Atlas › Modular forms › Weight 4 › Level 230 › 230.4.a

Newspace 230.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(230))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
230.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 5q^{3} + 4q^{4} - 5q^{5} + 10q^{6} + 12q^{7} + O(q^{8})$	+ + −	$-1$
230.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{3} + 4q^{4} - 5q^{5} - 8q^{6} + 3q^{7} + O(q^{8})$	+ + −	$-1$
230.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + 7q^{3} + 4q^{4} + 5q^{5} - 14q^{6} + 20q^{7} + O(q^{8})$	+ − −	$+1$
230.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 4q^{4} + 5q^{5} - 2q^{6} - 32q^{7} + O(q^{8})$	− − −	$-1$
230.4.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 4q^{4} - 5q^{5} + 2q^{6} - 18q^{7} + O(q^{8})$	− + +	$-1$
230.4.a.f	2	$\mathbb{Q}(\sqrt{73})$	$2q - 4q^{2} - 3q^{3} + 8q^{4} + 10q^{5} + 6q^{6} - 17q^{7} + O(q^{8})$	+ − +	$-1$
230.4.a.g	3	degree 3	$3q - 6q^{2} - q^{3} + 12q^{4} + 15q^{5} + 2q^{6} + 7q^{7} + O(q^{8})$	+ − −	$+1$
230.4.a.h	4	degree 4	$4q - 8q^{2} - 4q^{3} + 16q^{4} - 20q^{5} + 8q^{6} - q^{7} + O(q^{8})$	+ + +	$+1$
230.4.a.i	4	degree 4	$4q + 8q^{2} + 4q^{3} + 16q^{4} - 20q^{5} + 8q^{6} + 26q^{7} + O(q^{8})$	− + −	$+1$
230.4.a.j	4	degree 4	$4q + 8q^{2} + 14q^{3} + 16q^{4} + 20q^{5} + 28q^{6} + 8q^{7} + O(q^{8})$	− − +	$+1$

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 $T$
230.4.a.a	$x - 31$
230.4.a.b	$x - 13$
230.4.a.c	$x - 67$
230.4.a.d	$x + 97$
230.4.a.e	$x + 53$
230.4.a.f	$x^{2} + 54x + 437$
230.4.a.g	$x^{3} - 20x^{2} - 7653x + 12972$
230.4.a.h	$x^{4} + 7x^{3} - 14513x^{2} + 91533x + 40877636$
230.4.a.i	$x^{4} - 82x^{3} - 6504x^{2} + 570658x - 3003625$
230.4.a.j	$x^{4} - 38x^{3} - 4620x^{2} + 110078x - 611293$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(230))$	22
Old forms	82
Cusp forms $S_{4}(\Gamma_0(230))$	104
Eisenstein series $E_{4}(\Gamma_0(230))$	8
Modular forms $M_{4}(\Gamma_0(230))$	112

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(230)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(46))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(115))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.206 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, 5, 10, 23, 46, 115, 230]: [1, 2, 3, 7, 7, 20, 38, 112] (certified at ell = 2147483053)
T = [(3, 1), (7, 3)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 6 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(230, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 230:4 1000

← 229.4.a · 231.4.a →