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Properties

Label 189.2.bd.a.185.13
Level $189$
Weight $2$
Character 189.185
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 185.13
Character \(\chi\) \(=\) 189.185
Dual form 189.2.bd.a.47.13

$q$-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+(0.245063 - 0.0432112i) q^{2} +(1.36915 - 1.06086i) q^{3} +(-1.82120 + 0.662861i) q^{4} +(1.99870 - 0.727467i) q^{5} +(0.289688 - 0.319140i) q^{6} +(0.302346 - 2.62842i) q^{7} +(-0.848674 + 0.489982i) q^{8} +(0.749157 - 2.90496i) q^{9} +(0.458373 - 0.264642i) q^{10} +(-0.489168 + 1.34398i) q^{11} +(-1.79029 + 2.83959i) q^{12} +(1.30709 + 3.59121i) q^{13} +(-0.0394834 - 0.657193i) q^{14} +(1.96479 - 3.11635i) q^{15} +(2.78250 - 2.33479i) q^{16} +(0.109097 + 0.188961i) q^{17} +(0.0580640 - 0.744269i) q^{18} +(2.56992 + 1.48374i) q^{19} +(-3.15782 + 2.64972i) q^{20} +(-2.37442 - 3.91945i) q^{21} +(-0.0618021 + 0.350497i) q^{22} +(-7.61556 - 1.34283i) q^{23} +(-0.642162 + 1.57118i) q^{24} +(-0.364628 + 0.305959i) q^{25} +(0.475500 + 0.823590i) q^{26} +(-2.05604 - 4.77208i) q^{27} +(1.19165 + 4.98728i) q^{28} +(-1.78349 + 4.90009i) q^{29} +(0.346835 - 0.848603i) q^{30} +(1.32486 + 3.64001i) q^{31} +(1.84082 - 2.19380i) q^{32} +(0.756026 + 2.35905i) q^{33} +(0.0349009 + 0.0415932i) q^{34} +(-1.30779 - 5.47337i) q^{35} +(0.561220 + 5.78708i) q^{36} +6.01251 q^{37} +(0.693906 + 0.252561i) q^{38} +(5.59937 + 3.53027i) q^{39} +(-1.33980 + 1.59671i) q^{40} +(-10.0627 + 3.66253i) q^{41} +(-0.751248 - 0.857911i) q^{42} +(1.29579 + 7.34878i) q^{43} -2.77190i q^{44} +(-0.615920 - 6.35112i) q^{45} -1.92432 q^{46} +(-5.51793 - 2.00836i) q^{47} +(1.33278 - 6.14853i) q^{48} +(-6.81717 - 1.58938i) q^{49} +(-0.0761360 + 0.0907353i) q^{50} +(0.349832 + 0.142981i) q^{51} +(-4.76094 - 5.67387i) q^{52} +(12.1036 + 6.98800i) q^{53} +(-0.710066 - 1.08062i) q^{54} +3.04206i q^{55} +(1.03129 + 2.37882i) q^{56} +(5.09265 - 0.694850i) q^{57} +(-0.225328 + 1.27790i) q^{58} +(-1.21070 - 1.01590i) q^{59} +(-1.51255 + 6.97787i) q^{60} +(1.45500 - 3.99757i) q^{61} +(0.481963 + 0.834784i) q^{62} +(-7.40893 - 2.84740i) q^{63} +(-3.27598 + 5.67416i) q^{64} +(5.22497 + 6.22688i) q^{65} +(0.287211 + 0.545447i) q^{66} +(2.59043 - 14.6911i) q^{67} +(-0.323942 - 0.271820i) q^{68} +(-11.8514 + 6.24050i) q^{69} +(-0.557002 - 1.28481i) q^{70} +(-4.29452 - 2.47944i) q^{71} +(0.787586 + 2.83243i) q^{72} -2.46806i q^{73} +(1.47344 - 0.259808i) q^{74} +(-0.174652 + 0.805724i) q^{75} +(-5.66384 - 0.998688i) q^{76} +(3.38464 + 1.69209i) q^{77} +(1.52475 + 0.623182i) q^{78} +(-0.675969 - 3.83361i) q^{79} +(3.86290 - 6.69073i) q^{80} +(-7.87753 - 4.35254i) q^{81} +(-2.30773 + 1.33237i) q^{82} +(-0.141502 - 0.0515026i) q^{83} +(6.92235 + 5.56418i) q^{84} +(0.355515 + 0.298313i) q^{85} +(0.635099 + 1.74492i) q^{86} +(2.75644 + 8.60100i) q^{87} +(-0.243381 - 1.38028i) q^{88} +(3.40271 - 5.89366i) q^{89} +(-0.425379 - 1.52981i) q^{90} +(9.83439 - 2.34980i) q^{91} +(14.7595 - 2.60251i) q^{92} +(5.67547 + 3.57825i) q^{93} +(-1.43902 - 0.253739i) q^{94} +(6.21587 + 1.09603i) q^{95} +(0.193047 - 4.95649i) q^{96} +(12.2644 - 2.16255i) q^{97} +(-1.73932 - 0.0949204i) q^{98} +(3.53773 + 2.42786i) 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 \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.245063 − 0.0432112i 0.173286 − 0.0305549i −0.0863320 − 0.996266i \(-0.527515\pi\)
0.259618 + 0.965711i \(0.416403\pi\)
\(3\) 1.36915 − 1.06086i 0.790481 − 0.612487i
\(4\) −1.82120 + 0.662861i −0.910598 + 0.331431i
\(5\) 1.99870 − 0.727467i 0.893846 − 0.325333i 0.146062 − 0.989275i \(-0.453340\pi\)
0.747784 + 0.663942i \(0.231118\pi\)
\(6\) 0.289688 − 0.319140i 0.118264 − 0.130288i
\(7\) 0.302346 − 2.62842i 0.114276 − 0.993449i
\(8\) −0.848674 + 0.489982i −0.300052 + 0.173235i
\(9\) 0.749157 − 2.90496i 0.249719 − 0.968318i
\(10\) 0.458373 − 0.264642i 0.144950 − 0.0836870i
\(11\) −0.489168 + 1.34398i −0.147490 + 0.405225i −0.991334 − 0.131363i \(-0.958065\pi\)
0.843845 + 0.536588i \(0.180287\pi\)
\(12\) −1.79029 + 2.83959i −0.516813 + 0.819719i
\(13\) 1.30709 + 3.59121i 0.362522 + 0.996021i 0.978135 + 0.207972i \(0.0666864\pi\)
−0.615613 + 0.788049i \(0.711091\pi\)
\(14\) −0.0394834 − 0.657193i −0.0105524 − 0.175642i
\(15\) 1.96479 − 3.11635i 0.507305 − 0.804639i
\(16\) 2.78250 − 2.33479i 0.695625 − 0.583699i
\(17\) 0.109097 + 0.188961i 0.0264599 + 0.0458299i 0.878952 − 0.476910i \(-0.158243\pi\)
−0.852492 + 0.522740i \(0.824910\pi\)
\(18\) 0.0580640 − 0.744269i 0.0136858 − 0.175426i
\(19\) 2.56992 + 1.48374i 0.589579 + 0.340394i 0.764931 − 0.644112i \(-0.222773\pi\)
−0.175352 + 0.984506i \(0.556106\pi\)
\(20\) −3.15782 + 2.64972i −0.706109 + 0.592496i
\(21\) −2.37442 − 3.91945i −0.518142 − 0.855295i
\(22\) −0.0618021 + 0.350497i −0.0131762 + 0.0747262i
\(23\) −7.61556 − 1.34283i −1.58795 − 0.279999i −0.691246 − 0.722620i \(-0.742938\pi\)
−0.896709 + 0.442621i \(0.854049\pi\)
\(24\) −0.642162 + 1.57118i −0.131081 + 0.320717i
\(25\) −0.364628 + 0.305959i −0.0729256 + 0.0611919i
\(26\) 0.475500 + 0.823590i 0.0932532 + 0.161519i
\(27\) −2.05604 − 4.77208i −0.395684 − 0.918387i
\(28\) 1.19165 + 4.98728i 0.225200 + 0.942508i
\(29\) −1.78349 + 4.90009i −0.331185 + 0.909924i 0.656619 + 0.754223i \(0.271986\pi\)
−0.987804 + 0.155702i \(0.950236\pi\)
\(30\) 0.346835 − 0.848603i 0.0633231 − 0.154933i
\(31\) 1.32486 + 3.64001i 0.237951 + 0.653766i 0.999981 + 0.00622334i \(0.00198096\pi\)
−0.762029 + 0.647543i \(0.775797\pi\)
\(32\) 1.84082 − 2.19380i 0.325413 − 0.387813i
\(33\) 0.756026 + 2.35905i 0.131607 + 0.410658i
\(34\) 0.0349009 + 0.0415932i 0.00598545 + 0.00713318i
\(35\) −1.30779 − 5.47337i −0.221057 − 0.925168i
\(36\) 0.561220 + 5.78708i 0.0935367 + 0.964514i
\(37\) 6.01251 0.988450 0.494225 − 0.869334i \(-0.335452\pi\)
0.494225 + 0.869334i \(0.335452\pi\)
\(38\) 0.693906 + 0.252561i 0.112566 + 0.0409708i
\(39\) 5.59937 + 3.53027i 0.896617 + 0.565295i
\(40\) −1.33980 + 1.59671i −0.211841 + 0.252462i
\(41\) −10.0627 + 3.66253i −1.57153 + 0.571990i −0.973341 − 0.229364i \(-0.926335\pi\)
−0.598190 + 0.801354i \(0.704113\pi\)
\(42\) −0.751248 − 0.857911i −0.115920 − 0.132379i
\(43\) 1.29579 + 7.34878i 0.197606 + 1.12068i 0.908659 + 0.417540i \(0.137108\pi\)
−0.711053 + 0.703139i \(0.751781\pi\)
\(44\) − 2.77190i − 0.417880i
\(45\) −0.615920 − 6.35112i −0.0918159 − 0.946769i
\(46\) −1.92432 −0.283725
\(47\) −5.51793 − 2.00836i −0.804873 − 0.292950i −0.0933687 − 0.995632i \(-0.529764\pi\)
−0.711504 + 0.702682i \(0.751986\pi\)
\(48\) 1.33278 − 6.14853i 0.192370 − 0.887464i
\(49\) −6.81717 − 1.58938i −0.973882 − 0.227055i
\(50\) −0.0761360 + 0.0907353i −0.0107673 + 0.0128319i
\(51\) 0.349832 + 0.142981i 0.0489862 + 0.0200213i
\(52\) −4.76094 − 5.67387i −0.660224 − 0.786824i
\(53\) 12.1036 + 6.98800i 1.66255 + 0.959875i 0.971490 + 0.237080i \(0.0761904\pi\)
0.691063 + 0.722795i \(0.257143\pi\)
\(54\) −0.710066 − 1.08062i −0.0966277 − 0.147053i
\(55\) 3.04206i 0.410192i
\(56\) 1.03129 + 2.37882i 0.137811 + 0.317883i
\(57\) 5.09265 − 0.694850i 0.674538 − 0.0920351i
\(58\) −0.225328 + 1.27790i −0.0295870 + 0.167796i
\(59\) −1.21070 − 1.01590i −0.157620 − 0.132259i 0.560567 − 0.828109i \(-0.310583\pi\)
−0.718187 + 0.695850i \(0.755028\pi\)
\(60\) −1.51255 + 6.97787i −0.195269 + 0.900839i
\(61\) 1.45500 − 3.99757i 0.186293 − 0.511837i −0.811026 − 0.585010i \(-0.801090\pi\)
0.997319 + 0.0731735i \(0.0233127\pi\)
\(62\) 0.481963 + 0.834784i 0.0612093 + 0.106018i
\(63\) −7.40893 − 2.84740i −0.933438 − 0.358739i
\(64\) −3.27598 + 5.67416i −0.409497 + 0.709270i
\(65\) 5.22497 + 6.22688i 0.648078 + 0.772349i
\(66\) 0.287211 + 0.545447i 0.0353533 + 0.0671399i
\(67\) 2.59043 − 14.6911i 0.316471 − 1.79480i −0.247376 − 0.968920i \(-0.579568\pi\)
0.563848 − 0.825879i \(-0.309321\pi\)
\(68\) −0.323942 − 0.271820i −0.0392838 − 0.0329630i
\(69\) −11.8514 + 6.24050i −1.42674 + 0.751268i
\(70\) −0.557002 − 1.28481i −0.0665745 − 0.153564i
\(71\) −4.29452 − 2.47944i −0.509666 − 0.294256i 0.223031 − 0.974811i \(-0.428405\pi\)
−0.732696 + 0.680556i \(0.761738\pi\)
\(72\) 0.787586 + 2.83243i 0.0928179 + 0.333806i
\(73\) − 2.46806i − 0.288864i −0.989515 − 0.144432i \(-0.953864\pi\)
0.989515 − 0.144432i \(-0.0461355\pi\)
\(74\) 1.47344 − 0.259808i 0.171284 − 0.0302020i
\(75\) −0.174652 + 0.805724i −0.0201671 + 0.0930370i
\(76\) −5.66384 − 0.998688i −0.649687 − 0.114557i
\(77\) 3.38464 + 1.69209i 0.385716 + 0.192831i
\(78\) 1.52475 + 0.623182i 0.172643 + 0.0705615i
\(79\) −0.675969 − 3.83361i −0.0760524 − 0.431315i −0.998931 − 0.0462239i \(-0.985281\pi\)
0.922879 − 0.385091i \(-0.125830\pi\)
\(80\) 3.86290 − 6.69073i 0.431885 − 0.748047i
\(81\) −7.87753 − 4.35254i −0.875281 − 0.483615i
\(82\) −2.30773 + 1.33237i −0.254847 + 0.147136i
\(83\) −0.141502 − 0.0515026i −0.0155319 − 0.00565315i 0.334243 − 0.942487i \(-0.391520\pi\)
−0.349774 + 0.936834i \(0.613742\pi\)
\(84\) 6.92235 + 5.56418i 0.755290 + 0.607102i
\(85\) 0.355515 + 0.298313i 0.0385610 + 0.0323566i
\(86\) 0.635099 + 1.74492i 0.0684845 + 0.188160i
\(87\) 2.75644 + 8.60100i 0.295521 + 0.922124i
\(88\) −0.243381 − 1.38028i −0.0259445 − 0.147139i
\(89\) 3.40271 − 5.89366i 0.360686 − 0.624727i −0.627388 − 0.778707i \(-0.715876\pi\)
0.988074 + 0.153980i \(0.0492092\pi\)
\(90\) −0.425379 − 1.52981i −0.0448388 − 0.161256i
\(91\) 9.83439 − 2.34980i 1.03092 − 0.246326i
\(92\) 14.7595 − 2.60251i 1.53879 − 0.271330i
\(93\) 5.67547 + 3.57825i 0.588519 + 0.371047i
\(94\) −1.43902 − 0.253739i −0.148424 − 0.0261712i
\(95\) 6.21587 + 1.09603i 0.637735 + 0.112450i
\(96\) 0.193047 − 4.95649i 0.0197028 − 0.505870i
\(97\) 12.2644 − 2.16255i 1.24526 − 0.219573i 0.488092 − 0.872792i \(-0.337693\pi\)
0.757169 + 0.653219i \(0.226582\pi\)
\(98\) −1.73932 − 0.0949204i −0.175697 − 0.00958841i
\(99\) 3.53773 + 2.42786i 0.355556 + 0.244009i
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.185.13 yes 132
3.2 odd 2 567.2.bd.a.17.10 132
7.5 odd 6 189.2.ba.a.131.10 yes 132
21.5 even 6 567.2.ba.a.341.13 132
27.7 even 9 567.2.ba.a.143.13 132
27.20 odd 18 189.2.ba.a.101.10 ✓ 132
189.47 even 18 inner 189.2.bd.a.47.13 yes 132
189.61 odd 18 567.2.bd.a.467.10 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.10 ✓ 132 27.20 odd 18
189.2.ba.a.131.10 yes 132 7.5 odd 6
189.2.bd.a.47.13 yes 132 189.47 even 18 inner
189.2.bd.a.185.13 yes 132 1.1 even 1 trivial
567.2.ba.a.143.13 132 27.7 even 9
567.2.ba.a.341.13 132 21.5 even 6
567.2.bd.a.17.10 132 3.2 odd 2
567.2.bd.a.467.10 132 189.61 odd 18