Atlas › Modular forms › Weight 2 › Level 175 › 175.2.a

Newspace 175.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(175))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
175.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} + q^{7} + O(q^{8})$	−	175a1 rank 1
175.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} - q^{7} + O(q^{8})$	+	175b1 rank 1
175.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + 2q^{6} - q^{7} + O(q^{8})$	+	175c1 rank 0
175.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} + 4q^{6} - 2q^{7} + O(q^{8})$	+
175.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 2q^{3} - q^{4} + 4q^{6} + 2q^{7} + O(q^{8})$	−
175.2.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + q^{3} + 5q^{4} - 8q^{6} + 2q^{7} + O(q^{8})$	−

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 $T$
175.2.a.a	$x + 5$
175.2.a.b	$x + 3$
175.2.a.c	$x - 5$
175.2.a.d	$x^{2} - 5x - 55$
175.2.a.e	$x^{2} + 5x - 55$
175.2.a.f	$x^{2} - 4x - 13$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(175))$	9
Old forms	6
Cusp forms $S_{2}(\Gamma_0(175))$	15
Eisenstein series $E_{2}(\Gamma_0(175))$	11
Modular forms $M_{2}(\Gamma_0(175))$	26

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(175)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 2}$

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, 7, 25, 35, 175]: [0, 1, 1, 3, 6, 22] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 2 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(175, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 175:2 1000

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