Atlas › Modular forms › Weight 2 › Level 847 › 847.2.a

Newspace 847.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(847))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
847.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} - q^{4} - 2q^{5} - 2q^{6} + q^{7} + O(q^{8})$	−	847c1 rank 1
847.2.a.b	1	$\mathbb{Q}$	$q - 3q^{3} - 2q^{4} - q^{5} + q^{7} + O(q^{8})$	−	847b1 rank 1
847.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + 3q^{5} - q^{7} + O(q^{8})$	+	847a1 rank 0
847.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} - q^{3} + 3q^{4} + 2q^{5} - q^{6} + 2q^{7} + O(q^{8})$	−
847.2.a.e	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} - q^{3} + 3q^{4} + 7q^{6} + 2q^{7} + O(q^{8})$	−
847.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} + 6q^{4} - 4q^{5} + 10q^{6} - 2q^{7} + O(q^{8})$	+
847.2.a.g	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} - q^{3} + 3q^{4} - 7q^{6} - 2q^{7} + O(q^{8})$	+
847.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} - q^{3} + 3q^{4} + 2q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+
847.2.a.i	3	degree 3	$3q - 2q^{2} - q^{3} + 8q^{4} + q^{5} - 12q^{6} - 3q^{7} + O(q^{8})$	+
847.2.a.j	3	degree 3	$3q + 2q^{2} - q^{3} + 8q^{4} + q^{5} + 12q^{6} + 3q^{7} + O(q^{8})$	−
847.2.a.k	4	degree 4	$4q - 2q^{2} - 2q^{3} + 4q^{4} - 6q^{5} + q^{6} + 4q^{7} + O(q^{8})$	−
847.2.a.l	4	degree 4	$4q + 2q^{2} - 2q^{3} + 4q^{4} - 6q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+
847.2.a.m	6	degree 6	$6q - 4q^{2} - 2q^{3} + 4q^{4} - 4q^{5} + 6q^{6} - 6q^{7} + O(q^{8})$	+
847.2.a.n	6	degree 6	$6q + 4q^{2} - 2q^{3} + 4q^{4} - 4q^{5} - 6q^{6} + 6q^{7} + O(q^{8})$	−
847.2.a.o	8	degree 8	$8q - q^{2} + 4q^{3} + 7q^{4} + 10q^{5} + q^{6} - 8q^{7} + O(q^{8})$	+
847.2.a.p	8	degree 8	$8q + q^{2} + 4q^{3} + 7q^{4} + 10q^{5} - q^{6} + 8q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{13}$ 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 $T$
847.2.a.a	$x + 33$
847.2.a.b	$x - 14$
847.2.a.c	$x - 46$
847.2.a.d	$x^{2} - 18x - 239$
847.2.a.e	$x^{2} + 46x + 321$
847.2.a.f	$x^{2} + 28x - 409$
847.2.a.g	$x^{2} - 40x - 1173$
847.2.a.h	$x^{2} + 4x - 121$
847.2.a.i	$x^{3} + 56x^{2} + 415x - 4850$
847.2.a.j	$x^{3} - 60x^{2} - 225x + 33966$
847.2.a.k	$x^{4} + 38x^{3} - 638x^{2} - 12841x + 144439$
847.2.a.l	$x^{4} + 34x^{3} - 1686x^{2} - 74425x - 673805$
847.2.a.m	$x^{6} + 58x^{5} + 939x^{4} + 2268x^{3} - 35065x^{2} - 47350x + 109933$
847.2.a.n	$x^{6} - 6x^{5} - 3129x^{4} + 42444x^{3} + 670527x^{2} + 937818x - 4604391$
847.2.a.o	$x^{8} - 103x^{7} + 1813x^{6} + 167384x^{5} - 9478475x^{4} + 200517474x^{3} - 1945415892x^{2} + 7693531752x - 8546174729$
847.2.a.p	$x^{8} - 21x^{7} - 3591x^{6} + 63670x^{5} + 3142135x^{4} - 27778580x^{3} - 805511250x^{2} - 517825450x + 10712525275$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(847))$	55
Old forms	22
Cusp forms $S_{2}(\Gamma_0(847))$	77
Eisenstein series $E_{2}(\Gamma_0(847))$	23
Modular forms $M_{2}(\Gamma_0(847))$	100

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(847)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(77))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(121))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.343 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, 7, 11, 77, 121, 847]: [0, 1, 2, 10, 12, 90] (certified at ell = 2147482921)
T = [(2, 1), (3, 3), (5, 5), (13, 7)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 12 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(847, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 847:2 1000

← 846.2.a · 848.2.a →