Properties

Label 5070.2.b.ba.1351.3
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.17284886784.1
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 30 x^{5} + 185 x^{4} + 36 x^{3} + 8 x^{2} + 208 x + 2704\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.3
Root \(3.17270 + 3.17270i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.ba.1351.6

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +2.32258i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +2.32258i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.34541i q^{11} -1.00000 q^{12} +2.32258 q^{14} +1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000i q^{18} +4.02283i q^{19} -1.00000i q^{20} +2.32258i q^{21} +5.34541 q^{22} +4.93593 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -2.32258i q^{28} +4.29078 q^{29} +1.00000 q^{30} -3.47183i q^{31} -1.00000i q^{32} +5.34541i q^{33} +4.00000i q^{34} -2.32258 q^{35} -1.00000 q^{36} -3.14925i q^{37} +4.02283 q^{38} -1.00000 q^{40} +2.64516i q^{41} +2.32258 q^{42} -12.2508 q^{43} -5.34541i q^{44} +1.00000i q^{45} -4.93593i q^{46} +1.81894i q^{47} +1.00000 q^{48} +1.60562 q^{49} +1.00000i q^{50} -4.00000 q^{51} -5.48693 q^{53} -1.00000i q^{54} -5.34541 q^{55} -2.32258 q^{56} +4.02283i q^{57} -4.29078i q^{58} -6.78668i q^{59} -1.00000i q^{60} +0.535898 q^{61} -3.47183 q^{62} +2.32258i q^{63} -1.00000 q^{64} +5.34541 q^{66} +4.10926i q^{67} +4.00000 q^{68} +4.93593 q^{69} +2.32258i q^{70} -15.8719i q^{71} +1.00000i q^{72} +13.5734i q^{73} -3.14925 q^{74} -1.00000 q^{75} -4.02283i q^{76} -12.4151 q^{77} -7.96774 q^{79} +1.00000i q^{80} +1.00000 q^{81} +2.64516 q^{82} +11.3360i q^{83} -2.32258i q^{84} -4.00000i q^{85} +12.2508i q^{86} +4.29078 q^{87} -5.34541 q^{88} -1.73978i q^{89} +1.00000 q^{90} -4.93593 q^{92} -3.47183i q^{93} +1.81894 q^{94} -4.02283 q^{95} -1.00000i q^{96} +16.1093i q^{97} -1.60562i q^{98} +5.34541i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} - 8q^{4} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{3} - 8q^{4} + 8q^{9} + 8q^{10} - 8q^{12} + 4q^{14} + 8q^{16} - 32q^{17} - 4q^{22} - 8q^{23} - 8q^{25} + 8q^{27} + 16q^{29} + 8q^{30} - 4q^{35} - 8q^{36} - 8q^{40} + 4q^{42} - 28q^{43} + 8q^{48} - 28q^{49} - 32q^{51} + 16q^{53} + 4q^{55} - 4q^{56} + 32q^{61} - 8q^{62} - 8q^{64} - 4q^{66} + 32q^{68} - 8q^{69} - 20q^{74} - 8q^{75} + 16q^{77} - 20q^{79} + 8q^{81} - 8q^{82} + 16q^{87} + 4q^{88} + 8q^{90} + 8q^{92} + 16q^{94} + O(q^{100}) \)

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
\(7\) 2.32258i 0.877853i 0.898523 + 0.438926i \(0.144641\pi\)
−0.898523 + 0.438926i \(0.855359\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 5.34541i 1.61170i 0.592119 + 0.805850i \(0.298291\pi\)
−0.592119 + 0.805850i \(0.701709\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 2.32258 0.620736
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 4.02283i 0.922900i 0.887166 + 0.461450i \(0.152671\pi\)
−0.887166 + 0.461450i \(0.847329\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) 2.32258i 0.506828i
\(22\) 5.34541 1.13964
\(23\) 4.93593 1.02921 0.514607 0.857426i \(-0.327938\pi\)
0.514607 + 0.857426i \(0.327938\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 2.32258i − 0.438926i
\(29\) 4.29078 0.796777 0.398388 0.917217i \(-0.369570\pi\)
0.398388 + 0.917217i \(0.369570\pi\)
\(30\) 1.00000 0.182574
\(31\) − 3.47183i − 0.623560i −0.950154 0.311780i \(-0.899075\pi\)
0.950154 0.311780i \(-0.100925\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 5.34541i 0.930516i
\(34\) 4.00000i 0.685994i
\(35\) −2.32258 −0.392588
\(36\) −1.00000 −0.166667
\(37\) − 3.14925i − 0.517734i −0.965913 0.258867i \(-0.916651\pi\)
0.965913 0.258867i \(-0.0833491\pi\)
\(38\) 4.02283 0.652589
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.64516i 0.413104i 0.978436 + 0.206552i \(0.0662243\pi\)
−0.978436 + 0.206552i \(0.933776\pi\)
\(42\) 2.32258 0.358382
\(43\) −12.2508 −1.86823 −0.934113 0.356976i \(-0.883808\pi\)
−0.934113 + 0.356976i \(0.883808\pi\)
\(44\) − 5.34541i − 0.805850i
\(45\) 1.00000i 0.149071i
\(46\) − 4.93593i − 0.727764i
\(47\) 1.81894i 0.265320i 0.991162 + 0.132660i \(0.0423518\pi\)
−0.991162 + 0.132660i \(0.957648\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.60562 0.229375
\(50\) 1.00000i 0.141421i
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −5.48693 −0.753687 −0.376844 0.926277i \(-0.622991\pi\)
−0.376844 + 0.926277i \(0.622991\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −5.34541 −0.720774
\(56\) −2.32258 −0.310368
\(57\) 4.02283i 0.532836i
\(58\) − 4.29078i − 0.563406i
\(59\) − 6.78668i − 0.883551i −0.897126 0.441775i \(-0.854349\pi\)
0.897126 0.441775i \(-0.145651\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) 0.535898 0.0686148 0.0343074 0.999411i \(-0.489077\pi\)
0.0343074 + 0.999411i \(0.489077\pi\)
\(62\) −3.47183 −0.440923
\(63\) 2.32258i 0.292618i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 5.34541 0.657974
\(67\) 4.10926i 0.502026i 0.967984 + 0.251013i \(0.0807637\pi\)
−0.967984 + 0.251013i \(0.919236\pi\)
\(68\) 4.00000 0.485071
\(69\) 4.93593 0.594217
\(70\) 2.32258i 0.277601i
\(71\) − 15.8719i − 1.88364i −0.336112 0.941822i \(-0.609112\pi\)
0.336112 0.941822i \(-0.390888\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 13.5734i 1.58864i 0.607498 + 0.794321i \(0.292173\pi\)
−0.607498 + 0.794321i \(0.707827\pi\)
\(74\) −3.14925 −0.366093
\(75\) −1.00000 −0.115470
\(76\) − 4.02283i − 0.461450i
\(77\) −12.4151 −1.41484
\(78\) 0 0
\(79\) −7.96774 −0.896441 −0.448220 0.893923i \(-0.647942\pi\)
−0.448220 + 0.893923i \(0.647942\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 2.64516 0.292109
\(83\) 11.3360i 1.24428i 0.782904 + 0.622142i \(0.213737\pi\)
−0.782904 + 0.622142i \(0.786263\pi\)
\(84\) − 2.32258i − 0.253414i
\(85\) − 4.00000i − 0.433861i
\(86\) 12.2508i 1.32104i
\(87\) 4.29078 0.460019
\(88\) −5.34541 −0.569822
\(89\) − 1.73978i − 0.184417i −0.995740 0.0922083i \(-0.970607\pi\)
0.995740 0.0922083i \(-0.0293926\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −4.93593 −0.514607
\(93\) − 3.47183i − 0.360012i
\(94\) 1.81894 0.187610
\(95\) −4.02283 −0.412733
\(96\) − 1.00000i − 0.102062i
\(97\) 16.1093i 1.63565i 0.575469 + 0.817824i \(0.304820\pi\)
−0.575469 + 0.817824i \(0.695180\pi\)
\(98\) − 1.60562i − 0.162192i
\(99\) 5.34541i 0.537233i
\(100\) 1.00000 0.100000
\(101\) −12.6908 −1.26278 −0.631391 0.775464i \(-0.717516\pi\)
−0.631391 + 0.775464i \(0.717516\pi\)
\(102\) 4.00000i 0.396059i
\(103\) −4.79612 −0.472575 −0.236288 0.971683i \(-0.575931\pi\)
−0.236288 + 0.971683i \(0.575931\pi\)
\(104\) 0 0
\(105\) −2.32258 −0.226661
\(106\) 5.48693i 0.532938i
\(107\) 3.07180 0.296962 0.148481 0.988915i \(-0.452562\pi\)
0.148481 + 0.988915i \(0.452562\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.69081i 0.640864i 0.947272 + 0.320432i \(0.103828\pi\)
−0.947272 + 0.320432i \(0.896172\pi\)
\(110\) 5.34541i 0.509664i
\(111\) − 3.14925i − 0.298914i
\(112\) 2.32258i 0.219463i
\(113\) −7.10972 −0.668826 −0.334413 0.942427i \(-0.608538\pi\)
−0.334413 + 0.942427i \(0.608538\pi\)
\(114\) 4.02283 0.376772
\(115\) 4.93593i 0.460278i
\(116\) −4.29078 −0.398388
\(117\) 0 0
\(118\) −6.78668 −0.624765
\(119\) − 9.29032i − 0.851642i
\(120\) −1.00000 −0.0912871
\(121\) −17.5734 −1.59758
\(122\) − 0.535898i − 0.0485180i
\(123\) 2.64516i 0.238506i
\(124\) 3.47183i 0.311780i
\(125\) − 1.00000i − 0.0894427i
\(126\) 2.32258 0.206912
\(127\) 2.13209 0.189192 0.0945961 0.995516i \(-0.469844\pi\)
0.0945961 + 0.995516i \(0.469844\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −12.2508 −1.07862
\(130\) 0 0
\(131\) 1.25851 0.109957 0.0549785 0.998488i \(-0.482491\pi\)
0.0549785 + 0.998488i \(0.482491\pi\)
\(132\) − 5.34541i − 0.465258i
\(133\) −9.34333 −0.810170
\(134\) 4.10926 0.354986
\(135\) 1.00000i 0.0860663i
\(136\) − 4.00000i − 0.342997i
\(137\) − 19.7149i − 1.68436i −0.539199 0.842178i \(-0.681273\pi\)
0.539199 0.842178i \(-0.318727\pi\)
\(138\) − 4.93593i − 0.420175i
\(139\) 5.67742 0.481553 0.240776 0.970581i \(-0.422598\pi\)
0.240776 + 0.970581i \(0.422598\pi\)
\(140\) 2.32258 0.196294
\(141\) 1.81894i 0.153183i
\(142\) −15.8719 −1.33194
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.29078i 0.356329i
\(146\) 13.5734 1.12334
\(147\) 1.60562 0.132430
\(148\) 3.14925i 0.258867i
\(149\) 19.4775i 1.59566i 0.602884 + 0.797829i \(0.294018\pi\)
−0.602884 + 0.797829i \(0.705982\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) − 14.5170i − 1.18138i −0.806899 0.590690i \(-0.798856\pi\)
0.806899 0.590690i \(-0.201144\pi\)
\(152\) −4.02283 −0.326294
\(153\) −4.00000 −0.323381
\(154\) 12.4151i 1.00044i
\(155\) 3.47183 0.278864
\(156\) 0 0
\(157\) 24.3829 1.94596 0.972982 0.230879i \(-0.0741602\pi\)
0.972982 + 0.230879i \(0.0741602\pi\)
\(158\) 7.96774i 0.633879i
\(159\) −5.48693 −0.435142
\(160\) 1.00000 0.0790569
\(161\) 11.4641i 0.903498i
\(162\) − 1.00000i − 0.0785674i
\(163\) − 23.6267i − 1.85059i −0.379249 0.925295i \(-0.623818\pi\)
0.379249 0.925295i \(-0.376182\pi\)
\(164\) − 2.64516i − 0.206552i
\(165\) −5.34541 −0.416139
\(166\) 11.3360 0.879842
\(167\) 16.7471i 1.29593i 0.761669 + 0.647967i \(0.224380\pi\)
−0.761669 + 0.647967i \(0.775620\pi\)
\(168\) −2.32258 −0.179191
\(169\) 0 0
\(170\) −4.00000 −0.306786
\(171\) 4.02283i 0.307633i
\(172\) 12.2508 0.934113
\(173\) −16.1321 −1.22650 −0.613250 0.789889i \(-0.710138\pi\)
−0.613250 + 0.789889i \(0.710138\pi\)
\(174\) − 4.29078i − 0.325283i
\(175\) − 2.32258i − 0.175571i
\(176\) 5.34541i 0.402925i
\(177\) − 6.78668i − 0.510118i
\(178\) −1.73978 −0.130402
\(179\) −7.32824 −0.547738 −0.273869 0.961767i \(-0.588304\pi\)
−0.273869 + 0.961767i \(0.588304\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) −19.5734 −1.45488 −0.727438 0.686173i \(-0.759289\pi\)
−0.727438 + 0.686173i \(0.759289\pi\)
\(182\) 0 0
\(183\) 0.535898 0.0396147
\(184\) 4.93593i 0.363882i
\(185\) 3.14925 0.231538
\(186\) −3.47183 −0.254567
\(187\) − 21.3816i − 1.56358i
\(188\) − 1.81894i − 0.132660i
\(189\) 2.32258i 0.168943i
\(190\) 4.02283i 0.291846i
\(191\) −17.0375 −1.23279 −0.616394 0.787438i \(-0.711407\pi\)
−0.616394 + 0.787438i \(0.711407\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 6.15491i 0.443040i 0.975156 + 0.221520i \(0.0711019\pi\)
−0.975156 + 0.221520i \(0.928898\pi\)
\(194\) 16.1093 1.15658
\(195\) 0 0
\(196\) −1.60562 −0.114687
\(197\) − 13.7867i − 0.982260i −0.871086 0.491130i \(-0.836584\pi\)
0.871086 0.491130i \(-0.163416\pi\)
\(198\) 5.34541 0.379881
\(199\) −2.28304 −0.161841 −0.0809203 0.996721i \(-0.525786\pi\)
−0.0809203 + 0.996721i \(0.525786\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) 4.10926i 0.289845i
\(202\) 12.6908i 0.892922i
\(203\) 9.96567i 0.699453i
\(204\) 4.00000 0.280056
\(205\) −2.64516 −0.184746
\(206\) 4.79612i 0.334161i
\(207\) 4.93593 0.343071
\(208\) 0 0
\(209\) −21.5036 −1.48744
\(210\) 2.32258i 0.160273i
\(211\) 22.4775 1.54741 0.773707 0.633543i \(-0.218400\pi\)
0.773707 + 0.633543i \(0.218400\pi\)
\(212\) 5.48693 0.376844
\(213\) − 15.8719i − 1.08752i
\(214\) − 3.07180i − 0.209984i
\(215\) − 12.2508i − 0.835496i
\(216\) 1.00000i 0.0680414i
\(217\) 8.06361 0.547393
\(218\) 6.69081 0.453159
\(219\) 13.5734i 0.917203i
\(220\) 5.34541 0.360387
\(221\) 0 0
\(222\) −3.14925 −0.211364
\(223\) 1.37891i 0.0923389i 0.998934 + 0.0461694i \(0.0147014\pi\)
−0.998934 + 0.0461694i \(0.985299\pi\)
\(224\) 2.32258 0.155184
\(225\) −1.00000 −0.0666667
\(226\) 7.10972i 0.472931i
\(227\) 19.3205i 1.28235i 0.767396 + 0.641174i \(0.221552\pi\)
−0.767396 + 0.641174i \(0.778448\pi\)
\(228\) − 4.02283i − 0.266418i
\(229\) 15.7626i 1.04162i 0.853672 + 0.520811i \(0.174370\pi\)
−0.853672 + 0.520811i \(0.825630\pi\)
\(230\) 4.93593 0.325466
\(231\) −12.4151 −0.816856
\(232\) 4.29078i 0.281703i
\(233\) −16.5549 −1.08455 −0.542275 0.840201i \(-0.682437\pi\)
−0.542275 + 0.840201i \(0.682437\pi\)
\(234\) 0 0
\(235\) −1.81894 −0.118655
\(236\) 6.78668i 0.441775i
\(237\) −7.96774 −0.517560
\(238\) −9.29032 −0.602202
\(239\) − 26.2006i − 1.69477i −0.530976 0.847387i \(-0.678175\pi\)
0.530976 0.847387i \(-0.321825\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 15.6496i 1.00808i 0.863681 + 0.504039i \(0.168153\pi\)
−0.863681 + 0.504039i \(0.831847\pi\)
\(242\) 17.5734i 1.12966i
\(243\) 1.00000 0.0641500
\(244\) −0.535898 −0.0343074
\(245\) 1.60562i 0.102580i
\(246\) 2.64516 0.168649
\(247\) 0 0
\(248\) 3.47183 0.220462
\(249\) 11.3360i 0.718388i
\(250\) −1.00000 −0.0632456
\(251\) 28.3416 1.78891 0.894454 0.447160i \(-0.147565\pi\)
0.894454 + 0.447160i \(0.147565\pi\)
\(252\) − 2.32258i − 0.146309i
\(253\) 26.3846i 1.65878i
\(254\) − 2.13209i − 0.133779i
\(255\) − 4.00000i − 0.250490i
\(256\) 1.00000 0.0625000
\(257\) 12.5093 0.780309 0.390154 0.920750i \(-0.372422\pi\)
0.390154 + 0.920750i \(0.372422\pi\)
\(258\) 12.2508i 0.762700i
\(259\) 7.31439 0.454494
\(260\) 0 0
\(261\) 4.29078 0.265592
\(262\) − 1.25851i − 0.0777513i
\(263\) −10.7059 −0.660154 −0.330077 0.943954i \(-0.607075\pi\)
−0.330077 + 0.943954i \(0.607075\pi\)
\(264\) −5.34541 −0.328987
\(265\) − 5.48693i − 0.337059i
\(266\) 9.34333i 0.572877i
\(267\) − 1.73978i − 0.106473i
\(268\) − 4.10926i − 0.251013i
\(269\) 7.52522 0.458821 0.229410 0.973330i \(-0.426320\pi\)
0.229410 + 0.973330i \(0.426320\pi\)
\(270\) 1.00000 0.0608581
\(271\) 9.84868i 0.598264i 0.954212 + 0.299132i \(0.0966972\pi\)
−0.954212 + 0.299132i \(0.903303\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) −19.7149 −1.19102
\(275\) − 5.34541i − 0.322340i
\(276\) −4.93593 −0.297108
\(277\) −0.953101 −0.0572663 −0.0286331 0.999590i \(-0.509115\pi\)
−0.0286331 + 0.999590i \(0.509115\pi\)
\(278\) − 5.67742i − 0.340509i
\(279\) − 3.47183i − 0.207853i
\(280\) − 2.32258i − 0.138801i
\(281\) 5.57336i 0.332479i 0.986085 + 0.166239i \(0.0531625\pi\)
−0.986085 + 0.166239i \(0.946838\pi\)
\(282\) 1.81894 0.108316
\(283\) −22.0134 −1.30856 −0.654280 0.756252i \(-0.727028\pi\)
−0.654280 + 0.756252i \(0.727028\pi\)
\(284\) 15.8719i 0.941822i
\(285\) −4.02283 −0.238292
\(286\) 0 0
\(287\) −6.14359 −0.362645
\(288\) − 1.00000i − 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) 4.29078 0.251963
\(291\) 16.1093i 0.944342i
\(292\) − 13.5734i − 0.794321i
\(293\) − 6.84302i − 0.399773i −0.979819 0.199887i \(-0.935943\pi\)
0.979819 0.199887i \(-0.0640574\pi\)
\(294\) − 1.60562i − 0.0936419i
\(295\) 6.78668 0.395136
\(296\) 3.14925 0.183047
\(297\) 5.34541i 0.310172i
\(298\) 19.4775 1.12830
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) − 28.4534i − 1.64003i
\(302\) −14.5170 −0.835361
\(303\) −12.6908 −0.729068
\(304\) 4.02283i 0.230725i
\(305\) 0.535898i 0.0306855i
\(306\) 4.00000i 0.228665i
\(307\) 4.75442i 0.271349i 0.990753 + 0.135675i \(0.0433202\pi\)
−0.990753 + 0.135675i \(0.956680\pi\)
\(308\) 12.4151 0.707418
\(309\) −4.79612 −0.272842
\(310\) − 3.47183i − 0.197187i
\(311\) 1.93639 0.109803 0.0549013 0.998492i \(-0.482516\pi\)
0.0549013 + 0.998492i \(0.482516\pi\)
\(312\) 0 0
\(313\) 25.5545 1.44443 0.722213 0.691671i \(-0.243125\pi\)
0.722213 + 0.691671i \(0.243125\pi\)
\(314\) − 24.3829i − 1.37600i
\(315\) −2.32258 −0.130863
\(316\) 7.96774 0.448220
\(317\) 12.6667i 0.711435i 0.934594 + 0.355717i \(0.115763\pi\)
−0.934594 + 0.355717i \(0.884237\pi\)
\(318\) 5.48693i 0.307692i
\(319\) 22.9359i 1.28417i
\(320\) − 1.00000i − 0.0559017i
\(321\) 3.07180 0.171451
\(322\) 11.4641 0.638869
\(323\) − 16.0913i − 0.895344i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −23.6267 −1.30856
\(327\) 6.69081i 0.370003i
\(328\) −2.64516 −0.146054
\(329\) −4.22464 −0.232912
\(330\) 5.34541i 0.294255i
\(331\) 23.6981i 1.30256i 0.758836 + 0.651282i \(0.225769\pi\)
−0.758836 + 0.651282i \(0.774231\pi\)
\(332\) − 11.3360i − 0.622142i
\(333\) − 3.14925i − 0.172578i
\(334\) 16.7471 0.916363
\(335\) −4.10926 −0.224513
\(336\) 2.32258i 0.126707i
\(337\) 19.5554 1.06525 0.532625 0.846351i \(-0.321205\pi\)
0.532625 + 0.846351i \(0.321205\pi\)
\(338\) 0 0
\(339\) −7.10972 −0.386147
\(340\) 4.00000i 0.216930i
\(341\) 18.5584 1.00499
\(342\) 4.02283 0.217530
\(343\) 19.9872i 1.07921i
\(344\) − 12.2508i − 0.660518i
\(345\) 4.93593i 0.265742i
\(346\) 16.1321i 0.867266i
\(347\) −23.0375 −1.23672 −0.618358 0.785897i \(-0.712202\pi\)
−0.618358 + 0.785897i \(0.712202\pi\)
\(348\) −4.29078 −0.230010
\(349\) 15.3205i 0.820088i 0.912066 + 0.410044i \(0.134487\pi\)
−0.912066 + 0.410044i \(0.865513\pi\)
\(350\) −2.32258 −0.124147
\(351\) 0 0
\(352\) 5.34541 0.284911
\(353\) 28.4078i 1.51199i 0.654576 + 0.755996i \(0.272847\pi\)
−0.654576 + 0.755996i \(0.727153\pi\)
\(354\) −6.78668 −0.360708
\(355\) 15.8719 0.842391
\(356\) 1.73978i 0.0922083i
\(357\) − 9.29032i − 0.491696i
\(358\) 7.32824i 0.387309i
\(359\) − 23.5734i − 1.24415i −0.782956 0.622077i \(-0.786289\pi\)
0.782956 0.622077i \(-0.213711\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 2.81687 0.148256
\(362\) 19.5734i 1.02875i
\(363\) −17.5734 −0.922362
\(364\) 0 0
\(365\) −13.5734 −0.710462
\(366\) − 0.535898i − 0.0280119i
\(367\) 27.4909 1.43501 0.717506 0.696552i \(-0.245283\pi\)
0.717506 + 0.696552i \(0.245283\pi\)
\(368\) 4.93593 0.257303
\(369\) 2.64516i 0.137701i
\(370\) − 3.14925i − 0.163722i
\(371\) − 12.7438i − 0.661627i
\(372\) 3.47183i 0.180006i
\(373\) 26.3421 1.36394 0.681971 0.731379i \(-0.261123\pi\)
0.681971 + 0.731379i \(0.261123\pi\)
\(374\) −21.3816 −1.10562
\(375\) − 1.00000i − 0.0516398i
\(376\) −1.81894 −0.0938048
\(377\) 0 0
\(378\) 2.32258 0.119461
\(379\) 0.448551i 0.0230405i 0.999934 + 0.0115202i \(0.00366709\pi\)
−0.999934 + 0.0115202i \(0.996333\pi\)
\(380\) 4.02283 0.206367
\(381\) 2.13209 0.109230
\(382\) 17.0375i 0.871713i
\(383\) − 12.1227i − 0.619439i −0.950828 0.309719i \(-0.899765\pi\)
0.950828 0.309719i \(-0.100235\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) − 12.4151i − 0.632734i
\(386\) 6.15491 0.313277
\(387\) −12.2508 −0.622742
\(388\) − 16.1093i − 0.817824i
\(389\) 18.6195 0.944045 0.472022 0.881587i \(-0.343524\pi\)
0.472022 + 0.881587i \(0.343524\pi\)
\(390\) 0 0
\(391\) −19.7437 −0.998484
\(392\) 1.60562i 0.0810962i
\(393\) 1.25851 0.0634837
\(394\) −13.7867 −0.694563
\(395\) − 7.96774i − 0.400900i
\(396\) − 5.34541i − 0.268617i
\(397\) − 11.3042i − 0.567340i −0.958922 0.283670i \(-0.908448\pi\)
0.958922 0.283670i \(-0.0915520\pi\)
\(398\) 2.28304i 0.114439i
\(399\) −9.34333 −0.467752
\(400\) −1.00000 −0.0500000
\(401\) − 1.50580i − 0.0751959i −0.999293 0.0375980i \(-0.988029\pi\)
0.999293 0.0375980i \(-0.0119706\pi\)
\(402\) 4.10926 0.204951
\(403\) 0 0
\(404\) 12.6908 0.631391
\(405\) 1.00000i 0.0496904i
\(406\) 9.96567 0.494588
\(407\) 16.8340 0.834432
\(408\) − 4.00000i − 0.198030i
\(409\) − 11.1394i − 0.550806i −0.961329 0.275403i \(-0.911189\pi\)
0.961329 0.275403i \(-0.0888113\pi\)
\(410\) 2.64516i 0.130635i
\(411\) − 19.7149i − 0.972464i
\(412\) 4.79612 0.236288
\(413\) 15.7626 0.775627
\(414\) − 4.93593i − 0.242588i
\(415\) −11.3360 −0.556461
\(416\) 0 0
\(417\) 5.67742 0.278024
\(418\) 21.5036i 1.05178i
\(419\) −15.5098 −0.757701 −0.378851 0.925458i \(-0.623681\pi\)
−0.378851 + 0.925458i \(0.623681\pi\)
\(420\) 2.32258 0.113330
\(421\) 39.4452i 1.92244i 0.275778 + 0.961221i \(0.411065\pi\)
−0.275778 + 0.961221i \(0.588935\pi\)
\(422\) − 22.4775i − 1.09419i
\(423\) 1.81894i 0.0884400i
\(424\) − 5.48693i − 0.266469i
\(425\) 4.00000 0.194029
\(426\) −15.8719 −0.768995
\(427\) 1.24467i 0.0602336i
\(428\) −3.07180 −0.148481
\(429\) 0 0
\(430\) −12.2508 −0.590785
\(431\) 2.15491i 0.103799i 0.998652 + 0.0518993i \(0.0165275\pi\)
−0.998652 + 0.0518993i \(0.983473\pi\)
\(432\) 1.00000 0.0481125
\(433\) −1.33938 −0.0643664 −0.0321832 0.999482i \(-0.510246\pi\)
−0.0321832 + 0.999482i \(0.510246\pi\)
\(434\) − 8.06361i − 0.387066i
\(435\) 4.29078i 0.205727i
\(436\) − 6.69081i − 0.320432i
\(437\) 19.8564i 0.949861i
\(438\) 13.5734 0.648560
\(439\) 31.6981 1.51287 0.756434 0.654071i \(-0.226940\pi\)
0.756434 + 0.654071i \(0.226940\pi\)
\(440\) − 5.34541i − 0.254832i
\(441\) 1.60562 0.0764583
\(442\) 0 0
\(443\) −28.0904 −1.33461 −0.667307 0.744782i \(-0.732553\pi\)
−0.667307 + 0.744782i \(0.732553\pi\)
\(444\) 3.14925i 0.149457i
\(445\) 1.73978 0.0824736
\(446\) 1.37891 0.0652935
\(447\) 19.4775i 0.921254i
\(448\) − 2.32258i − 0.109732i
\(449\) − 28.9167i − 1.36466i −0.731043 0.682332i \(-0.760966\pi\)
0.731043 0.682332i \(-0.239034\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) −14.1395 −0.665801
\(452\) 7.10972 0.334413
\(453\) − 14.5170i − 0.682070i
\(454\) 19.3205 0.906756
\(455\) 0 0
\(456\) −4.02283 −0.188386
\(457\) 9.69900i 0.453700i 0.973930 + 0.226850i \(0.0728427\pi\)
−0.973930 + 0.226850i \(0.927157\pi\)
\(458\) 15.7626 0.736538
\(459\) −4.00000 −0.186704
\(460\) − 4.93593i − 0.230139i
\(461\) − 11.9979i − 0.558799i −0.960175 0.279400i \(-0.909865\pi\)
0.960175 0.279400i \(-0.0901354\pi\)
\(462\) 12.4151i 0.577604i
\(463\) − 6.42664i − 0.298671i −0.988787 0.149336i \(-0.952287\pi\)
0.988787 0.149336i \(-0.0477135\pi\)
\(464\) 4.29078 0.199194
\(465\) 3.47183 0.161002
\(466\) 16.5549i 0.766893i
\(467\) 26.2642 1.21536 0.607681 0.794182i \(-0.292100\pi\)
0.607681 + 0.794182i \(0.292100\pi\)
\(468\) 0 0
\(469\) −9.54409 −0.440705
\(470\) 1.81894i 0.0839015i
\(471\) 24.3829 1.12350
\(472\) 6.78668 0.312382
\(473\) − 65.4854i − 3.01102i
\(474\) 7.96774i 0.365970i
\(475\) − 4.02283i − 0.184580i
\(476\) 9.29032i 0.425821i
\(477\) −5.48693 −0.251229
\(478\) −26.2006 −1.19839
\(479\) − 25.6826i − 1.17347i −0.809779 0.586735i \(-0.800413\pi\)
0.809779 0.586735i \(-0.199587\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 15.6496 0.712818
\(483\) 11.4641i 0.521635i
\(484\) 17.5734 0.798789
\(485\) −16.1093 −0.731484
\(486\) − 1.00000i − 0.0453609i
\(487\) 9.37891i 0.424999i 0.977161 + 0.212500i \(0.0681604\pi\)
−0.977161 + 0.212500i \(0.931840\pi\)
\(488\) 0.535898i 0.0242590i
\(489\) − 23.6267i − 1.06844i
\(490\) 1.60562 0.0725347
\(491\) 7.73854 0.349235 0.174618 0.984636i \(-0.444131\pi\)
0.174618 + 0.984636i \(0.444131\pi\)
\(492\) − 2.64516i − 0.119253i
\(493\) −17.1631 −0.772987
\(494\) 0 0
\(495\) −5.34541 −0.240258
\(496\) − 3.47183i − 0.155890i
\(497\) 36.8637 1.65356
\(498\) 11.3360 0.507977
\(499\) − 3.18106i − 0.142404i −0.997462 0.0712019i \(-0.977317\pi\)
0.997462 0.0712019i \(-0.0226834\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 16.7471i 0.748207i
\(502\) − 28.3416i − 1.26495i
\(503\) −35.0758 −1.56395 −0.781975 0.623309i \(-0.785788\pi\)
−0.781975 + 0.623309i \(0.785788\pi\)
\(504\) −2.32258 −0.103456
\(505\) − 12.6908i − 0.564734i
\(506\) 26.3846 1.17294
\(507\) 0 0
\(508\) −2.13209 −0.0945961
\(509\) − 18.7867i − 0.832705i −0.909203 0.416353i \(-0.863308\pi\)
0.909203 0.416353i \(-0.136692\pi\)
\(510\) −4.00000 −0.177123
\(511\) −31.5252 −1.39459
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.02283i 0.177612i
\(514\) − 12.5093i − 0.551761i
\(515\) − 4.79612i − 0.211342i
\(516\) 12.2508 0.539311
\(517\) −9.72298 −0.427616
\(518\) − 7.31439i − 0.321376i
\(519\) −16.1321 −0.708120
\(520\) 0 0
\(521\) 5.28512 0.231545 0.115773 0.993276i \(-0.463066\pi\)
0.115773 + 0.993276i \(0.463066\pi\)
\(522\) − 4.29078i − 0.187802i
\(523\) −20.7523 −0.907437 −0.453718 0.891145i \(-0.649903\pi\)
−0.453718 + 0.891145i \(0.649903\pi\)
\(524\) −1.25851 −0.0549785
\(525\) − 2.32258i − 0.101366i
\(526\) 10.7059i 0.466800i
\(527\) 13.8873i 0.604942i
\(528\) 5.34541i 0.232629i
\(529\) 1.36345 0.0592805
\(530\) −5.48693 −0.238337
\(531\) − 6.78668i − 0.294517i
\(532\) 9.34333 0.405085
\(533\) 0 0
\(534\) −1.73978 −0.0752877
\(535\) 3.07180i 0.132805i
\(536\) −4.10926 −0.177493
\(537\) −7.32824 −0.316237
\(538\) − 7.52522i − 0.324435i
\(539\) 8.58271i 0.369683i
\(540\) − 1.00000i − 0.0430331i
\(541\) 28.7365i 1.23548i 0.786384 + 0.617739i \(0.211951\pi\)
−0.786384 + 0.617739i \(0.788049\pi\)
\(542\) 9.84868 0.423037
\(543\) −19.5734 −0.839973
\(544\) 4.00000i 0.171499i
\(545\) −6.69081 −0.286603
\(546\) 0 0
\(547\) 18.3768 0.785737 0.392869 0.919595i \(-0.371483\pi\)
0.392869 + 0.919595i \(0.371483\pi\)
\(548\) 19.7149i 0.842178i
\(549\) 0.535898 0.0228716
\(550\) −5.34541 −0.227929
\(551\) 17.2610i 0.735345i
\(552\) 4.93593i 0.210087i
\(553\) − 18.5057i − 0.786943i
\(554\) 0.953101i 0.0404934i
\(555\) 3.14925 0.133678
\(556\) −5.67742 −0.240776
\(557\) 26.5386i 1.12448i 0.826975 + 0.562238i \(0.190060\pi\)
−0.826975 + 0.562238i \(0.809940\pi\)
\(558\) −3.47183 −0.146974
\(559\) 0 0
\(560\) −2.32258 −0.0981469
\(561\) − 21.3816i − 0.902733i
\(562\) 5.57336 0.235098
\(563\) 6.06111 0.255446 0.127723 0.991810i \(-0.459233\pi\)
0.127723 + 0.991810i \(0.459233\pi\)
\(564\) − 1.81894i − 0.0765913i
\(565\) − 7.10972i − 0.299108i
\(566\) 22.0134i 0.925292i
\(567\) 2.32258i 0.0975392i
\(568\) 15.8719 0.665969
\(569\) 14.4964 0.607719 0.303860 0.952717i \(-0.401725\pi\)
0.303860 + 0.952717i \(0.401725\pi\)
\(570\) 4.02283i 0.168498i
\(571\) −2.90413 −0.121534 −0.0607670 0.998152i \(-0.519355\pi\)
−0.0607670 + 0.998152i \(0.519355\pi\)
\(572\) 0 0
\(573\) −17.0375 −0.711750
\(574\) 6.14359i 0.256429i
\(575\) −4.93593 −0.205843
\(576\) −1.00000 −0.0416667
\(577\) 18.8180i 0.783405i 0.920092 + 0.391702i \(0.128114\pi\)
−0.920092 + 0.391702i \(0.871886\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) 6.15491i 0.255789i
\(580\) − 4.29078i − 0.178165i
\(581\) −26.3287 −1.09230
\(582\) 16.1093 0.667750
\(583\) − 29.3299i − 1.21472i
\(584\) −13.5734 −0.561670
\(585\) 0 0
\(586\) −6.84302 −0.282682
\(587\) − 3.93639i − 0.162472i −0.996695 0.0812361i \(-0.974113\pi\)
0.996695 0.0812361i \(-0.0258868\pi\)
\(588\) −1.60562 −0.0662148
\(589\) 13.9666 0.575483
\(590\) − 6.78668i − 0.279403i
\(591\) − 13.7867i − 0.567108i
\(592\) − 3.14925i − 0.129434i
\(593\) − 20.1227i − 0.826338i −0.910654 0.413169i \(-0.864422\pi\)
0.910654 0.413169i \(-0.135578\pi\)
\(594\) 5.34541 0.219325
\(595\) 9.29032 0.380866
\(596\) − 19.4775i − 0.797829i
\(597\) −2.28304 −0.0934388
\(598\) 0 0
\(599\) 48.1172 1.96601 0.983007 0.183567i \(-0.0587644\pi\)
0.983007 + 0.183567i \(0.0587644\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −4.24217 −0.173042 −0.0865209 0.996250i \(-0.527575\pi\)
−0.0865209 + 0.996250i \(0.527575\pi\)
\(602\) −28.4534 −1.15967
\(603\) 4.10926i 0.167342i
\(604\) 14.5170i 0.590690i
\(605\) − 17.5734i − 0.714459i
\(606\) 12.6908i 0.515529i
\(607\) 12.0685 0.489844 0.244922 0.969543i \(-0.421238\pi\)
0.244922 + 0.969543i \(0.421238\pi\)
\(608\) 4.02283 0.163147
\(609\) 9.96567i 0.403829i
\(610\) 0.535898 0.0216979
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) 41.7421i 1.68595i 0.537954 + 0.842974i \(0.319197\pi\)
−0.537954 + 0.842974i \(0.680803\pi\)
\(614\) 4.75442 0.191873
\(615\) −2.64516 −0.106663
\(616\) − 12.4151i − 0.500220i
\(617\) 33.4586i 1.34699i 0.739190 + 0.673497i \(0.235208\pi\)
−0.739190 + 0.673497i \(0.764792\pi\)
\(618\) 4.79612i 0.192928i
\(619\) − 38.0978i − 1.53128i −0.643270 0.765639i \(-0.722423\pi\)
0.643270 0.765639i \(-0.277577\pi\)
\(620\) −3.47183 −0.139432
\(621\) 4.93593 0.198072
\(622\) − 1.93639i − 0.0776422i
\(623\) 4.04078 0.161891
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 25.5545i − 1.02136i
\(627\) −21.5036 −0.858773
\(628\) −24.3829 −0.972982
\(629\) 12.5970i 0.502276i
\(630\) 2.32258i 0.0925338i
\(631\) 24.8001i 0.987275i 0.869668 + 0.493638i \(0.164333\pi\)
−0.869668 + 0.493638i \(0.835667\pi\)
\(632\) − 7.96774i − 0.316940i
\(633\) 22.4775 0.893400
\(634\) 12.6667 0.503060
\(635\) 2.13209i 0.0846093i
\(636\) 5.48693 0.217571
\(637\) 0 0
\(638\) 22.9359 0.908042
\(639\) − 15.8719i − 0.627881i
\(640\) −1.00000 −0.0395285
\(641\) −4.08519 −0.161355 −0.0806776 0.996740i \(-0.525708\pi\)
−0.0806776 + 0.996740i \(0.525708\pi\)
\(642\) − 3.07180i − 0.121234i
\(643\) 36.5816i 1.44264i 0.692604 + 0.721318i \(0.256463\pi\)
−0.692604 + 0.721318i \(0.743537\pi\)
\(644\) − 11.4641i − 0.451749i
\(645\) − 12.2508i − 0.482374i
\(646\) −16.0913 −0.633104
\(647\) −17.2341 −0.677541 −0.338771 0.940869i \(-0.610011\pi\)
−0.338771 + 0.940869i \(0.610011\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 36.2776 1.42402
\(650\) 0 0
\(651\) 8.06361 0.316038
\(652\) 23.6267i 0.925295i
\(653\) −21.9274 −0.858085 −0.429042 0.903284i \(-0.641149\pi\)
−0.429042 + 0.903284i \(0.641149\pi\)
\(654\) 6.69081 0.261631
\(655\) 1.25851i 0.0491742i
\(656\) 2.64516i 0.103276i
\(657\) 13.5734i 0.529547i
\(658\) 4.22464i 0.164694i
\(659\) −48.4759 −1.88835 −0.944176 0.329441i \(-0.893140\pi\)
−0.944176 + 0.329441i \(0.893140\pi\)
\(660\) 5.34541 0.208070
\(661\) − 29.8221i − 1.15994i −0.814636 0.579972i \(-0.803063\pi\)
0.814636 0.579972i \(-0.196937\pi\)
\(662\) 23.6981 0.921052
\(663\) 0 0
\(664\) −11.3360 −0.439921
\(665\) − 9.34333i − 0.362319i
\(666\) −3.14925 −0.122031
\(667\) 21.1790 0.820054
\(668\) − 16.7471i − 0.647967i
\(669\) 1.37891i 0.0533119i
\(670\) 4.10926i 0.158755i
\(671\) 2.86459i 0.110586i
\(672\) 2.32258 0.0895955
\(673\) 30.6719 1.18232 0.591158 0.806556i \(-0.298671\pi\)
0.591158 + 0.806556i \(0.298671\pi\)
\(674\) − 19.5554i − 0.753246i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 21.0831 0.810290 0.405145 0.914252i \(-0.367221\pi\)
0.405145 + 0.914252i \(0.367221\pi\)
\(678\) 7.10972i 0.273047i
\(679\) −37.4150 −1.43586
\(680\) 4.00000 0.153393
\(681\) 19.3205i 0.740363i
\(682\) − 18.5584i − 0.710636i
\(683\) 13.3360i 0.510287i 0.966903 + 0.255143i \(0.0821227\pi\)
−0.966903 + 0.255143i \(0.917877\pi\)
\(684\) − 4.02283i − 0.153817i
\(685\) 19.7149 0.753267
\(686\) 19.9872 0.763117
\(687\) 15.7626i 0.601381i
\(688\) −12.2508 −0.467057
\(689\) 0 0
\(690\) 4.93593 0.187908
\(691\) − 34.9054i − 1.32786i −0.747793 0.663932i \(-0.768887\pi\)
0.747793 0.663932i \(-0.231113\pi\)
\(692\) 16.1321 0.613250
\(693\) −12.4151 −0.471612
\(694\) 23.0375i 0.874490i
\(695\) 5.67742i 0.215357i
\(696\) 4.29078i 0.162641i
\(697\) − 10.5806i − 0.400770i
\(698\) 15.3205 0.579890
\(699\) −16.5549 −0.626166
\(700\) 2.32258i 0.0877853i
\(701\) 39.6715 1.49837 0.749186 0.662360i \(-0.230445\pi\)
0.749186 + 0.662360i \(0.230445\pi\)
\(702\) 0 0
\(703\) 12.6689 0.477817
\(704\) − 5.34541i − 0.201463i
\(705\) −1.81894 −0.0685053
\(706\) 28.4078 1.06914
\(707\) − 29.4754i − 1.10854i
\(708\) 6.78668i 0.255059i
\(709\) − 2.83441i − 0.106448i −0.998583 0.0532242i \(-0.983050\pi\)
0.998583 0.0532242i \(-0.0169498\pi\)
\(710\) − 15.8719i − 0.595661i
\(711\) −7.96774 −0.298814
\(712\) 1.73978 0.0652011
\(713\) − 17.1367i − 0.641776i
\(714\) −9.29032 −0.347681
\(715\) 0 0
\(716\) 7.32824 0.273869
\(717\) − 26.2006i − 0.978478i
\(718\) −23.5734 −0.879750
\(719\) 43.7128 1.63021 0.815106 0.579311i \(-0.196678\pi\)
0.815106 + 0.579311i \(0.196678\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) − 11.1394i − 0.414852i
\(722\) − 2.81687i − 0.104833i
\(723\) 15.6496i 0.582014i
\(724\) 19.5734 0.727438
\(725\) −4.29078 −0.159355
\(726\) 17.5734i 0.652209i
\(727\) −16.2568 −0.602932 −0.301466 0.953477i \(-0.597476\pi\)
−0.301466 + 0.953477i \(0.597476\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 13.5734i 0.502373i
\(731\) 49.0031 1.81245
\(732\) −0.535898 −0.0198074
\(733\) − 4.96774i − 0.183488i −0.995783 0.0917438i \(-0.970756\pi\)
0.995783 0.0917438i \(-0.0292441\pi\)
\(734\) − 27.4909i − 1.01471i
\(735\) 1.60562i 0.0592243i
\(736\) − 4.93593i − 0.181941i
\(737\) −21.9657 −0.809116
\(738\) 2.64516 0.0973697
\(739\) 49.6875i 1.82778i 0.405957 + 0.913892i \(0.366938\pi\)
−0.405957 + 0.913892i \(0.633062\pi\)
\(740\) −3.14925 −0.115769
\(741\) 0 0
\(742\) −12.7438 −0.467841
\(743\) 21.5420i 0.790300i 0.918617 + 0.395150i \(0.129307\pi\)
−0.918617 + 0.395150i \(0.870693\pi\)
\(744\) 3.47183 0.127284
\(745\) −19.4775 −0.713600
\(746\) − 26.3421i − 0.964452i
\(747\) 11.3360i 0.414761i
\(748\) 21.3816i 0.781790i
\(749\) 7.13449i 0.260689i
\(750\) −1.00000 −0.0365148
\(751\) 6.34872 0.231668 0.115834 0.993269i \(-0.463046\pi\)
0.115834 + 0.993269i \(0.463046\pi\)
\(752\) 1.81894i 0.0663300i
\(753\) 28.3416 1.03283
\(754\) 0 0
\(755\) 14.5170 0.528329
\(756\) − 2.32258i − 0.0844714i
\(757\) −48.1107 −1.74861 −0.874307 0.485373i \(-0.838684\pi\)
−0.874307 + 0.485373i \(0.838684\pi\)
\(758\) 0.448551 0.0162921
\(759\) 26.3846i 0.957699i
\(760\) − 4.02283i − 0.145923i
\(761\) − 33.6866i − 1.22114i −0.791963 0.610569i \(-0.790941\pi\)
0.791963 0.610569i \(-0.209059\pi\)
\(762\) − 2.13209i − 0.0772374i
\(763\) −15.5399 −0.562584
\(764\) 17.0375 0.616394
\(765\) − 4.00000i − 0.144620i
\(766\) −12.1227 −0.438009
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) − 5.98886i − 0.215964i −0.994153 0.107982i \(-0.965561\pi\)
0.994153 0.107982i \(-0.0344389\pi\)
\(770\) −12.4151 −0.447410
\(771\) 12.5093 0.450511
\(772\) − 6.15491i − 0.221520i
\(773\) − 13.4882i − 0.485136i −0.970134 0.242568i \(-0.922010\pi\)
0.970134 0.242568i \(-0.0779897\pi\)
\(774\) 12.2508i 0.440345i
\(775\) 3.47183i 0.124712i
\(776\) −16.1093 −0.578289
\(777\) 7.31439 0.262402
\(778\) − 18.6195i − 0.667540i
\(779\) −10.6410 −0.381254
\(780\) 0 0
\(781\) 84.8416 3.03587
\(782\) 19.7437i 0.706035i
\(783\) 4.29078 0.153340
\(784\) 1.60562 0.0573437
\(785\) 24.3829i 0.870262i
\(786\) − 1.25851i − 0.0448897i
\(787\) 54.6487i 1.94802i 0.226510 + 0.974009i \(0.427268\pi\)
−0.226510 + 0.974009i \(0.572732\pi\)
\(788\) 13.7867i 0.491130i
\(789\) −10.7059 −0.381140
\(790\) −7.96774 −0.283479
\(791\) − 16.5129i − 0.587131i
\(792\) −5.34541 −0.189941
\(793\) 0 0
\(794\) −11.3042 −0.401170
\(795\) − 5.48693i − 0.194601i
\(796\) 2.28304 0.0809203
\(797\) 38.3244 1.35752 0.678760 0.734361i \(-0.262518\pi\)
0.678760 + 0.734361i \(0.262518\pi\)
\(798\) 9.34333i 0.330750i
\(799\) − 7.27577i − 0.257398i
\(800\) 1.00000i 0.0353553i
\(801\) − 1.73978i − 0.0614722i
\(802\) −1.50580 −0.0531716
\(803\) −72.5551 −2.56041
\(804\) − 4.10926i − 0.144922i
\(805\) −11.4641 −0.404056
\(806\) 0 0
\(807\) 7.52522 0.264900
\(808\) − 12.6908i − 0.446461i
\(809\) 11.3816 0.400157 0.200078 0.979780i \(-0.435880\pi\)
0.200078 + 0.979780i \(0.435880\pi\)
\(810\) 1.00000 0.0351364
\(811\) 10.8899i 0.382397i 0.981551 + 0.191198i \(0.0612374\pi\)
−0.981551 + 0.191198i \(0.938763\pi\)
\(812\) − 9.96567i − 0.349726i
\(813\) 9.84868i 0.345408i
\(814\) − 16.8340i − 0.590033i
\(815\) 23.6267 0.827609
\(816\) −4.00000 −0.140028
\(817\) − 49.2828i − 1.72419i
\(818\) −11.1394 −0.389479
\(819\) 0 0
\(820\) 2.64516 0.0923730
\(821\) − 1.68810i − 0.0589152i −0.999566 0.0294576i \(-0.990622\pi\)
0.999566 0.0294576i \(-0.00937800\pi\)
\(822\) −19.7149 −0.687636
\(823\) −6.77816 −0.236272 −0.118136 0.992997i \(-0.537692\pi\)
−0.118136 + 0.992997i \(0.537692\pi\)
\(824\) − 4.79612i − 0.167081i
\(825\) − 5.34541i − 0.186103i
\(826\) − 15.7626i − 0.548451i
\(827\) 30.2298i 1.05119i 0.850733 + 0.525597i \(0.176158\pi\)
−0.850733 + 0.525597i \(0.823842\pi\)
\(828\) −4.93593 −0.171536
\(829\) 10.4405 0.362612 0.181306 0.983427i \(-0.441968\pi\)
0.181306 + 0.983427i \(0.441968\pi\)
\(830\) 11.3360i 0.393477i
\(831\) −0.953101 −0.0330627
\(832\) 0 0
\(833\) −6.42249 −0.222526
\(834\) − 5.67742i − 0.196593i
\(835\) −16.7471 −0.579559
\(836\) 21.5036 0.743719
\(837\) − 3.47183i − 0.120004i
\(838\) 15.5098i 0.535776i
\(839\) − 11.1965i − 0.386547i −0.981145 0.193273i \(-0.938090\pi\)
0.981145 0.193273i \(-0.0619104\pi\)
\(840\) − 2.32258i − 0.0801366i
\(841\) −10.5892 −0.365146
\(842\) 39.4452 1.35937
\(843\) 5.57336i 0.191957i
\(844\) −22.4775 −0.773707
\(845\) 0 0
\(846\) 1.81894 0.0625365
\(847\) − 40.8155i − 1.40244i
\(848\) −5.48693 −0.188422
\(849\) −22.0134 −0.755498
\(850\) − 4.00000i − 0.137199i
\(851\) − 15.5445i − 0.532859i
\(852\) 15.8719i 0.543761i
\(853\) − 21.8185i − 0.747051i −0.927620 0.373525i \(-0.878149\pi\)
0.927620 0.373525i \(-0.121851\pi\)
\(854\) 1.24467 0.0425916
\(855\) −4.02283 −0.137578
\(856\) 3.07180i 0.104992i
\(857\) −9.42369 −0.321907 −0.160954 0.986962i \(-0.551457\pi\)
−0.160954 + 0.986962i \(0.551457\pi\)
\(858\) 0 0
\(859\) 24.4775 0.835161 0.417581 0.908640i \(-0.362878\pi\)
0.417581 + 0.908640i \(0.362878\pi\)
\(860\) 12.2508i 0.417748i
\(861\) −6.14359 −0.209373
\(862\) 2.15491 0.0733966
\(863\) 29.7873i 1.01397i 0.861954 + 0.506986i \(0.169240\pi\)
−0.861954 + 0.506986i \(0.830760\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) − 16.1321i − 0.548507i
\(866\) 1.33938i 0.0455139i
\(867\) −1.00000 −0.0339618
\(868\) −8.06361 −0.273697
\(869\) − 42.5908i − 1.44479i
\(870\) 4.29078 0.145471
\(871\) 0 0
\(872\) −6.69081 −0.226579
\(873\) 16.1093i 0.545216i
\(874\) 19.8564 0.671653
\(875\) 2.32258 0.0785175
\(876\) − 13.5734i − 0.458601i
\(877\) − 1.90979i − 0.0644890i −0.999480 0.0322445i \(-0.989734\pi\)
0.999480 0.0322445i \(-0.0102655\pi\)
\(878\) − 31.6981i − 1.06976i
\(879\) − 6.84302i − 0.230809i
\(880\) −5.34541 −0.180194
\(881\) −6.36823 −0.214551 −0.107276 0.994229i \(-0.534213\pi\)
−0.107276 + 0.994229i \(0.534213\pi\)
\(882\) − 1.60562i − 0.0540642i
\(883\) −34.9897 −1.17750 −0.588749 0.808316i \(-0.700379\pi\)
−0.588749 + 0.808316i \(0.700379\pi\)
\(884\) 0 0
\(885\) 6.78668 0.228132
\(886\) 28.0904i 0.943715i
\(887\) 21.4001 0.718546 0.359273 0.933233i \(-0.383025\pi\)
0.359273 + 0.933233i \(0.383025\pi\)
\(888\) 3.14925 0.105682
\(889\) 4.95194i 0.166083i
\(890\) − 1.73978i − 0.0583176i
\(891\) 5.34541i 0.179078i
\(892\) − 1.37891i − 0.0461694i
\(893\) −7.31729 −0.244864
\(894\) 19.4775 0.651425
\(895\) − 7.32824i − 0.244956i
\(896\) −2.32258 −0.0775919
\(897\) 0 0
\(898\) −28.9167 −0.964963
\(899\) − 14.8969i − 0.496838i
\(900\) 1.00000 0.0333333
\(901\) 21.9477 0.731184
\(902\) 14.1395i 0.470792i
\(903\) − 28.4534i − 0.946871i
\(904\) − 7.10972i − 0.236466i
\(905\) − 19.5734i − 0.650641i
\(906\) −14.5170 −0.482296
\(907\) −32.4693 −1.07813 −0.539063 0.842266i \(-0.681221\pi\)
−0.539063 + 0.842266i \(0.681221\pi\)
\(908\) − 19.3205i − 0.641174i
\(909\) −12.6908 −0.420928
\(910\) 0 0
\(911\) 34.4380 1.14098 0.570490 0.821304i \(-0.306753\pi\)
0.570490 + 0.821304i \(0.306753\pi\)
\(912\) 4.02283i 0.133209i
\(913\) −60.5954 −2.00541
\(914\) 9.69900 0.320814
\(915\) 0.535898i 0.0177163i
\(916\) − 15.7626i − 0.520811i
\(917\) 2.92300i 0.0965260i
\(918\) 4.00000i 0.132020i
\(919\) 36.4827 1.20345 0.601727 0.798702i \(-0.294480\pi\)
0.601727 + 0.798702i \(0.294480\pi\)
\(920\) −4.93593 −0.162733
\(921\) 4.75442i 0.156663i
\(922\) −11.9979 −0.395131
\(923\) 0 0
\(924\) 12.4151 0.408428
\(925\) 3.14925i 0.103547i
\(926\) −6.42664 −0.211192
\(927\) −4.79612 −0.157525
\(928\) − 4.29078i − 0.140852i
\(929\) − 49.9100i − 1.63749i −0.574155 0.818747i \(-0.694669\pi\)
0.574155 0.818747i \(-0.305331\pi\)
\(930\) − 3.47183i − 0.113846i
\(931\) 6.45914i 0.211690i
\(932\) 16.5549 0.542275
\(933\) 1.93639 0.0633946
\(934\) − 26.2642i − 0.859390i
\(935\) 21.3816 0.699254
\(936\) 0 0
\(937\) −15.8873 −0.519017 −0.259508 0.965741i \(-0.583560\pi\)
−0.259508 + 0.965741i \(0.583560\pi\)
\(938\) 9.54409i 0.311625i
\(939\) 25.5545 0.833939
\(940\) 1.81894 0.0593274
\(941\) − 6.99273i − 0.227956i −0.993483 0.113978i \(-0.963641\pi\)
0.993483 0.113978i \(-0.0363594\pi\)
\(942\) − 24.3829i − 0.794437i
\(943\) 13.0563i 0.425173i
\(944\) − 6.78668i − 0.220888i
\(945\) −2.32258 −0.0755535
\(946\) −65.4854 −2.12911
\(947\) − 52.3064i − 1.69973i −0.527000 0.849865i \(-0.676683\pi\)
0.527000 0.849865i \(-0.323317\pi\)
\(948\) 7.96774 0.258780
\(949\) 0 0
\(950\) −4.02283 −0.130518
\(951\) 12.6667i 0.410747i
\(952\) 9.29032 0.301101
\(953\) −28.8827 −0.935603 −0.467802 0.883833i \(-0.654954\pi\)
−0.467802 + 0.883833i \(0.654954\pi\)
\(954\) 5.48693i 0.177646i
\(955\) − 17.0375i − 0.551319i
\(956\) 26.2006i 0.847387i
\(957\) 22.9359i 0.741413i
\(958\) −25.6826 −0.829768
\(959\) 45.7894 1.47862
\(960\) − 1.00000i − 0.0322749i
\(961\) 18.9464 0.611173
\(962\) 0 0
\(963\) 3.07180 0.0989873
\(964\) − 15.6496i − 0.504039i
\(965\) −6.15491 −0.198134
\(966\) 11.4641 0.368851
\(967\) − 10.8610i − 0.349265i −0.984634 0.174633i \(-0.944126\pi\)
0.984634 0.174633i \(-0.0558738\pi\)
\(968\) − 17.5734i − 0.564829i
\(969\) − 16.0913i − 0.516927i
\(970\) 16.1093i 0.517237i
\(971\) −24.9979 −0.802222 −0.401111 0.916030i \(-0.631376\pi\)
−0.401111 + 0.916030i \(0.631376\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 13.1863i 0.422732i
\(974\) 9.37891 0.300520
\(975\) 0 0
\(976\) 0.535898 0.0171537
\(977\) − 42.7653i − 1.36818i −0.729396 0.684092i \(-0.760199\pi\)
0.729396 0.684092i \(-0.239801\pi\)
\(978\) −23.6267 −0.755500
\(979\) 9.29984 0.297224
\(980\) − 1.60562i − 0.0512898i
\(981\) 6.69081i 0.213621i
\(982\) − 7.73854i − 0.246947i
\(983\) 10.1288i 0.323058i 0.986868 + 0.161529i \(0.0516425\pi\)
−0.986868 + 0.161529i \(0.948358\pi\)
\(984\) −2.64516 −0.0843246
\(985\) 13.7867 0.439280
\(986\) 17.1631i 0.546584i
\(987\) −4.22464 −0.134472
\(988\) 0 0
\(989\) −60.4691 −1.92280
\(990\) 5.34541i 0.169888i
\(991\) 20.2483 0.643208 0.321604 0.946874i \(-0.395778\pi\)
0.321604 + 0.946874i \(0.395778\pi\)
\(992\) −3.47183 −0.110231
\(993\) 23.6981i 0.752036i
\(994\) − 36.8637i − 1.16924i
\(995\) − 2.28304i − 0.0723774i
\(996\) − 11.3360i − 0.359194i
\(997\) −20.7961 −0.658620 −0.329310 0.944222i \(-0.606816\pi\)
−0.329310 + 0.944222i \(0.606816\pi\)
\(998\) −3.18106 −0.100695
\(999\) − 3.14925i − 0.0996380i
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.ba.1351.3 8
13.5 odd 4 5070.2.a.bz.1.2 4
13.8 odd 4 5070.2.a.ca.1.3 4
13.9 even 3 390.2.bb.c.361.1 yes 8
13.10 even 6 390.2.bb.c.121.1 8
13.12 even 2 inner 5070.2.b.ba.1351.6 8
39.23 odd 6 1170.2.bs.f.901.3 8
39.35 odd 6 1170.2.bs.f.361.3 8
65.9 even 6 1950.2.bc.g.751.4 8
65.22 odd 12 1950.2.y.j.49.2 8
65.23 odd 12 1950.2.y.j.199.2 8
65.48 odd 12 1950.2.y.k.49.3 8
65.49 even 6 1950.2.bc.g.901.4 8
65.62 odd 12 1950.2.y.k.199.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.c.121.1 8 13.10 even 6
390.2.bb.c.361.1 yes 8 13.9 even 3
1170.2.bs.f.361.3 8 39.35 odd 6
1170.2.bs.f.901.3 8 39.23 odd 6
1950.2.y.j.49.2 8 65.22 odd 12
1950.2.y.j.199.2 8 65.23 odd 12
1950.2.y.k.49.3 8 65.48 odd 12
1950.2.y.k.199.3 8 65.62 odd 12
1950.2.bc.g.751.4 8 65.9 even 6
1950.2.bc.g.901.4 8 65.49 even 6
5070.2.a.bz.1.2 4 13.5 odd 4
5070.2.a.ca.1.3 4 13.8 odd 4
5070.2.b.ba.1351.3 8 1.1 even 1 trivial
5070.2.b.ba.1351.6 8 13.12 even 2 inner