Atlas › Modular forms › Weight 2 › Level 634 › 634.2.a

Newspace 634.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(634))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
634.2.a.a	4	degree 4	$4q + 4q^{2} - 3q^{3} + 4q^{4} - 7q^{5} - 3q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
634.2.a.b	6	degree 6	$6q - 6q^{2} - 4q^{3} + 6q^{4} - 7q^{5} + 4q^{6} + 3q^{7} + O(q^{8})$	+ +	$-1$
634.2.a.c	7	degree 7	$7q - 7q^{2} + 5q^{3} + 7q^{4} + 4q^{5} - 5q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
634.2.a.d	9	degree 9	$9q + 9q^{2} + 2q^{3} + 9q^{4} + 10q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
634.2.a.a	$x^{4} + 10x^{3} + 34x^{2} + 45x + 19$
634.2.a.b	$x^{6} + 11x^{5} + 31x^{4} + 13x^{3} - 60x^{2} - 82x - 29$
634.2.a.c	$x^{7} - 9x^{6} - 8x^{5} + 196x^{4} - 52x^{3} - 1149x^{2} - 55x + 13$
634.2.a.d	$x^{9} - 12x^{8} + 30x^{7} + 161x^{6} - 1026x^{5} + 2015x^{4} - 1470x^{3} + 187x^{2} + 54x - 5$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(634))$	26
Old forms	52
Cusp forms $S_{2}(\Gamma_0(634))$	78
Eisenstein series $E_{2}(\Gamma_0(634))$	3
Modular forms $M_{2}(\Gamma_0(634))$	81

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(634)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(317))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.217 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, 2, 317, 634]: [0, 1, 27, 81] (certified at ell = 2147479753)
T = [(3, 1), (5, 1)]; h of degree 26 = 1 x 26 (phi(ord eps) x dim S_k^new); 3 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(634, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 634:2 1000

← 633.2.a · 635.2.a →