Properties

Label 3234.2.a.bh.1.1
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: 3.3.2700.1
Defining polynomial: \(x^{3} - 15 x - 20\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.41883\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.41883 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.41883 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -4.41883 q^{10} -1.00000 q^{11} +1.00000 q^{12} -4.41883 q^{15} +1.00000 q^{16} -2.37683 q^{17} +1.00000 q^{18} +3.68842 q^{19} -4.41883 q^{20} -1.00000 q^{22} +4.73042 q^{23} +1.00000 q^{24} +14.5261 q^{25} +1.00000 q^{27} -6.52608 q^{29} -4.41883 q^{30} +3.37683 q^{31} +1.00000 q^{32} -1.00000 q^{33} -2.37683 q^{34} +1.00000 q^{36} +7.68842 q^{37} +3.68842 q^{38} -4.41883 q^{40} -12.1492 q^{41} +3.06525 q^{43} -1.00000 q^{44} -4.41883 q^{45} +4.73042 q^{46} +4.10725 q^{47} +1.00000 q^{48} +14.5261 q^{50} -2.37683 q^{51} +8.00000 q^{53} +1.00000 q^{54} +4.41883 q^{55} +3.68842 q^{57} -6.52608 q^{58} +12.5261 q^{59} -4.41883 q^{60} -3.79567 q^{61} +3.37683 q^{62} +1.00000 q^{64} -1.00000 q^{66} +10.1492 q^{67} -2.37683 q^{68} +4.73042 q^{69} +0.934749 q^{71} +1.00000 q^{72} +7.46083 q^{73} +7.68842 q^{74} +14.5261 q^{75} +3.68842 q^{76} -0.418833 q^{79} -4.41883 q^{80} +1.00000 q^{81} -12.1492 q^{82} +2.14925 q^{83} +10.5028 q^{85} +3.06525 q^{86} -6.52608 q^{87} -1.00000 q^{88} +12.8377 q^{89} -4.41883 q^{90} +4.73042 q^{92} +3.37683 q^{93} +4.10725 q^{94} -16.2985 q^{95} +1.00000 q^{96} -7.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} + 3q^{8} + 3q^{9} - 3q^{11} + 3q^{12} + 3q^{16} - 3q^{17} + 3q^{18} + 9q^{19} - 3q^{22} + 3q^{23} + 3q^{24} + 15q^{25} + 3q^{27} + 9q^{29} + 6q^{31} + 3q^{32} - 3q^{33} - 3q^{34} + 3q^{36} + 21q^{37} + 9q^{38} - 12q^{41} + 3q^{43} - 3q^{44} + 3q^{46} - 3q^{47} + 3q^{48} + 15q^{50} - 3q^{51} + 24q^{53} + 3q^{54} + 9q^{57} + 9q^{58} + 9q^{59} + 6q^{61} + 6q^{62} + 3q^{64} - 3q^{66} + 6q^{67} - 3q^{68} + 3q^{69} + 9q^{71} + 3q^{72} + 21q^{74} + 15q^{75} + 9q^{76} + 12q^{79} + 3q^{81} - 12q^{82} - 18q^{83} + 3q^{86} + 9q^{87} - 3q^{88} + 12q^{89} + 3q^{92} + 6q^{93} - 3q^{94} + 3q^{96} - 21q^{97} - 3q^{99} + O(q^{100}) \)

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
\(5\) −4.41883 −1.97616 −0.988081 0.153935i \(-0.950805\pi\)
−0.988081 + 0.153935i \(0.950805\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −4.41883 −1.39736
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −4.41883 −1.14094
\(16\) 1.00000 0.250000
\(17\) −2.37683 −0.576467 −0.288233 0.957560i \(-0.593068\pi\)
−0.288233 + 0.957560i \(0.593068\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.68842 0.846181 0.423090 0.906087i \(-0.360945\pi\)
0.423090 + 0.906087i \(0.360945\pi\)
\(20\) −4.41883 −0.988081
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 4.73042 0.986360 0.493180 0.869927i \(-0.335834\pi\)
0.493180 + 0.869927i \(0.335834\pi\)
\(24\) 1.00000 0.204124
\(25\) 14.5261 2.90522
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.52608 −1.21186 −0.605932 0.795517i \(-0.707199\pi\)
−0.605932 + 0.795517i \(0.707199\pi\)
\(30\) −4.41883 −0.806765
\(31\) 3.37683 0.606497 0.303249 0.952911i \(-0.401929\pi\)
0.303249 + 0.952911i \(0.401929\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −2.37683 −0.407624
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.68842 1.26397 0.631984 0.774981i \(-0.282241\pi\)
0.631984 + 0.774981i \(0.282241\pi\)
\(38\) 3.68842 0.598340
\(39\) 0 0
\(40\) −4.41883 −0.698679
\(41\) −12.1492 −1.89739 −0.948697 0.316187i \(-0.897597\pi\)
−0.948697 + 0.316187i \(0.897597\pi\)
\(42\) 0 0
\(43\) 3.06525 0.467446 0.233723 0.972303i \(-0.424909\pi\)
0.233723 + 0.972303i \(0.424909\pi\)
\(44\) −1.00000 −0.150756
\(45\) −4.41883 −0.658721
\(46\) 4.73042 0.697462
\(47\) 4.10725 0.599104 0.299552 0.954080i \(-0.403163\pi\)
0.299552 + 0.954080i \(0.403163\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 14.5261 2.05430
\(51\) −2.37683 −0.332823
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.41883 0.595835
\(56\) 0 0
\(57\) 3.68842 0.488543
\(58\) −6.52608 −0.856917
\(59\) 12.5261 1.63076 0.815379 0.578928i \(-0.196529\pi\)
0.815379 + 0.578928i \(0.196529\pi\)
\(60\) −4.41883 −0.570469
\(61\) −3.79567 −0.485985 −0.242993 0.970028i \(-0.578129\pi\)
−0.242993 + 0.970028i \(0.578129\pi\)
\(62\) 3.37683 0.428858
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) 10.1492 1.23993 0.619964 0.784630i \(-0.287147\pi\)
0.619964 + 0.784630i \(0.287147\pi\)
\(68\) −2.37683 −0.288233
\(69\) 4.73042 0.569475
\(70\) 0 0
\(71\) 0.934749 0.110934 0.0554672 0.998461i \(-0.482335\pi\)
0.0554672 + 0.998461i \(0.482335\pi\)
\(72\) 1.00000 0.117851
\(73\) 7.46083 0.873224 0.436612 0.899650i \(-0.356178\pi\)
0.436612 + 0.899650i \(0.356178\pi\)
\(74\) 7.68842 0.893760
\(75\) 14.5261 1.67733
\(76\) 3.68842 0.423090
\(77\) 0 0
\(78\) 0 0
\(79\) −0.418833 −0.0471224 −0.0235612 0.999722i \(-0.507500\pi\)
−0.0235612 + 0.999722i \(0.507500\pi\)
\(80\) −4.41883 −0.494041
\(81\) 1.00000 0.111111
\(82\) −12.1492 −1.34166
\(83\) 2.14925 0.235911 0.117955 0.993019i \(-0.462366\pi\)
0.117955 + 0.993019i \(0.462366\pi\)
\(84\) 0 0
\(85\) 10.5028 1.13919
\(86\) 3.06525 0.330534
\(87\) −6.52608 −0.699669
\(88\) −1.00000 −0.106600
\(89\) 12.8377 1.36079 0.680395 0.732846i \(-0.261808\pi\)
0.680395 + 0.732846i \(0.261808\pi\)
\(90\) −4.41883 −0.465786
\(91\) 0 0
\(92\) 4.73042 0.493180
\(93\) 3.37683 0.350161
\(94\) 4.10725 0.423630
\(95\) −16.2985 −1.67219
\(96\) 1.00000 0.102062
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 14.5261 1.45261
\(101\) 15.3637 1.52875 0.764375 0.644772i \(-0.223048\pi\)
0.764375 + 0.644772i \(0.223048\pi\)
\(102\) −2.37683 −0.235342
\(103\) 2.62317 0.258468 0.129234 0.991614i \(-0.458748\pi\)
0.129234 + 0.991614i \(0.458748\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) 10.9029 1.05402 0.527012 0.849858i \(-0.323312\pi\)
0.527012 + 0.849858i \(0.323312\pi\)
\(108\) 1.00000 0.0962250
\(109\) −11.7957 −1.12982 −0.564910 0.825153i \(-0.691089\pi\)
−0.564910 + 0.825153i \(0.691089\pi\)
\(110\) 4.41883 0.421319
\(111\) 7.68842 0.729752
\(112\) 0 0
\(113\) 5.37683 0.505810 0.252905 0.967491i \(-0.418614\pi\)
0.252905 + 0.967491i \(0.418614\pi\)
\(114\) 3.68842 0.345452
\(115\) −20.9029 −1.94921
\(116\) −6.52608 −0.605932
\(117\) 0 0
\(118\) 12.5261 1.15312
\(119\) 0 0
\(120\) −4.41883 −0.403382
\(121\) 1.00000 0.0909091
\(122\) −3.79567 −0.343643
\(123\) −12.1492 −1.09546
\(124\) 3.37683 0.303249
\(125\) −42.0942 −3.76502
\(126\) 0 0
\(127\) 0.730416 0.0648139 0.0324070 0.999475i \(-0.489683\pi\)
0.0324070 + 0.999475i \(0.489683\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.06525 0.269880
\(130\) 0 0
\(131\) −7.37683 −0.644517 −0.322258 0.946652i \(-0.604442\pi\)
−0.322258 + 0.946652i \(0.604442\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) 10.1492 0.876762
\(135\) −4.41883 −0.380313
\(136\) −2.37683 −0.203812
\(137\) −8.83767 −0.755053 −0.377526 0.925999i \(-0.623225\pi\)
−0.377526 + 0.925999i \(0.623225\pi\)
\(138\) 4.73042 0.402680
\(139\) −1.47392 −0.125016 −0.0625080 0.998044i \(-0.519910\pi\)
−0.0625080 + 0.998044i \(0.519910\pi\)
\(140\) 0 0
\(141\) 4.10725 0.345893
\(142\) 0.934749 0.0784424
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 28.8377 2.39484
\(146\) 7.46083 0.617463
\(147\) 0 0
\(148\) 7.68842 0.631984
\(149\) −9.36375 −0.767108 −0.383554 0.923518i \(-0.625300\pi\)
−0.383554 + 0.923518i \(0.625300\pi\)
\(150\) 14.5261 1.18605
\(151\) −13.5681 −1.10415 −0.552077 0.833793i \(-0.686165\pi\)
−0.552077 + 0.833793i \(0.686165\pi\)
\(152\) 3.68842 0.299170
\(153\) −2.37683 −0.192156
\(154\) 0 0
\(155\) −14.9217 −1.19854
\(156\) 0 0
\(157\) 18.7406 1.49566 0.747831 0.663890i \(-0.231096\pi\)
0.747831 + 0.663890i \(0.231096\pi\)
\(158\) −0.418833 −0.0333205
\(159\) 8.00000 0.634441
\(160\) −4.41883 −0.349339
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 18.7724 1.47037 0.735185 0.677867i \(-0.237096\pi\)
0.735185 + 0.677867i \(0.237096\pi\)
\(164\) −12.1492 −0.948697
\(165\) 4.41883 0.344006
\(166\) 2.14925 0.166814
\(167\) 23.8898 1.84865 0.924325 0.381606i \(-0.124629\pi\)
0.924325 + 0.381606i \(0.124629\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 10.5028 0.805530
\(171\) 3.68842 0.282060
\(172\) 3.06525 0.233723
\(173\) −11.3768 −0.864965 −0.432482 0.901642i \(-0.642362\pi\)
−0.432482 + 0.901642i \(0.642362\pi\)
\(174\) −6.52608 −0.494741
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 12.5261 0.941518
\(178\) 12.8377 0.962224
\(179\) 24.6101 1.83944 0.919722 0.392571i \(-0.128414\pi\)
0.919722 + 0.392571i \(0.128414\pi\)
\(180\) −4.41883 −0.329360
\(181\) −17.4608 −1.29785 −0.648927 0.760851i \(-0.724782\pi\)
−0.648927 + 0.760851i \(0.724782\pi\)
\(182\) 0 0
\(183\) −3.79567 −0.280584
\(184\) 4.73042 0.348731
\(185\) −33.9738 −2.49781
\(186\) 3.37683 0.247601
\(187\) 2.37683 0.173811
\(188\) 4.10725 0.299552
\(189\) 0 0
\(190\) −16.2985 −1.18242
\(191\) −16.8377 −1.21833 −0.609165 0.793043i \(-0.708495\pi\)
−0.609165 + 0.793043i \(0.708495\pi\)
\(192\) 1.00000 0.0721688
\(193\) 5.46083 0.393079 0.196540 0.980496i \(-0.437030\pi\)
0.196540 + 0.980496i \(0.437030\pi\)
\(194\) −7.00000 −0.502571
\(195\) 0 0
\(196\) 0 0
\(197\) −9.68842 −0.690271 −0.345136 0.938553i \(-0.612167\pi\)
−0.345136 + 0.938553i \(0.612167\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −20.2985 −1.43892 −0.719461 0.694533i \(-0.755611\pi\)
−0.719461 + 0.694533i \(0.755611\pi\)
\(200\) 14.5261 1.02715
\(201\) 10.1492 0.715873
\(202\) 15.3637 1.08099
\(203\) 0 0
\(204\) −2.37683 −0.166412
\(205\) 53.6855 3.74956
\(206\) 2.62317 0.182765
\(207\) 4.73042 0.328787
\(208\) 0 0
\(209\) −3.68842 −0.255133
\(210\) 0 0
\(211\) 22.5130 1.54986 0.774929 0.632048i \(-0.217785\pi\)
0.774929 + 0.632048i \(0.217785\pi\)
\(212\) 8.00000 0.549442
\(213\) 0.934749 0.0640480
\(214\) 10.9029 0.745308
\(215\) −13.5448 −0.923750
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −11.7957 −0.798903
\(219\) 7.46083 0.504156
\(220\) 4.41883 0.297918
\(221\) 0 0
\(222\) 7.68842 0.516013
\(223\) 7.46083 0.499614 0.249807 0.968296i \(-0.419633\pi\)
0.249807 + 0.968296i \(0.419633\pi\)
\(224\) 0 0
\(225\) 14.5261 0.968405
\(226\) 5.37683 0.357662
\(227\) 2.77241 0.184012 0.0920058 0.995758i \(-0.470672\pi\)
0.0920058 + 0.995758i \(0.470672\pi\)
\(228\) 3.68842 0.244271
\(229\) 2.53917 0.167793 0.0838965 0.996474i \(-0.473263\pi\)
0.0838965 + 0.996474i \(0.473263\pi\)
\(230\) −20.9029 −1.37830
\(231\) 0 0
\(232\) −6.52608 −0.428458
\(233\) −24.0522 −1.57571 −0.787855 0.615861i \(-0.788808\pi\)
−0.787855 + 0.615861i \(0.788808\pi\)
\(234\) 0 0
\(235\) −18.1492 −1.18393
\(236\) 12.5261 0.815379
\(237\) −0.418833 −0.0272061
\(238\) 0 0
\(239\) 7.59133 0.491042 0.245521 0.969391i \(-0.421041\pi\)
0.245521 + 0.969391i \(0.421041\pi\)
\(240\) −4.41883 −0.285234
\(241\) −16.2985 −1.04988 −0.524939 0.851140i \(-0.675912\pi\)
−0.524939 + 0.851140i \(0.675912\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) −3.79567 −0.242993
\(245\) 0 0
\(246\) −12.1492 −0.774608
\(247\) 0 0
\(248\) 3.37683 0.214429
\(249\) 2.14925 0.136203
\(250\) −42.0942 −2.66227
\(251\) 8.61008 0.543463 0.271732 0.962373i \(-0.412404\pi\)
0.271732 + 0.962373i \(0.412404\pi\)
\(252\) 0 0
\(253\) −4.73042 −0.297399
\(254\) 0.730416 0.0458304
\(255\) 10.5028 0.657713
\(256\) 1.00000 0.0625000
\(257\) −0.837665 −0.0522521 −0.0261261 0.999659i \(-0.508317\pi\)
−0.0261261 + 0.999659i \(0.508317\pi\)
\(258\) 3.06525 0.190834
\(259\) 0 0
\(260\) 0 0
\(261\) −6.52608 −0.403954
\(262\) −7.37683 −0.455742
\(263\) 10.8377 0.668279 0.334140 0.942524i \(-0.391554\pi\)
0.334140 + 0.942524i \(0.391554\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −35.3507 −2.17157
\(266\) 0 0
\(267\) 12.8377 0.785652
\(268\) 10.1492 0.619964
\(269\) −17.8797 −1.09014 −0.545071 0.838390i \(-0.683497\pi\)
−0.545071 + 0.838390i \(0.683497\pi\)
\(270\) −4.41883 −0.268922
\(271\) 8.83767 0.536850 0.268425 0.963301i \(-0.413497\pi\)
0.268425 + 0.963301i \(0.413497\pi\)
\(272\) −2.37683 −0.144117
\(273\) 0 0
\(274\) −8.83767 −0.533903
\(275\) −14.5261 −0.875956
\(276\) 4.73042 0.284738
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) −1.47392 −0.0883997
\(279\) 3.37683 0.202166
\(280\) 0 0
\(281\) −15.9217 −0.949807 −0.474903 0.880038i \(-0.657517\pi\)
−0.474903 + 0.880038i \(0.657517\pi\)
\(282\) 4.10725 0.244583
\(283\) 18.0840 1.07498 0.537491 0.843269i \(-0.319372\pi\)
0.537491 + 0.843269i \(0.319372\pi\)
\(284\) 0.934749 0.0554672
\(285\) −16.2985 −0.965440
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −11.3507 −0.667686
\(290\) 28.8377 1.69341
\(291\) −7.00000 −0.410347
\(292\) 7.46083 0.436612
\(293\) −25.1492 −1.46923 −0.734617 0.678482i \(-0.762638\pi\)
−0.734617 + 0.678482i \(0.762638\pi\)
\(294\) 0 0
\(295\) −55.3507 −3.22264
\(296\) 7.68842 0.446880
\(297\) −1.00000 −0.0580259
\(298\) −9.36375 −0.542427
\(299\) 0 0
\(300\) 14.5261 0.838664
\(301\) 0 0
\(302\) −13.5681 −0.780755
\(303\) 15.3637 0.882624
\(304\) 3.68842 0.211545
\(305\) 16.7724 0.960386
\(306\) −2.37683 −0.135875
\(307\) 3.86950 0.220844 0.110422 0.993885i \(-0.464780\pi\)
0.110422 + 0.993885i \(0.464780\pi\)
\(308\) 0 0
\(309\) 2.62317 0.149227
\(310\) −14.9217 −0.847494
\(311\) 13.5681 0.769375 0.384688 0.923047i \(-0.374309\pi\)
0.384688 + 0.923047i \(0.374309\pi\)
\(312\) 0 0
\(313\) −2.52608 −0.142783 −0.0713913 0.997448i \(-0.522744\pi\)
−0.0713913 + 0.997448i \(0.522744\pi\)
\(314\) 18.7406 1.05759
\(315\) 0 0
\(316\) −0.418833 −0.0235612
\(317\) −20.0102 −1.12388 −0.561941 0.827177i \(-0.689945\pi\)
−0.561941 + 0.827177i \(0.689945\pi\)
\(318\) 8.00000 0.448618
\(319\) 6.52608 0.365390
\(320\) −4.41883 −0.247020
\(321\) 10.9029 0.608541
\(322\) 0 0
\(323\) −8.76675 −0.487795
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 18.7724 1.03971
\(327\) −11.7957 −0.652302
\(328\) −12.1492 −0.670830
\(329\) 0 0
\(330\) 4.41883 0.243249
\(331\) −16.1492 −0.887643 −0.443821 0.896115i \(-0.646378\pi\)
−0.443821 + 0.896115i \(0.646378\pi\)
\(332\) 2.14925 0.117955
\(333\) 7.68842 0.421323
\(334\) 23.8898 1.30719
\(335\) −44.8478 −2.45030
\(336\) 0 0
\(337\) 23.1362 1.26031 0.630154 0.776471i \(-0.282992\pi\)
0.630154 + 0.776471i \(0.282992\pi\)
\(338\) −13.0000 −0.707107
\(339\) 5.37683 0.292030
\(340\) 10.5028 0.569596
\(341\) −3.37683 −0.182866
\(342\) 3.68842 0.199447
\(343\) 0 0
\(344\) 3.06525 0.165267
\(345\) −20.9029 −1.12538
\(346\) −11.3768 −0.611622
\(347\) 15.6566 0.840489 0.420245 0.907411i \(-0.361944\pi\)
0.420245 + 0.907411i \(0.361944\pi\)
\(348\) −6.52608 −0.349835
\(349\) 20.0102 1.07112 0.535560 0.844497i \(-0.320101\pi\)
0.535560 + 0.844497i \(0.320101\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −7.46083 −0.397100 −0.198550 0.980091i \(-0.563623\pi\)
−0.198550 + 0.980091i \(0.563623\pi\)
\(354\) 12.5261 0.665754
\(355\) −4.13050 −0.219224
\(356\) 12.8377 0.680395
\(357\) 0 0
\(358\) 24.6101 1.30068
\(359\) −11.2463 −0.593559 −0.296779 0.954946i \(-0.595913\pi\)
−0.296779 + 0.954946i \(0.595913\pi\)
\(360\) −4.41883 −0.232893
\(361\) −5.39558 −0.283978
\(362\) −17.4608 −0.917721
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −32.9682 −1.72563
\(366\) −3.79567 −0.198403
\(367\) 5.91600 0.308813 0.154406 0.988007i \(-0.450653\pi\)
0.154406 + 0.988007i \(0.450653\pi\)
\(368\) 4.73042 0.246590
\(369\) −12.1492 −0.632465
\(370\) −33.9738 −1.76622
\(371\) 0 0
\(372\) 3.37683 0.175081
\(373\) −23.1260 −1.19742 −0.598709 0.800966i \(-0.704320\pi\)
−0.598709 + 0.800966i \(0.704320\pi\)
\(374\) 2.37683 0.122903
\(375\) −42.0942 −2.17373
\(376\) 4.10725 0.211815
\(377\) 0 0
\(378\) 0 0
\(379\) −6.77241 −0.347876 −0.173938 0.984757i \(-0.555649\pi\)
−0.173938 + 0.984757i \(0.555649\pi\)
\(380\) −16.2985 −0.836095
\(381\) 0.730416 0.0374203
\(382\) −16.8377 −0.861490
\(383\) −25.3637 −1.29603 −0.648013 0.761629i \(-0.724400\pi\)
−0.648013 + 0.761629i \(0.724400\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 5.46083 0.277949
\(387\) 3.06525 0.155815
\(388\) −7.00000 −0.355371
\(389\) 25.2565 1.28056 0.640278 0.768144i \(-0.278819\pi\)
0.640278 + 0.768144i \(0.278819\pi\)
\(390\) 0 0
\(391\) −11.2434 −0.568604
\(392\) 0 0
\(393\) −7.37683 −0.372112
\(394\) −9.68842 −0.488095
\(395\) 1.85075 0.0931214
\(396\) −1.00000 −0.0502519
\(397\) 13.4739 0.676237 0.338118 0.941104i \(-0.390210\pi\)
0.338118 + 0.941104i \(0.390210\pi\)
\(398\) −20.2985 −1.01747
\(399\) 0 0
\(400\) 14.5261 0.726304
\(401\) 33.8898 1.69238 0.846189 0.532883i \(-0.178892\pi\)
0.846189 + 0.532883i \(0.178892\pi\)
\(402\) 10.1492 0.506199
\(403\) 0 0
\(404\) 15.3637 0.764375
\(405\) −4.41883 −0.219574
\(406\) 0 0
\(407\) −7.68842 −0.381101
\(408\) −2.37683 −0.117671
\(409\) 4.21450 0.208394 0.104197 0.994557i \(-0.466773\pi\)
0.104197 + 0.994557i \(0.466773\pi\)
\(410\) 53.6855 2.65134
\(411\) −8.83767 −0.435930
\(412\) 2.62317 0.129234
\(413\) 0 0
\(414\) 4.73042 0.232487
\(415\) −9.49717 −0.466198
\(416\) 0 0
\(417\) −1.47392 −0.0721781
\(418\) −3.68842 −0.180406
\(419\) 22.3116 1.08999 0.544996 0.838439i \(-0.316531\pi\)
0.544996 + 0.838439i \(0.316531\pi\)
\(420\) 0 0
\(421\) 18.3116 0.892452 0.446226 0.894920i \(-0.352768\pi\)
0.446226 + 0.894920i \(0.352768\pi\)
\(422\) 22.5130 1.09592
\(423\) 4.10725 0.199701
\(424\) 8.00000 0.388514
\(425\) −34.5261 −1.67476
\(426\) 0.934749 0.0452888
\(427\) 0 0
\(428\) 10.9029 0.527012
\(429\) 0 0
\(430\) −13.5448 −0.653190
\(431\) −28.8377 −1.38906 −0.694531 0.719463i \(-0.744388\pi\)
−0.694531 + 0.719463i \(0.744388\pi\)
\(432\) 1.00000 0.0481125
\(433\) −7.62317 −0.366346 −0.183173 0.983081i \(-0.558637\pi\)
−0.183173 + 0.983081i \(0.558637\pi\)
\(434\) 0 0
\(435\) 28.8377 1.38266
\(436\) −11.7957 −0.564910
\(437\) 17.4477 0.834639
\(438\) 7.46083 0.356492
\(439\) 33.3739 1.59285 0.796425 0.604737i \(-0.206722\pi\)
0.796425 + 0.604737i \(0.206722\pi\)
\(440\) 4.41883 0.210660
\(441\) 0 0
\(442\) 0 0
\(443\) −4.85075 −0.230466 −0.115233 0.993338i \(-0.536761\pi\)
−0.115233 + 0.993338i \(0.536761\pi\)
\(444\) 7.68842 0.364876
\(445\) −56.7275 −2.68914
\(446\) 7.46083 0.353281
\(447\) −9.36375 −0.442890
\(448\) 0 0
\(449\) 14.3450 0.676982 0.338491 0.940970i \(-0.390083\pi\)
0.338491 + 0.940970i \(0.390083\pi\)
\(450\) 14.5261 0.684766
\(451\) 12.1492 0.572086
\(452\) 5.37683 0.252905
\(453\) −13.5681 −0.637484
\(454\) 2.77241 0.130116
\(455\) 0 0
\(456\) 3.68842 0.172726
\(457\) −25.1827 −1.17800 −0.588998 0.808135i \(-0.700477\pi\)
−0.588998 + 0.808135i \(0.700477\pi\)
\(458\) 2.53917 0.118648
\(459\) −2.37683 −0.110941
\(460\) −20.9029 −0.974603
\(461\) −13.0187 −0.606344 −0.303172 0.952936i \(-0.598046\pi\)
−0.303172 + 0.952936i \(0.598046\pi\)
\(462\) 0 0
\(463\) −14.2145 −0.660604 −0.330302 0.943875i \(-0.607151\pi\)
−0.330302 + 0.943875i \(0.607151\pi\)
\(464\) −6.52608 −0.302966
\(465\) −14.9217 −0.691976
\(466\) −24.0522 −1.11420
\(467\) 2.39558 0.110854 0.0554271 0.998463i \(-0.482348\pi\)
0.0554271 + 0.998463i \(0.482348\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −18.1492 −0.837162
\(471\) 18.7406 0.863520
\(472\) 12.5261 0.576560
\(473\) −3.06525 −0.140940
\(474\) −0.418833 −0.0192376
\(475\) 53.5782 2.45834
\(476\) 0 0
\(477\) 8.00000 0.366295
\(478\) 7.59133 0.347219
\(479\) −10.8377 −0.495186 −0.247593 0.968864i \(-0.579640\pi\)
−0.247593 + 0.968864i \(0.579640\pi\)
\(480\) −4.41883 −0.201691
\(481\) 0 0
\(482\) −16.2985 −0.742376
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 30.9318 1.40454
\(486\) 1.00000 0.0453609
\(487\) −34.6435 −1.56985 −0.784923 0.619593i \(-0.787298\pi\)
−0.784923 + 0.619593i \(0.787298\pi\)
\(488\) −3.79567 −0.171822
\(489\) 18.7724 0.848918
\(490\) 0 0
\(491\) 38.1492 1.72165 0.860826 0.508900i \(-0.169948\pi\)
0.860826 + 0.508900i \(0.169948\pi\)
\(492\) −12.1492 −0.547730
\(493\) 15.5114 0.698599
\(494\) 0 0
\(495\) 4.41883 0.198612
\(496\) 3.37683 0.151624
\(497\) 0 0
\(498\) 2.14925 0.0963101
\(499\) −0.623166 −0.0278968 −0.0139484 0.999903i \(-0.504440\pi\)
−0.0139484 + 0.999903i \(0.504440\pi\)
\(500\) −42.0942 −1.88251
\(501\) 23.8898 1.06732
\(502\) 8.61008 0.384287
\(503\) −31.1362 −1.38829 −0.694146 0.719834i \(-0.744218\pi\)
−0.694146 + 0.719834i \(0.744218\pi\)
\(504\) 0 0
\(505\) −67.8898 −3.02106
\(506\) −4.73042 −0.210293
\(507\) −13.0000 −0.577350
\(508\) 0.730416 0.0324070
\(509\) 42.1043 1.86624 0.933121 0.359563i \(-0.117074\pi\)
0.933121 + 0.359563i \(0.117074\pi\)
\(510\) 10.5028 0.465073
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 3.68842 0.162848
\(514\) −0.837665 −0.0369478
\(515\) −11.5913 −0.510775
\(516\) 3.06525 0.134940
\(517\) −4.10725 −0.180637
\(518\) 0 0
\(519\) −11.3768 −0.499388
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −6.52608 −0.285639
\(523\) 25.2667 1.10483 0.552417 0.833568i \(-0.313706\pi\)
0.552417 + 0.833568i \(0.313706\pi\)
\(524\) −7.37683 −0.322258
\(525\) 0 0
\(526\) 10.8377 0.472545
\(527\) −8.02617 −0.349626
\(528\) −1.00000 −0.0435194
\(529\) −0.623166 −0.0270942
\(530\) −35.3507 −1.53553
\(531\) 12.5261 0.543586
\(532\) 0 0
\(533\) 0 0
\(534\) 12.8377 0.555540
\(535\) −48.1782 −2.08292
\(536\) 10.1492 0.438381
\(537\) 24.6101 1.06200
\(538\) −17.8797 −0.770847
\(539\) 0 0
\(540\) −4.41883 −0.190156
\(541\) −4.41883 −0.189980 −0.0949902 0.995478i \(-0.530282\pi\)
−0.0949902 + 0.995478i \(0.530282\pi\)
\(542\) 8.83767 0.379610
\(543\) −17.4608 −0.749316
\(544\) −2.37683 −0.101906
\(545\) 52.1231 2.23271
\(546\) 0 0
\(547\) −27.6622 −1.18275 −0.591376 0.806396i \(-0.701415\pi\)
−0.591376 + 0.806396i \(0.701415\pi\)
\(548\) −8.83767 −0.377526
\(549\) −3.79567 −0.161995
\(550\) −14.5261 −0.619394
\(551\) −24.0709 −1.02546
\(552\) 4.73042 0.201340
\(553\) 0 0
\(554\) −16.0000 −0.679775
\(555\) −33.9738 −1.44211
\(556\) −1.47392 −0.0625080
\(557\) −7.98691 −0.338416 −0.169208 0.985580i \(-0.554121\pi\)
−0.169208 + 0.985580i \(0.554121\pi\)
\(558\) 3.37683 0.142953
\(559\) 0 0
\(560\) 0 0
\(561\) 2.37683 0.100350
\(562\) −15.9217 −0.671615
\(563\) 27.3507 1.15269 0.576346 0.817205i \(-0.304478\pi\)
0.576346 + 0.817205i \(0.304478\pi\)
\(564\) 4.10725 0.172946
\(565\) −23.7593 −0.999562
\(566\) 18.0840 0.760127
\(567\) 0 0
\(568\) 0.934749 0.0392212
\(569\) −29.4477 −1.23451 −0.617257 0.786762i \(-0.711756\pi\)
−0.617257 + 0.786762i \(0.711756\pi\)
\(570\) −16.2985 −0.682669
\(571\) −14.5261 −0.607898 −0.303949 0.952688i \(-0.598305\pi\)
−0.303949 + 0.952688i \(0.598305\pi\)
\(572\) 0 0
\(573\) −16.8377 −0.703404
\(574\) 0 0
\(575\) 68.7144 2.86559
\(576\) 1.00000 0.0416667
\(577\) −13.3956 −0.557665 −0.278833 0.960340i \(-0.589947\pi\)
−0.278833 + 0.960340i \(0.589947\pi\)
\(578\) −11.3507 −0.472125
\(579\) 5.46083 0.226944
\(580\) 28.8377 1.19742
\(581\) 0 0
\(582\) −7.00000 −0.290159
\(583\) −8.00000 −0.331326
\(584\) 7.46083 0.308731
\(585\) 0 0
\(586\) −25.1492 −1.03891
\(587\) 11.2928 0.466105 0.233053 0.972464i \(-0.425129\pi\)
0.233053 + 0.972464i \(0.425129\pi\)
\(588\) 0 0
\(589\) 12.4552 0.513206
\(590\) −55.3507 −2.27875
\(591\) −9.68842 −0.398528
\(592\) 7.68842 0.315992
\(593\) 19.5782 0.803982 0.401991 0.915644i \(-0.368318\pi\)
0.401991 + 0.915644i \(0.368318\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −9.36375 −0.383554
\(597\) −20.2985 −0.830762
\(598\) 0 0
\(599\) 10.9318 0.446662 0.223331 0.974743i \(-0.428307\pi\)
0.223331 + 0.974743i \(0.428307\pi\)
\(600\) 14.5261 0.593025
\(601\) 41.2202 1.68141 0.840703 0.541497i \(-0.182142\pi\)
0.840703 + 0.541497i \(0.182142\pi\)
\(602\) 0 0
\(603\) 10.1492 0.413309
\(604\) −13.5681 −0.552077
\(605\) −4.41883 −0.179651
\(606\) 15.3637 0.624110
\(607\) 31.3405 1.27207 0.636036 0.771660i \(-0.280573\pi\)
0.636036 + 0.771660i \(0.280573\pi\)
\(608\) 3.68842 0.149585
\(609\) 0 0
\(610\) 16.7724 0.679095
\(611\) 0 0
\(612\) −2.37683 −0.0960778
\(613\) 17.0885 0.690198 0.345099 0.938566i \(-0.387845\pi\)
0.345099 + 0.938566i \(0.387845\pi\)
\(614\) 3.86950 0.156160
\(615\) 53.6855 2.16481
\(616\) 0 0
\(617\) −6.96817 −0.280528 −0.140264 0.990114i \(-0.544795\pi\)
−0.140264 + 0.990114i \(0.544795\pi\)
\(618\) 2.62317 0.105519
\(619\) 0.0187473 0.000753517 0 0.000376759 1.00000i \(-0.499880\pi\)
0.000376759 1.00000i \(0.499880\pi\)
\(620\) −14.9217 −0.599268
\(621\) 4.73042 0.189825
\(622\) 13.5681 0.544030
\(623\) 0 0
\(624\) 0 0
\(625\) 113.377 4.53507
\(626\) −2.52608 −0.100963
\(627\) −3.68842 −0.147301
\(628\) 18.7406 0.747831
\(629\) −18.2741 −0.728636
\(630\) 0 0
\(631\) −5.78550 −0.230317 −0.115159 0.993347i \(-0.536738\pi\)
−0.115159 + 0.993347i \(0.536738\pi\)
\(632\) −0.418833 −0.0166603
\(633\) 22.5130 0.894811
\(634\) −20.0102 −0.794705
\(635\) −3.22759 −0.128083
\(636\) 8.00000 0.317221
\(637\) 0 0
\(638\) 6.52608 0.258370
\(639\) 0.934749 0.0369781
\(640\) −4.41883 −0.174670
\(641\) 15.3768 0.607348 0.303674 0.952776i \(-0.401787\pi\)
0.303674 + 0.952776i \(0.401787\pi\)
\(642\) 10.9029 0.430304
\(643\) 35.8058 1.41204 0.706022 0.708190i \(-0.250488\pi\)
0.706022 + 0.708190i \(0.250488\pi\)
\(644\) 0 0
\(645\) −13.5448 −0.533327
\(646\) −8.76675 −0.344923
\(647\) −9.25650 −0.363910 −0.181955 0.983307i \(-0.558243\pi\)
−0.181955 + 0.983307i \(0.558243\pi\)
\(648\) 1.00000 0.0392837
\(649\) −12.5261 −0.491692
\(650\) 0 0
\(651\) 0 0
\(652\) 18.7724 0.735185
\(653\) 24.8478 0.972371 0.486185 0.873856i \(-0.338388\pi\)
0.486185 + 0.873856i \(0.338388\pi\)
\(654\) −11.7957 −0.461247
\(655\) 32.5970 1.27367
\(656\) −12.1492 −0.474348
\(657\) 7.46083 0.291075
\(658\) 0 0
\(659\) −6.60442 −0.257272 −0.128636 0.991692i \(-0.541060\pi\)
−0.128636 + 0.991692i \(0.541060\pi\)
\(660\) 4.41883 0.172003
\(661\) 13.2332 0.514714 0.257357 0.966316i \(-0.417148\pi\)
0.257357 + 0.966316i \(0.417148\pi\)
\(662\) −16.1492 −0.627658
\(663\) 0 0
\(664\) 2.14925 0.0834070
\(665\) 0 0
\(666\) 7.68842 0.297920
\(667\) −30.8711 −1.19533
\(668\) 23.8898 0.924325
\(669\) 7.46083 0.288452
\(670\) −44.8478 −1.73262
\(671\) 3.79567 0.146530
\(672\) 0 0
\(673\) 0.130501 0.00503045 0.00251523 0.999997i \(-0.499199\pi\)
0.00251523 + 0.999997i \(0.499199\pi\)
\(674\) 23.1362 0.891172
\(675\) 14.5261 0.559109
\(676\) −13.0000 −0.500000
\(677\) −28.1174 −1.08064 −0.540320 0.841460i \(-0.681697\pi\)
−0.540320 + 0.841460i \(0.681697\pi\)
\(678\) 5.37683 0.206496
\(679\) 0 0
\(680\) 10.5028 0.402765
\(681\) 2.77241 0.106239
\(682\) −3.37683 −0.129306
\(683\) 15.9029 0.608508 0.304254 0.952591i \(-0.401593\pi\)
0.304254 + 0.952591i \(0.401593\pi\)
\(684\) 3.68842 0.141030
\(685\) 39.0522 1.49211
\(686\) 0 0
\(687\) 2.53917 0.0968753
\(688\) 3.06525 0.116862
\(689\) 0 0
\(690\) −20.9029 −0.795760
\(691\) 15.9813 0.607956 0.303978 0.952679i \(-0.401685\pi\)
0.303978 + 0.952679i \(0.401685\pi\)
\(692\) −11.3768 −0.432482
\(693\) 0 0
\(694\) 15.6566 0.594316
\(695\) 6.51300 0.247052
\(696\) −6.52608 −0.247371
\(697\) 28.8767 1.09378
\(698\) 20.0102 0.757396
\(699\) −24.0522 −0.909736
\(700\) 0 0
\(701\) 13.9869 0.528278 0.264139 0.964485i \(-0.414912\pi\)
0.264139 + 0.964485i \(0.414912\pi\)
\(702\) 0 0
\(703\) 28.3581 1.06955
\(704\) −1.00000 −0.0376889
\(705\) −18.1492 −0.683540
\(706\) −7.46083 −0.280792
\(707\) 0 0
\(708\) 12.5261 0.470759
\(709\) 7.64191 0.286998 0.143499 0.989650i \(-0.454165\pi\)
0.143499 + 0.989650i \(0.454165\pi\)
\(710\) −4.13050 −0.155015
\(711\) −0.418833 −0.0157075
\(712\) 12.8377 0.481112
\(713\) 15.9738 0.598225
\(714\) 0 0
\(715\) 0 0
\(716\) 24.6101 0.919722
\(717\) 7.59133 0.283504
\(718\) −11.2463 −0.419709
\(719\) 19.4376 0.724899 0.362450 0.932003i \(-0.381940\pi\)
0.362450 + 0.932003i \(0.381940\pi\)
\(720\) −4.41883 −0.164680
\(721\) 0 0
\(722\) −5.39558 −0.200803
\(723\) −16.2985 −0.606148
\(724\) −17.4608 −0.648927
\(725\) −94.7984 −3.52072
\(726\) 1.00000 0.0371135
\(727\) 44.9217 1.66605 0.833026 0.553234i \(-0.186606\pi\)
0.833026 + 0.553234i \(0.186606\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −32.9682 −1.22021
\(731\) −7.28559 −0.269467
\(732\) −3.79567 −0.140292
\(733\) −25.0420 −0.924947 −0.462474 0.886633i \(-0.653038\pi\)
−0.462474 + 0.886633i \(0.653038\pi\)
\(734\) 5.91600 0.218364
\(735\) 0 0
\(736\) 4.73042 0.174365
\(737\) −10.1492 −0.373852
\(738\) −12.1492 −0.447220
\(739\) −6.62317 −0.243637 −0.121819 0.992552i \(-0.538873\pi\)
−0.121819 + 0.992552i \(0.538873\pi\)
\(740\) −33.9738 −1.24890
\(741\) 0 0
\(742\) 0 0
\(743\) −39.3972 −1.44534 −0.722671 0.691192i \(-0.757086\pi\)
−0.722671 + 0.691192i \(0.757086\pi\)
\(744\) 3.37683 0.123801
\(745\) 41.3768 1.51593
\(746\) −23.1260 −0.846703
\(747\) 2.14925 0.0786369
\(748\) 2.37683 0.0869056
\(749\) 0 0
\(750\) −42.0942 −1.53706
\(751\) −10.7072 −0.390710 −0.195355 0.980733i \(-0.562586\pi\)
−0.195355 + 0.980733i \(0.562586\pi\)
\(752\) 4.10725 0.149776
\(753\) 8.61008 0.313769
\(754\) 0 0
\(755\) 59.9551 2.18199
\(756\) 0 0
\(757\) −17.2797 −0.628043 −0.314022 0.949416i \(-0.601676\pi\)
−0.314022 + 0.949416i \(0.601676\pi\)
\(758\) −6.77241 −0.245985
\(759\) −4.73042 −0.171703
\(760\) −16.2985 −0.591209
\(761\) 32.7087 1.18569 0.592846 0.805316i \(-0.298004\pi\)
0.592846 + 0.805316i \(0.298004\pi\)
\(762\) 0.730416 0.0264602
\(763\) 0 0
\(764\) −16.8377 −0.609165
\(765\) 10.5028 0.379731
\(766\) −25.3637 −0.916429
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −12.2145 −0.440466 −0.220233 0.975447i \(-0.570682\pi\)
−0.220233 + 0.975447i \(0.570682\pi\)
\(770\) 0 0
\(771\) −0.837665 −0.0301678
\(772\) 5.46083 0.196540
\(773\) −0.204334 −0.00734937 −0.00367468 0.999993i \(-0.501170\pi\)
−0.00367468 + 0.999993i \(0.501170\pi\)
\(774\) 3.06525 0.110178
\(775\) 49.0522 1.76201
\(776\) −7.00000 −0.251285
\(777\) 0 0
\(778\) 25.2565 0.905489
\(779\) −44.8115 −1.60554
\(780\) 0 0
\(781\) −0.934749 −0.0334480
\(782\) −11.2434 −0.402064
\(783\) −6.52608 −0.233223
\(784\) 0 0
\(785\) −82.8115 −2.95567
\(786\) −7.37683 −0.263123
\(787\) −25.5579 −0.911041 −0.455521 0.890225i \(-0.650547\pi\)
−0.455521 + 0.890225i \(0.650547\pi\)
\(788\) −9.68842 −0.345136
\(789\) 10.8377 0.385831
\(790\) 1.85075 0.0658468
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) 0 0
\(794\) 13.4739 0.478171
\(795\) −35.3507 −1.25376
\(796\) −20.2985 −0.719461
\(797\) −30.0477 −1.06434 −0.532171 0.846637i \(-0.678624\pi\)
−0.532171 + 0.846637i \(0.678624\pi\)
\(798\) 0 0
\(799\) −9.76225 −0.345364
\(800\) 14.5261 0.513575
\(801\) 12.8377 0.453597
\(802\) 33.8898 1.19669
\(803\) −7.46083 −0.263287
\(804\) 10.1492 0.357936
\(805\) 0 0
\(806\) 0 0
\(807\) −17.8797 −0.629394
\(808\) 15.3637 0.540495
\(809\) 2.34342 0.0823901 0.0411951 0.999151i \(-0.486883\pi\)
0.0411951 + 0.999151i \(0.486883\pi\)
\(810\) −4.41883 −0.155262
\(811\) 11.8695 0.416794 0.208397 0.978044i \(-0.433175\pi\)
0.208397 + 0.978044i \(0.433175\pi\)
\(812\) 0 0
\(813\) 8.83767 0.309950
\(814\) −7.68842 −0.269479
\(815\) −82.9522 −2.90569
\(816\) −2.37683 −0.0832058
\(817\) 11.3059 0.395544
\(818\) 4.21450 0.147357
\(819\) 0 0
\(820\) 53.6855 1.87478
\(821\) −37.4812 −1.30810 −0.654051 0.756451i \(-0.726932\pi\)
−0.654051 + 0.756451i \(0.726932\pi\)
\(822\) −8.83767 −0.308249
\(823\) 1.29284 0.0450654 0.0225327 0.999746i \(-0.492827\pi\)
0.0225327 + 0.999746i \(0.492827\pi\)
\(824\) 2.62317 0.0913823
\(825\) −14.5261 −0.505733
\(826\) 0 0
\(827\) 43.3319 1.50680 0.753399 0.657563i \(-0.228413\pi\)
0.753399 + 0.657563i \(0.228413\pi\)
\(828\) 4.73042 0.164393
\(829\) −40.1174 −1.39334 −0.696668 0.717394i \(-0.745335\pi\)
−0.696668 + 0.717394i \(0.745335\pi\)
\(830\) −9.49717 −0.329652
\(831\) −16.0000 −0.555034
\(832\) 0 0
\(833\) 0 0
\(834\) −1.47392 −0.0510376
\(835\) −105.565 −3.65323
\(836\) −3.68842 −0.127567
\(837\) 3.37683 0.116720
\(838\) 22.3116 0.770741
\(839\) −54.6912 −1.88815 −0.944074 0.329733i \(-0.893041\pi\)
−0.944074 + 0.329733i \(0.893041\pi\)
\(840\) 0 0
\(841\) 13.5897 0.468612
\(842\) 18.3116 0.631059
\(843\) −15.9217 −0.548371
\(844\) 22.5130 0.774929
\(845\) 57.4448 1.97616
\(846\) 4.10725 0.141210
\(847\) 0 0
\(848\) 8.00000 0.274721
\(849\) 18.0840 0.620641
\(850\) −34.5261 −1.18423
\(851\) 36.3694 1.24673
\(852\) 0.934749 0.0320240
\(853\) 20.8275 0.713120 0.356560 0.934272i \(-0.383950\pi\)
0.356560 + 0.934272i \(0.383950\pi\)
\(854\) 0 0
\(855\) −16.2985 −0.557397
\(856\) 10.9029 0.372654
\(857\) 36.8433 1.25854 0.629272 0.777185i \(-0.283353\pi\)
0.629272 + 0.777185i \(0.283353\pi\)
\(858\) 0 0
\(859\) −38.5782 −1.31627 −0.658136 0.752899i \(-0.728655\pi\)
−0.658136 + 0.752899i \(0.728655\pi\)
\(860\) −13.5448 −0.461875
\(861\) 0 0
\(862\) −28.8377 −0.982215
\(863\) −17.2565 −0.587418 −0.293709 0.955895i \(-0.594890\pi\)
−0.293709 + 0.955895i \(0.594890\pi\)
\(864\) 1.00000 0.0340207
\(865\) 50.2723 1.70931
\(866\) −7.62317 −0.259046
\(867\) −11.3507 −0.385489
\(868\) 0 0
\(869\) 0.418833 0.0142079
\(870\) 28.8377 0.977688
\(871\) 0 0
\(872\) −11.7957 −0.399452
\(873\) −7.00000 −0.236914
\(874\) 17.4477 0.590179
\(875\) 0 0
\(876\) 7.46083 0.252078
\(877\) 0.250837 0.00847016 0.00423508 0.999991i \(-0.498652\pi\)
0.00423508 + 0.999991i \(0.498652\pi\)
\(878\) 33.3739 1.12632
\(879\) −25.1492 −0.848263
\(880\) 4.41883 0.148959
\(881\) −16.2985 −0.549110 −0.274555 0.961571i \(-0.588531\pi\)
−0.274555 + 0.961571i \(0.588531\pi\)
\(882\) 0 0
\(883\) −33.9551 −1.14268 −0.571340 0.820714i \(-0.693576\pi\)
−0.571340 + 0.820714i \(0.693576\pi\)
\(884\) 0 0
\(885\) −55.3507 −1.86059
\(886\) −4.85075 −0.162964
\(887\) −19.4608 −0.653431 −0.326715 0.945123i \(-0.605942\pi\)
−0.326715 + 0.945123i \(0.605942\pi\)
\(888\) 7.68842 0.258006
\(889\) 0 0
\(890\) −56.7275 −1.90151
\(891\) −1.00000 −0.0335013
\(892\) 7.46083 0.249807
\(893\) 15.1492 0.506950
\(894\) −9.36375 −0.313171
\(895\) −108.748 −3.63504
\(896\) 0 0
\(897\) 0 0
\(898\) 14.3450 0.478699
\(899\) −22.0375 −0.734992
\(900\) 14.5261 0.484203
\(901\) −19.0147 −0.633470
\(902\) 12.1492 0.404526
\(903\) 0 0
\(904\) 5.37683 0.178831
\(905\) 77.1565 2.56477
\(906\) −13.5681 −0.450769
\(907\) 0.772415 0.0256476 0.0128238 0.999918i \(-0.495918\pi\)
0.0128238 + 0.999918i \(0.495918\pi\)
\(908\) 2.77241 0.0920058
\(909\) 15.3637 0.509583
\(910\) 0 0
\(911\) −8.94491 −0.296358 −0.148179 0.988961i \(-0.547341\pi\)
−0.148179 + 0.988961i \(0.547341\pi\)
\(912\) 3.68842 0.122136
\(913\) −2.14925 −0.0711297
\(914\) −25.1827 −0.832969
\(915\) 16.7724 0.554479
\(916\) 2.53917 0.0838965
\(917\) 0 0
\(918\) −2.37683 −0.0784472
\(919\) 58.8812 1.94231 0.971157 0.238443i \(-0.0766369\pi\)
0.971157 + 0.238443i \(0.0766369\pi\)
\(920\) −20.9029 −0.689149
\(921\) 3.86950 0.127504
\(922\) −13.0187 −0.428750
\(923\) 0 0
\(924\) 0 0
\(925\) 111.683 3.67210
\(926\) −14.2145 −0.467117
\(927\) 2.62317 0.0861561
\(928\) −6.52608 −0.214229
\(929\) 22.5392 0.739486 0.369743 0.929134i \(-0.379446\pi\)
0.369743 + 0.929134i \(0.379446\pi\)
\(930\) −14.9217 −0.489301
\(931\) 0 0
\(932\) −24.0522 −0.787855
\(933\) 13.5681 0.444199
\(934\) 2.39558 0.0783858
\(935\) −10.5028 −0.343479
\(936\) 0 0
\(937\) 12.9478 0.422987 0.211494 0.977379i \(-0.432167\pi\)
0.211494 + 0.977379i \(0.432167\pi\)
\(938\) 0 0
\(939\) −2.52608 −0.0824356
\(940\) −18.1492 −0.591963
\(941\) −48.8246 −1.59164 −0.795818 0.605536i \(-0.792959\pi\)
−0.795818 + 0.605536i \(0.792959\pi\)
\(942\) 18.7406 0.610601
\(943\) −57.4710 −1.87151
\(944\) 12.5261 0.407689
\(945\) 0 0
\(946\) −3.06525 −0.0996599
\(947\) −31.1231 −1.01136 −0.505682 0.862720i \(-0.668759\pi\)
−0.505682 + 0.862720i \(0.668759\pi\)
\(948\) −0.418833 −0.0136031
\(949\) 0 0
\(950\) 53.5782 1.73831
\(951\) −20.0102 −0.648874
\(952\) 0 0
\(953\) 16.2797 0.527353 0.263676 0.964611i \(-0.415065\pi\)
0.263676 + 0.964611i \(0.415065\pi\)
\(954\) 8.00000 0.259010
\(955\) 74.4028 2.40762
\(956\) 7.59133 0.245521
\(957\) 6.52608 0.210958
\(958\) −10.8377 −0.350149
\(959\) 0 0
\(960\) −4.41883 −0.142617
\(961\) −19.5970 −0.632161
\(962\) 0 0
\(963\) 10.9029 0.351342
\(964\) −16.2985 −0.524939
\(965\) −24.1305 −0.776788
\(966\) 0 0
\(967\) 3.91308 0.125836 0.0629181 0.998019i \(-0.479959\pi\)
0.0629181 + 0.998019i \(0.479959\pi\)
\(968\) 1.00000 0.0321412
\(969\) −8.76675 −0.281629
\(970\) 30.9318 0.993161
\(971\) −39.3132 −1.26162 −0.630810 0.775938i \(-0.717277\pi\)
−0.630810 + 0.775938i \(0.717277\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −34.6435 −1.11005
\(975\) 0 0
\(976\) −3.79567 −0.121496
\(977\) −23.3507 −0.747054 −0.373527 0.927619i \(-0.621852\pi\)
−0.373527 + 0.927619i \(0.621852\pi\)
\(978\) 18.7724 0.600276
\(979\) −12.8377 −0.410294
\(980\) 0 0
\(981\) −11.7957 −0.376607
\(982\) 38.1492 1.21739
\(983\) 4.51592 0.144035 0.0720177 0.997403i \(-0.477056\pi\)
0.0720177 + 0.997403i \(0.477056\pi\)
\(984\) −12.1492 −0.387304
\(985\) 42.8115 1.36409
\(986\) 15.5114 0.493984
\(987\) 0 0
\(988\) 0 0
\(989\) 14.4999 0.461070
\(990\) 4.41883 0.140440
\(991\) 59.9273 1.90365 0.951827 0.306635i \(-0.0992032\pi\)
0.951827 + 0.306635i \(0.0992032\pi\)
\(992\) 3.37683 0.107215
\(993\) −16.1492 −0.512481
\(994\) 0 0
\(995\) 89.6957 2.84354
\(996\) 2.14925 0.0681015
\(997\) 21.5073 0.681144 0.340572 0.940218i \(-0.389379\pi\)
0.340572 + 0.940218i \(0.389379\pi\)
\(998\) −0.623166 −0.0197260
\(999\) 7.68842 0.243251
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.bh.1.1 3
3.2 odd 2 9702.2.a.dw.1.3 3
7.3 odd 6 462.2.i.g.331.1 yes 6
7.5 odd 6 462.2.i.g.67.1 6
7.6 odd 2 3234.2.a.bf.1.3 3
21.5 even 6 1386.2.k.v.991.3 6
21.17 even 6 1386.2.k.v.793.3 6
21.20 even 2 9702.2.a.dv.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.g.67.1 6 7.5 odd 6
462.2.i.g.331.1 yes 6 7.3 odd 6
1386.2.k.v.793.3 6 21.17 even 6
1386.2.k.v.991.3 6 21.5 even 6
3234.2.a.bf.1.3 3 7.6 odd 2
3234.2.a.bh.1.1 3 1.1 even 1 trivial
9702.2.a.dv.1.1 3 21.20 even 2
9702.2.a.dw.1.3 3 3.2 odd 2