Properties

Label 8470.2.a.cx.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: 6.6.745749504.1
Defining polynomial: \(x^{6} - 18 x^{4} - 4 x^{3} + 81 x^{2} + 36 x - 44\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.16996\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.16996 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.16996 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.63119 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.16996 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.16996 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.63119 q^{9} +1.00000 q^{10} +1.16996 q^{12} +1.43479 q^{13} -1.00000 q^{14} -1.16996 q^{15} +1.00000 q^{16} -7.28628 q^{17} +1.63119 q^{18} -4.33680 q^{19} -1.00000 q^{20} +1.16996 q^{21} +2.02643 q^{23} -1.16996 q^{24} +1.00000 q^{25} -1.43479 q^{26} -5.41832 q^{27} +1.00000 q^{28} +9.11632 q^{29} +1.16996 q^{30} +3.65509 q^{31} -1.00000 q^{32} +7.28628 q^{34} -1.00000 q^{35} -1.63119 q^{36} -3.11632 q^{37} +4.33680 q^{38} +1.67865 q^{39} +1.00000 q^{40} -2.05574 q^{41} -1.16996 q^{42} -0.705861 q^{43} +1.63119 q^{45} -2.02643 q^{46} +4.00288 q^{47} +1.16996 q^{48} +1.00000 q^{49} -1.00000 q^{50} -8.52468 q^{51} +1.43479 q^{52} -4.87480 q^{53} +5.41832 q^{54} -1.00000 q^{56} -5.07390 q^{57} -9.11632 q^{58} +12.3811 q^{59} -1.16996 q^{60} -6.62309 q^{61} -3.65509 q^{62} -1.63119 q^{63} +1.00000 q^{64} -1.43479 q^{65} -1.89373 q^{67} -7.28628 q^{68} +2.37085 q^{69} +1.00000 q^{70} +0.421681 q^{71} +1.63119 q^{72} -7.37947 q^{73} +3.11632 q^{74} +1.16996 q^{75} -4.33680 q^{76} -1.67865 q^{78} -3.98377 q^{79} -1.00000 q^{80} -1.44566 q^{81} +2.05574 q^{82} +7.15061 q^{83} +1.16996 q^{84} +7.28628 q^{85} +0.705861 q^{86} +10.6658 q^{87} +8.34280 q^{89} -1.63119 q^{90} +1.43479 q^{91} +2.02643 q^{92} +4.27632 q^{93} -4.00288 q^{94} +4.33680 q^{95} -1.16996 q^{96} +11.3050 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + 6q^{4} - 6q^{5} + 6q^{7} - 6q^{8} + 18q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + 6q^{4} - 6q^{5} + 6q^{7} - 6q^{8} + 18q^{9} + 6q^{10} - 6q^{14} + 6q^{16} + 6q^{17} - 18q^{18} - 6q^{20} + 6q^{25} - 12q^{27} + 6q^{28} + 12q^{29} - 6q^{32} - 6q^{34} - 6q^{35} + 18q^{36} + 24q^{37} - 24q^{39} + 6q^{40} + 12q^{41} - 18q^{43} - 18q^{45} + 24q^{47} + 6q^{49} - 6q^{50} - 12q^{51} + 36q^{53} + 12q^{54} - 6q^{56} - 12q^{57} - 12q^{58} + 30q^{59} + 36q^{61} + 18q^{63} + 6q^{64} - 12q^{67} + 6q^{68} + 6q^{70} + 6q^{71} - 18q^{72} - 6q^{73} - 24q^{74} + 24q^{78} - 24q^{79} - 6q^{80} + 54q^{81} - 12q^{82} + 24q^{83} - 6q^{85} + 18q^{86} - 24q^{87} + 36q^{89} + 18q^{90} - 24q^{94} - 6q^{98} + O(q^{100}) \)

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
\(3\) 1.16996 0.675478 0.337739 0.941240i \(-0.390338\pi\)
0.337739 + 0.941240i \(0.390338\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.16996 −0.477635
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −1.63119 −0.543729
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.16996 0.337739
\(13\) 1.43479 0.397939 0.198970 0.980006i \(-0.436240\pi\)
0.198970 + 0.980006i \(0.436240\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.16996 −0.302083
\(16\) 1.00000 0.250000
\(17\) −7.28628 −1.76718 −0.883591 0.468259i \(-0.844882\pi\)
−0.883591 + 0.468259i \(0.844882\pi\)
\(18\) 1.63119 0.384475
\(19\) −4.33680 −0.994931 −0.497466 0.867484i \(-0.665736\pi\)
−0.497466 + 0.867484i \(0.665736\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.16996 0.255307
\(22\) 0 0
\(23\) 2.02643 0.422541 0.211270 0.977428i \(-0.432240\pi\)
0.211270 + 0.977428i \(0.432240\pi\)
\(24\) −1.16996 −0.238818
\(25\) 1.00000 0.200000
\(26\) −1.43479 −0.281386
\(27\) −5.41832 −1.04276
\(28\) 1.00000 0.188982
\(29\) 9.11632 1.69286 0.846429 0.532502i \(-0.178748\pi\)
0.846429 + 0.532502i \(0.178748\pi\)
\(30\) 1.16996 0.213605
\(31\) 3.65509 0.656474 0.328237 0.944595i \(-0.393546\pi\)
0.328237 + 0.944595i \(0.393546\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.28628 1.24959
\(35\) −1.00000 −0.169031
\(36\) −1.63119 −0.271865
\(37\) −3.11632 −0.512320 −0.256160 0.966634i \(-0.582457\pi\)
−0.256160 + 0.966634i \(0.582457\pi\)
\(38\) 4.33680 0.703523
\(39\) 1.67865 0.268799
\(40\) 1.00000 0.158114
\(41\) −2.05574 −0.321053 −0.160527 0.987031i \(-0.551319\pi\)
−0.160527 + 0.987031i \(0.551319\pi\)
\(42\) −1.16996 −0.180529
\(43\) −0.705861 −0.107643 −0.0538214 0.998551i \(-0.517140\pi\)
−0.0538214 + 0.998551i \(0.517140\pi\)
\(44\) 0 0
\(45\) 1.63119 0.243163
\(46\) −2.02643 −0.298781
\(47\) 4.00288 0.583880 0.291940 0.956437i \(-0.405699\pi\)
0.291940 + 0.956437i \(0.405699\pi\)
\(48\) 1.16996 0.168870
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −8.52468 −1.19369
\(52\) 1.43479 0.198970
\(53\) −4.87480 −0.669606 −0.334803 0.942288i \(-0.608670\pi\)
−0.334803 + 0.942288i \(0.608670\pi\)
\(54\) 5.41832 0.737339
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −5.07390 −0.672054
\(58\) −9.11632 −1.19703
\(59\) 12.3811 1.61189 0.805944 0.591992i \(-0.201658\pi\)
0.805944 + 0.591992i \(0.201658\pi\)
\(60\) −1.16996 −0.151042
\(61\) −6.62309 −0.847999 −0.424000 0.905662i \(-0.639374\pi\)
−0.424000 + 0.905662i \(0.639374\pi\)
\(62\) −3.65509 −0.464197
\(63\) −1.63119 −0.205510
\(64\) 1.00000 0.125000
\(65\) −1.43479 −0.177964
\(66\) 0 0
\(67\) −1.89373 −0.231356 −0.115678 0.993287i \(-0.536904\pi\)
−0.115678 + 0.993287i \(0.536904\pi\)
\(68\) −7.28628 −0.883591
\(69\) 2.37085 0.285417
\(70\) 1.00000 0.119523
\(71\) 0.421681 0.0500443 0.0250222 0.999687i \(-0.492034\pi\)
0.0250222 + 0.999687i \(0.492034\pi\)
\(72\) 1.63119 0.192237
\(73\) −7.37947 −0.863702 −0.431851 0.901945i \(-0.642139\pi\)
−0.431851 + 0.901945i \(0.642139\pi\)
\(74\) 3.11632 0.362265
\(75\) 1.16996 0.135096
\(76\) −4.33680 −0.497466
\(77\) 0 0
\(78\) −1.67865 −0.190070
\(79\) −3.98377 −0.448209 −0.224105 0.974565i \(-0.571946\pi\)
−0.224105 + 0.974565i \(0.571946\pi\)
\(80\) −1.00000 −0.111803
\(81\) −1.44566 −0.160629
\(82\) 2.05574 0.227019
\(83\) 7.15061 0.784882 0.392441 0.919777i \(-0.371631\pi\)
0.392441 + 0.919777i \(0.371631\pi\)
\(84\) 1.16996 0.127653
\(85\) 7.28628 0.790308
\(86\) 0.705861 0.0761149
\(87\) 10.6658 1.14349
\(88\) 0 0
\(89\) 8.34280 0.884335 0.442168 0.896932i \(-0.354210\pi\)
0.442168 + 0.896932i \(0.354210\pi\)
\(90\) −1.63119 −0.171942
\(91\) 1.43479 0.150407
\(92\) 2.02643 0.211270
\(93\) 4.27632 0.443434
\(94\) −4.00288 −0.412865
\(95\) 4.33680 0.444947
\(96\) −1.16996 −0.119409
\(97\) 11.3050 1.14785 0.573926 0.818907i \(-0.305420\pi\)
0.573926 + 0.818907i \(0.305420\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −11.8936 −1.18345 −0.591727 0.806138i \(-0.701554\pi\)
−0.591727 + 0.806138i \(0.701554\pi\)
\(102\) 8.52468 0.844069
\(103\) −2.57520 −0.253742 −0.126871 0.991919i \(-0.540493\pi\)
−0.126871 + 0.991919i \(0.540493\pi\)
\(104\) −1.43479 −0.140693
\(105\) −1.16996 −0.114177
\(106\) 4.87480 0.473483
\(107\) 0.0210293 0.00203298 0.00101649 0.999999i \(-0.499676\pi\)
0.00101649 + 0.999999i \(0.499676\pi\)
\(108\) −5.41832 −0.521378
\(109\) 3.22024 0.308443 0.154222 0.988036i \(-0.450713\pi\)
0.154222 + 0.988036i \(0.450713\pi\)
\(110\) 0 0
\(111\) −3.64598 −0.346061
\(112\) 1.00000 0.0944911
\(113\) 16.8832 1.58824 0.794118 0.607764i \(-0.207933\pi\)
0.794118 + 0.607764i \(0.207933\pi\)
\(114\) 5.07390 0.475214
\(115\) −2.02643 −0.188966
\(116\) 9.11632 0.846429
\(117\) −2.34041 −0.216371
\(118\) −12.3811 −1.13978
\(119\) −7.28628 −0.667932
\(120\) 1.16996 0.106802
\(121\) 0 0
\(122\) 6.62309 0.599626
\(123\) −2.40514 −0.216865
\(124\) 3.65509 0.328237
\(125\) −1.00000 −0.0894427
\(126\) 1.63119 0.145318
\(127\) 3.83782 0.340551 0.170276 0.985396i \(-0.445534\pi\)
0.170276 + 0.985396i \(0.445534\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.825831 −0.0727103
\(130\) 1.43479 0.125839
\(131\) 19.4691 1.70102 0.850511 0.525958i \(-0.176293\pi\)
0.850511 + 0.525958i \(0.176293\pi\)
\(132\) 0 0
\(133\) −4.33680 −0.376049
\(134\) 1.89373 0.163594
\(135\) 5.41832 0.466334
\(136\) 7.28628 0.624793
\(137\) 16.1146 1.37677 0.688383 0.725347i \(-0.258321\pi\)
0.688383 + 0.725347i \(0.258321\pi\)
\(138\) −2.37085 −0.201820
\(139\) 16.9156 1.43476 0.717382 0.696680i \(-0.245340\pi\)
0.717382 + 0.696680i \(0.245340\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 4.68321 0.394398
\(142\) −0.421681 −0.0353867
\(143\) 0 0
\(144\) −1.63119 −0.135932
\(145\) −9.11632 −0.757069
\(146\) 7.37947 0.610729
\(147\) 1.16996 0.0964969
\(148\) −3.11632 −0.256160
\(149\) 5.12256 0.419656 0.209828 0.977738i \(-0.432710\pi\)
0.209828 + 0.977738i \(0.432710\pi\)
\(150\) −1.16996 −0.0955270
\(151\) −22.4051 −1.82330 −0.911652 0.410964i \(-0.865192\pi\)
−0.911652 + 0.410964i \(0.865192\pi\)
\(152\) 4.33680 0.351761
\(153\) 11.8853 0.960869
\(154\) 0 0
\(155\) −3.65509 −0.293584
\(156\) 1.67865 0.134400
\(157\) 14.9308 1.19161 0.595804 0.803130i \(-0.296834\pi\)
0.595804 + 0.803130i \(0.296834\pi\)
\(158\) 3.98377 0.316932
\(159\) −5.70334 −0.452304
\(160\) 1.00000 0.0790569
\(161\) 2.02643 0.159705
\(162\) 1.44566 0.113582
\(163\) 3.69439 0.289367 0.144684 0.989478i \(-0.453784\pi\)
0.144684 + 0.989478i \(0.453784\pi\)
\(164\) −2.05574 −0.160527
\(165\) 0 0
\(166\) −7.15061 −0.554995
\(167\) 9.33915 0.722685 0.361343 0.932433i \(-0.382318\pi\)
0.361343 + 0.932433i \(0.382318\pi\)
\(168\) −1.16996 −0.0902646
\(169\) −10.9414 −0.841644
\(170\) −7.28628 −0.558832
\(171\) 7.07414 0.540973
\(172\) −0.705861 −0.0538214
\(173\) −1.89895 −0.144375 −0.0721874 0.997391i \(-0.522998\pi\)
−0.0721874 + 0.997391i \(0.522998\pi\)
\(174\) −10.6658 −0.808568
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 14.4855 1.08880
\(178\) −8.34280 −0.625319
\(179\) 6.00288 0.448676 0.224338 0.974511i \(-0.427978\pi\)
0.224338 + 0.974511i \(0.427978\pi\)
\(180\) 1.63119 0.121582
\(181\) −6.73019 −0.500251 −0.250126 0.968213i \(-0.580472\pi\)
−0.250126 + 0.968213i \(0.580472\pi\)
\(182\) −1.43479 −0.106354
\(183\) −7.74876 −0.572805
\(184\) −2.02643 −0.149391
\(185\) 3.11632 0.229116
\(186\) −4.27632 −0.313555
\(187\) 0 0
\(188\) 4.00288 0.291940
\(189\) −5.41832 −0.394124
\(190\) −4.33680 −0.314625
\(191\) 15.9464 1.15384 0.576919 0.816801i \(-0.304255\pi\)
0.576919 + 0.816801i \(0.304255\pi\)
\(192\) 1.16996 0.0844348
\(193\) −11.7470 −0.845569 −0.422784 0.906230i \(-0.638947\pi\)
−0.422784 + 0.906230i \(0.638947\pi\)
\(194\) −11.3050 −0.811655
\(195\) −1.67865 −0.120211
\(196\) 1.00000 0.0714286
\(197\) 2.70088 0.192430 0.0962148 0.995361i \(-0.469326\pi\)
0.0962148 + 0.995361i \(0.469326\pi\)
\(198\) 0 0
\(199\) 14.5804 1.03358 0.516789 0.856113i \(-0.327127\pi\)
0.516789 + 0.856113i \(0.327127\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −2.21560 −0.156276
\(202\) 11.8936 0.836828
\(203\) 9.11632 0.639840
\(204\) −8.52468 −0.596847
\(205\) 2.05574 0.143579
\(206\) 2.57520 0.179423
\(207\) −3.30550 −0.229748
\(208\) 1.43479 0.0994849
\(209\) 0 0
\(210\) 1.16996 0.0807351
\(211\) −18.2347 −1.25533 −0.627666 0.778483i \(-0.715990\pi\)
−0.627666 + 0.778483i \(0.715990\pi\)
\(212\) −4.87480 −0.334803
\(213\) 0.493351 0.0338038
\(214\) −0.0210293 −0.00143754
\(215\) 0.705861 0.0481393
\(216\) 5.41832 0.368670
\(217\) 3.65509 0.248124
\(218\) −3.22024 −0.218102
\(219\) −8.63370 −0.583412
\(220\) 0 0
\(221\) −10.4543 −0.703232
\(222\) 3.64598 0.244702
\(223\) 1.41856 0.0949938 0.0474969 0.998871i \(-0.484876\pi\)
0.0474969 + 0.998871i \(0.484876\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.63119 −0.108746
\(226\) −16.8832 −1.12305
\(227\) −8.18601 −0.543325 −0.271662 0.962393i \(-0.587573\pi\)
−0.271662 + 0.962393i \(0.587573\pi\)
\(228\) −5.07390 −0.336027
\(229\) −1.88578 −0.124616 −0.0623080 0.998057i \(-0.519846\pi\)
−0.0623080 + 0.998057i \(0.519846\pi\)
\(230\) 2.02643 0.133619
\(231\) 0 0
\(232\) −9.11632 −0.598516
\(233\) −22.9570 −1.50396 −0.751982 0.659184i \(-0.770902\pi\)
−0.751982 + 0.659184i \(0.770902\pi\)
\(234\) 2.34041 0.152998
\(235\) −4.00288 −0.261119
\(236\) 12.3811 0.805944
\(237\) −4.66086 −0.302755
\(238\) 7.28628 0.472299
\(239\) −0.315230 −0.0203905 −0.0101953 0.999948i \(-0.503245\pi\)
−0.0101953 + 0.999948i \(0.503245\pi\)
\(240\) −1.16996 −0.0755208
\(241\) 5.61640 0.361784 0.180892 0.983503i \(-0.442102\pi\)
0.180892 + 0.983503i \(0.442102\pi\)
\(242\) 0 0
\(243\) 14.5636 0.934254
\(244\) −6.62309 −0.424000
\(245\) −1.00000 −0.0638877
\(246\) 2.40514 0.153346
\(247\) −6.22241 −0.395922
\(248\) −3.65509 −0.232099
\(249\) 8.36595 0.530170
\(250\) 1.00000 0.0632456
\(251\) 14.3205 0.903903 0.451951 0.892043i \(-0.350728\pi\)
0.451951 + 0.892043i \(0.350728\pi\)
\(252\) −1.63119 −0.102755
\(253\) 0 0
\(254\) −3.83782 −0.240806
\(255\) 8.52468 0.533836
\(256\) 1.00000 0.0625000
\(257\) 28.8932 1.80231 0.901153 0.433501i \(-0.142722\pi\)
0.901153 + 0.433501i \(0.142722\pi\)
\(258\) 0.825831 0.0514140
\(259\) −3.11632 −0.193639
\(260\) −1.43479 −0.0889820
\(261\) −14.8704 −0.920456
\(262\) −19.4691 −1.20280
\(263\) −21.9055 −1.35075 −0.675377 0.737473i \(-0.736019\pi\)
−0.675377 + 0.737473i \(0.736019\pi\)
\(264\) 0 0
\(265\) 4.87480 0.299457
\(266\) 4.33680 0.265907
\(267\) 9.76076 0.597349
\(268\) −1.89373 −0.115678
\(269\) −2.97854 −0.181605 −0.0908025 0.995869i \(-0.528943\pi\)
−0.0908025 + 0.995869i \(0.528943\pi\)
\(270\) −5.41832 −0.329748
\(271\) −0.202215 −0.0122837 −0.00614183 0.999981i \(-0.501955\pi\)
−0.00614183 + 0.999981i \(0.501955\pi\)
\(272\) −7.28628 −0.441796
\(273\) 1.67865 0.101597
\(274\) −16.1146 −0.973521
\(275\) 0 0
\(276\) 2.37085 0.142709
\(277\) 4.93157 0.296309 0.148155 0.988964i \(-0.452667\pi\)
0.148155 + 0.988964i \(0.452667\pi\)
\(278\) −16.9156 −1.01453
\(279\) −5.96214 −0.356944
\(280\) 1.00000 0.0597614
\(281\) 17.8483 1.06474 0.532370 0.846512i \(-0.321302\pi\)
0.532370 + 0.846512i \(0.321302\pi\)
\(282\) −4.68321 −0.278881
\(283\) 29.2721 1.74005 0.870023 0.493010i \(-0.164104\pi\)
0.870023 + 0.493010i \(0.164104\pi\)
\(284\) 0.421681 0.0250222
\(285\) 5.07390 0.300552
\(286\) 0 0
\(287\) −2.05574 −0.121347
\(288\) 1.63119 0.0961187
\(289\) 36.0899 2.12293
\(290\) 9.11632 0.535329
\(291\) 13.2265 0.775350
\(292\) −7.37947 −0.431851
\(293\) 29.9025 1.74692 0.873460 0.486895i \(-0.161871\pi\)
0.873460 + 0.486895i \(0.161871\pi\)
\(294\) −1.16996 −0.0682336
\(295\) −12.3811 −0.720858
\(296\) 3.11632 0.181132
\(297\) 0 0
\(298\) −5.12256 −0.296742
\(299\) 2.90751 0.168146
\(300\) 1.16996 0.0675478
\(301\) −0.705861 −0.0406851
\(302\) 22.4051 1.28927
\(303\) −13.9150 −0.799397
\(304\) −4.33680 −0.248733
\(305\) 6.62309 0.379237
\(306\) −11.8853 −0.679437
\(307\) −13.6023 −0.776324 −0.388162 0.921591i \(-0.626890\pi\)
−0.388162 + 0.921591i \(0.626890\pi\)
\(308\) 0 0
\(309\) −3.01289 −0.171397
\(310\) 3.65509 0.207595
\(311\) 29.5607 1.67623 0.838117 0.545490i \(-0.183656\pi\)
0.838117 + 0.545490i \(0.183656\pi\)
\(312\) −1.67865 −0.0950349
\(313\) 9.23518 0.522003 0.261002 0.965338i \(-0.415947\pi\)
0.261002 + 0.965338i \(0.415947\pi\)
\(314\) −14.9308 −0.842593
\(315\) 1.63119 0.0919070
\(316\) −3.98377 −0.224105
\(317\) −5.25554 −0.295180 −0.147590 0.989049i \(-0.547152\pi\)
−0.147590 + 0.989049i \(0.547152\pi\)
\(318\) 5.70334 0.319827
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0.0246035 0.00137324
\(322\) −2.02643 −0.112929
\(323\) 31.5992 1.75823
\(324\) −1.44566 −0.0803146
\(325\) 1.43479 0.0795879
\(326\) −3.69439 −0.204614
\(327\) 3.76756 0.208347
\(328\) 2.05574 0.113510
\(329\) 4.00288 0.220686
\(330\) 0 0
\(331\) −23.4150 −1.28700 −0.643501 0.765445i \(-0.722519\pi\)
−0.643501 + 0.765445i \(0.722519\pi\)
\(332\) 7.15061 0.392441
\(333\) 5.08330 0.278563
\(334\) −9.33915 −0.511016
\(335\) 1.89373 0.103466
\(336\) 1.16996 0.0638267
\(337\) −2.73973 −0.149242 −0.0746212 0.997212i \(-0.523775\pi\)
−0.0746212 + 0.997212i \(0.523775\pi\)
\(338\) 10.9414 0.595132
\(339\) 19.7527 1.07282
\(340\) 7.28628 0.395154
\(341\) 0 0
\(342\) −7.07414 −0.382526
\(343\) 1.00000 0.0539949
\(344\) 0.705861 0.0380575
\(345\) −2.37085 −0.127642
\(346\) 1.89895 0.102088
\(347\) −5.82295 −0.312592 −0.156296 0.987710i \(-0.549955\pi\)
−0.156296 + 0.987710i \(0.549955\pi\)
\(348\) 10.6658 0.571744
\(349\) −8.42897 −0.451193 −0.225596 0.974221i \(-0.572433\pi\)
−0.225596 + 0.974221i \(0.572433\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −7.77415 −0.414953
\(352\) 0 0
\(353\) −23.9658 −1.27557 −0.637786 0.770214i \(-0.720149\pi\)
−0.637786 + 0.770214i \(0.720149\pi\)
\(354\) −14.4855 −0.769894
\(355\) −0.421681 −0.0223805
\(356\) 8.34280 0.442168
\(357\) −8.52468 −0.451174
\(358\) −6.00288 −0.317262
\(359\) 16.0490 0.847032 0.423516 0.905889i \(-0.360796\pi\)
0.423516 + 0.905889i \(0.360796\pi\)
\(360\) −1.63119 −0.0859711
\(361\) −0.192129 −0.0101120
\(362\) 6.73019 0.353731
\(363\) 0 0
\(364\) 1.43479 0.0752035
\(365\) 7.37947 0.386259
\(366\) 7.74876 0.405034
\(367\) −2.17504 −0.113536 −0.0567680 0.998387i \(-0.518080\pi\)
−0.0567680 + 0.998387i \(0.518080\pi\)
\(368\) 2.02643 0.105635
\(369\) 3.35331 0.174566
\(370\) −3.11632 −0.162010
\(371\) −4.87480 −0.253087
\(372\) 4.27632 0.221717
\(373\) 16.5044 0.854566 0.427283 0.904118i \(-0.359471\pi\)
0.427283 + 0.904118i \(0.359471\pi\)
\(374\) 0 0
\(375\) −1.16996 −0.0604166
\(376\) −4.00288 −0.206433
\(377\) 13.0800 0.673655
\(378\) 5.41832 0.278688
\(379\) 22.3263 1.14683 0.573413 0.819267i \(-0.305619\pi\)
0.573413 + 0.819267i \(0.305619\pi\)
\(380\) 4.33680 0.222473
\(381\) 4.49010 0.230035
\(382\) −15.9464 −0.815887
\(383\) −21.4500 −1.09604 −0.548021 0.836465i \(-0.684619\pi\)
−0.548021 + 0.836465i \(0.684619\pi\)
\(384\) −1.16996 −0.0597044
\(385\) 0 0
\(386\) 11.7470 0.597907
\(387\) 1.15139 0.0585285
\(388\) 11.3050 0.573926
\(389\) −37.9771 −1.92552 −0.962759 0.270362i \(-0.912857\pi\)
−0.962759 + 0.270362i \(0.912857\pi\)
\(390\) 1.67865 0.0850018
\(391\) −14.7652 −0.746707
\(392\) −1.00000 −0.0505076
\(393\) 22.7781 1.14900
\(394\) −2.70088 −0.136068
\(395\) 3.98377 0.200445
\(396\) 0 0
\(397\) −24.9220 −1.25080 −0.625399 0.780305i \(-0.715064\pi\)
−0.625399 + 0.780305i \(0.715064\pi\)
\(398\) −14.5804 −0.730850
\(399\) −5.07390 −0.254013
\(400\) 1.00000 0.0500000
\(401\) −23.7936 −1.18819 −0.594097 0.804393i \(-0.702491\pi\)
−0.594097 + 0.804393i \(0.702491\pi\)
\(402\) 2.21560 0.110504
\(403\) 5.24430 0.261237
\(404\) −11.8936 −0.591727
\(405\) 1.44566 0.0718355
\(406\) −9.11632 −0.452435
\(407\) 0 0
\(408\) 8.52468 0.422034
\(409\) 17.4118 0.860958 0.430479 0.902601i \(-0.358345\pi\)
0.430479 + 0.902601i \(0.358345\pi\)
\(410\) −2.05574 −0.101526
\(411\) 18.8535 0.929976
\(412\) −2.57520 −0.126871
\(413\) 12.3811 0.609236
\(414\) 3.30550 0.162456
\(415\) −7.15061 −0.351010
\(416\) −1.43479 −0.0703464
\(417\) 19.7906 0.969151
\(418\) 0 0
\(419\) 6.77394 0.330929 0.165464 0.986216i \(-0.447088\pi\)
0.165464 + 0.986216i \(0.447088\pi\)
\(420\) −1.16996 −0.0570883
\(421\) 16.3820 0.798409 0.399205 0.916862i \(-0.369286\pi\)
0.399205 + 0.916862i \(0.369286\pi\)
\(422\) 18.2347 0.887653
\(423\) −6.52944 −0.317472
\(424\) 4.87480 0.236741
\(425\) −7.28628 −0.353437
\(426\) −0.493351 −0.0239029
\(427\) −6.62309 −0.320514
\(428\) 0.0210293 0.00101649
\(429\) 0 0
\(430\) −0.705861 −0.0340396
\(431\) 35.7762 1.72328 0.861638 0.507523i \(-0.169439\pi\)
0.861638 + 0.507523i \(0.169439\pi\)
\(432\) −5.41832 −0.260689
\(433\) 12.5667 0.603919 0.301959 0.953321i \(-0.402359\pi\)
0.301959 + 0.953321i \(0.402359\pi\)
\(434\) −3.65509 −0.175450
\(435\) −10.6658 −0.511384
\(436\) 3.22024 0.154222
\(437\) −8.78825 −0.420399
\(438\) 8.63370 0.412534
\(439\) 34.6171 1.65218 0.826092 0.563535i \(-0.190559\pi\)
0.826092 + 0.563535i \(0.190559\pi\)
\(440\) 0 0
\(441\) −1.63119 −0.0776756
\(442\) 10.4543 0.497260
\(443\) 18.9816 0.901841 0.450921 0.892564i \(-0.351096\pi\)
0.450921 + 0.892564i \(0.351096\pi\)
\(444\) −3.64598 −0.173030
\(445\) −8.34280 −0.395487
\(446\) −1.41856 −0.0671708
\(447\) 5.99320 0.283469
\(448\) 1.00000 0.0472456
\(449\) 13.7748 0.650072 0.325036 0.945702i \(-0.394624\pi\)
0.325036 + 0.945702i \(0.394624\pi\)
\(450\) 1.63119 0.0768949
\(451\) 0 0
\(452\) 16.8832 0.794118
\(453\) −26.2132 −1.23160
\(454\) 8.18601 0.384188
\(455\) −1.43479 −0.0672640
\(456\) 5.07390 0.237607
\(457\) −36.0763 −1.68758 −0.843789 0.536676i \(-0.819680\pi\)
−0.843789 + 0.536676i \(0.819680\pi\)
\(458\) 1.88578 0.0881168
\(459\) 39.4794 1.84274
\(460\) −2.02643 −0.0944830
\(461\) 23.2972 1.08506 0.542529 0.840037i \(-0.317467\pi\)
0.542529 + 0.840037i \(0.317467\pi\)
\(462\) 0 0
\(463\) −31.8729 −1.48126 −0.740629 0.671914i \(-0.765472\pi\)
−0.740629 + 0.671914i \(0.765472\pi\)
\(464\) 9.11632 0.423214
\(465\) −4.27632 −0.198310
\(466\) 22.9570 1.06346
\(467\) −11.5661 −0.535216 −0.267608 0.963528i \(-0.586233\pi\)
−0.267608 + 0.963528i \(0.586233\pi\)
\(468\) −2.34041 −0.108186
\(469\) −1.89373 −0.0874444
\(470\) 4.00288 0.184639
\(471\) 17.4685 0.804905
\(472\) −12.3811 −0.569889
\(473\) 0 0
\(474\) 4.66086 0.214080
\(475\) −4.33680 −0.198986
\(476\) −7.28628 −0.333966
\(477\) 7.95172 0.364084
\(478\) 0.315230 0.0144183
\(479\) 13.2561 0.605689 0.302844 0.953040i \(-0.402064\pi\)
0.302844 + 0.953040i \(0.402064\pi\)
\(480\) 1.16996 0.0534012
\(481\) −4.47127 −0.203872
\(482\) −5.61640 −0.255820
\(483\) 2.37085 0.107878
\(484\) 0 0
\(485\) −11.3050 −0.513335
\(486\) −14.5636 −0.660617
\(487\) 5.42232 0.245709 0.122854 0.992425i \(-0.460795\pi\)
0.122854 + 0.992425i \(0.460795\pi\)
\(488\) 6.62309 0.299813
\(489\) 4.32230 0.195461
\(490\) 1.00000 0.0451754
\(491\) 36.0238 1.62573 0.812866 0.582450i \(-0.197906\pi\)
0.812866 + 0.582450i \(0.197906\pi\)
\(492\) −2.40514 −0.108432
\(493\) −66.4241 −2.99159
\(494\) 6.22241 0.279959
\(495\) 0 0
\(496\) 3.65509 0.164119
\(497\) 0.421681 0.0189150
\(498\) −8.36595 −0.374887
\(499\) −3.26974 −0.146374 −0.0731869 0.997318i \(-0.523317\pi\)
−0.0731869 + 0.997318i \(0.523317\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 10.9265 0.488158
\(502\) −14.3205 −0.639156
\(503\) 41.9101 1.86868 0.934339 0.356384i \(-0.115991\pi\)
0.934339 + 0.356384i \(0.115991\pi\)
\(504\) 1.63119 0.0726589
\(505\) 11.8936 0.529257
\(506\) 0 0
\(507\) −12.8010 −0.568512
\(508\) 3.83782 0.170276
\(509\) 19.3235 0.856500 0.428250 0.903660i \(-0.359130\pi\)
0.428250 + 0.903660i \(0.359130\pi\)
\(510\) −8.52468 −0.377479
\(511\) −7.37947 −0.326449
\(512\) −1.00000 −0.0441942
\(513\) 23.4982 1.03747
\(514\) −28.8932 −1.27442
\(515\) 2.57520 0.113477
\(516\) −0.825831 −0.0363552
\(517\) 0 0
\(518\) 3.11632 0.136923
\(519\) −2.22170 −0.0975220
\(520\) 1.43479 0.0629197
\(521\) −22.0283 −0.965078 −0.482539 0.875874i \(-0.660285\pi\)
−0.482539 + 0.875874i \(0.660285\pi\)
\(522\) 14.8704 0.650861
\(523\) 20.3706 0.890745 0.445373 0.895345i \(-0.353071\pi\)
0.445373 + 0.895345i \(0.353071\pi\)
\(524\) 19.4691 0.850511
\(525\) 1.16996 0.0510613
\(526\) 21.9055 0.955127
\(527\) −26.6320 −1.16011
\(528\) 0 0
\(529\) −18.8936 −0.821459
\(530\) −4.87480 −0.211748
\(531\) −20.1960 −0.876431
\(532\) −4.33680 −0.188024
\(533\) −2.94956 −0.127760
\(534\) −9.76076 −0.422390
\(535\) −0.0210293 −0.000909178 0
\(536\) 1.89373 0.0817968
\(537\) 7.02314 0.303071
\(538\) 2.97854 0.128414
\(539\) 0 0
\(540\) 5.41832 0.233167
\(541\) −29.5866 −1.27203 −0.636013 0.771678i \(-0.719418\pi\)
−0.636013 + 0.771678i \(0.719418\pi\)
\(542\) 0.202215 0.00868587
\(543\) −7.87407 −0.337909
\(544\) 7.28628 0.312397
\(545\) −3.22024 −0.137940
\(546\) −1.67865 −0.0718397
\(547\) 31.5318 1.34820 0.674102 0.738638i \(-0.264531\pi\)
0.674102 + 0.738638i \(0.264531\pi\)
\(548\) 16.1146 0.688383
\(549\) 10.8035 0.461082
\(550\) 0 0
\(551\) −39.5357 −1.68428
\(552\) −2.37085 −0.100910
\(553\) −3.98377 −0.169407
\(554\) −4.93157 −0.209522
\(555\) 3.64598 0.154763
\(556\) 16.9156 0.717382
\(557\) 2.97938 0.126240 0.0631202 0.998006i \(-0.479895\pi\)
0.0631202 + 0.998006i \(0.479895\pi\)
\(558\) 5.96214 0.252398
\(559\) −1.01276 −0.0428353
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −17.8483 −0.752885
\(563\) −15.7256 −0.662754 −0.331377 0.943498i \(-0.607513\pi\)
−0.331377 + 0.943498i \(0.607513\pi\)
\(564\) 4.68321 0.197199
\(565\) −16.8832 −0.710281
\(566\) −29.2721 −1.23040
\(567\) −1.44566 −0.0607121
\(568\) −0.421681 −0.0176933
\(569\) −37.0851 −1.55469 −0.777345 0.629074i \(-0.783434\pi\)
−0.777345 + 0.629074i \(0.783434\pi\)
\(570\) −5.07390 −0.212522
\(571\) 14.5281 0.607982 0.303991 0.952675i \(-0.401681\pi\)
0.303991 + 0.952675i \(0.401681\pi\)
\(572\) 0 0
\(573\) 18.6566 0.779392
\(574\) 2.05574 0.0858051
\(575\) 2.02643 0.0845082
\(576\) −1.63119 −0.0679662
\(577\) 2.22693 0.0927082 0.0463541 0.998925i \(-0.485240\pi\)
0.0463541 + 0.998925i \(0.485240\pi\)
\(578\) −36.0899 −1.50114
\(579\) −13.7436 −0.571163
\(580\) −9.11632 −0.378535
\(581\) 7.15061 0.296657
\(582\) −13.2265 −0.548255
\(583\) 0 0
\(584\) 7.37947 0.305365
\(585\) 2.34041 0.0967642
\(586\) −29.9025 −1.23526
\(587\) −8.50637 −0.351095 −0.175548 0.984471i \(-0.556170\pi\)
−0.175548 + 0.984471i \(0.556170\pi\)
\(588\) 1.16996 0.0482484
\(589\) −15.8514 −0.653147
\(590\) 12.3811 0.509724
\(591\) 3.15993 0.129982
\(592\) −3.11632 −0.128080
\(593\) −24.9289 −1.02371 −0.511853 0.859073i \(-0.671041\pi\)
−0.511853 + 0.859073i \(0.671041\pi\)
\(594\) 0 0
\(595\) 7.28628 0.298708
\(596\) 5.12256 0.209828
\(597\) 17.0585 0.698160
\(598\) −2.90751 −0.118897
\(599\) 35.1526 1.43630 0.718149 0.695889i \(-0.244990\pi\)
0.718149 + 0.695889i \(0.244990\pi\)
\(600\) −1.16996 −0.0477635
\(601\) −34.7898 −1.41910 −0.709552 0.704653i \(-0.751103\pi\)
−0.709552 + 0.704653i \(0.751103\pi\)
\(602\) 0.705861 0.0287687
\(603\) 3.08903 0.125795
\(604\) −22.4051 −0.911652
\(605\) 0 0
\(606\) 13.9150 0.565259
\(607\) −14.7721 −0.599581 −0.299791 0.954005i \(-0.596917\pi\)
−0.299791 + 0.954005i \(0.596917\pi\)
\(608\) 4.33680 0.175881
\(609\) 10.6658 0.432198
\(610\) −6.62309 −0.268161
\(611\) 5.74329 0.232349
\(612\) 11.8853 0.480435
\(613\) −10.3889 −0.419602 −0.209801 0.977744i \(-0.567282\pi\)
−0.209801 + 0.977744i \(0.567282\pi\)
\(614\) 13.6023 0.548944
\(615\) 2.40514 0.0969848
\(616\) 0 0
\(617\) 35.7962 1.44110 0.720551 0.693402i \(-0.243889\pi\)
0.720551 + 0.693402i \(0.243889\pi\)
\(618\) 3.01289 0.121196
\(619\) 31.2901 1.25765 0.628827 0.777545i \(-0.283535\pi\)
0.628827 + 0.777545i \(0.283535\pi\)
\(620\) −3.65509 −0.146792
\(621\) −10.9799 −0.440607
\(622\) −29.5607 −1.18528
\(623\) 8.34280 0.334247
\(624\) 1.67865 0.0671998
\(625\) 1.00000 0.0400000
\(626\) −9.23518 −0.369112
\(627\) 0 0
\(628\) 14.9308 0.595804
\(629\) 22.7064 0.905362
\(630\) −1.63119 −0.0649881
\(631\) 47.2969 1.88286 0.941429 0.337211i \(-0.109484\pi\)
0.941429 + 0.337211i \(0.109484\pi\)
\(632\) 3.98377 0.158466
\(633\) −21.3340 −0.847949
\(634\) 5.25554 0.208724
\(635\) −3.83782 −0.152299
\(636\) −5.70334 −0.226152
\(637\) 1.43479 0.0568485
\(638\) 0 0
\(639\) −0.687841 −0.0272106
\(640\) 1.00000 0.0395285
\(641\) 3.57172 0.141074 0.0705371 0.997509i \(-0.477529\pi\)
0.0705371 + 0.997509i \(0.477529\pi\)
\(642\) −0.0246035 −0.000971025 0
\(643\) −8.20306 −0.323497 −0.161749 0.986832i \(-0.551713\pi\)
−0.161749 + 0.986832i \(0.551713\pi\)
\(644\) 2.02643 0.0798527
\(645\) 0.825831 0.0325171
\(646\) −31.5992 −1.24325
\(647\) −33.2763 −1.30823 −0.654113 0.756397i \(-0.726958\pi\)
−0.654113 + 0.756397i \(0.726958\pi\)
\(648\) 1.44566 0.0567910
\(649\) 0 0
\(650\) −1.43479 −0.0562771
\(651\) 4.27632 0.167602
\(652\) 3.69439 0.144684
\(653\) −20.5390 −0.803754 −0.401877 0.915694i \(-0.631642\pi\)
−0.401877 + 0.915694i \(0.631642\pi\)
\(654\) −3.76756 −0.147323
\(655\) −19.4691 −0.760720
\(656\) −2.05574 −0.0802634
\(657\) 12.0373 0.469620
\(658\) −4.00288 −0.156048
\(659\) −4.70889 −0.183432 −0.0917161 0.995785i \(-0.529235\pi\)
−0.0917161 + 0.995785i \(0.529235\pi\)
\(660\) 0 0
\(661\) −8.57967 −0.333710 −0.166855 0.985981i \(-0.553361\pi\)
−0.166855 + 0.985981i \(0.553361\pi\)
\(662\) 23.4150 0.910048
\(663\) −12.2311 −0.475018
\(664\) −7.15061 −0.277498
\(665\) 4.33680 0.168174
\(666\) −5.08330 −0.196974
\(667\) 18.4736 0.715302
\(668\) 9.33915 0.361343
\(669\) 1.65966 0.0641663
\(670\) −1.89373 −0.0731613
\(671\) 0 0
\(672\) −1.16996 −0.0451323
\(673\) 37.8109 1.45750 0.728751 0.684779i \(-0.240101\pi\)
0.728751 + 0.684779i \(0.240101\pi\)
\(674\) 2.73973 0.105530
\(675\) −5.41832 −0.208551
\(676\) −10.9414 −0.420822
\(677\) −19.9974 −0.768562 −0.384281 0.923216i \(-0.625551\pi\)
−0.384281 + 0.923216i \(0.625551\pi\)
\(678\) −19.7527 −0.758597
\(679\) 11.3050 0.433848
\(680\) −7.28628 −0.279416
\(681\) −9.57732 −0.367004
\(682\) 0 0
\(683\) −23.1577 −0.886106 −0.443053 0.896495i \(-0.646105\pi\)
−0.443053 + 0.896495i \(0.646105\pi\)
\(684\) 7.07414 0.270487
\(685\) −16.1146 −0.615709
\(686\) −1.00000 −0.0381802
\(687\) −2.20629 −0.0841754
\(688\) −0.705861 −0.0269107
\(689\) −6.99433 −0.266463
\(690\) 2.37085 0.0902568
\(691\) 4.71821 0.179489 0.0897446 0.995965i \(-0.471395\pi\)
0.0897446 + 0.995965i \(0.471395\pi\)
\(692\) −1.89895 −0.0721874
\(693\) 0 0
\(694\) 5.82295 0.221036
\(695\) −16.9156 −0.641646
\(696\) −10.6658 −0.404284
\(697\) 14.9787 0.567360
\(698\) 8.42897 0.319041
\(699\) −26.8588 −1.01589
\(700\) 1.00000 0.0377964
\(701\) −8.58415 −0.324219 −0.162109 0.986773i \(-0.551830\pi\)
−0.162109 + 0.986773i \(0.551830\pi\)
\(702\) 7.77415 0.293416
\(703\) 13.5149 0.509723
\(704\) 0 0
\(705\) −4.68321 −0.176380
\(706\) 23.9658 0.901965
\(707\) −11.8936 −0.447303
\(708\) 14.4855 0.544398
\(709\) −28.0704 −1.05421 −0.527103 0.849802i \(-0.676722\pi\)
−0.527103 + 0.849802i \(0.676722\pi\)
\(710\) 0.421681 0.0158254
\(711\) 6.49828 0.243704
\(712\) −8.34280 −0.312660
\(713\) 7.40681 0.277387
\(714\) 8.52468 0.319028
\(715\) 0 0
\(716\) 6.00288 0.224338
\(717\) −0.368807 −0.0137734
\(718\) −16.0490 −0.598942
\(719\) 30.4817 1.13678 0.568388 0.822761i \(-0.307567\pi\)
0.568388 + 0.822761i \(0.307567\pi\)
\(720\) 1.63119 0.0607908
\(721\) −2.57520 −0.0959054
\(722\) 0.192129 0.00715029
\(723\) 6.57098 0.244377
\(724\) −6.73019 −0.250126
\(725\) 9.11632 0.338572
\(726\) 0 0
\(727\) 1.25367 0.0464961 0.0232480 0.999730i \(-0.492599\pi\)
0.0232480 + 0.999730i \(0.492599\pi\)
\(728\) −1.43479 −0.0531769
\(729\) 21.3758 0.791697
\(730\) −7.37947 −0.273126
\(731\) 5.14310 0.190224
\(732\) −7.74876 −0.286402
\(733\) 7.95330 0.293762 0.146881 0.989154i \(-0.453077\pi\)
0.146881 + 0.989154i \(0.453077\pi\)
\(734\) 2.17504 0.0802820
\(735\) −1.16996 −0.0431547
\(736\) −2.02643 −0.0746954
\(737\) 0 0
\(738\) −3.35331 −0.123437
\(739\) 11.8074 0.434342 0.217171 0.976134i \(-0.430317\pi\)
0.217171 + 0.976134i \(0.430317\pi\)
\(740\) 3.11632 0.114558
\(741\) −7.27998 −0.267437
\(742\) 4.87480 0.178960
\(743\) −41.2546 −1.51348 −0.756742 0.653714i \(-0.773210\pi\)
−0.756742 + 0.653714i \(0.773210\pi\)
\(744\) −4.27632 −0.156778
\(745\) −5.12256 −0.187676
\(746\) −16.5044 −0.604270
\(747\) −11.6640 −0.426763
\(748\) 0 0
\(749\) 0.0210293 0.000768396 0
\(750\) 1.16996 0.0427210
\(751\) −18.2805 −0.667066 −0.333533 0.942738i \(-0.608241\pi\)
−0.333533 + 0.942738i \(0.608241\pi\)
\(752\) 4.00288 0.145970
\(753\) 16.7545 0.610567
\(754\) −13.0800 −0.476346
\(755\) 22.4051 0.815406
\(756\) −5.41832 −0.197062
\(757\) 46.8864 1.70411 0.852057 0.523449i \(-0.175355\pi\)
0.852057 + 0.523449i \(0.175355\pi\)
\(758\) −22.3263 −0.810928
\(759\) 0 0
\(760\) −4.33680 −0.157312
\(761\) 50.8809 1.84443 0.922216 0.386675i \(-0.126376\pi\)
0.922216 + 0.386675i \(0.126376\pi\)
\(762\) −4.49010 −0.162659
\(763\) 3.22024 0.116581
\(764\) 15.9464 0.576919
\(765\) −11.8853 −0.429714
\(766\) 21.4500 0.775019
\(767\) 17.7644 0.641434
\(768\) 1.16996 0.0422174
\(769\) 26.5666 0.958017 0.479008 0.877810i \(-0.340996\pi\)
0.479008 + 0.877810i \(0.340996\pi\)
\(770\) 0 0
\(771\) 33.8039 1.21742
\(772\) −11.7470 −0.422784
\(773\) −30.5526 −1.09890 −0.549450 0.835527i \(-0.685163\pi\)
−0.549450 + 0.835527i \(0.685163\pi\)
\(774\) −1.15139 −0.0413859
\(775\) 3.65509 0.131295
\(776\) −11.3050 −0.405827
\(777\) −3.64598 −0.130799
\(778\) 37.9771 1.36155
\(779\) 8.91536 0.319426
\(780\) −1.67865 −0.0601054
\(781\) 0 0
\(782\) 14.7652 0.528001
\(783\) −49.3951 −1.76524
\(784\) 1.00000 0.0357143
\(785\) −14.9308 −0.532903
\(786\) −22.7781 −0.812468
\(787\) 27.0074 0.962710 0.481355 0.876526i \(-0.340145\pi\)
0.481355 + 0.876526i \(0.340145\pi\)
\(788\) 2.70088 0.0962148
\(789\) −25.6287 −0.912405
\(790\) −3.98377 −0.141736
\(791\) 16.8832 0.600297
\(792\) 0 0
\(793\) −9.50274 −0.337452
\(794\) 24.9220 0.884447
\(795\) 5.70334 0.202277
\(796\) 14.5804 0.516789
\(797\) −37.5786 −1.33110 −0.665551 0.746352i \(-0.731803\pi\)
−0.665551 + 0.746352i \(0.731803\pi\)
\(798\) 5.07390 0.179614
\(799\) −29.1661 −1.03182
\(800\) −1.00000 −0.0353553
\(801\) −13.6087 −0.480839
\(802\) 23.7936 0.840180
\(803\) 0 0
\(804\) −2.21560 −0.0781380
\(805\) −2.02643 −0.0714224
\(806\) −5.24430 −0.184722
\(807\) −3.48478 −0.122670
\(808\) 11.8936 0.418414
\(809\) 37.3166 1.31198 0.655991 0.754769i \(-0.272251\pi\)
0.655991 + 0.754769i \(0.272251\pi\)
\(810\) −1.44566 −0.0507954
\(811\) −43.9822 −1.54443 −0.772213 0.635364i \(-0.780850\pi\)
−0.772213 + 0.635364i \(0.780850\pi\)
\(812\) 9.11632 0.319920
\(813\) −0.236584 −0.00829735
\(814\) 0 0
\(815\) −3.69439 −0.129409
\(816\) −8.52468 −0.298423
\(817\) 3.06118 0.107097
\(818\) −17.4118 −0.608789
\(819\) −2.34041 −0.0817807
\(820\) 2.05574 0.0717897
\(821\) −30.6054 −1.06814 −0.534068 0.845442i \(-0.679337\pi\)
−0.534068 + 0.845442i \(0.679337\pi\)
\(822\) −18.8535 −0.657592
\(823\) −20.5472 −0.716230 −0.358115 0.933677i \(-0.616580\pi\)
−0.358115 + 0.933677i \(0.616580\pi\)
\(824\) 2.57520 0.0897113
\(825\) 0 0
\(826\) −12.3811 −0.430795
\(827\) 45.3404 1.57664 0.788319 0.615266i \(-0.210952\pi\)
0.788319 + 0.615266i \(0.210952\pi\)
\(828\) −3.30550 −0.114874
\(829\) 41.2595 1.43300 0.716500 0.697587i \(-0.245743\pi\)
0.716500 + 0.697587i \(0.245743\pi\)
\(830\) 7.15061 0.248201
\(831\) 5.76975 0.200150
\(832\) 1.43479 0.0497424
\(833\) −7.28628 −0.252455
\(834\) −19.7906 −0.685293
\(835\) −9.33915 −0.323195
\(836\) 0 0
\(837\) −19.8045 −0.684542
\(838\) −6.77394 −0.234002
\(839\) −27.6981 −0.956243 −0.478122 0.878294i \(-0.658682\pi\)
−0.478122 + 0.878294i \(0.658682\pi\)
\(840\) 1.16996 0.0403675
\(841\) 54.1073 1.86577
\(842\) −16.3820 −0.564560
\(843\) 20.8818 0.719209
\(844\) −18.2347 −0.627666
\(845\) 10.9414 0.376395
\(846\) 6.52944 0.224487
\(847\) 0 0
\(848\) −4.87480 −0.167401
\(849\) 34.2473 1.17536
\(850\) 7.28628 0.249917
\(851\) −6.31502 −0.216476
\(852\) 0.493351 0.0169019
\(853\) 41.7746 1.43033 0.715167 0.698954i \(-0.246351\pi\)
0.715167 + 0.698954i \(0.246351\pi\)
\(854\) 6.62309 0.226637
\(855\) −7.07414 −0.241931
\(856\) −0.0210293 −0.000718768 0
\(857\) −45.5421 −1.55569 −0.777845 0.628457i \(-0.783687\pi\)
−0.777845 + 0.628457i \(0.783687\pi\)
\(858\) 0 0
\(859\) −7.44024 −0.253858 −0.126929 0.991912i \(-0.540512\pi\)
−0.126929 + 0.991912i \(0.540512\pi\)
\(860\) 0.705861 0.0240697
\(861\) −2.40514 −0.0819671
\(862\) −35.7762 −1.21854
\(863\) 9.39601 0.319844 0.159922 0.987130i \(-0.448876\pi\)
0.159922 + 0.987130i \(0.448876\pi\)
\(864\) 5.41832 0.184335
\(865\) 1.89895 0.0645664
\(866\) −12.5667 −0.427035
\(867\) 42.2238 1.43400
\(868\) 3.65509 0.124062
\(869\) 0 0
\(870\) 10.6658 0.361603
\(871\) −2.71711 −0.0920658
\(872\) −3.22024 −0.109051
\(873\) −18.4406 −0.624121
\(874\) 8.78825 0.297267
\(875\) −1.00000 −0.0338062
\(876\) −8.63370 −0.291706
\(877\) −9.96978 −0.336655 −0.168328 0.985731i \(-0.553837\pi\)
−0.168328 + 0.985731i \(0.553837\pi\)
\(878\) −34.6171 −1.16827
\(879\) 34.9848 1.18001
\(880\) 0 0
\(881\) −3.94025 −0.132750 −0.0663752 0.997795i \(-0.521143\pi\)
−0.0663752 + 0.997795i \(0.521143\pi\)
\(882\) 1.63119 0.0549250
\(883\) −41.4577 −1.39516 −0.697582 0.716505i \(-0.745741\pi\)
−0.697582 + 0.716505i \(0.745741\pi\)
\(884\) −10.4543 −0.351616
\(885\) −14.4855 −0.486924
\(886\) −18.9816 −0.637698
\(887\) 31.2649 1.04977 0.524886 0.851172i \(-0.324108\pi\)
0.524886 + 0.851172i \(0.324108\pi\)
\(888\) 3.64598 0.122351
\(889\) 3.83782 0.128716
\(890\) 8.34280 0.279651
\(891\) 0 0
\(892\) 1.41856 0.0474969
\(893\) −17.3597 −0.580920
\(894\) −5.99320 −0.200443
\(895\) −6.00288 −0.200654
\(896\) −1.00000 −0.0334077
\(897\) 3.40168 0.113579
\(898\) −13.7748 −0.459670
\(899\) 33.3210 1.11132
\(900\) −1.63119 −0.0543729
\(901\) 35.5192 1.18332
\(902\) 0 0
\(903\) −0.825831 −0.0274819
\(904\) −16.8832 −0.561526
\(905\) 6.73019 0.223719
\(906\) 26.2132 0.870874
\(907\) 11.8441 0.393277 0.196638 0.980476i \(-0.436997\pi\)
0.196638 + 0.980476i \(0.436997\pi\)
\(908\) −8.18601 −0.271662
\(909\) 19.4006 0.643479
\(910\) 1.43479 0.0475629
\(911\) 52.8126 1.74976 0.874879 0.484341i \(-0.160941\pi\)
0.874879 + 0.484341i \(0.160941\pi\)
\(912\) −5.07390 −0.168014
\(913\) 0 0
\(914\) 36.0763 1.19330
\(915\) 7.74876 0.256166
\(916\) −1.88578 −0.0623080
\(917\) 19.4691 0.642926
\(918\) −39.4794 −1.30301
\(919\) 26.4724 0.873243 0.436621 0.899645i \(-0.356175\pi\)
0.436621 + 0.899645i \(0.356175\pi\)
\(920\) 2.02643 0.0668096
\(921\) −15.9142 −0.524390
\(922\) −23.2972 −0.767252
\(923\) 0.605024 0.0199146
\(924\) 0 0
\(925\) −3.11632 −0.102464
\(926\) 31.8729 1.04741
\(927\) 4.20063 0.137967
\(928\) −9.11632 −0.299258
\(929\) 39.7722 1.30488 0.652442 0.757838i \(-0.273744\pi\)
0.652442 + 0.757838i \(0.273744\pi\)
\(930\) 4.27632 0.140226
\(931\) −4.33680 −0.142133
\(932\) −22.9570 −0.751982
\(933\) 34.5849 1.13226
\(934\) 11.5661 0.378455
\(935\) 0 0
\(936\) 2.34041 0.0764988
\(937\) 27.1898 0.888251 0.444126 0.895965i \(-0.353514\pi\)
0.444126 + 0.895965i \(0.353514\pi\)
\(938\) 1.89373 0.0618325
\(939\) 10.8048 0.352602
\(940\) −4.00288 −0.130559
\(941\) 39.2513 1.27956 0.639778 0.768560i \(-0.279026\pi\)
0.639778 + 0.768560i \(0.279026\pi\)
\(942\) −17.4685 −0.569153
\(943\) −4.16583 −0.135658
\(944\) 12.3811 0.402972
\(945\) 5.41832 0.176258
\(946\) 0 0
\(947\) −42.7008 −1.38759 −0.693795 0.720173i \(-0.744062\pi\)
−0.693795 + 0.720173i \(0.744062\pi\)
\(948\) −4.66086 −0.151378
\(949\) −10.5880 −0.343701
\(950\) 4.33680 0.140705
\(951\) −6.14878 −0.199388
\(952\) 7.28628 0.236150
\(953\) −22.7464 −0.736828 −0.368414 0.929662i \(-0.620099\pi\)
−0.368414 + 0.929662i \(0.620099\pi\)
\(954\) −7.95172 −0.257446
\(955\) −15.9464 −0.516012
\(956\) −0.315230 −0.0101953
\(957\) 0 0
\(958\) −13.2561 −0.428287
\(959\) 16.1146 0.520369
\(960\) −1.16996 −0.0377604
\(961\) −17.6403 −0.569042
\(962\) 4.47127 0.144159
\(963\) −0.0343028 −0.00110539
\(964\) 5.61640 0.180892
\(965\) 11.7470 0.378150
\(966\) −2.37085 −0.0762809
\(967\) −5.16517 −0.166101 −0.0830504 0.996545i \(-0.526466\pi\)
−0.0830504 + 0.996545i \(0.526466\pi\)
\(968\) 0 0
\(969\) 36.9698 1.18764
\(970\) 11.3050 0.362983
\(971\) 28.1375 0.902977 0.451488 0.892277i \(-0.350893\pi\)
0.451488 + 0.892277i \(0.350893\pi\)
\(972\) 14.5636 0.467127
\(973\) 16.9156 0.542289
\(974\) −5.42232 −0.173742
\(975\) 1.67865 0.0537599
\(976\) −6.62309 −0.212000
\(977\) 24.0697 0.770058 0.385029 0.922905i \(-0.374191\pi\)
0.385029 + 0.922905i \(0.374191\pi\)
\(978\) −4.32230 −0.138212
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −5.25282 −0.167710
\(982\) −36.0238 −1.14957
\(983\) 20.4577 0.652500 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(984\) 2.40514 0.0766732
\(985\) −2.70088 −0.0860572
\(986\) 66.4241 2.11537
\(987\) 4.68321 0.149068
\(988\) −6.22241 −0.197961
\(989\) −1.43038 −0.0454835
\(990\) 0 0
\(991\) −28.8003 −0.914870 −0.457435 0.889243i \(-0.651232\pi\)
−0.457435 + 0.889243i \(0.651232\pi\)
\(992\) −3.65509 −0.116049
\(993\) −27.3946 −0.869342
\(994\) −0.421681 −0.0133749
\(995\) −14.5804 −0.462230
\(996\) 8.36595 0.265085
\(997\) −18.0825 −0.572680 −0.286340 0.958128i \(-0.592439\pi\)
−0.286340 + 0.958128i \(0.592439\pi\)
\(998\) 3.26974 0.103502
\(999\) 16.8852 0.534224
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.cx.1.4 6
11.10 odd 2 8470.2.a.dd.1.4 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cx.1.4 6 1.1 even 1 trivial
8470.2.a.dd.1.4 yes 6 11.10 odd 2