Atlas › Modular forms › Weight 2 › Level 157 › 157.2.a

Newspace 157.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(157))$

Newforms

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

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 $T$
157.2.a.a	$x^{5} + 12x^{4} + 49x^{3} + 87x^{2} + 67x + 17$
157.2.a.b	$x^{7} - 10x^{6} + 26x^{5} + 37x^{4} - 282x^{3} + 476x^{2} - 317x + 67$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(157))$	12
Old forms	0
Cusp forms $S_{2}(\Gamma_0(157))$	12
Eisenstein series $E_{2}(\Gamma_0(157))$	1
Modular forms $M_{2}(\Gamma_0(157))$	13

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.16 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, 157]: [0, 13] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 12 = 1 x 12 (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(157, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 157:2 1000

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