Properties

Label 189.2.c.b
Level 189
Weight 2
Character orbit 189.c
Analytic conductor 1.509
Analytic rank 0
Dimension 4
CM no
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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -3 q^{4} -\beta_{3} q^{5} + ( -2 + \beta_{2} ) q^{7} + \beta_{1} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} -3 q^{4} -\beta_{3} q^{5} + ( -2 + \beta_{2} ) q^{7} + \beta_{1} q^{8} + 5 \beta_{2} q^{10} -\beta_{1} q^{11} -2 \beta_{2} q^{13} + ( 2 \beta_{1} + \beta_{3} ) q^{14} - q^{16} -3 \beta_{2} q^{19} + 3 \beta_{3} q^{20} -5 q^{22} -\beta_{1} q^{23} + 10 q^{25} -2 \beta_{3} q^{26} + ( 6 - 3 \beta_{2} ) q^{28} + 2 \beta_{1} q^{29} + \beta_{2} q^{31} + 3 \beta_{1} q^{32} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{35} - q^{37} -3 \beta_{3} q^{38} -5 \beta_{2} q^{40} -\beta_{3} q^{41} + 2 q^{43} + 3 \beta_{1} q^{44} -5 q^{46} -2 \beta_{3} q^{47} + ( 1 - 4 \beta_{2} ) q^{49} -10 \beta_{1} q^{50} + 6 \beta_{2} q^{52} -4 \beta_{1} q^{53} + 5 \beta_{2} q^{55} + ( -2 \beta_{1} - \beta_{3} ) q^{56} + 10 q^{58} + 2 \beta_{3} q^{59} -4 \beta_{2} q^{61} + \beta_{3} q^{62} + 13 q^{64} + 6 \beta_{1} q^{65} -10 q^{67} + ( -15 - 10 \beta_{2} ) q^{70} + 5 \beta_{1} q^{71} -6 \beta_{2} q^{73} + \beta_{1} q^{74} + 9 \beta_{2} q^{76} + ( 2 \beta_{1} + \beta_{3} ) q^{77} + 2 q^{79} + \beta_{3} q^{80} + 5 \beta_{2} q^{82} -2 \beta_{3} q^{83} -2 \beta_{1} q^{86} + 5 q^{88} + 3 \beta_{3} q^{89} + ( 6 + 4 \beta_{2} ) q^{91} + 3 \beta_{1} q^{92} + 10 \beta_{2} q^{94} + 9 \beta_{1} q^{95} + 8 \beta_{2} q^{97} + ( -\beta_{1} - 4 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{4} - 8q^{7} + O(q^{10}) \) \( 4q - 12q^{4} - 8q^{7} - 4q^{16} - 20q^{22} + 40q^{25} + 24q^{28} - 4q^{37} + 8q^{43} - 20q^{46} + 4q^{49} + 40q^{58} + 52q^{64} - 40q^{67} - 60q^{70} + 8q^{79} + 20q^{88} + 24q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/5\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{2} - 5 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 10 \nu \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(5 \beta_{2} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\(5 \beta_{1}\)

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
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 1.11803i
−1.93649 1.11803i
2.23607i 0 −3.00000 −3.87298 0 −2.00000 + 1.73205i 2.23607i 0 8.66025i
188.2 2.23607i 0 −3.00000 3.87298 0 −2.00000 1.73205i 2.23607i 0 8.66025i
188.3 2.23607i 0 −3.00000 −3.87298 0 −2.00000 1.73205i 2.23607i 0 8.66025i
188.4 2.23607i 0 −3.00000 3.87298 0 −2.00000 + 1.73205i 2.23607i 0 8.66025i
\(n\): 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 inner
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.b 4
3.b odd 2 1 inner 189.2.c.b 4
4.b odd 2 1 3024.2.k.i 4
7.b odd 2 1 inner 189.2.c.b 4
9.c even 3 1 567.2.o.c 4
9.c even 3 1 567.2.o.d 4
9.d odd 6 1 567.2.o.c 4
9.d odd 6 1 567.2.o.d 4
12.b even 2 1 3024.2.k.i 4
21.c even 2 1 inner 189.2.c.b 4
28.d even 2 1 3024.2.k.i 4
63.l odd 6 1 567.2.o.c 4
63.l odd 6 1 567.2.o.d 4
63.o even 6 1 567.2.o.c 4
63.o even 6 1 567.2.o.d 4
84.h odd 2 1 3024.2.k.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.c.b 4 1.a even 1 1 trivial
189.2.c.b 4 3.b odd 2 1 inner
189.2.c.b 4 7.b odd 2 1 inner
189.2.c.b 4 21.c even 2 1 inner
567.2.o.c 4 9.c even 3 1
567.2.o.c 4 9.d odd 6 1
567.2.o.c 4 63.l odd 6 1
567.2.o.c 4 63.o even 6 1
567.2.o.d 4 9.c even 3 1
567.2.o.d 4 9.d odd 6 1
567.2.o.d 4 63.l odd 6 1
567.2.o.d 4 63.o even 6 1
3024.2.k.i 4 4.b odd 2 1
3024.2.k.i 4 12.b even 2 1
3024.2.k.i 4 28.d even 2 1
3024.2.k.i 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} + 4 T^{4} )^{2} \)
$3$ 1
$5$ \( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + 4 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 17 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 17 T^{2} )^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 41 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 38 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 67 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 2 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 34 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 26 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 58 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 + 10 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 17 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 38 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 2 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 106 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 43 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \)
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