Atlas › Modular forms › Weight 2 › Level 221 › 221.2.a

Newspace 221.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(221))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
221.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 4q^{5} - 2q^{7} + O(q^{8})$	+ −	$+1$	221a1 rank 0
221.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - q^{4} + 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$	221b1 rank 0
221.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 3q^{3} - q^{4} + 4q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
221.2.a.d	2	$\mathbb{Q}(\sqrt{21})$	$2q - q^{2} + q^{3} + 7q^{4} - 2q^{5} + 10q^{6} - 5q^{7} + O(q^{8})$	+ −	$+1$
221.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} + 6q^{4} - 2q^{5} - 10q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
221.2.a.f	3	degree 3	$3q - 3q^{3} + 2q^{4} - 2q^{5} - 8q^{6} - 9q^{7} + O(q^{8})$	− −	$-1$
221.2.a.g	6	degree 6	$6q + q^{2} + q^{3} + 7q^{4} - 2q^{5} + 2q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ 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 $T$
221.2.a.a	$x - 5$
221.2.a.b	$x - 31$
221.2.a.c	$x^{2} + 17x - 139$
221.2.a.d	$x^{2} + 43x + 415$
221.2.a.e	$x^{2} - 24x + 99$
221.2.a.f	$x^{3} + 82x^{2} + 2108x + 16309$
221.2.a.g	$x^{6} - 43x^{5} - 1569x^{4} + 85694x^{3} - 122721x^{2} - 29977475x + 289332409$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(221))$	17
Old forms	2
Cusp forms $S_{2}(\Gamma_0(221))$	19
Eisenstein series $E_{2}(\Gamma_0(221))$	3
Modular forms $M_{2}(\Gamma_0(221))$	22

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(221)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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, 13, 17, 221]: [0, 1, 2, 22] (certified at ell = 2147483137)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 4 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(221, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 221:2 1000

← 220.2.a · 222.2.a →