Properties

Label 189.2.e.f
Level 189
Weight 2
Character orbit 189.e
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.e (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: 6.0.309123.1
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.

\(f(q)\) \(=\) \( q\) \( + ( 1 - \beta_{4} + \beta_{5} ) q^{2} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4} \) \( + \beta_{2} q^{5} \) \( + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{7} \) \( + ( -4 - \beta_{1} - 2 \beta_{3} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + ( 1 - \beta_{4} + \beta_{5} ) q^{2} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4} \) \( + \beta_{2} q^{5} \) \( + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{7} \) \( + ( -4 - \beta_{1} - 2 \beta_{3} ) q^{8} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{10} \) \( + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{11} \) \( + ( 1 + 2 \beta_{1} - \beta_{3} ) q^{13} \) \( + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{14} \) \( + ( -5 + \beta_{2} + 5 \beta_{4} - 4 \beta_{5} ) q^{16} \) \( + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{17} \) \( + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{19} \) \( + ( 5 + 2 \beta_{1} ) q^{20} \) \( + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{22} \) \( + ( 2 - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{23} \) \( + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{25} \) \( + ( 6 - \beta_{2} - 6 \beta_{4} ) q^{26} \) \( + ( -6 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{28} \) \( + ( -4 - \beta_{1} + 2 \beta_{3} ) q^{29} \) \( + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{31} \) \( + ( 4 \beta_{1} + \beta_{2} + \beta_{3} + 10 \beta_{4} - 4 \beta_{5} ) q^{32} \) \( + ( 6 + 3 \beta_{1} + 3 \beta_{3} ) q^{34} \) \( + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{35} \) \( + ( 3 - 2 \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{37} \) \( + ( -2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{38} \) \( + ( 9 - 9 \beta_{4} + 3 \beta_{5} ) q^{40} \) \( + ( -2 - 5 \beta_{1} + \beta_{3} ) q^{41} \) \( + ( -4 - 3 \beta_{1} ) q^{43} \) \( -\beta_{2} q^{44} \) \( + ( -3 \beta_{1} + 3 \beta_{5} ) q^{46} \) \( + ( 2 - \beta_{2} - 2 \beta_{4} - 4 \beta_{5} ) q^{47} \) \( + ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{49} \) \( + ( 3 \beta_{1} + \beta_{3} ) q^{50} \) \( + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{52} \) \( + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{53} \) \( + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{55} \) \( + ( -13 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 9 \beta_{4} - 6 \beta_{5} ) q^{56} \) \( + ( -5 - \beta_{2} + 5 \beta_{4} - 2 \beta_{5} ) q^{58} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{59} \) \( + ( -1 + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{61} \) \( + ( 8 + 2 \beta_{1} + \beta_{3} ) q^{62} \) \( + ( 13 + 3 \beta_{1} + 3 \beta_{3} ) q^{64} \) \( + ( 2 + 3 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{65} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{67} \) \( + ( 16 - 2 \beta_{2} - 16 \beta_{4} + 7 \beta_{5} ) q^{68} \) \( + ( 5 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{70} \) \( + ( 3 + 3 \beta_{1} + 3 \beta_{3} ) q^{71} \) \( + ( 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{5} ) q^{73} \) \( + ( -5 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 5 \beta_{5} ) q^{74} \) \( + ( -7 - 3 \beta_{1} ) q^{76} \) \( + ( -1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{77} \) \( + ( -2 + 3 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} ) q^{79} \) \( + ( -5 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} + 5 \beta_{5} ) q^{80} \) \( + ( -16 + 4 \beta_{2} + 16 \beta_{4} - \beta_{5} ) q^{82} \) \( + ( 6 \beta_{1} - 3 \beta_{3} ) q^{83} \) \( + ( -9 - 3 \beta_{1} + 3 \beta_{3} ) q^{85} \) \( + ( -13 + 3 \beta_{2} + 13 \beta_{4} - 4 \beta_{5} ) q^{86} \) \( + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{88} \) \( + ( -3 + 4 \beta_{2} + 3 \beta_{4} ) q^{89} \) \( + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 6 \beta_{4} ) q^{91} \) \( + ( -5 - 2 \beta_{1} - \beta_{3} ) q^{92} \) \( + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} + 3 \beta_{5} ) q^{94} \) \( + ( 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{95} \) \( + ( -6 - 2 \beta_{1} - 2 \beta_{3} ) q^{97} \) \( + ( -10 - \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut q^{10} \) \(\mathstrut +\mathstrut 7q^{11} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 17q^{14} \) \(\mathstrut -\mathstrut 10q^{16} \) \(\mathstrut -\mathstrut 5q^{19} \) \(\mathstrut +\mathstrut 26q^{20} \) \(\mathstrut -\mathstrut 8q^{22} \) \(\mathstrut +\mathstrut 6q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 17q^{26} \) \(\mathstrut -\mathstrut 30q^{28} \) \(\mathstrut -\mathstrut 26q^{29} \) \(\mathstrut +\mathstrut 8q^{31} \) \(\mathstrut +\mathstrut 25q^{32} \) \(\mathstrut +\mathstrut 24q^{34} \) \(\mathstrut -\mathstrut 10q^{35} \) \(\mathstrut +\mathstrut 8q^{37} \) \(\mathstrut -\mathstrut 7q^{38} \) \(\mathstrut +\mathstrut 24q^{40} \) \(\mathstrut -\mathstrut 4q^{41} \) \(\mathstrut -\mathstrut 18q^{43} \) \(\mathstrut -\mathstrut q^{44} \) \(\mathstrut +\mathstrut 3q^{46} \) \(\mathstrut +\mathstrut 9q^{47} \) \(\mathstrut +\mathstrut 12q^{49} \) \(\mathstrut -\mathstrut 8q^{50} \) \(\mathstrut -\mathstrut 9q^{52} \) \(\mathstrut +\mathstrut 24q^{53} \) \(\mathstrut +\mathstrut 8q^{55} \) \(\mathstrut -\mathstrut 48q^{56} \) \(\mathstrut -\mathstrut 14q^{58} \) \(\mathstrut -\mathstrut 15q^{59} \) \(\mathstrut +\mathstrut q^{61} \) \(\mathstrut +\mathstrut 42q^{62} \) \(\mathstrut +\mathstrut 66q^{64} \) \(\mathstrut +\mathstrut 10q^{65} \) \(\mathstrut -\mathstrut 14q^{67} \) \(\mathstrut +\mathstrut 39q^{68} \) \(\mathstrut +\mathstrut 26q^{70} \) \(\mathstrut +\mathstrut 6q^{71} \) \(\mathstrut -\mathstrut 7q^{73} \) \(\mathstrut -\mathstrut 36q^{76} \) \(\mathstrut -\mathstrut q^{77} \) \(\mathstrut -\mathstrut 6q^{79} \) \(\mathstrut -\mathstrut 16q^{80} \) \(\mathstrut -\mathstrut 43q^{82} \) \(\mathstrut -\mathstrut 6q^{83} \) \(\mathstrut -\mathstrut 54q^{85} \) \(\mathstrut -\mathstrut 32q^{86} \) \(\mathstrut -\mathstrut 9q^{88} \) \(\mathstrut -\mathstrut 5q^{89} \) \(\mathstrut -\mathstrut 33q^{91} \) \(\mathstrut -\mathstrut 24q^{92} \) \(\mathstrut +\mathstrut 27q^{94} \) \(\mathstrut +\mathstrut 16q^{95} \) \(\mathstrut -\mathstrut 28q^{97} \) \(\mathstrut -\mathstrut 49q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut +\mathstrut \) \(10\) \(x^{4}\mathstrut -\mathstrut \) \(15\) \(x^{3}\mathstrut +\mathstrut \) \(19\) \(x^{2}\mathstrut -\mathstrut \) \(12\) \(x\mathstrut +\mathstrut \) \(3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 2 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut -\mathstrut \) \(4\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(10\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut -\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(5\) \(\beta_{2}\mathstrut -\mathstrut \) \(23\) \(\beta_{1}\mathstrut +\mathstrut \) \(47\)\()/3\)

Character Values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\)

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 + 2.05195i
0.500000 1.41036i
0.500000 + 0.224437i
0.500000 2.05195i
0.500000 + 1.41036i
0.500000 0.224437i
−0.730252 1.26483i 0 −0.0665372 + 0.115246i −0.296790 0.514055i 0 2.32383 1.26483i −2.72665 0 −0.433463 + 0.750780i
109.2 0.380438 + 0.658939i 0 0.710533 1.23068i 1.59097 + 2.75564i 0 −2.56238 + 0.658939i 2.60301 0 −1.21053 + 2.09671i
109.3 1.34981 + 2.33795i 0 −2.64400 + 4.57954i −0.794182 1.37556i 0 1.23855 + 2.33795i −8.87636 0 2.14400 3.71351i
163.1 −0.730252 + 1.26483i 0 −0.0665372 0.115246i −0.296790 + 0.514055i 0 2.32383 + 1.26483i −2.72665 0 −0.433463 0.750780i
163.2 0.380438 0.658939i 0 0.710533 + 1.23068i 1.59097 2.75564i 0 −2.56238 0.658939i 2.60301 0 −1.21053 2.09671i
163.3 1.34981 2.33795i 0 −2.64400 4.57954i −0.794182 + 1.37556i 0 1.23855 2.33795i −8.87636 0 2.14400 + 3.71351i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.3
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{6} \) \(\mathstrut -\mathstrut 2 T_{2}^{5} \) \(\mathstrut +\mathstrut 7 T_{2}^{4} \) \(\mathstrut +\mathstrut 15 T_{2}^{2} \) \(\mathstrut -\mathstrut 9 T_{2} \) \(\mathstrut +\mathstrut 9 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).