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Properties

Label 164.2.i.b
Level $164$
Weight $2$
Character orbit 164.i
Analytic conductor $1.310$
Analytic rank $0$
Dimension $72$
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] = [164,2,Mod(3,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 164.i (of order \(8\), 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: \(1.30954659315\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 8 q^{2} - 8 q^{5} + 8 q^{6} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 8 q^{2} - 8 q^{5} + 8 q^{6} + 4 q^{8} - 8 q^{10} - 12 q^{12} - 16 q^{13} - 16 q^{14} + 8 q^{16} - 40 q^{17} - 8 q^{18} + 24 q^{20} - 8 q^{21} + 20 q^{22} - 28 q^{24} - 24 q^{26} - 20 q^{28} + 8 q^{29} + 24 q^{30} - 48 q^{32} + 32 q^{33} + 28 q^{34} - 16 q^{37} - 76 q^{38} - 48 q^{41} + 72 q^{42} - 20 q^{44} + 28 q^{46} + 48 q^{48} - 24 q^{49} - 60 q^{50} - 16 q^{52} + 24 q^{53} - 84 q^{54} + 64 q^{56} - 16 q^{57} - 20 q^{58} - 60 q^{60} - 32 q^{61} + 24 q^{62} + 56 q^{65} + 108 q^{68} - 32 q^{69} + 52 q^{70} - 8 q^{73} + 20 q^{74} - 8 q^{76} - 8 q^{77} + 64 q^{78} + 4 q^{80} + 8 q^{82} + 68 q^{84} - 24 q^{85} + 36 q^{88} + 32 q^{89} + 56 q^{90} + 64 q^{92} - 32 q^{93} + 108 q^{94} + 44 q^{96} + 48 q^{97} - 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.41311 + 0.0558260i −1.87447 − 0.776430i 1.99377 − 0.157777i 0.943371 + 0.943371i 2.69218 + 0.992537i −0.0973328 − 0.0403166i −2.80861 + 0.334260i 0.789462 + 0.789462i −1.38575 − 1.28042i
3.2 −1.38253 − 0.297686i 1.82981 + 0.757931i 1.82277 + 0.823117i −0.700759 − 0.700759i −2.30413 − 1.59257i 3.17088 + 1.31342i −2.27500 − 1.68059i 0.652413 + 0.652413i 0.760213 + 1.17743i
3.3 −1.28595 + 0.588506i 0.599280 + 0.248230i 1.30732 − 1.51358i −2.33065 − 2.33065i −0.916728 + 0.0334696i −3.82576 − 1.58468i −0.790396 + 2.71575i −1.82380 − 1.82380i 4.36870 + 1.62549i
3.4 −1.15286 + 0.819087i 1.81711 + 0.752670i 0.658192 − 1.88859i 1.86334 + 1.86334i −2.71138 + 0.620643i −0.905548 − 0.375090i 0.788117 + 2.71641i 0.614045 + 0.614045i −3.67442 − 0.621941i
3.5 −1.02800 − 0.971190i −2.79191 − 1.15645i 0.113579 + 1.99677i −2.77644 − 2.77644i 1.74696 + 3.90030i 1.41564 + 0.586376i 1.82249 − 2.16299i 4.33605 + 4.33605i 0.157737 + 5.55065i
3.6 −0.819087 + 1.15286i −1.81711 − 0.752670i −0.658192 − 1.88859i 1.86334 + 1.86334i 2.35610 − 1.47837i 0.905548 + 0.375090i 2.71641 + 0.788117i 0.614045 + 0.614045i −3.67442 + 0.621941i
3.7 −0.723027 − 1.21541i 2.77465 + 1.14930i −0.954463 + 1.75756i 1.29422 + 1.29422i −0.609276 − 4.20333i −3.09158 − 1.28057i 2.82626 − 0.110693i 4.25649 + 4.25649i 0.637258 − 2.50878i
3.8 −0.601931 − 1.27972i −0.337920 − 0.139971i −1.27536 + 1.54060i 1.59866 + 1.59866i 0.0242810 + 0.516695i 3.95387 + 1.63775i 2.73922 + 0.704761i −2.02672 − 2.02672i 1.08355 − 3.00812i
3.9 −0.588506 + 1.28595i −0.599280 − 0.248230i −1.30732 − 1.51358i −2.33065 − 2.33065i 0.671891 − 0.624558i 3.82576 + 1.58468i 2.71575 − 0.790396i −1.82380 − 1.82380i 4.36870 − 1.62549i
3.10 −0.0558260 + 1.41311i 1.87447 + 0.776430i −1.99377 − 0.157777i 0.943371 + 0.943371i −1.20183 + 2.60549i 0.0973328 + 0.0403166i 0.334260 − 2.80861i 0.789462 + 0.789462i −1.38575 + 1.28042i
3.11 −0.0496612 − 1.41334i −0.318493 − 0.131924i −1.99507 + 0.140376i −1.39473 − 1.39473i −0.170637 + 0.456691i −1.69916 − 0.703813i 0.297477 + 2.81274i −2.03729 − 2.03729i −1.90197 + 2.04050i
3.12 0.297686 + 1.38253i −1.82981 − 0.757931i −1.82277 + 0.823117i −0.700759 − 0.700759i 0.503153 − 2.75538i −3.17088 − 1.31342i −1.68059 − 2.27500i 0.652413 + 0.652413i 0.760213 − 1.17743i
3.13 0.687017 − 1.23613i 2.12038 + 0.878291i −1.05601 − 1.69848i −0.911225 − 0.911225i 2.54242 − 2.01766i 0.724542 + 0.300115i −2.82503 + 0.138482i 1.60331 + 1.60331i −1.75242 + 0.500362i
3.14 0.971190 + 1.02800i 2.79191 + 1.15645i −0.113579 + 1.99677i −2.77644 − 2.77644i 1.52264 + 3.99321i −1.41564 − 0.586376i −2.16299 + 1.82249i 4.33605 + 4.33605i 0.157737 − 5.55065i
3.15 1.21541 + 0.723027i −2.77465 − 1.14930i 0.954463 + 1.75756i 1.29422 + 1.29422i −2.54138 − 3.40302i 3.09158 + 1.28057i −0.110693 + 2.82626i 4.25649 + 4.25649i 0.637258 + 2.50878i
3.16 1.23613 − 0.687017i −2.12038 − 0.878291i 1.05601 − 1.69848i −0.911225 − 0.911225i −3.22446 + 0.371061i −0.724542 − 0.300115i 0.138482 − 2.82503i 1.60331 + 1.60331i −1.75242 − 0.500362i
3.17 1.27972 + 0.601931i 0.337920 + 0.139971i 1.27536 + 1.54060i 1.59866 + 1.59866i 0.348189 + 0.382528i −3.95387 − 1.63775i 0.704761 + 2.73922i −2.02672 − 2.02672i 1.08355 + 3.00812i
3.18 1.41334 + 0.0496612i 0.318493 + 0.131924i 1.99507 + 0.140376i −1.39473 − 1.39473i 0.443588 + 0.202271i 1.69916 + 0.703813i 2.81274 + 0.297477i −2.03729 − 2.03729i −1.90197 − 2.04050i
27.1 −1.41354 − 0.0434904i 0.710404 + 1.71507i 1.99622 + 0.122951i 2.34845 − 2.34845i −0.929599 − 2.45522i −0.184818 − 0.446191i −2.81639 − 0.260613i −0.315463 + 0.315463i −3.42178 + 3.21751i
27.2 −1.38931 − 0.264231i −0.420025 − 1.01403i 1.86036 + 0.734198i −2.33909 + 2.33909i 0.315606 + 1.51978i −1.47378 − 3.55802i −2.39062 − 1.51160i 1.26949 − 1.26949i 3.86778 − 2.63166i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
41.e odd 8 1 inner
164.i even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.2.i.b ✓ 72
4.b odd 2 1 inner 164.2.i.b ✓ 72
41.e odd 8 1 inner 164.2.i.b ✓ 72
164.i even 8 1 inner 164.2.i.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.i.b ✓ 72 1.a even 1 1 trivial
164.2.i.b ✓ 72 4.b odd 2 1 inner
164.2.i.b ✓ 72 41.e odd 8 1 inner
164.2.i.b ✓ 72 164.i even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} - 40 T_{3}^{66} + 15928 T_{3}^{64} - 6968 T_{3}^{62} + 800 T_{3}^{60} + 182952 T_{3}^{58} + \cdots + 92182675456 \) acting on \(S_{2}^{\mathrm{new}}(164, [\chi])\). Copy content Toggle raw display