Properties

Label 845.2.c.g.506.6
Level $845$
Weight $2$
Character 845.506
Analytic conductor $6.747$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
Defining polynomial: \(x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 506.6
Root \(-1.27597 - 0.609843i\) of defining polynomial
Character \(\chi\) \(=\) 845.506
Dual form 845.2.c.g.506.3

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.21969i 0.862449i 0.902245 + 0.431224i \(0.141918\pi\)
−0.902245 + 0.431224i \(0.858082\pi\)
\(3\) 2.33225 1.34652 0.673262 0.739404i \(-0.264893\pi\)
0.673262 + 0.739404i \(0.264893\pi\)
\(4\) 0.512364 0.256182
\(5\) − 1.00000i − 0.447214i
\(6\) 2.84461i 1.16131i
\(7\) − 3.60020i − 1.36075i −0.732866 0.680373i \(-0.761818\pi\)
0.732866 0.680373i \(-0.238182\pi\)
\(8\) 3.06430i 1.08339i
\(9\) 2.43937 0.813125
\(10\) 1.21969 0.385699
\(11\) − 5.37182i − 1.61966i −0.586662 0.809832i \(-0.699558\pi\)
0.586662 0.809832i \(-0.300442\pi\)
\(12\) 1.19496 0.344955
\(13\) 0 0
\(14\) 4.39111 1.17357
\(15\) − 2.33225i − 0.602183i
\(16\) −2.71276 −0.678189
\(17\) 1.13186 0.274515 0.137258 0.990535i \(-0.456171\pi\)
0.137258 + 0.990535i \(0.456171\pi\)
\(18\) 2.97527i 0.701278i
\(19\) 2.26795i 0.520303i 0.965568 + 0.260152i \(0.0837725\pi\)
−0.965568 + 0.260152i \(0.916227\pi\)
\(20\) − 0.512364i − 0.114568i
\(21\) − 8.39654i − 1.83228i
\(22\) 6.55193 1.39688
\(23\) 3.89287 0.811720 0.405860 0.913935i \(-0.366972\pi\)
0.405860 + 0.913935i \(0.366972\pi\)
\(24\) 7.14670i 1.45881i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.30752 −0.251632
\(28\) − 1.84461i − 0.348599i
\(29\) −0.0247279 −0.00459185 −0.00229593 0.999997i \(-0.500731\pi\)
−0.00229593 + 0.999997i \(0.500731\pi\)
\(30\) 2.84461 0.519352
\(31\) 5.46410i 0.981382i 0.871334 + 0.490691i \(0.163256\pi\)
−0.871334 + 0.490691i \(0.836744\pi\)
\(32\) 2.81988i 0.498490i
\(33\) − 12.5284i − 2.18091i
\(34\) 1.38051i 0.236755i
\(35\) −3.60020 −0.608544
\(36\) 1.24985 0.208308
\(37\) 8.70406i 1.43094i 0.698644 + 0.715470i \(0.253787\pi\)
−0.698644 + 0.715470i \(0.746213\pi\)
\(38\) −2.76619 −0.448735
\(39\) 0 0
\(40\) 3.06430 0.484508
\(41\) 3.73205i 0.582848i 0.956594 + 0.291424i \(0.0941291\pi\)
−0.956594 + 0.291424i \(0.905871\pi\)
\(42\) 10.2412 1.58024
\(43\) 1.13186 0.172606 0.0863031 0.996269i \(-0.472495\pi\)
0.0863031 + 0.996269i \(0.472495\pi\)
\(44\) − 2.75232i − 0.414929i
\(45\) − 2.43937i − 0.363640i
\(46\) 4.74809i 0.700067i
\(47\) − 2.58535i − 0.377113i −0.982062 0.188556i \(-0.939619\pi\)
0.982062 0.188556i \(-0.0603808\pi\)
\(48\) −6.32681 −0.913197
\(49\) −5.96141 −0.851630
\(50\) − 1.21969i − 0.172490i
\(51\) 2.63977 0.369641
\(52\) 0 0
\(53\) −4.43937 −0.609795 −0.304897 0.952385i \(-0.598622\pi\)
−0.304897 + 0.952385i \(0.598622\pi\)
\(54\) − 1.59476i − 0.217020i
\(55\) −5.37182 −0.724336
\(56\) 11.0321 1.47422
\(57\) 5.28942i 0.700600i
\(58\) − 0.0301603i − 0.00396024i
\(59\) 0.171425i 0.0223176i 0.999938 + 0.0111588i \(0.00355203\pi\)
−0.999938 + 0.0111588i \(0.996448\pi\)
\(60\) − 1.19496i − 0.154269i
\(61\) 3.36023 0.430234 0.215117 0.976588i \(-0.430987\pi\)
0.215117 + 0.976588i \(0.430987\pi\)
\(62\) −6.66449 −0.846391
\(63\) − 8.78222i − 1.10646i
\(64\) −8.86488 −1.10811
\(65\) 0 0
\(66\) 15.2807 1.88093
\(67\) 6.39980i 0.781861i 0.920420 + 0.390930i \(0.127847\pi\)
−0.920420 + 0.390930i \(0.872153\pi\)
\(68\) 0.579922 0.0703258
\(69\) 9.07914 1.09300
\(70\) − 4.39111i − 0.524838i
\(71\) − 10.7973i − 1.28141i −0.767788 0.640703i \(-0.778643\pi\)
0.767788 0.640703i \(-0.221357\pi\)
\(72\) 7.47497i 0.880933i
\(73\) 4.70308i 0.550454i 0.961379 + 0.275227i \(0.0887531\pi\)
−0.961379 + 0.275227i \(0.911247\pi\)
\(74\) −10.6162 −1.23411
\(75\) −2.33225 −0.269305
\(76\) 1.16202i 0.133292i
\(77\) −19.3396 −2.20395
\(78\) 0 0
\(79\) −11.9826 −1.34815 −0.674075 0.738663i \(-0.735457\pi\)
−0.674075 + 0.738663i \(0.735457\pi\)
\(80\) 2.71276i 0.303295i
\(81\) −10.3676 −1.15195
\(82\) −4.55193 −0.502677
\(83\) − 12.1286i − 1.33129i −0.746270 0.665643i \(-0.768157\pi\)
0.746270 0.665643i \(-0.231843\pi\)
\(84\) − 4.30209i − 0.469396i
\(85\) − 1.13186i − 0.122767i
\(86\) 1.38051i 0.148864i
\(87\) −0.0576715 −0.00618303
\(88\) 16.4608 1.75473
\(89\) 16.1540i 1.71232i 0.516707 + 0.856162i \(0.327158\pi\)
−0.516707 + 0.856162i \(0.672842\pi\)
\(90\) 2.97527 0.313621
\(91\) 0 0
\(92\) 1.99457 0.207948
\(93\) 12.7436i 1.32145i
\(94\) 3.15332 0.325240
\(95\) 2.26795 0.232687
\(96\) 6.57666i 0.671228i
\(97\) 12.1682i 1.23549i 0.786379 + 0.617745i \(0.211954\pi\)
−0.786379 + 0.617745i \(0.788046\pi\)
\(98\) − 7.27105i − 0.734487i
\(99\) − 13.1039i − 1.31699i
\(100\) −0.512364 −0.0512364
\(101\) 4.05441 0.403429 0.201714 0.979444i \(-0.435349\pi\)
0.201714 + 0.979444i \(0.435349\pi\)
\(102\) 3.21969i 0.318797i
\(103\) 17.9035 1.76408 0.882041 0.471173i \(-0.156169\pi\)
0.882041 + 0.471173i \(0.156169\pi\)
\(104\) 0 0
\(105\) −8.39654 −0.819419
\(106\) − 5.41465i − 0.525917i
\(107\) −9.13186 −0.882810 −0.441405 0.897308i \(-0.645520\pi\)
−0.441405 + 0.897308i \(0.645520\pi\)
\(108\) −0.669925 −0.0644636
\(109\) 7.37605i 0.706498i 0.935529 + 0.353249i \(0.114923\pi\)
−0.935529 + 0.353249i \(0.885077\pi\)
\(110\) − 6.55193i − 0.624702i
\(111\) 20.3000i 1.92679i
\(112\) 9.76645i 0.922843i
\(113\) 7.07588 0.665643 0.332821 0.942990i \(-0.391999\pi\)
0.332821 + 0.942990i \(0.391999\pi\)
\(114\) −6.45143 −0.604232
\(115\) − 3.89287i − 0.363012i
\(116\) −0.0126697 −0.00117635
\(117\) 0 0
\(118\) −0.209084 −0.0192478
\(119\) − 4.07490i − 0.373545i
\(120\) 7.14670 0.652401
\(121\) −17.8564 −1.62331
\(122\) 4.09843i 0.371055i
\(123\) 8.70406i 0.784819i
\(124\) 2.79961i 0.251412i
\(125\) 1.00000i 0.0894427i
\(126\) 10.7116 0.954262
\(127\) 11.4361 1.01479 0.507395 0.861713i \(-0.330608\pi\)
0.507395 + 0.861713i \(0.330608\pi\)
\(128\) − 5.17262i − 0.457199i
\(129\) 2.63977 0.232418
\(130\) 0 0
\(131\) −10.5680 −0.923328 −0.461664 0.887055i \(-0.652747\pi\)
−0.461664 + 0.887055i \(0.652747\pi\)
\(132\) − 6.41910i − 0.558711i
\(133\) 8.16506 0.708001
\(134\) −7.80576 −0.674315
\(135\) 1.30752i 0.112533i
\(136\) 3.46834i 0.297408i
\(137\) − 3.78672i − 0.323522i −0.986830 0.161761i \(-0.948283\pi\)
0.986830 0.161761i \(-0.0517173\pi\)
\(138\) 11.0737i 0.942656i
\(139\) 2.01386 0.170814 0.0854068 0.996346i \(-0.472781\pi\)
0.0854068 + 0.996346i \(0.472781\pi\)
\(140\) −1.84461 −0.155898
\(141\) − 6.02968i − 0.507791i
\(142\) 13.1694 1.10515
\(143\) 0 0
\(144\) −6.61742 −0.551452
\(145\) 0.0247279i 0.00205354i
\(146\) −5.73629 −0.474739
\(147\) −13.9035 −1.14674
\(148\) 4.45965i 0.366581i
\(149\) − 5.51780i − 0.452035i −0.974123 0.226018i \(-0.927429\pi\)
0.974123 0.226018i \(-0.0725707\pi\)
\(150\) − 2.84461i − 0.232261i
\(151\) 4.88961i 0.397911i 0.980009 + 0.198956i \(0.0637549\pi\)
−0.980009 + 0.198956i \(0.936245\pi\)
\(152\) −6.94967 −0.563693
\(153\) 2.76102 0.223215
\(154\) − 23.5882i − 1.90079i
\(155\) 5.46410 0.438887
\(156\) 0 0
\(157\) 10.0405 0.801323 0.400661 0.916226i \(-0.368780\pi\)
0.400661 + 0.916226i \(0.368780\pi\)
\(158\) − 14.6150i − 1.16271i
\(159\) −10.3537 −0.821103
\(160\) 2.81988 0.222931
\(161\) − 14.0151i − 1.10454i
\(162\) − 12.6452i − 0.993501i
\(163\) − 6.78124i − 0.531148i −0.964090 0.265574i \(-0.914439\pi\)
0.964090 0.265574i \(-0.0855615\pi\)
\(164\) 1.91217i 0.149315i
\(165\) −12.5284 −0.975335
\(166\) 14.7931 1.14817
\(167\) − 10.4898i − 0.811726i −0.913934 0.405863i \(-0.866971\pi\)
0.913934 0.405863i \(-0.133029\pi\)
\(168\) 25.7295 1.98507
\(169\) 0 0
\(170\) 1.38051 0.105880
\(171\) 5.53238i 0.423071i
\(172\) 0.579922 0.0442186
\(173\) 4.45845 0.338970 0.169485 0.985533i \(-0.445790\pi\)
0.169485 + 0.985533i \(0.445790\pi\)
\(174\) − 0.0703412i − 0.00533255i
\(175\) 3.60020i 0.272149i
\(176\) 14.5724i 1.09844i
\(177\) 0.399804i 0.0300511i
\(178\) −19.7029 −1.47679
\(179\) −18.6313 −1.39257 −0.696284 0.717766i \(-0.745165\pi\)
−0.696284 + 0.717766i \(0.745165\pi\)
\(180\) − 1.24985i − 0.0931581i
\(181\) −18.0900 −1.34462 −0.672310 0.740270i \(-0.734698\pi\)
−0.672310 + 0.740270i \(0.734698\pi\)
\(182\) 0 0
\(183\) 7.83690 0.579320
\(184\) 11.9289i 0.879412i
\(185\) 8.70406 0.639935
\(186\) −15.5432 −1.13969
\(187\) − 6.08012i − 0.444622i
\(188\) − 1.32464i − 0.0966095i
\(189\) 4.70732i 0.342407i
\(190\) 2.76619i 0.200680i
\(191\) 27.3363 1.97799 0.988994 0.147958i \(-0.0472701\pi\)
0.988994 + 0.147958i \(0.0472701\pi\)
\(192\) −20.6751 −1.49210
\(193\) − 21.7674i − 1.56685i −0.621486 0.783425i \(-0.713471\pi\)
0.621486 0.783425i \(-0.286529\pi\)
\(194\) −14.8413 −1.06555
\(195\) 0 0
\(196\) −3.05441 −0.218172
\(197\) − 1.69672i − 0.120886i −0.998172 0.0604432i \(-0.980749\pi\)
0.998172 0.0604432i \(-0.0192514\pi\)
\(198\) 15.9826 1.13583
\(199\) −25.3255 −1.79527 −0.897637 0.440735i \(-0.854718\pi\)
−0.897637 + 0.440735i \(0.854718\pi\)
\(200\) − 3.06430i − 0.216679i
\(201\) 14.9259i 1.05279i
\(202\) 4.94511i 0.347937i
\(203\) 0.0890252i 0.00624834i
\(204\) 1.35252 0.0946954
\(205\) 3.73205 0.260658
\(206\) 21.8366i 1.52143i
\(207\) 9.49617 0.660030
\(208\) 0 0
\(209\) 12.1830 0.842716
\(210\) − 10.2412i − 0.706707i
\(211\) −0.335507 −0.0230973 −0.0115486 0.999933i \(-0.503676\pi\)
−0.0115486 + 0.999933i \(0.503676\pi\)
\(212\) −2.27458 −0.156218
\(213\) − 25.1820i − 1.72544i
\(214\) − 11.1380i − 0.761378i
\(215\) − 1.13186i − 0.0771919i
\(216\) − 4.00663i − 0.272616i
\(217\) 19.6718 1.33541
\(218\) −8.99648 −0.609318
\(219\) 10.9688i 0.741200i
\(220\) −2.75232 −0.185562
\(221\) 0 0
\(222\) −24.7597 −1.66176
\(223\) − 12.2968i − 0.823452i −0.911308 0.411726i \(-0.864926\pi\)
0.911308 0.411726i \(-0.135074\pi\)
\(224\) 10.1521 0.678318
\(225\) −2.43937 −0.162625
\(226\) 8.63036i 0.574083i
\(227\) − 7.63227i − 0.506571i −0.967392 0.253286i \(-0.918489\pi\)
0.967392 0.253286i \(-0.0815112\pi\)
\(228\) 2.71011i 0.179481i
\(229\) 14.4008i 0.951631i 0.879545 + 0.475815i \(0.157847\pi\)
−0.879545 + 0.475815i \(0.842153\pi\)
\(230\) 4.74809 0.313080
\(231\) −45.1047 −2.96767
\(232\) − 0.0757736i − 0.00497478i
\(233\) −9.49617 −0.622115 −0.311057 0.950391i \(-0.600683\pi\)
−0.311057 + 0.950391i \(0.600683\pi\)
\(234\) 0 0
\(235\) −2.58535 −0.168650
\(236\) 0.0878318i 0.00571736i
\(237\) −27.9464 −1.81531
\(238\) 4.97010 0.322164
\(239\) − 19.9143i − 1.28815i −0.764962 0.644076i \(-0.777242\pi\)
0.764962 0.644076i \(-0.222758\pi\)
\(240\) 6.32681i 0.408394i
\(241\) − 23.2664i − 1.49872i −0.662163 0.749360i \(-0.730361\pi\)
0.662163 0.749360i \(-0.269639\pi\)
\(242\) − 21.7792i − 1.40002i
\(243\) −20.2572 −1.29950
\(244\) 1.72166 0.110218
\(245\) 5.96141i 0.380860i
\(246\) −10.6162 −0.676866
\(247\) 0 0
\(248\) −16.7436 −1.06322
\(249\) − 28.2869i − 1.79261i
\(250\) −1.21969 −0.0771398
\(251\) −11.8402 −0.747344 −0.373672 0.927561i \(-0.621901\pi\)
−0.373672 + 0.927561i \(0.621901\pi\)
\(252\) − 4.49969i − 0.283454i
\(253\) − 20.9118i − 1.31471i
\(254\) 13.9485i 0.875205i
\(255\) − 2.63977i − 0.165309i
\(256\) −11.4208 −0.713800
\(257\) 5.55002 0.346201 0.173100 0.984904i \(-0.444621\pi\)
0.173100 + 0.984904i \(0.444621\pi\)
\(258\) 3.21969i 0.200449i
\(259\) 31.3363 1.94714
\(260\) 0 0
\(261\) −0.0603205 −0.00373375
\(262\) − 12.8896i − 0.796323i
\(263\) 6.85967 0.422985 0.211493 0.977380i \(-0.432168\pi\)
0.211493 + 0.977380i \(0.432168\pi\)
\(264\) 38.3907 2.36279
\(265\) 4.43937i 0.272709i
\(266\) 9.95882i 0.610614i
\(267\) 37.6752i 2.30568i
\(268\) 3.27903i 0.200299i
\(269\) −1.42199 −0.0867001 −0.0433501 0.999060i \(-0.513803\pi\)
−0.0433501 + 0.999060i \(0.513803\pi\)
\(270\) −1.59476 −0.0970542
\(271\) 9.96947i 0.605602i 0.953054 + 0.302801i \(0.0979218\pi\)
−0.953054 + 0.302801i \(0.902078\pi\)
\(272\) −3.07045 −0.186173
\(273\) 0 0
\(274\) 4.61862 0.279021
\(275\) 5.37182i 0.323933i
\(276\) 4.65182 0.280007
\(277\) 17.5237 1.05290 0.526449 0.850206i \(-0.323523\pi\)
0.526449 + 0.850206i \(0.323523\pi\)
\(278\) 2.45628i 0.147318i
\(279\) 13.3290i 0.797986i
\(280\) − 11.0321i − 0.659292i
\(281\) − 10.7352i − 0.640406i −0.947349 0.320203i \(-0.896249\pi\)
0.947349 0.320203i \(-0.103751\pi\)
\(282\) 7.35433 0.437944
\(283\) −1.31838 −0.0783698 −0.0391849 0.999232i \(-0.512476\pi\)
−0.0391849 + 0.999232i \(0.512476\pi\)
\(284\) − 5.53216i − 0.328273i
\(285\) 5.28942 0.313318
\(286\) 0 0
\(287\) 13.4361 0.793109
\(288\) 6.87875i 0.405334i
\(289\) −15.7189 −0.924641
\(290\) −0.0301603 −0.00177107
\(291\) 28.3792i 1.66362i
\(292\) 2.40969i 0.141016i
\(293\) − 18.7427i − 1.09496i −0.836820 0.547479i \(-0.815588\pi\)
0.836820 0.547479i \(-0.184412\pi\)
\(294\) − 16.9579i − 0.989004i
\(295\) 0.171425 0.00998072
\(296\) −26.6718 −1.55027
\(297\) 7.02375i 0.407559i
\(298\) 6.72998 0.389857
\(299\) 0 0
\(300\) −1.19496 −0.0689910
\(301\) − 4.07490i − 0.234873i
\(302\) −5.96380 −0.343178
\(303\) 9.45589 0.543226
\(304\) − 6.15239i − 0.352864i
\(305\) − 3.36023i − 0.192406i
\(306\) 3.36758i 0.192512i
\(307\) − 14.3043i − 0.816387i −0.912895 0.408194i \(-0.866159\pi\)
0.912895 0.408194i \(-0.133841\pi\)
\(308\) −9.90891 −0.564612
\(309\) 41.7553 2.37538
\(310\) 6.66449i 0.378518i
\(311\) −2.76102 −0.156563 −0.0782815 0.996931i \(-0.524943\pi\)
−0.0782815 + 0.996931i \(0.524943\pi\)
\(312\) 0 0
\(313\) −16.3858 −0.926179 −0.463090 0.886311i \(-0.653259\pi\)
−0.463090 + 0.886311i \(0.653259\pi\)
\(314\) 12.2463i 0.691100i
\(315\) −8.78222 −0.494822
\(316\) −6.13946 −0.345372
\(317\) 1.78575i 0.100297i 0.998742 + 0.0501487i \(0.0159695\pi\)
−0.998742 + 0.0501487i \(0.984030\pi\)
\(318\) − 12.6283i − 0.708159i
\(319\) 0.132834i 0.00743725i
\(320\) 8.86488i 0.495562i
\(321\) −21.2977 −1.18872
\(322\) 17.0940 0.952613
\(323\) 2.56699i 0.142831i
\(324\) −5.31197 −0.295110
\(325\) 0 0
\(326\) 8.27099 0.458088
\(327\) 17.2028i 0.951316i
\(328\) −11.4361 −0.631454
\(329\) −9.30778 −0.513155
\(330\) − 15.2807i − 0.841176i
\(331\) 7.22440i 0.397089i 0.980092 + 0.198545i \(0.0636215\pi\)
−0.980092 + 0.198545i \(0.936379\pi\)
\(332\) − 6.21425i − 0.341052i
\(333\) 21.2325i 1.16353i
\(334\) 12.7943 0.700072
\(335\) 6.39980 0.349659
\(336\) 22.7778i 1.24263i
\(337\) 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
\(340\) − 0.579922i − 0.0314507i
\(341\) 29.3521 1.58951
\(342\) −6.74777 −0.364877
\(343\) − 3.73913i − 0.201894i
\(344\) 3.46834i 0.187000i
\(345\) − 9.07914i − 0.488804i
\(346\) 5.43792i 0.292344i
\(347\) −26.7072 −1.43372 −0.716858 0.697219i \(-0.754420\pi\)
−0.716858 + 0.697219i \(0.754420\pi\)
\(348\) −0.0295488 −0.00158398
\(349\) − 23.5711i − 1.26173i −0.775892 0.630865i \(-0.782700\pi\)
0.775892 0.630865i \(-0.217300\pi\)
\(350\) −4.39111 −0.234715
\(351\) 0 0
\(352\) 15.1479 0.807385
\(353\) 5.73727i 0.305364i 0.988275 + 0.152682i \(0.0487910\pi\)
−0.988275 + 0.152682i \(0.951209\pi\)
\(354\) −0.487636 −0.0259176
\(355\) −10.7973 −0.573063
\(356\) 8.27675i 0.438667i
\(357\) − 9.50367i − 0.502988i
\(358\) − 22.7243i − 1.20102i
\(359\) 24.7583i 1.30669i 0.757059 + 0.653347i \(0.226636\pi\)
−0.757059 + 0.653347i \(0.773364\pi\)
\(360\) 7.47497 0.393965
\(361\) 13.8564 0.729285
\(362\) − 22.0641i − 1.15967i
\(363\) −41.6455 −2.18582
\(364\) 0 0
\(365\) 4.70308 0.246171
\(366\) 9.55856i 0.499634i
\(367\) 26.0535 1.35998 0.679992 0.733220i \(-0.261983\pi\)
0.679992 + 0.733220i \(0.261983\pi\)
\(368\) −10.5604 −0.550499
\(369\) 9.10387i 0.473928i
\(370\) 10.6162i 0.551912i
\(371\) 15.9826i 0.829776i
\(372\) 6.52938i 0.338532i
\(373\) 13.2045 0.683702 0.341851 0.939754i \(-0.388946\pi\)
0.341851 + 0.939754i \(0.388946\pi\)
\(374\) 7.41584 0.383464
\(375\) 2.33225i 0.120437i
\(376\) 7.92229 0.408561
\(377\) 0 0
\(378\) −5.74146 −0.295309
\(379\) 25.9977i 1.33541i 0.744425 + 0.667707i \(0.232724\pi\)
−0.744425 + 0.667707i \(0.767276\pi\)
\(380\) 1.16202 0.0596101
\(381\) 26.6718 1.36644
\(382\) 33.3418i 1.70591i
\(383\) − 9.60020i − 0.490547i −0.969454 0.245274i \(-0.921122\pi\)
0.969454 0.245274i \(-0.0788778\pi\)
\(384\) − 12.0638i − 0.615629i
\(385\) 19.3396i 0.985637i
\(386\) 26.5494 1.35133
\(387\) 2.76102 0.140350
\(388\) 6.23453i 0.316510i
\(389\) −5.63129 −0.285518 −0.142759 0.989758i \(-0.545597\pi\)
−0.142759 + 0.989758i \(0.545597\pi\)
\(390\) 0 0
\(391\) 4.40617 0.222829
\(392\) − 18.2675i − 0.922650i
\(393\) −24.6471 −1.24328
\(394\) 2.06947 0.104258
\(395\) 11.9826i 0.602911i
\(396\) − 6.71395i − 0.337389i
\(397\) 16.7658i 0.841452i 0.907188 + 0.420726i \(0.138225\pi\)
−0.907188 + 0.420726i \(0.861775\pi\)
\(398\) − 30.8891i − 1.54833i
\(399\) 19.0429 0.953339
\(400\) 2.71276 0.135638
\(401\) 13.8780i 0.693036i 0.938043 + 0.346518i \(0.112636\pi\)
−0.938043 + 0.346518i \(0.887364\pi\)
\(402\) −18.2050 −0.907980
\(403\) 0 0
\(404\) 2.07733 0.103351
\(405\) 10.3676i 0.515169i
\(406\) −0.108583 −0.00538888
\(407\) 46.7566 2.31764
\(408\) 8.08903i 0.400466i
\(409\) 29.4251i 1.45498i 0.686120 + 0.727489i \(0.259313\pi\)
−0.686120 + 0.727489i \(0.740687\pi\)
\(410\) 4.55193i 0.224804i
\(411\) − 8.83157i − 0.435629i
\(412\) 9.17310 0.451926
\(413\) 0.617162 0.0303686
\(414\) 11.5824i 0.569242i
\(415\) −12.1286 −0.595369
\(416\) 0 0
\(417\) 4.69683 0.230005
\(418\) 14.8595i 0.726800i
\(419\) 6.96793 0.340406 0.170203 0.985409i \(-0.445558\pi\)
0.170203 + 0.985409i \(0.445558\pi\)
\(420\) −4.30209 −0.209920
\(421\) 7.12125i 0.347069i 0.984828 + 0.173534i \(0.0555188\pi\)
−0.984828 + 0.173534i \(0.944481\pi\)
\(422\) − 0.409213i − 0.0199202i
\(423\) − 6.30664i − 0.306640i
\(424\) − 13.6036i − 0.660647i
\(425\) −1.13186 −0.0549030
\(426\) 30.7142 1.48811
\(427\) − 12.0975i − 0.585439i
\(428\) −4.67883 −0.226160
\(429\) 0 0
\(430\) 1.38051 0.0665740
\(431\) 30.2144i 1.45537i 0.685909 + 0.727687i \(0.259405\pi\)
−0.685909 + 0.727687i \(0.740595\pi\)
\(432\) 3.54698 0.170654
\(433\) −1.20013 −0.0576745 −0.0288373 0.999584i \(-0.509180\pi\)
−0.0288373 + 0.999584i \(0.509180\pi\)
\(434\) 23.9935i 1.15172i
\(435\) 0.0576715i 0.00276514i
\(436\) 3.77922i 0.180992i
\(437\) 8.82884i 0.422341i
\(438\) −13.3784 −0.639247
\(439\) 16.5541 0.790084 0.395042 0.918663i \(-0.370730\pi\)
0.395042 + 0.918663i \(0.370730\pi\)
\(440\) − 16.4608i − 0.784740i
\(441\) −14.5421 −0.692481
\(442\) 0 0
\(443\) 4.55949 0.216628 0.108314 0.994117i \(-0.465455\pi\)
0.108314 + 0.994117i \(0.465455\pi\)
\(444\) 10.4010i 0.493610i
\(445\) 16.1540 0.765775
\(446\) 14.9982 0.710185
\(447\) − 12.8689i − 0.608676i
\(448\) 31.9153i 1.50786i
\(449\) − 13.8522i − 0.653724i −0.945072 0.326862i \(-0.894009\pi\)
0.945072 0.326862i \(-0.105991\pi\)
\(450\) − 2.97527i − 0.140256i
\(451\) 20.0479 0.944018
\(452\) 3.62542 0.170526
\(453\) 11.4038i 0.535796i
\(454\) 9.30897 0.436892
\(455\) 0 0
\(456\) −16.2083 −0.759025
\(457\) 40.1146i 1.87648i 0.345984 + 0.938240i \(0.387545\pi\)
−0.345984 + 0.938240i \(0.612455\pi\)
\(458\) −17.5644 −0.820733
\(459\) −1.47992 −0.0690768
\(460\) − 1.99457i − 0.0929972i
\(461\) − 7.53900i − 0.351126i −0.984468 0.175563i \(-0.943825\pi\)
0.984468 0.175563i \(-0.0561746\pi\)
\(462\) − 55.0136i − 2.55946i
\(463\) − 23.3031i − 1.08299i −0.840705 0.541494i \(-0.817859\pi\)
0.840705 0.541494i \(-0.182141\pi\)
\(464\) 0.0670807 0.00311414
\(465\) 12.7436 0.590972
\(466\) − 11.5824i − 0.536542i
\(467\) 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
\(470\) − 3.15332i − 0.145452i
\(471\) 23.4170 1.07900
\(472\) −0.525296 −0.0241787
\(473\) − 6.08012i − 0.279564i
\(474\) − 34.0859i − 1.56562i
\(475\) − 2.26795i − 0.104061i
\(476\) − 2.08783i − 0.0956956i
\(477\) −10.8293 −0.495839
\(478\) 24.2893 1.11096
\(479\) − 20.6448i − 0.943286i −0.881790 0.471643i \(-0.843661\pi\)
0.881790 0.471643i \(-0.156339\pi\)
\(480\) 6.57666 0.300182
\(481\) 0 0
\(482\) 28.3777 1.29257
\(483\) − 32.6867i − 1.48730i
\(484\) −9.14898 −0.415863
\(485\) 12.1682 0.552528
\(486\) − 24.7074i − 1.12075i
\(487\) 3.03605i 0.137576i 0.997631 + 0.0687882i \(0.0219133\pi\)
−0.997631 + 0.0687882i \(0.978087\pi\)
\(488\) 10.2968i 0.466112i
\(489\) − 15.8155i − 0.715203i
\(490\) −7.27105 −0.328473
\(491\) −10.6680 −0.481441 −0.240720 0.970595i \(-0.577384\pi\)
−0.240720 + 0.970595i \(0.577384\pi\)
\(492\) 4.45965i 0.201056i
\(493\) −0.0279884 −0.00126053
\(494\) 0 0
\(495\) −13.1039 −0.588975
\(496\) − 14.8228i − 0.665562i
\(497\) −38.8725 −1.74367
\(498\) 34.5011 1.54603
\(499\) 33.9143i 1.51821i 0.650966 + 0.759107i \(0.274364\pi\)
−0.650966 + 0.759107i \(0.725636\pi\)
\(500\) 0.512364i 0.0229136i
\(501\) − 24.4648i − 1.09301i
\(502\) − 14.4413i − 0.644546i
\(503\) −12.6276 −0.563037 −0.281518 0.959556i \(-0.590838\pi\)
−0.281518 + 0.959556i \(0.590838\pi\)
\(504\) 26.9113 1.19873
\(505\) − 4.05441i − 0.180419i
\(506\) 25.5058 1.13387
\(507\) 0 0
\(508\) 5.85945 0.259971
\(509\) − 24.1526i − 1.07055i −0.844679 0.535273i \(-0.820209\pi\)
0.844679 0.535273i \(-0.179791\pi\)
\(510\) 3.21969 0.142570
\(511\) 16.9320 0.749029
\(512\) − 24.2750i − 1.07281i
\(513\) − 2.96539i − 0.130925i
\(514\) 6.76929i 0.298581i
\(515\) − 17.9035i − 0.788921i
\(516\) 1.35252 0.0595414
\(517\) −13.8880 −0.610796
\(518\) 38.2205i 1.67931i
\(519\) 10.3982 0.456431
\(520\) 0 0
\(521\) −24.7521 −1.08441 −0.542205 0.840246i \(-0.682410\pi\)
−0.542205 + 0.840246i \(0.682410\pi\)
\(522\) − 0.0735722i − 0.00322017i
\(523\) 37.0326 1.61932 0.809662 0.586897i \(-0.199650\pi\)
0.809662 + 0.586897i \(0.199650\pi\)
\(524\) −5.41465 −0.236540
\(525\) 8.39654i 0.366455i
\(526\) 8.36665i 0.364803i
\(527\) 6.18457i 0.269404i
\(528\) 33.9865i 1.47907i
\(529\) −7.84554 −0.341111
\(530\) −5.41465 −0.235197
\(531\) 0.418169i 0.0181470i
\(532\) 4.18348 0.181377
\(533\) 0 0
\(534\) −45.9519 −1.98854
\(535\) 9.13186i 0.394805i
\(536\) −19.6109 −0.847062
\(537\) −43.4528 −1.87512
\(538\) − 1.73438i − 0.0747744i
\(539\) 32.0236i 1.37935i
\(540\) 0.669925i 0.0288290i
\(541\) 8.38144i 0.360346i 0.983635 + 0.180173i \(0.0576658\pi\)
−0.983635 + 0.180173i \(0.942334\pi\)
\(542\) −12.1596 −0.522301
\(543\) −42.1903 −1.81056
\(544\) 3.19170i 0.136843i
\(545\) 7.37605 0.315955
\(546\) 0 0
\(547\) −22.7842 −0.974181 −0.487091 0.873351i \(-0.661942\pi\)
−0.487091 + 0.873351i \(0.661942\pi\)
\(548\) − 1.94018i − 0.0828804i
\(549\) 8.19687 0.349834
\(550\) −6.55193 −0.279375
\(551\) − 0.0560816i − 0.00238915i
\(552\) 27.8212i 1.18415i
\(553\) 43.1398i 1.83449i
\(554\) 21.3735i 0.908071i
\(555\) 20.3000 0.861688
\(556\) 1.03183 0.0437594
\(557\) 28.1527i 1.19287i 0.802662 + 0.596435i \(0.203417\pi\)
−0.802662 + 0.596435i \(0.796583\pi\)
\(558\) −16.2572 −0.688222
\(559\) 0 0
\(560\) 9.76645 0.412708
\(561\) − 14.1803i − 0.598694i
\(562\) 13.0935 0.552317
\(563\) 18.1303 0.764101 0.382050 0.924142i \(-0.375218\pi\)
0.382050 + 0.924142i \(0.375218\pi\)
\(564\) − 3.08939i − 0.130087i
\(565\) − 7.07588i − 0.297684i
\(566\) − 1.60801i − 0.0675899i
\(567\) 37.3253i 1.56752i
\(568\) 33.0862 1.38827
\(569\) −40.5985 −1.70198 −0.850988 0.525185i \(-0.823996\pi\)
−0.850988 + 0.525185i \(0.823996\pi\)
\(570\) 6.45143i 0.270221i
\(571\) −24.7159 −1.03433 −0.517164 0.855886i \(-0.673012\pi\)
−0.517164 + 0.855886i \(0.673012\pi\)
\(572\) 0 0
\(573\) 63.7551 2.66341
\(574\) 16.3879i 0.684016i
\(575\) −3.89287 −0.162344
\(576\) −21.6248 −0.901032
\(577\) 23.0691i 0.960379i 0.877165 + 0.480189i \(0.159432\pi\)
−0.877165 + 0.480189i \(0.840568\pi\)
\(578\) − 19.1721i − 0.797456i
\(579\) − 50.7669i − 2.10980i
\(580\) 0.0126697i 0 0.000526079i
\(581\) −43.6653 −1.81154
\(582\) −34.6137 −1.43478
\(583\) 23.8475i 0.987662i
\(584\) −14.4116 −0.596358
\(585\) 0 0
\(586\) 22.8602 0.944345
\(587\) − 20.3523i − 0.840030i −0.907517 0.420015i \(-0.862025\pi\)
0.907517 0.420015i \(-0.137975\pi\)
\(588\) −7.12364 −0.293774
\(589\) −12.3923 −0.510616
\(590\) 0.209084i 0.00860786i
\(591\) − 3.95717i − 0.162776i
\(592\) − 23.6120i − 0.970447i
\(593\) − 10.3834i − 0.426395i −0.977009 0.213198i \(-0.931612\pi\)
0.977009 0.213198i \(-0.0683878\pi\)
\(594\) −8.56677 −0.351499
\(595\) −4.07490 −0.167055
\(596\) − 2.82712i − 0.115803i
\(597\) −59.0652 −2.41738
\(598\) 0 0
\(599\) −31.5965 −1.29100 −0.645499 0.763761i \(-0.723351\pi\)
−0.645499 + 0.763761i \(0.723351\pi\)
\(600\) − 7.14670i − 0.291763i
\(601\) −43.8845 −1.79009 −0.895044 0.445979i \(-0.852856\pi\)
−0.895044 + 0.445979i \(0.852856\pi\)
\(602\) 4.97010 0.202566
\(603\) 15.6115i 0.635750i
\(604\) 2.50526i 0.101938i
\(605\) 17.8564i 0.725966i
\(606\) 11.5332i 0.468505i
\(607\) 2.17540 0.0882968 0.0441484 0.999025i \(-0.485943\pi\)
0.0441484 + 0.999025i \(0.485943\pi\)
\(608\) −6.39535 −0.259366
\(609\) 0.207629i 0.00841354i
\(610\) 4.09843 0.165941
\(611\) 0 0
\(612\) 1.41465 0.0571837
\(613\) − 14.7620i − 0.596231i −0.954530 0.298116i \(-0.903642\pi\)
0.954530 0.298116i \(-0.0963581\pi\)
\(614\) 17.4467 0.704092
\(615\) 8.70406 0.350982
\(616\) − 59.2622i − 2.38774i
\(617\) − 20.2972i − 0.817134i −0.912728 0.408567i \(-0.866029\pi\)
0.912728 0.408567i \(-0.133971\pi\)
\(618\) 50.9284i 2.04864i
\(619\) − 9.94207i − 0.399605i −0.979836 0.199803i \(-0.935970\pi\)
0.979836 0.199803i \(-0.0640301\pi\)
\(620\) 2.79961 0.112435
\(621\) −5.09000 −0.204255
\(622\) − 3.36758i − 0.135028i
\(623\) 58.1577 2.33004
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 19.9855i − 0.798782i
\(627\) 28.4138 1.13474
\(628\) 5.14441 0.205284
\(629\) 9.85174i 0.392815i
\(630\) − 10.7116i − 0.426759i
\(631\) − 0.973420i − 0.0387512i −0.999812 0.0193756i \(-0.993832\pi\)
0.999812 0.0193756i \(-0.00616784\pi\)
\(632\) − 36.7183i − 1.46058i
\(633\) −0.782485 −0.0311010
\(634\) −2.17805 −0.0865014
\(635\) − 11.4361i − 0.453828i
\(636\) −5.30487 −0.210352
\(637\) 0 0
\(638\) −0.162015 −0.00641425
\(639\) − 26.3387i − 1.04194i
\(640\) −5.17262 −0.204466
\(641\) 12.6209 0.498497 0.249249 0.968440i \(-0.419816\pi\)
0.249249 + 0.968440i \(0.419816\pi\)
\(642\) − 25.9766i − 1.02521i
\(643\) 9.96043i 0.392801i 0.980524 + 0.196401i \(0.0629253\pi\)
−0.980524 + 0.196401i \(0.937075\pi\)
\(644\) − 7.18083i − 0.282964i
\(645\) − 2.63977i − 0.103941i
\(646\) −3.13092 −0.123185
\(647\) 36.2763 1.42617 0.713084 0.701079i \(-0.247298\pi\)
0.713084 + 0.701079i \(0.247298\pi\)
\(648\) − 31.7693i − 1.24802i
\(649\) 0.920861 0.0361470
\(650\) 0 0
\(651\) 45.8796 1.79816
\(652\) − 3.47447i − 0.136071i
\(653\) 13.7554 0.538289 0.269145 0.963100i \(-0.413259\pi\)
0.269145 + 0.963100i \(0.413259\pi\)
\(654\) −20.9820 −0.820461
\(655\) 10.5680i 0.412925i
\(656\) − 10.1241i − 0.395281i
\(657\) 11.4726i 0.447588i
\(658\) − 11.3526i − 0.442570i
\(659\) −2.58183 −0.100574 −0.0502869 0.998735i \(-0.516014\pi\)
−0.0502869 + 0.998735i \(0.516014\pi\)
\(660\) −6.41910 −0.249863
\(661\) − 24.8765i − 0.967582i −0.875183 0.483791i \(-0.839259\pi\)
0.875183 0.483791i \(-0.160741\pi\)
\(662\) −8.81151 −0.342469
\(663\) 0 0
\(664\) 37.1656 1.44231
\(665\) − 8.16506i − 0.316627i
\(666\) −25.8970 −1.00349
\(667\) −0.0962625 −0.00372730
\(668\) − 5.37460i − 0.207949i
\(669\) − 28.6791i − 1.10880i
\(670\) 7.80576i 0.301563i
\(671\) − 18.0506i − 0.696834i
\(672\) 23.6773 0.913370
\(673\) −43.3222 −1.66995 −0.834974 0.550289i \(-0.814517\pi\)
−0.834974 + 0.550289i \(0.814517\pi\)
\(674\) 5.32051i 0.204938i
\(675\) 1.30752 0.0503264
\(676\) 0 0
\(677\) −41.3625 −1.58969 −0.794845 0.606813i \(-0.792448\pi\)
−0.794845 + 0.606813i \(0.792448\pi\)
\(678\) 20.1281i 0.773016i
\(679\) 43.8078 1.68119
\(680\) 3.46834 0.133005
\(681\) − 17.8003i − 0.682110i
\(682\) 35.8004i 1.37087i
\(683\) − 2.62688i − 0.100515i −0.998736 0.0502574i \(-0.983996\pi\)
0.998736 0.0502574i \(-0.0160042\pi\)
\(684\) 2.83459i 0.108383i
\(685\) −3.78672 −0.144683
\(686\) 4.56057 0.174123
\(687\) 33.5862i 1.28139i
\(688\) −3.07045 −0.117060
\(689\) 0 0
\(690\) 11.0737 0.421569
\(691\) − 15.2753i − 0.581099i −0.956860 0.290550i \(-0.906162\pi\)
0.956860 0.290550i \(-0.0938382\pi\)
\(692\) 2.28435 0.0868380
\(693\) −47.1765 −1.79209
\(694\) − 32.5744i − 1.23651i
\(695\) − 2.01386i − 0.0763902i
\(696\) − 0.176723i − 0.00669865i
\(697\) 4.22414i 0.160001i
\(698\) 28.7493 1.08818
\(699\) −22.1474 −0.837692
\(700\) 1.84461i 0.0697197i
\(701\) 48.1947 1.82029 0.910144 0.414292i \(-0.135971\pi\)
0.910144 + 0.414292i \(0.135971\pi\)
\(702\) 0 0
\(703\) −19.7404 −0.744522
\(704\) 47.6205i 1.79477i
\(705\) −6.02968 −0.227091
\(706\) −6.99767 −0.263361
\(707\) − 14.5967i − 0.548964i
\(708\) 0.204845i 0.00769856i
\(709\) 38.8699i 1.45979i 0.683559 + 0.729896i \(0.260431\pi\)
−0.683559 + 0.729896i \(0.739569\pi\)
\(710\) − 13.1694i − 0.494237i
\(711\) −29.2301 −1.09621
\(712\) −49.5008 −1.85512
\(713\) 21.2711i 0.796607i
\(714\) 11.5915 0.433801
\(715\) 0 0
\(716\) −9.54600 −0.356751
\(717\) − 46.4452i − 1.73453i
\(718\) −30.1974 −1.12696
\(719\) −6.61660 −0.246758 −0.123379 0.992360i \(-0.539373\pi\)
−0.123379 + 0.992360i \(0.539373\pi\)
\(720\) 6.61742i 0.246617i
\(721\) − 64.4560i − 2.40047i
\(722\) 16.9005i 0.628971i
\(723\) − 54.2629i − 2.01806i
\(724\) −9.26867 −0.344467
\(725\) 0.0247279 0.000918370 0
\(726\) − 50.7945i − 1.88516i
\(727\) 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
\(730\) 5.73629i 0.212310i
\(731\) 1.28110 0.0473830
\(732\) 4.01534 0.148411
\(733\) 0.791131i 0.0292211i 0.999893 + 0.0146105i \(0.00465084\pi\)
−0.999893 + 0.0146105i \(0.995349\pi\)
\(734\) 31.7772i 1.17292i
\(735\) 13.9035i 0.512837i
\(736\) 10.9774i 0.404634i
\(737\) 34.3786 1.26635
\(738\) −11.1039 −0.408739
\(739\) − 31.1853i − 1.14717i −0.819146 0.573585i \(-0.805552\pi\)
0.819146 0.573585i \(-0.194448\pi\)
\(740\) 4.45965 0.163940
\(741\) 0 0
\(742\) −19.4938 −0.715639
\(743\) 5.56304i 0.204088i 0.994780 + 0.102044i \(0.0325383\pi\)
−0.994780 + 0.102044i \(0.967462\pi\)
\(744\) −39.0503 −1.43165
\(745\) −5.51780 −0.202156
\(746\) 16.1053i 0.589658i
\(747\) − 29.5862i − 1.08250i
\(748\) − 3.11523i − 0.113904i
\(749\) 32.8765i 1.20128i
\(750\) −2.84461 −0.103870
\(751\) 35.2097 1.28482 0.642410 0.766361i \(-0.277935\pi\)
0.642410 + 0.766361i \(0.277935\pi\)
\(752\) 7.01343i 0.255754i
\(753\) −27.6142 −1.00632
\(754\) 0 0
\(755\) 4.88961 0.177951
\(756\) 2.41186i 0.0877186i
\(757\) 50.0446 1.81890 0.909451 0.415810i \(-0.136502\pi\)
0.909451 + 0.415810i \(0.136502\pi\)
\(758\) −31.7091 −1.15173
\(759\) − 48.7715i − 1.77029i
\(760\) 6.94967i 0.252091i
\(761\) − 44.8209i − 1.62476i −0.583130 0.812379i \(-0.698172\pi\)
0.583130 0.812379i \(-0.301828\pi\)
\(762\) 32.5313i 1.17848i
\(763\) 26.5552 0.961364
\(764\) 14.0061 0.506725
\(765\) − 2.76102i − 0.0998248i
\(766\) 11.7092 0.423072
\(767\) 0 0
\(768\) −26.6361 −0.961148
\(769\) − 39.3633i − 1.41948i −0.704465 0.709739i \(-0.748813\pi\)
0.704465 0.709739i \(-0.251187\pi\)
\(770\) −23.5882 −0.850061
\(771\) 12.9440 0.466168
\(772\) − 11.1528i − 0.401399i
\(773\) 48.7805i 1.75451i 0.480022 + 0.877256i \(0.340629\pi\)
−0.480022 + 0.877256i \(0.659371\pi\)
\(774\) 3.36758i 0.121045i
\(775\) − 5.46410i − 0.196276i
\(776\) −37.2869 −1.33852
\(777\) 73.0840 2.62188
\(778\) − 6.86841i − 0.246244i
\(779\) −8.46410 −0.303258
\(780\) 0 0
\(781\) −58.0013 −2.07545
\(782\) 5.37415i 0.192179i
\(783\) 0.0323322 0.00115546
\(784\) 16.1718 0.577566
\(785\) − 10.0405i − 0.358363i
\(786\) − 30.0618i − 1.07227i
\(787\) − 39.8608i − 1.42088i −0.703756 0.710442i \(-0.748495\pi\)
0.703756 0.710442i \(-0.251505\pi\)
\(788\) − 0.869338i − 0.0309689i
\(789\) 15.9984 0.569559
\(790\) −14.6150 −0.519980
\(791\) − 25.4745i − 0.905771i
\(792\) 40.1541 1.42682
\(793\) 0 0
\(794\) −20.4490 −0.725709
\(795\) 10.3537i 0.367208i
\(796\) −12.9759 −0.459917
\(797\) −26.2118 −0.928470 −0.464235 0.885712i \(-0.653671\pi\)
−0.464235 + 0.885712i \(0.653671\pi\)
\(798\) 23.2264i 0.822206i
\(799\) − 2.92625i − 0.103523i
\(800\) − 2.81988i − 0.0996979i
\(801\) 39.4057i 1.39233i
\(802\) −16.9269 −0.597708
\(803\) 25.2641 0.891551
\(804\) 7.64750i 0.269707i
\(805\) −14.0151 −0.493968
\(806\) 0 0
\(807\) −3.31643 −0.116744
\(808\) 12.4239i 0.437072i
\(809\) 22.2136 0.780990 0.390495 0.920605i \(-0.372304\pi\)
0.390495 + 0.920605i \(0.372304\pi\)
\(810\) −12.6452 −0.444307
\(811\) 19.0950i 0.670515i 0.942127 + 0.335257i \(0.108823\pi\)
−0.942127 + 0.335257i \(0.891177\pi\)
\(812\) 0.0456133i 0.00160071i
\(813\) 23.2513i 0.815457i
\(814\) 57.0284i 1.99885i
\(815\) −6.78124 −0.237537
\(816\) −7.16104 −0.250686
\(817\) 2.56699i 0.0898076i
\(818\) −35.8894 −1.25484
\(819\) 0 0
\(820\) 1.91217 0.0667758
\(821\) − 34.1584i − 1.19214i −0.802934 0.596068i \(-0.796729\pi\)
0.802934 0.596068i \(-0.203271\pi\)
\(822\) 10.7718 0.375708
\(823\) −4.07940 −0.142199 −0.0710995 0.997469i \(-0.522651\pi\)
−0.0710995 + 0.997469i \(0.522651\pi\)
\(824\) 54.8616i 1.91119i
\(825\) 12.5284i 0.436183i
\(826\) 0.752744i 0.0261913i
\(827\) 54.8780i 1.90830i 0.299337 + 0.954148i \(0.403235\pi\)
−0.299337 + 0.954148i \(0.596765\pi\)
\(828\) 4.86550 0.169088
\(829\) −8.14950 −0.283044 −0.141522 0.989935i \(-0.545200\pi\)
−0.141522 + 0.989935i \(0.545200\pi\)
\(830\) − 14.7931i − 0.513476i
\(831\) 40.8697 1.41775
\(832\) 0 0
\(833\) −6.74745 −0.233785
\(834\) 5.72866i 0.198367i
\(835\) −10.4898 −0.363015
\(836\) 6.24213 0.215889
\(837\) − 7.14441i − 0.246947i
\(838\) 8.49869i 0.293582i
\(839\) 21.8865i 0.755606i 0.925886 + 0.377803i \(0.123320\pi\)
−0.925886 + 0.377803i \(0.876680\pi\)
\(840\) − 25.7295i − 0.887752i
\(841\) −28.9994 −0.999979
\(842\) −8.68570 −0.299329
\(843\) − 25.0370i − 0.862321i
\(844\) −0.171902 −0.00591710
\(845\) 0 0
\(846\) 7.69213 0.264461
\(847\) 64.2866i 2.20891i
\(848\) 12.0429 0.413556
\(849\) −3.07480 −0.105527
\(850\) − 1.38051i − 0.0473511i
\(851\) 33.8838i 1.16152i
\(852\) − 12.9024i − 0.442028i
\(853\) 19.2240i 0.658217i 0.944292 + 0.329108i \(0.106748\pi\)
−0.944292 + 0.329108i \(0.893252\pi\)
\(854\) 14.7552 0.504911
\(855\) 5.53238 0.189203
\(856\) − 27.9827i − 0.956430i
\(857\) 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 \(0.215098\pi\)
\(860\) − 0.579922i − 0.0197752i
\(861\) 31.3363 1.06794
\(862\) −36.8521 −1.25519
\(863\) 54.8186i 1.86605i 0.359814 + 0.933024i \(0.382840\pi\)
−0.359814 + 0.933024i \(0.617160\pi\)
\(864\) − 3.68705i − 0.125436i
\(865\) − 4.45845i − 0.151592i
\(866\) − 1.46378i − 0.0497413i
\(867\) −36.6604 −1.24505
\(868\) 10.0791 0.342108
\(869\) 64.3684i 2.18355i
\(870\) −0.0703412 −0.00238479
\(871\) 0 0
\(872\) −22.6024 −0.765415
\(873\) 29.6827i 1.00461i
\(874\) −10.7684 −0.364247
\(875\) 3.60020 0.121709
\(876\) 5.61999i 0.189882i
\(877\) − 27.3794i − 0.924537i −0.886740 0.462269i \(-0.847036\pi\)
0.886740 0.462269i \(-0.152964\pi\)
\(878\) 20.1908i 0.681407i
\(879\) − 43.7125i − 1.47439i
\(880\) 14.5724 0.491236
\(881\) 34.4426 1.16040 0.580200 0.814474i \(-0.302974\pi\)
0.580200 + 0.814474i \(0.302974\pi\)
\(882\) − 17.7368i − 0.597230i
\(883\) 17.3592 0.584183 0.292092 0.956390i \(-0.405649\pi\)
0.292092 + 0.956390i \(0.405649\pi\)
\(884\) 0 0
\(885\) 0.399804 0.0134393
\(886\) 5.56115i 0.186831i
\(887\) −31.1427 −1.04567 −0.522835 0.852434i \(-0.675126\pi\)
−0.522835 + 0.852434i \(0.675126\pi\)
\(888\) −62.2053 −2.08747
\(889\) − 41.1722i − 1.38087i
\(890\) 19.7029i 0.660442i
\(891\) 55.6927i 1.86578i
\(892\) − 6.30042i − 0.210954i
\(893\) 5.86345 0.196213
\(894\) 15.6960 0.524952
\(895\) 18.6313i 0.622775i
\(896\) −18.6224 −0.622132
\(897\) 0 0
\(898\) 16.8953 0.563804
\(899\) − 0.135116i − 0.00450636i
\(900\) −1.24985 −0.0416616
\(901\) −5.02473 −0.167398
\(902\) 24.4521i 0.814167i
\(903\) − 9.50367i − 0.316262i
\(904\) 21.6826i 0.721152i
\(905\) 18.0900i 0.601332i
\(906\) −13.9090 −0.462097
\(907\) −17.6057 −0.584587 −0.292294 0.956329i \(-0.594418\pi\)
−0.292294 + 0.956329i \(0.594418\pi\)
\(908\) − 3.91050i − 0.129774i
\(909\) 9.89022 0.328038
\(910\) 0 0
\(911\) 50.0232 1.65734 0.828671 0.559737i \(-0.189098\pi\)
0.828671 + 0.559737i \(0.189098\pi\)
\(912\) − 14.3489i − 0.475139i
\(913\) −65.1526 −2.15624
\(914\) −48.9272 −1.61837
\(915\) − 7.83690i − 0.259080i
\(916\) 7.37844i 0.243791i
\(917\) 38.0468i 1.25641i
\(918\) − 1.80504i − 0.0595752i
\(919\) −7.61556 −0.251214 −0.125607 0.992080i \(-0.540088\pi\)
−0.125607 + 0.992080i \(0.540088\pi\)
\(920\) 11.9289 0.393285
\(921\) − 33.3611i − 1.09928i
\(922\) 9.19522 0.302828
\(923\) 0 0
\(924\) −23.1100 −0.760264
\(925\) − 8.70406i − 0.286188i
\(926\) 28.4225 0.934022
\(927\) 43.6733 1.43442
\(928\) − 0.0697297i − 0.00228899i
\(929\) − 14.1239i − 0.463391i −0.972788 0.231695i \(-0.925573\pi\)
0.972788 0.231695i \(-0.0744273\pi\)
\(930\) 15.5432i 0.509683i
\(931\) − 13.5202i − 0.443106i
\(932\) −4.86550 −0.159375
\(933\) −6.43937 −0.210816
\(934\) 27.6012i 0.903138i
\(935\) −6.08012 −0.198841
\(936\) 0 0
\(937\) −23.9317 −0.781815 −0.390908 0.920430i \(-0.627839\pi\)
−0.390908 + 0.920430i \(0.627839\pi\)
\(938\) 28.1023i 0.917571i
\(939\) −38.2157 −1.24712
\(940\) −1.32464 −0.0432051
\(941\) − 25.3591i − 0.826683i −0.910576 0.413342i \(-0.864362\pi\)
0.910576 0.413342i \(-0.135638\pi\)
\(942\) 28.5614i 0.930582i
\(943\) 14.5284i 0.473110i
\(944\) − 0.465033i − 0.0151355i
\(945\) 4.70732 0.153129
\(946\) 7.41584 0.241110
\(947\) − 41.4223i − 1.34604i −0.739623 0.673021i \(-0.764996\pi\)
0.739623 0.673021i \(-0.235004\pi\)
\(948\) −14.3187 −0.465051
\(949\) 0 0
\(950\) 2.76619 0.0897470
\(951\) 4.16480i 0.135053i
\(952\) 12.4867 0.404696
\(953\) −24.3026 −0.787237 −0.393619 0.919274i \(-0.628777\pi\)
−0.393619 + 0.919274i \(0.628777\pi\)
\(954\) − 13.2083i − 0.427636i
\(955\) − 27.3363i − 0.884583i
\(956\) − 10.2034i − 0.330001i
\(957\) 0.309801i 0.0100144i
\(958\) 25.1802 0.813536
\(959\) −13.6329 −0.440231
\(960\) 20.6751i 0.667286i
\(961\) 1.14359 0.0368901
\(962\) 0 0
\(963\) −22.2760 −0.717834
\(964\) − 11.9209i − 0.383945i
\(965\) −21.7674 −0.700717
\(966\) 39.8675 1.28272
\(967\) 23.6784i 0.761445i 0.924689 + 0.380722i \(0.124325\pi\)
−0.924689 + 0.380722i \(0.875675\pi\)
\(968\) − 54.7173i − 1.75868i
\(969\) 5.98685i 0.192325i
\(970\) 14.8413i 0.476527i
\(971\) −16.9722 −0.544663 −0.272332 0.962203i \(-0.587795\pi\)
−0.272332 + 0.962203i \(0.587795\pi\)
\(972\) −10.3791 −0.332908
\(973\) − 7.25030i − 0.232434i
\(974\) −3.70303 −0.118653
\(975\) 0 0
\(976\) −9.11550 −0.291780
\(977\) 24.9994i 0.799802i 0.916558 + 0.399901i \(0.130956\pi\)
−0.916558 + 0.399901i \(0.869044\pi\)
\(978\) 19.2900 0.616826
\(979\) 86.7765 2.77339
\(980\) 3.05441i 0.0975696i
\(981\) 17.9930i 0.574471i
\(982\) − 13.0116i − 0.415218i
\(983\) 27.3418i 0.872068i 0.899930 + 0.436034i \(0.143617\pi\)
−0.899930 + 0.436034i \(0.856383\pi\)
\(984\) −26.6718 −0.850267
\(985\) −1.69672 −0.0540620
\(986\) − 0.0341370i − 0.00108715i
\(987\) −21.7080 −0.690975
\(988\) 0 0
\(989\) 4.40617 0.140108
\(990\) − 15.9826i − 0.507961i
\(991\) 16.0760 0.510672 0.255336 0.966852i \(-0.417814\pi\)
0.255336 + 0.966852i \(0.417814\pi\)
\(992\) −15.4081 −0.489208
\(993\) 16.8491i 0.534690i
\(994\) − 47.4123i − 1.50383i
\(995\) 25.3255i 0.802871i
\(996\) − 14.4932i − 0.459234i
\(997\) −34.5612 −1.09456 −0.547282 0.836948i \(-0.684337\pi\)
−0.547282 + 0.836948i \(0.684337\pi\)
\(998\) −41.3649 −1.30938
\(999\) − 11.3807i − 0.360070i
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.c.g.506.6 8
13.2 odd 12 845.2.e.m.191.2 8
13.3 even 3 845.2.m.g.316.2 8
13.4 even 6 845.2.m.g.361.2 8
13.5 odd 4 845.2.a.m.1.3 4
13.6 odd 12 845.2.e.m.146.2 8
13.7 odd 12 845.2.e.n.146.3 8
13.8 odd 4 845.2.a.l.1.2 4
13.9 even 3 65.2.m.a.36.3 8
13.10 even 6 65.2.m.a.56.3 yes 8
13.11 odd 12 845.2.e.n.191.3 8
13.12 even 2 inner 845.2.c.g.506.3 8
39.5 even 4 7605.2.a.cf.1.2 4
39.8 even 4 7605.2.a.cj.1.3 4
39.23 odd 6 585.2.bu.c.316.2 8
39.35 odd 6 585.2.bu.c.361.2 8
52.23 odd 6 1040.2.da.b.641.4 8
52.35 odd 6 1040.2.da.b.881.4 8
65.9 even 6 325.2.n.d.101.2 8
65.22 odd 12 325.2.m.c.49.3 8
65.23 odd 12 325.2.m.c.199.3 8
65.34 odd 4 4225.2.a.bl.1.3 4
65.44 odd 4 4225.2.a.bi.1.2 4
65.48 odd 12 325.2.m.b.49.2 8
65.49 even 6 325.2.n.d.251.2 8
65.62 odd 12 325.2.m.b.199.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.3 8 13.9 even 3
65.2.m.a.56.3 yes 8 13.10 even 6
325.2.m.b.49.2 8 65.48 odd 12
325.2.m.b.199.2 8 65.62 odd 12
325.2.m.c.49.3 8 65.22 odd 12
325.2.m.c.199.3 8 65.23 odd 12
325.2.n.d.101.2 8 65.9 even 6
325.2.n.d.251.2 8 65.49 even 6
585.2.bu.c.316.2 8 39.23 odd 6
585.2.bu.c.361.2 8 39.35 odd 6
845.2.a.l.1.2 4 13.8 odd 4
845.2.a.m.1.3 4 13.5 odd 4
845.2.c.g.506.3 8 13.12 even 2 inner
845.2.c.g.506.6 8 1.1 even 1 trivial
845.2.e.m.146.2 8 13.6 odd 12
845.2.e.m.191.2 8 13.2 odd 12
845.2.e.n.146.3 8 13.7 odd 12
845.2.e.n.191.3 8 13.11 odd 12
845.2.m.g.316.2 8 13.3 even 3
845.2.m.g.361.2 8 13.4 even 6
1040.2.da.b.641.4 8 52.23 odd 6
1040.2.da.b.881.4 8 52.35 odd 6
4225.2.a.bi.1.2 4 65.44 odd 4
4225.2.a.bl.1.3 4 65.34 odd 4
7605.2.a.cf.1.2 4 39.5 even 4
7605.2.a.cj.1.3 4 39.8 even 4