Properties

Label 3380.2.a.r.1.8
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 19 x^{7} + 16 x^{6} + 106 x^{5} - 87 x^{4} - 153 x^{3} + 149 x^{2} - 26 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-2.79462\) of defining polynomial
Character \(\chi\) \(=\) 3380.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.79462 q^{3} -1.00000 q^{5} -1.22908 q^{7} +4.80991 q^{9} +O(q^{10})\) \(q+2.79462 q^{3} -1.00000 q^{5} -1.22908 q^{7} +4.80991 q^{9} -3.73479 q^{11} -2.79462 q^{15} +4.95611 q^{17} +3.06501 q^{19} -3.43481 q^{21} -1.04385 q^{23} +1.00000 q^{25} +5.05802 q^{27} +9.02219 q^{29} +6.57334 q^{31} -10.4373 q^{33} +1.22908 q^{35} +0.619519 q^{37} +8.41141 q^{41} -8.50043 q^{43} -4.80991 q^{45} +5.65693 q^{47} -5.48936 q^{49} +13.8504 q^{51} +9.83977 q^{53} +3.73479 q^{55} +8.56555 q^{57} -12.1840 q^{59} +11.5333 q^{61} -5.91176 q^{63} +7.73375 q^{67} -2.91718 q^{69} +5.36427 q^{71} +14.3772 q^{73} +2.79462 q^{75} +4.59035 q^{77} -11.8261 q^{79} -0.294490 q^{81} -12.6989 q^{83} -4.95611 q^{85} +25.2136 q^{87} +13.5729 q^{89} +18.3700 q^{93} -3.06501 q^{95} -13.9803 q^{97} -17.9640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - q^{3} - 9q^{5} - q^{7} + 12q^{9} + O(q^{10}) \) \( 9q - q^{3} - 9q^{5} - q^{7} + 12q^{9} + 7q^{11} + q^{15} + 13q^{17} + 4q^{19} + 3q^{21} + 12q^{23} + 9q^{25} - 4q^{27} + 16q^{29} - 13q^{31} - 34q^{33} + q^{35} - q^{37} + 6q^{41} + q^{43} - 12q^{45} + 2q^{47} + 20q^{49} + 11q^{51} + 30q^{53} - 7q^{55} - 38q^{57} - 15q^{59} + 21q^{61} + 17q^{63} + 7q^{67} + 15q^{69} + 7q^{71} + 28q^{73} - q^{75} + 46q^{77} + 31q^{79} + 41q^{81} - 45q^{83} - 13q^{85} + 28q^{87} + 41q^{89} + 11q^{93} - 4q^{95} - 8q^{97} + 81q^{99} + O(q^{100}) \)

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
\(3\) 2.79462 1.61348 0.806738 0.590910i \(-0.201231\pi\)
0.806738 + 0.590910i \(0.201231\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.22908 −0.464548 −0.232274 0.972650i \(-0.574617\pi\)
−0.232274 + 0.972650i \(0.574617\pi\)
\(8\) 0 0
\(9\) 4.80991 1.60330
\(10\) 0 0
\(11\) −3.73479 −1.12608 −0.563040 0.826429i \(-0.690368\pi\)
−0.563040 + 0.826429i \(0.690368\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −2.79462 −0.721568
\(16\) 0 0
\(17\) 4.95611 1.20203 0.601016 0.799237i \(-0.294763\pi\)
0.601016 + 0.799237i \(0.294763\pi\)
\(18\) 0 0
\(19\) 3.06501 0.703162 0.351581 0.936158i \(-0.385644\pi\)
0.351581 + 0.936158i \(0.385644\pi\)
\(20\) 0 0
\(21\) −3.43481 −0.749537
\(22\) 0 0
\(23\) −1.04385 −0.217659 −0.108829 0.994060i \(-0.534710\pi\)
−0.108829 + 0.994060i \(0.534710\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.05802 0.973416
\(28\) 0 0
\(29\) 9.02219 1.67538 0.837689 0.546147i \(-0.183906\pi\)
0.837689 + 0.546147i \(0.183906\pi\)
\(30\) 0 0
\(31\) 6.57334 1.18061 0.590303 0.807182i \(-0.299008\pi\)
0.590303 + 0.807182i \(0.299008\pi\)
\(32\) 0 0
\(33\) −10.4373 −1.81690
\(34\) 0 0
\(35\) 1.22908 0.207752
\(36\) 0 0
\(37\) 0.619519 0.101848 0.0509241 0.998703i \(-0.483783\pi\)
0.0509241 + 0.998703i \(0.483783\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.41141 1.31364 0.656821 0.754047i \(-0.271901\pi\)
0.656821 + 0.754047i \(0.271901\pi\)
\(42\) 0 0
\(43\) −8.50043 −1.29630 −0.648152 0.761511i \(-0.724458\pi\)
−0.648152 + 0.761511i \(0.724458\pi\)
\(44\) 0 0
\(45\) −4.80991 −0.717019
\(46\) 0 0
\(47\) 5.65693 0.825148 0.412574 0.910924i \(-0.364630\pi\)
0.412574 + 0.910924i \(0.364630\pi\)
\(48\) 0 0
\(49\) −5.48936 −0.784195
\(50\) 0 0
\(51\) 13.8504 1.93945
\(52\) 0 0
\(53\) 9.83977 1.35160 0.675798 0.737087i \(-0.263799\pi\)
0.675798 + 0.737087i \(0.263799\pi\)
\(54\) 0 0
\(55\) 3.73479 0.503599
\(56\) 0 0
\(57\) 8.56555 1.13453
\(58\) 0 0
\(59\) −12.1840 −1.58622 −0.793110 0.609078i \(-0.791539\pi\)
−0.793110 + 0.609078i \(0.791539\pi\)
\(60\) 0 0
\(61\) 11.5333 1.47668 0.738342 0.674426i \(-0.235609\pi\)
0.738342 + 0.674426i \(0.235609\pi\)
\(62\) 0 0
\(63\) −5.91176 −0.744812
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.73375 0.944827 0.472414 0.881377i \(-0.343383\pi\)
0.472414 + 0.881377i \(0.343383\pi\)
\(68\) 0 0
\(69\) −2.91718 −0.351187
\(70\) 0 0
\(71\) 5.36427 0.636622 0.318311 0.947986i \(-0.396884\pi\)
0.318311 + 0.947986i \(0.396884\pi\)
\(72\) 0 0
\(73\) 14.3772 1.68273 0.841363 0.540471i \(-0.181754\pi\)
0.841363 + 0.540471i \(0.181754\pi\)
\(74\) 0 0
\(75\) 2.79462 0.322695
\(76\) 0 0
\(77\) 4.59035 0.523119
\(78\) 0 0
\(79\) −11.8261 −1.33054 −0.665271 0.746602i \(-0.731684\pi\)
−0.665271 + 0.746602i \(0.731684\pi\)
\(80\) 0 0
\(81\) −0.294490 −0.0327211
\(82\) 0 0
\(83\) −12.6989 −1.39389 −0.696943 0.717127i \(-0.745457\pi\)
−0.696943 + 0.717127i \(0.745457\pi\)
\(84\) 0 0
\(85\) −4.95611 −0.537565
\(86\) 0 0
\(87\) 25.2136 2.70318
\(88\) 0 0
\(89\) 13.5729 1.43872 0.719362 0.694635i \(-0.244434\pi\)
0.719362 + 0.694635i \(0.244434\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 18.3700 1.90488
\(94\) 0 0
\(95\) −3.06501 −0.314463
\(96\) 0 0
\(97\) −13.9803 −1.41948 −0.709742 0.704462i \(-0.751189\pi\)
−0.709742 + 0.704462i \(0.751189\pi\)
\(98\) 0 0
\(99\) −17.9640 −1.80545
\(100\) 0 0
\(101\) 7.43227 0.739538 0.369769 0.929124i \(-0.379437\pi\)
0.369769 + 0.929124i \(0.379437\pi\)
\(102\) 0 0
\(103\) 16.4706 1.62289 0.811447 0.584427i \(-0.198681\pi\)
0.811447 + 0.584427i \(0.198681\pi\)
\(104\) 0 0
\(105\) 3.43481 0.335203
\(106\) 0 0
\(107\) −8.76920 −0.847750 −0.423875 0.905721i \(-0.639330\pi\)
−0.423875 + 0.905721i \(0.639330\pi\)
\(108\) 0 0
\(109\) 7.17891 0.687615 0.343808 0.939040i \(-0.388283\pi\)
0.343808 + 0.939040i \(0.388283\pi\)
\(110\) 0 0
\(111\) 1.73132 0.164330
\(112\) 0 0
\(113\) −10.0341 −0.943931 −0.471966 0.881617i \(-0.656455\pi\)
−0.471966 + 0.881617i \(0.656455\pi\)
\(114\) 0 0
\(115\) 1.04385 0.0973399
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.09145 −0.558402
\(120\) 0 0
\(121\) 2.94864 0.268058
\(122\) 0 0
\(123\) 23.5067 2.11953
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 1.60663 0.142565 0.0712827 0.997456i \(-0.477291\pi\)
0.0712827 + 0.997456i \(0.477291\pi\)
\(128\) 0 0
\(129\) −23.7555 −2.09155
\(130\) 0 0
\(131\) −10.8166 −0.945051 −0.472525 0.881317i \(-0.656657\pi\)
−0.472525 + 0.881317i \(0.656657\pi\)
\(132\) 0 0
\(133\) −3.76714 −0.326653
\(134\) 0 0
\(135\) −5.05802 −0.435325
\(136\) 0 0
\(137\) −7.73123 −0.660523 −0.330262 0.943889i \(-0.607137\pi\)
−0.330262 + 0.943889i \(0.607137\pi\)
\(138\) 0 0
\(139\) 12.9186 1.09574 0.547869 0.836564i \(-0.315439\pi\)
0.547869 + 0.836564i \(0.315439\pi\)
\(140\) 0 0
\(141\) 15.8090 1.33136
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −9.02219 −0.749252
\(146\) 0 0
\(147\) −15.3407 −1.26528
\(148\) 0 0
\(149\) −17.0466 −1.39651 −0.698255 0.715849i \(-0.746040\pi\)
−0.698255 + 0.715849i \(0.746040\pi\)
\(150\) 0 0
\(151\) −2.41977 −0.196918 −0.0984588 0.995141i \(-0.531391\pi\)
−0.0984588 + 0.995141i \(0.531391\pi\)
\(152\) 0 0
\(153\) 23.8384 1.92722
\(154\) 0 0
\(155\) −6.57334 −0.527983
\(156\) 0 0
\(157\) 19.3659 1.54557 0.772785 0.634668i \(-0.218863\pi\)
0.772785 + 0.634668i \(0.218863\pi\)
\(158\) 0 0
\(159\) 27.4984 2.18077
\(160\) 0 0
\(161\) 1.28298 0.101113
\(162\) 0 0
\(163\) 8.19218 0.641661 0.320831 0.947137i \(-0.396038\pi\)
0.320831 + 0.947137i \(0.396038\pi\)
\(164\) 0 0
\(165\) 10.4373 0.812544
\(166\) 0 0
\(167\) 0.119894 0.00927770 0.00463885 0.999989i \(-0.498523\pi\)
0.00463885 + 0.999989i \(0.498523\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 14.7424 1.12738
\(172\) 0 0
\(173\) −13.3720 −1.01665 −0.508326 0.861165i \(-0.669735\pi\)
−0.508326 + 0.861165i \(0.669735\pi\)
\(174\) 0 0
\(175\) −1.22908 −0.0929097
\(176\) 0 0
\(177\) −34.0496 −2.55933
\(178\) 0 0
\(179\) 14.3046 1.06918 0.534588 0.845113i \(-0.320467\pi\)
0.534588 + 0.845113i \(0.320467\pi\)
\(180\) 0 0
\(181\) −9.65346 −0.717536 −0.358768 0.933427i \(-0.616803\pi\)
−0.358768 + 0.933427i \(0.616803\pi\)
\(182\) 0 0
\(183\) 32.2311 2.38259
\(184\) 0 0
\(185\) −0.619519 −0.0455479
\(186\) 0 0
\(187\) −18.5100 −1.35359
\(188\) 0 0
\(189\) −6.21670 −0.452199
\(190\) 0 0
\(191\) 2.60265 0.188321 0.0941607 0.995557i \(-0.469983\pi\)
0.0941607 + 0.995557i \(0.469983\pi\)
\(192\) 0 0
\(193\) 2.44821 0.176226 0.0881132 0.996110i \(-0.471916\pi\)
0.0881132 + 0.996110i \(0.471916\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −0.851216 −0.0606466 −0.0303233 0.999540i \(-0.509654\pi\)
−0.0303233 + 0.999540i \(0.509654\pi\)
\(198\) 0 0
\(199\) 9.06160 0.642360 0.321180 0.947018i \(-0.395921\pi\)
0.321180 + 0.947018i \(0.395921\pi\)
\(200\) 0 0
\(201\) 21.6129 1.52446
\(202\) 0 0
\(203\) −11.0890 −0.778294
\(204\) 0 0
\(205\) −8.41141 −0.587478
\(206\) 0 0
\(207\) −5.02085 −0.348973
\(208\) 0 0
\(209\) −11.4472 −0.791817
\(210\) 0 0
\(211\) −16.4317 −1.13121 −0.565603 0.824678i \(-0.691356\pi\)
−0.565603 + 0.824678i \(0.691356\pi\)
\(212\) 0 0
\(213\) 14.9911 1.02717
\(214\) 0 0
\(215\) 8.50043 0.579725
\(216\) 0 0
\(217\) −8.07915 −0.548449
\(218\) 0 0
\(219\) 40.1789 2.71504
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 23.2006 1.55363 0.776815 0.629729i \(-0.216834\pi\)
0.776815 + 0.629729i \(0.216834\pi\)
\(224\) 0 0
\(225\) 4.80991 0.320661
\(226\) 0 0
\(227\) −5.33774 −0.354278 −0.177139 0.984186i \(-0.556684\pi\)
−0.177139 + 0.984186i \(0.556684\pi\)
\(228\) 0 0
\(229\) −16.1535 −1.06745 −0.533726 0.845657i \(-0.679209\pi\)
−0.533726 + 0.845657i \(0.679209\pi\)
\(230\) 0 0
\(231\) 12.8283 0.844040
\(232\) 0 0
\(233\) −5.09722 −0.333930 −0.166965 0.985963i \(-0.553397\pi\)
−0.166965 + 0.985963i \(0.553397\pi\)
\(234\) 0 0
\(235\) −5.65693 −0.369017
\(236\) 0 0
\(237\) −33.0495 −2.14680
\(238\) 0 0
\(239\) −17.7527 −1.14833 −0.574163 0.818741i \(-0.694672\pi\)
−0.574163 + 0.818741i \(0.694672\pi\)
\(240\) 0 0
\(241\) −3.17008 −0.204203 −0.102101 0.994774i \(-0.532557\pi\)
−0.102101 + 0.994774i \(0.532557\pi\)
\(242\) 0 0
\(243\) −15.9970 −1.02621
\(244\) 0 0
\(245\) 5.48936 0.350703
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −35.4886 −2.24900
\(250\) 0 0
\(251\) −18.4648 −1.16549 −0.582744 0.812656i \(-0.698021\pi\)
−0.582744 + 0.812656i \(0.698021\pi\)
\(252\) 0 0
\(253\) 3.89858 0.245101
\(254\) 0 0
\(255\) −13.8504 −0.867348
\(256\) 0 0
\(257\) −6.02478 −0.375816 −0.187908 0.982187i \(-0.560171\pi\)
−0.187908 + 0.982187i \(0.560171\pi\)
\(258\) 0 0
\(259\) −0.761438 −0.0473134
\(260\) 0 0
\(261\) 43.3959 2.68614
\(262\) 0 0
\(263\) −28.3677 −1.74923 −0.874614 0.484820i \(-0.838885\pi\)
−0.874614 + 0.484820i \(0.838885\pi\)
\(264\) 0 0
\(265\) −9.83977 −0.604452
\(266\) 0 0
\(267\) 37.9311 2.32135
\(268\) 0 0
\(269\) 0.738744 0.0450420 0.0225210 0.999746i \(-0.492831\pi\)
0.0225210 + 0.999746i \(0.492831\pi\)
\(270\) 0 0
\(271\) −3.61053 −0.219324 −0.109662 0.993969i \(-0.534977\pi\)
−0.109662 + 0.993969i \(0.534977\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.73479 −0.225216
\(276\) 0 0
\(277\) −19.9027 −1.19583 −0.597917 0.801558i \(-0.704005\pi\)
−0.597917 + 0.801558i \(0.704005\pi\)
\(278\) 0 0
\(279\) 31.6172 1.89287
\(280\) 0 0
\(281\) 27.1022 1.61678 0.808391 0.588645i \(-0.200339\pi\)
0.808391 + 0.588645i \(0.200339\pi\)
\(282\) 0 0
\(283\) 3.45523 0.205392 0.102696 0.994713i \(-0.467253\pi\)
0.102696 + 0.994713i \(0.467253\pi\)
\(284\) 0 0
\(285\) −8.56555 −0.507379
\(286\) 0 0
\(287\) −10.3383 −0.610250
\(288\) 0 0
\(289\) 7.56298 0.444881
\(290\) 0 0
\(291\) −39.0696 −2.29030
\(292\) 0 0
\(293\) −13.8410 −0.808600 −0.404300 0.914626i \(-0.632485\pi\)
−0.404300 + 0.914626i \(0.632485\pi\)
\(294\) 0 0
\(295\) 12.1840 0.709379
\(296\) 0 0
\(297\) −18.8906 −1.09614
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 10.4477 0.602196
\(302\) 0 0
\(303\) 20.7704 1.19323
\(304\) 0 0
\(305\) −11.5333 −0.660393
\(306\) 0 0
\(307\) 13.4284 0.766401 0.383201 0.923665i \(-0.374822\pi\)
0.383201 + 0.923665i \(0.374822\pi\)
\(308\) 0 0
\(309\) 46.0290 2.61850
\(310\) 0 0
\(311\) 18.7934 1.06567 0.532837 0.846218i \(-0.321126\pi\)
0.532837 + 0.846218i \(0.321126\pi\)
\(312\) 0 0
\(313\) −22.1388 −1.25136 −0.625679 0.780081i \(-0.715178\pi\)
−0.625679 + 0.780081i \(0.715178\pi\)
\(314\) 0 0
\(315\) 5.91176 0.333090
\(316\) 0 0
\(317\) −21.7038 −1.21901 −0.609504 0.792783i \(-0.708631\pi\)
−0.609504 + 0.792783i \(0.708631\pi\)
\(318\) 0 0
\(319\) −33.6960 −1.88661
\(320\) 0 0
\(321\) −24.5066 −1.36782
\(322\) 0 0
\(323\) 15.1905 0.845223
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 20.0623 1.10945
\(328\) 0 0
\(329\) −6.95281 −0.383321
\(330\) 0 0
\(331\) −13.1827 −0.724588 −0.362294 0.932064i \(-0.618006\pi\)
−0.362294 + 0.932064i \(0.618006\pi\)
\(332\) 0 0
\(333\) 2.97983 0.163294
\(334\) 0 0
\(335\) −7.73375 −0.422540
\(336\) 0 0
\(337\) 18.9196 1.03062 0.515309 0.857004i \(-0.327677\pi\)
0.515309 + 0.857004i \(0.327677\pi\)
\(338\) 0 0
\(339\) −28.0416 −1.52301
\(340\) 0 0
\(341\) −24.5500 −1.32946
\(342\) 0 0
\(343\) 15.3504 0.828845
\(344\) 0 0
\(345\) 2.91718 0.157056
\(346\) 0 0
\(347\) −1.48879 −0.0799226 −0.0399613 0.999201i \(-0.512723\pi\)
−0.0399613 + 0.999201i \(0.512723\pi\)
\(348\) 0 0
\(349\) 15.5379 0.831725 0.415862 0.909428i \(-0.363480\pi\)
0.415862 + 0.909428i \(0.363480\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.4163 0.980199 0.490099 0.871667i \(-0.336960\pi\)
0.490099 + 0.871667i \(0.336960\pi\)
\(354\) 0 0
\(355\) −5.36427 −0.284706
\(356\) 0 0
\(357\) −17.0233 −0.900968
\(358\) 0 0
\(359\) −2.26366 −0.119472 −0.0597358 0.998214i \(-0.519026\pi\)
−0.0597358 + 0.998214i \(0.519026\pi\)
\(360\) 0 0
\(361\) −9.60571 −0.505564
\(362\) 0 0
\(363\) 8.24034 0.432505
\(364\) 0 0
\(365\) −14.3772 −0.752538
\(366\) 0 0
\(367\) 22.4716 1.17301 0.586504 0.809947i \(-0.300504\pi\)
0.586504 + 0.809947i \(0.300504\pi\)
\(368\) 0 0
\(369\) 40.4581 2.10617
\(370\) 0 0
\(371\) −12.0939 −0.627882
\(372\) 0 0
\(373\) 14.6760 0.759896 0.379948 0.925008i \(-0.375942\pi\)
0.379948 + 0.925008i \(0.375942\pi\)
\(374\) 0 0
\(375\) −2.79462 −0.144314
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −31.2728 −1.60638 −0.803188 0.595725i \(-0.796865\pi\)
−0.803188 + 0.595725i \(0.796865\pi\)
\(380\) 0 0
\(381\) 4.48992 0.230026
\(382\) 0 0
\(383\) 26.2659 1.34212 0.671062 0.741401i \(-0.265838\pi\)
0.671062 + 0.741401i \(0.265838\pi\)
\(384\) 0 0
\(385\) −4.59035 −0.233946
\(386\) 0 0
\(387\) −40.8863 −2.07837
\(388\) 0 0
\(389\) −18.8290 −0.954667 −0.477333 0.878722i \(-0.658397\pi\)
−0.477333 + 0.878722i \(0.658397\pi\)
\(390\) 0 0
\(391\) −5.17345 −0.261633
\(392\) 0 0
\(393\) −30.2283 −1.52482
\(394\) 0 0
\(395\) 11.8261 0.595037
\(396\) 0 0
\(397\) −23.8701 −1.19800 −0.599002 0.800747i \(-0.704436\pi\)
−0.599002 + 0.800747i \(0.704436\pi\)
\(398\) 0 0
\(399\) −10.5277 −0.527046
\(400\) 0 0
\(401\) −2.56312 −0.127996 −0.0639980 0.997950i \(-0.520385\pi\)
−0.0639980 + 0.997950i \(0.520385\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0.294490 0.0146333
\(406\) 0 0
\(407\) −2.31377 −0.114689
\(408\) 0 0
\(409\) −18.9442 −0.936732 −0.468366 0.883535i \(-0.655157\pi\)
−0.468366 + 0.883535i \(0.655157\pi\)
\(410\) 0 0
\(411\) −21.6059 −1.06574
\(412\) 0 0
\(413\) 14.9751 0.736876
\(414\) 0 0
\(415\) 12.6989 0.623365
\(416\) 0 0
\(417\) 36.1025 1.76795
\(418\) 0 0
\(419\) 11.9453 0.583567 0.291783 0.956484i \(-0.405751\pi\)
0.291783 + 0.956484i \(0.405751\pi\)
\(420\) 0 0
\(421\) 25.0406 1.22040 0.610201 0.792247i \(-0.291089\pi\)
0.610201 + 0.792247i \(0.291089\pi\)
\(422\) 0 0
\(423\) 27.2093 1.32296
\(424\) 0 0
\(425\) 4.95611 0.240406
\(426\) 0 0
\(427\) −14.1753 −0.685991
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.58228 −0.413394 −0.206697 0.978405i \(-0.566271\pi\)
−0.206697 + 0.978405i \(0.566271\pi\)
\(432\) 0 0
\(433\) 12.5767 0.604399 0.302200 0.953245i \(-0.402279\pi\)
0.302200 + 0.953245i \(0.402279\pi\)
\(434\) 0 0
\(435\) −25.2136 −1.20890
\(436\) 0 0
\(437\) −3.19943 −0.153049
\(438\) 0 0
\(439\) −33.2154 −1.58528 −0.792642 0.609687i \(-0.791295\pi\)
−0.792642 + 0.609687i \(0.791295\pi\)
\(440\) 0 0
\(441\) −26.4034 −1.25730
\(442\) 0 0
\(443\) −36.7404 −1.74559 −0.872795 0.488087i \(-0.837695\pi\)
−0.872795 + 0.488087i \(0.837695\pi\)
\(444\) 0 0
\(445\) −13.5729 −0.643417
\(446\) 0 0
\(447\) −47.6387 −2.25323
\(448\) 0 0
\(449\) 5.12272 0.241756 0.120878 0.992667i \(-0.461429\pi\)
0.120878 + 0.992667i \(0.461429\pi\)
\(450\) 0 0
\(451\) −31.4148 −1.47927
\(452\) 0 0
\(453\) −6.76233 −0.317722
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.0427 −1.12467 −0.562334 0.826910i \(-0.690097\pi\)
−0.562334 + 0.826910i \(0.690097\pi\)
\(458\) 0 0
\(459\) 25.0681 1.17008
\(460\) 0 0
\(461\) 4.22878 0.196954 0.0984770 0.995139i \(-0.468603\pi\)
0.0984770 + 0.995139i \(0.468603\pi\)
\(462\) 0 0
\(463\) 29.4028 1.36647 0.683233 0.730200i \(-0.260573\pi\)
0.683233 + 0.730200i \(0.260573\pi\)
\(464\) 0 0
\(465\) −18.3700 −0.851888
\(466\) 0 0
\(467\) −5.06469 −0.234366 −0.117183 0.993110i \(-0.537386\pi\)
−0.117183 + 0.993110i \(0.537386\pi\)
\(468\) 0 0
\(469\) −9.50539 −0.438918
\(470\) 0 0
\(471\) 54.1204 2.49374
\(472\) 0 0
\(473\) 31.7473 1.45974
\(474\) 0 0
\(475\) 3.06501 0.140632
\(476\) 0 0
\(477\) 47.3284 2.16702
\(478\) 0 0
\(479\) 7.18242 0.328173 0.164087 0.986446i \(-0.447532\pi\)
0.164087 + 0.986446i \(0.447532\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 3.58544 0.163143
\(484\) 0 0
\(485\) 13.9803 0.634813
\(486\) 0 0
\(487\) −19.0308 −0.862366 −0.431183 0.902264i \(-0.641904\pi\)
−0.431183 + 0.902264i \(0.641904\pi\)
\(488\) 0 0
\(489\) 22.8941 1.03530
\(490\) 0 0
\(491\) 9.89677 0.446635 0.223318 0.974746i \(-0.428311\pi\)
0.223318 + 0.974746i \(0.428311\pi\)
\(492\) 0 0
\(493\) 44.7149 2.01386
\(494\) 0 0
\(495\) 17.9640 0.807422
\(496\) 0 0
\(497\) −6.59311 −0.295742
\(498\) 0 0
\(499\) 11.7475 0.525890 0.262945 0.964811i \(-0.415306\pi\)
0.262945 + 0.964811i \(0.415306\pi\)
\(500\) 0 0
\(501\) 0.335059 0.0149693
\(502\) 0 0
\(503\) −7.50291 −0.334538 −0.167269 0.985911i \(-0.553495\pi\)
−0.167269 + 0.985911i \(0.553495\pi\)
\(504\) 0 0
\(505\) −7.43227 −0.330732
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.05609 0.135459 0.0677294 0.997704i \(-0.478425\pi\)
0.0677294 + 0.997704i \(0.478425\pi\)
\(510\) 0 0
\(511\) −17.6707 −0.781707
\(512\) 0 0
\(513\) 15.5029 0.684469
\(514\) 0 0
\(515\) −16.4706 −0.725780
\(516\) 0 0
\(517\) −21.1274 −0.929183
\(518\) 0 0
\(519\) −37.3696 −1.64034
\(520\) 0 0
\(521\) 4.66278 0.204280 0.102140 0.994770i \(-0.467431\pi\)
0.102140 + 0.994770i \(0.467431\pi\)
\(522\) 0 0
\(523\) 37.7910 1.65249 0.826243 0.563314i \(-0.190474\pi\)
0.826243 + 0.563314i \(0.190474\pi\)
\(524\) 0 0
\(525\) −3.43481 −0.149907
\(526\) 0 0
\(527\) 32.5782 1.41913
\(528\) 0 0
\(529\) −21.9104 −0.952625
\(530\) 0 0
\(531\) −58.6039 −2.54319
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 8.76920 0.379125
\(536\) 0 0
\(537\) 39.9760 1.72509
\(538\) 0 0
\(539\) 20.5016 0.883067
\(540\) 0 0
\(541\) 38.5091 1.65564 0.827818 0.560997i \(-0.189582\pi\)
0.827818 + 0.560997i \(0.189582\pi\)
\(542\) 0 0
\(543\) −26.9778 −1.15773
\(544\) 0 0
\(545\) −7.17891 −0.307511
\(546\) 0 0
\(547\) 6.31857 0.270163 0.135081 0.990835i \(-0.456870\pi\)
0.135081 + 0.990835i \(0.456870\pi\)
\(548\) 0 0
\(549\) 55.4740 2.36757
\(550\) 0 0
\(551\) 27.6531 1.17806
\(552\) 0 0
\(553\) 14.5352 0.618102
\(554\) 0 0
\(555\) −1.73132 −0.0734905
\(556\) 0 0
\(557\) −25.4127 −1.07677 −0.538385 0.842699i \(-0.680966\pi\)
−0.538385 + 0.842699i \(0.680966\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −51.7285 −2.18398
\(562\) 0 0
\(563\) 3.72041 0.156797 0.0783983 0.996922i \(-0.475019\pi\)
0.0783983 + 0.996922i \(0.475019\pi\)
\(564\) 0 0
\(565\) 10.0341 0.422139
\(566\) 0 0
\(567\) 0.361952 0.0152005
\(568\) 0 0
\(569\) 6.96568 0.292016 0.146008 0.989283i \(-0.453357\pi\)
0.146008 + 0.989283i \(0.453357\pi\)
\(570\) 0 0
\(571\) 9.94609 0.416231 0.208116 0.978104i \(-0.433267\pi\)
0.208116 + 0.978104i \(0.433267\pi\)
\(572\) 0 0
\(573\) 7.27343 0.303852
\(574\) 0 0
\(575\) −1.04385 −0.0435317
\(576\) 0 0
\(577\) 13.3817 0.557087 0.278544 0.960424i \(-0.410148\pi\)
0.278544 + 0.960424i \(0.410148\pi\)
\(578\) 0 0
\(579\) 6.84183 0.284337
\(580\) 0 0
\(581\) 15.6080 0.647527
\(582\) 0 0
\(583\) −36.7495 −1.52201
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −0.400413 −0.0165268 −0.00826340 0.999966i \(-0.502630\pi\)
−0.00826340 + 0.999966i \(0.502630\pi\)
\(588\) 0 0
\(589\) 20.1473 0.830157
\(590\) 0 0
\(591\) −2.37883 −0.0978518
\(592\) 0 0
\(593\) 16.8256 0.690945 0.345472 0.938429i \(-0.387719\pi\)
0.345472 + 0.938429i \(0.387719\pi\)
\(594\) 0 0
\(595\) 6.09145 0.249725
\(596\) 0 0
\(597\) 25.3237 1.03643
\(598\) 0 0
\(599\) −5.94610 −0.242951 −0.121475 0.992594i \(-0.538763\pi\)
−0.121475 + 0.992594i \(0.538763\pi\)
\(600\) 0 0
\(601\) 41.0898 1.67609 0.838045 0.545601i \(-0.183699\pi\)
0.838045 + 0.545601i \(0.183699\pi\)
\(602\) 0 0
\(603\) 37.1986 1.51485
\(604\) 0 0
\(605\) −2.94864 −0.119879
\(606\) 0 0
\(607\) 34.3906 1.39587 0.697935 0.716161i \(-0.254102\pi\)
0.697935 + 0.716161i \(0.254102\pi\)
\(608\) 0 0
\(609\) −30.9895 −1.25576
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 13.1645 0.531709 0.265854 0.964013i \(-0.414346\pi\)
0.265854 + 0.964013i \(0.414346\pi\)
\(614\) 0 0
\(615\) −23.5067 −0.947882
\(616\) 0 0
\(617\) −36.4559 −1.46766 −0.733830 0.679333i \(-0.762269\pi\)
−0.733830 + 0.679333i \(0.762269\pi\)
\(618\) 0 0
\(619\) 44.4340 1.78595 0.892976 0.450105i \(-0.148613\pi\)
0.892976 + 0.450105i \(0.148613\pi\)
\(620\) 0 0
\(621\) −5.27983 −0.211872
\(622\) 0 0
\(623\) −16.6822 −0.668357
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −31.9905 −1.27758
\(628\) 0 0
\(629\) 3.07040 0.122425
\(630\) 0 0
\(631\) −22.7501 −0.905666 −0.452833 0.891595i \(-0.649587\pi\)
−0.452833 + 0.891595i \(0.649587\pi\)
\(632\) 0 0
\(633\) −45.9204 −1.82517
\(634\) 0 0
\(635\) −1.60663 −0.0637572
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 25.8017 1.02070
\(640\) 0 0
\(641\) 36.2272 1.43089 0.715444 0.698670i \(-0.246225\pi\)
0.715444 + 0.698670i \(0.246225\pi\)
\(642\) 0 0
\(643\) 3.55026 0.140008 0.0700042 0.997547i \(-0.477699\pi\)
0.0700042 + 0.997547i \(0.477699\pi\)
\(644\) 0 0
\(645\) 23.7555 0.935372
\(646\) 0 0
\(647\) −13.2766 −0.521958 −0.260979 0.965344i \(-0.584045\pi\)
−0.260979 + 0.965344i \(0.584045\pi\)
\(648\) 0 0
\(649\) 45.5046 1.78621
\(650\) 0 0
\(651\) −22.5782 −0.884909
\(652\) 0 0
\(653\) 12.3080 0.481651 0.240825 0.970568i \(-0.422582\pi\)
0.240825 + 0.970568i \(0.422582\pi\)
\(654\) 0 0
\(655\) 10.8166 0.422640
\(656\) 0 0
\(657\) 69.1531 2.69792
\(658\) 0 0
\(659\) −13.0293 −0.507551 −0.253776 0.967263i \(-0.581673\pi\)
−0.253776 + 0.967263i \(0.581673\pi\)
\(660\) 0 0
\(661\) 27.5909 1.07316 0.536581 0.843849i \(-0.319716\pi\)
0.536581 + 0.843849i \(0.319716\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.76714 0.146083
\(666\) 0 0
\(667\) −9.41785 −0.364661
\(668\) 0 0
\(669\) 64.8370 2.50674
\(670\) 0 0
\(671\) −43.0743 −1.66287
\(672\) 0 0
\(673\) 11.5736 0.446131 0.223065 0.974803i \(-0.428394\pi\)
0.223065 + 0.974803i \(0.428394\pi\)
\(674\) 0 0
\(675\) 5.05802 0.194683
\(676\) 0 0
\(677\) 9.62321 0.369850 0.184925 0.982753i \(-0.440796\pi\)
0.184925 + 0.982753i \(0.440796\pi\)
\(678\) 0 0
\(679\) 17.1829 0.659419
\(680\) 0 0
\(681\) −14.9170 −0.571619
\(682\) 0 0
\(683\) −33.3432 −1.27584 −0.637921 0.770102i \(-0.720205\pi\)
−0.637921 + 0.770102i \(0.720205\pi\)
\(684\) 0 0
\(685\) 7.73123 0.295395
\(686\) 0 0
\(687\) −45.1429 −1.72231
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 43.1681 1.64219 0.821095 0.570791i \(-0.193363\pi\)
0.821095 + 0.570791i \(0.193363\pi\)
\(692\) 0 0
\(693\) 22.0792 0.838719
\(694\) 0 0
\(695\) −12.9186 −0.490029
\(696\) 0 0
\(697\) 41.6878 1.57904
\(698\) 0 0
\(699\) −14.2448 −0.538788
\(700\) 0 0
\(701\) 16.4423 0.621017 0.310508 0.950571i \(-0.399501\pi\)
0.310508 + 0.950571i \(0.399501\pi\)
\(702\) 0 0
\(703\) 1.89883 0.0716158
\(704\) 0 0
\(705\) −15.8090 −0.595400
\(706\) 0 0
\(707\) −9.13485 −0.343551
\(708\) 0 0
\(709\) −25.8445 −0.970611 −0.485306 0.874345i \(-0.661292\pi\)
−0.485306 + 0.874345i \(0.661292\pi\)
\(710\) 0 0
\(711\) −56.8826 −2.13326
\(712\) 0 0
\(713\) −6.86161 −0.256969
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −49.6120 −1.85279
\(718\) 0 0
\(719\) −7.66800 −0.285968 −0.142984 0.989725i \(-0.545670\pi\)
−0.142984 + 0.989725i \(0.545670\pi\)
\(720\) 0 0
\(721\) −20.2436 −0.753912
\(722\) 0 0
\(723\) −8.85917 −0.329476
\(724\) 0 0
\(725\) 9.02219 0.335076
\(726\) 0 0
\(727\) −35.4370 −1.31428 −0.657142 0.753767i \(-0.728235\pi\)
−0.657142 + 0.753767i \(0.728235\pi\)
\(728\) 0 0
\(729\) −43.8222 −1.62304
\(730\) 0 0
\(731\) −42.1290 −1.55820
\(732\) 0 0
\(733\) −48.0345 −1.77419 −0.887097 0.461582i \(-0.847282\pi\)
−0.887097 + 0.461582i \(0.847282\pi\)
\(734\) 0 0
\(735\) 15.3407 0.565850
\(736\) 0 0
\(737\) −28.8839 −1.06395
\(738\) 0 0
\(739\) −20.3284 −0.747794 −0.373897 0.927470i \(-0.621979\pi\)
−0.373897 + 0.927470i \(0.621979\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −39.5385 −1.45053 −0.725263 0.688472i \(-0.758282\pi\)
−0.725263 + 0.688472i \(0.758282\pi\)
\(744\) 0 0
\(745\) 17.0466 0.624538
\(746\) 0 0
\(747\) −61.0806 −2.23482
\(748\) 0 0
\(749\) 10.7780 0.393821
\(750\) 0 0
\(751\) −36.1294 −1.31838 −0.659191 0.751976i \(-0.729101\pi\)
−0.659191 + 0.751976i \(0.729101\pi\)
\(752\) 0 0
\(753\) −51.6021 −1.88049
\(754\) 0 0
\(755\) 2.41977 0.0880643
\(756\) 0 0
\(757\) −11.2361 −0.408382 −0.204191 0.978931i \(-0.565456\pi\)
−0.204191 + 0.978931i \(0.565456\pi\)
\(758\) 0 0
\(759\) 10.8950 0.395465
\(760\) 0 0
\(761\) 43.4661 1.57564 0.787822 0.615902i \(-0.211208\pi\)
0.787822 + 0.615902i \(0.211208\pi\)
\(762\) 0 0
\(763\) −8.82345 −0.319430
\(764\) 0 0
\(765\) −23.8384 −0.861880
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.98755 0.0716729 0.0358364 0.999358i \(-0.488590\pi\)
0.0358364 + 0.999358i \(0.488590\pi\)
\(770\) 0 0
\(771\) −16.8370 −0.606369
\(772\) 0 0
\(773\) −46.0457 −1.65615 −0.828074 0.560619i \(-0.810563\pi\)
−0.828074 + 0.560619i \(0.810563\pi\)
\(774\) 0 0
\(775\) 6.57334 0.236121
\(776\) 0 0
\(777\) −2.12793 −0.0763391
\(778\) 0 0
\(779\) 25.7811 0.923702
\(780\) 0 0
\(781\) −20.0344 −0.716887
\(782\) 0 0
\(783\) 45.6344 1.63084
\(784\) 0 0
\(785\) −19.3659 −0.691200
\(786\) 0 0
\(787\) −29.7703 −1.06120 −0.530598 0.847623i \(-0.678033\pi\)
−0.530598 + 0.847623i \(0.678033\pi\)
\(788\) 0 0
\(789\) −79.2770 −2.82234
\(790\) 0 0
\(791\) 12.3327 0.438502
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −27.4984 −0.975269
\(796\) 0 0
\(797\) −51.7725 −1.83388 −0.916938 0.399030i \(-0.869347\pi\)
−0.916938 + 0.399030i \(0.869347\pi\)
\(798\) 0 0
\(799\) 28.0363 0.991854
\(800\) 0 0
\(801\) 65.2844 2.30671
\(802\) 0 0
\(803\) −53.6958 −1.89489
\(804\) 0 0
\(805\) −1.28298 −0.0452191
\(806\) 0 0
\(807\) 2.06451 0.0726742
\(808\) 0 0
\(809\) −19.4923 −0.685312 −0.342656 0.939461i \(-0.611327\pi\)
−0.342656 + 0.939461i \(0.611327\pi\)
\(810\) 0 0
\(811\) 54.9207 1.92853 0.964264 0.264945i \(-0.0853538\pi\)
0.964264 + 0.264945i \(0.0853538\pi\)
\(812\) 0 0
\(813\) −10.0901 −0.353874
\(814\) 0 0
\(815\) −8.19218 −0.286960
\(816\) 0 0
\(817\) −26.0539 −0.911511
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 21.6079 0.754120 0.377060 0.926189i \(-0.376935\pi\)
0.377060 + 0.926189i \(0.376935\pi\)
\(822\) 0 0
\(823\) −26.0715 −0.908797 −0.454399 0.890799i \(-0.650146\pi\)
−0.454399 + 0.890799i \(0.650146\pi\)
\(824\) 0 0
\(825\) −10.4373 −0.363381
\(826\) 0 0
\(827\) 35.6861 1.24093 0.620463 0.784235i \(-0.286945\pi\)
0.620463 + 0.784235i \(0.286945\pi\)
\(828\) 0 0
\(829\) 45.9700 1.59661 0.798303 0.602256i \(-0.205732\pi\)
0.798303 + 0.602256i \(0.205732\pi\)
\(830\) 0 0
\(831\) −55.6204 −1.92945
\(832\) 0 0
\(833\) −27.2059 −0.942627
\(834\) 0 0
\(835\) −0.119894 −0.00414912
\(836\) 0 0
\(837\) 33.2480 1.14922
\(838\) 0 0
\(839\) 21.5917 0.745427 0.372714 0.927946i \(-0.378427\pi\)
0.372714 + 0.927946i \(0.378427\pi\)
\(840\) 0 0
\(841\) 52.3999 1.80689
\(842\) 0 0
\(843\) 75.7404 2.60864
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −3.62411 −0.124526
\(848\) 0 0
\(849\) 9.65605 0.331395
\(850\) 0 0
\(851\) −0.646687 −0.0221682
\(852\) 0 0
\(853\) −9.78391 −0.334995 −0.167497 0.985873i \(-0.553569\pi\)
−0.167497 + 0.985873i \(0.553569\pi\)
\(854\) 0 0
\(855\) −14.7424 −0.504180
\(856\) 0 0
\(857\) 43.4611 1.48460 0.742301 0.670067i \(-0.233735\pi\)
0.742301 + 0.670067i \(0.233735\pi\)
\(858\) 0 0
\(859\) −4.30641 −0.146933 −0.0734664 0.997298i \(-0.523406\pi\)
−0.0734664 + 0.997298i \(0.523406\pi\)
\(860\) 0 0
\(861\) −28.8916 −0.984624
\(862\) 0 0
\(863\) −20.2021 −0.687688 −0.343844 0.939027i \(-0.611729\pi\)
−0.343844 + 0.939027i \(0.611729\pi\)
\(864\) 0 0
\(865\) 13.3720 0.454660
\(866\) 0 0
\(867\) 21.1357 0.717805
\(868\) 0 0
\(869\) 44.1681 1.49830
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −67.2440 −2.27586
\(874\) 0 0
\(875\) 1.22908 0.0415505
\(876\) 0 0
\(877\) −28.9831 −0.978689 −0.489344 0.872091i \(-0.662764\pi\)
−0.489344 + 0.872091i \(0.662764\pi\)
\(878\) 0 0
\(879\) −38.6804 −1.30466
\(880\) 0 0
\(881\) −40.9072 −1.37820 −0.689100 0.724667i \(-0.741994\pi\)
−0.689100 + 0.724667i \(0.741994\pi\)
\(882\) 0 0
\(883\) −25.8593 −0.870234 −0.435117 0.900374i \(-0.643293\pi\)
−0.435117 + 0.900374i \(0.643293\pi\)
\(884\) 0 0
\(885\) 34.0496 1.14457
\(886\) 0 0
\(887\) −48.3313 −1.62281 −0.811404 0.584486i \(-0.801296\pi\)
−0.811404 + 0.584486i \(0.801296\pi\)
\(888\) 0 0
\(889\) −1.97468 −0.0662285
\(890\) 0 0
\(891\) 1.09986 0.0368466
\(892\) 0 0
\(893\) 17.3385 0.580212
\(894\) 0 0
\(895\) −14.3046 −0.478150
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 59.3059 1.97796
\(900\) 0 0
\(901\) 48.7669 1.62466
\(902\) 0 0
\(903\) 29.1974 0.971628
\(904\) 0 0
\(905\) 9.65346 0.320892
\(906\) 0 0
\(907\) −44.4104 −1.47462 −0.737311 0.675553i \(-0.763905\pi\)
−0.737311 + 0.675553i \(0.763905\pi\)
\(908\) 0 0
\(909\) 35.7485 1.18570
\(910\) 0 0
\(911\) −45.6542 −1.51259 −0.756296 0.654230i \(-0.772993\pi\)
−0.756296 + 0.654230i \(0.772993\pi\)
\(912\) 0 0
\(913\) 47.4277 1.56963
\(914\) 0 0
\(915\) −32.2311 −1.06553
\(916\) 0 0
\(917\) 13.2945 0.439022
\(918\) 0 0
\(919\) −26.9213 −0.888053 −0.444026 0.896014i \(-0.646450\pi\)
−0.444026 + 0.896014i \(0.646450\pi\)
\(920\) 0 0
\(921\) 37.5274 1.23657
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0.619519 0.0203696
\(926\) 0 0
\(927\) 79.2220 2.60199
\(928\) 0 0
\(929\) 7.00807 0.229927 0.114964 0.993370i \(-0.463325\pi\)
0.114964 + 0.993370i \(0.463325\pi\)
\(930\) 0 0
\(931\) −16.8250 −0.551416
\(932\) 0 0
\(933\) 52.5204 1.71944
\(934\) 0 0
\(935\) 18.5100 0.605342
\(936\) 0 0
\(937\) −34.8660 −1.13902 −0.569512 0.821983i \(-0.692868\pi\)
−0.569512 + 0.821983i \(0.692868\pi\)
\(938\) 0 0
\(939\) −61.8695 −2.01903
\(940\) 0 0
\(941\) −38.5412 −1.25641 −0.628204 0.778048i \(-0.716210\pi\)
−0.628204 + 0.778048i \(0.716210\pi\)
\(942\) 0 0
\(943\) −8.78029 −0.285926
\(944\) 0 0
\(945\) 6.21670 0.202229
\(946\) 0 0
\(947\) 21.8022 0.708475 0.354237 0.935156i \(-0.384740\pi\)
0.354237 + 0.935156i \(0.384740\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −60.6540 −1.96684
\(952\) 0 0
\(953\) 7.43281 0.240772 0.120386 0.992727i \(-0.461587\pi\)
0.120386 + 0.992727i \(0.461587\pi\)
\(954\) 0 0
\(955\) −2.60265 −0.0842199
\(956\) 0 0
\(957\) −94.1675 −3.04400
\(958\) 0 0
\(959\) 9.50229 0.306845
\(960\) 0 0
\(961\) 12.2088 0.393831
\(962\) 0 0
\(963\) −42.1791 −1.35920
\(964\) 0 0
\(965\) −2.44821 −0.0788108
\(966\) 0 0
\(967\) 10.0827 0.324237 0.162118 0.986771i \(-0.448167\pi\)
0.162118 + 0.986771i \(0.448167\pi\)
\(968\) 0 0
\(969\) 42.4517 1.36375
\(970\) 0 0
\(971\) −18.7318 −0.601131 −0.300565 0.953761i \(-0.597175\pi\)
−0.300565 + 0.953761i \(0.597175\pi\)
\(972\) 0 0
\(973\) −15.8779 −0.509023
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −34.3147 −1.09782 −0.548912 0.835880i \(-0.684958\pi\)
−0.548912 + 0.835880i \(0.684958\pi\)
\(978\) 0 0
\(979\) −50.6919 −1.62012
\(980\) 0 0
\(981\) 34.5299 1.10246
\(982\) 0 0
\(983\) −21.9179 −0.699074 −0.349537 0.936923i \(-0.613661\pi\)
−0.349537 + 0.936923i \(0.613661\pi\)
\(984\) 0 0
\(985\) 0.851216 0.0271220
\(986\) 0 0
\(987\) −19.4305 −0.618479
\(988\) 0 0
\(989\) 8.87321 0.282152
\(990\) 0 0
\(991\) −4.18345 −0.132892 −0.0664459 0.997790i \(-0.521166\pi\)
−0.0664459 + 0.997790i \(0.521166\pi\)
\(992\) 0 0
\(993\) −36.8407 −1.16911
\(994\) 0 0
\(995\) −9.06160 −0.287272
\(996\) 0 0
\(997\) −8.42055 −0.266682 −0.133341 0.991070i \(-0.542570\pi\)
−0.133341 + 0.991070i \(0.542570\pi\)
\(998\) 0 0
\(999\) 3.13354 0.0991407
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 3380.2.a.r.1.8 9
13.5 odd 4 3380.2.f.j.3041.16 18
13.8 odd 4 3380.2.f.j.3041.15 18
13.12 even 2 3380.2.a.s.1.8 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3380.2.a.r.1.8 9 1.1 even 1 trivial
3380.2.a.s.1.8 yes 9 13.12 even 2
3380.2.f.j.3041.15 18 13.8 odd 4
3380.2.f.j.3041.16 18 13.5 odd 4