Atlas › Modular forms › Weight 4 › Level 99 › 99.4.a

Newspace 99.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(99))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
99.4.a.a	1	$\mathbb{Q}$	$q + q^{2} - 7q^{4} + 4q^{5} - 26q^{7} + O(q^{8})$	+
99.4.a.b	1	$\mathbb{Q}$	$q + 5q^{2} + 17q^{4} + 14q^{5} - 32q^{7} + O(q^{8})$	−
99.4.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 8q^{4} - 2q^{5} + 20q^{7} + O(q^{8})$	−
99.4.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 10q^{4} - 20q^{5} - 16q^{7} + O(q^{8})$	−
99.4.a.e	2	$\mathbb{Q}(\sqrt{33})$	$2q - q^{2} + q^{4} - 16q^{5} + 2q^{7} + O(q^{8})$	+
99.4.a.f	2	$\mathbb{Q}(\sqrt{97})$	$2q - q^{2} + 33q^{4} + 14q^{5} + 24q^{7} + O(q^{8})$	−
99.4.a.g	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} + 10q^{4} + 20q^{5} - 16q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
99.4.a.a	$x - 13$
99.4.a.b	$x - 47$
99.4.a.c	$x^{2} + 8x - 1859$
99.4.a.d	$x^{2} + 62x + 913$
99.4.a.e	$x^{2} + 49x - 398$
99.4.a.f	$x^{2} - 41x - 186$
99.4.a.g	$x^{2} - 62x + 913$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(99))$	12
Old forms	20
Cusp forms $S_{4}(\Gamma_0(99))$	32
Eisenstein series $E_{4}(\Gamma_0(99))$	8
Modular forms $M_{4}(\Gamma_0(99))$	40

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(99)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.113 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, 3, 9, 11, 33, 99]: [1, 2, 4, 4, 14, 38] (certified at ell = 2147482951)
T = [(2, 1), (5, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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(99, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 99:4 1000

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