Properties

Label 714.2.a.f.1.1
Level $714$
Weight $2$
Character 714.1
Self dual yes
Analytic conductor $5.701$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 714.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} +2.00000 q^{30} +1.00000 q^{32} -4.00000 q^{33} +1.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} +10.0000 q^{41} -1.00000 q^{42} -4.00000 q^{43} +4.00000 q^{44} -2.00000 q^{45} +8.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -8.00000 q^{55} +1.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} +1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -4.00000 q^{66} -12.0000 q^{67} +1.00000 q^{68} -8.00000 q^{69} -2.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +4.00000 q^{77} +2.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -12.0000 q^{83} -1.00000 q^{84} -2.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} +4.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} -2.00000 q^{91} +8.00000 q^{92} -8.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 −0.447214
\(21\) −1.00000 −0.218218
\(22\) 4.00000 0.852803
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 1.00000 0.171499
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) −2.00000 −0.316228
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) −1.00000 −0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) −2.00000 −0.298142
\(46\) 8.00000 1.17954
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −1.00000 −0.140028
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) −8.00000 −1.07872
\(56\) 1.00000 0.133631
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 2.00000 0.258199
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −4.00000 −0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 1.00000 0.121268
\(69\) −8.00000 −0.963087
\(70\) −2.00000 −0.239046
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 4.00000 0.455842
\(78\) 2.00000 0.226455
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −1.00000 −0.109109
\(85\) −2.00000 −0.216930
\(86\) −4.00000 −0.431331
\(87\) −6.00000 −0.643268
\(88\) 4.00000 0.426401
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −2.00000 −0.210819
\(91\) −2.00000 −0.209657
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) 4.00000 0.402015
\(100\) −1.00000 −0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) 2.00000 0.195180
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −8.00000 −0.762770
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) −4.00000 −0.374634
\(115\) −16.0000 −1.49201
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) −4.00000 −0.368230
\(119\) 1.00000 0.0916698
\(120\) 2.00000 0.182574
\(121\) 5.00000 0.454545
\(122\) 6.00000 0.543214
\(123\) −10.0000 −0.901670
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 1.00000 0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 4.00000 0.350823
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −4.00000 −0.348155
\(133\) 4.00000 0.346844
\(134\) −12.0000 −1.03664
\(135\) 2.00000 0.172133
\(136\) 1.00000 0.0857493
\(137\) −22.0000 −1.87959 −0.939793 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) −8.00000 −0.681005
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) −12.0000 −0.996546
\(146\) −6.00000 −0.496564
\(147\) −1.00000 −0.0824786
\(148\) −2.00000 −0.164399
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 4.00000 0.324443
\(153\) 1.00000 0.0808452
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) −2.00000 −0.158114
\(161\) 8.00000 0.630488
\(162\) 1.00000 0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 10.0000 0.780869
\(165\) 8.00000 0.622799
\(166\) −12.0000 −0.931381
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −6.00000 −0.454859
\(175\) −1.00000 −0.0755929
\(176\) 4.00000 0.301511
\(177\) 4.00000 0.300658
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −2.00000 −0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −2.00000 −0.148250
\(183\) −6.00000 −0.443533
\(184\) 8.00000 0.589768
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) −8.00000 −0.580381
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 2.00000 0.143592
\(195\) −4.00000 −0.286446
\(196\) 1.00000 0.0714286
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 4.00000 0.284268
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 12.0000 0.846415
\(202\) −10.0000 −0.703598
\(203\) 6.00000 0.421117
\(204\) −1.00000 −0.0700140
\(205\) −20.0000 −1.39686
\(206\) 8.00000 0.557386
\(207\) 8.00000 0.556038
\(208\) −2.00000 −0.138675
\(209\) 16.0000 1.10674
\(210\) 2.00000 0.138013
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 6.00000 0.412082
\(213\) 8.00000 0.548151
\(214\) −12.0000 −0.820303
\(215\) 8.00000 0.545595
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 6.00000 0.405442
\(220\) −8.00000 −0.539360
\(221\) −2.00000 −0.134535
\(222\) 2.00000 0.134231
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 18.0000 1.19734
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) −4.00000 −0.264906
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −16.0000 −1.05501
\(231\) −4.00000 −0.263181
\(232\) 6.00000 0.393919
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 1.00000 0.0648204
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 2.00000 0.129099
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) 6.00000 0.384111
\(245\) −2.00000 −0.127775
\(246\) −10.0000 −0.637577
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 12.0000 0.758947
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 1.00000 0.0629941
\(253\) 32.0000 2.01182
\(254\) 16.0000 1.00393
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 4.00000 0.249029
\(259\) −2.00000 −0.124274
\(260\) 4.00000 0.248069
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) −4.00000 −0.246183
\(265\) −12.0000 −0.737154
\(266\) 4.00000 0.245256
\(267\) 6.00000 0.367194
\(268\) −12.0000 −0.733017
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 2.00000 0.121716
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 1.00000 0.0606339
\(273\) 2.00000 0.121046
\(274\) −22.0000 −1.32907
\(275\) −4.00000 −0.241209
\(276\) −8.00000 −0.481543
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −12.0000 −0.719712
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −8.00000 −0.474713
\(285\) 8.00000 0.473879
\(286\) −8.00000 −0.473050
\(287\) 10.0000 0.590281
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) −12.0000 −0.704664
\(291\) −2.00000 −0.117242
\(292\) −6.00000 −0.351123
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 8.00000 0.465778
\(296\) −2.00000 −0.116248
\(297\) −4.00000 −0.232104
\(298\) 22.0000 1.27443
\(299\) −16.0000 −0.925304
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) 8.00000 0.460348
\(303\) 10.0000 0.574485
\(304\) 4.00000 0.229416
\(305\) −12.0000 −0.687118
\(306\) 1.00000 0.0571662
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 4.00000 0.227921
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 2.00000 0.113228
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −2.00000 −0.112867
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) −6.00000 −0.336463
\(319\) 24.0000 1.34374
\(320\) −2.00000 −0.111803
\(321\) 12.0000 0.669775
\(322\) 8.00000 0.445823
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −20.0000 −1.10770
\(327\) 10.0000 0.553001
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) 8.00000 0.440386
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −12.0000 −0.658586
\(333\) −2.00000 −0.109599
\(334\) 8.00000 0.437741
\(335\) 24.0000 1.31126
\(336\) −1.00000 −0.0545545
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −9.00000 −0.489535
\(339\) −18.0000 −0.977626
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) 16.0000 0.861411
\(346\) 6.00000 0.322562
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) −6.00000 −0.321634
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 2.00000 0.106752
\(352\) 4.00000 0.213201
\(353\) 34.0000 1.80964 0.904819 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 4.00000 0.212598
\(355\) 16.0000 0.849192
\(356\) −6.00000 −0.317999
\(357\) −1.00000 −0.0529256
\(358\) −12.0000 −0.634220
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) −5.00000 −0.262432
\(364\) −2.00000 −0.104828
\(365\) 12.0000 0.628109
\(366\) −6.00000 −0.313625
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 8.00000 0.417029
\(369\) 10.0000 0.520579
\(370\) 4.00000 0.207950
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 4.00000 0.206835
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) −1.00000 −0.0514344
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −8.00000 −0.410391
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −8.00000 −0.407718
\(386\) −14.0000 −0.712581
\(387\) −4.00000 −0.203331
\(388\) 2.00000 0.101535
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) −4.00000 −0.202548
\(391\) 8.00000 0.404577
\(392\) 1.00000 0.0505076
\(393\) −12.0000 −0.605320
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −8.00000 −0.401004
\(399\) −4.00000 −0.200250
\(400\) −1.00000 −0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 12.0000 0.598506
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) −2.00000 −0.0993808
\(406\) 6.00000 0.297775
\(407\) −8.00000 −0.396545
\(408\) −1.00000 −0.0495074
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) −20.0000 −0.987730
\(411\) 22.0000 1.08518
\(412\) 8.00000 0.394132
\(413\) −4.00000 −0.196827
\(414\) 8.00000 0.393179
\(415\) 24.0000 1.17811
\(416\) −2.00000 −0.0980581
\(417\) 12.0000 0.587643
\(418\) 16.0000 0.782586
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 2.00000 0.0975900
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −1.00000 −0.0485071
\(426\) 8.00000 0.387601
\(427\) 6.00000 0.290360
\(428\) −12.0000 −0.580042
\(429\) 8.00000 0.386244
\(430\) 8.00000 0.385794
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) −10.0000 −0.478913
\(437\) 32.0000 1.53077
\(438\) 6.00000 0.286691
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) −8.00000 −0.381385
\(441\) 1.00000 0.0476190
\(442\) −2.00000 −0.0951303
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 2.00000 0.0949158
\(445\) 12.0000 0.568855
\(446\) −16.0000 −0.757622
\(447\) −22.0000 −1.04056
\(448\) 1.00000 0.0472456
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 40.0000 1.88353
\(452\) 18.0000 0.846649
\(453\) −8.00000 −0.375873
\(454\) −20.0000 −0.938647
\(455\) 4.00000 0.187523
\(456\) −4.00000 −0.187317
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 6.00000 0.280362
\(459\) −1.00000 −0.0466760
\(460\) −16.0000 −0.746004
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) −4.00000 −0.186097
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) −4.00000 −0.184115
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 1.00000 0.0458349
\(477\) 6.00000 0.274721
\(478\) 16.0000 0.731823
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 2.00000 0.0912871
\(481\) 4.00000 0.182384
\(482\) 18.0000 0.819878
\(483\) −8.00000 −0.364013
\(484\) 5.00000 0.227273
\(485\) −4.00000 −0.181631
\(486\) −1.00000 −0.0453609
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 6.00000 0.271607
\(489\) 20.0000 0.904431
\(490\) −2.00000 −0.0903508
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) −10.0000 −0.450835
\(493\) 6.00000 0.270226
\(494\) −8.00000 −0.359937
\(495\) −8.00000 −0.359573
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 12.0000 0.537733
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 12.0000 0.536656
\(501\) −8.00000 −0.357414
\(502\) −4.00000 −0.178529
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 1.00000 0.0445435
\(505\) 20.0000 0.889988
\(506\) 32.0000 1.42257
\(507\) 9.00000 0.399704
\(508\) 16.0000 0.709885
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 2.00000 0.0885615
\(511\) −6.00000 −0.265424
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 2.00000 0.0882162
\(515\) −16.0000 −0.705044
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) −2.00000 −0.0878750
\(519\) −6.00000 −0.263371
\(520\) 4.00000 0.175412
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 6.00000 0.262613
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 12.0000 0.524222
\(525\) 1.00000 0.0436436
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) −4.00000 −0.174078
\(529\) 41.0000 1.78261
\(530\) −12.0000 −0.521247
\(531\) −4.00000 −0.173585
\(532\) 4.00000 0.173422
\(533\) −20.0000 −0.866296
\(534\) 6.00000 0.259645
\(535\) 24.0000 1.03761
\(536\) −12.0000 −0.518321
\(537\) 12.0000 0.517838
\(538\) −26.0000 −1.12094
\(539\) 4.00000 0.172292
\(540\) 2.00000 0.0860663
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) 1.00000 0.0428746
\(545\) 20.0000 0.856706
\(546\) 2.00000 0.0855921
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −22.0000 −0.939793
\(549\) 6.00000 0.256074
\(550\) −4.00000 −0.170561
\(551\) 24.0000 1.02243
\(552\) −8.00000 −0.340503
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) −4.00000 −0.169791
\(556\) −12.0000 −0.508913
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) −2.00000 −0.0845154
\(561\) −4.00000 −0.168880
\(562\) 26.0000 1.09674
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) −36.0000 −1.51453
\(566\) 20.0000 0.840663
\(567\) 1.00000 0.0419961
\(568\) −8.00000 −0.335673
\(569\) −38.0000 −1.59304 −0.796521 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 8.00000 0.335083
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) 10.0000 0.417392
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 1.00000 0.0415945
\(579\) 14.0000 0.581820
\(580\) −12.0000 −0.498273
\(581\) −12.0000 −0.497844
\(582\) −2.00000 −0.0829027
\(583\) 24.0000 0.993978
\(584\) −6.00000 −0.248282
\(585\) 4.00000 0.165380
\(586\) 6.00000 0.247858
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) 8.00000 0.329355
\(591\) 2.00000 0.0822690
\(592\) −2.00000 −0.0821995
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) −4.00000 −0.164122
\(595\) −2.00000 −0.0819920
\(596\) 22.0000 0.901155
\(597\) 8.00000 0.327418
\(598\) −16.0000 −0.654289
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 1.00000 0.0408248
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −4.00000 −0.163028
\(603\) −12.0000 −0.488678
\(604\) 8.00000 0.325515
\(605\) −10.0000 −0.406558
\(606\) 10.0000 0.406222
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 4.00000 0.162221
\(609\) −6.00000 −0.243132
\(610\) −12.0000 −0.485866
\(611\) 0 0
\(612\) 1.00000 0.0404226
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) −28.0000 −1.12999
\(615\) 20.0000 0.806478
\(616\) 4.00000 0.161165
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) −8.00000 −0.321807
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 24.0000 0.962312
\(623\) −6.00000 −0.240385
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) 26.0000 1.03917
\(627\) −16.0000 −0.638978
\(628\) −2.00000 −0.0798087
\(629\) −2.00000 −0.0797452
\(630\) −2.00000 −0.0796819
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 22.0000 0.873732
\(635\) −32.0000 −1.26988
\(636\) −6.00000 −0.237915
\(637\) −2.00000 −0.0792429
\(638\) 24.0000 0.950169
\(639\) −8.00000 −0.316475
\(640\) −2.00000 −0.0790569
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 12.0000 0.473602
\(643\) −36.0000 −1.41970 −0.709851 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) 8.00000 0.315244
\(645\) −8.00000 −0.315000
\(646\) 4.00000 0.157378
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) −16.0000 −0.628055
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −20.0000 −0.783260
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 10.0000 0.391031
\(655\) −24.0000 −0.937758
\(656\) 10.0000 0.390434
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 8.00000 0.311400
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) −4.00000 −0.155464
\(663\) 2.00000 0.0776736
\(664\) −12.0000 −0.465690
\(665\) −8.00000 −0.310227
\(666\) −2.00000 −0.0774984
\(667\) 48.0000 1.85857
\(668\) 8.00000 0.309529
\(669\) 16.0000 0.618596
\(670\) 24.0000 0.927201
\(671\) 24.0000 0.926510
\(672\) −1.00000 −0.0385758
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −14.0000 −0.539260
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −18.0000 −0.691286
\(679\) 2.00000 0.0767530
\(680\) −2.00000 −0.0766965
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 4.00000 0.152944
\(685\) 44.0000 1.68115
\(686\) 1.00000 0.0381802
\(687\) −6.00000 −0.228914
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 16.0000 0.609110
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 6.00000 0.228086
\(693\) 4.00000 0.151947
\(694\) −28.0000 −1.06287
\(695\) 24.0000 0.910372
\(696\) −6.00000 −0.227429
\(697\) 10.0000 0.378777
\(698\) −18.0000 −0.681310
\(699\) −10.0000 −0.378235
\(700\) −1.00000 −0.0377964
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 2.00000 0.0754851
\(703\) −8.00000 −0.301726
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 34.0000 1.27961
\(707\) −10.0000 −0.376089
\(708\) 4.00000 0.150329
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 16.0000 0.600469
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) −1.00000 −0.0374241
\(715\) 16.0000 0.598366
\(716\) −12.0000 −0.448461
\(717\) −16.0000 −0.597531
\(718\) −24.0000 −0.895672
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 8.00000 0.297936
\(722\) −3.00000 −0.111648
\(723\) −18.0000 −0.669427
\(724\) −2.00000 −0.0743294
\(725\) −6.00000 −0.222834
\(726\) −5.00000 −0.185567
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) −4.00000 −0.147945
\(732\) −6.00000 −0.221766
\(733\) −50.0000 −1.84679 −0.923396 0.383849i \(-0.874598\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) −32.0000 −1.18114
\(735\) 2.00000 0.0737711
\(736\) 8.00000 0.294884
\(737\) −48.0000 −1.76810
\(738\) 10.0000 0.368105
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 4.00000 0.147043
\(741\) 8.00000 0.293887
\(742\) 6.00000 0.220267
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) −44.0000 −1.61204
\(746\) −26.0000 −0.951928
\(747\) −12.0000 −0.439057
\(748\) 4.00000 0.146254
\(749\) −12.0000 −0.438470
\(750\) −12.0000 −0.438178
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 4.00000 0.145768
\(754\) −12.0000 −0.437014
\(755\) −16.0000 −0.582300
\(756\) −1.00000 −0.0363696
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 20.0000 0.726433
\(759\) −32.0000 −1.16153
\(760\) −8.00000 −0.290191
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −16.0000 −0.579619
\(763\) −10.0000 −0.362024
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) −1.00000 −0.0360844
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) −8.00000 −0.288300
\(771\) −2.00000 −0.0720282
\(772\) −14.0000 −0.503871
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 2.00000 0.0717496
\(778\) −26.0000 −0.932145
\(779\) 40.0000 1.43315
\(780\) −4.00000 −0.143223
\(781\) −32.0000 −1.14505
\(782\) 8.00000 0.286079
\(783\) −6.00000 −0.214423
\(784\) 1.00000 0.0357143
\(785\) 4.00000 0.142766
\(786\) −12.0000 −0.428026
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 4.00000 0.142134
\(793\) −12.0000 −0.426132
\(794\) 22.0000 0.780751
\(795\) 12.0000 0.425596
\(796\) −8.00000 −0.283552
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) −30.0000 −1.05934
\(803\) −24.0000 −0.846942
\(804\) 12.0000 0.423207
\(805\) −16.0000 −0.563926
\(806\) 0 0
\(807\) 26.0000 0.915243
\(808\) −10.0000 −0.351799
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 6.00000 0.210559
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) 40.0000 1.40114
\(816\) −1.00000 −0.0350070
\(817\) −16.0000 −0.559769
\(818\) 26.0000 0.909069
\(819\) −2.00000 −0.0698857
\(820\) −20.0000 −0.698430
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 22.0000 0.767338
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 8.00000 0.278693
\(825\) 4.00000 0.139262
\(826\) −4.00000 −0.139178
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 8.00000 0.278019
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 24.0000 0.833052
\(831\) 2.00000 0.0693792
\(832\) −2.00000 −0.0693375
\(833\) 1.00000 0.0346479
\(834\) 12.0000 0.415526
\(835\) −16.0000 −0.553703
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) −20.0000 −0.690889
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 2.00000 0.0690066
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) −26.0000 −0.895488
\(844\) 12.0000 0.413057
\(845\) 18.0000 0.619219
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 6.00000 0.206041
\(849\) −20.0000 −0.686398
\(850\) −1.00000 −0.0342997
\(851\) −16.0000 −0.548473
\(852\) 8.00000 0.274075
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 6.00000 0.205316
\(855\) −8.00000 −0.273594
\(856\) −12.0000 −0.410152
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 8.00000 0.273115
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 8.00000 0.272798
\(861\) −10.0000 −0.340799
\(862\) 0 0
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −12.0000 −0.408012
\(866\) −14.0000 −0.475739
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) 12.0000 0.406838
\(871\) 24.0000 0.813209
\(872\) −10.0000 −0.338643
\(873\) 2.00000 0.0676897
\(874\) 32.0000 1.08242
\(875\) 12.0000 0.405674
\(876\) 6.00000 0.202721
\(877\) 54.0000 1.82345 0.911725 0.410801i \(-0.134751\pi\)
0.911725 + 0.410801i \(0.134751\pi\)
\(878\) 8.00000 0.269987
\(879\) −6.00000 −0.202375
\(880\) −8.00000 −0.269680
\(881\) 34.0000 1.14549 0.572745 0.819734i \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) 1.00000 0.0336718
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −2.00000 −0.0672673
\(885\) −8.00000 −0.268917
\(886\) 12.0000 0.403148
\(887\) 40.0000 1.34307 0.671534 0.740973i \(-0.265636\pi\)
0.671534 + 0.740973i \(0.265636\pi\)
\(888\) 2.00000 0.0671156
\(889\) 16.0000 0.536623
\(890\) 12.0000 0.402241
\(891\) 4.00000 0.134005
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −22.0000 −0.735790
\(895\) 24.0000 0.802232
\(896\) 1.00000 0.0334077
\(897\) 16.0000 0.534224
\(898\) 18.0000 0.600668
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) 6.00000 0.199889
\(902\) 40.0000 1.33185
\(903\) 4.00000 0.133112
\(904\) 18.0000 0.598671
\(905\) 4.00000 0.132964
\(906\) −8.00000 −0.265782
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −20.0000 −0.663723
\(909\) −10.0000 −0.331679
\(910\) 4.00000 0.132599
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) −4.00000 −0.132453
\(913\) −48.0000 −1.58857
\(914\) 10.0000 0.330771
\(915\) 12.0000 0.396708
\(916\) 6.00000 0.198246
\(917\) 12.0000 0.396275
\(918\) −1.00000 −0.0330049
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) −16.0000 −0.527504
\(921\) 28.0000 0.922631
\(922\) −2.00000 −0.0658665
\(923\) 16.0000 0.526646
\(924\) −4.00000 −0.131590
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 6.00000 0.196960
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 10.0000 0.327561
\(933\) −24.0000 −0.785725
\(934\) 36.0000 1.17796
\(935\) −8.00000 −0.261628
\(936\) −2.00000 −0.0653720
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) −12.0000 −0.391814
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) 2.00000 0.0651635
\(943\) 80.0000 2.60516
\(944\) −4.00000 −0.130189
\(945\) 2.00000 0.0650600
\(946\) −16.0000 −0.520205
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) −4.00000 −0.129777
\(951\) −22.0000 −0.713399
\(952\) 1.00000 0.0324102
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) −24.0000 −0.775810
\(958\) −16.0000 −0.516937
\(959\) −22.0000 −0.710417
\(960\) 2.00000 0.0645497
\(961\) −31.0000 −1.00000
\(962\) 4.00000 0.128965
\(963\) −12.0000 −0.386695
\(964\) 18.0000 0.579741
\(965\) 28.0000 0.901352
\(966\) −8.00000 −0.257396
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 5.00000 0.160706
\(969\) −4.00000 −0.128499
\(970\) −4.00000 −0.128432
\(971\) −52.0000 −1.66876 −0.834380 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −12.0000 −0.384702
\(974\) 24.0000 0.769010
\(975\) −2.00000 −0.0640513
\(976\) 6.00000 0.192055
\(977\) 50.0000 1.59964 0.799821 0.600239i \(-0.204928\pi\)
0.799821 + 0.600239i \(0.204928\pi\)
\(978\) 20.0000 0.639529
\(979\) −24.0000 −0.767043
\(980\) −2.00000 −0.0638877
\(981\) −10.0000 −0.319275
\(982\) −20.0000 −0.638226
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −10.0000 −0.318788
\(985\) 4.00000 0.127451
\(986\) 6.00000 0.191079
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) −32.0000 −1.01754
\(990\) −8.00000 −0.254257
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) −8.00000 −0.253745
\(995\) 16.0000 0.507234
\(996\) 12.0000 0.380235
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) −20.0000 −0.633089
\(999\) 2.00000 0.0632772
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 714.2.a.f.1.1 1
3.2 odd 2 2142.2.a.h.1.1 1
4.3 odd 2 5712.2.a.o.1.1 1
7.6 odd 2 4998.2.a.bq.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
714.2.a.f.1.1 1 1.1 even 1 trivial
2142.2.a.h.1.1 1 3.2 odd 2
4998.2.a.bq.1.1 1 7.6 odd 2
5712.2.a.o.1.1 1 4.3 odd 2