Atlas › Modular forms › Weight 2 › Level 592 › 592.2.a

Newspace 592.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(592))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
592.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + q^{7} + O(q^{8})$	−	592e1 rank 1
592.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 4q^{5} + 3q^{7} + O(q^{8})$	−	592d1 rank 1
592.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} - q^{7} + O(q^{8})$	+	592a1 rank 1
592.2.a.d	1	$\mathbb{Q}$	$q + q^{3} + 3q^{7} + O(q^{8})$	−	592b1 rank 0
592.2.a.e	1	$\mathbb{Q}$	$q + 3q^{3} - 2q^{5} + q^{7} + O(q^{8})$	+	592c1 rank 0
592.2.a.f	2	$\mathbb{Q}(\sqrt{13})$	$2q - 3q^{3} - q^{5} - 2q^{7} + O(q^{8})$	−
592.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{3} + q^{5} + 2q^{7} + O(q^{8})$	+
592.2.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} + 4q^{5} - q^{7} + O(q^{8})$	+
592.2.a.i	3	degree 3	$3q - 2q^{3} - q^{5} - 7q^{7} + O(q^{8})$	+
592.2.a.j	4	degree 4	$4q - 2q^{3} + 5q^{5} + q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ 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 $T$
592.2.a.a	$x + 53$
592.2.a.b	$x + 31$
592.2.a.c	$x + 71$
592.2.a.d	$x - 37$
592.2.a.e	$x - 19$
592.2.a.f	$x^{2} + 18x - 1492$
592.2.a.g	$x^{2} - 58x + 596$
592.2.a.h	$x^{2} - 37x - 172$
592.2.a.i	$x^{3} + 13x^{2} - 458x - 6412$
592.2.a.j	$x^{4} - 35x^{3} - 2654x^{2} + 23132x + 277664$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(592))$	18
Old forms	53
Cusp forms $S_{2}(\Gamma_0(592))$	71
Eisenstein series $E_{2}(\Gamma_0(592))$	11
Modular forms $M_{2}(\Gamma_0(592))$	82

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(592)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(74))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(148))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(296))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.381 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, 2, 4, 8, 16, 37, 74, 148, 296, 592]: [0, 1, 2, 3, 4, 3, 11, 22, 42, 80] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 18 = 1 x 18 (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(592, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 592:2 1000

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