Atlas › Modular forms › Weight 2 › Level 990 › 990.2.a

Newspace 990.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(990))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
990.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + O(q^{8})$	+ + +	990a1 rank 1
990.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + O(q^{8})$	+ + +	990c1 rank 0
990.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + O(q^{8})$	+ + −	990e1 rank 1
990.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + 3q^{7} + O(q^{8})$	+ + +	990d1 rank 0
990.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - 4q^{7} + O(q^{8})$	+ − +	990b1 rank 0
990.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - q^{7} + O(q^{8})$	+ − −	990f1 rank 0
990.2.a.g	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} + 4q^{7} + O(q^{8})$	+ − −	990g1 rank 0
990.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} - 4q^{7} + O(q^{8})$	− + +	990j1 rank 1
990.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} - 4q^{7} + O(q^{8})$	− + −	990h1 rank 1
990.2.a.j	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{5} + O(q^{8})$	− − +	990k1 rank 0
990.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{5} + O(q^{8})$	− − −	990i1 rank 0
990.2.a.l	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{5} + 5q^{7} + O(q^{8})$	− − +	990l1 rank 0
990.2.a.m	2	$\mathbb{Q}(\sqrt{33})$	$2q + 2q^{2} + 2q^{4} - 2q^{5} + q^{7} + O(q^{8})$	− + −

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 2T_{13} + 3T_{17} + 4T_{19} + 5T_{23} + 6T_{29} + 7T_{31} + 8T_{37}$ 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 $T$
990.2.a.a	$x + 88$
990.2.a.b	$x - 98$
990.2.a.c	$x + 30$
990.2.a.d	$x - 35$
990.2.a.e	$x - 152$
990.2.a.f	$x - 107$
990.2.a.g	$x + 78$
990.2.a.h	$x + 182$
990.2.a.i	$x + 16$
990.2.a.j	$x - 50$
990.2.a.k	$x - 8$
990.2.a.l	$x + 85$
990.2.a.m	$x^{2} - 205x + 10102$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(990))$	14
Old forms	187
Cusp forms $S_{2}(\Gamma_0(990))$	201
Eisenstein series $E_{2}(\Gamma_0(990))$	31
Modular forms $M_{2}(\Gamma_0(990))$	232

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.634 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, 9, 10, 11, 15, 18, 22, 30, 33, 45, 55, 66, 90, 99, 110, 165, 198, 330, 495, 990]: [0, 1, 1, 1, 3, 2, 3, 2, 4, 5, 5, 10, 6, 8, 8, 16, 22, 14, 22, 28, 40, 80, 76, 224] (certified at ell = 2147482921)
T = [(7, 1), (13, 2), (17, 3), (19, 4), (23, 5), (29, 6), (31, 7), (37, 8)]; h of degree 14 = 1 x 14 (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(990, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 990:2 1000

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