Atlas › Modular forms › Weight 2 › Level 98 › 98.2.a

Newspace 98.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(98))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
98.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - 2q^{6} + O(q^{8})$	+			98a1 rank 0
98.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + O(q^{8})$	−

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 $T$
98.2.a.a	$x - 2$
98.2.a.b	$x^{2} - 2$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(98))$	3
Old forms	4
Cusp forms $S_{2}(\Gamma_0(98))$	7
Eisenstein series $E_{2}(\Gamma_0(98))$	15
Modular forms $M_{2}(\Gamma_0(98))$	22

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.091 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]: [0, 1, 1, 4, 5, 16] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 98:2 1000

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