Atlas › Modular forms › Weight 2 › Level 968 › 968.2.a

Newspace 968.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(968))$

Newforms

Label	Dim	Coefficient field	Trace form	Elliptic curve
968.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 3q^{5} + 2q^{7} + O(q^{8})$	968e1 rank 1
968.2.a.b	1	$\mathbb{Q}$	$q + 3q^{5} - 4q^{7} + O(q^{8})$	968d1 rank 1
968.2.a.c	1	$\mathbb{Q}$	$q + 3q^{5} + 4q^{7} + O(q^{8})$	968b1 rank 0
968.2.a.d	1	$\mathbb{Q}$	$q + q^{3} + q^{5} - 4q^{7} + O(q^{8})$	968a1 rank 1
968.2.a.e	1	$\mathbb{Q}$	$q + q^{3} + q^{5} + 4q^{7} + O(q^{8})$	968c1 rank 0
968.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} - 2q^{7} + O(q^{8})$
968.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} + 2q^{7} + O(q^{8})$
968.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{3} - q^{5} - q^{7} + O(q^{8})$
968.2.a.i	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{3} - q^{5} + q^{7} + O(q^{8})$
968.2.a.j	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} + 3q^{5} + 2q^{7} + O(q^{8})$
968.2.a.k	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} - 4q^{5} - 2q^{7} + O(q^{8})$
968.2.a.l	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} - 4q^{5} + 2q^{7} + O(q^{8})$
968.2.a.m	4	degree 4	$4q + 2q^{3} + q^{5} - q^{7} + O(q^{8})$
968.2.a.n	4	degree 4	$4q + 2q^{3} + q^{5} + q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
968.2.a.a	$x - 74$
968.2.a.b	$x + 73$
968.2.a.c	$x + 27$
968.2.a.d	$x + 106$
968.2.a.e	$x - 70$
968.2.a.f	$x^{2} + 84x - 441$
968.2.a.g	$x^{2} - 164x + 1279$
968.2.a.h	$x^{2} + 67x - 1879$
968.2.a.i	$x^{2} + 51x - 2601$
968.2.a.j	$x^{2} - 128x + 3484$
968.2.a.k	$x^{2} + 62x + 853$
968.2.a.l	$x^{2} - 134x + 157$
968.2.a.m	$x^{4} - 31x^{3} - 11368x^{2} - 130499x - 193099$
968.2.a.n	$x^{4} + 51x^{3} + 450x^{2} + 1033x - 211$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(968))$	27
Old forms	82
Cusp forms $S_{2}(\Gamma_0(968))$	109
Eisenstein series $E_{2}(\Gamma_0(968))$	47
Modular forms $M_{2}(\Gamma_0(968))$	156

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(968)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(88))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(121))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(242))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(484))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.771 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, 4, 8, 11, 22, 44, 88, 121, 242, 484, 968]: [0, 1, 2, 3, 2, 5, 9, 16, 12, 35, 69, 136] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 8 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(968, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 968:2 1000

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