Atlas › Modular forms › Weight 2 › Level 513 › 513.2.a

Newspace 513.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(513))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
513.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - 2q^{7} + O(q^{8})$	+	513b1 rank 1
513.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - 2q^{7} + O(q^{8})$	+	513a1 rank 1
513.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{4} - 8q^{7} + O(q^{8})$	−
513.2.a.d	3	degree 3	$3q - 3q^{2} + 3q^{4} - 3q^{5} - 3q^{7} + O(q^{8})$	−
513.2.a.e	3	degree 3	$3q - q^{2} + 3q^{4} - 5q^{5} + q^{7} + O(q^{8})$	+
513.2.a.f	3	degree 3	$3q + q^{2} + 3q^{4} + 5q^{5} + q^{7} + O(q^{8})$	+
513.2.a.g	3	degree 3	$3q + 3q^{2} + 3q^{4} + 3q^{5} - 3q^{7} + O(q^{8})$	−
513.2.a.h	4	degree 4	$4q + 4q^{4} + 8q^{7} + O(q^{8})$	−
513.2.a.i	4	degree 4	$4q + 8q^{4} + 12q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
513.2.a.a	$x - 24$
513.2.a.b	$x + 44$
513.2.a.c	$x^{2} + 40x - 368$
513.2.a.d	$x^{3} + 27x^{2} - 921x - 24571$
513.2.a.e	$x^{3} + 67x^{2} + 527x - 16843$
513.2.a.f	$x^{3} - 77x^{2} + 1927x - 15747$
513.2.a.g	$x^{3} + 3x^{2} - 1521x - 22699$
513.2.a.h	$x^{4} - 40x^{3} - 276x^{2} + 9680x - 26300$
513.2.a.i	$x^{4} - 60x^{3} - 848x^{2} + 52680x + 453028$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(513))$	24
Old forms	31
Cusp forms $S_{2}(\Gamma_0(513))$	55
Eisenstein series $E_{2}(\Gamma_0(513))$	11
Modular forms $M_{2}(\Gamma_0(513))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(513)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(171))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.342 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, 9, 19, 27, 57, 171, 513]: [0, 1, 2, 2, 4, 8, 22, 62] (certified at ell = 2147483647)
T = [(2, 1), (5, 3), (7, 5), (11, 7)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 6 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(513, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 513:2 1000

← 512.2.a · 514.2.a →