Atlas › Modular forms › Weight 2 › Level 137 › 137.2.a

Newspace 137.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(137))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
137.2.a.a	4	degree 4	$4q - 3q^{2} - 5q^{3} + q^{4} - 2q^{5} + q^{6} - 13q^{7} + O(q^{8})$	+	$-1$
137.2.a.b	7	degree 7	$7q + 3q^{3} + 6q^{4} - 2q^{5} - q^{6} + 15q^{7} + O(q^{8})$	−	$+1$

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 $T$
137.2.a.a	$x^{4} + 8x^{3} + 18x^{2} + 13x + 1$
137.2.a.b	$x^{7} - 3x^{6} - 17x^{5} + 58x^{4} + 22x^{3} - 112x^{2} - 54x + 7$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(137))$	11
Old forms	0
Cusp forms $S_{2}(\Gamma_0(137))$	11
Eisenstein series $E_{2}(\Gamma_0(137))$	1
Modular forms $M_{2}(\Gamma_0(137))$	12

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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, 137]: [0, 12] (certified at ell = 2147483249)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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(137, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 137:2 1000

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