Atlas › Modular forms › Weight 8 › Level 36 › 36.8.a

Newspace 36.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(36))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
36.8.a.a	1	$\mathbb{Q}$	$q - 270q^{5} + 1112q^{7} + O(q^{8})$
36.8.a.b	1	$\mathbb{Q}$	$q - 508q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$
36.8.a.c	1	$\mathbb{Q}$	$q + 378q^{5} - 832q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
36.8.a.a	$x - 3066$
36.8.a.b	$x + 1524$
36.8.a.c	$x + 2118$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(36))$	3
Old forms	33
Cusp forms $S_{8}(\Gamma_0(36))$	36
Eisenstein series $E_{8}(\Gamma_0(36))$	12
Modular forms $M_{8}(\Gamma_0(36))$	48

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(36)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 6}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 6}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 3}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(18))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.293 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, 36]: [1, 3, 3, 5, 9, 8, 17, 23, 45] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; h of degree 3 = 1 x 3 (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(36, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 36:8 1000

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