Properties

Label 189.2.h.b
Level 189
Weight 2
Character orbit 189.h
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.h (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.5091725982\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.991381711347.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( \beta_{1} - \beta_{5} ) q^{2} \) \( + ( 1 + \beta_{3} ) q^{4} \) \( + ( \beta_{6} - \beta_{9} ) q^{5} \) \( + ( -1 + \beta_{1} + \beta_{4} - \beta_{5} ) q^{7} \) \( + ( 1 - \beta_{4} - \beta_{8} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + ( \beta_{1} - \beta_{5} ) q^{2} \) \( + ( 1 + \beta_{3} ) q^{4} \) \( + ( \beta_{6} - \beta_{9} ) q^{5} \) \( + ( -1 + \beta_{1} + \beta_{4} - \beta_{5} ) q^{7} \) \( + ( 1 - \beta_{4} - \beta_{8} ) q^{8} \) \( + ( 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{10} \) \( + ( -1 + \beta_{4} - \beta_{6} + \beta_{7} ) q^{11} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{6} ) q^{13} \) \( + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{14} \) \( + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{16} \) \( + ( -\beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{17} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{19} \) \( + ( -\beta_{3} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{20} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} ) q^{22} \) \( + ( -2 \beta_{3} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{23} \) \( + ( -\beta_{1} + 2 \beta_{2} + \beta_{7} ) q^{25} \) \( + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{26} \) \( + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{28} \) \( + ( 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{29} \) \( + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{31} \) \( + ( 1 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{8} + 2 \beta_{9} ) q^{32} \) \( + ( \beta_{3} + 4 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{34} \) \( + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{35} \) \( + ( 2 \beta_{1} - 2 \beta_{4} ) q^{37} \) \( + ( -5 + 2 \beta_{1} - \beta_{4} - 5 \beta_{6} - 3 \beta_{7} ) q^{38} \) \( + ( -2 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{40} \) \( + ( -\beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{7} ) q^{41} \) \( + ( \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{43} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{6} - \beta_{7} ) q^{44} \) \( + ( 2 \beta_{3} + 5 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{46} \) \( + ( 4 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} ) q^{47} \) \( + ( 2 \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{49} \) \( + ( -6 + 3 \beta_{1} + 2 \beta_{2} + \beta_{4} - 6 \beta_{6} - \beta_{7} ) q^{50} \) \( + ( -1 - 2 \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{52} \) \( + ( \beta_{5} - 5 \beta_{6} + 2 \beta_{9} ) q^{53} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{55} \) \( + ( -4 - \beta_{1} + \beta_{3} + \beta_{4} - 5 \beta_{6} + \beta_{9} ) q^{56} \) \( + ( -2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{58} \) \( + ( 6 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{9} ) q^{59} \) \( + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{61} \) \( + ( 3 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} + 2 \beta_{9} ) q^{62} \) \( + ( -6 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{9} ) q^{64} \) \( + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{65} \) \( + ( -1 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + \beta_{8} + \beta_{9} ) q^{67} \) \( + ( -3 \beta_{3} - 2 \beta_{5} + 7 \beta_{6} + 3 \beta_{7} - \beta_{9} ) q^{68} \) \( + ( 5 + 2 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{70} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{8} - 2 \beta_{9} ) q^{71} \) \( + ( -\beta_{3} - 4 \beta_{6} + \beta_{7} - 3 \beta_{8} ) q^{73} \) \( + ( 10 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 10 \beta_{6} + 4 \beta_{7} ) q^{74} \) \( + ( 3 - 5 \beta_{1} + \beta_{2} + 3 \beta_{6} + \beta_{7} ) q^{76} \) \( + ( \beta_{1} + \beta_{2} - \beta_{4} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{77} \) \( + ( 3 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{5} + \beta_{9} ) q^{79} \) \( + ( 2 \beta_{5} + 3 \beta_{6} - \beta_{8} - \beta_{9} ) q^{80} \) \( + ( 2 - 4 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{82} \) \( + ( -2 \beta_{3} + \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} ) q^{83} \) \( + ( -2 + \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{85} \) \( + ( \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} - 4 \beta_{9} ) q^{86} \) \( + ( -4 - \beta_{2} + 2 \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{88} \) \( + ( -7 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - 7 \beta_{6} ) q^{89} \) \( + ( -\beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{91} \) \( + ( -2 \beta_{5} + 5 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{92} \) \( + ( -3 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{4} - 4 \beta_{5} + 4 \beta_{8} - 2 \beta_{9} ) q^{94} \) \( + ( -2 + 2 \beta_{3} - \beta_{4} - \beta_{8} ) q^{95} \) \( + ( -4 \beta_{3} + \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{97} \) \( + ( -5 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} + 5 \beta_{6} + 3 \beta_{7} - \beta_{9} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut -\mathstrut 7q^{10} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut -\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut 20q^{14} \) \(\mathstrut -\mathstrut 4q^{16} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut +\mathstrut q^{19} \) \(\mathstrut -\mathstrut 5q^{20} \) \(\mathstrut -\mathstrut q^{22} \) \(\mathstrut -\mathstrut 3q^{23} \) \(\mathstrut -\mathstrut q^{25} \) \(\mathstrut -\mathstrut 11q^{26} \) \(\mathstrut -\mathstrut 2q^{28} \) \(\mathstrut -\mathstrut 7q^{29} \) \(\mathstrut +\mathstrut 6q^{31} \) \(\mathstrut -\mathstrut 4q^{32} \) \(\mathstrut +\mathstrut 3q^{34} \) \(\mathstrut -\mathstrut 5q^{35} \) \(\mathstrut -\mathstrut 20q^{38} \) \(\mathstrut -\mathstrut 3q^{40} \) \(\mathstrut -\mathstrut 5q^{41} \) \(\mathstrut -\mathstrut 7q^{43} \) \(\mathstrut +\mathstrut 10q^{44} \) \(\mathstrut +\mathstrut 3q^{46} \) \(\mathstrut +\mathstrut 54q^{47} \) \(\mathstrut -\mathstrut 8q^{49} \) \(\mathstrut -\mathstrut 19q^{50} \) \(\mathstrut -\mathstrut 10q^{52} \) \(\mathstrut +\mathstrut 21q^{53} \) \(\mathstrut +\mathstrut 4q^{55} \) \(\mathstrut -\mathstrut 18q^{56} \) \(\mathstrut -\mathstrut 10q^{58} \) \(\mathstrut +\mathstrut 60q^{59} \) \(\mathstrut +\mathstrut 28q^{61} \) \(\mathstrut +\mathstrut 12q^{62} \) \(\mathstrut -\mathstrut 50q^{64} \) \(\mathstrut -\mathstrut 22q^{65} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut -\mathstrut 27q^{68} \) \(\mathstrut +\mathstrut 40q^{70} \) \(\mathstrut +\mathstrut 6q^{71} \) \(\mathstrut +\mathstrut 15q^{73} \) \(\mathstrut +\mathstrut 36q^{74} \) \(\mathstrut +\mathstrut 5q^{76} \) \(\mathstrut -\mathstrut 11q^{77} \) \(\mathstrut +\mathstrut 8q^{79} \) \(\mathstrut -\mathstrut 20q^{80} \) \(\mathstrut -\mathstrut 5q^{82} \) \(\mathstrut -\mathstrut 9q^{83} \) \(\mathstrut -\mathstrut 6q^{85} \) \(\mathstrut +\mathstrut 8q^{86} \) \(\mathstrut -\mathstrut 18q^{88} \) \(\mathstrut -\mathstrut 28q^{89} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 27q^{92} \) \(\mathstrut +\mathstrut 6q^{94} \) \(\mathstrut -\mathstrut 28q^{95} \) \(\mathstrut -\mathstrut 12q^{97} \) \(\mathstrut -\mathstrut 59q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(2\) \(x^{9}\mathstrut +\mathstrut \) \(9\) \(x^{8}\mathstrut -\mathstrut \) \(8\) \(x^{7}\mathstrut +\mathstrut \) \(40\) \(x^{6}\mathstrut -\mathstrut \) \(36\) \(x^{5}\mathstrut +\mathstrut \) \(90\) \(x^{4}\mathstrut -\mathstrut \) \(3\) \(x^{3}\mathstrut +\mathstrut \) \(36\) \(x^{2}\mathstrut -\mathstrut \) \(9\) \(x\mathstrut +\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{9} - 9 \nu^{8} - 3 \nu^{7} - 61 \nu^{6} - 72 \nu^{5} - 282 \nu^{4} - 204 \nu^{3} - 387 \nu^{2} - 873 \nu - 117 \)\()/189\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{9} - 12 \nu^{8} + 48 \nu^{7} - 23 \nu^{6} + 204 \nu^{5} - 240 \nu^{4} + 303 \nu^{3} - 108 \nu^{2} + 36 \nu - 1557 \)\()/567\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{9} - \nu^{8} + 12 \nu^{7} + 8 \nu^{6} + 68 \nu^{5} + 30 \nu^{4} + 123 \nu^{3} + 204 \nu^{2} + 270 \nu + 63 \)\()/63\)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180 \)\()/567\)
\(\beta_{6}\)\(=\)\((\)\( 20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu - 63 \)\()/567\)
\(\beta_{7}\)\(=\)\((\)\( -53 \nu^{9} + 60 \nu^{8} - 375 \nu^{7} - 11 \nu^{6} - 1668 \nu^{5} - 69 \nu^{4} - 2757 \nu^{3} - 4401 \nu^{2} - 1071 \nu - 1368 \)\()/567\)
\(\beta_{8}\)\(=\)\((\)\( -82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720 \)\()/567\)
\(\beta_{9}\)\(=\)\((\)\( -91 \nu^{9} + 174 \nu^{8} - 813 \nu^{7} + 704 \nu^{6} - 3633 \nu^{5} + 3174 \nu^{4} - 8070 \nu^{3} + 648 \nu^{2} - 3222 \nu + 801 \)\()/567\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(7\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(9\) \(\beta_{6}\mathstrut -\mathstrut \) \(17\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\)
\(\nu^{6}\)\(=\)\(9\) \(\beta_{9}\mathstrut -\mathstrut \) \(10\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(24\) \(\beta_{3}\mathstrut -\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(70\)
\(\nu^{7}\)\(=\)\(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(65\) \(\beta_{6}\mathstrut -\mathstrut \) \(43\) \(\beta_{4}\mathstrut -\mathstrut \) \(19\) \(\beta_{2}\mathstrut +\mathstrut \) \(75\) \(\beta_{1}\mathstrut +\mathstrut \) \(65\)
\(\nu^{8}\)\(=\)\(-\)\(62\) \(\beta_{9}\mathstrut +\mathstrut \) \(73\) \(\beta_{8}\mathstrut +\mathstrut \) \(118\) \(\beta_{7}\mathstrut +\mathstrut \) \(360\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(118\) \(\beta_{3}\)
\(\nu^{9}\)\(=\)\(-\)\(135\) \(\beta_{9}\mathstrut +\mathstrut \) \(253\) \(\beta_{8}\mathstrut +\mathstrut \) \(343\) \(\beta_{5}\mathstrut +\mathstrut \) \(253\) \(\beta_{4}\mathstrut -\mathstrut \) \(87\) \(\beta_{3}\mathstrut +\mathstrut \) \(135\) \(\beta_{2}\mathstrut -\mathstrut \) \(343\) \(\beta_{1}\mathstrut -\mathstrut \) \(430\)

Character Values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1 - \beta_{6}\) \(-1 - \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−1.02682 + 1.77851i
−0.335166 + 0.580525i
0.247934 0.429435i
0.920620 1.59456i
1.19343 2.06709i
−1.02682 1.77851i
−0.335166 0.580525i
0.247934 + 0.429435i
0.920620 + 1.59456i
1.19343 + 2.06709i
−2.05365 0 2.21746 −0.0731228 0.126652i 0 −2.33035 1.25278i −0.446582 0 0.150168 + 0.260099i
37.2 −0.670333 0 −1.55065 0.712469 + 1.23403i 0 −2.36039 + 1.19522i 2.38012 0 −0.477591 0.827212i
37.3 0.495868 0 −1.75411 −1.84629 3.19787i 0 0.926641 2.47817i −1.86155 0 −0.915516 1.58572i
37.4 1.84124 0 1.39017 0.667377 + 1.15593i 0 1.90267 1.83844i −1.12285 0 1.22880 + 2.12835i
37.5 2.38687 0 3.69714 −1.46043 2.52954i 0 −0.138560 + 2.64212i 4.05086 0 −3.48586 6.03769i
46.1 −2.05365 0 2.21746 −0.0731228 + 0.126652i 0 −2.33035 + 1.25278i −0.446582 0 0.150168 0.260099i
46.2 −0.670333 0 −1.55065 0.712469 1.23403i 0 −2.36039 1.19522i 2.38012 0 −0.477591 + 0.827212i
46.3 0.495868 0 −1.75411 −1.84629 + 3.19787i 0 0.926641 + 2.47817i −1.86155 0 −0.915516 + 1.58572i
46.4 1.84124 0 1.39017 0.667377 1.15593i 0 1.90267 + 1.83844i −1.12285 0 1.22880 2.12835i
46.5 2.38687 0 3.69714 −1.46043 + 2.52954i 0 −0.138560 2.64212i 4.05086 0 −3.48586 + 6.03769i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.5
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
63.h Even 1 yes

Hecke kernels

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