Properties

Label 5070.2.a.bf.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.60555 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.60555 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} -2.60555 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.60555 q^{17} +1.00000 q^{18} +2.60555 q^{19} +1.00000 q^{20} +2.60555 q^{21} +8.60555 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.60555 q^{28} -2.60555 q^{29} -1.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} +2.60555 q^{34} -2.60555 q^{35} +1.00000 q^{36} -5.21110 q^{37} +2.60555 q^{38} +1.00000 q^{40} -11.2111 q^{41} +2.60555 q^{42} -8.00000 q^{43} +1.00000 q^{45} +8.60555 q^{46} -5.21110 q^{47} -1.00000 q^{48} -0.211103 q^{49} +1.00000 q^{50} -2.60555 q^{51} +6.00000 q^{53} -1.00000 q^{54} -2.60555 q^{56} -2.60555 q^{57} -2.60555 q^{58} -5.21110 q^{59} -1.00000 q^{60} +3.21110 q^{61} +6.00000 q^{62} -2.60555 q^{63} +1.00000 q^{64} +11.2111 q^{67} +2.60555 q^{68} -8.60555 q^{69} -2.60555 q^{70} -5.21110 q^{71} +1.00000 q^{72} +8.60555 q^{73} -5.21110 q^{74} -1.00000 q^{75} +2.60555 q^{76} +14.4222 q^{79} +1.00000 q^{80} +1.00000 q^{81} -11.2111 q^{82} +17.2111 q^{83} +2.60555 q^{84} +2.60555 q^{85} -8.00000 q^{86} +2.60555 q^{87} -0.788897 q^{89} +1.00000 q^{90} +8.60555 q^{92} -6.00000 q^{93} -5.21110 q^{94} +2.60555 q^{95} -1.00000 q^{96} +8.60555 q^{97} -0.211103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + 2q^{10} - 2q^{12} + 2q^{14} - 2q^{15} + 2q^{16} - 2q^{17} + 2q^{18} - 2q^{19} + 2q^{20} - 2q^{21} + 10q^{23} - 2q^{24} + 2q^{25} - 2q^{27} + 2q^{28} + 2q^{29} - 2q^{30} + 12q^{31} + 2q^{32} - 2q^{34} + 2q^{35} + 2q^{36} + 4q^{37} - 2q^{38} + 2q^{40} - 8q^{41} - 2q^{42} - 16q^{43} + 2q^{45} + 10q^{46} + 4q^{47} - 2q^{48} + 14q^{49} + 2q^{50} + 2q^{51} + 12q^{53} - 2q^{54} + 2q^{56} + 2q^{57} + 2q^{58} + 4q^{59} - 2q^{60} - 8q^{61} + 12q^{62} + 2q^{63} + 2q^{64} + 8q^{67} - 2q^{68} - 10q^{69} + 2q^{70} + 4q^{71} + 2q^{72} + 10q^{73} + 4q^{74} - 2q^{75} - 2q^{76} + 2q^{80} + 2q^{81} - 8q^{82} + 20q^{83} - 2q^{84} - 2q^{85} - 16q^{86} - 2q^{87} - 16q^{89} + 2q^{90} + 10q^{92} - 12q^{93} + 4q^{94} - 2q^{95} - 2q^{96} + 10q^{97} + 14q^{98} + O(q^{100}) \)

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
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) −2.60555 −0.984806 −0.492403 0.870367i \(-0.663881\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −2.60555 −0.696363
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 2.60555 0.631939 0.315970 0.948769i \(-0.397670\pi\)
0.315970 + 0.948769i \(0.397670\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.60555 0.597754 0.298877 0.954292i \(-0.403388\pi\)
0.298877 + 0.954292i \(0.403388\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.60555 0.568578
\(22\) 0 0
\(23\) 8.60555 1.79438 0.897191 0.441643i \(-0.145604\pi\)
0.897191 + 0.441643i \(0.145604\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −2.60555 −0.492403
\(29\) −2.60555 −0.483839 −0.241919 0.970296i \(-0.577777\pi\)
−0.241919 + 0.970296i \(0.577777\pi\)
\(30\) −1.00000 −0.182574
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.60555 0.446848
\(35\) −2.60555 −0.440419
\(36\) 1.00000 0.166667
\(37\) −5.21110 −0.856700 −0.428350 0.903613i \(-0.640905\pi\)
−0.428350 + 0.903613i \(0.640905\pi\)
\(38\) 2.60555 0.422676
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −11.2111 −1.75088 −0.875440 0.483327i \(-0.839428\pi\)
−0.875440 + 0.483327i \(0.839428\pi\)
\(42\) 2.60555 0.402045
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 8.60555 1.26882
\(47\) −5.21110 −0.760117 −0.380059 0.924962i \(-0.624096\pi\)
−0.380059 + 0.924962i \(0.624096\pi\)
\(48\) −1.00000 −0.144338
\(49\) −0.211103 −0.0301575
\(50\) 1.00000 0.141421
\(51\) −2.60555 −0.364850
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.60555 −0.348181
\(57\) −2.60555 −0.345114
\(58\) −2.60555 −0.342126
\(59\) −5.21110 −0.678428 −0.339214 0.940709i \(-0.610161\pi\)
−0.339214 + 0.940709i \(0.610161\pi\)
\(60\) −1.00000 −0.129099
\(61\) 3.21110 0.411140 0.205570 0.978642i \(-0.434095\pi\)
0.205570 + 0.978642i \(0.434095\pi\)
\(62\) 6.00000 0.762001
\(63\) −2.60555 −0.328269
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 11.2111 1.36965 0.684827 0.728706i \(-0.259878\pi\)
0.684827 + 0.728706i \(0.259878\pi\)
\(68\) 2.60555 0.315970
\(69\) −8.60555 −1.03599
\(70\) −2.60555 −0.311423
\(71\) −5.21110 −0.618444 −0.309222 0.950990i \(-0.600069\pi\)
−0.309222 + 0.950990i \(0.600069\pi\)
\(72\) 1.00000 0.117851
\(73\) 8.60555 1.00720 0.503602 0.863936i \(-0.332008\pi\)
0.503602 + 0.863936i \(0.332008\pi\)
\(74\) −5.21110 −0.605778
\(75\) −1.00000 −0.115470
\(76\) 2.60555 0.298877
\(77\) 0 0
\(78\) 0 0
\(79\) 14.4222 1.62262 0.811312 0.584613i \(-0.198754\pi\)
0.811312 + 0.584613i \(0.198754\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −11.2111 −1.23806
\(83\) 17.2111 1.88916 0.944582 0.328276i \(-0.106467\pi\)
0.944582 + 0.328276i \(0.106467\pi\)
\(84\) 2.60555 0.284289
\(85\) 2.60555 0.282612
\(86\) −8.00000 −0.862662
\(87\) 2.60555 0.279344
\(88\) 0 0
\(89\) −0.788897 −0.0836230 −0.0418115 0.999126i \(-0.513313\pi\)
−0.0418115 + 0.999126i \(0.513313\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 8.60555 0.897191
\(93\) −6.00000 −0.622171
\(94\) −5.21110 −0.537484
\(95\) 2.60555 0.267324
\(96\) −1.00000 −0.102062
\(97\) 8.60555 0.873761 0.436881 0.899519i \(-0.356083\pi\)
0.436881 + 0.899519i \(0.356083\pi\)
\(98\) −0.211103 −0.0213246
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 14.6056 1.45331 0.726653 0.687004i \(-0.241075\pi\)
0.726653 + 0.687004i \(0.241075\pi\)
\(102\) −2.60555 −0.257988
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 2.60555 0.254276
\(106\) 6.00000 0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 8.60555 0.824262 0.412131 0.911125i \(-0.364785\pi\)
0.412131 + 0.911125i \(0.364785\pi\)
\(110\) 0 0
\(111\) 5.21110 0.494616
\(112\) −2.60555 −0.246201
\(113\) 7.81665 0.735329 0.367664 0.929959i \(-0.380157\pi\)
0.367664 + 0.929959i \(0.380157\pi\)
\(114\) −2.60555 −0.244032
\(115\) 8.60555 0.802472
\(116\) −2.60555 −0.241919
\(117\) 0 0
\(118\) −5.21110 −0.479721
\(119\) −6.78890 −0.622337
\(120\) −1.00000 −0.0912871
\(121\) −11.0000 −1.00000
\(122\) 3.21110 0.290720
\(123\) 11.2111 1.01087
\(124\) 6.00000 0.538816
\(125\) 1.00000 0.0894427
\(126\) −2.60555 −0.232121
\(127\) 13.2111 1.17230 0.586148 0.810204i \(-0.300644\pi\)
0.586148 + 0.810204i \(0.300644\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 15.3944 1.34502 0.672510 0.740088i \(-0.265216\pi\)
0.672510 + 0.740088i \(0.265216\pi\)
\(132\) 0 0
\(133\) −6.78890 −0.588672
\(134\) 11.2111 0.968492
\(135\) −1.00000 −0.0860663
\(136\) 2.60555 0.223424
\(137\) 11.2111 0.957829 0.478915 0.877862i \(-0.341030\pi\)
0.478915 + 0.877862i \(0.341030\pi\)
\(138\) −8.60555 −0.732553
\(139\) 2.78890 0.236551 0.118276 0.992981i \(-0.462263\pi\)
0.118276 + 0.992981i \(0.462263\pi\)
\(140\) −2.60555 −0.220209
\(141\) 5.21110 0.438854
\(142\) −5.21110 −0.437306
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −2.60555 −0.216379
\(146\) 8.60555 0.712200
\(147\) 0.211103 0.0174114
\(148\) −5.21110 −0.428350
\(149\) 0.788897 0.0646290 0.0323145 0.999478i \(-0.489712\pi\)
0.0323145 + 0.999478i \(0.489712\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) 2.60555 0.211338
\(153\) 2.60555 0.210646
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −8.42221 −0.672165 −0.336083 0.941833i \(-0.609102\pi\)
−0.336083 + 0.941833i \(0.609102\pi\)
\(158\) 14.4222 1.14737
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) −22.4222 −1.76712
\(162\) 1.00000 0.0785674
\(163\) 4.42221 0.346374 0.173187 0.984889i \(-0.444594\pi\)
0.173187 + 0.984889i \(0.444594\pi\)
\(164\) −11.2111 −0.875440
\(165\) 0 0
\(166\) 17.2111 1.33584
\(167\) 5.21110 0.403247 0.201624 0.979463i \(-0.435378\pi\)
0.201624 + 0.979463i \(0.435378\pi\)
\(168\) 2.60555 0.201023
\(169\) 0 0
\(170\) 2.60555 0.199837
\(171\) 2.60555 0.199251
\(172\) −8.00000 −0.609994
\(173\) −16.4222 −1.24856 −0.624279 0.781202i \(-0.714607\pi\)
−0.624279 + 0.781202i \(0.714607\pi\)
\(174\) 2.60555 0.197526
\(175\) −2.60555 −0.196961
\(176\) 0 0
\(177\) 5.21110 0.391690
\(178\) −0.788897 −0.0591304
\(179\) −1.81665 −0.135783 −0.0678915 0.997693i \(-0.521627\pi\)
−0.0678915 + 0.997693i \(0.521627\pi\)
\(180\) 1.00000 0.0745356
\(181\) 20.4222 1.51797 0.758985 0.651108i \(-0.225695\pi\)
0.758985 + 0.651108i \(0.225695\pi\)
\(182\) 0 0
\(183\) −3.21110 −0.237372
\(184\) 8.60555 0.634410
\(185\) −5.21110 −0.383128
\(186\) −6.00000 −0.439941
\(187\) 0 0
\(188\) −5.21110 −0.380059
\(189\) 2.60555 0.189526
\(190\) 2.60555 0.189027
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 13.8167 0.994545 0.497272 0.867595i \(-0.334335\pi\)
0.497272 + 0.867595i \(0.334335\pi\)
\(194\) 8.60555 0.617843
\(195\) 0 0
\(196\) −0.211103 −0.0150788
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 6.42221 0.455258 0.227629 0.973748i \(-0.426903\pi\)
0.227629 + 0.973748i \(0.426903\pi\)
\(200\) 1.00000 0.0707107
\(201\) −11.2111 −0.790770
\(202\) 14.6056 1.02764
\(203\) 6.78890 0.476487
\(204\) −2.60555 −0.182425
\(205\) −11.2111 −0.783017
\(206\) 4.00000 0.278693
\(207\) 8.60555 0.598127
\(208\) 0 0
\(209\) 0 0
\(210\) 2.60555 0.179800
\(211\) −2.78890 −0.191996 −0.0959978 0.995382i \(-0.530604\pi\)
−0.0959978 + 0.995382i \(0.530604\pi\)
\(212\) 6.00000 0.412082
\(213\) 5.21110 0.357059
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) −1.00000 −0.0680414
\(217\) −15.6333 −1.06126
\(218\) 8.60555 0.582841
\(219\) −8.60555 −0.581509
\(220\) 0 0
\(221\) 0 0
\(222\) 5.21110 0.349746
\(223\) −19.8167 −1.32702 −0.663511 0.748167i \(-0.730934\pi\)
−0.663511 + 0.748167i \(0.730934\pi\)
\(224\) −2.60555 −0.174091
\(225\) 1.00000 0.0666667
\(226\) 7.81665 0.519956
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −2.60555 −0.172557
\(229\) −1.81665 −0.120048 −0.0600239 0.998197i \(-0.519118\pi\)
−0.0600239 + 0.998197i \(0.519118\pi\)
\(230\) 8.60555 0.567433
\(231\) 0 0
\(232\) −2.60555 −0.171063
\(233\) −19.8167 −1.29823 −0.649116 0.760689i \(-0.724861\pi\)
−0.649116 + 0.760689i \(0.724861\pi\)
\(234\) 0 0
\(235\) −5.21110 −0.339935
\(236\) −5.21110 −0.339214
\(237\) −14.4222 −0.936823
\(238\) −6.78890 −0.440059
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −22.4222 −1.44434 −0.722171 0.691715i \(-0.756855\pi\)
−0.722171 + 0.691715i \(0.756855\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) 3.21110 0.205570
\(245\) −0.211103 −0.0134868
\(246\) 11.2111 0.714794
\(247\) 0 0
\(248\) 6.00000 0.381000
\(249\) −17.2111 −1.09071
\(250\) 1.00000 0.0632456
\(251\) 20.6056 1.30061 0.650305 0.759673i \(-0.274641\pi\)
0.650305 + 0.759673i \(0.274641\pi\)
\(252\) −2.60555 −0.164134
\(253\) 0 0
\(254\) 13.2111 0.828938
\(255\) −2.60555 −0.163166
\(256\) 1.00000 0.0625000
\(257\) 21.3944 1.33455 0.667275 0.744812i \(-0.267461\pi\)
0.667275 + 0.744812i \(0.267461\pi\)
\(258\) 8.00000 0.498058
\(259\) 13.5778 0.843683
\(260\) 0 0
\(261\) −2.60555 −0.161280
\(262\) 15.3944 0.951072
\(263\) 13.8167 0.851971 0.425986 0.904730i \(-0.359927\pi\)
0.425986 + 0.904730i \(0.359927\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −6.78890 −0.416254
\(267\) 0.788897 0.0482797
\(268\) 11.2111 0.684827
\(269\) 4.18335 0.255063 0.127532 0.991835i \(-0.459295\pi\)
0.127532 + 0.991835i \(0.459295\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −28.4222 −1.72653 −0.863263 0.504754i \(-0.831583\pi\)
−0.863263 + 0.504754i \(0.831583\pi\)
\(272\) 2.60555 0.157985
\(273\) 0 0
\(274\) 11.2111 0.677287
\(275\) 0 0
\(276\) −8.60555 −0.517993
\(277\) −12.4222 −0.746378 −0.373189 0.927755i \(-0.621736\pi\)
−0.373189 + 0.927755i \(0.621736\pi\)
\(278\) 2.78890 0.167267
\(279\) 6.00000 0.359211
\(280\) −2.60555 −0.155711
\(281\) −12.7889 −0.762922 −0.381461 0.924385i \(-0.624579\pi\)
−0.381461 + 0.924385i \(0.624579\pi\)
\(282\) 5.21110 0.310317
\(283\) 18.4222 1.09509 0.547543 0.836777i \(-0.315563\pi\)
0.547543 + 0.836777i \(0.315563\pi\)
\(284\) −5.21110 −0.309222
\(285\) −2.60555 −0.154340
\(286\) 0 0
\(287\) 29.2111 1.72428
\(288\) 1.00000 0.0589256
\(289\) −10.2111 −0.600653
\(290\) −2.60555 −0.153003
\(291\) −8.60555 −0.504466
\(292\) 8.60555 0.503602
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0.211103 0.0123118
\(295\) −5.21110 −0.303402
\(296\) −5.21110 −0.302889
\(297\) 0 0
\(298\) 0.788897 0.0456996
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 20.8444 1.20145
\(302\) −6.00000 −0.345261
\(303\) −14.6056 −0.839067
\(304\) 2.60555 0.149439
\(305\) 3.21110 0.183867
\(306\) 2.60555 0.148949
\(307\) 23.2111 1.32473 0.662364 0.749182i \(-0.269553\pi\)
0.662364 + 0.749182i \(0.269553\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 6.00000 0.340777
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 32.4222 1.83261 0.916306 0.400480i \(-0.131156\pi\)
0.916306 + 0.400480i \(0.131156\pi\)
\(314\) −8.42221 −0.475293
\(315\) −2.60555 −0.146806
\(316\) 14.4222 0.811312
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −22.4222 −1.24954
\(323\) 6.78890 0.377744
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 4.42221 0.244923
\(327\) −8.60555 −0.475888
\(328\) −11.2111 −0.619030
\(329\) 13.5778 0.748568
\(330\) 0 0
\(331\) −9.39445 −0.516366 −0.258183 0.966096i \(-0.583124\pi\)
−0.258183 + 0.966096i \(0.583124\pi\)
\(332\) 17.2111 0.944582
\(333\) −5.21110 −0.285567
\(334\) 5.21110 0.285139
\(335\) 11.2111 0.612528
\(336\) 2.60555 0.142144
\(337\) 29.6333 1.61423 0.807115 0.590395i \(-0.201028\pi\)
0.807115 + 0.590395i \(0.201028\pi\)
\(338\) 0 0
\(339\) −7.81665 −0.424542
\(340\) 2.60555 0.141306
\(341\) 0 0
\(342\) 2.60555 0.140892
\(343\) 18.7889 1.01451
\(344\) −8.00000 −0.431331
\(345\) −8.60555 −0.463307
\(346\) −16.4222 −0.882863
\(347\) 15.6333 0.839240 0.419620 0.907700i \(-0.362163\pi\)
0.419620 + 0.907700i \(0.362163\pi\)
\(348\) 2.60555 0.139672
\(349\) −13.8167 −0.739589 −0.369794 0.929114i \(-0.620572\pi\)
−0.369794 + 0.929114i \(0.620572\pi\)
\(350\) −2.60555 −0.139273
\(351\) 0 0
\(352\) 0 0
\(353\) −23.2111 −1.23540 −0.617701 0.786413i \(-0.711936\pi\)
−0.617701 + 0.786413i \(0.711936\pi\)
\(354\) 5.21110 0.276967
\(355\) −5.21110 −0.276577
\(356\) −0.788897 −0.0418115
\(357\) 6.78890 0.359307
\(358\) −1.81665 −0.0960131
\(359\) 27.6333 1.45843 0.729215 0.684285i \(-0.239885\pi\)
0.729215 + 0.684285i \(0.239885\pi\)
\(360\) 1.00000 0.0527046
\(361\) −12.2111 −0.642690
\(362\) 20.4222 1.07337
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 8.60555 0.450435
\(366\) −3.21110 −0.167847
\(367\) 23.6333 1.23365 0.616824 0.787101i \(-0.288419\pi\)
0.616824 + 0.787101i \(0.288419\pi\)
\(368\) 8.60555 0.448595
\(369\) −11.2111 −0.583627
\(370\) −5.21110 −0.270912
\(371\) −15.6333 −0.811641
\(372\) −6.00000 −0.311086
\(373\) 8.42221 0.436085 0.218043 0.975939i \(-0.430033\pi\)
0.218043 + 0.975939i \(0.430033\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −5.21110 −0.268742
\(377\) 0 0
\(378\) 2.60555 0.134015
\(379\) −1.02776 −0.0527923 −0.0263961 0.999652i \(-0.508403\pi\)
−0.0263961 + 0.999652i \(0.508403\pi\)
\(380\) 2.60555 0.133662
\(381\) −13.2111 −0.676825
\(382\) −12.0000 −0.613973
\(383\) 15.6333 0.798825 0.399412 0.916771i \(-0.369214\pi\)
0.399412 + 0.916771i \(0.369214\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 13.8167 0.703249
\(387\) −8.00000 −0.406663
\(388\) 8.60555 0.436881
\(389\) 2.60555 0.132107 0.0660533 0.997816i \(-0.478959\pi\)
0.0660533 + 0.997816i \(0.478959\pi\)
\(390\) 0 0
\(391\) 22.4222 1.13394
\(392\) −0.211103 −0.0106623
\(393\) −15.3944 −0.776547
\(394\) −6.00000 −0.302276
\(395\) 14.4222 0.725660
\(396\) 0 0
\(397\) −39.6333 −1.98914 −0.994569 0.104076i \(-0.966811\pi\)
−0.994569 + 0.104076i \(0.966811\pi\)
\(398\) 6.42221 0.321916
\(399\) 6.78890 0.339870
\(400\) 1.00000 0.0500000
\(401\) −23.2111 −1.15911 −0.579554 0.814934i \(-0.696773\pi\)
−0.579554 + 0.814934i \(0.696773\pi\)
\(402\) −11.2111 −0.559159
\(403\) 0 0
\(404\) 14.6056 0.726653
\(405\) 1.00000 0.0496904
\(406\) 6.78890 0.336927
\(407\) 0 0
\(408\) −2.60555 −0.128994
\(409\) −29.2111 −1.44440 −0.722198 0.691686i \(-0.756868\pi\)
−0.722198 + 0.691686i \(0.756868\pi\)
\(410\) −11.2111 −0.553677
\(411\) −11.2111 −0.553003
\(412\) 4.00000 0.197066
\(413\) 13.5778 0.668120
\(414\) 8.60555 0.422940
\(415\) 17.2111 0.844860
\(416\) 0 0
\(417\) −2.78890 −0.136573
\(418\) 0 0
\(419\) 25.8167 1.26123 0.630613 0.776097i \(-0.282804\pi\)
0.630613 + 0.776097i \(0.282804\pi\)
\(420\) 2.60555 0.127138
\(421\) 1.81665 0.0885383 0.0442691 0.999020i \(-0.485904\pi\)
0.0442691 + 0.999020i \(0.485904\pi\)
\(422\) −2.78890 −0.135761
\(423\) −5.21110 −0.253372
\(424\) 6.00000 0.291386
\(425\) 2.60555 0.126388
\(426\) 5.21110 0.252479
\(427\) −8.36669 −0.404893
\(428\) 0 0
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −4.78890 −0.230140 −0.115070 0.993357i \(-0.536709\pi\)
−0.115070 + 0.993357i \(0.536709\pi\)
\(434\) −15.6333 −0.750423
\(435\) 2.60555 0.124927
\(436\) 8.60555 0.412131
\(437\) 22.4222 1.07260
\(438\) −8.60555 −0.411189
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −0.211103 −0.0100525
\(442\) 0 0
\(443\) −27.6333 −1.31290 −0.656449 0.754370i \(-0.727942\pi\)
−0.656449 + 0.754370i \(0.727942\pi\)
\(444\) 5.21110 0.247308
\(445\) −0.788897 −0.0373973
\(446\) −19.8167 −0.938346
\(447\) −0.788897 −0.0373136
\(448\) −2.60555 −0.123101
\(449\) 9.63331 0.454624 0.227312 0.973822i \(-0.427006\pi\)
0.227312 + 0.973822i \(0.427006\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 7.81665 0.367664
\(453\) 6.00000 0.281905
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) −2.60555 −0.122016
\(457\) −12.2389 −0.572510 −0.286255 0.958154i \(-0.592410\pi\)
−0.286255 + 0.958154i \(0.592410\pi\)
\(458\) −1.81665 −0.0848867
\(459\) −2.60555 −0.121617
\(460\) 8.60555 0.401236
\(461\) 9.63331 0.448668 0.224334 0.974512i \(-0.427979\pi\)
0.224334 + 0.974512i \(0.427979\pi\)
\(462\) 0 0
\(463\) 38.6056 1.79415 0.897076 0.441876i \(-0.145687\pi\)
0.897076 + 0.441876i \(0.145687\pi\)
\(464\) −2.60555 −0.120960
\(465\) −6.00000 −0.278243
\(466\) −19.8167 −0.917989
\(467\) 1.57779 0.0730116 0.0365058 0.999333i \(-0.488377\pi\)
0.0365058 + 0.999333i \(0.488377\pi\)
\(468\) 0 0
\(469\) −29.2111 −1.34884
\(470\) −5.21110 −0.240370
\(471\) 8.42221 0.388075
\(472\) −5.21110 −0.239860
\(473\) 0 0
\(474\) −14.4222 −0.662434
\(475\) 2.60555 0.119551
\(476\) −6.78890 −0.311169
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) −34.4222 −1.57279 −0.786395 0.617724i \(-0.788055\pi\)
−0.786395 + 0.617724i \(0.788055\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −22.4222 −1.02130
\(483\) 22.4222 1.02025
\(484\) −11.0000 −0.500000
\(485\) 8.60555 0.390758
\(486\) −1.00000 −0.0453609
\(487\) −37.0278 −1.67789 −0.838944 0.544218i \(-0.816826\pi\)
−0.838944 + 0.544218i \(0.816826\pi\)
\(488\) 3.21110 0.145360
\(489\) −4.42221 −0.199979
\(490\) −0.211103 −0.00953664
\(491\) 13.8167 0.623537 0.311768 0.950158i \(-0.399079\pi\)
0.311768 + 0.950158i \(0.399079\pi\)
\(492\) 11.2111 0.505436
\(493\) −6.78890 −0.305757
\(494\) 0 0
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 13.5778 0.609047
\(498\) −17.2111 −0.771248
\(499\) −13.0278 −0.583202 −0.291601 0.956540i \(-0.594188\pi\)
−0.291601 + 0.956540i \(0.594188\pi\)
\(500\) 1.00000 0.0447214
\(501\) −5.21110 −0.232815
\(502\) 20.6056 0.919671
\(503\) −41.4500 −1.84816 −0.924081 0.382196i \(-0.875168\pi\)
−0.924081 + 0.382196i \(0.875168\pi\)
\(504\) −2.60555 −0.116060
\(505\) 14.6056 0.649939
\(506\) 0 0
\(507\) 0 0
\(508\) 13.2111 0.586148
\(509\) 9.63331 0.426989 0.213494 0.976944i \(-0.431515\pi\)
0.213494 + 0.976944i \(0.431515\pi\)
\(510\) −2.60555 −0.115376
\(511\) −22.4222 −0.991900
\(512\) 1.00000 0.0441942
\(513\) −2.60555 −0.115038
\(514\) 21.3944 0.943669
\(515\) 4.00000 0.176261
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 13.5778 0.596574
\(519\) 16.4222 0.720855
\(520\) 0 0
\(521\) −21.6333 −0.947772 −0.473886 0.880586i \(-0.657149\pi\)
−0.473886 + 0.880586i \(0.657149\pi\)
\(522\) −2.60555 −0.114042
\(523\) −24.8444 −1.08637 −0.543185 0.839613i \(-0.682782\pi\)
−0.543185 + 0.839613i \(0.682782\pi\)
\(524\) 15.3944 0.672510
\(525\) 2.60555 0.113716
\(526\) 13.8167 0.602435
\(527\) 15.6333 0.680998
\(528\) 0 0
\(529\) 51.0555 2.21980
\(530\) 6.00000 0.260623
\(531\) −5.21110 −0.226143
\(532\) −6.78890 −0.294336
\(533\) 0 0
\(534\) 0.788897 0.0341389
\(535\) 0 0
\(536\) 11.2111 0.484246
\(537\) 1.81665 0.0783944
\(538\) 4.18335 0.180357
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −43.0278 −1.84991 −0.924954 0.380079i \(-0.875897\pi\)
−0.924954 + 0.380079i \(0.875897\pi\)
\(542\) −28.4222 −1.22084
\(543\) −20.4222 −0.876401
\(544\) 2.60555 0.111712
\(545\) 8.60555 0.368621
\(546\) 0 0
\(547\) −14.4222 −0.616649 −0.308324 0.951281i \(-0.599768\pi\)
−0.308324 + 0.951281i \(0.599768\pi\)
\(548\) 11.2111 0.478915
\(549\) 3.21110 0.137047
\(550\) 0 0
\(551\) −6.78890 −0.289217
\(552\) −8.60555 −0.366277
\(553\) −37.5778 −1.59797
\(554\) −12.4222 −0.527769
\(555\) 5.21110 0.221199
\(556\) 2.78890 0.118276
\(557\) −40.4222 −1.71274 −0.856372 0.516360i \(-0.827287\pi\)
−0.856372 + 0.516360i \(0.827287\pi\)
\(558\) 6.00000 0.254000
\(559\) 0 0
\(560\) −2.60555 −0.110105
\(561\) 0 0
\(562\) −12.7889 −0.539467
\(563\) 38.0555 1.60385 0.801924 0.597426i \(-0.203810\pi\)
0.801924 + 0.597426i \(0.203810\pi\)
\(564\) 5.21110 0.219427
\(565\) 7.81665 0.328849
\(566\) 18.4222 0.774343
\(567\) −2.60555 −0.109423
\(568\) −5.21110 −0.218653
\(569\) 9.63331 0.403849 0.201925 0.979401i \(-0.435280\pi\)
0.201925 + 0.979401i \(0.435280\pi\)
\(570\) −2.60555 −0.109135
\(571\) −42.0555 −1.75997 −0.879984 0.475003i \(-0.842447\pi\)
−0.879984 + 0.475003i \(0.842447\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 29.2111 1.21925
\(575\) 8.60555 0.358876
\(576\) 1.00000 0.0416667
\(577\) −44.6056 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(578\) −10.2111 −0.424726
\(579\) −13.8167 −0.574201
\(580\) −2.60555 −0.108190
\(581\) −44.8444 −1.86046
\(582\) −8.60555 −0.356712
\(583\) 0 0
\(584\) 8.60555 0.356100
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 22.4222 0.925463 0.462732 0.886498i \(-0.346869\pi\)
0.462732 + 0.886498i \(0.346869\pi\)
\(588\) 0.211103 0.00870572
\(589\) 15.6333 0.644159
\(590\) −5.21110 −0.214538
\(591\) 6.00000 0.246807
\(592\) −5.21110 −0.214175
\(593\) −4.42221 −0.181598 −0.0907991 0.995869i \(-0.528942\pi\)
−0.0907991 + 0.995869i \(0.528942\pi\)
\(594\) 0 0
\(595\) −6.78890 −0.278318
\(596\) 0.788897 0.0323145
\(597\) −6.42221 −0.262843
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −41.6333 −1.69826 −0.849129 0.528185i \(-0.822873\pi\)
−0.849129 + 0.528185i \(0.822873\pi\)
\(602\) 20.8444 0.849555
\(603\) 11.2111 0.456551
\(604\) −6.00000 −0.244137
\(605\) −11.0000 −0.447214
\(606\) −14.6056 −0.593310
\(607\) 2.78890 0.113198 0.0565989 0.998397i \(-0.481974\pi\)
0.0565989 + 0.998397i \(0.481974\pi\)
\(608\) 2.60555 0.105669
\(609\) −6.78890 −0.275100
\(610\) 3.21110 0.130014
\(611\) 0 0
\(612\) 2.60555 0.105323
\(613\) −18.7889 −0.758876 −0.379438 0.925217i \(-0.623883\pi\)
−0.379438 + 0.925217i \(0.623883\pi\)
\(614\) 23.2111 0.936724
\(615\) 11.2111 0.452075
\(616\) 0 0
\(617\) −16.4222 −0.661133 −0.330567 0.943783i \(-0.607240\pi\)
−0.330567 + 0.943783i \(0.607240\pi\)
\(618\) −4.00000 −0.160904
\(619\) −4.18335 −0.168143 −0.0840714 0.996460i \(-0.526792\pi\)
−0.0840714 + 0.996460i \(0.526792\pi\)
\(620\) 6.00000 0.240966
\(621\) −8.60555 −0.345329
\(622\) −12.0000 −0.481156
\(623\) 2.05551 0.0823524
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 32.4222 1.29585
\(627\) 0 0
\(628\) −8.42221 −0.336083
\(629\) −13.5778 −0.541382
\(630\) −2.60555 −0.103808
\(631\) 11.2111 0.446307 0.223153 0.974783i \(-0.428365\pi\)
0.223153 + 0.974783i \(0.428365\pi\)
\(632\) 14.4222 0.573685
\(633\) 2.78890 0.110849
\(634\) −18.0000 −0.714871
\(635\) 13.2111 0.524267
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) −5.21110 −0.206148
\(640\) 1.00000 0.0395285
\(641\) 28.4222 1.12261 0.561305 0.827609i \(-0.310300\pi\)
0.561305 + 0.827609i \(0.310300\pi\)
\(642\) 0 0
\(643\) −33.6333 −1.32637 −0.663184 0.748456i \(-0.730795\pi\)
−0.663184 + 0.748456i \(0.730795\pi\)
\(644\) −22.4222 −0.883559
\(645\) 8.00000 0.315000
\(646\) 6.78890 0.267106
\(647\) 27.3944 1.07699 0.538493 0.842630i \(-0.318994\pi\)
0.538493 + 0.842630i \(0.318994\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 15.6333 0.612718
\(652\) 4.42221 0.173187
\(653\) 24.7889 0.970065 0.485032 0.874496i \(-0.338808\pi\)
0.485032 + 0.874496i \(0.338808\pi\)
\(654\) −8.60555 −0.336504
\(655\) 15.3944 0.601511
\(656\) −11.2111 −0.437720
\(657\) 8.60555 0.335735
\(658\) 13.5778 0.529318
\(659\) 24.2389 0.944212 0.472106 0.881542i \(-0.343494\pi\)
0.472106 + 0.881542i \(0.343494\pi\)
\(660\) 0 0
\(661\) 0.238859 0.00929054 0.00464527 0.999989i \(-0.498521\pi\)
0.00464527 + 0.999989i \(0.498521\pi\)
\(662\) −9.39445 −0.365126
\(663\) 0 0
\(664\) 17.2111 0.667920
\(665\) −6.78890 −0.263262
\(666\) −5.21110 −0.201926
\(667\) −22.4222 −0.868191
\(668\) 5.21110 0.201624
\(669\) 19.8167 0.766156
\(670\) 11.2111 0.433123
\(671\) 0 0
\(672\) 2.60555 0.100511
\(673\) −5.63331 −0.217148 −0.108574 0.994088i \(-0.534628\pi\)
−0.108574 + 0.994088i \(0.534628\pi\)
\(674\) 29.6333 1.14143
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 44.0555 1.69319 0.846595 0.532237i \(-0.178648\pi\)
0.846595 + 0.532237i \(0.178648\pi\)
\(678\) −7.81665 −0.300197
\(679\) −22.4222 −0.860485
\(680\) 2.60555 0.0999183
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) 5.21110 0.199397 0.0996986 0.995018i \(-0.468212\pi\)
0.0996986 + 0.995018i \(0.468212\pi\)
\(684\) 2.60555 0.0996257
\(685\) 11.2111 0.428354
\(686\) 18.7889 0.717363
\(687\) 1.81665 0.0693097
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) −8.60555 −0.327608
\(691\) −30.2389 −1.15034 −0.575170 0.818034i \(-0.695064\pi\)
−0.575170 + 0.818034i \(0.695064\pi\)
\(692\) −16.4222 −0.624279
\(693\) 0 0
\(694\) 15.6333 0.593432
\(695\) 2.78890 0.105789
\(696\) 2.60555 0.0987632
\(697\) −29.2111 −1.10645
\(698\) −13.8167 −0.522968
\(699\) 19.8167 0.749535
\(700\) −2.60555 −0.0984806
\(701\) −10.9722 −0.414416 −0.207208 0.978297i \(-0.566438\pi\)
−0.207208 + 0.978297i \(0.566438\pi\)
\(702\) 0 0
\(703\) −13.5778 −0.512096
\(704\) 0 0
\(705\) 5.21110 0.196261
\(706\) −23.2111 −0.873561
\(707\) −38.0555 −1.43122
\(708\) 5.21110 0.195845
\(709\) 8.60555 0.323188 0.161594 0.986857i \(-0.448336\pi\)
0.161594 + 0.986857i \(0.448336\pi\)
\(710\) −5.21110 −0.195569
\(711\) 14.4222 0.540875
\(712\) −0.788897 −0.0295652
\(713\) 51.6333 1.93368
\(714\) 6.78890 0.254068
\(715\) 0 0
\(716\) −1.81665 −0.0678915
\(717\) 0 0
\(718\) 27.6333 1.03127
\(719\) 8.36669 0.312025 0.156012 0.987755i \(-0.450136\pi\)
0.156012 + 0.987755i \(0.450136\pi\)
\(720\) 1.00000 0.0372678
\(721\) −10.4222 −0.388143
\(722\) −12.2111 −0.454450
\(723\) 22.4222 0.833891
\(724\) 20.4222 0.758985
\(725\) −2.60555 −0.0967677
\(726\) 11.0000 0.408248
\(727\) −14.4222 −0.534890 −0.267445 0.963573i \(-0.586179\pi\)
−0.267445 + 0.963573i \(0.586179\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 8.60555 0.318506
\(731\) −20.8444 −0.770958
\(732\) −3.21110 −0.118686
\(733\) 38.0555 1.40561 0.702806 0.711381i \(-0.251930\pi\)
0.702806 + 0.711381i \(0.251930\pi\)
\(734\) 23.6333 0.872321
\(735\) 0.211103 0.00778663
\(736\) 8.60555 0.317205
\(737\) 0 0
\(738\) −11.2111 −0.412686
\(739\) 30.2389 1.11235 0.556177 0.831064i \(-0.312268\pi\)
0.556177 + 0.831064i \(0.312268\pi\)
\(740\) −5.21110 −0.191564
\(741\) 0 0
\(742\) −15.6333 −0.573917
\(743\) −20.8444 −0.764707 −0.382354 0.924016i \(-0.624886\pi\)
−0.382354 + 0.924016i \(0.624886\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0.788897 0.0289030
\(746\) 8.42221 0.308359
\(747\) 17.2111 0.629721
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) −18.4222 −0.672236 −0.336118 0.941820i \(-0.609114\pi\)
−0.336118 + 0.941820i \(0.609114\pi\)
\(752\) −5.21110 −0.190029
\(753\) −20.6056 −0.750908
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) 2.60555 0.0947630
\(757\) 27.2111 0.989004 0.494502 0.869176i \(-0.335350\pi\)
0.494502 + 0.869176i \(0.335350\pi\)
\(758\) −1.02776 −0.0373298
\(759\) 0 0
\(760\) 2.60555 0.0945133
\(761\) 9.63331 0.349207 0.174604 0.984639i \(-0.444136\pi\)
0.174604 + 0.984639i \(0.444136\pi\)
\(762\) −13.2111 −0.478588
\(763\) −22.4222 −0.811738
\(764\) −12.0000 −0.434145
\(765\) 2.60555 0.0942039
\(766\) 15.6333 0.564854
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 44.8444 1.61713 0.808565 0.588406i \(-0.200244\pi\)
0.808565 + 0.588406i \(0.200244\pi\)
\(770\) 0 0
\(771\) −21.3944 −0.770502
\(772\) 13.8167 0.497272
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) −8.00000 −0.287554
\(775\) 6.00000 0.215526
\(776\) 8.60555 0.308921
\(777\) −13.5778 −0.487101
\(778\) 2.60555 0.0934135
\(779\) −29.2111 −1.04660
\(780\) 0 0
\(781\) 0 0
\(782\) 22.4222 0.801816
\(783\) 2.60555 0.0931148
\(784\) −0.211103 −0.00753938
\(785\) −8.42221 −0.300601
\(786\) −15.3944 −0.549102
\(787\) 37.2666 1.32841 0.664206 0.747550i \(-0.268770\pi\)
0.664206 + 0.747550i \(0.268770\pi\)
\(788\) −6.00000 −0.213741
\(789\) −13.8167 −0.491886
\(790\) 14.4222 0.513119
\(791\) −20.3667 −0.724156
\(792\) 0 0
\(793\) 0 0
\(794\) −39.6333 −1.40653
\(795\) −6.00000 −0.212798
\(796\) 6.42221 0.227629
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 6.78890 0.240324
\(799\) −13.5778 −0.480348
\(800\) 1.00000 0.0353553
\(801\) −0.788897 −0.0278743
\(802\) −23.2111 −0.819613
\(803\) 0 0
\(804\) −11.2111 −0.395385
\(805\) −22.4222 −0.790279
\(806\) 0 0
\(807\) −4.18335 −0.147261
\(808\) 14.6056 0.513822
\(809\) 50.8444 1.78759 0.893797 0.448471i \(-0.148031\pi\)
0.893797 + 0.448471i \(0.148031\pi\)
\(810\) 1.00000 0.0351364
\(811\) −18.2389 −0.640453 −0.320226 0.947341i \(-0.603759\pi\)
−0.320226 + 0.947341i \(0.603759\pi\)
\(812\) 6.78890 0.238244
\(813\) 28.4222 0.996810
\(814\) 0 0
\(815\) 4.42221 0.154903
\(816\) −2.60555 −0.0912125
\(817\) −20.8444 −0.729254
\(818\) −29.2111 −1.02134
\(819\) 0 0
\(820\) −11.2111 −0.391509
\(821\) 11.2111 0.391270 0.195635 0.980677i \(-0.437323\pi\)
0.195635 + 0.980677i \(0.437323\pi\)
\(822\) −11.2111 −0.391032
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 13.5778 0.472432
\(827\) −15.6333 −0.543623 −0.271812 0.962350i \(-0.587623\pi\)
−0.271812 + 0.962350i \(0.587623\pi\)
\(828\) 8.60555 0.299064
\(829\) 10.8444 0.376642 0.188321 0.982108i \(-0.439695\pi\)
0.188321 + 0.982108i \(0.439695\pi\)
\(830\) 17.2111 0.597406
\(831\) 12.4222 0.430922
\(832\) 0 0
\(833\) −0.550039 −0.0190577
\(834\) −2.78890 −0.0965716
\(835\) 5.21110 0.180338
\(836\) 0 0
\(837\) −6.00000 −0.207390
\(838\) 25.8167 0.891822
\(839\) −10.4222 −0.359814 −0.179907 0.983684i \(-0.557580\pi\)
−0.179907 + 0.983684i \(0.557580\pi\)
\(840\) 2.60555 0.0899001
\(841\) −22.2111 −0.765900
\(842\) 1.81665 0.0626060
\(843\) 12.7889 0.440473
\(844\) −2.78890 −0.0959978
\(845\) 0 0
\(846\) −5.21110 −0.179161
\(847\) 28.6611 0.984806
\(848\) 6.00000 0.206041
\(849\) −18.4222 −0.632248
\(850\) 2.60555 0.0893697
\(851\) −44.8444 −1.53725
\(852\) 5.21110 0.178529
\(853\) 29.2111 1.00017 0.500085 0.865977i \(-0.333302\pi\)
0.500085 + 0.865977i \(0.333302\pi\)
\(854\) −8.36669 −0.286302
\(855\) 2.60555 0.0891080
\(856\) 0 0
\(857\) −13.0278 −0.445020 −0.222510 0.974930i \(-0.571425\pi\)
−0.222510 + 0.974930i \(0.571425\pi\)
\(858\) 0 0
\(859\) 10.7889 0.368112 0.184056 0.982916i \(-0.441077\pi\)
0.184056 + 0.982916i \(0.441077\pi\)
\(860\) −8.00000 −0.272798
\(861\) −29.2111 −0.995512
\(862\) 12.0000 0.408722
\(863\) 8.36669 0.284806 0.142403 0.989809i \(-0.454517\pi\)
0.142403 + 0.989809i \(0.454517\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −16.4222 −0.558372
\(866\) −4.78890 −0.162733
\(867\) 10.2111 0.346787
\(868\) −15.6333 −0.530629
\(869\) 0 0
\(870\) 2.60555 0.0883365
\(871\) 0 0
\(872\) 8.60555 0.291421
\(873\) 8.60555 0.291254
\(874\) 22.4222 0.758442
\(875\) −2.60555 −0.0880837
\(876\) −8.60555 −0.290755
\(877\) 32.8444 1.10908 0.554538 0.832158i \(-0.312895\pi\)
0.554538 + 0.832158i \(0.312895\pi\)
\(878\) −8.00000 −0.269987
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) −24.7889 −0.835159 −0.417580 0.908640i \(-0.637122\pi\)
−0.417580 + 0.908640i \(0.637122\pi\)
\(882\) −0.211103 −0.00710819
\(883\) −38.4222 −1.29301 −0.646505 0.762910i \(-0.723770\pi\)
−0.646505 + 0.762910i \(0.723770\pi\)
\(884\) 0 0
\(885\) 5.21110 0.175169
\(886\) −27.6333 −0.928359
\(887\) 43.0278 1.44473 0.722365 0.691512i \(-0.243055\pi\)
0.722365 + 0.691512i \(0.243055\pi\)
\(888\) 5.21110 0.174873
\(889\) −34.4222 −1.15448
\(890\) −0.788897 −0.0264439
\(891\) 0 0
\(892\) −19.8167 −0.663511
\(893\) −13.5778 −0.454364
\(894\) −0.788897 −0.0263847
\(895\) −1.81665 −0.0607240
\(896\) −2.60555 −0.0870454
\(897\) 0 0
\(898\) 9.63331 0.321468
\(899\) −15.6333 −0.521400
\(900\) 1.00000 0.0333333
\(901\) 15.6333 0.520821
\(902\) 0 0
\(903\) −20.8444 −0.693659
\(904\) 7.81665 0.259978
\(905\) 20.4222 0.678857
\(906\) 6.00000 0.199337
\(907\) −50.4222 −1.67424 −0.837121 0.547018i \(-0.815763\pi\)
−0.837121 + 0.547018i \(0.815763\pi\)
\(908\) 24.0000 0.796468
\(909\) 14.6056 0.484436
\(910\) 0 0
\(911\) 15.6333 0.517955 0.258977 0.965883i \(-0.416615\pi\)
0.258977 + 0.965883i \(0.416615\pi\)
\(912\) −2.60555 −0.0862784
\(913\) 0 0
\(914\) −12.2389 −0.404825
\(915\) −3.21110 −0.106156
\(916\) −1.81665 −0.0600239
\(917\) −40.1110 −1.32458
\(918\) −2.60555 −0.0859960
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 8.60555 0.283717
\(921\) −23.2111 −0.764832
\(922\) 9.63331 0.317256
\(923\) 0 0
\(924\) 0 0
\(925\) −5.21110 −0.171340
\(926\) 38.6056 1.26866
\(927\) 4.00000 0.131377
\(928\) −2.60555 −0.0855314
\(929\) 24.7889 0.813297 0.406649 0.913585i \(-0.366697\pi\)
0.406649 + 0.913585i \(0.366697\pi\)
\(930\) −6.00000 −0.196748
\(931\) −0.550039 −0.0180268
\(932\) −19.8167 −0.649116
\(933\) 12.0000 0.392862
\(934\) 1.57779 0.0516270
\(935\) 0 0
\(936\) 0 0
\(937\) −53.6333 −1.75212 −0.876062 0.482199i \(-0.839838\pi\)
−0.876062 + 0.482199i \(0.839838\pi\)
\(938\) −29.2111 −0.953776
\(939\) −32.4222 −1.05806
\(940\) −5.21110 −0.169967
\(941\) −54.0000 −1.76035 −0.880175 0.474650i \(-0.842575\pi\)
−0.880175 + 0.474650i \(0.842575\pi\)
\(942\) 8.42221 0.274410
\(943\) −96.4777 −3.14175
\(944\) −5.21110 −0.169607
\(945\) 2.60555 0.0847586
\(946\) 0 0
\(947\) −27.6333 −0.897962 −0.448981 0.893541i \(-0.648213\pi\)
−0.448981 + 0.893541i \(0.648213\pi\)
\(948\) −14.4222 −0.468411
\(949\) 0 0
\(950\) 2.60555 0.0845352
\(951\) 18.0000 0.583690
\(952\) −6.78890 −0.220029
\(953\) −30.2389 −0.979533 −0.489766 0.871854i \(-0.662918\pi\)
−0.489766 + 0.871854i \(0.662918\pi\)
\(954\) 6.00000 0.194257
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) −34.4222 −1.11213
\(959\) −29.2111 −0.943276
\(960\) −1.00000 −0.0322749
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) −22.4222 −0.722171
\(965\) 13.8167 0.444774
\(966\) 22.4222 0.721423
\(967\) 42.2389 1.35831 0.679155 0.733995i \(-0.262346\pi\)
0.679155 + 0.733995i \(0.262346\pi\)
\(968\) −11.0000 −0.353553
\(969\) −6.78890 −0.218091
\(970\) 8.60555 0.276308
\(971\) −16.9722 −0.544665 −0.272333 0.962203i \(-0.587795\pi\)
−0.272333 + 0.962203i \(0.587795\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −7.26662 −0.232957
\(974\) −37.0278 −1.18645
\(975\) 0 0
\(976\) 3.21110 0.102785
\(977\) −38.8444 −1.24274 −0.621371 0.783516i \(-0.713424\pi\)
−0.621371 + 0.783516i \(0.713424\pi\)
\(978\) −4.42221 −0.141407
\(979\) 0 0
\(980\) −0.211103 −0.00674342
\(981\) 8.60555 0.274754
\(982\) 13.8167 0.440907
\(983\) 13.5778 0.433064 0.216532 0.976275i \(-0.430525\pi\)
0.216532 + 0.976275i \(0.430525\pi\)
\(984\) 11.2111 0.357397
\(985\) −6.00000 −0.191176
\(986\) −6.78890 −0.216203
\(987\) −13.5778 −0.432186
\(988\) 0 0
\(989\) −68.8444 −2.18912
\(990\) 0 0
\(991\) 6.42221 0.204008 0.102004 0.994784i \(-0.467475\pi\)
0.102004 + 0.994784i \(0.467475\pi\)
\(992\) 6.00000 0.190500
\(993\) 9.39445 0.298124
\(994\) 13.5778 0.430662
\(995\) 6.42221 0.203598
\(996\) −17.2111 −0.545355
\(997\) 12.4222 0.393415 0.196708 0.980462i \(-0.436975\pi\)
0.196708 + 0.980462i \(0.436975\pi\)
\(998\) −13.0278 −0.412386
\(999\) 5.21110 0.164872
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 5070.2.a.bf.1.1 2
13.5 odd 4 390.2.b.c.181.1 4
13.8 odd 4 390.2.b.c.181.4 yes 4
13.12 even 2 5070.2.a.z.1.2 2
39.5 even 4 1170.2.b.d.181.3 4
39.8 even 4 1170.2.b.d.181.2 4
52.31 even 4 3120.2.g.q.961.2 4
52.47 even 4 3120.2.g.q.961.3 4
65.8 even 4 1950.2.f.n.649.2 4
65.18 even 4 1950.2.f.m.649.1 4
65.34 odd 4 1950.2.b.k.1351.1 4
65.44 odd 4 1950.2.b.k.1351.4 4
65.47 even 4 1950.2.f.m.649.3 4
65.57 even 4 1950.2.f.n.649.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.1 4 13.5 odd 4
390.2.b.c.181.4 yes 4 13.8 odd 4
1170.2.b.d.181.2 4 39.8 even 4
1170.2.b.d.181.3 4 39.5 even 4
1950.2.b.k.1351.1 4 65.34 odd 4
1950.2.b.k.1351.4 4 65.44 odd 4
1950.2.f.m.649.1 4 65.18 even 4
1950.2.f.m.649.3 4 65.47 even 4
1950.2.f.n.649.2 4 65.8 even 4
1950.2.f.n.649.4 4 65.57 even 4
3120.2.g.q.961.2 4 52.31 even 4
3120.2.g.q.961.3 4 52.47 even 4
5070.2.a.z.1.2 2 13.12 even 2
5070.2.a.bf.1.1 2 1.1 even 1 trivial