Atlas › Modular forms › Weight 2 › Level 299 › 299.2.a

Newspace 299.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(299))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
299.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{3} - q^{4} - q^{5} + 3q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
299.2.a.b	2	$\mathbb{Q}(\sqrt{21})$	$2q - q^{2} - q^{3} + 7q^{4} + 3q^{5} + 11q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
299.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 6q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	− +	$+1$
299.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{3} - q^{4} - 3q^{5} - 3q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
299.2.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + q^{3} + 5q^{4} + q^{5} - 8q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
299.2.a.f	3	degree 3	$3q - q^{3} - 6q^{4} + q^{5} + 2q^{7} + O(q^{8})$	− +	$+1$
299.2.a.g	10	degree 10	$10q + q^{2} + 3q^{3} + 19q^{4} + 3q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
299.2.a.a	$x^{2} + 4x - 16$
299.2.a.b	$x^{2} + 4x - 80$
299.2.a.c	$x^{2} - 5$
299.2.a.d	$x^{2} + 2x - 4$
299.2.a.e	$x^{2} - 4x - 13$
299.2.a.f	$x^{3} + 3x^{2} - 81x - 135$
299.2.a.g	$x^{10} - 10x^{9} - 184x^{8} + 1872x^{7} + 10918x^{6} - 121792x^{5} - 173628x^{4} + 3089160x^{3} - 2523311x^{2} - 20669838x + 33020188$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(299))$	23
Old forms	4
Cusp forms $S_{2}(\Gamma_0(299))$	27
Eisenstein series $E_{2}(\Gamma_0(299))$	3
Modular forms $M_{2}(\Gamma_0(299))$	30

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(299)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.184 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, 13, 23, 299]: [0, 1, 3, 30] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 3 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(299, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 299:2 1000

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