Properties

Label 8470.2.a.dc.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.19898000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 7 x^{3} + 24 x^{2} - 15 x - 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.68692\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.68692 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.68692 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.154300 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.68692 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.68692 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.154300 q^{9} +1.00000 q^{10} -1.68692 q^{12} -0.0425741 q^{13} -1.00000 q^{14} -1.68692 q^{15} +1.00000 q^{16} +4.72038 q^{17} -0.154300 q^{18} +1.71340 q^{19} +1.00000 q^{20} +1.68692 q^{21} -6.95966 q^{23} -1.68692 q^{24} +1.00000 q^{25} -0.0425741 q^{26} +5.32105 q^{27} -1.00000 q^{28} +1.34277 q^{29} -1.68692 q^{30} +6.01245 q^{31} +1.00000 q^{32} +4.72038 q^{34} -1.00000 q^{35} -0.154300 q^{36} +8.48459 q^{37} +1.71340 q^{38} +0.0718191 q^{39} +1.00000 q^{40} -3.91371 q^{41} +1.68692 q^{42} -6.42455 q^{43} -0.154300 q^{45} -6.95966 q^{46} +4.59129 q^{47} -1.68692 q^{48} +1.00000 q^{49} +1.00000 q^{50} -7.96291 q^{51} -0.0425741 q^{52} -14.1024 q^{53} +5.32105 q^{54} -1.00000 q^{56} -2.89036 q^{57} +1.34277 q^{58} -10.2177 q^{59} -1.68692 q^{60} -13.8661 q^{61} +6.01245 q^{62} +0.154300 q^{63} +1.00000 q^{64} -0.0425741 q^{65} +6.39600 q^{67} +4.72038 q^{68} +11.7404 q^{69} -1.00000 q^{70} +11.3041 q^{71} -0.154300 q^{72} +1.98728 q^{73} +8.48459 q^{74} -1.68692 q^{75} +1.71340 q^{76} +0.0718191 q^{78} +8.87065 q^{79} +1.00000 q^{80} -8.51329 q^{81} -3.91371 q^{82} +10.0807 q^{83} +1.68692 q^{84} +4.72038 q^{85} -6.42455 q^{86} -2.26515 q^{87} +2.93362 q^{89} -0.154300 q^{90} +0.0425741 q^{91} -6.95966 q^{92} -10.1425 q^{93} +4.59129 q^{94} +1.71340 q^{95} -1.68692 q^{96} -17.4812 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} - q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} + 6q^{8} + 3q^{9} + O(q^{10}) \) \( 6q + 6q^{2} - q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} + 6q^{8} + 3q^{9} + 6q^{10} - q^{12} + 9q^{13} - 6q^{14} - q^{15} + 6q^{16} + 9q^{17} + 3q^{18} + 12q^{19} + 6q^{20} + q^{21} + 4q^{23} - q^{24} + 6q^{25} + 9q^{26} - 4q^{27} - 6q^{28} + 15q^{29} - q^{30} + 8q^{31} + 6q^{32} + 9q^{34} - 6q^{35} + 3q^{36} - 4q^{37} + 12q^{38} - 19q^{39} + 6q^{40} + 4q^{41} + q^{42} + 30q^{43} + 3q^{45} + 4q^{46} - 7q^{47} - q^{48} + 6q^{49} + 6q^{50} + 16q^{51} + 9q^{52} - 6q^{53} - 4q^{54} - 6q^{56} - 14q^{57} + 15q^{58} + 4q^{59} - q^{60} - 14q^{61} + 8q^{62} - 3q^{63} + 6q^{64} + 9q^{65} + 18q^{67} + 9q^{68} - 10q^{69} - 6q^{70} + 23q^{71} + 3q^{72} + 23q^{73} - 4q^{74} - q^{75} + 12q^{76} - 19q^{78} + 21q^{79} + 6q^{80} - 18q^{81} + 4q^{82} + 25q^{83} + q^{84} + 9q^{85} + 30q^{86} + 14q^{87} - 18q^{89} + 3q^{90} - 9q^{91} + 4q^{92} - 24q^{93} - 7q^{94} + 12q^{95} - q^{96} + 7q^{97} + 6q^{98} + O(q^{100}) \)

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
\(3\) −1.68692 −0.973944 −0.486972 0.873418i \(-0.661899\pi\)
−0.486972 + 0.873418i \(0.661899\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.68692 −0.688682
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −0.154300 −0.0514332
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −1.68692 −0.486972
\(13\) −0.0425741 −0.0118079 −0.00590397 0.999983i \(-0.501879\pi\)
−0.00590397 + 0.999983i \(0.501879\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.68692 −0.435561
\(16\) 1.00000 0.250000
\(17\) 4.72038 1.14486 0.572430 0.819953i \(-0.306001\pi\)
0.572430 + 0.819953i \(0.306001\pi\)
\(18\) −0.154300 −0.0363688
\(19\) 1.71340 0.393080 0.196540 0.980496i \(-0.437029\pi\)
0.196540 + 0.980496i \(0.437029\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.68692 0.368116
\(22\) 0 0
\(23\) −6.95966 −1.45119 −0.725595 0.688122i \(-0.758435\pi\)
−0.725595 + 0.688122i \(0.758435\pi\)
\(24\) −1.68692 −0.344341
\(25\) 1.00000 0.200000
\(26\) −0.0425741 −0.00834947
\(27\) 5.32105 1.02404
\(28\) −1.00000 −0.188982
\(29\) 1.34277 0.249346 0.124673 0.992198i \(-0.460212\pi\)
0.124673 + 0.992198i \(0.460212\pi\)
\(30\) −1.68692 −0.307988
\(31\) 6.01245 1.07987 0.539934 0.841707i \(-0.318449\pi\)
0.539934 + 0.841707i \(0.318449\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.72038 0.809539
\(35\) −1.00000 −0.169031
\(36\) −0.154300 −0.0257166
\(37\) 8.48459 1.39486 0.697429 0.716654i \(-0.254327\pi\)
0.697429 + 0.716654i \(0.254327\pi\)
\(38\) 1.71340 0.277950
\(39\) 0.0718191 0.0115003
\(40\) 1.00000 0.158114
\(41\) −3.91371 −0.611219 −0.305609 0.952157i \(-0.598860\pi\)
−0.305609 + 0.952157i \(0.598860\pi\)
\(42\) 1.68692 0.260297
\(43\) −6.42455 −0.979735 −0.489867 0.871797i \(-0.662955\pi\)
−0.489867 + 0.871797i \(0.662955\pi\)
\(44\) 0 0
\(45\) −0.154300 −0.0230016
\(46\) −6.95966 −1.02615
\(47\) 4.59129 0.669709 0.334854 0.942270i \(-0.391313\pi\)
0.334854 + 0.942270i \(0.391313\pi\)
\(48\) −1.68692 −0.243486
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −7.96291 −1.11503
\(52\) −0.0425741 −0.00590397
\(53\) −14.1024 −1.93711 −0.968554 0.248804i \(-0.919963\pi\)
−0.968554 + 0.248804i \(0.919963\pi\)
\(54\) 5.32105 0.724104
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −2.89036 −0.382838
\(58\) 1.34277 0.176315
\(59\) −10.2177 −1.33023 −0.665117 0.746739i \(-0.731619\pi\)
−0.665117 + 0.746739i \(0.731619\pi\)
\(60\) −1.68692 −0.217780
\(61\) −13.8661 −1.77538 −0.887688 0.460446i \(-0.847690\pi\)
−0.887688 + 0.460446i \(0.847690\pi\)
\(62\) 6.01245 0.763582
\(63\) 0.154300 0.0194399
\(64\) 1.00000 0.125000
\(65\) −0.0425741 −0.00528067
\(66\) 0 0
\(67\) 6.39600 0.781396 0.390698 0.920519i \(-0.372234\pi\)
0.390698 + 0.920519i \(0.372234\pi\)
\(68\) 4.72038 0.572430
\(69\) 11.7404 1.41338
\(70\) −1.00000 −0.119523
\(71\) 11.3041 1.34155 0.670774 0.741662i \(-0.265962\pi\)
0.670774 + 0.741662i \(0.265962\pi\)
\(72\) −0.154300 −0.0181844
\(73\) 1.98728 0.232594 0.116297 0.993214i \(-0.462898\pi\)
0.116297 + 0.993214i \(0.462898\pi\)
\(74\) 8.48459 0.986313
\(75\) −1.68692 −0.194789
\(76\) 1.71340 0.196540
\(77\) 0 0
\(78\) 0.0718191 0.00813192
\(79\) 8.87065 0.998026 0.499013 0.866595i \(-0.333696\pi\)
0.499013 + 0.866595i \(0.333696\pi\)
\(80\) 1.00000 0.111803
\(81\) −8.51329 −0.945921
\(82\) −3.91371 −0.432197
\(83\) 10.0807 1.10650 0.553249 0.833016i \(-0.313388\pi\)
0.553249 + 0.833016i \(0.313388\pi\)
\(84\) 1.68692 0.184058
\(85\) 4.72038 0.511997
\(86\) −6.42455 −0.692777
\(87\) −2.26515 −0.242849
\(88\) 0 0
\(89\) 2.93362 0.310963 0.155481 0.987839i \(-0.450307\pi\)
0.155481 + 0.987839i \(0.450307\pi\)
\(90\) −0.154300 −0.0162646
\(91\) 0.0425741 0.00446298
\(92\) −6.95966 −0.725595
\(93\) −10.1425 −1.05173
\(94\) 4.59129 0.473556
\(95\) 1.71340 0.175791
\(96\) −1.68692 −0.172171
\(97\) −17.4812 −1.77495 −0.887474 0.460857i \(-0.847542\pi\)
−0.887474 + 0.460857i \(0.847542\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 5.70914 0.568081 0.284040 0.958812i \(-0.408325\pi\)
0.284040 + 0.958812i \(0.408325\pi\)
\(102\) −7.96291 −0.788445
\(103\) 13.4052 1.32085 0.660425 0.750892i \(-0.270376\pi\)
0.660425 + 0.750892i \(0.270376\pi\)
\(104\) −0.0425741 −0.00417474
\(105\) 1.68692 0.164627
\(106\) −14.1024 −1.36974
\(107\) 15.9669 1.54358 0.771790 0.635878i \(-0.219362\pi\)
0.771790 + 0.635878i \(0.219362\pi\)
\(108\) 5.32105 0.512019
\(109\) −15.9501 −1.52774 −0.763871 0.645369i \(-0.776703\pi\)
−0.763871 + 0.645369i \(0.776703\pi\)
\(110\) 0 0
\(111\) −14.3128 −1.35851
\(112\) −1.00000 −0.0944911
\(113\) 8.65087 0.813805 0.406903 0.913472i \(-0.366609\pi\)
0.406903 + 0.913472i \(0.366609\pi\)
\(114\) −2.89036 −0.270707
\(115\) −6.95966 −0.648992
\(116\) 1.34277 0.124673
\(117\) 0.00656917 0.000607320 0
\(118\) −10.2177 −0.940618
\(119\) −4.72038 −0.432717
\(120\) −1.68692 −0.153994
\(121\) 0 0
\(122\) −13.8661 −1.25538
\(123\) 6.60212 0.595293
\(124\) 6.01245 0.539934
\(125\) 1.00000 0.0894427
\(126\) 0.154300 0.0137461
\(127\) 1.82824 0.162230 0.0811149 0.996705i \(-0.474152\pi\)
0.0811149 + 0.996705i \(0.474152\pi\)
\(128\) 1.00000 0.0883883
\(129\) 10.8377 0.954207
\(130\) −0.0425741 −0.00373400
\(131\) 1.61678 0.141259 0.0706294 0.997503i \(-0.477499\pi\)
0.0706294 + 0.997503i \(0.477499\pi\)
\(132\) 0 0
\(133\) −1.71340 −0.148570
\(134\) 6.39600 0.552530
\(135\) 5.32105 0.457963
\(136\) 4.72038 0.404769
\(137\) 10.2667 0.877144 0.438572 0.898696i \(-0.355484\pi\)
0.438572 + 0.898696i \(0.355484\pi\)
\(138\) 11.7404 0.999409
\(139\) 1.94758 0.165191 0.0825957 0.996583i \(-0.473679\pi\)
0.0825957 + 0.996583i \(0.473679\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −7.74515 −0.652259
\(142\) 11.3041 0.948617
\(143\) 0 0
\(144\) −0.154300 −0.0128583
\(145\) 1.34277 0.111511
\(146\) 1.98728 0.164469
\(147\) −1.68692 −0.139135
\(148\) 8.48459 0.697429
\(149\) 6.46920 0.529978 0.264989 0.964251i \(-0.414632\pi\)
0.264989 + 0.964251i \(0.414632\pi\)
\(150\) −1.68692 −0.137736
\(151\) 23.6110 1.92143 0.960717 0.277531i \(-0.0895160\pi\)
0.960717 + 0.277531i \(0.0895160\pi\)
\(152\) 1.71340 0.138975
\(153\) −0.728353 −0.0588839
\(154\) 0 0
\(155\) 6.01245 0.482932
\(156\) 0.0718191 0.00575013
\(157\) 11.4891 0.916932 0.458466 0.888712i \(-0.348399\pi\)
0.458466 + 0.888712i \(0.348399\pi\)
\(158\) 8.87065 0.705711
\(159\) 23.7896 1.88663
\(160\) 1.00000 0.0790569
\(161\) 6.95966 0.548498
\(162\) −8.51329 −0.668867
\(163\) −14.1296 −1.10672 −0.553359 0.832943i \(-0.686654\pi\)
−0.553359 + 0.832943i \(0.686654\pi\)
\(164\) −3.91371 −0.305609
\(165\) 0 0
\(166\) 10.0807 0.782412
\(167\) 15.6015 1.20728 0.603641 0.797256i \(-0.293716\pi\)
0.603641 + 0.797256i \(0.293716\pi\)
\(168\) 1.68692 0.130149
\(169\) −12.9982 −0.999861
\(170\) 4.72038 0.362037
\(171\) −0.264377 −0.0202174
\(172\) −6.42455 −0.489867
\(173\) −6.21176 −0.472271 −0.236136 0.971720i \(-0.575881\pi\)
−0.236136 + 0.971720i \(0.575881\pi\)
\(174\) −2.26515 −0.171721
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 17.2365 1.29557
\(178\) 2.93362 0.219884
\(179\) 15.9071 1.18895 0.594475 0.804114i \(-0.297360\pi\)
0.594475 + 0.804114i \(0.297360\pi\)
\(180\) −0.154300 −0.0115008
\(181\) −6.59938 −0.490528 −0.245264 0.969456i \(-0.578875\pi\)
−0.245264 + 0.969456i \(0.578875\pi\)
\(182\) 0.0425741 0.00315580
\(183\) 23.3911 1.72912
\(184\) −6.95966 −0.513073
\(185\) 8.48459 0.623799
\(186\) −10.1425 −0.743686
\(187\) 0 0
\(188\) 4.59129 0.334854
\(189\) −5.32105 −0.387050
\(190\) 1.71340 0.124303
\(191\) 17.1690 1.24231 0.621153 0.783689i \(-0.286664\pi\)
0.621153 + 0.783689i \(0.286664\pi\)
\(192\) −1.68692 −0.121743
\(193\) 10.2185 0.735541 0.367771 0.929917i \(-0.380121\pi\)
0.367771 + 0.929917i \(0.380121\pi\)
\(194\) −17.4812 −1.25508
\(195\) 0.0718191 0.00514308
\(196\) 1.00000 0.0714286
\(197\) 7.33268 0.522432 0.261216 0.965280i \(-0.415876\pi\)
0.261216 + 0.965280i \(0.415876\pi\)
\(198\) 0 0
\(199\) −8.90352 −0.631154 −0.315577 0.948900i \(-0.602198\pi\)
−0.315577 + 0.948900i \(0.602198\pi\)
\(200\) 1.00000 0.0707107
\(201\) −10.7895 −0.761036
\(202\) 5.70914 0.401694
\(203\) −1.34277 −0.0942441
\(204\) −7.96291 −0.557515
\(205\) −3.91371 −0.273345
\(206\) 13.4052 0.933982
\(207\) 1.07387 0.0746394
\(208\) −0.0425741 −0.00295198
\(209\) 0 0
\(210\) 1.68692 0.116409
\(211\) −4.54767 −0.313075 −0.156537 0.987672i \(-0.550033\pi\)
−0.156537 + 0.987672i \(0.550033\pi\)
\(212\) −14.1024 −0.968554
\(213\) −19.0691 −1.30659
\(214\) 15.9669 1.09148
\(215\) −6.42455 −0.438151
\(216\) 5.32105 0.362052
\(217\) −6.01245 −0.408152
\(218\) −15.9501 −1.08028
\(219\) −3.35239 −0.226533
\(220\) 0 0
\(221\) −0.200966 −0.0135184
\(222\) −14.3128 −0.960614
\(223\) 18.0702 1.21007 0.605034 0.796200i \(-0.293160\pi\)
0.605034 + 0.796200i \(0.293160\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −0.154300 −0.0102866
\(226\) 8.65087 0.575447
\(227\) −0.970501 −0.0644144 −0.0322072 0.999481i \(-0.510254\pi\)
−0.0322072 + 0.999481i \(0.510254\pi\)
\(228\) −2.89036 −0.191419
\(229\) −2.46292 −0.162755 −0.0813773 0.996683i \(-0.525932\pi\)
−0.0813773 + 0.996683i \(0.525932\pi\)
\(230\) −6.95966 −0.458907
\(231\) 0 0
\(232\) 1.34277 0.0881573
\(233\) 6.65041 0.435683 0.217841 0.975984i \(-0.430098\pi\)
0.217841 + 0.975984i \(0.430098\pi\)
\(234\) 0.00656917 0.000429440 0
\(235\) 4.59129 0.299503
\(236\) −10.2177 −0.665117
\(237\) −14.9641 −0.972021
\(238\) −4.72038 −0.305977
\(239\) −21.0529 −1.36180 −0.680900 0.732377i \(-0.738411\pi\)
−0.680900 + 0.732377i \(0.738411\pi\)
\(240\) −1.68692 −0.108890
\(241\) 7.93244 0.510973 0.255487 0.966813i \(-0.417764\pi\)
0.255487 + 0.966813i \(0.417764\pi\)
\(242\) 0 0
\(243\) −1.60191 −0.102763
\(244\) −13.8661 −0.887688
\(245\) 1.00000 0.0638877
\(246\) 6.60212 0.420936
\(247\) −0.0729463 −0.00464146
\(248\) 6.01245 0.381791
\(249\) −17.0053 −1.07767
\(250\) 1.00000 0.0632456
\(251\) 18.9431 1.19568 0.597839 0.801617i \(-0.296026\pi\)
0.597839 + 0.801617i \(0.296026\pi\)
\(252\) 0.154300 0.00971997
\(253\) 0 0
\(254\) 1.82824 0.114714
\(255\) −7.96291 −0.498657
\(256\) 1.00000 0.0625000
\(257\) −26.8598 −1.67547 −0.837736 0.546076i \(-0.816121\pi\)
−0.837736 + 0.546076i \(0.816121\pi\)
\(258\) 10.8377 0.674726
\(259\) −8.48459 −0.527207
\(260\) −0.0425741 −0.00264033
\(261\) −0.207189 −0.0128247
\(262\) 1.61678 0.0998850
\(263\) 13.3750 0.824740 0.412370 0.911017i \(-0.364701\pi\)
0.412370 + 0.911017i \(0.364701\pi\)
\(264\) 0 0
\(265\) −14.1024 −0.866301
\(266\) −1.71340 −0.105055
\(267\) −4.94878 −0.302860
\(268\) 6.39600 0.390698
\(269\) −4.05979 −0.247530 −0.123765 0.992312i \(-0.539497\pi\)
−0.123765 + 0.992312i \(0.539497\pi\)
\(270\) 5.32105 0.323829
\(271\) −23.0226 −1.39852 −0.699261 0.714866i \(-0.746488\pi\)
−0.699261 + 0.714866i \(0.746488\pi\)
\(272\) 4.72038 0.286215
\(273\) −0.0718191 −0.00434669
\(274\) 10.2667 0.620235
\(275\) 0 0
\(276\) 11.7404 0.706689
\(277\) 22.2862 1.33905 0.669524 0.742790i \(-0.266498\pi\)
0.669524 + 0.742790i \(0.266498\pi\)
\(278\) 1.94758 0.116808
\(279\) −0.927720 −0.0555411
\(280\) −1.00000 −0.0597614
\(281\) 2.24959 0.134199 0.0670996 0.997746i \(-0.478625\pi\)
0.0670996 + 0.997746i \(0.478625\pi\)
\(282\) −7.74515 −0.461217
\(283\) 3.71034 0.220557 0.110279 0.993901i \(-0.464826\pi\)
0.110279 + 0.993901i \(0.464826\pi\)
\(284\) 11.3041 0.670774
\(285\) −2.89036 −0.171210
\(286\) 0 0
\(287\) 3.91371 0.231019
\(288\) −0.154300 −0.00909220
\(289\) 5.28199 0.310705
\(290\) 1.34277 0.0788503
\(291\) 29.4894 1.72870
\(292\) 1.98728 0.116297
\(293\) −1.85170 −0.108177 −0.0540886 0.998536i \(-0.517225\pi\)
−0.0540886 + 0.998536i \(0.517225\pi\)
\(294\) −1.68692 −0.0983832
\(295\) −10.2177 −0.594899
\(296\) 8.48459 0.493157
\(297\) 0 0
\(298\) 6.46920 0.374751
\(299\) 0.296302 0.0171356
\(300\) −1.68692 −0.0973944
\(301\) 6.42455 0.370305
\(302\) 23.6110 1.35866
\(303\) −9.63087 −0.553279
\(304\) 1.71340 0.0982700
\(305\) −13.8661 −0.793972
\(306\) −0.728353 −0.0416372
\(307\) 6.66980 0.380666 0.190333 0.981720i \(-0.439043\pi\)
0.190333 + 0.981720i \(0.439043\pi\)
\(308\) 0 0
\(309\) −22.6134 −1.28643
\(310\) 6.01245 0.341484
\(311\) 15.0969 0.856067 0.428034 0.903763i \(-0.359207\pi\)
0.428034 + 0.903763i \(0.359207\pi\)
\(312\) 0.0718191 0.00406596
\(313\) 7.33744 0.414736 0.207368 0.978263i \(-0.433510\pi\)
0.207368 + 0.978263i \(0.433510\pi\)
\(314\) 11.4891 0.648369
\(315\) 0.154300 0.00869380
\(316\) 8.87065 0.499013
\(317\) 22.4900 1.26316 0.631581 0.775310i \(-0.282406\pi\)
0.631581 + 0.775310i \(0.282406\pi\)
\(318\) 23.7896 1.33405
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −26.9349 −1.50336
\(322\) 6.95966 0.387847
\(323\) 8.08788 0.450022
\(324\) −8.51329 −0.472961
\(325\) −0.0425741 −0.00236159
\(326\) −14.1296 −0.782568
\(327\) 26.9065 1.48793
\(328\) −3.91371 −0.216099
\(329\) −4.59129 −0.253126
\(330\) 0 0
\(331\) 10.3154 0.566987 0.283494 0.958974i \(-0.408507\pi\)
0.283494 + 0.958974i \(0.408507\pi\)
\(332\) 10.0807 0.553249
\(333\) −1.30917 −0.0717420
\(334\) 15.6015 0.853677
\(335\) 6.39600 0.349451
\(336\) 1.68692 0.0920291
\(337\) 27.2073 1.48208 0.741038 0.671463i \(-0.234334\pi\)
0.741038 + 0.671463i \(0.234334\pi\)
\(338\) −12.9982 −0.707008
\(339\) −14.5933 −0.792600
\(340\) 4.72038 0.255999
\(341\) 0 0
\(342\) −0.264377 −0.0142958
\(343\) −1.00000 −0.0539949
\(344\) −6.42455 −0.346389
\(345\) 11.7404 0.632082
\(346\) −6.21176 −0.333946
\(347\) 6.65299 0.357151 0.178576 0.983926i \(-0.442851\pi\)
0.178576 + 0.983926i \(0.442851\pi\)
\(348\) −2.26515 −0.121425
\(349\) 25.4581 1.36274 0.681371 0.731938i \(-0.261384\pi\)
0.681371 + 0.731938i \(0.261384\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −0.226539 −0.0120918
\(352\) 0 0
\(353\) −29.0584 −1.54662 −0.773311 0.634026i \(-0.781401\pi\)
−0.773311 + 0.634026i \(0.781401\pi\)
\(354\) 17.2365 0.916109
\(355\) 11.3041 0.599958
\(356\) 2.93362 0.155481
\(357\) 7.96291 0.421442
\(358\) 15.9071 0.840715
\(359\) 28.3224 1.49480 0.747400 0.664375i \(-0.231302\pi\)
0.747400 + 0.664375i \(0.231302\pi\)
\(360\) −0.154300 −0.00813231
\(361\) −16.0643 −0.845488
\(362\) −6.59938 −0.346856
\(363\) 0 0
\(364\) 0.0425741 0.00223149
\(365\) 1.98728 0.104019
\(366\) 23.3911 1.22267
\(367\) −27.5184 −1.43645 −0.718224 0.695812i \(-0.755044\pi\)
−0.718224 + 0.695812i \(0.755044\pi\)
\(368\) −6.95966 −0.362798
\(369\) 0.603884 0.0314370
\(370\) 8.48459 0.441093
\(371\) 14.1024 0.732158
\(372\) −10.1425 −0.525866
\(373\) 22.9306 1.18730 0.593650 0.804724i \(-0.297687\pi\)
0.593650 + 0.804724i \(0.297687\pi\)
\(374\) 0 0
\(375\) −1.68692 −0.0871122
\(376\) 4.59129 0.236778
\(377\) −0.0571673 −0.00294427
\(378\) −5.32105 −0.273685
\(379\) 15.4140 0.791764 0.395882 0.918301i \(-0.370439\pi\)
0.395882 + 0.918301i \(0.370439\pi\)
\(380\) 1.71340 0.0878954
\(381\) −3.08409 −0.158003
\(382\) 17.1690 0.878443
\(383\) −30.8927 −1.57854 −0.789270 0.614046i \(-0.789541\pi\)
−0.789270 + 0.614046i \(0.789541\pi\)
\(384\) −1.68692 −0.0860853
\(385\) 0 0
\(386\) 10.2185 0.520106
\(387\) 0.991306 0.0503909
\(388\) −17.4812 −0.887474
\(389\) 0.452651 0.0229503 0.0114752 0.999934i \(-0.496347\pi\)
0.0114752 + 0.999934i \(0.496347\pi\)
\(390\) 0.0718191 0.00363670
\(391\) −32.8523 −1.66141
\(392\) 1.00000 0.0505076
\(393\) −2.72738 −0.137578
\(394\) 7.33268 0.369415
\(395\) 8.87065 0.446331
\(396\) 0 0
\(397\) 24.1263 1.21086 0.605431 0.795897i \(-0.293001\pi\)
0.605431 + 0.795897i \(0.293001\pi\)
\(398\) −8.90352 −0.446293
\(399\) 2.89036 0.144699
\(400\) 1.00000 0.0500000
\(401\) −11.2261 −0.560603 −0.280301 0.959912i \(-0.590434\pi\)
−0.280301 + 0.959912i \(0.590434\pi\)
\(402\) −10.7895 −0.538134
\(403\) −0.255975 −0.0127510
\(404\) 5.70914 0.284040
\(405\) −8.51329 −0.423029
\(406\) −1.34277 −0.0666406
\(407\) 0 0
\(408\) −7.96291 −0.394223
\(409\) 30.4772 1.50700 0.753500 0.657448i \(-0.228364\pi\)
0.753500 + 0.657448i \(0.228364\pi\)
\(410\) −3.91371 −0.193284
\(411\) −17.3191 −0.854289
\(412\) 13.4052 0.660425
\(413\) 10.2177 0.502782
\(414\) 1.07387 0.0527780
\(415\) 10.0807 0.494841
\(416\) −0.0425741 −0.00208737
\(417\) −3.28541 −0.160887
\(418\) 0 0
\(419\) 11.8710 0.579937 0.289968 0.957036i \(-0.406355\pi\)
0.289968 + 0.957036i \(0.406355\pi\)
\(420\) 1.68692 0.0823133
\(421\) 23.7676 1.15836 0.579181 0.815199i \(-0.303373\pi\)
0.579181 + 0.815199i \(0.303373\pi\)
\(422\) −4.54767 −0.221377
\(423\) −0.708435 −0.0344453
\(424\) −14.1024 −0.684871
\(425\) 4.72038 0.228972
\(426\) −19.0691 −0.923900
\(427\) 13.8661 0.671029
\(428\) 15.9669 0.771790
\(429\) 0 0
\(430\) −6.42455 −0.309819
\(431\) −27.1939 −1.30989 −0.654943 0.755678i \(-0.727307\pi\)
−0.654943 + 0.755678i \(0.727307\pi\)
\(432\) 5.32105 0.256009
\(433\) −2.21065 −0.106237 −0.0531185 0.998588i \(-0.516916\pi\)
−0.0531185 + 0.998588i \(0.516916\pi\)
\(434\) −6.01245 −0.288607
\(435\) −2.26515 −0.108606
\(436\) −15.9501 −0.763871
\(437\) −11.9247 −0.570434
\(438\) −3.35239 −0.160183
\(439\) 6.47985 0.309267 0.154633 0.987972i \(-0.450580\pi\)
0.154633 + 0.987972i \(0.450580\pi\)
\(440\) 0 0
\(441\) −0.154300 −0.00734760
\(442\) −0.200966 −0.00955898
\(443\) 17.0792 0.811456 0.405728 0.913994i \(-0.367018\pi\)
0.405728 + 0.913994i \(0.367018\pi\)
\(444\) −14.3128 −0.679257
\(445\) 2.93362 0.139067
\(446\) 18.0702 0.855647
\(447\) −10.9130 −0.516169
\(448\) −1.00000 −0.0472456
\(449\) 13.1328 0.619774 0.309887 0.950773i \(-0.399709\pi\)
0.309887 + 0.950773i \(0.399709\pi\)
\(450\) −0.154300 −0.00727376
\(451\) 0 0
\(452\) 8.65087 0.406903
\(453\) −39.8298 −1.87137
\(454\) −0.970501 −0.0455479
\(455\) 0.0425741 0.00199591
\(456\) −2.89036 −0.135354
\(457\) −15.4508 −0.722758 −0.361379 0.932419i \(-0.617694\pi\)
−0.361379 + 0.932419i \(0.617694\pi\)
\(458\) −2.46292 −0.115085
\(459\) 25.1174 1.17238
\(460\) −6.95966 −0.324496
\(461\) 33.8352 1.57586 0.787932 0.615763i \(-0.211152\pi\)
0.787932 + 0.615763i \(0.211152\pi\)
\(462\) 0 0
\(463\) 31.7915 1.47748 0.738738 0.673993i \(-0.235422\pi\)
0.738738 + 0.673993i \(0.235422\pi\)
\(464\) 1.34277 0.0623366
\(465\) −10.1425 −0.470349
\(466\) 6.65041 0.308074
\(467\) −17.0295 −0.788032 −0.394016 0.919104i \(-0.628915\pi\)
−0.394016 + 0.919104i \(0.628915\pi\)
\(468\) 0.00656917 0.000303660 0
\(469\) −6.39600 −0.295340
\(470\) 4.59129 0.211781
\(471\) −19.3812 −0.893040
\(472\) −10.2177 −0.470309
\(473\) 0 0
\(474\) −14.9641 −0.687323
\(475\) 1.71340 0.0786160
\(476\) −4.72038 −0.216358
\(477\) 2.17599 0.0996317
\(478\) −21.0529 −0.962938
\(479\) −29.9718 −1.36945 −0.684724 0.728802i \(-0.740077\pi\)
−0.684724 + 0.728802i \(0.740077\pi\)
\(480\) −1.68692 −0.0769970
\(481\) −0.361224 −0.0164704
\(482\) 7.93244 0.361313
\(483\) −11.7404 −0.534207
\(484\) 0 0
\(485\) −17.4812 −0.793781
\(486\) −1.60191 −0.0726641
\(487\) 0.733047 0.0332175 0.0166088 0.999862i \(-0.494713\pi\)
0.0166088 + 0.999862i \(0.494713\pi\)
\(488\) −13.8661 −0.627690
\(489\) 23.8356 1.07788
\(490\) 1.00000 0.0451754
\(491\) −10.1494 −0.458035 −0.229018 0.973422i \(-0.573551\pi\)
−0.229018 + 0.973422i \(0.573551\pi\)
\(492\) 6.60212 0.297646
\(493\) 6.33839 0.285467
\(494\) −0.0729463 −0.00328201
\(495\) 0 0
\(496\) 6.01245 0.269967
\(497\) −11.3041 −0.507057
\(498\) −17.0053 −0.762025
\(499\) −11.1673 −0.499919 −0.249959 0.968256i \(-0.580417\pi\)
−0.249959 + 0.968256i \(0.580417\pi\)
\(500\) 1.00000 0.0447214
\(501\) −26.3185 −1.17582
\(502\) 18.9431 0.845471
\(503\) 7.46249 0.332736 0.166368 0.986064i \(-0.446796\pi\)
0.166368 + 0.986064i \(0.446796\pi\)
\(504\) 0.154300 0.00687305
\(505\) 5.70914 0.254053
\(506\) 0 0
\(507\) 21.9269 0.973808
\(508\) 1.82824 0.0811149
\(509\) −30.8240 −1.36625 −0.683125 0.730302i \(-0.739380\pi\)
−0.683125 + 0.730302i \(0.739380\pi\)
\(510\) −7.96291 −0.352603
\(511\) −1.98728 −0.0879122
\(512\) 1.00000 0.0441942
\(513\) 9.11707 0.402529
\(514\) −26.8598 −1.18474
\(515\) 13.4052 0.590702
\(516\) 10.8377 0.477103
\(517\) 0 0
\(518\) −8.48459 −0.372791
\(519\) 10.4787 0.459966
\(520\) −0.0425741 −0.00186700
\(521\) −14.5848 −0.638970 −0.319485 0.947591i \(-0.603510\pi\)
−0.319485 + 0.947591i \(0.603510\pi\)
\(522\) −0.207189 −0.00906843
\(523\) 9.58569 0.419153 0.209576 0.977792i \(-0.432792\pi\)
0.209576 + 0.977792i \(0.432792\pi\)
\(524\) 1.61678 0.0706294
\(525\) 1.68692 0.0736232
\(526\) 13.3750 0.583179
\(527\) 28.3811 1.23630
\(528\) 0 0
\(529\) 25.4369 1.10595
\(530\) −14.1024 −0.612567
\(531\) 1.57659 0.0684183
\(532\) −1.71340 −0.0742852
\(533\) 0.166623 0.00721723
\(534\) −4.94878 −0.214155
\(535\) 15.9669 0.690310
\(536\) 6.39600 0.276265
\(537\) −26.8340 −1.15797
\(538\) −4.05979 −0.175030
\(539\) 0 0
\(540\) 5.32105 0.228982
\(541\) −1.71177 −0.0735948 −0.0367974 0.999323i \(-0.511716\pi\)
−0.0367974 + 0.999323i \(0.511716\pi\)
\(542\) −23.0226 −0.988905
\(543\) 11.1326 0.477747
\(544\) 4.72038 0.202385
\(545\) −15.9501 −0.683227
\(546\) −0.0718191 −0.00307358
\(547\) −17.4770 −0.747262 −0.373631 0.927577i \(-0.621887\pi\)
−0.373631 + 0.927577i \(0.621887\pi\)
\(548\) 10.2667 0.438572
\(549\) 2.13954 0.0913133
\(550\) 0 0
\(551\) 2.30070 0.0980131
\(552\) 11.7404 0.499705
\(553\) −8.87065 −0.377218
\(554\) 22.2862 0.946850
\(555\) −14.3128 −0.607546
\(556\) 1.94758 0.0825957
\(557\) 25.6895 1.08850 0.544250 0.838923i \(-0.316814\pi\)
0.544250 + 0.838923i \(0.316814\pi\)
\(558\) −0.927720 −0.0392735
\(559\) 0.273520 0.0115686
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 2.24959 0.0948932
\(563\) 19.3434 0.815225 0.407613 0.913155i \(-0.366361\pi\)
0.407613 + 0.913155i \(0.366361\pi\)
\(564\) −7.74515 −0.326129
\(565\) 8.65087 0.363945
\(566\) 3.71034 0.155957
\(567\) 8.51329 0.357525
\(568\) 11.3041 0.474309
\(569\) −1.51921 −0.0636886 −0.0318443 0.999493i \(-0.510138\pi\)
−0.0318443 + 0.999493i \(0.510138\pi\)
\(570\) −2.89036 −0.121064
\(571\) −4.97687 −0.208276 −0.104138 0.994563i \(-0.533208\pi\)
−0.104138 + 0.994563i \(0.533208\pi\)
\(572\) 0 0
\(573\) −28.9628 −1.20994
\(574\) 3.91371 0.163355
\(575\) −6.95966 −0.290238
\(576\) −0.154300 −0.00642915
\(577\) −31.6102 −1.31595 −0.657976 0.753039i \(-0.728587\pi\)
−0.657976 + 0.753039i \(0.728587\pi\)
\(578\) 5.28199 0.219702
\(579\) −17.2377 −0.716376
\(580\) 1.34277 0.0557556
\(581\) −10.0807 −0.418217
\(582\) 29.4894 1.22238
\(583\) 0 0
\(584\) 1.98728 0.0822343
\(585\) 0.00656917 0.000271602 0
\(586\) −1.85170 −0.0764928
\(587\) 7.11885 0.293826 0.146913 0.989149i \(-0.453066\pi\)
0.146913 + 0.989149i \(0.453066\pi\)
\(588\) −1.68692 −0.0695674
\(589\) 10.3017 0.424475
\(590\) −10.2177 −0.420657
\(591\) −12.3697 −0.508820
\(592\) 8.48459 0.348714
\(593\) 18.4209 0.756456 0.378228 0.925713i \(-0.376534\pi\)
0.378228 + 0.925713i \(0.376534\pi\)
\(594\) 0 0
\(595\) −4.72038 −0.193517
\(596\) 6.46920 0.264989
\(597\) 15.0195 0.614709
\(598\) 0.296302 0.0121167
\(599\) −1.22319 −0.0499783 −0.0249892 0.999688i \(-0.507955\pi\)
−0.0249892 + 0.999688i \(0.507955\pi\)
\(600\) −1.68692 −0.0688682
\(601\) 20.6888 0.843916 0.421958 0.906615i \(-0.361343\pi\)
0.421958 + 0.906615i \(0.361343\pi\)
\(602\) 6.42455 0.261845
\(603\) −0.986901 −0.0401897
\(604\) 23.6110 0.960717
\(605\) 0 0
\(606\) −9.63087 −0.391227
\(607\) 10.3033 0.418199 0.209100 0.977894i \(-0.432947\pi\)
0.209100 + 0.977894i \(0.432947\pi\)
\(608\) 1.71340 0.0694874
\(609\) 2.26515 0.0917885
\(610\) −13.8661 −0.561423
\(611\) −0.195470 −0.00790788
\(612\) −0.728353 −0.0294419
\(613\) 42.7503 1.72667 0.863333 0.504634i \(-0.168373\pi\)
0.863333 + 0.504634i \(0.168373\pi\)
\(614\) 6.66980 0.269171
\(615\) 6.60212 0.266223
\(616\) 0 0
\(617\) 10.3582 0.417004 0.208502 0.978022i \(-0.433141\pi\)
0.208502 + 0.978022i \(0.433141\pi\)
\(618\) −22.6134 −0.909646
\(619\) 14.2901 0.574368 0.287184 0.957875i \(-0.407281\pi\)
0.287184 + 0.957875i \(0.407281\pi\)
\(620\) 6.01245 0.241466
\(621\) −37.0327 −1.48607
\(622\) 15.0969 0.605331
\(623\) −2.93362 −0.117533
\(624\) 0.0718191 0.00287507
\(625\) 1.00000 0.0400000
\(626\) 7.33744 0.293263
\(627\) 0 0
\(628\) 11.4891 0.458466
\(629\) 40.0505 1.59692
\(630\) 0.154300 0.00614745
\(631\) −11.6481 −0.463703 −0.231851 0.972751i \(-0.574478\pi\)
−0.231851 + 0.972751i \(0.574478\pi\)
\(632\) 8.87065 0.352855
\(633\) 7.67156 0.304917
\(634\) 22.4900 0.893190
\(635\) 1.82824 0.0725514
\(636\) 23.7896 0.943317
\(637\) −0.0425741 −0.00168685
\(638\) 0 0
\(639\) −1.74422 −0.0690001
\(640\) 1.00000 0.0395285
\(641\) 18.0247 0.711931 0.355965 0.934499i \(-0.384152\pi\)
0.355965 + 0.934499i \(0.384152\pi\)
\(642\) −26.9349 −1.06304
\(643\) −10.6620 −0.420469 −0.210235 0.977651i \(-0.567423\pi\)
−0.210235 + 0.977651i \(0.567423\pi\)
\(644\) 6.95966 0.274249
\(645\) 10.8377 0.426734
\(646\) 8.08788 0.318214
\(647\) −21.7314 −0.854348 −0.427174 0.904170i \(-0.640491\pi\)
−0.427174 + 0.904170i \(0.640491\pi\)
\(648\) −8.51329 −0.334434
\(649\) 0 0
\(650\) −0.0425741 −0.00166989
\(651\) 10.1425 0.397517
\(652\) −14.1296 −0.553359
\(653\) 1.06200 0.0415592 0.0207796 0.999784i \(-0.493385\pi\)
0.0207796 + 0.999784i \(0.493385\pi\)
\(654\) 26.9065 1.05213
\(655\) 1.61678 0.0631728
\(656\) −3.91371 −0.152805
\(657\) −0.306637 −0.0119630
\(658\) −4.59129 −0.178987
\(659\) −33.2641 −1.29578 −0.647892 0.761732i \(-0.724349\pi\)
−0.647892 + 0.761732i \(0.724349\pi\)
\(660\) 0 0
\(661\) −8.99126 −0.349719 −0.174860 0.984593i \(-0.555947\pi\)
−0.174860 + 0.984593i \(0.555947\pi\)
\(662\) 10.3154 0.400920
\(663\) 0.339014 0.0131662
\(664\) 10.0807 0.391206
\(665\) −1.71340 −0.0664427
\(666\) −1.30917 −0.0507293
\(667\) −9.34524 −0.361849
\(668\) 15.6015 0.603641
\(669\) −30.4829 −1.17854
\(670\) 6.39600 0.247099
\(671\) 0 0
\(672\) 1.68692 0.0650744
\(673\) 9.38976 0.361949 0.180974 0.983488i \(-0.442075\pi\)
0.180974 + 0.983488i \(0.442075\pi\)
\(674\) 27.2073 1.04799
\(675\) 5.32105 0.204807
\(676\) −12.9982 −0.499930
\(677\) −37.7060 −1.44916 −0.724579 0.689191i \(-0.757966\pi\)
−0.724579 + 0.689191i \(0.757966\pi\)
\(678\) −14.5933 −0.560453
\(679\) 17.4812 0.670868
\(680\) 4.72038 0.181018
\(681\) 1.63716 0.0627360
\(682\) 0 0
\(683\) 11.5317 0.441248 0.220624 0.975359i \(-0.429191\pi\)
0.220624 + 0.975359i \(0.429191\pi\)
\(684\) −0.264377 −0.0101087
\(685\) 10.2667 0.392271
\(686\) −1.00000 −0.0381802
\(687\) 4.15476 0.158514
\(688\) −6.42455 −0.244934
\(689\) 0.600395 0.0228732
\(690\) 11.7404 0.446949
\(691\) −12.7280 −0.484196 −0.242098 0.970252i \(-0.577835\pi\)
−0.242098 + 0.970252i \(0.577835\pi\)
\(692\) −6.21176 −0.236136
\(693\) 0 0
\(694\) 6.65299 0.252544
\(695\) 1.94758 0.0738758
\(696\) −2.26515 −0.0858603
\(697\) −18.4742 −0.699760
\(698\) 25.4581 0.963604
\(699\) −11.2187 −0.424331
\(700\) −1.00000 −0.0377964
\(701\) −11.1988 −0.422972 −0.211486 0.977381i \(-0.567830\pi\)
−0.211486 + 0.977381i \(0.567830\pi\)
\(702\) −0.226539 −0.00855017
\(703\) 14.5375 0.548291
\(704\) 0 0
\(705\) −7.74515 −0.291699
\(706\) −29.0584 −1.09363
\(707\) −5.70914 −0.214714
\(708\) 17.2365 0.647787
\(709\) −28.4744 −1.06938 −0.534688 0.845049i \(-0.679571\pi\)
−0.534688 + 0.845049i \(0.679571\pi\)
\(710\) 11.3041 0.424234
\(711\) −1.36874 −0.0513317
\(712\) 2.93362 0.109942
\(713\) −41.8447 −1.56709
\(714\) 7.96291 0.298004
\(715\) 0 0
\(716\) 15.9071 0.594475
\(717\) 35.5146 1.32632
\(718\) 28.3224 1.05698
\(719\) −44.5968 −1.66318 −0.831589 0.555391i \(-0.812569\pi\)
−0.831589 + 0.555391i \(0.812569\pi\)
\(720\) −0.154300 −0.00575041
\(721\) −13.4052 −0.499234
\(722\) −16.0643 −0.597850
\(723\) −13.3814 −0.497659
\(724\) −6.59938 −0.245264
\(725\) 1.34277 0.0498693
\(726\) 0 0
\(727\) 50.4565 1.87133 0.935664 0.352892i \(-0.114802\pi\)
0.935664 + 0.352892i \(0.114802\pi\)
\(728\) 0.0425741 0.00157790
\(729\) 28.2422 1.04601
\(730\) 1.98728 0.0735526
\(731\) −30.3263 −1.12166
\(732\) 23.3911 0.864558
\(733\) −38.0995 −1.40724 −0.703618 0.710578i \(-0.748434\pi\)
−0.703618 + 0.710578i \(0.748434\pi\)
\(734\) −27.5184 −1.01572
\(735\) −1.68692 −0.0622230
\(736\) −6.95966 −0.256537
\(737\) 0 0
\(738\) 0.603884 0.0222293
\(739\) −0.758286 −0.0278940 −0.0139470 0.999903i \(-0.504440\pi\)
−0.0139470 + 0.999903i \(0.504440\pi\)
\(740\) 8.48459 0.311900
\(741\) 0.123055 0.00452053
\(742\) 14.1024 0.517714
\(743\) 34.7125 1.27348 0.636739 0.771079i \(-0.280283\pi\)
0.636739 + 0.771079i \(0.280283\pi\)
\(744\) −10.1425 −0.371843
\(745\) 6.46920 0.237013
\(746\) 22.9306 0.839547
\(747\) −1.55544 −0.0569107
\(748\) 0 0
\(749\) −15.9669 −0.583418
\(750\) −1.68692 −0.0615976
\(751\) −46.0088 −1.67889 −0.839443 0.543448i \(-0.817119\pi\)
−0.839443 + 0.543448i \(0.817119\pi\)
\(752\) 4.59129 0.167427
\(753\) −31.9555 −1.16452
\(754\) −0.0571673 −0.00208191
\(755\) 23.6110 0.859291
\(756\) −5.32105 −0.193525
\(757\) −23.1505 −0.841420 −0.420710 0.907195i \(-0.638219\pi\)
−0.420710 + 0.907195i \(0.638219\pi\)
\(758\) 15.4140 0.559862
\(759\) 0 0
\(760\) 1.71340 0.0621514
\(761\) −7.19624 −0.260864 −0.130432 0.991457i \(-0.541636\pi\)
−0.130432 + 0.991457i \(0.541636\pi\)
\(762\) −3.08409 −0.111725
\(763\) 15.9501 0.577432
\(764\) 17.1690 0.621153
\(765\) −0.728353 −0.0263337
\(766\) −30.8927 −1.11620
\(767\) 0.435011 0.0157073
\(768\) −1.68692 −0.0608715
\(769\) −8.64296 −0.311673 −0.155837 0.987783i \(-0.549807\pi\)
−0.155837 + 0.987783i \(0.549807\pi\)
\(770\) 0 0
\(771\) 45.3104 1.63181
\(772\) 10.2185 0.367771
\(773\) −39.4190 −1.41780 −0.708901 0.705308i \(-0.750809\pi\)
−0.708901 + 0.705308i \(0.750809\pi\)
\(774\) 0.991306 0.0356318
\(775\) 6.01245 0.215974
\(776\) −17.4812 −0.627539
\(777\) 14.3128 0.513470
\(778\) 0.452651 0.0162283
\(779\) −6.70574 −0.240258
\(780\) 0.0718191 0.00257154
\(781\) 0 0
\(782\) −32.8523 −1.17479
\(783\) 7.14496 0.255340
\(784\) 1.00000 0.0357143
\(785\) 11.4891 0.410064
\(786\) −2.72738 −0.0972824
\(787\) 44.2814 1.57846 0.789230 0.614098i \(-0.210480\pi\)
0.789230 + 0.614098i \(0.210480\pi\)
\(788\) 7.33268 0.261216
\(789\) −22.5626 −0.803250
\(790\) 8.87065 0.315603
\(791\) −8.65087 −0.307589
\(792\) 0 0
\(793\) 0.590338 0.0209635
\(794\) 24.1263 0.856209
\(795\) 23.7896 0.843729
\(796\) −8.90352 −0.315577
\(797\) −20.9781 −0.743081 −0.371540 0.928417i \(-0.621170\pi\)
−0.371540 + 0.928417i \(0.621170\pi\)
\(798\) 2.89036 0.102318
\(799\) 21.6726 0.766723
\(800\) 1.00000 0.0353553
\(801\) −0.452656 −0.0159938
\(802\) −11.2261 −0.396406
\(803\) 0 0
\(804\) −10.7895 −0.380518
\(805\) 6.95966 0.245296
\(806\) −0.255975 −0.00901633
\(807\) 6.84854 0.241080
\(808\) 5.70914 0.200847
\(809\) 24.2672 0.853190 0.426595 0.904443i \(-0.359713\pi\)
0.426595 + 0.904443i \(0.359713\pi\)
\(810\) −8.51329 −0.299127
\(811\) 37.0315 1.30035 0.650175 0.759784i \(-0.274695\pi\)
0.650175 + 0.759784i \(0.274695\pi\)
\(812\) −1.34277 −0.0471221
\(813\) 38.8373 1.36208
\(814\) 0 0
\(815\) −14.1296 −0.494940
\(816\) −7.96291 −0.278757
\(817\) −11.0078 −0.385114
\(818\) 30.4772 1.06561
\(819\) −0.00656917 −0.000229546 0
\(820\) −3.91371 −0.136673
\(821\) −22.0348 −0.769019 −0.384509 0.923121i \(-0.625629\pi\)
−0.384509 + 0.923121i \(0.625629\pi\)
\(822\) −17.3191 −0.604074
\(823\) 25.6954 0.895684 0.447842 0.894113i \(-0.352193\pi\)
0.447842 + 0.894113i \(0.352193\pi\)
\(824\) 13.4052 0.466991
\(825\) 0 0
\(826\) 10.2177 0.355520
\(827\) 16.8897 0.587313 0.293657 0.955911i \(-0.405128\pi\)
0.293657 + 0.955911i \(0.405128\pi\)
\(828\) 1.07387 0.0373197
\(829\) −20.4637 −0.710732 −0.355366 0.934727i \(-0.615644\pi\)
−0.355366 + 0.934727i \(0.615644\pi\)
\(830\) 10.0807 0.349905
\(831\) −37.5951 −1.30416
\(832\) −0.0425741 −0.00147599
\(833\) 4.72038 0.163551
\(834\) −3.28541 −0.113764
\(835\) 15.6015 0.539913
\(836\) 0 0
\(837\) 31.9926 1.10583
\(838\) 11.8710 0.410077
\(839\) 34.0393 1.17517 0.587584 0.809163i \(-0.300079\pi\)
0.587584 + 0.809163i \(0.300079\pi\)
\(840\) 1.68692 0.0582043
\(841\) −27.1970 −0.937826
\(842\) 23.7676 0.819085
\(843\) −3.79488 −0.130703
\(844\) −4.54767 −0.156537
\(845\) −12.9982 −0.447151
\(846\) −0.708435 −0.0243565
\(847\) 0 0
\(848\) −14.1024 −0.484277
\(849\) −6.25906 −0.214810
\(850\) 4.72038 0.161908
\(851\) −59.0499 −2.02420
\(852\) −19.0691 −0.653296
\(853\) 20.6791 0.708040 0.354020 0.935238i \(-0.384814\pi\)
0.354020 + 0.935238i \(0.384814\pi\)
\(854\) 13.8661 0.474489
\(855\) −0.264377 −0.00904149
\(856\) 15.9669 0.545738
\(857\) 44.5639 1.52227 0.761136 0.648592i \(-0.224642\pi\)
0.761136 + 0.648592i \(0.224642\pi\)
\(858\) 0 0
\(859\) 0.992443 0.0338617 0.0169309 0.999857i \(-0.494610\pi\)
0.0169309 + 0.999857i \(0.494610\pi\)
\(860\) −6.42455 −0.219075
\(861\) −6.60212 −0.225000
\(862\) −27.1939 −0.926229
\(863\) 34.0157 1.15791 0.578954 0.815360i \(-0.303461\pi\)
0.578954 + 0.815360i \(0.303461\pi\)
\(864\) 5.32105 0.181026
\(865\) −6.21176 −0.211206
\(866\) −2.21065 −0.0751209
\(867\) −8.91030 −0.302610
\(868\) −6.01245 −0.204076
\(869\) 0 0
\(870\) −2.26515 −0.0767957
\(871\) −0.272304 −0.00922667
\(872\) −15.9501 −0.540138
\(873\) 2.69735 0.0912914
\(874\) −11.9247 −0.403358
\(875\) −1.00000 −0.0338062
\(876\) −3.35239 −0.113267
\(877\) −37.7370 −1.27429 −0.637144 0.770745i \(-0.719884\pi\)
−0.637144 + 0.770745i \(0.719884\pi\)
\(878\) 6.47985 0.218685
\(879\) 3.12366 0.105359
\(880\) 0 0
\(881\) 12.1850 0.410522 0.205261 0.978707i \(-0.434196\pi\)
0.205261 + 0.978707i \(0.434196\pi\)
\(882\) −0.154300 −0.00519554
\(883\) −23.9895 −0.807311 −0.403656 0.914911i \(-0.632261\pi\)
−0.403656 + 0.914911i \(0.632261\pi\)
\(884\) −0.200966 −0.00675922
\(885\) 17.2365 0.579398
\(886\) 17.0792 0.573786
\(887\) 42.9265 1.44133 0.720666 0.693282i \(-0.243836\pi\)
0.720666 + 0.693282i \(0.243836\pi\)
\(888\) −14.3128 −0.480307
\(889\) −1.82824 −0.0613171
\(890\) 2.93362 0.0983351
\(891\) 0 0
\(892\) 18.0702 0.605034
\(893\) 7.86670 0.263249
\(894\) −10.9130 −0.364986
\(895\) 15.9071 0.531715
\(896\) −1.00000 −0.0334077
\(897\) −0.499837 −0.0166891
\(898\) 13.1328 0.438246
\(899\) 8.07335 0.269261
\(900\) −0.154300 −0.00514332
\(901\) −66.5685 −2.21772
\(902\) 0 0
\(903\) −10.8377 −0.360656
\(904\) 8.65087 0.287724
\(905\) −6.59938 −0.219371
\(906\) −39.8298 −1.32326
\(907\) 17.8773 0.593606 0.296803 0.954939i \(-0.404079\pi\)
0.296803 + 0.954939i \(0.404079\pi\)
\(908\) −0.970501 −0.0322072
\(909\) −0.880919 −0.0292182
\(910\) 0.0425741 0.00141132
\(911\) −26.8557 −0.889769 −0.444885 0.895588i \(-0.646755\pi\)
−0.444885 + 0.895588i \(0.646755\pi\)
\(912\) −2.89036 −0.0957095
\(913\) 0 0
\(914\) −15.4508 −0.511067
\(915\) 23.3911 0.773284
\(916\) −2.46292 −0.0813773
\(917\) −1.61678 −0.0533908
\(918\) 25.1174 0.828997
\(919\) −16.2819 −0.537091 −0.268546 0.963267i \(-0.586543\pi\)
−0.268546 + 0.963267i \(0.586543\pi\)
\(920\) −6.95966 −0.229453
\(921\) −11.2514 −0.370747
\(922\) 33.8352 1.11430
\(923\) −0.481261 −0.0158409
\(924\) 0 0
\(925\) 8.48459 0.278972
\(926\) 31.7915 1.04473
\(927\) −2.06841 −0.0679356
\(928\) 1.34277 0.0440786
\(929\) −36.6780 −1.20337 −0.601684 0.798734i \(-0.705503\pi\)
−0.601684 + 0.798734i \(0.705503\pi\)
\(930\) −10.1425 −0.332587
\(931\) 1.71340 0.0561543
\(932\) 6.65041 0.217841
\(933\) −25.4673 −0.833761
\(934\) −17.0295 −0.557223
\(935\) 0 0
\(936\) 0.00656917 0.000214720 0
\(937\) 29.0963 0.950536 0.475268 0.879841i \(-0.342351\pi\)
0.475268 + 0.879841i \(0.342351\pi\)
\(938\) −6.39600 −0.208837
\(939\) −12.3777 −0.403930
\(940\) 4.59129 0.149751
\(941\) 5.63123 0.183573 0.0917865 0.995779i \(-0.470742\pi\)
0.0917865 + 0.995779i \(0.470742\pi\)
\(942\) −19.3812 −0.631475
\(943\) 27.2381 0.886995
\(944\) −10.2177 −0.332559
\(945\) −5.32105 −0.173094
\(946\) 0 0
\(947\) −25.4396 −0.826677 −0.413338 0.910577i \(-0.635637\pi\)
−0.413338 + 0.910577i \(0.635637\pi\)
\(948\) −14.9641 −0.486011
\(949\) −0.0846068 −0.00274645
\(950\) 1.71340 0.0555899
\(951\) −37.9388 −1.23025
\(952\) −4.72038 −0.152988
\(953\) 36.6858 1.18837 0.594184 0.804329i \(-0.297475\pi\)
0.594184 + 0.804329i \(0.297475\pi\)
\(954\) 2.17599 0.0704503
\(955\) 17.1690 0.555576
\(956\) −21.0529 −0.680900
\(957\) 0 0
\(958\) −29.9718 −0.968346
\(959\) −10.2667 −0.331529
\(960\) −1.68692 −0.0544451
\(961\) 5.14959 0.166116
\(962\) −0.361224 −0.0116463
\(963\) −2.46369 −0.0793913
\(964\) 7.93244 0.255487
\(965\) 10.2185 0.328944
\(966\) −11.7404 −0.377741
\(967\) −23.5537 −0.757438 −0.378719 0.925512i \(-0.623635\pi\)
−0.378719 + 0.925512i \(0.623635\pi\)
\(968\) 0 0
\(969\) −13.6436 −0.438296
\(970\) −17.4812 −0.561288
\(971\) 7.95989 0.255445 0.127722 0.991810i \(-0.459233\pi\)
0.127722 + 0.991810i \(0.459233\pi\)
\(972\) −1.60191 −0.0513813
\(973\) −1.94758 −0.0624365
\(974\) 0.733047 0.0234884
\(975\) 0.0718191 0.00230005
\(976\) −13.8661 −0.443844
\(977\) 16.8029 0.537574 0.268787 0.963200i \(-0.413377\pi\)
0.268787 + 0.963200i \(0.413377\pi\)
\(978\) 23.8356 0.762178
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 2.46109 0.0785767
\(982\) −10.1494 −0.323880
\(983\) −6.64006 −0.211785 −0.105893 0.994378i \(-0.533770\pi\)
−0.105893 + 0.994378i \(0.533770\pi\)
\(984\) 6.60212 0.210468
\(985\) 7.33268 0.233639
\(986\) 6.33839 0.201856
\(987\) 7.74515 0.246531
\(988\) −0.0729463 −0.00232073
\(989\) 44.7127 1.42178
\(990\) 0 0
\(991\) −49.2695 −1.56510 −0.782548 0.622590i \(-0.786080\pi\)
−0.782548 + 0.622590i \(0.786080\pi\)
\(992\) 6.01245 0.190896
\(993\) −17.4013 −0.552214
\(994\) −11.3041 −0.358544
\(995\) −8.90352 −0.282261
\(996\) −17.0053 −0.538833
\(997\) −32.5523 −1.03094 −0.515471 0.856907i \(-0.672383\pi\)
−0.515471 + 0.856907i \(0.672383\pi\)
\(998\) −11.1673 −0.353496
\(999\) 45.1469 1.42839
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 8470.2.a.dc.1.2 6
11.2 odd 10 770.2.n.j.631.1 yes 12
11.6 odd 10 770.2.n.j.421.1 12
11.10 odd 2 8470.2.a.cw.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.j.421.1 12 11.6 odd 10
770.2.n.j.631.1 yes 12 11.2 odd 10
8470.2.a.cw.1.2 6 11.10 odd 2
8470.2.a.dc.1.2 6 1.1 even 1 trivial