Properties

Label 8034.2.a.ba.1.2
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $14$
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: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} - 4126 x^{5} - 22759 x^{4} + 2997 x^{3} + 10504 x^{2} - 506 x - 1492\)
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 1.2
Root \(3.43788\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.43788 q^{5} +1.00000 q^{6} -0.875663 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.43788 q^{5} +1.00000 q^{6} -0.875663 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.43788 q^{10} -4.12894 q^{11} -1.00000 q^{12} -1.00000 q^{13} +0.875663 q^{14} +3.43788 q^{15} +1.00000 q^{16} +2.91275 q^{17} -1.00000 q^{18} +6.27054 q^{19} -3.43788 q^{20} +0.875663 q^{21} +4.12894 q^{22} -2.22770 q^{23} +1.00000 q^{24} +6.81900 q^{25} +1.00000 q^{26} -1.00000 q^{27} -0.875663 q^{28} -3.54973 q^{29} -3.43788 q^{30} +4.78943 q^{31} -1.00000 q^{32} +4.12894 q^{33} -2.91275 q^{34} +3.01042 q^{35} +1.00000 q^{36} +1.66087 q^{37} -6.27054 q^{38} +1.00000 q^{39} +3.43788 q^{40} -2.47869 q^{41} -0.875663 q^{42} -7.01489 q^{43} -4.12894 q^{44} -3.43788 q^{45} +2.22770 q^{46} +7.14668 q^{47} -1.00000 q^{48} -6.23321 q^{49} -6.81900 q^{50} -2.91275 q^{51} -1.00000 q^{52} -10.0017 q^{53} +1.00000 q^{54} +14.1948 q^{55} +0.875663 q^{56} -6.27054 q^{57} +3.54973 q^{58} -7.72440 q^{59} +3.43788 q^{60} -2.25787 q^{61} -4.78943 q^{62} -0.875663 q^{63} +1.00000 q^{64} +3.43788 q^{65} -4.12894 q^{66} -5.69363 q^{67} +2.91275 q^{68} +2.22770 q^{69} -3.01042 q^{70} -0.740352 q^{71} -1.00000 q^{72} +5.84278 q^{73} -1.66087 q^{74} -6.81900 q^{75} +6.27054 q^{76} +3.61556 q^{77} -1.00000 q^{78} -15.8644 q^{79} -3.43788 q^{80} +1.00000 q^{81} +2.47869 q^{82} -15.7650 q^{83} +0.875663 q^{84} -10.0137 q^{85} +7.01489 q^{86} +3.54973 q^{87} +4.12894 q^{88} +3.84817 q^{89} +3.43788 q^{90} +0.875663 q^{91} -2.22770 q^{92} -4.78943 q^{93} -7.14668 q^{94} -21.5574 q^{95} +1.00000 q^{96} +10.5202 q^{97} +6.23321 q^{98} -4.12894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + q^{10} - q^{11} - 14q^{12} - 14q^{13} - 5q^{14} + q^{15} + 14q^{16} - 12q^{17} - 14q^{18} + 10q^{19} - q^{20} - 5q^{21} + q^{22} - q^{23} + 14q^{24} + 19q^{25} + 14q^{26} - 14q^{27} + 5q^{28} + 6q^{29} - q^{30} + 20q^{31} - 14q^{32} + q^{33} + 12q^{34} - 16q^{35} + 14q^{36} - 3q^{37} - 10q^{38} + 14q^{39} + q^{40} + q^{41} + 5q^{42} + 6q^{43} - q^{44} - q^{45} + q^{46} - 13q^{47} - 14q^{48} + 9q^{49} - 19q^{50} + 12q^{51} - 14q^{52} - 27q^{53} + 14q^{54} + 10q^{55} - 5q^{56} - 10q^{57} - 6q^{58} - 6q^{59} + q^{60} - 4q^{61} - 20q^{62} + 5q^{63} + 14q^{64} + q^{65} - q^{66} + 13q^{67} - 12q^{68} + q^{69} + 16q^{70} + 18q^{71} - 14q^{72} + 11q^{73} + 3q^{74} - 19q^{75} + 10q^{76} - 15q^{77} - 14q^{78} + 33q^{79} - q^{80} + 14q^{81} - q^{82} - 25q^{83} - 5q^{84} + 25q^{85} - 6q^{86} - 6q^{87} + q^{88} + 3q^{89} + q^{90} - 5q^{91} - q^{92} - 20q^{93} + 13q^{94} + 30q^{95} + 14q^{96} + 11q^{97} - 9q^{98} - q^{99} + O(q^{100}) \)

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
\(5\) −3.43788 −1.53747 −0.768733 0.639570i \(-0.779112\pi\)
−0.768733 + 0.639570i \(0.779112\pi\)
\(6\) 1.00000 0.408248
\(7\) −0.875663 −0.330969 −0.165485 0.986212i \(-0.552919\pi\)
−0.165485 + 0.986212i \(0.552919\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.43788 1.08715
\(11\) −4.12894 −1.24492 −0.622462 0.782650i \(-0.713867\pi\)
−0.622462 + 0.782650i \(0.713867\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 0.875663 0.234031
\(15\) 3.43788 0.887656
\(16\) 1.00000 0.250000
\(17\) 2.91275 0.706446 0.353223 0.935539i \(-0.385086\pi\)
0.353223 + 0.935539i \(0.385086\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.27054 1.43856 0.719281 0.694719i \(-0.244471\pi\)
0.719281 + 0.694719i \(0.244471\pi\)
\(20\) −3.43788 −0.768733
\(21\) 0.875663 0.191085
\(22\) 4.12894 0.880294
\(23\) −2.22770 −0.464508 −0.232254 0.972655i \(-0.574610\pi\)
−0.232254 + 0.972655i \(0.574610\pi\)
\(24\) 1.00000 0.204124
\(25\) 6.81900 1.36380
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) −0.875663 −0.165485
\(29\) −3.54973 −0.659168 −0.329584 0.944126i \(-0.606909\pi\)
−0.329584 + 0.944126i \(0.606909\pi\)
\(30\) −3.43788 −0.627668
\(31\) 4.78943 0.860206 0.430103 0.902780i \(-0.358477\pi\)
0.430103 + 0.902780i \(0.358477\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.12894 0.718757
\(34\) −2.91275 −0.499533
\(35\) 3.01042 0.508854
\(36\) 1.00000 0.166667
\(37\) 1.66087 0.273045 0.136522 0.990637i \(-0.456407\pi\)
0.136522 + 0.990637i \(0.456407\pi\)
\(38\) −6.27054 −1.01722
\(39\) 1.00000 0.160128
\(40\) 3.43788 0.543576
\(41\) −2.47869 −0.387107 −0.193553 0.981090i \(-0.562001\pi\)
−0.193553 + 0.981090i \(0.562001\pi\)
\(42\) −0.875663 −0.135118
\(43\) −7.01489 −1.06976 −0.534880 0.844928i \(-0.679643\pi\)
−0.534880 + 0.844928i \(0.679643\pi\)
\(44\) −4.12894 −0.622462
\(45\) −3.43788 −0.512488
\(46\) 2.22770 0.328457
\(47\) 7.14668 1.04245 0.521225 0.853419i \(-0.325475\pi\)
0.521225 + 0.853419i \(0.325475\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.23321 −0.890459
\(50\) −6.81900 −0.964352
\(51\) −2.91275 −0.407867
\(52\) −1.00000 −0.138675
\(53\) −10.0017 −1.37383 −0.686917 0.726736i \(-0.741036\pi\)
−0.686917 + 0.726736i \(0.741036\pi\)
\(54\) 1.00000 0.136083
\(55\) 14.1948 1.91403
\(56\) 0.875663 0.117015
\(57\) −6.27054 −0.830554
\(58\) 3.54973 0.466102
\(59\) −7.72440 −1.00563 −0.502816 0.864394i \(-0.667703\pi\)
−0.502816 + 0.864394i \(0.667703\pi\)
\(60\) 3.43788 0.443828
\(61\) −2.25787 −0.289091 −0.144545 0.989498i \(-0.546172\pi\)
−0.144545 + 0.989498i \(0.546172\pi\)
\(62\) −4.78943 −0.608258
\(63\) −0.875663 −0.110323
\(64\) 1.00000 0.125000
\(65\) 3.43788 0.426416
\(66\) −4.12894 −0.508238
\(67\) −5.69363 −0.695588 −0.347794 0.937571i \(-0.613069\pi\)
−0.347794 + 0.937571i \(0.613069\pi\)
\(68\) 2.91275 0.353223
\(69\) 2.22770 0.268184
\(70\) −3.01042 −0.359814
\(71\) −0.740352 −0.0878636 −0.0439318 0.999035i \(-0.513988\pi\)
−0.0439318 + 0.999035i \(0.513988\pi\)
\(72\) −1.00000 −0.117851
\(73\) 5.84278 0.683845 0.341923 0.939728i \(-0.388922\pi\)
0.341923 + 0.939728i \(0.388922\pi\)
\(74\) −1.66087 −0.193072
\(75\) −6.81900 −0.787390
\(76\) 6.27054 0.719281
\(77\) 3.61556 0.412032
\(78\) −1.00000 −0.113228
\(79\) −15.8644 −1.78488 −0.892442 0.451162i \(-0.851010\pi\)
−0.892442 + 0.451162i \(0.851010\pi\)
\(80\) −3.43788 −0.384366
\(81\) 1.00000 0.111111
\(82\) 2.47869 0.273726
\(83\) −15.7650 −1.73044 −0.865219 0.501395i \(-0.832820\pi\)
−0.865219 + 0.501395i \(0.832820\pi\)
\(84\) 0.875663 0.0955426
\(85\) −10.0137 −1.08614
\(86\) 7.01489 0.756435
\(87\) 3.54973 0.380571
\(88\) 4.12894 0.440147
\(89\) 3.84817 0.407905 0.203952 0.978981i \(-0.434621\pi\)
0.203952 + 0.978981i \(0.434621\pi\)
\(90\) 3.43788 0.362384
\(91\) 0.875663 0.0917944
\(92\) −2.22770 −0.232254
\(93\) −4.78943 −0.496640
\(94\) −7.14668 −0.737124
\(95\) −21.5574 −2.21174
\(96\) 1.00000 0.102062
\(97\) 10.5202 1.06817 0.534083 0.845432i \(-0.320657\pi\)
0.534083 + 0.845432i \(0.320657\pi\)
\(98\) 6.23321 0.629650
\(99\) −4.12894 −0.414975
\(100\) 6.81900 0.681900
\(101\) −2.19696 −0.218606 −0.109303 0.994008i \(-0.534862\pi\)
−0.109303 + 0.994008i \(0.534862\pi\)
\(102\) 2.91275 0.288405
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) −3.01042 −0.293787
\(106\) 10.0017 0.971447
\(107\) −5.52530 −0.534151 −0.267076 0.963676i \(-0.586057\pi\)
−0.267076 + 0.963676i \(0.586057\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −11.0915 −1.06238 −0.531188 0.847254i \(-0.678254\pi\)
−0.531188 + 0.847254i \(0.678254\pi\)
\(110\) −14.1948 −1.35342
\(111\) −1.66087 −0.157642
\(112\) −0.875663 −0.0827424
\(113\) −15.7340 −1.48013 −0.740064 0.672536i \(-0.765205\pi\)
−0.740064 + 0.672536i \(0.765205\pi\)
\(114\) 6.27054 0.587290
\(115\) 7.65856 0.714165
\(116\) −3.54973 −0.329584
\(117\) −1.00000 −0.0924500
\(118\) 7.72440 0.711089
\(119\) −2.55059 −0.233812
\(120\) −3.43788 −0.313834
\(121\) 6.04818 0.549835
\(122\) 2.25787 0.204418
\(123\) 2.47869 0.223496
\(124\) 4.78943 0.430103
\(125\) −6.25349 −0.559329
\(126\) 0.875663 0.0780102
\(127\) −3.96860 −0.352156 −0.176078 0.984376i \(-0.556341\pi\)
−0.176078 + 0.984376i \(0.556341\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.01489 0.617627
\(130\) −3.43788 −0.301522
\(131\) 9.42019 0.823046 0.411523 0.911399i \(-0.364997\pi\)
0.411523 + 0.911399i \(0.364997\pi\)
\(132\) 4.12894 0.359378
\(133\) −5.49088 −0.476120
\(134\) 5.69363 0.491855
\(135\) 3.43788 0.295885
\(136\) −2.91275 −0.249766
\(137\) −11.3092 −0.966209 −0.483104 0.875563i \(-0.660491\pi\)
−0.483104 + 0.875563i \(0.660491\pi\)
\(138\) −2.22770 −0.189635
\(139\) 2.37115 0.201118 0.100559 0.994931i \(-0.467937\pi\)
0.100559 + 0.994931i \(0.467937\pi\)
\(140\) 3.01042 0.254427
\(141\) −7.14668 −0.601859
\(142\) 0.740352 0.0621290
\(143\) 4.12894 0.345280
\(144\) 1.00000 0.0833333
\(145\) 12.2035 1.01345
\(146\) −5.84278 −0.483552
\(147\) 6.23321 0.514107
\(148\) 1.66087 0.136522
\(149\) 4.79584 0.392890 0.196445 0.980515i \(-0.437060\pi\)
0.196445 + 0.980515i \(0.437060\pi\)
\(150\) 6.81900 0.556769
\(151\) 7.33233 0.596697 0.298348 0.954457i \(-0.403564\pi\)
0.298348 + 0.954457i \(0.403564\pi\)
\(152\) −6.27054 −0.508608
\(153\) 2.91275 0.235482
\(154\) −3.61556 −0.291350
\(155\) −16.4655 −1.32254
\(156\) 1.00000 0.0800641
\(157\) −8.80210 −0.702484 −0.351242 0.936285i \(-0.614241\pi\)
−0.351242 + 0.936285i \(0.614241\pi\)
\(158\) 15.8644 1.26210
\(159\) 10.0017 0.793183
\(160\) 3.43788 0.271788
\(161\) 1.95072 0.153738
\(162\) −1.00000 −0.0785674
\(163\) −5.78137 −0.452832 −0.226416 0.974031i \(-0.572701\pi\)
−0.226416 + 0.974031i \(0.572701\pi\)
\(164\) −2.47869 −0.193553
\(165\) −14.1948 −1.10506
\(166\) 15.7650 1.22360
\(167\) 11.7593 0.909962 0.454981 0.890501i \(-0.349646\pi\)
0.454981 + 0.890501i \(0.349646\pi\)
\(168\) −0.875663 −0.0675588
\(169\) 1.00000 0.0769231
\(170\) 10.0137 0.768014
\(171\) 6.27054 0.479521
\(172\) −7.01489 −0.534880
\(173\) 4.87706 0.370796 0.185398 0.982664i \(-0.440643\pi\)
0.185398 + 0.982664i \(0.440643\pi\)
\(174\) −3.54973 −0.269104
\(175\) −5.97114 −0.451376
\(176\) −4.12894 −0.311231
\(177\) 7.72440 0.580601
\(178\) −3.84817 −0.288432
\(179\) −1.99720 −0.149278 −0.0746389 0.997211i \(-0.523780\pi\)
−0.0746389 + 0.997211i \(0.523780\pi\)
\(180\) −3.43788 −0.256244
\(181\) 10.2805 0.764146 0.382073 0.924132i \(-0.375210\pi\)
0.382073 + 0.924132i \(0.375210\pi\)
\(182\) −0.875663 −0.0649084
\(183\) 2.25787 0.166907
\(184\) 2.22770 0.164228
\(185\) −5.70985 −0.419797
\(186\) 4.78943 0.351178
\(187\) −12.0266 −0.879471
\(188\) 7.14668 0.521225
\(189\) 0.875663 0.0636951
\(190\) 21.5574 1.56394
\(191\) 21.0062 1.51995 0.759976 0.649951i \(-0.225211\pi\)
0.759976 + 0.649951i \(0.225211\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 3.91167 0.281568 0.140784 0.990040i \(-0.455038\pi\)
0.140784 + 0.990040i \(0.455038\pi\)
\(194\) −10.5202 −0.755308
\(195\) −3.43788 −0.246191
\(196\) −6.23321 −0.445230
\(197\) 22.7100 1.61802 0.809012 0.587793i \(-0.200003\pi\)
0.809012 + 0.587793i \(0.200003\pi\)
\(198\) 4.12894 0.293431
\(199\) −5.93474 −0.420702 −0.210351 0.977626i \(-0.567461\pi\)
−0.210351 + 0.977626i \(0.567461\pi\)
\(200\) −6.81900 −0.482176
\(201\) 5.69363 0.401598
\(202\) 2.19696 0.154578
\(203\) 3.10837 0.218165
\(204\) −2.91275 −0.203933
\(205\) 8.52144 0.595163
\(206\) −1.00000 −0.0696733
\(207\) −2.22770 −0.154836
\(208\) −1.00000 −0.0693375
\(209\) −25.8907 −1.79090
\(210\) 3.01042 0.207739
\(211\) 3.86201 0.265871 0.132936 0.991125i \(-0.457560\pi\)
0.132936 + 0.991125i \(0.457560\pi\)
\(212\) −10.0017 −0.686917
\(213\) 0.740352 0.0507281
\(214\) 5.52530 0.377702
\(215\) 24.1163 1.64472
\(216\) 1.00000 0.0680414
\(217\) −4.19392 −0.284702
\(218\) 11.0915 0.751214
\(219\) −5.84278 −0.394818
\(220\) 14.1948 0.957013
\(221\) −2.91275 −0.195933
\(222\) 1.66087 0.111470
\(223\) 2.77489 0.185820 0.0929102 0.995674i \(-0.470383\pi\)
0.0929102 + 0.995674i \(0.470383\pi\)
\(224\) 0.875663 0.0585077
\(225\) 6.81900 0.454600
\(226\) 15.7340 1.04661
\(227\) −26.0660 −1.73006 −0.865031 0.501719i \(-0.832701\pi\)
−0.865031 + 0.501719i \(0.832701\pi\)
\(228\) −6.27054 −0.415277
\(229\) −18.3397 −1.21192 −0.605960 0.795495i \(-0.707211\pi\)
−0.605960 + 0.795495i \(0.707211\pi\)
\(230\) −7.65856 −0.504991
\(231\) −3.61556 −0.237887
\(232\) 3.54973 0.233051
\(233\) −8.09808 −0.530523 −0.265261 0.964177i \(-0.585458\pi\)
−0.265261 + 0.964177i \(0.585458\pi\)
\(234\) 1.00000 0.0653720
\(235\) −24.5694 −1.60273
\(236\) −7.72440 −0.502816
\(237\) 15.8644 1.03050
\(238\) 2.55059 0.165330
\(239\) 21.0662 1.36266 0.681330 0.731977i \(-0.261402\pi\)
0.681330 + 0.731977i \(0.261402\pi\)
\(240\) 3.43788 0.221914
\(241\) 8.02129 0.516696 0.258348 0.966052i \(-0.416822\pi\)
0.258348 + 0.966052i \(0.416822\pi\)
\(242\) −6.04818 −0.388792
\(243\) −1.00000 −0.0641500
\(244\) −2.25787 −0.144545
\(245\) 21.4290 1.36905
\(246\) −2.47869 −0.158036
\(247\) −6.27054 −0.398985
\(248\) −4.78943 −0.304129
\(249\) 15.7650 0.999068
\(250\) 6.25349 0.395505
\(251\) 18.0515 1.13940 0.569699 0.821853i \(-0.307060\pi\)
0.569699 + 0.821853i \(0.307060\pi\)
\(252\) −0.875663 −0.0551616
\(253\) 9.19806 0.578277
\(254\) 3.96860 0.249012
\(255\) 10.0137 0.627081
\(256\) 1.00000 0.0625000
\(257\) −2.46728 −0.153905 −0.0769524 0.997035i \(-0.524519\pi\)
−0.0769524 + 0.997035i \(0.524519\pi\)
\(258\) −7.01489 −0.436728
\(259\) −1.45436 −0.0903694
\(260\) 3.43788 0.213208
\(261\) −3.54973 −0.219723
\(262\) −9.42019 −0.581981
\(263\) −11.8755 −0.732274 −0.366137 0.930561i \(-0.619320\pi\)
−0.366137 + 0.930561i \(0.619320\pi\)
\(264\) −4.12894 −0.254119
\(265\) 34.3845 2.11222
\(266\) 5.49088 0.336668
\(267\) −3.84817 −0.235504
\(268\) −5.69363 −0.347794
\(269\) 4.75452 0.289888 0.144944 0.989440i \(-0.453700\pi\)
0.144944 + 0.989440i \(0.453700\pi\)
\(270\) −3.43788 −0.209223
\(271\) −1.68670 −0.102460 −0.0512299 0.998687i \(-0.516314\pi\)
−0.0512299 + 0.998687i \(0.516314\pi\)
\(272\) 2.91275 0.176611
\(273\) −0.875663 −0.0529975
\(274\) 11.3092 0.683213
\(275\) −28.1553 −1.69783
\(276\) 2.22770 0.134092
\(277\) −17.6902 −1.06290 −0.531450 0.847089i \(-0.678353\pi\)
−0.531450 + 0.847089i \(0.678353\pi\)
\(278\) −2.37115 −0.142212
\(279\) 4.78943 0.286735
\(280\) −3.01042 −0.179907
\(281\) −31.3662 −1.87115 −0.935574 0.353130i \(-0.885117\pi\)
−0.935574 + 0.353130i \(0.885117\pi\)
\(282\) 7.14668 0.425579
\(283\) 6.42432 0.381886 0.190943 0.981601i \(-0.438845\pi\)
0.190943 + 0.981601i \(0.438845\pi\)
\(284\) −0.740352 −0.0439318
\(285\) 21.5574 1.27695
\(286\) −4.12894 −0.244150
\(287\) 2.17050 0.128120
\(288\) −1.00000 −0.0589256
\(289\) −8.51589 −0.500935
\(290\) −12.2035 −0.716616
\(291\) −10.5202 −0.616706
\(292\) 5.84278 0.341923
\(293\) 13.7022 0.800492 0.400246 0.916408i \(-0.368925\pi\)
0.400246 + 0.916408i \(0.368925\pi\)
\(294\) −6.23321 −0.363528
\(295\) 26.5555 1.54612
\(296\) −1.66087 −0.0965359
\(297\) 4.12894 0.239586
\(298\) −4.79584 −0.277815
\(299\) 2.22770 0.128831
\(300\) −6.81900 −0.393695
\(301\) 6.14268 0.354058
\(302\) −7.33233 −0.421928
\(303\) 2.19696 0.126212
\(304\) 6.27054 0.359640
\(305\) 7.76229 0.444467
\(306\) −2.91275 −0.166511
\(307\) 10.4205 0.594728 0.297364 0.954764i \(-0.403892\pi\)
0.297364 + 0.954764i \(0.403892\pi\)
\(308\) 3.61556 0.206016
\(309\) −1.00000 −0.0568880
\(310\) 16.4655 0.935175
\(311\) −2.65273 −0.150423 −0.0752113 0.997168i \(-0.523963\pi\)
−0.0752113 + 0.997168i \(0.523963\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 0.504119 0.0284945 0.0142472 0.999899i \(-0.495465\pi\)
0.0142472 + 0.999899i \(0.495465\pi\)
\(314\) 8.80210 0.496732
\(315\) 3.01042 0.169618
\(316\) −15.8644 −0.892442
\(317\) −31.0342 −1.74305 −0.871527 0.490347i \(-0.836870\pi\)
−0.871527 + 0.490347i \(0.836870\pi\)
\(318\) −10.0017 −0.560865
\(319\) 14.6566 0.820614
\(320\) −3.43788 −0.192183
\(321\) 5.52530 0.308392
\(322\) −1.95072 −0.108709
\(323\) 18.2645 1.01627
\(324\) 1.00000 0.0555556
\(325\) −6.81900 −0.378250
\(326\) 5.78137 0.320200
\(327\) 11.0915 0.613363
\(328\) 2.47869 0.136863
\(329\) −6.25808 −0.345019
\(330\) 14.1948 0.781398
\(331\) 24.6859 1.35686 0.678430 0.734665i \(-0.262660\pi\)
0.678430 + 0.734665i \(0.262660\pi\)
\(332\) −15.7650 −0.865219
\(333\) 1.66087 0.0910149
\(334\) −11.7593 −0.643440
\(335\) 19.5740 1.06944
\(336\) 0.875663 0.0477713
\(337\) 28.0255 1.52665 0.763323 0.646018i \(-0.223567\pi\)
0.763323 + 0.646018i \(0.223567\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 15.7340 0.854552
\(340\) −10.0137 −0.543068
\(341\) −19.7753 −1.07089
\(342\) −6.27054 −0.339072
\(343\) 11.5878 0.625684
\(344\) 7.01489 0.378217
\(345\) −7.65856 −0.412323
\(346\) −4.87706 −0.262192
\(347\) 2.12054 0.113837 0.0569183 0.998379i \(-0.481873\pi\)
0.0569183 + 0.998379i \(0.481873\pi\)
\(348\) 3.54973 0.190285
\(349\) −36.3447 −1.94549 −0.972744 0.231883i \(-0.925511\pi\)
−0.972744 + 0.231883i \(0.925511\pi\)
\(350\) 5.97114 0.319171
\(351\) 1.00000 0.0533761
\(352\) 4.12894 0.220073
\(353\) 25.4726 1.35577 0.677884 0.735169i \(-0.262897\pi\)
0.677884 + 0.735169i \(0.262897\pi\)
\(354\) −7.72440 −0.410547
\(355\) 2.54524 0.135087
\(356\) 3.84817 0.203952
\(357\) 2.55059 0.134991
\(358\) 1.99720 0.105555
\(359\) −28.2455 −1.49074 −0.745369 0.666652i \(-0.767727\pi\)
−0.745369 + 0.666652i \(0.767727\pi\)
\(360\) 3.43788 0.181192
\(361\) 20.3197 1.06946
\(362\) −10.2805 −0.540333
\(363\) −6.04818 −0.317447
\(364\) 0.875663 0.0458972
\(365\) −20.0868 −1.05139
\(366\) −2.25787 −0.118021
\(367\) 8.99275 0.469418 0.234709 0.972066i \(-0.424586\pi\)
0.234709 + 0.972066i \(0.424586\pi\)
\(368\) −2.22770 −0.116127
\(369\) −2.47869 −0.129036
\(370\) 5.70985 0.296841
\(371\) 8.75808 0.454697
\(372\) −4.78943 −0.248320
\(373\) −5.98913 −0.310105 −0.155053 0.987906i \(-0.549555\pi\)
−0.155053 + 0.987906i \(0.549555\pi\)
\(374\) 12.0266 0.621880
\(375\) 6.25349 0.322929
\(376\) −7.14668 −0.368562
\(377\) 3.54973 0.182820
\(378\) −0.875663 −0.0450392
\(379\) 34.3213 1.76296 0.881482 0.472217i \(-0.156546\pi\)
0.881482 + 0.472217i \(0.156546\pi\)
\(380\) −21.5574 −1.10587
\(381\) 3.96860 0.203317
\(382\) −21.0062 −1.07477
\(383\) 1.97543 0.100940 0.0504699 0.998726i \(-0.483928\pi\)
0.0504699 + 0.998726i \(0.483928\pi\)
\(384\) 1.00000 0.0510310
\(385\) −12.4299 −0.633484
\(386\) −3.91167 −0.199099
\(387\) −7.01489 −0.356587
\(388\) 10.5202 0.534083
\(389\) 14.2832 0.724188 0.362094 0.932142i \(-0.382062\pi\)
0.362094 + 0.932142i \(0.382062\pi\)
\(390\) 3.43788 0.174084
\(391\) −6.48874 −0.328150
\(392\) 6.23321 0.314825
\(393\) −9.42019 −0.475186
\(394\) −22.7100 −1.14412
\(395\) 54.5399 2.74420
\(396\) −4.12894 −0.207487
\(397\) 28.1499 1.41280 0.706401 0.707812i \(-0.250318\pi\)
0.706401 + 0.707812i \(0.250318\pi\)
\(398\) 5.93474 0.297482
\(399\) 5.49088 0.274888
\(400\) 6.81900 0.340950
\(401\) −33.3899 −1.66741 −0.833707 0.552207i \(-0.813786\pi\)
−0.833707 + 0.552207i \(0.813786\pi\)
\(402\) −5.69363 −0.283972
\(403\) −4.78943 −0.238578
\(404\) −2.19696 −0.109303
\(405\) −3.43788 −0.170829
\(406\) −3.10837 −0.154266
\(407\) −6.85762 −0.339920
\(408\) 2.91275 0.144203
\(409\) −14.4536 −0.714684 −0.357342 0.933974i \(-0.616317\pi\)
−0.357342 + 0.933974i \(0.616317\pi\)
\(410\) −8.52144 −0.420844
\(411\) 11.3092 0.557841
\(412\) 1.00000 0.0492665
\(413\) 6.76397 0.332833
\(414\) 2.22770 0.109486
\(415\) 54.1982 2.66049
\(416\) 1.00000 0.0490290
\(417\) −2.37115 −0.116116
\(418\) 25.8907 1.26636
\(419\) −20.8682 −1.01948 −0.509739 0.860329i \(-0.670258\pi\)
−0.509739 + 0.860329i \(0.670258\pi\)
\(420\) −3.01042 −0.146893
\(421\) −4.12536 −0.201058 −0.100529 0.994934i \(-0.532053\pi\)
−0.100529 + 0.994934i \(0.532053\pi\)
\(422\) −3.86201 −0.188000
\(423\) 7.14668 0.347484
\(424\) 10.0017 0.485724
\(425\) 19.8620 0.963450
\(426\) −0.740352 −0.0358702
\(427\) 1.97713 0.0956803
\(428\) −5.52530 −0.267076
\(429\) −4.12894 −0.199347
\(430\) −24.1163 −1.16299
\(431\) 29.8416 1.43742 0.718709 0.695311i \(-0.244734\pi\)
0.718709 + 0.695311i \(0.244734\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −33.2387 −1.59735 −0.798675 0.601763i \(-0.794465\pi\)
−0.798675 + 0.601763i \(0.794465\pi\)
\(434\) 4.19392 0.201315
\(435\) −12.2035 −0.585115
\(436\) −11.0915 −0.531188
\(437\) −13.9689 −0.668223
\(438\) 5.84278 0.279179
\(439\) 21.2514 1.01427 0.507137 0.861866i \(-0.330704\pi\)
0.507137 + 0.861866i \(0.330704\pi\)
\(440\) −14.1948 −0.676711
\(441\) −6.23321 −0.296820
\(442\) 2.91275 0.138545
\(443\) −5.44412 −0.258658 −0.129329 0.991602i \(-0.541282\pi\)
−0.129329 + 0.991602i \(0.541282\pi\)
\(444\) −1.66087 −0.0788212
\(445\) −13.2295 −0.627140
\(446\) −2.77489 −0.131395
\(447\) −4.79584 −0.226835
\(448\) −0.875663 −0.0413712
\(449\) 37.7400 1.78106 0.890530 0.454924i \(-0.150333\pi\)
0.890530 + 0.454924i \(0.150333\pi\)
\(450\) −6.81900 −0.321451
\(451\) 10.2344 0.481918
\(452\) −15.7340 −0.740064
\(453\) −7.33233 −0.344503
\(454\) 26.0660 1.22334
\(455\) −3.01042 −0.141131
\(456\) 6.27054 0.293645
\(457\) 11.0662 0.517657 0.258829 0.965923i \(-0.416664\pi\)
0.258829 + 0.965923i \(0.416664\pi\)
\(458\) 18.3397 0.856957
\(459\) −2.91275 −0.135956
\(460\) 7.65856 0.357082
\(461\) 39.3826 1.83423 0.917115 0.398624i \(-0.130512\pi\)
0.917115 + 0.398624i \(0.130512\pi\)
\(462\) 3.61556 0.168211
\(463\) −24.5365 −1.14031 −0.570154 0.821538i \(-0.693117\pi\)
−0.570154 + 0.821538i \(0.693117\pi\)
\(464\) −3.54973 −0.164792
\(465\) 16.4655 0.763567
\(466\) 8.09808 0.375136
\(467\) −37.4980 −1.73520 −0.867601 0.497261i \(-0.834339\pi\)
−0.867601 + 0.497261i \(0.834339\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 4.98570 0.230218
\(470\) 24.5694 1.13330
\(471\) 8.80210 0.405580
\(472\) 7.72440 0.355544
\(473\) 28.9641 1.33177
\(474\) −15.8644 −0.728676
\(475\) 42.7588 1.96191
\(476\) −2.55059 −0.116906
\(477\) −10.0017 −0.457945
\(478\) −21.0662 −0.963545
\(479\) 15.1467 0.692073 0.346036 0.938221i \(-0.387527\pi\)
0.346036 + 0.938221i \(0.387527\pi\)
\(480\) −3.43788 −0.156917
\(481\) −1.66087 −0.0757290
\(482\) −8.02129 −0.365360
\(483\) −1.95072 −0.0887606
\(484\) 6.04818 0.274917
\(485\) −36.1672 −1.64227
\(486\) 1.00000 0.0453609
\(487\) 11.9348 0.540819 0.270410 0.962745i \(-0.412841\pi\)
0.270410 + 0.962745i \(0.412841\pi\)
\(488\) 2.25787 0.102209
\(489\) 5.78137 0.261442
\(490\) −21.4290 −0.968065
\(491\) −20.5208 −0.926090 −0.463045 0.886335i \(-0.653243\pi\)
−0.463045 + 0.886335i \(0.653243\pi\)
\(492\) 2.47869 0.111748
\(493\) −10.3395 −0.465667
\(494\) 6.27054 0.282125
\(495\) 14.1948 0.638009
\(496\) 4.78943 0.215052
\(497\) 0.648299 0.0290802
\(498\) −15.7650 −0.706448
\(499\) −27.9203 −1.24988 −0.624942 0.780671i \(-0.714877\pi\)
−0.624942 + 0.780671i \(0.714877\pi\)
\(500\) −6.25349 −0.279664
\(501\) −11.7593 −0.525367
\(502\) −18.0515 −0.805677
\(503\) −12.8948 −0.574950 −0.287475 0.957788i \(-0.592816\pi\)
−0.287475 + 0.957788i \(0.592816\pi\)
\(504\) 0.875663 0.0390051
\(505\) 7.55289 0.336099
\(506\) −9.19806 −0.408903
\(507\) −1.00000 −0.0444116
\(508\) −3.96860 −0.176078
\(509\) 2.25904 0.100130 0.0500650 0.998746i \(-0.484057\pi\)
0.0500650 + 0.998746i \(0.484057\pi\)
\(510\) −10.0137 −0.443413
\(511\) −5.11630 −0.226332
\(512\) −1.00000 −0.0441942
\(513\) −6.27054 −0.276851
\(514\) 2.46728 0.108827
\(515\) −3.43788 −0.151491
\(516\) 7.01489 0.308813
\(517\) −29.5083 −1.29777
\(518\) 1.45436 0.0639008
\(519\) −4.87706 −0.214079
\(520\) −3.43788 −0.150761
\(521\) 19.9571 0.874336 0.437168 0.899380i \(-0.355982\pi\)
0.437168 + 0.899380i \(0.355982\pi\)
\(522\) 3.54973 0.155367
\(523\) 15.6375 0.683780 0.341890 0.939740i \(-0.388933\pi\)
0.341890 + 0.939740i \(0.388933\pi\)
\(524\) 9.42019 0.411523
\(525\) 5.97114 0.260602
\(526\) 11.8755 0.517796
\(527\) 13.9504 0.607689
\(528\) 4.12894 0.179689
\(529\) −18.0373 −0.784232
\(530\) −34.3845 −1.49357
\(531\) −7.72440 −0.335210
\(532\) −5.49088 −0.238060
\(533\) 2.47869 0.107364
\(534\) 3.84817 0.166526
\(535\) 18.9953 0.821239
\(536\) 5.69363 0.245927
\(537\) 1.99720 0.0861856
\(538\) −4.75452 −0.204982
\(539\) 25.7366 1.10855
\(540\) 3.43788 0.147943
\(541\) 30.9568 1.33094 0.665468 0.746426i \(-0.268232\pi\)
0.665468 + 0.746426i \(0.268232\pi\)
\(542\) 1.68670 0.0724500
\(543\) −10.2805 −0.441180
\(544\) −2.91275 −0.124883
\(545\) 38.1313 1.63337
\(546\) 0.875663 0.0374749
\(547\) −33.0068 −1.41127 −0.705634 0.708576i \(-0.749338\pi\)
−0.705634 + 0.708576i \(0.749338\pi\)
\(548\) −11.3092 −0.483104
\(549\) −2.25787 −0.0963637
\(550\) 28.1553 1.20054
\(551\) −22.2587 −0.948254
\(552\) −2.22770 −0.0948173
\(553\) 13.8919 0.590742
\(554\) 17.6902 0.751584
\(555\) 5.70985 0.242370
\(556\) 2.37115 0.100559
\(557\) 21.6305 0.916512 0.458256 0.888820i \(-0.348474\pi\)
0.458256 + 0.888820i \(0.348474\pi\)
\(558\) −4.78943 −0.202753
\(559\) 7.01489 0.296698
\(560\) 3.01042 0.127213
\(561\) 12.0266 0.507763
\(562\) 31.3662 1.32310
\(563\) −19.9428 −0.840489 −0.420245 0.907411i \(-0.638056\pi\)
−0.420245 + 0.907411i \(0.638056\pi\)
\(564\) −7.14668 −0.300930
\(565\) 54.0915 2.27565
\(566\) −6.42432 −0.270034
\(567\) −0.875663 −0.0367744
\(568\) 0.740352 0.0310645
\(569\) −41.1857 −1.72659 −0.863296 0.504697i \(-0.831604\pi\)
−0.863296 + 0.504697i \(0.831604\pi\)
\(570\) −21.5574 −0.902938
\(571\) 32.1447 1.34522 0.672608 0.739999i \(-0.265174\pi\)
0.672608 + 0.739999i \(0.265174\pi\)
\(572\) 4.12894 0.172640
\(573\) −21.0062 −0.877545
\(574\) −2.17050 −0.0905949
\(575\) −15.1907 −0.633496
\(576\) 1.00000 0.0416667
\(577\) −13.0790 −0.544487 −0.272244 0.962228i \(-0.587766\pi\)
−0.272244 + 0.962228i \(0.587766\pi\)
\(578\) 8.51589 0.354214
\(579\) −3.91167 −0.162564
\(580\) 12.2035 0.506724
\(581\) 13.8049 0.572722
\(582\) 10.5202 0.436077
\(583\) 41.2963 1.71032
\(584\) −5.84278 −0.241776
\(585\) 3.43788 0.142139
\(586\) −13.7022 −0.566033
\(587\) 1.01916 0.0420651 0.0210326 0.999779i \(-0.493305\pi\)
0.0210326 + 0.999779i \(0.493305\pi\)
\(588\) 6.23321 0.257053
\(589\) 30.0323 1.23746
\(590\) −26.5555 −1.09327
\(591\) −22.7100 −0.934166
\(592\) 1.66087 0.0682612
\(593\) 8.55788 0.351430 0.175715 0.984441i \(-0.443776\pi\)
0.175715 + 0.984441i \(0.443776\pi\)
\(594\) −4.12894 −0.169413
\(595\) 8.76860 0.359478
\(596\) 4.79584 0.196445
\(597\) 5.93474 0.242893
\(598\) −2.22770 −0.0910975
\(599\) 30.6520 1.25241 0.626203 0.779660i \(-0.284608\pi\)
0.626203 + 0.779660i \(0.284608\pi\)
\(600\) 6.81900 0.278384
\(601\) 33.7400 1.37628 0.688141 0.725577i \(-0.258427\pi\)
0.688141 + 0.725577i \(0.258427\pi\)
\(602\) −6.14268 −0.250357
\(603\) −5.69363 −0.231863
\(604\) 7.33233 0.298348
\(605\) −20.7929 −0.845352
\(606\) −2.19696 −0.0892456
\(607\) 0.587676 0.0238530 0.0119265 0.999929i \(-0.496204\pi\)
0.0119265 + 0.999929i \(0.496204\pi\)
\(608\) −6.27054 −0.254304
\(609\) −3.10837 −0.125957
\(610\) −7.76229 −0.314286
\(611\) −7.14668 −0.289124
\(612\) 2.91275 0.117741
\(613\) 36.0967 1.45793 0.728967 0.684549i \(-0.240001\pi\)
0.728967 + 0.684549i \(0.240001\pi\)
\(614\) −10.4205 −0.420536
\(615\) −8.52144 −0.343618
\(616\) −3.61556 −0.145675
\(617\) −21.3445 −0.859296 −0.429648 0.902996i \(-0.641362\pi\)
−0.429648 + 0.902996i \(0.641362\pi\)
\(618\) 1.00000 0.0402259
\(619\) −6.44665 −0.259113 −0.129556 0.991572i \(-0.541355\pi\)
−0.129556 + 0.991572i \(0.541355\pi\)
\(620\) −16.4655 −0.661269
\(621\) 2.22770 0.0893946
\(622\) 2.65273 0.106365
\(623\) −3.36970 −0.135004
\(624\) 1.00000 0.0400320
\(625\) −12.5963 −0.503851
\(626\) −0.504119 −0.0201486
\(627\) 25.8907 1.03398
\(628\) −8.80210 −0.351242
\(629\) 4.83769 0.192891
\(630\) −3.01042 −0.119938
\(631\) −19.0214 −0.757231 −0.378616 0.925554i \(-0.623600\pi\)
−0.378616 + 0.925554i \(0.623600\pi\)
\(632\) 15.8644 0.631052
\(633\) −3.86201 −0.153501
\(634\) 31.0342 1.23253
\(635\) 13.6436 0.541428
\(636\) 10.0017 0.396592
\(637\) 6.23321 0.246969
\(638\) −14.6566 −0.580262
\(639\) −0.740352 −0.0292879
\(640\) 3.43788 0.135894
\(641\) 46.6464 1.84242 0.921211 0.389062i \(-0.127201\pi\)
0.921211 + 0.389062i \(0.127201\pi\)
\(642\) −5.52530 −0.218066
\(643\) −23.0133 −0.907555 −0.453777 0.891115i \(-0.649924\pi\)
−0.453777 + 0.891115i \(0.649924\pi\)
\(644\) 1.95072 0.0768690
\(645\) −24.1163 −0.949579
\(646\) −18.2645 −0.718608
\(647\) 4.69070 0.184411 0.0922053 0.995740i \(-0.470608\pi\)
0.0922053 + 0.995740i \(0.470608\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 31.8936 1.25193
\(650\) 6.81900 0.267463
\(651\) 4.19392 0.164373
\(652\) −5.78137 −0.226416
\(653\) 2.14583 0.0839730 0.0419865 0.999118i \(-0.486631\pi\)
0.0419865 + 0.999118i \(0.486631\pi\)
\(654\) −11.0915 −0.433713
\(655\) −32.3854 −1.26540
\(656\) −2.47869 −0.0967767
\(657\) 5.84278 0.227948
\(658\) 6.25808 0.243966
\(659\) 38.9252 1.51631 0.758155 0.652074i \(-0.226101\pi\)
0.758155 + 0.652074i \(0.226101\pi\)
\(660\) −14.1948 −0.552532
\(661\) 29.4704 1.14627 0.573133 0.819462i \(-0.305728\pi\)
0.573133 + 0.819462i \(0.305728\pi\)
\(662\) −24.6859 −0.959446
\(663\) 2.91275 0.113122
\(664\) 15.7650 0.611802
\(665\) 18.8770 0.732018
\(666\) −1.66087 −0.0643572
\(667\) 7.90774 0.306189
\(668\) 11.7593 0.454981
\(669\) −2.77489 −0.107284
\(670\) −19.5740 −0.756209
\(671\) 9.32263 0.359896
\(672\) −0.875663 −0.0337794
\(673\) 8.78181 0.338514 0.169257 0.985572i \(-0.445863\pi\)
0.169257 + 0.985572i \(0.445863\pi\)
\(674\) −28.0255 −1.07950
\(675\) −6.81900 −0.262463
\(676\) 1.00000 0.0384615
\(677\) −23.2590 −0.893917 −0.446958 0.894555i \(-0.647493\pi\)
−0.446958 + 0.894555i \(0.647493\pi\)
\(678\) −15.7340 −0.604260
\(679\) −9.21217 −0.353531
\(680\) 10.0137 0.384007
\(681\) 26.0660 0.998852
\(682\) 19.7753 0.757234
\(683\) 34.6559 1.32607 0.663036 0.748588i \(-0.269268\pi\)
0.663036 + 0.748588i \(0.269268\pi\)
\(684\) 6.27054 0.239760
\(685\) 38.8796 1.48551
\(686\) −11.5878 −0.442426
\(687\) 18.3397 0.699702
\(688\) −7.01489 −0.267440
\(689\) 10.0017 0.381033
\(690\) 7.65856 0.291557
\(691\) 6.75173 0.256848 0.128424 0.991719i \(-0.459008\pi\)
0.128424 + 0.991719i \(0.459008\pi\)
\(692\) 4.87706 0.185398
\(693\) 3.61556 0.137344
\(694\) −2.12054 −0.0804947
\(695\) −8.15172 −0.309212
\(696\) −3.54973 −0.134552
\(697\) −7.21981 −0.273470
\(698\) 36.3447 1.37567
\(699\) 8.09808 0.306297
\(700\) −5.97114 −0.225688
\(701\) −41.2012 −1.55615 −0.778074 0.628173i \(-0.783803\pi\)
−0.778074 + 0.628173i \(0.783803\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 10.4145 0.392791
\(704\) −4.12894 −0.155615
\(705\) 24.5694 0.925338
\(706\) −25.4726 −0.958672
\(707\) 1.92380 0.0723519
\(708\) 7.72440 0.290301
\(709\) 20.8885 0.784485 0.392243 0.919862i \(-0.371699\pi\)
0.392243 + 0.919862i \(0.371699\pi\)
\(710\) −2.54524 −0.0955211
\(711\) −15.8644 −0.594961
\(712\) −3.84817 −0.144216
\(713\) −10.6694 −0.399573
\(714\) −2.55059 −0.0954533
\(715\) −14.1948 −0.530855
\(716\) −1.99720 −0.0746389
\(717\) −21.0662 −0.786732
\(718\) 28.2455 1.05411
\(719\) 24.7262 0.922131 0.461066 0.887366i \(-0.347467\pi\)
0.461066 + 0.887366i \(0.347467\pi\)
\(720\) −3.43788 −0.128122
\(721\) −0.875663 −0.0326114
\(722\) −20.3197 −0.756222
\(723\) −8.02129 −0.298315
\(724\) 10.2805 0.382073
\(725\) −24.2056 −0.898973
\(726\) 6.04818 0.224469
\(727\) −37.6271 −1.39551 −0.697757 0.716335i \(-0.745818\pi\)
−0.697757 + 0.716335i \(0.745818\pi\)
\(728\) −0.875663 −0.0324542
\(729\) 1.00000 0.0370370
\(730\) 20.0868 0.743444
\(731\) −20.4326 −0.755728
\(732\) 2.25787 0.0834534
\(733\) 35.7815 1.32162 0.660809 0.750554i \(-0.270213\pi\)
0.660809 + 0.750554i \(0.270213\pi\)
\(734\) −8.99275 −0.331929
\(735\) −21.4290 −0.790421
\(736\) 2.22770 0.0821142
\(737\) 23.5087 0.865953
\(738\) 2.47869 0.0912419
\(739\) 25.6956 0.945228 0.472614 0.881269i \(-0.343310\pi\)
0.472614 + 0.881269i \(0.343310\pi\)
\(740\) −5.70985 −0.209898
\(741\) 6.27054 0.230354
\(742\) −8.75808 −0.321519
\(743\) −23.2477 −0.852875 −0.426437 0.904517i \(-0.640232\pi\)
−0.426437 + 0.904517i \(0.640232\pi\)
\(744\) 4.78943 0.175589
\(745\) −16.4875 −0.604055
\(746\) 5.98913 0.219278
\(747\) −15.7650 −0.576812
\(748\) −12.0266 −0.439735
\(749\) 4.83830 0.176788
\(750\) −6.25349 −0.228345
\(751\) 23.7486 0.866597 0.433299 0.901250i \(-0.357350\pi\)
0.433299 + 0.901250i \(0.357350\pi\)
\(752\) 7.14668 0.260613
\(753\) −18.0515 −0.657832
\(754\) −3.54973 −0.129274
\(755\) −25.2076 −0.917400
\(756\) 0.875663 0.0318475
\(757\) −33.9998 −1.23574 −0.617871 0.786279i \(-0.712005\pi\)
−0.617871 + 0.786279i \(0.712005\pi\)
\(758\) −34.3213 −1.24660
\(759\) −9.19806 −0.333868
\(760\) 21.5574 0.781968
\(761\) 45.5259 1.65031 0.825157 0.564904i \(-0.191087\pi\)
0.825157 + 0.564904i \(0.191087\pi\)
\(762\) −3.96860 −0.143767
\(763\) 9.71245 0.351614
\(764\) 21.0062 0.759976
\(765\) −10.0137 −0.362045
\(766\) −1.97543 −0.0713752
\(767\) 7.72440 0.278912
\(768\) −1.00000 −0.0360844
\(769\) −6.84969 −0.247006 −0.123503 0.992344i \(-0.539413\pi\)
−0.123503 + 0.992344i \(0.539413\pi\)
\(770\) 12.4299 0.447941
\(771\) 2.46728 0.0888570
\(772\) 3.91167 0.140784
\(773\) 23.3368 0.839367 0.419684 0.907671i \(-0.362141\pi\)
0.419684 + 0.907671i \(0.362141\pi\)
\(774\) 7.01489 0.252145
\(775\) 32.6591 1.17315
\(776\) −10.5202 −0.377654
\(777\) 1.45436 0.0521748
\(778\) −14.2832 −0.512078
\(779\) −15.5427 −0.556877
\(780\) −3.43788 −0.123096
\(781\) 3.05687 0.109383
\(782\) 6.48874 0.232037
\(783\) 3.54973 0.126857
\(784\) −6.23321 −0.222615
\(785\) 30.2606 1.08005
\(786\) 9.42019 0.336007
\(787\) −33.8268 −1.20579 −0.602897 0.797819i \(-0.705987\pi\)
−0.602897 + 0.797819i \(0.705987\pi\)
\(788\) 22.7100 0.809012
\(789\) 11.8755 0.422779
\(790\) −54.5399 −1.94044
\(791\) 13.7777 0.489877
\(792\) 4.12894 0.146716
\(793\) 2.25787 0.0801794
\(794\) −28.1499 −0.999001
\(795\) −34.3845 −1.21949
\(796\) −5.93474 −0.210351
\(797\) 0.440113 0.0155896 0.00779480 0.999970i \(-0.497519\pi\)
0.00779480 + 0.999970i \(0.497519\pi\)
\(798\) −5.49088 −0.194375
\(799\) 20.8165 0.736435
\(800\) −6.81900 −0.241088
\(801\) 3.84817 0.135968
\(802\) 33.3899 1.17904
\(803\) −24.1245 −0.851335
\(804\) 5.69363 0.200799
\(805\) −6.70632 −0.236367
\(806\) 4.78943 0.168700
\(807\) −4.75452 −0.167367
\(808\) 2.19696 0.0772889
\(809\) −8.41166 −0.295738 −0.147869 0.989007i \(-0.547241\pi\)
−0.147869 + 0.989007i \(0.547241\pi\)
\(810\) 3.43788 0.120795
\(811\) −4.01487 −0.140981 −0.0704906 0.997512i \(-0.522456\pi\)
−0.0704906 + 0.997512i \(0.522456\pi\)
\(812\) 3.10837 0.109082
\(813\) 1.68670 0.0591552
\(814\) 6.85762 0.240360
\(815\) 19.8756 0.696213
\(816\) −2.91275 −0.101967
\(817\) −43.9872 −1.53892
\(818\) 14.4536 0.505358
\(819\) 0.875663 0.0305981
\(820\) 8.52144 0.297582
\(821\) −17.0208 −0.594029 −0.297015 0.954873i \(-0.595991\pi\)
−0.297015 + 0.954873i \(0.595991\pi\)
\(822\) −11.3092 −0.394453
\(823\) −3.76801 −0.131345 −0.0656723 0.997841i \(-0.520919\pi\)
−0.0656723 + 0.997841i \(0.520919\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 28.1553 0.980240
\(826\) −6.76397 −0.235349
\(827\) 27.3459 0.950909 0.475455 0.879740i \(-0.342284\pi\)
0.475455 + 0.879740i \(0.342284\pi\)
\(828\) −2.22770 −0.0774180
\(829\) −26.5720 −0.922884 −0.461442 0.887170i \(-0.652668\pi\)
−0.461442 + 0.887170i \(0.652668\pi\)
\(830\) −54.1982 −1.88125
\(831\) 17.6902 0.613666
\(832\) −1.00000 −0.0346688
\(833\) −18.1558 −0.629061
\(834\) 2.37115 0.0821062
\(835\) −40.4270 −1.39903
\(836\) −25.8907 −0.895450
\(837\) −4.78943 −0.165547
\(838\) 20.8682 0.720879
\(839\) 8.05851 0.278211 0.139105 0.990278i \(-0.455577\pi\)
0.139105 + 0.990278i \(0.455577\pi\)
\(840\) 3.01042 0.103869
\(841\) −16.3994 −0.565497
\(842\) 4.12536 0.142169
\(843\) 31.3662 1.08031
\(844\) 3.86201 0.132936
\(845\) −3.43788 −0.118267
\(846\) −7.14668 −0.245708
\(847\) −5.29617 −0.181978
\(848\) −10.0017 −0.343458
\(849\) −6.42432 −0.220482
\(850\) −19.8620 −0.681262
\(851\) −3.69991 −0.126831
\(852\) 0.740352 0.0253640
\(853\) −15.1334 −0.518158 −0.259079 0.965856i \(-0.583419\pi\)
−0.259079 + 0.965856i \(0.583419\pi\)
\(854\) −1.97713 −0.0676562
\(855\) −21.5574 −0.737246
\(856\) 5.52530 0.188851
\(857\) 12.5579 0.428971 0.214485 0.976727i \(-0.431193\pi\)
0.214485 + 0.976727i \(0.431193\pi\)
\(858\) 4.12894 0.140960
\(859\) −11.0310 −0.376374 −0.188187 0.982133i \(-0.560261\pi\)
−0.188187 + 0.982133i \(0.560261\pi\)
\(860\) 24.1163 0.822360
\(861\) −2.17050 −0.0739704
\(862\) −29.8416 −1.01641
\(863\) −16.8235 −0.572680 −0.286340 0.958128i \(-0.592439\pi\)
−0.286340 + 0.958128i \(0.592439\pi\)
\(864\) 1.00000 0.0340207
\(865\) −16.7667 −0.570085
\(866\) 33.2387 1.12950
\(867\) 8.51589 0.289215
\(868\) −4.19392 −0.142351
\(869\) 65.5032 2.22204
\(870\) 12.2035 0.413739
\(871\) 5.69363 0.192921
\(872\) 11.0915 0.375607
\(873\) 10.5202 0.356056
\(874\) 13.9689 0.472505
\(875\) 5.47595 0.185121
\(876\) −5.84278 −0.197409
\(877\) 57.9498 1.95683 0.978413 0.206659i \(-0.0662591\pi\)
0.978413 + 0.206659i \(0.0662591\pi\)
\(878\) −21.2514 −0.717200
\(879\) −13.7022 −0.462164
\(880\) 14.1948 0.478507
\(881\) 21.0322 0.708591 0.354296 0.935133i \(-0.384721\pi\)
0.354296 + 0.935133i \(0.384721\pi\)
\(882\) 6.23321 0.209883
\(883\) 48.3885 1.62840 0.814201 0.580584i \(-0.197176\pi\)
0.814201 + 0.580584i \(0.197176\pi\)
\(884\) −2.91275 −0.0979664
\(885\) −26.5555 −0.892655
\(886\) 5.44412 0.182899
\(887\) −3.38532 −0.113668 −0.0568339 0.998384i \(-0.518101\pi\)
−0.0568339 + 0.998384i \(0.518101\pi\)
\(888\) 1.66087 0.0557350
\(889\) 3.47515 0.116553
\(890\) 13.2295 0.443455
\(891\) −4.12894 −0.138325
\(892\) 2.77489 0.0929102
\(893\) 44.8136 1.49963
\(894\) 4.79584 0.160397
\(895\) 6.86613 0.229509
\(896\) 0.875663 0.0292538
\(897\) −2.22770 −0.0743808
\(898\) −37.7400 −1.25940
\(899\) −17.0012 −0.567021
\(900\) 6.81900 0.227300
\(901\) −29.1323 −0.970539
\(902\) −10.2344 −0.340768
\(903\) −6.14268 −0.204416
\(904\) 15.7340 0.523304
\(905\) −35.3432 −1.17485
\(906\) 7.33233 0.243600
\(907\) −14.3909 −0.477841 −0.238921 0.971039i \(-0.576794\pi\)
−0.238921 + 0.971039i \(0.576794\pi\)
\(908\) −26.0660 −0.865031
\(909\) −2.19696 −0.0728687
\(910\) 3.01042 0.0997945
\(911\) −3.49541 −0.115808 −0.0579040 0.998322i \(-0.518442\pi\)
−0.0579040 + 0.998322i \(0.518442\pi\)
\(912\) −6.27054 −0.207638
\(913\) 65.0929 2.15426
\(914\) −11.0662 −0.366039
\(915\) −7.76229 −0.256613
\(916\) −18.3397 −0.605960
\(917\) −8.24891 −0.272403
\(918\) 2.91275 0.0961351
\(919\) 14.9357 0.492684 0.246342 0.969183i \(-0.420771\pi\)
0.246342 + 0.969183i \(0.420771\pi\)
\(920\) −7.65856 −0.252495
\(921\) −10.4205 −0.343367
\(922\) −39.3826 −1.29700
\(923\) 0.740352 0.0243690
\(924\) −3.61556 −0.118943
\(925\) 11.3254 0.372378
\(926\) 24.5365 0.806320
\(927\) 1.00000 0.0328443
\(928\) 3.54973 0.116526
\(929\) 32.3954 1.06286 0.531430 0.847103i \(-0.321655\pi\)
0.531430 + 0.847103i \(0.321655\pi\)
\(930\) −16.4655 −0.539924
\(931\) −39.0856 −1.28098
\(932\) −8.09808 −0.265261
\(933\) 2.65273 0.0868465
\(934\) 37.4980 1.22697
\(935\) 41.3459 1.35216
\(936\) 1.00000 0.0326860
\(937\) −20.9785 −0.685336 −0.342668 0.939456i \(-0.611331\pi\)
−0.342668 + 0.939456i \(0.611331\pi\)
\(938\) −4.98570 −0.162789
\(939\) −0.504119 −0.0164513
\(940\) −24.5694 −0.801366
\(941\) −36.2449 −1.18155 −0.590775 0.806837i \(-0.701178\pi\)
−0.590775 + 0.806837i \(0.701178\pi\)
\(942\) −8.80210 −0.286788
\(943\) 5.52179 0.179814
\(944\) −7.72440 −0.251408
\(945\) −3.01042 −0.0979290
\(946\) −28.9641 −0.941704
\(947\) 40.0744 1.30224 0.651122 0.758973i \(-0.274299\pi\)
0.651122 + 0.758973i \(0.274299\pi\)
\(948\) 15.8644 0.515252
\(949\) −5.84278 −0.189665
\(950\) −42.7588 −1.38728
\(951\) 31.0342 1.00635
\(952\) 2.55059 0.0826650
\(953\) −28.7014 −0.929730 −0.464865 0.885381i \(-0.653897\pi\)
−0.464865 + 0.885381i \(0.653897\pi\)
\(954\) 10.0017 0.323816
\(955\) −72.2166 −2.33687
\(956\) 21.0662 0.681330
\(957\) −14.6566 −0.473782
\(958\) −15.1467 −0.489369
\(959\) 9.90303 0.319786
\(960\) 3.43788 0.110957
\(961\) −8.06140 −0.260045
\(962\) 1.66087 0.0535485
\(963\) −5.52530 −0.178050
\(964\) 8.02129 0.258348
\(965\) −13.4478 −0.432902
\(966\) 1.95072 0.0627632
\(967\) −13.8179 −0.444353 −0.222177 0.975006i \(-0.571316\pi\)
−0.222177 + 0.975006i \(0.571316\pi\)
\(968\) −6.04818 −0.194396
\(969\) −18.2645 −0.586741
\(970\) 36.1672 1.16126
\(971\) −30.1735 −0.968314 −0.484157 0.874981i \(-0.660874\pi\)
−0.484157 + 0.874981i \(0.660874\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −2.07633 −0.0665640
\(974\) −11.9348 −0.382417
\(975\) 6.81900 0.218383
\(976\) −2.25787 −0.0722727
\(977\) −6.27177 −0.200652 −0.100326 0.994955i \(-0.531989\pi\)
−0.100326 + 0.994955i \(0.531989\pi\)
\(978\) −5.78137 −0.184868
\(979\) −15.8889 −0.507810
\(980\) 21.4290 0.684525
\(981\) −11.0915 −0.354126
\(982\) 20.5208 0.654845
\(983\) 26.0909 0.832171 0.416086 0.909325i \(-0.363402\pi\)
0.416086 + 0.909325i \(0.363402\pi\)
\(984\) −2.47869 −0.0790178
\(985\) −78.0743 −2.48765
\(986\) 10.3395 0.329276
\(987\) 6.25808 0.199197
\(988\) −6.27054 −0.199493
\(989\) 15.6271 0.496912
\(990\) −14.1948 −0.451140
\(991\) 17.8943 0.568431 0.284216 0.958760i \(-0.408267\pi\)
0.284216 + 0.958760i \(0.408267\pi\)
\(992\) −4.78943 −0.152064
\(993\) −24.6859 −0.783384
\(994\) −0.648299 −0.0205628
\(995\) 20.4029 0.646815
\(996\) 15.7650 0.499534
\(997\) 44.3111 1.40335 0.701673 0.712499i \(-0.252437\pi\)
0.701673 + 0.712499i \(0.252437\pi\)
\(998\) 27.9203 0.883802
\(999\) −1.66087 −0.0525475
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.ba.1.2 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.ba.1.2 14 1.1 even 1 trivial