Atlas › Modular forms › Weight 2 › Level 198 › 198.2.a

Newspace 198.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(198))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
198.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	+ −	198a1 rank 1
198.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{7} + O(q^{8})$	+ −	198d1 rank 0
198.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 4q^{5} - 2q^{7} + O(q^{8})$	+ +	198e1 rank 0
198.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{7} + O(q^{8})$	− +	198c1 rank 0
198.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{7} + O(q^{8})$	− −	198b1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
198.2.a.a	$x - 220$
198.2.a.b	$x + 242$
198.2.a.c	$x + 48$
198.2.a.d	$x - 34$
198.2.a.e	$x + 8$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(198))$	5
Old forms	24
Cusp forms $S_{2}(\Gamma_0(198))$	29
Eisenstein series $E_{2}(\Gamma_0(198))$	15
Modular forms $M_{2}(\Gamma_0(198))$	44

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.362 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]: [0, 1, 1, 3, 2, 2, 5, 5, 6, 16, 14, 40] (certified at ell = 2147482951)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 5 = 1 x 5 (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(198, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 198:2 1000

← 197.2.a · 199.2.a →