↑

Properties

Label 189.2.bd.a.47.8
Level $189$
Weight $2$
Character 189.47
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(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 47.8
Character \(\chi\) \(=\) 189.47
Dual form 189.2.bd.a.185.8

$q$-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+(-0.910598 - 0.160563i) q^{2} +(-1.05237 - 1.37569i) q^{3} +(-1.07598 - 0.391623i) q^{4} +(-0.473927 - 0.172495i) q^{5} +(0.737399 + 1.42167i) q^{6} +(-2.46079 + 0.971863i) q^{7} +(2.51844 + 1.45402i) q^{8} +(-0.785044 + 2.89546i) q^{9} +(0.403861 + 0.233169i) q^{10} +(1.66156 + 4.56510i) q^{11} +(0.593571 + 1.89234i) q^{12} +(1.60911 - 4.42100i) q^{13} +(2.39684 - 0.489865i) q^{14} +(0.261446 + 0.833505i) q^{15} +(-0.305533 - 0.256373i) q^{16} +(-2.38696 + 4.13434i) q^{17} +(1.17976 - 2.51055i) q^{18} +(-5.51758 + 3.18557i) q^{19} +(0.442381 + 0.371202i) q^{20} +(3.92664 + 2.36252i) q^{21} +(-0.780028 - 4.42376i) q^{22} +(-3.53298 + 0.622960i) q^{23} +(-0.650041 - 4.99475i) q^{24} +(-3.63537 - 3.05044i) q^{25} +(-2.17510 + 3.76739i) q^{26} +(4.80941 - 1.96711i) q^{27} +(3.02836 - 0.0819996i) q^{28} +(0.997075 + 2.73944i) q^{29} +(-0.104242 - 0.800967i) q^{30} +(-0.268468 + 0.737610i) q^{31} +(-3.50145 - 4.17286i) q^{32} +(4.53159 - 7.08995i) q^{33} +(2.83739 - 3.38147i) q^{34} +(1.33388 - 0.0361178i) q^{35} +(1.97862 - 2.80801i) q^{36} -5.47451 q^{37} +(5.53578 - 2.01486i) q^{38} +(-7.77529 + 2.43888i) q^{39} +(-0.942744 - 1.12352i) q^{40} +(-5.44849 - 1.98309i) q^{41} +(-3.19626 - 2.78178i) q^{42} +(-1.36729 + 7.75429i) q^{43} -5.56264i q^{44} +(0.871508 - 1.23682i) q^{45} +3.31715 q^{46} +(7.22256 - 2.62880i) q^{47} +(-0.0311561 + 0.690117i) q^{48} +(5.11096 - 4.78310i) q^{49} +(2.82057 + 3.36143i) q^{50} +(8.19953 - 1.06713i) q^{51} +(-3.46273 + 4.12672i) q^{52} +(4.26125 - 2.46023i) q^{53} +(-4.69529 + 1.01904i) q^{54} -2.45014i q^{55} +(-7.61045 - 1.13046i) q^{56} +(10.1889 + 4.23808i) q^{57} +(-0.468082 - 2.65462i) q^{58} +(-4.08557 + 3.42820i) q^{59} +(0.0451109 - 0.999221i) q^{60} +(-2.45624 - 6.74846i) q^{61} +(0.362900 - 0.628561i) q^{62} +(-0.882166 - 7.88808i) q^{63} +(2.91725 + 5.05283i) q^{64} +(-1.52520 + 1.81767i) q^{65} +(-5.26484 + 5.72849i) q^{66} +(1.51964 + 8.61830i) q^{67} +(4.18742 - 3.51366i) q^{68} +(4.57499 + 4.20470i) q^{69} +(-1.22043 - 0.181283i) q^{70} +(-11.3110 + 6.53043i) q^{71} +(-6.18714 + 6.15057i) q^{72} -9.73306i q^{73} +(4.98508 + 0.879004i) q^{74} +(-0.370709 + 8.21132i) q^{75} +(7.18433 - 1.26679i) q^{76} +(-8.52540 - 9.61893i) q^{77} +(7.47176 - 0.972412i) q^{78} +(-0.825069 + 4.67920i) q^{79} +(0.100577 + 0.174205i) q^{80} +(-7.76741 - 4.54613i) q^{81} +(4.64298 + 2.68063i) q^{82} +(-2.75845 + 1.00400i) q^{83} +(-3.29975 - 4.07978i) q^{84} +(1.84440 - 1.54764i) q^{85} +(2.49011 - 6.84151i) q^{86} +(2.71933 - 4.25456i) q^{87} +(-2.45321 + 13.9128i) q^{88} +(5.60470 + 9.70762i) q^{89} +(-0.992182 + 0.986316i) q^{90} +(0.336922 + 12.4430i) q^{91} +(4.04537 + 0.713308i) q^{92} +(1.29725 - 0.406908i) q^{93} +(-6.99893 + 1.23410i) q^{94} +(3.16443 - 0.557974i) q^{95} +(-2.05575 + 9.20829i) q^{96} +(2.41733 + 0.426241i) q^{97} +(-5.42202 + 3.53485i) q^{98} +(-14.5225 + 1.22718i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.910598 − 0.160563i −0.643890 − 0.113535i −0.157838 − 0.987465i \(-0.550452\pi\)
−0.486052 + 0.873930i \(0.661564\pi\)
\(3\) −1.05237 − 1.37569i −0.607585 − 0.794255i
\(4\) −1.07598 − 0.391623i −0.537988 − 0.195812i
\(5\) −0.473927 − 0.172495i −0.211947 − 0.0771423i 0.233865 − 0.972269i \(-0.424863\pi\)
−0.445812 + 0.895127i \(0.647085\pi\)
\(6\) 0.737399 + 1.42167i 0.301042 + 0.580395i
\(7\) −2.46079 + 0.971863i −0.930091 + 0.367330i
\(8\) 2.51844 + 1.45402i 0.890401 + 0.514073i
\(9\) −0.785044 + 2.89546i −0.261681 + 0.965154i
\(10\) 0.403861 + 0.233169i 0.127712 + 0.0737346i
\(11\) 1.66156 + 4.56510i 0.500979 + 1.37643i 0.890320 + 0.455335i \(0.150481\pi\)
−0.389341 + 0.921094i \(0.627297\pi\)
\(12\) 0.593571 + 1.89234i 0.171349 + 0.546272i
\(13\) 1.60911 − 4.42100i 0.446287 − 1.22616i −0.489003 − 0.872282i \(-0.662639\pi\)
0.935290 − 0.353882i \(-0.115138\pi\)
\(14\) 2.39684 − 0.489865i 0.640581 − 0.130922i
\(15\) 0.261446 + 0.833505i 0.0675050 + 0.215210i
\(16\) −0.305533 − 0.256373i −0.0763833 − 0.0640932i
\(17\) −2.38696 + 4.13434i −0.578923 + 1.00272i 0.416680 + 0.909053i \(0.363194\pi\)
−0.995603 + 0.0936714i \(0.970140\pi\)
\(18\) 1.17976 − 2.51055i 0.278073 − 0.591743i
\(19\) −5.51758 + 3.18557i −1.26582 + 0.730821i −0.974194 − 0.225712i \(-0.927529\pi\)
−0.291625 + 0.956533i \(0.594196\pi\)
\(20\) 0.442381 + 0.371202i 0.0989195 + 0.0830033i
\(21\) 3.92664 + 2.36252i 0.856862 + 0.515545i
\(22\) −0.780028 − 4.42376i −0.166302 − 0.943148i
\(23\) −3.53298 + 0.622960i −0.736677 + 0.129896i −0.529383 − 0.848383i \(-0.677577\pi\)
−0.207294 + 0.978279i \(0.566466\pi\)
\(24\) −0.650041 − 4.99475i −0.132689 − 1.01955i
\(25\) −3.63537 − 3.05044i −0.727074 − 0.610087i
\(26\) −2.17510 + 3.76739i −0.426573 + 0.738846i
\(27\) 4.80941 − 1.96711i 0.925572 − 0.378571i
\(28\) 3.02836 − 0.0819996i 0.572305 − 0.0154965i
\(29\) 0.997075 + 2.73944i 0.185152 + 0.508701i 0.997191 − 0.0749022i \(-0.0238645\pi\)
−0.812039 + 0.583604i \(0.801642\pi\)
\(30\) −0.104242 − 0.800967i −0.0190319 − 0.146236i
\(31\) −0.268468 + 0.737610i −0.0482183 + 0.132479i −0.961464 − 0.274930i \(-0.911345\pi\)
0.913246 + 0.407409i \(0.133568\pi\)
\(32\) −3.50145 − 4.17286i −0.618975 − 0.737665i
\(33\) 4.53159 − 7.08995i 0.788848 − 1.23420i
\(34\) 2.83739 − 3.38147i 0.486608 − 0.579916i
\(35\) 1.33388 − 0.0361178i 0.225466 − 0.00610501i
\(36\) 1.97862 − 2.80801i 0.329770 − 0.468001i
\(37\) −5.47451 −0.900004 −0.450002 − 0.893028i \(-0.648577\pi\)
−0.450002 + 0.893028i \(0.648577\pi\)
\(38\) 5.53578 − 2.01486i 0.898022 − 0.326853i
\(39\) −7.77529 + 2.43888i −1.24504 + 0.390533i
\(40\) −0.942744 − 1.12352i −0.149061 − 0.177644i
\(41\) −5.44849 − 1.98309i −0.850912 − 0.309707i −0.120500 − 0.992713i \(-0.538450\pi\)
−0.730412 + 0.683007i \(0.760672\pi\)
\(42\) −3.19626 − 2.78178i −0.493193 − 0.429238i
\(43\) −1.36729 + 7.75429i −0.208510 + 1.18252i 0.683310 + 0.730128i \(0.260540\pi\)
−0.891820 + 0.452390i \(0.850571\pi\)
\(44\) − 5.56264i − 0.838600i
\(45\) 0.871508 − 1.23682i 0.129917 − 0.184375i
\(46\) 3.31715 0.489087
\(47\) 7.22256 − 2.62880i 1.05352 − 0.383449i 0.243529 − 0.969894i \(-0.421695\pi\)
0.809989 + 0.586445i \(0.199473\pi\)
\(48\) −0.0311561 + 0.690117i −0.00449700 + 0.0996099i
\(49\) 5.11096 − 4.78310i 0.730138 − 0.683300i
\(50\) 2.82057 + 3.36143i 0.398889 + 0.475378i
\(51\) 8.19953 − 1.06713i 1.14816 − 0.149428i
\(52\) −3.46273 + 4.12672i −0.480194 + 0.572273i
\(53\) 4.26125 − 2.46023i 0.585328 − 0.337939i −0.177920 − 0.984045i \(-0.556937\pi\)
0.763248 + 0.646106i \(0.223604\pi\)
\(54\) −4.69529 + 1.01904i −0.638948 + 0.138673i
\(55\) − 2.45014i − 0.330376i
\(56\) −7.61045 − 1.13046i −1.01699 − 0.151064i
\(57\) 10.1889 + 4.23808i 1.34955 + 0.561347i
\(58\) −0.468082 − 2.65462i −0.0614621 − 0.348569i
\(59\) −4.08557 + 3.42820i −0.531896 + 0.446313i −0.868755 − 0.495241i \(-0.835080\pi\)
0.336860 + 0.941555i \(0.390635\pi\)
\(60\) 0.0451109 − 0.999221i 0.00582380 − 0.128999i
\(61\) −2.45624 − 6.74846i −0.314489 − 0.864052i −0.991736 − 0.128296i \(-0.959049\pi\)
0.677247 − 0.735756i \(-0.263173\pi\)
\(62\) 0.362900 − 0.628561i 0.0460883 − 0.0798273i
\(63\) −0.882166 − 7.88808i −0.111143 − 0.993804i
\(64\) 2.91725 + 5.05283i 0.364656 + 0.631603i
\(65\) −1.52520 + 1.81767i −0.189178 + 0.225454i
\(66\) −5.26484 + 5.72849i −0.648057 + 0.705129i
\(67\) 1.51964 + 8.61830i 0.185653 + 1.05289i 0.925113 + 0.379692i \(0.123970\pi\)
−0.739460 + 0.673201i \(0.764919\pi\)
\(68\) 4.18742 − 3.51366i 0.507799 − 0.426094i
\(69\) 4.57499 + 4.20470i 0.550764 + 0.506187i
\(70\) −1.22043 − 0.181283i −0.145869 − 0.0216674i
\(71\) −11.3110 + 6.53043i −1.34237 + 0.775019i −0.987155 − 0.159764i \(-0.948927\pi\)
−0.355217 + 0.934784i \(0.615593\pi\)
\(72\) −6.18714 + 6.15057i −0.729162 + 0.724851i
\(73\) − 9.73306i − 1.13917i −0.821933 − 0.569584i \(-0.807104\pi\)
0.821933 − 0.569584i \(-0.192896\pi\)
\(74\) 4.98508 + 0.879004i 0.579504 + 0.102182i
\(75\) −0.370709 + 8.21132i −0.0428058 + 0.948162i
\(76\) 7.18433 − 1.26679i 0.824099 − 0.145311i
\(77\) −8.52540 − 9.61893i −0.971560 − 1.09618i
\(78\) 7.47176 − 0.972412i 0.846011 − 0.110104i
\(79\) −0.825069 + 4.67920i −0.0928275 + 0.526451i 0.902564 + 0.430556i \(0.141682\pi\)
−0.995391 + 0.0958952i \(0.969429\pi\)
\(80\) 0.100577 + 0.174205i 0.0112449 + 0.0194767i
\(81\) −7.76741 − 4.54613i −0.863046 − 0.505126i
\(82\) 4.64298 + 2.68063i 0.512731 + 0.296026i
\(83\) −2.75845 + 1.00400i −0.302780 + 0.110203i −0.488942 − 0.872316i \(-0.662617\pi\)
0.186163 + 0.982519i \(0.440395\pi\)
\(84\) −3.29975 − 4.07978i −0.360032 − 0.445141i
\(85\) 1.84440 − 1.54764i 0.200053 − 0.167865i
\(86\) 2.49011 − 6.84151i 0.268515 − 0.737739i
\(87\) 2.71933 − 4.25456i 0.291543 − 0.456137i
\(88\) −2.45321 + 13.9128i −0.261513 + 1.48311i
\(89\) 5.60470 + 9.70762i 0.594097 + 1.02901i 0.993674 + 0.112306i \(0.0358236\pi\)
−0.399577 + 0.916700i \(0.630843\pi\)
\(90\) −0.992182 + 0.986316i −0.104585 + 0.103967i
\(91\) 0.336922 + 12.4430i 0.0353190 + 1.30438i
\(92\) 4.04537 + 0.713308i 0.421759 + 0.0743675i
\(93\) 1.29725 − 0.406908i 0.134519 − 0.0421944i
\(94\) −6.99893 + 1.23410i −0.721885 + 0.127288i
\(95\) 3.16443 − 0.557974i 0.324663 − 0.0572469i
\(96\) −2.05575 + 9.20829i −0.209815 + 0.939818i
\(97\) 2.41733 + 0.426241i 0.245443 + 0.0432783i 0.295016 − 0.955492i \(-0.404675\pi\)
−0.0495730 + 0.998771i \(0.515786\pi\)
\(98\) −5.42202 + 3.53485i −0.547707 + 0.357074i
\(99\) −14.5225 + 1.22718i −1.45956 + 0.123336i
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.47.8 yes 132
3.2 odd 2 567.2.bd.a.467.15 132
7.3 odd 6 189.2.ba.a.101.15 ✓ 132
21.17 even 6 567.2.ba.a.143.8 132
27.4 even 9 567.2.ba.a.341.8 132
27.23 odd 18 189.2.ba.a.131.15 yes 132
189.31 odd 18 567.2.bd.a.17.15 132
189.185 even 18 inner 189.2.bd.a.185.8 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.15 ✓ 132 7.3 odd 6
189.2.ba.a.131.15 yes 132 27.23 odd 18
189.2.bd.a.47.8 yes 132 1.1 even 1 trivial
189.2.bd.a.185.8 yes 132 189.185 even 18 inner
567.2.ba.a.143.8 132 21.17 even 6
567.2.ba.a.341.8 132 27.4 even 9
567.2.bd.a.17.15 132 189.31 odd 18
567.2.bd.a.467.15 132 3.2 odd 2