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Properties

Label 164.2.o.b
Level $164$
Weight $2$
Character orbit 164.o
Analytic conductor $1.310$
Analytic rank $0$
Dimension $288$
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(7,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 39]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 164.o (of order \(40\), degree \(16\), 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: \(288\)
Relative dimension: \(18\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9} - 12 q^{10} - 8 q^{12} - 24 q^{13} - 4 q^{14} - 28 q^{16} - 12 q^{18} - 44 q^{20} - 32 q^{21} - 40 q^{22} + 8 q^{24} - 40 q^{25} + 4 q^{26} - 48 q^{29} - 44 q^{30} + 108 q^{32} - 72 q^{33} + 12 q^{34} - 20 q^{36} - 24 q^{37} + 56 q^{38} + 8 q^{41} - 112 q^{42} - 40 q^{45} - 48 q^{46} - 68 q^{48} - 16 q^{49} - 60 q^{50} - 124 q^{52} - 64 q^{53} + 64 q^{54} - 84 q^{56} - 24 q^{57} + 40 q^{60} - 8 q^{61} - 44 q^{62} - 20 q^{64} + 64 q^{65} + 60 q^{66} - 28 q^{68} - 8 q^{69} + 128 q^{70} + 160 q^{72} - 32 q^{73} + 80 q^{74} + 288 q^{76} - 32 q^{77} + 116 q^{78} + 176 q^{80} + 212 q^{82} + 152 q^{84} - 56 q^{85} + 180 q^{86} + 144 q^{88} - 72 q^{89} + 224 q^{90} + 36 q^{92} - 8 q^{93} + 52 q^{94} + 136 q^{96} - 88 q^{97} + 104 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.41078 + 0.0984343i −1.75033 − 0.725011i 1.98062 − 0.277739i 0.186813 + 1.17949i 2.54071 + 0.850542i 0.145426 − 1.84782i −2.76689 + 0.586791i 0.416702 + 0.416702i −0.379654 − 1.64561i
7.2 −1.35891 − 0.391616i 0.746289 + 0.309123i 1.69327 + 1.06434i 0.150350 + 0.949272i −0.893082 − 0.712329i −0.400684 + 5.09117i −1.88419 − 2.10946i −1.65993 − 1.65993i 0.167438 − 1.34885i
7.3 −1.34854 − 0.425962i 2.36643 + 0.980207i 1.63711 + 1.14885i −0.320933 − 2.02629i −2.77369 − 2.32986i 0.339440 − 4.31299i −1.71834 − 2.24662i 2.51786 + 2.51786i −0.430333 + 2.86924i
7.4 −1.21959 + 0.715958i 0.361903 + 0.149905i 0.974809 − 1.74635i −0.336807 − 2.12652i −0.548700 + 0.0762842i 0.102503 − 1.30243i 0.0614463 + 2.82776i −2.01282 − 2.01282i 1.93326 + 2.35234i
7.5 −0.905780 + 1.08608i 0.603023 + 0.249780i −0.359125 − 1.96749i 0.581147 + 3.66922i −0.817487 + 0.428683i 0.0648794 − 0.824371i 2.46214 + 1.39208i −1.82007 − 1.82007i −4.51144 − 2.69234i
7.6 −0.810858 − 1.15867i −2.26723 − 0.939117i −0.685018 + 1.87903i 0.334666 + 2.11300i 0.750277 + 3.38845i −0.0162305 + 0.206228i 2.73262 − 0.729918i 2.13706 + 2.13706i 2.17689 − 2.10111i
7.7 −0.606870 + 1.27738i 2.84485 + 1.17838i −1.26342 − 1.55041i −0.359268 − 2.26833i −3.23169 + 2.91885i −0.245346 + 3.11742i 2.74720 − 0.672970i 4.58329 + 4.58329i 3.11555 + 0.917658i
7.8 −0.460771 − 1.33705i 2.26723 + 0.939117i −1.57538 + 1.23214i 0.334666 + 2.11300i 0.210969 − 3.46410i 0.0162305 − 0.206228i 2.37332 + 1.53862i 2.13706 + 2.13706i 2.67097 − 1.42107i
7.9 −0.000945666 1.41421i −1.22351 − 0.506793i −2.00000 − 0.00267475i −0.450201 − 2.84246i 0.717870 − 1.72982i 0.187651 − 2.38434i 0.00567399 − 2.82842i −0.881193 − 0.881193i 4.02027 − 0.633993i
7.10 0.448041 − 1.34136i −2.36643 − 0.980207i −1.59852 − 1.20197i −0.320933 − 2.02629i −2.37507 + 2.73507i −0.339440 + 4.31299i −2.32848 + 1.60567i 2.51786 + 2.51786i −2.86179 − 0.477373i
7.11 0.464226 + 1.33585i 1.84094 + 0.762540i −1.56899 + 1.24027i 0.261972 + 1.65403i −0.164028 + 2.81320i 0.216665 − 2.75299i −2.38518 − 1.52016i 0.686255 + 0.686255i −2.08792 + 1.11780i
7.12 0.481923 − 1.32957i −0.746289 − 0.309123i −1.53550 − 1.28150i 0.150350 + 0.949272i −0.770654 + 0.843268i 0.400684 − 5.09117i −2.44383 + 1.42397i −1.65993 − 1.65993i 1.33458 + 0.257576i
7.13 0.807859 + 1.16076i −1.84094 − 0.762540i −0.694726 + 1.87546i 0.261972 + 1.65403i −0.602091 − 2.75291i −0.216665 + 2.75299i −2.73820 + 0.708699i 0.686255 + 0.686255i −1.70829 + 1.64031i
7.14 0.908873 − 1.08349i 1.75033 + 0.725011i −0.347900 − 1.96951i 0.186813 + 1.17949i 2.37637 − 1.23752i −0.145426 + 1.84782i −2.45014 − 1.41309i 0.416702 + 0.416702i 1.44775 + 0.869596i
7.15 1.14468 + 0.830489i 1.22351 + 0.506793i 0.620577 + 1.90128i −0.450201 − 2.84246i 0.979635 + 1.59622i −0.187651 + 2.38434i −0.868634 + 2.69174i −0.881193 − 0.881193i 1.84529 − 3.62759i
7.16 1.29608 − 0.565841i −0.361903 − 0.149905i 1.35965 − 1.46675i −0.336807 − 2.12652i −0.553878 + 0.0104905i −0.102503 + 1.30243i 0.932265 − 2.67037i −2.01282 − 2.01282i −1.63980 − 2.56556i
7.17 1.39013 + 0.259859i −2.84485 − 1.17838i 1.86495 + 0.722478i −0.359268 − 2.26833i −3.64851 − 2.37736i 0.245346 − 3.11742i 2.40478 + 1.48896i 4.58329 + 4.58329i 0.0900149 − 3.24664i
7.18 1.41106 − 0.0944117i −0.603023 − 0.249780i 1.98217 − 0.266441i 0.581147 + 3.66922i −0.874483 − 0.295522i −0.0648794 + 0.824371i 2.77181 − 0.563104i −1.82007 − 1.82007i 1.16645 + 5.12262i
11.1 −1.38810 − 0.270514i 0.869335 − 2.09876i 1.85364 + 0.751000i 1.95666 − 3.84017i −1.77447 + 2.67812i 0.861690 + 3.58920i −2.36989 − 1.54390i −1.52773 − 1.52773i −3.75486 + 4.80123i
11.2 −1.26038 − 0.641434i −1.13045 + 2.72914i 1.17712 + 1.61690i −0.430197 + 0.844309i 3.17535 − 2.71465i −0.0257188 − 0.107126i −0.446489 − 2.79296i −4.04896 − 4.04896i 1.08378 − 0.788209i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
41.h odd 40 1 inner
164.o even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.2.o.b ✓ 288
4.b odd 2 1 inner 164.2.o.b ✓ 288
41.h odd 40 1 inner 164.2.o.b ✓ 288
164.o even 40 1 inner 164.2.o.b ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.o.b ✓ 288 1.a even 1 1 trivial
164.2.o.b ✓ 288 4.b odd 2 1 inner
164.2.o.b ✓ 288 41.h odd 40 1 inner
164.2.o.b ✓ 288 164.o even 40 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{288} + 20 T_{3}^{286} + 200 T_{3}^{284} + 480 T_{3}^{282} + 55562 T_{3}^{280} + \cdots + 72\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(164, [\chi])\). Copy content Toggle raw display