Properties

Label 189.2.s.a
Level 189
Weight 2
Character orbit 189.s
Analytic conductor 1.509
Analytic rank 0
Dimension 2
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.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.500000 + 0.866025i
0.500000 0.866025i
1.50000 + 0.866025i 0 0.500000 + 0.866025i 3.00000 0 −2.50000 + 0.866025i 1.73205i 0 4.50000 + 2.59808i
89.1 1.50000 0.866025i 0 0.500000 0.866025i 3.00000 0 −2.50000 0.866025i 1.73205i 0 4.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.s.a 2
3.b odd 2 1 63.2.s.a yes 2
4.b odd 2 1 3024.2.df.a 2
7.b odd 2 1 1323.2.s.a 2
7.c even 3 1 1323.2.i.a 2
7.c even 3 1 1323.2.o.a 2
7.d odd 6 1 189.2.i.a 2
7.d odd 6 1 1323.2.o.b 2
9.c even 3 1 63.2.i.a 2
9.c even 3 1 567.2.p.a 2
9.d odd 6 1 189.2.i.a 2
9.d odd 6 1 567.2.p.b 2
12.b even 2 1 1008.2.df.a 2
21.c even 2 1 441.2.s.a 2
21.g even 6 1 63.2.i.a 2
21.g even 6 1 441.2.o.a 2
21.h odd 6 1 441.2.i.a 2
21.h odd 6 1 441.2.o.b 2
28.f even 6 1 3024.2.ca.a 2
36.f odd 6 1 1008.2.ca.a 2
36.h even 6 1 3024.2.ca.a 2
63.g even 3 1 441.2.s.a 2
63.h even 3 1 441.2.o.a 2
63.i even 6 1 567.2.p.a 2
63.i even 6 1 1323.2.o.a 2
63.j odd 6 1 1323.2.o.b 2
63.k odd 6 1 63.2.s.a yes 2
63.l odd 6 1 441.2.i.a 2
63.n odd 6 1 1323.2.s.a 2
63.o even 6 1 1323.2.i.a 2
63.s even 6 1 inner 189.2.s.a 2
63.t odd 6 1 441.2.o.b 2
63.t odd 6 1 567.2.p.b 2
84.j odd 6 1 1008.2.ca.a 2
252.n even 6 1 1008.2.df.a 2
252.bn odd 6 1 3024.2.df.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.i.a 2 9.c even 3 1
63.2.i.a 2 21.g even 6 1
63.2.s.a yes 2 3.b odd 2 1
63.2.s.a yes 2 63.k odd 6 1
189.2.i.a 2 7.d odd 6 1
189.2.i.a 2 9.d odd 6 1
189.2.s.a 2 1.a even 1 1 trivial
189.2.s.a 2 63.s even 6 1 inner
441.2.i.a 2 21.h odd 6 1
441.2.i.a 2 63.l odd 6 1
441.2.o.a 2 21.g even 6 1
441.2.o.a 2 63.h even 3 1
441.2.o.b 2 21.h odd 6 1
441.2.o.b 2 63.t odd 6 1
441.2.s.a 2 21.c even 2 1
441.2.s.a 2 63.g even 3 1
567.2.p.a 2 9.c even 3 1
567.2.p.a 2 63.i even 6 1
567.2.p.b 2 9.d odd 6 1
567.2.p.b 2 63.t odd 6 1
1008.2.ca.a 2 36.f odd 6 1
1008.2.ca.a 2 84.j odd 6 1
1008.2.df.a 2 12.b even 2 1
1008.2.df.a 2 252.n even 6 1
1323.2.i.a 2 7.c even 3 1
1323.2.i.a 2 63.o even 6 1
1323.2.o.a 2 7.c even 3 1
1323.2.o.a 2 63.i even 6 1
1323.2.o.b 2 7.d odd 6 1
1323.2.o.b 2 63.j odd 6 1
1323.2.s.a 2 7.b odd 2 1
1323.2.s.a 2 63.n odd 6 1
3024.2.ca.a 2 28.f even 6 1
3024.2.ca.a 2 36.h even 6 1
3024.2.df.a 2 4.b odd 2 1
3024.2.df.a 2 252.bn odd 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} \)
$3$ 1
$5$ \( ( 1 - 3 T + 5 T^{2} )^{2} \)
$7$ \( 1 + 5 T + 7 T^{2} \)
$11$ \( 1 - 19 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - 19 T^{2} + 529 T^{4} \)
$29$ \( 1 - 9 T + 56 T^{2} - 261 T^{3} + 841 T^{4} \)
$31$ \( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} \)
$37$ \( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \)
$43$ \( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 15 T + 128 T^{2} - 795 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 24 T + 253 T^{2} - 1464 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 130 T^{2} + 5041 T^{4} \)
$73$ \( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 3 T + 100 T^{2} + 291 T^{3} + 9409 T^{4} \)
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