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Properties

Label 189.2.bd.a.47.7
Level $189$
Weight $2$
Character 189.47
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 47.7
Character \(\chi\) \(=\) 189.47
Dual form 189.2.bd.a.185.7

$q$-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+(-1.38665 - 0.244504i) q^{2} +(-0.458387 + 1.67029i) q^{3} +(-0.0163608 - 0.00595485i) q^{4} +(1.98299 + 0.721749i) q^{5} +(1.04402 - 2.20404i) q^{6} +(-2.18605 + 1.49037i) q^{7} +(2.46004 + 1.42030i) q^{8} +(-2.57976 - 1.53128i) q^{9} +(-2.57325 - 1.48566i) q^{10} +(-1.26664 - 3.48007i) q^{11} +(0.0174459 - 0.0245978i) q^{12} +(-2.27684 + 6.25557i) q^{13} +(3.39569 - 1.53213i) q^{14} +(-2.11451 + 2.98133i) q^{15} +(-3.03727 - 2.54857i) q^{16} +(-1.61887 + 2.80397i) q^{17} +(3.20283 + 2.75412i) q^{18} +(-4.61385 + 2.66381i) q^{19} +(-0.0281454 - 0.0236168i) q^{20} +(-1.48730 - 4.33451i) q^{21} +(0.905502 + 5.13536i) q^{22} +(0.689782 - 0.121627i) q^{23} +(-3.49997 + 3.45793i) q^{24} +(-0.418902 - 0.351501i) q^{25} +(4.68670 - 8.11761i) q^{26} +(3.74022 - 3.60704i) q^{27} +(0.0446405 - 0.0113661i) q^{28} +(1.50083 + 4.12351i) q^{29} +(3.66104 - 3.61707i) q^{30} +(-0.513951 + 1.41207i) q^{31} +(-0.0633067 - 0.0754460i) q^{32} +(6.39336 - 0.520445i) q^{33} +(2.93040 - 3.49232i) q^{34} +(-5.41058 + 1.37761i) q^{35} +(0.0330885 + 0.0404152i) q^{36} +6.49662 q^{37} +(7.04913 - 2.56567i) q^{38} +(-9.40496 - 6.67047i) q^{39} +(3.85312 + 4.59197i) q^{40} +(10.4849 + 3.81619i) q^{41} +(1.00256 + 6.37411i) q^{42} +(1.29121 - 7.32280i) q^{43} +0.0644796i q^{44} +(-4.01044 - 4.89845i) q^{45} -0.986227 q^{46} +(-5.63412 + 2.05065i) q^{47} +(5.64911 - 3.90490i) q^{48} +(2.55760 - 6.51603i) q^{49} +(0.494929 + 0.589833i) q^{50} +(-3.94139 - 3.98930i) q^{51} +(0.0745020 - 0.0887880i) q^{52} +(0.766350 - 0.442452i) q^{53} +(-6.06833 + 4.08721i) q^{54} -7.81514i q^{55} +(-7.49453 + 0.561517i) q^{56} +(-2.33441 - 8.92755i) q^{57} +(-1.07292 - 6.08484i) q^{58} +(4.77604 - 4.00757i) q^{59} +(0.0523485 - 0.0361855i) q^{60} +(2.54368 + 6.98869i) q^{61} +(1.05793 - 1.83239i) q^{62} +(7.92166 - 0.497345i) q^{63} +(4.03421 + 6.98746i) q^{64} +(-9.02990 + 10.7614i) q^{65} +(-8.99262 - 0.841528i) q^{66} +(1.82662 + 10.3593i) q^{67} +(0.0431834 - 0.0362352i) q^{68} +(-0.113034 + 1.20789i) q^{69} +(7.83942 - 0.587358i) q^{70} +(-3.83248 + 2.21268i) q^{71} +(-4.17142 - 7.43105i) q^{72} +3.06138i q^{73} +(-9.00855 - 1.58845i) q^{74} +(0.779129 - 0.538566i) q^{75} +(0.0913491 - 0.0161073i) q^{76} +(7.95554 + 5.71984i) q^{77} +(11.4105 + 11.5492i) q^{78} +(-1.65307 + 9.37505i) q^{79} +(-4.18344 - 7.24593i) q^{80} +(4.31035 + 7.90069i) q^{81} +(-13.6058 - 7.85533i) q^{82} +(11.4377 - 4.16299i) q^{83} +(-0.00147789 + 0.0797727i) q^{84} +(-5.23397 + 4.39182i) q^{85} +(-3.58091 + 9.83847i) q^{86} +(-7.57543 + 0.616671i) q^{87} +(1.82677 - 10.3601i) q^{88} +(-2.17916 - 3.77442i) q^{89} +(4.36339 + 7.77303i) q^{90} +(-4.34583 - 17.0683i) q^{91} +(-0.0120097 - 0.00211763i) q^{92} +(-2.12298 - 1.50572i) q^{93} +(8.31397 - 1.46598i) q^{94} +(-11.0718 + 1.95226i) q^{95} +(0.155036 - 0.0711573i) q^{96} +(-6.85869 - 1.20937i) q^{97} +(-5.13970 + 8.41014i) q^{98} +(-2.06134 + 10.9174i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38665 − 0.244504i −0.980512 − 0.172891i −0.339654 − 0.940550i \(-0.610310\pi\)
−0.640858 + 0.767660i \(0.721421\pi\)
\(3\) −0.458387 + 1.67029i −0.264650 + 0.964345i
\(4\) −0.0163608 − 0.00595485i −0.00818041 − 0.00297743i
\(5\) 1.98299 + 0.721749i 0.886819 + 0.322776i 0.744958 − 0.667111i \(-0.232469\pi\)
0.141861 + 0.989887i \(0.454691\pi\)
\(6\) 1.04402 − 2.20404i 0.426219 − 0.899796i
\(7\) −2.18605 + 1.49037i −0.826248 + 0.563307i
\(8\) 2.46004 + 1.42030i 0.869754 + 0.502153i
\(9\) −2.57976 − 1.53128i −0.859921 − 0.510428i
\(10\) −2.57325 − 1.48566i −0.813732 − 0.469808i
\(11\) −1.26664 − 3.48007i −0.381907 − 1.04928i −0.970552 − 0.240890i \(-0.922561\pi\)
0.588645 − 0.808392i \(-0.299662\pi\)
\(12\) 0.0174459 − 0.0245978i 0.00503621 − 0.00710076i
\(13\) −2.27684 + 6.25557i −0.631482 + 1.73498i 0.0454806 + 0.998965i \(0.485518\pi\)
−0.676963 + 0.736017i \(0.736704\pi\)
\(14\) 3.39569 − 1.53213i 0.907536 − 0.409478i
\(15\) −2.11451 + 2.98133i −0.545964 + 0.769777i
\(16\) −3.03727 − 2.54857i −0.759318 − 0.637143i
\(17\) −1.61887 + 2.80397i −0.392635 + 0.680063i −0.992796 − 0.119816i \(-0.961770\pi\)
0.600161 + 0.799879i \(0.295103\pi\)
\(18\) 3.20283 + 2.75412i 0.754914 + 0.649153i
\(19\) −4.61385 + 2.66381i −1.05849 + 0.611120i −0.925014 − 0.379932i \(-0.875947\pi\)
−0.133476 + 0.991052i \(0.542614\pi\)
\(20\) −0.0281454 − 0.0236168i −0.00629351 − 0.00528088i
\(21\) −1.48730 − 4.33451i −0.324555 − 0.945867i
\(22\) 0.905502 + 5.13536i 0.193054 + 1.09486i
\(23\) 0.689782 − 0.121627i 0.143829 − 0.0253610i −0.101270 − 0.994859i \(-0.532291\pi\)
0.245099 + 0.969498i \(0.421179\pi\)
\(24\) −3.49997 + 3.45793i −0.714428 + 0.705848i
\(25\) −0.418902 − 0.351501i −0.0837805 − 0.0703002i
\(26\) 4.68670 − 8.11761i 0.919138 − 1.59199i
\(27\) 3.74022 − 3.60704i 0.719806 − 0.694175i
\(28\) 0.0446405 − 0.0113661i 0.00843625 − 0.00214799i
\(29\) 1.50083 + 4.12351i 0.278698 + 0.765716i 0.997511 + 0.0705128i \(0.0224636\pi\)
−0.718813 + 0.695204i \(0.755314\pi\)
\(30\) 3.66104 − 3.61707i 0.668411 − 0.660383i
\(31\) −0.513951 + 1.41207i −0.0923084 + 0.253615i −0.977251 − 0.212084i \(-0.931975\pi\)
0.884943 + 0.465699i \(0.154197\pi\)
\(32\) −0.0633067 − 0.0754460i −0.0111911 − 0.0133371i
\(33\) 6.39336 − 0.520445i 1.11294 − 0.0905978i
\(34\) 2.93040 − 3.49232i 0.502560 − 0.598927i
\(35\) −5.41058 + 1.37761i −0.914554 + 0.232859i
\(36\) 0.0330885 + 0.0404152i 0.00551475 + 0.00673586i
\(37\) 6.49662 1.06804 0.534019 − 0.845473i \(-0.320681\pi\)
0.534019 + 0.845473i \(0.320681\pi\)
\(38\) 7.04913 − 2.56567i 1.14352 − 0.416207i
\(39\) −9.40496 − 6.67047i −1.50600 − 1.06813i
\(40\) 3.85312 + 4.59197i 0.609232 + 0.726054i
\(41\) 10.4849 + 3.81619i 1.63746 + 0.595988i 0.986593 − 0.163200i \(-0.0521817\pi\)
0.650871 + 0.759188i \(0.274404\pi\)
\(42\) 1.00256 + 6.37411i 0.154699 + 0.983546i
\(43\) 1.29121 − 7.32280i 0.196907 − 1.11672i −0.712769 − 0.701399i \(-0.752559\pi\)
0.909676 − 0.415318i \(-0.136330\pi\)
\(44\) 0.0644796i 0.00972066i
\(45\) −4.01044 − 4.89845i −0.597841 − 0.730219i
\(46\) −0.986227 −0.145411
\(47\) −5.63412 + 2.05065i −0.821821 + 0.299119i −0.718498 − 0.695529i \(-0.755170\pi\)
−0.103324 + 0.994648i \(0.532948\pi\)
\(48\) 5.64911 − 3.90490i 0.815379 − 0.563624i
\(49\) 2.55760 − 6.51603i 0.365371 − 0.930862i
\(50\) 0.494929 + 0.589833i 0.0699935 + 0.0834150i
\(51\) −3.94139 − 3.98930i −0.551905 − 0.558614i
\(52\) 0.0745020 − 0.0887880i 0.0103316 − 0.0123127i
\(53\) 0.766350 − 0.442452i 0.105266 − 0.0607755i −0.446443 − 0.894812i \(-0.647309\pi\)
0.551709 + 0.834037i \(0.313976\pi\)
\(54\) −6.06833 + 4.08721i −0.825795 + 0.556199i
\(55\) − 7.81514i − 1.05379i
\(56\) −7.49453 + 0.561517i −1.00150 + 0.0750358i
\(57\) −2.33441 − 8.92755i −0.309201 − 1.18248i
\(58\) −1.07292 − 6.08484i −0.140881 − 0.798978i
\(59\) 4.77604 − 4.00757i 0.621787 − 0.521742i −0.276577 − 0.960992i \(-0.589200\pi\)
0.898365 + 0.439250i \(0.144756\pi\)
\(60\) 0.0523485 − 0.0361855i 0.00675816 − 0.00467152i
\(61\) 2.54368 + 6.98869i 0.325684 + 0.894811i 0.989190 + 0.146639i \(0.0468456\pi\)
−0.663506 + 0.748171i \(0.730932\pi\)
\(62\) 1.05793 − 1.83239i 0.134357 − 0.232713i
\(63\) 7.92166 − 0.497345i 0.998035 − 0.0626596i
\(64\) 4.03421 + 6.98746i 0.504277 + 0.873433i
\(65\) −9.02990 + 10.7614i −1.12002 + 1.33479i
\(66\) −8.99262 − 0.841528i −1.10692 − 0.103585i
\(67\) 1.82662 + 10.3593i 0.223157 + 1.26559i 0.866177 + 0.499738i \(0.166570\pi\)
−0.643019 + 0.765850i \(0.722319\pi\)
\(68\) 0.0431834 − 0.0362352i 0.00523675 − 0.00439416i
\(69\) −0.113034 + 1.20789i −0.0136077 + 0.145413i
\(70\) 7.83942 − 0.587358i 0.936990 − 0.0702027i
\(71\) −3.83248 + 2.21268i −0.454832 + 0.262597i −0.709869 − 0.704334i \(-0.751246\pi\)
0.255037 + 0.966931i \(0.417912\pi\)
\(72\) −4.17142 − 7.43105i −0.491607 − 0.875758i
\(73\) 3.06138i 0.358307i 0.983821 + 0.179154i \(0.0573359\pi\)
−0.983821 + 0.179154i \(0.942664\pi\)
\(74\) −9.00855 − 1.58845i −1.04722 − 0.184654i
\(75\) 0.779129 − 0.538566i 0.0899661 − 0.0621883i
\(76\) 0.0913491 − 0.0161073i 0.0104785 − 0.00184763i
\(77\) 7.95554 + 5.71984i 0.906618 + 0.651836i
\(78\) 11.4105 + 11.5492i 1.29198 + 1.30769i
\(79\) −1.65307 + 9.37505i −0.185985 + 1.05478i 0.738697 + 0.674037i \(0.235441\pi\)
−0.924683 + 0.380739i \(0.875670\pi\)
\(80\) −4.18344 − 7.24593i −0.467723 − 0.810120i
\(81\) 4.31035 + 7.90069i 0.478927 + 0.877855i
\(82\) −13.6058 − 7.85533i −1.50251 − 0.867476i
\(83\) 11.4377 − 4.16299i 1.25545 − 0.456947i 0.373212 − 0.927746i \(-0.378256\pi\)
0.882241 + 0.470799i \(0.156034\pi\)
\(84\) −0.00147789 + 0.0797727i −0.000161251 + 0.00870392i
\(85\) −5.23397 + 4.39182i −0.567704 + 0.476360i
\(86\) −3.58091 + 9.83847i −0.386140 + 1.06091i
\(87\) −7.57543 + 0.616671i −0.812172 + 0.0661140i
\(88\) 1.82677 − 10.3601i 0.194734 − 1.10439i
\(89\) −2.17916 − 3.77442i −0.230991 − 0.400088i 0.727109 − 0.686522i \(-0.240863\pi\)
−0.958100 + 0.286434i \(0.907530\pi\)
\(90\) 4.36339 + 7.77303i 0.459942 + 0.819349i
\(91\) −4.34583 − 17.0683i −0.455567 − 1.78924i
\(92\) −0.0120097 − 0.00211763i −0.00125210 − 0.000220778i
\(93\) −2.12298 − 1.50572i −0.220143 − 0.156136i
\(94\) 8.31397 − 1.46598i 0.857520 − 0.151204i
\(95\) −11.0718 + 1.95226i −1.13594 + 0.200298i
\(96\) 0.155036 − 0.0711573i 0.0158233 − 0.00726246i
\(97\) −6.85869 − 1.20937i −0.696394 − 0.122793i −0.185765 − 0.982594i \(-0.559476\pi\)
−0.510629 + 0.859801i \(0.670587\pi\)
\(98\) −5.13970 + 8.41014i −0.519188 + 0.849552i
\(99\) −2.06134 + 10.9174i −0.207172 + 1.09724i
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.47.7 yes 132
3.2 odd 2 567.2.bd.a.467.16 132
7.3 odd 6 189.2.ba.a.101.16 ✓ 132
21.17 even 6 567.2.ba.a.143.7 132
27.4 even 9 567.2.ba.a.341.7 132
27.23 odd 18 189.2.ba.a.131.16 yes 132
189.31 odd 18 567.2.bd.a.17.16 132
189.185 even 18 inner 189.2.bd.a.185.7 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.16 ✓ 132 7.3 odd 6
189.2.ba.a.131.16 yes 132 27.23 odd 18
189.2.bd.a.47.7 yes 132 1.1 even 1 trivial
189.2.bd.a.185.7 yes 132 189.185 even 18 inner
567.2.ba.a.143.7 132 21.17 even 6
567.2.ba.a.341.7 132 27.4 even 9
567.2.bd.a.17.16 132 189.31 odd 18
567.2.bd.a.467.16 132 3.2 odd 2