Properties

Label 3234.2.a.bf.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 \(-2.80560\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

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

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
\(5\) −2.80560 −1.25470 −0.627352 0.778736i \(-0.715861\pi\)
−0.627352 + 0.778736i \(0.715861\pi\)
\(6\) −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 \(-0.0833337\pi\)
0.965926 + 0.258820i \(0.0833337\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.48261 −1.48721 −0.743607 0.668617i \(-0.766886\pi\)
−0.743607 + 0.668617i \(0.766886\pi\)
\(20\) −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 \(-0.797889\pi\)
−0.805101 + 0.593138i \(0.797889\pi\)
\(32\) 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 \(-0.630212\pi\)
−0.397760 + 0.917489i \(0.630212\pi\)
\(42\) 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 \(-0.492501\pi\)
0.0235565 + 0.999723i \(0.492501\pi\)
\(48\) −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 \(-0.518065\pi\)
−0.0567238 + 0.998390i \(0.518065\pi\)
\(60\) 2.80560 0.362202
\(61\) 2.15962 0.276511 0.138256 0.990397i \(-0.455850\pi\)
0.138256 + 0.990397i \(0.455850\pi\)
\(62\) −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 \(-0.236722\pi\)
0.735980 + 0.677003i \(0.236722\pi\)
\(74\) 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 \(-0.189262\pi\)
0.828381 + 0.560165i \(0.189262\pi\)
\(84\) 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 \(-0.472785\pi\)
0.0853937 + 0.996347i \(0.472785\pi\)
\(90\) −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 \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 2.87141 0.287141
\(101\) 10.7398 1.06865 0.534325 0.845279i \(-0.320566\pi\)
0.534325 + 0.845279i \(0.320566\pi\)
\(102\) −7.96523 −0.788675
\(103\) 2.96523 0.292172 0.146086 0.989272i \(-0.453332\pi\)
0.146086 + 0.989272i \(0.453332\pi\)
\(104\) 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 \(-0.308340\pi\)
0.566389 + 0.824138i \(0.308340\pi\)
\(132\) 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 \(-0.312038\pi\)
0.556776 + 0.830662i \(0.312038\pi\)
\(140\) 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 \(-0.477441\pi\)
0.0708132 + 0.997490i \(0.477441\pi\)
\(158\) 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 \(-0.319714\pi\)
0.536584 + 0.843847i \(0.319714\pi\)
\(168\) 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 \(-0.276890\pi\)
0.644921 + 0.764249i \(0.276890\pi\)
\(174\) −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 \(-0.530526\pi\)
−0.0957523 + 0.995405i \(0.530526\pi\)
\(182\) 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 \(-0.667722\pi\)
−0.502867 + 0.864364i \(0.667722\pi\)
\(200\) 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 \(-0.361648\pi\)
0.421090 + 0.907019i \(0.361648\pi\)
\(224\) 0 0
\(225\) 2.87141 0.191427
\(226\) 10.9652 0.729396
\(227\) 20.0590 1.33137 0.665683 0.746235i \(-0.268140\pi\)
0.665683 + 0.746235i \(0.268140\pi\)
\(228\) 6.48261 0.429322
\(229\) −22.5764 −1.49189 −0.745946 0.666006i \(-0.768002\pi\)
−0.745946 + 0.666006i \(0.768002\pi\)
\(230\) 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 \(-0.699213\pi\)
−0.585784 + 0.810467i \(0.699213\pi\)
\(242\) 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 \(-0.140001\pi\)
0.904825 + 0.425784i \(0.140001\pi\)
\(252\) 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 \(-0.639558\pi\)
−0.424522 + 0.905418i \(0.639558\pi\)
\(258\) −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 \(-0.592331\pi\)
−0.286016 + 0.958225i \(0.592331\pi\)
\(270\) 2.80560 0.170744
\(271\) 5.61121 0.340856 0.170428 0.985370i \(-0.445485\pi\)
0.170428 + 0.985370i \(0.445485\pi\)
\(272\) 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 \(-0.428039\pi\)
0.224152 + 0.974554i \(0.428039\pi\)
\(284\) −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 \(-0.425819\pi\)
0.230942 + 0.972968i \(0.425819\pi\)
\(294\) 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 \(-0.380043\pi\)
0.367998 + 0.929827i \(0.380043\pi\)
\(308\) 0 0
\(309\) −2.96523 −0.168686
\(310\) 25.1529 1.42859
\(311\) 10.8994 0.618049 0.309025 0.951054i \(-0.399997\pi\)
0.309025 + 0.951054i \(0.399997\pi\)
\(312\) 0 0
\(313\) −9.12859 −0.515979 −0.257989 0.966148i \(-0.583060\pi\)
−0.257989 + 0.966148i \(0.583060\pi\)
\(314\) 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 \(-0.581953\pi\)
−0.254627 + 0.967039i \(0.581953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −12.5764 −0.669376 −0.334688 0.942329i \(-0.608631\pi\)
−0.334688 + 0.942329i \(0.608631\pi\)
\(354\) 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 \(-0.807831\pi\)
−0.823231 + 0.567707i \(0.807831\pi\)
\(368\) −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 \(-0.506017\pi\)
−0.0189010 + 0.999821i \(0.506017\pi\)
\(384\) −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 \(-0.717185\pi\)
−0.630584 + 0.776121i \(0.717185\pi\)
\(398\) −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 \(-0.463357\pi\)
0.114864 + 0.993381i \(0.463357\pi\)
\(410\) 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 \(-0.658183\pi\)
−0.476743 + 0.879043i \(0.658183\pi\)
\(420\) 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 \(-0.484431\pi\)
0.0488925 + 0.998804i \(0.484431\pi\)
\(434\) 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 \(-0.475470\pi\)
0.0769880 + 0.997032i \(0.475470\pi\)
\(440\) 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 \(-0.662561\pi\)
−0.488789 + 0.872402i \(0.662561\pi\)
\(462\) 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 \(-0.294319\pi\)
0.602130 + 0.798398i \(0.294319\pi\)
\(468\) 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 \(-0.526290\pi\)
−0.0825001 + 0.996591i \(0.526290\pi\)
\(480\) 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 \(-0.629881\pi\)
−0.396806 + 0.917903i \(0.629881\pi\)
\(504\) 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 \(-0.468100\pi\)
0.100048 + 0.994983i \(0.468100\pi\)
\(510\) 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 \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 5.12859 0.224472
\(523\) 6.90317 0.301854 0.150927 0.988545i \(-0.451774\pi\)
0.150927 + 0.988545i \(0.451774\pi\)
\(524\) 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 \(-0.278292\pi\)
0.641548 + 0.767083i \(0.278292\pi\)
\(564\) −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 \(-0.601245\pi\)
−0.312734 + 0.949841i \(0.601245\pi\)
\(578\) 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 \(-0.840607\pi\)
−0.877223 + 0.480082i \(0.840607\pi\)
\(588\) 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 \(-0.397691\pi\)
0.315909 + 0.948789i \(0.397691\pi\)
\(594\) 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 \(-0.261983\pi\)
0.679994 + 0.733218i \(0.261983\pi\)
\(602\) 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 \(-0.395017\pi\)
0.323867 + 0.946103i \(0.395017\pi\)
\(608\) −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 \(-0.260642\pi\)
0.683077 + 0.730347i \(0.260642\pi\)
\(620\) 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 \(-0.654590\pi\)
−0.466791 + 0.884368i \(0.654590\pi\)
\(644\) 0 0
\(645\) 32.1181 1.26465
\(646\) −51.6355 −2.03157
\(647\) −12.4168 −0.488155 −0.244078 0.969756i \(-0.578485\pi\)
−0.244078 + 0.969756i \(0.578485\pi\)
\(648\) 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 \(-0.304479\pi\)
0.576343 + 0.817208i \(0.304479\pi\)
\(662\) 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 \(-0.418425\pi\)
0.253479 + 0.967341i \(0.418425\pi\)
\(678\) −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 \(-0.899782\pi\)
−0.950845 + 0.309667i \(0.899782\pi\)
\(692\) 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 \(-0.366780\pi\)
0.406410 + 0.913691i \(0.366780\pi\)
\(720\) −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 \(-0.528650\pi\)
−0.0898852 + 0.995952i \(0.528650\pi\)
\(728\) 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 \(-0.427485\pi\)
0.225847 + 0.974163i \(0.427485\pi\)
\(734\) −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 \(-0.584709\pi\)
−0.262992 + 0.964798i \(0.584709\pi\)
\(762\) 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 \(-0.480739\pi\)
0.0604745 + 0.998170i \(0.480739\pi\)
\(770\) 0 0
\(771\) 13.6112 0.490196
\(772\) −14.5764 −0.524617
\(773\) 1.84038 0.0661938 0.0330969 0.999452i \(-0.489463\pi\)
0.0330969 + 0.999452i \(0.489463\pi\)
\(774\) 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 \(-0.433788\pi\)
0.206515 + 0.978443i \(0.433788\pi\)
\(788\) −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 \(-0.828520\pi\)
−0.858365 + 0.513039i \(0.828520\pi\)
\(798\) 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 \(-0.472606\pi\)
0.0859552 + 0.996299i \(0.472606\pi\)
\(812\) 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 \(-0.355880\pi\)
0.437454 + 0.899241i \(0.355880\pi\)
\(830\) −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 \(-0.835918\pi\)
−0.870057 + 0.492950i \(0.835918\pi\)
\(840\) 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 \(-0.593289\pi\)
−0.288897 + 0.957360i \(0.593289\pi\)
\(854\) 0 0
\(855\) 18.1876 0.622004
\(856\) 4.83663 0.165313
\(857\) 43.3057 1.47930 0.739648 0.672994i \(-0.234992\pi\)
0.739648 + 0.672994i \(0.234992\pi\)
\(858\) 0 0
\(859\) 3.61422 0.123316 0.0616578 0.998097i \(-0.480361\pi\)
0.0616578 + 0.998097i \(0.480361\pi\)
\(860\) −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 \(-0.599117\pi\)
−0.306379 + 0.951910i \(0.599117\pi\)
\(882\) 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 \(-0.503080\pi\)
−0.00967734 + 0.999953i \(0.503080\pi\)
\(888\) −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 \(-0.746124\pi\)
−0.698444 + 0.715665i \(0.746124\pi\)
\(930\) −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 \(-0.701754\pi\)
−0.592235 + 0.805765i \(0.701754\pi\)
\(938\) 0 0
\(939\) 9.12859 0.297900
\(940\) −0.906181 −0.0295564
\(941\) 2.68377 0.0874884 0.0437442 0.999043i \(-0.486071\pi\)
0.0437442 + 0.999043i \(0.486071\pi\)
\(942\) −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 \(-0.207569\pi\)
0.794814 + 0.606854i \(0.207569\pi\)
\(972\) −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 \(-0.517053\pi\)
−0.0535480 + 0.998565i \(0.517053\pi\)
\(984\) 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 \(-0.744394\pi\)
−0.694544 + 0.719450i \(0.744394\pi\)
\(998\) 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.bf.1.1 3
3.2 odd 2 9702.2.a.dv.1.3 3
7.2 even 3 462.2.i.g.67.3 6
7.4 even 3 462.2.i.g.331.3 yes 6
7.6 odd 2 3234.2.a.bh.1.3 3
21.2 odd 6 1386.2.k.v.991.1 6
21.11 odd 6 1386.2.k.v.793.1 6
21.20 even 2 9702.2.a.dw.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.g.67.3 6 7.2 even 3
462.2.i.g.331.3 yes 6 7.4 even 3
1386.2.k.v.793.1 6 21.11 odd 6
1386.2.k.v.991.1 6 21.2 odd 6
3234.2.a.bf.1.1 3 1.1 even 1 trivial
3234.2.a.bh.1.3 3 7.6 odd 2
9702.2.a.dv.1.3 3 3.2 odd 2
9702.2.a.dw.1.1 3 21.20 even 2