Properties

Label 845.2.a.l.1.2
Level $845$
Weight $2$
Character 845.1
Self dual yes
Analytic conductor $6.747$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.74735897080\)
Analytic rank: \(1\)
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 1.2
Root \(1.21969\) of defining polynomial
Character \(\chi\) \(=\) 845.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.21969 q^{2} +2.33225 q^{3} -0.512364 q^{4} +1.00000 q^{5} -2.84461 q^{6} -3.60020 q^{7} +3.06430 q^{8} +2.43937 q^{9} +O(q^{10})\) \(q-1.21969 q^{2} +2.33225 q^{3} -0.512364 q^{4} +1.00000 q^{5} -2.84461 q^{6} -3.60020 q^{7} +3.06430 q^{8} +2.43937 q^{9} -1.21969 q^{10} -5.37182 q^{11} -1.19496 q^{12} +4.39111 q^{14} +2.33225 q^{15} -2.71276 q^{16} -1.13186 q^{17} -2.97527 q^{18} -2.26795 q^{19} -0.512364 q^{20} -8.39654 q^{21} +6.55193 q^{22} -3.89287 q^{23} +7.14670 q^{24} +1.00000 q^{25} -1.30752 q^{27} +1.84461 q^{28} -0.0247279 q^{29} -2.84461 q^{30} -5.46410 q^{31} -2.81988 q^{32} -12.5284 q^{33} +1.38051 q^{34} -3.60020 q^{35} -1.24985 q^{36} +8.70406 q^{37} +2.76619 q^{38} +3.06430 q^{40} -3.73205 q^{41} +10.2412 q^{42} -1.13186 q^{43} +2.75232 q^{44} +2.43937 q^{45} +4.74809 q^{46} -2.58535 q^{47} -6.32681 q^{48} +5.96141 q^{49} -1.21969 q^{50} -2.63977 q^{51} -4.43937 q^{53} +1.59476 q^{54} -5.37182 q^{55} -11.0321 q^{56} -5.28942 q^{57} +0.0301603 q^{58} +0.171425 q^{59} -1.19496 q^{60} +3.36023 q^{61} +6.66449 q^{62} -8.78222 q^{63} +8.86488 q^{64} +15.2807 q^{66} -6.39980 q^{67} +0.579922 q^{68} -9.07914 q^{69} +4.39111 q^{70} +10.7973 q^{71} +7.47497 q^{72} +4.70308 q^{73} -10.6162 q^{74} +2.33225 q^{75} +1.16202 q^{76} +19.3396 q^{77} -11.9826 q^{79} -2.71276 q^{80} -10.3676 q^{81} +4.55193 q^{82} +12.1286 q^{83} +4.30209 q^{84} -1.13186 q^{85} +1.38051 q^{86} -0.0576715 q^{87} -16.4608 q^{88} +16.1540 q^{89} -2.97527 q^{90} +1.99457 q^{92} -12.7436 q^{93} +3.15332 q^{94} -2.26795 q^{95} -6.57666 q^{96} -12.1682 q^{97} -7.27105 q^{98} -13.1039 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} + 4 q^{6} - 10 q^{7} - 6 q^{8} + 4 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} + 4 q^{6} - 10 q^{7} - 6 q^{8} + 4 q^{9} - 2 q^{10} - 10 q^{12} + 2 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{17} - 20 q^{18} - 16 q^{19} + 2 q^{20} - 4 q^{21} + 12 q^{22} - 10 q^{23} + 24 q^{24} + 4 q^{25} - 2 q^{27} - 8 q^{28} + 8 q^{29} + 4 q^{30} - 8 q^{31} - 4 q^{32} - 18 q^{33} + 4 q^{34} - 10 q^{35} + 20 q^{36} + 2 q^{37} + 8 q^{38} - 6 q^{40} - 8 q^{41} - 4 q^{42} - 2 q^{43} - 12 q^{44} + 4 q^{45} - 16 q^{46} - 8 q^{47} - 28 q^{48} + 12 q^{49} - 2 q^{50} + 4 q^{51} - 12 q^{53} + 16 q^{54} + 12 q^{56} + 14 q^{57} - 22 q^{58} - 12 q^{59} - 10 q^{60} + 28 q^{61} + 4 q^{62} - 4 q^{63} + 4 q^{64} + 6 q^{66} - 30 q^{67} + 14 q^{68} - 16 q^{69} + 2 q^{70} - 4 q^{71} - 12 q^{72} + 8 q^{73} - 10 q^{74} - 2 q^{75} - 20 q^{76} + 18 q^{77} - 8 q^{79} + 2 q^{80} - 8 q^{81} + 4 q^{82} + 12 q^{83} + 28 q^{84} - 2 q^{85} + 4 q^{86} - 22 q^{87} - 18 q^{88} + 12 q^{89} - 20 q^{90} + 22 q^{92} - 8 q^{93} - 32 q^{94} - 16 q^{95} - 4 q^{96} - 2 q^{97} + 24 q^{98} - 24 q^{99} + O(q^{100}) \)

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 \(-0.641918\pi\)
−0.431224 + 0.902245i \(0.641918\pi\)
\(3\) 2.33225 1.34652 0.673262 0.739404i \(-0.264893\pi\)
0.673262 + 0.739404i \(0.264893\pi\)
\(4\) −0.512364 −0.256182
\(5\) 1.00000 0.447214
\(6\) −2.84461 −1.16131
\(7\) −3.60020 −1.36075 −0.680373 0.732866i \(-0.738182\pi\)
−0.680373 + 0.732866i \(0.738182\pi\)
\(8\) 3.06430 1.08339
\(9\) 2.43937 0.813125
\(10\) −1.21969 −0.385699
\(11\) −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
\(15\) 2.33225 0.602183
\(16\) −2.71276 −0.678189
\(17\) −1.13186 −0.274515 −0.137258 0.990535i \(-0.543829\pi\)
−0.137258 + 0.990535i \(0.543829\pi\)
\(18\) −2.97527 −0.701278
\(19\) −2.26795 −0.520303 −0.260152 0.965568i \(-0.583773\pi\)
−0.260152 + 0.965568i \(0.583773\pi\)
\(20\) −0.512364 −0.114568
\(21\) −8.39654 −1.83228
\(22\) 6.55193 1.39688
\(23\) −3.89287 −0.811720 −0.405860 0.913935i \(-0.633028\pi\)
−0.405860 + 0.913935i \(0.633028\pi\)
\(24\) 7.14670 1.45881
\(25\) 1.00000 0.200000
\(26\) 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 \(0.500731\pi\)
\(30\) −2.84461 −0.519352
\(31\) −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
\(35\) −3.60020 −0.608544
\(36\) −1.24985 −0.208308
\(37\) 8.70406 1.43094 0.715470 0.698644i \(-0.246213\pi\)
0.715470 + 0.698644i \(0.246213\pi\)
\(38\) 2.76619 0.448735
\(39\) 0 0
\(40\) 3.06430 0.484508
\(41\) −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 \(-0.527505\pi\)
−0.0863031 + 0.996269i \(0.527505\pi\)
\(44\) 2.75232 0.414929
\(45\) 2.43937 0.363640
\(46\) 4.74809 0.700067
\(47\) −2.58535 −0.377113 −0.188556 0.982062i \(-0.560381\pi\)
−0.188556 + 0.982062i \(0.560381\pi\)
\(48\) −6.32681 −0.913197
\(49\) 5.96141 0.851630
\(50\) −1.21969 −0.172490
\(51\) −2.63977 −0.369641
\(52\) 0 0
\(53\) −4.43937 −0.609795 −0.304897 0.952385i \(-0.598622\pi\)
−0.304897 + 0.952385i \(0.598622\pi\)
\(54\) 1.59476 0.217020
\(55\) −5.37182 −0.724336
\(56\) −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 \(0.496448\pi\)
\(60\) −1.19496 −0.154269
\(61\) 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 \(-0.627847\pi\)
−0.390930 + 0.920420i \(0.627847\pi\)
\(68\) 0.579922 0.0703258
\(69\) −9.07914 −1.09300
\(70\) 4.39111 0.524838
\(71\) 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 \(-0.411247\pi\)
0.275227 + 0.961379i \(0.411247\pi\)
\(74\) −10.6162 −1.23411
\(75\) 2.33225 0.269305
\(76\) 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 \(0.735457\pi\)
\(80\) −2.71276 −0.303295
\(81\) −10.3676 −1.15195
\(82\) 4.55193 0.502677
\(83\) 12.1286 1.33129 0.665643 0.746270i \(-0.268157\pi\)
0.665643 + 0.746270i \(0.268157\pi\)
\(84\) 4.30209 0.469396
\(85\) −1.13186 −0.122767
\(86\) 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 \(0.172842\pi\)
\(90\) −2.97527 −0.313621
\(91\) 0 0
\(92\) 1.99457 0.207948
\(93\) −12.7436 −1.32145
\(94\) 3.15332 0.325240
\(95\) −2.26795 −0.232687
\(96\) −6.57666 −0.671228
\(97\) −12.1682 −1.23549 −0.617745 0.786379i \(-0.711954\pi\)
−0.617745 + 0.786379i \(0.711954\pi\)
\(98\) −7.27105 −0.734487
\(99\) −13.1039 −1.31699
\(100\) −0.512364 −0.0512364
\(101\) −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 \(-0.843831\pi\)
−0.882041 + 0.471173i \(0.843831\pi\)
\(104\) 0 0
\(105\) −8.39654 −0.819419
\(106\) 5.41465 0.525917
\(107\) −9.13186 −0.882810 −0.441405 0.897308i \(-0.645520\pi\)
−0.441405 + 0.897308i \(0.645520\pi\)
\(108\) 0.669925 0.0644636
\(109\) −7.37605 −0.706498 −0.353249 0.935529i \(-0.614923\pi\)
−0.353249 + 0.935529i \(0.614923\pi\)
\(110\) 6.55193 0.624702
\(111\) 20.3000 1.92679
\(112\) 9.76645 0.922843
\(113\) 7.07588 0.665643 0.332821 0.942990i \(-0.391999\pi\)
0.332821 + 0.942990i \(0.391999\pi\)
\(114\) 6.45143 0.604232
\(115\) −3.89287 −0.363012
\(116\) 0.0126697 0.00117635
\(117\) 0 0
\(118\) −0.209084 −0.0192478
\(119\) 4.07490 0.373545
\(120\) 7.14670 0.652401
\(121\) 17.8564 1.62331
\(122\) −4.09843 −0.371055
\(123\) −8.70406 −0.784819
\(124\) 2.79961 0.251412
\(125\) 1.00000 0.0894427
\(126\) 10.7116 0.954262
\(127\) −11.4361 −1.01479 −0.507395 0.861713i \(-0.669392\pi\)
−0.507395 + 0.861713i \(0.669392\pi\)
\(128\) −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
\(135\) −1.30752 −0.112533
\(136\) −3.46834 −0.297408
\(137\) −3.78672 −0.323522 −0.161761 0.986830i \(-0.551717\pi\)
−0.161761 + 0.986830i \(0.551717\pi\)
\(138\) 11.0737 0.942656
\(139\) 2.01386 0.170814 0.0854068 0.996346i \(-0.472781\pi\)
0.0854068 + 0.996346i \(0.472781\pi\)
\(140\) 1.84461 0.155898
\(141\) −6.02968 −0.507791
\(142\) −13.1694 −1.10515
\(143\) 0 0
\(144\) −6.61742 −0.551452
\(145\) −0.0247279 −0.00205354
\(146\) −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 \(0.427429\pi\)
\(150\) −2.84461 −0.232261
\(151\) 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
\(155\) −5.46410 −0.438887
\(156\) 0 0
\(157\) 10.0405 0.801323 0.400661 0.916226i \(-0.368780\pi\)
0.400661 + 0.916226i \(0.368780\pi\)
\(158\) 14.6150 1.16271
\(159\) −10.3537 −0.821103
\(160\) −2.81988 −0.222931
\(161\) 14.0151 1.10454
\(162\) 12.6452 0.993501
\(163\) −6.78124 −0.531148 −0.265574 0.964090i \(-0.585561\pi\)
−0.265574 + 0.964090i \(0.585561\pi\)
\(164\) 1.91217 0.149315
\(165\) −12.5284 −0.975335
\(166\) −14.7931 −1.14817
\(167\) −10.4898 −0.811726 −0.405863 0.913934i \(-0.633029\pi\)
−0.405863 + 0.913934i \(0.633029\pi\)
\(168\) −25.7295 −1.98507
\(169\) 0 0
\(170\) 1.38051 0.105880
\(171\) −5.53238 −0.423071
\(172\) 0.579922 0.0442186
\(173\) −4.45845 −0.338970 −0.169485 0.985533i \(-0.554210\pi\)
−0.169485 + 0.985533i \(0.554210\pi\)
\(174\) 0.0703412 0.00533255
\(175\) −3.60020 −0.272149
\(176\) 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 \(0.254835\pi\)
\(180\) −1.24985 −0.0931581
\(181\) 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
\(185\) 8.70406 0.639935
\(186\) 15.5432 1.13969
\(187\) 6.08012 0.444622
\(188\) 1.32464 0.0966095
\(189\) 4.70732 0.342407
\(190\) 2.76619 0.200680
\(191\) 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 \(-0.786529\pi\)
−0.783425 + 0.621486i \(0.786529\pi\)
\(194\) 14.8413 1.06555
\(195\) 0 0
\(196\) −3.05441 −0.218172
\(197\) 1.69672 0.120886 0.0604432 0.998172i \(-0.480749\pi\)
0.0604432 + 0.998172i \(0.480749\pi\)
\(198\) 15.9826 1.13583
\(199\) 25.3255 1.79527 0.897637 0.440735i \(-0.145282\pi\)
0.897637 + 0.440735i \(0.145282\pi\)
\(200\) 3.06430 0.216679
\(201\) −14.9259 −1.05279
\(202\) 4.94511 0.347937
\(203\) 0.0890252 0.00624834
\(204\) 1.35252 0.0946954
\(205\) −3.73205 −0.260658
\(206\) 21.8366 1.52143
\(207\) −9.49617 −0.660030
\(208\) 0 0
\(209\) 12.1830 0.842716
\(210\) 10.2412 0.706707
\(211\) −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
\(215\) −1.13186 −0.0771919
\(216\) −4.00663 −0.272616
\(217\) 19.6718 1.33541
\(218\) 8.99648 0.609318
\(219\) 10.9688 0.741200
\(220\) 2.75232 0.185562
\(221\) 0 0
\(222\) −24.7597 −1.66176
\(223\) 12.2968 0.823452 0.411726 0.911308i \(-0.364926\pi\)
0.411726 + 0.911308i \(0.364926\pi\)
\(224\) 10.1521 0.678318
\(225\) 2.43937 0.162625
\(226\) −8.63036 −0.574083
\(227\) 7.63227 0.506571 0.253286 0.967392i \(-0.418489\pi\)
0.253286 + 0.967392i \(0.418489\pi\)
\(228\) 2.71011 0.179481
\(229\) 14.4008 0.951631 0.475815 0.879545i \(-0.342153\pi\)
0.475815 + 0.879545i \(0.342153\pi\)
\(230\) 4.74809 0.313080
\(231\) 45.1047 2.96767
\(232\) −0.0757736 −0.00497478
\(233\) 9.49617 0.622115 0.311057 0.950391i \(-0.399317\pi\)
0.311057 + 0.950391i \(0.399317\pi\)
\(234\) 0 0
\(235\) −2.58535 −0.168650
\(236\) −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 \(0.277242\pi\)
\(240\) −6.32681 −0.408394
\(241\) −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
\(245\) 5.96141 0.380860
\(246\) 10.6162 0.676866
\(247\) 0 0
\(248\) −16.7436 −1.06322
\(249\) 28.2869 1.79261
\(250\) −1.21969 −0.0771398
\(251\) 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
\(255\) −2.63977 −0.165309
\(256\) −11.4208 −0.713800
\(257\) −5.55002 −0.346201 −0.173100 0.984904i \(-0.555379\pi\)
−0.173100 + 0.984904i \(0.555379\pi\)
\(258\) 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 \(-0.432168\pi\)
0.211493 + 0.977380i \(0.432168\pi\)
\(264\) −38.3907 −2.36279
\(265\) −4.43937 −0.272709
\(266\) −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 \(0.513803\pi\)
\(270\) 1.59476 0.0970542
\(271\) 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
\(275\) −5.37182 −0.323933
\(276\) 4.65182 0.280007
\(277\) −17.5237 −1.05290 −0.526449 0.850206i \(-0.676477\pi\)
−0.526449 + 0.850206i \(0.676477\pi\)
\(278\) −2.45628 −0.147318
\(279\) −13.3290 −0.797986
\(280\) −11.0321 −0.659292
\(281\) −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 \(-0.487524\pi\)
0.0391849 + 0.999232i \(0.487524\pi\)
\(284\) −5.53216 −0.328273
\(285\) −5.28942 −0.313318
\(286\) 0 0
\(287\) 13.4361 0.793109
\(288\) −6.87875 −0.405334
\(289\) −15.7189 −0.924641
\(290\) 0.0301603 0.00177107
\(291\) −28.3792 −1.66362
\(292\) −2.40969 −0.141016
\(293\) −18.7427 −1.09496 −0.547479 0.836820i \(-0.684412\pi\)
−0.547479 + 0.836820i \(0.684412\pi\)
\(294\) −16.9579 −0.989004
\(295\) 0.171425 0.00998072
\(296\) 26.6718 1.55027
\(297\) 7.02375 0.407559
\(298\) −6.72998 −0.389857
\(299\) 0 0
\(300\) −1.19496 −0.0689910
\(301\) 4.07490 0.234873
\(302\) −5.96380 −0.343178
\(303\) −9.45589 −0.543226
\(304\) 6.15239 0.352864
\(305\) 3.36023 0.192406
\(306\) 3.36758 0.192512
\(307\) −14.3043 −0.816387 −0.408194 0.912895i \(-0.633841\pi\)
−0.408194 + 0.912895i \(0.633841\pi\)
\(308\) −9.90891 −0.564612
\(309\) −41.7553 −2.37538
\(310\) 6.66449 0.378518
\(311\) 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 \(-0.653259\pi\)
−0.463090 + 0.886311i \(0.653259\pi\)
\(314\) −12.2463 −0.691100
\(315\) −8.78222 −0.494822
\(316\) 6.13946 0.345372
\(317\) −1.78575 −0.100297 −0.0501487 0.998742i \(-0.515970\pi\)
−0.0501487 + 0.998742i \(0.515970\pi\)
\(318\) 12.6283 0.708159
\(319\) 0.132834 0.00743725
\(320\) 8.86488 0.495562
\(321\) −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
\(330\) 15.2807 0.841176
\(331\) −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
\(335\) −6.39980 −0.349659
\(336\) 22.7778 1.24263
\(337\) −4.36219 −0.237624 −0.118812 0.992917i \(-0.537909\pi\)
−0.118812 + 0.992917i \(0.537909\pi\)
\(338\) 0 0
\(339\) 16.5027 0.896303
\(340\) 0.579922 0.0314507
\(341\) 29.3521 1.58951
\(342\) 6.74777 0.364877
\(343\) 3.73913 0.201894
\(344\) −3.46834 −0.187000
\(345\) −9.07914 −0.488804
\(346\) 5.43792 0.292344
\(347\) −26.7072 −1.43372 −0.716858 0.697219i \(-0.754420\pi\)
−0.716858 + 0.697219i \(0.754420\pi\)
\(348\) 0.0295488 0.00158398
\(349\) −23.5711 −1.26173 −0.630865 0.775892i \(-0.717300\pi\)
−0.630865 + 0.775892i \(0.717300\pi\)
\(350\) 4.39111 0.234715
\(351\) 0 0
\(352\) 15.1479 0.807385
\(353\) −5.73727 −0.305364 −0.152682 0.988275i \(-0.548791\pi\)
−0.152682 + 0.988275i \(0.548791\pi\)
\(354\) −0.487636 −0.0259176
\(355\) 10.7973 0.573063
\(356\) −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 \(0.273364\pi\)
\(360\) 7.47497 0.393965
\(361\) −13.8564 −0.729285
\(362\) −22.0641 −1.15967
\(363\) 41.6455 2.18582
\(364\) 0 0
\(365\) 4.70308 0.246171
\(366\) −9.55856 −0.499634
\(367\) 26.0535 1.35998 0.679992 0.733220i \(-0.261983\pi\)
0.679992 + 0.733220i \(0.261983\pi\)
\(368\) 10.5604 0.550499
\(369\) −9.10387 −0.473928
\(370\) −10.6162 −0.551912
\(371\) 15.9826 0.829776
\(372\) 6.52938 0.338532
\(373\) 13.2045 0.683702 0.341851 0.939754i \(-0.388946\pi\)
0.341851 + 0.939754i \(0.388946\pi\)
\(374\) −7.41584 −0.383464
\(375\) 2.33225 0.120437
\(376\) −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 \(0.732724\pi\)
\(380\) 1.16202 0.0596101
\(381\) −26.6718 −1.36644
\(382\) −33.3418 −1.70591
\(383\) 9.60020 0.490547 0.245274 0.969454i \(-0.421122\pi\)
0.245274 + 0.969454i \(0.421122\pi\)
\(384\) −12.0638 −0.615629
\(385\) 19.3396 0.985637
\(386\) 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
\(395\) −11.9826 −0.602911
\(396\) 6.71395 0.337389
\(397\) 16.7658 0.841452 0.420726 0.907188i \(-0.361775\pi\)
0.420726 + 0.907188i \(0.361775\pi\)
\(398\) −30.8891 −1.54833
\(399\) 19.0429 0.953339
\(400\) −2.71276 −0.135638
\(401\) 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
\(405\) −10.3676 −0.515169
\(406\) −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 \(0.759313\pi\)
\(410\) 4.55193 0.224804
\(411\) −8.83157 −0.435629
\(412\) 9.17310 0.451926
\(413\) −0.617162 −0.0303686
\(414\) 11.5824 0.569242
\(415\) 12.1286 0.595369
\(416\) 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 \(0.445558\pi\)
\(420\) 4.30209 0.209920
\(421\) −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
\(425\) −1.13186 −0.0549030
\(426\) −30.7142 −1.48811
\(427\) −12.0975 −0.585439
\(428\) 4.67883 0.226160
\(429\) 0 0
\(430\) 1.38051 0.0665740
\(431\) −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 \(-0.490820\pi\)
0.0288373 + 0.999584i \(0.490820\pi\)
\(434\) −23.9935 −1.15172
\(435\) −0.0576715 −0.00276514
\(436\) 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 \(0.629270\pi\)
\(440\) −16.4608 −0.784740
\(441\) 14.5421 0.692481
\(442\) 0 0
\(443\) 4.55949 0.216628 0.108314 0.994117i \(-0.465455\pi\)
0.108314 + 0.994117i \(0.465455\pi\)
\(444\) −10.4010 −0.493610
\(445\) 16.1540 0.765775
\(446\) −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 \(0.605991\pi\)
\(450\) −2.97527 −0.140256
\(451\) 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 \(-0.887545\pi\)
−0.938240 + 0.345984i \(0.887545\pi\)
\(458\) −17.5644 −0.820733
\(459\) 1.47992 0.0690768
\(460\) 1.99457 0.0929972
\(461\) 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 \(-0.682141\pi\)
−0.541494 + 0.840705i \(0.682141\pi\)
\(464\) 0.0670807 0.00311414
\(465\) −12.7436 −0.590972
\(466\) −11.5824 −0.536542
\(467\) −22.6297 −1.04718 −0.523589 0.851971i \(-0.675407\pi\)
−0.523589 + 0.851971i \(0.675407\pi\)
\(468\) 0 0
\(469\) 23.0405 1.06391
\(470\) 3.15332 0.145452
\(471\) 23.4170 1.07900
\(472\) 0.525296 0.0241787
\(473\) 6.08012 0.279564
\(474\) 34.0859 1.56562
\(475\) −2.26795 −0.104061
\(476\) −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 \(0.656339\pi\)
\(480\) −6.57666 −0.300182
\(481\) 0 0
\(482\) 28.3777 1.29257
\(483\) 32.6867 1.48730
\(484\) −9.14898 −0.415863
\(485\) −12.1682 −0.552528
\(486\) 24.7074 1.12075
\(487\) −3.03605 −0.137576 −0.0687882 0.997631i \(-0.521913\pi\)
−0.0687882 + 0.997631i \(0.521913\pi\)
\(488\) 10.2968 0.466112
\(489\) −15.8155 −0.715203
\(490\) −7.27105 −0.328473
\(491\) 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
\(495\) −13.1039 −0.588975
\(496\) 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 \(0.774364\pi\)
\(500\) −0.512364 −0.0229136
\(501\) −24.4648 −1.09301
\(502\) −14.4413 −0.644546
\(503\) −12.6276 −0.563037 −0.281518 0.959556i \(-0.590838\pi\)
−0.281518 + 0.959556i \(0.590838\pi\)
\(504\) −26.9113 −1.19873
\(505\) −4.05441 −0.180419
\(506\) −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 \(0.320209\pi\)
\(510\) 3.21969 0.142570
\(511\) −16.9320 −0.749029
\(512\) 24.2750 1.07281
\(513\) 2.96539 0.130925
\(514\) 6.76929 0.298581
\(515\) −17.9035 −0.788921
\(516\) 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 \(-0.199650\pi\)
0.809662 + 0.586897i \(0.199650\pi\)
\(524\) 5.41465 0.236540
\(525\) −8.39654 −0.366455
\(526\) −8.36665 −0.364803
\(527\) 6.18457 0.269404
\(528\) 33.9865 1.47907
\(529\) −7.84554 −0.341111
\(530\) 5.41465 0.235197
\(531\) 0.418169 0.0181470
\(532\) −4.18348 −0.181377
\(533\) 0 0
\(534\) −45.9519 −1.98854
\(535\) −9.13186 −0.394805
\(536\) −19.6109 −0.847062
\(537\) 43.4528 1.87512
\(538\) 1.73438 0.0747744
\(539\) −32.0236 −1.37935
\(540\) 0.669925 0.0288290
\(541\) 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
\(545\) −7.37605 −0.315955
\(546\) 0 0
\(547\) −22.7842 −0.974181 −0.487091 0.873351i \(-0.661942\pi\)
−0.487091 + 0.873351i \(0.661942\pi\)
\(548\) 1.94018 0.0828804
\(549\) 8.19687 0.349834
\(550\) 6.55193 0.279375
\(551\) 0.0560816 0.00238915
\(552\) −27.8212 −1.18415
\(553\) 43.1398 1.83449
\(554\) 21.3735 0.908071
\(555\) 20.3000 0.861688
\(556\) −1.03183 −0.0437594
\(557\) 28.1527 1.19287 0.596435 0.802662i \(-0.296583\pi\)
0.596435 + 0.802662i \(0.296583\pi\)
\(558\) 16.2572 0.688222
\(559\) 0 0
\(560\) 9.76645 0.412708
\(561\) 14.1803 0.598694
\(562\) 13.0935 0.552317
\(563\) −18.1303 −0.764101 −0.382050 0.924142i \(-0.624782\pi\)
−0.382050 + 0.924142i \(0.624782\pi\)
\(564\) 3.08939 0.130087
\(565\) 7.07588 0.297684
\(566\) −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 \(0.176004\pi\)
\(570\) 6.45143 0.270221
\(571\) 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
\(575\) −3.89287 −0.162344
\(576\) 21.6248 0.901032
\(577\) −23.0691 −0.960379 −0.480189 0.877165i \(-0.659432\pi\)
−0.480189 + 0.877165i \(0.659432\pi\)
\(578\) 19.1721 0.797456
\(579\) −50.7669 −2.10980
\(580\) 0.0126697 0.000526079 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 \(-0.362025\pi\)
0.420015 + 0.907517i \(0.362025\pi\)
\(588\) −7.12364 −0.293774
\(589\) 12.3923 0.510616
\(590\) −0.209084 −0.00860786
\(591\) 3.95717 0.162776
\(592\) −23.6120 −0.970447
\(593\) −10.3834 −0.426395 −0.213198 0.977009i \(-0.568388\pi\)
−0.213198 + 0.977009i \(0.568388\pi\)
\(594\) −8.56677 −0.351499
\(595\) 4.07490 0.167055
\(596\) −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 \(0.723351\pi\)
\(600\) 7.14670 0.291763
\(601\) −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
\(605\) 17.8564 0.725966
\(606\) 11.5332 0.468505
\(607\) 2.17540 0.0882968 0.0441484 0.999025i \(-0.485943\pi\)
0.0441484 + 0.999025i \(0.485943\pi\)
\(608\) 6.39535 0.259366
\(609\) 0.207629 0.00841354
\(610\) −4.09843 −0.165941
\(611\) 0 0
\(612\) 1.41465 0.0571837
\(613\) 14.7620 0.596231 0.298116 0.954530i \(-0.403642\pi\)
0.298116 + 0.954530i \(0.403642\pi\)
\(614\) 17.4467 0.704092
\(615\) −8.70406 −0.350982
\(616\) 59.2622 2.38774
\(617\) 20.2972 0.817134 0.408567 0.912728i \(-0.366029\pi\)
0.408567 + 0.912728i \(0.366029\pi\)
\(618\) 50.9284 2.04864
\(619\) −9.94207 −0.399605 −0.199803 0.979836i \(-0.564030\pi\)
−0.199803 + 0.979836i \(0.564030\pi\)
\(620\) 2.79961 0.112435
\(621\) 5.09000 0.204255
\(622\) −3.36758 −0.135028
\(623\) −58.1577 −2.33004
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 19.9855 0.798782
\(627\) 28.4138 1.13474
\(628\) −5.14441 −0.205284
\(629\) −9.85174 −0.392815
\(630\) 10.7116 0.426759
\(631\) −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
\(635\) −11.4361 −0.453828
\(636\) 5.30487 0.210352
\(637\) 0 0
\(638\) −0.162015 −0.00641425
\(639\) 26.3387 1.04194
\(640\) −5.17262 −0.204466
\(641\) −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 \(-0.562925\pi\)
−0.196401 + 0.980524i \(0.562925\pi\)
\(644\) −7.18083 −0.282964
\(645\) −2.63977 −0.103941
\(646\) −3.13092 −0.123185
\(647\) −36.2763 −1.42617 −0.713084 0.701079i \(-0.752702\pi\)
−0.713084 + 0.701079i \(0.752702\pi\)
\(648\) −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 \(-0.413259\pi\)
0.269145 + 0.963100i \(0.413259\pi\)
\(654\) 20.9820 0.820461
\(655\) −10.5680 −0.412925
\(656\) 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 \(0.516014\pi\)
\(660\) 6.41910 0.249863
\(661\) −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
\(665\) 8.16506 0.316627
\(666\) −25.8970 −1.00349
\(667\) 0.0962625 0.00372730
\(668\) 5.37460 0.207949
\(669\) 28.6791 1.10880
\(670\) 7.80576 0.301563
\(671\) −18.0506 −0.696834
\(672\) 23.6773 0.913370
\(673\) 43.3222 1.66995 0.834974 0.550289i \(-0.185483\pi\)
0.834974 + 0.550289i \(0.185483\pi\)
\(674\) 5.32051 0.204938
\(675\) −1.30752 −0.0503264
\(676\) 0 0
\(677\) −41.3625 −1.58969 −0.794845 0.606813i \(-0.792448\pi\)
−0.794845 + 0.606813i \(0.792448\pi\)
\(678\) −20.1281 −0.773016
\(679\) 43.8078 1.68119
\(680\) −3.46834 −0.133005
\(681\) 17.8003 0.682110
\(682\) −35.8004 −1.37087
\(683\) −2.62688 −0.100515 −0.0502574 0.998736i \(-0.516004\pi\)
−0.0502574 + 0.998736i \(0.516004\pi\)
\(684\) 2.83459 0.108383
\(685\) −3.78672 −0.144683
\(686\) −4.56057 −0.174123
\(687\) 33.5862 1.28139
\(688\) 3.07045 0.117060
\(689\) 0 0
\(690\) 11.0737 0.421569
\(691\) 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
\(695\) 2.01386 0.0763902
\(696\) −0.176723 −0.00669865
\(697\) 4.22414 0.160001
\(698\) 28.7493 1.08818
\(699\) 22.1474 0.837692
\(700\) 1.84461 0.0697197
\(701\) −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
\(705\) −6.02968 −0.227091
\(706\) 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 \(0.239569\pi\)
\(710\) −13.1694 −0.494237
\(711\) −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 \(0.460627\pi\)
\(720\) −6.61742 −0.246617
\(721\) 64.4560 2.40047
\(722\) 16.9005 0.628971
\(723\) −54.2629 −2.01806
\(724\) −9.26867 −0.344467
\(725\) −0.0247279 −0.000918370 0
\(726\) −50.7945 −1.88516
\(727\) −18.3735 −0.681435 −0.340717 0.940166i \(-0.610670\pi\)
−0.340717 + 0.940166i \(0.610670\pi\)
\(728\) 0 0
\(729\) −16.1420 −0.597853
\(730\) −5.73629 −0.212310
\(731\) 1.28110 0.0473830
\(732\) −4.01534 −0.148411
\(733\) −0.791131 −0.0292211 −0.0146105 0.999893i \(-0.504651\pi\)
−0.0146105 + 0.999893i \(0.504651\pi\)
\(734\) −31.7772 −1.17292
\(735\) 13.9035 0.512837
\(736\) 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 \(0.694448\pi\)
\(740\) −4.45965 −0.163940
\(741\) 0 0
\(742\) −19.4938 −0.715639
\(743\) −5.56304 −0.204088 −0.102044 0.994780i \(-0.532538\pi\)
−0.102044 + 0.994780i \(0.532538\pi\)
\(744\) −39.0503 −1.43165
\(745\) 5.51780 0.202156
\(746\) −16.1053 −0.589658
\(747\) 29.5862 1.08250
\(748\) −3.11523 −0.113904
\(749\) 32.8765 1.20128
\(750\) −2.84461 −0.103870
\(751\) −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
\(755\) 4.88961 0.177951
\(756\) −2.41186 −0.0877186
\(757\) 50.0446 1.81890 0.909451 0.415810i \(-0.136502\pi\)
0.909451 + 0.415810i \(0.136502\pi\)
\(758\) 31.7091 1.15173
\(759\) 48.7715 1.77029
\(760\) −6.94967 −0.252091
\(761\) −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
\(765\) −2.76102 −0.0998248
\(766\) −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 \(0.248813\pi\)
\(770\) −23.5882 −0.850061
\(771\) −12.9440 −0.466168
\(772\) 11.1528 0.401399
\(773\) −48.7805 −1.75451 −0.877256 0.480022i \(-0.840629\pi\)
−0.877256 + 0.480022i \(0.840629\pi\)
\(774\) 3.36758 0.121045
\(775\) −5.46410 −0.196276
\(776\) −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
\(785\) 10.0405 0.358363
\(786\) 30.0618 1.07227
\(787\) −39.8608 −1.42088 −0.710442 0.703756i \(-0.751505\pi\)
−0.710442 + 0.703756i \(0.751505\pi\)
\(788\) −0.869338 −0.0309689
\(789\) 15.9984 0.569559
\(790\) 14.6150 0.519980
\(791\) −25.4745 −0.905771
\(792\) −40.1541 −1.42682
\(793\) 0 0
\(794\) −20.4490 −0.725709
\(795\) −10.3537 −0.367208
\(796\) −12.9759 −0.459917
\(797\) 26.2118 0.928470 0.464235 0.885712i \(-0.346329\pi\)
0.464235 + 0.885712i \(0.346329\pi\)
\(798\) −23.2264 −0.822206
\(799\) 2.92625 0.103523
\(800\) −2.81988 −0.0996979
\(801\) 39.4057 1.39233
\(802\) −16.9269 −0.597708
\(803\) −25.2641 −0.891551
\(804\) 7.64750 0.269707
\(805\) 14.0151 0.493968
\(806\) 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 \(0.372304\pi\)
\(810\) 12.6452 0.444307
\(811\) −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
\(815\) −6.78124 −0.237537
\(816\) 7.16104 0.250686
\(817\) 2.56699 0.0898076
\(818\) 35.8894 1.25484
\(819\) 0 0
\(820\) 1.91217 0.0667758
\(821\) 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 \(-0.477349\pi\)
0.0710995 + 0.997469i \(0.477349\pi\)
\(824\) −54.8616 −1.91119
\(825\) −12.5284 −0.436183
\(826\) 0.752744 0.0261913
\(827\) 54.8780 1.90830 0.954148 0.299337i \(-0.0967654\pi\)
0.954148 + 0.299337i \(0.0967654\pi\)
\(828\) 4.86550 0.169088
\(829\) 8.14950 0.283044 0.141522 0.989935i \(-0.454800\pi\)
0.141522 + 0.989935i \(0.454800\pi\)
\(830\) −14.7931 −0.513476
\(831\) −40.8697 −1.41775
\(832\) 0 0
\(833\) −6.74745 −0.233785
\(834\) −5.72866 −0.198367
\(835\) −10.4898 −0.363015
\(836\) −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 \(0.376680\pi\)
\(840\) −25.7295 −0.887752
\(841\) −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
\(850\) 1.38051 0.0473511
\(851\) −33.8838 −1.16152
\(852\) −12.9024 −0.442028
\(853\) 19.2240 0.658217 0.329108 0.944292i \(-0.393252\pi\)
0.329108 + 0.944292i \(0.393252\pi\)
\(854\) 14.7552 0.504911
\(855\) −5.53238 −0.189203
\(856\) −27.9827 −0.956430
\(857\) −27.8197 −0.950302 −0.475151 0.879904i \(-0.657607\pi\)
−0.475151 + 0.879904i \(0.657607\pi\)
\(858\) 0 0
\(859\) 45.7355 1.56048 0.780238 0.625482i \(-0.215098\pi\)
0.780238 + 0.625482i \(0.215098\pi\)
\(860\) 0.579922 0.0197752
\(861\) 31.3363 1.06794
\(862\) 36.8521 1.25519
\(863\) −54.8186 −1.86605 −0.933024 0.359814i \(-0.882840\pi\)
−0.933024 + 0.359814i \(0.882840\pi\)
\(864\) 3.68705 0.125436
\(865\) −4.45845 −0.151592
\(866\) −1.46378 −0.0497413
\(867\) −36.6604 −1.24505
\(868\) −10.0791 −0.342108
\(869\) 64.3684 2.18355
\(870\) 0.0703412 0.00238479
\(871\) 0 0
\(872\) −22.6024 −0.765415
\(873\) −29.6827 −1.00461
\(874\) −10.7684 −0.364247
\(875\) −3.60020 −0.121709
\(876\) −5.61999 −0.189882
\(877\) 27.3794 0.924537 0.462269 0.886740i \(-0.347036\pi\)
0.462269 + 0.886740i \(0.347036\pi\)
\(878\) 20.1908 0.681407
\(879\) −43.7125 −1.47439
\(880\) 14.5724 0.491236
\(881\) −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 \(-0.594351\pi\)
−0.292092 + 0.956390i \(0.594351\pi\)
\(884\) 0 0
\(885\) 0.399804 0.0134393
\(886\) −5.56115 −0.186831
\(887\) −31.1427 −1.04567 −0.522835 0.852434i \(-0.675126\pi\)
−0.522835 + 0.852434i \(0.675126\pi\)
\(888\) 62.2053 2.08747
\(889\) 41.1722 1.38087
\(890\) −19.7029 −0.660442
\(891\) 55.6927 1.86578
\(892\) −6.30042 −0.210954
\(893\) 5.86345 0.196213
\(894\) −15.6960 −0.524952
\(895\) 18.6313 0.622775
\(896\) 18.6224 0.622132
\(897\) 0 0
\(898\) 16.8953 0.563804
\(899\) 0.135116 0.00450636
\(900\) −1.24985 −0.0416616
\(901\) 5.02473 0.167398
\(902\) −24.4521 −0.814167
\(903\) 9.50367 0.316262
\(904\) 21.6826 0.721152
\(905\) 18.0900 0.601332
\(906\) −13.9090 −0.462097
\(907\) 17.6057 0.584587 0.292294 0.956329i \(-0.405582\pi\)
0.292294 + 0.956329i \(0.405582\pi\)
\(908\) −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
\(915\) 7.83690 0.259080
\(916\) −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 \(0.540088\pi\)
\(920\) −11.9289 −0.393285
\(921\) −33.3611 −1.09928
\(922\) −9.19522 −0.302828
\(923\) 0 0
\(924\) −23.1100 −0.760264
\(925\) 8.70406 0.286188
\(926\) 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 \(0.425573\pi\)
\(930\) 15.5432 0.509683
\(931\) −13.5202 −0.443106
\(932\) −4.86550 −0.159375
\(933\) 6.43937 0.210816
\(934\) 27.6012 0.903138
\(935\) 6.08012 0.198841
\(936\) 0 0
\(937\) −23.9317 −0.781815 −0.390908 0.920430i \(-0.627839\pi\)
−0.390908 + 0.920430i \(0.627839\pi\)
\(938\) −28.1023 −0.917571
\(939\) −38.2157 −1.24712
\(940\) 1.32464 0.0432051
\(941\) 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
\(945\) 4.70732 0.153129
\(946\) −7.41584 −0.241110
\(947\) −41.4223 −1.34604 −0.673021 0.739623i \(-0.735004\pi\)
−0.673021 + 0.739623i \(0.735004\pi\)
\(948\) 14.3187 0.465051
\(949\) 0 0
\(950\) 2.76619 0.0897470
\(951\) −4.16480 −0.135053
\(952\) 12.4867 0.404696
\(953\) 24.3026 0.787237 0.393619 0.919274i \(-0.371223\pi\)
0.393619 + 0.919274i \(0.371223\pi\)
\(954\) 13.2083 0.427636
\(955\) 27.3363 0.884583
\(956\) −10.2034 −0.330001
\(957\) 0.309801 0.0100144
\(958\) 25.1802 0.813536
\(959\) 13.6329 0.440231
\(960\) 20.6751 0.667286
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) −22.2760 −0.717834
\(964\) 11.9209 0.383945
\(965\) −21.7674 −0.700717
\(966\) −39.8675 −1.28272
\(967\) −23.6784 −0.761445 −0.380722 0.924689i \(-0.624325\pi\)
−0.380722 + 0.924689i \(0.624325\pi\)
\(968\) 54.7173 1.75868
\(969\) 5.98685 0.192325
\(970\) 14.8413 0.476527
\(971\) −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 \(-0.630956\pi\)
−0.399901 + 0.916558i \(0.630956\pi\)
\(978\) 19.2900 0.616826
\(979\) −86.7765 −2.77339
\(980\) −3.05441 −0.0975696
\(981\) −17.9930 −0.574471
\(982\) −13.0116 −0.415218
\(983\) 27.3418 0.872068 0.436034 0.899930i \(-0.356383\pi\)
0.436034 + 0.899930i \(0.356383\pi\)
\(984\) −26.6718 −0.850267
\(985\) 1.69672 0.0540620
\(986\) −0.0341370 −0.00108715
\(987\) 21.7080 0.690975
\(988\) 0 0
\(989\) 4.40617 0.140108
\(990\) 15.9826 0.507961
\(991\) 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
\(995\) 25.3255 0.802871
\(996\) −14.4932 −0.459234
\(997\) −34.5612 −1.09456 −0.547282 0.836948i \(-0.684337\pi\)
−0.547282 + 0.836948i \(0.684337\pi\)
\(998\) 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 845.2.a.l.1.2 4
3.2 odd 2 7605.2.a.cj.1.3 4
5.4 even 2 4225.2.a.bl.1.3 4
13.2 odd 12 845.2.m.g.316.2 8
13.3 even 3 845.2.e.n.191.3 8
13.4 even 6 845.2.e.m.146.2 8
13.5 odd 4 845.2.c.g.506.6 8
13.6 odd 12 65.2.m.a.36.3 8
13.7 odd 12 845.2.m.g.361.2 8
13.8 odd 4 845.2.c.g.506.3 8
13.9 even 3 845.2.e.n.146.3 8
13.10 even 6 845.2.e.m.191.2 8
13.11 odd 12 65.2.m.a.56.3 yes 8
13.12 even 2 845.2.a.m.1.3 4
39.11 even 12 585.2.bu.c.316.2 8
39.32 even 12 585.2.bu.c.361.2 8
39.38 odd 2 7605.2.a.cf.1.2 4
52.11 even 12 1040.2.da.b.641.4 8
52.19 even 12 1040.2.da.b.881.4 8
65.19 odd 12 325.2.n.d.101.2 8
65.24 odd 12 325.2.n.d.251.2 8
65.32 even 12 325.2.m.c.49.3 8
65.37 even 12 325.2.m.b.199.2 8
65.58 even 12 325.2.m.b.49.2 8
65.63 even 12 325.2.m.c.199.3 8
65.64 even 2 4225.2.a.bi.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.3 8 13.6 odd 12
65.2.m.a.56.3 yes 8 13.11 odd 12
325.2.m.b.49.2 8 65.58 even 12
325.2.m.b.199.2 8 65.37 even 12
325.2.m.c.49.3 8 65.32 even 12
325.2.m.c.199.3 8 65.63 even 12
325.2.n.d.101.2 8 65.19 odd 12
325.2.n.d.251.2 8 65.24 odd 12
585.2.bu.c.316.2 8 39.11 even 12
585.2.bu.c.361.2 8 39.32 even 12
845.2.a.l.1.2 4 1.1 even 1 trivial
845.2.a.m.1.3 4 13.12 even 2
845.2.c.g.506.3 8 13.8 odd 4
845.2.c.g.506.6 8 13.5 odd 4
845.2.e.m.146.2 8 13.4 even 6
845.2.e.m.191.2 8 13.10 even 6
845.2.e.n.146.3 8 13.9 even 3
845.2.e.n.191.3 8 13.3 even 3
845.2.m.g.316.2 8 13.2 odd 12
845.2.m.g.361.2 8 13.7 odd 12
1040.2.da.b.641.4 8 52.11 even 12
1040.2.da.b.881.4 8 52.19 even 12
4225.2.a.bi.1.2 4 65.64 even 2
4225.2.a.bl.1.3 4 5.4 even 2
7605.2.a.cf.1.2 4 39.38 odd 2
7605.2.a.cj.1.3 4 3.2 odd 2