Atlas › Modular forms › Weight 2 › Level 243 › 243.2.a

Newspace 243.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(243))$

Newforms

Label	Dim	Coefficient field	Trace form	CM	Elliptic curve
243.2.a.a	1	$\mathbb{Q}$	$q - 2q^{4} - 4q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	243a1 rank 1
243.2.a.b	1	$\mathbb{Q}$	$q - 2q^{4} + 5q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	243b1 rank 0
243.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{4} - 2q^{7} + O(q^{8})$
243.2.a.d	2	$\mathbb{Q}(\sqrt{6})$	$2q + 8q^{4} + 4q^{7} + O(q^{8})$
243.2.a.e	3	degree 3	$3q - 3q^{2} + 3q^{4} - 6q^{5} - 3q^{7} + O(q^{8})$
243.2.a.f	3	degree 3	$3q + 3q^{2} + 3q^{4} + 6q^{5} - 3q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
243.2.a.a	$x + 20$
243.2.a.b	$x - 25$
243.2.a.c	$x^{2} + 10x - 122$
243.2.a.d	$x^{2} - 20x - 50$
243.2.a.e	$x^{3} + 57x^{2} + 720x - 1369$
243.2.a.f	$x^{3} - 27x^{2} - 2010x + 1979$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(243))$	12
Old forms	7
Cusp forms $S_{2}(\Gamma_0(243))$	19
Eisenstein series $E_{2}(\Gamma_0(243))$	17
Modular forms $M_{2}(\Gamma_0(243))$	36

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(243)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(81))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 9, 27, 81, 243]: [0, 1, 2, 4, 10, 28] (certified at ell = 2147483179)
T = [(2, 1), (5, 3), (7, 5), (11, 7)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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(243, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 243:2 1000

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