Properties

Label 189.2.c.c
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: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -\beta_{2} q^{5} + ( 1 - \beta_{3} ) q^{7} -2 \beta_{1} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} -\beta_{2} q^{5} + ( 1 - \beta_{3} ) q^{7} -2 \beta_{1} q^{8} + \beta_{3} q^{10} -\beta_{1} q^{11} -\beta_{3} q^{13} + ( -\beta_{1} - 2 \beta_{2} ) q^{14} -4 q^{16} + 3 \beta_{2} q^{17} + 3 \beta_{3} q^{19} -2 q^{22} + 2 \beta_{1} q^{23} -2 q^{25} -2 \beta_{2} q^{26} + 5 \beta_{1} q^{29} -\beta_{3} q^{31} -3 \beta_{3} q^{34} + ( 3 \beta_{1} - \beta_{2} ) q^{35} + 5 q^{37} + 6 \beta_{2} q^{38} + 2 \beta_{3} q^{40} + 5 \beta_{2} q^{41} + 5 q^{43} + 4 q^{46} -5 \beta_{2} q^{47} + ( -5 - 2 \beta_{3} ) q^{49} + 2 \beta_{1} q^{50} + 8 \beta_{1} q^{53} + \beta_{3} q^{55} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{56} + 10 q^{58} + 5 \beta_{2} q^{59} + \beta_{3} q^{61} -2 \beta_{2} q^{62} -8 q^{64} + 3 \beta_{1} q^{65} + 2 q^{67} + ( 6 + \beta_{3} ) q^{70} -10 \beta_{1} q^{71} -5 \beta_{1} q^{74} + ( -\beta_{1} - 2 \beta_{2} ) q^{77} -13 q^{79} + 4 \beta_{2} q^{80} -5 \beta_{3} q^{82} + \beta_{2} q^{83} -9 q^{85} -5 \beta_{1} q^{86} -4 q^{88} -6 \beta_{2} q^{89} + ( -6 - \beta_{3} ) q^{91} + 5 \beta_{3} q^{94} -9 \beta_{1} q^{95} + 7 \beta_{3} q^{97} + ( 5 \beta_{1} - 4 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} - 16q^{16} - 8q^{22} - 8q^{25} + 20q^{37} + 20q^{43} + 16q^{46} - 20q^{49} + 40q^{58} - 32q^{64} + 8q^{67} + 24q^{70} - 52q^{79} - 36q^{85} - 16q^{88} - 24q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

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

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
0.517638i
1.93185i
0.517638i
1.93185i
1.41421i 0 0 −1.73205 0 1.00000 2.44949i 2.82843i 0 2.44949i
188.2 1.41421i 0 0 1.73205 0 1.00000 + 2.44949i 2.82843i 0 2.44949i
188.3 1.41421i 0 0 −1.73205 0 1.00000 + 2.44949i 2.82843i 0 2.44949i
188.4 1.41421i 0 0 1.73205 0 1.00000 2.44949i 2.82843i 0 2.44949i
\(n\): 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.c 4
3.b odd 2 1 inner 189.2.c.c 4
4.b odd 2 1 3024.2.k.g 4
7.b odd 2 1 inner 189.2.c.c 4
9.c even 3 2 567.2.o.e 8
9.d odd 6 2 567.2.o.e 8
12.b even 2 1 3024.2.k.g 4
21.c even 2 1 inner 189.2.c.c 4
28.d even 2 1 3024.2.k.g 4
63.l odd 6 2 567.2.o.e 8
63.o even 6 2 567.2.o.e 8
84.h odd 2 1 3024.2.k.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.c.c 4 1.a even 1 1 trivial
189.2.c.c 4 3.b odd 2 1 inner
189.2.c.c 4 7.b odd 2 1 inner
189.2.c.c 4 21.c even 2 1 inner
567.2.o.e 8 9.c even 3 2
567.2.o.e 8 9.d odd 6 2
567.2.o.e 8 63.l odd 6 2
567.2.o.e 8 63.o even 6 2
3024.2.k.g 4 4.b odd 2 1
3024.2.k.g 4 12.b even 2 1
3024.2.k.g 4 28.d even 2 1
3024.2.k.g 4 84.h odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \)
$3$ 1
$5$ \( ( 1 + 7 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 20 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 7 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 16 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 38 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 8 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 56 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 5 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 7 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 19 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 22 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 43 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 116 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 2 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 + 58 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 + 13 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 163 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 100 T^{2} + 9409 T^{4} )^{2} \)
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