Atlas › Modular forms › Weight 2 › Level 575 › 575.2.a

Newspace 575.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(575))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
575.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 2q^{3} + 2q^{4} + 4q^{6} - q^{7} + O(q^{8})$	−	575e1 rank 1
575.2.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - q^{7} + O(q^{8})$	+	575b1 rank 1
575.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{7} + O(q^{8})$	−	575d1 rank 1
575.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{7} + O(q^{8})$	+	575a1 rank 1
575.2.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{6} + q^{7} + O(q^{8})$	+	575c1 rank 0
575.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} - 5q^{6} - 2q^{7} + O(q^{8})$	+
575.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{6} + 2q^{7} + O(q^{8})$	−
575.2.a.h	4	degree 4	$4q - 2q^{2} + 2q^{3} + 4q^{4} - q^{6} + 3q^{7} + O(q^{8})$	−
575.2.a.i	4	degree 4	$4q - 2q^{3} + 2q^{4} - 6q^{6} - 6q^{7} + O(q^{8})$	−
575.2.a.j	4	degree 4	$4q + 2q^{3} + 2q^{4} - 6q^{6} + 6q^{7} + O(q^{8})$	+
575.2.a.k	7	degree 7	$7q - q^{2} + 11q^{4} + 5q^{6} + 3q^{7} + O(q^{8})$	+
575.2.a.l	7	degree 7	$7q + q^{2} + 11q^{4} + 5q^{6} - 3q^{7} + O(q^{8})$	−

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 $T$
575.2.a.a	$x + 4$
575.2.a.b	$x + 2$
575.2.a.c	$x + 1$
575.2.a.d	$x - 1$
575.2.a.e	$x - 4$
575.2.a.f	$x^{2} - x - 1$
575.2.a.g	$x^{2} - 5x + 5$
575.2.a.h	$x^{4} - 14x^{2} - 17x - 2$
575.2.a.i	$x^{4} + 2x^{3} - 5x^{2} + 1$
575.2.a.j	$x^{4} - 2x^{3} - 5x^{2} + 1$
575.2.a.k	$x^{7} + x^{6} - 35x^{5} - 11x^{4} + 303x^{3} + 74x^{2} - 658x + 7$
575.2.a.l	$x^{7} - x^{6} - 35x^{5} + 11x^{4} + 303x^{3} - 74x^{2} - 658x - 7$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(575))$	35
Old forms	20
Cusp forms $S_{2}(\Gamma_0(575))$	55
Eisenstein series $E_{2}(\Gamma_0(575))$	11
Modular forms $M_{2}(\Gamma_0(575))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(575)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(115))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.318 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, 23, 25, 115, 575]: [0, 1, 3, 3, 14, 62] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 35 = 1 x 35 (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(575, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 575:2 1000

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