Atlas › Modular forms › Weight 4 › Level 220 › 220.4.a

Newspace 220.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(220))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
220.4.a.a	1	$\mathbb{Q}$	$q - 5q^{3} + 5q^{5} + 11q^{7} + O(q^{8})$	− +
220.4.a.b	1	$\mathbb{Q}$	$q + 5q^{3} - 5q^{5} - 19q^{7} + O(q^{8})$	+ +
220.4.a.c	1	$\mathbb{Q}$	$q + 8q^{3} + 5q^{5} + 24q^{7} + O(q^{8})$	− +
220.4.a.d	2	$\mathbb{Q}(\sqrt{97})$	$2q - 9q^{3} + 10q^{5} - 15q^{7} + O(q^{8})$	− −
220.4.a.e	2	$\mathbb{Q}(\sqrt{6})$	$2q - 8q^{3} - 10q^{5} + 36q^{7} + O(q^{8})$	+ +
220.4.a.f	3	degree 3	$3q - 3q^{3} - 15q^{5} - 5q^{7} + O(q^{8})$	+ −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7}$ 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 $T$
220.4.a.a	$x - 28$
220.4.a.b	$x + 52$
220.4.a.c	$x - 80$
220.4.a.d	$x^{2} + 54x + 632$
220.4.a.e	$x^{2} - 100x + 1900$
220.4.a.f	$x^{3} + 18x^{2} - 4092x + 19688$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(220))$	10
Old forms	92
Cusp forms $S_{4}(\Gamma_0(220))$	102
Eisenstein series $E_{4}(\Gamma_0(220))$	12
Modular forms $M_{4}(\Gamma_0(220))$	114

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(220)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(22))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(110))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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, 5, 10, 11, 20, 22, 44, 55, 110, 220]: [1, 2, 3, 3, 7, 4, 12, 11, 21, 20, 58, 114] (certified at ell = 2147482921)
T = [(3, 1), (7, 3)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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(220, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 220:4 1000

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