Atlas › Modular forms › Weight 2 › Level 231 › 231.2.a

Newspace 231.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(231))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
231.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	231a1 rank 0
231.2.a.b	2	$\mathbb{Q}(\sqrt{21})$	$2q - q^{2} - 2q^{3} + 7q^{4} + 6q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
231.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 2q^{3} - q^{4} + 2q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$
231.2.a.d	3	degree 3	$3q - 3q^{3} + 6q^{4} - 3q^{7} + O(q^{8})$	+ + −	$+1$
231.2.a.e	3	degree 3	$3q + 2q^{2} + 3q^{3} + 6q^{4} + 4q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	− + +	$+1$

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 $T$
231.2.a.a	$x - 31$
231.2.a.b	$x^{2} - x - 2315$
231.2.a.c	$x^{2} - 19x - 461$
231.2.a.d	$x^{3} - 30x^{2} - 7596x + 341991$
231.2.a.e	$x^{3} - 88x^{2} + 344x + 32609$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(231))$	11
Old forms	18
Cusp forms $S_{2}(\Gamma_0(231))$	29
Eisenstein series $E_{2}(\Gamma_0(231))$	7
Modular forms $M_{2}(\Gamma_0(231))$	36

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.116 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]: [0, 1, 1, 2, 4, 6, 10, 36] (certified at ell = 2147482951)
T = [(2, 1), (5, 3), (17, 5), (19, 7), (23, 9)]; h of degree 11 = 1 x 11 (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(231, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 231:2 1000

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