Atlas › Modular forms › Weight 2 › Level 630 › 630.2.a

Newspace 630.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(630))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
630.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} - q^{7} + O(q^{8})$	+ + +	630c1 rank 0
630.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + q^{7} + O(q^{8})$	+ + −	630d1 rank 1
630.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + q^{7} + O(q^{8})$	+ + −	630a1 rank 0
630.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - q^{7} + O(q^{8})$	+ − +	630e1 rank 1
630.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - q^{7} + O(q^{8})$	+ − +	630b1 rank 0
630.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} + q^{7} + O(q^{8})$	+ − −	630f1 rank 0
630.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} - q^{7} + O(q^{8})$	− + +	630g1 rank 0
630.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} + q^{7} + O(q^{8})$	− + −	630i1 rank 0
630.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{5} - q^{7} + O(q^{8})$	− − +	630j1 rank 0
630.2.a.j	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{5} + q^{7} + O(q^{8})$	− − −	630h1 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 $T$
630.2.a.a	$x - 212$
630.2.a.b	$x + 252$
630.2.a.c	$x - 538$
630.2.a.d	$x + 140$
630.2.a.e	$x - 298$
630.2.a.f	$x - 436$
630.2.a.g	$x + 250$
630.2.a.h	$x + 224$
630.2.a.i	$x + 32$
630.2.a.j	$x + 146$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(630))$	10
Old forms	119
Cusp forms $S_{2}(\Gamma_0(630))$	129
Eisenstein series $E_{2}(\Gamma_0(630))$	31
Modular forms $M_{2}(\Gamma_0(630))$	160

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.349 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, 9, 10, 14, 15, 18, 21, 30, 35, 42, 45, 63, 70, 90, 105, 126, 210, 315, 630]: [0, 1, 1, 1, 3, 1, 2, 3, 4, 4, 5, 4, 10, 6, 12, 8, 10, 16, 22, 20, 28, 56, 52, 152] (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 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 4 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(630, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 630:2 1000

← 629.2.a · 631.2.a →