Properties

Label 2070.2.a.u.1.2
Level $2070$
Weight $2$
Character 2070.1
Self dual yes
Analytic conductor $16.529$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.61803 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.61803 q^{7} -1.00000 q^{8} +1.00000 q^{10} -3.85410 q^{11} +4.09017 q^{13} -1.61803 q^{14} +1.00000 q^{16} +5.09017 q^{17} -4.85410 q^{19} -1.00000 q^{20} +3.85410 q^{22} -1.00000 q^{23} +1.00000 q^{25} -4.09017 q^{26} +1.61803 q^{28} +4.76393 q^{29} -2.09017 q^{31} -1.00000 q^{32} -5.09017 q^{34} -1.61803 q^{35} -2.47214 q^{37} +4.85410 q^{38} +1.00000 q^{40} +12.3262 q^{41} -3.85410 q^{44} +1.00000 q^{46} -9.70820 q^{47} -4.38197 q^{49} -1.00000 q^{50} +4.09017 q^{52} +8.47214 q^{53} +3.85410 q^{55} -1.61803 q^{56} -4.76393 q^{58} +11.7082 q^{59} +6.32624 q^{61} +2.09017 q^{62} +1.00000 q^{64} -4.09017 q^{65} +5.52786 q^{67} +5.09017 q^{68} +1.61803 q^{70} -7.09017 q^{71} -1.23607 q^{73} +2.47214 q^{74} -4.85410 q^{76} -6.23607 q^{77} +10.4721 q^{79} -1.00000 q^{80} -12.3262 q^{82} -10.9443 q^{83} -5.09017 q^{85} +3.85410 q^{88} +1.52786 q^{89} +6.61803 q^{91} -1.00000 q^{92} +9.70820 q^{94} +4.85410 q^{95} +14.6180 q^{97} +4.38197 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} - 3 q^{13} - q^{14} + 2 q^{16} - q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 2 q^{23} + 2 q^{25} + 3 q^{26} + q^{28} + 14 q^{29} + 7 q^{31} - 2 q^{32} + q^{34} - q^{35} + 4 q^{37} + 3 q^{38} + 2 q^{40} + 9 q^{41} - q^{44} + 2 q^{46} - 6 q^{47} - 11 q^{49} - 2 q^{50} - 3 q^{52} + 8 q^{53} + q^{55} - q^{56} - 14 q^{58} + 10 q^{59} - 3 q^{61} - 7 q^{62} + 2 q^{64} + 3 q^{65} + 20 q^{67} - q^{68} + q^{70} - 3 q^{71} + 2 q^{73} - 4 q^{74} - 3 q^{76} - 8 q^{77} + 12 q^{79} - 2 q^{80} - 9 q^{82} - 4 q^{83} + q^{85} + q^{88} + 12 q^{89} + 11 q^{91} - 2 q^{92} + 6 q^{94} + 3 q^{95} + 27 q^{97} + 11 q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.61803 0.611559 0.305780 0.952102i \(-0.401083\pi\)
0.305780 + 0.952102i \(0.401083\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −3.85410 −1.16206 −0.581028 0.813884i \(-0.697349\pi\)
−0.581028 + 0.813884i \(0.697349\pi\)
\(12\) 0 0
\(13\) 4.09017 1.13441 0.567205 0.823577i \(-0.308025\pi\)
0.567205 + 0.823577i \(0.308025\pi\)
\(14\) −1.61803 −0.432438
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.09017 1.23455 0.617274 0.786748i \(-0.288237\pi\)
0.617274 + 0.786748i \(0.288237\pi\)
\(18\) 0 0
\(19\) −4.85410 −1.11361 −0.556804 0.830644i \(-0.687972\pi\)
−0.556804 + 0.830644i \(0.687972\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 3.85410 0.821697
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.09017 −0.802148
\(27\) 0 0
\(28\) 1.61803 0.305780
\(29\) 4.76393 0.884640 0.442320 0.896857i \(-0.354156\pi\)
0.442320 + 0.896857i \(0.354156\pi\)
\(30\) 0 0
\(31\) −2.09017 −0.375406 −0.187703 0.982226i \(-0.560104\pi\)
−0.187703 + 0.982226i \(0.560104\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −5.09017 −0.872957
\(35\) −1.61803 −0.273498
\(36\) 0 0
\(37\) −2.47214 −0.406417 −0.203208 0.979136i \(-0.565137\pi\)
−0.203208 + 0.979136i \(0.565137\pi\)
\(38\) 4.85410 0.787439
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 12.3262 1.92503 0.962517 0.271220i \(-0.0874270\pi\)
0.962517 + 0.271220i \(0.0874270\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −3.85410 −0.581028
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −9.70820 −1.41609 −0.708044 0.706169i \(-0.750422\pi\)
−0.708044 + 0.706169i \(0.750422\pi\)
\(48\) 0 0
\(49\) −4.38197 −0.625995
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 4.09017 0.567205
\(53\) 8.47214 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 0 0
\(55\) 3.85410 0.519687
\(56\) −1.61803 −0.216219
\(57\) 0 0
\(58\) −4.76393 −0.625535
\(59\) 11.7082 1.52428 0.762139 0.647413i \(-0.224149\pi\)
0.762139 + 0.647413i \(0.224149\pi\)
\(60\) 0 0
\(61\) 6.32624 0.809992 0.404996 0.914319i \(-0.367273\pi\)
0.404996 + 0.914319i \(0.367273\pi\)
\(62\) 2.09017 0.265452
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.09017 −0.507323
\(66\) 0 0
\(67\) 5.52786 0.675336 0.337668 0.941265i \(-0.390362\pi\)
0.337668 + 0.941265i \(0.390362\pi\)
\(68\) 5.09017 0.617274
\(69\) 0 0
\(70\) 1.61803 0.193392
\(71\) −7.09017 −0.841448 −0.420724 0.907189i \(-0.638224\pi\)
−0.420724 + 0.907189i \(0.638224\pi\)
\(72\) 0 0
\(73\) −1.23607 −0.144671 −0.0723354 0.997380i \(-0.523045\pi\)
−0.0723354 + 0.997380i \(0.523045\pi\)
\(74\) 2.47214 0.287380
\(75\) 0 0
\(76\) −4.85410 −0.556804
\(77\) −6.23607 −0.710666
\(78\) 0 0
\(79\) 10.4721 1.17821 0.589104 0.808057i \(-0.299481\pi\)
0.589104 + 0.808057i \(0.299481\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −12.3262 −1.36121
\(83\) −10.9443 −1.20129 −0.600645 0.799516i \(-0.705089\pi\)
−0.600645 + 0.799516i \(0.705089\pi\)
\(84\) 0 0
\(85\) −5.09017 −0.552106
\(86\) 0 0
\(87\) 0 0
\(88\) 3.85410 0.410849
\(89\) 1.52786 0.161953 0.0809766 0.996716i \(-0.474196\pi\)
0.0809766 + 0.996716i \(0.474196\pi\)
\(90\) 0 0
\(91\) 6.61803 0.693758
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 9.70820 1.00132
\(95\) 4.85410 0.498020
\(96\) 0 0
\(97\) 14.6180 1.48424 0.742118 0.670269i \(-0.233821\pi\)
0.742118 + 0.670269i \(0.233821\pi\)
\(98\) 4.38197 0.442645
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 13.7082 1.36402 0.682009 0.731344i \(-0.261107\pi\)
0.682009 + 0.731344i \(0.261107\pi\)
\(102\) 0 0
\(103\) −3.56231 −0.351004 −0.175502 0.984479i \(-0.556155\pi\)
−0.175502 + 0.984479i \(0.556155\pi\)
\(104\) −4.09017 −0.401074
\(105\) 0 0
\(106\) −8.47214 −0.822887
\(107\) 4.18034 0.404129 0.202064 0.979372i \(-0.435235\pi\)
0.202064 + 0.979372i \(0.435235\pi\)
\(108\) 0 0
\(109\) 8.56231 0.820120 0.410060 0.912059i \(-0.365508\pi\)
0.410060 + 0.912059i \(0.365508\pi\)
\(110\) −3.85410 −0.367474
\(111\) 0 0
\(112\) 1.61803 0.152890
\(113\) −18.9443 −1.78213 −0.891064 0.453878i \(-0.850040\pi\)
−0.891064 + 0.453878i \(0.850040\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 4.76393 0.442320
\(117\) 0 0
\(118\) −11.7082 −1.07783
\(119\) 8.23607 0.754999
\(120\) 0 0
\(121\) 3.85410 0.350373
\(122\) −6.32624 −0.572751
\(123\) 0 0
\(124\) −2.09017 −0.187703
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.18034 −0.548416 −0.274208 0.961670i \(-0.588416\pi\)
−0.274208 + 0.961670i \(0.588416\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.09017 0.358732
\(131\) 14.9443 1.30569 0.652844 0.757493i \(-0.273576\pi\)
0.652844 + 0.757493i \(0.273576\pi\)
\(132\) 0 0
\(133\) −7.85410 −0.681037
\(134\) −5.52786 −0.477535
\(135\) 0 0
\(136\) −5.09017 −0.436478
\(137\) −5.32624 −0.455051 −0.227526 0.973772i \(-0.573064\pi\)
−0.227526 + 0.973772i \(0.573064\pi\)
\(138\) 0 0
\(139\) 17.2361 1.46194 0.730972 0.682407i \(-0.239067\pi\)
0.730972 + 0.682407i \(0.239067\pi\)
\(140\) −1.61803 −0.136749
\(141\) 0 0
\(142\) 7.09017 0.594994
\(143\) −15.7639 −1.31825
\(144\) 0 0
\(145\) −4.76393 −0.395623
\(146\) 1.23607 0.102298
\(147\) 0 0
\(148\) −2.47214 −0.203208
\(149\) 1.14590 0.0938756 0.0469378 0.998898i \(-0.485054\pi\)
0.0469378 + 0.998898i \(0.485054\pi\)
\(150\) 0 0
\(151\) 17.5623 1.42920 0.714600 0.699533i \(-0.246609\pi\)
0.714600 + 0.699533i \(0.246609\pi\)
\(152\) 4.85410 0.393720
\(153\) 0 0
\(154\) 6.23607 0.502517
\(155\) 2.09017 0.167886
\(156\) 0 0
\(157\) 9.70820 0.774799 0.387400 0.921912i \(-0.373373\pi\)
0.387400 + 0.921912i \(0.373373\pi\)
\(158\) −10.4721 −0.833118
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −1.61803 −0.127519
\(162\) 0 0
\(163\) 3.61803 0.283386 0.141693 0.989911i \(-0.454745\pi\)
0.141693 + 0.989911i \(0.454745\pi\)
\(164\) 12.3262 0.962517
\(165\) 0 0
\(166\) 10.9443 0.849440
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 3.72949 0.286884
\(170\) 5.09017 0.390398
\(171\) 0 0
\(172\) 0 0
\(173\) 21.5623 1.63935 0.819676 0.572828i \(-0.194154\pi\)
0.819676 + 0.572828i \(0.194154\pi\)
\(174\) 0 0
\(175\) 1.61803 0.122312
\(176\) −3.85410 −0.290514
\(177\) 0 0
\(178\) −1.52786 −0.114518
\(179\) 20.1803 1.50835 0.754175 0.656674i \(-0.228037\pi\)
0.754175 + 0.656674i \(0.228037\pi\)
\(180\) 0 0
\(181\) −18.8541 −1.40141 −0.700707 0.713449i \(-0.747132\pi\)
−0.700707 + 0.713449i \(0.747132\pi\)
\(182\) −6.61803 −0.490561
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 2.47214 0.181755
\(186\) 0 0
\(187\) −19.6180 −1.43461
\(188\) −9.70820 −0.708044
\(189\) 0 0
\(190\) −4.85410 −0.352154
\(191\) 0.291796 0.0211136 0.0105568 0.999944i \(-0.496640\pi\)
0.0105568 + 0.999944i \(0.496640\pi\)
\(192\) 0 0
\(193\) 5.23607 0.376900 0.188450 0.982083i \(-0.439654\pi\)
0.188450 + 0.982083i \(0.439654\pi\)
\(194\) −14.6180 −1.04951
\(195\) 0 0
\(196\) −4.38197 −0.312998
\(197\) 2.43769 0.173679 0.0868393 0.996222i \(-0.472323\pi\)
0.0868393 + 0.996222i \(0.472323\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −13.7082 −0.964506
\(203\) 7.70820 0.541010
\(204\) 0 0
\(205\) −12.3262 −0.860902
\(206\) 3.56231 0.248198
\(207\) 0 0
\(208\) 4.09017 0.283602
\(209\) 18.7082 1.29407
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 8.47214 0.581869
\(213\) 0 0
\(214\) −4.18034 −0.285762
\(215\) 0 0
\(216\) 0 0
\(217\) −3.38197 −0.229583
\(218\) −8.56231 −0.579913
\(219\) 0 0
\(220\) 3.85410 0.259844
\(221\) 20.8197 1.40048
\(222\) 0 0
\(223\) −3.05573 −0.204627 −0.102313 0.994752i \(-0.532624\pi\)
−0.102313 + 0.994752i \(0.532624\pi\)
\(224\) −1.61803 −0.108109
\(225\) 0 0
\(226\) 18.9443 1.26015
\(227\) 23.2361 1.54223 0.771116 0.636695i \(-0.219699\pi\)
0.771116 + 0.636695i \(0.219699\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −4.76393 −0.312767
\(233\) −19.7082 −1.29113 −0.645564 0.763706i \(-0.723378\pi\)
−0.645564 + 0.763706i \(0.723378\pi\)
\(234\) 0 0
\(235\) 9.70820 0.633293
\(236\) 11.7082 0.762139
\(237\) 0 0
\(238\) −8.23607 −0.533865
\(239\) −24.3607 −1.57576 −0.787881 0.615828i \(-0.788822\pi\)
−0.787881 + 0.615828i \(0.788822\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −3.85410 −0.247751
\(243\) 0 0
\(244\) 6.32624 0.404996
\(245\) 4.38197 0.279954
\(246\) 0 0
\(247\) −19.8541 −1.26329
\(248\) 2.09017 0.132726
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −12.8541 −0.811344 −0.405672 0.914019i \(-0.632962\pi\)
−0.405672 + 0.914019i \(0.632962\pi\)
\(252\) 0 0
\(253\) 3.85410 0.242305
\(254\) 6.18034 0.387789
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 30.1803 1.88260 0.941299 0.337574i \(-0.109606\pi\)
0.941299 + 0.337574i \(0.109606\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) −4.09017 −0.253662
\(261\) 0 0
\(262\) −14.9443 −0.923260
\(263\) 21.7426 1.34071 0.670354 0.742041i \(-0.266142\pi\)
0.670354 + 0.742041i \(0.266142\pi\)
\(264\) 0 0
\(265\) −8.47214 −0.520439
\(266\) 7.85410 0.481566
\(267\) 0 0
\(268\) 5.52786 0.337668
\(269\) −8.18034 −0.498764 −0.249382 0.968405i \(-0.580227\pi\)
−0.249382 + 0.968405i \(0.580227\pi\)
\(270\) 0 0
\(271\) −14.6738 −0.891368 −0.445684 0.895190i \(-0.647039\pi\)
−0.445684 + 0.895190i \(0.647039\pi\)
\(272\) 5.09017 0.308637
\(273\) 0 0
\(274\) 5.32624 0.321770
\(275\) −3.85410 −0.232411
\(276\) 0 0
\(277\) 2.58359 0.155233 0.0776165 0.996983i \(-0.475269\pi\)
0.0776165 + 0.996983i \(0.475269\pi\)
\(278\) −17.2361 −1.03375
\(279\) 0 0
\(280\) 1.61803 0.0966960
\(281\) −27.2361 −1.62477 −0.812384 0.583123i \(-0.801831\pi\)
−0.812384 + 0.583123i \(0.801831\pi\)
\(282\) 0 0
\(283\) −9.05573 −0.538307 −0.269154 0.963097i \(-0.586744\pi\)
−0.269154 + 0.963097i \(0.586744\pi\)
\(284\) −7.09017 −0.420724
\(285\) 0 0
\(286\) 15.7639 0.932141
\(287\) 19.9443 1.17727
\(288\) 0 0
\(289\) 8.90983 0.524108
\(290\) 4.76393 0.279748
\(291\) 0 0
\(292\) −1.23607 −0.0723354
\(293\) −15.8885 −0.928219 −0.464109 0.885778i \(-0.653626\pi\)
−0.464109 + 0.885778i \(0.653626\pi\)
\(294\) 0 0
\(295\) −11.7082 −0.681678
\(296\) 2.47214 0.143690
\(297\) 0 0
\(298\) −1.14590 −0.0663801
\(299\) −4.09017 −0.236541
\(300\) 0 0
\(301\) 0 0
\(302\) −17.5623 −1.01060
\(303\) 0 0
\(304\) −4.85410 −0.278402
\(305\) −6.32624 −0.362239
\(306\) 0 0
\(307\) 27.4508 1.56670 0.783351 0.621579i \(-0.213509\pi\)
0.783351 + 0.621579i \(0.213509\pi\)
\(308\) −6.23607 −0.355333
\(309\) 0 0
\(310\) −2.09017 −0.118714
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) −11.7984 −0.666884 −0.333442 0.942771i \(-0.608210\pi\)
−0.333442 + 0.942771i \(0.608210\pi\)
\(314\) −9.70820 −0.547866
\(315\) 0 0
\(316\) 10.4721 0.589104
\(317\) −0.0901699 −0.00506445 −0.00253222 0.999997i \(-0.500806\pi\)
−0.00253222 + 0.999997i \(0.500806\pi\)
\(318\) 0 0
\(319\) −18.3607 −1.02800
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 1.61803 0.0901695
\(323\) −24.7082 −1.37480
\(324\) 0 0
\(325\) 4.09017 0.226882
\(326\) −3.61803 −0.200384
\(327\) 0 0
\(328\) −12.3262 −0.680603
\(329\) −15.7082 −0.866021
\(330\) 0 0
\(331\) 14.7639 0.811499 0.405750 0.913984i \(-0.367011\pi\)
0.405750 + 0.913984i \(0.367011\pi\)
\(332\) −10.9443 −0.600645
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) −5.52786 −0.302019
\(336\) 0 0
\(337\) 29.3262 1.59750 0.798751 0.601662i \(-0.205494\pi\)
0.798751 + 0.601662i \(0.205494\pi\)
\(338\) −3.72949 −0.202858
\(339\) 0 0
\(340\) −5.09017 −0.276053
\(341\) 8.05573 0.436242
\(342\) 0 0
\(343\) −18.4164 −0.994393
\(344\) 0 0
\(345\) 0 0
\(346\) −21.5623 −1.15920
\(347\) 8.61803 0.462640 0.231320 0.972878i \(-0.425696\pi\)
0.231320 + 0.972878i \(0.425696\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −1.61803 −0.0864876
\(351\) 0 0
\(352\) 3.85410 0.205424
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) 7.09017 0.376307
\(356\) 1.52786 0.0809766
\(357\) 0 0
\(358\) −20.1803 −1.06656
\(359\) 18.3607 0.969040 0.484520 0.874780i \(-0.338994\pi\)
0.484520 + 0.874780i \(0.338994\pi\)
\(360\) 0 0
\(361\) 4.56231 0.240121
\(362\) 18.8541 0.990950
\(363\) 0 0
\(364\) 6.61803 0.346879
\(365\) 1.23607 0.0646988
\(366\) 0 0
\(367\) −2.47214 −0.129044 −0.0645222 0.997916i \(-0.520552\pi\)
−0.0645222 + 0.997916i \(0.520552\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −2.47214 −0.128520
\(371\) 13.7082 0.711694
\(372\) 0 0
\(373\) −2.18034 −0.112894 −0.0564469 0.998406i \(-0.517977\pi\)
−0.0564469 + 0.998406i \(0.517977\pi\)
\(374\) 19.6180 1.01442
\(375\) 0 0
\(376\) 9.70820 0.500662
\(377\) 19.4853 1.00354
\(378\) 0 0
\(379\) 33.4508 1.71825 0.859127 0.511762i \(-0.171007\pi\)
0.859127 + 0.511762i \(0.171007\pi\)
\(380\) 4.85410 0.249010
\(381\) 0 0
\(382\) −0.291796 −0.0149296
\(383\) −17.8885 −0.914062 −0.457031 0.889451i \(-0.651087\pi\)
−0.457031 + 0.889451i \(0.651087\pi\)
\(384\) 0 0
\(385\) 6.23607 0.317819
\(386\) −5.23607 −0.266509
\(387\) 0 0
\(388\) 14.6180 0.742118
\(389\) −5.67376 −0.287671 −0.143836 0.989602i \(-0.545944\pi\)
−0.143836 + 0.989602i \(0.545944\pi\)
\(390\) 0 0
\(391\) −5.09017 −0.257421
\(392\) 4.38197 0.221323
\(393\) 0 0
\(394\) −2.43769 −0.122809
\(395\) −10.4721 −0.526910
\(396\) 0 0
\(397\) −8.32624 −0.417882 −0.208941 0.977928i \(-0.567002\pi\)
−0.208941 + 0.977928i \(0.567002\pi\)
\(398\) −2.00000 −0.100251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −11.7082 −0.584680 −0.292340 0.956314i \(-0.594434\pi\)
−0.292340 + 0.956314i \(0.594434\pi\)
\(402\) 0 0
\(403\) −8.54915 −0.425864
\(404\) 13.7082 0.682009
\(405\) 0 0
\(406\) −7.70820 −0.382552
\(407\) 9.52786 0.472279
\(408\) 0 0
\(409\) 21.2148 1.04900 0.524502 0.851409i \(-0.324252\pi\)
0.524502 + 0.851409i \(0.324252\pi\)
\(410\) 12.3262 0.608750
\(411\) 0 0
\(412\) −3.56231 −0.175502
\(413\) 18.9443 0.932187
\(414\) 0 0
\(415\) 10.9443 0.537233
\(416\) −4.09017 −0.200537
\(417\) 0 0
\(418\) −18.7082 −0.915048
\(419\) −5.52786 −0.270054 −0.135027 0.990842i \(-0.543112\pi\)
−0.135027 + 0.990842i \(0.543112\pi\)
\(420\) 0 0
\(421\) −28.7426 −1.40083 −0.700415 0.713735i \(-0.747002\pi\)
−0.700415 + 0.713735i \(0.747002\pi\)
\(422\) −14.0000 −0.681509
\(423\) 0 0
\(424\) −8.47214 −0.411443
\(425\) 5.09017 0.246910
\(426\) 0 0
\(427\) 10.2361 0.495358
\(428\) 4.18034 0.202064
\(429\) 0 0
\(430\) 0 0
\(431\) −34.6525 −1.66915 −0.834576 0.550894i \(-0.814287\pi\)
−0.834576 + 0.550894i \(0.814287\pi\)
\(432\) 0 0
\(433\) −29.5066 −1.41800 −0.708998 0.705211i \(-0.750852\pi\)
−0.708998 + 0.705211i \(0.750852\pi\)
\(434\) 3.38197 0.162340
\(435\) 0 0
\(436\) 8.56231 0.410060
\(437\) 4.85410 0.232203
\(438\) 0 0
\(439\) −15.6180 −0.745408 −0.372704 0.927950i \(-0.621569\pi\)
−0.372704 + 0.927950i \(0.621569\pi\)
\(440\) −3.85410 −0.183737
\(441\) 0 0
\(442\) −20.8197 −0.990290
\(443\) −13.9098 −0.660876 −0.330438 0.943828i \(-0.607196\pi\)
−0.330438 + 0.943828i \(0.607196\pi\)
\(444\) 0 0
\(445\) −1.52786 −0.0724277
\(446\) 3.05573 0.144693
\(447\) 0 0
\(448\) 1.61803 0.0764449
\(449\) −18.5623 −0.876009 −0.438005 0.898973i \(-0.644315\pi\)
−0.438005 + 0.898973i \(0.644315\pi\)
\(450\) 0 0
\(451\) −47.5066 −2.23700
\(452\) −18.9443 −0.891064
\(453\) 0 0
\(454\) −23.2361 −1.09052
\(455\) −6.61803 −0.310258
\(456\) 0 0
\(457\) 33.7771 1.58003 0.790013 0.613090i \(-0.210074\pi\)
0.790013 + 0.613090i \(0.210074\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 1.00000 0.0466252
\(461\) 34.7639 1.61912 0.809559 0.587039i \(-0.199706\pi\)
0.809559 + 0.587039i \(0.199706\pi\)
\(462\) 0 0
\(463\) 2.00000 0.0929479 0.0464739 0.998920i \(-0.485202\pi\)
0.0464739 + 0.998920i \(0.485202\pi\)
\(464\) 4.76393 0.221160
\(465\) 0 0
\(466\) 19.7082 0.912965
\(467\) −23.1246 −1.07008 −0.535040 0.844827i \(-0.679703\pi\)
−0.535040 + 0.844827i \(0.679703\pi\)
\(468\) 0 0
\(469\) 8.94427 0.413008
\(470\) −9.70820 −0.447806
\(471\) 0 0
\(472\) −11.7082 −0.538914
\(473\) 0 0
\(474\) 0 0
\(475\) −4.85410 −0.222721
\(476\) 8.23607 0.377500
\(477\) 0 0
\(478\) 24.3607 1.11423
\(479\) −3.88854 −0.177672 −0.0888361 0.996046i \(-0.528315\pi\)
−0.0888361 + 0.996046i \(0.528315\pi\)
\(480\) 0 0
\(481\) −10.1115 −0.461043
\(482\) 0 0
\(483\) 0 0
\(484\) 3.85410 0.175186
\(485\) −14.6180 −0.663771
\(486\) 0 0
\(487\) −42.1803 −1.91137 −0.955687 0.294385i \(-0.904885\pi\)
−0.955687 + 0.294385i \(0.904885\pi\)
\(488\) −6.32624 −0.286375
\(489\) 0 0
\(490\) −4.38197 −0.197957
\(491\) 16.1803 0.730209 0.365104 0.930967i \(-0.381033\pi\)
0.365104 + 0.930967i \(0.381033\pi\)
\(492\) 0 0
\(493\) 24.2492 1.09213
\(494\) 19.8541 0.893278
\(495\) 0 0
\(496\) −2.09017 −0.0938514
\(497\) −11.4721 −0.514596
\(498\) 0 0
\(499\) −32.3607 −1.44866 −0.724331 0.689452i \(-0.757851\pi\)
−0.724331 + 0.689452i \(0.757851\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 12.8541 0.573707
\(503\) −20.6738 −0.921797 −0.460899 0.887453i \(-0.652473\pi\)
−0.460899 + 0.887453i \(0.652473\pi\)
\(504\) 0 0
\(505\) −13.7082 −0.610007
\(506\) −3.85410 −0.171336
\(507\) 0 0
\(508\) −6.18034 −0.274208
\(509\) −5.34752 −0.237025 −0.118512 0.992953i \(-0.537813\pi\)
−0.118512 + 0.992953i \(0.537813\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −30.1803 −1.33120
\(515\) 3.56231 0.156974
\(516\) 0 0
\(517\) 37.4164 1.64557
\(518\) 4.00000 0.175750
\(519\) 0 0
\(520\) 4.09017 0.179366
\(521\) 24.4721 1.07214 0.536072 0.844172i \(-0.319908\pi\)
0.536072 + 0.844172i \(0.319908\pi\)
\(522\) 0 0
\(523\) −26.0000 −1.13690 −0.568450 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) 14.9443 0.652844
\(525\) 0 0
\(526\) −21.7426 −0.948024
\(527\) −10.6393 −0.463456
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 8.47214 0.368006
\(531\) 0 0
\(532\) −7.85410 −0.340519
\(533\) 50.4164 2.18378
\(534\) 0 0
\(535\) −4.18034 −0.180732
\(536\) −5.52786 −0.238767
\(537\) 0 0
\(538\) 8.18034 0.352679
\(539\) 16.8885 0.727441
\(540\) 0 0
\(541\) −30.8328 −1.32561 −0.662803 0.748794i \(-0.730633\pi\)
−0.662803 + 0.748794i \(0.730633\pi\)
\(542\) 14.6738 0.630292
\(543\) 0 0
\(544\) −5.09017 −0.218239
\(545\) −8.56231 −0.366769
\(546\) 0 0
\(547\) −36.9230 −1.57871 −0.789356 0.613935i \(-0.789586\pi\)
−0.789356 + 0.613935i \(0.789586\pi\)
\(548\) −5.32624 −0.227526
\(549\) 0 0
\(550\) 3.85410 0.164339
\(551\) −23.1246 −0.985142
\(552\) 0 0
\(553\) 16.9443 0.720544
\(554\) −2.58359 −0.109766
\(555\) 0 0
\(556\) 17.2361 0.730972
\(557\) −30.8328 −1.30643 −0.653214 0.757173i \(-0.726580\pi\)
−0.653214 + 0.757173i \(0.726580\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1.61803 −0.0683744
\(561\) 0 0
\(562\) 27.2361 1.14888
\(563\) −21.8885 −0.922492 −0.461246 0.887272i \(-0.652597\pi\)
−0.461246 + 0.887272i \(0.652597\pi\)
\(564\) 0 0
\(565\) 18.9443 0.796992
\(566\) 9.05573 0.380641
\(567\) 0 0
\(568\) 7.09017 0.297497
\(569\) 2.00000 0.0838444 0.0419222 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 0 0
\(571\) −30.9787 −1.29642 −0.648209 0.761462i \(-0.724482\pi\)
−0.648209 + 0.761462i \(0.724482\pi\)
\(572\) −15.7639 −0.659123
\(573\) 0 0
\(574\) −19.9443 −0.832458
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −12.4721 −0.519222 −0.259611 0.965713i \(-0.583594\pi\)
−0.259611 + 0.965713i \(0.583594\pi\)
\(578\) −8.90983 −0.370600
\(579\) 0 0
\(580\) −4.76393 −0.197812
\(581\) −17.7082 −0.734660
\(582\) 0 0
\(583\) −32.6525 −1.35233
\(584\) 1.23607 0.0511489
\(585\) 0 0
\(586\) 15.8885 0.656350
\(587\) −11.3820 −0.469784 −0.234892 0.972021i \(-0.575474\pi\)
−0.234892 + 0.972021i \(0.575474\pi\)
\(588\) 0 0
\(589\) 10.1459 0.418054
\(590\) 11.7082 0.482019
\(591\) 0 0
\(592\) −2.47214 −0.101604
\(593\) −34.7639 −1.42758 −0.713792 0.700358i \(-0.753024\pi\)
−0.713792 + 0.700358i \(0.753024\pi\)
\(594\) 0 0
\(595\) −8.23607 −0.337646
\(596\) 1.14590 0.0469378
\(597\) 0 0
\(598\) 4.09017 0.167259
\(599\) 20.6180 0.842430 0.421215 0.906961i \(-0.361604\pi\)
0.421215 + 0.906961i \(0.361604\pi\)
\(600\) 0 0
\(601\) 0.270510 0.0110343 0.00551716 0.999985i \(-0.498244\pi\)
0.00551716 + 0.999985i \(0.498244\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 17.5623 0.714600
\(605\) −3.85410 −0.156692
\(606\) 0 0
\(607\) 17.5279 0.711434 0.355717 0.934594i \(-0.384237\pi\)
0.355717 + 0.934594i \(0.384237\pi\)
\(608\) 4.85410 0.196860
\(609\) 0 0
\(610\) 6.32624 0.256142
\(611\) −39.7082 −1.60642
\(612\) 0 0
\(613\) −43.3050 −1.74907 −0.874535 0.484962i \(-0.838833\pi\)
−0.874535 + 0.484962i \(0.838833\pi\)
\(614\) −27.4508 −1.10783
\(615\) 0 0
\(616\) 6.23607 0.251258
\(617\) −22.9098 −0.922315 −0.461158 0.887318i \(-0.652566\pi\)
−0.461158 + 0.887318i \(0.652566\pi\)
\(618\) 0 0
\(619\) 21.7984 0.876151 0.438075 0.898938i \(-0.355660\pi\)
0.438075 + 0.898938i \(0.355660\pi\)
\(620\) 2.09017 0.0839432
\(621\) 0 0
\(622\) 4.00000 0.160385
\(623\) 2.47214 0.0990440
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 11.7984 0.471558
\(627\) 0 0
\(628\) 9.70820 0.387400
\(629\) −12.5836 −0.501741
\(630\) 0 0
\(631\) −16.0689 −0.639692 −0.319846 0.947470i \(-0.603631\pi\)
−0.319846 + 0.947470i \(0.603631\pi\)
\(632\) −10.4721 −0.416559
\(633\) 0 0
\(634\) 0.0901699 0.00358111
\(635\) 6.18034 0.245259
\(636\) 0 0
\(637\) −17.9230 −0.710135
\(638\) 18.3607 0.726906
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 44.3607 1.75214 0.876071 0.482183i \(-0.160156\pi\)
0.876071 + 0.482183i \(0.160156\pi\)
\(642\) 0 0
\(643\) −21.7082 −0.856088 −0.428044 0.903758i \(-0.640797\pi\)
−0.428044 + 0.903758i \(0.640797\pi\)
\(644\) −1.61803 −0.0637595
\(645\) 0 0
\(646\) 24.7082 0.972131
\(647\) −44.2492 −1.73962 −0.869808 0.493390i \(-0.835758\pi\)
−0.869808 + 0.493390i \(0.835758\pi\)
\(648\) 0 0
\(649\) −45.1246 −1.77130
\(650\) −4.09017 −0.160430
\(651\) 0 0
\(652\) 3.61803 0.141693
\(653\) −21.0344 −0.823141 −0.411571 0.911378i \(-0.635020\pi\)
−0.411571 + 0.911378i \(0.635020\pi\)
\(654\) 0 0
\(655\) −14.9443 −0.583921
\(656\) 12.3262 0.481259
\(657\) 0 0
\(658\) 15.7082 0.612370
\(659\) −34.2492 −1.33416 −0.667080 0.744986i \(-0.732456\pi\)
−0.667080 + 0.744986i \(0.732456\pi\)
\(660\) 0 0
\(661\) 34.3262 1.33514 0.667568 0.744549i \(-0.267335\pi\)
0.667568 + 0.744549i \(0.267335\pi\)
\(662\) −14.7639 −0.573817
\(663\) 0 0
\(664\) 10.9443 0.424720
\(665\) 7.85410 0.304569
\(666\) 0 0
\(667\) −4.76393 −0.184460
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) 5.52786 0.213560
\(671\) −24.3820 −0.941255
\(672\) 0 0
\(673\) 6.94427 0.267682 0.133841 0.991003i \(-0.457269\pi\)
0.133841 + 0.991003i \(0.457269\pi\)
\(674\) −29.3262 −1.12960
\(675\) 0 0
\(676\) 3.72949 0.143442
\(677\) 33.0557 1.27043 0.635217 0.772333i \(-0.280910\pi\)
0.635217 + 0.772333i \(0.280910\pi\)
\(678\) 0 0
\(679\) 23.6525 0.907699
\(680\) 5.09017 0.195199
\(681\) 0 0
\(682\) −8.05573 −0.308470
\(683\) 11.4377 0.437651 0.218826 0.975764i \(-0.429777\pi\)
0.218826 + 0.975764i \(0.429777\pi\)
\(684\) 0 0
\(685\) 5.32624 0.203505
\(686\) 18.4164 0.703142
\(687\) 0 0
\(688\) 0 0
\(689\) 34.6525 1.32015
\(690\) 0 0
\(691\) −24.7639 −0.942064 −0.471032 0.882116i \(-0.656118\pi\)
−0.471032 + 0.882116i \(0.656118\pi\)
\(692\) 21.5623 0.819676
\(693\) 0 0
\(694\) −8.61803 −0.327136
\(695\) −17.2361 −0.653801
\(696\) 0 0
\(697\) 62.7426 2.37655
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) 1.61803 0.0611559
\(701\) 48.3394 1.82575 0.912877 0.408235i \(-0.133856\pi\)
0.912877 + 0.408235i \(0.133856\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) −3.85410 −0.145257
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 22.1803 0.834178
\(708\) 0 0
\(709\) −14.9098 −0.559950 −0.279975 0.960007i \(-0.590326\pi\)
−0.279975 + 0.960007i \(0.590326\pi\)
\(710\) −7.09017 −0.266089
\(711\) 0 0
\(712\) −1.52786 −0.0572591
\(713\) 2.09017 0.0782775
\(714\) 0 0
\(715\) 15.7639 0.589538
\(716\) 20.1803 0.754175
\(717\) 0 0
\(718\) −18.3607 −0.685214
\(719\) −1.72949 −0.0644991 −0.0322495 0.999480i \(-0.510267\pi\)
−0.0322495 + 0.999480i \(0.510267\pi\)
\(720\) 0 0
\(721\) −5.76393 −0.214660
\(722\) −4.56231 −0.169791
\(723\) 0 0
\(724\) −18.8541 −0.700707
\(725\) 4.76393 0.176928
\(726\) 0 0
\(727\) 52.7984 1.95818 0.979092 0.203420i \(-0.0652056\pi\)
0.979092 + 0.203420i \(0.0652056\pi\)
\(728\) −6.61803 −0.245281
\(729\) 0 0
\(730\) −1.23607 −0.0457489
\(731\) 0 0
\(732\) 0 0
\(733\) −2.58359 −0.0954272 −0.0477136 0.998861i \(-0.515193\pi\)
−0.0477136 + 0.998861i \(0.515193\pi\)
\(734\) 2.47214 0.0912482
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −21.3050 −0.784778
\(738\) 0 0
\(739\) 21.8885 0.805183 0.402592 0.915380i \(-0.368110\pi\)
0.402592 + 0.915380i \(0.368110\pi\)
\(740\) 2.47214 0.0908775
\(741\) 0 0
\(742\) −13.7082 −0.503244
\(743\) 44.6312 1.63736 0.818680 0.574250i \(-0.194706\pi\)
0.818680 + 0.574250i \(0.194706\pi\)
\(744\) 0 0
\(745\) −1.14590 −0.0419825
\(746\) 2.18034 0.0798279
\(747\) 0 0
\(748\) −19.6180 −0.717306
\(749\) 6.76393 0.247149
\(750\) 0 0
\(751\) 29.0132 1.05871 0.529353 0.848402i \(-0.322435\pi\)
0.529353 + 0.848402i \(0.322435\pi\)
\(752\) −9.70820 −0.354022
\(753\) 0 0
\(754\) −19.4853 −0.709612
\(755\) −17.5623 −0.639158
\(756\) 0 0
\(757\) 17.8885 0.650170 0.325085 0.945685i \(-0.394607\pi\)
0.325085 + 0.945685i \(0.394607\pi\)
\(758\) −33.4508 −1.21499
\(759\) 0 0
\(760\) −4.85410 −0.176077
\(761\) 35.8673 1.30019 0.650094 0.759854i \(-0.274730\pi\)
0.650094 + 0.759854i \(0.274730\pi\)
\(762\) 0 0
\(763\) 13.8541 0.501552
\(764\) 0.291796 0.0105568
\(765\) 0 0
\(766\) 17.8885 0.646339
\(767\) 47.8885 1.72916
\(768\) 0 0
\(769\) −33.4164 −1.20503 −0.602513 0.798109i \(-0.705834\pi\)
−0.602513 + 0.798109i \(0.705834\pi\)
\(770\) −6.23607 −0.224732
\(771\) 0 0
\(772\) 5.23607 0.188450
\(773\) −11.0557 −0.397647 −0.198823 0.980035i \(-0.563712\pi\)
−0.198823 + 0.980035i \(0.563712\pi\)
\(774\) 0 0
\(775\) −2.09017 −0.0750811
\(776\) −14.6180 −0.524757
\(777\) 0 0
\(778\) 5.67376 0.203414
\(779\) −59.8328 −2.14373
\(780\) 0 0
\(781\) 27.3262 0.977810
\(782\) 5.09017 0.182024
\(783\) 0 0
\(784\) −4.38197 −0.156499
\(785\) −9.70820 −0.346501
\(786\) 0 0
\(787\) −43.1246 −1.53723 −0.768613 0.639714i \(-0.779053\pi\)
−0.768613 + 0.639714i \(0.779053\pi\)
\(788\) 2.43769 0.0868393
\(789\) 0 0
\(790\) 10.4721 0.372582
\(791\) −30.6525 −1.08988
\(792\) 0 0
\(793\) 25.8754 0.918862
\(794\) 8.32624 0.295487
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) −0.291796 −0.0103359 −0.00516797 0.999987i \(-0.501645\pi\)
−0.00516797 + 0.999987i \(0.501645\pi\)
\(798\) 0 0
\(799\) −49.4164 −1.74823
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 11.7082 0.413431
\(803\) 4.76393 0.168116
\(804\) 0 0
\(805\) 1.61803 0.0570282
\(806\) 8.54915 0.301131
\(807\) 0 0
\(808\) −13.7082 −0.482253
\(809\) 4.25735 0.149681 0.0748403 0.997196i \(-0.476155\pi\)
0.0748403 + 0.997196i \(0.476155\pi\)
\(810\) 0 0
\(811\) 44.1803 1.55138 0.775691 0.631113i \(-0.217402\pi\)
0.775691 + 0.631113i \(0.217402\pi\)
\(812\) 7.70820 0.270505
\(813\) 0 0
\(814\) −9.52786 −0.333951
\(815\) −3.61803 −0.126734
\(816\) 0 0
\(817\) 0 0
\(818\) −21.2148 −0.741757
\(819\) 0 0
\(820\) −12.3262 −0.430451
\(821\) 50.9443 1.77797 0.888984 0.457939i \(-0.151412\pi\)
0.888984 + 0.457939i \(0.151412\pi\)
\(822\) 0 0
\(823\) −1.41641 −0.0493729 −0.0246864 0.999695i \(-0.507859\pi\)
−0.0246864 + 0.999695i \(0.507859\pi\)
\(824\) 3.56231 0.124099
\(825\) 0 0
\(826\) −18.9443 −0.659156
\(827\) −8.29180 −0.288334 −0.144167 0.989553i \(-0.546050\pi\)
−0.144167 + 0.989553i \(0.546050\pi\)
\(828\) 0 0
\(829\) 1.05573 0.0366670 0.0183335 0.999832i \(-0.494164\pi\)
0.0183335 + 0.999832i \(0.494164\pi\)
\(830\) −10.9443 −0.379881
\(831\) 0 0
\(832\) 4.09017 0.141801
\(833\) −22.3050 −0.772821
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 18.7082 0.647037
\(837\) 0 0
\(838\) 5.52786 0.190957
\(839\) 43.0132 1.48498 0.742490 0.669858i \(-0.233645\pi\)
0.742490 + 0.669858i \(0.233645\pi\)
\(840\) 0 0
\(841\) −6.30495 −0.217412
\(842\) 28.7426 0.990537
\(843\) 0 0
\(844\) 14.0000 0.481900
\(845\) −3.72949 −0.128298
\(846\) 0 0
\(847\) 6.23607 0.214274
\(848\) 8.47214 0.290934
\(849\) 0 0
\(850\) −5.09017 −0.174591
\(851\) 2.47214 0.0847437
\(852\) 0 0
\(853\) 13.7984 0.472447 0.236224 0.971699i \(-0.424090\pi\)
0.236224 + 0.971699i \(0.424090\pi\)
\(854\) −10.2361 −0.350271
\(855\) 0 0
\(856\) −4.18034 −0.142881
\(857\) 33.4164 1.14148 0.570741 0.821130i \(-0.306656\pi\)
0.570741 + 0.821130i \(0.306656\pi\)
\(858\) 0 0
\(859\) 34.0689 1.16242 0.581208 0.813755i \(-0.302580\pi\)
0.581208 + 0.813755i \(0.302580\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 34.6525 1.18027
\(863\) −37.2361 −1.26753 −0.633765 0.773525i \(-0.718491\pi\)
−0.633765 + 0.773525i \(0.718491\pi\)
\(864\) 0 0
\(865\) −21.5623 −0.733140
\(866\) 29.5066 1.00267
\(867\) 0 0
\(868\) −3.38197 −0.114791
\(869\) −40.3607 −1.36914
\(870\) 0 0
\(871\) 22.6099 0.766107
\(872\) −8.56231 −0.289956
\(873\) 0 0
\(874\) −4.85410 −0.164192
\(875\) −1.61803 −0.0546995
\(876\) 0 0
\(877\) −23.7426 −0.801732 −0.400866 0.916137i \(-0.631291\pi\)
−0.400866 + 0.916137i \(0.631291\pi\)
\(878\) 15.6180 0.527083
\(879\) 0 0
\(880\) 3.85410 0.129922
\(881\) −35.4164 −1.19321 −0.596605 0.802535i \(-0.703484\pi\)
−0.596605 + 0.802535i \(0.703484\pi\)
\(882\) 0 0
\(883\) −4.56231 −0.153534 −0.0767669 0.997049i \(-0.524460\pi\)
−0.0767669 + 0.997049i \(0.524460\pi\)
\(884\) 20.8197 0.700241
\(885\) 0 0
\(886\) 13.9098 0.467310
\(887\) 58.8328 1.97541 0.987706 0.156321i \(-0.0499634\pi\)
0.987706 + 0.156321i \(0.0499634\pi\)
\(888\) 0 0
\(889\) −10.0000 −0.335389
\(890\) 1.52786 0.0512141
\(891\) 0 0
\(892\) −3.05573 −0.102313
\(893\) 47.1246 1.57697
\(894\) 0 0
\(895\) −20.1803 −0.674554
\(896\) −1.61803 −0.0540547
\(897\) 0 0
\(898\) 18.5623 0.619432
\(899\) −9.95743 −0.332099
\(900\) 0 0
\(901\) 43.1246 1.43669
\(902\) 47.5066 1.58180
\(903\) 0 0
\(904\) 18.9443 0.630077
\(905\) 18.8541 0.626732
\(906\) 0 0
\(907\) −33.1246 −1.09988 −0.549942 0.835203i \(-0.685350\pi\)
−0.549942 + 0.835203i \(0.685350\pi\)
\(908\) 23.2361 0.771116
\(909\) 0 0
\(910\) 6.61803 0.219386
\(911\) 22.0689 0.731175 0.365587 0.930777i \(-0.380868\pi\)
0.365587 + 0.930777i \(0.380868\pi\)
\(912\) 0 0
\(913\) 42.1803 1.39597
\(914\) −33.7771 −1.11725
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 24.1803 0.798505
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) −34.7639 −1.14489
\(923\) −29.0000 −0.954547
\(924\) 0 0
\(925\) −2.47214 −0.0812833
\(926\) −2.00000 −0.0657241
\(927\) 0 0
\(928\) −4.76393 −0.156384
\(929\) 12.4721 0.409198 0.204599 0.978846i \(-0.434411\pi\)
0.204599 + 0.978846i \(0.434411\pi\)
\(930\) 0 0
\(931\) 21.2705 0.697113
\(932\) −19.7082 −0.645564
\(933\) 0 0
\(934\) 23.1246 0.756660
\(935\) 19.6180 0.641578
\(936\) 0 0
\(937\) −12.2016 −0.398610 −0.199305 0.979938i \(-0.563868\pi\)
−0.199305 + 0.979938i \(0.563868\pi\)
\(938\) −8.94427 −0.292041
\(939\) 0 0
\(940\) 9.70820 0.316647
\(941\) 60.5066 1.97246 0.986229 0.165385i \(-0.0528868\pi\)
0.986229 + 0.165385i \(0.0528868\pi\)
\(942\) 0 0
\(943\) −12.3262 −0.401398
\(944\) 11.7082 0.381070
\(945\) 0 0
\(946\) 0 0
\(947\) −5.68692 −0.184800 −0.0924000 0.995722i \(-0.529454\pi\)
−0.0924000 + 0.995722i \(0.529454\pi\)
\(948\) 0 0
\(949\) −5.05573 −0.164116
\(950\) 4.85410 0.157488
\(951\) 0 0
\(952\) −8.23607 −0.266932
\(953\) 20.7984 0.673725 0.336863 0.941554i \(-0.390634\pi\)
0.336863 + 0.941554i \(0.390634\pi\)
\(954\) 0 0
\(955\) −0.291796 −0.00944230
\(956\) −24.3607 −0.787881
\(957\) 0 0
\(958\) 3.88854 0.125633
\(959\) −8.61803 −0.278291
\(960\) 0 0
\(961\) −26.6312 −0.859071
\(962\) 10.1115 0.326006
\(963\) 0 0
\(964\) 0 0
\(965\) −5.23607 −0.168555
\(966\) 0 0
\(967\) 50.5410 1.62529 0.812645 0.582759i \(-0.198027\pi\)
0.812645 + 0.582759i \(0.198027\pi\)
\(968\) −3.85410 −0.123876
\(969\) 0 0
\(970\) 14.6180 0.469357
\(971\) −0.729490 −0.0234105 −0.0117052 0.999931i \(-0.503726\pi\)
−0.0117052 + 0.999931i \(0.503726\pi\)
\(972\) 0 0
\(973\) 27.8885 0.894066
\(974\) 42.1803 1.35155
\(975\) 0 0
\(976\) 6.32624 0.202498
\(977\) 3.43769 0.109982 0.0549908 0.998487i \(-0.482487\pi\)
0.0549908 + 0.998487i \(0.482487\pi\)
\(978\) 0 0
\(979\) −5.88854 −0.188199
\(980\) 4.38197 0.139977
\(981\) 0 0
\(982\) −16.1803 −0.516335
\(983\) −19.2705 −0.614634 −0.307317 0.951607i \(-0.599431\pi\)
−0.307317 + 0.951607i \(0.599431\pi\)
\(984\) 0 0
\(985\) −2.43769 −0.0776714
\(986\) −24.2492 −0.772253
\(987\) 0 0
\(988\) −19.8541 −0.631643
\(989\) 0 0
\(990\) 0 0
\(991\) −10.5066 −0.333752 −0.166876 0.985978i \(-0.553368\pi\)
−0.166876 + 0.985978i \(0.553368\pi\)
\(992\) 2.09017 0.0663630
\(993\) 0 0
\(994\) 11.4721 0.363874
\(995\) −2.00000 −0.0634043
\(996\) 0 0
\(997\) −41.1935 −1.30461 −0.652306 0.757956i \(-0.726198\pi\)
−0.652306 + 0.757956i \(0.726198\pi\)
\(998\) 32.3607 1.02436
\(999\) 0 0
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 2070.2.a.u.1.2 2
3.2 odd 2 230.2.a.c.1.1 2
12.11 even 2 1840.2.a.l.1.2 2
15.2 even 4 1150.2.b.i.599.4 4
15.8 even 4 1150.2.b.i.599.1 4
15.14 odd 2 1150.2.a.j.1.2 2
24.5 odd 2 7360.2.a.bh.1.2 2
24.11 even 2 7360.2.a.bn.1.1 2
60.59 even 2 9200.2.a.bu.1.1 2
69.68 even 2 5290.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.1 2 3.2 odd 2
1150.2.a.j.1.2 2 15.14 odd 2
1150.2.b.i.599.1 4 15.8 even 4
1150.2.b.i.599.4 4 15.2 even 4
1840.2.a.l.1.2 2 12.11 even 2
2070.2.a.u.1.2 2 1.1 even 1 trivial
5290.2.a.o.1.1 2 69.68 even 2
7360.2.a.bh.1.2 2 24.5 odd 2
7360.2.a.bn.1.1 2 24.11 even 2
9200.2.a.bu.1.1 2 60.59 even 2