Properties

Label 189.3.b.b
Level $189$
Weight $3$
Character orbit 189.b
Analytic conductor $5.150$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 189.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.14987699641\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{3} q^{5} + \beta_{2} q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{3} q^{5} + \beta_{2} q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( 1 + 2 \beta_{2} ) q^{10} + ( 3 \beta_{1} - 4 \beta_{3} ) q^{11} + ( -3 + 5 \beta_{2} ) q^{13} + ( -2 \beta_{1} + \beta_{3} ) q^{14} + ( -9 + 4 \beta_{2} ) q^{16} + ( \beta_{1} - \beta_{3} ) q^{17} + ( 6 - \beta_{2} ) q^{19} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{20} + ( -8 + 11 \beta_{2} ) q^{22} + ( -3 \beta_{1} + 6 \beta_{3} ) q^{23} + ( 9 - 5 \beta_{2} ) q^{25} + ( -13 \beta_{1} + 5 \beta_{3} ) q^{26} + 7 q^{28} + ( 17 \beta_{1} + 3 \beta_{3} ) q^{29} + ( 16 + 7 \beta_{2} ) q^{31} + ( -9 \beta_{1} + 8 \beta_{3} ) q^{32} + ( -3 + 3 \beta_{2} ) q^{34} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{35} + ( -23 - 14 \beta_{2} ) q^{37} + ( 8 \beta_{1} - \beta_{3} ) q^{38} + ( 18 + 9 \beta_{2} ) q^{40} + ( -20 \beta_{1} - 9 \beta_{3} ) q^{41} + ( -7 - 15 \beta_{2} ) q^{43} + ( -18 \beta_{1} - 5 \beta_{3} ) q^{44} + ( 6 - 15 \beta_{2} ) q^{46} + ( -11 \beta_{1} - \beta_{3} ) q^{47} + 7 q^{49} + ( 19 \beta_{1} - 5 \beta_{3} ) q^{50} + ( 35 - 3 \beta_{2} ) q^{52} + ( 28 \beta_{1} + 2 \beta_{3} ) q^{53} + ( -61 - 14 \beta_{2} ) q^{55} + ( -\beta_{1} + 4 \beta_{3} ) q^{56} + ( -71 + 11 \beta_{2} ) q^{58} + ( 41 \beta_{1} + \beta_{3} ) q^{59} + ( 24 - 34 \beta_{2} ) q^{61} + ( 2 \beta_{1} + 7 \beta_{3} ) q^{62} + ( -8 - 9 \beta_{2} ) q^{64} + ( -15 \beta_{1} - 7 \beta_{3} ) q^{65} + ( 39 - \beta_{2} ) q^{67} + ( -5 \beta_{1} - \beta_{3} ) q^{68} + ( 14 + \beta_{2} ) q^{70} + ( 42 \beta_{1} - 9 \beta_{3} ) q^{71} + ( 49 - 23 \beta_{2} ) q^{73} + ( 5 \beta_{1} - 14 \beta_{3} ) q^{74} + ( -7 + 6 \beta_{2} ) q^{76} + ( -18 \beta_{1} - 5 \beta_{3} ) q^{77} + ( -121 - 9 \beta_{2} ) q^{79} + ( -12 \beta_{1} + \beta_{3} ) q^{80} + ( 89 - 2 \beta_{2} ) q^{82} + ( -31 \beta_{1} + 25 \beta_{3} ) q^{83} + ( -15 - 3 \beta_{2} ) q^{85} + ( 23 \beta_{1} - 15 \beta_{3} ) q^{86} + ( 45 + 36 \beta_{2} ) q^{88} + ( -10 \beta_{1} - 15 \beta_{3} ) q^{89} + ( 35 - 3 \beta_{2} ) q^{91} + ( 24 \beta_{1} + 9 \beta_{3} ) q^{92} + ( 45 - 9 \beta_{2} ) q^{94} + ( 3 \beta_{1} - 4 \beta_{3} ) q^{95} + ( 4 - 38 \beta_{2} ) q^{97} + 7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 4q^{10} - 12q^{13} - 36q^{16} + 24q^{19} - 32q^{22} + 36q^{25} + 28q^{28} + 64q^{31} - 12q^{34} - 92q^{37} + 72q^{40} - 28q^{43} + 24q^{46} + 28q^{49} + 140q^{52} - 244q^{55} - 284q^{58} + 96q^{61} - 32q^{64} + 156q^{67} + 56q^{70} + 196q^{73} - 28q^{76} - 484q^{79} + 356q^{82} - 60q^{85} + 180q^{88} + 140q^{91} + 180q^{94} + 16q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 6 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 6 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1
2.57794i
1.16372i
1.16372i
2.57794i
2.57794i 0 −2.64575 1.66471i 0 −2.64575 3.49117i 0 −4.29150
134.2 1.16372i 0 2.64575 5.40636i 0 2.64575 7.73381i 0 6.29150
134.3 1.16372i 0 2.64575 5.40636i 0 2.64575 7.73381i 0 6.29150
134.4 2.57794i 0 −2.64575 1.66471i 0 −2.64575 3.49117i 0 −4.29150
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.b.b 4
3.b odd 2 1 inner 189.3.b.b 4
4.b odd 2 1 3024.3.d.c 4
9.c even 3 2 567.3.r.b 8
9.d odd 6 2 567.3.r.b 8
12.b even 2 1 3024.3.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.3.b.b 4 1.a even 1 1 trivial
189.3.b.b 4 3.b odd 2 1 inner
567.3.r.b 8 9.c even 3 2
567.3.r.b 8 9.d odd 6 2
3024.3.d.c 4 4.b odd 2 1
3024.3.d.c 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 8 T_{2}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 8 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 81 + 32 T^{2} + T^{4} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( 68121 + 536 T^{2} + T^{4} \)
$13$ \( ( -166 + 6 T + T^{2} )^{2} \)
$17$ \( ( 18 + T^{2} )^{2} \)
$19$ \( ( 29 - 12 T + T^{2} )^{2} \)
$23$ \( 263169 + 1152 T^{2} + T^{4} \)
$29$ \( 1954404 + 2804 T^{2} + T^{4} \)
$31$ \( ( -87 - 32 T + T^{2} )^{2} \)
$37$ \( ( -843 + 46 T + T^{2} )^{2} \)
$41$ \( 6922161 + 6512 T^{2} + T^{4} \)
$43$ \( ( -1526 + 14 T + T^{2} )^{2} \)
$47$ \( 236196 + 1044 T^{2} + T^{4} \)
$53$ \( 8928144 + 6624 T^{2} + T^{4} \)
$59$ \( 30536676 + 13644 T^{2} + T^{4} \)
$61$ \( ( -7516 - 48 T + T^{2} )^{2} \)
$67$ \( ( 1514 - 78 T + T^{2} )^{2} \)
$71$ \( 729 + 15192 T^{2} + T^{4} \)
$73$ \( ( -1302 - 98 T + T^{2} )^{2} \)
$79$ \( ( 14074 + 242 T + T^{2} )^{2} \)
$83$ \( 145009764 + 24588 T^{2} + T^{4} \)
$89$ \( 5625 + 8600 T^{2} + T^{4} \)
$97$ \( ( -10092 - 8 T + T^{2} )^{2} \)
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