Atlas › Modular forms › Weight 4 › Level 224 › 224.4.a

Newspace 224.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(224))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
224.4.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + 7q^{7} + O(q^{8})$	−
224.4.a.b	1	$\mathbb{Q}$	$q + 2q^{3} - 7q^{7} + O(q^{8})$	+
224.4.a.c	2	$\mathbb{Q}(\sqrt{37})$	$2q - 6q^{3} - 6q^{5} + 14q^{7} + O(q^{8})$	−
224.4.a.d	2	$\mathbb{Q}(\sqrt{37})$	$2q + 6q^{3} - 6q^{5} - 14q^{7} + O(q^{8})$	+
224.4.a.e	3	degree 3	$3q - 8q^{3} + 10q^{5} - 21q^{7} + O(q^{8})$	+
224.4.a.f	3	degree 3	$3q - 6q^{5} - 21q^{7} + O(q^{8})$	+
224.4.a.g	3	degree 3	$3q - 6q^{5} + 21q^{7} + O(q^{8})$	−
224.4.a.h	3	degree 3	$3q + 8q^{3} + 10q^{5} + 21q^{7} + O(q^{8})$	−

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 $T$
224.4.a.a	$x + 492$
224.4.a.b	$x + 688$
224.4.a.c	$x^{2} - 718x - 205044$
224.4.a.d	$x^{2} - 1170x + 122852$
224.4.a.e	$x^{3} - 1874x^{2} + 728508x + 107920576$
224.4.a.f	$x^{3} + 654x^{2} - 262092x - 97238656$
224.4.a.g	$x^{3} + 654x^{2} - 1354092x - 585897456$
224.4.a.h	$x^{3} - 2130x^{2} + 920092x - 84952016$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(224))$	18
Old forms	70
Cusp forms $S_{4}(\Gamma_0(224))$	88
Eisenstein series $E_{4}(\Gamma_0(224))$	16
Modular forms $M_{4}(\Gamma_0(224))$	104

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(224)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 5}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(28))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(112))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.314 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, 32, 56, 112, 224]: [1, 2, 3, 3, 5, 8, 8, 15, 14, 28, 52, 100] (certified at ell = 2147483497)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 7 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(224, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 224:4 1000

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