Atlas › Modular forms › Weight 2 › Level 845 › 845.2.a

Newspace 845.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(845))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
845.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} - q^{4} + q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	−	845a1 rank 0
845.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{4} - 2q^{5} + 5q^{6} - 4q^{7} + O(q^{8})$	+
845.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + 2q^{3} + 3q^{4} - 2q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+
845.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{4} + 2q^{5} + 6q^{6} - 4q^{7} + O(q^{8})$	−
845.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} + 2q^{5} - 5q^{6} + 4q^{7} + O(q^{8})$	−
845.2.a.f	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} + 2q^{3} + 3q^{4} + 2q^{5} + q^{6} + 2q^{7} + O(q^{8})$	−
845.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 2q^{5} - 4q^{6} - 4q^{7} + O(q^{8})$	+
845.2.a.h	3	degree 3	$3q - q^{2} - 5q^{3} - q^{4} + 3q^{5} + 4q^{6} - 5q^{7} + O(q^{8})$	−
845.2.a.i	3	degree 3	$3q - q^{2} - 2q^{3} + 5q^{4} + 3q^{5} - 10q^{6} - 2q^{7} + O(q^{8})$	−
845.2.a.j	3	degree 3	$3q + q^{2} - 5q^{3} - q^{4} - 3q^{5} - 4q^{6} + 5q^{7} + O(q^{8})$	+
845.2.a.k	3	degree 3	$3q + q^{2} - 2q^{3} + 5q^{4} - 3q^{5} + 10q^{6} + 2q^{7} + O(q^{8})$	+
845.2.a.l	4	degree 4	$4q - 2q^{2} - 2q^{3} + 2q^{4} + 4q^{5} + 4q^{6} - 10q^{7} + O(q^{8})$	−
845.2.a.m	4	degree 4	$4q + 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} - 4q^{6} + 10q^{7} + O(q^{8})$	+
845.2.a.n	9	degree 9	$9q - 3q^{2} + 7q^{3} + 17q^{4} - 9q^{5} + 2q^{6} - 7q^{7} + O(q^{8})$	+
845.2.a.o	9	degree 9	$9q + 3q^{2} + 7q^{3} + 17q^{4} + 9q^{5} - 2q^{6} + 7q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
845.2.a.a	$x + 5$
845.2.a.b	$x^{2} + x - 61$
845.2.a.c	$x^{2} - 5x + 3$
845.2.a.d	$x^{2} - 6x - 39$
845.2.a.e	$x^{2} - x - 31$
845.2.a.f	$x^{2} - 7x + 9$
845.2.a.g	$x^{2} - 2x - 7$
845.2.a.h	$x^{3} + 16x^{2} + 55x - 43$
845.2.a.i	$x^{3} + 7x^{2} - 5x - 27$
845.2.a.j	$x^{3} + 14x^{2} + 49x + 49$
845.2.a.k	$x^{3} + 5x^{2} - 77x - 57$
845.2.a.l	$x^{4} + 8x^{3} - 57x^{2} - 220x + 517$
845.2.a.m	$x^{4} + 4x^{3} - 57x^{2} - 320x - 299$
845.2.a.n	$x^{9} - 18x^{8} - 55x^{7} + 2279x^{6} - 4679x^{5} - 77988x^{4} + 282570x^{3} + 444367x^{2} - 1591177x - 1195649$
845.2.a.o	$x^{9} - 24x^{8} + 83x^{7} + 1963x^{6} - 16751x^{5} + 12438x^{4} + 176328x^{3} - 151543x^{2} - 365887x + 25523$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(845))$	51
Old forms	26
Cusp forms $S_{2}(\Gamma_0(845))$	77
Eisenstein series $E_{2}(\Gamma_0(845))$	27
Modular forms $M_{2}(\Gamma_0(845))$	104

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(845)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(169))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 5, 13, 65, 169, 845]: [0, 1, 1, 8, 15, 92] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 51 = 1 x 51 (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(845, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 845:2 1000

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