Properties

Label 8470.2.a.cy.1.3
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.13298000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 3 x^{3} + 26 x^{2} + 13 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 1.3
Root \(-1.24667\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.418877 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.418877 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.82454 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.418877 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.418877 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.82454 q^{9} -1.00000 q^{10} -0.418877 q^{12} +5.30073 q^{13} +1.00000 q^{14} -0.418877 q^{15} +1.00000 q^{16} -1.47618 q^{17} +2.82454 q^{18} -0.952490 q^{19} +1.00000 q^{20} +0.418877 q^{21} +2.37872 q^{23} +0.418877 q^{24} +1.00000 q^{25} -5.30073 q^{26} +2.43977 q^{27} -1.00000 q^{28} -5.97768 q^{29} +0.418877 q^{30} +2.34330 q^{31} -1.00000 q^{32} +1.47618 q^{34} -1.00000 q^{35} -2.82454 q^{36} +0.0224026 q^{37} +0.952490 q^{38} -2.22035 q^{39} -1.00000 q^{40} +1.53186 q^{41} -0.418877 q^{42} -3.83101 q^{43} -2.82454 q^{45} -2.37872 q^{46} -1.50443 q^{47} -0.418877 q^{48} +1.00000 q^{49} -1.00000 q^{50} +0.618340 q^{51} +5.30073 q^{52} -11.3487 q^{53} -2.43977 q^{54} +1.00000 q^{56} +0.398977 q^{57} +5.97768 q^{58} +0.768866 q^{59} -0.418877 q^{60} -8.06011 q^{61} -2.34330 q^{62} +2.82454 q^{63} +1.00000 q^{64} +5.30073 q^{65} +12.1841 q^{67} -1.47618 q^{68} -0.996391 q^{69} +1.00000 q^{70} -11.9348 q^{71} +2.82454 q^{72} -7.28500 q^{73} -0.0224026 q^{74} -0.418877 q^{75} -0.952490 q^{76} +2.22035 q^{78} +12.8107 q^{79} +1.00000 q^{80} +7.45166 q^{81} -1.53186 q^{82} -6.13526 q^{83} +0.418877 q^{84} -1.47618 q^{85} +3.83101 q^{86} +2.50392 q^{87} +13.4596 q^{89} +2.82454 q^{90} -5.30073 q^{91} +2.37872 q^{92} -0.981556 q^{93} +1.50443 q^{94} -0.952490 q^{95} +0.418877 q^{96} +18.4862 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} - 6 q^{8} + 15 q^{9} + O(q^{10}) \) \( 6 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} - 6 q^{8} + 15 q^{9} - 6 q^{10} + q^{12} - 2 q^{13} + 6 q^{14} + q^{15} + 6 q^{16} - 7 q^{17} - 15 q^{18} - 11 q^{19} + 6 q^{20} - q^{21} - 6 q^{23} - q^{24} + 6 q^{25} + 2 q^{26} + 4 q^{27} - 6 q^{28} - 2 q^{29} - q^{30} - 6 q^{32} + 7 q^{34} - 6 q^{35} + 15 q^{36} - 14 q^{37} + 11 q^{38} - 20 q^{39} - 6 q^{40} - 13 q^{41} + q^{42} - 19 q^{43} + 15 q^{45} + 6 q^{46} + 22 q^{47} + q^{48} + 6 q^{49} - 6 q^{50} + 14 q^{51} - 2 q^{52} - 10 q^{53} - 4 q^{54} + 6 q^{56} - 32 q^{57} + 2 q^{58} - 7 q^{59} + q^{60} - 22 q^{61} - 15 q^{63} + 6 q^{64} - 2 q^{65} + 5 q^{67} - 7 q^{68} - 36 q^{69} + 6 q^{70} + 8 q^{71} - 15 q^{72} - 13 q^{73} + 14 q^{74} + q^{75} - 11 q^{76} + 20 q^{78} + 16 q^{79} + 6 q^{80} + 18 q^{81} + 13 q^{82} + 5 q^{83} - q^{84} - 7 q^{85} + 19 q^{86} - 14 q^{87} + q^{89} - 15 q^{90} + 2 q^{91} - 6 q^{92} - 42 q^{93} - 22 q^{94} - 11 q^{95} - q^{96} - 3 q^{97} - 6 q^{98} + O(q^{100}) \)

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
\(3\) −0.418877 −0.241839 −0.120919 0.992662i \(-0.538584\pi\)
−0.120919 + 0.992662i \(0.538584\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0.418877 0.171006
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.82454 −0.941514
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −0.418877 −0.120919
\(13\) 5.30073 1.47016 0.735078 0.677982i \(-0.237145\pi\)
0.735078 + 0.677982i \(0.237145\pi\)
\(14\) 1.00000 0.267261
\(15\) −0.418877 −0.108154
\(16\) 1.00000 0.250000
\(17\) −1.47618 −0.358027 −0.179014 0.983847i \(-0.557291\pi\)
−0.179014 + 0.983847i \(0.557291\pi\)
\(18\) 2.82454 0.665751
\(19\) −0.952490 −0.218516 −0.109258 0.994013i \(-0.534848\pi\)
−0.109258 + 0.994013i \(0.534848\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.418877 0.0914065
\(22\) 0 0
\(23\) 2.37872 0.495997 0.247999 0.968760i \(-0.420227\pi\)
0.247999 + 0.968760i \(0.420227\pi\)
\(24\) 0.418877 0.0855030
\(25\) 1.00000 0.200000
\(26\) −5.30073 −1.03956
\(27\) 2.43977 0.469534
\(28\) −1.00000 −0.188982
\(29\) −5.97768 −1.11003 −0.555014 0.831841i \(-0.687287\pi\)
−0.555014 + 0.831841i \(0.687287\pi\)
\(30\) 0.418877 0.0764762
\(31\) 2.34330 0.420869 0.210435 0.977608i \(-0.432512\pi\)
0.210435 + 0.977608i \(0.432512\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.47618 0.253163
\(35\) −1.00000 −0.169031
\(36\) −2.82454 −0.470757
\(37\) 0.0224026 0.00368297 0.00184148 0.999998i \(-0.499414\pi\)
0.00184148 + 0.999998i \(0.499414\pi\)
\(38\) 0.952490 0.154514
\(39\) −2.22035 −0.355541
\(40\) −1.00000 −0.158114
\(41\) 1.53186 0.239236 0.119618 0.992820i \(-0.461833\pi\)
0.119618 + 0.992820i \(0.461833\pi\)
\(42\) −0.418877 −0.0646342
\(43\) −3.83101 −0.584223 −0.292112 0.956384i \(-0.594358\pi\)
−0.292112 + 0.956384i \(0.594358\pi\)
\(44\) 0 0
\(45\) −2.82454 −0.421058
\(46\) −2.37872 −0.350723
\(47\) −1.50443 −0.219443 −0.109722 0.993962i \(-0.534996\pi\)
−0.109722 + 0.993962i \(0.534996\pi\)
\(48\) −0.418877 −0.0604597
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0.618340 0.0865849
\(52\) 5.30073 0.735078
\(53\) −11.3487 −1.55887 −0.779434 0.626484i \(-0.784493\pi\)
−0.779434 + 0.626484i \(0.784493\pi\)
\(54\) −2.43977 −0.332010
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0.398977 0.0528457
\(58\) 5.97768 0.784908
\(59\) 0.768866 0.100098 0.0500489 0.998747i \(-0.484062\pi\)
0.0500489 + 0.998747i \(0.484062\pi\)
\(60\) −0.418877 −0.0540768
\(61\) −8.06011 −1.03199 −0.515996 0.856591i \(-0.672578\pi\)
−0.515996 + 0.856591i \(0.672578\pi\)
\(62\) −2.34330 −0.297600
\(63\) 2.82454 0.355859
\(64\) 1.00000 0.125000
\(65\) 5.30073 0.657474
\(66\) 0 0
\(67\) 12.1841 1.48852 0.744260 0.667890i \(-0.232802\pi\)
0.744260 + 0.667890i \(0.232802\pi\)
\(68\) −1.47618 −0.179014
\(69\) −0.996391 −0.119951
\(70\) 1.00000 0.119523
\(71\) −11.9348 −1.41640 −0.708199 0.706013i \(-0.750492\pi\)
−0.708199 + 0.706013i \(0.750492\pi\)
\(72\) 2.82454 0.332875
\(73\) −7.28500 −0.852644 −0.426322 0.904571i \(-0.640191\pi\)
−0.426322 + 0.904571i \(0.640191\pi\)
\(74\) −0.0224026 −0.00260425
\(75\) −0.418877 −0.0483678
\(76\) −0.952490 −0.109258
\(77\) 0 0
\(78\) 2.22035 0.251406
\(79\) 12.8107 1.44131 0.720657 0.693292i \(-0.243840\pi\)
0.720657 + 0.693292i \(0.243840\pi\)
\(80\) 1.00000 0.111803
\(81\) 7.45166 0.827962
\(82\) −1.53186 −0.169166
\(83\) −6.13526 −0.673433 −0.336716 0.941606i \(-0.609316\pi\)
−0.336716 + 0.941606i \(0.609316\pi\)
\(84\) 0.418877 0.0457033
\(85\) −1.47618 −0.160115
\(86\) 3.83101 0.413108
\(87\) 2.50392 0.268448
\(88\) 0 0
\(89\) 13.4596 1.42671 0.713357 0.700801i \(-0.247174\pi\)
0.713357 + 0.700801i \(0.247174\pi\)
\(90\) 2.82454 0.297733
\(91\) −5.30073 −0.555667
\(92\) 2.37872 0.247999
\(93\) −0.981556 −0.101783
\(94\) 1.50443 0.155170
\(95\) −0.952490 −0.0977235
\(96\) 0.418877 0.0427515
\(97\) 18.4862 1.87699 0.938495 0.345292i \(-0.112220\pi\)
0.938495 + 0.345292i \(0.112220\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −8.92131 −0.887704 −0.443852 0.896100i \(-0.646388\pi\)
−0.443852 + 0.896100i \(0.646388\pi\)
\(102\) −0.618340 −0.0612248
\(103\) −12.3386 −1.21576 −0.607881 0.794028i \(-0.707980\pi\)
−0.607881 + 0.794028i \(0.707980\pi\)
\(104\) −5.30073 −0.519779
\(105\) 0.418877 0.0408782
\(106\) 11.3487 1.10229
\(107\) −9.65719 −0.933596 −0.466798 0.884364i \(-0.654593\pi\)
−0.466798 + 0.884364i \(0.654593\pi\)
\(108\) 2.43977 0.234767
\(109\) 5.91826 0.566867 0.283433 0.958992i \(-0.408526\pi\)
0.283433 + 0.958992i \(0.408526\pi\)
\(110\) 0 0
\(111\) −0.00938394 −0.000890684 0
\(112\) −1.00000 −0.0944911
\(113\) 12.5957 1.18490 0.592452 0.805606i \(-0.298160\pi\)
0.592452 + 0.805606i \(0.298160\pi\)
\(114\) −0.398977 −0.0373676
\(115\) 2.37872 0.221817
\(116\) −5.97768 −0.555014
\(117\) −14.9721 −1.38417
\(118\) −0.768866 −0.0707799
\(119\) 1.47618 0.135322
\(120\) 0.418877 0.0382381
\(121\) 0 0
\(122\) 8.06011 0.729729
\(123\) −0.641661 −0.0578566
\(124\) 2.34330 0.210435
\(125\) 1.00000 0.0894427
\(126\) −2.82454 −0.251630
\(127\) 2.54206 0.225572 0.112786 0.993619i \(-0.464023\pi\)
0.112786 + 0.993619i \(0.464023\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.60472 0.141288
\(130\) −5.30073 −0.464904
\(131\) −7.80495 −0.681921 −0.340961 0.940078i \(-0.610752\pi\)
−0.340961 + 0.940078i \(0.610752\pi\)
\(132\) 0 0
\(133\) 0.952490 0.0825914
\(134\) −12.1841 −1.05254
\(135\) 2.43977 0.209982
\(136\) 1.47618 0.126582
\(137\) 19.7473 1.68713 0.843564 0.537029i \(-0.180453\pi\)
0.843564 + 0.537029i \(0.180453\pi\)
\(138\) 0.996391 0.0848184
\(139\) −13.2995 −1.12805 −0.564023 0.825759i \(-0.690747\pi\)
−0.564023 + 0.825759i \(0.690747\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0.630170 0.0530699
\(142\) 11.9348 1.00155
\(143\) 0 0
\(144\) −2.82454 −0.235378
\(145\) −5.97768 −0.496420
\(146\) 7.28500 0.602911
\(147\) −0.418877 −0.0345484
\(148\) 0.0224026 0.00184148
\(149\) −3.95398 −0.323923 −0.161961 0.986797i \(-0.551782\pi\)
−0.161961 + 0.986797i \(0.551782\pi\)
\(150\) 0.418877 0.0342012
\(151\) 9.41563 0.766233 0.383117 0.923700i \(-0.374851\pi\)
0.383117 + 0.923700i \(0.374851\pi\)
\(152\) 0.952490 0.0772572
\(153\) 4.16954 0.337088
\(154\) 0 0
\(155\) 2.34330 0.188218
\(156\) −2.22035 −0.177771
\(157\) −8.62372 −0.688248 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(158\) −12.8107 −1.01916
\(159\) 4.75372 0.376995
\(160\) −1.00000 −0.0790569
\(161\) −2.37872 −0.187469
\(162\) −7.45166 −0.585458
\(163\) −10.9905 −0.860842 −0.430421 0.902628i \(-0.641635\pi\)
−0.430421 + 0.902628i \(0.641635\pi\)
\(164\) 1.53186 0.119618
\(165\) 0 0
\(166\) 6.13526 0.476189
\(167\) −4.26261 −0.329851 −0.164925 0.986306i \(-0.552738\pi\)
−0.164925 + 0.986306i \(0.552738\pi\)
\(168\) −0.418877 −0.0323171
\(169\) 15.0977 1.16136
\(170\) 1.47618 0.113218
\(171\) 2.69035 0.205736
\(172\) −3.83101 −0.292112
\(173\) −21.8753 −1.66315 −0.831575 0.555413i \(-0.812560\pi\)
−0.831575 + 0.555413i \(0.812560\pi\)
\(174\) −2.50392 −0.189821
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −0.322061 −0.0242076
\(178\) −13.4596 −1.00884
\(179\) 12.0148 0.898028 0.449014 0.893525i \(-0.351775\pi\)
0.449014 + 0.893525i \(0.351775\pi\)
\(180\) −2.82454 −0.210529
\(181\) −21.0621 −1.56554 −0.782768 0.622314i \(-0.786193\pi\)
−0.782768 + 0.622314i \(0.786193\pi\)
\(182\) 5.30073 0.392916
\(183\) 3.37620 0.249576
\(184\) −2.37872 −0.175361
\(185\) 0.0224026 0.00164707
\(186\) 0.981556 0.0719711
\(187\) 0 0
\(188\) −1.50443 −0.109722
\(189\) −2.43977 −0.177467
\(190\) 0.952490 0.0691009
\(191\) −7.60935 −0.550593 −0.275296 0.961359i \(-0.588776\pi\)
−0.275296 + 0.961359i \(0.588776\pi\)
\(192\) −0.418877 −0.0302299
\(193\) −20.6268 −1.48475 −0.742375 0.669985i \(-0.766301\pi\)
−0.742375 + 0.669985i \(0.766301\pi\)
\(194\) −18.4862 −1.32723
\(195\) −2.22035 −0.159003
\(196\) 1.00000 0.0714286
\(197\) 10.1376 0.722273 0.361136 0.932513i \(-0.382389\pi\)
0.361136 + 0.932513i \(0.382389\pi\)
\(198\) 0 0
\(199\) 22.6636 1.60658 0.803290 0.595588i \(-0.203081\pi\)
0.803290 + 0.595588i \(0.203081\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −5.10363 −0.359982
\(202\) 8.92131 0.627702
\(203\) 5.97768 0.419551
\(204\) 0.618340 0.0432925
\(205\) 1.53186 0.106990
\(206\) 12.3386 0.859673
\(207\) −6.71879 −0.466988
\(208\) 5.30073 0.367539
\(209\) 0 0
\(210\) −0.418877 −0.0289053
\(211\) −13.2771 −0.914033 −0.457016 0.889458i \(-0.651082\pi\)
−0.457016 + 0.889458i \(0.651082\pi\)
\(212\) −11.3487 −0.779434
\(213\) 4.99921 0.342540
\(214\) 9.65719 0.660152
\(215\) −3.83101 −0.261272
\(216\) −2.43977 −0.166005
\(217\) −2.34330 −0.159074
\(218\) −5.91826 −0.400835
\(219\) 3.05152 0.206203
\(220\) 0 0
\(221\) −7.82485 −0.526356
\(222\) 0.00938394 0.000629809 0
\(223\) −10.6151 −0.710841 −0.355420 0.934707i \(-0.615662\pi\)
−0.355420 + 0.934707i \(0.615662\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.82454 −0.188303
\(226\) −12.5957 −0.837853
\(227\) 24.3926 1.61899 0.809497 0.587124i \(-0.199740\pi\)
0.809497 + 0.587124i \(0.199740\pi\)
\(228\) 0.398977 0.0264229
\(229\) 11.2804 0.745433 0.372716 0.927945i \(-0.378426\pi\)
0.372716 + 0.927945i \(0.378426\pi\)
\(230\) −2.37872 −0.156848
\(231\) 0 0
\(232\) 5.97768 0.392454
\(233\) −9.18604 −0.601798 −0.300899 0.953656i \(-0.597287\pi\)
−0.300899 + 0.953656i \(0.597287\pi\)
\(234\) 14.9721 0.978758
\(235\) −1.50443 −0.0981379
\(236\) 0.768866 0.0500489
\(237\) −5.36610 −0.348566
\(238\) −1.47618 −0.0956868
\(239\) 3.05287 0.197474 0.0987368 0.995114i \(-0.468520\pi\)
0.0987368 + 0.995114i \(0.468520\pi\)
\(240\) −0.418877 −0.0270384
\(241\) 5.27801 0.339987 0.169993 0.985445i \(-0.445625\pi\)
0.169993 + 0.985445i \(0.445625\pi\)
\(242\) 0 0
\(243\) −10.4406 −0.669767
\(244\) −8.06011 −0.515996
\(245\) 1.00000 0.0638877
\(246\) 0.641661 0.0409108
\(247\) −5.04889 −0.321253
\(248\) −2.34330 −0.148800
\(249\) 2.56992 0.162862
\(250\) −1.00000 −0.0632456
\(251\) −2.11801 −0.133688 −0.0668438 0.997763i \(-0.521293\pi\)
−0.0668438 + 0.997763i \(0.521293\pi\)
\(252\) 2.82454 0.177929
\(253\) 0 0
\(254\) −2.54206 −0.159503
\(255\) 0.618340 0.0387219
\(256\) 1.00000 0.0625000
\(257\) −26.8253 −1.67331 −0.836657 0.547727i \(-0.815493\pi\)
−0.836657 + 0.547727i \(0.815493\pi\)
\(258\) −1.60472 −0.0999056
\(259\) −0.0224026 −0.00139203
\(260\) 5.30073 0.328737
\(261\) 16.8842 1.04511
\(262\) 7.80495 0.482191
\(263\) 8.68185 0.535346 0.267673 0.963510i \(-0.413745\pi\)
0.267673 + 0.963510i \(0.413745\pi\)
\(264\) 0 0
\(265\) −11.3487 −0.697147
\(266\) −0.952490 −0.0584009
\(267\) −5.63791 −0.345035
\(268\) 12.1841 0.744260
\(269\) −11.4629 −0.698903 −0.349452 0.936954i \(-0.613632\pi\)
−0.349452 + 0.936954i \(0.613632\pi\)
\(270\) −2.43977 −0.148480
\(271\) −17.1909 −1.04427 −0.522137 0.852862i \(-0.674865\pi\)
−0.522137 + 0.852862i \(0.674865\pi\)
\(272\) −1.47618 −0.0895068
\(273\) 2.22035 0.134382
\(274\) −19.7473 −1.19298
\(275\) 0 0
\(276\) −0.996391 −0.0599757
\(277\) −15.2055 −0.913610 −0.456805 0.889567i \(-0.651006\pi\)
−0.456805 + 0.889567i \(0.651006\pi\)
\(278\) 13.2995 0.797649
\(279\) −6.61875 −0.396254
\(280\) 1.00000 0.0597614
\(281\) 2.39475 0.142859 0.0714293 0.997446i \(-0.477244\pi\)
0.0714293 + 0.997446i \(0.477244\pi\)
\(282\) −0.630170 −0.0375261
\(283\) 12.0220 0.714632 0.357316 0.933983i \(-0.383692\pi\)
0.357316 + 0.933983i \(0.383692\pi\)
\(284\) −11.9348 −0.708199
\(285\) 0.398977 0.0236333
\(286\) 0 0
\(287\) −1.53186 −0.0904228
\(288\) 2.82454 0.166438
\(289\) −14.8209 −0.871816
\(290\) 5.97768 0.351022
\(291\) −7.74345 −0.453929
\(292\) −7.28500 −0.426322
\(293\) −11.2424 −0.656790 −0.328395 0.944540i \(-0.606508\pi\)
−0.328395 + 0.944540i \(0.606508\pi\)
\(294\) 0.418877 0.0244294
\(295\) 0.768866 0.0447651
\(296\) −0.0224026 −0.00130213
\(297\) 0 0
\(298\) 3.95398 0.229048
\(299\) 12.6089 0.729194
\(300\) −0.418877 −0.0241839
\(301\) 3.83101 0.220816
\(302\) −9.41563 −0.541809
\(303\) 3.73694 0.214681
\(304\) −0.952490 −0.0546291
\(305\) −8.06011 −0.461521
\(306\) −4.16954 −0.238357
\(307\) −25.1111 −1.43317 −0.716584 0.697501i \(-0.754295\pi\)
−0.716584 + 0.697501i \(0.754295\pi\)
\(308\) 0 0
\(309\) 5.16837 0.294018
\(310\) −2.34330 −0.133091
\(311\) −2.62754 −0.148994 −0.0744971 0.997221i \(-0.523735\pi\)
−0.0744971 + 0.997221i \(0.523735\pi\)
\(312\) 2.22035 0.125703
\(313\) 3.90115 0.220506 0.110253 0.993904i \(-0.464834\pi\)
0.110253 + 0.993904i \(0.464834\pi\)
\(314\) 8.62372 0.486665
\(315\) 2.82454 0.159145
\(316\) 12.8107 0.720657
\(317\) −27.5825 −1.54919 −0.774594 0.632459i \(-0.782046\pi\)
−0.774594 + 0.632459i \(0.782046\pi\)
\(318\) −4.75372 −0.266576
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 4.04518 0.225780
\(322\) 2.37872 0.132561
\(323\) 1.40605 0.0782348
\(324\) 7.45166 0.413981
\(325\) 5.30073 0.294031
\(326\) 10.9905 0.608708
\(327\) −2.47903 −0.137090
\(328\) −1.53186 −0.0845828
\(329\) 1.50443 0.0829417
\(330\) 0 0
\(331\) 33.4595 1.83910 0.919551 0.392972i \(-0.128553\pi\)
0.919551 + 0.392972i \(0.128553\pi\)
\(332\) −6.13526 −0.336716
\(333\) −0.0632771 −0.00346756
\(334\) 4.26261 0.233240
\(335\) 12.1841 0.665687
\(336\) 0.418877 0.0228516
\(337\) −15.9275 −0.867627 −0.433813 0.901003i \(-0.642832\pi\)
−0.433813 + 0.901003i \(0.642832\pi\)
\(338\) −15.0977 −0.821206
\(339\) −5.27605 −0.286556
\(340\) −1.47618 −0.0800573
\(341\) 0 0
\(342\) −2.69035 −0.145477
\(343\) −1.00000 −0.0539949
\(344\) 3.83101 0.206554
\(345\) −0.996391 −0.0536439
\(346\) 21.8753 1.17602
\(347\) −19.0799 −1.02426 −0.512132 0.858907i \(-0.671144\pi\)
−0.512132 + 0.858907i \(0.671144\pi\)
\(348\) 2.50392 0.134224
\(349\) 17.3451 0.928460 0.464230 0.885715i \(-0.346331\pi\)
0.464230 + 0.885715i \(0.346331\pi\)
\(350\) 1.00000 0.0534522
\(351\) 12.9325 0.690288
\(352\) 0 0
\(353\) −23.2547 −1.23772 −0.618861 0.785501i \(-0.712405\pi\)
−0.618861 + 0.785501i \(0.712405\pi\)
\(354\) 0.322061 0.0171173
\(355\) −11.9348 −0.633433
\(356\) 13.4596 0.713357
\(357\) −0.618340 −0.0327260
\(358\) −12.0148 −0.635002
\(359\) 0.200151 0.0105636 0.00528179 0.999986i \(-0.498319\pi\)
0.00528179 + 0.999986i \(0.498319\pi\)
\(360\) 2.82454 0.148866
\(361\) −18.0928 −0.952251
\(362\) 21.0621 1.10700
\(363\) 0 0
\(364\) −5.30073 −0.277834
\(365\) −7.28500 −0.381314
\(366\) −3.37620 −0.176477
\(367\) 13.0368 0.680516 0.340258 0.940332i \(-0.389486\pi\)
0.340258 + 0.940332i \(0.389486\pi\)
\(368\) 2.37872 0.123999
\(369\) −4.32680 −0.225244
\(370\) −0.0224026 −0.00116466
\(371\) 11.3487 0.589197
\(372\) −0.981556 −0.0508913
\(373\) −16.1910 −0.838336 −0.419168 0.907909i \(-0.637678\pi\)
−0.419168 + 0.907909i \(0.637678\pi\)
\(374\) 0 0
\(375\) −0.418877 −0.0216307
\(376\) 1.50443 0.0775848
\(377\) −31.6861 −1.63192
\(378\) 2.43977 0.125488
\(379\) 38.0576 1.95489 0.977443 0.211198i \(-0.0677366\pi\)
0.977443 + 0.211198i \(0.0677366\pi\)
\(380\) −0.952490 −0.0488617
\(381\) −1.06481 −0.0545520
\(382\) 7.60935 0.389328
\(383\) −3.26994 −0.167086 −0.0835431 0.996504i \(-0.526624\pi\)
−0.0835431 + 0.996504i \(0.526624\pi\)
\(384\) 0.418877 0.0213757
\(385\) 0 0
\(386\) 20.6268 1.04988
\(387\) 10.8208 0.550054
\(388\) 18.4862 0.938495
\(389\) −30.0096 −1.52155 −0.760775 0.649016i \(-0.775181\pi\)
−0.760775 + 0.649016i \(0.775181\pi\)
\(390\) 2.22035 0.112432
\(391\) −3.51143 −0.177580
\(392\) −1.00000 −0.0505076
\(393\) 3.26931 0.164915
\(394\) −10.1376 −0.510724
\(395\) 12.8107 0.644575
\(396\) 0 0
\(397\) 15.5047 0.778158 0.389079 0.921204i \(-0.372793\pi\)
0.389079 + 0.921204i \(0.372793\pi\)
\(398\) −22.6636 −1.13602
\(399\) −0.398977 −0.0199738
\(400\) 1.00000 0.0500000
\(401\) −5.70472 −0.284880 −0.142440 0.989803i \(-0.545495\pi\)
−0.142440 + 0.989803i \(0.545495\pi\)
\(402\) 5.10363 0.254546
\(403\) 12.4212 0.618744
\(404\) −8.92131 −0.443852
\(405\) 7.45166 0.370276
\(406\) −5.97768 −0.296667
\(407\) 0 0
\(408\) −0.618340 −0.0306124
\(409\) −1.28797 −0.0636860 −0.0318430 0.999493i \(-0.510138\pi\)
−0.0318430 + 0.999493i \(0.510138\pi\)
\(410\) −1.53186 −0.0756532
\(411\) −8.27171 −0.408013
\(412\) −12.3386 −0.607881
\(413\) −0.768866 −0.0378334
\(414\) 6.71879 0.330211
\(415\) −6.13526 −0.301168
\(416\) −5.30073 −0.259889
\(417\) 5.57084 0.272805
\(418\) 0 0
\(419\) 36.2889 1.77283 0.886414 0.462893i \(-0.153189\pi\)
0.886414 + 0.462893i \(0.153189\pi\)
\(420\) 0.418877 0.0204391
\(421\) −13.4042 −0.653279 −0.326639 0.945149i \(-0.605916\pi\)
−0.326639 + 0.945149i \(0.605916\pi\)
\(422\) 13.2771 0.646319
\(423\) 4.24931 0.206609
\(424\) 11.3487 0.551143
\(425\) −1.47618 −0.0716054
\(426\) −4.99921 −0.242213
\(427\) 8.06011 0.390056
\(428\) −9.65719 −0.466798
\(429\) 0 0
\(430\) 3.83101 0.184748
\(431\) −2.38684 −0.114970 −0.0574851 0.998346i \(-0.518308\pi\)
−0.0574851 + 0.998346i \(0.518308\pi\)
\(432\) 2.43977 0.117383
\(433\) −22.6678 −1.08935 −0.544673 0.838648i \(-0.683346\pi\)
−0.544673 + 0.838648i \(0.683346\pi\)
\(434\) 2.34330 0.112482
\(435\) 2.50392 0.120054
\(436\) 5.91826 0.283433
\(437\) −2.26571 −0.108383
\(438\) −3.05152 −0.145807
\(439\) −17.7876 −0.848958 −0.424479 0.905438i \(-0.639543\pi\)
−0.424479 + 0.905438i \(0.639543\pi\)
\(440\) 0 0
\(441\) −2.82454 −0.134502
\(442\) 7.82485 0.372190
\(443\) −24.0914 −1.14462 −0.572309 0.820038i \(-0.693952\pi\)
−0.572309 + 0.820038i \(0.693952\pi\)
\(444\) −0.00938394 −0.000445342 0
\(445\) 13.4596 0.638046
\(446\) 10.6151 0.502640
\(447\) 1.65623 0.0783371
\(448\) −1.00000 −0.0472456
\(449\) −6.06955 −0.286440 −0.143220 0.989691i \(-0.545746\pi\)
−0.143220 + 0.989691i \(0.545746\pi\)
\(450\) 2.82454 0.133150
\(451\) 0 0
\(452\) 12.5957 0.592452
\(453\) −3.94399 −0.185305
\(454\) −24.3926 −1.14480
\(455\) −5.30073 −0.248502
\(456\) −0.398977 −0.0186838
\(457\) −4.92065 −0.230178 −0.115089 0.993355i \(-0.536715\pi\)
−0.115089 + 0.993355i \(0.536715\pi\)
\(458\) −11.2804 −0.527101
\(459\) −3.60155 −0.168106
\(460\) 2.37872 0.110908
\(461\) 33.6918 1.56918 0.784591 0.620013i \(-0.212873\pi\)
0.784591 + 0.620013i \(0.212873\pi\)
\(462\) 0 0
\(463\) 2.26163 0.105107 0.0525533 0.998618i \(-0.483264\pi\)
0.0525533 + 0.998618i \(0.483264\pi\)
\(464\) −5.97768 −0.277507
\(465\) −0.981556 −0.0455185
\(466\) 9.18604 0.425535
\(467\) 13.3924 0.619727 0.309863 0.950781i \(-0.399717\pi\)
0.309863 + 0.950781i \(0.399717\pi\)
\(468\) −14.9721 −0.692087
\(469\) −12.1841 −0.562608
\(470\) 1.50443 0.0693940
\(471\) 3.61228 0.166445
\(472\) −0.768866 −0.0353899
\(473\) 0 0
\(474\) 5.36610 0.246473
\(475\) −0.952490 −0.0437033
\(476\) 1.47618 0.0676608
\(477\) 32.0550 1.46770
\(478\) −3.05287 −0.139635
\(479\) −16.6637 −0.761385 −0.380692 0.924702i \(-0.624314\pi\)
−0.380692 + 0.924702i \(0.624314\pi\)
\(480\) 0.418877 0.0191190
\(481\) 0.118750 0.00541454
\(482\) −5.27801 −0.240407
\(483\) 0.996391 0.0453374
\(484\) 0 0
\(485\) 18.4862 0.839416
\(486\) 10.4406 0.473597
\(487\) −12.6959 −0.575308 −0.287654 0.957734i \(-0.592875\pi\)
−0.287654 + 0.957734i \(0.592875\pi\)
\(488\) 8.06011 0.364864
\(489\) 4.60367 0.208185
\(490\) −1.00000 −0.0451754
\(491\) −15.4392 −0.696761 −0.348380 0.937353i \(-0.613268\pi\)
−0.348380 + 0.937353i \(0.613268\pi\)
\(492\) −0.641661 −0.0289283
\(493\) 8.82416 0.397420
\(494\) 5.04889 0.227160
\(495\) 0 0
\(496\) 2.34330 0.105217
\(497\) 11.9348 0.535348
\(498\) −2.56992 −0.115161
\(499\) −5.96950 −0.267231 −0.133616 0.991033i \(-0.542659\pi\)
−0.133616 + 0.991033i \(0.542659\pi\)
\(500\) 1.00000 0.0447214
\(501\) 1.78551 0.0797707
\(502\) 2.11801 0.0945314
\(503\) 1.00487 0.0448052 0.0224026 0.999749i \(-0.492868\pi\)
0.0224026 + 0.999749i \(0.492868\pi\)
\(504\) −2.82454 −0.125815
\(505\) −8.92131 −0.396993
\(506\) 0 0
\(507\) −6.32408 −0.280862
\(508\) 2.54206 0.112786
\(509\) 19.2453 0.853034 0.426517 0.904480i \(-0.359740\pi\)
0.426517 + 0.904480i \(0.359740\pi\)
\(510\) −0.618340 −0.0273806
\(511\) 7.28500 0.322269
\(512\) −1.00000 −0.0441942
\(513\) −2.32386 −0.102601
\(514\) 26.8253 1.18321
\(515\) −12.3386 −0.543705
\(516\) 1.60472 0.0706439
\(517\) 0 0
\(518\) 0.0224026 0.000984314 0
\(519\) 9.16307 0.402214
\(520\) −5.30073 −0.232452
\(521\) 4.02400 0.176295 0.0881473 0.996107i \(-0.471905\pi\)
0.0881473 + 0.996107i \(0.471905\pi\)
\(522\) −16.8842 −0.739002
\(523\) 1.07196 0.0468734 0.0234367 0.999725i \(-0.492539\pi\)
0.0234367 + 0.999725i \(0.492539\pi\)
\(524\) −7.80495 −0.340961
\(525\) 0.418877 0.0182813
\(526\) −8.68185 −0.378547
\(527\) −3.45914 −0.150683
\(528\) 0 0
\(529\) −17.3417 −0.753987
\(530\) 11.3487 0.492957
\(531\) −2.17170 −0.0942436
\(532\) 0.952490 0.0412957
\(533\) 8.11997 0.351715
\(534\) 5.63791 0.243976
\(535\) −9.65719 −0.417517
\(536\) −12.1841 −0.526272
\(537\) −5.03273 −0.217178
\(538\) 11.4629 0.494199
\(539\) 0 0
\(540\) 2.43977 0.104991
\(541\) 31.8711 1.37025 0.685123 0.728428i \(-0.259749\pi\)
0.685123 + 0.728428i \(0.259749\pi\)
\(542\) 17.1909 0.738413
\(543\) 8.82245 0.378607
\(544\) 1.47618 0.0632909
\(545\) 5.91826 0.253511
\(546\) −2.22035 −0.0950224
\(547\) 7.60840 0.325312 0.162656 0.986683i \(-0.447994\pi\)
0.162656 + 0.986683i \(0.447994\pi\)
\(548\) 19.7473 0.843564
\(549\) 22.7661 0.971635
\(550\) 0 0
\(551\) 5.69369 0.242559
\(552\) 0.996391 0.0424092
\(553\) −12.8107 −0.544765
\(554\) 15.2055 0.646020
\(555\) −0.00938394 −0.000398326 0
\(556\) −13.2995 −0.564023
\(557\) −25.1526 −1.06575 −0.532875 0.846194i \(-0.678888\pi\)
−0.532875 + 0.846194i \(0.678888\pi\)
\(558\) 6.61875 0.280194
\(559\) −20.3071 −0.858900
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −2.39475 −0.101016
\(563\) −36.3670 −1.53268 −0.766342 0.642432i \(-0.777925\pi\)
−0.766342 + 0.642432i \(0.777925\pi\)
\(564\) 0.630170 0.0265349
\(565\) 12.5957 0.529905
\(566\) −12.0220 −0.505321
\(567\) −7.45166 −0.312940
\(568\) 11.9348 0.500773
\(569\) −23.7370 −0.995109 −0.497554 0.867433i \(-0.665769\pi\)
−0.497554 + 0.867433i \(0.665769\pi\)
\(570\) −0.398977 −0.0167113
\(571\) 42.7475 1.78893 0.894464 0.447139i \(-0.147557\pi\)
0.894464 + 0.447139i \(0.147557\pi\)
\(572\) 0 0
\(573\) 3.18738 0.133155
\(574\) 1.53186 0.0639386
\(575\) 2.37872 0.0991994
\(576\) −2.82454 −0.117689
\(577\) 25.6758 1.06890 0.534449 0.845201i \(-0.320519\pi\)
0.534449 + 0.845201i \(0.320519\pi\)
\(578\) 14.8209 0.616467
\(579\) 8.64010 0.359070
\(580\) −5.97768 −0.248210
\(581\) 6.13526 0.254534
\(582\) 7.74345 0.320976
\(583\) 0 0
\(584\) 7.28500 0.301455
\(585\) −14.9721 −0.619021
\(586\) 11.2424 0.464421
\(587\) −37.8416 −1.56189 −0.780945 0.624600i \(-0.785262\pi\)
−0.780945 + 0.624600i \(0.785262\pi\)
\(588\) −0.418877 −0.0172742
\(589\) −2.23197 −0.0919668
\(590\) −0.768866 −0.0316537
\(591\) −4.24640 −0.174674
\(592\) 0.0224026 0.000920741 0
\(593\) −8.15354 −0.334826 −0.167413 0.985887i \(-0.553541\pi\)
−0.167413 + 0.985887i \(0.553541\pi\)
\(594\) 0 0
\(595\) 1.47618 0.0605177
\(596\) −3.95398 −0.161961
\(597\) −9.49326 −0.388533
\(598\) −12.6089 −0.515618
\(599\) −10.0438 −0.410378 −0.205189 0.978722i \(-0.565781\pi\)
−0.205189 + 0.978722i \(0.565781\pi\)
\(600\) 0.418877 0.0171006
\(601\) −19.4237 −0.792307 −0.396154 0.918184i \(-0.629655\pi\)
−0.396154 + 0.918184i \(0.629655\pi\)
\(602\) −3.83101 −0.156140
\(603\) −34.4144 −1.40146
\(604\) 9.41563 0.383117
\(605\) 0 0
\(606\) −3.73694 −0.151803
\(607\) −14.8851 −0.604169 −0.302084 0.953281i \(-0.597682\pi\)
−0.302084 + 0.953281i \(0.597682\pi\)
\(608\) 0.952490 0.0386286
\(609\) −2.50392 −0.101464
\(610\) 8.06011 0.326345
\(611\) −7.97455 −0.322616
\(612\) 4.16954 0.168544
\(613\) 18.1069 0.731332 0.365666 0.930746i \(-0.380841\pi\)
0.365666 + 0.930746i \(0.380841\pi\)
\(614\) 25.1111 1.01340
\(615\) −0.641661 −0.0258743
\(616\) 0 0
\(617\) −33.3189 −1.34137 −0.670684 0.741743i \(-0.733999\pi\)
−0.670684 + 0.741743i \(0.733999\pi\)
\(618\) −5.16837 −0.207902
\(619\) −13.0672 −0.525214 −0.262607 0.964903i \(-0.584582\pi\)
−0.262607 + 0.964903i \(0.584582\pi\)
\(620\) 2.34330 0.0941092
\(621\) 5.80352 0.232887
\(622\) 2.62754 0.105355
\(623\) −13.4596 −0.539247
\(624\) −2.22035 −0.0888853
\(625\) 1.00000 0.0400000
\(626\) −3.90115 −0.155921
\(627\) 0 0
\(628\) −8.62372 −0.344124
\(629\) −0.0330704 −0.00131860
\(630\) −2.82454 −0.112532
\(631\) 39.3161 1.56515 0.782575 0.622557i \(-0.213906\pi\)
0.782575 + 0.622557i \(0.213906\pi\)
\(632\) −12.8107 −0.509581
\(633\) 5.56147 0.221049
\(634\) 27.5825 1.09544
\(635\) 2.54206 0.100879
\(636\) 4.75372 0.188497
\(637\) 5.30073 0.210022
\(638\) 0 0
\(639\) 33.7103 1.33356
\(640\) −1.00000 −0.0395285
\(641\) 33.0127 1.30392 0.651962 0.758251i \(-0.273946\pi\)
0.651962 + 0.758251i \(0.273946\pi\)
\(642\) −4.04518 −0.159650
\(643\) 0.128880 0.00508254 0.00254127 0.999997i \(-0.499191\pi\)
0.00254127 + 0.999997i \(0.499191\pi\)
\(644\) −2.37872 −0.0937347
\(645\) 1.60472 0.0631858
\(646\) −1.40605 −0.0553203
\(647\) 31.9843 1.25743 0.628716 0.777635i \(-0.283581\pi\)
0.628716 + 0.777635i \(0.283581\pi\)
\(648\) −7.45166 −0.292729
\(649\) 0 0
\(650\) −5.30073 −0.207912
\(651\) 0.981556 0.0384702
\(652\) −10.9905 −0.430421
\(653\) −30.0198 −1.17477 −0.587384 0.809309i \(-0.699842\pi\)
−0.587384 + 0.809309i \(0.699842\pi\)
\(654\) 2.47903 0.0969376
\(655\) −7.80495 −0.304965
\(656\) 1.53186 0.0598091
\(657\) 20.5768 0.802776
\(658\) −1.50443 −0.0586486
\(659\) 19.3178 0.752515 0.376258 0.926515i \(-0.377211\pi\)
0.376258 + 0.926515i \(0.377211\pi\)
\(660\) 0 0
\(661\) −0.389498 −0.0151497 −0.00757487 0.999971i \(-0.502411\pi\)
−0.00757487 + 0.999971i \(0.502411\pi\)
\(662\) −33.4595 −1.30044
\(663\) 3.27765 0.127293
\(664\) 6.13526 0.238094
\(665\) 0.952490 0.0369360
\(666\) 0.0632771 0.00245194
\(667\) −14.2192 −0.550571
\(668\) −4.26261 −0.164925
\(669\) 4.44643 0.171909
\(670\) −12.1841 −0.470712
\(671\) 0 0
\(672\) −0.418877 −0.0161585
\(673\) −31.7397 −1.22347 −0.611737 0.791061i \(-0.709529\pi\)
−0.611737 + 0.791061i \(0.709529\pi\)
\(674\) 15.9275 0.613505
\(675\) 2.43977 0.0939067
\(676\) 15.0977 0.580681
\(677\) −42.1706 −1.62075 −0.810374 0.585913i \(-0.800736\pi\)
−0.810374 + 0.585913i \(0.800736\pi\)
\(678\) 5.27605 0.202625
\(679\) −18.4862 −0.709436
\(680\) 1.47618 0.0566091
\(681\) −10.2175 −0.391536
\(682\) 0 0
\(683\) −22.4212 −0.857923 −0.428961 0.903323i \(-0.641120\pi\)
−0.428961 + 0.903323i \(0.641120\pi\)
\(684\) 2.69035 0.102868
\(685\) 19.7473 0.754507
\(686\) 1.00000 0.0381802
\(687\) −4.72512 −0.180275
\(688\) −3.83101 −0.146056
\(689\) −60.1565 −2.29178
\(690\) 0.996391 0.0379320
\(691\) −21.3409 −0.811844 −0.405922 0.913908i \(-0.633050\pi\)
−0.405922 + 0.913908i \(0.633050\pi\)
\(692\) −21.8753 −0.831575
\(693\) 0 0
\(694\) 19.0799 0.724264
\(695\) −13.2995 −0.504478
\(696\) −2.50392 −0.0949107
\(697\) −2.26131 −0.0856531
\(698\) −17.3451 −0.656520
\(699\) 3.84782 0.145538
\(700\) −1.00000 −0.0377964
\(701\) 22.4894 0.849412 0.424706 0.905331i \(-0.360377\pi\)
0.424706 + 0.905331i \(0.360377\pi\)
\(702\) −12.9325 −0.488107
\(703\) −0.0213383 −0.000804788 0
\(704\) 0 0
\(705\) 0.630170 0.0237336
\(706\) 23.2547 0.875201
\(707\) 8.92131 0.335521
\(708\) −0.322061 −0.0121038
\(709\) −38.6442 −1.45131 −0.725656 0.688058i \(-0.758464\pi\)
−0.725656 + 0.688058i \(0.758464\pi\)
\(710\) 11.9348 0.447905
\(711\) −36.1843 −1.35702
\(712\) −13.4596 −0.504419
\(713\) 5.57405 0.208750
\(714\) 0.618340 0.0231408
\(715\) 0 0
\(716\) 12.0148 0.449014
\(717\) −1.27878 −0.0477568
\(718\) −0.200151 −0.00746958
\(719\) 17.1230 0.638580 0.319290 0.947657i \(-0.396556\pi\)
0.319290 + 0.947657i \(0.396556\pi\)
\(720\) −2.82454 −0.105264
\(721\) 12.3386 0.459514
\(722\) 18.0928 0.673343
\(723\) −2.21084 −0.0822220
\(724\) −21.0621 −0.782768
\(725\) −5.97768 −0.222006
\(726\) 0 0
\(727\) −21.9967 −0.815811 −0.407906 0.913024i \(-0.633741\pi\)
−0.407906 + 0.913024i \(0.633741\pi\)
\(728\) 5.30073 0.196458
\(729\) −17.9816 −0.665987
\(730\) 7.28500 0.269630
\(731\) 5.65527 0.209168
\(732\) 3.37620 0.124788
\(733\) −13.0302 −0.481283 −0.240641 0.970614i \(-0.577358\pi\)
−0.240641 + 0.970614i \(0.577358\pi\)
\(734\) −13.0368 −0.481198
\(735\) −0.418877 −0.0154505
\(736\) −2.37872 −0.0876807
\(737\) 0 0
\(738\) 4.32680 0.159272
\(739\) 17.3424 0.637949 0.318974 0.947763i \(-0.396662\pi\)
0.318974 + 0.947763i \(0.396662\pi\)
\(740\) 0.0224026 0.000823536 0
\(741\) 2.11487 0.0776915
\(742\) −11.3487 −0.416625
\(743\) 27.3421 1.00308 0.501542 0.865133i \(-0.332766\pi\)
0.501542 + 0.865133i \(0.332766\pi\)
\(744\) 0.981556 0.0359856
\(745\) −3.95398 −0.144863
\(746\) 16.1910 0.592793
\(747\) 17.3293 0.634046
\(748\) 0 0
\(749\) 9.65719 0.352866
\(750\) 0.418877 0.0152952
\(751\) −37.6779 −1.37488 −0.687442 0.726239i \(-0.741267\pi\)
−0.687442 + 0.726239i \(0.741267\pi\)
\(752\) −1.50443 −0.0548608
\(753\) 0.887186 0.0323309
\(754\) 31.6861 1.15394
\(755\) 9.41563 0.342670
\(756\) −2.43977 −0.0887335
\(757\) −8.49638 −0.308806 −0.154403 0.988008i \(-0.549345\pi\)
−0.154403 + 0.988008i \(0.549345\pi\)
\(758\) −38.0576 −1.38231
\(759\) 0 0
\(760\) 0.952490 0.0345505
\(761\) −14.7525 −0.534777 −0.267388 0.963589i \(-0.586161\pi\)
−0.267388 + 0.963589i \(0.586161\pi\)
\(762\) 1.06481 0.0385741
\(763\) −5.91826 −0.214256
\(764\) −7.60935 −0.275296
\(765\) 4.16954 0.150750
\(766\) 3.26994 0.118148
\(767\) 4.07555 0.147160
\(768\) −0.418877 −0.0151149
\(769\) −39.2157 −1.41415 −0.707077 0.707136i \(-0.749987\pi\)
−0.707077 + 0.707136i \(0.749987\pi\)
\(770\) 0 0
\(771\) 11.2365 0.404672
\(772\) −20.6268 −0.742375
\(773\) 1.26797 0.0456058 0.0228029 0.999740i \(-0.492741\pi\)
0.0228029 + 0.999740i \(0.492741\pi\)
\(774\) −10.8208 −0.388947
\(775\) 2.34330 0.0841739
\(776\) −18.4862 −0.663616
\(777\) 0.00938394 0.000336647 0
\(778\) 30.0096 1.07590
\(779\) −1.45908 −0.0522770
\(780\) −2.22035 −0.0795014
\(781\) 0 0
\(782\) 3.51143 0.125568
\(783\) −14.5842 −0.521195
\(784\) 1.00000 0.0357143
\(785\) −8.62372 −0.307794
\(786\) −3.26931 −0.116613
\(787\) −24.7449 −0.882060 −0.441030 0.897492i \(-0.645387\pi\)
−0.441030 + 0.897492i \(0.645387\pi\)
\(788\) 10.1376 0.361136
\(789\) −3.63663 −0.129467
\(790\) −12.8107 −0.455783
\(791\) −12.5957 −0.447851
\(792\) 0 0
\(793\) −42.7245 −1.51719
\(794\) −15.5047 −0.550241
\(795\) 4.75372 0.168597
\(796\) 22.6636 0.803290
\(797\) −24.3368 −0.862054 −0.431027 0.902339i \(-0.641849\pi\)
−0.431027 + 0.902339i \(0.641849\pi\)
\(798\) 0.398977 0.0141236
\(799\) 2.22081 0.0785666
\(800\) −1.00000 −0.0353553
\(801\) −38.0172 −1.34327
\(802\) 5.70472 0.201441
\(803\) 0 0
\(804\) −5.10363 −0.179991
\(805\) −2.37872 −0.0838388
\(806\) −12.4212 −0.437518
\(807\) 4.80153 0.169022
\(808\) 8.92131 0.313851
\(809\) 41.5305 1.46013 0.730067 0.683376i \(-0.239489\pi\)
0.730067 + 0.683376i \(0.239489\pi\)
\(810\) −7.45166 −0.261825
\(811\) −44.5784 −1.56536 −0.782680 0.622424i \(-0.786148\pi\)
−0.782680 + 0.622424i \(0.786148\pi\)
\(812\) 5.97768 0.209776
\(813\) 7.20089 0.252546
\(814\) 0 0
\(815\) −10.9905 −0.384980
\(816\) 0.618340 0.0216462
\(817\) 3.64900 0.127662
\(818\) 1.28797 0.0450328
\(819\) 14.9721 0.523168
\(820\) 1.53186 0.0534949
\(821\) 18.0614 0.630348 0.315174 0.949034i \(-0.397937\pi\)
0.315174 + 0.949034i \(0.397937\pi\)
\(822\) 8.27171 0.288509
\(823\) 55.8537 1.94694 0.973468 0.228823i \(-0.0734877\pi\)
0.973468 + 0.228823i \(0.0734877\pi\)
\(824\) 12.3386 0.429836
\(825\) 0 0
\(826\) 0.768866 0.0267523
\(827\) 48.4371 1.68432 0.842161 0.539227i \(-0.181283\pi\)
0.842161 + 0.539227i \(0.181283\pi\)
\(828\) −6.71879 −0.233494
\(829\) 15.6816 0.544643 0.272321 0.962206i \(-0.412209\pi\)
0.272321 + 0.962206i \(0.412209\pi\)
\(830\) 6.13526 0.212958
\(831\) 6.36924 0.220946
\(832\) 5.30073 0.183770
\(833\) −1.47618 −0.0511467
\(834\) −5.57084 −0.192903
\(835\) −4.26261 −0.147514
\(836\) 0 0
\(837\) 5.71711 0.197612
\(838\) −36.2889 −1.25358
\(839\) −22.7942 −0.786945 −0.393472 0.919336i \(-0.628726\pi\)
−0.393472 + 0.919336i \(0.628726\pi\)
\(840\) −0.418877 −0.0144526
\(841\) 6.73269 0.232162
\(842\) 13.4042 0.461938
\(843\) −1.00310 −0.0345488
\(844\) −13.2771 −0.457016
\(845\) 15.0977 0.519377
\(846\) −4.24931 −0.146094
\(847\) 0 0
\(848\) −11.3487 −0.389717
\(849\) −5.03573 −0.172826
\(850\) 1.47618 0.0506327
\(851\) 0.0532895 0.00182674
\(852\) 4.99921 0.171270
\(853\) −13.4043 −0.458953 −0.229477 0.973314i \(-0.573701\pi\)
−0.229477 + 0.973314i \(0.573701\pi\)
\(854\) −8.06011 −0.275811
\(855\) 2.69035 0.0920080
\(856\) 9.65719 0.330076
\(857\) −6.20533 −0.211970 −0.105985 0.994368i \(-0.533800\pi\)
−0.105985 + 0.994368i \(0.533800\pi\)
\(858\) 0 0
\(859\) −34.2020 −1.16696 −0.583479 0.812128i \(-0.698309\pi\)
−0.583479 + 0.812128i \(0.698309\pi\)
\(860\) −3.83101 −0.130636
\(861\) 0.641661 0.0218678
\(862\) 2.38684 0.0812962
\(863\) −12.9621 −0.441233 −0.220617 0.975361i \(-0.570807\pi\)
−0.220617 + 0.975361i \(0.570807\pi\)
\(864\) −2.43977 −0.0830026
\(865\) −21.8753 −0.743783
\(866\) 22.6678 0.770284
\(867\) 6.20813 0.210839
\(868\) −2.34330 −0.0795368
\(869\) 0 0
\(870\) −2.50392 −0.0848907
\(871\) 64.5844 2.18836
\(872\) −5.91826 −0.200418
\(873\) −52.2151 −1.76721
\(874\) 2.26571 0.0766387
\(875\) −1.00000 −0.0338062
\(876\) 3.05152 0.103101
\(877\) −6.05485 −0.204458 −0.102229 0.994761i \(-0.532597\pi\)
−0.102229 + 0.994761i \(0.532597\pi\)
\(878\) 17.7876 0.600304
\(879\) 4.70920 0.158837
\(880\) 0 0
\(881\) −19.9937 −0.673605 −0.336803 0.941575i \(-0.609346\pi\)
−0.336803 + 0.941575i \(0.609346\pi\)
\(882\) 2.82454 0.0951073
\(883\) −7.40727 −0.249275 −0.124637 0.992202i \(-0.539777\pi\)
−0.124637 + 0.992202i \(0.539777\pi\)
\(884\) −7.82485 −0.263178
\(885\) −0.322061 −0.0108260
\(886\) 24.0914 0.809367
\(887\) −57.8294 −1.94172 −0.970860 0.239647i \(-0.922968\pi\)
−0.970860 + 0.239647i \(0.922968\pi\)
\(888\) 0.00938394 0.000314904 0
\(889\) −2.54206 −0.0852580
\(890\) −13.4596 −0.451166
\(891\) 0 0
\(892\) −10.6151 −0.355420
\(893\) 1.43295 0.0479519
\(894\) −1.65623 −0.0553927
\(895\) 12.0148 0.401610
\(896\) 1.00000 0.0334077
\(897\) −5.28160 −0.176347
\(898\) 6.06955 0.202543
\(899\) −14.0075 −0.467177
\(900\) −2.82454 −0.0941514
\(901\) 16.7528 0.558117
\(902\) 0 0
\(903\) −1.60472 −0.0534018
\(904\) −12.5957 −0.418927
\(905\) −21.0621 −0.700129
\(906\) 3.94399 0.131030
\(907\) −15.8050 −0.524795 −0.262398 0.964960i \(-0.584513\pi\)
−0.262398 + 0.964960i \(0.584513\pi\)
\(908\) 24.3926 0.809497
\(909\) 25.1986 0.835786
\(910\) 5.30073 0.175717
\(911\) −36.0360 −1.19393 −0.596963 0.802269i \(-0.703626\pi\)
−0.596963 + 0.802269i \(0.703626\pi\)
\(912\) 0.398977 0.0132114
\(913\) 0 0
\(914\) 4.92065 0.162761
\(915\) 3.37620 0.111614
\(916\) 11.2804 0.372716
\(917\) 7.80495 0.257742
\(918\) 3.60155 0.118869
\(919\) 40.4661 1.33485 0.667427 0.744675i \(-0.267396\pi\)
0.667427 + 0.744675i \(0.267396\pi\)
\(920\) −2.37872 −0.0784240
\(921\) 10.5185 0.346596
\(922\) −33.6918 −1.10958
\(923\) −63.2630 −2.08233
\(924\) 0 0
\(925\) 0.0224026 0.000736593 0
\(926\) −2.26163 −0.0743216
\(927\) 34.8510 1.14466
\(928\) 5.97768 0.196227
\(929\) 29.6298 0.972122 0.486061 0.873925i \(-0.338433\pi\)
0.486061 + 0.873925i \(0.338433\pi\)
\(930\) 0.981556 0.0321865
\(931\) −0.952490 −0.0312166
\(932\) −9.18604 −0.300899
\(933\) 1.10062 0.0360326
\(934\) −13.3924 −0.438213
\(935\) 0 0
\(936\) 14.9721 0.489379
\(937\) 8.28639 0.270705 0.135352 0.990798i \(-0.456783\pi\)
0.135352 + 0.990798i \(0.456783\pi\)
\(938\) 12.1841 0.397824
\(939\) −1.63410 −0.0533269
\(940\) −1.50443 −0.0490690
\(941\) −30.8591 −1.00598 −0.502988 0.864293i \(-0.667766\pi\)
−0.502988 + 0.864293i \(0.667766\pi\)
\(942\) −3.61228 −0.117694
\(943\) 3.64386 0.118661
\(944\) 0.768866 0.0250245
\(945\) −2.43977 −0.0793657
\(946\) 0 0
\(947\) 12.2671 0.398627 0.199313 0.979936i \(-0.436129\pi\)
0.199313 + 0.979936i \(0.436129\pi\)
\(948\) −5.36610 −0.174283
\(949\) −38.6158 −1.25352
\(950\) 0.952490 0.0309029
\(951\) 11.5537 0.374654
\(952\) −1.47618 −0.0478434
\(953\) 7.17969 0.232573 0.116286 0.993216i \(-0.462901\pi\)
0.116286 + 0.993216i \(0.462901\pi\)
\(954\) −32.0550 −1.03782
\(955\) −7.60935 −0.246233
\(956\) 3.05287 0.0987368
\(957\) 0 0
\(958\) 16.6637 0.538380
\(959\) −19.7473 −0.637674
\(960\) −0.418877 −0.0135192
\(961\) −25.5089 −0.822869
\(962\) −0.118750 −0.00382866
\(963\) 27.2771 0.878994
\(964\) 5.27801 0.169993
\(965\) −20.6268 −0.664000
\(966\) −0.996391 −0.0320584
\(967\) −5.07937 −0.163341 −0.0816707 0.996659i \(-0.526026\pi\)
−0.0816707 + 0.996659i \(0.526026\pi\)
\(968\) 0 0
\(969\) −0.588963 −0.0189202
\(970\) −18.4862 −0.593557
\(971\) −40.3117 −1.29366 −0.646832 0.762633i \(-0.723906\pi\)
−0.646832 + 0.762633i \(0.723906\pi\)
\(972\) −10.4406 −0.334884
\(973\) 13.2995 0.426361
\(974\) 12.6959 0.406804
\(975\) −2.22035 −0.0711082
\(976\) −8.06011 −0.257998
\(977\) −28.1223 −0.899711 −0.449855 0.893101i \(-0.648524\pi\)
−0.449855 + 0.893101i \(0.648524\pi\)
\(978\) −4.60367 −0.147209
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −16.7164 −0.533713
\(982\) 15.4392 0.492684
\(983\) 23.1675 0.738927 0.369464 0.929245i \(-0.379541\pi\)
0.369464 + 0.929245i \(0.379541\pi\)
\(984\) 0.641661 0.0204554
\(985\) 10.1376 0.323010
\(986\) −8.82416 −0.281019
\(987\) −0.630170 −0.0200585
\(988\) −5.04889 −0.160627
\(989\) −9.11289 −0.289773
\(990\) 0 0
\(991\) −33.5898 −1.06701 −0.533507 0.845795i \(-0.679126\pi\)
−0.533507 + 0.845795i \(0.679126\pi\)
\(992\) −2.34330 −0.0743999
\(993\) −14.0154 −0.444766
\(994\) −11.9348 −0.378548
\(995\) 22.6636 0.718484
\(996\) 2.56992 0.0814311
\(997\) 22.6238 0.716502 0.358251 0.933625i \(-0.383373\pi\)
0.358251 + 0.933625i \(0.383373\pi\)
\(998\) 5.96950 0.188961
\(999\) 0.0546572 0.00172928
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.cy.1.3 6
11.7 odd 10 770.2.n.i.71.2 12
11.8 odd 10 770.2.n.i.141.2 yes 12
11.10 odd 2 8470.2.a.de.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.i.71.2 12 11.7 odd 10
770.2.n.i.141.2 yes 12 11.8 odd 10
8470.2.a.cy.1.3 6 1.1 even 1 trivial
8470.2.a.de.1.3 6 11.10 odd 2