Atlas › Modular forms › Weight 2 › Level 930 › 930.2.a

Newspace 930.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(930))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
930.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+ + + −	$+1$	930b1 rank 0
930.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + O(q^{8})$	+ + + +	$-1$	930a1 rank 1
930.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+ + − −	$-1$	930e1 rank 1
930.2.a.d	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ + − −	$-1$	930d1 rank 1
930.2.a.e	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ + − +	$+1$	930c1 rank 0
930.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − + +	$+1$	930f1 rank 0
930.2.a.g	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 4q^{7} + O(q^{8})$	+ − + +	$+1$	930g1 rank 0
930.2.a.h	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ − − +	$-1$	930h1 rank 1
930.2.a.i	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} + O(q^{8})$	+ − − −	$+1$	930i1 rank 0
930.2.a.j	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 4q^{7} + O(q^{8})$	+ − − −	$+1$	930j1 rank 0
930.2.a.k	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + + +	$+1$	930k1 rank 0
930.2.a.l	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + O(q^{8})$	− + + −	$-1$	930m1 rank 1
930.2.a.m	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 3q^{7} + O(q^{8})$	− + + +	$+1$	930l1 rank 0
930.2.a.n	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− − + −	$+1$	930n1 rank 0
930.2.a.o	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + O(q^{8})$	− − − +	$+1$	930o1 rank 0
930.2.a.p	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + q^{7} + O(q^{8})$	− + − −	$+1$
930.2.a.q	2	$\mathbb{Q}(\sqrt{65})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - q^{7} + O(q^{8})$	− − + −	$+1$
930.2.a.r	2	$\mathbb{Q}(\sqrt{33})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− − − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53}$ 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 $T$
930.2.a.a	$x - 216$
930.2.a.b	$x + 98$
930.2.a.c	$x + 164$
930.2.a.d	$x + 272$
930.2.a.e	$x - 90$
930.2.a.f	$x - 244$
930.2.a.g	$x - 58$
930.2.a.h	$x - 28$
930.2.a.i	$x - 78$
930.2.a.j	$x + 52$
930.2.a.k	$x - 80$
930.2.a.l	$x + 132$
930.2.a.m	$x - 250$
930.2.a.n	$x + 156$
930.2.a.o	$x + 54$
930.2.a.p	$x^{2} - 462x + 42736$
930.2.a.q	$x^{2} - 58x - 13784$
930.2.a.r	$x^{2} + 58x - 6584$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(930))$	21
Old forms	164
Cusp forms $S_{2}(\Gamma_0(930))$	185
Eisenstein series $E_{2}(\Gamma_0(930))$	15
Modular forms $M_{2}(\Gamma_0(930))$	200

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.387 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, 10, 15, 30, 31, 62, 93, 155, 186, 310, 465, 930]: [0, 1, 1, 1, 3, 3, 4, 10, 3, 10, 12, 18, 36, 52, 68, 200] (certified at ell = 2147482921)
T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 7 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(930, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 930:2 1000

← 929.2.a · 931.2.a →