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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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
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

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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \(S_{2}^{\mathrm{new}}(189, [\chi])\).

Hecke characteristic polynomials

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