Atlas › Modular forms › Weight 8 › Level 33 › 33.8.a

Newspace 33.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(33))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
33.8.a.a	1	$\mathbb{Q}$	$q + 10q^{2} + 27q^{3} - 28q^{4} - 410q^{5} + 270q^{6} - 1028q^{7} + O(q^{8})$	− +	$-1$
33.8.a.b	2	$\mathbb{Q}(\sqrt{177})$	$2q - 19q^{2} + 54q^{3} + 13q^{4} - 34q^{5} - 513q^{6} - 166q^{7} + O(q^{8})$	− +	$-1$
33.8.a.c	2	$\mathbb{Q}(\sqrt{97})$	$2q + q^{2} - 54q^{3} + 181q^{4} - 194q^{5} - 27q^{6} - 418q^{7} + O(q^{8})$	+ −	$-1$
33.8.a.d	3	degree 3	$3q + 9q^{2} - 81q^{3} - 15q^{4} - 444q^{5} - 243q^{6} + 1614q^{7} + O(q^{8})$	+ +	$+1$
33.8.a.e	4	degree 4	$4q + 15q^{2} + 108q^{3} + 565q^{4} + 306q^{5} + 405q^{6} + 890q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
33.8.a.a	$x + 400$
33.8.a.b	$x^{2} + 53x - 66602$
33.8.a.c	$x^{2} + 193x - 27572$
33.8.a.d	$x^{3} + 435x^{2} - 40200x - 1533356$
33.8.a.e	$x^{4} - 321x^{3} - 132750x^{2} + 38510636x + 1253935800$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(33))$	12
Old forms	14
Cusp forms $S_{8}(\Gamma_0(33))$	26
Eisenstein series $E_{8}(\Gamma_0(33))$	4
Modular forms $M_{8}(\Gamma_0(33))$	30

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(33)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 3, 11, 33]: [1, 3, 8, 30] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 4 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(33, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 33:8 1000

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