Atlas › Modular forms › Weight 6 › Level 80 › 80.6.a

Newspace 80.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(80))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
80.6.a.a	1	$\mathbb{Q}$	$q - 24q^{3} + 25q^{5} + 172q^{7} + O(q^{8})$	−
80.6.a.b	1	$\mathbb{Q}$	$q - 22q^{3} - 25q^{5} - 218q^{7} + O(q^{8})$	+
80.6.a.c	1	$\mathbb{Q}$	$q - 6q^{3} - 25q^{5} + 118q^{7} + O(q^{8})$	+
80.6.a.d	1	$\mathbb{Q}$	$q + 2q^{3} - 25q^{5} + 62q^{7} + O(q^{8})$	+
80.6.a.e	1	$\mathbb{Q}$	$q + 4q^{3} + 25q^{5} - 192q^{7} + O(q^{8})$	−
80.6.a.f	1	$\mathbb{Q}$	$q + 8q^{3} + 25q^{5} + 108q^{7} + O(q^{8})$	−
80.6.a.g	1	$\mathbb{Q}$	$q + 18q^{3} - 25q^{5} - 242q^{7} + O(q^{8})$	+
80.6.a.h	1	$\mathbb{Q}$	$q + 26q^{3} - 25q^{5} + 22q^{7} + O(q^{8})$	+
80.6.a.i	2	$\mathbb{Q}(\sqrt{129})$	$2q + 12q^{3} + 50q^{5} - 52q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
80.6.a.a	$x - 148$
80.6.a.b	$x + 240$
80.6.a.c	$x - 112$
80.6.a.d	$x - 64$
80.6.a.e	$x + 188$
80.6.a.f	$x - 116$
80.6.a.g	$x + 224$
80.6.a.h	$x - 48$
80.6.a.i	$x^{2} + 40x - 7856$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(80))$	10
Old forms	44
Cusp forms $S_{6}(\Gamma_0(80))$	54
Eisenstein series $E_{6}(\Gamma_0(80))$	12
Modular forms $M_{6}(\Gamma_0(80))$	66

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(80)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 6}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 5}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.281 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, 4, 5, 8, 10, 16, 20, 40, 80]: [1, 2, 4, 3, 7, 9, 12, 18, 34, 64] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 10 = 1 x 10 (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(80, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 80:6 1000

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