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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,6,Mod(289,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.289");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.f (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \( T_{3}^{8} - 1340T_{3}^{6} + 487876T_{3}^{4} - 49871616T_{3}^{2} + 897122304 \) acting on \(S_{6}^{\mathrm{new}}(320, [\chi])\). Copy content Toggle raw display