Atlas › Modular forms › Weight 4 › Level 88 › 88.4.a

Newspace 88.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(88))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
88.4.a.a	1	$\mathbb{Q}$	$q - q^{3} - 7q^{5} - 6q^{7} + O(q^{8})$	+
88.4.a.b	1	$\mathbb{Q}$	$q + 7q^{3} + 9q^{5} + 2q^{7} + O(q^{8})$	+
88.4.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} - 6q^{5} - 56q^{7} + O(q^{8})$	−
88.4.a.d	3	degree 3	$3q + 2q^{3} + 8q^{5} + 24q^{7} + O(q^{8})$	−

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 $T$
88.4.a.a	$x + 8$
88.4.a.b	$x - 16$
88.4.a.c	$x^{2} + 8x - 164$
88.4.a.d	$x^{3} - 10x^{2} - 212x + 72$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(88))$	7
Old forms	25
Cusp forms $S_{4}(\Gamma_0(88))$	32
Eisenstein series $E_{4}(\Gamma_0(88))$	8
Modular forms $M_{4}(\Gamma_0(88))$	40

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.109 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, 3, 5, 4, 11, 21, 40] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (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(88, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 88:4 1000

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