Atlas › Modular forms › Weight 2 › Level 928 › 928.2.a

Newspace 928.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(928))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
928.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + O(q^{8})$	+	928b1 rank 1
928.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - q^{5} + O(q^{8})$	+	928a1 rank 1
928.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	−
928.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 6q^{5} + O(q^{8})$	+
928.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} + 2q^{5} + 4q^{7} + O(q^{8})$	−
928.2.a.f	4	degree 4	$4q - 8q^{5} + O(q^{8})$	−
928.2.a.g	5	degree 5	$5q - 4q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	+
928.2.a.h	5	degree 5	$5q + 4q^{3} - 2q^{5} + 4q^{7} + O(q^{8})$	+
928.2.a.i	6	degree 6	$6q + 4q^{5} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ 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 $T$
928.2.a.a	$x - 6$
928.2.a.b	$x + 72$
928.2.a.c	$x^{2} + 100x + 2162$
928.2.a.d	$x^{2} - 46x - 316$
928.2.a.e	$x^{2} - 128x + 3518$
928.2.a.f	$x^{4} + 168x^{3} + 8964x^{2} + 172152x + 1075356$
928.2.a.g	$x^{5} + 152x^{4} - 588x^{3} - 621400x^{2} - 3230760x + 99878864$
928.2.a.h	$x^{5} - 156x^{4} + 5416x^{3} + 96216x^{2} - 3597416x - 21816272$
928.2.a.i	$x^{6} - 124x^{5} - 20684x^{4} + 1643424x^{3} + 150893380x^{2} - 1492745456x - 164903206768$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(928))$	28
Old forms	85
Cusp forms $S_{2}(\Gamma_0(928))$	113
Eisenstein series $E_{2}(\Gamma_0(928))$	15
Modular forms $M_{2}(\Gamma_0(928))$	128

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(928)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(116))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(232))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(464))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.747 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, 2, 4, 8, 16, 29, 32, 58, 116, 232, 464, 928]: [0, 1, 2, 3, 4, 3, 6, 9, 18, 34, 64, 124] (certified at ell = 2147483353)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 8 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(928, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 928:2 1000

← 927.2.a · 929.2.a →