Atlas › Modular forms › Weight 2 › Level 288 › 288.2.a

Newspace 288.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(288))$

Newforms

Label	Dim	Coefficient field	Trace form	CM	Elliptic curve
288.2.a.a	1	$\mathbb{Q}$	$q - 4q^{5} + O(q^{8})$	$\mathbb{Q}(\sqrt{-4})$	288a1 rank 1
288.2.a.b	1	$\mathbb{Q}$	$q - 2q^{5} - 4q^{7} + O(q^{8})$		288b1 rank 1
288.2.a.c	1	$\mathbb{Q}$	$q - 2q^{5} + 4q^{7} + O(q^{8})$		288c1 rank 0
288.2.a.d	1	$\mathbb{Q}$	$q + 2q^{5} + O(q^{8})$	$\mathbb{Q}(\sqrt{-4})$	288d1 rank 0
288.2.a.e	1	$\mathbb{Q}$	$q + 4q^{5} + O(q^{8})$	$\mathbb{Q}(\sqrt{-4})$	288e1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{11} + 7T_{23} + 8T_{29}$ 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 $T$
288.2.a.a	$x + 22$
288.2.a.b	$x + 52$
288.2.a.c	$x - 52$
288.2.a.d	$x - 92$
288.2.a.e	$x + 14$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(288))$	5
Old forms	28
Cusp forms $S_{2}(\Gamma_0(288))$	33
Eisenstein series $E_{2}(\Gamma_0(288))$	31
Modular forms $M_{2}(\Gamma_0(288))$	64

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(288)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(48))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(72))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(96))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(144))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.417 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, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288]: [0, 1, 1, 2, 3, 3, 2, 5, 4, 5, 8, 6, 9, 12, 16, 20, 28, 52] (certified at ell = 2147483497)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (19, 5), (11, 6), (23, 7), (29, 8)]; h of degree 5 = 1 x 5 (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(288, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 288:2 1000

← 287.2.a · 289.2.a →