Properties

Label 189.2.o.a
Level 189
Weight 2
Character orbit 189.o
Analytic conductor 1.509
Analytic rank 0
Dimension 12
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.o (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.5091725982\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{2} \) \( + ( -\beta_{3} - \beta_{5} - \beta_{8} ) q^{4} \) \( + \beta_{10} q^{5} \) \( + ( \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{7} \) \( + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{8} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{2} \) \( + ( -\beta_{3} - \beta_{5} - \beta_{8} ) q^{4} \) \( + \beta_{10} q^{5} \) \( + ( \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{7} \) \( + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{8} ) q^{8} \) \( + ( \beta_{2} + \beta_{6} - \beta_{9} - \beta_{11} ) q^{10} \) \( + ( \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} ) q^{11} \) \( + ( -\beta_{2} - 2 \beta_{7} - \beta_{10} ) q^{13} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} + 2 \beta_{9} - \beta_{10} ) q^{14} \) \( + ( -\beta_{3} + \beta_{4} + \beta_{8} ) q^{16} \) \( + ( \beta_{6} + \beta_{11} ) q^{17} \) \( + ( \beta_{2} - \beta_{6} + \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{19} \) \( + ( 2 \beta_{2} - \beta_{9} ) q^{20} \) \( + ( -1 + 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{8} ) q^{22} \) \( + ( -1 - \beta_{1} - \beta_{5} ) q^{23} \) \( + ( \beta_{1} + \beta_{3} ) q^{25} \) \( + ( -2 \beta_{2} - \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{11} ) q^{26} \) \( + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{28} \) \( + ( -4 + \beta_{1} - \beta_{3} + \beta_{5} ) q^{29} \) \( + ( 2 \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{31} \) \( + ( 1 + \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + 2 \beta_{8} ) q^{32} \) \( + ( \beta_{7} - \beta_{10} ) q^{34} \) \( + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{35} \) \( + ( -1 + \beta_{1} - \beta_{4} ) q^{37} \) \( + ( 4 \beta_{2} + 2 \beta_{6} - 2 \beta_{9} - \beta_{11} ) q^{38} \) \( + ( \beta_{2} - 2 \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{40} \) \( + ( -\beta_{2} + 2 \beta_{9} - 2 \beta_{10} ) q^{41} \) \( + ( -2 \beta_{1} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{8} ) q^{43} \) \( + ( -2 + 3 \beta_{3} + 7 \beta_{5} ) q^{44} \) \( + ( -2 - 3 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{46} \) \( + ( -2 \beta_{2} - 2 \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{47} \) \( + ( -1 + 2 \beta_{1} - 4 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{49} \) \( + ( 2 + \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{8} ) q^{50} \) \( + ( -\beta_{7} - 5 \beta_{9} + \beta_{10} - \beta_{11} ) q^{52} \) \( + ( -1 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 4 \beta_{8} ) q^{53} \) \( + ( -2 \beta_{2} + \beta_{6} + 2 \beta_{9} - \beta_{11} ) q^{55} \) \( + ( -6 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{56} \) \( + ( 2 - 2 \beta_{1} + 3 \beta_{3} + \beta_{5} - \beta_{8} ) q^{58} \) \( + ( 2 \beta_{2} - \beta_{6} - 4 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{59} \) \( + ( \beta_{9} + \beta_{11} ) q^{61} \) \( + ( \beta_{2} + \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{11} ) q^{62} \) \( + ( -2 + 7 \beta_{1} - 4 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{64} \) \( + ( 10 - 2 \beta_{1} + \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{8} ) q^{65} \) \( + ( 1 + 2 \beta_{1} - \beta_{3} - 2 \beta_{5} + 3 \beta_{8} ) q^{67} \) \( + ( -\beta_{2} + \beta_{6} + 2 \beta_{9} - 2 \beta_{11} ) q^{68} \) \( + ( 2 \beta_{1} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{70} \) \( + ( 2 + 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 7 \beta_{5} + 4 \beta_{8} ) q^{71} \) \( + ( 3 \beta_{2} + 3 \beta_{6} - \beta_{7} - 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{73} \) \( + ( 2 - \beta_{1} + \beta_{3} ) q^{74} \) \( + ( 4 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{76} \) \( + ( 1 + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{77} \) \( + ( -\beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{8} ) q^{79} \) \( + ( \beta_{6} - \beta_{7} + \beta_{11} ) q^{80} \) \( + ( -5 \beta_{2} - 2 \beta_{6} - \beta_{7} + 5 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{82} \) \( + ( -4 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{83} \) \( + ( 3 - 6 \beta_{1} + 6 \beta_{3} + 3 \beta_{5} - 6 \beta_{8} ) q^{85} \) \( + ( -5 - 3 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{8} ) q^{86} \) \( + ( \beta_{1} + 2 \beta_{3} - \beta_{4} + 6 \beta_{5} - \beta_{8} ) q^{88} \) \( + ( 3 \beta_{2} + 3 \beta_{9} ) q^{89} \) \( + ( -5 \beta_{1} + 4 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{10} + 2 \beta_{11} ) q^{91} \) \( + ( -2 - 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + 2 \beta_{8} ) q^{92} \) \( + ( -2 \beta_{2} + \beta_{6} - 4 \beta_{7} - 2 \beta_{10} ) q^{94} \) \( + ( -6 + 3 \beta_{1} + 3 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} - 3 \beta_{8} ) q^{95} \) \( + ( -\beta_{7} + \beta_{10} - \beta_{11} ) q^{97} \) \( + ( 4 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - 9 \beta_{5} - 2 \beta_{7} - 4 \beta_{8} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 12q^{14} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 10q^{22} \) \(\mathstrut -\mathstrut 24q^{23} \) \(\mathstrut -\mathstrut 8q^{28} \) \(\mathstrut -\mathstrut 30q^{29} \) \(\mathstrut +\mathstrut 12q^{32} \) \(\mathstrut -\mathstrut 4q^{37} \) \(\mathstrut -\mathstrut 10q^{43} \) \(\mathstrut -\mathstrut 40q^{46} \) \(\mathstrut +\mathstrut 6q^{49} \) \(\mathstrut +\mathstrut 36q^{50} \) \(\mathstrut -\mathstrut 42q^{56} \) \(\mathstrut +\mathstrut 2q^{58} \) \(\mathstrut +\mathstrut 16q^{64} \) \(\mathstrut +\mathstrut 78q^{65} \) \(\mathstrut +\mathstrut 12q^{67} \) \(\mathstrut +\mathstrut 18q^{70} \) \(\mathstrut +\mathstrut 12q^{74} \) \(\mathstrut +\mathstrut 24q^{77} \) \(\mathstrut -\mathstrut 6q^{79} \) \(\mathstrut -\mathstrut 6q^{85} \) \(\mathstrut -\mathstrut 96q^{86} \) \(\mathstrut +\mathstrut 34q^{88} \) \(\mathstrut -\mathstrut 24q^{91} \) \(\mathstrut -\mathstrut 30q^{92} \) \(\mathstrut -\mathstrut 72q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(7\) \(x^{10}\mathstrut +\mathstrut \) \(37\) \(x^{8}\mathstrut -\mathstrut \) \(78\) \(x^{6}\mathstrut +\mathstrut \) \(123\) \(x^{4}\mathstrut -\mathstrut \) \(36\) \(x^{2}\mathstrut +\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -28 \nu^{10} + 148 \nu^{8} + 446 \nu^{6} - 3807 \nu^{4} + 17052 \nu^{2} + 4446 \)\()/12897\)
\(\beta_{2}\)\(=\)\((\)\( -49 \nu^{11} + 259 \nu^{9} - 1369 \nu^{7} + 861 \nu^{5} - 252 \nu^{3} - 15864 \nu \)\()/4299\)
\(\beta_{3}\)\(=\)\((\)\( -164 \nu^{10} + 1481 \nu^{8} - 7214 \nu^{6} + 17007 \nu^{4} - 16197 \nu^{2} - 3438 \)\()/12897\)
\(\beta_{4}\)\(=\)\((\)\( -175 \nu^{10} + 925 \nu^{8} - 3661 \nu^{6} - 1224 \nu^{4} + 16296 \nu^{2} - 30249 \)\()/12897\)
\(\beta_{5}\)\(=\)\((\)\( 148 \nu^{10} - 987 \nu^{8} + 5217 \nu^{6} - 10175 \nu^{4} + 17343 \nu^{2} - 777 \)\()/4299\)
\(\beta_{6}\)\(=\)\((\)\( 70 \nu^{11} - 370 \nu^{9} + 1751 \nu^{7} - 1230 \nu^{5} + 360 \nu^{3} + 8947 \nu \)\()/1433\)
\(\beta_{7}\)\(=\)\((\)\( -730 \nu^{11} + 5701 \nu^{9} - 30748 \nu^{7} + 76698 \nu^{5} - 122898 \nu^{3} + 66168 \nu \)\()/12897\)
\(\beta_{8}\)\(=\)\((\)\( -877 \nu^{10} + 6478 \nu^{8} - 34855 \nu^{6} + 79281 \nu^{4} - 123654 \nu^{2} + 31473 \)\()/12897\)
\(\beta_{9}\)\(=\)\((\)\( 543 \nu^{11} - 3689 \nu^{9} + 19499 \nu^{7} - 39839 \nu^{5} + 64821 \nu^{3} - 18972 \nu \)\()/4299\)
\(\beta_{10}\)\(=\)\((\)\( 2237 \nu^{11} - 15509 \nu^{9} + 81362 \nu^{7} - 167049 \nu^{5} + 249792 \nu^{3} - 42912 \nu \)\()/12897\)
\(\beta_{11}\)\(=\)\((\)\( -890 \nu^{11} + 6342 \nu^{9} - 33522 \nu^{7} + 71935 \nu^{5} - 111438 \nu^{3} + 32616 \nu \)\()/4299\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(4\) \(\beta_{11}\mathstrut -\mathstrut \) \(8\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(4\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{2}\)\()/3\)
\(\nu^{4}\)\(=\)\(4\) \(\beta_{8}\mathstrut +\mathstrut \) \(6\) \(\beta_{5}\mathstrut -\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut -\mathstrut \) \(12\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(19\) \(\beta_{11}\mathstrut -\mathstrut \) \(16\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(16\) \(\beta_{7}\)\()/3\)
\(\nu^{6}\)\(=\)\(-\)\(7\) \(\beta_{5}\mathstrut -\mathstrut \) \(16\) \(\beta_{4}\mathstrut -\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(30\) \(\beta_{1}\mathstrut -\mathstrut \) \(51\)
\(\nu^{7}\)\(=\)\((\)\(67\) \(\beta_{10}\mathstrut +\mathstrut \) \(134\) \(\beta_{7}\mathstrut -\mathstrut \) \(88\) \(\beta_{6}\mathstrut -\mathstrut \) \(23\) \(\beta_{2}\)\()/3\)
\(\nu^{8}\)\(=\)\(-\)\(67\) \(\beta_{8}\mathstrut -\mathstrut \) \(118\) \(\beta_{5}\mathstrut -\mathstrut \) \(67\) \(\beta_{4}\mathstrut +\mathstrut \) \(104\) \(\beta_{3}\mathstrut +\mathstrut \) \(37\) \(\beta_{1}\)
\(\nu^{9}\)\(=\)\((\)\(400\) \(\beta_{11}\mathstrut +\mathstrut \) \(578\) \(\beta_{10}\mathstrut +\mathstrut \) \(134\) \(\beta_{9}\mathstrut +\mathstrut \) \(289\) \(\beta_{7}\mathstrut -\mathstrut \) \(400\) \(\beta_{6}\mathstrut -\mathstrut \) \(134\) \(\beta_{2}\)\()/3\)
\(\nu^{10}\)\(=\)\(-\)\(289\) \(\beta_{8}\mathstrut -\mathstrut \) \(333\) \(\beta_{5}\mathstrut +\mathstrut \) \(645\) \(\beta_{3}\mathstrut -\mathstrut \) \(467\) \(\beta_{1}\mathstrut +\mathstrut \) \(978\)
\(\nu^{11}\)\(=\)\((\)\(1801\) \(\beta_{11}\mathstrut +\mathstrut \) \(1267\) \(\beta_{10}\mathstrut +\mathstrut \) \(668\) \(\beta_{9}\mathstrut -\mathstrut \) \(1267\) \(\beta_{7}\)\()/3\)

Character Values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(1 - \beta_{5}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
62.1
−1.82904 + 1.05600i
1.82904 1.05600i
0.474636 0.274031i
−0.474636 + 0.274031i
−1.29589 + 0.748185i
1.29589 0.748185i
−1.82904 1.05600i
1.82904 + 1.05600i
0.474636 + 0.274031i
−0.474636 0.274031i
−1.29589 0.748185i
1.29589 + 0.748185i
−1.02704 0.592963i 0 −0.296790 0.514055i −1.41899 2.45776i 0 −2.07253 + 1.64457i 3.07579i 0 3.36562i
62.2 −1.02704 0.592963i 0 −0.296790 0.514055i 1.41899 + 2.45776i 0 −0.387972 + 2.61715i 3.07579i 0 3.36562i
62.3 0.555632 + 0.320794i 0 −0.794182 1.37556i −1.10552 1.91482i 0 −0.906161 2.48573i 2.30225i 0 1.41858i
62.4 0.555632 + 0.320794i 0 −0.794182 1.37556i 1.10552 + 1.91482i 0 2.60579 0.458109i 2.30225i 0 1.41858i
62.5 1.97141 + 1.13819i 0 1.59097 + 2.75564i −0.717144 1.24213i 0 2.16235 + 1.52455i 2.69056i 0 3.26499i
62.6 1.97141 + 1.13819i 0 1.59097 + 2.75564i 0.717144 + 1.24213i 0 −2.40147 1.11037i 2.69056i 0 3.26499i
125.1 −1.02704 + 0.592963i 0 −0.296790 + 0.514055i −1.41899 + 2.45776i 0 −2.07253 1.64457i 3.07579i 0 3.36562i
125.2 −1.02704 + 0.592963i 0 −0.296790 + 0.514055i 1.41899 2.45776i 0 −0.387972 2.61715i 3.07579i 0 3.36562i
125.3 0.555632 0.320794i 0 −0.794182 + 1.37556i −1.10552 + 1.91482i 0 −0.906161 + 2.48573i 2.30225i 0 1.41858i
125.4 0.555632 0.320794i 0 −0.794182 + 1.37556i 1.10552 1.91482i 0 2.60579 + 0.458109i 2.30225i 0 1.41858i
125.5 1.97141 1.13819i 0 1.59097 2.75564i −0.717144 + 1.24213i 0 2.16235 1.52455i 2.69056i 0 3.26499i
125.6 1.97141 1.13819i 0 1.59097 2.75564i 0.717144 1.24213i 0 −2.40147 + 1.11037i 2.69056i 0 3.26499i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.b Odd 1 yes
9.d Odd 1 yes
63.o Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(189, [\chi])\).