Atlas › Modular forms › Weight 6 › Level 48 › 48.6.a

Newspace 48.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(48))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
48.6.a.a	1	$\mathbb{Q}$	$q - 9q^{3} + 6q^{5} + 40q^{7} + O(q^{8})$	+
48.6.a.b	1	$\mathbb{Q}$	$q - 9q^{3} + 38q^{5} - 120q^{7} + O(q^{8})$	+
48.6.a.c	1	$\mathbb{Q}$	$q + 9q^{3} - 66q^{5} - 176q^{7} + O(q^{8})$	−
48.6.a.d	1	$\mathbb{Q}$	$q + 9q^{3} - 34q^{5} + 240q^{7} + O(q^{8})$	−
48.6.a.e	1	$\mathbb{Q}$	$q + 9q^{3} + 94q^{5} - 144q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
48.6.a.a	$x - 126$
48.6.a.b	$x + 322$
48.6.a.c	$x + 594$
48.6.a.d	$x - 686$
48.6.a.e	$x + 338$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(48))$	5
Old forms	29
Cusp forms $S_{6}(\Gamma_0(48))$	34
Eisenstein series $E_{6}(\Gamma_0(48))$	12
Modular forms $M_{6}(\Gamma_0(48))$	46

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(48)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 5}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 6}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.134 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, 3, 4, 7, 7, 13, 12, 24, 44] (certified at ell = 2147483629)
T = [(5, 1), (7, 3)]; h of degree 5 = 1 x 5 (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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 48:6 1000

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