Properties

Label 189.2.i.b
Level 189
Weight 2
Character orbit 189.i
Analytic conductor 1.509
Analytic rank 0
Dimension 10
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.i (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.5091725982\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: 10.0.288778218147.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -\beta_{3} - \beta_{5} ) q^{2} \) \( + ( -1 - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{4} \) \( + ( \beta_{1} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{5} \) \( + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{7} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + ( -\beta_{3} - \beta_{5} ) q^{2} \) \( + ( -1 - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{4} \) \( + ( \beta_{1} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{5} \) \( + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{7} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{8} \) \( + ( -1 - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{10} \) \( + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{11} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{13} \) \( + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{9} ) q^{14} \) \( + ( 1 - \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{16} \) \( + ( -3 - \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{6} + \beta_{7} - \beta_{9} ) q^{17} \) \( + ( 2 \beta_{4} + \beta_{5} + 3 \beta_{7} + \beta_{8} - \beta_{9} ) q^{19} \) \( + ( \beta_{2} + 2 \beta_{3} + \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{20} \) \( + ( -3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{22} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{23} \) \( + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{25} \) \( + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{26} \) \( + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{28} \) \( + ( 1 + \beta_{1} + \beta_{2} + 4 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} - \beta_{6} + 4 \beta_{7} + 3 \beta_{8} - 5 \beta_{9} ) q^{29} \) \( + ( 4 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{8} - \beta_{9} ) q^{31} \) \( + ( -3 - 2 \beta_{1} + \beta_{2} - 6 \beta_{6} ) q^{32} \) \( + ( \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} ) q^{34} \) \( + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{35} \) \( + ( 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{37} \) \( + ( 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} ) q^{38} \) \( + ( 1 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{40} \) \( + ( 5 \beta_{1} - \beta_{2} + 4 \beta_{4} + 4 \beta_{8} - 4 \beta_{9} ) q^{41} \) \( + ( 2 - 6 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{43} \) \( + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{44} \) \( + ( -3 - \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{46} \) \( + ( -3 - 4 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{9} ) q^{47} \) \( + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{49} \) \( + ( -\beta_{1} - \beta_{2} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} ) q^{50} \) \( + ( -2 + \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{6} + 3 \beta_{7} + 3 \beta_{8} ) q^{52} \) \( + ( -1 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{53} \) \( + ( -1 + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{55} \) \( + ( 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} - \beta_{9} ) q^{56} \) \( + ( 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - 7 \beta_{5} + 3 \beta_{7} + 7 \beta_{8} - 3 \beta_{9} ) q^{58} \) \( + ( 3 - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + \beta_{8} + \beta_{9} ) q^{59} \) \( + ( 1 + 4 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{61} \) \( + ( 3 - \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - \beta_{5} - 6 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} ) q^{62} \) \( + ( -1 - \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + \beta_{5} + 6 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{64} \) \( + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - \beta_{8} + \beta_{9} ) q^{65} \) \( + ( 2 + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{67} \) \( + ( 6 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} - 6 \beta_{8} + 3 \beta_{9} ) q^{68} \) \( + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + \beta_{6} - 5 \beta_{7} - 4 \beta_{8} ) q^{70} \) \( + ( -2 \beta_{1} + \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} ) q^{71} \) \( + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} ) q^{73} \) \( + ( 4 + 4 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} ) q^{74} \) \( + ( -3 \beta_{1} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} - 6 \beta_{8} ) q^{76} \) \( + ( -6 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} - 3 \beta_{9} ) q^{77} \) \( + ( -4 - \beta_{2} - 4 \beta_{3} - 10 \beta_{4} - 6 \beta_{5} - 8 \beta_{7} - 3 \beta_{8} + 5 \beta_{9} ) q^{79} \) \( + ( -6 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 6 \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{80} \) \( + ( 2 - \beta_{1} - 3 \beta_{2} - \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{82} \) \( + ( -3 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{83} \) \( + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{85} \) \( + ( -5 - 3 \beta_{1} + 5 \beta_{3} + 3 \beta_{4} + \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{86} \) \( + ( -3 \beta_{1} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - 3 \beta_{8} + 2 \beta_{9} ) q^{88} \) \( + ( 3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} - \beta_{5} - 6 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} ) q^{89} \) \( + ( -4 - 3 \beta_{1} - 2 \beta_{3} - 6 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{91} \) \( + ( -3 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{4} + 3 \beta_{6} + 6 \beta_{9} ) q^{92} \) \( + ( 1 - 4 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} + 8 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{94} \) \( + ( 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} ) q^{95} \) \( + ( 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 8 \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} - 7 \beta_{9} ) q^{97} \) \( + ( 9 + 2 \beta_{1} + \beta_{2} - 5 \beta_{3} - \beta_{4} - 3 \beta_{5} + 6 \beta_{6} - 5 \beta_{7} - 4 \beta_{8} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 15q^{10} \) \(\mathstrut +\mathstrut 12q^{11} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 12q^{14} \) \(\mathstrut +\mathstrut 12q^{16} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut +\mathstrut 3q^{19} \) \(\mathstrut -\mathstrut 3q^{20} \) \(\mathstrut +\mathstrut 5q^{22} \) \(\mathstrut +\mathstrut 15q^{23} \) \(\mathstrut +\mathstrut 7q^{25} \) \(\mathstrut +\mathstrut 3q^{26} \) \(\mathstrut +\mathstrut 2q^{28} \) \(\mathstrut +\mathstrut 15q^{29} \) \(\mathstrut -\mathstrut 3q^{34} \) \(\mathstrut -\mathstrut 15q^{35} \) \(\mathstrut +\mathstrut 6q^{37} \) \(\mathstrut -\mathstrut 18q^{38} \) \(\mathstrut +\mathstrut 15q^{40} \) \(\mathstrut -\mathstrut 9q^{41} \) \(\mathstrut +\mathstrut 3q^{43} \) \(\mathstrut +\mathstrut 24q^{44} \) \(\mathstrut -\mathstrut 13q^{46} \) \(\mathstrut -\mathstrut 30q^{47} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 3q^{50} \) \(\mathstrut -\mathstrut 12q^{52} \) \(\mathstrut -\mathstrut 9q^{53} \) \(\mathstrut +\mathstrut 30q^{56} \) \(\mathstrut +\mathstrut 8q^{58} \) \(\mathstrut +\mathstrut 36q^{59} \) \(\mathstrut +\mathstrut 12q^{62} \) \(\mathstrut +\mathstrut 6q^{64} \) \(\mathstrut +\mathstrut 20q^{67} \) \(\mathstrut +\mathstrut 27q^{68} \) \(\mathstrut +\mathstrut 6q^{70} \) \(\mathstrut +\mathstrut 3q^{73} \) \(\mathstrut +\mathstrut 30q^{74} \) \(\mathstrut -\mathstrut 9q^{76} \) \(\mathstrut -\mathstrut 39q^{77} \) \(\mathstrut -\mathstrut 40q^{79} \) \(\mathstrut -\mathstrut 30q^{80} \) \(\mathstrut +\mathstrut 9q^{82} \) \(\mathstrut -\mathstrut 15q^{83} \) \(\mathstrut +\mathstrut 18q^{85} \) \(\mathstrut -\mathstrut 54q^{86} \) \(\mathstrut -\mathstrut 8q^{88} \) \(\mathstrut +\mathstrut 24q^{89} \) \(\mathstrut -\mathstrut 24q^{91} \) \(\mathstrut -\mathstrut 39q^{92} \) \(\mathstrut -\mathstrut 6q^{97} \) \(\mathstrut +\mathstrut 45q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(x^{9}\mathstrut +\mathstrut \) \(7\) \(x^{8}\mathstrut -\mathstrut \) \(4\) \(x^{7}\mathstrut +\mathstrut \) \(34\) \(x^{6}\mathstrut -\mathstrut \) \(19\) \(x^{5}\mathstrut +\mathstrut \) \(64\) \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut +\mathstrut \) \(64\) \(x^{2}\mathstrut -\mathstrut \) \(21\) \(x\mathstrut +\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -339 \nu^{9} + 1348 \nu^{8} - 4381 \nu^{7} + 7882 \nu^{6} - 19883 \nu^{5} + 36059 \nu^{4} - 75410 \nu^{3} + 44484 \nu^{2} - 15165 \nu + 29709 \)\()/72795\)
\(\beta_{3}\)\(=\)\((\)\( 658 \nu^{9} + 2394 \nu^{8} + 4352 \nu^{7} + 10326 \nu^{6} + 25351 \nu^{5} + 51907 \nu^{4} + 47450 \nu^{3} + 30472 \nu^{2} + 130790 \nu + 98232 \)\()/72795\)
\(\beta_{4}\)\(=\)\((\)\( -4192 \nu^{9} - 796 \nu^{8} - 21678 \nu^{7} - 20279 \nu^{6} - 85319 \nu^{5} - 118353 \nu^{4} - 2560 \nu^{3} - 414508 \nu^{2} + 81750 \nu - 398583 \)\()/218385\)
\(\beta_{5}\)\(=\)\((\)\( 8236 \nu^{9} - 9272 \nu^{8} + 54399 \nu^{7} - 28438 \nu^{6} + 233822 \nu^{5} - 150966 \nu^{4} + 361225 \nu^{3} + 82264 \nu^{2} + 31515 \nu - 336546 \)\()/218385\)
\(\beta_{6}\)\(=\)\((\)\( 3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + 175205 \nu^{3} + 72109 \nu^{2} + 166780 \nu - 54156 \)\()/72795\)
\(\beta_{7}\)\(=\)\((\)\( -840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} - 51488 \nu^{3} - 30640 \nu^{2} - 51320 \nu + 11514 \)\()/14559\)
\(\beta_{8}\)\(=\)\((\)\( 3085 \nu^{9} - 1373 \nu^{8} + 17808 \nu^{7} + 1181 \nu^{6} + 84554 \nu^{5} + 5736 \nu^{4} + 111910 \nu^{3} + 124546 \nu^{2} + 106440 \nu + 17856 \)\()/43677\)
\(\beta_{9}\)\(=\)\((\)\( -18476 \nu^{9} + 18997 \nu^{8} - 128469 \nu^{7} + 65033 \nu^{6} - 601717 \nu^{5} + 295851 \nu^{4} - 1019855 \nu^{3} - 222374 \nu^{2} - 668685 \nu + 178101 \)\()/218385\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\)
\(\nu^{4}\)\(=\)\(-\)\(5\) \(\beta_{9}\mathstrut +\mathstrut \) \(5\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut -\mathstrut \) \(12\)
\(\nu^{5}\)\(=\)\(-\)\(5\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut -\mathstrut \) \(11\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(6\) \(\beta_{9}\mathstrut -\mathstrut \) \(8\) \(\beta_{8}\mathstrut -\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut -\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(22\) \(\beta_{2}\mathstrut +\mathstrut \) \(51\)
\(\nu^{7}\)\(=\)\(\beta_{9}\mathstrut -\mathstrut \) \(31\) \(\beta_{8}\mathstrut -\mathstrut \) \(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(31\) \(\beta_{5}\mathstrut -\mathstrut \) \(30\) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(43\) \(\beta_{1}\)
\(\nu^{8}\)\(=\)\(75\) \(\beta_{9}\mathstrut -\mathstrut \) \(66\) \(\beta_{8}\mathstrut +\mathstrut \) \(38\) \(\beta_{7}\mathstrut +\mathstrut \) \(222\) \(\beta_{6}\mathstrut +\mathstrut \) \(47\) \(\beta_{5}\mathstrut -\mathstrut \) \(37\) \(\beta_{4}\mathstrut +\mathstrut \) \(76\) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut -\mathstrut \) \(112\) \(\beta_{1}\)
\(\nu^{9}\)\(=\)\(95\) \(\beta_{9}\mathstrut +\mathstrut \) \(189\) \(\beta_{8}\mathstrut +\mathstrut \) \(94\) \(\beta_{7}\mathstrut +\mathstrut \) \(37\) \(\beta_{5}\mathstrut +\mathstrut \) \(84\) \(\beta_{4}\mathstrut +\mathstrut \) \(47\) \(\beta_{3}\mathstrut -\mathstrut \) \(194\) \(\beta_{2}\mathstrut -\mathstrut \) \(3\)

Character Values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-\beta_{6}\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
0.827154 1.43267i
−1.04536 + 1.81062i
−0.539982 + 0.935277i
0.187540 0.324828i
1.07065 1.85442i
1.07065 + 1.85442i
0.187540 + 0.324828i
−0.539982 0.935277i
−1.04536 1.81062i
0.827154 + 1.43267i
2.09548i 0 −2.39104 1.04492 1.80985i 0 2.60068 0.486271i 0.819421i 0 −3.79250 2.18960i
143.2 1.51009i 0 −0.280386 0.387938 0.671929i 0 −2.46849 0.952131i 2.59678i 0 −1.01468 0.585823i
143.3 0.293869i 0 1.91364 −1.53014 + 2.65027i 0 −1.41763 + 2.23391i 1.15010i 0 −0.778834 0.449660i
143.4 0.718167i 0 1.48424 0.723774 1.25361i 0 0.182786 2.63943i 2.50226i 0 0.900304 + 0.519791i
143.5 2.59354i 0 −4.72645 −0.626493 + 1.08512i 0 −1.89735 + 1.84393i 7.07116i 0 −2.81429 1.62483i
152.1 2.59354i 0 −4.72645 −0.626493 1.08512i 0 −1.89735 1.84393i 7.07116i 0 −2.81429 + 1.62483i
152.2 0.718167i 0 1.48424 0.723774 + 1.25361i 0 0.182786 + 2.63943i 2.50226i 0 0.900304 0.519791i
152.3 0.293869i 0 1.91364 −1.53014 2.65027i 0 −1.41763 2.23391i 1.15010i 0 −0.778834 + 0.449660i
152.4 1.51009i 0 −0.280386 0.387938 + 0.671929i 0 −2.46849 + 0.952131i 2.59678i 0 −1.01468 + 0.585823i
152.5 2.09548i 0 −2.39104 1.04492 + 1.80985i 0 2.60068 + 0.486271i 0.819421i 0 −3.79250 + 2.18960i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 152.5
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
63.i Even 1 no

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{10} \) \(\mathstrut +\mathstrut 14 T_{2}^{8} \) \(\mathstrut +\mathstrut 63 T_{2}^{6} \) \(\mathstrut +\mathstrut 101 T_{2}^{4} \) \(\mathstrut +\mathstrut 43 T_{2}^{2} \) \(\mathstrut +\mathstrut 3 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).