↑

Properties

Label 189.2.ba.a.101.3
Level $189$
Weight $2$
Character 189.101
Analytic conductor $1.509$
Analytic rank $0$
Dimension $132$
Inner twists $2$

Related objects

Downloads

Learn more

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(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 101.3
Character \(\chi\) \(=\) 189.101
Dual form 189.2.ba.a.131.3

$q$-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+(-1.44973 + 1.72772i) q^{2} +(0.462924 + 1.66904i) q^{3} +(-0.536009 - 3.03986i) q^{4} +(-2.61524 + 2.19445i) q^{5} +(-3.55476 - 1.61986i) q^{6} +(-0.126746 - 2.64271i) q^{7} +(2.12267 + 1.22552i) q^{8} +(-2.57140 + 1.54528i) q^{9} -7.69979i q^{10} +(1.63538 - 1.94897i) q^{11} +(4.82552 - 2.30185i) q^{12} +(-2.11236 + 5.80365i) q^{13} +(4.74963 + 3.61225i) q^{14} +(-4.87329 - 3.34909i) q^{15} +(0.606533 - 0.220760i) q^{16} -2.39853 q^{17} +(1.05803 - 6.68292i) q^{18} +1.24485i q^{19} +(8.07261 + 6.77373i) q^{20} +(4.35213 - 1.43492i) q^{21} +(0.996420 + 5.65098i) q^{22} +(0.769210 - 2.11339i) q^{23} +(-1.06281 + 4.11015i) q^{24} +(1.15565 - 6.55401i) q^{25} +(-6.96475 - 12.0633i) q^{26} +(-3.76950 - 3.57643i) q^{27} +(-7.96554 + 1.80181i) q^{28} +(0.685884 + 1.88445i) q^{29} +(12.8513 - 3.56442i) q^{30} +(-1.55132 + 0.273539i) q^{31} +(-2.17451 + 5.97442i) q^{32} +(4.00997 + 1.82729i) q^{33} +(3.47723 - 4.14400i) q^{34} +(6.13078 + 6.63320i) q^{35} +(6.07573 + 6.98842i) q^{36} +(-5.01223 + 8.68144i) q^{37} +(-2.15077 - 1.80471i) q^{38} +(-10.6644 - 0.838960i) q^{39} +(-8.24065 + 1.45305i) q^{40} +(-5.39584 - 1.96393i) q^{41} +(-3.83027 + 9.59952i) q^{42} +(-0.110284 + 0.625454i) q^{43} +(-6.80118 - 3.92666i) q^{44} +(3.33380 - 9.68410i) q^{45} +(2.53620 + 4.39283i) q^{46} +(-2.09655 + 11.8901i) q^{47} +(0.649236 + 0.910134i) q^{48} +(-6.96787 + 0.669906i) q^{49} +(9.64814 + 11.4982i) q^{50} +(-1.11034 - 4.00325i) q^{51} +(18.7745 + 3.31045i) q^{52} +(4.15350 + 2.39802i) q^{53} +(11.6439 - 1.32779i) q^{54} +8.68580i q^{55} +(2.96967 - 5.76493i) q^{56} +(-2.07772 + 0.576273i) q^{57} +(-4.25016 - 1.54693i) q^{58} +(2.59194 + 0.943390i) q^{59} +(-7.56863 + 16.6093i) q^{60} +(7.26032 + 1.28019i) q^{61} +(1.77639 - 3.07681i) q^{62} +(4.40965 + 6.59962i) q^{63} +(-6.52423 - 11.3003i) q^{64} +(-7.21149 - 19.8134i) q^{65} +(-8.97045 + 4.27904i) q^{66} +(-6.14145 + 5.15329i) q^{67} +(1.28564 + 7.29120i) q^{68} +(3.88342 + 0.305506i) q^{69} +(-20.3483 + 0.975916i) q^{70} +(4.45847 - 2.57410i) q^{71} +(-7.35201 + 0.128804i) q^{72} +(11.6132 - 6.70488i) q^{73} +(-7.73274 - 21.2455i) q^{74} +(11.4739 - 1.10518i) q^{75} +(3.78418 - 0.667254i) q^{76} +(-5.35785 - 4.07482i) q^{77} +(16.9100 - 17.2089i) q^{78} +(0.740362 + 0.621237i) q^{79} +(-1.10178 + 1.90835i) q^{80} +(4.22422 - 7.94707i) q^{81} +(11.2156 - 6.47536i) q^{82} +(-2.38230 + 0.867085i) q^{83} +(-6.69474 - 12.4607i) q^{84} +(6.27275 - 5.26346i) q^{85} +(-0.920729 - 1.09728i) q^{86} +(-2.82772 + 2.01713i) q^{87} +(5.85988 - 2.13282i) q^{88} -9.95066 q^{89} +(11.8983 + 19.7992i) q^{90} +(15.6051 + 4.84676i) q^{91} +(-6.83671 - 1.20550i) q^{92} +(-1.17469 - 2.46259i) q^{93} +(-17.5034 - 20.8598i) q^{94} +(-2.73177 - 3.25560i) q^{95} +(-10.9782 - 0.863647i) q^{96} +(7.93719 + 1.39954i) q^{97} +(8.94414 - 13.0097i) q^{98} +(-1.19352 + 7.53871i) 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 \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.44973 + 1.72772i −1.02512 + 1.22169i −0.0502858 + 0.998735i \(0.516013\pi\)
−0.974830 + 0.222950i \(0.928431\pi\)
\(3\) 0.462924 + 1.66904i 0.267269 + 0.963622i
\(4\) −0.536009 − 3.03986i −0.268005 − 1.51993i
\(5\) −2.61524 + 2.19445i −1.16957 + 0.981388i −0.999991 − 0.00412812i \(-0.998686\pi\)
−0.169581 + 0.985516i \(0.554242\pi\)
\(6\) −3.55476 − 1.61986i −1.45122 − 0.661305i
\(7\) −0.126746 − 2.64271i −0.0479054 − 0.998852i
\(8\) 2.12267 + 1.22552i 0.750477 + 0.433288i
\(9\) −2.57140 + 1.54528i −0.857134 + 0.515093i
\(10\) − 7.69979i − 2.43489i
\(11\) 1.63538 − 1.94897i 0.493086 − 0.587637i −0.460913 − 0.887445i \(-0.652478\pi\)
0.954000 + 0.299808i \(0.0969226\pi\)
\(12\) 4.82552 − 2.30185i 1.39301 − 0.664486i
\(13\) −2.11236 + 5.80365i −0.585862 + 1.60964i 0.192131 + 0.981369i \(0.438460\pi\)
−0.777993 + 0.628273i \(0.783762\pi\)
\(14\) 4.74963 + 3.61225i 1.26939 + 0.965413i
\(15\) −4.87329 − 3.34909i −1.25828 − 0.864731i
\(16\) 0.606533 − 0.220760i 0.151633 − 0.0551900i
\(17\) −2.39853 −0.581730 −0.290865 − 0.956764i \(-0.593943\pi\)
−0.290865 + 0.956764i \(0.593943\pi\)
\(18\) 1.05803 − 6.68292i 0.249380 − 1.57518i
\(19\) 1.24485i 0.285589i 0.989752 + 0.142795i \(0.0456089\pi\)
−0.989752 + 0.142795i \(0.954391\pi\)
\(20\) 8.07261 + 6.77373i 1.80509 + 1.51465i
\(21\) 4.35213 − 1.43492i 0.949712 − 0.313125i
\(22\) 0.996420 + 5.65098i 0.212437 + 1.20479i
\(23\) 0.769210 − 2.11339i 0.160391 − 0.440672i −0.833300 − 0.552821i \(-0.813551\pi\)
0.993691 + 0.112149i \(0.0357735\pi\)
\(24\) −1.06281 + 4.11015i −0.216946 + 0.838980i
\(25\) 1.15565 − 6.55401i 0.231130 − 1.31080i
\(26\) −6.96475 − 12.0633i −1.36590 − 2.36581i
\(27\) −3.76950 − 3.57643i −0.725441 − 0.688284i
\(28\) −7.96554 + 1.80181i −1.50535 + 0.340510i
\(29\) 0.685884 + 1.88445i 0.127366 + 0.349934i 0.986943 − 0.161072i \(-0.0514952\pi\)
−0.859577 + 0.511006i \(0.829273\pi\)
\(30\) 12.8513 − 3.56442i 2.34631 − 0.650771i
\(31\) −1.55132 + 0.273539i −0.278625 + 0.0491291i −0.311214 − 0.950340i \(-0.600736\pi\)
0.0325896 + 0.999469i \(0.489625\pi\)
\(32\) −2.17451 + 5.97442i −0.384403 + 1.05614i
\(33\) 4.00997 + 1.82729i 0.698047 + 0.318091i
\(34\) 3.47723 − 4.14400i 0.596340 − 0.710691i
\(35\) 6.13078 + 6.63320i 1.03629 + 1.12122i
\(36\) 6.07573 + 6.98842i 1.01262 + 1.16474i
\(37\) −5.01223 + 8.68144i −0.824006 + 1.42722i 0.0786706 + 0.996901i \(0.474932\pi\)
−0.902677 + 0.430320i \(0.858401\pi\)
\(38\) −2.15077 − 1.80471i −0.348900 − 0.292762i
\(39\) −10.6644 − 0.838960i −1.70767 − 0.134341i
\(40\) −8.24065 + 1.45305i −1.30296 + 0.229747i
\(41\) −5.39584 − 1.96393i −0.842689 − 0.306714i −0.115633 − 0.993292i \(-0.536890\pi\)
−0.727056 + 0.686578i \(0.759112\pi\)
\(42\) −3.83027 + 9.59952i −0.591024 + 1.48124i
\(43\) −0.110284 + 0.625454i −0.0168182 + 0.0953809i −0.992061 − 0.125754i \(-0.959865\pi\)
0.975243 + 0.221135i \(0.0709761\pi\)
\(44\) −6.80118 − 3.92666i −1.02532 − 0.591967i
\(45\) 3.33380 − 9.68410i 0.496974 − 1.44362i
\(46\) 2.53620 + 4.39283i 0.373943 + 0.647687i
\(47\) −2.09655 + 11.8901i −0.305813 + 1.73435i 0.313839 + 0.949476i \(0.398385\pi\)
−0.619653 + 0.784876i \(0.712726\pi\)
\(48\) 0.649236 + 0.910134i 0.0937092 + 0.131366i
\(49\) −6.96787 + 0.669906i −0.995410 + 0.0957009i
\(50\) 9.64814 + 11.4982i 1.36445 + 1.62609i
\(51\) −1.11034 − 4.00325i −0.155479 − 0.560568i
\(52\) 18.7745 + 3.31045i 2.60356 + 0.459077i
\(53\) 4.15350 + 2.39802i 0.570527 + 0.329394i 0.757360 − 0.652998i \(-0.226489\pi\)
−0.186833 + 0.982392i \(0.559822\pi\)
\(54\) 11.6439 − 1.32779i 1.58453 − 0.180689i
\(55\) 8.68580i 1.17119i
\(56\) 2.96967 − 5.76493i 0.396838 − 0.770372i
\(57\) −2.07772 + 0.576273i −0.275200 + 0.0763293i
\(58\) −4.25016 − 1.54693i −0.558074 − 0.203122i
\(59\) 2.59194 + 0.943390i 0.337442 + 0.122819i 0.505183 − 0.863012i \(-0.331425\pi\)
−0.167741 + 0.985831i \(0.553647\pi\)
\(60\) −7.56863 + 16.6093i −0.977106 + 2.14425i
\(61\) 7.26032 + 1.28019i 0.929589 + 0.163912i 0.617886 − 0.786268i \(-0.287989\pi\)
0.311703 + 0.950179i \(0.399100\pi\)
\(62\) 1.77639 − 3.07681i 0.225602 − 0.390755i
\(63\) 4.40965 + 6.59962i 0.555563 + 0.831474i
\(64\) −6.52423 − 11.3003i −0.815529 − 1.41254i
\(65\) −7.21149 − 19.8134i −0.894476 − 2.45755i
\(66\) −8.97045 + 4.27904i −1.10419 + 0.526713i
\(67\) −6.14145 + 5.15329i −0.750298 + 0.629574i −0.935582 − 0.353111i \(-0.885124\pi\)
0.185284 + 0.982685i \(0.440680\pi\)
\(68\) 1.28564 + 7.29120i 0.155906 + 0.884188i
\(69\) 3.88342 + 0.305506i 0.467509 + 0.0367786i
\(70\) −20.3483 + 0.975916i −2.43209 + 0.116644i
\(71\) 4.45847 − 2.57410i 0.529123 − 0.305489i −0.211536 − 0.977370i \(-0.567847\pi\)
0.740659 + 0.671881i \(0.234513\pi\)
\(72\) −7.35201 + 0.128804i −0.866443 + 0.0151797i
\(73\) 11.6132 − 6.70488i 1.35922 − 0.784747i 0.369702 − 0.929150i \(-0.379460\pi\)
0.989519 + 0.144403i \(0.0461263\pi\)
\(74\) −7.73274 − 21.2455i −0.898912 − 2.46974i
\(75\) 11.4739 − 1.10518i 1.32489 − 0.127616i
\(76\) 3.78418 − 0.667254i 0.434076 − 0.0765392i
\(77\) −5.35785 − 4.07482i −0.610584 − 0.464369i
\(78\) 16.9100 − 17.2089i 1.91468 − 1.94852i
\(79\) 0.740362 + 0.621237i 0.0832972 + 0.0698946i 0.683485 − 0.729964i \(-0.260463\pi\)
−0.600188 + 0.799859i \(0.704908\pi\)
\(80\) −1.10178 + 1.90835i −0.123183 + 0.213360i
\(81\) 4.22422 − 7.94707i 0.469358 − 0.883008i
\(82\) 11.2156 − 6.47536i 1.23856 − 0.715084i
\(83\) −2.38230 + 0.867085i −0.261491 + 0.0951749i −0.469439 − 0.882965i \(-0.655544\pi\)
0.207948 + 0.978140i \(0.433322\pi\)
\(84\) −6.69474 − 12.4607i −0.730456 − 1.35958i
\(85\) 6.27275 − 5.26346i 0.680375 − 0.570903i
\(86\) −0.920729 − 1.09728i −0.0992848 − 0.118323i
\(87\) −2.82772 + 2.01713i −0.303163 + 0.216259i
\(88\) 5.85988 − 2.13282i 0.624666 − 0.227360i
\(89\) −9.95066 −1.05477 −0.527384 − 0.849627i \(-0.676827\pi\)
−0.527384 + 0.849627i \(0.676827\pi\)
\(90\) 11.8983 + 19.7992i 1.25419 + 2.08702i
\(91\) 15.6051 + 4.84676i 1.63586 + 0.508079i
\(92\) −6.83671 − 1.20550i −0.712776 − 0.125682i
\(93\) −1.17469 − 2.46259i −0.121810 − 0.255358i
\(94\) −17.5034 − 20.8598i −1.80534 − 2.15152i
\(95\) −2.73177 − 3.25560i −0.280274 − 0.334017i
\(96\) −10.9782 − 0.863647i −1.12046 − 0.0881456i
\(97\) 7.93719 + 1.39954i 0.805899 + 0.142102i 0.561397 − 0.827547i \(-0.310264\pi\)
0.244503 + 0.969649i \(0.421375\pi\)
\(98\) 8.94414 − 13.0097i 0.903494 − 1.31418i
\(99\) −1.19352 + 7.53871i −0.119953 + 0.757669i
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.101.3 ✓ 132
3.2 odd 2 567.2.ba.a.143.20 132
7.5 odd 6 189.2.bd.a.47.20 yes 132
21.5 even 6 567.2.bd.a.467.3 132
27.4 even 9 567.2.bd.a.17.3 132
27.23 odd 18 189.2.bd.a.185.20 yes 132
189.131 even 18 inner 189.2.ba.a.131.3 yes 132
189.166 odd 18 567.2.ba.a.341.20 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.3 ✓ 132 1.1 even 1 trivial
189.2.ba.a.131.3 yes 132 189.131 even 18 inner
189.2.bd.a.47.20 yes 132 7.5 odd 6
189.2.bd.a.185.20 yes 132 27.23 odd 18
567.2.ba.a.143.20 132 3.2 odd 2
567.2.ba.a.341.20 132 189.166 odd 18
567.2.bd.a.17.3 132 27.4 even 9
567.2.bd.a.467.3 132 21.5 even 6