Atlas › Modular forms › Weight 2 › Level 593 › 593.2.a

Newspace 593.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(593))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
593.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} - q^{4} + 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	−	$+1$	593b1 rank 0
593.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} - 2q^{5} + q^{6} - q^{7} + O(q^{8})$	+	$-1$	593a1 rank 1
593.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + 6q^{3} + 3q^{4} - q^{5} - 3q^{6} + 8q^{7} + O(q^{8})$	−	$+1$
593.2.a.d	18	degree 18	$18q - 6q^{2} - 9q^{3} + 10q^{4} - q^{5} - 5q^{6} - 30q^{7} + O(q^{8})$	+	$-1$
593.2.a.e	27	degree 27	$27q + 6q^{2} + 4q^{3} + 36q^{4} - q^{6} + 25q^{7} + O(q^{8})$	−	$+1$

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 $T$
593.2.a.a	$x + 3$
593.2.a.b	$x - 2$
593.2.a.c	$x^{2} - 5x + 3$
593.2.a.d	$x^{18} + 15x^{17} + 61x^{16} - 113x^{15} - 1300x^{14} - 1495x^{13} + 7397x^{12} + 15445x^{11} - 18320x^{10} - 54011x^{9} + 23983x^{8} + 91671x^{7} - 21656x^{6} - 77181x^{5} + 15088x^{4} + 27942x^{3} - 4625x^{2} - 3006x + 67$
593.2.a.e	$x^{27} - 10x^{26} - 45x^{25} + 729x^{24} - 11x^{23} - 20850x^{22} + 32354x^{21} + 302710x^{20} - 722805x^{19} - 2454715x^{18} + 7363467x^{17} + 11782258x^{16} - 41160388x^{15} - 36892120x^{14} + 133247690x^{13} + 90609661x^{12} - 250557870x^{11} - 185843187x^{10} + 247370815x^{9} + 253712426x^{8} - 68843042x^{7} - 158094214x^{6} - 52384006x^{5} + 7626956x^{4} + 7915166x^{3} + 1703338x^{2} + 150920x + 4802$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(593))$	49
Old forms	0
Cusp forms $S_{2}(\Gamma_0(593))$	49
Eisenstein series $E_{2}(\Gamma_0(593))$	1
Modular forms $M_{2}(\Gamma_0(593))$	50

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.222 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, 593]: [0, 50] (certified at ell = 2147470529)
T = [(2, 1), (3, 1)]; h of degree 49 = 1 x 49 (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(593, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 593:2 1000

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