Atlas › Modular forms › Weight 10 › Level 33 › 33.10.a

Newspace 33.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(33))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
33.10.a.a	3	degree 3	$3q - 51q^{2} + 243q^{3} + 1425q^{4} - 1362q^{5} - 4131q^{6} - 15090q^{7} + O(q^{8})$	− −	$-1$
33.10.a.b	3	degree 3	$3q + 3q^{2} - 243q^{3} - 315q^{4} + 2388q^{5} - 243q^{6} - 228q^{7} + O(q^{8})$	+ +	$-1$
33.10.a.c	4	degree 4	$4q - 3q^{2} + 324q^{3} + 709q^{4} + 2388q^{5} - 243q^{6} + 5372q^{7} + O(q^{8})$	− +	$+1$
33.10.a.d	4	degree 4	$4q + 19q^{2} - 324q^{3} + 1937q^{4} + 1138q^{5} - 1539q^{6} + 8122q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
33.10.a.a	$x^{3} + 1413x^{2} - 1055154x + 104055784$
33.10.a.b	$x^{3} - 2391x^{2} + 1293552x + 146425460$
33.10.a.c	$x^{4} - 2385x^{3} - 3246402x^{2} + 10250147072x - 5302309177728$
33.10.a.d	$x^{4} - 1157x^{3} - 4118808x^{2} + 3761411332x - 650348382608$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(33))$	14
Old forms	20
Cusp forms $S_{10}(\Gamma_0(33))$	34
Eisenstein series $E_{10}(\Gamma_0(33))$	4
Modular forms $M_{10}(\Gamma_0(33))$	38

Decomposition of the old subspace

$S_{10}^{\mathrm{old}}(\Gamma_0(33)) =$ $S_{10}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.327 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, 4, 10, 38] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 6 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, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 33:10 1000

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