Properties

Label 197.8.a.b.1.20
Level 197197
Weight 88
Character 197.1
Self dual yes
Analytic conductor 61.54061.540
Analytic rank 00
Dimension 6060
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [197,8,Mod(1,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: N N == 197 197
Weight: k k == 8 8
Character orbit: [χ][\chi] == 197.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 61.539850020461.5398500204
Analytic rank: 00
Dimension: 6060
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.20
Character χ\chi == 197.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q8.25431q214.1669q359.8663q4+11.5658q5+116.938q6664.362q7+1550.71q81986.30q995.4680q103556.40q11+848.118q126714.69q13+5483.85q14163.852q155137.13q16+13719.6q17+16395.5q1834381.8q19692.405q20+9411.91q21+29355.7q2236355.9q2321968.6q2477991.2q25+55425.2q26+59122.5q27+39772.9q2846569.7q29+1352.48q30287094.q31156087.q32+50383.0q33113246.q347683.90q35+118913.q36374998.q37+283798.q38+95126.1q39+17935.2q40+8647.96q4177688.9q42326648.q43+212909.q4422973.2q45+300093.q46+322881.q47+72776.9q48382167.q49+643764.q50194363.q51+401984.q521.15879e6q53488016.q5441132.8q551.03023e6q56+487082.q57+384401.q58381147.q59+9809.20q60+681676.q61+2.36976e6q62+1.31962e6q63+1.94594e6q6477661.1q65415877.q663.84661e6q67821342.q68+515049.q69+63425.3q70+3.36103e6q713.08017e6q723.33642e6q73+3.09535e6q74+1.10489e6q75+2.05831e6q76+2.36274e6q77785201.q78+2.91848e6q7959415.2q80+3.50646e6q8171383.0q82+6.89244e6q83563457.q84+158679.q85+2.69626e6q86+659746.q875.51494e6q88+432784.q89+189628.q90+4.46098e6q91+2.17650e6q92+4.06722e6q932.66516e6q94397654.q95+2.21126e6q966.90655e6q97+3.15452e6q98+7.06408e6q99+O(q100)q-8.25431 q^{2} -14.1669 q^{3} -59.8663 q^{4} +11.5658 q^{5} +116.938 q^{6} -664.362 q^{7} +1550.71 q^{8} -1986.30 q^{9} -95.4680 q^{10} -3556.40 q^{11} +848.118 q^{12} -6714.69 q^{13} +5483.85 q^{14} -163.852 q^{15} -5137.13 q^{16} +13719.6 q^{17} +16395.5 q^{18} -34381.8 q^{19} -692.405 q^{20} +9411.91 q^{21} +29355.7 q^{22} -36355.9 q^{23} -21968.6 q^{24} -77991.2 q^{25} +55425.2 q^{26} +59122.5 q^{27} +39772.9 q^{28} -46569.7 q^{29} +1352.48 q^{30} -287094. q^{31} -156087. q^{32} +50383.0 q^{33} -113246. q^{34} -7683.90 q^{35} +118913. q^{36} -374998. q^{37} +283798. q^{38} +95126.1 q^{39} +17935.2 q^{40} +8647.96 q^{41} -77688.9 q^{42} -326648. q^{43} +212909. q^{44} -22973.2 q^{45} +300093. q^{46} +322881. q^{47} +72776.9 q^{48} -382167. q^{49} +643764. q^{50} -194363. q^{51} +401984. q^{52} -1.15879e6 q^{53} -488016. q^{54} -41132.8 q^{55} -1.03023e6 q^{56} +487082. q^{57} +384401. q^{58} -381147. q^{59} +9809.20 q^{60} +681676. q^{61} +2.36976e6 q^{62} +1.31962e6 q^{63} +1.94594e6 q^{64} -77661.1 q^{65} -415877. q^{66} -3.84661e6 q^{67} -821342. q^{68} +515049. q^{69} +63425.3 q^{70} +3.36103e6 q^{71} -3.08017e6 q^{72} -3.33642e6 q^{73} +3.09535e6 q^{74} +1.10489e6 q^{75} +2.05831e6 q^{76} +2.36274e6 q^{77} -785201. q^{78} +2.91848e6 q^{79} -59415.2 q^{80} +3.50646e6 q^{81} -71383.0 q^{82} +6.89244e6 q^{83} -563457. q^{84} +158679. q^{85} +2.69626e6 q^{86} +659746. q^{87} -5.51494e6 q^{88} +432784. q^{89} +189628. q^{90} +4.46098e6 q^{91} +2.17650e6 q^{92} +4.06722e6 q^{93} -2.66516e6 q^{94} -397654. q^{95} +2.21126e6 q^{96} -6.90655e6 q^{97} +3.15452e6 q^{98} +7.06408e6 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 60q+16q2+296q3+4224q4+554q5+1200q6+4959q7+2571q8+47384q9+16237q10+12452q11+38656q12+36460q13567q14+55139q15+319488q16++64628303q99+O(q100) 60 q + 16 q^{2} + 296 q^{3} + 4224 q^{4} + 554 q^{5} + 1200 q^{6} + 4959 q^{7} + 2571 q^{8} + 47384 q^{9} + 16237 q^{10} + 12452 q^{11} + 38656 q^{12} + 36460 q^{13} - 567 q^{14} + 55139 q^{15} + 319488 q^{16}+ \cdots + 64628303 q^{99}+O(q^{100}) Copy content Toggle raw display

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 −8.25431 −0.729585 −0.364792 0.931089i 0.618860π-0.618860\pi
−0.364792 + 0.931089i 0.618860π0.618860\pi
33 −14.1669 −0.302935 −0.151467 0.988462i 0.548400π-0.548400\pi
−0.151467 + 0.988462i 0.548400π0.548400\pi
44 −59.8663 −0.467706
55 11.5658 0.0413792 0.0206896 0.999786i 0.493414π-0.493414\pi
0.0206896 + 0.999786i 0.493414π0.493414\pi
66 116.938 0.221017
77 −664.362 −0.732085 −0.366042 0.930598i 0.619287π-0.619287\pi
−0.366042 + 0.930598i 0.619287π0.619287\pi
88 1550.71 1.07082
99 −1986.30 −0.908231
1010 −95.4680 −0.0301896
1111 −3556.40 −0.805632 −0.402816 0.915281i 0.631969π-0.631969\pi
−0.402816 + 0.915281i 0.631969π0.631969\pi
1212 848.118 0.141684
1313 −6714.69 −0.847666 −0.423833 0.905740i 0.639316π-0.639316\pi
−0.423833 + 0.905740i 0.639316π0.639316\pi
1414 5483.85 0.534118
1515 −163.852 −0.0125352
1616 −5137.13 −0.313545
1717 13719.6 0.677282 0.338641 0.940916i 0.390033π-0.390033\pi
0.338641 + 0.940916i 0.390033π0.390033\pi
1818 16395.5 0.662631
1919 −34381.8 −1.14998 −0.574991 0.818160i 0.694994π-0.694994\pi
−0.574991 + 0.818160i 0.694994π0.694994\pi
2020 −692.405 −0.0193533
2121 9411.91 0.221774
2222 29355.7 0.587777
2323 −36355.9 −0.623057 −0.311529 0.950237i 0.600841π-0.600841\pi
−0.311529 + 0.950237i 0.600841π0.600841\pi
2424 −21968.6 −0.324387
2525 −77991.2 −0.998288
2626 55425.2 0.618444
2727 59122.5 0.578069
2828 39772.9 0.342400
2929 −46569.7 −0.354577 −0.177288 0.984159i 0.556733π-0.556733\pi
−0.177288 + 0.984159i 0.556733π0.556733\pi
3030 1352.48 0.00914549
3131 −287094. −1.73084 −0.865422 0.501044i 0.832950π-0.832950\pi
−0.865422 + 0.501044i 0.832950π0.832950\pi
3232 −156087. −0.842058
3333 50383.0 0.244054
3434 −113246. −0.494135
3535 −7683.90 −0.0302931
3636 118913. 0.424785
3737 −374998. −1.21709 −0.608545 0.793519i 0.708247π-0.708247\pi
−0.608545 + 0.793519i 0.708247π0.708247\pi
3838 283798. 0.839009
3939 95126.1 0.256787
4040 17935.2 0.0443095
4141 8647.96 0.0195961 0.00979807 0.999952i 0.496881π-0.496881\pi
0.00979807 + 0.999952i 0.496881π0.496881\pi
4242 −77688.9 −0.161803
4343 −326648. −0.626528 −0.313264 0.949666i 0.601423π-0.601423\pi
−0.313264 + 0.949666i 0.601423π0.601423\pi
4444 212909. 0.376799
4545 −22973.2 −0.0375819
4646 300093. 0.454573
4747 322881. 0.453629 0.226814 0.973938i 0.427169π-0.427169\pi
0.226814 + 0.973938i 0.427169π0.427169\pi
4848 72776.9 0.0949838
4949 −382167. −0.464052
5050 643764. 0.728336
5151 −194363. −0.205172
5252 401984. 0.396458
5353 −1.15879e6 −1.06915 −0.534577 0.845120i 0.679529π-0.679529\pi
−0.534577 + 0.845120i 0.679529π0.679529\pi
5454 −488016. −0.421751
5555 −41132.8 −0.0333364
5656 −1.03023e6 −0.783928
5757 487082. 0.348369
5858 384401. 0.258694
5959 −381147. −0.241607 −0.120804 0.992676i 0.538547π-0.538547\pi
−0.120804 + 0.992676i 0.538547π0.538547\pi
6060 9809.20 0.00586279
6161 681676. 0.384524 0.192262 0.981344i 0.438418π-0.438418\pi
0.192262 + 0.981344i 0.438418π0.438418\pi
6262 2.36976e6 1.26280
6363 1.31962e6 0.664902
6464 1.94594e6 0.927898
6565 −77661.1 −0.0350757
6666 −415877. −0.178058
6767 −3.84661e6 −1.56249 −0.781244 0.624226i 0.785414π-0.785414\pi
−0.781244 + 0.624226i 0.785414π0.785414\pi
6868 −821342. −0.316769
6969 515049. 0.188746
7070 63425.3 0.0221014
7171 3.36103e6 1.11447 0.557234 0.830355i 0.311863π-0.311863\pi
0.557234 + 0.830355i 0.311863π0.311863\pi
7272 −3.08017e6 −0.972548
7373 −3.33642e6 −1.00381 −0.501904 0.864923i 0.667367π-0.667367\pi
−0.501904 + 0.864923i 0.667367π0.667367\pi
7474 3.09535e6 0.887971
7575 1.10489e6 0.302416
7676 2.05831e6 0.537853
7777 2.36274e6 0.589791
7878 −785201. −0.187348
7979 2.91848e6 0.665981 0.332991 0.942930i 0.391942π-0.391942\pi
0.332991 + 0.942930i 0.391942π0.391942\pi
8080 −59415.2 −0.0129743
8181 3.50646e6 0.733113
8282 −71383.0 −0.0142970
8383 6.89244e6 1.32312 0.661560 0.749892i 0.269895π-0.269895\pi
0.661560 + 0.749892i 0.269895π0.269895\pi
8484 −563457. −0.103725
8585 158679. 0.0280254
8686 2.69626e6 0.457106
8787 659746. 0.107414
8888 −5.51494e6 −0.862683
8989 432784. 0.0650738 0.0325369 0.999471i 0.489641π-0.489641\pi
0.0325369 + 0.999471i 0.489641π0.489641\pi
9090 189628. 0.0274192
9191 4.46098e6 0.620563
9292 2.17650e6 0.291408
9393 4.06722e6 0.524333
9494 −2.66516e6 −0.330961
9595 −397654. −0.0475853
9696 2.21126e6 0.255089
9797 −6.90655e6 −0.768352 −0.384176 0.923260i 0.625514π-0.625514\pi
−0.384176 + 0.923260i 0.625514π0.625514\pi
9898 3.15452e6 0.338565
9999 7.06408e6 0.731699
100100 4.66905e6 0.466905
101101 1.36825e6 0.132142 0.0660711 0.997815i 0.478954π-0.478954\pi
0.0660711 + 0.997815i 0.478954π0.478954\pi
102102 1.60434e6 0.149691
103103 1.35466e6 0.122152 0.0610760 0.998133i 0.480547π-0.480547\pi
0.0610760 + 0.998133i 0.480547π0.480547\pi
104104 −1.04125e7 −0.907694
105105 108857. 0.00917683
106106 9.56504e6 0.780039
107107 1.27997e7 1.01008 0.505039 0.863096i 0.331478π-0.331478\pi
0.505039 + 0.863096i 0.331478π0.331478\pi
108108 −3.53945e6 −0.270366
109109 1.20453e7 0.890891 0.445446 0.895309i 0.353045π-0.353045\pi
0.445446 + 0.895309i 0.353045π0.353045\pi
110110 339523. 0.0243217
111111 5.31254e6 0.368699
112112 3.41291e6 0.229542
113113 8.88122e6 0.579026 0.289513 0.957174i 0.406507π-0.406507\pi
0.289513 + 0.957174i 0.406507π0.406507\pi
114114 −4.02053e6 −0.254165
115115 −420487. −0.0257816
116116 2.78796e6 0.165838
117117 1.33374e7 0.769876
118118 3.14611e6 0.176273
119119 −9.11477e6 −0.495828
120120 −254086. −0.0134229
121121 −6.83917e6 −0.350957
122122 −5.62676e6 −0.280543
123123 −122514. −0.00593635
124124 1.71873e7 0.809526
125125 −1.80562e6 −0.0826876
126126 −1.08926e7 −0.485102
127127 −3.50062e6 −0.151646 −0.0758232 0.997121i 0.524158π-0.524158\pi
−0.0758232 + 0.997121i 0.524158π0.524158\pi
128128 3.91672e6 0.165078
129129 4.62758e6 0.189797
130130 641039. 0.0255907
131131 1.48152e7 0.575783 0.287892 0.957663i 0.407046π-0.407046\pi
0.287892 + 0.957663i 0.407046π0.407046\pi
132132 −3.01625e6 −0.114145
133133 2.28419e7 0.841884
134134 3.17511e7 1.13997
135135 683802. 0.0239200
136136 2.12751e7 0.725245
137137 1.09864e7 0.365034 0.182517 0.983203i 0.441576π-0.441576\pi
0.182517 + 0.983203i 0.441576π0.441576\pi
138138 −4.25138e6 −0.137706
139139 1.25769e7 0.397211 0.198606 0.980079i 0.436359π-0.436359\pi
0.198606 + 0.980079i 0.436359π0.436359\pi
140140 460007. 0.0141683
141141 −4.57421e6 −0.137420
142142 −2.77430e7 −0.813099
143143 2.38802e7 0.682906
144144 1.02039e7 0.284771
145145 −538618. −0.0146721
146146 2.75398e7 0.732363
147147 5.41410e6 0.140577
148148 2.24498e7 0.569240
149149 4.33466e7 1.07350 0.536751 0.843741i 0.319651π-0.319651\pi
0.536751 + 0.843741i 0.319651π0.319651\pi
150150 −9.12011e6 −0.220638
151151 −1.97642e7 −0.467153 −0.233577 0.972338i 0.575043π-0.575043\pi
−0.233577 + 0.972338i 0.575043π0.575043\pi
152152 −5.33161e7 −1.23142
153153 −2.72512e7 −0.615129
154154 −1.95028e7 −0.430302
155155 −3.32048e6 −0.0716209
156156 −5.69485e6 −0.120101
157157 −4.88463e7 −1.00736 −0.503678 0.863891i 0.668020π-0.668020\pi
−0.503678 + 0.863891i 0.668020π0.668020\pi
158158 −2.40900e7 −0.485890
159159 1.64165e7 0.323884
160160 −1.80528e6 −0.0348437
161161 2.41535e7 0.456131
162162 −2.89434e7 −0.534868
163163 −4.25840e7 −0.770176 −0.385088 0.922880i 0.625829π-0.625829\pi
−0.385088 + 0.922880i 0.625829π0.625829\pi
164164 −517722. −0.00916523
165165 582722. 0.0100988
166166 −5.68923e7 −0.965329
167167 −8.93370e7 −1.48431 −0.742154 0.670230i 0.766196π-0.766196\pi
−0.742154 + 0.670230i 0.766196π0.766196\pi
168168 1.45951e7 0.237479
169169 −1.76614e7 −0.281463
170170 −1.30978e6 −0.0204469
171171 6.82926e7 1.04445
172172 1.95552e7 0.293031
173173 2.27349e7 0.333835 0.166918 0.985971i 0.446619π-0.446619\pi
0.166918 + 0.985971i 0.446619π0.446619\pi
174174 −5.44575e6 −0.0783674
175175 5.18144e7 0.730831
176176 1.82697e7 0.252602
177177 5.39965e6 0.0731913
178178 −3.57234e6 −0.0474769
179179 −3.65563e6 −0.0476406 −0.0238203 0.999716i 0.507583π-0.507583\pi
−0.0238203 + 0.999716i 0.507583π0.507583\pi
180180 1.37532e6 0.0175773
181181 −7.40252e7 −0.927908 −0.463954 0.885859i 0.653570π-0.653570\pi
−0.463954 + 0.885859i 0.653570π0.653570\pi
182182 −3.68224e7 −0.452753
183183 −9.65720e6 −0.116486
184184 −5.63774e7 −0.667180
185185 −4.33717e6 −0.0503622
186186 −3.35721e7 −0.382545
187187 −4.87924e7 −0.545640
188188 −1.93297e7 −0.212165
189189 −3.92787e7 −0.423196
190190 3.28236e6 0.0347175
191191 −7.21721e7 −0.749467 −0.374734 0.927132i 0.622266π-0.622266\pi
−0.374734 + 0.927132i 0.622266π0.622266\pi
192192 −2.75679e7 −0.281093
193193 −1.98181e7 −0.198432 −0.0992160 0.995066i 0.531633π-0.531633\pi
−0.0992160 + 0.995066i 0.531633π0.531633\pi
194194 5.70088e7 0.560578
195195 1.10021e6 0.0106257
196196 2.28789e7 0.217040
197197 −7.64537e6 −0.0712470
198198 −5.83091e7 −0.533837
199199 −9.86165e7 −0.887082 −0.443541 0.896254i 0.646278π-0.646278\pi
−0.443541 + 0.896254i 0.646278π0.646278\pi
200200 −1.20942e8 −1.06898
201201 5.44944e7 0.473332
202202 −1.12940e7 −0.0964089
203203 3.09391e7 0.259580
204204 1.16358e7 0.0959603
205205 100021. 0.000810872 0
206206 −1.11818e7 −0.0891202
207207 7.22138e7 0.565880
208208 3.44942e7 0.265782
209209 1.22276e8 0.926462
210210 −898537. −0.00669527
211211 −5.66199e7 −0.414936 −0.207468 0.978242i 0.566522π-0.566522\pi
−0.207468 + 0.978242i 0.566522π0.566522\pi
212212 6.93727e7 0.500050
213213 −4.76152e7 −0.337611
214214 −1.05652e8 −0.736938
215215 −3.77796e6 −0.0259252
216216 9.16818e7 0.619006
217217 1.90734e8 1.26712
218218 −9.94256e7 −0.649981
219219 4.72666e7 0.304088
220220 2.46247e6 0.0155916
221221 −9.21228e7 −0.574109
222222 −4.38514e7 −0.268997
223223 −1.29085e7 −0.0779490 −0.0389745 0.999240i 0.512409π-0.512409\pi
−0.0389745 + 0.999240i 0.512409π0.512409\pi
224224 1.03698e8 0.616458
225225 1.54914e8 0.906675
226226 −7.33083e7 −0.422448
227227 9.61523e7 0.545593 0.272797 0.962072i 0.412051π-0.412051\pi
0.272797 + 0.962072i 0.412051π0.412051\pi
228228 −2.91598e7 −0.162934
229229 −2.53548e8 −1.39520 −0.697598 0.716489i 0.745748π-0.745748\pi
−0.697598 + 0.716489i 0.745748π0.745748\pi
230230 3.47083e6 0.0188099
231231 −3.34726e7 −0.178668
232232 −7.22160e7 −0.379687
233233 −1.26742e8 −0.656410 −0.328205 0.944607i 0.606444π-0.606444\pi
−0.328205 + 0.944607i 0.606444π0.606444\pi
234234 −1.10091e8 −0.561690
235235 3.73439e6 0.0187708
236236 2.28179e7 0.113001
237237 −4.13457e7 −0.201749
238238 7.52361e7 0.361749
239239 −1.57899e8 −0.748146 −0.374073 0.927399i 0.622039π-0.622039\pi
−0.374073 + 0.927399i 0.622039π0.622039\pi
240240 841726. 0.00393035
241241 −1.51861e7 −0.0698854 −0.0349427 0.999389i 0.511125π-0.511125\pi
−0.0349427 + 0.999389i 0.511125π0.511125\pi
242242 5.64526e7 0.256053
243243 −1.78976e8 −0.800155
244244 −4.08094e7 −0.179844
245245 −4.42008e6 −0.0192021
246246 1.01127e6 0.00433107
247247 2.30863e8 0.974800
248248 −4.45198e8 −1.85342
249249 −9.76441e7 −0.400819
250250 1.49041e7 0.0603276
251251 −1.35485e8 −0.540797 −0.270399 0.962748i 0.587155π-0.587155\pi
−0.270399 + 0.962748i 0.587155π0.587155\pi
252252 −7.90009e7 −0.310978
253253 1.29296e8 0.501955
254254 2.88952e7 0.110639
255255 −2.24798e6 −0.00848987
256256 −2.81411e8 −1.04834
257257 2.50898e8 0.922000 0.461000 0.887400i 0.347491π-0.347491\pi
0.461000 + 0.887400i 0.347491π0.347491\pi
258258 −3.81975e7 −0.138473
259259 2.49134e8 0.891013
260260 4.64929e6 0.0164051
261261 9.25014e7 0.322038
262262 −1.22290e8 −0.420083
263263 −2.64736e8 −0.897362 −0.448681 0.893692i 0.648106π-0.648106\pi
−0.448681 + 0.893692i 0.648106π0.648106\pi
264264 7.81294e7 0.261337
265265 −1.34024e7 −0.0442408
266266 −1.88544e8 −0.614226
267267 −6.13119e6 −0.0197131
268268 2.30283e8 0.730785
269269 −2.75003e8 −0.861398 −0.430699 0.902496i 0.641733π-0.641733\pi
−0.430699 + 0.902496i 0.641733π0.641733\pi
270270 −5.64431e6 −0.0174517
271271 5.61337e8 1.71329 0.856646 0.515905i 0.172544π-0.172544\pi
0.856646 + 0.515905i 0.172544π0.172544\pi
272272 −7.04793e7 −0.212359
273273 −6.31981e7 −0.187990
274274 −9.06851e7 −0.266323
275275 2.77368e8 0.804252
276276 −3.08341e7 −0.0882775
277277 5.96151e8 1.68530 0.842650 0.538462i 0.180994π-0.180994\pi
0.842650 + 0.538462i 0.180994π0.180994\pi
278278 −1.03814e8 −0.289799
279279 5.70255e8 1.57201
280280 −1.19155e7 −0.0324383
281281 6.92463e8 1.86176 0.930881 0.365322i 0.119041π-0.119041\pi
0.930881 + 0.365322i 0.119041π0.119041\pi
282282 3.77570e7 0.100259
283283 6.08991e7 0.159720 0.0798598 0.996806i 0.474553π-0.474553\pi
0.0798598 + 0.996806i 0.474553π0.474553\pi
284284 −2.01212e8 −0.521243
285285 5.63351e6 0.0144152
286286 −1.97114e8 −0.498238
287287 −5.74537e6 −0.0143460
288288 3.10036e8 0.764783
289289 −2.22112e8 −0.541289
290290 4.44592e6 0.0107046
291291 9.78441e7 0.232760
292292 1.99739e8 0.469487
293293 −6.64918e8 −1.54430 −0.772150 0.635440i 0.780819π-0.780819\pi
−0.772150 + 0.635440i 0.780819π0.780819\pi
294294 −4.46897e7 −0.102563
295295 −4.40828e6 −0.00999752
296296 −5.81512e8 −1.30328
297297 −2.10264e8 −0.465711
298298 −3.57796e8 −0.783211
299299 2.44119e8 0.528144
300300 −6.61458e7 −0.141442
301301 2.17013e8 0.458672
302302 1.63140e8 0.340828
303303 −1.93838e7 −0.0400304
304304 1.76624e8 0.360571
305305 7.88415e6 0.0159113
306306 2.24940e8 0.448789
307307 4.87322e8 0.961241 0.480620 0.876929i 0.340412π-0.340412\pi
0.480620 + 0.876929i 0.340412π0.340412\pi
308308 −1.41448e8 −0.275849
309309 −1.91913e7 −0.0370041
310310 2.74083e7 0.0522536
311311 7.96951e8 1.50235 0.751173 0.660105i 0.229488π-0.229488\pi
0.751173 + 0.660105i 0.229488π0.229488\pi
312312 1.47513e8 0.274972
313313 −8.04789e7 −0.148346 −0.0741732 0.997245i 0.523632π-0.523632\pi
−0.0741732 + 0.997245i 0.523632π0.523632\pi
314314 4.03193e8 0.734952
315315 1.52625e7 0.0275131
316316 −1.74719e8 −0.311483
317317 −8.24931e8 −1.45449 −0.727244 0.686379i 0.759199π-0.759199\pi
−0.727244 + 0.686379i 0.759199π0.759199\pi
318318 −1.35507e8 −0.236301
319319 1.65621e8 0.285658
320320 2.25065e7 0.0383957
321321 −1.81331e8 −0.305988
322322 −1.99370e8 −0.332786
323323 −4.71704e8 −0.778862
324324 −2.09919e8 −0.342881
325325 5.23687e8 0.846214
326326 3.51502e8 0.561909
327327 −1.70644e8 −0.269882
328328 1.34105e7 0.0209839
329329 −2.14510e8 −0.332094
330330 −4.80997e6 −0.00736790
331331 2.72216e8 0.412588 0.206294 0.978490i 0.433860π-0.433860\pi
0.206294 + 0.978490i 0.433860π0.433860\pi
332332 −4.12625e8 −0.618831
333333 7.44859e8 1.10540
334334 7.37416e8 1.08293
335335 −4.44893e7 −0.0646545
336336 −4.83502e7 −0.0695362
337337 −8.19304e8 −1.16611 −0.583056 0.812432i 0.698143π-0.698143\pi
−0.583056 + 0.812432i 0.698143π0.698143\pi
338338 1.45783e8 0.205351
339339 −1.25819e8 −0.175407
340340 −9.49950e6 −0.0131076
341341 1.02102e9 1.39442
342342 −5.63708e8 −0.762014
343343 8.01027e8 1.07181
344344 −5.06536e8 −0.670897
345345 5.95698e6 0.00781015
346346 −1.87661e8 −0.243561
347347 1.08036e8 0.138809 0.0694043 0.997589i 0.477890π-0.477890\pi
0.0694043 + 0.997589i 0.477890π0.477890\pi
348348 −3.94966e7 −0.0502380
349349 −2.32572e8 −0.292866 −0.146433 0.989221i 0.546779π-0.546779\pi
−0.146433 + 0.989221i 0.546779π0.546779\pi
350350 −4.27692e8 −0.533203
351351 −3.96990e8 −0.490009
352352 5.55109e8 0.678389
353353 −6.96581e8 −0.842869 −0.421435 0.906859i 0.638473π-0.638473\pi
−0.421435 + 0.906859i 0.638473π0.638473\pi
354354 −4.45704e7 −0.0533993
355355 3.88731e7 0.0461158
356356 −2.59092e7 −0.0304354
357357 1.29128e8 0.150204
358358 3.01747e7 0.0347579
359359 −5.53880e8 −0.631808 −0.315904 0.948791i 0.602308π-0.602308\pi
−0.315904 + 0.948791i 0.602308π0.602308\pi
360360 −3.56248e7 −0.0402433
361361 2.88236e8 0.322458
362362 6.11027e8 0.676987
363363 9.68895e7 0.106317
364364 −2.67063e8 −0.290241
365365 −3.85885e7 −0.0415368
366366 7.97136e7 0.0849862
367367 −9.17961e7 −0.0969378 −0.0484689 0.998825i 0.515434π-0.515434\pi
−0.0484689 + 0.998825i 0.515434π0.515434\pi
368368 1.86765e8 0.195357
369369 −1.71775e7 −0.0177978
370370 3.58003e7 0.0367435
371371 7.69858e8 0.782712
372372 −2.43489e8 −0.245233
373373 1.60597e8 0.160235 0.0801175 0.996785i 0.474470π-0.474470\pi
0.0801175 + 0.996785i 0.474470π0.474470\pi
374374 4.02747e8 0.398091
375375 2.55799e7 0.0250489
376376 5.00694e8 0.485753
377377 3.12701e8 0.300563
378378 3.24219e8 0.308757
379379 −5.89008e8 −0.555756 −0.277878 0.960616i 0.589631π-0.589631\pi
−0.277878 + 0.960616i 0.589631π0.589631\pi
380380 2.38061e7 0.0222559
381381 4.95928e7 0.0459390
382382 5.95731e8 0.546800
383383 1.20792e9 1.09861 0.549305 0.835622i 0.314893π-0.314893\pi
0.549305 + 0.835622i 0.314893π0.314893\pi
384384 −5.54876e7 −0.0500077
385385 2.73270e7 0.0244051
386386 1.63585e8 0.144773
387387 6.48822e8 0.569032
388388 4.13470e8 0.359363
389389 3.81729e8 0.328800 0.164400 0.986394i 0.447431π-0.447431\pi
0.164400 + 0.986394i 0.447431π0.447431\pi
390390 −9.08150e6 −0.00775232
391391 −4.98788e8 −0.421986
392392 −5.92629e8 −0.496914
393393 −2.09885e8 −0.174425
394394 6.31073e7 0.0519808
395395 3.37547e7 0.0275578
396396 −4.22901e8 −0.342220
397397 3.24022e8 0.259901 0.129951 0.991520i 0.458518π-0.458518\pi
0.129951 + 0.991520i 0.458518π0.458518\pi
398398 8.14011e8 0.647201
399399 −3.23598e8 −0.255036
400400 4.00651e8 0.313009
401401 1.18362e9 0.916658 0.458329 0.888783i 0.348448π-0.348448\pi
0.458329 + 0.888783i 0.348448π0.348448\pi
402402 −4.49814e8 −0.345336
403403 1.92775e9 1.46718
404404 −8.19123e7 −0.0618037
405405 4.05551e7 0.0303356
406406 −2.55381e8 −0.189386
407407 1.33364e9 0.980527
408408 −3.01401e8 −0.219702
409409 −2.85737e8 −0.206507 −0.103254 0.994655i 0.532925π-0.532925\pi
−0.103254 + 0.994655i 0.532925π0.532925\pi
410410 −825604. −0.000591600 0
411411 −1.55643e8 −0.110581
412412 −8.10986e7 −0.0571312
413413 2.53219e8 0.176877
414414 −5.96075e8 −0.412857
415415 7.97168e7 0.0547497
416416 1.04808e9 0.713784
417417 −1.78175e8 −0.120329
418418 −1.00930e9 −0.675933
419419 −2.31695e9 −1.53875 −0.769376 0.638797i 0.779433π-0.779433\pi
−0.769376 + 0.638797i 0.779433π0.779433\pi
420420 −6.51685e6 −0.00429206
421421 1.99117e9 1.30054 0.650268 0.759705i 0.274657π-0.274657\pi
0.650268 + 0.759705i 0.274657π0.274657\pi
422422 4.67358e8 0.302731
423423 −6.41339e8 −0.411999
424424 −1.79695e9 −1.14487
425425 −1.07001e9 −0.676123
426426 3.93030e8 0.246316
427427 −4.52879e8 −0.281504
428428 −7.66268e8 −0.472419
429429 −3.38307e8 −0.206876
430430 3.11845e7 0.0189147
431431 −2.00750e8 −0.120777 −0.0603886 0.998175i 0.519234π-0.519234\pi
−0.0603886 + 0.998175i 0.519234π0.519234\pi
432432 −3.03720e8 −0.181251
433433 −2.15542e9 −1.27592 −0.637962 0.770068i 0.720222π-0.720222\pi
−0.637962 + 0.770068i 0.720222π0.720222\pi
434434 −1.57438e9 −0.924475
435435 7.63052e6 0.00444469
436436 −7.21108e8 −0.416675
437437 1.24998e9 0.716504
438438 −3.90153e8 −0.221858
439439 1.03334e9 0.582929 0.291465 0.956582i 0.405857π-0.405857\pi
0.291465 + 0.956582i 0.405857π0.405857\pi
440440 −6.37849e7 −0.0356972
441441 7.59098e8 0.421466
442442 7.60411e8 0.418861
443443 −1.59156e9 −0.869780 −0.434890 0.900483i 0.643213π-0.643213\pi
−0.434890 + 0.900483i 0.643213π0.643213\pi
444444 −3.18043e8 −0.172443
445445 5.00551e6 0.00269270
446446 1.06551e8 0.0568704
447447 −6.14085e8 −0.325201
448448 −1.29281e9 −0.679300
449449 −1.37310e9 −0.715877 −0.357939 0.933745i 0.616520π-0.616520\pi
−0.357939 + 0.933745i 0.616520π0.616520\pi
450450 −1.27871e9 −0.661497
451451 −3.07556e7 −0.0157873
452452 −5.31686e8 −0.270814
453453 2.79996e8 0.141517
454454 −7.93671e8 −0.398057
455455 5.15950e7 0.0256784
456456 7.55321e8 0.373040
457457 2.25601e9 1.10569 0.552847 0.833283i 0.313541π-0.313541\pi
0.552847 + 0.833283i 0.313541π0.313541\pi
458458 2.09286e9 1.01791
459459 8.11137e8 0.391516
460460 2.51730e7 0.0120582
461461 −1.88925e9 −0.898125 −0.449062 0.893500i 0.648242π-0.648242\pi
−0.449062 + 0.893500i 0.648242π0.648242\pi
462462 2.76293e8 0.130354
463463 −2.88415e8 −0.135047 −0.0675233 0.997718i 0.521510π-0.521510\pi
−0.0675233 + 0.997718i 0.521510π0.521510\pi
464464 2.39235e8 0.111176
465465 4.70408e7 0.0216965
466466 1.04617e9 0.478907
467467 7.47402e8 0.339582 0.169791 0.985480i 0.445691π-0.445691\pi
0.169791 + 0.985480i 0.445691π0.445691\pi
468468 −7.98461e8 −0.360075
469469 2.55554e9 1.14387
470470 −3.08248e7 −0.0136949
471471 6.91999e8 0.305163
472472 −5.91047e8 −0.258717
473473 1.16169e9 0.504751
474474 3.41280e8 0.147193
475475 2.68148e9 1.14801
476476 5.45668e8 0.231902
477477 2.30171e9 0.971039
478478 1.30335e9 0.545836
479479 −9.92413e8 −0.412590 −0.206295 0.978490i 0.566141π-0.566141\pi
−0.206295 + 0.978490i 0.566141π0.566141\pi
480480 2.55751e7 0.0105554
481481 2.51800e9 1.03169
482482 1.25351e8 0.0509873
483483 −3.42179e8 −0.138178
484484 4.09436e8 0.164145
485485 −7.98800e7 −0.0317938
486486 1.47733e9 0.583781
487487 −3.61900e9 −1.41983 −0.709917 0.704286i 0.751267π-0.751267\pi
−0.709917 + 0.704286i 0.751267π0.751267\pi
488488 1.05708e9 0.411754
489489 6.03282e8 0.233313
490490 3.64847e7 0.0140096
491491 4.95756e7 0.0189009 0.00945045 0.999955i 0.496992π-0.496992\pi
0.00945045 + 0.999955i 0.496992π0.496992\pi
492492 7.33449e6 0.00277647
493493 −6.38917e8 −0.240149
494494 −1.90562e9 −0.711199
495495 8.17021e7 0.0302771
496496 1.47484e9 0.542698
497497 −2.23294e9 −0.815885
498498 8.05985e8 0.292432
499499 2.11782e8 0.0763022 0.0381511 0.999272i 0.487853π-0.487853\pi
0.0381511 + 0.999272i 0.487853π0.487853\pi
500500 1.08096e8 0.0386735
501501 1.26562e9 0.449648
502502 1.11834e9 0.394557
503503 −5.16907e9 −1.81102 −0.905512 0.424321i 0.860513π-0.860513\pi
−0.905512 + 0.424321i 0.860513π0.860513\pi
504504 2.04635e9 0.711987
505505 1.58250e7 0.00546794
506506 −1.06725e9 −0.366219
507507 2.50206e8 0.0852649
508508 2.09569e8 0.0709259
509509 −1.88195e8 −0.0632550 −0.0316275 0.999500i 0.510069π-0.510069\pi
−0.0316275 + 0.999500i 0.510069π0.510069\pi
510510 1.85555e7 0.00619408
511511 2.21659e9 0.734873
512512 1.82151e9 0.599773
513513 −2.03274e9 −0.664769
514514 −2.07099e9 −0.672677
515515 1.56678e7 0.00505455
516516 −2.77036e8 −0.0887693
517517 −1.14830e9 −0.365458
518518 −2.05643e9 −0.650070
519519 −3.22082e8 −0.101130
520520 −1.20430e8 −0.0375596
521521 −3.82610e9 −1.18529 −0.592645 0.805464i 0.701916π-0.701916\pi
−0.592645 + 0.805464i 0.701916π0.701916\pi
522522 −7.63536e8 −0.234954
523523 −3.91879e9 −1.19783 −0.598916 0.800812i 0.704402π-0.704402\pi
−0.598916 + 0.800812i 0.704402π0.704402\pi
524524 −8.86934e8 −0.269297
525525 −7.34047e8 −0.221394
526526 2.18521e9 0.654702
527527 −3.93881e9 −1.17227
528528 −2.58824e8 −0.0765220
529529 −2.08307e9 −0.611800
530530 1.10628e8 0.0322774
531531 7.57072e8 0.219435
532532 −1.36746e9 −0.393754
533533 −5.80684e7 −0.0166110
534534 5.06088e7 0.0143824
535535 1.48039e8 0.0417962
536536 −5.96497e9 −1.67314
537537 5.17889e7 0.0144320
538538 2.26996e9 0.628463
539539 1.35914e9 0.373855
540540 −4.09367e7 −0.0111875
541541 −5.44753e9 −1.47914 −0.739570 0.673080i 0.764971π-0.764971\pi
−0.739570 + 0.673080i 0.764971π0.764971\pi
542542 −4.63345e9 −1.24999
543543 1.04870e9 0.281095
544544 −2.14145e9 −0.570311
545545 1.39314e8 0.0368644
546546 5.21657e8 0.137155
547547 3.56310e9 0.930834 0.465417 0.885092i 0.345904π-0.345904\pi
0.465417 + 0.885092i 0.345904π0.345904\pi
548548 −6.57715e8 −0.170728
549549 −1.35401e9 −0.349236
550550 −2.28948e9 −0.586770
551551 1.60115e9 0.407757
552552 7.98691e8 0.202112
553553 −1.93893e9 −0.487555
554554 −4.92082e9 −1.22957
555555 6.14440e7 0.0152565
556556 −7.52933e8 −0.185778
557557 5.36288e8 0.131494 0.0657468 0.997836i 0.479057π-0.479057\pi
0.0657468 + 0.997836i 0.479057π0.479057\pi
558558 −4.70706e9 −1.14691
559559 2.19334e9 0.531087
560560 3.94732e7 0.00949826
561561 6.91235e8 0.165293
562562 −5.71580e9 −1.35831
563563 1.04961e9 0.247884 0.123942 0.992289i 0.460446π-0.460446\pi
0.123942 + 0.992289i 0.460446π0.460446\pi
564564 2.73841e8 0.0642721
565565 1.02719e8 0.0239596
566566 −5.02680e8 −0.116529
567567 −2.32956e9 −0.536701
568568 5.21197e9 1.19339
569569 −3.50902e9 −0.798532 −0.399266 0.916835i 0.630735π-0.630735\pi
−0.399266 + 0.916835i 0.630735π0.630735\pi
570570 −4.65008e7 −0.0105171
571571 −5.00548e9 −1.12517 −0.562587 0.826738i 0.690194π-0.690194\pi
−0.562587 + 0.826738i 0.690194π0.690194\pi
572572 −1.42962e9 −0.319399
573573 1.02245e9 0.227040
574574 4.74241e7 0.0104666
575575 2.83544e9 0.621990
576576 −3.86523e9 −0.842746
577577 1.34452e9 0.291375 0.145688 0.989331i 0.453461π-0.453461\pi
0.145688 + 0.989331i 0.453461π0.453461\pi
578578 1.83338e9 0.394916
579579 2.80760e8 0.0601119
580580 3.22451e7 0.00686223
581581 −4.57907e9 −0.968636
582582 −8.07635e8 −0.169818
583583 4.12114e9 0.861345
584584 −5.17381e9 −1.07489
585585 1.54258e8 0.0318568
586586 5.48844e9 1.12670
587587 7.52304e9 1.53518 0.767591 0.640940i 0.221455π-0.221455\pi
0.767591 + 0.640940i 0.221455π0.221455\pi
588588 −3.24123e8 −0.0657489
589589 9.87080e9 1.99044
590590 3.63873e7 0.00729404
591591 1.08311e8 0.0215832
592592 1.92641e9 0.381613
593593 −7.56574e9 −1.48991 −0.744955 0.667114i 0.767529π-0.767529\pi
−0.744955 + 0.667114i 0.767529π0.767529\pi
594594 1.73558e9 0.339776
595595 −1.05420e8 −0.0205170
596596 −2.59500e9 −0.502083
597597 1.39709e9 0.268728
598598 −2.01503e9 −0.385326
599599 5.39804e9 1.02622 0.513112 0.858322i 0.328493π-0.328493\pi
0.513112 + 0.858322i 0.328493π0.328493\pi
600600 1.71336e9 0.323832
601601 −5.41597e9 −1.01769 −0.508845 0.860858i 0.669927π-0.669927\pi
−0.508845 + 0.860858i 0.669927π0.669927\pi
602602 −1.79129e9 −0.334640
603603 7.64053e9 1.41910
604604 1.18321e9 0.218490
605605 −7.91007e7 −0.0145223
606606 1.60000e8 0.0292056
607607 5.29850e9 0.961595 0.480798 0.876832i 0.340347π-0.340347\pi
0.480798 + 0.876832i 0.340347π0.340347\pi
608608 5.36655e9 0.968351
609609 −4.38310e8 −0.0786359
610610 −6.50782e7 −0.0116086
611611 −2.16805e9 −0.384525
612612 1.63143e9 0.287699
613613 −7.31057e8 −0.128186 −0.0640928 0.997944i 0.520415π-0.520415\pi
−0.0640928 + 0.997944i 0.520415π0.520415\pi
614614 −4.02251e9 −0.701307
615615 −1.41698e6 −0.000245641 0
616616 3.66391e9 0.631557
617617 3.25921e9 0.558617 0.279308 0.960201i 0.409895π-0.409895\pi
0.279308 + 0.960201i 0.409895π0.409895\pi
618618 1.58411e8 0.0269976
619619 −7.52009e9 −1.27440 −0.637200 0.770698i 0.719907π-0.719907\pi
−0.637200 + 0.770698i 0.719907π0.719907\pi
620620 1.98785e8 0.0334975
621621 −2.14946e9 −0.360170
622622 −6.57828e9 −1.09609
623623 −2.87525e8 −0.0476396
624624 −4.88675e8 −0.0805145
625625 6.07218e9 0.994866
626626 6.64298e8 0.108231
627627 −1.73226e9 −0.280657
628628 2.92425e9 0.471146
629629 −5.14482e9 −0.824314
630630 −1.25982e8 −0.0200731
631631 9.37784e8 0.148594 0.0742968 0.997236i 0.476329π-0.476329\pi
0.0742968 + 0.997236i 0.476329π0.476329\pi
632632 4.52571e9 0.713143
633633 8.02126e8 0.125698
634634 6.80924e9 1.06117
635635 −4.04876e7 −0.00627501
636636 −9.82793e8 −0.151482
637637 2.56613e9 0.393361
638638 −1.36708e9 −0.208412
639639 −6.67601e9 −1.01219
640640 4.53002e7 0.00683078
641641 −8.70085e9 −1.30484 −0.652422 0.757856i 0.726247π-0.726247\pi
−0.652422 + 0.757856i 0.726247π0.726247\pi
642642 1.49676e9 0.223244
643643 7.11195e9 1.05499 0.527497 0.849557i 0.323131π-0.323131\pi
0.527497 + 0.849557i 0.323131π0.323131\pi
644644 −1.44598e9 −0.213335
645645 5.35218e7 0.00785366
646646 3.89359e9 0.568246
647647 −8.41505e9 −1.22150 −0.610748 0.791825i 0.709131π-0.709131\pi
−0.610748 + 0.791825i 0.709131π0.709131\pi
648648 5.43749e9 0.785029
649649 1.35551e9 0.194647
650650 −4.32268e9 −0.617385
651651 −2.70210e9 −0.383856
652652 2.54935e9 0.360216
653653 −6.82007e9 −0.958502 −0.479251 0.877678i 0.659092π-0.659092\pi
−0.479251 + 0.877678i 0.659092π0.659092\pi
654654 1.40855e9 0.196902
655655 1.71351e8 0.0238255
656656 −4.44257e7 −0.00614428
657657 6.62713e9 0.911689
658658 1.77063e9 0.242291
659659 1.07296e10 1.46045 0.730224 0.683208i 0.239416π-0.239416\pi
0.730224 + 0.683208i 0.239416π0.239416\pi
660660 −3.48855e7 −0.00472325
661661 8.40206e9 1.13157 0.565784 0.824553i 0.308574π-0.308574\pi
0.565784 + 0.824553i 0.308574π0.308574\pi
662662 −2.24696e9 −0.301018
663663 1.30509e9 0.173918
664664 1.06881e10 1.41682
665665 2.64186e8 0.0348365
666666 −6.14829e9 −0.806482
667667 1.69309e9 0.220922
668668 5.34828e9 0.694219
669669 1.82874e8 0.0236135
670670 3.67229e8 0.0471709
671671 −2.42431e9 −0.309785
672672 −1.46908e9 −0.186746
673673 9.11306e9 1.15242 0.576211 0.817301i 0.304531π-0.304531\pi
0.576211 + 0.817301i 0.304531π0.304531\pi
674674 6.76279e9 0.850778
675675 −4.61104e9 −0.577080
676676 1.05732e9 0.131642
677677 2.44302e9 0.302598 0.151299 0.988488i 0.451654π-0.451654\pi
0.151299 + 0.988488i 0.451654π0.451654\pi
678678 1.03855e9 0.127974
679679 4.58844e9 0.562498
680680 2.46064e8 0.0300101
681681 −1.36218e9 −0.165279
682682 −8.42783e9 −1.01735
683683 8.03044e9 0.964422 0.482211 0.876055i 0.339834π-0.339834\pi
0.482211 + 0.876055i 0.339834π0.339834\pi
684684 −4.08843e9 −0.488495
685685 1.27067e8 0.0151048
686686 −6.61193e9 −0.781976
687687 3.59197e9 0.422653
688688 1.67803e9 0.196445
689689 7.78094e9 0.906285
690690 −4.91707e7 −0.00569816
691691 −2.28558e8 −0.0263526 −0.0131763 0.999913i 0.504194π-0.504194\pi
−0.0131763 + 0.999913i 0.504194π0.504194\pi
692692 −1.36106e9 −0.156137
693693 −4.69311e9 −0.535666
694694 −8.91765e8 −0.101273
695695 1.45462e8 0.0164363
696696 1.02307e9 0.115020
697697 1.18647e8 0.0132721
698698 1.91972e9 0.213671
699699 1.79554e9 0.198849
700700 −3.10194e9 −0.341814
701701 1.42983e10 1.56773 0.783864 0.620932i 0.213246π-0.213246\pi
0.783864 + 0.620932i 0.213246π0.213246\pi
702702 3.27688e9 0.357504
703703 1.28931e10 1.39963
704704 −6.92056e9 −0.747544
705705 −5.29046e7 −0.00568632
706706 5.74980e9 0.614945
707707 −9.09014e8 −0.0967392
708708 −3.23258e8 −0.0342320
709709 1.32839e10 1.39979 0.699894 0.714246i 0.253230π-0.253230\pi
0.699894 + 0.714246i 0.253230π0.253230\pi
710710 −3.20871e8 −0.0336454
711711 −5.79698e9 −0.604864
712712 6.71122e8 0.0696821
713713 1.04376e10 1.07841
714714 −1.06586e9 −0.109586
715715 2.76194e8 0.0282581
716716 2.18850e8 0.0222818
717717 2.23693e9 0.226639
718718 4.57190e9 0.460957
719719 −7.10546e9 −0.712920 −0.356460 0.934311i 0.616016π-0.616016\pi
−0.356460 + 0.934311i 0.616016π0.616016\pi
720720 1.18016e8 0.0117836
721721 −8.99985e8 −0.0894256
722722 −2.37919e9 −0.235260
723723 2.15139e8 0.0211707
724724 4.43162e9 0.433988
725725 3.63203e9 0.353970
726726 −7.99756e8 −0.0775674
727727 9.98976e9 0.964239 0.482119 0.876106i 0.339867π-0.339867\pi
0.482119 + 0.876106i 0.339867π0.339867\pi
728728 6.91768e9 0.664509
729729 −5.13309e9 −0.490719
730730 3.18521e8 0.0303046
731731 −4.48148e9 −0.424337
732732 5.78141e8 0.0544810
733733 −3.14329e9 −0.294795 −0.147398 0.989077i 0.547090π-0.547090\pi
−0.147398 + 0.989077i 0.547090π0.547090\pi
734734 7.57714e8 0.0707243
735735 6.26186e7 0.00581698
736736 5.67469e9 0.524650
737737 1.36801e10 1.25879
738738 1.41788e8 0.0129850
739739 −1.52632e10 −1.39120 −0.695599 0.718430i 0.744861π-0.744861\pi
−0.695599 + 0.718430i 0.744861π0.744861\pi
740740 2.59650e8 0.0235547
741741 −3.27061e9 −0.295301
742742 −6.35465e9 −0.571055
743743 1.35283e10 1.20999 0.604994 0.796230i 0.293175π-0.293175\pi
0.604994 + 0.796230i 0.293175π0.293175\pi
744744 6.30706e9 0.561464
745745 5.01340e8 0.0444207
746746 −1.32562e9 −0.116905
747747 −1.36904e10 −1.20170
748748 2.92102e9 0.255199
749749 −8.50360e9 −0.739463
750750 −2.11144e8 −0.0182753
751751 −1.78250e10 −1.53565 −0.767823 0.640662i 0.778660π-0.778660\pi
−0.767823 + 0.640662i 0.778660π0.778660\pi
752752 −1.65868e9 −0.142233
753753 1.91940e9 0.163826
754754 −2.58113e9 −0.219286
755755 −2.28589e8 −0.0193304
756756 2.35147e9 0.197931
757757 −8.14422e9 −0.682361 −0.341180 0.939998i 0.610827π-0.610827\pi
−0.341180 + 0.939998i 0.610827π0.610827\pi
758758 4.86186e9 0.405471
759759 −1.83172e9 −0.152060
760760 −6.16645e8 −0.0509551
761761 1.58821e10 1.30636 0.653180 0.757203i 0.273435π-0.273435\pi
0.653180 + 0.757203i 0.273435π0.273435\pi
762762 −4.09355e8 −0.0335164
763763 −8.00243e9 −0.652208
764764 4.32068e9 0.350530
765765 −3.15183e8 −0.0254535
766766 −9.97056e9 −0.801529
767767 2.55929e9 0.204802
768768 3.98670e9 0.317577
769769 −2.05162e9 −0.162687 −0.0813437 0.996686i 0.525921π-0.525921\pi
−0.0813437 + 0.996686i 0.525921π0.525921\pi
770770 −2.25566e8 −0.0178056
771771 −3.55443e9 −0.279306
772772 1.18644e9 0.0928078
773773 −6.64726e9 −0.517624 −0.258812 0.965928i 0.583331π-0.583331\pi
−0.258812 + 0.965928i 0.583331π0.583331\pi
774774 −5.35558e9 −0.415157
775775 2.23908e10 1.72788
776776 −1.07100e10 −0.822763
777777 −3.52945e9 −0.269919
778778 −3.15091e9 −0.239888
779779 −2.97333e8 −0.0225352
780780 −6.58658e7 −0.00496968
781781 −1.19532e10 −0.897851
782782 4.11716e9 0.307874
783783 −2.75332e9 −0.204970
784784 1.96324e9 0.145501
785785 −5.64949e8 −0.0416836
786786 1.73246e9 0.127258
787787 8.88179e9 0.649515 0.324757 0.945797i 0.394717π-0.394717\pi
0.324757 + 0.945797i 0.394717π0.394717\pi
788788 4.57701e8 0.0333227
789789 3.75048e9 0.271842
790790 −2.78622e8 −0.0201057
791791 −5.90034e9 −0.423896
792792 1.09543e10 0.783515
793793 −4.57725e9 −0.325948
794794 −2.67458e9 −0.189620
795795 1.89870e8 0.0134021
796796 5.90381e9 0.414893
797797 −1.89614e10 −1.32668 −0.663341 0.748317i 0.730862π-0.730862\pi
−0.663341 + 0.748317i 0.730862π0.730862\pi
798798 2.67108e9 0.186070
799799 4.42980e9 0.307235
800800 1.21734e10 0.840616
801801 −8.59639e8 −0.0591020
802802 −9.76997e9 −0.668780
803803 1.18657e10 0.808700
804804 −3.26238e9 −0.221380
805805 2.79355e8 0.0188743
806806 −1.59122e10 −1.07043
807807 3.89592e9 0.260947
808808 2.12176e9 0.141500
809809 8.92048e9 0.592336 0.296168 0.955136i 0.404291π-0.404291\pi
0.296168 + 0.955136i 0.404291π0.404291\pi
810810 −3.34755e8 −0.0221324
811811 2.13219e10 1.40363 0.701815 0.712359i 0.252373π-0.252373\pi
0.701815 + 0.712359i 0.252373π0.252373\pi
812812 −1.85221e9 −0.121407
813813 −7.95238e9 −0.519016
814814 −1.10083e10 −0.715378
815815 −4.92520e8 −0.0318693
816816 9.98470e8 0.0643308
817817 1.12308e10 0.720496
818818 2.35856e9 0.150665
819819 −8.86085e9 −0.563614
820820 −5.98789e6 −0.000379250 0
821821 −2.80733e9 −0.177049 −0.0885244 0.996074i 0.528215π-0.528215\pi
−0.0885244 + 0.996074i 0.528215π0.528215\pi
822822 1.28472e9 0.0806785
823823 −2.51291e10 −1.57137 −0.785684 0.618628i 0.787689π-0.787689\pi
−0.785684 + 0.618628i 0.787689π0.787689\pi
824824 2.10068e9 0.130802
825825 −3.92944e9 −0.243636
826826 −2.09015e9 −0.129047
827827 −5.59383e9 −0.343906 −0.171953 0.985105i 0.555008π-0.555008\pi
−0.171953 + 0.985105i 0.555008π0.555008\pi
828828 −4.32318e9 −0.264665
829829 −2.79239e10 −1.70230 −0.851148 0.524925i 0.824093π-0.824093\pi
−0.851148 + 0.524925i 0.824093π0.824093\pi
830830 −6.58007e8 −0.0399445
831831 −8.44559e9 −0.510536
832832 −1.30664e10 −0.786547
833833 −5.24317e9 −0.314294
834834 1.47071e9 0.0877903
835835 −1.03326e9 −0.0614195
836836 −7.32019e9 −0.433312
837837 −1.69737e10 −1.00055
838838 1.91249e10 1.12265
839839 1.27967e9 0.0748050 0.0374025 0.999300i 0.488092π-0.488092\pi
0.0374025 + 0.999300i 0.488092π0.488092\pi
840840 1.68805e8 0.00982669
841841 −1.50811e10 −0.874275
842842 −1.64358e10 −0.948851
843843 −9.81002e9 −0.563992
844844 3.38963e9 0.194068
845845 −2.04269e8 −0.0116467
846846 5.29381e9 0.300588
847847 4.54368e9 0.256931
848848 5.95287e9 0.335228
849849 −8.62749e8 −0.0483846
850850 8.83217e9 0.493289
851851 1.36334e10 0.758317
852852 2.85055e9 0.157903
853853 −2.24153e10 −1.23658 −0.618290 0.785950i 0.712174π-0.712174\pi
−0.618290 + 0.785950i 0.712174π0.712174\pi
854854 3.73821e9 0.205381
855855 7.89861e8 0.0432184
856856 1.98485e10 1.08161
857857 −7.10390e9 −0.385535 −0.192768 0.981244i 0.561746π-0.561746\pi
−0.192768 + 0.981244i 0.561746π0.561746\pi
858858 2.79249e9 0.150934
859859 −2.60820e10 −1.40399 −0.701995 0.712182i 0.747707π-0.747707\pi
−0.701995 + 0.712182i 0.747707π0.747707\pi
860860 2.26173e8 0.0121254
861861 8.13939e7 0.00434591
862862 1.65705e9 0.0881173
863863 2.10561e10 1.11517 0.557585 0.830120i 0.311728π-0.311728\pi
0.557585 + 0.830120i 0.311728π0.311728\pi
864864 −9.22826e9 −0.486768
865865 2.62948e8 0.0138138
866866 1.77915e10 0.930895
867867 3.14662e9 0.163975
868868 −1.14186e10 −0.592641
869869 −1.03793e10 −0.536535
870870 −6.29847e7 −0.00324278
871871 2.58288e10 1.32447
872872 1.86787e10 0.953981
873873 1.37185e10 0.697840
874874 −1.03177e10 −0.522751
875875 1.19958e9 0.0605343
876876 −2.82968e9 −0.142224
877877 −1.35854e10 −0.680101 −0.340051 0.940407i 0.610444π-0.610444\pi
−0.340051 + 0.940407i 0.610444π0.610444\pi
878878 −8.52948e9 −0.425296
879879 9.41980e9 0.467822
880880 2.11304e8 0.0104525
881881 −6.40798e9 −0.315723 −0.157861 0.987461i 0.550460π-0.550460\pi
−0.157861 + 0.987461i 0.550460π0.550460\pi
882882 −6.26583e9 −0.307495
883883 −2.46152e10 −1.20321 −0.601603 0.798795i 0.705471π-0.705471\pi
−0.601603 + 0.798795i 0.705471π0.705471\pi
884884 5.51506e9 0.268514
885885 6.24515e7 0.00302860
886886 1.31372e10 0.634579
887887 −1.55262e10 −0.747020 −0.373510 0.927626i 0.621846π-0.621846\pi
−0.373510 + 0.927626i 0.621846π0.621846\pi
888888 8.23820e9 0.394809
889889 2.32568e9 0.111018
890890 −4.13171e7 −0.00196456
891891 −1.24704e10 −0.590619
892892 7.72788e8 0.0364572
893893 −1.11012e10 −0.521665
894894 5.06885e9 0.237262
895895 −4.22805e7 −0.00197133
896896 −2.60212e9 −0.120851
897897 −3.45840e9 −0.159993
898898 1.13340e10 0.522293
899899 1.33699e10 0.613717
900900 −9.27414e9 −0.424057
901901 −1.58982e10 −0.724120
902902 2.53867e8 0.0115182
903903 −3.07439e9 −0.138948
904904 1.37722e10 0.620030
905905 −8.56164e8 −0.0383961
906906 −2.31117e9 −0.103249
907907 5.42068e9 0.241228 0.120614 0.992699i 0.461514π-0.461514\pi
0.120614 + 0.992699i 0.461514π0.461514\pi
908908 −5.75629e9 −0.255177
909909 −2.71776e9 −0.120016
910910 −4.25881e8 −0.0187346
911911 6.21006e9 0.272133 0.136067 0.990700i 0.456554π-0.456554\pi
0.136067 + 0.990700i 0.456554π0.456554\pi
912912 −2.50220e9 −0.109230
913913 −2.45123e10 −1.06595
914914 −1.86218e10 −0.806698
915915 −1.11694e8 −0.00482008
916916 1.51790e10 0.652541
917917 −9.84267e9 −0.421522
918918 −6.69538e9 −0.285644
919919 2.46977e10 1.04967 0.524835 0.851204i 0.324127π-0.324127\pi
0.524835 + 0.851204i 0.324127π0.324127\pi
920920 −6.52052e8 −0.0276074
921921 −6.90383e9 −0.291193
922922 1.55945e10 0.655258
923923 −2.25683e10 −0.944696
924924 2.00388e9 0.0835641
925925 2.92466e10 1.21501
926926 2.38066e9 0.0985280
927927 −2.69076e9 −0.110942
928928 7.26893e9 0.298574
929929 −1.03175e10 −0.422201 −0.211100 0.977464i 0.567705π-0.567705\pi
−0.211100 + 0.977464i 0.567705π0.567705\pi
930930 −3.88289e8 −0.0158294
931931 1.31396e10 0.533651
932932 7.58758e9 0.307007
933933 −1.12903e10 −0.455113
934934 −6.16929e9 −0.247754
935935 −5.64325e8 −0.0225782
936936 2.06824e10 0.824395
937937 −3.69314e9 −0.146659 −0.0733293 0.997308i 0.523362π-0.523362\pi
−0.0733293 + 0.997308i 0.523362π0.523362\pi
938938 −2.10942e10 −0.834553
939939 1.14013e9 0.0449393
940940 −2.23564e8 −0.00877921
941941 −4.47084e10 −1.74914 −0.874572 0.484896i 0.838858π-0.838858\pi
−0.874572 + 0.484896i 0.838858π0.838858\pi
942942 −5.71197e9 −0.222642
943943 −3.14405e8 −0.0122095
944944 1.95800e9 0.0757549
945945 −4.54292e8 −0.0175115
946946 −9.58898e9 −0.368259
947947 −3.44488e9 −0.131810 −0.0659050 0.997826i 0.520993π-0.520993\pi
−0.0659050 + 0.997826i 0.520993π0.520993\pi
948948 2.47521e9 0.0943591
949949 2.24030e10 0.850894
950950 −2.21338e10 −0.837573
951951 1.16867e10 0.440615
952952 −1.41343e10 −0.530941
953953 −1.90963e10 −0.714702 −0.357351 0.933970i 0.616320π-0.616320\pi
−0.357351 + 0.933970i 0.616320π0.616320\pi
954954 −1.89990e10 −0.708455
955955 −8.34731e8 −0.0310124
956956 9.45282e9 0.349912
957957 −2.34632e9 −0.0865359
958958 8.19169e9 0.301019
959959 −7.29894e9 −0.267236
960960 −3.18846e8 −0.0116314
961961 5.49102e10 1.99582
962962 −2.07843e10 −0.752702
963963 −2.54240e10 −0.917384
964964 9.09136e8 0.0326858
965965 −2.29213e8 −0.00821096
966966 2.82445e9 0.100812
967967 −2.14635e10 −0.763322 −0.381661 0.924302i 0.624648π-0.624648\pi
−0.381661 + 0.924302i 0.624648π0.624648\pi
968968 −1.06055e10 −0.375811
969969 6.68256e9 0.235944
970970 6.59355e8 0.0231963
971971 −2.34216e10 −0.821012 −0.410506 0.911858i 0.634648π-0.634648\pi
−0.410506 + 0.911858i 0.634648π0.634648\pi
972972 1.07147e10 0.374237
973973 −8.35560e9 −0.290792
974974 2.98724e10 1.03589
975975 −7.41900e9 −0.256348
976976 −3.50186e9 −0.120566
977977 −4.40383e10 −1.51077 −0.755387 0.655279i 0.772551π-0.772551\pi
−0.755387 + 0.655279i 0.772551π0.772551\pi
978978 −4.97967e9 −0.170222
979979 −1.53915e9 −0.0524255
980980 2.64614e8 0.00898094
981981 −2.39256e10 −0.809135
982982 −4.09212e8 −0.0137898
983983 3.83832e10 1.28885 0.644427 0.764666i 0.277096π-0.277096\pi
0.644427 + 0.764666i 0.277096π0.277096\pi
984984 −1.89984e8 −0.00635674
985985 −8.84252e7 −0.00294815
986986 5.27382e9 0.175209
987987 3.03893e9 0.100603
988988 −1.38209e10 −0.455920
989989 1.18756e10 0.390363
990990 −6.74394e8 −0.0220897
991991 3.50069e10 1.14260 0.571302 0.820740i 0.306438π-0.306438\pi
0.571302 + 0.820740i 0.306438π0.306438\pi
992992 4.48116e10 1.45747
993993 −3.85645e9 −0.124987
994994 1.84314e10 0.595257
995995 −1.14058e9 −0.0367067
996996 5.84560e9 0.187465
997997 4.69124e10 1.49918 0.749592 0.661900i 0.230250π-0.230250\pi
0.749592 + 0.661900i 0.230250π0.230250\pi
998998 −1.74811e9 −0.0556690
999999 −2.21708e10 −0.703563
Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 197.8.a.b.1.20 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
197.8.a.b.1.20 60 1.1 even 1 trivial