Properties

Label 8034.2.a.l.1.1
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -2.56155 q^{11} -1.00000 q^{12} -1.00000 q^{13} +0.561553 q^{14} +2.00000 q^{15} +1.00000 q^{16} +3.43845 q^{17} -1.00000 q^{18} -2.00000 q^{19} -2.00000 q^{20} +0.561553 q^{21} +2.56155 q^{22} +3.12311 q^{23} +1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -0.561553 q^{28} +6.00000 q^{29} -2.00000 q^{30} +1.12311 q^{31} -1.00000 q^{32} +2.56155 q^{33} -3.43845 q^{34} +1.12311 q^{35} +1.00000 q^{36} -9.12311 q^{37} +2.00000 q^{38} +1.00000 q^{39} +2.00000 q^{40} +2.00000 q^{41} -0.561553 q^{42} +1.12311 q^{43} -2.56155 q^{44} -2.00000 q^{45} -3.12311 q^{46} -10.2462 q^{47} -1.00000 q^{48} -6.68466 q^{49} +1.00000 q^{50} -3.43845 q^{51} -1.00000 q^{52} +6.56155 q^{53} +1.00000 q^{54} +5.12311 q^{55} +0.561553 q^{56} +2.00000 q^{57} -6.00000 q^{58} -8.00000 q^{59} +2.00000 q^{60} +11.1231 q^{61} -1.12311 q^{62} -0.561553 q^{63} +1.00000 q^{64} +2.00000 q^{65} -2.56155 q^{66} +11.6847 q^{67} +3.43845 q^{68} -3.12311 q^{69} -1.12311 q^{70} +2.24621 q^{71} -1.00000 q^{72} +3.68466 q^{73} +9.12311 q^{74} +1.00000 q^{75} -2.00000 q^{76} +1.43845 q^{77} -1.00000 q^{78} +7.36932 q^{79} -2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +11.3693 q^{83} +0.561553 q^{84} -6.87689 q^{85} -1.12311 q^{86} -6.00000 q^{87} +2.56155 q^{88} +2.00000 q^{89} +2.00000 q^{90} +0.561553 q^{91} +3.12311 q^{92} -1.12311 q^{93} +10.2462 q^{94} +4.00000 q^{95} +1.00000 q^{96} +4.24621 q^{97} +6.68466 q^{98} -2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} + 3q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} + 3q^{7} - 2q^{8} + 2q^{9} + 4q^{10} - q^{11} - 2q^{12} - 2q^{13} - 3q^{14} + 4q^{15} + 2q^{16} + 11q^{17} - 2q^{18} - 4q^{19} - 4q^{20} - 3q^{21} + q^{22} - 2q^{23} + 2q^{24} - 2q^{25} + 2q^{26} - 2q^{27} + 3q^{28} + 12q^{29} - 4q^{30} - 6q^{31} - 2q^{32} + q^{33} - 11q^{34} - 6q^{35} + 2q^{36} - 10q^{37} + 4q^{38} + 2q^{39} + 4q^{40} + 4q^{41} + 3q^{42} - 6q^{43} - q^{44} - 4q^{45} + 2q^{46} - 4q^{47} - 2q^{48} - q^{49} + 2q^{50} - 11q^{51} - 2q^{52} + 9q^{53} + 2q^{54} + 2q^{55} - 3q^{56} + 4q^{57} - 12q^{58} - 16q^{59} + 4q^{60} + 14q^{61} + 6q^{62} + 3q^{63} + 2q^{64} + 4q^{65} - q^{66} + 11q^{67} + 11q^{68} + 2q^{69} + 6q^{70} - 12q^{71} - 2q^{72} - 5q^{73} + 10q^{74} + 2q^{75} - 4q^{76} + 7q^{77} - 2q^{78} - 10q^{79} - 4q^{80} + 2q^{81} - 4q^{82} - 2q^{83} - 3q^{84} - 22q^{85} + 6q^{86} - 12q^{87} + q^{88} + 4q^{89} + 4q^{90} - 3q^{91} - 2q^{92} + 6q^{93} + 4q^{94} + 8q^{95} + 2q^{96} - 8q^{97} + q^{98} - q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) −0.561553 −0.212247 −0.106124 0.994353i \(-0.533844\pi\)
−0.106124 + 0.994353i \(0.533844\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −2.56155 −0.772337 −0.386169 0.922428i \(-0.626202\pi\)
−0.386169 + 0.922428i \(0.626202\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 0.561553 0.150081
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 3.43845 0.833946 0.416973 0.908919i \(-0.363091\pi\)
0.416973 + 0.908919i \(0.363091\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0.561553 0.122541
\(22\) 2.56155 0.546125
\(23\) 3.12311 0.651213 0.325606 0.945505i \(-0.394432\pi\)
0.325606 + 0.945505i \(0.394432\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) −0.561553 −0.106124
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −2.00000 −0.365148
\(31\) 1.12311 0.201716 0.100858 0.994901i \(-0.467841\pi\)
0.100858 + 0.994901i \(0.467841\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.56155 0.445909
\(34\) −3.43845 −0.589689
\(35\) 1.12311 0.189839
\(36\) 1.00000 0.166667
\(37\) −9.12311 −1.49983 −0.749915 0.661535i \(-0.769905\pi\)
−0.749915 + 0.661535i \(0.769905\pi\)
\(38\) 2.00000 0.324443
\(39\) 1.00000 0.160128
\(40\) 2.00000 0.316228
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −0.561553 −0.0866495
\(43\) 1.12311 0.171272 0.0856360 0.996326i \(-0.472708\pi\)
0.0856360 + 0.996326i \(0.472708\pi\)
\(44\) −2.56155 −0.386169
\(45\) −2.00000 −0.298142
\(46\) −3.12311 −0.460477
\(47\) −10.2462 −1.49456 −0.747282 0.664507i \(-0.768641\pi\)
−0.747282 + 0.664507i \(0.768641\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.68466 −0.954951
\(50\) 1.00000 0.141421
\(51\) −3.43845 −0.481479
\(52\) −1.00000 −0.138675
\(53\) 6.56155 0.901299 0.450649 0.892701i \(-0.351193\pi\)
0.450649 + 0.892701i \(0.351193\pi\)
\(54\) 1.00000 0.136083
\(55\) 5.12311 0.690799
\(56\) 0.561553 0.0750407
\(57\) 2.00000 0.264906
\(58\) −6.00000 −0.787839
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 2.00000 0.258199
\(61\) 11.1231 1.42417 0.712084 0.702094i \(-0.247752\pi\)
0.712084 + 0.702094i \(0.247752\pi\)
\(62\) −1.12311 −0.142635
\(63\) −0.561553 −0.0707490
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) −2.56155 −0.315305
\(67\) 11.6847 1.42751 0.713754 0.700396i \(-0.246993\pi\)
0.713754 + 0.700396i \(0.246993\pi\)
\(68\) 3.43845 0.416973
\(69\) −3.12311 −0.375978
\(70\) −1.12311 −0.134237
\(71\) 2.24621 0.266576 0.133288 0.991077i \(-0.457446\pi\)
0.133288 + 0.991077i \(0.457446\pi\)
\(72\) −1.00000 −0.117851
\(73\) 3.68466 0.431257 0.215628 0.976476i \(-0.430820\pi\)
0.215628 + 0.976476i \(0.430820\pi\)
\(74\) 9.12311 1.06054
\(75\) 1.00000 0.115470
\(76\) −2.00000 −0.229416
\(77\) 1.43845 0.163926
\(78\) −1.00000 −0.113228
\(79\) 7.36932 0.829113 0.414556 0.910024i \(-0.363937\pi\)
0.414556 + 0.910024i \(0.363937\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 11.3693 1.24794 0.623972 0.781446i \(-0.285518\pi\)
0.623972 + 0.781446i \(0.285518\pi\)
\(84\) 0.561553 0.0612704
\(85\) −6.87689 −0.745904
\(86\) −1.12311 −0.121108
\(87\) −6.00000 −0.643268
\(88\) 2.56155 0.273062
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 2.00000 0.210819
\(91\) 0.561553 0.0588667
\(92\) 3.12311 0.325606
\(93\) −1.12311 −0.116461
\(94\) 10.2462 1.05682
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) 4.24621 0.431137 0.215569 0.976489i \(-0.430839\pi\)
0.215569 + 0.976489i \(0.430839\pi\)
\(98\) 6.68466 0.675252
\(99\) −2.56155 −0.257446
\(100\) −1.00000 −0.100000
\(101\) −3.68466 −0.366637 −0.183319 0.983054i \(-0.558684\pi\)
−0.183319 + 0.983054i \(0.558684\pi\)
\(102\) 3.43845 0.340457
\(103\) −1.00000 −0.0985329
\(104\) 1.00000 0.0980581
\(105\) −1.12311 −0.109604
\(106\) −6.56155 −0.637314
\(107\) 8.56155 0.827677 0.413838 0.910350i \(-0.364188\pi\)
0.413838 + 0.910350i \(0.364188\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.24621 −0.598279 −0.299139 0.954209i \(-0.596700\pi\)
−0.299139 + 0.954209i \(0.596700\pi\)
\(110\) −5.12311 −0.488469
\(111\) 9.12311 0.865927
\(112\) −0.561553 −0.0530618
\(113\) 4.00000 0.376288 0.188144 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) −2.00000 −0.187317
\(115\) −6.24621 −0.582462
\(116\) 6.00000 0.557086
\(117\) −1.00000 −0.0924500
\(118\) 8.00000 0.736460
\(119\) −1.93087 −0.177003
\(120\) −2.00000 −0.182574
\(121\) −4.43845 −0.403495
\(122\) −11.1231 −1.00704
\(123\) −2.00000 −0.180334
\(124\) 1.12311 0.100858
\(125\) 12.0000 1.07331
\(126\) 0.561553 0.0500271
\(127\) −11.6847 −1.03685 −0.518423 0.855124i \(-0.673481\pi\)
−0.518423 + 0.855124i \(0.673481\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.12311 −0.0988839
\(130\) −2.00000 −0.175412
\(131\) 21.3693 1.86705 0.933523 0.358518i \(-0.116718\pi\)
0.933523 + 0.358518i \(0.116718\pi\)
\(132\) 2.56155 0.222955
\(133\) 1.12311 0.0973856
\(134\) −11.6847 −1.00940
\(135\) 2.00000 0.172133
\(136\) −3.43845 −0.294844
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 3.12311 0.265856
\(139\) −12.8078 −1.08634 −0.543170 0.839623i \(-0.682776\pi\)
−0.543170 + 0.839623i \(0.682776\pi\)
\(140\) 1.12311 0.0949197
\(141\) 10.2462 0.862887
\(142\) −2.24621 −0.188498
\(143\) 2.56155 0.214208
\(144\) 1.00000 0.0833333
\(145\) −12.0000 −0.996546
\(146\) −3.68466 −0.304945
\(147\) 6.68466 0.551341
\(148\) −9.12311 −0.749915
\(149\) 6.31534 0.517373 0.258686 0.965961i \(-0.416710\pi\)
0.258686 + 0.965961i \(0.416710\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −7.36932 −0.599707 −0.299853 0.953985i \(-0.596938\pi\)
−0.299853 + 0.953985i \(0.596938\pi\)
\(152\) 2.00000 0.162221
\(153\) 3.43845 0.277982
\(154\) −1.43845 −0.115913
\(155\) −2.24621 −0.180420
\(156\) 1.00000 0.0800641
\(157\) −6.80776 −0.543319 −0.271659 0.962393i \(-0.587572\pi\)
−0.271659 + 0.962393i \(0.587572\pi\)
\(158\) −7.36932 −0.586271
\(159\) −6.56155 −0.520365
\(160\) 2.00000 0.158114
\(161\) −1.75379 −0.138218
\(162\) −1.00000 −0.0785674
\(163\) −22.4924 −1.76174 −0.880871 0.473356i \(-0.843042\pi\)
−0.880871 + 0.473356i \(0.843042\pi\)
\(164\) 2.00000 0.156174
\(165\) −5.12311 −0.398833
\(166\) −11.3693 −0.882430
\(167\) −3.19224 −0.247023 −0.123511 0.992343i \(-0.539416\pi\)
−0.123511 + 0.992343i \(0.539416\pi\)
\(168\) −0.561553 −0.0433247
\(169\) 1.00000 0.0769231
\(170\) 6.87689 0.527434
\(171\) −2.00000 −0.152944
\(172\) 1.12311 0.0856360
\(173\) 20.8078 1.58199 0.790993 0.611826i \(-0.209565\pi\)
0.790993 + 0.611826i \(0.209565\pi\)
\(174\) 6.00000 0.454859
\(175\) 0.561553 0.0424494
\(176\) −2.56155 −0.193084
\(177\) 8.00000 0.601317
\(178\) −2.00000 −0.149906
\(179\) −10.8078 −0.807810 −0.403905 0.914801i \(-0.632347\pi\)
−0.403905 + 0.914801i \(0.632347\pi\)
\(180\) −2.00000 −0.149071
\(181\) 15.9309 1.18413 0.592066 0.805889i \(-0.298312\pi\)
0.592066 + 0.805889i \(0.298312\pi\)
\(182\) −0.561553 −0.0416251
\(183\) −11.1231 −0.822244
\(184\) −3.12311 −0.230238
\(185\) 18.2462 1.34149
\(186\) 1.12311 0.0823501
\(187\) −8.80776 −0.644087
\(188\) −10.2462 −0.747282
\(189\) 0.561553 0.0408470
\(190\) −4.00000 −0.290191
\(191\) −3.68466 −0.266613 −0.133306 0.991075i \(-0.542559\pi\)
−0.133306 + 0.991075i \(0.542559\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −0.630683 −0.0453976 −0.0226988 0.999742i \(-0.507226\pi\)
−0.0226988 + 0.999742i \(0.507226\pi\)
\(194\) −4.24621 −0.304860
\(195\) −2.00000 −0.143223
\(196\) −6.68466 −0.477476
\(197\) 18.4924 1.31753 0.658765 0.752349i \(-0.271079\pi\)
0.658765 + 0.752349i \(0.271079\pi\)
\(198\) 2.56155 0.182042
\(199\) −19.6847 −1.39541 −0.697704 0.716386i \(-0.745795\pi\)
−0.697704 + 0.716386i \(0.745795\pi\)
\(200\) 1.00000 0.0707107
\(201\) −11.6847 −0.824172
\(202\) 3.68466 0.259252
\(203\) −3.36932 −0.236480
\(204\) −3.43845 −0.240739
\(205\) −4.00000 −0.279372
\(206\) 1.00000 0.0696733
\(207\) 3.12311 0.217071
\(208\) −1.00000 −0.0693375
\(209\) 5.12311 0.354373
\(210\) 1.12311 0.0775017
\(211\) 8.49242 0.584642 0.292321 0.956320i \(-0.405572\pi\)
0.292321 + 0.956320i \(0.405572\pi\)
\(212\) 6.56155 0.450649
\(213\) −2.24621 −0.153908
\(214\) −8.56155 −0.585256
\(215\) −2.24621 −0.153190
\(216\) 1.00000 0.0680414
\(217\) −0.630683 −0.0428136
\(218\) 6.24621 0.423047
\(219\) −3.68466 −0.248986
\(220\) 5.12311 0.345400
\(221\) −3.43845 −0.231295
\(222\) −9.12311 −0.612303
\(223\) −5.68466 −0.380673 −0.190336 0.981719i \(-0.560958\pi\)
−0.190336 + 0.981719i \(0.560958\pi\)
\(224\) 0.561553 0.0375203
\(225\) −1.00000 −0.0666667
\(226\) −4.00000 −0.266076
\(227\) −4.80776 −0.319103 −0.159551 0.987190i \(-0.551005\pi\)
−0.159551 + 0.987190i \(0.551005\pi\)
\(228\) 2.00000 0.132453
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 6.24621 0.411863
\(231\) −1.43845 −0.0946429
\(232\) −6.00000 −0.393919
\(233\) 24.4924 1.60455 0.802276 0.596953i \(-0.203622\pi\)
0.802276 + 0.596953i \(0.203622\pi\)
\(234\) 1.00000 0.0653720
\(235\) 20.4924 1.33678
\(236\) −8.00000 −0.520756
\(237\) −7.36932 −0.478689
\(238\) 1.93087 0.125160
\(239\) −2.56155 −0.165693 −0.0828465 0.996562i \(-0.526401\pi\)
−0.0828465 + 0.996562i \(0.526401\pi\)
\(240\) 2.00000 0.129099
\(241\) −8.31534 −0.535638 −0.267819 0.963469i \(-0.586303\pi\)
−0.267819 + 0.963469i \(0.586303\pi\)
\(242\) 4.43845 0.285314
\(243\) −1.00000 −0.0641500
\(244\) 11.1231 0.712084
\(245\) 13.3693 0.854134
\(246\) 2.00000 0.127515
\(247\) 2.00000 0.127257
\(248\) −1.12311 −0.0713173
\(249\) −11.3693 −0.720501
\(250\) −12.0000 −0.758947
\(251\) 5.75379 0.363176 0.181588 0.983375i \(-0.441876\pi\)
0.181588 + 0.983375i \(0.441876\pi\)
\(252\) −0.561553 −0.0353745
\(253\) −8.00000 −0.502956
\(254\) 11.6847 0.733161
\(255\) 6.87689 0.430648
\(256\) 1.00000 0.0625000
\(257\) 21.1231 1.31762 0.658812 0.752308i \(-0.271059\pi\)
0.658812 + 0.752308i \(0.271059\pi\)
\(258\) 1.12311 0.0699215
\(259\) 5.12311 0.318334
\(260\) 2.00000 0.124035
\(261\) 6.00000 0.371391
\(262\) −21.3693 −1.32020
\(263\) 0.315342 0.0194448 0.00972240 0.999953i \(-0.496905\pi\)
0.00972240 + 0.999953i \(0.496905\pi\)
\(264\) −2.56155 −0.157653
\(265\) −13.1231 −0.806146
\(266\) −1.12311 −0.0688620
\(267\) −2.00000 −0.122398
\(268\) 11.6847 0.713754
\(269\) −11.7538 −0.716641 −0.358321 0.933599i \(-0.616651\pi\)
−0.358321 + 0.933599i \(0.616651\pi\)
\(270\) −2.00000 −0.121716
\(271\) −11.3693 −0.690637 −0.345318 0.938486i \(-0.612229\pi\)
−0.345318 + 0.938486i \(0.612229\pi\)
\(272\) 3.43845 0.208486
\(273\) −0.561553 −0.0339867
\(274\) 6.00000 0.362473
\(275\) 2.56155 0.154467
\(276\) −3.12311 −0.187989
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 12.8078 0.768159
\(279\) 1.12311 0.0672386
\(280\) −1.12311 −0.0671184
\(281\) −22.4924 −1.34179 −0.670893 0.741554i \(-0.734089\pi\)
−0.670893 + 0.741554i \(0.734089\pi\)
\(282\) −10.2462 −0.610153
\(283\) 21.6155 1.28491 0.642455 0.766324i \(-0.277916\pi\)
0.642455 + 0.766324i \(0.277916\pi\)
\(284\) 2.24621 0.133288
\(285\) −4.00000 −0.236940
\(286\) −2.56155 −0.151468
\(287\) −1.12311 −0.0662948
\(288\) −1.00000 −0.0589256
\(289\) −5.17708 −0.304534
\(290\) 12.0000 0.704664
\(291\) −4.24621 −0.248917
\(292\) 3.68466 0.215628
\(293\) −7.61553 −0.444904 −0.222452 0.974944i \(-0.571406\pi\)
−0.222452 + 0.974944i \(0.571406\pi\)
\(294\) −6.68466 −0.389857
\(295\) 16.0000 0.931556
\(296\) 9.12311 0.530270
\(297\) 2.56155 0.148636
\(298\) −6.31534 −0.365838
\(299\) −3.12311 −0.180614
\(300\) 1.00000 0.0577350
\(301\) −0.630683 −0.0363520
\(302\) 7.36932 0.424057
\(303\) 3.68466 0.211678
\(304\) −2.00000 −0.114708
\(305\) −22.2462 −1.27381
\(306\) −3.43845 −0.196563
\(307\) 21.4384 1.22356 0.611778 0.791029i \(-0.290454\pi\)
0.611778 + 0.791029i \(0.290454\pi\)
\(308\) 1.43845 0.0819631
\(309\) 1.00000 0.0568880
\(310\) 2.24621 0.127576
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −16.2462 −0.918290 −0.459145 0.888361i \(-0.651844\pi\)
−0.459145 + 0.888361i \(0.651844\pi\)
\(314\) 6.80776 0.384184
\(315\) 1.12311 0.0632798
\(316\) 7.36932 0.414556
\(317\) −13.0540 −0.733184 −0.366592 0.930382i \(-0.619476\pi\)
−0.366592 + 0.930382i \(0.619476\pi\)
\(318\) 6.56155 0.367954
\(319\) −15.3693 −0.860517
\(320\) −2.00000 −0.111803
\(321\) −8.56155 −0.477859
\(322\) 1.75379 0.0977348
\(323\) −6.87689 −0.382641
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) 22.4924 1.24574
\(327\) 6.24621 0.345416
\(328\) −2.00000 −0.110432
\(329\) 5.75379 0.317217
\(330\) 5.12311 0.282018
\(331\) 21.9309 1.20543 0.602715 0.797957i \(-0.294086\pi\)
0.602715 + 0.797957i \(0.294086\pi\)
\(332\) 11.3693 0.623972
\(333\) −9.12311 −0.499943
\(334\) 3.19224 0.174671
\(335\) −23.3693 −1.27680
\(336\) 0.561553 0.0306352
\(337\) −27.9309 −1.52149 −0.760746 0.649050i \(-0.775167\pi\)
−0.760746 + 0.649050i \(0.775167\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −4.00000 −0.217250
\(340\) −6.87689 −0.372952
\(341\) −2.87689 −0.155793
\(342\) 2.00000 0.108148
\(343\) 7.68466 0.414933
\(344\) −1.12311 −0.0605538
\(345\) 6.24621 0.336285
\(346\) −20.8078 −1.11863
\(347\) −16.8769 −0.905999 −0.453000 0.891511i \(-0.649646\pi\)
−0.453000 + 0.891511i \(0.649646\pi\)
\(348\) −6.00000 −0.321634
\(349\) 1.75379 0.0938782 0.0469391 0.998898i \(-0.485053\pi\)
0.0469391 + 0.998898i \(0.485053\pi\)
\(350\) −0.561553 −0.0300163
\(351\) 1.00000 0.0533761
\(352\) 2.56155 0.136531
\(353\) −14.3153 −0.761929 −0.380964 0.924590i \(-0.624408\pi\)
−0.380964 + 0.924590i \(0.624408\pi\)
\(354\) −8.00000 −0.425195
\(355\) −4.49242 −0.238433
\(356\) 2.00000 0.106000
\(357\) 1.93087 0.102192
\(358\) 10.8078 0.571208
\(359\) 19.0540 1.00563 0.502815 0.864394i \(-0.332298\pi\)
0.502815 + 0.864394i \(0.332298\pi\)
\(360\) 2.00000 0.105409
\(361\) −15.0000 −0.789474
\(362\) −15.9309 −0.837308
\(363\) 4.43845 0.232958
\(364\) 0.561553 0.0294334
\(365\) −7.36932 −0.385728
\(366\) 11.1231 0.581414
\(367\) −13.7538 −0.717942 −0.358971 0.933349i \(-0.616872\pi\)
−0.358971 + 0.933349i \(0.616872\pi\)
\(368\) 3.12311 0.162803
\(369\) 2.00000 0.104116
\(370\) −18.2462 −0.948575
\(371\) −3.68466 −0.191298
\(372\) −1.12311 −0.0582303
\(373\) −29.3693 −1.52069 −0.760343 0.649522i \(-0.774969\pi\)
−0.760343 + 0.649522i \(0.774969\pi\)
\(374\) 8.80776 0.455439
\(375\) −12.0000 −0.619677
\(376\) 10.2462 0.528408
\(377\) −6.00000 −0.309016
\(378\) −0.561553 −0.0288832
\(379\) −12.8078 −0.657891 −0.328945 0.944349i \(-0.606693\pi\)
−0.328945 + 0.944349i \(0.606693\pi\)
\(380\) 4.00000 0.205196
\(381\) 11.6847 0.598623
\(382\) 3.68466 0.188524
\(383\) 4.49242 0.229552 0.114776 0.993391i \(-0.463385\pi\)
0.114776 + 0.993391i \(0.463385\pi\)
\(384\) 1.00000 0.0510310
\(385\) −2.87689 −0.146620
\(386\) 0.630683 0.0321009
\(387\) 1.12311 0.0570907
\(388\) 4.24621 0.215569
\(389\) 2.56155 0.129876 0.0649379 0.997889i \(-0.479315\pi\)
0.0649379 + 0.997889i \(0.479315\pi\)
\(390\) 2.00000 0.101274
\(391\) 10.7386 0.543076
\(392\) 6.68466 0.337626
\(393\) −21.3693 −1.07794
\(394\) −18.4924 −0.931635
\(395\) −14.7386 −0.741581
\(396\) −2.56155 −0.128723
\(397\) 17.6155 0.884098 0.442049 0.896991i \(-0.354252\pi\)
0.442049 + 0.896991i \(0.354252\pi\)
\(398\) 19.6847 0.986703
\(399\) −1.12311 −0.0562256
\(400\) −1.00000 −0.0500000
\(401\) 5.36932 0.268131 0.134065 0.990972i \(-0.457197\pi\)
0.134065 + 0.990972i \(0.457197\pi\)
\(402\) 11.6847 0.582778
\(403\) −1.12311 −0.0559459
\(404\) −3.68466 −0.183319
\(405\) −2.00000 −0.0993808
\(406\) 3.36932 0.167216
\(407\) 23.3693 1.15837
\(408\) 3.43845 0.170229
\(409\) 21.3693 1.05664 0.528322 0.849044i \(-0.322821\pi\)
0.528322 + 0.849044i \(0.322821\pi\)
\(410\) 4.00000 0.197546
\(411\) 6.00000 0.295958
\(412\) −1.00000 −0.0492665
\(413\) 4.49242 0.221058
\(414\) −3.12311 −0.153492
\(415\) −22.7386 −1.11620
\(416\) 1.00000 0.0490290
\(417\) 12.8078 0.627199
\(418\) −5.12311 −0.250579
\(419\) −19.9309 −0.973687 −0.486843 0.873489i \(-0.661852\pi\)
−0.486843 + 0.873489i \(0.661852\pi\)
\(420\) −1.12311 −0.0548019
\(421\) 17.0540 0.831160 0.415580 0.909557i \(-0.363579\pi\)
0.415580 + 0.909557i \(0.363579\pi\)
\(422\) −8.49242 −0.413405
\(423\) −10.2462 −0.498188
\(424\) −6.56155 −0.318657
\(425\) −3.43845 −0.166789
\(426\) 2.24621 0.108829
\(427\) −6.24621 −0.302275
\(428\) 8.56155 0.413838
\(429\) −2.56155 −0.123673
\(430\) 2.24621 0.108322
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −22.4924 −1.08092 −0.540458 0.841371i \(-0.681749\pi\)
−0.540458 + 0.841371i \(0.681749\pi\)
\(434\) 0.630683 0.0302738
\(435\) 12.0000 0.575356
\(436\) −6.24621 −0.299139
\(437\) −6.24621 −0.298797
\(438\) 3.68466 0.176060
\(439\) −11.0540 −0.527577 −0.263789 0.964580i \(-0.584972\pi\)
−0.263789 + 0.964580i \(0.584972\pi\)
\(440\) −5.12311 −0.244234
\(441\) −6.68466 −0.318317
\(442\) 3.43845 0.163550
\(443\) −12.4924 −0.593533 −0.296766 0.954950i \(-0.595908\pi\)
−0.296766 + 0.954950i \(0.595908\pi\)
\(444\) 9.12311 0.432963
\(445\) −4.00000 −0.189618
\(446\) 5.68466 0.269176
\(447\) −6.31534 −0.298705
\(448\) −0.561553 −0.0265309
\(449\) −26.3153 −1.24190 −0.620949 0.783851i \(-0.713253\pi\)
−0.620949 + 0.783851i \(0.713253\pi\)
\(450\) 1.00000 0.0471405
\(451\) −5.12311 −0.241238
\(452\) 4.00000 0.188144
\(453\) 7.36932 0.346241
\(454\) 4.80776 0.225640
\(455\) −1.12311 −0.0526520
\(456\) −2.00000 −0.0936586
\(457\) 8.63068 0.403726 0.201863 0.979414i \(-0.435300\pi\)
0.201863 + 0.979414i \(0.435300\pi\)
\(458\) 6.00000 0.280362
\(459\) −3.43845 −0.160493
\(460\) −6.24621 −0.291231
\(461\) 8.24621 0.384064 0.192032 0.981389i \(-0.438492\pi\)
0.192032 + 0.981389i \(0.438492\pi\)
\(462\) 1.43845 0.0669226
\(463\) 13.1231 0.609882 0.304941 0.952371i \(-0.401363\pi\)
0.304941 + 0.952371i \(0.401363\pi\)
\(464\) 6.00000 0.278543
\(465\) 2.24621 0.104166
\(466\) −24.4924 −1.13459
\(467\) −38.1771 −1.76662 −0.883312 0.468785i \(-0.844692\pi\)
−0.883312 + 0.468785i \(0.844692\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −6.56155 −0.302984
\(470\) −20.4924 −0.945245
\(471\) 6.80776 0.313685
\(472\) 8.00000 0.368230
\(473\) −2.87689 −0.132280
\(474\) 7.36932 0.338484
\(475\) 2.00000 0.0917663
\(476\) −1.93087 −0.0885013
\(477\) 6.56155 0.300433
\(478\) 2.56155 0.117163
\(479\) −5.12311 −0.234081 −0.117040 0.993127i \(-0.537341\pi\)
−0.117040 + 0.993127i \(0.537341\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 9.12311 0.415978
\(482\) 8.31534 0.378753
\(483\) 1.75379 0.0798002
\(484\) −4.43845 −0.201748
\(485\) −8.49242 −0.385621
\(486\) 1.00000 0.0453609
\(487\) 5.61553 0.254464 0.127232 0.991873i \(-0.459391\pi\)
0.127232 + 0.991873i \(0.459391\pi\)
\(488\) −11.1231 −0.503519
\(489\) 22.4924 1.01714
\(490\) −13.3693 −0.603964
\(491\) 15.4384 0.696727 0.348364 0.937359i \(-0.386737\pi\)
0.348364 + 0.937359i \(0.386737\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 20.6307 0.929159
\(494\) −2.00000 −0.0899843
\(495\) 5.12311 0.230266
\(496\) 1.12311 0.0504289
\(497\) −1.26137 −0.0565800
\(498\) 11.3693 0.509471
\(499\) 22.7386 1.01792 0.508961 0.860790i \(-0.330030\pi\)
0.508961 + 0.860790i \(0.330030\pi\)
\(500\) 12.0000 0.536656
\(501\) 3.19224 0.142619
\(502\) −5.75379 −0.256804
\(503\) −10.6307 −0.473999 −0.236999 0.971510i \(-0.576164\pi\)
−0.236999 + 0.971510i \(0.576164\pi\)
\(504\) 0.561553 0.0250136
\(505\) 7.36932 0.327930
\(506\) 8.00000 0.355643
\(507\) −1.00000 −0.0444116
\(508\) −11.6847 −0.518423
\(509\) 7.93087 0.351530 0.175765 0.984432i \(-0.443760\pi\)
0.175765 + 0.984432i \(0.443760\pi\)
\(510\) −6.87689 −0.304514
\(511\) −2.06913 −0.0915329
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) −21.1231 −0.931700
\(515\) 2.00000 0.0881305
\(516\) −1.12311 −0.0494420
\(517\) 26.2462 1.15431
\(518\) −5.12311 −0.225096
\(519\) −20.8078 −0.913359
\(520\) −2.00000 −0.0877058
\(521\) 6.87689 0.301282 0.150641 0.988589i \(-0.451866\pi\)
0.150641 + 0.988589i \(0.451866\pi\)
\(522\) −6.00000 −0.262613
\(523\) −26.5616 −1.16146 −0.580728 0.814098i \(-0.697232\pi\)
−0.580728 + 0.814098i \(0.697232\pi\)
\(524\) 21.3693 0.933523
\(525\) −0.561553 −0.0245082
\(526\) −0.315342 −0.0137495
\(527\) 3.86174 0.168220
\(528\) 2.56155 0.111477
\(529\) −13.2462 −0.575922
\(530\) 13.1231 0.570031
\(531\) −8.00000 −0.347170
\(532\) 1.12311 0.0486928
\(533\) −2.00000 −0.0866296
\(534\) 2.00000 0.0865485
\(535\) −17.1231 −0.740296
\(536\) −11.6847 −0.504700
\(537\) 10.8078 0.466389
\(538\) 11.7538 0.506742
\(539\) 17.1231 0.737544
\(540\) 2.00000 0.0860663
\(541\) −16.5616 −0.712037 −0.356018 0.934479i \(-0.615866\pi\)
−0.356018 + 0.934479i \(0.615866\pi\)
\(542\) 11.3693 0.488354
\(543\) −15.9309 −0.683659
\(544\) −3.43845 −0.147422
\(545\) 12.4924 0.535117
\(546\) 0.561553 0.0240322
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −6.00000 −0.256307
\(549\) 11.1231 0.474723
\(550\) −2.56155 −0.109225
\(551\) −12.0000 −0.511217
\(552\) 3.12311 0.132928
\(553\) −4.13826 −0.175977
\(554\) −6.00000 −0.254916
\(555\) −18.2462 −0.774509
\(556\) −12.8078 −0.543170
\(557\) 4.87689 0.206641 0.103320 0.994648i \(-0.467053\pi\)
0.103320 + 0.994648i \(0.467053\pi\)
\(558\) −1.12311 −0.0475449
\(559\) −1.12311 −0.0475023
\(560\) 1.12311 0.0474599
\(561\) 8.80776 0.371864
\(562\) 22.4924 0.948786
\(563\) 33.6155 1.41673 0.708363 0.705849i \(-0.249434\pi\)
0.708363 + 0.705849i \(0.249434\pi\)
\(564\) 10.2462 0.431443
\(565\) −8.00000 −0.336563
\(566\) −21.6155 −0.908568
\(567\) −0.561553 −0.0235830
\(568\) −2.24621 −0.0942489
\(569\) 0.630683 0.0264396 0.0132198 0.999913i \(-0.495792\pi\)
0.0132198 + 0.999913i \(0.495792\pi\)
\(570\) 4.00000 0.167542
\(571\) −26.2462 −1.09837 −0.549185 0.835701i \(-0.685062\pi\)
−0.549185 + 0.835701i \(0.685062\pi\)
\(572\) 2.56155 0.107104
\(573\) 3.68466 0.153929
\(574\) 1.12311 0.0468775
\(575\) −3.12311 −0.130243
\(576\) 1.00000 0.0416667
\(577\) −20.8078 −0.866239 −0.433119 0.901337i \(-0.642587\pi\)
−0.433119 + 0.901337i \(0.642587\pi\)
\(578\) 5.17708 0.215338
\(579\) 0.630683 0.0262103
\(580\) −12.0000 −0.498273
\(581\) −6.38447 −0.264873
\(582\) 4.24621 0.176011
\(583\) −16.8078 −0.696106
\(584\) −3.68466 −0.152472
\(585\) 2.00000 0.0826898
\(586\) 7.61553 0.314595
\(587\) −36.4924 −1.50620 −0.753102 0.657904i \(-0.771443\pi\)
−0.753102 + 0.657904i \(0.771443\pi\)
\(588\) 6.68466 0.275671
\(589\) −2.24621 −0.0925535
\(590\) −16.0000 −0.658710
\(591\) −18.4924 −0.760677
\(592\) −9.12311 −0.374957
\(593\) 13.6847 0.561962 0.280981 0.959713i \(-0.409340\pi\)
0.280981 + 0.959713i \(0.409340\pi\)
\(594\) −2.56155 −0.105102
\(595\) 3.86174 0.158316
\(596\) 6.31534 0.258686
\(597\) 19.6847 0.805639
\(598\) 3.12311 0.127713
\(599\) 9.93087 0.405764 0.202882 0.979203i \(-0.434969\pi\)
0.202882 + 0.979203i \(0.434969\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 3.75379 0.153120 0.0765601 0.997065i \(-0.475606\pi\)
0.0765601 + 0.997065i \(0.475606\pi\)
\(602\) 0.630683 0.0257047
\(603\) 11.6847 0.475836
\(604\) −7.36932 −0.299853
\(605\) 8.87689 0.360897
\(606\) −3.68466 −0.149679
\(607\) −14.8769 −0.603835 −0.301917 0.953334i \(-0.597627\pi\)
−0.301917 + 0.953334i \(0.597627\pi\)
\(608\) 2.00000 0.0811107
\(609\) 3.36932 0.136532
\(610\) 22.2462 0.900723
\(611\) 10.2462 0.414517
\(612\) 3.43845 0.138991
\(613\) −43.7926 −1.76877 −0.884383 0.466761i \(-0.845421\pi\)
−0.884383 + 0.466761i \(0.845421\pi\)
\(614\) −21.4384 −0.865185
\(615\) 4.00000 0.161296
\(616\) −1.43845 −0.0579567
\(617\) −12.2462 −0.493014 −0.246507 0.969141i \(-0.579283\pi\)
−0.246507 + 0.969141i \(0.579283\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −6.49242 −0.260952 −0.130476 0.991451i \(-0.541651\pi\)
−0.130476 + 0.991451i \(0.541651\pi\)
\(620\) −2.24621 −0.0902100
\(621\) −3.12311 −0.125326
\(622\) 18.0000 0.721734
\(623\) −1.12311 −0.0449963
\(624\) 1.00000 0.0400320
\(625\) −19.0000 −0.760000
\(626\) 16.2462 0.649329
\(627\) −5.12311 −0.204597
\(628\) −6.80776 −0.271659
\(629\) −31.3693 −1.25078
\(630\) −1.12311 −0.0447456
\(631\) −37.0540 −1.47510 −0.737548 0.675295i \(-0.764016\pi\)
−0.737548 + 0.675295i \(0.764016\pi\)
\(632\) −7.36932 −0.293136
\(633\) −8.49242 −0.337543
\(634\) 13.0540 0.518440
\(635\) 23.3693 0.927383
\(636\) −6.56155 −0.260182
\(637\) 6.68466 0.264856
\(638\) 15.3693 0.608477
\(639\) 2.24621 0.0888587
\(640\) 2.00000 0.0790569
\(641\) −38.8078 −1.53281 −0.766407 0.642355i \(-0.777958\pi\)
−0.766407 + 0.642355i \(0.777958\pi\)
\(642\) 8.56155 0.337898
\(643\) −24.7386 −0.975596 −0.487798 0.872956i \(-0.662200\pi\)
−0.487798 + 0.872956i \(0.662200\pi\)
\(644\) −1.75379 −0.0691090
\(645\) 2.24621 0.0884445
\(646\) 6.87689 0.270568
\(647\) −8.24621 −0.324192 −0.162096 0.986775i \(-0.551825\pi\)
−0.162096 + 0.986775i \(0.551825\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 20.4924 0.804398
\(650\) −1.00000 −0.0392232
\(651\) 0.630683 0.0247184
\(652\) −22.4924 −0.880871
\(653\) 18.8769 0.738710 0.369355 0.929288i \(-0.379579\pi\)
0.369355 + 0.929288i \(0.379579\pi\)
\(654\) −6.24621 −0.244246
\(655\) −42.7386 −1.66994
\(656\) 2.00000 0.0780869
\(657\) 3.68466 0.143752
\(658\) −5.75379 −0.224306
\(659\) 21.5464 0.839328 0.419664 0.907679i \(-0.362148\pi\)
0.419664 + 0.907679i \(0.362148\pi\)
\(660\) −5.12311 −0.199417
\(661\) −18.2462 −0.709695 −0.354848 0.934924i \(-0.615467\pi\)
−0.354848 + 0.934924i \(0.615467\pi\)
\(662\) −21.9309 −0.852367
\(663\) 3.43845 0.133538
\(664\) −11.3693 −0.441215
\(665\) −2.24621 −0.0871043
\(666\) 9.12311 0.353513
\(667\) 18.7386 0.725563
\(668\) −3.19224 −0.123511
\(669\) 5.68466 0.219782
\(670\) 23.3693 0.902835
\(671\) −28.4924 −1.09994
\(672\) −0.561553 −0.0216624
\(673\) −1.68466 −0.0649388 −0.0324694 0.999473i \(-0.510337\pi\)
−0.0324694 + 0.999473i \(0.510337\pi\)
\(674\) 27.9309 1.07586
\(675\) 1.00000 0.0384900
\(676\) 1.00000 0.0384615
\(677\) −24.2462 −0.931858 −0.465929 0.884822i \(-0.654280\pi\)
−0.465929 + 0.884822i \(0.654280\pi\)
\(678\) 4.00000 0.153619
\(679\) −2.38447 −0.0915076
\(680\) 6.87689 0.263717
\(681\) 4.80776 0.184234
\(682\) 2.87689 0.110162
\(683\) 5.43845 0.208096 0.104048 0.994572i \(-0.466820\pi\)
0.104048 + 0.994572i \(0.466820\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 12.0000 0.458496
\(686\) −7.68466 −0.293402
\(687\) 6.00000 0.228914
\(688\) 1.12311 0.0428180
\(689\) −6.56155 −0.249975
\(690\) −6.24621 −0.237789
\(691\) −21.7538 −0.827553 −0.413777 0.910378i \(-0.635791\pi\)
−0.413777 + 0.910378i \(0.635791\pi\)
\(692\) 20.8078 0.790993
\(693\) 1.43845 0.0546421
\(694\) 16.8769 0.640638
\(695\) 25.6155 0.971652
\(696\) 6.00000 0.227429
\(697\) 6.87689 0.260481
\(698\) −1.75379 −0.0663819
\(699\) −24.4924 −0.926388
\(700\) 0.561553 0.0212247
\(701\) −17.3693 −0.656030 −0.328015 0.944673i \(-0.606380\pi\)
−0.328015 + 0.944673i \(0.606380\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 18.2462 0.688169
\(704\) −2.56155 −0.0965422
\(705\) −20.4924 −0.771789
\(706\) 14.3153 0.538765
\(707\) 2.06913 0.0778177
\(708\) 8.00000 0.300658
\(709\) 8.73863 0.328186 0.164093 0.986445i \(-0.447530\pi\)
0.164093 + 0.986445i \(0.447530\pi\)
\(710\) 4.49242 0.168598
\(711\) 7.36932 0.276371
\(712\) −2.00000 −0.0749532
\(713\) 3.50758 0.131360
\(714\) −1.93087 −0.0722610
\(715\) −5.12311 −0.191593
\(716\) −10.8078 −0.403905
\(717\) 2.56155 0.0956629
\(718\) −19.0540 −0.711088
\(719\) 11.1922 0.417400 0.208700 0.977980i \(-0.433077\pi\)
0.208700 + 0.977980i \(0.433077\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0.561553 0.0209133
\(722\) 15.0000 0.558242
\(723\) 8.31534 0.309251
\(724\) 15.9309 0.592066
\(725\) −6.00000 −0.222834
\(726\) −4.43845 −0.164726
\(727\) −21.9309 −0.813371 −0.406685 0.913568i \(-0.633316\pi\)
−0.406685 + 0.913568i \(0.633316\pi\)
\(728\) −0.561553 −0.0208125
\(729\) 1.00000 0.0370370
\(730\) 7.36932 0.272751
\(731\) 3.86174 0.142832
\(732\) −11.1231 −0.411122
\(733\) 46.1080 1.70304 0.851518 0.524325i \(-0.175682\pi\)
0.851518 + 0.524325i \(0.175682\pi\)
\(734\) 13.7538 0.507662
\(735\) −13.3693 −0.493135
\(736\) −3.12311 −0.115119
\(737\) −29.9309 −1.10252
\(738\) −2.00000 −0.0736210
\(739\) −25.3693 −0.933225 −0.466613 0.884462i \(-0.654526\pi\)
−0.466613 + 0.884462i \(0.654526\pi\)
\(740\) 18.2462 0.670744
\(741\) −2.00000 −0.0734718
\(742\) 3.68466 0.135268
\(743\) 6.73863 0.247216 0.123608 0.992331i \(-0.460553\pi\)
0.123608 + 0.992331i \(0.460553\pi\)
\(744\) 1.12311 0.0411750
\(745\) −12.6307 −0.462752
\(746\) 29.3693 1.07529
\(747\) 11.3693 0.415982
\(748\) −8.80776 −0.322044
\(749\) −4.80776 −0.175672
\(750\) 12.0000 0.438178
\(751\) −18.7386 −0.683782 −0.341891 0.939740i \(-0.611067\pi\)
−0.341891 + 0.939740i \(0.611067\pi\)
\(752\) −10.2462 −0.373641
\(753\) −5.75379 −0.209680
\(754\) 6.00000 0.218507
\(755\) 14.7386 0.536394
\(756\) 0.561553 0.0204235
\(757\) −1.50758 −0.0547938 −0.0273969 0.999625i \(-0.508722\pi\)
−0.0273969 + 0.999625i \(0.508722\pi\)
\(758\) 12.8078 0.465199
\(759\) 8.00000 0.290382
\(760\) −4.00000 −0.145095
\(761\) −20.2462 −0.733925 −0.366962 0.930236i \(-0.619602\pi\)
−0.366962 + 0.930236i \(0.619602\pi\)
\(762\) −11.6847 −0.423291
\(763\) 3.50758 0.126983
\(764\) −3.68466 −0.133306
\(765\) −6.87689 −0.248635
\(766\) −4.49242 −0.162318
\(767\) 8.00000 0.288863
\(768\) −1.00000 −0.0360844
\(769\) 5.93087 0.213873 0.106936 0.994266i \(-0.465896\pi\)
0.106936 + 0.994266i \(0.465896\pi\)
\(770\) 2.87689 0.103676
\(771\) −21.1231 −0.760730
\(772\) −0.630683 −0.0226988
\(773\) −45.0540 −1.62048 −0.810239 0.586099i \(-0.800663\pi\)
−0.810239 + 0.586099i \(0.800663\pi\)
\(774\) −1.12311 −0.0403692
\(775\) −1.12311 −0.0403431
\(776\) −4.24621 −0.152430
\(777\) −5.12311 −0.183790
\(778\) −2.56155 −0.0918361
\(779\) −4.00000 −0.143315
\(780\) −2.00000 −0.0716115
\(781\) −5.75379 −0.205887
\(782\) −10.7386 −0.384013
\(783\) −6.00000 −0.214423
\(784\) −6.68466 −0.238738
\(785\) 13.6155 0.485959
\(786\) 21.3693 0.762218
\(787\) 45.8617 1.63479 0.817397 0.576074i \(-0.195416\pi\)
0.817397 + 0.576074i \(0.195416\pi\)
\(788\) 18.4924 0.658765
\(789\) −0.315342 −0.0112265
\(790\) 14.7386 0.524377
\(791\) −2.24621 −0.0798661
\(792\) 2.56155 0.0910208
\(793\) −11.1231 −0.394993
\(794\) −17.6155 −0.625152
\(795\) 13.1231 0.465429
\(796\) −19.6847 −0.697704
\(797\) 32.7386 1.15966 0.579831 0.814737i \(-0.303119\pi\)
0.579831 + 0.814737i \(0.303119\pi\)
\(798\) 1.12311 0.0397575
\(799\) −35.2311 −1.24639
\(800\) 1.00000 0.0353553
\(801\) 2.00000 0.0706665
\(802\) −5.36932 −0.189597
\(803\) −9.43845 −0.333076
\(804\) −11.6847 −0.412086
\(805\) 3.50758 0.123626
\(806\) 1.12311 0.0395597
\(807\) 11.7538 0.413753
\(808\) 3.68466 0.129626
\(809\) −23.3693 −0.821621 −0.410811 0.911721i \(-0.634754\pi\)
−0.410811 + 0.911721i \(0.634754\pi\)
\(810\) 2.00000 0.0702728
\(811\) −39.2311 −1.37759 −0.688794 0.724957i \(-0.741860\pi\)
−0.688794 + 0.724957i \(0.741860\pi\)
\(812\) −3.36932 −0.118240
\(813\) 11.3693 0.398739
\(814\) −23.3693 −0.819094
\(815\) 44.9848 1.57575
\(816\) −3.43845 −0.120370
\(817\) −2.24621 −0.0785850
\(818\) −21.3693 −0.747161
\(819\) 0.561553 0.0196222
\(820\) −4.00000 −0.139686
\(821\) 2.17708 0.0759806 0.0379903 0.999278i \(-0.487904\pi\)
0.0379903 + 0.999278i \(0.487904\pi\)
\(822\) −6.00000 −0.209274
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 1.00000 0.0348367
\(825\) −2.56155 −0.0891818
\(826\) −4.49242 −0.156311
\(827\) 1.75379 0.0609852 0.0304926 0.999535i \(-0.490292\pi\)
0.0304926 + 0.999535i \(0.490292\pi\)
\(828\) 3.12311 0.108535
\(829\) 49.2311 1.70987 0.854933 0.518739i \(-0.173598\pi\)
0.854933 + 0.518739i \(0.173598\pi\)
\(830\) 22.7386 0.789269
\(831\) −6.00000 −0.208138
\(832\) −1.00000 −0.0346688
\(833\) −22.9848 −0.796378
\(834\) −12.8078 −0.443497
\(835\) 6.38447 0.220944
\(836\) 5.12311 0.177186
\(837\) −1.12311 −0.0388202
\(838\) 19.9309 0.688500
\(839\) −46.2462 −1.59660 −0.798298 0.602262i \(-0.794266\pi\)
−0.798298 + 0.602262i \(0.794266\pi\)
\(840\) 1.12311 0.0387508
\(841\) 7.00000 0.241379
\(842\) −17.0540 −0.587719
\(843\) 22.4924 0.774680
\(844\) 8.49242 0.292321
\(845\) −2.00000 −0.0688021
\(846\) 10.2462 0.352272
\(847\) 2.49242 0.0856407
\(848\) 6.56155 0.225325
\(849\) −21.6155 −0.741843
\(850\) 3.43845 0.117938
\(851\) −28.4924 −0.976708
\(852\) −2.24621 −0.0769539
\(853\) 23.3002 0.797783 0.398892 0.916998i \(-0.369395\pi\)
0.398892 + 0.916998i \(0.369395\pi\)
\(854\) 6.24621 0.213741
\(855\) 4.00000 0.136797
\(856\) −8.56155 −0.292628
\(857\) 35.9309 1.22738 0.613688 0.789549i \(-0.289685\pi\)
0.613688 + 0.789549i \(0.289685\pi\)
\(858\) 2.56155 0.0874500
\(859\) 5.61553 0.191599 0.0957997 0.995401i \(-0.469459\pi\)
0.0957997 + 0.995401i \(0.469459\pi\)
\(860\) −2.24621 −0.0765952
\(861\) 1.12311 0.0382753
\(862\) 32.0000 1.08992
\(863\) −29.7538 −1.01283 −0.506415 0.862290i \(-0.669030\pi\)
−0.506415 + 0.862290i \(0.669030\pi\)
\(864\) 1.00000 0.0340207
\(865\) −41.6155 −1.41497
\(866\) 22.4924 0.764324
\(867\) 5.17708 0.175823
\(868\) −0.630683 −0.0214068
\(869\) −18.8769 −0.640355
\(870\) −12.0000 −0.406838
\(871\) −11.6847 −0.395920
\(872\) 6.24621 0.211523
\(873\) 4.24621 0.143712
\(874\) 6.24621 0.211281
\(875\) −6.73863 −0.227807
\(876\) −3.68466 −0.124493
\(877\) −36.9848 −1.24889 −0.624445 0.781069i \(-0.714675\pi\)
−0.624445 + 0.781069i \(0.714675\pi\)
\(878\) 11.0540 0.373054
\(879\) 7.61553 0.256865
\(880\) 5.12311 0.172700
\(881\) 2.73863 0.0922669 0.0461335 0.998935i \(-0.485310\pi\)
0.0461335 + 0.998935i \(0.485310\pi\)
\(882\) 6.68466 0.225084
\(883\) −38.9157 −1.30962 −0.654809 0.755794i \(-0.727251\pi\)
−0.654809 + 0.755794i \(0.727251\pi\)
\(884\) −3.43845 −0.115647
\(885\) −16.0000 −0.537834
\(886\) 12.4924 0.419691
\(887\) 31.1231 1.04501 0.522506 0.852636i \(-0.324997\pi\)
0.522506 + 0.852636i \(0.324997\pi\)
\(888\) −9.12311 −0.306151
\(889\) 6.56155 0.220067
\(890\) 4.00000 0.134080
\(891\) −2.56155 −0.0858152
\(892\) −5.68466 −0.190336
\(893\) 20.4924 0.685753
\(894\) 6.31534 0.211217
\(895\) 21.6155 0.722527
\(896\) 0.561553 0.0187602
\(897\) 3.12311 0.104277
\(898\) 26.3153 0.878154
\(899\) 6.73863 0.224746
\(900\) −1.00000 −0.0333333
\(901\) 22.5616 0.751634
\(902\) 5.12311 0.170581
\(903\) 0.630683 0.0209878
\(904\) −4.00000 −0.133038
\(905\) −31.8617 −1.05912
\(906\) −7.36932 −0.244829
\(907\) −55.5464 −1.84439 −0.922194 0.386728i \(-0.873605\pi\)
−0.922194 + 0.386728i \(0.873605\pi\)
\(908\) −4.80776 −0.159551
\(909\) −3.68466 −0.122212
\(910\) 1.12311 0.0372306
\(911\) 10.7386 0.355787 0.177893 0.984050i \(-0.443072\pi\)
0.177893 + 0.984050i \(0.443072\pi\)
\(912\) 2.00000 0.0662266
\(913\) −29.1231 −0.963834
\(914\) −8.63068 −0.285478
\(915\) 22.2462 0.735437
\(916\) −6.00000 −0.198246
\(917\) −12.0000 −0.396275
\(918\) 3.43845 0.113486
\(919\) −32.8078 −1.08223 −0.541114 0.840949i \(-0.681997\pi\)
−0.541114 + 0.840949i \(0.681997\pi\)
\(920\) 6.24621 0.205931
\(921\) −21.4384 −0.706421
\(922\) −8.24621 −0.271575
\(923\) −2.24621 −0.0739349
\(924\) −1.43845 −0.0473214
\(925\) 9.12311 0.299966
\(926\) −13.1231 −0.431252
\(927\) −1.00000 −0.0328443
\(928\) −6.00000 −0.196960
\(929\) 0.384472 0.0126141 0.00630706 0.999980i \(-0.497992\pi\)
0.00630706 + 0.999980i \(0.497992\pi\)
\(930\) −2.24621 −0.0736562
\(931\) 13.3693 0.438162
\(932\) 24.4924 0.802276
\(933\) 18.0000 0.589294
\(934\) 38.1771 1.24919
\(935\) 17.6155 0.576089
\(936\) 1.00000 0.0326860
\(937\) 48.7386 1.59222 0.796111 0.605151i \(-0.206887\pi\)
0.796111 + 0.605151i \(0.206887\pi\)
\(938\) 6.56155 0.214242
\(939\) 16.2462 0.530175
\(940\) 20.4924 0.668389
\(941\) −19.4384 −0.633675 −0.316838 0.948480i \(-0.602621\pi\)
−0.316838 + 0.948480i \(0.602621\pi\)
\(942\) −6.80776 −0.221809
\(943\) 6.24621 0.203405
\(944\) −8.00000 −0.260378
\(945\) −1.12311 −0.0365346
\(946\) 2.87689 0.0935359
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −7.36932 −0.239344
\(949\) −3.68466 −0.119609
\(950\) −2.00000 −0.0648886
\(951\) 13.0540 0.423304
\(952\) 1.93087 0.0625798
\(953\) 13.1922 0.427338 0.213669 0.976906i \(-0.431459\pi\)
0.213669 + 0.976906i \(0.431459\pi\)
\(954\) −6.56155 −0.212438
\(955\) 7.36932 0.238465
\(956\) −2.56155 −0.0828465
\(957\) 15.3693 0.496819
\(958\) 5.12311 0.165520
\(959\) 3.36932 0.108801
\(960\) 2.00000 0.0645497
\(961\) −29.7386 −0.959311
\(962\) −9.12311 −0.294141
\(963\) 8.56155 0.275892
\(964\) −8.31534 −0.267819
\(965\) 1.26137 0.0406048
\(966\) −1.75379 −0.0564272
\(967\) 17.1231 0.550642 0.275321 0.961352i \(-0.411216\pi\)
0.275321 + 0.961352i \(0.411216\pi\)
\(968\) 4.43845 0.142657
\(969\) 6.87689 0.220918
\(970\) 8.49242 0.272675
\(971\) 13.6155 0.436943 0.218472 0.975843i \(-0.429893\pi\)
0.218472 + 0.975843i \(0.429893\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 7.19224 0.230572
\(974\) −5.61553 −0.179933
\(975\) −1.00000 −0.0320256
\(976\) 11.1231 0.356042
\(977\) −15.1231 −0.483831 −0.241916 0.970297i \(-0.577776\pi\)
−0.241916 + 0.970297i \(0.577776\pi\)
\(978\) −22.4924 −0.719228
\(979\) −5.12311 −0.163735
\(980\) 13.3693 0.427067
\(981\) −6.24621 −0.199426
\(982\) −15.4384 −0.492661
\(983\) 20.8078 0.663665 0.331832 0.943338i \(-0.392333\pi\)
0.331832 + 0.943338i \(0.392333\pi\)
\(984\) 2.00000 0.0637577
\(985\) −36.9848 −1.17844
\(986\) −20.6307 −0.657015
\(987\) −5.75379 −0.183145
\(988\) 2.00000 0.0636285
\(989\) 3.50758 0.111534
\(990\) −5.12311 −0.162823
\(991\) −55.8617 −1.77451 −0.887253 0.461283i \(-0.847389\pi\)
−0.887253 + 0.461283i \(0.847389\pi\)
\(992\) −1.12311 −0.0356586
\(993\) −21.9309 −0.695955
\(994\) 1.26137 0.0400081
\(995\) 39.3693 1.24809
\(996\) −11.3693 −0.360251
\(997\) 50.4924 1.59911 0.799556 0.600592i \(-0.205068\pi\)
0.799556 + 0.600592i \(0.205068\pi\)
\(998\) −22.7386 −0.719779
\(999\) 9.12311 0.288642
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 8034.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.l.1.1 2 1.1 even 1 trivial