Atlas › Modular forms › Weight 8 › Level 45 › 45.8.a

Newspace 45.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(45))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
45.8.a.a	1	$\mathbb{Q}$	$q - 10q^{2} - 28q^{4} - 125q^{5} - 1170q^{7} + O(q^{8})$	+
45.8.a.b	1	$\mathbb{Q}$	$q - 5q^{2} - 103q^{4} + 125q^{5} + 930q^{7} + O(q^{8})$	−
45.8.a.c	1	$\mathbb{Q}$	$q + 5q^{2} - 103q^{4} - 125q^{5} + 930q^{7} + O(q^{8})$	+
45.8.a.d	1	$\mathbb{Q}$	$q + 10q^{2} - 28q^{4} + 125q^{5} - 1170q^{7} + O(q^{8})$	−
45.8.a.e	1	$\mathbb{Q}$	$q + 13q^{2} + 41q^{4} + 125q^{5} + 1380q^{7} + O(q^{8})$	−
45.8.a.f	1	$\mathbb{Q}$	$q + 14q^{2} + 68q^{4} - 125q^{5} - 1644q^{7} + O(q^{8})$	+
45.8.a.g	1	$\mathbb{Q}$	$q + 22q^{2} + 356q^{4} + 125q^{5} - 420q^{7} + O(q^{8})$	−
45.8.a.h	2	$\mathbb{Q}(\sqrt{19})$	$2q - 20q^{2} + 96q^{4} + 250q^{5} - 100q^{7} + O(q^{8})$	−
45.8.a.i	2	$\mathbb{Q}(\sqrt{601})$	$2q - 7q^{2} + 69q^{4} - 250q^{5} + 1304q^{7} + O(q^{8})$	+

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 $T$
45.8.a.a	$x + 1180$
45.8.a.b	$x - 925$
45.8.a.c	$x - 935$
45.8.a.d	$x + 1160$
45.8.a.e	$x - 1393$
45.8.a.f	$x + 1630$
45.8.a.g	$x + 398$
45.8.a.h	$x^{2} + 120x - 226300$
45.8.a.i	$x^{2} - 1297x - 67610$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(45))$	11
Old forms	27
Cusp forms $S_{8}(\Gamma_0(45))$	38
Eisenstein series $E_{8}(\Gamma_0(45))$	8
Modular forms $M_{8}(\Gamma_0(45))$	46

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(45)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 3}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.298 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, 5, 8, 16, 44] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 4 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, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 45:8 1000

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