Properties

Label 5070.2.b.s.1351.5
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 1351.5
Root \(1.80194i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.s.1351.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +1.35690i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +1.35690i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.19806i q^{11} +1.00000 q^{12} -1.35690 q^{14} -1.00000i q^{15} +1.00000 q^{16} +2.04892 q^{17} +1.00000i q^{18} +5.34481i q^{19} -1.00000i q^{20} -1.35690i q^{21} -3.19806 q^{22} +8.32304 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -1.35690i q^{28} +8.07069 q^{29} +1.00000 q^{30} +0.185981i q^{31} +1.00000i q^{32} -3.19806i q^{33} +2.04892i q^{34} -1.35690 q^{35} -1.00000 q^{36} -2.46681i q^{37} -5.34481 q^{38} +1.00000 q^{40} -12.2349i q^{41} +1.35690 q^{42} +2.91185 q^{43} -3.19806i q^{44} +1.00000i q^{45} +8.32304i q^{46} +13.0978i q^{47} -1.00000 q^{48} +5.15883 q^{49} -1.00000i q^{50} -2.04892 q^{51} +5.74094 q^{53} -1.00000i q^{54} -3.19806 q^{55} +1.35690 q^{56} -5.34481i q^{57} +8.07069i q^{58} -1.62565i q^{59} +1.00000i q^{60} +5.28382 q^{61} -0.185981 q^{62} +1.35690i q^{63} -1.00000 q^{64} +3.19806 q^{66} -1.55496i q^{67} -2.04892 q^{68} -8.32304 q^{69} -1.35690i q^{70} +2.32304i q^{71} -1.00000i q^{72} -5.11529i q^{73} +2.46681 q^{74} +1.00000 q^{75} -5.34481i q^{76} -4.33944 q^{77} -1.42327 q^{79} +1.00000i q^{80} +1.00000 q^{81} +12.2349 q^{82} -11.1196i q^{83} +1.35690i q^{84} +2.04892i q^{85} +2.91185i q^{86} -8.07069 q^{87} +3.19806 q^{88} +3.37867i q^{89} -1.00000 q^{90} -8.32304 q^{92} -0.185981i q^{93} -13.0978 q^{94} -5.34481 q^{95} -1.00000i q^{96} +4.74094i q^{97} +5.15883i q^{98} +3.19806i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} - 6q^{10} + 6q^{12} + 6q^{16} - 6q^{17} - 28q^{22} + 10q^{23} - 6q^{25} - 6q^{27} + 24q^{29} + 6q^{30} - 6q^{36} + 14q^{38} + 6q^{40} + 10q^{43} - 6q^{48} + 14q^{49} + 6q^{51} + 6q^{53} - 28q^{55} - 34q^{61} + 28q^{62} - 6q^{64} + 28q^{66} + 6q^{68} - 10q^{69} + 8q^{74} + 6q^{75} + 14q^{77} - 14q^{79} + 6q^{81} + 26q^{82} - 24q^{87} + 28q^{88} - 6q^{90} - 10q^{92} - 42q^{94} + 14q^{95} + O(q^{100}) \)

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
\(7\) 1.35690i 0.512858i 0.966563 + 0.256429i \(0.0825461\pi\)
−0.966563 + 0.256429i \(0.917454\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 3.19806i 0.964252i 0.876102 + 0.482126i \(0.160135\pi\)
−0.876102 + 0.482126i \(0.839865\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.35690 −0.362646
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) 2.04892 0.496935 0.248468 0.968640i \(-0.420073\pi\)
0.248468 + 0.968640i \(0.420073\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 5.34481i 1.22618i 0.790011 + 0.613092i \(0.210075\pi\)
−0.790011 + 0.613092i \(0.789925\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) − 1.35690i − 0.296099i
\(22\) −3.19806 −0.681829
\(23\) 8.32304 1.73547 0.867737 0.497023i \(-0.165574\pi\)
0.867737 + 0.497023i \(0.165574\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) − 1.35690i − 0.256429i
\(29\) 8.07069 1.49869 0.749345 0.662180i \(-0.230369\pi\)
0.749345 + 0.662180i \(0.230369\pi\)
\(30\) 1.00000 0.182574
\(31\) 0.185981i 0.0334031i 0.999861 + 0.0167016i \(0.00531652\pi\)
−0.999861 + 0.0167016i \(0.994683\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 3.19806i − 0.556711i
\(34\) 2.04892i 0.351386i
\(35\) −1.35690 −0.229357
\(36\) −1.00000 −0.166667
\(37\) − 2.46681i − 0.405541i −0.979226 0.202771i \(-0.935005\pi\)
0.979226 0.202771i \(-0.0649946\pi\)
\(38\) −5.34481 −0.867043
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) − 12.2349i − 1.91077i −0.295363 0.955385i \(-0.595441\pi\)
0.295363 0.955385i \(-0.404559\pi\)
\(42\) 1.35690 0.209374
\(43\) 2.91185 0.444054 0.222027 0.975041i \(-0.428733\pi\)
0.222027 + 0.975041i \(0.428733\pi\)
\(44\) − 3.19806i − 0.482126i
\(45\) 1.00000i 0.149071i
\(46\) 8.32304i 1.22717i
\(47\) 13.0978i 1.91052i 0.295775 + 0.955258i \(0.404422\pi\)
−0.295775 + 0.955258i \(0.595578\pi\)
\(48\) −1.00000 −0.144338
\(49\) 5.15883 0.736976
\(50\) − 1.00000i − 0.141421i
\(51\) −2.04892 −0.286906
\(52\) 0 0
\(53\) 5.74094 0.788579 0.394289 0.918986i \(-0.370991\pi\)
0.394289 + 0.918986i \(0.370991\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −3.19806 −0.431227
\(56\) 1.35690 0.181323
\(57\) − 5.34481i − 0.707938i
\(58\) 8.07069i 1.05973i
\(59\) − 1.62565i − 0.211641i −0.994385 0.105821i \(-0.966253\pi\)
0.994385 0.105821i \(-0.0337469\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 5.28382 0.676523 0.338262 0.941052i \(-0.390161\pi\)
0.338262 + 0.941052i \(0.390161\pi\)
\(62\) −0.185981 −0.0236196
\(63\) 1.35690i 0.170953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.19806 0.393654
\(67\) − 1.55496i − 0.189968i −0.995479 0.0949842i \(-0.969720\pi\)
0.995479 0.0949842i \(-0.0302801\pi\)
\(68\) −2.04892 −0.248468
\(69\) −8.32304 −1.00198
\(70\) − 1.35690i − 0.162180i
\(71\) 2.32304i 0.275695i 0.990453 + 0.137847i \(0.0440183\pi\)
−0.990453 + 0.137847i \(0.955982\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 5.11529i − 0.598700i −0.954143 0.299350i \(-0.903230\pi\)
0.954143 0.299350i \(-0.0967698\pi\)
\(74\) 2.46681 0.286761
\(75\) 1.00000 0.115470
\(76\) − 5.34481i − 0.613092i
\(77\) −4.33944 −0.494525
\(78\) 0 0
\(79\) −1.42327 −0.160131 −0.0800653 0.996790i \(-0.525513\pi\)
−0.0800653 + 0.996790i \(0.525513\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 12.2349 1.35112
\(83\) − 11.1196i − 1.22054i −0.792195 0.610268i \(-0.791062\pi\)
0.792195 0.610268i \(-0.208938\pi\)
\(84\) 1.35690i 0.148049i
\(85\) 2.04892i 0.222236i
\(86\) 2.91185i 0.313993i
\(87\) −8.07069 −0.865269
\(88\) 3.19806 0.340915
\(89\) 3.37867i 0.358138i 0.983837 + 0.179069i \(0.0573085\pi\)
−0.983837 + 0.179069i \(0.942691\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −8.32304 −0.867737
\(93\) − 0.185981i − 0.0192853i
\(94\) −13.0978 −1.35094
\(95\) −5.34481 −0.548366
\(96\) − 1.00000i − 0.102062i
\(97\) 4.74094i 0.481369i 0.970603 + 0.240685i \(0.0773720\pi\)
−0.970603 + 0.240685i \(0.922628\pi\)
\(98\) 5.15883i 0.521121i
\(99\) 3.19806i 0.321417i
\(100\) 1.00000 0.100000
\(101\) 12.5918 1.25293 0.626465 0.779449i \(-0.284501\pi\)
0.626465 + 0.779449i \(0.284501\pi\)
\(102\) − 2.04892i − 0.202873i
\(103\) 9.00000 0.886796 0.443398 0.896325i \(-0.353773\pi\)
0.443398 + 0.896325i \(0.353773\pi\)
\(104\) 0 0
\(105\) 1.35690 0.132419
\(106\) 5.74094i 0.557609i
\(107\) −5.08815 −0.491890 −0.245945 0.969284i \(-0.579098\pi\)
−0.245945 + 0.969284i \(0.579098\pi\)
\(108\) 1.00000 0.0962250
\(109\) − 6.43296i − 0.616166i −0.951360 0.308083i \(-0.900313\pi\)
0.951360 0.308083i \(-0.0996874\pi\)
\(110\) − 3.19806i − 0.304923i
\(111\) 2.46681i 0.234139i
\(112\) 1.35690i 0.128215i
\(113\) −18.6571 −1.75511 −0.877556 0.479473i \(-0.840828\pi\)
−0.877556 + 0.479473i \(0.840828\pi\)
\(114\) 5.34481 0.500588
\(115\) 8.32304i 0.776128i
\(116\) −8.07069 −0.749345
\(117\) 0 0
\(118\) 1.62565 0.149653
\(119\) 2.78017i 0.254858i
\(120\) −1.00000 −0.0912871
\(121\) 0.772398 0.0702180
\(122\) 5.28382i 0.478374i
\(123\) 12.2349i 1.10318i
\(124\) − 0.185981i − 0.0167016i
\(125\) − 1.00000i − 0.0894427i
\(126\) −1.35690 −0.120882
\(127\) 16.3763 1.45316 0.726580 0.687082i \(-0.241109\pi\)
0.726580 + 0.687082i \(0.241109\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −2.91185 −0.256374
\(130\) 0 0
\(131\) 3.00969 0.262958 0.131479 0.991319i \(-0.458027\pi\)
0.131479 + 0.991319i \(0.458027\pi\)
\(132\) 3.19806i 0.278356i
\(133\) −7.25236 −0.628859
\(134\) 1.55496 0.134328
\(135\) − 1.00000i − 0.0860663i
\(136\) − 2.04892i − 0.175693i
\(137\) 20.2892i 1.73342i 0.498810 + 0.866711i \(0.333771\pi\)
−0.498810 + 0.866711i \(0.666229\pi\)
\(138\) − 8.32304i − 0.708505i
\(139\) −0.153457 −0.0130160 −0.00650802 0.999979i \(-0.502072\pi\)
−0.00650802 + 0.999979i \(0.502072\pi\)
\(140\) 1.35690 0.114679
\(141\) − 13.0978i − 1.10304i
\(142\) −2.32304 −0.194946
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 8.07069i 0.670234i
\(146\) 5.11529 0.423345
\(147\) −5.15883 −0.425493
\(148\) 2.46681i 0.202771i
\(149\) 10.7724i 0.882509i 0.897382 + 0.441255i \(0.145466\pi\)
−0.897382 + 0.441255i \(0.854534\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 17.3817i 1.41450i 0.706964 + 0.707249i \(0.250064\pi\)
−0.706964 + 0.707249i \(0.749936\pi\)
\(152\) 5.34481 0.433522
\(153\) 2.04892 0.165645
\(154\) − 4.33944i − 0.349682i
\(155\) −0.185981 −0.0149383
\(156\) 0 0
\(157\) 15.5429 1.24046 0.620228 0.784421i \(-0.287040\pi\)
0.620228 + 0.784421i \(0.287040\pi\)
\(158\) − 1.42327i − 0.113229i
\(159\) −5.74094 −0.455286
\(160\) −1.00000 −0.0790569
\(161\) 11.2935i 0.890053i
\(162\) 1.00000i 0.0785674i
\(163\) − 2.00239i − 0.156840i −0.996920 0.0784198i \(-0.975013\pi\)
0.996920 0.0784198i \(-0.0249874\pi\)
\(164\) 12.2349i 0.955385i
\(165\) 3.19806 0.248969
\(166\) 11.1196 0.863049
\(167\) − 8.24027i − 0.637652i −0.947813 0.318826i \(-0.896712\pi\)
0.947813 0.318826i \(-0.103288\pi\)
\(168\) −1.35690 −0.104687
\(169\) 0 0
\(170\) −2.04892 −0.157145
\(171\) 5.34481i 0.408728i
\(172\) −2.91185 −0.222027
\(173\) −12.7463 −0.969084 −0.484542 0.874768i \(-0.661014\pi\)
−0.484542 + 0.874768i \(0.661014\pi\)
\(174\) − 8.07069i − 0.611837i
\(175\) − 1.35690i − 0.102572i
\(176\) 3.19806i 0.241063i
\(177\) 1.62565i 0.122191i
\(178\) −3.37867 −0.253242
\(179\) −16.0422 −1.19905 −0.599526 0.800356i \(-0.704644\pi\)
−0.599526 + 0.800356i \(0.704644\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) −3.85086 −0.286232 −0.143116 0.989706i \(-0.545712\pi\)
−0.143116 + 0.989706i \(0.545712\pi\)
\(182\) 0 0
\(183\) −5.28382 −0.390591
\(184\) − 8.32304i − 0.613583i
\(185\) 2.46681 0.181364
\(186\) 0.185981 0.0136368
\(187\) 6.55257i 0.479171i
\(188\) − 13.0978i − 0.955258i
\(189\) − 1.35690i − 0.0986997i
\(190\) − 5.34481i − 0.387754i
\(191\) 11.8280 0.855845 0.427923 0.903815i \(-0.359246\pi\)
0.427923 + 0.903815i \(0.359246\pi\)
\(192\) 1.00000 0.0721688
\(193\) 0.872625i 0.0628129i 0.999507 + 0.0314065i \(0.00999864\pi\)
−0.999507 + 0.0314065i \(0.990001\pi\)
\(194\) −4.74094 −0.340380
\(195\) 0 0
\(196\) −5.15883 −0.368488
\(197\) − 10.6213i − 0.756739i −0.925655 0.378369i \(-0.876485\pi\)
0.925655 0.378369i \(-0.123515\pi\)
\(198\) −3.19806 −0.227276
\(199\) −12.3424 −0.874931 −0.437466 0.899235i \(-0.644124\pi\)
−0.437466 + 0.899235i \(0.644124\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 1.55496i 0.109678i
\(202\) 12.5918i 0.885956i
\(203\) 10.9511i 0.768615i
\(204\) 2.04892 0.143453
\(205\) 12.2349 0.854522
\(206\) 9.00000i 0.627060i
\(207\) 8.32304 0.578492
\(208\) 0 0
\(209\) −17.0930 −1.18235
\(210\) 1.35690i 0.0936347i
\(211\) 6.75733 0.465194 0.232597 0.972573i \(-0.425278\pi\)
0.232597 + 0.972573i \(0.425278\pi\)
\(212\) −5.74094 −0.394289
\(213\) − 2.32304i − 0.159172i
\(214\) − 5.08815i − 0.347819i
\(215\) 2.91185i 0.198587i
\(216\) 1.00000i 0.0680414i
\(217\) −0.252356 −0.0171311
\(218\) 6.43296 0.435695
\(219\) 5.11529i 0.345659i
\(220\) 3.19806 0.215613
\(221\) 0 0
\(222\) −2.46681 −0.165562
\(223\) − 21.5972i − 1.44625i −0.690715 0.723127i \(-0.742704\pi\)
0.690715 0.723127i \(-0.257296\pi\)
\(224\) −1.35690 −0.0906614
\(225\) −1.00000 −0.0666667
\(226\) − 18.6571i − 1.24105i
\(227\) − 10.2862i − 0.682720i −0.939933 0.341360i \(-0.889113\pi\)
0.939933 0.341360i \(-0.110887\pi\)
\(228\) 5.34481i 0.353969i
\(229\) 4.25906i 0.281447i 0.990049 + 0.140723i \(0.0449428\pi\)
−0.990049 + 0.140723i \(0.955057\pi\)
\(230\) −8.32304 −0.548805
\(231\) 4.33944 0.285514
\(232\) − 8.07069i − 0.529867i
\(233\) −19.5133 −1.27836 −0.639181 0.769057i \(-0.720726\pi\)
−0.639181 + 0.769057i \(0.720726\pi\)
\(234\) 0 0
\(235\) −13.0978 −0.854409
\(236\) 1.62565i 0.105821i
\(237\) 1.42327 0.0924514
\(238\) −2.78017 −0.180211
\(239\) 7.16315i 0.463345i 0.972794 + 0.231673i \(0.0744198\pi\)
−0.972794 + 0.231673i \(0.925580\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) 17.9584i 1.15680i 0.815753 + 0.578400i \(0.196323\pi\)
−0.815753 + 0.578400i \(0.803677\pi\)
\(242\) 0.772398i 0.0496516i
\(243\) −1.00000 −0.0641500
\(244\) −5.28382 −0.338262
\(245\) 5.15883i 0.329586i
\(246\) −12.2349 −0.780069
\(247\) 0 0
\(248\) 0.185981 0.0118098
\(249\) 11.1196i 0.704676i
\(250\) 1.00000 0.0632456
\(251\) 6.68425 0.421906 0.210953 0.977496i \(-0.432343\pi\)
0.210953 + 0.977496i \(0.432343\pi\)
\(252\) − 1.35690i − 0.0854764i
\(253\) 26.6176i 1.67343i
\(254\) 16.3763i 1.02754i
\(255\) − 2.04892i − 0.128308i
\(256\) 1.00000 0.0625000
\(257\) −9.91484 −0.618471 −0.309235 0.950986i \(-0.600073\pi\)
−0.309235 + 0.950986i \(0.600073\pi\)
\(258\) − 2.91185i − 0.181284i
\(259\) 3.34721 0.207985
\(260\) 0 0
\(261\) 8.07069 0.499563
\(262\) 3.00969i 0.185939i
\(263\) 13.8509 0.854080 0.427040 0.904233i \(-0.359556\pi\)
0.427040 + 0.904233i \(0.359556\pi\)
\(264\) −3.19806 −0.196827
\(265\) 5.74094i 0.352663i
\(266\) − 7.25236i − 0.444671i
\(267\) − 3.37867i − 0.206771i
\(268\) 1.55496i 0.0949842i
\(269\) −17.8756 −1.08990 −0.544948 0.838470i \(-0.683450\pi\)
−0.544948 + 0.838470i \(0.683450\pi\)
\(270\) 1.00000 0.0608581
\(271\) 31.2083i 1.89577i 0.318610 + 0.947886i \(0.396784\pi\)
−0.318610 + 0.947886i \(0.603216\pi\)
\(272\) 2.04892 0.124234
\(273\) 0 0
\(274\) −20.2892 −1.22571
\(275\) − 3.19806i − 0.192850i
\(276\) 8.32304 0.500988
\(277\) −22.9245 −1.37740 −0.688701 0.725046i \(-0.741819\pi\)
−0.688701 + 0.725046i \(0.741819\pi\)
\(278\) − 0.153457i − 0.00920373i
\(279\) 0.185981i 0.0111344i
\(280\) 1.35690i 0.0810900i
\(281\) 28.0683i 1.67441i 0.546886 + 0.837207i \(0.315813\pi\)
−0.546886 + 0.837207i \(0.684187\pi\)
\(282\) 13.0978 0.779965
\(283\) 24.8364 1.47637 0.738185 0.674599i \(-0.235683\pi\)
0.738185 + 0.674599i \(0.235683\pi\)
\(284\) − 2.32304i − 0.137847i
\(285\) 5.34481 0.316599
\(286\) 0 0
\(287\) 16.6015 0.979955
\(288\) 1.00000i 0.0589256i
\(289\) −12.8019 −0.753055
\(290\) −8.07069 −0.473927
\(291\) − 4.74094i − 0.277919i
\(292\) 5.11529i 0.299350i
\(293\) − 28.4349i − 1.66118i −0.556882 0.830592i \(-0.688002\pi\)
0.556882 0.830592i \(-0.311998\pi\)
\(294\) − 5.15883i − 0.300869i
\(295\) 1.62565 0.0946488
\(296\) −2.46681 −0.143381
\(297\) − 3.19806i − 0.185570i
\(298\) −10.7724 −0.624028
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 3.95108i 0.227737i
\(302\) −17.3817 −1.00020
\(303\) −12.5918 −0.723380
\(304\) 5.34481i 0.306546i
\(305\) 5.28382i 0.302550i
\(306\) 2.04892i 0.117129i
\(307\) − 10.1347i − 0.578416i −0.957266 0.289208i \(-0.906608\pi\)
0.957266 0.289208i \(-0.0933920\pi\)
\(308\) 4.33944 0.247262
\(309\) −9.00000 −0.511992
\(310\) − 0.185981i − 0.0105630i
\(311\) −20.0218 −1.13533 −0.567665 0.823259i \(-0.692153\pi\)
−0.567665 + 0.823259i \(0.692153\pi\)
\(312\) 0 0
\(313\) 8.63640 0.488158 0.244079 0.969755i \(-0.421514\pi\)
0.244079 + 0.969755i \(0.421514\pi\)
\(314\) 15.5429i 0.877135i
\(315\) −1.35690 −0.0764524
\(316\) 1.42327 0.0800653
\(317\) − 17.6256i − 0.989955i −0.868906 0.494977i \(-0.835176\pi\)
0.868906 0.494977i \(-0.164824\pi\)
\(318\) − 5.74094i − 0.321936i
\(319\) 25.8106i 1.44511i
\(320\) − 1.00000i − 0.0559017i
\(321\) 5.08815 0.283993
\(322\) −11.2935 −0.629362
\(323\) 10.9511i 0.609335i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 2.00239 0.110902
\(327\) 6.43296i 0.355744i
\(328\) −12.2349 −0.675559
\(329\) −17.7724 −0.979824
\(330\) 3.19806i 0.176048i
\(331\) 31.5163i 1.73229i 0.499790 + 0.866147i \(0.333411\pi\)
−0.499790 + 0.866147i \(0.666589\pi\)
\(332\) 11.1196i 0.610268i
\(333\) − 2.46681i − 0.135180i
\(334\) 8.24027 0.450888
\(335\) 1.55496 0.0849564
\(336\) − 1.35690i − 0.0740247i
\(337\) 11.8116 0.643420 0.321710 0.946838i \(-0.395742\pi\)
0.321710 + 0.946838i \(0.395742\pi\)
\(338\) 0 0
\(339\) 18.6571 1.01331
\(340\) − 2.04892i − 0.111118i
\(341\) −0.594778 −0.0322090
\(342\) −5.34481 −0.289014
\(343\) 16.4983i 0.890823i
\(344\) − 2.91185i − 0.156997i
\(345\) − 8.32304i − 0.448098i
\(346\) − 12.7463i − 0.685246i
\(347\) −17.4101 −0.934624 −0.467312 0.884092i \(-0.654778\pi\)
−0.467312 + 0.884092i \(0.654778\pi\)
\(348\) 8.07069 0.432634
\(349\) 18.6799i 0.999914i 0.866050 + 0.499957i \(0.166651\pi\)
−0.866050 + 0.499957i \(0.833349\pi\)
\(350\) 1.35690 0.0725291
\(351\) 0 0
\(352\) −3.19806 −0.170457
\(353\) − 25.3991i − 1.35186i −0.736967 0.675929i \(-0.763743\pi\)
0.736967 0.675929i \(-0.236257\pi\)
\(354\) −1.62565 −0.0864021
\(355\) −2.32304 −0.123294
\(356\) − 3.37867i − 0.179069i
\(357\) − 2.78017i − 0.147142i
\(358\) − 16.0422i − 0.847857i
\(359\) 34.0103i 1.79499i 0.441021 + 0.897497i \(0.354617\pi\)
−0.441021 + 0.897497i \(0.645383\pi\)
\(360\) 1.00000 0.0527046
\(361\) −9.56704 −0.503528
\(362\) − 3.85086i − 0.202396i
\(363\) −0.772398 −0.0405404
\(364\) 0 0
\(365\) 5.11529 0.267747
\(366\) − 5.28382i − 0.276189i
\(367\) −25.6437 −1.33859 −0.669295 0.742997i \(-0.733404\pi\)
−0.669295 + 0.742997i \(0.733404\pi\)
\(368\) 8.32304 0.433869
\(369\) − 12.2349i − 0.636923i
\(370\) 2.46681i 0.128243i
\(371\) 7.78986i 0.404429i
\(372\) 0.185981i 0.00964265i
\(373\) −16.3612 −0.847151 −0.423576 0.905861i \(-0.639225\pi\)
−0.423576 + 0.905861i \(0.639225\pi\)
\(374\) −6.55257 −0.338825
\(375\) 1.00000i 0.0516398i
\(376\) 13.0978 0.675469
\(377\) 0 0
\(378\) 1.35690 0.0697912
\(379\) − 22.8823i − 1.17539i −0.809084 0.587693i \(-0.800036\pi\)
0.809084 0.587693i \(-0.199964\pi\)
\(380\) 5.34481 0.274183
\(381\) −16.3763 −0.838982
\(382\) 11.8280i 0.605174i
\(383\) − 11.9191i − 0.609040i −0.952506 0.304520i \(-0.901504\pi\)
0.952506 0.304520i \(-0.0984961\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) − 4.33944i − 0.221158i
\(386\) −0.872625 −0.0444155
\(387\) 2.91185 0.148018
\(388\) − 4.74094i − 0.240685i
\(389\) −8.07606 −0.409473 −0.204736 0.978817i \(-0.565634\pi\)
−0.204736 + 0.978817i \(0.565634\pi\)
\(390\) 0 0
\(391\) 17.0532 0.862419
\(392\) − 5.15883i − 0.260560i
\(393\) −3.00969 −0.151819
\(394\) 10.6213 0.535095
\(395\) − 1.42327i − 0.0716126i
\(396\) − 3.19806i − 0.160709i
\(397\) − 20.5230i − 1.03002i −0.857184 0.515011i \(-0.827788\pi\)
0.857184 0.515011i \(-0.172212\pi\)
\(398\) − 12.3424i − 0.618670i
\(399\) 7.25236 0.363072
\(400\) −1.00000 −0.0500000
\(401\) − 18.3274i − 0.915224i −0.889152 0.457612i \(-0.848705\pi\)
0.889152 0.457612i \(-0.151295\pi\)
\(402\) −1.55496 −0.0775543
\(403\) 0 0
\(404\) −12.5918 −0.626465
\(405\) 1.00000i 0.0496904i
\(406\) −10.9511 −0.543493
\(407\) 7.88902 0.391044
\(408\) 2.04892i 0.101437i
\(409\) − 9.00000i − 0.445021i −0.974930 0.222511i \(-0.928575\pi\)
0.974930 0.222511i \(-0.0714252\pi\)
\(410\) 12.2349i 0.604239i
\(411\) − 20.2892i − 1.00079i
\(412\) −9.00000 −0.443398
\(413\) 2.20583 0.108542
\(414\) 8.32304i 0.409055i
\(415\) 11.1196 0.545840
\(416\) 0 0
\(417\) 0.153457 0.00751481
\(418\) − 17.0930i − 0.836048i
\(419\) −14.4185 −0.704389 −0.352195 0.935927i \(-0.614564\pi\)
−0.352195 + 0.935927i \(0.614564\pi\)
\(420\) −1.35690 −0.0662097
\(421\) 19.9801i 0.973773i 0.873465 + 0.486886i \(0.161867\pi\)
−0.873465 + 0.486886i \(0.838133\pi\)
\(422\) 6.75733i 0.328942i
\(423\) 13.0978i 0.636839i
\(424\) − 5.74094i − 0.278805i
\(425\) −2.04892 −0.0993871
\(426\) 2.32304 0.112552
\(427\) 7.16959i 0.346961i
\(428\) 5.08815 0.245945
\(429\) 0 0
\(430\) −2.91185 −0.140422
\(431\) 21.1812i 1.02026i 0.860097 + 0.510131i \(0.170403\pi\)
−0.860097 + 0.510131i \(0.829597\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −31.0291 −1.49116 −0.745581 0.666415i \(-0.767828\pi\)
−0.745581 + 0.666415i \(0.767828\pi\)
\(434\) − 0.252356i − 0.0121135i
\(435\) − 8.07069i − 0.386960i
\(436\) 6.43296i 0.308083i
\(437\) 44.4851i 2.12801i
\(438\) −5.11529 −0.244418
\(439\) −7.33406 −0.350036 −0.175018 0.984565i \(-0.555998\pi\)
−0.175018 + 0.984565i \(0.555998\pi\)
\(440\) 3.19806i 0.152462i
\(441\) 5.15883 0.245659
\(442\) 0 0
\(443\) −18.1360 −0.861667 −0.430834 0.902431i \(-0.641780\pi\)
−0.430834 + 0.902431i \(0.641780\pi\)
\(444\) − 2.46681i − 0.117070i
\(445\) −3.37867 −0.160164
\(446\) 21.5972 1.02266
\(447\) − 10.7724i − 0.509517i
\(448\) − 1.35690i − 0.0641073i
\(449\) 4.55496i 0.214962i 0.994207 + 0.107481i \(0.0342784\pi\)
−0.994207 + 0.107481i \(0.965722\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) 39.1280 1.84246
\(452\) 18.6571 0.877556
\(453\) − 17.3817i − 0.816661i
\(454\) 10.2862 0.482756
\(455\) 0 0
\(456\) −5.34481 −0.250294
\(457\) 9.98121i 0.466901i 0.972369 + 0.233451i \(0.0750018\pi\)
−0.972369 + 0.233451i \(0.924998\pi\)
\(458\) −4.25906 −0.199013
\(459\) −2.04892 −0.0956353
\(460\) − 8.32304i − 0.388064i
\(461\) − 16.1672i − 0.752981i −0.926420 0.376491i \(-0.877131\pi\)
0.926420 0.376491i \(-0.122869\pi\)
\(462\) 4.33944i 0.201889i
\(463\) − 3.88902i − 0.180738i −0.995908 0.0903690i \(-0.971195\pi\)
0.995908 0.0903690i \(-0.0288046\pi\)
\(464\) 8.07069 0.374672
\(465\) 0.185981 0.00862465
\(466\) − 19.5133i − 0.903938i
\(467\) 0.872625 0.0403803 0.0201901 0.999796i \(-0.493573\pi\)
0.0201901 + 0.999796i \(0.493573\pi\)
\(468\) 0 0
\(469\) 2.10992 0.0974269
\(470\) − 13.0978i − 0.604158i
\(471\) −15.5429 −0.716178
\(472\) −1.62565 −0.0748264
\(473\) 9.31229i 0.428180i
\(474\) 1.42327i 0.0653730i
\(475\) − 5.34481i − 0.245237i
\(476\) − 2.78017i − 0.127429i
\(477\) 5.74094 0.262860
\(478\) −7.16315 −0.327635
\(479\) 31.6305i 1.44524i 0.691247 + 0.722618i \(0.257062\pi\)
−0.691247 + 0.722618i \(0.742938\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −17.9584 −0.817982
\(483\) − 11.2935i − 0.513872i
\(484\) −0.772398 −0.0351090
\(485\) −4.74094 −0.215275
\(486\) − 1.00000i − 0.0453609i
\(487\) 21.7748i 0.986710i 0.869828 + 0.493355i \(0.164230\pi\)
−0.869828 + 0.493355i \(0.835770\pi\)
\(488\) − 5.28382i − 0.239187i
\(489\) 2.00239i 0.0905513i
\(490\) −5.15883 −0.233052
\(491\) −42.9590 −1.93871 −0.969356 0.245662i \(-0.920995\pi\)
−0.969356 + 0.245662i \(0.920995\pi\)
\(492\) − 12.2349i − 0.551592i
\(493\) 16.5362 0.744752
\(494\) 0 0
\(495\) −3.19806 −0.143742
\(496\) 0.185981i 0.00835078i
\(497\) −3.15213 −0.141392
\(498\) −11.1196 −0.498281
\(499\) − 36.0978i − 1.61596i −0.589209 0.807981i \(-0.700561\pi\)
0.589209 0.807981i \(-0.299439\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 8.24027i 0.368148i
\(502\) 6.68425i 0.298333i
\(503\) −21.4534 −0.956560 −0.478280 0.878207i \(-0.658740\pi\)
−0.478280 + 0.878207i \(0.658740\pi\)
\(504\) 1.35690 0.0604409
\(505\) 12.5918i 0.560327i
\(506\) −26.6176 −1.18330
\(507\) 0 0
\(508\) −16.3763 −0.726580
\(509\) 0.545860i 0.0241948i 0.999927 + 0.0120974i \(0.00385082\pi\)
−0.999927 + 0.0120974i \(0.996149\pi\)
\(510\) 2.04892 0.0907276
\(511\) 6.94092 0.307048
\(512\) 1.00000i 0.0441942i
\(513\) − 5.34481i − 0.235979i
\(514\) − 9.91484i − 0.437325i
\(515\) 9.00000i 0.396587i
\(516\) 2.91185 0.128187
\(517\) −41.8877 −1.84222
\(518\) 3.34721i 0.147068i
\(519\) 12.7463 0.559501
\(520\) 0 0
\(521\) 29.6136 1.29739 0.648697 0.761047i \(-0.275314\pi\)
0.648697 + 0.761047i \(0.275314\pi\)
\(522\) 8.07069i 0.353244i
\(523\) 44.4543 1.94385 0.971924 0.235293i \(-0.0756050\pi\)
0.971924 + 0.235293i \(0.0756050\pi\)
\(524\) −3.00969 −0.131479
\(525\) 1.35690i 0.0592198i
\(526\) 13.8509i 0.603926i
\(527\) 0.381059i 0.0165992i
\(528\) − 3.19806i − 0.139178i
\(529\) 46.2731 2.01187
\(530\) −5.74094 −0.249370
\(531\) − 1.62565i − 0.0705470i
\(532\) 7.25236 0.314430
\(533\) 0 0
\(534\) 3.37867 0.146209
\(535\) − 5.08815i − 0.219980i
\(536\) −1.55496 −0.0671640
\(537\) 16.0422 0.692273
\(538\) − 17.8756i − 0.770672i
\(539\) 16.4983i 0.710631i
\(540\) 1.00000i 0.0430331i
\(541\) − 18.7651i − 0.806775i −0.915029 0.403387i \(-0.867833\pi\)
0.915029 0.403387i \(-0.132167\pi\)
\(542\) −31.2083 −1.34051
\(543\) 3.85086 0.165256
\(544\) 2.04892i 0.0878466i
\(545\) 6.43296 0.275558
\(546\) 0 0
\(547\) 32.0116 1.36872 0.684359 0.729145i \(-0.260082\pi\)
0.684359 + 0.729145i \(0.260082\pi\)
\(548\) − 20.2892i − 0.866711i
\(549\) 5.28382 0.225508
\(550\) 3.19806 0.136366
\(551\) 43.1363i 1.83767i
\(552\) 8.32304i 0.354252i
\(553\) − 1.93123i − 0.0821243i
\(554\) − 22.9245i − 0.973970i
\(555\) −2.46681 −0.104710
\(556\) 0.153457 0.00650802
\(557\) 8.23490i 0.348924i 0.984664 + 0.174462i \(0.0558186\pi\)
−0.984664 + 0.174462i \(0.944181\pi\)
\(558\) −0.185981 −0.00787319
\(559\) 0 0
\(560\) −1.35690 −0.0573393
\(561\) − 6.55257i − 0.276650i
\(562\) −28.0683 −1.18399
\(563\) 30.2295 1.27402 0.637011 0.770855i \(-0.280170\pi\)
0.637011 + 0.770855i \(0.280170\pi\)
\(564\) 13.0978i 0.551518i
\(565\) − 18.6571i − 0.784910i
\(566\) 24.8364i 1.04395i
\(567\) 1.35690i 0.0569843i
\(568\) 2.32304 0.0974728
\(569\) −44.3110 −1.85761 −0.928806 0.370566i \(-0.879164\pi\)
−0.928806 + 0.370566i \(0.879164\pi\)
\(570\) 5.34481i 0.223870i
\(571\) −34.4873 −1.44325 −0.721623 0.692286i \(-0.756604\pi\)
−0.721623 + 0.692286i \(0.756604\pi\)
\(572\) 0 0
\(573\) −11.8280 −0.494123
\(574\) 16.6015i 0.692932i
\(575\) −8.32304 −0.347095
\(576\) −1.00000 −0.0416667
\(577\) − 14.1715i − 0.589968i −0.955502 0.294984i \(-0.904686\pi\)
0.955502 0.294984i \(-0.0953142\pi\)
\(578\) − 12.8019i − 0.532490i
\(579\) − 0.872625i − 0.0362651i
\(580\) − 8.07069i − 0.335117i
\(581\) 15.0881 0.625962
\(582\) 4.74094 0.196518
\(583\) 18.3599i 0.760389i
\(584\) −5.11529 −0.211672
\(585\) 0 0
\(586\) 28.4349 1.17463
\(587\) − 1.12737i − 0.0465317i −0.999729 0.0232659i \(-0.992594\pi\)
0.999729 0.0232659i \(-0.00740642\pi\)
\(588\) 5.15883 0.212747
\(589\) −0.994032 −0.0409584
\(590\) 1.62565i 0.0669268i
\(591\) 10.6213i 0.436903i
\(592\) − 2.46681i − 0.101385i
\(593\) − 18.3913i − 0.755242i −0.925960 0.377621i \(-0.876742\pi\)
0.925960 0.377621i \(-0.123258\pi\)
\(594\) 3.19806 0.131218
\(595\) −2.78017 −0.113976
\(596\) − 10.7724i − 0.441255i
\(597\) 12.3424 0.505142
\(598\) 0 0
\(599\) 13.1226 0.536174 0.268087 0.963395i \(-0.413608\pi\)
0.268087 + 0.963395i \(0.413608\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) 25.5754 1.04324 0.521621 0.853177i \(-0.325327\pi\)
0.521621 + 0.853177i \(0.325327\pi\)
\(602\) −3.95108 −0.161034
\(603\) − 1.55496i − 0.0633228i
\(604\) − 17.3817i − 0.707249i
\(605\) 0.772398i 0.0314024i
\(606\) − 12.5918i − 0.511507i
\(607\) 44.4782 1.80531 0.902656 0.430362i \(-0.141614\pi\)
0.902656 + 0.430362i \(0.141614\pi\)
\(608\) −5.34481 −0.216761
\(609\) − 10.9511i − 0.443760i
\(610\) −5.28382 −0.213935
\(611\) 0 0
\(612\) −2.04892 −0.0828226
\(613\) − 21.7657i − 0.879108i −0.898216 0.439554i \(-0.855137\pi\)
0.898216 0.439554i \(-0.144863\pi\)
\(614\) 10.1347 0.409002
\(615\) −12.2349 −0.493359
\(616\) 4.33944i 0.174841i
\(617\) − 17.3569i − 0.698762i −0.936981 0.349381i \(-0.886392\pi\)
0.936981 0.349381i \(-0.113608\pi\)
\(618\) − 9.00000i − 0.362033i
\(619\) − 36.8049i − 1.47931i −0.672984 0.739657i \(-0.734988\pi\)
0.672984 0.739657i \(-0.265012\pi\)
\(620\) 0.185981 0.00746916
\(621\) −8.32304 −0.333992
\(622\) − 20.0218i − 0.802800i
\(623\) −4.58450 −0.183674
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 8.63640i 0.345180i
\(627\) 17.0930 0.682631
\(628\) −15.5429 −0.620228
\(629\) − 5.05429i − 0.201528i
\(630\) − 1.35690i − 0.0540600i
\(631\) − 11.7554i − 0.467976i −0.972240 0.233988i \(-0.924822\pi\)
0.972240 0.233988i \(-0.0751776\pi\)
\(632\) 1.42327i 0.0566147i
\(633\) −6.75733 −0.268580
\(634\) 17.6256 0.700004
\(635\) 16.3763i 0.649873i
\(636\) 5.74094 0.227643
\(637\) 0 0
\(638\) −25.8106 −1.02185
\(639\) 2.32304i 0.0918982i
\(640\) 1.00000 0.0395285
\(641\) −33.0858 −1.30681 −0.653404 0.757009i \(-0.726660\pi\)
−0.653404 + 0.757009i \(0.726660\pi\)
\(642\) 5.08815i 0.200813i
\(643\) − 10.3461i − 0.408012i −0.978970 0.204006i \(-0.934604\pi\)
0.978970 0.204006i \(-0.0653962\pi\)
\(644\) − 11.2935i − 0.445026i
\(645\) − 2.91185i − 0.114654i
\(646\) −10.9511 −0.430865
\(647\) 11.0030 0.432572 0.216286 0.976330i \(-0.430606\pi\)
0.216286 + 0.976330i \(0.430606\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 5.19892 0.204075
\(650\) 0 0
\(651\) 0.252356 0.00989063
\(652\) 2.00239i 0.0784198i
\(653\) −30.8224 −1.20617 −0.603086 0.797676i \(-0.706063\pi\)
−0.603086 + 0.797676i \(0.706063\pi\)
\(654\) −6.43296 −0.251549
\(655\) 3.00969i 0.117598i
\(656\) − 12.2349i − 0.477693i
\(657\) − 5.11529i − 0.199567i
\(658\) − 17.7724i − 0.692840i
\(659\) −33.2010 −1.29333 −0.646665 0.762774i \(-0.723837\pi\)
−0.646665 + 0.762774i \(0.723837\pi\)
\(660\) −3.19806 −0.124484
\(661\) − 37.4252i − 1.45567i −0.685752 0.727836i \(-0.740526\pi\)
0.685752 0.727836i \(-0.259474\pi\)
\(662\) −31.5163 −1.22492
\(663\) 0 0
\(664\) −11.1196 −0.431524
\(665\) − 7.25236i − 0.281234i
\(666\) 2.46681 0.0955870
\(667\) 67.1727 2.60094
\(668\) 8.24027i 0.318826i
\(669\) 21.5972i 0.834995i
\(670\) 1.55496i 0.0600733i
\(671\) 16.8980i 0.652339i
\(672\) 1.35690 0.0523434
\(673\) 34.9409 1.34687 0.673437 0.739245i \(-0.264817\pi\)
0.673437 + 0.739245i \(0.264817\pi\)
\(674\) 11.8116i 0.454967i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 29.3913 1.12960 0.564800 0.825228i \(-0.308953\pi\)
0.564800 + 0.825228i \(0.308953\pi\)
\(678\) 18.6571i 0.716522i
\(679\) −6.43296 −0.246874
\(680\) 2.04892 0.0785724
\(681\) 10.2862i 0.394168i
\(682\) − 0.594778i − 0.0227752i
\(683\) − 28.9855i − 1.10910i −0.832150 0.554550i \(-0.812890\pi\)
0.832150 0.554550i \(-0.187110\pi\)
\(684\) − 5.34481i − 0.204364i
\(685\) −20.2892 −0.775210
\(686\) −16.4983 −0.629907
\(687\) − 4.25906i − 0.162493i
\(688\) 2.91185 0.111013
\(689\) 0 0
\(690\) 8.32304 0.316853
\(691\) − 1.37627i − 0.0523559i −0.999657 0.0261780i \(-0.991666\pi\)
0.999657 0.0261780i \(-0.00833365\pi\)
\(692\) 12.7463 0.484542
\(693\) −4.33944 −0.164842
\(694\) − 17.4101i − 0.660879i
\(695\) − 0.153457i − 0.00582095i
\(696\) 8.07069i 0.305919i
\(697\) − 25.0683i − 0.949529i
\(698\) −18.6799 −0.707046
\(699\) 19.5133 0.738062
\(700\) 1.35690i 0.0512858i
\(701\) −9.21014 −0.347862 −0.173931 0.984758i \(-0.555647\pi\)
−0.173931 + 0.984758i \(0.555647\pi\)
\(702\) 0 0
\(703\) 13.1847 0.497269
\(704\) − 3.19806i − 0.120532i
\(705\) 13.0978 0.493293
\(706\) 25.3991 0.955908
\(707\) 17.0858i 0.642576i
\(708\) − 1.62565i − 0.0610955i
\(709\) 46.1613i 1.73363i 0.498634 + 0.866813i \(0.333835\pi\)
−0.498634 + 0.866813i \(0.666165\pi\)
\(710\) − 2.32304i − 0.0871823i
\(711\) −1.42327 −0.0533769
\(712\) 3.37867 0.126621
\(713\) 1.54793i 0.0579703i
\(714\) 2.78017 0.104045
\(715\) 0 0
\(716\) 16.0422 0.599526
\(717\) − 7.16315i − 0.267513i
\(718\) −34.0103 −1.26925
\(719\) 46.3588 1.72889 0.864446 0.502726i \(-0.167669\pi\)
0.864446 + 0.502726i \(0.167669\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 12.2121i 0.454801i
\(722\) − 9.56704i − 0.356048i
\(723\) − 17.9584i − 0.667879i
\(724\) 3.85086 0.143116
\(725\) −8.07069 −0.299738
\(726\) − 0.772398i − 0.0286664i
\(727\) 1.67217 0.0620174 0.0310087 0.999519i \(-0.490128\pi\)
0.0310087 + 0.999519i \(0.490128\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 5.11529i 0.189325i
\(731\) 5.96615 0.220666
\(732\) 5.28382 0.195295
\(733\) 33.6189i 1.24174i 0.783912 + 0.620872i \(0.213221\pi\)
−0.783912 + 0.620872i \(0.786779\pi\)
\(734\) − 25.6437i − 0.946526i
\(735\) − 5.15883i − 0.190286i
\(736\) 8.32304i 0.306791i
\(737\) 4.97285 0.183177
\(738\) 12.2349 0.450373
\(739\) − 31.8437i − 1.17139i −0.810532 0.585694i \(-0.800822\pi\)
0.810532 0.585694i \(-0.199178\pi\)
\(740\) −2.46681 −0.0906818
\(741\) 0 0
\(742\) −7.78986 −0.285975
\(743\) − 23.4010i − 0.858500i −0.903186 0.429250i \(-0.858778\pi\)
0.903186 0.429250i \(-0.141222\pi\)
\(744\) −0.185981 −0.00681838
\(745\) −10.7724 −0.394670
\(746\) − 16.3612i − 0.599026i
\(747\) − 11.1196i − 0.406845i
\(748\) − 6.55257i − 0.239586i
\(749\) − 6.90408i − 0.252270i
\(750\) −1.00000 −0.0365148
\(751\) 39.0146 1.42366 0.711831 0.702350i \(-0.247866\pi\)
0.711831 + 0.702350i \(0.247866\pi\)
\(752\) 13.0978i 0.477629i
\(753\) −6.68425 −0.243588
\(754\) 0 0
\(755\) −17.3817 −0.632583
\(756\) 1.35690i 0.0493498i
\(757\) 43.0224 1.56367 0.781837 0.623483i \(-0.214283\pi\)
0.781837 + 0.623483i \(0.214283\pi\)
\(758\) 22.8823 0.831123
\(759\) − 26.6176i − 0.966158i
\(760\) 5.34481i 0.193877i
\(761\) 14.9903i 0.543398i 0.962382 + 0.271699i \(0.0875856\pi\)
−0.962382 + 0.271699i \(0.912414\pi\)
\(762\) − 16.3763i − 0.593250i
\(763\) 8.72886 0.316006
\(764\) −11.8280 −0.427923
\(765\) 2.04892i 0.0740788i
\(766\) 11.9191 0.430656
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 19.9138i 0.718109i 0.933317 + 0.359055i \(0.116901\pi\)
−0.933317 + 0.359055i \(0.883099\pi\)
\(770\) 4.33944 0.156382
\(771\) 9.91484 0.357074
\(772\) − 0.872625i − 0.0314065i
\(773\) − 25.5566i − 0.919208i −0.888124 0.459604i \(-0.847991\pi\)
0.888124 0.459604i \(-0.152009\pi\)
\(774\) 2.91185i 0.104664i
\(775\) − 0.185981i − 0.00668062i
\(776\) 4.74094 0.170190
\(777\) −3.34721 −0.120080
\(778\) − 8.07606i − 0.289541i
\(779\) 65.3933 2.34296
\(780\) 0 0
\(781\) −7.42924 −0.265839
\(782\) 17.0532i 0.609822i
\(783\) −8.07069 −0.288423
\(784\) 5.15883 0.184244
\(785\) 15.5429i 0.554749i
\(786\) − 3.00969i − 0.107352i
\(787\) 28.6297i 1.02054i 0.860015 + 0.510269i \(0.170454\pi\)
−0.860015 + 0.510269i \(0.829546\pi\)
\(788\) 10.6213i 0.378369i
\(789\) −13.8509 −0.493103
\(790\) 1.42327 0.0506377
\(791\) − 25.3157i − 0.900124i
\(792\) 3.19806 0.113638
\(793\) 0 0
\(794\) 20.5230 0.728335
\(795\) − 5.74094i − 0.203610i
\(796\) 12.3424 0.437466
\(797\) −1.00431 −0.0355746 −0.0177873 0.999842i \(-0.505662\pi\)
−0.0177873 + 0.999842i \(0.505662\pi\)
\(798\) 7.25236i 0.256731i
\(799\) 26.8364i 0.949403i
\(800\) − 1.00000i − 0.0353553i
\(801\) 3.37867i 0.119379i
\(802\) 18.3274 0.647161
\(803\) 16.3590 0.577297
\(804\) − 1.55496i − 0.0548391i
\(805\) −11.2935 −0.398044
\(806\) 0 0
\(807\) 17.8756 0.629251
\(808\) − 12.5918i − 0.442978i
\(809\) 49.8743 1.75349 0.876743 0.480959i \(-0.159711\pi\)
0.876743 + 0.480959i \(0.159711\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 7.95646i 0.279389i 0.990195 + 0.139695i \(0.0446121\pi\)
−0.990195 + 0.139695i \(0.955388\pi\)
\(812\) − 10.9511i − 0.384308i
\(813\) − 31.2083i − 1.09452i
\(814\) 7.88902i 0.276510i
\(815\) 2.00239 0.0701408
\(816\) −2.04892 −0.0717265
\(817\) 15.5633i 0.544492i
\(818\) 9.00000 0.314678
\(819\) 0 0
\(820\) −12.2349 −0.427261
\(821\) 52.9071i 1.84647i 0.384236 + 0.923235i \(0.374465\pi\)
−0.384236 + 0.923235i \(0.625535\pi\)
\(822\) 20.2892 0.707667
\(823\) −21.8006 −0.759921 −0.379961 0.925003i \(-0.624063\pi\)
−0.379961 + 0.925003i \(0.624063\pi\)
\(824\) − 9.00000i − 0.313530i
\(825\) 3.19806i 0.111342i
\(826\) 2.20583i 0.0767507i
\(827\) 27.3139i 0.949799i 0.880040 + 0.474899i \(0.157516\pi\)
−0.880040 + 0.474899i \(0.842484\pi\)
\(828\) −8.32304 −0.289246
\(829\) −40.8471 −1.41868 −0.709340 0.704867i \(-0.751007\pi\)
−0.709340 + 0.704867i \(0.751007\pi\)
\(830\) 11.1196i 0.385967i
\(831\) 22.9245 0.795243
\(832\) 0 0
\(833\) 10.5700 0.366230
\(834\) 0.153457i 0.00531377i
\(835\) 8.24027 0.285166
\(836\) 17.0930 0.591175
\(837\) − 0.185981i − 0.00642843i
\(838\) − 14.4185i − 0.498078i
\(839\) − 43.0823i − 1.48737i −0.668532 0.743683i \(-0.733077\pi\)
0.668532 0.743683i \(-0.266923\pi\)
\(840\) − 1.35690i − 0.0468174i
\(841\) 36.1360 1.24607
\(842\) −19.9801 −0.688561
\(843\) − 28.0683i − 0.966723i
\(844\) −6.75733 −0.232597
\(845\) 0 0
\(846\) −13.0978 −0.450313
\(847\) 1.04806i 0.0360119i
\(848\) 5.74094 0.197145
\(849\) −24.8364 −0.852382
\(850\) − 2.04892i − 0.0702773i
\(851\) − 20.5314i − 0.703807i
\(852\) 2.32304i 0.0795862i
\(853\) 24.6813i 0.845071i 0.906346 + 0.422535i \(0.138860\pi\)
−0.906346 + 0.422535i \(0.861140\pi\)
\(854\) −7.16959 −0.245338
\(855\) −5.34481 −0.182789
\(856\) 5.08815i 0.173909i
\(857\) −43.1075 −1.47252 −0.736262 0.676696i \(-0.763411\pi\)
−0.736262 + 0.676696i \(0.763411\pi\)
\(858\) 0 0
\(859\) −42.1323 −1.43753 −0.718767 0.695251i \(-0.755293\pi\)
−0.718767 + 0.695251i \(0.755293\pi\)
\(860\) − 2.91185i − 0.0992934i
\(861\) −16.6015 −0.565777
\(862\) −21.1812 −0.721434
\(863\) 47.9342i 1.63170i 0.578264 + 0.815850i \(0.303730\pi\)
−0.578264 + 0.815850i \(0.696270\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) − 12.7463i − 0.433388i
\(866\) − 31.0291i − 1.05441i
\(867\) 12.8019 0.434777
\(868\) 0.252356 0.00856553
\(869\) − 4.55171i − 0.154406i
\(870\) 8.07069 0.273622
\(871\) 0 0
\(872\) −6.43296 −0.217848
\(873\) 4.74094i 0.160456i
\(874\) −44.4851 −1.50473
\(875\) 1.35690 0.0458715
\(876\) − 5.11529i − 0.172830i
\(877\) − 38.6674i − 1.30570i −0.757485 0.652852i \(-0.773572\pi\)
0.757485 0.652852i \(-0.226428\pi\)
\(878\) − 7.33406i − 0.247513i
\(879\) 28.4349i 0.959085i
\(880\) −3.19806 −0.107807
\(881\) −25.4980 −0.859050 −0.429525 0.903055i \(-0.641319\pi\)
−0.429525 + 0.903055i \(0.641319\pi\)
\(882\) 5.15883i 0.173707i
\(883\) 31.8377 1.07142 0.535712 0.844401i \(-0.320043\pi\)
0.535712 + 0.844401i \(0.320043\pi\)
\(884\) 0 0
\(885\) −1.62565 −0.0546455
\(886\) − 18.1360i − 0.609291i
\(887\) −7.83446 −0.263055 −0.131528 0.991312i \(-0.541988\pi\)
−0.131528 + 0.991312i \(0.541988\pi\)
\(888\) 2.46681 0.0827808
\(889\) 22.2209i 0.745265i
\(890\) − 3.37867i − 0.113253i
\(891\) 3.19806i 0.107139i
\(892\) 21.5972i 0.723127i
\(893\) −70.0055 −2.34264
\(894\) 10.7724 0.360283
\(895\) − 16.0422i − 0.536232i
\(896\) 1.35690 0.0453307
\(897\) 0 0
\(898\) −4.55496 −0.152001
\(899\) 1.50099i 0.0500609i
\(900\) 1.00000 0.0333333
\(901\) 11.7627 0.391873
\(902\) 39.1280i 1.30282i
\(903\) − 3.95108i − 0.131484i
\(904\) 18.6571i 0.620526i
\(905\) − 3.85086i − 0.128007i
\(906\) 17.3817 0.577467
\(907\) −33.1089 −1.09936 −0.549681 0.835375i \(-0.685251\pi\)
−0.549681 + 0.835375i \(0.685251\pi\)
\(908\) 10.2862i 0.341360i
\(909\) 12.5918 0.417643
\(910\) 0 0
\(911\) −14.0556 −0.465684 −0.232842 0.972515i \(-0.574802\pi\)
−0.232842 + 0.972515i \(0.574802\pi\)
\(912\) − 5.34481i − 0.176984i
\(913\) 35.5612 1.17690
\(914\) −9.98121 −0.330149
\(915\) − 5.28382i − 0.174678i
\(916\) − 4.25906i − 0.140723i
\(917\) 4.08383i 0.134860i
\(918\) − 2.04892i − 0.0676243i
\(919\) −27.1830 −0.896684 −0.448342 0.893862i \(-0.647985\pi\)
−0.448342 + 0.893862i \(0.647985\pi\)
\(920\) 8.32304 0.274403
\(921\) 10.1347i 0.333949i
\(922\) 16.1672 0.532438
\(923\) 0 0
\(924\) −4.33944 −0.142757
\(925\) 2.46681i 0.0811083i
\(926\) 3.88902 0.127801
\(927\) 9.00000 0.295599
\(928\) 8.07069i 0.264933i
\(929\) − 50.3521i − 1.65200i −0.563671 0.826000i \(-0.690611\pi\)
0.563671 0.826000i \(-0.309389\pi\)
\(930\) 0.185981i 0.00609855i
\(931\) 27.5730i 0.903669i
\(932\) 19.5133 0.639181
\(933\) 20.0218 0.655483
\(934\) 0.872625i 0.0285532i
\(935\) −6.55257 −0.214292
\(936\) 0 0
\(937\) 26.6394 0.870271 0.435135 0.900365i \(-0.356701\pi\)
0.435135 + 0.900365i \(0.356701\pi\)
\(938\) 2.10992i 0.0688912i
\(939\) −8.63640 −0.281838
\(940\) 13.0978 0.427204
\(941\) − 25.7198i − 0.838440i −0.907885 0.419220i \(-0.862304\pi\)
0.907885 0.419220i \(-0.137696\pi\)
\(942\) − 15.5429i − 0.506414i
\(943\) − 101.832i − 3.31609i
\(944\) − 1.62565i − 0.0529103i
\(945\) 1.35690 0.0441398
\(946\) −9.31229 −0.302769
\(947\) − 14.4862i − 0.470738i −0.971906 0.235369i \(-0.924370\pi\)
0.971906 0.235369i \(-0.0756298\pi\)
\(948\) −1.42327 −0.0462257
\(949\) 0 0
\(950\) 5.34481 0.173409
\(951\) 17.6256i 0.571551i
\(952\) 2.78017 0.0901057
\(953\) 29.3690 0.951354 0.475677 0.879620i \(-0.342203\pi\)
0.475677 + 0.879620i \(0.342203\pi\)
\(954\) 5.74094i 0.185870i
\(955\) 11.8280i 0.382746i
\(956\) − 7.16315i − 0.231673i
\(957\) − 25.8106i − 0.834337i
\(958\) −31.6305 −1.02194
\(959\) −27.5303 −0.889000
\(960\) 1.00000i 0.0322749i
\(961\) 30.9654 0.998884
\(962\) 0 0
\(963\) −5.08815 −0.163963
\(964\) − 17.9584i − 0.578400i
\(965\) −0.872625 −0.0280908
\(966\) 11.2935 0.363363
\(967\) − 21.9360i − 0.705415i −0.935734 0.352707i \(-0.885261\pi\)
0.935734 0.352707i \(-0.114739\pi\)
\(968\) − 0.772398i − 0.0248258i
\(969\) − 10.9511i − 0.351799i
\(970\) − 4.74094i − 0.152222i
\(971\) 34.9754 1.12241 0.561206 0.827676i \(-0.310337\pi\)
0.561206 + 0.827676i \(0.310337\pi\)
\(972\) 1.00000 0.0320750
\(973\) − 0.208225i − 0.00667538i
\(974\) −21.7748 −0.697709
\(975\) 0 0
\(976\) 5.28382 0.169131
\(977\) − 7.72289i − 0.247077i −0.992340 0.123539i \(-0.960576\pi\)
0.992340 0.123539i \(-0.0394243\pi\)
\(978\) −2.00239 −0.0640295
\(979\) −10.8052 −0.345335
\(980\) − 5.15883i − 0.164793i
\(981\) − 6.43296i − 0.205389i
\(982\) − 42.9590i − 1.37088i
\(983\) − 29.0489i − 0.926517i −0.886223 0.463258i \(-0.846680\pi\)
0.886223 0.463258i \(-0.153320\pi\)
\(984\) 12.2349 0.390034
\(985\) 10.6213 0.338424
\(986\) 16.5362i 0.526619i
\(987\) 17.7724 0.565702
\(988\) 0 0
\(989\) 24.2355 0.770644
\(990\) − 3.19806i − 0.101641i
\(991\) 37.7192 1.19819 0.599094 0.800678i \(-0.295527\pi\)
0.599094 + 0.800678i \(0.295527\pi\)
\(992\) −0.185981 −0.00590489
\(993\) − 31.5163i − 1.00014i
\(994\) − 3.15213i − 0.0999795i
\(995\) − 12.3424i − 0.391281i
\(996\) − 11.1196i − 0.352338i
\(997\) −7.18060 −0.227412 −0.113706 0.993514i \(-0.536272\pi\)
−0.113706 + 0.993514i \(0.536272\pi\)
\(998\) 36.0978 1.14266
\(999\) 2.46681i 0.0780465i
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.s.1351.5 6
13.5 odd 4 5070.2.a.bt.1.2 yes 3
13.8 odd 4 5070.2.a.bk.1.2 3
13.12 even 2 inner 5070.2.b.s.1351.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bk.1.2 3 13.8 odd 4
5070.2.a.bt.1.2 yes 3 13.5 odd 4
5070.2.b.s.1351.2 6 13.12 even 2 inner
5070.2.b.s.1351.5 6 1.1 even 1 trivial