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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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 \( T_{2}^{2} - 3T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3 \) Copy content Toggle raw display
$13$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$23$ \( T^{2} + 27 \) Copy content Toggle raw display
$29$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$31$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$37$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 24T + 192 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$89$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
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