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Properties

Label 189.3.j.b
Level $189$
Weight $3$
Character orbit 189.j
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(44,189)]
 
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([5, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.44");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 189.j (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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 24 q^{4} - 12 q^{5} + 25 q^{10} - 24 q^{11} - 18 q^{13} + 60 q^{14} - 24 q^{16} + 6 q^{17} + 3 q^{19} + 39 q^{20} - 59 q^{22} - 81 q^{23} + 57 q^{25} - 3 q^{26} + 34 q^{28} + 63 q^{29} + 58 q^{31}+ \cdots + 483 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 − 3.69072i 0 −9.62138 −5.05096 + 2.91617i 0 3.74467 + 5.91417i 20.7469i 0 10.7628 + 18.6416i
44.2 − 3.22356i 0 −6.39137 4.79167 − 2.76647i 0 −6.99696 − 0.206413i 7.70873i 0 −8.91790 − 15.4463i
44.3 − 2.41355i 0 −1.82523 −5.93347 + 3.42569i 0 −6.55070 + 2.46747i − 5.24892i 0 8.26808 + 14.3207i
44.4 − 1.46555i 0 1.85217 −0.998268 + 0.576350i 0 −4.05935 − 5.70277i − 8.57663i 0 0.844669 + 1.46301i
44.5 − 1.29088i 0 2.33363 −4.18841 + 2.41818i 0 6.74202 + 1.88284i − 8.17595i 0 3.12158 + 5.40673i
44.6 − 0.513687i 0 3.73613 6.08350 − 3.51231i 0 1.23968 + 6.88935i − 3.97395i 0 −1.80423 − 3.12502i
44.7 0.0767494i 0 3.99411 3.53076 − 2.03848i 0 2.97142 − 6.33803i 0.613543i 0 0.156452 + 0.270983i
44.8 1.28539i 0 2.34778 −6.82011 + 3.93759i 0 −6.26023 − 3.13201i 8.15936i 0 −5.06133 − 8.76649i
44.9 2.15495i 0 −0.643801 −1.62693 + 0.939308i 0 6.99124 − 0.350157i 7.23243i 0 −2.02416 − 3.50595i
44.10 2.74500i 0 −3.53503 −2.32531 + 1.34252i 0 0.122457 + 6.99893i 1.27635i 0 −3.68521 − 6.38297i
44.11 2.87176i 0 −4.24700 6.53753 − 3.77444i 0 2.05575 − 6.69133i − 0.709334i 0 10.8393 + 18.7742i
116.1 − 2.87176i 0 −4.24700 6.53753 + 3.77444i 0 2.05575 + 6.69133i 0.709334i 0 10.8393 − 18.7742i
116.2 − 2.74500i 0 −3.53503 −2.32531 − 1.34252i 0 0.122457 − 6.99893i − 1.27635i 0 −3.68521 + 6.38297i
116.3 − 2.15495i 0 −0.643801 −1.62693 − 0.939308i 0 6.99124 + 0.350157i − 7.23243i 0 −2.02416 + 3.50595i
116.4 − 1.28539i 0 2.34778 −6.82011 − 3.93759i 0 −6.26023 + 3.13201i − 8.15936i 0 −5.06133 + 8.76649i
116.5 − 0.0767494i 0 3.99411 3.53076 + 2.03848i 0 2.97142 + 6.33803i − 0.613543i 0 0.156452 − 0.270983i
116.6 0.513687i 0 3.73613 6.08350 + 3.51231i 0 1.23968 − 6.88935i 3.97395i 0 −1.80423 + 3.12502i
116.7 1.29088i 0 2.33363 −4.18841 − 2.41818i 0 6.74202 − 1.88284i 8.17595i 0 3.12158 − 5.40673i
116.8 1.46555i 0 1.85217 −0.998268 − 0.576350i 0 −4.05935 + 5.70277i 8.57663i 0 0.844669 − 1.46301i
116.9 2.41355i 0 −1.82523 −5.93347 − 3.42569i 0 −6.55070 − 2.46747i 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 44.11
Significant digits:
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Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} + 56 T_{2}^{20} + 1326 T_{2}^{18} + 17369 T_{2}^{16} + 138193 T_{2}^{14} + 690216 T_{2}^{12} + \cdots + 2187 \) acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display