Atlas › Modular forms › Weight 2 › Level 635 › 635.2.a

Newspace 635.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(635))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
635.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− −	$-1$	635b1 rank 1
635.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + q^{5} - q^{7} + O(q^{8})$	− −	$-1$	635a1 rank 1
635.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 2q^{3} - q^{4} + 2q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
635.2.a.d	4	degree 4	$4q + q^{3} - 8q^{4} - 4q^{5} - q^{7} + O(q^{8})$	+ −	$+1$
635.2.a.e	8	degree 8	$8q - 2q^{2} - 2q^{3} + 2q^{4} - 8q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
635.2.a.f	9	degree 9	$9q + 3q^{2} + 3q^{3} + 25q^{4} - 9q^{5} + q^{6} + 12q^{7} + O(q^{8})$	+ −	$+1$
635.2.a.g	18	degree 18	$18q + q^{2} + 27q^{4} + 18q^{5} + 5q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$

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 $T$
635.2.a.a	$x + 3$
635.2.a.b	$x - 1$
635.2.a.c	$x^{2} + x - 1$
635.2.a.d	$x^{4} - x^{3} - 6x^{2} + 2x + 1$
635.2.a.e	$x^{8} + 4x^{7} - 12x^{6} - 35x^{5} + 60x^{4} + 35x^{3} - 50x^{2} + 14x - 1$
635.2.a.f	$x^{9} - 6x^{8} - 22x^{7} + 141x^{6} + 202x^{5} - 1042x^{4} - 1005x^{3} + 2232x^{2} + 984x - 1481$
635.2.a.g	$x^{18} - x^{17} - 80x^{16} + 103x^{15} + 2396x^{14} - 3561x^{13} - 34074x^{12} + 52776x^{11} + 247345x^{10} - 348367x^{9} - 992820x^{8} + 1041545x^{7} + 2305894x^{6} - 1197943x^{5} - 2916555x^{4} - 22994x^{3} + 1411228x^{2} + 511278x + 9533$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(635))$	43
Old forms	20
Cusp forms $S_{2}(\Gamma_0(635))$	63
Eisenstein series $E_{2}(\Gamma_0(635))$	3
Modular forms $M_{2}(\Gamma_0(635))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(635)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(127))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.22 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, 5, 127, 635]: [0, 1, 11, 66] (certified at ell = 2147483269)
T = [(2, 1), (3, 1)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 5 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(635, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 635:2 1000

← 634.2.a · 636.2.a →