Atlas › Modular forms › Weight 2 › Level 202 › 202.2.a

Newspace 202.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(202))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
202.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{5} + q^{7} + O(q^{8})$	+ −	$+1$	202a1 rank 0
202.2.a.b	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} - 3q^{5} + 3q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
202.2.a.c	4	degree 4	$4q + 4q^{2} - q^{3} + 4q^{4} + 3q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
202.2.a.a	$x - 2$
202.2.a.b	$x^{3} + 6x^{2} + 9x + 3$
202.2.a.c	$x^{4} - 2x^{3} - 9x^{2} + 17x + 6$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(202))$	8
Old forms	16
Cusp forms $S_{2}(\Gamma_0(202))$	24
Eisenstein series $E_{2}(\Gamma_0(202))$	3
Modular forms $M_{2}(\Gamma_0(202))$	27

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(202)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(101))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.175 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, 101, 202]: [0, 1, 9, 27] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(202, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 202:2 1000

← 201.2.a · 203.2.a →