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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 47.17
Character \(\chi\) \(=\) 189.47
Dual form 189.2.bd.a.185.17

$q$-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+(1.65422 + 0.291684i) q^{2} +(-0.549790 + 1.64248i) q^{3} +(0.771989 + 0.280981i) q^{4} +(3.52188 + 1.28186i) q^{5} +(-1.38856 + 2.55666i) q^{6} +(-2.17331 - 1.50888i) q^{7} +(-1.71431 - 0.989760i) q^{8} +(-2.39546 - 1.80604i) q^{9} +(5.45209 + 3.14776i) q^{10} +(1.63515 + 4.49253i) q^{11} +(-0.885937 + 1.11349i) q^{12} +(1.10865 - 3.04598i) q^{13} +(-3.15503 - 3.12994i) q^{14} +(-4.04173 + 5.07986i) q^{15} +(-3.80582 - 3.19347i) q^{16} +(1.93415 - 3.35005i) q^{17} +(-3.43584 - 3.68630i) q^{18} +(-1.31749 + 0.760652i) q^{19} +(2.35868 + 1.97917i) q^{20} +(3.67316 - 2.74005i) q^{21} +(1.39450 + 7.90859i) q^{22} +(-0.358638 + 0.0632375i) q^{23} +(2.56817 - 2.27156i) q^{24} +(6.93028 + 5.81520i) 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} +(-8.16763 + 7.22431i) q^{30} +(-0.240377 + 0.660432i) q^{31} +(-2.81938 - 3.36000i) q^{32} +(-8.27786 + 0.215742i) q^{33} +(4.17668 - 4.97757i) q^{34} +(-5.71999 - 8.09998i) q^{35} +(-1.34181 - 2.06732i) q^{36} -5.50394 q^{37} +(-2.40129 + 0.873998i) q^{38} +(4.39343 + 3.49558i) q^{39} +(-4.76888 - 5.68333i) q^{40} +(-0.793900 - 0.288956i) q^{41} +(6.87546 - 3.46125i) q^{42} +(0.0136704 - 0.0775288i) q^{43} +3.92763i q^{44} +(-6.12145 - 9.43130i) q^{45} -0.611712 q^{46} +(3.34124 - 1.21611i) q^{47} +(7.33760 - 4.49524i) q^{48} +(2.44658 + 6.55852i) q^{49} +(9.76803 + 11.6411i) q^{50} +(4.43900 + 5.01863i) q^{51} +(1.71173 - 2.03996i) q^{52} +(-7.25668 + 4.18964i) q^{53} +(7.94366 - 3.61659i) q^{54} +17.9182i q^{55} +(2.23232 + 4.73775i) q^{56} +(-0.525011 - 2.58214i) q^{57} +(-1.95717 - 11.0997i) q^{58} +(1.11005 - 0.931445i) q^{59} +(-4.54752 + 2.78595i) q^{60} +(0.979516 + 2.69120i) q^{61} +(-0.590275 + 1.02239i) q^{62} +(2.48100 + 7.53954i) q^{63} +(1.28433 + 2.22452i) q^{64} +(7.80906 - 9.30647i) q^{65} +(-13.7564 - 2.05764i) q^{66} +(0.702586 + 3.98456i) q^{67} +(2.43445 - 2.04274i) q^{68} +(0.0933094 - 0.623821i) q^{69} +(-7.09950 - 15.0676i) q^{70} +(-14.1956 + 8.19580i) q^{71} +(2.31903 + 5.46704i) q^{72} +12.0494i q^{73} +(-9.10474 - 1.60541i) q^{74} +(-13.3615 + 8.18569i) q^{75} +(-1.23082 + 0.217026i) q^{76} +(3.22498 - 12.2309i) q^{77} +(6.24812 + 7.06396i) q^{78} +(0.228017 - 1.29315i) q^{79} +(-9.31009 - 16.1256i) q^{80} +(2.47647 + 8.65258i) q^{81} +(-1.22900 - 0.709566i) q^{82} +(12.7610 - 4.64462i) q^{83} +(3.60555 - 1.08320i) q^{84} +(11.1062 - 9.31918i) q^{85} +(0.0452278 - 0.124262i) q^{86} +(11.6180 - 0.302793i) q^{87} +(1.64337 - 9.32000i) q^{88} +(4.36439 + 7.55934i) q^{89} +(-7.37529 - 17.3870i) q^{90} +(-7.00545 + 4.94706i) q^{91} +(-0.294633 - 0.0519517i) q^{92} +(-0.952586 - 0.757913i) q^{93} +(5.88187 - 1.03713i) q^{94} +(-5.61509 + 0.990092i) q^{95} +(7.06880 - 2.78347i) q^{96} +(4.18960 + 0.738740i) q^{97} +(2.13418 + 11.5629i) q^{98} +(4.19674 - 13.7148i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} + 18 q^{6} - 6 q^{7} - 18 q^{8} - 15 q^{9} - 9 q^{10} + 9 q^{11} - 9 q^{12} - 42 q^{14} - 24 q^{15} - 15 q^{16} - 9 q^{17} - 3 q^{18} - 9 q^{19} - 18 q^{20}+ \cdots + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.65422 + 0.291684i 1.16971 + 0.206252i 0.724566 − 0.689206i \(-0.242040\pi\)
0.445147 + 0.895458i \(0.353152\pi\)
\(3\) −0.549790 + 1.64248i −0.317422 + 0.948285i
\(4\) 0.771989 + 0.280981i 0.385995 + 0.140491i
\(5\) 3.52188 + 1.28186i 1.57503 + 0.573266i 0.974117 − 0.226045i \(-0.0725796\pi\)
0.600918 + 0.799311i \(0.294802\pi\)
\(6\) −1.38856 + 2.55666i −0.566877 + 1.04375i
\(7\) −2.17331 − 1.50888i −0.821435 − 0.570302i
\(8\) −1.71431 − 0.989760i −0.606101 − 0.349933i
\(9\) −2.39546 − 1.80604i −0.798487 − 0.602012i
\(10\) 5.45209 + 3.14776i 1.72410 + 0.995410i
\(11\) 1.63515 + 4.49253i 0.493015 + 1.35455i 0.897908 + 0.440184i \(0.145087\pi\)
−0.404893 + 0.914364i \(0.632691\pi\)
\(12\) −0.885937 + 1.11349i −0.255748 + 0.321438i
\(13\) 1.10865 − 3.04598i 0.307483 − 0.844804i −0.685662 − 0.727920i \(-0.740487\pi\)
0.993146 − 0.116884i \(-0.0372905\pi\)
\(14\) −3.15503 − 3.12994i −0.843217 − 0.836512i
\(15\) −4.04173 + 5.07986i −1.04357 + 1.31161i
\(16\) −3.80582 − 3.19347i −0.951456 − 0.798366i
\(17\) 1.93415 − 3.35005i 0.469101 − 0.812507i −0.530275 − 0.847826i \(-0.677911\pi\)
0.999376 + 0.0353188i \(0.0112446\pi\)
\(18\) −3.43584 − 3.68630i −0.809834 − 0.868870i
\(19\) −1.31749 + 0.760652i −0.302252 + 0.174506i −0.643454 − 0.765485i \(-0.722499\pi\)
0.341202 + 0.939990i \(0.389166\pi\)
\(20\) 2.35868 + 1.97917i 0.527417 + 0.442555i
\(21\) 3.67316 − 2.74005i 0.801550 − 0.597928i
\(22\) 1.39450 + 7.90859i 0.297308 + 1.68612i
\(23\) −0.358638 + 0.0632375i −0.0747811 + 0.0131859i −0.210914 − 0.977505i \(-0.567644\pi\)
0.136132 + 0.990691i \(0.456533\pi\)
\(24\) 2.56817 − 2.27156i 0.524226 − 0.463680i
\(25\) 6.93028 + 5.81520i 1.38606 + 1.16304i
\(26\) 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 −0.948127 − 0.317892i \(-0.897025\pi\)
0.521970 − 0.852964i \(-0.325197\pi\)
\(30\) −8.16763 + 7.22431i −1.49120 + 1.31897i
\(31\) −0.240377 + 0.660432i −0.0431731 + 0.118617i −0.959406 − 0.282030i \(-0.908992\pi\)
0.916233 + 0.400647i \(0.131215\pi\)
\(32\) −2.81938 − 3.36000i −0.498400 − 0.593970i
\(33\) −8.27786 + 0.215742i −1.44099 + 0.0375559i
\(34\) 4.17668 − 4.97757i 0.716294 − 0.853647i
\(35\) −5.71999 − 8.09998i −0.966855 − 1.36915i
\(36\) −1.34181 − 2.06732i −0.223635 − 0.344553i
\(37\) −5.50394 −0.904841 −0.452421 − 0.891805i \(-0.649439\pi\)
−0.452421 + 0.891805i \(0.649439\pi\)
\(38\) −2.40129 + 0.873998i −0.389541 + 0.141781i
\(39\) 4.39343 + 3.49558i 0.703512 + 0.559741i
\(40\) −4.76888 − 5.68333i −0.754026 − 0.898614i
\(41\) −0.793900 − 0.288956i −0.123986 − 0.0451273i 0.279282 − 0.960209i \(-0.409904\pi\)
−0.403268 + 0.915082i \(0.632126\pi\)
\(42\) 6.87546 − 3.46125i 1.06091 − 0.534083i
\(43\) 0.0136704 − 0.0775288i 0.00208472 − 0.0118230i −0.983748 − 0.179557i \(-0.942534\pi\)
0.985832 + 0.167734i \(0.0536448\pi\)
\(44\) 3.92763i 0.592112i
\(45\) −6.12145 − 9.43130i −0.912532 − 1.40594i
\(46\) −0.611712 −0.0901920
\(47\) 3.34124 − 1.21611i 0.487369 − 0.177388i −0.0866351 − 0.996240i \(-0.527611\pi\)
0.574004 + 0.818852i \(0.305389\pi\)
\(48\) 7.33760 − 4.49524i 1.05909 − 0.648832i
\(49\) 2.44658 + 6.55852i 0.349512 + 0.936932i
\(50\) 9.76803 + 11.6411i 1.38141 + 1.64630i
\(51\) 4.43900 + 5.01863i 0.621585 + 0.702749i
\(52\) 1.71173 − 2.03996i 0.237374 − 0.282891i
\(53\) −7.25668 + 4.18964i −0.996781 + 0.575492i −0.907294 − 0.420496i \(-0.861856\pi\)
−0.0894868 + 0.995988i \(0.528523\pi\)
\(54\) 7.94366 − 3.61659i 1.08100 − 0.492155i
\(55\) 17.9182i 2.41609i
\(56\) 2.23232 + 4.73775i 0.298306 + 0.633108i
\(57\) −0.525011 − 2.58214i −0.0695394 − 0.342013i
\(58\) −1.95717 − 11.0997i −0.256989 − 1.45746i
\(59\) 1.11005 − 0.931445i 0.144517 − 0.121264i −0.567663 − 0.823261i \(-0.692152\pi\)
0.712180 + 0.701997i \(0.247708\pi\)
\(60\) −4.54752 + 2.78595i −0.587082 + 0.359665i
\(61\) 0.979516 + 2.69120i 0.125414 + 0.344573i 0.986471 − 0.163936i \(-0.0524191\pi\)
−0.861057 + 0.508509i \(0.830197\pi\)
\(62\) −0.590275 + 1.02239i −0.0749650 + 0.129843i
\(63\) 2.48100 + 7.53954i 0.312577 + 0.949892i
\(64\) 1.28433 + 2.22452i 0.160541 + 0.278066i
\(65\) 7.80906 − 9.30647i 0.968594 − 1.15433i
\(66\) −13.7564 − 2.05764i −1.69329 − 0.253277i
\(67\) 0.702586 + 3.98456i 0.0858345 + 0.486792i 0.997173 + 0.0751348i \(0.0239387\pi\)
−0.911339 + 0.411657i \(0.864950\pi\)
\(68\) 2.43445 − 2.04274i 0.295220 − 0.247719i
\(69\) 0.0933094 − 0.623821i 0.0112331 − 0.0750993i
\(70\) −7.09950 − 15.0676i −0.848553 − 1.80092i
\(71\) −14.1956 + 8.19580i −1.68470 + 0.972663i −0.726240 + 0.687441i \(0.758734\pi\)
−0.958461 + 0.285222i \(0.907933\pi\)
\(72\) 2.31903 + 5.46704i 0.273300 + 0.644297i
\(73\) 12.0494i 1.41027i 0.709072 + 0.705136i \(0.249114\pi\)
−0.709072 + 0.705136i \(0.750886\pi\)
\(74\) −9.10474 − 1.60541i −1.05840 − 0.186625i
\(75\) −13.3615 + 8.18569i −1.54286 + 0.945202i
\(76\) −1.23082 + 0.217026i −0.141184 + 0.0248946i
\(77\) 3.22498 − 12.2309i 0.367521 − 1.39384i
\(78\) 6.24812 + 7.06396i 0.707460 + 0.799836i
\(79\) 0.228017 − 1.29315i 0.0256539 − 0.145491i −0.969290 − 0.245919i \(-0.920910\pi\)
0.994944 + 0.100428i \(0.0320214\pi\)
\(80\) −9.31009 − 16.1256i −1.04090 − 1.80289i
\(81\) 2.47647 + 8.65258i 0.275163 + 0.961398i
\(82\) −1.22900 − 0.709566i −0.135721 − 0.0783584i
\(83\) 12.7610 − 4.64462i 1.40070 − 0.509814i 0.472315 − 0.881430i \(-0.343419\pi\)
0.928386 + 0.371616i \(0.121196\pi\)
\(84\) 3.60555 − 1.08320i 0.393397 − 0.118187i
\(85\) 11.1062 − 9.31918i 1.20463 − 1.01081i
\(86\) 0.0452278 − 0.124262i 0.00487704 − 0.0133996i
\(87\) 11.6180 − 0.302793i 1.24558 − 0.0324628i
\(88\) 1.64337 − 9.32000i 0.175184 − 0.993516i
\(89\) 4.36439 + 7.55934i 0.462624 + 0.801289i 0.999091 − 0.0426330i \(-0.0135746\pi\)
−0.536467 + 0.843922i \(0.680241\pi\)
\(90\) −7.37529 − 17.3870i −0.777423 − 1.83275i
\(91\) −7.00545 + 4.94706i −0.734371 + 0.518593i
\(92\) −0.294633 − 0.0519517i −0.0307176 − 0.00541634i
\(93\) −0.952586 − 0.757913i −0.0987786 − 0.0785919i
\(94\) 5.88187 − 1.03713i 0.606669 − 0.106972i
\(95\) −5.61509 + 0.990092i −0.576096 + 0.101581i
\(96\) 7.06880 − 2.78347i 0.721456 − 0.284086i
\(97\) 4.18960 + 0.738740i 0.425390 + 0.0750077i 0.382245 − 0.924061i \(-0.375151\pi\)
0.0431451 + 0.999069i \(0.486262\pi\)
\(98\) 2.13418 + 11.5629i 0.215584 + 1.16803i
\(99\) 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.bd.a.47.17 yes 132
3.2 odd 2 567.2.bd.a.467.6 132
7.3 odd 6 189.2.ba.a.101.6 ✓ 132
21.17 even 6 567.2.ba.a.143.17 132
27.4 even 9 567.2.ba.a.341.17 132
27.23 odd 18 189.2.ba.a.131.6 yes 132
189.31 odd 18 567.2.bd.a.17.6 132
189.185 even 18 inner 189.2.bd.a.185.17 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.6 ✓ 132 7.3 odd 6
189.2.ba.a.131.6 yes 132 27.23 odd 18
189.2.bd.a.47.17 yes 132 1.1 even 1 trivial
189.2.bd.a.185.17 yes 132 189.185 even 18 inner
567.2.ba.a.143.17 132 21.17 even 6
567.2.ba.a.341.17 132 27.4 even 9
567.2.bd.a.17.6 132 189.31 odd 18
567.2.bd.a.467.6 132 3.2 odd 2