Properties

Label 8470.2.a.de.1.4
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: 6.6.13298000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 3 x^{3} + 26 x^{2} + 13 x - 1\)
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 1.4
Root \(2.42266\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.17142 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.17142 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.62777 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.17142 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.17142 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.62777 q^{9} +1.00000 q^{10} +1.17142 q^{12} +2.71483 q^{13} +1.00000 q^{14} +1.17142 q^{15} +1.00000 q^{16} -5.34260 q^{17} -1.62777 q^{18} +3.97605 q^{19} +1.00000 q^{20} +1.17142 q^{21} +5.49704 q^{23} +1.17142 q^{24} +1.00000 q^{25} +2.71483 q^{26} -5.42107 q^{27} +1.00000 q^{28} -4.99939 q^{29} +1.17142 q^{30} -2.87010 q^{31} +1.00000 q^{32} -5.34260 q^{34} +1.00000 q^{35} -1.62777 q^{36} +0.0405417 q^{37} +3.97605 q^{38} +3.18021 q^{39} +1.00000 q^{40} +5.13012 q^{41} +1.17142 q^{42} +12.2705 q^{43} -1.62777 q^{45} +5.49704 q^{46} +9.89983 q^{47} +1.17142 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.25844 q^{51} +2.71483 q^{52} +4.55880 q^{53} -5.42107 q^{54} +1.00000 q^{56} +4.65762 q^{57} -4.99939 q^{58} -0.584705 q^{59} +1.17142 q^{60} +7.25963 q^{61} -2.87010 q^{62} -1.62777 q^{63} +1.00000 q^{64} +2.71483 q^{65} -14.4148 q^{67} -5.34260 q^{68} +6.43935 q^{69} +1.00000 q^{70} +8.20815 q^{71} -1.62777 q^{72} +14.5897 q^{73} +0.0405417 q^{74} +1.17142 q^{75} +3.97605 q^{76} +3.18021 q^{78} +6.75342 q^{79} +1.00000 q^{80} -1.46704 q^{81} +5.13012 q^{82} -15.5794 q^{83} +1.17142 q^{84} -5.34260 q^{85} +12.2705 q^{86} -5.85640 q^{87} +2.73408 q^{89} -1.62777 q^{90} +2.71483 q^{91} +5.49704 q^{92} -3.36210 q^{93} +9.89983 q^{94} +3.97605 q^{95} +1.17142 q^{96} -1.32780 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + q^{3} + 6q^{4} + 6q^{5} + q^{6} + 6q^{7} + 6q^{8} + 15q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + q^{3} + 6q^{4} + 6q^{5} + q^{6} + 6q^{7} + 6q^{8} + 15q^{9} + 6q^{10} + q^{12} + 2q^{13} + 6q^{14} + q^{15} + 6q^{16} + 7q^{17} + 15q^{18} + 11q^{19} + 6q^{20} + q^{21} - 6q^{23} + q^{24} + 6q^{25} + 2q^{26} + 4q^{27} + 6q^{28} + 2q^{29} + q^{30} + 6q^{32} + 7q^{34} + 6q^{35} + 15q^{36} - 14q^{37} + 11q^{38} + 20q^{39} + 6q^{40} + 13q^{41} + q^{42} + 19q^{43} + 15q^{45} - 6q^{46} + 22q^{47} + q^{48} + 6q^{49} + 6q^{50} - 14q^{51} + 2q^{52} - 10q^{53} + 4q^{54} + 6q^{56} + 32q^{57} + 2q^{58} - 7q^{59} + q^{60} + 22q^{61} + 15q^{63} + 6q^{64} + 2q^{65} + 5q^{67} + 7q^{68} - 36q^{69} + 6q^{70} + 8q^{71} + 15q^{72} + 13q^{73} - 14q^{74} + q^{75} + 11q^{76} + 20q^{78} - 16q^{79} + 6q^{80} + 18q^{81} + 13q^{82} - 5q^{83} + q^{84} + 7q^{85} + 19q^{86} + 14q^{87} + q^{89} + 15q^{90} + 2q^{91} - 6q^{92} - 42q^{93} + 22q^{94} + 11q^{95} + q^{96} - 3q^{97} + 6q^{98} + O(q^{100}) \)

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
\(3\) 1.17142 0.676320 0.338160 0.941089i \(-0.390195\pi\)
0.338160 + 0.941089i \(0.390195\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.17142 0.478231
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −1.62777 −0.542591
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.17142 0.338160
\(13\) 2.71483 0.752958 0.376479 0.926425i \(-0.377135\pi\)
0.376479 + 0.926425i \(0.377135\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.17142 0.302460
\(16\) 1.00000 0.250000
\(17\) −5.34260 −1.29577 −0.647885 0.761738i \(-0.724346\pi\)
−0.647885 + 0.761738i \(0.724346\pi\)
\(18\) −1.62777 −0.383670
\(19\) 3.97605 0.912167 0.456084 0.889937i \(-0.349252\pi\)
0.456084 + 0.889937i \(0.349252\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.17142 0.255625
\(22\) 0 0
\(23\) 5.49704 1.14621 0.573106 0.819481i \(-0.305738\pi\)
0.573106 + 0.819481i \(0.305738\pi\)
\(24\) 1.17142 0.239115
\(25\) 1.00000 0.200000
\(26\) 2.71483 0.532422
\(27\) −5.42107 −1.04329
\(28\) 1.00000 0.188982
\(29\) −4.99939 −0.928364 −0.464182 0.885740i \(-0.653652\pi\)
−0.464182 + 0.885740i \(0.653652\pi\)
\(30\) 1.17142 0.213871
\(31\) −2.87010 −0.515485 −0.257743 0.966214i \(-0.582979\pi\)
−0.257743 + 0.966214i \(0.582979\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −5.34260 −0.916248
\(35\) 1.00000 0.169031
\(36\) −1.62777 −0.271295
\(37\) 0.0405417 0.00666501 0.00333251 0.999994i \(-0.498939\pi\)
0.00333251 + 0.999994i \(0.498939\pi\)
\(38\) 3.97605 0.645000
\(39\) 3.18021 0.509241
\(40\) 1.00000 0.158114
\(41\) 5.13012 0.801191 0.400595 0.916255i \(-0.368803\pi\)
0.400595 + 0.916255i \(0.368803\pi\)
\(42\) 1.17142 0.180754
\(43\) 12.2705 1.87123 0.935617 0.353017i \(-0.114844\pi\)
0.935617 + 0.353017i \(0.114844\pi\)
\(44\) 0 0
\(45\) −1.62777 −0.242654
\(46\) 5.49704 0.810495
\(47\) 9.89983 1.44404 0.722019 0.691873i \(-0.243214\pi\)
0.722019 + 0.691873i \(0.243214\pi\)
\(48\) 1.17142 0.169080
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −6.25844 −0.876356
\(52\) 2.71483 0.376479
\(53\) 4.55880 0.626199 0.313100 0.949720i \(-0.398633\pi\)
0.313100 + 0.949720i \(0.398633\pi\)
\(54\) −5.42107 −0.737714
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 4.65762 0.616917
\(58\) −4.99939 −0.656453
\(59\) −0.584705 −0.0761221 −0.0380610 0.999275i \(-0.512118\pi\)
−0.0380610 + 0.999275i \(0.512118\pi\)
\(60\) 1.17142 0.151230
\(61\) 7.25963 0.929500 0.464750 0.885442i \(-0.346144\pi\)
0.464750 + 0.885442i \(0.346144\pi\)
\(62\) −2.87010 −0.364503
\(63\) −1.62777 −0.205080
\(64\) 1.00000 0.125000
\(65\) 2.71483 0.336733
\(66\) 0 0
\(67\) −14.4148 −1.76105 −0.880526 0.473998i \(-0.842811\pi\)
−0.880526 + 0.473998i \(0.842811\pi\)
\(68\) −5.34260 −0.647885
\(69\) 6.43935 0.775207
\(70\) 1.00000 0.119523
\(71\) 8.20815 0.974128 0.487064 0.873366i \(-0.338068\pi\)
0.487064 + 0.873366i \(0.338068\pi\)
\(72\) −1.62777 −0.191835
\(73\) 14.5897 1.70760 0.853801 0.520600i \(-0.174292\pi\)
0.853801 + 0.520600i \(0.174292\pi\)
\(74\) 0.0405417 0.00471288
\(75\) 1.17142 0.135264
\(76\) 3.97605 0.456084
\(77\) 0 0
\(78\) 3.18021 0.360088
\(79\) 6.75342 0.759819 0.379910 0.925024i \(-0.375955\pi\)
0.379910 + 0.925024i \(0.375955\pi\)
\(80\) 1.00000 0.111803
\(81\) −1.46704 −0.163005
\(82\) 5.13012 0.566527
\(83\) −15.5794 −1.71006 −0.855031 0.518577i \(-0.826462\pi\)
−0.855031 + 0.518577i \(0.826462\pi\)
\(84\) 1.17142 0.127813
\(85\) −5.34260 −0.579486
\(86\) 12.2705 1.32316
\(87\) −5.85640 −0.627872
\(88\) 0 0
\(89\) 2.73408 0.289812 0.144906 0.989445i \(-0.453712\pi\)
0.144906 + 0.989445i \(0.453712\pi\)
\(90\) −1.62777 −0.171582
\(91\) 2.71483 0.284591
\(92\) 5.49704 0.573106
\(93\) −3.36210 −0.348633
\(94\) 9.89983 1.02109
\(95\) 3.97605 0.407934
\(96\) 1.17142 0.119558
\(97\) −1.32780 −0.134818 −0.0674088 0.997725i \(-0.521473\pi\)
−0.0674088 + 0.997725i \(0.521473\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 12.9533 1.28890 0.644449 0.764647i \(-0.277087\pi\)
0.644449 + 0.764647i \(0.277087\pi\)
\(102\) −6.25844 −0.619678
\(103\) 9.24014 0.910458 0.455229 0.890374i \(-0.349557\pi\)
0.455229 + 0.890374i \(0.349557\pi\)
\(104\) 2.71483 0.266211
\(105\) 1.17142 0.114319
\(106\) 4.55880 0.442790
\(107\) 10.0022 0.966947 0.483473 0.875359i \(-0.339375\pi\)
0.483473 + 0.875359i \(0.339375\pi\)
\(108\) −5.42107 −0.521643
\(109\) −0.535064 −0.0512498 −0.0256249 0.999672i \(-0.508158\pi\)
−0.0256249 + 0.999672i \(0.508158\pi\)
\(110\) 0 0
\(111\) 0.0474914 0.00450768
\(112\) 1.00000 0.0944911
\(113\) −1.55217 −0.146016 −0.0730078 0.997331i \(-0.523260\pi\)
−0.0730078 + 0.997331i \(0.523260\pi\)
\(114\) 4.65762 0.436226
\(115\) 5.49704 0.512602
\(116\) −4.99939 −0.464182
\(117\) −4.41912 −0.408548
\(118\) −0.584705 −0.0538264
\(119\) −5.34260 −0.489755
\(120\) 1.17142 0.106936
\(121\) 0 0
\(122\) 7.25963 0.657256
\(123\) 6.00954 0.541862
\(124\) −2.87010 −0.257743
\(125\) 1.00000 0.0894427
\(126\) −1.62777 −0.145013
\(127\) −17.2777 −1.53315 −0.766575 0.642155i \(-0.778041\pi\)
−0.766575 + 0.642155i \(0.778041\pi\)
\(128\) 1.00000 0.0883883
\(129\) 14.3739 1.26555
\(130\) 2.71483 0.238106
\(131\) 11.0180 0.962651 0.481325 0.876542i \(-0.340156\pi\)
0.481325 + 0.876542i \(0.340156\pi\)
\(132\) 0 0
\(133\) 3.97605 0.344767
\(134\) −14.4148 −1.24525
\(135\) −5.42107 −0.466572
\(136\) −5.34260 −0.458124
\(137\) 9.48971 0.810760 0.405380 0.914148i \(-0.367139\pi\)
0.405380 + 0.914148i \(0.367139\pi\)
\(138\) 6.43935 0.548154
\(139\) −10.8603 −0.921162 −0.460581 0.887618i \(-0.652359\pi\)
−0.460581 + 0.887618i \(0.652359\pi\)
\(140\) 1.00000 0.0845154
\(141\) 11.5969 0.976633
\(142\) 8.20815 0.688813
\(143\) 0 0
\(144\) −1.62777 −0.135648
\(145\) −4.99939 −0.415177
\(146\) 14.5897 1.20746
\(147\) 1.17142 0.0966172
\(148\) 0.0405417 0.00333251
\(149\) 12.3315 1.01024 0.505118 0.863050i \(-0.331449\pi\)
0.505118 + 0.863050i \(0.331449\pi\)
\(150\) 1.17142 0.0956462
\(151\) −8.70759 −0.708613 −0.354307 0.935129i \(-0.615283\pi\)
−0.354307 + 0.935129i \(0.615283\pi\)
\(152\) 3.97605 0.322500
\(153\) 8.69653 0.703073
\(154\) 0 0
\(155\) −2.87010 −0.230532
\(156\) 3.18021 0.254620
\(157\) −19.1275 −1.52654 −0.763269 0.646081i \(-0.776407\pi\)
−0.763269 + 0.646081i \(0.776407\pi\)
\(158\) 6.75342 0.537273
\(159\) 5.34028 0.423511
\(160\) 1.00000 0.0790569
\(161\) 5.49704 0.433228
\(162\) −1.46704 −0.115262
\(163\) −18.6244 −1.45878 −0.729389 0.684099i \(-0.760195\pi\)
−0.729389 + 0.684099i \(0.760195\pi\)
\(164\) 5.13012 0.400595
\(165\) 0 0
\(166\) −15.5794 −1.20920
\(167\) 1.66091 0.128525 0.0642624 0.997933i \(-0.479531\pi\)
0.0642624 + 0.997933i \(0.479531\pi\)
\(168\) 1.17142 0.0903771
\(169\) −5.62971 −0.433054
\(170\) −5.34260 −0.409759
\(171\) −6.47209 −0.494933
\(172\) 12.2705 0.935617
\(173\) −10.4055 −0.791115 −0.395557 0.918441i \(-0.629449\pi\)
−0.395557 + 0.918441i \(0.629449\pi\)
\(174\) −5.85640 −0.443972
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −0.684935 −0.0514829
\(178\) 2.73408 0.204928
\(179\) −25.4215 −1.90010 −0.950048 0.312104i \(-0.898966\pi\)
−0.950048 + 0.312104i \(0.898966\pi\)
\(180\) −1.62777 −0.121327
\(181\) −9.34641 −0.694713 −0.347357 0.937733i \(-0.612921\pi\)
−0.347357 + 0.937733i \(0.612921\pi\)
\(182\) 2.71483 0.201236
\(183\) 8.50409 0.628640
\(184\) 5.49704 0.405247
\(185\) 0.0405417 0.00298068
\(186\) −3.36210 −0.246521
\(187\) 0 0
\(188\) 9.89983 0.722019
\(189\) −5.42107 −0.394325
\(190\) 3.97605 0.288453
\(191\) 4.55912 0.329886 0.164943 0.986303i \(-0.447256\pi\)
0.164943 + 0.986303i \(0.447256\pi\)
\(192\) 1.17142 0.0845401
\(193\) 15.8146 1.13836 0.569181 0.822212i \(-0.307260\pi\)
0.569181 + 0.822212i \(0.307260\pi\)
\(194\) −1.32780 −0.0953304
\(195\) 3.18021 0.227739
\(196\) 1.00000 0.0714286
\(197\) 14.6162 1.04136 0.520679 0.853753i \(-0.325679\pi\)
0.520679 + 0.853753i \(0.325679\pi\)
\(198\) 0 0
\(199\) 16.4255 1.16438 0.582188 0.813054i \(-0.302197\pi\)
0.582188 + 0.813054i \(0.302197\pi\)
\(200\) 1.00000 0.0707107
\(201\) −16.8858 −1.19104
\(202\) 12.9533 0.911389
\(203\) −4.99939 −0.350889
\(204\) −6.25844 −0.438178
\(205\) 5.13012 0.358303
\(206\) 9.24014 0.643791
\(207\) −8.94793 −0.621924
\(208\) 2.71483 0.188239
\(209\) 0 0
\(210\) 1.17142 0.0808358
\(211\) −4.78958 −0.329728 −0.164864 0.986316i \(-0.552719\pi\)
−0.164864 + 0.986316i \(0.552719\pi\)
\(212\) 4.55880 0.313100
\(213\) 9.61520 0.658823
\(214\) 10.0022 0.683735
\(215\) 12.2705 0.836841
\(216\) −5.42107 −0.368857
\(217\) −2.87010 −0.194835
\(218\) −0.535064 −0.0362391
\(219\) 17.0907 1.15489
\(220\) 0 0
\(221\) −14.5042 −0.975661
\(222\) 0.0474914 0.00318741
\(223\) −23.2488 −1.55685 −0.778426 0.627737i \(-0.783981\pi\)
−0.778426 + 0.627737i \(0.783981\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.62777 −0.108518
\(226\) −1.55217 −0.103249
\(227\) −17.0503 −1.13167 −0.565834 0.824519i \(-0.691446\pi\)
−0.565834 + 0.824519i \(0.691446\pi\)
\(228\) 4.65762 0.308459
\(229\) 10.5051 0.694196 0.347098 0.937829i \(-0.387167\pi\)
0.347098 + 0.937829i \(0.387167\pi\)
\(230\) 5.49704 0.362464
\(231\) 0 0
\(232\) −4.99939 −0.328226
\(233\) 21.0739 1.38060 0.690300 0.723523i \(-0.257479\pi\)
0.690300 + 0.723523i \(0.257479\pi\)
\(234\) −4.41912 −0.288887
\(235\) 9.89983 0.645794
\(236\) −0.584705 −0.0380610
\(237\) 7.91110 0.513881
\(238\) −5.34260 −0.346309
\(239\) 1.21833 0.0788074 0.0394037 0.999223i \(-0.487454\pi\)
0.0394037 + 0.999223i \(0.487454\pi\)
\(240\) 1.17142 0.0756149
\(241\) −1.70364 −0.109741 −0.0548706 0.998493i \(-0.517475\pi\)
−0.0548706 + 0.998493i \(0.517475\pi\)
\(242\) 0 0
\(243\) 14.5447 0.933042
\(244\) 7.25963 0.464750
\(245\) 1.00000 0.0638877
\(246\) 6.00954 0.383154
\(247\) 10.7943 0.686824
\(248\) −2.87010 −0.182252
\(249\) −18.2500 −1.15655
\(250\) 1.00000 0.0632456
\(251\) −19.2825 −1.21710 −0.608551 0.793515i \(-0.708249\pi\)
−0.608551 + 0.793515i \(0.708249\pi\)
\(252\) −1.62777 −0.102540
\(253\) 0 0
\(254\) −17.2777 −1.08410
\(255\) −6.25844 −0.391918
\(256\) 1.00000 0.0625000
\(257\) −9.96978 −0.621898 −0.310949 0.950427i \(-0.600647\pi\)
−0.310949 + 0.950427i \(0.600647\pi\)
\(258\) 14.3739 0.894881
\(259\) 0.0405417 0.00251914
\(260\) 2.71483 0.168367
\(261\) 8.13787 0.503722
\(262\) 11.0180 0.680697
\(263\) 15.3900 0.948985 0.474493 0.880260i \(-0.342632\pi\)
0.474493 + 0.880260i \(0.342632\pi\)
\(264\) 0 0
\(265\) 4.55880 0.280045
\(266\) 3.97605 0.243787
\(267\) 3.20276 0.196006
\(268\) −14.4148 −0.880526
\(269\) −16.6521 −1.01530 −0.507649 0.861564i \(-0.669485\pi\)
−0.507649 + 0.861564i \(0.669485\pi\)
\(270\) −5.42107 −0.329916
\(271\) −2.85449 −0.173398 −0.0866989 0.996235i \(-0.527632\pi\)
−0.0866989 + 0.996235i \(0.527632\pi\)
\(272\) −5.34260 −0.323943
\(273\) 3.18021 0.192475
\(274\) 9.48971 0.573294
\(275\) 0 0
\(276\) 6.43935 0.387603
\(277\) 25.0469 1.50492 0.752462 0.658636i \(-0.228866\pi\)
0.752462 + 0.658636i \(0.228866\pi\)
\(278\) −10.8603 −0.651360
\(279\) 4.67187 0.279698
\(280\) 1.00000 0.0597614
\(281\) 4.37825 0.261184 0.130592 0.991436i \(-0.458312\pi\)
0.130592 + 0.991436i \(0.458312\pi\)
\(282\) 11.5969 0.690584
\(283\) −7.46206 −0.443573 −0.221787 0.975095i \(-0.571189\pi\)
−0.221787 + 0.975095i \(0.571189\pi\)
\(284\) 8.20815 0.487064
\(285\) 4.65762 0.275894
\(286\) 0 0
\(287\) 5.13012 0.302822
\(288\) −1.62777 −0.0959174
\(289\) 11.5434 0.679022
\(290\) −4.99939 −0.293575
\(291\) −1.55541 −0.0911799
\(292\) 14.5897 0.853801
\(293\) 5.85672 0.342153 0.171077 0.985258i \(-0.445275\pi\)
0.171077 + 0.985258i \(0.445275\pi\)
\(294\) 1.17142 0.0683187
\(295\) −0.584705 −0.0340428
\(296\) 0.0405417 0.00235644
\(297\) 0 0
\(298\) 12.3315 0.714344
\(299\) 14.9235 0.863050
\(300\) 1.17142 0.0676320
\(301\) 12.2705 0.707260
\(302\) −8.70759 −0.501065
\(303\) 15.1737 0.871709
\(304\) 3.97605 0.228042
\(305\) 7.25963 0.415685
\(306\) 8.69653 0.497148
\(307\) 34.7874 1.98542 0.992712 0.120513i \(-0.0384540\pi\)
0.992712 + 0.120513i \(0.0384540\pi\)
\(308\) 0 0
\(309\) 10.8241 0.615761
\(310\) −2.87010 −0.163011
\(311\) 19.5779 1.11016 0.555082 0.831796i \(-0.312687\pi\)
0.555082 + 0.831796i \(0.312687\pi\)
\(312\) 3.18021 0.180044
\(313\) −12.3630 −0.698796 −0.349398 0.936974i \(-0.613614\pi\)
−0.349398 + 0.936974i \(0.613614\pi\)
\(314\) −19.1275 −1.07942
\(315\) −1.62777 −0.0917146
\(316\) 6.75342 0.379910
\(317\) 5.15560 0.289568 0.144784 0.989463i \(-0.453751\pi\)
0.144784 + 0.989463i \(0.453751\pi\)
\(318\) 5.34028 0.299468
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 11.7168 0.653966
\(322\) 5.49704 0.306338
\(323\) −21.2424 −1.18196
\(324\) −1.46704 −0.0815024
\(325\) 2.71483 0.150592
\(326\) −18.6244 −1.03151
\(327\) −0.626785 −0.0346613
\(328\) 5.13012 0.283264
\(329\) 9.89983 0.545795
\(330\) 0 0
\(331\) −32.5061 −1.78670 −0.893349 0.449364i \(-0.851651\pi\)
−0.893349 + 0.449364i \(0.851651\pi\)
\(332\) −15.5794 −0.855031
\(333\) −0.0659926 −0.00361637
\(334\) 1.66091 0.0908807
\(335\) −14.4148 −0.787566
\(336\) 1.17142 0.0639063
\(337\) −27.1277 −1.47774 −0.738870 0.673848i \(-0.764640\pi\)
−0.738870 + 0.673848i \(0.764640\pi\)
\(338\) −5.62971 −0.306216
\(339\) −1.81824 −0.0987534
\(340\) −5.34260 −0.289743
\(341\) 0 0
\(342\) −6.47209 −0.349971
\(343\) 1.00000 0.0539949
\(344\) 12.2705 0.661581
\(345\) 6.43935 0.346683
\(346\) −10.4055 −0.559403
\(347\) 11.1959 0.601027 0.300514 0.953778i \(-0.402842\pi\)
0.300514 + 0.953778i \(0.402842\pi\)
\(348\) −5.85640 −0.313936
\(349\) −28.5841 −1.53007 −0.765035 0.643989i \(-0.777278\pi\)
−0.765035 + 0.643989i \(0.777278\pi\)
\(350\) 1.00000 0.0534522
\(351\) −14.7173 −0.785550
\(352\) 0 0
\(353\) 4.00343 0.213081 0.106541 0.994308i \(-0.466023\pi\)
0.106541 + 0.994308i \(0.466023\pi\)
\(354\) −0.684935 −0.0364039
\(355\) 8.20815 0.435643
\(356\) 2.73408 0.144906
\(357\) −6.25844 −0.331232
\(358\) −25.4215 −1.34357
\(359\) −9.77677 −0.515998 −0.257999 0.966145i \(-0.583063\pi\)
−0.257999 + 0.966145i \(0.583063\pi\)
\(360\) −1.62777 −0.0857911
\(361\) −3.19107 −0.167951
\(362\) −9.34641 −0.491237
\(363\) 0 0
\(364\) 2.71483 0.142296
\(365\) 14.5897 0.763662
\(366\) 8.50409 0.444516
\(367\) −13.1201 −0.684865 −0.342432 0.939543i \(-0.611251\pi\)
−0.342432 + 0.939543i \(0.611251\pi\)
\(368\) 5.49704 0.286553
\(369\) −8.35067 −0.434719
\(370\) 0.0405417 0.00210766
\(371\) 4.55880 0.236681
\(372\) −3.36210 −0.174317
\(373\) −15.2552 −0.789886 −0.394943 0.918706i \(-0.629236\pi\)
−0.394943 + 0.918706i \(0.629236\pi\)
\(374\) 0 0
\(375\) 1.17142 0.0604919
\(376\) 9.89983 0.510545
\(377\) −13.5725 −0.699019
\(378\) −5.42107 −0.278830
\(379\) 26.6376 1.36828 0.684141 0.729349i \(-0.260177\pi\)
0.684141 + 0.729349i \(0.260177\pi\)
\(380\) 3.97605 0.203967
\(381\) −20.2395 −1.03690
\(382\) 4.55912 0.233265
\(383\) 6.81998 0.348485 0.174242 0.984703i \(-0.444252\pi\)
0.174242 + 0.984703i \(0.444252\pi\)
\(384\) 1.17142 0.0597788
\(385\) 0 0
\(386\) 15.8146 0.804944
\(387\) −19.9736 −1.01531
\(388\) −1.32780 −0.0674088
\(389\) 23.0806 1.17023 0.585117 0.810949i \(-0.301048\pi\)
0.585117 + 0.810949i \(0.301048\pi\)
\(390\) 3.18021 0.161036
\(391\) −29.3685 −1.48523
\(392\) 1.00000 0.0505076
\(393\) 12.9068 0.651060
\(394\) 14.6162 0.736351
\(395\) 6.75342 0.339801
\(396\) 0 0
\(397\) −15.6435 −0.785123 −0.392561 0.919726i \(-0.628411\pi\)
−0.392561 + 0.919726i \(0.628411\pi\)
\(398\) 16.4255 0.823338
\(399\) 4.65762 0.233173
\(400\) 1.00000 0.0500000
\(401\) −1.51369 −0.0755900 −0.0377950 0.999286i \(-0.512033\pi\)
−0.0377950 + 0.999286i \(0.512033\pi\)
\(402\) −16.8858 −0.842189
\(403\) −7.79183 −0.388139
\(404\) 12.9533 0.644449
\(405\) −1.46704 −0.0728980
\(406\) −4.99939 −0.248116
\(407\) 0 0
\(408\) −6.25844 −0.309839
\(409\) −14.2110 −0.702689 −0.351345 0.936246i \(-0.614275\pi\)
−0.351345 + 0.936246i \(0.614275\pi\)
\(410\) 5.13012 0.253359
\(411\) 11.1164 0.548334
\(412\) 9.24014 0.455229
\(413\) −0.584705 −0.0287714
\(414\) −8.94793 −0.439767
\(415\) −15.5794 −0.764763
\(416\) 2.71483 0.133105
\(417\) −12.7220 −0.623001
\(418\) 0 0
\(419\) 6.61451 0.323140 0.161570 0.986861i \(-0.448344\pi\)
0.161570 + 0.986861i \(0.448344\pi\)
\(420\) 1.17142 0.0571595
\(421\) −38.2913 −1.86620 −0.933101 0.359614i \(-0.882908\pi\)
−0.933101 + 0.359614i \(0.882908\pi\)
\(422\) −4.78958 −0.233153
\(423\) −16.1147 −0.783522
\(424\) 4.55880 0.221395
\(425\) −5.34260 −0.259154
\(426\) 9.61520 0.465858
\(427\) 7.25963 0.351318
\(428\) 10.0022 0.483473
\(429\) 0 0
\(430\) 12.2705 0.591736
\(431\) 6.73123 0.324232 0.162116 0.986772i \(-0.448168\pi\)
0.162116 + 0.986772i \(0.448168\pi\)
\(432\) −5.42107 −0.260821
\(433\) −25.7699 −1.23842 −0.619212 0.785224i \(-0.712548\pi\)
−0.619212 + 0.785224i \(0.712548\pi\)
\(434\) −2.87010 −0.137769
\(435\) −5.85640 −0.280793
\(436\) −0.535064 −0.0256249
\(437\) 21.8565 1.04554
\(438\) 17.0907 0.816627
\(439\) 13.8131 0.659262 0.329631 0.944110i \(-0.393076\pi\)
0.329631 + 0.944110i \(0.393076\pi\)
\(440\) 0 0
\(441\) −1.62777 −0.0775129
\(442\) −14.5042 −0.689896
\(443\) −28.1206 −1.33605 −0.668025 0.744139i \(-0.732860\pi\)
−0.668025 + 0.744139i \(0.732860\pi\)
\(444\) 0.0474914 0.00225384
\(445\) 2.73408 0.129608
\(446\) −23.2488 −1.10086
\(447\) 14.4454 0.683243
\(448\) 1.00000 0.0472456
\(449\) 27.8576 1.31468 0.657341 0.753593i \(-0.271681\pi\)
0.657341 + 0.753593i \(0.271681\pi\)
\(450\) −1.62777 −0.0767339
\(451\) 0 0
\(452\) −1.55217 −0.0730078
\(453\) −10.2003 −0.479250
\(454\) −17.0503 −0.800210
\(455\) 2.71483 0.127273
\(456\) 4.65762 0.218113
\(457\) −38.0839 −1.78149 −0.890745 0.454504i \(-0.849817\pi\)
−0.890745 + 0.454504i \(0.849817\pi\)
\(458\) 10.5051 0.490871
\(459\) 28.9626 1.35186
\(460\) 5.49704 0.256301
\(461\) 13.9251 0.648556 0.324278 0.945962i \(-0.394879\pi\)
0.324278 + 0.945962i \(0.394879\pi\)
\(462\) 0 0
\(463\) −2.93872 −0.136574 −0.0682870 0.997666i \(-0.521753\pi\)
−0.0682870 + 0.997666i \(0.521753\pi\)
\(464\) −4.99939 −0.232091
\(465\) −3.36210 −0.155914
\(466\) 21.0739 0.976231
\(467\) −31.8095 −1.47197 −0.735985 0.676998i \(-0.763280\pi\)
−0.735985 + 0.676998i \(0.763280\pi\)
\(468\) −4.41912 −0.204274
\(469\) −14.4148 −0.665615
\(470\) 9.89983 0.456645
\(471\) −22.4063 −1.03243
\(472\) −0.584705 −0.0269132
\(473\) 0 0
\(474\) 7.91110 0.363369
\(475\) 3.97605 0.182433
\(476\) −5.34260 −0.244878
\(477\) −7.42069 −0.339770
\(478\) 1.21833 0.0557252
\(479\) 37.7725 1.72587 0.862936 0.505314i \(-0.168623\pi\)
0.862936 + 0.505314i \(0.168623\pi\)
\(480\) 1.17142 0.0534678
\(481\) 0.110064 0.00501847
\(482\) −1.70364 −0.0775988
\(483\) 6.43935 0.293001
\(484\) 0 0
\(485\) −1.32780 −0.0602922
\(486\) 14.5447 0.659760
\(487\) −9.53146 −0.431912 −0.215956 0.976403i \(-0.569287\pi\)
−0.215956 + 0.976403i \(0.569287\pi\)
\(488\) 7.25963 0.328628
\(489\) −21.8170 −0.986601
\(490\) 1.00000 0.0451754
\(491\) −22.2193 −1.00274 −0.501371 0.865232i \(-0.667171\pi\)
−0.501371 + 0.865232i \(0.667171\pi\)
\(492\) 6.00954 0.270931
\(493\) 26.7098 1.20295
\(494\) 10.7943 0.485658
\(495\) 0 0
\(496\) −2.87010 −0.128871
\(497\) 8.20815 0.368186
\(498\) −18.2500 −0.817804
\(499\) −10.8366 −0.485112 −0.242556 0.970137i \(-0.577986\pi\)
−0.242556 + 0.970137i \(0.577986\pi\)
\(500\) 1.00000 0.0447214
\(501\) 1.94562 0.0869239
\(502\) −19.2825 −0.860621
\(503\) −12.2494 −0.546173 −0.273087 0.961989i \(-0.588045\pi\)
−0.273087 + 0.961989i \(0.588045\pi\)
\(504\) −1.62777 −0.0725067
\(505\) 12.9533 0.576413
\(506\) 0 0
\(507\) −6.59476 −0.292884
\(508\) −17.2777 −0.766575
\(509\) −0.295071 −0.0130788 −0.00653939 0.999979i \(-0.502082\pi\)
−0.00653939 + 0.999979i \(0.502082\pi\)
\(510\) −6.25844 −0.277128
\(511\) 14.5897 0.645413
\(512\) 1.00000 0.0441942
\(513\) −21.5544 −0.951651
\(514\) −9.96978 −0.439748
\(515\) 9.24014 0.407169
\(516\) 14.3739 0.632777
\(517\) 0 0
\(518\) 0.0405417 0.00178130
\(519\) −12.1892 −0.535047
\(520\) 2.71483 0.119053
\(521\) 27.5697 1.20785 0.603925 0.797041i \(-0.293603\pi\)
0.603925 + 0.797041i \(0.293603\pi\)
\(522\) 8.13787 0.356185
\(523\) 24.2495 1.06036 0.530178 0.847887i \(-0.322125\pi\)
0.530178 + 0.847887i \(0.322125\pi\)
\(524\) 11.0180 0.481325
\(525\) 1.17142 0.0511250
\(526\) 15.3900 0.671034
\(527\) 15.3338 0.667951
\(528\) 0 0
\(529\) 7.21747 0.313803
\(530\) 4.55880 0.198022
\(531\) 0.951766 0.0413031
\(532\) 3.97605 0.172383
\(533\) 13.9274 0.603263
\(534\) 3.20276 0.138597
\(535\) 10.0022 0.432432
\(536\) −14.4148 −0.622626
\(537\) −29.7793 −1.28507
\(538\) −16.6521 −0.717924
\(539\) 0 0
\(540\) −5.42107 −0.233286
\(541\) 19.6382 0.844310 0.422155 0.906524i \(-0.361274\pi\)
0.422155 + 0.906524i \(0.361274\pi\)
\(542\) −2.85449 −0.122611
\(543\) −10.9486 −0.469849
\(544\) −5.34260 −0.229062
\(545\) −0.535064 −0.0229196
\(546\) 3.18021 0.136100
\(547\) −0.502119 −0.0214691 −0.0107345 0.999942i \(-0.503417\pi\)
−0.0107345 + 0.999942i \(0.503417\pi\)
\(548\) 9.48971 0.405380
\(549\) −11.8170 −0.504338
\(550\) 0 0
\(551\) −19.8778 −0.846823
\(552\) 6.43935 0.274077
\(553\) 6.75342 0.287185
\(554\) 25.0469 1.06414
\(555\) 0.0474914 0.00201590
\(556\) −10.8603 −0.460581
\(557\) 2.50897 0.106308 0.0531541 0.998586i \(-0.483073\pi\)
0.0531541 + 0.998586i \(0.483073\pi\)
\(558\) 4.67187 0.197776
\(559\) 33.3123 1.40896
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 4.37825 0.184685
\(563\) −17.3706 −0.732086 −0.366043 0.930598i \(-0.619288\pi\)
−0.366043 + 0.930598i \(0.619288\pi\)
\(564\) 11.5969 0.488317
\(565\) −1.55217 −0.0653002
\(566\) −7.46206 −0.313654
\(567\) −1.46704 −0.0616100
\(568\) 8.20815 0.344406
\(569\) −31.7612 −1.33150 −0.665750 0.746175i \(-0.731888\pi\)
−0.665750 + 0.746175i \(0.731888\pi\)
\(570\) 4.65762 0.195086
\(571\) 7.43290 0.311057 0.155529 0.987831i \(-0.450292\pi\)
0.155529 + 0.987831i \(0.450292\pi\)
\(572\) 0 0
\(573\) 5.34065 0.223109
\(574\) 5.13012 0.214127
\(575\) 5.49704 0.229243
\(576\) −1.62777 −0.0678238
\(577\) −18.4342 −0.767426 −0.383713 0.923452i \(-0.625355\pi\)
−0.383713 + 0.923452i \(0.625355\pi\)
\(578\) 11.5434 0.480141
\(579\) 18.5256 0.769898
\(580\) −4.99939 −0.207589
\(581\) −15.5794 −0.646343
\(582\) −1.55541 −0.0644739
\(583\) 0 0
\(584\) 14.5897 0.603728
\(585\) −4.41912 −0.182708
\(586\) 5.85672 0.241939
\(587\) 10.1616 0.419413 0.209707 0.977764i \(-0.432749\pi\)
0.209707 + 0.977764i \(0.432749\pi\)
\(588\) 1.17142 0.0483086
\(589\) −11.4117 −0.470209
\(590\) −0.584705 −0.0240719
\(591\) 17.1217 0.704292
\(592\) 0.0405417 0.00166625
\(593\) −22.2216 −0.912532 −0.456266 0.889844i \(-0.650813\pi\)
−0.456266 + 0.889844i \(0.650813\pi\)
\(594\) 0 0
\(595\) −5.34260 −0.219025
\(596\) 12.3315 0.505118
\(597\) 19.2412 0.787491
\(598\) 14.9235 0.610268
\(599\) 9.45290 0.386235 0.193117 0.981176i \(-0.438140\pi\)
0.193117 + 0.981176i \(0.438140\pi\)
\(600\) 1.17142 0.0478231
\(601\) 26.9039 1.09743 0.548716 0.836009i \(-0.315117\pi\)
0.548716 + 0.836009i \(0.315117\pi\)
\(602\) 12.2705 0.500108
\(603\) 23.4641 0.955530
\(604\) −8.70759 −0.354307
\(605\) 0 0
\(606\) 15.1737 0.616391
\(607\) −17.1534 −0.696237 −0.348118 0.937451i \(-0.613179\pi\)
−0.348118 + 0.937451i \(0.613179\pi\)
\(608\) 3.97605 0.161250
\(609\) −5.85640 −0.237313
\(610\) 7.25963 0.293934
\(611\) 26.8763 1.08730
\(612\) 8.69653 0.351537
\(613\) 37.6479 1.52059 0.760293 0.649580i \(-0.225055\pi\)
0.760293 + 0.649580i \(0.225055\pi\)
\(614\) 34.7874 1.40391
\(615\) 6.00954 0.242328
\(616\) 0 0
\(617\) 22.0199 0.886488 0.443244 0.896401i \(-0.353828\pi\)
0.443244 + 0.896401i \(0.353828\pi\)
\(618\) 10.8241 0.435409
\(619\) 30.3761 1.22092 0.610459 0.792047i \(-0.290985\pi\)
0.610459 + 0.792047i \(0.290985\pi\)
\(620\) −2.87010 −0.115266
\(621\) −29.7999 −1.19583
\(622\) 19.5779 0.785004
\(623\) 2.73408 0.109539
\(624\) 3.18021 0.127310
\(625\) 1.00000 0.0400000
\(626\) −12.3630 −0.494123
\(627\) 0 0
\(628\) −19.1275 −0.763269
\(629\) −0.216598 −0.00863633
\(630\) −1.62777 −0.0648520
\(631\) 46.9052 1.86727 0.933634 0.358229i \(-0.116619\pi\)
0.933634 + 0.358229i \(0.116619\pi\)
\(632\) 6.75342 0.268637
\(633\) −5.61062 −0.223002
\(634\) 5.15560 0.204755
\(635\) −17.2777 −0.685645
\(636\) 5.34028 0.211756
\(637\) 2.71483 0.107565
\(638\) 0 0
\(639\) −13.3610 −0.528553
\(640\) 1.00000 0.0395285
\(641\) −25.1252 −0.992385 −0.496192 0.868213i \(-0.665269\pi\)
−0.496192 + 0.868213i \(0.665269\pi\)
\(642\) 11.7168 0.462424
\(643\) −24.8911 −0.981608 −0.490804 0.871270i \(-0.663297\pi\)
−0.490804 + 0.871270i \(0.663297\pi\)
\(644\) 5.49704 0.216614
\(645\) 14.3739 0.565973
\(646\) −21.2424 −0.835772
\(647\) 40.7004 1.60010 0.800049 0.599934i \(-0.204807\pi\)
0.800049 + 0.599934i \(0.204807\pi\)
\(648\) −1.46704 −0.0576309
\(649\) 0 0
\(650\) 2.71483 0.106484
\(651\) −3.36210 −0.131771
\(652\) −18.6244 −0.729389
\(653\) −13.3072 −0.520752 −0.260376 0.965507i \(-0.583847\pi\)
−0.260376 + 0.965507i \(0.583847\pi\)
\(654\) −0.626785 −0.0245093
\(655\) 11.0180 0.430510
\(656\) 5.13012 0.200298
\(657\) −23.7488 −0.926528
\(658\) 9.89983 0.385936
\(659\) −17.1669 −0.668726 −0.334363 0.942444i \(-0.608521\pi\)
−0.334363 + 0.942444i \(0.608521\pi\)
\(660\) 0 0
\(661\) 1.82979 0.0711705 0.0355853 0.999367i \(-0.488670\pi\)
0.0355853 + 0.999367i \(0.488670\pi\)
\(662\) −32.5061 −1.26339
\(663\) −16.9906 −0.659859
\(664\) −15.5794 −0.604598
\(665\) 3.97605 0.154184
\(666\) −0.0659926 −0.00255716
\(667\) −27.4819 −1.06410
\(668\) 1.66091 0.0642624
\(669\) −27.2341 −1.05293
\(670\) −14.4148 −0.556894
\(671\) 0 0
\(672\) 1.17142 0.0451886
\(673\) −19.8563 −0.765402 −0.382701 0.923872i \(-0.625006\pi\)
−0.382701 + 0.923872i \(0.625006\pi\)
\(674\) −27.1277 −1.04492
\(675\) −5.42107 −0.208657
\(676\) −5.62971 −0.216527
\(677\) −24.8180 −0.953835 −0.476918 0.878948i \(-0.658246\pi\)
−0.476918 + 0.878948i \(0.658246\pi\)
\(678\) −1.81824 −0.0698292
\(679\) −1.32780 −0.0509562
\(680\) −5.34260 −0.204879
\(681\) −19.9731 −0.765370
\(682\) 0 0
\(683\) 7.25911 0.277762 0.138881 0.990309i \(-0.455649\pi\)
0.138881 + 0.990309i \(0.455649\pi\)
\(684\) −6.47209 −0.247467
\(685\) 9.48971 0.362583
\(686\) 1.00000 0.0381802
\(687\) 12.3059 0.469499
\(688\) 12.2705 0.467808
\(689\) 12.3764 0.471502
\(690\) 6.43935 0.245142
\(691\) −20.0175 −0.761500 −0.380750 0.924678i \(-0.624334\pi\)
−0.380750 + 0.924678i \(0.624334\pi\)
\(692\) −10.4055 −0.395557
\(693\) 0 0
\(694\) 11.1959 0.424990
\(695\) −10.8603 −0.411956
\(696\) −5.85640 −0.221986
\(697\) −27.4082 −1.03816
\(698\) −28.5841 −1.08192
\(699\) 24.6865 0.933728
\(700\) 1.00000 0.0377964
\(701\) −41.0871 −1.55184 −0.775920 0.630831i \(-0.782714\pi\)
−0.775920 + 0.630831i \(0.782714\pi\)
\(702\) −14.7173 −0.555468
\(703\) 0.161196 0.00607961
\(704\) 0 0
\(705\) 11.5969 0.436764
\(706\) 4.00343 0.150671
\(707\) 12.9533 0.487158
\(708\) −0.684935 −0.0257415
\(709\) −14.8981 −0.559510 −0.279755 0.960071i \(-0.590253\pi\)
−0.279755 + 0.960071i \(0.590253\pi\)
\(710\) 8.20815 0.308046
\(711\) −10.9930 −0.412271
\(712\) 2.73408 0.102464
\(713\) −15.7771 −0.590856
\(714\) −6.25844 −0.234216
\(715\) 0 0
\(716\) −25.4215 −0.950048
\(717\) 1.42718 0.0532990
\(718\) −9.77677 −0.364866
\(719\) 35.3705 1.31910 0.659548 0.751662i \(-0.270748\pi\)
0.659548 + 0.751662i \(0.270748\pi\)
\(720\) −1.62777 −0.0606635
\(721\) 9.24014 0.344121
\(722\) −3.19107 −0.118759
\(723\) −1.99568 −0.0742203
\(724\) −9.34641 −0.347357
\(725\) −4.99939 −0.185673
\(726\) 0 0
\(727\) 1.10139 0.0408482 0.0204241 0.999791i \(-0.493498\pi\)
0.0204241 + 0.999791i \(0.493498\pi\)
\(728\) 2.71483 0.100618
\(729\) 21.4391 0.794040
\(730\) 14.5897 0.539991
\(731\) −65.5564 −2.42469
\(732\) 8.50409 0.314320
\(733\) 12.9270 0.477471 0.238735 0.971085i \(-0.423267\pi\)
0.238735 + 0.971085i \(0.423267\pi\)
\(734\) −13.1201 −0.484272
\(735\) 1.17142 0.0432085
\(736\) 5.49704 0.202624
\(737\) 0 0
\(738\) −8.35067 −0.307392
\(739\) −32.1959 −1.18435 −0.592173 0.805810i \(-0.701730\pi\)
−0.592173 + 0.805810i \(0.701730\pi\)
\(740\) 0.0405417 0.00149034
\(741\) 12.6447 0.464513
\(742\) 4.55880 0.167359
\(743\) 26.4833 0.971577 0.485788 0.874077i \(-0.338533\pi\)
0.485788 + 0.874077i \(0.338533\pi\)
\(744\) −3.36210 −0.123260
\(745\) 12.3315 0.451791
\(746\) −15.2552 −0.558534
\(747\) 25.3597 0.927864
\(748\) 0 0
\(749\) 10.0022 0.365472
\(750\) 1.17142 0.0427743
\(751\) −38.6078 −1.40882 −0.704409 0.709795i \(-0.748788\pi\)
−0.704409 + 0.709795i \(0.748788\pi\)
\(752\) 9.89983 0.361010
\(753\) −22.5879 −0.823151
\(754\) −13.5725 −0.494281
\(755\) −8.70759 −0.316902
\(756\) −5.42107 −0.197162
\(757\) −38.9080 −1.41413 −0.707067 0.707146i \(-0.749982\pi\)
−0.707067 + 0.707146i \(0.749982\pi\)
\(758\) 26.6376 0.967522
\(759\) 0 0
\(760\) 3.97605 0.144226
\(761\) 9.88051 0.358168 0.179084 0.983834i \(-0.442687\pi\)
0.179084 + 0.983834i \(0.442687\pi\)
\(762\) −20.2395 −0.733199
\(763\) −0.535064 −0.0193706
\(764\) 4.55912 0.164943
\(765\) 8.69653 0.314424
\(766\) 6.81998 0.246416
\(767\) −1.58737 −0.0573167
\(768\) 1.17142 0.0422700
\(769\) −20.8801 −0.752955 −0.376478 0.926426i \(-0.622865\pi\)
−0.376478 + 0.926426i \(0.622865\pi\)
\(770\) 0 0
\(771\) −11.6788 −0.420602
\(772\) 15.8146 0.569181
\(773\) −37.3833 −1.34459 −0.672293 0.740286i \(-0.734690\pi\)
−0.672293 + 0.740286i \(0.734690\pi\)
\(774\) −19.9736 −0.717935
\(775\) −2.87010 −0.103097
\(776\) −1.32780 −0.0476652
\(777\) 0.0474914 0.00170374
\(778\) 23.0806 0.827481
\(779\) 20.3976 0.730820
\(780\) 3.18021 0.113870
\(781\) 0 0
\(782\) −29.3685 −1.05022
\(783\) 27.1021 0.968549
\(784\) 1.00000 0.0357143
\(785\) −19.1275 −0.682688
\(786\) 12.9068 0.460369
\(787\) −16.4247 −0.585477 −0.292739 0.956193i \(-0.594567\pi\)
−0.292739 + 0.956193i \(0.594567\pi\)
\(788\) 14.6162 0.520679
\(789\) 18.0281 0.641818
\(790\) 6.75342 0.240276
\(791\) −1.55217 −0.0551887
\(792\) 0 0
\(793\) 19.7086 0.699875
\(794\) −15.6435 −0.555166
\(795\) 5.34028 0.189400
\(796\) 16.4255 0.582188
\(797\) 31.5634 1.11803 0.559016 0.829157i \(-0.311179\pi\)
0.559016 + 0.829157i \(0.311179\pi\)
\(798\) 4.65762 0.164878
\(799\) −52.8908 −1.87114
\(800\) 1.00000 0.0353553
\(801\) −4.45046 −0.157249
\(802\) −1.51369 −0.0534502
\(803\) 0 0
\(804\) −16.8858 −0.595518
\(805\) 5.49704 0.193745
\(806\) −7.79183 −0.274456
\(807\) −19.5067 −0.686667
\(808\) 12.9533 0.455695
\(809\) −6.30494 −0.221670 −0.110835 0.993839i \(-0.535353\pi\)
−0.110835 + 0.993839i \(0.535353\pi\)
\(810\) −1.46704 −0.0515466
\(811\) 28.7394 1.00918 0.504588 0.863361i \(-0.331645\pi\)
0.504588 + 0.863361i \(0.331645\pi\)
\(812\) −4.99939 −0.175444
\(813\) −3.34381 −0.117273
\(814\) 0 0
\(815\) −18.6244 −0.652385
\(816\) −6.25844 −0.219089
\(817\) 48.7881 1.70688
\(818\) −14.2110 −0.496876
\(819\) −4.41912 −0.154417
\(820\) 5.13012 0.179152
\(821\) 40.1131 1.39996 0.699978 0.714165i \(-0.253193\pi\)
0.699978 + 0.714165i \(0.253193\pi\)
\(822\) 11.1164 0.387731
\(823\) −16.7750 −0.584739 −0.292370 0.956305i \(-0.594444\pi\)
−0.292370 + 0.956305i \(0.594444\pi\)
\(824\) 9.24014 0.321895
\(825\) 0 0
\(826\) −0.584705 −0.0203445
\(827\) 7.08065 0.246218 0.123109 0.992393i \(-0.460713\pi\)
0.123109 + 0.992393i \(0.460713\pi\)
\(828\) −8.94793 −0.310962
\(829\) −17.7789 −0.617485 −0.308743 0.951146i \(-0.599908\pi\)
−0.308743 + 0.951146i \(0.599908\pi\)
\(830\) −15.5794 −0.540769
\(831\) 29.3405 1.01781
\(832\) 2.71483 0.0941197
\(833\) −5.34260 −0.185110
\(834\) −12.7220 −0.440528
\(835\) 1.66091 0.0574780
\(836\) 0 0
\(837\) 15.5590 0.537798
\(838\) 6.61451 0.228494
\(839\) −7.03094 −0.242735 −0.121368 0.992608i \(-0.538728\pi\)
−0.121368 + 0.992608i \(0.538728\pi\)
\(840\) 1.17142 0.0404179
\(841\) −4.00606 −0.138140
\(842\) −38.2913 −1.31960
\(843\) 5.12877 0.176644
\(844\) −4.78958 −0.164864
\(845\) −5.62971 −0.193668
\(846\) −16.1147 −0.554034
\(847\) 0 0
\(848\) 4.55880 0.156550
\(849\) −8.74121 −0.299998
\(850\) −5.34260 −0.183250
\(851\) 0.222859 0.00763952
\(852\) 9.61520 0.329411
\(853\) 48.2865 1.65330 0.826648 0.562719i \(-0.190245\pi\)
0.826648 + 0.562719i \(0.190245\pi\)
\(854\) 7.25963 0.248419
\(855\) −6.47209 −0.221341
\(856\) 10.0022 0.341867
\(857\) −30.6816 −1.04806 −0.524032 0.851699i \(-0.675573\pi\)
−0.524032 + 0.851699i \(0.675573\pi\)
\(858\) 0 0
\(859\) 12.1328 0.413965 0.206983 0.978345i \(-0.433636\pi\)
0.206983 + 0.978345i \(0.433636\pi\)
\(860\) 12.2705 0.418421
\(861\) 6.00954 0.204804
\(862\) 6.73123 0.229267
\(863\) −42.0830 −1.43252 −0.716261 0.697832i \(-0.754148\pi\)
−0.716261 + 0.697832i \(0.754148\pi\)
\(864\) −5.42107 −0.184429
\(865\) −10.4055 −0.353797
\(866\) −25.7699 −0.875697
\(867\) 13.5222 0.459237
\(868\) −2.87010 −0.0974176
\(869\) 0 0
\(870\) −5.85640 −0.198550
\(871\) −39.1338 −1.32600
\(872\) −0.535064 −0.0181196
\(873\) 2.16135 0.0731507
\(874\) 21.8565 0.739307
\(875\) 1.00000 0.0338062
\(876\) 17.0907 0.577443
\(877\) −34.3045 −1.15838 −0.579190 0.815193i \(-0.696631\pi\)
−0.579190 + 0.815193i \(0.696631\pi\)
\(878\) 13.8131 0.466169
\(879\) 6.86069 0.231405
\(880\) 0 0
\(881\) 0.874683 0.0294688 0.0147344 0.999891i \(-0.495310\pi\)
0.0147344 + 0.999891i \(0.495310\pi\)
\(882\) −1.62777 −0.0548099
\(883\) 49.3208 1.65978 0.829888 0.557930i \(-0.188404\pi\)
0.829888 + 0.557930i \(0.188404\pi\)
\(884\) −14.5042 −0.487830
\(885\) −0.684935 −0.0230239
\(886\) −28.1206 −0.944730
\(887\) 10.2517 0.344218 0.172109 0.985078i \(-0.444942\pi\)
0.172109 + 0.985078i \(0.444942\pi\)
\(888\) 0.0474914 0.00159371
\(889\) −17.2777 −0.579476
\(890\) 2.73408 0.0916466
\(891\) 0 0
\(892\) −23.2488 −0.778426
\(893\) 39.3622 1.31721
\(894\) 14.4454 0.483126
\(895\) −25.4215 −0.849749
\(896\) 1.00000 0.0334077
\(897\) 17.4817 0.583698
\(898\) 27.8576 0.929621
\(899\) 14.3488 0.478558
\(900\) −1.62777 −0.0542591
\(901\) −24.3558 −0.811411
\(902\) 0 0
\(903\) 14.3739 0.478334
\(904\) −1.55217 −0.0516243
\(905\) −9.34641 −0.310685
\(906\) −10.2003 −0.338881
\(907\) 54.0073 1.79328 0.896642 0.442757i \(-0.146000\pi\)
0.896642 + 0.442757i \(0.146000\pi\)
\(908\) −17.0503 −0.565834
\(909\) −21.0850 −0.699344
\(910\) 2.71483 0.0899957
\(911\) −21.9362 −0.726779 −0.363390 0.931637i \(-0.618381\pi\)
−0.363390 + 0.931637i \(0.618381\pi\)
\(912\) 4.65762 0.154229
\(913\) 0 0
\(914\) −38.0839 −1.25970
\(915\) 8.50409 0.281136
\(916\) 10.5051 0.347098
\(917\) 11.0180 0.363848
\(918\) 28.9626 0.955909
\(919\) −32.5378 −1.07332 −0.536662 0.843798i \(-0.680315\pi\)
−0.536662 + 0.843798i \(0.680315\pi\)
\(920\) 5.49704 0.181232
\(921\) 40.7507 1.34278
\(922\) 13.9251 0.458598
\(923\) 22.2837 0.733478
\(924\) 0 0
\(925\) 0.0405417 0.00133300
\(926\) −2.93872 −0.0965725
\(927\) −15.0408 −0.494006
\(928\) −4.99939 −0.164113
\(929\) 1.52103 0.0499034 0.0249517 0.999689i \(-0.492057\pi\)
0.0249517 + 0.999689i \(0.492057\pi\)
\(930\) −3.36210 −0.110248
\(931\) 3.97605 0.130310
\(932\) 21.0739 0.690300
\(933\) 22.9340 0.750826
\(934\) −31.8095 −1.04084
\(935\) 0 0
\(936\) −4.41912 −0.144443
\(937\) 35.3146 1.15368 0.576839 0.816858i \(-0.304286\pi\)
0.576839 + 0.816858i \(0.304286\pi\)
\(938\) −14.4148 −0.470661
\(939\) −14.4822 −0.472610
\(940\) 9.89983 0.322897
\(941\) 17.8323 0.581316 0.290658 0.956827i \(-0.406126\pi\)
0.290658 + 0.956827i \(0.406126\pi\)
\(942\) −22.4063 −0.730037
\(943\) 28.2005 0.918335
\(944\) −0.584705 −0.0190305
\(945\) −5.42107 −0.176347
\(946\) 0 0
\(947\) −26.5867 −0.863953 −0.431977 0.901885i \(-0.642184\pi\)
−0.431977 + 0.901885i \(0.642184\pi\)
\(948\) 7.91110 0.256941
\(949\) 39.6087 1.28575
\(950\) 3.97605 0.129000
\(951\) 6.03939 0.195840
\(952\) −5.34260 −0.173155
\(953\) 12.9284 0.418791 0.209396 0.977831i \(-0.432850\pi\)
0.209396 + 0.977831i \(0.432850\pi\)
\(954\) −7.42069 −0.240254
\(955\) 4.55912 0.147530
\(956\) 1.21833 0.0394037
\(957\) 0 0
\(958\) 37.7725 1.22038
\(959\) 9.48971 0.306439
\(960\) 1.17142 0.0378075
\(961\) −22.7625 −0.734275
\(962\) 0.110064 0.00354860
\(963\) −16.2813 −0.524656
\(964\) −1.70364 −0.0548706
\(965\) 15.8146 0.509091
\(966\) 6.43935 0.207183
\(967\) 6.45564 0.207599 0.103800 0.994598i \(-0.466900\pi\)
0.103800 + 0.994598i \(0.466900\pi\)
\(968\) 0 0
\(969\) −24.8838 −0.799384
\(970\) −1.32780 −0.0426331
\(971\) 16.5435 0.530907 0.265454 0.964124i \(-0.414478\pi\)
0.265454 + 0.964124i \(0.414478\pi\)
\(972\) 14.5447 0.466521
\(973\) −10.8603 −0.348167
\(974\) −9.53146 −0.305408
\(975\) 3.18021 0.101848
\(976\) 7.25963 0.232375
\(977\) −24.7794 −0.792762 −0.396381 0.918086i \(-0.629734\pi\)
−0.396381 + 0.918086i \(0.629734\pi\)
\(978\) −21.8170 −0.697632
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 0.870962 0.0278077
\(982\) −22.2193 −0.709046
\(983\) −46.4963 −1.48300 −0.741501 0.670952i \(-0.765886\pi\)
−0.741501 + 0.670952i \(0.765886\pi\)
\(984\) 6.00954 0.191577
\(985\) 14.6162 0.465709
\(986\) 26.7098 0.850612
\(987\) 11.5969 0.369133
\(988\) 10.7943 0.343412
\(989\) 67.4514 2.14483
\(990\) 0 0
\(991\) 47.0722 1.49530 0.747649 0.664094i \(-0.231182\pi\)
0.747649 + 0.664094i \(0.231182\pi\)
\(992\) −2.87010 −0.0911258
\(993\) −38.0784 −1.20838
\(994\) 8.20815 0.260347
\(995\) 16.4255 0.520725
\(996\) −18.2500 −0.578275
\(997\) 56.2531 1.78155 0.890776 0.454442i \(-0.150161\pi\)
0.890776 + 0.454442i \(0.150161\pi\)
\(998\) −10.8366 −0.343026
\(999\) −0.219779 −0.00695351
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.de.1.4 6
11.5 even 5 770.2.n.i.421.2 12
11.9 even 5 770.2.n.i.631.2 yes 12
11.10 odd 2 8470.2.a.cy.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.i.421.2 12 11.5 even 5
770.2.n.i.631.2 yes 12 11.9 even 5
8470.2.a.cy.1.4 6 11.10 odd 2
8470.2.a.de.1.4 6 1.1 even 1 trivial