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Properties

Label 164.4.k.a
Level $164$
Weight $4$
Character orbit 164.k
Analytic conductor $9.676$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [164,4,Mod(25,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 164.k (of order \(10\), degree \(4\), 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: \(9.67631324094\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 306 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 306 q^{9} - 125 q^{11} + 60 q^{15} + 40 q^{17} - 225 q^{19} - 408 q^{21} - 336 q^{23} - 270 q^{25} - 210 q^{29} - 298 q^{31} - 131 q^{33} - 410 q^{35} + 353 q^{37} + 536 q^{39} - 738 q^{41} - 113 q^{43} + 16 q^{45} + 60 q^{47} + 514 q^{49} - 1354 q^{51} + 100 q^{53} + 1952 q^{57} - 838 q^{59} - 426 q^{61} + 2360 q^{63} + 2635 q^{65} + 3795 q^{67} + 1790 q^{69} - 750 q^{71} + 1828 q^{73} + 755 q^{75} - 348 q^{77} + 2796 q^{81} - 2594 q^{83} - 2372 q^{87} - 770 q^{89} + 4984 q^{91} + 120 q^{93} - 2050 q^{95} + 5335 q^{97} + 3265 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 − 9.50359i 0 −4.89702 − 3.55789i 0 −15.9962 + 5.19748i 0 −63.3183 0
25.2 0 − 5.64605i 0 4.93248 + 3.58366i 0 16.7687 − 5.44847i 0 −4.87793 0
25.3 0 − 5.05229i 0 −13.4279 − 9.75595i 0 24.9766 − 8.11537i 0 1.47435 0
25.4 0 − 4.11688i 0 8.82256 + 6.40996i 0 3.29997 − 1.07222i 0 10.0513 0
25.5 0 − 2.02741i 0 4.15080 + 3.01573i 0 −31.5864 + 10.2630i 0 22.8896 0
25.6 0 1.29820i 0 −11.8134 − 8.58296i 0 −11.9025 + 3.86736i 0 25.3147 0
25.7 0 3.56868i 0 −8.18421 − 5.94617i 0 −2.89086 + 0.939299i 0 14.2645 0
25.8 0 4.66306i 0 15.2790 + 11.1008i 0 −7.31666 + 2.37733i 0 5.25583 0
25.9 0 5.32457i 0 2.56548 + 1.86393i 0 33.2174 − 10.7930i 0 −1.35106 0
25.10 0 9.58960i 0 −3.01792 − 2.19264i 0 −8.56997 + 2.78455i 0 −64.9604 0
45.1 0 − 8.71237i 0 −1.76701 + 5.43830i 0 −11.1752 + 15.3813i 0 −48.9053 0
45.2 0 − 7.58356i 0 4.45047 − 13.6971i 0 19.5240 − 26.8724i 0 −30.5103 0
45.3 0 − 4.90443i 0 −4.75038 + 14.6202i 0 7.61206 − 10.4771i 0 2.94654 0
45.4 0 − 4.17033i 0 4.00668 − 12.3313i 0 −2.11280 + 2.90802i 0 9.60837 0
45.5 0 − 0.683869i 0 1.78678 − 5.49914i 0 −17.0737 + 23.5000i 0 26.5323 0
45.6 0 1.24677i 0 −3.39514 + 10.4492i 0 3.56576 − 4.90785i 0 25.4456 0
45.7 0 3.27232i 0 2.70571 − 8.32732i 0 8.89877 − 12.2481i 0 16.2919 0
45.8 0 5.76927i 0 1.30150 − 4.00562i 0 12.2125 − 16.8090i 0 −6.28446 0
45.9 0 7.17560i 0 −4.22561 + 13.0051i 0 −8.07963 + 11.1207i 0 −24.4892 0
45.10 0 9.76617i 0 5.47718 − 16.8570i 0 −13.3717 + 18.4045i 0 −68.3780 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.4.k.a ✓ 40
41.f even 10 1 inner 164.4.k.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.4.k.a ✓ 40 1.a even 1 1 trivial
164.4.k.a ✓ 40 41.f even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(164, [\chi])\).