Atlas › Modular forms › Weight 2 › Level 633 › 633.2.a

Newspace 633.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(633))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
633.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 3q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ +	$-1$	633a1 rank 1
633.2.a.b	4	degree 4	$4q - 2q^{2} + 4q^{3} - 3q^{5} - 2q^{6} - 11q^{7} + O(q^{8})$	− −	$-1$
633.2.a.c	8	degree 8	$8q - 3q^{2} - 8q^{3} + 9q^{4} - 2q^{5} + 3q^{6} - 11q^{7} + O(q^{8})$	+ +	$-1$
633.2.a.d	8	degree 8	$8q + 4q^{2} - 8q^{3} + 8q^{4} + 3q^{5} - 4q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$
633.2.a.e	14	degree 14	$14q + 3q^{2} + 14q^{3} + 23q^{4} - q^{5} + 3q^{6} + 13q^{7} + O(q^{8})$	− +	$+1$

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 $T$
633.2.a.a	$x + 4$
633.2.a.b	$x^{4} + 5x^{3} + 4x^{2} - 10x - 11$
633.2.a.c	$x^{8} + 5x^{7} - 17x^{6} - 42x^{5} + 94x^{4} + 56x^{3} - 159x^{2} + 65x - 4$
633.2.a.d	$x^{8} - 7x^{7} - 10x^{6} + 128x^{5} - 114x^{4} - 393x^{3} + 770x^{2} - 480x + 101$
633.2.a.e	$x^{14} - 2x^{13} - 78x^{12} + 213x^{11} + 2033x^{10} - 6936x^{9} - 19061x^{8} + 84163x^{7} + 30317x^{6} - 318258x^{5} + 71599x^{4} + 351040x^{3} - 34375x^{2} - 39344x - 24$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(633))$	35
Old forms	34
Cusp forms $S_{2}(\Gamma_0(633))$	69
Eisenstein series $E_{2}(\Gamma_0(633))$	3
Modular forms $M_{2}(\Gamma_0(633))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(633)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(211))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.219 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, 211, 633]: [0, 1, 18, 72] (certified at ell = 2147482681)
T = [(2, 1), (5, 1)]; h of degree 35 = 1 x 35 (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(633, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 633:2 1000

← 632.2.a · 634.2.a →