Atlas › Modular forms › Weight 6 › Level 84 › 84.6.a

Newspace 84.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(84))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
84.6.a.a	1	$\mathbb{Q}$	$q - 9q^{3} + 6q^{5} + 49q^{7} + O(q^{8})$	+ −
84.6.a.b	1	$\mathbb{Q}$	$q + 9q^{3} - 34q^{5} - 49q^{7} + O(q^{8})$	− +
84.6.a.c	2	$\mathbb{Q}(\sqrt{5569})$	$2q - 18q^{3} - 6q^{5} - 98q^{7} + O(q^{8})$	+ +
84.6.a.d	2	$\mathbb{Q}(\sqrt{505})$	$2q + 18q^{3} + 78q^{5} + 98q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{11}$ 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 $T$
84.6.a.a	$x + 102$
84.6.a.b	$x + 366$
84.6.a.c	$x^{2} + 96x - 198180$
84.6.a.d	$x^{2} - 252x - 56844$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(84))$	6
Old forms	68
Cusp forms $S_{6}(\Gamma_0(84))$	74
Eisenstein series $E_{6}(\Gamma_0(84))$	12
Modular forms $M_{6}(\Gamma_0(84))$	86

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(84)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 6}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 6}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(28))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(42))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 7, 12, 14, 21, 28, 42, 84]: [1, 2, 3, 4, 7, 5, 13, 12, 16, 23, 44, 86] (certified at ell = 2147483647)
T = [(5, 1), (11, 1)]; h of degree 6 = 1 x 6 (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(84, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 84:6 1000

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