Properties

Label 3380.2.a.s.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: \( x^{9} - x^{8} - 19x^{7} + 16x^{6} + 106x^{5} - 87x^{4} - 153x^{3} + 149x^{2} - 26x + 1 \) Copy content Toggle raw display
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

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

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 \(-0.425383\pi\)
0.232274 + 0.972650i \(0.425383\pi\)
\(8\) 0 0
\(9\) 4.80991 1.60330
\(10\) 0 0
\(11\) 3.73479 1.12608 0.563040 0.826429i \(-0.309632\pi\)
0.563040 + 0.826429i \(0.309632\pi\)
\(12\) 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 \(-0.614356\pi\)
−0.351581 + 0.936158i \(0.614356\pi\)
\(20\) 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 \(-0.700992\pi\)
−0.590303 + 0.807182i \(0.700992\pi\)
\(32\) 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 \(-0.516217\pi\)
−0.0509241 + 0.998703i \(0.516217\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.41141 −1.31364 −0.656821 0.754047i \(-0.728099\pi\)
−0.656821 + 0.754047i \(0.728099\pi\)
\(42\) 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 \(-0.635370\pi\)
−0.412574 + 0.910924i \(0.635370\pi\)
\(48\) 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 \(-0.208461\pi\)
0.793110 + 0.609078i \(0.208461\pi\)
\(60\) 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 \(-0.656617\pi\)
−0.472414 + 0.881377i \(0.656617\pi\)
\(68\) 0 0
\(69\) −2.91718 −0.351187
\(70\) 0 0
\(71\) −5.36427 −0.636622 −0.318311 0.947986i \(-0.603116\pi\)
−0.318311 + 0.947986i \(0.603116\pi\)
\(72\) 0 0
\(73\) −14.3772 −1.68273 −0.841363 0.540471i \(-0.818246\pi\)
−0.841363 + 0.540471i \(0.818246\pi\)
\(74\) 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 \(-0.254543\pi\)
0.696943 + 0.717127i \(0.254543\pi\)
\(84\) 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 \(-0.755566\pi\)
−0.719362 + 0.694635i \(0.755566\pi\)
\(90\) 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 \(-0.248811\pi\)
0.709742 + 0.704462i \(0.248811\pi\)
\(98\) 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 \(-0.611717\pi\)
−0.343808 + 0.939040i \(0.611717\pi\)
\(110\) 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 \(-0.392863\pi\)
0.330262 + 0.943889i \(0.392863\pi\)
\(138\) 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 \(-0.253960\pi\)
0.698255 + 0.715849i \(0.253960\pi\)
\(150\) 0 0
\(151\) 2.41977 0.196918 0.0984588 0.995141i \(-0.468609\pi\)
0.0984588 + 0.995141i \(0.468609\pi\)
\(152\) 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 \(-0.603962\pi\)
−0.320831 + 0.947137i \(0.603962\pi\)
\(164\) 0 0
\(165\) 10.4373 0.812544
\(166\) 0 0
\(167\) −0.119894 −0.00927770 −0.00463885 0.999989i \(-0.501477\pi\)
−0.00463885 + 0.999989i \(0.501477\pi\)
\(168\) 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 \(-0.528084\pi\)
−0.0881132 + 0.996110i \(0.528084\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.851216 0.0606466 0.0303233 0.999540i \(-0.490346\pi\)
0.0303233 + 0.999540i \(0.490346\pi\)
\(198\) 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 \(-0.783166\pi\)
−0.776815 + 0.629729i \(0.783166\pi\)
\(224\) 0 0
\(225\) 4.80991 0.320661
\(226\) 0 0
\(227\) 5.33774 0.354278 0.177139 0.984186i \(-0.443316\pi\)
0.177139 + 0.984186i \(0.443316\pi\)
\(228\) 0 0
\(229\) 16.1535 1.06745 0.533726 0.845657i \(-0.320791\pi\)
0.533726 + 0.845657i \(0.320791\pi\)
\(230\) 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 \(-0.305328\pi\)
0.574163 + 0.818741i \(0.305328\pi\)
\(240\) 0 0
\(241\) 3.17008 0.204203 0.102101 0.994774i \(-0.467443\pi\)
0.102101 + 0.994774i \(0.467443\pi\)
\(242\) 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 \(-0.465023\pi\)
0.109662 + 0.993969i \(0.465023\pi\)
\(272\) 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 \(-0.799661\pi\)
−0.808391 + 0.588645i \(0.799661\pi\)
\(282\) 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 \(-0.367515\pi\)
0.404300 + 0.914626i \(0.367515\pi\)
\(294\) 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 \(-0.625178\pi\)
−0.383201 + 0.923665i \(0.625178\pi\)
\(308\) 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 \(-0.291369\pi\)
0.609504 + 0.792783i \(0.291369\pi\)
\(318\) 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 \(-0.381994\pi\)
0.362294 + 0.932064i \(0.381994\pi\)
\(332\) 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 \(-0.636520\pi\)
−0.415862 + 0.909428i \(0.636520\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.4163 −0.980199 −0.490099 0.871667i \(-0.663040\pi\)
−0.490099 + 0.871667i \(0.663040\pi\)
\(354\) 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 \(-0.480974\pi\)
0.0597358 + 0.998214i \(0.480974\pi\)
\(360\) 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 \(-0.203135\pi\)
0.803188 + 0.595725i \(0.203135\pi\)
\(380\) 0 0
\(381\) 4.48992 0.230026
\(382\) 0 0
\(383\) −26.2659 −1.34212 −0.671062 0.741401i \(-0.734162\pi\)
−0.671062 + 0.741401i \(0.734162\pi\)
\(384\) 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 \(-0.295564\pi\)
0.599002 + 0.800747i \(0.295564\pi\)
\(398\) 0 0
\(399\) −10.5277 −0.527046
\(400\) 0 0
\(401\) 2.56312 0.127996 0.0639980 0.997950i \(-0.479615\pi\)
0.0639980 + 0.997950i \(0.479615\pi\)
\(402\) 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 \(-0.344843\pi\)
0.468366 + 0.883535i \(0.344843\pi\)
\(410\) 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 \(-0.708911\pi\)
−0.610201 + 0.792247i \(0.708911\pi\)
\(422\) 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 \(-0.433729\pi\)
0.206697 + 0.978405i \(0.433729\pi\)
\(432\) 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 \(-0.538571\pi\)
−0.120878 + 0.992667i \(0.538571\pi\)
\(450\) 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 \(-0.309903\pi\)
0.562334 + 0.826910i \(0.309903\pi\)
\(458\) 0 0
\(459\) 25.0681 1.17008
\(460\) 0 0
\(461\) −4.22878 −0.196954 −0.0984770 0.995139i \(-0.531397\pi\)
−0.0984770 + 0.995139i \(0.531397\pi\)
\(462\) 0 0
\(463\) −29.4028 −1.36647 −0.683233 0.730200i \(-0.739427\pi\)
−0.683233 + 0.730200i \(0.739427\pi\)
\(464\) 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 \(-0.552468\pi\)
−0.164087 + 0.986446i \(0.552468\pi\)
\(480\) 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 \(-0.358096\pi\)
0.431183 + 0.902264i \(0.358096\pi\)
\(488\) 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 \(-0.584694\pi\)
−0.262945 + 0.964811i \(0.584694\pi\)
\(500\) 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 \(-0.521575\pi\)
−0.0677294 + 0.997704i \(0.521575\pi\)
\(510\) 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 \(-0.810418\pi\)
−0.827818 + 0.560997i \(0.810418\pi\)
\(542\) 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 \(-0.319034\pi\)
0.538385 + 0.842699i \(0.319034\pi\)
\(558\) 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 \(-0.589852\pi\)
−0.278544 + 0.960424i \(0.589852\pi\)
\(578\) 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 \(-0.497370\pi\)
0.00826340 + 0.999966i \(0.497370\pi\)
\(588\) 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 \(-0.612281\pi\)
−0.345472 + 0.938429i \(0.612281\pi\)
\(594\) 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 \(-0.585654\pi\)
−0.265854 + 0.964013i \(0.585654\pi\)
\(614\) 0 0
\(615\) −23.5067 −0.947882
\(616\) 0 0
\(617\) 36.4559 1.46766 0.733830 0.679333i \(-0.237731\pi\)
0.733830 + 0.679333i \(0.237731\pi\)
\(618\) 0 0
\(619\) −44.4340 −1.78595 −0.892976 0.450105i \(-0.851387\pi\)
−0.892976 + 0.450105i \(0.851387\pi\)
\(620\) 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 \(-0.350413\pi\)
0.452833 + 0.891595i \(0.350413\pi\)
\(632\) 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 \(-0.522301\pi\)
−0.0700042 + 0.997547i \(0.522301\pi\)
\(644\) 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 \(-0.680284\pi\)
−0.536581 + 0.843849i \(0.680284\pi\)
\(662\) 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 \(-0.279795\pi\)
0.637921 + 0.770102i \(0.279795\pi\)
\(684\) 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 \(-0.806637\pi\)
−0.821095 + 0.570791i \(0.806637\pi\)
\(692\) 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 \(-0.338708\pi\)
0.485306 + 0.874345i \(0.338708\pi\)
\(710\) 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 \(-0.152718\pi\)
0.887097 + 0.461582i \(0.152718\pi\)
\(734\) 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 \(-0.378021\pi\)
0.373897 + 0.927470i \(0.378021\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 39.5385 1.45053 0.725263 0.688472i \(-0.241718\pi\)
0.725263 + 0.688472i \(0.241718\pi\)
\(744\) 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 \(-0.788792\pi\)
−0.787822 + 0.615902i \(0.788792\pi\)
\(762\) 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 \(-0.511410\pi\)
−0.0358364 + 0.999358i \(0.511410\pi\)
\(770\) 0 0
\(771\) −16.8370 −0.606369
\(772\) 0 0
\(773\) 46.0457 1.65615 0.828074 0.560619i \(-0.189437\pi\)
0.828074 + 0.560619i \(0.189437\pi\)
\(774\) 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 \(-0.321967\pi\)
0.530598 + 0.847623i \(0.321967\pi\)
\(788\) 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 \(-0.914646\pi\)
−0.964264 + 0.264945i \(0.914646\pi\)
\(812\) 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 \(-0.623065\pi\)
−0.377060 + 0.926189i \(0.623065\pi\)
\(822\) 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 \(-0.713055\pi\)
−0.620463 + 0.784235i \(0.713055\pi\)
\(828\) 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 \(-0.621573\pi\)
−0.372714 + 0.927946i \(0.621573\pi\)
\(840\) 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 \(-0.446431\pi\)
0.167497 + 0.985873i \(0.446431\pi\)
\(854\) 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 \(-0.388271\pi\)
0.343844 + 0.939027i \(0.388271\pi\)
\(864\) 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 \(-0.337236\pi\)
0.489344 + 0.872091i \(0.337236\pi\)
\(878\) 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 \(-0.536675\pi\)
−0.114964 + 0.993370i \(0.536675\pi\)
\(930\) 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 \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(942\) 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 \(-0.615260\pi\)
−0.354237 + 0.935156i \(0.615260\pi\)
\(948\) 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 \(-0.551833\pi\)
−0.162118 + 0.986771i \(0.551833\pi\)
\(968\) 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 \(-0.315042\pi\)
0.548912 + 0.835880i \(0.315042\pi\)
\(978\) 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 \(-0.386339\pi\)
0.349537 + 0.936923i \(0.386339\pi\)
\(984\) 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.s.1.8 yes 9
13.5 odd 4 3380.2.f.j.3041.15 18
13.8 odd 4 3380.2.f.j.3041.16 18
13.12 even 2 3380.2.a.r.1.8 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3380.2.a.r.1.8 9 13.12 even 2
3380.2.a.s.1.8 yes 9 1.1 even 1 trivial
3380.2.f.j.3041.15 18 13.5 odd 4
3380.2.f.j.3041.16 18 13.8 odd 4