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Properties

Label 189.2.be.a.104.10
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 104.10
Character \(\chi\) \(=\) 189.104
Dual form 189.2.be.a.20.10

$q$-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+(-0.234777 + 0.645045i) q^{2} +(0.440723 - 1.67504i) q^{3} +(1.17113 + 0.982692i) q^{4} +(0.231854 - 1.31491i) q^{5} +(0.977005 + 0.677547i) q^{6} +(1.38106 - 2.25670i) q^{7} +(-2.09779 + 1.21116i) q^{8} +(-2.61153 - 1.47646i) q^{9} +(0.793741 + 0.458267i) q^{10} +(0.216508 - 0.0381762i) q^{11} +(2.16219 - 1.52859i) q^{12} +(0.673991 + 1.85178i) q^{13} +(1.13143 + 1.42066i) q^{14} +(-2.10034 - 0.967875i) q^{15} +(0.242207 + 1.37362i) q^{16} +(-0.469803 + 0.813723i) q^{17} +(1.56551 - 1.33791i) q^{18} +(3.04150 - 1.75601i) q^{19} +(1.56368 - 1.31208i) q^{20} +(-3.17140 - 3.30790i) q^{21} +(-0.0262057 + 0.148620i) q^{22} +(1.17579 - 1.40126i) q^{23} +(1.10420 + 4.04766i) q^{24} +(3.02323 + 1.10037i) q^{25} -1.35272 q^{26} +(-3.62409 + 3.72371i) q^{27} +(3.83503 - 1.28572i) q^{28} +(-1.33239 + 3.66070i) q^{29} +(1.11744 - 1.12758i) q^{30} +(-5.95195 + 7.09326i) q^{31} +(-5.71394 - 1.00752i) q^{32} +(0.0314733 - 0.379485i) q^{33} +(-0.414589 - 0.494088i) q^{34} +(-2.64715 - 2.33919i) q^{35} +(-1.60753 - 4.29544i) q^{36} +(-1.56504 + 2.71072i) q^{37} +(0.418631 + 2.37418i) q^{38} +(3.39885 - 0.312844i) q^{39} +(1.10618 + 3.03921i) q^{40} +(-10.8664 + 3.95506i) q^{41} +(2.87832 - 1.26907i) q^{42} +(-0.261843 - 1.48498i) q^{43} +(0.291074 + 0.168051i) q^{44} +(-2.54690 + 3.09160i) q^{45} +(0.627824 + 1.08742i) q^{46} +(5.18484 - 4.35059i) q^{47} +(2.40762 + 0.199681i) q^{48} +(-3.18536 - 6.23325i) q^{49} +(-1.41957 + 1.69178i) q^{50} +(1.15597 + 1.14557i) q^{51} +(-1.03040 + 2.83099i) q^{52} -3.77077i q^{53} +(-1.55111 - 3.21194i) q^{54} -0.293540i q^{55} +(-0.163947 + 6.40675i) q^{56} +(-1.60093 - 5.86856i) q^{57} +(-2.04850 - 1.71890i) q^{58} +(-1.72130 + 9.76195i) q^{59} +(-1.50865 - 3.19750i) q^{60} +(-6.04827 - 7.20805i) q^{61} +(-3.17809 - 5.50461i) q^{62} +(-6.93858 + 3.85435i) q^{63} +(0.596586 - 1.03332i) q^{64} +(2.59119 - 0.456896i) q^{65} +(0.237395 + 0.109396i) q^{66} +(-6.36714 + 2.31745i) q^{67} +(-1.34984 + 0.491301i) q^{68} +(-1.82897 - 2.58707i) q^{69} +(2.13037 - 1.15834i) q^{70} +(11.8706 + 6.85351i) q^{71} +(7.26665 - 0.0656782i) q^{72} +(7.68223 - 4.43534i) q^{73} +(-1.38110 - 1.64593i) q^{74} +(3.17557 - 4.57908i) q^{75} +(5.28760 + 0.932347i) q^{76} +(0.212858 - 0.541316i) q^{77} +(-0.596173 + 2.26586i) q^{78} +(-7.91776 - 2.88183i) q^{79} +1.86235 q^{80} +(4.64015 + 7.71162i) q^{81} -7.93789i q^{82} +(16.1714 + 5.88589i) q^{83} +(-0.463457 - 6.99048i) q^{84} +(0.961047 + 0.806414i) q^{85} +(1.01936 + 0.179740i) q^{86} +(5.54462 + 3.84516i) q^{87} +(-0.407950 + 0.342311i) q^{88} +(-3.46240 - 5.99705i) q^{89} +(-1.39627 - 2.36870i) q^{90} +(5.10972 + 1.03641i) q^{91} +(2.75401 - 0.485606i) q^{92} +(9.25834 + 13.0959i) q^{93} +(1.58905 + 4.36587i) q^{94} +(-1.60381 - 4.40644i) q^{95} +(-4.20591 + 9.12705i) q^{96} +(-4.42872 + 0.780904i) q^{97} +(4.76858 - 0.591277i) q^{98} +(-0.621782 - 0.219967i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.234777 + 0.645045i −0.166012 + 0.456116i −0.994605 − 0.103737i \(-0.966920\pi\)
0.828592 + 0.559852i \(0.189142\pi\)
\(3\) 0.440723 − 1.67504i 0.254451 − 0.967086i
\(4\) 1.17113 + 0.982692i 0.585563 + 0.491346i
\(5\) 0.231854 − 1.31491i 0.103688 − 0.588045i −0.888048 − 0.459751i \(-0.847939\pi\)
0.991736 − 0.128294i \(-0.0409502\pi\)
\(6\) 0.977005 + 0.677547i 0.398861 + 0.276607i
\(7\) 1.38106 − 2.25670i 0.521990 − 0.852951i
\(8\) −2.09779 + 1.21116i −0.741680 + 0.428209i
\(9\) −2.61153 − 1.47646i −0.870509 − 0.492152i
\(10\) 0.793741 + 0.458267i 0.251003 + 0.144917i
\(11\) 0.216508 − 0.0381762i 0.0652796 − 0.0115106i −0.140913 − 0.990022i \(-0.545004\pi\)
0.206193 + 0.978511i \(0.433893\pi\)
\(12\) 2.16219 − 1.52859i 0.624171 − 0.441266i
\(13\) 0.673991 + 1.85178i 0.186932 + 0.513590i 0.997390 − 0.0722062i \(-0.0230040\pi\)
−0.810458 + 0.585797i \(0.800782\pi\)
\(14\) 1.13143 + 1.42066i 0.302387 + 0.379689i
\(15\) −2.10034 − 0.967875i −0.542307 − 0.249904i
\(16\) 0.242207 + 1.37362i 0.0605517 + 0.343406i
\(17\) −0.469803 + 0.813723i −0.113944 + 0.197357i −0.917357 − 0.398065i \(-0.869682\pi\)
0.803413 + 0.595422i \(0.203015\pi\)
\(18\) 1.56551 − 1.33791i 0.368994 − 0.315349i
\(19\) 3.04150 − 1.75601i 0.697768 − 0.402857i −0.108747 − 0.994069i \(-0.534684\pi\)
0.806516 + 0.591213i \(0.201351\pi\)
\(20\) 1.56368 − 1.31208i 0.349650 − 0.293391i
\(21\) −3.17140 − 3.30790i −0.692056 − 0.721844i
\(22\) −0.0262057 + 0.148620i −0.00558708 + 0.0316859i
\(23\) 1.17579 − 1.40126i 0.245170 − 0.292182i −0.629400 − 0.777082i \(-0.716699\pi\)
0.874570 + 0.484899i \(0.161144\pi\)
\(24\) 1.10420 + 4.04766i 0.225393 + 0.826226i
\(25\) 3.02323 + 1.10037i 0.604647 + 0.220073i
\(26\) −1.35272 −0.265289
\(27\) −3.62409 + 3.72371i −0.697456 + 0.716628i
\(28\) 3.83503 − 1.28572i 0.724753 − 0.242979i
\(29\) −1.33239 + 3.66070i −0.247418 + 0.679776i 0.752361 + 0.658751i \(0.228915\pi\)
−0.999779 + 0.0210245i \(0.993307\pi\)
\(30\) 1.11744 − 1.12758i 0.204015 − 0.205867i
\(31\) −5.95195 + 7.09326i −1.06900 + 1.27399i −0.108985 + 0.994043i \(0.534760\pi\)
−0.960017 + 0.279943i \(0.909684\pi\)
\(32\) −5.71394 − 1.00752i −1.01009 − 0.178106i
\(33\) 0.0314733 − 0.379485i 0.00547879 − 0.0660598i
\(34\) −0.414589 − 0.494088i −0.0711014 − 0.0847353i
\(35\) −2.64715 − 2.33919i −0.447450 − 0.395395i
\(36\) −1.60753 − 4.29544i −0.267921 − 0.715907i
\(37\) −1.56504 + 2.71072i −0.257290 + 0.445640i −0.965515 − 0.260347i \(-0.916163\pi\)
0.708225 + 0.705987i \(0.249496\pi\)
\(38\) 0.418631 + 2.37418i 0.0679110 + 0.385142i
\(39\) 3.39885 − 0.312844i 0.544251 − 0.0500951i
\(40\) 1.10618 + 3.03921i 0.174903 + 0.480542i
\(41\) −10.8664 + 3.95506i −1.69705 + 0.617676i −0.995484 − 0.0949267i \(-0.969738\pi\)
−0.701567 + 0.712603i \(0.747516\pi\)
\(42\) 2.87832 − 1.26907i 0.444134 − 0.195822i
\(43\) −0.261843 − 1.48498i −0.0399307 − 0.226458i 0.958311 − 0.285726i \(-0.0922346\pi\)
−0.998242 + 0.0592676i \(0.981123\pi\)
\(44\) 0.291074 + 0.168051i 0.0438810 + 0.0253347i
\(45\) −2.54690 + 3.09160i −0.379670 + 0.460868i
\(46\) 0.627824 + 1.08742i 0.0925676 + 0.160332i
\(47\) 5.18484 − 4.35059i 0.756286 − 0.634599i −0.180871 − 0.983507i \(-0.557892\pi\)
0.937157 + 0.348907i \(0.113447\pi\)
\(48\) 2.40762 + 0.199681i 0.347510 + 0.0288214i
\(49\) −3.18536 − 6.23325i −0.455052 − 0.890465i
\(50\) −1.41957 + 1.69178i −0.200758 + 0.239254i
\(51\) 1.15597 + 1.14557i 0.161868 + 0.160411i
\(52\) −1.03040 + 2.83099i −0.142890 + 0.392588i
\(53\) − 3.77077i − 0.517954i −0.965883 − 0.258977i \(-0.916615\pi\)
0.965883 − 0.258977i \(-0.0833855\pi\)
\(54\) −1.55111 − 3.21194i −0.211079 − 0.437090i
\(55\) − 0.293540i − 0.0395809i
\(56\) −0.163947 + 6.40675i −0.0219083 + 0.856138i
\(57\) −1.60093 − 5.86856i −0.212049 − 0.777309i
\(58\) −2.04850 − 1.71890i −0.268982 − 0.225703i
\(59\) −1.72130 + 9.76195i −0.224094 + 1.27090i 0.640317 + 0.768111i \(0.278803\pi\)
−0.864410 + 0.502787i \(0.832308\pi\)
\(60\) −1.50865 − 3.19750i −0.194765 − 0.412795i
\(61\) −6.04827 − 7.20805i −0.774402 − 0.922896i 0.224264 − 0.974528i \(-0.428002\pi\)
−0.998666 + 0.0516319i \(0.983558\pi\)
\(62\) −3.17809 − 5.50461i −0.403617 − 0.699086i
\(63\) −6.93858 + 3.85435i −0.874179 + 0.485603i
\(64\) 0.596586 − 1.03332i 0.0745733 − 0.129165i
\(65\) 2.59119 − 0.456896i 0.321397 − 0.0566710i
\(66\) 0.237395 + 0.109396i 0.0292214 + 0.0134657i
\(67\) −6.36714 + 2.31745i −0.777871 + 0.283122i −0.700284 − 0.713864i \(-0.746943\pi\)
−0.0775861 + 0.996986i \(0.524721\pi\)
\(68\) −1.34984 + 0.491301i −0.163692 + 0.0595790i
\(69\) −1.82897 − 2.58707i −0.220182 − 0.311447i
\(70\) 2.13037 − 1.15834i 0.254628 − 0.138448i
\(71\) 11.8706 + 6.85351i 1.40879 + 0.813362i 0.995271 − 0.0971354i \(-0.0309680\pi\)
0.413514 + 0.910498i \(0.364301\pi\)
\(72\) 7.26665 − 0.0656782i 0.856383 − 0.00774025i
\(73\) 7.68223 − 4.43534i 0.899137 − 0.519117i 0.0222170 − 0.999753i \(-0.492928\pi\)
0.876920 + 0.480636i \(0.159594\pi\)
\(74\) −1.38110 − 1.64593i −0.160550 − 0.191336i
\(75\) 3.17557 − 4.57908i 0.366683 − 0.528747i
\(76\) 5.28760 + 0.932347i 0.606529 + 0.106948i
\(77\) 0.212858 − 0.541316i 0.0242574 − 0.0616887i
\(78\) −0.596173 + 2.26586i −0.0675033 + 0.256558i
\(79\) −7.91776 − 2.88183i −0.890818 − 0.324231i −0.144251 − 0.989541i \(-0.546077\pi\)
−0.746567 + 0.665310i \(0.768299\pi\)
\(80\) 1.86235 0.208217
\(81\) 4.64015 + 7.71162i 0.515572 + 0.856846i
\(82\) − 7.93789i − 0.876594i
\(83\) 16.1714 + 5.88589i 1.77504 + 0.646061i 0.999898 + 0.0142697i \(0.00454236\pi\)
0.775139 + 0.631791i \(0.217680\pi\)
\(84\) −0.463457 − 6.99048i −0.0505673 − 0.762724i
\(85\) 0.961047 + 0.806414i 0.104240 + 0.0874679i
\(86\) 1.01936 + 0.179740i 0.109920 + 0.0193819i
\(87\) 5.54462 + 3.84516i 0.594445 + 0.412244i
\(88\) −0.407950 + 0.342311i −0.0434876 + 0.0364904i
\(89\) −3.46240 − 5.99705i −0.367014 − 0.635686i 0.622084 − 0.782951i \(-0.286286\pi\)
−0.989097 + 0.147265i \(0.952953\pi\)
\(90\) −1.39627 − 2.36870i −0.147179 − 0.249683i
\(91\) 5.10972 + 1.03641i 0.535644 + 0.108646i
\(92\) 2.75401 − 0.485606i 0.287125 − 0.0506279i
\(93\) 9.25834 + 13.0959i 0.960045 + 1.35798i
\(94\) 1.58905 + 4.36587i 0.163898 + 0.450305i
\(95\) −1.60381 − 4.40644i −0.164548 − 0.452091i
\(96\) −4.20591 + 9.12705i −0.429264 + 0.931526i
\(97\) −4.42872 + 0.780904i −0.449669 + 0.0792887i −0.393897 − 0.919155i \(-0.628873\pi\)
−0.0557722 + 0.998444i \(0.517762\pi\)
\(98\) 4.76858 − 0.591277i 0.481699 − 0.0597280i
\(99\) −0.621782 − 0.219967i −0.0624914 − 0.0221075i
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.104.10 yes 132
3.2 odd 2 567.2.be.a.503.13 132
7.6 odd 2 inner 189.2.be.a.104.9 yes 132
21.20 even 2 567.2.be.a.503.14 132
27.7 even 9 567.2.be.a.62.14 132
27.20 odd 18 inner 189.2.be.a.20.9 ✓ 132
189.20 even 18 inner 189.2.be.a.20.10 yes 132
189.34 odd 18 567.2.be.a.62.13 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.be.a.20.9 ✓ 132 27.20 odd 18 inner
189.2.be.a.20.10 yes 132 189.20 even 18 inner
189.2.be.a.104.9 yes 132 7.6 odd 2 inner
189.2.be.a.104.10 yes 132 1.1 even 1 trivial
567.2.be.a.62.13 132 189.34 odd 18
567.2.be.a.62.14 132 27.7 even 9
567.2.be.a.503.13 132 3.2 odd 2
567.2.be.a.503.14 132 21.20 even 2