Atlas › Modular forms › Weight 6 › Level 33 › 33.6.a

Newspace 33.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(33))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
33.6.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 9q^{3} - 28q^{4} + 46q^{5} + 18q^{6} + 148q^{7} + O(q^{8})$	+ −	$+1$
33.6.a.b	1	$\mathbb{Q}$	$q + q^{2} + 9q^{3} - 31q^{4} - 92q^{5} + 9q^{6} - 26q^{7} + O(q^{8})$	− −	$-1$
33.6.a.c	2	$\mathbb{Q}(\sqrt{177})$	$2q - 5q^{2} - 18q^{3} + 37q^{4} + 58q^{5} + 45q^{6} - 286q^{7} + O(q^{8})$	+ +	$-1$
33.6.a.d	2	$\mathbb{Q}(\sqrt{313})$	$2q + q^{2} - 18q^{3} + 93q^{4} - 38q^{5} - 9q^{6} - 18q^{7} + O(q^{8})$	+ −	$+1$
33.6.a.e	2	$\mathbb{Q}(\sqrt{33})$	$2q + 13q^{2} + 18q^{3} + 37q^{4} + 58q^{5} + 117q^{6} + 146q^{7} + O(q^{8})$	− +	$+1$

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 $T$
33.6.a.a	$x - 44$
33.6.a.b	$x + 91$
33.6.a.c	$x^{2} - 53x - 2882$
33.6.a.d	$x^{2} + 37x - 9126$
33.6.a.e	$x^{2} - 71x + 592$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(33))$	8
Old forms	10
Cusp forms $S_{6}(\Gamma_0(33))$	18
Eisenstein series $E_{6}(\Gamma_0(33))$	4
Modular forms $M_{6}(\Gamma_0(33))$	22

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(33)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.227 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, 6, 22] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(33, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 33:6 1000

← 32.6.a · 34.6.a →