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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [189,2,Mod(5,189)] 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("189.5"); 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([5, 15])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.ba (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 131.6
Character \(\chi\) \(=\) 189.131
Dual form 189.2.ba.a.101.6

$q$-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+(-1.07972 - 1.28676i) q^{2} +(-1.69732 - 0.345106i) q^{3} +(-0.142658 + 0.809053i) q^{4} +(2.87107 + 2.40911i) q^{5} +(1.38856 + 2.55666i) q^{6} +(-1.86335 + 1.87828i) q^{7} +(-1.71431 + 0.989760i) q^{8} +(2.76180 + 1.17151i) q^{9} -6.29553i q^{10} +(3.07307 + 3.66234i) q^{11} +(0.521346 - 1.32399i) q^{12} +(-1.10865 - 3.04598i) q^{13} +(4.42878 + 0.369659i) q^{14} +(-4.04173 - 5.07986i) q^{15} +(4.66853 + 1.69921i) q^{16} +3.86831 q^{17} +(-1.47452 - 4.81867i) q^{18} -1.52130i q^{19} +(-2.35868 + 1.97917i) q^{20} +(3.81090 - 2.54500i) q^{21} +(1.39450 - 7.90859i) q^{22} +(0.124554 + 0.342208i) q^{23} +(3.25131 - 1.08832i) q^{24} +(1.57097 + 8.90940i) q^{25} +(-2.72242 + 4.71536i) q^{26} +(-4.28337 - 2.94155i) q^{27} +(-1.25381 - 1.77550i) q^{28} +(-2.29492 + 6.30525i) q^{29} +(-2.17262 + 10.6855i) q^{30} +(-0.692139 - 0.122043i) q^{31} +(-1.50016 - 4.12165i) q^{32} +(-3.95209 - 7.27671i) q^{33} +(-4.17668 - 4.97757i) q^{34} +(-9.87478 + 0.903668i) q^{35} +(-1.34181 + 2.06732i) q^{36} +(2.75197 + 4.76655i) q^{37} +(-1.95755 + 1.64258i) q^{38} +(0.830544 + 5.55262i) q^{39} +(-7.30635 - 1.28831i) q^{40} +(0.793900 - 0.288956i) q^{41} +(-7.38949 - 2.15583i) q^{42} +(0.0136704 + 0.0775288i) q^{43} +(-3.40143 + 1.96381i) q^{44} +(5.10702 + 10.0170i) q^{45} +(0.305856 - 0.529758i) q^{46} +(0.617436 + 3.50165i) q^{47} +(-7.33760 - 4.49524i) q^{48} +(-0.0558886 - 6.99978i) q^{49} +(9.76803 - 11.6411i) q^{50} +(-6.56576 - 1.33498i) q^{51} +(2.62252 - 0.462421i) q^{52} +(7.25668 - 4.18964i) q^{53} +(0.839774 + 8.68770i) q^{54} +17.9182i q^{55} +(1.33531 - 5.06423i) q^{56} +(-0.525011 + 2.58214i) q^{57} +(10.5912 - 3.85488i) q^{58} +(1.36168 - 0.495612i) q^{59} +(4.68646 - 2.54529i) q^{60} +(2.82040 - 0.497313i) q^{61} +(0.590275 + 1.02239i) q^{62} +(-7.34662 + 3.00451i) q^{63} +(1.28433 - 2.22452i) q^{64} +(4.15511 - 11.4161i) q^{65} +(-5.09621 + 12.9422i) q^{66} +(3.09944 + 2.60074i) q^{67} +(-0.551844 + 3.12967i) q^{68} +(-0.0933094 - 0.623821i) q^{69} +(11.8248 + 11.7307i) q^{70} +(-14.1956 - 8.19580i) q^{71} +(-5.89411 + 0.725182i) q^{72} +(-10.4351 - 6.02468i) q^{73} +(3.16204 - 8.68764i) q^{74} +(0.408251 - 15.6643i) q^{75} +(1.23082 + 0.217026i) q^{76} +(-12.6051 - 1.05212i) q^{77} +(6.24812 - 7.06396i) q^{78} +(1.00589 - 0.844043i) q^{79} +(9.31009 + 16.1256i) q^{80} +(6.25512 + 6.47097i) q^{81} +(-1.22900 - 0.709566i) q^{82} +(-12.7610 - 4.64462i) q^{83} +(1.51538 + 3.44629i) q^{84} +(11.1062 + 9.31918i) q^{85} +(0.0850005 - 0.101300i) q^{86} +(6.07120 - 9.91005i) q^{87} +(-8.89304 - 3.23680i) q^{88} +8.72878 q^{89} +(7.37529 - 17.3870i) q^{90} +(7.78701 + 3.59337i) q^{91} +(-0.294633 + 0.0519517i) q^{92} +(1.13267 + 0.446007i) q^{93} +(3.83912 - 4.57528i) q^{94} +(3.66499 - 4.36777i) q^{95} +(1.12385 + 7.51349i) q^{96} +(-4.18960 + 0.738740i) q^{97} +(-8.94667 + 7.62970i) q^{98} +(4.19674 + 13.7148i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} - 18 q^{6} - 6 q^{7} - 18 q^{8} + 3 q^{9} - 9 q^{11} - 9 q^{12} + 3 q^{14} - 24 q^{15} + 3 q^{16} - 18 q^{17} - 3 q^{18} + 18 q^{20} - 21 q^{21} - 12 q^{22}+ \cdots + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.07972 − 1.28676i −0.763476 − 0.909875i 0.234587 − 0.972095i \(-0.424626\pi\)
−0.998062 + 0.0622204i \(0.980182\pi\)
\(3\) −1.69732 − 0.345106i −0.979949 − 0.199247i
\(4\) −0.142658 + 0.809053i −0.0713289 + 0.404527i
\(5\) 2.87107 + 2.40911i 1.28398 + 1.07739i 0.992682 + 0.120755i \(0.0385315\pi\)
0.291298 + 0.956632i \(0.405913\pi\)
\(6\) 1.38856 + 2.55666i 0.566877 + 1.04375i
\(7\) −1.86335 + 1.87828i −0.704278 + 0.709924i
\(8\) −1.71431 + 0.989760i −0.606101 + 0.349933i
\(9\) 2.76180 + 1.17151i 0.920601 + 0.390504i
\(10\) − 6.29553i − 1.99082i
\(11\) 3.07307 + 3.66234i 0.926565 + 1.10424i 0.994309 + 0.106535i \(0.0339756\pi\)
−0.0677435 + 0.997703i \(0.521580\pi\)
\(12\) 0.521346 − 1.32399i 0.150499 − 0.382203i
\(13\) −1.10865 − 3.04598i −0.307483 − 0.844804i −0.993146 − 0.116884i \(-0.962709\pi\)
0.685662 − 0.727920i \(-0.259513\pi\)
\(14\) 4.42878 + 0.369659i 1.18364 + 0.0987955i
\(15\) −4.04173 − 5.07986i −1.04357 − 1.31161i
\(16\) 4.66853 + 1.69921i 1.16713 + 0.424802i
\(17\) 3.86831 0.938202 0.469101 − 0.883144i \(-0.344578\pi\)
0.469101 + 0.883144i \(0.344578\pi\)
\(18\) −1.47452 − 4.81867i −0.347547 − 1.13577i
\(19\) − 1.52130i − 0.349011i −0.984656 − 0.174506i \(-0.944167\pi\)
0.984656 − 0.174506i \(-0.0558327\pi\)
\(20\) −2.35868 + 1.97917i −0.527417 + 0.442555i
\(21\) 3.81090 − 2.54500i 0.831607 − 0.555364i
\(22\) 1.39450 − 7.90859i 0.297308 − 1.68612i
\(23\) 0.124554 + 0.342208i 0.0259712 + 0.0713553i 0.952001 − 0.306096i \(-0.0990228\pi\)
−0.926029 + 0.377451i \(0.876801\pi\)
\(24\) 3.25131 − 1.08832i 0.663672 − 0.222152i
\(25\) 1.57097 + 8.90940i 0.314194 + 1.78188i
\(26\) −2.72242 + 4.71536i −0.533910 + 0.924758i
\(27\) −4.28337 − 2.94155i −0.824336 − 0.566101i
\(28\) −1.25381 − 1.77550i −0.236948 − 0.335537i
\(29\) −2.29492 + 6.30525i −0.426157 + 1.17086i 0.521970 + 0.852964i \(0.325197\pi\)
−0.948127 + 0.317892i \(0.897025\pi\)
\(30\) −2.17262 + 10.6855i −0.396665 + 1.95090i
\(31\) −0.692139 − 0.122043i −0.124312 − 0.0219195i 0.111146 − 0.993804i \(-0.464548\pi\)
−0.235458 + 0.971885i \(0.575659\pi\)
\(32\) −1.50016 − 4.12165i −0.265193 − 0.728612i
\(33\) −3.95209 − 7.27671i −0.687971 − 1.26671i
\(34\) −4.17668 − 4.97757i −0.716294 − 0.853647i
\(35\) −9.87478 + 0.903668i −1.66914 + 0.152748i
\(36\) −1.34181 + 2.06732i −0.223635 + 0.344553i
\(37\) 2.75197 + 4.76655i 0.452421 + 0.783616i 0.998536 − 0.0540945i \(-0.0172272\pi\)
−0.546115 + 0.837710i \(0.683894\pi\)
\(38\) −1.95755 + 1.64258i −0.317556 + 0.266461i
\(39\) 0.830544 + 5.55262i 0.132993 + 0.889130i
\(40\) −7.30635 − 1.28831i −1.15524 − 0.203699i
\(41\) 0.793900 − 0.288956i 0.123986 − 0.0451273i −0.279282 − 0.960209i \(-0.590096\pi\)
0.403268 + 0.915082i \(0.367874\pi\)
\(42\) −7.38949 − 2.15583i −1.14022 − 0.332652i
\(43\) 0.0136704 + 0.0775288i 0.00208472 + 0.0118230i 0.985832 − 0.167734i \(-0.0536448\pi\)
−0.983748 + 0.179557i \(0.942534\pi\)
\(44\) −3.40143 + 1.96381i −0.512784 + 0.296056i
\(45\) 5.10702 + 10.0170i 0.761309 + 1.49324i
\(46\) 0.305856 − 0.529758i 0.0450960 − 0.0781086i
\(47\) 0.617436 + 3.50165i 0.0900622 + 0.510768i 0.996149 + 0.0876764i \(0.0279441\pi\)
−0.906087 + 0.423092i \(0.860945\pi\)
\(48\) −7.33760 − 4.49524i −1.05909 − 0.648832i
\(49\) −0.0558886 − 6.99978i −0.00798409 − 0.999968i
\(50\) 9.76803 − 11.6411i 1.38141 − 1.64630i
\(51\) −6.56576 − 1.33498i −0.919391 − 0.186934i
\(52\) 2.62252 − 0.462421i 0.363678 − 0.0641262i
\(53\) 7.25668 − 4.18964i 0.996781 − 0.575492i 0.0894868 − 0.995988i \(-0.471477\pi\)
0.907294 + 0.420496i \(0.138144\pi\)
\(54\) 0.839774 + 8.68770i 0.114279 + 1.18225i
\(55\) 17.9182i 2.41609i
\(56\) 1.33531 − 5.06423i 0.178438 − 0.676736i
\(57\) −0.525011 + 2.58214i −0.0695394 + 0.342013i
\(58\) 10.5912 − 3.85488i 1.39069 − 0.506170i
\(59\) 1.36168 − 0.495612i 0.177276 − 0.0645231i −0.251857 − 0.967764i \(-0.581041\pi\)
0.429133 + 0.903241i \(0.358819\pi\)
\(60\) 4.68646 − 2.54529i 0.605019 − 0.328595i
\(61\) 2.82040 − 0.497313i 0.361116 − 0.0636745i 0.00985324 − 0.999951i \(-0.496864\pi\)
0.351263 + 0.936277i \(0.385752\pi\)
\(62\) 0.590275 + 1.02239i 0.0749650 + 0.129843i
\(63\) −7.34662 + 3.00451i −0.925588 + 0.378533i
\(64\) 1.28433 − 2.22452i 0.160541 − 0.278066i
\(65\) 4.15511 − 11.4161i 0.515378 − 1.41599i
\(66\) −5.09621 + 12.9422i −0.627301 + 1.59307i
\(67\) 3.09944 + 2.60074i 0.378657 + 0.317731i 0.812175 − 0.583414i \(-0.198284\pi\)
−0.433518 + 0.901145i \(0.642728\pi\)
\(68\) −0.551844 + 3.12967i −0.0669210 + 0.379528i
\(69\) −0.0933094 − 0.623821i −0.0112331 − 0.0750993i
\(70\) 11.8248 + 11.7307i 1.41333 + 1.40209i
\(71\) −14.1956 − 8.19580i −1.68470 − 0.972663i −0.958461 − 0.285222i \(-0.907933\pi\)
−0.726240 − 0.687441i \(-0.758734\pi\)
\(72\) −5.89411 + 0.725182i −0.694628 + 0.0854635i
\(73\) −10.4351 − 6.02468i −1.22133 − 0.705136i −0.256129 − 0.966643i \(-0.582447\pi\)
−0.965202 + 0.261507i \(0.915781\pi\)
\(74\) 3.16204 − 8.68764i 0.367580 − 1.00992i
\(75\) 0.408251 − 15.6643i 0.0471407 − 1.80875i
\(76\) 1.23082 + 0.217026i 0.141184 + 0.0248946i
\(77\) −12.6051 − 1.05212i −1.43648 − 0.119900i
\(78\) 6.24812 − 7.06396i 0.707460 − 0.799836i
\(79\) 1.00589 − 0.844043i 0.113172 − 0.0949623i −0.584446 − 0.811433i \(-0.698688\pi\)
0.697617 + 0.716471i \(0.254244\pi\)
\(80\) 9.31009 + 16.1256i 1.04090 + 1.80289i
\(81\) 6.25512 + 6.47097i 0.695013 + 0.718997i
\(82\) −1.22900 − 0.709566i −0.135721 − 0.0783584i
\(83\) −12.7610 − 4.64462i −1.40070 − 0.509814i −0.472315 − 0.881430i \(-0.656581\pi\)
−0.928386 + 0.371616i \(0.878804\pi\)
\(84\) 1.51538 + 3.44629i 0.165342 + 0.376021i
\(85\) 11.1062 + 9.31918i 1.20463 + 1.01081i
\(86\) 0.0850005 − 0.101300i 0.00916584 − 0.0109234i
\(87\) 6.07120 − 9.91005i 0.650901 − 1.06247i
\(88\) −8.89304 − 3.23680i −0.948002 − 0.345044i
\(89\) 8.72878 0.925248 0.462624 − 0.886555i \(-0.346908\pi\)
0.462624 + 0.886555i \(0.346908\pi\)
\(90\) 7.37529 − 17.3870i 0.777423 − 1.83275i
\(91\) 7.78701 + 3.59337i 0.816300 + 0.376687i
\(92\) −0.294633 + 0.0519517i −0.0307176 + 0.00541634i
\(93\) 1.13267 + 0.446007i 0.117452 + 0.0462488i
\(94\) 3.83912 − 4.57528i 0.395975 − 0.471904i
\(95\) 3.66499 − 4.36777i 0.376020 − 0.448123i
\(96\) 1.12385 + 7.51349i 0.114702 + 0.766842i
\(97\) −4.18960 + 0.738740i −0.425390 + 0.0750077i −0.382245 − 0.924061i \(-0.624849\pi\)
−0.0431451 + 0.999069i \(0.513738\pi\)
\(98\) −8.94667 + 7.62970i −0.903750 + 0.770716i
\(99\) 4.19674 + 13.7148i 0.421788 + 1.37839i
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.ba.a.131.6 yes 132
3.2 odd 2 567.2.ba.a.341.17 132
7.3 odd 6 189.2.bd.a.185.17 yes 132
21.17 even 6 567.2.bd.a.17.6 132
27.7 even 9 567.2.bd.a.467.6 132
27.20 odd 18 189.2.bd.a.47.17 yes 132
189.101 even 18 inner 189.2.ba.a.101.6 ✓ 132
189.115 odd 18 567.2.ba.a.143.17 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.6 ✓ 132 189.101 even 18 inner
189.2.ba.a.131.6 yes 132 1.1 even 1 trivial
189.2.bd.a.47.17 yes 132 27.20 odd 18
189.2.bd.a.185.17 yes 132 7.3 odd 6
567.2.ba.a.143.17 132 189.115 odd 18
567.2.ba.a.341.17 132 3.2 odd 2
567.2.bd.a.17.6 132 21.17 even 6
567.2.bd.a.467.6 132 27.7 even 9