Properties

Label 5070.2.b.t.1351.1
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 1351.1
Root \(-0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.t.1351.6

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -1.69202i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -1.69202i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.15883i q^{11} +1.00000 q^{12} -1.69202 q^{14} +1.00000i q^{15} +1.00000 q^{16} -2.35690 q^{17} -1.00000i q^{18} -0.198062i q^{19} +1.00000i q^{20} +1.69202i q^{21} +2.15883 q^{22} +3.74094 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +1.69202i q^{28} +1.29590 q^{29} +1.00000 q^{30} -1.44504i q^{31} -1.00000i q^{32} -2.15883i q^{33} +2.35690i q^{34} -1.69202 q^{35} -1.00000 q^{36} +0.801938i q^{37} -0.198062 q^{38} +1.00000 q^{40} +1.89977i q^{41} +1.69202 q^{42} -12.5429 q^{43} -2.15883i q^{44} -1.00000i q^{45} -3.74094i q^{46} +8.87800i q^{47} -1.00000 q^{48} +4.13706 q^{49} +1.00000i q^{50} +2.35690 q^{51} -1.00969 q^{53} +1.00000i q^{54} +2.15883 q^{55} +1.69202 q^{56} +0.198062i q^{57} -1.29590i q^{58} +3.73125i q^{59} -1.00000i q^{60} -6.32304 q^{61} -1.44504 q^{62} -1.69202i q^{63} -1.00000 q^{64} -2.15883 q^{66} +7.56465i q^{67} +2.35690 q^{68} -3.74094 q^{69} +1.69202i q^{70} +4.18060i q^{71} +1.00000i q^{72} +11.9366i q^{73} +0.801938 q^{74} +1.00000 q^{75} +0.198062i q^{76} +3.65279 q^{77} +9.40581 q^{79} -1.00000i q^{80} +1.00000 q^{81} +1.89977 q^{82} -8.43296i q^{83} -1.69202i q^{84} +2.35690i q^{85} +12.5429i q^{86} -1.29590 q^{87} -2.15883 q^{88} -2.41119i q^{89} -1.00000 q^{90} -3.74094 q^{92} +1.44504i q^{93} +8.87800 q^{94} -0.198062 q^{95} +1.00000i q^{96} +0.0881460i q^{97} -4.13706i q^{98} +2.15883i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} - 6q^{10} + 6q^{12} + 6q^{16} - 6q^{17} - 4q^{22} - 6q^{23} - 6q^{25} - 6q^{27} - 20q^{29} + 6q^{30} - 6q^{36} - 10q^{38} + 6q^{40} - 38q^{43} - 6q^{48} + 14q^{49} + 6q^{51} + 38q^{53} - 4q^{55} + 2q^{61} - 8q^{62} - 6q^{64} + 4q^{66} + 6q^{68} + 6q^{69} - 4q^{74} + 6q^{75} - 14q^{77} + 30q^{79} + 6q^{81} - 34q^{82} + 20q^{87} + 4q^{88} - 6q^{90} + 6q^{92} + 14q^{94} - 10q^{95} + O(q^{100}) \)

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
\(7\) − 1.69202i − 0.639524i −0.947498 0.319762i \(-0.896397\pi\)
0.947498 0.319762i \(-0.103603\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 2.15883i 0.650913i 0.945557 + 0.325456i \(0.105518\pi\)
−0.945557 + 0.325456i \(0.894482\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.69202 −0.452212
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) −2.35690 −0.571631 −0.285816 0.958285i \(-0.592264\pi\)
−0.285816 + 0.958285i \(0.592264\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) − 0.198062i − 0.0454386i −0.999742 0.0227193i \(-0.992768\pi\)
0.999742 0.0227193i \(-0.00723240\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 1.69202i 0.369229i
\(22\) 2.15883 0.460265
\(23\) 3.74094 0.780040 0.390020 0.920806i \(-0.372468\pi\)
0.390020 + 0.920806i \(0.372468\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.69202i 0.319762i
\(29\) 1.29590 0.240642 0.120321 0.992735i \(-0.461608\pi\)
0.120321 + 0.992735i \(0.461608\pi\)
\(30\) 1.00000 0.182574
\(31\) − 1.44504i − 0.259537i −0.991544 0.129769i \(-0.958577\pi\)
0.991544 0.129769i \(-0.0414234\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 2.15883i − 0.375805i
\(34\) 2.35690i 0.404204i
\(35\) −1.69202 −0.286004
\(36\) −1.00000 −0.166667
\(37\) 0.801938i 0.131838i 0.997825 + 0.0659189i \(0.0209979\pi\)
−0.997825 + 0.0659189i \(0.979002\pi\)
\(38\) −0.198062 −0.0321299
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 1.89977i 0.296695i 0.988935 + 0.148347i \(0.0473953\pi\)
−0.988935 + 0.148347i \(0.952605\pi\)
\(42\) 1.69202 0.261085
\(43\) −12.5429 −1.91277 −0.956385 0.292108i \(-0.905643\pi\)
−0.956385 + 0.292108i \(0.905643\pi\)
\(44\) − 2.15883i − 0.325456i
\(45\) − 1.00000i − 0.149071i
\(46\) − 3.74094i − 0.551571i
\(47\) 8.87800i 1.29499i 0.762070 + 0.647495i \(0.224183\pi\)
−0.762070 + 0.647495i \(0.775817\pi\)
\(48\) −1.00000 −0.144338
\(49\) 4.13706 0.591009
\(50\) 1.00000i 0.141421i
\(51\) 2.35690 0.330031
\(52\) 0 0
\(53\) −1.00969 −0.138691 −0.0693457 0.997593i \(-0.522091\pi\)
−0.0693457 + 0.997593i \(0.522091\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 2.15883 0.291097
\(56\) 1.69202 0.226106
\(57\) 0.198062i 0.0262340i
\(58\) − 1.29590i − 0.170160i
\(59\) 3.73125i 0.485767i 0.970055 + 0.242884i \(0.0780933\pi\)
−0.970055 + 0.242884i \(0.921907\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) −6.32304 −0.809583 −0.404791 0.914409i \(-0.632656\pi\)
−0.404791 + 0.914409i \(0.632656\pi\)
\(62\) −1.44504 −0.183521
\(63\) − 1.69202i − 0.213175i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.15883 −0.265734
\(67\) 7.56465i 0.924169i 0.886836 + 0.462084i \(0.152898\pi\)
−0.886836 + 0.462084i \(0.847102\pi\)
\(68\) 2.35690 0.285816
\(69\) −3.74094 −0.450356
\(70\) 1.69202i 0.202235i
\(71\) 4.18060i 0.496146i 0.968741 + 0.248073i \(0.0797974\pi\)
−0.968741 + 0.248073i \(0.920203\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 11.9366i 1.39707i 0.715574 + 0.698537i \(0.246165\pi\)
−0.715574 + 0.698537i \(0.753835\pi\)
\(74\) 0.801938 0.0932234
\(75\) 1.00000 0.115470
\(76\) 0.198062i 0.0227193i
\(77\) 3.65279 0.416274
\(78\) 0 0
\(79\) 9.40581 1.05824 0.529118 0.848548i \(-0.322523\pi\)
0.529118 + 0.848548i \(0.322523\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) 1.89977 0.209795
\(83\) − 8.43296i − 0.925638i −0.886453 0.462819i \(-0.846838\pi\)
0.886453 0.462819i \(-0.153162\pi\)
\(84\) − 1.69202i − 0.184615i
\(85\) 2.35690i 0.255641i
\(86\) 12.5429i 1.35253i
\(87\) −1.29590 −0.138935
\(88\) −2.15883 −0.230132
\(89\) − 2.41119i − 0.255586i −0.991801 0.127793i \(-0.959211\pi\)
0.991801 0.127793i \(-0.0407893\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −3.74094 −0.390020
\(93\) 1.44504i 0.149844i
\(94\) 8.87800 0.915696
\(95\) −0.198062 −0.0203208
\(96\) 1.00000i 0.102062i
\(97\) 0.0881460i 0.00894987i 0.999990 + 0.00447494i \(0.00142442\pi\)
−0.999990 + 0.00447494i \(0.998576\pi\)
\(98\) − 4.13706i − 0.417907i
\(99\) 2.15883i 0.216971i
\(100\) 1.00000 0.100000
\(101\) 11.1099 1.10548 0.552739 0.833354i \(-0.313583\pi\)
0.552739 + 0.833354i \(0.313583\pi\)
\(102\) − 2.35690i − 0.233367i
\(103\) −9.54825 −0.940817 −0.470409 0.882449i \(-0.655893\pi\)
−0.470409 + 0.882449i \(0.655893\pi\)
\(104\) 0 0
\(105\) 1.69202 0.165124
\(106\) 1.00969i 0.0980696i
\(107\) 18.1468 1.75431 0.877156 0.480205i \(-0.159438\pi\)
0.877156 + 0.480205i \(0.159438\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.24698i 0.215222i 0.994193 + 0.107611i \(0.0343200\pi\)
−0.994193 + 0.107611i \(0.965680\pi\)
\(110\) − 2.15883i − 0.205837i
\(111\) − 0.801938i − 0.0761166i
\(112\) − 1.69202i − 0.159881i
\(113\) −11.8726 −1.11688 −0.558441 0.829544i \(-0.688600\pi\)
−0.558441 + 0.829544i \(0.688600\pi\)
\(114\) 0.198062 0.0185502
\(115\) − 3.74094i − 0.348844i
\(116\) −1.29590 −0.120321
\(117\) 0 0
\(118\) 3.73125 0.343489
\(119\) 3.98792i 0.365572i
\(120\) −1.00000 −0.0912871
\(121\) 6.33944 0.576312
\(122\) 6.32304i 0.572462i
\(123\) − 1.89977i − 0.171297i
\(124\) 1.44504i 0.129769i
\(125\) 1.00000i 0.0894427i
\(126\) −1.69202 −0.150737
\(127\) −13.0097 −1.15442 −0.577212 0.816595i \(-0.695859\pi\)
−0.577212 + 0.816595i \(0.695859\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 12.5429 1.10434
\(130\) 0 0
\(131\) −7.82908 −0.684030 −0.342015 0.939694i \(-0.611109\pi\)
−0.342015 + 0.939694i \(0.611109\pi\)
\(132\) 2.15883i 0.187902i
\(133\) −0.335126 −0.0290591
\(134\) 7.56465 0.653486
\(135\) 1.00000i 0.0860663i
\(136\) − 2.35690i − 0.202102i
\(137\) 5.36227i 0.458130i 0.973411 + 0.229065i \(0.0735668\pi\)
−0.973411 + 0.229065i \(0.926433\pi\)
\(138\) 3.74094i 0.318450i
\(139\) 3.12200 0.264804 0.132402 0.991196i \(-0.457731\pi\)
0.132402 + 0.991196i \(0.457731\pi\)
\(140\) 1.69202 0.143002
\(141\) − 8.87800i − 0.747663i
\(142\) 4.18060 0.350828
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 1.29590i − 0.107618i
\(146\) 11.9366 0.987881
\(147\) −4.13706 −0.341219
\(148\) − 0.801938i − 0.0659189i
\(149\) − 4.24160i − 0.347486i −0.984791 0.173743i \(-0.944414\pi\)
0.984791 0.173743i \(-0.0555861\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) 22.9148i 1.86478i 0.361450 + 0.932392i \(0.382282\pi\)
−0.361450 + 0.932392i \(0.617718\pi\)
\(152\) 0.198062 0.0160650
\(153\) −2.35690 −0.190544
\(154\) − 3.65279i − 0.294350i
\(155\) −1.44504 −0.116069
\(156\) 0 0
\(157\) 16.0707 1.28258 0.641290 0.767298i \(-0.278399\pi\)
0.641290 + 0.767298i \(0.278399\pi\)
\(158\) − 9.40581i − 0.748286i
\(159\) 1.00969 0.0800735
\(160\) −1.00000 −0.0790569
\(161\) − 6.32975i − 0.498854i
\(162\) − 1.00000i − 0.0785674i
\(163\) − 0.488582i − 0.0382687i −0.999817 0.0191344i \(-0.993909\pi\)
0.999817 0.0191344i \(-0.00609103\pi\)
\(164\) − 1.89977i − 0.148347i
\(165\) −2.15883 −0.168065
\(166\) −8.43296 −0.654525
\(167\) 14.0610i 1.08807i 0.839062 + 0.544036i \(0.183105\pi\)
−0.839062 + 0.544036i \(0.816895\pi\)
\(168\) −1.69202 −0.130542
\(169\) 0 0
\(170\) 2.35690 0.180766
\(171\) − 0.198062i − 0.0151462i
\(172\) 12.5429 0.956385
\(173\) −21.7942 −1.65698 −0.828490 0.560004i \(-0.810800\pi\)
−0.828490 + 0.560004i \(0.810800\pi\)
\(174\) 1.29590i 0.0982417i
\(175\) 1.69202i 0.127905i
\(176\) 2.15883i 0.162728i
\(177\) − 3.73125i − 0.280458i
\(178\) −2.41119 −0.180726
\(179\) 14.1293 1.05607 0.528036 0.849222i \(-0.322928\pi\)
0.528036 + 0.849222i \(0.322928\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 5.63773 0.419049 0.209524 0.977803i \(-0.432808\pi\)
0.209524 + 0.977803i \(0.432808\pi\)
\(182\) 0 0
\(183\) 6.32304 0.467413
\(184\) 3.74094i 0.275786i
\(185\) 0.801938 0.0589596
\(186\) 1.44504 0.105956
\(187\) − 5.08815i − 0.372082i
\(188\) − 8.87800i − 0.647495i
\(189\) 1.69202i 0.123076i
\(190\) 0.198062i 0.0143689i
\(191\) −2.59419 −0.187709 −0.0938544 0.995586i \(-0.529919\pi\)
−0.0938544 + 0.995586i \(0.529919\pi\)
\(192\) 1.00000 0.0721688
\(193\) − 3.98361i − 0.286746i −0.989669 0.143373i \(-0.954205\pi\)
0.989669 0.143373i \(-0.0457949\pi\)
\(194\) 0.0881460 0.00632851
\(195\) 0 0
\(196\) −4.13706 −0.295505
\(197\) 6.65040i 0.473821i 0.971531 + 0.236911i \(0.0761349\pi\)
−0.971531 + 0.236911i \(0.923865\pi\)
\(198\) 2.15883 0.153422
\(199\) 1.01639 0.0720502 0.0360251 0.999351i \(-0.488530\pi\)
0.0360251 + 0.999351i \(0.488530\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 7.56465i − 0.533569i
\(202\) − 11.1099i − 0.781691i
\(203\) − 2.19269i − 0.153896i
\(204\) −2.35690 −0.165016
\(205\) 1.89977 0.132686
\(206\) 9.54825i 0.665258i
\(207\) 3.74094 0.260013
\(208\) 0 0
\(209\) 0.427583 0.0295766
\(210\) − 1.69202i − 0.116761i
\(211\) 10.1414 0.698161 0.349081 0.937093i \(-0.386494\pi\)
0.349081 + 0.937093i \(0.386494\pi\)
\(212\) 1.00969 0.0693457
\(213\) − 4.18060i − 0.286450i
\(214\) − 18.1468i − 1.24049i
\(215\) 12.5429i 0.855417i
\(216\) − 1.00000i − 0.0680414i
\(217\) −2.44504 −0.165980
\(218\) 2.24698 0.152185
\(219\) − 11.9366i − 0.806601i
\(220\) −2.15883 −0.145549
\(221\) 0 0
\(222\) −0.801938 −0.0538225
\(223\) 16.8388i 1.12761i 0.825909 + 0.563804i \(0.190663\pi\)
−0.825909 + 0.563804i \(0.809337\pi\)
\(224\) −1.69202 −0.113053
\(225\) −1.00000 −0.0666667
\(226\) 11.8726i 0.789755i
\(227\) − 18.3056i − 1.21498i −0.794326 0.607492i \(-0.792176\pi\)
0.794326 0.607492i \(-0.207824\pi\)
\(228\) − 0.198062i − 0.0131170i
\(229\) 5.14244i 0.339822i 0.985459 + 0.169911i \(0.0543480\pi\)
−0.985459 + 0.169911i \(0.945652\pi\)
\(230\) −3.74094 −0.246670
\(231\) −3.65279 −0.240336
\(232\) 1.29590i 0.0850798i
\(233\) 10.9215 0.715494 0.357747 0.933819i \(-0.383545\pi\)
0.357747 + 0.933819i \(0.383545\pi\)
\(234\) 0 0
\(235\) 8.87800 0.579137
\(236\) − 3.73125i − 0.242884i
\(237\) −9.40581 −0.610973
\(238\) 3.98792 0.258498
\(239\) − 19.9041i − 1.28749i −0.765241 0.643744i \(-0.777380\pi\)
0.765241 0.643744i \(-0.222620\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 18.2228i 1.17383i 0.809647 + 0.586917i \(0.199659\pi\)
−0.809647 + 0.586917i \(0.800341\pi\)
\(242\) − 6.33944i − 0.407514i
\(243\) −1.00000 −0.0641500
\(244\) 6.32304 0.404791
\(245\) − 4.13706i − 0.264307i
\(246\) −1.89977 −0.121125
\(247\) 0 0
\(248\) 1.44504 0.0917603
\(249\) 8.43296i 0.534417i
\(250\) 1.00000 0.0632456
\(251\) 17.0030 1.07322 0.536609 0.843831i \(-0.319705\pi\)
0.536609 + 0.843831i \(0.319705\pi\)
\(252\) 1.69202i 0.106587i
\(253\) 8.07606i 0.507738i
\(254\) 13.0097i 0.816300i
\(255\) − 2.35690i − 0.147595i
\(256\) 1.00000 0.0625000
\(257\) 7.65519 0.477517 0.238759 0.971079i \(-0.423260\pi\)
0.238759 + 0.971079i \(0.423260\pi\)
\(258\) − 12.5429i − 0.780885i
\(259\) 1.35690 0.0843134
\(260\) 0 0
\(261\) 1.29590 0.0802140
\(262\) 7.82908i 0.483682i
\(263\) 21.0411 1.29745 0.648726 0.761022i \(-0.275302\pi\)
0.648726 + 0.761022i \(0.275302\pi\)
\(264\) 2.15883 0.132867
\(265\) 1.00969i 0.0620247i
\(266\) 0.335126i 0.0205479i
\(267\) 2.41119i 0.147562i
\(268\) − 7.56465i − 0.462084i
\(269\) 24.3183 1.48271 0.741355 0.671113i \(-0.234183\pi\)
0.741355 + 0.671113i \(0.234183\pi\)
\(270\) 1.00000 0.0608581
\(271\) 10.9922i 0.667730i 0.942621 + 0.333865i \(0.108353\pi\)
−0.942621 + 0.333865i \(0.891647\pi\)
\(272\) −2.35690 −0.142908
\(273\) 0 0
\(274\) 5.36227 0.323947
\(275\) − 2.15883i − 0.130183i
\(276\) 3.74094 0.225178
\(277\) 11.4601 0.688571 0.344286 0.938865i \(-0.388121\pi\)
0.344286 + 0.938865i \(0.388121\pi\)
\(278\) − 3.12200i − 0.187245i
\(279\) − 1.44504i − 0.0865124i
\(280\) − 1.69202i − 0.101118i
\(281\) − 9.88231i − 0.589529i −0.955570 0.294765i \(-0.904759\pi\)
0.955570 0.294765i \(-0.0952413\pi\)
\(282\) −8.87800 −0.528677
\(283\) 21.3207 1.26738 0.633691 0.773587i \(-0.281539\pi\)
0.633691 + 0.773587i \(0.281539\pi\)
\(284\) − 4.18060i − 0.248073i
\(285\) 0.198062 0.0117322
\(286\) 0 0
\(287\) 3.21446 0.189743
\(288\) − 1.00000i − 0.0589256i
\(289\) −11.4450 −0.673238
\(290\) −1.29590 −0.0760977
\(291\) − 0.0881460i − 0.00516721i
\(292\) − 11.9366i − 0.698537i
\(293\) − 18.7090i − 1.09299i −0.837462 0.546496i \(-0.815961\pi\)
0.837462 0.546496i \(-0.184039\pi\)
\(294\) 4.13706i 0.241278i
\(295\) 3.73125 0.217242
\(296\) −0.801938 −0.0466117
\(297\) − 2.15883i − 0.125268i
\(298\) −4.24160 −0.245709
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 21.2228i 1.22326i
\(302\) 22.9148 1.31860
\(303\) −11.1099 −0.638248
\(304\) − 0.198062i − 0.0113596i
\(305\) 6.32304i 0.362056i
\(306\) 2.35690i 0.134735i
\(307\) 17.2241i 0.983034i 0.870868 + 0.491517i \(0.163557\pi\)
−0.870868 + 0.491517i \(0.836443\pi\)
\(308\) −3.65279 −0.208137
\(309\) 9.54825 0.543181
\(310\) 1.44504i 0.0820729i
\(311\) 13.5200 0.766651 0.383326 0.923613i \(-0.374779\pi\)
0.383326 + 0.923613i \(0.374779\pi\)
\(312\) 0 0
\(313\) 24.3303 1.37523 0.687616 0.726074i \(-0.258657\pi\)
0.687616 + 0.726074i \(0.258657\pi\)
\(314\) − 16.0707i − 0.906921i
\(315\) −1.69202 −0.0953346
\(316\) −9.40581 −0.529118
\(317\) − 34.7821i − 1.95356i −0.214253 0.976778i \(-0.568732\pi\)
0.214253 0.976778i \(-0.431268\pi\)
\(318\) − 1.00969i − 0.0566205i
\(319\) 2.79763i 0.156637i
\(320\) 1.00000i 0.0559017i
\(321\) −18.1468 −1.01285
\(322\) −6.32975 −0.352743
\(323\) 0.466812i 0.0259741i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −0.488582 −0.0270601
\(327\) − 2.24698i − 0.124258i
\(328\) −1.89977 −0.104897
\(329\) 15.0218 0.828177
\(330\) 2.15883i 0.118840i
\(331\) 9.66919i 0.531467i 0.964047 + 0.265733i \(0.0856140\pi\)
−0.964047 + 0.265733i \(0.914386\pi\)
\(332\) 8.43296i 0.462819i
\(333\) 0.801938i 0.0439459i
\(334\) 14.0610 0.769384
\(335\) 7.56465 0.413301
\(336\) 1.69202i 0.0923073i
\(337\) 17.3612 0.945725 0.472863 0.881136i \(-0.343221\pi\)
0.472863 + 0.881136i \(0.343221\pi\)
\(338\) 0 0
\(339\) 11.8726 0.644832
\(340\) − 2.35690i − 0.127821i
\(341\) 3.11960 0.168936
\(342\) −0.198062 −0.0107100
\(343\) − 18.8442i − 1.01749i
\(344\) − 12.5429i − 0.676267i
\(345\) 3.74094i 0.201405i
\(346\) 21.7942i 1.17166i
\(347\) 8.86294 0.475787 0.237894 0.971291i \(-0.423543\pi\)
0.237894 + 0.971291i \(0.423543\pi\)
\(348\) 1.29590 0.0694674
\(349\) 2.78554i 0.149107i 0.997217 + 0.0745534i \(0.0237531\pi\)
−0.997217 + 0.0745534i \(0.976247\pi\)
\(350\) 1.69202 0.0904424
\(351\) 0 0
\(352\) 2.15883 0.115066
\(353\) − 21.0640i − 1.12112i −0.828113 0.560561i \(-0.810585\pi\)
0.828113 0.560561i \(-0.189415\pi\)
\(354\) −3.73125 −0.198314
\(355\) 4.18060 0.221883
\(356\) 2.41119i 0.127793i
\(357\) − 3.98792i − 0.211063i
\(358\) − 14.1293i − 0.746756i
\(359\) 17.3274i 0.914503i 0.889337 + 0.457251i \(0.151166\pi\)
−0.889337 + 0.457251i \(0.848834\pi\)
\(360\) 1.00000 0.0527046
\(361\) 18.9608 0.997935
\(362\) − 5.63773i − 0.296312i
\(363\) −6.33944 −0.332734
\(364\) 0 0
\(365\) 11.9366 0.624791
\(366\) − 6.32304i − 0.330511i
\(367\) −30.8049 −1.60800 −0.804002 0.594627i \(-0.797300\pi\)
−0.804002 + 0.594627i \(0.797300\pi\)
\(368\) 3.74094 0.195010
\(369\) 1.89977i 0.0988982i
\(370\) − 0.801938i − 0.0416908i
\(371\) 1.70841i 0.0886965i
\(372\) − 1.44504i − 0.0749219i
\(373\) −25.6668 −1.32898 −0.664488 0.747299i \(-0.731350\pi\)
−0.664488 + 0.747299i \(0.731350\pi\)
\(374\) −5.08815 −0.263102
\(375\) − 1.00000i − 0.0516398i
\(376\) −8.87800 −0.457848
\(377\) 0 0
\(378\) 1.69202 0.0870282
\(379\) − 5.46144i − 0.280535i −0.990114 0.140268i \(-0.955204\pi\)
0.990114 0.140268i \(-0.0447963\pi\)
\(380\) 0.198062 0.0101604
\(381\) 13.0097 0.666507
\(382\) 2.59419i 0.132730i
\(383\) 25.9221i 1.32456i 0.749257 + 0.662280i \(0.230411\pi\)
−0.749257 + 0.662280i \(0.769589\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) − 3.65279i − 0.186164i
\(386\) −3.98361 −0.202760
\(387\) −12.5429 −0.637590
\(388\) − 0.0881460i − 0.00447494i
\(389\) 32.0006 1.62249 0.811247 0.584703i \(-0.198789\pi\)
0.811247 + 0.584703i \(0.198789\pi\)
\(390\) 0 0
\(391\) −8.81700 −0.445895
\(392\) 4.13706i 0.208953i
\(393\) 7.82908 0.394925
\(394\) 6.65040 0.335042
\(395\) − 9.40581i − 0.473258i
\(396\) − 2.15883i − 0.108485i
\(397\) 0.457123i 0.0229424i 0.999934 + 0.0114712i \(0.00365147\pi\)
−0.999934 + 0.0114712i \(0.996349\pi\)
\(398\) − 1.01639i − 0.0509472i
\(399\) 0.335126 0.0167773
\(400\) −1.00000 −0.0500000
\(401\) 23.2553i 1.16132i 0.814147 + 0.580658i \(0.197205\pi\)
−0.814147 + 0.580658i \(0.802795\pi\)
\(402\) −7.56465 −0.377290
\(403\) 0 0
\(404\) −11.1099 −0.552739
\(405\) − 1.00000i − 0.0496904i
\(406\) −2.19269 −0.108821
\(407\) −1.73125 −0.0858149
\(408\) 2.35690i 0.116684i
\(409\) − 39.0629i − 1.93154i −0.259406 0.965768i \(-0.583527\pi\)
0.259406 0.965768i \(-0.416473\pi\)
\(410\) − 1.89977i − 0.0938231i
\(411\) − 5.36227i − 0.264501i
\(412\) 9.54825 0.470409
\(413\) 6.31336 0.310660
\(414\) − 3.74094i − 0.183857i
\(415\) −8.43296 −0.413958
\(416\) 0 0
\(417\) −3.12200 −0.152885
\(418\) − 0.427583i − 0.0209138i
\(419\) −18.1269 −0.885557 −0.442779 0.896631i \(-0.646007\pi\)
−0.442779 + 0.896631i \(0.646007\pi\)
\(420\) −1.69202 −0.0825622
\(421\) 20.4916i 0.998698i 0.866401 + 0.499349i \(0.166427\pi\)
−0.866401 + 0.499349i \(0.833573\pi\)
\(422\) − 10.1414i − 0.493674i
\(423\) 8.87800i 0.431663i
\(424\) − 1.00969i − 0.0490348i
\(425\) 2.35690 0.114326
\(426\) −4.18060 −0.202551
\(427\) 10.6987i 0.517748i
\(428\) −18.1468 −0.877156
\(429\) 0 0
\(430\) 12.5429 0.604871
\(431\) 22.2989i 1.07410i 0.843551 + 0.537050i \(0.180461\pi\)
−0.843551 + 0.537050i \(0.819539\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −7.47650 −0.359298 −0.179649 0.983731i \(-0.557496\pi\)
−0.179649 + 0.983731i \(0.557496\pi\)
\(434\) 2.44504i 0.117366i
\(435\) 1.29590i 0.0621335i
\(436\) − 2.24698i − 0.107611i
\(437\) − 0.740939i − 0.0354439i
\(438\) −11.9366 −0.570353
\(439\) 19.3924 0.925549 0.462774 0.886476i \(-0.346854\pi\)
0.462774 + 0.886476i \(0.346854\pi\)
\(440\) 2.15883i 0.102918i
\(441\) 4.13706 0.197003
\(442\) 0 0
\(443\) 10.0935 0.479558 0.239779 0.970828i \(-0.422925\pi\)
0.239779 + 0.970828i \(0.422925\pi\)
\(444\) 0.801938i 0.0380583i
\(445\) −2.41119 −0.114301
\(446\) 16.8388 0.797339
\(447\) 4.24160i 0.200621i
\(448\) 1.69202i 0.0799405i
\(449\) − 2.78448i − 0.131408i −0.997839 0.0657039i \(-0.979071\pi\)
0.997839 0.0657039i \(-0.0209293\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) −4.10129 −0.193122
\(452\) 11.8726 0.558441
\(453\) − 22.9148i − 1.07663i
\(454\) −18.3056 −0.859124
\(455\) 0 0
\(456\) −0.198062 −0.00927512
\(457\) 13.1056i 0.613054i 0.951862 + 0.306527i \(0.0991670\pi\)
−0.951862 + 0.306527i \(0.900833\pi\)
\(458\) 5.14244 0.240290
\(459\) 2.35690 0.110010
\(460\) 3.74094i 0.174422i
\(461\) − 16.2174i − 0.755321i −0.925944 0.377661i \(-0.876729\pi\)
0.925944 0.377661i \(-0.123271\pi\)
\(462\) 3.65279i 0.169943i
\(463\) 15.5415i 0.722277i 0.932512 + 0.361139i \(0.117612\pi\)
−0.932512 + 0.361139i \(0.882388\pi\)
\(464\) 1.29590 0.0601605
\(465\) 1.44504 0.0670122
\(466\) − 10.9215i − 0.505931i
\(467\) 10.1578 0.470045 0.235023 0.971990i \(-0.424484\pi\)
0.235023 + 0.971990i \(0.424484\pi\)
\(468\) 0 0
\(469\) 12.7995 0.591028
\(470\) − 8.87800i − 0.409512i
\(471\) −16.0707 −0.740498
\(472\) −3.73125 −0.171745
\(473\) − 27.0780i − 1.24505i
\(474\) 9.40581i 0.432023i
\(475\) 0.198062i 0.00908772i
\(476\) − 3.98792i − 0.182786i
\(477\) −1.00969 −0.0462305
\(478\) −19.9041 −0.910392
\(479\) 31.4263i 1.43590i 0.696094 + 0.717951i \(0.254920\pi\)
−0.696094 + 0.717951i \(0.745080\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 18.2228 0.830027
\(483\) 6.32975i 0.288014i
\(484\) −6.33944 −0.288156
\(485\) 0.0881460 0.00400250
\(486\) 1.00000i 0.0453609i
\(487\) 24.1323i 1.09354i 0.837284 + 0.546769i \(0.184142\pi\)
−0.837284 + 0.546769i \(0.815858\pi\)
\(488\) − 6.32304i − 0.286231i
\(489\) 0.488582i 0.0220945i
\(490\) −4.13706 −0.186893
\(491\) 19.1099 0.862418 0.431209 0.902252i \(-0.358087\pi\)
0.431209 + 0.902252i \(0.358087\pi\)
\(492\) 1.89977i 0.0856484i
\(493\) −3.05429 −0.137558
\(494\) 0 0
\(495\) 2.15883 0.0970324
\(496\) − 1.44504i − 0.0648843i
\(497\) 7.07367 0.317298
\(498\) 8.43296 0.377890
\(499\) 34.9724i 1.56558i 0.622287 + 0.782789i \(0.286204\pi\)
−0.622287 + 0.782789i \(0.713796\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 14.0610i − 0.628199i
\(502\) − 17.0030i − 0.758880i
\(503\) 29.5579 1.31792 0.658962 0.752176i \(-0.270996\pi\)
0.658962 + 0.752176i \(0.270996\pi\)
\(504\) 1.69202 0.0753686
\(505\) − 11.1099i − 0.494385i
\(506\) 8.07606 0.359025
\(507\) 0 0
\(508\) 13.0097 0.577212
\(509\) − 24.4838i − 1.08523i −0.839983 0.542613i \(-0.817435\pi\)
0.839983 0.542613i \(-0.182565\pi\)
\(510\) −2.35690 −0.104365
\(511\) 20.1970 0.893463
\(512\) − 1.00000i − 0.0441942i
\(513\) 0.198062i 0.00874466i
\(514\) − 7.65519i − 0.337656i
\(515\) 9.54825i 0.420746i
\(516\) −12.5429 −0.552169
\(517\) −19.1661 −0.842925
\(518\) − 1.35690i − 0.0596186i
\(519\) 21.7942 0.956658
\(520\) 0 0
\(521\) −33.8713 −1.48393 −0.741964 0.670439i \(-0.766106\pi\)
−0.741964 + 0.670439i \(0.766106\pi\)
\(522\) − 1.29590i − 0.0567199i
\(523\) −42.5870 −1.86220 −0.931100 0.364764i \(-0.881150\pi\)
−0.931100 + 0.364764i \(0.881150\pi\)
\(524\) 7.82908 0.342015
\(525\) − 1.69202i − 0.0738459i
\(526\) − 21.0411i − 0.917438i
\(527\) 3.40581i 0.148360i
\(528\) − 2.15883i − 0.0939512i
\(529\) −9.00538 −0.391538
\(530\) 1.00969 0.0438581
\(531\) 3.73125i 0.161922i
\(532\) 0.335126 0.0145295
\(533\) 0 0
\(534\) 2.41119 0.104342
\(535\) − 18.1468i − 0.784553i
\(536\) −7.56465 −0.326743
\(537\) −14.1293 −0.609724
\(538\) − 24.3183i − 1.04843i
\(539\) 8.93123i 0.384695i
\(540\) − 1.00000i − 0.0430331i
\(541\) − 3.23298i − 0.138997i −0.997582 0.0694983i \(-0.977860\pi\)
0.997582 0.0694983i \(-0.0221398\pi\)
\(542\) 10.9922 0.472157
\(543\) −5.63773 −0.241938
\(544\) 2.35690i 0.101051i
\(545\) 2.24698 0.0962500
\(546\) 0 0
\(547\) −5.51573 −0.235836 −0.117918 0.993023i \(-0.537622\pi\)
−0.117918 + 0.993023i \(0.537622\pi\)
\(548\) − 5.36227i − 0.229065i
\(549\) −6.32304 −0.269861
\(550\) −2.15883 −0.0920530
\(551\) − 0.256668i − 0.0109344i
\(552\) − 3.74094i − 0.159225i
\(553\) − 15.9148i − 0.676768i
\(554\) − 11.4601i − 0.486893i
\(555\) −0.801938 −0.0340404
\(556\) −3.12200 −0.132402
\(557\) 30.3502i 1.28598i 0.765875 + 0.642989i \(0.222306\pi\)
−0.765875 + 0.642989i \(0.777694\pi\)
\(558\) −1.44504 −0.0611735
\(559\) 0 0
\(560\) −1.69202 −0.0715010
\(561\) 5.08815i 0.214822i
\(562\) −9.88231 −0.416860
\(563\) −6.34375 −0.267357 −0.133679 0.991025i \(-0.542679\pi\)
−0.133679 + 0.991025i \(0.542679\pi\)
\(564\) 8.87800i 0.373831i
\(565\) 11.8726i 0.499485i
\(566\) − 21.3207i − 0.896174i
\(567\) − 1.69202i − 0.0710582i
\(568\) −4.18060 −0.175414
\(569\) −24.7275 −1.03663 −0.518316 0.855189i \(-0.673441\pi\)
−0.518316 + 0.855189i \(0.673441\pi\)
\(570\) − 0.198062i − 0.00829592i
\(571\) −13.6262 −0.570240 −0.285120 0.958492i \(-0.592033\pi\)
−0.285120 + 0.958492i \(0.592033\pi\)
\(572\) 0 0
\(573\) 2.59419 0.108374
\(574\) − 3.21446i − 0.134169i
\(575\) −3.74094 −0.156008
\(576\) −1.00000 −0.0416667
\(577\) 16.2828i 0.677860i 0.940812 + 0.338930i \(0.110065\pi\)
−0.940812 + 0.338930i \(0.889935\pi\)
\(578\) 11.4450i 0.476051i
\(579\) 3.98361i 0.165553i
\(580\) 1.29590i 0.0538092i
\(581\) −14.2687 −0.591967
\(582\) −0.0881460 −0.00365377
\(583\) − 2.17975i − 0.0902760i
\(584\) −11.9366 −0.493940
\(585\) 0 0
\(586\) −18.7090 −0.772862
\(587\) 11.0344i 0.455440i 0.973727 + 0.227720i \(0.0731271\pi\)
−0.973727 + 0.227720i \(0.926873\pi\)
\(588\) 4.13706 0.170610
\(589\) −0.286208 −0.0117930
\(590\) − 3.73125i − 0.153613i
\(591\) − 6.65040i − 0.273561i
\(592\) 0.801938i 0.0329594i
\(593\) − 25.4905i − 1.04677i −0.852096 0.523385i \(-0.824669\pi\)
0.852096 0.523385i \(-0.175331\pi\)
\(594\) −2.15883 −0.0885780
\(595\) 3.98792 0.163489
\(596\) 4.24160i 0.173743i
\(597\) −1.01639 −0.0415982
\(598\) 0 0
\(599\) −0.501729 −0.0205001 −0.0102500 0.999947i \(-0.503263\pi\)
−0.0102500 + 0.999947i \(0.503263\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −18.3744 −0.749506 −0.374753 0.927125i \(-0.622272\pi\)
−0.374753 + 0.927125i \(0.622272\pi\)
\(602\) 21.2228 0.864977
\(603\) 7.56465i 0.308056i
\(604\) − 22.9148i − 0.932392i
\(605\) − 6.33944i − 0.257735i
\(606\) 11.1099i 0.451309i
\(607\) −17.1395 −0.695669 −0.347835 0.937556i \(-0.613083\pi\)
−0.347835 + 0.937556i \(0.613083\pi\)
\(608\) −0.198062 −0.00803249
\(609\) 2.19269i 0.0888521i
\(610\) 6.32304 0.256013
\(611\) 0 0
\(612\) 2.35690 0.0952719
\(613\) − 1.15751i − 0.0467512i −0.999727 0.0233756i \(-0.992559\pi\)
0.999727 0.0233756i \(-0.00744136\pi\)
\(614\) 17.2241 0.695110
\(615\) −1.89977 −0.0766062
\(616\) 3.65279i 0.147175i
\(617\) − 7.71618i − 0.310642i −0.987864 0.155321i \(-0.950359\pi\)
0.987864 0.155321i \(-0.0496412\pi\)
\(618\) − 9.54825i − 0.384087i
\(619\) 16.1142i 0.647686i 0.946111 + 0.323843i \(0.104975\pi\)
−0.946111 + 0.323843i \(0.895025\pi\)
\(620\) 1.44504 0.0580343
\(621\) −3.74094 −0.150119
\(622\) − 13.5200i − 0.542104i
\(623\) −4.07979 −0.163453
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 24.3303i − 0.972436i
\(627\) −0.427583 −0.0170760
\(628\) −16.0707 −0.641290
\(629\) − 1.89008i − 0.0753626i
\(630\) 1.69202i 0.0674117i
\(631\) − 7.77586i − 0.309552i −0.987950 0.154776i \(-0.950534\pi\)
0.987950 0.154776i \(-0.0494656\pi\)
\(632\) 9.40581i 0.374143i
\(633\) −10.1414 −0.403083
\(634\) −34.7821 −1.38137
\(635\) 13.0097i 0.516274i
\(636\) −1.00969 −0.0400368
\(637\) 0 0
\(638\) 2.79763 0.110759
\(639\) 4.18060i 0.165382i
\(640\) 1.00000 0.0395285
\(641\) 35.1293 1.38752 0.693762 0.720204i \(-0.255952\pi\)
0.693762 + 0.720204i \(0.255952\pi\)
\(642\) 18.1468i 0.716195i
\(643\) 14.5459i 0.573633i 0.957986 + 0.286816i \(0.0925970\pi\)
−0.957986 + 0.286816i \(0.907403\pi\)
\(644\) 6.32975i 0.249427i
\(645\) − 12.5429i − 0.493875i
\(646\) 0.466812 0.0183665
\(647\) −36.2234 −1.42409 −0.712045 0.702134i \(-0.752231\pi\)
−0.712045 + 0.702134i \(0.752231\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −8.05515 −0.316192
\(650\) 0 0
\(651\) 2.44504 0.0958287
\(652\) 0.488582i 0.0191344i
\(653\) 19.4926 0.762806 0.381403 0.924409i \(-0.375441\pi\)
0.381403 + 0.924409i \(0.375441\pi\)
\(654\) −2.24698 −0.0878639
\(655\) 7.82908i 0.305908i
\(656\) 1.89977i 0.0741737i
\(657\) 11.9366i 0.465691i
\(658\) − 15.0218i − 0.585610i
\(659\) 38.3400 1.49352 0.746758 0.665096i \(-0.231609\pi\)
0.746758 + 0.665096i \(0.231609\pi\)
\(660\) 2.15883 0.0840325
\(661\) 13.5985i 0.528920i 0.964397 + 0.264460i \(0.0851938\pi\)
−0.964397 + 0.264460i \(0.914806\pi\)
\(662\) 9.66919 0.375804
\(663\) 0 0
\(664\) 8.43296 0.327262
\(665\) 0.335126i 0.0129956i
\(666\) 0.801938 0.0310745
\(667\) 4.84787 0.187710
\(668\) − 14.0610i − 0.544036i
\(669\) − 16.8388i − 0.651025i
\(670\) − 7.56465i − 0.292248i
\(671\) − 13.6504i − 0.526968i
\(672\) 1.69202 0.0652711
\(673\) 49.7090 1.91614 0.958071 0.286532i \(-0.0925025\pi\)
0.958071 + 0.286532i \(0.0925025\pi\)
\(674\) − 17.3612i − 0.668729i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −50.2573 −1.93154 −0.965772 0.259394i \(-0.916477\pi\)
−0.965772 + 0.259394i \(0.916477\pi\)
\(678\) − 11.8726i − 0.455965i
\(679\) 0.149145 0.00572366
\(680\) −2.35690 −0.0903828
\(681\) 18.3056i 0.701472i
\(682\) − 3.11960i − 0.119456i
\(683\) − 21.5415i − 0.824264i −0.911124 0.412132i \(-0.864784\pi\)
0.911124 0.412132i \(-0.135216\pi\)
\(684\) 0.198062i 0.00757310i
\(685\) 5.36227 0.204882
\(686\) −18.8442 −0.719473
\(687\) − 5.14244i − 0.196196i
\(688\) −12.5429 −0.478193
\(689\) 0 0
\(690\) 3.74094 0.142415
\(691\) 50.8165i 1.93315i 0.256380 + 0.966576i \(0.417470\pi\)
−0.256380 + 0.966576i \(0.582530\pi\)
\(692\) 21.7942 0.828490
\(693\) 3.65279 0.138758
\(694\) − 8.86294i − 0.336432i
\(695\) − 3.12200i − 0.118424i
\(696\) − 1.29590i − 0.0491208i
\(697\) − 4.47757i − 0.169600i
\(698\) 2.78554 0.105434
\(699\) −10.9215 −0.413091
\(700\) − 1.69202i − 0.0639524i
\(701\) 13.2024 0.498647 0.249323 0.968420i \(-0.419792\pi\)
0.249323 + 0.968420i \(0.419792\pi\)
\(702\) 0 0
\(703\) 0.158834 0.00599052
\(704\) − 2.15883i − 0.0813641i
\(705\) −8.87800 −0.334365
\(706\) −21.0640 −0.792753
\(707\) − 18.7982i − 0.706980i
\(708\) 3.73125i 0.140229i
\(709\) 0.430567i 0.0161703i 0.999967 + 0.00808515i \(0.00257361\pi\)
−0.999967 + 0.00808515i \(0.997426\pi\)
\(710\) − 4.18060i − 0.156895i
\(711\) 9.40581 0.352746
\(712\) 2.41119 0.0903632
\(713\) − 5.40581i − 0.202449i
\(714\) −3.98792 −0.149244
\(715\) 0 0
\(716\) −14.1293 −0.528036
\(717\) 19.9041i 0.743332i
\(718\) 17.3274 0.646651
\(719\) −8.24459 −0.307471 −0.153736 0.988112i \(-0.549130\pi\)
−0.153736 + 0.988112i \(0.549130\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) 16.1558i 0.601675i
\(722\) − 18.9608i − 0.705647i
\(723\) − 18.2228i − 0.677714i
\(724\) −5.63773 −0.209524
\(725\) −1.29590 −0.0481284
\(726\) 6.33944i 0.235279i
\(727\) −21.2664 −0.788726 −0.394363 0.918955i \(-0.629035\pi\)
−0.394363 + 0.918955i \(0.629035\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 11.9366i − 0.441794i
\(731\) 29.5623 1.09340
\(732\) −6.32304 −0.233706
\(733\) − 38.9178i − 1.43746i −0.695288 0.718731i \(-0.744723\pi\)
0.695288 0.718731i \(-0.255277\pi\)
\(734\) 30.8049i 1.13703i
\(735\) 4.13706i 0.152598i
\(736\) − 3.74094i − 0.137893i
\(737\) −16.3308 −0.601553
\(738\) 1.89977 0.0699316
\(739\) 4.48486i 0.164978i 0.996592 + 0.0824891i \(0.0262870\pi\)
−0.996592 + 0.0824891i \(0.973713\pi\)
\(740\) −0.801938 −0.0294798
\(741\) 0 0
\(742\) 1.70841 0.0627179
\(743\) − 26.8558i − 0.985242i −0.870244 0.492621i \(-0.836039\pi\)
0.870244 0.492621i \(-0.163961\pi\)
\(744\) −1.44504 −0.0529778
\(745\) −4.24160 −0.155400
\(746\) 25.6668i 0.939728i
\(747\) − 8.43296i − 0.308546i
\(748\) 5.08815i 0.186041i
\(749\) − 30.7047i − 1.12193i
\(750\) −1.00000 −0.0365148
\(751\) 5.64012 0.205811 0.102905 0.994691i \(-0.467186\pi\)
0.102905 + 0.994691i \(0.467186\pi\)
\(752\) 8.87800i 0.323747i
\(753\) −17.0030 −0.619623
\(754\) 0 0
\(755\) 22.9148 0.833956
\(756\) − 1.69202i − 0.0615382i
\(757\) −35.6644 −1.29624 −0.648122 0.761536i \(-0.724445\pi\)
−0.648122 + 0.761536i \(0.724445\pi\)
\(758\) −5.46144 −0.198368
\(759\) − 8.07606i − 0.293143i
\(760\) − 0.198062i − 0.00718447i
\(761\) 2.68425i 0.0973041i 0.998816 + 0.0486520i \(0.0154925\pi\)
−0.998816 + 0.0486520i \(0.984507\pi\)
\(762\) − 13.0097i − 0.471291i
\(763\) 3.80194 0.137639
\(764\) 2.59419 0.0938544
\(765\) 2.35690i 0.0852137i
\(766\) 25.9221 0.936605
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) − 48.7644i − 1.75849i −0.476372 0.879244i \(-0.658048\pi\)
0.476372 0.879244i \(-0.341952\pi\)
\(770\) −3.65279 −0.131638
\(771\) −7.65519 −0.275695
\(772\) 3.98361i 0.143373i
\(773\) 46.6152i 1.67663i 0.545184 + 0.838316i \(0.316460\pi\)
−0.545184 + 0.838316i \(0.683540\pi\)
\(774\) 12.5429i 0.450844i
\(775\) 1.44504i 0.0519074i
\(776\) −0.0881460 −0.00316426
\(777\) −1.35690 −0.0486784
\(778\) − 32.0006i − 1.14728i
\(779\) 0.376273 0.0134814
\(780\) 0 0
\(781\) −9.02523 −0.322948
\(782\) 8.81700i 0.315295i
\(783\) −1.29590 −0.0463116
\(784\) 4.13706 0.147752
\(785\) − 16.0707i − 0.573587i
\(786\) − 7.82908i − 0.279254i
\(787\) − 27.4045i − 0.976864i −0.872602 0.488432i \(-0.837569\pi\)
0.872602 0.488432i \(-0.162431\pi\)
\(788\) − 6.65040i − 0.236911i
\(789\) −21.0411 −0.749085
\(790\) −9.40581 −0.334644
\(791\) 20.0887i 0.714273i
\(792\) −2.15883 −0.0767108
\(793\) 0 0
\(794\) 0.457123 0.0162227
\(795\) − 1.00969i − 0.0358100i
\(796\) −1.01639 −0.0360251
\(797\) 5.70278 0.202003 0.101001 0.994886i \(-0.467795\pi\)
0.101001 + 0.994886i \(0.467795\pi\)
\(798\) − 0.335126i − 0.0118633i
\(799\) − 20.9245i − 0.740257i
\(800\) 1.00000i 0.0353553i
\(801\) − 2.41119i − 0.0851952i
\(802\) 23.2553 0.821175
\(803\) −25.7692 −0.909374
\(804\) 7.56465i 0.266785i
\(805\) −6.32975 −0.223094
\(806\) 0 0
\(807\) −24.3183 −0.856043
\(808\) 11.1099i 0.390845i
\(809\) −29.6165 −1.04126 −0.520631 0.853782i \(-0.674303\pi\)
−0.520631 + 0.853782i \(0.674303\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 48.8901i 1.71676i 0.513012 + 0.858382i \(0.328530\pi\)
−0.513012 + 0.858382i \(0.671470\pi\)
\(812\) 2.19269i 0.0769482i
\(813\) − 10.9922i − 0.385514i
\(814\) 1.73125i 0.0606803i
\(815\) −0.488582 −0.0171143
\(816\) 2.35690 0.0825079
\(817\) 2.48427i 0.0869136i
\(818\) −39.0629 −1.36580
\(819\) 0 0
\(820\) −1.89977 −0.0663429
\(821\) 19.7942i 0.690821i 0.938452 + 0.345411i \(0.112260\pi\)
−0.938452 + 0.345411i \(0.887740\pi\)
\(822\) −5.36227 −0.187031
\(823\) −49.6118 −1.72936 −0.864679 0.502325i \(-0.832478\pi\)
−0.864679 + 0.502325i \(0.832478\pi\)
\(824\) − 9.54825i − 0.332629i
\(825\) 2.15883i 0.0751609i
\(826\) − 6.31336i − 0.219670i
\(827\) 26.0683i 0.906483i 0.891388 + 0.453242i \(0.149733\pi\)
−0.891388 + 0.453242i \(0.850267\pi\)
\(828\) −3.74094 −0.130007
\(829\) −46.2215 −1.60534 −0.802669 0.596424i \(-0.796588\pi\)
−0.802669 + 0.596424i \(0.796588\pi\)
\(830\) 8.43296i 0.292712i
\(831\) −11.4601 −0.397547
\(832\) 0 0
\(833\) −9.75063 −0.337839
\(834\) 3.12200i 0.108106i
\(835\) 14.0610 0.486601
\(836\) −0.427583 −0.0147883
\(837\) 1.44504i 0.0499480i
\(838\) 18.1269i 0.626183i
\(839\) 33.2693i 1.14859i 0.818650 + 0.574293i \(0.194723\pi\)
−0.818650 + 0.574293i \(0.805277\pi\)
\(840\) 1.69202i 0.0583803i
\(841\) −27.3207 −0.942091
\(842\) 20.4916 0.706186
\(843\) 9.88231i 0.340365i
\(844\) −10.1414 −0.349081
\(845\) 0 0
\(846\) 8.87800 0.305232
\(847\) − 10.7265i − 0.368566i
\(848\) −1.00969 −0.0346729
\(849\) −21.3207 −0.731723
\(850\) − 2.35690i − 0.0808409i
\(851\) 3.00000i 0.102839i
\(852\) 4.18060i 0.143225i
\(853\) − 39.3002i − 1.34561i −0.739818 0.672807i \(-0.765089\pi\)
0.739818 0.672807i \(-0.234911\pi\)
\(854\) 10.6987 0.366103
\(855\) −0.198062 −0.00677359
\(856\) 18.1468i 0.620243i
\(857\) 39.7566 1.35806 0.679030 0.734111i \(-0.262401\pi\)
0.679030 + 0.734111i \(0.262401\pi\)
\(858\) 0 0
\(859\) −3.43860 −0.117324 −0.0586618 0.998278i \(-0.518683\pi\)
−0.0586618 + 0.998278i \(0.518683\pi\)
\(860\) − 12.5429i − 0.427709i
\(861\) −3.21446 −0.109548
\(862\) 22.2989 0.759503
\(863\) − 21.2301i − 0.722681i −0.932434 0.361341i \(-0.882319\pi\)
0.932434 0.361341i \(-0.117681\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 21.7942i 0.741024i
\(866\) 7.47650i 0.254062i
\(867\) 11.4450 0.388694
\(868\) 2.44504 0.0829901
\(869\) 20.3056i 0.688820i
\(870\) 1.29590 0.0439350
\(871\) 0 0
\(872\) −2.24698 −0.0760923
\(873\) 0.0881460i 0.00298329i
\(874\) −0.740939 −0.0250626
\(875\) 1.69202 0.0572008
\(876\) 11.9366i 0.403301i
\(877\) 5.74227i 0.193903i 0.995289 + 0.0969513i \(0.0309091\pi\)
−0.995289 + 0.0969513i \(0.969091\pi\)
\(878\) − 19.3924i − 0.654462i
\(879\) 18.7090i 0.631039i
\(880\) 2.15883 0.0727743
\(881\) −12.6069 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(882\) − 4.13706i − 0.139302i
\(883\) −25.8086 −0.868530 −0.434265 0.900785i \(-0.642992\pi\)
−0.434265 + 0.900785i \(0.642992\pi\)
\(884\) 0 0
\(885\) −3.73125 −0.125425
\(886\) − 10.0935i − 0.339099i
\(887\) −20.2392 −0.679566 −0.339783 0.940504i \(-0.610354\pi\)
−0.339783 + 0.940504i \(0.610354\pi\)
\(888\) 0.801938 0.0269113
\(889\) 22.0127i 0.738281i
\(890\) 2.41119i 0.0808233i
\(891\) 2.15883i 0.0723236i
\(892\) − 16.8388i − 0.563804i
\(893\) 1.75840 0.0588425
\(894\) 4.24160 0.141860
\(895\) − 14.1293i − 0.472290i
\(896\) 1.69202 0.0565265
\(897\) 0 0
\(898\) −2.78448 −0.0929193
\(899\) − 1.87263i − 0.0624556i
\(900\) 1.00000 0.0333333
\(901\) 2.37973 0.0792803
\(902\) 4.10129i 0.136558i
\(903\) − 21.2228i − 0.706251i
\(904\) − 11.8726i − 0.394878i
\(905\) − 5.63773i − 0.187404i
\(906\) −22.9148 −0.761294
\(907\) 3.52350 0.116996 0.0584979 0.998288i \(-0.481369\pi\)
0.0584979 + 0.998288i \(0.481369\pi\)
\(908\) 18.3056i 0.607492i
\(909\) 11.1099 0.368493
\(910\) 0 0
\(911\) −34.7426 −1.15107 −0.575537 0.817776i \(-0.695207\pi\)
−0.575537 + 0.817776i \(0.695207\pi\)
\(912\) 0.198062i 0.00655850i
\(913\) 18.2054 0.602509
\(914\) 13.1056 0.433495
\(915\) − 6.32304i − 0.209033i
\(916\) − 5.14244i − 0.169911i
\(917\) 13.2470i 0.437454i
\(918\) − 2.35690i − 0.0777892i
\(919\) −24.1430 −0.796405 −0.398203 0.917298i \(-0.630366\pi\)
−0.398203 + 0.917298i \(0.630366\pi\)
\(920\) 3.74094 0.123335
\(921\) − 17.2241i − 0.567555i
\(922\) −16.2174 −0.534093
\(923\) 0 0
\(924\) 3.65279 0.120168
\(925\) − 0.801938i − 0.0263676i
\(926\) 15.5415 0.510727
\(927\) −9.54825 −0.313606
\(928\) − 1.29590i − 0.0425399i
\(929\) 56.9700i 1.86912i 0.355800 + 0.934562i \(0.384209\pi\)
−0.355800 + 0.934562i \(0.615791\pi\)
\(930\) − 1.44504i − 0.0473848i
\(931\) − 0.819396i − 0.0268546i
\(932\) −10.9215 −0.357747
\(933\) −13.5200 −0.442626
\(934\) − 10.1578i − 0.332372i
\(935\) −5.08815 −0.166400
\(936\) 0 0
\(937\) −45.7966 −1.49611 −0.748054 0.663638i \(-0.769012\pi\)
−0.748054 + 0.663638i \(0.769012\pi\)
\(938\) − 12.7995i − 0.417920i
\(939\) −24.3303 −0.793991
\(940\) −8.87800 −0.289569
\(941\) 7.10262i 0.231539i 0.993276 + 0.115769i \(0.0369334\pi\)
−0.993276 + 0.115769i \(0.963067\pi\)
\(942\) 16.0707i 0.523611i
\(943\) 7.10693i 0.231434i
\(944\) 3.73125i 0.121442i
\(945\) 1.69202 0.0550415
\(946\) −27.0780 −0.880381
\(947\) − 11.2524i − 0.365652i −0.983145 0.182826i \(-0.941475\pi\)
0.983145 0.182826i \(-0.0585246\pi\)
\(948\) 9.40581 0.305487
\(949\) 0 0
\(950\) 0.198062 0.00642599
\(951\) 34.7821i 1.12789i
\(952\) −3.98792 −0.129249
\(953\) 1.61702 0.0523805 0.0261902 0.999657i \(-0.491662\pi\)
0.0261902 + 0.999657i \(0.491662\pi\)
\(954\) 1.00969i 0.0326899i
\(955\) 2.59419i 0.0839459i
\(956\) 19.9041i 0.643744i
\(957\) − 2.79763i − 0.0904344i
\(958\) 31.4263 1.01534
\(959\) 9.07308 0.292985
\(960\) − 1.00000i − 0.0322749i
\(961\) 28.9119 0.932640
\(962\) 0 0
\(963\) 18.1468 0.584771
\(964\) − 18.2228i − 0.586917i
\(965\) −3.98361 −0.128237
\(966\) 6.32975 0.203656
\(967\) 24.1588i 0.776896i 0.921471 + 0.388448i \(0.126989\pi\)
−0.921471 + 0.388448i \(0.873011\pi\)
\(968\) 6.33944i 0.203757i
\(969\) − 0.466812i − 0.0149962i
\(970\) − 0.0881460i − 0.00283020i
\(971\) 43.2669 1.38850 0.694251 0.719733i \(-0.255736\pi\)
0.694251 + 0.719733i \(0.255736\pi\)
\(972\) 1.00000 0.0320750
\(973\) − 5.28249i − 0.169349i
\(974\) 24.1323 0.773248
\(975\) 0 0
\(976\) −6.32304 −0.202396
\(977\) − 24.9782i − 0.799124i −0.916706 0.399562i \(-0.869162\pi\)
0.916706 0.399562i \(-0.130838\pi\)
\(978\) 0.488582 0.0156231
\(979\) 5.20536 0.166364
\(980\) 4.13706i 0.132154i
\(981\) 2.24698i 0.0717405i
\(982\) − 19.1099i − 0.609822i
\(983\) − 19.3134i − 0.616000i −0.951386 0.308000i \(-0.900340\pi\)
0.951386 0.308000i \(-0.0996597\pi\)
\(984\) 1.89977 0.0605625
\(985\) 6.65040 0.211899
\(986\) 3.05429i 0.0972685i
\(987\) −15.0218 −0.478148
\(988\) 0 0
\(989\) −46.9221 −1.49204
\(990\) − 2.15883i − 0.0686122i
\(991\) −27.5090 −0.873853 −0.436926 0.899497i \(-0.643933\pi\)
−0.436926 + 0.899497i \(0.643933\pi\)
\(992\) −1.44504 −0.0458801
\(993\) − 9.66919i − 0.306842i
\(994\) − 7.07367i − 0.224363i
\(995\) − 1.01639i − 0.0322218i
\(996\) − 8.43296i − 0.267209i
\(997\) −23.1933 −0.734538 −0.367269 0.930115i \(-0.619707\pi\)
−0.367269 + 0.930115i \(0.619707\pi\)
\(998\) 34.9724 1.10703
\(999\) − 0.801938i − 0.0253722i
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.1351.1 6
13.5 odd 4 5070.2.a.bj.1.3 3
13.8 odd 4 5070.2.a.bu.1.1 yes 3
13.12 even 2 inner 5070.2.b.t.1351.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bj.1.3 3 13.5 odd 4
5070.2.a.bu.1.1 yes 3 13.8 odd 4
5070.2.b.t.1351.1 6 1.1 even 1 trivial
5070.2.b.t.1351.6 6 13.12 even 2 inner