Atlas › Modular forms › Weight 2 › Level 196 › 196.2.a

Newspace 196.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(196))$

Newforms

Label	Dim	Coefficient field	Trace form	Elliptic curve
196.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 3q^{5} + O(q^{8})$	196a1 rank 1
196.2.a.b	1	$\mathbb{Q}$	$q + q^{3} + 3q^{5} + O(q^{8})$	196b1 rank 0
196.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
196.2.a.a	$x + 10$
196.2.a.b	$x - 10$
196.2.a.c	$x^{2} - 2$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(196))$	4
Old forms	13
Cusp forms $S_{2}(\Gamma_0(196))$	17
Eisenstein series $E_{2}(\Gamma_0(196))$	23
Modular forms $M_{2}(\Gamma_0(196))$	40

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(196)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(98))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.205 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, 2, 4, 7, 14, 28, 49, 98, 196]: [0, 1, 2, 1, 4, 7, 5, 16, 31] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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(196, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 196:2 1000

← 195.2.a · 197.2.a →