Atlas › Modular forms › Weight 2 › Level 245 › 245.2.a

Newspace 245.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(245))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
245.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 2q^{4} + q^{5} + 6q^{6} + O(q^{8})$	−	245a1 rank 1
245.2.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 2q^{4} - q^{5} - 6q^{6} + O(q^{8})$	+	245b1 rank 0
245.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} + q^{5} + O(q^{8})$	−	245c1 rank 1
245.2.a.d	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} + q^{3} + 5q^{4} - 2q^{5} + 8q^{6} + O(q^{8})$	+
245.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{5} - 4q^{6} + O(q^{8})$	+
245.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} + 2q^{5} + 4q^{6} + O(q^{8})$	−
245.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + O(q^{8})$	+
245.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
245.2.a.a	$x + 11$
245.2.a.b	$x - 7$
245.2.a.c	$x + 3$
245.2.a.d	$x^{2} - 2x - 67$
245.2.a.e	$x^{2} + 6x + 1$
245.2.a.f	$x^{2} - 6x - 23$
245.2.a.g	$x^{2} + 4x - 28$
245.2.a.h	$x^{2} - 8x + 8$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(245))$	13
Old forms	8
Cusp forms $S_{2}(\Gamma_0(245))$	21
Eisenstein series $E_{2}(\Gamma_0(245))$	15
Modular forms $M_{2}(\Gamma_0(245))$	36

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(245)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.194 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, 5, 7, 35, 49, 245]: [0, 1, 1, 6, 5, 30] (certified at ell = 2147483353)
T = [(2, 1), (3, 3)]; h of degree 13 = 1 x 13 (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(245, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 245:2 1000

← 244.2.a · 246.2.a →