Atlas › Modular forms › Weight 2 › Level 525 › 525.2.a

Newspace 525.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(525))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
525.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{6} - q^{7} + O(q^{8})$	+ +	525a1 rank 1
525.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{6} - q^{7} + O(q^{8})$	− +	525d1 rank 1
525.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{6} + q^{7} + O(q^{8})$	+ −	525c1 rank 1
525.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{6} + q^{7} + O(q^{8})$	+ −	525b1 rank 0
525.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} - 2q^{3} + 3q^{4} + 3q^{6} + 2q^{7} + O(q^{8})$	+ −
525.2.a.f	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} - 2q^{3} + 3q^{4} + q^{6} - 2q^{7} + O(q^{8})$	+ +
525.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} + 6q^{4} - 2q^{7} + O(q^{8})$	− +
525.2.a.h	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} + 2q^{3} + 3q^{4} + q^{6} + 2q^{7} + O(q^{8})$	− −
525.2.a.i	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{6} - 2q^{7} + O(q^{8})$	− +
525.2.a.j	3	degree 3	$3q - q^{2} + 3q^{3} + 5q^{4} - q^{6} + 3q^{7} + O(q^{8})$	− −
525.2.a.k	3	degree 3	$3q + q^{2} - 3q^{3} + 5q^{4} - q^{6} - 3q^{7} + O(q^{8})$	+ +

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{11}$ 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 $T$
525.2.a.a	$x + 1$
525.2.a.b	$x + 19$
525.2.a.c	$x + 17$
525.2.a.d	$x - 13$
525.2.a.e	$x^{2} + 9x - 191$
525.2.a.f	$x^{2} + 19x + 87$
525.2.a.g	$x^{2} - 12x - 209$
525.2.a.h	$x^{2} + 17x + 69$
525.2.a.i	$x^{2} + 3x - 149$
525.2.a.j	$x^{3} - 17x^{2} + 91x - 151$
525.2.a.k	$x^{3} - 19x^{2} + 115x - 221$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(525))$	20
Old forms	49
Cusp forms $S_{2}(\Gamma_0(525))$	69
Eisenstein series $E_{2}(\Gamma_0(525))$	23
Modular forms $M_{2}(\Gamma_0(525))$	92

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(525)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(105))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(175))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.365 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, 7, 15, 21, 25, 35, 75, 105, 175, 525]: [0, 1, 1, 1, 4, 4, 3, 6, 12, 20, 22, 84] (certified at ell = 2147482921)
T = [(2, 1), (11, 3)]; h of degree 20 = 1 x 20 (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(525, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 525:2 1000

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