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Properties

Label 189.2.bd.a.185.12
Level $189$
Weight $2$
Character 189.185
Analytic conductor $1.509$
Analytic rank $0$
Dimension $132$
Inner twists $2$

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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(47,189)] 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("189.47"); 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, 15])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.bd (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 185.12
Character \(\chi\) \(=\) 189.185
Dual form 189.2.bd.a.47.12

$q$-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+(0.0159182 - 0.00280680i) q^{2} +(-0.402650 + 1.68460i) q^{3} +(-1.87914 + 0.683951i) q^{4} +(3.75147 - 1.36542i) q^{5} +(-0.00168110 + 0.0279459i) q^{6} +(0.157938 + 2.64103i) q^{7} +(-0.0559891 + 0.0323253i) q^{8} +(-2.67575 - 1.35661i) q^{9} +(0.0558840 - 0.0322646i) q^{10} +(-1.76436 + 4.84753i) q^{11} +(-0.395548 - 3.44099i) q^{12} +(-0.220791 - 0.606618i) q^{13} +(0.00992693 + 0.0415971i) q^{14} +(0.789663 + 6.86951i) q^{15} +(3.06298 - 2.57014i) q^{16} +(1.69383 + 2.93380i) q^{17} +(-0.0464007 - 0.0140844i) q^{18} +(0.328436 + 0.189623i) q^{19} +(-6.11565 + 5.13164i) q^{20} +(-4.51267 - 0.797349i) q^{21} +(-0.0144792 + 0.0821158i) q^{22} +(-0.993035 - 0.175099i) q^{23} +(-0.0319112 - 0.107335i) q^{24} +(8.37891 - 7.03074i) q^{25} +(-0.00521724 - 0.00903653i) q^{26} +(3.36272 - 3.96132i) q^{27} +(-2.10312 - 4.85485i) q^{28} +(1.97120 - 5.41582i) q^{29} +(0.0318513 + 0.107133i) q^{30} +(-2.20741 - 6.06480i) q^{31} +(0.124656 - 0.148560i) q^{32} +(-7.45572 - 4.92408i) q^{33} +(0.0351973 + 0.0419465i) q^{34} +(4.19863 + 9.69210i) q^{35} +(5.95595 + 0.719172i) q^{36} +6.16814 q^{37} +(0.00576033 + 0.00209659i) q^{38} +(1.11081 - 0.127690i) q^{39} +(-0.165904 + 0.197716i) q^{40} +(0.266252 - 0.0969077i) q^{41} +(-0.0740714 - 2.61403e-5i) q^{42} +(-1.45052 - 8.22633i) q^{43} -10.3159i q^{44} +(-11.8903 - 1.43574i) q^{45} -0.0162988 q^{46} +(6.91258 + 2.51597i) q^{47} +(3.09635 + 6.19475i) q^{48} +(-6.95011 + 0.834239i) q^{49} +(0.113643 - 0.135434i) q^{50} +(-5.62430 + 1.67213i) q^{51} +(0.829794 + 0.988910i) q^{52} +(1.71989 + 0.992981i) q^{53} +(0.0424097 - 0.0724954i) q^{54} +20.5944i q^{55} +(-0.0942150 - 0.142764i) q^{56} +(-0.451683 + 0.476932i) q^{57} +(0.0161767 - 0.0917427i) q^{58} +(-2.62371 - 2.20155i) q^{59} +(-6.18229 - 12.3687i) q^{60} +(-2.75545 + 7.57053i) q^{61} +(-0.0521605 - 0.0903447i) q^{62} +(3.16024 - 7.28100i) q^{63} +(-3.99687 + 6.92277i) q^{64} +(-1.65658 - 1.97424i) q^{65} +(-0.132502 - 0.0574556i) q^{66} +(1.63911 - 9.29584i) q^{67} +(-5.18952 - 4.35453i) q^{68} +(0.694817 - 1.60236i) q^{69} +(0.0940381 + 0.142496i) q^{70} +(8.57363 + 4.94999i) q^{71} +(0.193665 - 0.0105392i) q^{72} -7.08887i q^{73} +(0.0981854 - 0.0173127i) q^{74} +(8.47022 + 16.9460i) q^{75} +(-0.746871 - 0.131693i) q^{76} +(-13.0811 - 3.89411i) q^{77} +(0.0173236 - 0.00515040i) q^{78} +(-0.222353 - 1.26102i) q^{79} +(7.98133 - 13.8241i) q^{80} +(5.31924 + 7.25987i) q^{81} +(0.00396624 - 0.00228991i) q^{82} +(-8.07527 - 2.93916i) q^{83} +(9.02529 - 1.58812i) q^{84} +(10.3602 + 8.69327i) q^{85} +(-0.0461793 - 0.126877i) q^{86} +(8.32979 + 5.50136i) q^{87} +(-0.0579132 - 0.328442i) q^{88} +(-2.67849 + 4.63928i) q^{89} +(-0.193302 + 0.0105194i) q^{90} +(1.56723 - 0.678924i) q^{91} +(1.98581 - 0.350152i) q^{92} +(11.1056 - 1.27661i) q^{93} +(0.117097 + 0.0206474i) q^{94} +(1.49103 + 0.262910i) q^{95} +(0.200071 + 0.269813i) q^{96} +(-7.39427 + 1.30381i) q^{97} +(-0.108291 + 0.0327871i) q^{98} +(11.2971 - 10.5772i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} + 18 q^{6} - 6 q^{7} - 18 q^{8} - 15 q^{9} - 9 q^{10} + 9 q^{11} - 9 q^{12} - 42 q^{14} - 24 q^{15} - 15 q^{16} - 9 q^{17} - 3 q^{18} - 9 q^{19} - 18 q^{20}+ \cdots + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0159182 − 0.00280680i 0.0112558 − 0.00198471i −0.168017 − 0.985784i \(-0.553736\pi\)
0.179273 + 0.983799i \(0.442625\pi\)
\(3\) −0.402650 + 1.68460i −0.232470 + 0.972604i
\(4\) −1.87914 + 0.683951i −0.939570 + 0.341975i
\(5\) 3.75147 − 1.36542i 1.67771 − 0.610636i 0.684715 − 0.728811i \(-0.259927\pi\)
0.992993 + 0.118175i \(0.0377045\pi\)
\(6\) −0.00168110 + 0.0279459i −0.000686308 + 0.0114088i
\(7\) 0.157938 + 2.64103i 0.0596949 + 0.998217i
\(8\) −0.0559891 + 0.0323253i −0.0197951 + 0.0114287i
\(9\) −2.67575 − 1.35661i −0.891916 − 0.452202i
\(10\) 0.0558840 − 0.0322646i 0.0176721 − 0.0102030i
\(11\) −1.76436 + 4.84753i −0.531973 + 1.46158i 0.324746 + 0.945801i \(0.394721\pi\)
−0.856719 + 0.515783i \(0.827501\pi\)
\(12\) −0.395548 − 3.44099i −0.114185 − 0.993328i
\(13\) −0.220791 − 0.606618i −0.0612364 − 0.168246i 0.905301 − 0.424770i \(-0.139645\pi\)
−0.966538 + 0.256524i \(0.917423\pi\)
\(14\) 0.00992693 + 0.0415971i 0.00265308 + 0.0111173i
\(15\) 0.789663 + 6.86951i 0.203890 + 1.77370i
\(16\) 3.06298 − 2.57014i 0.765744 − 0.642536i
\(17\) 1.69383 + 2.93380i 0.410815 + 0.711552i 0.994979 − 0.100084i \(-0.0319111\pi\)
−0.584165 + 0.811635i \(0.698578\pi\)
\(18\) −0.0464007 − 0.0140844i −0.0109367 − 0.00331972i
\(19\) 0.328436 + 0.189623i 0.0753485 + 0.0435025i 0.537201 − 0.843454i \(-0.319482\pi\)
−0.461852 + 0.886957i \(0.652815\pi\)
\(20\) −6.11565 + 5.13164i −1.36750 + 1.14747i
\(21\) −4.51267 − 0.797349i −0.984746 − 0.173996i
\(22\) −0.0144792 + 0.0821158i −0.00308698 + 0.0175072i
\(23\) −0.993035 − 0.175099i −0.207062 − 0.0365106i 0.0691549 − 0.997606i \(-0.477970\pi\)
−0.276217 + 0.961095i \(0.589081\pi\)
\(24\) −0.0319112 − 0.107335i −0.00651385 − 0.0219097i
\(25\) 8.37891 − 7.03074i 1.67578 − 1.40615i
\(26\) −0.00521724 − 0.00903653i −0.00102318 − 0.00177221i
\(27\) 3.36272 − 3.96132i 0.647157 − 0.762357i
\(28\) −2.10312 − 4.85485i −0.397453 − 0.917480i
\(29\) 1.97120 − 5.41582i 0.366042 − 1.00569i −0.610810 − 0.791777i \(-0.709156\pi\)
0.976852 − 0.213916i \(-0.0686218\pi\)
\(30\) 0.0318513 + 0.107133i 0.00581523 + 0.0195598i
\(31\) −2.20741 − 6.06480i −0.396462 − 1.08927i −0.963995 − 0.265920i \(-0.914324\pi\)
0.567533 − 0.823351i \(-0.307898\pi\)
\(32\) 0.124656 − 0.148560i 0.0220363 − 0.0262619i
\(33\) −7.45572 − 4.92408i −1.29787 − 0.857173i
\(34\) 0.0351973 + 0.0419465i 0.00603628 + 0.00719376i
\(35\) 4.19863 + 9.69210i 0.709697 + 1.63826i
\(36\) 5.95595 + 0.719172i 0.992659 + 0.119862i
\(37\) 6.16814 1.01404 0.507018 − 0.861936i \(-0.330748\pi\)
0.507018 + 0.861936i \(0.330748\pi\)
\(38\) 0.00576033 + 0.00209659i 0.000934449 + 0.000340112i
\(39\) 1.11081 − 0.127690i 0.177872 − 0.0204467i
\(40\) −0.165904 + 0.197716i −0.0262317 + 0.0312617i
\(41\) 0.266252 − 0.0969077i 0.0415816 − 0.0151344i −0.321146 − 0.947030i \(-0.604068\pi\)
0.362727 + 0.931895i \(0.381846\pi\)
\(42\) −0.0740714 2.61403e-5i −0.0114295 4.03353e-6i
\(43\) −1.45052 − 8.22633i −0.221203 − 1.25450i −0.869812 − 0.493383i \(-0.835760\pi\)
0.648610 − 0.761121i \(-0.275351\pi\)
\(44\) − 10.3159i − 1.55518i
\(45\) −11.8903 − 1.43574i −1.77250 − 0.214027i
\(46\) −0.0162988 −0.00240312
\(47\) 6.91258 + 2.51597i 1.00830 + 0.366992i 0.792781 − 0.609507i \(-0.208632\pi\)
0.215522 + 0.976499i \(0.430855\pi\)
\(48\) 3.09635 + 6.19475i 0.446920 + 0.894136i
\(49\) −6.95011 + 0.834239i −0.992873 + 0.119177i
\(50\) 0.113643 − 0.135434i 0.0160715 − 0.0191533i
\(51\) −5.62430 + 1.67213i −0.787560 + 0.234145i
\(52\) 0.829794 + 0.988910i 0.115072 + 0.137137i
\(53\) 1.71989 + 0.992981i 0.236246 + 0.136396i 0.613450 − 0.789734i \(-0.289781\pi\)
−0.377204 + 0.926130i \(0.623115\pi\)
\(54\) 0.0424097 − 0.0724954i 0.00577123 − 0.00986538i
\(55\) 20.5944i 2.77695i
\(56\) −0.0942150 − 0.142764i −0.0125900 − 0.0190776i
\(57\) −0.451683 + 0.476932i −0.0598269 + 0.0631712i
\(58\) 0.0161767 − 0.0917427i 0.00212411 − 0.0120464i
\(59\) −2.62371 − 2.20155i −0.341578 − 0.286618i 0.455820 − 0.890072i \(-0.349346\pi\)
−0.797398 + 0.603454i \(0.793791\pi\)
\(60\) −6.18229 − 12.3687i −0.798130 − 1.59679i
\(61\) −2.75545 + 7.57053i −0.352799 + 0.969307i 0.628668 + 0.777674i \(0.283601\pi\)
−0.981467 + 0.191633i \(0.938622\pi\)
\(62\) −0.0521605 − 0.0903447i −0.00662439 − 0.0114738i
\(63\) 3.16024 − 7.28100i 0.398153 − 0.917319i
\(64\) −3.99687 + 6.92277i −0.499608 + 0.865347i
\(65\) −1.65658 − 1.97424i −0.205474 − 0.244874i
\(66\) −0.132502 − 0.0574556i −0.0163099 − 0.00707230i
\(67\) 1.63911 − 9.29584i 0.200249 − 1.13567i −0.704494 − 0.709710i \(-0.748826\pi\)
0.904743 − 0.425958i \(-0.140063\pi\)
\(68\) −5.18952 − 4.35453i −0.629322 − 0.528064i
\(69\) 0.694817 − 1.60236i 0.0836461 − 0.192902i
\(70\) 0.0940381 + 0.142496i 0.0112397 + 0.0170315i
\(71\) 8.57363 + 4.94999i 1.01750 + 0.587456i 0.913380 − 0.407109i \(-0.133463\pi\)
0.104123 + 0.994564i \(0.466796\pi\)
\(72\) 0.193665 − 0.0105392i 0.0228237 − 0.00124206i
\(73\) − 7.08887i − 0.829690i −0.909892 − 0.414845i \(-0.863836\pi\)
0.909892 − 0.414845i \(-0.136164\pi\)
\(74\) 0.0981854 − 0.0173127i 0.0114138 − 0.00201256i
\(75\) 8.47022 + 16.9460i 0.978056 + 1.95676i
\(76\) −0.746871 − 0.131693i −0.0856719 − 0.0151063i
\(77\) −13.0811 − 3.89411i −1.49073 − 0.443775i
\(78\) 0.0173236 − 0.00515040i 0.00196152 − 0.000583168i
\(79\) −0.222353 − 1.26102i −0.0250166 − 0.141876i 0.969741 − 0.244135i \(-0.0785041\pi\)
−0.994758 + 0.102259i \(0.967393\pi\)
\(80\) 7.98133 − 13.8241i 0.892340 − 1.54558i
\(81\) 5.31924 + 7.25987i 0.591027 + 0.806652i
\(82\) 0.00396624 − 0.00228991i 0.000437998 − 0.000252878i
\(83\) −8.07527 − 2.93916i −0.886376 − 0.322614i −0.141596 − 0.989925i \(-0.545223\pi\)
−0.744780 + 0.667310i \(0.767446\pi\)
\(84\) 9.02529 − 1.58812i 0.984740 − 0.173278i
\(85\) 10.3602 + 8.69327i 1.12373 + 0.942918i
\(86\) −0.0461793 − 0.126877i −0.00497964 − 0.0136815i
\(87\) 8.32979 + 5.50136i 0.893047 + 0.589807i
\(88\) −0.0579132 − 0.328442i −0.00617356 − 0.0350120i
\(89\) −2.67849 + 4.63928i −0.283920 + 0.491763i −0.972347 − 0.233543i \(-0.924968\pi\)
0.688427 + 0.725306i \(0.258302\pi\)
\(90\) −0.193302 + 0.0105194i −0.0203758 + 0.00110885i
\(91\) 1.56723 − 0.678924i 0.164290 − 0.0711706i
\(92\) 1.98581 − 0.350152i 0.207035 − 0.0365059i
\(93\) 11.1056 − 1.27661i 1.15159 − 0.132378i
\(94\) 0.117097 + 0.0206474i 0.0120777 + 0.00212962i
\(95\) 1.49103 + 0.262910i 0.152977 + 0.0269739i
\(96\) 0.200071 + 0.269813i 0.0204196 + 0.0275377i
\(97\) −7.39427 + 1.30381i −0.750774 + 0.132382i −0.535925 − 0.844265i \(-0.680037\pi\)
−0.214849 + 0.976647i \(0.568926\pi\)
\(98\) −0.108291 + 0.0327871i −0.0109391 + 0.00331200i
\(99\) 11.2971 − 10.5772i 1.13541 − 1.06305i
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.bd.a.185.12 yes 132
3.2 odd 2 567.2.bd.a.17.11 132
7.5 odd 6 189.2.ba.a.131.11 yes 132
21.5 even 6 567.2.ba.a.341.12 132
27.7 even 9 567.2.ba.a.143.12 132
27.20 odd 18 189.2.ba.a.101.11 ✓ 132
189.47 even 18 inner 189.2.bd.a.47.12 yes 132
189.61 odd 18 567.2.bd.a.467.11 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.11 ✓ 132 27.20 odd 18
189.2.ba.a.131.11 yes 132 7.5 odd 6
189.2.bd.a.47.12 yes 132 189.47 even 18 inner
189.2.bd.a.185.12 yes 132 1.1 even 1 trivial
567.2.ba.a.143.12 132 27.7 even 9
567.2.ba.a.341.12 132 21.5 even 6
567.2.bd.a.17.11 132 3.2 odd 2
567.2.bd.a.467.11 132 189.61 odd 18