Atlas › Modular forms › Weight 4 › Level 90 › 90.4.a

Newspace 90.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(90))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
90.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} - 5q^{5} - 4q^{7} + O(q^{8})$	+ +
90.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} - 5q^{5} + 14q^{7} + O(q^{8})$	+ +
90.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} + 5q^{5} - 4q^{7} + O(q^{8})$	+ −
90.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{4} - 5q^{5} + 32q^{7} + O(q^{8})$	− +
90.4.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{4} + 5q^{5} + 14q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31}$ 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 $T$
90.4.a.a	$x - 88$
90.4.a.b	$x - 412$
90.4.a.c	$x - 1528$
90.4.a.d	$x + 2396$
90.4.a.e	$x + 3440$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(90))$	5
Old forms	41
Cusp forms $S_{4}(\Gamma_0(90))$	46
Eisenstein series $E_{4}(\Gamma_0(90))$	16
Modular forms $M_{4}(\Gamma_0(90))$	62

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(90)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(18))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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, 5, 6, 9, 10, 15, 18, 30, 45, 90]: [1, 2, 2, 3, 5, 4, 7, 8, 11, 22, 20, 58] (certified at ell = 2147483629)
T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 5 = 1 x 5 (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(90, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 90:4 1000

← 89.4.a · 91.4.a →