Atlas › Modular forms › Weight 4 › Level 198 › 198.4.a

Newspace 198.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(198))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
198.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} - 10q^{5} + 16q^{7} + O(q^{8})$	+ +
198.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} + 3q^{5} - 10q^{7} + O(q^{8})$	+ +
198.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} + 8q^{5} - 22q^{7} + O(q^{8})$	+ −
198.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{4} - 14q^{5} - 8q^{7} + O(q^{8})$	− −
198.4.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{4} - 8q^{5} - 22q^{7} + O(q^{8})$	− +
198.4.a.f	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{4} + 14q^{7} + O(q^{8})$	− +
198.4.a.g	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{4} + 19q^{5} + 14q^{7} + O(q^{8})$	− +
198.4.a.h	2	$\mathbb{Q}(\sqrt{97})$	$2q - 4q^{2} + 8q^{4} - 10q^{5} - 2q^{7} + O(q^{8})$	+ −
198.4.a.i	2	$\mathbb{Q}(\sqrt{70})$	$2q - 4q^{2} + 8q^{4} + 8q^{5} + 36q^{7} + O(q^{8})$	+ +
198.4.a.j	2	$\mathbb{Q}(\sqrt{70})$	$2q + 4q^{2} + 8q^{4} - 8q^{5} + 36q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17}$ 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 $T$
198.4.a.a	$x - 158$
198.4.a.b	$x + 401$
198.4.a.c	$x + 146$
198.4.a.d	$x + 1198$
198.4.a.e	$x + 526$
198.4.a.f	$x - 232$
198.4.a.g	$x - 23$
198.4.a.h	$x^{2} - 614x - 53288$
198.4.a.i	$x^{2} + 228x - 88084$
198.4.a.j	$x^{2} - 1044x - 70516$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(198))$	13
Old forms	87
Cusp forms $S_{4}(\Gamma_0(198))$	100
Eisenstein series $E_{4}(\Gamma_0(198))$	16
Modular forms $M_{4}(\Gamma_0(198))$	116

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(198)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(18))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(22))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(66))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(99))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.288 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, 6, 9, 11, 18, 22, 33, 66, 99, 198]: [1, 2, 2, 5, 4, 4, 11, 11, 14, 40, 38, 112] (certified at ell = 2147482951)
T = [(5, 1), (7, 3), (13, 5), (17, 7)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 5 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(198, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 198:4 1000

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