Atlas › Modular forms › Weight 2 › Level 324 › 324.2.a

Newspace 324.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(324))$

Newforms

Label	Dim	Coefficient field	Trace form	Elliptic curve
324.2.a.a	1	$\mathbb{Q}$	$q - 3q^{5} - q^{7} + O(q^{8})$	324c1 rank 1
324.2.a.b	1	$\mathbb{Q}$	$q - 3q^{5} + 2q^{7} + O(q^{8})$	324d1 rank 0
324.2.a.c	1	$\mathbb{Q}$	$q + 3q^{5} - q^{7} + O(q^{8})$	324a1 rank 0
324.2.a.d	1	$\mathbb{Q}$	$q + 3q^{5} + 2q^{7} + O(q^{8})$	324b1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
324.2.a.a	$x + 28$
324.2.a.b	$x - 68$
324.2.a.c	$x - 8$
324.2.a.d	$x - 14$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(324))$	4
Old forms	33
Cusp forms $S_{2}(\Gamma_0(324))$	37
Eisenstein series $E_{2}(\Gamma_0(324))$	35
Modular forms $M_{2}(\Gamma_0(324))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(324)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(54))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(81))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(108))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(162))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.234 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, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324]: [0, 1, 1, 2, 3, 2, 5, 5, 4, 9, 11, 10, 21, 29, 57] (certified at ell = 2147483179)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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(324, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 324:2 1000

← 323.2.a · 325.2.a →