Atlas › Modular forms › Weight 4 › Level 204 › 204.4.a

Newspace 204.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(204))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
204.4.a.a	1	$\mathbb{Q}$	$q + 3q^{3} - 3q^{5} - 16q^{7} + O(q^{8})$	− −
204.4.a.b	2	$\mathbb{Q}(\sqrt{201})$	$2q - 6q^{3} + 3q^{5} - 2q^{7} + O(q^{8})$	+ +
204.4.a.c	2	$\mathbb{Q}(\sqrt{217})$	$2q + 6q^{3} + 19q^{5} + 18q^{7} + O(q^{8})$	− +
204.4.a.d	3	degree 3	$3q - 9q^{3} - 19q^{5} - 8q^{7} + O(q^{8})$	+ −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{7}$ 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 $T$
204.4.a.a	$x + 19$
204.4.a.b	$x^{2} - x - 50$
204.4.a.c	$x^{2} - 37x - 146$
204.4.a.d	$x^{3} + 27x^{2} - 1200x - 1604$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(204))$	8
Old forms	94
Cusp forms $S_{4}(\Gamma_0(204))$	102
Eisenstein series $E_{4}(\Gamma_0(204))$	12
Modular forms $M_{4}(\Gamma_0(204))$	114

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(204)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(51))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(68))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(102))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.16 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, 12, 17, 34, 51, 68, 102, 204]: [1, 2, 2, 3, 5, 9, 6, 16, 20, 30, 58, 114] (certified at ell = 2147483489)
T = [(5, 1), (7, 1)]; h of degree 8 = 1 x 8 (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(204, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 204:4 1000

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