Atlas › Modular forms › Weight 2 › Level 533 › 533.2.a

Newspace 533.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(533))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
533.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 6q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
533.2.a.b	7	degree 7	$7q - 2q^{2} - 8q^{3} + 2q^{4} - 2q^{5} + 5q^{6} - 10q^{7} + O(q^{8})$	− −	$-1$
533.2.a.c	8	degree 8	$8q + q^{2} + 4q^{3} + 5q^{4} + 4q^{5} - 5q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$
533.2.a.d	11	degree 11	$11q - 2q^{2} - 8q^{3} + 10q^{4} - 4q^{5} - 7q^{6} - 18q^{7} + O(q^{8})$	+ +	$-1$
533.2.a.e	13	degree 13	$13q + 10q^{3} + 16q^{4} + q^{6} + 10q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
533.2.a.a	$x^{2} - 4x - 4$
533.2.a.b	$x^{7} + 10x^{6} + 24x^{5} - 34x^{4} - 166x^{3} - 64x^{2} + 187x + 103$
533.2.a.c	$x^{8} - 5x^{7} - 12x^{6} + 74x^{5} + 4x^{4} - 214x^{3} + 39x^{2} + 70x - 21$
533.2.a.d	$x^{11} + 10x^{10} + 16x^{9} - 86x^{8} - 187x^{7} + 240x^{6} + 405x^{5} - 259x^{4} - 226x^{3} + 102x^{2} - 5x - 1$
533.2.a.e	$x^{13} - 10x^{12} + 2x^{11} + 272x^{10} - 769x^{9} - 1302x^{8} + 7205x^{7} - 2055x^{6} - 20182x^{5} + 15170x^{4} + 21447x^{3} - 18261x^{2} - 7616x + 6272$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(533))$	41
Old forms	6
Cusp forms $S_{2}(\Gamma_0(533))$	47
Eisenstein series $E_{2}(\Gamma_0(533))$	3
Modular forms $M_{2}(\Gamma_0(533))$	50

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(533)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.197 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, 13, 41, 533]: [0, 1, 4, 50] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (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(533, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 533:2 1000

← 532.2.a · 534.2.a →