Properties

Label 189.2.c.b
Level 189
Weight 2
Character orbit 189.c
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.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.5091725982\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( -3 q^{4} \) \( -\beta_{3} q^{5} \) \( + ( -2 + \beta_{2} ) q^{7} \) \( + \beta_{1} q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( -3 q^{4} \) \( -\beta_{3} q^{5} \) \( + ( -2 + \beta_{2} ) q^{7} \) \( + \beta_{1} q^{8} \) \( + 5 \beta_{2} q^{10} \) \( -\beta_{1} q^{11} \) \( -2 \beta_{2} q^{13} \) \( + ( 2 \beta_{1} + \beta_{3} ) q^{14} \) \(- q^{16}\) \( -3 \beta_{2} q^{19} \) \( + 3 \beta_{3} q^{20} \) \( -5 q^{22} \) \( -\beta_{1} q^{23} \) \( + 10 q^{25} \) \( -2 \beta_{3} q^{26} \) \( + ( 6 - 3 \beta_{2} ) q^{28} \) \( + 2 \beta_{1} q^{29} \) \( + \beta_{2} q^{31} \) \( + 3 \beta_{1} q^{32} \) \( + ( -3 \beta_{1} + 2 \beta_{3} ) q^{35} \) \(- q^{37}\) \( -3 \beta_{3} q^{38} \) \( -5 \beta_{2} q^{40} \) \( -\beta_{3} q^{41} \) \( + 2 q^{43} \) \( + 3 \beta_{1} q^{44} \) \( -5 q^{46} \) \( -2 \beta_{3} q^{47} \) \( + ( 1 - 4 \beta_{2} ) q^{49} \) \( -10 \beta_{1} q^{50} \) \( + 6 \beta_{2} q^{52} \) \( -4 \beta_{1} q^{53} \) \( + 5 \beta_{2} q^{55} \) \( + ( -2 \beta_{1} - \beta_{3} ) q^{56} \) \( + 10 q^{58} \) \( + 2 \beta_{3} q^{59} \) \( -4 \beta_{2} q^{61} \) \( + \beta_{3} q^{62} \) \( + 13 q^{64} \) \( + 6 \beta_{1} q^{65} \) \( -10 q^{67} \) \( + ( -15 - 10 \beta_{2} ) q^{70} \) \( + 5 \beta_{1} q^{71} \) \( -6 \beta_{2} q^{73} \) \( + \beta_{1} q^{74} \) \( + 9 \beta_{2} q^{76} \) \( + ( 2 \beta_{1} + \beta_{3} ) q^{77} \) \( + 2 q^{79} \) \( + \beta_{3} q^{80} \) \( + 5 \beta_{2} q^{82} \) \( -2 \beta_{3} q^{83} \) \( -2 \beta_{1} q^{86} \) \( + 5 q^{88} \) \( + 3 \beta_{3} q^{89} \) \( + ( 6 + 4 \beta_{2} ) q^{91} \) \( + 3 \beta_{1} q^{92} \) \( + 10 \beta_{2} q^{94} \) \( + 9 \beta_{1} q^{95} \) \( + 8 \beta_{2} q^{97} \) \( + ( -\beta_{1} - 4 \beta_{3} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut -\mathstrut 4q^{16} \) \(\mathstrut -\mathstrut 20q^{22} \) \(\mathstrut +\mathstrut 40q^{25} \) \(\mathstrut +\mathstrut 24q^{28} \) \(\mathstrut -\mathstrut 4q^{37} \) \(\mathstrut +\mathstrut 8q^{43} \) \(\mathstrut -\mathstrut 20q^{46} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 40q^{58} \) \(\mathstrut +\mathstrut 52q^{64} \) \(\mathstrut -\mathstrut 40q^{67} \) \(\mathstrut -\mathstrut 60q^{70} \) \(\mathstrut +\mathstrut 8q^{79} \) \(\mathstrut +\mathstrut 20q^{88} \) \(\mathstrut +\mathstrut 24q^{91} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/5\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{2} - 5 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 10 \nu \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\)\()/2\)
\(\nu^{3}\)\(=\)\(5\) \(\beta_{1}\)

Character Values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 1.11803i
−1.93649 1.11803i
2.23607i 0 −3.00000 −3.87298 0 −2.00000 + 1.73205i 2.23607i 0 8.66025i
188.2 2.23607i 0 −3.00000 3.87298 0 −2.00000 1.73205i 2.23607i 0 8.66025i
188.3 2.23607i 0 −3.00000 −3.87298 0 −2.00000 1.73205i 2.23607i 0 8.66025i
188.4 2.23607i 0 −3.00000 3.87298 0 −2.00000 + 1.73205i 2.23607i 0 8.66025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
7.b Odd 1 yes
21.c Even 1 yes

Hecke kernels

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