Properties

Label 538.8.a.b.1.20
Level 538538
Weight 88
Character 538.1
Self dual yes
Analytic conductor 168.063168.063
Analytic rank 00
Dimension 3737
CM no
Inner twists 11

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,8,Mod(1,538)]
 
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([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: N N == 538=2269 538 = 2 \cdot 269
Weight: k k == 8 8
Character orbit: [χ][\chi] == 538.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: 168.063143710168.063143710
Analytic rank: 00
Dimension: 3737
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.20
Character χ\chi == 538.1

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) == q8.00000q2+3.68486q3+64.0000q424.6615q529.4789q6+748.149q7512.000q82173.42q9+197.292q103107.98q11+235.831q1211419.5q135985.19q1490.8741q15+4096.00q1625777.9q17+17387.4q186067.56q191578.33q20+2756.83q21+24863.8q22104371.q231886.65q2477516.8q25+91355.7q2616067.6q27+47881.6q28191769.q29+726.992q30+99256.8q3132768.0q3211452.5q33+206223.q3418450.5q35139099.q36116870.q37+48540.4q3842079.1q39+12626.7q40+541105.q4122054.6q42215032.q43198910.q44+53599.7q45+834967.q46+24313.5q47+15093.2q48263816.q49+620135.q5094988.1q51730846.q52+1.58913e6q53+128540.q54+76647.2q55383052.q5622358.1q57+1.53415e6q58+2.40011e6q595815.94q60835584.q61794054.q621.62604e6q63+262144.q64+281621.q65+91619.7q66+1.39127e6q671.64979e6q68384592.q69+147604.q70975591.q71+1.11279e6q72+28301.8q73+934958.q74285639.q75388324.q762.32523e6q77+336633.q785.86093e6q79101013.q80+4.69407e6q814.32884e6q821.29009e6q83+176437.q84+635721.q85+1.72026e6q86706641.q87+1.59128e6q88+7.32857e6q89428798.q908.54346e6q916.67974e6q92+365748.q93194508.q94+149635.q95120746.q961.82237e6q97+2.11052e6q98+6.75494e6q99+O(q100)q-8.00000 q^{2} +3.68486 q^{3} +64.0000 q^{4} -24.6615 q^{5} -29.4789 q^{6} +748.149 q^{7} -512.000 q^{8} -2173.42 q^{9} +197.292 q^{10} -3107.98 q^{11} +235.831 q^{12} -11419.5 q^{13} -5985.19 q^{14} -90.8741 q^{15} +4096.00 q^{16} -25777.9 q^{17} +17387.4 q^{18} -6067.56 q^{19} -1578.33 q^{20} +2756.83 q^{21} +24863.8 q^{22} -104371. q^{23} -1886.65 q^{24} -77516.8 q^{25} +91355.7 q^{26} -16067.6 q^{27} +47881.6 q^{28} -191769. q^{29} +726.992 q^{30} +99256.8 q^{31} -32768.0 q^{32} -11452.5 q^{33} +206223. q^{34} -18450.5 q^{35} -139099. q^{36} -116870. q^{37} +48540.4 q^{38} -42079.1 q^{39} +12626.7 q^{40} +541105. q^{41} -22054.6 q^{42} -215032. q^{43} -198910. q^{44} +53599.7 q^{45} +834967. q^{46} +24313.5 q^{47} +15093.2 q^{48} -263816. q^{49} +620135. q^{50} -94988.1 q^{51} -730846. q^{52} +1.58913e6 q^{53} +128540. q^{54} +76647.2 q^{55} -383052. q^{56} -22358.1 q^{57} +1.53415e6 q^{58} +2.40011e6 q^{59} -5815.94 q^{60} -835584. q^{61} -794054. q^{62} -1.62604e6 q^{63} +262144. q^{64} +281621. q^{65} +91619.7 q^{66} +1.39127e6 q^{67} -1.64979e6 q^{68} -384592. q^{69} +147604. q^{70} -975591. q^{71} +1.11279e6 q^{72} +28301.8 q^{73} +934958. q^{74} -285639. q^{75} -388324. q^{76} -2.32523e6 q^{77} +336633. q^{78} -5.86093e6 q^{79} -101013. q^{80} +4.69407e6 q^{81} -4.32884e6 q^{82} -1.29009e6 q^{83} +176437. q^{84} +635721. q^{85} +1.72026e6 q^{86} -706641. q^{87} +1.59128e6 q^{88} +7.32857e6 q^{89} -428798. q^{90} -8.54346e6 q^{91} -6.67974e6 q^{92} +365748. q^{93} -194508. q^{94} +149635. q^{95} -120746. q^{96} -1.82237e6 q^{97} +2.11052e6 q^{98} +6.75494e6 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 37q296q2+80q3+2368q4+624q5640q6+453q718944q8+26707q94992q10+12312q11+5120q122673q133624q14+38182q15+151552q16+31108161q99+O(q100) 37 q - 296 q^{2} + 80 q^{3} + 2368 q^{4} + 624 q^{5} - 640 q^{6} + 453 q^{7} - 18944 q^{8} + 26707 q^{9} - 4992 q^{10} + 12312 q^{11} + 5120 q^{12} - 2673 q^{13} - 3624 q^{14} + 38182 q^{15} + 151552 q^{16}+ \cdots - 31108161 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 −8.00000 −0.707107
33 3.68486 0.0787947 0.0393973 0.999224i 0.487456π-0.487456\pi
0.0393973 + 0.999224i 0.487456π0.487456\pi
44 64.0000 0.500000
55 −24.6615 −0.0882315 −0.0441158 0.999026i 0.514047π-0.514047\pi
−0.0441158 + 0.999026i 0.514047π0.514047\pi
66 −29.4789 −0.0557162
77 748.149 0.824414 0.412207 0.911090i 0.364758π-0.364758\pi
0.412207 + 0.911090i 0.364758π0.364758\pi
88 −512.000 −0.353553
99 −2173.42 −0.993791
1010 197.292 0.0623891
1111 −3107.98 −0.704050 −0.352025 0.935991i 0.614507π-0.614507\pi
−0.352025 + 0.935991i 0.614507π0.614507\pi
1212 235.831 0.0393973
1313 −11419.5 −1.44160 −0.720799 0.693145i 0.756225π-0.756225\pi
−0.720799 + 0.693145i 0.756225π0.756225\pi
1414 −5985.19 −0.582948
1515 −90.8741 −0.00695217
1616 4096.00 0.250000
1717 −25777.9 −1.27256 −0.636278 0.771460i 0.719527π-0.719527\pi
−0.636278 + 0.771460i 0.719527π0.719527\pi
1818 17387.4 0.702717
1919 −6067.56 −0.202944 −0.101472 0.994838i 0.532355π-0.532355\pi
−0.101472 + 0.994838i 0.532355π0.532355\pi
2020 −1578.33 −0.0441158
2121 2756.83 0.0649594
2222 24863.8 0.497838
2323 −104371. −1.78868 −0.894339 0.447390i 0.852354π-0.852354\pi
−0.894339 + 0.447390i 0.852354π0.852354\pi
2424 −1886.65 −0.0278581
2525 −77516.8 −0.992215
2626 91355.7 1.01936
2727 −16067.6 −0.157100
2828 47881.6 0.412207
2929 −191769. −1.46011 −0.730054 0.683390i 0.760505π-0.760505\pi
−0.730054 + 0.683390i 0.760505π0.760505\pi
3030 726.992 0.00491593
3131 99256.8 0.598404 0.299202 0.954190i 0.403280π-0.403280\pi
0.299202 + 0.954190i 0.403280π0.403280\pi
3232 −32768.0 −0.176777
3333 −11452.5 −0.0554753
3434 206223. 0.899832
3535 −18450.5 −0.0727393
3636 −139099. −0.496896
3737 −116870. −0.379312 −0.189656 0.981851i 0.560737π-0.560737\pi
−0.189656 + 0.981851i 0.560737π0.560737\pi
3838 48540.4 0.143503
3939 −42079.1 −0.113590
4040 12626.7 0.0311945
4141 541105. 1.22613 0.613067 0.790031i 0.289936π-0.289936\pi
0.613067 + 0.790031i 0.289936π0.289936\pi
4242 −22054.6 −0.0459332
4343 −215032. −0.412443 −0.206222 0.978505i 0.566117π-0.566117\pi
−0.206222 + 0.978505i 0.566117π0.566117\pi
4444 −198910. −0.352025
4545 53599.7 0.0876837
4646 834967. 1.26479
4747 24313.5 0.0341589 0.0170795 0.999854i 0.494563π-0.494563\pi
0.0170795 + 0.999854i 0.494563π0.494563\pi
4848 15093.2 0.0196987
4949 −263816. −0.320342
5050 620135. 0.701602
5151 −94988.1 −0.100271
5252 −730846. −0.720799
5353 1.58913e6 1.46620 0.733102 0.680118i 0.238072π-0.238072\pi
0.733102 + 0.680118i 0.238072π0.238072\pi
5454 128540. 0.111087
5555 76647.2 0.0621193
5656 −383052. −0.291474
5757 −22358.1 −0.0159909
5858 1.53415e6 1.03245
5959 2.40011e6 1.52142 0.760710 0.649092i 0.224851π-0.224851\pi
0.760710 + 0.649092i 0.224851π0.224851\pi
6060 −5815.94 −0.00347609
6161 −835584. −0.471341 −0.235671 0.971833i 0.575729π-0.575729\pi
−0.235671 + 0.971833i 0.575729π0.575729\pi
6262 −794054. −0.423135
6363 −1.62604e6 −0.819295
6464 262144. 0.125000
6565 281621. 0.127194
6666 91619.7 0.0392270
6767 1.39127e6 0.565131 0.282565 0.959248i 0.408815π-0.408815\pi
0.282565 + 0.959248i 0.408815π0.408815\pi
6868 −1.64979e6 −0.636278
6969 −384592. −0.140938
7070 147604. 0.0514344
7171 −975591. −0.323492 −0.161746 0.986832i 0.551713π-0.551713\pi
−0.161746 + 0.986832i 0.551713π0.551713\pi
7272 1.11279e6 0.351358
7373 28301.8 0.00851499 0.00425750 0.999991i 0.498645π-0.498645\pi
0.00425750 + 0.999991i 0.498645π0.498645\pi
7474 934958. 0.268214
7575 −285639. −0.0781813
7676 −388324. −0.101472
7777 −2.32523e6 −0.580428
7878 336633. 0.0803204
7979 −5.86093e6 −1.33743 −0.668716 0.743518i 0.733156π-0.733156\pi
−0.668716 + 0.743518i 0.733156π0.733156\pi
8080 −101013. −0.0220579
8181 4.69407e6 0.981413
8282 −4.32884e6 −0.867008
8383 −1.29009e6 −0.247654 −0.123827 0.992304i 0.539517π-0.539517\pi
−0.123827 + 0.992304i 0.539517π0.539517\pi
8484 176437. 0.0324797
8585 635721. 0.112279
8686 1.72026e6 0.291641
8787 −706641. −0.115049
8888 1.59128e6 0.248919
8989 7.32857e6 1.10193 0.550965 0.834528i 0.314260π-0.314260\pi
0.550965 + 0.834528i 0.314260π0.314260\pi
9090 −428798. −0.0620017
9191 −8.54346e6 −1.18847
9292 −6.67974e6 −0.894339
9393 365748. 0.0471510
9494 −194508. −0.0241540
9595 149635. 0.0179061
9696 −120746. −0.0139291
9797 −1.82237e6 −0.202738 −0.101369 0.994849i 0.532322π-0.532322\pi
−0.101369 + 0.994849i 0.532322π0.532322\pi
9898 2.11052e6 0.226516
9999 6.75494e6 0.699678
100100 −4.96108e6 −0.496108
101101 −38631.5 −0.00373093 −0.00186546 0.999998i 0.500594π-0.500594\pi
−0.00186546 + 0.999998i 0.500594π0.500594\pi
102102 759905. 0.0709020
103103 8.66453e6 0.781294 0.390647 0.920541i 0.372251π-0.372251\pi
0.390647 + 0.920541i 0.372251π0.372251\pi
104104 5.84676e6 0.509682
105105 −67987.4 −0.00573147
106106 −1.27131e7 −1.03676
107107 1.15310e7 0.909963 0.454981 0.890501i 0.349646π-0.349646\pi
0.454981 + 0.890501i 0.349646π0.349646\pi
108108 −1.02832e6 −0.0785501
109109 −1.66184e7 −1.22912 −0.614562 0.788868i 0.710667π-0.710667\pi
−0.614562 + 0.788868i 0.710667π0.710667\pi
110110 −613178. −0.0439250
111111 −430649. −0.0298877
112112 3.06442e6 0.206103
113113 1.57905e7 1.02949 0.514744 0.857344i 0.327887π-0.327887\pi
0.514744 + 0.857344i 0.327887π0.327887\pi
114114 178865. 0.0113073
115115 2.57394e6 0.157818
116116 −1.22732e7 −0.730054
117117 2.48193e7 1.43265
118118 −1.92009e7 −1.07581
119119 −1.92857e7 −1.04911
120120 46527.5 0.00245796
121121 −9.82766e6 −0.504314
122122 6.68467e6 0.333289
123123 1.99390e6 0.0966128
124124 6.35243e6 0.299202
125125 3.83835e6 0.175776
126126 1.30084e7 0.579329
127127 −1.60162e7 −0.693817 −0.346909 0.937899i 0.612769π-0.612769\pi
−0.346909 + 0.937899i 0.612769π0.612769\pi
128128 −2.09715e6 −0.0883883
129129 −792365. −0.0324983
130130 −2.25296e6 −0.0899399
131131 −3.15372e7 −1.22567 −0.612836 0.790210i 0.709971π-0.709971\pi
−0.612836 + 0.790210i 0.709971π0.709971\pi
132132 −732958. −0.0277377
133133 −4.53944e6 −0.167310
134134 −1.11301e7 −0.399608
135135 396249. 0.0138612
136136 1.31983e7 0.449916
137137 2.48135e7 0.824454 0.412227 0.911081i 0.364751π-0.364751\pi
0.412227 + 0.911081i 0.364751π0.364751\pi
138138 3.07674e6 0.0996584
139139 −4.67053e7 −1.47508 −0.737538 0.675305i 0.764012π-0.764012\pi
−0.737538 + 0.675305i 0.764012π0.764012\pi
140140 −1.18083e6 −0.0363696
141141 89591.8 0.00269154
142142 7.80473e6 0.228743
143143 3.54914e7 1.01496
144144 −8.90234e6 −0.248448
145145 4.72930e6 0.128827
146146 −226415. −0.00602101
147147 −972124. −0.0252413
148148 −7.47967e6 −0.189656
149149 1.69085e7 0.418749 0.209374 0.977836i 0.432857π-0.432857\pi
0.209374 + 0.977836i 0.432857π0.432857\pi
150150 2.28511e6 0.0552825
151151 −3.36590e7 −0.795576 −0.397788 0.917477i 0.630222π-0.630222\pi
−0.397788 + 0.917477i 0.630222π0.630222\pi
152152 3.10659e6 0.0717515
153153 5.60263e7 1.26465
154154 1.86018e7 0.410425
155155 −2.44782e6 −0.0527981
156156 −2.69306e6 −0.0567951
157157 −3.80232e7 −0.784150 −0.392075 0.919933i 0.628243π-0.628243\pi
−0.392075 + 0.919933i 0.628243π0.628243\pi
158158 4.68875e7 0.945708
159159 5.85574e6 0.115529
160160 808107. 0.0155973
161161 −7.80850e7 −1.47461
162162 −3.75525e7 −0.693964
163163 3.95545e7 0.715385 0.357692 0.933840i 0.383564π-0.383564\pi
0.357692 + 0.933840i 0.383564π0.383564\pi
164164 3.46307e7 0.613067
165165 282434. 0.00489467
166166 1.03207e7 0.175118
167167 −2.38141e7 −0.395664 −0.197832 0.980236i 0.563390π-0.563390\pi
−0.197832 + 0.980236i 0.563390π0.563390\pi
168168 −1.41150e6 −0.0229666
169169 6.76556e7 1.07820
170170 −5.08577e6 −0.0793936
171171 1.31874e7 0.201684
172172 −1.37621e7 −0.206222
173173 −1.12844e8 −1.65698 −0.828489 0.560005i 0.810799π-0.810799\pi
−0.828489 + 0.560005i 0.810799π0.810799\pi
174174 5.65313e6 0.0813517
175175 −5.79942e7 −0.817996
176176 −1.27303e7 −0.176012
177177 8.84407e6 0.119880
178178 −5.86285e7 −0.779182
179179 9.77776e7 1.27425 0.637123 0.770762i 0.280124π-0.280124\pi
0.637123 + 0.770762i 0.280124π0.280124\pi
180180 3.43038e6 0.0438419
181181 1.06192e8 1.33112 0.665562 0.746343i 0.268192π-0.268192\pi
0.665562 + 0.746343i 0.268192π0.268192\pi
182182 6.83477e7 0.840377
183183 −3.07901e6 −0.0371392
184184 5.34379e7 0.632393
185185 2.88218e6 0.0334672
186186 −2.92598e6 −0.0333408
187187 8.01171e7 0.895942
188188 1.55606e6 0.0170795
189189 −1.20209e7 −0.129515
190190 −1.19708e6 −0.0126615
191191 −1.16068e8 −1.20530 −0.602649 0.798006i 0.705888π-0.705888\pi
−0.602649 + 0.798006i 0.705888π0.705888\pi
192192 965964. 0.00984933
193193 670722. 0.00671571 0.00335786 0.999994i 0.498931π-0.498931\pi
0.00335786 + 0.999994i 0.498931π0.498931\pi
194194 1.45789e7 0.143357
195195 1.03773e6 0.0100222
196196 −1.68842e7 −0.160171
197197 −1.15034e8 −1.07200 −0.535999 0.844218i 0.680065π-0.680065\pi
−0.535999 + 0.844218i 0.680065π0.680065\pi
198198 −5.40395e7 −0.494747
199199 1.03306e8 0.929263 0.464632 0.885504i 0.346187π-0.346187\pi
0.464632 + 0.885504i 0.346187π0.346187\pi
200200 3.96886e7 0.350801
201201 5.12663e6 0.0445293
202202 309052. 0.00263816
203203 −1.43472e8 −1.20373
204204 −6.07924e6 −0.0501353
205205 −1.33444e7 −0.108184
206206 −6.93162e7 −0.552458
207207 2.26842e8 1.77757
208208 −4.67741e7 −0.360399
209209 1.88578e7 0.142883
210210 543899. 0.00405276
211211 −1.98712e8 −1.45625 −0.728123 0.685447i 0.759607π-0.759607\pi
−0.728123 + 0.685447i 0.759607π0.759607\pi
212212 1.01705e8 0.733102
213213 −3.59492e6 −0.0254895
214214 −9.22480e7 −0.643441
215215 5.30301e6 0.0363905
216216 8.22659e6 0.0555433
217217 7.42589e7 0.493332
218218 1.32947e8 0.869123
219219 104288. 0.000670936 0
220220 4.90542e6 0.0310597
221221 2.94370e8 1.83451
222222 3.44519e6 0.0211338
223223 −2.64676e8 −1.59826 −0.799132 0.601156i 0.794707π-0.794707\pi
−0.799132 + 0.601156i 0.794707π0.794707\pi
224224 −2.45154e7 −0.145737
225225 1.68477e8 0.986055
226226 −1.26324e8 −0.727958
227227 −6.60375e7 −0.374714 −0.187357 0.982292i 0.559992π-0.559992\pi
−0.187357 + 0.982292i 0.559992π0.559992\pi
228228 −1.43092e6 −0.00799545
229229 3.47562e8 1.91253 0.956266 0.292499i 0.0944868π-0.0944868\pi
0.956266 + 0.292499i 0.0944868π0.0944868\pi
230230 −2.05915e7 −0.111594
231231 −8.56815e6 −0.0457346
232232 9.81856e7 0.516226
233233 1.79098e8 0.927568 0.463784 0.885948i 0.346491π-0.346491\pi
0.463784 + 0.885948i 0.346491π0.346491\pi
234234 −1.98554e8 −1.01303
235235 −599606. −0.00301390
236236 1.53607e8 0.760710
237237 −2.15967e7 −0.105383
238238 1.54286e8 0.741834
239239 8.63960e7 0.409356 0.204678 0.978829i 0.434385π-0.434385\pi
0.204678 + 0.978829i 0.434385π0.434385\pi
240240 −372220. −0.00173804
241241 −1.71501e8 −0.789236 −0.394618 0.918845i 0.629123π-0.629123\pi
−0.394618 + 0.918845i 0.629123π0.629123\pi
242242 7.86213e7 0.356604
243243 5.24367e7 0.234430
244244 −5.34773e7 −0.235671
245245 6.50608e6 0.0282643
246246 −1.59512e7 −0.0683156
247247 6.92882e7 0.292563
248248 −5.08195e7 −0.211568
249249 −4.75380e6 −0.0195139
250250 −3.07068e7 −0.124293
251251 −5.98726e7 −0.238985 −0.119492 0.992835i 0.538127π-0.538127\pi
−0.119492 + 0.992835i 0.538127π0.538127\pi
252252 −1.04067e8 −0.409648
253253 3.24382e8 1.25932
254254 1.28129e8 0.490603
255255 2.34254e6 0.00884702
256256 1.67772e7 0.0625000
257257 5.36341e8 1.97095 0.985474 0.169828i 0.0543213π-0.0543213\pi
0.985474 + 0.169828i 0.0543213π0.0543213\pi
258258 6.33892e6 0.0229798
259259 −8.74360e7 −0.312710
260260 1.80237e7 0.0635971
261261 4.16794e8 1.45104
262262 2.52298e8 0.866681
263263 −1.69744e8 −0.575373 −0.287686 0.957725i 0.592886π-0.592886\pi
−0.287686 + 0.957725i 0.592886π0.592886\pi
264264 5.86366e6 0.0196135
265265 −3.91903e7 −0.129365
266266 3.63155e7 0.118306
267267 2.70048e7 0.0868262
268268 8.90412e7 0.282565
269269 −1.94651e7 −0.0609711
270270 −3.16999e6 −0.00980134
271271 4.51187e8 1.37710 0.688548 0.725191i 0.258248π-0.258248\pi
0.688548 + 0.725191i 0.258248π0.258248\pi
272272 −1.05586e8 −0.318139
273273 −3.14815e7 −0.0936453
274274 −1.98508e8 −0.582977
275275 2.40920e8 0.698569
276276 −2.46139e7 −0.0704691
277277 2.68847e8 0.760022 0.380011 0.924982i 0.375920π-0.375920\pi
0.380011 + 0.924982i 0.375920π0.375920\pi
278278 3.73643e8 1.04304
279279 −2.15727e8 −0.594688
280280 9.44663e6 0.0257172
281281 −2.97092e8 −0.798765 −0.399383 0.916784i 0.630776π-0.630776\pi
−0.399383 + 0.916784i 0.630776π0.630776\pi
282282 −716734. −0.00190321
283283 1.76958e8 0.464106 0.232053 0.972703i 0.425456π-0.425456\pi
0.232053 + 0.972703i 0.425456π0.425456\pi
284284 −6.24378e7 −0.161746
285285 551383. 0.00141090
286286 −2.83931e8 −0.717682
287287 4.04827e8 1.01084
288288 7.12187e7 0.175679
289289 2.54162e8 0.619397
290290 −3.78344e7 −0.0910948
291291 −6.71517e6 −0.0159747
292292 1.81132e6 0.00425750
293293 6.77138e8 1.57268 0.786341 0.617793i 0.211973π-0.211973\pi
0.786341 + 0.617793i 0.211973π0.211973\pi
294294 7.77699e6 0.0178483
295295 −5.91902e7 −0.134237
296296 5.98373e7 0.134107
297297 4.99376e7 0.110606
298298 −1.35268e8 −0.296100
299299 1.19186e9 2.57855
300300 −1.82809e7 −0.0390906
301301 −1.60876e8 −0.340024
302302 2.69272e8 0.562557
303303 −142352. −0.000293977 0
304304 −2.48527e7 −0.0507360
305305 2.06067e7 0.0415871
306306 −4.48210e8 −0.894246
307307 4.99174e8 0.984619 0.492309 0.870420i 0.336153π-0.336153\pi
0.492309 + 0.870420i 0.336153π0.336153\pi
308308 −1.48815e8 −0.290214
309309 3.19276e7 0.0615618
310310 1.95825e7 0.0373339
311311 6.33910e8 1.19500 0.597498 0.801870i 0.296162π-0.296162\pi
0.597498 + 0.801870i 0.296162π0.296162\pi
312312 2.15445e7 0.0401602
313313 4.52801e8 0.834646 0.417323 0.908758i 0.362968π-0.362968\pi
0.417323 + 0.908758i 0.362968π0.362968\pi
314314 3.04185e8 0.554478
315315 4.01006e7 0.0722876
316316 −3.75100e8 −0.668716
317317 −6.01806e8 −1.06108 −0.530541 0.847659i 0.678011π-0.678011\pi
−0.530541 + 0.847659i 0.678011π0.678011\pi
318318 −4.68459e7 −0.0816914
319319 5.96013e8 1.02799
320320 −6.46485e6 −0.0110289
321321 4.24901e7 0.0717002
322322 6.24680e8 1.04271
323323 1.56409e8 0.258257
324324 3.00420e8 0.490706
325325 8.85200e8 1.43037
326326 −3.16436e8 −0.505853
327327 −6.12364e7 −0.0968485
328328 −2.77046e8 −0.433504
329329 1.81901e7 0.0281611
330330 −2.25947e6 −0.00346106
331331 −7.89167e8 −1.19611 −0.598055 0.801455i 0.704060π-0.704060\pi
−0.598055 + 0.801455i 0.704060π0.704060\pi
332332 −8.25657e7 −0.123827
333333 2.54007e8 0.376957
334334 1.90513e8 0.279777
335335 −3.43107e7 −0.0498623
336336 1.12920e7 0.0162398
337337 4.03204e8 0.573879 0.286940 0.957949i 0.407362π-0.407362\pi
0.286940 + 0.957949i 0.407362π0.407362\pi
338338 −5.41245e8 −0.762404
339339 5.81858e7 0.0811182
340340 4.06861e7 0.0561397
341341 −3.08488e8 −0.421306
342342 −1.05499e8 −0.142612
343343 −8.13507e8 −1.08851
344344 1.10097e8 0.145821
345345 9.48461e6 0.0124352
346346 9.02752e8 1.17166
347347 −1.62205e8 −0.208406 −0.104203 0.994556i 0.533229π-0.533229\pi
−0.104203 + 0.994556i 0.533229π0.533229\pi
348348 −4.52250e7 −0.0575243
349349 −1.21123e9 −1.52524 −0.762620 0.646846i 0.776087π-0.776087\pi
−0.762620 + 0.646846i 0.776087π0.776087\pi
350350 4.63953e8 0.578410
351351 1.83483e8 0.226475
352352 1.01842e8 0.124460
353353 4.00820e7 0.0484995 0.0242498 0.999706i 0.492280π-0.492280\pi
0.0242498 + 0.999706i 0.492280π0.492280\pi
354354 −7.07526e7 −0.0847678
355355 2.40595e7 0.0285422
356356 4.69028e8 0.550965
357357 −7.10653e7 −0.0826644
358358 −7.82220e8 −0.901029
359359 −8.92123e8 −1.01764 −0.508820 0.860873i 0.669918π-0.669918\pi
−0.508820 + 0.860873i 0.669918π0.669918\pi
360360 −2.74431e7 −0.0310009
361361 −8.57057e8 −0.958814
362362 −8.49539e8 −0.941246
363363 −3.62136e7 −0.0397373
364364 −5.46782e8 −0.594236
365365 −697964. −0.000751291 0
366366 2.46321e7 0.0262614
367367 −1.35690e9 −1.43291 −0.716454 0.697635i 0.754236π-0.754236\pi
−0.716454 + 0.697635i 0.754236π0.754236\pi
368368 −4.27503e8 −0.447169
369369 −1.17605e9 −1.21852
370370 −2.30574e7 −0.0236649
371371 1.18891e9 1.20876
372372 2.34078e7 0.0235755
373373 −7.96472e8 −0.794675 −0.397337 0.917673i 0.630066π-0.630066\pi
−0.397337 + 0.917673i 0.630066π0.630066\pi
374374 −6.40937e8 −0.633527
375375 1.41438e7 0.0138502
376376 −1.24485e7 −0.0120770
377377 2.18990e9 2.10489
378378 9.61674e7 0.0915813
379379 9.71067e8 0.916246 0.458123 0.888889i 0.348522π-0.348522\pi
0.458123 + 0.888889i 0.348522π0.348522\pi
380380 9.57662e6 0.00895303
381381 −5.90173e7 −0.0546691
382382 9.28542e8 0.852275
383383 1.98419e8 0.180463 0.0902313 0.995921i 0.471239π-0.471239\pi
0.0902313 + 0.995921i 0.471239π0.471239\pi
384384 −7.72772e6 −0.00696453
385385 5.73436e7 0.0512120
386386 −5.36578e6 −0.00474873
387387 4.67356e8 0.409883
388388 −1.16631e8 −0.101369
389389 7.03196e8 0.605694 0.302847 0.953039i 0.402063π-0.402063\pi
0.302847 + 0.953039i 0.402063π0.402063\pi
390390 −8.30186e6 −0.00708679
391391 2.69047e9 2.27619
392392 1.35074e8 0.113258
393393 −1.16210e8 −0.0965764
394394 9.20272e8 0.758017
395395 1.44539e8 0.118004
396396 4.32316e8 0.349839
397397 −2.35998e8 −0.189296 −0.0946479 0.995511i 0.530173π-0.530173\pi
−0.0946479 + 0.995511i 0.530173π0.530173\pi
398398 −8.26446e8 −0.657088
399399 −1.67272e7 −0.0131831
400400 −3.17509e8 −0.248054
401401 −1.19841e9 −0.928109 −0.464055 0.885807i 0.653606π-0.653606\pi
−0.464055 + 0.885807i 0.653606π0.653606\pi
402402 −4.10130e7 −0.0314870
403403 −1.13346e9 −0.862657
404404 −2.47241e6 −0.00186546
405405 −1.15763e8 −0.0865915
406406 1.14777e9 0.851167
407407 3.63228e8 0.267054
408408 4.86339e7 0.0354510
409409 1.89843e9 1.37202 0.686012 0.727590i 0.259360π-0.259360\pi
0.686012 + 0.727590i 0.259360π0.259360\pi
410410 1.06755e8 0.0764974
411411 9.14344e7 0.0649626
412412 5.54530e8 0.390647
413413 1.79564e9 1.25428
414414 −1.81474e9 −1.25693
415415 3.18155e7 0.0218509
416416 3.74193e8 0.254841
417417 −1.72103e8 −0.116228
418418 −1.50863e8 −0.101033
419419 9.69389e8 0.643797 0.321898 0.946774i 0.395679π-0.395679\pi
0.321898 + 0.946774i 0.395679π0.395679\pi
420420 −4.35119e6 −0.00286573
421421 −1.58828e9 −1.03739 −0.518693 0.854960i 0.673581π-0.673581\pi
−0.518693 + 0.854960i 0.673581π0.673581\pi
422422 1.58969e9 1.02972
423423 −5.28434e7 −0.0339469
424424 −8.13636e8 −0.518382
425425 1.99822e9 1.26265
426426 2.87594e7 0.0180238
427427 −6.25141e8 −0.388580
428428 7.37984e8 0.454981
429429 1.30781e8 0.0799731
430430 −4.24241e7 −0.0257320
431431 −4.46524e8 −0.268642 −0.134321 0.990938i 0.542885π-0.542885\pi
−0.134321 + 0.990938i 0.542885π0.542885\pi
432432 −6.58127e7 −0.0392750
433433 −1.34341e9 −0.795242 −0.397621 0.917550i 0.630164π-0.630164\pi
−0.397621 + 0.917550i 0.630164π0.630164\pi
434434 −5.94071e8 −0.348838
435435 1.74268e7 0.0101509
436436 −1.06358e9 −0.614562
437437 6.33276e8 0.363001
438438 −834306. −0.000474423 0
439439 1.04086e8 0.0587176 0.0293588 0.999569i 0.490653π-0.490653\pi
0.0293588 + 0.999569i 0.490653π0.490653\pi
440440 −3.92434e7 −0.0219625
441441 5.73382e8 0.318353
442442 −2.35496e9 −1.29720
443443 −7.56464e8 −0.413405 −0.206702 0.978404i 0.566273π-0.566273\pi
−0.206702 + 0.978404i 0.566273π0.566273\pi
444444 −2.75615e7 −0.0149439
445445 −1.80733e8 −0.0972250
446446 2.11741e9 1.13014
447447 6.23055e7 0.0329952
448448 1.96123e8 0.103052
449449 −1.56287e9 −0.814816 −0.407408 0.913246i 0.633567π-0.633567\pi
−0.407408 + 0.913246i 0.633567π0.633567\pi
450450 −1.34781e9 −0.697246
451451 −1.68174e9 −0.863259
452452 1.01059e9 0.514744
453453 −1.24029e8 −0.0626872
454454 5.28300e8 0.264963
455455 2.10694e8 0.104861
456456 1.14474e7 0.00565364
457457 −1.55277e8 −0.0761028 −0.0380514 0.999276i 0.512115π-0.512115\pi
−0.0380514 + 0.999276i 0.512115π0.512115\pi
458458 −2.78050e9 −1.35236
459459 4.14188e8 0.199919
460460 1.64732e8 0.0789089
461461 2.03586e9 0.967822 0.483911 0.875117i 0.339216π-0.339216\pi
0.483911 + 0.875117i 0.339216π0.339216\pi
462462 6.85452e7 0.0323393
463463 2.28804e9 1.07135 0.535673 0.844425i 0.320058π-0.320058\pi
0.535673 + 0.844425i 0.320058π0.320058\pi
464464 −7.85485e8 −0.365027
465465 −9.01987e6 −0.00416021
466466 −1.43279e9 −0.655890
467467 −8.86358e8 −0.402717 −0.201359 0.979518i 0.564536π-0.564536\pi
−0.201359 + 0.979518i 0.564536π0.564536\pi
468468 1.58844e9 0.716323
469469 1.04088e9 0.465902
470470 4.79684e6 0.00213115
471471 −1.40110e8 −0.0617869
472472 −1.22886e9 −0.537903
473473 6.68315e8 0.290380
474474 1.72774e8 0.0745167
475475 4.70338e8 0.201364
476476 −1.23429e9 −0.524556
477477 −3.45386e9 −1.45710
478478 −6.91168e8 −0.289458
479479 −1.03497e9 −0.430282 −0.215141 0.976583i 0.569021π-0.569021\pi
−0.215141 + 0.976583i 0.569021π0.569021\pi
480480 2.97776e6 0.00122898
481481 1.33459e9 0.546815
482482 1.37201e9 0.558074
483483 −2.87733e8 −0.116191
484484 −6.28970e8 −0.252157
485485 4.49422e7 0.0178879
486486 −4.19494e8 −0.165767
487487 −3.72598e9 −1.46180 −0.730901 0.682483i 0.760900π-0.760900\pi
−0.730901 + 0.682483i 0.760900π0.760900\pi
488488 4.27819e8 0.166644
489489 1.45753e8 0.0563685
490490 −5.20486e7 −0.0199859
491491 −9.75689e8 −0.371986 −0.185993 0.982551i 0.559550π-0.559550\pi
−0.185993 + 0.982551i 0.559550π0.559550\pi
492492 1.27609e8 0.0483064
493493 4.94340e9 1.85807
494494 −5.54306e8 −0.206874
495495 −1.66587e8 −0.0617337
496496 4.06556e8 0.149601
497497 −7.29888e8 −0.266691
498498 3.80304e7 0.0137984
499499 −4.10187e9 −1.47785 −0.738924 0.673789i 0.764666π-0.764666\pi
−0.738924 + 0.673789i 0.764666π0.764666\pi
500500 2.45655e8 0.0878881
501501 −8.77517e7 −0.0311762
502502 4.78981e8 0.168988
503503 3.71584e9 1.30188 0.650938 0.759131i 0.274376π-0.274376\pi
0.650938 + 0.759131i 0.274376π0.274376\pi
504504 8.32535e8 0.289665
505505 952709. 0.000329185 0
506506 −2.59506e9 −0.890472
507507 2.49302e8 0.0849566
508508 −1.02503e9 −0.346909
509509 3.24034e8 0.108913 0.0544563 0.998516i 0.482657π-0.482657\pi
0.0544563 + 0.998516i 0.482657π0.482657\pi
510510 −1.87404e7 −0.00625579
511511 2.11740e7 0.00701988
512512 −1.34218e8 −0.0441942
513513 9.74908e7 0.0318825
514514 −4.29073e9 −1.39367
515515 −2.13680e8 −0.0689348
516516 −5.07113e7 −0.0162492
517517 −7.55657e7 −0.0240496
518518 6.99488e8 0.221119
519519 −4.15814e8 −0.130561
520520 −1.44190e8 −0.0449700
521521 3.91250e9 1.21205 0.606027 0.795444i 0.292762π-0.292762\pi
0.606027 + 0.795444i 0.292762π0.292762\pi
522522 −3.33435e9 −1.02604
523523 −3.44008e9 −1.05151 −0.525754 0.850637i 0.676217π-0.676217\pi
−0.525754 + 0.850637i 0.676217π0.676217\pi
524524 −2.01838e9 −0.612836
525525 −2.13700e8 −0.0644537
526526 1.35795e9 0.406850
527527 −2.55863e9 −0.761502
528528 −4.69093e7 −0.0138688
529529 7.48847e9 2.19937
530530 3.13523e8 0.0914752
531531 −5.21645e9 −1.51197
532532 −2.90524e8 −0.0836549
533533 −6.17913e9 −1.76759
534534 −2.16038e8 −0.0613954
535535 −2.84371e8 −0.0802874
536536 −7.12329e8 −0.199804
537537 3.60297e8 0.100404
538538 1.55721e8 0.0431131
539539 8.19932e8 0.225537
540540 2.53600e7 0.00693059
541541 −5.46865e9 −1.48488 −0.742438 0.669915i 0.766331π-0.766331\pi
−0.742438 + 0.669915i 0.766331π0.766331\pi
542542 −3.60950e9 −0.973754
543543 3.91304e8 0.104885
544544 8.44691e8 0.224958
545545 4.09834e8 0.108448
546546 2.51852e8 0.0662172
547547 −1.30590e9 −0.341157 −0.170579 0.985344i 0.554564π-0.554564\pi
−0.170579 + 0.985344i 0.554564π0.554564\pi
548548 1.58807e9 0.412227
549549 1.81608e9 0.468415
550550 −1.92736e9 −0.493963
551551 1.16357e9 0.296320
552552 1.96911e8 0.0498292
553553 −4.38485e9 −1.10260
554554 −2.15078e9 −0.537417
555555 1.06204e7 0.00263704
556556 −2.98914e9 −0.737538
557557 −4.95942e9 −1.21601 −0.608005 0.793933i 0.708030π-0.708030\pi
−0.608005 + 0.793933i 0.708030π0.708030\pi
558558 1.72581e9 0.420508
559559 2.45555e9 0.594577
560560 −7.55730e7 −0.0181848
561561 2.95221e8 0.0705954
562562 2.37674e9 0.564812
563563 −5.82990e8 −0.137683 −0.0688417 0.997628i 0.521930π-0.521930\pi
−0.0688417 + 0.997628i 0.521930π0.521930\pi
564564 5.73387e6 0.00134577
565565 −3.89417e8 −0.0908333
566566 −1.41566e9 −0.328173
567567 3.51186e9 0.809090
568568 4.99503e8 0.114372
569569 3.07710e8 0.0700243 0.0350121 0.999387i 0.488853π-0.488853\pi
0.0350121 + 0.999387i 0.488853π0.488853\pi
570570 −4.41107e6 −0.000997658 0
571571 −2.07222e9 −0.465811 −0.232906 0.972499i 0.574823π-0.574823\pi
−0.232906 + 0.972499i 0.574823π0.574823\pi
572572 2.27145e9 0.507478
573573 −4.27693e8 −0.0949711
574574 −3.23862e9 −0.714773
575575 8.09050e9 1.77475
576576 −5.69749e8 −0.124224
577577 4.82434e9 1.04550 0.522748 0.852487i 0.324907π-0.324907\pi
0.522748 + 0.852487i 0.324907π0.324907\pi
578578 −2.03330e9 −0.437980
579579 2.47152e6 0.000529162 0
580580 3.02675e8 0.0644137
581581 −9.65179e8 −0.204170
582582 5.37214e7 0.0112958
583583 −4.93899e9 −1.03228
584584 −1.44905e7 −0.00301050
585585 −6.12080e8 −0.126405
586586 −5.41711e9 −1.11205
587587 7.62822e9 1.55665 0.778323 0.627863i 0.216070π-0.216070\pi
0.778323 + 0.627863i 0.216070π0.216070\pi
588588 −6.22159e7 −0.0126206
589589 −6.02246e8 −0.121442
590590 4.73521e8 0.0949200
591591 −4.23884e8 −0.0844678
592592 −4.78699e8 −0.0948279
593593 −6.75477e9 −1.33021 −0.665104 0.746751i 0.731613π-0.731613\pi
−0.665104 + 0.746751i 0.731613π0.731613\pi
594594 −3.99500e8 −0.0782104
595595 4.75614e8 0.0925647
596596 1.08214e9 0.209374
597597 3.80667e8 0.0732210
598598 −9.53488e9 −1.82331
599599 4.26968e9 0.811712 0.405856 0.913937i 0.366974π-0.366974\pi
0.405856 + 0.913937i 0.366974π0.366974\pi
600600 1.46247e8 0.0276413
601601 −3.39925e9 −0.638737 −0.319368 0.947631i 0.603471π-0.603471\pi
−0.319368 + 0.947631i 0.603471π0.603471\pi
602602 1.28701e9 0.240433
603603 −3.02381e9 −0.561622
604604 −2.15418e9 −0.397788
605605 2.42364e8 0.0444964
606606 1.13881e6 0.000207873 0
607607 −2.06229e9 −0.374273 −0.187137 0.982334i 0.559921π-0.559921\pi
−0.187137 + 0.982334i 0.559921π0.559921\pi
608608 1.98822e8 0.0358758
609609 −5.28673e8 −0.0948477
610610 −1.64854e8 −0.0294066
611611 −2.77647e8 −0.0492434
612612 3.58568e9 0.632327
613613 1.92019e9 0.336692 0.168346 0.985728i 0.446157π-0.446157\pi
0.168346 + 0.985728i 0.446157π0.446157\pi
614614 −3.99340e9 −0.696231
615615 −4.91724e7 −0.00852429
616616 1.19052e9 0.205212
617617 −1.30859e9 −0.224287 −0.112144 0.993692i 0.535772π-0.535772\pi
−0.112144 + 0.993692i 0.535772π0.535772\pi
618618 −2.55421e8 −0.0435308
619619 −2.62837e9 −0.445420 −0.222710 0.974885i 0.571490π-0.571490\pi
−0.222710 + 0.974885i 0.571490π0.571490\pi
620620 −1.56660e8 −0.0263990
621621 1.67699e9 0.281002
622622 −5.07128e9 −0.844990
623623 5.48286e9 0.908446
624624 −1.72356e8 −0.0283975
625625 5.96134e9 0.976706
626626 −3.62241e9 −0.590184
627627 6.94885e7 0.0112584
628628 −2.43348e9 −0.392075
629629 3.01266e9 0.482695
630630 −3.20805e8 −0.0511151
631631 −4.66029e8 −0.0738431 −0.0369215 0.999318i 0.511755π-0.511755\pi
−0.0369215 + 0.999318i 0.511755π0.511755\pi
632632 3.00080e9 0.472854
633633 −7.32225e8 −0.114744
634634 4.81444e9 0.750298
635635 3.94982e8 0.0612166
636636 3.74767e8 0.0577646
637637 3.01263e9 0.461804
638638 −4.76810e9 −0.726897
639639 2.12037e9 0.321484
640640 5.17188e7 0.00779864
641641 −2.89227e9 −0.433746 −0.216873 0.976200i 0.569586π-0.569586\pi
−0.216873 + 0.976200i 0.569586π0.569586\pi
642642 −3.39921e8 −0.0506997
643643 2.10954e9 0.312932 0.156466 0.987683i 0.449990π-0.449990\pi
0.156466 + 0.987683i 0.449990π0.449990\pi
644644 −4.99744e9 −0.737305
645645 1.95409e7 0.00286738
646646 −1.25127e9 −0.182616
647647 −6.06566e9 −0.880467 −0.440234 0.897883i 0.645104π-0.645104\pi
−0.440234 + 0.897883i 0.645104π0.645104\pi
648648 −2.40336e9 −0.346982
649649 −7.45948e9 −1.07115
650650 −7.08160e9 −1.01143
651651 2.73634e8 0.0388719
652652 2.53149e9 0.357692
653653 2.96594e9 0.416836 0.208418 0.978040i 0.433169π-0.433169\pi
0.208418 + 0.978040i 0.433169π0.433169\pi
654654 4.89892e8 0.0684822
655655 7.77754e8 0.108143
656656 2.21637e9 0.306533
657657 −6.15118e7 −0.00846213
658658 −1.45521e8 −0.0199129
659659 −1.59311e8 −0.0216844 −0.0108422 0.999941i 0.503451π-0.503451\pi
−0.0108422 + 0.999941i 0.503451π0.503451\pi
660660 1.80758e7 0.00244734
661661 1.36437e10 1.83750 0.918749 0.394843i 0.129201π-0.129201\pi
0.918749 + 0.394843i 0.129201π0.129201\pi
662662 6.31334e9 0.845777
663663 1.08471e9 0.144550
664664 6.60525e8 0.0875591
665665 1.11949e8 0.0147620
666666 −2.03206e9 −0.266549
667667 2.00151e10 2.61166
668668 −1.52410e9 −0.197832
669669 −9.75296e8 −0.125935
670670 2.74486e8 0.0352580
671671 2.59697e9 0.331848
672672 −9.03357e7 −0.0114833
673673 −9.28685e9 −1.17440 −0.587199 0.809442i 0.699769π-0.699769\pi
−0.587199 + 0.809442i 0.699769π0.699769\pi
674674 −3.22563e9 −0.405794
675675 1.24551e9 0.155877
676676 4.32996e9 0.539101
677677 −1.05420e10 −1.30576 −0.652880 0.757461i 0.726440π-0.726440\pi
−0.652880 + 0.757461i 0.726440π0.726440\pi
678678 −4.65486e8 −0.0573592
679679 −1.36340e9 −0.167140
680680 −3.25489e8 −0.0396968
681681 −2.43339e8 −0.0295255
682682 2.46790e9 0.297908
683683 −9.02461e9 −1.08382 −0.541909 0.840437i 0.682298π-0.682298\pi
−0.541909 + 0.840437i 0.682298π0.682298\pi
684684 8.43991e8 0.100842
685685 −6.11938e8 −0.0727428
686686 6.50805e9 0.769691
687687 1.28072e9 0.150697
688688 −8.80773e8 −0.103111
689689 −1.81470e10 −2.11368
690690 −7.58769e7 −0.00879301
691691 −1.90822e9 −0.220016 −0.110008 0.993931i 0.535088π-0.535088\pi
−0.110008 + 0.993931i 0.535088π0.535088\pi
692692 −7.22201e9 −0.828489
693693 5.05371e9 0.576824
694694 1.29764e9 0.147365
695695 1.15182e9 0.130148
696696 3.61800e8 0.0406758
697697 −1.39486e10 −1.56032
698698 9.68986e9 1.07851
699699 6.59953e8 0.0730874
700700 −3.71163e9 −0.408998
701701 −9.34534e9 −1.02467 −0.512333 0.858787i 0.671219π-0.671219\pi
−0.512333 + 0.858787i 0.671219π0.671219\pi
702702 −1.46786e9 −0.160142
703703 7.09114e8 0.0769790
704704 −8.14737e8 −0.0880062
705705 −2.20946e6 −0.000237479 0
706706 −3.20656e8 −0.0342944
707707 −2.89021e7 −0.00307583
708708 5.66021e8 0.0599399
709709 1.04561e10 1.10181 0.550907 0.834567i 0.314282π-0.314282\pi
0.550907 + 0.834567i 0.314282π0.314282\pi
710710 −1.92476e8 −0.0201824
711711 1.27383e10 1.32913
712712 −3.75223e9 −0.389591
713713 −1.03595e10 −1.07035
714714 5.68522e8 0.0584526
715715 −8.75270e8 −0.0895511
716716 6.25776e9 0.637123
717717 3.18357e8 0.0322551
718718 7.13698e9 0.719580
719719 9.06156e9 0.909185 0.454592 0.890700i 0.349785π-0.349785\pi
0.454592 + 0.890700i 0.349785π0.349785\pi
720720 2.19545e8 0.0219209
721721 6.48236e9 0.644110
722722 6.85645e9 0.677984
723723 −6.31958e8 −0.0621876
724724 6.79631e9 0.665562
725725 1.48653e10 1.44874
726726 2.89709e8 0.0280985
727727 1.30418e10 1.25883 0.629415 0.777070i 0.283295π-0.283295\pi
0.629415 + 0.777070i 0.283295π0.283295\pi
728728 4.37425e9 0.420188
729729 −1.00727e10 −0.962941
730730 5.58371e6 0.000531243 0
731731 5.54309e9 0.524857
732732 −1.97057e8 −0.0185696
733733 −7.38068e9 −0.692201 −0.346101 0.938197i 0.612494π-0.612494\pi
−0.346101 + 0.938197i 0.612494π0.612494\pi
734734 1.08552e10 1.01322
735735 2.39740e7 0.00222707
736736 3.42003e9 0.316197
737737 −4.32403e9 −0.397880
738738 9.40839e9 0.861625
739739 1.54298e10 1.40638 0.703192 0.711000i 0.251757π-0.251757\pi
0.703192 + 0.711000i 0.251757π0.251757\pi
740740 1.84459e8 0.0167336
741741 2.55318e8 0.0230524
742742 −9.51127e9 −0.854722
743743 −1.84855e10 −1.65337 −0.826686 0.562664i 0.809777π-0.809777\pi
−0.826686 + 0.562664i 0.809777π0.809777\pi
744744 −1.87263e8 −0.0166704
745745 −4.16988e8 −0.0369468
746746 6.37177e9 0.561920
747747 2.80391e9 0.246117
748748 5.12750e9 0.447971
749749 8.62691e9 0.750186
750750 −1.13150e8 −0.00979359
751751 −6.42980e9 −0.553934 −0.276967 0.960879i 0.589329π-0.589329\pi
−0.276967 + 0.960879i 0.589329π0.589329\pi
752752 9.95880e7 0.00853974
753753 −2.20622e8 −0.0188307
754754 −1.75192e10 −1.48838
755755 8.30080e8 0.0701949
756756 −7.69339e8 −0.0647577
757757 8.89856e9 0.745563 0.372781 0.927919i 0.378404π-0.378404\pi
0.372781 + 0.927919i 0.378404π0.378404\pi
758758 −7.76854e9 −0.647884
759759 1.19530e9 0.0992275
760760 −7.66130e7 −0.00633075
761761 7.31378e9 0.601583 0.300792 0.953690i 0.402749π-0.402749\pi
0.300792 + 0.953690i 0.402749π0.402749\pi
762762 4.72139e8 0.0386569
763763 −1.24330e10 −1.01331
764764 −7.42833e9 −0.602649
765765 −1.38169e9 −0.111582
766766 −1.58735e9 −0.127606
767767 −2.74080e10 −2.19327
768768 6.18217e7 0.00492467
769769 −1.66255e10 −1.31836 −0.659178 0.751987i 0.729096π-0.729096\pi
−0.659178 + 0.751987i 0.729096π0.729096\pi
770770 −4.58748e8 −0.0362124
771771 1.97634e9 0.155300
772772 4.29262e7 0.00335786
773773 −2.09583e9 −0.163203 −0.0816013 0.996665i 0.526003π-0.526003\pi
−0.0816013 + 0.996665i 0.526003π0.526003\pi
774774 −3.73885e9 −0.289831
775775 −7.69407e9 −0.593745
776776 9.33052e8 0.0716786
777777 −3.22190e8 −0.0246399
778778 −5.62557e9 −0.428290
779779 −3.28318e9 −0.248837
780780 6.64149e7 0.00501112
781781 3.03211e9 0.227754
782782 −2.15237e10 −1.60951
783783 3.08125e9 0.229383
784784 −1.08059e9 −0.0800855
785785 9.37706e8 0.0691868
786786 9.29683e8 0.0682898
787787 1.42203e10 1.03991 0.519956 0.854193i 0.325948π-0.325948\pi
0.519956 + 0.854193i 0.325948π0.325948\pi
788788 −7.36217e9 −0.535999
789789 −6.25483e8 −0.0453363
790790 −1.15631e9 −0.0834412
791791 1.18137e10 0.848724
792792 −3.45853e9 −0.247374
793793 9.54191e9 0.679484
794794 1.88798e9 0.133852
795795 −1.44411e8 −0.0101933
796796 6.61157e9 0.464632
797797 −1.26917e10 −0.888006 −0.444003 0.896025i 0.646442π-0.646442\pi
−0.444003 + 0.896025i 0.646442π0.646442\pi
798798 1.33818e8 0.00932187
799799 −6.26751e8 −0.0434691
800800 2.54007e9 0.175401
801801 −1.59281e10 −1.09509
802802 9.58726e9 0.656272
803803 −8.79614e7 −0.00599498
804804 3.28104e8 0.0222646
805805 1.92569e9 0.130107
806806 9.06767e9 0.609991
807807 −7.17262e7 −0.00480420
808808 1.97793e7 0.00131908
809809 1.18581e10 0.787401 0.393701 0.919239i 0.371195π-0.371195\pi
0.393701 + 0.919239i 0.371195π0.371195\pi
810810 9.26100e8 0.0612295
811811 −2.38522e9 −0.157020 −0.0785101 0.996913i 0.525016π-0.525016\pi
−0.0785101 + 0.996913i 0.525016π0.525016\pi
812812 −9.18219e9 −0.601866
813813 1.66256e9 0.108508
814814 −2.90583e9 −0.188836
815815 −9.75472e8 −0.0631195
816816 −3.89071e8 −0.0250676
817817 1.30472e9 0.0837029
818818 −1.51874e10 −0.970168
819819 1.85685e10 1.18109
820820 −8.54044e8 −0.0540918
821821 2.62041e10 1.65260 0.826300 0.563230i 0.190441π-0.190441\pi
0.826300 + 0.563230i 0.190441π0.190441\pi
822822 −7.31475e8 −0.0459355
823823 8.01435e8 0.0501152 0.0250576 0.999686i 0.492023π-0.492023\pi
0.0250576 + 0.999686i 0.492023π0.492023\pi
824824 −4.43624e9 −0.276229
825825 8.87758e8 0.0550435
826826 −1.43651e10 −0.886909
827827 1.70111e10 1.04583 0.522916 0.852384i 0.324844π-0.324844\pi
0.522916 + 0.852384i 0.324844π0.324844\pi
828828 1.45179e10 0.888786
829829 −1.46657e9 −0.0894049 −0.0447025 0.999000i 0.514234π-0.514234\pi
−0.0447025 + 0.999000i 0.514234π0.514234\pi
830830 −2.54524e8 −0.0154509
831831 9.90664e8 0.0598857
832832 −2.99354e9 −0.180200
833833 6.80062e9 0.407653
834834 1.37682e9 0.0821857
835835 5.87290e8 0.0349100
836836 1.20690e9 0.0714413
837837 −1.59481e9 −0.0940093
838838 −7.75511e9 −0.455233
839839 1.59346e10 0.931483 0.465741 0.884921i 0.345788π-0.345788\pi
0.465741 + 0.884921i 0.345788π0.345788\pi
840840 3.48095e7 0.00202638
841841 1.95254e10 1.13191
842842 1.27063e10 0.733543
843843 −1.09474e9 −0.0629385
844844 −1.27175e10 −0.728123
845845 −1.66849e9 −0.0951314
846846 4.22747e8 0.0240041
847847 −7.35256e9 −0.415764
848848 6.50909e9 0.366551
849849 6.52065e8 0.0365691
850850 −1.59858e10 −0.892827
851851 1.21978e10 0.678466
852852 −2.30075e8 −0.0127447
853853 3.20330e10 1.76716 0.883581 0.468278i 0.155125π-0.155125\pi
0.883581 + 0.468278i 0.155125π0.155125\pi
854854 5.00113e9 0.274768
855855 −3.25219e8 −0.0177949
856856 −5.90387e9 −0.321720
857857 1.97007e10 1.06918 0.534589 0.845112i 0.320467π-0.320467\pi
0.534589 + 0.845112i 0.320467π0.320467\pi
858858 −1.04625e9 −0.0565495
859859 −2.28489e10 −1.22995 −0.614976 0.788546i 0.710834π-0.710834\pi
−0.614976 + 0.788546i 0.710834π0.710834\pi
860860 3.39393e8 0.0181952
861861 1.49173e9 0.0796489
862862 3.57219e9 0.189959
863863 −2.12966e10 −1.12791 −0.563953 0.825807i 0.690720π-0.690720\pi
−0.563953 + 0.825807i 0.690720π0.690720\pi
864864 5.26502e8 0.0277716
865865 2.78290e9 0.146198
866866 1.07472e10 0.562321
867867 9.36554e8 0.0488052
868868 4.75257e9 0.246666
869869 1.82156e10 0.941619
870870 −1.39414e8 −0.00717778
871871 −1.58875e10 −0.814691
872872 8.50861e9 0.434561
873873 3.96077e9 0.201479
874874 −5.06621e9 −0.256681
875875 2.87166e9 0.144912
876876 6.67445e6 0.000335468 0
877877 −2.28649e9 −0.114465 −0.0572324 0.998361i 0.518228π-0.518228\pi
−0.0572324 + 0.998361i 0.518228π0.518228\pi
878878 −8.32692e8 −0.0415196
879879 2.49516e9 0.123919
880880 3.13947e8 0.0155298
881881 3.33656e10 1.64393 0.821964 0.569540i 0.192879π-0.192879\pi
0.821964 + 0.569540i 0.192879π0.192879\pi
882882 −4.58706e9 −0.225110
883883 1.39278e10 0.680802 0.340401 0.940280i 0.389437π-0.389437\pi
0.340401 + 0.940280i 0.389437π0.389437\pi
884884 1.88397e10 0.917256
885885 −2.18108e8 −0.0105772
886886 6.05171e9 0.292321
887887 3.75152e10 1.80499 0.902495 0.430700i 0.141733π-0.141733\pi
0.902495 + 0.430700i 0.141733π0.141733\pi
888888 2.20492e8 0.0105669
889889 −1.19825e10 −0.571993
890890 1.44587e9 0.0687484
891891 −1.45890e10 −0.690963
892892 −1.69393e10 −0.799132
893893 −1.47523e8 −0.00693235
894894 −4.98444e8 −0.0233311
895895 −2.41134e9 −0.112429
896896 −1.56898e9 −0.0728686
897897 4.39184e9 0.203176
898898 1.25029e10 0.576162
899899 −1.90343e10 −0.873734
900900 1.07825e10 0.493027
901901 −4.09645e10 −1.86583
902902 1.34539e10 0.610416
903903 −5.92807e8 −0.0267921
904904 −8.08474e9 −0.363979
905905 −2.61886e9 −0.117447
906906 9.92230e8 0.0443265
907907 2.62576e10 1.16850 0.584252 0.811572i 0.301388π-0.301388\pi
0.584252 + 0.811572i 0.301388π0.301388\pi
908908 −4.22640e9 −0.187357
909909 8.39625e7 0.00370776
910910 −1.68555e9 −0.0741477
911911 −2.63441e10 −1.15443 −0.577217 0.816591i 0.695861π-0.695861\pi
−0.577217 + 0.816591i 0.695861π0.695861\pi
912912 −9.15788e7 −0.00399773
913913 4.00956e9 0.174361
914914 1.24222e9 0.0538128
915915 7.59329e7 0.00327685
916916 2.22440e10 0.956266
917917 −2.35946e10 −1.01046
918918 −3.31350e9 −0.141364
919919 3.39271e10 1.44192 0.720962 0.692975i 0.243700π-0.243700\pi
0.720962 + 0.692975i 0.243700π0.243700\pi
920920 −1.31786e9 −0.0557970
921921 1.83939e9 0.0775827
922922 −1.62869e10 −0.684353
923923 1.11407e10 0.466345
924924 −5.48362e8 −0.0228673
925925 9.05937e9 0.376359
926926 −1.83043e10 −0.757556
927927 −1.88317e10 −0.776443
928928 6.28388e9 0.258113
929929 −4.45761e9 −0.182409 −0.0912046 0.995832i 0.529072π-0.529072\pi
−0.0912046 + 0.995832i 0.529072π0.529072\pi
930930 7.21589e7 0.00294171
931931 1.60072e9 0.0650115
932932 1.14623e10 0.463784
933933 2.33587e9 0.0941593
934934 7.09086e9 0.284764
935935 −1.97581e9 −0.0790503
936936 −1.27075e10 −0.506517
937937 −3.30717e10 −1.31331 −0.656655 0.754191i 0.728029π-0.728029\pi
−0.656655 + 0.754191i 0.728029π0.728029\pi
938938 −8.32701e9 −0.329442
939939 1.66851e9 0.0657656
940940 −3.83748e7 −0.00150695
941941 6.22284e9 0.243459 0.121729 0.992563i 0.461156π-0.461156\pi
0.121729 + 0.992563i 0.461156π0.461156\pi
942942 1.12088e9 0.0436899
943943 −5.64756e10 −2.19316
944944 9.83085e9 0.380355
945945 2.96454e8 0.0114273
946946 −5.34652e9 −0.205330
947947 4.02764e10 1.54108 0.770542 0.637390i 0.219986π-0.219986\pi
0.770542 + 0.637390i 0.219986π0.219986\pi
948948 −1.38219e9 −0.0526913
949949 −3.23192e8 −0.0122752
950950 −3.76270e9 −0.142386
951951 −2.21757e9 −0.0836076
952952 9.87430e9 0.370917
953953 −4.05662e10 −1.51824 −0.759118 0.650953i 0.774370π-0.774370\pi
−0.759118 + 0.650953i 0.774370π0.774370\pi
954954 2.76308e10 1.03033
955955 2.86240e9 0.106345
956956 5.52934e9 0.204678
957957 2.19622e9 0.0810000
958958 8.27975e9 0.304255
959959 1.85642e10 0.679691
960960 −2.38221e7 −0.000869022 0
961961 −1.76607e10 −0.641913
962962 −1.06767e10 −0.386656
963963 −2.50617e10 −0.904313
964964 −1.09761e10 −0.394618
965965 −1.65410e7 −0.000592538 0
966966 2.30186e9 0.0821598
967967 −1.51573e10 −0.539051 −0.269526 0.962993i 0.586867π-0.586867\pi
−0.269526 + 0.962993i 0.586867π0.586867\pi
968968 5.03176e9 0.178302
969969 5.76345e8 0.0203493
970970 −3.59538e8 −0.0126486
971971 −1.62700e10 −0.570322 −0.285161 0.958480i 0.592047π-0.592047\pi
−0.285161 + 0.958480i 0.592047π0.592047\pi
972972 3.35595e9 0.117215
973973 −3.49426e10 −1.21607
974974 2.98078e10 1.03365
975975 3.26184e9 0.112706
976976 −3.42255e9 −0.117835
977977 3.14263e9 0.107811 0.0539054 0.998546i 0.482833π-0.482833\pi
0.0539054 + 0.998546i 0.482833π0.482833\pi
978978 −1.16602e9 −0.0398585
979979 −2.27770e10 −0.775813
980980 4.16389e8 0.0141321
981981 3.61188e10 1.22149
982982 7.80552e9 0.263034
983983 −4.22390e10 −1.41833 −0.709163 0.705044i 0.750927π-0.750927\pi
−0.709163 + 0.705044i 0.750927π0.750927\pi
984984 −1.02088e9 −0.0341578
985985 2.83690e9 0.0945840
986986 −3.95472e10 −1.31385
987987 6.70280e7 0.00221894
988988 4.43445e9 0.146282
989989 2.24431e10 0.737728
990990 1.33269e9 0.0436523
991991 2.20014e10 0.718111 0.359056 0.933316i 0.383099π-0.383099\pi
0.359056 + 0.933316i 0.383099π0.383099\pi
992992 −3.25245e9 −0.105784
993993 −2.90797e9 −0.0942471
994994 5.83910e9 0.188579
995995 −2.54767e9 −0.0819903
996996 −3.04243e8 −0.00975693
997997 4.63423e10 1.48097 0.740483 0.672076i 0.234597π-0.234597\pi
0.740483 + 0.672076i 0.234597π0.234597\pi
998998 3.28149e10 1.04500
999999 1.87781e9 0.0595899
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 538.8.a.b.1.20 37
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.8.a.b.1.20 37 1.1 even 1 trivial