Atlas › Modular forms › Weight 4 › Level 210 › 210.4.a

Newspace 210.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(210))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
210.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} - 5q^{5} + 6q^{6} + 7q^{7} + O(q^{8})$	+ + + −	$-1$
210.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} + 5q^{5} + 6q^{6} - 7q^{7} + O(q^{8})$	+ + − +	$-1$
210.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} + 5q^{5} + 6q^{6} + 7q^{7} + O(q^{8})$	+ + − −	$+1$
210.4.a.d	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} - 5q^{5} - 6q^{6} - 7q^{7} + O(q^{8})$	+ − + +	$-1$
210.4.a.e	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} - 5q^{5} - 6q^{6} + 7q^{7} + O(q^{8})$	+ − + −	$+1$
210.4.a.f	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} + 5q^{5} - 6q^{6} - 7q^{7} + O(q^{8})$	+ − − +	$+1$
210.4.a.g	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} - 5q^{5} - 6q^{6} - 7q^{7} + O(q^{8})$	− + + +	$-1$
210.4.a.h	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} - 5q^{5} - 6q^{6} + 7q^{7} + O(q^{8})$	− + + −	$+1$
210.4.a.i	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} + 5q^{5} - 6q^{6} - 7q^{7} + O(q^{8})$	− + − +	$+1$
210.4.a.j	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 4q^{4} - 5q^{5} + 6q^{6} - 7q^{7} + O(q^{8})$	− − + +	$+1$
210.4.a.k	2	$\mathbb{Q}(\sqrt{106})$	$2q + 4q^{2} + 6q^{3} + 8q^{4} + 10q^{5} + 12q^{6} + 14q^{7} + O(q^{8})$	− − − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{11} + 3T_{13} + 5T_{19} + 7T_{23} + 9T_{29}$ 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 $T$
210.4.a.a	$x + 2468$
210.4.a.b	$x + 3080$
210.4.a.c	$x - 2632$
210.4.a.d	$x - 204$
210.4.a.e	$x - 484$
210.4.a.f	$x - 304$
210.4.a.g	$x + 1984$
210.4.a.h	$x - 2944$
210.4.a.i	$x - 956$
210.4.a.j	$x + 2512$
210.4.a.k	$x^{2} + 216x - 157936$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(210))$	12
Old forms	124
Cusp forms $S_{4}(\Gamma_0(210))$	136
Eisenstein series $E_{4}(\Gamma_0(210))$	16
Modular forms $M_{4}(\Gamma_0(210))$	152

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.6 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, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210]: [1, 2, 2, 3, 5, 3, 7, 8, 8, 10, 22, 14, 28, 40, 52, 152] (certified at ell = 2147483629)
T = [(11, 1), (13, 3), (19, 5), (23, 7), (29, 9)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 5 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(210, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 210:4 1000

← 209.4.a · 211.4.a →