Atlas › Modular forms › Weight 2 › Level 824 › 824.2.a

Newspace 824.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(824))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
824.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 4q^{3} + 2q^{5} - q^{7} + O(q^{8})$	+
824.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 5q^{5} + O(q^{8})$	−
824.2.a.c	2	$\mathbb{Q}(\sqrt{21})$	$2q - q^{7} + O(q^{8})$	−
824.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} - q^{5} - 4q^{7} + O(q^{8})$	+
824.2.a.e	9	degree 9	$9q + 7q^{5} + q^{7} + O(q^{8})$	+
824.2.a.f	9	degree 9	$9q + 4q^{3} - 5q^{5} + 5q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
824.2.a.a	$x^{2} + 35x + 245$
824.2.a.b	$x^{2} + 10x - 1100$
824.2.a.c	$x^{2} + 113x + 2935$
824.2.a.d	$x^{2} + 68x + 836$
824.2.a.e	$x^{9} - 159x^{8} - 3422x^{7} + 1428807x^{6} - 30314200x^{5} - 3625686447x^{4} + 139937303251x^{3} + 1677757391290x^{2} - 118330251693084x + 966600405539240$
824.2.a.f	$x^{9} - 83x^{8} - 19622x^{7} + 1746633x^{6} + 79393018x^{5} - 7090876847x^{4} - 141825530167x^{3} + 9155769423932x^{2} + 103677897787744x - 2986150020336632$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(824))$	26
Old forms	75
Cusp forms $S_{2}(\Gamma_0(824))$	101
Eisenstein series $E_{2}(\Gamma_0(824))$	7
Modular forms $M_{2}(\Gamma_0(824))$	108

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(824)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(103))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(206))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(412))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.394 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, 103, 206, 412, 824]: [0, 1, 2, 3, 9, 28, 55, 108] (certified at ell = 2147482501)
T = [(3, 1), (5, 3), (13, 5), (17, 7), (19, 9)]; h of degree 26 = 1 x 26 (phi(ord eps) x dim S_k^new); 7 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(824, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 824:2 1000

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