Properties

Label 8470.2.a.bl.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.73205 q^{12} +1.73205 q^{13} +1.00000 q^{14} +1.73205 q^{15} +1.00000 q^{16} +1.26795 q^{17} -1.73205 q^{19} -1.00000 q^{20} +1.73205 q^{21} +2.46410 q^{23} +1.73205 q^{24} +1.00000 q^{25} -1.73205 q^{26} +5.19615 q^{27} -1.00000 q^{28} +2.53590 q^{29} -1.73205 q^{30} -3.26795 q^{31} -1.00000 q^{32} -1.26795 q^{34} +1.00000 q^{35} -7.46410 q^{37} +1.73205 q^{38} -3.00000 q^{39} +1.00000 q^{40} -6.73205 q^{41} -1.73205 q^{42} -7.66025 q^{43} -2.46410 q^{46} -2.73205 q^{47} -1.73205 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.19615 q^{51} +1.73205 q^{52} -1.26795 q^{53} -5.19615 q^{54} +1.00000 q^{56} +3.00000 q^{57} -2.53590 q^{58} -1.00000 q^{59} +1.73205 q^{60} +4.00000 q^{61} +3.26795 q^{62} +1.00000 q^{64} -1.73205 q^{65} +14.1962 q^{67} +1.26795 q^{68} -4.26795 q^{69} -1.00000 q^{70} +2.92820 q^{71} +9.66025 q^{73} +7.46410 q^{74} -1.73205 q^{75} -1.73205 q^{76} +3.00000 q^{78} -0.0717968 q^{79} -1.00000 q^{80} -9.00000 q^{81} +6.73205 q^{82} +11.9282 q^{83} +1.73205 q^{84} -1.26795 q^{85} +7.66025 q^{86} -4.39230 q^{87} -4.73205 q^{89} -1.73205 q^{91} +2.46410 q^{92} +5.66025 q^{93} +2.73205 q^{94} +1.73205 q^{95} +1.73205 q^{96} +1.46410 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} - 2q^{8} + 2q^{10} + 2q^{14} + 2q^{16} + 6q^{17} - 2q^{20} - 2q^{23} + 2q^{25} - 2q^{28} + 12q^{29} - 10q^{31} - 2q^{32} - 6q^{34} + 2q^{35} - 8q^{37} - 6q^{39} + 2q^{40} - 10q^{41} + 2q^{43} + 2q^{46} - 2q^{47} + 2q^{49} - 2q^{50} + 6q^{51} - 6q^{53} + 2q^{56} + 6q^{57} - 12q^{58} - 2q^{59} + 8q^{61} + 10q^{62} + 2q^{64} + 18q^{67} + 6q^{68} - 12q^{69} - 2q^{70} - 8q^{71} + 2q^{73} + 8q^{74} + 6q^{78} - 14q^{79} - 2q^{80} - 18q^{81} + 10q^{82} + 10q^{83} - 6q^{85} - 2q^{86} + 12q^{87} - 6q^{89} - 2q^{92} - 6q^{93} + 2q^{94} - 4q^{97} - 2q^{98} + O(q^{100}) \)

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
\(3\) −1.73205 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.73205 0.707107
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −1.73205 −0.500000
\(13\) 1.73205 0.480384 0.240192 0.970725i \(-0.422790\pi\)
0.240192 + 0.970725i \(0.422790\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.73205 0.447214
\(16\) 1.00000 0.250000
\(17\) 1.26795 0.307523 0.153761 0.988108i \(-0.450861\pi\)
0.153761 + 0.988108i \(0.450861\pi\)
\(18\) 0 0
\(19\) −1.73205 −0.397360 −0.198680 0.980064i \(-0.563665\pi\)
−0.198680 + 0.980064i \(0.563665\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.73205 0.377964
\(22\) 0 0
\(23\) 2.46410 0.513801 0.256900 0.966438i \(-0.417299\pi\)
0.256900 + 0.966438i \(0.417299\pi\)
\(24\) 1.73205 0.353553
\(25\) 1.00000 0.200000
\(26\) −1.73205 −0.339683
\(27\) 5.19615 1.00000
\(28\) −1.00000 −0.188982
\(29\) 2.53590 0.470905 0.235452 0.971886i \(-0.424343\pi\)
0.235452 + 0.971886i \(0.424343\pi\)
\(30\) −1.73205 −0.316228
\(31\) −3.26795 −0.586941 −0.293471 0.955968i \(-0.594810\pi\)
−0.293471 + 0.955968i \(0.594810\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.26795 −0.217451
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −7.46410 −1.22709 −0.613545 0.789659i \(-0.710257\pi\)
−0.613545 + 0.789659i \(0.710257\pi\)
\(38\) 1.73205 0.280976
\(39\) −3.00000 −0.480384
\(40\) 1.00000 0.158114
\(41\) −6.73205 −1.05137 −0.525685 0.850679i \(-0.676191\pi\)
−0.525685 + 0.850679i \(0.676191\pi\)
\(42\) −1.73205 −0.267261
\(43\) −7.66025 −1.16818 −0.584089 0.811690i \(-0.698548\pi\)
−0.584089 + 0.811690i \(0.698548\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.46410 −0.363312
\(47\) −2.73205 −0.398511 −0.199255 0.979948i \(-0.563852\pi\)
−0.199255 + 0.979948i \(0.563852\pi\)
\(48\) −1.73205 −0.250000
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −2.19615 −0.307523
\(52\) 1.73205 0.240192
\(53\) −1.26795 −0.174166 −0.0870831 0.996201i \(-0.527755\pi\)
−0.0870831 + 0.996201i \(0.527755\pi\)
\(54\) −5.19615 −0.707107
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 3.00000 0.397360
\(58\) −2.53590 −0.332980
\(59\) −1.00000 −0.130189 −0.0650945 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) 1.73205 0.223607
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 3.26795 0.415030
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.73205 −0.214834
\(66\) 0 0
\(67\) 14.1962 1.73434 0.867168 0.498016i \(-0.165938\pi\)
0.867168 + 0.498016i \(0.165938\pi\)
\(68\) 1.26795 0.153761
\(69\) −4.26795 −0.513801
\(70\) −1.00000 −0.119523
\(71\) 2.92820 0.347514 0.173757 0.984789i \(-0.444409\pi\)
0.173757 + 0.984789i \(0.444409\pi\)
\(72\) 0 0
\(73\) 9.66025 1.13065 0.565324 0.824869i \(-0.308751\pi\)
0.565324 + 0.824869i \(0.308751\pi\)
\(74\) 7.46410 0.867684
\(75\) −1.73205 −0.200000
\(76\) −1.73205 −0.198680
\(77\) 0 0
\(78\) 3.00000 0.339683
\(79\) −0.0717968 −0.00807777 −0.00403888 0.999992i \(-0.501286\pi\)
−0.00403888 + 0.999992i \(0.501286\pi\)
\(80\) −1.00000 −0.111803
\(81\) −9.00000 −1.00000
\(82\) 6.73205 0.743431
\(83\) 11.9282 1.30929 0.654645 0.755936i \(-0.272818\pi\)
0.654645 + 0.755936i \(0.272818\pi\)
\(84\) 1.73205 0.188982
\(85\) −1.26795 −0.137528
\(86\) 7.66025 0.826026
\(87\) −4.39230 −0.470905
\(88\) 0 0
\(89\) −4.73205 −0.501596 −0.250798 0.968039i \(-0.580693\pi\)
−0.250798 + 0.968039i \(0.580693\pi\)
\(90\) 0 0
\(91\) −1.73205 −0.181568
\(92\) 2.46410 0.256900
\(93\) 5.66025 0.586941
\(94\) 2.73205 0.281790
\(95\) 1.73205 0.177705
\(96\) 1.73205 0.176777
\(97\) 1.46410 0.148657 0.0743285 0.997234i \(-0.476319\pi\)
0.0743285 + 0.997234i \(0.476319\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 7.92820 0.788886 0.394443 0.918920i \(-0.370938\pi\)
0.394443 + 0.918920i \(0.370938\pi\)
\(102\) 2.19615 0.217451
\(103\) 1.80385 0.177738 0.0888692 0.996043i \(-0.471675\pi\)
0.0888692 + 0.996043i \(0.471675\pi\)
\(104\) −1.73205 −0.169842
\(105\) −1.73205 −0.169031
\(106\) 1.26795 0.123154
\(107\) 0.535898 0.0518073 0.0259036 0.999664i \(-0.491754\pi\)
0.0259036 + 0.999664i \(0.491754\pi\)
\(108\) 5.19615 0.500000
\(109\) 0.196152 0.0187880 0.00939400 0.999956i \(-0.497010\pi\)
0.00939400 + 0.999956i \(0.497010\pi\)
\(110\) 0 0
\(111\) 12.9282 1.22709
\(112\) −1.00000 −0.0944911
\(113\) −4.66025 −0.438400 −0.219200 0.975680i \(-0.570345\pi\)
−0.219200 + 0.975680i \(0.570345\pi\)
\(114\) −3.00000 −0.280976
\(115\) −2.46410 −0.229779
\(116\) 2.53590 0.235452
\(117\) 0 0
\(118\) 1.00000 0.0920575
\(119\) −1.26795 −0.116233
\(120\) −1.73205 −0.158114
\(121\) 0 0
\(122\) −4.00000 −0.362143
\(123\) 11.6603 1.05137
\(124\) −3.26795 −0.293471
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 13.1962 1.17097 0.585485 0.810684i \(-0.300904\pi\)
0.585485 + 0.810684i \(0.300904\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 13.2679 1.16818
\(130\) 1.73205 0.151911
\(131\) 4.26795 0.372892 0.186446 0.982465i \(-0.440303\pi\)
0.186446 + 0.982465i \(0.440303\pi\)
\(132\) 0 0
\(133\) 1.73205 0.150188
\(134\) −14.1962 −1.22636
\(135\) −5.19615 −0.447214
\(136\) −1.26795 −0.108726
\(137\) 12.1244 1.03585 0.517927 0.855425i \(-0.326704\pi\)
0.517927 + 0.855425i \(0.326704\pi\)
\(138\) 4.26795 0.363312
\(139\) 1.33975 0.113636 0.0568179 0.998385i \(-0.481905\pi\)
0.0568179 + 0.998385i \(0.481905\pi\)
\(140\) 1.00000 0.0845154
\(141\) 4.73205 0.398511
\(142\) −2.92820 −0.245729
\(143\) 0 0
\(144\) 0 0
\(145\) −2.53590 −0.210595
\(146\) −9.66025 −0.799488
\(147\) −1.73205 −0.142857
\(148\) −7.46410 −0.613545
\(149\) 0.928203 0.0760414 0.0380207 0.999277i \(-0.487895\pi\)
0.0380207 + 0.999277i \(0.487895\pi\)
\(150\) 1.73205 0.141421
\(151\) −13.0000 −1.05792 −0.528962 0.848645i \(-0.677419\pi\)
−0.528962 + 0.848645i \(0.677419\pi\)
\(152\) 1.73205 0.140488
\(153\) 0 0
\(154\) 0 0
\(155\) 3.26795 0.262488
\(156\) −3.00000 −0.240192
\(157\) −10.4641 −0.835126 −0.417563 0.908648i \(-0.637116\pi\)
−0.417563 + 0.908648i \(0.637116\pi\)
\(158\) 0.0717968 0.00571184
\(159\) 2.19615 0.174166
\(160\) 1.00000 0.0790569
\(161\) −2.46410 −0.194198
\(162\) 9.00000 0.707107
\(163\) 8.92820 0.699311 0.349655 0.936878i \(-0.386299\pi\)
0.349655 + 0.936878i \(0.386299\pi\)
\(164\) −6.73205 −0.525685
\(165\) 0 0
\(166\) −11.9282 −0.925808
\(167\) 6.73205 0.520942 0.260471 0.965482i \(-0.416122\pi\)
0.260471 + 0.965482i \(0.416122\pi\)
\(168\) −1.73205 −0.133631
\(169\) −10.0000 −0.769231
\(170\) 1.26795 0.0972473
\(171\) 0 0
\(172\) −7.66025 −0.584089
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 4.39230 0.332980
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 1.73205 0.130189
\(178\) 4.73205 0.354682
\(179\) −9.12436 −0.681986 −0.340993 0.940066i \(-0.610763\pi\)
−0.340993 + 0.940066i \(0.610763\pi\)
\(180\) 0 0
\(181\) −4.12436 −0.306561 −0.153280 0.988183i \(-0.548984\pi\)
−0.153280 + 0.988183i \(0.548984\pi\)
\(182\) 1.73205 0.128388
\(183\) −6.92820 −0.512148
\(184\) −2.46410 −0.181656
\(185\) 7.46410 0.548772
\(186\) −5.66025 −0.415030
\(187\) 0 0
\(188\) −2.73205 −0.199255
\(189\) −5.19615 −0.377964
\(190\) −1.73205 −0.125656
\(191\) −18.2679 −1.32182 −0.660911 0.750464i \(-0.729830\pi\)
−0.660911 + 0.750464i \(0.729830\pi\)
\(192\) −1.73205 −0.125000
\(193\) 21.7846 1.56809 0.784045 0.620704i \(-0.213153\pi\)
0.784045 + 0.620704i \(0.213153\pi\)
\(194\) −1.46410 −0.105116
\(195\) 3.00000 0.214834
\(196\) 1.00000 0.0714286
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 0 0
\(199\) 8.39230 0.594915 0.297457 0.954735i \(-0.403861\pi\)
0.297457 + 0.954735i \(0.403861\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −24.5885 −1.73434
\(202\) −7.92820 −0.557826
\(203\) −2.53590 −0.177985
\(204\) −2.19615 −0.153761
\(205\) 6.73205 0.470187
\(206\) −1.80385 −0.125680
\(207\) 0 0
\(208\) 1.73205 0.120096
\(209\) 0 0
\(210\) 1.73205 0.119523
\(211\) 1.46410 0.100793 0.0503965 0.998729i \(-0.483952\pi\)
0.0503965 + 0.998729i \(0.483952\pi\)
\(212\) −1.26795 −0.0870831
\(213\) −5.07180 −0.347514
\(214\) −0.535898 −0.0366333
\(215\) 7.66025 0.522425
\(216\) −5.19615 −0.353553
\(217\) 3.26795 0.221843
\(218\) −0.196152 −0.0132851
\(219\) −16.7321 −1.13065
\(220\) 0 0
\(221\) 2.19615 0.147729
\(222\) −12.9282 −0.867684
\(223\) 11.1244 0.744942 0.372471 0.928044i \(-0.378511\pi\)
0.372471 + 0.928044i \(0.378511\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 4.66025 0.309995
\(227\) 7.46410 0.495410 0.247705 0.968836i \(-0.420324\pi\)
0.247705 + 0.968836i \(0.420324\pi\)
\(228\) 3.00000 0.198680
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 2.46410 0.162478
\(231\) 0 0
\(232\) −2.53590 −0.166490
\(233\) 22.8564 1.49737 0.748686 0.662924i \(-0.230685\pi\)
0.748686 + 0.662924i \(0.230685\pi\)
\(234\) 0 0
\(235\) 2.73205 0.178219
\(236\) −1.00000 −0.0650945
\(237\) 0.124356 0.00807777
\(238\) 1.26795 0.0821889
\(239\) 2.46410 0.159389 0.0796947 0.996819i \(-0.474605\pi\)
0.0796947 + 0.996819i \(0.474605\pi\)
\(240\) 1.73205 0.111803
\(241\) 10.9282 0.703947 0.351974 0.936010i \(-0.385511\pi\)
0.351974 + 0.936010i \(0.385511\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) −1.00000 −0.0638877
\(246\) −11.6603 −0.743431
\(247\) −3.00000 −0.190885
\(248\) 3.26795 0.207515
\(249\) −20.6603 −1.30929
\(250\) 1.00000 0.0632456
\(251\) −10.3923 −0.655956 −0.327978 0.944685i \(-0.606367\pi\)
−0.327978 + 0.944685i \(0.606367\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −13.1962 −0.828000
\(255\) 2.19615 0.137528
\(256\) 1.00000 0.0625000
\(257\) −19.2679 −1.20190 −0.600951 0.799286i \(-0.705211\pi\)
−0.600951 + 0.799286i \(0.705211\pi\)
\(258\) −13.2679 −0.826026
\(259\) 7.46410 0.463797
\(260\) −1.73205 −0.107417
\(261\) 0 0
\(262\) −4.26795 −0.263675
\(263\) 17.7321 1.09341 0.546703 0.837327i \(-0.315883\pi\)
0.546703 + 0.837327i \(0.315883\pi\)
\(264\) 0 0
\(265\) 1.26795 0.0778895
\(266\) −1.73205 −0.106199
\(267\) 8.19615 0.501596
\(268\) 14.1962 0.867168
\(269\) 16.5167 1.00704 0.503519 0.863984i \(-0.332038\pi\)
0.503519 + 0.863984i \(0.332038\pi\)
\(270\) 5.19615 0.316228
\(271\) 32.3923 1.96769 0.983846 0.179016i \(-0.0572913\pi\)
0.983846 + 0.179016i \(0.0572913\pi\)
\(272\) 1.26795 0.0768807
\(273\) 3.00000 0.181568
\(274\) −12.1244 −0.732459
\(275\) 0 0
\(276\) −4.26795 −0.256900
\(277\) −4.58846 −0.275694 −0.137847 0.990454i \(-0.544018\pi\)
−0.137847 + 0.990454i \(0.544018\pi\)
\(278\) −1.33975 −0.0803526
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −7.73205 −0.461255 −0.230628 0.973042i \(-0.574078\pi\)
−0.230628 + 0.973042i \(0.574078\pi\)
\(282\) −4.73205 −0.281790
\(283\) 17.7846 1.05719 0.528593 0.848876i \(-0.322720\pi\)
0.528593 + 0.848876i \(0.322720\pi\)
\(284\) 2.92820 0.173757
\(285\) −3.00000 −0.177705
\(286\) 0 0
\(287\) 6.73205 0.397380
\(288\) 0 0
\(289\) −15.3923 −0.905430
\(290\) 2.53590 0.148913
\(291\) −2.53590 −0.148657
\(292\) 9.66025 0.565324
\(293\) 5.33975 0.311951 0.155976 0.987761i \(-0.450148\pi\)
0.155976 + 0.987761i \(0.450148\pi\)
\(294\) 1.73205 0.101015
\(295\) 1.00000 0.0582223
\(296\) 7.46410 0.433842
\(297\) 0 0
\(298\) −0.928203 −0.0537694
\(299\) 4.26795 0.246822
\(300\) −1.73205 −0.100000
\(301\) 7.66025 0.441530
\(302\) 13.0000 0.748066
\(303\) −13.7321 −0.788886
\(304\) −1.73205 −0.0993399
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) −3.07180 −0.175317 −0.0876584 0.996151i \(-0.527938\pi\)
−0.0876584 + 0.996151i \(0.527938\pi\)
\(308\) 0 0
\(309\) −3.12436 −0.177738
\(310\) −3.26795 −0.185607
\(311\) −32.1962 −1.82568 −0.912838 0.408322i \(-0.866114\pi\)
−0.912838 + 0.408322i \(0.866114\pi\)
\(312\) 3.00000 0.169842
\(313\) −16.5359 −0.934664 −0.467332 0.884082i \(-0.654785\pi\)
−0.467332 + 0.884082i \(0.654785\pi\)
\(314\) 10.4641 0.590523
\(315\) 0 0
\(316\) −0.0717968 −0.00403888
\(317\) 21.6603 1.21656 0.608281 0.793722i \(-0.291860\pi\)
0.608281 + 0.793722i \(0.291860\pi\)
\(318\) −2.19615 −0.123154
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −0.928203 −0.0518073
\(322\) 2.46410 0.137319
\(323\) −2.19615 −0.122197
\(324\) −9.00000 −0.500000
\(325\) 1.73205 0.0960769
\(326\) −8.92820 −0.494487
\(327\) −0.339746 −0.0187880
\(328\) 6.73205 0.371715
\(329\) 2.73205 0.150623
\(330\) 0 0
\(331\) −1.80385 −0.0991484 −0.0495742 0.998770i \(-0.515786\pi\)
−0.0495742 + 0.998770i \(0.515786\pi\)
\(332\) 11.9282 0.654645
\(333\) 0 0
\(334\) −6.73205 −0.368361
\(335\) −14.1962 −0.775619
\(336\) 1.73205 0.0944911
\(337\) −15.7846 −0.859842 −0.429921 0.902866i \(-0.641459\pi\)
−0.429921 + 0.902866i \(0.641459\pi\)
\(338\) 10.0000 0.543928
\(339\) 8.07180 0.438400
\(340\) −1.26795 −0.0687642
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 7.66025 0.413013
\(345\) 4.26795 0.229779
\(346\) 10.0000 0.537603
\(347\) −8.53590 −0.458231 −0.229116 0.973399i \(-0.573583\pi\)
−0.229116 + 0.973399i \(0.573583\pi\)
\(348\) −4.39230 −0.235452
\(349\) −3.00000 −0.160586 −0.0802932 0.996771i \(-0.525586\pi\)
−0.0802932 + 0.996771i \(0.525586\pi\)
\(350\) 1.00000 0.0534522
\(351\) 9.00000 0.480384
\(352\) 0 0
\(353\) 7.66025 0.407714 0.203857 0.979001i \(-0.434652\pi\)
0.203857 + 0.979001i \(0.434652\pi\)
\(354\) −1.73205 −0.0920575
\(355\) −2.92820 −0.155413
\(356\) −4.73205 −0.250798
\(357\) 2.19615 0.116233
\(358\) 9.12436 0.482237
\(359\) −14.0000 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) 0 0
\(361\) −16.0000 −0.842105
\(362\) 4.12436 0.216771
\(363\) 0 0
\(364\) −1.73205 −0.0907841
\(365\) −9.66025 −0.505641
\(366\) 6.92820 0.362143
\(367\) −4.33975 −0.226533 −0.113266 0.993565i \(-0.536131\pi\)
−0.113266 + 0.993565i \(0.536131\pi\)
\(368\) 2.46410 0.128450
\(369\) 0 0
\(370\) −7.46410 −0.388040
\(371\) 1.26795 0.0658286
\(372\) 5.66025 0.293471
\(373\) −12.9282 −0.669397 −0.334698 0.942325i \(-0.608634\pi\)
−0.334698 + 0.942325i \(0.608634\pi\)
\(374\) 0 0
\(375\) 1.73205 0.0894427
\(376\) 2.73205 0.140895
\(377\) 4.39230 0.226215
\(378\) 5.19615 0.267261
\(379\) −6.39230 −0.328351 −0.164175 0.986431i \(-0.552496\pi\)
−0.164175 + 0.986431i \(0.552496\pi\)
\(380\) 1.73205 0.0888523
\(381\) −22.8564 −1.17097
\(382\) 18.2679 0.934670
\(383\) −26.5885 −1.35861 −0.679303 0.733858i \(-0.737718\pi\)
−0.679303 + 0.733858i \(0.737718\pi\)
\(384\) 1.73205 0.0883883
\(385\) 0 0
\(386\) −21.7846 −1.10881
\(387\) 0 0
\(388\) 1.46410 0.0743285
\(389\) −10.8756 −0.551417 −0.275709 0.961241i \(-0.588912\pi\)
−0.275709 + 0.961241i \(0.588912\pi\)
\(390\) −3.00000 −0.151911
\(391\) 3.12436 0.158005
\(392\) −1.00000 −0.0505076
\(393\) −7.39230 −0.372892
\(394\) 24.0000 1.20910
\(395\) 0.0717968 0.00361249
\(396\) 0 0
\(397\) −9.85641 −0.494679 −0.247339 0.968929i \(-0.579556\pi\)
−0.247339 + 0.968929i \(0.579556\pi\)
\(398\) −8.39230 −0.420668
\(399\) −3.00000 −0.150188
\(400\) 1.00000 0.0500000
\(401\) −30.2487 −1.51055 −0.755274 0.655409i \(-0.772496\pi\)
−0.755274 + 0.655409i \(0.772496\pi\)
\(402\) 24.5885 1.22636
\(403\) −5.66025 −0.281957
\(404\) 7.92820 0.394443
\(405\) 9.00000 0.447214
\(406\) 2.53590 0.125855
\(407\) 0 0
\(408\) 2.19615 0.108726
\(409\) 25.1769 1.24492 0.622459 0.782652i \(-0.286134\pi\)
0.622459 + 0.782652i \(0.286134\pi\)
\(410\) −6.73205 −0.332472
\(411\) −21.0000 −1.03585
\(412\) 1.80385 0.0888692
\(413\) 1.00000 0.0492068
\(414\) 0 0
\(415\) −11.9282 −0.585532
\(416\) −1.73205 −0.0849208
\(417\) −2.32051 −0.113636
\(418\) 0 0
\(419\) −9.39230 −0.458844 −0.229422 0.973327i \(-0.573684\pi\)
−0.229422 + 0.973327i \(0.573684\pi\)
\(420\) −1.73205 −0.0845154
\(421\) −10.2487 −0.499492 −0.249746 0.968311i \(-0.580347\pi\)
−0.249746 + 0.968311i \(0.580347\pi\)
\(422\) −1.46410 −0.0712714
\(423\) 0 0
\(424\) 1.26795 0.0615771
\(425\) 1.26795 0.0615046
\(426\) 5.07180 0.245729
\(427\) −4.00000 −0.193574
\(428\) 0.535898 0.0259036
\(429\) 0 0
\(430\) −7.66025 −0.369410
\(431\) −18.4641 −0.889384 −0.444692 0.895683i \(-0.646687\pi\)
−0.444692 + 0.895683i \(0.646687\pi\)
\(432\) 5.19615 0.250000
\(433\) 12.1962 0.586110 0.293055 0.956096i \(-0.405328\pi\)
0.293055 + 0.956096i \(0.405328\pi\)
\(434\) −3.26795 −0.156867
\(435\) 4.39230 0.210595
\(436\) 0.196152 0.00939400
\(437\) −4.26795 −0.204164
\(438\) 16.7321 0.799488
\(439\) −13.8038 −0.658822 −0.329411 0.944187i \(-0.606850\pi\)
−0.329411 + 0.944187i \(0.606850\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.19615 −0.104460
\(443\) −18.9282 −0.899306 −0.449653 0.893203i \(-0.648452\pi\)
−0.449653 + 0.893203i \(0.648452\pi\)
\(444\) 12.9282 0.613545
\(445\) 4.73205 0.224321
\(446\) −11.1244 −0.526754
\(447\) −1.60770 −0.0760414
\(448\) −1.00000 −0.0472456
\(449\) −27.5359 −1.29950 −0.649750 0.760148i \(-0.725126\pi\)
−0.649750 + 0.760148i \(0.725126\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −4.66025 −0.219200
\(453\) 22.5167 1.05792
\(454\) −7.46410 −0.350308
\(455\) 1.73205 0.0811998
\(456\) −3.00000 −0.140488
\(457\) −19.9282 −0.932202 −0.466101 0.884732i \(-0.654342\pi\)
−0.466101 + 0.884732i \(0.654342\pi\)
\(458\) 0 0
\(459\) 6.58846 0.307523
\(460\) −2.46410 −0.114889
\(461\) 38.2487 1.78142 0.890710 0.454572i \(-0.150208\pi\)
0.890710 + 0.454572i \(0.150208\pi\)
\(462\) 0 0
\(463\) 35.0000 1.62659 0.813294 0.581853i \(-0.197672\pi\)
0.813294 + 0.581853i \(0.197672\pi\)
\(464\) 2.53590 0.117726
\(465\) −5.66025 −0.262488
\(466\) −22.8564 −1.05880
\(467\) −5.87564 −0.271892 −0.135946 0.990716i \(-0.543407\pi\)
−0.135946 + 0.990716i \(0.543407\pi\)
\(468\) 0 0
\(469\) −14.1962 −0.655517
\(470\) −2.73205 −0.126020
\(471\) 18.1244 0.835126
\(472\) 1.00000 0.0460287
\(473\) 0 0
\(474\) −0.124356 −0.00571184
\(475\) −1.73205 −0.0794719
\(476\) −1.26795 −0.0581164
\(477\) 0 0
\(478\) −2.46410 −0.112705
\(479\) −15.2679 −0.697610 −0.348805 0.937195i \(-0.613413\pi\)
−0.348805 + 0.937195i \(0.613413\pi\)
\(480\) −1.73205 −0.0790569
\(481\) −12.9282 −0.589475
\(482\) −10.9282 −0.497766
\(483\) 4.26795 0.194198
\(484\) 0 0
\(485\) −1.46410 −0.0664814
\(486\) 0 0
\(487\) 22.7128 1.02922 0.514608 0.857426i \(-0.327938\pi\)
0.514608 + 0.857426i \(0.327938\pi\)
\(488\) −4.00000 −0.181071
\(489\) −15.4641 −0.699311
\(490\) 1.00000 0.0451754
\(491\) −17.5167 −0.790516 −0.395258 0.918570i \(-0.629345\pi\)
−0.395258 + 0.918570i \(0.629345\pi\)
\(492\) 11.6603 0.525685
\(493\) 3.21539 0.144814
\(494\) 3.00000 0.134976
\(495\) 0 0
\(496\) −3.26795 −0.146735
\(497\) −2.92820 −0.131348
\(498\) 20.6603 0.925808
\(499\) 1.60770 0.0719703 0.0359852 0.999352i \(-0.488543\pi\)
0.0359852 + 0.999352i \(0.488543\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −11.6603 −0.520942
\(502\) 10.3923 0.463831
\(503\) −28.9808 −1.29219 −0.646094 0.763258i \(-0.723599\pi\)
−0.646094 + 0.763258i \(0.723599\pi\)
\(504\) 0 0
\(505\) −7.92820 −0.352800
\(506\) 0 0
\(507\) 17.3205 0.769231
\(508\) 13.1962 0.585485
\(509\) −35.1962 −1.56004 −0.780021 0.625753i \(-0.784792\pi\)
−0.780021 + 0.625753i \(0.784792\pi\)
\(510\) −2.19615 −0.0972473
\(511\) −9.66025 −0.427344
\(512\) −1.00000 −0.0441942
\(513\) −9.00000 −0.397360
\(514\) 19.2679 0.849873
\(515\) −1.80385 −0.0794870
\(516\) 13.2679 0.584089
\(517\) 0 0
\(518\) −7.46410 −0.327954
\(519\) 17.3205 0.760286
\(520\) 1.73205 0.0759555
\(521\) −14.5359 −0.636829 −0.318415 0.947952i \(-0.603150\pi\)
−0.318415 + 0.947952i \(0.603150\pi\)
\(522\) 0 0
\(523\) 31.2487 1.36641 0.683205 0.730226i \(-0.260585\pi\)
0.683205 + 0.730226i \(0.260585\pi\)
\(524\) 4.26795 0.186446
\(525\) 1.73205 0.0755929
\(526\) −17.7321 −0.773154
\(527\) −4.14359 −0.180498
\(528\) 0 0
\(529\) −16.9282 −0.736009
\(530\) −1.26795 −0.0550762
\(531\) 0 0
\(532\) 1.73205 0.0750939
\(533\) −11.6603 −0.505062
\(534\) −8.19615 −0.354682
\(535\) −0.535898 −0.0231689
\(536\) −14.1962 −0.613180
\(537\) 15.8038 0.681986
\(538\) −16.5167 −0.712084
\(539\) 0 0
\(540\) −5.19615 −0.223607
\(541\) −24.2487 −1.04253 −0.521267 0.853394i \(-0.674540\pi\)
−0.521267 + 0.853394i \(0.674540\pi\)
\(542\) −32.3923 −1.39137
\(543\) 7.14359 0.306561
\(544\) −1.26795 −0.0543629
\(545\) −0.196152 −0.00840225
\(546\) −3.00000 −0.128388
\(547\) 26.5885 1.13684 0.568420 0.822738i \(-0.307555\pi\)
0.568420 + 0.822738i \(0.307555\pi\)
\(548\) 12.1244 0.517927
\(549\) 0 0
\(550\) 0 0
\(551\) −4.39230 −0.187118
\(552\) 4.26795 0.181656
\(553\) 0.0717968 0.00305311
\(554\) 4.58846 0.194945
\(555\) −12.9282 −0.548772
\(556\) 1.33975 0.0568179
\(557\) −40.0526 −1.69708 −0.848541 0.529130i \(-0.822518\pi\)
−0.848541 + 0.529130i \(0.822518\pi\)
\(558\) 0 0
\(559\) −13.2679 −0.561174
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 7.73205 0.326157
\(563\) 42.4641 1.78965 0.894824 0.446419i \(-0.147301\pi\)
0.894824 + 0.446419i \(0.147301\pi\)
\(564\) 4.73205 0.199255
\(565\) 4.66025 0.196058
\(566\) −17.7846 −0.747543
\(567\) 9.00000 0.377964
\(568\) −2.92820 −0.122865
\(569\) 14.1244 0.592124 0.296062 0.955169i \(-0.404327\pi\)
0.296062 + 0.955169i \(0.404327\pi\)
\(570\) 3.00000 0.125656
\(571\) 0.679492 0.0284359 0.0142179 0.999899i \(-0.495474\pi\)
0.0142179 + 0.999899i \(0.495474\pi\)
\(572\) 0 0
\(573\) 31.6410 1.32182
\(574\) −6.73205 −0.280990
\(575\) 2.46410 0.102760
\(576\) 0 0
\(577\) 40.1962 1.67339 0.836694 0.547671i \(-0.184485\pi\)
0.836694 + 0.547671i \(0.184485\pi\)
\(578\) 15.3923 0.640235
\(579\) −37.7321 −1.56809
\(580\) −2.53590 −0.105297
\(581\) −11.9282 −0.494865
\(582\) 2.53590 0.105116
\(583\) 0 0
\(584\) −9.66025 −0.399744
\(585\) 0 0
\(586\) −5.33975 −0.220583
\(587\) −36.1244 −1.49101 −0.745506 0.666499i \(-0.767792\pi\)
−0.745506 + 0.666499i \(0.767792\pi\)
\(588\) −1.73205 −0.0714286
\(589\) 5.66025 0.233227
\(590\) −1.00000 −0.0411693
\(591\) 41.5692 1.70993
\(592\) −7.46410 −0.306773
\(593\) −2.92820 −0.120247 −0.0601234 0.998191i \(-0.519149\pi\)
−0.0601234 + 0.998191i \(0.519149\pi\)
\(594\) 0 0
\(595\) 1.26795 0.0519808
\(596\) 0.928203 0.0380207
\(597\) −14.5359 −0.594915
\(598\) −4.26795 −0.174529
\(599\) 2.66025 0.108695 0.0543475 0.998522i \(-0.482692\pi\)
0.0543475 + 0.998522i \(0.482692\pi\)
\(600\) 1.73205 0.0707107
\(601\) 5.41154 0.220741 0.110371 0.993890i \(-0.464796\pi\)
0.110371 + 0.993890i \(0.464796\pi\)
\(602\) −7.66025 −0.312209
\(603\) 0 0
\(604\) −13.0000 −0.528962
\(605\) 0 0
\(606\) 13.7321 0.557826
\(607\) 36.5359 1.48295 0.741473 0.670983i \(-0.234127\pi\)
0.741473 + 0.670983i \(0.234127\pi\)
\(608\) 1.73205 0.0702439
\(609\) 4.39230 0.177985
\(610\) 4.00000 0.161955
\(611\) −4.73205 −0.191438
\(612\) 0 0
\(613\) −16.9808 −0.685847 −0.342923 0.939363i \(-0.611417\pi\)
−0.342923 + 0.939363i \(0.611417\pi\)
\(614\) 3.07180 0.123968
\(615\) −11.6603 −0.470187
\(616\) 0 0
\(617\) 2.39230 0.0963106 0.0481553 0.998840i \(-0.484666\pi\)
0.0481553 + 0.998840i \(0.484666\pi\)
\(618\) 3.12436 0.125680
\(619\) −0.607695 −0.0244253 −0.0122127 0.999925i \(-0.503888\pi\)
−0.0122127 + 0.999925i \(0.503888\pi\)
\(620\) 3.26795 0.131244
\(621\) 12.8038 0.513801
\(622\) 32.1962 1.29095
\(623\) 4.73205 0.189586
\(624\) −3.00000 −0.120096
\(625\) 1.00000 0.0400000
\(626\) 16.5359 0.660907
\(627\) 0 0
\(628\) −10.4641 −0.417563
\(629\) −9.46410 −0.377358
\(630\) 0 0
\(631\) 6.53590 0.260190 0.130095 0.991502i \(-0.458472\pi\)
0.130095 + 0.991502i \(0.458472\pi\)
\(632\) 0.0717968 0.00285592
\(633\) −2.53590 −0.100793
\(634\) −21.6603 −0.860239
\(635\) −13.1962 −0.523673
\(636\) 2.19615 0.0870831
\(637\) 1.73205 0.0686264
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −20.3205 −0.802612 −0.401306 0.915944i \(-0.631444\pi\)
−0.401306 + 0.915944i \(0.631444\pi\)
\(642\) 0.928203 0.0366333
\(643\) −40.2487 −1.58725 −0.793627 0.608404i \(-0.791810\pi\)
−0.793627 + 0.608404i \(0.791810\pi\)
\(644\) −2.46410 −0.0970992
\(645\) −13.2679 −0.522425
\(646\) 2.19615 0.0864065
\(647\) −28.4449 −1.11828 −0.559141 0.829072i \(-0.688869\pi\)
−0.559141 + 0.829072i \(0.688869\pi\)
\(648\) 9.00000 0.353553
\(649\) 0 0
\(650\) −1.73205 −0.0679366
\(651\) −5.66025 −0.221843
\(652\) 8.92820 0.349655
\(653\) −4.05256 −0.158589 −0.0792944 0.996851i \(-0.525267\pi\)
−0.0792944 + 0.996851i \(0.525267\pi\)
\(654\) 0.339746 0.0132851
\(655\) −4.26795 −0.166763
\(656\) −6.73205 −0.262842
\(657\) 0 0
\(658\) −2.73205 −0.106506
\(659\) −14.8756 −0.579473 −0.289736 0.957106i \(-0.593568\pi\)
−0.289736 + 0.957106i \(0.593568\pi\)
\(660\) 0 0
\(661\) −5.58846 −0.217366 −0.108683 0.994076i \(-0.534663\pi\)
−0.108683 + 0.994076i \(0.534663\pi\)
\(662\) 1.80385 0.0701085
\(663\) −3.80385 −0.147729
\(664\) −11.9282 −0.462904
\(665\) −1.73205 −0.0671660
\(666\) 0 0
\(667\) 6.24871 0.241951
\(668\) 6.73205 0.260471
\(669\) −19.2679 −0.744942
\(670\) 14.1962 0.548445
\(671\) 0 0
\(672\) −1.73205 −0.0668153
\(673\) 33.7846 1.30230 0.651150 0.758949i \(-0.274287\pi\)
0.651150 + 0.758949i \(0.274287\pi\)
\(674\) 15.7846 0.608000
\(675\) 5.19615 0.200000
\(676\) −10.0000 −0.384615
\(677\) −25.7321 −0.988963 −0.494482 0.869188i \(-0.664642\pi\)
−0.494482 + 0.869188i \(0.664642\pi\)
\(678\) −8.07180 −0.309995
\(679\) −1.46410 −0.0561871
\(680\) 1.26795 0.0486236
\(681\) −12.9282 −0.495410
\(682\) 0 0
\(683\) 4.39230 0.168067 0.0840334 0.996463i \(-0.473220\pi\)
0.0840334 + 0.996463i \(0.473220\pi\)
\(684\) 0 0
\(685\) −12.1244 −0.463248
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −7.66025 −0.292044
\(689\) −2.19615 −0.0836667
\(690\) −4.26795 −0.162478
\(691\) −18.6410 −0.709138 −0.354569 0.935030i \(-0.615372\pi\)
−0.354569 + 0.935030i \(0.615372\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) 8.53590 0.324018
\(695\) −1.33975 −0.0508195
\(696\) 4.39230 0.166490
\(697\) −8.53590 −0.323320
\(698\) 3.00000 0.113552
\(699\) −39.5885 −1.49737
\(700\) −1.00000 −0.0377964
\(701\) −7.41154 −0.279930 −0.139965 0.990156i \(-0.544699\pi\)
−0.139965 + 0.990156i \(0.544699\pi\)
\(702\) −9.00000 −0.339683
\(703\) 12.9282 0.487596
\(704\) 0 0
\(705\) −4.73205 −0.178219
\(706\) −7.66025 −0.288297
\(707\) −7.92820 −0.298171
\(708\) 1.73205 0.0650945
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 2.92820 0.109894
\(711\) 0 0
\(712\) 4.73205 0.177341
\(713\) −8.05256 −0.301571
\(714\) −2.19615 −0.0821889
\(715\) 0 0
\(716\) −9.12436 −0.340993
\(717\) −4.26795 −0.159389
\(718\) 14.0000 0.522475
\(719\) 32.0000 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 0 0
\(721\) −1.80385 −0.0671788
\(722\) 16.0000 0.595458
\(723\) −18.9282 −0.703947
\(724\) −4.12436 −0.153280
\(725\) 2.53590 0.0941809
\(726\) 0 0
\(727\) 51.7128 1.91792 0.958961 0.283538i \(-0.0915082\pi\)
0.958961 + 0.283538i \(0.0915082\pi\)
\(728\) 1.73205 0.0641941
\(729\) 27.0000 1.00000
\(730\) 9.66025 0.357542
\(731\) −9.71281 −0.359241
\(732\) −6.92820 −0.256074
\(733\) −26.5167 −0.979415 −0.489708 0.871887i \(-0.662896\pi\)
−0.489708 + 0.871887i \(0.662896\pi\)
\(734\) 4.33975 0.160183
\(735\) 1.73205 0.0638877
\(736\) −2.46410 −0.0908280
\(737\) 0 0
\(738\) 0 0
\(739\) 8.33975 0.306783 0.153391 0.988166i \(-0.450981\pi\)
0.153391 + 0.988166i \(0.450981\pi\)
\(740\) 7.46410 0.274386
\(741\) 5.19615 0.190885
\(742\) −1.26795 −0.0465479
\(743\) 23.1769 0.850279 0.425139 0.905128i \(-0.360225\pi\)
0.425139 + 0.905128i \(0.360225\pi\)
\(744\) −5.66025 −0.207515
\(745\) −0.928203 −0.0340067
\(746\) 12.9282 0.473335
\(747\) 0 0
\(748\) 0 0
\(749\) −0.535898 −0.0195813
\(750\) −1.73205 −0.0632456
\(751\) −30.6603 −1.11881 −0.559404 0.828895i \(-0.688970\pi\)
−0.559404 + 0.828895i \(0.688970\pi\)
\(752\) −2.73205 −0.0996276
\(753\) 18.0000 0.655956
\(754\) −4.39230 −0.159958
\(755\) 13.0000 0.473118
\(756\) −5.19615 −0.188982
\(757\) −26.9808 −0.980632 −0.490316 0.871545i \(-0.663119\pi\)
−0.490316 + 0.871545i \(0.663119\pi\)
\(758\) 6.39230 0.232179
\(759\) 0 0
\(760\) −1.73205 −0.0628281
\(761\) −21.5167 −0.779978 −0.389989 0.920819i \(-0.627521\pi\)
−0.389989 + 0.920819i \(0.627521\pi\)
\(762\) 22.8564 0.828000
\(763\) −0.196152 −0.00710119
\(764\) −18.2679 −0.660911
\(765\) 0 0
\(766\) 26.5885 0.960680
\(767\) −1.73205 −0.0625407
\(768\) −1.73205 −0.0625000
\(769\) −30.3923 −1.09597 −0.547987 0.836487i \(-0.684606\pi\)
−0.547987 + 0.836487i \(0.684606\pi\)
\(770\) 0 0
\(771\) 33.3731 1.20190
\(772\) 21.7846 0.784045
\(773\) −6.60770 −0.237662 −0.118831 0.992914i \(-0.537915\pi\)
−0.118831 + 0.992914i \(0.537915\pi\)
\(774\) 0 0
\(775\) −3.26795 −0.117388
\(776\) −1.46410 −0.0525582
\(777\) −12.9282 −0.463797
\(778\) 10.8756 0.389911
\(779\) 11.6603 0.417772
\(780\) 3.00000 0.107417
\(781\) 0 0
\(782\) −3.12436 −0.111727
\(783\) 13.1769 0.470905
\(784\) 1.00000 0.0357143
\(785\) 10.4641 0.373480
\(786\) 7.39230 0.263675
\(787\) −25.7128 −0.916563 −0.458281 0.888807i \(-0.651535\pi\)
−0.458281 + 0.888807i \(0.651535\pi\)
\(788\) −24.0000 −0.854965
\(789\) −30.7128 −1.09341
\(790\) −0.0717968 −0.00255441
\(791\) 4.66025 0.165700
\(792\) 0 0
\(793\) 6.92820 0.246028
\(794\) 9.85641 0.349791
\(795\) −2.19615 −0.0778895
\(796\) 8.39230 0.297457
\(797\) −11.6795 −0.413709 −0.206854 0.978372i \(-0.566323\pi\)
−0.206854 + 0.978372i \(0.566323\pi\)
\(798\) 3.00000 0.106199
\(799\) −3.46410 −0.122551
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 30.2487 1.06812
\(803\) 0 0
\(804\) −24.5885 −0.867168
\(805\) 2.46410 0.0868482
\(806\) 5.66025 0.199374
\(807\) −28.6077 −1.00704
\(808\) −7.92820 −0.278913
\(809\) 3.85641 0.135584 0.0677920 0.997699i \(-0.478405\pi\)
0.0677920 + 0.997699i \(0.478405\pi\)
\(810\) −9.00000 −0.316228
\(811\) 7.32051 0.257058 0.128529 0.991706i \(-0.458974\pi\)
0.128529 + 0.991706i \(0.458974\pi\)
\(812\) −2.53590 −0.0889926
\(813\) −56.1051 −1.96769
\(814\) 0 0
\(815\) −8.92820 −0.312741
\(816\) −2.19615 −0.0768807
\(817\) 13.2679 0.464187
\(818\) −25.1769 −0.880290
\(819\) 0 0
\(820\) 6.73205 0.235093
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 21.0000 0.732459
\(823\) −49.7128 −1.73288 −0.866440 0.499281i \(-0.833597\pi\)
−0.866440 + 0.499281i \(0.833597\pi\)
\(824\) −1.80385 −0.0628400
\(825\) 0 0
\(826\) −1.00000 −0.0347945
\(827\) −36.9808 −1.28595 −0.642974 0.765888i \(-0.722299\pi\)
−0.642974 + 0.765888i \(0.722299\pi\)
\(828\) 0 0
\(829\) 38.3731 1.33275 0.666376 0.745616i \(-0.267845\pi\)
0.666376 + 0.745616i \(0.267845\pi\)
\(830\) 11.9282 0.414034
\(831\) 7.94744 0.275694
\(832\) 1.73205 0.0600481
\(833\) 1.26795 0.0439318
\(834\) 2.32051 0.0803526
\(835\) −6.73205 −0.232972
\(836\) 0 0
\(837\) −16.9808 −0.586941
\(838\) 9.39230 0.324452
\(839\) 5.66025 0.195414 0.0977068 0.995215i \(-0.468849\pi\)
0.0977068 + 0.995215i \(0.468849\pi\)
\(840\) 1.73205 0.0597614
\(841\) −22.5692 −0.778249
\(842\) 10.2487 0.353194
\(843\) 13.3923 0.461255
\(844\) 1.46410 0.0503965
\(845\) 10.0000 0.344010
\(846\) 0 0
\(847\) 0 0
\(848\) −1.26795 −0.0435416
\(849\) −30.8038 −1.05719
\(850\) −1.26795 −0.0434903
\(851\) −18.3923 −0.630480
\(852\) −5.07180 −0.173757
\(853\) −45.7321 −1.56584 −0.782918 0.622125i \(-0.786269\pi\)
−0.782918 + 0.622125i \(0.786269\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) −0.535898 −0.0183166
\(857\) −28.3923 −0.969863 −0.484931 0.874552i \(-0.661155\pi\)
−0.484931 + 0.874552i \(0.661155\pi\)
\(858\) 0 0
\(859\) −10.0000 −0.341196 −0.170598 0.985341i \(-0.554570\pi\)
−0.170598 + 0.985341i \(0.554570\pi\)
\(860\) 7.66025 0.261212
\(861\) −11.6603 −0.397380
\(862\) 18.4641 0.628890
\(863\) −38.6410 −1.31536 −0.657678 0.753299i \(-0.728461\pi\)
−0.657678 + 0.753299i \(0.728461\pi\)
\(864\) −5.19615 −0.176777
\(865\) 10.0000 0.340010
\(866\) −12.1962 −0.414442
\(867\) 26.6603 0.905430
\(868\) 3.26795 0.110921
\(869\) 0 0
\(870\) −4.39230 −0.148913
\(871\) 24.5885 0.833148
\(872\) −0.196152 −0.00664256
\(873\) 0 0
\(874\) 4.26795 0.144366
\(875\) 1.00000 0.0338062
\(876\) −16.7321 −0.565324
\(877\) −19.1244 −0.645784 −0.322892 0.946436i \(-0.604655\pi\)
−0.322892 + 0.946436i \(0.604655\pi\)
\(878\) 13.8038 0.465857
\(879\) −9.24871 −0.311951
\(880\) 0 0
\(881\) −33.3205 −1.12260 −0.561298 0.827614i \(-0.689698\pi\)
−0.561298 + 0.827614i \(0.689698\pi\)
\(882\) 0 0
\(883\) 34.3923 1.15739 0.578697 0.815543i \(-0.303562\pi\)
0.578697 + 0.815543i \(0.303562\pi\)
\(884\) 2.19615 0.0738646
\(885\) −1.73205 −0.0582223
\(886\) 18.9282 0.635905
\(887\) 12.9808 0.435851 0.217926 0.975965i \(-0.430071\pi\)
0.217926 + 0.975965i \(0.430071\pi\)
\(888\) −12.9282 −0.433842
\(889\) −13.1962 −0.442585
\(890\) −4.73205 −0.158619
\(891\) 0 0
\(892\) 11.1244 0.372471
\(893\) 4.73205 0.158352
\(894\) 1.60770 0.0537694
\(895\) 9.12436 0.304994
\(896\) 1.00000 0.0334077
\(897\) −7.39230 −0.246822
\(898\) 27.5359 0.918885
\(899\) −8.28719 −0.276393
\(900\) 0 0
\(901\) −1.60770 −0.0535601
\(902\) 0 0
\(903\) −13.2679 −0.441530
\(904\) 4.66025 0.154998
\(905\) 4.12436 0.137098
\(906\) −22.5167 −0.748066
\(907\) −14.9282 −0.495683 −0.247841 0.968801i \(-0.579721\pi\)
−0.247841 + 0.968801i \(0.579721\pi\)
\(908\) 7.46410 0.247705
\(909\) 0 0
\(910\) −1.73205 −0.0574169
\(911\) −49.7321 −1.64770 −0.823848 0.566811i \(-0.808177\pi\)
−0.823848 + 0.566811i \(0.808177\pi\)
\(912\) 3.00000 0.0993399
\(913\) 0 0
\(914\) 19.9282 0.659166
\(915\) 6.92820 0.229039
\(916\) 0 0
\(917\) −4.26795 −0.140940
\(918\) −6.58846 −0.217451
\(919\) 5.71281 0.188448 0.0942242 0.995551i \(-0.469963\pi\)
0.0942242 + 0.995551i \(0.469963\pi\)
\(920\) 2.46410 0.0812390
\(921\) 5.32051 0.175317
\(922\) −38.2487 −1.25965
\(923\) 5.07180 0.166940
\(924\) 0 0
\(925\) −7.46410 −0.245418
\(926\) −35.0000 −1.15017
\(927\) 0 0
\(928\) −2.53590 −0.0832449
\(929\) 31.3205 1.02759 0.513796 0.857912i \(-0.328239\pi\)
0.513796 + 0.857912i \(0.328239\pi\)
\(930\) 5.66025 0.185607
\(931\) −1.73205 −0.0567657
\(932\) 22.8564 0.748686
\(933\) 55.7654 1.82568
\(934\) 5.87564 0.192257
\(935\) 0 0
\(936\) 0 0
\(937\) −50.6410 −1.65437 −0.827185 0.561930i \(-0.810059\pi\)
−0.827185 + 0.561930i \(0.810059\pi\)
\(938\) 14.1962 0.463521
\(939\) 28.6410 0.934664
\(940\) 2.73205 0.0891097
\(941\) 2.28719 0.0745602 0.0372801 0.999305i \(-0.488131\pi\)
0.0372801 + 0.999305i \(0.488131\pi\)
\(942\) −18.1244 −0.590523
\(943\) −16.5885 −0.540194
\(944\) −1.00000 −0.0325472
\(945\) 5.19615 0.169031
\(946\) 0 0
\(947\) 3.26795 0.106194 0.0530970 0.998589i \(-0.483091\pi\)
0.0530970 + 0.998589i \(0.483091\pi\)
\(948\) 0.124356 0.00403888
\(949\) 16.7321 0.543145
\(950\) 1.73205 0.0561951
\(951\) −37.5167 −1.21656
\(952\) 1.26795 0.0410945
\(953\) −6.07180 −0.196685 −0.0983424 0.995153i \(-0.531354\pi\)
−0.0983424 + 0.995153i \(0.531354\pi\)
\(954\) 0 0
\(955\) 18.2679 0.591137
\(956\) 2.46410 0.0796947
\(957\) 0 0
\(958\) 15.2679 0.493285
\(959\) −12.1244 −0.391516
\(960\) 1.73205 0.0559017
\(961\) −20.3205 −0.655500
\(962\) 12.9282 0.416822
\(963\) 0 0
\(964\) 10.9282 0.351974
\(965\) −21.7846 −0.701271
\(966\) −4.26795 −0.137319
\(967\) −52.4974 −1.68820 −0.844102 0.536183i \(-0.819866\pi\)
−0.844102 + 0.536183i \(0.819866\pi\)
\(968\) 0 0
\(969\) 3.80385 0.122197
\(970\) 1.46410 0.0470095
\(971\) −6.60770 −0.212051 −0.106026 0.994363i \(-0.533813\pi\)
−0.106026 + 0.994363i \(0.533813\pi\)
\(972\) 0 0
\(973\) −1.33975 −0.0429503
\(974\) −22.7128 −0.727765
\(975\) −3.00000 −0.0960769
\(976\) 4.00000 0.128037
\(977\) −40.5167 −1.29624 −0.648121 0.761537i \(-0.724445\pi\)
−0.648121 + 0.761537i \(0.724445\pi\)
\(978\) 15.4641 0.494487
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 17.5167 0.558979
\(983\) 23.0718 0.735876 0.367938 0.929850i \(-0.380064\pi\)
0.367938 + 0.929850i \(0.380064\pi\)
\(984\) −11.6603 −0.371715
\(985\) 24.0000 0.764704
\(986\) −3.21539 −0.102399
\(987\) −4.73205 −0.150623
\(988\) −3.00000 −0.0954427
\(989\) −18.8756 −0.600211
\(990\) 0 0
\(991\) 24.5167 0.778797 0.389399 0.921069i \(-0.372683\pi\)
0.389399 + 0.921069i \(0.372683\pi\)
\(992\) 3.26795 0.103757
\(993\) 3.12436 0.0991484
\(994\) 2.92820 0.0928770
\(995\) −8.39230 −0.266054
\(996\) −20.6603 −0.654645
\(997\) 3.19615 0.101223 0.0506116 0.998718i \(-0.483883\pi\)
0.0506116 + 0.998718i \(0.483883\pi\)
\(998\) −1.60770 −0.0508907
\(999\) −38.7846 −1.22709
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 8470.2.a.bl.1.1 2
11.10 odd 2 8470.2.a.ca.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.bl.1.1 2 1.1 even 1 trivial
8470.2.a.ca.1.1 yes 2 11.10 odd 2