Properties

Label 189.2.p.c
Level 189
Weight 2
Character orbit 189.p
Analytic conductor 1.509
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 189.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 3 \beta_{2} q^{4} + ( -3 + 2 \beta_{2} ) q^{7} + \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 3 \beta_{2} q^{4} + ( -3 + 2 \beta_{2} ) q^{7} + \beta_{3} q^{8} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{11} + ( -1 + 2 \beta_{2} ) q^{13} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{14} + ( 1 - \beta_{2} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -2 - 2 \beta_{2} ) q^{19} + 10 q^{22} -2 \beta_{1} q^{23} + 5 \beta_{2} q^{25} + ( -\beta_{1} + 2 \beta_{3} ) q^{26} + ( -6 - 3 \beta_{2} ) q^{28} + 2 \beta_{3} q^{29} + ( -2 + \beta_{2} ) q^{31} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{32} + ( 10 - 20 \beta_{2} ) q^{34} + ( -5 + 5 \beta_{2} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{38} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{41} -7 q^{43} + 6 \beta_{1} q^{44} -10 \beta_{2} q^{46} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{47} + ( 5 - 8 \beta_{2} ) q^{49} + 5 \beta_{3} q^{50} + ( -6 + 3 \beta_{2} ) q^{52} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -2 \beta_{1} - \beta_{3} ) q^{56} + ( -10 + 10 \beta_{2} ) q^{58} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{59} + ( 5 + 5 \beta_{2} ) q^{61} + ( -2 \beta_{1} + \beta_{3} ) q^{62} + 13 q^{64} + \beta_{2} q^{67} + ( 6 \beta_{1} - 12 \beta_{3} ) q^{68} -4 \beta_{3} q^{71} + ( 8 - 4 \beta_{2} ) q^{73} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{74} + ( 6 - 12 \beta_{2} ) q^{76} + ( -2 \beta_{1} + 6 \beta_{3} ) q^{77} + ( -11 + 11 \beta_{2} ) q^{79} + ( 10 + 10 \beta_{2} ) q^{82} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{83} -7 \beta_{1} q^{86} + 10 \beta_{2} q^{88} + ( -4 \beta_{1} + 8 \beta_{3} ) q^{89} + ( -1 - 4 \beta_{2} ) q^{91} -6 \beta_{3} q^{92} + ( -20 + 10 \beta_{2} ) q^{94} + ( 1 - 2 \beta_{2} ) q^{97} + ( 5 \beta_{1} - 8 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} - 8q^{7} + O(q^{10}) \) \( 4q + 6q^{4} - 8q^{7} + 2q^{16} - 12q^{19} + 40q^{22} + 10q^{25} - 30q^{28} - 6q^{31} - 10q^{37} - 28q^{43} - 20q^{46} + 4q^{49} - 18q^{52} - 20q^{58} + 30q^{61} + 52q^{64} + 2q^{67} + 24q^{73} - 22q^{79} + 60q^{82} + 20q^{88} - 12q^{91} - 60q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3}\)

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.93649 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 1.11803i
−1.93649 1.11803i 0 1.50000 + 2.59808i 0 0 −2.00000 + 1.73205i 2.23607i 0 0
26.2 1.93649 + 1.11803i 0 1.50000 + 2.59808i 0 0 −2.00000 + 1.73205i 2.23607i 0 0
80.1 −1.93649 + 1.11803i 0 1.50000 2.59808i 0 0 −2.00000 1.73205i 2.23607i 0 0
80.2 1.93649 1.11803i 0 1.50000 2.59808i 0 0 −2.00000 1.73205i 2.23607i 0 0
\(n\): 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
7.d odd 6 1 inner
21.g 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.p.c 4
3.b odd 2 1 inner 189.2.p.c 4
7.c even 3 1 1323.2.c.b 4
7.d odd 6 1 inner 189.2.p.c 4
7.d odd 6 1 1323.2.c.b 4
9.c even 3 1 567.2.i.c 4
9.c even 3 1 567.2.s.e 4
9.d odd 6 1 567.2.i.c 4
9.d odd 6 1 567.2.s.e 4
21.g even 6 1 inner 189.2.p.c 4
21.g even 6 1 1323.2.c.b 4
21.h odd 6 1 1323.2.c.b 4
63.i even 6 1 567.2.s.e 4
63.k odd 6 1 567.2.i.c 4
63.s even 6 1 567.2.i.c 4
63.t odd 6 1 567.2.s.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.c 4 1.a even 1 1 trivial
189.2.p.c 4 3.b odd 2 1 inner
189.2.p.c 4 7.d odd 6 1 inner
189.2.p.c 4 21.g even 6 1 inner
567.2.i.c 4 9.c even 3 1
567.2.i.c 4 9.d odd 6 1
567.2.i.c 4 63.k odd 6 1
567.2.i.c 4 63.s even 6 1
567.2.s.e 4 9.c even 3 1
567.2.s.e 4 9.d odd 6 1
567.2.s.e 4 63.i even 6 1
567.2.s.e 4 63.t odd 6 1
1323.2.c.b 4 7.c even 3 1
1323.2.c.b 4 7.d odd 6 1
1323.2.c.b 4 21.g even 6 1
1323.2.c.b 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5 T_{2}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8} \)
$3$ 1
$5$ \( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + 4 T + 7 T^{2} )^{2} \)
$11$ \( 1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 26 T^{2} + 387 T^{4} + 7514 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 26 T^{2} + 147 T^{4} + 13754 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 38 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 22 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 7 T + 43 T^{2} )^{4} \)
$47$ \( 1 - 34 T^{2} - 1053 T^{4} - 75106 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 86 T^{2} + 4587 T^{4} + 241574 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 58 T^{2} - 117 T^{4} - 201898 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{2}( 1 - T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - T - 66 T^{2} - 67 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 62 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 106 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( 1 + 62 T^{2} - 4077 T^{4} + 491102 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 - 191 T^{2} + 9409 T^{4} )^{2} \)
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