Atlas › Modular forms › Weight 2 › Level 599 › 599.2.a

Newspace 599.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(599))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
599.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 3q^{3} - q^{4} - 2q^{5} + q^{6} + O(q^{8})$	+	$-1$
599.2.a.b	11	degree 11	$11q - 3q^{2} - q^{3} + 5q^{4} - 2q^{5} - 9q^{6} - 8q^{7} + O(q^{8})$	+	$-1$
599.2.a.c	37	degree 37	$37q + q^{2} + 4q^{3} + 47q^{4} + 6q^{5} + 8q^{6} + 12q^{7} + O(q^{8})$	−	$+1$

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 $T$
599.2.a.a	$x^{2} + 2x - 4$
599.2.a.b	$x^{11} + 4x^{10} - 10x^{9} - 46x^{8} + 28x^{7} + 147x^{6} - 53x^{5} - 130x^{4} + 86x^{3} - 5x^{2} - 6x + 1$
599.2.a.c	$x^{37} - 5x^{36} - 140x^{35} + 720x^{34} + 8503x^{33} - 44991x^{32} - 297030x^{31} + 1613223x^{30} + 6704963x^{29} - 37073173x^{28} - 104844093x^{27} + 578847565x^{26} + 1196385071x^{25} - 6356681407x^{24} - 10390984350x^{23} + 50086690359x^{22} + 70731555877x^{21} - 285484119961x^{20} - 381023743595x^{19} + 1172385899121x^{18} + 1604123335160x^{17} - 3400075844553x^{16} - 5121156887506x^{15} + 6638797681253x^{14} + 11885849152391x^{13} - 7758284305057x^{12} - 18983435215631x^{11} + 3299396094546x^{10} + 19270135316239x^{9} + 3457053578571x^{8} - 10809350342055x^{7} - 4938041777852x^{6} + 2355386131976x^{5} + 1848416194658x^{4} + 77052534928x^{3} - 133981873631x^{2} - 11985208190x + 2578689028$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(599))$	50
Old forms	0
Cusp forms $S_{2}(\Gamma_0(599))$	50
Eisenstein series $E_{2}(\Gamma_0(599))$	1
Modular forms $M_{2}(\Gamma_0(599))$	51

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.218 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, 599]: [0, 51] (certified at ell = 2147480791)
T = [(2, 1), (3, 1)]; h of degree 50 = 1 x 50 (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(599, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 599:2 1000

← 598.2.a · 600.2.a →