Properties

Label 3234.2.a.bg.1.3
Level 3234
Weight 2
Character 3234.1
Self dual yes
Analytic conductor 25.824
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.523976\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.20147 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.20147 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.20147 q^{10} +1.00000 q^{11} -1.00000 q^{12} +6.40294 q^{13} -3.20147 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -1.15352 q^{19} +3.20147 q^{20} +1.00000 q^{22} +1.95205 q^{23} -1.00000 q^{24} +5.24943 q^{25} +6.40294 q^{26} -1.00000 q^{27} +7.24943 q^{29} -3.20147 q^{30} -10.4989 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +5.15352 q^{37} -1.15352 q^{38} -6.40294 q^{39} +3.20147 q^{40} -8.24943 q^{41} +5.15352 q^{43} +1.00000 q^{44} +3.20147 q^{45} +1.95205 q^{46} +6.04795 q^{47} -1.00000 q^{48} +5.24943 q^{50} +1.00000 q^{51} +6.40294 q^{52} -6.40294 q^{53} -1.00000 q^{54} +3.20147 q^{55} +1.15352 q^{57} +7.24943 q^{58} -11.5565 q^{59} -3.20147 q^{60} -9.70032 q^{61} -10.4989 q^{62} +1.00000 q^{64} +20.4989 q^{65} -1.00000 q^{66} +6.24943 q^{67} -1.00000 q^{68} -1.95205 q^{69} +5.24943 q^{71} +1.00000 q^{72} +2.09591 q^{73} +5.15352 q^{74} -5.24943 q^{75} -1.15352 q^{76} -6.40294 q^{78} -5.60442 q^{79} +3.20147 q^{80} +1.00000 q^{81} -8.24943 q^{82} -6.55646 q^{83} -3.20147 q^{85} +5.15352 q^{86} -7.24943 q^{87} +1.00000 q^{88} +18.4029 q^{89} +3.20147 q^{90} +1.95205 q^{92} +10.4989 q^{93} +6.04795 q^{94} -3.69296 q^{95} -1.00000 q^{96} +5.49885 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 3q^{8} + 3q^{9} + 3q^{11} - 3q^{12} + 3q^{16} - 3q^{17} + 3q^{18} + 3q^{19} + 3q^{22} + 9q^{23} - 3q^{24} + 3q^{25} - 3q^{27} + 9q^{29} - 6q^{31} + 3q^{32} - 3q^{33} - 3q^{34} + 3q^{36} + 9q^{37} + 3q^{38} - 12q^{41} + 9q^{43} + 3q^{44} + 9q^{46} + 15q^{47} - 3q^{48} + 3q^{50} + 3q^{51} - 3q^{54} - 3q^{57} + 9q^{58} - 9q^{59} + 6q^{61} - 6q^{62} + 3q^{64} + 36q^{65} - 3q^{66} + 6q^{67} - 3q^{68} - 9q^{69} + 3q^{71} + 3q^{72} + 9q^{74} - 3q^{75} + 3q^{76} + 12q^{79} + 3q^{81} - 12q^{82} + 6q^{83} + 9q^{86} - 9q^{87} + 3q^{88} + 36q^{89} + 9q^{92} + 6q^{93} + 15q^{94} - 24q^{95} - 3q^{96} - 9q^{97} + 3q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 3.20147 1.43174 0.715871 0.698233i \(-0.246030\pi\)
0.715871 + 0.698233i \(0.246030\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.20147 1.01239
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) 6.40294 1.77586 0.887929 0.459981i \(-0.152144\pi\)
0.887929 + 0.459981i \(0.152144\pi\)
\(14\) 0 0
\(15\) −3.20147 −0.826617
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.15352 −0.264636 −0.132318 0.991207i \(-0.542242\pi\)
−0.132318 + 0.991207i \(0.542242\pi\)
\(20\) 3.20147 0.715871
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 1.95205 0.407030 0.203515 0.979072i \(-0.434763\pi\)
0.203515 + 0.979072i \(0.434763\pi\)
\(24\) −1.00000 −0.204124
\(25\) 5.24943 1.04989
\(26\) 6.40294 1.25572
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.24943 1.34618 0.673092 0.739559i \(-0.264966\pi\)
0.673092 + 0.739559i \(0.264966\pi\)
\(30\) −3.20147 −0.584506
\(31\) −10.4989 −1.88565 −0.942825 0.333289i \(-0.891841\pi\)
−0.942825 + 0.333289i \(0.891841\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 5.15352 0.847233 0.423617 0.905842i \(-0.360760\pi\)
0.423617 + 0.905842i \(0.360760\pi\)
\(38\) −1.15352 −0.187126
\(39\) −6.40294 −1.02529
\(40\) 3.20147 0.506197
\(41\) −8.24943 −1.28834 −0.644172 0.764881i \(-0.722798\pi\)
−0.644172 + 0.764881i \(0.722798\pi\)
\(42\) 0 0
\(43\) 5.15352 0.785904 0.392952 0.919559i \(-0.371454\pi\)
0.392952 + 0.919559i \(0.371454\pi\)
\(44\) 1.00000 0.150756
\(45\) 3.20147 0.477247
\(46\) 1.95205 0.287814
\(47\) 6.04795 0.882185 0.441092 0.897462i \(-0.354591\pi\)
0.441092 + 0.897462i \(0.354591\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 5.24943 0.742381
\(51\) 1.00000 0.140028
\(52\) 6.40294 0.887929
\(53\) −6.40294 −0.879512 −0.439756 0.898117i \(-0.644935\pi\)
−0.439756 + 0.898117i \(0.644935\pi\)
\(54\) −1.00000 −0.136083
\(55\) 3.20147 0.431686
\(56\) 0 0
\(57\) 1.15352 0.152787
\(58\) 7.24943 0.951896
\(59\) −11.5565 −1.50452 −0.752262 0.658864i \(-0.771037\pi\)
−0.752262 + 0.658864i \(0.771037\pi\)
\(60\) −3.20147 −0.413308
\(61\) −9.70032 −1.24200 −0.621000 0.783811i \(-0.713273\pi\)
−0.621000 + 0.783811i \(0.713273\pi\)
\(62\) −10.4989 −1.33336
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 20.4989 2.54257
\(66\) −1.00000 −0.123091
\(67\) 6.24943 0.763489 0.381744 0.924268i \(-0.375323\pi\)
0.381744 + 0.924268i \(0.375323\pi\)
\(68\) −1.00000 −0.121268
\(69\) −1.95205 −0.234999
\(70\) 0 0
\(71\) 5.24943 0.622992 0.311496 0.950247i \(-0.399170\pi\)
0.311496 + 0.950247i \(0.399170\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.09591 0.245307 0.122654 0.992450i \(-0.460860\pi\)
0.122654 + 0.992450i \(0.460860\pi\)
\(74\) 5.15352 0.599084
\(75\) −5.24943 −0.606151
\(76\) −1.15352 −0.132318
\(77\) 0 0
\(78\) −6.40294 −0.724991
\(79\) −5.60442 −0.630546 −0.315273 0.949001i \(-0.602096\pi\)
−0.315273 + 0.949001i \(0.602096\pi\)
\(80\) 3.20147 0.357935
\(81\) 1.00000 0.111111
\(82\) −8.24943 −0.910997
\(83\) −6.55646 −0.719665 −0.359833 0.933017i \(-0.617166\pi\)
−0.359833 + 0.933017i \(0.617166\pi\)
\(84\) 0 0
\(85\) −3.20147 −0.347248
\(86\) 5.15352 0.555718
\(87\) −7.24943 −0.777220
\(88\) 1.00000 0.106600
\(89\) 18.4029 1.95071 0.975354 0.220645i \(-0.0708163\pi\)
0.975354 + 0.220645i \(0.0708163\pi\)
\(90\) 3.20147 0.337465
\(91\) 0 0
\(92\) 1.95205 0.203515
\(93\) 10.4989 1.08868
\(94\) 6.04795 0.623799
\(95\) −3.69296 −0.378890
\(96\) −1.00000 −0.102062
\(97\) 5.49885 0.558324 0.279162 0.960244i \(-0.409943\pi\)
0.279162 + 0.960244i \(0.409943\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 5.24943 0.524943
\(101\) 19.9594 1.98604 0.993018 0.117965i \(-0.0376372\pi\)
0.993018 + 0.117965i \(0.0376372\pi\)
\(102\) 1.00000 0.0990148
\(103\) −0.307039 −0.0302535 −0.0151267 0.999886i \(-0.504815\pi\)
−0.0151267 + 0.999886i \(0.504815\pi\)
\(104\) 6.40294 0.627860
\(105\) 0 0
\(106\) −6.40294 −0.621909
\(107\) −9.05531 −0.875410 −0.437705 0.899119i \(-0.644209\pi\)
−0.437705 + 0.899119i \(0.644209\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −9.70032 −0.929122 −0.464561 0.885541i \(-0.653788\pi\)
−0.464561 + 0.885541i \(0.653788\pi\)
\(110\) 3.20147 0.305248
\(111\) −5.15352 −0.489150
\(112\) 0 0
\(113\) −12.4989 −1.17579 −0.587896 0.808936i \(-0.700044\pi\)
−0.587896 + 0.808936i \(0.700044\pi\)
\(114\) 1.15352 0.108037
\(115\) 6.24943 0.582762
\(116\) 7.24943 0.673092
\(117\) 6.40294 0.591952
\(118\) −11.5565 −1.06386
\(119\) 0 0
\(120\) −3.20147 −0.292253
\(121\) 1.00000 0.0909091
\(122\) −9.70032 −0.878226
\(123\) 8.24943 0.743826
\(124\) −10.4989 −0.942825
\(125\) 0.798528 0.0714225
\(126\) 0 0
\(127\) 8.35499 0.741386 0.370693 0.928756i \(-0.379120\pi\)
0.370693 + 0.928756i \(0.379120\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.15352 −0.453742
\(130\) 20.4989 1.79787
\(131\) −15.3047 −1.33718 −0.668591 0.743631i \(-0.733102\pi\)
−0.668591 + 0.743631i \(0.733102\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) 6.24943 0.539868
\(135\) −3.20147 −0.275539
\(136\) −1.00000 −0.0857493
\(137\) −6.40294 −0.547040 −0.273520 0.961866i \(-0.588188\pi\)
−0.273520 + 0.961866i \(0.588188\pi\)
\(138\) −1.95205 −0.166169
\(139\) −21.7483 −1.84466 −0.922332 0.386398i \(-0.873719\pi\)
−0.922332 + 0.386398i \(0.873719\pi\)
\(140\) 0 0
\(141\) −6.04795 −0.509330
\(142\) 5.24943 0.440522
\(143\) 6.40294 0.535441
\(144\) 1.00000 0.0833333
\(145\) 23.2088 1.92739
\(146\) 2.09591 0.173458
\(147\) 0 0
\(148\) 5.15352 0.423617
\(149\) 6.84648 0.560886 0.280443 0.959871i \(-0.409519\pi\)
0.280443 + 0.959871i \(0.409519\pi\)
\(150\) −5.24943 −0.428614
\(151\) 14.8538 1.20879 0.604394 0.796685i \(-0.293415\pi\)
0.604394 + 0.796685i \(0.293415\pi\)
\(152\) −1.15352 −0.0935628
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) −33.6118 −2.69976
\(156\) −6.40294 −0.512646
\(157\) 9.46056 0.755035 0.377517 0.926002i \(-0.376778\pi\)
0.377517 + 0.926002i \(0.376778\pi\)
\(158\) −5.60442 −0.445863
\(159\) 6.40294 0.507787
\(160\) 3.20147 0.253099
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 13.0553 1.02257 0.511286 0.859411i \(-0.329169\pi\)
0.511286 + 0.859411i \(0.329169\pi\)
\(164\) −8.24943 −0.644172
\(165\) −3.20147 −0.249234
\(166\) −6.55646 −0.508880
\(167\) 6.70998 0.519234 0.259617 0.965712i \(-0.416404\pi\)
0.259617 + 0.965712i \(0.416404\pi\)
\(168\) 0 0
\(169\) 27.9977 2.15367
\(170\) −3.20147 −0.245542
\(171\) −1.15352 −0.0882118
\(172\) 5.15352 0.392952
\(173\) 5.50115 0.418245 0.209122 0.977889i \(-0.432939\pi\)
0.209122 + 0.977889i \(0.432939\pi\)
\(174\) −7.24943 −0.549578
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 11.5565 0.868637
\(178\) 18.4029 1.37936
\(179\) 1.65237 0.123504 0.0617520 0.998092i \(-0.480331\pi\)
0.0617520 + 0.998092i \(0.480331\pi\)
\(180\) 3.20147 0.238624
\(181\) −24.9018 −1.85094 −0.925468 0.378826i \(-0.876328\pi\)
−0.925468 + 0.378826i \(0.876328\pi\)
\(182\) 0 0
\(183\) 9.70032 0.717068
\(184\) 1.95205 0.143907
\(185\) 16.4989 1.21302
\(186\) 10.4989 0.769813
\(187\) −1.00000 −0.0731272
\(188\) 6.04795 0.441092
\(189\) 0 0
\(190\) −3.69296 −0.267916
\(191\) −4.80589 −0.347742 −0.173871 0.984768i \(-0.555628\pi\)
−0.173871 + 0.984768i \(0.555628\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 8.70998 0.626958 0.313479 0.949595i \(-0.398505\pi\)
0.313479 + 0.949595i \(0.398505\pi\)
\(194\) 5.49885 0.394794
\(195\) −20.4989 −1.46795
\(196\) 0 0
\(197\) 0.654669 0.0466433 0.0233216 0.999728i \(-0.492576\pi\)
0.0233216 + 0.999728i \(0.492576\pi\)
\(198\) 1.00000 0.0710669
\(199\) 7.69296 0.545340 0.272670 0.962108i \(-0.412093\pi\)
0.272670 + 0.962108i \(0.412093\pi\)
\(200\) 5.24943 0.371190
\(201\) −6.24943 −0.440800
\(202\) 19.9594 1.40434
\(203\) 0 0
\(204\) 1.00000 0.0700140
\(205\) −26.4103 −1.84458
\(206\) −0.307039 −0.0213924
\(207\) 1.95205 0.135677
\(208\) 6.40294 0.443964
\(209\) −1.15352 −0.0797906
\(210\) 0 0
\(211\) 14.5948 1.00474 0.502372 0.864651i \(-0.332461\pi\)
0.502372 + 0.864651i \(0.332461\pi\)
\(212\) −6.40294 −0.439756
\(213\) −5.24943 −0.359685
\(214\) −9.05531 −0.619009
\(215\) 16.4989 1.12521
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −9.70032 −0.656989
\(219\) −2.09591 −0.141628
\(220\) 3.20147 0.215843
\(221\) −6.40294 −0.430709
\(222\) −5.15352 −0.345882
\(223\) 14.9018 0.997898 0.498949 0.866631i \(-0.333719\pi\)
0.498949 + 0.866631i \(0.333719\pi\)
\(224\) 0 0
\(225\) 5.24943 0.349962
\(226\) −12.4989 −0.831411
\(227\) 0.249425 0.0165549 0.00827746 0.999966i \(-0.497365\pi\)
0.00827746 + 0.999966i \(0.497365\pi\)
\(228\) 1.15352 0.0763937
\(229\) 4.09591 0.270665 0.135333 0.990800i \(-0.456790\pi\)
0.135333 + 0.990800i \(0.456790\pi\)
\(230\) 6.24943 0.412075
\(231\) 0 0
\(232\) 7.24943 0.475948
\(233\) 11.3070 0.740749 0.370374 0.928883i \(-0.379229\pi\)
0.370374 + 0.928883i \(0.379229\pi\)
\(234\) 6.40294 0.418574
\(235\) 19.3624 1.26306
\(236\) −11.5565 −0.752262
\(237\) 5.60442 0.364046
\(238\) 0 0
\(239\) 6.59476 0.426579 0.213290 0.976989i \(-0.431582\pi\)
0.213290 + 0.976989i \(0.431582\pi\)
\(240\) −3.20147 −0.206654
\(241\) −17.1129 −1.10234 −0.551170 0.834393i \(-0.685819\pi\)
−0.551170 + 0.834393i \(0.685819\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −9.70032 −0.621000
\(245\) 0 0
\(246\) 8.24943 0.525964
\(247\) −7.38592 −0.469955
\(248\) −10.4989 −0.666678
\(249\) 6.55646 0.415499
\(250\) 0.798528 0.0505033
\(251\) −7.95941 −0.502393 −0.251197 0.967936i \(-0.580824\pi\)
−0.251197 + 0.967936i \(0.580824\pi\)
\(252\) 0 0
\(253\) 1.95205 0.122724
\(254\) 8.35499 0.524239
\(255\) 3.20147 0.200484
\(256\) 1.00000 0.0625000
\(257\) 22.4029 1.39746 0.698729 0.715387i \(-0.253749\pi\)
0.698729 + 0.715387i \(0.253749\pi\)
\(258\) −5.15352 −0.320844
\(259\) 0 0
\(260\) 20.4989 1.27128
\(261\) 7.24943 0.448728
\(262\) −15.3047 −0.945530
\(263\) −25.0170 −1.54262 −0.771308 0.636462i \(-0.780397\pi\)
−0.771308 + 0.636462i \(0.780397\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −20.4989 −1.25923
\(266\) 0 0
\(267\) −18.4029 −1.12624
\(268\) 6.24943 0.381744
\(269\) −8.10327 −0.494065 −0.247032 0.969007i \(-0.579455\pi\)
−0.247032 + 0.969007i \(0.579455\pi\)
\(270\) −3.20147 −0.194835
\(271\) 4.80589 0.291937 0.145968 0.989289i \(-0.453370\pi\)
0.145968 + 0.989289i \(0.453370\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) −6.40294 −0.386816
\(275\) 5.24943 0.316552
\(276\) −1.95205 −0.117499
\(277\) −11.2088 −0.673474 −0.336737 0.941599i \(-0.609323\pi\)
−0.336737 + 0.941599i \(0.609323\pi\)
\(278\) −21.7483 −1.30437
\(279\) −10.4989 −0.628550
\(280\) 0 0
\(281\) −17.4989 −1.04389 −0.521947 0.852978i \(-0.674794\pi\)
−0.521947 + 0.852978i \(0.674794\pi\)
\(282\) −6.04795 −0.360150
\(283\) −27.4006 −1.62880 −0.814400 0.580304i \(-0.802934\pi\)
−0.814400 + 0.580304i \(0.802934\pi\)
\(284\) 5.24943 0.311496
\(285\) 3.69296 0.218752
\(286\) 6.40294 0.378614
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −16.0000 −0.941176
\(290\) 23.2088 1.36287
\(291\) −5.49885 −0.322348
\(292\) 2.09591 0.122654
\(293\) 19.7483 1.15371 0.576853 0.816848i \(-0.304280\pi\)
0.576853 + 0.816848i \(0.304280\pi\)
\(294\) 0 0
\(295\) −36.9977 −2.15409
\(296\) 5.15352 0.299542
\(297\) −1.00000 −0.0580259
\(298\) 6.84648 0.396606
\(299\) 12.4989 0.722827
\(300\) −5.24943 −0.303076
\(301\) 0 0
\(302\) 14.8538 0.854743
\(303\) −19.9594 −1.14664
\(304\) −1.15352 −0.0661589
\(305\) −31.0553 −1.77822
\(306\) −1.00000 −0.0571662
\(307\) −4.30704 −0.245816 −0.122908 0.992418i \(-0.539222\pi\)
−0.122908 + 0.992418i \(0.539222\pi\)
\(308\) 0 0
\(309\) 0.307039 0.0174668
\(310\) −33.6118 −1.90902
\(311\) 14.7579 0.836846 0.418423 0.908252i \(-0.362583\pi\)
0.418423 + 0.908252i \(0.362583\pi\)
\(312\) −6.40294 −0.362495
\(313\) −20.7506 −1.17289 −0.586446 0.809988i \(-0.699473\pi\)
−0.586446 + 0.809988i \(0.699473\pi\)
\(314\) 9.46056 0.533890
\(315\) 0 0
\(316\) −5.60442 −0.315273
\(317\) 16.1992 0.909836 0.454918 0.890533i \(-0.349669\pi\)
0.454918 + 0.890533i \(0.349669\pi\)
\(318\) 6.40294 0.359059
\(319\) 7.24943 0.405890
\(320\) 3.20147 0.178968
\(321\) 9.05531 0.505418
\(322\) 0 0
\(323\) 1.15352 0.0641835
\(324\) 1.00000 0.0555556
\(325\) 33.6118 1.86445
\(326\) 13.0553 0.723067
\(327\) 9.70032 0.536429
\(328\) −8.24943 −0.455498
\(329\) 0 0
\(330\) −3.20147 −0.176235
\(331\) 21.2471 1.16785 0.583924 0.811808i \(-0.301517\pi\)
0.583924 + 0.811808i \(0.301517\pi\)
\(332\) −6.55646 −0.359833
\(333\) 5.15352 0.282411
\(334\) 6.70998 0.367154
\(335\) 20.0074 1.09312
\(336\) 0 0
\(337\) −25.5159 −1.38994 −0.694969 0.719040i \(-0.744582\pi\)
−0.694969 + 0.719040i \(0.744582\pi\)
\(338\) 27.9977 1.52287
\(339\) 12.4989 0.678844
\(340\) −3.20147 −0.173624
\(341\) −10.4989 −0.568545
\(342\) −1.15352 −0.0623752
\(343\) 0 0
\(344\) 5.15352 0.277859
\(345\) −6.24943 −0.336458
\(346\) 5.50115 0.295744
\(347\) 3.05531 0.164018 0.0820089 0.996632i \(-0.473866\pi\)
0.0820089 + 0.996632i \(0.473866\pi\)
\(348\) −7.24943 −0.388610
\(349\) 4.00736 0.214509 0.107255 0.994232i \(-0.465794\pi\)
0.107255 + 0.994232i \(0.465794\pi\)
\(350\) 0 0
\(351\) −6.40294 −0.341764
\(352\) 1.00000 0.0533002
\(353\) 1.09821 0.0584516 0.0292258 0.999573i \(-0.490696\pi\)
0.0292258 + 0.999573i \(0.490696\pi\)
\(354\) 11.5565 0.614219
\(355\) 16.8059 0.891964
\(356\) 18.4029 0.975354
\(357\) 0 0
\(358\) 1.65237 0.0873305
\(359\) 15.6118 0.823958 0.411979 0.911193i \(-0.364838\pi\)
0.411979 + 0.911193i \(0.364838\pi\)
\(360\) 3.20147 0.168732
\(361\) −17.6694 −0.929968
\(362\) −24.9018 −1.30881
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 6.70998 0.351217
\(366\) 9.70032 0.507044
\(367\) −0.791166 −0.0412985 −0.0206493 0.999787i \(-0.506573\pi\)
−0.0206493 + 0.999787i \(0.506573\pi\)
\(368\) 1.95205 0.101757
\(369\) −8.24943 −0.429448
\(370\) 16.4989 0.857734
\(371\) 0 0
\(372\) 10.4989 0.544340
\(373\) 15.9115 0.823864 0.411932 0.911215i \(-0.364854\pi\)
0.411932 + 0.911215i \(0.364854\pi\)
\(374\) −1.00000 −0.0517088
\(375\) −0.798528 −0.0412358
\(376\) 6.04795 0.311899
\(377\) 46.4177 2.39063
\(378\) 0 0
\(379\) −17.0553 −0.876073 −0.438036 0.898957i \(-0.644326\pi\)
−0.438036 + 0.898957i \(0.644326\pi\)
\(380\) −3.69296 −0.189445
\(381\) −8.35499 −0.428039
\(382\) −4.80589 −0.245891
\(383\) −13.4412 −0.686815 −0.343408 0.939186i \(-0.611581\pi\)
−0.343408 + 0.939186i \(0.611581\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 8.70998 0.443327
\(387\) 5.15352 0.261968
\(388\) 5.49885 0.279162
\(389\) −7.20147 −0.365129 −0.182565 0.983194i \(-0.558440\pi\)
−0.182565 + 0.983194i \(0.558440\pi\)
\(390\) −20.4989 −1.03800
\(391\) −1.95205 −0.0987193
\(392\) 0 0
\(393\) 15.3047 0.772022
\(394\) 0.654669 0.0329818
\(395\) −17.9424 −0.902779
\(396\) 1.00000 0.0502519
\(397\) −12.9424 −0.649560 −0.324780 0.945790i \(-0.605290\pi\)
−0.324780 + 0.945790i \(0.605290\pi\)
\(398\) 7.69296 0.385613
\(399\) 0 0
\(400\) 5.24943 0.262471
\(401\) −16.9018 −0.844035 −0.422018 0.906588i \(-0.638678\pi\)
−0.422018 + 0.906588i \(0.638678\pi\)
\(402\) −6.24943 −0.311693
\(403\) −67.2236 −3.34864
\(404\) 19.9594 0.993018
\(405\) 3.20147 0.159082
\(406\) 0 0
\(407\) 5.15352 0.255450
\(408\) 1.00000 0.0495074
\(409\) 37.0936 1.83416 0.917080 0.398702i \(-0.130539\pi\)
0.917080 + 0.398702i \(0.130539\pi\)
\(410\) −26.4103 −1.30431
\(411\) 6.40294 0.315834
\(412\) −0.307039 −0.0151267
\(413\) 0 0
\(414\) 1.95205 0.0959379
\(415\) −20.9903 −1.03038
\(416\) 6.40294 0.313930
\(417\) 21.7483 1.06502
\(418\) −1.15352 −0.0564205
\(419\) −22.4583 −1.09716 −0.548579 0.836099i \(-0.684831\pi\)
−0.548579 + 0.836099i \(0.684831\pi\)
\(420\) 0 0
\(421\) −23.3453 −1.13778 −0.568891 0.822413i \(-0.692627\pi\)
−0.568891 + 0.822413i \(0.692627\pi\)
\(422\) 14.5948 0.710462
\(423\) 6.04795 0.294062
\(424\) −6.40294 −0.310954
\(425\) −5.24943 −0.254635
\(426\) −5.24943 −0.254335
\(427\) 0 0
\(428\) −9.05531 −0.437705
\(429\) −6.40294 −0.309137
\(430\) 16.4989 0.795645
\(431\) 31.8229 1.53286 0.766428 0.642330i \(-0.222032\pi\)
0.766428 + 0.642330i \(0.222032\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −30.3047 −1.45635 −0.728176 0.685390i \(-0.759632\pi\)
−0.728176 + 0.685390i \(0.759632\pi\)
\(434\) 0 0
\(435\) −23.2088 −1.11278
\(436\) −9.70032 −0.464561
\(437\) −2.25172 −0.107715
\(438\) −2.09591 −0.100146
\(439\) 10.6620 0.508871 0.254435 0.967090i \(-0.418110\pi\)
0.254435 + 0.967090i \(0.418110\pi\)
\(440\) 3.20147 0.152624
\(441\) 0 0
\(442\) −6.40294 −0.304557
\(443\) −36.0553 −1.71304 −0.856520 0.516114i \(-0.827378\pi\)
−0.856520 + 0.516114i \(0.827378\pi\)
\(444\) −5.15352 −0.244575
\(445\) 58.9165 2.79291
\(446\) 14.9018 0.705620
\(447\) −6.84648 −0.323827
\(448\) 0 0
\(449\) 23.4006 1.10434 0.552172 0.833730i \(-0.313799\pi\)
0.552172 + 0.833730i \(0.313799\pi\)
\(450\) 5.24943 0.247460
\(451\) −8.24943 −0.388450
\(452\) −12.4989 −0.587896
\(453\) −14.8538 −0.697894
\(454\) 0.249425 0.0117061
\(455\) 0 0
\(456\) 1.15352 0.0540185
\(457\) −18.6141 −0.870730 −0.435365 0.900254i \(-0.643381\pi\)
−0.435365 + 0.900254i \(0.643381\pi\)
\(458\) 4.09591 0.191389
\(459\) 1.00000 0.0466760
\(460\) 6.24943 0.291381
\(461\) 31.4412 1.46436 0.732182 0.681109i \(-0.238502\pi\)
0.732182 + 0.681109i \(0.238502\pi\)
\(462\) 0 0
\(463\) −22.0959 −1.02688 −0.513442 0.858124i \(-0.671630\pi\)
−0.513442 + 0.858124i \(0.671630\pi\)
\(464\) 7.24943 0.336546
\(465\) 33.6118 1.55871
\(466\) 11.3070 0.523788
\(467\) −21.0576 −0.974430 −0.487215 0.873282i \(-0.661987\pi\)
−0.487215 + 0.873282i \(0.661987\pi\)
\(468\) 6.40294 0.295976
\(469\) 0 0
\(470\) 19.3624 0.893119
\(471\) −9.46056 −0.435920
\(472\) −11.5565 −0.531929
\(473\) 5.15352 0.236959
\(474\) 5.60442 0.257419
\(475\) −6.05531 −0.277837
\(476\) 0 0
\(477\) −6.40294 −0.293171
\(478\) 6.59476 0.301637
\(479\) 21.2088 0.969056 0.484528 0.874776i \(-0.338991\pi\)
0.484528 + 0.874776i \(0.338991\pi\)
\(480\) −3.20147 −0.146127
\(481\) 32.9977 1.50457
\(482\) −17.1129 −0.779473
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 17.6044 0.799375
\(486\) −1.00000 −0.0453609
\(487\) −38.7100 −1.75412 −0.877058 0.480384i \(-0.840497\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(488\) −9.70032 −0.439113
\(489\) −13.0553 −0.590382
\(490\) 0 0
\(491\) 19.1705 0.865154 0.432577 0.901597i \(-0.357604\pi\)
0.432577 + 0.901597i \(0.357604\pi\)
\(492\) 8.24943 0.371913
\(493\) −7.24943 −0.326498
\(494\) −7.38592 −0.332308
\(495\) 3.20147 0.143895
\(496\) −10.4989 −0.471412
\(497\) 0 0
\(498\) 6.55646 0.293802
\(499\) −27.9188 −1.24982 −0.624909 0.780698i \(-0.714864\pi\)
−0.624909 + 0.780698i \(0.714864\pi\)
\(500\) 0.798528 0.0357112
\(501\) −6.70998 −0.299780
\(502\) −7.95941 −0.355246
\(503\) 8.70998 0.388359 0.194179 0.980966i \(-0.437796\pi\)
0.194179 + 0.980966i \(0.437796\pi\)
\(504\) 0 0
\(505\) 63.8995 2.84349
\(506\) 1.95205 0.0867791
\(507\) −27.9977 −1.24342
\(508\) 8.35499 0.370693
\(509\) 1.78887 0.0792901 0.0396451 0.999214i \(-0.487377\pi\)
0.0396451 + 0.999214i \(0.487377\pi\)
\(510\) 3.20147 0.141764
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.15352 0.0509291
\(514\) 22.4029 0.988152
\(515\) −0.982977 −0.0433151
\(516\) −5.15352 −0.226871
\(517\) 6.04795 0.265989
\(518\) 0 0
\(519\) −5.50115 −0.241474
\(520\) 20.4989 0.898934
\(521\) −23.6118 −1.03445 −0.517225 0.855849i \(-0.673035\pi\)
−0.517225 + 0.855849i \(0.673035\pi\)
\(522\) 7.24943 0.317299
\(523\) −43.4006 −1.89778 −0.948889 0.315610i \(-0.897791\pi\)
−0.948889 + 0.315610i \(0.897791\pi\)
\(524\) −15.3047 −0.668591
\(525\) 0 0
\(526\) −25.0170 −1.09079
\(527\) 10.4989 0.457337
\(528\) −1.00000 −0.0435194
\(529\) −19.1895 −0.834327
\(530\) −20.4989 −0.890413
\(531\) −11.5565 −0.501508
\(532\) 0 0
\(533\) −52.8206 −2.28791
\(534\) −18.4029 −0.796373
\(535\) −28.9903 −1.25336
\(536\) 6.24943 0.269934
\(537\) −1.65237 −0.0713050
\(538\) −8.10327 −0.349357
\(539\) 0 0
\(540\) −3.20147 −0.137769
\(541\) 16.9903 0.730472 0.365236 0.930915i \(-0.380988\pi\)
0.365236 + 0.930915i \(0.380988\pi\)
\(542\) 4.80589 0.206431
\(543\) 24.9018 1.06864
\(544\) −1.00000 −0.0428746
\(545\) −31.0553 −1.33026
\(546\) 0 0
\(547\) −1.34533 −0.0575222 −0.0287611 0.999586i \(-0.509156\pi\)
−0.0287611 + 0.999586i \(0.509156\pi\)
\(548\) −6.40294 −0.273520
\(549\) −9.70032 −0.414000
\(550\) 5.24943 0.223836
\(551\) −8.36235 −0.356248
\(552\) −1.95205 −0.0830846
\(553\) 0 0
\(554\) −11.2088 −0.476218
\(555\) −16.4989 −0.700337
\(556\) −21.7483 −0.922332
\(557\) −16.6547 −0.705681 −0.352840 0.935683i \(-0.614784\pi\)
−0.352840 + 0.935683i \(0.614784\pi\)
\(558\) −10.4989 −0.444452
\(559\) 32.9977 1.39565
\(560\) 0 0
\(561\) 1.00000 0.0422200
\(562\) −17.4989 −0.738144
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −6.04795 −0.254665
\(565\) −40.0147 −1.68343
\(566\) −27.4006 −1.15174
\(567\) 0 0
\(568\) 5.24943 0.220261
\(569\) −10.5542 −0.442454 −0.221227 0.975222i \(-0.571006\pi\)
−0.221227 + 0.975222i \(0.571006\pi\)
\(570\) 3.69296 0.154681
\(571\) 4.75057 0.198805 0.0994027 0.995047i \(-0.468307\pi\)
0.0994027 + 0.995047i \(0.468307\pi\)
\(572\) 6.40294 0.267721
\(573\) 4.80589 0.200769
\(574\) 0 0
\(575\) 10.2471 0.427335
\(576\) 1.00000 0.0416667
\(577\) −37.5542 −1.56340 −0.781700 0.623654i \(-0.785647\pi\)
−0.781700 + 0.623654i \(0.785647\pi\)
\(578\) −16.0000 −0.665512
\(579\) −8.70998 −0.361975
\(580\) 23.2088 0.963694
\(581\) 0 0
\(582\) −5.49885 −0.227935
\(583\) −6.40294 −0.265183
\(584\) 2.09591 0.0867292
\(585\) 20.4989 0.847523
\(586\) 19.7483 0.815794
\(587\) 17.0982 0.705718 0.352859 0.935676i \(-0.385209\pi\)
0.352859 + 0.935676i \(0.385209\pi\)
\(588\) 0 0
\(589\) 12.1106 0.499010
\(590\) −36.9977 −1.52317
\(591\) −0.654669 −0.0269295
\(592\) 5.15352 0.211808
\(593\) 32.2471 1.32423 0.662115 0.749402i \(-0.269659\pi\)
0.662115 + 0.749402i \(0.269659\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 6.84648 0.280443
\(597\) −7.69296 −0.314852
\(598\) 12.4989 0.511116
\(599\) 3.39328 0.138646 0.0693229 0.997594i \(-0.477916\pi\)
0.0693229 + 0.997594i \(0.477916\pi\)
\(600\) −5.24943 −0.214307
\(601\) −30.4989 −1.24407 −0.622037 0.782988i \(-0.713695\pi\)
−0.622037 + 0.782988i \(0.713695\pi\)
\(602\) 0 0
\(603\) 6.24943 0.254496
\(604\) 14.8538 0.604394
\(605\) 3.20147 0.130158
\(606\) −19.9594 −0.810796
\(607\) −12.9903 −0.527262 −0.263631 0.964624i \(-0.584920\pi\)
−0.263631 + 0.964624i \(0.584920\pi\)
\(608\) −1.15352 −0.0467814
\(609\) 0 0
\(610\) −31.0553 −1.25739
\(611\) 38.7247 1.56663
\(612\) −1.00000 −0.0404226
\(613\) 20.1992 0.815837 0.407918 0.913018i \(-0.366255\pi\)
0.407918 + 0.913018i \(0.366255\pi\)
\(614\) −4.30704 −0.173818
\(615\) 26.4103 1.06497
\(616\) 0 0
\(617\) 17.0936 0.688163 0.344081 0.938940i \(-0.388190\pi\)
0.344081 + 0.938940i \(0.388190\pi\)
\(618\) 0.307039 0.0123509
\(619\) −41.6694 −1.67483 −0.837417 0.546564i \(-0.815935\pi\)
−0.837417 + 0.546564i \(0.815935\pi\)
\(620\) −33.6118 −1.34988
\(621\) −1.95205 −0.0783330
\(622\) 14.7579 0.591739
\(623\) 0 0
\(624\) −6.40294 −0.256323
\(625\) −23.6907 −0.947626
\(626\) −20.7506 −0.829360
\(627\) 1.15352 0.0460671
\(628\) 9.46056 0.377517
\(629\) −5.15352 −0.205484
\(630\) 0 0
\(631\) −23.8995 −0.951424 −0.475712 0.879601i \(-0.657810\pi\)
−0.475712 + 0.879601i \(0.657810\pi\)
\(632\) −5.60442 −0.222932
\(633\) −14.5948 −0.580089
\(634\) 16.1992 0.643351
\(635\) 26.7483 1.06147
\(636\) 6.40294 0.253893
\(637\) 0 0
\(638\) 7.24943 0.287007
\(639\) 5.24943 0.207664
\(640\) 3.20147 0.126549
\(641\) −14.3070 −0.565094 −0.282547 0.959253i \(-0.591179\pi\)
−0.282547 + 0.959253i \(0.591179\pi\)
\(642\) 9.05531 0.357385
\(643\) 35.4966 1.39985 0.699924 0.714218i \(-0.253217\pi\)
0.699924 + 0.714218i \(0.253217\pi\)
\(644\) 0 0
\(645\) −16.4989 −0.649642
\(646\) 1.15352 0.0453846
\(647\) −23.8156 −0.936286 −0.468143 0.883653i \(-0.655077\pi\)
−0.468143 + 0.883653i \(0.655077\pi\)
\(648\) 1.00000 0.0392837
\(649\) −11.5565 −0.453631
\(650\) 33.6118 1.31836
\(651\) 0 0
\(652\) 13.0553 0.511286
\(653\) −13.7962 −0.539888 −0.269944 0.962876i \(-0.587005\pi\)
−0.269944 + 0.962876i \(0.587005\pi\)
\(654\) 9.70032 0.379313
\(655\) −48.9977 −1.91450
\(656\) −8.24943 −0.322086
\(657\) 2.09591 0.0817691
\(658\) 0 0
\(659\) 18.9447 0.737980 0.368990 0.929433i \(-0.379704\pi\)
0.368990 + 0.929433i \(0.379704\pi\)
\(660\) −3.20147 −0.124617
\(661\) −3.95941 −0.154003 −0.0770016 0.997031i \(-0.524535\pi\)
−0.0770016 + 0.997031i \(0.524535\pi\)
\(662\) 21.2471 0.825793
\(663\) 6.40294 0.248670
\(664\) −6.55646 −0.254440
\(665\) 0 0
\(666\) 5.15352 0.199695
\(667\) 14.1512 0.547937
\(668\) 6.70998 0.259617
\(669\) −14.9018 −0.576137
\(670\) 20.0074 0.772952
\(671\) −9.70032 −0.374477
\(672\) 0 0
\(673\) −8.69066 −0.335000 −0.167500 0.985872i \(-0.553569\pi\)
−0.167500 + 0.985872i \(0.553569\pi\)
\(674\) −25.5159 −0.982835
\(675\) −5.24943 −0.202050
\(676\) 27.9977 1.07683
\(677\) 31.4606 1.20913 0.604564 0.796557i \(-0.293347\pi\)
0.604564 + 0.796557i \(0.293347\pi\)
\(678\) 12.4989 0.480015
\(679\) 0 0
\(680\) −3.20147 −0.122771
\(681\) −0.249425 −0.00955799
\(682\) −10.4989 −0.402022
\(683\) 24.7460 0.946878 0.473439 0.880826i \(-0.343012\pi\)
0.473439 + 0.880826i \(0.343012\pi\)
\(684\) −1.15352 −0.0441059
\(685\) −20.4989 −0.783221
\(686\) 0 0
\(687\) −4.09591 −0.156269
\(688\) 5.15352 0.196476
\(689\) −40.9977 −1.56189
\(690\) −6.24943 −0.237912
\(691\) −25.5542 −0.972126 −0.486063 0.873924i \(-0.661567\pi\)
−0.486063 + 0.873924i \(0.661567\pi\)
\(692\) 5.50115 0.209122
\(693\) 0 0
\(694\) 3.05531 0.115978
\(695\) −69.6265 −2.64108
\(696\) −7.24943 −0.274789
\(697\) 8.24943 0.312469
\(698\) 4.00736 0.151681
\(699\) −11.3070 −0.427671
\(700\) 0 0
\(701\) −14.1512 −0.534484 −0.267242 0.963629i \(-0.586112\pi\)
−0.267242 + 0.963629i \(0.586112\pi\)
\(702\) −6.40294 −0.241664
\(703\) −5.94469 −0.224208
\(704\) 1.00000 0.0376889
\(705\) −19.3624 −0.729228
\(706\) 1.09821 0.0413315
\(707\) 0 0
\(708\) 11.5565 0.434319
\(709\) −10.7506 −0.403746 −0.201873 0.979412i \(-0.564703\pi\)
−0.201873 + 0.979412i \(0.564703\pi\)
\(710\) 16.8059 0.630714
\(711\) −5.60442 −0.210182
\(712\) 18.4029 0.689680
\(713\) −20.4943 −0.767516
\(714\) 0 0
\(715\) 20.4989 0.766614
\(716\) 1.65237 0.0617520
\(717\) −6.59476 −0.246286
\(718\) 15.6118 0.582626
\(719\) −2.75794 −0.102854 −0.0514268 0.998677i \(-0.516377\pi\)
−0.0514268 + 0.998677i \(0.516377\pi\)
\(720\) 3.20147 0.119312
\(721\) 0 0
\(722\) −17.6694 −0.657587
\(723\) 17.1129 0.636437
\(724\) −24.9018 −0.925468
\(725\) 38.0553 1.41334
\(726\) −1.00000 −0.0371135
\(727\) −14.6141 −0.542006 −0.271003 0.962578i \(-0.587355\pi\)
−0.271003 + 0.962578i \(0.587355\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.70998 0.248348
\(731\) −5.15352 −0.190610
\(732\) 9.70032 0.358534
\(733\) −7.10557 −0.262450 −0.131225 0.991353i \(-0.541891\pi\)
−0.131225 + 0.991353i \(0.541891\pi\)
\(734\) −0.791166 −0.0292025
\(735\) 0 0
\(736\) 1.95205 0.0719534
\(737\) 6.24943 0.230201
\(738\) −8.24943 −0.303666
\(739\) −41.4966 −1.52648 −0.763238 0.646118i \(-0.776391\pi\)
−0.763238 + 0.646118i \(0.776391\pi\)
\(740\) 16.4989 0.606510
\(741\) 7.38592 0.271329
\(742\) 0 0
\(743\) −31.9041 −1.17045 −0.585224 0.810872i \(-0.698993\pi\)
−0.585224 + 0.810872i \(0.698993\pi\)
\(744\) 10.4989 0.384907
\(745\) 21.9188 0.803043
\(746\) 15.9115 0.582560
\(747\) −6.55646 −0.239888
\(748\) −1.00000 −0.0365636
\(749\) 0 0
\(750\) −0.798528 −0.0291581
\(751\) 13.5159 0.493201 0.246601 0.969117i \(-0.420686\pi\)
0.246601 + 0.969117i \(0.420686\pi\)
\(752\) 6.04795 0.220546
\(753\) 7.95941 0.290057
\(754\) 46.4177 1.69043
\(755\) 47.5542 1.73067
\(756\) 0 0
\(757\) −18.5542 −0.674363 −0.337181 0.941440i \(-0.609474\pi\)
−0.337181 + 0.941440i \(0.609474\pi\)
\(758\) −17.0553 −0.619477
\(759\) −1.95205 −0.0708548
\(760\) −3.69296 −0.133958
\(761\) −15.7506 −0.570958 −0.285479 0.958385i \(-0.592153\pi\)
−0.285479 + 0.958385i \(0.592153\pi\)
\(762\) −8.35499 −0.302669
\(763\) 0 0
\(764\) −4.80589 −0.173871
\(765\) −3.20147 −0.115749
\(766\) −13.4412 −0.485652
\(767\) −73.9954 −2.67182
\(768\) −1.00000 −0.0360844
\(769\) −41.2854 −1.48879 −0.744395 0.667739i \(-0.767262\pi\)
−0.744395 + 0.667739i \(0.767262\pi\)
\(770\) 0 0
\(771\) −22.4029 −0.806822
\(772\) 8.70998 0.313479
\(773\) −4.90916 −0.176570 −0.0882850 0.996095i \(-0.528139\pi\)
−0.0882850 + 0.996095i \(0.528139\pi\)
\(774\) 5.15352 0.185239
\(775\) −55.1129 −1.97971
\(776\) 5.49885 0.197397
\(777\) 0 0
\(778\) −7.20147 −0.258185
\(779\) 9.51587 0.340942
\(780\) −20.4989 −0.733977
\(781\) 5.24943 0.187839
\(782\) −1.95205 −0.0698051
\(783\) −7.24943 −0.259073
\(784\) 0 0
\(785\) 30.2877 1.08101
\(786\) 15.3047 0.545902
\(787\) −0.769897 −0.0274439 −0.0137219 0.999906i \(-0.504368\pi\)
−0.0137219 + 0.999906i \(0.504368\pi\)
\(788\) 0.654669 0.0233216
\(789\) 25.0170 0.890630
\(790\) −17.9424 −0.638361
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) −62.1106 −2.20561
\(794\) −12.9424 −0.459308
\(795\) 20.4989 0.727019
\(796\) 7.69296 0.272670
\(797\) −28.9092 −1.02401 −0.512007 0.858981i \(-0.671098\pi\)
−0.512007 + 0.858981i \(0.671098\pi\)
\(798\) 0 0
\(799\) −6.04795 −0.213961
\(800\) 5.24943 0.185595
\(801\) 18.4029 0.650236
\(802\) −16.9018 −0.596823
\(803\) 2.09591 0.0739629
\(804\) −6.24943 −0.220400
\(805\) 0 0
\(806\) −67.2236 −2.36785
\(807\) 8.10327 0.285249
\(808\) 19.9594 0.702170
\(809\) −20.2494 −0.711932 −0.355966 0.934499i \(-0.615848\pi\)
−0.355966 + 0.934499i \(0.615848\pi\)
\(810\) 3.20147 0.112488
\(811\) 48.4989 1.70302 0.851512 0.524334i \(-0.175686\pi\)
0.851512 + 0.524334i \(0.175686\pi\)
\(812\) 0 0
\(813\) −4.80589 −0.168550
\(814\) 5.15352 0.180631
\(815\) 41.7962 1.46406
\(816\) 1.00000 0.0350070
\(817\) −5.94469 −0.207978
\(818\) 37.0936 1.29695
\(819\) 0 0
\(820\) −26.4103 −0.922288
\(821\) 31.3047 1.09254 0.546271 0.837608i \(-0.316047\pi\)
0.546271 + 0.837608i \(0.316047\pi\)
\(822\) 6.40294 0.223328
\(823\) 0.287717 0.0100292 0.00501459 0.999987i \(-0.498404\pi\)
0.00501459 + 0.999987i \(0.498404\pi\)
\(824\) −0.307039 −0.0106962
\(825\) −5.24943 −0.182762
\(826\) 0 0
\(827\) 37.2471 1.29521 0.647605 0.761976i \(-0.275771\pi\)
0.647605 + 0.761976i \(0.275771\pi\)
\(828\) 1.95205 0.0678383
\(829\) −49.1535 −1.70717 −0.853586 0.520952i \(-0.825577\pi\)
−0.853586 + 0.520952i \(0.825577\pi\)
\(830\) −20.9903 −0.728585
\(831\) 11.2088 0.388830
\(832\) 6.40294 0.221982
\(833\) 0 0
\(834\) 21.7483 0.753081
\(835\) 21.4818 0.743409
\(836\) −1.15352 −0.0398953
\(837\) 10.4989 0.362893
\(838\) −22.4583 −0.775808
\(839\) 2.81785 0.0972830 0.0486415 0.998816i \(-0.484511\pi\)
0.0486415 + 0.998816i \(0.484511\pi\)
\(840\) 0 0
\(841\) 23.5542 0.812213
\(842\) −23.3453 −0.804533
\(843\) 17.4989 0.602692
\(844\) 14.5948 0.502372
\(845\) 89.6339 3.08350
\(846\) 6.04795 0.207933
\(847\) 0 0
\(848\) −6.40294 −0.219878
\(849\) 27.4006 0.940388
\(850\) −5.24943 −0.180054
\(851\) 10.0599 0.344849
\(852\) −5.24943 −0.179842
\(853\) −2.39558 −0.0820232 −0.0410116 0.999159i \(-0.513058\pi\)
−0.0410116 + 0.999159i \(0.513058\pi\)
\(854\) 0 0
\(855\) −3.69296 −0.126297
\(856\) −9.05531 −0.309504
\(857\) −0.808189 −0.0276072 −0.0138036 0.999905i \(-0.504394\pi\)
−0.0138036 + 0.999905i \(0.504394\pi\)
\(858\) −6.40294 −0.218593
\(859\) −6.55646 −0.223704 −0.111852 0.993725i \(-0.535678\pi\)
−0.111852 + 0.993725i \(0.535678\pi\)
\(860\) 16.4989 0.562606
\(861\) 0 0
\(862\) 31.8229 1.08389
\(863\) −20.6067 −0.701461 −0.350730 0.936476i \(-0.614067\pi\)
−0.350730 + 0.936476i \(0.614067\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 17.6118 0.598818
\(866\) −30.3047 −1.02980
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) −5.60442 −0.190117
\(870\) −23.2088 −0.786853
\(871\) 40.0147 1.35585
\(872\) −9.70032 −0.328494
\(873\) 5.49885 0.186108
\(874\) −2.25172 −0.0761657
\(875\) 0 0
\(876\) −2.09591 −0.0708141
\(877\) 26.6021 0.898290 0.449145 0.893459i \(-0.351729\pi\)
0.449145 + 0.893459i \(0.351729\pi\)
\(878\) 10.6620 0.359826
\(879\) −19.7483 −0.666093
\(880\) 3.20147 0.107922
\(881\) 8.49885 0.286334 0.143167 0.989699i \(-0.454271\pi\)
0.143167 + 0.989699i \(0.454271\pi\)
\(882\) 0 0
\(883\) 46.2448 1.55626 0.778131 0.628102i \(-0.216168\pi\)
0.778131 + 0.628102i \(0.216168\pi\)
\(884\) −6.40294 −0.215354
\(885\) 36.9977 1.24366
\(886\) −36.0553 −1.21130
\(887\) −18.2877 −0.614041 −0.307021 0.951703i \(-0.599332\pi\)
−0.307021 + 0.951703i \(0.599332\pi\)
\(888\) −5.15352 −0.172941
\(889\) 0 0
\(890\) 58.9165 1.97489
\(891\) 1.00000 0.0335013
\(892\) 14.9018 0.498949
\(893\) −6.97643 −0.233457
\(894\) −6.84648 −0.228981
\(895\) 5.29002 0.176826
\(896\) 0 0
\(897\) −12.4989 −0.417324
\(898\) 23.4006 0.780890
\(899\) −76.1106 −2.53843
\(900\) 5.24943 0.174981
\(901\) 6.40294 0.213313
\(902\) −8.24943 −0.274676
\(903\) 0 0
\(904\) −12.4989 −0.415706
\(905\) −79.7224 −2.65006
\(906\) −14.8538 −0.493486
\(907\) −31.6694 −1.05156 −0.525782 0.850619i \(-0.676227\pi\)
−0.525782 + 0.850619i \(0.676227\pi\)
\(908\) 0.249425 0.00827746
\(909\) 19.9594 0.662012
\(910\) 0 0
\(911\) 58.7727 1.94723 0.973613 0.228207i \(-0.0732864\pi\)
0.973613 + 0.228207i \(0.0732864\pi\)
\(912\) 1.15352 0.0381968
\(913\) −6.55646 −0.216987
\(914\) −18.6141 −0.615699
\(915\) 31.0553 1.02666
\(916\) 4.09591 0.135333
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) −32.9691 −1.08755 −0.543775 0.839231i \(-0.683005\pi\)
−0.543775 + 0.839231i \(0.683005\pi\)
\(920\) 6.24943 0.206037
\(921\) 4.30704 0.141922
\(922\) 31.4412 1.03546
\(923\) 33.6118 1.10635
\(924\) 0 0
\(925\) 27.0530 0.889498
\(926\) −22.0959 −0.726117
\(927\) −0.307039 −0.0100845
\(928\) 7.24943 0.237974
\(929\) −41.8995 −1.37468 −0.687339 0.726337i \(-0.741221\pi\)
−0.687339 + 0.726337i \(0.741221\pi\)
\(930\) 33.6118 1.10217
\(931\) 0 0
\(932\) 11.3070 0.370374
\(933\) −14.7579 −0.483153
\(934\) −21.0576 −0.689026
\(935\) −3.20147 −0.104699
\(936\) 6.40294 0.209287
\(937\) 33.4966 1.09428 0.547142 0.837040i \(-0.315716\pi\)
0.547142 + 0.837040i \(0.315716\pi\)
\(938\) 0 0
\(939\) 20.7506 0.677169
\(940\) 19.3624 0.631530
\(941\) 45.7483 1.49135 0.745676 0.666309i \(-0.232127\pi\)
0.745676 + 0.666309i \(0.232127\pi\)
\(942\) −9.46056 −0.308242
\(943\) −16.1033 −0.524395
\(944\) −11.5565 −0.376131
\(945\) 0 0
\(946\) 5.15352 0.167555
\(947\) −45.0530 −1.46403 −0.732013 0.681291i \(-0.761419\pi\)
−0.732013 + 0.681291i \(0.761419\pi\)
\(948\) 5.60442 0.182023
\(949\) 13.4200 0.435631
\(950\) −6.05531 −0.196460
\(951\) −16.1992 −0.525294
\(952\) 0 0
\(953\) 37.1705 1.20407 0.602036 0.798469i \(-0.294356\pi\)
0.602036 + 0.798469i \(0.294356\pi\)
\(954\) −6.40294 −0.207303
\(955\) −15.3859 −0.497877
\(956\) 6.59476 0.213290
\(957\) −7.24943 −0.234341
\(958\) 21.2088 0.685226
\(959\) 0 0
\(960\) −3.20147 −0.103327
\(961\) 79.2259 2.55567
\(962\) 32.9977 1.06389
\(963\) −9.05531 −0.291803
\(964\) −17.1129 −0.551170
\(965\) 27.8848 0.897643
\(966\) 0 0
\(967\) 42.7579 1.37500 0.687501 0.726183i \(-0.258708\pi\)
0.687501 + 0.726183i \(0.258708\pi\)
\(968\) 1.00000 0.0321412
\(969\) −1.15352 −0.0370564
\(970\) 17.6044 0.565244
\(971\) −0.690661 −0.0221644 −0.0110822 0.999939i \(-0.503528\pi\)
−0.0110822 + 0.999939i \(0.503528\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −38.7100 −1.24035
\(975\) −33.6118 −1.07644
\(976\) −9.70032 −0.310500
\(977\) −33.1895 −1.06183 −0.530913 0.847426i \(-0.678151\pi\)
−0.530913 + 0.847426i \(0.678151\pi\)
\(978\) −13.0553 −0.417463
\(979\) 18.4029 0.588161
\(980\) 0 0
\(981\) −9.70032 −0.309707
\(982\) 19.1705 0.611757
\(983\) 42.6620 1.36071 0.680354 0.732884i \(-0.261826\pi\)
0.680354 + 0.732884i \(0.261826\pi\)
\(984\) 8.24943 0.262982
\(985\) 2.09591 0.0667811
\(986\) −7.24943 −0.230869
\(987\) 0 0
\(988\) −7.38592 −0.234977
\(989\) 10.0599 0.319887
\(990\) 3.20147 0.101749
\(991\) −24.0147 −0.762853 −0.381426 0.924399i \(-0.624567\pi\)
−0.381426 + 0.924399i \(0.624567\pi\)
\(992\) −10.4989 −0.333339
\(993\) −21.2471 −0.674257
\(994\) 0 0
\(995\) 24.6288 0.780785
\(996\) 6.55646 0.207750
\(997\) −33.5971 −1.06403 −0.532015 0.846735i \(-0.678565\pi\)
−0.532015 + 0.846735i \(0.678565\pi\)
\(998\) −27.9188 −0.883755
\(999\) −5.15352 −0.163050
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 3234.2.a.bg.1.3 3
3.2 odd 2 9702.2.a.du.1.1 3
7.3 odd 6 462.2.i.f.331.3 yes 6
7.5 odd 6 462.2.i.f.67.3 6
7.6 odd 2 3234.2.a.bi.1.1 3
21.5 even 6 1386.2.k.w.991.1 6
21.17 even 6 1386.2.k.w.793.1 6
21.20 even 2 9702.2.a.dt.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.f.67.3 6 7.5 odd 6
462.2.i.f.331.3 yes 6 7.3 odd 6
1386.2.k.w.793.1 6 21.17 even 6
1386.2.k.w.991.1 6 21.5 even 6
3234.2.a.bg.1.3 3 1.1 even 1 trivial
3234.2.a.bi.1.1 3 7.6 odd 2
9702.2.a.dt.1.3 3 21.20 even 2
9702.2.a.du.1.1 3 3.2 odd 2