Atlas › Modular forms › Weight 2 › Level 238 › 238.2.a

Newspace 238.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(238))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
238.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{5} - q^{7} + O(q^{8})$	+ + +	$-1$	238b1 rank 1
238.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + 4q^{5} - 2q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	238e1 rank 0
238.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 4q^{5} - 2q^{6} + q^{7} + O(q^{8})$	− − +	$-1$	238a1 rank 1
238.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} + q^{7} + O(q^{8})$	− − −	$+1$	238c1 rank 0
238.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	238d1 rank 0
238.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{19}$ 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 $T$
238.2.a.a	$x + 34$
238.2.a.b	$x + 88$
238.2.a.c	$x + 58$
238.2.a.d	$x - 28$
238.2.a.e	$x + 22$
238.2.a.f	$x^{2} + 6x - 4196$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(238))$	7
Old forms	26
Cusp forms $S_{2}(\Gamma_0(238))$	33
Eisenstein series $E_{2}(\Gamma_0(238))$	7
Modular forms $M_{2}(\Gamma_0(238))$	40

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(238)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(119))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.339 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, 7, 14, 17, 34, 119, 238]: [0, 1, 1, 4, 2, 6, 14, 40] (certified at ell = 2147483137)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (19, 9)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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(238, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 238:2 1000

← 237.2.a · 239.2.a →