Properties

Label 5070.2.b.t.1351.3
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.3
Root \(1.24698i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.t.1351.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +3.04892i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +3.04892i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.13706i q^{11} +1.00000 q^{12} +3.04892 q^{14} +1.00000i q^{15} +1.00000 q^{16} -2.69202 q^{17} -1.00000i q^{18} -1.55496i q^{19} +1.00000i q^{20} -3.04892i q^{21} +1.13706 q^{22} -5.40581 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -3.04892i q^{28} -6.15883 q^{29} +1.00000 q^{30} +0.246980i q^{31} -1.00000i q^{32} -1.13706i q^{33} +2.69202i q^{34} +3.04892 q^{35} -1.00000 q^{36} -0.554958i q^{37} -1.55496 q^{38} +1.00000 q^{40} -8.26875i q^{41} -3.04892 q^{42} -2.03923 q^{43} -1.13706i q^{44} -1.00000i q^{45} +5.40581i q^{46} -6.70171i q^{47} -1.00000 q^{48} -2.29590 q^{49} +1.00000i q^{50} +2.69202 q^{51} +5.77479 q^{53} +1.00000i q^{54} +1.13706 q^{55} -3.04892 q^{56} +1.55496i q^{57} +6.15883i q^{58} +1.36898i q^{59} -1.00000i q^{60} +10.9487 q^{61} +0.246980 q^{62} +3.04892i q^{63} -1.00000 q^{64} -1.13706 q^{66} +2.47219i q^{67} +2.69202 q^{68} +5.40581 q^{69} -3.04892i q^{70} +8.57002i q^{71} +1.00000i q^{72} -14.8334i q^{73} -0.554958 q^{74} +1.00000 q^{75} +1.55496i q^{76} -3.46681 q^{77} +5.33513 q^{79} -1.00000i q^{80} +1.00000 q^{81} -8.26875 q^{82} +5.45473i q^{83} +3.04892i q^{84} +2.69202i q^{85} +2.03923i q^{86} +6.15883 q^{87} -1.13706 q^{88} +16.8877i q^{89} -1.00000 q^{90} +5.40581 q^{92} -0.246980i q^{93} -6.70171 q^{94} -1.55496 q^{95} +1.00000i q^{96} -1.93900i q^{97} +2.29590i q^{98} +1.13706i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} - 6q^{10} + 6q^{12} + 6q^{16} - 6q^{17} - 4q^{22} - 6q^{23} - 6q^{25} - 6q^{27} - 20q^{29} + 6q^{30} - 6q^{36} - 10q^{38} + 6q^{40} - 38q^{43} - 6q^{48} + 14q^{49} + 6q^{51} + 38q^{53} - 4q^{55} + 2q^{61} - 8q^{62} - 6q^{64} + 4q^{66} + 6q^{68} + 6q^{69} - 4q^{74} + 6q^{75} - 14q^{77} + 30q^{79} + 6q^{81} - 34q^{82} + 20q^{87} + 4q^{88} - 6q^{90} + 6q^{92} + 14q^{94} - 10q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) 3.04892i 1.15238i 0.817315 + 0.576191i \(0.195462\pi\)
−0.817315 + 0.576191i \(0.804538\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.13706i 0.342837i 0.985198 + 0.171419i \(0.0548351\pi\)
−0.985198 + 0.171419i \(0.945165\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 3.04892 0.814857
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) −2.69202 −0.652911 −0.326456 0.945213i \(-0.605854\pi\)
−0.326456 + 0.945213i \(0.605854\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) − 1.55496i − 0.356732i −0.983964 0.178366i \(-0.942919\pi\)
0.983964 0.178366i \(-0.0570811\pi\)
\(20\) 1.00000i 0.223607i
\(21\) − 3.04892i − 0.665328i
\(22\) 1.13706 0.242423
\(23\) −5.40581 −1.12719 −0.563595 0.826051i \(-0.690582\pi\)
−0.563595 + 0.826051i \(0.690582\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) − 3.04892i − 0.576191i
\(29\) −6.15883 −1.14367 −0.571833 0.820370i \(-0.693768\pi\)
−0.571833 + 0.820370i \(0.693768\pi\)
\(30\) 1.00000 0.182574
\(31\) 0.246980i 0.0443588i 0.999754 + 0.0221794i \(0.00706051\pi\)
−0.999754 + 0.0221794i \(0.992939\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 1.13706i − 0.197937i
\(34\) 2.69202i 0.461678i
\(35\) 3.04892 0.515361
\(36\) −1.00000 −0.166667
\(37\) − 0.554958i − 0.0912346i −0.998959 0.0456173i \(-0.985475\pi\)
0.998959 0.0456173i \(-0.0145255\pi\)
\(38\) −1.55496 −0.252248
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) − 8.26875i − 1.29136i −0.763607 0.645681i \(-0.776574\pi\)
0.763607 0.645681i \(-0.223426\pi\)
\(42\) −3.04892 −0.470458
\(43\) −2.03923 −0.310979 −0.155490 0.987838i \(-0.549696\pi\)
−0.155490 + 0.987838i \(0.549696\pi\)
\(44\) − 1.13706i − 0.171419i
\(45\) − 1.00000i − 0.149071i
\(46\) 5.40581i 0.797044i
\(47\) − 6.70171i − 0.977545i −0.872411 0.488772i \(-0.837445\pi\)
0.872411 0.488772i \(-0.162555\pi\)
\(48\) −1.00000 −0.144338
\(49\) −2.29590 −0.327985
\(50\) 1.00000i 0.141421i
\(51\) 2.69202 0.376958
\(52\) 0 0
\(53\) 5.77479 0.793229 0.396614 0.917985i \(-0.370185\pi\)
0.396614 + 0.917985i \(0.370185\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 1.13706 0.153322
\(56\) −3.04892 −0.407429
\(57\) 1.55496i 0.205959i
\(58\) 6.15883i 0.808694i
\(59\) 1.36898i 0.178226i 0.996022 + 0.0891128i \(0.0284032\pi\)
−0.996022 + 0.0891128i \(0.971597\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) 10.9487 1.40184 0.700918 0.713242i \(-0.252774\pi\)
0.700918 + 0.713242i \(0.252774\pi\)
\(62\) 0.246980 0.0313664
\(63\) 3.04892i 0.384127i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −1.13706 −0.139963
\(67\) 2.47219i 0.302026i 0.988532 + 0.151013i \(0.0482535\pi\)
−0.988532 + 0.151013i \(0.951746\pi\)
\(68\) 2.69202 0.326456
\(69\) 5.40581 0.650783
\(70\) − 3.04892i − 0.364415i
\(71\) 8.57002i 1.01707i 0.861040 + 0.508537i \(0.169814\pi\)
−0.861040 + 0.508537i \(0.830186\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 14.8334i − 1.73612i −0.496461 0.868059i \(-0.665368\pi\)
0.496461 0.868059i \(-0.334632\pi\)
\(74\) −0.554958 −0.0645126
\(75\) 1.00000 0.115470
\(76\) 1.55496i 0.178366i
\(77\) −3.46681 −0.395080
\(78\) 0 0
\(79\) 5.33513 0.600249 0.300124 0.953900i \(-0.402972\pi\)
0.300124 + 0.953900i \(0.402972\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) −8.26875 −0.913131
\(83\) 5.45473i 0.598734i 0.954138 + 0.299367i \(0.0967756\pi\)
−0.954138 + 0.299367i \(0.903224\pi\)
\(84\) 3.04892i 0.332664i
\(85\) 2.69202i 0.291991i
\(86\) 2.03923i 0.219896i
\(87\) 6.15883 0.660296
\(88\) −1.13706 −0.121211
\(89\) 16.8877i 1.79009i 0.445974 + 0.895046i \(0.352857\pi\)
−0.445974 + 0.895046i \(0.647143\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 5.40581 0.563595
\(93\) − 0.246980i − 0.0256106i
\(94\) −6.70171 −0.691229
\(95\) −1.55496 −0.159535
\(96\) 1.00000i 0.102062i
\(97\) − 1.93900i − 0.196876i −0.995143 0.0984379i \(-0.968615\pi\)
0.995143 0.0984379i \(-0.0313846\pi\)
\(98\) 2.29590i 0.231921i
\(99\) 1.13706i 0.114279i
\(100\) 1.00000 0.100000
\(101\) 14.4940 1.44220 0.721101 0.692830i \(-0.243636\pi\)
0.721101 + 0.692830i \(0.243636\pi\)
\(102\) − 2.69202i − 0.266550i
\(103\) 16.1836 1.59462 0.797308 0.603572i \(-0.206257\pi\)
0.797308 + 0.603572i \(0.206257\pi\)
\(104\) 0 0
\(105\) −3.04892 −0.297544
\(106\) − 5.77479i − 0.560897i
\(107\) 4.92931 0.476535 0.238267 0.971200i \(-0.423421\pi\)
0.238267 + 0.971200i \(0.423421\pi\)
\(108\) 1.00000 0.0962250
\(109\) − 0.801938i − 0.0768117i −0.999262 0.0384059i \(-0.987772\pi\)
0.999262 0.0384059i \(-0.0122280\pi\)
\(110\) − 1.13706i − 0.108415i
\(111\) 0.554958i 0.0526743i
\(112\) 3.04892i 0.288096i
\(113\) −11.5211 −1.08381 −0.541907 0.840438i \(-0.682298\pi\)
−0.541907 + 0.840438i \(0.682298\pi\)
\(114\) 1.55496 0.145635
\(115\) 5.40581i 0.504095i
\(116\) 6.15883 0.571833
\(117\) 0 0
\(118\) 1.36898 0.126025
\(119\) − 8.20775i − 0.752403i
\(120\) −1.00000 −0.0912871
\(121\) 9.70709 0.882462
\(122\) − 10.9487i − 0.991248i
\(123\) 8.26875i 0.745568i
\(124\) − 0.246980i − 0.0221794i
\(125\) 1.00000i 0.0894427i
\(126\) 3.04892 0.271619
\(127\) −6.22521 −0.552398 −0.276199 0.961100i \(-0.589075\pi\)
−0.276199 + 0.961100i \(0.589075\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 2.03923 0.179544
\(130\) 0 0
\(131\) 3.34481 0.292238 0.146119 0.989267i \(-0.453322\pi\)
0.146119 + 0.989267i \(0.453322\pi\)
\(132\) 1.13706i 0.0989687i
\(133\) 4.74094 0.411092
\(134\) 2.47219 0.213565
\(135\) 1.00000i 0.0860663i
\(136\) − 2.69202i − 0.230839i
\(137\) − 9.53079i − 0.814271i −0.913368 0.407135i \(-0.866528\pi\)
0.913368 0.407135i \(-0.133472\pi\)
\(138\) − 5.40581i − 0.460173i
\(139\) 18.7017 1.58626 0.793129 0.609053i \(-0.208451\pi\)
0.793129 + 0.609053i \(0.208451\pi\)
\(140\) −3.04892 −0.257681
\(141\) 6.70171i 0.564386i
\(142\) 8.57002 0.719180
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 6.15883i 0.511463i
\(146\) −14.8334 −1.22762
\(147\) 2.29590 0.189362
\(148\) 0.554958i 0.0456173i
\(149\) − 16.4209i − 1.34525i −0.739983 0.672625i \(-0.765167\pi\)
0.739983 0.672625i \(-0.234833\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) − 9.26636i − 0.754085i −0.926196 0.377043i \(-0.876941\pi\)
0.926196 0.377043i \(-0.123059\pi\)
\(152\) 1.55496 0.126124
\(153\) −2.69202 −0.217637
\(154\) 3.46681i 0.279364i
\(155\) 0.246980 0.0198379
\(156\) 0 0
\(157\) 17.0761 1.36282 0.681409 0.731903i \(-0.261367\pi\)
0.681409 + 0.731903i \(0.261367\pi\)
\(158\) − 5.33513i − 0.424440i
\(159\) −5.77479 −0.457971
\(160\) −1.00000 −0.0790569
\(161\) − 16.4819i − 1.29895i
\(162\) − 1.00000i − 0.0785674i
\(163\) − 9.61894i − 0.753414i −0.926333 0.376707i \(-0.877056\pi\)
0.926333 0.376707i \(-0.122944\pi\)
\(164\) 8.26875i 0.645681i
\(165\) −1.13706 −0.0885203
\(166\) 5.45473 0.423369
\(167\) 21.8509i 1.69087i 0.534078 + 0.845435i \(0.320659\pi\)
−0.534078 + 0.845435i \(0.679341\pi\)
\(168\) 3.04892 0.235229
\(169\) 0 0
\(170\) 2.69202 0.206469
\(171\) − 1.55496i − 0.118911i
\(172\) 2.03923 0.155490
\(173\) −16.6853 −1.26856 −0.634281 0.773103i \(-0.718704\pi\)
−0.634281 + 0.773103i \(0.718704\pi\)
\(174\) − 6.15883i − 0.466900i
\(175\) − 3.04892i − 0.230476i
\(176\) 1.13706i 0.0857094i
\(177\) − 1.36898i − 0.102899i
\(178\) 16.8877 1.26579
\(179\) 3.94438 0.294817 0.147408 0.989076i \(-0.452907\pi\)
0.147408 + 0.989076i \(0.452907\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 20.5308 1.52604 0.763021 0.646374i \(-0.223715\pi\)
0.763021 + 0.646374i \(0.223715\pi\)
\(182\) 0 0
\(183\) −10.9487 −0.809350
\(184\) − 5.40581i − 0.398522i
\(185\) −0.554958 −0.0408013
\(186\) −0.246980 −0.0181094
\(187\) − 3.06100i − 0.223842i
\(188\) 6.70171i 0.488772i
\(189\) − 3.04892i − 0.221776i
\(190\) 1.55496i 0.112809i
\(191\) −6.66487 −0.482253 −0.241127 0.970494i \(-0.577517\pi\)
−0.241127 + 0.970494i \(0.577517\pi\)
\(192\) 1.00000 0.0721688
\(193\) 16.6558i 1.19891i 0.800409 + 0.599455i \(0.204616\pi\)
−0.800409 + 0.599455i \(0.795384\pi\)
\(194\) −1.93900 −0.139212
\(195\) 0 0
\(196\) 2.29590 0.163993
\(197\) − 19.4494i − 1.38571i −0.721077 0.692855i \(-0.756353\pi\)
0.721077 0.692855i \(-0.243647\pi\)
\(198\) 1.13706 0.0808076
\(199\) 21.6558 1.53514 0.767569 0.640967i \(-0.221466\pi\)
0.767569 + 0.640967i \(0.221466\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 2.47219i − 0.174375i
\(202\) − 14.4940i − 1.01979i
\(203\) − 18.7778i − 1.31794i
\(204\) −2.69202 −0.188479
\(205\) −8.26875 −0.577515
\(206\) − 16.1836i − 1.12756i
\(207\) −5.40581 −0.375730
\(208\) 0 0
\(209\) 1.76809 0.122301
\(210\) 3.04892i 0.210395i
\(211\) 12.1521 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(212\) −5.77479 −0.396614
\(213\) − 8.57002i − 0.587208i
\(214\) − 4.92931i − 0.336961i
\(215\) 2.03923i 0.139074i
\(216\) − 1.00000i − 0.0680414i
\(217\) −0.753020 −0.0511184
\(218\) −0.801938 −0.0543141
\(219\) 14.8334i 1.00235i
\(220\) −1.13706 −0.0766608
\(221\) 0 0
\(222\) 0.554958 0.0372464
\(223\) − 1.11960i − 0.0749743i −0.999297 0.0374871i \(-0.988065\pi\)
0.999297 0.0374871i \(-0.0119353\pi\)
\(224\) 3.04892 0.203714
\(225\) −1.00000 −0.0666667
\(226\) 11.5211i 0.766373i
\(227\) − 4.06638i − 0.269895i −0.990853 0.134947i \(-0.956913\pi\)
0.990853 0.134947i \(-0.0430866\pi\)
\(228\) − 1.55496i − 0.102980i
\(229\) − 16.5187i − 1.09159i −0.837920 0.545794i \(-0.816228\pi\)
0.837920 0.545794i \(-0.183772\pi\)
\(230\) 5.40581 0.356449
\(231\) 3.46681 0.228099
\(232\) − 6.15883i − 0.404347i
\(233\) 6.16421 0.403831 0.201915 0.979403i \(-0.435283\pi\)
0.201915 + 0.979403i \(0.435283\pi\)
\(234\) 0 0
\(235\) −6.70171 −0.437171
\(236\) − 1.36898i − 0.0891128i
\(237\) −5.33513 −0.346554
\(238\) −8.20775 −0.532029
\(239\) − 18.1793i − 1.17592i −0.808890 0.587960i \(-0.799931\pi\)
0.808890 0.587960i \(-0.200069\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) − 9.21744i − 0.593747i −0.954917 0.296874i \(-0.904056\pi\)
0.954917 0.296874i \(-0.0959441\pi\)
\(242\) − 9.70709i − 0.623995i
\(243\) −1.00000 −0.0641500
\(244\) −10.9487 −0.700918
\(245\) 2.29590i 0.146679i
\(246\) 8.26875 0.527196
\(247\) 0 0
\(248\) −0.246980 −0.0156832
\(249\) − 5.45473i − 0.345680i
\(250\) 1.00000 0.0632456
\(251\) −17.2054 −1.08599 −0.542996 0.839735i \(-0.682710\pi\)
−0.542996 + 0.839735i \(0.682710\pi\)
\(252\) − 3.04892i − 0.192064i
\(253\) − 6.14675i − 0.386443i
\(254\) 6.22521i 0.390604i
\(255\) − 2.69202i − 0.168581i
\(256\) 1.00000 0.0625000
\(257\) 19.5157 1.21736 0.608679 0.793417i \(-0.291700\pi\)
0.608679 + 0.793417i \(0.291700\pi\)
\(258\) − 2.03923i − 0.126957i
\(259\) 1.69202 0.105137
\(260\) 0 0
\(261\) −6.15883 −0.381222
\(262\) − 3.34481i − 0.206643i
\(263\) 12.8834 0.794423 0.397212 0.917727i \(-0.369978\pi\)
0.397212 + 0.917727i \(0.369978\pi\)
\(264\) 1.13706 0.0699814
\(265\) − 5.77479i − 0.354743i
\(266\) − 4.74094i − 0.290686i
\(267\) − 16.8877i − 1.03351i
\(268\) − 2.47219i − 0.151013i
\(269\) −30.9138 −1.88485 −0.942423 0.334423i \(-0.891459\pi\)
−0.942423 + 0.334423i \(0.891459\pi\)
\(270\) 1.00000 0.0608581
\(271\) 7.24027i 0.439815i 0.975521 + 0.219908i \(0.0705756\pi\)
−0.975521 + 0.219908i \(0.929424\pi\)
\(272\) −2.69202 −0.163228
\(273\) 0 0
\(274\) −9.53079 −0.575776
\(275\) − 1.13706i − 0.0685675i
\(276\) −5.40581 −0.325392
\(277\) −12.2446 −0.735706 −0.367853 0.929884i \(-0.619907\pi\)
−0.367853 + 0.929884i \(0.619907\pi\)
\(278\) − 18.7017i − 1.12165i
\(279\) 0.246980i 0.0147863i
\(280\) 3.04892i 0.182208i
\(281\) − 2.74632i − 0.163831i −0.996639 0.0819157i \(-0.973896\pi\)
0.996639 0.0819157i \(-0.0261038\pi\)
\(282\) 6.70171 0.399081
\(283\) −14.9312 −0.887570 −0.443785 0.896133i \(-0.646365\pi\)
−0.443785 + 0.896133i \(0.646365\pi\)
\(284\) − 8.57002i − 0.508537i
\(285\) 1.55496 0.0921078
\(286\) 0 0
\(287\) 25.2107 1.48814
\(288\) − 1.00000i − 0.0589256i
\(289\) −9.75302 −0.573707
\(290\) 6.15883 0.361659
\(291\) 1.93900i 0.113666i
\(292\) 14.8334i 0.868059i
\(293\) 18.5810i 1.08552i 0.839889 + 0.542758i \(0.182620\pi\)
−0.839889 + 0.542758i \(0.817380\pi\)
\(294\) − 2.29590i − 0.133899i
\(295\) 1.36898 0.0797049
\(296\) 0.554958 0.0322563
\(297\) − 1.13706i − 0.0659791i
\(298\) −16.4209 −0.951236
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) − 6.21744i − 0.358367i
\(302\) −9.26636 −0.533219
\(303\) −14.4940 −0.832656
\(304\) − 1.55496i − 0.0891830i
\(305\) − 10.9487i − 0.626920i
\(306\) 2.69202i 0.153893i
\(307\) 32.4359i 1.85122i 0.378482 + 0.925609i \(0.376446\pi\)
−0.378482 + 0.925609i \(0.623554\pi\)
\(308\) 3.46681 0.197540
\(309\) −16.1836 −0.920652
\(310\) − 0.246980i − 0.0140275i
\(311\) 21.2771 1.20651 0.603257 0.797547i \(-0.293869\pi\)
0.603257 + 0.797547i \(0.293869\pi\)
\(312\) 0 0
\(313\) −18.7060 −1.05733 −0.528663 0.848832i \(-0.677307\pi\)
−0.528663 + 0.848832i \(0.677307\pi\)
\(314\) − 17.0761i − 0.963658i
\(315\) 3.04892 0.171787
\(316\) −5.33513 −0.300124
\(317\) − 17.4776i − 0.981638i −0.871262 0.490819i \(-0.836698\pi\)
0.871262 0.490819i \(-0.163302\pi\)
\(318\) 5.77479i 0.323834i
\(319\) − 7.00298i − 0.392092i
\(320\) 1.00000i 0.0559017i
\(321\) −4.92931 −0.275127
\(322\) −16.4819 −0.918499
\(323\) 4.18598i 0.232914i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −9.61894 −0.532744
\(327\) 0.801938i 0.0443473i
\(328\) 8.26875 0.456565
\(329\) 20.4330 1.12651
\(330\) 1.13706i 0.0625933i
\(331\) 23.1890i 1.27458i 0.770624 + 0.637290i \(0.219945\pi\)
−0.770624 + 0.637290i \(0.780055\pi\)
\(332\) − 5.45473i − 0.299367i
\(333\) − 0.554958i − 0.0304115i
\(334\) 21.8509 1.19563
\(335\) 2.47219 0.135070
\(336\) − 3.04892i − 0.166332i
\(337\) 26.1400 1.42394 0.711970 0.702210i \(-0.247803\pi\)
0.711970 + 0.702210i \(0.247803\pi\)
\(338\) 0 0
\(339\) 11.5211 0.625741
\(340\) − 2.69202i − 0.145995i
\(341\) −0.280831 −0.0152079
\(342\) −1.55496 −0.0840825
\(343\) 14.3424i 0.774418i
\(344\) − 2.03923i − 0.109948i
\(345\) − 5.40581i − 0.291039i
\(346\) 16.6853i 0.897008i
\(347\) 15.2959 0.821127 0.410563 0.911832i \(-0.365332\pi\)
0.410563 + 0.911832i \(0.365332\pi\)
\(348\) −6.15883 −0.330148
\(349\) − 19.2107i − 1.02833i −0.857692 0.514164i \(-0.828102\pi\)
0.857692 0.514164i \(-0.171898\pi\)
\(350\) −3.04892 −0.162971
\(351\) 0 0
\(352\) 1.13706 0.0606057
\(353\) 5.35450i 0.284991i 0.989795 + 0.142496i \(0.0455127\pi\)
−0.989795 + 0.142496i \(0.954487\pi\)
\(354\) −1.36898 −0.0727603
\(355\) 8.57002 0.454850
\(356\) − 16.8877i − 0.895046i
\(357\) 8.20775i 0.434400i
\(358\) − 3.94438i − 0.208467i
\(359\) 8.49934i 0.448578i 0.974523 + 0.224289i \(0.0720059\pi\)
−0.974523 + 0.224289i \(0.927994\pi\)
\(360\) 1.00000 0.0527046
\(361\) 16.5821 0.872742
\(362\) − 20.5308i − 1.07907i
\(363\) −9.70709 −0.509490
\(364\) 0 0
\(365\) −14.8334 −0.776415
\(366\) 10.9487i 0.572297i
\(367\) 4.76032 0.248486 0.124243 0.992252i \(-0.460350\pi\)
0.124243 + 0.992252i \(0.460350\pi\)
\(368\) −5.40581 −0.281797
\(369\) − 8.26875i − 0.430454i
\(370\) 0.554958i 0.0288509i
\(371\) 17.6069i 0.914103i
\(372\) 0.246980i 0.0128053i
\(373\) −20.2064 −1.04625 −0.523124 0.852256i \(-0.675234\pi\)
−0.523124 + 0.852256i \(0.675234\pi\)
\(374\) −3.06100 −0.158280
\(375\) − 1.00000i − 0.0516398i
\(376\) 6.70171 0.345614
\(377\) 0 0
\(378\) −3.04892 −0.156819
\(379\) − 24.4088i − 1.25380i −0.779101 0.626898i \(-0.784324\pi\)
0.779101 0.626898i \(-0.215676\pi\)
\(380\) 1.55496 0.0797677
\(381\) 6.22521 0.318927
\(382\) 6.66487i 0.341005i
\(383\) − 32.0237i − 1.63633i −0.574981 0.818167i \(-0.694990\pi\)
0.574981 0.818167i \(-0.305010\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 3.46681i 0.176685i
\(386\) 16.6558 0.847757
\(387\) −2.03923 −0.103660
\(388\) 1.93900i 0.0984379i
\(389\) −21.1879 −1.07427 −0.537135 0.843497i \(-0.680493\pi\)
−0.537135 + 0.843497i \(0.680493\pi\)
\(390\) 0 0
\(391\) 14.5526 0.735955
\(392\) − 2.29590i − 0.115960i
\(393\) −3.34481 −0.168724
\(394\) −19.4494 −0.979844
\(395\) − 5.33513i − 0.268439i
\(396\) − 1.13706i − 0.0571396i
\(397\) 10.9608i 0.550105i 0.961429 + 0.275053i \(0.0886953\pi\)
−0.961429 + 0.275053i \(0.911305\pi\)
\(398\) − 21.6558i − 1.08551i
\(399\) −4.74094 −0.237344
\(400\) −1.00000 −0.0500000
\(401\) − 29.2301i − 1.45968i −0.683617 0.729841i \(-0.739594\pi\)
0.683617 0.729841i \(-0.260406\pi\)
\(402\) −2.47219 −0.123302
\(403\) 0 0
\(404\) −14.4940 −0.721101
\(405\) − 1.00000i − 0.0496904i
\(406\) −18.7778 −0.931925
\(407\) 0.631023 0.0312786
\(408\) 2.69202i 0.133275i
\(409\) − 36.3163i − 1.79573i −0.440274 0.897864i \(-0.645119\pi\)
0.440274 0.897864i \(-0.354881\pi\)
\(410\) 8.26875i 0.408364i
\(411\) 9.53079i 0.470119i
\(412\) −16.1836 −0.797308
\(413\) −4.17390 −0.205384
\(414\) 5.40581i 0.265681i
\(415\) 5.45473 0.267762
\(416\) 0 0
\(417\) −18.7017 −0.915827
\(418\) − 1.76809i − 0.0864799i
\(419\) 11.0382 0.539250 0.269625 0.962965i \(-0.413100\pi\)
0.269625 + 0.962965i \(0.413100\pi\)
\(420\) 3.04892 0.148772
\(421\) − 4.58642i − 0.223528i −0.993735 0.111764i \(-0.964350\pi\)
0.993735 0.111764i \(-0.0356501\pi\)
\(422\) − 12.1521i − 0.591556i
\(423\) − 6.70171i − 0.325848i
\(424\) 5.77479i 0.280449i
\(425\) 2.69202 0.130582
\(426\) −8.57002 −0.415219
\(427\) 33.3817i 1.61545i
\(428\) −4.92931 −0.238267
\(429\) 0 0
\(430\) 2.03923 0.0983403
\(431\) − 19.3642i − 0.932740i −0.884590 0.466370i \(-0.845561\pi\)
0.884590 0.466370i \(-0.154439\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −4.41119 −0.211988 −0.105994 0.994367i \(-0.533802\pi\)
−0.105994 + 0.994367i \(0.533802\pi\)
\(434\) 0.753020i 0.0361461i
\(435\) − 6.15883i − 0.295293i
\(436\) 0.801938i 0.0384059i
\(437\) 8.40581i 0.402105i
\(438\) 14.8334 0.708767
\(439\) −39.5260 −1.88647 −0.943237 0.332121i \(-0.892236\pi\)
−0.943237 + 0.332121i \(0.892236\pi\)
\(440\) 1.13706i 0.0542074i
\(441\) −2.29590 −0.109328
\(442\) 0 0
\(443\) −7.16182 −0.340268 −0.170134 0.985421i \(-0.554420\pi\)
−0.170134 + 0.985421i \(0.554420\pi\)
\(444\) − 0.554958i − 0.0263371i
\(445\) 16.8877 0.800553
\(446\) −1.11960 −0.0530148
\(447\) 16.4209i 0.776681i
\(448\) − 3.04892i − 0.144048i
\(449\) − 4.46011i − 0.210485i −0.994447 0.105243i \(-0.966438\pi\)
0.994447 0.105243i \(-0.0335620\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 9.40209 0.442727
\(452\) 11.5211 0.541907
\(453\) 9.26636i 0.435371i
\(454\) −4.06638 −0.190844
\(455\) 0 0
\(456\) −1.55496 −0.0728176
\(457\) 8.04593i 0.376373i 0.982133 + 0.188186i \(0.0602609\pi\)
−0.982133 + 0.188186i \(0.939739\pi\)
\(458\) −16.5187 −0.771869
\(459\) 2.69202 0.125653
\(460\) − 5.40581i − 0.252047i
\(461\) − 4.00538i − 0.186549i −0.995640 0.0932745i \(-0.970267\pi\)
0.995640 0.0932745i \(-0.0297334\pi\)
\(462\) − 3.46681i − 0.161291i
\(463\) − 37.6142i − 1.74808i −0.485856 0.874039i \(-0.661492\pi\)
0.485856 0.874039i \(-0.338508\pi\)
\(464\) −6.15883 −0.285917
\(465\) −0.246980 −0.0114534
\(466\) − 6.16421i − 0.285552i
\(467\) 32.8079 1.51817 0.759084 0.650992i \(-0.225647\pi\)
0.759084 + 0.650992i \(0.225647\pi\)
\(468\) 0 0
\(469\) −7.53750 −0.348049
\(470\) 6.70171i 0.309127i
\(471\) −17.0761 −0.786824
\(472\) −1.36898 −0.0630123
\(473\) − 2.31873i − 0.106615i
\(474\) 5.33513i 0.245050i
\(475\) 1.55496i 0.0713464i
\(476\) 8.20775i 0.376202i
\(477\) 5.77479 0.264410
\(478\) −18.1793 −0.831501
\(479\) − 9.88530i − 0.451671i −0.974165 0.225835i \(-0.927489\pi\)
0.974165 0.225835i \(-0.0725112\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −9.21744 −0.419843
\(483\) 16.4819i 0.749951i
\(484\) −9.70709 −0.441231
\(485\) −1.93900 −0.0880455
\(486\) 1.00000i 0.0453609i
\(487\) − 20.2610i − 0.918113i −0.888407 0.459056i \(-0.848188\pi\)
0.888407 0.459056i \(-0.151812\pi\)
\(488\) 10.9487i 0.495624i
\(489\) 9.61894i 0.434984i
\(490\) 2.29590 0.103718
\(491\) 22.4940 1.01514 0.507569 0.861611i \(-0.330544\pi\)
0.507569 + 0.861611i \(0.330544\pi\)
\(492\) − 8.26875i − 0.372784i
\(493\) 16.5797 0.746713
\(494\) 0 0
\(495\) 1.13706 0.0511072
\(496\) 0.246980i 0.0110897i
\(497\) −26.1293 −1.17206
\(498\) −5.45473 −0.244432
\(499\) 15.2728i 0.683704i 0.939754 + 0.341852i \(0.111054\pi\)
−0.939754 + 0.341852i \(0.888946\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 21.8509i − 0.976224i
\(502\) 17.2054i 0.767913i
\(503\) −2.95838 −0.131908 −0.0659538 0.997823i \(-0.521009\pi\)
−0.0659538 + 0.997823i \(0.521009\pi\)
\(504\) −3.04892 −0.135810
\(505\) − 14.4940i − 0.644973i
\(506\) −6.14675 −0.273256
\(507\) 0 0
\(508\) 6.22521 0.276199
\(509\) 4.34614i 0.192639i 0.995350 + 0.0963197i \(0.0307071\pi\)
−0.995350 + 0.0963197i \(0.969293\pi\)
\(510\) −2.69202 −0.119205
\(511\) 45.2258 2.00067
\(512\) − 1.00000i − 0.0441942i
\(513\) 1.55496i 0.0686531i
\(514\) − 19.5157i − 0.860802i
\(515\) − 16.1836i − 0.713134i
\(516\) −2.03923 −0.0897720
\(517\) 7.62027 0.335139
\(518\) − 1.69202i − 0.0743432i
\(519\) 16.6853 0.732404
\(520\) 0 0
\(521\) 9.13228 0.400092 0.200046 0.979786i \(-0.435891\pi\)
0.200046 + 0.979786i \(0.435891\pi\)
\(522\) 6.15883i 0.269565i
\(523\) 10.2828 0.449633 0.224817 0.974401i \(-0.427822\pi\)
0.224817 + 0.974401i \(0.427822\pi\)
\(524\) −3.34481 −0.146119
\(525\) 3.04892i 0.133066i
\(526\) − 12.8834i − 0.561742i
\(527\) − 0.664874i − 0.0289624i
\(528\) − 1.13706i − 0.0494843i
\(529\) 6.22282 0.270557
\(530\) −5.77479 −0.250841
\(531\) 1.36898i 0.0594086i
\(532\) −4.74094 −0.205546
\(533\) 0 0
\(534\) −16.8877 −0.730802
\(535\) − 4.92931i − 0.213113i
\(536\) −2.47219 −0.106782
\(537\) −3.94438 −0.170212
\(538\) 30.9138i 1.33279i
\(539\) − 2.61058i − 0.112446i
\(540\) − 1.00000i − 0.0430331i
\(541\) 1.47517i 0.0634226i 0.999497 + 0.0317113i \(0.0100957\pi\)
−0.999497 + 0.0317113i \(0.989904\pi\)
\(542\) 7.24027 0.310996
\(543\) −20.5308 −0.881061
\(544\) 2.69202i 0.115419i
\(545\) −0.801938 −0.0343512
\(546\) 0 0
\(547\) −4.82908 −0.206477 −0.103238 0.994657i \(-0.532920\pi\)
−0.103238 + 0.994657i \(0.532920\pi\)
\(548\) 9.53079i 0.407135i
\(549\) 10.9487 0.467279
\(550\) −1.13706 −0.0484845
\(551\) 9.57673i 0.407982i
\(552\) 5.40581i 0.230087i
\(553\) 16.2664i 0.691716i
\(554\) 12.2446i 0.520223i
\(555\) 0.554958 0.0235567
\(556\) −18.7017 −0.793129
\(557\) 3.26145i 0.138192i 0.997610 + 0.0690961i \(0.0220115\pi\)
−0.997610 + 0.0690961i \(0.977988\pi\)
\(558\) 0.246980 0.0104555
\(559\) 0 0
\(560\) 3.04892 0.128840
\(561\) 3.06100i 0.129235i
\(562\) −2.74632 −0.115846
\(563\) −18.1551 −0.765147 −0.382573 0.923925i \(-0.624962\pi\)
−0.382573 + 0.923925i \(0.624962\pi\)
\(564\) − 6.70171i − 0.282193i
\(565\) 11.5211i 0.484697i
\(566\) 14.9312i 0.627606i
\(567\) 3.04892i 0.128042i
\(568\) −8.57002 −0.359590
\(569\) 39.2669 1.64616 0.823078 0.567928i \(-0.192255\pi\)
0.823078 + 0.567928i \(0.192255\pi\)
\(570\) − 1.55496i − 0.0651300i
\(571\) 36.8649 1.54275 0.771373 0.636383i \(-0.219570\pi\)
0.771373 + 0.636383i \(0.219570\pi\)
\(572\) 0 0
\(573\) 6.66487 0.278429
\(574\) − 25.2107i − 1.05228i
\(575\) 5.40581 0.225438
\(576\) −1.00000 −0.0416667
\(577\) 20.3043i 0.845277i 0.906298 + 0.422639i \(0.138896\pi\)
−0.906298 + 0.422639i \(0.861104\pi\)
\(578\) 9.75302i 0.405672i
\(579\) − 16.6558i − 0.692190i
\(580\) − 6.15883i − 0.255732i
\(581\) −16.6310 −0.689971
\(582\) 1.93900 0.0803742
\(583\) 6.56630i 0.271948i
\(584\) 14.8334 0.613810
\(585\) 0 0
\(586\) 18.5810 0.767576
\(587\) − 24.5472i − 1.01317i −0.862190 0.506585i \(-0.830907\pi\)
0.862190 0.506585i \(-0.169093\pi\)
\(588\) −2.29590 −0.0946812
\(589\) 0.384043 0.0158242
\(590\) − 1.36898i − 0.0563599i
\(591\) 19.4494i 0.800040i
\(592\) − 0.554958i − 0.0228086i
\(593\) − 24.0844i − 0.989029i −0.869169 0.494514i \(-0.835346\pi\)
0.869169 0.494514i \(-0.164654\pi\)
\(594\) −1.13706 −0.0466543
\(595\) −8.20775 −0.336485
\(596\) 16.4209i 0.672625i
\(597\) −21.6558 −0.886312
\(598\) 0 0
\(599\) 1.84415 0.0753499 0.0376750 0.999290i \(-0.488005\pi\)
0.0376750 + 0.999290i \(0.488005\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −15.6770 −0.639476 −0.319738 0.947506i \(-0.603595\pi\)
−0.319738 + 0.947506i \(0.603595\pi\)
\(602\) −6.21744 −0.253404
\(603\) 2.47219i 0.100675i
\(604\) 9.26636i 0.377043i
\(605\) − 9.70709i − 0.394649i
\(606\) 14.4940i 0.588777i
\(607\) −29.6866 −1.20494 −0.602472 0.798140i \(-0.705817\pi\)
−0.602472 + 0.798140i \(0.705817\pi\)
\(608\) −1.55496 −0.0630619
\(609\) 18.7778i 0.760914i
\(610\) −10.9487 −0.443299
\(611\) 0 0
\(612\) 2.69202 0.108819
\(613\) 42.5163i 1.71722i 0.512631 + 0.858609i \(0.328671\pi\)
−0.512631 + 0.858609i \(0.671329\pi\)
\(614\) 32.4359 1.30901
\(615\) 8.26875 0.333428
\(616\) − 3.46681i − 0.139682i
\(617\) − 27.3666i − 1.10174i −0.834592 0.550869i \(-0.814297\pi\)
0.834592 0.550869i \(-0.185703\pi\)
\(618\) 16.1836i 0.650999i
\(619\) 27.9420i 1.12308i 0.827449 + 0.561542i \(0.189792\pi\)
−0.827449 + 0.561542i \(0.810208\pi\)
\(620\) −0.246980 −0.00991894
\(621\) 5.40581 0.216928
\(622\) − 21.2771i − 0.853134i
\(623\) −51.4892 −2.06287
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 18.7060i 0.747643i
\(627\) −1.76809 −0.0706105
\(628\) −17.0761 −0.681409
\(629\) 1.49396i 0.0595681i
\(630\) − 3.04892i − 0.121472i
\(631\) 7.43594i 0.296020i 0.988986 + 0.148010i \(0.0472868\pi\)
−0.988986 + 0.148010i \(0.952713\pi\)
\(632\) 5.33513i 0.212220i
\(633\) −12.1521 −0.483004
\(634\) −17.4776 −0.694123
\(635\) 6.22521i 0.247040i
\(636\) 5.77479 0.228985
\(637\) 0 0
\(638\) −7.00298 −0.277251
\(639\) 8.57002i 0.339025i
\(640\) 1.00000 0.0395285
\(641\) 24.9444 0.985244 0.492622 0.870243i \(-0.336039\pi\)
0.492622 + 0.870243i \(0.336039\pi\)
\(642\) 4.92931i 0.194544i
\(643\) − 30.1661i − 1.18964i −0.803860 0.594818i \(-0.797224\pi\)
0.803860 0.594818i \(-0.202776\pi\)
\(644\) 16.4819i 0.649477i
\(645\) − 2.03923i − 0.0802946i
\(646\) 4.18598 0.164695
\(647\) 44.4053 1.74575 0.872877 0.487940i \(-0.162252\pi\)
0.872877 + 0.487940i \(0.162252\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −1.55661 −0.0611024
\(650\) 0 0
\(651\) 0.753020 0.0295132
\(652\) 9.61894i 0.376707i
\(653\) −29.2573 −1.14493 −0.572463 0.819931i \(-0.694012\pi\)
−0.572463 + 0.819931i \(0.694012\pi\)
\(654\) 0.801938 0.0313582
\(655\) − 3.34481i − 0.130693i
\(656\) − 8.26875i − 0.322840i
\(657\) − 14.8334i − 0.578706i
\(658\) − 20.4330i − 0.796560i
\(659\) −11.4808 −0.447229 −0.223614 0.974678i \(-0.571786\pi\)
−0.223614 + 0.974678i \(0.571786\pi\)
\(660\) 1.13706 0.0442601
\(661\) 26.1129i 1.01567i 0.861453 + 0.507837i \(0.169555\pi\)
−0.861453 + 0.507837i \(0.830445\pi\)
\(662\) 23.1890 0.901265
\(663\) 0 0
\(664\) −5.45473 −0.211685
\(665\) − 4.74094i − 0.183846i
\(666\) −0.554958 −0.0215042
\(667\) 33.2935 1.28913
\(668\) − 21.8509i − 0.845435i
\(669\) 1.11960i 0.0432864i
\(670\) − 2.47219i − 0.0955090i
\(671\) 12.4494i 0.480602i
\(672\) −3.04892 −0.117615
\(673\) 12.4190 0.478716 0.239358 0.970931i \(-0.423063\pi\)
0.239358 + 0.970931i \(0.423063\pi\)
\(674\) − 26.1400i − 1.00688i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 12.7646 0.490585 0.245292 0.969449i \(-0.421116\pi\)
0.245292 + 0.969449i \(0.421116\pi\)
\(678\) − 11.5211i − 0.442465i
\(679\) 5.91185 0.226876
\(680\) −2.69202 −0.103234
\(681\) 4.06638i 0.155824i
\(682\) 0.280831i 0.0107536i
\(683\) 31.6142i 1.20968i 0.796346 + 0.604841i \(0.206763\pi\)
−0.796346 + 0.604841i \(0.793237\pi\)
\(684\) 1.55496i 0.0594553i
\(685\) −9.53079 −0.364153
\(686\) 14.3424 0.547596
\(687\) 16.5187i 0.630228i
\(688\) −2.03923 −0.0777449
\(689\) 0 0
\(690\) −5.40581 −0.205796
\(691\) − 2.06962i − 0.0787322i −0.999225 0.0393661i \(-0.987466\pi\)
0.999225 0.0393661i \(-0.0125339\pi\)
\(692\) 16.6853 0.634281
\(693\) −3.46681 −0.131693
\(694\) − 15.2959i − 0.580624i
\(695\) − 18.7017i − 0.709396i
\(696\) 6.15883i 0.233450i
\(697\) 22.2597i 0.843144i
\(698\) −19.2107 −0.727137
\(699\) −6.16421 −0.233152
\(700\) 3.04892i 0.115238i
\(701\) 23.0030 0.868811 0.434405 0.900717i \(-0.356959\pi\)
0.434405 + 0.900717i \(0.356959\pi\)
\(702\) 0 0
\(703\) −0.862937 −0.0325463
\(704\) − 1.13706i − 0.0428547i
\(705\) 6.70171 0.252401
\(706\) 5.35450 0.201519
\(707\) 44.1909i 1.66197i
\(708\) 1.36898i 0.0514493i
\(709\) − 32.4373i − 1.21821i −0.793091 0.609104i \(-0.791529\pi\)
0.793091 0.609104i \(-0.208471\pi\)
\(710\) − 8.57002i − 0.321627i
\(711\) 5.33513 0.200083
\(712\) −16.8877 −0.632893
\(713\) − 1.33513i − 0.0500008i
\(714\) 8.20775 0.307167
\(715\) 0 0
\(716\) −3.94438 −0.147408
\(717\) 18.1793i 0.678917i
\(718\) 8.49934 0.317192
\(719\) 13.7845 0.514074 0.257037 0.966402i \(-0.417254\pi\)
0.257037 + 0.966402i \(0.417254\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) 49.3424i 1.83761i
\(722\) − 16.5821i − 0.617122i
\(723\) 9.21744i 0.342800i
\(724\) −20.5308 −0.763021
\(725\) 6.15883 0.228733
\(726\) 9.70709i 0.360264i
\(727\) −4.64848 −0.172403 −0.0862013 0.996278i \(-0.527473\pi\)
−0.0862013 + 0.996278i \(0.527473\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 14.8334i 0.549009i
\(731\) 5.48965 0.203042
\(732\) 10.9487 0.404675
\(733\) 27.4717i 1.01469i 0.861743 + 0.507345i \(0.169373\pi\)
−0.861743 + 0.507345i \(0.830627\pi\)
\(734\) − 4.76032i − 0.175706i
\(735\) − 2.29590i − 0.0846854i
\(736\) 5.40581i 0.199261i
\(737\) −2.81104 −0.103546
\(738\) −8.26875 −0.304377
\(739\) − 48.0170i − 1.76633i −0.469059 0.883167i \(-0.655407\pi\)
0.469059 0.883167i \(-0.344593\pi\)
\(740\) 0.554958 0.0204007
\(741\) 0 0
\(742\) 17.6069 0.646368
\(743\) 23.6517i 0.867698i 0.900986 + 0.433849i \(0.142845\pi\)
−0.900986 + 0.433849i \(0.857155\pi\)
\(744\) 0.246980 0.00905471
\(745\) −16.4209 −0.601614
\(746\) 20.2064i 0.739810i
\(747\) 5.45473i 0.199578i
\(748\) 3.06100i 0.111921i
\(749\) 15.0291i 0.549150i
\(750\) −1.00000 −0.0365148
\(751\) 39.5133 1.44186 0.720931 0.693007i \(-0.243714\pi\)
0.720931 + 0.693007i \(0.243714\pi\)
\(752\) − 6.70171i − 0.244386i
\(753\) 17.2054 0.626998
\(754\) 0 0
\(755\) −9.26636 −0.337237
\(756\) 3.04892i 0.110888i
\(757\) −11.2239 −0.407939 −0.203969 0.978977i \(-0.565384\pi\)
−0.203969 + 0.978977i \(0.565384\pi\)
\(758\) −24.4088 −0.886567
\(759\) 6.14675i 0.223113i
\(760\) − 1.55496i − 0.0564043i
\(761\) − 5.80864i − 0.210563i −0.994442 0.105282i \(-0.966426\pi\)
0.994442 0.105282i \(-0.0335744\pi\)
\(762\) − 6.22521i − 0.225516i
\(763\) 2.44504 0.0885165
\(764\) 6.66487 0.241127
\(765\) 2.69202i 0.0973302i
\(766\) −32.0237 −1.15706
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 31.8316i 1.14788i 0.818899 + 0.573938i \(0.194585\pi\)
−0.818899 + 0.573938i \(0.805415\pi\)
\(770\) 3.46681 0.124935
\(771\) −19.5157 −0.702842
\(772\) − 16.6558i − 0.599455i
\(773\) − 39.7434i − 1.42947i −0.699394 0.714736i \(-0.746547\pi\)
0.699394 0.714736i \(-0.253453\pi\)
\(774\) 2.03923i 0.0732986i
\(775\) − 0.246980i − 0.00887177i
\(776\) 1.93900 0.0696061
\(777\) −1.69202 −0.0607009
\(778\) 21.1879i 0.759623i
\(779\) −12.8576 −0.460670
\(780\) 0 0
\(781\) −9.74466 −0.348691
\(782\) − 14.5526i − 0.520399i
\(783\) 6.15883 0.220099
\(784\) −2.29590 −0.0819963
\(785\) − 17.0761i − 0.609471i
\(786\) 3.34481i 0.119306i
\(787\) 19.3183i 0.688622i 0.938856 + 0.344311i \(0.111887\pi\)
−0.938856 + 0.344311i \(0.888113\pi\)
\(788\) 19.4494i 0.692855i
\(789\) −12.8834 −0.458660
\(790\) −5.33513 −0.189815
\(791\) − 35.1269i − 1.24897i
\(792\) −1.13706 −0.0404038
\(793\) 0 0
\(794\) 10.9608 0.388983
\(795\) 5.77479i 0.204811i
\(796\) −21.6558 −0.767569
\(797\) −29.4946 −1.04475 −0.522375 0.852716i \(-0.674954\pi\)
−0.522375 + 0.852716i \(0.674954\pi\)
\(798\) 4.74094i 0.167827i
\(799\) 18.0411i 0.638250i
\(800\) 1.00000i 0.0353553i
\(801\) 16.8877i 0.596697i
\(802\) −29.2301 −1.03215
\(803\) 16.8665 0.595206
\(804\) 2.47219i 0.0871874i
\(805\) −16.4819 −0.580910
\(806\) 0 0
\(807\) 30.9138 1.08822
\(808\) 14.4940i 0.509896i
\(809\) 14.0901 0.495380 0.247690 0.968839i \(-0.420329\pi\)
0.247690 + 0.968839i \(0.420329\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 45.5060i 1.59793i 0.601375 + 0.798967i \(0.294620\pi\)
−0.601375 + 0.798967i \(0.705380\pi\)
\(812\) 18.7778i 0.658971i
\(813\) − 7.24027i − 0.253928i
\(814\) − 0.631023i − 0.0221173i
\(815\) −9.61894 −0.336937
\(816\) 2.69202 0.0942396
\(817\) 3.17092i 0.110936i
\(818\) −36.3163 −1.26977
\(819\) 0 0
\(820\) 8.26875 0.288757
\(821\) 14.6853i 0.512521i 0.966608 + 0.256261i \(0.0824905\pi\)
−0.966608 + 0.256261i \(0.917509\pi\)
\(822\) 9.53079 0.332425
\(823\) 32.0551 1.11737 0.558686 0.829379i \(-0.311306\pi\)
0.558686 + 0.829379i \(0.311306\pi\)
\(824\) 16.1836i 0.563782i
\(825\) 1.13706i 0.0395875i
\(826\) 4.17390i 0.145229i
\(827\) 8.09352i 0.281439i 0.990049 + 0.140720i \(0.0449416\pi\)
−0.990049 + 0.140720i \(0.955058\pi\)
\(828\) 5.40581 0.187865
\(829\) 23.8708 0.829068 0.414534 0.910034i \(-0.363945\pi\)
0.414534 + 0.910034i \(0.363945\pi\)
\(830\) − 5.45473i − 0.189336i
\(831\) 12.2446 0.424760
\(832\) 0 0
\(833\) 6.18060 0.214145
\(834\) 18.7017i 0.647587i
\(835\) 21.8509 0.756180
\(836\) −1.76809 −0.0611505
\(837\) − 0.246980i − 0.00853686i
\(838\) − 11.0382i − 0.381307i
\(839\) − 17.5569i − 0.606131i −0.952970 0.303065i \(-0.901990\pi\)
0.952970 0.303065i \(-0.0980100\pi\)
\(840\) − 3.04892i − 0.105198i
\(841\) 8.93123 0.307973
\(842\) −4.58642 −0.158058
\(843\) 2.74632i 0.0945881i
\(844\) −12.1521 −0.418294
\(845\) 0 0
\(846\) −6.70171 −0.230410
\(847\) 29.5961i 1.01693i
\(848\) 5.77479 0.198307
\(849\) 14.9312 0.512439
\(850\) − 2.69202i − 0.0923356i
\(851\) 3.00000i 0.102839i
\(852\) 8.57002i 0.293604i
\(853\) − 40.2892i − 1.37948i −0.724059 0.689738i \(-0.757726\pi\)
0.724059 0.689738i \(-0.242274\pi\)
\(854\) 33.3817 1.14230
\(855\) −1.55496 −0.0531784
\(856\) 4.92931i 0.168480i
\(857\) −44.5913 −1.52321 −0.761605 0.648041i \(-0.775588\pi\)
−0.761605 + 0.648041i \(0.775588\pi\)
\(858\) 0 0
\(859\) −40.6467 −1.38685 −0.693423 0.720530i \(-0.743898\pi\)
−0.693423 + 0.720530i \(0.743898\pi\)
\(860\) − 2.03923i − 0.0695371i
\(861\) −25.2107 −0.859180
\(862\) −19.3642 −0.659547
\(863\) 31.9748i 1.08843i 0.838945 + 0.544217i \(0.183173\pi\)
−0.838945 + 0.544217i \(0.816827\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 16.6853i 0.567318i
\(866\) 4.41119i 0.149898i
\(867\) 9.75302 0.331230
\(868\) 0.753020 0.0255592
\(869\) 6.06638i 0.205788i
\(870\) −6.15883 −0.208804
\(871\) 0 0
\(872\) 0.801938 0.0271570
\(873\) − 1.93900i − 0.0656252i
\(874\) 8.40581 0.284331
\(875\) −3.04892 −0.103072
\(876\) − 14.8334i − 0.501174i
\(877\) 39.2476i 1.32530i 0.748931 + 0.662648i \(0.230567\pi\)
−0.748931 + 0.662648i \(0.769433\pi\)
\(878\) 39.5260i 1.33394i
\(879\) − 18.5810i − 0.626723i
\(880\) 1.13706 0.0383304
\(881\) 24.3153 0.819202 0.409601 0.912265i \(-0.365668\pi\)
0.409601 + 0.912265i \(0.365668\pi\)
\(882\) 2.29590i 0.0773069i
\(883\) −51.8756 −1.74575 −0.872877 0.487941i \(-0.837748\pi\)
−0.872877 + 0.487941i \(0.837748\pi\)
\(884\) 0 0
\(885\) −1.36898 −0.0460177
\(886\) 7.16182i 0.240606i
\(887\) −13.4383 −0.451215 −0.225608 0.974218i \(-0.572437\pi\)
−0.225608 + 0.974218i \(0.572437\pi\)
\(888\) −0.554958 −0.0186232
\(889\) − 18.9801i − 0.636574i
\(890\) − 16.8877i − 0.566077i
\(891\) 1.13706i 0.0380931i
\(892\) 1.11960i 0.0374871i
\(893\) −10.4209 −0.348721
\(894\) 16.4209 0.549196
\(895\) − 3.94438i − 0.131846i
\(896\) −3.04892 −0.101857
\(897\) 0 0
\(898\) −4.46011 −0.148836
\(899\) − 1.52111i − 0.0507317i
\(900\) 1.00000 0.0333333
\(901\) −15.5459 −0.517908
\(902\) − 9.40209i − 0.313055i
\(903\) 6.21744i 0.206903i
\(904\) − 11.5211i − 0.383186i
\(905\) − 20.5308i − 0.682467i
\(906\) 9.26636 0.307854
\(907\) 6.58881 0.218778 0.109389 0.993999i \(-0.465111\pi\)
0.109389 + 0.993999i \(0.465111\pi\)
\(908\) 4.06638i 0.134947i
\(909\) 14.4940 0.480734
\(910\) 0 0
\(911\) 51.2646 1.69847 0.849235 0.528015i \(-0.177063\pi\)
0.849235 + 0.528015i \(0.177063\pi\)
\(912\) 1.55496i 0.0514898i
\(913\) −6.20237 −0.205269
\(914\) 8.04593 0.266136
\(915\) 10.9487i 0.361953i
\(916\) 16.5187i 0.545794i
\(917\) 10.1981i 0.336770i
\(918\) − 2.69202i − 0.0888499i
\(919\) 50.7066 1.67266 0.836328 0.548229i \(-0.184698\pi\)
0.836328 + 0.548229i \(0.184698\pi\)
\(920\) −5.40581 −0.178224
\(921\) − 32.4359i − 1.06880i
\(922\) −4.00538 −0.131910
\(923\) 0 0
\(924\) −3.46681 −0.114050
\(925\) 0.554958i 0.0182469i
\(926\) −37.6142 −1.23608
\(927\) 16.1836 0.531539
\(928\) 6.15883i 0.202174i
\(929\) 18.2903i 0.600084i 0.953926 + 0.300042i \(0.0970007\pi\)
−0.953926 + 0.300042i \(0.902999\pi\)
\(930\) 0.246980i 0.00809878i
\(931\) 3.57002i 0.117003i
\(932\) −6.16421 −0.201915
\(933\) −21.2771 −0.696581
\(934\) − 32.8079i − 1.07351i
\(935\) −3.06100 −0.100105
\(936\) 0 0
\(937\) −59.6679 −1.94926 −0.974632 0.223814i \(-0.928149\pi\)
−0.974632 + 0.223814i \(0.928149\pi\)
\(938\) 7.53750i 0.246108i
\(939\) 18.7060 0.610448
\(940\) 6.70171 0.218586
\(941\) 36.2513i 1.18176i 0.806760 + 0.590879i \(0.201219\pi\)
−0.806760 + 0.590879i \(0.798781\pi\)
\(942\) 17.0761i 0.556368i
\(943\) 44.6993i 1.45561i
\(944\) 1.36898i 0.0445564i
\(945\) −3.04892 −0.0991813
\(946\) −2.31873 −0.0753885
\(947\) 7.02475i 0.228274i 0.993465 + 0.114137i \(0.0364103\pi\)
−0.993465 + 0.114137i \(0.963590\pi\)
\(948\) 5.33513 0.173277
\(949\) 0 0
\(950\) 1.55496 0.0504495
\(951\) 17.4776i 0.566749i
\(952\) 8.20775 0.266015
\(953\) −12.5730 −0.407280 −0.203640 0.979046i \(-0.565277\pi\)
−0.203640 + 0.979046i \(0.565277\pi\)
\(954\) − 5.77479i − 0.186966i
\(955\) 6.66487i 0.215670i
\(956\) 18.1793i 0.587960i
\(957\) 7.00298i 0.226374i
\(958\) −9.88530 −0.319379
\(959\) 29.0586 0.938351
\(960\) − 1.00000i − 0.0322749i
\(961\) 30.9390 0.998032
\(962\) 0 0
\(963\) 4.92931 0.158845
\(964\) 9.21744i 0.296874i
\(965\) 16.6558 0.536168
\(966\) 16.4819 0.530296
\(967\) 23.1371i 0.744038i 0.928225 + 0.372019i \(0.121334\pi\)
−0.928225 + 0.372019i \(0.878666\pi\)
\(968\) 9.70709i 0.311998i
\(969\) − 4.18598i − 0.134473i
\(970\) 1.93900i 0.0622576i
\(971\) −26.5394 −0.851690 −0.425845 0.904796i \(-0.640023\pi\)
−0.425845 + 0.904796i \(0.640023\pi\)
\(972\) 1.00000 0.0320750
\(973\) 57.0200i 1.82798i
\(974\) −20.2610 −0.649204
\(975\) 0 0
\(976\) 10.9487 0.350459
\(977\) − 19.5670i − 0.626005i −0.949752 0.313003i \(-0.898665\pi\)
0.949752 0.313003i \(-0.101335\pi\)
\(978\) 9.61894 0.307580
\(979\) −19.2024 −0.613711
\(980\) − 2.29590i − 0.0733397i
\(981\) − 0.801938i − 0.0256039i
\(982\) − 22.4940i − 0.717811i
\(983\) − 8.82610i − 0.281509i −0.990045 0.140754i \(-0.955047\pi\)
0.990045 0.140754i \(-0.0449528\pi\)
\(984\) −8.26875 −0.263598
\(985\) −19.4494 −0.619708
\(986\) − 16.5797i − 0.528006i
\(987\) −20.4330 −0.650388
\(988\) 0 0
\(989\) 11.0237 0.350533
\(990\) − 1.13706i − 0.0361382i
\(991\) 0.601483 0.0191067 0.00955336 0.999954i \(-0.496959\pi\)
0.00955336 + 0.999954i \(0.496959\pi\)
\(992\) 0.246980 0.00784161
\(993\) − 23.1890i − 0.735880i
\(994\) 26.1293i 0.828771i
\(995\) − 21.6558i − 0.686534i
\(996\) 5.45473i 0.172840i
\(997\) 13.4101 0.424703 0.212351 0.977193i \(-0.431888\pi\)
0.212351 + 0.977193i \(0.431888\pi\)
\(998\) 15.2728 0.483452
\(999\) 0.554958i 0.0175581i
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 5070.2.b.t.1351.3 6
13.5 odd 4 5070.2.a.bj.1.1 3
13.8 odd 4 5070.2.a.bu.1.3 yes 3
13.12 even 2 inner 5070.2.b.t.1351.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bj.1.1 3 13.5 odd 4
5070.2.a.bu.1.3 yes 3 13.8 odd 4
5070.2.b.t.1351.3 6 1.1 even 1 trivial
5070.2.b.t.1351.4 6 13.12 even 2 inner