Atlas › Modular forms › Weight 2 › Level 695 › 695.2.a

Newspace 695.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(695))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
695.2.a.a	6	degree 6	$6q - 4q^{2} - 4q^{3} + 2q^{4} + 6q^{5} + 3q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$
695.2.a.b	6	degree 6	$6q + 2q^{4} - 6q^{5} - 5q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
695.2.a.c	17	degree 17	$17q + 5q^{2} + 6q^{3} + 19q^{4} + 17q^{5} + 3q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$
695.2.a.d	18	degree 18	$18q - 2q^{2} + 2q^{3} + 26q^{4} - 18q^{5} + 7q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
695.2.a.a	$x^{6} + 8x^{5} + 14x^{4} - 14x^{3} - 27x^{2} + 18x + 1$
695.2.a.b	$x^{6} - 10x^{4} + 4x^{3} + 9x^{2} - 1$
695.2.a.c	$x^{17} - 11x^{16} - 5x^{15} + 431x^{14} - 978x^{13} - 5180x^{12} + 18527x^{11} + 23533x^{10} - 136027x^{9} - 10577x^{8} + 477607x^{7} - 243557x^{6} - 764724x^{5} + 723392x^{4} + 321244x^{3} - 617544x^{2} + 255097x - 34223$
695.2.a.d	$x^{18} - 82x^{16} - 6x^{15} + 2555x^{14} + 644x^{13} - 38859x^{12} - 18522x^{11} + 303416x^{10} + 219190x^{9} - 1141508x^{8} - 1122192x^{7} + 1595675x^{6} + 2088644x^{5} - 19344x^{4} - 694994x^{3} - 196545x^{2} + 3742x + 2203$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(695))$	47
Old forms	22
Cusp forms $S_{2}(\Gamma_0(695))$	69
Eisenstein series $E_{2}(\Gamma_0(695))$	3
Modular forms $M_{2}(\Gamma_0(695))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(695)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(139))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.226 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, 139, 695]: [0, 1, 12, 72] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 47 = 1 x 47 (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(695, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 695:2 1000

← 694.2.a · 696.2.a →