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Properties

Label 189.2.be.a.20.15
Level $189$
Weight $2$
Character 189.20
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 20.15
Character \(\chi\) \(=\) 189.20
Dual form 189.2.be.a.104.15

$q$-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} +(1.90985 - 1.83097i) 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} +(3.27192 + 1.61055i) 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} +(-4.55336 + 0.516662i) 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} +(5.64470 + 4.13250i) 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} +(-2.81579 - 5.65413i) 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} +(0.295069 - 6.99378i) 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} +(7.43787 - 1.82582i) 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} +(6.81459 + 4.06956i) 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} +(-2.69151 + 9.25944i) 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} +(-1.52226 + 1.01876i) 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} +(-0.315910 + 0.332592i) 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} +(0.536208 + 1.21620i) 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} +(9.19775 - 2.91489i) 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 \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\) \(-1\)

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 0.987234 + 0.159279i \(0.0509169\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(3\) −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 0.896907 + 0.442218i \(0.145808\pi\)
−0.691570 + 0.722310i \(0.743081\pi\)
\(6\) 0.710690 − 2.27917i 0.290138 − 0.930468i
\(7\) 1.90985 − 1.83097i 0.721856 − 0.692043i
\(8\) 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 0.0699143 − 0.997553i \(-0.477727\pi\)
−0.275485 + 0.961305i \(0.588839\pi\)
\(12\) −0.173187 + 0.00811128i −0.0499947 + 0.00234153i
\(13\) −0.171823 + 0.472079i −0.0476550 + 0.130931i −0.961237 − 0.275724i \(-0.911083\pi\)
0.913582 + 0.406655i \(0.133305\pi\)
\(14\) 3.27192 + 1.61055i 0.874457 + 0.430437i
\(15\) 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 0.994514 + 0.104604i \(0.0333574\pi\)
−0.406668 + 0.913576i \(0.633309\pi\)
\(18\) −3.37219 + 2.39321i −0.794832 + 0.564086i
\(19\) 1.03438 + 0.597199i 0.237303 + 0.137007i 0.613936 − 0.789355i \(-0.289585\pi\)
−0.376634 + 0.926362i \(0.622918\pi\)
\(20\) 0.202753 + 0.170130i 0.0453370 + 0.0380422i
\(21\) −4.55336 + 0.516662i −0.993624 + 0.112745i
\(22\) −0.165708 − 0.939774i −0.0353290 − 0.200361i
\(23\) −4.74912 − 5.65978i −0.990259 − 1.18014i −0.983636 − 0.180167i \(-0.942336\pi\)
−0.00662300 − 0.999978i \(-0.502108\pi\)
\(24\) −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 −0.958152 − 0.286259i \(-0.907588\pi\)
0.549984 − 0.835175i \(-0.314634\pi\)
\(30\) 6.26119 + 0.804131i 1.14313 + 0.146814i
\(31\) −5.17613 − 6.16868i −0.929661 − 1.10793i −0.993932 − 0.109994i \(-0.964917\pi\)
0.0642712 − 0.997932i \(-0.479528\pi\)
\(32\) 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
\(35\) 5.64470 + 4.13250i 0.954128 + 0.698519i
\(36\) 0.247075 + 0.170681i 0.0411792 + 0.0284469i
\(37\) −3.71414 − 6.43307i −0.610600 − 1.05759i −0.991139 − 0.132826i \(-0.957595\pi\)
0.380539 − 0.924765i \(-0.375738\pi\)
\(38\) −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 \(-0.476012\pi\)
−0.583288 + 0.812265i \(0.698234\pi\)
\(42\) −2.81579 − 5.65413i −0.434486 − 0.872452i
\(43\) −1.81376 + 10.2863i −0.276596 + 1.56865i 0.457250 + 0.889338i \(0.348834\pi\)
−0.733846 + 0.679315i \(0.762277\pi\)
\(44\) −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 \(-0.187748\pi\)
−0.403458 + 0.914998i \(0.632192\pi\)
\(48\) 4.82549 − 4.44995i 0.696499 − 0.642295i
\(49\) 0.295069 − 6.99378i 0.0421527 − 0.999111i
\(50\) −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 \(0.251355\pi\)
−0.704090 + 0.710111i \(0.748645\pi\)
\(54\) 7.15804 + 0.244393i 0.974086 + 0.0332577i
\(55\) − 1.83059i − 0.246836i
\(56\) 7.43787 − 1.82582i 0.993927 − 0.243986i
\(57\) −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 −0.965538 − 0.260262i \(-0.916191\pi\)
0.818294 − 0.574799i \(-0.194920\pi\)
\(60\) −0.100640 − 0.447248i −0.0129926 − 0.0577395i
\(61\) −2.36159 + 2.81444i −0.302371 + 0.360352i −0.895739 − 0.444579i \(-0.853353\pi\)
0.593369 + 0.804931i \(0.297798\pi\)
\(62\) 5.54975 − 9.61245i 0.704819 − 1.22078i
\(63\) 6.81459 + 4.06956i 0.858558 + 0.512716i
\(64\) 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 0.136434 − 0.990649i \(-0.456436\pi\)
−0.532263 + 0.846579i \(0.678658\pi\)
\(68\) 0.455968 + 0.165959i 0.0552942 + 0.0201254i
\(69\) 0.598694 + 12.7829i 0.0720743 + 1.53888i
\(70\) −2.69151 + 9.25944i −0.321697 + 1.10671i
\(71\) 0.286623 − 0.165482i 0.0340159 − 0.0196391i −0.482896 − 0.875678i \(-0.660415\pi\)
0.516912 + 0.856039i \(0.327082\pi\)
\(72\) −2.19443 + 8.40230i −0.258616 + 0.990221i
\(73\) −3.13921 − 1.81243i −0.367417 − 0.212128i 0.304912 − 0.952380i \(-0.401373\pi\)
−0.672329 + 0.740252i \(0.734706\pi\)
\(74\) 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
\(77\) −1.52226 + 1.01876i −0.173478 + 0.116098i
\(78\) 0.953836 + 0.727115i 0.108001 + 0.0823295i
\(79\) 9.71572 − 3.53623i 1.09310 − 0.397857i 0.268334 − 0.963326i \(-0.413527\pi\)
0.824770 + 0.565469i \(0.191305\pi\)
\(80\) −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 \(-0.882627\pi\)
−0.482866 + 0.875694i \(0.660404\pi\)
\(84\) −0.315910 + 0.332592i −0.0344686 + 0.0362888i
\(85\) −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 −0.486266 − 0.873811i \(-0.661642\pi\)
0.999875 − 0.0157861i \(-0.00502510\pi\)
\(90\) −7.78018 − 7.68221i −0.820103 − 0.809776i
\(91\) 0.536208 + 1.21620i 0.0562099 + 0.127493i
\(92\) −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 \(-0.391799\pi\)
−0.00914255 + 0.999958i \(0.502910\pi\)
\(98\) 9.19775 − 2.91489i 0.929113 − 0.294448i
\(99\) −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.20.15 ✓ 132
3.2 odd 2 567.2.be.a.62.7 132
7.6 odd 2 inner 189.2.be.a.20.16 yes 132
21.20 even 2 567.2.be.a.62.8 132
27.4 even 9 567.2.be.a.503.8 132
27.23 odd 18 inner 189.2.be.a.104.16 yes 132
189.104 even 18 inner 189.2.be.a.104.15 yes 132
189.139 odd 18 567.2.be.a.503.7 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.be.a.20.15 ✓ 132 1.1 even 1 trivial
189.2.be.a.20.16 yes 132 7.6 odd 2 inner
189.2.be.a.104.15 yes 132 189.104 even 18 inner
189.2.be.a.104.16 yes 132 27.23 odd 18 inner
567.2.be.a.62.7 132 3.2 odd 2
567.2.be.a.62.8 132 21.20 even 2
567.2.be.a.503.7 132 189.139 odd 18
567.2.be.a.503.8 132 27.4 even 9