Atlas › Modular forms › Weight 4 › Level 98 › 98.4.a

Newspace 98.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(98))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
98.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 8q^{3} + 4q^{4} + 14q^{5} + 16q^{6} + O(q^{8})$	+
98.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 4q^{4} + 7q^{5} + 2q^{6} + O(q^{8})$	+
98.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 4q^{4} - 7q^{5} - 2q^{6} + O(q^{8})$	+
98.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - 5q^{3} + 4q^{4} - 9q^{5} - 10q^{6} + O(q^{8})$	−
98.4.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{3} + 4q^{4} + 12q^{5} + 4q^{6} + O(q^{8})$	−
98.4.a.f	1	$\mathbb{Q}$	$q + 2q^{2} + 5q^{3} + 4q^{4} + 9q^{5} + 10q^{6} + O(q^{8})$	−
98.4.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{2} + 8q^{4} + O(q^{8})$	+
98.4.a.h	2	$\mathbb{Q}(\sqrt{22})$	$2q + 4q^{2} + 8q^{4} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
98.4.a.a	$x - 34$
98.4.a.b	$x - 20$
98.4.a.c	$x + 20$
98.4.a.d	$x + 32$
98.4.a.e	$x - 38$
98.4.a.f	$x - 32$
98.4.a.g	$x^{2} - 4418$
98.4.a.h	$x^{2} - 352$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(98))$	10
Old forms	24
Cusp forms $S_{4}(\Gamma_0(98))$	34
Eisenstein series $E_{4}(\Gamma_0(98))$	16
Modular forms $M_{4}(\Gamma_0(98))$	50

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.117 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, 3, 8, 15, 44] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 98:4 1000

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