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

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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:

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