Atlas › Modular forms › Weight 2 › Level 870 › 870.2.a

Newspace 870.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(870))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
870.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + O(q^{8})$	+ + − −	$-1$	870a1 rank 1
870.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ − + −	$-1$	870b1 rank 1
870.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+ − − +	$-1$	870c1 rank 1
870.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + O(q^{8})$	+ − − −	$+1$	870d1 rank 0
870.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + + −	$-1$	870e1 rank 1
870.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− + − +	$-1$	870f1 rank 1
870.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 4q^{7} + O(q^{8})$	− + − −	$+1$	870g1 rank 0
870.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− − − +	$+1$	870i1 rank 0
870.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− − − +	$+1$	870h1 rank 0
870.2.a.j	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+ + + −	$+1$
870.2.a.k	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	+ + − +	$+1$
870.2.a.l	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + O(q^{8})$	− + + +	$+1$
870.2.a.m	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− − + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = 9399T_{7} + 2198T_{11} + 5531T_{19} + 9022T_{23} + 6143T_{31} + 7295T_{13} + 377T_{17} + 2724T_{37}$ 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 $T$
870.2.a.a	$x + 22332$
870.2.a.b	$x + 30222$
870.2.a.c	$x + 131232$
870.2.a.d	$x - 40724$
870.2.a.e	$x + 113282$
870.2.a.f	$x + 124840$
870.2.a.g	$x - 28588$
870.2.a.h	$x - 56950$
870.2.a.i	$x + 29918$
870.2.a.j	$x^{2} - 59986x + 767539452$
870.2.a.k	$x^{2} - 74996x + 1387641116$
870.2.a.l	$x^{2} - 234504x + 12386237832$
870.2.a.m	$x^{2} - 17294x + 72754484$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(870))$	17
Old forms	156
Cusp forms $S_{2}(\Gamma_0(870))$	173
Eisenstein series $E_{2}(\Gamma_0(870))$	15
Modular forms $M_{2}(\Gamma_0(870))$	188

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.322 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, 5, 6, 10, 15, 29, 30, 58, 87, 145, 174, 290, 435, 870]: [0, 1, 1, 1, 3, 3, 4, 3, 10, 9, 12, 16, 34, 48, 64, 188] (certified at ell = 2147483549)
T = [(7, 9399), (11, 2198), (19, 5531), (23, 9022), (31, 6143), (13, 7295), (17, 377), (37, 2724)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 11 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(870, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 870:2 1000

← 869.2.a · 871.2.a →