Atlas › Modular forms › Weight 2 › Level 589 › 589.2.a

Newspace 589.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(589))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
589.2.a.a	3	degree 3	$3q + q^{2} + 2q^{3} + 3q^{4} - 4q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$
589.2.a.b	9	degree 9	$9q - 4q^{2} - 5q^{3} + 8q^{4} - 10q^{5} - q^{6} - q^{7} + O(q^{8})$	− −	$-1$
589.2.a.c	9	degree 9	$9q + 7q^{3} + 10q^{4} + 5q^{6} + q^{7} + O(q^{8})$	− +	$+1$
589.2.a.d	10	degree 10	$10q - 7q^{3} + 6q^{4} - 7q^{5} - 5q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
589.2.a.e	14	degree 14	$14q + 7q^{3} + 18q^{4} + 9q^{5} + 9q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
589.2.a.a	$x^{3} - 3x^{2} - 4x + 8$
589.2.a.b	$x^{9} + 9x^{8} + 13x^{7} - 75x^{6} - 166x^{5} + 209x^{4} + 423x^{3} - 255x^{2} - 44x + 8$
589.2.a.c	$x^{9} - 7x^{8} - 10x^{7} + 122x^{6} - 33x^{5} - 600x^{4} + 467x^{3} + 742x^{2} - 855x + 181$
589.2.a.d	$x^{10} + 7x^{9} + x^{8} - 63x^{7} - 48x^{6} + 155x^{5} + 67x^{4} - 103x^{3} - 32x^{2} + 16x + 4$
589.2.a.e	$x^{14} - 7x^{13} - 39x^{12} + 371x^{11} + 131x^{10} - 6124x^{9} + 8206x^{8} + 31156x^{7} - 71630x^{6} - 10125x^{5} + 102319x^{4} - 45679x^{3} - 14328x^{2} + 6152x - 388$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(589))$	45
Old forms	6
Cusp forms $S_{2}(\Gamma_0(589))$	51
Eisenstein series $E_{2}(\Gamma_0(589))$	3
Modular forms $M_{2}(\Gamma_0(589))$	54

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(589)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.201 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, 19, 31, 589]: [0, 2, 3, 54] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 45 = 1 x 45 (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(589, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 589:2 1000

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