Properties

Label 4225.2.a.bl.1.3
Level $4225$
Weight $2$
Character 4225.1
Self dual yes
Analytic conductor $33.737$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.7367948540\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.21969\) of defining polynomial
Character \(\chi\) \(=\) 4225.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.21969 q^{2} -2.33225 q^{3} -0.512364 q^{4} -2.84461 q^{6} +3.60020 q^{7} -3.06430 q^{8} +2.43937 q^{9} +O(q^{10})\) \(q+1.21969 q^{2} -2.33225 q^{3} -0.512364 q^{4} -2.84461 q^{6} +3.60020 q^{7} -3.06430 q^{8} +2.43937 q^{9} -5.37182 q^{11} +1.19496 q^{12} +4.39111 q^{14} -2.71276 q^{16} +1.13186 q^{17} +2.97527 q^{18} -2.26795 q^{19} -8.39654 q^{21} -6.55193 q^{22} +3.89287 q^{23} +7.14670 q^{24} +1.30752 q^{27} -1.84461 q^{28} -0.0247279 q^{29} -5.46410 q^{31} +2.81988 q^{32} +12.5284 q^{33} +1.38051 q^{34} -1.24985 q^{36} -8.70406 q^{37} -2.76619 q^{38} -3.73205 q^{41} -10.2412 q^{42} +1.13186 q^{43} +2.75232 q^{44} +4.74809 q^{46} +2.58535 q^{47} +6.32681 q^{48} +5.96141 q^{49} -2.63977 q^{51} +4.43937 q^{53} +1.59476 q^{54} -11.0321 q^{56} +5.28942 q^{57} -0.0301603 q^{58} +0.171425 q^{59} +3.36023 q^{61} -6.66449 q^{62} +8.78222 q^{63} +8.86488 q^{64} +15.2807 q^{66} +6.39980 q^{67} -0.579922 q^{68} -9.07914 q^{69} +10.7973 q^{71} -7.47497 q^{72} -4.70308 q^{73} -10.6162 q^{74} +1.16202 q^{76} -19.3396 q^{77} -11.9826 q^{79} -10.3676 q^{81} -4.55193 q^{82} -12.1286 q^{83} +4.30209 q^{84} +1.38051 q^{86} +0.0576715 q^{87} +16.4608 q^{88} +16.1540 q^{89} -1.99457 q^{92} +12.7436 q^{93} +3.15332 q^{94} -6.57666 q^{96} +12.1682 q^{97} +7.27105 q^{98} -13.1039 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{6} + 10q^{7} + 6q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{6} + 10q^{7} + 6q^{8} + 4q^{9} + 10q^{12} + 2q^{14} + 2q^{16} + 2q^{17} + 20q^{18} - 16q^{19} - 4q^{21} - 12q^{22} + 10q^{23} + 24q^{24} + 2q^{27} + 8q^{28} + 8q^{29} - 8q^{31} + 4q^{32} + 18q^{33} + 4q^{34} + 20q^{36} - 2q^{37} - 8q^{38} - 8q^{41} + 4q^{42} + 2q^{43} - 12q^{44} - 16q^{46} + 8q^{47} + 28q^{48} + 12q^{49} + 4q^{51} + 12q^{53} + 16q^{54} + 12q^{56} - 14q^{57} + 22q^{58} - 12q^{59} + 28q^{61} - 4q^{62} + 4q^{63} + 4q^{64} + 6q^{66} + 30q^{67} - 14q^{68} - 16q^{69} - 4q^{71} + 12q^{72} - 8q^{73} - 10q^{74} - 20q^{76} - 18q^{77} - 8q^{79} - 8q^{81} - 4q^{82} - 12q^{83} + 28q^{84} + 4q^{86} + 22q^{87} + 18q^{88} + 12q^{89} - 22q^{92} + 8q^{93} - 32q^{94} - 4q^{96} + 2q^{97} - 24q^{98} - 24q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.21969 0.862449 0.431224 0.902245i \(-0.358082\pi\)
0.431224 + 0.902245i \(0.358082\pi\)
\(3\) −2.33225 −1.34652 −0.673262 0.739404i \(-0.735107\pi\)
−0.673262 + 0.739404i \(0.735107\pi\)
\(4\) −0.512364 −0.256182
\(5\) 0 0
\(6\) −2.84461 −1.16131
\(7\) 3.60020 1.36075 0.680373 0.732866i \(-0.261818\pi\)
0.680373 + 0.732866i \(0.261818\pi\)
\(8\) −3.06430 −1.08339
\(9\) 2.43937 0.813125
\(10\) 0 0
\(11\) −5.37182 −1.61966 −0.809832 0.586662i \(-0.800442\pi\)
−0.809832 + 0.586662i \(0.800442\pi\)
\(12\) 1.19496 0.344955
\(13\) 0 0
\(14\) 4.39111 1.17357
\(15\) 0 0
\(16\) −2.71276 −0.678189
\(17\) 1.13186 0.274515 0.137258 0.990535i \(-0.456171\pi\)
0.137258 + 0.990535i \(0.456171\pi\)
\(18\) 2.97527 0.701278
\(19\) −2.26795 −0.520303 −0.260152 0.965568i \(-0.583773\pi\)
−0.260152 + 0.965568i \(0.583773\pi\)
\(20\) 0 0
\(21\) −8.39654 −1.83228
\(22\) −6.55193 −1.39688
\(23\) 3.89287 0.811720 0.405860 0.913935i \(-0.366972\pi\)
0.405860 + 0.913935i \(0.366972\pi\)
\(24\) 7.14670 1.45881
\(25\) 0 0
\(26\) 0 0
\(27\) 1.30752 0.251632
\(28\) −1.84461 −0.348599
\(29\) −0.0247279 −0.00459185 −0.00229593 0.999997i \(-0.500731\pi\)
−0.00229593 + 0.999997i \(0.500731\pi\)
\(30\) 0 0
\(31\) −5.46410 −0.981382 −0.490691 0.871334i \(-0.663256\pi\)
−0.490691 + 0.871334i \(0.663256\pi\)
\(32\) 2.81988 0.498490
\(33\) 12.5284 2.18091
\(34\) 1.38051 0.236755
\(35\) 0 0
\(36\) −1.24985 −0.208308
\(37\) −8.70406 −1.43094 −0.715470 0.698644i \(-0.753787\pi\)
−0.715470 + 0.698644i \(0.753787\pi\)
\(38\) −2.76619 −0.448735
\(39\) 0 0
\(40\) 0 0
\(41\) −3.73205 −0.582848 −0.291424 0.956594i \(-0.594129\pi\)
−0.291424 + 0.956594i \(0.594129\pi\)
\(42\) −10.2412 −1.58024
\(43\) 1.13186 0.172606 0.0863031 0.996269i \(-0.472495\pi\)
0.0863031 + 0.996269i \(0.472495\pi\)
\(44\) 2.75232 0.414929
\(45\) 0 0
\(46\) 4.74809 0.700067
\(47\) 2.58535 0.377113 0.188556 0.982062i \(-0.439619\pi\)
0.188556 + 0.982062i \(0.439619\pi\)
\(48\) 6.32681 0.913197
\(49\) 5.96141 0.851630
\(50\) 0 0
\(51\) −2.63977 −0.369641
\(52\) 0 0
\(53\) 4.43937 0.609795 0.304897 0.952385i \(-0.401378\pi\)
0.304897 + 0.952385i \(0.401378\pi\)
\(54\) 1.59476 0.217020
\(55\) 0 0
\(56\) −11.0321 −1.47422
\(57\) 5.28942 0.700600
\(58\) −0.0301603 −0.00396024
\(59\) 0.171425 0.0223176 0.0111588 0.999938i \(-0.496448\pi\)
0.0111588 + 0.999938i \(0.496448\pi\)
\(60\) 0 0
\(61\) 3.36023 0.430234 0.215117 0.976588i \(-0.430987\pi\)
0.215117 + 0.976588i \(0.430987\pi\)
\(62\) −6.66449 −0.846391
\(63\) 8.78222 1.10646
\(64\) 8.86488 1.10811
\(65\) 0 0
\(66\) 15.2807 1.88093
\(67\) 6.39980 0.781861 0.390930 0.920420i \(-0.372153\pi\)
0.390930 + 0.920420i \(0.372153\pi\)
\(68\) −0.579922 −0.0703258
\(69\) −9.07914 −1.09300
\(70\) 0 0
\(71\) 10.7973 1.28141 0.640703 0.767788i \(-0.278643\pi\)
0.640703 + 0.767788i \(0.278643\pi\)
\(72\) −7.47497 −0.880933
\(73\) −4.70308 −0.550454 −0.275227 0.961379i \(-0.588753\pi\)
−0.275227 + 0.961379i \(0.588753\pi\)
\(74\) −10.6162 −1.23411
\(75\) 0 0
\(76\) 1.16202 0.133292
\(77\) −19.3396 −2.20395
\(78\) 0 0
\(79\) −11.9826 −1.34815 −0.674075 0.738663i \(-0.735457\pi\)
−0.674075 + 0.738663i \(0.735457\pi\)
\(80\) 0 0
\(81\) −10.3676 −1.15195
\(82\) −4.55193 −0.502677
\(83\) −12.1286 −1.33129 −0.665643 0.746270i \(-0.731843\pi\)
−0.665643 + 0.746270i \(0.731843\pi\)
\(84\) 4.30209 0.469396
\(85\) 0 0
\(86\) 1.38051 0.148864
\(87\) 0.0576715 0.00618303
\(88\) 16.4608 1.75473
\(89\) 16.1540 1.71232 0.856162 0.516707i \(-0.172842\pi\)
0.856162 + 0.516707i \(0.172842\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.99457 −0.207948
\(93\) 12.7436 1.32145
\(94\) 3.15332 0.325240
\(95\) 0 0
\(96\) −6.57666 −0.671228
\(97\) 12.1682 1.23549 0.617745 0.786379i \(-0.288046\pi\)
0.617745 + 0.786379i \(0.288046\pi\)
\(98\) 7.27105 0.734487
\(99\) −13.1039 −1.31699
\(100\) 0 0
\(101\) −4.05441 −0.403429 −0.201714 0.979444i \(-0.564651\pi\)
−0.201714 + 0.979444i \(0.564651\pi\)
\(102\) −3.21969 −0.318797
\(103\) 17.9035 1.76408 0.882041 0.471173i \(-0.156169\pi\)
0.882041 + 0.471173i \(0.156169\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 5.41465 0.525917
\(107\) 9.13186 0.882810 0.441405 0.897308i \(-0.354480\pi\)
0.441405 + 0.897308i \(0.354480\pi\)
\(108\) −0.669925 −0.0644636
\(109\) −7.37605 −0.706498 −0.353249 0.935529i \(-0.614923\pi\)
−0.353249 + 0.935529i \(0.614923\pi\)
\(110\) 0 0
\(111\) 20.3000 1.92679
\(112\) −9.76645 −0.922843
\(113\) −7.07588 −0.665643 −0.332821 0.942990i \(-0.608001\pi\)
−0.332821 + 0.942990i \(0.608001\pi\)
\(114\) 6.45143 0.604232
\(115\) 0 0
\(116\) 0.0126697 0.00117635
\(117\) 0 0
\(118\) 0.209084 0.0192478
\(119\) 4.07490 0.373545
\(120\) 0 0
\(121\) 17.8564 1.62331
\(122\) 4.09843 0.371055
\(123\) 8.70406 0.784819
\(124\) 2.79961 0.251412
\(125\) 0 0
\(126\) 10.7116 0.954262
\(127\) 11.4361 1.01479 0.507395 0.861713i \(-0.330608\pi\)
0.507395 + 0.861713i \(0.330608\pi\)
\(128\) 5.17262 0.457199
\(129\) −2.63977 −0.232418
\(130\) 0 0
\(131\) −10.5680 −0.923328 −0.461664 0.887055i \(-0.652747\pi\)
−0.461664 + 0.887055i \(0.652747\pi\)
\(132\) −6.41910 −0.558711
\(133\) −8.16506 −0.708001
\(134\) 7.80576 0.674315
\(135\) 0 0
\(136\) −3.46834 −0.297408
\(137\) 3.78672 0.323522 0.161761 0.986830i \(-0.448283\pi\)
0.161761 + 0.986830i \(0.448283\pi\)
\(138\) −11.0737 −0.942656
\(139\) 2.01386 0.170814 0.0854068 0.996346i \(-0.472781\pi\)
0.0854068 + 0.996346i \(0.472781\pi\)
\(140\) 0 0
\(141\) −6.02968 −0.507791
\(142\) 13.1694 1.10515
\(143\) 0 0
\(144\) −6.61742 −0.551452
\(145\) 0 0
\(146\) −5.73629 −0.474739
\(147\) −13.9035 −1.14674
\(148\) 4.45965 0.366581
\(149\) 5.51780 0.452035 0.226018 0.974123i \(-0.427429\pi\)
0.226018 + 0.974123i \(0.427429\pi\)
\(150\) 0 0
\(151\) 4.88961 0.397911 0.198956 0.980009i \(-0.436245\pi\)
0.198956 + 0.980009i \(0.436245\pi\)
\(152\) 6.94967 0.563693
\(153\) 2.76102 0.223215
\(154\) −23.5882 −1.90079
\(155\) 0 0
\(156\) 0 0
\(157\) −10.0405 −0.801323 −0.400661 0.916226i \(-0.631220\pi\)
−0.400661 + 0.916226i \(0.631220\pi\)
\(158\) −14.6150 −1.16271
\(159\) −10.3537 −0.821103
\(160\) 0 0
\(161\) 14.0151 1.10454
\(162\) −12.6452 −0.993501
\(163\) 6.78124 0.531148 0.265574 0.964090i \(-0.414439\pi\)
0.265574 + 0.964090i \(0.414439\pi\)
\(164\) 1.91217 0.149315
\(165\) 0 0
\(166\) −14.7931 −1.14817
\(167\) 10.4898 0.811726 0.405863 0.913934i \(-0.366971\pi\)
0.405863 + 0.913934i \(0.366971\pi\)
\(168\) 25.7295 1.98507
\(169\) 0 0
\(170\) 0 0
\(171\) −5.53238 −0.423071
\(172\) −0.579922 −0.0442186
\(173\) 4.45845 0.338970 0.169485 0.985533i \(-0.445790\pi\)
0.169485 + 0.985533i \(0.445790\pi\)
\(174\) 0.0703412 0.00533255
\(175\) 0 0
\(176\) 14.5724 1.09844
\(177\) −0.399804 −0.0300511
\(178\) 19.7029 1.47679
\(179\) 18.6313 1.39257 0.696284 0.717766i \(-0.254835\pi\)
0.696284 + 0.717766i \(0.254835\pi\)
\(180\) 0 0
\(181\) 18.0900 1.34462 0.672310 0.740270i \(-0.265302\pi\)
0.672310 + 0.740270i \(0.265302\pi\)
\(182\) 0 0
\(183\) −7.83690 −0.579320
\(184\) −11.9289 −0.879412
\(185\) 0 0
\(186\) 15.5432 1.13969
\(187\) −6.08012 −0.444622
\(188\) −1.32464 −0.0966095
\(189\) 4.70732 0.342407
\(190\) 0 0
\(191\) 27.3363 1.97799 0.988994 0.147958i \(-0.0472701\pi\)
0.988994 + 0.147958i \(0.0472701\pi\)
\(192\) −20.6751 −1.49210
\(193\) 21.7674 1.56685 0.783425 0.621486i \(-0.213471\pi\)
0.783425 + 0.621486i \(0.213471\pi\)
\(194\) 14.8413 1.06555
\(195\) 0 0
\(196\) −3.05441 −0.218172
\(197\) −1.69672 −0.120886 −0.0604432 0.998172i \(-0.519251\pi\)
−0.0604432 + 0.998172i \(0.519251\pi\)
\(198\) −15.9826 −1.13583
\(199\) 25.3255 1.79527 0.897637 0.440735i \(-0.145282\pi\)
0.897637 + 0.440735i \(0.145282\pi\)
\(200\) 0 0
\(201\) −14.9259 −1.05279
\(202\) −4.94511 −0.347937
\(203\) −0.0890252 −0.00624834
\(204\) 1.35252 0.0946954
\(205\) 0 0
\(206\) 21.8366 1.52143
\(207\) 9.49617 0.660030
\(208\) 0 0
\(209\) 12.1830 0.842716
\(210\) 0 0
\(211\) −0.335507 −0.0230973 −0.0115486 0.999933i \(-0.503676\pi\)
−0.0115486 + 0.999933i \(0.503676\pi\)
\(212\) −2.27458 −0.156218
\(213\) −25.1820 −1.72544
\(214\) 11.1380 0.761378
\(215\) 0 0
\(216\) −4.00663 −0.272616
\(217\) −19.6718 −1.33541
\(218\) −8.99648 −0.609318
\(219\) 10.9688 0.741200
\(220\) 0 0
\(221\) 0 0
\(222\) 24.7597 1.66176
\(223\) −12.2968 −0.823452 −0.411726 0.911308i \(-0.635074\pi\)
−0.411726 + 0.911308i \(0.635074\pi\)
\(224\) 10.1521 0.678318
\(225\) 0 0
\(226\) −8.63036 −0.574083
\(227\) −7.63227 −0.506571 −0.253286 0.967392i \(-0.581511\pi\)
−0.253286 + 0.967392i \(0.581511\pi\)
\(228\) −2.71011 −0.179481
\(229\) 14.4008 0.951631 0.475815 0.879545i \(-0.342153\pi\)
0.475815 + 0.879545i \(0.342153\pi\)
\(230\) 0 0
\(231\) 45.1047 2.96767
\(232\) 0.0757736 0.00497478
\(233\) −9.49617 −0.622115 −0.311057 0.950391i \(-0.600683\pi\)
−0.311057 + 0.950391i \(0.600683\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.0878318 −0.00571736
\(237\) 27.9464 1.81531
\(238\) 4.97010 0.322164
\(239\) 19.9143 1.28815 0.644076 0.764962i \(-0.277242\pi\)
0.644076 + 0.764962i \(0.277242\pi\)
\(240\) 0 0
\(241\) −23.2664 −1.49872 −0.749360 0.662163i \(-0.769639\pi\)
−0.749360 + 0.662163i \(0.769639\pi\)
\(242\) 21.7792 1.40002
\(243\) 20.2572 1.29950
\(244\) −1.72166 −0.110218
\(245\) 0 0
\(246\) 10.6162 0.676866
\(247\) 0 0
\(248\) 16.7436 1.06322
\(249\) 28.2869 1.79261
\(250\) 0 0
\(251\) 11.8402 0.747344 0.373672 0.927561i \(-0.378099\pi\)
0.373672 + 0.927561i \(0.378099\pi\)
\(252\) −4.49969 −0.283454
\(253\) −20.9118 −1.31471
\(254\) 13.9485 0.875205
\(255\) 0 0
\(256\) −11.4208 −0.713800
\(257\) 5.55002 0.346201 0.173100 0.984904i \(-0.444621\pi\)
0.173100 + 0.984904i \(0.444621\pi\)
\(258\) −3.21969 −0.200449
\(259\) −31.3363 −1.94714
\(260\) 0 0
\(261\) −0.0603205 −0.00373375
\(262\) −12.8896 −0.796323
\(263\) −6.85967 −0.422985 −0.211493 0.977380i \(-0.567832\pi\)
−0.211493 + 0.977380i \(0.567832\pi\)
\(264\) −38.3907 −2.36279
\(265\) 0 0
\(266\) −9.95882 −0.610614
\(267\) −37.6752 −2.30568
\(268\) −3.27903 −0.200299
\(269\) −1.42199 −0.0867001 −0.0433501 0.999060i \(-0.513803\pi\)
−0.0433501 + 0.999060i \(0.513803\pi\)
\(270\) 0 0
\(271\) 9.96947 0.605602 0.302801 0.953054i \(-0.402078\pi\)
0.302801 + 0.953054i \(0.402078\pi\)
\(272\) −3.07045 −0.186173
\(273\) 0 0
\(274\) 4.61862 0.279021
\(275\) 0 0
\(276\) 4.65182 0.280007
\(277\) 17.5237 1.05290 0.526449 0.850206i \(-0.323523\pi\)
0.526449 + 0.850206i \(0.323523\pi\)
\(278\) 2.45628 0.147318
\(279\) −13.3290 −0.797986
\(280\) 0 0
\(281\) −10.7352 −0.640406 −0.320203 0.947349i \(-0.603751\pi\)
−0.320203 + 0.947349i \(0.603751\pi\)
\(282\) −7.35433 −0.437944
\(283\) −1.31838 −0.0783698 −0.0391849 0.999232i \(-0.512476\pi\)
−0.0391849 + 0.999232i \(0.512476\pi\)
\(284\) −5.53216 −0.328273
\(285\) 0 0
\(286\) 0 0
\(287\) −13.4361 −0.793109
\(288\) 6.87875 0.405334
\(289\) −15.7189 −0.924641
\(290\) 0 0
\(291\) −28.3792 −1.66362
\(292\) 2.40969 0.141016
\(293\) 18.7427 1.09496 0.547479 0.836820i \(-0.315588\pi\)
0.547479 + 0.836820i \(0.315588\pi\)
\(294\) −16.9579 −0.989004
\(295\) 0 0
\(296\) 26.6718 1.55027
\(297\) −7.02375 −0.407559
\(298\) 6.72998 0.389857
\(299\) 0 0
\(300\) 0 0
\(301\) 4.07490 0.234873
\(302\) 5.96380 0.343178
\(303\) 9.45589 0.543226
\(304\) 6.15239 0.352864
\(305\) 0 0
\(306\) 3.36758 0.192512
\(307\) 14.3043 0.816387 0.408194 0.912895i \(-0.366159\pi\)
0.408194 + 0.912895i \(0.366159\pi\)
\(308\) 9.90891 0.564612
\(309\) −41.7553 −2.37538
\(310\) 0 0
\(311\) 2.76102 0.156563 0.0782815 0.996931i \(-0.475057\pi\)
0.0782815 + 0.996931i \(0.475057\pi\)
\(312\) 0 0
\(313\) 16.3858 0.926179 0.463090 0.886311i \(-0.346741\pi\)
0.463090 + 0.886311i \(0.346741\pi\)
\(314\) −12.2463 −0.691100
\(315\) 0 0
\(316\) 6.13946 0.345372
\(317\) 1.78575 0.100297 0.0501487 0.998742i \(-0.484030\pi\)
0.0501487 + 0.998742i \(0.484030\pi\)
\(318\) −12.6283 −0.708159
\(319\) 0.132834 0.00743725
\(320\) 0 0
\(321\) −21.2977 −1.18872
\(322\) 17.0940 0.952613
\(323\) −2.56699 −0.142831
\(324\) 5.31197 0.295110
\(325\) 0 0
\(326\) 8.27099 0.458088
\(327\) 17.2028 0.951316
\(328\) 11.4361 0.631454
\(329\) 9.30778 0.513155
\(330\) 0 0
\(331\) −7.22440 −0.397089 −0.198545 0.980092i \(-0.563621\pi\)
−0.198545 + 0.980092i \(0.563621\pi\)
\(332\) 6.21425 0.341052
\(333\) −21.2325 −1.16353
\(334\) 12.7943 0.700072
\(335\) 0 0
\(336\) 22.7778 1.24263
\(337\) 4.36219 0.237624 0.118812 0.992917i \(-0.462091\pi\)
0.118812 + 0.992917i \(0.462091\pi\)
\(338\) 0 0
\(339\) 16.5027 0.896303
\(340\) 0 0
\(341\) 29.3521 1.58951
\(342\) −6.74777 −0.364877
\(343\) −3.73913 −0.201894
\(344\) −3.46834 −0.187000
\(345\) 0 0
\(346\) 5.43792 0.292344
\(347\) 26.7072 1.43372 0.716858 0.697219i \(-0.245580\pi\)
0.716858 + 0.697219i \(0.245580\pi\)
\(348\) −0.0295488 −0.00158398
\(349\) −23.5711 −1.26173 −0.630865 0.775892i \(-0.717300\pi\)
−0.630865 + 0.775892i \(0.717300\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −15.1479 −0.807385
\(353\) 5.73727 0.305364 0.152682 0.988275i \(-0.451209\pi\)
0.152682 + 0.988275i \(0.451209\pi\)
\(354\) −0.487636 −0.0259176
\(355\) 0 0
\(356\) −8.27675 −0.438667
\(357\) −9.50367 −0.502988
\(358\) 22.7243 1.20102
\(359\) 24.7583 1.30669 0.653347 0.757059i \(-0.273364\pi\)
0.653347 + 0.757059i \(0.273364\pi\)
\(360\) 0 0
\(361\) −13.8564 −0.729285
\(362\) 22.0641 1.15967
\(363\) −41.6455 −2.18582
\(364\) 0 0
\(365\) 0 0
\(366\) −9.55856 −0.499634
\(367\) −26.0535 −1.35998 −0.679992 0.733220i \(-0.738017\pi\)
−0.679992 + 0.733220i \(0.738017\pi\)
\(368\) −10.5604 −0.550499
\(369\) −9.10387 −0.473928
\(370\) 0 0
\(371\) 15.9826 0.829776
\(372\) −6.52938 −0.338532
\(373\) −13.2045 −0.683702 −0.341851 0.939754i \(-0.611054\pi\)
−0.341851 + 0.939754i \(0.611054\pi\)
\(374\) −7.41584 −0.383464
\(375\) 0 0
\(376\) −7.92229 −0.408561
\(377\) 0 0
\(378\) 5.74146 0.295309
\(379\) −25.9977 −1.33541 −0.667707 0.744425i \(-0.732724\pi\)
−0.667707 + 0.744425i \(0.732724\pi\)
\(380\) 0 0
\(381\) −26.6718 −1.36644
\(382\) 33.3418 1.70591
\(383\) −9.60020 −0.490547 −0.245274 0.969454i \(-0.578878\pi\)
−0.245274 + 0.969454i \(0.578878\pi\)
\(384\) −12.0638 −0.615629
\(385\) 0 0
\(386\) 26.5494 1.35133
\(387\) 2.76102 0.140350
\(388\) −6.23453 −0.316510
\(389\) 5.63129 0.285518 0.142759 0.989758i \(-0.454403\pi\)
0.142759 + 0.989758i \(0.454403\pi\)
\(390\) 0 0
\(391\) 4.40617 0.222829
\(392\) −18.2675 −0.922650
\(393\) 24.6471 1.24328
\(394\) −2.06947 −0.104258
\(395\) 0 0
\(396\) 6.71395 0.337389
\(397\) −16.7658 −0.841452 −0.420726 0.907188i \(-0.638225\pi\)
−0.420726 + 0.907188i \(0.638225\pi\)
\(398\) 30.8891 1.54833
\(399\) 19.0429 0.953339
\(400\) 0 0
\(401\) 13.8780 0.693036 0.346518 0.938043i \(-0.387364\pi\)
0.346518 + 0.938043i \(0.387364\pi\)
\(402\) −18.2050 −0.907980
\(403\) 0 0
\(404\) 2.07733 0.103351
\(405\) 0 0
\(406\) −0.108583 −0.00538888
\(407\) 46.7566 2.31764
\(408\) 8.08903 0.400466
\(409\) −29.4251 −1.45498 −0.727489 0.686120i \(-0.759313\pi\)
−0.727489 + 0.686120i \(0.759313\pi\)
\(410\) 0 0
\(411\) −8.83157 −0.435629
\(412\) −9.17310 −0.451926
\(413\) 0.617162 0.0303686
\(414\) 11.5824 0.569242
\(415\) 0 0
\(416\) 0 0
\(417\) −4.69683 −0.230005
\(418\) 14.8595 0.726800
\(419\) 6.96793 0.340406 0.170203 0.985409i \(-0.445558\pi\)
0.170203 + 0.985409i \(0.445558\pi\)
\(420\) 0 0
\(421\) −7.12125 −0.347069 −0.173534 0.984828i \(-0.555519\pi\)
−0.173534 + 0.984828i \(0.555519\pi\)
\(422\) −0.409213 −0.0199202
\(423\) 6.30664 0.306640
\(424\) −13.6036 −0.660647
\(425\) 0 0
\(426\) −30.7142 −1.48811
\(427\) 12.0975 0.585439
\(428\) −4.67883 −0.226160
\(429\) 0 0
\(430\) 0 0
\(431\) −30.2144 −1.45537 −0.727687 0.685909i \(-0.759405\pi\)
−0.727687 + 0.685909i \(0.759405\pi\)
\(432\) −3.54698 −0.170654
\(433\) −1.20013 −0.0576745 −0.0288373 0.999584i \(-0.509180\pi\)
−0.0288373 + 0.999584i \(0.509180\pi\)
\(434\) −23.9935 −1.15172
\(435\) 0 0
\(436\) 3.77922 0.180992
\(437\) −8.82884 −0.422341
\(438\) 13.3784 0.639247
\(439\) −16.5541 −0.790084 −0.395042 0.918663i \(-0.629270\pi\)
−0.395042 + 0.918663i \(0.629270\pi\)
\(440\) 0 0
\(441\) 14.5421 0.692481
\(442\) 0 0
\(443\) −4.55949 −0.216628 −0.108314 0.994117i \(-0.534545\pi\)
−0.108314 + 0.994117i \(0.534545\pi\)
\(444\) −10.4010 −0.493610
\(445\) 0 0
\(446\) −14.9982 −0.710185
\(447\) −12.8689 −0.608676
\(448\) 31.9153 1.50786
\(449\) −13.8522 −0.653724 −0.326862 0.945072i \(-0.605991\pi\)
−0.326862 + 0.945072i \(0.605991\pi\)
\(450\) 0 0
\(451\) 20.0479 0.944018
\(452\) 3.62542 0.170526
\(453\) −11.4038 −0.535796
\(454\) −9.30897 −0.436892
\(455\) 0 0
\(456\) −16.2083 −0.759025
\(457\) 40.1146 1.87648 0.938240 0.345984i \(-0.112455\pi\)
0.938240 + 0.345984i \(0.112455\pi\)
\(458\) 17.5644 0.820733
\(459\) 1.47992 0.0690768
\(460\) 0 0
\(461\) 7.53900 0.351126 0.175563 0.984468i \(-0.443825\pi\)
0.175563 + 0.984468i \(0.443825\pi\)
\(462\) 55.0136 2.55946
\(463\) 23.3031 1.08299 0.541494 0.840705i \(-0.317859\pi\)
0.541494 + 0.840705i \(0.317859\pi\)
\(464\) 0.0670807 0.00311414
\(465\) 0 0
\(466\) −11.5824 −0.536542
\(467\) 22.6297 1.04718 0.523589 0.851971i \(-0.324593\pi\)
0.523589 + 0.851971i \(0.324593\pi\)
\(468\) 0 0
\(469\) 23.0405 1.06391
\(470\) 0 0
\(471\) 23.4170 1.07900
\(472\) −0.525296 −0.0241787
\(473\) −6.08012 −0.279564
\(474\) 34.0859 1.56562
\(475\) 0 0
\(476\) −2.08783 −0.0956956
\(477\) 10.8293 0.495839
\(478\) 24.2893 1.11096
\(479\) −20.6448 −0.943286 −0.471643 0.881790i \(-0.656339\pi\)
−0.471643 + 0.881790i \(0.656339\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −28.3777 −1.29257
\(483\) −32.6867 −1.48730
\(484\) −9.14898 −0.415863
\(485\) 0 0
\(486\) 24.7074 1.12075
\(487\) 3.03605 0.137576 0.0687882 0.997631i \(-0.478087\pi\)
0.0687882 + 0.997631i \(0.478087\pi\)
\(488\) −10.2968 −0.466112
\(489\) −15.8155 −0.715203
\(490\) 0 0
\(491\) 10.6680 0.481441 0.240720 0.970595i \(-0.422616\pi\)
0.240720 + 0.970595i \(0.422616\pi\)
\(492\) −4.45965 −0.201056
\(493\) −0.0279884 −0.00126053
\(494\) 0 0
\(495\) 0 0
\(496\) 14.8228 0.665562
\(497\) 38.8725 1.74367
\(498\) 34.5011 1.54603
\(499\) −33.9143 −1.51821 −0.759107 0.650966i \(-0.774364\pi\)
−0.759107 + 0.650966i \(0.774364\pi\)
\(500\) 0 0
\(501\) −24.4648 −1.09301
\(502\) 14.4413 0.644546
\(503\) 12.6276 0.563037 0.281518 0.959556i \(-0.409162\pi\)
0.281518 + 0.959556i \(0.409162\pi\)
\(504\) −26.9113 −1.19873
\(505\) 0 0
\(506\) −25.5058 −1.13387
\(507\) 0 0
\(508\) −5.85945 −0.259971
\(509\) 24.1526 1.07055 0.535273 0.844679i \(-0.320209\pi\)
0.535273 + 0.844679i \(0.320209\pi\)
\(510\) 0 0
\(511\) −16.9320 −0.749029
\(512\) −24.2750 −1.07281
\(513\) −2.96539 −0.130925
\(514\) 6.76929 0.298581
\(515\) 0 0
\(516\) 1.35252 0.0595414
\(517\) −13.8880 −0.610796
\(518\) −38.2205 −1.67931
\(519\) −10.3982 −0.456431
\(520\) 0 0
\(521\) −24.7521 −1.08441 −0.542205 0.840246i \(-0.682410\pi\)
−0.542205 + 0.840246i \(0.682410\pi\)
\(522\) −0.0735722 −0.00322017
\(523\) −37.0326 −1.61932 −0.809662 0.586897i \(-0.800350\pi\)
−0.809662 + 0.586897i \(0.800350\pi\)
\(524\) 5.41465 0.236540
\(525\) 0 0
\(526\) −8.36665 −0.364803
\(527\) −6.18457 −0.269404
\(528\) −33.9865 −1.47907
\(529\) −7.84554 −0.341111
\(530\) 0 0
\(531\) 0.418169 0.0181470
\(532\) 4.18348 0.181377
\(533\) 0 0
\(534\) −45.9519 −1.98854
\(535\) 0 0
\(536\) −19.6109 −0.847062
\(537\) −43.4528 −1.87512
\(538\) −1.73438 −0.0747744
\(539\) −32.0236 −1.37935
\(540\) 0 0
\(541\) 8.38144 0.360346 0.180173 0.983635i \(-0.442334\pi\)
0.180173 + 0.983635i \(0.442334\pi\)
\(542\) 12.1596 0.522301
\(543\) −42.1903 −1.81056
\(544\) 3.19170 0.136843
\(545\) 0 0
\(546\) 0 0
\(547\) 22.7842 0.974181 0.487091 0.873351i \(-0.338058\pi\)
0.487091 + 0.873351i \(0.338058\pi\)
\(548\) −1.94018 −0.0828804
\(549\) 8.19687 0.349834
\(550\) 0 0
\(551\) 0.0560816 0.00238915
\(552\) 27.8212 1.18415
\(553\) −43.1398 −1.83449
\(554\) 21.3735 0.908071
\(555\) 0 0
\(556\) −1.03183 −0.0437594
\(557\) −28.1527 −1.19287 −0.596435 0.802662i \(-0.703417\pi\)
−0.596435 + 0.802662i \(0.703417\pi\)
\(558\) −16.2572 −0.688222
\(559\) 0 0
\(560\) 0 0
\(561\) 14.1803 0.598694
\(562\) −13.0935 −0.552317
\(563\) 18.1303 0.764101 0.382050 0.924142i \(-0.375218\pi\)
0.382050 + 0.924142i \(0.375218\pi\)
\(564\) 3.08939 0.130087
\(565\) 0 0
\(566\) −1.60801 −0.0675899
\(567\) −37.3253 −1.56752
\(568\) −33.0862 −1.38827
\(569\) 40.5985 1.70198 0.850988 0.525185i \(-0.176004\pi\)
0.850988 + 0.525185i \(0.176004\pi\)
\(570\) 0 0
\(571\) 24.7159 1.03433 0.517164 0.855886i \(-0.326988\pi\)
0.517164 + 0.855886i \(0.326988\pi\)
\(572\) 0 0
\(573\) −63.7551 −2.66341
\(574\) −16.3879 −0.684016
\(575\) 0 0
\(576\) 21.6248 0.901032
\(577\) 23.0691 0.960379 0.480189 0.877165i \(-0.340568\pi\)
0.480189 + 0.877165i \(0.340568\pi\)
\(578\) −19.1721 −0.797456
\(579\) −50.7669 −2.10980
\(580\) 0 0
\(581\) −43.6653 −1.81154
\(582\) −34.6137 −1.43478
\(583\) −23.8475 −0.987662
\(584\) 14.4116 0.596358
\(585\) 0 0
\(586\) 22.8602 0.944345
\(587\) −20.3523 −0.840030 −0.420015 0.907517i \(-0.637975\pi\)
−0.420015 + 0.907517i \(0.637975\pi\)
\(588\) 7.12364 0.293774
\(589\) 12.3923 0.510616
\(590\) 0 0
\(591\) 3.95717 0.162776
\(592\) 23.6120 0.970447
\(593\) 10.3834 0.426395 0.213198 0.977009i \(-0.431612\pi\)
0.213198 + 0.977009i \(0.431612\pi\)
\(594\) −8.56677 −0.351499
\(595\) 0 0
\(596\) −2.82712 −0.115803
\(597\) −59.0652 −2.41738
\(598\) 0 0
\(599\) −31.5965 −1.29100 −0.645499 0.763761i \(-0.723351\pi\)
−0.645499 + 0.763761i \(0.723351\pi\)
\(600\) 0 0
\(601\) −43.8845 −1.79009 −0.895044 0.445979i \(-0.852856\pi\)
−0.895044 + 0.445979i \(0.852856\pi\)
\(602\) 4.97010 0.202566
\(603\) 15.6115 0.635750
\(604\) −2.50526 −0.101938
\(605\) 0 0
\(606\) 11.5332 0.468505
\(607\) −2.17540 −0.0882968 −0.0441484 0.999025i \(-0.514057\pi\)
−0.0441484 + 0.999025i \(0.514057\pi\)
\(608\) −6.39535 −0.259366
\(609\) 0.207629 0.00841354
\(610\) 0 0
\(611\) 0 0
\(612\) −1.41465 −0.0571837
\(613\) −14.7620 −0.596231 −0.298116 0.954530i \(-0.596358\pi\)
−0.298116 + 0.954530i \(0.596358\pi\)
\(614\) 17.4467 0.704092
\(615\) 0 0
\(616\) 59.2622 2.38774
\(617\) −20.2972 −0.817134 −0.408567 0.912728i \(-0.633971\pi\)
−0.408567 + 0.912728i \(0.633971\pi\)
\(618\) −50.9284 −2.04864
\(619\) −9.94207 −0.399605 −0.199803 0.979836i \(-0.564030\pi\)
−0.199803 + 0.979836i \(0.564030\pi\)
\(620\) 0 0
\(621\) 5.09000 0.204255
\(622\) 3.36758 0.135028
\(623\) 58.1577 2.33004
\(624\) 0 0
\(625\) 0 0
\(626\) 19.9855 0.798782
\(627\) −28.4138 −1.13474
\(628\) 5.14441 0.205284
\(629\) −9.85174 −0.392815
\(630\) 0 0
\(631\) −0.973420 −0.0387512 −0.0193756 0.999812i \(-0.506168\pi\)
−0.0193756 + 0.999812i \(0.506168\pi\)
\(632\) 36.7183 1.46058
\(633\) 0.782485 0.0311010
\(634\) 2.17805 0.0865014
\(635\) 0 0
\(636\) 5.30487 0.210352
\(637\) 0 0
\(638\) 0.162015 0.00641425
\(639\) 26.3387 1.04194
\(640\) 0 0
\(641\) −12.6209 −0.498497 −0.249249 0.968440i \(-0.580184\pi\)
−0.249249 + 0.968440i \(0.580184\pi\)
\(642\) −25.9766 −1.02521
\(643\) 9.96043 0.392801 0.196401 0.980524i \(-0.437075\pi\)
0.196401 + 0.980524i \(0.437075\pi\)
\(644\) −7.18083 −0.282964
\(645\) 0 0
\(646\) −3.13092 −0.123185
\(647\) 36.2763 1.42617 0.713084 0.701079i \(-0.247298\pi\)
0.713084 + 0.701079i \(0.247298\pi\)
\(648\) 31.7693 1.24802
\(649\) −0.920861 −0.0361470
\(650\) 0 0
\(651\) 45.8796 1.79816
\(652\) −3.47447 −0.136071
\(653\) −13.7554 −0.538289 −0.269145 0.963100i \(-0.586741\pi\)
−0.269145 + 0.963100i \(0.586741\pi\)
\(654\) 20.9820 0.820461
\(655\) 0 0
\(656\) 10.1241 0.395281
\(657\) −11.4726 −0.447588
\(658\) 11.3526 0.442570
\(659\) −2.58183 −0.100574 −0.0502869 0.998735i \(-0.516014\pi\)
−0.0502869 + 0.998735i \(0.516014\pi\)
\(660\) 0 0
\(661\) −24.8765 −0.967582 −0.483791 0.875183i \(-0.660741\pi\)
−0.483791 + 0.875183i \(0.660741\pi\)
\(662\) −8.81151 −0.342469
\(663\) 0 0
\(664\) 37.1656 1.44231
\(665\) 0 0
\(666\) −25.8970 −1.00349
\(667\) −0.0962625 −0.00372730
\(668\) −5.37460 −0.207949
\(669\) 28.6791 1.10880
\(670\) 0 0
\(671\) −18.0506 −0.696834
\(672\) −23.6773 −0.913370
\(673\) −43.3222 −1.66995 −0.834974 0.550289i \(-0.814517\pi\)
−0.834974 + 0.550289i \(0.814517\pi\)
\(674\) 5.32051 0.204938
\(675\) 0 0
\(676\) 0 0
\(677\) 41.3625 1.58969 0.794845 0.606813i \(-0.207552\pi\)
0.794845 + 0.606813i \(0.207552\pi\)
\(678\) 20.1281 0.773016
\(679\) 43.8078 1.68119
\(680\) 0 0
\(681\) 17.8003 0.682110
\(682\) 35.8004 1.37087
\(683\) 2.62688 0.100515 0.0502574 0.998736i \(-0.483996\pi\)
0.0502574 + 0.998736i \(0.483996\pi\)
\(684\) 2.83459 0.108383
\(685\) 0 0
\(686\) −4.56057 −0.174123
\(687\) −33.5862 −1.28139
\(688\) −3.07045 −0.117060
\(689\) 0 0
\(690\) 0 0
\(691\) 15.2753 0.581099 0.290550 0.956860i \(-0.406162\pi\)
0.290550 + 0.956860i \(0.406162\pi\)
\(692\) −2.28435 −0.0868380
\(693\) −47.1765 −1.79209
\(694\) 32.5744 1.23651
\(695\) 0 0
\(696\) −0.176723 −0.00669865
\(697\) −4.22414 −0.160001
\(698\) −28.7493 −1.08818
\(699\) 22.1474 0.837692
\(700\) 0 0
\(701\) −48.1947 −1.82029 −0.910144 0.414292i \(-0.864029\pi\)
−0.910144 + 0.414292i \(0.864029\pi\)
\(702\) 0 0
\(703\) 19.7404 0.744522
\(704\) −47.6205 −1.79477
\(705\) 0 0
\(706\) 6.99767 0.263361
\(707\) −14.5967 −0.548964
\(708\) 0.204845 0.00769856
\(709\) 38.8699 1.45979 0.729896 0.683559i \(-0.239569\pi\)
0.729896 + 0.683559i \(0.239569\pi\)
\(710\) 0 0
\(711\) −29.2301 −1.09621
\(712\) −49.5008 −1.85512
\(713\) −21.2711 −0.796607
\(714\) −11.5915 −0.433801
\(715\) 0 0
\(716\) −9.54600 −0.356751
\(717\) −46.4452 −1.73453
\(718\) 30.1974 1.12696
\(719\) 6.61660 0.246758 0.123379 0.992360i \(-0.460627\pi\)
0.123379 + 0.992360i \(0.460627\pi\)
\(720\) 0 0
\(721\) 64.4560 2.40047
\(722\) −16.9005 −0.628971
\(723\) 54.2629 2.01806
\(724\) −9.26867 −0.344467
\(725\) 0 0
\(726\) −50.7945 −1.88516
\(727\) 18.3735 0.681435 0.340717 0.940166i \(-0.389330\pi\)
0.340717 + 0.940166i \(0.389330\pi\)
\(728\) 0 0
\(729\) −16.1420 −0.597853
\(730\) 0 0
\(731\) 1.28110 0.0473830
\(732\) 4.01534 0.148411
\(733\) 0.791131 0.0292211 0.0146105 0.999893i \(-0.495349\pi\)
0.0146105 + 0.999893i \(0.495349\pi\)
\(734\) −31.7772 −1.17292
\(735\) 0 0
\(736\) 10.9774 0.404634
\(737\) −34.3786 −1.26635
\(738\) −11.1039 −0.408739
\(739\) −31.1853 −1.14717 −0.573585 0.819146i \(-0.694448\pi\)
−0.573585 + 0.819146i \(0.694448\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 19.4938 0.715639
\(743\) 5.56304 0.204088 0.102044 0.994780i \(-0.467462\pi\)
0.102044 + 0.994780i \(0.467462\pi\)
\(744\) −39.0503 −1.43165
\(745\) 0 0
\(746\) −16.1053 −0.589658
\(747\) −29.5862 −1.08250
\(748\) 3.11523 0.113904
\(749\) 32.8765 1.20128
\(750\) 0 0
\(751\) −35.2097 −1.28482 −0.642410 0.766361i \(-0.722065\pi\)
−0.642410 + 0.766361i \(0.722065\pi\)
\(752\) −7.01343 −0.255754
\(753\) −27.6142 −1.00632
\(754\) 0 0
\(755\) 0 0
\(756\) −2.41186 −0.0877186
\(757\) −50.0446 −1.81890 −0.909451 0.415810i \(-0.863498\pi\)
−0.909451 + 0.415810i \(0.863498\pi\)
\(758\) −31.7091 −1.15173
\(759\) 48.7715 1.77029
\(760\) 0 0
\(761\) −44.8209 −1.62476 −0.812379 0.583130i \(-0.801828\pi\)
−0.812379 + 0.583130i \(0.801828\pi\)
\(762\) −32.5313 −1.17848
\(763\) −26.5552 −0.961364
\(764\) −14.0061 −0.506725
\(765\) 0 0
\(766\) −11.7092 −0.423072
\(767\) 0 0
\(768\) 26.6361 0.961148
\(769\) 39.3633 1.41948 0.709739 0.704465i \(-0.248813\pi\)
0.709739 + 0.704465i \(0.248813\pi\)
\(770\) 0 0
\(771\) −12.9440 −0.466168
\(772\) −11.1528 −0.401399
\(773\) 48.7805 1.75451 0.877256 0.480022i \(-0.159371\pi\)
0.877256 + 0.480022i \(0.159371\pi\)
\(774\) 3.36758 0.121045
\(775\) 0 0
\(776\) −37.2869 −1.33852
\(777\) 73.0840 2.62188
\(778\) 6.86841 0.246244
\(779\) 8.46410 0.303258
\(780\) 0 0
\(781\) −58.0013 −2.07545
\(782\) 5.37415 0.192179
\(783\) −0.0323322 −0.00115546
\(784\) −16.1718 −0.577566
\(785\) 0 0
\(786\) 30.0618 1.07227
\(787\) 39.8608 1.42088 0.710442 0.703756i \(-0.248495\pi\)
0.710442 + 0.703756i \(0.248495\pi\)
\(788\) 0.869338 0.0309689
\(789\) 15.9984 0.569559
\(790\) 0 0
\(791\) −25.4745 −0.905771
\(792\) 40.1541 1.42682
\(793\) 0 0
\(794\) −20.4490 −0.725709
\(795\) 0 0
\(796\) −12.9759 −0.459917
\(797\) −26.2118 −0.928470 −0.464235 0.885712i \(-0.653671\pi\)
−0.464235 + 0.885712i \(0.653671\pi\)
\(798\) 23.2264 0.822206
\(799\) 2.92625 0.103523
\(800\) 0 0
\(801\) 39.4057 1.39233
\(802\) 16.9269 0.597708
\(803\) 25.2641 0.891551
\(804\) 7.64750 0.269707
\(805\) 0 0
\(806\) 0 0
\(807\) 3.31643 0.116744
\(808\) 12.4239 0.437072
\(809\) 22.2136 0.780990 0.390495 0.920605i \(-0.372304\pi\)
0.390495 + 0.920605i \(0.372304\pi\)
\(810\) 0 0
\(811\) −19.0950 −0.670515 −0.335257 0.942127i \(-0.608823\pi\)
−0.335257 + 0.942127i \(0.608823\pi\)
\(812\) 0.0456133 0.00160071
\(813\) −23.2513 −0.815457
\(814\) 57.0284 1.99885
\(815\) 0 0
\(816\) 7.16104 0.250686
\(817\) −2.56699 −0.0898076
\(818\) −35.8894 −1.25484
\(819\) 0 0
\(820\) 0 0
\(821\) 34.1584 1.19214 0.596068 0.802934i \(-0.296729\pi\)
0.596068 + 0.802934i \(0.296729\pi\)
\(822\) −10.7718 −0.375708
\(823\) −4.07940 −0.142199 −0.0710995 0.997469i \(-0.522651\pi\)
−0.0710995 + 0.997469i \(0.522651\pi\)
\(824\) −54.8616 −1.91119
\(825\) 0 0
\(826\) 0.752744 0.0261913
\(827\) −54.8780 −1.90830 −0.954148 0.299337i \(-0.903235\pi\)
−0.954148 + 0.299337i \(0.903235\pi\)
\(828\) −4.86550 −0.169088
\(829\) 8.14950 0.283044 0.141522 0.989935i \(-0.454800\pi\)
0.141522 + 0.989935i \(0.454800\pi\)
\(830\) 0 0
\(831\) −40.8697 −1.41775
\(832\) 0 0
\(833\) 6.74745 0.233785
\(834\) −5.72866 −0.198367
\(835\) 0 0
\(836\) −6.24213 −0.215889
\(837\) −7.14441 −0.246947
\(838\) 8.49869 0.293582
\(839\) 21.8865 0.755606 0.377803 0.925886i \(-0.376680\pi\)
0.377803 + 0.925886i \(0.376680\pi\)
\(840\) 0 0
\(841\) −28.9994 −0.999979
\(842\) −8.68570 −0.299329
\(843\) 25.0370 0.862321
\(844\) 0.171902 0.00591710
\(845\) 0 0
\(846\) 7.69213 0.264461
\(847\) 64.2866 2.20891
\(848\) −12.0429 −0.413556
\(849\) 3.07480 0.105527
\(850\) 0 0
\(851\) −33.8838 −1.16152
\(852\) 12.9024 0.442028
\(853\) −19.2240 −0.658217 −0.329108 0.944292i \(-0.606748\pi\)
−0.329108 + 0.944292i \(0.606748\pi\)
\(854\) 14.7552 0.504911
\(855\) 0 0
\(856\) −27.9827 −0.956430
\(857\) 27.8197 0.950302 0.475151 0.879904i \(-0.342393\pi\)
0.475151 + 0.879904i \(0.342393\pi\)
\(858\) 0 0
\(859\) 45.7355 1.56048 0.780238 0.625482i \(-0.215098\pi\)
0.780238 + 0.625482i \(0.215098\pi\)
\(860\) 0 0
\(861\) 31.3363 1.06794
\(862\) −36.8521 −1.25519
\(863\) 54.8186 1.86605 0.933024 0.359814i \(-0.117160\pi\)
0.933024 + 0.359814i \(0.117160\pi\)
\(864\) 3.68705 0.125436
\(865\) 0 0
\(866\) −1.46378 −0.0497413
\(867\) 36.6604 1.24505
\(868\) 10.0791 0.342108
\(869\) 64.3684 2.18355
\(870\) 0 0
\(871\) 0 0
\(872\) 22.6024 0.765415
\(873\) 29.6827 1.00461
\(874\) −10.7684 −0.364247
\(875\) 0 0
\(876\) −5.61999 −0.189882
\(877\) −27.3794 −0.924537 −0.462269 0.886740i \(-0.652964\pi\)
−0.462269 + 0.886740i \(0.652964\pi\)
\(878\) −20.1908 −0.681407
\(879\) −43.7125 −1.47439
\(880\) 0 0
\(881\) −34.4426 −1.16040 −0.580200 0.814474i \(-0.697026\pi\)
−0.580200 + 0.814474i \(0.697026\pi\)
\(882\) 17.7368 0.597230
\(883\) 17.3592 0.584183 0.292092 0.956390i \(-0.405649\pi\)
0.292092 + 0.956390i \(0.405649\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −5.56115 −0.186831
\(887\) 31.1427 1.04567 0.522835 0.852434i \(-0.324874\pi\)
0.522835 + 0.852434i \(0.324874\pi\)
\(888\) −62.2053 −2.08747
\(889\) 41.1722 1.38087
\(890\) 0 0
\(891\) 55.6927 1.86578
\(892\) 6.30042 0.210954
\(893\) −5.86345 −0.196213
\(894\) −15.6960 −0.524952
\(895\) 0 0
\(896\) 18.6224 0.622132
\(897\) 0 0
\(898\) −16.8953 −0.563804
\(899\) 0.135116 0.00450636
\(900\) 0 0
\(901\) 5.02473 0.167398
\(902\) 24.4521 0.814167
\(903\) −9.50367 −0.316262
\(904\) 21.6826 0.721152
\(905\) 0 0
\(906\) −13.9090 −0.462097
\(907\) −17.6057 −0.584587 −0.292294 0.956329i \(-0.594418\pi\)
−0.292294 + 0.956329i \(0.594418\pi\)
\(908\) 3.91050 0.129774
\(909\) −9.89022 −0.328038
\(910\) 0 0
\(911\) 50.0232 1.65734 0.828671 0.559737i \(-0.189098\pi\)
0.828671 + 0.559737i \(0.189098\pi\)
\(912\) −14.3489 −0.475139
\(913\) 65.1526 2.15624
\(914\) 48.9272 1.61837
\(915\) 0 0
\(916\) −7.37844 −0.243791
\(917\) −38.0468 −1.25641
\(918\) 1.80504 0.0595752
\(919\) −7.61556 −0.251214 −0.125607 0.992080i \(-0.540088\pi\)
−0.125607 + 0.992080i \(0.540088\pi\)
\(920\) 0 0
\(921\) −33.3611 −1.09928
\(922\) 9.19522 0.302828
\(923\) 0 0
\(924\) −23.1100 −0.760264
\(925\) 0 0
\(926\) 28.4225 0.934022
\(927\) 43.6733 1.43442
\(928\) −0.0697297 −0.00228899
\(929\) 14.1239 0.463391 0.231695 0.972788i \(-0.425573\pi\)
0.231695 + 0.972788i \(0.425573\pi\)
\(930\) 0 0
\(931\) −13.5202 −0.443106
\(932\) 4.86550 0.159375
\(933\) −6.43937 −0.210816
\(934\) 27.6012 0.903138
\(935\) 0 0
\(936\) 0 0
\(937\) 23.9317 0.781815 0.390908 0.920430i \(-0.372161\pi\)
0.390908 + 0.920430i \(0.372161\pi\)
\(938\) 28.1023 0.917571
\(939\) −38.2157 −1.24712
\(940\) 0 0
\(941\) 25.3591 0.826683 0.413342 0.910576i \(-0.364362\pi\)
0.413342 + 0.910576i \(0.364362\pi\)
\(942\) 28.5614 0.930582
\(943\) −14.5284 −0.473110
\(944\) −0.465033 −0.0151355
\(945\) 0 0
\(946\) −7.41584 −0.241110
\(947\) 41.4223 1.34604 0.673021 0.739623i \(-0.264996\pi\)
0.673021 + 0.739623i \(0.264996\pi\)
\(948\) −14.3187 −0.465051
\(949\) 0 0
\(950\) 0 0
\(951\) −4.16480 −0.135053
\(952\) −12.4867 −0.404696
\(953\) −24.3026 −0.787237 −0.393619 0.919274i \(-0.628777\pi\)
−0.393619 + 0.919274i \(0.628777\pi\)
\(954\) 13.2083 0.427636
\(955\) 0 0
\(956\) −10.2034 −0.330001
\(957\) −0.309801 −0.0100144
\(958\) −25.1802 −0.813536
\(959\) 13.6329 0.440231
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) 22.2760 0.717834
\(964\) 11.9209 0.383945
\(965\) 0 0
\(966\) −39.8675 −1.28272
\(967\) 23.6784 0.761445 0.380722 0.924689i \(-0.375675\pi\)
0.380722 + 0.924689i \(0.375675\pi\)
\(968\) −54.7173 −1.75868
\(969\) 5.98685 0.192325
\(970\) 0 0
\(971\) −16.9722 −0.544663 −0.272332 0.962203i \(-0.587795\pi\)
−0.272332 + 0.962203i \(0.587795\pi\)
\(972\) −10.3791 −0.332908
\(973\) 7.25030 0.232434
\(974\) 3.70303 0.118653
\(975\) 0 0
\(976\) −9.11550 −0.291780
\(977\) 24.9994 0.799802 0.399901 0.916558i \(-0.369044\pi\)
0.399901 + 0.916558i \(0.369044\pi\)
\(978\) −19.2900 −0.616826
\(979\) −86.7765 −2.77339
\(980\) 0 0
\(981\) −17.9930 −0.574471
\(982\) 13.0116 0.415218
\(983\) −27.3418 −0.872068 −0.436034 0.899930i \(-0.643617\pi\)
−0.436034 + 0.899930i \(0.643617\pi\)
\(984\) −26.6718 −0.850267
\(985\) 0 0
\(986\) −0.0341370 −0.00108715
\(987\) −21.7080 −0.690975
\(988\) 0 0
\(989\) 4.40617 0.140108
\(990\) 0 0
\(991\) 16.0760 0.510672 0.255336 0.966852i \(-0.417814\pi\)
0.255336 + 0.966852i \(0.417814\pi\)
\(992\) −15.4081 −0.489208
\(993\) 16.8491 0.534690
\(994\) 47.4123 1.50383
\(995\) 0 0
\(996\) −14.4932 −0.459234
\(997\) 34.5612 1.09456 0.547282 0.836948i \(-0.315663\pi\)
0.547282 + 0.836948i \(0.315663\pi\)
\(998\) −41.3649 −1.30938
\(999\) −11.3807 −0.360070
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 4225.2.a.bl.1.3 4
5.4 even 2 845.2.a.l.1.2 4
13.6 odd 12 325.2.n.d.101.2 8
13.11 odd 12 325.2.n.d.251.2 8
13.12 even 2 4225.2.a.bi.1.2 4
15.14 odd 2 7605.2.a.cj.1.3 4
65.4 even 6 845.2.e.m.146.2 8
65.9 even 6 845.2.e.n.146.3 8
65.19 odd 12 65.2.m.a.36.3 8
65.24 odd 12 65.2.m.a.56.3 yes 8
65.29 even 6 845.2.e.n.191.3 8
65.32 even 12 325.2.m.b.49.2 8
65.34 odd 4 845.2.c.g.506.3 8
65.37 even 12 325.2.m.c.199.3 8
65.44 odd 4 845.2.c.g.506.6 8
65.49 even 6 845.2.e.m.191.2 8
65.54 odd 12 845.2.m.g.316.2 8
65.58 even 12 325.2.m.c.49.3 8
65.59 odd 12 845.2.m.g.361.2 8
65.63 even 12 325.2.m.b.199.2 8
65.64 even 2 845.2.a.m.1.3 4
195.89 even 12 585.2.bu.c.316.2 8
195.149 even 12 585.2.bu.c.361.2 8
195.194 odd 2 7605.2.a.cf.1.2 4
260.19 even 12 1040.2.da.b.881.4 8
260.219 even 12 1040.2.da.b.641.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.3 8 65.19 odd 12
65.2.m.a.56.3 yes 8 65.24 odd 12
325.2.m.b.49.2 8 65.32 even 12
325.2.m.b.199.2 8 65.63 even 12
325.2.m.c.49.3 8 65.58 even 12
325.2.m.c.199.3 8 65.37 even 12
325.2.n.d.101.2 8 13.6 odd 12
325.2.n.d.251.2 8 13.11 odd 12
585.2.bu.c.316.2 8 195.89 even 12
585.2.bu.c.361.2 8 195.149 even 12
845.2.a.l.1.2 4 5.4 even 2
845.2.a.m.1.3 4 65.64 even 2
845.2.c.g.506.3 8 65.34 odd 4
845.2.c.g.506.6 8 65.44 odd 4
845.2.e.m.146.2 8 65.4 even 6
845.2.e.m.191.2 8 65.49 even 6
845.2.e.n.146.3 8 65.9 even 6
845.2.e.n.191.3 8 65.29 even 6
845.2.m.g.316.2 8 65.54 odd 12
845.2.m.g.361.2 8 65.59 odd 12
1040.2.da.b.641.4 8 260.219 even 12
1040.2.da.b.881.4 8 260.19 even 12
4225.2.a.bi.1.2 4 13.12 even 2
4225.2.a.bl.1.3 4 1.1 even 1 trivial
7605.2.a.cf.1.2 4 195.194 odd 2
7605.2.a.cj.1.3 4 15.14 odd 2