Atlas › Modular forms › Weight 2 › Level 99 › 99.2.a

Newspace 99.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(99))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
99.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - 4q^{5} - 2q^{7} + O(q^{8})$	+	99a1 rank 1
99.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	+	99b1 rank 0
99.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + 4q^{5} - 2q^{7} + O(q^{8})$	−	99c1 rank 0
99.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} - q^{5} - 2q^{7} + O(q^{8})$	+	99d1 rank 0

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 $T$
99.2.a.a	$x + 13$
99.2.a.b	$x - 5$
99.2.a.c	$x - 13$
99.2.a.d	$x + 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(99))$	4
Old forms	5
Cusp forms $S_{2}(\Gamma_0(99))$	9
Eisenstein series $E_{2}(\Gamma_0(99))$	7
Modular forms $M_{2}(\Gamma_0(99))$	16

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.19 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]: [0, 1, 2, 2, 6, 14] (certified at ell = 2147482951)
T = [(2, 1), (5, 3)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 99:2 1000

← 98.2.a · 100.2.a →