Atlas › Modular forms › Weight 2 › Level 545 › 545.2.a

Newspace 545.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(545))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
545.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + q^{5} - 4q^{7} + O(q^{8})$	− −	$-1$	545a1 rank 1
545.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} - 8q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
545.2.a.c	5	degree 5	$5q - 3q^{2} - 7q^{3} + 3q^{4} + 5q^{5} - 8q^{7} + O(q^{8})$	− −	$-1$
545.2.a.d	5	degree 5	$5q + q^{2} + q^{3} - q^{4} - 5q^{5} - 2q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
545.2.a.e	11	degree 11	$11q - 3q^{2} + q^{3} + 21q^{4} - 11q^{5} + 12q^{6} + 8q^{7} + O(q^{8})$	+ −	$+1$
545.2.a.f	13	degree 13	$13q + 3q^{2} + 9q^{3} + 19q^{4} + 13q^{5} - 6q^{6} + 16q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{11}$ 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 $T$
545.2.a.a	$x - 9$
545.2.a.b	$x^{2} - 10x - 367$
545.2.a.c	$x^{5} + 120x^{4} + 4887x^{3} + 75834x^{2} + 356318x + 251513$
545.2.a.d	$x^{5} + 64x^{4} + 677x^{3} - 16696x^{2} - 195180x + 1151041$
545.2.a.e	$x^{11} - 124x^{10} + 2218x^{9} + 216956x^{8} - 7086996x^{7} - 96132105x^{6} + 4566688088x^{5} + 10214380813x^{4} - 932062833529x^{3} - 2029818059419x^{2} + 66509005655354x + 301254796615347$
545.2.a.f	$x^{13} - 110x^{12} + 2253x^{11} + 127438x^{10} - 5083832x^{9} - 13648453x^{8} + 2591591050x^{7} - 21510959312x^{6} - 342591453507x^{5} + 5131062976508x^{4} - 9788850843627x^{3} - 101054305189210x^{2} + 368809465603326x - 169626754112977$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(545))$	37
Old forms	16
Cusp forms $S_{2}(\Gamma_0(545))$	53
Eisenstein series $E_{2}(\Gamma_0(545))$	3
Modular forms $M_{2}(\Gamma_0(545))$	56

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(545)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(109))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 109, 545]: [0, 1, 9, 56] (certified at ell = 2147482801)
T = [(2, 1), (3, 3), (7, 5), (11, 7)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 8 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(545, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 545:2 1000

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