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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 101.20
Character \(\chi\) \(=\) 189.101
Dual form 189.2.ba.a.131.20

$q$-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+(1.47978 - 1.76353i) q^{2} +(1.66401 - 0.480715i) q^{3} +(-0.573000 - 3.24965i) q^{4} +(-1.41208 + 1.18487i) q^{5} +(1.61460 - 3.64587i) q^{6} +(-2.40040 + 1.11270i) q^{7} +(-2.59137 - 1.49613i) q^{8} +(2.53783 - 1.59982i) q^{9} +4.24358i q^{10} +(-0.0789393 + 0.0940762i) q^{11} +(-2.51563 - 5.13198i) q^{12} +(0.0159063 - 0.0437022i) q^{13} +(-1.58977 + 5.87972i) q^{14} +(-1.78012 + 2.65044i) q^{15} +(-0.271551 + 0.0988366i) q^{16} +0.157246 q^{17} +(0.934081 - 6.84292i) q^{18} +7.59626i q^{19} +(4.65954 + 3.90982i) q^{20} +(-3.45938 + 3.00544i) q^{21} +(0.0490936 + 0.278423i) q^{22} +(0.332628 - 0.913889i) q^{23} +(-5.03126 - 1.24385i) q^{24} +(-0.278205 + 1.57778i) q^{25} +(-0.0535324 - 0.0927208i) q^{26} +(3.45390 - 3.88209i) q^{27} +(4.99131 + 7.16286i) q^{28} +(-3.33678 - 9.16772i) q^{29} +(2.03995 + 7.06135i) q^{30} +(-4.49600 + 0.792766i) q^{31} +(1.81929 - 4.99845i) q^{32} +(-0.0861316 + 0.194490i) q^{33} +(0.232689 - 0.277308i) q^{34} +(2.07113 - 4.41538i) q^{35} +(-6.65304 - 7.33034i) q^{36} +(-4.68718 + 8.11843i) q^{37} +(13.3962 + 11.2408i) q^{38} +(0.00545988 - 0.0803671i) q^{39} +(5.43192 - 0.957794i) q^{40} +(-5.76925 - 2.09983i) q^{41} +(0.181079 + 10.5481i) q^{42} +(1.67890 - 9.52153i) q^{43} +(0.350947 + 0.202619i) q^{44} +(-1.68802 + 5.26607i) q^{45} +(-1.11945 - 1.93895i) q^{46} +(0.723937 - 4.10565i) q^{47} +(-0.404351 + 0.295003i) q^{48} +(4.52380 - 5.34184i) q^{49} +(2.37078 + 2.82539i) q^{50} +(0.261658 - 0.0755904i) q^{51} +(-0.151131 - 0.0266485i) q^{52} +(0.141686 + 0.0818025i) q^{53} +(-1.73517 - 11.8357i) q^{54} -0.226376i q^{55} +(7.88504 + 0.707881i) q^{56} +(3.65163 + 12.6402i) q^{57} +(-21.1052 - 7.68168i) q^{58} +(11.1248 + 4.04911i) q^{59} +(9.63300 + 4.26604i) q^{60} +(2.98946 + 0.527123i) q^{61} +(-5.25501 + 9.10194i) q^{62} +(-4.31166 + 6.66405i) q^{63} +(-6.41175 - 11.1055i) q^{64} +(0.0293206 + 0.0805578i) q^{65} +(0.215534 + 0.439698i) q^{66} +(-7.53281 + 6.32078i) q^{67} +(-0.0901020 - 0.510994i) q^{68} +(0.114175 - 1.68062i) q^{69} +(-4.72184 - 10.1863i) q^{70} +(-0.863276 + 0.498413i) q^{71} +(-8.96997 + 0.348821i) q^{72} +(7.95454 - 4.59256i) q^{73} +(7.38112 + 20.2795i) q^{74} +(0.295527 + 2.75917i) q^{75} +(24.6852 - 4.35266i) q^{76} +(0.0848069 - 0.313656i) q^{77} +(-0.133650 - 0.128554i) q^{78} +(1.99133 + 1.67093i) q^{79} +(0.266342 - 0.461318i) q^{80} +(3.88113 - 8.12015i) q^{81} +(-12.2403 + 7.06695i) q^{82} +(6.02764 - 2.19388i) q^{83} +(11.7489 + 9.51964i) q^{84} +(-0.222043 + 0.186316i) q^{85} +(-14.3071 - 17.0505i) q^{86} +(-9.95947 - 13.6511i) q^{87} +(0.345310 - 0.125683i) q^{88} -10.6041 q^{89} +(6.78899 + 10.7695i) q^{90} +(0.0104460 + 0.122602i) q^{91} +(-3.16041 - 0.557266i) q^{92} +(-7.10027 + 3.48046i) q^{93} +(-6.16918 - 7.35214i) q^{94} +(-9.00059 - 10.7265i) q^{95} +(0.624474 - 9.19200i) q^{96} +(-13.2580 - 2.33775i) q^{97} +(-2.72628 - 15.8826i) q^{98} +(-0.0498289 + 0.365038i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.47978 − 1.76353i 1.04636 − 1.24700i 0.0781303 − 0.996943i \(-0.475105\pi\)
0.968230 − 0.250061i \(-0.0804506\pi\)
\(3\) 1.66401 − 0.480715i 0.960714 − 0.277541i
\(4\) −0.573000 − 3.24965i −0.286500 − 1.62482i
\(5\) −1.41208 + 1.18487i −0.631499 + 0.529891i −0.901394 − 0.432999i \(-0.857455\pi\)
0.269895 + 0.962890i \(0.413011\pi\)
\(6\) 1.61460 − 3.64587i 0.659159 − 1.48842i
\(7\) −2.40040 + 1.11270i −0.907264 + 0.420561i
\(8\) −2.59137 − 1.49613i −0.916186 − 0.528960i
\(9\) 2.53783 − 1.59982i 0.845942 − 0.533275i
\(10\) 4.24358i 1.34194i
\(11\) −0.0789393 + 0.0940762i −0.0238011 + 0.0283650i −0.777814 − 0.628495i \(-0.783671\pi\)
0.754013 + 0.656860i \(0.228116\pi\)
\(12\) −2.51563 − 5.13198i −0.726200 − 1.48147i
\(13\) 0.0159063 − 0.0437022i 0.00441162 − 0.0121208i −0.937467 − 0.348073i \(-0.886836\pi\)
0.941879 + 0.335953i \(0.109058\pi\)
\(14\) −1.58977 + 5.87972i −0.424884 + 1.57142i
\(15\) −1.78012 + 2.65044i −0.459624 + 0.684340i
\(16\) −0.271551 + 0.0988366i −0.0678878 + 0.0247091i
\(17\) 0.157246 0.0381377 0.0190689 − 0.999818i \(-0.493930\pi\)
0.0190689 + 0.999818i \(0.493930\pi\)
\(18\) 0.934081 − 6.84292i 0.220165 − 1.61289i
\(19\) 7.59626i 1.74270i 0.490661 + 0.871350i \(0.336755\pi\)
−0.490661 + 0.871350i \(0.663245\pi\)
\(20\) 4.65954 + 3.90982i 1.04190 + 0.874261i
\(21\) −3.45938 + 3.00544i −0.754898 + 0.655842i
\(22\) 0.0490936 + 0.278423i 0.0104668 + 0.0593601i
\(23\) 0.332628 − 0.913889i 0.0693578 − 0.190559i −0.900171 − 0.435537i \(-0.856559\pi\)
0.969529 + 0.244978i \(0.0787807\pi\)
\(24\) −5.03126 − 1.24385i −1.02700 − 0.253900i
\(25\) −0.278205 + 1.57778i −0.0556411 + 0.315556i
\(26\) −0.0535324 − 0.0927208i −0.0104986 − 0.0181840i
\(27\) 3.45390 − 3.88209i 0.664703 − 0.747108i
\(28\) 4.99131 + 7.16286i 0.943269 + 1.35365i
\(29\) −3.33678 − 9.16772i −0.619624 − 1.70240i −0.707906 − 0.706307i \(-0.750360\pi\)
0.0882816 − 0.996096i \(-0.471862\pi\)
\(30\) 2.03995 + 7.06135i 0.372443 + 1.28922i
\(31\) −4.49600 + 0.792766i −0.807505 + 0.142385i −0.562136 − 0.827045i \(-0.690020\pi\)
−0.245369 + 0.969430i \(0.578909\pi\)
\(32\) 1.81929 − 4.99845i 0.321607 − 0.883609i
\(33\) −0.0861316 + 0.194490i −0.0149936 + 0.0338564i
\(34\) 0.232689 − 0.277308i 0.0399058 − 0.0475579i
\(35\) 2.07113 − 4.41538i 0.350085 − 0.746335i
\(36\) −6.65304 − 7.33034i −1.10884 − 1.22172i
\(37\) −4.68718 + 8.11843i −0.770567 + 1.33466i 0.166685 + 0.986010i \(0.446694\pi\)
−0.937252 + 0.348652i \(0.886640\pi\)
\(38\) 13.3962 + 11.2408i 2.17315 + 1.82349i
\(39\) 0.00545988 − 0.0803671i 0.000874280 − 0.0128690i
\(40\) 5.43192 − 0.957794i 0.858862 − 0.151441i
\(41\) −5.76925 − 2.09983i −0.901005 − 0.327939i −0.150350 − 0.988633i \(-0.548040\pi\)
−0.750656 + 0.660694i \(0.770262\pi\)
\(42\) 0.181079 + 10.5481i 0.0279411 + 1.62761i
\(43\) 1.67890 − 9.52153i 0.256030 − 1.45202i −0.537384 − 0.843338i \(-0.680587\pi\)
0.793414 − 0.608682i \(-0.208301\pi\)
\(44\) 0.350947 + 0.202619i 0.0529072 + 0.0305460i
\(45\) −1.68802 + 5.26607i −0.251635 + 0.785020i
\(46\) −1.11945 − 1.93895i −0.165055 − 0.285883i
\(47\) 0.723937 − 4.10565i 0.105597 − 0.598871i −0.885383 − 0.464862i \(-0.846104\pi\)
0.990980 − 0.134009i \(-0.0427851\pi\)
\(48\) −0.404351 + 0.295003i −0.0583630 + 0.0425801i
\(49\) 4.52380 − 5.34184i 0.646257 − 0.763120i
\(50\) 2.37078 + 2.82539i 0.335279 + 0.399570i
\(51\) 0.261658 − 0.0755904i 0.0366395 − 0.0105848i
\(52\) −0.151131 − 0.0266485i −0.0209581 − 0.00369548i
\(53\) 0.141686 + 0.0818025i 0.0194621 + 0.0112364i 0.509699 − 0.860353i \(-0.329757\pi\)
−0.490237 + 0.871589i \(0.663090\pi\)
\(54\) −1.73517 − 11.8357i −0.236127 − 1.61063i
\(55\) − 0.226376i − 0.0305245i
\(56\) 7.88504 + 0.707881i 1.05368 + 0.0945946i
\(57\) 3.65163 + 12.6402i 0.483671 + 1.67424i
\(58\) −21.1052 − 7.68168i −2.77125 − 1.00865i
\(59\) 11.1248 + 4.04911i 1.44833 + 0.527150i 0.942125 − 0.335262i \(-0.108825\pi\)
0.506207 + 0.862412i \(0.331047\pi\)
\(60\) 9.63300 + 4.26604i 1.24361 + 0.550744i
\(61\) 2.98946 + 0.527123i 0.382762 + 0.0674912i 0.361718 − 0.932287i \(-0.382190\pi\)
0.0210431 + 0.999779i \(0.493301\pi\)
\(62\) −5.25501 + 9.10194i −0.667387 + 1.15595i
\(63\) −4.31166 + 6.66405i −0.543219 + 0.839591i
\(64\) −6.41175 − 11.1055i −0.801469 − 1.38818i
\(65\) 0.0293206 + 0.0805578i 0.00363678 + 0.00999196i
\(66\) 0.215534 + 0.439698i 0.0265304 + 0.0541231i
\(67\) −7.53281 + 6.32078i −0.920279 + 0.772206i −0.974047 − 0.226348i \(-0.927321\pi\)
0.0537674 + 0.998553i \(0.482877\pi\)
\(68\) −0.0901020 − 0.510994i −0.0109265 − 0.0619671i
\(69\) 0.114175 − 1.68062i 0.0137451 − 0.202322i
\(70\) −4.72184 − 10.1863i −0.564367 − 1.21749i
\(71\) −0.863276 + 0.498413i −0.102452 + 0.0591507i −0.550351 − 0.834934i \(-0.685506\pi\)
0.447899 + 0.894084i \(0.352173\pi\)
\(72\) −8.96997 + 0.348821i −1.05712 + 0.0411089i
\(73\) 7.95454 − 4.59256i 0.931009 − 0.537518i 0.0438782 − 0.999037i \(-0.486029\pi\)
0.887130 + 0.461519i \(0.152695\pi\)
\(74\) 7.38112 + 20.2795i 0.858037 + 2.35744i
\(75\) 0.295527 + 2.75917i 0.0341246 + 0.318602i
\(76\) 24.6852 − 4.35266i 2.83158 − 0.499284i
\(77\) 0.0848069 − 0.313656i 0.00966465 − 0.0357444i
\(78\) −0.133650 − 0.128554i −0.0151329 − 0.0145559i
\(79\) 1.99133 + 1.67093i 0.224043 + 0.187994i 0.747899 − 0.663812i \(-0.231063\pi\)
−0.523856 + 0.851807i \(0.675507\pi\)
\(80\) 0.266342 − 0.461318i 0.0297780 − 0.0515769i
\(81\) 3.88113 − 8.12015i 0.431236 − 0.902239i
\(82\) −12.2403 + 7.06695i −1.35172 + 0.780414i
\(83\) 6.02764 − 2.19388i 0.661619 − 0.240810i 0.0106840 − 0.999943i \(-0.496599\pi\)
0.650935 + 0.759133i \(0.274377\pi\)
\(84\) 11.7489 + 9.51964i 1.28191 + 1.03868i
\(85\) −0.222043 + 0.186316i −0.0240840 + 0.0202088i
\(86\) −14.3071 − 17.0505i −1.54277 − 1.83861i
\(87\) −9.95947 − 13.6511i −1.06777 − 1.46355i
\(88\) 0.345310 − 0.125683i 0.0368102 − 0.0133978i
\(89\) −10.6041 −1.12403 −0.562014 − 0.827128i \(-0.689973\pi\)
−0.562014 + 0.827128i \(0.689973\pi\)
\(90\) 6.78899 + 10.7695i 0.715622 + 1.13520i
\(91\) 0.0104460 + 0.122602i 0.00109504 + 0.0128521i
\(92\) −3.16041 − 0.557266i −0.329496 − 0.0580990i
\(93\) −7.10027 + 3.48046i −0.736264 + 0.360907i
\(94\) −6.16918 − 7.35214i −0.636302 − 0.758315i
\(95\) −9.00059 − 10.7265i −0.923441 − 1.10051i
\(96\) 0.624474 − 9.19200i 0.0637351 − 0.938155i
\(97\) −13.2580 − 2.33775i −1.34615 − 0.237363i −0.546313 − 0.837581i \(-0.683969\pi\)
−0.799836 + 0.600218i \(0.795080\pi\)
\(98\) −2.72628 − 15.8826i −0.275396 − 1.60438i
\(99\) −0.0498289 + 0.365038i −0.00500799 + 0.0366877i
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.101.20 ✓ 132
3.2 odd 2 567.2.ba.a.143.3 132
7.5 odd 6 189.2.bd.a.47.3 yes 132
21.5 even 6 567.2.bd.a.467.20 132
27.4 even 9 567.2.bd.a.17.20 132
27.23 odd 18 189.2.bd.a.185.3 yes 132
189.131 even 18 inner 189.2.ba.a.131.20 yes 132
189.166 odd 18 567.2.ba.a.341.3 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.20 ✓ 132 1.1 even 1 trivial
189.2.ba.a.131.20 yes 132 189.131 even 18 inner
189.2.bd.a.47.3 yes 132 7.5 odd 6
189.2.bd.a.185.3 yes 132 27.23 odd 18
567.2.ba.a.143.3 132 3.2 odd 2
567.2.ba.a.341.3 132 189.166 odd 18
567.2.bd.a.17.20 132 27.4 even 9
567.2.bd.a.467.20 132 21.5 even 6