Properties

Label 4560.2.a.bq.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1016.1
Defining polynomial: \(x^{3} - x^{2} - 6 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.85577\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.29966 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.29966 q^{7} +1.00000 q^{9} +5.01121 q^{11} +3.29966 q^{13} +1.00000 q^{15} +5.71155 q^{17} +1.00000 q^{19} +1.29966 q^{21} +3.71155 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.41188 q^{29} -7.71155 q^{31} -5.01121 q^{33} +1.29966 q^{35} +3.29966 q^{37} -3.29966 q^{39} -7.01121 q^{41} -5.29966 q^{43} -1.00000 q^{45} -3.71155 q^{47} -5.31087 q^{49} -5.71155 q^{51} +5.71155 q^{53} -5.01121 q^{55} -1.00000 q^{57} -6.59933 q^{59} +7.11222 q^{61} -1.29966 q^{63} -3.29966 q^{65} -11.4231 q^{67} -3.71155 q^{69} +10.0224 q^{71} +10.0000 q^{73} -1.00000 q^{75} -6.51289 q^{77} +11.4231 q^{79} +1.00000 q^{81} -16.5353 q^{83} -5.71155 q^{85} -4.41188 q^{87} -9.61054 q^{89} -4.28845 q^{91} +7.71155 q^{93} -1.00000 q^{95} +11.2997 q^{97} +5.01121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{9} - 4 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} + 3 q^{19} - 4 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} + 6 q^{37} - 6 q^{39} - 2 q^{41} - 12 q^{43} - 3 q^{45} + 4 q^{47} + 7 q^{49} - 2 q^{51} + 2 q^{53} + 4 q^{55} - 3 q^{57} - 12 q^{59} + 14 q^{61} - 6 q^{65} - 4 q^{67} + 4 q^{69} - 8 q^{71} + 30 q^{73} - 3 q^{75} - 20 q^{77} + 4 q^{79} + 3 q^{81} - 12 q^{83} - 2 q^{85} - 2 q^{87} - 2 q^{89} - 28 q^{91} + 8 q^{93} - 3 q^{95} + 30 q^{97} - 4 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.29966 −0.491227 −0.245613 0.969368i \(-0.578989\pi\)
−0.245613 + 0.969368i \(0.578989\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.01121 1.51094 0.755468 0.655185i \(-0.227409\pi\)
0.755468 + 0.655185i \(0.227409\pi\)
\(12\) 0 0
\(13\) 3.29966 0.915162 0.457581 0.889168i \(-0.348716\pi\)
0.457581 + 0.889168i \(0.348716\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 5.71155 1.38525 0.692627 0.721296i \(-0.256453\pi\)
0.692627 + 0.721296i \(0.256453\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.29966 0.283610
\(22\) 0 0
\(23\) 3.71155 0.773911 0.386955 0.922098i \(-0.373527\pi\)
0.386955 + 0.922098i \(0.373527\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.41188 0.819266 0.409633 0.912250i \(-0.365657\pi\)
0.409633 + 0.912250i \(0.365657\pi\)
\(30\) 0 0
\(31\) −7.71155 −1.38503 −0.692517 0.721401i \(-0.743498\pi\)
−0.692517 + 0.721401i \(0.743498\pi\)
\(32\) 0 0
\(33\) −5.01121 −0.872340
\(34\) 0 0
\(35\) 1.29966 0.219683
\(36\) 0 0
\(37\) 3.29966 0.542461 0.271231 0.962514i \(-0.412569\pi\)
0.271231 + 0.962514i \(0.412569\pi\)
\(38\) 0 0
\(39\) −3.29966 −0.528369
\(40\) 0 0
\(41\) −7.01121 −1.09497 −0.547483 0.836817i \(-0.684414\pi\)
−0.547483 + 0.836817i \(0.684414\pi\)
\(42\) 0 0
\(43\) −5.29966 −0.808191 −0.404096 0.914717i \(-0.632414\pi\)
−0.404096 + 0.914717i \(0.632414\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −3.71155 −0.541384 −0.270692 0.962666i \(-0.587253\pi\)
−0.270692 + 0.962666i \(0.587253\pi\)
\(48\) 0 0
\(49\) −5.31087 −0.758696
\(50\) 0 0
\(51\) −5.71155 −0.799776
\(52\) 0 0
\(53\) 5.71155 0.784541 0.392271 0.919850i \(-0.371690\pi\)
0.392271 + 0.919850i \(0.371690\pi\)
\(54\) 0 0
\(55\) −5.01121 −0.675711
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −6.59933 −0.859159 −0.429580 0.903029i \(-0.641338\pi\)
−0.429580 + 0.903029i \(0.641338\pi\)
\(60\) 0 0
\(61\) 7.11222 0.910626 0.455313 0.890331i \(-0.349527\pi\)
0.455313 + 0.890331i \(0.349527\pi\)
\(62\) 0 0
\(63\) −1.29966 −0.163742
\(64\) 0 0
\(65\) −3.29966 −0.409273
\(66\) 0 0
\(67\) −11.4231 −1.39555 −0.697776 0.716316i \(-0.745827\pi\)
−0.697776 + 0.716316i \(0.745827\pi\)
\(68\) 0 0
\(69\) −3.71155 −0.446818
\(70\) 0 0
\(71\) 10.0224 1.18944 0.594721 0.803932i \(-0.297262\pi\)
0.594721 + 0.803932i \(0.297262\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −6.51289 −0.742213
\(78\) 0 0
\(79\) 11.4231 1.28520 0.642599 0.766203i \(-0.277856\pi\)
0.642599 + 0.766203i \(0.277856\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.5353 −1.81499 −0.907493 0.420068i \(-0.862006\pi\)
−0.907493 + 0.420068i \(0.862006\pi\)
\(84\) 0 0
\(85\) −5.71155 −0.619504
\(86\) 0 0
\(87\) −4.41188 −0.473003
\(88\) 0 0
\(89\) −9.61054 −1.01871 −0.509357 0.860555i \(-0.670117\pi\)
−0.509357 + 0.860555i \(0.670117\pi\)
\(90\) 0 0
\(91\) −4.28845 −0.449552
\(92\) 0 0
\(93\) 7.71155 0.799650
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 11.2997 1.14731 0.573654 0.819098i \(-0.305526\pi\)
0.573654 + 0.819098i \(0.305526\pi\)
\(98\) 0 0
\(99\) 5.01121 0.503645
\(100\) 0 0
\(101\) 4.59933 0.457650 0.228825 0.973468i \(-0.426512\pi\)
0.228825 + 0.973468i \(0.426512\pi\)
\(102\) 0 0
\(103\) 11.4231 1.12555 0.562775 0.826610i \(-0.309734\pi\)
0.562775 + 0.826610i \(0.309734\pi\)
\(104\) 0 0
\(105\) −1.29966 −0.126834
\(106\) 0 0
\(107\) 11.4231 1.10431 0.552156 0.833741i \(-0.313805\pi\)
0.552156 + 0.833741i \(0.313805\pi\)
\(108\) 0 0
\(109\) 7.40067 0.708856 0.354428 0.935083i \(-0.384676\pi\)
0.354428 + 0.935083i \(0.384676\pi\)
\(110\) 0 0
\(111\) −3.29966 −0.313190
\(112\) 0 0
\(113\) 7.11222 0.669061 0.334531 0.942385i \(-0.391422\pi\)
0.334531 + 0.942385i \(0.391422\pi\)
\(114\) 0 0
\(115\) −3.71155 −0.346103
\(116\) 0 0
\(117\) 3.29966 0.305054
\(118\) 0 0
\(119\) −7.42309 −0.680474
\(120\) 0 0
\(121\) 14.1122 1.28293
\(122\) 0 0
\(123\) 7.01121 0.632179
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 19.4231 1.72352 0.861760 0.507316i \(-0.169362\pi\)
0.861760 + 0.507316i \(0.169362\pi\)
\(128\) 0 0
\(129\) 5.29966 0.466609
\(130\) 0 0
\(131\) −2.41188 −0.210727 −0.105364 0.994434i \(-0.533601\pi\)
−0.105364 + 0.994434i \(0.533601\pi\)
\(132\) 0 0
\(133\) −1.29966 −0.112695
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 7.19866 0.615023 0.307511 0.951544i \(-0.400504\pi\)
0.307511 + 0.951544i \(0.400504\pi\)
\(138\) 0 0
\(139\) −22.0224 −1.86792 −0.933959 0.357381i \(-0.883670\pi\)
−0.933959 + 0.357381i \(0.883670\pi\)
\(140\) 0 0
\(141\) 3.71155 0.312568
\(142\) 0 0
\(143\) 16.5353 1.38275
\(144\) 0 0
\(145\) −4.41188 −0.366387
\(146\) 0 0
\(147\) 5.31087 0.438033
\(148\) 0 0
\(149\) −8.22443 −0.673772 −0.336886 0.941545i \(-0.609374\pi\)
−0.336886 + 0.941545i \(0.609374\pi\)
\(150\) 0 0
\(151\) 5.48711 0.446535 0.223267 0.974757i \(-0.428328\pi\)
0.223267 + 0.974757i \(0.428328\pi\)
\(152\) 0 0
\(153\) 5.71155 0.461751
\(154\) 0 0
\(155\) 7.71155 0.619406
\(156\) 0 0
\(157\) −12.0224 −0.959493 −0.479747 0.877407i \(-0.659271\pi\)
−0.479747 + 0.877407i \(0.659271\pi\)
\(158\) 0 0
\(159\) −5.71155 −0.452955
\(160\) 0 0
\(161\) −4.82376 −0.380166
\(162\) 0 0
\(163\) 10.1234 0.792928 0.396464 0.918050i \(-0.370237\pi\)
0.396464 + 0.918050i \(0.370237\pi\)
\(164\) 0 0
\(165\) 5.01121 0.390122
\(166\) 0 0
\(167\) 6.88778 0.532993 0.266496 0.963836i \(-0.414134\pi\)
0.266496 + 0.963836i \(0.414134\pi\)
\(168\) 0 0
\(169\) −2.11222 −0.162478
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 21.5095 1.63534 0.817670 0.575688i \(-0.195266\pi\)
0.817670 + 0.575688i \(0.195266\pi\)
\(174\) 0 0
\(175\) −1.29966 −0.0982454
\(176\) 0 0
\(177\) 6.59933 0.496036
\(178\) 0 0
\(179\) −19.2211 −1.43665 −0.718325 0.695707i \(-0.755091\pi\)
−0.718325 + 0.695707i \(0.755091\pi\)
\(180\) 0 0
\(181\) 12.0224 0.893619 0.446810 0.894629i \(-0.352560\pi\)
0.446810 + 0.894629i \(0.352560\pi\)
\(182\) 0 0
\(183\) −7.11222 −0.525750
\(184\) 0 0
\(185\) −3.29966 −0.242596
\(186\) 0 0
\(187\) 28.6217 2.09303
\(188\) 0 0
\(189\) 1.29966 0.0945367
\(190\) 0 0
\(191\) −4.43430 −0.320855 −0.160427 0.987048i \(-0.551287\pi\)
−0.160427 + 0.987048i \(0.551287\pi\)
\(192\) 0 0
\(193\) 24.1234 1.73644 0.868221 0.496178i \(-0.165263\pi\)
0.868221 + 0.496178i \(0.165263\pi\)
\(194\) 0 0
\(195\) 3.29966 0.236294
\(196\) 0 0
\(197\) −7.48711 −0.533435 −0.266717 0.963775i \(-0.585939\pi\)
−0.266717 + 0.963775i \(0.585939\pi\)
\(198\) 0 0
\(199\) 3.17624 0.225158 0.112579 0.993643i \(-0.464089\pi\)
0.112579 + 0.993643i \(0.464089\pi\)
\(200\) 0 0
\(201\) 11.4231 0.805723
\(202\) 0 0
\(203\) −5.73396 −0.402445
\(204\) 0 0
\(205\) 7.01121 0.489684
\(206\) 0 0
\(207\) 3.71155 0.257970
\(208\) 0 0
\(209\) 5.01121 0.346633
\(210\) 0 0
\(211\) 18.8462 1.29742 0.648712 0.761034i \(-0.275308\pi\)
0.648712 + 0.761034i \(0.275308\pi\)
\(212\) 0 0
\(213\) −10.0224 −0.686725
\(214\) 0 0
\(215\) 5.29966 0.361434
\(216\) 0 0
\(217\) 10.0224 0.680366
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 18.8462 1.26773
\(222\) 0 0
\(223\) −3.42309 −0.229227 −0.114614 0.993410i \(-0.536563\pi\)
−0.114614 + 0.993410i \(0.536563\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −2.31087 −0.153378 −0.0766890 0.997055i \(-0.524435\pi\)
−0.0766890 + 0.997055i \(0.524435\pi\)
\(228\) 0 0
\(229\) 11.7340 0.775402 0.387701 0.921785i \(-0.373269\pi\)
0.387701 + 0.921785i \(0.373269\pi\)
\(230\) 0 0
\(231\) 6.51289 0.428517
\(232\) 0 0
\(233\) −18.0448 −1.18216 −0.591078 0.806614i \(-0.701298\pi\)
−0.591078 + 0.806614i \(0.701298\pi\)
\(234\) 0 0
\(235\) 3.71155 0.242115
\(236\) 0 0
\(237\) −11.4231 −0.742009
\(238\) 0 0
\(239\) −5.01121 −0.324148 −0.162074 0.986779i \(-0.551818\pi\)
−0.162074 + 0.986779i \(0.551818\pi\)
\(240\) 0 0
\(241\) −18.0448 −1.16237 −0.581185 0.813771i \(-0.697411\pi\)
−0.581185 + 0.813771i \(0.697411\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 5.31087 0.339299
\(246\) 0 0
\(247\) 3.29966 0.209953
\(248\) 0 0
\(249\) 16.5353 1.04788
\(250\) 0 0
\(251\) 7.61054 0.480373 0.240186 0.970727i \(-0.422791\pi\)
0.240186 + 0.970727i \(0.422791\pi\)
\(252\) 0 0
\(253\) 18.5993 1.16933
\(254\) 0 0
\(255\) 5.71155 0.357671
\(256\) 0 0
\(257\) −16.8878 −1.05343 −0.526715 0.850042i \(-0.676577\pi\)
−0.526715 + 0.850042i \(0.676577\pi\)
\(258\) 0 0
\(259\) −4.28845 −0.266472
\(260\) 0 0
\(261\) 4.41188 0.273089
\(262\) 0 0
\(263\) 27.5095 1.69631 0.848155 0.529748i \(-0.177713\pi\)
0.848155 + 0.529748i \(0.177713\pi\)
\(264\) 0 0
\(265\) −5.71155 −0.350857
\(266\) 0 0
\(267\) 9.61054 0.588155
\(268\) 0 0
\(269\) −5.61054 −0.342080 −0.171040 0.985264i \(-0.554713\pi\)
−0.171040 + 0.985264i \(0.554713\pi\)
\(270\) 0 0
\(271\) −22.6442 −1.37554 −0.687768 0.725931i \(-0.741409\pi\)
−0.687768 + 0.725931i \(0.741409\pi\)
\(272\) 0 0
\(273\) 4.28845 0.259549
\(274\) 0 0
\(275\) 5.01121 0.302187
\(276\) 0 0
\(277\) −22.0448 −1.32455 −0.662273 0.749263i \(-0.730408\pi\)
−0.662273 + 0.749263i \(0.730408\pi\)
\(278\) 0 0
\(279\) −7.71155 −0.461678
\(280\) 0 0
\(281\) −17.2356 −1.02819 −0.514096 0.857733i \(-0.671873\pi\)
−0.514096 + 0.857733i \(0.671873\pi\)
\(282\) 0 0
\(283\) −2.49832 −0.148510 −0.0742549 0.997239i \(-0.523658\pi\)
−0.0742549 + 0.997239i \(0.523658\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 9.11222 0.537877
\(288\) 0 0
\(289\) 15.6217 0.918926
\(290\) 0 0
\(291\) −11.2997 −0.662398
\(292\) 0 0
\(293\) −4.31087 −0.251844 −0.125922 0.992040i \(-0.540189\pi\)
−0.125922 + 0.992040i \(0.540189\pi\)
\(294\) 0 0
\(295\) 6.59933 0.384228
\(296\) 0 0
\(297\) −5.01121 −0.290780
\(298\) 0 0
\(299\) 12.2469 0.708254
\(300\) 0 0
\(301\) 6.88778 0.397005
\(302\) 0 0
\(303\) −4.59933 −0.264225
\(304\) 0 0
\(305\) −7.11222 −0.407244
\(306\) 0 0
\(307\) 30.0224 1.71347 0.856735 0.515757i \(-0.172489\pi\)
0.856735 + 0.515757i \(0.172489\pi\)
\(308\) 0 0
\(309\) −11.4231 −0.649837
\(310\) 0 0
\(311\) 30.6587 1.73850 0.869249 0.494375i \(-0.164603\pi\)
0.869249 + 0.494375i \(0.164603\pi\)
\(312\) 0 0
\(313\) 20.0224 1.13173 0.565867 0.824497i \(-0.308542\pi\)
0.565867 + 0.824497i \(0.308542\pi\)
\(314\) 0 0
\(315\) 1.29966 0.0732278
\(316\) 0 0
\(317\) −20.1089 −1.12943 −0.564713 0.825287i \(-0.691013\pi\)
−0.564713 + 0.825287i \(0.691013\pi\)
\(318\) 0 0
\(319\) 22.1089 1.23786
\(320\) 0 0
\(321\) −11.4231 −0.637575
\(322\) 0 0
\(323\) 5.71155 0.317799
\(324\) 0 0
\(325\) 3.29966 0.183032
\(326\) 0 0
\(327\) −7.40067 −0.409258
\(328\) 0 0
\(329\) 4.82376 0.265943
\(330\) 0 0
\(331\) −28.3333 −1.55734 −0.778669 0.627435i \(-0.784105\pi\)
−0.778669 + 0.627435i \(0.784105\pi\)
\(332\) 0 0
\(333\) 3.29966 0.180820
\(334\) 0 0
\(335\) 11.4231 0.624110
\(336\) 0 0
\(337\) 2.92477 0.159322 0.0796612 0.996822i \(-0.474616\pi\)
0.0796612 + 0.996822i \(0.474616\pi\)
\(338\) 0 0
\(339\) −7.11222 −0.386283
\(340\) 0 0
\(341\) −38.6442 −2.09270
\(342\) 0 0
\(343\) 16.0000 0.863919
\(344\) 0 0
\(345\) 3.71155 0.199823
\(346\) 0 0
\(347\) −1.11222 −0.0597069 −0.0298535 0.999554i \(-0.509504\pi\)
−0.0298535 + 0.999554i \(0.509504\pi\)
\(348\) 0 0
\(349\) 24.2244 1.29670 0.648352 0.761341i \(-0.275458\pi\)
0.648352 + 0.761341i \(0.275458\pi\)
\(350\) 0 0
\(351\) −3.29966 −0.176123
\(352\) 0 0
\(353\) 20.2244 1.07644 0.538219 0.842805i \(-0.319097\pi\)
0.538219 + 0.842805i \(0.319097\pi\)
\(354\) 0 0
\(355\) −10.0224 −0.531935
\(356\) 0 0
\(357\) 7.42309 0.392872
\(358\) 0 0
\(359\) 2.41188 0.127294 0.0636471 0.997972i \(-0.479727\pi\)
0.0636471 + 0.997972i \(0.479727\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −14.1122 −0.740699
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) −23.5689 −1.23029 −0.615144 0.788415i \(-0.710902\pi\)
−0.615144 + 0.788415i \(0.710902\pi\)
\(368\) 0 0
\(369\) −7.01121 −0.364989
\(370\) 0 0
\(371\) −7.42309 −0.385388
\(372\) 0 0
\(373\) −4.12343 −0.213503 −0.106751 0.994286i \(-0.534045\pi\)
−0.106751 + 0.994286i \(0.534045\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 14.5577 0.749761
\(378\) 0 0
\(379\) 22.5577 1.15871 0.579356 0.815074i \(-0.303304\pi\)
0.579356 + 0.815074i \(0.303304\pi\)
\(380\) 0 0
\(381\) −19.4231 −0.995075
\(382\) 0 0
\(383\) −19.4679 −0.994765 −0.497382 0.867531i \(-0.665705\pi\)
−0.497382 + 0.867531i \(0.665705\pi\)
\(384\) 0 0
\(385\) 6.51289 0.331928
\(386\) 0 0
\(387\) −5.29966 −0.269397
\(388\) 0 0
\(389\) −23.8204 −1.20774 −0.603871 0.797082i \(-0.706376\pi\)
−0.603871 + 0.797082i \(0.706376\pi\)
\(390\) 0 0
\(391\) 21.1987 1.07206
\(392\) 0 0
\(393\) 2.41188 0.121663
\(394\) 0 0
\(395\) −11.4231 −0.574758
\(396\) 0 0
\(397\) −25.4231 −1.27595 −0.637974 0.770058i \(-0.720227\pi\)
−0.637974 + 0.770058i \(0.720227\pi\)
\(398\) 0 0
\(399\) 1.29966 0.0650646
\(400\) 0 0
\(401\) 28.8316 1.43978 0.719891 0.694087i \(-0.244192\pi\)
0.719891 + 0.694087i \(0.244192\pi\)
\(402\) 0 0
\(403\) −25.4455 −1.26753
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 16.5353 0.819625
\(408\) 0 0
\(409\) 9.22107 0.455953 0.227976 0.973667i \(-0.426789\pi\)
0.227976 + 0.973667i \(0.426789\pi\)
\(410\) 0 0
\(411\) −7.19866 −0.355084
\(412\) 0 0
\(413\) 8.57691 0.422042
\(414\) 0 0
\(415\) 16.5353 0.811686
\(416\) 0 0
\(417\) 22.0224 1.07844
\(418\) 0 0
\(419\) −23.0336 −1.12527 −0.562633 0.826707i \(-0.690212\pi\)
−0.562633 + 0.826707i \(0.690212\pi\)
\(420\) 0 0
\(421\) −17.4679 −0.851335 −0.425667 0.904880i \(-0.639961\pi\)
−0.425667 + 0.904880i \(0.639961\pi\)
\(422\) 0 0
\(423\) −3.71155 −0.180461
\(424\) 0 0
\(425\) 5.71155 0.277051
\(426\) 0 0
\(427\) −9.24349 −0.447324
\(428\) 0 0
\(429\) −16.5353 −0.798332
\(430\) 0 0
\(431\) −17.4455 −0.840321 −0.420160 0.907450i \(-0.638026\pi\)
−0.420160 + 0.907450i \(0.638026\pi\)
\(432\) 0 0
\(433\) 18.5207 0.890050 0.445025 0.895518i \(-0.353195\pi\)
0.445025 + 0.895518i \(0.353195\pi\)
\(434\) 0 0
\(435\) 4.41188 0.211533
\(436\) 0 0
\(437\) 3.71155 0.177547
\(438\) 0 0
\(439\) 0.621746 0.0296743 0.0148372 0.999890i \(-0.495277\pi\)
0.0148372 + 0.999890i \(0.495277\pi\)
\(440\) 0 0
\(441\) −5.31087 −0.252899
\(442\) 0 0
\(443\) 8.91020 0.423336 0.211668 0.977342i \(-0.432110\pi\)
0.211668 + 0.977342i \(0.432110\pi\)
\(444\) 0 0
\(445\) 9.61054 0.455583
\(446\) 0 0
\(447\) 8.22443 0.389002
\(448\) 0 0
\(449\) −30.4343 −1.43628 −0.718142 0.695897i \(-0.755007\pi\)
−0.718142 + 0.695897i \(0.755007\pi\)
\(450\) 0 0
\(451\) −35.1346 −1.65443
\(452\) 0 0
\(453\) −5.48711 −0.257807
\(454\) 0 0
\(455\) 4.28845 0.201046
\(456\) 0 0
\(457\) 11.4455 0.535398 0.267699 0.963503i \(-0.413737\pi\)
0.267699 + 0.963503i \(0.413737\pi\)
\(458\) 0 0
\(459\) −5.71155 −0.266592
\(460\) 0 0
\(461\) 22.0448 1.02673 0.513365 0.858170i \(-0.328399\pi\)
0.513365 + 0.858170i \(0.328399\pi\)
\(462\) 0 0
\(463\) 16.5207 0.767784 0.383892 0.923378i \(-0.374584\pi\)
0.383892 + 0.923378i \(0.374584\pi\)
\(464\) 0 0
\(465\) −7.71155 −0.357614
\(466\) 0 0
\(467\) −0.910201 −0.0421191 −0.0210595 0.999778i \(-0.506704\pi\)
−0.0210595 + 0.999778i \(0.506704\pi\)
\(468\) 0 0
\(469\) 14.8462 0.685533
\(470\) 0 0
\(471\) 12.0224 0.553964
\(472\) 0 0
\(473\) −26.5577 −1.22113
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 5.71155 0.261514
\(478\) 0 0
\(479\) 9.63296 0.440141 0.220070 0.975484i \(-0.429371\pi\)
0.220070 + 0.975484i \(0.429371\pi\)
\(480\) 0 0
\(481\) 10.8878 0.496440
\(482\) 0 0
\(483\) 4.82376 0.219489
\(484\) 0 0
\(485\) −11.2997 −0.513091
\(486\) 0 0
\(487\) 30.5993 1.38659 0.693294 0.720655i \(-0.256159\pi\)
0.693294 + 0.720655i \(0.256159\pi\)
\(488\) 0 0
\(489\) −10.1234 −0.457797
\(490\) 0 0
\(491\) 26.4119 1.19195 0.595976 0.803002i \(-0.296765\pi\)
0.595976 + 0.803002i \(0.296765\pi\)
\(492\) 0 0
\(493\) 25.1987 1.13489
\(494\) 0 0
\(495\) −5.01121 −0.225237
\(496\) 0 0
\(497\) −13.0258 −0.584286
\(498\) 0 0
\(499\) 3.22107 0.144195 0.0720976 0.997398i \(-0.477031\pi\)
0.0720976 + 0.997398i \(0.477031\pi\)
\(500\) 0 0
\(501\) −6.88778 −0.307723
\(502\) 0 0
\(503\) 31.9584 1.42495 0.712477 0.701695i \(-0.247573\pi\)
0.712477 + 0.701695i \(0.247573\pi\)
\(504\) 0 0
\(505\) −4.59933 −0.204667
\(506\) 0 0
\(507\) 2.11222 0.0938068
\(508\) 0 0
\(509\) −16.4119 −0.727444 −0.363722 0.931508i \(-0.618494\pi\)
−0.363722 + 0.931508i \(0.618494\pi\)
\(510\) 0 0
\(511\) −12.9966 −0.574938
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −11.4231 −0.503361
\(516\) 0 0
\(517\) −18.5993 −0.817998
\(518\) 0 0
\(519\) −21.5095 −0.944164
\(520\) 0 0
\(521\) −7.01121 −0.307167 −0.153583 0.988136i \(-0.549081\pi\)
−0.153583 + 0.988136i \(0.549081\pi\)
\(522\) 0 0
\(523\) 17.4007 0.760878 0.380439 0.924806i \(-0.375773\pi\)
0.380439 + 0.924806i \(0.375773\pi\)
\(524\) 0 0
\(525\) 1.29966 0.0567220
\(526\) 0 0
\(527\) −44.0448 −1.91862
\(528\) 0 0
\(529\) −9.22443 −0.401062
\(530\) 0 0
\(531\) −6.59933 −0.286386
\(532\) 0 0
\(533\) −23.1346 −1.00207
\(534\) 0 0
\(535\) −11.4231 −0.493863
\(536\) 0 0
\(537\) 19.2211 0.829451
\(538\) 0 0
\(539\) −26.6139 −1.14634
\(540\) 0 0
\(541\) 20.8462 0.896247 0.448124 0.893972i \(-0.352092\pi\)
0.448124 + 0.893972i \(0.352092\pi\)
\(542\) 0 0
\(543\) −12.0224 −0.515931
\(544\) 0 0
\(545\) −7.40067 −0.317010
\(546\) 0 0
\(547\) 30.0224 1.28367 0.641833 0.766844i \(-0.278174\pi\)
0.641833 + 0.766844i \(0.278174\pi\)
\(548\) 0 0
\(549\) 7.11222 0.303542
\(550\) 0 0
\(551\) 4.41188 0.187952
\(552\) 0 0
\(553\) −14.8462 −0.631324
\(554\) 0 0
\(555\) 3.29966 0.140063
\(556\) 0 0
\(557\) −34.0448 −1.44253 −0.721263 0.692661i \(-0.756438\pi\)
−0.721263 + 0.692661i \(0.756438\pi\)
\(558\) 0 0
\(559\) −17.4871 −0.739626
\(560\) 0 0
\(561\) −28.6217 −1.20841
\(562\) 0 0
\(563\) 30.9326 1.30365 0.651827 0.758367i \(-0.274003\pi\)
0.651827 + 0.758367i \(0.274003\pi\)
\(564\) 0 0
\(565\) −7.11222 −0.299213
\(566\) 0 0
\(567\) −1.29966 −0.0545808
\(568\) 0 0
\(569\) 39.2581 1.64578 0.822892 0.568198i \(-0.192359\pi\)
0.822892 + 0.568198i \(0.192359\pi\)
\(570\) 0 0
\(571\) 41.4903 1.73632 0.868158 0.496287i \(-0.165304\pi\)
0.868158 + 0.496287i \(0.165304\pi\)
\(572\) 0 0
\(573\) 4.43430 0.185246
\(574\) 0 0
\(575\) 3.71155 0.154782
\(576\) 0 0
\(577\) −28.4713 −1.18528 −0.592638 0.805469i \(-0.701913\pi\)
−0.592638 + 0.805469i \(0.701913\pi\)
\(578\) 0 0
\(579\) −24.1234 −1.00254
\(580\) 0 0
\(581\) 21.4903 0.891570
\(582\) 0 0
\(583\) 28.6217 1.18539
\(584\) 0 0
\(585\) −3.29966 −0.136424
\(586\) 0 0
\(587\) −6.51289 −0.268816 −0.134408 0.990926i \(-0.542913\pi\)
−0.134408 + 0.990926i \(0.542913\pi\)
\(588\) 0 0
\(589\) −7.71155 −0.317749
\(590\) 0 0
\(591\) 7.48711 0.307979
\(592\) 0 0
\(593\) 21.4679 0.881582 0.440791 0.897610i \(-0.354698\pi\)
0.440791 + 0.897610i \(0.354698\pi\)
\(594\) 0 0
\(595\) 7.42309 0.304317
\(596\) 0 0
\(597\) −3.17624 −0.129995
\(598\) 0 0
\(599\) 25.2435 1.03142 0.515711 0.856763i \(-0.327528\pi\)
0.515711 + 0.856763i \(0.327528\pi\)
\(600\) 0 0
\(601\) 27.4455 1.11953 0.559763 0.828653i \(-0.310892\pi\)
0.559763 + 0.828653i \(0.310892\pi\)
\(602\) 0 0
\(603\) −11.4231 −0.465184
\(604\) 0 0
\(605\) −14.1122 −0.573743
\(606\) 0 0
\(607\) 35.2211 1.42958 0.714790 0.699340i \(-0.246522\pi\)
0.714790 + 0.699340i \(0.246522\pi\)
\(608\) 0 0
\(609\) 5.73396 0.232352
\(610\) 0 0
\(611\) −12.2469 −0.495455
\(612\) 0 0
\(613\) −11.4455 −0.462280 −0.231140 0.972921i \(-0.574245\pi\)
−0.231140 + 0.972921i \(0.574245\pi\)
\(614\) 0 0
\(615\) −7.01121 −0.282719
\(616\) 0 0
\(617\) −20.6858 −0.832778 −0.416389 0.909187i \(-0.636704\pi\)
−0.416389 + 0.909187i \(0.636704\pi\)
\(618\) 0 0
\(619\) −1.40067 −0.0562978 −0.0281489 0.999604i \(-0.508961\pi\)
−0.0281489 + 0.999604i \(0.508961\pi\)
\(620\) 0 0
\(621\) −3.71155 −0.148939
\(622\) 0 0
\(623\) 12.4905 0.500420
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −5.01121 −0.200128
\(628\) 0 0
\(629\) 18.8462 0.751446
\(630\) 0 0
\(631\) 23.5960 0.939341 0.469670 0.882842i \(-0.344373\pi\)
0.469670 + 0.882842i \(0.344373\pi\)
\(632\) 0 0
\(633\) −18.8462 −0.749068
\(634\) 0 0
\(635\) −19.4231 −0.770782
\(636\) 0 0
\(637\) −17.5241 −0.694330
\(638\) 0 0
\(639\) 10.0224 0.396481
\(640\) 0 0
\(641\) 39.6330 1.56541 0.782704 0.622394i \(-0.213840\pi\)
0.782704 + 0.622394i \(0.213840\pi\)
\(642\) 0 0
\(643\) −7.14920 −0.281937 −0.140969 0.990014i \(-0.545022\pi\)
−0.140969 + 0.990014i \(0.545022\pi\)
\(644\) 0 0
\(645\) −5.29966 −0.208674
\(646\) 0 0
\(647\) 31.9584 1.25641 0.628207 0.778046i \(-0.283789\pi\)
0.628207 + 0.778046i \(0.283789\pi\)
\(648\) 0 0
\(649\) −33.0706 −1.29814
\(650\) 0 0
\(651\) −10.0224 −0.392810
\(652\) 0 0
\(653\) 11.4871 0.449525 0.224763 0.974414i \(-0.427839\pi\)
0.224763 + 0.974414i \(0.427839\pi\)
\(654\) 0 0
\(655\) 2.41188 0.0942400
\(656\) 0 0
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) 24.0448 0.936654 0.468327 0.883555i \(-0.344857\pi\)
0.468327 + 0.883555i \(0.344857\pi\)
\(660\) 0 0
\(661\) 6.82376 0.265414 0.132707 0.991155i \(-0.457633\pi\)
0.132707 + 0.991155i \(0.457633\pi\)
\(662\) 0 0
\(663\) −18.8462 −0.731925
\(664\) 0 0
\(665\) 1.29966 0.0503988
\(666\) 0 0
\(667\) 16.3749 0.634038
\(668\) 0 0
\(669\) 3.42309 0.132344
\(670\) 0 0
\(671\) 35.6408 1.37590
\(672\) 0 0
\(673\) −9.69698 −0.373791 −0.186895 0.982380i \(-0.559843\pi\)
−0.186895 + 0.982380i \(0.559843\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −21.0898 −0.810547 −0.405273 0.914196i \(-0.632824\pi\)
−0.405273 + 0.914196i \(0.632824\pi\)
\(678\) 0 0
\(679\) −14.6858 −0.563588
\(680\) 0 0
\(681\) 2.31087 0.0885529
\(682\) 0 0
\(683\) −16.6217 −0.636013 −0.318007 0.948088i \(-0.603013\pi\)
−0.318007 + 0.948088i \(0.603013\pi\)
\(684\) 0 0
\(685\) −7.19866 −0.275047
\(686\) 0 0
\(687\) −11.7340 −0.447679
\(688\) 0 0
\(689\) 18.8462 0.717982
\(690\) 0 0
\(691\) −2.55449 −0.0971774 −0.0485887 0.998819i \(-0.515472\pi\)
−0.0485887 + 0.998819i \(0.515472\pi\)
\(692\) 0 0
\(693\) −6.51289 −0.247404
\(694\) 0 0
\(695\) 22.0224 0.835358
\(696\) 0 0
\(697\) −40.0448 −1.51681
\(698\) 0 0
\(699\) 18.0448 0.682518
\(700\) 0 0
\(701\) −45.0191 −1.70035 −0.850173 0.526503i \(-0.823503\pi\)
−0.850173 + 0.526503i \(0.823503\pi\)
\(702\) 0 0
\(703\) 3.29966 0.124449
\(704\) 0 0
\(705\) −3.71155 −0.139785
\(706\) 0 0
\(707\) −5.97758 −0.224810
\(708\) 0 0
\(709\) −25.3815 −0.953222 −0.476611 0.879114i \(-0.658135\pi\)
−0.476611 + 0.879114i \(0.658135\pi\)
\(710\) 0 0
\(711\) 11.4231 0.428399
\(712\) 0 0
\(713\) −28.6217 −1.07189
\(714\) 0 0
\(715\) −16.5353 −0.618385
\(716\) 0 0
\(717\) 5.01121 0.187147
\(718\) 0 0
\(719\) −9.63296 −0.359249 −0.179624 0.983735i \(-0.557488\pi\)
−0.179624 + 0.983735i \(0.557488\pi\)
\(720\) 0 0
\(721\) −14.8462 −0.552901
\(722\) 0 0
\(723\) 18.0448 0.671095
\(724\) 0 0
\(725\) 4.41188 0.163853
\(726\) 0 0
\(727\) −16.5207 −0.612720 −0.306360 0.951916i \(-0.599111\pi\)
−0.306360 + 0.951916i \(0.599111\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −30.2693 −1.11955
\(732\) 0 0
\(733\) −14.6217 −0.540067 −0.270033 0.962851i \(-0.587035\pi\)
−0.270033 + 0.962851i \(0.587035\pi\)
\(734\) 0 0
\(735\) −5.31087 −0.195895
\(736\) 0 0
\(737\) −57.2435 −2.10859
\(738\) 0 0
\(739\) −34.8462 −1.28184 −0.640919 0.767609i \(-0.721446\pi\)
−0.640919 + 0.767609i \(0.721446\pi\)
\(740\) 0 0
\(741\) −3.29966 −0.121216
\(742\) 0 0
\(743\) −4.62175 −0.169555 −0.0847777 0.996400i \(-0.527018\pi\)
−0.0847777 + 0.996400i \(0.527018\pi\)
\(744\) 0 0
\(745\) 8.22443 0.301320
\(746\) 0 0
\(747\) −16.5353 −0.604995
\(748\) 0 0
\(749\) −14.8462 −0.542468
\(750\) 0 0
\(751\) −5.48711 −0.200228 −0.100114 0.994976i \(-0.531921\pi\)
−0.100114 + 0.994976i \(0.531921\pi\)
\(752\) 0 0
\(753\) −7.61054 −0.277343
\(754\) 0 0
\(755\) −5.48711 −0.199696
\(756\) 0 0
\(757\) 38.2917 1.39174 0.695868 0.718170i \(-0.255020\pi\)
0.695868 + 0.718170i \(0.255020\pi\)
\(758\) 0 0
\(759\) −18.5993 −0.675113
\(760\) 0 0
\(761\) −19.7756 −0.716864 −0.358432 0.933556i \(-0.616688\pi\)
−0.358432 + 0.933556i \(0.616688\pi\)
\(762\) 0 0
\(763\) −9.61839 −0.348209
\(764\) 0 0
\(765\) −5.71155 −0.206501
\(766\) 0 0
\(767\) −21.7756 −0.786270
\(768\) 0 0
\(769\) −9.95516 −0.358992 −0.179496 0.983759i \(-0.557447\pi\)
−0.179496 + 0.983759i \(0.557447\pi\)
\(770\) 0 0
\(771\) 16.8878 0.608199
\(772\) 0 0
\(773\) 54.3781 1.95585 0.977923 0.208967i \(-0.0670102\pi\)
0.977923 + 0.208967i \(0.0670102\pi\)
\(774\) 0 0
\(775\) −7.71155 −0.277007
\(776\) 0 0
\(777\) 4.28845 0.153847
\(778\) 0 0
\(779\) −7.01121 −0.251203
\(780\) 0 0
\(781\) 50.2244 1.79717
\(782\) 0 0
\(783\) −4.41188 −0.157668
\(784\) 0 0
\(785\) 12.0224 0.429099
\(786\) 0 0
\(787\) −13.4455 −0.479281 −0.239640 0.970862i \(-0.577030\pi\)
−0.239640 + 0.970862i \(0.577030\pi\)
\(788\) 0 0
\(789\) −27.5095 −0.979365
\(790\) 0 0
\(791\) −9.24349 −0.328661
\(792\) 0 0
\(793\) 23.4679 0.833371
\(794\) 0 0
\(795\) 5.71155 0.202568
\(796\) 0 0
\(797\) −19.7340 −0.699013 −0.349506 0.936934i \(-0.613651\pi\)
−0.349506 + 0.936934i \(0.613651\pi\)
\(798\) 0 0
\(799\) −21.1987 −0.749955
\(800\) 0 0
\(801\) −9.61054 −0.339572
\(802\) 0 0
\(803\) 50.1121 1.76842
\(804\) 0 0
\(805\) 4.82376 0.170015
\(806\) 0 0
\(807\) 5.61054 0.197500
\(808\) 0 0
\(809\) 15.7756 0.554639 0.277320 0.960778i \(-0.410554\pi\)
0.277320 + 0.960778i \(0.410554\pi\)
\(810\) 0 0
\(811\) 5.64752 0.198311 0.0991557 0.995072i \(-0.468386\pi\)
0.0991557 + 0.995072i \(0.468386\pi\)
\(812\) 0 0
\(813\) 22.6442 0.794166
\(814\) 0 0
\(815\) −10.1234 −0.354608
\(816\) 0 0
\(817\) −5.29966 −0.185412
\(818\) 0 0
\(819\) −4.28845 −0.149851
\(820\) 0 0
\(821\) −49.6699 −1.73349 −0.866746 0.498749i \(-0.833793\pi\)
−0.866746 + 0.498749i \(0.833793\pi\)
\(822\) 0 0
\(823\) 46.7900 1.63100 0.815499 0.578759i \(-0.196463\pi\)
0.815499 + 0.578759i \(0.196463\pi\)
\(824\) 0 0
\(825\) −5.01121 −0.174468
\(826\) 0 0
\(827\) −16.0864 −0.559380 −0.279690 0.960090i \(-0.590232\pi\)
−0.279690 + 0.960090i \(0.590232\pi\)
\(828\) 0 0
\(829\) 18.8686 0.655334 0.327667 0.944793i \(-0.393738\pi\)
0.327667 + 0.944793i \(0.393738\pi\)
\(830\) 0 0
\(831\) 22.0448 0.764727
\(832\) 0 0
\(833\) −30.3333 −1.05099
\(834\) 0 0
\(835\) −6.88778 −0.238362
\(836\) 0 0
\(837\) 7.71155 0.266550
\(838\) 0 0
\(839\) −18.0224 −0.622203 −0.311101 0.950377i \(-0.600698\pi\)
−0.311101 + 0.950377i \(0.600698\pi\)
\(840\) 0 0
\(841\) −9.53531 −0.328804
\(842\) 0 0
\(843\) 17.2356 0.593627
\(844\) 0 0
\(845\) 2.11222 0.0726625
\(846\) 0 0
\(847\) −18.3411 −0.630209
\(848\) 0 0
\(849\) 2.49832 0.0857421
\(850\) 0 0
\(851\) 12.2469 0.419817
\(852\) 0 0
\(853\) 41.0930 1.40700 0.703499 0.710696i \(-0.251620\pi\)
0.703499 + 0.710696i \(0.251620\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 0.640931 0.0218938 0.0109469 0.999940i \(-0.496515\pi\)
0.0109469 + 0.999940i \(0.496515\pi\)
\(858\) 0 0
\(859\) −47.4679 −1.61958 −0.809792 0.586717i \(-0.800420\pi\)
−0.809792 + 0.586717i \(0.800420\pi\)
\(860\) 0 0
\(861\) −9.11222 −0.310544
\(862\) 0 0
\(863\) −49.0706 −1.67038 −0.835192 0.549959i \(-0.814643\pi\)
−0.835192 + 0.549959i \(0.814643\pi\)
\(864\) 0 0
\(865\) −21.5095 −0.731346
\(866\) 0 0
\(867\) −15.6217 −0.530542
\(868\) 0 0
\(869\) 57.2435 1.94185
\(870\) 0 0
\(871\) −37.6924 −1.27716
\(872\) 0 0
\(873\) 11.2997 0.382436
\(874\) 0 0
\(875\) 1.29966 0.0439367
\(876\) 0 0
\(877\) 49.1940 1.66116 0.830582 0.556896i \(-0.188008\pi\)
0.830582 + 0.556896i \(0.188008\pi\)
\(878\) 0 0
\(879\) 4.31087 0.145402
\(880\) 0 0
\(881\) 25.2211 0.849720 0.424860 0.905259i \(-0.360323\pi\)
0.424860 + 0.905259i \(0.360323\pi\)
\(882\) 0 0
\(883\) −7.12007 −0.239609 −0.119805 0.992797i \(-0.538227\pi\)
−0.119805 + 0.992797i \(0.538227\pi\)
\(884\) 0 0
\(885\) −6.59933 −0.221834
\(886\) 0 0
\(887\) −10.8013 −0.362674 −0.181337 0.983421i \(-0.558042\pi\)
−0.181337 + 0.983421i \(0.558042\pi\)
\(888\) 0 0
\(889\) −25.2435 −0.846640
\(890\) 0 0
\(891\) 5.01121 0.167882
\(892\) 0 0
\(893\) −3.71155 −0.124202
\(894\) 0 0
\(895\) 19.2211 0.642490
\(896\) 0 0
\(897\) −12.2469 −0.408910
\(898\) 0 0
\(899\) −34.0224 −1.13471
\(900\) 0 0
\(901\) 32.6217 1.08679
\(902\) 0 0
\(903\) −6.88778 −0.229211
\(904\) 0 0
\(905\) −12.0224 −0.399639
\(906\) 0 0
\(907\) 3.62511 0.120370 0.0601848 0.998187i \(-0.480831\pi\)
0.0601848 + 0.998187i \(0.480831\pi\)
\(908\) 0 0
\(909\) 4.59933 0.152550
\(910\) 0 0
\(911\) 5.77557 0.191353 0.0956765 0.995412i \(-0.469499\pi\)
0.0956765 + 0.995412i \(0.469499\pi\)
\(912\) 0 0
\(913\) −82.8619 −2.74233
\(914\) 0 0
\(915\) 7.11222 0.235123
\(916\) 0 0
\(917\) 3.13464 0.103515
\(918\) 0 0
\(919\) −40.6666 −1.34147 −0.670733 0.741699i \(-0.734021\pi\)
−0.670733 + 0.741699i \(0.734021\pi\)
\(920\) 0 0
\(921\) −30.0224 −0.989272
\(922\) 0 0
\(923\) 33.0706 1.08853
\(924\) 0 0
\(925\) 3.29966 0.108492
\(926\) 0 0
\(927\) 11.4231 0.375184
\(928\) 0 0
\(929\) 30.4197 0.998039 0.499020 0.866591i \(-0.333694\pi\)
0.499020 + 0.866591i \(0.333694\pi\)
\(930\) 0 0
\(931\) −5.31087 −0.174057
\(932\) 0 0
\(933\) −30.6587 −1.00372
\(934\) 0 0
\(935\) −28.6217 −0.936031
\(936\) 0 0
\(937\) −25.6699 −0.838600 −0.419300 0.907848i \(-0.637725\pi\)
−0.419300 + 0.907848i \(0.637725\pi\)
\(938\) 0 0
\(939\) −20.0224 −0.653407
\(940\) 0 0
\(941\) −51.3029 −1.67243 −0.836213 0.548404i \(-0.815236\pi\)
−0.836213 + 0.548404i \(0.815236\pi\)
\(942\) 0 0
\(943\) −26.0224 −0.847407
\(944\) 0 0
\(945\) −1.29966 −0.0422781
\(946\) 0 0
\(947\) −6.31087 −0.205076 −0.102538 0.994729i \(-0.532696\pi\)
−0.102538 + 0.994729i \(0.532696\pi\)
\(948\) 0 0
\(949\) 32.9966 1.07112
\(950\) 0 0
\(951\) 20.1089 0.652074
\(952\) 0 0
\(953\) 58.0032 1.87891 0.939455 0.342674i \(-0.111332\pi\)
0.939455 + 0.342674i \(0.111332\pi\)
\(954\) 0 0
\(955\) 4.43430 0.143491
\(956\) 0 0
\(957\) −22.1089 −0.714678
\(958\) 0 0
\(959\) −9.35584 −0.302116
\(960\) 0 0
\(961\) 28.4679 0.918320
\(962\) 0 0
\(963\) 11.4231 0.368104
\(964\) 0 0
\(965\) −24.1234 −0.776561
\(966\) 0 0
\(967\) 27.6970 0.890675 0.445337 0.895363i \(-0.353084\pi\)
0.445337 + 0.895363i \(0.353084\pi\)
\(968\) 0 0
\(969\) −5.71155 −0.183481
\(970\) 0 0
\(971\) −30.3973 −0.975496 −0.487748 0.872984i \(-0.662182\pi\)
−0.487748 + 0.872984i \(0.662182\pi\)
\(972\) 0 0
\(973\) 28.6217 0.917571
\(974\) 0 0
\(975\) −3.29966 −0.105674
\(976\) 0 0
\(977\) −9.17947 −0.293677 −0.146839 0.989160i \(-0.546910\pi\)
−0.146839 + 0.989160i \(0.546910\pi\)
\(978\) 0 0
\(979\) −48.1604 −1.53921
\(980\) 0 0
\(981\) 7.40067 0.236285
\(982\) 0 0
\(983\) −60.5801 −1.93221 −0.966103 0.258156i \(-0.916885\pi\)
−0.966103 + 0.258156i \(0.916885\pi\)
\(984\) 0 0
\(985\) 7.48711 0.238559
\(986\) 0 0
\(987\) −4.82376 −0.153542
\(988\) 0 0
\(989\) −19.6699 −0.625468
\(990\) 0 0
\(991\) 43.4231 1.37938 0.689690 0.724105i \(-0.257747\pi\)
0.689690 + 0.724105i \(0.257747\pi\)
\(992\) 0 0
\(993\) 28.3333 0.899130
\(994\) 0 0
\(995\) −3.17624 −0.100694
\(996\) 0 0
\(997\) −25.6251 −0.811555 −0.405778 0.913972i \(-0.632999\pi\)
−0.405778 + 0.913972i \(0.632999\pi\)
\(998\) 0 0
\(999\) −3.29966 −0.104397
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 4560.2.a.bq.1.2 3
4.3 odd 2 2280.2.a.t.1.2 3
12.11 even 2 6840.2.a.bn.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.t.1.2 3 4.3 odd 2
4560.2.a.bq.1.2 3 1.1 even 1 trivial
6840.2.a.bn.1.2 3 12.11 even 2