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Properties

Label 1008.2.v.d
Level $1008$
Weight $2$
Character orbit 1008.v
Analytic conductor $8.049$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(323,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.v (of order \(4\), degree \(2\), 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{4} - 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{4} - 36 q^{7} - 8 q^{10} - 16 q^{13} - 16 q^{19} + 32 q^{22} + 8 q^{28} + 40 q^{34} + 20 q^{37} - 24 q^{40} + 36 q^{43} - 64 q^{46} + 36 q^{49} - 48 q^{52} + 32 q^{55} + 112 q^{61} - 32 q^{64} - 36 q^{67} + 8 q^{70} + 8 q^{76} - 48 q^{82} - 96 q^{85} + 8 q^{88} + 16 q^{91} - 16 q^{94} + 56 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1 −1.41230 − 0.0735033i 0 1.98919 + 0.207618i 2.53823 + 2.53823i 0 −1.00000 −2.79408 − 0.439432i 0 −3.39817 − 3.77131i
323.2 −1.32282 + 0.500135i 0 1.49973 − 1.32318i 1.18126 + 1.18126i 0 −1.00000 −1.32211 + 2.50040i 0 −2.15338 − 0.971808i
323.3 −1.13296 − 0.846410i 0 0.567180 + 1.91789i −2.98923 − 2.98923i 0 −1.00000 0.980732 − 2.65295i 0 0.856554 + 5.91679i
323.4 −1.07785 + 0.915557i 0 0.323512 − 1.97366i −2.01396 − 2.01396i 0 −1.00000 1.45830 + 2.42350i 0 4.01464 + 0.326848i
323.5 −1.06730 − 0.927825i 0 0.278280 + 1.98055i 2.28967 + 2.28967i 0 −1.00000 1.54059 − 2.37204i 0 −0.319362 − 4.56819i
323.6 −0.575817 − 1.29168i 0 −1.33687 + 1.48754i 0.270063 + 0.270063i 0 −1.00000 2.69122 + 0.870254i 0 0.193328 − 0.504342i
323.7 −0.411956 + 1.35288i 0 −1.66058 − 1.11466i 1.68827 + 1.68827i 0 −1.00000 2.19209 − 1.78739i 0 −2.97953 + 1.58854i
323.8 −0.400836 − 1.35622i 0 −1.67866 + 1.08724i 0.495166 + 0.495166i 0 −1.00000 2.14741 + 1.84083i 0 0.473073 − 0.870033i
323.9 −0.0954430 + 1.41099i 0 −1.98178 − 0.269338i −0.871498 − 0.871498i 0 −1.00000 0.569181 − 2.77057i 0 1.31285 − 1.14650i
323.10 0.0954430 − 1.41099i 0 −1.98178 − 0.269338i 0.871498 + 0.871498i 0 −1.00000 −0.569181 + 2.77057i 0 1.31285 − 1.14650i
323.11 0.400836 + 1.35622i 0 −1.67866 + 1.08724i −0.495166 − 0.495166i 0 −1.00000 −2.14741 − 1.84083i 0 0.473073 − 0.870033i
323.12 0.411956 − 1.35288i 0 −1.66058 − 1.11466i −1.68827 − 1.68827i 0 −1.00000 −2.19209 + 1.78739i 0 −2.97953 + 1.58854i
323.13 0.575817 + 1.29168i 0 −1.33687 + 1.48754i −0.270063 − 0.270063i 0 −1.00000 −2.69122 − 0.870254i 0 0.193328 − 0.504342i
323.14 1.06730 + 0.927825i 0 0.278280 + 1.98055i −2.28967 − 2.28967i 0 −1.00000 −1.54059 + 2.37204i 0 −0.319362 − 4.56819i
323.15 1.07785 − 0.915557i 0 0.323512 − 1.97366i 2.01396 + 2.01396i 0 −1.00000 −1.45830 − 2.42350i 0 4.01464 + 0.326848i
323.16 1.13296 + 0.846410i 0 0.567180 + 1.91789i 2.98923 + 2.98923i 0 −1.00000 −0.980732 + 2.65295i 0 0.856554 + 5.91679i
323.17 1.32282 − 0.500135i 0 1.49973 − 1.32318i −1.18126 − 1.18126i 0 −1.00000 1.32211 − 2.50040i 0 −2.15338 − 0.971808i
323.18 1.41230 + 0.0735033i 0 1.98919 + 0.207618i −2.53823 − 2.53823i 0 −1.00000 2.79408 + 0.439432i 0 −3.39817 − 3.77131i
827.1 −1.41230 + 0.0735033i 0 1.98919 − 0.207618i 2.53823 − 2.53823i 0 −1.00000 −2.79408 + 0.439432i 0 −3.39817 + 3.77131i
827.2 −1.32282 − 0.500135i 0 1.49973 + 1.32318i 1.18126 − 1.18126i 0 −1.00000 −1.32211 − 2.50040i 0 −2.15338 + 0.971808i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.v.d ✓ 36
3.b odd 2 1 inner 1008.2.v.d ✓ 36
4.b odd 2 1 4032.2.v.d 36
12.b even 2 1 4032.2.v.d 36
16.e even 4 1 4032.2.v.d 36
16.f odd 4 1 inner 1008.2.v.d ✓ 36
48.i odd 4 1 4032.2.v.d 36
48.k even 4 1 inner 1008.2.v.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.d ✓ 36 1.a even 1 1 trivial
1008.2.v.d ✓ 36 3.b odd 2 1 inner
1008.2.v.d ✓ 36 16.f odd 4 1 inner
1008.2.v.d ✓ 36 48.k even 4 1 inner
4032.2.v.d 36 4.b odd 2 1
4032.2.v.d 36 12.b even 2 1
4032.2.v.d 36 16.e even 4 1
4032.2.v.d 36 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{36} + 704 T_{5}^{32} + 174256 T_{5}^{28} + 19305408 T_{5}^{24} + 985890144 T_{5}^{20} + \cdots + 1146228736 \) Copy content Toggle raw display
\( T_{11}^{36} + 2208 T_{11}^{32} + 1643312 T_{11}^{28} + 477101376 T_{11}^{24} + 50827259744 T_{11}^{20} + \cdots + 6423507767296 \) Copy content Toggle raw display