Atlas › Modular forms › Weight 4 › Level 45 › 45.4.a

Newspace 45.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(45))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
45.4.a.a	1	$\mathbb{Q}$	$q - 5q^{2} + 17q^{4} + 5q^{5} - 30q^{7} + O(q^{8})$	−
45.4.a.b	1	$\mathbb{Q}$	$q - 3q^{2} + q^{4} + 5q^{5} + 20q^{7} + O(q^{8})$	−
45.4.a.c	1	$\mathbb{Q}$	$q - q^{2} - 7q^{4} - 5q^{5} - 24q^{7} + O(q^{8})$	+
45.4.a.d	1	$\mathbb{Q}$	$q + 4q^{2} + 8q^{4} + 5q^{5} + 6q^{7} + O(q^{8})$	−
45.4.a.e	1	$\mathbb{Q}$	$q + 5q^{2} + 17q^{4} - 5q^{5} - 30q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7}$ 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 $T$
45.4.a.a	$x + 95$
45.4.a.b	$x - 57$
45.4.a.c	$x + 73$
45.4.a.d	$x - 22$
45.4.a.e	$x + 85$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(45))$	5
Old forms	9
Cusp forms $S_{4}(\Gamma_0(45))$	14
Eisenstein series $E_{4}(\Gamma_0(45))$	8
Modular forms $M_{4}(\Gamma_0(45))$	22

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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, 2, 3, 4, 8, 20] (certified at ell = 2147483629)
T = [(2, 1), (7, 3)]; h of degree 5 = 1 x 5 (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(45, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 45:4 1000

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