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Properties

Label 164.3.d.c
Level $164$
Weight $3$
Character orbit 164.d
Analytic conductor $4.469$
Analytic rank $0$
Dimension $32$
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] = [164,3,Mod(163,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.163");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 164.d (of order \(2\), degree \(1\), 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: \(4.46867633551\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} - 34 q^{4} - 8 q^{5} - 14 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} - 34 q^{4} - 8 q^{5} - 14 q^{8} + 24 q^{9} + 16 q^{10} - 90 q^{16} + 14 q^{18} + 8 q^{20} - 56 q^{21} + 16 q^{25} + 18 q^{32} - 40 q^{33} - 154 q^{36} - 24 q^{37} - 120 q^{40} + 208 q^{41} + 132 q^{42} + 312 q^{45} + 16 q^{46} - 152 q^{49} + 194 q^{50} + 104 q^{57} + 96 q^{61} - 32 q^{62} - 154 q^{64} + 84 q^{66} + 26 q^{72} - 136 q^{73} - 48 q^{74} - 120 q^{77} + 48 q^{78} - 248 q^{80} - 344 q^{81} + 222 q^{82} - 84 q^{84} + 672 q^{86} - 112 q^{90} - 256 q^{92} + 510 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1 −1.83775 − 0.789089i −0.994532 2.75468 + 2.90030i −1.69735 1.82770 + 0.784775i 6.08835 −2.77382 − 7.50373i −8.01091 3.11931 + 1.33936i
163.2 −1.83775 − 0.789089i 0.994532 2.75468 + 2.90030i −1.69735 −1.82770 − 0.784775i −6.08835 −2.77382 − 7.50373i −8.01091 3.11931 + 1.33936i
163.3 −1.83775 + 0.789089i −0.994532 2.75468 − 2.90030i −1.69735 1.82770 − 0.784775i 6.08835 −2.77382 + 7.50373i −8.01091 3.11931 − 1.33936i
163.4 −1.83775 + 0.789089i 0.994532 2.75468 − 2.90030i −1.69735 −1.82770 + 0.784775i −6.08835 −2.77382 + 7.50373i −8.01091 3.11931 − 1.33936i
163.5 −1.43715 − 1.39090i −4.50897 0.130779 + 3.99786i 3.66935 6.48005 + 6.27154i 2.75944 5.37269 − 5.92741i 11.3308 −5.27339 − 5.10371i
163.6 −1.43715 − 1.39090i 4.50897 0.130779 + 3.99786i 3.66935 −6.48005 − 6.27154i −2.75944 5.37269 − 5.92741i 11.3308 −5.27339 − 5.10371i
163.7 −1.43715 + 1.39090i −4.50897 0.130779 − 3.99786i 3.66935 6.48005 − 6.27154i 2.75944 5.37269 + 5.92741i 11.3308 −5.27339 + 5.10371i
163.8 −1.43715 + 1.39090i 4.50897 0.130779 − 3.99786i 3.66935 −6.48005 + 6.27154i −2.75944 5.37269 + 5.92741i 11.3308 −5.27339 + 5.10371i
163.9 −0.920996 − 1.77532i −3.01133 −2.30353 + 3.27013i −7.77019 2.77343 + 5.34609i 2.28597 7.92707 + 1.07774i 0.0681311 7.15632 + 13.7946i
163.10 −0.920996 − 1.77532i 3.01133 −2.30353 + 3.27013i −7.77019 −2.77343 − 5.34609i −2.28597 7.92707 + 1.07774i 0.0681311 7.15632 + 13.7946i
163.11 −0.920996 + 1.77532i −3.01133 −2.30353 − 3.27013i −7.77019 2.77343 − 5.34609i 2.28597 7.92707 − 1.07774i 0.0681311 7.15632 − 13.7946i
163.12 −0.920996 + 1.77532i 3.01133 −2.30353 − 3.27013i −7.77019 −2.77343 + 5.34609i −2.28597 7.92707 − 1.07774i 0.0681311 7.15632 − 13.7946i
163.13 −0.417115 − 1.95602i −0.771966 −3.65203 + 1.63177i 4.30571 0.321999 + 1.50998i −9.44612 4.71509 + 6.46281i −8.40407 −1.79598 − 8.42205i
163.14 −0.417115 − 1.95602i 0.771966 −3.65203 + 1.63177i 4.30571 −0.321999 − 1.50998i 9.44612 4.71509 + 6.46281i −8.40407 −1.79598 − 8.42205i
163.15 −0.417115 + 1.95602i −0.771966 −3.65203 − 1.63177i 4.30571 0.321999 − 1.50998i −9.44612 4.71509 − 6.46281i −8.40407 −1.79598 + 8.42205i
163.16 −0.417115 + 1.95602i 0.771966 −3.65203 − 1.63177i 4.30571 −0.321999 + 1.50998i 9.44612 4.71509 − 6.46281i −8.40407 −1.79598 + 8.42205i
163.17 0.392436 − 1.96112i −5.16771 −3.69199 − 1.53923i 0.943183 −2.02800 + 10.1345i −4.65695 −4.46748 + 6.63638i 17.7052 0.370139 − 1.84970i
163.18 0.392436 − 1.96112i 5.16771 −3.69199 − 1.53923i 0.943183 2.02800 − 10.1345i 4.65695 −4.46748 + 6.63638i 17.7052 0.370139 − 1.84970i
163.19 0.392436 + 1.96112i −5.16771 −3.69199 + 1.53923i 0.943183 −2.02800 − 10.1345i −4.65695 −4.46748 − 6.63638i 17.7052 0.370139 + 1.84970i
163.20 0.392436 + 1.96112i 5.16771 −3.69199 + 1.53923i 0.943183 2.02800 + 10.1345i 4.65695 −4.46748 − 6.63638i 17.7052 0.370139 + 1.84970i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.32
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.3.d.c ✓ 32
4.b odd 2 1 inner 164.3.d.c ✓ 32
41.b even 2 1 inner 164.3.d.c ✓ 32
164.d odd 2 1 inner 164.3.d.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.3.d.c ✓ 32 1.a even 1 1 trivial
164.3.d.c ✓ 32 4.b odd 2 1 inner
164.3.d.c ✓ 32 41.b even 2 1 inner
164.3.d.c ✓ 32 164.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 78 T_{3}^{14} + 2356 T_{3}^{12} - 35444 T_{3}^{10} + 285264 T_{3}^{8} - 1212680 T_{3}^{6} + \cdots + 619520 \) acting on \(S_{3}^{\mathrm{new}}(164, [\chi])\). Copy content Toggle raw display