Atlas › Modular forms › Weight 2 › Level 475 › 475.2.a

Newspace 475.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(475))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
475.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 2q^{7} + O(q^{8})$	−	475c1 rank 1
475.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} - 2q^{4} + q^{7} + O(q^{8})$	−	475a1 rank 0
475.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - 2q^{7} + O(q^{8})$	−	475b1 rank 1
475.2.a.d	3	degree 3	$3q - 4q^{2} - 2q^{3} + 4q^{4} + 5q^{6} + O(q^{8})$	+
475.2.a.e	3	degree 3	$3q - 2q^{2} - 2q^{3} + 4q^{4} - 3q^{6} - 4q^{7} + O(q^{8})$	−
475.2.a.f	3	degree 3	$3q - q^{2} - 2q^{3} + q^{4} - 4q^{6} + O(q^{8})$	+
475.2.a.g	3	degree 3	$3q + 2q^{2} + 2q^{3} + 4q^{4} - 3q^{6} + 4q^{7} + O(q^{8})$	−
475.2.a.h	3	degree 3	$3q + 4q^{2} + 2q^{3} + 4q^{4} + 5q^{6} + O(q^{8})$	+
475.2.a.i	4	degree 4	$4q + 2q^{2} - 2q^{3} + 8q^{4} - 4q^{7} + O(q^{8})$	−
475.2.a.j	6	degree 6	$6q + 8q^{4} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
475.2.a.a	$x + 1$
475.2.a.b	$x - 2$
475.2.a.c	$x - 1$
475.2.a.d	$x^{3} + 6x^{2} + 5x - 13$
475.2.a.e	$x^{3} + 4x^{2} + x - 1$
475.2.a.f	$x^{3} + 3x^{2} - x - 1$
475.2.a.g	$x^{3} - 4x^{2} + x + 1$
475.2.a.h	$x^{3} - 6x^{2} + 5x + 13$
475.2.a.i	$x^{4} - 18x^{2} - 34x - 11$
475.2.a.j	$x^{6} - 26x^{4} + 171x^{2} - 324$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(475))$	28
Old forms	17
Cusp forms $S_{2}(\Gamma_0(475))$	45
Eisenstein series $E_{2}(\Gamma_0(475))$	11
Modular forms $M_{2}(\Gamma_0(475))$	56

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(475)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(95))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 5, 19, 25, 95, 475]: [0, 1, 2, 3, 12, 52] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (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(475, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 475:2 1000

← 474.2.a · 476.2.a →