Atlas › Modular forms › Weight 2 › Level 588 › 588.2.a

Newspace 588.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(588))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
588.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + O(q^{8})$	+	588a1 rank 0
588.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + O(q^{8})$	+	588c1 rank 1
588.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + O(q^{8})$	+	588b1 rank 1
588.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 4q^{5} + O(q^{8})$	−	588f1 rank 0
588.2.a.e	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + O(q^{8})$	−	588e1 rank 0
588.2.a.f	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + O(q^{8})$	−	588d1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
588.2.a.a	$x - 36$
588.2.a.b	$x + 90$
588.2.a.c	$x - 8$
588.2.a.d	$x - 96$
588.2.a.e	$x - 102$
588.2.a.f	$x + 24$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(588))$	6
Old forms	83
Cusp forms $S_{2}(\Gamma_0(588))$	89
Eisenstein series $E_{2}(\Gamma_0(588))$	47
Modular forms $M_{2}(\Gamma_0(588))$	136

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.376 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, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588]: [0, 1, 1, 2, 3, 1, 5, 4, 4, 7, 12, 5, 22, 16, 20, 31, 60, 118] (certified at ell = 2147483647)
T = [(5, 1), (11, 3), (13, 5), (17, 7), (19, 9)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 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(588, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 588:2 1000

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