Properties

Label 5070.2.a.bk.1.2
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.35690 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.35690 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.19806 q^{11} -1.00000 q^{12} -1.35690 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.04892 q^{17} -1.00000 q^{18} -5.34481 q^{19} -1.00000 q^{20} -1.35690 q^{21} -3.19806 q^{22} -8.32304 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +1.35690 q^{28} +8.07069 q^{29} -1.00000 q^{30} -0.185981 q^{31} -1.00000 q^{32} -3.19806 q^{33} +2.04892 q^{34} -1.35690 q^{35} +1.00000 q^{36} -2.46681 q^{37} +5.34481 q^{38} +1.00000 q^{40} +12.2349 q^{41} +1.35690 q^{42} -2.91185 q^{43} +3.19806 q^{44} -1.00000 q^{45} +8.32304 q^{46} +13.0978 q^{47} -1.00000 q^{48} -5.15883 q^{49} -1.00000 q^{50} +2.04892 q^{51} +5.74094 q^{53} +1.00000 q^{54} -3.19806 q^{55} -1.35690 q^{56} +5.34481 q^{57} -8.07069 q^{58} -1.62565 q^{59} +1.00000 q^{60} +5.28382 q^{61} +0.185981 q^{62} +1.35690 q^{63} +1.00000 q^{64} +3.19806 q^{66} +1.55496 q^{67} -2.04892 q^{68} +8.32304 q^{69} +1.35690 q^{70} -2.32304 q^{71} -1.00000 q^{72} -5.11529 q^{73} +2.46681 q^{74} -1.00000 q^{75} -5.34481 q^{76} +4.33944 q^{77} -1.42327 q^{79} -1.00000 q^{80} +1.00000 q^{81} -12.2349 q^{82} +11.1196 q^{83} -1.35690 q^{84} +2.04892 q^{85} +2.91185 q^{86} -8.07069 q^{87} -3.19806 q^{88} +3.37867 q^{89} +1.00000 q^{90} -8.32304 q^{92} +0.185981 q^{93} -13.0978 q^{94} +5.34481 q^{95} +1.00000 q^{96} -4.74094 q^{97} +5.15883 q^{98} +3.19806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} - 3q^{5} + 3q^{6} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} - 3q^{5} + 3q^{6} - 3q^{8} + 3q^{9} + 3q^{10} + 14q^{11} - 3q^{12} + 3q^{15} + 3q^{16} + 3q^{17} - 3q^{18} + 7q^{19} - 3q^{20} - 14q^{22} - 5q^{23} + 3q^{24} + 3q^{25} - 3q^{27} + 12q^{29} - 3q^{30} + 14q^{31} - 3q^{32} - 14q^{33} - 3q^{34} + 3q^{36} - 4q^{37} - 7q^{38} + 3q^{40} + 13q^{41} - 5q^{43} + 14q^{44} - 3q^{45} + 5q^{46} + 21q^{47} - 3q^{48} - 7q^{49} - 3q^{50} - 3q^{51} + 3q^{53} + 3q^{54} - 14q^{55} - 7q^{57} - 12q^{58} + 7q^{59} + 3q^{60} - 17q^{61} - 14q^{62} + 3q^{64} + 14q^{66} + 5q^{67} + 3q^{68} + 5q^{69} + 13q^{71} - 3q^{72} - 13q^{73} + 4q^{74} - 3q^{75} + 7q^{76} - 7q^{77} - 7q^{79} - 3q^{80} + 3q^{81} - 13q^{82} + 12q^{83} - 3q^{85} + 5q^{86} - 12q^{87} - 14q^{88} + 3q^{89} + 3q^{90} - 5q^{92} - 14q^{93} - 21q^{94} - 7q^{95} + 3q^{96} + 7q^{98} + 14q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 1.35690 0.512858 0.256429 0.966563i \(-0.417454\pi\)
0.256429 + 0.966563i \(0.417454\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 3.19806 0.964252 0.482126 0.876102i \(-0.339865\pi\)
0.482126 + 0.876102i \(0.339865\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.35690 −0.362646
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −2.04892 −0.496935 −0.248468 0.968640i \(-0.579927\pi\)
−0.248468 + 0.968640i \(0.579927\pi\)
\(18\) −1.00000 −0.235702
\(19\) −5.34481 −1.22618 −0.613092 0.790011i \(-0.710075\pi\)
−0.613092 + 0.790011i \(0.710075\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.35690 −0.296099
\(22\) −3.19806 −0.681829
\(23\) −8.32304 −1.73547 −0.867737 0.497023i \(-0.834426\pi\)
−0.867737 + 0.497023i \(0.834426\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.35690 0.256429
\(29\) 8.07069 1.49869 0.749345 0.662180i \(-0.230369\pi\)
0.749345 + 0.662180i \(0.230369\pi\)
\(30\) −1.00000 −0.182574
\(31\) −0.185981 −0.0334031 −0.0167016 0.999861i \(-0.505317\pi\)
−0.0167016 + 0.999861i \(0.505317\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.19806 −0.556711
\(34\) 2.04892 0.351386
\(35\) −1.35690 −0.229357
\(36\) 1.00000 0.166667
\(37\) −2.46681 −0.405541 −0.202771 0.979226i \(-0.564995\pi\)
−0.202771 + 0.979226i \(0.564995\pi\)
\(38\) 5.34481 0.867043
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 12.2349 1.91077 0.955385 0.295363i \(-0.0954407\pi\)
0.955385 + 0.295363i \(0.0954407\pi\)
\(42\) 1.35690 0.209374
\(43\) −2.91185 −0.444054 −0.222027 0.975041i \(-0.571267\pi\)
−0.222027 + 0.975041i \(0.571267\pi\)
\(44\) 3.19806 0.482126
\(45\) −1.00000 −0.149071
\(46\) 8.32304 1.22717
\(47\) 13.0978 1.91052 0.955258 0.295775i \(-0.0955777\pi\)
0.955258 + 0.295775i \(0.0955777\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.15883 −0.736976
\(50\) −1.00000 −0.141421
\(51\) 2.04892 0.286906
\(52\) 0 0
\(53\) 5.74094 0.788579 0.394289 0.918986i \(-0.370991\pi\)
0.394289 + 0.918986i \(0.370991\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.19806 −0.431227
\(56\) −1.35690 −0.181323
\(57\) 5.34481 0.707938
\(58\) −8.07069 −1.05973
\(59\) −1.62565 −0.211641 −0.105821 0.994385i \(-0.533747\pi\)
−0.105821 + 0.994385i \(0.533747\pi\)
\(60\) 1.00000 0.129099
\(61\) 5.28382 0.676523 0.338262 0.941052i \(-0.390161\pi\)
0.338262 + 0.941052i \(0.390161\pi\)
\(62\) 0.185981 0.0236196
\(63\) 1.35690 0.170953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.19806 0.393654
\(67\) 1.55496 0.189968 0.0949842 0.995479i \(-0.469720\pi\)
0.0949842 + 0.995479i \(0.469720\pi\)
\(68\) −2.04892 −0.248468
\(69\) 8.32304 1.00198
\(70\) 1.35690 0.162180
\(71\) −2.32304 −0.275695 −0.137847 0.990453i \(-0.544018\pi\)
−0.137847 + 0.990453i \(0.544018\pi\)
\(72\) −1.00000 −0.117851
\(73\) −5.11529 −0.598700 −0.299350 0.954143i \(-0.596770\pi\)
−0.299350 + 0.954143i \(0.596770\pi\)
\(74\) 2.46681 0.286761
\(75\) −1.00000 −0.115470
\(76\) −5.34481 −0.613092
\(77\) 4.33944 0.494525
\(78\) 0 0
\(79\) −1.42327 −0.160131 −0.0800653 0.996790i \(-0.525513\pi\)
−0.0800653 + 0.996790i \(0.525513\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −12.2349 −1.35112
\(83\) 11.1196 1.22054 0.610268 0.792195i \(-0.291062\pi\)
0.610268 + 0.792195i \(0.291062\pi\)
\(84\) −1.35690 −0.148049
\(85\) 2.04892 0.222236
\(86\) 2.91185 0.313993
\(87\) −8.07069 −0.865269
\(88\) −3.19806 −0.340915
\(89\) 3.37867 0.358138 0.179069 0.983837i \(-0.442691\pi\)
0.179069 + 0.983837i \(0.442691\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −8.32304 −0.867737
\(93\) 0.185981 0.0192853
\(94\) −13.0978 −1.35094
\(95\) 5.34481 0.548366
\(96\) 1.00000 0.102062
\(97\) −4.74094 −0.481369 −0.240685 0.970603i \(-0.577372\pi\)
−0.240685 + 0.970603i \(0.577372\pi\)
\(98\) 5.15883 0.521121
\(99\) 3.19806 0.321417
\(100\) 1.00000 0.100000
\(101\) −12.5918 −1.25293 −0.626465 0.779449i \(-0.715499\pi\)
−0.626465 + 0.779449i \(0.715499\pi\)
\(102\) −2.04892 −0.202873
\(103\) −9.00000 −0.886796 −0.443398 0.896325i \(-0.646227\pi\)
−0.443398 + 0.896325i \(0.646227\pi\)
\(104\) 0 0
\(105\) 1.35690 0.132419
\(106\) −5.74094 −0.557609
\(107\) −5.08815 −0.491890 −0.245945 0.969284i \(-0.579098\pi\)
−0.245945 + 0.969284i \(0.579098\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.43296 0.616166 0.308083 0.951360i \(-0.400313\pi\)
0.308083 + 0.951360i \(0.400313\pi\)
\(110\) 3.19806 0.304923
\(111\) 2.46681 0.234139
\(112\) 1.35690 0.128215
\(113\) −18.6571 −1.75511 −0.877556 0.479473i \(-0.840828\pi\)
−0.877556 + 0.479473i \(0.840828\pi\)
\(114\) −5.34481 −0.500588
\(115\) 8.32304 0.776128
\(116\) 8.07069 0.749345
\(117\) 0 0
\(118\) 1.62565 0.149653
\(119\) −2.78017 −0.254858
\(120\) −1.00000 −0.0912871
\(121\) −0.772398 −0.0702180
\(122\) −5.28382 −0.478374
\(123\) −12.2349 −1.10318
\(124\) −0.185981 −0.0167016
\(125\) −1.00000 −0.0894427
\(126\) −1.35690 −0.120882
\(127\) −16.3763 −1.45316 −0.726580 0.687082i \(-0.758891\pi\)
−0.726580 + 0.687082i \(0.758891\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.91185 0.256374
\(130\) 0 0
\(131\) 3.00969 0.262958 0.131479 0.991319i \(-0.458027\pi\)
0.131479 + 0.991319i \(0.458027\pi\)
\(132\) −3.19806 −0.278356
\(133\) −7.25236 −0.628859
\(134\) −1.55496 −0.134328
\(135\) 1.00000 0.0860663
\(136\) 2.04892 0.175693
\(137\) 20.2892 1.73342 0.866711 0.498810i \(-0.166229\pi\)
0.866711 + 0.498810i \(0.166229\pi\)
\(138\) −8.32304 −0.708505
\(139\) −0.153457 −0.0130160 −0.00650802 0.999979i \(-0.502072\pi\)
−0.00650802 + 0.999979i \(0.502072\pi\)
\(140\) −1.35690 −0.114679
\(141\) −13.0978 −1.10304
\(142\) 2.32304 0.194946
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −8.07069 −0.670234
\(146\) 5.11529 0.423345
\(147\) 5.15883 0.425493
\(148\) −2.46681 −0.202771
\(149\) −10.7724 −0.882509 −0.441255 0.897382i \(-0.645466\pi\)
−0.441255 + 0.897382i \(0.645466\pi\)
\(150\) 1.00000 0.0816497
\(151\) 17.3817 1.41450 0.707249 0.706964i \(-0.249936\pi\)
0.707249 + 0.706964i \(0.249936\pi\)
\(152\) 5.34481 0.433522
\(153\) −2.04892 −0.165645
\(154\) −4.33944 −0.349682
\(155\) 0.185981 0.0149383
\(156\) 0 0
\(157\) 15.5429 1.24046 0.620228 0.784421i \(-0.287040\pi\)
0.620228 + 0.784421i \(0.287040\pi\)
\(158\) 1.42327 0.113229
\(159\) −5.74094 −0.455286
\(160\) 1.00000 0.0790569
\(161\) −11.2935 −0.890053
\(162\) −1.00000 −0.0785674
\(163\) −2.00239 −0.156840 −0.0784198 0.996920i \(-0.524987\pi\)
−0.0784198 + 0.996920i \(0.524987\pi\)
\(164\) 12.2349 0.955385
\(165\) 3.19806 0.248969
\(166\) −11.1196 −0.863049
\(167\) −8.24027 −0.637652 −0.318826 0.947813i \(-0.603288\pi\)
−0.318826 + 0.947813i \(0.603288\pi\)
\(168\) 1.35690 0.104687
\(169\) 0 0
\(170\) −2.04892 −0.157145
\(171\) −5.34481 −0.408728
\(172\) −2.91185 −0.222027
\(173\) 12.7463 0.969084 0.484542 0.874768i \(-0.338986\pi\)
0.484542 + 0.874768i \(0.338986\pi\)
\(174\) 8.07069 0.611837
\(175\) 1.35690 0.102572
\(176\) 3.19806 0.241063
\(177\) 1.62565 0.122191
\(178\) −3.37867 −0.253242
\(179\) 16.0422 1.19905 0.599526 0.800356i \(-0.295356\pi\)
0.599526 + 0.800356i \(0.295356\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 3.85086 0.286232 0.143116 0.989706i \(-0.454288\pi\)
0.143116 + 0.989706i \(0.454288\pi\)
\(182\) 0 0
\(183\) −5.28382 −0.390591
\(184\) 8.32304 0.613583
\(185\) 2.46681 0.181364
\(186\) −0.185981 −0.0136368
\(187\) −6.55257 −0.479171
\(188\) 13.0978 0.955258
\(189\) −1.35690 −0.0986997
\(190\) −5.34481 −0.387754
\(191\) 11.8280 0.855845 0.427923 0.903815i \(-0.359246\pi\)
0.427923 + 0.903815i \(0.359246\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 0.872625 0.0628129 0.0314065 0.999507i \(-0.490001\pi\)
0.0314065 + 0.999507i \(0.490001\pi\)
\(194\) 4.74094 0.340380
\(195\) 0 0
\(196\) −5.15883 −0.368488
\(197\) 10.6213 0.756739 0.378369 0.925655i \(-0.376485\pi\)
0.378369 + 0.925655i \(0.376485\pi\)
\(198\) −3.19806 −0.227276
\(199\) 12.3424 0.874931 0.437466 0.899235i \(-0.355876\pi\)
0.437466 + 0.899235i \(0.355876\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −1.55496 −0.109678
\(202\) 12.5918 0.885956
\(203\) 10.9511 0.768615
\(204\) 2.04892 0.143453
\(205\) −12.2349 −0.854522
\(206\) 9.00000 0.627060
\(207\) −8.32304 −0.578492
\(208\) 0 0
\(209\) −17.0930 −1.18235
\(210\) −1.35690 −0.0936347
\(211\) 6.75733 0.465194 0.232597 0.972573i \(-0.425278\pi\)
0.232597 + 0.972573i \(0.425278\pi\)
\(212\) 5.74094 0.394289
\(213\) 2.32304 0.159172
\(214\) 5.08815 0.347819
\(215\) 2.91185 0.198587
\(216\) 1.00000 0.0680414
\(217\) −0.252356 −0.0171311
\(218\) −6.43296 −0.435695
\(219\) 5.11529 0.345659
\(220\) −3.19806 −0.215613
\(221\) 0 0
\(222\) −2.46681 −0.165562
\(223\) 21.5972 1.44625 0.723127 0.690715i \(-0.242704\pi\)
0.723127 + 0.690715i \(0.242704\pi\)
\(224\) −1.35690 −0.0906614
\(225\) 1.00000 0.0666667
\(226\) 18.6571 1.24105
\(227\) 10.2862 0.682720 0.341360 0.939933i \(-0.389113\pi\)
0.341360 + 0.939933i \(0.389113\pi\)
\(228\) 5.34481 0.353969
\(229\) 4.25906 0.281447 0.140723 0.990049i \(-0.455057\pi\)
0.140723 + 0.990049i \(0.455057\pi\)
\(230\) −8.32304 −0.548805
\(231\) −4.33944 −0.285514
\(232\) −8.07069 −0.529867
\(233\) 19.5133 1.27836 0.639181 0.769057i \(-0.279274\pi\)
0.639181 + 0.769057i \(0.279274\pi\)
\(234\) 0 0
\(235\) −13.0978 −0.854409
\(236\) −1.62565 −0.105821
\(237\) 1.42327 0.0924514
\(238\) 2.78017 0.180211
\(239\) −7.16315 −0.463345 −0.231673 0.972794i \(-0.574420\pi\)
−0.231673 + 0.972794i \(0.574420\pi\)
\(240\) 1.00000 0.0645497
\(241\) 17.9584 1.15680 0.578400 0.815753i \(-0.303677\pi\)
0.578400 + 0.815753i \(0.303677\pi\)
\(242\) 0.772398 0.0496516
\(243\) −1.00000 −0.0641500
\(244\) 5.28382 0.338262
\(245\) 5.15883 0.329586
\(246\) 12.2349 0.780069
\(247\) 0 0
\(248\) 0.185981 0.0118098
\(249\) −11.1196 −0.704676
\(250\) 1.00000 0.0632456
\(251\) −6.68425 −0.421906 −0.210953 0.977496i \(-0.567657\pi\)
−0.210953 + 0.977496i \(0.567657\pi\)
\(252\) 1.35690 0.0854764
\(253\) −26.6176 −1.67343
\(254\) 16.3763 1.02754
\(255\) −2.04892 −0.128308
\(256\) 1.00000 0.0625000
\(257\) 9.91484 0.618471 0.309235 0.950986i \(-0.399927\pi\)
0.309235 + 0.950986i \(0.399927\pi\)
\(258\) −2.91185 −0.181284
\(259\) −3.34721 −0.207985
\(260\) 0 0
\(261\) 8.07069 0.499563
\(262\) −3.00969 −0.185939
\(263\) 13.8509 0.854080 0.427040 0.904233i \(-0.359556\pi\)
0.427040 + 0.904233i \(0.359556\pi\)
\(264\) 3.19806 0.196827
\(265\) −5.74094 −0.352663
\(266\) 7.25236 0.444671
\(267\) −3.37867 −0.206771
\(268\) 1.55496 0.0949842
\(269\) −17.8756 −1.08990 −0.544948 0.838470i \(-0.683450\pi\)
−0.544948 + 0.838470i \(0.683450\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 31.2083 1.89577 0.947886 0.318610i \(-0.103216\pi\)
0.947886 + 0.318610i \(0.103216\pi\)
\(272\) −2.04892 −0.124234
\(273\) 0 0
\(274\) −20.2892 −1.22571
\(275\) 3.19806 0.192850
\(276\) 8.32304 0.500988
\(277\) 22.9245 1.37740 0.688701 0.725046i \(-0.258181\pi\)
0.688701 + 0.725046i \(0.258181\pi\)
\(278\) 0.153457 0.00920373
\(279\) −0.185981 −0.0111344
\(280\) 1.35690 0.0810900
\(281\) 28.0683 1.67441 0.837207 0.546886i \(-0.184187\pi\)
0.837207 + 0.546886i \(0.184187\pi\)
\(282\) 13.0978 0.779965
\(283\) −24.8364 −1.47637 −0.738185 0.674599i \(-0.764317\pi\)
−0.738185 + 0.674599i \(0.764317\pi\)
\(284\) −2.32304 −0.137847
\(285\) −5.34481 −0.316599
\(286\) 0 0
\(287\) 16.6015 0.979955
\(288\) −1.00000 −0.0589256
\(289\) −12.8019 −0.753055
\(290\) 8.07069 0.473927
\(291\) 4.74094 0.277919
\(292\) −5.11529 −0.299350
\(293\) −28.4349 −1.66118 −0.830592 0.556882i \(-0.811998\pi\)
−0.830592 + 0.556882i \(0.811998\pi\)
\(294\) −5.15883 −0.300869
\(295\) 1.62565 0.0946488
\(296\) 2.46681 0.143381
\(297\) −3.19806 −0.185570
\(298\) 10.7724 0.624028
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −3.95108 −0.227737
\(302\) −17.3817 −1.00020
\(303\) 12.5918 0.723380
\(304\) −5.34481 −0.306546
\(305\) −5.28382 −0.302550
\(306\) 2.04892 0.117129
\(307\) −10.1347 −0.578416 −0.289208 0.957266i \(-0.593392\pi\)
−0.289208 + 0.957266i \(0.593392\pi\)
\(308\) 4.33944 0.247262
\(309\) 9.00000 0.511992
\(310\) −0.185981 −0.0105630
\(311\) 20.0218 1.13533 0.567665 0.823259i \(-0.307847\pi\)
0.567665 + 0.823259i \(0.307847\pi\)
\(312\) 0 0
\(313\) 8.63640 0.488158 0.244079 0.969755i \(-0.421514\pi\)
0.244079 + 0.969755i \(0.421514\pi\)
\(314\) −15.5429 −0.877135
\(315\) −1.35690 −0.0764524
\(316\) −1.42327 −0.0800653
\(317\) 17.6256 0.989955 0.494977 0.868906i \(-0.335176\pi\)
0.494977 + 0.868906i \(0.335176\pi\)
\(318\) 5.74094 0.321936
\(319\) 25.8106 1.44511
\(320\) −1.00000 −0.0559017
\(321\) 5.08815 0.283993
\(322\) 11.2935 0.629362
\(323\) 10.9511 0.609335
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 2.00239 0.110902
\(327\) −6.43296 −0.355744
\(328\) −12.2349 −0.675559
\(329\) 17.7724 0.979824
\(330\) −3.19806 −0.176048
\(331\) −31.5163 −1.73229 −0.866147 0.499790i \(-0.833411\pi\)
−0.866147 + 0.499790i \(0.833411\pi\)
\(332\) 11.1196 0.610268
\(333\) −2.46681 −0.135180
\(334\) 8.24027 0.450888
\(335\) −1.55496 −0.0849564
\(336\) −1.35690 −0.0740247
\(337\) −11.8116 −0.643420 −0.321710 0.946838i \(-0.604258\pi\)
−0.321710 + 0.946838i \(0.604258\pi\)
\(338\) 0 0
\(339\) 18.6571 1.01331
\(340\) 2.04892 0.111118
\(341\) −0.594778 −0.0322090
\(342\) 5.34481 0.289014
\(343\) −16.4983 −0.890823
\(344\) 2.91185 0.156997
\(345\) −8.32304 −0.448098
\(346\) −12.7463 −0.685246
\(347\) −17.4101 −0.934624 −0.467312 0.884092i \(-0.654778\pi\)
−0.467312 + 0.884092i \(0.654778\pi\)
\(348\) −8.07069 −0.432634
\(349\) 18.6799 0.999914 0.499957 0.866050i \(-0.333349\pi\)
0.499957 + 0.866050i \(0.333349\pi\)
\(350\) −1.35690 −0.0725291
\(351\) 0 0
\(352\) −3.19806 −0.170457
\(353\) 25.3991 1.35186 0.675929 0.736967i \(-0.263743\pi\)
0.675929 + 0.736967i \(0.263743\pi\)
\(354\) −1.62565 −0.0864021
\(355\) 2.32304 0.123294
\(356\) 3.37867 0.179069
\(357\) 2.78017 0.147142
\(358\) −16.0422 −0.847857
\(359\) 34.0103 1.79499 0.897497 0.441021i \(-0.145383\pi\)
0.897497 + 0.441021i \(0.145383\pi\)
\(360\) 1.00000 0.0527046
\(361\) 9.56704 0.503528
\(362\) −3.85086 −0.202396
\(363\) 0.772398 0.0405404
\(364\) 0 0
\(365\) 5.11529 0.267747
\(366\) 5.28382 0.276189
\(367\) −25.6437 −1.33859 −0.669295 0.742997i \(-0.733404\pi\)
−0.669295 + 0.742997i \(0.733404\pi\)
\(368\) −8.32304 −0.433869
\(369\) 12.2349 0.636923
\(370\) −2.46681 −0.128243
\(371\) 7.78986 0.404429
\(372\) 0.185981 0.00964265
\(373\) −16.3612 −0.847151 −0.423576 0.905861i \(-0.639225\pi\)
−0.423576 + 0.905861i \(0.639225\pi\)
\(374\) 6.55257 0.338825
\(375\) 1.00000 0.0516398
\(376\) −13.0978 −0.675469
\(377\) 0 0
\(378\) 1.35690 0.0697912
\(379\) 22.8823 1.17539 0.587693 0.809084i \(-0.300036\pi\)
0.587693 + 0.809084i \(0.300036\pi\)
\(380\) 5.34481 0.274183
\(381\) 16.3763 0.838982
\(382\) −11.8280 −0.605174
\(383\) 11.9191 0.609040 0.304520 0.952506i \(-0.401504\pi\)
0.304520 + 0.952506i \(0.401504\pi\)
\(384\) 1.00000 0.0510310
\(385\) −4.33944 −0.221158
\(386\) −0.872625 −0.0444155
\(387\) −2.91185 −0.148018
\(388\) −4.74094 −0.240685
\(389\) 8.07606 0.409473 0.204736 0.978817i \(-0.434366\pi\)
0.204736 + 0.978817i \(0.434366\pi\)
\(390\) 0 0
\(391\) 17.0532 0.862419
\(392\) 5.15883 0.260560
\(393\) −3.00969 −0.151819
\(394\) −10.6213 −0.535095
\(395\) 1.42327 0.0716126
\(396\) 3.19806 0.160709
\(397\) −20.5230 −1.03002 −0.515011 0.857184i \(-0.672212\pi\)
−0.515011 + 0.857184i \(0.672212\pi\)
\(398\) −12.3424 −0.618670
\(399\) 7.25236 0.363072
\(400\) 1.00000 0.0500000
\(401\) −18.3274 −0.915224 −0.457612 0.889152i \(-0.651295\pi\)
−0.457612 + 0.889152i \(0.651295\pi\)
\(402\) 1.55496 0.0775543
\(403\) 0 0
\(404\) −12.5918 −0.626465
\(405\) −1.00000 −0.0496904
\(406\) −10.9511 −0.543493
\(407\) −7.88902 −0.391044
\(408\) −2.04892 −0.101437
\(409\) 9.00000 0.445021 0.222511 0.974930i \(-0.428575\pi\)
0.222511 + 0.974930i \(0.428575\pi\)
\(410\) 12.2349 0.604239
\(411\) −20.2892 −1.00079
\(412\) −9.00000 −0.443398
\(413\) −2.20583 −0.108542
\(414\) 8.32304 0.409055
\(415\) −11.1196 −0.545840
\(416\) 0 0
\(417\) 0.153457 0.00751481
\(418\) 17.0930 0.836048
\(419\) −14.4185 −0.704389 −0.352195 0.935927i \(-0.614564\pi\)
−0.352195 + 0.935927i \(0.614564\pi\)
\(420\) 1.35690 0.0662097
\(421\) −19.9801 −0.973773 −0.486886 0.873465i \(-0.661867\pi\)
−0.486886 + 0.873465i \(0.661867\pi\)
\(422\) −6.75733 −0.328942
\(423\) 13.0978 0.636839
\(424\) −5.74094 −0.278805
\(425\) −2.04892 −0.0993871
\(426\) −2.32304 −0.112552
\(427\) 7.16959 0.346961
\(428\) −5.08815 −0.245945
\(429\) 0 0
\(430\) −2.91185 −0.140422
\(431\) −21.1812 −1.02026 −0.510131 0.860097i \(-0.670403\pi\)
−0.510131 + 0.860097i \(0.670403\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 31.0291 1.49116 0.745581 0.666415i \(-0.232172\pi\)
0.745581 + 0.666415i \(0.232172\pi\)
\(434\) 0.252356 0.0121135
\(435\) 8.07069 0.386960
\(436\) 6.43296 0.308083
\(437\) 44.4851 2.12801
\(438\) −5.11529 −0.244418
\(439\) 7.33406 0.350036 0.175018 0.984565i \(-0.444002\pi\)
0.175018 + 0.984565i \(0.444002\pi\)
\(440\) 3.19806 0.152462
\(441\) −5.15883 −0.245659
\(442\) 0 0
\(443\) −18.1360 −0.861667 −0.430834 0.902431i \(-0.641780\pi\)
−0.430834 + 0.902431i \(0.641780\pi\)
\(444\) 2.46681 0.117070
\(445\) −3.37867 −0.160164
\(446\) −21.5972 −1.02266
\(447\) 10.7724 0.509517
\(448\) 1.35690 0.0641073
\(449\) 4.55496 0.214962 0.107481 0.994207i \(-0.465722\pi\)
0.107481 + 0.994207i \(0.465722\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 39.1280 1.84246
\(452\) −18.6571 −0.877556
\(453\) −17.3817 −0.816661
\(454\) −10.2862 −0.482756
\(455\) 0 0
\(456\) −5.34481 −0.250294
\(457\) −9.98121 −0.466901 −0.233451 0.972369i \(-0.575002\pi\)
−0.233451 + 0.972369i \(0.575002\pi\)
\(458\) −4.25906 −0.199013
\(459\) 2.04892 0.0956353
\(460\) 8.32304 0.388064
\(461\) 16.1672 0.752981 0.376491 0.926420i \(-0.377131\pi\)
0.376491 + 0.926420i \(0.377131\pi\)
\(462\) 4.33944 0.201889
\(463\) −3.88902 −0.180738 −0.0903690 0.995908i \(-0.528805\pi\)
−0.0903690 + 0.995908i \(0.528805\pi\)
\(464\) 8.07069 0.374672
\(465\) −0.185981 −0.00862465
\(466\) −19.5133 −0.903938
\(467\) −0.872625 −0.0403803 −0.0201901 0.999796i \(-0.506427\pi\)
−0.0201901 + 0.999796i \(0.506427\pi\)
\(468\) 0 0
\(469\) 2.10992 0.0974269
\(470\) 13.0978 0.604158
\(471\) −15.5429 −0.716178
\(472\) 1.62565 0.0748264
\(473\) −9.31229 −0.428180
\(474\) −1.42327 −0.0653730
\(475\) −5.34481 −0.245237
\(476\) −2.78017 −0.127429
\(477\) 5.74094 0.262860
\(478\) 7.16315 0.327635
\(479\) 31.6305 1.44524 0.722618 0.691247i \(-0.242938\pi\)
0.722618 + 0.691247i \(0.242938\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −17.9584 −0.817982
\(483\) 11.2935 0.513872
\(484\) −0.772398 −0.0351090
\(485\) 4.74094 0.215275
\(486\) 1.00000 0.0453609
\(487\) −21.7748 −0.986710 −0.493355 0.869828i \(-0.664230\pi\)
−0.493355 + 0.869828i \(0.664230\pi\)
\(488\) −5.28382 −0.239187
\(489\) 2.00239 0.0905513
\(490\) −5.15883 −0.233052
\(491\) 42.9590 1.93871 0.969356 0.245662i \(-0.0790053\pi\)
0.969356 + 0.245662i \(0.0790053\pi\)
\(492\) −12.2349 −0.551592
\(493\) −16.5362 −0.744752
\(494\) 0 0
\(495\) −3.19806 −0.143742
\(496\) −0.185981 −0.00835078
\(497\) −3.15213 −0.141392
\(498\) 11.1196 0.498281
\(499\) 36.0978 1.61596 0.807981 0.589209i \(-0.200561\pi\)
0.807981 + 0.589209i \(0.200561\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 8.24027 0.368148
\(502\) 6.68425 0.298333
\(503\) −21.4534 −0.956560 −0.478280 0.878207i \(-0.658740\pi\)
−0.478280 + 0.878207i \(0.658740\pi\)
\(504\) −1.35690 −0.0604409
\(505\) 12.5918 0.560327
\(506\) 26.6176 1.18330
\(507\) 0 0
\(508\) −16.3763 −0.726580
\(509\) −0.545860 −0.0241948 −0.0120974 0.999927i \(-0.503851\pi\)
−0.0120974 + 0.999927i \(0.503851\pi\)
\(510\) 2.04892 0.0907276
\(511\) −6.94092 −0.307048
\(512\) −1.00000 −0.0441942
\(513\) 5.34481 0.235979
\(514\) −9.91484 −0.437325
\(515\) 9.00000 0.396587
\(516\) 2.91185 0.128187
\(517\) 41.8877 1.84222
\(518\) 3.34721 0.147068
\(519\) −12.7463 −0.559501
\(520\) 0 0
\(521\) 29.6136 1.29739 0.648697 0.761047i \(-0.275314\pi\)
0.648697 + 0.761047i \(0.275314\pi\)
\(522\) −8.07069 −0.353244
\(523\) 44.4543 1.94385 0.971924 0.235293i \(-0.0756050\pi\)
0.971924 + 0.235293i \(0.0756050\pi\)
\(524\) 3.00969 0.131479
\(525\) −1.35690 −0.0592198
\(526\) −13.8509 −0.603926
\(527\) 0.381059 0.0165992
\(528\) −3.19806 −0.139178
\(529\) 46.2731 2.01187
\(530\) 5.74094 0.249370
\(531\) −1.62565 −0.0705470
\(532\) −7.25236 −0.314430
\(533\) 0 0
\(534\) 3.37867 0.146209
\(535\) 5.08815 0.219980
\(536\) −1.55496 −0.0671640
\(537\) −16.0422 −0.692273
\(538\) 17.8756 0.770672
\(539\) −16.4983 −0.710631
\(540\) 1.00000 0.0430331
\(541\) −18.7651 −0.806775 −0.403387 0.915029i \(-0.632167\pi\)
−0.403387 + 0.915029i \(0.632167\pi\)
\(542\) −31.2083 −1.34051
\(543\) −3.85086 −0.165256
\(544\) 2.04892 0.0878466
\(545\) −6.43296 −0.275558
\(546\) 0 0
\(547\) 32.0116 1.36872 0.684359 0.729145i \(-0.260082\pi\)
0.684359 + 0.729145i \(0.260082\pi\)
\(548\) 20.2892 0.866711
\(549\) 5.28382 0.225508
\(550\) −3.19806 −0.136366
\(551\) −43.1363 −1.83767
\(552\) −8.32304 −0.354252
\(553\) −1.93123 −0.0821243
\(554\) −22.9245 −0.973970
\(555\) −2.46681 −0.104710
\(556\) −0.153457 −0.00650802
\(557\) 8.23490 0.348924 0.174462 0.984664i \(-0.444181\pi\)
0.174462 + 0.984664i \(0.444181\pi\)
\(558\) 0.185981 0.00787319
\(559\) 0 0
\(560\) −1.35690 −0.0573393
\(561\) 6.55257 0.276650
\(562\) −28.0683 −1.18399
\(563\) −30.2295 −1.27402 −0.637011 0.770855i \(-0.719830\pi\)
−0.637011 + 0.770855i \(0.719830\pi\)
\(564\) −13.0978 −0.551518
\(565\) 18.6571 0.784910
\(566\) 24.8364 1.04395
\(567\) 1.35690 0.0569843
\(568\) 2.32304 0.0974728
\(569\) 44.3110 1.85761 0.928806 0.370566i \(-0.120836\pi\)
0.928806 + 0.370566i \(0.120836\pi\)
\(570\) 5.34481 0.223870
\(571\) 34.4873 1.44325 0.721623 0.692286i \(-0.243396\pi\)
0.721623 + 0.692286i \(0.243396\pi\)
\(572\) 0 0
\(573\) −11.8280 −0.494123
\(574\) −16.6015 −0.692932
\(575\) −8.32304 −0.347095
\(576\) 1.00000 0.0416667
\(577\) 14.1715 0.589968 0.294984 0.955502i \(-0.404686\pi\)
0.294984 + 0.955502i \(0.404686\pi\)
\(578\) 12.8019 0.532490
\(579\) −0.872625 −0.0362651
\(580\) −8.07069 −0.335117
\(581\) 15.0881 0.625962
\(582\) −4.74094 −0.196518
\(583\) 18.3599 0.760389
\(584\) 5.11529 0.211672
\(585\) 0 0
\(586\) 28.4349 1.17463
\(587\) 1.12737 0.0465317 0.0232659 0.999729i \(-0.492594\pi\)
0.0232659 + 0.999729i \(0.492594\pi\)
\(588\) 5.15883 0.212747
\(589\) 0.994032 0.0409584
\(590\) −1.62565 −0.0669268
\(591\) −10.6213 −0.436903
\(592\) −2.46681 −0.101385
\(593\) −18.3913 −0.755242 −0.377621 0.925960i \(-0.623258\pi\)
−0.377621 + 0.925960i \(0.623258\pi\)
\(594\) 3.19806 0.131218
\(595\) 2.78017 0.113976
\(596\) −10.7724 −0.441255
\(597\) −12.3424 −0.505142
\(598\) 0 0
\(599\) 13.1226 0.536174 0.268087 0.963395i \(-0.413608\pi\)
0.268087 + 0.963395i \(0.413608\pi\)
\(600\) 1.00000 0.0408248
\(601\) 25.5754 1.04324 0.521621 0.853177i \(-0.325327\pi\)
0.521621 + 0.853177i \(0.325327\pi\)
\(602\) 3.95108 0.161034
\(603\) 1.55496 0.0633228
\(604\) 17.3817 0.707249
\(605\) 0.772398 0.0314024
\(606\) −12.5918 −0.511507
\(607\) 44.4782 1.80531 0.902656 0.430362i \(-0.141614\pi\)
0.902656 + 0.430362i \(0.141614\pi\)
\(608\) 5.34481 0.216761
\(609\) −10.9511 −0.443760
\(610\) 5.28382 0.213935
\(611\) 0 0
\(612\) −2.04892 −0.0828226
\(613\) 21.7657 0.879108 0.439554 0.898216i \(-0.355137\pi\)
0.439554 + 0.898216i \(0.355137\pi\)
\(614\) 10.1347 0.409002
\(615\) 12.2349 0.493359
\(616\) −4.33944 −0.174841
\(617\) 17.3569 0.698762 0.349381 0.936981i \(-0.386392\pi\)
0.349381 + 0.936981i \(0.386392\pi\)
\(618\) −9.00000 −0.362033
\(619\) −36.8049 −1.47931 −0.739657 0.672984i \(-0.765012\pi\)
−0.739657 + 0.672984i \(0.765012\pi\)
\(620\) 0.185981 0.00746916
\(621\) 8.32304 0.333992
\(622\) −20.0218 −0.802800
\(623\) 4.58450 0.183674
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −8.63640 −0.345180
\(627\) 17.0930 0.682631
\(628\) 15.5429 0.620228
\(629\) 5.05429 0.201528
\(630\) 1.35690 0.0540600
\(631\) −11.7554 −0.467976 −0.233988 0.972240i \(-0.575178\pi\)
−0.233988 + 0.972240i \(0.575178\pi\)
\(632\) 1.42327 0.0566147
\(633\) −6.75733 −0.268580
\(634\) −17.6256 −0.700004
\(635\) 16.3763 0.649873
\(636\) −5.74094 −0.227643
\(637\) 0 0
\(638\) −25.8106 −1.02185
\(639\) −2.32304 −0.0918982
\(640\) 1.00000 0.0395285
\(641\) 33.0858 1.30681 0.653404 0.757009i \(-0.273340\pi\)
0.653404 + 0.757009i \(0.273340\pi\)
\(642\) −5.08815 −0.200813
\(643\) 10.3461 0.408012 0.204006 0.978970i \(-0.434604\pi\)
0.204006 + 0.978970i \(0.434604\pi\)
\(644\) −11.2935 −0.445026
\(645\) −2.91185 −0.114654
\(646\) −10.9511 −0.430865
\(647\) −11.0030 −0.432572 −0.216286 0.976330i \(-0.569394\pi\)
−0.216286 + 0.976330i \(0.569394\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −5.19892 −0.204075
\(650\) 0 0
\(651\) 0.252356 0.00989063
\(652\) −2.00239 −0.0784198
\(653\) −30.8224 −1.20617 −0.603086 0.797676i \(-0.706063\pi\)
−0.603086 + 0.797676i \(0.706063\pi\)
\(654\) 6.43296 0.251549
\(655\) −3.00969 −0.117598
\(656\) 12.2349 0.477693
\(657\) −5.11529 −0.199567
\(658\) −17.7724 −0.692840
\(659\) −33.2010 −1.29333 −0.646665 0.762774i \(-0.723837\pi\)
−0.646665 + 0.762774i \(0.723837\pi\)
\(660\) 3.19806 0.124484
\(661\) −37.4252 −1.45567 −0.727836 0.685752i \(-0.759474\pi\)
−0.727836 + 0.685752i \(0.759474\pi\)
\(662\) 31.5163 1.22492
\(663\) 0 0
\(664\) −11.1196 −0.431524
\(665\) 7.25236 0.281234
\(666\) 2.46681 0.0955870
\(667\) −67.1727 −2.60094
\(668\) −8.24027 −0.318826
\(669\) −21.5972 −0.834995
\(670\) 1.55496 0.0600733
\(671\) 16.8980 0.652339
\(672\) 1.35690 0.0523434
\(673\) −34.9409 −1.34687 −0.673437 0.739245i \(-0.735183\pi\)
−0.673437 + 0.739245i \(0.735183\pi\)
\(674\) 11.8116 0.454967
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 29.3913 1.12960 0.564800 0.825228i \(-0.308953\pi\)
0.564800 + 0.825228i \(0.308953\pi\)
\(678\) −18.6571 −0.716522
\(679\) −6.43296 −0.246874
\(680\) −2.04892 −0.0785724
\(681\) −10.2862 −0.394168
\(682\) 0.594778 0.0227752
\(683\) −28.9855 −1.10910 −0.554550 0.832150i \(-0.687110\pi\)
−0.554550 + 0.832150i \(0.687110\pi\)
\(684\) −5.34481 −0.204364
\(685\) −20.2892 −0.775210
\(686\) 16.4983 0.629907
\(687\) −4.25906 −0.162493
\(688\) −2.91185 −0.111013
\(689\) 0 0
\(690\) 8.32304 0.316853
\(691\) 1.37627 0.0523559 0.0261780 0.999657i \(-0.491666\pi\)
0.0261780 + 0.999657i \(0.491666\pi\)
\(692\) 12.7463 0.484542
\(693\) 4.33944 0.164842
\(694\) 17.4101 0.660879
\(695\) 0.153457 0.00582095
\(696\) 8.07069 0.305919
\(697\) −25.0683 −0.949529
\(698\) −18.6799 −0.707046
\(699\) −19.5133 −0.738062
\(700\) 1.35690 0.0512858
\(701\) 9.21014 0.347862 0.173931 0.984758i \(-0.444353\pi\)
0.173931 + 0.984758i \(0.444353\pi\)
\(702\) 0 0
\(703\) 13.1847 0.497269
\(704\) 3.19806 0.120532
\(705\) 13.0978 0.493293
\(706\) −25.3991 −0.955908
\(707\) −17.0858 −0.642576
\(708\) 1.62565 0.0610955
\(709\) 46.1613 1.73363 0.866813 0.498634i \(-0.166165\pi\)
0.866813 + 0.498634i \(0.166165\pi\)
\(710\) −2.32304 −0.0871823
\(711\) −1.42327 −0.0533769
\(712\) −3.37867 −0.126621
\(713\) 1.54793 0.0579703
\(714\) −2.78017 −0.104045
\(715\) 0 0
\(716\) 16.0422 0.599526
\(717\) 7.16315 0.267513
\(718\) −34.0103 −1.26925
\(719\) −46.3588 −1.72889 −0.864446 0.502726i \(-0.832331\pi\)
−0.864446 + 0.502726i \(0.832331\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −12.2121 −0.454801
\(722\) −9.56704 −0.356048
\(723\) −17.9584 −0.667879
\(724\) 3.85086 0.143116
\(725\) 8.07069 0.299738
\(726\) −0.772398 −0.0286664
\(727\) −1.67217 −0.0620174 −0.0310087 0.999519i \(-0.509872\pi\)
−0.0310087 + 0.999519i \(0.509872\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −5.11529 −0.189325
\(731\) 5.96615 0.220666
\(732\) −5.28382 −0.195295
\(733\) −33.6189 −1.24174 −0.620872 0.783912i \(-0.713221\pi\)
−0.620872 + 0.783912i \(0.713221\pi\)
\(734\) 25.6437 0.946526
\(735\) −5.15883 −0.190286
\(736\) 8.32304 0.306791
\(737\) 4.97285 0.183177
\(738\) −12.2349 −0.450373
\(739\) −31.8437 −1.17139 −0.585694 0.810532i \(-0.699178\pi\)
−0.585694 + 0.810532i \(0.699178\pi\)
\(740\) 2.46681 0.0906818
\(741\) 0 0
\(742\) −7.78986 −0.285975
\(743\) 23.4010 0.858500 0.429250 0.903186i \(-0.358778\pi\)
0.429250 + 0.903186i \(0.358778\pi\)
\(744\) −0.185981 −0.00681838
\(745\) 10.7724 0.394670
\(746\) 16.3612 0.599026
\(747\) 11.1196 0.406845
\(748\) −6.55257 −0.239586
\(749\) −6.90408 −0.252270
\(750\) −1.00000 −0.0365148
\(751\) −39.0146 −1.42366 −0.711831 0.702350i \(-0.752134\pi\)
−0.711831 + 0.702350i \(0.752134\pi\)
\(752\) 13.0978 0.477629
\(753\) 6.68425 0.243588
\(754\) 0 0
\(755\) −17.3817 −0.632583
\(756\) −1.35690 −0.0493498
\(757\) 43.0224 1.56367 0.781837 0.623483i \(-0.214283\pi\)
0.781837 + 0.623483i \(0.214283\pi\)
\(758\) −22.8823 −0.831123
\(759\) 26.6176 0.966158
\(760\) −5.34481 −0.193877
\(761\) 14.9903 0.543398 0.271699 0.962382i \(-0.412414\pi\)
0.271699 + 0.962382i \(0.412414\pi\)
\(762\) −16.3763 −0.593250
\(763\) 8.72886 0.316006
\(764\) 11.8280 0.427923
\(765\) 2.04892 0.0740788
\(766\) −11.9191 −0.430656
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −19.9138 −0.718109 −0.359055 0.933317i \(-0.616901\pi\)
−0.359055 + 0.933317i \(0.616901\pi\)
\(770\) 4.33944 0.156382
\(771\) −9.91484 −0.357074
\(772\) 0.872625 0.0314065
\(773\) 25.5566 0.919208 0.459604 0.888124i \(-0.347991\pi\)
0.459604 + 0.888124i \(0.347991\pi\)
\(774\) 2.91185 0.104664
\(775\) −0.185981 −0.00668062
\(776\) 4.74094 0.170190
\(777\) 3.34721 0.120080
\(778\) −8.07606 −0.289541
\(779\) −65.3933 −2.34296
\(780\) 0 0
\(781\) −7.42924 −0.265839
\(782\) −17.0532 −0.609822
\(783\) −8.07069 −0.288423
\(784\) −5.15883 −0.184244
\(785\) −15.5429 −0.554749
\(786\) 3.00969 0.107352
\(787\) 28.6297 1.02054 0.510269 0.860015i \(-0.329546\pi\)
0.510269 + 0.860015i \(0.329546\pi\)
\(788\) 10.6213 0.378369
\(789\) −13.8509 −0.493103
\(790\) −1.42327 −0.0506377
\(791\) −25.3157 −0.900124
\(792\) −3.19806 −0.113638
\(793\) 0 0
\(794\) 20.5230 0.728335
\(795\) 5.74094 0.203610
\(796\) 12.3424 0.437466
\(797\) 1.00431 0.0355746 0.0177873 0.999842i \(-0.494338\pi\)
0.0177873 + 0.999842i \(0.494338\pi\)
\(798\) −7.25236 −0.256731
\(799\) −26.8364 −0.949403
\(800\) −1.00000 −0.0353553
\(801\) 3.37867 0.119379
\(802\) 18.3274 0.647161
\(803\) −16.3590 −0.577297
\(804\) −1.55496 −0.0548391
\(805\) 11.2935 0.398044
\(806\) 0 0
\(807\) 17.8756 0.629251
\(808\) 12.5918 0.442978
\(809\) 49.8743 1.75349 0.876743 0.480959i \(-0.159711\pi\)
0.876743 + 0.480959i \(0.159711\pi\)
\(810\) 1.00000 0.0351364
\(811\) −7.95646 −0.279389 −0.139695 0.990195i \(-0.544612\pi\)
−0.139695 + 0.990195i \(0.544612\pi\)
\(812\) 10.9511 0.384308
\(813\) −31.2083 −1.09452
\(814\) 7.88902 0.276510
\(815\) 2.00239 0.0701408
\(816\) 2.04892 0.0717265
\(817\) 15.5633 0.544492
\(818\) −9.00000 −0.314678
\(819\) 0 0
\(820\) −12.2349 −0.427261
\(821\) −52.9071 −1.84647 −0.923235 0.384236i \(-0.874465\pi\)
−0.923235 + 0.384236i \(0.874465\pi\)
\(822\) 20.2892 0.707667
\(823\) 21.8006 0.759921 0.379961 0.925003i \(-0.375937\pi\)
0.379961 + 0.925003i \(0.375937\pi\)
\(824\) 9.00000 0.313530
\(825\) −3.19806 −0.111342
\(826\) 2.20583 0.0767507
\(827\) 27.3139 0.949799 0.474899 0.880040i \(-0.342484\pi\)
0.474899 + 0.880040i \(0.342484\pi\)
\(828\) −8.32304 −0.289246
\(829\) 40.8471 1.41868 0.709340 0.704867i \(-0.248993\pi\)
0.709340 + 0.704867i \(0.248993\pi\)
\(830\) 11.1196 0.385967
\(831\) −22.9245 −0.795243
\(832\) 0 0
\(833\) 10.5700 0.366230
\(834\) −0.153457 −0.00531377
\(835\) 8.24027 0.285166
\(836\) −17.0930 −0.591175
\(837\) 0.185981 0.00642843
\(838\) 14.4185 0.498078
\(839\) −43.0823 −1.48737 −0.743683 0.668532i \(-0.766923\pi\)
−0.743683 + 0.668532i \(0.766923\pi\)
\(840\) −1.35690 −0.0468174
\(841\) 36.1360 1.24607
\(842\) 19.9801 0.688561
\(843\) −28.0683 −0.966723
\(844\) 6.75733 0.232597
\(845\) 0 0
\(846\) −13.0978 −0.450313
\(847\) −1.04806 −0.0360119
\(848\) 5.74094 0.197145
\(849\) 24.8364 0.852382
\(850\) 2.04892 0.0702773
\(851\) 20.5314 0.703807
\(852\) 2.32304 0.0795862
\(853\) 24.6813 0.845071 0.422535 0.906346i \(-0.361140\pi\)
0.422535 + 0.906346i \(0.361140\pi\)
\(854\) −7.16959 −0.245338
\(855\) 5.34481 0.182789
\(856\) 5.08815 0.173909
\(857\) 43.1075 1.47252 0.736262 0.676696i \(-0.236589\pi\)
0.736262 + 0.676696i \(0.236589\pi\)
\(858\) 0 0
\(859\) −42.1323 −1.43753 −0.718767 0.695251i \(-0.755293\pi\)
−0.718767 + 0.695251i \(0.755293\pi\)
\(860\) 2.91185 0.0992934
\(861\) −16.6015 −0.565777
\(862\) 21.1812 0.721434
\(863\) −47.9342 −1.63170 −0.815850 0.578264i \(-0.803730\pi\)
−0.815850 + 0.578264i \(0.803730\pi\)
\(864\) 1.00000 0.0340207
\(865\) −12.7463 −0.433388
\(866\) −31.0291 −1.05441
\(867\) 12.8019 0.434777
\(868\) −0.252356 −0.00856553
\(869\) −4.55171 −0.154406
\(870\) −8.07069 −0.273622
\(871\) 0 0
\(872\) −6.43296 −0.217848
\(873\) −4.74094 −0.160456
\(874\) −44.4851 −1.50473
\(875\) −1.35690 −0.0458715
\(876\) 5.11529 0.172830
\(877\) 38.6674 1.30570 0.652852 0.757485i \(-0.273572\pi\)
0.652852 + 0.757485i \(0.273572\pi\)
\(878\) −7.33406 −0.247513
\(879\) 28.4349 0.959085
\(880\) −3.19806 −0.107807
\(881\) 25.4980 0.859050 0.429525 0.903055i \(-0.358681\pi\)
0.429525 + 0.903055i \(0.358681\pi\)
\(882\) 5.15883 0.173707
\(883\) −31.8377 −1.07142 −0.535712 0.844401i \(-0.679957\pi\)
−0.535712 + 0.844401i \(0.679957\pi\)
\(884\) 0 0
\(885\) −1.62565 −0.0546455
\(886\) 18.1360 0.609291
\(887\) −7.83446 −0.263055 −0.131528 0.991312i \(-0.541988\pi\)
−0.131528 + 0.991312i \(0.541988\pi\)
\(888\) −2.46681 −0.0827808
\(889\) −22.2209 −0.745265
\(890\) 3.37867 0.113253
\(891\) 3.19806 0.107139
\(892\) 21.5972 0.723127
\(893\) −70.0055 −2.34264
\(894\) −10.7724 −0.360283
\(895\) −16.0422 −0.536232
\(896\) −1.35690 −0.0453307
\(897\) 0 0
\(898\) −4.55496 −0.152001
\(899\) −1.50099 −0.0500609
\(900\) 1.00000 0.0333333
\(901\) −11.7627 −0.391873
\(902\) −39.1280 −1.30282
\(903\) 3.95108 0.131484
\(904\) 18.6571 0.620526
\(905\) −3.85086 −0.128007
\(906\) 17.3817 0.577467
\(907\) 33.1089 1.09936 0.549681 0.835375i \(-0.314749\pi\)
0.549681 + 0.835375i \(0.314749\pi\)
\(908\) 10.2862 0.341360
\(909\) −12.5918 −0.417643
\(910\) 0 0
\(911\) −14.0556 −0.465684 −0.232842 0.972515i \(-0.574802\pi\)
−0.232842 + 0.972515i \(0.574802\pi\)
\(912\) 5.34481 0.176984
\(913\) 35.5612 1.17690
\(914\) 9.98121 0.330149
\(915\) 5.28382 0.174678
\(916\) 4.25906 0.140723
\(917\) 4.08383 0.134860
\(918\) −2.04892 −0.0676243
\(919\) −27.1830 −0.896684 −0.448342 0.893862i \(-0.647985\pi\)
−0.448342 + 0.893862i \(0.647985\pi\)
\(920\) −8.32304 −0.274403
\(921\) 10.1347 0.333949
\(922\) −16.1672 −0.532438
\(923\) 0 0
\(924\) −4.33944 −0.142757
\(925\) −2.46681 −0.0811083
\(926\) 3.88902 0.127801
\(927\) −9.00000 −0.295599
\(928\) −8.07069 −0.264933
\(929\) 50.3521 1.65200 0.826000 0.563671i \(-0.190611\pi\)
0.826000 + 0.563671i \(0.190611\pi\)
\(930\) 0.185981 0.00609855
\(931\) 27.5730 0.903669
\(932\) 19.5133 0.639181
\(933\) −20.0218 −0.655483
\(934\) 0.872625 0.0285532
\(935\) 6.55257 0.214292
\(936\) 0 0
\(937\) 26.6394 0.870271 0.435135 0.900365i \(-0.356701\pi\)
0.435135 + 0.900365i \(0.356701\pi\)
\(938\) −2.10992 −0.0688912
\(939\) −8.63640 −0.281838
\(940\) −13.0978 −0.427204
\(941\) 25.7198 0.838440 0.419220 0.907885i \(-0.362304\pi\)
0.419220 + 0.907885i \(0.362304\pi\)
\(942\) 15.5429 0.506414
\(943\) −101.832 −3.31609
\(944\) −1.62565 −0.0529103
\(945\) 1.35690 0.0441398
\(946\) 9.31229 0.302769
\(947\) −14.4862 −0.470738 −0.235369 0.971906i \(-0.575630\pi\)
−0.235369 + 0.971906i \(0.575630\pi\)
\(948\) 1.42327 0.0462257
\(949\) 0 0
\(950\) 5.34481 0.173409
\(951\) −17.6256 −0.571551
\(952\) 2.78017 0.0901057
\(953\) −29.3690 −0.951354 −0.475677 0.879620i \(-0.657797\pi\)
−0.475677 + 0.879620i \(0.657797\pi\)
\(954\) −5.74094 −0.185870
\(955\) −11.8280 −0.382746
\(956\) −7.16315 −0.231673
\(957\) −25.8106 −0.834337
\(958\) −31.6305 −1.02194
\(959\) 27.5303 0.889000
\(960\) 1.00000 0.0322749
\(961\) −30.9654 −0.998884
\(962\) 0 0
\(963\) −5.08815 −0.163963
\(964\) 17.9584 0.578400
\(965\) −0.872625 −0.0280908
\(966\) −11.2935 −0.363363
\(967\) 21.9360 0.705415 0.352707 0.935734i \(-0.385261\pi\)
0.352707 + 0.935734i \(0.385261\pi\)
\(968\) 0.772398 0.0248258
\(969\) −10.9511 −0.351799
\(970\) −4.74094 −0.152222
\(971\) 34.9754 1.12241 0.561206 0.827676i \(-0.310337\pi\)
0.561206 + 0.827676i \(0.310337\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −0.208225 −0.00667538
\(974\) 21.7748 0.697709
\(975\) 0 0
\(976\) 5.28382 0.169131
\(977\) 7.72289 0.247077 0.123539 0.992340i \(-0.460576\pi\)
0.123539 + 0.992340i \(0.460576\pi\)
\(978\) −2.00239 −0.0640295
\(979\) 10.8052 0.345335
\(980\) 5.15883 0.164793
\(981\) 6.43296 0.205389
\(982\) −42.9590 −1.37088
\(983\) −29.0489 −0.926517 −0.463258 0.886223i \(-0.653320\pi\)
−0.463258 + 0.886223i \(0.653320\pi\)
\(984\) 12.2349 0.390034
\(985\) −10.6213 −0.338424
\(986\) 16.5362 0.526619
\(987\) −17.7724 −0.565702
\(988\) 0 0
\(989\) 24.2355 0.770644
\(990\) 3.19806 0.101641
\(991\) 37.7192 1.19819 0.599094 0.800678i \(-0.295527\pi\)
0.599094 + 0.800678i \(0.295527\pi\)
\(992\) 0.185981 0.00590489
\(993\) 31.5163 1.00014
\(994\) 3.15213 0.0999795
\(995\) −12.3424 −0.391281
\(996\) −11.1196 −0.352338
\(997\) −7.18060 −0.227412 −0.113706 0.993514i \(-0.536272\pi\)
−0.113706 + 0.993514i \(0.536272\pi\)
\(998\) −36.0978 −1.14266
\(999\) 2.46681 0.0780465
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.a.bk.1.2 3
13.5 odd 4 5070.2.b.s.1351.5 6
13.8 odd 4 5070.2.b.s.1351.2 6
13.12 even 2 5070.2.a.bt.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bk.1.2 3 1.1 even 1 trivial
5070.2.a.bt.1.2 yes 3 13.12 even 2
5070.2.b.s.1351.2 6 13.8 odd 4
5070.2.b.s.1351.5 6 13.5 odd 4