Atlas › Modular forms › Weight 2 › Level 725 › 725.2.a

Newspace 725.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(725))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
725.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + 2q^{7} + O(q^{8})$	+	725a1 rank 1
725.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 6q^{6} + O(q^{8})$	−
725.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 4q^{3} + 2q^{4} + 4q^{6} + 4q^{7} + O(q^{8})$	−
725.2.a.d	3	degree 3	$3q - 3q^{2} + 2q^{3} + 5q^{4} + 4q^{6} + 2q^{7} + O(q^{8})$	−
725.2.a.e	3	degree 3	$3q - q^{2} - 2q^{3} + q^{4} - 4q^{6} - 4q^{7} + O(q^{8})$	+
725.2.a.f	4	degree 4	$4q - 4q^{6} + O(q^{8})$	−
725.2.a.g	4	degree 4	$4q + 4q^{4} - 6q^{6} + O(q^{8})$	+
725.2.a.h	5	degree 5	$5q - 2q^{2} - 6q^{3} + 4q^{4} - q^{6} - 6q^{7} + O(q^{8})$	−
725.2.a.i	5	degree 5	$5q - 6q^{3} + 4q^{4} - q^{6} - 10q^{7} + O(q^{8})$	+
725.2.a.j	5	degree 5	$5q + 6q^{3} + 4q^{4} - q^{6} + 10q^{7} + O(q^{8})$	+
725.2.a.k	5	degree 5	$5q + 2q^{2} + 6q^{3} + 4q^{4} - q^{6} + 6q^{7} + O(q^{8})$	−
725.2.a.l	6	degree 6	$6q + 14q^{4} + 14q^{6} + O(q^{8})$	+

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 $T$
725.2.a.a	$x - 1$
725.2.a.b	$x^{2} + 4x - 4$
725.2.a.c	$x^{2} - 14x + 47$
725.2.a.d	$x^{3} - 3x^{2} - 67x - 17$
725.2.a.e	$x^{3} + 7x^{2} - 21x - 25$
725.2.a.f	$x^{4} - 28x^{2} + 121$
725.2.a.g	$x^{4} - 51x^{2} + 576$
725.2.a.h	$x^{5} + 20x^{4} + 79x^{3} - 408x^{2} - 2296x - 2285$
725.2.a.i	$x^{5} + 18x^{4} + 59x^{3} - 424x^{2} - 1736x + 2499$
725.2.a.j	$x^{5} - 18x^{4} + 59x^{3} + 424x^{2} - 1736x - 2499$
725.2.a.k	$x^{5} - 20x^{4} + 79x^{3} + 408x^{2} - 2296x + 2285$
725.2.a.l	$x^{6} - 190x^{4} + 9107x^{2} - 87616$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(725))$	45
Old forms	24
Cusp forms $S_{2}(\Gamma_0(725))$	69
Eisenstein series $E_{2}(\Gamma_0(725))$	11
Modular forms $M_{2}(\Gamma_0(725))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(725)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(145))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.248 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, 25, 29, 145, 725]: [0, 1, 3, 3, 16, 76] (certified at ell = 2147482681)
T = [(2, 1), (3, 3)]; h of degree 45 = 1 x 45 (phi(ord eps) x dim S_k^new); 6 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(725, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 725:2 1000

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