Properties

Label 189.2.e.f
Level $189$
Weight $2$
Character orbit 189.e
Analytic conductor $1.509$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Defining polynomial: \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{4} + \beta_{5} ) q^{2} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4} + \beta_{2} q^{5} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{7} + ( -4 - \beta_{1} - 2 \beta_{3} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{4} + \beta_{5} ) q^{2} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4} + \beta_{2} q^{5} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{7} + ( -4 - \beta_{1} - 2 \beta_{3} ) q^{8} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{10} + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{11} + ( 1 + 2 \beta_{1} - \beta_{3} ) q^{13} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{14} + ( -5 + \beta_{2} + 5 \beta_{4} - 4 \beta_{5} ) q^{16} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{17} + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{19} + ( 5 + 2 \beta_{1} ) q^{20} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{22} + ( 2 - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{23} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{25} + ( 6 - \beta_{2} - 6 \beta_{4} ) q^{26} + ( -6 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{28} + ( -4 - \beta_{1} + 2 \beta_{3} ) q^{29} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{31} + ( 4 \beta_{1} + \beta_{2} + \beta_{3} + 10 \beta_{4} - 4 \beta_{5} ) q^{32} + ( 6 + 3 \beta_{1} + 3 \beta_{3} ) q^{34} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{35} + ( 3 - 2 \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{37} + ( -2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{38} + ( 9 - 9 \beta_{4} + 3 \beta_{5} ) q^{40} + ( -2 - 5 \beta_{1} + \beta_{3} ) q^{41} + ( -4 - 3 \beta_{1} ) q^{43} -\beta_{2} q^{44} + ( -3 \beta_{1} + 3 \beta_{5} ) q^{46} + ( 2 - \beta_{2} - 2 \beta_{4} - 4 \beta_{5} ) q^{47} + ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{49} + ( 3 \beta_{1} + \beta_{3} ) q^{50} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{52} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{53} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{55} + ( -13 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 9 \beta_{4} - 6 \beta_{5} ) q^{56} + ( -5 - \beta_{2} + 5 \beta_{4} - 2 \beta_{5} ) q^{58} + ( \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{59} + ( -1 + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{61} + ( 8 + 2 \beta_{1} + \beta_{3} ) q^{62} + ( 13 + 3 \beta_{1} + 3 \beta_{3} ) q^{64} + ( 2 + 3 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{65} + ( \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{67} + ( 16 - 2 \beta_{2} - 16 \beta_{4} + 7 \beta_{5} ) q^{68} + ( 5 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{70} + ( 3 + 3 \beta_{1} + 3 \beta_{3} ) q^{71} + ( 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{5} ) q^{73} + ( -5 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 5 \beta_{5} ) q^{74} + ( -7 - 3 \beta_{1} ) q^{76} + ( -1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{77} + ( -2 + 3 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} ) q^{79} + ( -5 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} + 5 \beta_{5} ) q^{80} + ( -16 + 4 \beta_{2} + 16 \beta_{4} - \beta_{5} ) q^{82} + ( 6 \beta_{1} - 3 \beta_{3} ) q^{83} + ( -9 - 3 \beta_{1} + 3 \beta_{3} ) q^{85} + ( -13 + 3 \beta_{2} + 13 \beta_{4} - 4 \beta_{5} ) q^{86} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{88} + ( -3 + 4 \beta_{2} + 3 \beta_{4} ) q^{89} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 6 \beta_{4} ) q^{91} + ( -5 - 2 \beta_{1} - \beta_{3} ) q^{92} + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} + 3 \beta_{5} ) q^{94} + ( 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{95} + ( -6 - 2 \beta_{1} - 2 \beta_{3} ) q^{97} + ( -10 - \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} - 4q^{4} + q^{5} + 2q^{7} - 18q^{8} + O(q^{10}) \) \( 6q + 2q^{2} - 4q^{4} + q^{5} + 2q^{7} - 18q^{8} + q^{10} + 7q^{11} + 4q^{13} - 17q^{14} - 10q^{16} - 5q^{19} + 26q^{20} - 8q^{22} + 6q^{23} + 2q^{25} + 17q^{26} - 30q^{28} - 26q^{29} + 8q^{31} + 25q^{32} + 24q^{34} - 10q^{35} + 8q^{37} - 7q^{38} + 24q^{40} - 4q^{41} - 18q^{43} - q^{44} + 3q^{46} + 9q^{47} + 12q^{49} - 8q^{50} - 9q^{52} + 24q^{53} + 8q^{55} - 48q^{56} - 14q^{58} - 15q^{59} + q^{61} + 42q^{62} + 66q^{64} + 10q^{65} - 14q^{67} + 39q^{68} + 26q^{70} + 6q^{71} - 7q^{73} - 36q^{76} - q^{77} - 6q^{79} - 16q^{80} - 43q^{82} - 6q^{83} - 54q^{85} - 32q^{86} - 9q^{88} - 5q^{89} - 33q^{91} - 24q^{92} + 27q^{94} + 16q^{95} - 28q^{97} - 49q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 2 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 4\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} - 10\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(16 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} + 5\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-14 \beta_{5} - 16 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 23 \beta_{1} + 47\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 + 2.05195i
0.500000 1.41036i
0.500000 + 0.224437i
0.500000 2.05195i
0.500000 + 1.41036i
0.500000 0.224437i
−0.730252 1.26483i 0 −0.0665372 + 0.115246i −0.296790 0.514055i 0 2.32383 1.26483i −2.72665 0 −0.433463 + 0.750780i
109.2 0.380438 + 0.658939i 0 0.710533 1.23068i 1.59097 + 2.75564i 0 −2.56238 + 0.658939i 2.60301 0 −1.21053 + 2.09671i
109.3 1.34981 + 2.33795i 0 −2.64400 + 4.57954i −0.794182 1.37556i 0 1.23855 + 2.33795i −8.87636 0 2.14400 3.71351i
163.1 −0.730252 + 1.26483i 0 −0.0665372 0.115246i −0.296790 + 0.514055i 0 2.32383 + 1.26483i −2.72665 0 −0.433463 0.750780i
163.2 0.380438 0.658939i 0 0.710533 + 1.23068i 1.59097 2.75564i 0 −2.56238 0.658939i 2.60301 0 −1.21053 2.09671i
163.3 1.34981 2.33795i 0 −2.64400 4.57954i −0.794182 + 1.37556i 0 1.23855 2.33795i −8.87636 0 2.14400 + 3.71351i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.e.f yes 6
3.b odd 2 1 189.2.e.e 6
7.c even 3 1 inner 189.2.e.f yes 6
7.c even 3 1 1323.2.a.x 3
7.d odd 6 1 1323.2.a.y 3
9.c even 3 1 567.2.g.i 6
9.c even 3 1 567.2.h.h 6
9.d odd 6 1 567.2.g.h 6
9.d odd 6 1 567.2.h.i 6
21.g even 6 1 1323.2.a.z 3
21.h odd 6 1 189.2.e.e 6
21.h odd 6 1 1323.2.a.ba 3
63.g even 3 1 567.2.h.h 6
63.h even 3 1 567.2.g.i 6
63.j odd 6 1 567.2.g.h 6
63.n odd 6 1 567.2.h.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.e 6 3.b odd 2 1
189.2.e.e 6 21.h odd 6 1
189.2.e.f yes 6 1.a even 1 1 trivial
189.2.e.f yes 6 7.c even 3 1 inner
567.2.g.h 6 9.d odd 6 1
567.2.g.h 6 63.j odd 6 1
567.2.g.i 6 9.c even 3 1
567.2.g.i 6 63.h even 3 1
567.2.h.h 6 9.c even 3 1
567.2.h.h 6 63.g even 3 1
567.2.h.i 6 9.d odd 6 1
567.2.h.i 6 63.n odd 6 1
1323.2.a.x 3 7.c even 3 1
1323.2.a.y 3 7.d odd 6 1
1323.2.a.z 3 21.g even 6 1
1323.2.a.ba 3 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 2 T_{2}^{5} + 7 T_{2}^{4} + 15 T_{2}^{2} - 9 T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 - 9 T + 15 T^{2} + 7 T^{4} - 2 T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( 9 + 18 T + 33 T^{2} + 12 T^{3} + 7 T^{4} - T^{5} + T^{6} \)
$7$ \( 343 - 98 T - 28 T^{2} + 31 T^{3} - 4 T^{4} - 2 T^{5} + T^{6} \)
$11$ \( 9 - 36 T + 123 T^{2} - 78 T^{3} + 37 T^{4} - 7 T^{5} + T^{6} \)
$13$ \( ( 47 - 19 T - 2 T^{2} + T^{3} )^{2} \)
$17$ \( 81 + 297 T + 1089 T^{2} + 18 T^{3} + 33 T^{4} + T^{6} \)
$19$ \( 841 + 116 T + 161 T^{2} + 38 T^{3} + 29 T^{4} + 5 T^{5} + T^{6} \)
$23$ \( 81 + 27 T + 63 T^{2} - 36 T^{3} + 33 T^{4} - 6 T^{5} + T^{6} \)
$29$ \( ( 9 + 30 T + 13 T^{2} + T^{3} )^{2} \)
$31$ \( 4761 + 69 T + 553 T^{2} - 146 T^{3} + 63 T^{4} - 8 T^{5} + T^{6} \)
$37$ \( 8649 - 465 T + 769 T^{2} - 146 T^{3} + 69 T^{4} - 8 T^{5} + T^{6} \)
$41$ \( ( -387 - 105 T + 2 T^{2} + T^{3} )^{2} \)
$43$ \( ( -101 - 12 T + 9 T^{2} + T^{3} )^{2} \)
$47$ \( 81 + 378 T + 1683 T^{2} + 396 T^{3} + 123 T^{4} - 9 T^{5} + T^{6} \)
$53$ \( 59049 - 40095 T + 21393 T^{2} - 3474 T^{3} + 411 T^{4} - 24 T^{5} + T^{6} \)
$59$ \( 6561 + 5346 T + 3141 T^{2} + 828 T^{3} + 159 T^{4} + 15 T^{5} + T^{6} \)
$61$ \( 14641 - 5929 T + 2522 T^{2} - 193 T^{3} + 50 T^{4} - T^{5} + T^{6} \)
$67$ \( 961 + 1643 T + 2375 T^{2} + 680 T^{3} + 143 T^{4} + 14 T^{5} + T^{6} \)
$71$ \( ( -243 - 108 T - 3 T^{2} + T^{3} )^{2} \)
$73$ \( 962361 + 131454 T + 24823 T^{2} + 1024 T^{3} + 183 T^{4} + 7 T^{5} + T^{6} \)
$79$ \( 16129 + 8763 T + 5523 T^{2} - 160 T^{3} + 105 T^{4} + 6 T^{5} + T^{6} \)
$83$ \( ( 729 - 180 T + 3 T^{2} + T^{3} )^{2} \)
$89$ \( 239121 + 45477 T + 11094 T^{2} + 513 T^{3} + 118 T^{4} + 5 T^{5} + T^{6} \)
$97$ \( ( -24 + 16 T + 14 T^{2} + T^{3} )^{2} \)
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