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Properties

Label 189.2.be.a.104.18
Level $189$
Weight $2$
Character 189.104
Analytic conductor $1.509$
Analytic rank $0$
Dimension $132$
Inner twists $4$

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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(20,189)] 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("189.20"); 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, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.be (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 104.18
Character \(\chi\) \(=\) 189.104
Dual form 189.2.be.a.20.18

$q$-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+(0.492726 - 1.35375i) q^{2} +(0.333137 + 1.69971i) q^{3} +(-0.0577819 - 0.0484848i) q^{4} +(-0.440273 + 2.49691i) q^{5} +(2.46514 + 0.386507i) q^{6} +(-2.34762 + 1.22011i) q^{7} +(2.40115 - 1.38630i) q^{8} +(-2.77804 + 1.13247i) q^{9} +(3.16327 + 1.82631i) q^{10} +(2.06906 - 0.364832i) q^{11} +(0.0631609 - 0.114365i) q^{12} +(-0.0331071 - 0.0909611i) q^{13} +(0.494989 + 3.77928i) q^{14} +(-4.39070 + 0.0834760i) q^{15} +(-0.719801 - 4.08219i) q^{16} +(3.94280 - 6.82913i) q^{17} +(0.164277 + 4.31878i) q^{18} +(-1.53773 + 0.887809i) q^{19} +(0.146502 - 0.122930i) q^{20} +(-2.85591 - 3.58382i) q^{21} +(0.525589 - 2.98077i) q^{22} +(3.82906 - 4.56329i) q^{23} +(3.15622 + 3.61943i) q^{24} +(-1.34226 - 0.488541i) q^{25} -0.139452 q^{26} +(-2.85034 - 4.34460i) q^{27} +(0.194807 + 0.0433240i) q^{28} +(0.573096 - 1.57457i) q^{29} +(-2.05041 + 5.98506i) q^{30} +(-0.916092 + 1.09176i) q^{31} +(-0.419987 - 0.0740550i) q^{32} +(1.30939 + 3.39527i) q^{33} +(-7.30224 - 8.70247i) q^{34} +(-2.01290 - 6.39899i) q^{35} +(0.215428 + 0.0692562i) q^{36} +(-1.88241 + 3.26042i) q^{37} +(0.444195 + 2.51916i) q^{38} +(0.143578 - 0.0865751i) q^{39} +(2.40431 + 6.60580i) q^{40} +(-2.60026 + 0.946417i) q^{41} +(-6.25879 + 2.10036i) q^{42} +(1.03518 + 5.87080i) q^{43} +(-0.137243 - 0.0792375i) q^{44} +(-1.60459 - 7.43511i) q^{45} +(-4.29090 - 7.43206i) q^{46} +(-3.23438 + 2.71396i) q^{47} +(6.69876 - 2.58338i) q^{48} +(4.02268 - 5.72870i) q^{49} +(-1.32273 + 1.57637i) q^{50} +(12.9210 + 4.42659i) q^{51} +(-0.00249724 + 0.00686110i) q^{52} +9.25188i q^{53} +(-7.28596 + 1.71797i) q^{54} +5.32689i q^{55} +(-3.94555 + 6.18417i) q^{56} +(-2.02129 - 2.31794i) q^{57} +(-1.84920 - 1.55166i) q^{58} +(-1.28271 + 7.27458i) q^{59} +(0.257750 + 0.208059i) q^{60} +(-7.80852 - 9.30583i) q^{61} +(1.02659 + 1.77810i) q^{62} +(5.14006 - 6.04813i) q^{63} +(3.83798 - 6.64757i) q^{64} +(0.241698 - 0.0426179i) q^{65} +(5.24153 - 0.0996521i) q^{66} +(-10.6299 + 3.86898i) q^{67} +(-0.558931 + 0.203434i) q^{68} +(9.03188 + 4.98809i) q^{69} +(-9.65446 - 0.427973i) q^{70} +(-11.9234 - 6.88396i) q^{71} +(-5.10053 + 6.57043i) q^{72} +(12.0252 - 6.94274i) q^{73} +(3.48630 + 4.15481i) q^{74} +(0.383224 - 2.44420i) q^{75} +(0.131898 + 0.0232572i) q^{76} +(-4.41225 + 3.38097i) q^{77} +(-0.0464565 - 0.237028i) q^{78} +(-5.82433 - 2.11988i) q^{79} +10.5098 q^{80} +(6.43501 - 6.29211i) q^{81} +3.98644i q^{82} +(8.67692 + 3.15814i) q^{83} +(-0.00874090 + 0.345548i) q^{84} +(15.3158 + 12.8515i) q^{85} +(8.45769 + 1.49132i) q^{86} +(2.86723 + 0.449552i) q^{87} +(4.46236 - 3.74436i) q^{88} +(-6.28476 - 10.8855i) q^{89} +(-10.8559 - 1.49126i) q^{90} +(0.188705 + 0.173148i) q^{91} +(-0.442501 + 0.0780248i) q^{92} +(-2.16085 - 1.19339i) q^{93} +(2.08038 + 5.71579i) q^{94} +(-1.53976 - 4.23045i) q^{95} +(-0.0140409 - 0.738527i) q^{96} +(15.0854 - 2.65997i) q^{97} +(-5.77318 - 8.26840i) q^{98} +(-5.33478 + 3.35668i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q - 12 q^{2} - 12 q^{4} - 6 q^{7} - 18 q^{8} - 6 q^{9} - 18 q^{11} + 3 q^{14} - 24 q^{15} - 24 q^{16} - 12 q^{18} - 12 q^{21} - 12 q^{22} + 12 q^{23} - 12 q^{25} - 12 q^{28} - 48 q^{29} + 42 q^{30}+ \cdots - 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(e\left(\frac{11}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.492726 − 1.35375i 0.348410 − 0.957249i −0.634461 − 0.772955i \(-0.718778\pi\)
0.982871 − 0.184294i \(-0.0589998\pi\)
\(3\) 0.333137 + 1.69971i 0.192337 + 0.981329i
\(4\) −0.0577819 − 0.0484848i −0.0288910 − 0.0242424i
\(5\) −0.440273 + 2.49691i −0.196896 + 1.11665i 0.712797 + 0.701370i \(0.247428\pi\)
−0.909693 + 0.415282i \(0.863683\pi\)
\(6\) 2.46514 + 0.386507i 1.00639 + 0.157791i
\(7\) −2.34762 + 1.22011i −0.887319 + 0.461157i
\(8\) 2.40115 − 1.38630i 0.848933 − 0.490132i
\(9\) −2.77804 + 1.13247i −0.926013 + 0.377491i
\(10\) 3.16327 + 1.82631i 1.00031 + 0.577531i
\(11\) 2.06906 − 0.364832i 0.623846 − 0.110001i 0.147215 − 0.989104i \(-0.452969\pi\)
0.476631 + 0.879104i \(0.341858\pi\)
\(12\) 0.0631609 − 0.114365i 0.0182330 − 0.0330142i
\(13\) −0.0331071 − 0.0909611i −0.00918227 − 0.0252281i 0.935017 − 0.354603i \(-0.115384\pi\)
−0.944199 + 0.329375i \(0.893162\pi\)
\(14\) 0.494989 + 3.77928i 0.132291 + 1.01006i
\(15\) −4.39070 + 0.0834760i −1.13367 + 0.0215534i
\(16\) −0.719801 − 4.08219i −0.179950 − 1.02055i
\(17\) 3.94280 − 6.82913i 0.956269 − 1.65631i 0.224831 − 0.974398i \(-0.427817\pi\)
0.731438 − 0.681908i \(-0.238850\pi\)
\(18\) 0.164277 + 4.31878i 0.0387204 + 1.01795i
\(19\) −1.53773 + 0.887809i −0.352780 + 0.203677i −0.665909 − 0.746033i \(-0.731956\pi\)
0.313129 + 0.949711i \(0.398623\pi\)
\(20\) 0.146502 − 0.122930i 0.0327588 − 0.0274879i
\(21\) −2.85591 − 3.58382i −0.623211 − 0.782054i
\(22\) 0.525589 − 2.98077i 0.112056 − 0.635501i
\(23\) 3.82906 − 4.56329i 0.798414 − 0.951513i −0.201193 − 0.979552i \(-0.564482\pi\)
0.999607 + 0.0280391i \(0.00892629\pi\)
\(24\) 3.15622 + 3.61943i 0.644261 + 0.738813i
\(25\) −1.34226 − 0.488541i −0.268451 − 0.0977083i
\(26\) −0.139452 −0.0273487
\(27\) −2.85034 − 4.34460i −0.548549 − 0.836118i
\(28\) 0.194807 + 0.0433240i 0.0368150 + 0.00818746i
\(29\) 0.573096 − 1.57457i 0.106421 − 0.292390i −0.875040 − 0.484051i \(-0.839165\pi\)
0.981461 + 0.191661i \(0.0613873\pi\)
\(30\) −2.05041 + 5.98506i −0.374351 + 1.09272i
\(31\) −0.916092 + 1.09176i −0.164535 + 0.196085i −0.842012 − 0.539459i \(-0.818629\pi\)
0.677477 + 0.735544i \(0.263073\pi\)
\(32\) −0.419987 − 0.0740550i −0.0742439 − 0.0130912i
\(33\) 1.30939 + 3.39527i 0.227936 + 0.591041i
\(34\) −7.30224 − 8.70247i −1.25232 − 1.49246i
\(35\) −2.01290 − 6.39899i −0.340243 − 1.08163i
\(36\) 0.215428 + 0.0692562i 0.0359047 + 0.0115427i
\(37\) −1.88241 + 3.26042i −0.309466 + 0.536010i −0.978246 − 0.207450i \(-0.933483\pi\)
0.668780 + 0.743460i \(0.266817\pi\)
\(38\) 0.444195 + 2.51916i 0.0720580 + 0.408661i
\(39\) 0.143578 − 0.0865751i 0.0229910 − 0.0138631i
\(40\) 2.40431 + 6.60580i 0.380155 + 1.04447i
\(41\) −2.60026 + 0.946417i −0.406092 + 0.147806i −0.536987 − 0.843591i \(-0.680437\pi\)
0.130894 + 0.991396i \(0.458215\pi\)
\(42\) −6.25879 + 2.10036i −0.965753 + 0.324092i
\(43\) 1.03518 + 5.87080i 0.157864 + 0.895289i 0.956121 + 0.292973i \(0.0946445\pi\)
−0.798257 + 0.602317i \(0.794244\pi\)
\(44\) −0.137243 − 0.0792375i −0.0206902 − 0.0119455i
\(45\) −1.60459 − 7.43511i −0.239198 − 1.10836i
\(46\) −4.29090 − 7.43206i −0.632659 − 1.09580i
\(47\) −3.23438 + 2.71396i −0.471782 + 0.395872i −0.847444 − 0.530885i \(-0.821860\pi\)
0.375662 + 0.926757i \(0.377415\pi\)
\(48\) 6.69876 − 2.58338i 0.966883 − 0.372879i
\(49\) 4.02268 − 5.72870i 0.574668 − 0.818386i
\(50\) −1.32273 + 1.57637i −0.187062 + 0.222932i
\(51\) 12.9210 + 4.42659i 1.80931 + 0.619846i
\(52\) −0.00249724 + 0.00686110i −0.000346304 + 0.000951464i
\(53\) 9.25188i 1.27084i 0.772165 + 0.635422i \(0.219174\pi\)
−0.772165 + 0.635422i \(0.780826\pi\)
\(54\) −7.28596 + 1.71797i −0.991493 + 0.233786i
\(55\) 5.32689i 0.718278i
\(56\) −3.94555 + 6.18417i −0.527246 + 0.826395i
\(57\) −2.02129 − 2.31794i −0.267727 − 0.307018i
\(58\) −1.84920 − 1.55166i −0.242812 − 0.203743i
\(59\) −1.28271 + 7.27458i −0.166994 + 0.947070i 0.779991 + 0.625791i \(0.215224\pi\)
−0.946985 + 0.321279i \(0.895887\pi\)
\(60\) 0.257750 + 0.208059i 0.0332754 + 0.0268603i
\(61\) −7.80852 − 9.30583i −0.999778 − 1.19149i −0.981462 − 0.191657i \(-0.938614\pi\)
−0.0183160 − 0.999832i \(-0.505831\pi\)
\(62\) 1.02659 + 1.77810i 0.130377 + 0.225819i
\(63\) 5.14006 − 6.04813i 0.647586 − 0.761992i
\(64\) 3.83798 − 6.64757i 0.479747 − 0.830947i
\(65\) 0.241698 − 0.0426179i 0.0299789 − 0.00528609i
\(66\) 5.24153 − 0.0996521i 0.645188 − 0.0122663i
\(67\) −10.6299 + 3.86898i −1.29865 + 0.472671i −0.896558 − 0.442926i \(-0.853940\pi\)
−0.402096 + 0.915597i \(0.631718\pi\)
\(68\) −0.558931 + 0.203434i −0.0677804 + 0.0246700i
\(69\) 9.03188 + 4.98809i 1.08731 + 0.600496i
\(70\) −9.65446 − 0.427973i −1.15393 − 0.0511526i
\(71\) −11.9234 − 6.88396i −1.41504 − 0.816976i −0.419185 − 0.907901i \(-0.637684\pi\)
−0.995858 + 0.0909251i \(0.971018\pi\)
\(72\) −5.10053 + 6.57043i −0.601103 + 0.774333i
\(73\) 12.0252 − 6.94274i 1.40744 − 0.812587i 0.412301 − 0.911048i \(-0.364725\pi\)
0.995141 + 0.0984612i \(0.0313920\pi\)
\(74\) 3.48630 + 4.15481i 0.405274 + 0.482987i
\(75\) 0.383224 − 2.44420i 0.0442510 − 0.282232i
\(76\) 0.131898 + 0.0232572i 0.0151298 + 0.00266779i
\(77\) −4.41225 + 3.38097i −0.502823 + 0.385297i
\(78\) −0.0464565 − 0.237028i −0.00526016 − 0.0268381i
\(79\) −5.82433 − 2.11988i −0.655288 − 0.238505i −0.00708756 − 0.999975i \(-0.502256\pi\)
−0.648201 + 0.761469i \(0.724478\pi\)
\(80\) 10.5098 1.17503
\(81\) 6.43501 − 6.29211i 0.715001 − 0.699123i
\(82\) 3.98644i 0.440228i
\(83\) 8.67692 + 3.15814i 0.952416 + 0.346651i 0.771057 − 0.636766i \(-0.219728\pi\)
0.181359 + 0.983417i \(0.441951\pi\)
\(84\) −0.00874090 + 0.345548i −0.000953710 + 0.0377024i
\(85\) 15.3158 + 12.8515i 1.66123 + 1.39394i
\(86\) 8.45769 + 1.49132i 0.912016 + 0.160813i
\(87\) 2.86723 + 0.449552i 0.307400 + 0.0481970i
\(88\) 4.46236 − 3.74436i 0.475689 − 0.399150i
\(89\) −6.28476 − 10.8855i −0.666183 − 1.15386i −0.978963 − 0.204037i \(-0.934594\pi\)
0.312781 − 0.949825i \(-0.398740\pi\)
\(90\) −10.8559 − 1.49126i −1.14432 − 0.157192i
\(91\) 0.188705 + 0.173148i 0.0197817 + 0.0181509i
\(92\) −0.442501 + 0.0780248i −0.0461339 + 0.00813465i
\(93\) −2.16085 − 1.19339i −0.224070 − 0.123749i
\(94\) 2.08038 + 5.71579i 0.214575 + 0.589539i
\(95\) −1.53976 − 4.23045i −0.157976 − 0.434035i
\(96\) −0.0140409 − 0.738527i −0.00143304 − 0.0753756i
\(97\) 15.0854 − 2.65997i 1.53169 − 0.270079i 0.656676 − 0.754172i \(-0.271962\pi\)
0.875016 + 0.484094i \(0.160851\pi\)
\(98\) −5.77318 − 8.26840i −0.583179 − 0.835235i
\(99\) −5.33478 + 3.35668i −0.536165 + 0.337359i
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.be.a.104.18 yes 132
3.2 odd 2 567.2.be.a.503.6 132
7.6 odd 2 inner 189.2.be.a.104.17 yes 132
21.20 even 2 567.2.be.a.503.5 132
27.7 even 9 567.2.be.a.62.5 132
27.20 odd 18 inner 189.2.be.a.20.17 ✓ 132
189.20 even 18 inner 189.2.be.a.20.18 yes 132
189.34 odd 18 567.2.be.a.62.6 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.be.a.20.17 ✓ 132 27.20 odd 18 inner
189.2.be.a.20.18 yes 132 189.20 even 18 inner
189.2.be.a.104.17 yes 132 7.6 odd 2 inner
189.2.be.a.104.18 yes 132 1.1 even 1 trivial
567.2.be.a.62.5 132 27.7 even 9
567.2.be.a.62.6 132 189.34 odd 18
567.2.be.a.503.5 132 21.20 even 2
567.2.be.a.503.6 132 3.2 odd 2