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Properties

Label 189.2.be.a.104.16
Level $189$
Weight $2$
Character 189.104
Analytic conductor $1.509$
Analytic rank $0$
Dimension $132$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [189,2,Mod(20,189)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("189.20"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(189, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([7, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.be (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 104.16
Character \(\chi\) \(=\) 189.104
Dual form 189.2.be.a.20.16

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.471430 - 1.29524i) q^{2} +(1.37746 - 1.05005i) q^{3} +(0.0766803 + 0.0643424i) q^{4} +(-0.459149 + 2.60396i) q^{5} +(-0.710690 - 2.27917i) q^{6} +(-1.47152 - 2.19878i) q^{7} +(2.50689 - 1.44736i) q^{8} +(0.794802 - 2.89280i) q^{9} +(3.15631 + 1.82230i) q^{10} +(-0.681802 + 0.120220i) q^{11} +(0.173187 + 0.00811128i) q^{12} +(0.171823 + 0.472079i) q^{13} +(-3.54167 + 0.869398i) q^{14} +(2.10182 + 4.06899i) q^{15} +(-0.658089 - 3.73221i) q^{16} +(-2.42375 + 4.19806i) q^{17} +(-3.37219 - 2.39321i) q^{18} +(-1.03438 + 0.597199i) q^{19} +(-0.202753 + 0.170130i) q^{20} +(-4.33578 - 1.48358i) q^{21} +(-0.165708 + 0.939774i) q^{22} +(-4.74912 + 5.65978i) q^{23} +(1.93336 - 4.62603i) q^{24} +(-1.87135 - 0.681115i) q^{25} +0.692459 q^{26} +(-1.94277 - 4.81930i) q^{27} +(0.0286387 - 0.263284i) q^{28} +(-2.19806 + 6.03911i) q^{29} +(6.26119 - 0.804131i) q^{30} +(5.17613 - 6.16868i) q^{31} +(0.557110 + 0.0982336i) q^{32} +(-0.812919 + 0.881523i) q^{33} +(4.29488 + 5.11844i) q^{34} +(6.40120 - 2.82220i) q^{35} +(0.247075 - 0.170681i) q^{36} +(-3.71414 + 6.43307i) q^{37} +(0.285881 + 1.62131i) q^{38} +(0.732384 + 0.469849i) q^{39} +(2.61782 + 7.19241i) q^{40} +(3.25278 - 1.18391i) q^{41} +(-3.96561 + 4.91649i) q^{42} +(-1.81376 - 10.2863i) q^{43} +(-0.0600160 - 0.0346503i) q^{44} +(7.16781 + 3.39786i) q^{45} +(5.09191 + 8.81945i) q^{46} +(-2.93133 + 2.45968i) q^{47} +(-4.82549 - 4.44995i) q^{48} +(-2.66929 + 6.47108i) q^{49} +(-1.76442 + 2.10275i) q^{50} +(1.06954 + 8.32773i) q^{51} +(-0.0171993 + 0.0472546i) q^{52} -10.3394i q^{53} +(-7.15804 + 0.244393i) q^{54} -1.83059i q^{55} +(-6.87135 - 3.38231i) q^{56} +(-0.797730 + 1.90876i) q^{57} +(6.78588 + 5.69403i) q^{58} +(1.13100 - 6.41423i) q^{59} +(-0.100640 + 0.447248i) q^{60} +(2.36159 + 2.81444i) q^{61} +(-5.54975 - 9.61245i) q^{62} +(-7.53020 + 2.50920i) q^{63} +(4.17966 - 7.23938i) q^{64} +(-1.30817 + 0.230665i) q^{65} +(0.758552 + 1.46850i) q^{66} +(-3.24000 + 1.17926i) q^{67} +(-0.455968 + 0.165959i) q^{68} +(-0.598694 + 12.7829i) q^{69} +(-0.637725 - 9.62158i) q^{70} +(0.286623 + 0.165482i) q^{71} +(-2.19443 - 8.40230i) q^{72} +(3.13921 - 1.81243i) q^{73} +(6.58144 + 7.84345i) q^{74} +(-3.29291 + 1.02679i) q^{75} +(-0.117742 - 0.0207610i) q^{76} +(1.26762 + 1.32223i) q^{77} +(0.953836 - 0.727115i) q^{78} +(9.71572 + 3.53623i) q^{79} +10.0207 q^{80} +(-7.73658 - 4.59840i) q^{81} -4.77127i q^{82} +(12.8972 + 4.69419i) q^{83} +(-0.237012 - 0.392736i) q^{84} +(-9.81874 - 8.23890i) q^{85} +(-14.1784 - 2.50003i) q^{86} +(3.31361 + 10.6267i) q^{87} +(-1.53520 + 1.28819i) q^{88} +(-4.84538 - 8.39244i) q^{89} +(7.78018 - 7.68221i) q^{90} +(0.785159 - 1.07247i) q^{91} +(-0.728328 + 0.128424i) q^{92} +(0.652525 - 13.9323i) q^{93} +(1.80397 + 4.95636i) q^{94} +(-1.08015 - 2.96769i) q^{95} +(0.870548 - 0.449679i) q^{96} +(-3.19371 + 0.563137i) q^{97} +(7.12324 + 6.50804i) q^{98} +(-0.194125 + 2.06787i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q - 12 q^{2} - 12 q^{4} - 6 q^{7} - 18 q^{8} - 6 q^{9} - 18 q^{11} + 3 q^{14} - 24 q^{15} - 24 q^{16} - 12 q^{18} - 12 q^{21} - 12 q^{22} + 12 q^{23} - 12 q^{25} - 12 q^{28} - 48 q^{29} + 42 q^{30}+ \cdots - 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(e\left(\frac{11}{18}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.471430 − 1.29524i 0.333351 − 0.915875i −0.653882 − 0.756596i \(-0.726861\pi\)
0.987234 − 0.159279i \(-0.0509169\pi\)
\(3\) 1.37746 − 1.05005i 0.795278 − 0.606245i
\(4\) 0.0766803 + 0.0643424i 0.0383402 + 0.0321712i
\(5\) −0.459149 + 2.60396i −0.205338 + 1.16453i 0.691570 + 0.722310i \(0.256919\pi\)
−0.896907 + 0.442218i \(0.854192\pi\)
\(6\) −0.710690 − 2.27917i −0.290138 − 0.930468i
\(7\) −1.47152 − 2.19878i −0.556180 − 0.831062i
\(8\) 2.50689 − 1.44736i 0.886321 − 0.511718i
\(9\) 0.794802 − 2.89280i 0.264934 − 0.964267i
\(10\) 3.15631 + 1.82230i 0.998113 + 0.576261i
\(11\) −0.681802 + 0.120220i −0.205571 + 0.0362477i −0.275485 − 0.961305i \(-0.588839\pi\)
0.0699143 + 0.997553i \(0.477727\pi\)
\(12\) 0.173187 + 0.00811128i 0.0499947 + 0.00234153i
\(13\) 0.171823 + 0.472079i 0.0476550 + 0.130931i 0.961237 − 0.275724i \(-0.0889174\pi\)
−0.913582 + 0.406655i \(0.866695\pi\)
\(14\) −3.54167 + 0.869398i −0.946552 + 0.232356i
\(15\) 2.10182 + 4.06899i 0.542689 + 1.05061i
\(16\) −0.658089 − 3.73221i −0.164522 − 0.933052i
\(17\) −2.42375 + 4.19806i −0.587846 + 1.01818i 0.406668 + 0.913576i \(0.366691\pi\)
−0.994514 + 0.104604i \(0.966643\pi\)
\(18\) −3.37219 − 2.39321i −0.794832 − 0.564086i
\(19\) −1.03438 + 0.597199i −0.237303 + 0.137007i −0.613936 − 0.789355i \(-0.710415\pi\)
0.376634 + 0.926362i \(0.377082\pi\)
\(20\) −0.202753 + 0.170130i −0.0453370 + 0.0380422i
\(21\) −4.33578 − 1.48358i −0.946145 − 0.323743i
\(22\) −0.165708 + 0.939774i −0.0353290 + 0.200361i
\(23\) −4.74912 + 5.65978i −0.990259 + 1.18014i −0.00662300 + 0.999978i \(0.502108\pi\)
−0.983636 + 0.180167i \(0.942336\pi\)
\(24\) 1.93336 − 4.62603i 0.394645 − 0.944285i
\(25\) −1.87135 − 0.681115i −0.374270 − 0.136223i
\(26\) 0.692459 0.135802
\(27\) −1.94277 − 4.81930i −0.373886 − 0.927475i
\(28\) 0.0286387 − 0.263284i 0.00541221 − 0.0497560i
\(29\) −2.19806 + 6.03911i −0.408169 + 1.12143i 0.549984 + 0.835175i \(0.314634\pi\)
−0.958152 + 0.286259i \(0.907588\pi\)
\(30\) 6.26119 − 0.804131i 1.14313 − 0.146814i
\(31\) 5.17613 − 6.16868i 0.929661 − 1.10793i −0.0642712 − 0.997932i \(-0.520472\pi\)
0.993932 − 0.109994i \(-0.0350833\pi\)
\(32\) 0.557110 + 0.0982336i 0.0984841 + 0.0173654i
\(33\) −0.812919 + 0.881523i −0.141511 + 0.153453i
\(34\) 4.29488 + 5.11844i 0.736566 + 0.877806i
\(35\) 6.40120 − 2.82220i 1.08200 − 0.477039i
\(36\) 0.247075 − 0.170681i 0.0411792 − 0.0284469i
\(37\) −3.71414 + 6.43307i −0.610600 + 1.05759i 0.380539 + 0.924765i \(0.375738\pi\)
−0.991139 + 0.132826i \(0.957595\pi\)
\(38\) 0.285881 + 1.62131i 0.0463759 + 0.263011i
\(39\) 0.732384 + 0.469849i 0.117275 + 0.0752360i
\(40\) 2.61782 + 7.19241i 0.413914 + 1.13722i
\(41\) 3.25278 − 1.18391i 0.507998 − 0.184896i −0.0752897 − 0.997162i \(-0.523988\pi\)
0.583288 + 0.812265i \(0.301766\pi\)
\(42\) −3.96561 + 4.91649i −0.611907 + 0.758630i
\(43\) −1.81376 − 10.2863i −0.276596 − 1.56865i −0.733846 − 0.679315i \(-0.762277\pi\)
0.457250 − 0.889338i \(-0.348834\pi\)
\(44\) −0.0600160 − 0.0346503i −0.00904776 − 0.00522373i
\(45\) 7.16781 + 3.39786i 1.06851 + 0.506523i
\(46\) 5.09191 + 8.81945i 0.750761 + 1.30036i
\(47\) −2.93133 + 2.45968i −0.427579 + 0.358781i −0.831037 − 0.556217i \(-0.812252\pi\)
0.403458 + 0.914998i \(0.367808\pi\)
\(48\) −4.82549 − 4.44995i −0.696499 − 0.642295i
\(49\) −2.66929 + 6.47108i −0.381327 + 0.924440i
\(50\) −1.76442 + 2.10275i −0.249527 + 0.297374i
\(51\) 1.06954 + 8.32773i 0.149765 + 1.16611i
\(52\) −0.0171993 + 0.0472546i −0.00238511 + 0.00655304i
\(53\) − 10.3394i − 1.42022i −0.704090 − 0.710111i \(-0.748645\pi\)
0.704090 − 0.710111i \(-0.251355\pi\)
\(54\) −7.15804 + 0.244393i −0.974086 + 0.0332577i
\(55\) − 1.83059i − 0.246836i
\(56\) −6.87135 − 3.38231i −0.918223 − 0.451980i
\(57\) −0.797730 + 1.90876i −0.105662 + 0.252822i
\(58\) 6.78588 + 5.69403i 0.891030 + 0.747663i
\(59\) 1.13100 − 6.41423i 0.147244 − 0.835061i −0.818294 − 0.574799i \(-0.805080\pi\)
0.965538 − 0.260262i \(-0.0838088\pi\)
\(60\) −0.100640 + 0.447248i −0.0129926 + 0.0577395i
\(61\) 2.36159 + 2.81444i 0.302371 + 0.360352i 0.895739 − 0.444579i \(-0.146647\pi\)
−0.593369 + 0.804931i \(0.702202\pi\)
\(62\) −5.54975 − 9.61245i −0.704819 − 1.22078i
\(63\) −7.53020 + 2.50920i −0.948716 + 0.316130i
\(64\) 4.17966 − 7.23938i 0.522457 − 0.904922i
\(65\) −1.30817 + 0.230665i −0.162258 + 0.0286105i
\(66\) 0.758552 + 1.46850i 0.0933712 + 0.180760i
\(67\) −3.24000 + 1.17926i −0.395829 + 0.144070i −0.532263 − 0.846579i \(-0.678658\pi\)
0.136434 + 0.990649i \(0.456436\pi\)
\(68\) −0.455968 + 0.165959i −0.0552942 + 0.0201254i
\(69\) −0.598694 + 12.7829i −0.0720743 + 1.53888i
\(70\) −0.637725 − 9.62158i −0.0762227 − 1.15000i
\(71\) 0.286623 + 0.165482i 0.0340159 + 0.0196391i 0.516912 − 0.856039i \(-0.327082\pi\)
−0.482896 + 0.875678i \(0.660415\pi\)
\(72\) −2.19443 − 8.40230i −0.258616 − 0.990221i
\(73\) 3.13921 − 1.81243i 0.367417 − 0.212128i −0.304912 − 0.952380i \(-0.598627\pi\)
0.672329 + 0.740252i \(0.265294\pi\)
\(74\) 6.58144 + 7.84345i 0.765077 + 0.911783i
\(75\) −3.29291 + 1.02679i −0.380233 + 0.118564i
\(76\) −0.117742 − 0.0207610i −0.0135059 − 0.00238145i
\(77\) 1.26762 + 1.32223i 0.144459 + 0.150682i
\(78\) 0.953836 − 0.727115i 0.108001 − 0.0823295i
\(79\) 9.71572 + 3.53623i 1.09310 + 0.397857i 0.824770 − 0.565469i \(-0.191305\pi\)
0.268334 + 0.963326i \(0.413527\pi\)
\(80\) 10.0207 1.12035
\(81\) −7.73658 − 4.59840i −0.859620 − 0.510934i
\(82\) − 4.77127i − 0.526898i
\(83\) 12.8972 + 4.69419i 1.41565 + 0.515254i 0.932782 − 0.360440i \(-0.117373\pi\)
0.482866 + 0.875694i \(0.339596\pi\)
\(84\) −0.237012 − 0.392736i −0.0258601 − 0.0428510i
\(85\) −9.81874 − 8.23890i −1.06499 − 0.893634i
\(86\) −14.1784 − 2.50003i −1.52889 − 0.269585i
\(87\) 3.31361 + 10.6267i 0.355256 + 1.13930i
\(88\) −1.53520 + 1.28819i −0.163653 + 0.137321i
\(89\) −4.84538 − 8.39244i −0.513609 − 0.889597i −0.999875 − 0.0157861i \(-0.994975\pi\)
0.486266 − 0.873811i \(-0.338358\pi\)
\(90\) 7.78018 − 7.68221i 0.820103 − 0.809776i
\(91\) 0.785159 − 1.07247i 0.0823070 − 0.112426i
\(92\) −0.728328 + 0.128424i −0.0759334 + 0.0133891i
\(93\) 0.652525 − 13.9323i 0.0676637 − 1.44471i
\(94\) 1.80397 + 4.95636i 0.186065 + 0.511209i
\(95\) −1.08015 − 2.96769i −0.110821 − 0.304478i
\(96\) 0.870548 − 0.449679i 0.0888500 − 0.0458952i
\(97\) −3.19371 + 0.563137i −0.324272 + 0.0571779i −0.333415 − 0.942780i \(-0.608201\pi\)
0.00914255 + 0.999958i \(0.497090\pi\)
\(98\) 7.12324 + 6.50804i 0.719556 + 0.657411i
\(99\) −0.194125 + 2.06787i −0.0195103 + 0.207828i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 189.2.be.a.104.16 yes 132
3.2 odd 2 567.2.be.a.503.8 132
7.6 odd 2 inner 189.2.be.a.104.15 yes 132
21.20 even 2 567.2.be.a.503.7 132
27.7 even 9 567.2.be.a.62.7 132
27.20 odd 18 inner 189.2.be.a.20.15 ✓ 132
189.20 even 18 inner 189.2.be.a.20.16 yes 132
189.34 odd 18 567.2.be.a.62.8 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.be.a.20.15 ✓ 132 27.20 odd 18 inner
189.2.be.a.20.16 yes 132 189.20 even 18 inner
189.2.be.a.104.15 yes 132 7.6 odd 2 inner
189.2.be.a.104.16 yes 132 1.1 even 1 trivial
567.2.be.a.62.7 132 27.7 even 9
567.2.be.a.62.8 132 189.34 odd 18
567.2.be.a.503.7 132 21.20 even 2
567.2.be.a.503.8 132 3.2 odd 2