Atlas › Modular forms › Weight 2 › Level 224 › 224.2.a

Newspace 224.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(224))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
224.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - q^{7} + O(q^{8})$	+	224a1 rank 1
224.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} + q^{7} + O(q^{8})$	−	224b1 rank 0
224.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	−
224.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} + 2q^{5} - 2q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
224.2.a.a	$x + 68$
224.2.a.b	$x + 24$
224.2.a.c	$x^{2} - 26x + 44$
224.2.a.d	$x^{2} - 70x - 1420$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(224))$	6
Old forms	19
Cusp forms $S_{2}(\Gamma_0(224))$	25
Eisenstein series $E_{2}(\Gamma_0(224))$	15
Modular forms $M_{2}(\Gamma_0(224))$	40

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(224)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(112))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.226 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, 8, 14, 16, 28, 32, 56, 112, 224]: [0, 1, 2, 1, 3, 4, 4, 7, 6, 12, 20, 36] (certified at ell = 2147483497)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 6 = 1 x 6 (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(224, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 224:2 1000

← 223.2.a · 225.2.a →