Atlas › Modular forms › Weight 4 › Level 192 › 192.4.a

Newspace 192.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(192))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
192.4.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 14q^{5} - 24q^{7} + O(q^{8})$	+
192.4.a.b	1	$\mathbb{Q}$	$q - 3q^{3} - 10q^{5} + 4q^{7} + O(q^{8})$	+
192.4.a.c	1	$\mathbb{Q}$	$q - 3q^{3} - 6q^{5} + 16q^{7} + O(q^{8})$	+
192.4.a.d	1	$\mathbb{Q}$	$q - 3q^{3} - 2q^{5} + 12q^{7} + O(q^{8})$	+
192.4.a.e	1	$\mathbb{Q}$	$q - 3q^{3} + 14q^{5} - 36q^{7} + O(q^{8})$	+
192.4.a.f	1	$\mathbb{Q}$	$q - 3q^{3} + 18q^{5} + 8q^{7} + O(q^{8})$	+
192.4.a.g	1	$\mathbb{Q}$	$q + 3q^{3} - 14q^{5} + 24q^{7} + O(q^{8})$	−
192.4.a.h	1	$\mathbb{Q}$	$q + 3q^{3} - 10q^{5} - 4q^{7} + O(q^{8})$	−
192.4.a.i	1	$\mathbb{Q}$	$q + 3q^{3} - 6q^{5} - 16q^{7} + O(q^{8})$	−
192.4.a.j	1	$\mathbb{Q}$	$q + 3q^{3} - 2q^{5} - 12q^{7} + O(q^{8})$	−
192.4.a.k	1	$\mathbb{Q}$	$q + 3q^{3} + 14q^{5} + 36q^{7} + O(q^{8})$	−
192.4.a.l	1	$\mathbb{Q}$	$q + 3q^{3} + 18q^{5} - 8q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
192.4.a.a	$x - 30$
192.4.a.b	$x - 1542$
192.4.a.c	$x + 850$
192.4.a.d	$x + 874$
192.4.a.e	$x + 842$
192.4.a.f	$x - 1118$
192.4.a.g	$x - 1830$
192.4.a.h	$x + 1002$
192.4.a.i	$x + 1306$
192.4.a.j	$x - 1430$
192.4.a.k	$x - 22$
192.4.a.l	$x + 730$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(192))$	12
Old forms	72
Cusp forms $S_{4}(\Gamma_0(192))$	84
Eisenstein series $E_{4}(\Gamma_0(192))$	24
Modular forms $M_{4}(\Gamma_0(192))$	108

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.32 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, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192]: [1, 2, 2, 3, 5, 5, 9, 8, 16, 14, 28, 26, 52, 100] (certified at ell = 2147483489)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 12 = 1 x 12 (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(192, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 192:4 1000

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