Atlas › Modular forms › Weight 4 › Level 80 › 80.4.a

Newspace 80.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(80))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
80.4.a.a	1	$\mathbb{Q}$	$q - 10q^{3} - 5q^{5} + 18q^{7} + O(q^{8})$	+
80.4.a.b	1	$\mathbb{Q}$	$q - 4q^{3} + 5q^{5} - 16q^{7} + O(q^{8})$	−
80.4.a.c	1	$\mathbb{Q}$	$q - 4q^{3} + 5q^{5} + 16q^{7} + O(q^{8})$	−
80.4.a.d	1	$\mathbb{Q}$	$q - 2q^{3} - 5q^{5} - 6q^{7} + O(q^{8})$	+
80.4.a.e	1	$\mathbb{Q}$	$q + 6q^{3} - 5q^{5} + 34q^{7} + O(q^{8})$	+
80.4.a.f	1	$\mathbb{Q}$	$q + 8q^{3} + 5q^{5} + 4q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
80.4.a.a	$x - 1606$
80.4.a.b	$x - 1306$
80.4.a.c	$x + 1446$
80.4.a.d	$x + 586$
80.4.a.e	$x + 1114$
80.4.a.f	$x + 966$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(80))$	6
Old forms	24
Cusp forms $S_{4}(\Gamma_0(80))$	30
Eisenstein series $E_{4}(\Gamma_0(80))$	12
Modular forms $M_{4}(\Gamma_0(80))$	42

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.174 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, 3, 3, 5, 7, 8, 12, 22, 40] (certified at ell = 2147483629)
T = [(3, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 6 = 1 x 6 (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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 80:4 1000

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