, 48]: [1, 2, 2, 3, 5, 5, 9, 8, 16, 28] (certified at ell = 2147483629)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (19, 5), (11, 6), (23, 7), (29, 8)]; h of degree 3 = 1 x 3 (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(48, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 48:4 1000

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