Atlas › Modular forms › Weight 4 › Level 32 › 32.4.a

Newspace 32.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(32))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
32.4.a.a	1	$\mathbb{Q}$	$q - 8q^{3} - 10q^{5} - 16q^{7} + O(q^{8})$
32.4.a.b	1	$\mathbb{Q}$	$q + 22q^{5} + O(q^{8})$	$\mathbb{Q}(\sqrt{-4})$
32.4.a.c	1	$\mathbb{Q}$	$q + 8q^{3} - 10q^{5} + 16q^{7} + O(q^{8})$

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 $T$
32.4.a.a	$x + 18$
32.4.a.b	$x - 22$
32.4.a.c	$x + 2$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(32))$	3
Old forms	5
Cusp forms $S_{4}(\Gamma_0(32))$	8
Eisenstein series $E_{4}(\Gamma_0(32))$	8
Modular forms $M_{4}(\Gamma_0(32))$	16

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(32)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.1 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, 8, 16, 32]: [1, 2, 3, 5, 8, 14] (certified at ell = 2147483497)
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(32, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 32:4 1000

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