Atlas › Modular forms › Weight 2 › Level 197 › 197.2.a

Newspace 197.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(197))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
197.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - 3q^{7} + O(q^{8})$	+	$-1$	197a1 rank 1
197.2.a.b	5	degree 5	$5q - 8q^{3} - 4q^{5} - 2q^{6} - 10q^{7} + O(q^{8})$	+	$-1$
197.2.a.c	10	degree 10	$10q + 10q^{3} + 10q^{4} + 2q^{5} - 4q^{6} + 11q^{7} + O(q^{8})$	−	$+1$

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 $T$
197.2.a.a	$x + 2$
197.2.a.b	$x^{5} + 8x^{4} + 15x^{3} + 5x^{2} - 5x - 1$
197.2.a.c	$x^{10} - 10x^{9} + 18x^{8} + 113x^{7} - 463x^{6} + 182x^{5} + 1642x^{4} - 3208x^{3} + 2475x^{2} - 863x + 112$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(197))$	16
Old forms	0
Cusp forms $S_{2}(\Gamma_0(197))$	16
Eisenstein series $E_{2}(\Gamma_0(197))$	1
Modular forms $M_{2}(\Gamma_0(197))$	17

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.247 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, 197]: [0, 17] (certified at ell = 2147479489)
T = [(2, 1), (3, 1)]; h of degree 16 = 1 x 16 (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(197, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 197:2 1000

← 196.2.a · 198.2.a →