Atlas › Modular forms › Weight 6 › Level 96 › 96.6.a

Newspace 96.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(96))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
96.6.a.a	1	$\mathbb{Q}$	$q - 9q^{3} - 86q^{5} - 180q^{7} + O(q^{8})$	+
96.6.a.b	1	$\mathbb{Q}$	$q - 9q^{3} - 14q^{5} + 100q^{7} + O(q^{8})$	+
96.6.a.c	1	$\mathbb{Q}$	$q - 9q^{3} + 26q^{5} - 36q^{7} + O(q^{8})$	+
96.6.a.d	1	$\mathbb{Q}$	$q + 9q^{3} - 86q^{5} + 180q^{7} + O(q^{8})$	−
96.6.a.e	1	$\mathbb{Q}$	$q + 9q^{3} - 14q^{5} - 100q^{7} + O(q^{8})$	−
96.6.a.f	1	$\mathbb{Q}$	$q + 9q^{3} + 26q^{5} + 36q^{7} + O(q^{8})$	−
96.6.a.g	2	$\mathbb{Q}(\sqrt{31})$	$2q - 18q^{3} + 36q^{5} - 120q^{7} + O(q^{8})$	+
96.6.a.h	2	$\mathbb{Q}(\sqrt{31})$	$2q + 18q^{3} + 36q^{5} + 120q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
96.6.a.a	$x - 4062$
96.6.a.b	$x + 22146$
96.6.a.c	$x + 13794$
96.6.a.d	$x + 29202$
96.6.a.e	$x - 3102$
96.6.a.f	$x - 9102$
96.6.a.g	$x^{2} - 20476x - 108630012$
96.6.a.h	$x^{2} - 19900x + 95161476$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(96))$	10
Old forms	62
Cusp forms $S_{6}(\Gamma_0(96))$	72
Eisenstein series $E_{6}(\Gamma_0(96))$	16
Modular forms $M_{6}(\Gamma_0(96))$	88

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(96)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 6}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 8}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 5}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 6}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(48))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.491 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, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96]: [1, 2, 3, 4, 7, 7, 13, 12, 24, 22, 44, 84] (certified at ell = 2147483497)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 6 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(96, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 96:6 1000

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