Properties

Label 189.2.c.a
Level 189
Weight 2
Character orbit 189.c
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 q^{4} + ( -1 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 2 q^{4} + ( -1 + 3 \zeta_{6} ) q^{7} + ( 3 - 6 \zeta_{6} ) q^{13} + 4 q^{16} + ( -3 + 6 \zeta_{6} ) q^{19} -5 q^{25} + ( -2 + 6 \zeta_{6} ) q^{28} + ( 6 - 12 \zeta_{6} ) q^{31} -11 q^{37} -8 q^{43} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 6 - 12 \zeta_{6} ) q^{52} + ( -9 + 18 \zeta_{6} ) q^{61} + 8 q^{64} + 5 q^{67} + ( 9 - 18 \zeta_{6} ) q^{73} + ( -6 + 12 \zeta_{6} ) q^{76} + 17 q^{79} + ( 15 - 3 \zeta_{6} ) q^{91} + ( 3 - 6 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{4} + q^{7} + O(q^{10}) \) \( 2q + 4q^{4} + q^{7} + 8q^{16} - 10q^{25} + 2q^{28} - 22q^{37} - 16q^{43} - 13q^{49} + 16q^{64} + 10q^{67} + 34q^{79} + 27q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 2.00000 0 0 0.500000 2.59808i 0 0 0
188.2 0 0 2.00000 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.c.a 2
3.b odd 2 1 CM 189.2.c.a 2
4.b odd 2 1 3024.2.k.b 2
7.b odd 2 1 inner 189.2.c.a 2
9.c even 3 1 567.2.o.a 2
9.c even 3 1 567.2.o.b 2
9.d odd 6 1 567.2.o.a 2
9.d odd 6 1 567.2.o.b 2
12.b even 2 1 3024.2.k.b 2
21.c even 2 1 inner 189.2.c.a 2
28.d even 2 1 3024.2.k.b 2
63.l odd 6 1 567.2.o.a 2
63.l odd 6 1 567.2.o.b 2
63.o even 6 1 567.2.o.a 2
63.o even 6 1 567.2.o.b 2
84.h odd 2 1 3024.2.k.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.c.a 2 1.a even 1 1 trivial
189.2.c.a 2 3.b odd 2 1 CM
189.2.c.a 2 7.b odd 2 1 inner
189.2.c.a 2 21.c even 2 1 inner
567.2.o.a 2 9.c even 3 1
567.2.o.a 2 9.d odd 6 1
567.2.o.a 2 63.l odd 6 1
567.2.o.a 2 63.o even 6 1
567.2.o.b 2 9.c even 3 1
567.2.o.b 2 9.d odd 6 1
567.2.o.b 2 63.l odd 6 1
567.2.o.b 2 63.o even 6 1
3024.2.k.b 2 4.b odd 2 1
3024.2.k.b 2 12.b even 2 1
3024.2.k.b 2 28.d even 2 1
3024.2.k.b 2 84.h odd 2 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

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{2} \)
$3$ 1
$5$ \( ( 1 + 5 T^{2} )^{2} \)
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \)
$37$ \( ( 1 + 11 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 8 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - T + 61 T^{2} )( 1 + T + 61 T^{2} ) \)
$67$ \( ( 1 - 5 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 71 T^{2} )^{2} \)
$73$ \( ( 1 - 7 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \)
$79$ \( ( 1 - 17 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \)
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