Properties

Label 538.4.b.a.537.17
Level 538538
Weight 44
Character 538.537
Analytic conductor 31.74331.743
Analytic rank 00
Dimension 6868
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,4,Mod(537,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.537");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 538=2269 538 = 2 \cdot 269
Weight: k k == 4 4
Character orbit: [χ][\chi] == 538.b (of order 22, degree 11, minimal)

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: no
Analytic conductor: 31.743027583131.7430275831
Analytic rank: 00
Dimension: 6868
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 537.17
Character χ\chi == 538.537
Dual form 538.4.b.a.537.52

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) == q2.00000iq20.122336iq34.00000q4+7.26358q50.244671q6+31.0626iq7+8.00000iq8+26.9850q914.5272iq103.20731q11+0.489343iq1210.0275q13+62.1251q140.888595iq15+16.0000q1627.1710iq1753.9701iq18+41.3494iq1929.0543q20+3.80006q21+6.41461iq22160.747q23+0.978685q2472.2404q25+20.0551iq266.60430iq27124.250iq28+265.809iq291.77719q30+85.9313iq3132.0000iq32+0.392368iq3354.3420q34+225.625iq35107.940q36320.328q37+82.6987q38+1.22673iq39+58.1086iq40357.805q417.60012iq42+263.079q43+12.8292q44+196.008q45+321.494iq46+577.976q471.95737iq48621.883q49+144.481iq503.32398q51+40.1102q52482.084q5313.2086q5423.2965q55248.501q56+5.05850q57+531.617q58377.297iq59+3.55438iq60+695.295q61+171.863q62+838.225iq6364.0000q6472.8359q65+0.784736q66+383.312q67+108.684iq68+19.6651iq69+451.251q70+1054.15iq71+215.880iq72+685.580q73+640.656iq74+8.83758iq75165.397iq7699.6271iq77+2.45345q78707.008q79+116.217q80+727.788q81+715.609iq8266.6066iq8315.2002q84197.359iq85526.158iq86+32.5179q8725.6584iq881138.10q89392.016iq90311.481iq91+642.988q92+10.5125q931155.95iq94+300.344iq953.91474q96+29.1175q97+1243.77iq9886.5492q99+O(q100)q-2.00000i q^{2} -0.122336i q^{3} -4.00000 q^{4} +7.26358 q^{5} -0.244671 q^{6} +31.0626i q^{7} +8.00000i q^{8} +26.9850 q^{9} -14.5272i q^{10} -3.20731 q^{11} +0.489343i q^{12} -10.0275 q^{13} +62.1251 q^{14} -0.888595i q^{15} +16.0000 q^{16} -27.1710i q^{17} -53.9701i q^{18} +41.3494i q^{19} -29.0543 q^{20} +3.80006 q^{21} +6.41461i q^{22} -160.747 q^{23} +0.978685 q^{24} -72.2404 q^{25} +20.0551i q^{26} -6.60430i q^{27} -124.250i q^{28} +265.809i q^{29} -1.77719 q^{30} +85.9313i q^{31} -32.0000i q^{32} +0.392368i q^{33} -54.3420 q^{34} +225.625i q^{35} -107.940 q^{36} -320.328 q^{37} +82.6987 q^{38} +1.22673i q^{39} +58.1086i q^{40} -357.805 q^{41} -7.60012i q^{42} +263.079 q^{43} +12.8292 q^{44} +196.008 q^{45} +321.494i q^{46} +577.976 q^{47} -1.95737i q^{48} -621.883 q^{49} +144.481i q^{50} -3.32398 q^{51} +40.1102 q^{52} -482.084 q^{53} -13.2086 q^{54} -23.2965 q^{55} -248.501 q^{56} +5.05850 q^{57} +531.617 q^{58} -377.297i q^{59} +3.55438i q^{60} +695.295 q^{61} +171.863 q^{62} +838.225i q^{63} -64.0000 q^{64} -72.8359 q^{65} +0.784736 q^{66} +383.312 q^{67} +108.684i q^{68} +19.6651i q^{69} +451.251 q^{70} +1054.15i q^{71} +215.880i q^{72} +685.580 q^{73} +640.656i q^{74} +8.83758i q^{75} -165.397i q^{76} -99.6271i q^{77} +2.45345 q^{78} -707.008 q^{79} +116.217 q^{80} +727.788 q^{81} +715.609i q^{82} -66.6066i q^{83} -15.2002 q^{84} -197.359i q^{85} -526.158i q^{86} +32.5179 q^{87} -25.6584i q^{88} -1138.10 q^{89} -392.016i q^{90} -311.481i q^{91} +642.988 q^{92} +10.5125 q^{93} -1155.95i q^{94} +300.344i q^{95} -3.91474 q^{96} +29.1175 q^{97} +1243.77i q^{98} -86.5492 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 68q272q4+38q54q6594q9+18q11114q13+8q14+1088q16152q2020q21224q23+16q24+1098q25+384q30600q34+2376q36++3370q99+O(q100) 68 q - 272 q^{4} + 38 q^{5} - 4 q^{6} - 594 q^{9} + 18 q^{11} - 114 q^{13} + 8 q^{14} + 1088 q^{16} - 152 q^{20} - 20 q^{21} - 224 q^{23} + 16 q^{24} + 1098 q^{25} + 384 q^{30} - 600 q^{34} + 2376 q^{36}+ \cdots + 3370 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/538Z)×\left(\mathbb{Z}/538\mathbb{Z}\right)^\times.

nn 271271
χ(n)\chi(n) 1-1

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 2.00000i 0.707107i
33 0.122336i 0.0235435i −0.999931 0.0117718i 0.996253π-0.996253\pi
0.999931 0.0117718i 0.00374715π-0.00374715\pi
44 −4.00000 −0.500000
55 7.26358 0.649674 0.324837 0.945770i 0.394690π-0.394690\pi
0.324837 + 0.945770i 0.394690π0.394690\pi
66 −0.244671 −0.0166478
77 31.0626i 1.67722i 0.544731 + 0.838611i 0.316632π0.316632\pi
−0.544731 + 0.838611i 0.683368π0.683368\pi
88 8.00000i 0.353553i
99 26.9850 0.999446
1010 14.5272i 0.459389i
1111 −3.20731 −0.0879126 −0.0439563 0.999033i 0.513996π-0.513996\pi
−0.0439563 + 0.999033i 0.513996π0.513996\pi
1212 0.489343i 0.0117718i
1313 −10.0275 −0.213934 −0.106967 0.994263i 0.534114π-0.534114\pi
−0.106967 + 0.994263i 0.534114π0.534114\pi
1414 62.1251 1.18597
1515 0.888595i 0.0152956i
1616 16.0000 0.250000
1717 27.1710i 0.387643i −0.981037 0.193821i 0.937912π-0.937912\pi
0.981037 0.193821i 0.0620883π-0.0620883\pi
1818 53.9701i 0.706715i
1919 41.3494i 0.499273i 0.968340 + 0.249637i 0.0803112π0.0803112\pi
−0.968340 + 0.249637i 0.919689π0.919689\pi
2020 −29.0543 −0.324837
2121 3.80006 0.0394877
2222 6.41461i 0.0621636i
2323 −160.747 −1.45731 −0.728654 0.684882i 0.759854π-0.759854\pi
−0.728654 + 0.684882i 0.759854π0.759854\pi
2424 0.978685 0.00832389
2525 −72.2404 −0.577923
2626 20.0551i 0.151274i
2727 6.60430i 0.0470740i
2828 124.250i 0.838611i
2929 265.809i 1.70205i 0.525127 + 0.851024i 0.324018π0.324018\pi
−0.525127 + 0.851024i 0.675982π0.675982\pi
3030 −1.77719 −0.0108156
3131 85.9313i 0.497862i 0.968521 + 0.248931i 0.0800792π0.0800792\pi
−0.968521 + 0.248931i 0.919921π0.919921\pi
3232 32.0000i 0.176777i
3333 0.392368i 0.00206977i
3434 −54.3420 −0.274105
3535 225.625i 1.08965i
3636 −107.940 −0.499723
3737 −320.328 −1.42329 −0.711644 0.702541i 0.752049π-0.752049\pi
−0.711644 + 0.702541i 0.752049π0.752049\pi
3838 82.6987 0.353040
3939 1.22673i 0.00503676i
4040 58.1086i 0.229695i
4141 −357.805 −1.36292 −0.681460 0.731856i 0.738654π-0.738654\pi
−0.681460 + 0.731856i 0.738654π0.738654\pi
4242 7.60012i 0.0279220i
4343 263.079 0.933004 0.466502 0.884520i 0.345514π-0.345514\pi
0.466502 + 0.884520i 0.345514π0.345514\pi
4444 12.8292 0.0439563
4545 196.008 0.649314
4646 321.494i 1.03047i
4747 577.976 1.79375 0.896877 0.442280i 0.145830π-0.145830\pi
0.896877 + 0.442280i 0.145830π0.145830\pi
4848 1.95737i 0.00588588i
4949 −621.883 −1.81307
5050 144.481i 0.408653i
5151 −3.32398 −0.00912648
5252 40.1102 0.106967
5353 −482.084 −1.24942 −0.624711 0.780856i 0.714783π-0.714783\pi
−0.624711 + 0.780856i 0.714783π0.714783\pi
5454 −13.2086 −0.0332863
5555 −23.2965 −0.0571146
5656 −248.501 −0.592987
5757 5.05850 0.0117546
5858 531.617 1.20353
5959 377.297i 0.832539i −0.909241 0.416270i 0.863337π-0.863337\pi
0.909241 0.416270i 0.136663π-0.136663\pi
6060 3.55438i 0.00764781i
6161 695.295 1.45940 0.729700 0.683767i 0.239660π-0.239660\pi
0.729700 + 0.683767i 0.239660π0.239660\pi
6262 171.863 0.352041
6363 838.225i 1.67629i
6464 −64.0000 −0.125000
6565 −72.8359 −0.138987
6666 0.784736 0.00146355
6767 383.312 0.698940 0.349470 0.936948i 0.386362π-0.386362\pi
0.349470 + 0.936948i 0.386362π0.386362\pi
6868 108.684i 0.193821i
6969 19.6651i 0.0343102i
7070 451.251 0.770497
7171 1054.15i 1.76204i 0.473081 + 0.881019i 0.343142π0.343142\pi
−0.473081 + 0.881019i 0.656858π0.656858\pi
7272 215.880i 0.353357i
7373 685.580 1.09919 0.549596 0.835430i 0.314782π-0.314782\pi
0.549596 + 0.835430i 0.314782π0.314782\pi
7474 640.656i 1.00642i
7575 8.83758i 0.0136063i
7676 165.397i 0.249637i
7777 99.6271i 0.147449i
7878 2.45345 0.00356153
7979 −707.008 −1.00689 −0.503447 0.864026i 0.667935π-0.667935\pi
−0.503447 + 0.864026i 0.667935π0.667935\pi
8080 116.217 0.162419
8181 727.788 0.998337
8282 715.609i 0.963729i
8383 66.6066i 0.0880847i −0.999030 0.0440423i 0.985976π-0.985976\pi
0.999030 0.0440423i 0.0140236π-0.0140236\pi
8484 −15.2002 −0.0197438
8585 197.359i 0.251842i
8686 526.158i 0.659734i
8787 32.5179 0.0400722
8888 25.6584i 0.0310818i
8989 −1138.10 −1.35549 −0.677746 0.735296i 0.737043π-0.737043\pi
−0.677746 + 0.735296i 0.737043π0.737043\pi
9090 392.016i 0.459134i
9191 311.481i 0.358815i
9292 642.988 0.728654
9393 10.5125 0.0117214
9494 1155.95i 1.26838i
9595 300.344i 0.324365i
9696 −3.91474 −0.00416194
9797 29.1175 0.0304787 0.0152393 0.999884i 0.495149π-0.495149\pi
0.0152393 + 0.999884i 0.495149π0.495149\pi
9898 1243.77i 1.28203i
9999 −86.5492 −0.0878639
100100 288.962 0.288962
101101 691.301i 0.681059i 0.940234 + 0.340530i 0.110606π0.110606\pi
−0.940234 + 0.340530i 0.889394π0.889394\pi
102102 6.64796i 0.00645339i
103103 495.942 0.474433 0.237217 0.971457i 0.423765π-0.423765\pi
0.237217 + 0.971457i 0.423765π0.423765\pi
104104 80.2204i 0.0756371i
105105 27.6020 0.0256541
106106 964.169i 0.883475i
107107 1846.33i 1.66814i 0.551657 + 0.834071i 0.313996π0.313996\pi
−0.551657 + 0.834071i 0.686004π0.686004\pi
108108 26.4172i 0.0235370i
109109 1176.18i 1.03356i −0.856118 0.516780i 0.827130π-0.827130\pi
0.856118 0.516780i 0.172870π-0.172870\pi
110110 46.5930i 0.0403861i
111111 39.1876i 0.0335092i
112112 497.001i 0.419305i
113113 1643.91i 1.36855i 0.729225 + 0.684274i 0.239881π0.239881\pi
−0.729225 + 0.684274i 0.760119π0.760119\pi
114114 10.1170i 0.00831179i
115115 −1167.60 −0.946776
116116 1063.23i 0.851024i
117117 −270.594 −0.213815
118118 −754.593 −0.588694
119119 844.000 0.650163
120120 7.10876 0.00540782
121121 −1320.71 −0.992271
122122 1390.59i 1.03195i
123123 43.7723i 0.0320879i
124124 343.725i 0.248931i
125125 −1432.67 −1.02514
126126 1676.45 1.18532
127127 1520.34 1.06227 0.531136 0.847287i 0.321765π-0.321765\pi
0.531136 + 0.847287i 0.321765π0.321765\pi
128128 128.000i 0.0883883i
129129 32.1840i 0.0219662i
130130 145.672i 0.0982789i
131131 1277.37 0.851939 0.425969 0.904738i 0.359933π-0.359933\pi
0.425969 + 0.904738i 0.359933π0.359933\pi
132132 1.56947i 0.00103489i
133133 −1284.42 −0.837392
134134 766.623i 0.494225i
135135 47.9708i 0.0305828i
136136 217.368 0.137052
137137 2021.91i 1.26090i 0.776230 + 0.630450i 0.217129π0.217129\pi
−0.776230 + 0.630450i 0.782871π0.782871\pi
138138 39.3302 0.0242609
139139 269.209i 0.164274i −0.996621 0.0821368i 0.973826π-0.973826\pi
0.996621 0.0821368i 0.0261744π-0.0261744\pi
140140 902.502i 0.544824i
141141 70.7071i 0.0422313i
142142 2108.30 1.24595
143143 32.1614 0.0188075
144144 431.761 0.249861
145145 1930.72i 1.10578i
146146 1371.16i 0.777247i
147147 76.0785i 0.0426861i
148148 1281.31 0.711644
149149 280.768 0.154372 0.0771860 0.997017i 0.475406π-0.475406\pi
0.0771860 + 0.997017i 0.475406π0.475406\pi
150150 17.6752 0.00962114
151151 2271.80 1.22435 0.612174 0.790723i 0.290295π-0.290295\pi
0.612174 + 0.790723i 0.290295π0.290295\pi
152152 −330.795 −0.176520
153153 733.210i 0.387428i
154154 −199.254 −0.104262
155155 624.169i 0.323448i
156156 4.90691i 0.00251838i
157157 162.060i 0.0823808i −0.999151 0.0411904i 0.986885π-0.986885\pi
0.999151 0.0411904i 0.0131150π-0.0131150\pi
158158 1414.02i 0.711981i
159159 58.9761i 0.0294158i
160160 232.435i 0.114847i
161161 4993.22i 2.44423i
162162 1455.58i 0.705931i
163163 2598.91i 1.24885i −0.781085 0.624425i 0.785333π-0.785333\pi
0.781085 0.624425i 0.214667π-0.214667\pi
164164 1431.22 0.681460
165165 2.85000i 0.00134468i
166166 −133.213 −0.0622853
167167 436.551i 0.202283i −0.994872 0.101142i 0.967750π-0.967750\pi
0.994872 0.101142i 0.0322495π-0.0322495\pi
168168 30.4005i 0.0139610i
169169 −2096.45 −0.954232
170170 −394.717 −0.178079
171171 1115.81i 0.498997i
172172 −1052.32 −0.466502
173173 4047.07 1.77857 0.889286 0.457351i 0.151202π-0.151202\pi
0.889286 + 0.457351i 0.151202π0.151202\pi
174174 65.0357i 0.0283353i
175175 2243.97i 0.969305i
176176 −51.3169 −0.0219782
177177 −46.1568 −0.0196009
178178 2276.21i 0.958478i
179179 1628.93i 0.680179i −0.940393 0.340089i 0.889543π-0.889543\pi
0.940393 0.340089i 0.110457π-0.110457\pi
180180 −784.032 −0.324657
181181 3241.10i 1.33099i 0.746403 + 0.665495i 0.231779π0.231779\pi
−0.746403 + 0.665495i 0.768221π0.768221\pi
182182 −622.963 −0.253720
183183 85.0594i 0.0343594i
184184 1285.98i 0.515236i
185185 −2326.73 −0.924673
186186 21.0249i 0.00828829i
187187 87.1456i 0.0340787i
188188 −2311.90 −0.896877
189189 205.146 0.0789535
190190 600.689 0.229361
191191 −3075.70 −1.16518 −0.582591 0.812765i 0.697961π-0.697961\pi
−0.582591 + 0.812765i 0.697961π0.697961\pi
192192 7.82948i 0.00294294i
193193 1969.70i 0.734621i −0.930098 0.367310i 0.880279π-0.880279\pi
0.930098 0.367310i 0.119721π-0.119721\pi
194194 58.2350i 0.0215517i
195195 8.91043i 0.00327225i
196196 2487.53 0.906535
197197 1090.38i 0.394347i 0.980369 + 0.197174i 0.0631762π0.0631762\pi
−0.980369 + 0.197174i 0.936824π0.936824\pi
198198 173.098i 0.0621292i
199199 1851.20 0.659436 0.329718 0.944079i 0.393046π-0.393046\pi
0.329718 + 0.944079i 0.393046π0.393046\pi
200200 577.923i 0.204327i
201201 46.8927i 0.0164555i
202202 1382.60 0.481582
203203 −8256.70 −2.85471
204204 13.2959 0.00456324
205205 −2598.94 −0.885454
206206 991.884i 0.335475i
207207 −4337.77 −1.45650
208208 −160.441 −0.0534835
209209 132.620i 0.0438924i
210210 55.2041i 0.0181402i
211211 −2123.86 −0.692952 −0.346476 0.938059i 0.612622π-0.612622\pi
−0.346476 + 0.938059i 0.612622π0.612622\pi
212212 1928.34 0.624711
213213 128.960 0.0414846
214214 3692.65 1.17955
215215 1910.90 0.606149
216216 52.8344 0.0166432
217217 −2669.25 −0.835024
218218 −2352.37 −0.730838
219219 83.8709i 0.0258789i
220220 93.1861 0.0285573
221221 272.458i 0.0829300i
222222 78.3751 0.0236946
223223 2167.13i 0.650770i 0.945582 + 0.325385i 0.105494π0.105494\pi
−0.945582 + 0.325385i 0.894506π0.894506\pi
224224 994.002 0.296494
225225 −1949.41 −0.577603
226226 3287.82 0.967709
227227 4523.56i 1.32264i 0.750104 + 0.661320i 0.230004π0.230004\pi
−0.750104 + 0.661320i 0.769996π0.769996\pi
228228 −20.2340 −0.00587732
229229 493.569i 0.142428i −0.997461 0.0712139i 0.977313π-0.977313\pi
0.997461 0.0712139i 0.0226873π-0.0226873\pi
230230 2335.20i 0.669471i
231231 −12.1880 −0.00347147
232232 −2126.47 −0.601765
233233 −417.669 −0.117435 −0.0587177 0.998275i 0.518701π-0.518701\pi
−0.0587177 + 0.998275i 0.518701π0.518701\pi
234234 541.188i 0.151190i
235235 4198.17 1.16536
236236 1509.19i 0.416270i
237237 86.4923i 0.0237058i
238238 1688.00i 0.459735i
239239 −1920.77 −0.519850 −0.259925 0.965629i 0.583698π-0.583698\pi
−0.259925 + 0.965629i 0.583698π0.583698\pi
240240 14.2175i 0.00382390i
241241 3148.64i 0.841584i −0.907157 0.420792i 0.861752π-0.861752\pi
0.907157 0.420792i 0.138248π-0.138248\pi
242242 2641.43i 0.701642i
243243 267.350i 0.0705783i
244244 −2781.18 −0.729700
245245 −4517.10 −1.17791
246246 87.5445 0.0226896
247247 414.633i 0.106812i
248248 −687.450 −0.176021
249249 −8.14837 −0.00207382
250250 2865.34i 0.724881i
251251 1308.03i 0.328934i 0.986383 + 0.164467i 0.0525904π0.0525904\pi
−0.986383 + 0.164467i 0.947410π0.947410\pi
252252 3352.90i 0.838146i
253253 515.565 0.128116
254254 3040.68i 0.751139i
255255 −24.1440 −0.00592924
256256 256.000 0.0625000
257257 8167.52i 1.98240i −0.132382 0.991199i 0.542263π-0.542263\pi
0.132382 0.991199i 0.457737π-0.457737\pi
258258 −64.3679 −0.0155325
259259 9950.22i 2.38717i
260260 291.344 0.0694937
261261 7172.85i 1.70110i
262262 2554.73i 0.602412i
263263 −185.643 −0.0435256 −0.0217628 0.999763i 0.506928π-0.506928\pi
−0.0217628 + 0.999763i 0.506928π0.506928\pi
264264 −3.13894 −0.000731775
265265 −3501.66 −0.811718
266266 2568.83i 0.592125i
267267 139.231i 0.0319131i
268268 −1533.25 −0.349470
269269 4197.58 + 1358.48i 0.951416 + 0.307910i
270270 −95.9417 −0.0216253
271271 5940.84i 1.33166i −0.746103 0.665831i 0.768077π-0.768077\pi
0.746103 0.665831i 0.231923π-0.231923\pi
272272 434.736i 0.0969107i
273273 −38.1053 −0.00844776
274274 4043.82 0.891591
275275 231.697 0.0508068
276276 78.6604i 0.0171551i
277277 1194.15i 0.259022i 0.991578 + 0.129511i 0.0413408π0.0413408\pi
−0.991578 + 0.129511i 0.958659π0.958659\pi
278278 −538.418 −0.116159
279279 2318.86i 0.497586i
280280 −1805.00 −0.385249
281281 6228.31i 1.32224i −0.750280 0.661120i 0.770081π-0.770081\pi
0.750280 0.661120i 0.229919π-0.229919\pi
282282 −141.414 −0.0298620
283283 −5882.70 −1.23566 −0.617828 0.786314i 0.711987π-0.711987\pi
−0.617828 + 0.786314i 0.711987π0.711987\pi
284284 4216.60i 0.881019i
285285 36.7428 0.00763669
286286 64.3228i 0.0132989i
287287 11114.3i 2.28592i
288288 863.521i 0.176679i
289289 4174.74 0.849733
290290 3861.44 0.781902
291291 3.56211i 0.000717575i
292292 −2742.32 −0.549596
293293 −2627.25 −0.523841 −0.261921 0.965089i 0.584356π-0.584356\pi
−0.261921 + 0.965089i 0.584356π0.584356\pi
294294 152.157 0.0301836
295295 2740.52i 0.540879i
296296 2562.62i 0.503208i
297297 21.1820i 0.00413840i
298298 561.537i 0.109158i
299299 1611.90 0.311768
300300 35.3503i 0.00680317i
301301 8171.91i 1.56485i
302302 4543.60i 0.865745i
303303 84.5708 0.0160345
304304 661.590i 0.124818i
305305 5050.33 0.948135
306306 −1466.42 −0.273953
307307 2129.41 0.395870 0.197935 0.980215i 0.436577π-0.436577\pi
0.197935 + 0.980215i 0.436577π0.436577\pi
308308 398.509i 0.0737245i
309309 60.6714i 0.0111698i
310310 1248.34 0.228712
311311 6693.56i 1.22044i −0.792232 0.610220i 0.791081π-0.791081\pi
0.792232 0.610220i 0.208919π-0.208919\pi
312312 −9.81382 −0.00178076
313313 6519.74 1.17737 0.588686 0.808362i 0.299646π-0.299646\pi
0.588686 + 0.808362i 0.299646π0.299646\pi
314314 −324.120 −0.0582520
315315 6088.51i 1.08904i
316316 2828.03 0.503447
317317 5992.74i 1.06178i −0.847439 0.530892i 0.821857π-0.821857\pi
0.847439 0.530892i 0.178143π-0.178143\pi
318318 117.952 0.0208001
319319 852.529i 0.149632i
320320 −464.869 −0.0812093
321321 225.872 0.0392739
322322 −9986.44 −1.72833
323323 1123.50 0.193540
324324 −2911.15 −0.499169
325325 724.394 0.123637
326326 −5197.82 −0.883070
327327 −143.889 −0.0243336
328328 2862.44i 0.481865i
329329 17953.4i 3.00852i
330330 5.69999 0.000950831
331331 7466.81 1.23992 0.619959 0.784634i 0.287149π-0.287149\pi
0.619959 + 0.784634i 0.287149π0.287149\pi
332332 266.427i 0.0440423i
333333 −8644.07 −1.42250
334334 −873.102 −0.143036
335335 2784.21 0.454083
336336 60.8010 0.00987192
337337 308.183i 0.0498154i 0.999690 + 0.0249077i 0.00792919π0.00792919\pi
−0.999690 + 0.0249077i 0.992071π0.992071\pi
338338 4192.90i 0.674744i
339339 201.109 0.0322204
340340 789.434i 0.125921i
341341 275.608i 0.0437683i
342342 2231.63 0.352844
343343 8662.83i 1.36370i
344344 2104.63i 0.329867i
345345 142.839i 0.0222904i
346346 8094.14i 1.25764i
347347 2712.41 0.419625 0.209812 0.977742i 0.432715π-0.432715\pi
0.209812 + 0.977742i 0.432715π0.432715\pi
348348 −130.071 −0.0200361
349349 −5057.24 −0.775668 −0.387834 0.921729i 0.626777π-0.626777\pi
−0.387834 + 0.921729i 0.626777π0.626777\pi
350350 −4487.95 −0.685402
351351 66.2249i 0.0100707i
352352 102.634i 0.0155409i
353353 7714.10 1.16312 0.581558 0.813505i 0.302443π-0.302443\pi
0.581558 + 0.813505i 0.302443π0.302443\pi
354354 92.3137i 0.0138599i
355355 7656.91i 1.14475i
356356 4552.42 0.677746
357357 103.251i 0.0153071i
358358 −3257.86 −0.480959
359359 12270.4i 1.80392i 0.431823 + 0.901959i 0.357871π0.357871\pi
−0.431823 + 0.901959i 0.642129π0.642129\pi
360360 1568.06i 0.229567i
361361 5149.23 0.750726
362362 6482.20 0.941151
363363 161.570i 0.0233616i
364364 1245.93i 0.179407i
365365 4979.76 0.714117
366366 −170.119 −0.0242958
367367 2970.62i 0.422520i 0.977430 + 0.211260i 0.0677567π0.0677567\pi
−0.977430 + 0.211260i 0.932243π0.932243\pi
368368 −2571.95 −0.364327
369369 −9655.37 −1.36216
370370 4653.46i 0.653843i
371371 14974.8i 2.09556i
372372 −42.0498 −0.00586071
373373 9243.34i 1.28312i 0.767075 + 0.641558i 0.221712π0.221712\pi
−0.767075 + 0.641558i 0.778288π0.778288\pi
374374 174.291 0.0240973
375375 175.267i 0.0241353i
376376 4623.81i 0.634188i
377377 2665.41i 0.364126i
378378 410.293i 0.0558285i
379379 13810.3i 1.87174i −0.352349 0.935869i 0.614617π-0.614617\pi
0.352349 0.935869i 0.385383π-0.385383\pi
380380 1201.38i 0.162183i
381381 185.992i 0.0250096i
382382 6151.40i 0.823908i
383383 13891.0i 1.85326i 0.375975 + 0.926630i 0.377308π0.377308\pi
−0.375975 + 0.926630i 0.622692π0.622692\pi
384384 15.6590 0.00208097
385385 723.650i 0.0957938i
386386 −3939.39 −0.519455
387387 7099.20 0.932487
388388 −116.470 −0.0152393
389389 −7854.08 −1.02370 −0.511848 0.859076i 0.671039π-0.671039\pi
−0.511848 + 0.859076i 0.671039π0.671039\pi
390390 17.8209 0.00231383
391391 4367.66i 0.564915i
392392 4975.07i 0.641017i
393393 156.267i 0.0200576i
394394 2180.76 0.278846
395395 −5135.41 −0.654153
396396 346.197 0.0439319
397397 9957.88i 1.25887i 0.777053 + 0.629436i 0.216714π0.216714\pi
−0.777053 + 0.629436i 0.783286π0.783286\pi
398398 3702.39i 0.466292i
399399 157.130i 0.0197151i
400400 −1155.85 −0.144481
401401 8394.48i 1.04539i −0.852520 0.522694i 0.824927π-0.824927\pi
0.852520 0.522694i 0.175073π-0.175073\pi
402402 −93.7854 −0.0116358
403403 861.680i 0.106510i
404404 2765.20i 0.340530i
405405 5286.35 0.648594
406406 16513.4i 2.01859i
407407 1027.39 0.125125
408408 26.5918i 0.00322670i
409409 3999.23i 0.483495i −0.970339 0.241747i 0.922279π-0.922279\pi
0.970339 0.241747i 0.0777205π-0.0777205\pi
410410 5197.88i 0.626110i
411411 247.351 0.0296860
412412 −1983.77 −0.237217
413413 11719.8 1.39635
414414 8675.53i 1.02990i
415415 483.803i 0.0572264i
416416 320.882i 0.0378185i
417417 −32.9339 −0.00386758
418418 −265.240 −0.0310366
419419 12862.8 1.49974 0.749870 0.661585i 0.230116π-0.230116\pi
0.749870 + 0.661585i 0.230116π0.230116\pi
420420 −110.408 −0.0128271
421421 6298.92 0.729194 0.364597 0.931165i 0.381207π-0.381207\pi
0.364597 + 0.931165i 0.381207π0.381207\pi
422422 4247.73i 0.489991i
423423 15596.7 1.79276
424424 3856.68i 0.441738i
425425 1962.84i 0.224028i
426426 257.921i 0.0293340i
427427 21597.7i 2.44774i
428428 7385.31i 0.834071i
429429 3.93449i 0.000442795i
430430 3821.79i 0.428612i
431431 6881.85i 0.769112i −0.923102 0.384556i 0.874355π-0.874355\pi
0.923102 0.384556i 0.125645π-0.125645\pi
432432 105.669i 0.0117685i
433433 −8331.33 −0.924661 −0.462330 0.886708i 0.652987π-0.652987\pi
−0.462330 + 0.886708i 0.652987π0.652987\pi
434434 5338.49i 0.590451i
435435 236.196 0.0260339
436436 4704.74i 0.516780i
437437 6646.79i 0.727595i
438438 −167.742 −0.0182991
439439 16612.5 1.80609 0.903045 0.429545i 0.141326π-0.141326\pi
0.903045 + 0.429545i 0.141326π0.141326\pi
440440 186.372i 0.0201931i
441441 −16781.5 −1.81207
442442 544.917 0.0586404
443443 4223.25i 0.452941i −0.974018 0.226470i 0.927281π-0.927281\pi
0.974018 0.226470i 0.0727186π-0.0727186\pi
444444 156.750i 0.0167546i
445445 −8266.71 −0.880629
446446 4334.26 0.460164
447447 34.3480i 0.00363446i
448448 1988.00i 0.209653i
449449 −7728.23 −0.812289 −0.406145 0.913809i 0.633127π-0.633127\pi
−0.406145 + 0.913809i 0.633127π0.633127\pi
450450 3898.82i 0.408427i
451451 1147.59 0.119818
452452 6575.63i 0.684274i
453453 277.922i 0.0288255i
454454 9047.13 0.935248
455455 2262.47i 0.233113i
456456 40.4680i 0.00415590i
457457 17690.5 1.81078 0.905388 0.424586i 0.139580π-0.139580\pi
0.905388 + 0.424586i 0.139580π0.139580\pi
458458 −987.138 −0.100712
459459 −179.445 −0.0182479
460460 4670.40 0.473388
461461 7931.13i 0.801279i −0.916236 0.400639i 0.868788π-0.868788\pi
0.916236 0.400639i 0.131212π-0.131212\pi
462462 24.3759i 0.00245470i
463463 427.435i 0.0429040i 0.999770 + 0.0214520i 0.00682891π0.00682891\pi
−0.999770 + 0.0214520i 0.993171π0.993171\pi
464464 4252.94i 0.425512i
465465 76.3581 0.00761510
466466 835.339i 0.0830393i
467467 13725.1i 1.36000i 0.733212 + 0.680000i 0.238020π0.238020\pi
−0.733212 + 0.680000i 0.761980π0.761980\pi
468468 1082.38 0.106908
469469 11906.6i 1.17228i
470470 8396.35i 0.824031i
471471 −19.8257 −0.00193953
472472 3018.37 0.294347
473473 −843.775 −0.0820229
474474 172.985 0.0167625
475475 2987.09i 0.288542i
476476 −3376.00 −0.325081
477477 −13009.1 −1.24873
478478 3841.54i 0.367590i
479479 7925.52i 0.756005i −0.925805 0.378002i 0.876611π-0.876611\pi
0.925805 0.378002i 0.123389π-0.123389\pi
480480 −28.4350 −0.00270391
481481 3212.11 0.304489
482482 −6297.28 −0.595090
483483 −610.849 −0.0575457
484484 5282.85 0.496136
485485 211.497 0.0198012
486486 −534.701 −0.0499064
487487 21429.7 1.99399 0.996993 0.0774867i 0.0246895π-0.0246895\pi
0.996993 + 0.0774867i 0.0246895π0.0246895\pi
488488 5562.36i 0.515976i
489489 −317.940 −0.0294023
490490 9034.20i 0.832905i
491491 −5643.24 −0.518688 −0.259344 0.965785i 0.583506π-0.583506\pi
−0.259344 + 0.965785i 0.583506π0.583506\pi
492492 175.089i 0.0160440i
493493 7222.28 0.659787
494494 −829.265 −0.0755271
495495 −628.657 −0.0570829
496496 1374.90i 0.124465i
497497 −32744.6 −2.95533
498498 16.2967i 0.00146641i
499499 19020.6i 1.70637i 0.521604 + 0.853187i 0.325334π0.325334\pi
−0.521604 + 0.853187i 0.674666π0.674666\pi
500500 5730.69 0.512568
501501 −53.4058 −0.00476246
502502 2616.07 0.232591
503503 10333.1i 0.915965i −0.888961 0.457982i 0.848572π-0.848572\pi
0.888961 0.457982i 0.151428π-0.151428\pi
504504 −6705.80 −0.592659
505505 5021.32i 0.442467i
506506 1031.13i 0.0905915i
507507 256.470i 0.0224660i
508508 −6081.36 −0.531136
509509 11741.1i 1.02243i 0.859453 + 0.511215i 0.170805π0.170805\pi
−0.859453 + 0.511215i 0.829195π0.829195\pi
510510 48.2880i 0.00419260i
511511 21295.9i 1.84359i
512512 512.000i 0.0441942i
513513 273.083 0.0235028
514514 −16335.0 −1.40177
515515 3602.31 0.308227
516516 128.736i 0.0109831i
517517 −1853.74 −0.157694
518518 −19900.4 −1.68798
519519 495.101i 0.0418739i
520520 582.687i 0.0491395i
521521 23455.7i 1.97238i −0.165605 0.986192i 0.552958π-0.552958\pi
0.165605 0.986192i 0.447042π-0.447042\pi
522522 14345.7 1.20286
523523 4737.20i 0.396067i −0.980195 0.198034i 0.936544π-0.936544\pi
0.980195 0.198034i 0.0634555π-0.0634555\pi
524524 −5109.46 −0.425969
525525 −274.518 −0.0228208
526526 371.286i 0.0307772i
527527 2334.84 0.192993
528528 6.27789i 0.000517443i
529529 13672.6 1.12375
530530 7003.32i 0.573971i
531531 10181.4i 0.832078i
532532 5137.67 0.418696
533533 3587.90 0.291575
534534 278.462 0.0225659
535535 13410.9i 1.08375i
536536 3066.49i 0.247112i
537537 −199.276 −0.0160138
538538 2716.95 8395.15i 0.217725 0.672752i
539539 1994.57 0.159392
540540 191.883i 0.0152914i
541541 16504.9i 1.31165i −0.754915 0.655823i 0.772322π-0.772322\pi
0.754915 0.655823i 0.227678π-0.227678\pi
542542 −11881.7 −0.941627
543543 396.502 0.0313362
544544 −869.471 −0.0685262
545545 8543.31i 0.671478i
546546 76.2106i 0.00597347i
547547 2430.36 0.189972 0.0949861 0.995479i 0.469719π-0.469719\pi
0.0949861 + 0.995479i 0.469719π0.469719\pi
548548 8087.63i 0.630450i
549549 18762.6 1.45859
550550 463.394i 0.0359258i
551551 −10991.0 −0.849787
552552 −157.321 −0.0121305
553553 21961.5i 1.68878i
554554 2388.29 0.183157
555555 284.642i 0.0217701i
556556 1076.84i 0.0821368i
557557 19699.4i 1.49854i 0.662263 + 0.749272i 0.269596π0.269596\pi
−0.662263 + 0.749272i 0.730404π0.730404\pi
558558 4637.72 0.351846
559559 −2638.04 −0.199601
560560 3610.01i 0.272412i
561561 10.6610 0.000802333
562562 −12456.6 −0.934965
563563 −10922.1 −0.817606 −0.408803 0.912623i 0.634054π-0.634054\pi
−0.408803 + 0.912623i 0.634054π0.634054\pi
564564 282.828i 0.0211156i
565565 11940.7i 0.889110i
566566 11765.4i 0.873740i
567567 22607.0i 1.67443i
568568 −8433.21 −0.622975
569569 2918.32i 0.215013i −0.994204 0.107506i 0.965713π-0.965713\pi
0.994204 0.107506i 0.0342866π-0.0342866\pi
570570 73.4857i 0.00539996i
571571 7609.77i 0.557722i −0.960332 0.278861i 0.910043π-0.910043\pi
0.960332 0.278861i 0.0899568π-0.0899568\pi
572572 −128.646 −0.00940375
573573 376.268i 0.0274325i
574574 −22228.7 −1.61639
575575 11612.4 0.842212
576576 −1727.04 −0.124931
577577 16744.6i 1.20813i 0.796937 + 0.604063i 0.206452π0.206452\pi
−0.796937 + 0.604063i 0.793548π0.793548\pi
578578 8349.48i 0.600852i
579579 −240.964 −0.0172956
580580 7722.89i 0.552889i
581581 2068.97 0.147737
582582 −7.12421 −0.000507402
583583 1546.19 0.109840
584584 5484.64i 0.388623i
585585 −1965.48 −0.138910
586586 5254.50i 0.370412i
587587 6105.40 0.429296 0.214648 0.976691i 0.431140π-0.431140\pi
0.214648 + 0.976691i 0.431140π0.431140\pi
588588 304.314i 0.0213430i
589589 −3553.20 −0.248569
590590 −5481.05 −0.382460
591591 133.392 0.00928432
592592 −5125.25 −0.355822
593593 −24079.3 −1.66749 −0.833743 0.552152i 0.813807π-0.813807\pi
−0.833743 + 0.552152i 0.813807π0.813807\pi
594594 42.3640 0.00292629
595595 6130.46 0.422394
596596 −1123.07 −0.0771860
597597 226.467i 0.0155254i
598598 3223.80i 0.220453i
599599 −13036.2 −0.889224 −0.444612 0.895723i 0.646658π-0.646658\pi
−0.444612 + 0.895723i 0.646658π0.646658\pi
600600 −70.7006 −0.00481057
601601 23316.0i 1.58250i 0.611495 + 0.791248i 0.290568π0.290568\pi
−0.611495 + 0.791248i 0.709432π0.709432\pi
602602 16343.8 1.10652
603603 10343.7 0.698552
604604 −9087.20 −0.612174
605605 −9593.11 −0.644653
606606 169.142i 0.0113381i
607607 5393.06i 0.360622i −0.983610 0.180311i 0.942290π-0.942290\pi
0.983610 0.180311i 0.0577104π-0.0577104\pi
608608 1323.18 0.0882599
609609 1010.09i 0.0672099i
610610 10100.7i 0.670433i
611611 −5795.68 −0.383745
612612 2932.84i 0.193714i
613613 3038.78i 0.200221i 0.994976 + 0.100110i 0.0319196π0.0319196\pi
−0.994976 + 0.100110i 0.968080π0.968080\pi
614614 4258.82i 0.279922i
615615 317.943i 0.0208467i
616616 797.017 0.0521311
617617 13293.5 0.867385 0.433692 0.901061i 0.357210π-0.357210\pi
0.433692 + 0.901061i 0.357210π0.357210\pi
618618 −121.343 −0.00789826
619619 7323.81 0.475555 0.237778 0.971320i 0.423581π-0.423581\pi
0.237778 + 0.971320i 0.423581π0.423581\pi
620620 2496.67i 0.161724i
621621 1061.62i 0.0686013i
622622 −13387.1 −0.862982
623623 35352.5i 2.27346i
624624 19.6276i 0.00125919i
625625 −1376.27 −0.0880814
626626 13039.5i 0.832527i
627627 −16.2242 −0.00103338
628628 648.240i 0.0411904i
629629 8703.63i 0.551727i
630630 12177.0 0.770070
631631 −10301.6 −0.649922 −0.324961 0.945727i 0.605351π-0.605351\pi
−0.324961 + 0.945727i 0.605351π0.605351\pi
632632 5656.06i 0.355991i
633633 259.824i 0.0163145i
634634 −11985.5 −0.750795
635635 11043.1 0.690130
636636 235.905i 0.0147079i
637637 6235.97 0.387877
638638 −1705.06 −0.105805
639639 28446.3i 1.76106i
640640 929.738i 0.0574236i
641641 −28814.6 −1.77552 −0.887761 0.460304i 0.847740π-0.847740\pi
−0.887761 + 0.460304i 0.847740π0.847740\pi
642642 451.743i 0.0277709i
643643 21689.4 1.33024 0.665121 0.746736i 0.268380π-0.268380\pi
0.665121 + 0.746736i 0.268380π0.268380\pi
644644 19972.9i 1.22211i
645645 233.771i 0.0142709i
646646 2247.00i 0.136853i
647647 8098.38i 0.492087i 0.969259 + 0.246044i 0.0791306π0.0791306\pi
−0.969259 + 0.246044i 0.920869π0.920869\pi
648648 5822.30i 0.352966i
649649 1210.11i 0.0731907i
650650 1448.79i 0.0874249i
651651 326.544i 0.0196594i
652652 10395.6i 0.624425i
653653 −8898.08 −0.533244 −0.266622 0.963801i 0.585908π-0.585908\pi
−0.266622 + 0.963801i 0.585908π0.585908\pi
654654 287.779i 0.0172065i
655655 9278.25 0.553483
656656 −5724.87 −0.340730
657657 18500.4 1.09858
658658 35906.8 2.12735
659659 29685.7 1.75477 0.877383 0.479790i 0.159287π-0.159287\pi
0.877383 + 0.479790i 0.159287π0.159287\pi
660660 11.4000i 0.000672339i
661661 20652.3i 1.21525i −0.794224 0.607625i 0.792123π-0.792123\pi
0.794224 0.607625i 0.207877π-0.207877\pi
662662 14933.6i 0.876755i
663663 33.3314 0.00195246
664664 532.853 0.0311426
665665 −9329.47 −0.544032
666666 17288.1i 1.00586i
667667 42727.9i 2.48041i
668668 1746.20i 0.101142i
669669 265.117 0.0153214
670670 5568.43i 0.321085i
671671 −2230.02 −0.128300
672672 121.602i 0.00698050i
673673 10310.0i 0.590525i −0.955416 0.295262i 0.904593π-0.904593\pi
0.955416 0.295262i 0.0954070π-0.0954070\pi
674674 616.366 0.0352248
675675 477.097i 0.0272051i
676676 8385.79 0.477116
677677 23022.6i 1.30699i −0.756932 0.653494i 0.773303π-0.773303\pi
0.756932 0.653494i 0.226697π-0.226697\pi
678678 402.217i 0.0227833i
679679 904.464i 0.0511195i
680680 1578.87 0.0890395
681681 553.393 0.0311396
682682 −551.216 −0.0309489
683683 16684.0i 0.934691i 0.884075 + 0.467346i 0.154790π0.154790\pi
−0.884075 + 0.467346i 0.845210π0.845210\pi
684684 4463.25i 0.249498i
685685 14686.3i 0.819174i
686686 −17325.7 −0.964281
687687 −60.3811 −0.00335325
688688 4209.27 0.233251
689689 4834.13 0.267294
690690 285.678 0.0157617
691691 778.602i 0.0428645i −0.999770 0.0214323i 0.993177π-0.993177\pi
0.999770 0.0214323i 0.00682263π-0.00682263\pi
692692 −16188.3 −0.889286
693693 2688.44i 0.147367i
694694 5424.82i 0.296719i
695695 1955.42i 0.106724i
696696 260.143i 0.0141677i
697697 9721.90i 0.528326i
698698 10114.5i 0.548480i
699699 51.0959i 0.00276484i
700700 8975.89i 0.484653i
701701 15688.2i 0.845271i −0.906300 0.422636i 0.861105π-0.861105\pi
0.906300 0.422636i 0.138895π-0.138895\pi
702702 132.450 0.00712108
703703 13245.4i 0.710609i
704704 205.268 0.0109891
705705 513.586i 0.0274366i
706706 15428.2i 0.822448i
707707 −21473.6 −1.14229
708708 184.627 0.00980045
709709 20712.4i 1.09714i 0.836106 + 0.548568i 0.184827π0.184827\pi
−0.836106 + 0.548568i 0.815173π0.815173\pi
710710 15313.8 0.809461
711711 −19078.6 −1.00634
712712 9104.84i 0.479239i
713713 13813.2i 0.725538i
714714 −206.503 −0.0108238
715715 233.607 0.0122187
716716 6515.72i 0.340089i
717717 234.979i 0.0122391i
718718 24540.8 1.27556
719719 17274.8i 0.896024i −0.894028 0.448012i 0.852132π-0.852132\pi
0.894028 0.448012i 0.147868π-0.147868\pi
720720 3136.13 0.162329
721721 15405.2i 0.795729i
722722 10298.5i 0.530844i
723723 −385.191 −0.0198138
724724 12964.4i 0.665495i
725725 19202.1i 0.983653i
726726 323.141 0.0165191
727727 −9015.71 −0.459937 −0.229968 0.973198i 0.573862π-0.573862\pi
−0.229968 + 0.973198i 0.573862π0.573862\pi
728728 2491.85 0.126860
729729 19617.6 0.996676
730730 9959.53i 0.504957i
731731 7148.12i 0.361673i
732732 340.238i 0.0171797i
733733 19576.5i 0.986458i 0.869899 + 0.493229i 0.164184π0.164184\pi
−0.869899 + 0.493229i 0.835816π0.835816\pi
734734 5941.23 0.298767
735735 552.602i 0.0277320i
736736 5143.91i 0.257618i
737737 −1229.40 −0.0614456
738738 19310.7i 0.963195i
739739 3744.57i 0.186396i −0.995648 0.0931978i 0.970291π-0.970291\pi
0.995648 0.0931978i 0.0297089π-0.0297089\pi
740740 9306.92 0.462337
741741 −50.7244 −0.00251472
742742 −29949.6 −1.48178
743743 33774.0 1.66763 0.833814 0.552046i 0.186153π-0.186153\pi
0.833814 + 0.552046i 0.186153π0.186153\pi
744744 84.0997i 0.00414414i
745745 2039.38 0.100292
746746 18486.7 0.907300
747747 1797.38i 0.0880359i
748748 348.582i 0.0170394i
749749 −57351.7 −2.79784
750750 350.534 0.0170662
751751 −8064.59 −0.391852 −0.195926 0.980619i 0.562771π-0.562771\pi
−0.195926 + 0.980619i 0.562771π0.562771\pi
752752 9247.61 0.448438
753753 160.019 0.00774426
754754 −5330.82 −0.257476
755755 16501.4 0.795428
756756 −820.586 −0.0394767
757757 5247.19i 0.251932i −0.992035 0.125966i 0.959797π-0.959797\pi
0.992035 0.125966i 0.0402030π-0.0402030\pi
758758 −27620.6 −1.32352
759759 63.0720i 0.00301630i
760760 −2402.75 −0.114680
761761 26230.0i 1.24946i −0.780842 0.624729i 0.785210π-0.785210\pi
0.780842 0.624729i 0.214790π-0.214790\pi
762762 −371.984 −0.0176845
763763 36535.3 1.73351
764764 12302.8 0.582591
765765 5325.73i 0.251702i
766766 27782.1 1.31045
767767 3783.36i 0.178108i
768768 31.3179i 0.00147147i
769769 −22118.8 −1.03722 −0.518611 0.855010i 0.673551π-0.673551\pi
−0.518611 + 0.855010i 0.673551π0.673551\pi
770770 −1447.30 −0.0677364
771771 −999.180 −0.0466726
772772 7878.78i 0.367310i
773773 14551.5 0.677076 0.338538 0.940953i 0.390068π-0.390068\pi
0.338538 + 0.940953i 0.390068π0.390068\pi
774774 14198.4i 0.659368i
775775 6207.71i 0.287726i
776776 232.940i 0.0107758i
777777 −1217.27 −0.0562023
778778 15708.2i 0.723862i
779779 14795.0i 0.680469i
780780 35.6417i 0.00163613i
781781 3380.98i 0.154905i
782782 8735.31 0.399455
783783 1755.48 0.0801222
784784 −9950.13 −0.453268
785785 1177.14i 0.0535207i
786786 −312.535 −0.0141829
787787 −15917.0 −0.720940 −0.360470 0.932771i 0.617384π-0.617384\pi
−0.360470 + 0.932771i 0.617384π0.617384\pi
788788 4361.52i 0.197174i
789789 22.7108i 0.00102475i
790790 10270.8i 0.462556i
791791 −51064.0 −2.29536
792792 692.394i 0.0310646i
793793 −6972.10 −0.312215
794794 19915.8 0.890156
795795 428.378i 0.0191107i
796796 −7404.79 −0.329718
797797 27816.6i 1.23628i −0.786068 0.618140i 0.787886π-0.787886\pi
0.786068 0.618140i 0.212114π-0.212114\pi
798798 314.260 0.0139407
799799 15704.2i 0.695336i
800800 2311.69i 0.102163i
801801 −30711.8 −1.35474
802802 −16789.0 −0.739201
803803 −2198.86 −0.0966329
804804 187.571i 0.00822775i
805805 36268.6i 1.58795i
806806 −1723.36 −0.0753136
807807 166.190 513.513i 0.00724928 0.0223997i
808808 −5530.41 −0.240791
809809 125.653i 0.00546073i −0.999996 0.00273037i 0.999131π-0.999131\pi
0.999996 0.00273037i 0.000869104π-0.000869104\pi
810810 10572.7i 0.458625i
811811 −23561.6 −1.02017 −0.510086 0.860123i 0.670387π-0.670387\pi
−0.510086 + 0.860123i 0.670387π0.670387\pi
812812 33026.8 1.42736
813813 −726.777 −0.0313520
814814 2054.78i 0.0884767i
815815 18877.4i 0.811345i
816816 −53.1837 −0.00228162
817817 10878.2i 0.465824i
818818 −7998.47 −0.341882
819819 8405.34i 0.358616i
820820 10395.8 0.442727
821821 −13635.4 −0.579631 −0.289816 0.957083i 0.593594π-0.593594\pi
−0.289816 + 0.957083i 0.593594π0.593594\pi
822822 494.703i 0.0209912i
823823 10832.5 0.458804 0.229402 0.973332i 0.426323π-0.426323\pi
0.229402 + 0.973332i 0.426323π0.426323\pi
824824 3967.54i 0.167737i
825825 28.3448i 0.00119617i
826826 23439.6i 0.987371i
827827 −32051.0 −1.34767 −0.673835 0.738882i 0.735354π-0.735354\pi
−0.673835 + 0.738882i 0.735354π0.735354\pi
828828 17351.1 0.728250
829829 8792.02i 0.368347i 0.982894 + 0.184173i 0.0589608π0.0589608\pi
−0.982894 + 0.184173i 0.941039π0.941039\pi
830830 −967.605 −0.0404651
831831 146.087 0.00609830
832832 641.763 0.0267417
833833 16897.2i 0.702824i
834834 65.8678i 0.00273479i
835835 3170.92i 0.131418i
836836 530.480i 0.0219462i
837837 567.515 0.0234363
838838 25725.7i 1.06048i
839839 20675.3i 0.850765i −0.905014 0.425382i 0.860140π-0.860140\pi
0.905014 0.425382i 0.139860π-0.139860\pi
840840 220.816i 0.00907011i
841841 −46265.2 −1.89697
842842 12597.8i 0.515618i
843843 −761.944 −0.0311302
844844 8495.46 0.346476
845845 −15227.7 −0.619940
846846 31193.4i 1.26767i
847847 41024.7i 1.66426i
848848 −7713.35 −0.312356
849849 719.665i 0.0290917i
850850 3925.68 0.158412
851851 51491.8 2.07417
852852 −515.841 −0.0207423
853853 31131.7i 1.24962i −0.780776 0.624811i 0.785176π-0.785176\pi
0.780776 0.624811i 0.214824π-0.214824\pi
854854 43195.3 1.73081
855855 8104.80i 0.324185i
856856 −14770.6 −0.589777
857857 44684.1i 1.78107i 0.454910 + 0.890537i 0.349671π0.349671\pi
−0.454910 + 0.890537i 0.650329π0.650329\pi
858858 −7.86898 −0.000313103
859859 30638.1 1.21695 0.608474 0.793574i 0.291782π-0.291782\pi
0.608474 + 0.793574i 0.291782π0.291782\pi
860860 −7643.58 −0.303075
861861 −1359.68 −0.0538185
862862 −13763.7 −0.543844
863863 26688.4 1.05271 0.526353 0.850266i 0.323559π-0.323559\pi
0.526353 + 0.850266i 0.323559π0.323559\pi
864864 −211.337 −0.00832158
865865 29396.2 1.15549
866866 16662.7i 0.653834i
867867 510.719i 0.0200057i
868868 10677.0 0.417512
869869 2267.59 0.0885186
870870 472.392i 0.0184087i
871871 −3843.68 −0.149527
872872 9409.48 0.365419
873873 785.736 0.0304618
874874 −13293.6 −0.514487
875875 44502.5i 1.71938i
876876 335.484i 0.0129394i
877877 −22694.3 −0.873809 −0.436905 0.899508i 0.643925π-0.643925\pi
−0.436905 + 0.899508i 0.643925π0.643925\pi
878878 33225.1i 1.27710i
879879 321.406i 0.0123331i
880880 −372.744 −0.0142786
881881 8706.87i 0.332965i −0.986044 0.166482i 0.946759π-0.946759\pi
0.986044 0.166482i 0.0532409π-0.0532409\pi
882882 33563.1i 1.28132i
883883 21582.4i 0.822543i 0.911513 + 0.411272i 0.134915π0.134915\pi
−0.911513 + 0.411272i 0.865085π0.865085\pi
884884 1089.83i 0.0414650i
885885 −335.264 −0.0127342
886886 −8446.50 −0.320277
887887 15786.3 0.597579 0.298790 0.954319i 0.403417π-0.403417\pi
0.298790 + 0.954319i 0.403417π0.403417\pi
888888 −313.500 −0.0118473
889889 47225.7i 1.78166i
890890 16533.4i 0.622699i
891891 −2334.24 −0.0877665
892892 8668.52i 0.325385i
893893 23898.9i 0.895573i
894894 −68.6960 −0.00256995
895895 11831.9i 0.441895i
896896 −3976.01 −0.148247
897897 197.193i 0.00734011i
898898 15456.5i 0.574375i
899899 −22841.3 −0.847385
900900 7797.64 0.288801
901901 13098.7i 0.484330i
902902 2295.18i 0.0847240i
903903 999.717 0.0368422
904904 −13151.3 −0.483855
905905 23542.0i 0.864709i
906906 −555.845 −0.0203827
907907 9745.82 0.356786 0.178393 0.983959i 0.442910π-0.442910\pi
0.178393 + 0.983959i 0.442910π0.442910\pi
908908 18094.3i 0.661320i
909909 18654.8i 0.680682i
910910 −4524.94 −0.164836
911911 646.928i 0.0235276i 0.999931 + 0.0117638i 0.00374463π0.00374463\pi
−0.999931 + 0.0117638i 0.996255π0.996255\pi
912912 80.9360 0.00293866
913913 213.628i 0.00774376i
914914 35380.9i 1.28041i
915915 617.836i 0.0223224i
916916 1974.28i 0.0712139i
917917 39678.3i 1.42889i
918918 358.890i 0.0129032i
919919 10915.2i 0.391795i 0.980624 + 0.195897i 0.0627620π0.0627620\pi
−0.980624 + 0.195897i 0.937238π0.937238\pi
920920 9340.79i 0.334736i
921921 260.503i 0.00932016i
922922 −15862.3 −0.566590
923923 10570.6i 0.376960i
924924 48.7518 0.00173573
925925 23140.6 0.822551
926926 854.869 0.0303377
927927 13383.0 0.474170
928928 8505.87 0.300882
929929 31007.7i 1.09508i 0.836780 + 0.547540i 0.184436π0.184436\pi
−0.836780 + 0.547540i 0.815564π0.815564\pi
930930 152.716i 0.00538469i
931931 25714.5i 0.905218i
932932 1670.68 0.0587177
933933 −818.861 −0.0287335
934934 27450.1 0.961665
935935 632.989i 0.0221401i
936936 2164.75i 0.0755952i
937937 8644.64i 0.301396i 0.988580 + 0.150698i 0.0481521π0.0481521\pi
−0.988580 + 0.150698i 0.951848π0.951848\pi
938938 23813.3 0.828925
939939 797.596i 0.0277195i
940940 −16792.7 −0.582678
941941 18563.7i 0.643102i 0.946892 + 0.321551i 0.104204π0.104204\pi
−0.946892 + 0.321551i 0.895796π0.895796\pi
942942 39.6514i 0.00137146i
943943 57516.0 1.98619
944944 6036.74i 0.208135i
945945 1490.10 0.0512940
946946 1687.55i 0.0579989i
947947 56236.7i 1.92972i 0.262758 + 0.964862i 0.415368π0.415368\pi
−0.262758 + 0.964862i 0.584632π0.584632\pi
948948 345.969i 0.0118529i
949949 −6874.69 −0.235155
950950 −5974.19 −0.204030
951951 −733.126 −0.0249981
952952 6752.00i 0.229867i
953953 18539.4i 0.630169i −0.949064 0.315084i 0.897967π-0.897967\pi
0.949064 0.315084i 0.102033π-0.102033\pi
954954 26018.1i 0.882985i
955955 −22340.6 −0.756989
956956 7683.08 0.259925
957957 −104.295 −0.00352285
958958 −15851.0 −0.534576
959959 −62805.6 −2.11481
960960 56.8701i 0.00191195i
961961 22406.8 0.752134
962962 6424.21i 0.215307i
963963 49823.2i 1.66722i
964964 12594.6i 0.420792i
965965 14307.0i 0.477264i
966966 1221.70i 0.0406910i
967967 39149.7i 1.30193i 0.759106 + 0.650967i 0.225637π0.225637\pi
−0.759106 + 0.650967i 0.774363π0.774363\pi
968968 10565.7i 0.350821i
969969 137.444i 0.00455661i
970970 422.994i 0.0140016i
971971 −32454.4 −1.07262 −0.536308 0.844022i 0.680181π-0.680181\pi
−0.536308 + 0.844022i 0.680181π0.680181\pi
972972 1069.40i 0.0352892i
973973 8362.33 0.275523
974974 42859.4i 1.40996i
975975 88.6193i 0.00291086i
976976 11124.7 0.364850
977977 22864.2 0.748712 0.374356 0.927285i 0.377864π-0.377864\pi
0.374356 + 0.927285i 0.377864π0.377864\pi
978978 635.879i 0.0207906i
979979 3650.25 0.119165
980980 18068.4 0.588953
981981 31739.4i 1.03299i
982982 11286.5i 0.366768i
983983 25559.0 0.829304 0.414652 0.909980i 0.363903π-0.363903\pi
0.414652 + 0.909980i 0.363903π0.363903\pi
984984 −350.178 −0.0113448
985985 7920.06i 0.256197i
986986 14444.6i 0.466540i
987987 2196.34 0.0708312
988988 1658.53i 0.0534058i
989989 −42289.2 −1.35967
990990 1257.31i 0.0403637i
991991 8692.11i 0.278622i 0.990249 + 0.139311i 0.0444887π0.0444887\pi
−0.990249 + 0.139311i 0.955511π0.955511\pi
992992 2749.80 0.0880103
993993 913.457i 0.0291920i
994994 65489.3i 2.08973i
995995 13446.3 0.428419
996996 32.5935 0.00103691
997997 −12135.9 −0.385505 −0.192752 0.981247i 0.561741π-0.561741\pi
−0.192752 + 0.981247i 0.561741π0.561741\pi
998998 38041.3 1.20659
999999 2115.54i 0.0669998i
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 538.4.b.a.537.17 68
269.268 even 2 inner 538.4.b.a.537.52 yes 68
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.4.b.a.537.17 68 1.1 even 1 trivial
538.4.b.a.537.52 yes 68 269.268 even 2 inner