# Properties

 Label 320.6.f.d Level 320 Weight 6 Character orbit 320.f Analytic conductor 51.323 Analytic rank 0 Dimension 32 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 320.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$51.3228223402$$ Analytic rank: $$0$$ Dimension: $$32$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 2944q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 2944q^{9} - 46528q^{25} - 36768q^{41} - 113920q^{49} + 25376q^{65} + 633472q^{81} + 968704q^{89} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1 0 −28.5405 0 5.81388 + 55.5985i 0 92.3750i 0 571.559 0
289.2 0 −28.5405 0 −5.81388 55.5985i 0 92.3750i 0 571.559 0
289.3 0 −28.5405 0 −5.81388 + 55.5985i 0 92.3750i 0 571.559 0
289.4 0 −28.5405 0 5.81388 55.5985i 0 92.3750i 0 571.559 0
289.5 0 −19.3064 0 −19.7709 52.2887i 0 226.751i 0 129.737 0
289.6 0 −19.3064 0 19.7709 52.2887i 0 226.751i 0 129.737 0
289.7 0 −19.3064 0 19.7709 + 52.2887i 0 226.751i 0 129.737 0
289.8 0 −19.3064 0 −19.7709 + 52.2887i 0 226.751i 0 129.737 0
289.9 0 −11.4004 0 −32.3114 + 45.6177i 0 121.355i 0 −113.031 0
289.10 0 −11.4004 0 32.3114 45.6177i 0 121.355i 0 −113.031 0
289.11 0 −11.4004 0 32.3114 + 45.6177i 0 121.355i 0 −113.031 0
289.12 0 −11.4004 0 −32.3114 45.6177i 0 121.355i 0 −113.031 0
289.13 0 −4.76807 0 43.2814 35.3796i 0 82.4139i 0 −220.266 0
289.14 0 −4.76807 0 −43.2814 35.3796i 0 82.4139i 0 −220.266 0
289.15 0 −4.76807 0 −43.2814 + 35.3796i 0 82.4139i 0 −220.266 0
289.16 0 −4.76807 0 43.2814 + 35.3796i 0 82.4139i 0 −220.266 0
289.17 0 4.76807 0 43.2814 + 35.3796i 0 82.4139i 0 −220.266 0
289.18 0 4.76807 0 −43.2814 + 35.3796i 0 82.4139i 0 −220.266 0
289.19 0 4.76807 0 −43.2814 35.3796i 0 82.4139i 0 −220.266 0
289.20 0 4.76807 0 43.2814 35.3796i 0 82.4139i 0 −220.266 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.e odd 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 320.6.f.d 32
4.b odd 2 1 inner 320.6.f.d 32
5.b even 2 1 inner 320.6.f.d 32
8.b even 2 1 inner 320.6.f.d 32
8.d odd 2 1 inner 320.6.f.d 32
20.d odd 2 1 inner 320.6.f.d 32
40.e odd 2 1 inner 320.6.f.d 32
40.f even 2 1 inner 320.6.f.d 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.6.f.d 32 1.a even 1 1 trivial
320.6.f.d 32 4.b odd 2 1 inner
320.6.f.d 32 5.b even 2 1 inner
320.6.f.d 32 8.b even 2 1 inner
320.6.f.d 32 8.d odd 2 1 inner
320.6.f.d 32 20.d odd 2 1 inner
320.6.f.d 32 40.e odd 2 1 inner
320.6.f.d 32 40.f even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 1340 T_{3}^{6} + 487876 T_{3}^{4} - 49871616 T_{3}^{2} + 897122304$$ acting on $$S_{6}^{\mathrm{new}}(320, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database