Atlas › Modular forms › Weight 4 › Level 222 › 222.4.a

Newspace 222.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(222))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
222.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} - 2q^{5} + 6q^{6} + O(q^{8})$	+ + −	$-1$
222.4.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} - 6q^{6} - 16q^{7} + O(q^{8})$	− + −	$+1$
222.4.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 4q^{4} - 16q^{5} + 6q^{6} - 24q^{7} + O(q^{8})$	− − −	$-1$
222.4.a.d	2	$\mathbb{Q}(\sqrt{11})$	$2q - 4q^{2} + 6q^{3} + 8q^{4} - 14q^{5} - 12q^{6} + 4q^{7} + O(q^{8})$	+ − +	$-1$
222.4.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{2} - 6q^{3} + 8q^{4} - 4q^{5} - 12q^{6} - 20q^{7} + O(q^{8})$	− + +	$-1$
222.4.a.f	2	$\mathbb{Q}(\sqrt{97})$	$2q + 4q^{2} - 6q^{3} + 8q^{4} - 4q^{5} - 12q^{6} + 17q^{7} + O(q^{8})$	− + −	$+1$
222.4.a.g	3	degree 3	$3q - 6q^{2} - 9q^{3} + 12q^{4} - 12q^{5} + 18q^{6} + 21q^{7} + O(q^{8})$	+ + +	$+1$
222.4.a.h	3	degree 3	$3q - 6q^{2} + 9q^{3} + 12q^{4} + 6q^{5} - 18q^{6} - 3q^{7} + O(q^{8})$	+ − −	$+1$
222.4.a.i	3	degree 3	$3q + 6q^{2} + 9q^{3} + 12q^{4} + 14q^{5} + 18q^{6} + 25q^{7} + O(q^{8})$	− − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{7}$ 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 $T$
222.4.a.a	$x + 2$
222.4.a.b	$x + 16$
222.4.a.c	$x + 40$
222.4.a.d	$x^{2} + 10x + 14$
222.4.a.e	$x^{2} + 24x + 46$
222.4.a.f	$x^{2} - 13x - 176$
222.4.a.g	$x^{3} - 9x^{2} - 2750x + 48934$
222.4.a.h	$x^{3} - 3x^{2} - 1374x + 8180$
222.4.a.i	$x^{3} - 39x^{2} + 160x + 2190$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(222))$	18
Old forms	92
Cusp forms $S_{4}(\Gamma_0(222))$	110
Eisenstein series $E_{4}(\Gamma_0(222))$	8
Modular forms $M_{4}(\Gamma_0(222))$	118

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(222)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(74))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(111))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.193 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, 6, 37, 74, 111, 222]: [1, 2, 2, 5, 11, 31, 40, 118] (certified at ell = 2147483629)
T = [(5, 1), (7, 1)]; h of degree 18 = 1 x 18 (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(222, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 222:4 1000

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