Atlas › Modular forms › Weight 6 › Level 88 › 88.6.a

Newspace 88.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(88))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
88.6.a.a	2	$\mathbb{Q}(\sqrt{37})$	$2q - 14q^{3} + 18q^{5} + 48q^{7} + O(q^{8})$	−
88.6.a.b	3	degree 3	$3q + 14q^{3} + 56q^{5} + 112q^{7} + O(q^{8})$	−
88.6.a.c	4	degree 4	$4q - 13q^{3} - 19q^{5} + 58q^{7} + O(q^{8})$	+
88.6.a.d	4	degree 4	$4q - 5q^{3} + 93q^{5} - 94q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
88.6.a.a	$x^{2} - 4x - 144$
88.6.a.b	$x^{3} - 70x^{2} - 5720x + 155936$
88.6.a.c	$x^{4} + 32x^{3} - 3224x^{2} + 52800x - 246576$
88.6.a.d	$x^{4} - 88x^{3} - 13960x^{2} + 1343648x - 23071856$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(88))$	13
Old forms	43
Cusp forms $S_{6}(\Gamma_0(88))$	56
Eisenstein series $E_{6}(\Gamma_0(88))$	8
Modular forms $M_{6}(\Gamma_0(88))$	64

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(88)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(22))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.271 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, 4, 8, 11, 22, 44, 88]: [1, 2, 4, 7, 6, 17, 33, 64] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 13 = 1 x 13 (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(88, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 88:6 1000

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