Properties

Label 189.2.e.c
Level 189
Weight 2
Character orbit 189.e
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.e (of order \(3\), degree \(2\), 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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Character values

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

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

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 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i −2.00000 3.46410i 0 2.00000 + 1.73205i 3.00000 0 2.00000 3.46410i
163.1 0.500000 0.866025i 0 0.500000 + 0.866025i −2.00000 + 3.46410i 0 2.00000 1.73205i 3.00000 0 2.00000 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
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.c yes 2
3.b odd 2 1 189.2.e.a 2
7.c even 3 1 inner 189.2.e.c yes 2
7.c even 3 1 1323.2.a.g 1
7.d odd 6 1 1323.2.a.d 1
9.c even 3 1 567.2.g.e 2
9.c even 3 1 567.2.h.b 2
9.d odd 6 1 567.2.g.b 2
9.d odd 6 1 567.2.h.e 2
21.g even 6 1 1323.2.a.p 1
21.h odd 6 1 189.2.e.a 2
21.h odd 6 1 1323.2.a.m 1
63.g even 3 1 567.2.h.b 2
63.h even 3 1 567.2.g.e 2
63.j odd 6 1 567.2.g.b 2
63.n odd 6 1 567.2.h.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.a 2 3.b odd 2 1
189.2.e.a 2 21.h odd 6 1
189.2.e.c yes 2 1.a even 1 1 trivial
189.2.e.c yes 2 7.c even 3 1 inner
567.2.g.b 2 9.d odd 6 1
567.2.g.b 2 63.j odd 6 1
567.2.g.e 2 9.c even 3 1
567.2.g.e 2 63.h even 3 1
567.2.h.b 2 9.c even 3 1
567.2.h.b 2 63.g even 3 1
567.2.h.e 2 9.d odd 6 1
567.2.h.e 2 63.n odd 6 1
1323.2.a.d 1 7.d odd 6 1
1323.2.a.g 1 7.c even 3 1
1323.2.a.m 1 21.h odd 6 1
1323.2.a.p 1 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \)
$3$ 1
$5$ \( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \)
$7$ \( 1 - 4 T + 7 T^{2} \)
$11$ \( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - T + 13 T^{2} )^{2} \)
$17$ \( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \)
$19$ \( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 2 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \)
$37$ \( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + T + 43 T^{2} )^{2} \)
$47$ \( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 4 T - 73 T^{2} + 356 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 9 T + 97 T^{2} )^{2} \)
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