Atlas › Modular forms › Weight 4 › Level 112 › 112.4.a

Newspace 112.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(112))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
112.4.a.a	1	$\mathbb{Q}$	$q - 8q^{3} - 14q^{5} + 7q^{7} + O(q^{8})$	−
112.4.a.b	1	$\mathbb{Q}$	$q - 6q^{3} + 8q^{5} + 7q^{7} + O(q^{8})$	−
112.4.a.c	1	$\mathbb{Q}$	$q - 4q^{3} + 6q^{5} - 7q^{7} + O(q^{8})$	+
112.4.a.d	1	$\mathbb{Q}$	$q + 2q^{3} - 16q^{5} + 7q^{7} + O(q^{8})$	−
112.4.a.e	1	$\mathbb{Q}$	$q + 2q^{3} - 12q^{5} - 7q^{7} + O(q^{8})$	+
112.4.a.f	1	$\mathbb{Q}$	$q + 2q^{3} + 16q^{5} + 7q^{7} + O(q^{8})$	−
112.4.a.g	1	$\mathbb{Q}$	$q + 10q^{3} - 8q^{5} + 7q^{7} + O(q^{8})$	−
112.4.a.h	2	$\mathbb{Q}(\sqrt{57})$	$2q + 2q^{3} + 22q^{5} - 14q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
112.4.a.a	$x - 882$
112.4.a.b	$x + 1268$
112.4.a.c	$x + 770$
112.4.a.d	$x + 156$
112.4.a.e	$x + 908$
112.4.a.f	$x - 772$
112.4.a.g	$x + 420$
112.4.a.h	$x^{2} - 254x - 737696$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(112))$	9
Old forms	33
Cusp forms $S_{4}(\Gamma_0(112))$	42
Eisenstein series $E_{4}(\Gamma_0(112))$	12
Modular forms $M_{4}(\Gamma_0(112))$	54

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(112)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 5}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(28))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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, 8, 14, 16, 28, 56, 112]: [1, 2, 3, 3, 5, 8, 8, 15, 28, 52] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 4 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(112, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 112:4 1000

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