Properties

Label 189.2.e.b
Level $189$
Weight $2$
Character orbit 189.e
Analytic conductor $1.509$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [189,2,Mod(109,189)]
 
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([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.e (of order \(3\), degree \(2\), 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: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{4} + ( - 3 \zeta_{6} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{4} + ( - 3 \zeta_{6} + 1) q^{7} + 2 q^{13} - 4 \zeta_{6} q^{16} + 7 \zeta_{6} q^{19} + ( - 5 \zeta_{6} + 5) q^{25} + ( - 2 \zeta_{6} - 4) q^{28} + (11 \zeta_{6} - 11) q^{31} + 10 \zeta_{6} q^{37} - 13 q^{43} + (3 \zeta_{6} - 8) q^{49} + ( - 4 \zeta_{6} + 4) q^{52} + 13 \zeta_{6} q^{61} - 8 q^{64} + ( - 16 \zeta_{6} + 16) q^{67} + ( - 7 \zeta_{6} + 7) q^{73} + 14 q^{76} + 4 \zeta_{6} q^{79} + ( - 6 \zeta_{6} + 2) q^{91} + 5 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - q^{7} + 4 q^{13} - 4 q^{16} + 7 q^{19} + 5 q^{25} - 10 q^{28} - 11 q^{31} + 10 q^{37} - 26 q^{43} - 13 q^{49} + 4 q^{52} + 13 q^{61} - 16 q^{64} + 16 q^{67} + 7 q^{73} + 28 q^{76} + 4 q^{79} - 2 q^{91} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 1.00000 1.73205i 0 0 −0.500000 2.59808i 0 0 0
163.1 0 0 1.00000 + 1.73205i 0 0 −0.500000 + 2.59808i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.e.b 2
3.b odd 2 1 CM 189.2.e.b 2
7.c even 3 1 inner 189.2.e.b 2
7.c even 3 1 1323.2.a.k 1
7.d odd 6 1 1323.2.a.j 1
9.c even 3 1 567.2.g.c 2
9.c even 3 1 567.2.h.d 2
9.d odd 6 1 567.2.g.c 2
9.d odd 6 1 567.2.h.d 2
21.g even 6 1 1323.2.a.j 1
21.h odd 6 1 inner 189.2.e.b 2
21.h odd 6 1 1323.2.a.k 1
63.g even 3 1 567.2.h.d 2
63.h even 3 1 567.2.g.c 2
63.j odd 6 1 567.2.g.c 2
63.n odd 6 1 567.2.h.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.b 2 1.a even 1 1 trivial
189.2.e.b 2 3.b odd 2 1 CM
189.2.e.b 2 7.c even 3 1 inner
189.2.e.b 2 21.h odd 6 1 inner
567.2.g.c 2 9.c even 3 1
567.2.g.c 2 9.d odd 6 1
567.2.g.c 2 63.h even 3 1
567.2.g.c 2 63.j odd 6 1
567.2.h.d 2 9.c even 3 1
567.2.h.d 2 9.d odd 6 1
567.2.h.d 2 63.g even 3 1
567.2.h.d 2 63.n odd 6 1
1323.2.a.j 1 7.d odd 6 1
1323.2.a.j 1 21.g even 6 1
1323.2.a.k 1 7.c even 3 1
1323.2.a.k 1 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 13)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 5)^{2} \) Copy content Toggle raw display
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