Atlas › Modular forms › Weight 4 › Level 196 › 196.4.a

Newspace 196.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(196))$

Newforms

Label	Dim	Coefficient field	Trace form
196.4.a.a	1	$\mathbb{Q}$	$q - 4q^{3} - 20q^{5} + O(q^{8})$
196.4.a.b	1	$\mathbb{Q}$	$q - 4q^{3} - 6q^{5} + O(q^{8})$
196.4.a.c	1	$\mathbb{Q}$	$q + 4q^{3} + 20q^{5} + O(q^{8})$
196.4.a.d	1	$\mathbb{Q}$	$q + 10q^{3} + 8q^{5} + O(q^{8})$
196.4.a.e	2	$\mathbb{Q}(\sqrt{37})$	$2q - 14q^{5} + O(q^{8})$
196.4.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q + O(q^{8})$
196.4.a.g	2	$\mathbb{Q}(\sqrt{37})$	$2q + 14q^{5} + O(q^{8})$

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 $T$
196.4.a.a	$x + 24$
196.4.a.b	$x + 10$
196.4.a.c	$x - 24$
196.4.a.d	$x - 18$
196.4.a.e	$x^{2} + 14x + 12$
196.4.a.f	$x^{2} - 2$
196.4.a.g	$x^{2} - 14x + 12$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(196))$	10
Old forms	62
Cusp forms $S_{4}(\Gamma_0(196))$	72
Eisenstein series $E_{4}(\Gamma_0(196))$	24
Modular forms $M_{4}(\Gamma_0(196))$	96

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(196)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(28))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(98))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 4, 7, 14, 28, 49, 98, 196]: [1, 2, 3, 3, 8, 15, 15, 44, 87] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (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(196, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 196:4 1000

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