Atlas › Modular forms › Weight 2 › Level 571 › 571.2.a

Newspace 571.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(571))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
571.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 4q^{6} - 4q^{7} + O(q^{8})$	−	$+1$	571b1 rank 2
571.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} - 2q^{4} - 2q^{5} + 2q^{7} + O(q^{8})$	−	$+1$	571a1 rank 0
571.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 4q^{3} + 6q^{4} + q^{5} + 4q^{7} + O(q^{8})$	−	$+1$
571.2.a.d	2	$\mathbb{Q}(\sqrt{37})$	$2q + 2q^{2} - 4q^{3} - 2q^{4} + q^{5} - 4q^{6} + 4q^{7} + O(q^{8})$	−	$+1$
571.2.a.e	3	degree 3	$3q - 2q^{2} + 2q^{4} - 4q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+	$-1$
571.2.a.f	4	degree 4	$4q + 2q^{2} + 2q^{3} + 4q^{4} + 4q^{5} + 12q^{6} - 2q^{7} + O(q^{8})$	−	$+1$
571.2.a.g	6	degree 6	$6q - q^{2} - 3q^{3} + 9q^{4} - 9q^{5} - 8q^{6} - 3q^{7} + O(q^{8})$	+	$-1$
571.2.a.h	10	degree 10	$10q - q^{2} - 5q^{3} + q^{4} - 13q^{5} + 4q^{6} - 5q^{7} + O(q^{8})$	+	$-1$
571.2.a.i	18	degree 18	$18q + 4q^{3} + 22q^{4} + 26q^{5} - 6q^{6} - 4q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ 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 $T$
571.2.a.a	$x + 46$
571.2.a.b	$x - 10$
571.2.a.c	$x^{2} - 45x + 495$
571.2.a.d	$x^{2} - 23x - 99$
571.2.a.e	$x^{3} - 6x^{2} - 88x + 460$
571.2.a.f	$x^{4} - 14x^{3} - 640x^{2} + 3592x + 91860$
571.2.a.g	$x^{6} + 76x^{5} + 1574x^{4} + 4352x^{3} - 135371x^{2} - 625636x + 2640511$
571.2.a.h	$x^{10} + 116x^{9} + 4283x^{8} + 21851x^{7} - 1892153x^{6} - 30503241x^{5} + 124599657x^{4} + 4183470151x^{3} + 10844738910x^{2} - 12783182636x - 41511368420$
571.2.a.i	$x^{18} - 114x^{17} - 282x^{16} + 445195x^{15} - 10656264x^{14} - 565499038x^{13} + 23232216942x^{12} + 174516573544x^{11} - 18576625693213x^{10} + 149155995753815x^{9} + 5352702534744041x^{8} - 103152844806741442x^{7} + 31197361895123547x^{6} + 13460034295649199593x^{5} - 110277016448480109745x^{4} + 10075032992020957949x^{3} + 3265027640220622988591x^{2} - 12215237689437400357901x + 9714647710614367915166$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(571))$	47
Old forms	0
Cusp forms $S_{2}(\Gamma_0(571))$	47
Eisenstein series $E_{2}(\Gamma_0(571))$	1
Modular forms $M_{2}(\Gamma_0(571))$	48

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.312 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, 571]: [0, 48] (certified at ell = 2147477851)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 10 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(571, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 571:2 1000

← 570.2.a · 572.2.a →