Atlas › Modular forms › Weight 2 › Level 45 › 45.2.a

Newspace 45.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(45))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
45.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{5} + O(q^{8})$	+			45a1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{7}$ 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 $T$
45.2.a.a	$x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(45))$	1
Old forms	2
Cusp forms $S_{2}(\Gamma_0(45))$	3
Eisenstein series $E_{2}(\Gamma_0(45))$	7
Modular forms $M_{2}(\Gamma_0(45))$	10

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(45)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 5, 9, 15, 45]: [0, 1, 1, 2, 4, 8] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 1 = 1 x 1 (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(45, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 45:2 1000

← 44.2.a · 46.2.a →