Properties

Label 189.2.e.d
Level 189
Weight 2
Character orbit 189.e
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -4 + 4 \beta_{1} ) q^{4} -\beta_{2} q^{5} + ( 1 - 3 \beta_{1} ) q^{7} + 2 \beta_{3} q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -4 + 4 \beta_{1} ) q^{4} -\beta_{2} q^{5} + ( 1 - 3 \beta_{1} ) q^{7} + 2 \beta_{3} q^{8} + ( -6 + 6 \beta_{1} ) q^{10} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{11} -4 q^{13} + ( 2 \beta_{2} - 3 \beta_{3} ) q^{14} -4 \beta_{1} q^{16} + ( -\beta_{2} + \beta_{3} ) q^{17} + \beta_{1} q^{19} + 4 \beta_{3} q^{20} + 12 q^{22} -\beta_{2} q^{23} + ( -1 + \beta_{1} ) q^{25} + 4 \beta_{2} q^{26} + ( 8 + 4 \beta_{1} ) q^{28} -3 \beta_{3} q^{29} + ( 7 - 7 \beta_{1} ) q^{31} -6 q^{34} + ( 2 \beta_{2} - 3 \beta_{3} ) q^{35} -8 \beta_{1} q^{37} + ( -\beta_{2} + \beta_{3} ) q^{38} -12 \beta_{1} q^{40} + 3 \beta_{3} q^{41} - q^{43} -8 \beta_{2} q^{44} + ( -6 + 6 \beta_{1} ) q^{46} -\beta_{2} q^{47} + ( -8 + 3 \beta_{1} ) q^{49} + \beta_{3} q^{50} + ( 16 - 16 \beta_{1} ) q^{52} + ( -\beta_{2} + \beta_{3} ) q^{53} + 12 q^{55} + ( -6 \beta_{2} + 2 \beta_{3} ) q^{56} + 18 \beta_{1} q^{58} + ( -4 \beta_{2} + 4 \beta_{3} ) q^{59} -5 \beta_{1} q^{61} -7 \beta_{3} q^{62} -8 q^{64} + 4 \beta_{2} q^{65} + ( -2 + 2 \beta_{1} ) q^{67} + 4 \beta_{2} q^{68} + ( 12 + 6 \beta_{1} ) q^{70} + ( 1 - \beta_{1} ) q^{73} + ( 8 \beta_{2} - 8 \beta_{3} ) q^{74} -4 q^{76} + ( 2 \beta_{2} + 4 \beta_{3} ) q^{77} + 4 \beta_{1} q^{79} + ( 4 \beta_{2} - 4 \beta_{3} ) q^{80} -18 \beta_{1} q^{82} + 6 \beta_{3} q^{83} -6 q^{85} + \beta_{2} q^{86} + ( -24 + 24 \beta_{1} ) q^{88} -\beta_{2} q^{89} + ( -4 + 12 \beta_{1} ) q^{91} + 4 \beta_{3} q^{92} + ( -6 + 6 \beta_{1} ) q^{94} + ( -\beta_{2} + \beta_{3} ) q^{95} - q^{97} + ( 5 \beta_{2} + 3 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} - 2q^{7} + O(q^{10}) \) \( 4q - 8q^{4} - 2q^{7} - 12q^{10} - 16q^{13} - 8q^{16} + 2q^{19} + 48q^{22} - 2q^{25} + 40q^{28} + 14q^{31} - 24q^{34} - 16q^{37} - 24q^{40} - 4q^{43} - 12q^{46} - 26q^{49} + 32q^{52} + 48q^{55} + 36q^{58} - 10q^{61} - 32q^{64} - 4q^{67} + 60q^{70} + 2q^{73} - 16q^{76} + 8q^{79} - 36q^{82} - 24q^{85} - 48q^{88} + 8q^{91} - 12q^{94} - 4q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + 4 \beta_{2}\)\()/3\)

Character values

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

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

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
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
−1.22474 2.12132i 0 −2.00000 + 3.46410i −1.22474 2.12132i 0 −0.500000 2.59808i 4.89898 0 −3.00000 + 5.19615i
109.2 1.22474 + 2.12132i 0 −2.00000 + 3.46410i 1.22474 + 2.12132i 0 −0.500000 2.59808i −4.89898 0 −3.00000 + 5.19615i
163.1 −1.22474 + 2.12132i 0 −2.00000 3.46410i −1.22474 + 2.12132i 0 −0.500000 + 2.59808i 4.89898 0 −3.00000 5.19615i
163.2 1.22474 2.12132i 0 −2.00000 3.46410i 1.22474 2.12132i 0 −0.500000 + 2.59808i −4.89898 0 −3.00000 5.19615i
\(n\): 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.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.e.d 4
3.b odd 2 1 inner 189.2.e.d 4
7.c even 3 1 inner 189.2.e.d 4
7.c even 3 1 1323.2.a.v 2
7.d odd 6 1 1323.2.a.u 2
9.c even 3 1 567.2.g.g 4
9.c even 3 1 567.2.h.g 4
9.d odd 6 1 567.2.g.g 4
9.d odd 6 1 567.2.h.g 4
21.g even 6 1 1323.2.a.u 2
21.h odd 6 1 inner 189.2.e.d 4
21.h odd 6 1 1323.2.a.v 2
63.g even 3 1 567.2.h.g 4
63.h even 3 1 567.2.g.g 4
63.j odd 6 1 567.2.g.g 4
63.n odd 6 1 567.2.h.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.d 4 1.a even 1 1 trivial
189.2.e.d 4 3.b odd 2 1 inner
189.2.e.d 4 7.c even 3 1 inner
189.2.e.d 4 21.h odd 6 1 inner
567.2.g.g 4 9.c even 3 1
567.2.g.g 4 9.d odd 6 1
567.2.g.g 4 63.h even 3 1
567.2.g.g 4 63.j odd 6 1
567.2.h.g 4 9.c even 3 1
567.2.h.g 4 9.d odd 6 1
567.2.h.g 4 63.g even 3 1
567.2.h.g 4 63.n odd 6 1
1323.2.a.u 2 7.d odd 6 1
1323.2.a.u 2 21.g even 6 1
1323.2.a.v 2 7.c even 3 1
1323.2.a.v 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 6 T_{2}^{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{2}( 1 - 2 T^{2} + 4 T^{4} ) \)
$3$ 1
$5$ \( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \)
$7$ \( ( 1 + T + 7 T^{2} )^{2} \)
$11$ \( 1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 + 4 T + 13 T^{2} )^{4} \)
$17$ \( 1 - 28 T^{2} + 495 T^{4} - 8092 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 40 T^{2} + 1071 T^{4} - 21160 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 4 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 28 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + T + 43 T^{2} )^{4} \)
$47$ \( 1 - 88 T^{2} + 5535 T^{4} - 194392 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 100 T^{2} + 7191 T^{4} - 280900 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 22 T^{2} - 2997 T^{4} - 76582 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( ( 1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 50 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 172 T^{2} + 21663 T^{4} - 1362412 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 + T + 97 T^{2} )^{4} \)
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