↑

Properties

Label 189.2.ba.a.101.11
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 101.11
Character \(\chi\) \(=\) 189.101
Dual form 189.2.ba.a.131.11

$q$-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+(-0.0103898 + 0.0123821i) q^{2} +(1.25758 - 1.19100i) q^{3} +(0.347251 + 1.96936i) q^{4} +(3.05823 - 2.56616i) q^{5} +(0.00168110 + 0.0279459i) q^{6} +(-2.57348 + 0.614149i) q^{7} +(-0.0559891 - 0.0323253i) q^{8} +(0.163018 - 2.99557i) q^{9} +0.0645293i q^{10} +(-3.31590 + 3.95174i) q^{11} +(2.78221 + 2.06305i) q^{12} +(0.220791 - 0.606618i) q^{13} +(0.0191336 - 0.0382461i) q^{14} +(0.789663 - 6.86951i) q^{15} +(-3.75730 + 1.36754i) q^{16} +3.38766 q^{17} +(0.0353978 + 0.0331420i) q^{18} -0.379246i q^{19} +(6.11565 + 5.13164i) q^{20} +(-2.50491 + 3.83737i) q^{21} +(-0.0144792 - 0.0821158i) q^{22} +(0.344877 - 0.947543i) q^{23} +(-0.108910 + 0.0260315i) q^{24} +(1.89935 - 10.7717i) q^{25} +(0.00521724 + 0.00903653i) q^{26} +(-3.36272 - 3.96132i) q^{27} +(-2.10312 - 4.85485i) q^{28} +(1.97120 + 5.41582i) q^{29} +(0.0768546 + 0.0811507i) q^{30} +(-6.35598 + 1.12073i) q^{31} +(0.0663282 - 0.182235i) q^{32} +(0.536522 + 8.91889i) q^{33} +(-0.0351973 + 0.0419465i) q^{34} +(-6.29429 + 8.48217i) q^{35} +(5.95595 - 0.719172i) q^{36} +(-3.08407 + 5.34176i) q^{37} +(0.00469587 + 0.00394030i) q^{38} +(-0.444822 - 1.02583i) q^{39} +(-0.254179 + 0.0448186i) q^{40} +(-0.266252 - 0.0969077i) q^{41} +(-0.0214892 - 0.0708858i) q^{42} +(-1.45052 + 8.22633i) q^{43} +(-8.93384 - 5.15796i) q^{44} +(-7.18855 - 9.57945i) q^{45} +(0.00814938 + 0.0141151i) q^{46} +(1.27739 - 7.24446i) q^{47} +(-3.09635 + 6.19475i) q^{48} +(6.24564 - 3.16101i) q^{49} +(0.113643 + 0.135434i) q^{50} +(4.26026 - 4.03472i) q^{51} +(1.27132 + 0.224168i) q^{52} +(-1.71989 - 0.992981i) q^{53} +(0.0839877 - 0.000480176i) q^{54} +20.5944i q^{55} +(0.163940 + 0.0488030i) q^{56} +(-0.451683 - 0.476932i) q^{57} +(-0.0875398 - 0.0318619i) q^{58} +(-3.21846 - 1.17142i) q^{59} +(13.8027 - 0.830314i) q^{60} +(-7.93400 - 1.39898i) q^{61} +(0.0521605 - 0.0903447i) q^{62} +(1.42020 + 7.80916i) q^{63} +(-3.99687 - 6.92277i) q^{64} +(-0.881448 - 2.42176i) q^{65} +(-0.116009 - 0.0860225i) q^{66} +(7.23088 - 6.06743i) q^{67} +(1.17637 + 6.67152i) q^{68} +(-0.694817 - 1.60236i) q^{69} +(-0.0396306 - 0.166065i) q^{70} +(8.57363 - 4.94999i) q^{71} +(-0.105960 + 0.162449i) q^{72} +(-6.13914 + 3.54444i) q^{73} +(-0.0340994 - 0.0936874i) q^{74} +(-10.4406 - 15.8084i) q^{75} +(0.746871 - 0.131693i) q^{76} +(6.10647 - 12.2062i) q^{77} +(0.0173236 + 0.00515040i) q^{78} +(-0.980903 - 0.823075i) q^{79} +(-7.98133 + 13.8241i) q^{80} +(-8.94685 - 0.976665i) q^{81} +(0.00396624 - 0.00228991i) q^{82} +(8.07527 - 2.93916i) q^{83} +(-8.42699 - 3.60053i) q^{84} +(10.3602 - 8.69327i) q^{85} +(-0.0867887 - 0.103431i) q^{86} +(8.92921 + 4.46313i) q^{87} +(0.313396 - 0.114067i) q^{88} -5.35698 q^{89} +(0.193302 + 0.0105194i) q^{90} +(-0.195648 + 1.69672i) q^{91} +(1.98581 + 0.350152i) q^{92} +(-6.65836 + 8.97940i) q^{93} +(0.0764298 + 0.0910855i) q^{94} +(-0.973203 - 1.15982i) q^{95} +(-0.133630 - 0.308173i) q^{96} +(7.39427 + 1.30381i) q^{97} +(-0.0257512 + 0.110177i) q^{98} +(11.2971 + 10.5772i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0103898 + 0.0123821i −0.00734672 + 0.00875548i −0.769705 − 0.638399i \(-0.779597\pi\)
0.762359 + 0.647155i \(0.224041\pi\)
\(3\) 1.25758 − 1.19100i 0.726065 − 0.687627i
\(4\) 0.347251 + 1.96936i 0.173625 + 0.984679i
\(5\) 3.05823 − 2.56616i 1.36768 − 1.14762i 0.394154 − 0.919045i \(-0.371038\pi\)
0.973526 − 0.228575i \(-0.0734066\pi\)
\(6\) 0.00168110 + 0.0279459i 0.000686308 + 0.0114088i
\(7\) −2.57348 + 0.614149i −0.972686 + 0.232127i
\(8\) −0.0559891 − 0.0323253i −0.0197951 − 0.0114287i
\(9\) 0.163018 − 2.99557i 0.0543394 − 0.998523i
\(10\) 0.0645293i 0.0204059i
\(11\) −3.31590 + 3.95174i −0.999782 + 1.19149i −0.0183214 + 0.999832i \(0.505832\pi\)
−0.981461 + 0.191662i \(0.938612\pi\)
\(12\) 2.78221 + 2.06305i 0.803155 + 0.595551i
\(13\) 0.220791 − 0.606618i 0.0612364 − 0.168246i −0.905301 − 0.424770i \(-0.860355\pi\)
0.966538 + 0.256524i \(0.0825773\pi\)
\(14\) 0.0191336 − 0.0382461i 0.00511367 − 0.0102217i
\(15\) 0.789663 − 6.86951i 0.203890 − 1.77370i
\(16\) −3.75730 + 1.36754i −0.939324 + 0.341886i
\(17\) 3.38766 0.821629 0.410815 − 0.911719i \(-0.365244\pi\)
0.410815 + 0.911719i \(0.365244\pi\)
\(18\) 0.0353978 + 0.0331420i 0.00834333 + 0.00781164i
\(19\) − 0.379246i − 0.0870049i −0.999053 − 0.0435025i \(-0.986148\pi\)
0.999053 − 0.0435025i \(-0.0138516\pi\)
\(20\) 6.11565 + 5.13164i 1.36750 + 1.14747i
\(21\) −2.50491 + 3.83737i −0.546616 + 0.837383i
\(22\) −0.0144792 − 0.0821158i −0.00308698 − 0.0175072i
\(23\) 0.344877 − 0.947543i 0.0719119 − 0.197576i −0.898530 − 0.438913i \(-0.855364\pi\)
0.970441 + 0.241337i \(0.0775858\pi\)
\(24\) −0.108910 + 0.0260315i −0.0222312 + 0.00531367i
\(25\) 1.89935 − 10.7717i 0.379869 − 2.15434i
\(26\) 0.00521724 + 0.00903653i 0.00102318 + 0.00177221i
\(27\) −3.36272 − 3.96132i −0.647157 − 0.762357i
\(28\) −2.10312 − 4.85485i −0.397453 − 0.917480i
\(29\) 1.97120 + 5.41582i 0.366042 + 1.00569i 0.976852 + 0.213916i \(0.0686218\pi\)
−0.610810 + 0.791777i \(0.709156\pi\)
\(30\) 0.0768546 + 0.0811507i 0.0140317 + 0.0148160i
\(31\) −6.35598 + 1.12073i −1.14157 + 0.201289i −0.712291 − 0.701884i \(-0.752342\pi\)
−0.429276 + 0.903173i \(0.641231\pi\)
\(32\) 0.0663282 − 0.182235i 0.0117253 − 0.0322150i
\(33\) 0.536522 + 8.91889i 0.0933966 + 1.55258i
\(34\) −0.0351973 + 0.0419465i −0.00603628 + 0.00719376i
\(35\) −6.29429 + 8.48217i −1.06393 + 1.43375i
\(36\) 5.95595 − 0.719172i 0.992659 − 0.119862i
\(37\) −3.08407 + 5.34176i −0.507018 + 0.878181i 0.492949 + 0.870058i \(0.335919\pi\)
−0.999967 + 0.00812261i \(0.997414\pi\)
\(38\) 0.00469587 + 0.00394030i 0.000761770 + 0.000639201i
\(39\) −0.444822 − 1.02583i −0.0712286 − 0.164265i
\(40\) −0.254179 + 0.0448186i −0.0401892 + 0.00708645i
\(41\) −0.266252 − 0.0969077i −0.0415816 − 0.0151344i 0.321146 − 0.947030i \(-0.395932\pi\)
−0.362727 + 0.931895i \(0.618154\pi\)
\(42\) −0.0214892 − 0.0708858i −0.00331586 − 0.0109379i
\(43\) −1.45052 + 8.22633i −0.221203 + 1.25450i 0.648610 + 0.761121i \(0.275351\pi\)
−0.869812 + 0.493383i \(0.835760\pi\)
\(44\) −8.93384 − 5.15796i −1.34683 − 0.777591i
\(45\) −7.18855 − 9.57945i −1.07161 − 1.42802i
\(46\) 0.00814938 + 0.0141151i 0.00120156 + 0.00208116i
\(47\) 1.27739 − 7.24446i 0.186327 − 1.05671i −0.737912 − 0.674897i \(-0.764188\pi\)
0.924239 − 0.381815i \(-0.124701\pi\)
\(48\) −3.09635 + 6.19475i −0.446920 + 0.894136i
\(49\) 6.24564 − 3.16101i 0.892235 − 0.451572i
\(50\) 0.113643 + 0.135434i 0.0160715 + 0.0191533i
\(51\) 4.26026 − 4.03472i 0.596556 − 0.564974i
\(52\) 1.27132 + 0.224168i 0.176300 + 0.0310865i
\(53\) −1.71989 − 0.992981i −0.236246 − 0.136396i 0.377204 − 0.926130i \(-0.376885\pi\)
−0.613450 + 0.789734i \(0.710219\pi\)
\(54\) 0.0839877 0.000480176i 0.0114293 6.53437e-5i
\(55\) 20.5944i 2.77695i
\(56\) 0.163940 + 0.0488030i 0.0219074 + 0.00652158i
\(57\) −0.451683 − 0.476932i −0.0598269 − 0.0631712i
\(58\) −0.0875398 − 0.0318619i −0.0114945 − 0.00418367i
\(59\) −3.21846 − 1.17142i −0.419007 − 0.152506i 0.123908 − 0.992294i \(-0.460457\pi\)
−0.542915 + 0.839788i \(0.682679\pi\)
\(60\) 13.8027 − 0.830314i 1.78192 − 0.107193i
\(61\) −7.93400 − 1.39898i −1.01584 − 0.179121i −0.359152 − 0.933279i \(-0.616934\pi\)
−0.656692 + 0.754158i \(0.728045\pi\)
\(62\) 0.0521605 − 0.0903447i 0.00662439 − 0.0114738i
\(63\) 1.42020 + 7.80916i 0.178928 + 0.983862i
\(64\) −3.99687 − 6.92277i −0.499608 − 0.865347i
\(65\) −0.881448 − 2.42176i −0.109330 − 0.300382i
\(66\) −0.116009 − 0.0860225i −0.0142797 − 0.0105886i
\(67\) 7.23088 − 6.06743i 0.883393 − 0.741255i −0.0834809 − 0.996509i \(-0.526604\pi\)
0.966874 + 0.255255i \(0.0821593\pi\)
\(68\) 1.17637 + 6.67152i 0.142656 + 0.809041i
\(69\) −0.694817 − 1.60236i −0.0836461 − 0.192902i
\(70\) −0.0396306 − 0.166065i −0.00473676 − 0.0198486i
\(71\) 8.57363 − 4.94999i 1.01750 − 0.587456i 0.104123 − 0.994564i \(-0.466796\pi\)
0.913380 + 0.407109i \(0.133463\pi\)
\(72\) −0.105960 + 0.162449i −0.0124875 + 0.0191449i
\(73\) −6.13914 + 3.54444i −0.718533 + 0.414845i −0.814212 − 0.580567i \(-0.802831\pi\)
0.0956798 + 0.995412i \(0.469498\pi\)
\(74\) −0.0340994 − 0.0936874i −0.00396398 − 0.0108909i
\(75\) −10.4406 − 15.8084i −1.20558 − 1.82540i
\(76\) 0.746871 − 0.131693i 0.0856719 − 0.0151063i
\(77\) 6.10647 − 12.2062i 0.695896 − 1.39103i
\(78\) 0.0173236 + 0.00515040i 0.00196152 + 0.000583168i
\(79\) −0.980903 − 0.823075i −0.110360 − 0.0926032i 0.585938 − 0.810356i \(-0.300726\pi\)
−0.696298 + 0.717753i \(0.745171\pi\)
\(80\) −7.98133 + 13.8241i −0.892340 + 1.54558i
\(81\) −8.94685 − 0.976665i −0.994094 − 0.108518i
\(82\) 0.00396624 − 0.00228991i 0.000437998 − 0.000252878i
\(83\) 8.07527 − 2.93916i 0.886376 − 0.322614i 0.141596 − 0.989925i \(-0.454777\pi\)
0.744780 + 0.667310i \(0.232554\pi\)
\(84\) −8.42699 − 3.60053i −0.919460 − 0.392850i
\(85\) 10.3602 − 8.69327i 1.12373 − 0.942918i
\(86\) −0.0867887 − 0.103431i −0.00935867 − 0.0111532i
\(87\) 8.92921 + 4.46313i 0.957312 + 0.478498i
\(88\) 0.313396 − 0.114067i 0.0334081 − 0.0121595i
\(89\) −5.35698 −0.567839 −0.283920 − 0.958848i \(-0.591635\pi\)
−0.283920 + 0.958848i \(0.591635\pi\)
\(90\) 0.193302 + 0.0105194i 0.0203758 + 0.00110885i
\(91\) −0.195648 + 1.69672i −0.0205095 + 0.177865i
\(92\) 1.98581 + 0.350152i 0.207035 + 0.0365059i
\(93\) −6.65836 + 8.97940i −0.690440 + 0.931121i
\(94\) 0.0764298 + 0.0910855i 0.00788314 + 0.00939476i
\(95\) −0.973203 − 1.15982i −0.0998486 − 0.118995i
\(96\) −0.133630 − 0.308173i −0.0136385 − 0.0314528i
\(97\) 7.39427 + 1.30381i 0.750774 + 0.132382i 0.535925 − 0.844265i \(-0.319963\pi\)
0.214849 + 0.976647i \(0.431074\pi\)
\(98\) −0.0257512 + 0.110177i −0.00260127 + 0.0111295i
\(99\) 11.2971 + 10.5772i 1.13541 + 1.06305i
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.101.11 ✓ 132
3.2 odd 2 567.2.ba.a.143.12 132
7.5 odd 6 189.2.bd.a.47.12 yes 132
21.5 even 6 567.2.bd.a.467.11 132
27.4 even 9 567.2.bd.a.17.11 132
27.23 odd 18 189.2.bd.a.185.12 yes 132
189.131 even 18 inner 189.2.ba.a.131.11 yes 132
189.166 odd 18 567.2.ba.a.341.12 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.11 ✓ 132 1.1 even 1 trivial
189.2.ba.a.131.11 yes 132 189.131 even 18 inner
189.2.bd.a.47.12 yes 132 7.5 odd 6
189.2.bd.a.185.12 yes 132 27.23 odd 18
567.2.ba.a.143.12 132 3.2 odd 2
567.2.ba.a.341.12 132 189.166 odd 18
567.2.bd.a.17.11 132 27.4 even 9
567.2.bd.a.467.11 132 21.5 even 6