Properties

Label 189.2.f.b
Level 189
Weight 2
Character orbit 189.f
Analytic conductor 1.509
Analytic rank 0
Dimension 6
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.f (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.5091725982\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{2} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{4} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{5} \) \( -\beta_{1} q^{7} \) \( + ( -2 - \beta_{3} + 2 \beta_{4} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{2} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{4} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{5} \) \( -\beta_{1} q^{7} \) \( + ( -2 - \beta_{3} + 2 \beta_{4} ) q^{8} \) \( + ( -\beta_{3} + \beta_{4} ) q^{10} \) \( + ( 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} \) \( + ( 1 - \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} ) q^{13} \) \( + ( 1 - \beta_{1} - \beta_{5} ) q^{14} \) \( + ( -\beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{16} \) \( + ( -2 + \beta_{3} ) q^{17} \) \( + ( -1 - \beta_{3} - 2 \beta_{4} ) q^{19} \) \( + ( -2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{20} \) \( + ( -3 + 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} ) q^{22} \) \( + ( 4 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{23} \) \( + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{25} \) \( + ( 1 - \beta_{3} - 7 \beta_{4} ) q^{26} \) \( + ( 1 - \beta_{3} - 2 \beta_{4} ) q^{28} \) \( + ( 3 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} ) q^{29} \) \( + ( 1 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{5} ) q^{31} \) \( -3 \beta_{5} q^{32} \) \( + ( -3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{34} \) \( + ( -1 - \beta_{4} ) q^{35} \) \( + ( -1 + 6 \beta_{3} + 3 \beta_{4} ) q^{37} \) \( + ( 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{38} \) \( + ( 3 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{40} \) \( + ( -\beta_{2} + \beta_{3} + \beta_{5} ) q^{41} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{43} \) \( + ( -5 + 3 \beta_{3} + 7 \beta_{4} ) q^{44} \) \( + ( -4 \beta_{3} - 5 \beta_{4} ) q^{46} \) \( + ( \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{47} \) \( + ( -1 + \beta_{1} ) q^{49} \) \( + ( -2 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 5 \beta_{5} ) q^{50} \) \( + ( 7 \beta_{1} + 5 \beta_{2} - 10 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} ) q^{52} \) \( + ( -2 + \beta_{3} + 3 \beta_{4} ) q^{53} \) \( + ( \beta_{3} + 2 \beta_{4} ) q^{55} \) \( + ( 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{56} \) \( + ( 3 - 3 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} - 6 \beta_{5} ) q^{58} \) \( + ( -1 + \beta_{1} - 5 \beta_{5} ) q^{59} \) \( + ( -2 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{61} \) \( + ( 10 - \beta_{3} - \beta_{4} ) q^{62} \) \( + ( 4 + 3 \beta_{4} ) q^{64} \) \( + ( 5 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{65} \) \( + ( 4 - 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{67} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} ) q^{68} \) \( + ( 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{70} \) \( + ( -3 + 3 \beta_{3} - 3 \beta_{4} ) q^{71} \) \( + ( -7 + 5 \beta_{3} + 4 \beta_{4} ) q^{73} \) \( + ( -10 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{74} \) \( + ( -5 + 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{76} \) \( + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{77} \) \( + ( 7 \beta_{1} - 3 \beta_{2} ) q^{79} \) \( + ( 5 - 3 \beta_{3} - \beta_{4} ) q^{80} \) \( -3 q^{82} \) \( + ( -6 \beta_{1} - \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{83} \) \( + ( -3 + 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} ) q^{85} \) \( + ( 2 - 2 \beta_{1} + \beta_{5} ) q^{86} \) \( + ( -9 \beta_{1} - 7 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} ) q^{88} \) \( + ( -4 - 4 \beta_{3} + 3 \beta_{4} ) q^{89} \) \( + ( -1 - 2 \beta_{3} + 2 \beta_{4} ) q^{91} \) \( + ( \beta_{1} + 2 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} ) q^{92} \) \( + ( 6 - 6 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{94} \) \( + ( -4 + 4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{95} \) \( + ( \beta_{1} - 8 \beta_{2} + 7 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} ) q^{97} \) \( + ( -1 + \beta_{3} + \beta_{4} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 6q^{11} \) \(\mathstrut +\mathstrut 3q^{13} \) \(\mathstrut +\mathstrut 3q^{14} \) \(\mathstrut -\mathstrut 3q^{16} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut 6q^{20} \) \(\mathstrut -\mathstrut 9q^{22} \) \(\mathstrut +\mathstrut 12q^{23} \) \(\mathstrut +\mathstrut 6q^{25} \) \(\mathstrut +\mathstrut 6q^{26} \) \(\mathstrut +\mathstrut 6q^{28} \) \(\mathstrut +\mathstrut 9q^{29} \) \(\mathstrut +\mathstrut 3q^{31} \) \(\mathstrut -\mathstrut 9q^{34} \) \(\mathstrut -\mathstrut 6q^{35} \) \(\mathstrut -\mathstrut 6q^{37} \) \(\mathstrut +\mathstrut 6q^{38} \) \(\mathstrut +\mathstrut 9q^{40} \) \(\mathstrut +\mathstrut 3q^{43} \) \(\mathstrut -\mathstrut 30q^{44} \) \(\mathstrut +\mathstrut 3q^{47} \) \(\mathstrut -\mathstrut 3q^{49} \) \(\mathstrut -\mathstrut 6q^{50} \) \(\mathstrut +\mathstrut 21q^{52} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 6q^{56} \) \(\mathstrut +\mathstrut 9q^{58} \) \(\mathstrut -\mathstrut 3q^{59} \) \(\mathstrut -\mathstrut 6q^{61} \) \(\mathstrut +\mathstrut 60q^{62} \) \(\mathstrut +\mathstrut 24q^{64} \) \(\mathstrut +\mathstrut 15q^{65} \) \(\mathstrut +\mathstrut 12q^{67} \) \(\mathstrut +\mathstrut 6q^{68} \) \(\mathstrut -\mathstrut 18q^{71} \) \(\mathstrut -\mathstrut 42q^{73} \) \(\mathstrut -\mathstrut 30q^{74} \) \(\mathstrut -\mathstrut 15q^{76} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut +\mathstrut 21q^{79} \) \(\mathstrut +\mathstrut 30q^{80} \) \(\mathstrut -\mathstrut 18q^{82} \) \(\mathstrut -\mathstrut 18q^{83} \) \(\mathstrut -\mathstrut 9q^{85} \) \(\mathstrut +\mathstrut 6q^{86} \) \(\mathstrut -\mathstrut 27q^{88} \) \(\mathstrut -\mathstrut 24q^{89} \) \(\mathstrut -\mathstrut 6q^{91} \) \(\mathstrut +\mathstrut 3q^{92} \) \(\mathstrut +\mathstrut 18q^{94} \) \(\mathstrut -\mathstrut 12q^{95} \) \(\mathstrut +\mathstrut 3q^{97} \) \(\mathstrut -\mathstrut 6q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring:

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \zeta_{18}^{3} \)
\(\beta_{2}\)\(=\)\( \zeta_{18}^{5} + \zeta_{18} \)
\(\beta_{3}\)\(=\)\( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \)
\(\beta_{4}\)\(=\)\( -\zeta_{18}^{5} + \zeta_{18}^{4} \)
\(\beta_{5}\)\(=\)\( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \)
\(1\)\(=\)\(\beta_0\)
\(\zeta_{18}\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\)\()/3\)
\(\zeta_{18}^{2}\)\(=\)\((\)\(-\)\(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\)\()/3\)
\(\zeta_{18}^{3}\)\(=\)\(\beta_{1}\)
\(\zeta_{18}^{4}\)\(=\)\((\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/3\)
\(\zeta_{18}^{5}\)\(=\)\((\)\(-\)\(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/3\)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
0.939693 + 0.342020i
−0.173648 0.984808i
−0.766044 + 0.642788i
0.939693 0.342020i
−0.173648 + 0.984808i
−0.766044 0.642788i
−0.439693 0.761570i 0 0.613341 1.06234i 0.673648 1.16679i 0 −0.500000 0.866025i −2.83750 0 −1.18479
64.2 0.673648 + 1.16679i 0 0.0923963 0.160035i 1.26604 2.19285i 0 −0.500000 0.866025i 2.94356 0 3.41147
64.3 1.26604 + 2.19285i 0 −2.20574 + 3.82045i −0.439693 + 0.761570i 0 −0.500000 0.866025i −6.10607 0 −2.22668
127.1 −0.439693 + 0.761570i 0 0.613341 + 1.06234i 0.673648 + 1.16679i 0 −0.500000 + 0.866025i −2.83750 0 −1.18479
127.2 0.673648 1.16679i 0 0.0923963 + 0.160035i 1.26604 + 2.19285i 0 −0.500000 + 0.866025i 2.94356 0 3.41147
127.3 1.26604 2.19285i 0 −2.20574 3.82045i −0.439693 0.761570i 0 −0.500000 + 0.866025i −6.10607 0 −2.22668
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.3
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{6} \) \(\mathstrut -\mathstrut 3 T_{2}^{5} \) \(\mathstrut +\mathstrut 9 T_{2}^{4} \) \(\mathstrut -\mathstrut 6 T_{2}^{3} \) \(\mathstrut +\mathstrut 9 T_{2}^{2} \) \(\mathstrut +\mathstrut 9 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).