Properties

Label 91.8.a.e.1.9
Level $91$
Weight $8$
Character 91.1
Self dual yes
Analytic conductor $28.427$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4270373191\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 6 x^{11} - 1243 x^{10} + 5598 x^{9} + 567554 x^{8} - 1739560 x^{7} - 117081910 x^{6} + 186018392 x^{5} + 10752389517 x^{4} + 491049966 x^{3} + \cdots + 59402280000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-12.7078\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+13.7078 q^{2} +50.2321 q^{3} +59.9043 q^{4} +544.453 q^{5} +688.573 q^{6} +343.000 q^{7} -933.443 q^{8} +336.267 q^{9} +O(q^{10})\) \(q+13.7078 q^{2} +50.2321 q^{3} +59.9043 q^{4} +544.453 q^{5} +688.573 q^{6} +343.000 q^{7} -933.443 q^{8} +336.267 q^{9} +7463.26 q^{10} +3185.05 q^{11} +3009.12 q^{12} +2197.00 q^{13} +4701.78 q^{14} +27349.0 q^{15} -20463.2 q^{16} +18610.0 q^{17} +4609.48 q^{18} -48240.1 q^{19} +32615.1 q^{20} +17229.6 q^{21} +43660.1 q^{22} -101451. q^{23} -46888.8 q^{24} +218304. q^{25} +30116.1 q^{26} -92966.3 q^{27} +20547.2 q^{28} +163486. q^{29} +374895. q^{30} -76991.3 q^{31} -161025. q^{32} +159992. q^{33} +255102. q^{34} +186747. q^{35} +20143.8 q^{36} +224246. q^{37} -661266. q^{38} +110360. q^{39} -508216. q^{40} -71118.9 q^{41} +236181. q^{42} -366021. q^{43} +190798. q^{44} +183081. q^{45} -1.39068e6 q^{46} +545639. q^{47} -1.02791e6 q^{48} +117649. q^{49} +2.99247e6 q^{50} +934819. q^{51} +131610. q^{52} +1.16583e6 q^{53} -1.27436e6 q^{54} +1.73411e6 q^{55} -320171. q^{56} -2.42320e6 q^{57} +2.24103e6 q^{58} +67871.3 q^{59} +1.63832e6 q^{60} -2.67077e6 q^{61} -1.05538e6 q^{62} +115339. q^{63} +411985. q^{64} +1.19616e6 q^{65} +2.19314e6 q^{66} -803728. q^{67} +1.11482e6 q^{68} -5.09611e6 q^{69} +2.55990e6 q^{70} -3.09022e6 q^{71} -313886. q^{72} +1.19301e6 q^{73} +3.07393e6 q^{74} +1.09659e7 q^{75} -2.88979e6 q^{76} +1.09247e6 q^{77} +1.51279e6 q^{78} -466693. q^{79} -1.11413e7 q^{80} -5.40531e6 q^{81} -974885. q^{82} -3.98882e6 q^{83} +1.03213e6 q^{84} +1.01323e7 q^{85} -5.01736e6 q^{86} +8.21223e6 q^{87} -2.97306e6 q^{88} -387371. q^{89} +2.50964e6 q^{90} +753571. q^{91} -6.07737e6 q^{92} -3.86744e6 q^{93} +7.47952e6 q^{94} -2.62644e7 q^{95} -8.08865e6 q^{96} +8.39907e6 q^{97} +1.61271e6 q^{98} +1.07103e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9} + 6668 q^{10} + 12168 q^{11} - 183 q^{12} + 26364 q^{13} + 2058 q^{14} - 28790 q^{15} + 85914 q^{16} + 82710 q^{17} - 44965 q^{18} - 10302 q^{19} + 141318 q^{20} + 28126 q^{21} - 97457 q^{22} + 98376 q^{23} - 519981 q^{24} + 272736 q^{25} + 13182 q^{26} + 306652 q^{27} + 338198 q^{28} + 350592 q^{29} + 231528 q^{30} + 55092 q^{31} + 114420 q^{32} + 609912 q^{33} + 812002 q^{34} + 351918 q^{35} + 1472143 q^{36} + 376310 q^{37} + 2825424 q^{38} + 180154 q^{39} + 2169290 q^{40} + 1387272 q^{41} + 105987 q^{42} + 568708 q^{43} + 3392031 q^{44} + 3556226 q^{45} - 1736829 q^{46} + 1359444 q^{47} + 4151249 q^{48} + 1411788 q^{49} + 3983712 q^{50} + 2709260 q^{51} + 2166242 q^{52} + 2061780 q^{53} + 2196651 q^{54} - 2112846 q^{55} + 78204 q^{56} + 2359902 q^{57} + 670268 q^{58} + 395964 q^{59} - 1052376 q^{60} + 444006 q^{61} + 2854353 q^{62} + 3739386 q^{63} + 12026858 q^{64} + 2254122 q^{65} - 4605681 q^{66} - 3094010 q^{67} + 4668954 q^{68} + 3839892 q^{69} + 2287124 q^{70} + 5694366 q^{71} - 9780585 q^{72} + 7052346 q^{73} - 4436259 q^{74} - 16288696 q^{75} - 3051830 q^{76} + 4173624 q^{77} + 678873 q^{78} + 4304160 q^{79} + 3807018 q^{80} - 6689556 q^{81} - 4733665 q^{82} + 2704554 q^{83} - 62769 q^{84} + 9301878 q^{85} + 1510998 q^{86} + 16231802 q^{87} - 70453923 q^{88} - 10986042 q^{89} - 12851300 q^{90} + 9042852 q^{91} - 16505451 q^{92} - 47230934 q^{93} - 24306151 q^{94} - 21839424 q^{95} - 86512741 q^{96} - 24462382 q^{97} + 705894 q^{98} + 11555078 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 13.7078 1.21161 0.605806 0.795613i \(-0.292851\pi\)
0.605806 + 0.795613i \(0.292851\pi\)
\(3\) 50.2321 1.07413 0.537065 0.843541i \(-0.319533\pi\)
0.537065 + 0.843541i \(0.319533\pi\)
\(4\) 59.9043 0.468002
\(5\) 544.453 1.94789 0.973947 0.226778i \(-0.0728191\pi\)
0.973947 + 0.226778i \(0.0728191\pi\)
\(6\) 688.573 1.30143
\(7\) 343.000 0.377964
\(8\) −933.443 −0.644574
\(9\) 336.267 0.153757
\(10\) 7463.26 2.36009
\(11\) 3185.05 0.721509 0.360755 0.932661i \(-0.382519\pi\)
0.360755 + 0.932661i \(0.382519\pi\)
\(12\) 3009.12 0.502696
\(13\) 2197.00 0.277350
\(14\) 4701.78 0.457946
\(15\) 27349.0 2.09229
\(16\) −20463.2 −1.24898
\(17\) 18610.0 0.918702 0.459351 0.888255i \(-0.348082\pi\)
0.459351 + 0.888255i \(0.348082\pi\)
\(18\) 4609.48 0.186294
\(19\) −48240.1 −1.61351 −0.806753 0.590889i \(-0.798777\pi\)
−0.806753 + 0.590889i \(0.798777\pi\)
\(20\) 32615.1 0.911619
\(21\) 17229.6 0.405983
\(22\) 43660.1 0.874189
\(23\) −101451. −1.73864 −0.869321 0.494249i \(-0.835443\pi\)
−0.869321 + 0.494249i \(0.835443\pi\)
\(24\) −46888.8 −0.692357
\(25\) 218304. 2.79429
\(26\) 30116.1 0.336041
\(27\) −92966.3 −0.908976
\(28\) 20547.2 0.176888
\(29\) 163486. 1.24476 0.622381 0.782714i \(-0.286165\pi\)
0.622381 + 0.782714i \(0.286165\pi\)
\(30\) 374895. 2.53504
\(31\) −76991.3 −0.464169 −0.232084 0.972696i \(-0.574555\pi\)
−0.232084 + 0.972696i \(0.574555\pi\)
\(32\) −161025. −0.868699
\(33\) 159992. 0.774995
\(34\) 255102. 1.11311
\(35\) 186747. 0.736234
\(36\) 20143.8 0.0719586
\(37\) 224246. 0.727811 0.363906 0.931436i \(-0.381443\pi\)
0.363906 + 0.931436i \(0.381443\pi\)
\(38\) −661266. −1.95494
\(39\) 110360. 0.297910
\(40\) −508216. −1.25556
\(41\) −71118.9 −0.161154 −0.0805771 0.996748i \(-0.525676\pi\)
−0.0805771 + 0.996748i \(0.525676\pi\)
\(42\) 236181. 0.491894
\(43\) −366021. −0.702048 −0.351024 0.936366i \(-0.614166\pi\)
−0.351024 + 0.936366i \(0.614166\pi\)
\(44\) 190798. 0.337668
\(45\) 183081. 0.299502
\(46\) −1.39068e6 −2.10656
\(47\) 545639. 0.766590 0.383295 0.923626i \(-0.374789\pi\)
0.383295 + 0.923626i \(0.374789\pi\)
\(48\) −1.02791e6 −1.34156
\(49\) 117649. 0.142857
\(50\) 2.99247e6 3.38559
\(51\) 934819. 0.986806
\(52\) 131610. 0.129801
\(53\) 1.16583e6 1.07565 0.537825 0.843057i \(-0.319246\pi\)
0.537825 + 0.843057i \(0.319246\pi\)
\(54\) −1.27436e6 −1.10133
\(55\) 1.73411e6 1.40542
\(56\) −320171. −0.243626
\(57\) −2.42320e6 −1.73312
\(58\) 2.24103e6 1.50817
\(59\) 67871.3 0.0430233 0.0215117 0.999769i \(-0.493152\pi\)
0.0215117 + 0.999769i \(0.493152\pi\)
\(60\) 1.63832e6 0.979198
\(61\) −2.67077e6 −1.50655 −0.753273 0.657708i \(-0.771526\pi\)
−0.753273 + 0.657708i \(0.771526\pi\)
\(62\) −1.05538e6 −0.562392
\(63\) 115339. 0.0581147
\(64\) 411985. 0.196450
\(65\) 1.19616e6 0.540248
\(66\) 2.19314e6 0.938993
\(67\) −803728. −0.326473 −0.163237 0.986587i \(-0.552193\pi\)
−0.163237 + 0.986587i \(0.552193\pi\)
\(68\) 1.11482e6 0.429955
\(69\) −5.09611e6 −1.86753
\(70\) 2.55990e6 0.892030
\(71\) −3.09022e6 −1.02467 −0.512337 0.858785i \(-0.671220\pi\)
−0.512337 + 0.858785i \(0.671220\pi\)
\(72\) −313886. −0.0991078
\(73\) 1.19301e6 0.358935 0.179468 0.983764i \(-0.442563\pi\)
0.179468 + 0.983764i \(0.442563\pi\)
\(74\) 3.07393e6 0.881825
\(75\) 1.09659e7 3.00143
\(76\) −2.88979e6 −0.755125
\(77\) 1.09247e6 0.272705
\(78\) 1.51279e6 0.360952
\(79\) −466693. −0.106497 −0.0532484 0.998581i \(-0.516958\pi\)
−0.0532484 + 0.998581i \(0.516958\pi\)
\(80\) −1.11413e7 −2.43287
\(81\) −5.40531e6 −1.13012
\(82\) −974885. −0.195256
\(83\) −3.98882e6 −0.765721 −0.382861 0.923806i \(-0.625061\pi\)
−0.382861 + 0.923806i \(0.625061\pi\)
\(84\) 1.03213e6 0.190001
\(85\) 1.01323e7 1.78953
\(86\) −5.01736e6 −0.850610
\(87\) 8.21223e6 1.33704
\(88\) −2.97306e6 −0.465066
\(89\) −387371. −0.0582454 −0.0291227 0.999576i \(-0.509271\pi\)
−0.0291227 + 0.999576i \(0.509271\pi\)
\(90\) 2.50964e6 0.362880
\(91\) 753571. 0.104828
\(92\) −6.07737e6 −0.813688
\(93\) −3.86744e6 −0.498578
\(94\) 7.47952e6 0.928809
\(95\) −2.62644e7 −3.14294
\(96\) −8.08865e6 −0.933097
\(97\) 8.39907e6 0.934394 0.467197 0.884153i \(-0.345264\pi\)
0.467197 + 0.884153i \(0.345264\pi\)
\(98\) 1.61271e6 0.173087
\(99\) 1.07103e6 0.110937
\(100\) 1.30773e7 1.30773
\(101\) −1.65322e7 −1.59663 −0.798315 0.602239i \(-0.794275\pi\)
−0.798315 + 0.602239i \(0.794275\pi\)
\(102\) 1.28143e7 1.19563
\(103\) −1.04478e7 −0.942095 −0.471048 0.882108i \(-0.656124\pi\)
−0.471048 + 0.882108i \(0.656124\pi\)
\(104\) −2.05078e6 −0.178773
\(105\) 9.38071e6 0.790812
\(106\) 1.59810e7 1.30327
\(107\) −9.15329e6 −0.722327 −0.361164 0.932502i \(-0.617620\pi\)
−0.361164 + 0.932502i \(0.617620\pi\)
\(108\) −5.56908e6 −0.425403
\(109\) 1.76810e7 1.30772 0.653859 0.756617i \(-0.273149\pi\)
0.653859 + 0.756617i \(0.273149\pi\)
\(110\) 2.37708e7 1.70283
\(111\) 1.12644e7 0.781765
\(112\) −7.01889e6 −0.472069
\(113\) −1.51600e7 −0.988380 −0.494190 0.869354i \(-0.664535\pi\)
−0.494190 + 0.869354i \(0.664535\pi\)
\(114\) −3.32168e7 −2.09986
\(115\) −5.52354e7 −3.38669
\(116\) 9.79349e6 0.582552
\(117\) 738778. 0.0426445
\(118\) 930367. 0.0521276
\(119\) 6.38322e6 0.347237
\(120\) −2.55288e7 −1.34864
\(121\) −9.34263e6 −0.479425
\(122\) −3.66104e7 −1.82535
\(123\) −3.57245e6 −0.173101
\(124\) −4.61211e6 −0.217232
\(125\) 7.63207e7 3.49508
\(126\) 1.58105e6 0.0704124
\(127\) −2.11379e7 −0.915690 −0.457845 0.889032i \(-0.651378\pi\)
−0.457845 + 0.889032i \(0.651378\pi\)
\(128\) 2.62587e7 1.10672
\(129\) −1.83860e7 −0.754092
\(130\) 1.63968e7 0.654571
\(131\) 1.89064e7 0.734783 0.367392 0.930066i \(-0.380251\pi\)
0.367392 + 0.930066i \(0.380251\pi\)
\(132\) 9.58420e6 0.362700
\(133\) −1.65464e7 −0.609848
\(134\) −1.10174e7 −0.395559
\(135\) −5.06157e7 −1.77059
\(136\) −1.73714e7 −0.592172
\(137\) −4.59503e7 −1.52675 −0.763373 0.645958i \(-0.776458\pi\)
−0.763373 + 0.645958i \(0.776458\pi\)
\(138\) −6.98566e7 −2.26272
\(139\) 3.80955e7 1.20315 0.601577 0.798815i \(-0.294539\pi\)
0.601577 + 0.798815i \(0.294539\pi\)
\(140\) 1.11870e7 0.344559
\(141\) 2.74086e7 0.823418
\(142\) −4.23602e7 −1.24151
\(143\) 6.99755e6 0.200111
\(144\) −6.88110e6 −0.192039
\(145\) 8.90102e7 2.42466
\(146\) 1.63536e7 0.434890
\(147\) 5.90976e6 0.153447
\(148\) 1.34333e7 0.340618
\(149\) −3.25299e7 −0.805621 −0.402811 0.915283i \(-0.631967\pi\)
−0.402811 + 0.915283i \(0.631967\pi\)
\(150\) 1.50318e8 3.63657
\(151\) 7.90428e7 1.86829 0.934143 0.356900i \(-0.116166\pi\)
0.934143 + 0.356900i \(0.116166\pi\)
\(152\) 4.50294e7 1.04002
\(153\) 6.25791e6 0.141257
\(154\) 1.49754e7 0.330412
\(155\) −4.19181e7 −0.904151
\(156\) 6.61104e6 0.139423
\(157\) 1.42411e6 0.0293693 0.0146847 0.999892i \(-0.495326\pi\)
0.0146847 + 0.999892i \(0.495326\pi\)
\(158\) −6.39734e6 −0.129033
\(159\) 5.85623e7 1.15539
\(160\) −8.76707e7 −1.69213
\(161\) −3.47978e7 −0.657145
\(162\) −7.40950e7 −1.36926
\(163\) 9.91388e7 1.79303 0.896514 0.443015i \(-0.146091\pi\)
0.896514 + 0.443015i \(0.146091\pi\)
\(164\) −4.26033e6 −0.0754206
\(165\) 8.71080e7 1.50961
\(166\) −5.46780e7 −0.927756
\(167\) −3.00881e7 −0.499905 −0.249952 0.968258i \(-0.580415\pi\)
−0.249952 + 0.968258i \(0.580415\pi\)
\(168\) −1.60829e7 −0.261686
\(169\) 4.82681e6 0.0769231
\(170\) 1.38891e8 2.16822
\(171\) −1.62215e7 −0.248088
\(172\) −2.19263e7 −0.328560
\(173\) 4.37139e7 0.641886 0.320943 0.947099i \(-0.396000\pi\)
0.320943 + 0.947099i \(0.396000\pi\)
\(174\) 1.12572e8 1.61997
\(175\) 7.48782e7 1.05614
\(176\) −6.51764e7 −0.901148
\(177\) 3.40932e6 0.0462127
\(178\) −5.31001e6 −0.0705708
\(179\) 1.33672e8 1.74203 0.871013 0.491260i \(-0.163463\pi\)
0.871013 + 0.491260i \(0.163463\pi\)
\(180\) 1.09674e7 0.140168
\(181\) 4.30566e7 0.539716 0.269858 0.962900i \(-0.413023\pi\)
0.269858 + 0.962900i \(0.413023\pi\)
\(182\) 1.03298e7 0.127011
\(183\) −1.34158e8 −1.61823
\(184\) 9.46990e7 1.12068
\(185\) 1.22091e8 1.41770
\(186\) −5.30142e7 −0.604083
\(187\) 5.92737e7 0.662852
\(188\) 3.26861e7 0.358766
\(189\) −3.18874e7 −0.343561
\(190\) −3.60028e8 −3.80802
\(191\) −1.71611e8 −1.78208 −0.891042 0.453921i \(-0.850025\pi\)
−0.891042 + 0.453921i \(0.850025\pi\)
\(192\) 2.06949e7 0.211013
\(193\) 1.81584e7 0.181814 0.0909069 0.995859i \(-0.471023\pi\)
0.0909069 + 0.995859i \(0.471023\pi\)
\(194\) 1.15133e8 1.13212
\(195\) 6.00858e7 0.580297
\(196\) 7.04768e6 0.0668575
\(197\) 7.06759e7 0.658627 0.329313 0.944221i \(-0.393183\pi\)
0.329313 + 0.944221i \(0.393183\pi\)
\(198\) 1.46814e7 0.134413
\(199\) −9.03017e6 −0.0812289 −0.0406144 0.999175i \(-0.512932\pi\)
−0.0406144 + 0.999175i \(0.512932\pi\)
\(200\) −2.03774e8 −1.80113
\(201\) −4.03730e7 −0.350675
\(202\) −2.26620e8 −1.93450
\(203\) 5.60756e7 0.470476
\(204\) 5.59997e7 0.461828
\(205\) −3.87209e7 −0.313911
\(206\) −1.43217e8 −1.14145
\(207\) −3.41147e7 −0.267328
\(208\) −4.49577e7 −0.346404
\(209\) −1.53647e8 −1.16416
\(210\) 1.28589e8 0.958157
\(211\) 7.31185e7 0.535845 0.267922 0.963441i \(-0.413663\pi\)
0.267922 + 0.963441i \(0.413663\pi\)
\(212\) 6.98384e7 0.503406
\(213\) −1.55228e8 −1.10063
\(214\) −1.25472e8 −0.875180
\(215\) −1.99281e8 −1.36751
\(216\) 8.67788e7 0.585902
\(217\) −2.64080e7 −0.175439
\(218\) 2.42368e8 1.58445
\(219\) 5.99277e7 0.385543
\(220\) 1.03881e8 0.657741
\(221\) 4.08861e7 0.254802
\(222\) 1.54410e8 0.947195
\(223\) 1.37129e8 0.828060 0.414030 0.910263i \(-0.364121\pi\)
0.414030 + 0.910263i \(0.364121\pi\)
\(224\) −5.52317e7 −0.328338
\(225\) 7.34082e7 0.429641
\(226\) −2.07810e8 −1.19753
\(227\) −9.17915e7 −0.520849 −0.260425 0.965494i \(-0.583863\pi\)
−0.260425 + 0.965494i \(0.583863\pi\)
\(228\) −1.45160e8 −0.811103
\(229\) −9.94686e7 −0.547346 −0.273673 0.961823i \(-0.588239\pi\)
−0.273673 + 0.961823i \(0.588239\pi\)
\(230\) −7.57157e8 −4.10335
\(231\) 5.48772e7 0.292921
\(232\) −1.52605e8 −0.802342
\(233\) 1.68551e8 0.872944 0.436472 0.899718i \(-0.356228\pi\)
0.436472 + 0.899718i \(0.356228\pi\)
\(234\) 1.01270e7 0.0516686
\(235\) 2.97075e8 1.49323
\(236\) 4.06578e6 0.0201350
\(237\) −2.34430e7 −0.114391
\(238\) 8.75001e7 0.420716
\(239\) −4.75224e7 −0.225167 −0.112584 0.993642i \(-0.535913\pi\)
−0.112584 + 0.993642i \(0.535913\pi\)
\(240\) −5.59649e8 −2.61322
\(241\) 1.39642e8 0.642622 0.321311 0.946974i \(-0.395877\pi\)
0.321311 + 0.946974i \(0.395877\pi\)
\(242\) −1.28067e8 −0.580876
\(243\) −6.82029e7 −0.304916
\(244\) −1.59991e8 −0.705067
\(245\) 6.40543e7 0.278270
\(246\) −4.89706e7 −0.209731
\(247\) −1.05983e8 −0.447506
\(248\) 7.18671e7 0.299191
\(249\) −2.00367e8 −0.822485
\(250\) 1.04619e9 4.23468
\(251\) −1.74028e8 −0.694643 −0.347322 0.937746i \(-0.612909\pi\)
−0.347322 + 0.937746i \(0.612909\pi\)
\(252\) 6.90933e6 0.0271978
\(253\) −3.23127e8 −1.25445
\(254\) −2.89754e8 −1.10946
\(255\) 5.08965e8 1.92219
\(256\) 3.07215e8 1.14447
\(257\) 3.63317e8 1.33512 0.667559 0.744557i \(-0.267339\pi\)
0.667559 + 0.744557i \(0.267339\pi\)
\(258\) −2.52032e8 −0.913666
\(259\) 7.69164e7 0.275087
\(260\) 7.16553e7 0.252838
\(261\) 5.49747e7 0.191391
\(262\) 2.59165e8 0.890272
\(263\) 1.43176e8 0.485318 0.242659 0.970112i \(-0.421980\pi\)
0.242659 + 0.970112i \(0.421980\pi\)
\(264\) −1.49343e8 −0.499542
\(265\) 6.34741e8 2.09525
\(266\) −2.26814e8 −0.738899
\(267\) −1.94585e7 −0.0625632
\(268\) −4.81468e7 −0.152790
\(269\) −6.42270e7 −0.201180 −0.100590 0.994928i \(-0.532073\pi\)
−0.100590 + 0.994928i \(0.532073\pi\)
\(270\) −6.93831e8 −2.14526
\(271\) 4.55860e8 1.39136 0.695679 0.718352i \(-0.255103\pi\)
0.695679 + 0.718352i \(0.255103\pi\)
\(272\) −3.80820e8 −1.14744
\(273\) 3.78535e7 0.112600
\(274\) −6.29879e8 −1.84982
\(275\) 6.95308e8 2.01610
\(276\) −3.05279e8 −0.874008
\(277\) 4.81356e8 1.36078 0.680389 0.732851i \(-0.261811\pi\)
0.680389 + 0.732851i \(0.261811\pi\)
\(278\) 5.22206e8 1.45776
\(279\) −2.58896e7 −0.0713692
\(280\) −1.74318e8 −0.474558
\(281\) 5.23317e8 1.40700 0.703498 0.710697i \(-0.251620\pi\)
0.703498 + 0.710697i \(0.251620\pi\)
\(282\) 3.75712e8 0.997662
\(283\) −2.76148e6 −0.00724252 −0.00362126 0.999993i \(-0.501153\pi\)
−0.00362126 + 0.999993i \(0.501153\pi\)
\(284\) −1.85118e8 −0.479550
\(285\) −1.31932e9 −3.37592
\(286\) 9.59212e7 0.242456
\(287\) −2.43938e7 −0.0609106
\(288\) −5.41475e7 −0.133569
\(289\) −6.40074e7 −0.155987
\(290\) 1.22014e9 2.93775
\(291\) 4.21903e8 1.00366
\(292\) 7.14667e7 0.167982
\(293\) 5.25307e8 1.22005 0.610023 0.792384i \(-0.291160\pi\)
0.610023 + 0.792384i \(0.291160\pi\)
\(294\) 8.10099e7 0.185918
\(295\) 3.69527e7 0.0838049
\(296\) −2.09321e8 −0.469129
\(297\) −2.96102e8 −0.655834
\(298\) −4.45914e8 −0.976100
\(299\) −2.22888e8 −0.482212
\(300\) 6.56902e8 1.40468
\(301\) −1.25545e8 −0.265349
\(302\) 1.08350e9 2.26364
\(303\) −8.30445e8 −1.71499
\(304\) 9.87148e8 2.01523
\(305\) −1.45411e9 −2.93459
\(306\) 8.57823e7 0.171148
\(307\) −5.59927e8 −1.10445 −0.552227 0.833694i \(-0.686222\pi\)
−0.552227 + 0.833694i \(0.686222\pi\)
\(308\) 6.54438e7 0.127626
\(309\) −5.24816e8 −1.01193
\(310\) −5.74606e8 −1.09548
\(311\) −6.45480e8 −1.21681 −0.608403 0.793628i \(-0.708189\pi\)
−0.608403 + 0.793628i \(0.708189\pi\)
\(312\) −1.03015e8 −0.192025
\(313\) −5.61337e8 −1.03471 −0.517355 0.855771i \(-0.673083\pi\)
−0.517355 + 0.855771i \(0.673083\pi\)
\(314\) 1.95214e7 0.0355842
\(315\) 6.27968e7 0.113201
\(316\) −2.79569e7 −0.0498407
\(317\) −9.48152e8 −1.67175 −0.835874 0.548922i \(-0.815038\pi\)
−0.835874 + 0.548922i \(0.815038\pi\)
\(318\) 8.02761e8 1.39988
\(319\) 5.20710e8 0.898107
\(320\) 2.24306e8 0.382663
\(321\) −4.59789e8 −0.775874
\(322\) −4.77002e8 −0.796204
\(323\) −8.97747e8 −1.48233
\(324\) −3.23801e8 −0.528897
\(325\) 4.79613e8 0.774996
\(326\) 1.35898e9 2.17245
\(327\) 8.88154e8 1.40466
\(328\) 6.63855e7 0.103876
\(329\) 1.87154e8 0.289744
\(330\) 1.19406e9 1.82906
\(331\) 1.27938e8 0.193911 0.0969556 0.995289i \(-0.469090\pi\)
0.0969556 + 0.995289i \(0.469090\pi\)
\(332\) −2.38947e8 −0.358359
\(333\) 7.54065e7 0.111906
\(334\) −4.12442e8 −0.605690
\(335\) −4.37592e8 −0.635935
\(336\) −3.52574e8 −0.507063
\(337\) 1.12221e8 0.159724 0.0798620 0.996806i \(-0.474552\pi\)
0.0798620 + 0.996806i \(0.474552\pi\)
\(338\) 6.61650e7 0.0932009
\(339\) −7.61518e8 −1.06165
\(340\) 6.06966e8 0.837506
\(341\) −2.45221e8 −0.334902
\(342\) −2.22362e8 −0.300586
\(343\) 4.03536e7 0.0539949
\(344\) 3.41660e8 0.452522
\(345\) −2.77459e9 −3.63774
\(346\) 5.99222e8 0.777716
\(347\) 1.44603e9 1.85791 0.928954 0.370196i \(-0.120710\pi\)
0.928954 + 0.370196i \(0.120710\pi\)
\(348\) 4.91948e8 0.625737
\(349\) −3.90752e7 −0.0492054 −0.0246027 0.999697i \(-0.507832\pi\)
−0.0246027 + 0.999697i \(0.507832\pi\)
\(350\) 1.02642e9 1.27963
\(351\) −2.04247e8 −0.252104
\(352\) −5.12874e8 −0.626775
\(353\) 7.97761e8 0.965298 0.482649 0.875814i \(-0.339675\pi\)
0.482649 + 0.875814i \(0.339675\pi\)
\(354\) 4.67343e7 0.0559918
\(355\) −1.68248e9 −1.99595
\(356\) −2.32052e7 −0.0272590
\(357\) 3.20643e8 0.372978
\(358\) 1.83235e9 2.11066
\(359\) 1.01654e9 1.15956 0.579779 0.814774i \(-0.303139\pi\)
0.579779 + 0.814774i \(0.303139\pi\)
\(360\) −1.70896e8 −0.193051
\(361\) 1.43323e9 1.60340
\(362\) 5.90213e8 0.653926
\(363\) −4.69300e8 −0.514965
\(364\) 4.51421e7 0.0490600
\(365\) 6.49540e8 0.699167
\(366\) −1.83902e9 −1.96066
\(367\) 8.06715e8 0.851901 0.425950 0.904747i \(-0.359940\pi\)
0.425950 + 0.904747i \(0.359940\pi\)
\(368\) 2.07602e9 2.17152
\(369\) −2.39149e7 −0.0247786
\(370\) 1.67361e9 1.71770
\(371\) 3.99881e8 0.406557
\(372\) −2.31676e8 −0.233336
\(373\) −1.13638e9 −1.13381 −0.566907 0.823782i \(-0.691860\pi\)
−0.566907 + 0.823782i \(0.691860\pi\)
\(374\) 8.12513e8 0.803119
\(375\) 3.83375e9 3.75417
\(376\) −5.09323e8 −0.494124
\(377\) 3.59178e8 0.345235
\(378\) −4.37107e8 −0.416262
\(379\) 1.89687e8 0.178979 0.0894893 0.995988i \(-0.471477\pi\)
0.0894893 + 0.995988i \(0.471477\pi\)
\(380\) −1.57335e9 −1.47090
\(381\) −1.06180e9 −0.983570
\(382\) −2.35241e9 −2.15919
\(383\) 3.11890e7 0.0283665 0.0141833 0.999899i \(-0.495485\pi\)
0.0141833 + 0.999899i \(0.495485\pi\)
\(384\) 1.31903e9 1.18876
\(385\) 5.94799e8 0.531200
\(386\) 2.48912e8 0.220288
\(387\) −1.23081e8 −0.107945
\(388\) 5.03140e8 0.437299
\(389\) −1.97500e9 −1.70115 −0.850575 0.525854i \(-0.823746\pi\)
−0.850575 + 0.525854i \(0.823746\pi\)
\(390\) 8.23645e8 0.703095
\(391\) −1.88801e9 −1.59729
\(392\) −1.09819e8 −0.0920821
\(393\) 9.49708e8 0.789253
\(394\) 9.68812e8 0.798000
\(395\) −2.54092e8 −0.207444
\(396\) 6.41590e7 0.0519188
\(397\) −1.01548e9 −0.814524 −0.407262 0.913311i \(-0.633516\pi\)
−0.407262 + 0.913311i \(0.633516\pi\)
\(398\) −1.23784e8 −0.0984178
\(399\) −8.31158e8 −0.655056
\(400\) −4.46720e9 −3.49000
\(401\) 1.88819e9 1.46231 0.731157 0.682210i \(-0.238981\pi\)
0.731157 + 0.682210i \(0.238981\pi\)
\(402\) −5.53426e8 −0.424882
\(403\) −1.69150e8 −0.128737
\(404\) −9.90347e8 −0.747227
\(405\) −2.94293e9 −2.20134
\(406\) 7.68674e8 0.570034
\(407\) 7.14235e8 0.525123
\(408\) −8.72600e8 −0.636070
\(409\) −2.45154e9 −1.77177 −0.885886 0.463904i \(-0.846448\pi\)
−0.885886 + 0.463904i \(0.846448\pi\)
\(410\) −5.30779e8 −0.380338
\(411\) −2.30818e9 −1.63992
\(412\) −6.25869e8 −0.440903
\(413\) 2.32799e7 0.0162613
\(414\) −4.67637e8 −0.323898
\(415\) −2.17172e9 −1.49154
\(416\) −3.53773e8 −0.240934
\(417\) 1.91362e9 1.29235
\(418\) −2.10617e9 −1.41051
\(419\) 1.83732e8 0.122021 0.0610105 0.998137i \(-0.480568\pi\)
0.0610105 + 0.998137i \(0.480568\pi\)
\(420\) 5.61945e8 0.370102
\(421\) −3.42562e8 −0.223744 −0.111872 0.993723i \(-0.535685\pi\)
−0.111872 + 0.993723i \(0.535685\pi\)
\(422\) 1.00230e9 0.649235
\(423\) 1.83480e8 0.117869
\(424\) −1.08824e9 −0.693336
\(425\) 4.06263e9 2.56712
\(426\) −2.12784e9 −1.33354
\(427\) −9.16074e8 −0.569421
\(428\) −5.48322e8 −0.338051
\(429\) 3.51502e8 0.214945
\(430\) −2.73171e9 −1.65690
\(431\) −8.39710e8 −0.505195 −0.252597 0.967571i \(-0.581285\pi\)
−0.252597 + 0.967571i \(0.581285\pi\)
\(432\) 1.90239e9 1.13529
\(433\) 1.04393e9 0.617963 0.308981 0.951068i \(-0.400012\pi\)
0.308981 + 0.951068i \(0.400012\pi\)
\(434\) −3.61997e8 −0.212564
\(435\) 4.47117e9 2.60441
\(436\) 1.05917e9 0.612015
\(437\) 4.89402e9 2.80531
\(438\) 8.21478e8 0.467129
\(439\) −6.15666e8 −0.347312 −0.173656 0.984806i \(-0.555558\pi\)
−0.173656 + 0.984806i \(0.555558\pi\)
\(440\) −1.61869e9 −0.905899
\(441\) 3.95614e7 0.0219653
\(442\) 5.60460e8 0.308721
\(443\) 2.86221e7 0.0156419 0.00782093 0.999969i \(-0.497510\pi\)
0.00782093 + 0.999969i \(0.497510\pi\)
\(444\) 6.74784e8 0.365868
\(445\) −2.10905e8 −0.113456
\(446\) 1.87974e9 1.00329
\(447\) −1.63405e9 −0.865343
\(448\) 1.41311e8 0.0742511
\(449\) −1.95784e9 −1.02074 −0.510370 0.859955i \(-0.670491\pi\)
−0.510370 + 0.859955i \(0.670491\pi\)
\(450\) 1.00627e9 0.520558
\(451\) −2.26517e8 −0.116274
\(452\) −9.08148e8 −0.462564
\(453\) 3.97049e9 2.00678
\(454\) −1.25826e9 −0.631067
\(455\) 4.10284e8 0.204195
\(456\) 2.26192e9 1.11712
\(457\) 1.67574e9 0.821299 0.410649 0.911793i \(-0.365302\pi\)
0.410649 + 0.911793i \(0.365302\pi\)
\(458\) −1.36350e9 −0.663171
\(459\) −1.73010e9 −0.835078
\(460\) −3.30884e9 −1.58498
\(461\) −1.66083e9 −0.789537 −0.394768 0.918781i \(-0.629175\pi\)
−0.394768 + 0.918781i \(0.629175\pi\)
\(462\) 7.52247e8 0.354906
\(463\) 6.87351e8 0.321844 0.160922 0.986967i \(-0.448553\pi\)
0.160922 + 0.986967i \(0.448553\pi\)
\(464\) −3.34544e9 −1.55468
\(465\) −2.10564e9 −0.971177
\(466\) 2.31047e9 1.05767
\(467\) 1.76187e9 0.800506 0.400253 0.916405i \(-0.368922\pi\)
0.400253 + 0.916405i \(0.368922\pi\)
\(468\) 4.42560e7 0.0199577
\(469\) −2.75679e8 −0.123395
\(470\) 4.07225e9 1.80922
\(471\) 7.15360e7 0.0315465
\(472\) −6.33540e7 −0.0277317
\(473\) −1.16580e9 −0.506534
\(474\) −3.21352e8 −0.138598
\(475\) −1.05310e10 −4.50860
\(476\) 3.82383e8 0.162508
\(477\) 3.92031e8 0.165389
\(478\) −6.51428e8 −0.272815
\(479\) 3.42957e8 0.142582 0.0712911 0.997456i \(-0.477288\pi\)
0.0712911 + 0.997456i \(0.477288\pi\)
\(480\) −4.40389e9 −1.81757
\(481\) 4.92669e8 0.201859
\(482\) 1.91418e9 0.778608
\(483\) −1.74797e9 −0.705859
\(484\) −5.59664e8 −0.224372
\(485\) 4.57290e9 1.82010
\(486\) −9.34913e8 −0.369440
\(487\) −6.48627e6 −0.00254474 −0.00127237 0.999999i \(-0.500405\pi\)
−0.00127237 + 0.999999i \(0.500405\pi\)
\(488\) 2.49301e9 0.971080
\(489\) 4.97995e9 1.92595
\(490\) 8.78045e8 0.337156
\(491\) −2.70060e9 −1.02962 −0.514808 0.857306i \(-0.672137\pi\)
−0.514808 + 0.857306i \(0.672137\pi\)
\(492\) −2.14005e8 −0.0810115
\(493\) 3.04246e9 1.14357
\(494\) −1.45280e9 −0.542203
\(495\) 5.83123e8 0.216094
\(496\) 1.57549e9 0.579736
\(497\) −1.05995e9 −0.387290
\(498\) −2.74659e9 −0.996532
\(499\) −4.53628e8 −0.163436 −0.0817180 0.996655i \(-0.526041\pi\)
−0.0817180 + 0.996655i \(0.526041\pi\)
\(500\) 4.57194e9 1.63571
\(501\) −1.51139e9 −0.536963
\(502\) −2.38555e9 −0.841638
\(503\) 2.63093e9 0.921767 0.460884 0.887461i \(-0.347533\pi\)
0.460884 + 0.887461i \(0.347533\pi\)
\(504\) −1.07663e8 −0.0374592
\(505\) −9.00097e9 −3.11007
\(506\) −4.42937e9 −1.51990
\(507\) 2.42461e8 0.0826254
\(508\) −1.26625e9 −0.428545
\(509\) 2.69397e9 0.905485 0.452742 0.891641i \(-0.350446\pi\)
0.452742 + 0.891641i \(0.350446\pi\)
\(510\) 6.97679e9 2.32895
\(511\) 4.09204e8 0.135665
\(512\) 8.50138e8 0.279927
\(513\) 4.48470e9 1.46664
\(514\) 4.98028e9 1.61764
\(515\) −5.68834e9 −1.83510
\(516\) −1.10140e9 −0.352917
\(517\) 1.73789e9 0.553101
\(518\) 1.05436e9 0.333298
\(519\) 2.19584e9 0.689469
\(520\) −1.11655e9 −0.348230
\(521\) −2.33917e9 −0.724651 −0.362326 0.932052i \(-0.618017\pi\)
−0.362326 + 0.932052i \(0.618017\pi\)
\(522\) 7.53584e8 0.231891
\(523\) 1.44074e9 0.440381 0.220191 0.975457i \(-0.429332\pi\)
0.220191 + 0.975457i \(0.429332\pi\)
\(524\) 1.13257e9 0.343880
\(525\) 3.76129e9 1.13443
\(526\) 1.96263e9 0.588016
\(527\) −1.43281e9 −0.426433
\(528\) −3.27395e9 −0.967950
\(529\) 6.88753e9 2.02287
\(530\) 8.70091e9 2.53863
\(531\) 2.28228e7 0.00661514
\(532\) −9.91198e8 −0.285410
\(533\) −1.56248e8 −0.0446961
\(534\) −2.66733e8 −0.0758023
\(535\) −4.98353e9 −1.40702
\(536\) 7.50235e8 0.210436
\(537\) 6.71463e9 1.87116
\(538\) −8.80412e8 −0.243752
\(539\) 3.74718e8 0.103073
\(540\) −3.03210e9 −0.828639
\(541\) −2.49039e9 −0.676202 −0.338101 0.941110i \(-0.609785\pi\)
−0.338101 + 0.941110i \(0.609785\pi\)
\(542\) 6.24885e9 1.68579
\(543\) 2.16283e9 0.579725
\(544\) −2.99668e9 −0.798076
\(545\) 9.62646e9 2.54729
\(546\) 5.18889e8 0.136427
\(547\) −3.38305e9 −0.883798 −0.441899 0.897065i \(-0.645695\pi\)
−0.441899 + 0.897065i \(0.645695\pi\)
\(548\) −2.75262e9 −0.714520
\(549\) −8.98091e8 −0.231642
\(550\) 9.53116e9 2.44273
\(551\) −7.88656e9 −2.00843
\(552\) 4.75693e9 1.20376
\(553\) −1.60076e8 −0.0402520
\(554\) 6.59834e9 1.64873
\(555\) 6.13291e9 1.52279
\(556\) 2.28208e9 0.563079
\(557\) −3.56492e9 −0.874092 −0.437046 0.899439i \(-0.643975\pi\)
−0.437046 + 0.899439i \(0.643975\pi\)
\(558\) −3.54890e8 −0.0864717
\(559\) −8.04149e8 −0.194713
\(560\) −3.82145e9 −0.919539
\(561\) 2.97744e9 0.711989
\(562\) 7.17354e9 1.70473
\(563\) 1.98385e9 0.468522 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(564\) 1.64189e9 0.385361
\(565\) −8.25389e9 −1.92526
\(566\) −3.78539e7 −0.00877512
\(567\) −1.85402e9 −0.427144
\(568\) 2.88455e9 0.660478
\(569\) −2.67463e9 −0.608653 −0.304327 0.952568i \(-0.598431\pi\)
−0.304327 + 0.952568i \(0.598431\pi\)
\(570\) −1.80850e10 −4.09031
\(571\) −3.17840e8 −0.0714468 −0.0357234 0.999362i \(-0.511374\pi\)
−0.0357234 + 0.999362i \(0.511374\pi\)
\(572\) 4.19184e8 0.0936522
\(573\) −8.62038e9 −1.91419
\(574\) −3.34386e8 −0.0737999
\(575\) −2.21472e10 −4.85826
\(576\) 1.38537e8 0.0302055
\(577\) 7.42047e9 1.60811 0.804056 0.594554i \(-0.202671\pi\)
0.804056 + 0.594554i \(0.202671\pi\)
\(578\) −8.77402e8 −0.188995
\(579\) 9.12135e8 0.195292
\(580\) 5.33209e9 1.13475
\(581\) −1.36816e9 −0.289415
\(582\) 5.78337e9 1.21605
\(583\) 3.71324e9 0.776091
\(584\) −1.11361e9 −0.231360
\(585\) 4.02229e8 0.0830669
\(586\) 7.20081e9 1.47822
\(587\) 6.74146e9 1.37569 0.687845 0.725858i \(-0.258557\pi\)
0.687845 + 0.725858i \(0.258557\pi\)
\(588\) 3.54020e8 0.0718137
\(589\) 3.71407e9 0.748939
\(590\) 5.06541e8 0.101539
\(591\) 3.55020e9 0.707451
\(592\) −4.58880e9 −0.909019
\(593\) 2.47701e9 0.487794 0.243897 0.969801i \(-0.421574\pi\)
0.243897 + 0.969801i \(0.421574\pi\)
\(594\) −4.05892e9 −0.794616
\(595\) 3.47536e9 0.676380
\(596\) −1.94868e9 −0.377033
\(597\) −4.53605e8 −0.0872504
\(598\) −3.05531e9 −0.584254
\(599\) 4.69524e9 0.892614 0.446307 0.894880i \(-0.352739\pi\)
0.446307 + 0.894880i \(0.352739\pi\)
\(600\) −1.02360e10 −1.93465
\(601\) −1.29838e9 −0.243973 −0.121986 0.992532i \(-0.538926\pi\)
−0.121986 + 0.992532i \(0.538926\pi\)
\(602\) −1.72095e9 −0.321500
\(603\) −2.70267e8 −0.0501975
\(604\) 4.73501e9 0.874362
\(605\) −5.08662e9 −0.933868
\(606\) −1.13836e10 −2.07790
\(607\) −2.74419e9 −0.498028 −0.249014 0.968500i \(-0.580107\pi\)
−0.249014 + 0.968500i \(0.580107\pi\)
\(608\) 7.76788e9 1.40165
\(609\) 2.81679e9 0.505353
\(610\) −1.99327e10 −3.55558
\(611\) 1.19877e9 0.212614
\(612\) 3.74876e8 0.0661085
\(613\) −2.10437e9 −0.368986 −0.184493 0.982834i \(-0.559064\pi\)
−0.184493 + 0.982834i \(0.559064\pi\)
\(614\) −7.67538e9 −1.33817
\(615\) −1.94503e9 −0.337182
\(616\) −1.01976e9 −0.175779
\(617\) −4.96827e9 −0.851545 −0.425772 0.904830i \(-0.639998\pi\)
−0.425772 + 0.904830i \(0.639998\pi\)
\(618\) −7.19408e9 −1.22607
\(619\) 8.25149e9 1.39835 0.699174 0.714952i \(-0.253551\pi\)
0.699174 + 0.714952i \(0.253551\pi\)
\(620\) −2.51108e9 −0.423145
\(621\) 9.43154e9 1.58038
\(622\) −8.84812e9 −1.47430
\(623\) −1.32868e8 −0.0220147
\(624\) −2.25832e9 −0.372083
\(625\) 2.44980e10 4.01375
\(626\) −7.69471e9 −1.25367
\(627\) −7.71802e9 −1.25046
\(628\) 8.53102e7 0.0137449
\(629\) 4.17322e9 0.668642
\(630\) 8.60808e8 0.137156
\(631\) −1.03604e10 −1.64162 −0.820811 0.571200i \(-0.806478\pi\)
−0.820811 + 0.571200i \(0.806478\pi\)
\(632\) 4.35631e8 0.0686451
\(633\) 3.67290e9 0.575567
\(634\) −1.29971e10 −2.02551
\(635\) −1.15086e10 −1.78367
\(636\) 3.50813e9 0.540724
\(637\) 2.58475e8 0.0396214
\(638\) 7.13779e9 1.08816
\(639\) −1.03914e9 −0.157551
\(640\) 1.42966e10 2.15577
\(641\) 4.44804e9 0.667061 0.333531 0.942739i \(-0.391760\pi\)
0.333531 + 0.942739i \(0.391760\pi\)
\(642\) −6.30271e9 −0.940058
\(643\) 6.76162e9 1.00303 0.501513 0.865150i \(-0.332777\pi\)
0.501513 + 0.865150i \(0.332777\pi\)
\(644\) −2.08454e9 −0.307545
\(645\) −1.00103e10 −1.46889
\(646\) −1.23062e10 −1.79601
\(647\) −1.96787e9 −0.285648 −0.142824 0.989748i \(-0.545618\pi\)
−0.142824 + 0.989748i \(0.545618\pi\)
\(648\) 5.04555e9 0.728444
\(649\) 2.16173e8 0.0310417
\(650\) 6.57445e9 0.938994
\(651\) −1.32653e9 −0.188445
\(652\) 5.93884e9 0.839141
\(653\) 9.11572e9 1.28114 0.640568 0.767902i \(-0.278699\pi\)
0.640568 + 0.767902i \(0.278699\pi\)
\(654\) 1.21747e10 1.70190
\(655\) 1.02936e10 1.43128
\(656\) 1.45532e9 0.201278
\(657\) 4.01171e8 0.0551888
\(658\) 2.56548e9 0.351057
\(659\) 2.16900e9 0.295230 0.147615 0.989045i \(-0.452840\pi\)
0.147615 + 0.989045i \(0.452840\pi\)
\(660\) 5.21814e9 0.706500
\(661\) −1.00456e9 −0.135292 −0.0676459 0.997709i \(-0.521549\pi\)
−0.0676459 + 0.997709i \(0.521549\pi\)
\(662\) 1.75376e9 0.234945
\(663\) 2.05380e9 0.273691
\(664\) 3.72333e9 0.493564
\(665\) −9.00870e9 −1.18792
\(666\) 1.03366e9 0.135587
\(667\) −1.65858e10 −2.16420
\(668\) −1.80241e9 −0.233957
\(669\) 6.88827e9 0.889444
\(670\) −5.99843e9 −0.770506
\(671\) −8.50654e9 −1.08699
\(672\) −2.77441e9 −0.352677
\(673\) 1.05631e10 1.33579 0.667894 0.744257i \(-0.267196\pi\)
0.667894 + 0.744257i \(0.267196\pi\)
\(674\) 1.53831e9 0.193523
\(675\) −2.02949e10 −2.53994
\(676\) 2.89147e8 0.0360002
\(677\) 4.37092e9 0.541393 0.270697 0.962665i \(-0.412746\pi\)
0.270697 + 0.962665i \(0.412746\pi\)
\(678\) −1.04387e10 −1.28631
\(679\) 2.88088e9 0.353168
\(680\) −9.45788e9 −1.15349
\(681\) −4.61088e9 −0.559460
\(682\) −3.36145e9 −0.405771
\(683\) 2.59419e9 0.311551 0.155776 0.987792i \(-0.450212\pi\)
0.155776 + 0.987792i \(0.450212\pi\)
\(684\) −9.71739e8 −0.116106
\(685\) −2.50178e10 −2.97394
\(686\) 5.53160e8 0.0654209
\(687\) −4.99652e9 −0.587921
\(688\) 7.48998e9 0.876842
\(689\) 2.56133e9 0.298331
\(690\) −3.80336e10 −4.40753
\(691\) −5.26255e9 −0.606769 −0.303384 0.952868i \(-0.598117\pi\)
−0.303384 + 0.952868i \(0.598117\pi\)
\(692\) 2.61865e9 0.300404
\(693\) 3.67362e8 0.0419303
\(694\) 1.98219e10 2.25106
\(695\) 2.07412e10 2.34362
\(696\) −7.66565e9 −0.861820
\(697\) −1.32352e9 −0.148053
\(698\) −5.35636e8 −0.0596178
\(699\) 8.46669e9 0.937656
\(700\) 4.48552e9 0.494277
\(701\) −8.38476e8 −0.0919343 −0.0459671 0.998943i \(-0.514637\pi\)
−0.0459671 + 0.998943i \(0.514637\pi\)
\(702\) −2.79978e9 −0.305453
\(703\) −1.08177e10 −1.17433
\(704\) 1.31219e9 0.141740
\(705\) 1.49227e10 1.60393
\(706\) 1.09356e10 1.16957
\(707\) −5.67053e9 −0.603470
\(708\) 2.04233e8 0.0216276
\(709\) 9.93910e8 0.104733 0.0523667 0.998628i \(-0.483324\pi\)
0.0523667 + 0.998628i \(0.483324\pi\)
\(710\) −2.30631e10 −2.41832
\(711\) −1.56933e8 −0.0163746
\(712\) 3.61589e8 0.0375435
\(713\) 7.81087e9 0.807023
\(714\) 4.39531e9 0.451904
\(715\) 3.80984e9 0.389794
\(716\) 8.00753e9 0.815273
\(717\) −2.38715e9 −0.241859
\(718\) 1.39345e10 1.40493
\(719\) 1.44874e10 1.45358 0.726791 0.686859i \(-0.241011\pi\)
0.726791 + 0.686859i \(0.241011\pi\)
\(720\) −3.74643e9 −0.374071
\(721\) −3.58360e9 −0.356079
\(722\) 1.96465e10 1.94270
\(723\) 7.01450e9 0.690260
\(724\) 2.57928e9 0.252588
\(725\) 3.56895e10 3.47822
\(726\) −6.43308e9 −0.623937
\(727\) −2.04576e9 −0.197462 −0.0987311 0.995114i \(-0.531478\pi\)
−0.0987311 + 0.995114i \(0.531478\pi\)
\(728\) −7.03416e8 −0.0675698
\(729\) 8.39543e9 0.802596
\(730\) 8.90378e9 0.847119
\(731\) −6.81165e9 −0.644973
\(732\) −8.03667e9 −0.757334
\(733\) −3.60556e9 −0.338150 −0.169075 0.985603i \(-0.554078\pi\)
−0.169075 + 0.985603i \(0.554078\pi\)
\(734\) 1.10583e10 1.03217
\(735\) 3.21758e9 0.298899
\(736\) 1.63362e10 1.51036
\(737\) −2.55991e9 −0.235553
\(738\) −3.27821e8 −0.0300220
\(739\) −1.83001e10 −1.66800 −0.834002 0.551761i \(-0.813956\pi\)
−0.834002 + 0.551761i \(0.813956\pi\)
\(740\) 7.31380e9 0.663487
\(741\) −5.32378e9 −0.480680
\(742\) 5.48149e9 0.492589
\(743\) 8.66858e8 0.0775331 0.0387665 0.999248i \(-0.487657\pi\)
0.0387665 + 0.999248i \(0.487657\pi\)
\(744\) 3.61004e9 0.321371
\(745\) −1.77110e10 −1.56926
\(746\) −1.55772e10 −1.37374
\(747\) −1.34130e9 −0.117735
\(748\) 3.55075e9 0.310216
\(749\) −3.13958e9 −0.273014
\(750\) 5.25523e10 4.54860
\(751\) 4.42366e9 0.381103 0.190552 0.981677i \(-0.438972\pi\)
0.190552 + 0.981677i \(0.438972\pi\)
\(752\) −1.11655e10 −0.957452
\(753\) −8.74181e9 −0.746138
\(754\) 4.92354e9 0.418291
\(755\) 4.30351e10 3.63922
\(756\) −1.91019e9 −0.160787
\(757\) 3.88102e9 0.325170 0.162585 0.986695i \(-0.448017\pi\)
0.162585 + 0.986695i \(0.448017\pi\)
\(758\) 2.60020e9 0.216853
\(759\) −1.62314e10 −1.34744
\(760\) 2.45164e10 2.02586
\(761\) −2.25509e10 −1.85488 −0.927442 0.373967i \(-0.877997\pi\)
−0.927442 + 0.373967i \(0.877997\pi\)
\(762\) −1.45550e10 −1.19171
\(763\) 6.06458e9 0.494271
\(764\) −1.02802e10 −0.834019
\(765\) 3.40714e9 0.275153
\(766\) 4.27534e8 0.0343692
\(767\) 1.49113e8 0.0119325
\(768\) 1.54321e10 1.22931
\(769\) −4.41602e9 −0.350178 −0.175089 0.984553i \(-0.556021\pi\)
−0.175089 + 0.984553i \(0.556021\pi\)
\(770\) 8.15340e9 0.643608
\(771\) 1.82502e10 1.43409
\(772\) 1.08777e9 0.0850893
\(773\) −1.44577e10 −1.12583 −0.562913 0.826516i \(-0.690320\pi\)
−0.562913 + 0.826516i \(0.690320\pi\)
\(774\) −1.68717e9 −0.130787
\(775\) −1.68075e10 −1.29702
\(776\) −7.84006e9 −0.602287
\(777\) 3.86368e9 0.295479
\(778\) −2.70729e10 −2.06113
\(779\) 3.43078e9 0.260023
\(780\) 3.59940e9 0.271581
\(781\) −9.84251e9 −0.739311
\(782\) −2.58804e10 −1.93530
\(783\) −1.51986e10 −1.13146
\(784\) −2.40748e9 −0.178425
\(785\) 7.75360e8 0.0572083
\(786\) 1.30184e10 0.956268
\(787\) 2.10751e10 1.54120 0.770598 0.637321i \(-0.219958\pi\)
0.770598 + 0.637321i \(0.219958\pi\)
\(788\) 4.23379e9 0.308239
\(789\) 7.19205e9 0.521295
\(790\) −3.48305e9 −0.251342
\(791\) −5.19987e9 −0.373572
\(792\) −9.99742e8 −0.0715072
\(793\) −5.86768e9 −0.417840
\(794\) −1.39200e10 −0.986886
\(795\) 3.18844e10 2.25057
\(796\) −5.40946e8 −0.0380153
\(797\) −1.76476e10 −1.23476 −0.617380 0.786665i \(-0.711806\pi\)
−0.617380 + 0.786665i \(0.711806\pi\)
\(798\) −1.13934e10 −0.793674
\(799\) 1.01543e10 0.704267
\(800\) −3.51524e10 −2.42740
\(801\) −1.30260e8 −0.00895564
\(802\) 2.58830e10 1.77176
\(803\) 3.79981e9 0.258975
\(804\) −2.41852e9 −0.164117
\(805\) −1.89457e10 −1.28005
\(806\) −2.31868e9 −0.155980
\(807\) −3.22626e9 −0.216094
\(808\) 1.54318e10 1.02915
\(809\) −5.90754e9 −0.392272 −0.196136 0.980577i \(-0.562839\pi\)
−0.196136 + 0.980577i \(0.562839\pi\)
\(810\) −4.03412e10 −2.66717
\(811\) −2.45675e10 −1.61729 −0.808646 0.588295i \(-0.799799\pi\)
−0.808646 + 0.588295i \(0.799799\pi\)
\(812\) 3.35917e9 0.220184
\(813\) 2.28988e10 1.49450
\(814\) 9.79060e9 0.636245
\(815\) 5.39764e10 3.49263
\(816\) −1.91294e10 −1.23250
\(817\) 1.76569e10 1.13276
\(818\) −3.36053e10 −2.14670
\(819\) 2.53401e8 0.0161181
\(820\) −2.31955e9 −0.146911
\(821\) −1.65259e10 −1.04223 −0.521115 0.853487i \(-0.674484\pi\)
−0.521115 + 0.853487i \(0.674484\pi\)
\(822\) −3.16401e10 −1.98695
\(823\) 9.90852e9 0.619597 0.309799 0.950802i \(-0.399738\pi\)
0.309799 + 0.950802i \(0.399738\pi\)
\(824\) 9.75244e9 0.607251
\(825\) 3.49268e10 2.16556
\(826\) 3.19116e8 0.0197024
\(827\) −1.09902e10 −0.675671 −0.337835 0.941205i \(-0.609695\pi\)
−0.337835 + 0.941205i \(0.609695\pi\)
\(828\) −2.04361e9 −0.125110
\(829\) −1.40135e10 −0.854294 −0.427147 0.904182i \(-0.640481\pi\)
−0.427147 + 0.904182i \(0.640481\pi\)
\(830\) −2.97696e10 −1.80717
\(831\) 2.41795e10 1.46165
\(832\) 9.05132e8 0.0544854
\(833\) 2.18945e9 0.131243
\(834\) 2.62315e10 1.56582
\(835\) −1.63816e10 −0.973761
\(836\) −9.20412e9 −0.544829
\(837\) 7.15760e9 0.421918
\(838\) 2.51856e9 0.147842
\(839\) 4.20630e9 0.245886 0.122943 0.992414i \(-0.460767\pi\)
0.122943 + 0.992414i \(0.460767\pi\)
\(840\) −8.75636e9 −0.509737
\(841\) 9.47766e9 0.549433
\(842\) −4.69578e9 −0.271091
\(843\) 2.62874e10 1.51130
\(844\) 4.38011e9 0.250777
\(845\) 2.62797e9 0.149838
\(846\) 2.51511e9 0.142811
\(847\) −3.20452e9 −0.181206
\(848\) −2.38567e10 −1.34346
\(849\) −1.38715e8 −0.00777941
\(850\) 5.56898e10 3.11035
\(851\) −2.27500e10 −1.26540
\(852\) −9.29885e9 −0.515099
\(853\) −5.04693e9 −0.278423 −0.139212 0.990263i \(-0.544457\pi\)
−0.139212 + 0.990263i \(0.544457\pi\)
\(854\) −1.25574e10 −0.689916
\(855\) −8.83185e9 −0.483248
\(856\) 8.54408e9 0.465594
\(857\) −2.34862e10 −1.27462 −0.637310 0.770607i \(-0.719953\pi\)
−0.637310 + 0.770607i \(0.719953\pi\)
\(858\) 4.81833e9 0.260430
\(859\) −3.57907e10 −1.92661 −0.963305 0.268408i \(-0.913503\pi\)
−0.963305 + 0.268408i \(0.913503\pi\)
\(860\) −1.19378e10 −0.640000
\(861\) −1.22535e9 −0.0654259
\(862\) −1.15106e10 −0.612100
\(863\) −1.96959e10 −1.04313 −0.521565 0.853212i \(-0.674651\pi\)
−0.521565 + 0.853212i \(0.674651\pi\)
\(864\) 1.49699e10 0.789627
\(865\) 2.38001e10 1.25032
\(866\) 1.43100e10 0.748731
\(867\) −3.21523e9 −0.167550
\(868\) −1.58195e9 −0.0821060
\(869\) −1.48644e9 −0.0768384
\(870\) 6.12900e10 3.15553
\(871\) −1.76579e9 −0.0905474
\(872\) −1.65042e10 −0.842921
\(873\) 2.82433e9 0.143670
\(874\) 6.70863e10 3.39894
\(875\) 2.61780e10 1.32102
\(876\) 3.58993e9 0.180435
\(877\) −1.17028e10 −0.585857 −0.292929 0.956134i \(-0.594630\pi\)
−0.292929 + 0.956134i \(0.594630\pi\)
\(878\) −8.43944e9 −0.420807
\(879\) 2.63873e10 1.31049
\(880\) −3.54855e10 −1.75534
\(881\) −4.39739e9 −0.216660 −0.108330 0.994115i \(-0.534550\pi\)
−0.108330 + 0.994115i \(0.534550\pi\)
\(882\) 5.42301e8 0.0266134
\(883\) −3.98937e10 −1.95003 −0.975017 0.222131i \(-0.928699\pi\)
−0.975017 + 0.222131i \(0.928699\pi\)
\(884\) 2.44925e9 0.119248
\(885\) 1.85621e9 0.0900174
\(886\) 3.92346e8 0.0189518
\(887\) 3.19786e10 1.53860 0.769302 0.638885i \(-0.220604\pi\)
0.769302 + 0.638885i \(0.220604\pi\)
\(888\) −1.05146e10 −0.503906
\(889\) −7.25029e9 −0.346098
\(890\) −2.89105e9 −0.137464
\(891\) −1.72162e10 −0.815389
\(892\) 8.21461e9 0.387534
\(893\) −2.63217e10 −1.23690
\(894\) −2.23992e10 −1.04846
\(895\) 7.27781e10 3.39328
\(896\) 9.00673e9 0.418301
\(897\) −1.11962e10 −0.517959
\(898\) −2.68377e10 −1.23674
\(899\) −1.25870e10 −0.577780
\(900\) 4.39747e9 0.201073
\(901\) 2.16961e10 0.988201
\(902\) −3.10506e9 −0.140879
\(903\) −6.30641e9 −0.285020
\(904\) 1.41510e10 0.637084
\(905\) 2.34423e10 1.05131
\(906\) 5.44267e10 2.43144
\(907\) −3.40207e10 −1.51397 −0.756985 0.653432i \(-0.773329\pi\)
−0.756985 + 0.653432i \(0.773329\pi\)
\(908\) −5.49871e9 −0.243759
\(909\) −5.55921e9 −0.245493
\(910\) 5.62410e9 0.247405
\(911\) −9.45648e9 −0.414396 −0.207198 0.978299i \(-0.566434\pi\)
−0.207198 + 0.978299i \(0.566434\pi\)
\(912\) 4.95865e10 2.16462
\(913\) −1.27046e10 −0.552475
\(914\) 2.29708e10 0.995095
\(915\) −7.30429e10 −3.15213
\(916\) −5.95860e9 −0.256159
\(917\) 6.48489e9 0.277722
\(918\) −2.37159e10 −1.01179
\(919\) −4.30221e9 −0.182847 −0.0914234 0.995812i \(-0.529142\pi\)
−0.0914234 + 0.995812i \(0.529142\pi\)
\(920\) 5.15591e10 2.18297
\(921\) −2.81263e10 −1.18633
\(922\) −2.27664e10 −0.956612
\(923\) −6.78922e9 −0.284193
\(924\) 3.28738e9 0.137088
\(925\) 4.89538e10 2.03371
\(926\) 9.42208e9 0.389950
\(927\) −3.51325e9 −0.144854
\(928\) −2.63253e10 −1.08132
\(929\) 1.94090e10 0.794235 0.397118 0.917768i \(-0.370010\pi\)
0.397118 + 0.917768i \(0.370010\pi\)
\(930\) −2.88637e10 −1.17669
\(931\) −5.67540e9 −0.230501
\(932\) 1.00969e10 0.408540
\(933\) −3.24238e10 −1.30701
\(934\) 2.41514e10 0.969902
\(935\) 3.22717e10 1.29116
\(936\) −6.89607e8 −0.0274876
\(937\) 3.16846e10 1.25823 0.629114 0.777313i \(-0.283418\pi\)
0.629114 + 0.777313i \(0.283418\pi\)
\(938\) −3.77896e9 −0.149507
\(939\) −2.81972e10 −1.11141
\(940\) 1.77961e10 0.698837
\(941\) 1.04030e10 0.407001 0.203501 0.979075i \(-0.434768\pi\)
0.203501 + 0.979075i \(0.434768\pi\)
\(942\) 9.80602e8 0.0382221
\(943\) 7.21510e9 0.280189
\(944\) −1.38887e9 −0.0537351
\(945\) −1.73612e10 −0.669219
\(946\) −1.59805e10 −0.613723
\(947\) −2.96702e10 −1.13526 −0.567631 0.823283i \(-0.692140\pi\)
−0.567631 + 0.823283i \(0.692140\pi\)
\(948\) −1.40433e9 −0.0535355
\(949\) 2.62105e9 0.0995507
\(950\) −1.44357e11 −5.46267
\(951\) −4.76277e10 −1.79568
\(952\) −5.95838e9 −0.223820
\(953\) 3.63188e10 1.35927 0.679637 0.733549i \(-0.262137\pi\)
0.679637 + 0.733549i \(0.262137\pi\)
\(954\) 5.37388e9 0.200387
\(955\) −9.34340e10 −3.47131
\(956\) −2.84679e9 −0.105379
\(957\) 2.61564e10 0.964685
\(958\) 4.70119e9 0.172754
\(959\) −1.57610e10 −0.577055
\(960\) 1.12674e10 0.411031
\(961\) −2.15849e10 −0.784547
\(962\) 6.75341e9 0.244574
\(963\) −3.07795e9 −0.111063
\(964\) 8.36514e9 0.300748
\(965\) 9.88639e9 0.354154
\(966\) −2.39608e10 −0.855227
\(967\) −1.26084e10 −0.448403 −0.224202 0.974543i \(-0.571977\pi\)
−0.224202 + 0.974543i \(0.571977\pi\)
\(968\) 8.72082e9 0.309025
\(969\) −4.50957e10 −1.59222
\(970\) 6.26844e10 2.20525
\(971\) −4.35074e10 −1.52509 −0.762546 0.646935i \(-0.776051\pi\)
−0.762546 + 0.646935i \(0.776051\pi\)
\(972\) −4.08565e9 −0.142702
\(973\) 1.30667e10 0.454750
\(974\) −8.89126e7 −0.00308324
\(975\) 2.40920e10 0.832447
\(976\) 5.46526e10 1.88164
\(977\) −1.29363e10 −0.443793 −0.221897 0.975070i \(-0.571225\pi\)
−0.221897 + 0.975070i \(0.571225\pi\)
\(978\) 6.82643e10 2.33350
\(979\) −1.23380e9 −0.0420246
\(980\) 3.83713e9 0.130231
\(981\) 5.94553e9 0.201071
\(982\) −3.70193e10 −1.24749
\(983\) −3.80363e10 −1.27721 −0.638604 0.769536i \(-0.720488\pi\)
−0.638604 + 0.769536i \(0.720488\pi\)
\(984\) 3.33468e9 0.111576
\(985\) 3.84797e10 1.28293
\(986\) 4.17055e10 1.38556
\(987\) 9.40115e9 0.311223
\(988\) −6.34887e9 −0.209434
\(989\) 3.71333e10 1.22061
\(990\) 7.99334e9 0.261821
\(991\) 4.33456e10 1.41477 0.707387 0.706827i \(-0.249874\pi\)
0.707387 + 0.706827i \(0.249874\pi\)
\(992\) 1.23976e10 0.403223
\(993\) 6.42662e9 0.208286
\(994\) −1.45295e10 −0.469245
\(995\) −4.91650e9 −0.158225
\(996\) −1.20028e10 −0.384925
\(997\) −4.73692e9 −0.151378 −0.0756890 0.997131i \(-0.524116\pi\)
−0.0756890 + 0.997131i \(0.524116\pi\)
\(998\) −6.21825e9 −0.198021
\(999\) −2.08473e10 −0.661563
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 91.8.a.e.1.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.e.1.9 12 1.1 even 1 trivial