Properties

Label 8470.2.a.cn.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.14637 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.14637 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.68585 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.14637 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.14637 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.68585 q^{9} +1.00000 q^{10} +1.14637 q^{12} +1.00000 q^{14} +1.14637 q^{15} +1.00000 q^{16} -2.68585 q^{17} -1.68585 q^{18} +5.53948 q^{19} +1.00000 q^{20} +1.14637 q^{21} -1.83221 q^{23} +1.14637 q^{24} +1.00000 q^{25} -5.37169 q^{27} +1.00000 q^{28} +3.83221 q^{29} +1.14637 q^{30} +9.66442 q^{31} +1.00000 q^{32} -2.68585 q^{34} +1.00000 q^{35} -1.68585 q^{36} -1.83221 q^{37} +5.53948 q^{38} +1.00000 q^{40} -1.43910 q^{41} +1.14637 q^{42} +4.97858 q^{43} -1.68585 q^{45} -1.83221 q^{46} -0.393115 q^{47} +1.14637 q^{48} +1.00000 q^{49} +1.00000 q^{50} -3.07896 q^{51} +13.4966 q^{53} -5.37169 q^{54} +1.00000 q^{56} +6.35027 q^{57} +3.83221 q^{58} +0.685846 q^{59} +1.14637 q^{60} +2.39312 q^{61} +9.66442 q^{62} -1.68585 q^{63} +1.00000 q^{64} +3.37169 q^{67} -2.68585 q^{68} -2.10038 q^{69} +1.00000 q^{70} -0.585462 q^{71} -1.68585 q^{72} -8.97858 q^{73} -1.83221 q^{74} +1.14637 q^{75} +5.53948 q^{76} -13.2039 q^{79} +1.00000 q^{80} -1.10038 q^{81} -1.43910 q^{82} +9.37169 q^{83} +1.14637 q^{84} -2.68585 q^{85} +4.97858 q^{86} +4.39312 q^{87} +6.58546 q^{89} -1.68585 q^{90} -1.83221 q^{92} +11.0790 q^{93} -0.393115 q^{94} +5.53948 q^{95} +1.14637 q^{96} +10.1678 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} + 2q^{6} + 3q^{7} + 3q^{8} + 7q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} + 2q^{6} + 3q^{7} + 3q^{8} + 7q^{9} + 3q^{10} + 2q^{12} + 3q^{14} + 2q^{15} + 3q^{16} + 4q^{17} + 7q^{18} + 6q^{19} + 3q^{20} + 2q^{21} + 8q^{23} + 2q^{24} + 3q^{25} + 8q^{27} + 3q^{28} - 2q^{29} + 2q^{30} + 2q^{31} + 3q^{32} + 4q^{34} + 3q^{35} + 7q^{36} + 8q^{37} + 6q^{38} + 3q^{40} + 2q^{42} + 7q^{45} + 8q^{46} + 8q^{47} + 2q^{48} + 3q^{49} + 3q^{50} + 12q^{51} + 8q^{54} + 3q^{56} - 20q^{57} - 2q^{58} - 10q^{59} + 2q^{60} - 2q^{61} + 2q^{62} + 7q^{63} + 3q^{64} - 14q^{67} + 4q^{68} + 3q^{70} + 4q^{71} + 7q^{72} - 12q^{73} + 8q^{74} + 2q^{75} + 6q^{76} - 2q^{79} + 3q^{80} + 3q^{81} + 4q^{83} + 2q^{84} + 4q^{85} + 4q^{87} + 14q^{89} + 7q^{90} + 8q^{92} + 12q^{93} + 8q^{94} + 6q^{95} + 2q^{96} + 44q^{97} + 3q^{98} + O(q^{100}) \)

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
\(3\) 1.14637 0.661854 0.330927 0.943656i \(-0.392639\pi\)
0.330927 + 0.943656i \(0.392639\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.14637 0.468002
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −1.68585 −0.561949
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.14637 0.330927
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.14637 0.295990
\(16\) 1.00000 0.250000
\(17\) −2.68585 −0.651413 −0.325707 0.945471i \(-0.605602\pi\)
−0.325707 + 0.945471i \(0.605602\pi\)
\(18\) −1.68585 −0.397358
\(19\) 5.53948 1.27084 0.635422 0.772165i \(-0.280826\pi\)
0.635422 + 0.772165i \(0.280826\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.14637 0.250157
\(22\) 0 0
\(23\) −1.83221 −0.382043 −0.191021 0.981586i \(-0.561180\pi\)
−0.191021 + 0.981586i \(0.561180\pi\)
\(24\) 1.14637 0.234001
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.37169 −1.03378
\(28\) 1.00000 0.188982
\(29\) 3.83221 0.711624 0.355812 0.934558i \(-0.384204\pi\)
0.355812 + 0.934558i \(0.384204\pi\)
\(30\) 1.14637 0.209297
\(31\) 9.66442 1.73578 0.867891 0.496755i \(-0.165475\pi\)
0.867891 + 0.496755i \(0.165475\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.68585 −0.460619
\(35\) 1.00000 0.169031
\(36\) −1.68585 −0.280974
\(37\) −1.83221 −0.301214 −0.150607 0.988594i \(-0.548123\pi\)
−0.150607 + 0.988594i \(0.548123\pi\)
\(38\) 5.53948 0.898622
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −1.43910 −0.224749 −0.112375 0.993666i \(-0.535846\pi\)
−0.112375 + 0.993666i \(0.535846\pi\)
\(42\) 1.14637 0.176888
\(43\) 4.97858 0.759226 0.379613 0.925145i \(-0.376057\pi\)
0.379613 + 0.925145i \(0.376057\pi\)
\(44\) 0 0
\(45\) −1.68585 −0.251311
\(46\) −1.83221 −0.270145
\(47\) −0.393115 −0.0573417 −0.0286709 0.999589i \(-0.509127\pi\)
−0.0286709 + 0.999589i \(0.509127\pi\)
\(48\) 1.14637 0.165464
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −3.07896 −0.431141
\(52\) 0 0
\(53\) 13.4966 1.85391 0.926953 0.375178i \(-0.122418\pi\)
0.926953 + 0.375178i \(0.122418\pi\)
\(54\) −5.37169 −0.730995
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 6.35027 0.841114
\(58\) 3.83221 0.503194
\(59\) 0.685846 0.0892896 0.0446448 0.999003i \(-0.485784\pi\)
0.0446448 + 0.999003i \(0.485784\pi\)
\(60\) 1.14637 0.147995
\(61\) 2.39312 0.306407 0.153204 0.988195i \(-0.451041\pi\)
0.153204 + 0.988195i \(0.451041\pi\)
\(62\) 9.66442 1.22738
\(63\) −1.68585 −0.212397
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.37169 0.411918 0.205959 0.978561i \(-0.433969\pi\)
0.205959 + 0.978561i \(0.433969\pi\)
\(68\) −2.68585 −0.325707
\(69\) −2.10038 −0.252857
\(70\) 1.00000 0.119523
\(71\) −0.585462 −0.0694816 −0.0347408 0.999396i \(-0.511061\pi\)
−0.0347408 + 0.999396i \(0.511061\pi\)
\(72\) −1.68585 −0.198679
\(73\) −8.97858 −1.05086 −0.525431 0.850836i \(-0.676096\pi\)
−0.525431 + 0.850836i \(0.676096\pi\)
\(74\) −1.83221 −0.212990
\(75\) 1.14637 0.132371
\(76\) 5.53948 0.635422
\(77\) 0 0
\(78\) 0 0
\(79\) −13.2039 −1.48556 −0.742778 0.669538i \(-0.766492\pi\)
−0.742778 + 0.669538i \(0.766492\pi\)
\(80\) 1.00000 0.111803
\(81\) −1.10038 −0.122265
\(82\) −1.43910 −0.158922
\(83\) 9.37169 1.02868 0.514338 0.857587i \(-0.328038\pi\)
0.514338 + 0.857587i \(0.328038\pi\)
\(84\) 1.14637 0.125079
\(85\) −2.68585 −0.291321
\(86\) 4.97858 0.536854
\(87\) 4.39312 0.470991
\(88\) 0 0
\(89\) 6.58546 0.698058 0.349029 0.937112i \(-0.386512\pi\)
0.349029 + 0.937112i \(0.386512\pi\)
\(90\) −1.68585 −0.177704
\(91\) 0 0
\(92\) −1.83221 −0.191021
\(93\) 11.0790 1.14883
\(94\) −0.393115 −0.0405467
\(95\) 5.53948 0.568339
\(96\) 1.14637 0.117000
\(97\) 10.1678 1.03238 0.516191 0.856473i \(-0.327349\pi\)
0.516191 + 0.856473i \(0.327349\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −1.02142 −0.101635 −0.0508177 0.998708i \(-0.516183\pi\)
−0.0508177 + 0.998708i \(0.516183\pi\)
\(102\) −3.07896 −0.304863
\(103\) 2.68585 0.264644 0.132322 0.991207i \(-0.457757\pi\)
0.132322 + 0.991207i \(0.457757\pi\)
\(104\) 0 0
\(105\) 1.14637 0.111874
\(106\) 13.4966 1.31091
\(107\) 0.393115 0.0380039 0.0190019 0.999819i \(-0.493951\pi\)
0.0190019 + 0.999819i \(0.493951\pi\)
\(108\) −5.37169 −0.516891
\(109\) 4.41767 0.423136 0.211568 0.977363i \(-0.432143\pi\)
0.211568 + 0.977363i \(0.432143\pi\)
\(110\) 0 0
\(111\) −2.10038 −0.199360
\(112\) 1.00000 0.0944911
\(113\) 11.3717 1.06976 0.534879 0.844928i \(-0.320357\pi\)
0.534879 + 0.844928i \(0.320357\pi\)
\(114\) 6.35027 0.594757
\(115\) −1.83221 −0.170855
\(116\) 3.83221 0.355812
\(117\) 0 0
\(118\) 0.685846 0.0631373
\(119\) −2.68585 −0.246211
\(120\) 1.14637 0.104648
\(121\) 0 0
\(122\) 2.39312 0.216662
\(123\) −1.64973 −0.148751
\(124\) 9.66442 0.867891
\(125\) 1.00000 0.0894427
\(126\) −1.68585 −0.150187
\(127\) 16.4507 1.45976 0.729880 0.683576i \(-0.239576\pi\)
0.729880 + 0.683576i \(0.239576\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.70727 0.502497
\(130\) 0 0
\(131\) −5.20390 −0.454667 −0.227334 0.973817i \(-0.573001\pi\)
−0.227334 + 0.973817i \(0.573001\pi\)
\(132\) 0 0
\(133\) 5.53948 0.480334
\(134\) 3.37169 0.291270
\(135\) −5.37169 −0.462322
\(136\) −2.68585 −0.230309
\(137\) 1.41454 0.120852 0.0604261 0.998173i \(-0.480754\pi\)
0.0604261 + 0.998173i \(0.480754\pi\)
\(138\) −2.10038 −0.178797
\(139\) −13.2039 −1.11994 −0.559970 0.828513i \(-0.689187\pi\)
−0.559970 + 0.828513i \(0.689187\pi\)
\(140\) 1.00000 0.0845154
\(141\) −0.450654 −0.0379519
\(142\) −0.585462 −0.0491309
\(143\) 0 0
\(144\) −1.68585 −0.140487
\(145\) 3.83221 0.318248
\(146\) −8.97858 −0.743072
\(147\) 1.14637 0.0945506
\(148\) −1.83221 −0.150607
\(149\) −0.417674 −0.0342172 −0.0171086 0.999854i \(-0.505446\pi\)
−0.0171086 + 0.999854i \(0.505446\pi\)
\(150\) 1.14637 0.0936004
\(151\) −6.91117 −0.562423 −0.281212 0.959646i \(-0.590736\pi\)
−0.281212 + 0.959646i \(0.590736\pi\)
\(152\) 5.53948 0.449311
\(153\) 4.52792 0.366061
\(154\) 0 0
\(155\) 9.66442 0.776265
\(156\) 0 0
\(157\) 14.2499 1.13726 0.568632 0.822592i \(-0.307473\pi\)
0.568632 + 0.822592i \(0.307473\pi\)
\(158\) −13.2039 −1.05045
\(159\) 15.4721 1.22702
\(160\) 1.00000 0.0790569
\(161\) −1.83221 −0.144399
\(162\) −1.10038 −0.0864543
\(163\) 0.628308 0.0492129 0.0246064 0.999697i \(-0.492167\pi\)
0.0246064 + 0.999697i \(0.492167\pi\)
\(164\) −1.43910 −0.112375
\(165\) 0 0
\(166\) 9.37169 0.727384
\(167\) −18.5426 −1.43487 −0.717435 0.696625i \(-0.754684\pi\)
−0.717435 + 0.696625i \(0.754684\pi\)
\(168\) 1.14637 0.0884440
\(169\) −13.0000 −1.00000
\(170\) −2.68585 −0.205995
\(171\) −9.33871 −0.714149
\(172\) 4.97858 0.379613
\(173\) 10.3503 0.786916 0.393458 0.919343i \(-0.371279\pi\)
0.393458 + 0.919343i \(0.371279\pi\)
\(174\) 4.39312 0.333041
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 0.786230 0.0590967
\(178\) 6.58546 0.493601
\(179\) 12.6430 0.944982 0.472491 0.881335i \(-0.343355\pi\)
0.472491 + 0.881335i \(0.343355\pi\)
\(180\) −1.68585 −0.125656
\(181\) −2.58546 −0.192176 −0.0960879 0.995373i \(-0.530633\pi\)
−0.0960879 + 0.995373i \(0.530633\pi\)
\(182\) 0 0
\(183\) 2.74338 0.202797
\(184\) −1.83221 −0.135072
\(185\) −1.83221 −0.134707
\(186\) 11.0790 0.812349
\(187\) 0 0
\(188\) −0.393115 −0.0286709
\(189\) −5.37169 −0.390733
\(190\) 5.53948 0.401876
\(191\) 2.87819 0.208259 0.104129 0.994564i \(-0.466794\pi\)
0.104129 + 0.994564i \(0.466794\pi\)
\(192\) 1.14637 0.0827318
\(193\) −11.1793 −0.804707 −0.402353 0.915484i \(-0.631808\pi\)
−0.402353 + 0.915484i \(0.631808\pi\)
\(194\) 10.1678 0.730005
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 15.6216 1.11299 0.556496 0.830851i \(-0.312146\pi\)
0.556496 + 0.830851i \(0.312146\pi\)
\(198\) 0 0
\(199\) −21.7220 −1.53983 −0.769915 0.638147i \(-0.779701\pi\)
−0.769915 + 0.638147i \(0.779701\pi\)
\(200\) 1.00000 0.0707107
\(201\) 3.86519 0.272630
\(202\) −1.02142 −0.0718671
\(203\) 3.83221 0.268969
\(204\) −3.07896 −0.215570
\(205\) −1.43910 −0.100511
\(206\) 2.68585 0.187132
\(207\) 3.08883 0.214688
\(208\) 0 0
\(209\) 0 0
\(210\) 1.14637 0.0791067
\(211\) −18.5426 −1.27653 −0.638263 0.769818i \(-0.720347\pi\)
−0.638263 + 0.769818i \(0.720347\pi\)
\(212\) 13.4966 0.926953
\(213\) −0.671153 −0.0459867
\(214\) 0.393115 0.0268728
\(215\) 4.97858 0.339536
\(216\) −5.37169 −0.365497
\(217\) 9.66442 0.656064
\(218\) 4.41767 0.299203
\(219\) −10.2927 −0.695518
\(220\) 0 0
\(221\) 0 0
\(222\) −2.10038 −0.140969
\(223\) 1.31415 0.0880022 0.0440011 0.999031i \(-0.485989\pi\)
0.0440011 + 0.999031i \(0.485989\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.68585 −0.112390
\(226\) 11.3717 0.756434
\(227\) −6.87819 −0.456522 −0.228261 0.973600i \(-0.573304\pi\)
−0.228261 + 0.973600i \(0.573304\pi\)
\(228\) 6.35027 0.420557
\(229\) 14.0575 0.928948 0.464474 0.885587i \(-0.346243\pi\)
0.464474 + 0.885587i \(0.346243\pi\)
\(230\) −1.83221 −0.120812
\(231\) 0 0
\(232\) 3.83221 0.251597
\(233\) −4.04285 −0.264856 −0.132428 0.991193i \(-0.542277\pi\)
−0.132428 + 0.991193i \(0.542277\pi\)
\(234\) 0 0
\(235\) −0.393115 −0.0256440
\(236\) 0.685846 0.0446448
\(237\) −15.1365 −0.983221
\(238\) −2.68585 −0.174098
\(239\) 24.4177 1.57945 0.789724 0.613462i \(-0.210224\pi\)
0.789724 + 0.613462i \(0.210224\pi\)
\(240\) 1.14637 0.0739976
\(241\) −3.73183 −0.240388 −0.120194 0.992750i \(-0.538352\pi\)
−0.120194 + 0.992750i \(0.538352\pi\)
\(242\) 0 0
\(243\) 14.8536 0.952861
\(244\) 2.39312 0.153204
\(245\) 1.00000 0.0638877
\(246\) −1.64973 −0.105183
\(247\) 0 0
\(248\) 9.66442 0.613691
\(249\) 10.7434 0.680834
\(250\) 1.00000 0.0632456
\(251\) 23.5640 1.48735 0.743674 0.668542i \(-0.233081\pi\)
0.743674 + 0.668542i \(0.233081\pi\)
\(252\) −1.68585 −0.106198
\(253\) 0 0
\(254\) 16.4507 1.03221
\(255\) −3.07896 −0.192812
\(256\) 1.00000 0.0625000
\(257\) −14.0821 −0.878417 −0.439209 0.898385i \(-0.644741\pi\)
−0.439209 + 0.898385i \(0.644741\pi\)
\(258\) 5.70727 0.355319
\(259\) −1.83221 −0.113848
\(260\) 0 0
\(261\) −6.46052 −0.399896
\(262\) −5.20390 −0.321498
\(263\) 15.6644 0.965910 0.482955 0.875645i \(-0.339564\pi\)
0.482955 + 0.875645i \(0.339564\pi\)
\(264\) 0 0
\(265\) 13.4966 0.829092
\(266\) 5.53948 0.339647
\(267\) 7.54935 0.462012
\(268\) 3.37169 0.205959
\(269\) 22.0575 1.34487 0.672436 0.740155i \(-0.265248\pi\)
0.672436 + 0.740155i \(0.265248\pi\)
\(270\) −5.37169 −0.326911
\(271\) −13.0361 −0.791888 −0.395944 0.918275i \(-0.629583\pi\)
−0.395944 + 0.918275i \(0.629583\pi\)
\(272\) −2.68585 −0.162853
\(273\) 0 0
\(274\) 1.41454 0.0854554
\(275\) 0 0
\(276\) −2.10038 −0.126428
\(277\) −18.2499 −1.09653 −0.548265 0.836305i \(-0.684711\pi\)
−0.548265 + 0.836305i \(0.684711\pi\)
\(278\) −13.2039 −0.791918
\(279\) −16.2927 −0.975420
\(280\) 1.00000 0.0597614
\(281\) −18.4078 −1.09812 −0.549059 0.835784i \(-0.685014\pi\)
−0.549059 + 0.835784i \(0.685014\pi\)
\(282\) −0.450654 −0.0268360
\(283\) −9.62158 −0.571943 −0.285972 0.958238i \(-0.592316\pi\)
−0.285972 + 0.958238i \(0.592316\pi\)
\(284\) −0.585462 −0.0347408
\(285\) 6.35027 0.376157
\(286\) 0 0
\(287\) −1.43910 −0.0849472
\(288\) −1.68585 −0.0993394
\(289\) −9.78623 −0.575661
\(290\) 3.83221 0.225035
\(291\) 11.6560 0.683287
\(292\) −8.97858 −0.525431
\(293\) −31.9143 −1.86445 −0.932227 0.361874i \(-0.882137\pi\)
−0.932227 + 0.361874i \(0.882137\pi\)
\(294\) 1.14637 0.0668574
\(295\) 0.685846 0.0399315
\(296\) −1.83221 −0.106495
\(297\) 0 0
\(298\) −0.417674 −0.0241952
\(299\) 0 0
\(300\) 1.14637 0.0661854
\(301\) 4.97858 0.286960
\(302\) −6.91117 −0.397693
\(303\) −1.17092 −0.0672678
\(304\) 5.53948 0.317711
\(305\) 2.39312 0.137029
\(306\) 4.52792 0.258844
\(307\) −9.03612 −0.515718 −0.257859 0.966183i \(-0.583017\pi\)
−0.257859 + 0.966183i \(0.583017\pi\)
\(308\) 0 0
\(309\) 3.07896 0.175156
\(310\) 9.66442 0.548902
\(311\) −13.6644 −0.774838 −0.387419 0.921904i \(-0.626633\pi\)
−0.387419 + 0.921904i \(0.626633\pi\)
\(312\) 0 0
\(313\) −4.71040 −0.266248 −0.133124 0.991099i \(-0.542501\pi\)
−0.133124 + 0.991099i \(0.542501\pi\)
\(314\) 14.2499 0.804168
\(315\) −1.68585 −0.0949867
\(316\) −13.2039 −0.742778
\(317\) 20.9112 1.17449 0.587244 0.809410i \(-0.300213\pi\)
0.587244 + 0.809410i \(0.300213\pi\)
\(318\) 15.4721 0.867631
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 0.450654 0.0251530
\(322\) −1.83221 −0.102105
\(323\) −14.8782 −0.827845
\(324\) −1.10038 −0.0611325
\(325\) 0 0
\(326\) 0.628308 0.0347987
\(327\) 5.06427 0.280055
\(328\) −1.43910 −0.0794608
\(329\) −0.393115 −0.0216731
\(330\) 0 0
\(331\) −15.4721 −0.850422 −0.425211 0.905094i \(-0.639800\pi\)
−0.425211 + 0.905094i \(0.639800\pi\)
\(332\) 9.37169 0.514338
\(333\) 3.08883 0.169267
\(334\) −18.5426 −1.01461
\(335\) 3.37169 0.184215
\(336\) 1.14637 0.0625394
\(337\) −35.9572 −1.95871 −0.979356 0.202145i \(-0.935209\pi\)
−0.979356 + 0.202145i \(0.935209\pi\)
\(338\) −13.0000 −0.707107
\(339\) 13.0361 0.708025
\(340\) −2.68585 −0.145660
\(341\) 0 0
\(342\) −9.33871 −0.504980
\(343\) 1.00000 0.0539949
\(344\) 4.97858 0.268427
\(345\) −2.10038 −0.113081
\(346\) 10.3503 0.556434
\(347\) −2.23519 −0.119991 −0.0599957 0.998199i \(-0.519109\pi\)
−0.0599957 + 0.998199i \(0.519109\pi\)
\(348\) 4.39312 0.235496
\(349\) 12.0147 0.643132 0.321566 0.946887i \(-0.395791\pi\)
0.321566 + 0.946887i \(0.395791\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 0 0
\(353\) 31.2039 1.66082 0.830408 0.557156i \(-0.188107\pi\)
0.830408 + 0.557156i \(0.188107\pi\)
\(354\) 0.786230 0.0417877
\(355\) −0.585462 −0.0310731
\(356\) 6.58546 0.349029
\(357\) −3.07896 −0.162956
\(358\) 12.6430 0.668203
\(359\) 20.8683 1.10139 0.550694 0.834707i \(-0.314363\pi\)
0.550694 + 0.834707i \(0.314363\pi\)
\(360\) −1.68585 −0.0888519
\(361\) 11.6858 0.615045
\(362\) −2.58546 −0.135889
\(363\) 0 0
\(364\) 0 0
\(365\) −8.97858 −0.469960
\(366\) 2.74338 0.143399
\(367\) −33.2285 −1.73451 −0.867256 0.497863i \(-0.834118\pi\)
−0.867256 + 0.497863i \(0.834118\pi\)
\(368\) −1.83221 −0.0955106
\(369\) 2.42610 0.126297
\(370\) −1.83221 −0.0952521
\(371\) 13.4966 0.700710
\(372\) 11.0790 0.574417
\(373\) −9.66442 −0.500405 −0.250202 0.968194i \(-0.580497\pi\)
−0.250202 + 0.968194i \(0.580497\pi\)
\(374\) 0 0
\(375\) 1.14637 0.0591981
\(376\) −0.393115 −0.0202734
\(377\) 0 0
\(378\) −5.37169 −0.276290
\(379\) −3.21377 −0.165080 −0.0825401 0.996588i \(-0.526303\pi\)
−0.0825401 + 0.996588i \(0.526303\pi\)
\(380\) 5.53948 0.284169
\(381\) 18.8585 0.966148
\(382\) 2.87819 0.147261
\(383\) −24.6430 −1.25920 −0.629599 0.776920i \(-0.716781\pi\)
−0.629599 + 0.776920i \(0.716781\pi\)
\(384\) 1.14637 0.0585002
\(385\) 0 0
\(386\) −11.1793 −0.569014
\(387\) −8.39312 −0.426646
\(388\) 10.1678 0.516191
\(389\) −6.92104 −0.350911 −0.175455 0.984487i \(-0.556140\pi\)
−0.175455 + 0.984487i \(0.556140\pi\)
\(390\) 0 0
\(391\) 4.92104 0.248868
\(392\) 1.00000 0.0505076
\(393\) −5.96558 −0.300923
\(394\) 15.6216 0.787004
\(395\) −13.2039 −0.664361
\(396\) 0 0
\(397\) 6.92104 0.347357 0.173678 0.984802i \(-0.444435\pi\)
0.173678 + 0.984802i \(0.444435\pi\)
\(398\) −21.7220 −1.08882
\(399\) 6.35027 0.317911
\(400\) 1.00000 0.0500000
\(401\) 18.6430 0.930987 0.465494 0.885051i \(-0.345877\pi\)
0.465494 + 0.885051i \(0.345877\pi\)
\(402\) 3.86519 0.192778
\(403\) 0 0
\(404\) −1.02142 −0.0508177
\(405\) −1.10038 −0.0546785
\(406\) 3.83221 0.190189
\(407\) 0 0
\(408\) −3.07896 −0.152431
\(409\) 33.6890 1.66581 0.832906 0.553414i \(-0.186675\pi\)
0.832906 + 0.553414i \(0.186675\pi\)
\(410\) −1.43910 −0.0710719
\(411\) 1.62158 0.0799865
\(412\) 2.68585 0.132322
\(413\) 0.685846 0.0337483
\(414\) 3.08883 0.151808
\(415\) 9.37169 0.460038
\(416\) 0 0
\(417\) −15.1365 −0.741238
\(418\) 0 0
\(419\) 21.0214 1.02696 0.513482 0.858100i \(-0.328355\pi\)
0.513482 + 0.858100i \(0.328355\pi\)
\(420\) 1.14637 0.0559369
\(421\) −10.8353 −0.528083 −0.264041 0.964511i \(-0.585056\pi\)
−0.264041 + 0.964511i \(0.585056\pi\)
\(422\) −18.5426 −0.902640
\(423\) 0.662732 0.0322231
\(424\) 13.4966 0.655455
\(425\) −2.68585 −0.130283
\(426\) −0.671153 −0.0325175
\(427\) 2.39312 0.115811
\(428\) 0.393115 0.0190019
\(429\) 0 0
\(430\) 4.97858 0.240088
\(431\) −32.6676 −1.57354 −0.786770 0.617246i \(-0.788248\pi\)
−0.786770 + 0.617246i \(0.788248\pi\)
\(432\) −5.37169 −0.258446
\(433\) −16.3748 −0.786924 −0.393462 0.919341i \(-0.628723\pi\)
−0.393462 + 0.919341i \(0.628723\pi\)
\(434\) 9.66442 0.463907
\(435\) 4.39312 0.210634
\(436\) 4.41767 0.211568
\(437\) −10.1495 −0.485516
\(438\) −10.2927 −0.491806
\(439\) 0.450654 0.0215085 0.0107543 0.999942i \(-0.496577\pi\)
0.0107543 + 0.999942i \(0.496577\pi\)
\(440\) 0 0
\(441\) −1.68585 −0.0802784
\(442\) 0 0
\(443\) 10.7862 0.512469 0.256235 0.966615i \(-0.417518\pi\)
0.256235 + 0.966615i \(0.417518\pi\)
\(444\) −2.10038 −0.0996798
\(445\) 6.58546 0.312181
\(446\) 1.31415 0.0622270
\(447\) −0.478807 −0.0226468
\(448\) 1.00000 0.0472456
\(449\) −19.8139 −0.935077 −0.467538 0.883973i \(-0.654859\pi\)
−0.467538 + 0.883973i \(0.654859\pi\)
\(450\) −1.68585 −0.0794716
\(451\) 0 0
\(452\) 11.3717 0.534879
\(453\) −7.92273 −0.372242
\(454\) −6.87819 −0.322810
\(455\) 0 0
\(456\) 6.35027 0.297379
\(457\) 23.9572 1.12067 0.560334 0.828267i \(-0.310673\pi\)
0.560334 + 0.828267i \(0.310673\pi\)
\(458\) 14.0575 0.656866
\(459\) 14.4275 0.673420
\(460\) −1.83221 −0.0854273
\(461\) 26.6430 1.24089 0.620444 0.784251i \(-0.286952\pi\)
0.620444 + 0.784251i \(0.286952\pi\)
\(462\) 0 0
\(463\) 2.75325 0.127954 0.0639772 0.997951i \(-0.479622\pi\)
0.0639772 + 0.997951i \(0.479622\pi\)
\(464\) 3.83221 0.177906
\(465\) 11.0790 0.513775
\(466\) −4.04285 −0.187281
\(467\) −20.1396 −0.931951 −0.465976 0.884798i \(-0.654297\pi\)
−0.465976 + 0.884798i \(0.654297\pi\)
\(468\) 0 0
\(469\) 3.37169 0.155690
\(470\) −0.393115 −0.0181331
\(471\) 16.3356 0.752704
\(472\) 0.685846 0.0315686
\(473\) 0 0
\(474\) −15.1365 −0.695242
\(475\) 5.53948 0.254169
\(476\) −2.68585 −0.123106
\(477\) −22.7533 −1.04180
\(478\) 24.4177 1.11684
\(479\) −4.58546 −0.209515 −0.104758 0.994498i \(-0.533407\pi\)
−0.104758 + 0.994498i \(0.533407\pi\)
\(480\) 1.14637 0.0523242
\(481\) 0 0
\(482\) −3.73183 −0.169980
\(483\) −2.10038 −0.0955708
\(484\) 0 0
\(485\) 10.1678 0.461695
\(486\) 14.8536 0.673775
\(487\) 33.1611 1.50267 0.751335 0.659920i \(-0.229410\pi\)
0.751335 + 0.659920i \(0.229410\pi\)
\(488\) 2.39312 0.108331
\(489\) 0.720270 0.0325717
\(490\) 1.00000 0.0451754
\(491\) 23.9143 1.07924 0.539619 0.841909i \(-0.318568\pi\)
0.539619 + 0.841909i \(0.318568\pi\)
\(492\) −1.64973 −0.0743756
\(493\) −10.2927 −0.463561
\(494\) 0 0
\(495\) 0 0
\(496\) 9.66442 0.433945
\(497\) −0.585462 −0.0262616
\(498\) 10.7434 0.481423
\(499\) −39.3864 −1.76318 −0.881588 0.472019i \(-0.843525\pi\)
−0.881588 + 0.472019i \(0.843525\pi\)
\(500\) 1.00000 0.0447214
\(501\) −21.2566 −0.949676
\(502\) 23.5640 1.05171
\(503\) −23.5296 −1.04913 −0.524567 0.851369i \(-0.675773\pi\)
−0.524567 + 0.851369i \(0.675773\pi\)
\(504\) −1.68585 −0.0750936
\(505\) −1.02142 −0.0454527
\(506\) 0 0
\(507\) −14.9028 −0.661854
\(508\) 16.4507 0.729880
\(509\) −20.5426 −0.910535 −0.455268 0.890355i \(-0.650456\pi\)
−0.455268 + 0.890355i \(0.650456\pi\)
\(510\) −3.07896 −0.136339
\(511\) −8.97858 −0.397189
\(512\) 1.00000 0.0441942
\(513\) −29.7564 −1.31378
\(514\) −14.0821 −0.621135
\(515\) 2.68585 0.118353
\(516\) 5.70727 0.251249
\(517\) 0 0
\(518\) −1.83221 −0.0805028
\(519\) 11.8652 0.520824
\(520\) 0 0
\(521\) −6.53635 −0.286362 −0.143181 0.989696i \(-0.545733\pi\)
−0.143181 + 0.989696i \(0.545733\pi\)
\(522\) −6.46052 −0.282769
\(523\) −37.7367 −1.65011 −0.825054 0.565053i \(-0.808856\pi\)
−0.825054 + 0.565053i \(0.808856\pi\)
\(524\) −5.20390 −0.227334
\(525\) 1.14637 0.0500315
\(526\) 15.6644 0.683001
\(527\) −25.9572 −1.13071
\(528\) 0 0
\(529\) −19.6430 −0.854043
\(530\) 13.4966 0.586256
\(531\) −1.15623 −0.0501762
\(532\) 5.53948 0.240167
\(533\) 0 0
\(534\) 7.54935 0.326692
\(535\) 0.393115 0.0169959
\(536\) 3.37169 0.145635
\(537\) 14.4935 0.625441
\(538\) 22.0575 0.950968
\(539\) 0 0
\(540\) −5.37169 −0.231161
\(541\) −2.99686 −0.128845 −0.0644226 0.997923i \(-0.520521\pi\)
−0.0644226 + 0.997923i \(0.520521\pi\)
\(542\) −13.0361 −0.559949
\(543\) −2.96388 −0.127192
\(544\) −2.68585 −0.115155
\(545\) 4.41767 0.189232
\(546\) 0 0
\(547\) 0.0856914 0.00366390 0.00183195 0.999998i \(-0.499417\pi\)
0.00183195 + 0.999998i \(0.499417\pi\)
\(548\) 1.41454 0.0604261
\(549\) −4.03442 −0.172185
\(550\) 0 0
\(551\) 21.2285 0.904363
\(552\) −2.10038 −0.0893983
\(553\) −13.2039 −0.561487
\(554\) −18.2499 −0.775363
\(555\) −2.10038 −0.0891563
\(556\) −13.2039 −0.559970
\(557\) −27.2860 −1.15614 −0.578072 0.815985i \(-0.696195\pi\)
−0.578072 + 0.815985i \(0.696195\pi\)
\(558\) −16.2927 −0.689726
\(559\) 0 0
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −18.4078 −0.776487
\(563\) 29.0852 1.22580 0.612898 0.790162i \(-0.290004\pi\)
0.612898 + 0.790162i \(0.290004\pi\)
\(564\) −0.450654 −0.0189759
\(565\) 11.3717 0.478411
\(566\) −9.62158 −0.404425
\(567\) −1.10038 −0.0462118
\(568\) −0.585462 −0.0245654
\(569\) −23.9143 −1.00254 −0.501270 0.865291i \(-0.667134\pi\)
−0.501270 + 0.865291i \(0.667134\pi\)
\(570\) 6.35027 0.265984
\(571\) −15.2797 −0.639437 −0.319718 0.947513i \(-0.603588\pi\)
−0.319718 + 0.947513i \(0.603588\pi\)
\(572\) 0 0
\(573\) 3.29946 0.137837
\(574\) −1.43910 −0.0600667
\(575\) −1.83221 −0.0764085
\(576\) −1.68585 −0.0702436
\(577\) −9.58233 −0.398917 −0.199459 0.979906i \(-0.563918\pi\)
−0.199459 + 0.979906i \(0.563918\pi\)
\(578\) −9.78623 −0.407054
\(579\) −12.8156 −0.532599
\(580\) 3.83221 0.159124
\(581\) 9.37169 0.388803
\(582\) 11.6560 0.483157
\(583\) 0 0
\(584\) −8.97858 −0.371536
\(585\) 0 0
\(586\) −31.9143 −1.31837
\(587\) 0.176210 0.00727296 0.00363648 0.999993i \(-0.498842\pi\)
0.00363648 + 0.999993i \(0.498842\pi\)
\(588\) 1.14637 0.0472753
\(589\) 53.5359 2.20591
\(590\) 0.685846 0.0282358
\(591\) 17.9080 0.736638
\(592\) −1.83221 −0.0753034
\(593\) 0.863500 0.0354597 0.0177299 0.999843i \(-0.494356\pi\)
0.0177299 + 0.999843i \(0.494356\pi\)
\(594\) 0 0
\(595\) −2.68585 −0.110109
\(596\) −0.417674 −0.0171086
\(597\) −24.9013 −1.01914
\(598\) 0 0
\(599\) 23.4145 0.956692 0.478346 0.878171i \(-0.341236\pi\)
0.478346 + 0.878171i \(0.341236\pi\)
\(600\) 1.14637 0.0468002
\(601\) −13.1892 −0.537999 −0.269000 0.963140i \(-0.586693\pi\)
−0.269000 + 0.963140i \(0.586693\pi\)
\(602\) 4.97858 0.202912
\(603\) −5.68415 −0.231477
\(604\) −6.91117 −0.281212
\(605\) 0 0
\(606\) −1.17092 −0.0475655
\(607\) 11.8077 0.479258 0.239629 0.970865i \(-0.422974\pi\)
0.239629 + 0.970865i \(0.422974\pi\)
\(608\) 5.53948 0.224656
\(609\) 4.39312 0.178018
\(610\) 2.39312 0.0968944
\(611\) 0 0
\(612\) 4.52792 0.183030
\(613\) 39.4868 1.59486 0.797428 0.603414i \(-0.206193\pi\)
0.797428 + 0.603414i \(0.206193\pi\)
\(614\) −9.03612 −0.364668
\(615\) −1.64973 −0.0665236
\(616\) 0 0
\(617\) −38.3650 −1.54452 −0.772258 0.635310i \(-0.780873\pi\)
−0.772258 + 0.635310i \(0.780873\pi\)
\(618\) 3.07896 0.123854
\(619\) −5.52119 −0.221915 −0.110958 0.993825i \(-0.535392\pi\)
−0.110958 + 0.993825i \(0.535392\pi\)
\(620\) 9.66442 0.388133
\(621\) 9.84208 0.394949
\(622\) −13.6644 −0.547893
\(623\) 6.58546 0.263841
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −4.71040 −0.188266
\(627\) 0 0
\(628\) 14.2499 0.568632
\(629\) 4.92104 0.196215
\(630\) −1.68585 −0.0671657
\(631\) −46.2730 −1.84210 −0.921050 0.389445i \(-0.872667\pi\)
−0.921050 + 0.389445i \(0.872667\pi\)
\(632\) −13.2039 −0.525223
\(633\) −21.2566 −0.844875
\(634\) 20.9112 0.830489
\(635\) 16.4507 0.652824
\(636\) 15.4721 0.613508
\(637\) 0 0
\(638\) 0 0
\(639\) 0.986999 0.0390451
\(640\) 1.00000 0.0395285
\(641\) −39.0852 −1.54377 −0.771887 0.635760i \(-0.780687\pi\)
−0.771887 + 0.635760i \(0.780687\pi\)
\(642\) 0.450654 0.0177859
\(643\) 40.3404 1.59087 0.795435 0.606039i \(-0.207242\pi\)
0.795435 + 0.606039i \(0.207242\pi\)
\(644\) −1.83221 −0.0721993
\(645\) 5.70727 0.224723
\(646\) −14.8782 −0.585375
\(647\) −6.48508 −0.254955 −0.127477 0.991841i \(-0.540688\pi\)
−0.127477 + 0.991841i \(0.540688\pi\)
\(648\) −1.10038 −0.0432272
\(649\) 0 0
\(650\) 0 0
\(651\) 11.0790 0.434219
\(652\) 0.628308 0.0246064
\(653\) 18.7533 0.733871 0.366936 0.930246i \(-0.380407\pi\)
0.366936 + 0.930246i \(0.380407\pi\)
\(654\) 5.06427 0.198029
\(655\) −5.20390 −0.203333
\(656\) −1.43910 −0.0561873
\(657\) 15.1365 0.590531
\(658\) −0.393115 −0.0153252
\(659\) −6.37842 −0.248468 −0.124234 0.992253i \(-0.539647\pi\)
−0.124234 + 0.992253i \(0.539647\pi\)
\(660\) 0 0
\(661\) 30.6148 1.19078 0.595390 0.803437i \(-0.296998\pi\)
0.595390 + 0.803437i \(0.296998\pi\)
\(662\) −15.4721 −0.601339
\(663\) 0 0
\(664\) 9.37169 0.363692
\(665\) 5.53948 0.214812
\(666\) 3.08883 0.119690
\(667\) −7.02142 −0.271871
\(668\) −18.5426 −0.717435
\(669\) 1.50650 0.0582447
\(670\) 3.37169 0.130260
\(671\) 0 0
\(672\) 1.14637 0.0442220
\(673\) −5.91431 −0.227980 −0.113990 0.993482i \(-0.536363\pi\)
−0.113990 + 0.993482i \(0.536363\pi\)
\(674\) −35.9572 −1.38502
\(675\) −5.37169 −0.206757
\(676\) −13.0000 −0.500000
\(677\) −29.9656 −1.15167 −0.575835 0.817566i \(-0.695323\pi\)
−0.575835 + 0.817566i \(0.695323\pi\)
\(678\) 13.0361 0.500649
\(679\) 10.1678 0.390204
\(680\) −2.68585 −0.102997
\(681\) −7.88492 −0.302151
\(682\) 0 0
\(683\) 33.0790 1.26573 0.632866 0.774262i \(-0.281879\pi\)
0.632866 + 0.774262i \(0.281879\pi\)
\(684\) −9.33871 −0.357075
\(685\) 1.41454 0.0540467
\(686\) 1.00000 0.0381802
\(687\) 16.1151 0.614829
\(688\) 4.97858 0.189806
\(689\) 0 0
\(690\) −2.10038 −0.0799603
\(691\) 35.9656 1.36819 0.684097 0.729391i \(-0.260196\pi\)
0.684097 + 0.729391i \(0.260196\pi\)
\(692\) 10.3503 0.393458
\(693\) 0 0
\(694\) −2.23519 −0.0848467
\(695\) −13.2039 −0.500853
\(696\) 4.39312 0.166521
\(697\) 3.86519 0.146405
\(698\) 12.0147 0.454763
\(699\) −4.63458 −0.175296
\(700\) 1.00000 0.0377964
\(701\) −4.95402 −0.187111 −0.0935554 0.995614i \(-0.529823\pi\)
−0.0935554 + 0.995614i \(0.529823\pi\)
\(702\) 0 0
\(703\) −10.1495 −0.382796
\(704\) 0 0
\(705\) −0.450654 −0.0169726
\(706\) 31.2039 1.17437
\(707\) −1.02142 −0.0384146
\(708\) 0.786230 0.0295483
\(709\) −23.5725 −0.885282 −0.442641 0.896699i \(-0.645958\pi\)
−0.442641 + 0.896699i \(0.645958\pi\)
\(710\) −0.585462 −0.0219720
\(711\) 22.2598 0.834806
\(712\) 6.58546 0.246801
\(713\) −17.7073 −0.663142
\(714\) −3.07896 −0.115227
\(715\) 0 0
\(716\) 12.6430 0.472491
\(717\) 27.9916 1.04536
\(718\) 20.8683 0.778799
\(719\) 43.2944 1.61461 0.807305 0.590135i \(-0.200925\pi\)
0.807305 + 0.590135i \(0.200925\pi\)
\(720\) −1.68585 −0.0628278
\(721\) 2.68585 0.100026
\(722\) 11.6858 0.434902
\(723\) −4.27804 −0.159102
\(724\) −2.58546 −0.0960879
\(725\) 3.83221 0.142325
\(726\) 0 0
\(727\) 39.8370 1.47747 0.738737 0.673994i \(-0.235423\pi\)
0.738737 + 0.673994i \(0.235423\pi\)
\(728\) 0 0
\(729\) 20.3288 0.752920
\(730\) −8.97858 −0.332312
\(731\) −13.3717 −0.494570
\(732\) 2.74338 0.101398
\(733\) 13.6791 0.505250 0.252625 0.967564i \(-0.418706\pi\)
0.252625 + 0.967564i \(0.418706\pi\)
\(734\) −33.2285 −1.22648
\(735\) 1.14637 0.0422843
\(736\) −1.83221 −0.0675362
\(737\) 0 0
\(738\) 2.42610 0.0893058
\(739\) 1.25662 0.0462253 0.0231127 0.999733i \(-0.492642\pi\)
0.0231127 + 0.999733i \(0.492642\pi\)
\(740\) −1.83221 −0.0673534
\(741\) 0 0
\(742\) 13.4966 0.495477
\(743\) −1.90804 −0.0699991 −0.0349996 0.999387i \(-0.511143\pi\)
−0.0349996 + 0.999387i \(0.511143\pi\)
\(744\) 11.0790 0.406174
\(745\) −0.417674 −0.0153024
\(746\) −9.66442 −0.353840
\(747\) −15.7992 −0.578064
\(748\) 0 0
\(749\) 0.393115 0.0143641
\(750\) 1.14637 0.0418593
\(751\) −41.0852 −1.49922 −0.749611 0.661879i \(-0.769759\pi\)
−0.749611 + 0.661879i \(0.769759\pi\)
\(752\) −0.393115 −0.0143354
\(753\) 27.0130 0.984408
\(754\) 0 0
\(755\) −6.91117 −0.251523
\(756\) −5.37169 −0.195367
\(757\) 4.07583 0.148138 0.0740692 0.997253i \(-0.476401\pi\)
0.0740692 + 0.997253i \(0.476401\pi\)
\(758\) −3.21377 −0.116729
\(759\) 0 0
\(760\) 5.53948 0.200938
\(761\) 5.10352 0.185002 0.0925012 0.995713i \(-0.470514\pi\)
0.0925012 + 0.995713i \(0.470514\pi\)
\(762\) 18.8585 0.683170
\(763\) 4.41767 0.159931
\(764\) 2.87819 0.104129
\(765\) 4.52792 0.163707
\(766\) −24.6430 −0.890388
\(767\) 0 0
\(768\) 1.14637 0.0413659
\(769\) −21.8406 −0.787593 −0.393797 0.919198i \(-0.628839\pi\)
−0.393797 + 0.919198i \(0.628839\pi\)
\(770\) 0 0
\(771\) −16.1432 −0.581384
\(772\) −11.1793 −0.402353
\(773\) 34.0294 1.22395 0.611976 0.790876i \(-0.290375\pi\)
0.611976 + 0.790876i \(0.290375\pi\)
\(774\) −8.39312 −0.301684
\(775\) 9.66442 0.347156
\(776\) 10.1678 0.365002
\(777\) −2.10038 −0.0753509
\(778\) −6.92104 −0.248131
\(779\) −7.97185 −0.285621
\(780\) 0 0
\(781\) 0 0
\(782\) 4.92104 0.175976
\(783\) −20.5855 −0.735664
\(784\) 1.00000 0.0357143
\(785\) 14.2499 0.508600
\(786\) −5.96558 −0.212785
\(787\) −16.4998 −0.588153 −0.294077 0.955782i \(-0.595012\pi\)
−0.294077 + 0.955782i \(0.595012\pi\)
\(788\) 15.6216 0.556496
\(789\) 17.9572 0.639292
\(790\) −13.2039 −0.469774
\(791\) 11.3717 0.404331
\(792\) 0 0
\(793\) 0 0
\(794\) 6.92104 0.245618
\(795\) 15.4721 0.548738
\(796\) −21.7220 −0.769915
\(797\) 32.3221 1.14491 0.572454 0.819937i \(-0.305991\pi\)
0.572454 + 0.819937i \(0.305991\pi\)
\(798\) 6.35027 0.224797
\(799\) 1.05585 0.0373532
\(800\) 1.00000 0.0353553
\(801\) −11.1021 −0.392273
\(802\) 18.6430 0.658307
\(803\) 0 0
\(804\) 3.86519 0.136315
\(805\) −1.83221 −0.0645770
\(806\) 0 0
\(807\) 25.2860 0.890109
\(808\) −1.02142 −0.0359335
\(809\) −24.6514 −0.866698 −0.433349 0.901226i \(-0.642668\pi\)
−0.433349 + 0.901226i \(0.642668\pi\)
\(810\) −1.10038 −0.0386636
\(811\) −6.66129 −0.233909 −0.116955 0.993137i \(-0.537313\pi\)
−0.116955 + 0.993137i \(0.537313\pi\)
\(812\) 3.83221 0.134484
\(813\) −14.9442 −0.524114
\(814\) 0 0
\(815\) 0.628308 0.0220087
\(816\) −3.07896 −0.107785
\(817\) 27.5787 0.964858
\(818\) 33.6890 1.17791
\(819\) 0 0
\(820\) −1.43910 −0.0502554
\(821\) −42.2400 −1.47419 −0.737093 0.675791i \(-0.763802\pi\)
−0.737093 + 0.675791i \(0.763802\pi\)
\(822\) 1.62158 0.0565590
\(823\) −26.2829 −0.916163 −0.458082 0.888910i \(-0.651463\pi\)
−0.458082 + 0.888910i \(0.651463\pi\)
\(824\) 2.68585 0.0935659
\(825\) 0 0
\(826\) 0.685846 0.0238636
\(827\) 0.986999 0.0343213 0.0171607 0.999853i \(-0.494537\pi\)
0.0171607 + 0.999853i \(0.494537\pi\)
\(828\) 3.08883 0.107344
\(829\) −1.07054 −0.0371814 −0.0185907 0.999827i \(-0.505918\pi\)
−0.0185907 + 0.999827i \(0.505918\pi\)
\(830\) 9.37169 0.325296
\(831\) −20.9210 −0.725743
\(832\) 0 0
\(833\) −2.68585 −0.0930591
\(834\) −15.1365 −0.524134
\(835\) −18.5426 −0.641694
\(836\) 0 0
\(837\) −51.9143 −1.79442
\(838\) 21.0214 0.726173
\(839\) −30.4507 −1.05127 −0.525637 0.850709i \(-0.676173\pi\)
−0.525637 + 0.850709i \(0.676173\pi\)
\(840\) 1.14637 0.0395534
\(841\) −14.3142 −0.493592
\(842\) −10.8353 −0.373411
\(843\) −21.1021 −0.726794
\(844\) −18.5426 −0.638263
\(845\) −13.0000 −0.447214
\(846\) 0.662732 0.0227852
\(847\) 0 0
\(848\) 13.4966 0.463476
\(849\) −11.0298 −0.378543
\(850\) −2.68585 −0.0921238
\(851\) 3.35700 0.115076
\(852\) −0.671153 −0.0229933
\(853\) −6.43596 −0.220363 −0.110182 0.993911i \(-0.535143\pi\)
−0.110182 + 0.993911i \(0.535143\pi\)
\(854\) 2.39312 0.0818907
\(855\) −9.33871 −0.319377
\(856\) 0.393115 0.0134364
\(857\) 1.26504 0.0432128 0.0216064 0.999767i \(-0.493122\pi\)
0.0216064 + 0.999767i \(0.493122\pi\)
\(858\) 0 0
\(859\) 1.13650 0.0387769 0.0193884 0.999812i \(-0.493828\pi\)
0.0193884 + 0.999812i \(0.493828\pi\)
\(860\) 4.97858 0.169768
\(861\) −1.64973 −0.0562227
\(862\) −32.6676 −1.11266
\(863\) 44.7764 1.52421 0.762103 0.647456i \(-0.224167\pi\)
0.762103 + 0.647456i \(0.224167\pi\)
\(864\) −5.37169 −0.182749
\(865\) 10.3503 0.351920
\(866\) −16.3748 −0.556439
\(867\) −11.2186 −0.381004
\(868\) 9.66442 0.328032
\(869\) 0 0
\(870\) 4.39312 0.148941
\(871\) 0 0
\(872\) 4.41767 0.149601
\(873\) −17.1413 −0.580146
\(874\) −10.1495 −0.343312
\(875\) 1.00000 0.0338062
\(876\) −10.2927 −0.347759
\(877\) −53.6644 −1.81212 −0.906059 0.423151i \(-0.860924\pi\)
−0.906059 + 0.423151i \(0.860924\pi\)
\(878\) 0.450654 0.0152088
\(879\) −36.5855 −1.23400
\(880\) 0 0
\(881\) 4.42754 0.149168 0.0745838 0.997215i \(-0.476237\pi\)
0.0745838 + 0.997215i \(0.476237\pi\)
\(882\) −1.68585 −0.0567654
\(883\) 31.7073 1.06703 0.533517 0.845789i \(-0.320870\pi\)
0.533517 + 0.845789i \(0.320870\pi\)
\(884\) 0 0
\(885\) 0.786230 0.0264288
\(886\) 10.7862 0.362370
\(887\) 45.9865 1.54408 0.772038 0.635576i \(-0.219237\pi\)
0.772038 + 0.635576i \(0.219237\pi\)
\(888\) −2.10038 −0.0704843
\(889\) 16.4507 0.551737
\(890\) 6.58546 0.220745
\(891\) 0 0
\(892\) 1.31415 0.0440011
\(893\) −2.17765 −0.0728724
\(894\) −0.478807 −0.0160137
\(895\) 12.6430 0.422609
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −19.8139 −0.661199
\(899\) 37.0361 1.23522
\(900\) −1.68585 −0.0561949
\(901\) −36.2499 −1.20766
\(902\) 0 0
\(903\) 5.70727 0.189926
\(904\) 11.3717 0.378217
\(905\) −2.58546 −0.0859437
\(906\) −7.92273 −0.263215
\(907\) −11.5065 −0.382067 −0.191034 0.981584i \(-0.561184\pi\)
−0.191034 + 0.981584i \(0.561184\pi\)
\(908\) −6.87819 −0.228261
\(909\) 1.72196 0.0571139
\(910\) 0 0
\(911\) 16.8353 0.557780 0.278890 0.960323i \(-0.410034\pi\)
0.278890 + 0.960323i \(0.410034\pi\)
\(912\) 6.35027 0.210278
\(913\) 0 0
\(914\) 23.9572 0.792432
\(915\) 2.74338 0.0906935
\(916\) 14.0575 0.464474
\(917\) −5.20390 −0.171848
\(918\) 14.4275 0.476180
\(919\) 48.6184 1.60377 0.801887 0.597475i \(-0.203829\pi\)
0.801887 + 0.597475i \(0.203829\pi\)
\(920\) −1.83221 −0.0604062
\(921\) −10.3587 −0.341330
\(922\) 26.6430 0.877440
\(923\) 0 0
\(924\) 0 0
\(925\) −1.83221 −0.0602427
\(926\) 2.75325 0.0904774
\(927\) −4.52792 −0.148717
\(928\) 3.83221 0.125799
\(929\) −5.57873 −0.183032 −0.0915161 0.995804i \(-0.529171\pi\)
−0.0915161 + 0.995804i \(0.529171\pi\)
\(930\) 11.0790 0.363293
\(931\) 5.53948 0.181549
\(932\) −4.04285 −0.132428
\(933\) −15.6644 −0.512830
\(934\) −20.1396 −0.658989
\(935\) 0 0
\(936\) 0 0
\(937\) 20.6430 0.674377 0.337189 0.941437i \(-0.390524\pi\)
0.337189 + 0.941437i \(0.390524\pi\)
\(938\) 3.37169 0.110090
\(939\) −5.39985 −0.176217
\(940\) −0.393115 −0.0128220
\(941\) −45.0214 −1.46766 −0.733828 0.679335i \(-0.762268\pi\)
−0.733828 + 0.679335i \(0.762268\pi\)
\(942\) 16.3356 0.532242
\(943\) 2.63673 0.0858637
\(944\) 0.685846 0.0223224
\(945\) −5.37169 −0.174741
\(946\) 0 0
\(947\) −6.45065 −0.209618 −0.104809 0.994492i \(-0.533423\pi\)
−0.104809 + 0.994492i \(0.533423\pi\)
\(948\) −15.1365 −0.491611
\(949\) 0 0
\(950\) 5.53948 0.179724
\(951\) 23.9718 0.777340
\(952\) −2.68585 −0.0870488
\(953\) −28.5510 −0.924859 −0.462429 0.886656i \(-0.653022\pi\)
−0.462429 + 0.886656i \(0.653022\pi\)
\(954\) −22.7533 −0.736664
\(955\) 2.87819 0.0931361
\(956\) 24.4177 0.789724
\(957\) 0 0
\(958\) −4.58546 −0.148150
\(959\) 1.41454 0.0456778
\(960\) 1.14637 0.0369988
\(961\) 62.4011 2.01294
\(962\) 0 0
\(963\) −0.662732 −0.0213562
\(964\) −3.73183 −0.120194
\(965\) −11.1793 −0.359876
\(966\) −2.10038 −0.0675788
\(967\) −1.90804 −0.0613583 −0.0306792 0.999529i \(-0.509767\pi\)
−0.0306792 + 0.999529i \(0.509767\pi\)
\(968\) 0 0
\(969\) −17.0558 −0.547913
\(970\) 10.1678 0.326468
\(971\) −31.8996 −1.02371 −0.511854 0.859073i \(-0.671041\pi\)
−0.511854 + 0.859073i \(0.671041\pi\)
\(972\) 14.8536 0.476431
\(973\) −13.2039 −0.423298
\(974\) 33.1611 1.06255
\(975\) 0 0
\(976\) 2.39312 0.0766018
\(977\) 28.5426 0.913159 0.456580 0.889683i \(-0.349074\pi\)
0.456580 + 0.889683i \(0.349074\pi\)
\(978\) 0.720270 0.0230317
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −7.44752 −0.237781
\(982\) 23.9143 0.763136
\(983\) −36.0869 −1.15099 −0.575497 0.817804i \(-0.695191\pi\)
−0.575497 + 0.817804i \(0.695191\pi\)
\(984\) −1.64973 −0.0525915
\(985\) 15.6216 0.497745
\(986\) −10.2927 −0.327787
\(987\) −0.450654 −0.0143445
\(988\) 0 0
\(989\) −9.12181 −0.290057
\(990\) 0 0
\(991\) −57.2369 −1.81819 −0.909095 0.416589i \(-0.863225\pi\)
−0.909095 + 0.416589i \(0.863225\pi\)
\(992\) 9.66442 0.306846
\(993\) −17.7367 −0.562856
\(994\) −0.585462 −0.0185697
\(995\) −21.7220 −0.688632
\(996\) 10.7434 0.340417
\(997\) −34.0638 −1.07881 −0.539406 0.842046i \(-0.681351\pi\)
−0.539406 + 0.842046i \(0.681351\pi\)
\(998\) −39.3864 −1.24675
\(999\) 9.84208 0.311390
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.cn.1.2 yes 3
11.10 odd 2 8470.2.a.ch.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.ch.1.2 3 11.10 odd 2
8470.2.a.cn.1.2 yes 3 1.1 even 1 trivial