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Properties

Label 1008.2.v.e
Level $1008$
Weight $2$
Character orbit 1008.v
Analytic conductor $8.049$
Analytic rank $0$
Dimension $40$
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: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 40 q^{7} + 48 q^{10} - 24 q^{13} + 12 q^{16} - 32 q^{19} - 8 q^{22} - 56 q^{34} - 8 q^{37} + 32 q^{43} - 52 q^{46} + 40 q^{49} - 8 q^{52} + 48 q^{55} + 56 q^{58} - 24 q^{61} + 48 q^{64} + 48 q^{70}+ \cdots + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −1.40729 − 0.139782i 0 1.96092 + 0.393427i −1.17902 − 1.17902i 0 1.00000 −2.70459 − 0.827766i 0 1.49442 + 1.82403i
323.2 −1.39680 + 0.221233i 0 1.90211 − 0.618037i −2.51504 − 2.51504i 0 1.00000 −2.52014 + 1.28408i 0 4.06942 + 2.95660i
323.3 −1.37088 − 0.347400i 0 1.75863 + 0.952488i 0.111394 + 0.111394i 0 1.00000 −2.07997 − 1.91669i 0 −0.114009 − 0.191406i
323.4 −1.13420 − 0.844744i 0 0.572816 + 1.91622i 2.12043 + 2.12043i 0 1.00000 0.969023 − 2.65725i 0 −0.613769 − 4.19620i
323.5 −1.11731 + 0.866958i 0 0.496766 − 1.93732i 0.925496 + 0.925496i 0 1.00000 1.12454 + 2.59527i 0 −1.83643 − 0.231700i
323.6 −0.890883 − 1.09833i 0 −0.412656 + 1.95697i −1.65702 − 1.65702i 0 1.00000 2.51702 − 1.29019i 0 −0.343745 + 3.29617i
323.7 −0.614527 + 1.27372i 0 −1.24471 − 1.56547i −2.62814 − 2.62814i 0 1.00000 2.75887 − 0.623393i 0 4.96257 − 1.73245i
323.8 −0.579268 + 1.29014i 0 −1.32890 − 1.49467i −0.667815 − 0.667815i 0 1.00000 2.69811 − 0.848644i 0 1.24842 − 0.474728i
323.9 −0.351970 − 1.36971i 0 −1.75223 + 0.964197i 2.96859 + 2.96859i 0 1.00000 1.93741 + 2.06069i 0 3.02127 − 5.11098i
323.10 −0.153718 − 1.40583i 0 −1.95274 + 0.432203i 0.0893433 + 0.0893433i 0 1.00000 0.907777 + 2.67879i 0 0.111868 − 0.139335i
323.11 0.153718 + 1.40583i 0 −1.95274 + 0.432203i −0.0893433 − 0.0893433i 0 1.00000 −0.907777 − 2.67879i 0 0.111868 − 0.139335i
323.12 0.351970 + 1.36971i 0 −1.75223 + 0.964197i −2.96859 − 2.96859i 0 1.00000 −1.93741 − 2.06069i 0 3.02127 − 5.11098i
323.13 0.579268 − 1.29014i 0 −1.32890 − 1.49467i 0.667815 + 0.667815i 0 1.00000 −2.69811 + 0.848644i 0 1.24842 − 0.474728i
323.14 0.614527 − 1.27372i 0 −1.24471 − 1.56547i 2.62814 + 2.62814i 0 1.00000 −2.75887 + 0.623393i 0 4.96257 − 1.73245i
323.15 0.890883 + 1.09833i 0 −0.412656 + 1.95697i 1.65702 + 1.65702i 0 1.00000 −2.51702 + 1.29019i 0 −0.343745 + 3.29617i
323.16 1.11731 − 0.866958i 0 0.496766 − 1.93732i −0.925496 − 0.925496i 0 1.00000 −1.12454 − 2.59527i 0 −1.83643 − 0.231700i
323.17 1.13420 + 0.844744i 0 0.572816 + 1.91622i −2.12043 − 2.12043i 0 1.00000 −0.969023 + 2.65725i 0 −0.613769 − 4.19620i
323.18 1.37088 + 0.347400i 0 1.75863 + 0.952488i −0.111394 − 0.111394i 0 1.00000 2.07997 + 1.91669i 0 −0.114009 − 0.191406i
323.19 1.39680 − 0.221233i 0 1.90211 − 0.618037i 2.51504 + 2.51504i 0 1.00000 2.52014 − 1.28408i 0 4.06942 + 2.95660i
323.20 1.40729 + 0.139782i 0 1.96092 + 0.393427i 1.17902 + 1.17902i 0 1.00000 2.70459 + 0.827766i 0 1.49442 + 1.82403i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.20
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.e ✓ 40
3.b odd 2 1 inner 1008.2.v.e ✓ 40
4.b odd 2 1 4032.2.v.e 40
12.b even 2 1 4032.2.v.e 40
16.e even 4 1 4032.2.v.e 40
16.f odd 4 1 inner 1008.2.v.e ✓ 40
48.i odd 4 1 4032.2.v.e 40
48.k even 4 1 inner 1008.2.v.e ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.e ✓ 40 1.a even 1 1 trivial
1008.2.v.e ✓ 40 3.b odd 2 1 inner
1008.2.v.e ✓ 40 16.f odd 4 1 inner
1008.2.v.e ✓ 40 48.k even 4 1 inner
4032.2.v.e 40 4.b odd 2 1
4032.2.v.e 40 12.b even 2 1
4032.2.v.e 40 16.e even 4 1
4032.2.v.e 40 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 \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{40} + 784 T_{5}^{36} + 224304 T_{5}^{32} + 29085120 T_{5}^{28} + 1705059168 T_{5}^{24} + \cdots + 65536 \) Copy content Toggle raw display
\( T_{11}^{40} + 3664 T_{11}^{36} + 4437680 T_{11}^{32} + 2045565248 T_{11}^{28} + 297547122016 T_{11}^{24} + \cdots + 426337261060096 \) Copy content Toggle raw display