Atlas › Modular forms › Weight 6 › Level 45 › 45.6.a

Newspace 45.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(45))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
45.6.a.a	1	$\mathbb{Q}$	$q - 7q^{2} + 17q^{4} + 25q^{5} + 12q^{7} + O(q^{8})$	−
45.6.a.b	1	$\mathbb{Q}$	$q - 2q^{2} - 28q^{4} - 25q^{5} + 192q^{7} + O(q^{8})$	+
45.6.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - 28q^{4} + 25q^{5} - 132q^{7} + O(q^{8})$	−
45.6.a.d	2	$\mathbb{Q}(\sqrt{145})$	$2q - 5q^{2} + 21q^{4} - 50q^{5} + 80q^{7} + O(q^{8})$	+
45.6.a.e	2	$\mathbb{Q}(\sqrt{409})$	$2q + q^{2} + 141q^{4} - 50q^{5} - 112q^{7} + O(q^{8})$	+
45.6.a.f	2	$\mathbb{Q}(\sqrt{145})$	$2q + 5q^{2} + 21q^{4} + 50q^{5} + 80q^{7} + O(q^{8})$	−

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 $T$
45.6.a.a	$x - 5$
45.6.a.b	$x - 190$
45.6.a.c	$x + 130$
45.6.a.d	$x^{2} - 75x - 11680$
45.6.a.e	$x^{2} + 111x - 26470$
45.6.a.f	$x^{2} - 85x - 14180$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(45))$	9
Old forms	17
Cusp forms $S_{6}(\Gamma_0(45))$	26
Eisenstein series $E_{6}(\Gamma_0(45))$	8
Modular forms $M_{6}(\Gamma_0(45))$	34

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(45)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.244 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]: [1, 3, 3, 6, 12, 32] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 9 = 1 x 9 (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(45, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 45:6 1000

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