Atlas › Modular forms › Weight 4 › Level 72 › 72.4.a

Newspace 72.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(72))$

Newforms

Label	Dim	Coefficient field	Trace form
72.4.a.a	1	$\mathbb{Q}$	$q - 16q^{5} - 12q^{7} + O(q^{8})$
72.4.a.b	1	$\mathbb{Q}$	$q - 14q^{5} - 24q^{7} + O(q^{8})$
72.4.a.c	1	$\mathbb{Q}$	$q + 2q^{5} + 24q^{7} + O(q^{8})$
72.4.a.d	1	$\mathbb{Q}$	$q + 16q^{5} - 12q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
72.4.a.a	$x + 52$
72.4.a.b	$x + 86$
72.4.a.c	$x - 74$
72.4.a.d	$x + 20$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(72))$	4
Old forms	24
Cusp forms $S_{4}(\Gamma_0(72))$	28
Eisenstein series $E_{4}(\Gamma_0(72))$	16
Modular forms $M_{4}(\Gamma_0(72))$	44

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(72)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(18))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.114 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, 9, 12, 18, 24, 36, 72]: [1, 2, 2, 3, 5, 5, 4, 9, 11, 16, 21, 40] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; h of degree 4 = 1 x 4 (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(72, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 72:4 1000

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