Properties

Label 3234.2.a.bh.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: 3.3.2700.1
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 1.3
Root \(-2.80560\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.80560 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.80560 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.80560 q^{10} -1.00000 q^{11} +1.00000 q^{12} +2.80560 q^{15} +1.00000 q^{16} -7.96523 q^{17} +1.00000 q^{18} +6.48261 q^{19} +2.80560 q^{20} -1.00000 q^{22} -5.28822 q^{23} +1.00000 q^{24} +2.87141 q^{25} +1.00000 q^{27} +5.12859 q^{29} +2.80560 q^{30} +8.96523 q^{31} +1.00000 q^{32} -1.00000 q^{33} -7.96523 q^{34} +1.00000 q^{36} +10.4826 q^{37} +6.48261 q^{38} +2.80560 q^{40} +5.09382 q^{41} +11.4478 q^{43} -1.00000 q^{44} +2.80560 q^{45} -5.28822 q^{46} -0.322990 q^{47} +1.00000 q^{48} +2.87141 q^{50} -7.96523 q^{51} +8.00000 q^{53} +1.00000 q^{54} -2.80560 q^{55} +6.48261 q^{57} +5.12859 q^{58} +0.871407 q^{59} +2.80560 q^{60} -2.15962 q^{61} +8.96523 q^{62} +1.00000 q^{64} -1.00000 q^{66} -7.09382 q^{67} -7.96523 q^{68} -5.28822 q^{69} -7.44784 q^{71} +1.00000 q^{72} -12.5764 q^{73} +10.4826 q^{74} +2.87141 q^{75} +6.48261 q^{76} +6.80560 q^{79} +2.80560 q^{80} +1.00000 q^{81} +5.09382 q^{82} -15.0938 q^{83} -22.3473 q^{85} +11.4478 q^{86} +5.12859 q^{87} -1.00000 q^{88} -1.61121 q^{89} +2.80560 q^{90} -5.28822 q^{92} +8.96523 q^{93} -0.322990 q^{94} +18.1876 q^{95} +1.00000 q^{96} -7.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{8} + 3 q^{9} - 3 q^{11} + 3 q^{12} + 3 q^{16} - 3 q^{17} + 3 q^{18} + 9 q^{19} - 3 q^{22} + 3 q^{23} + 3 q^{24} + 15 q^{25} + 3 q^{27} + 9 q^{29} + 6 q^{31} + 3 q^{32} - 3 q^{33} - 3 q^{34} + 3 q^{36} + 21 q^{37} + 9 q^{38} - 12 q^{41} + 3 q^{43} - 3 q^{44} + 3 q^{46} - 3 q^{47} + 3 q^{48} + 15 q^{50} - 3 q^{51} + 24 q^{53} + 3 q^{54} + 9 q^{57} + 9 q^{58} + 9 q^{59} + 6 q^{61} + 6 q^{62} + 3 q^{64} - 3 q^{66} + 6 q^{67} - 3 q^{68} + 3 q^{69} + 9 q^{71} + 3 q^{72} + 21 q^{74} + 15 q^{75} + 9 q^{76} + 12 q^{79} + 3 q^{81} - 12 q^{82} - 18 q^{83} + 3 q^{86} + 9 q^{87} - 3 q^{88} + 12 q^{89} + 3 q^{92} + 6 q^{93} - 3 q^{94} + 3 q^{96} - 21 q^{97} - 3 q^{99} + O(q^{100}) \)

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
\(5\) 2.80560 1.25470 0.627352 0.778736i \(-0.284139\pi\)
0.627352 + 0.778736i \(0.284139\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.80560 0.887210
\(11\) −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
\(15\) 2.80560 0.724404
\(16\) 1.00000 0.250000
\(17\) −7.96523 −1.93185 −0.965926 0.258820i \(-0.916666\pi\)
−0.965926 + 0.258820i \(0.916666\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.48261 1.48721 0.743607 0.668617i \(-0.233114\pi\)
0.743607 + 0.668617i \(0.233114\pi\)
\(20\) 2.80560 0.627352
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −5.28822 −1.10267 −0.551335 0.834284i \(-0.685881\pi\)
−0.551335 + 0.834284i \(0.685881\pi\)
\(24\) 1.00000 0.204124
\(25\) 2.87141 0.574281
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.12859 0.952356 0.476178 0.879349i \(-0.342022\pi\)
0.476178 + 0.879349i \(0.342022\pi\)
\(30\) 2.80560 0.512231
\(31\) 8.96523 1.61020 0.805101 0.593138i \(-0.202111\pi\)
0.805101 + 0.593138i \(0.202111\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −7.96523 −1.36602
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.4826 1.72333 0.861665 0.507477i \(-0.169422\pi\)
0.861665 + 0.507477i \(0.169422\pi\)
\(38\) 6.48261 1.05162
\(39\) 0 0
\(40\) 2.80560 0.443605
\(41\) 5.09382 0.795521 0.397760 0.917489i \(-0.369788\pi\)
0.397760 + 0.917489i \(0.369788\pi\)
\(42\) 0 0
\(43\) 11.4478 1.74578 0.872890 0.487918i \(-0.162243\pi\)
0.872890 + 0.487918i \(0.162243\pi\)
\(44\) −1.00000 −0.150756
\(45\) 2.80560 0.418235
\(46\) −5.28822 −0.779705
\(47\) −0.322990 −0.0471129 −0.0235565 0.999723i \(-0.507499\pi\)
−0.0235565 + 0.999723i \(0.507499\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 2.87141 0.406078
\(51\) −7.96523 −1.11535
\(52\) 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
\(55\) −2.80560 −0.378307
\(56\) 0 0
\(57\) 6.48261 0.858643
\(58\) 5.12859 0.673417
\(59\) 0.871407 0.113448 0.0567238 0.998390i \(-0.481935\pi\)
0.0567238 + 0.998390i \(0.481935\pi\)
\(60\) 2.80560 0.362202
\(61\) −2.15962 −0.276511 −0.138256 0.990397i \(-0.544150\pi\)
−0.138256 + 0.990397i \(0.544150\pi\)
\(62\) 8.96523 1.13858
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) −7.09382 −0.866648 −0.433324 0.901238i \(-0.642659\pi\)
−0.433324 + 0.901238i \(0.642659\pi\)
\(68\) −7.96523 −0.965926
\(69\) −5.28822 −0.636626
\(70\) 0 0
\(71\) −7.44784 −0.883896 −0.441948 0.897041i \(-0.645712\pi\)
−0.441948 + 0.897041i \(0.645712\pi\)
\(72\) 1.00000 0.117851
\(73\) −12.5764 −1.47196 −0.735980 0.677003i \(-0.763278\pi\)
−0.735980 + 0.677003i \(0.763278\pi\)
\(74\) 10.4826 1.21858
\(75\) 2.87141 0.331562
\(76\) 6.48261 0.743607
\(77\) 0 0
\(78\) 0 0
\(79\) 6.80560 0.765690 0.382845 0.923813i \(-0.374944\pi\)
0.382845 + 0.923813i \(0.374944\pi\)
\(80\) 2.80560 0.313676
\(81\) 1.00000 0.111111
\(82\) 5.09382 0.562518
\(83\) −15.0938 −1.65676 −0.828381 0.560165i \(-0.810738\pi\)
−0.828381 + 0.560165i \(0.810738\pi\)
\(84\) 0 0
\(85\) −22.3473 −2.42390
\(86\) 11.4478 1.23445
\(87\) 5.12859 0.549843
\(88\) −1.00000 −0.106600
\(89\) −1.61121 −0.170787 −0.0853937 0.996347i \(-0.527215\pi\)
−0.0853937 + 0.996347i \(0.527215\pi\)
\(90\) 2.80560 0.295737
\(91\) 0 0
\(92\) −5.28822 −0.551335
\(93\) 8.96523 0.929651
\(94\) −0.322990 −0.0333139
\(95\) 18.1876 1.86601
\(96\) 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
\(100\) 2.87141 0.287141
\(101\) −10.7398 −1.06865 −0.534325 0.845279i \(-0.679434\pi\)
−0.534325 + 0.845279i \(0.679434\pi\)
\(102\) −7.96523 −0.788675
\(103\) −2.96523 −0.292172 −0.146086 0.989272i \(-0.546668\pi\)
−0.146086 + 0.989272i \(0.546668\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) 4.83663 0.467575 0.233787 0.972288i \(-0.424888\pi\)
0.233787 + 0.972288i \(0.424888\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.1596 −0.973115 −0.486558 0.873648i \(-0.661748\pi\)
−0.486558 + 0.873648i \(0.661748\pi\)
\(110\) −2.80560 −0.267504
\(111\) 10.4826 0.994966
\(112\) 0 0
\(113\) 10.9652 1.03152 0.515761 0.856733i \(-0.327509\pi\)
0.515761 + 0.856733i \(0.327509\pi\)
\(114\) 6.48261 0.607152
\(115\) −14.8366 −1.38352
\(116\) 5.12859 0.476178
\(117\) 0 0
\(118\) 0.871407 0.0802195
\(119\) 0 0
\(120\) 2.80560 0.256115
\(121\) 1.00000 0.0909091
\(122\) −2.15962 −0.195523
\(123\) 5.09382 0.459294
\(124\) 8.96523 0.805101
\(125\) −5.97199 −0.534151
\(126\) 0 0
\(127\) −9.28822 −0.824196 −0.412098 0.911140i \(-0.635204\pi\)
−0.412098 + 0.911140i \(0.635204\pi\)
\(128\) 1.00000 0.0883883
\(129\) 11.4478 1.00793
\(130\) 0 0
\(131\) −12.9652 −1.13278 −0.566389 0.824138i \(-0.691660\pi\)
−0.566389 + 0.824138i \(0.691660\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −7.09382 −0.612813
\(135\) 2.80560 0.241468
\(136\) −7.96523 −0.683012
\(137\) 5.61121 0.479398 0.239699 0.970847i \(-0.422951\pi\)
0.239699 + 0.970847i \(0.422951\pi\)
\(138\) −5.28822 −0.450163
\(139\) −13.1286 −1.11355 −0.556776 0.830662i \(-0.687962\pi\)
−0.556776 + 0.830662i \(0.687962\pi\)
\(140\) 0 0
\(141\) −0.322990 −0.0272007
\(142\) −7.44784 −0.625009
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 14.3888 1.19492
\(146\) −12.5764 −1.04083
\(147\) 0 0
\(148\) 10.4826 0.861665
\(149\) 16.7398 1.37138 0.685689 0.727895i \(-0.259501\pi\)
0.685689 + 0.727895i \(0.259501\pi\)
\(150\) 2.87141 0.234449
\(151\) 10.8994 0.886982 0.443491 0.896279i \(-0.353740\pi\)
0.443491 + 0.896279i \(0.353740\pi\)
\(152\) 6.48261 0.525809
\(153\) −7.96523 −0.643950
\(154\) 0 0
\(155\) 25.1529 2.02033
\(156\) 0 0
\(157\) −1.77457 −0.141626 −0.0708132 0.997490i \(-0.522559\pi\)
−0.0708132 + 0.997490i \(0.522559\pi\)
\(158\) 6.80560 0.541425
\(159\) 8.00000 0.634441
\(160\) 2.80560 0.221802
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.05904 −0.317929 −0.158964 0.987284i \(-0.550816\pi\)
−0.158964 + 0.987284i \(0.550816\pi\)
\(164\) 5.09382 0.397760
\(165\) −2.80560 −0.218416
\(166\) −15.0938 −1.17151
\(167\) −13.8684 −1.07317 −0.536584 0.843847i \(-0.680286\pi\)
−0.536584 + 0.843847i \(0.680286\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −22.3473 −1.71396
\(171\) 6.48261 0.495738
\(172\) 11.4478 0.872890
\(173\) −16.9652 −1.28984 −0.644921 0.764249i \(-0.723110\pi\)
−0.644921 + 0.764249i \(0.723110\pi\)
\(174\) 5.12859 0.388798
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0.871407 0.0654990
\(178\) −1.61121 −0.120765
\(179\) −12.6703 −0.947019 −0.473509 0.880789i \(-0.657013\pi\)
−0.473509 + 0.880789i \(0.657013\pi\)
\(180\) 2.80560 0.209117
\(181\) 2.57643 0.191505 0.0957523 0.995405i \(-0.469474\pi\)
0.0957523 + 0.995405i \(0.469474\pi\)
\(182\) 0 0
\(183\) −2.15962 −0.159644
\(184\) −5.28822 −0.389852
\(185\) 29.4100 2.16227
\(186\) 8.96523 0.657362
\(187\) 7.96523 0.582475
\(188\) −0.322990 −0.0235565
\(189\) 0 0
\(190\) 18.1876 1.31947
\(191\) −2.38879 −0.172847 −0.0864235 0.996258i \(-0.527544\pi\)
−0.0864235 + 0.996258i \(0.527544\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.5764 −1.04923 −0.524617 0.851338i \(-0.675792\pi\)
−0.524617 + 0.851338i \(0.675792\pi\)
\(194\) −7.00000 −0.502571
\(195\) 0 0
\(196\) 0 0
\(197\) −12.4826 −0.889349 −0.444675 0.895692i \(-0.646681\pi\)
−0.444675 + 0.895692i \(0.646681\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 14.1876 1.00573 0.502867 0.864364i \(-0.332278\pi\)
0.502867 + 0.864364i \(0.332278\pi\)
\(200\) 2.87141 0.203039
\(201\) −7.09382 −0.500359
\(202\) −10.7398 −0.755650
\(203\) 0 0
\(204\) −7.96523 −0.557677
\(205\) 14.2912 0.998143
\(206\) −2.96523 −0.206597
\(207\) −5.28822 −0.367556
\(208\) 0 0
\(209\) −6.48261 −0.448412
\(210\) 0 0
\(211\) −20.8336 −1.43425 −0.717123 0.696947i \(-0.754541\pi\)
−0.717123 + 0.696947i \(0.754541\pi\)
\(212\) 8.00000 0.549442
\(213\) −7.44784 −0.510318
\(214\) 4.83663 0.330625
\(215\) 32.1181 2.19044
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −10.1596 −0.688096
\(219\) −12.5764 −0.849836
\(220\) −2.80560 −0.189154
\(221\) 0 0
\(222\) 10.4826 0.703547
\(223\) −12.5764 −0.842180 −0.421090 0.907019i \(-0.638352\pi\)
−0.421090 + 0.907019i \(0.638352\pi\)
\(224\) 0 0
\(225\) 2.87141 0.191427
\(226\) 10.9652 0.729396
\(227\) −20.0590 −1.33137 −0.665683 0.746235i \(-0.731860\pi\)
−0.665683 + 0.746235i \(0.731860\pi\)
\(228\) 6.48261 0.429322
\(229\) 22.5764 1.49189 0.745946 0.666006i \(-0.231998\pi\)
0.745946 + 0.666006i \(0.231998\pi\)
\(230\) −14.8366 −0.978299
\(231\) 0 0
\(232\) 5.12859 0.336709
\(233\) −0.742815 −0.0486634 −0.0243317 0.999704i \(-0.507746\pi\)
−0.0243317 + 0.999704i \(0.507746\pi\)
\(234\) 0 0
\(235\) −0.906181 −0.0591128
\(236\) 0.871407 0.0567238
\(237\) 6.80560 0.442071
\(238\) 0 0
\(239\) 4.31925 0.279389 0.139694 0.990195i \(-0.455388\pi\)
0.139694 + 0.990195i \(0.455388\pi\)
\(240\) 2.80560 0.181101
\(241\) 18.1876 1.17157 0.585784 0.810467i \(-0.300787\pi\)
0.585784 + 0.810467i \(0.300787\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) −2.15962 −0.138256
\(245\) 0 0
\(246\) 5.09382 0.324770
\(247\) 0 0
\(248\) 8.96523 0.569292
\(249\) −15.0938 −0.956532
\(250\) −5.97199 −0.377702
\(251\) −28.6703 −1.80965 −0.904825 0.425784i \(-0.859999\pi\)
−0.904825 + 0.425784i \(0.859999\pi\)
\(252\) 0 0
\(253\) 5.28822 0.332467
\(254\) −9.28822 −0.582794
\(255\) −22.3473 −1.39944
\(256\) 1.00000 0.0625000
\(257\) 13.6112 0.849044 0.424522 0.905418i \(-0.360442\pi\)
0.424522 + 0.905418i \(0.360442\pi\)
\(258\) 11.4478 0.712711
\(259\) 0 0
\(260\) 0 0
\(261\) 5.12859 0.317452
\(262\) −12.9652 −0.800994
\(263\) −3.61121 −0.222676 −0.111338 0.993783i \(-0.535514\pi\)
−0.111338 + 0.993783i \(0.535514\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 22.4448 1.37877
\(266\) 0 0
\(267\) −1.61121 −0.0986042
\(268\) −7.09382 −0.433324
\(269\) 9.38203 0.572033 0.286016 0.958225i \(-0.407669\pi\)
0.286016 + 0.958225i \(0.407669\pi\)
\(270\) 2.80560 0.170744
\(271\) −5.61121 −0.340856 −0.170428 0.985370i \(-0.554515\pi\)
−0.170428 + 0.985370i \(0.554515\pi\)
\(272\) −7.96523 −0.482963
\(273\) 0 0
\(274\) 5.61121 0.338985
\(275\) −2.87141 −0.173152
\(276\) −5.28822 −0.318313
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) −13.1286 −0.787401
\(279\) 8.96523 0.536734
\(280\) 0 0
\(281\) 24.1529 1.44084 0.720420 0.693539i \(-0.243949\pi\)
0.720420 + 0.693539i \(0.243949\pi\)
\(282\) −0.322990 −0.0192338
\(283\) −7.54166 −0.448305 −0.224152 0.974554i \(-0.571961\pi\)
−0.224152 + 0.974554i \(0.571961\pi\)
\(284\) −7.44784 −0.441948
\(285\) 18.1876 1.07734
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 46.4448 2.73205
\(290\) 14.3888 0.844939
\(291\) −7.00000 −0.410347
\(292\) −12.5764 −0.735980
\(293\) −7.90618 −0.461884 −0.230942 0.972968i \(-0.574181\pi\)
−0.230942 + 0.972968i \(0.574181\pi\)
\(294\) 0 0
\(295\) 2.44482 0.142343
\(296\) 10.4826 0.609289
\(297\) −1.00000 −0.0580259
\(298\) 16.7398 0.969710
\(299\) 0 0
\(300\) 2.87141 0.165781
\(301\) 0 0
\(302\) 10.8994 0.627191
\(303\) −10.7398 −0.616985
\(304\) 6.48261 0.371803
\(305\) −6.05904 −0.346940
\(306\) −7.96523 −0.455342
\(307\) −12.8957 −0.735995 −0.367998 0.929827i \(-0.619957\pi\)
−0.367998 + 0.929827i \(0.619957\pi\)
\(308\) 0 0
\(309\) −2.96523 −0.168686
\(310\) 25.1529 1.42859
\(311\) −10.8994 −0.618049 −0.309025 0.951054i \(-0.600003\pi\)
−0.309025 + 0.951054i \(0.600003\pi\)
\(312\) 0 0
\(313\) 9.12859 0.515979 0.257989 0.966148i \(-0.416940\pi\)
0.257989 + 0.966148i \(0.416940\pi\)
\(314\) −1.77457 −0.100145
\(315\) 0 0
\(316\) 6.80560 0.382845
\(317\) −9.51364 −0.534339 −0.267170 0.963649i \(-0.586088\pi\)
−0.267170 + 0.963649i \(0.586088\pi\)
\(318\) 8.00000 0.448618
\(319\) −5.12859 −0.287146
\(320\) 2.80560 0.156838
\(321\) 4.83663 0.269955
\(322\) 0 0
\(323\) −51.6355 −2.87307
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.05904 −0.224810
\(327\) −10.1596 −0.561828
\(328\) 5.09382 0.281259
\(329\) 0 0
\(330\) −2.80560 −0.154443
\(331\) 1.09382 0.0601217 0.0300609 0.999548i \(-0.490430\pi\)
0.0300609 + 0.999548i \(0.490430\pi\)
\(332\) −15.0938 −0.828381
\(333\) 10.4826 0.574444
\(334\) −13.8684 −0.758845
\(335\) −19.9024 −1.08739
\(336\) 0 0
\(337\) −25.7988 −1.40535 −0.702676 0.711510i \(-0.748012\pi\)
−0.702676 + 0.711510i \(0.748012\pi\)
\(338\) −13.0000 −0.707107
\(339\) 10.9652 0.595549
\(340\) −22.3473 −1.21195
\(341\) −8.96523 −0.485494
\(342\) 6.48261 0.350540
\(343\) 0 0
\(344\) 11.4478 0.617226
\(345\) −14.8366 −0.798777
\(346\) −16.9652 −0.912056
\(347\) 20.7671 1.11484 0.557418 0.830232i \(-0.311792\pi\)
0.557418 + 0.830232i \(0.311792\pi\)
\(348\) 5.12859 0.274921
\(349\) 9.51364 0.509254 0.254627 0.967039i \(-0.418047\pi\)
0.254627 + 0.967039i \(0.418047\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 12.5764 0.669376 0.334688 0.942329i \(-0.391369\pi\)
0.334688 + 0.942329i \(0.391369\pi\)
\(354\) 0.871407 0.0463148
\(355\) −20.8957 −1.10903
\(356\) −1.61121 −0.0853937
\(357\) 0 0
\(358\) −12.6703 −0.669644
\(359\) −0.0695483 −0.00367062 −0.00183531 0.999998i \(-0.500584\pi\)
−0.00183531 + 0.999998i \(0.500584\pi\)
\(360\) 2.80560 0.147868
\(361\) 23.0243 1.21180
\(362\) 2.57643 0.135414
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −35.2845 −1.84687
\(366\) −2.15962 −0.112885
\(367\) 31.5417 1.64646 0.823231 0.567707i \(-0.192169\pi\)
0.823231 + 0.567707i \(0.192169\pi\)
\(368\) −5.28822 −0.275667
\(369\) 5.09382 0.265174
\(370\) 29.4100 1.52896
\(371\) 0 0
\(372\) 8.96523 0.464825
\(373\) 15.3125 0.792850 0.396425 0.918067i \(-0.370251\pi\)
0.396425 + 0.918067i \(0.370251\pi\)
\(374\) 7.96523 0.411872
\(375\) −5.97199 −0.308392
\(376\) −0.322990 −0.0166569
\(377\) 0 0
\(378\) 0 0
\(379\) 16.0590 0.824898 0.412449 0.910981i \(-0.364674\pi\)
0.412449 + 0.910981i \(0.364674\pi\)
\(380\) 18.1876 0.933006
\(381\) −9.28822 −0.475850
\(382\) −2.38879 −0.122221
\(383\) 0.739798 0.0378019 0.0189010 0.999821i \(-0.493983\pi\)
0.0189010 + 0.999821i \(0.493983\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.5764 −0.741921
\(387\) 11.4478 0.581926
\(388\) −7.00000 −0.355371
\(389\) 3.58319 0.181675 0.0908375 0.995866i \(-0.471046\pi\)
0.0908375 + 0.995866i \(0.471046\pi\)
\(390\) 0 0
\(391\) 42.1218 2.13019
\(392\) 0 0
\(393\) −12.9652 −0.654009
\(394\) −12.4826 −0.628865
\(395\) 19.0938 0.960714
\(396\) −1.00000 −0.0502519
\(397\) 25.1286 1.26117 0.630584 0.776121i \(-0.282815\pi\)
0.630584 + 0.776121i \(0.282815\pi\)
\(398\) 14.1876 0.711162
\(399\) 0 0
\(400\) 2.87141 0.143570
\(401\) −3.86839 −0.193178 −0.0965891 0.995324i \(-0.530793\pi\)
−0.0965891 + 0.995324i \(0.530793\pi\)
\(402\) −7.09382 −0.353808
\(403\) 0 0
\(404\) −10.7398 −0.534325
\(405\) 2.80560 0.139412
\(406\) 0 0
\(407\) −10.4826 −0.519604
\(408\) −7.96523 −0.394337
\(409\) −4.64598 −0.229729 −0.114864 0.993381i \(-0.536643\pi\)
−0.114864 + 0.993381i \(0.536643\pi\)
\(410\) 14.2912 0.705794
\(411\) 5.61121 0.276780
\(412\) −2.96523 −0.146086
\(413\) 0 0
\(414\) −5.28822 −0.259902
\(415\) −42.3473 −2.07875
\(416\) 0 0
\(417\) −13.1286 −0.642910
\(418\) −6.48261 −0.317075
\(419\) 19.5174 0.953487 0.476743 0.879043i \(-0.341817\pi\)
0.476743 + 0.879043i \(0.341817\pi\)
\(420\) 0 0
\(421\) 15.5174 0.756271 0.378136 0.925750i \(-0.376565\pi\)
0.378136 + 0.925750i \(0.376565\pi\)
\(422\) −20.8336 −1.01416
\(423\) −0.322990 −0.0157043
\(424\) 8.00000 0.388514
\(425\) −22.8714 −1.10943
\(426\) −7.44784 −0.360849
\(427\) 0 0
\(428\) 4.83663 0.233787
\(429\) 0 0
\(430\) 32.1181 1.54887
\(431\) −14.3888 −0.693084 −0.346542 0.938034i \(-0.612644\pi\)
−0.346542 + 0.938034i \(0.612644\pi\)
\(432\) 1.00000 0.0481125
\(433\) −2.03477 −0.0977850 −0.0488925 0.998804i \(-0.515569\pi\)
−0.0488925 + 0.998804i \(0.515569\pi\)
\(434\) 0 0
\(435\) 14.3888 0.689890
\(436\) −10.1596 −0.486558
\(437\) −34.2815 −1.63990
\(438\) −12.5764 −0.600925
\(439\) −3.22616 −0.153976 −0.0769880 0.997032i \(-0.524530\pi\)
−0.0769880 + 0.997032i \(0.524530\pi\)
\(440\) −2.80560 −0.133752
\(441\) 0 0
\(442\) 0 0
\(443\) −22.0938 −1.04971 −0.524854 0.851192i \(-0.675880\pi\)
−0.524854 + 0.851192i \(0.675880\pi\)
\(444\) 10.4826 0.497483
\(445\) −4.52040 −0.214288
\(446\) −12.5764 −0.595511
\(447\) 16.7398 0.791765
\(448\) 0 0
\(449\) 22.2497 1.05003 0.525014 0.851094i \(-0.324060\pi\)
0.525014 + 0.851094i \(0.324060\pi\)
\(450\) 2.87141 0.135359
\(451\) −5.09382 −0.239859
\(452\) 10.9652 0.515761
\(453\) 10.8994 0.512099
\(454\) −20.0590 −0.941418
\(455\) 0 0
\(456\) 6.48261 0.303576
\(457\) −18.6385 −0.871872 −0.435936 0.899978i \(-0.643583\pi\)
−0.435936 + 0.899978i \(0.643583\pi\)
\(458\) 22.5764 1.05493
\(459\) −7.96523 −0.371785
\(460\) −14.8366 −0.691762
\(461\) 20.9895 0.977578 0.488789 0.872402i \(-0.337439\pi\)
0.488789 + 0.872402i \(0.337439\pi\)
\(462\) 0 0
\(463\) −5.35402 −0.248822 −0.124411 0.992231i \(-0.539704\pi\)
−0.124411 + 0.992231i \(0.539704\pi\)
\(464\) 5.12859 0.238089
\(465\) 25.1529 1.16644
\(466\) −0.742815 −0.0344102
\(467\) −26.0243 −1.20426 −0.602130 0.798398i \(-0.705681\pi\)
−0.602130 + 0.798398i \(0.705681\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −0.906181 −0.0417990
\(471\) −1.77457 −0.0817680
\(472\) 0.871407 0.0401098
\(473\) −11.4478 −0.526372
\(474\) 6.80560 0.312592
\(475\) 18.6142 0.854079
\(476\) 0 0
\(477\) 8.00000 0.366295
\(478\) 4.31925 0.197558
\(479\) 3.61121 0.165000 0.0825001 0.996591i \(-0.473710\pi\)
0.0825001 + 0.996591i \(0.473710\pi\)
\(480\) 2.80560 0.128058
\(481\) 0 0
\(482\) 18.1876 0.828424
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −19.6392 −0.891771
\(486\) 1.00000 0.0453609
\(487\) −8.06206 −0.365327 −0.182663 0.983176i \(-0.558472\pi\)
−0.182663 + 0.983176i \(0.558472\pi\)
\(488\) −2.15962 −0.0977615
\(489\) −4.05904 −0.183556
\(490\) 0 0
\(491\) 20.9062 0.943483 0.471741 0.881737i \(-0.343626\pi\)
0.471741 + 0.881737i \(0.343626\pi\)
\(492\) 5.09382 0.229647
\(493\) −40.8504 −1.83981
\(494\) 0 0
\(495\) −2.80560 −0.126102
\(496\) 8.96523 0.402551
\(497\) 0 0
\(498\) −15.0938 −0.676370
\(499\) 4.96523 0.222274 0.111137 0.993805i \(-0.464551\pi\)
0.111137 + 0.993805i \(0.464551\pi\)
\(500\) −5.97199 −0.267075
\(501\) −13.8684 −0.619594
\(502\) −28.6703 −1.27962
\(503\) 17.7988 0.793611 0.396806 0.917903i \(-0.370119\pi\)
0.396806 + 0.917903i \(0.370119\pi\)
\(504\) 0 0
\(505\) −30.1316 −1.34084
\(506\) 5.28822 0.235090
\(507\) −13.0000 −0.577350
\(508\) −9.28822 −0.412098
\(509\) −4.51437 −0.200096 −0.100048 0.994983i \(-0.531900\pi\)
−0.100048 + 0.994983i \(0.531900\pi\)
\(510\) −22.3473 −0.989553
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 6.48261 0.286214
\(514\) 13.6112 0.600365
\(515\) −8.31925 −0.366590
\(516\) 11.4478 0.503963
\(517\) 0.322990 0.0142051
\(518\) 0 0
\(519\) −16.9652 −0.744691
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 5.12859 0.224472
\(523\) −6.90317 −0.301854 −0.150927 0.988545i \(-0.548226\pi\)
−0.150927 + 0.988545i \(0.548226\pi\)
\(524\) −12.9652 −0.566389
\(525\) 0 0
\(526\) −3.61121 −0.157456
\(527\) −71.4100 −3.11067
\(528\) −1.00000 −0.0435194
\(529\) 4.96523 0.215879
\(530\) 22.4448 0.974941
\(531\) 0.871407 0.0378159
\(532\) 0 0
\(533\) 0 0
\(534\) −1.61121 −0.0697237
\(535\) 13.5697 0.586668
\(536\) −7.09382 −0.306406
\(537\) −12.6703 −0.546762
\(538\) 9.38203 0.404488
\(539\) 0 0
\(540\) 2.80560 0.120734
\(541\) 2.80560 0.120622 0.0603111 0.998180i \(-0.480791\pi\)
0.0603111 + 0.998180i \(0.480791\pi\)
\(542\) −5.61121 −0.241022
\(543\) 2.57643 0.110565
\(544\) −7.96523 −0.341506
\(545\) −28.5039 −1.22097
\(546\) 0 0
\(547\) 32.9274 1.40788 0.703938 0.710262i \(-0.251423\pi\)
0.703938 + 0.710262i \(0.251423\pi\)
\(548\) 5.61121 0.239699
\(549\) −2.15962 −0.0921705
\(550\) −2.87141 −0.122437
\(551\) 33.2467 1.41636
\(552\) −5.28822 −0.225081
\(553\) 0 0
\(554\) −16.0000 −0.679775
\(555\) 29.4100 1.24839
\(556\) −13.1286 −0.556776
\(557\) 23.7050 1.00441 0.502207 0.864747i \(-0.332522\pi\)
0.502207 + 0.864747i \(0.332522\pi\)
\(558\) 8.96523 0.379528
\(559\) 0 0
\(560\) 0 0
\(561\) 7.96523 0.336292
\(562\) 24.1529 1.01883
\(563\) −30.4448 −1.28310 −0.641548 0.767083i \(-0.721708\pi\)
−0.641548 + 0.767083i \(0.721708\pi\)
\(564\) −0.322990 −0.0136003
\(565\) 30.7641 1.29425
\(566\) −7.54166 −0.317000
\(567\) 0 0
\(568\) −7.44784 −0.312504
\(569\) 22.2815 0.934087 0.467044 0.884234i \(-0.345319\pi\)
0.467044 + 0.884234i \(0.345319\pi\)
\(570\) 18.1876 0.761796
\(571\) −2.87141 −0.120165 −0.0600823 0.998193i \(-0.519136\pi\)
−0.0600823 + 0.998193i \(0.519136\pi\)
\(572\) 0 0
\(573\) −2.38879 −0.0997933
\(574\) 0 0
\(575\) −15.1846 −0.633242
\(576\) 1.00000 0.0416667
\(577\) 15.0243 0.625469 0.312734 0.949841i \(-0.398755\pi\)
0.312734 + 0.949841i \(0.398755\pi\)
\(578\) 46.4448 1.93185
\(579\) −14.5764 −0.605776
\(580\) 14.3888 0.597462
\(581\) 0 0
\(582\) −7.00000 −0.290159
\(583\) −8.00000 −0.331326
\(584\) −12.5764 −0.520416
\(585\) 0 0
\(586\) −7.90618 −0.326601
\(587\) 42.5069 1.75445 0.877223 0.480082i \(-0.159393\pi\)
0.877223 + 0.480082i \(0.159393\pi\)
\(588\) 0 0
\(589\) 58.1181 2.39471
\(590\) 2.44482 0.100652
\(591\) −12.4826 −0.513466
\(592\) 10.4826 0.430833
\(593\) −15.3858 −0.631818 −0.315909 0.948789i \(-0.602309\pi\)
−0.315909 + 0.948789i \(0.602309\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 16.7398 0.685689
\(597\) 14.1876 0.580661
\(598\) 0 0
\(599\) −39.6392 −1.61961 −0.809807 0.586696i \(-0.800428\pi\)
−0.809807 + 0.586696i \(0.800428\pi\)
\(600\) 2.87141 0.117225
\(601\) −33.3405 −1.35999 −0.679994 0.733218i \(-0.738017\pi\)
−0.679994 + 0.733218i \(0.738017\pi\)
\(602\) 0 0
\(603\) −7.09382 −0.288883
\(604\) 10.8994 0.443491
\(605\) 2.80560 0.114064
\(606\) −10.7398 −0.436274
\(607\) −15.9585 −0.647734 −0.323867 0.946103i \(-0.604983\pi\)
−0.323867 + 0.946103i \(0.604983\pi\)
\(608\) 6.48261 0.262905
\(609\) 0 0
\(610\) −6.05904 −0.245324
\(611\) 0 0
\(612\) −7.96523 −0.321975
\(613\) 46.6665 1.88484 0.942421 0.334428i \(-0.108543\pi\)
0.942421 + 0.334428i \(0.108543\pi\)
\(614\) −12.8957 −0.520427
\(615\) 14.2912 0.576278
\(616\) 0 0
\(617\) −9.28447 −0.373779 −0.186889 0.982381i \(-0.559841\pi\)
−0.186889 + 0.982381i \(0.559841\pi\)
\(618\) −2.96523 −0.119279
\(619\) −33.9895 −1.36615 −0.683077 0.730347i \(-0.739358\pi\)
−0.683077 + 0.730347i \(0.739358\pi\)
\(620\) 25.1529 1.01016
\(621\) −5.28822 −0.212209
\(622\) −10.8994 −0.437027
\(623\) 0 0
\(624\) 0 0
\(625\) −31.1121 −1.24448
\(626\) 9.12859 0.364852
\(627\) −6.48261 −0.258891
\(628\) −1.77457 −0.0708132
\(629\) −83.4964 −3.32922
\(630\) 0 0
\(631\) −14.6460 −0.583047 −0.291524 0.956564i \(-0.594162\pi\)
−0.291524 + 0.956564i \(0.594162\pi\)
\(632\) 6.80560 0.270712
\(633\) −20.8336 −0.828062
\(634\) −9.51364 −0.377835
\(635\) −26.0590 −1.03412
\(636\) 8.00000 0.317221
\(637\) 0 0
\(638\) −5.12859 −0.203043
\(639\) −7.44784 −0.294632
\(640\) 2.80560 0.110901
\(641\) 20.9652 0.828077 0.414038 0.910259i \(-0.364118\pi\)
0.414038 + 0.910259i \(0.364118\pi\)
\(642\) 4.83663 0.190887
\(643\) 23.6733 0.933582 0.466791 0.884368i \(-0.345410\pi\)
0.466791 + 0.884368i \(0.345410\pi\)
\(644\) 0 0
\(645\) 32.1181 1.26465
\(646\) −51.6355 −2.03157
\(647\) 12.4168 0.488155 0.244078 0.969756i \(-0.421515\pi\)
0.244078 + 0.969756i \(0.421515\pi\)
\(648\) 1.00000 0.0392837
\(649\) −0.871407 −0.0342057
\(650\) 0 0
\(651\) 0 0
\(652\) −4.05904 −0.158964
\(653\) −0.0975625 −0.00381792 −0.00190896 0.999998i \(-0.500608\pi\)
−0.00190896 + 0.999998i \(0.500608\pi\)
\(654\) −10.1596 −0.397273
\(655\) −36.3753 −1.42130
\(656\) 5.09382 0.198880
\(657\) −12.5764 −0.490653
\(658\) 0 0
\(659\) −35.0243 −1.36435 −0.682176 0.731188i \(-0.738966\pi\)
−0.682176 + 0.731188i \(0.738966\pi\)
\(660\) −2.80560 −0.109208
\(661\) −29.6355 −1.15269 −0.576343 0.817208i \(-0.695521\pi\)
−0.576343 + 0.817208i \(0.695521\pi\)
\(662\) 1.09382 0.0425125
\(663\) 0 0
\(664\) −15.0938 −0.585754
\(665\) 0 0
\(666\) 10.4826 0.406193
\(667\) −27.1211 −1.05013
\(668\) −13.8684 −0.536584
\(669\) −12.5764 −0.486233
\(670\) −19.9024 −0.768898
\(671\) 2.15962 0.0833713
\(672\) 0 0
\(673\) 16.8957 0.651281 0.325640 0.945494i \(-0.394420\pi\)
0.325640 + 0.945494i \(0.394420\pi\)
\(674\) −25.7988 −0.993734
\(675\) 2.87141 0.110521
\(676\) −13.0000 −0.500000
\(677\) −13.1907 −0.506958 −0.253479 0.967341i \(-0.581575\pi\)
−0.253479 + 0.967341i \(0.581575\pi\)
\(678\) 10.9652 0.421117
\(679\) 0 0
\(680\) −22.3473 −0.856978
\(681\) −20.0590 −0.768664
\(682\) −8.96523 −0.343296
\(683\) 9.83663 0.376388 0.188194 0.982132i \(-0.439737\pi\)
0.188194 + 0.982132i \(0.439737\pi\)
\(684\) 6.48261 0.247869
\(685\) 15.7428 0.601502
\(686\) 0 0
\(687\) 22.5764 0.861345
\(688\) 11.4478 0.436445
\(689\) 0 0
\(690\) −14.8366 −0.564821
\(691\) 49.9895 1.90169 0.950845 0.309667i \(-0.100218\pi\)
0.950845 + 0.309667i \(0.100218\pi\)
\(692\) −16.9652 −0.644921
\(693\) 0 0
\(694\) 20.7671 0.788308
\(695\) −36.8336 −1.39718
\(696\) 5.12859 0.194399
\(697\) −40.5734 −1.53683
\(698\) 9.51364 0.360097
\(699\) −0.742815 −0.0280958
\(700\) 0 0
\(701\) −17.7050 −0.668710 −0.334355 0.942447i \(-0.608518\pi\)
−0.334355 + 0.942447i \(0.608518\pi\)
\(702\) 0 0
\(703\) 67.9547 2.56296
\(704\) −1.00000 −0.0376889
\(705\) −0.906181 −0.0341288
\(706\) 12.5764 0.473320
\(707\) 0 0
\(708\) 0.871407 0.0327495
\(709\) −31.9547 −1.20008 −0.600042 0.799968i \(-0.704850\pi\)
−0.600042 + 0.799968i \(0.704850\pi\)
\(710\) −20.8957 −0.784201
\(711\) 6.80560 0.255230
\(712\) −1.61121 −0.0603825
\(713\) −47.4100 −1.77552
\(714\) 0 0
\(715\) 0 0
\(716\) −12.6703 −0.473509
\(717\) 4.31925 0.161305
\(718\) −0.0695483 −0.00259552
\(719\) −21.7951 −0.812820 −0.406410 0.913691i \(-0.633220\pi\)
−0.406410 + 0.913691i \(0.633220\pi\)
\(720\) 2.80560 0.104559
\(721\) 0 0
\(722\) 23.0243 0.856875
\(723\) 18.1876 0.676406
\(724\) 2.57643 0.0957523
\(725\) 14.7263 0.546920
\(726\) 1.00000 0.0371135
\(727\) 4.84714 0.179770 0.0898852 0.995952i \(-0.471350\pi\)
0.0898852 + 0.995952i \(0.471350\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −35.2845 −1.30594
\(731\) −91.1846 −3.37259
\(732\) −2.15962 −0.0798220
\(733\) −12.2292 −0.451695 −0.225847 0.974163i \(-0.572515\pi\)
−0.225847 + 0.974163i \(0.572515\pi\)
\(734\) 31.5417 1.16422
\(735\) 0 0
\(736\) −5.28822 −0.194926
\(737\) 7.09382 0.261304
\(738\) 5.09382 0.187506
\(739\) −1.03477 −0.0380648 −0.0190324 0.999819i \(-0.506059\pi\)
−0.0190324 + 0.999819i \(0.506059\pi\)
\(740\) 29.4100 1.08113
\(741\) 0 0
\(742\) 0 0
\(743\) −23.9925 −0.880200 −0.440100 0.897949i \(-0.645057\pi\)
−0.440100 + 0.897949i \(0.645057\pi\)
\(744\) 8.96523 0.328681
\(745\) 46.9652 1.72067
\(746\) 15.3125 0.560630
\(747\) −15.0938 −0.552254
\(748\) 7.96523 0.291238
\(749\) 0 0
\(750\) −5.97199 −0.218066
\(751\) 20.5069 0.748307 0.374153 0.927367i \(-0.377933\pi\)
0.374153 + 0.927367i \(0.377933\pi\)
\(752\) −0.322990 −0.0117782
\(753\) −28.6703 −1.04480
\(754\) 0 0
\(755\) 30.5794 1.11290
\(756\) 0 0
\(757\) −16.8019 −0.610674 −0.305337 0.952244i \(-0.598769\pi\)
−0.305337 + 0.952244i \(0.598769\pi\)
\(758\) 16.0590 0.583291
\(759\) 5.28822 0.191950
\(760\) 18.1876 0.659735
\(761\) 14.5099 0.525983 0.262992 0.964798i \(-0.415291\pi\)
0.262992 + 0.964798i \(0.415291\pi\)
\(762\) −9.28822 −0.336477
\(763\) 0 0
\(764\) −2.38879 −0.0864235
\(765\) −22.3473 −0.807967
\(766\) 0.739798 0.0267300
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −3.35402 −0.120949 −0.0604745 0.998170i \(-0.519261\pi\)
−0.0604745 + 0.998170i \(0.519261\pi\)
\(770\) 0 0
\(771\) 13.6112 0.490196
\(772\) −14.5764 −0.524617
\(773\) −1.84038 −0.0661938 −0.0330969 0.999452i \(-0.510537\pi\)
−0.0330969 + 0.999452i \(0.510537\pi\)
\(774\) 11.4478 0.411484
\(775\) 25.7428 0.924709
\(776\) −7.00000 −0.251285
\(777\) 0 0
\(778\) 3.58319 0.128464
\(779\) 33.0213 1.18311
\(780\) 0 0
\(781\) 7.44784 0.266505
\(782\) 42.1218 1.50627
\(783\) 5.12859 0.183281
\(784\) 0 0
\(785\) −4.97875 −0.177699
\(786\) −12.9652 −0.462454
\(787\) −11.5869 −0.413030 −0.206515 0.978443i \(-0.566212\pi\)
−0.206515 + 0.978443i \(0.566212\pi\)
\(788\) −12.4826 −0.444675
\(789\) −3.61121 −0.128562
\(790\) 19.0938 0.679328
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) 0 0
\(794\) 25.1286 0.891780
\(795\) 22.4448 0.796036
\(796\) 14.1876 0.502867
\(797\) 48.4653 1.71673 0.858365 0.513039i \(-0.171480\pi\)
0.858365 + 0.513039i \(0.171480\pi\)
\(798\) 0 0
\(799\) 2.57269 0.0910151
\(800\) 2.87141 0.101520
\(801\) −1.61121 −0.0569292
\(802\) −3.86839 −0.136598
\(803\) 12.5764 0.443813
\(804\) −7.09382 −0.250180
\(805\) 0 0
\(806\) 0 0
\(807\) 9.38203 0.330263
\(808\) −10.7398 −0.377825
\(809\) −2.76708 −0.0972855 −0.0486428 0.998816i \(-0.515490\pi\)
−0.0486428 + 0.998816i \(0.515490\pi\)
\(810\) 2.80560 0.0985788
\(811\) −4.89568 −0.171910 −0.0859552 0.996299i \(-0.527394\pi\)
−0.0859552 + 0.996299i \(0.527394\pi\)
\(812\) 0 0
\(813\) −5.61121 −0.196794
\(814\) −10.4826 −0.367415
\(815\) −11.3881 −0.398907
\(816\) −7.96523 −0.278839
\(817\) 74.2119 2.59635
\(818\) −4.64598 −0.162443
\(819\) 0 0
\(820\) 14.2912 0.499071
\(821\) 3.54915 0.123866 0.0619330 0.998080i \(-0.480273\pi\)
0.0619330 + 0.998080i \(0.480273\pi\)
\(822\) 5.61121 0.195713
\(823\) 32.5069 1.13312 0.566559 0.824021i \(-0.308274\pi\)
0.566559 + 0.824021i \(0.308274\pi\)
\(824\) −2.96523 −0.103299
\(825\) −2.87141 −0.0999696
\(826\) 0 0
\(827\) 19.5447 0.679635 0.339817 0.940491i \(-0.389635\pi\)
0.339817 + 0.940491i \(0.389635\pi\)
\(828\) −5.28822 −0.183778
\(829\) −25.1907 −0.874908 −0.437454 0.899241i \(-0.644120\pi\)
−0.437454 + 0.899241i \(0.644120\pi\)
\(830\) −42.3473 −1.46989
\(831\) −16.0000 −0.555034
\(832\) 0 0
\(833\) 0 0
\(834\) −13.1286 −0.454606
\(835\) −38.9092 −1.34651
\(836\) −6.48261 −0.224206
\(837\) 8.96523 0.309884
\(838\) 19.5174 0.674217
\(839\) 50.4033 1.74011 0.870057 0.492950i \(-0.164082\pi\)
0.870057 + 0.492950i \(0.164082\pi\)
\(840\) 0 0
\(841\) −2.69754 −0.0930185
\(842\) 15.5174 0.534764
\(843\) 24.1529 0.831869
\(844\) −20.8336 −0.717123
\(845\) −36.4728 −1.25470
\(846\) −0.322990 −0.0111046
\(847\) 0 0
\(848\) 8.00000 0.274721
\(849\) −7.54166 −0.258829
\(850\) −22.8714 −0.784483
\(851\) −55.4343 −1.90026
\(852\) −7.44784 −0.255159
\(853\) 16.8752 0.577794 0.288897 0.957360i \(-0.406711\pi\)
0.288897 + 0.957360i \(0.406711\pi\)
\(854\) 0 0
\(855\) 18.1876 0.622004
\(856\) 4.83663 0.165313
\(857\) −43.3057 −1.47930 −0.739648 0.672994i \(-0.765008\pi\)
−0.739648 + 0.672994i \(0.765008\pi\)
\(858\) 0 0
\(859\) −3.61422 −0.123316 −0.0616578 0.998097i \(-0.519639\pi\)
−0.0616578 + 0.998097i \(0.519639\pi\)
\(860\) 32.1181 1.09522
\(861\) 0 0
\(862\) −14.3888 −0.490084
\(863\) 4.41681 0.150350 0.0751750 0.997170i \(-0.476048\pi\)
0.0751750 + 0.997170i \(0.476048\pi\)
\(864\) 1.00000 0.0340207
\(865\) −47.5977 −1.61837
\(866\) −2.03477 −0.0691444
\(867\) 46.4448 1.57735
\(868\) 0 0
\(869\) −6.80560 −0.230864
\(870\) 14.3888 0.487826
\(871\) 0 0
\(872\) −10.1596 −0.344048
\(873\) −7.00000 −0.236914
\(874\) −34.2815 −1.15959
\(875\) 0 0
\(876\) −12.5764 −0.424918
\(877\) 44.2777 1.49515 0.747576 0.664176i \(-0.231218\pi\)
0.747576 + 0.664176i \(0.231218\pi\)
\(878\) −3.22616 −0.108877
\(879\) −7.90618 −0.266669
\(880\) −2.80560 −0.0945769
\(881\) 18.1876 0.612757 0.306379 0.951910i \(-0.400883\pi\)
0.306379 + 0.951910i \(0.400883\pi\)
\(882\) 0 0
\(883\) −4.57945 −0.154111 −0.0770553 0.997027i \(-0.524552\pi\)
−0.0770553 + 0.997027i \(0.524552\pi\)
\(884\) 0 0
\(885\) 2.44482 0.0821818
\(886\) −22.0938 −0.742256
\(887\) 0.576432 0.0193547 0.00967734 0.999953i \(-0.496920\pi\)
0.00967734 + 0.999953i \(0.496920\pi\)
\(888\) 10.4826 0.351773
\(889\) 0 0
\(890\) −4.52040 −0.151524
\(891\) −1.00000 −0.0335013
\(892\) −12.5764 −0.421090
\(893\) −2.09382 −0.0700670
\(894\) 16.7398 0.559863
\(895\) −35.5477 −1.18823
\(896\) 0 0
\(897\) 0 0
\(898\) 22.2497 0.742482
\(899\) 45.9790 1.53349
\(900\) 2.87141 0.0957136
\(901\) −63.7218 −2.12288
\(902\) −5.09382 −0.169606
\(903\) 0 0
\(904\) 10.9652 0.364698
\(905\) 7.22844 0.240282
\(906\) 10.8994 0.362109
\(907\) −22.0590 −0.732459 −0.366229 0.930525i \(-0.619351\pi\)
−0.366229 + 0.930525i \(0.619351\pi\)
\(908\) −20.0590 −0.665683
\(909\) −10.7398 −0.356217
\(910\) 0 0
\(911\) 9.93420 0.329135 0.164567 0.986366i \(-0.447377\pi\)
0.164567 + 0.986366i \(0.447377\pi\)
\(912\) 6.48261 0.214661
\(913\) 15.0938 0.499532
\(914\) −18.6385 −0.616507
\(915\) −6.05904 −0.200306
\(916\) 22.5764 0.745946
\(917\) 0 0
\(918\) −7.96523 −0.262892
\(919\) 44.6347 1.47236 0.736182 0.676783i \(-0.236627\pi\)
0.736182 + 0.676783i \(0.236627\pi\)
\(920\) −14.8366 −0.489149
\(921\) −12.8957 −0.424927
\(922\) 20.9895 0.691252
\(923\) 0 0
\(924\) 0 0
\(925\) 30.0999 0.989677
\(926\) −5.35402 −0.175944
\(927\) −2.96523 −0.0973908
\(928\) 5.12859 0.168354
\(929\) 42.5764 1.39689 0.698444 0.715665i \(-0.253876\pi\)
0.698444 + 0.715665i \(0.253876\pi\)
\(930\) 25.1529 0.824795
\(931\) 0 0
\(932\) −0.742815 −0.0243317
\(933\) −10.8994 −0.356831
\(934\) −26.0243 −0.851540
\(935\) 22.3473 0.730834
\(936\) 0 0
\(937\) 36.2572 1.18447 0.592235 0.805765i \(-0.298246\pi\)
0.592235 + 0.805765i \(0.298246\pi\)
\(938\) 0 0
\(939\) 9.12859 0.297900
\(940\) −0.906181 −0.0295564
\(941\) −2.68377 −0.0874884 −0.0437442 0.999043i \(-0.513929\pi\)
−0.0437442 + 0.999043i \(0.513929\pi\)
\(942\) −1.77457 −0.0578187
\(943\) −26.9372 −0.877196
\(944\) 0.871407 0.0283619
\(945\) 0 0
\(946\) −11.4478 −0.372201
\(947\) 49.5039 1.60866 0.804330 0.594183i \(-0.202525\pi\)
0.804330 + 0.594183i \(0.202525\pi\)
\(948\) 6.80560 0.221036
\(949\) 0 0
\(950\) 18.6142 0.603925
\(951\) −9.51364 −0.308501
\(952\) 0 0
\(953\) 15.8019 0.511872 0.255936 0.966694i \(-0.417616\pi\)
0.255936 + 0.966694i \(0.417616\pi\)
\(954\) 8.00000 0.259010
\(955\) −6.70201 −0.216872
\(956\) 4.31925 0.139694
\(957\) −5.12859 −0.165784
\(958\) 3.61121 0.116673
\(959\) 0 0
\(960\) 2.80560 0.0905504
\(961\) 49.3753 1.59275
\(962\) 0 0
\(963\) 4.83663 0.155858
\(964\) 18.1876 0.585784
\(965\) −40.8957 −1.31648
\(966\) 0 0
\(967\) −12.6497 −0.406788 −0.203394 0.979097i \(-0.565197\pi\)
−0.203394 + 0.979097i \(0.565197\pi\)
\(968\) 1.00000 0.0321412
\(969\) −51.6355 −1.65877
\(970\) −19.6392 −0.630577
\(971\) −49.5342 −1.58963 −0.794814 0.606854i \(-0.792431\pi\)
−0.794814 + 0.606854i \(0.792431\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −8.06206 −0.258325
\(975\) 0 0
\(976\) −2.15962 −0.0691278
\(977\) 34.4448 1.10199 0.550994 0.834509i \(-0.314249\pi\)
0.550994 + 0.834509i \(0.314249\pi\)
\(978\) −4.05904 −0.129794
\(979\) 1.61121 0.0514944
\(980\) 0 0
\(981\) −10.1596 −0.324372
\(982\) 20.9062 0.667143
\(983\) 3.35776 0.107096 0.0535480 0.998565i \(-0.482947\pi\)
0.0535480 + 0.998565i \(0.482947\pi\)
\(984\) 5.09382 0.162385
\(985\) −35.0213 −1.11587
\(986\) −40.8504 −1.30094
\(987\) 0 0
\(988\) 0 0
\(989\) −60.5386 −1.92502
\(990\) −2.80560 −0.0891679
\(991\) −45.8474 −1.45639 −0.728195 0.685370i \(-0.759641\pi\)
−0.728195 + 0.685370i \(0.759641\pi\)
\(992\) 8.96523 0.284646
\(993\) 1.09382 0.0347113
\(994\) 0 0
\(995\) 39.8049 1.26190
\(996\) −15.0938 −0.478266
\(997\) 43.8609 1.38909 0.694544 0.719450i \(-0.255606\pi\)
0.694544 + 0.719450i \(0.255606\pi\)
\(998\) 4.96523 0.157171
\(999\) 10.4826 0.331655
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.1.3 3
3.2 odd 2 9702.2.a.dw.1.1 3
7.3 odd 6 462.2.i.g.331.3 yes 6
7.5 odd 6 462.2.i.g.67.3 6
7.6 odd 2 3234.2.a.bf.1.1 3
21.5 even 6 1386.2.k.v.991.1 6
21.17 even 6 1386.2.k.v.793.1 6
21.20 even 2 9702.2.a.dv.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.g.67.3 6 7.5 odd 6
462.2.i.g.331.3 yes 6 7.3 odd 6
1386.2.k.v.793.1 6 21.17 even 6
1386.2.k.v.991.1 6 21.5 even 6
3234.2.a.bf.1.1 3 7.6 odd 2
3234.2.a.bh.1.3 3 1.1 even 1 trivial
9702.2.a.dv.1.3 3 21.20 even 2
9702.2.a.dw.1.1 3 3.2 odd 2