Properties

Label 8470.2.a.cf.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.381966 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.381966 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} -1.00000 q^{10} +0.381966 q^{12} +2.00000 q^{13} +1.00000 q^{14} -0.381966 q^{15} +1.00000 q^{16} +0.618034 q^{17} -2.85410 q^{18} +0.145898 q^{19} -1.00000 q^{20} +0.381966 q^{21} +6.00000 q^{23} +0.381966 q^{24} +1.00000 q^{25} +2.00000 q^{26} -2.23607 q^{27} +1.00000 q^{28} -1.23607 q^{29} -0.381966 q^{30} +3.23607 q^{31} +1.00000 q^{32} +0.618034 q^{34} -1.00000 q^{35} -2.85410 q^{36} +2.47214 q^{37} +0.145898 q^{38} +0.763932 q^{39} -1.00000 q^{40} +5.32624 q^{41} +0.381966 q^{42} -4.85410 q^{43} +2.85410 q^{45} +6.00000 q^{46} +4.76393 q^{47} +0.381966 q^{48} +1.00000 q^{49} +1.00000 q^{50} +0.236068 q^{51} +2.00000 q^{52} -3.23607 q^{53} -2.23607 q^{54} +1.00000 q^{56} +0.0557281 q^{57} -1.23607 q^{58} -5.38197 q^{59} -0.381966 q^{60} -12.4721 q^{61} +3.23607 q^{62} -2.85410 q^{63} +1.00000 q^{64} -2.00000 q^{65} -5.09017 q^{67} +0.618034 q^{68} +2.29180 q^{69} -1.00000 q^{70} +3.70820 q^{71} -2.85410 q^{72} +7.14590 q^{73} +2.47214 q^{74} +0.381966 q^{75} +0.145898 q^{76} +0.763932 q^{78} +0.472136 q^{79} -1.00000 q^{80} +7.70820 q^{81} +5.32624 q^{82} -5.32624 q^{83} +0.381966 q^{84} -0.618034 q^{85} -4.85410 q^{86} -0.472136 q^{87} +1.90983 q^{89} +2.85410 q^{90} +2.00000 q^{91} +6.00000 q^{92} +1.23607 q^{93} +4.76393 q^{94} -0.145898 q^{95} +0.381966 q^{96} +10.5623 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 3q^{3} + 2q^{4} - 2q^{5} + 3q^{6} + 2q^{7} + 2q^{8} + q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 3q^{3} + 2q^{4} - 2q^{5} + 3q^{6} + 2q^{7} + 2q^{8} + q^{9} - 2q^{10} + 3q^{12} + 4q^{13} + 2q^{14} - 3q^{15} + 2q^{16} - q^{17} + q^{18} + 7q^{19} - 2q^{20} + 3q^{21} + 12q^{23} + 3q^{24} + 2q^{25} + 4q^{26} + 2q^{28} + 2q^{29} - 3q^{30} + 2q^{31} + 2q^{32} - q^{34} - 2q^{35} + q^{36} - 4q^{37} + 7q^{38} + 6q^{39} - 2q^{40} - 5q^{41} + 3q^{42} - 3q^{43} - q^{45} + 12q^{46} + 14q^{47} + 3q^{48} + 2q^{49} + 2q^{50} - 4q^{51} + 4q^{52} - 2q^{53} + 2q^{56} + 18q^{57} + 2q^{58} - 13q^{59} - 3q^{60} - 16q^{61} + 2q^{62} + q^{63} + 2q^{64} - 4q^{65} + q^{67} - q^{68} + 18q^{69} - 2q^{70} - 6q^{71} + q^{72} + 21q^{73} - 4q^{74} + 3q^{75} + 7q^{76} + 6q^{78} - 8q^{79} - 2q^{80} + 2q^{81} - 5q^{82} + 5q^{83} + 3q^{84} + q^{85} - 3q^{86} + 8q^{87} + 15q^{89} - q^{90} + 4q^{91} + 12q^{92} - 2q^{93} + 14q^{94} - 7q^{95} + 3q^{96} + q^{97} + 2q^{98} + O(q^{100}) \)

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
\(3\) 0.381966 0.220528 0.110264 0.993902i \(-0.464830\pi\)
0.110264 + 0.993902i \(0.464830\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0.381966 0.155937
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −2.85410 −0.951367
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 0.381966 0.110264
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) −0.381966 −0.0986232
\(16\) 1.00000 0.250000
\(17\) 0.618034 0.149895 0.0749476 0.997187i \(-0.476121\pi\)
0.0749476 + 0.997187i \(0.476121\pi\)
\(18\) −2.85410 −0.672718
\(19\) 0.145898 0.0334713 0.0167357 0.999860i \(-0.494673\pi\)
0.0167357 + 0.999860i \(0.494673\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0.381966 0.0833518
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0.381966 0.0779685
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −2.23607 −0.430331
\(28\) 1.00000 0.188982
\(29\) −1.23607 −0.229532 −0.114766 0.993393i \(-0.536612\pi\)
−0.114766 + 0.993393i \(0.536612\pi\)
\(30\) −0.381966 −0.0697371
\(31\) 3.23607 0.581215 0.290607 0.956842i \(-0.406143\pi\)
0.290607 + 0.956842i \(0.406143\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.618034 0.105992
\(35\) −1.00000 −0.169031
\(36\) −2.85410 −0.475684
\(37\) 2.47214 0.406417 0.203208 0.979136i \(-0.434863\pi\)
0.203208 + 0.979136i \(0.434863\pi\)
\(38\) 0.145898 0.0236678
\(39\) 0.763932 0.122327
\(40\) −1.00000 −0.158114
\(41\) 5.32624 0.831819 0.415909 0.909406i \(-0.363463\pi\)
0.415909 + 0.909406i \(0.363463\pi\)
\(42\) 0.381966 0.0589386
\(43\) −4.85410 −0.740244 −0.370122 0.928983i \(-0.620684\pi\)
−0.370122 + 0.928983i \(0.620684\pi\)
\(44\) 0 0
\(45\) 2.85410 0.425464
\(46\) 6.00000 0.884652
\(47\) 4.76393 0.694891 0.347445 0.937700i \(-0.387049\pi\)
0.347445 + 0.937700i \(0.387049\pi\)
\(48\) 0.381966 0.0551320
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0.236068 0.0330561
\(52\) 2.00000 0.277350
\(53\) −3.23607 −0.444508 −0.222254 0.974989i \(-0.571341\pi\)
−0.222254 + 0.974989i \(0.571341\pi\)
\(54\) −2.23607 −0.304290
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0.0557281 0.00738137
\(58\) −1.23607 −0.162304
\(59\) −5.38197 −0.700672 −0.350336 0.936624i \(-0.613933\pi\)
−0.350336 + 0.936624i \(0.613933\pi\)
\(60\) −0.381966 −0.0493116
\(61\) −12.4721 −1.59689 −0.798447 0.602066i \(-0.794345\pi\)
−0.798447 + 0.602066i \(0.794345\pi\)
\(62\) 3.23607 0.410981
\(63\) −2.85410 −0.359583
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −5.09017 −0.621863 −0.310932 0.950432i \(-0.600641\pi\)
−0.310932 + 0.950432i \(0.600641\pi\)
\(68\) 0.618034 0.0749476
\(69\) 2.29180 0.275900
\(70\) −1.00000 −0.119523
\(71\) 3.70820 0.440083 0.220041 0.975491i \(-0.429381\pi\)
0.220041 + 0.975491i \(0.429381\pi\)
\(72\) −2.85410 −0.336359
\(73\) 7.14590 0.836364 0.418182 0.908363i \(-0.362667\pi\)
0.418182 + 0.908363i \(0.362667\pi\)
\(74\) 2.47214 0.287380
\(75\) 0.381966 0.0441056
\(76\) 0.145898 0.0167357
\(77\) 0 0
\(78\) 0.763932 0.0864983
\(79\) 0.472136 0.0531194 0.0265597 0.999647i \(-0.491545\pi\)
0.0265597 + 0.999647i \(0.491545\pi\)
\(80\) −1.00000 −0.111803
\(81\) 7.70820 0.856467
\(82\) 5.32624 0.588185
\(83\) −5.32624 −0.584631 −0.292315 0.956322i \(-0.594426\pi\)
−0.292315 + 0.956322i \(0.594426\pi\)
\(84\) 0.381966 0.0416759
\(85\) −0.618034 −0.0670352
\(86\) −4.85410 −0.523431
\(87\) −0.472136 −0.0506183
\(88\) 0 0
\(89\) 1.90983 0.202442 0.101221 0.994864i \(-0.467725\pi\)
0.101221 + 0.994864i \(0.467725\pi\)
\(90\) 2.85410 0.300849
\(91\) 2.00000 0.209657
\(92\) 6.00000 0.625543
\(93\) 1.23607 0.128174
\(94\) 4.76393 0.491362
\(95\) −0.145898 −0.0149688
\(96\) 0.381966 0.0389842
\(97\) 10.5623 1.07244 0.536220 0.844078i \(-0.319852\pi\)
0.536220 + 0.844078i \(0.319852\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −8.94427 −0.889988 −0.444994 0.895533i \(-0.646794\pi\)
−0.444994 + 0.895533i \(0.646794\pi\)
\(102\) 0.236068 0.0233742
\(103\) 0.944272 0.0930419 0.0465209 0.998917i \(-0.485187\pi\)
0.0465209 + 0.998917i \(0.485187\pi\)
\(104\) 2.00000 0.196116
\(105\) −0.381966 −0.0372761
\(106\) −3.23607 −0.314315
\(107\) 17.6180 1.70320 0.851600 0.524192i \(-0.175633\pi\)
0.851600 + 0.524192i \(0.175633\pi\)
\(108\) −2.23607 −0.215166
\(109\) 8.47214 0.811483 0.405742 0.913988i \(-0.367013\pi\)
0.405742 + 0.913988i \(0.367013\pi\)
\(110\) 0 0
\(111\) 0.944272 0.0896263
\(112\) 1.00000 0.0944911
\(113\) 6.32624 0.595122 0.297561 0.954703i \(-0.403827\pi\)
0.297561 + 0.954703i \(0.403827\pi\)
\(114\) 0.0557281 0.00521941
\(115\) −6.00000 −0.559503
\(116\) −1.23607 −0.114766
\(117\) −5.70820 −0.527724
\(118\) −5.38197 −0.495450
\(119\) 0.618034 0.0566551
\(120\) −0.381966 −0.0348686
\(121\) 0 0
\(122\) −12.4721 −1.12917
\(123\) 2.03444 0.183439
\(124\) 3.23607 0.290607
\(125\) −1.00000 −0.0894427
\(126\) −2.85410 −0.254264
\(127\) 15.4164 1.36798 0.683992 0.729489i \(-0.260242\pi\)
0.683992 + 0.729489i \(0.260242\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.85410 −0.163245
\(130\) −2.00000 −0.175412
\(131\) 16.7984 1.46768 0.733840 0.679322i \(-0.237726\pi\)
0.733840 + 0.679322i \(0.237726\pi\)
\(132\) 0 0
\(133\) 0.145898 0.0126510
\(134\) −5.09017 −0.439724
\(135\) 2.23607 0.192450
\(136\) 0.618034 0.0529960
\(137\) 10.0902 0.862061 0.431031 0.902337i \(-0.358150\pi\)
0.431031 + 0.902337i \(0.358150\pi\)
\(138\) 2.29180 0.195091
\(139\) 9.52786 0.808143 0.404071 0.914727i \(-0.367595\pi\)
0.404071 + 0.914727i \(0.367595\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 1.81966 0.153243
\(142\) 3.70820 0.311186
\(143\) 0 0
\(144\) −2.85410 −0.237842
\(145\) 1.23607 0.102650
\(146\) 7.14590 0.591399
\(147\) 0.381966 0.0315040
\(148\) 2.47214 0.203208
\(149\) 1.81966 0.149072 0.0745362 0.997218i \(-0.476252\pi\)
0.0745362 + 0.997218i \(0.476252\pi\)
\(150\) 0.381966 0.0311874
\(151\) 15.7082 1.27832 0.639158 0.769076i \(-0.279283\pi\)
0.639158 + 0.769076i \(0.279283\pi\)
\(152\) 0.145898 0.0118339
\(153\) −1.76393 −0.142605
\(154\) 0 0
\(155\) −3.23607 −0.259927
\(156\) 0.763932 0.0611635
\(157\) 13.7082 1.09403 0.547017 0.837122i \(-0.315763\pi\)
0.547017 + 0.837122i \(0.315763\pi\)
\(158\) 0.472136 0.0375611
\(159\) −1.23607 −0.0980266
\(160\) −1.00000 −0.0790569
\(161\) 6.00000 0.472866
\(162\) 7.70820 0.605614
\(163\) −3.85410 −0.301877 −0.150938 0.988543i \(-0.548229\pi\)
−0.150938 + 0.988543i \(0.548229\pi\)
\(164\) 5.32624 0.415909
\(165\) 0 0
\(166\) −5.32624 −0.413396
\(167\) −2.47214 −0.191300 −0.0956498 0.995415i \(-0.530493\pi\)
−0.0956498 + 0.995415i \(0.530493\pi\)
\(168\) 0.381966 0.0294693
\(169\) −9.00000 −0.692308
\(170\) −0.618034 −0.0474010
\(171\) −0.416408 −0.0318435
\(172\) −4.85410 −0.370122
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) −0.472136 −0.0357925
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −2.05573 −0.154518
\(178\) 1.90983 0.143148
\(179\) −8.61803 −0.644142 −0.322071 0.946715i \(-0.604379\pi\)
−0.322071 + 0.946715i \(0.604379\pi\)
\(180\) 2.85410 0.212732
\(181\) 12.4721 0.927047 0.463523 0.886085i \(-0.346585\pi\)
0.463523 + 0.886085i \(0.346585\pi\)
\(182\) 2.00000 0.148250
\(183\) −4.76393 −0.352160
\(184\) 6.00000 0.442326
\(185\) −2.47214 −0.181755
\(186\) 1.23607 0.0906329
\(187\) 0 0
\(188\) 4.76393 0.347445
\(189\) −2.23607 −0.162650
\(190\) −0.145898 −0.0105846
\(191\) −10.4721 −0.757737 −0.378869 0.925450i \(-0.623687\pi\)
−0.378869 + 0.925450i \(0.623687\pi\)
\(192\) 0.381966 0.0275660
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 10.5623 0.758329
\(195\) −0.763932 −0.0547063
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 7.41641 0.525735 0.262868 0.964832i \(-0.415332\pi\)
0.262868 + 0.964832i \(0.415332\pi\)
\(200\) 1.00000 0.0707107
\(201\) −1.94427 −0.137138
\(202\) −8.94427 −0.629317
\(203\) −1.23607 −0.0867550
\(204\) 0.236068 0.0165281
\(205\) −5.32624 −0.372001
\(206\) 0.944272 0.0657905
\(207\) −17.1246 −1.19024
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) −0.381966 −0.0263582
\(211\) −22.2705 −1.53317 −0.766583 0.642146i \(-0.778044\pi\)
−0.766583 + 0.642146i \(0.778044\pi\)
\(212\) −3.23607 −0.222254
\(213\) 1.41641 0.0970507
\(214\) 17.6180 1.20434
\(215\) 4.85410 0.331047
\(216\) −2.23607 −0.152145
\(217\) 3.23607 0.219679
\(218\) 8.47214 0.573805
\(219\) 2.72949 0.184442
\(220\) 0 0
\(221\) 1.23607 0.0831469
\(222\) 0.944272 0.0633754
\(223\) 19.7082 1.31976 0.659879 0.751371i \(-0.270607\pi\)
0.659879 + 0.751371i \(0.270607\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.85410 −0.190273
\(226\) 6.32624 0.420815
\(227\) 9.27051 0.615305 0.307653 0.951499i \(-0.400457\pi\)
0.307653 + 0.951499i \(0.400457\pi\)
\(228\) 0.0557281 0.00369068
\(229\) −29.1246 −1.92461 −0.962304 0.271975i \(-0.912323\pi\)
−0.962304 + 0.271975i \(0.912323\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) −1.23607 −0.0811518
\(233\) 22.0902 1.44718 0.723588 0.690233i \(-0.242492\pi\)
0.723588 + 0.690233i \(0.242492\pi\)
\(234\) −5.70820 −0.373157
\(235\) −4.76393 −0.310765
\(236\) −5.38197 −0.350336
\(237\) 0.180340 0.0117143
\(238\) 0.618034 0.0400612
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −0.381966 −0.0246558
\(241\) 27.2705 1.75665 0.878324 0.478066i \(-0.158662\pi\)
0.878324 + 0.478066i \(0.158662\pi\)
\(242\) 0 0
\(243\) 9.65248 0.619207
\(244\) −12.4721 −0.798447
\(245\) −1.00000 −0.0638877
\(246\) 2.03444 0.129711
\(247\) 0.291796 0.0185665
\(248\) 3.23607 0.205491
\(249\) −2.03444 −0.128928
\(250\) −1.00000 −0.0632456
\(251\) −26.8328 −1.69367 −0.846836 0.531854i \(-0.821496\pi\)
−0.846836 + 0.531854i \(0.821496\pi\)
\(252\) −2.85410 −0.179792
\(253\) 0 0
\(254\) 15.4164 0.967311
\(255\) −0.236068 −0.0147832
\(256\) 1.00000 0.0625000
\(257\) 9.43769 0.588707 0.294354 0.955697i \(-0.404896\pi\)
0.294354 + 0.955697i \(0.404896\pi\)
\(258\) −1.85410 −0.115431
\(259\) 2.47214 0.153611
\(260\) −2.00000 −0.124035
\(261\) 3.52786 0.218369
\(262\) 16.7984 1.03781
\(263\) −2.18034 −0.134446 −0.0672228 0.997738i \(-0.521414\pi\)
−0.0672228 + 0.997738i \(0.521414\pi\)
\(264\) 0 0
\(265\) 3.23607 0.198790
\(266\) 0.145898 0.00894558
\(267\) 0.729490 0.0446441
\(268\) −5.09017 −0.310932
\(269\) 10.1803 0.620706 0.310353 0.950621i \(-0.399553\pi\)
0.310353 + 0.950621i \(0.399553\pi\)
\(270\) 2.23607 0.136083
\(271\) 17.4164 1.05797 0.528986 0.848631i \(-0.322573\pi\)
0.528986 + 0.848631i \(0.322573\pi\)
\(272\) 0.618034 0.0374738
\(273\) 0.763932 0.0462353
\(274\) 10.0902 0.609569
\(275\) 0 0
\(276\) 2.29180 0.137950
\(277\) 19.7082 1.18415 0.592076 0.805882i \(-0.298309\pi\)
0.592076 + 0.805882i \(0.298309\pi\)
\(278\) 9.52786 0.571443
\(279\) −9.23607 −0.552949
\(280\) −1.00000 −0.0597614
\(281\) 5.03444 0.300330 0.150165 0.988661i \(-0.452020\pi\)
0.150165 + 0.988661i \(0.452020\pi\)
\(282\) 1.81966 0.108359
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 3.70820 0.220041
\(285\) −0.0557281 −0.00330105
\(286\) 0 0
\(287\) 5.32624 0.314398
\(288\) −2.85410 −0.168180
\(289\) −16.6180 −0.977531
\(290\) 1.23607 0.0725844
\(291\) 4.03444 0.236503
\(292\) 7.14590 0.418182
\(293\) 7.52786 0.439783 0.219891 0.975524i \(-0.429430\pi\)
0.219891 + 0.975524i \(0.429430\pi\)
\(294\) 0.381966 0.0222767
\(295\) 5.38197 0.313350
\(296\) 2.47214 0.143690
\(297\) 0 0
\(298\) 1.81966 0.105410
\(299\) 12.0000 0.693978
\(300\) 0.381966 0.0220528
\(301\) −4.85410 −0.279786
\(302\) 15.7082 0.903906
\(303\) −3.41641 −0.196268
\(304\) 0.145898 0.00836783
\(305\) 12.4721 0.714152
\(306\) −1.76393 −0.100837
\(307\) −7.43769 −0.424492 −0.212246 0.977216i \(-0.568078\pi\)
−0.212246 + 0.977216i \(0.568078\pi\)
\(308\) 0 0
\(309\) 0.360680 0.0205184
\(310\) −3.23607 −0.183796
\(311\) 11.1246 0.630819 0.315409 0.948956i \(-0.397858\pi\)
0.315409 + 0.948956i \(0.397858\pi\)
\(312\) 0.763932 0.0432491
\(313\) 5.85410 0.330893 0.165447 0.986219i \(-0.447093\pi\)
0.165447 + 0.986219i \(0.447093\pi\)
\(314\) 13.7082 0.773599
\(315\) 2.85410 0.160810
\(316\) 0.472136 0.0265597
\(317\) −25.2361 −1.41740 −0.708699 0.705511i \(-0.750718\pi\)
−0.708699 + 0.705511i \(0.750718\pi\)
\(318\) −1.23607 −0.0693153
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 6.72949 0.375604
\(322\) 6.00000 0.334367
\(323\) 0.0901699 0.00501719
\(324\) 7.70820 0.428234
\(325\) 2.00000 0.110940
\(326\) −3.85410 −0.213459
\(327\) 3.23607 0.178955
\(328\) 5.32624 0.294092
\(329\) 4.76393 0.262644
\(330\) 0 0
\(331\) −23.0344 −1.26609 −0.633044 0.774116i \(-0.718195\pi\)
−0.633044 + 0.774116i \(0.718195\pi\)
\(332\) −5.32624 −0.292315
\(333\) −7.05573 −0.386652
\(334\) −2.47214 −0.135269
\(335\) 5.09017 0.278106
\(336\) 0.381966 0.0208380
\(337\) −13.3820 −0.728962 −0.364481 0.931211i \(-0.618754\pi\)
−0.364481 + 0.931211i \(0.618754\pi\)
\(338\) −9.00000 −0.489535
\(339\) 2.41641 0.131241
\(340\) −0.618034 −0.0335176
\(341\) 0 0
\(342\) −0.416408 −0.0225168
\(343\) 1.00000 0.0539949
\(344\) −4.85410 −0.261716
\(345\) −2.29180 −0.123386
\(346\) 12.0000 0.645124
\(347\) −10.3262 −0.554341 −0.277171 0.960821i \(-0.589397\pi\)
−0.277171 + 0.960821i \(0.589397\pi\)
\(348\) −0.472136 −0.0253091
\(349\) 14.4721 0.774676 0.387338 0.921938i \(-0.373395\pi\)
0.387338 + 0.921938i \(0.373395\pi\)
\(350\) 1.00000 0.0534522
\(351\) −4.47214 −0.238705
\(352\) 0 0
\(353\) 27.4508 1.46106 0.730531 0.682880i \(-0.239273\pi\)
0.730531 + 0.682880i \(0.239273\pi\)
\(354\) −2.05573 −0.109261
\(355\) −3.70820 −0.196811
\(356\) 1.90983 0.101221
\(357\) 0.236068 0.0124940
\(358\) −8.61803 −0.455477
\(359\) −12.6525 −0.667772 −0.333886 0.942613i \(-0.608360\pi\)
−0.333886 + 0.942613i \(0.608360\pi\)
\(360\) 2.85410 0.150424
\(361\) −18.9787 −0.998880
\(362\) 12.4721 0.655521
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −7.14590 −0.374033
\(366\) −4.76393 −0.249015
\(367\) 5.81966 0.303784 0.151892 0.988397i \(-0.451463\pi\)
0.151892 + 0.988397i \(0.451463\pi\)
\(368\) 6.00000 0.312772
\(369\) −15.2016 −0.791365
\(370\) −2.47214 −0.128520
\(371\) −3.23607 −0.168008
\(372\) 1.23607 0.0640871
\(373\) −15.5279 −0.804002 −0.402001 0.915639i \(-0.631685\pi\)
−0.402001 + 0.915639i \(0.631685\pi\)
\(374\) 0 0
\(375\) −0.381966 −0.0197246
\(376\) 4.76393 0.245681
\(377\) −2.47214 −0.127321
\(378\) −2.23607 −0.115011
\(379\) −21.1459 −1.08619 −0.543096 0.839671i \(-0.682748\pi\)
−0.543096 + 0.839671i \(0.682748\pi\)
\(380\) −0.145898 −0.00748441
\(381\) 5.88854 0.301679
\(382\) −10.4721 −0.535801
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0.381966 0.0194921
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 13.8541 0.704244
\(388\) 10.5623 0.536220
\(389\) 20.8328 1.05627 0.528133 0.849162i \(-0.322892\pi\)
0.528133 + 0.849162i \(0.322892\pi\)
\(390\) −0.763932 −0.0386832
\(391\) 3.70820 0.187532
\(392\) 1.00000 0.0505076
\(393\) 6.41641 0.323665
\(394\) 6.00000 0.302276
\(395\) −0.472136 −0.0237557
\(396\) 0 0
\(397\) 0.944272 0.0473916 0.0236958 0.999719i \(-0.492457\pi\)
0.0236958 + 0.999719i \(0.492457\pi\)
\(398\) 7.41641 0.371751
\(399\) 0.0557281 0.00278989
\(400\) 1.00000 0.0500000
\(401\) 11.5066 0.574611 0.287306 0.957839i \(-0.407240\pi\)
0.287306 + 0.957839i \(0.407240\pi\)
\(402\) −1.94427 −0.0969715
\(403\) 6.47214 0.322400
\(404\) −8.94427 −0.444994
\(405\) −7.70820 −0.383024
\(406\) −1.23607 −0.0613450
\(407\) 0 0
\(408\) 0.236068 0.0116871
\(409\) −7.88854 −0.390063 −0.195032 0.980797i \(-0.562481\pi\)
−0.195032 + 0.980797i \(0.562481\pi\)
\(410\) −5.32624 −0.263044
\(411\) 3.85410 0.190109
\(412\) 0.944272 0.0465209
\(413\) −5.38197 −0.264829
\(414\) −17.1246 −0.841629
\(415\) 5.32624 0.261455
\(416\) 2.00000 0.0980581
\(417\) 3.63932 0.178218
\(418\) 0 0
\(419\) −3.32624 −0.162497 −0.0812487 0.996694i \(-0.525891\pi\)
−0.0812487 + 0.996694i \(0.525891\pi\)
\(420\) −0.381966 −0.0186380
\(421\) 38.3607 1.86959 0.934793 0.355194i \(-0.115585\pi\)
0.934793 + 0.355194i \(0.115585\pi\)
\(422\) −22.2705 −1.08411
\(423\) −13.5967 −0.661096
\(424\) −3.23607 −0.157157
\(425\) 0.618034 0.0299791
\(426\) 1.41641 0.0686252
\(427\) −12.4721 −0.603569
\(428\) 17.6180 0.851600
\(429\) 0 0
\(430\) 4.85410 0.234086
\(431\) 26.7639 1.28917 0.644587 0.764531i \(-0.277030\pi\)
0.644587 + 0.764531i \(0.277030\pi\)
\(432\) −2.23607 −0.107583
\(433\) −39.4508 −1.89589 −0.947943 0.318439i \(-0.896841\pi\)
−0.947943 + 0.318439i \(0.896841\pi\)
\(434\) 3.23607 0.155336
\(435\) 0.472136 0.0226372
\(436\) 8.47214 0.405742
\(437\) 0.875388 0.0418755
\(438\) 2.72949 0.130420
\(439\) 2.65248 0.126596 0.0632979 0.997995i \(-0.479838\pi\)
0.0632979 + 0.997995i \(0.479838\pi\)
\(440\) 0 0
\(441\) −2.85410 −0.135910
\(442\) 1.23607 0.0587938
\(443\) −1.03444 −0.0491478 −0.0245739 0.999698i \(-0.507823\pi\)
−0.0245739 + 0.999698i \(0.507823\pi\)
\(444\) 0.944272 0.0448132
\(445\) −1.90983 −0.0905346
\(446\) 19.7082 0.933211
\(447\) 0.695048 0.0328747
\(448\) 1.00000 0.0472456
\(449\) −31.4508 −1.48426 −0.742129 0.670257i \(-0.766184\pi\)
−0.742129 + 0.670257i \(0.766184\pi\)
\(450\) −2.85410 −0.134544
\(451\) 0 0
\(452\) 6.32624 0.297561
\(453\) 6.00000 0.281905
\(454\) 9.27051 0.435087
\(455\) −2.00000 −0.0937614
\(456\) 0.0557281 0.00260971
\(457\) 2.27051 0.106210 0.0531050 0.998589i \(-0.483088\pi\)
0.0531050 + 0.998589i \(0.483088\pi\)
\(458\) −29.1246 −1.36090
\(459\) −1.38197 −0.0645046
\(460\) −6.00000 −0.279751
\(461\) −10.9443 −0.509726 −0.254863 0.966977i \(-0.582030\pi\)
−0.254863 + 0.966977i \(0.582030\pi\)
\(462\) 0 0
\(463\) 19.4164 0.902357 0.451178 0.892434i \(-0.351004\pi\)
0.451178 + 0.892434i \(0.351004\pi\)
\(464\) −1.23607 −0.0573830
\(465\) −1.23607 −0.0573213
\(466\) 22.0902 1.02331
\(467\) 17.8885 0.827783 0.413892 0.910326i \(-0.364169\pi\)
0.413892 + 0.910326i \(0.364169\pi\)
\(468\) −5.70820 −0.263862
\(469\) −5.09017 −0.235042
\(470\) −4.76393 −0.219744
\(471\) 5.23607 0.241265
\(472\) −5.38197 −0.247725
\(473\) 0 0
\(474\) 0.180340 0.00828329
\(475\) 0.145898 0.00669426
\(476\) 0.618034 0.0283275
\(477\) 9.23607 0.422891
\(478\) 0 0
\(479\) −2.11146 −0.0964749 −0.0482374 0.998836i \(-0.515360\pi\)
−0.0482374 + 0.998836i \(0.515360\pi\)
\(480\) −0.381966 −0.0174343
\(481\) 4.94427 0.225439
\(482\) 27.2705 1.24214
\(483\) 2.29180 0.104280
\(484\) 0 0
\(485\) −10.5623 −0.479610
\(486\) 9.65248 0.437845
\(487\) 26.8328 1.21591 0.607955 0.793971i \(-0.291990\pi\)
0.607955 + 0.793971i \(0.291990\pi\)
\(488\) −12.4721 −0.564587
\(489\) −1.47214 −0.0665723
\(490\) −1.00000 −0.0451754
\(491\) 5.50658 0.248508 0.124254 0.992250i \(-0.460346\pi\)
0.124254 + 0.992250i \(0.460346\pi\)
\(492\) 2.03444 0.0917197
\(493\) −0.763932 −0.0344058
\(494\) 0.291796 0.0131285
\(495\) 0 0
\(496\) 3.23607 0.145304
\(497\) 3.70820 0.166336
\(498\) −2.03444 −0.0911655
\(499\) −39.7984 −1.78162 −0.890810 0.454376i \(-0.849862\pi\)
−0.890810 + 0.454376i \(0.849862\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −0.944272 −0.0421870
\(502\) −26.8328 −1.19761
\(503\) 23.1246 1.03108 0.515538 0.856867i \(-0.327592\pi\)
0.515538 + 0.856867i \(0.327592\pi\)
\(504\) −2.85410 −0.127132
\(505\) 8.94427 0.398015
\(506\) 0 0
\(507\) −3.43769 −0.152673
\(508\) 15.4164 0.683992
\(509\) −34.4721 −1.52795 −0.763975 0.645246i \(-0.776755\pi\)
−0.763975 + 0.645246i \(0.776755\pi\)
\(510\) −0.236068 −0.0104533
\(511\) 7.14590 0.316116
\(512\) 1.00000 0.0441942
\(513\) −0.326238 −0.0144038
\(514\) 9.43769 0.416279
\(515\) −0.944272 −0.0416096
\(516\) −1.85410 −0.0816223
\(517\) 0 0
\(518\) 2.47214 0.108619
\(519\) 4.58359 0.201197
\(520\) −2.00000 −0.0877058
\(521\) 11.2016 0.490752 0.245376 0.969428i \(-0.421089\pi\)
0.245376 + 0.969428i \(0.421089\pi\)
\(522\) 3.52786 0.154410
\(523\) −40.9787 −1.79187 −0.895937 0.444181i \(-0.853495\pi\)
−0.895937 + 0.444181i \(0.853495\pi\)
\(524\) 16.7984 0.733840
\(525\) 0.381966 0.0166704
\(526\) −2.18034 −0.0950673
\(527\) 2.00000 0.0871214
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 3.23607 0.140566
\(531\) 15.3607 0.666597
\(532\) 0.145898 0.00632548
\(533\) 10.6525 0.461410
\(534\) 0.729490 0.0315681
\(535\) −17.6180 −0.761694
\(536\) −5.09017 −0.219862
\(537\) −3.29180 −0.142051
\(538\) 10.1803 0.438906
\(539\) 0 0
\(540\) 2.23607 0.0962250
\(541\) −38.8328 −1.66955 −0.834777 0.550589i \(-0.814403\pi\)
−0.834777 + 0.550589i \(0.814403\pi\)
\(542\) 17.4164 0.748099
\(543\) 4.76393 0.204440
\(544\) 0.618034 0.0264980
\(545\) −8.47214 −0.362906
\(546\) 0.763932 0.0326933
\(547\) −29.0902 −1.24381 −0.621903 0.783094i \(-0.713640\pi\)
−0.621903 + 0.783094i \(0.713640\pi\)
\(548\) 10.0902 0.431031
\(549\) 35.5967 1.51923
\(550\) 0 0
\(551\) −0.180340 −0.00768274
\(552\) 2.29180 0.0975453
\(553\) 0.472136 0.0200773
\(554\) 19.7082 0.837321
\(555\) −0.944272 −0.0400821
\(556\) 9.52786 0.404071
\(557\) 15.8197 0.670301 0.335150 0.942165i \(-0.391213\pi\)
0.335150 + 0.942165i \(0.391213\pi\)
\(558\) −9.23607 −0.390994
\(559\) −9.70820 −0.410613
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 5.03444 0.212365
\(563\) 41.5066 1.74929 0.874647 0.484761i \(-0.161093\pi\)
0.874647 + 0.484761i \(0.161093\pi\)
\(564\) 1.81966 0.0766215
\(565\) −6.32624 −0.266147
\(566\) −12.0000 −0.504398
\(567\) 7.70820 0.323714
\(568\) 3.70820 0.155593
\(569\) 3.09017 0.129547 0.0647733 0.997900i \(-0.479368\pi\)
0.0647733 + 0.997900i \(0.479368\pi\)
\(570\) −0.0557281 −0.00233419
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) −4.00000 −0.167102
\(574\) 5.32624 0.222313
\(575\) 6.00000 0.250217
\(576\) −2.85410 −0.118921
\(577\) −26.2148 −1.09134 −0.545668 0.838002i \(-0.683724\pi\)
−0.545668 + 0.838002i \(0.683724\pi\)
\(578\) −16.6180 −0.691219
\(579\) −0.763932 −0.0317479
\(580\) 1.23607 0.0513249
\(581\) −5.32624 −0.220970
\(582\) 4.03444 0.167233
\(583\) 0 0
\(584\) 7.14590 0.295699
\(585\) 5.70820 0.236005
\(586\) 7.52786 0.310973
\(587\) −39.7426 −1.64035 −0.820177 0.572109i \(-0.806125\pi\)
−0.820177 + 0.572109i \(0.806125\pi\)
\(588\) 0.381966 0.0157520
\(589\) 0.472136 0.0194540
\(590\) 5.38197 0.221572
\(591\) 2.29180 0.0942719
\(592\) 2.47214 0.101604
\(593\) 0.909830 0.0373622 0.0186811 0.999825i \(-0.494053\pi\)
0.0186811 + 0.999825i \(0.494053\pi\)
\(594\) 0 0
\(595\) −0.618034 −0.0253369
\(596\) 1.81966 0.0745362
\(597\) 2.83282 0.115939
\(598\) 12.0000 0.490716
\(599\) −20.7639 −0.848391 −0.424196 0.905571i \(-0.639443\pi\)
−0.424196 + 0.905571i \(0.639443\pi\)
\(600\) 0.381966 0.0155937
\(601\) −17.9098 −0.730557 −0.365279 0.930898i \(-0.619026\pi\)
−0.365279 + 0.930898i \(0.619026\pi\)
\(602\) −4.85410 −0.197838
\(603\) 14.5279 0.591620
\(604\) 15.7082 0.639158
\(605\) 0 0
\(606\) −3.41641 −0.138782
\(607\) 19.5967 0.795407 0.397704 0.917514i \(-0.369807\pi\)
0.397704 + 0.917514i \(0.369807\pi\)
\(608\) 0.145898 0.00591695
\(609\) −0.472136 −0.0191319
\(610\) 12.4721 0.504982
\(611\) 9.52786 0.385456
\(612\) −1.76393 −0.0713027
\(613\) −10.5836 −0.427467 −0.213734 0.976892i \(-0.568562\pi\)
−0.213734 + 0.976892i \(0.568562\pi\)
\(614\) −7.43769 −0.300161
\(615\) −2.03444 −0.0820366
\(616\) 0 0
\(617\) −6.03444 −0.242937 −0.121469 0.992595i \(-0.538760\pi\)
−0.121469 + 0.992595i \(0.538760\pi\)
\(618\) 0.360680 0.0145087
\(619\) 34.3262 1.37969 0.689844 0.723958i \(-0.257679\pi\)
0.689844 + 0.723958i \(0.257679\pi\)
\(620\) −3.23607 −0.129964
\(621\) −13.4164 −0.538382
\(622\) 11.1246 0.446056
\(623\) 1.90983 0.0765157
\(624\) 0.763932 0.0305818
\(625\) 1.00000 0.0400000
\(626\) 5.85410 0.233977
\(627\) 0 0
\(628\) 13.7082 0.547017
\(629\) 1.52786 0.0609199
\(630\) 2.85410 0.113710
\(631\) −10.2918 −0.409710 −0.204855 0.978792i \(-0.565672\pi\)
−0.204855 + 0.978792i \(0.565672\pi\)
\(632\) 0.472136 0.0187806
\(633\) −8.50658 −0.338106
\(634\) −25.2361 −1.00225
\(635\) −15.4164 −0.611781
\(636\) −1.23607 −0.0490133
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) −10.5836 −0.418680
\(640\) −1.00000 −0.0395285
\(641\) 24.6738 0.974555 0.487278 0.873247i \(-0.337990\pi\)
0.487278 + 0.873247i \(0.337990\pi\)
\(642\) 6.72949 0.265592
\(643\) 31.9787 1.26112 0.630559 0.776142i \(-0.282826\pi\)
0.630559 + 0.776142i \(0.282826\pi\)
\(644\) 6.00000 0.236433
\(645\) 1.85410 0.0730052
\(646\) 0.0901699 0.00354769
\(647\) −22.1803 −0.871999 −0.436000 0.899947i \(-0.643605\pi\)
−0.436000 + 0.899947i \(0.643605\pi\)
\(648\) 7.70820 0.302807
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 1.23607 0.0484453
\(652\) −3.85410 −0.150938
\(653\) −26.8328 −1.05005 −0.525025 0.851087i \(-0.675944\pi\)
−0.525025 + 0.851087i \(0.675944\pi\)
\(654\) 3.23607 0.126540
\(655\) −16.7984 −0.656367
\(656\) 5.32624 0.207955
\(657\) −20.3951 −0.795689
\(658\) 4.76393 0.185717
\(659\) −7.09017 −0.276194 −0.138097 0.990419i \(-0.544099\pi\)
−0.138097 + 0.990419i \(0.544099\pi\)
\(660\) 0 0
\(661\) −10.1115 −0.393290 −0.196645 0.980475i \(-0.563005\pi\)
−0.196645 + 0.980475i \(0.563005\pi\)
\(662\) −23.0344 −0.895259
\(663\) 0.472136 0.0183362
\(664\) −5.32624 −0.206698
\(665\) −0.145898 −0.00565768
\(666\) −7.05573 −0.273404
\(667\) −7.41641 −0.287164
\(668\) −2.47214 −0.0956498
\(669\) 7.52786 0.291044
\(670\) 5.09017 0.196650
\(671\) 0 0
\(672\) 0.381966 0.0147347
\(673\) −32.1459 −1.23913 −0.619567 0.784944i \(-0.712692\pi\)
−0.619567 + 0.784944i \(0.712692\pi\)
\(674\) −13.3820 −0.515454
\(675\) −2.23607 −0.0860663
\(676\) −9.00000 −0.346154
\(677\) 35.5967 1.36809 0.684047 0.729438i \(-0.260218\pi\)
0.684047 + 0.729438i \(0.260218\pi\)
\(678\) 2.41641 0.0928016
\(679\) 10.5623 0.405344
\(680\) −0.618034 −0.0237005
\(681\) 3.54102 0.135692
\(682\) 0 0
\(683\) −48.9443 −1.87280 −0.936400 0.350934i \(-0.885864\pi\)
−0.936400 + 0.350934i \(0.885864\pi\)
\(684\) −0.416408 −0.0159218
\(685\) −10.0902 −0.385526
\(686\) 1.00000 0.0381802
\(687\) −11.1246 −0.424430
\(688\) −4.85410 −0.185061
\(689\) −6.47214 −0.246569
\(690\) −2.29180 −0.0872472
\(691\) 27.8541 1.05962 0.529810 0.848116i \(-0.322263\pi\)
0.529810 + 0.848116i \(0.322263\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) −10.3262 −0.391979
\(695\) −9.52786 −0.361412
\(696\) −0.472136 −0.0178963
\(697\) 3.29180 0.124686
\(698\) 14.4721 0.547778
\(699\) 8.43769 0.319143
\(700\) 1.00000 0.0377964
\(701\) 10.1803 0.384506 0.192253 0.981345i \(-0.438421\pi\)
0.192253 + 0.981345i \(0.438421\pi\)
\(702\) −4.47214 −0.168790
\(703\) 0.360680 0.0136033
\(704\) 0 0
\(705\) −1.81966 −0.0685324
\(706\) 27.4508 1.03313
\(707\) −8.94427 −0.336384
\(708\) −2.05573 −0.0772590
\(709\) −7.05573 −0.264983 −0.132492 0.991184i \(-0.542298\pi\)
−0.132492 + 0.991184i \(0.542298\pi\)
\(710\) −3.70820 −0.139166
\(711\) −1.34752 −0.0505361
\(712\) 1.90983 0.0715739
\(713\) 19.4164 0.727150
\(714\) 0.236068 0.00883462
\(715\) 0 0
\(716\) −8.61803 −0.322071
\(717\) 0 0
\(718\) −12.6525 −0.472186
\(719\) −12.3607 −0.460976 −0.230488 0.973075i \(-0.574032\pi\)
−0.230488 + 0.973075i \(0.574032\pi\)
\(720\) 2.85410 0.106366
\(721\) 0.944272 0.0351665
\(722\) −18.9787 −0.706315
\(723\) 10.4164 0.387390
\(724\) 12.4721 0.463523
\(725\) −1.23607 −0.0459064
\(726\) 0 0
\(727\) −14.8328 −0.550119 −0.275059 0.961427i \(-0.588698\pi\)
−0.275059 + 0.961427i \(0.588698\pi\)
\(728\) 2.00000 0.0741249
\(729\) −19.4377 −0.719915
\(730\) −7.14590 −0.264482
\(731\) −3.00000 −0.110959
\(732\) −4.76393 −0.176080
\(733\) −8.76393 −0.323703 −0.161852 0.986815i \(-0.551747\pi\)
−0.161852 + 0.986815i \(0.551747\pi\)
\(734\) 5.81966 0.214808
\(735\) −0.381966 −0.0140890
\(736\) 6.00000 0.221163
\(737\) 0 0
\(738\) −15.2016 −0.559580
\(739\) −40.3262 −1.48342 −0.741712 0.670718i \(-0.765986\pi\)
−0.741712 + 0.670718i \(0.765986\pi\)
\(740\) −2.47214 −0.0908775
\(741\) 0.111456 0.00409445
\(742\) −3.23607 −0.118800
\(743\) −47.8885 −1.75686 −0.878430 0.477871i \(-0.841409\pi\)
−0.878430 + 0.477871i \(0.841409\pi\)
\(744\) 1.23607 0.0453165
\(745\) −1.81966 −0.0666672
\(746\) −15.5279 −0.568515
\(747\) 15.2016 0.556198
\(748\) 0 0
\(749\) 17.6180 0.643749
\(750\) −0.381966 −0.0139474
\(751\) −1.34752 −0.0491719 −0.0245859 0.999698i \(-0.507827\pi\)
−0.0245859 + 0.999698i \(0.507827\pi\)
\(752\) 4.76393 0.173723
\(753\) −10.2492 −0.373502
\(754\) −2.47214 −0.0900299
\(755\) −15.7082 −0.571680
\(756\) −2.23607 −0.0813250
\(757\) −15.8885 −0.577479 −0.288739 0.957408i \(-0.593236\pi\)
−0.288739 + 0.957408i \(0.593236\pi\)
\(758\) −21.1459 −0.768054
\(759\) 0 0
\(760\) −0.145898 −0.00529228
\(761\) −7.20163 −0.261059 −0.130529 0.991444i \(-0.541668\pi\)
−0.130529 + 0.991444i \(0.541668\pi\)
\(762\) 5.88854 0.213319
\(763\) 8.47214 0.306712
\(764\) −10.4721 −0.378869
\(765\) 1.76393 0.0637751
\(766\) −24.0000 −0.867155
\(767\) −10.7639 −0.388663
\(768\) 0.381966 0.0137830
\(769\) 27.8885 1.00569 0.502843 0.864378i \(-0.332287\pi\)
0.502843 + 0.864378i \(0.332287\pi\)
\(770\) 0 0
\(771\) 3.60488 0.129827
\(772\) −2.00000 −0.0719816
\(773\) 6.65248 0.239273 0.119636 0.992818i \(-0.461827\pi\)
0.119636 + 0.992818i \(0.461827\pi\)
\(774\) 13.8541 0.497975
\(775\) 3.23607 0.116243
\(776\) 10.5623 0.379165
\(777\) 0.944272 0.0338756
\(778\) 20.8328 0.746893
\(779\) 0.777088 0.0278421
\(780\) −0.763932 −0.0273532
\(781\) 0 0
\(782\) 3.70820 0.132605
\(783\) 2.76393 0.0987749
\(784\) 1.00000 0.0357143
\(785\) −13.7082 −0.489267
\(786\) 6.41641 0.228866
\(787\) −36.1033 −1.28694 −0.643472 0.765469i \(-0.722507\pi\)
−0.643472 + 0.765469i \(0.722507\pi\)
\(788\) 6.00000 0.213741
\(789\) −0.832816 −0.0296490
\(790\) −0.472136 −0.0167978
\(791\) 6.32624 0.224935
\(792\) 0 0
\(793\) −24.9443 −0.885797
\(794\) 0.944272 0.0335110
\(795\) 1.23607 0.0438388
\(796\) 7.41641 0.262868
\(797\) 38.3607 1.35880 0.679402 0.733766i \(-0.262239\pi\)
0.679402 + 0.733766i \(0.262239\pi\)
\(798\) 0.0557281 0.00197275
\(799\) 2.94427 0.104161
\(800\) 1.00000 0.0353553
\(801\) −5.45085 −0.192596
\(802\) 11.5066 0.406311
\(803\) 0 0
\(804\) −1.94427 −0.0685692
\(805\) −6.00000 −0.211472
\(806\) 6.47214 0.227971
\(807\) 3.88854 0.136883
\(808\) −8.94427 −0.314658
\(809\) 51.8115 1.82160 0.910798 0.412852i \(-0.135467\pi\)
0.910798 + 0.412852i \(0.135467\pi\)
\(810\) −7.70820 −0.270839
\(811\) 34.9230 1.22631 0.613156 0.789962i \(-0.289900\pi\)
0.613156 + 0.789962i \(0.289900\pi\)
\(812\) −1.23607 −0.0433775
\(813\) 6.65248 0.233313
\(814\) 0 0
\(815\) 3.85410 0.135003
\(816\) 0.236068 0.00826403
\(817\) −0.708204 −0.0247769
\(818\) −7.88854 −0.275816
\(819\) −5.70820 −0.199461
\(820\) −5.32624 −0.186000
\(821\) 48.9443 1.70817 0.854083 0.520136i \(-0.174119\pi\)
0.854083 + 0.520136i \(0.174119\pi\)
\(822\) 3.85410 0.134427
\(823\) 50.8328 1.77192 0.885960 0.463761i \(-0.153500\pi\)
0.885960 + 0.463761i \(0.153500\pi\)
\(824\) 0.944272 0.0328953
\(825\) 0 0
\(826\) −5.38197 −0.187263
\(827\) 1.74265 0.0605977 0.0302989 0.999541i \(-0.490354\pi\)
0.0302989 + 0.999541i \(0.490354\pi\)
\(828\) −17.1246 −0.595121
\(829\) 11.8197 0.410514 0.205257 0.978708i \(-0.434197\pi\)
0.205257 + 0.978708i \(0.434197\pi\)
\(830\) 5.32624 0.184876
\(831\) 7.52786 0.261139
\(832\) 2.00000 0.0693375
\(833\) 0.618034 0.0214136
\(834\) 3.63932 0.126019
\(835\) 2.47214 0.0855518
\(836\) 0 0
\(837\) −7.23607 −0.250115
\(838\) −3.32624 −0.114903
\(839\) −4.47214 −0.154395 −0.0771976 0.997016i \(-0.524597\pi\)
−0.0771976 + 0.997016i \(0.524597\pi\)
\(840\) −0.381966 −0.0131791
\(841\) −27.4721 −0.947315
\(842\) 38.3607 1.32200
\(843\) 1.92299 0.0662311
\(844\) −22.2705 −0.766583
\(845\) 9.00000 0.309609
\(846\) −13.5967 −0.467466
\(847\) 0 0
\(848\) −3.23607 −0.111127
\(849\) −4.58359 −0.157308
\(850\) 0.618034 0.0211984
\(851\) 14.8328 0.508462
\(852\) 1.41641 0.0485253
\(853\) −17.7082 −0.606317 −0.303159 0.952940i \(-0.598041\pi\)
−0.303159 + 0.952940i \(0.598041\pi\)
\(854\) −12.4721 −0.426788
\(855\) 0.416408 0.0142408
\(856\) 17.6180 0.602172
\(857\) 24.2705 0.829065 0.414532 0.910035i \(-0.363945\pi\)
0.414532 + 0.910035i \(0.363945\pi\)
\(858\) 0 0
\(859\) −27.5623 −0.940414 −0.470207 0.882556i \(-0.655821\pi\)
−0.470207 + 0.882556i \(0.655821\pi\)
\(860\) 4.85410 0.165524
\(861\) 2.03444 0.0693336
\(862\) 26.7639 0.911583
\(863\) 11.2361 0.382480 0.191240 0.981543i \(-0.438749\pi\)
0.191240 + 0.981543i \(0.438749\pi\)
\(864\) −2.23607 −0.0760726
\(865\) −12.0000 −0.408012
\(866\) −39.4508 −1.34059
\(867\) −6.34752 −0.215573
\(868\) 3.23607 0.109839
\(869\) 0 0
\(870\) 0.472136 0.0160069
\(871\) −10.1803 −0.344948
\(872\) 8.47214 0.286903
\(873\) −30.1459 −1.02028
\(874\) 0.875388 0.0296104
\(875\) −1.00000 −0.0338062
\(876\) 2.72949 0.0922209
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 2.65248 0.0895167
\(879\) 2.87539 0.0969844
\(880\) 0 0
\(881\) 10.8541 0.365684 0.182842 0.983142i \(-0.441470\pi\)
0.182842 + 0.983142i \(0.441470\pi\)
\(882\) −2.85410 −0.0961026
\(883\) 22.5623 0.759282 0.379641 0.925134i \(-0.376048\pi\)
0.379641 + 0.925134i \(0.376048\pi\)
\(884\) 1.23607 0.0415735
\(885\) 2.05573 0.0691025
\(886\) −1.03444 −0.0347528
\(887\) 40.3607 1.35518 0.677589 0.735440i \(-0.263025\pi\)
0.677589 + 0.735440i \(0.263025\pi\)
\(888\) 0.944272 0.0316877
\(889\) 15.4164 0.517050
\(890\) −1.90983 −0.0640176
\(891\) 0 0
\(892\) 19.7082 0.659879
\(893\) 0.695048 0.0232589
\(894\) 0.695048 0.0232459
\(895\) 8.61803 0.288069
\(896\) 1.00000 0.0334077
\(897\) 4.58359 0.153042
\(898\) −31.4508 −1.04953
\(899\) −4.00000 −0.133407
\(900\) −2.85410 −0.0951367
\(901\) −2.00000 −0.0666297
\(902\) 0 0
\(903\) −1.85410 −0.0617007
\(904\) 6.32624 0.210408
\(905\) −12.4721 −0.414588
\(906\) 6.00000 0.199337
\(907\) −11.2705 −0.374231 −0.187116 0.982338i \(-0.559914\pi\)
−0.187116 + 0.982338i \(0.559914\pi\)
\(908\) 9.27051 0.307653
\(909\) 25.5279 0.846706
\(910\) −2.00000 −0.0662994
\(911\) −2.40325 −0.0796233 −0.0398116 0.999207i \(-0.512676\pi\)
−0.0398116 + 0.999207i \(0.512676\pi\)
\(912\) 0.0557281 0.00184534
\(913\) 0 0
\(914\) 2.27051 0.0751018
\(915\) 4.76393 0.157491
\(916\) −29.1246 −0.962304
\(917\) 16.7984 0.554731
\(918\) −1.38197 −0.0456117
\(919\) 44.8328 1.47890 0.739449 0.673213i \(-0.235086\pi\)
0.739449 + 0.673213i \(0.235086\pi\)
\(920\) −6.00000 −0.197814
\(921\) −2.84095 −0.0936124
\(922\) −10.9443 −0.360430
\(923\) 7.41641 0.244114
\(924\) 0 0
\(925\) 2.47214 0.0812833
\(926\) 19.4164 0.638063
\(927\) −2.69505 −0.0885170
\(928\) −1.23607 −0.0405759
\(929\) −45.7426 −1.50077 −0.750384 0.661002i \(-0.770131\pi\)
−0.750384 + 0.661002i \(0.770131\pi\)
\(930\) −1.23607 −0.0405323
\(931\) 0.145898 0.00478161
\(932\) 22.0902 0.723588
\(933\) 4.24922 0.139113
\(934\) 17.8885 0.585331
\(935\) 0 0
\(936\) −5.70820 −0.186578
\(937\) −40.3951 −1.31965 −0.659826 0.751419i \(-0.729370\pi\)
−0.659826 + 0.751419i \(0.729370\pi\)
\(938\) −5.09017 −0.166200
\(939\) 2.23607 0.0729713
\(940\) −4.76393 −0.155382
\(941\) 11.8197 0.385310 0.192655 0.981267i \(-0.438290\pi\)
0.192655 + 0.981267i \(0.438290\pi\)
\(942\) 5.23607 0.170600
\(943\) 31.9574 1.04068
\(944\) −5.38197 −0.175168
\(945\) 2.23607 0.0727393
\(946\) 0 0
\(947\) −53.0902 −1.72520 −0.862599 0.505888i \(-0.831165\pi\)
−0.862599 + 0.505888i \(0.831165\pi\)
\(948\) 0.180340 0.00585717
\(949\) 14.2918 0.463931
\(950\) 0.145898 0.00473356
\(951\) −9.63932 −0.312576
\(952\) 0.618034 0.0200306
\(953\) −26.4508 −0.856827 −0.428414 0.903583i \(-0.640927\pi\)
−0.428414 + 0.903583i \(0.640927\pi\)
\(954\) 9.23607 0.299029
\(955\) 10.4721 0.338870
\(956\) 0 0
\(957\) 0 0
\(958\) −2.11146 −0.0682181
\(959\) 10.0902 0.325829
\(960\) −0.381966 −0.0123279
\(961\) −20.5279 −0.662189
\(962\) 4.94427 0.159410
\(963\) −50.2837 −1.62037
\(964\) 27.2705 0.878324
\(965\) 2.00000 0.0643823
\(966\) 2.29180 0.0737373
\(967\) −49.0132 −1.57616 −0.788078 0.615575i \(-0.788924\pi\)
−0.788078 + 0.615575i \(0.788924\pi\)
\(968\) 0 0
\(969\) 0.0344419 0.00110643
\(970\) −10.5623 −0.339135
\(971\) 26.8328 0.861106 0.430553 0.902565i \(-0.358319\pi\)
0.430553 + 0.902565i \(0.358319\pi\)
\(972\) 9.65248 0.309603
\(973\) 9.52786 0.305449
\(974\) 26.8328 0.859779
\(975\) 0.763932 0.0244654
\(976\) −12.4721 −0.399223
\(977\) −19.3050 −0.617620 −0.308810 0.951124i \(-0.599931\pi\)
−0.308810 + 0.951124i \(0.599931\pi\)
\(978\) −1.47214 −0.0470737
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −24.1803 −0.772019
\(982\) 5.50658 0.175722
\(983\) 16.6525 0.531131 0.265566 0.964093i \(-0.414441\pi\)
0.265566 + 0.964093i \(0.414441\pi\)
\(984\) 2.03444 0.0648556
\(985\) −6.00000 −0.191176
\(986\) −0.763932 −0.0243286
\(987\) 1.81966 0.0579204
\(988\) 0.291796 0.00928327
\(989\) −29.1246 −0.926109
\(990\) 0 0
\(991\) −10.6525 −0.338387 −0.169194 0.985583i \(-0.554116\pi\)
−0.169194 + 0.985583i \(0.554116\pi\)
\(992\) 3.23607 0.102745
\(993\) −8.79837 −0.279208
\(994\) 3.70820 0.117617
\(995\) −7.41641 −0.235116
\(996\) −2.03444 −0.0644638
\(997\) 17.1246 0.542342 0.271171 0.962531i \(-0.412589\pi\)
0.271171 + 0.962531i \(0.412589\pi\)
\(998\) −39.7984 −1.25980
\(999\) −5.52786 −0.174894
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.cf.1.1 2
11.2 odd 10 770.2.n.b.631.1 yes 4
11.6 odd 10 770.2.n.b.421.1 4
11.10 odd 2 8470.2.a.bt.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.b.421.1 4 11.6 odd 10
770.2.n.b.631.1 yes 4 11.2 odd 10
8470.2.a.bt.1.1 2 11.10 odd 2
8470.2.a.cf.1.1 2 1.1 even 1 trivial