Atlas › Modular forms › Weight 2 › Level 275 › 275.2.a

Newspace 275.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(275))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
275.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + O(q^{8})$	+	275a1 rank 1
275.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + 2q^{6} + 2q^{7} + O(q^{8})$	−	275b1 rank 0
275.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 8q^{6} + 4q^{7} + O(q^{8})$	−
275.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 3q^{3} - q^{4} + 4q^{6} - q^{7} + O(q^{8})$	−
275.2.a.e	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} - q^{3} + 3q^{4} - 6q^{6} - 5q^{7} + O(q^{8})$	+
275.2.a.f	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} + q^{3} + 3q^{4} - 6q^{6} + 5q^{7} + O(q^{8})$	+
275.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 3q^{3} - q^{4} + 4q^{6} + q^{7} + O(q^{8})$	−
275.2.a.h	4	degree 4	$4q + 6q^{4} + 8q^{6} + O(q^{8})$	+

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 $T$
275.2.a.a	$x + 1$
275.2.a.b	$x - 5$
275.2.a.c	$x^{2} + 2x - 49$
275.2.a.d	$x^{2} + 10x + 5$
275.2.a.e	$x^{2} + 4x - 9$
275.2.a.f	$x^{2} - 4x - 9$
275.2.a.g	$x^{2} - 10x + 5$
275.2.a.h	$x^{4} - 94x^{2} + 1681$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(275))$	16
Old forms	9
Cusp forms $S_{2}(\Gamma_0(275))$	25
Eisenstein series $E_{2}(\Gamma_0(275))$	11
Modular forms $M_{2}(\Gamma_0(275))$	36

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(275)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.19 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, 11, 25, 55, 275]: [0, 1, 2, 3, 8, 32] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 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(275, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 275:2 1000

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