Atlas › Modular forms › Weight 4 › Level 243 › 243.4.a

Newspace 243.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(243))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
243.4.a.a	1	$\mathbb{Q}$	$q - 3q^{2} + q^{4} + 3q^{5} - 10q^{7} + O(q^{8})$
243.4.a.b	1	$\mathbb{Q}$	$q - 8q^{4} - 37q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$
243.4.a.c	1	$\mathbb{Q}$	$q - 8q^{4} + 17q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$
243.4.a.d	1	$\mathbb{Q}$	$q + 3q^{2} + q^{4} - 3q^{5} - 10q^{7} + O(q^{8})$
243.4.a.e	2	$\mathbb{Q}(\sqrt{6})$	$2q - 4q^{4} - 8q^{7} + O(q^{8})$
243.4.a.f	2	$\mathbb{Q}(\sqrt{15})$	$2q + 14q^{4} + 46q^{7} + O(q^{8})$
243.4.a.g	2	$\mathbb{Q}(\sqrt{21})$	$2q + 26q^{4} - 50q^{7} + O(q^{8})$
243.4.a.h	4	degree 4	$4q + 22q^{4} + 50q^{7} + O(q^{8})$
243.4.a.i	4	degree 4	$4q + 28q^{4} - 16q^{7} + O(q^{8})$
243.4.a.j	9	degree 9	$9q - 6q^{2} + 36q^{4} - 30q^{5} + 18q^{7} + O(q^{8})$
243.4.a.k	9	degree 9	$9q + 6q^{2} + 36q^{4} + 30q^{5} + 18q^{7} + O(q^{8})$

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 $T$
243.4.a.a	$x + 128$
243.4.a.b	$x + 185$
243.4.a.c	$x - 85$
243.4.a.d	$x - 28$
243.4.a.e	$x^{2} + 40x - 183350$
243.4.a.f	$x^{2} - 230x - 75710$
243.4.a.g	$x^{2} + 250x + 2500$
243.4.a.h	$x^{4} - 250x^{3} - 272829x^{2} + 20295320x + 15639706372$
243.4.a.i	$x^{4} + 80x^{3} - 216870x^{2} + 9012440x + 6020690692$
243.4.a.j	$x^{9} + 804x^{8} + 32025x^{7} - 119283618x^{6} - 25122518178x^{5} + 3817377372504x^{4} + 1433114302365210x^{3} + 48693185966449317x^{2} - 14052030456002681058x - 987552519253260515911$
243.4.a.k	$x^{9} - 984x^{8} + 105135x^{7} + 153499218x^{6} - 39247977468x^{5} - 5318971769244x^{4} + 1830079973170830x^{3} + 79445699767223853x^{2} - 22905472743992356518x - 1246504433750598374119$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(243))$	36
Old forms	36
Cusp forms $S_{4}(\Gamma_0(243))$	72
Eisenstein series $E_{4}(\Gamma_0(243))$	18
Modular forms $M_{4}(\Gamma_0(243))$	90

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(243)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(81))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.235 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]: [1, 2, 4, 10, 28, 82] (certified at ell = 2147483179)
T = [(2, 1), (5, 3), (7, 5), (11, 7)]; h of degree 36 = 1 x 36 (phi(ord eps) x dim S_k^new); 12 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 243:4 1000

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