Atlas › Modular forms › Weight 2 › Level 840 › 840.2.a

Newspace 840.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(840))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
840.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - q^{5} - q^{7} + O(q^{8})$	+ + +	840a1 rank 1
840.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + q^{7} + O(q^{8})$	+ + −	840e1 rank 1
840.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + q^{7} + O(q^{8})$	+ + −	840b1 rank 0
840.2.a.d	1	$\mathbb{Q}$	$q - q^{3} + q^{5} - q^{7} + O(q^{8})$	+ − +	840f1 rank 1
840.2.a.e	1	$\mathbb{Q}$	$q - q^{3} + q^{5} - q^{7} + O(q^{8})$	+ − +	840c1 rank 0
840.2.a.f	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + q^{7} + O(q^{8})$	+ − −	840g1 rank 0
840.2.a.g	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - q^{7} + O(q^{8})$	− + +	840h1 rank 1
840.2.a.h	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - q^{7} + O(q^{8})$	− + +	840d1 rank 0
840.2.a.i	1	$\mathbb{Q}$	$q + q^{3} - q^{5} + q^{7} + O(q^{8})$	− + −	840i1 rank 0
840.2.a.j	1	$\mathbb{Q}$	$q + q^{3} + q^{5} - q^{7} + O(q^{8})$	− − +	840j1 rank 0
840.2.a.k	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	− − −

Hecke characteristic polynomials

The eigenvalue of $T = 6665T_{11} + 1937T_{13} + 5856T_{17} + 9453T_{19} + 4431T_{23} + 7435T_{29} + 859T_{31} + 3987T_{37}$ 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 $T$
840.2.a.a	$x + 80882$
840.2.a.b	$x + 110398$
840.2.a.c	$x - 79706$
840.2.a.d	$x + 107062$
840.2.a.e	$x - 55658$
840.2.a.f	$x - 38066$
840.2.a.g	$x + 67822$
840.2.a.h	$x - 123978$
840.2.a.i	$x - 79678$
840.2.a.j	$x - 56338$
840.2.a.k	$x^{2} - 56148x + 700712388$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(840))$	12
Old forms	165
Cusp forms $S_{2}(\Gamma_0(840))$	177
Eisenstein series $E_{2}(\Gamma_0(840))$	31
Modular forms $M_{2}(\Gamma_0(840))$	208

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 3.547 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, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840]: [0, 1, 1, 2, 1, 3, 1, 3, 3, 5, 4, 4, 6, 4, 8, 7, 10, 6, 10, 12, 12, 18, 16, 22, 20, 32, 30, 40, 56, 56, 108, 208] (certified at ell = 2147483629)
T = [(11, 6665), (13, 1937), (17, 5856), (19, 9453), (23, 4431), (29, 7435), (31, 859), (37, 3987)]; h of degree 12 = 1 x 12 (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(840, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 840:2 1000

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