Atlas › Modular forms › Weight 2 › Level 966 › 966.2.a

Newspace 966.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(966))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
966.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 4q^{5} + q^{6} + q^{7} + O(q^{8})$	+ + − −	$-1$	966d1 rank 1
966.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + + +	$-1$	966a1 rank 1
966.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} + q^{7} + O(q^{8})$	+ + − −	$-1$	966c1 rank 1
966.2.a.d	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 3q^{5} + q^{6} + q^{7} + O(q^{8})$	+ + − +	$+1$	966b1 rank 0
966.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} + O(q^{8})$	+ − + −	$-1$	966e1 rank 1
966.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} + O(q^{8})$	+ − − −	$+1$	966f1 rank 0
966.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + O(q^{8})$	− + − +	$-1$	966g1 rank 1
966.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + O(q^{8})$	− + − −	$+1$	966h1 rank 0
966.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} + q^{7} + O(q^{8})$	− − − +	$+1$	966k1 rank 0
966.2.a.j	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} - q^{7} + O(q^{8})$	− − + −	$+1$	966i1 rank 0
966.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 3q^{5} + q^{6} - q^{7} + O(q^{8})$	− − + −	$+1$	966j1 rank 0
966.2.a.l	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+ + + −	$+1$
966.2.a.m	2	$\mathbb{Q}(\sqrt{41})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ − + +	$+1$
966.2.a.n	2	$\mathbb{Q}(\sqrt{41})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+ − − −	$+1$
966.2.a.o	2	$\mathbb{Q}(\sqrt{33})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− + + +	$+1$
966.2.a.p	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− − − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{11} + 3T_{19} + 4T_{29} + 5T_{31} + 6T_{13} + 7T_{17} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53}$ 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 $T$
966.2.a.a	$x + 250$
966.2.a.b	$x - 10$
966.2.a.c	$x + 46$
966.2.a.d	$x - 69$
966.2.a.e	$x + 34$
966.2.a.f	$x + 2$
966.2.a.g	$x + 162$
966.2.a.h	$x - 107$
966.2.a.i	$x + 89$
966.2.a.j	$x - 126$
966.2.a.k	$x + 199$
966.2.a.l	$x^{2} - 345x + 14962$
966.2.a.m	$x^{2} - 345x - 3546$
966.2.a.n	$x^{2} - 67x - 47678$
966.2.a.o	$x^{2} - 283x + 18166$
966.2.a.p	$x^{2} + 68x - 30844$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(966))$	21
Old forms	164
Cusp forms $S_{2}(\Gamma_0(966))$	185
Eisenstein series $E_{2}(\Gamma_0(966))$	15
Modular forms $M_{2}(\Gamma_0(966))$	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, 6, 7, 14, 21, 23, 42, 46, 69, 138, 161, 322, 483, 966]: [0, 1, 1, 3, 1, 4, 4, 3, 12, 8, 10, 28, 18, 52, 68, 200] (certified at ell = 2147483647)
T = [(5, 1), (11, 2), (19, 3), (29, 4), (31, 5), (13, 6), (17, 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(966, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 966:2 1000

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