Atlas › Modular forms › Weight 2 › Level 210 › 210.2.a

Newspace 210.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(210))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
210.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + + +	$-1$	210d1 rank 1
210.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − − −	$+1$	210b1 rank 0
210.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + O(q^{8})$	− + − −	$+1$	210c1 rank 0
210.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + O(q^{8})$	− − + −	$+1$	210a1 rank 0
210.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + O(q^{8})$	− − − +	$+1$	210e1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{11} + 2T_{13} + 3T_{19} + 4T_{23} + 5T_{29} + 6T_{17} + 7T_{31} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53} + 13T_{59} + 14T_{61} + 15T_{67} + 16T_{71}$ 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 $T$
210.2.a.a	$x + 64$
210.2.a.b	$x + 416$
210.2.a.c	$x - 140$
210.2.a.d	$x + 20$
210.2.a.e	$x + 148$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(210))$	5
Old forms	36
Cusp forms $S_{2}(\Gamma_0(210))$	41
Eisenstein series $E_{2}(\Gamma_0(210))$	15
Modular forms $M_{2}(\Gamma_0(210))$	56

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(210)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(42))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(70))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(105))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.604 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]: [0, 1, 1, 1, 3, 1, 3, 4, 4, 4, 10, 6, 12, 16, 20, 56] (certified at ell = 2147483629)
T = [(11, 1), (13, 2), (19, 3), (23, 4), (29, 5), (17, 6), (31, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12), (59, 13), (61, 14), (67, 15), (71, 16)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 2 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, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 210:2 1000

← 209.2.a · 211.2.a →