Properties

Label 189.2.p.d
Level 189
Weight 2
Character orbit 189.p
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.p (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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 - \beta_{3} - \beta_{7} + \beta_{9} ) q^{4} \) \( + ( \beta_{5} + \beta_{6} ) q^{5} \) \( + ( -1 + \beta_{4} - \beta_{9} ) q^{7} \) \( + ( \beta_{8} - \beta_{10} + \beta_{11} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 - \beta_{3} - \beta_{7} + \beta_{9} ) q^{4} \) \( + ( \beta_{5} + \beta_{6} ) q^{5} \) \( + ( -1 + \beta_{4} - \beta_{9} ) q^{7} \) \( + ( \beta_{8} - \beta_{10} + \beta_{11} ) q^{8} \) \( + ( -2 \beta_{2} - 2 \beta_{4} + \beta_{7} + \beta_{9} ) q^{10} \) \( + ( \beta_{1} - \beta_{10} + \beta_{11} ) q^{11} \) \( + ( -1 + \beta_{2} + 2 \beta_{3} - \beta_{9} ) q^{13} \) \( + ( \beta_{6} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{14} \) \( + ( 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{16} \) \( + ( -\beta_{1} + \beta_{11} ) q^{17} \) \( + ( 2 \beta_{7} - \beta_{9} ) q^{19} \) \( + ( \beta_{8} + \beta_{10} ) q^{20} \) \( + ( -3 + \beta_{2} + 2 \beta_{4} + 2 \beta_{7} - \beta_{9} ) q^{22} \) \( + ( -\beta_{1} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{23} \) \( + ( -4 + \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{7} + 2 \beta_{9} ) q^{25} \) \( + ( -2 \beta_{1} - \beta_{5} - \beta_{6} - 4 \beta_{11} ) q^{26} \) \( + ( 2 - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{7} - 2 \beta_{9} ) q^{28} \) \( + ( \beta_{8} - \beta_{10} - \beta_{11} ) q^{29} \) \( + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{31} \) \( + ( -2 \beta_{5} - \beta_{6} ) q^{32} \) \( + ( 3 - 6 \beta_{3} - \beta_{7} + 2 \beta_{9} ) q^{34} \) \( + ( 3 \beta_{1} - \beta_{6} - \beta_{8} + 2 \beta_{10} + 3 \beta_{11} ) q^{35} \) \( + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{7} + \beta_{9} ) q^{37} \) \( + ( 2 \beta_{1} - 2 \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{38} \) \( + ( \beta_{4} - 3 \beta_{7} + \beta_{9} ) q^{40} \) \( + ( 2 \beta_{1} - \beta_{5} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{41} \) \( + ( 1 + \beta_{2} + 2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{43} \) \( + ( 6 \beta_{1} - \beta_{5} + \beta_{6} - 2 \beta_{8} ) q^{44} \) \( + ( 3 + \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{46} \) \( + ( 2 \beta_{1} - \beta_{5} - \beta_{6} + 4 \beta_{11} ) q^{47} \) \( + ( 4 - \beta_{2} - 3 \beta_{3} + \beta_{7} + 2 \beta_{9} ) q^{49} \) \( + ( -\beta_{5} - 2 \beta_{6} + 3 \beta_{8} - 3 \beta_{10} - \beta_{11} ) q^{50} \) \( + ( 4 + 4 \beta_{3} - \beta_{7} - \beta_{9} ) q^{52} \) \( + ( -2 \beta_{1} + 2 \beta_{5} + \beta_{6} - \beta_{10} - 2 \beta_{11} ) q^{53} \) \( + ( -\beta_{2} - \beta_{7} + 3 \beta_{9} ) q^{55} \) \( + ( -4 \beta_{1} - \beta_{6} - \beta_{8} + 2 \beta_{10} - 4 \beta_{11} ) q^{56} \) \( + ( 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{7} ) q^{58} \) \( + ( \beta_{1} - \beta_{6} - \beta_{11} ) q^{59} \) \( + ( 2 - \beta_{3} + 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{9} ) q^{61} \) \( + ( -4 \beta_{1} + \beta_{5} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{62} \) \( + ( 2 - \beta_{7} ) q^{64} \) \( + ( -3 \beta_{1} - \beta_{5} + \beta_{6} - 3 \beta_{8} ) q^{65} \) \( + ( 7 + 2 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} - \beta_{7} + \beta_{9} ) q^{67} \) \( + ( 3 \beta_{1} + \beta_{8} - 2 \beta_{10} + 6 \beta_{11} ) q^{68} \) \( + ( -9 - \beta_{2} - 3 \beta_{4} + 2 \beta_{7} ) q^{70} \) \( + ( \beta_{5} + 2 \beta_{6} - \beta_{11} ) q^{71} \) \( + ( -3 + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{7} - 2 \beta_{9} ) q^{73} \) \( + ( -\beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{10} - \beta_{11} ) q^{74} \) \( + ( -6 - 3 \beta_{2} + 12 \beta_{3} + 3 \beta_{7} - 3 \beta_{9} ) q^{76} \) \( + ( -4 \beta_{1} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{10} ) q^{77} \) \( + ( -4 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} + \beta_{9} ) q^{79} \) \( + ( -3 \beta_{1} + \beta_{6} - 2 \beta_{8} + \beta_{10} + 3 \beta_{11} ) q^{80} \) \( + ( -6 + 3 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{82} \) \( + ( 4 \beta_{1} + \beta_{5} + 2 \beta_{11} ) q^{83} \) \( + ( -2 \beta_{2} - 4 \beta_{4} + \beta_{7} + 2 \beta_{9} ) q^{85} \) \( + ( -2 \beta_{1} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{86} \) \( + ( -12 - 2 \beta_{2} + 12 \beta_{3} + 2 \beta_{4} + 5 \beta_{7} - 5 \beta_{9} ) q^{88} \) \( + ( -2 \beta_{1} + \beta_{8} - 2 \beta_{10} - 4 \beta_{11} ) q^{89} \) \( + ( -2 + 4 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 2 \beta_{7} ) q^{91} \) \( + ( \beta_{5} + 2 \beta_{6} + 3 \beta_{8} - 3 \beta_{10} ) q^{92} \) \( + ( -6 + 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + \beta_{7} + \beta_{9} ) q^{94} \) \( + ( 2 \beta_{5} + \beta_{6} - 3 \beta_{10} ) q^{95} \) \( + ( 4 + 2 \beta_{2} - 8 \beta_{3} - 2 \beta_{9} ) q^{97} \) \( + ( 4 \beta_{1} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} + 7 \beta_{11} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut -\mathstrut 6q^{10} \) \(\mathstrut -\mathstrut 4q^{16} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut 40q^{22} \) \(\mathstrut -\mathstrut 24q^{25} \) \(\mathstrut +\mathstrut 28q^{28} \) \(\mathstrut -\mathstrut 12q^{31} \) \(\mathstrut +\mathstrut 8q^{37} \) \(\mathstrut +\mathstrut 12q^{40} \) \(\mathstrut +\mathstrut 20q^{43} \) \(\mathstrut +\mathstrut 14q^{46} \) \(\mathstrut +\mathstrut 24q^{49} \) \(\mathstrut +\mathstrut 78q^{52} \) \(\mathstrut +\mathstrut 20q^{58} \) \(\mathstrut +\mathstrut 18q^{61} \) \(\mathstrut +\mathstrut 28q^{64} \) \(\mathstrut +\mathstrut 36q^{67} \) \(\mathstrut -\mathstrut 120q^{70} \) \(\mathstrut -\mathstrut 42q^{73} \) \(\mathstrut -\mathstrut 36q^{79} \) \(\mathstrut -\mathstrut 54q^{82} \) \(\mathstrut -\mathstrut 12q^{85} \) \(\mathstrut -\mathstrut 74q^{88} \) \(\mathstrut +\mathstrut 6q^{91} \) \(\mathstrut -\mathstrut 114q^{94} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(9\) \(x^{10}\mathstrut +\mathstrut \) \(59\) \(x^{8}\mathstrut -\mathstrut \) \(180\) \(x^{6}\mathstrut +\mathstrut \) \(403\) \(x^{4}\mathstrut -\mathstrut \) \(198\) \(x^{2}\mathstrut +\mathstrut \) \(81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 81 \nu^{11} - 531 \nu^{9} + 3481 \nu^{7} - 3627 \nu^{5} + 1782 \nu^{3} + 76298 \nu \)\()/21995\)
\(\beta_{2}\)\(=\)\((\)\( 117 \nu^{10} - 767 \nu^{8} + 7472 \nu^{6} - 27234 \nu^{4} + 90554 \nu^{2} - 60864 \)\()/21995\)
\(\beta_{3}\)\(=\)\((\)\( -1298 \nu^{10} + 10953 \nu^{8} - 71803 \nu^{6} + 202311 \nu^{4} - 490451 \nu^{2} + 240966 \)\()/197955\)
\(\beta_{4}\)\(=\)\((\)\( 461 \nu^{10} - 5466 \nu^{8} + 28501 \nu^{6} - 98847 \nu^{4} + 142112 \nu^{2} - 186237 \)\()/65985\)
\(\beta_{5}\)\(=\)\((\)\( 1298 \nu^{11} - 10953 \nu^{9} + 71803 \nu^{7} - 202311 \nu^{5} + 490451 \nu^{3} - 438921 \nu \)\()/197955\)
\(\beta_{6}\)\(=\)\((\)\( 1298 \nu^{11} - 10953 \nu^{9} + 71803 \nu^{7} - 202311 \nu^{5} + 490451 \nu^{3} + 154944 \nu \)\()/197955\)
\(\beta_{7}\)\(=\)\((\)\( -288 \nu^{10} + 1888 \nu^{8} - 9933 \nu^{6} + 12896 \nu^{4} - 6336 \nu^{2} - 68439 \)\()/21995\)
\(\beta_{8}\)\(=\)\((\)\( -288 \nu^{11} + 1888 \nu^{9} - 9933 \nu^{7} + 12896 \nu^{5} - 6336 \nu^{3} - 90434 \nu \)\()/21995\)
\(\beta_{9}\)\(=\)\((\)\( 1138 \nu^{10} - 12348 \nu^{8} + 80948 \nu^{6} - 273351 \nu^{4} + 552916 \nu^{2} - 271656 \)\()/65985\)
\(\beta_{10}\)\(=\)\((\)\( 4712 \nu^{11} - 47997 \nu^{9} + 314647 \nu^{7} - 1022364 \nu^{5} + 2149199 \nu^{3} - 1055934 \nu \)\()/197955\)
\(\beta_{11}\)\(=\)\((\)\( -5192 \nu^{11} + 43812 \nu^{9} - 287212 \nu^{7} + 809244 \nu^{5} - 1763849 \nu^{3} + 172044 \nu \)\()/197955\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(9\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(9\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\)\()/3\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(12\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\)
\(\nu^{5}\)\(=\)\((\)\(18\) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(17\) \(\beta_{6}\mathstrut +\mathstrut \) \(34\) \(\beta_{5}\mathstrut +\mathstrut \) \(18\) \(\beta_{1}\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(32\) \(\beta_{9}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(64\) \(\beta_{4}\mathstrut -\mathstrut \) \(32\) \(\beta_{2}\mathstrut -\mathstrut \) \(153\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(27\) \(\beta_{8}\mathstrut -\mathstrut \) \(74\) \(\beta_{6}\mathstrut +\mathstrut \) \(74\) \(\beta_{5}\mathstrut +\mathstrut \) \(96\) \(\beta_{1}\)\()/3\)
\(\nu^{8}\)\(=\)\(51\) \(\beta_{9}\mathstrut -\mathstrut \) \(51\) \(\beta_{7}\mathstrut -\mathstrut \) \(55\) \(\beta_{4}\mathstrut +\mathstrut \) \(222\) \(\beta_{3}\mathstrut +\mathstrut \) \(55\) \(\beta_{2}\mathstrut -\mathstrut \) \(222\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(495\) \(\beta_{11}\mathstrut -\mathstrut \) \(177\) \(\beta_{10}\mathstrut +\mathstrut \) \(177\) \(\beta_{8}\mathstrut -\mathstrut \) \(656\) \(\beta_{6}\mathstrut -\mathstrut \) \(328\) \(\beta_{5}\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(191\) \(\beta_{9}\mathstrut -\mathstrut \) \(835\) \(\beta_{7}\mathstrut +\mathstrut \) \(835\) \(\beta_{4}\mathstrut +\mathstrut \) \(2952\) \(\beta_{3}\mathstrut +\mathstrut \) \(1670\) \(\beta_{2}\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(2505\) \(\beta_{11}\mathstrut -\mathstrut \) \(1026\) \(\beta_{10}\mathstrut -\mathstrut \) \(1477\) \(\beta_{6}\mathstrut -\mathstrut \) \(2954\) \(\beta_{5}\mathstrut -\mathstrut \) \(2505\) \(\beta_{1}\)\()/3\)

Character Values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.617942 + 0.356769i
−1.90412 1.09935i
1.65604 + 0.956115i
−1.65604 0.956115i
1.90412 + 1.09935i
−0.617942 0.356769i
0.617942 0.356769i
−1.90412 + 1.09935i
1.65604 0.956115i
−1.65604 + 0.956115i
1.90412 1.09935i
−0.617942 + 0.356769i
−2.15715 1.24543i 0 2.10220 + 3.64112i −0.617942 + 1.07031i 0 −1.53189 2.15715i 5.49086i 0 2.66599 1.53921i
26.2 −1.58850 0.917122i 0 0.682224 + 1.18165i 1.90412 3.29804i 0 2.11581 + 1.58850i 1.16576i 0 −6.04940 + 3.49262i
26.3 −0.568650 0.328310i 0 −0.784425 1.35866i −1.65604 + 2.86834i 0 −2.58392 + 0.568650i 2.34338i 0 1.88341 1.08739i
26.4 0.568650 + 0.328310i 0 −0.784425 1.35866i 1.65604 2.86834i 0 −2.58392 + 0.568650i 2.34338i 0 1.88341 1.08739i
26.5 1.58850 + 0.917122i 0 0.682224 + 1.18165i −1.90412 + 3.29804i 0 2.11581 + 1.58850i 1.16576i 0 −6.04940 + 3.49262i
26.6 2.15715 + 1.24543i 0 2.10220 + 3.64112i 0.617942 1.07031i 0 −1.53189 2.15715i 5.49086i 0 2.66599 1.53921i
80.1 −2.15715 + 1.24543i 0 2.10220 3.64112i −0.617942 1.07031i 0 −1.53189 + 2.15715i 5.49086i 0 2.66599 + 1.53921i
80.2 −1.58850 + 0.917122i 0 0.682224 1.18165i 1.90412 + 3.29804i 0 2.11581 1.58850i 1.16576i 0 −6.04940 3.49262i
80.3 −0.568650 + 0.328310i 0 −0.784425 + 1.35866i −1.65604 2.86834i 0 −2.58392 0.568650i 2.34338i 0 1.88341 + 1.08739i
80.4 0.568650 0.328310i 0 −0.784425 + 1.35866i 1.65604 + 2.86834i 0 −2.58392 0.568650i 2.34338i 0 1.88341 + 1.08739i
80.5 1.58850 0.917122i 0 0.682224 1.18165i −1.90412 3.29804i 0 2.11581 1.58850i 1.16576i 0 −6.04940 3.49262i
80.6 2.15715 1.24543i 0 2.10220 3.64112i 0.617942 + 1.07031i 0 −1.53189 + 2.15715i 5.49086i 0 2.66599 + 1.53921i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
7.d Odd 1 yes
21.g Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{12} \) \(\mathstrut -\mathstrut 10 T_{2}^{10} \) \(\mathstrut +\mathstrut 75 T_{2}^{8} \) \(\mathstrut -\mathstrut 232 T_{2}^{6} \) \(\mathstrut +\mathstrut 535 T_{2}^{4} \) \(\mathstrut -\mathstrut 225 T_{2}^{2} \) \(\mathstrut +\mathstrut 81 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).