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Properties

Label 189.3.n.b
Level $189$
Weight $3$
Character orbit 189.n
Analytic conductor $5.150$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [189,3,Mod(170,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.170");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 189.n (of order \(6\), degree \(2\), not 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: \(5.14987699641\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 6 q^{2} + 12 q^{4} + 3 q^{7} + 25 q^{10} - 18 q^{13} + 90 q^{14} + 12 q^{16} - 6 q^{17} + 3 q^{19} + 39 q^{20} - 59 q^{22} - 114 q^{25} + 3 q^{26} + 34 q^{28} + 63 q^{29} - 29 q^{31} - 246 q^{32}+ \cdots - 483 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
170.1 −2.48702 + 1.43588i 0 2.12350 − 3.67801i − 7.54889i 0 −6.82274 + 1.56534i 0.709334i 0 10.8393 + 18.7742i
170.2 −2.37724 + 1.37250i 0 1.76751 − 3.06142i 2.68504i 0 6.00002 − 3.60552i − 1.27635i 0 −3.68521 − 6.38297i
170.3 −1.86624 + 1.07747i 0 0.321900 − 0.557548i 1.87862i 0 −3.79886 − 5.87951i − 7.23243i 0 −2.02416 − 3.50595i
170.4 −1.11318 + 0.642694i 0 −1.17389 + 2.03324i 7.87519i 0 0.417718 + 6.98753i − 8.15936i 0 −5.06133 − 8.76649i
170.5 −0.0664669 + 0.0383747i 0 −1.99705 + 3.45900i − 4.07697i 0 −6.97461 + 0.595686i − 0.613543i 0 0.156452 + 0.270983i
170.6 0.444866 − 0.256844i 0 −1.86806 + 3.23558i − 7.02462i 0 5.34652 − 4.51827i 3.97395i 0 −1.80423 − 3.12502i
170.7 1.11793 − 0.645440i 0 −1.16681 + 2.02098i 4.83636i 0 −1.74042 − 6.78019i 8.17595i 0 3.12158 + 5.40673i
170.8 1.26920 − 0.732774i 0 −0.926086 + 1.60403i 1.15270i 0 −2.90907 + 6.36689i 8.57663i 0 0.844669 + 1.46301i
170.9 2.09020 − 1.20678i 0 0.912615 − 1.58070i 6.85138i 0 5.41224 + 4.43934i 5.24892i 0 8.26808 + 14.3207i
170.10 2.79169 − 1.61178i 0 3.19568 − 5.53509i − 5.53294i 0 3.31972 + 6.16275i − 7.70873i 0 −8.91790 − 15.4463i
170.11 3.19625 − 1.84536i 0 4.81069 − 8.33236i 5.83234i 0 3.24949 − 6.20007i − 20.7469i 0 10.7628 + 18.6416i
179.1 −2.48702 − 1.43588i 0 2.12350 + 3.67801i 7.54889i 0 −6.82274 − 1.56534i − 0.709334i 0 10.8393 − 18.7742i
179.2 −2.37724 − 1.37250i 0 1.76751 + 3.06142i − 2.68504i 0 6.00002 + 3.60552i 1.27635i 0 −3.68521 + 6.38297i
179.3 −1.86624 − 1.07747i 0 0.321900 + 0.557548i − 1.87862i 0 −3.79886 + 5.87951i 7.23243i 0 −2.02416 + 3.50595i
179.4 −1.11318 − 0.642694i 0 −1.17389 − 2.03324i − 7.87519i 0 0.417718 − 6.98753i 8.15936i 0 −5.06133 + 8.76649i
179.5 −0.0664669 − 0.0383747i 0 −1.99705 − 3.45900i 4.07697i 0 −6.97461 − 0.595686i 0.613543i 0 0.156452 − 0.270983i
179.6 0.444866 + 0.256844i 0 −1.86806 − 3.23558i 7.02462i 0 5.34652 + 4.51827i − 3.97395i 0 −1.80423 + 3.12502i
179.7 1.11793 + 0.645440i 0 −1.16681 − 2.02098i − 4.83636i 0 −1.74042 + 6.78019i − 8.17595i 0 3.12158 − 5.40673i
179.8 1.26920 + 0.732774i 0 −0.926086 − 1.60403i − 1.15270i 0 −2.90907 − 6.36689i − 8.57663i 0 0.844669 − 1.46301i
179.9 2.09020 + 1.20678i 0 0.912615 + 1.58070i − 6.85138i 0 5.41224 − 4.43934i − 5.24892i 0 8.26808 − 14.3207i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 170.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.n.b 22
3.b odd 2 1 63.3.n.b yes 22
7.c even 3 1 189.3.j.b 22
9.c even 3 1 63.3.j.b ✓ 22
9.d odd 6 1 189.3.j.b 22
21.c even 2 1 441.3.n.f 22
21.g even 6 1 441.3.j.f 22
21.g even 6 1 441.3.r.f 22
21.h odd 6 1 63.3.j.b ✓ 22
21.h odd 6 1 441.3.r.g 22
63.g even 3 1 63.3.n.b yes 22
63.h even 3 1 441.3.r.g 22
63.k odd 6 1 441.3.n.f 22
63.l odd 6 1 441.3.j.f 22
63.n odd 6 1 inner 189.3.n.b 22
63.t odd 6 1 441.3.r.f 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.j.b ✓ 22 9.c even 3 1
63.3.j.b ✓ 22 21.h odd 6 1
63.3.n.b yes 22 3.b odd 2 1
63.3.n.b yes 22 63.g even 3 1
189.3.j.b 22 7.c even 3 1
189.3.j.b 22 9.d odd 6 1
189.3.n.b 22 1.a even 1 1 trivial
189.3.n.b 22 63.n odd 6 1 inner
441.3.j.f 22 21.g even 6 1
441.3.j.f 22 63.l odd 6 1
441.3.n.f 22 21.c even 2 1
441.3.n.f 22 63.k odd 6 1
441.3.r.f 22 21.g even 6 1
441.3.r.f 22 63.t odd 6 1
441.3.r.g 22 21.h odd 6 1
441.3.r.g 22 63.h even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} - 6 T_{2}^{21} - 10 T_{2}^{20} + 132 T_{2}^{19} + 63 T_{2}^{18} - 1884 T_{2}^{17} + \cdots + 2187 \) acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display