Properties

Label 8034.2.a.bb.1.3
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
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: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.75223\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.75223 q^{5} -1.00000 q^{6} +1.91373 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.75223 q^{5} -1.00000 q^{6} +1.91373 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.75223 q^{10} -4.14756 q^{11} +1.00000 q^{12} -1.00000 q^{13} -1.91373 q^{14} -3.75223 q^{15} +1.00000 q^{16} -4.83878 q^{17} -1.00000 q^{18} +2.13299 q^{19} -3.75223 q^{20} +1.91373 q^{21} +4.14756 q^{22} +2.08206 q^{23} -1.00000 q^{24} +9.07922 q^{25} +1.00000 q^{26} +1.00000 q^{27} +1.91373 q^{28} -2.19238 q^{29} +3.75223 q^{30} +5.15289 q^{31} -1.00000 q^{32} -4.14756 q^{33} +4.83878 q^{34} -7.18074 q^{35} +1.00000 q^{36} +9.63112 q^{37} -2.13299 q^{38} -1.00000 q^{39} +3.75223 q^{40} +7.88650 q^{41} -1.91373 q^{42} +1.60304 q^{43} -4.14756 q^{44} -3.75223 q^{45} -2.08206 q^{46} -9.35222 q^{47} +1.00000 q^{48} -3.33765 q^{49} -9.07922 q^{50} -4.83878 q^{51} -1.00000 q^{52} +12.8161 q^{53} -1.00000 q^{54} +15.5626 q^{55} -1.91373 q^{56} +2.13299 q^{57} +2.19238 q^{58} -7.21379 q^{59} -3.75223 q^{60} +10.2293 q^{61} -5.15289 q^{62} +1.91373 q^{63} +1.00000 q^{64} +3.75223 q^{65} +4.14756 q^{66} -8.07141 q^{67} -4.83878 q^{68} +2.08206 q^{69} +7.18074 q^{70} +9.46244 q^{71} -1.00000 q^{72} +4.11647 q^{73} -9.63112 q^{74} +9.07922 q^{75} +2.13299 q^{76} -7.93729 q^{77} +1.00000 q^{78} -8.08441 q^{79} -3.75223 q^{80} +1.00000 q^{81} -7.88650 q^{82} +0.397774 q^{83} +1.91373 q^{84} +18.1562 q^{85} -1.60304 q^{86} -2.19238 q^{87} +4.14756 q^{88} -12.6791 q^{89} +3.75223 q^{90} -1.91373 q^{91} +2.08206 q^{92} +5.15289 q^{93} +9.35222 q^{94} -8.00347 q^{95} -1.00000 q^{96} -14.9290 q^{97} +3.33765 q^{98} -4.14756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) \)

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
\(5\) −3.75223 −1.67805 −0.839024 0.544095i \(-0.816873\pi\)
−0.839024 + 0.544095i \(0.816873\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.91373 0.723321 0.361660 0.932310i \(-0.382210\pi\)
0.361660 + 0.932310i \(0.382210\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.75223 1.18656
\(11\) −4.14756 −1.25054 −0.625268 0.780410i \(-0.715010\pi\)
−0.625268 + 0.780410i \(0.715010\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) −1.91373 −0.511465
\(15\) −3.75223 −0.968821
\(16\) 1.00000 0.250000
\(17\) −4.83878 −1.17358 −0.586789 0.809740i \(-0.699608\pi\)
−0.586789 + 0.809740i \(0.699608\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.13299 0.489342 0.244671 0.969606i \(-0.421320\pi\)
0.244671 + 0.969606i \(0.421320\pi\)
\(20\) −3.75223 −0.839024
\(21\) 1.91373 0.417610
\(22\) 4.14756 0.884262
\(23\) 2.08206 0.434140 0.217070 0.976156i \(-0.430350\pi\)
0.217070 + 0.976156i \(0.430350\pi\)
\(24\) −1.00000 −0.204124
\(25\) 9.07922 1.81584
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 1.91373 0.361660
\(29\) −2.19238 −0.407114 −0.203557 0.979063i \(-0.565250\pi\)
−0.203557 + 0.979063i \(0.565250\pi\)
\(30\) 3.75223 0.685060
\(31\) 5.15289 0.925487 0.462743 0.886492i \(-0.346865\pi\)
0.462743 + 0.886492i \(0.346865\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.14756 −0.721997
\(34\) 4.83878 0.829845
\(35\) −7.18074 −1.21377
\(36\) 1.00000 0.166667
\(37\) 9.63112 1.58335 0.791674 0.610944i \(-0.209210\pi\)
0.791674 + 0.610944i \(0.209210\pi\)
\(38\) −2.13299 −0.346017
\(39\) −1.00000 −0.160128
\(40\) 3.75223 0.593279
\(41\) 7.88650 1.23166 0.615832 0.787878i \(-0.288820\pi\)
0.615832 + 0.787878i \(0.288820\pi\)
\(42\) −1.91373 −0.295295
\(43\) 1.60304 0.244461 0.122230 0.992502i \(-0.460995\pi\)
0.122230 + 0.992502i \(0.460995\pi\)
\(44\) −4.14756 −0.625268
\(45\) −3.75223 −0.559349
\(46\) −2.08206 −0.306983
\(47\) −9.35222 −1.36416 −0.682081 0.731277i \(-0.738925\pi\)
−0.682081 + 0.731277i \(0.738925\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.33765 −0.476807
\(50\) −9.07922 −1.28400
\(51\) −4.83878 −0.677565
\(52\) −1.00000 −0.138675
\(53\) 12.8161 1.76042 0.880212 0.474580i \(-0.157400\pi\)
0.880212 + 0.474580i \(0.157400\pi\)
\(54\) −1.00000 −0.136083
\(55\) 15.5626 2.09846
\(56\) −1.91373 −0.255733
\(57\) 2.13299 0.282522
\(58\) 2.19238 0.287873
\(59\) −7.21379 −0.939156 −0.469578 0.882891i \(-0.655594\pi\)
−0.469578 + 0.882891i \(0.655594\pi\)
\(60\) −3.75223 −0.484411
\(61\) 10.2293 1.30973 0.654864 0.755747i \(-0.272726\pi\)
0.654864 + 0.755747i \(0.272726\pi\)
\(62\) −5.15289 −0.654418
\(63\) 1.91373 0.241107
\(64\) 1.00000 0.125000
\(65\) 3.75223 0.465407
\(66\) 4.14756 0.510529
\(67\) −8.07141 −0.986079 −0.493040 0.870007i \(-0.664114\pi\)
−0.493040 + 0.870007i \(0.664114\pi\)
\(68\) −4.83878 −0.586789
\(69\) 2.08206 0.250651
\(70\) 7.18074 0.858263
\(71\) 9.46244 1.12298 0.561492 0.827482i \(-0.310227\pi\)
0.561492 + 0.827482i \(0.310227\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.11647 0.481797 0.240898 0.970550i \(-0.422558\pi\)
0.240898 + 0.970550i \(0.422558\pi\)
\(74\) −9.63112 −1.11960
\(75\) 9.07922 1.04838
\(76\) 2.13299 0.244671
\(77\) −7.93729 −0.904538
\(78\) 1.00000 0.113228
\(79\) −8.08441 −0.909567 −0.454783 0.890602i \(-0.650283\pi\)
−0.454783 + 0.890602i \(0.650283\pi\)
\(80\) −3.75223 −0.419512
\(81\) 1.00000 0.111111
\(82\) −7.88650 −0.870918
\(83\) 0.397774 0.0436614 0.0218307 0.999762i \(-0.493051\pi\)
0.0218307 + 0.999762i \(0.493051\pi\)
\(84\) 1.91373 0.208805
\(85\) 18.1562 1.96932
\(86\) −1.60304 −0.172860
\(87\) −2.19238 −0.235048
\(88\) 4.14756 0.442131
\(89\) −12.6791 −1.34399 −0.671993 0.740557i \(-0.734562\pi\)
−0.671993 + 0.740557i \(0.734562\pi\)
\(90\) 3.75223 0.395520
\(91\) −1.91373 −0.200613
\(92\) 2.08206 0.217070
\(93\) 5.15289 0.534330
\(94\) 9.35222 0.964608
\(95\) −8.00347 −0.821139
\(96\) −1.00000 −0.102062
\(97\) −14.9290 −1.51581 −0.757905 0.652365i \(-0.773777\pi\)
−0.757905 + 0.652365i \(0.773777\pi\)
\(98\) 3.33765 0.337153
\(99\) −4.14756 −0.416845
\(100\) 9.07922 0.907922
\(101\) 16.7182 1.66353 0.831763 0.555130i \(-0.187332\pi\)
0.831763 + 0.555130i \(0.187332\pi\)
\(102\) 4.83878 0.479111
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) −7.18074 −0.700769
\(106\) −12.8161 −1.24481
\(107\) 15.6347 1.51146 0.755729 0.654884i \(-0.227283\pi\)
0.755729 + 0.654884i \(0.227283\pi\)
\(108\) 1.00000 0.0962250
\(109\) −20.5955 −1.97269 −0.986346 0.164686i \(-0.947339\pi\)
−0.986346 + 0.164686i \(0.947339\pi\)
\(110\) −15.5626 −1.48383
\(111\) 9.63112 0.914146
\(112\) 1.91373 0.180830
\(113\) −0.0306698 −0.00288518 −0.00144259 0.999999i \(-0.500459\pi\)
−0.00144259 + 0.999999i \(0.500459\pi\)
\(114\) −2.13299 −0.199773
\(115\) −7.81237 −0.728507
\(116\) −2.19238 −0.203557
\(117\) −1.00000 −0.0924500
\(118\) 7.21379 0.664083
\(119\) −9.26011 −0.848873
\(120\) 3.75223 0.342530
\(121\) 6.20223 0.563839
\(122\) −10.2293 −0.926117
\(123\) 7.88650 0.711101
\(124\) 5.15289 0.462743
\(125\) −15.3062 −1.36903
\(126\) −1.91373 −0.170488
\(127\) −4.17744 −0.370688 −0.185344 0.982674i \(-0.559340\pi\)
−0.185344 + 0.982674i \(0.559340\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.60304 0.141139
\(130\) −3.75223 −0.329092
\(131\) 0.693576 0.0605981 0.0302990 0.999541i \(-0.490354\pi\)
0.0302990 + 0.999541i \(0.490354\pi\)
\(132\) −4.14756 −0.360998
\(133\) 4.08196 0.353951
\(134\) 8.07141 0.697263
\(135\) −3.75223 −0.322940
\(136\) 4.83878 0.414922
\(137\) −15.5620 −1.32955 −0.664775 0.747044i \(-0.731473\pi\)
−0.664775 + 0.747044i \(0.731473\pi\)
\(138\) −2.08206 −0.177237
\(139\) −14.8728 −1.26149 −0.630747 0.775988i \(-0.717252\pi\)
−0.630747 + 0.775988i \(0.717252\pi\)
\(140\) −7.18074 −0.606884
\(141\) −9.35222 −0.787599
\(142\) −9.46244 −0.794070
\(143\) 4.14756 0.346836
\(144\) 1.00000 0.0833333
\(145\) 8.22631 0.683158
\(146\) −4.11647 −0.340682
\(147\) −3.33765 −0.275285
\(148\) 9.63112 0.791674
\(149\) 6.78344 0.555721 0.277861 0.960621i \(-0.410375\pi\)
0.277861 + 0.960621i \(0.410375\pi\)
\(150\) −9.07922 −0.741315
\(151\) −19.0254 −1.54826 −0.774132 0.633024i \(-0.781813\pi\)
−0.774132 + 0.633024i \(0.781813\pi\)
\(152\) −2.13299 −0.173008
\(153\) −4.83878 −0.391193
\(154\) 7.93729 0.639605
\(155\) −19.3348 −1.55301
\(156\) −1.00000 −0.0800641
\(157\) −24.8181 −1.98070 −0.990352 0.138577i \(-0.955747\pi\)
−0.990352 + 0.138577i \(0.955747\pi\)
\(158\) 8.08441 0.643161
\(159\) 12.8161 1.01638
\(160\) 3.75223 0.296640
\(161\) 3.98450 0.314022
\(162\) −1.00000 −0.0785674
\(163\) −11.4078 −0.893525 −0.446762 0.894653i \(-0.647423\pi\)
−0.446762 + 0.894653i \(0.647423\pi\)
\(164\) 7.88650 0.615832
\(165\) 15.5626 1.21155
\(166\) −0.397774 −0.0308733
\(167\) −8.21066 −0.635360 −0.317680 0.948198i \(-0.602904\pi\)
−0.317680 + 0.948198i \(0.602904\pi\)
\(168\) −1.91373 −0.147647
\(169\) 1.00000 0.0769231
\(170\) −18.1562 −1.39252
\(171\) 2.13299 0.163114
\(172\) 1.60304 0.122230
\(173\) 9.58909 0.729045 0.364523 0.931195i \(-0.381232\pi\)
0.364523 + 0.931195i \(0.381232\pi\)
\(174\) 2.19238 0.166204
\(175\) 17.3752 1.31344
\(176\) −4.14756 −0.312634
\(177\) −7.21379 −0.542222
\(178\) 12.6791 0.950342
\(179\) 8.14337 0.608664 0.304332 0.952566i \(-0.401567\pi\)
0.304332 + 0.952566i \(0.401567\pi\)
\(180\) −3.75223 −0.279675
\(181\) 1.68358 0.125139 0.0625697 0.998041i \(-0.480070\pi\)
0.0625697 + 0.998041i \(0.480070\pi\)
\(182\) 1.91373 0.141855
\(183\) 10.2293 0.756171
\(184\) −2.08206 −0.153492
\(185\) −36.1382 −2.65693
\(186\) −5.15289 −0.377828
\(187\) 20.0691 1.46760
\(188\) −9.35222 −0.682081
\(189\) 1.91373 0.139203
\(190\) 8.00347 0.580633
\(191\) −21.2826 −1.53995 −0.769976 0.638072i \(-0.779732\pi\)
−0.769976 + 0.638072i \(0.779732\pi\)
\(192\) 1.00000 0.0721688
\(193\) 19.4142 1.39747 0.698733 0.715382i \(-0.253747\pi\)
0.698733 + 0.715382i \(0.253747\pi\)
\(194\) 14.9290 1.07184
\(195\) 3.75223 0.268703
\(196\) −3.33765 −0.238403
\(197\) −11.6487 −0.829933 −0.414966 0.909837i \(-0.636207\pi\)
−0.414966 + 0.909837i \(0.636207\pi\)
\(198\) 4.14756 0.294754
\(199\) 15.3303 1.08674 0.543370 0.839494i \(-0.317148\pi\)
0.543370 + 0.839494i \(0.317148\pi\)
\(200\) −9.07922 −0.641998
\(201\) −8.07141 −0.569313
\(202\) −16.7182 −1.17629
\(203\) −4.19561 −0.294474
\(204\) −4.83878 −0.338783
\(205\) −29.5919 −2.06679
\(206\) −1.00000 −0.0696733
\(207\) 2.08206 0.144713
\(208\) −1.00000 −0.0693375
\(209\) −8.84670 −0.611939
\(210\) 7.18074 0.495518
\(211\) −11.7754 −0.810654 −0.405327 0.914172i \(-0.632842\pi\)
−0.405327 + 0.914172i \(0.632842\pi\)
\(212\) 12.8161 0.880212
\(213\) 9.46244 0.648356
\(214\) −15.6347 −1.06876
\(215\) −6.01496 −0.410217
\(216\) −1.00000 −0.0680414
\(217\) 9.86123 0.669424
\(218\) 20.5955 1.39490
\(219\) 4.11647 0.278165
\(220\) 15.5626 1.04923
\(221\) 4.83878 0.325492
\(222\) −9.63112 −0.646399
\(223\) −20.7006 −1.38621 −0.693107 0.720835i \(-0.743759\pi\)
−0.693107 + 0.720835i \(0.743759\pi\)
\(224\) −1.91373 −0.127866
\(225\) 9.07922 0.605282
\(226\) 0.0306698 0.00204013
\(227\) 16.2577 1.07906 0.539530 0.841967i \(-0.318602\pi\)
0.539530 + 0.841967i \(0.318602\pi\)
\(228\) 2.13299 0.141261
\(229\) −13.6150 −0.899703 −0.449851 0.893103i \(-0.648523\pi\)
−0.449851 + 0.893103i \(0.648523\pi\)
\(230\) 7.81237 0.515133
\(231\) −7.93729 −0.522236
\(232\) 2.19238 0.143937
\(233\) −19.0339 −1.24695 −0.623477 0.781842i \(-0.714280\pi\)
−0.623477 + 0.781842i \(0.714280\pi\)
\(234\) 1.00000 0.0653720
\(235\) 35.0917 2.28913
\(236\) −7.21379 −0.469578
\(237\) −8.08441 −0.525139
\(238\) 9.26011 0.600244
\(239\) 7.92027 0.512320 0.256160 0.966634i \(-0.417543\pi\)
0.256160 + 0.966634i \(0.417543\pi\)
\(240\) −3.75223 −0.242205
\(241\) 17.3089 1.11496 0.557482 0.830189i \(-0.311767\pi\)
0.557482 + 0.830189i \(0.311767\pi\)
\(242\) −6.20223 −0.398694
\(243\) 1.00000 0.0641500
\(244\) 10.2293 0.654864
\(245\) 12.5236 0.800105
\(246\) −7.88650 −0.502825
\(247\) −2.13299 −0.135719
\(248\) −5.15289 −0.327209
\(249\) 0.397774 0.0252079
\(250\) 15.3062 0.968048
\(251\) 6.93300 0.437607 0.218804 0.975769i \(-0.429785\pi\)
0.218804 + 0.975769i \(0.429785\pi\)
\(252\) 1.91373 0.120553
\(253\) −8.63547 −0.542907
\(254\) 4.17744 0.262116
\(255\) 18.1562 1.13699
\(256\) 1.00000 0.0625000
\(257\) 18.6390 1.16267 0.581335 0.813664i \(-0.302531\pi\)
0.581335 + 0.813664i \(0.302531\pi\)
\(258\) −1.60304 −0.0998006
\(259\) 18.4313 1.14527
\(260\) 3.75223 0.232703
\(261\) −2.19238 −0.135705
\(262\) −0.693576 −0.0428493
\(263\) −19.2932 −1.18967 −0.594835 0.803847i \(-0.702783\pi\)
−0.594835 + 0.803847i \(0.702783\pi\)
\(264\) 4.14756 0.255264
\(265\) −48.0889 −2.95408
\(266\) −4.08196 −0.250281
\(267\) −12.6791 −0.775951
\(268\) −8.07141 −0.493040
\(269\) −25.0184 −1.52540 −0.762698 0.646755i \(-0.776126\pi\)
−0.762698 + 0.646755i \(0.776126\pi\)
\(270\) 3.75223 0.228353
\(271\) 8.50393 0.516577 0.258289 0.966068i \(-0.416841\pi\)
0.258289 + 0.966068i \(0.416841\pi\)
\(272\) −4.83878 −0.293394
\(273\) −1.91373 −0.115824
\(274\) 15.5620 0.940134
\(275\) −37.6566 −2.27078
\(276\) 2.08206 0.125325
\(277\) 16.2159 0.974321 0.487160 0.873313i \(-0.338033\pi\)
0.487160 + 0.873313i \(0.338033\pi\)
\(278\) 14.8728 0.892011
\(279\) 5.15289 0.308496
\(280\) 7.18074 0.429131
\(281\) 18.6296 1.11135 0.555673 0.831401i \(-0.312461\pi\)
0.555673 + 0.831401i \(0.312461\pi\)
\(282\) 9.35222 0.556917
\(283\) 2.72112 0.161754 0.0808769 0.996724i \(-0.474228\pi\)
0.0808769 + 0.996724i \(0.474228\pi\)
\(284\) 9.46244 0.561492
\(285\) −8.00347 −0.474085
\(286\) −4.14756 −0.245250
\(287\) 15.0926 0.890888
\(288\) −1.00000 −0.0589256
\(289\) 6.41383 0.377284
\(290\) −8.22631 −0.483065
\(291\) −14.9290 −0.875153
\(292\) 4.11647 0.240898
\(293\) 29.3839 1.71662 0.858312 0.513128i \(-0.171513\pi\)
0.858312 + 0.513128i \(0.171513\pi\)
\(294\) 3.33765 0.194656
\(295\) 27.0678 1.57595
\(296\) −9.63112 −0.559798
\(297\) −4.14756 −0.240666
\(298\) −6.78344 −0.392954
\(299\) −2.08206 −0.120409
\(300\) 9.07922 0.524189
\(301\) 3.06777 0.176823
\(302\) 19.0254 1.09479
\(303\) 16.7182 0.960438
\(304\) 2.13299 0.122335
\(305\) −38.3827 −2.19778
\(306\) 4.83878 0.276615
\(307\) −7.72568 −0.440928 −0.220464 0.975395i \(-0.570757\pi\)
−0.220464 + 0.975395i \(0.570757\pi\)
\(308\) −7.93729 −0.452269
\(309\) 1.00000 0.0568880
\(310\) 19.3348 1.09814
\(311\) 23.8089 1.35008 0.675040 0.737781i \(-0.264126\pi\)
0.675040 + 0.737781i \(0.264126\pi\)
\(312\) 1.00000 0.0566139
\(313\) −7.23019 −0.408675 −0.204337 0.978901i \(-0.565504\pi\)
−0.204337 + 0.978901i \(0.565504\pi\)
\(314\) 24.8181 1.40057
\(315\) −7.18074 −0.404589
\(316\) −8.08441 −0.454783
\(317\) 15.4352 0.866929 0.433464 0.901171i \(-0.357291\pi\)
0.433464 + 0.901171i \(0.357291\pi\)
\(318\) −12.8161 −0.718690
\(319\) 9.09301 0.509111
\(320\) −3.75223 −0.209756
\(321\) 15.6347 0.872641
\(322\) −3.98450 −0.222047
\(323\) −10.3211 −0.574281
\(324\) 1.00000 0.0555556
\(325\) −9.07922 −0.503625
\(326\) 11.4078 0.631818
\(327\) −20.5955 −1.13893
\(328\) −7.88650 −0.435459
\(329\) −17.8976 −0.986727
\(330\) −15.5626 −0.856692
\(331\) −31.3126 −1.72110 −0.860548 0.509370i \(-0.829879\pi\)
−0.860548 + 0.509370i \(0.829879\pi\)
\(332\) 0.397774 0.0218307
\(333\) 9.63112 0.527782
\(334\) 8.21066 0.449267
\(335\) 30.2858 1.65469
\(336\) 1.91373 0.104402
\(337\) 21.7554 1.18509 0.592547 0.805536i \(-0.298122\pi\)
0.592547 + 0.805536i \(0.298122\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −0.0306698 −0.00166576
\(340\) 18.1562 0.984660
\(341\) −21.3719 −1.15735
\(342\) −2.13299 −0.115339
\(343\) −19.7834 −1.06821
\(344\) −1.60304 −0.0864299
\(345\) −7.81237 −0.420604
\(346\) −9.58909 −0.515513
\(347\) −5.69161 −0.305542 −0.152771 0.988262i \(-0.548820\pi\)
−0.152771 + 0.988262i \(0.548820\pi\)
\(348\) −2.19238 −0.117524
\(349\) −9.10237 −0.487238 −0.243619 0.969871i \(-0.578335\pi\)
−0.243619 + 0.969871i \(0.578335\pi\)
\(350\) −17.3752 −0.928741
\(351\) −1.00000 −0.0533761
\(352\) 4.14756 0.221066
\(353\) −20.7573 −1.10480 −0.552400 0.833579i \(-0.686288\pi\)
−0.552400 + 0.833579i \(0.686288\pi\)
\(354\) 7.21379 0.383409
\(355\) −35.5052 −1.88442
\(356\) −12.6791 −0.671993
\(357\) −9.26011 −0.490097
\(358\) −8.14337 −0.430390
\(359\) −33.7756 −1.78261 −0.891304 0.453406i \(-0.850209\pi\)
−0.891304 + 0.453406i \(0.850209\pi\)
\(360\) 3.75223 0.197760
\(361\) −14.4503 −0.760545
\(362\) −1.68358 −0.0884869
\(363\) 6.20223 0.325533
\(364\) −1.91373 −0.100307
\(365\) −15.4459 −0.808478
\(366\) −10.2293 −0.534694
\(367\) −7.16383 −0.373949 −0.186975 0.982365i \(-0.559868\pi\)
−0.186975 + 0.982365i \(0.559868\pi\)
\(368\) 2.08206 0.108535
\(369\) 7.88650 0.410555
\(370\) 36.1382 1.87873
\(371\) 24.5265 1.27335
\(372\) 5.15289 0.267165
\(373\) −31.8515 −1.64921 −0.824604 0.565710i \(-0.808602\pi\)
−0.824604 + 0.565710i \(0.808602\pi\)
\(374\) −20.0691 −1.03775
\(375\) −15.3062 −0.790408
\(376\) 9.35222 0.482304
\(377\) 2.19238 0.112913
\(378\) −1.91373 −0.0984315
\(379\) 0.748216 0.0384333 0.0192166 0.999815i \(-0.493883\pi\)
0.0192166 + 0.999815i \(0.493883\pi\)
\(380\) −8.00347 −0.410569
\(381\) −4.17744 −0.214017
\(382\) 21.2826 1.08891
\(383\) 15.9351 0.814247 0.407124 0.913373i \(-0.366532\pi\)
0.407124 + 0.913373i \(0.366532\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 29.7825 1.51786
\(386\) −19.4142 −0.988158
\(387\) 1.60304 0.0814869
\(388\) −14.9290 −0.757905
\(389\) −5.36783 −0.272160 −0.136080 0.990698i \(-0.543450\pi\)
−0.136080 + 0.990698i \(0.543450\pi\)
\(390\) −3.75223 −0.190002
\(391\) −10.0746 −0.509497
\(392\) 3.33765 0.168577
\(393\) 0.693576 0.0349863
\(394\) 11.6487 0.586851
\(395\) 30.3345 1.52630
\(396\) −4.14756 −0.208423
\(397\) −5.30293 −0.266147 −0.133073 0.991106i \(-0.542485\pi\)
−0.133073 + 0.991106i \(0.542485\pi\)
\(398\) −15.3303 −0.768441
\(399\) 4.08196 0.204354
\(400\) 9.07922 0.453961
\(401\) −30.6283 −1.52951 −0.764753 0.644323i \(-0.777139\pi\)
−0.764753 + 0.644323i \(0.777139\pi\)
\(402\) 8.07141 0.402565
\(403\) −5.15289 −0.256684
\(404\) 16.7182 0.831763
\(405\) −3.75223 −0.186450
\(406\) 4.19561 0.208225
\(407\) −39.9456 −1.98003
\(408\) 4.83878 0.239556
\(409\) −25.3342 −1.25270 −0.626348 0.779543i \(-0.715451\pi\)
−0.626348 + 0.779543i \(0.715451\pi\)
\(410\) 29.5919 1.46144
\(411\) −15.5620 −0.767616
\(412\) 1.00000 0.0492665
\(413\) −13.8052 −0.679311
\(414\) −2.08206 −0.102328
\(415\) −1.49254 −0.0732659
\(416\) 1.00000 0.0490290
\(417\) −14.8728 −0.728324
\(418\) 8.84670 0.432706
\(419\) 29.4907 1.44071 0.720356 0.693604i \(-0.243978\pi\)
0.720356 + 0.693604i \(0.243978\pi\)
\(420\) −7.18074 −0.350384
\(421\) 27.7454 1.35223 0.676114 0.736797i \(-0.263663\pi\)
0.676114 + 0.736797i \(0.263663\pi\)
\(422\) 11.7754 0.573219
\(423\) −9.35222 −0.454721
\(424\) −12.8161 −0.622404
\(425\) −43.9324 −2.13103
\(426\) −9.46244 −0.458457
\(427\) 19.5761 0.947353
\(428\) 15.6347 0.755729
\(429\) 4.14756 0.200246
\(430\) 6.01496 0.290067
\(431\) −2.87789 −0.138623 −0.0693115 0.997595i \(-0.522080\pi\)
−0.0693115 + 0.997595i \(0.522080\pi\)
\(432\) 1.00000 0.0481125
\(433\) −16.0609 −0.771838 −0.385919 0.922533i \(-0.626116\pi\)
−0.385919 + 0.922533i \(0.626116\pi\)
\(434\) −9.86123 −0.473354
\(435\) 8.22631 0.394421
\(436\) −20.5955 −0.986346
\(437\) 4.44102 0.212443
\(438\) −4.11647 −0.196693
\(439\) 26.0727 1.24438 0.622190 0.782866i \(-0.286243\pi\)
0.622190 + 0.782866i \(0.286243\pi\)
\(440\) −15.5626 −0.741917
\(441\) −3.33765 −0.158936
\(442\) −4.83878 −0.230157
\(443\) −17.8236 −0.846827 −0.423413 0.905937i \(-0.639168\pi\)
−0.423413 + 0.905937i \(0.639168\pi\)
\(444\) 9.63112 0.457073
\(445\) 47.5751 2.25527
\(446\) 20.7006 0.980202
\(447\) 6.78344 0.320846
\(448\) 1.91373 0.0904151
\(449\) −31.6248 −1.49247 −0.746234 0.665684i \(-0.768140\pi\)
−0.746234 + 0.665684i \(0.768140\pi\)
\(450\) −9.07922 −0.427999
\(451\) −32.7097 −1.54024
\(452\) −0.0306698 −0.00144259
\(453\) −19.0254 −0.893890
\(454\) −16.2577 −0.763010
\(455\) 7.18074 0.336638
\(456\) −2.13299 −0.0998865
\(457\) −30.9954 −1.44990 −0.724951 0.688800i \(-0.758138\pi\)
−0.724951 + 0.688800i \(0.758138\pi\)
\(458\) 13.6150 0.636186
\(459\) −4.83878 −0.225855
\(460\) −7.81237 −0.364254
\(461\) −36.7779 −1.71292 −0.856459 0.516215i \(-0.827341\pi\)
−0.856459 + 0.516215i \(0.827341\pi\)
\(462\) 7.93729 0.369276
\(463\) −10.4965 −0.487813 −0.243907 0.969799i \(-0.578429\pi\)
−0.243907 + 0.969799i \(0.578429\pi\)
\(464\) −2.19238 −0.101779
\(465\) −19.3348 −0.896631
\(466\) 19.0339 0.881729
\(467\) 16.3515 0.756656 0.378328 0.925672i \(-0.376499\pi\)
0.378328 + 0.925672i \(0.376499\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −15.4465 −0.713252
\(470\) −35.0917 −1.61866
\(471\) −24.8181 −1.14356
\(472\) 7.21379 0.332042
\(473\) −6.64868 −0.305707
\(474\) 8.08441 0.371329
\(475\) 19.3659 0.888569
\(476\) −9.26011 −0.424437
\(477\) 12.8161 0.586808
\(478\) −7.92027 −0.362265
\(479\) −35.5629 −1.62491 −0.812456 0.583022i \(-0.801870\pi\)
−0.812456 + 0.583022i \(0.801870\pi\)
\(480\) 3.75223 0.171265
\(481\) −9.63112 −0.439141
\(482\) −17.3089 −0.788399
\(483\) 3.98450 0.181301
\(484\) 6.20223 0.281919
\(485\) 56.0170 2.54360
\(486\) −1.00000 −0.0453609
\(487\) 1.39658 0.0632850 0.0316425 0.999499i \(-0.489926\pi\)
0.0316425 + 0.999499i \(0.489926\pi\)
\(488\) −10.2293 −0.463059
\(489\) −11.4078 −0.515877
\(490\) −12.5236 −0.565759
\(491\) 2.49644 0.112663 0.0563315 0.998412i \(-0.482060\pi\)
0.0563315 + 0.998412i \(0.482060\pi\)
\(492\) 7.88650 0.355551
\(493\) 10.6084 0.477780
\(494\) 2.13299 0.0959678
\(495\) 15.5626 0.699486
\(496\) 5.15289 0.231372
\(497\) 18.1085 0.812279
\(498\) −0.397774 −0.0178247
\(499\) 18.8673 0.844616 0.422308 0.906452i \(-0.361220\pi\)
0.422308 + 0.906452i \(0.361220\pi\)
\(500\) −15.3062 −0.684513
\(501\) −8.21066 −0.366825
\(502\) −6.93300 −0.309435
\(503\) −11.0795 −0.494010 −0.247005 0.969014i \(-0.579446\pi\)
−0.247005 + 0.969014i \(0.579446\pi\)
\(504\) −1.91373 −0.0852442
\(505\) −62.7307 −2.79148
\(506\) 8.63547 0.383893
\(507\) 1.00000 0.0444116
\(508\) −4.17744 −0.185344
\(509\) −0.550164 −0.0243856 −0.0121928 0.999926i \(-0.503881\pi\)
−0.0121928 + 0.999926i \(0.503881\pi\)
\(510\) −18.1562 −0.803971
\(511\) 7.87780 0.348493
\(512\) −1.00000 −0.0441942
\(513\) 2.13299 0.0941739
\(514\) −18.6390 −0.822132
\(515\) −3.75223 −0.165343
\(516\) 1.60304 0.0705697
\(517\) 38.7889 1.70593
\(518\) −18.4313 −0.809827
\(519\) 9.58909 0.420915
\(520\) −3.75223 −0.164546
\(521\) 25.6635 1.12434 0.562169 0.827022i \(-0.309967\pi\)
0.562169 + 0.827022i \(0.309967\pi\)
\(522\) 2.19238 0.0959578
\(523\) 37.4839 1.63906 0.819528 0.573039i \(-0.194236\pi\)
0.819528 + 0.573039i \(0.194236\pi\)
\(524\) 0.693576 0.0302990
\(525\) 17.3752 0.758314
\(526\) 19.2932 0.841224
\(527\) −24.9337 −1.08613
\(528\) −4.14756 −0.180499
\(529\) −18.6650 −0.811523
\(530\) 48.0889 2.08885
\(531\) −7.21379 −0.313052
\(532\) 4.08196 0.176976
\(533\) −7.88650 −0.341602
\(534\) 12.6791 0.548680
\(535\) −58.6648 −2.53630
\(536\) 8.07141 0.348632
\(537\) 8.14337 0.351412
\(538\) 25.0184 1.07862
\(539\) 13.8431 0.596264
\(540\) −3.75223 −0.161470
\(541\) 26.1597 1.12469 0.562346 0.826902i \(-0.309899\pi\)
0.562346 + 0.826902i \(0.309899\pi\)
\(542\) −8.50393 −0.365275
\(543\) 1.68358 0.0722492
\(544\) 4.83878 0.207461
\(545\) 77.2791 3.31027
\(546\) 1.91373 0.0819000
\(547\) 7.54438 0.322574 0.161287 0.986908i \(-0.448435\pi\)
0.161287 + 0.986908i \(0.448435\pi\)
\(548\) −15.5620 −0.664775
\(549\) 10.2293 0.436576
\(550\) 37.6566 1.60568
\(551\) −4.67632 −0.199218
\(552\) −2.08206 −0.0886184
\(553\) −15.4713 −0.657909
\(554\) −16.2159 −0.688949
\(555\) −36.1382 −1.53398
\(556\) −14.8728 −0.630747
\(557\) −36.7923 −1.55894 −0.779471 0.626438i \(-0.784512\pi\)
−0.779471 + 0.626438i \(0.784512\pi\)
\(558\) −5.15289 −0.218139
\(559\) −1.60304 −0.0678012
\(560\) −7.18074 −0.303442
\(561\) 20.0691 0.847319
\(562\) −18.6296 −0.785841
\(563\) −5.96214 −0.251274 −0.125637 0.992076i \(-0.540097\pi\)
−0.125637 + 0.992076i \(0.540097\pi\)
\(564\) −9.35222 −0.393800
\(565\) 0.115080 0.00484146
\(566\) −2.72112 −0.114377
\(567\) 1.91373 0.0803690
\(568\) −9.46244 −0.397035
\(569\) 5.23201 0.219337 0.109669 0.993968i \(-0.465021\pi\)
0.109669 + 0.993968i \(0.465021\pi\)
\(570\) 8.00347 0.335229
\(571\) 9.05233 0.378828 0.189414 0.981897i \(-0.439341\pi\)
0.189414 + 0.981897i \(0.439341\pi\)
\(572\) 4.14756 0.173418
\(573\) −21.2826 −0.889092
\(574\) −15.0926 −0.629953
\(575\) 18.9035 0.788330
\(576\) 1.00000 0.0416667
\(577\) 28.5282 1.18764 0.593822 0.804596i \(-0.297618\pi\)
0.593822 + 0.804596i \(0.297618\pi\)
\(578\) −6.41383 −0.266780
\(579\) 19.4142 0.806828
\(580\) 8.22631 0.341579
\(581\) 0.761232 0.0315812
\(582\) 14.9290 0.618827
\(583\) −53.1554 −2.20147
\(584\) −4.11647 −0.170341
\(585\) 3.75223 0.155136
\(586\) −29.3839 −1.21384
\(587\) 1.97365 0.0814612 0.0407306 0.999170i \(-0.487031\pi\)
0.0407306 + 0.999170i \(0.487031\pi\)
\(588\) −3.33765 −0.137642
\(589\) 10.9911 0.452879
\(590\) −27.0678 −1.11436
\(591\) −11.6487 −0.479162
\(592\) 9.63112 0.395837
\(593\) −6.79289 −0.278951 −0.139475 0.990226i \(-0.544542\pi\)
−0.139475 + 0.990226i \(0.544542\pi\)
\(594\) 4.14756 0.170176
\(595\) 34.7461 1.42445
\(596\) 6.78344 0.277861
\(597\) 15.3303 0.627429
\(598\) 2.08206 0.0851418
\(599\) −1.22559 −0.0500763 −0.0250382 0.999686i \(-0.507971\pi\)
−0.0250382 + 0.999686i \(0.507971\pi\)
\(600\) −9.07922 −0.370658
\(601\) −13.5109 −0.551121 −0.275560 0.961284i \(-0.588863\pi\)
−0.275560 + 0.961284i \(0.588863\pi\)
\(602\) −3.06777 −0.125033
\(603\) −8.07141 −0.328693
\(604\) −19.0254 −0.774132
\(605\) −23.2722 −0.946149
\(606\) −16.7182 −0.679132
\(607\) −27.2985 −1.10801 −0.554005 0.832513i \(-0.686901\pi\)
−0.554005 + 0.832513i \(0.686901\pi\)
\(608\) −2.13299 −0.0865042
\(609\) −4.19561 −0.170015
\(610\) 38.3827 1.55407
\(611\) 9.35222 0.378350
\(612\) −4.83878 −0.195596
\(613\) 5.48355 0.221479 0.110739 0.993849i \(-0.464678\pi\)
0.110739 + 0.993849i \(0.464678\pi\)
\(614\) 7.72568 0.311783
\(615\) −29.5919 −1.19326
\(616\) 7.93729 0.319803
\(617\) 26.7890 1.07848 0.539241 0.842151i \(-0.318711\pi\)
0.539241 + 0.842151i \(0.318711\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −7.65296 −0.307598 −0.153799 0.988102i \(-0.549151\pi\)
−0.153799 + 0.988102i \(0.549151\pi\)
\(620\) −19.3348 −0.776505
\(621\) 2.08206 0.0835503
\(622\) −23.8089 −0.954651
\(623\) −24.2644 −0.972134
\(624\) −1.00000 −0.0400320
\(625\) 12.0362 0.481447
\(626\) 7.23019 0.288977
\(627\) −8.84670 −0.353303
\(628\) −24.8181 −0.990352
\(629\) −46.6029 −1.85818
\(630\) 7.18074 0.286088
\(631\) 29.8512 1.18836 0.594178 0.804334i \(-0.297477\pi\)
0.594178 + 0.804334i \(0.297477\pi\)
\(632\) 8.08441 0.321580
\(633\) −11.7754 −0.468031
\(634\) −15.4352 −0.613011
\(635\) 15.6747 0.622032
\(636\) 12.8161 0.508191
\(637\) 3.33765 0.132242
\(638\) −9.09301 −0.359996
\(639\) 9.46244 0.374328
\(640\) 3.75223 0.148320
\(641\) 33.8726 1.33789 0.668943 0.743314i \(-0.266747\pi\)
0.668943 + 0.743314i \(0.266747\pi\)
\(642\) −15.6347 −0.617050
\(643\) 43.0406 1.69736 0.848678 0.528910i \(-0.177399\pi\)
0.848678 + 0.528910i \(0.177399\pi\)
\(644\) 3.98450 0.157011
\(645\) −6.01496 −0.236839
\(646\) 10.3211 0.406078
\(647\) 18.4015 0.723439 0.361720 0.932287i \(-0.382190\pi\)
0.361720 + 0.932287i \(0.382190\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 29.9196 1.17445
\(650\) 9.07922 0.356116
\(651\) 9.86123 0.386492
\(652\) −11.4078 −0.446762
\(653\) −12.1955 −0.477246 −0.238623 0.971112i \(-0.576696\pi\)
−0.238623 + 0.971112i \(0.576696\pi\)
\(654\) 20.5955 0.805348
\(655\) −2.60246 −0.101686
\(656\) 7.88650 0.307916
\(657\) 4.11647 0.160599
\(658\) 17.8976 0.697721
\(659\) −27.2559 −1.06174 −0.530869 0.847454i \(-0.678134\pi\)
−0.530869 + 0.847454i \(0.678134\pi\)
\(660\) 15.5626 0.605773
\(661\) −48.4571 −1.88476 −0.942381 0.334541i \(-0.891419\pi\)
−0.942381 + 0.334541i \(0.891419\pi\)
\(662\) 31.3126 1.21700
\(663\) 4.83878 0.187923
\(664\) −0.397774 −0.0154366
\(665\) −15.3165 −0.593947
\(666\) −9.63112 −0.373199
\(667\) −4.56467 −0.176745
\(668\) −8.21066 −0.317680
\(669\) −20.7006 −0.800331
\(670\) −30.2858 −1.17004
\(671\) −42.4266 −1.63786
\(672\) −1.91373 −0.0738236
\(673\) 3.73117 0.143826 0.0719130 0.997411i \(-0.477090\pi\)
0.0719130 + 0.997411i \(0.477090\pi\)
\(674\) −21.7554 −0.837989
\(675\) 9.07922 0.349459
\(676\) 1.00000 0.0384615
\(677\) 38.5407 1.48124 0.740620 0.671924i \(-0.234532\pi\)
0.740620 + 0.671924i \(0.234532\pi\)
\(678\) 0.0306698 0.00117787
\(679\) −28.5700 −1.09642
\(680\) −18.1562 −0.696260
\(681\) 16.2577 0.622995
\(682\) 21.3719 0.818373
\(683\) −41.1026 −1.57275 −0.786373 0.617752i \(-0.788043\pi\)
−0.786373 + 0.617752i \(0.788043\pi\)
\(684\) 2.13299 0.0815570
\(685\) 58.3921 2.23105
\(686\) 19.7834 0.755335
\(687\) −13.6150 −0.519444
\(688\) 1.60304 0.0611151
\(689\) −12.8161 −0.488254
\(690\) 7.81237 0.297412
\(691\) 35.5935 1.35404 0.677020 0.735965i \(-0.263271\pi\)
0.677020 + 0.735965i \(0.263271\pi\)
\(692\) 9.58909 0.364523
\(693\) −7.93729 −0.301513
\(694\) 5.69161 0.216051
\(695\) 55.8062 2.11685
\(696\) 2.19238 0.0831019
\(697\) −38.1611 −1.44545
\(698\) 9.10237 0.344530
\(699\) −19.0339 −0.719929
\(700\) 17.3752 0.656719
\(701\) −4.01572 −0.151672 −0.0758358 0.997120i \(-0.524162\pi\)
−0.0758358 + 0.997120i \(0.524162\pi\)
\(702\) 1.00000 0.0377426
\(703\) 20.5431 0.774798
\(704\) −4.14756 −0.156317
\(705\) 35.0917 1.32163
\(706\) 20.7573 0.781211
\(707\) 31.9941 1.20326
\(708\) −7.21379 −0.271111
\(709\) −37.7839 −1.41900 −0.709501 0.704704i \(-0.751080\pi\)
−0.709501 + 0.704704i \(0.751080\pi\)
\(710\) 35.5052 1.33249
\(711\) −8.08441 −0.303189
\(712\) 12.6791 0.475171
\(713\) 10.7286 0.401791
\(714\) 9.26011 0.346551
\(715\) −15.5626 −0.582008
\(716\) 8.14337 0.304332
\(717\) 7.92027 0.295788
\(718\) 33.7756 1.26049
\(719\) −28.2566 −1.05379 −0.526897 0.849929i \(-0.676645\pi\)
−0.526897 + 0.849929i \(0.676645\pi\)
\(720\) −3.75223 −0.139837
\(721\) 1.91373 0.0712709
\(722\) 14.4503 0.537786
\(723\) 17.3089 0.643725
\(724\) 1.68358 0.0625697
\(725\) −19.9051 −0.739257
\(726\) −6.20223 −0.230186
\(727\) −41.8677 −1.55279 −0.776394 0.630248i \(-0.782953\pi\)
−0.776394 + 0.630248i \(0.782953\pi\)
\(728\) 1.91373 0.0709275
\(729\) 1.00000 0.0370370
\(730\) 15.4459 0.571680
\(731\) −7.75674 −0.286893
\(732\) 10.2293 0.378086
\(733\) −45.1539 −1.66780 −0.833900 0.551916i \(-0.813897\pi\)
−0.833900 + 0.551916i \(0.813897\pi\)
\(734\) 7.16383 0.264422
\(735\) 12.5236 0.461941
\(736\) −2.08206 −0.0767458
\(737\) 33.4766 1.23313
\(738\) −7.88650 −0.290306
\(739\) 31.9727 1.17614 0.588068 0.808812i \(-0.299889\pi\)
0.588068 + 0.808812i \(0.299889\pi\)
\(740\) −36.1382 −1.32847
\(741\) −2.13299 −0.0783574
\(742\) −24.5265 −0.900396
\(743\) −27.6910 −1.01589 −0.507943 0.861391i \(-0.669594\pi\)
−0.507943 + 0.861391i \(0.669594\pi\)
\(744\) −5.15289 −0.188914
\(745\) −25.4530 −0.932527
\(746\) 31.8515 1.16617
\(747\) 0.397774 0.0145538
\(748\) 20.0691 0.733800
\(749\) 29.9205 1.09327
\(750\) 15.3062 0.558903
\(751\) 34.4297 1.25636 0.628179 0.778069i \(-0.283801\pi\)
0.628179 + 0.778069i \(0.283801\pi\)
\(752\) −9.35222 −0.341040
\(753\) 6.93300 0.252653
\(754\) −2.19238 −0.0798417
\(755\) 71.3876 2.59806
\(756\) 1.91373 0.0696016
\(757\) −11.4723 −0.416969 −0.208485 0.978026i \(-0.566853\pi\)
−0.208485 + 0.978026i \(0.566853\pi\)
\(758\) −0.748216 −0.0271764
\(759\) −8.63547 −0.313448
\(760\) 8.00347 0.290316
\(761\) 34.5110 1.25102 0.625512 0.780215i \(-0.284890\pi\)
0.625512 + 0.780215i \(0.284890\pi\)
\(762\) 4.17744 0.151333
\(763\) −39.4142 −1.42689
\(764\) −21.2826 −0.769976
\(765\) 18.1562 0.656440
\(766\) −15.9351 −0.575760
\(767\) 7.21379 0.260475
\(768\) 1.00000 0.0360844
\(769\) −0.332765 −0.0119998 −0.00599991 0.999982i \(-0.501910\pi\)
−0.00599991 + 0.999982i \(0.501910\pi\)
\(770\) −29.7825 −1.07329
\(771\) 18.6390 0.671268
\(772\) 19.4142 0.698733
\(773\) 9.97635 0.358824 0.179412 0.983774i \(-0.442580\pi\)
0.179412 + 0.983774i \(0.442580\pi\)
\(774\) −1.60304 −0.0576199
\(775\) 46.7842 1.68054
\(776\) 14.9290 0.535920
\(777\) 18.4313 0.661221
\(778\) 5.36783 0.192446
\(779\) 16.8218 0.602705
\(780\) 3.75223 0.134351
\(781\) −39.2460 −1.40433
\(782\) 10.0746 0.360269
\(783\) −2.19238 −0.0783492
\(784\) −3.33765 −0.119202
\(785\) 93.1233 3.32371
\(786\) −0.693576 −0.0247391
\(787\) 21.4623 0.765049 0.382525 0.923945i \(-0.375055\pi\)
0.382525 + 0.923945i \(0.375055\pi\)
\(788\) −11.6487 −0.414966
\(789\) −19.2932 −0.686857
\(790\) −30.3345 −1.07925
\(791\) −0.0586937 −0.00208691
\(792\) 4.14756 0.147377
\(793\) −10.2293 −0.363253
\(794\) 5.30293 0.188194
\(795\) −48.0889 −1.70554
\(796\) 15.3303 0.543370
\(797\) 0.719890 0.0254998 0.0127499 0.999919i \(-0.495941\pi\)
0.0127499 + 0.999919i \(0.495941\pi\)
\(798\) −4.08196 −0.144500
\(799\) 45.2534 1.60095
\(800\) −9.07922 −0.320999
\(801\) −12.6791 −0.447996
\(802\) 30.6283 1.08152
\(803\) −17.0733 −0.602504
\(804\) −8.07141 −0.284657
\(805\) −14.9507 −0.526945
\(806\) 5.15289 0.181503
\(807\) −25.0184 −0.880688
\(808\) −16.7182 −0.588146
\(809\) 7.41244 0.260608 0.130304 0.991474i \(-0.458405\pi\)
0.130304 + 0.991474i \(0.458405\pi\)
\(810\) 3.75223 0.131840
\(811\) −49.5990 −1.74165 −0.870827 0.491589i \(-0.836416\pi\)
−0.870827 + 0.491589i \(0.836416\pi\)
\(812\) −4.19561 −0.147237
\(813\) 8.50393 0.298246
\(814\) 39.9456 1.40009
\(815\) 42.8045 1.49938
\(816\) −4.83878 −0.169391
\(817\) 3.41926 0.119625
\(818\) 25.3342 0.885790
\(819\) −1.91373 −0.0668710
\(820\) −29.5919 −1.03340
\(821\) −16.0854 −0.561385 −0.280692 0.959798i \(-0.590564\pi\)
−0.280692 + 0.959798i \(0.590564\pi\)
\(822\) 15.5620 0.542787
\(823\) 40.7960 1.42206 0.711029 0.703163i \(-0.248230\pi\)
0.711029 + 0.703163i \(0.248230\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −37.6566 −1.31103
\(826\) 13.8052 0.480345
\(827\) −1.22302 −0.0425286 −0.0212643 0.999774i \(-0.506769\pi\)
−0.0212643 + 0.999774i \(0.506769\pi\)
\(828\) 2.08206 0.0723566
\(829\) 20.6281 0.716442 0.358221 0.933637i \(-0.383383\pi\)
0.358221 + 0.933637i \(0.383383\pi\)
\(830\) 1.49254 0.0518068
\(831\) 16.2159 0.562524
\(832\) −1.00000 −0.0346688
\(833\) 16.1502 0.559570
\(834\) 14.8728 0.515003
\(835\) 30.8083 1.06616
\(836\) −8.84670 −0.305970
\(837\) 5.15289 0.178110
\(838\) −29.4907 −1.01874
\(839\) 3.31359 0.114398 0.0571989 0.998363i \(-0.481783\pi\)
0.0571989 + 0.998363i \(0.481783\pi\)
\(840\) 7.18074 0.247759
\(841\) −24.1935 −0.834258
\(842\) −27.7454 −0.956169
\(843\) 18.6296 0.641636
\(844\) −11.7754 −0.405327
\(845\) −3.75223 −0.129081
\(846\) 9.35222 0.321536
\(847\) 11.8694 0.407836
\(848\) 12.8161 0.440106
\(849\) 2.72112 0.0933886
\(850\) 43.9324 1.50687
\(851\) 20.0526 0.687394
\(852\) 9.46244 0.324178
\(853\) 4.81021 0.164698 0.0823492 0.996604i \(-0.473758\pi\)
0.0823492 + 0.996604i \(0.473758\pi\)
\(854\) −19.5761 −0.669880
\(855\) −8.00347 −0.273713
\(856\) −15.6347 −0.534381
\(857\) −29.0740 −0.993150 −0.496575 0.867994i \(-0.665409\pi\)
−0.496575 + 0.867994i \(0.665409\pi\)
\(858\) −4.14756 −0.141595
\(859\) 23.2416 0.792992 0.396496 0.918036i \(-0.370226\pi\)
0.396496 + 0.918036i \(0.370226\pi\)
\(860\) −6.01496 −0.205108
\(861\) 15.0926 0.514355
\(862\) 2.87789 0.0980213
\(863\) −18.9313 −0.644428 −0.322214 0.946667i \(-0.604427\pi\)
−0.322214 + 0.946667i \(0.604427\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −35.9805 −1.22337
\(866\) 16.0609 0.545772
\(867\) 6.41383 0.217825
\(868\) 9.86123 0.334712
\(869\) 33.5305 1.13745
\(870\) −8.22631 −0.278898
\(871\) 8.07141 0.273489
\(872\) 20.5955 0.697452
\(873\) −14.9290 −0.505270
\(874\) −4.44102 −0.150220
\(875\) −29.2918 −0.990245
\(876\) 4.11647 0.139083
\(877\) −7.28841 −0.246112 −0.123056 0.992400i \(-0.539270\pi\)
−0.123056 + 0.992400i \(0.539270\pi\)
\(878\) −26.0727 −0.879910
\(879\) 29.3839 0.991093
\(880\) 15.5626 0.524615
\(881\) 14.5805 0.491230 0.245615 0.969367i \(-0.421010\pi\)
0.245615 + 0.969367i \(0.421010\pi\)
\(882\) 3.33765 0.112384
\(883\) 14.8464 0.499621 0.249810 0.968295i \(-0.419632\pi\)
0.249810 + 0.968295i \(0.419632\pi\)
\(884\) 4.83878 0.162746
\(885\) 27.0678 0.909874
\(886\) 17.8236 0.598797
\(887\) −1.25625 −0.0421808 −0.0210904 0.999778i \(-0.506714\pi\)
−0.0210904 + 0.999778i \(0.506714\pi\)
\(888\) −9.63112 −0.323199
\(889\) −7.99448 −0.268126
\(890\) −47.5751 −1.59472
\(891\) −4.14756 −0.138948
\(892\) −20.7006 −0.693107
\(893\) −19.9482 −0.667541
\(894\) −6.78344 −0.226872
\(895\) −30.5558 −1.02137
\(896\) −1.91373 −0.0639331
\(897\) −2.08206 −0.0695180
\(898\) 31.6248 1.05533
\(899\) −11.2971 −0.376779
\(900\) 9.07922 0.302641
\(901\) −62.0143 −2.06599
\(902\) 32.7097 1.08911
\(903\) 3.06777 0.102089
\(904\) 0.0306698 0.00102006
\(905\) −6.31717 −0.209990
\(906\) 19.0254 0.632076
\(907\) −45.0966 −1.49741 −0.748705 0.662904i \(-0.769324\pi\)
−0.748705 + 0.662904i \(0.769324\pi\)
\(908\) 16.2577 0.539530
\(909\) 16.7182 0.554509
\(910\) −7.18074 −0.238039
\(911\) −33.9383 −1.12442 −0.562212 0.826993i \(-0.690050\pi\)
−0.562212 + 0.826993i \(0.690050\pi\)
\(912\) 2.13299 0.0706304
\(913\) −1.64979 −0.0546001
\(914\) 30.9954 1.02524
\(915\) −38.3827 −1.26889
\(916\) −13.6150 −0.449851
\(917\) 1.32732 0.0438318
\(918\) 4.83878 0.159704
\(919\) 2.80843 0.0926415 0.0463207 0.998927i \(-0.485250\pi\)
0.0463207 + 0.998927i \(0.485250\pi\)
\(920\) 7.81237 0.257566
\(921\) −7.72568 −0.254570
\(922\) 36.7779 1.21122
\(923\) −9.46244 −0.311460
\(924\) −7.93729 −0.261118
\(925\) 87.4431 2.87511
\(926\) 10.4965 0.344936
\(927\) 1.00000 0.0328443
\(928\) 2.19238 0.0719683
\(929\) −6.37691 −0.209220 −0.104610 0.994513i \(-0.533359\pi\)
−0.104610 + 0.994513i \(0.533359\pi\)
\(930\) 19.3348 0.634014
\(931\) −7.11917 −0.233322
\(932\) −19.0339 −0.623477
\(933\) 23.8089 0.779469
\(934\) −16.3515 −0.535036
\(935\) −75.3040 −2.46270
\(936\) 1.00000 0.0326860
\(937\) 2.86601 0.0936283 0.0468142 0.998904i \(-0.485093\pi\)
0.0468142 + 0.998904i \(0.485093\pi\)
\(938\) 15.4465 0.504345
\(939\) −7.23019 −0.235948
\(940\) 35.0917 1.14456
\(941\) −9.15333 −0.298390 −0.149195 0.988808i \(-0.547668\pi\)
−0.149195 + 0.988808i \(0.547668\pi\)
\(942\) 24.8181 0.808619
\(943\) 16.4202 0.534714
\(944\) −7.21379 −0.234789
\(945\) −7.18074 −0.233590
\(946\) 6.64868 0.216167
\(947\) −35.7862 −1.16290 −0.581448 0.813583i \(-0.697514\pi\)
−0.581448 + 0.813583i \(0.697514\pi\)
\(948\) −8.08441 −0.262569
\(949\) −4.11647 −0.133626
\(950\) −19.3659 −0.628313
\(951\) 15.4352 0.500521
\(952\) 9.26011 0.300122
\(953\) −44.3315 −1.43604 −0.718019 0.696024i \(-0.754951\pi\)
−0.718019 + 0.696024i \(0.754951\pi\)
\(954\) −12.8161 −0.414936
\(955\) 79.8571 2.58411
\(956\) 7.92027 0.256160
\(957\) 9.09301 0.293935
\(958\) 35.5629 1.14899
\(959\) −29.7814 −0.961691
\(960\) −3.75223 −0.121103
\(961\) −4.44772 −0.143475
\(962\) 9.63112 0.310520
\(963\) 15.6347 0.503820
\(964\) 17.3089 0.557482
\(965\) −72.8466 −2.34502
\(966\) −3.98450 −0.128199
\(967\) −53.2003 −1.71081 −0.855404 0.517962i \(-0.826691\pi\)
−0.855404 + 0.517962i \(0.826691\pi\)
\(968\) −6.20223 −0.199347
\(969\) −10.3211 −0.331561
\(970\) −56.0170 −1.79860
\(971\) 27.6174 0.886284 0.443142 0.896451i \(-0.353864\pi\)
0.443142 + 0.896451i \(0.353864\pi\)
\(972\) 1.00000 0.0320750
\(973\) −28.4625 −0.912465
\(974\) −1.39658 −0.0447493
\(975\) −9.07922 −0.290768
\(976\) 10.2293 0.327432
\(977\) −30.5303 −0.976750 −0.488375 0.872634i \(-0.662410\pi\)
−0.488375 + 0.872634i \(0.662410\pi\)
\(978\) 11.4078 0.364780
\(979\) 52.5875 1.68070
\(980\) 12.5236 0.400052
\(981\) −20.5955 −0.657564
\(982\) −2.49644 −0.0796647
\(983\) 15.9005 0.507147 0.253573 0.967316i \(-0.418394\pi\)
0.253573 + 0.967316i \(0.418394\pi\)
\(984\) −7.88650 −0.251412
\(985\) 43.7084 1.39267
\(986\) −10.6084 −0.337842
\(987\) −17.8976 −0.569687
\(988\) −2.13299 −0.0678595
\(989\) 3.33762 0.106130
\(990\) −15.5626 −0.494611
\(991\) −49.8508 −1.58356 −0.791781 0.610805i \(-0.790846\pi\)
−0.791781 + 0.610805i \(0.790846\pi\)
\(992\) −5.15289 −0.163604
\(993\) −31.3126 −0.993675
\(994\) −18.1085 −0.574368
\(995\) −57.5230 −1.82360
\(996\) 0.397774 0.0126040
\(997\) −31.7453 −1.00538 −0.502691 0.864466i \(-0.667657\pi\)
−0.502691 + 0.864466i \(0.667657\pi\)
\(998\) −18.8673 −0.597233
\(999\) 9.63112 0.304715
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.bb.1.3 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bb.1.3 14 1.1 even 1 trivial