Properties

Label 8085.2.a.ba.1.2
Level $8085$
Weight $2$
Character 8085.1
Self dual yes
Analytic conductor $64.559$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8085.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.5590500342\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8085.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +1.00000 q^{5} +0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +1.00000 q^{5} +0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +0.414214 q^{10} -1.00000 q^{11} -1.82843 q^{12} -5.65685 q^{13} +1.00000 q^{15} +3.00000 q^{16} +6.82843 q^{17} +0.414214 q^{18} +1.17157 q^{19} -1.82843 q^{20} -0.414214 q^{22} -4.00000 q^{23} -1.58579 q^{24} +1.00000 q^{25} -2.34315 q^{26} +1.00000 q^{27} +0.828427 q^{29} +0.414214 q^{30} +4.41421 q^{32} -1.00000 q^{33} +2.82843 q^{34} -1.82843 q^{36} +0.343146 q^{37} +0.485281 q^{38} -5.65685 q^{39} -1.58579 q^{40} +0.828427 q^{41} -3.17157 q^{43} +1.82843 q^{44} +1.00000 q^{45} -1.65685 q^{46} +4.00000 q^{47} +3.00000 q^{48} +0.414214 q^{50} +6.82843 q^{51} +10.3431 q^{52} -13.3137 q^{53} +0.414214 q^{54} -1.00000 q^{55} +1.17157 q^{57} +0.343146 q^{58} +4.00000 q^{59} -1.82843 q^{60} +0.343146 q^{61} -4.17157 q^{64} -5.65685 q^{65} -0.414214 q^{66} +5.65685 q^{67} -12.4853 q^{68} -4.00000 q^{69} +13.6569 q^{71} -1.58579 q^{72} +11.3137 q^{73} +0.142136 q^{74} +1.00000 q^{75} -2.14214 q^{76} -2.34315 q^{78} -8.48528 q^{79} +3.00000 q^{80} +1.00000 q^{81} +0.343146 q^{82} +10.0000 q^{83} +6.82843 q^{85} -1.31371 q^{86} +0.828427 q^{87} +1.58579 q^{88} +7.65685 q^{89} +0.414214 q^{90} +7.31371 q^{92} +1.65685 q^{94} +1.17157 q^{95} +4.41421 q^{96} -0.343146 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 6q^{8} + 2q^{9} - 2q^{10} - 2q^{11} + 2q^{12} + 2q^{15} + 6q^{16} + 8q^{17} - 2q^{18} + 8q^{19} + 2q^{20} + 2q^{22} - 8q^{23} - 6q^{24} + 2q^{25} - 16q^{26} + 2q^{27} - 4q^{29} - 2q^{30} + 6q^{32} - 2q^{33} + 2q^{36} + 12q^{37} - 16q^{38} - 6q^{40} - 4q^{41} - 12q^{43} - 2q^{44} + 2q^{45} + 8q^{46} + 8q^{47} + 6q^{48} - 2q^{50} + 8q^{51} + 32q^{52} - 4q^{53} - 2q^{54} - 2q^{55} + 8q^{57} + 12q^{58} + 8q^{59} + 2q^{60} + 12q^{61} - 14q^{64} + 2q^{66} - 8q^{68} - 8q^{69} + 16q^{71} - 6q^{72} - 28q^{74} + 2q^{75} + 24q^{76} - 16q^{78} + 6q^{80} + 2q^{81} + 12q^{82} + 20q^{83} + 8q^{85} + 20q^{86} - 4q^{87} + 6q^{88} + 4q^{89} - 2q^{90} - 8q^{92} - 8q^{94} + 8q^{95} + 6q^{96} - 12q^{97} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.82843 −0.914214
\(5\) 1.00000 0.447214
\(6\) 0.414214 0.169102
\(7\) 0 0
\(8\) −1.58579 −0.560660
\(9\) 1.00000 0.333333
\(10\) 0.414214 0.130986
\(11\) −1.00000 −0.301511
\(12\) −1.82843 −0.527821
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 3.00000 0.750000
\(17\) 6.82843 1.65614 0.828068 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) 0.414214 0.0976311
\(19\) 1.17157 0.268777 0.134389 0.990929i \(-0.457093\pi\)
0.134389 + 0.990929i \(0.457093\pi\)
\(20\) −1.82843 −0.408849
\(21\) 0 0
\(22\) −0.414214 −0.0883106
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.58579 −0.323697
\(25\) 1.00000 0.200000
\(26\) −2.34315 −0.459529
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.828427 0.153835 0.0769175 0.997037i \(-0.475492\pi\)
0.0769175 + 0.997037i \(0.475492\pi\)
\(30\) 0.414214 0.0756247
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 4.41421 0.780330
\(33\) −1.00000 −0.174078
\(34\) 2.82843 0.485071
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) 0.343146 0.0564128 0.0282064 0.999602i \(-0.491020\pi\)
0.0282064 + 0.999602i \(0.491020\pi\)
\(38\) 0.485281 0.0787230
\(39\) −5.65685 −0.905822
\(40\) −1.58579 −0.250735
\(41\) 0.828427 0.129379 0.0646893 0.997905i \(-0.479394\pi\)
0.0646893 + 0.997905i \(0.479394\pi\)
\(42\) 0 0
\(43\) −3.17157 −0.483660 −0.241830 0.970319i \(-0.577748\pi\)
−0.241830 + 0.970319i \(0.577748\pi\)
\(44\) 1.82843 0.275646
\(45\) 1.00000 0.149071
\(46\) −1.65685 −0.244290
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 3.00000 0.433013
\(49\) 0 0
\(50\) 0.414214 0.0585786
\(51\) 6.82843 0.956171
\(52\) 10.3431 1.43434
\(53\) −13.3137 −1.82878 −0.914389 0.404836i \(-0.867329\pi\)
−0.914389 + 0.404836i \(0.867329\pi\)
\(54\) 0.414214 0.0563673
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 1.17157 0.155179
\(58\) 0.343146 0.0450572
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −1.82843 −0.236049
\(61\) 0.343146 0.0439353 0.0219677 0.999759i \(-0.493007\pi\)
0.0219677 + 0.999759i \(0.493007\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) −5.65685 −0.701646
\(66\) −0.414214 −0.0509862
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) −12.4853 −1.51406
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 13.6569 1.62077 0.810385 0.585897i \(-0.199258\pi\)
0.810385 + 0.585897i \(0.199258\pi\)
\(72\) −1.58579 −0.186887
\(73\) 11.3137 1.32417 0.662085 0.749429i \(-0.269672\pi\)
0.662085 + 0.749429i \(0.269672\pi\)
\(74\) 0.142136 0.0165229
\(75\) 1.00000 0.115470
\(76\) −2.14214 −0.245720
\(77\) 0 0
\(78\) −2.34315 −0.265309
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) 0.343146 0.0378941
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 0 0
\(85\) 6.82843 0.740647
\(86\) −1.31371 −0.141661
\(87\) 0.828427 0.0888167
\(88\) 1.58579 0.169045
\(89\) 7.65685 0.811625 0.405812 0.913956i \(-0.366989\pi\)
0.405812 + 0.913956i \(0.366989\pi\)
\(90\) 0.414214 0.0436619
\(91\) 0 0
\(92\) 7.31371 0.762507
\(93\) 0 0
\(94\) 1.65685 0.170891
\(95\) 1.17157 0.120201
\(96\) 4.41421 0.450524
\(97\) −0.343146 −0.0348412 −0.0174206 0.999848i \(-0.505545\pi\)
−0.0174206 + 0.999848i \(0.505545\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −1.82843 −0.182843
\(101\) −4.82843 −0.480446 −0.240223 0.970718i \(-0.577221\pi\)
−0.240223 + 0.970718i \(0.577221\pi\)
\(102\) 2.82843 0.280056
\(103\) −19.3137 −1.90304 −0.951518 0.307593i \(-0.900477\pi\)
−0.951518 + 0.307593i \(0.900477\pi\)
\(104\) 8.97056 0.879636
\(105\) 0 0
\(106\) −5.51472 −0.535637
\(107\) −5.31371 −0.513696 −0.256848 0.966452i \(-0.582684\pi\)
−0.256848 + 0.966452i \(0.582684\pi\)
\(108\) −1.82843 −0.175940
\(109\) −5.31371 −0.508961 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(110\) −0.414214 −0.0394937
\(111\) 0.343146 0.0325700
\(112\) 0 0
\(113\) 14.9706 1.40831 0.704156 0.710045i \(-0.251326\pi\)
0.704156 + 0.710045i \(0.251326\pi\)
\(114\) 0.485281 0.0454508
\(115\) −4.00000 −0.373002
\(116\) −1.51472 −0.140638
\(117\) −5.65685 −0.522976
\(118\) 1.65685 0.152526
\(119\) 0 0
\(120\) −1.58579 −0.144762
\(121\) 1.00000 0.0909091
\(122\) 0.142136 0.0128684
\(123\) 0.828427 0.0746968
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 2.48528 0.220533 0.110267 0.993902i \(-0.464830\pi\)
0.110267 + 0.993902i \(0.464830\pi\)
\(128\) −10.5563 −0.933058
\(129\) −3.17157 −0.279241
\(130\) −2.34315 −0.205507
\(131\) 19.3137 1.68745 0.843723 0.536778i \(-0.180359\pi\)
0.843723 + 0.536778i \(0.180359\pi\)
\(132\) 1.82843 0.159144
\(133\) 0 0
\(134\) 2.34315 0.202417
\(135\) 1.00000 0.0860663
\(136\) −10.8284 −0.928530
\(137\) 9.31371 0.795724 0.397862 0.917445i \(-0.369752\pi\)
0.397862 + 0.917445i \(0.369752\pi\)
\(138\) −1.65685 −0.141041
\(139\) 16.4853 1.39826 0.699132 0.714993i \(-0.253570\pi\)
0.699132 + 0.714993i \(0.253570\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 5.65685 0.474713
\(143\) 5.65685 0.473050
\(144\) 3.00000 0.250000
\(145\) 0.828427 0.0687971
\(146\) 4.68629 0.387840
\(147\) 0 0
\(148\) −0.627417 −0.0515734
\(149\) 18.4853 1.51437 0.757187 0.653199i \(-0.226573\pi\)
0.757187 + 0.653199i \(0.226573\pi\)
\(150\) 0.414214 0.0338204
\(151\) −0.485281 −0.0394916 −0.0197458 0.999805i \(-0.506286\pi\)
−0.0197458 + 0.999805i \(0.506286\pi\)
\(152\) −1.85786 −0.150693
\(153\) 6.82843 0.552046
\(154\) 0 0
\(155\) 0 0
\(156\) 10.3431 0.828114
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −3.51472 −0.279616
\(159\) −13.3137 −1.05585
\(160\) 4.41421 0.348974
\(161\) 0 0
\(162\) 0.414214 0.0325437
\(163\) 15.3137 1.19946 0.599731 0.800202i \(-0.295274\pi\)
0.599731 + 0.800202i \(0.295274\pi\)
\(164\) −1.51472 −0.118280
\(165\) −1.00000 −0.0778499
\(166\) 4.14214 0.321492
\(167\) −9.31371 −0.720716 −0.360358 0.932814i \(-0.617346\pi\)
−0.360358 + 0.932814i \(0.617346\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 2.82843 0.216930
\(171\) 1.17157 0.0895924
\(172\) 5.79899 0.442169
\(173\) 2.82843 0.215041 0.107521 0.994203i \(-0.465709\pi\)
0.107521 + 0.994203i \(0.465709\pi\)
\(174\) 0.343146 0.0260138
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 4.00000 0.300658
\(178\) 3.17157 0.237719
\(179\) −6.34315 −0.474109 −0.237054 0.971496i \(-0.576182\pi\)
−0.237054 + 0.971496i \(0.576182\pi\)
\(180\) −1.82843 −0.136283
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0.343146 0.0253661
\(184\) 6.34315 0.467623
\(185\) 0.343146 0.0252286
\(186\) 0 0
\(187\) −6.82843 −0.499344
\(188\) −7.31371 −0.533407
\(189\) 0 0
\(190\) 0.485281 0.0352060
\(191\) −5.65685 −0.409316 −0.204658 0.978834i \(-0.565608\pi\)
−0.204658 + 0.978834i \(0.565608\pi\)
\(192\) −4.17157 −0.301057
\(193\) 2.34315 0.168663 0.0843317 0.996438i \(-0.473124\pi\)
0.0843317 + 0.996438i \(0.473124\pi\)
\(194\) −0.142136 −0.0102047
\(195\) −5.65685 −0.405096
\(196\) 0 0
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) −0.414214 −0.0294369
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) −1.58579 −0.112132
\(201\) 5.65685 0.399004
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) −12.4853 −0.874145
\(205\) 0.828427 0.0578599
\(206\) −8.00000 −0.557386
\(207\) −4.00000 −0.278019
\(208\) −16.9706 −1.17670
\(209\) −1.17157 −0.0810394
\(210\) 0 0
\(211\) 6.82843 0.470088 0.235044 0.971985i \(-0.424477\pi\)
0.235044 + 0.971985i \(0.424477\pi\)
\(212\) 24.3431 1.67189
\(213\) 13.6569 0.935752
\(214\) −2.20101 −0.150458
\(215\) −3.17157 −0.216299
\(216\) −1.58579 −0.107899
\(217\) 0 0
\(218\) −2.20101 −0.149071
\(219\) 11.3137 0.764510
\(220\) 1.82843 0.123273
\(221\) −38.6274 −2.59836
\(222\) 0.142136 0.00953952
\(223\) 17.6569 1.18239 0.591195 0.806529i \(-0.298656\pi\)
0.591195 + 0.806529i \(0.298656\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 6.20101 0.412485
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) −2.14214 −0.141866
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) −1.65685 −0.109250
\(231\) 0 0
\(232\) −1.31371 −0.0862492
\(233\) −13.1716 −0.862898 −0.431449 0.902137i \(-0.641998\pi\)
−0.431449 + 0.902137i \(0.641998\pi\)
\(234\) −2.34315 −0.153176
\(235\) 4.00000 0.260931
\(236\) −7.31371 −0.476082
\(237\) −8.48528 −0.551178
\(238\) 0 0
\(239\) 6.34315 0.410304 0.205152 0.978730i \(-0.434231\pi\)
0.205152 + 0.978730i \(0.434231\pi\)
\(240\) 3.00000 0.193649
\(241\) 23.6569 1.52387 0.761936 0.647652i \(-0.224249\pi\)
0.761936 + 0.647652i \(0.224249\pi\)
\(242\) 0.414214 0.0266267
\(243\) 1.00000 0.0641500
\(244\) −0.627417 −0.0401663
\(245\) 0 0
\(246\) 0.343146 0.0218782
\(247\) −6.62742 −0.421692
\(248\) 0 0
\(249\) 10.0000 0.633724
\(250\) 0.414214 0.0261972
\(251\) 12.9706 0.818695 0.409347 0.912379i \(-0.365756\pi\)
0.409347 + 0.912379i \(0.365756\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 1.02944 0.0645926
\(255\) 6.82843 0.427613
\(256\) 3.97056 0.248160
\(257\) 27.6569 1.72519 0.862594 0.505898i \(-0.168839\pi\)
0.862594 + 0.505898i \(0.168839\pi\)
\(258\) −1.31371 −0.0817879
\(259\) 0 0
\(260\) 10.3431 0.641455
\(261\) 0.828427 0.0512784
\(262\) 8.00000 0.494242
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 1.58579 0.0975984
\(265\) −13.3137 −0.817855
\(266\) 0 0
\(267\) 7.65685 0.468592
\(268\) −10.3431 −0.631808
\(269\) 24.6274 1.50156 0.750780 0.660552i \(-0.229678\pi\)
0.750780 + 0.660552i \(0.229678\pi\)
\(270\) 0.414214 0.0252082
\(271\) −27.7990 −1.68867 −0.844334 0.535817i \(-0.820004\pi\)
−0.844334 + 0.535817i \(0.820004\pi\)
\(272\) 20.4853 1.24210
\(273\) 0 0
\(274\) 3.85786 0.233062
\(275\) −1.00000 −0.0603023
\(276\) 7.31371 0.440234
\(277\) −13.6569 −0.820561 −0.410280 0.911959i \(-0.634569\pi\)
−0.410280 + 0.911959i \(0.634569\pi\)
\(278\) 6.82843 0.409542
\(279\) 0 0
\(280\) 0 0
\(281\) −16.8284 −1.00390 −0.501950 0.864897i \(-0.667384\pi\)
−0.501950 + 0.864897i \(0.667384\pi\)
\(282\) 1.65685 0.0986642
\(283\) 3.17157 0.188530 0.0942652 0.995547i \(-0.469950\pi\)
0.0942652 + 0.995547i \(0.469950\pi\)
\(284\) −24.9706 −1.48173
\(285\) 1.17157 0.0693980
\(286\) 2.34315 0.138553
\(287\) 0 0
\(288\) 4.41421 0.260110
\(289\) 29.6274 1.74279
\(290\) 0.343146 0.0201502
\(291\) −0.343146 −0.0201156
\(292\) −20.6863 −1.21057
\(293\) 1.17157 0.0684440 0.0342220 0.999414i \(-0.489105\pi\)
0.0342220 + 0.999414i \(0.489105\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) −0.544156 −0.0316284
\(297\) −1.00000 −0.0580259
\(298\) 7.65685 0.443550
\(299\) 22.6274 1.30858
\(300\) −1.82843 −0.105564
\(301\) 0 0
\(302\) −0.201010 −0.0115668
\(303\) −4.82843 −0.277386
\(304\) 3.51472 0.201583
\(305\) 0.343146 0.0196485
\(306\) 2.82843 0.161690
\(307\) 8.82843 0.503865 0.251932 0.967745i \(-0.418934\pi\)
0.251932 + 0.967745i \(0.418934\pi\)
\(308\) 0 0
\(309\) −19.3137 −1.09872
\(310\) 0 0
\(311\) −19.3137 −1.09518 −0.547590 0.836747i \(-0.684455\pi\)
−0.547590 + 0.836747i \(0.684455\pi\)
\(312\) 8.97056 0.507858
\(313\) −4.34315 −0.245489 −0.122745 0.992438i \(-0.539170\pi\)
−0.122745 + 0.992438i \(0.539170\pi\)
\(314\) −7.45584 −0.420758
\(315\) 0 0
\(316\) 15.5147 0.872771
\(317\) 30.2843 1.70093 0.850467 0.526028i \(-0.176319\pi\)
0.850467 + 0.526028i \(0.176319\pi\)
\(318\) −5.51472 −0.309250
\(319\) −0.828427 −0.0463830
\(320\) −4.17157 −0.233198
\(321\) −5.31371 −0.296582
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) −1.82843 −0.101579
\(325\) −5.65685 −0.313786
\(326\) 6.34315 0.351314
\(327\) −5.31371 −0.293849
\(328\) −1.31371 −0.0725374
\(329\) 0 0
\(330\) −0.414214 −0.0228017
\(331\) 17.6569 0.970508 0.485254 0.874373i \(-0.338727\pi\)
0.485254 + 0.874373i \(0.338727\pi\)
\(332\) −18.2843 −1.00348
\(333\) 0.343146 0.0188043
\(334\) −3.85786 −0.211093
\(335\) 5.65685 0.309067
\(336\) 0 0
\(337\) −19.3137 −1.05208 −0.526042 0.850458i \(-0.676325\pi\)
−0.526042 + 0.850458i \(0.676325\pi\)
\(338\) 7.87006 0.428075
\(339\) 14.9706 0.813089
\(340\) −12.4853 −0.677109
\(341\) 0 0
\(342\) 0.485281 0.0262410
\(343\) 0 0
\(344\) 5.02944 0.271169
\(345\) −4.00000 −0.215353
\(346\) 1.17157 0.0629841
\(347\) −6.68629 −0.358939 −0.179469 0.983764i \(-0.557438\pi\)
−0.179469 + 0.983764i \(0.557438\pi\)
\(348\) −1.51472 −0.0811974
\(349\) 22.9706 1.22959 0.614793 0.788688i \(-0.289240\pi\)
0.614793 + 0.788688i \(0.289240\pi\)
\(350\) 0 0
\(351\) −5.65685 −0.301941
\(352\) −4.41421 −0.235278
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 1.65685 0.0880608
\(355\) 13.6569 0.724831
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −2.62742 −0.138863
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) −1.58579 −0.0835783
\(361\) −17.6274 −0.927759
\(362\) 5.79899 0.304788
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 11.3137 0.592187
\(366\) 0.142136 0.00742955
\(367\) 1.65685 0.0864871 0.0432435 0.999065i \(-0.486231\pi\)
0.0432435 + 0.999065i \(0.486231\pi\)
\(368\) −12.0000 −0.625543
\(369\) 0.828427 0.0431262
\(370\) 0.142136 0.00738928
\(371\) 0 0
\(372\) 0 0
\(373\) −34.6274 −1.79294 −0.896470 0.443105i \(-0.853877\pi\)
−0.896470 + 0.443105i \(0.853877\pi\)
\(374\) −2.82843 −0.146254
\(375\) 1.00000 0.0516398
\(376\) −6.34315 −0.327123
\(377\) −4.68629 −0.241356
\(378\) 0 0
\(379\) −0.686292 −0.0352524 −0.0176262 0.999845i \(-0.505611\pi\)
−0.0176262 + 0.999845i \(0.505611\pi\)
\(380\) −2.14214 −0.109889
\(381\) 2.48528 0.127325
\(382\) −2.34315 −0.119886
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) −10.5563 −0.538701
\(385\) 0 0
\(386\) 0.970563 0.0494003
\(387\) −3.17157 −0.161220
\(388\) 0.627417 0.0318523
\(389\) −12.3431 −0.625822 −0.312911 0.949782i \(-0.601304\pi\)
−0.312911 + 0.949782i \(0.601304\pi\)
\(390\) −2.34315 −0.118650
\(391\) −27.3137 −1.38131
\(392\) 0 0
\(393\) 19.3137 0.974248
\(394\) 3.51472 0.177069
\(395\) −8.48528 −0.426941
\(396\) 1.82843 0.0918819
\(397\) −18.9706 −0.952105 −0.476053 0.879417i \(-0.657933\pi\)
−0.476053 + 0.879417i \(0.657933\pi\)
\(398\) 4.28427 0.214751
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) −29.3137 −1.46386 −0.731928 0.681382i \(-0.761379\pi\)
−0.731928 + 0.681382i \(0.761379\pi\)
\(402\) 2.34315 0.116865
\(403\) 0 0
\(404\) 8.82843 0.439231
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −0.343146 −0.0170091
\(408\) −10.8284 −0.536087
\(409\) 8.34315 0.412542 0.206271 0.978495i \(-0.433867\pi\)
0.206271 + 0.978495i \(0.433867\pi\)
\(410\) 0.343146 0.0169468
\(411\) 9.31371 0.459411
\(412\) 35.3137 1.73978
\(413\) 0 0
\(414\) −1.65685 −0.0814299
\(415\) 10.0000 0.490881
\(416\) −24.9706 −1.22428
\(417\) 16.4853 0.807288
\(418\) −0.485281 −0.0237359
\(419\) 3.02944 0.147998 0.0739988 0.997258i \(-0.476424\pi\)
0.0739988 + 0.997258i \(0.476424\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 2.82843 0.137686
\(423\) 4.00000 0.194487
\(424\) 21.1127 1.02532
\(425\) 6.82843 0.331227
\(426\) 5.65685 0.274075
\(427\) 0 0
\(428\) 9.71573 0.469627
\(429\) 5.65685 0.273115
\(430\) −1.31371 −0.0633526
\(431\) 10.3431 0.498212 0.249106 0.968476i \(-0.419863\pi\)
0.249106 + 0.968476i \(0.419863\pi\)
\(432\) 3.00000 0.144338
\(433\) 4.34315 0.208718 0.104359 0.994540i \(-0.466721\pi\)
0.104359 + 0.994540i \(0.466721\pi\)
\(434\) 0 0
\(435\) 0.828427 0.0397200
\(436\) 9.71573 0.465299
\(437\) −4.68629 −0.224176
\(438\) 4.68629 0.223920
\(439\) 3.51472 0.167748 0.0838742 0.996476i \(-0.473271\pi\)
0.0838742 + 0.996476i \(0.473271\pi\)
\(440\) 1.58579 0.0755994
\(441\) 0 0
\(442\) −16.0000 −0.761042
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −0.627417 −0.0297759
\(445\) 7.65685 0.362970
\(446\) 7.31371 0.346314
\(447\) 18.4853 0.874324
\(448\) 0 0
\(449\) −2.97056 −0.140190 −0.0700948 0.997540i \(-0.522330\pi\)
−0.0700948 + 0.997540i \(0.522330\pi\)
\(450\) 0.414214 0.0195262
\(451\) −0.828427 −0.0390091
\(452\) −27.3726 −1.28750
\(453\) −0.485281 −0.0228005
\(454\) 5.79899 0.272160
\(455\) 0 0
\(456\) −1.85786 −0.0870025
\(457\) −0.686292 −0.0321034 −0.0160517 0.999871i \(-0.505110\pi\)
−0.0160517 + 0.999871i \(0.505110\pi\)
\(458\) 0.828427 0.0387099
\(459\) 6.82843 0.318724
\(460\) 7.31371 0.341003
\(461\) 28.1421 1.31071 0.655355 0.755321i \(-0.272519\pi\)
0.655355 + 0.755321i \(0.272519\pi\)
\(462\) 0 0
\(463\) −28.9706 −1.34638 −0.673188 0.739471i \(-0.735076\pi\)
−0.673188 + 0.739471i \(0.735076\pi\)
\(464\) 2.48528 0.115376
\(465\) 0 0
\(466\) −5.45584 −0.252737
\(467\) −22.6274 −1.04707 −0.523536 0.852004i \(-0.675387\pi\)
−0.523536 + 0.852004i \(0.675387\pi\)
\(468\) 10.3431 0.478112
\(469\) 0 0
\(470\) 1.65685 0.0764250
\(471\) −18.0000 −0.829396
\(472\) −6.34315 −0.291967
\(473\) 3.17157 0.145829
\(474\) −3.51472 −0.161436
\(475\) 1.17157 0.0537555
\(476\) 0 0
\(477\) −13.3137 −0.609593
\(478\) 2.62742 0.120175
\(479\) −3.02944 −0.138419 −0.0692093 0.997602i \(-0.522048\pi\)
−0.0692093 + 0.997602i \(0.522048\pi\)
\(480\) 4.41421 0.201480
\(481\) −1.94113 −0.0885077
\(482\) 9.79899 0.446332
\(483\) 0 0
\(484\) −1.82843 −0.0831103
\(485\) −0.343146 −0.0155814
\(486\) 0.414214 0.0187891
\(487\) 20.9706 0.950267 0.475133 0.879914i \(-0.342400\pi\)
0.475133 + 0.879914i \(0.342400\pi\)
\(488\) −0.544156 −0.0246328
\(489\) 15.3137 0.692510
\(490\) 0 0
\(491\) 25.6569 1.15788 0.578939 0.815371i \(-0.303467\pi\)
0.578939 + 0.815371i \(0.303467\pi\)
\(492\) −1.51472 −0.0682888
\(493\) 5.65685 0.254772
\(494\) −2.74517 −0.123511
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) 0 0
\(498\) 4.14214 0.185614
\(499\) −33.6569 −1.50669 −0.753344 0.657627i \(-0.771560\pi\)
−0.753344 + 0.657627i \(0.771560\pi\)
\(500\) −1.82843 −0.0817697
\(501\) −9.31371 −0.416106
\(502\) 5.37258 0.239790
\(503\) 5.31371 0.236927 0.118463 0.992958i \(-0.462203\pi\)
0.118463 + 0.992958i \(0.462203\pi\)
\(504\) 0 0
\(505\) −4.82843 −0.214862
\(506\) 1.65685 0.0736562
\(507\) 19.0000 0.843820
\(508\) −4.54416 −0.201614
\(509\) −41.3137 −1.83120 −0.915599 0.402093i \(-0.868283\pi\)
−0.915599 + 0.402093i \(0.868283\pi\)
\(510\) 2.82843 0.125245
\(511\) 0 0
\(512\) 22.7574 1.00574
\(513\) 1.17157 0.0517262
\(514\) 11.4558 0.505296
\(515\) −19.3137 −0.851064
\(516\) 5.79899 0.255286
\(517\) −4.00000 −0.175920
\(518\) 0 0
\(519\) 2.82843 0.124154
\(520\) 8.97056 0.393385
\(521\) −12.6274 −0.553217 −0.276609 0.960983i \(-0.589211\pi\)
−0.276609 + 0.960983i \(0.589211\pi\)
\(522\) 0.343146 0.0150191
\(523\) 26.4853 1.15812 0.579060 0.815285i \(-0.303420\pi\)
0.579060 + 0.815285i \(0.303420\pi\)
\(524\) −35.3137 −1.54269
\(525\) 0 0
\(526\) 7.45584 0.325090
\(527\) 0 0
\(528\) −3.00000 −0.130558
\(529\) −7.00000 −0.304348
\(530\) −5.51472 −0.239544
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −4.68629 −0.202986
\(534\) 3.17157 0.137247
\(535\) −5.31371 −0.229732
\(536\) −8.97056 −0.387469
\(537\) −6.34315 −0.273727
\(538\) 10.2010 0.439797
\(539\) 0 0
\(540\) −1.82843 −0.0786830
\(541\) −5.31371 −0.228454 −0.114227 0.993455i \(-0.536439\pi\)
−0.114227 + 0.993455i \(0.536439\pi\)
\(542\) −11.5147 −0.494600
\(543\) 14.0000 0.600798
\(544\) 30.1421 1.29233
\(545\) −5.31371 −0.227614
\(546\) 0 0
\(547\) 20.1421 0.861216 0.430608 0.902539i \(-0.358299\pi\)
0.430608 + 0.902539i \(0.358299\pi\)
\(548\) −17.0294 −0.727462
\(549\) 0.343146 0.0146451
\(550\) −0.414214 −0.0176621
\(551\) 0.970563 0.0413474
\(552\) 6.34315 0.269982
\(553\) 0 0
\(554\) −5.65685 −0.240337
\(555\) 0.343146 0.0145657
\(556\) −30.1421 −1.27831
\(557\) 10.8284 0.458815 0.229408 0.973330i \(-0.426321\pi\)
0.229408 + 0.973330i \(0.426321\pi\)
\(558\) 0 0
\(559\) 17.9411 0.758829
\(560\) 0 0
\(561\) −6.82843 −0.288296
\(562\) −6.97056 −0.294035
\(563\) −20.3431 −0.857361 −0.428681 0.903456i \(-0.641021\pi\)
−0.428681 + 0.903456i \(0.641021\pi\)
\(564\) −7.31371 −0.307963
\(565\) 14.9706 0.629816
\(566\) 1.31371 0.0552193
\(567\) 0 0
\(568\) −21.6569 −0.908701
\(569\) 15.4558 0.647943 0.323971 0.946067i \(-0.394982\pi\)
0.323971 + 0.946067i \(0.394982\pi\)
\(570\) 0.485281 0.0203262
\(571\) 0.485281 0.0203084 0.0101542 0.999948i \(-0.496768\pi\)
0.0101542 + 0.999948i \(0.496768\pi\)
\(572\) −10.3431 −0.432469
\(573\) −5.65685 −0.236318
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) −4.17157 −0.173816
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 12.2721 0.510451
\(579\) 2.34315 0.0973778
\(580\) −1.51472 −0.0628953
\(581\) 0 0
\(582\) −0.142136 −0.00589171
\(583\) 13.3137 0.551397
\(584\) −17.9411 −0.742409
\(585\) −5.65685 −0.233882
\(586\) 0.485281 0.0200468
\(587\) 30.6274 1.26413 0.632064 0.774916i \(-0.282208\pi\)
0.632064 + 0.774916i \(0.282208\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 1.65685 0.0682116
\(591\) 8.48528 0.349038
\(592\) 1.02944 0.0423096
\(593\) 17.1716 0.705152 0.352576 0.935783i \(-0.385306\pi\)
0.352576 + 0.935783i \(0.385306\pi\)
\(594\) −0.414214 −0.0169954
\(595\) 0 0
\(596\) −33.7990 −1.38446
\(597\) 10.3431 0.423317
\(598\) 9.37258 0.383273
\(599\) 4.68629 0.191477 0.0957383 0.995407i \(-0.469479\pi\)
0.0957383 + 0.995407i \(0.469479\pi\)
\(600\) −1.58579 −0.0647395
\(601\) −17.3137 −0.706241 −0.353120 0.935578i \(-0.614879\pi\)
−0.353120 + 0.935578i \(0.614879\pi\)
\(602\) 0 0
\(603\) 5.65685 0.230365
\(604\) 0.887302 0.0361038
\(605\) 1.00000 0.0406558
\(606\) −2.00000 −0.0812444
\(607\) −18.4853 −0.750294 −0.375147 0.926965i \(-0.622408\pi\)
−0.375147 + 0.926965i \(0.622408\pi\)
\(608\) 5.17157 0.209735
\(609\) 0 0
\(610\) 0.142136 0.00575490
\(611\) −22.6274 −0.915407
\(612\) −12.4853 −0.504688
\(613\) 21.9411 0.886194 0.443097 0.896474i \(-0.353880\pi\)
0.443097 + 0.896474i \(0.353880\pi\)
\(614\) 3.65685 0.147579
\(615\) 0.828427 0.0334054
\(616\) 0 0
\(617\) 11.6569 0.469287 0.234644 0.972081i \(-0.424608\pi\)
0.234644 + 0.972081i \(0.424608\pi\)
\(618\) −8.00000 −0.321807
\(619\) 25.6569 1.03124 0.515618 0.856819i \(-0.327562\pi\)
0.515618 + 0.856819i \(0.327562\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −8.00000 −0.320771
\(623\) 0 0
\(624\) −16.9706 −0.679366
\(625\) 1.00000 0.0400000
\(626\) −1.79899 −0.0719021
\(627\) −1.17157 −0.0467881
\(628\) 32.9117 1.31332
\(629\) 2.34315 0.0934273
\(630\) 0 0
\(631\) 34.3431 1.36718 0.683590 0.729867i \(-0.260418\pi\)
0.683590 + 0.729867i \(0.260418\pi\)
\(632\) 13.4558 0.535245
\(633\) 6.82843 0.271406
\(634\) 12.5442 0.498192
\(635\) 2.48528 0.0986254
\(636\) 24.3431 0.965269
\(637\) 0 0
\(638\) −0.343146 −0.0135853
\(639\) 13.6569 0.540257
\(640\) −10.5563 −0.417276
\(641\) −26.9706 −1.06527 −0.532637 0.846344i \(-0.678799\pi\)
−0.532637 + 0.846344i \(0.678799\pi\)
\(642\) −2.20101 −0.0868669
\(643\) 29.9411 1.18076 0.590381 0.807124i \(-0.298977\pi\)
0.590381 + 0.807124i \(0.298977\pi\)
\(644\) 0 0
\(645\) −3.17157 −0.124881
\(646\) 3.31371 0.130376
\(647\) −27.3137 −1.07381 −0.536906 0.843642i \(-0.680407\pi\)
−0.536906 + 0.843642i \(0.680407\pi\)
\(648\) −1.58579 −0.0622956
\(649\) −4.00000 −0.157014
\(650\) −2.34315 −0.0919057
\(651\) 0 0
\(652\) −28.0000 −1.09656
\(653\) −26.9706 −1.05544 −0.527720 0.849418i \(-0.676953\pi\)
−0.527720 + 0.849418i \(0.676953\pi\)
\(654\) −2.20101 −0.0860663
\(655\) 19.3137 0.754649
\(656\) 2.48528 0.0970339
\(657\) 11.3137 0.441390
\(658\) 0 0
\(659\) 7.31371 0.284902 0.142451 0.989802i \(-0.454502\pi\)
0.142451 + 0.989802i \(0.454502\pi\)
\(660\) 1.82843 0.0711714
\(661\) 13.3137 0.517843 0.258922 0.965898i \(-0.416633\pi\)
0.258922 + 0.965898i \(0.416633\pi\)
\(662\) 7.31371 0.284255
\(663\) −38.6274 −1.50016
\(664\) −15.8579 −0.615404
\(665\) 0 0
\(666\) 0.142136 0.00550764
\(667\) −3.31371 −0.128307
\(668\) 17.0294 0.658889
\(669\) 17.6569 0.682653
\(670\) 2.34315 0.0905236
\(671\) −0.343146 −0.0132470
\(672\) 0 0
\(673\) 29.6569 1.14319 0.571594 0.820537i \(-0.306325\pi\)
0.571594 + 0.820537i \(0.306325\pi\)
\(674\) −8.00000 −0.308148
\(675\) 1.00000 0.0384900
\(676\) −34.7401 −1.33616
\(677\) 21.4558 0.824615 0.412308 0.911045i \(-0.364723\pi\)
0.412308 + 0.911045i \(0.364723\pi\)
\(678\) 6.20101 0.238148
\(679\) 0 0
\(680\) −10.8284 −0.415251
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) −2.14214 −0.0819066
\(685\) 9.31371 0.355859
\(686\) 0 0
\(687\) 2.00000 0.0763048
\(688\) −9.51472 −0.362745
\(689\) 75.3137 2.86922
\(690\) −1.65685 −0.0630754
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −5.17157 −0.196594
\(693\) 0 0
\(694\) −2.76955 −0.105131
\(695\) 16.4853 0.625322
\(696\) −1.31371 −0.0497960
\(697\) 5.65685 0.214269
\(698\) 9.51472 0.360137
\(699\) −13.1716 −0.498195
\(700\) 0 0
\(701\) 7.85786 0.296787 0.148394 0.988928i \(-0.452590\pi\)
0.148394 + 0.988928i \(0.452590\pi\)
\(702\) −2.34315 −0.0884363
\(703\) 0.402020 0.0151625
\(704\) 4.17157 0.157222
\(705\) 4.00000 0.150649
\(706\) −10.7696 −0.405317
\(707\) 0 0
\(708\) −7.31371 −0.274866
\(709\) 29.3137 1.10090 0.550450 0.834868i \(-0.314456\pi\)
0.550450 + 0.834868i \(0.314456\pi\)
\(710\) 5.65685 0.212298
\(711\) −8.48528 −0.318223
\(712\) −12.1421 −0.455046
\(713\) 0 0
\(714\) 0 0
\(715\) 5.65685 0.211554
\(716\) 11.5980 0.433437
\(717\) 6.34315 0.236889
\(718\) 4.97056 0.185500
\(719\) −31.5980 −1.17841 −0.589203 0.807985i \(-0.700558\pi\)
−0.589203 + 0.807985i \(0.700558\pi\)
\(720\) 3.00000 0.111803
\(721\) 0 0
\(722\) −7.30152 −0.271734
\(723\) 23.6569 0.879808
\(724\) −25.5980 −0.951341
\(725\) 0.828427 0.0307670
\(726\) 0.414214 0.0153729
\(727\) 33.9411 1.25881 0.629403 0.777079i \(-0.283299\pi\)
0.629403 + 0.777079i \(0.283299\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.68629 0.173447
\(731\) −21.6569 −0.801008
\(732\) −0.627417 −0.0231900
\(733\) 17.6569 0.652171 0.326085 0.945340i \(-0.394270\pi\)
0.326085 + 0.945340i \(0.394270\pi\)
\(734\) 0.686292 0.0253315
\(735\) 0 0
\(736\) −17.6569 −0.650840
\(737\) −5.65685 −0.208373
\(738\) 0.343146 0.0126314
\(739\) 47.1127 1.73307 0.866534 0.499118i \(-0.166342\pi\)
0.866534 + 0.499118i \(0.166342\pi\)
\(740\) −0.627417 −0.0230643
\(741\) −6.62742 −0.243464
\(742\) 0 0
\(743\) 47.6569 1.74836 0.874180 0.485602i \(-0.161399\pi\)
0.874180 + 0.485602i \(0.161399\pi\)
\(744\) 0 0
\(745\) 18.4853 0.677248
\(746\) −14.3431 −0.525140
\(747\) 10.0000 0.365881
\(748\) 12.4853 0.456507
\(749\) 0 0
\(750\) 0.414214 0.0151249
\(751\) −36.2843 −1.32403 −0.662016 0.749490i \(-0.730299\pi\)
−0.662016 + 0.749490i \(0.730299\pi\)
\(752\) 12.0000 0.437595
\(753\) 12.9706 0.472674
\(754\) −1.94113 −0.0706916
\(755\) −0.485281 −0.0176612
\(756\) 0 0
\(757\) 8.62742 0.313569 0.156784 0.987633i \(-0.449887\pi\)
0.156784 + 0.987633i \(0.449887\pi\)
\(758\) −0.284271 −0.0103252
\(759\) 4.00000 0.145191
\(760\) −1.85786 −0.0673918
\(761\) 23.1716 0.839969 0.419984 0.907531i \(-0.362036\pi\)
0.419984 + 0.907531i \(0.362036\pi\)
\(762\) 1.02944 0.0372926
\(763\) 0 0
\(764\) 10.3431 0.374202
\(765\) 6.82843 0.246882
\(766\) 3.31371 0.119729
\(767\) −22.6274 −0.817029
\(768\) 3.97056 0.143275
\(769\) −33.3137 −1.20132 −0.600662 0.799503i \(-0.705096\pi\)
−0.600662 + 0.799503i \(0.705096\pi\)
\(770\) 0 0
\(771\) 27.6569 0.996037
\(772\) −4.28427 −0.154194
\(773\) 7.65685 0.275398 0.137699 0.990474i \(-0.456029\pi\)
0.137699 + 0.990474i \(0.456029\pi\)
\(774\) −1.31371 −0.0472203
\(775\) 0 0
\(776\) 0.544156 0.0195341
\(777\) 0 0
\(778\) −5.11270 −0.183299
\(779\) 0.970563 0.0347740
\(780\) 10.3431 0.370344
\(781\) −13.6569 −0.488681
\(782\) −11.3137 −0.404577
\(783\) 0.828427 0.0296056
\(784\) 0 0
\(785\) −18.0000 −0.642448
\(786\) 8.00000 0.285351
\(787\) −8.14214 −0.290236 −0.145118 0.989414i \(-0.546356\pi\)
−0.145118 + 0.989414i \(0.546356\pi\)
\(788\) −15.5147 −0.552689
\(789\) 18.0000 0.640817
\(790\) −3.51472 −0.125048
\(791\) 0 0
\(792\) 1.58579 0.0563485
\(793\) −1.94113 −0.0689314
\(794\) −7.85786 −0.278865
\(795\) −13.3137 −0.472189
\(796\) −18.9117 −0.670307
\(797\) −1.02944 −0.0364645 −0.0182323 0.999834i \(-0.505804\pi\)
−0.0182323 + 0.999834i \(0.505804\pi\)
\(798\) 0 0
\(799\) 27.3137 0.966290
\(800\) 4.41421 0.156066
\(801\) 7.65685 0.270542
\(802\) −12.1421 −0.428754
\(803\) −11.3137 −0.399252
\(804\) −10.3431 −0.364775
\(805\) 0 0
\(806\) 0 0
\(807\) 24.6274 0.866926
\(808\) 7.65685 0.269367
\(809\) −56.4264 −1.98385 −0.991923 0.126838i \(-0.959517\pi\)
−0.991923 + 0.126838i \(0.959517\pi\)
\(810\) 0.414214 0.0145540
\(811\) 16.4853 0.578877 0.289438 0.957197i \(-0.406532\pi\)
0.289438 + 0.957197i \(0.406532\pi\)
\(812\) 0 0
\(813\) −27.7990 −0.974953
\(814\) −0.142136 −0.00498185
\(815\) 15.3137 0.536416
\(816\) 20.4853 0.717128
\(817\) −3.71573 −0.129997
\(818\) 3.45584 0.120831
\(819\) 0 0
\(820\) −1.51472 −0.0528963
\(821\) −7.17157 −0.250290 −0.125145 0.992138i \(-0.539940\pi\)
−0.125145 + 0.992138i \(0.539940\pi\)
\(822\) 3.85786 0.134558
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 30.6274 1.06696
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 18.6863 0.649786 0.324893 0.945751i \(-0.394672\pi\)
0.324893 + 0.945751i \(0.394672\pi\)
\(828\) 7.31371 0.254169
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 4.14214 0.143776
\(831\) −13.6569 −0.473751
\(832\) 23.5980 0.818113
\(833\) 0 0
\(834\) 6.82843 0.236449
\(835\) −9.31371 −0.322314
\(836\) 2.14214 0.0740873
\(837\) 0 0
\(838\) 1.25483 0.0433475
\(839\) −22.6274 −0.781185 −0.390593 0.920564i \(-0.627730\pi\)
−0.390593 + 0.920564i \(0.627730\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) −2.48528 −0.0856485
\(843\) −16.8284 −0.579602
\(844\) −12.4853 −0.429761
\(845\) 19.0000 0.653620
\(846\) 1.65685 0.0569638
\(847\) 0 0
\(848\) −39.9411 −1.37158
\(849\) 3.17157 0.108848
\(850\) 2.82843 0.0970143
\(851\) −1.37258 −0.0470515
\(852\) −24.9706 −0.855477
\(853\) 31.3137 1.07216 0.536080 0.844167i \(-0.319904\pi\)
0.536080 + 0.844167i \(0.319904\pi\)
\(854\) 0 0
\(855\) 1.17157 0.0400669
\(856\) 8.42641 0.288009
\(857\) 11.5147 0.393335 0.196668 0.980470i \(-0.436988\pi\)
0.196668 + 0.980470i \(0.436988\pi\)
\(858\) 2.34315 0.0799937
\(859\) 19.0294 0.649276 0.324638 0.945838i \(-0.394758\pi\)
0.324638 + 0.945838i \(0.394758\pi\)
\(860\) 5.79899 0.197744
\(861\) 0 0
\(862\) 4.28427 0.145923
\(863\) −43.3137 −1.47442 −0.737208 0.675666i \(-0.763856\pi\)
−0.737208 + 0.675666i \(0.763856\pi\)
\(864\) 4.41421 0.150175
\(865\) 2.82843 0.0961694
\(866\) 1.79899 0.0611322
\(867\) 29.6274 1.00620
\(868\) 0 0
\(869\) 8.48528 0.287843
\(870\) 0.343146 0.0116337
\(871\) −32.0000 −1.08428
\(872\) 8.42641 0.285354
\(873\) −0.343146 −0.0116137
\(874\) −1.94113 −0.0656595
\(875\) 0 0
\(876\) −20.6863 −0.698925
\(877\) −42.6274 −1.43943 −0.719713 0.694272i \(-0.755727\pi\)
−0.719713 + 0.694272i \(0.755727\pi\)
\(878\) 1.45584 0.0491324
\(879\) 1.17157 0.0395162
\(880\) −3.00000 −0.101130
\(881\) 13.0294 0.438973 0.219486 0.975616i \(-0.429562\pi\)
0.219486 + 0.975616i \(0.429562\pi\)
\(882\) 0 0
\(883\) −50.6274 −1.70375 −0.851874 0.523747i \(-0.824534\pi\)
−0.851874 + 0.523747i \(0.824534\pi\)
\(884\) 70.6274 2.37546
\(885\) 4.00000 0.134459
\(886\) 4.97056 0.166989
\(887\) 4.34315 0.145829 0.0729143 0.997338i \(-0.476770\pi\)
0.0729143 + 0.997338i \(0.476770\pi\)
\(888\) −0.544156 −0.0182607
\(889\) 0 0
\(890\) 3.17157 0.106311
\(891\) −1.00000 −0.0335013
\(892\) −32.2843 −1.08096
\(893\) 4.68629 0.156821
\(894\) 7.65685 0.256084
\(895\) −6.34315 −0.212028
\(896\) 0 0
\(897\) 22.6274 0.755507
\(898\) −1.23045 −0.0410606
\(899\) 0 0
\(900\) −1.82843 −0.0609476
\(901\) −90.9117 −3.02871
\(902\) −0.343146 −0.0114255
\(903\) 0 0
\(904\) −23.7401 −0.789584
\(905\) 14.0000 0.465376
\(906\) −0.201010 −0.00667811
\(907\) 7.02944 0.233409 0.116704 0.993167i \(-0.462767\pi\)
0.116704 + 0.993167i \(0.462767\pi\)
\(908\) −25.5980 −0.849499
\(909\) −4.82843 −0.160149
\(910\) 0 0
\(911\) −15.0294 −0.497947 −0.248974 0.968510i \(-0.580093\pi\)
−0.248974 + 0.968510i \(0.580093\pi\)
\(912\) 3.51472 0.116384
\(913\) −10.0000 −0.330952
\(914\) −0.284271 −0.00940286
\(915\) 0.343146 0.0113440
\(916\) −3.65685 −0.120826
\(917\) 0 0
\(918\) 2.82843 0.0933520
\(919\) 28.4853 0.939643 0.469821 0.882762i \(-0.344318\pi\)
0.469821 + 0.882762i \(0.344318\pi\)
\(920\) 6.34315 0.209127
\(921\) 8.82843 0.290907
\(922\) 11.6569 0.383898
\(923\) −77.2548 −2.54287
\(924\) 0 0
\(925\) 0.343146 0.0112826
\(926\) −12.0000 −0.394344
\(927\) −19.3137 −0.634345
\(928\) 3.65685 0.120042
\(929\) −33.5980 −1.10231 −0.551157 0.834402i \(-0.685813\pi\)
−0.551157 + 0.834402i \(0.685813\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 24.0833 0.788873
\(933\) −19.3137 −0.632302
\(934\) −9.37258 −0.306680
\(935\) −6.82843 −0.223313
\(936\) 8.97056 0.293212
\(937\) −44.9706 −1.46912 −0.734562 0.678541i \(-0.762612\pi\)
−0.734562 + 0.678541i \(0.762612\pi\)
\(938\) 0 0
\(939\) −4.34315 −0.141733
\(940\) −7.31371 −0.238547
\(941\) −38.7696 −1.26385 −0.631926 0.775029i \(-0.717735\pi\)
−0.631926 + 0.775029i \(0.717735\pi\)
\(942\) −7.45584 −0.242925
\(943\) −3.31371 −0.107909
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 1.31371 0.0427123
\(947\) −38.6274 −1.25522 −0.627611 0.778527i \(-0.715967\pi\)
−0.627611 + 0.778527i \(0.715967\pi\)
\(948\) 15.5147 0.503895
\(949\) −64.0000 −2.07753
\(950\) 0.485281 0.0157446
\(951\) 30.2843 0.982035
\(952\) 0 0
\(953\) 27.7990 0.900498 0.450249 0.892903i \(-0.351335\pi\)
0.450249 + 0.892903i \(0.351335\pi\)
\(954\) −5.51472 −0.178546
\(955\) −5.65685 −0.183052
\(956\) −11.5980 −0.375105
\(957\) −0.828427 −0.0267792
\(958\) −1.25483 −0.0405418
\(959\) 0 0
\(960\) −4.17157 −0.134637
\(961\) −31.0000 −1.00000
\(962\) −0.804041 −0.0259233
\(963\) −5.31371 −0.171232
\(964\) −43.2548 −1.39314
\(965\) 2.34315 0.0754285
\(966\) 0 0
\(967\) −39.4558 −1.26881 −0.634407 0.772999i \(-0.718756\pi\)
−0.634407 + 0.772999i \(0.718756\pi\)
\(968\) −1.58579 −0.0509691
\(969\) 8.00000 0.256997
\(970\) −0.142136 −0.00456370
\(971\) −10.6274 −0.341050 −0.170525 0.985353i \(-0.554546\pi\)
−0.170525 + 0.985353i \(0.554546\pi\)
\(972\) −1.82843 −0.0586468
\(973\) 0 0
\(974\) 8.68629 0.278327
\(975\) −5.65685 −0.181164
\(976\) 1.02944 0.0329515
\(977\) 25.3137 0.809857 0.404929 0.914348i \(-0.367296\pi\)
0.404929 + 0.914348i \(0.367296\pi\)
\(978\) 6.34315 0.202831
\(979\) −7.65685 −0.244714
\(980\) 0 0
\(981\) −5.31371 −0.169654
\(982\) 10.6274 0.339135
\(983\) −14.6274 −0.466542 −0.233271 0.972412i \(-0.574943\pi\)
−0.233271 + 0.972412i \(0.574943\pi\)
\(984\) −1.31371 −0.0418795
\(985\) 8.48528 0.270364
\(986\) 2.34315 0.0746210
\(987\) 0 0
\(988\) 12.1177 0.385517
\(989\) 12.6863 0.403401
\(990\) −0.414214 −0.0131646
\(991\) 14.6274 0.464655 0.232328 0.972638i \(-0.425366\pi\)
0.232328 + 0.972638i \(0.425366\pi\)
\(992\) 0 0
\(993\) 17.6569 0.560323
\(994\) 0 0
\(995\) 10.3431 0.327900
\(996\) −18.2843 −0.579359
\(997\) 16.6863 0.528460 0.264230 0.964460i \(-0.414882\pi\)
0.264230 + 0.964460i \(0.414882\pi\)
\(998\) −13.9411 −0.441299
\(999\) 0.343146 0.0108567
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 8085.2.a.ba.1.2 2
7.6 odd 2 165.2.a.a.1.2 2
21.20 even 2 495.2.a.d.1.1 2
28.27 even 2 2640.2.a.bb.1.2 2
35.13 even 4 825.2.c.e.199.2 4
35.27 even 4 825.2.c.e.199.3 4
35.34 odd 2 825.2.a.g.1.1 2
77.76 even 2 1815.2.a.k.1.1 2
84.83 odd 2 7920.2.a.cg.1.2 2
105.62 odd 4 2475.2.c.m.199.2 4
105.83 odd 4 2475.2.c.m.199.3 4
105.104 even 2 2475.2.a.m.1.2 2
231.230 odd 2 5445.2.a.m.1.2 2
385.384 even 2 9075.2.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.2 2 7.6 odd 2
495.2.a.d.1.1 2 21.20 even 2
825.2.a.g.1.1 2 35.34 odd 2
825.2.c.e.199.2 4 35.13 even 4
825.2.c.e.199.3 4 35.27 even 4
1815.2.a.k.1.1 2 77.76 even 2
2475.2.a.m.1.2 2 105.104 even 2
2475.2.c.m.199.2 4 105.62 odd 4
2475.2.c.m.199.3 4 105.83 odd 4
2640.2.a.bb.1.2 2 28.27 even 2
5445.2.a.m.1.2 2 231.230 odd 2
7920.2.a.cg.1.2 2 84.83 odd 2
8085.2.a.ba.1.2 2 1.1 even 1 trivial
9075.2.a.v.1.2 2 385.384 even 2