Atlas › Modular forms › Weight 4 › Level 225 › 225.4.a

Newspace 225.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(225))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
225.4.a.a	1	$\mathbb{Q}$	$q - 5q^{2} + 17q^{4} + 30q^{7} + O(q^{8})$
225.4.a.b	1	$\mathbb{Q}$	$q - 4q^{2} + 8q^{4} - 6q^{7} + O(q^{8})$
225.4.a.c	1	$\mathbb{Q}$	$q - q^{2} - 7q^{4} + 6q^{7} + O(q^{8})$
225.4.a.d	1	$\mathbb{Q}$	$q - 8q^{4} - 20q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$
225.4.a.e	1	$\mathbb{Q}$	$q + q^{2} - 7q^{4} - 6q^{7} + O(q^{8})$
225.4.a.f	1	$\mathbb{Q}$	$q + q^{2} - 7q^{4} + 24q^{7} + O(q^{8})$
225.4.a.g	1	$\mathbb{Q}$	$q + 3q^{2} + q^{4} - 20q^{7} + O(q^{8})$
225.4.a.h	1	$\mathbb{Q}$	$q + 5q^{2} + 17q^{4} + 30q^{7} + O(q^{8})$
225.4.a.i	2	$\mathbb{Q}(\sqrt{41})$	$2q - 3q^{2} + 9q^{4} + 6q^{7} + O(q^{8})$
225.4.a.j	2	$\mathbb{Q}(\sqrt{19})$	$2q - 2q^{2} + 24q^{4} - 26q^{7} + O(q^{8})$
225.4.a.k	2	$\mathbb{Q}(\sqrt{5})$	$2q - 6q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-15})$
225.4.a.l	2	$\mathbb{Q}(\sqrt{10})$	$2q + 4q^{4} - 30q^{7} + O(q^{8})$
225.4.a.m	2	$\mathbb{Q}(\sqrt{10})$	$2q + 4q^{4} + 30q^{7} + O(q^{8})$
225.4.a.n	2	$\mathbb{Q}(\sqrt{19})$	$2q + 2q^{2} + 24q^{4} + 26q^{7} + O(q^{8})$
225.4.a.o	2	$\mathbb{Q}(\sqrt{41})$	$2q + 3q^{2} + 9q^{4} - 6q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7}$ 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 $T$
225.4.a.a	$x - 85$
225.4.a.b	$x + 22$
225.4.a.c	$x - 17$
225.4.a.d	$x + 60$
225.4.a.e	$x + 17$
225.4.a.f	$x - 73$
225.4.a.g	$x + 57$
225.4.a.h	$x - 95$
225.4.a.i	$x^{2} - 15x - 3644$
225.4.a.j	$x^{2} + 80x - 699$
225.4.a.k	$x^{2} - 5$
225.4.a.l	$x^{2} + 90x + 2015$
225.4.a.m	$x^{2} - 90x + 2015$
225.4.a.n	$x^{2} - 80x - 699$
225.4.a.o	$x^{2} + 15x - 3644$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(225))$	22
Old forms	56
Cusp forms $S_{4}(\Gamma_0(225))$	78
Eisenstein series $E_{4}(\Gamma_0(225))$	24
Modular forms $M_{4}(\Gamma_0(225))$	102

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(225)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(25))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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, 3, 5, 9, 15, 25, 45, 75, 225]: [1, 2, 3, 4, 8, 9, 20, 32, 92] (certified at ell = 2147482921)
T = [(2, 1), (7, 3)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 5 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(225, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 225:4 1000

← 224.4.a · 226.4.a →