Properties

Label 668.2.a.a.1.1
Level 668
Weight 2
Character 668.1
Self dual yes
Analytic conductor 5.334
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 668 = 2^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 668.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.33400685502\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 668.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.30278 q^{3} -3.00000 q^{5} -0.302776 q^{7} -1.30278 q^{9} +O(q^{10})\) \(q-1.30278 q^{3} -3.00000 q^{5} -0.302776 q^{7} -1.30278 q^{9} +2.69722 q^{13} +3.90833 q^{15} +2.30278 q^{17} +2.00000 q^{19} +0.394449 q^{21} -2.30278 q^{23} +4.00000 q^{25} +5.60555 q^{27} +7.60555 q^{29} +6.60555 q^{31} +0.908327 q^{35} +0.394449 q^{37} -3.51388 q^{39} -6.21110 q^{41} +9.60555 q^{43} +3.90833 q^{45} +1.60555 q^{47} -6.90833 q^{49} -3.00000 q^{51} -4.60555 q^{53} -2.60555 q^{57} +7.81665 q^{59} +6.81665 q^{61} +0.394449 q^{63} -8.09167 q^{65} +8.21110 q^{67} +3.00000 q^{69} +3.90833 q^{71} -3.09167 q^{73} -5.21110 q^{75} +6.39445 q^{79} -3.39445 q^{81} -1.60555 q^{83} -6.90833 q^{85} -9.90833 q^{87} -13.8167 q^{89} -0.816654 q^{91} -8.60555 q^{93} -6.00000 q^{95} +4.51388 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 6q^{5} + 3q^{7} + q^{9} + O(q^{10}) \) \( 2q + q^{3} - 6q^{5} + 3q^{7} + q^{9} + 9q^{13} - 3q^{15} + q^{17} + 4q^{19} + 8q^{21} - q^{23} + 8q^{25} + 4q^{27} + 8q^{29} + 6q^{31} - 9q^{35} + 8q^{37} + 11q^{39} + 2q^{41} + 12q^{43} - 3q^{45} - 4q^{47} - 3q^{49} - 6q^{51} - 2q^{53} + 2q^{57} - 6q^{59} - 8q^{61} + 8q^{63} - 27q^{65} + 2q^{67} + 6q^{69} - 3q^{71} - 17q^{73} + 4q^{75} + 20q^{79} - 14q^{81} + 4q^{83} - 3q^{85} - 9q^{87} - 6q^{89} + 20q^{91} - 10q^{93} - 12q^{95} - 9q^{97} + O(q^{100}) \)

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
\(3\) −1.30278 −0.752158 −0.376079 0.926588i \(-0.622728\pi\)
−0.376079 + 0.926588i \(0.622728\pi\)
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) −0.302776 −0.114438 −0.0572192 0.998362i \(-0.518223\pi\)
−0.0572192 + 0.998362i \(0.518223\pi\)
\(8\) 0 0
\(9\) −1.30278 −0.434259
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 2.69722 0.748075 0.374038 0.927413i \(-0.377973\pi\)
0.374038 + 0.927413i \(0.377973\pi\)
\(14\) 0 0
\(15\) 3.90833 1.00913
\(16\) 0 0
\(17\) 2.30278 0.558505 0.279253 0.960218i \(-0.409913\pi\)
0.279253 + 0.960218i \(0.409913\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0.394449 0.0860758
\(22\) 0 0
\(23\) −2.30278 −0.480162 −0.240081 0.970753i \(-0.577174\pi\)
−0.240081 + 0.970753i \(0.577174\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 5.60555 1.07879
\(28\) 0 0
\(29\) 7.60555 1.41232 0.706158 0.708055i \(-0.250427\pi\)
0.706158 + 0.708055i \(0.250427\pi\)
\(30\) 0 0
\(31\) 6.60555 1.18639 0.593196 0.805058i \(-0.297866\pi\)
0.593196 + 0.805058i \(0.297866\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.908327 0.153535
\(36\) 0 0
\(37\) 0.394449 0.0648470 0.0324235 0.999474i \(-0.489677\pi\)
0.0324235 + 0.999474i \(0.489677\pi\)
\(38\) 0 0
\(39\) −3.51388 −0.562671
\(40\) 0 0
\(41\) −6.21110 −0.970011 −0.485006 0.874511i \(-0.661182\pi\)
−0.485006 + 0.874511i \(0.661182\pi\)
\(42\) 0 0
\(43\) 9.60555 1.46483 0.732416 0.680857i \(-0.238392\pi\)
0.732416 + 0.680857i \(0.238392\pi\)
\(44\) 0 0
\(45\) 3.90833 0.582619
\(46\) 0 0
\(47\) 1.60555 0.234194 0.117097 0.993120i \(-0.462641\pi\)
0.117097 + 0.993120i \(0.462641\pi\)
\(48\) 0 0
\(49\) −6.90833 −0.986904
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) −4.60555 −0.632621 −0.316311 0.948656i \(-0.602444\pi\)
−0.316311 + 0.948656i \(0.602444\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.60555 −0.345114
\(58\) 0 0
\(59\) 7.81665 1.01764 0.508821 0.860872i \(-0.330082\pi\)
0.508821 + 0.860872i \(0.330082\pi\)
\(60\) 0 0
\(61\) 6.81665 0.872783 0.436392 0.899757i \(-0.356256\pi\)
0.436392 + 0.899757i \(0.356256\pi\)
\(62\) 0 0
\(63\) 0.394449 0.0496959
\(64\) 0 0
\(65\) −8.09167 −1.00365
\(66\) 0 0
\(67\) 8.21110 1.00315 0.501573 0.865115i \(-0.332755\pi\)
0.501573 + 0.865115i \(0.332755\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 3.90833 0.463833 0.231917 0.972736i \(-0.425500\pi\)
0.231917 + 0.972736i \(0.425500\pi\)
\(72\) 0 0
\(73\) −3.09167 −0.361853 −0.180926 0.983497i \(-0.557910\pi\)
−0.180926 + 0.983497i \(0.557910\pi\)
\(74\) 0 0
\(75\) −5.21110 −0.601726
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.39445 0.719432 0.359716 0.933062i \(-0.382874\pi\)
0.359716 + 0.933062i \(0.382874\pi\)
\(80\) 0 0
\(81\) −3.39445 −0.377161
\(82\) 0 0
\(83\) −1.60555 −0.176232 −0.0881161 0.996110i \(-0.528085\pi\)
−0.0881161 + 0.996110i \(0.528085\pi\)
\(84\) 0 0
\(85\) −6.90833 −0.749313
\(86\) 0 0
\(87\) −9.90833 −1.06228
\(88\) 0 0
\(89\) −13.8167 −1.46456 −0.732281 0.681002i \(-0.761544\pi\)
−0.732281 + 0.681002i \(0.761544\pi\)
\(90\) 0 0
\(91\) −0.816654 −0.0856086
\(92\) 0 0
\(93\) −8.60555 −0.892354
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 4.51388 0.458315 0.229157 0.973389i \(-0.426403\pi\)
0.229157 + 0.973389i \(0.426403\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.8167 −1.07630 −0.538149 0.842850i \(-0.680876\pi\)
−0.538149 + 0.842850i \(0.680876\pi\)
\(102\) 0 0
\(103\) −3.30278 −0.325432 −0.162716 0.986673i \(-0.552025\pi\)
−0.162716 + 0.986673i \(0.552025\pi\)
\(104\) 0 0
\(105\) −1.18335 −0.115483
\(106\) 0 0
\(107\) 7.60555 0.735256 0.367628 0.929973i \(-0.380170\pi\)
0.367628 + 0.929973i \(0.380170\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −0.513878 −0.0487752
\(112\) 0 0
\(113\) 9.42221 0.886366 0.443183 0.896431i \(-0.353849\pi\)
0.443183 + 0.896431i \(0.353849\pi\)
\(114\) 0 0
\(115\) 6.90833 0.644205
\(116\) 0 0
\(117\) −3.51388 −0.324858
\(118\) 0 0
\(119\) −0.697224 −0.0639145
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 8.09167 0.729602
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 14.2111 1.26103 0.630516 0.776176i \(-0.282843\pi\)
0.630516 + 0.776176i \(0.282843\pi\)
\(128\) 0 0
\(129\) −12.5139 −1.10179
\(130\) 0 0
\(131\) −20.0278 −1.74983 −0.874917 0.484274i \(-0.839084\pi\)
−0.874917 + 0.484274i \(0.839084\pi\)
\(132\) 0 0
\(133\) −0.605551 −0.0525080
\(134\) 0 0
\(135\) −16.8167 −1.44735
\(136\) 0 0
\(137\) −2.78890 −0.238272 −0.119136 0.992878i \(-0.538012\pi\)
−0.119136 + 0.992878i \(0.538012\pi\)
\(138\) 0 0
\(139\) 5.90833 0.501138 0.250569 0.968099i \(-0.419382\pi\)
0.250569 + 0.968099i \(0.419382\pi\)
\(140\) 0 0
\(141\) −2.09167 −0.176151
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −22.8167 −1.89482
\(146\) 0 0
\(147\) 9.00000 0.742307
\(148\) 0 0
\(149\) 17.5139 1.43479 0.717396 0.696665i \(-0.245334\pi\)
0.717396 + 0.696665i \(0.245334\pi\)
\(150\) 0 0
\(151\) 19.5139 1.58802 0.794008 0.607907i \(-0.207991\pi\)
0.794008 + 0.607907i \(0.207991\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) −19.8167 −1.59171
\(156\) 0 0
\(157\) 0.816654 0.0651761 0.0325880 0.999469i \(-0.489625\pi\)
0.0325880 + 0.999469i \(0.489625\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0.697224 0.0549490
\(162\) 0 0
\(163\) −5.39445 −0.422526 −0.211263 0.977429i \(-0.567758\pi\)
−0.211263 + 0.977429i \(0.567758\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −5.72498 −0.440383
\(170\) 0 0
\(171\) −2.60555 −0.199251
\(172\) 0 0
\(173\) 5.78890 0.440122 0.220061 0.975486i \(-0.429374\pi\)
0.220061 + 0.975486i \(0.429374\pi\)
\(174\) 0 0
\(175\) −1.21110 −0.0915507
\(176\) 0 0
\(177\) −10.1833 −0.765427
\(178\) 0 0
\(179\) 0.908327 0.0678915 0.0339458 0.999424i \(-0.489193\pi\)
0.0339458 + 0.999424i \(0.489193\pi\)
\(180\) 0 0
\(181\) −7.21110 −0.535997 −0.267999 0.963419i \(-0.586362\pi\)
−0.267999 + 0.963419i \(0.586362\pi\)
\(182\) 0 0
\(183\) −8.88057 −0.656471
\(184\) 0 0
\(185\) −1.18335 −0.0870013
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.69722 −0.123455
\(190\) 0 0
\(191\) −11.3028 −0.817840 −0.408920 0.912570i \(-0.634095\pi\)
−0.408920 + 0.912570i \(0.634095\pi\)
\(192\) 0 0
\(193\) 5.90833 0.425291 0.212645 0.977129i \(-0.431792\pi\)
0.212645 + 0.977129i \(0.431792\pi\)
\(194\) 0 0
\(195\) 10.5416 0.754902
\(196\) 0 0
\(197\) 3.69722 0.263416 0.131708 0.991289i \(-0.457954\pi\)
0.131708 + 0.991289i \(0.457954\pi\)
\(198\) 0 0
\(199\) −10.6972 −0.758306 −0.379153 0.925334i \(-0.623785\pi\)
−0.379153 + 0.925334i \(0.623785\pi\)
\(200\) 0 0
\(201\) −10.6972 −0.754524
\(202\) 0 0
\(203\) −2.30278 −0.161623
\(204\) 0 0
\(205\) 18.6333 1.30141
\(206\) 0 0
\(207\) 3.00000 0.208514
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 13.5139 0.930334 0.465167 0.885223i \(-0.345994\pi\)
0.465167 + 0.885223i \(0.345994\pi\)
\(212\) 0 0
\(213\) −5.09167 −0.348876
\(214\) 0 0
\(215\) −28.8167 −1.96528
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) 4.02776 0.272171
\(220\) 0 0
\(221\) 6.21110 0.417804
\(222\) 0 0
\(223\) −0.513878 −0.0344118 −0.0172059 0.999852i \(-0.505477\pi\)
−0.0172059 + 0.999852i \(0.505477\pi\)
\(224\) 0 0
\(225\) −5.21110 −0.347407
\(226\) 0 0
\(227\) 1.18335 0.0785414 0.0392707 0.999229i \(-0.487497\pi\)
0.0392707 + 0.999229i \(0.487497\pi\)
\(228\) 0 0
\(229\) 8.69722 0.574729 0.287364 0.957821i \(-0.407221\pi\)
0.287364 + 0.957821i \(0.407221\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.90833 0.452580 0.226290 0.974060i \(-0.427340\pi\)
0.226290 + 0.974060i \(0.427340\pi\)
\(234\) 0 0
\(235\) −4.81665 −0.314204
\(236\) 0 0
\(237\) −8.33053 −0.541126
\(238\) 0 0
\(239\) 11.0917 0.717461 0.358730 0.933441i \(-0.383210\pi\)
0.358730 + 0.933441i \(0.383210\pi\)
\(240\) 0 0
\(241\) 2.69722 0.173743 0.0868717 0.996220i \(-0.472313\pi\)
0.0868717 + 0.996220i \(0.472313\pi\)
\(242\) 0 0
\(243\) −12.3944 −0.795104
\(244\) 0 0
\(245\) 20.7250 1.32407
\(246\) 0 0
\(247\) 5.39445 0.343241
\(248\) 0 0
\(249\) 2.09167 0.132554
\(250\) 0 0
\(251\) −10.3944 −0.656092 −0.328046 0.944662i \(-0.606390\pi\)
−0.328046 + 0.944662i \(0.606390\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 9.00000 0.563602
\(256\) 0 0
\(257\) 19.1194 1.19264 0.596319 0.802748i \(-0.296629\pi\)
0.596319 + 0.802748i \(0.296629\pi\)
\(258\) 0 0
\(259\) −0.119429 −0.00742099
\(260\) 0 0
\(261\) −9.90833 −0.613310
\(262\) 0 0
\(263\) 12.2111 0.752969 0.376484 0.926423i \(-0.377133\pi\)
0.376484 + 0.926423i \(0.377133\pi\)
\(264\) 0 0
\(265\) 13.8167 0.848750
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) 0 0
\(269\) 5.30278 0.323316 0.161658 0.986847i \(-0.448316\pi\)
0.161658 + 0.986847i \(0.448316\pi\)
\(270\) 0 0
\(271\) 14.4222 0.876087 0.438043 0.898954i \(-0.355672\pi\)
0.438043 + 0.898954i \(0.355672\pi\)
\(272\) 0 0
\(273\) 1.06392 0.0643912
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 25.5139 1.53298 0.766490 0.642256i \(-0.222001\pi\)
0.766490 + 0.642256i \(0.222001\pi\)
\(278\) 0 0
\(279\) −8.60555 −0.515201
\(280\) 0 0
\(281\) −15.4222 −0.920012 −0.460006 0.887916i \(-0.652153\pi\)
−0.460006 + 0.887916i \(0.652153\pi\)
\(282\) 0 0
\(283\) −10.4222 −0.619536 −0.309768 0.950812i \(-0.600251\pi\)
−0.309768 + 0.950812i \(0.600251\pi\)
\(284\) 0 0
\(285\) 7.81665 0.463019
\(286\) 0 0
\(287\) 1.88057 0.111007
\(288\) 0 0
\(289\) −11.6972 −0.688072
\(290\) 0 0
\(291\) −5.88057 −0.344725
\(292\) 0 0
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) −23.4500 −1.36531
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.21110 −0.359197
\(300\) 0 0
\(301\) −2.90833 −0.167633
\(302\) 0 0
\(303\) 14.0917 0.809545
\(304\) 0 0
\(305\) −20.4500 −1.17096
\(306\) 0 0
\(307\) 25.9361 1.48025 0.740125 0.672469i \(-0.234766\pi\)
0.740125 + 0.672469i \(0.234766\pi\)
\(308\) 0 0
\(309\) 4.30278 0.244776
\(310\) 0 0
\(311\) 6.21110 0.352199 0.176100 0.984372i \(-0.443652\pi\)
0.176100 + 0.984372i \(0.443652\pi\)
\(312\) 0 0
\(313\) 10.5139 0.594280 0.297140 0.954834i \(-0.403967\pi\)
0.297140 + 0.954834i \(0.403967\pi\)
\(314\) 0 0
\(315\) −1.18335 −0.0666740
\(316\) 0 0
\(317\) 28.5416 1.60306 0.801529 0.597956i \(-0.204020\pi\)
0.801529 + 0.597956i \(0.204020\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −9.90833 −0.553029
\(322\) 0 0
\(323\) 4.60555 0.256260
\(324\) 0 0
\(325\) 10.7889 0.598460
\(326\) 0 0
\(327\) −2.60555 −0.144087
\(328\) 0 0
\(329\) −0.486122 −0.0268008
\(330\) 0 0
\(331\) 9.18335 0.504762 0.252381 0.967628i \(-0.418786\pi\)
0.252381 + 0.967628i \(0.418786\pi\)
\(332\) 0 0
\(333\) −0.513878 −0.0281604
\(334\) 0 0
\(335\) −24.6333 −1.34586
\(336\) 0 0
\(337\) 27.6056 1.50377 0.751885 0.659294i \(-0.229145\pi\)
0.751885 + 0.659294i \(0.229145\pi\)
\(338\) 0 0
\(339\) −12.2750 −0.666688
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 4.21110 0.227378
\(344\) 0 0
\(345\) −9.00000 −0.484544
\(346\) 0 0
\(347\) 21.9083 1.17610 0.588050 0.808824i \(-0.299896\pi\)
0.588050 + 0.808824i \(0.299896\pi\)
\(348\) 0 0
\(349\) −36.4500 −1.95112 −0.975561 0.219729i \(-0.929483\pi\)
−0.975561 + 0.219729i \(0.929483\pi\)
\(350\) 0 0
\(351\) 15.1194 0.807015
\(352\) 0 0
\(353\) −4.18335 −0.222657 −0.111329 0.993784i \(-0.535511\pi\)
−0.111329 + 0.993784i \(0.535511\pi\)
\(354\) 0 0
\(355\) −11.7250 −0.622297
\(356\) 0 0
\(357\) 0.908327 0.0480738
\(358\) 0 0
\(359\) 1.18335 0.0624546 0.0312273 0.999512i \(-0.490058\pi\)
0.0312273 + 0.999512i \(0.490058\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 14.3305 0.752158
\(364\) 0 0
\(365\) 9.27502 0.485477
\(366\) 0 0
\(367\) 18.8167 0.982221 0.491111 0.871097i \(-0.336591\pi\)
0.491111 + 0.871097i \(0.336591\pi\)
\(368\) 0 0
\(369\) 8.09167 0.421236
\(370\) 0 0
\(371\) 1.39445 0.0723962
\(372\) 0 0
\(373\) −16.4861 −0.853619 −0.426810 0.904342i \(-0.640363\pi\)
−0.426810 + 0.904342i \(0.640363\pi\)
\(374\) 0 0
\(375\) −3.90833 −0.201825
\(376\) 0 0
\(377\) 20.5139 1.05652
\(378\) 0 0
\(379\) −0.0277564 −0.00142575 −0.000712875 1.00000i \(-0.500227\pi\)
−0.000712875 1.00000i \(0.500227\pi\)
\(380\) 0 0
\(381\) −18.5139 −0.948495
\(382\) 0 0
\(383\) −27.8444 −1.42278 −0.711391 0.702796i \(-0.751935\pi\)
−0.711391 + 0.702796i \(0.751935\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −12.5139 −0.636116
\(388\) 0 0
\(389\) −4.81665 −0.244214 −0.122107 0.992517i \(-0.538965\pi\)
−0.122107 + 0.992517i \(0.538965\pi\)
\(390\) 0 0
\(391\) −5.30278 −0.268173
\(392\) 0 0
\(393\) 26.0917 1.31615
\(394\) 0 0
\(395\) −19.1833 −0.965219
\(396\) 0 0
\(397\) 3.11943 0.156560 0.0782798 0.996931i \(-0.475057\pi\)
0.0782798 + 0.996931i \(0.475057\pi\)
\(398\) 0 0
\(399\) 0.788897 0.0394943
\(400\) 0 0
\(401\) −21.6972 −1.08351 −0.541754 0.840537i \(-0.682240\pi\)
−0.541754 + 0.840537i \(0.682240\pi\)
\(402\) 0 0
\(403\) 17.8167 0.887511
\(404\) 0 0
\(405\) 10.1833 0.506015
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 19.0917 0.944022 0.472011 0.881593i \(-0.343528\pi\)
0.472011 + 0.881593i \(0.343528\pi\)
\(410\) 0 0
\(411\) 3.63331 0.179218
\(412\) 0 0
\(413\) −2.36669 −0.116457
\(414\) 0 0
\(415\) 4.81665 0.236440
\(416\) 0 0
\(417\) −7.69722 −0.376935
\(418\) 0 0
\(419\) −12.2111 −0.596551 −0.298276 0.954480i \(-0.596411\pi\)
−0.298276 + 0.954480i \(0.596411\pi\)
\(420\) 0 0
\(421\) 12.6056 0.614357 0.307178 0.951652i \(-0.400615\pi\)
0.307178 + 0.951652i \(0.400615\pi\)
\(422\) 0 0
\(423\) −2.09167 −0.101701
\(424\) 0 0
\(425\) 9.21110 0.446804
\(426\) 0 0
\(427\) −2.06392 −0.0998799
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.0278 0.820198 0.410099 0.912041i \(-0.365494\pi\)
0.410099 + 0.912041i \(0.365494\pi\)
\(432\) 0 0
\(433\) −29.3305 −1.40954 −0.704768 0.709438i \(-0.748949\pi\)
−0.704768 + 0.709438i \(0.748949\pi\)
\(434\) 0 0
\(435\) 29.7250 1.42520
\(436\) 0 0
\(437\) −4.60555 −0.220313
\(438\) 0 0
\(439\) −22.2111 −1.06008 −0.530039 0.847973i \(-0.677823\pi\)
−0.530039 + 0.847973i \(0.677823\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 0.275019 0.0130666 0.00653328 0.999979i \(-0.497920\pi\)
0.00653328 + 0.999979i \(0.497920\pi\)
\(444\) 0 0
\(445\) 41.4500 1.96492
\(446\) 0 0
\(447\) −22.8167 −1.07919
\(448\) 0 0
\(449\) 3.97224 0.187462 0.0937309 0.995598i \(-0.470121\pi\)
0.0937309 + 0.995598i \(0.470121\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −25.4222 −1.19444
\(454\) 0 0
\(455\) 2.44996 0.114856
\(456\) 0 0
\(457\) 6.88057 0.321860 0.160930 0.986966i \(-0.448551\pi\)
0.160930 + 0.986966i \(0.448551\pi\)
\(458\) 0 0
\(459\) 12.9083 0.602509
\(460\) 0 0
\(461\) −21.6972 −1.01054 −0.505270 0.862961i \(-0.668607\pi\)
−0.505270 + 0.862961i \(0.668607\pi\)
\(462\) 0 0
\(463\) 1.78890 0.0831371 0.0415686 0.999136i \(-0.486765\pi\)
0.0415686 + 0.999136i \(0.486765\pi\)
\(464\) 0 0
\(465\) 25.8167 1.19722
\(466\) 0 0
\(467\) −13.1194 −0.607095 −0.303547 0.952816i \(-0.598171\pi\)
−0.303547 + 0.952816i \(0.598171\pi\)
\(468\) 0 0
\(469\) −2.48612 −0.114798
\(470\) 0 0
\(471\) −1.06392 −0.0490227
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 3.63331 0.166010 0.0830050 0.996549i \(-0.473548\pi\)
0.0830050 + 0.996549i \(0.473548\pi\)
\(480\) 0 0
\(481\) 1.06392 0.0485104
\(482\) 0 0
\(483\) −0.908327 −0.0413303
\(484\) 0 0
\(485\) −13.5416 −0.614894
\(486\) 0 0
\(487\) 30.8167 1.39644 0.698218 0.715885i \(-0.253977\pi\)
0.698218 + 0.715885i \(0.253977\pi\)
\(488\) 0 0
\(489\) 7.02776 0.317806
\(490\) 0 0
\(491\) 0.633308 0.0285808 0.0142904 0.999898i \(-0.495451\pi\)
0.0142904 + 0.999898i \(0.495451\pi\)
\(492\) 0 0
\(493\) 17.5139 0.788785
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.18335 −0.0530803
\(498\) 0 0
\(499\) 0.669468 0.0299695 0.0149848 0.999888i \(-0.495230\pi\)
0.0149848 + 0.999888i \(0.495230\pi\)
\(500\) 0 0
\(501\) 1.30278 0.0582037
\(502\) 0 0
\(503\) −0.486122 −0.0216751 −0.0108376 0.999941i \(-0.503450\pi\)
−0.0108376 + 0.999941i \(0.503450\pi\)
\(504\) 0 0
\(505\) 32.4500 1.44400
\(506\) 0 0
\(507\) 7.45837 0.331238
\(508\) 0 0
\(509\) 24.9083 1.10404 0.552021 0.833830i \(-0.313857\pi\)
0.552021 + 0.833830i \(0.313857\pi\)
\(510\) 0 0
\(511\) 0.936083 0.0414099
\(512\) 0 0
\(513\) 11.2111 0.494982
\(514\) 0 0
\(515\) 9.90833 0.436613
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −7.54163 −0.331041
\(520\) 0 0
\(521\) −43.8167 −1.91964 −0.959821 0.280612i \(-0.909463\pi\)
−0.959821 + 0.280612i \(0.909463\pi\)
\(522\) 0 0
\(523\) −8.81665 −0.385525 −0.192763 0.981245i \(-0.561745\pi\)
−0.192763 + 0.981245i \(0.561745\pi\)
\(524\) 0 0
\(525\) 1.57779 0.0688606
\(526\) 0 0
\(527\) 15.2111 0.662606
\(528\) 0 0
\(529\) −17.6972 −0.769445
\(530\) 0 0
\(531\) −10.1833 −0.441920
\(532\) 0 0
\(533\) −16.7527 −0.725642
\(534\) 0 0
\(535\) −22.8167 −0.986450
\(536\) 0 0
\(537\) −1.18335 −0.0510652
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −35.6056 −1.53080 −0.765401 0.643554i \(-0.777459\pi\)
−0.765401 + 0.643554i \(0.777459\pi\)
\(542\) 0 0
\(543\) 9.39445 0.403154
\(544\) 0 0
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −34.3583 −1.46905 −0.734527 0.678579i \(-0.762596\pi\)
−0.734527 + 0.678579i \(0.762596\pi\)
\(548\) 0 0
\(549\) −8.88057 −0.379014
\(550\) 0 0
\(551\) 15.2111 0.648015
\(552\) 0 0
\(553\) −1.93608 −0.0823306
\(554\) 0 0
\(555\) 1.54163 0.0654387
\(556\) 0 0
\(557\) 10.8167 0.458316 0.229158 0.973389i \(-0.426403\pi\)
0.229158 + 0.973389i \(0.426403\pi\)
\(558\) 0 0
\(559\) 25.9083 1.09581
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 23.5139 0.990992 0.495496 0.868610i \(-0.334986\pi\)
0.495496 + 0.868610i \(0.334986\pi\)
\(564\) 0 0
\(565\) −28.2666 −1.18919
\(566\) 0 0
\(567\) 1.02776 0.0431617
\(568\) 0 0
\(569\) 36.2111 1.51805 0.759024 0.651062i \(-0.225676\pi\)
0.759024 + 0.651062i \(0.225676\pi\)
\(570\) 0 0
\(571\) −14.5416 −0.608548 −0.304274 0.952584i \(-0.598414\pi\)
−0.304274 + 0.952584i \(0.598414\pi\)
\(572\) 0 0
\(573\) 14.7250 0.615145
\(574\) 0 0
\(575\) −9.21110 −0.384130
\(576\) 0 0
\(577\) −24.5139 −1.02053 −0.510263 0.860018i \(-0.670452\pi\)
−0.510263 + 0.860018i \(0.670452\pi\)
\(578\) 0 0
\(579\) −7.69722 −0.319886
\(580\) 0 0
\(581\) 0.486122 0.0201677
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 10.5416 0.435843
\(586\) 0 0
\(587\) −42.9083 −1.77102 −0.885508 0.464624i \(-0.846190\pi\)
−0.885508 + 0.464624i \(0.846190\pi\)
\(588\) 0 0
\(589\) 13.2111 0.544354
\(590\) 0 0
\(591\) −4.81665 −0.198131
\(592\) 0 0
\(593\) −28.3944 −1.16602 −0.583010 0.812465i \(-0.698125\pi\)
−0.583010 + 0.812465i \(0.698125\pi\)
\(594\) 0 0
\(595\) 2.09167 0.0857502
\(596\) 0 0
\(597\) 13.9361 0.570366
\(598\) 0 0
\(599\) −26.0917 −1.06608 −0.533038 0.846091i \(-0.678950\pi\)
−0.533038 + 0.846091i \(0.678950\pi\)
\(600\) 0 0
\(601\) 24.5416 1.00107 0.500537 0.865715i \(-0.333136\pi\)
0.500537 + 0.865715i \(0.333136\pi\)
\(602\) 0 0
\(603\) −10.6972 −0.435625
\(604\) 0 0
\(605\) 33.0000 1.34164
\(606\) 0 0
\(607\) 13.7250 0.557080 0.278540 0.960425i \(-0.410150\pi\)
0.278540 + 0.960425i \(0.410150\pi\)
\(608\) 0 0
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) 4.33053 0.175195
\(612\) 0 0
\(613\) −38.3305 −1.54816 −0.774078 0.633090i \(-0.781786\pi\)
−0.774078 + 0.633090i \(0.781786\pi\)
\(614\) 0 0
\(615\) −24.2750 −0.978863
\(616\) 0 0
\(617\) −38.4500 −1.54794 −0.773969 0.633224i \(-0.781731\pi\)
−0.773969 + 0.633224i \(0.781731\pi\)
\(618\) 0 0
\(619\) 25.0278 1.00595 0.502975 0.864301i \(-0.332239\pi\)
0.502975 + 0.864301i \(0.332239\pi\)
\(620\) 0 0
\(621\) −12.9083 −0.517993
\(622\) 0 0
\(623\) 4.18335 0.167602
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.908327 0.0362174
\(630\) 0 0
\(631\) −33.9361 −1.35097 −0.675487 0.737372i \(-0.736067\pi\)
−0.675487 + 0.737372i \(0.736067\pi\)
\(632\) 0 0
\(633\) −17.6056 −0.699758
\(634\) 0 0
\(635\) −42.6333 −1.69185
\(636\) 0 0
\(637\) −18.6333 −0.738279
\(638\) 0 0
\(639\) −5.09167 −0.201423
\(640\) 0 0
\(641\) −11.3028 −0.446433 −0.223216 0.974769i \(-0.571656\pi\)
−0.223216 + 0.974769i \(0.571656\pi\)
\(642\) 0 0
\(643\) −12.5139 −0.493499 −0.246750 0.969079i \(-0.579363\pi\)
−0.246750 + 0.969079i \(0.579363\pi\)
\(644\) 0 0
\(645\) 37.5416 1.47820
\(646\) 0 0
\(647\) −35.9361 −1.41279 −0.706397 0.707816i \(-0.749680\pi\)
−0.706397 + 0.707816i \(0.749680\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 2.60555 0.102120
\(652\) 0 0
\(653\) −13.1194 −0.513403 −0.256701 0.966491i \(-0.582636\pi\)
−0.256701 + 0.966491i \(0.582636\pi\)
\(654\) 0 0
\(655\) 60.0833 2.34765
\(656\) 0 0
\(657\) 4.02776 0.157138
\(658\) 0 0
\(659\) −19.8167 −0.771947 −0.385974 0.922510i \(-0.626134\pi\)
−0.385974 + 0.922510i \(0.626134\pi\)
\(660\) 0 0
\(661\) −38.1194 −1.48267 −0.741337 0.671133i \(-0.765808\pi\)
−0.741337 + 0.671133i \(0.765808\pi\)
\(662\) 0 0
\(663\) −8.09167 −0.314255
\(664\) 0 0
\(665\) 1.81665 0.0704468
\(666\) 0 0
\(667\) −17.5139 −0.678140
\(668\) 0 0
\(669\) 0.669468 0.0258831
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 41.6333 1.60485 0.802423 0.596756i \(-0.203544\pi\)
0.802423 + 0.596756i \(0.203544\pi\)
\(674\) 0 0
\(675\) 22.4222 0.863031
\(676\) 0 0
\(677\) −13.8167 −0.531017 −0.265509 0.964108i \(-0.585540\pi\)
−0.265509 + 0.964108i \(0.585540\pi\)
\(678\) 0 0
\(679\) −1.36669 −0.0524488
\(680\) 0 0
\(681\) −1.54163 −0.0590756
\(682\) 0 0
\(683\) −49.5416 −1.89566 −0.947829 0.318779i \(-0.896727\pi\)
−0.947829 + 0.318779i \(0.896727\pi\)
\(684\) 0 0
\(685\) 8.36669 0.319675
\(686\) 0 0
\(687\) −11.3305 −0.432287
\(688\) 0 0
\(689\) −12.4222 −0.473248
\(690\) 0 0
\(691\) −42.4500 −1.61487 −0.807436 0.589955i \(-0.799146\pi\)
−0.807436 + 0.589955i \(0.799146\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −17.7250 −0.672347
\(696\) 0 0
\(697\) −14.3028 −0.541756
\(698\) 0 0
\(699\) −9.00000 −0.340411
\(700\) 0 0
\(701\) 8.02776 0.303204 0.151602 0.988442i \(-0.451557\pi\)
0.151602 + 0.988442i \(0.451557\pi\)
\(702\) 0 0
\(703\) 0.788897 0.0297538
\(704\) 0 0
\(705\) 6.27502 0.236331
\(706\) 0 0
\(707\) 3.27502 0.123170
\(708\) 0 0
\(709\) 50.1472 1.88332 0.941659 0.336570i \(-0.109267\pi\)
0.941659 + 0.336570i \(0.109267\pi\)
\(710\) 0 0
\(711\) −8.33053 −0.312419
\(712\) 0 0
\(713\) −15.2111 −0.569660
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −14.4500 −0.539644
\(718\) 0 0
\(719\) −22.8806 −0.853301 −0.426651 0.904417i \(-0.640307\pi\)
−0.426651 + 0.904417i \(0.640307\pi\)
\(720\) 0 0
\(721\) 1.00000 0.0372419
\(722\) 0 0
\(723\) −3.51388 −0.130683
\(724\) 0 0
\(725\) 30.4222 1.12985
\(726\) 0 0
\(727\) 28.7250 1.06535 0.532675 0.846320i \(-0.321187\pi\)
0.532675 + 0.846320i \(0.321187\pi\)
\(728\) 0 0
\(729\) 26.3305 0.975205
\(730\) 0 0
\(731\) 22.1194 0.818117
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 0 0
\(735\) −27.0000 −0.995910
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −3.93608 −0.144791 −0.0723956 0.997376i \(-0.523064\pi\)
−0.0723956 + 0.997376i \(0.523064\pi\)
\(740\) 0 0
\(741\) −7.02776 −0.258171
\(742\) 0 0
\(743\) 52.1194 1.91208 0.956038 0.293242i \(-0.0947342\pi\)
0.956038 + 0.293242i \(0.0947342\pi\)
\(744\) 0 0
\(745\) −52.5416 −1.92498
\(746\) 0 0
\(747\) 2.09167 0.0765303
\(748\) 0 0
\(749\) −2.30278 −0.0841416
\(750\) 0 0
\(751\) −23.6056 −0.861379 −0.430689 0.902500i \(-0.641730\pi\)
−0.430689 + 0.902500i \(0.641730\pi\)
\(752\) 0 0
\(753\) 13.5416 0.493485
\(754\) 0 0
\(755\) −58.5416 −2.13055
\(756\) 0 0
\(757\) 22.7250 0.825953 0.412977 0.910742i \(-0.364489\pi\)
0.412977 + 0.910742i \(0.364489\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.69722 −0.134024 −0.0670121 0.997752i \(-0.521347\pi\)
−0.0670121 + 0.997752i \(0.521347\pi\)
\(762\) 0 0
\(763\) −0.605551 −0.0219224
\(764\) 0 0
\(765\) 9.00000 0.325396
\(766\) 0 0
\(767\) 21.0833 0.761273
\(768\) 0 0
\(769\) −14.8167 −0.534302 −0.267151 0.963655i \(-0.586082\pi\)
−0.267151 + 0.963655i \(0.586082\pi\)
\(770\) 0 0
\(771\) −24.9083 −0.897051
\(772\) 0 0
\(773\) 10.6056 0.381455 0.190728 0.981643i \(-0.438915\pi\)
0.190728 + 0.981643i \(0.438915\pi\)
\(774\) 0 0
\(775\) 26.4222 0.949114
\(776\) 0 0
\(777\) 0.155590 0.00558175
\(778\) 0 0
\(779\) −12.4222 −0.445072
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 42.6333 1.52359
\(784\) 0 0
\(785\) −2.44996 −0.0874429
\(786\) 0 0
\(787\) 43.4500 1.54882 0.774412 0.632682i \(-0.218046\pi\)
0.774412 + 0.632682i \(0.218046\pi\)
\(788\) 0 0
\(789\) −15.9083 −0.566351
\(790\) 0 0
\(791\) −2.85281 −0.101434
\(792\) 0 0
\(793\) 18.3860 0.652908
\(794\) 0 0
\(795\) −18.0000 −0.638394
\(796\) 0 0
\(797\) 36.6333 1.29762 0.648809 0.760951i \(-0.275267\pi\)
0.648809 + 0.760951i \(0.275267\pi\)
\(798\) 0 0
\(799\) 3.69722 0.130798
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −2.09167 −0.0737218
\(806\) 0 0
\(807\) −6.90833 −0.243185
\(808\) 0 0
\(809\) 23.7250 0.834126 0.417063 0.908878i \(-0.363059\pi\)
0.417063 + 0.908878i \(0.363059\pi\)
\(810\) 0 0
\(811\) −37.8444 −1.32890 −0.664448 0.747334i \(-0.731333\pi\)
−0.664448 + 0.747334i \(0.731333\pi\)
\(812\) 0 0
\(813\) −18.7889 −0.658955
\(814\) 0 0
\(815\) 16.1833 0.566878
\(816\) 0 0
\(817\) 19.2111 0.672111
\(818\) 0 0
\(819\) 1.06392 0.0371763
\(820\) 0 0
\(821\) 14.7250 0.513905 0.256953 0.966424i \(-0.417282\pi\)
0.256953 + 0.966424i \(0.417282\pi\)
\(822\) 0 0
\(823\) −21.3028 −0.742568 −0.371284 0.928519i \(-0.621082\pi\)
−0.371284 + 0.928519i \(0.621082\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.45837 0.0507123 0.0253562 0.999678i \(-0.491928\pi\)
0.0253562 + 0.999678i \(0.491928\pi\)
\(828\) 0 0
\(829\) 37.8722 1.31535 0.657677 0.753300i \(-0.271539\pi\)
0.657677 + 0.753300i \(0.271539\pi\)
\(830\) 0 0
\(831\) −33.2389 −1.15304
\(832\) 0 0
\(833\) −15.9083 −0.551191
\(834\) 0 0
\(835\) 3.00000 0.103819
\(836\) 0 0
\(837\) 37.0278 1.27987
\(838\) 0 0
\(839\) 40.3944 1.39457 0.697286 0.716793i \(-0.254391\pi\)
0.697286 + 0.716793i \(0.254391\pi\)
\(840\) 0 0
\(841\) 28.8444 0.994635
\(842\) 0 0
\(843\) 20.0917 0.691994
\(844\) 0 0
\(845\) 17.1749 0.590836
\(846\) 0 0
\(847\) 3.33053 0.114438
\(848\) 0 0
\(849\) 13.5778 0.465989
\(850\) 0 0
\(851\) −0.908327 −0.0311370
\(852\) 0 0
\(853\) 40.7250 1.39440 0.697198 0.716878i \(-0.254430\pi\)
0.697198 + 0.716878i \(0.254430\pi\)
\(854\) 0 0
\(855\) 7.81665 0.267324
\(856\) 0 0
\(857\) 25.5416 0.872486 0.436243 0.899829i \(-0.356309\pi\)
0.436243 + 0.899829i \(0.356309\pi\)
\(858\) 0 0
\(859\) −41.3944 −1.41236 −0.706180 0.708032i \(-0.749583\pi\)
−0.706180 + 0.708032i \(0.749583\pi\)
\(860\) 0 0
\(861\) −2.44996 −0.0834945
\(862\) 0 0
\(863\) −11.5139 −0.391937 −0.195968 0.980610i \(-0.562785\pi\)
−0.195968 + 0.980610i \(0.562785\pi\)
\(864\) 0 0
\(865\) −17.3667 −0.590485
\(866\) 0 0
\(867\) 15.2389 0.517539
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 22.1472 0.750429
\(872\) 0 0
\(873\) −5.88057 −0.199027
\(874\) 0 0
\(875\) −0.908327 −0.0307071
\(876\) 0 0
\(877\) 19.5139 0.658937 0.329468 0.944167i \(-0.393130\pi\)
0.329468 + 0.944167i \(0.393130\pi\)
\(878\) 0 0
\(879\) −23.4500 −0.790948
\(880\) 0 0
\(881\) −14.7889 −0.498251 −0.249125 0.968471i \(-0.580143\pi\)
−0.249125 + 0.968471i \(0.580143\pi\)
\(882\) 0 0
\(883\) −27.9361 −0.940124 −0.470062 0.882633i \(-0.655768\pi\)
−0.470062 + 0.882633i \(0.655768\pi\)
\(884\) 0 0
\(885\) 30.5500 1.02693
\(886\) 0 0
\(887\) 10.8167 0.363188 0.181594 0.983374i \(-0.441874\pi\)
0.181594 + 0.983374i \(0.441874\pi\)
\(888\) 0 0
\(889\) −4.30278 −0.144310
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.21110 0.107455
\(894\) 0 0
\(895\) −2.72498 −0.0910861
\(896\) 0 0
\(897\) 8.09167 0.270173
\(898\) 0 0
\(899\) 50.2389 1.67556
\(900\) 0 0
\(901\) −10.6056 −0.353322
\(902\) 0 0
\(903\) 3.78890 0.126087
\(904\) 0 0
\(905\) 21.6333 0.719115
\(906\) 0 0
\(907\) −4.63331 −0.153846 −0.0769232 0.997037i \(-0.524510\pi\)
−0.0769232 + 0.997037i \(0.524510\pi\)
\(908\) 0 0
\(909\) 14.0917 0.467391
\(910\) 0 0
\(911\) 7.18335 0.237995 0.118997 0.992895i \(-0.462032\pi\)
0.118997 + 0.992895i \(0.462032\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 26.6417 0.880748
\(916\) 0 0
\(917\) 6.06392 0.200248
\(918\) 0 0
\(919\) −0.0916731 −0.00302402 −0.00151201 0.999999i \(-0.500481\pi\)
−0.00151201 + 0.999999i \(0.500481\pi\)
\(920\) 0 0
\(921\) −33.7889 −1.11338
\(922\) 0 0
\(923\) 10.5416 0.346982
\(924\) 0 0
\(925\) 1.57779 0.0518776
\(926\) 0 0
\(927\) 4.30278 0.141322
\(928\) 0 0
\(929\) 15.4861 0.508083 0.254042 0.967193i \(-0.418240\pi\)
0.254042 + 0.967193i \(0.418240\pi\)
\(930\) 0 0
\(931\) −13.8167 −0.452823
\(932\) 0 0
\(933\) −8.09167 −0.264909
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −51.4500 −1.68080 −0.840398 0.541969i \(-0.817679\pi\)
−0.840398 + 0.541969i \(0.817679\pi\)
\(938\) 0 0
\(939\) −13.6972 −0.446992
\(940\) 0 0
\(941\) 59.2389 1.93113 0.965566 0.260159i \(-0.0837750\pi\)
0.965566 + 0.260159i \(0.0837750\pi\)
\(942\) 0 0
\(943\) 14.3028 0.465762
\(944\) 0 0
\(945\) 5.09167 0.165632
\(946\) 0 0
\(947\) 14.0917 0.457918 0.228959 0.973436i \(-0.426468\pi\)
0.228959 + 0.973436i \(0.426468\pi\)
\(948\) 0 0
\(949\) −8.33894 −0.270693
\(950\) 0 0
\(951\) −37.1833 −1.20575
\(952\) 0 0
\(953\) 42.3583 1.37212 0.686060 0.727545i \(-0.259339\pi\)
0.686060 + 0.727545i \(0.259339\pi\)
\(954\) 0 0
\(955\) 33.9083 1.09725
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0.844410 0.0272674
\(960\) 0 0
\(961\) 12.6333 0.407526
\(962\) 0 0
\(963\) −9.90833 −0.319291
\(964\) 0 0
\(965\) −17.7250 −0.570587
\(966\) 0 0
\(967\) −25.4861 −0.819578 −0.409789 0.912180i \(-0.634398\pi\)
−0.409789 + 0.912180i \(0.634398\pi\)
\(968\) 0 0
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) 49.4777 1.58782 0.793908 0.608038i \(-0.208043\pi\)
0.793908 + 0.608038i \(0.208043\pi\)
\(972\) 0 0
\(973\) −1.78890 −0.0573494
\(974\) 0 0
\(975\) −14.0555 −0.450137
\(976\) 0 0
\(977\) −42.5694 −1.36192 −0.680958 0.732323i \(-0.738436\pi\)
−0.680958 + 0.732323i \(0.738436\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −2.60555 −0.0831888
\(982\) 0 0
\(983\) −31.2666 −0.997250 −0.498625 0.866818i \(-0.666162\pi\)
−0.498625 + 0.866818i \(0.666162\pi\)
\(984\) 0 0
\(985\) −11.0917 −0.353410
\(986\) 0 0
\(987\) 0.633308 0.0201584
\(988\) 0 0
\(989\) −22.1194 −0.703357
\(990\) 0 0
\(991\) −19.6333 −0.623673 −0.311836 0.950136i \(-0.600944\pi\)
−0.311836 + 0.950136i \(0.600944\pi\)
\(992\) 0 0
\(993\) −11.9638 −0.379661
\(994\) 0 0
\(995\) 32.0917 1.01737
\(996\) 0 0
\(997\) −24.7889 −0.785072 −0.392536 0.919737i \(-0.628402\pi\)
−0.392536 + 0.919737i \(0.628402\pi\)
\(998\) 0 0
\(999\) 2.21110 0.0699562
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 668.2.a.a.1.1 2
3.2 odd 2 6012.2.a.a.1.1 2
4.3 odd 2 2672.2.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
668.2.a.a.1.1 2 1.1 even 1 trivial
2672.2.a.c.1.2 2 4.3 odd 2
6012.2.a.a.1.1 2 3.2 odd 2