Atlas › Modular forms › Weight 2 › Level 627 › 627.2.a

Newspace 627.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(627))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
627.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + 2q^{7} + O(q^{8})$	− − −	$+1$	627b1 rank 0
627.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + 4q^{5} + 2q^{7} + O(q^{8})$	− + +	$+1$	627a1 rank 0
627.2.a.c	3	degree 3	$3q - 2q^{2} - 3q^{3} + 4q^{4} + q^{5} + 2q^{6} - 5q^{7} + O(q^{8})$	+ − −	$-1$
627.2.a.d	3	degree 3	$3q - 2q^{2} + 3q^{3} - 7q^{5} - 2q^{6} + q^{7} + O(q^{8})$	− + −	$-1$
627.2.a.e	3	degree 3	$3q - 2q^{2} + 3q^{3} + 4q^{4} - 3q^{5} - 2q^{6} - 7q^{7} + O(q^{8})$	− − +	$-1$
627.2.a.f	3	degree 3	$3q + 2q^{2} - 3q^{3} - 3q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$
627.2.a.g	3	degree 3	$3q + 2q^{2} + 3q^{3} + 4q^{4} + 5q^{5} + 2q^{6} - q^{7} + O(q^{8})$	− − −	$+1$
627.2.a.h	4	degree 4	$4q + q^{2} + 4q^{3} + 7q^{4} + 3q^{5} + q^{6} - 5q^{7} + O(q^{8})$	− + +	$+1$
627.2.a.i	5	degree 5	$5q + q^{2} - 5q^{3} + 5q^{4} + 3q^{5} - q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$
627.2.a.j	5	degree 5	$5q + q^{2} - 5q^{3} + 9q^{4} + 7q^{5} - q^{6} + 7q^{7} + O(q^{8})$	+ − +	$+1$

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 $T$
627.2.a.a	$x$
627.2.a.b	$x - 4$
627.2.a.c	$x^{3} + x^{2} - 4x + 1$
627.2.a.d	$x^{3} + 9x^{2} + 20x - 1$
627.2.a.e	$x^{3} + 5x^{2} + 2x - 11$
627.2.a.f	$x^{3} + x^{2} - 2x - 1$
627.2.a.g	$x^{3} - 7x^{2} + 12x - 5$
627.2.a.h	$x^{4} - 4x^{3} - 11x^{2} + 35x - 13$
627.2.a.i	$x^{5} - 4x^{4} - 25x^{3} + 93x^{2} + 159x - 520$
627.2.a.j	$x^{5} - 8x^{4} + 9x^{3} + 25x^{2} - 39x + 4$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(627))$	31
Old forms	46
Cusp forms $S_{2}(\Gamma_0(627))$	77
Eisenstein series $E_{2}(\Gamma_0(627))$	7
Modular forms $M_{2}(\Gamma_0(627))$	84

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(627)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(209))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.245 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, 11, 19, 33, 57, 209, 627]: [0, 1, 2, 2, 6, 8, 22, 84] (certified at ell = 2147482801)
T = [(2, 1), (5, 1)]; h of degree 31 = 1 x 31 (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(627, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 627:2 1000

← 626.2.a · 628.2.a →