Atlas › Modular forms › Weight 2 › Level 828 › 828.2.a

Newspace 828.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(828))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
828.2.a.a	1	$\mathbb{Q}$	$q - 2q^{5} + 2q^{7} + O(q^{8})$	−	828b1 rank 1
828.2.a.b	1	$\mathbb{Q}$	$q + 2q^{7} + O(q^{8})$	−	828d1 rank 0
828.2.a.c	1	$\mathbb{Q}$	$q + 2q^{5} - 4q^{7} + O(q^{8})$	+	828c1 rank 1
828.2.a.d	1	$\mathbb{Q}$	$q + 2q^{5} + 2q^{7} + O(q^{8})$	+	828a1 rank 0
828.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{5} + O(q^{8})$	+
828.2.a.f	2	$\mathbb{Q}(\sqrt{10})$	$2q + 4q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
828.2.a.a	$x + 48$
828.2.a.b	$x - 53$
828.2.a.c	$x + 91$
828.2.a.d	$x - 32$
828.2.a.e	$x^{2} + 40x - 1778$
828.2.a.f	$x^{2} + 4x - 246$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(828))$	8
Old forms	125
Cusp forms $S_{2}(\Gamma_0(828))$	133
Eisenstein series $E_{2}(\Gamma_0(828))$	23
Modular forms $M_{2}(\Gamma_0(828))$	156

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.601 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, 3, 4, 6, 9, 12, 18, 23, 36, 46, 69, 92, 138, 207, 276, 414, 828]: [0, 1, 1, 2, 3, 2, 5, 5, 3, 9, 8, 10, 15, 28, 26, 54, 76, 150] (certified at ell = 2147483647)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 8 = 1 x 8 (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(828, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 828:2 1000

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