Properties

Label 189.3.b.a
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: 4.0.1166592.2
Defining polynomial: \(x^{4} + 20 x^{2} + 93\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -6 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( -3 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -6 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( -3 \beta_{1} + \beta_{3} ) q^{8} + ( 7 - 10 \beta_{2} ) q^{10} -\beta_{1} q^{11} + ( 9 - \beta_{2} ) q^{13} + ( -\beta_{1} + \beta_{3} ) q^{14} + ( 3 - 8 \beta_{2} ) q^{16} + ( -6 \beta_{1} + \beta_{3} ) q^{17} + ( -12 - 7 \beta_{2} ) q^{19} + ( 13 \beta_{1} - 6 \beta_{3} ) q^{20} + ( 10 - \beta_{2} ) q^{22} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{23} + ( -45 + 7 \beta_{2} ) q^{25} + ( 10 \beta_{1} - \beta_{3} ) q^{26} + ( 7 - 6 \beta_{2} ) q^{28} + ( -10 \beta_{1} - \beta_{3} ) q^{29} + ( 10 + 13 \beta_{2} ) q^{31} + ( -\beta_{1} - 4 \beta_{3} ) q^{32} + ( 57 - 15 \beta_{2} ) q^{34} + 7 \beta_{1} q^{35} + ( 13 + 22 \beta_{2} ) q^{37} + ( -5 \beta_{1} - 7 \beta_{3} ) q^{38} + ( -84 + 27 \beta_{2} ) q^{40} + ( 7 \beta_{1} + \beta_{3} ) q^{41} + ( 35 - 15 \beta_{2} ) q^{43} + ( 7 \beta_{1} - \beta_{3} ) q^{44} + ( -36 - 15 \beta_{2} ) q^{46} + ( 6 \beta_{1} + 5 \beta_{3} ) q^{47} + 7 q^{49} + ( -52 \beta_{1} + 7 \beta_{3} ) q^{50} + ( -61 + 15 \beta_{2} ) q^{52} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -7 + 10 \beta_{2} ) q^{55} + ( 9 \beta_{1} - 2 \beta_{3} ) q^{56} + ( 103 - \beta_{2} ) q^{58} + ( 12 \beta_{1} + 7 \beta_{3} ) q^{59} + ( 72 - 10 \beta_{2} ) q^{61} + ( -3 \beta_{1} + 13 \beta_{3} ) q^{62} + ( 34 + 3 \beta_{2} ) q^{64} + ( -16 \beta_{1} + 9 \beta_{3} ) q^{65} + ( 33 + 23 \beta_{2} ) q^{67} + ( 48 \beta_{1} - 11 \beta_{3} ) q^{68} + ( -70 + 7 \beta_{2} ) q^{70} + ( 3 \beta_{1} - 9 \beta_{3} ) q^{71} + ( -47 - 5 \beta_{2} ) q^{73} + ( -9 \beta_{1} + 22 \beta_{3} ) q^{74} + ( 23 + 30 \beta_{2} ) q^{76} + ( \beta_{1} - \beta_{3} ) q^{77} + ( -19 + 15 \beta_{2} ) q^{79} + ( -59 \beta_{1} + 3 \beta_{3} ) q^{80} + ( -73 - 2 \beta_{2} ) q^{82} + ( 24 \beta_{1} + 3 \beta_{3} ) q^{83} + ( -105 + 57 \beta_{2} ) q^{85} + ( 50 \beta_{1} - 15 \beta_{3} ) q^{86} + ( -27 + 12 \beta_{2} ) q^{88} + ( -\beta_{1} - 5 \beta_{3} ) q^{89} + ( -7 + 9 \beta_{2} ) q^{91} + ( -9 \beta_{1} - 7 \beta_{3} ) q^{92} + ( -75 - 39 \beta_{2} ) q^{94} + ( -37 \beta_{1} - 12 \beta_{3} ) q^{95} + ( 40 + 22 \beta_{2} ) q^{97} + 7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 24q^{4} + O(q^{10}) \) \( 4q - 24q^{4} + 28q^{10} + 36q^{13} + 12q^{16} - 48q^{19} + 40q^{22} - 180q^{25} + 28q^{28} + 40q^{31} + 228q^{34} + 52q^{37} - 336q^{40} + 140q^{43} - 144q^{46} + 28q^{49} - 244q^{52} - 28q^{55} + 412q^{58} + 288q^{61} + 136q^{64} + 132q^{67} - 280q^{70} - 188q^{73} + 92q^{76} - 76q^{79} - 292q^{82} - 420q^{85} - 108q^{88} - 28q^{91} - 300q^{94} + 160q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 20 x^{2} + 93\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 10 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 11 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 10\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 11 \beta_{1}\)

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
3.55609i
2.71187i
2.71187i
3.55609i
3.55609i 0 −8.64575 9.40852i 0 −2.64575 16.5207i 0 33.4575
134.2 2.71187i 0 −3.35425 7.17494i 0 2.64575 1.75119i 0 −19.4575
134.3 2.71187i 0 −3.35425 7.17494i 0 2.64575 1.75119i 0 −19.4575
134.4 3.55609i 0 −8.64575 9.40852i 0 −2.64575 16.5207i 0 33.4575
\(n\): 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.a 4
3.b odd 2 1 inner 189.3.b.a 4
4.b odd 2 1 3024.3.d.f 4
9.c even 3 2 567.3.r.d 8
9.d odd 6 2 567.3.r.d 8
12.b even 2 1 3024.3.d.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.3.b.a 4 1.a even 1 1 trivial
189.3.b.a 4 3.b odd 2 1 inner
567.3.r.d 8 9.c even 3 2
567.3.r.d 8 9.d odd 6 2
3024.3.d.f 4 4.b odd 2 1
3024.3.d.f 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 20 T_{2}^{2} + 93 \) acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 93 + 20 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 4557 + 140 T^{2} + T^{4} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( 93 + 20 T^{2} + T^{4} \)
$13$ \( ( 74 - 18 T + T^{2} )^{2} \)
$17$ \( 30132 + 780 T^{2} + T^{4} \)
$19$ \( ( -199 + 24 T + T^{2} )^{2} \)
$23$ \( 837 + 780 T^{2} + T^{4} \)
$29$ \( 1208628 + 2252 T^{2} + T^{4} \)
$31$ \( ( -1083 - 20 T + T^{2} )^{2} \)
$37$ \( ( -3219 - 26 T + T^{2} )^{2} \)
$41$ \( 302157 + 1196 T^{2} + T^{4} \)
$43$ \( ( -350 - 70 T + T^{2} )^{2} \)
$47$ \( 271188 + 4380 T^{2} + T^{4} \)
$53$ \( 120528 + 1392 T^{2} + T^{4} \)
$59$ \( 30132 + 10356 T^{2} + T^{4} \)
$61$ \( ( 4484 - 144 T + T^{2} )^{2} \)
$67$ \( ( -2614 - 66 T + T^{2} )^{2} \)
$71$ \( 26222373 + 10548 T^{2} + T^{4} \)
$73$ \( ( 2034 + 94 T + T^{2} )^{2} \)
$79$ \( ( -1214 + 38 T + T^{2} )^{2} \)
$83$ \( 41250708 + 13572 T^{2} + T^{4} \)
$89$ \( 1796853 + 3380 T^{2} + T^{4} \)
$97$ \( ( -1788 - 80 T + T^{2} )^{2} \)
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