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: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{4} - \beta_{3} q^{5} + (\beta_{2} - 2) q^{7} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 3 q^{4} - \beta_{3} q^{5} + (\beta_{2} - 2) q^{7} + \beta_1 q^{8} + 5 \beta_{2} q^{10} - \beta_1 q^{11} - 2 \beta_{2} q^{13} + (\beta_{3} + 2 \beta_1) q^{14} - q^{16} - 3 \beta_{2} q^{19} + 3 \beta_{3} q^{20} - 5 q^{22} - \beta_1 q^{23} + 10 q^{25} - 2 \beta_{3} q^{26} + ( - 3 \beta_{2} + 6) q^{28} + 2 \beta_1 q^{29} + \beta_{2} q^{31} + 3 \beta_1 q^{32} + (2 \beta_{3} - 3 \beta_1) q^{35} - q^{37} - 3 \beta_{3} q^{38} - 5 \beta_{2} q^{40} - \beta_{3} q^{41} + 2 q^{43} + 3 \beta_1 q^{44} - 5 q^{46} - 2 \beta_{3} q^{47} + ( - 4 \beta_{2} + 1) q^{49} - 10 \beta_1 q^{50} + 6 \beta_{2} q^{52} - 4 \beta_1 q^{53} + 5 \beta_{2} q^{55} + ( - \beta_{3} - 2 \beta_1) q^{56} + 10 q^{58} + 2 \beta_{3} q^{59} - 4 \beta_{2} q^{61} + \beta_{3} q^{62} + 13 q^{64} + 6 \beta_1 q^{65} - 10 q^{67} + ( - 10 \beta_{2} - 15) q^{70} + 5 \beta_1 q^{71} - 6 \beta_{2} q^{73} + \beta_1 q^{74} + 9 \beta_{2} q^{76} + (\beta_{3} + 2 \beta_1) q^{77} + 2 q^{79} + \beta_{3} q^{80} + 5 \beta_{2} q^{82} - 2 \beta_{3} q^{83} - 2 \beta_1 q^{86} + 5 q^{88} + 3 \beta_{3} q^{89} + (4 \beta_{2} + 6) q^{91} + 3 \beta_1 q^{92} + 10 \beta_{2} q^{94} + 9 \beta_1 q^{95} + 8 \beta_{2} q^{97} + ( - 4 \beta_{3} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{4} - 8 q^{7} - 4 q^{16} - 20 q^{22} + 40 q^{25} + 24 q^{28} - 4 q^{37} + 8 q^{43} - 20 q^{46} + 4 q^{49} + 40 q^{58} + 52 q^{64} - 40 q^{67} - 60 q^{70} + 8 q^{79} + 20 q^{88} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{2} - 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 10\nu ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_1 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 15)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 15)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 60)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 60)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$67$ \( (T + 10)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 125)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$79$ \( (T - 2)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 60)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 135)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
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