Properties

Label 5070.2.a.by.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -0.801938 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -0.801938 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.04892 q^{11} +1.00000 q^{12} -0.801938 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.24698 q^{17} +1.00000 q^{18} +7.40581 q^{19} +1.00000 q^{20} -0.801938 q^{21} -3.04892 q^{22} -0.356896 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -0.801938 q^{28} +0.307979 q^{29} +1.00000 q^{30} +3.44504 q^{31} +1.00000 q^{32} -3.04892 q^{33} -1.24698 q^{34} -0.801938 q^{35} +1.00000 q^{36} +7.29590 q^{37} +7.40581 q^{38} +1.00000 q^{40} +3.40581 q^{41} -0.801938 q^{42} +6.54288 q^{43} -3.04892 q^{44} +1.00000 q^{45} -0.356896 q^{46} +0.603875 q^{47} +1.00000 q^{48} -6.35690 q^{49} +1.00000 q^{50} -1.24698 q^{51} -6.29590 q^{53} +1.00000 q^{54} -3.04892 q^{55} -0.801938 q^{56} +7.40581 q^{57} +0.307979 q^{58} +3.11529 q^{59} +1.00000 q^{60} +11.0368 q^{61} +3.44504 q^{62} -0.801938 q^{63} +1.00000 q^{64} -3.04892 q^{66} +5.96077 q^{67} -1.24698 q^{68} -0.356896 q^{69} -0.801938 q^{70} +6.67456 q^{71} +1.00000 q^{72} +10.1371 q^{73} +7.29590 q^{74} +1.00000 q^{75} +7.40581 q^{76} +2.44504 q^{77} -13.6799 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.40581 q^{82} -3.93900 q^{83} -0.801938 q^{84} -1.24698 q^{85} +6.54288 q^{86} +0.307979 q^{87} -3.04892 q^{88} -8.05861 q^{89} +1.00000 q^{90} -0.356896 q^{92} +3.44504 q^{93} +0.603875 q^{94} +7.40581 q^{95} +1.00000 q^{96} +9.73556 q^{97} -6.35690 q^{98} -3.04892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} + 2q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} + 2q^{7} + 3q^{8} + 3q^{9} + 3q^{10} + 3q^{12} + 2q^{14} + 3q^{15} + 3q^{16} + q^{17} + 3q^{18} + 9q^{19} + 3q^{20} + 2q^{21} + 3q^{23} + 3q^{24} + 3q^{25} + 3q^{27} + 2q^{28} + 6q^{29} + 3q^{30} + 10q^{31} + 3q^{32} + q^{34} + 2q^{35} + 3q^{36} + 8q^{37} + 9q^{38} + 3q^{40} - 3q^{41} + 2q^{42} + q^{43} + 3q^{45} + 3q^{46} - 7q^{47} + 3q^{48} - 15q^{49} + 3q^{50} + q^{51} - 5q^{53} + 3q^{54} + 2q^{56} + 9q^{57} + 6q^{58} + 7q^{59} + 3q^{60} + 5q^{61} + 10q^{62} + 2q^{63} + 3q^{64} + 5q^{67} + q^{68} + 3q^{69} + 2q^{70} - q^{71} + 3q^{72} + 25q^{73} + 8q^{74} + 3q^{75} + 9q^{76} + 7q^{77} - 17q^{79} + 3q^{80} + 3q^{81} - 3q^{82} - 2q^{83} + 2q^{84} + q^{85} + q^{86} + 6q^{87} + 7q^{89} + 3q^{90} + 3q^{92} + 10q^{93} - 7q^{94} + 9q^{95} + 3q^{96} + 18q^{97} - 15q^{98} + O(q^{100}) \)

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
\(7\) −0.801938 −0.303104 −0.151552 0.988449i \(-0.548427\pi\)
−0.151552 + 0.988449i \(0.548427\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −3.04892 −0.919283 −0.459642 0.888105i \(-0.652022\pi\)
−0.459642 + 0.888105i \(0.652022\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −0.801938 −0.214327
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −1.24698 −0.302437 −0.151218 0.988500i \(-0.548320\pi\)
−0.151218 + 0.988500i \(0.548320\pi\)
\(18\) 1.00000 0.235702
\(19\) 7.40581 1.69901 0.849505 0.527581i \(-0.176901\pi\)
0.849505 + 0.527581i \(0.176901\pi\)
\(20\) 1.00000 0.223607
\(21\) −0.801938 −0.174997
\(22\) −3.04892 −0.650031
\(23\) −0.356896 −0.0744179 −0.0372090 0.999308i \(-0.511847\pi\)
−0.0372090 + 0.999308i \(0.511847\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −0.801938 −0.151552
\(29\) 0.307979 0.0571902 0.0285951 0.999591i \(-0.490897\pi\)
0.0285951 + 0.999591i \(0.490897\pi\)
\(30\) 1.00000 0.182574
\(31\) 3.44504 0.618748 0.309374 0.950940i \(-0.399881\pi\)
0.309374 + 0.950940i \(0.399881\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.04892 −0.530748
\(34\) −1.24698 −0.213855
\(35\) −0.801938 −0.135552
\(36\) 1.00000 0.166667
\(37\) 7.29590 1.19944 0.599719 0.800211i \(-0.295279\pi\)
0.599719 + 0.800211i \(0.295279\pi\)
\(38\) 7.40581 1.20138
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 3.40581 0.531899 0.265949 0.963987i \(-0.414315\pi\)
0.265949 + 0.963987i \(0.414315\pi\)
\(42\) −0.801938 −0.123742
\(43\) 6.54288 0.997779 0.498890 0.866666i \(-0.333741\pi\)
0.498890 + 0.866666i \(0.333741\pi\)
\(44\) −3.04892 −0.459642
\(45\) 1.00000 0.149071
\(46\) −0.356896 −0.0526214
\(47\) 0.603875 0.0880843 0.0440421 0.999030i \(-0.485976\pi\)
0.0440421 + 0.999030i \(0.485976\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.35690 −0.908128
\(50\) 1.00000 0.141421
\(51\) −1.24698 −0.174612
\(52\) 0 0
\(53\) −6.29590 −0.864808 −0.432404 0.901680i \(-0.642335\pi\)
−0.432404 + 0.901680i \(0.642335\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.04892 −0.411116
\(56\) −0.801938 −0.107163
\(57\) 7.40581 0.980924
\(58\) 0.307979 0.0404396
\(59\) 3.11529 0.405577 0.202788 0.979223i \(-0.435000\pi\)
0.202788 + 0.979223i \(0.435000\pi\)
\(60\) 1.00000 0.129099
\(61\) 11.0368 1.41312 0.706561 0.707652i \(-0.250246\pi\)
0.706561 + 0.707652i \(0.250246\pi\)
\(62\) 3.44504 0.437521
\(63\) −0.801938 −0.101035
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.04892 −0.375296
\(67\) 5.96077 0.728224 0.364112 0.931355i \(-0.381373\pi\)
0.364112 + 0.931355i \(0.381373\pi\)
\(68\) −1.24698 −0.151218
\(69\) −0.356896 −0.0429652
\(70\) −0.801938 −0.0958499
\(71\) 6.67456 0.792125 0.396062 0.918224i \(-0.370376\pi\)
0.396062 + 0.918224i \(0.370376\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.1371 1.18645 0.593227 0.805035i \(-0.297854\pi\)
0.593227 + 0.805035i \(0.297854\pi\)
\(74\) 7.29590 0.848131
\(75\) 1.00000 0.115470
\(76\) 7.40581 0.849505
\(77\) 2.44504 0.278638
\(78\) 0 0
\(79\) −13.6799 −1.53911 −0.769557 0.638578i \(-0.779523\pi\)
−0.769557 + 0.638578i \(0.779523\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 3.40581 0.376109
\(83\) −3.93900 −0.432362 −0.216181 0.976353i \(-0.569360\pi\)
−0.216181 + 0.976353i \(0.569360\pi\)
\(84\) −0.801938 −0.0874986
\(85\) −1.24698 −0.135254
\(86\) 6.54288 0.705537
\(87\) 0.307979 0.0330188
\(88\) −3.04892 −0.325016
\(89\) −8.05861 −0.854211 −0.427105 0.904202i \(-0.640467\pi\)
−0.427105 + 0.904202i \(0.640467\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −0.356896 −0.0372090
\(93\) 3.44504 0.357234
\(94\) 0.603875 0.0622850
\(95\) 7.40581 0.759820
\(96\) 1.00000 0.102062
\(97\) 9.73556 0.988497 0.494248 0.869321i \(-0.335443\pi\)
0.494248 + 0.869321i \(0.335443\pi\)
\(98\) −6.35690 −0.642143
\(99\) −3.04892 −0.306428
\(100\) 1.00000 0.100000
\(101\) 7.65817 0.762016 0.381008 0.924572i \(-0.375577\pi\)
0.381008 + 0.924572i \(0.375577\pi\)
\(102\) −1.24698 −0.123469
\(103\) 2.32975 0.229557 0.114778 0.993391i \(-0.463384\pi\)
0.114778 + 0.993391i \(0.463384\pi\)
\(104\) 0 0
\(105\) −0.801938 −0.0782611
\(106\) −6.29590 −0.611512
\(107\) 3.77479 0.364923 0.182461 0.983213i \(-0.441594\pi\)
0.182461 + 0.983213i \(0.441594\pi\)
\(108\) 1.00000 0.0962250
\(109\) −1.05861 −0.101396 −0.0506980 0.998714i \(-0.516145\pi\)
−0.0506980 + 0.998714i \(0.516145\pi\)
\(110\) −3.04892 −0.290703
\(111\) 7.29590 0.692496
\(112\) −0.801938 −0.0757760
\(113\) 11.0368 1.03826 0.519129 0.854696i \(-0.326256\pi\)
0.519129 + 0.854696i \(0.326256\pi\)
\(114\) 7.40581 0.693618
\(115\) −0.356896 −0.0332807
\(116\) 0.307979 0.0285951
\(117\) 0 0
\(118\) 3.11529 0.286786
\(119\) 1.00000 0.0916698
\(120\) 1.00000 0.0912871
\(121\) −1.70410 −0.154918
\(122\) 11.0368 0.999228
\(123\) 3.40581 0.307092
\(124\) 3.44504 0.309374
\(125\) 1.00000 0.0894427
\(126\) −0.801938 −0.0714423
\(127\) 15.9541 1.41569 0.707847 0.706366i \(-0.249667\pi\)
0.707847 + 0.706366i \(0.249667\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.54288 0.576068
\(130\) 0 0
\(131\) −17.0911 −1.49326 −0.746629 0.665240i \(-0.768329\pi\)
−0.746629 + 0.665240i \(0.768329\pi\)
\(132\) −3.04892 −0.265374
\(133\) −5.93900 −0.514977
\(134\) 5.96077 0.514932
\(135\) 1.00000 0.0860663
\(136\) −1.24698 −0.106928
\(137\) −16.4373 −1.40433 −0.702165 0.712014i \(-0.747783\pi\)
−0.702165 + 0.712014i \(0.747783\pi\)
\(138\) −0.356896 −0.0303810
\(139\) −8.10992 −0.687874 −0.343937 0.938993i \(-0.611761\pi\)
−0.343937 + 0.938993i \(0.611761\pi\)
\(140\) −0.801938 −0.0677761
\(141\) 0.603875 0.0508555
\(142\) 6.67456 0.560117
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0.307979 0.0255762
\(146\) 10.1371 0.838949
\(147\) −6.35690 −0.524308
\(148\) 7.29590 0.599719
\(149\) 11.0532 0.905516 0.452758 0.891633i \(-0.350440\pi\)
0.452758 + 0.891633i \(0.350440\pi\)
\(150\) 1.00000 0.0816497
\(151\) 6.89546 0.561145 0.280572 0.959833i \(-0.409476\pi\)
0.280572 + 0.959833i \(0.409476\pi\)
\(152\) 7.40581 0.600691
\(153\) −1.24698 −0.100812
\(154\) 2.44504 0.197027
\(155\) 3.44504 0.276712
\(156\) 0 0
\(157\) −20.8388 −1.66312 −0.831558 0.555439i \(-0.812550\pi\)
−0.831558 + 0.555439i \(0.812550\pi\)
\(158\) −13.6799 −1.08832
\(159\) −6.29590 −0.499297
\(160\) 1.00000 0.0790569
\(161\) 0.286208 0.0225564
\(162\) 1.00000 0.0785674
\(163\) −4.71917 −0.369634 −0.184817 0.982773i \(-0.559169\pi\)
−0.184817 + 0.982773i \(0.559169\pi\)
\(164\) 3.40581 0.265949
\(165\) −3.04892 −0.237358
\(166\) −3.93900 −0.305726
\(167\) −14.5308 −1.12443 −0.562213 0.826992i \(-0.690050\pi\)
−0.562213 + 0.826992i \(0.690050\pi\)
\(168\) −0.801938 −0.0618708
\(169\) 0 0
\(170\) −1.24698 −0.0956390
\(171\) 7.40581 0.566337
\(172\) 6.54288 0.498890
\(173\) 11.4523 0.870705 0.435353 0.900260i \(-0.356624\pi\)
0.435353 + 0.900260i \(0.356624\pi\)
\(174\) 0.307979 0.0233478
\(175\) −0.801938 −0.0606208
\(176\) −3.04892 −0.229821
\(177\) 3.11529 0.234160
\(178\) −8.05861 −0.604018
\(179\) −4.32975 −0.323621 −0.161810 0.986822i \(-0.551733\pi\)
−0.161810 + 0.986822i \(0.551733\pi\)
\(180\) 1.00000 0.0745356
\(181\) −24.8321 −1.84575 −0.922877 0.385096i \(-0.874168\pi\)
−0.922877 + 0.385096i \(0.874168\pi\)
\(182\) 0 0
\(183\) 11.0368 0.815866
\(184\) −0.356896 −0.0263107
\(185\) 7.29590 0.536405
\(186\) 3.44504 0.252603
\(187\) 3.80194 0.278025
\(188\) 0.603875 0.0440421
\(189\) −0.801938 −0.0583324
\(190\) 7.40581 0.537274
\(191\) 22.1933 1.60585 0.802925 0.596081i \(-0.203276\pi\)
0.802925 + 0.596081i \(0.203276\pi\)
\(192\) 1.00000 0.0721688
\(193\) 25.4306 1.83053 0.915266 0.402850i \(-0.131980\pi\)
0.915266 + 0.402850i \(0.131980\pi\)
\(194\) 9.73556 0.698973
\(195\) 0 0
\(196\) −6.35690 −0.454064
\(197\) −10.6310 −0.757429 −0.378714 0.925514i \(-0.623634\pi\)
−0.378714 + 0.925514i \(0.623634\pi\)
\(198\) −3.04892 −0.216677
\(199\) 11.0465 0.783067 0.391534 0.920164i \(-0.371945\pi\)
0.391534 + 0.920164i \(0.371945\pi\)
\(200\) 1.00000 0.0707107
\(201\) 5.96077 0.420440
\(202\) 7.65817 0.538827
\(203\) −0.246980 −0.0173346
\(204\) −1.24698 −0.0873060
\(205\) 3.40581 0.237872
\(206\) 2.32975 0.162321
\(207\) −0.356896 −0.0248060
\(208\) 0 0
\(209\) −22.5797 −1.56187
\(210\) −0.801938 −0.0553390
\(211\) −21.3491 −1.46973 −0.734867 0.678211i \(-0.762756\pi\)
−0.734867 + 0.678211i \(0.762756\pi\)
\(212\) −6.29590 −0.432404
\(213\) 6.67456 0.457334
\(214\) 3.77479 0.258039
\(215\) 6.54288 0.446220
\(216\) 1.00000 0.0680414
\(217\) −2.76271 −0.187545
\(218\) −1.05861 −0.0716978
\(219\) 10.1371 0.684999
\(220\) −3.04892 −0.205558
\(221\) 0 0
\(222\) 7.29590 0.489669
\(223\) −12.7603 −0.854494 −0.427247 0.904135i \(-0.640517\pi\)
−0.427247 + 0.904135i \(0.640517\pi\)
\(224\) −0.801938 −0.0535817
\(225\) 1.00000 0.0666667
\(226\) 11.0368 0.734159
\(227\) 14.1836 0.941398 0.470699 0.882294i \(-0.344002\pi\)
0.470699 + 0.882294i \(0.344002\pi\)
\(228\) 7.40581 0.490462
\(229\) −4.71140 −0.311338 −0.155669 0.987809i \(-0.549753\pi\)
−0.155669 + 0.987809i \(0.549753\pi\)
\(230\) −0.356896 −0.0235330
\(231\) 2.44504 0.160872
\(232\) 0.307979 0.0202198
\(233\) 0.0556221 0.00364393 0.00182196 0.999998i \(-0.499420\pi\)
0.00182196 + 0.999998i \(0.499420\pi\)
\(234\) 0 0
\(235\) 0.603875 0.0393925
\(236\) 3.11529 0.202788
\(237\) −13.6799 −0.888608
\(238\) 1.00000 0.0648204
\(239\) 21.0489 1.36154 0.680771 0.732497i \(-0.261645\pi\)
0.680771 + 0.732497i \(0.261645\pi\)
\(240\) 1.00000 0.0645497
\(241\) −12.0030 −0.773180 −0.386590 0.922252i \(-0.626347\pi\)
−0.386590 + 0.922252i \(0.626347\pi\)
\(242\) −1.70410 −0.109544
\(243\) 1.00000 0.0641500
\(244\) 11.0368 0.706561
\(245\) −6.35690 −0.406127
\(246\) 3.40581 0.217147
\(247\) 0 0
\(248\) 3.44504 0.218760
\(249\) −3.93900 −0.249624
\(250\) 1.00000 0.0632456
\(251\) −2.25906 −0.142591 −0.0712953 0.997455i \(-0.522713\pi\)
−0.0712953 + 0.997455i \(0.522713\pi\)
\(252\) −0.801938 −0.0505173
\(253\) 1.08815 0.0684112
\(254\) 15.9541 1.00105
\(255\) −1.24698 −0.0780889
\(256\) 1.00000 0.0625000
\(257\) −17.0043 −1.06070 −0.530350 0.847779i \(-0.677939\pi\)
−0.530350 + 0.847779i \(0.677939\pi\)
\(258\) 6.54288 0.407342
\(259\) −5.85086 −0.363554
\(260\) 0 0
\(261\) 0.307979 0.0190634
\(262\) −17.0911 −1.05589
\(263\) −28.5773 −1.76215 −0.881077 0.472974i \(-0.843181\pi\)
−0.881077 + 0.472974i \(0.843181\pi\)
\(264\) −3.04892 −0.187648
\(265\) −6.29590 −0.386754
\(266\) −5.93900 −0.364144
\(267\) −8.05861 −0.493179
\(268\) 5.96077 0.364112
\(269\) −3.28083 −0.200036 −0.100018 0.994986i \(-0.531890\pi\)
−0.100018 + 0.994986i \(0.531890\pi\)
\(270\) 1.00000 0.0608581
\(271\) −8.70948 −0.529063 −0.264532 0.964377i \(-0.585217\pi\)
−0.264532 + 0.964377i \(0.585217\pi\)
\(272\) −1.24698 −0.0756092
\(273\) 0 0
\(274\) −16.4373 −0.993012
\(275\) −3.04892 −0.183857
\(276\) −0.356896 −0.0214826
\(277\) −8.91185 −0.535461 −0.267731 0.963494i \(-0.586274\pi\)
−0.267731 + 0.963494i \(0.586274\pi\)
\(278\) −8.10992 −0.486400
\(279\) 3.44504 0.206249
\(280\) −0.801938 −0.0479249
\(281\) −1.90648 −0.113731 −0.0568654 0.998382i \(-0.518111\pi\)
−0.0568654 + 0.998382i \(0.518111\pi\)
\(282\) 0.603875 0.0359603
\(283\) −14.9245 −0.887171 −0.443586 0.896232i \(-0.646294\pi\)
−0.443586 + 0.896232i \(0.646294\pi\)
\(284\) 6.67456 0.396062
\(285\) 7.40581 0.438683
\(286\) 0 0
\(287\) −2.73125 −0.161221
\(288\) 1.00000 0.0589256
\(289\) −15.4450 −0.908532
\(290\) 0.307979 0.0180851
\(291\) 9.73556 0.570709
\(292\) 10.1371 0.593227
\(293\) −12.8901 −0.753047 −0.376523 0.926407i \(-0.622881\pi\)
−0.376523 + 0.926407i \(0.622881\pi\)
\(294\) −6.35690 −0.370742
\(295\) 3.11529 0.181379
\(296\) 7.29590 0.424065
\(297\) −3.04892 −0.176916
\(298\) 11.0532 0.640296
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −5.24698 −0.302431
\(302\) 6.89546 0.396789
\(303\) 7.65817 0.439950
\(304\) 7.40581 0.424753
\(305\) 11.0368 0.631967
\(306\) −1.24698 −0.0712851
\(307\) 19.2892 1.10089 0.550446 0.834871i \(-0.314458\pi\)
0.550446 + 0.834871i \(0.314458\pi\)
\(308\) 2.44504 0.139319
\(309\) 2.32975 0.132535
\(310\) 3.44504 0.195665
\(311\) 1.23729 0.0701603 0.0350802 0.999385i \(-0.488831\pi\)
0.0350802 + 0.999385i \(0.488831\pi\)
\(312\) 0 0
\(313\) 8.09113 0.457338 0.228669 0.973504i \(-0.426563\pi\)
0.228669 + 0.973504i \(0.426563\pi\)
\(314\) −20.8388 −1.17600
\(315\) −0.801938 −0.0451841
\(316\) −13.6799 −0.769557
\(317\) −21.0368 −1.18155 −0.590773 0.806838i \(-0.701177\pi\)
−0.590773 + 0.806838i \(0.701177\pi\)
\(318\) −6.29590 −0.353056
\(319\) −0.939001 −0.0525740
\(320\) 1.00000 0.0559017
\(321\) 3.77479 0.210688
\(322\) 0.286208 0.0159498
\(323\) −9.23490 −0.513843
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.71917 −0.261371
\(327\) −1.05861 −0.0585410
\(328\) 3.40581 0.188055
\(329\) −0.484271 −0.0266987
\(330\) −3.04892 −0.167837
\(331\) −29.4252 −1.61735 −0.808677 0.588252i \(-0.799816\pi\)
−0.808677 + 0.588252i \(0.799816\pi\)
\(332\) −3.93900 −0.216181
\(333\) 7.29590 0.399813
\(334\) −14.5308 −0.795090
\(335\) 5.96077 0.325672
\(336\) −0.801938 −0.0437493
\(337\) −17.5483 −0.955914 −0.477957 0.878383i \(-0.658623\pi\)
−0.477957 + 0.878383i \(0.658623\pi\)
\(338\) 0 0
\(339\) 11.0368 0.599439
\(340\) −1.24698 −0.0676270
\(341\) −10.5036 −0.568804
\(342\) 7.40581 0.400461
\(343\) 10.7114 0.578361
\(344\) 6.54288 0.352768
\(345\) −0.356896 −0.0192146
\(346\) 11.4523 0.615681
\(347\) 1.27844 0.0686302 0.0343151 0.999411i \(-0.489075\pi\)
0.0343151 + 0.999411i \(0.489075\pi\)
\(348\) 0.307979 0.0165094
\(349\) 18.0911 0.968396 0.484198 0.874958i \(-0.339111\pi\)
0.484198 + 0.874958i \(0.339111\pi\)
\(350\) −0.801938 −0.0428654
\(351\) 0 0
\(352\) −3.04892 −0.162508
\(353\) −18.9071 −1.00632 −0.503161 0.864193i \(-0.667830\pi\)
−0.503161 + 0.864193i \(0.667830\pi\)
\(354\) 3.11529 0.165576
\(355\) 6.67456 0.354249
\(356\) −8.05861 −0.427105
\(357\) 1.00000 0.0529256
\(358\) −4.32975 −0.228834
\(359\) 32.7308 1.72746 0.863732 0.503951i \(-0.168121\pi\)
0.863732 + 0.503951i \(0.168121\pi\)
\(360\) 1.00000 0.0527046
\(361\) 35.8461 1.88664
\(362\) −24.8321 −1.30514
\(363\) −1.70410 −0.0894422
\(364\) 0 0
\(365\) 10.1371 0.530598
\(366\) 11.0368 0.576905
\(367\) −3.10321 −0.161986 −0.0809932 0.996715i \(-0.525809\pi\)
−0.0809932 + 0.996715i \(0.525809\pi\)
\(368\) −0.356896 −0.0186045
\(369\) 3.40581 0.177300
\(370\) 7.29590 0.379296
\(371\) 5.04892 0.262127
\(372\) 3.44504 0.178617
\(373\) 13.7366 0.711256 0.355628 0.934628i \(-0.384267\pi\)
0.355628 + 0.934628i \(0.384267\pi\)
\(374\) 3.80194 0.196594
\(375\) 1.00000 0.0516398
\(376\) 0.603875 0.0311425
\(377\) 0 0
\(378\) −0.801938 −0.0412472
\(379\) 34.0097 1.74696 0.873480 0.486860i \(-0.161858\pi\)
0.873480 + 0.486860i \(0.161858\pi\)
\(380\) 7.40581 0.379910
\(381\) 15.9541 0.817352
\(382\) 22.1933 1.13551
\(383\) 38.6902 1.97698 0.988489 0.151293i \(-0.0483438\pi\)
0.988489 + 0.151293i \(0.0483438\pi\)
\(384\) 1.00000 0.0510310
\(385\) 2.44504 0.124611
\(386\) 25.4306 1.29438
\(387\) 6.54288 0.332593
\(388\) 9.73556 0.494248
\(389\) −11.2567 −0.570736 −0.285368 0.958418i \(-0.592116\pi\)
−0.285368 + 0.958418i \(0.592116\pi\)
\(390\) 0 0
\(391\) 0.445042 0.0225067
\(392\) −6.35690 −0.321072
\(393\) −17.0911 −0.862133
\(394\) −10.6310 −0.535583
\(395\) −13.6799 −0.688312
\(396\) −3.04892 −0.153214
\(397\) −23.4403 −1.17643 −0.588216 0.808704i \(-0.700170\pi\)
−0.588216 + 0.808704i \(0.700170\pi\)
\(398\) 11.0465 0.553712
\(399\) −5.93900 −0.297322
\(400\) 1.00000 0.0500000
\(401\) −37.6950 −1.88240 −0.941199 0.337852i \(-0.890300\pi\)
−0.941199 + 0.337852i \(0.890300\pi\)
\(402\) 5.96077 0.297296
\(403\) 0 0
\(404\) 7.65817 0.381008
\(405\) 1.00000 0.0496904
\(406\) −0.246980 −0.0122574
\(407\) −22.2446 −1.10262
\(408\) −1.24698 −0.0617347
\(409\) −1.19567 −0.0591220 −0.0295610 0.999563i \(-0.509411\pi\)
−0.0295610 + 0.999563i \(0.509411\pi\)
\(410\) 3.40581 0.168201
\(411\) −16.4373 −0.810791
\(412\) 2.32975 0.114778
\(413\) −2.49827 −0.122932
\(414\) −0.356896 −0.0175405
\(415\) −3.93900 −0.193358
\(416\) 0 0
\(417\) −8.10992 −0.397144
\(418\) −22.5797 −1.10441
\(419\) 1.77048 0.0864935 0.0432468 0.999064i \(-0.486230\pi\)
0.0432468 + 0.999064i \(0.486230\pi\)
\(420\) −0.801938 −0.0391306
\(421\) 1.80194 0.0878211 0.0439105 0.999035i \(-0.486018\pi\)
0.0439105 + 0.999035i \(0.486018\pi\)
\(422\) −21.3491 −1.03926
\(423\) 0.603875 0.0293614
\(424\) −6.29590 −0.305756
\(425\) −1.24698 −0.0604874
\(426\) 6.67456 0.323384
\(427\) −8.85086 −0.428323
\(428\) 3.77479 0.182461
\(429\) 0 0
\(430\) 6.54288 0.315526
\(431\) −0.567040 −0.0273133 −0.0136567 0.999907i \(-0.504347\pi\)
−0.0136567 + 0.999907i \(0.504347\pi\)
\(432\) 1.00000 0.0481125
\(433\) 17.5743 0.844569 0.422284 0.906463i \(-0.361228\pi\)
0.422284 + 0.906463i \(0.361228\pi\)
\(434\) −2.76271 −0.132614
\(435\) 0.307979 0.0147664
\(436\) −1.05861 −0.0506980
\(437\) −2.64310 −0.126437
\(438\) 10.1371 0.484368
\(439\) 3.00969 0.143645 0.0718223 0.997417i \(-0.477119\pi\)
0.0718223 + 0.997417i \(0.477119\pi\)
\(440\) −3.04892 −0.145351
\(441\) −6.35690 −0.302709
\(442\) 0 0
\(443\) 38.4499 1.82681 0.913406 0.407050i \(-0.133442\pi\)
0.913406 + 0.407050i \(0.133442\pi\)
\(444\) 7.29590 0.346248
\(445\) −8.05861 −0.382015
\(446\) −12.7603 −0.604218
\(447\) 11.0532 0.522800
\(448\) −0.801938 −0.0378880
\(449\) 26.9909 1.27378 0.636890 0.770955i \(-0.280221\pi\)
0.636890 + 0.770955i \(0.280221\pi\)
\(450\) 1.00000 0.0471405
\(451\) −10.3840 −0.488965
\(452\) 11.0368 0.519129
\(453\) 6.89546 0.323977
\(454\) 14.1836 0.665669
\(455\) 0 0
\(456\) 7.40581 0.346809
\(457\) 19.2664 0.901242 0.450621 0.892715i \(-0.351203\pi\)
0.450621 + 0.892715i \(0.351203\pi\)
\(458\) −4.71140 −0.220149
\(459\) −1.24698 −0.0582040
\(460\) −0.356896 −0.0166404
\(461\) −18.5942 −0.866018 −0.433009 0.901390i \(-0.642548\pi\)
−0.433009 + 0.901390i \(0.642548\pi\)
\(462\) 2.44504 0.113754
\(463\) 5.79417 0.269278 0.134639 0.990895i \(-0.457013\pi\)
0.134639 + 0.990895i \(0.457013\pi\)
\(464\) 0.307979 0.0142975
\(465\) 3.44504 0.159760
\(466\) 0.0556221 0.00257664
\(467\) 11.2457 0.520387 0.260193 0.965557i \(-0.416214\pi\)
0.260193 + 0.965557i \(0.416214\pi\)
\(468\) 0 0
\(469\) −4.78017 −0.220728
\(470\) 0.603875 0.0278547
\(471\) −20.8388 −0.960200
\(472\) 3.11529 0.143393
\(473\) −19.9487 −0.917242
\(474\) −13.6799 −0.628340
\(475\) 7.40581 0.339802
\(476\) 1.00000 0.0458349
\(477\) −6.29590 −0.288269
\(478\) 21.0489 0.962755
\(479\) −35.2452 −1.61039 −0.805197 0.593008i \(-0.797940\pi\)
−0.805197 + 0.593008i \(0.797940\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −12.0030 −0.546721
\(483\) 0.286208 0.0130229
\(484\) −1.70410 −0.0774592
\(485\) 9.73556 0.442069
\(486\) 1.00000 0.0453609
\(487\) 8.49827 0.385093 0.192547 0.981288i \(-0.438325\pi\)
0.192547 + 0.981288i \(0.438325\pi\)
\(488\) 11.0368 0.499614
\(489\) −4.71917 −0.213408
\(490\) −6.35690 −0.287175
\(491\) −25.5254 −1.15195 −0.575973 0.817469i \(-0.695377\pi\)
−0.575973 + 0.817469i \(0.695377\pi\)
\(492\) 3.40581 0.153546
\(493\) −0.384043 −0.0172964
\(494\) 0 0
\(495\) −3.04892 −0.137039
\(496\) 3.44504 0.154687
\(497\) −5.35258 −0.240096
\(498\) −3.93900 −0.176511
\(499\) −29.4577 −1.31871 −0.659354 0.751832i \(-0.729170\pi\)
−0.659354 + 0.751832i \(0.729170\pi\)
\(500\) 1.00000 0.0447214
\(501\) −14.5308 −0.649188
\(502\) −2.25906 −0.100827
\(503\) 14.4480 0.644206 0.322103 0.946705i \(-0.395610\pi\)
0.322103 + 0.946705i \(0.395610\pi\)
\(504\) −0.801938 −0.0357211
\(505\) 7.65817 0.340784
\(506\) 1.08815 0.0483740
\(507\) 0 0
\(508\) 15.9541 0.707847
\(509\) 4.54288 0.201359 0.100680 0.994919i \(-0.467898\pi\)
0.100680 + 0.994919i \(0.467898\pi\)
\(510\) −1.24698 −0.0552172
\(511\) −8.12929 −0.359619
\(512\) 1.00000 0.0441942
\(513\) 7.40581 0.326975
\(514\) −17.0043 −0.750028
\(515\) 2.32975 0.102661
\(516\) 6.54288 0.288034
\(517\) −1.84117 −0.0809744
\(518\) −5.85086 −0.257072
\(519\) 11.4523 0.502702
\(520\) 0 0
\(521\) −30.0911 −1.31832 −0.659158 0.752004i \(-0.729087\pi\)
−0.659158 + 0.752004i \(0.729087\pi\)
\(522\) 0.307979 0.0134799
\(523\) −3.18359 −0.139209 −0.0696043 0.997575i \(-0.522174\pi\)
−0.0696043 + 0.997575i \(0.522174\pi\)
\(524\) −17.0911 −0.746629
\(525\) −0.801938 −0.0349994
\(526\) −28.5773 −1.24603
\(527\) −4.29590 −0.187132
\(528\) −3.04892 −0.132687
\(529\) −22.8726 −0.994462
\(530\) −6.29590 −0.273476
\(531\) 3.11529 0.135192
\(532\) −5.93900 −0.257488
\(533\) 0 0
\(534\) −8.05861 −0.348730
\(535\) 3.77479 0.163198
\(536\) 5.96077 0.257466
\(537\) −4.32975 −0.186842
\(538\) −3.28083 −0.141447
\(539\) 19.3817 0.834827
\(540\) 1.00000 0.0430331
\(541\) 21.2825 0.915006 0.457503 0.889208i \(-0.348744\pi\)
0.457503 + 0.889208i \(0.348744\pi\)
\(542\) −8.70948 −0.374104
\(543\) −24.8321 −1.06565
\(544\) −1.24698 −0.0534638
\(545\) −1.05861 −0.0453457
\(546\) 0 0
\(547\) 27.5545 1.17814 0.589072 0.808080i \(-0.299493\pi\)
0.589072 + 0.808080i \(0.299493\pi\)
\(548\) −16.4373 −0.702165
\(549\) 11.0368 0.471041
\(550\) −3.04892 −0.130006
\(551\) 2.28083 0.0971667
\(552\) −0.356896 −0.0151905
\(553\) 10.9705 0.466511
\(554\) −8.91185 −0.378628
\(555\) 7.29590 0.309694
\(556\) −8.10992 −0.343937
\(557\) −3.88040 −0.164418 −0.0822088 0.996615i \(-0.526197\pi\)
−0.0822088 + 0.996615i \(0.526197\pi\)
\(558\) 3.44504 0.145840
\(559\) 0 0
\(560\) −0.801938 −0.0338881
\(561\) 3.80194 0.160518
\(562\) −1.90648 −0.0804199
\(563\) −38.6286 −1.62800 −0.814001 0.580863i \(-0.802715\pi\)
−0.814001 + 0.580863i \(0.802715\pi\)
\(564\) 0.603875 0.0254277
\(565\) 11.0368 0.464323
\(566\) −14.9245 −0.627325
\(567\) −0.801938 −0.0336782
\(568\) 6.67456 0.280058
\(569\) −27.6746 −1.16018 −0.580089 0.814553i \(-0.696982\pi\)
−0.580089 + 0.814553i \(0.696982\pi\)
\(570\) 7.40581 0.310195
\(571\) −10.9065 −0.456422 −0.228211 0.973612i \(-0.573288\pi\)
−0.228211 + 0.973612i \(0.573288\pi\)
\(572\) 0 0
\(573\) 22.1933 0.927137
\(574\) −2.73125 −0.114000
\(575\) −0.356896 −0.0148836
\(576\) 1.00000 0.0416667
\(577\) 29.9758 1.24791 0.623955 0.781460i \(-0.285525\pi\)
0.623955 + 0.781460i \(0.285525\pi\)
\(578\) −15.4450 −0.642429
\(579\) 25.4306 1.05686
\(580\) 0.307979 0.0127881
\(581\) 3.15883 0.131051
\(582\) 9.73556 0.403552
\(583\) 19.1957 0.795003
\(584\) 10.1371 0.419475
\(585\) 0 0
\(586\) −12.8901 −0.532484
\(587\) 32.1551 1.32718 0.663592 0.748095i \(-0.269031\pi\)
0.663592 + 0.748095i \(0.269031\pi\)
\(588\) −6.35690 −0.262154
\(589\) 25.5133 1.05126
\(590\) 3.11529 0.128255
\(591\) −10.6310 −0.437302
\(592\) 7.29590 0.299860
\(593\) −3.88876 −0.159692 −0.0798460 0.996807i \(-0.525443\pi\)
−0.0798460 + 0.996807i \(0.525443\pi\)
\(594\) −3.04892 −0.125099
\(595\) 1.00000 0.0409960
\(596\) 11.0532 0.452758
\(597\) 11.0465 0.452104
\(598\) 0 0
\(599\) −37.2301 −1.52118 −0.760591 0.649232i \(-0.775090\pi\)
−0.760591 + 0.649232i \(0.775090\pi\)
\(600\) 1.00000 0.0408248
\(601\) −4.04593 −0.165037 −0.0825185 0.996590i \(-0.526296\pi\)
−0.0825185 + 0.996590i \(0.526296\pi\)
\(602\) −5.24698 −0.213851
\(603\) 5.96077 0.242741
\(604\) 6.89546 0.280572
\(605\) −1.70410 −0.0692816
\(606\) 7.65817 0.311092
\(607\) −6.36658 −0.258412 −0.129206 0.991618i \(-0.541243\pi\)
−0.129206 + 0.991618i \(0.541243\pi\)
\(608\) 7.40581 0.300345
\(609\) −0.246980 −0.0100081
\(610\) 11.0368 0.446868
\(611\) 0 0
\(612\) −1.24698 −0.0504062
\(613\) −42.2519 −1.70654 −0.853269 0.521471i \(-0.825383\pi\)
−0.853269 + 0.521471i \(0.825383\pi\)
\(614\) 19.2892 0.778448
\(615\) 3.40581 0.137336
\(616\) 2.44504 0.0985135
\(617\) 0.978230 0.0393820 0.0196910 0.999806i \(-0.493732\pi\)
0.0196910 + 0.999806i \(0.493732\pi\)
\(618\) 2.32975 0.0937162
\(619\) 19.0392 0.765251 0.382626 0.923903i \(-0.375020\pi\)
0.382626 + 0.923903i \(0.375020\pi\)
\(620\) 3.44504 0.138356
\(621\) −0.356896 −0.0143217
\(622\) 1.23729 0.0496108
\(623\) 6.46250 0.258915
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 8.09113 0.323387
\(627\) −22.5797 −0.901747
\(628\) −20.8388 −0.831558
\(629\) −9.09783 −0.362754
\(630\) −0.801938 −0.0319500
\(631\) 37.2664 1.48355 0.741775 0.670649i \(-0.233984\pi\)
0.741775 + 0.670649i \(0.233984\pi\)
\(632\) −13.6799 −0.544159
\(633\) −21.3491 −0.848552
\(634\) −21.0368 −0.835479
\(635\) 15.9541 0.633118
\(636\) −6.29590 −0.249649
\(637\) 0 0
\(638\) −0.939001 −0.0371754
\(639\) 6.67456 0.264042
\(640\) 1.00000 0.0395285
\(641\) −29.7017 −1.17315 −0.586574 0.809896i \(-0.699523\pi\)
−0.586574 + 0.809896i \(0.699523\pi\)
\(642\) 3.77479 0.148979
\(643\) −21.5907 −0.851455 −0.425728 0.904851i \(-0.639982\pi\)
−0.425728 + 0.904851i \(0.639982\pi\)
\(644\) 0.286208 0.0112782
\(645\) 6.54288 0.257626
\(646\) −9.23490 −0.363342
\(647\) −29.3623 −1.15435 −0.577175 0.816620i \(-0.695845\pi\)
−0.577175 + 0.816620i \(0.695845\pi\)
\(648\) 1.00000 0.0392837
\(649\) −9.49827 −0.372840
\(650\) 0 0
\(651\) −2.76271 −0.108279
\(652\) −4.71917 −0.184817
\(653\) 33.4470 1.30888 0.654440 0.756114i \(-0.272904\pi\)
0.654440 + 0.756114i \(0.272904\pi\)
\(654\) −1.05861 −0.0413948
\(655\) −17.0911 −0.667806
\(656\) 3.40581 0.132975
\(657\) 10.1371 0.395485
\(658\) −0.484271 −0.0188788
\(659\) 39.3715 1.53370 0.766848 0.641829i \(-0.221824\pi\)
0.766848 + 0.641829i \(0.221824\pi\)
\(660\) −3.04892 −0.118679
\(661\) −10.1274 −0.393909 −0.196955 0.980413i \(-0.563105\pi\)
−0.196955 + 0.980413i \(0.563105\pi\)
\(662\) −29.4252 −1.14364
\(663\) 0 0
\(664\) −3.93900 −0.152863
\(665\) −5.93900 −0.230305
\(666\) 7.29590 0.282710
\(667\) −0.109916 −0.00425597
\(668\) −14.5308 −0.562213
\(669\) −12.7603 −0.493342
\(670\) 5.96077 0.230285
\(671\) −33.6504 −1.29906
\(672\) −0.801938 −0.0309354
\(673\) −10.1400 −0.390870 −0.195435 0.980717i \(-0.562612\pi\)
−0.195435 + 0.980717i \(0.562612\pi\)
\(674\) −17.5483 −0.675933
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 22.9952 0.883778 0.441889 0.897070i \(-0.354309\pi\)
0.441889 + 0.897070i \(0.354309\pi\)
\(678\) 11.0368 0.423867
\(679\) −7.80731 −0.299617
\(680\) −1.24698 −0.0478195
\(681\) 14.1836 0.543516
\(682\) −10.5036 −0.402205
\(683\) −4.10800 −0.157188 −0.0785941 0.996907i \(-0.525043\pi\)
−0.0785941 + 0.996907i \(0.525043\pi\)
\(684\) 7.40581 0.283168
\(685\) −16.4373 −0.628036
\(686\) 10.7114 0.408963
\(687\) −4.71140 −0.179751
\(688\) 6.54288 0.249445
\(689\) 0 0
\(690\) −0.356896 −0.0135868
\(691\) 30.1618 1.14741 0.573705 0.819062i \(-0.305506\pi\)
0.573705 + 0.819062i \(0.305506\pi\)
\(692\) 11.4523 0.435353
\(693\) 2.44504 0.0928795
\(694\) 1.27844 0.0485289
\(695\) −8.10992 −0.307627
\(696\) 0.307979 0.0116739
\(697\) −4.24698 −0.160866
\(698\) 18.0911 0.684759
\(699\) 0.0556221 0.00210382
\(700\) −0.801938 −0.0303104
\(701\) −37.5066 −1.41661 −0.708303 0.705909i \(-0.750539\pi\)
−0.708303 + 0.705909i \(0.750539\pi\)
\(702\) 0 0
\(703\) 54.0320 2.03786
\(704\) −3.04892 −0.114910
\(705\) 0.603875 0.0227433
\(706\) −18.9071 −0.711577
\(707\) −6.14138 −0.230970
\(708\) 3.11529 0.117080
\(709\) −4.20211 −0.157814 −0.0789068 0.996882i \(-0.525143\pi\)
−0.0789068 + 0.996882i \(0.525143\pi\)
\(710\) 6.67456 0.250492
\(711\) −13.6799 −0.513038
\(712\) −8.05861 −0.302009
\(713\) −1.22952 −0.0460459
\(714\) 1.00000 0.0374241
\(715\) 0 0
\(716\) −4.32975 −0.161810
\(717\) 21.0489 0.786086
\(718\) 32.7308 1.22150
\(719\) −28.3532 −1.05740 −0.528698 0.848810i \(-0.677319\pi\)
−0.528698 + 0.848810i \(0.677319\pi\)
\(720\) 1.00000 0.0372678
\(721\) −1.86831 −0.0695796
\(722\) 35.8461 1.33405
\(723\) −12.0030 −0.446396
\(724\) −24.8321 −0.922877
\(725\) 0.307979 0.0114380
\(726\) −1.70410 −0.0632452
\(727\) 10.2892 0.381605 0.190803 0.981628i \(-0.438891\pi\)
0.190803 + 0.981628i \(0.438891\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 10.1371 0.375190
\(731\) −8.15883 −0.301765
\(732\) 11.0368 0.407933
\(733\) −48.8963 −1.80603 −0.903013 0.429613i \(-0.858650\pi\)
−0.903013 + 0.429613i \(0.858650\pi\)
\(734\) −3.10321 −0.114542
\(735\) −6.35690 −0.234478
\(736\) −0.356896 −0.0131554
\(737\) −18.1739 −0.669444
\(738\) 3.40581 0.125370
\(739\) −23.7496 −0.873642 −0.436821 0.899549i \(-0.643896\pi\)
−0.436821 + 0.899549i \(0.643896\pi\)
\(740\) 7.29590 0.268203
\(741\) 0 0
\(742\) 5.04892 0.185352
\(743\) −51.5153 −1.88991 −0.944956 0.327198i \(-0.893896\pi\)
−0.944956 + 0.327198i \(0.893896\pi\)
\(744\) 3.44504 0.126301
\(745\) 11.0532 0.404959
\(746\) 13.7366 0.502934
\(747\) −3.93900 −0.144121
\(748\) 3.80194 0.139013
\(749\) −3.02715 −0.110610
\(750\) 1.00000 0.0365148
\(751\) 43.8939 1.60171 0.800856 0.598857i \(-0.204378\pi\)
0.800856 + 0.598857i \(0.204378\pi\)
\(752\) 0.603875 0.0220211
\(753\) −2.25906 −0.0823248
\(754\) 0 0
\(755\) 6.89546 0.250952
\(756\) −0.801938 −0.0291662
\(757\) −18.6136 −0.676521 −0.338261 0.941052i \(-0.609839\pi\)
−0.338261 + 0.941052i \(0.609839\pi\)
\(758\) 34.0097 1.23529
\(759\) 1.08815 0.0394972
\(760\) 7.40581 0.268637
\(761\) −44.8364 −1.62532 −0.812659 0.582740i \(-0.801981\pi\)
−0.812659 + 0.582740i \(0.801981\pi\)
\(762\) 15.9541 0.577955
\(763\) 0.848936 0.0307336
\(764\) 22.1933 0.802925
\(765\) −1.24698 −0.0450846
\(766\) 38.6902 1.39793
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 39.5303 1.42550 0.712750 0.701418i \(-0.247449\pi\)
0.712750 + 0.701418i \(0.247449\pi\)
\(770\) 2.44504 0.0881132
\(771\) −17.0043 −0.612395
\(772\) 25.4306 0.915266
\(773\) 49.8665 1.79357 0.896787 0.442463i \(-0.145895\pi\)
0.896787 + 0.442463i \(0.145895\pi\)
\(774\) 6.54288 0.235179
\(775\) 3.44504 0.123750
\(776\) 9.73556 0.349486
\(777\) −5.85086 −0.209898
\(778\) −11.2567 −0.403571
\(779\) 25.2228 0.903701
\(780\) 0 0
\(781\) −20.3502 −0.728187
\(782\) 0.445042 0.0159147
\(783\) 0.307979 0.0110063
\(784\) −6.35690 −0.227032
\(785\) −20.8388 −0.743768
\(786\) −17.0911 −0.609620
\(787\) −26.0398 −0.928219 −0.464110 0.885778i \(-0.653626\pi\)
−0.464110 + 0.885778i \(0.653626\pi\)
\(788\) −10.6310 −0.378714
\(789\) −28.5773 −1.01738
\(790\) −13.6799 −0.486710
\(791\) −8.85086 −0.314700
\(792\) −3.04892 −0.108339
\(793\) 0 0
\(794\) −23.4403 −0.831864
\(795\) −6.29590 −0.223292
\(796\) 11.0465 0.391534
\(797\) −23.9855 −0.849611 −0.424805 0.905285i \(-0.639657\pi\)
−0.424805 + 0.905285i \(0.639657\pi\)
\(798\) −5.93900 −0.210238
\(799\) −0.753020 −0.0266399
\(800\) 1.00000 0.0353553
\(801\) −8.05861 −0.284737
\(802\) −37.6950 −1.33106
\(803\) −30.9071 −1.09069
\(804\) 5.96077 0.210220
\(805\) 0.286208 0.0100875
\(806\) 0 0
\(807\) −3.28083 −0.115491
\(808\) 7.65817 0.269413
\(809\) 52.4825 1.84519 0.922593 0.385776i \(-0.126066\pi\)
0.922593 + 0.385776i \(0.126066\pi\)
\(810\) 1.00000 0.0351364
\(811\) 3.63666 0.127700 0.0638502 0.997959i \(-0.479662\pi\)
0.0638502 + 0.997959i \(0.479662\pi\)
\(812\) −0.246980 −0.00866728
\(813\) −8.70948 −0.305455
\(814\) −22.2446 −0.779672
\(815\) −4.71917 −0.165305
\(816\) −1.24698 −0.0436530
\(817\) 48.4553 1.69524
\(818\) −1.19567 −0.0418056
\(819\) 0 0
\(820\) 3.40581 0.118936
\(821\) 7.98984 0.278847 0.139424 0.990233i \(-0.455475\pi\)
0.139424 + 0.990233i \(0.455475\pi\)
\(822\) −16.4373 −0.573316
\(823\) 28.1395 0.980880 0.490440 0.871475i \(-0.336836\pi\)
0.490440 + 0.871475i \(0.336836\pi\)
\(824\) 2.32975 0.0811606
\(825\) −3.04892 −0.106150
\(826\) −2.49827 −0.0869260
\(827\) −30.9256 −1.07539 −0.537694 0.843140i \(-0.680704\pi\)
−0.537694 + 0.843140i \(0.680704\pi\)
\(828\) −0.356896 −0.0124030
\(829\) −31.9172 −1.10853 −0.554266 0.832340i \(-0.687001\pi\)
−0.554266 + 0.832340i \(0.687001\pi\)
\(830\) −3.93900 −0.136725
\(831\) −8.91185 −0.309149
\(832\) 0 0
\(833\) 7.92692 0.274651
\(834\) −8.10992 −0.280823
\(835\) −14.5308 −0.502859
\(836\) −22.5797 −0.780936
\(837\) 3.44504 0.119078
\(838\) 1.77048 0.0611602
\(839\) 24.9866 0.862633 0.431316 0.902201i \(-0.358049\pi\)
0.431316 + 0.902201i \(0.358049\pi\)
\(840\) −0.801938 −0.0276695
\(841\) −28.9051 −0.996729
\(842\) 1.80194 0.0620989
\(843\) −1.90648 −0.0656626
\(844\) −21.3491 −0.734867
\(845\) 0 0
\(846\) 0.603875 0.0207617
\(847\) 1.36658 0.0469564
\(848\) −6.29590 −0.216202
\(849\) −14.9245 −0.512208
\(850\) −1.24698 −0.0427710
\(851\) −2.60388 −0.0892597
\(852\) 6.67456 0.228667
\(853\) 35.4416 1.21350 0.606748 0.794894i \(-0.292474\pi\)
0.606748 + 0.794894i \(0.292474\pi\)
\(854\) −8.85086 −0.302870
\(855\) 7.40581 0.253273
\(856\) 3.77479 0.129020
\(857\) −14.6541 −0.500575 −0.250288 0.968172i \(-0.580525\pi\)
−0.250288 + 0.968172i \(0.580525\pi\)
\(858\) 0 0
\(859\) 46.3153 1.58026 0.790128 0.612941i \(-0.210014\pi\)
0.790128 + 0.612941i \(0.210014\pi\)
\(860\) 6.54288 0.223110
\(861\) −2.73125 −0.0930808
\(862\) −0.567040 −0.0193134
\(863\) −40.3642 −1.37401 −0.687007 0.726651i \(-0.741076\pi\)
−0.687007 + 0.726651i \(0.741076\pi\)
\(864\) 1.00000 0.0340207
\(865\) 11.4523 0.389391
\(866\) 17.5743 0.597200
\(867\) −15.4450 −0.524541
\(868\) −2.76271 −0.0937725
\(869\) 41.7090 1.41488
\(870\) 0.307979 0.0104414
\(871\) 0 0
\(872\) −1.05861 −0.0358489
\(873\) 9.73556 0.329499
\(874\) −2.64310 −0.0894043
\(875\) −0.801938 −0.0271104
\(876\) 10.1371 0.342500
\(877\) −35.0103 −1.18221 −0.591107 0.806593i \(-0.701309\pi\)
−0.591107 + 0.806593i \(0.701309\pi\)
\(878\) 3.00969 0.101572
\(879\) −12.8901 −0.434772
\(880\) −3.04892 −0.102779
\(881\) 23.8689 0.804164 0.402082 0.915604i \(-0.368287\pi\)
0.402082 + 0.915604i \(0.368287\pi\)
\(882\) −6.35690 −0.214048
\(883\) 26.1806 0.881047 0.440524 0.897741i \(-0.354793\pi\)
0.440524 + 0.897741i \(0.354793\pi\)
\(884\) 0 0
\(885\) 3.11529 0.104719
\(886\) 38.4499 1.29175
\(887\) 25.8625 0.868377 0.434188 0.900822i \(-0.357035\pi\)
0.434188 + 0.900822i \(0.357035\pi\)
\(888\) 7.29590 0.244834
\(889\) −12.7942 −0.429103
\(890\) −8.05861 −0.270125
\(891\) −3.04892 −0.102143
\(892\) −12.7603 −0.427247
\(893\) 4.47219 0.149656
\(894\) 11.0532 0.369675
\(895\) −4.32975 −0.144728
\(896\) −0.801938 −0.0267909
\(897\) 0 0
\(898\) 26.9909 0.900698
\(899\) 1.06100 0.0353863
\(900\) 1.00000 0.0333333
\(901\) 7.85086 0.261550
\(902\) −10.3840 −0.345751
\(903\) −5.24698 −0.174609
\(904\) 11.0368 0.367080
\(905\) −24.8321 −0.825446
\(906\) 6.89546 0.229086
\(907\) 12.3980 0.411670 0.205835 0.978587i \(-0.434009\pi\)
0.205835 + 0.978587i \(0.434009\pi\)
\(908\) 14.1836 0.470699
\(909\) 7.65817 0.254005
\(910\) 0 0
\(911\) 5.34183 0.176983 0.0884914 0.996077i \(-0.471795\pi\)
0.0884914 + 0.996077i \(0.471795\pi\)
\(912\) 7.40581 0.245231
\(913\) 12.0097 0.397463
\(914\) 19.2664 0.637274
\(915\) 11.0368 0.364867
\(916\) −4.71140 −0.155669
\(917\) 13.7060 0.452613
\(918\) −1.24698 −0.0411565
\(919\) 32.3327 1.06656 0.533279 0.845939i \(-0.320960\pi\)
0.533279 + 0.845939i \(0.320960\pi\)
\(920\) −0.356896 −0.0117665
\(921\) 19.2892 0.635600
\(922\) −18.5942 −0.612367
\(923\) 0 0
\(924\) 2.44504 0.0804360
\(925\) 7.29590 0.239888
\(926\) 5.79417 0.190408
\(927\) 2.32975 0.0765190
\(928\) 0.307979 0.0101099
\(929\) 19.3381 0.634463 0.317231 0.948348i \(-0.397247\pi\)
0.317231 + 0.948348i \(0.397247\pi\)
\(930\) 3.44504 0.112967
\(931\) −47.0780 −1.54292
\(932\) 0.0556221 0.00182196
\(933\) 1.23729 0.0405071
\(934\) 11.2457 0.367969
\(935\) 3.80194 0.124337
\(936\) 0 0
\(937\) −40.4271 −1.32070 −0.660348 0.750960i \(-0.729591\pi\)
−0.660348 + 0.750960i \(0.729591\pi\)
\(938\) −4.78017 −0.156078
\(939\) 8.09113 0.264044
\(940\) 0.603875 0.0196962
\(941\) −9.38537 −0.305954 −0.152977 0.988230i \(-0.548886\pi\)
−0.152977 + 0.988230i \(0.548886\pi\)
\(942\) −20.8388 −0.678964
\(943\) −1.21552 −0.0395828
\(944\) 3.11529 0.101394
\(945\) −0.801938 −0.0260870
\(946\) −19.9487 −0.648588
\(947\) −24.4252 −0.793712 −0.396856 0.917881i \(-0.629899\pi\)
−0.396856 + 0.917881i \(0.629899\pi\)
\(948\) −13.6799 −0.444304
\(949\) 0 0
\(950\) 7.40581 0.240276
\(951\) −21.0368 −0.682166
\(952\) 1.00000 0.0324102
\(953\) 57.1657 1.85178 0.925889 0.377797i \(-0.123318\pi\)
0.925889 + 0.377797i \(0.123318\pi\)
\(954\) −6.29590 −0.203837
\(955\) 22.1933 0.718158
\(956\) 21.0489 0.680771
\(957\) −0.939001 −0.0303536
\(958\) −35.2452 −1.13872
\(959\) 13.1817 0.425658
\(960\) 1.00000 0.0322749
\(961\) −19.1317 −0.617151
\(962\) 0 0
\(963\) 3.77479 0.121641
\(964\) −12.0030 −0.386590
\(965\) 25.4306 0.818639
\(966\) 0.286208 0.00920860
\(967\) −5.76271 −0.185316 −0.0926581 0.995698i \(-0.529536\pi\)
−0.0926581 + 0.995698i \(0.529536\pi\)
\(968\) −1.70410 −0.0547719
\(969\) −9.23490 −0.296668
\(970\) 9.73556 0.312590
\(971\) −1.72694 −0.0554201 −0.0277100 0.999616i \(-0.508822\pi\)
−0.0277100 + 0.999616i \(0.508822\pi\)
\(972\) 1.00000 0.0320750
\(973\) 6.50365 0.208497
\(974\) 8.49827 0.272302
\(975\) 0 0
\(976\) 11.0368 0.353281
\(977\) 26.4747 0.847001 0.423500 0.905896i \(-0.360801\pi\)
0.423500 + 0.905896i \(0.360801\pi\)
\(978\) −4.71917 −0.150902
\(979\) 24.5700 0.785261
\(980\) −6.35690 −0.203064
\(981\) −1.05861 −0.0337987
\(982\) −25.5254 −0.814549
\(983\) 6.81727 0.217437 0.108719 0.994073i \(-0.465325\pi\)
0.108719 + 0.994073i \(0.465325\pi\)
\(984\) 3.40581 0.108573
\(985\) −10.6310 −0.338733
\(986\) −0.384043 −0.0122304
\(987\) −0.484271 −0.0154145
\(988\) 0 0
\(989\) −2.33513 −0.0742527
\(990\) −3.04892 −0.0969010
\(991\) 8.31203 0.264040 0.132020 0.991247i \(-0.457854\pi\)
0.132020 + 0.991247i \(0.457854\pi\)
\(992\) 3.44504 0.109380
\(993\) −29.4252 −0.933780
\(994\) −5.35258 −0.169774
\(995\) 11.0465 0.350198
\(996\) −3.93900 −0.124812
\(997\) 49.6002 1.57085 0.785426 0.618955i \(-0.212444\pi\)
0.785426 + 0.618955i \(0.212444\pi\)
\(998\) −29.4577 −0.932468
\(999\) 7.29590 0.230832
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.by.1.1 yes 3
13.5 odd 4 5070.2.b.x.1351.1 6
13.8 odd 4 5070.2.b.x.1351.6 6
13.12 even 2 5070.2.a.bn.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bn.1.3 3 13.12 even 2
5070.2.a.by.1.1 yes 3 1.1 even 1 trivial
5070.2.b.x.1351.1 6 13.5 odd 4
5070.2.b.x.1351.6 6 13.8 odd 4