Properties

Label 189.3.d.b
Level $189$
Weight $3$
Character orbit 189.d
Analytic conductor $5.150$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 189.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.14987699641\)
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: \( 3 \)
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.

\(f(q)\) \(=\) \( q -4 q^{4} + ( 5 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -4 q^{4} + ( 5 + 3 \zeta_{6} ) q^{7} + ( -15 + 30 \zeta_{6} ) q^{13} + 16 q^{16} + ( -21 + 42 \zeta_{6} ) q^{19} + 25 q^{25} + ( -20 - 12 \zeta_{6} ) q^{28} + ( 24 - 48 \zeta_{6} ) q^{31} -47 q^{37} + 22 q^{43} + ( 16 + 39 \zeta_{6} ) q^{49} + ( 60 - 120 \zeta_{6} ) q^{52} + ( -9 + 18 \zeta_{6} ) q^{61} -64 q^{64} -109 q^{67} + ( 63 - 126 \zeta_{6} ) q^{73} + ( 84 - 168 \zeta_{6} ) q^{76} + 131 q^{79} + ( -165 + 195 \zeta_{6} ) q^{91} + ( 57 - 114 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{4} + 13q^{7} + O(q^{10}) \) \( 2q - 8q^{4} + 13q^{7} + 32q^{16} + 50q^{25} - 52q^{28} - 94q^{37} + 44q^{43} + 71q^{49} - 128q^{64} - 218q^{67} + 262q^{79} - 135q^{91} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 −4.00000 0 0 6.50000 2.59808i 0 0 0
55.2 0 0 −4.00000 0 0 6.50000 + 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 189.3.d.b 2
3.b odd 2 1 CM 189.3.d.b 2
7.b odd 2 1 inner 189.3.d.b 2
21.c even 2 1 inner 189.3.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.3.d.b 2 1.a even 1 1 trivial
189.3.d.b 2 3.b odd 2 1 CM
189.3.d.b 2 7.b odd 2 1 inner
189.3.d.b 2 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 49 - 13 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 675 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 1323 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 1728 + T^{2} \)
$37$ \( ( 47 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -22 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 243 + T^{2} \)
$67$ \( ( 109 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 11907 + T^{2} \)
$79$ \( ( -131 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 9747 + T^{2} \)
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