Atlas › Modular forms › Weight 2 › Level 425 › 425.2.a

Newspace 425.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(425))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
425.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} - q^{4} + 2q^{6} + 2q^{7} + O(q^{8})$	+	425d1 rank 1
425.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{6} - q^{7} + O(q^{8})$	+	425c1 rank 1
425.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{6} + q^{7} + O(q^{8})$	−	425b1 rank 1
425.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - 4q^{7} + O(q^{8})$	+	425a1 rank 1
425.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{4} - 6q^{6} + 2q^{7} + O(q^{8})$	−
425.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 4q^{3} + 2q^{4} + 4q^{7} + O(q^{8})$	−
425.2.a.g	4	degree 4	$4q - 2q^{2} - 4q^{3} + 4q^{4} - 10q^{7} + O(q^{8})$	−
425.2.a.h	4	degree 4	$4q + 2q^{2} + 4q^{3} + 4q^{4} + 10q^{7} + O(q^{8})$	+
425.2.a.i	5	degree 5	$5q - q^{2} + q^{3} + 11q^{4} + 3q^{6} + q^{7} + O(q^{8})$	−
425.2.a.j	5	degree 5	$5q + q^{2} - q^{3} + 11q^{4} + 3q^{6} - q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
425.2.a.a	$x + 7$
425.2.a.b	$x - 2$
425.2.a.c	$x + 2$
425.2.a.d	$x - 1$
425.2.a.e	$x^{2} + 6x - 3$
425.2.a.f	$x^{2} - 14x + 41$
425.2.a.g	$x^{4} + 14x^{3} + 20x^{2} - 254x - 637$
425.2.a.h	$x^{4} - 14x^{3} + 20x^{2} + 254x - 637$
425.2.a.i	$x^{5} - 2x^{4} - 112x^{3} + 390x^{2} + 2604x - 10650$
425.2.a.j	$x^{5} + 2x^{4} - 112x^{3} - 390x^{2} + 2604x + 10650$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(425))$	26
Old forms	13
Cusp forms $S_{2}(\Gamma_0(425))$	39
Eisenstein series $E_{2}(\Gamma_0(425))$	11
Modular forms $M_{2}(\Gamma_0(425))$	50

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(425)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(85))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.308 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, 17, 25, 85, 425]: [0, 1, 2, 3, 10, 46] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 26 = 1 x 26 (phi(ord eps) x dim S_k^new); 4 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(425, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 425:2 1000

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