Atlas › Modular forms › Weight 6 › Level 99 › 99.6.a

Newspace 99.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(99))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
99.6.a.a	1	$\mathbb{Q}$	$q - q^{2} - 31q^{4} + 92q^{5} - 26q^{7} + O(q^{8})$	+
99.6.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - 28q^{4} - 46q^{5} + 148q^{7} + O(q^{8})$	+
99.6.a.c	1	$\mathbb{Q}$	$q + 4q^{2} - 16q^{4} + 19q^{5} + 10q^{7} + O(q^{8})$	−
99.6.a.d	2	$\mathbb{Q}(\sqrt{33})$	$2q - 13q^{2} + 37q^{4} - 58q^{5} + 146q^{7} + O(q^{8})$	−
99.6.a.e	2	$\mathbb{Q}(\sqrt{313})$	$2q - q^{2} + 93q^{4} + 38q^{5} - 18q^{7} + O(q^{8})$	+
99.6.a.f	2	$\mathbb{Q}(\sqrt{177})$	$2q + 5q^{2} + 37q^{4} - 58q^{5} - 286q^{7} + O(q^{8})$	−
99.6.a.g	3	degree 3	$3q + 84q^{4} - 24q^{5} + 84q^{7} + O(q^{8})$	+
99.6.a.h	5	degree 5	$5q - 4q^{2} + 102q^{4} - 100q^{5} - 18q^{7} + O(q^{8})$	+
99.6.a.i	5	degree 5	$5q + 4q^{2} + 102q^{4} + 100q^{5} - 18q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
99.6.a.a	$x - 91$
99.6.a.b	$x + 44$
99.6.a.c	$x - 23$
99.6.a.d	$x^{2} + 71x + 592$
99.6.a.e	$x^{2} - 37x - 9126$
99.6.a.f	$x^{2} + 53x - 2882$
99.6.a.g	$x^{3} + 24x^{2} - 4047x + 6992$
99.6.a.h	$x^{5} + 104x^{4} - 2559x^{3} - 344038x^{2} + 1640980x + 118906920$
99.6.a.i	$x^{5} - 104x^{4} - 2559x^{3} + 344038x^{2} + 1640980x - 118906920$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(99))$	22
Old forms	34
Cusp forms $S_{6}(\Gamma_0(99))$	56
Eisenstein series $E_{6}(\Gamma_0(99))$	8
Modular forms $M_{6}(\Gamma_0(99))$	64

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(99)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.283 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, 3, 6, 6, 22, 62] (certified at ell = 2147482951)
T = [(2, 1), (5, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 5 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 99:6 1000

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