Properties

Label 3450.2.a.bl.1.2
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -2.89898 q^{13} +1.44949 q^{14} +1.00000 q^{16} -5.44949 q^{17} +1.00000 q^{18} +6.89898 q^{19} +1.44949 q^{21} +2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -2.89898 q^{26} +1.00000 q^{27} +1.44949 q^{28} +5.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -5.44949 q^{34} +1.00000 q^{36} +8.34847 q^{37} +6.89898 q^{38} -2.89898 q^{39} -4.89898 q^{41} +1.44949 q^{42} +10.8990 q^{43} +2.00000 q^{44} +1.00000 q^{46} -3.89898 q^{47} +1.00000 q^{48} -4.89898 q^{49} -5.44949 q^{51} -2.89898 q^{52} +0.898979 q^{53} +1.00000 q^{54} +1.44949 q^{56} +6.89898 q^{57} +5.00000 q^{58} +10.0000 q^{59} -4.89898 q^{61} +2.00000 q^{62} +1.44949 q^{63} +1.00000 q^{64} +2.00000 q^{66} -2.00000 q^{67} -5.44949 q^{68} +1.00000 q^{69} +10.7980 q^{71} +1.00000 q^{72} +2.10102 q^{73} +8.34847 q^{74} +6.89898 q^{76} +2.89898 q^{77} -2.89898 q^{78} +10.0000 q^{79} +1.00000 q^{81} -4.89898 q^{82} -2.55051 q^{83} +1.44949 q^{84} +10.8990 q^{86} +5.00000 q^{87} +2.00000 q^{88} -10.3485 q^{89} -4.20204 q^{91} +1.00000 q^{92} +2.00000 q^{93} -3.89898 q^{94} +1.00000 q^{96} -12.6969 q^{97} -4.89898 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + 4q^{11} + 2q^{12} + 4q^{13} - 2q^{14} + 2q^{16} - 6q^{17} + 2q^{18} + 4q^{19} - 2q^{21} + 4q^{22} + 2q^{23} + 2q^{24} + 4q^{26} + 2q^{27} - 2q^{28} + 10q^{29} + 4q^{31} + 2q^{32} + 4q^{33} - 6q^{34} + 2q^{36} + 2q^{37} + 4q^{38} + 4q^{39} - 2q^{42} + 12q^{43} + 4q^{44} + 2q^{46} + 2q^{47} + 2q^{48} - 6q^{51} + 4q^{52} - 8q^{53} + 2q^{54} - 2q^{56} + 4q^{57} + 10q^{58} + 20q^{59} + 4q^{62} - 2q^{63} + 2q^{64} + 4q^{66} - 4q^{67} - 6q^{68} + 2q^{69} + 2q^{71} + 2q^{72} + 14q^{73} + 2q^{74} + 4q^{76} - 4q^{77} + 4q^{78} + 20q^{79} + 2q^{81} - 10q^{83} - 2q^{84} + 12q^{86} + 10q^{87} + 4q^{88} - 6q^{89} - 28q^{91} + 2q^{92} + 4q^{93} + 2q^{94} + 2q^{96} + 4q^{97} + 4q^{99} + O(q^{100}) \)

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
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 1.44949 0.547856 0.273928 0.961750i \(-0.411677\pi\)
0.273928 + 0.961750i \(0.411677\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.89898 −0.804032 −0.402016 0.915633i \(-0.631690\pi\)
−0.402016 + 0.915633i \(0.631690\pi\)
\(14\) 1.44949 0.387392
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.44949 −1.32170 −0.660848 0.750520i \(-0.729803\pi\)
−0.660848 + 0.750520i \(0.729803\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.89898 1.58273 0.791367 0.611341i \(-0.209370\pi\)
0.791367 + 0.611341i \(0.209370\pi\)
\(20\) 0 0
\(21\) 1.44949 0.316305
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.89898 −0.568537
\(27\) 1.00000 0.192450
\(28\) 1.44949 0.273928
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) −5.44949 −0.934580
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.34847 1.37248 0.686240 0.727375i \(-0.259260\pi\)
0.686240 + 0.727375i \(0.259260\pi\)
\(38\) 6.89898 1.11916
\(39\) −2.89898 −0.464208
\(40\) 0 0
\(41\) −4.89898 −0.765092 −0.382546 0.923936i \(-0.624953\pi\)
−0.382546 + 0.923936i \(0.624953\pi\)
\(42\) 1.44949 0.223661
\(43\) 10.8990 1.66208 0.831039 0.556214i \(-0.187746\pi\)
0.831039 + 0.556214i \(0.187746\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −3.89898 −0.568725 −0.284362 0.958717i \(-0.591782\pi\)
−0.284362 + 0.958717i \(0.591782\pi\)
\(48\) 1.00000 0.144338
\(49\) −4.89898 −0.699854
\(50\) 0 0
\(51\) −5.44949 −0.763081
\(52\) −2.89898 −0.402016
\(53\) 0.898979 0.123484 0.0617422 0.998092i \(-0.480334\pi\)
0.0617422 + 0.998092i \(0.480334\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 1.44949 0.193696
\(57\) 6.89898 0.913792
\(58\) 5.00000 0.656532
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −4.89898 −0.627250 −0.313625 0.949547i \(-0.601543\pi\)
−0.313625 + 0.949547i \(0.601543\pi\)
\(62\) 2.00000 0.254000
\(63\) 1.44949 0.182619
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −5.44949 −0.660848
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 10.7980 1.28148 0.640741 0.767757i \(-0.278627\pi\)
0.640741 + 0.767757i \(0.278627\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.10102 0.245906 0.122953 0.992413i \(-0.460764\pi\)
0.122953 + 0.992413i \(0.460764\pi\)
\(74\) 8.34847 0.970490
\(75\) 0 0
\(76\) 6.89898 0.791367
\(77\) 2.89898 0.330369
\(78\) −2.89898 −0.328245
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −4.89898 −0.541002
\(83\) −2.55051 −0.279955 −0.139977 0.990155i \(-0.544703\pi\)
−0.139977 + 0.990155i \(0.544703\pi\)
\(84\) 1.44949 0.158152
\(85\) 0 0
\(86\) 10.8990 1.17527
\(87\) 5.00000 0.536056
\(88\) 2.00000 0.213201
\(89\) −10.3485 −1.09694 −0.548468 0.836172i \(-0.684789\pi\)
−0.548468 + 0.836172i \(0.684789\pi\)
\(90\) 0 0
\(91\) −4.20204 −0.440494
\(92\) 1.00000 0.104257
\(93\) 2.00000 0.207390
\(94\) −3.89898 −0.402149
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −12.6969 −1.28918 −0.644589 0.764529i \(-0.722972\pi\)
−0.644589 + 0.764529i \(0.722972\pi\)
\(98\) −4.89898 −0.494872
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −13.6969 −1.36290 −0.681448 0.731866i \(-0.738650\pi\)
−0.681448 + 0.731866i \(0.738650\pi\)
\(102\) −5.44949 −0.539580
\(103\) −13.2474 −1.30531 −0.652655 0.757655i \(-0.726345\pi\)
−0.652655 + 0.757655i \(0.726345\pi\)
\(104\) −2.89898 −0.284268
\(105\) 0 0
\(106\) 0.898979 0.0873166
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 1.00000 0.0962250
\(109\) 3.44949 0.330401 0.165201 0.986260i \(-0.447173\pi\)
0.165201 + 0.986260i \(0.447173\pi\)
\(110\) 0 0
\(111\) 8.34847 0.792402
\(112\) 1.44949 0.136964
\(113\) 14.3485 1.34979 0.674895 0.737914i \(-0.264189\pi\)
0.674895 + 0.737914i \(0.264189\pi\)
\(114\) 6.89898 0.646149
\(115\) 0 0
\(116\) 5.00000 0.464238
\(117\) −2.89898 −0.268011
\(118\) 10.0000 0.920575
\(119\) −7.89898 −0.724098
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −4.89898 −0.443533
\(123\) −4.89898 −0.441726
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) 1.44949 0.129131
\(127\) 1.79796 0.159543 0.0797715 0.996813i \(-0.474581\pi\)
0.0797715 + 0.996813i \(0.474581\pi\)
\(128\) 1.00000 0.0883883
\(129\) 10.8990 0.959602
\(130\) 0 0
\(131\) 8.89898 0.777507 0.388754 0.921342i \(-0.372906\pi\)
0.388754 + 0.921342i \(0.372906\pi\)
\(132\) 2.00000 0.174078
\(133\) 10.0000 0.867110
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) −5.44949 −0.467290
\(137\) −22.3485 −1.90936 −0.954679 0.297636i \(-0.903802\pi\)
−0.954679 + 0.297636i \(0.903802\pi\)
\(138\) 1.00000 0.0851257
\(139\) 15.6969 1.33140 0.665698 0.746221i \(-0.268134\pi\)
0.665698 + 0.746221i \(0.268134\pi\)
\(140\) 0 0
\(141\) −3.89898 −0.328353
\(142\) 10.7980 0.906145
\(143\) −5.79796 −0.484850
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.10102 0.173882
\(147\) −4.89898 −0.404061
\(148\) 8.34847 0.686240
\(149\) 20.0000 1.63846 0.819232 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 0 0
\(151\) −18.6969 −1.52154 −0.760768 0.649024i \(-0.775177\pi\)
−0.760768 + 0.649024i \(0.775177\pi\)
\(152\) 6.89898 0.559581
\(153\) −5.44949 −0.440565
\(154\) 2.89898 0.233606
\(155\) 0 0
\(156\) −2.89898 −0.232104
\(157\) −8.89898 −0.710216 −0.355108 0.934825i \(-0.615556\pi\)
−0.355108 + 0.934825i \(0.615556\pi\)
\(158\) 10.0000 0.795557
\(159\) 0.898979 0.0712937
\(160\) 0 0
\(161\) 1.44949 0.114236
\(162\) 1.00000 0.0785674
\(163\) 17.7980 1.39404 0.697022 0.717050i \(-0.254508\pi\)
0.697022 + 0.717050i \(0.254508\pi\)
\(164\) −4.89898 −0.382546
\(165\) 0 0
\(166\) −2.55051 −0.197958
\(167\) −7.00000 −0.541676 −0.270838 0.962625i \(-0.587301\pi\)
−0.270838 + 0.962625i \(0.587301\pi\)
\(168\) 1.44949 0.111831
\(169\) −4.59592 −0.353532
\(170\) 0 0
\(171\) 6.89898 0.527578
\(172\) 10.8990 0.831039
\(173\) −19.7980 −1.50521 −0.752605 0.658472i \(-0.771203\pi\)
−0.752605 + 0.658472i \(0.771203\pi\)
\(174\) 5.00000 0.379049
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 10.0000 0.751646
\(178\) −10.3485 −0.775651
\(179\) 3.10102 0.231781 0.115891 0.993262i \(-0.463028\pi\)
0.115891 + 0.993262i \(0.463028\pi\)
\(180\) 0 0
\(181\) −18.3485 −1.36383 −0.681915 0.731431i \(-0.738853\pi\)
−0.681915 + 0.731431i \(0.738853\pi\)
\(182\) −4.20204 −0.311476
\(183\) −4.89898 −0.362143
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 2.00000 0.146647
\(187\) −10.8990 −0.797012
\(188\) −3.89898 −0.284362
\(189\) 1.44949 0.105435
\(190\) 0 0
\(191\) 2.00000 0.144715 0.0723575 0.997379i \(-0.476948\pi\)
0.0723575 + 0.997379i \(0.476948\pi\)
\(192\) 1.00000 0.0721688
\(193\) 9.69694 0.698001 0.349000 0.937123i \(-0.386521\pi\)
0.349000 + 0.937123i \(0.386521\pi\)
\(194\) −12.6969 −0.911587
\(195\) 0 0
\(196\) −4.89898 −0.349927
\(197\) −3.89898 −0.277791 −0.138895 0.990307i \(-0.544355\pi\)
−0.138895 + 0.990307i \(0.544355\pi\)
\(198\) 2.00000 0.142134
\(199\) 24.1464 1.71169 0.855847 0.517228i \(-0.173036\pi\)
0.855847 + 0.517228i \(0.173036\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) −13.6969 −0.963713
\(203\) 7.24745 0.508671
\(204\) −5.44949 −0.381541
\(205\) 0 0
\(206\) −13.2474 −0.922993
\(207\) 1.00000 0.0695048
\(208\) −2.89898 −0.201008
\(209\) 13.7980 0.954425
\(210\) 0 0
\(211\) 10.7980 0.743362 0.371681 0.928360i \(-0.378781\pi\)
0.371681 + 0.928360i \(0.378781\pi\)
\(212\) 0.898979 0.0617422
\(213\) 10.7980 0.739864
\(214\) −2.00000 −0.136717
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 2.89898 0.196796
\(218\) 3.44949 0.233629
\(219\) 2.10102 0.141974
\(220\) 0 0
\(221\) 15.7980 1.06269
\(222\) 8.34847 0.560313
\(223\) 24.6969 1.65383 0.826915 0.562327i \(-0.190094\pi\)
0.826915 + 0.562327i \(0.190094\pi\)
\(224\) 1.44949 0.0968481
\(225\) 0 0
\(226\) 14.3485 0.954446
\(227\) −4.75255 −0.315438 −0.157719 0.987484i \(-0.550414\pi\)
−0.157719 + 0.987484i \(0.550414\pi\)
\(228\) 6.89898 0.456896
\(229\) 3.10102 0.204921 0.102461 0.994737i \(-0.467328\pi\)
0.102461 + 0.994737i \(0.467328\pi\)
\(230\) 0 0
\(231\) 2.89898 0.190739
\(232\) 5.00000 0.328266
\(233\) −26.6969 −1.74897 −0.874487 0.485048i \(-0.838802\pi\)
−0.874487 + 0.485048i \(0.838802\pi\)
\(234\) −2.89898 −0.189512
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) 10.0000 0.649570
\(238\) −7.89898 −0.512015
\(239\) −8.10102 −0.524011 −0.262006 0.965066i \(-0.584384\pi\)
−0.262006 + 0.965066i \(0.584384\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) −4.89898 −0.313625
\(245\) 0 0
\(246\) −4.89898 −0.312348
\(247\) −20.0000 −1.27257
\(248\) 2.00000 0.127000
\(249\) −2.55051 −0.161632
\(250\) 0 0
\(251\) −5.24745 −0.331216 −0.165608 0.986192i \(-0.552959\pi\)
−0.165608 + 0.986192i \(0.552959\pi\)
\(252\) 1.44949 0.0913093
\(253\) 2.00000 0.125739
\(254\) 1.79796 0.112814
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −8.20204 −0.511629 −0.255815 0.966726i \(-0.582344\pi\)
−0.255815 + 0.966726i \(0.582344\pi\)
\(258\) 10.8990 0.678541
\(259\) 12.1010 0.751921
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 8.89898 0.549781
\(263\) −19.7980 −1.22079 −0.610397 0.792095i \(-0.708990\pi\)
−0.610397 + 0.792095i \(0.708990\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 10.0000 0.613139
\(267\) −10.3485 −0.633316
\(268\) −2.00000 −0.122169
\(269\) 3.79796 0.231566 0.115783 0.993275i \(-0.463062\pi\)
0.115783 + 0.993275i \(0.463062\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −5.44949 −0.330424
\(273\) −4.20204 −0.254319
\(274\) −22.3485 −1.35012
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 25.5959 1.53791 0.768955 0.639303i \(-0.220777\pi\)
0.768955 + 0.639303i \(0.220777\pi\)
\(278\) 15.6969 0.941440
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −0.752551 −0.0448934 −0.0224467 0.999748i \(-0.507146\pi\)
−0.0224467 + 0.999748i \(0.507146\pi\)
\(282\) −3.89898 −0.232181
\(283\) 11.5959 0.689306 0.344653 0.938730i \(-0.387997\pi\)
0.344653 + 0.938730i \(0.387997\pi\)
\(284\) 10.7980 0.640741
\(285\) 0 0
\(286\) −5.79796 −0.342841
\(287\) −7.10102 −0.419160
\(288\) 1.00000 0.0589256
\(289\) 12.6969 0.746879
\(290\) 0 0
\(291\) −12.6969 −0.744308
\(292\) 2.10102 0.122953
\(293\) 7.79796 0.455562 0.227781 0.973712i \(-0.426853\pi\)
0.227781 + 0.973712i \(0.426853\pi\)
\(294\) −4.89898 −0.285714
\(295\) 0 0
\(296\) 8.34847 0.485245
\(297\) 2.00000 0.116052
\(298\) 20.0000 1.15857
\(299\) −2.89898 −0.167652
\(300\) 0 0
\(301\) 15.7980 0.910579
\(302\) −18.6969 −1.07589
\(303\) −13.6969 −0.786869
\(304\) 6.89898 0.395684
\(305\) 0 0
\(306\) −5.44949 −0.311527
\(307\) −20.1010 −1.14723 −0.573613 0.819126i \(-0.694459\pi\)
−0.573613 + 0.819126i \(0.694459\pi\)
\(308\) 2.89898 0.165185
\(309\) −13.2474 −0.753621
\(310\) 0 0
\(311\) 14.5959 0.827659 0.413829 0.910355i \(-0.364191\pi\)
0.413829 + 0.910355i \(0.364191\pi\)
\(312\) −2.89898 −0.164122
\(313\) −12.8990 −0.729093 −0.364547 0.931185i \(-0.618776\pi\)
−0.364547 + 0.931185i \(0.618776\pi\)
\(314\) −8.89898 −0.502198
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 10.5959 0.595126 0.297563 0.954702i \(-0.403826\pi\)
0.297563 + 0.954702i \(0.403826\pi\)
\(318\) 0.898979 0.0504123
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) 1.44949 0.0807769
\(323\) −37.5959 −2.09189
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 17.7980 0.985738
\(327\) 3.44949 0.190757
\(328\) −4.89898 −0.270501
\(329\) −5.65153 −0.311579
\(330\) 0 0
\(331\) −6.79796 −0.373650 −0.186825 0.982393i \(-0.559820\pi\)
−0.186825 + 0.982393i \(0.559820\pi\)
\(332\) −2.55051 −0.139977
\(333\) 8.34847 0.457493
\(334\) −7.00000 −0.383023
\(335\) 0 0
\(336\) 1.44949 0.0790761
\(337\) −16.4949 −0.898534 −0.449267 0.893397i \(-0.648315\pi\)
−0.449267 + 0.893397i \(0.648315\pi\)
\(338\) −4.59592 −0.249985
\(339\) 14.3485 0.779302
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 6.89898 0.373054
\(343\) −17.2474 −0.931275
\(344\) 10.8990 0.587634
\(345\) 0 0
\(346\) −19.7980 −1.06434
\(347\) −32.6969 −1.75526 −0.877632 0.479335i \(-0.840878\pi\)
−0.877632 + 0.479335i \(0.840878\pi\)
\(348\) 5.00000 0.268028
\(349\) 13.1010 0.701282 0.350641 0.936510i \(-0.385964\pi\)
0.350641 + 0.936510i \(0.385964\pi\)
\(350\) 0 0
\(351\) −2.89898 −0.154736
\(352\) 2.00000 0.106600
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) −10.3485 −0.548468
\(357\) −7.89898 −0.418058
\(358\) 3.10102 0.163894
\(359\) −6.20204 −0.327331 −0.163666 0.986516i \(-0.552332\pi\)
−0.163666 + 0.986516i \(0.552332\pi\)
\(360\) 0 0
\(361\) 28.5959 1.50505
\(362\) −18.3485 −0.964374
\(363\) −7.00000 −0.367405
\(364\) −4.20204 −0.220247
\(365\) 0 0
\(366\) −4.89898 −0.256074
\(367\) −8.20204 −0.428143 −0.214072 0.976818i \(-0.568673\pi\)
−0.214072 + 0.976818i \(0.568673\pi\)
\(368\) 1.00000 0.0521286
\(369\) −4.89898 −0.255031
\(370\) 0 0
\(371\) 1.30306 0.0676516
\(372\) 2.00000 0.103695
\(373\) −3.24745 −0.168147 −0.0840733 0.996460i \(-0.526793\pi\)
−0.0840733 + 0.996460i \(0.526793\pi\)
\(374\) −10.8990 −0.563573
\(375\) 0 0
\(376\) −3.89898 −0.201075
\(377\) −14.4949 −0.746525
\(378\) 1.44949 0.0745537
\(379\) −38.2929 −1.96697 −0.983486 0.180984i \(-0.942072\pi\)
−0.983486 + 0.180984i \(0.942072\pi\)
\(380\) 0 0
\(381\) 1.79796 0.0921122
\(382\) 2.00000 0.102329
\(383\) −9.79796 −0.500652 −0.250326 0.968162i \(-0.580538\pi\)
−0.250326 + 0.968162i \(0.580538\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 9.69694 0.493561
\(387\) 10.8990 0.554026
\(388\) −12.6969 −0.644589
\(389\) −23.7980 −1.20660 −0.603302 0.797513i \(-0.706149\pi\)
−0.603302 + 0.797513i \(0.706149\pi\)
\(390\) 0 0
\(391\) −5.44949 −0.275593
\(392\) −4.89898 −0.247436
\(393\) 8.89898 0.448894
\(394\) −3.89898 −0.196428
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 24.1464 1.21035
\(399\) 10.0000 0.500626
\(400\) 0 0
\(401\) −18.6969 −0.933681 −0.466840 0.884342i \(-0.654608\pi\)
−0.466840 + 0.884342i \(0.654608\pi\)
\(402\) −2.00000 −0.0997509
\(403\) −5.79796 −0.288817
\(404\) −13.6969 −0.681448
\(405\) 0 0
\(406\) 7.24745 0.359685
\(407\) 16.6969 0.827637
\(408\) −5.44949 −0.269790
\(409\) 25.6969 1.27063 0.635316 0.772252i \(-0.280870\pi\)
0.635316 + 0.772252i \(0.280870\pi\)
\(410\) 0 0
\(411\) −22.3485 −1.10237
\(412\) −13.2474 −0.652655
\(413\) 14.4949 0.713247
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −2.89898 −0.142134
\(417\) 15.6969 0.768682
\(418\) 13.7980 0.674880
\(419\) 14.1464 0.691098 0.345549 0.938401i \(-0.387693\pi\)
0.345549 + 0.938401i \(0.387693\pi\)
\(420\) 0 0
\(421\) −12.4949 −0.608964 −0.304482 0.952518i \(-0.598483\pi\)
−0.304482 + 0.952518i \(0.598483\pi\)
\(422\) 10.7980 0.525636
\(423\) −3.89898 −0.189575
\(424\) 0.898979 0.0436583
\(425\) 0 0
\(426\) 10.7980 0.523163
\(427\) −7.10102 −0.343642
\(428\) −2.00000 −0.0966736
\(429\) −5.79796 −0.279928
\(430\) 0 0
\(431\) 11.3031 0.544449 0.272225 0.962234i \(-0.412241\pi\)
0.272225 + 0.962234i \(0.412241\pi\)
\(432\) 1.00000 0.0481125
\(433\) −16.6969 −0.802404 −0.401202 0.915990i \(-0.631407\pi\)
−0.401202 + 0.915990i \(0.631407\pi\)
\(434\) 2.89898 0.139155
\(435\) 0 0
\(436\) 3.44949 0.165201
\(437\) 6.89898 0.330023
\(438\) 2.10102 0.100391
\(439\) −10.6969 −0.510537 −0.255269 0.966870i \(-0.582164\pi\)
−0.255269 + 0.966870i \(0.582164\pi\)
\(440\) 0 0
\(441\) −4.89898 −0.233285
\(442\) 15.7980 0.751432
\(443\) −5.30306 −0.251956 −0.125978 0.992033i \(-0.540207\pi\)
−0.125978 + 0.992033i \(0.540207\pi\)
\(444\) 8.34847 0.396201
\(445\) 0 0
\(446\) 24.6969 1.16943
\(447\) 20.0000 0.945968
\(448\) 1.44949 0.0684820
\(449\) 3.79796 0.179237 0.0896184 0.995976i \(-0.471435\pi\)
0.0896184 + 0.995976i \(0.471435\pi\)
\(450\) 0 0
\(451\) −9.79796 −0.461368
\(452\) 14.3485 0.674895
\(453\) −18.6969 −0.878459
\(454\) −4.75255 −0.223048
\(455\) 0 0
\(456\) 6.89898 0.323074
\(457\) 0.404082 0.0189022 0.00945108 0.999955i \(-0.496992\pi\)
0.00945108 + 0.999955i \(0.496992\pi\)
\(458\) 3.10102 0.144901
\(459\) −5.44949 −0.254360
\(460\) 0 0
\(461\) 21.4949 1.00112 0.500559 0.865703i \(-0.333128\pi\)
0.500559 + 0.865703i \(0.333128\pi\)
\(462\) 2.89898 0.134873
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) −26.6969 −1.23671
\(467\) −29.9444 −1.38566 −0.692830 0.721100i \(-0.743637\pi\)
−0.692830 + 0.721100i \(0.743637\pi\)
\(468\) −2.89898 −0.134005
\(469\) −2.89898 −0.133862
\(470\) 0 0
\(471\) −8.89898 −0.410043
\(472\) 10.0000 0.460287
\(473\) 21.7980 1.00227
\(474\) 10.0000 0.459315
\(475\) 0 0
\(476\) −7.89898 −0.362049
\(477\) 0.898979 0.0411614
\(478\) −8.10102 −0.370532
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) −24.2020 −1.10352
\(482\) −8.00000 −0.364390
\(483\) 1.44949 0.0659541
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −15.1010 −0.684293 −0.342146 0.939647i \(-0.611154\pi\)
−0.342146 + 0.939647i \(0.611154\pi\)
\(488\) −4.89898 −0.221766
\(489\) 17.7980 0.804852
\(490\) 0 0
\(491\) 25.1010 1.13279 0.566397 0.824133i \(-0.308337\pi\)
0.566397 + 0.824133i \(0.308337\pi\)
\(492\) −4.89898 −0.220863
\(493\) −27.2474 −1.22716
\(494\) −20.0000 −0.899843
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 15.6515 0.702067
\(498\) −2.55051 −0.114291
\(499\) 21.8990 0.980333 0.490166 0.871629i \(-0.336936\pi\)
0.490166 + 0.871629i \(0.336936\pi\)
\(500\) 0 0
\(501\) −7.00000 −0.312737
\(502\) −5.24745 −0.234205
\(503\) 14.6969 0.655304 0.327652 0.944798i \(-0.393743\pi\)
0.327652 + 0.944798i \(0.393743\pi\)
\(504\) 1.44949 0.0645654
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) −4.59592 −0.204112
\(508\) 1.79796 0.0797715
\(509\) −15.0000 −0.664863 −0.332432 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 0 0
\(511\) 3.04541 0.134721
\(512\) 1.00000 0.0441942
\(513\) 6.89898 0.304597
\(514\) −8.20204 −0.361777
\(515\) 0 0
\(516\) 10.8990 0.479801
\(517\) −7.79796 −0.342954
\(518\) 12.1010 0.531688
\(519\) −19.7980 −0.869034
\(520\) 0 0
\(521\) 26.1464 1.14550 0.572748 0.819732i \(-0.305877\pi\)
0.572748 + 0.819732i \(0.305877\pi\)
\(522\) 5.00000 0.218844
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 8.89898 0.388754
\(525\) 0 0
\(526\) −19.7980 −0.863232
\(527\) −10.8990 −0.474767
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 10.0000 0.433555
\(533\) 14.2020 0.615159
\(534\) −10.3485 −0.447822
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) 3.10102 0.133819
\(538\) 3.79796 0.163742
\(539\) −9.79796 −0.422028
\(540\) 0 0
\(541\) −15.5959 −0.670521 −0.335260 0.942125i \(-0.608824\pi\)
−0.335260 + 0.942125i \(0.608824\pi\)
\(542\) −8.00000 −0.343629
\(543\) −18.3485 −0.787408
\(544\) −5.44949 −0.233645
\(545\) 0 0
\(546\) −4.20204 −0.179831
\(547\) −10.7980 −0.461687 −0.230844 0.972991i \(-0.574149\pi\)
−0.230844 + 0.972991i \(0.574149\pi\)
\(548\) −22.3485 −0.954679
\(549\) −4.89898 −0.209083
\(550\) 0 0
\(551\) 34.4949 1.46953
\(552\) 1.00000 0.0425628
\(553\) 14.4949 0.616386
\(554\) 25.5959 1.08747
\(555\) 0 0
\(556\) 15.6969 0.665698
\(557\) 28.6969 1.21593 0.607964 0.793964i \(-0.291986\pi\)
0.607964 + 0.793964i \(0.291986\pi\)
\(558\) 2.00000 0.0846668
\(559\) −31.5959 −1.33636
\(560\) 0 0
\(561\) −10.8990 −0.460155
\(562\) −0.752551 −0.0317445
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) −3.89898 −0.164177
\(565\) 0 0
\(566\) 11.5959 0.487413
\(567\) 1.44949 0.0608728
\(568\) 10.7980 0.453072
\(569\) −44.4949 −1.86532 −0.932662 0.360753i \(-0.882520\pi\)
−0.932662 + 0.360753i \(0.882520\pi\)
\(570\) 0 0
\(571\) 32.6969 1.36832 0.684162 0.729330i \(-0.260168\pi\)
0.684162 + 0.729330i \(0.260168\pi\)
\(572\) −5.79796 −0.242425
\(573\) 2.00000 0.0835512
\(574\) −7.10102 −0.296391
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 12.6969 0.528123
\(579\) 9.69694 0.402991
\(580\) 0 0
\(581\) −3.69694 −0.153375
\(582\) −12.6969 −0.526305
\(583\) 1.79796 0.0744639
\(584\) 2.10102 0.0869408
\(585\) 0 0
\(586\) 7.79796 0.322131
\(587\) 4.89898 0.202203 0.101101 0.994876i \(-0.467763\pi\)
0.101101 + 0.994876i \(0.467763\pi\)
\(588\) −4.89898 −0.202031
\(589\) 13.7980 0.568535
\(590\) 0 0
\(591\) −3.89898 −0.160383
\(592\) 8.34847 0.343120
\(593\) −43.5959 −1.79027 −0.895135 0.445795i \(-0.852921\pi\)
−0.895135 + 0.445795i \(0.852921\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) 24.1464 0.988248
\(598\) −2.89898 −0.118548
\(599\) −21.3939 −0.874130 −0.437065 0.899430i \(-0.643982\pi\)
−0.437065 + 0.899430i \(0.643982\pi\)
\(600\) 0 0
\(601\) 37.0000 1.50926 0.754631 0.656150i \(-0.227816\pi\)
0.754631 + 0.656150i \(0.227816\pi\)
\(602\) 15.7980 0.643877
\(603\) −2.00000 −0.0814463
\(604\) −18.6969 −0.760768
\(605\) 0 0
\(606\) −13.6969 −0.556400
\(607\) 18.0000 0.730597 0.365299 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) 6.89898 0.279791
\(609\) 7.24745 0.293681
\(610\) 0 0
\(611\) 11.3031 0.457273
\(612\) −5.44949 −0.220283
\(613\) −0.146428 −0.00591418 −0.00295709 0.999996i \(-0.500941\pi\)
−0.00295709 + 0.999996i \(0.500941\pi\)
\(614\) −20.1010 −0.811211
\(615\) 0 0
\(616\) 2.89898 0.116803
\(617\) 12.4949 0.503026 0.251513 0.967854i \(-0.419072\pi\)
0.251513 + 0.967854i \(0.419072\pi\)
\(618\) −13.2474 −0.532891
\(619\) −44.4949 −1.78840 −0.894200 0.447667i \(-0.852255\pi\)
−0.894200 + 0.447667i \(0.852255\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 14.5959 0.585243
\(623\) −15.0000 −0.600962
\(624\) −2.89898 −0.116052
\(625\) 0 0
\(626\) −12.8990 −0.515547
\(627\) 13.7980 0.551037
\(628\) −8.89898 −0.355108
\(629\) −45.4949 −1.81400
\(630\) 0 0
\(631\) 15.4495 0.615034 0.307517 0.951543i \(-0.400502\pi\)
0.307517 + 0.951543i \(0.400502\pi\)
\(632\) 10.0000 0.397779
\(633\) 10.7980 0.429180
\(634\) 10.5959 0.420818
\(635\) 0 0
\(636\) 0.898979 0.0356469
\(637\) 14.2020 0.562705
\(638\) 10.0000 0.395904
\(639\) 10.7980 0.427161
\(640\) 0 0
\(641\) −24.5505 −0.969687 −0.484843 0.874601i \(-0.661123\pi\)
−0.484843 + 0.874601i \(0.661123\pi\)
\(642\) −2.00000 −0.0789337
\(643\) −19.7980 −0.780755 −0.390378 0.920655i \(-0.627656\pi\)
−0.390378 + 0.920655i \(0.627656\pi\)
\(644\) 1.44949 0.0571179
\(645\) 0 0
\(646\) −37.5959 −1.47919
\(647\) 20.5959 0.809709 0.404855 0.914381i \(-0.367322\pi\)
0.404855 + 0.914381i \(0.367322\pi\)
\(648\) 1.00000 0.0392837
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 2.89898 0.113620
\(652\) 17.7980 0.697022
\(653\) −0.303062 −0.0118597 −0.00592986 0.999982i \(-0.501888\pi\)
−0.00592986 + 0.999982i \(0.501888\pi\)
\(654\) 3.44949 0.134886
\(655\) 0 0
\(656\) −4.89898 −0.191273
\(657\) 2.10102 0.0819686
\(658\) −5.65153 −0.220320
\(659\) −23.4495 −0.913462 −0.456731 0.889605i \(-0.650980\pi\)
−0.456731 + 0.889605i \(0.650980\pi\)
\(660\) 0 0
\(661\) −21.4495 −0.834288 −0.417144 0.908840i \(-0.636969\pi\)
−0.417144 + 0.908840i \(0.636969\pi\)
\(662\) −6.79796 −0.264210
\(663\) 15.7980 0.613542
\(664\) −2.55051 −0.0989790
\(665\) 0 0
\(666\) 8.34847 0.323497
\(667\) 5.00000 0.193601
\(668\) −7.00000 −0.270838
\(669\) 24.6969 0.954839
\(670\) 0 0
\(671\) −9.79796 −0.378246
\(672\) 1.44949 0.0559153
\(673\) 26.5959 1.02520 0.512599 0.858628i \(-0.328683\pi\)
0.512599 + 0.858628i \(0.328683\pi\)
\(674\) −16.4949 −0.635360
\(675\) 0 0
\(676\) −4.59592 −0.176766
\(677\) −33.3939 −1.28343 −0.641715 0.766943i \(-0.721777\pi\)
−0.641715 + 0.766943i \(0.721777\pi\)
\(678\) 14.3485 0.551050
\(679\) −18.4041 −0.706284
\(680\) 0 0
\(681\) −4.75255 −0.182118
\(682\) 4.00000 0.153168
\(683\) 7.10102 0.271713 0.135856 0.990729i \(-0.456621\pi\)
0.135856 + 0.990729i \(0.456621\pi\)
\(684\) 6.89898 0.263789
\(685\) 0 0
\(686\) −17.2474 −0.658511
\(687\) 3.10102 0.118311
\(688\) 10.8990 0.415520
\(689\) −2.60612 −0.0992854
\(690\) 0 0
\(691\) −9.20204 −0.350062 −0.175031 0.984563i \(-0.556003\pi\)
−0.175031 + 0.984563i \(0.556003\pi\)
\(692\) −19.7980 −0.752605
\(693\) 2.89898 0.110123
\(694\) −32.6969 −1.24116
\(695\) 0 0
\(696\) 5.00000 0.189525
\(697\) 26.6969 1.01122
\(698\) 13.1010 0.495881
\(699\) −26.6969 −1.00977
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) −2.89898 −0.109415
\(703\) 57.5959 2.17227
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) −19.8536 −0.746670
\(708\) 10.0000 0.375823
\(709\) 27.9444 1.04947 0.524737 0.851265i \(-0.324164\pi\)
0.524737 + 0.851265i \(0.324164\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) −10.3485 −0.387825
\(713\) 2.00000 0.0749006
\(714\) −7.89898 −0.295612
\(715\) 0 0
\(716\) 3.10102 0.115891
\(717\) −8.10102 −0.302538
\(718\) −6.20204 −0.231458
\(719\) 28.7980 1.07398 0.536991 0.843588i \(-0.319561\pi\)
0.536991 + 0.843588i \(0.319561\pi\)
\(720\) 0 0
\(721\) −19.2020 −0.715121
\(722\) 28.5959 1.06423
\(723\) −8.00000 −0.297523
\(724\) −18.3485 −0.681915
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) 39.3939 1.46104 0.730519 0.682892i \(-0.239278\pi\)
0.730519 + 0.682892i \(0.239278\pi\)
\(728\) −4.20204 −0.155738
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −59.3939 −2.19676
\(732\) −4.89898 −0.181071
\(733\) −40.1464 −1.48284 −0.741421 0.671040i \(-0.765848\pi\)
−0.741421 + 0.671040i \(0.765848\pi\)
\(734\) −8.20204 −0.302743
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −4.00000 −0.147342
\(738\) −4.89898 −0.180334
\(739\) 12.5959 0.463348 0.231674 0.972793i \(-0.425580\pi\)
0.231674 + 0.972793i \(0.425580\pi\)
\(740\) 0 0
\(741\) −20.0000 −0.734718
\(742\) 1.30306 0.0478369
\(743\) −24.2929 −0.891218 −0.445609 0.895228i \(-0.647013\pi\)
−0.445609 + 0.895228i \(0.647013\pi\)
\(744\) 2.00000 0.0733236
\(745\) 0 0
\(746\) −3.24745 −0.118898
\(747\) −2.55051 −0.0933183
\(748\) −10.8990 −0.398506
\(749\) −2.89898 −0.105926
\(750\) 0 0
\(751\) −25.2474 −0.921292 −0.460646 0.887584i \(-0.652382\pi\)
−0.460646 + 0.887584i \(0.652382\pi\)
\(752\) −3.89898 −0.142181
\(753\) −5.24745 −0.191228
\(754\) −14.4949 −0.527873
\(755\) 0 0
\(756\) 1.44949 0.0527174
\(757\) 10.7526 0.390808 0.195404 0.980723i \(-0.437398\pi\)
0.195404 + 0.980723i \(0.437398\pi\)
\(758\) −38.2929 −1.39086
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) −50.0908 −1.81579 −0.907895 0.419197i \(-0.862312\pi\)
−0.907895 + 0.419197i \(0.862312\pi\)
\(762\) 1.79796 0.0651332
\(763\) 5.00000 0.181012
\(764\) 2.00000 0.0723575
\(765\) 0 0
\(766\) −9.79796 −0.354015
\(767\) −28.9898 −1.04676
\(768\) 1.00000 0.0360844
\(769\) 48.2929 1.74148 0.870742 0.491739i \(-0.163639\pi\)
0.870742 + 0.491739i \(0.163639\pi\)
\(770\) 0 0
\(771\) −8.20204 −0.295389
\(772\) 9.69694 0.349000
\(773\) 10.8990 0.392009 0.196005 0.980603i \(-0.437203\pi\)
0.196005 + 0.980603i \(0.437203\pi\)
\(774\) 10.8990 0.391756
\(775\) 0 0
\(776\) −12.6969 −0.455794
\(777\) 12.1010 0.434122
\(778\) −23.7980 −0.853198
\(779\) −33.7980 −1.21094
\(780\) 0 0
\(781\) 21.5959 0.772763
\(782\) −5.44949 −0.194873
\(783\) 5.00000 0.178685
\(784\) −4.89898 −0.174964
\(785\) 0 0
\(786\) 8.89898 0.317416
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −3.89898 −0.138895
\(789\) −19.7980 −0.704826
\(790\) 0 0
\(791\) 20.7980 0.739490
\(792\) 2.00000 0.0710669
\(793\) 14.2020 0.504329
\(794\) 28.0000 0.993683
\(795\) 0 0
\(796\) 24.1464 0.855847
\(797\) 34.8990 1.23619 0.618093 0.786105i \(-0.287906\pi\)
0.618093 + 0.786105i \(0.287906\pi\)
\(798\) 10.0000 0.353996
\(799\) 21.2474 0.751681
\(800\) 0 0
\(801\) −10.3485 −0.365645
\(802\) −18.6969 −0.660212
\(803\) 4.20204 0.148287
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) −5.79796 −0.204224
\(807\) 3.79796 0.133694
\(808\) −13.6969 −0.481857
\(809\) 11.3939 0.400587 0.200294 0.979736i \(-0.435810\pi\)
0.200294 + 0.979736i \(0.435810\pi\)
\(810\) 0 0
\(811\) 45.7980 1.60818 0.804092 0.594505i \(-0.202652\pi\)
0.804092 + 0.594505i \(0.202652\pi\)
\(812\) 7.24745 0.254336
\(813\) −8.00000 −0.280572
\(814\) 16.6969 0.585227
\(815\) 0 0
\(816\) −5.44949 −0.190770
\(817\) 75.1918 2.63063
\(818\) 25.6969 0.898472
\(819\) −4.20204 −0.146831
\(820\) 0 0
\(821\) −51.7980 −1.80776 −0.903881 0.427785i \(-0.859294\pi\)
−0.903881 + 0.427785i \(0.859294\pi\)
\(822\) −22.3485 −0.779492
\(823\) 56.0908 1.95520 0.977601 0.210465i \(-0.0674977\pi\)
0.977601 + 0.210465i \(0.0674977\pi\)
\(824\) −13.2474 −0.461497
\(825\) 0 0
\(826\) 14.4949 0.504342
\(827\) −14.7526 −0.512996 −0.256498 0.966545i \(-0.582569\pi\)
−0.256498 + 0.966545i \(0.582569\pi\)
\(828\) 1.00000 0.0347524
\(829\) −33.1010 −1.14965 −0.574823 0.818278i \(-0.694929\pi\)
−0.574823 + 0.818278i \(0.694929\pi\)
\(830\) 0 0
\(831\) 25.5959 0.887913
\(832\) −2.89898 −0.100504
\(833\) 26.6969 0.924994
\(834\) 15.6969 0.543541
\(835\) 0 0
\(836\) 13.7980 0.477212
\(837\) 2.00000 0.0691301
\(838\) 14.1464 0.488680
\(839\) −9.30306 −0.321177 −0.160589 0.987021i \(-0.551339\pi\)
−0.160589 + 0.987021i \(0.551339\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −12.4949 −0.430603
\(843\) −0.752551 −0.0259192
\(844\) 10.7980 0.371681
\(845\) 0 0
\(846\) −3.89898 −0.134050
\(847\) −10.1464 −0.348635
\(848\) 0.898979 0.0308711
\(849\) 11.5959 0.397971
\(850\) 0 0
\(851\) 8.34847 0.286182
\(852\) 10.7980 0.369932
\(853\) −44.9898 −1.54042 −0.770211 0.637790i \(-0.779849\pi\)
−0.770211 + 0.637790i \(0.779849\pi\)
\(854\) −7.10102 −0.242992
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) 42.4949 1.45160 0.725799 0.687907i \(-0.241470\pi\)
0.725799 + 0.687907i \(0.241470\pi\)
\(858\) −5.79796 −0.197939
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −7.10102 −0.242002
\(862\) 11.3031 0.384984
\(863\) 20.3939 0.694216 0.347108 0.937825i \(-0.387164\pi\)
0.347108 + 0.937825i \(0.387164\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −16.6969 −0.567385
\(867\) 12.6969 0.431211
\(868\) 2.89898 0.0983978
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) 5.79796 0.196456
\(872\) 3.44949 0.116814
\(873\) −12.6969 −0.429726
\(874\) 6.89898 0.233361
\(875\) 0 0
\(876\) 2.10102 0.0709869
\(877\) −9.59592 −0.324031 −0.162016 0.986788i \(-0.551799\pi\)
−0.162016 + 0.986788i \(0.551799\pi\)
\(878\) −10.6969 −0.361004
\(879\) 7.79796 0.263019
\(880\) 0 0
\(881\) −31.1010 −1.04782 −0.523910 0.851774i \(-0.675527\pi\)
−0.523910 + 0.851774i \(0.675527\pi\)
\(882\) −4.89898 −0.164957
\(883\) −48.5959 −1.63538 −0.817691 0.575657i \(-0.804746\pi\)
−0.817691 + 0.575657i \(0.804746\pi\)
\(884\) 15.7980 0.531343
\(885\) 0 0
\(886\) −5.30306 −0.178160
\(887\) −10.1010 −0.339159 −0.169580 0.985517i \(-0.554241\pi\)
−0.169580 + 0.985517i \(0.554241\pi\)
\(888\) 8.34847 0.280156
\(889\) 2.60612 0.0874066
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 24.6969 0.826915
\(893\) −26.8990 −0.900140
\(894\) 20.0000 0.668900
\(895\) 0 0
\(896\) 1.44949 0.0484241
\(897\) −2.89898 −0.0967941
\(898\) 3.79796 0.126740
\(899\) 10.0000 0.333519
\(900\) 0 0
\(901\) −4.89898 −0.163209
\(902\) −9.79796 −0.326236
\(903\) 15.7980 0.525723
\(904\) 14.3485 0.477223
\(905\) 0 0
\(906\) −18.6969 −0.621164
\(907\) 2.49490 0.0828417 0.0414209 0.999142i \(-0.486812\pi\)
0.0414209 + 0.999142i \(0.486812\pi\)
\(908\) −4.75255 −0.157719
\(909\) −13.6969 −0.454299
\(910\) 0 0
\(911\) 21.3031 0.705802 0.352901 0.935661i \(-0.385195\pi\)
0.352901 + 0.935661i \(0.385195\pi\)
\(912\) 6.89898 0.228448
\(913\) −5.10102 −0.168819
\(914\) 0.404082 0.0133658
\(915\) 0 0
\(916\) 3.10102 0.102461
\(917\) 12.8990 0.425962
\(918\) −5.44949 −0.179860
\(919\) −27.9444 −0.921800 −0.460900 0.887452i \(-0.652473\pi\)
−0.460900 + 0.887452i \(0.652473\pi\)
\(920\) 0 0
\(921\) −20.1010 −0.662351
\(922\) 21.4949 0.707897
\(923\) −31.3031 −1.03035
\(924\) 2.89898 0.0953694
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) −13.2474 −0.435103
\(928\) 5.00000 0.164133
\(929\) −3.79796 −0.124607 −0.0623035 0.998057i \(-0.519845\pi\)
−0.0623035 + 0.998057i \(0.519845\pi\)
\(930\) 0 0
\(931\) −33.7980 −1.10768
\(932\) −26.6969 −0.874487
\(933\) 14.5959 0.477849
\(934\) −29.9444 −0.979810
\(935\) 0 0
\(936\) −2.89898 −0.0947561
\(937\) −15.7980 −0.516097 −0.258048 0.966132i \(-0.583079\pi\)
−0.258048 + 0.966132i \(0.583079\pi\)
\(938\) −2.89898 −0.0946550
\(939\) −12.8990 −0.420942
\(940\) 0 0
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) −8.89898 −0.289944
\(943\) −4.89898 −0.159533
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) 21.7980 0.708713
\(947\) 0.404082 0.0131309 0.00656545 0.999978i \(-0.497910\pi\)
0.00656545 + 0.999978i \(0.497910\pi\)
\(948\) 10.0000 0.324785
\(949\) −6.09082 −0.197716
\(950\) 0 0
\(951\) 10.5959 0.343596
\(952\) −7.89898 −0.256007
\(953\) −30.8434 −0.999115 −0.499557 0.866281i \(-0.666504\pi\)
−0.499557 + 0.866281i \(0.666504\pi\)
\(954\) 0.898979 0.0291055
\(955\) 0 0
\(956\) −8.10102 −0.262006
\(957\) 10.0000 0.323254
\(958\) −30.0000 −0.969256
\(959\) −32.3939 −1.04605
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −24.2020 −0.780305
\(963\) −2.00000 −0.0644491
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) 1.44949 0.0466366
\(967\) −29.5959 −0.951741 −0.475870 0.879515i \(-0.657867\pi\)
−0.475870 + 0.879515i \(0.657867\pi\)
\(968\) −7.00000 −0.224989
\(969\) −37.5959 −1.20775
\(970\) 0 0
\(971\) 26.1464 0.839079 0.419539 0.907737i \(-0.362192\pi\)
0.419539 + 0.907737i \(0.362192\pi\)
\(972\) 1.00000 0.0320750
\(973\) 22.7526 0.729413
\(974\) −15.1010 −0.483868
\(975\) 0 0
\(976\) −4.89898 −0.156813
\(977\) −24.7526 −0.791904 −0.395952 0.918271i \(-0.629585\pi\)
−0.395952 + 0.918271i \(0.629585\pi\)
\(978\) 17.7980 0.569116
\(979\) −20.6969 −0.661477
\(980\) 0 0
\(981\) 3.44949 0.110134
\(982\) 25.1010 0.801006
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −4.89898 −0.156174
\(985\) 0 0
\(986\) −27.2474 −0.867736
\(987\) −5.65153 −0.179890
\(988\) −20.0000 −0.636285
\(989\) 10.8990 0.346567
\(990\) 0 0
\(991\) 18.2020 0.578207 0.289103 0.957298i \(-0.406643\pi\)
0.289103 + 0.957298i \(0.406643\pi\)
\(992\) 2.00000 0.0635001
\(993\) −6.79796 −0.215727
\(994\) 15.6515 0.496436
\(995\) 0 0
\(996\) −2.55051 −0.0808160
\(997\) 21.1010 0.668276 0.334138 0.942524i \(-0.391555\pi\)
0.334138 + 0.942524i \(0.391555\pi\)
\(998\) 21.8990 0.693200
\(999\) 8.34847 0.264134
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 3450.2.a.bl.1.2 yes 2
5.2 odd 4 3450.2.d.y.2899.4 4
5.3 odd 4 3450.2.d.y.2899.1 4
5.4 even 2 3450.2.a.bf.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.bf.1.1 2 5.4 even 2
3450.2.a.bl.1.2 yes 2 1.1 even 1 trivial
3450.2.d.y.2899.1 4 5.3 odd 4
3450.2.d.y.2899.4 4 5.2 odd 4