Atlas › Modular forms › Weight 4 › Level 231 › 231.4.a

Newspace 231.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(231))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
231.4.a.a	1	$\mathbb{Q}$	$q - 3q^{2} - 3q^{3} + q^{4} - 4q^{5} + 9q^{6} - 7q^{7} + O(q^{8})$	+ + −	$-1$
231.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} - 4q^{4} + q^{5} - 6q^{6} - 7q^{7} + O(q^{8})$	− + +	$-1$
231.4.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} - 4q^{4} + 11q^{5} - 6q^{6} - 7q^{7} + O(q^{8})$	+ + −	$-1$
231.4.a.d	1	$\mathbb{Q}$	$q + 3q^{2} + 3q^{3} + q^{4} - 14q^{5} + 9q^{6} - 7q^{7} + O(q^{8})$	− + +	$-1$
231.4.a.e	1	$\mathbb{Q}$	$q + 5q^{2} + 3q^{3} + 17q^{4} - 6q^{5} + 15q^{6} + 7q^{7} + O(q^{8})$	− − +	$+1$
231.4.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q - 3q^{2} + 6q^{3} - 3q^{4} - 19q^{5} - 9q^{6} + 14q^{7} + O(q^{8})$	− − −	$-1$
231.4.a.g	2	$\mathbb{Q}(\sqrt{17})$	$2q - 3q^{2} + 6q^{3} - 3q^{4} + 25q^{5} - 9q^{6} + 14q^{7} + O(q^{8})$	− − +	$+1$
231.4.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q + 3q^{2} - 6q^{3} - 3q^{4} + q^{5} - 9q^{6} + 14q^{7} + O(q^{8})$	+ − +	$-1$
231.4.a.i	2	$\mathbb{Q}(\sqrt{37})$	$2q + 3q^{2} + 6q^{3} + 7q^{4} + 2q^{5} + 9q^{6} + 14q^{7} + O(q^{8})$	− − +	$+1$
231.4.a.j	5	degree 5	$5q - q^{2} - 15q^{3} + 21q^{4} + 7q^{5} + 3q^{6} - 35q^{7} + O(q^{8})$	+ + +	$+1$
231.4.a.k	5	degree 5	$5q - q^{2} - 15q^{3} + 21q^{4} + 21q^{5} + 3q^{6} + 35q^{7} + O(q^{8})$	+ − −	$+1$
231.4.a.l	5	degree 5	$5q + 5q^{2} + 15q^{3} + 21q^{4} + 7q^{5} + 15q^{6} - 35q^{7} + O(q^{8})$	− + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{17} + 7T_{19} + 9T_{23}$ 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 $T$
231.4.a.a	$x - 1063$
231.4.a.b	$x + 1176$
231.4.a.c	$x + 1902$
231.4.a.d	$x + 1741$
231.4.a.e	$x + 1743$
231.4.a.f	$x^{2} + 3153x + 1991158$
231.4.a.g	$x^{2} - 257x - 1654286$
231.4.a.h	$x^{2} + 1555x + 546328$
231.4.a.i	$x^{2} - 2545x + 1455633$
231.4.a.j	$x^{5} - 153x^{4} - 5880578x^{3} + 7363701169x^{2} - 2754720616153x + 321713014539222$
231.4.a.k	$x^{5} - 901x^{4} - 5520942x^{3} + 1196150079x^{2} + 4949551927595x + 718146242962312$
231.4.a.l	$x^{5} - 1231x^{4} - 1678554x^{3} + 1647501531x^{2} + 576611076645x - 353026826363736$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(231))$	28
Old forms	64
Cusp forms $S_{4}(\Gamma_0(231))$	92
Eisenstein series $E_{4}(\Gamma_0(231))$	8
Modular forms $M_{4}(\Gamma_0(231))$	100

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(231)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(77))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.381 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, 3, 7, 11, 21, 33, 77, 231]: [1, 2, 3, 4, 10, 14, 26, 100] (certified at ell = 2147482951)
T = [(2, 1), (5, 3), (17, 5), (19, 7), (23, 9)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 11 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(231, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 231:4 1000

← 230.4.a · 232.4.a →