Atlas › Modular forms › Weight 2 › Level 880 › 880.2.a

Newspace 880.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(880))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
880.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} + q^{5} - q^{7} + O(q^{8})$	− −	880d1 rank 1
880.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} + q^{5} + O(q^{8})$	− +	880h1 rank 1
880.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - q^{5} - 5q^{7} + O(q^{8})$	+ +	880e1 rank 0
880.2.a.d	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + q^{7} + O(q^{8})$	+ −	880f1 rank 1
880.2.a.e	1	$\mathbb{Q}$	$q - q^{5} + 2q^{7} + O(q^{8})$	+ +	880a1 rank 1
880.2.a.f	1	$\mathbb{Q}$	$q - q^{5} + 2q^{7} + O(q^{8})$	+ −	880b1 rank 0
880.2.a.g	1	$\mathbb{Q}$	$q + q^{5} - 4q^{7} + O(q^{8})$	− −	880c1 rank 1
880.2.a.h	1	$\mathbb{Q}$	$q + q^{5} + O(q^{8})$	− −	880i1 rank 0
880.2.a.i	1	$\mathbb{Q}$	$q + q^{3} + q^{5} - 3q^{7} + O(q^{8})$	− +	880g1 rank 1
880.2.a.j	1	$\mathbb{Q}$	$q + 2q^{3} + q^{5} + 4q^{7} + O(q^{8})$	− −	880j1 rank 0
880.2.a.k	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} - 2q^{5} - 5q^{7} + O(q^{8})$	+ −
880.2.a.l	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} - 2q^{5} - 3q^{7} + O(q^{8})$	+ +
880.2.a.m	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{5} + 4q^{7} + O(q^{8})$	+ +
880.2.a.n	2	$\mathbb{Q}(\sqrt{33})$	$2q + q^{3} + 2q^{5} - q^{7} + O(q^{8})$	− −
880.2.a.o	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} + 2q^{5} + q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47} + 13T_{53} + 14T_{59} + 15T_{61} + 16T_{67}$ 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 $T$
880.2.a.a	$x + 624$
880.2.a.b	$x + 498$
880.2.a.c	$x + 148$
880.2.a.d	$x + 334$
880.2.a.e	$x + 242$
880.2.a.f	$x - 474$
880.2.a.g	$x + 150$
880.2.a.h	$x - 258$
880.2.a.i	$x - 150$
880.2.a.j	$x - 146$
880.2.a.k	$x^{2} - 48x - 36$
880.2.a.l	$x^{2} + 474x + 11952$
880.2.a.m	$x^{2} - 460x - 103900$
880.2.a.n	$x^{2} + 50x - 60392$
880.2.a.o	$x^{2} - 676x - 17404$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(880))$	20
Old forms	113
Cusp forms $S_{2}(\Gamma_0(880))$	133
Eisenstein series $E_{2}(\Gamma_0(880))$	23
Modular forms $M_{2}(\Gamma_0(880))$	156

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 3.261 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, 4, 5, 8, 10, 11, 16, 20, 22, 40, 44, 55, 80, 88, 110, 176, 220, 440, 880]: [0, 1, 2, 1, 3, 3, 2, 4, 6, 5, 10, 9, 8, 16, 16, 22, 28, 42, 80, 152] (certified at ell = 2147482921)
T = [(3, 1), (7, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12), (53, 13), (59, 14), (61, 15), (67, 16)]; h of degree 20 = 1 x 20 (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(880, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 880:2 1000

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