Atlas › Modular forms › Weight 2 › Level 234 › 234.2.a

Newspace 234.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(234))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
234.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ +	234c1 rank 1
234.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} + q^{7} + O(q^{8})$	+ +	234a1 rank 0
234.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	− −	234d1 rank 0
234.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} - 2q^{7} + O(q^{8})$	− +	234b1 rank 0
234.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 3q^{5} - q^{7} + O(q^{8})$	− −	234e1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
234.2.a.a	$x + 8$
234.2.a.b	$x - 4$
234.2.a.c	$x - 10$
234.2.a.d	$x + 4$
234.2.a.e	$x$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(234))$	5
Old forms	30
Cusp forms $S_{2}(\Gamma_0(234))$	35
Eisenstein series $E_{2}(\Gamma_0(234))$	15
Modular forms $M_{2}(\Gamma_0(234))$	50

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(234)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(78))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(117))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.22 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, 9, 13, 18, 26, 39, 78, 117, 234]: [0, 1, 1, 3, 2, 1, 5, 5, 6, 18, 16, 46] (certified at ell = 2147483629)
T = [(5, 1), (7, 3)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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(234, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 234:2 1000

← 233.2.a · 235.2.a →