Atlas › Modular forms › Weight 2 › Level 528 › 528.2.a

Newspace 528.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(528))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
528.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 4q^{5} + 2q^{7} + O(q^{8})$	+ +	528f1 rank 0
528.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	+ −	528b1 rank 0
528.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - 2q^{7} + O(q^{8})$	+ +	528a1 rank 1
528.2.a.d	1	$\mathbb{Q}$	$q - q^{3} - 2q^{7} + O(q^{8})$	+ −	528g1 rank 1
528.2.a.e	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	+ +	528e1 rank 0
528.2.a.f	1	$\mathbb{Q}$	$q - q^{3} + 4q^{5} + 2q^{7} + O(q^{8})$	+ −	528c1 rank 0
528.2.a.g	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	− +	528h1 rank 1
528.2.a.h	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} - 2q^{7} + O(q^{8})$	− −	528i1 rank 0
528.2.a.i	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + O(q^{8})$	− +	528d1 rank 0
528.2.a.j	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + 4q^{7} + O(q^{8})$	− −	528j1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{19} + 5T_{23} + 6T_{29} + 7T_{17} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47} + 13T_{53} + 14T_{59} + 15T_{61} + 16T_{67}$ 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 $T$
528.2.a.a	$x - 258$
528.2.a.b	$x - 272$
528.2.a.c	$x + 290$
528.2.a.d	$x - 38$
528.2.a.e	$x - 120$
528.2.a.f	$x - 194$
528.2.a.g	$x - 184$
528.2.a.h	$x + 52$
528.2.a.i	$x - 204$
528.2.a.j	$x + 316$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(528))$	10
Old forms	75
Cusp forms $S_{2}(\Gamma_0(528))$	85
Eisenstein series $E_{2}(\Gamma_0(528))$	23
Modular forms $M_{2}(\Gamma_0(528))$	108

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.665 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, 2, 3, 4, 6, 8, 11, 12, 16, 22, 24, 33, 44, 48, 66, 88, 132, 176, 264, 528]: [0, 1, 1, 2, 3, 3, 2, 5, 4, 5, 8, 6, 9, 12, 16, 16, 30, 28, 56, 104] (certified at ell = 2147482921)
T = [(5, 1), (7, 2), (13, 3), (19, 4), (23, 5), (29, 6), (17, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12), (53, 13), (59, 14), (61, 15), (67, 16)]; h of degree 10 = 1 x 10 (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(528, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 528:2 1000

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