# Properties

 Label 160.6.f.a Level 160 Weight 6 Character orbit 160.f Analytic conductor 25.661 Analytic rank 0 Dimension 28 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 160.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.6614111701$$ Analytic rank: $$0$$ Dimension: $$28$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$28q + 1940q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + 1940q^{9} + 488q^{15} + 1556q^{25} - 4368q^{31} - 23360q^{39} - 2480q^{41} - 38420q^{49} + 48776q^{55} + 37200q^{65} + 69232q^{71} + 35984q^{79} + 122596q^{81} - 178744q^{89} - 89416q^{95} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 0 −28.9041 0 −40.5064 38.5257i 0 128.100i 0 592.448 0
49.2 0 −28.9041 0 −40.5064 + 38.5257i 0 128.100i 0 592.448 0
49.3 0 −21.6833 0 36.8367 42.0483i 0 236.448i 0 227.167 0
49.4 0 −21.6833 0 36.8367 + 42.0483i 0 236.448i 0 227.167 0
49.5 0 −21.4357 0 48.3867 + 27.9951i 0 39.9262i 0 216.491 0
49.6 0 −21.4357 0 48.3867 27.9951i 0 39.9262i 0 216.491 0
49.7 0 −16.0077 0 −19.1184 + 52.5308i 0 20.5525i 0 13.2465 0
49.8 0 −16.0077 0 −19.1184 52.5308i 0 20.5525i 0 13.2465 0
49.9 0 −10.5561 0 −52.7686 18.4521i 0 47.9937i 0 −131.568 0
49.10 0 −10.5561 0 −52.7686 + 18.4521i 0 47.9937i 0 −131.568 0
49.11 0 −7.17847 0 1.28331 + 55.8870i 0 146.905i 0 −191.470 0
49.12 0 −7.17847 0 1.28331 55.8870i 0 146.905i 0 −191.470 0
49.13 0 −1.29818 0 51.3939 21.9924i 0 170.399i 0 −241.315 0
49.14 0 −1.29818 0 51.3939 + 21.9924i 0 170.399i 0 −241.315 0
49.15 0 1.29818 0 −51.3939 + 21.9924i 0 170.399i 0 −241.315 0
49.16 0 1.29818 0 −51.3939 21.9924i 0 170.399i 0 −241.315 0
49.17 0 7.17847 0 −1.28331 55.8870i 0 146.905i 0 −191.470 0
49.18 0 7.17847 0 −1.28331 + 55.8870i 0 146.905i 0 −191.470 0
49.19 0 10.5561 0 52.7686 + 18.4521i 0 47.9937i 0 −131.568 0
49.20 0 10.5561 0 52.7686 18.4521i 0 47.9937i 0 −131.568 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.6.f.a 28
4.b odd 2 1 40.6.f.a 28
5.b even 2 1 inner 160.6.f.a 28
5.c odd 4 2 800.6.d.e 28
8.b even 2 1 inner 160.6.f.a 28
8.d odd 2 1 40.6.f.a 28
20.d odd 2 1 40.6.f.a 28
20.e even 4 2 200.6.d.e 28
40.e odd 2 1 40.6.f.a 28
40.f even 2 1 inner 160.6.f.a 28
40.i odd 4 2 800.6.d.e 28
40.k even 4 2 200.6.d.e 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.f.a 28 4.b odd 2 1
40.6.f.a 28 8.d odd 2 1
40.6.f.a 28 20.d odd 2 1
40.6.f.a 28 40.e odd 2 1
160.6.f.a 28 1.a even 1 1 trivial
160.6.f.a 28 5.b even 2 1 inner
160.6.f.a 28 8.b even 2 1 inner
160.6.f.a 28 40.f even 2 1 inner
200.6.d.e 28 20.e even 4 2
200.6.d.e 28 40.k even 4 2
800.6.d.e 28 5.c odd 4 2
800.6.d.e 28 40.i odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(160, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database