Properties

Label 189.1.q.a
Level $189$
Weight $1$
Character orbit 189.q
Analytic conductor $0.094$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 189.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0943232873876\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.1323.1
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.107163.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{4} + q^{7} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{4} + q^{7} - q^{13} -\zeta_{6} q^{16} -2 \zeta_{6} q^{19} + \zeta_{6}^{2} q^{25} + \zeta_{6}^{2} q^{28} -\zeta_{6}^{2} q^{31} + \zeta_{6} q^{37} - q^{43} + q^{49} -\zeta_{6}^{2} q^{52} + \zeta_{6} q^{61} + q^{64} -\zeta_{6}^{2} q^{67} + 2 \zeta_{6}^{2} q^{73} + 2 q^{76} + \zeta_{6} q^{79} - q^{91} - q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{4} + 2q^{7} + O(q^{10}) \) \( 2q - q^{4} + 2q^{7} - 2q^{13} - q^{16} - 2q^{19} - q^{25} - q^{28} + q^{31} + q^{37} - 2q^{43} + 2q^{49} + q^{52} + q^{61} + 2q^{64} + q^{67} - 2q^{73} + 4q^{76} + q^{79} - 2q^{91} - 2q^{97} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 −0.500000 + 0.866025i 0 0 1.00000 0 0 0
107.1 0 0 −0.500000 0.866025i 0 0 1.00000 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 189.1.q.a 2
3.b odd 2 1 CM 189.1.q.a 2
4.b odd 2 1 3024.1.dc.a 2
7.b odd 2 1 1323.1.q.a 2
7.c even 3 1 inner 189.1.q.a 2
7.c even 3 1 1323.1.b.a 1
7.d odd 6 1 1323.1.b.b 1
7.d odd 6 1 1323.1.q.a 2
9.c even 3 1 567.1.j.a 2
9.c even 3 1 567.1.n.a 2
9.d odd 6 1 567.1.j.a 2
9.d odd 6 1 567.1.n.a 2
12.b even 2 1 3024.1.dc.a 2
21.c even 2 1 1323.1.q.a 2
21.g even 6 1 1323.1.b.b 1
21.g even 6 1 1323.1.q.a 2
21.h odd 6 1 inner 189.1.q.a 2
21.h odd 6 1 1323.1.b.a 1
28.g odd 6 1 3024.1.dc.a 2
63.g even 3 1 567.1.j.a 2
63.g even 3 1 3969.1.r.b 2
63.h even 3 1 567.1.n.a 2
63.h even 3 1 3969.1.r.b 2
63.i even 6 1 3969.1.n.a 2
63.i even 6 1 3969.1.r.a 2
63.j odd 6 1 567.1.n.a 2
63.j odd 6 1 3969.1.r.b 2
63.k odd 6 1 3969.1.j.a 2
63.k odd 6 1 3969.1.r.a 2
63.l odd 6 1 3969.1.j.a 2
63.l odd 6 1 3969.1.n.a 2
63.n odd 6 1 567.1.j.a 2
63.n odd 6 1 3969.1.r.b 2
63.o even 6 1 3969.1.j.a 2
63.o even 6 1 3969.1.n.a 2
63.s even 6 1 3969.1.j.a 2
63.s even 6 1 3969.1.r.a 2
63.t odd 6 1 3969.1.n.a 2
63.t odd 6 1 3969.1.r.a 2
84.n even 6 1 3024.1.dc.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.1.q.a 2 1.a even 1 1 trivial
189.1.q.a 2 3.b odd 2 1 CM
189.1.q.a 2 7.c even 3 1 inner
189.1.q.a 2 21.h odd 6 1 inner
567.1.j.a 2 9.c even 3 1
567.1.j.a 2 9.d odd 6 1
567.1.j.a 2 63.g even 3 1
567.1.j.a 2 63.n odd 6 1
567.1.n.a 2 9.c even 3 1
567.1.n.a 2 9.d odd 6 1
567.1.n.a 2 63.h even 3 1
567.1.n.a 2 63.j odd 6 1
1323.1.b.a 1 7.c even 3 1
1323.1.b.a 1 21.h odd 6 1
1323.1.b.b 1 7.d odd 6 1
1323.1.b.b 1 21.g even 6 1
1323.1.q.a 2 7.b odd 2 1
1323.1.q.a 2 7.d odd 6 1
1323.1.q.a 2 21.c even 2 1
1323.1.q.a 2 21.g even 6 1
3024.1.dc.a 2 4.b odd 2 1
3024.1.dc.a 2 12.b even 2 1
3024.1.dc.a 2 28.g odd 6 1
3024.1.dc.a 2 84.n even 6 1
3969.1.j.a 2 63.k odd 6 1
3969.1.j.a 2 63.l odd 6 1
3969.1.j.a 2 63.o even 6 1
3969.1.j.a 2 63.s even 6 1
3969.1.n.a 2 63.i even 6 1
3969.1.n.a 2 63.l odd 6 1
3969.1.n.a 2 63.o even 6 1
3969.1.n.a 2 63.t odd 6 1
3969.1.r.a 2 63.i even 6 1
3969.1.r.a 2 63.k odd 6 1
3969.1.r.a 2 63.s even 6 1
3969.1.r.a 2 63.t odd 6 1
3969.1.r.b 2 63.g even 3 1
3969.1.r.b 2 63.h even 3 1
3969.1.r.b 2 63.j odd 6 1
3969.1.r.b 2 63.n odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(189, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 1 - T + T^{2} \)
$37$ \( 1 - T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 1 - T + T^{2} \)
$67$ \( 1 - T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 + 2 T + T^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 1 + T )^{2} \)
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