Properties

Label 8470.2.a.ck.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 \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.289169 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.289169 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.91638 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.289169 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.289169 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.91638 q^{9} -1.00000 q^{10} +0.289169 q^{12} +0.289169 q^{13} +1.00000 q^{14} -0.289169 q^{15} +1.00000 q^{16} -0.916382 q^{17} -2.91638 q^{18} -3.20555 q^{19} -1.00000 q^{20} +0.289169 q^{21} -5.33804 q^{23} +0.289169 q^{24} +1.00000 q^{25} +0.289169 q^{26} -1.71083 q^{27} +1.00000 q^{28} +9.83276 q^{29} -0.289169 q^{30} +1.00000 q^{32} -0.916382 q^{34} -1.00000 q^{35} -2.91638 q^{36} -9.25443 q^{37} -3.20555 q^{38} +0.0836184 q^{39} -1.00000 q^{40} -0.578337 q^{41} +0.289169 q^{42} +5.75971 q^{43} +2.91638 q^{45} -5.33804 q^{46} +11.2544 q^{47} +0.289169 q^{48} +1.00000 q^{49} +1.00000 q^{50} -0.264989 q^{51} +0.289169 q^{52} +4.91638 q^{53} -1.71083 q^{54} +1.00000 q^{56} -0.926944 q^{57} +9.83276 q^{58} +7.20555 q^{59} -0.289169 q^{60} +9.49472 q^{61} -2.91638 q^{63} +1.00000 q^{64} -0.289169 q^{65} +10.7491 q^{67} -0.916382 q^{68} -1.54359 q^{69} -1.00000 q^{70} +1.66196 q^{71} -2.91638 q^{72} +10.3380 q^{73} -9.25443 q^{74} +0.289169 q^{75} -3.20555 q^{76} +0.0836184 q^{78} -2.42166 q^{79} -1.00000 q^{80} +8.25443 q^{81} -0.578337 q^{82} +2.28917 q^{83} +0.289169 q^{84} +0.916382 q^{85} +5.75971 q^{86} +2.84333 q^{87} +3.42166 q^{89} +2.91638 q^{90} +0.289169 q^{91} -5.33804 q^{92} +11.2544 q^{94} +3.20555 q^{95} +0.289169 q^{96} -6.33804 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} + 3q^{8} + 5q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} + 3q^{8} + 5q^{9} - 3q^{10} + 3q^{14} + 3q^{16} + 11q^{17} + 5q^{18} + 5q^{19} - 3q^{20} - 4q^{23} + 3q^{25} - 6q^{27} + 3q^{28} + 2q^{29} + 3q^{32} + 11q^{34} - 3q^{35} + 5q^{36} - 2q^{37} + 5q^{38} + 14q^{39} - 3q^{40} + 7q^{43} - 5q^{45} - 4q^{46} + 8q^{47} + 3q^{49} + 3q^{50} - 6q^{51} + q^{53} - 6q^{54} + 3q^{56} - 20q^{57} + 2q^{58} + 7q^{59} + 13q^{61} + 5q^{63} + 3q^{64} - 9q^{67} + 11q^{68} + 22q^{69} - 3q^{70} + 17q^{71} + 5q^{72} + 19q^{73} - 2q^{74} + 5q^{76} + 14q^{78} - 9q^{79} - 3q^{80} - q^{81} + 6q^{83} - 11q^{85} + 7q^{86} + 12q^{87} + 12q^{89} - 5q^{90} - 4q^{92} + 8q^{94} - 5q^{95} - 7q^{97} + 3q^{98} + O(q^{100}) \)

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
\(3\) 0.289169 0.166952 0.0834758 0.996510i \(-0.473398\pi\)
0.0834758 + 0.996510i \(0.473398\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0.289169 0.118053
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −2.91638 −0.972127
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 0.289169 0.0834758
\(13\) 0.289169 0.0802009 0.0401005 0.999196i \(-0.487232\pi\)
0.0401005 + 0.999196i \(0.487232\pi\)
\(14\) 1.00000 0.267261
\(15\) −0.289169 −0.0746630
\(16\) 1.00000 0.250000
\(17\) −0.916382 −0.222255 −0.111128 0.993806i \(-0.535446\pi\)
−0.111128 + 0.993806i \(0.535446\pi\)
\(18\) −2.91638 −0.687398
\(19\) −3.20555 −0.735404 −0.367702 0.929944i \(-0.619855\pi\)
−0.367702 + 0.929944i \(0.619855\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0.289169 0.0631018
\(22\) 0 0
\(23\) −5.33804 −1.11306 −0.556530 0.830828i \(-0.687867\pi\)
−0.556530 + 0.830828i \(0.687867\pi\)
\(24\) 0.289169 0.0590263
\(25\) 1.00000 0.200000
\(26\) 0.289169 0.0567106
\(27\) −1.71083 −0.329250
\(28\) 1.00000 0.188982
\(29\) 9.83276 1.82590 0.912949 0.408073i \(-0.133799\pi\)
0.912949 + 0.408073i \(0.133799\pi\)
\(30\) −0.289169 −0.0527947
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.916382 −0.157158
\(35\) −1.00000 −0.169031
\(36\) −2.91638 −0.486064
\(37\) −9.25443 −1.52142 −0.760709 0.649093i \(-0.775149\pi\)
−0.760709 + 0.649093i \(0.775149\pi\)
\(38\) −3.20555 −0.520009
\(39\) 0.0836184 0.0133897
\(40\) −1.00000 −0.158114
\(41\) −0.578337 −0.0903211 −0.0451605 0.998980i \(-0.514380\pi\)
−0.0451605 + 0.998980i \(0.514380\pi\)
\(42\) 0.289169 0.0446197
\(43\) 5.75971 0.878347 0.439174 0.898402i \(-0.355271\pi\)
0.439174 + 0.898402i \(0.355271\pi\)
\(44\) 0 0
\(45\) 2.91638 0.434748
\(46\) −5.33804 −0.787052
\(47\) 11.2544 1.64163 0.820813 0.571196i \(-0.193521\pi\)
0.820813 + 0.571196i \(0.193521\pi\)
\(48\) 0.289169 0.0417379
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −0.264989 −0.0371058
\(52\) 0.289169 0.0401005
\(53\) 4.91638 0.675317 0.337658 0.941269i \(-0.390365\pi\)
0.337658 + 0.941269i \(0.390365\pi\)
\(54\) −1.71083 −0.232815
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −0.926944 −0.122777
\(58\) 9.83276 1.29111
\(59\) 7.20555 0.938083 0.469041 0.883176i \(-0.344600\pi\)
0.469041 + 0.883176i \(0.344600\pi\)
\(60\) −0.289169 −0.0373315
\(61\) 9.49472 1.21567 0.607837 0.794062i \(-0.292037\pi\)
0.607837 + 0.794062i \(0.292037\pi\)
\(62\) 0 0
\(63\) −2.91638 −0.367430
\(64\) 1.00000 0.125000
\(65\) −0.289169 −0.0358669
\(66\) 0 0
\(67\) 10.7491 1.31322 0.656609 0.754232i \(-0.271990\pi\)
0.656609 + 0.754232i \(0.271990\pi\)
\(68\) −0.916382 −0.111128
\(69\) −1.54359 −0.185827
\(70\) −1.00000 −0.119523
\(71\) 1.66196 0.197238 0.0986189 0.995125i \(-0.468558\pi\)
0.0986189 + 0.995125i \(0.468558\pi\)
\(72\) −2.91638 −0.343699
\(73\) 10.3380 1.20998 0.604988 0.796234i \(-0.293178\pi\)
0.604988 + 0.796234i \(0.293178\pi\)
\(74\) −9.25443 −1.07581
\(75\) 0.289169 0.0333903
\(76\) −3.20555 −0.367702
\(77\) 0 0
\(78\) 0.0836184 0.00946792
\(79\) −2.42166 −0.272458 −0.136229 0.990677i \(-0.543498\pi\)
−0.136229 + 0.990677i \(0.543498\pi\)
\(80\) −1.00000 −0.111803
\(81\) 8.25443 0.917158
\(82\) −0.578337 −0.0638666
\(83\) 2.28917 0.251269 0.125634 0.992077i \(-0.459903\pi\)
0.125634 + 0.992077i \(0.459903\pi\)
\(84\) 0.289169 0.0315509
\(85\) 0.916382 0.0993955
\(86\) 5.75971 0.621085
\(87\) 2.84333 0.304837
\(88\) 0 0
\(89\) 3.42166 0.362696 0.181348 0.983419i \(-0.441954\pi\)
0.181348 + 0.983419i \(0.441954\pi\)
\(90\) 2.91638 0.307414
\(91\) 0.289169 0.0303131
\(92\) −5.33804 −0.556530
\(93\) 0 0
\(94\) 11.2544 1.16081
\(95\) 3.20555 0.328883
\(96\) 0.289169 0.0295131
\(97\) −6.33804 −0.643531 −0.321765 0.946819i \(-0.604276\pi\)
−0.321765 + 0.946819i \(0.604276\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −4.28917 −0.426788 −0.213394 0.976966i \(-0.568452\pi\)
−0.213394 + 0.976966i \(0.568452\pi\)
\(102\) −0.264989 −0.0262378
\(103\) 5.49472 0.541411 0.270705 0.962662i \(-0.412743\pi\)
0.270705 + 0.962662i \(0.412743\pi\)
\(104\) 0.289169 0.0283553
\(105\) −0.289169 −0.0282200
\(106\) 4.91638 0.477521
\(107\) −2.50528 −0.242195 −0.121097 0.992641i \(-0.538641\pi\)
−0.121097 + 0.992641i \(0.538641\pi\)
\(108\) −1.71083 −0.164625
\(109\) 7.25443 0.694848 0.347424 0.937708i \(-0.387056\pi\)
0.347424 + 0.937708i \(0.387056\pi\)
\(110\) 0 0
\(111\) −2.67609 −0.254003
\(112\) 1.00000 0.0944911
\(113\) −7.74914 −0.728978 −0.364489 0.931208i \(-0.618756\pi\)
−0.364489 + 0.931208i \(0.618756\pi\)
\(114\) −0.926944 −0.0868163
\(115\) 5.33804 0.497775
\(116\) 9.83276 0.912949
\(117\) −0.843326 −0.0779655
\(118\) 7.20555 0.663325
\(119\) −0.916382 −0.0840046
\(120\) −0.289169 −0.0263974
\(121\) 0 0
\(122\) 9.49472 0.859611
\(123\) −0.167237 −0.0150792
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) −2.91638 −0.259812
\(127\) 13.3380 1.18356 0.591780 0.806099i \(-0.298425\pi\)
0.591780 + 0.806099i \(0.298425\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.66553 0.146641
\(130\) −0.289169 −0.0253618
\(131\) 11.8086 1.03172 0.515860 0.856673i \(-0.327472\pi\)
0.515860 + 0.856673i \(0.327472\pi\)
\(132\) 0 0
\(133\) −3.20555 −0.277956
\(134\) 10.7491 0.928585
\(135\) 1.71083 0.147245
\(136\) −0.916382 −0.0785791
\(137\) −2.18137 −0.186367 −0.0931835 0.995649i \(-0.529704\pi\)
−0.0931835 + 0.995649i \(0.529704\pi\)
\(138\) −1.54359 −0.131399
\(139\) −16.2927 −1.38193 −0.690966 0.722887i \(-0.742815\pi\)
−0.690966 + 0.722887i \(0.742815\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 3.25443 0.274072
\(142\) 1.66196 0.139468
\(143\) 0 0
\(144\) −2.91638 −0.243032
\(145\) −9.83276 −0.816567
\(146\) 10.3380 0.855583
\(147\) 0.289169 0.0238502
\(148\) −9.25443 −0.760709
\(149\) −1.73501 −0.142138 −0.0710688 0.997471i \(-0.522641\pi\)
−0.0710688 + 0.997471i \(0.522641\pi\)
\(150\) 0.289169 0.0236105
\(151\) 6.18137 0.503033 0.251516 0.967853i \(-0.419071\pi\)
0.251516 + 0.967853i \(0.419071\pi\)
\(152\) −3.20555 −0.260004
\(153\) 2.67252 0.216060
\(154\) 0 0
\(155\) 0 0
\(156\) 0.0836184 0.00669483
\(157\) 5.54359 0.442427 0.221214 0.975225i \(-0.428998\pi\)
0.221214 + 0.975225i \(0.428998\pi\)
\(158\) −2.42166 −0.192657
\(159\) 1.42166 0.112745
\(160\) −1.00000 −0.0790569
\(161\) −5.33804 −0.420697
\(162\) 8.25443 0.648529
\(163\) −21.0141 −1.64595 −0.822977 0.568075i \(-0.807688\pi\)
−0.822977 + 0.568075i \(0.807688\pi\)
\(164\) −0.578337 −0.0451605
\(165\) 0 0
\(166\) 2.28917 0.177674
\(167\) −9.01413 −0.697535 −0.348767 0.937209i \(-0.613400\pi\)
−0.348767 + 0.937209i \(0.613400\pi\)
\(168\) 0.289169 0.0223098
\(169\) −12.9164 −0.993568
\(170\) 0.916382 0.0702833
\(171\) 9.34861 0.714906
\(172\) 5.75971 0.439174
\(173\) −16.9894 −1.29168 −0.645841 0.763472i \(-0.723493\pi\)
−0.645841 + 0.763472i \(0.723493\pi\)
\(174\) 2.84333 0.215552
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 2.08362 0.156614
\(178\) 3.42166 0.256464
\(179\) 5.32391 0.397928 0.198964 0.980007i \(-0.436242\pi\)
0.198964 + 0.980007i \(0.436242\pi\)
\(180\) 2.91638 0.217374
\(181\) 25.4494 1.89164 0.945820 0.324691i \(-0.105260\pi\)
0.945820 + 0.324691i \(0.105260\pi\)
\(182\) 0.289169 0.0214346
\(183\) 2.74557 0.202959
\(184\) −5.33804 −0.393526
\(185\) 9.25443 0.680399
\(186\) 0 0
\(187\) 0 0
\(188\) 11.2544 0.820813
\(189\) −1.71083 −0.124445
\(190\) 3.20555 0.232555
\(191\) 4.08362 0.295480 0.147740 0.989026i \(-0.452800\pi\)
0.147740 + 0.989026i \(0.452800\pi\)
\(192\) 0.289169 0.0208689
\(193\) 19.5819 1.40954 0.704768 0.709438i \(-0.251051\pi\)
0.704768 + 0.709438i \(0.251051\pi\)
\(194\) −6.33804 −0.455045
\(195\) −0.0836184 −0.00598804
\(196\) 1.00000 0.0714286
\(197\) −8.43580 −0.601026 −0.300513 0.953778i \(-0.597158\pi\)
−0.300513 + 0.953778i \(0.597158\pi\)
\(198\) 0 0
\(199\) 10.9894 0.779021 0.389510 0.921022i \(-0.372644\pi\)
0.389510 + 0.921022i \(0.372644\pi\)
\(200\) 1.00000 0.0707107
\(201\) 3.10831 0.219244
\(202\) −4.28917 −0.301785
\(203\) 9.83276 0.690125
\(204\) −0.264989 −0.0185529
\(205\) 0.578337 0.0403928
\(206\) 5.49472 0.382835
\(207\) 15.5678 1.08204
\(208\) 0.289169 0.0200502
\(209\) 0 0
\(210\) −0.289169 −0.0199545
\(211\) 3.66553 0.252345 0.126173 0.992008i \(-0.459731\pi\)
0.126173 + 0.992008i \(0.459731\pi\)
\(212\) 4.91638 0.337658
\(213\) 0.480585 0.0329292
\(214\) −2.50528 −0.171258
\(215\) −5.75971 −0.392809
\(216\) −1.71083 −0.116407
\(217\) 0 0
\(218\) 7.25443 0.491332
\(219\) 2.98944 0.202007
\(220\) 0 0
\(221\) −0.264989 −0.0178251
\(222\) −2.67609 −0.179607
\(223\) 14.0731 0.942402 0.471201 0.882026i \(-0.343821\pi\)
0.471201 + 0.882026i \(0.343821\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.91638 −0.194425
\(226\) −7.74914 −0.515465
\(227\) 3.25443 0.216004 0.108002 0.994151i \(-0.465555\pi\)
0.108002 + 0.994151i \(0.465555\pi\)
\(228\) −0.926944 −0.0613884
\(229\) 15.1814 1.00321 0.501607 0.865096i \(-0.332743\pi\)
0.501607 + 0.865096i \(0.332743\pi\)
\(230\) 5.33804 0.351980
\(231\) 0 0
\(232\) 9.83276 0.645553
\(233\) 3.50528 0.229639 0.114819 0.993386i \(-0.463371\pi\)
0.114819 + 0.993386i \(0.463371\pi\)
\(234\) −0.843326 −0.0551299
\(235\) −11.2544 −0.734158
\(236\) 7.20555 0.469041
\(237\) −0.700269 −0.0454873
\(238\) −0.916382 −0.0594002
\(239\) −26.6408 −1.72325 −0.861626 0.507544i \(-0.830554\pi\)
−0.861626 + 0.507544i \(0.830554\pi\)
\(240\) −0.289169 −0.0186657
\(241\) 3.58890 0.231181 0.115591 0.993297i \(-0.463124\pi\)
0.115591 + 0.993297i \(0.463124\pi\)
\(242\) 0 0
\(243\) 7.51941 0.482371
\(244\) 9.49472 0.607837
\(245\) −1.00000 −0.0638877
\(246\) −0.167237 −0.0106626
\(247\) −0.926944 −0.0589801
\(248\) 0 0
\(249\) 0.661956 0.0419497
\(250\) −1.00000 −0.0632456
\(251\) −10.0731 −0.635806 −0.317903 0.948123i \(-0.602979\pi\)
−0.317903 + 0.948123i \(0.602979\pi\)
\(252\) −2.91638 −0.183715
\(253\) 0 0
\(254\) 13.3380 0.836903
\(255\) 0.264989 0.0165942
\(256\) 1.00000 0.0625000
\(257\) 14.7491 0.920026 0.460013 0.887912i \(-0.347845\pi\)
0.460013 + 0.887912i \(0.347845\pi\)
\(258\) 1.66553 0.103691
\(259\) −9.25443 −0.575042
\(260\) −0.289169 −0.0179335
\(261\) −28.6761 −1.77501
\(262\) 11.8086 0.729537
\(263\) −6.90582 −0.425831 −0.212916 0.977071i \(-0.568296\pi\)
−0.212916 + 0.977071i \(0.568296\pi\)
\(264\) 0 0
\(265\) −4.91638 −0.302011
\(266\) −3.20555 −0.196545
\(267\) 0.989437 0.0605526
\(268\) 10.7491 0.656609
\(269\) 29.0383 1.77050 0.885249 0.465118i \(-0.153988\pi\)
0.885249 + 0.465118i \(0.153988\pi\)
\(270\) 1.71083 0.104118
\(271\) 4.26499 0.259080 0.129540 0.991574i \(-0.458650\pi\)
0.129540 + 0.991574i \(0.458650\pi\)
\(272\) −0.916382 −0.0555638
\(273\) 0.0836184 0.00506082
\(274\) −2.18137 −0.131781
\(275\) 0 0
\(276\) −1.54359 −0.0929135
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −16.2927 −0.977174
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 15.1461 0.903541 0.451771 0.892134i \(-0.350793\pi\)
0.451771 + 0.892134i \(0.350793\pi\)
\(282\) 3.25443 0.193798
\(283\) −18.5330 −1.10167 −0.550837 0.834613i \(-0.685691\pi\)
−0.550837 + 0.834613i \(0.685691\pi\)
\(284\) 1.66196 0.0986189
\(285\) 0.926944 0.0549074
\(286\) 0 0
\(287\) −0.578337 −0.0341382
\(288\) −2.91638 −0.171849
\(289\) −16.1602 −0.950603
\(290\) −9.83276 −0.577400
\(291\) −1.83276 −0.107438
\(292\) 10.3380 0.604988
\(293\) −4.02418 −0.235095 −0.117548 0.993067i \(-0.537503\pi\)
−0.117548 + 0.993067i \(0.537503\pi\)
\(294\) 0.289169 0.0168647
\(295\) −7.20555 −0.419523
\(296\) −9.25443 −0.537903
\(297\) 0 0
\(298\) −1.73501 −0.100507
\(299\) −1.54359 −0.0892684
\(300\) 0.289169 0.0166952
\(301\) 5.75971 0.331984
\(302\) 6.18137 0.355698
\(303\) −1.24029 −0.0712530
\(304\) −3.20555 −0.183851
\(305\) −9.49472 −0.543666
\(306\) 2.67252 0.152778
\(307\) 34.2439 1.95440 0.977200 0.212320i \(-0.0681018\pi\)
0.977200 + 0.212320i \(0.0681018\pi\)
\(308\) 0 0
\(309\) 1.58890 0.0903894
\(310\) 0 0
\(311\) −6.89169 −0.390792 −0.195396 0.980724i \(-0.562599\pi\)
−0.195396 + 0.980724i \(0.562599\pi\)
\(312\) 0.0836184 0.00473396
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 5.54359 0.312843
\(315\) 2.91638 0.164319
\(316\) −2.42166 −0.136229
\(317\) 29.7386 1.67029 0.835143 0.550034i \(-0.185385\pi\)
0.835143 + 0.550034i \(0.185385\pi\)
\(318\) 1.42166 0.0797229
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −0.724449 −0.0404348
\(322\) −5.33804 −0.297478
\(323\) 2.93751 0.163447
\(324\) 8.25443 0.458579
\(325\) 0.289169 0.0160402
\(326\) −21.0141 −1.16387
\(327\) 2.09775 0.116006
\(328\) −0.578337 −0.0319333
\(329\) 11.2544 0.620477
\(330\) 0 0
\(331\) 25.2544 1.38811 0.694054 0.719923i \(-0.255823\pi\)
0.694054 + 0.719923i \(0.255823\pi\)
\(332\) 2.28917 0.125634
\(333\) 26.9894 1.47901
\(334\) −9.01413 −0.493231
\(335\) −10.7491 −0.587289
\(336\) 0.289169 0.0157754
\(337\) −13.9164 −0.758074 −0.379037 0.925382i \(-0.623745\pi\)
−0.379037 + 0.925382i \(0.623745\pi\)
\(338\) −12.9164 −0.702559
\(339\) −2.24081 −0.121704
\(340\) 0.916382 0.0496978
\(341\) 0 0
\(342\) 9.34861 0.505515
\(343\) 1.00000 0.0539949
\(344\) 5.75971 0.310543
\(345\) 1.54359 0.0831043
\(346\) −16.9894 −0.913358
\(347\) 21.0836 1.13183 0.565914 0.824464i \(-0.308523\pi\)
0.565914 + 0.824464i \(0.308523\pi\)
\(348\) 2.84333 0.152418
\(349\) −29.2786 −1.56725 −0.783624 0.621236i \(-0.786631\pi\)
−0.783624 + 0.621236i \(0.786631\pi\)
\(350\) 1.00000 0.0534522
\(351\) −0.494719 −0.0264061
\(352\) 0 0
\(353\) −19.0141 −1.01202 −0.506010 0.862528i \(-0.668880\pi\)
−0.506010 + 0.862528i \(0.668880\pi\)
\(354\) 2.08362 0.110743
\(355\) −1.66196 −0.0882074
\(356\) 3.42166 0.181348
\(357\) −0.264989 −0.0140247
\(358\) 5.32391 0.281377
\(359\) −26.4842 −1.39778 −0.698890 0.715229i \(-0.746322\pi\)
−0.698890 + 0.715229i \(0.746322\pi\)
\(360\) 2.91638 0.153707
\(361\) −8.72445 −0.459182
\(362\) 25.4494 1.33759
\(363\) 0 0
\(364\) 0.289169 0.0151566
\(365\) −10.3380 −0.541118
\(366\) 2.74557 0.143513
\(367\) 13.2297 0.690586 0.345293 0.938495i \(-0.387779\pi\)
0.345293 + 0.938495i \(0.387779\pi\)
\(368\) −5.33804 −0.278265
\(369\) 1.68665 0.0878036
\(370\) 9.25443 0.481115
\(371\) 4.91638 0.255246
\(372\) 0 0
\(373\) 30.8469 1.59719 0.798596 0.601868i \(-0.205577\pi\)
0.798596 + 0.601868i \(0.205577\pi\)
\(374\) 0 0
\(375\) −0.289169 −0.0149326
\(376\) 11.2544 0.580403
\(377\) 2.84333 0.146439
\(378\) −1.71083 −0.0879957
\(379\) −34.0071 −1.74683 −0.873415 0.486977i \(-0.838100\pi\)
−0.873415 + 0.486977i \(0.838100\pi\)
\(380\) 3.20555 0.164441
\(381\) 3.85694 0.197597
\(382\) 4.08362 0.208936
\(383\) 2.09418 0.107008 0.0535038 0.998568i \(-0.482961\pi\)
0.0535038 + 0.998568i \(0.482961\pi\)
\(384\) 0.289169 0.0147566
\(385\) 0 0
\(386\) 19.5819 0.996693
\(387\) −16.7975 −0.853865
\(388\) −6.33804 −0.321765
\(389\) 8.94108 0.453331 0.226665 0.973973i \(-0.427218\pi\)
0.226665 + 0.973973i \(0.427218\pi\)
\(390\) −0.0836184 −0.00423418
\(391\) 4.89169 0.247383
\(392\) 1.00000 0.0505076
\(393\) 3.41467 0.172247
\(394\) −8.43580 −0.424989
\(395\) 2.42166 0.121847
\(396\) 0 0
\(397\) −31.4983 −1.58085 −0.790427 0.612556i \(-0.790141\pi\)
−0.790427 + 0.612556i \(0.790141\pi\)
\(398\) 10.9894 0.550851
\(399\) −0.926944 −0.0464053
\(400\) 1.00000 0.0500000
\(401\) −0.508852 −0.0254109 −0.0127054 0.999919i \(-0.504044\pi\)
−0.0127054 + 0.999919i \(0.504044\pi\)
\(402\) 3.10831 0.155029
\(403\) 0 0
\(404\) −4.28917 −0.213394
\(405\) −8.25443 −0.410166
\(406\) 9.83276 0.487992
\(407\) 0 0
\(408\) −0.264989 −0.0131189
\(409\) −0.745574 −0.0368663 −0.0184331 0.999830i \(-0.505868\pi\)
−0.0184331 + 0.999830i \(0.505868\pi\)
\(410\) 0.578337 0.0285620
\(411\) −0.630784 −0.0311143
\(412\) 5.49472 0.270705
\(413\) 7.20555 0.354562
\(414\) 15.5678 0.765114
\(415\) −2.28917 −0.112371
\(416\) 0.289169 0.0141777
\(417\) −4.71135 −0.230716
\(418\) 0 0
\(419\) 12.6272 0.616880 0.308440 0.951244i \(-0.400193\pi\)
0.308440 + 0.951244i \(0.400193\pi\)
\(420\) −0.289169 −0.0141100
\(421\) 1.35218 0.0659011 0.0329506 0.999457i \(-0.489510\pi\)
0.0329506 + 0.999457i \(0.489510\pi\)
\(422\) 3.66553 0.178435
\(423\) −32.8222 −1.59587
\(424\) 4.91638 0.238761
\(425\) −0.916382 −0.0444510
\(426\) 0.480585 0.0232844
\(427\) 9.49472 0.459482
\(428\) −2.50528 −0.121097
\(429\) 0 0
\(430\) −5.75971 −0.277758
\(431\) −7.33447 −0.353289 −0.176645 0.984275i \(-0.556524\pi\)
−0.176645 + 0.984275i \(0.556524\pi\)
\(432\) −1.71083 −0.0823124
\(433\) 30.6066 1.47086 0.735430 0.677601i \(-0.236980\pi\)
0.735430 + 0.677601i \(0.236980\pi\)
\(434\) 0 0
\(435\) −2.84333 −0.136327
\(436\) 7.25443 0.347424
\(437\) 17.1114 0.818548
\(438\) 2.98944 0.142841
\(439\) −36.5855 −1.74613 −0.873065 0.487604i \(-0.837871\pi\)
−0.873065 + 0.487604i \(0.837871\pi\)
\(440\) 0 0
\(441\) −2.91638 −0.138875
\(442\) −0.264989 −0.0126042
\(443\) 40.0766 1.90410 0.952049 0.305946i \(-0.0989726\pi\)
0.952049 + 0.305946i \(0.0989726\pi\)
\(444\) −2.67609 −0.127002
\(445\) −3.42166 −0.162202
\(446\) 14.0731 0.666379
\(447\) −0.501711 −0.0237301
\(448\) 1.00000 0.0472456
\(449\) 12.6655 0.597723 0.298862 0.954296i \(-0.403393\pi\)
0.298862 + 0.954296i \(0.403393\pi\)
\(450\) −2.91638 −0.137480
\(451\) 0 0
\(452\) −7.74914 −0.364489
\(453\) 1.78746 0.0839821
\(454\) 3.25443 0.152738
\(455\) −0.289169 −0.0135564
\(456\) −0.926944 −0.0434081
\(457\) −6.39697 −0.299237 −0.149619 0.988744i \(-0.547805\pi\)
−0.149619 + 0.988744i \(0.547805\pi\)
\(458\) 15.1814 0.709379
\(459\) 1.56777 0.0731774
\(460\) 5.33804 0.248888
\(461\) 27.4736 1.27957 0.639786 0.768553i \(-0.279023\pi\)
0.639786 + 0.768553i \(0.279023\pi\)
\(462\) 0 0
\(463\) 30.1119 1.39942 0.699709 0.714428i \(-0.253313\pi\)
0.699709 + 0.714428i \(0.253313\pi\)
\(464\) 9.83276 0.456475
\(465\) 0 0
\(466\) 3.50528 0.162379
\(467\) 37.5719 1.73862 0.869309 0.494269i \(-0.164564\pi\)
0.869309 + 0.494269i \(0.164564\pi\)
\(468\) −0.843326 −0.0389827
\(469\) 10.7491 0.496349
\(470\) −11.2544 −0.519128
\(471\) 1.60303 0.0738639
\(472\) 7.20555 0.331662
\(473\) 0 0
\(474\) −0.700269 −0.0321644
\(475\) −3.20555 −0.147081
\(476\) −0.916382 −0.0420023
\(477\) −14.3380 −0.656494
\(478\) −26.6408 −1.21852
\(479\) 20.3627 0.930397 0.465199 0.885206i \(-0.345983\pi\)
0.465199 + 0.885206i \(0.345983\pi\)
\(480\) −0.289169 −0.0131987
\(481\) −2.67609 −0.122019
\(482\) 3.58890 0.163470
\(483\) −1.54359 −0.0702360
\(484\) 0 0
\(485\) 6.33804 0.287796
\(486\) 7.51941 0.341088
\(487\) −6.18137 −0.280105 −0.140052 0.990144i \(-0.544727\pi\)
−0.140052 + 0.990144i \(0.544727\pi\)
\(488\) 9.49472 0.429806
\(489\) −6.07663 −0.274795
\(490\) −1.00000 −0.0451754
\(491\) −11.5889 −0.523000 −0.261500 0.965204i \(-0.584217\pi\)
−0.261500 + 0.965204i \(0.584217\pi\)
\(492\) −0.167237 −0.00753962
\(493\) −9.01056 −0.405815
\(494\) −0.926944 −0.0417052
\(495\) 0 0
\(496\) 0 0
\(497\) 1.66196 0.0745489
\(498\) 0.661956 0.0296629
\(499\) 10.6761 0.477927 0.238964 0.971029i \(-0.423192\pi\)
0.238964 + 0.971029i \(0.423192\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −2.60660 −0.116454
\(502\) −10.0731 −0.449582
\(503\) 6.02470 0.268628 0.134314 0.990939i \(-0.457117\pi\)
0.134314 + 0.990939i \(0.457117\pi\)
\(504\) −2.91638 −0.129906
\(505\) 4.28917 0.190865
\(506\) 0 0
\(507\) −3.73501 −0.165878
\(508\) 13.3380 0.591780
\(509\) 14.1255 0.626102 0.313051 0.949736i \(-0.398649\pi\)
0.313051 + 0.949736i \(0.398649\pi\)
\(510\) 0.264989 0.0117339
\(511\) 10.3380 0.457328
\(512\) 1.00000 0.0441942
\(513\) 5.48416 0.242131
\(514\) 14.7491 0.650557
\(515\) −5.49472 −0.242126
\(516\) 1.66553 0.0733207
\(517\) 0 0
\(518\) −9.25443 −0.406616
\(519\) −4.91281 −0.215648
\(520\) −0.289169 −0.0126809
\(521\) 20.6761 0.905836 0.452918 0.891552i \(-0.350383\pi\)
0.452918 + 0.891552i \(0.350383\pi\)
\(522\) −28.6761 −1.25512
\(523\) −19.0630 −0.833567 −0.416784 0.909006i \(-0.636843\pi\)
−0.416784 + 0.909006i \(0.636843\pi\)
\(524\) 11.8086 0.515860
\(525\) 0.289169 0.0126204
\(526\) −6.90582 −0.301108
\(527\) 0 0
\(528\) 0 0
\(529\) 5.49472 0.238901
\(530\) −4.91638 −0.213554
\(531\) −21.0141 −0.911936
\(532\) −3.20555 −0.138978
\(533\) −0.167237 −0.00724383
\(534\) 0.989437 0.0428171
\(535\) 2.50528 0.108313
\(536\) 10.7491 0.464292
\(537\) 1.53951 0.0664347
\(538\) 29.0383 1.25193
\(539\) 0 0
\(540\) 1.71083 0.0736225
\(541\) −23.2333 −0.998878 −0.499439 0.866349i \(-0.666460\pi\)
−0.499439 + 0.866349i \(0.666460\pi\)
\(542\) 4.26499 0.183197
\(543\) 7.35917 0.315812
\(544\) −0.916382 −0.0392895
\(545\) −7.25443 −0.310745
\(546\) 0.0836184 0.00357854
\(547\) −11.8575 −0.506988 −0.253494 0.967337i \(-0.581580\pi\)
−0.253494 + 0.967337i \(0.581580\pi\)
\(548\) −2.18137 −0.0931835
\(549\) −27.6902 −1.18179
\(550\) 0 0
\(551\) −31.5194 −1.34277
\(552\) −1.54359 −0.0656997
\(553\) −2.42166 −0.102980
\(554\) −10.0000 −0.424859
\(555\) 2.67609 0.113594
\(556\) −16.2927 −0.690966
\(557\) −6.16724 −0.261314 −0.130657 0.991428i \(-0.541709\pi\)
−0.130657 + 0.991428i \(0.541709\pi\)
\(558\) 0 0
\(559\) 1.66553 0.0704443
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 15.1461 0.638900
\(563\) −9.44584 −0.398095 −0.199047 0.979990i \(-0.563785\pi\)
−0.199047 + 0.979990i \(0.563785\pi\)
\(564\) 3.25443 0.137036
\(565\) 7.74914 0.326009
\(566\) −18.5330 −0.779001
\(567\) 8.25443 0.346653
\(568\) 1.66196 0.0697341
\(569\) −33.0978 −1.38753 −0.693765 0.720201i \(-0.744049\pi\)
−0.693765 + 0.720201i \(0.744049\pi\)
\(570\) 0.926944 0.0388254
\(571\) −11.4005 −0.477098 −0.238549 0.971131i \(-0.576672\pi\)
−0.238549 + 0.971131i \(0.576672\pi\)
\(572\) 0 0
\(573\) 1.18085 0.0493309
\(574\) −0.578337 −0.0241393
\(575\) −5.33804 −0.222612
\(576\) −2.91638 −0.121516
\(577\) −7.73858 −0.322161 −0.161081 0.986941i \(-0.551498\pi\)
−0.161081 + 0.986941i \(0.551498\pi\)
\(578\) −16.1602 −0.672178
\(579\) 5.66247 0.235324
\(580\) −9.83276 −0.408283
\(581\) 2.28917 0.0949707
\(582\) −1.83276 −0.0759705
\(583\) 0 0
\(584\) 10.3380 0.427791
\(585\) 0.843326 0.0348672
\(586\) −4.02418 −0.166237
\(587\) 5.61308 0.231677 0.115838 0.993268i \(-0.463045\pi\)
0.115838 + 0.993268i \(0.463045\pi\)
\(588\) 0.289169 0.0119251
\(589\) 0 0
\(590\) −7.20555 −0.296648
\(591\) −2.43937 −0.100342
\(592\) −9.25443 −0.380355
\(593\) 40.3169 1.65562 0.827809 0.561010i \(-0.189587\pi\)
0.827809 + 0.561010i \(0.189587\pi\)
\(594\) 0 0
\(595\) 0.916382 0.0375680
\(596\) −1.73501 −0.0710688
\(597\) 3.17780 0.130059
\(598\) −1.54359 −0.0631223
\(599\) 19.2439 0.786283 0.393141 0.919478i \(-0.371388\pi\)
0.393141 + 0.919478i \(0.371388\pi\)
\(600\) 0.289169 0.0118053
\(601\) −31.6655 −1.29166 −0.645832 0.763480i \(-0.723489\pi\)
−0.645832 + 0.763480i \(0.723489\pi\)
\(602\) 5.75971 0.234748
\(603\) −31.3486 −1.27661
\(604\) 6.18137 0.251516
\(605\) 0 0
\(606\) −1.24029 −0.0503834
\(607\) −29.0141 −1.17765 −0.588824 0.808262i \(-0.700409\pi\)
−0.588824 + 0.808262i \(0.700409\pi\)
\(608\) −3.20555 −0.130002
\(609\) 2.84333 0.115217
\(610\) −9.49472 −0.384430
\(611\) 3.25443 0.131660
\(612\) 2.67252 0.108030
\(613\) −2.91638 −0.117792 −0.0588958 0.998264i \(-0.518758\pi\)
−0.0588958 + 0.998264i \(0.518758\pi\)
\(614\) 34.2439 1.38197
\(615\) 0.167237 0.00674364
\(616\) 0 0
\(617\) 14.1955 0.571489 0.285745 0.958306i \(-0.407759\pi\)
0.285745 + 0.958306i \(0.407759\pi\)
\(618\) 1.58890 0.0639149
\(619\) −40.2680 −1.61851 −0.809255 0.587458i \(-0.800129\pi\)
−0.809255 + 0.587458i \(0.800129\pi\)
\(620\) 0 0
\(621\) 9.13249 0.366474
\(622\) −6.89169 −0.276331
\(623\) 3.42166 0.137086
\(624\) 0.0836184 0.00334742
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) 5.54359 0.221214
\(629\) 8.48059 0.338143
\(630\) 2.91638 0.116191
\(631\) −10.9647 −0.436499 −0.218250 0.975893i \(-0.570035\pi\)
−0.218250 + 0.975893i \(0.570035\pi\)
\(632\) −2.42166 −0.0963286
\(633\) 1.05995 0.0421294
\(634\) 29.7386 1.18107
\(635\) −13.3380 −0.529304
\(636\) 1.42166 0.0563726
\(637\) 0.289169 0.0114573
\(638\) 0 0
\(639\) −4.84690 −0.191740
\(640\) −1.00000 −0.0395285
\(641\) 1.57834 0.0623406 0.0311703 0.999514i \(-0.490077\pi\)
0.0311703 + 0.999514i \(0.490077\pi\)
\(642\) −0.724449 −0.0285917
\(643\) −24.7456 −0.975870 −0.487935 0.872880i \(-0.662250\pi\)
−0.487935 + 0.872880i \(0.662250\pi\)
\(644\) −5.33804 −0.210348
\(645\) −1.66553 −0.0655800
\(646\) 2.93751 0.115575
\(647\) −18.1013 −0.711636 −0.355818 0.934555i \(-0.615798\pi\)
−0.355818 + 0.934555i \(0.615798\pi\)
\(648\) 8.25443 0.324264
\(649\) 0 0
\(650\) 0.289169 0.0113421
\(651\) 0 0
\(652\) −21.0141 −0.822977
\(653\) −45.0908 −1.76454 −0.882269 0.470746i \(-0.843985\pi\)
−0.882269 + 0.470746i \(0.843985\pi\)
\(654\) 2.09775 0.0820286
\(655\) −11.8086 −0.461400
\(656\) −0.578337 −0.0225803
\(657\) −30.1497 −1.17625
\(658\) 11.2544 0.438743
\(659\) −6.31335 −0.245933 −0.122967 0.992411i \(-0.539241\pi\)
−0.122967 + 0.992411i \(0.539241\pi\)
\(660\) 0 0
\(661\) 46.2680 1.79962 0.899809 0.436284i \(-0.143706\pi\)
0.899809 + 0.436284i \(0.143706\pi\)
\(662\) 25.2544 0.981541
\(663\) −0.0766264 −0.00297592
\(664\) 2.28917 0.0888370
\(665\) 3.20555 0.124306
\(666\) 26.9894 1.04582
\(667\) −52.4877 −2.03233
\(668\) −9.01413 −0.348767
\(669\) 4.06949 0.157335
\(670\) −10.7491 −0.415276
\(671\) 0 0
\(672\) 0.289169 0.0111549
\(673\) −6.01413 −0.231828 −0.115914 0.993259i \(-0.536980\pi\)
−0.115914 + 0.993259i \(0.536980\pi\)
\(674\) −13.9164 −0.536039
\(675\) −1.71083 −0.0658499
\(676\) −12.9164 −0.496784
\(677\) −3.03474 −0.116635 −0.0583173 0.998298i \(-0.518574\pi\)
−0.0583173 + 0.998298i \(0.518574\pi\)
\(678\) −2.24081 −0.0860577
\(679\) −6.33804 −0.243232
\(680\) 0.916382 0.0351416
\(681\) 0.941078 0.0360622
\(682\) 0 0
\(683\) −33.7386 −1.29097 −0.645485 0.763773i \(-0.723345\pi\)
−0.645485 + 0.763773i \(0.723345\pi\)
\(684\) 9.34861 0.357453
\(685\) 2.18137 0.0833459
\(686\) 1.00000 0.0381802
\(687\) 4.38997 0.167488
\(688\) 5.75971 0.219587
\(689\) 1.42166 0.0541610
\(690\) 1.54359 0.0587636
\(691\) 15.2791 0.581245 0.290623 0.956838i \(-0.406138\pi\)
0.290623 + 0.956838i \(0.406138\pi\)
\(692\) −16.9894 −0.645841
\(693\) 0 0
\(694\) 21.0836 0.800323
\(695\) 16.2927 0.618019
\(696\) 2.84333 0.107776
\(697\) 0.529977 0.0200743
\(698\) −29.2786 −1.10821
\(699\) 1.01362 0.0383385
\(700\) 1.00000 0.0377964
\(701\) 16.4111 0.619839 0.309919 0.950763i \(-0.399698\pi\)
0.309919 + 0.950763i \(0.399698\pi\)
\(702\) −0.494719 −0.0186720
\(703\) 29.6655 1.11886
\(704\) 0 0
\(705\) −3.25443 −0.122569
\(706\) −19.0141 −0.715606
\(707\) −4.28917 −0.161311
\(708\) 2.08362 0.0783072
\(709\) 23.3522 0.877009 0.438505 0.898729i \(-0.355508\pi\)
0.438505 + 0.898729i \(0.355508\pi\)
\(710\) −1.66196 −0.0623721
\(711\) 7.06249 0.264864
\(712\) 3.42166 0.128232
\(713\) 0 0
\(714\) −0.264989 −0.00991695
\(715\) 0 0
\(716\) 5.32391 0.198964
\(717\) −7.70369 −0.287700
\(718\) −26.4842 −0.988380
\(719\) 28.3416 1.05696 0.528482 0.848945i \(-0.322761\pi\)
0.528482 + 0.848945i \(0.322761\pi\)
\(720\) 2.91638 0.108687
\(721\) 5.49472 0.204634
\(722\) −8.72445 −0.324690
\(723\) 1.03780 0.0385961
\(724\) 25.4494 0.945820
\(725\) 9.83276 0.365180
\(726\) 0 0
\(727\) −0.411100 −0.0152469 −0.00762343 0.999971i \(-0.502427\pi\)
−0.00762343 + 0.999971i \(0.502427\pi\)
\(728\) 0.289169 0.0107173
\(729\) −22.5889 −0.836626
\(730\) −10.3380 −0.382628
\(731\) −5.27809 −0.195217
\(732\) 2.74557 0.101479
\(733\) 34.1502 1.26137 0.630683 0.776040i \(-0.282775\pi\)
0.630683 + 0.776040i \(0.282775\pi\)
\(734\) 13.2297 0.488318
\(735\) −0.289169 −0.0106661
\(736\) −5.33804 −0.196763
\(737\) 0 0
\(738\) 1.68665 0.0620865
\(739\) −42.9200 −1.57884 −0.789418 0.613856i \(-0.789618\pi\)
−0.789418 + 0.613856i \(0.789618\pi\)
\(740\) 9.25443 0.340199
\(741\) −0.268043 −0.00984681
\(742\) 4.91638 0.180486
\(743\) 10.9894 0.403163 0.201582 0.979472i \(-0.435392\pi\)
0.201582 + 0.979472i \(0.435392\pi\)
\(744\) 0 0
\(745\) 1.73501 0.0635659
\(746\) 30.8469 1.12939
\(747\) −6.67609 −0.244265
\(748\) 0 0
\(749\) −2.50528 −0.0915410
\(750\) −0.289169 −0.0105589
\(751\) 23.0036 0.839412 0.419706 0.907660i \(-0.362133\pi\)
0.419706 + 0.907660i \(0.362133\pi\)
\(752\) 11.2544 0.410407
\(753\) −2.91281 −0.106149
\(754\) 2.84333 0.103548
\(755\) −6.18137 −0.224963
\(756\) −1.71083 −0.0622223
\(757\) −29.0836 −1.05706 −0.528531 0.848914i \(-0.677257\pi\)
−0.528531 + 0.848914i \(0.677257\pi\)
\(758\) −34.0071 −1.23519
\(759\) 0 0
\(760\) 3.20555 0.116278
\(761\) 1.90225 0.0689564 0.0344782 0.999405i \(-0.489023\pi\)
0.0344782 + 0.999405i \(0.489023\pi\)
\(762\) 3.85694 0.139722
\(763\) 7.25443 0.262628
\(764\) 4.08362 0.147740
\(765\) −2.67252 −0.0966251
\(766\) 2.09418 0.0756658
\(767\) 2.08362 0.0752351
\(768\) 0.289169 0.0104345
\(769\) 9.05892 0.326673 0.163337 0.986570i \(-0.447774\pi\)
0.163337 + 0.986570i \(0.447774\pi\)
\(770\) 0 0
\(771\) 4.26499 0.153600
\(772\) 19.5819 0.704768
\(773\) 5.49523 0.197650 0.0988249 0.995105i \(-0.468492\pi\)
0.0988249 + 0.995105i \(0.468492\pi\)
\(774\) −16.7975 −0.603774
\(775\) 0 0
\(776\) −6.33804 −0.227523
\(777\) −2.67609 −0.0960042
\(778\) 8.94108 0.320553
\(779\) 1.85389 0.0664224
\(780\) −0.0836184 −0.00299402
\(781\) 0 0
\(782\) 4.89169 0.174926
\(783\) −16.8222 −0.601176
\(784\) 1.00000 0.0357143
\(785\) −5.54359 −0.197859
\(786\) 3.41467 0.121797
\(787\) 1.42166 0.0506768 0.0253384 0.999679i \(-0.491934\pi\)
0.0253384 + 0.999679i \(0.491934\pi\)
\(788\) −8.43580 −0.300513
\(789\) −1.99695 −0.0710931
\(790\) 2.42166 0.0861589
\(791\) −7.74914 −0.275528
\(792\) 0 0
\(793\) 2.74557 0.0974982
\(794\) −31.4983 −1.11783
\(795\) −1.42166 −0.0504212
\(796\) 10.9894 0.389510
\(797\) −14.0524 −0.497763 −0.248882 0.968534i \(-0.580063\pi\)
−0.248882 + 0.968534i \(0.580063\pi\)
\(798\) −0.926944 −0.0328135
\(799\) −10.3133 −0.364860
\(800\) 1.00000 0.0353553
\(801\) −9.97887 −0.352586
\(802\) −0.508852 −0.0179682
\(803\) 0 0
\(804\) 3.10831 0.109622
\(805\) 5.33804 0.188141
\(806\) 0 0
\(807\) 8.39697 0.295587
\(808\) −4.28917 −0.150892
\(809\) −17.8363 −0.627092 −0.313546 0.949573i \(-0.601517\pi\)
−0.313546 + 0.949573i \(0.601517\pi\)
\(810\) −8.25443 −0.290031
\(811\) 45.4288 1.59522 0.797611 0.603173i \(-0.206097\pi\)
0.797611 + 0.603173i \(0.206097\pi\)
\(812\) 9.83276 0.345062
\(813\) 1.23330 0.0432537
\(814\) 0 0
\(815\) 21.0141 0.736093
\(816\) −0.264989 −0.00927646
\(817\) −18.4630 −0.645940
\(818\) −0.745574 −0.0260684
\(819\) −0.843326 −0.0294682
\(820\) 0.578337 0.0201964
\(821\) −53.4005 −1.86369 −0.931846 0.362855i \(-0.881802\pi\)
−0.931846 + 0.362855i \(0.881802\pi\)
\(822\) −0.630784 −0.0220011
\(823\) −30.1672 −1.05156 −0.525782 0.850619i \(-0.676227\pi\)
−0.525782 + 0.850619i \(0.676227\pi\)
\(824\) 5.49472 0.191418
\(825\) 0 0
\(826\) 7.20555 0.250713
\(827\) −3.54411 −0.123241 −0.0616204 0.998100i \(-0.519627\pi\)
−0.0616204 + 0.998100i \(0.519627\pi\)
\(828\) 15.5678 0.541018
\(829\) 34.2680 1.19018 0.595089 0.803660i \(-0.297117\pi\)
0.595089 + 0.803660i \(0.297117\pi\)
\(830\) −2.28917 −0.0794582
\(831\) −2.89169 −0.100311
\(832\) 0.289169 0.0100251
\(833\) −0.916382 −0.0317507
\(834\) −4.71135 −0.163141
\(835\) 9.01413 0.311947
\(836\) 0 0
\(837\) 0 0
\(838\) 12.6272 0.436200
\(839\) 36.7244 1.26787 0.633934 0.773387i \(-0.281439\pi\)
0.633934 + 0.773387i \(0.281439\pi\)
\(840\) −0.289169 −0.00997726
\(841\) 67.6832 2.33390
\(842\) 1.35218 0.0465991
\(843\) 4.37978 0.150848
\(844\) 3.66553 0.126173
\(845\) 12.9164 0.444337
\(846\) −32.8222 −1.12845
\(847\) 0 0
\(848\) 4.91638 0.168829
\(849\) −5.35917 −0.183926
\(850\) −0.916382 −0.0314316
\(851\) 49.4005 1.69343
\(852\) 0.480585 0.0164646
\(853\) −34.5542 −1.18311 −0.591556 0.806264i \(-0.701486\pi\)
−0.591556 + 0.806264i \(0.701486\pi\)
\(854\) 9.49472 0.324903
\(855\) −9.34861 −0.319716
\(856\) −2.50528 −0.0856288
\(857\) 14.5783 0.497987 0.248993 0.968505i \(-0.419900\pi\)
0.248993 + 0.968505i \(0.419900\pi\)
\(858\) 0 0
\(859\) −17.0625 −0.582165 −0.291082 0.956698i \(-0.594015\pi\)
−0.291082 + 0.956698i \(0.594015\pi\)
\(860\) −5.75971 −0.196404
\(861\) −0.167237 −0.00569942
\(862\) −7.33447 −0.249813
\(863\) −8.48059 −0.288682 −0.144341 0.989528i \(-0.546106\pi\)
−0.144341 + 0.989528i \(0.546106\pi\)
\(864\) −1.71083 −0.0582037
\(865\) 16.9894 0.577658
\(866\) 30.6066 1.04005
\(867\) −4.67303 −0.158705
\(868\) 0 0
\(869\) 0 0
\(870\) −2.84333 −0.0963978
\(871\) 3.10831 0.105321
\(872\) 7.25443 0.245666
\(873\) 18.4842 0.625594
\(874\) 17.1114 0.578801
\(875\) −1.00000 −0.0338062
\(876\) 2.98944 0.101004
\(877\) 19.8575 0.670539 0.335269 0.942122i \(-0.391173\pi\)
0.335269 + 0.942122i \(0.391173\pi\)
\(878\) −36.5855 −1.23470
\(879\) −1.16367 −0.0392495
\(880\) 0 0
\(881\) 41.1849 1.38756 0.693778 0.720189i \(-0.255945\pi\)
0.693778 + 0.720189i \(0.255945\pi\)
\(882\) −2.91638 −0.0981997
\(883\) 6.24029 0.210003 0.105001 0.994472i \(-0.466515\pi\)
0.105001 + 0.994472i \(0.466515\pi\)
\(884\) −0.264989 −0.00891254
\(885\) −2.08362 −0.0700401
\(886\) 40.0766 1.34640
\(887\) −33.8116 −1.13528 −0.567642 0.823276i \(-0.692144\pi\)
−0.567642 + 0.823276i \(0.692144\pi\)
\(888\) −2.67609 −0.0898037
\(889\) 13.3380 0.447344
\(890\) −3.42166 −0.114694
\(891\) 0 0
\(892\) 14.0731 0.471201
\(893\) −36.0766 −1.20726
\(894\) −0.501711 −0.0167797
\(895\) −5.32391 −0.177959
\(896\) 1.00000 0.0334077
\(897\) −0.446359 −0.0149035
\(898\) 12.6655 0.422654
\(899\) 0 0
\(900\) −2.91638 −0.0972127
\(901\) −4.50528 −0.150093
\(902\) 0 0
\(903\) 1.66553 0.0554252
\(904\) −7.74914 −0.257733
\(905\) −25.4494 −0.845967
\(906\) 1.78746 0.0593843
\(907\) −23.6408 −0.784981 −0.392490 0.919756i \(-0.628386\pi\)
−0.392490 + 0.919756i \(0.628386\pi\)
\(908\) 3.25443 0.108002
\(909\) 12.5089 0.414892
\(910\) −0.289169 −0.00958584
\(911\) −41.8222 −1.38563 −0.692816 0.721115i \(-0.743630\pi\)
−0.692816 + 0.721115i \(0.743630\pi\)
\(912\) −0.926944 −0.0306942
\(913\) 0 0
\(914\) −6.39697 −0.211593
\(915\) −2.74557 −0.0907659
\(916\) 15.1814 0.501607
\(917\) 11.8086 0.389954
\(918\) 1.56777 0.0517443
\(919\) −45.9789 −1.51670 −0.758352 0.651846i \(-0.773995\pi\)
−0.758352 + 0.651846i \(0.773995\pi\)
\(920\) 5.33804 0.175990
\(921\) 9.90225 0.326290
\(922\) 27.4736 0.904795
\(923\) 0.480585 0.0158187
\(924\) 0 0
\(925\) −9.25443 −0.304284
\(926\) 30.1119 0.989538
\(927\) −16.0247 −0.526320
\(928\) 9.83276 0.322776
\(929\) −39.5266 −1.29682 −0.648412 0.761290i \(-0.724567\pi\)
−0.648412 + 0.761290i \(0.724567\pi\)
\(930\) 0 0
\(931\) −3.20555 −0.105058
\(932\) 3.50528 0.114819
\(933\) −1.99286 −0.0652433
\(934\) 37.5719 1.22939
\(935\) 0 0
\(936\) −0.843326 −0.0275650
\(937\) 39.0661 1.27623 0.638116 0.769940i \(-0.279714\pi\)
0.638116 + 0.769940i \(0.279714\pi\)
\(938\) 10.7491 0.350972
\(939\) −2.89169 −0.0943666
\(940\) −11.2544 −0.367079
\(941\) 47.8152 1.55873 0.779366 0.626569i \(-0.215542\pi\)
0.779366 + 0.626569i \(0.215542\pi\)
\(942\) 1.60303 0.0522296
\(943\) 3.08719 0.100533
\(944\) 7.20555 0.234521
\(945\) 1.71083 0.0556534
\(946\) 0 0
\(947\) 19.7844 0.642907 0.321453 0.946925i \(-0.395829\pi\)
0.321453 + 0.946925i \(0.395829\pi\)
\(948\) −0.700269 −0.0227437
\(949\) 2.98944 0.0970412
\(950\) −3.20555 −0.104002
\(951\) 8.59946 0.278857
\(952\) −0.916382 −0.0297001
\(953\) 42.6902 1.38287 0.691436 0.722438i \(-0.256978\pi\)
0.691436 + 0.722438i \(0.256978\pi\)
\(954\) −14.3380 −0.464211
\(955\) −4.08362 −0.132143
\(956\) −26.6408 −0.861626
\(957\) 0 0
\(958\) 20.3627 0.657890
\(959\) −2.18137 −0.0704401
\(960\) −0.289169 −0.00933287
\(961\) −31.0000 −1.00000
\(962\) −2.67609 −0.0862806
\(963\) 7.30636 0.235444
\(964\) 3.58890 0.115591
\(965\) −19.5819 −0.630364
\(966\) −1.54359 −0.0496643
\(967\) 8.34162 0.268248 0.134124 0.990965i \(-0.457178\pi\)
0.134124 + 0.990965i \(0.457178\pi\)
\(968\) 0 0
\(969\) 0.849435 0.0272878
\(970\) 6.33804 0.203502
\(971\) −24.9688 −0.801288 −0.400644 0.916234i \(-0.631214\pi\)
−0.400644 + 0.916234i \(0.631214\pi\)
\(972\) 7.51941 0.241185
\(973\) −16.2927 −0.522321
\(974\) −6.18137 −0.198064
\(975\) 0.0836184 0.00267793
\(976\) 9.49472 0.303919
\(977\) 29.6725 0.949308 0.474654 0.880172i \(-0.342573\pi\)
0.474654 + 0.880172i \(0.342573\pi\)
\(978\) −6.07663 −0.194309
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −21.1567 −0.675481
\(982\) −11.5889 −0.369817
\(983\) 32.2439 1.02842 0.514210 0.857664i \(-0.328085\pi\)
0.514210 + 0.857664i \(0.328085\pi\)
\(984\) −0.167237 −0.00533132
\(985\) 8.43580 0.268787
\(986\) −9.01056 −0.286955
\(987\) 3.25443 0.103590
\(988\) −0.926944 −0.0294900
\(989\) −30.7456 −0.977652
\(990\) 0 0
\(991\) 5.74557 0.182514 0.0912571 0.995827i \(-0.470911\pi\)
0.0912571 + 0.995827i \(0.470911\pi\)
\(992\) 0 0
\(993\) 7.30279 0.231747
\(994\) 1.66196 0.0527140
\(995\) −10.9894 −0.348389
\(996\) 0.661956 0.0209749
\(997\) 12.6025 0.399126 0.199563 0.979885i \(-0.436048\pi\)
0.199563 + 0.979885i \(0.436048\pi\)
\(998\) 10.6761 0.337946
\(999\) 15.8328 0.500926
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.ck.1.2 yes 3
11.10 odd 2 8470.2.a.cg.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cg.1.2 3 11.10 odd 2
8470.2.a.ck.1.2 yes 3 1.1 even 1 trivial