Properties

Label 189.2.h.b
Level $189$
Weight $2$
Character orbit 189.h
Analytic conductor $1.509$
Analytic rank $0$
Dimension $10$
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.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.991381711347.1
Defining polynomial: \(x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{5} ) q^{2} + ( 1 + \beta_{3} ) q^{4} + ( \beta_{6} - \beta_{9} ) q^{5} + ( -1 + \beta_{1} + \beta_{4} - \beta_{5} ) q^{7} + ( 1 - \beta_{4} - \beta_{8} ) q^{8} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{5} ) q^{2} + ( 1 + \beta_{3} ) q^{4} + ( \beta_{6} - \beta_{9} ) q^{5} + ( -1 + \beta_{1} + \beta_{4} - \beta_{5} ) q^{7} + ( 1 - \beta_{4} - \beta_{8} ) q^{8} + ( 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{10} + ( -1 + \beta_{4} - \beta_{6} + \beta_{7} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} - \beta_{6} ) q^{13} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{14} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{16} + ( -\beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{17} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{19} + ( -\beta_{3} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{20} + ( -1 + \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} ) q^{22} + ( -2 \beta_{3} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{23} + ( -\beta_{1} + 2 \beta_{2} + \beta_{7} ) q^{25} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{26} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{28} + ( 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{29} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{31} + ( 1 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{8} + 2 \beta_{9} ) q^{32} + ( \beta_{3} + 4 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{34} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{35} + ( 2 \beta_{1} - 2 \beta_{4} ) q^{37} + ( -5 + 2 \beta_{1} - \beta_{4} - 5 \beta_{6} - 3 \beta_{7} ) q^{38} + ( -2 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{40} + ( -\beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{7} ) q^{41} + ( \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{43} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{6} - \beta_{7} ) q^{44} + ( 2 \beta_{3} + 5 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{46} + ( 4 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} ) q^{47} + ( 2 \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{49} + ( -6 + 3 \beta_{1} + 2 \beta_{2} + \beta_{4} - 6 \beta_{6} - \beta_{7} ) q^{50} + ( -1 - 2 \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{52} + ( \beta_{5} - 5 \beta_{6} + 2 \beta_{9} ) q^{53} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{55} + ( -4 - \beta_{1} + \beta_{3} + \beta_{4} - 5 \beta_{6} + \beta_{9} ) q^{56} + ( -2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{58} + ( 6 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{9} ) q^{59} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{61} + ( 3 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} + 2 \beta_{9} ) q^{62} + ( -6 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{9} ) q^{64} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{65} + ( -1 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + \beta_{8} + \beta_{9} ) q^{67} + ( -3 \beta_{3} - 2 \beta_{5} + 7 \beta_{6} + 3 \beta_{7} - \beta_{9} ) q^{68} + ( 5 + 2 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{70} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{8} - 2 \beta_{9} ) q^{71} + ( -\beta_{3} - 4 \beta_{6} + \beta_{7} - 3 \beta_{8} ) q^{73} + ( 10 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 10 \beta_{6} + 4 \beta_{7} ) q^{74} + ( 3 - 5 \beta_{1} + \beta_{2} + 3 \beta_{6} + \beta_{7} ) q^{76} + ( \beta_{1} + \beta_{2} - \beta_{4} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{77} + ( 3 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{5} + \beta_{9} ) q^{79} + ( 2 \beta_{5} + 3 \beta_{6} - \beta_{8} - \beta_{9} ) q^{80} + ( 2 - 4 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{82} + ( -2 \beta_{3} + \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} ) q^{83} + ( -2 + \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{85} + ( \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} - 4 \beta_{9} ) q^{86} + ( -4 - \beta_{2} + 2 \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{88} + ( -7 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - 7 \beta_{6} ) q^{89} + ( -\beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{91} + ( -2 \beta_{5} + 5 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{92} + ( -3 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{4} - 4 \beta_{5} + 4 \beta_{8} - 2 \beta_{9} ) q^{94} + ( -2 + 2 \beta_{3} - \beta_{4} - \beta_{8} ) q^{95} + ( -4 \beta_{3} + \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{97} + ( -5 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} + 5 \beta_{6} + 3 \beta_{7} - \beta_{9} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 4q^{2} + 8q^{4} - 4q^{5} - 4q^{7} + 6q^{8} + O(q^{10}) \) \( 10q + 4q^{2} + 8q^{4} - 4q^{5} - 4q^{7} + 6q^{8} - 7q^{10} - 4q^{11} - 8q^{13} + 20q^{14} - 4q^{16} - 12q^{17} + q^{19} - 5q^{20} - q^{22} - 3q^{23} - q^{25} - 11q^{26} - 2q^{28} - 7q^{29} + 6q^{31} - 4q^{32} + 3q^{34} - 5q^{35} - 20q^{38} - 3q^{40} - 5q^{41} - 7q^{43} + 10q^{44} + 3q^{46} + 54q^{47} - 8q^{49} - 19q^{50} - 10q^{52} + 21q^{53} + 4q^{55} - 18q^{56} - 10q^{58} + 60q^{59} + 28q^{61} + 12q^{62} - 50q^{64} - 22q^{65} + 4q^{67} - 27q^{68} + 40q^{70} + 6q^{71} + 15q^{73} + 36q^{74} + 5q^{76} - 11q^{77} + 8q^{79} - 20q^{80} - 5q^{82} - 9q^{83} - 6q^{85} + 8q^{86} - 18q^{88} - 28q^{89} - 4q^{91} - 27q^{92} + 6q^{94} - 28q^{95} - 12q^{97} - 59q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{9} - 9 \nu^{8} - 3 \nu^{7} - 61 \nu^{6} - 72 \nu^{5} - 282 \nu^{4} - 204 \nu^{3} - 387 \nu^{2} - 873 \nu - 117 \)\()/189\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{9} - 12 \nu^{8} + 48 \nu^{7} - 23 \nu^{6} + 204 \nu^{5} - 240 \nu^{4} + 303 \nu^{3} - 108 \nu^{2} + 36 \nu - 1557 \)\()/567\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{9} - \nu^{8} + 12 \nu^{7} + 8 \nu^{6} + 68 \nu^{5} + 30 \nu^{4} + 123 \nu^{3} + 204 \nu^{2} + 270 \nu + 63 \)\()/63\)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180 \)\()/567\)
\(\beta_{6}\)\(=\)\((\)\( 20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu - 63 \)\()/567\)
\(\beta_{7}\)\(=\)\((\)\( -53 \nu^{9} + 60 \nu^{8} - 375 \nu^{7} - 11 \nu^{6} - 1668 \nu^{5} - 69 \nu^{4} - 2757 \nu^{3} - 4401 \nu^{2} - 1071 \nu - 1368 \)\()/567\)
\(\beta_{8}\)\(=\)\((\)\( -82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720 \)\()/567\)
\(\beta_{9}\)\(=\)\((\)\( -91 \nu^{9} + 174 \nu^{8} - 813 \nu^{7} + 704 \nu^{6} - 3633 \nu^{5} + 3174 \nu^{4} - 8070 \nu^{3} + 648 \nu^{2} - 3222 \nu + 801 \)\()/567\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + 3 \beta_{6} - \beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{8} + 4 \beta_{5} + \beta_{4} - 4 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-5 \beta_{7} - 14 \beta_{6} + \beta_{4} + \beta_{2} - 14\)
\(\nu^{5}\)\(=\)\(2 \beta_{9} - 7 \beta_{8} - \beta_{7} - 9 \beta_{6} - 17 \beta_{5} + \beta_{3}\)
\(\nu^{6}\)\(=\)\(9 \beta_{9} - 10 \beta_{8} - \beta_{5} - 10 \beta_{4} + 24 \beta_{3} - 9 \beta_{2} + \beta_{1} + 70\)
\(\nu^{7}\)\(=\)\(11 \beta_{7} + 65 \beta_{6} - 43 \beta_{4} - 19 \beta_{2} + 75 \beta_{1} + 65\)
\(\nu^{8}\)\(=\)\(-62 \beta_{9} + 73 \beta_{8} + 118 \beta_{7} + 360 \beta_{6} + 14 \beta_{5} - 118 \beta_{3}\)
\(\nu^{9}\)\(=\)\(-135 \beta_{9} + 253 \beta_{8} + 343 \beta_{5} + 253 \beta_{4} - 87 \beta_{3} + 135 \beta_{2} - 343 \beta_{1} - 430\)

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1 - \beta_{6}\) \(-1 - \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−1.02682 + 1.77851i
−0.335166 + 0.580525i
0.247934 0.429435i
0.920620 1.59456i
1.19343 2.06709i
−1.02682 1.77851i
−0.335166 0.580525i
0.247934 + 0.429435i
0.920620 + 1.59456i
1.19343 + 2.06709i
−2.05365 0 2.21746 −0.0731228 0.126652i 0 −2.33035 1.25278i −0.446582 0 0.150168 + 0.260099i
37.2 −0.670333 0 −1.55065 0.712469 + 1.23403i 0 −2.36039 + 1.19522i 2.38012 0 −0.477591 0.827212i
37.3 0.495868 0 −1.75411 −1.84629 3.19787i 0 0.926641 2.47817i −1.86155 0 −0.915516 1.58572i
37.4 1.84124 0 1.39017 0.667377 + 1.15593i 0 1.90267 1.83844i −1.12285 0 1.22880 + 2.12835i
37.5 2.38687 0 3.69714 −1.46043 2.52954i 0 −0.138560 + 2.64212i 4.05086 0 −3.48586 6.03769i
46.1 −2.05365 0 2.21746 −0.0731228 + 0.126652i 0 −2.33035 + 1.25278i −0.446582 0 0.150168 0.260099i
46.2 −0.670333 0 −1.55065 0.712469 1.23403i 0 −2.36039 1.19522i 2.38012 0 −0.477591 + 0.827212i
46.3 0.495868 0 −1.75411 −1.84629 + 3.19787i 0 0.926641 + 2.47817i −1.86155 0 −0.915516 + 1.58572i
46.4 1.84124 0 1.39017 0.667377 1.15593i 0 1.90267 + 1.83844i −1.12285 0 1.22880 2.12835i
46.5 2.38687 0 3.69714 −1.46043 + 2.52954i 0 −0.138560 2.64212i 4.05086 0 −3.48586 + 6.03769i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h 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.h.b 10
3.b odd 2 1 63.2.h.b yes 10
4.b odd 2 1 3024.2.q.i 10
7.b odd 2 1 1323.2.h.f 10
7.c even 3 1 189.2.g.b 10
7.c even 3 1 1323.2.f.e 10
7.d odd 6 1 1323.2.f.f 10
7.d odd 6 1 1323.2.g.f 10
9.c even 3 1 189.2.g.b 10
9.c even 3 1 567.2.e.e 10
9.d odd 6 1 63.2.g.b 10
9.d odd 6 1 567.2.e.f 10
12.b even 2 1 1008.2.q.i 10
21.c even 2 1 441.2.h.f 10
21.g even 6 1 441.2.f.f 10
21.g even 6 1 441.2.g.f 10
21.h odd 6 1 63.2.g.b 10
21.h odd 6 1 441.2.f.e 10
28.g odd 6 1 3024.2.t.i 10
36.f odd 6 1 3024.2.t.i 10
36.h even 6 1 1008.2.t.i 10
63.g even 3 1 567.2.e.e 10
63.g even 3 1 1323.2.f.e 10
63.h even 3 1 inner 189.2.h.b 10
63.h even 3 1 3969.2.a.bc 5
63.i even 6 1 441.2.h.f 10
63.i even 6 1 3969.2.a.ba 5
63.j odd 6 1 63.2.h.b yes 10
63.j odd 6 1 3969.2.a.z 5
63.k odd 6 1 1323.2.f.f 10
63.l odd 6 1 1323.2.g.f 10
63.n odd 6 1 441.2.f.e 10
63.n odd 6 1 567.2.e.f 10
63.o even 6 1 441.2.g.f 10
63.s even 6 1 441.2.f.f 10
63.t odd 6 1 1323.2.h.f 10
63.t odd 6 1 3969.2.a.bb 5
84.n even 6 1 1008.2.t.i 10
252.u odd 6 1 3024.2.q.i 10
252.bb even 6 1 1008.2.q.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 9.d odd 6 1
63.2.g.b 10 21.h odd 6 1
63.2.h.b yes 10 3.b odd 2 1
63.2.h.b yes 10 63.j odd 6 1
189.2.g.b 10 7.c even 3 1
189.2.g.b 10 9.c even 3 1
189.2.h.b 10 1.a even 1 1 trivial
189.2.h.b 10 63.h even 3 1 inner
441.2.f.e 10 21.h odd 6 1
441.2.f.e 10 63.n odd 6 1
441.2.f.f 10 21.g even 6 1
441.2.f.f 10 63.s even 6 1
441.2.g.f 10 21.g even 6 1
441.2.g.f 10 63.o even 6 1
441.2.h.f 10 21.c even 2 1
441.2.h.f 10 63.i even 6 1
567.2.e.e 10 9.c even 3 1
567.2.e.e 10 63.g even 3 1
567.2.e.f 10 9.d odd 6 1
567.2.e.f 10 63.n odd 6 1
1008.2.q.i 10 12.b even 2 1
1008.2.q.i 10 252.bb even 6 1
1008.2.t.i 10 36.h even 6 1
1008.2.t.i 10 84.n even 6 1
1323.2.f.e 10 7.c even 3 1
1323.2.f.e 10 63.g even 3 1
1323.2.f.f 10 7.d odd 6 1
1323.2.f.f 10 63.k odd 6 1
1323.2.g.f 10 7.d odd 6 1
1323.2.g.f 10 63.l odd 6 1
1323.2.h.f 10 7.b odd 2 1
1323.2.h.f 10 63.t odd 6 1
3024.2.q.i 10 4.b odd 2 1
3024.2.q.i 10 252.u odd 6 1
3024.2.t.i 10 28.g odd 6 1
3024.2.t.i 10 36.f odd 6 1
3969.2.a.z 5 63.j odd 6 1
3969.2.a.ba 5 63.i even 6 1
3969.2.a.bb 5 63.t odd 6 1
3969.2.a.bc 5 63.h even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 2 T_{2}^{4} - 5 T_{2}^{3} + 9 T_{2}^{2} + 3 T_{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).