Atlas › Modular forms › Weight 2 › Level 219 › 219.2.a

Newspace 219.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(219))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
219.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} - q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+ +	$-1$	219a1 rank 1
219.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - 3q^{5} - 4q^{7} + O(q^{8})$	− −	$-1$	219b1 rank 1
219.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - 4q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ +	$-1$	219c1 rank 1
219.2.a.d	4	degree 4	$4q + q^{2} - 4q^{3} + 5q^{4} + 9q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+ −	$+1$
219.2.a.e	6	degree 6	$6q - q^{2} + 6q^{3} + 7q^{4} + 5q^{5} - q^{6} + 8q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
219.2.a.a	$x + 5$
219.2.a.b	$x + 9$
219.2.a.c	$x + 11$
219.2.a.d	$x^{4} - 28x^{3} + 252x^{2} - 698x - 191$
219.2.a.e	$x^{6} - 14x^{5} - 69x^{4} + 1306x^{3} + 125x^{2} - 20684x - 21961$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(219))$	13
Old forms	10
Cusp forms $S_{2}(\Gamma_0(219))$	23
Eisenstein series $E_{2}(\Gamma_0(219))$	3
Modular forms $M_{2}(\Gamma_0(219))$	26

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(219)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(73))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.371 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, 3, 73, 219]: [0, 1, 6, 26] (certified at ell = 2147482873)
T = [(2, 1), (5, 3)]; h of degree 13 = 1 x 13 (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(219, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 219:2 1000

← 218.2.a · 220.2.a →