Atlas › Modular forms › Weight 8 › Level 30 › 30.8.a

Newspace 30.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(30))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
30.8.a.a	1	$\mathbb{Q}$	$q - 8q^{2} - 27q^{3} + 64q^{4} - 125q^{5} + 216q^{6} - 1084q^{7} + O(q^{8})$	+ + +	$+1$
30.8.a.b	1	$\mathbb{Q}$	$q - 8q^{2} - 27q^{3} + 64q^{4} + 125q^{5} + 216q^{6} + 416q^{7} + O(q^{8})$	+ + −	$-1$
30.8.a.c	1	$\mathbb{Q}$	$q - 8q^{2} + 27q^{3} + 64q^{4} + 125q^{5} - 216q^{6} - 232q^{7} + O(q^{8})$	+ − −	$+1$
30.8.a.d	1	$\mathbb{Q}$	$q + 8q^{2} - 27q^{3} + 64q^{4} - 125q^{5} - 216q^{6} - 988q^{7} + O(q^{8})$	− + +	$-1$
30.8.a.e	1	$\mathbb{Q}$	$q + 8q^{2} - 27q^{3} + 64q^{4} + 125q^{5} - 216q^{6} + 512q^{7} + O(q^{8})$	− + −	$+1$
30.8.a.f	1	$\mathbb{Q}$	$q + 8q^{2} + 27q^{3} + 64q^{4} - 125q^{5} + 216q^{6} + 1604q^{7} + O(q^{8})$	− − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + T_{11}$ 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 $T$
30.8.a.a	$x - 5468$
30.8.a.b	$x + 3532$
30.8.a.c	$x + 292$
30.8.a.d	$x + 9028$
30.8.a.e	$x - 5972$
30.8.a.f	$x + 604$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(30))$	6
Old forms	32
Cusp forms $S_{8}(\Gamma_0(30))$	38
Eisenstein series $E_{8}(\Gamma_0(30))$	8
Modular forms $M_{8}(\Gamma_0(30))$	46

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(30)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 5, 6, 10, 15, 30]: [1, 3, 3, 5, 9, 13, 16, 46] (certified at ell = 2147483629)
T = [(7, 1), (11, 1)]; h of degree 6 = 1 x 6 (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(30, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 30:8 1000

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