Atlas › Modular forms › Weight 2 › Level 264 › 264.2.a

Newspace 264.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(264))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
264.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + O(q^{8})$	+ −	264b1 rank 0
264.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} + 4q^{7} + O(q^{8})$	− +	264c1 rank 0
264.2.a.c	1	$\mathbb{Q}$	$q + q^{3} + 2q^{7} + O(q^{8})$	− −	264a1 rank 0
264.2.a.d	1	$\mathbb{Q}$	$q + q^{3} + 4q^{5} - 2q^{7} + O(q^{8})$	− +	264d1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{19} + 9T_{23} + 11T_{29}$ 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 $T$
264.2.a.a	$x - 70$
264.2.a.b	$x + 82$
264.2.a.c	$x + 22$
264.2.a.d	$x - 38$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(264))$	4
Old forms	37
Cusp forms $S_{2}(\Gamma_0(264))$	41
Eisenstein series $E_{2}(\Gamma_0(264))$	15
Modular forms $M_{2}(\Gamma_0(264))$	56

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(264)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(66))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(88))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(132))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.164 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, 11, 12, 22, 24, 33, 44, 66, 88, 132, 264]: [0, 1, 1, 2, 3, 3, 2, 5, 5, 8, 6, 9, 16, 16, 30, 56] (certified at ell = 2147483171)
T = [(5, 1), (7, 3), (13, 5), (19, 7), (23, 9), (29, 11)]; h of degree 4 = 1 x 4 (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(264, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 264:2 1000

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