# Properties

 Label 160.6.o.a Level 160 Weight 6 Character orbit 160.o Analytic conductor 25.661 Analytic rank 0 Dimension 56 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 160.o (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.6614111701$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$28$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$56q + 4q^{3} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q + 4q^{3} + 8q^{11} - 408q^{17} - 3120q^{25} - 968q^{27} - 976q^{33} + 4780q^{35} - 8q^{41} - 1308q^{43} - 20872q^{51} + 968q^{57} + 17680q^{65} - 89252q^{67} - 25184q^{73} + 127740q^{75} - 67792q^{81} + 126444q^{83} - 329432q^{91} + 212576q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 0 −20.2357 + 20.2357i 0 −6.76469 + 55.4909i 0 37.6042 37.6042i 0 575.965i 0
47.2 0 −20.2357 + 20.2357i 0 6.76469 55.4909i 0 −37.6042 + 37.6042i 0 575.965i 0
47.3 0 −17.2238 + 17.2238i 0 −54.8737 10.6715i 0 −14.7411 + 14.7411i 0 350.316i 0
47.4 0 −17.2238 + 17.2238i 0 54.8737 + 10.6715i 0 14.7411 14.7411i 0 350.316i 0
47.5 0 −11.5932 + 11.5932i 0 48.9682 + 26.9650i 0 −75.8554 + 75.8554i 0 25.8041i 0
47.6 0 −11.5932 + 11.5932i 0 −48.9682 26.9650i 0 75.8554 75.8554i 0 25.8041i 0
47.7 0 −10.6033 + 10.6033i 0 15.3629 53.7493i 0 159.941 159.941i 0 18.1393i 0
47.8 0 −10.6033 + 10.6033i 0 −15.3629 + 53.7493i 0 −159.941 + 159.941i 0 18.1393i 0
47.9 0 −10.0657 + 10.0657i 0 −23.9125 + 50.5291i 0 96.6855 96.6855i 0 40.3635i 0
47.10 0 −10.0657 + 10.0657i 0 23.9125 50.5291i 0 −96.6855 + 96.6855i 0 40.3635i 0
47.11 0 −4.45554 + 4.45554i 0 49.9881 + 25.0237i 0 169.279 169.279i 0 203.296i 0
47.12 0 −4.45554 + 4.45554i 0 −49.9881 25.0237i 0 −169.279 + 169.279i 0 203.296i 0
47.13 0 −2.84747 + 2.84747i 0 51.8149 20.9812i 0 −44.8183 + 44.8183i 0 226.784i 0
47.14 0 −2.84747 + 2.84747i 0 −51.8149 + 20.9812i 0 44.8183 44.8183i 0 226.784i 0
47.15 0 3.12794 3.12794i 0 34.4559 + 44.0204i 0 −19.1929 + 19.1929i 0 223.432i 0
47.16 0 3.12794 3.12794i 0 −34.4559 44.0204i 0 19.1929 19.1929i 0 223.432i 0
47.17 0 5.85746 5.85746i 0 13.4330 54.2637i 0 100.487 100.487i 0 174.380i 0
47.18 0 5.85746 5.85746i 0 −13.4330 + 54.2637i 0 −100.487 + 100.487i 0 174.380i 0
47.19 0 7.52871 7.52871i 0 50.5551 23.8576i 0 −51.0903 + 51.0903i 0 129.637i 0
47.20 0 7.52871 7.52871i 0 −50.5551 + 23.8576i 0 51.0903 51.0903i 0 129.637i 0
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 143.28 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.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.6.o.a 56
4.b odd 2 1 40.6.k.a 56
5.c odd 4 1 inner 160.6.o.a 56
8.b even 2 1 40.6.k.a 56
8.d odd 2 1 inner 160.6.o.a 56
20.e even 4 1 40.6.k.a 56
40.i odd 4 1 40.6.k.a 56
40.k even 4 1 inner 160.6.o.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.k.a 56 4.b odd 2 1
40.6.k.a 56 8.b even 2 1
40.6.k.a 56 20.e even 4 1
40.6.k.a 56 40.i odd 4 1
160.6.o.a 56 1.a even 1 1 trivial
160.6.o.a 56 5.c odd 4 1 inner
160.6.o.a 56 8.d odd 2 1 inner
160.6.o.a 56 40.k even 4 1 inner

## 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