Atlas › Modular forms › Weight 2 › Level 499 › 499.2.a

Newspace 499.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(499))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
499.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{4} - 5q^{5} + 5q^{6} - 2q^{7} + O(q^{8})$	+	$-1$
499.2.a.b	16	degree 16	$16q - 5q^{2} - 5q^{3} + 15q^{4} - 20q^{5} - 18q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
499.2.a.c	23	degree 23	$23q + 4q^{2} + q^{3} + 22q^{4} + 25q^{5} + 7q^{6} + 2q^{7} + O(q^{8})$	−	$+1$

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 $T$
499.2.a.a	$x^{2} + x - 11$
499.2.a.b	$x^{16} + 10x^{15} + 18x^{14} - 96x^{13} - 292x^{12} + 312x^{11} + 1397x^{10} - 403x^{9} - 3020x^{8} + 280x^{7} + 3107x^{6} - 404x^{5} - 1326x^{4} + 354x^{3} + 77x^{2} - 19x - 1$
499.2.a.c	$x^{23} - 5x^{22} - 76x^{21} + 413x^{20} + 2183x^{19} - 13368x^{18} - 30099x^{17} + 221930x^{16} + 213247x^{15} - 2120635x^{14} - 742125x^{13} + 12369792x^{12} + 927282x^{11} - 45228179x^{10} + 317812x^{9} + 103288800x^{8} + 3155052x^{7} - 141212714x^{6} - 19347436x^{5} + 103479426x^{4} + 27732357x^{3} - 30205912x^{2} - 12093780x - 437688$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(499))$	41
Old forms	0
Cusp forms $S_{2}(\Gamma_0(499))$	41
Eisenstein series $E_{2}(\Gamma_0(499))$	1
Modular forms $M_{2}(\Gamma_0(499))$	42

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.196 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, 499]: [0, 42] (certified at ell = 2147475601)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 4 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(499, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 499:2 1000

← 498.2.a · 500.2.a →