Atlas › Modular forms › Weight 6 › Level 98 › 98.6.a

Newspace 98.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(98))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
98.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} - 10q^{3} + 16q^{4} - 84q^{5} + 40q^{6} + O(q^{8})$	+
98.6.a.b	1	$\mathbb{Q}$	$q + 4q^{2} - 8q^{3} + 16q^{4} - 10q^{5} - 32q^{6} + O(q^{8})$	−
98.6.a.c	2	$\mathbb{Q}(\sqrt{130})$	$2q - 8q^{2} - 14q^{3} + 32q^{4} - 42q^{5} + 56q^{6} + O(q^{8})$	+
98.6.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q - 8q^{2} + 32q^{4} + O(q^{8})$	+
98.6.a.e	2	$\mathbb{Q}(\sqrt{46})$	$2q - 8q^{2} + 32q^{4} + O(q^{8})$	+
98.6.a.f	2	$\mathbb{Q}(\sqrt{130})$	$2q - 8q^{2} + 14q^{3} + 32q^{4} + 42q^{5} - 56q^{6} + O(q^{8})$	+
98.6.a.g	2	$\mathbb{Q}(\sqrt{79})$	$2q + 8q^{2} - 14q^{3} + 32q^{4} - 70q^{5} - 56q^{6} + O(q^{8})$	−
98.6.a.h	2	$\mathbb{Q}(\sqrt{79})$	$2q + 8q^{2} + 14q^{3} + 32q^{4} + 70q^{5} + 56q^{6} + O(q^{8})$	−
98.6.a.i	4	degree 4	$4q + 16q^{2} + 64q^{4} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
98.6.a.a	$x + 94$
98.6.a.b	$x + 18$
98.6.a.c	$x^{2} + 56x + 264$
98.6.a.d	$x^{2} - 2$
98.6.a.e	$x^{2} - 6624$
98.6.a.f	$x^{2} - 56x + 264$
98.6.a.g	$x^{2} + 84x - 1080$
98.6.a.h	$x^{2} - 84x - 1080$
98.6.a.i	$x^{4} - 14452x^{2} + 29833444$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(98))$	18
Old forms	44
Cusp forms $S_{6}(\Gamma_0(98))$	62
Eisenstein series $E_{6}(\Gamma_0(98))$	16
Modular forms $M_{6}(\Gamma_0(98))$	78

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(98)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.293 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, 7, 14, 49, 98]: [1, 2, 5, 12, 25, 72] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 18 = 1 x 18 (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(98, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 98:6 1000

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