Properties

Label 690.2.a.l.1.1
Level $690$
Weight $2$
Character 690.1
Self dual yes
Analytic conductor $5.510$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 690.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.12311 q^{11} -1.00000 q^{12} +2.00000 q^{13} -5.12311 q^{14} -1.00000 q^{15} +1.00000 q^{16} +7.12311 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +5.12311 q^{21} +5.12311 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -5.12311 q^{28} +2.00000 q^{29} -1.00000 q^{30} +1.00000 q^{32} -5.12311 q^{33} +7.12311 q^{34} -5.12311 q^{35} +1.00000 q^{36} -7.12311 q^{37} +4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} +5.12311 q^{42} +5.12311 q^{44} +1.00000 q^{45} +1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +19.2462 q^{49} +1.00000 q^{50} -7.12311 q^{51} +2.00000 q^{52} -4.24621 q^{53} -1.00000 q^{54} +5.12311 q^{55} -5.12311 q^{56} -4.00000 q^{57} +2.00000 q^{58} -14.2462 q^{59} -1.00000 q^{60} +0.876894 q^{61} -5.12311 q^{63} +1.00000 q^{64} +2.00000 q^{65} -5.12311 q^{66} -8.00000 q^{67} +7.12311 q^{68} -1.00000 q^{69} -5.12311 q^{70} +6.24621 q^{71} +1.00000 q^{72} +12.2462 q^{73} -7.12311 q^{74} -1.00000 q^{75} +4.00000 q^{76} -26.2462 q^{77} -2.00000 q^{78} -5.12311 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +11.3693 q^{83} +5.12311 q^{84} +7.12311 q^{85} -2.00000 q^{87} +5.12311 q^{88} +3.12311 q^{89} +1.00000 q^{90} -10.2462 q^{91} +1.00000 q^{92} -8.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} +0.246211 q^{97} +19.2462 q^{98} +5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + 2q^{10} + 2q^{11} - 2q^{12} + 4q^{13} - 2q^{14} - 2q^{15} + 2q^{16} + 6q^{17} + 2q^{18} + 8q^{19} + 2q^{20} + 2q^{21} + 2q^{22} + 2q^{23} - 2q^{24} + 2q^{25} + 4q^{26} - 2q^{27} - 2q^{28} + 4q^{29} - 2q^{30} + 2q^{32} - 2q^{33} + 6q^{34} - 2q^{35} + 2q^{36} - 6q^{37} + 8q^{38} - 4q^{39} + 2q^{40} + 4q^{41} + 2q^{42} + 2q^{44} + 2q^{45} + 2q^{46} - 16q^{47} - 2q^{48} + 22q^{49} + 2q^{50} - 6q^{51} + 4q^{52} + 8q^{53} - 2q^{54} + 2q^{55} - 2q^{56} - 8q^{57} + 4q^{58} - 12q^{59} - 2q^{60} + 10q^{61} - 2q^{63} + 2q^{64} + 4q^{65} - 2q^{66} - 16q^{67} + 6q^{68} - 2q^{69} - 2q^{70} - 4q^{71} + 2q^{72} + 8q^{73} - 6q^{74} - 2q^{75} + 8q^{76} - 36q^{77} - 4q^{78} - 2q^{79} + 2q^{80} + 2q^{81} + 4q^{82} - 2q^{83} + 2q^{84} + 6q^{85} - 4q^{87} + 2q^{88} - 2q^{89} + 2q^{90} - 4q^{91} + 2q^{92} - 16q^{94} + 8q^{95} - 2q^{96} - 16q^{97} + 22q^{98} + 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) −5.12311 −1.93635 −0.968176 0.250270i \(-0.919480\pi\)
−0.968176 + 0.250270i \(0.919480\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 5.12311 1.54467 0.772337 0.635213i \(-0.219088\pi\)
0.772337 + 0.635213i \(0.219088\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −5.12311 −1.36921
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 7.12311 1.72761 0.863803 0.503829i \(-0.168076\pi\)
0.863803 + 0.503829i \(0.168076\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 5.12311 1.11795
\(22\) 5.12311 1.09225
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) −5.12311 −0.968176
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.12311 −0.891818
\(34\) 7.12311 1.22160
\(35\) −5.12311 −0.865963
\(36\) 1.00000 0.166667
\(37\) −7.12311 −1.17103 −0.585516 0.810661i \(-0.699108\pi\)
−0.585516 + 0.810661i \(0.699108\pi\)
\(38\) 4.00000 0.648886
\(39\) −2.00000 −0.320256
\(40\) 1.00000 0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 5.12311 0.790512
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 5.12311 0.772337
\(45\) 1.00000 0.149071
\(46\) 1.00000 0.147442
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 19.2462 2.74946
\(50\) 1.00000 0.141421
\(51\) −7.12311 −0.997434
\(52\) 2.00000 0.277350
\(53\) −4.24621 −0.583262 −0.291631 0.956531i \(-0.594198\pi\)
−0.291631 + 0.956531i \(0.594198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 5.12311 0.690799
\(56\) −5.12311 −0.684604
\(57\) −4.00000 −0.529813
\(58\) 2.00000 0.262613
\(59\) −14.2462 −1.85470 −0.927349 0.374197i \(-0.877918\pi\)
−0.927349 + 0.374197i \(0.877918\pi\)
\(60\) −1.00000 −0.129099
\(61\) 0.876894 0.112275 0.0561374 0.998423i \(-0.482122\pi\)
0.0561374 + 0.998423i \(0.482122\pi\)
\(62\) 0 0
\(63\) −5.12311 −0.645451
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) −5.12311 −0.630611
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 7.12311 0.863803
\(69\) −1.00000 −0.120386
\(70\) −5.12311 −0.612328
\(71\) 6.24621 0.741289 0.370644 0.928775i \(-0.379137\pi\)
0.370644 + 0.928775i \(0.379137\pi\)
\(72\) 1.00000 0.117851
\(73\) 12.2462 1.43331 0.716655 0.697428i \(-0.245672\pi\)
0.716655 + 0.697428i \(0.245672\pi\)
\(74\) −7.12311 −0.828044
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) −26.2462 −2.99103
\(78\) −2.00000 −0.226455
\(79\) −5.12311 −0.576394 −0.288197 0.957571i \(-0.593056\pi\)
−0.288197 + 0.957571i \(0.593056\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 11.3693 1.24794 0.623972 0.781446i \(-0.285518\pi\)
0.623972 + 0.781446i \(0.285518\pi\)
\(84\) 5.12311 0.558977
\(85\) 7.12311 0.772609
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 5.12311 0.546125
\(89\) 3.12311 0.331049 0.165524 0.986206i \(-0.447068\pi\)
0.165524 + 0.986206i \(0.447068\pi\)
\(90\) 1.00000 0.105409
\(91\) −10.2462 −1.07409
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 4.00000 0.410391
\(96\) −1.00000 −0.102062
\(97\) 0.246211 0.0249990 0.0124995 0.999922i \(-0.496021\pi\)
0.0124995 + 0.999922i \(0.496021\pi\)
\(98\) 19.2462 1.94416
\(99\) 5.12311 0.514891
\(100\) 1.00000 0.100000
\(101\) −16.2462 −1.61656 −0.808279 0.588799i \(-0.799601\pi\)
−0.808279 + 0.588799i \(0.799601\pi\)
\(102\) −7.12311 −0.705293
\(103\) 15.3693 1.51438 0.757192 0.653192i \(-0.226571\pi\)
0.757192 + 0.653192i \(0.226571\pi\)
\(104\) 2.00000 0.196116
\(105\) 5.12311 0.499964
\(106\) −4.24621 −0.412428
\(107\) 11.3693 1.09911 0.549557 0.835456i \(-0.314797\pi\)
0.549557 + 0.835456i \(0.314797\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 11.1231 1.06540 0.532700 0.846304i \(-0.321177\pi\)
0.532700 + 0.846304i \(0.321177\pi\)
\(110\) 5.12311 0.488469
\(111\) 7.12311 0.676095
\(112\) −5.12311 −0.484088
\(113\) −0.876894 −0.0824913 −0.0412456 0.999149i \(-0.513133\pi\)
−0.0412456 + 0.999149i \(0.513133\pi\)
\(114\) −4.00000 −0.374634
\(115\) 1.00000 0.0932505
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) −14.2462 −1.31147
\(119\) −36.4924 −3.34525
\(120\) −1.00000 −0.0912871
\(121\) 15.2462 1.38602
\(122\) 0.876894 0.0793903
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) −5.12311 −0.456403
\(127\) −22.2462 −1.97403 −0.987016 0.160622i \(-0.948650\pi\)
−0.987016 + 0.160622i \(0.948650\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −5.12311 −0.445909
\(133\) −20.4924 −1.77692
\(134\) −8.00000 −0.691095
\(135\) −1.00000 −0.0860663
\(136\) 7.12311 0.610801
\(137\) −19.1231 −1.63380 −0.816899 0.576781i \(-0.804308\pi\)
−0.816899 + 0.576781i \(0.804308\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −16.4924 −1.39887 −0.699435 0.714697i \(-0.746565\pi\)
−0.699435 + 0.714697i \(0.746565\pi\)
\(140\) −5.12311 −0.432981
\(141\) 8.00000 0.673722
\(142\) 6.24621 0.524170
\(143\) 10.2462 0.856831
\(144\) 1.00000 0.0833333
\(145\) 2.00000 0.166091
\(146\) 12.2462 1.01350
\(147\) −19.2462 −1.58740
\(148\) −7.12311 −0.585516
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −10.2462 −0.833825 −0.416912 0.908947i \(-0.636888\pi\)
−0.416912 + 0.908947i \(0.636888\pi\)
\(152\) 4.00000 0.324443
\(153\) 7.12311 0.575869
\(154\) −26.2462 −2.11498
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 0.876894 0.0699838 0.0349919 0.999388i \(-0.488859\pi\)
0.0349919 + 0.999388i \(0.488859\pi\)
\(158\) −5.12311 −0.407572
\(159\) 4.24621 0.336746
\(160\) 1.00000 0.0790569
\(161\) −5.12311 −0.403757
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 2.00000 0.156174
\(165\) −5.12311 −0.398833
\(166\) 11.3693 0.882430
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 5.12311 0.395256
\(169\) −9.00000 −0.692308
\(170\) 7.12311 0.546317
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 3.75379 0.285395 0.142698 0.989766i \(-0.454422\pi\)
0.142698 + 0.989766i \(0.454422\pi\)
\(174\) −2.00000 −0.151620
\(175\) −5.12311 −0.387270
\(176\) 5.12311 0.386169
\(177\) 14.2462 1.07081
\(178\) 3.12311 0.234087
\(179\) 24.4924 1.83065 0.915325 0.402716i \(-0.131934\pi\)
0.915325 + 0.402716i \(0.131934\pi\)
\(180\) 1.00000 0.0745356
\(181\) 19.1231 1.42141 0.710705 0.703491i \(-0.248376\pi\)
0.710705 + 0.703491i \(0.248376\pi\)
\(182\) −10.2462 −0.759500
\(183\) −0.876894 −0.0648219
\(184\) 1.00000 0.0737210
\(185\) −7.12311 −0.523701
\(186\) 0 0
\(187\) 36.4924 2.66859
\(188\) −8.00000 −0.583460
\(189\) 5.12311 0.372651
\(190\) 4.00000 0.290191
\(191\) −20.4924 −1.48278 −0.741390 0.671075i \(-0.765833\pi\)
−0.741390 + 0.671075i \(0.765833\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −8.24621 −0.593575 −0.296788 0.954944i \(-0.595915\pi\)
−0.296788 + 0.954944i \(0.595915\pi\)
\(194\) 0.246211 0.0176769
\(195\) −2.00000 −0.143223
\(196\) 19.2462 1.37473
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 5.12311 0.364083
\(199\) 10.8769 0.771043 0.385521 0.922699i \(-0.374022\pi\)
0.385521 + 0.922699i \(0.374022\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.00000 0.564276
\(202\) −16.2462 −1.14308
\(203\) −10.2462 −0.719143
\(204\) −7.12311 −0.498717
\(205\) 2.00000 0.139686
\(206\) 15.3693 1.07083
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) 20.4924 1.41749
\(210\) 5.12311 0.353528
\(211\) 16.4924 1.13539 0.567693 0.823241i \(-0.307836\pi\)
0.567693 + 0.823241i \(0.307836\pi\)
\(212\) −4.24621 −0.291631
\(213\) −6.24621 −0.427983
\(214\) 11.3693 0.777191
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 11.1231 0.753352
\(219\) −12.2462 −0.827522
\(220\) 5.12311 0.345400
\(221\) 14.2462 0.958304
\(222\) 7.12311 0.478072
\(223\) 6.24621 0.418277 0.209139 0.977886i \(-0.432934\pi\)
0.209139 + 0.977886i \(0.432934\pi\)
\(224\) −5.12311 −0.342302
\(225\) 1.00000 0.0666667
\(226\) −0.876894 −0.0583301
\(227\) 19.3693 1.28559 0.642793 0.766040i \(-0.277775\pi\)
0.642793 + 0.766040i \(0.277775\pi\)
\(228\) −4.00000 −0.264906
\(229\) −1.36932 −0.0904870 −0.0452435 0.998976i \(-0.514406\pi\)
−0.0452435 + 0.998976i \(0.514406\pi\)
\(230\) 1.00000 0.0659380
\(231\) 26.2462 1.72687
\(232\) 2.00000 0.131306
\(233\) −20.7386 −1.35863 −0.679317 0.733845i \(-0.737724\pi\)
−0.679317 + 0.733845i \(0.737724\pi\)
\(234\) 2.00000 0.130744
\(235\) −8.00000 −0.521862
\(236\) −14.2462 −0.927349
\(237\) 5.12311 0.332781
\(238\) −36.4924 −2.36545
\(239\) −1.75379 −0.113443 −0.0567216 0.998390i \(-0.518065\pi\)
−0.0567216 + 0.998390i \(0.518065\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −2.49242 −0.160551 −0.0802755 0.996773i \(-0.525580\pi\)
−0.0802755 + 0.996773i \(0.525580\pi\)
\(242\) 15.2462 0.980064
\(243\) −1.00000 −0.0641500
\(244\) 0.876894 0.0561374
\(245\) 19.2462 1.22960
\(246\) −2.00000 −0.127515
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) −11.3693 −0.720501
\(250\) 1.00000 0.0632456
\(251\) −7.36932 −0.465147 −0.232574 0.972579i \(-0.574715\pi\)
−0.232574 + 0.972579i \(0.574715\pi\)
\(252\) −5.12311 −0.322725
\(253\) 5.12311 0.322087
\(254\) −22.2462 −1.39585
\(255\) −7.12311 −0.446066
\(256\) 1.00000 0.0625000
\(257\) −8.24621 −0.514385 −0.257192 0.966360i \(-0.582797\pi\)
−0.257192 + 0.966360i \(0.582797\pi\)
\(258\) 0 0
\(259\) 36.4924 2.26753
\(260\) 2.00000 0.124035
\(261\) 2.00000 0.123797
\(262\) −4.00000 −0.247121
\(263\) 10.2462 0.631808 0.315904 0.948791i \(-0.397692\pi\)
0.315904 + 0.948791i \(0.397692\pi\)
\(264\) −5.12311 −0.315305
\(265\) −4.24621 −0.260843
\(266\) −20.4924 −1.25647
\(267\) −3.12311 −0.191131
\(268\) −8.00000 −0.488678
\(269\) −16.2462 −0.990549 −0.495274 0.868737i \(-0.664933\pi\)
−0.495274 + 0.868737i \(0.664933\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 10.2462 0.622413 0.311207 0.950342i \(-0.399267\pi\)
0.311207 + 0.950342i \(0.399267\pi\)
\(272\) 7.12311 0.431902
\(273\) 10.2462 0.620129
\(274\) −19.1231 −1.15527
\(275\) 5.12311 0.308935
\(276\) −1.00000 −0.0601929
\(277\) 20.2462 1.21648 0.608238 0.793754i \(-0.291876\pi\)
0.608238 + 0.793754i \(0.291876\pi\)
\(278\) −16.4924 −0.989150
\(279\) 0 0
\(280\) −5.12311 −0.306164
\(281\) −12.8769 −0.768171 −0.384086 0.923298i \(-0.625483\pi\)
−0.384086 + 0.923298i \(0.625483\pi\)
\(282\) 8.00000 0.476393
\(283\) −2.24621 −0.133523 −0.0667617 0.997769i \(-0.521267\pi\)
−0.0667617 + 0.997769i \(0.521267\pi\)
\(284\) 6.24621 0.370644
\(285\) −4.00000 −0.236940
\(286\) 10.2462 0.605871
\(287\) −10.2462 −0.604815
\(288\) 1.00000 0.0589256
\(289\) 33.7386 1.98463
\(290\) 2.00000 0.117444
\(291\) −0.246211 −0.0144332
\(292\) 12.2462 0.716655
\(293\) 11.7538 0.686664 0.343332 0.939214i \(-0.388444\pi\)
0.343332 + 0.939214i \(0.388444\pi\)
\(294\) −19.2462 −1.12246
\(295\) −14.2462 −0.829446
\(296\) −7.12311 −0.414022
\(297\) −5.12311 −0.297273
\(298\) −10.0000 −0.579284
\(299\) 2.00000 0.115663
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −10.2462 −0.589603
\(303\) 16.2462 0.933320
\(304\) 4.00000 0.229416
\(305\) 0.876894 0.0502108
\(306\) 7.12311 0.407201
\(307\) 0.492423 0.0281040 0.0140520 0.999901i \(-0.495527\pi\)
0.0140520 + 0.999901i \(0.495527\pi\)
\(308\) −26.2462 −1.49552
\(309\) −15.3693 −0.874330
\(310\) 0 0
\(311\) 26.7386 1.51621 0.758104 0.652133i \(-0.226126\pi\)
0.758104 + 0.652133i \(0.226126\pi\)
\(312\) −2.00000 −0.113228
\(313\) 10.4924 0.593067 0.296533 0.955022i \(-0.404169\pi\)
0.296533 + 0.955022i \(0.404169\pi\)
\(314\) 0.876894 0.0494860
\(315\) −5.12311 −0.288654
\(316\) −5.12311 −0.288197
\(317\) 20.7386 1.16480 0.582399 0.812903i \(-0.302114\pi\)
0.582399 + 0.812903i \(0.302114\pi\)
\(318\) 4.24621 0.238116
\(319\) 10.2462 0.573678
\(320\) 1.00000 0.0559017
\(321\) −11.3693 −0.634573
\(322\) −5.12311 −0.285500
\(323\) 28.4924 1.58536
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −12.0000 −0.664619
\(327\) −11.1231 −0.615109
\(328\) 2.00000 0.110432
\(329\) 40.9848 2.25957
\(330\) −5.12311 −0.282018
\(331\) −32.4924 −1.78595 −0.892973 0.450111i \(-0.851384\pi\)
−0.892973 + 0.450111i \(0.851384\pi\)
\(332\) 11.3693 0.623972
\(333\) −7.12311 −0.390344
\(334\) −8.00000 −0.437741
\(335\) −8.00000 −0.437087
\(336\) 5.12311 0.279488
\(337\) −6.49242 −0.353665 −0.176832 0.984241i \(-0.556585\pi\)
−0.176832 + 0.984241i \(0.556585\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0.876894 0.0476264
\(340\) 7.12311 0.386305
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) −62.7386 −3.38757
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 3.75379 0.201805
\(347\) −34.7386 −1.86487 −0.932434 0.361341i \(-0.882319\pi\)
−0.932434 + 0.361341i \(0.882319\pi\)
\(348\) −2.00000 −0.107211
\(349\) −24.7386 −1.32423 −0.662114 0.749403i \(-0.730341\pi\)
−0.662114 + 0.749403i \(0.730341\pi\)
\(350\) −5.12311 −0.273842
\(351\) −2.00000 −0.106752
\(352\) 5.12311 0.273062
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 14.2462 0.757178
\(355\) 6.24621 0.331514
\(356\) 3.12311 0.165524
\(357\) 36.4924 1.93138
\(358\) 24.4924 1.29446
\(359\) −12.4924 −0.659325 −0.329662 0.944099i \(-0.606935\pi\)
−0.329662 + 0.944099i \(0.606935\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 19.1231 1.00509
\(363\) −15.2462 −0.800219
\(364\) −10.2462 −0.537047
\(365\) 12.2462 0.640996
\(366\) −0.876894 −0.0458360
\(367\) 13.1231 0.685021 0.342510 0.939514i \(-0.388723\pi\)
0.342510 + 0.939514i \(0.388723\pi\)
\(368\) 1.00000 0.0521286
\(369\) 2.00000 0.104116
\(370\) −7.12311 −0.370313
\(371\) 21.7538 1.12940
\(372\) 0 0
\(373\) 13.3693 0.692237 0.346118 0.938191i \(-0.387500\pi\)
0.346118 + 0.938191i \(0.387500\pi\)
\(374\) 36.4924 1.88698
\(375\) −1.00000 −0.0516398
\(376\) −8.00000 −0.412568
\(377\) 4.00000 0.206010
\(378\) 5.12311 0.263504
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 4.00000 0.205196
\(381\) 22.2462 1.13971
\(382\) −20.4924 −1.04848
\(383\) 13.7538 0.702786 0.351393 0.936228i \(-0.385708\pi\)
0.351393 + 0.936228i \(0.385708\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −26.2462 −1.33763
\(386\) −8.24621 −0.419721
\(387\) 0 0
\(388\) 0.246211 0.0124995
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) −2.00000 −0.101274
\(391\) 7.12311 0.360231
\(392\) 19.2462 0.972080
\(393\) 4.00000 0.201773
\(394\) −10.0000 −0.503793
\(395\) −5.12311 −0.257771
\(396\) 5.12311 0.257446
\(397\) −34.4924 −1.73113 −0.865563 0.500801i \(-0.833039\pi\)
−0.865563 + 0.500801i \(0.833039\pi\)
\(398\) 10.8769 0.545209
\(399\) 20.4924 1.02590
\(400\) 1.00000 0.0500000
\(401\) −17.3693 −0.867382 −0.433691 0.901062i \(-0.642789\pi\)
−0.433691 + 0.901062i \(0.642789\pi\)
\(402\) 8.00000 0.399004
\(403\) 0 0
\(404\) −16.2462 −0.808279
\(405\) 1.00000 0.0496904
\(406\) −10.2462 −0.508511
\(407\) −36.4924 −1.80886
\(408\) −7.12311 −0.352646
\(409\) −16.2462 −0.803323 −0.401662 0.915788i \(-0.631567\pi\)
−0.401662 + 0.915788i \(0.631567\pi\)
\(410\) 2.00000 0.0987730
\(411\) 19.1231 0.943273
\(412\) 15.3693 0.757192
\(413\) 72.9848 3.59135
\(414\) 1.00000 0.0491473
\(415\) 11.3693 0.558098
\(416\) 2.00000 0.0980581
\(417\) 16.4924 0.807637
\(418\) 20.4924 1.00232
\(419\) −1.61553 −0.0789237 −0.0394619 0.999221i \(-0.512564\pi\)
−0.0394619 + 0.999221i \(0.512564\pi\)
\(420\) 5.12311 0.249982
\(421\) 13.3693 0.651581 0.325790 0.945442i \(-0.394370\pi\)
0.325790 + 0.945442i \(0.394370\pi\)
\(422\) 16.4924 0.802839
\(423\) −8.00000 −0.388973
\(424\) −4.24621 −0.206214
\(425\) 7.12311 0.345521
\(426\) −6.24621 −0.302630
\(427\) −4.49242 −0.217404
\(428\) 11.3693 0.549557
\(429\) −10.2462 −0.494692
\(430\) 0 0
\(431\) 26.2462 1.26424 0.632118 0.774872i \(-0.282186\pi\)
0.632118 + 0.774872i \(0.282186\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 10.4924 0.504234 0.252117 0.967697i \(-0.418873\pi\)
0.252117 + 0.967697i \(0.418873\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 11.1231 0.532700
\(437\) 4.00000 0.191346
\(438\) −12.2462 −0.585147
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 5.12311 0.244234
\(441\) 19.2462 0.916486
\(442\) 14.2462 0.677623
\(443\) −24.4924 −1.16367 −0.581835 0.813307i \(-0.697665\pi\)
−0.581835 + 0.813307i \(0.697665\pi\)
\(444\) 7.12311 0.338048
\(445\) 3.12311 0.148049
\(446\) 6.24621 0.295767
\(447\) 10.0000 0.472984
\(448\) −5.12311 −0.242044
\(449\) −10.4924 −0.495168 −0.247584 0.968866i \(-0.579637\pi\)
−0.247584 + 0.968866i \(0.579637\pi\)
\(450\) 1.00000 0.0471405
\(451\) 10.2462 0.482475
\(452\) −0.876894 −0.0412456
\(453\) 10.2462 0.481409
\(454\) 19.3693 0.909047
\(455\) −10.2462 −0.480350
\(456\) −4.00000 −0.187317
\(457\) −32.7386 −1.53145 −0.765724 0.643169i \(-0.777619\pi\)
−0.765724 + 0.643169i \(0.777619\pi\)
\(458\) −1.36932 −0.0639840
\(459\) −7.12311 −0.332478
\(460\) 1.00000 0.0466252
\(461\) 32.7386 1.52479 0.762395 0.647112i \(-0.224023\pi\)
0.762395 + 0.647112i \(0.224023\pi\)
\(462\) 26.2462 1.22108
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −20.7386 −0.960699
\(467\) −3.36932 −0.155913 −0.0779567 0.996957i \(-0.524840\pi\)
−0.0779567 + 0.996957i \(0.524840\pi\)
\(468\) 2.00000 0.0924500
\(469\) 40.9848 1.89250
\(470\) −8.00000 −0.369012
\(471\) −0.876894 −0.0404052
\(472\) −14.2462 −0.655735
\(473\) 0 0
\(474\) 5.12311 0.235312
\(475\) 4.00000 0.183533
\(476\) −36.4924 −1.67263
\(477\) −4.24621 −0.194421
\(478\) −1.75379 −0.0802164
\(479\) −20.4924 −0.936323 −0.468161 0.883643i \(-0.655083\pi\)
−0.468161 + 0.883643i \(0.655083\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −14.2462 −0.649571
\(482\) −2.49242 −0.113527
\(483\) 5.12311 0.233109
\(484\) 15.2462 0.693010
\(485\) 0.246211 0.0111799
\(486\) −1.00000 −0.0453609
\(487\) 40.4924 1.83489 0.917443 0.397866i \(-0.130249\pi\)
0.917443 + 0.397866i \(0.130249\pi\)
\(488\) 0.876894 0.0396951
\(489\) 12.0000 0.542659
\(490\) 19.2462 0.869455
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 14.2462 0.641617
\(494\) 8.00000 0.359937
\(495\) 5.12311 0.230266
\(496\) 0 0
\(497\) −32.0000 −1.43540
\(498\) −11.3693 −0.509471
\(499\) −28.9848 −1.29754 −0.648770 0.760985i \(-0.724716\pi\)
−0.648770 + 0.760985i \(0.724716\pi\)
\(500\) 1.00000 0.0447214
\(501\) 8.00000 0.357414
\(502\) −7.36932 −0.328909
\(503\) −6.73863 −0.300461 −0.150230 0.988651i \(-0.548002\pi\)
−0.150230 + 0.988651i \(0.548002\pi\)
\(504\) −5.12311 −0.228201
\(505\) −16.2462 −0.722947
\(506\) 5.12311 0.227750
\(507\) 9.00000 0.399704
\(508\) −22.2462 −0.987016
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) −7.12311 −0.315416
\(511\) −62.7386 −2.77539
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −8.24621 −0.363725
\(515\) 15.3693 0.677253
\(516\) 0 0
\(517\) −40.9848 −1.80251
\(518\) 36.4924 1.60338
\(519\) −3.75379 −0.164773
\(520\) 2.00000 0.0877058
\(521\) 8.87689 0.388904 0.194452 0.980912i \(-0.437707\pi\)
0.194452 + 0.980912i \(0.437707\pi\)
\(522\) 2.00000 0.0875376
\(523\) −28.4924 −1.24589 −0.622943 0.782267i \(-0.714063\pi\)
−0.622943 + 0.782267i \(0.714063\pi\)
\(524\) −4.00000 −0.174741
\(525\) 5.12311 0.223591
\(526\) 10.2462 0.446756
\(527\) 0 0
\(528\) −5.12311 −0.222955
\(529\) 1.00000 0.0434783
\(530\) −4.24621 −0.184444
\(531\) −14.2462 −0.618233
\(532\) −20.4924 −0.888459
\(533\) 4.00000 0.173259
\(534\) −3.12311 −0.135150
\(535\) 11.3693 0.491538
\(536\) −8.00000 −0.345547
\(537\) −24.4924 −1.05693
\(538\) −16.2462 −0.700424
\(539\) 98.6004 4.24702
\(540\) −1.00000 −0.0430331
\(541\) −15.7538 −0.677308 −0.338654 0.940911i \(-0.609972\pi\)
−0.338654 + 0.940911i \(0.609972\pi\)
\(542\) 10.2462 0.440112
\(543\) −19.1231 −0.820651
\(544\) 7.12311 0.305401
\(545\) 11.1231 0.476461
\(546\) 10.2462 0.438497
\(547\) 0.492423 0.0210545 0.0105272 0.999945i \(-0.496649\pi\)
0.0105272 + 0.999945i \(0.496649\pi\)
\(548\) −19.1231 −0.816899
\(549\) 0.876894 0.0374249
\(550\) 5.12311 0.218450
\(551\) 8.00000 0.340811
\(552\) −1.00000 −0.0425628
\(553\) 26.2462 1.11610
\(554\) 20.2462 0.860179
\(555\) 7.12311 0.302359
\(556\) −16.4924 −0.699435
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −5.12311 −0.216491
\(561\) −36.4924 −1.54071
\(562\) −12.8769 −0.543179
\(563\) 11.3693 0.479160 0.239580 0.970877i \(-0.422990\pi\)
0.239580 + 0.970877i \(0.422990\pi\)
\(564\) 8.00000 0.336861
\(565\) −0.876894 −0.0368912
\(566\) −2.24621 −0.0944153
\(567\) −5.12311 −0.215150
\(568\) 6.24621 0.262085
\(569\) 27.1231 1.13706 0.568530 0.822663i \(-0.307512\pi\)
0.568530 + 0.822663i \(0.307512\pi\)
\(570\) −4.00000 −0.167542
\(571\) 34.7386 1.45377 0.726883 0.686761i \(-0.240968\pi\)
0.726883 + 0.686761i \(0.240968\pi\)
\(572\) 10.2462 0.428416
\(573\) 20.4924 0.856083
\(574\) −10.2462 −0.427669
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 14.4924 0.603327 0.301664 0.953414i \(-0.402458\pi\)
0.301664 + 0.953414i \(0.402458\pi\)
\(578\) 33.7386 1.40334
\(579\) 8.24621 0.342701
\(580\) 2.00000 0.0830455
\(581\) −58.2462 −2.41646
\(582\) −0.246211 −0.0102058
\(583\) −21.7538 −0.900950
\(584\) 12.2462 0.506752
\(585\) 2.00000 0.0826898
\(586\) 11.7538 0.485545
\(587\) −9.75379 −0.402582 −0.201291 0.979531i \(-0.564514\pi\)
−0.201291 + 0.979531i \(0.564514\pi\)
\(588\) −19.2462 −0.793700
\(589\) 0 0
\(590\) −14.2462 −0.586507
\(591\) 10.0000 0.411345
\(592\) −7.12311 −0.292758
\(593\) 7.75379 0.318410 0.159205 0.987246i \(-0.449107\pi\)
0.159205 + 0.987246i \(0.449107\pi\)
\(594\) −5.12311 −0.210204
\(595\) −36.4924 −1.49604
\(596\) −10.0000 −0.409616
\(597\) −10.8769 −0.445162
\(598\) 2.00000 0.0817861
\(599\) −6.24621 −0.255213 −0.127607 0.991825i \(-0.540730\pi\)
−0.127607 + 0.991825i \(0.540730\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) −10.2462 −0.416912
\(605\) 15.2462 0.619847
\(606\) 16.2462 0.659957
\(607\) 26.7386 1.08529 0.542644 0.839963i \(-0.317423\pi\)
0.542644 + 0.839963i \(0.317423\pi\)
\(608\) 4.00000 0.162221
\(609\) 10.2462 0.415197
\(610\) 0.876894 0.0355044
\(611\) −16.0000 −0.647291
\(612\) 7.12311 0.287934
\(613\) 43.1231 1.74173 0.870863 0.491526i \(-0.163561\pi\)
0.870863 + 0.491526i \(0.163561\pi\)
\(614\) 0.492423 0.0198726
\(615\) −2.00000 −0.0806478
\(616\) −26.2462 −1.05749
\(617\) 40.1080 1.61469 0.807343 0.590083i \(-0.200905\pi\)
0.807343 + 0.590083i \(0.200905\pi\)
\(618\) −15.3693 −0.618245
\(619\) −36.0000 −1.44696 −0.723481 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 26.7386 1.07212
\(623\) −16.0000 −0.641026
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 10.4924 0.419362
\(627\) −20.4924 −0.818389
\(628\) 0.876894 0.0349919
\(629\) −50.7386 −2.02308
\(630\) −5.12311 −0.204109
\(631\) −39.3693 −1.56727 −0.783634 0.621223i \(-0.786636\pi\)
−0.783634 + 0.621223i \(0.786636\pi\)
\(632\) −5.12311 −0.203786
\(633\) −16.4924 −0.655515
\(634\) 20.7386 0.823636
\(635\) −22.2462 −0.882814
\(636\) 4.24621 0.168373
\(637\) 38.4924 1.52513
\(638\) 10.2462 0.405651
\(639\) 6.24621 0.247096
\(640\) 1.00000 0.0395285
\(641\) −9.36932 −0.370066 −0.185033 0.982732i \(-0.559239\pi\)
−0.185033 + 0.982732i \(0.559239\pi\)
\(642\) −11.3693 −0.448711
\(643\) 36.4924 1.43912 0.719560 0.694430i \(-0.244343\pi\)
0.719560 + 0.694430i \(0.244343\pi\)
\(644\) −5.12311 −0.201879
\(645\) 0 0
\(646\) 28.4924 1.12102
\(647\) 11.5076 0.452410 0.226205 0.974080i \(-0.427368\pi\)
0.226205 + 0.974080i \(0.427368\pi\)
\(648\) 1.00000 0.0392837
\(649\) −72.9848 −2.86491
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) 10.4924 0.410600 0.205300 0.978699i \(-0.434183\pi\)
0.205300 + 0.978699i \(0.434183\pi\)
\(654\) −11.1231 −0.434948
\(655\) −4.00000 −0.156293
\(656\) 2.00000 0.0780869
\(657\) 12.2462 0.477770
\(658\) 40.9848 1.59776
\(659\) −7.36932 −0.287068 −0.143534 0.989645i \(-0.545847\pi\)
−0.143534 + 0.989645i \(0.545847\pi\)
\(660\) −5.12311 −0.199417
\(661\) 33.8617 1.31707 0.658535 0.752551i \(-0.271177\pi\)
0.658535 + 0.752551i \(0.271177\pi\)
\(662\) −32.4924 −1.26285
\(663\) −14.2462 −0.553277
\(664\) 11.3693 0.441215
\(665\) −20.4924 −0.794662
\(666\) −7.12311 −0.276015
\(667\) 2.00000 0.0774403
\(668\) −8.00000 −0.309529
\(669\) −6.24621 −0.241492
\(670\) −8.00000 −0.309067
\(671\) 4.49242 0.173428
\(672\) 5.12311 0.197628
\(673\) −46.9848 −1.81113 −0.905566 0.424205i \(-0.860554\pi\)
−0.905566 + 0.424205i \(0.860554\pi\)
\(674\) −6.49242 −0.250079
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −46.4924 −1.78685 −0.893424 0.449213i \(-0.851704\pi\)
−0.893424 + 0.449213i \(0.851704\pi\)
\(678\) 0.876894 0.0336769
\(679\) −1.26137 −0.0484068
\(680\) 7.12311 0.273159
\(681\) −19.3693 −0.742234
\(682\) 0 0
\(683\) 17.7538 0.679330 0.339665 0.940547i \(-0.389686\pi\)
0.339665 + 0.940547i \(0.389686\pi\)
\(684\) 4.00000 0.152944
\(685\) −19.1231 −0.730656
\(686\) −62.7386 −2.39537
\(687\) 1.36932 0.0522427
\(688\) 0 0
\(689\) −8.49242 −0.323536
\(690\) −1.00000 −0.0380693
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 3.75379 0.142698
\(693\) −26.2462 −0.997011
\(694\) −34.7386 −1.31866
\(695\) −16.4924 −0.625593
\(696\) −2.00000 −0.0758098
\(697\) 14.2462 0.539614
\(698\) −24.7386 −0.936371
\(699\) 20.7386 0.784407
\(700\) −5.12311 −0.193635
\(701\) −36.2462 −1.36900 −0.684500 0.729013i \(-0.739980\pi\)
−0.684500 + 0.729013i \(0.739980\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −28.4924 −1.07461
\(704\) 5.12311 0.193084
\(705\) 8.00000 0.301297
\(706\) −14.0000 −0.526897
\(707\) 83.2311 3.13023
\(708\) 14.2462 0.535405
\(709\) 13.3693 0.502095 0.251048 0.967975i \(-0.419225\pi\)
0.251048 + 0.967975i \(0.419225\pi\)
\(710\) 6.24621 0.234416
\(711\) −5.12311 −0.192131
\(712\) 3.12311 0.117043
\(713\) 0 0
\(714\) 36.4924 1.36569
\(715\) 10.2462 0.383187
\(716\) 24.4924 0.915325
\(717\) 1.75379 0.0654964
\(718\) −12.4924 −0.466213
\(719\) −9.75379 −0.363755 −0.181877 0.983321i \(-0.558217\pi\)
−0.181877 + 0.983321i \(0.558217\pi\)
\(720\) 1.00000 0.0372678
\(721\) −78.7386 −2.93238
\(722\) −3.00000 −0.111648
\(723\) 2.49242 0.0926942
\(724\) 19.1231 0.710705
\(725\) 2.00000 0.0742781
\(726\) −15.2462 −0.565840
\(727\) 27.8617 1.03333 0.516667 0.856186i \(-0.327172\pi\)
0.516667 + 0.856186i \(0.327172\pi\)
\(728\) −10.2462 −0.379750
\(729\) 1.00000 0.0370370
\(730\) 12.2462 0.453253
\(731\) 0 0
\(732\) −0.876894 −0.0324109
\(733\) −13.8617 −0.511995 −0.255998 0.966677i \(-0.582404\pi\)
−0.255998 + 0.966677i \(0.582404\pi\)
\(734\) 13.1231 0.484383
\(735\) −19.2462 −0.709907
\(736\) 1.00000 0.0368605
\(737\) −40.9848 −1.50970
\(738\) 2.00000 0.0736210
\(739\) 40.4924 1.48954 0.744769 0.667322i \(-0.232560\pi\)
0.744769 + 0.667322i \(0.232560\pi\)
\(740\) −7.12311 −0.261851
\(741\) −8.00000 −0.293887
\(742\) 21.7538 0.798607
\(743\) −14.7386 −0.540708 −0.270354 0.962761i \(-0.587141\pi\)
−0.270354 + 0.962761i \(0.587141\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 13.3693 0.489485
\(747\) 11.3693 0.415982
\(748\) 36.4924 1.33430
\(749\) −58.2462 −2.12827
\(750\) −1.00000 −0.0365148
\(751\) 43.8617 1.60054 0.800269 0.599641i \(-0.204690\pi\)
0.800269 + 0.599641i \(0.204690\pi\)
\(752\) −8.00000 −0.291730
\(753\) 7.36932 0.268553
\(754\) 4.00000 0.145671
\(755\) −10.2462 −0.372898
\(756\) 5.12311 0.186326
\(757\) 9.86174 0.358431 0.179216 0.983810i \(-0.442644\pi\)
0.179216 + 0.983810i \(0.442644\pi\)
\(758\) 4.00000 0.145287
\(759\) −5.12311 −0.185957
\(760\) 4.00000 0.145095
\(761\) −48.2462 −1.74892 −0.874462 0.485094i \(-0.838785\pi\)
−0.874462 + 0.485094i \(0.838785\pi\)
\(762\) 22.2462 0.805895
\(763\) −56.9848 −2.06299
\(764\) −20.4924 −0.741390
\(765\) 7.12311 0.257536
\(766\) 13.7538 0.496945
\(767\) −28.4924 −1.02880
\(768\) −1.00000 −0.0360844
\(769\) −20.7386 −0.747854 −0.373927 0.927458i \(-0.621989\pi\)
−0.373927 + 0.927458i \(0.621989\pi\)
\(770\) −26.2462 −0.945848
\(771\) 8.24621 0.296980
\(772\) −8.24621 −0.296788
\(773\) −36.2462 −1.30369 −0.651843 0.758354i \(-0.726004\pi\)
−0.651843 + 0.758354i \(0.726004\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.246211 0.00883847
\(777\) −36.4924 −1.30916
\(778\) −18.0000 −0.645331
\(779\) 8.00000 0.286630
\(780\) −2.00000 −0.0716115
\(781\) 32.0000 1.14505
\(782\) 7.12311 0.254722
\(783\) −2.00000 −0.0714742
\(784\) 19.2462 0.687365
\(785\) 0.876894 0.0312977
\(786\) 4.00000 0.142675
\(787\) 48.0000 1.71102 0.855508 0.517790i \(-0.173245\pi\)
0.855508 + 0.517790i \(0.173245\pi\)
\(788\) −10.0000 −0.356235
\(789\) −10.2462 −0.364775
\(790\) −5.12311 −0.182272
\(791\) 4.49242 0.159732
\(792\) 5.12311 0.182042
\(793\) 1.75379 0.0622789
\(794\) −34.4924 −1.22409
\(795\) 4.24621 0.150598
\(796\) 10.8769 0.385521
\(797\) −22.4924 −0.796722 −0.398361 0.917229i \(-0.630421\pi\)
−0.398361 + 0.917229i \(0.630421\pi\)
\(798\) 20.4924 0.725424
\(799\) −56.9848 −2.01598
\(800\) 1.00000 0.0353553
\(801\) 3.12311 0.110350
\(802\) −17.3693 −0.613332
\(803\) 62.7386 2.21400
\(804\) 8.00000 0.282138
\(805\) −5.12311 −0.180566
\(806\) 0 0
\(807\) 16.2462 0.571894
\(808\) −16.2462 −0.571540
\(809\) 4.24621 0.149289 0.0746444 0.997210i \(-0.476218\pi\)
0.0746444 + 0.997210i \(0.476218\pi\)
\(810\) 1.00000 0.0351364
\(811\) −36.0000 −1.26413 −0.632065 0.774915i \(-0.717793\pi\)
−0.632065 + 0.774915i \(0.717793\pi\)
\(812\) −10.2462 −0.359572
\(813\) −10.2462 −0.359350
\(814\) −36.4924 −1.27906
\(815\) −12.0000 −0.420342
\(816\) −7.12311 −0.249359
\(817\) 0 0
\(818\) −16.2462 −0.568035
\(819\) −10.2462 −0.358032
\(820\) 2.00000 0.0698430
\(821\) −19.7538 −0.689412 −0.344706 0.938711i \(-0.612021\pi\)
−0.344706 + 0.938711i \(0.612021\pi\)
\(822\) 19.1231 0.666995
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 15.3693 0.535416
\(825\) −5.12311 −0.178364
\(826\) 72.9848 2.53947
\(827\) 17.1231 0.595429 0.297714 0.954655i \(-0.403776\pi\)
0.297714 + 0.954655i \(0.403776\pi\)
\(828\) 1.00000 0.0347524
\(829\) 32.2462 1.11996 0.559979 0.828507i \(-0.310809\pi\)
0.559979 + 0.828507i \(0.310809\pi\)
\(830\) 11.3693 0.394635
\(831\) −20.2462 −0.702333
\(832\) 2.00000 0.0693375
\(833\) 137.093 4.74998
\(834\) 16.4924 0.571086
\(835\) −8.00000 −0.276851
\(836\) 20.4924 0.708745
\(837\) 0 0
\(838\) −1.61553 −0.0558075
\(839\) −26.2462 −0.906120 −0.453060 0.891480i \(-0.649668\pi\)
−0.453060 + 0.891480i \(0.649668\pi\)
\(840\) 5.12311 0.176764
\(841\) −25.0000 −0.862069
\(842\) 13.3693 0.460737
\(843\) 12.8769 0.443504
\(844\) 16.4924 0.567693
\(845\) −9.00000 −0.309609
\(846\) −8.00000 −0.275046
\(847\) −78.1080 −2.68382
\(848\) −4.24621 −0.145815
\(849\) 2.24621 0.0770898
\(850\) 7.12311 0.244321
\(851\) −7.12311 −0.244177
\(852\) −6.24621 −0.213992
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) −4.49242 −0.153728
\(855\) 4.00000 0.136797
\(856\) 11.3693 0.388595
\(857\) 24.7386 0.845056 0.422528 0.906350i \(-0.361143\pi\)
0.422528 + 0.906350i \(0.361143\pi\)
\(858\) −10.2462 −0.349800
\(859\) 48.4924 1.65454 0.827270 0.561804i \(-0.189893\pi\)
0.827270 + 0.561804i \(0.189893\pi\)
\(860\) 0 0
\(861\) 10.2462 0.349190
\(862\) 26.2462 0.893950
\(863\) 48.9848 1.66746 0.833732 0.552170i \(-0.186200\pi\)
0.833732 + 0.552170i \(0.186200\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 3.75379 0.127633
\(866\) 10.4924 0.356547
\(867\) −33.7386 −1.14582
\(868\) 0 0
\(869\) −26.2462 −0.890342
\(870\) −2.00000 −0.0678064
\(871\) −16.0000 −0.542139
\(872\) 11.1231 0.376676
\(873\) 0.246211 0.00833299
\(874\) 4.00000 0.135302
\(875\) −5.12311 −0.173193
\(876\) −12.2462 −0.413761
\(877\) 3.26137 0.110129 0.0550643 0.998483i \(-0.482464\pi\)
0.0550643 + 0.998483i \(0.482464\pi\)
\(878\) 0 0
\(879\) −11.7538 −0.396445
\(880\) 5.12311 0.172700
\(881\) 31.6155 1.06515 0.532577 0.846381i \(-0.321224\pi\)
0.532577 + 0.846381i \(0.321224\pi\)
\(882\) 19.2462 0.648054
\(883\) −20.9848 −0.706196 −0.353098 0.935586i \(-0.614872\pi\)
−0.353098 + 0.935586i \(0.614872\pi\)
\(884\) 14.2462 0.479152
\(885\) 14.2462 0.478881
\(886\) −24.4924 −0.822839
\(887\) 52.4924 1.76252 0.881262 0.472629i \(-0.156695\pi\)
0.881262 + 0.472629i \(0.156695\pi\)
\(888\) 7.12311 0.239036
\(889\) 113.970 3.82242
\(890\) 3.12311 0.104687
\(891\) 5.12311 0.171630
\(892\) 6.24621 0.209139
\(893\) −32.0000 −1.07084
\(894\) 10.0000 0.334450
\(895\) 24.4924 0.818691
\(896\) −5.12311 −0.171151
\(897\) −2.00000 −0.0667781
\(898\) −10.4924 −0.350137
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −30.2462 −1.00765
\(902\) 10.2462 0.341162
\(903\) 0 0
\(904\) −0.876894 −0.0291651
\(905\) 19.1231 0.635674
\(906\) 10.2462 0.340408
\(907\) 37.7538 1.25359 0.626797 0.779183i \(-0.284366\pi\)
0.626797 + 0.779183i \(0.284366\pi\)
\(908\) 19.3693 0.642793
\(909\) −16.2462 −0.538853
\(910\) −10.2462 −0.339659
\(911\) −13.7538 −0.455683 −0.227842 0.973698i \(-0.573167\pi\)
−0.227842 + 0.973698i \(0.573167\pi\)
\(912\) −4.00000 −0.132453
\(913\) 58.2462 1.92767
\(914\) −32.7386 −1.08290
\(915\) −0.876894 −0.0289892
\(916\) −1.36932 −0.0452435
\(917\) 20.4924 0.676719
\(918\) −7.12311 −0.235098
\(919\) 10.8769 0.358796 0.179398 0.983777i \(-0.442585\pi\)
0.179398 + 0.983777i \(0.442585\pi\)
\(920\) 1.00000 0.0329690
\(921\) −0.492423 −0.0162259
\(922\) 32.7386 1.07819
\(923\) 12.4924 0.411193
\(924\) 26.2462 0.863437
\(925\) −7.12311 −0.234206
\(926\) −36.0000 −1.18303
\(927\) 15.3693 0.504795
\(928\) 2.00000 0.0656532
\(929\) −30.9848 −1.01658 −0.508290 0.861186i \(-0.669722\pi\)
−0.508290 + 0.861186i \(0.669722\pi\)
\(930\) 0 0
\(931\) 76.9848 2.52308
\(932\) −20.7386 −0.679317
\(933\) −26.7386 −0.875384
\(934\) −3.36932 −0.110247
\(935\) 36.4924 1.19343
\(936\) 2.00000 0.0653720
\(937\) −16.7386 −0.546827 −0.273414 0.961897i \(-0.588153\pi\)
−0.273414 + 0.961897i \(0.588153\pi\)
\(938\) 40.9848 1.33820
\(939\) −10.4924 −0.342407
\(940\) −8.00000 −0.260931
\(941\) −6.49242 −0.211647 −0.105823 0.994385i \(-0.533748\pi\)
−0.105823 + 0.994385i \(0.533748\pi\)
\(942\) −0.876894 −0.0285708
\(943\) 2.00000 0.0651290
\(944\) −14.2462 −0.463675
\(945\) 5.12311 0.166655
\(946\) 0 0
\(947\) 22.2462 0.722905 0.361452 0.932391i \(-0.382281\pi\)
0.361452 + 0.932391i \(0.382281\pi\)
\(948\) 5.12311 0.166391
\(949\) 24.4924 0.795058
\(950\) 4.00000 0.129777
\(951\) −20.7386 −0.672496
\(952\) −36.4924 −1.18273
\(953\) −33.8617 −1.09689 −0.548445 0.836187i \(-0.684780\pi\)
−0.548445 + 0.836187i \(0.684780\pi\)
\(954\) −4.24621 −0.137476
\(955\) −20.4924 −0.663119
\(956\) −1.75379 −0.0567216
\(957\) −10.2462 −0.331213
\(958\) −20.4924 −0.662080
\(959\) 97.9697 3.16361
\(960\) −1.00000 −0.0322749
\(961\) −31.0000 −1.00000
\(962\) −14.2462 −0.459316
\(963\) 11.3693 0.366371
\(964\) −2.49242 −0.0802755
\(965\) −8.24621 −0.265455
\(966\) 5.12311 0.164833
\(967\) −4.98485 −0.160302 −0.0801509 0.996783i \(-0.525540\pi\)
−0.0801509 + 0.996783i \(0.525540\pi\)
\(968\) 15.2462 0.490032
\(969\) −28.4924 −0.915308
\(970\) 0.246211 0.00790537
\(971\) 34.8769 1.11925 0.559626 0.828745i \(-0.310945\pi\)
0.559626 + 0.828745i \(0.310945\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 84.4924 2.70870
\(974\) 40.4924 1.29746
\(975\) −2.00000 −0.0640513
\(976\) 0.876894 0.0280687
\(977\) 21.8617 0.699419 0.349710 0.936858i \(-0.386280\pi\)
0.349710 + 0.936858i \(0.386280\pi\)
\(978\) 12.0000 0.383718
\(979\) 16.0000 0.511362
\(980\) 19.2462 0.614798
\(981\) 11.1231 0.355133
\(982\) −12.0000 −0.382935
\(983\) −26.2462 −0.837124 −0.418562 0.908188i \(-0.637466\pi\)
−0.418562 + 0.908188i \(0.637466\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −10.0000 −0.318626
\(986\) 14.2462 0.453692
\(987\) −40.9848 −1.30456
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) 5.12311 0.162823
\(991\) −37.4773 −1.19050 −0.595252 0.803539i \(-0.702948\pi\)
−0.595252 + 0.803539i \(0.702948\pi\)
\(992\) 0 0
\(993\) 32.4924 1.03112
\(994\) −32.0000 −1.01498
\(995\) 10.8769 0.344821
\(996\) −11.3693 −0.360251
\(997\) 20.2462 0.641204 0.320602 0.947214i \(-0.396115\pi\)
0.320602 + 0.947214i \(0.396115\pi\)
\(998\) −28.9848 −0.917499
\(999\) 7.12311 0.225365
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 690.2.a.l.1.1 2
3.2 odd 2 2070.2.a.t.1.1 2
4.3 odd 2 5520.2.a.bs.1.2 2
5.2 odd 4 3450.2.d.v.2899.3 4
5.3 odd 4 3450.2.d.v.2899.2 4
5.4 even 2 3450.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.l.1.1 2 1.1 even 1 trivial
2070.2.a.t.1.1 2 3.2 odd 2
3450.2.a.bi.1.2 2 5.4 even 2
3450.2.d.v.2899.2 4 5.3 odd 4
3450.2.d.v.2899.3 4 5.2 odd 4
5520.2.a.bs.1.2 2 4.3 odd 2