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Properties

Label 189.2.w.a.25.7
Level $189$
Weight $2$
Character 189.25
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(25,189)] 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("189.25"); 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([10, 12])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.w (of order \(9\), 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 \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 25.7
Character \(\chi\) \(=\) 189.25
Dual form 189.2.w.a.121.7

$q$-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+(-1.30907 + 0.476464i) q^{2} +(-1.39125 - 1.03171i) q^{3} +(-0.0454330 + 0.0381228i) q^{4} +(1.40139 + 0.510065i) q^{5} +(2.31282 + 0.687701i) q^{6} +(-2.64009 - 0.173054i) q^{7} +(1.43440 - 2.48445i) q^{8} +(0.871157 + 2.87073i) q^{9} -2.07755 q^{10} +(4.49413 - 1.63573i) q^{11} +(0.102540 - 0.00616480i) q^{12} +(0.371663 + 2.10780i) q^{13} +(3.53852 - 1.03137i) q^{14} +(-1.42345 - 2.15546i) q^{15} +(-0.673384 + 3.81895i) q^{16} +5.84823 q^{17} +(-2.50821 - 3.34292i) q^{18} +7.81006 q^{19} +(-0.0831146 + 0.0302512i) q^{20} +(3.49448 + 2.96456i) q^{21} +(-5.10378 + 4.28258i) q^{22} +(-0.451305 - 2.55948i) q^{23} +(-4.55883 + 1.97661i) q^{24} +(-2.12649 - 1.78433i) q^{25} +(-1.49083 - 2.58219i) q^{26} +(1.74976 - 4.89268i) q^{27} +(0.126544 - 0.0927851i) q^{28} +(-1.22445 + 6.94422i) q^{29} +(2.89040 + 2.14343i) q^{30} +(1.19198 - 1.00019i) q^{31} +(0.0582395 + 0.330293i) q^{32} +(-7.94006 - 2.36092i) q^{33} +(-7.65576 + 2.78647i) q^{34} +(-3.61153 - 1.58913i) q^{35} +(-0.149019 - 0.0972148i) q^{36} +(-3.52981 + 6.11382i) q^{37} +(-10.2239 + 3.72121i) q^{38} +(1.65756 - 3.31593i) q^{39} +(3.27739 - 2.75005i) q^{40} +(-1.33423 - 7.56680i) q^{41} +(-5.98704 - 2.21583i) q^{42} +(-6.86262 - 5.75842i) q^{43} +(-0.141823 + 0.245645i) q^{44} +(-0.243426 + 4.46737i) q^{45} +(1.81029 + 3.13551i) q^{46} +(4.90812 + 4.11840i) q^{47} +(4.87689 - 4.61838i) q^{48} +(6.94010 + 0.913753i) q^{49} +(3.63390 + 1.32263i) q^{50} +(-8.13636 - 6.03367i) q^{51} +(-0.0972411 - 0.0815949i) q^{52} +(3.57502 - 6.19211i) q^{53} +(0.0406280 + 7.23858i) q^{54} +7.13238 q^{55} +(-4.21688 + 6.31093i) q^{56} +(-10.8658 - 8.05770i) q^{57} +(-1.70577 - 9.67390i) q^{58} +(0.143277 + 0.812567i) q^{59} +(0.146844 + 0.0436629i) q^{60} +(6.10769 + 5.12496i) q^{61} +(-1.08383 + 1.87725i) q^{62} +(-1.80314 - 7.72973i) q^{63} +(-4.11148 - 7.12129i) q^{64} +(-0.554272 + 3.14343i) q^{65} +(11.5190 - 0.692533i) q^{66} +(0.431518 + 0.157060i) q^{67} +(-0.265703 + 0.222951i) q^{68} +(-2.01275 + 4.02649i) q^{69} +(5.48492 + 0.359528i) q^{70} +(2.15632 + 3.73486i) q^{71} +(8.38177 + 1.95342i) q^{72} +(2.27530 + 3.94093i) q^{73} +(1.70777 - 9.68527i) q^{74} +(1.11756 + 4.67637i) q^{75} +(-0.354834 + 0.297741i) q^{76} +(-12.1480 + 3.54074i) q^{77} +(-0.589949 + 5.13056i) q^{78} +(-5.54499 + 2.01821i) q^{79} +(-2.89159 + 5.00838i) q^{80} +(-7.48217 + 5.00171i) q^{81} +(5.35191 + 9.26979i) q^{82} +(-0.177225 + 1.00509i) q^{83} +(-0.271782 - 0.00146936i) q^{84} +(8.19567 + 2.98298i) q^{85} +(11.7274 + 4.26841i) q^{86} +(8.86793 - 8.39787i) q^{87} +(2.38248 - 13.5117i) q^{88} -1.93404 q^{89} +(-1.80988 - 5.96410i) q^{90} +(-0.616458 - 5.62910i) q^{91} +(0.118079 + 0.0990797i) q^{92} +(-2.69024 + 0.161739i) q^{93} +(-8.38736 - 3.05275i) q^{94} +(10.9450 + 3.98364i) q^{95} +(0.259740 - 0.519606i) q^{96} +(8.10490 + 6.80081i) q^{97} +(-9.52048 + 2.11054i) q^{98} +(8.61084 + 11.4765i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} + 3 q^{9} - 6 q^{10} + 3 q^{11} - 3 q^{12} - 12 q^{13} + 15 q^{14} - 9 q^{16} - 54 q^{17} - 3 q^{18} - 6 q^{19} - 18 q^{20} - 21 q^{21}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.30907 + 0.476464i −0.925655 + 0.336911i −0.760486 − 0.649354i \(-0.775039\pi\)
−0.165169 + 0.986265i \(0.552817\pi\)
\(3\) −1.39125 − 1.03171i −0.803239 − 0.595657i
\(4\) −0.0454330 + 0.0381228i −0.0227165 + 0.0190614i
\(5\) 1.40139 + 0.510065i 0.626722 + 0.228108i 0.635804 − 0.771851i \(-0.280669\pi\)
−0.00908190 + 0.999959i \(0.502891\pi\)
\(6\) 2.31282 + 0.687701i 0.944205 + 0.280753i
\(7\) −2.64009 − 0.173054i −0.997859 − 0.0654081i
\(8\) 1.43440 − 2.48445i 0.507136 − 0.878386i
\(9\) 0.871157 + 2.87073i 0.290386 + 0.956910i
\(10\) −2.07755 −0.656980
\(11\) 4.49413 − 1.63573i 1.35503 − 0.493191i 0.440518 − 0.897744i \(-0.354795\pi\)
0.914515 + 0.404553i \(0.132573\pi\)
\(12\) 0.102540 − 0.00616480i 0.0296008 − 0.00177963i
\(13\) 0.371663 + 2.10780i 0.103081 + 0.584599i 0.991970 + 0.126477i \(0.0403669\pi\)
−0.888889 + 0.458123i \(0.848522\pi\)
\(14\) 3.53852 − 1.03137i 0.945709 − 0.275644i
\(15\) −1.42345 − 2.15546i −0.367533 − 0.556537i
\(16\) −0.673384 + 3.81895i −0.168346 + 0.954738i
\(17\) 5.84823 1.41840 0.709202 − 0.705005i \(-0.249055\pi\)
0.709202 + 0.705005i \(0.249055\pi\)
\(18\) −2.50821 − 3.34292i −0.591190 − 0.787934i
\(19\) 7.81006 1.79175 0.895875 − 0.444306i \(-0.146550\pi\)
0.895875 + 0.444306i \(0.146550\pi\)
\(20\) −0.0831146 + 0.0302512i −0.0185850 + 0.00676438i
\(21\) 3.49448 + 2.96456i 0.762558 + 0.646920i
\(22\) −5.10378 + 4.28258i −1.08813 + 0.913050i
\(23\) −0.451305 − 2.55948i −0.0941036 − 0.533688i −0.995018 − 0.0996919i \(-0.968214\pi\)
0.900915 − 0.433996i \(-0.142897\pi\)
\(24\) −4.55883 + 1.97661i −0.930568 + 0.403475i
\(25\) −2.12649 − 1.78433i −0.425297 − 0.356867i
\(26\) −1.49083 − 2.58219i −0.292375 − 0.506408i
\(27\) 1.74976 − 4.89268i 0.336741 − 0.941597i
\(28\) 0.126544 − 0.0927851i 0.0239146 − 0.0175347i
\(29\) −1.22445 + 6.94422i −0.227375 + 1.28951i 0.630717 + 0.776013i \(0.282761\pi\)
−0.858092 + 0.513496i \(0.828350\pi\)
\(30\) 2.89040 + 2.14343i 0.527712 + 0.391335i
\(31\) 1.19198 − 1.00019i 0.214085 − 0.179639i −0.529439 − 0.848348i \(-0.677597\pi\)
0.743524 + 0.668709i \(0.233153\pi\)
\(32\) 0.0582395 + 0.330293i 0.0102954 + 0.0583881i
\(33\) −7.94006 − 2.36092i −1.38219 − 0.410984i
\(34\) −7.65576 + 2.78647i −1.31295 + 0.477876i
\(35\) −3.61153 − 1.58913i −0.610460 − 0.268612i
\(36\) −0.149019 − 0.0972148i −0.0248366 − 0.0162025i
\(37\) −3.52981 + 6.11382i −0.580298 + 1.00511i 0.415146 + 0.909755i \(0.363731\pi\)
−0.995444 + 0.0953507i \(0.969603\pi\)
\(38\) −10.2239 + 3.72121i −1.65854 + 0.603660i
\(39\) 1.65756 − 3.31593i 0.265422 − 0.530974i
\(40\) 3.27739 − 2.75005i 0.518200 − 0.434822i
\(41\) −1.33423 − 7.56680i −0.208372 − 1.18174i −0.892045 − 0.451947i \(-0.850730\pi\)
0.683673 − 0.729789i \(-0.260382\pi\)
\(42\) −5.98704 − 2.21583i −0.923820 − 0.341910i
\(43\) −6.86262 − 5.75842i −1.04654 − 0.878152i −0.0538147 − 0.998551i \(-0.517138\pi\)
−0.992726 + 0.120399i \(0.961582\pi\)
\(44\) −0.141823 + 0.245645i −0.0213807 + 0.0370324i
\(45\) −0.243426 + 4.46737i −0.0362877 + 0.665956i
\(46\) 1.81029 + 3.13551i 0.266913 + 0.462306i
\(47\) 4.90812 + 4.11840i 0.715923 + 0.600730i 0.926254 − 0.376900i \(-0.123010\pi\)
−0.210332 + 0.977630i \(0.567454\pi\)
\(48\) 4.87689 − 4.61838i 0.703918 − 0.666606i
\(49\) 6.94010 + 0.913753i 0.991444 + 0.130536i
\(50\) 3.63390 + 1.32263i 0.513911 + 0.187048i
\(51\) −8.13636 − 6.03367i −1.13932 − 0.844882i
\(52\) −0.0972411 − 0.0815949i −0.0134849 − 0.0113152i
\(53\) 3.57502 − 6.19211i 0.491066 − 0.850552i −0.508881 − 0.860837i \(-0.669941\pi\)
0.999947 + 0.0102850i \(0.00327388\pi\)
\(54\) 0.0406280 + 7.23858i 0.00552877 + 0.985046i
\(55\) 7.13238 0.961730
\(56\) −4.21688 + 6.31093i −0.563504 + 0.843334i
\(57\) −10.8658 − 8.05770i −1.43920 − 1.06727i
\(58\) −1.70577 − 9.67390i −0.223978 − 1.27025i
\(59\) 0.143277 + 0.812567i 0.0186531 + 0.105787i 0.992713 − 0.120504i \(-0.0384510\pi\)
−0.974060 + 0.226291i \(0.927340\pi\)
\(60\) 0.146844 + 0.0436629i 0.0189574 + 0.00563686i
\(61\) 6.10769 + 5.12496i 0.782010 + 0.656184i 0.943754 − 0.330649i \(-0.107267\pi\)
−0.161744 + 0.986833i \(0.551712\pi\)
\(62\) −1.08383 + 1.87725i −0.137647 + 0.238411i
\(63\) −1.80314 − 7.72973i −0.227174 − 0.973854i
\(64\) −4.11148 − 7.12129i −0.513935 − 0.890161i
\(65\) −0.554272 + 3.14343i −0.0687490 + 0.389895i
\(66\) 11.5190 − 0.692533i 1.41789 − 0.0852449i
\(67\) 0.431518 + 0.157060i 0.0527183 + 0.0191879i 0.368245 − 0.929729i \(-0.379959\pi\)
−0.315526 + 0.948917i \(0.602181\pi\)
\(68\) −0.265703 + 0.222951i −0.0322212 + 0.0270368i
\(69\) −2.01275 + 4.02649i −0.242307 + 0.484732i
\(70\) 5.48492 + 0.359528i 0.655574 + 0.0429719i
\(71\) 2.15632 + 3.73486i 0.255909 + 0.443247i 0.965142 − 0.261727i \(-0.0842921\pi\)
−0.709233 + 0.704974i \(0.750959\pi\)
\(72\) 8.38177 + 1.95342i 0.987801 + 0.230213i
\(73\) 2.27530 + 3.94093i 0.266304 + 0.461251i 0.967904 − 0.251319i \(-0.0808643\pi\)
−0.701601 + 0.712570i \(0.747531\pi\)
\(74\) 1.70777 − 9.68527i 0.198525 − 1.12589i
\(75\) 1.11756 + 4.67637i 0.129045 + 0.539981i
\(76\) −0.354834 + 0.297741i −0.0407023 + 0.0341533i
\(77\) −12.1480 + 3.54074i −1.38439 + 0.403505i
\(78\) −0.589949 + 5.13056i −0.0667986 + 0.580922i
\(79\) −5.54499 + 2.01821i −0.623860 + 0.227067i −0.634557 − 0.772876i \(-0.718817\pi\)
0.0106965 + 0.999943i \(0.496595\pi\)
\(80\) −2.89159 + 5.00838i −0.323290 + 0.559954i
\(81\) −7.48217 + 5.00171i −0.831352 + 0.555746i
\(82\) 5.35191 + 9.26979i 0.591020 + 1.02368i
\(83\) −0.177225 + 1.00509i −0.0194529 + 0.110323i −0.992988 − 0.118214i \(-0.962283\pi\)
0.973535 + 0.228537i \(0.0733942\pi\)
\(84\) −0.271782 − 0.00146936i −0.0296538 − 0.000160320i
\(85\) 8.19567 + 2.98298i 0.888945 + 0.323550i
\(86\) 11.7274 + 4.26841i 1.26459 + 0.460274i
\(87\) 8.86793 − 8.39787i 0.950741 − 0.900346i
\(88\) 2.38248 − 13.5117i 0.253974 − 1.44036i
\(89\) −1.93404 −0.205007 −0.102504 − 0.994733i \(-0.532685\pi\)
−0.102504 + 0.994733i \(0.532685\pi\)
\(90\) −1.80988 − 5.96410i −0.190778 − 0.628671i
\(91\) −0.616458 − 5.62910i −0.0646224 − 0.590090i
\(92\) 0.118079 + 0.0990797i 0.0123105 + 0.0103298i
\(93\) −2.69024 + 0.161739i −0.278965 + 0.0167716i
\(94\) −8.38736 − 3.05275i −0.865090 − 0.314867i
\(95\) 10.9450 + 3.98364i 1.12293 + 0.408713i
\(96\) 0.259740 − 0.519606i 0.0265096 − 0.0530321i
\(97\) 8.10490 + 6.80081i 0.822927 + 0.690518i 0.953656 − 0.300900i \(-0.0972871\pi\)
−0.130728 + 0.991418i \(0.541732\pi\)
\(98\) −9.52048 + 2.11054i −0.961714 + 0.213197i
\(99\) 8.61084 + 11.4765i 0.865422 + 1.15343i
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.w.a.25.7 yes 132
3.2 odd 2 567.2.w.a.235.16 132
7.2 even 3 189.2.u.a.79.16 yes 132
21.2 odd 6 567.2.u.a.478.7 132
27.13 even 9 189.2.u.a.67.16 ✓ 132
27.14 odd 18 567.2.u.a.172.7 132
189.121 even 9 inner 189.2.w.a.121.7 yes 132
189.149 odd 18 567.2.w.a.415.16 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.u.a.67.16 ✓ 132 27.13 even 9
189.2.u.a.79.16 yes 132 7.2 even 3
189.2.w.a.25.7 yes 132 1.1 even 1 trivial
189.2.w.a.121.7 yes 132 189.121 even 9 inner
567.2.u.a.172.7 132 27.14 odd 18
567.2.u.a.478.7 132 21.2 odd 6
567.2.w.a.235.16 132 3.2 odd 2
567.2.w.a.415.16 132 189.149 odd 18