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 \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.5091725982\)
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: \( 3 \)
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.

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

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

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

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

Hecke kernels

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