Properties

Label 8004.2.a.i.1.1
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.28632\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -4.28632 q^{5} +1.70119 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -4.28632 q^{5} +1.70119 q^{7} +1.00000 q^{9} +4.51446 q^{11} -1.48859 q^{13} +4.28632 q^{15} -1.55072 q^{17} -1.31263 q^{19} -1.70119 q^{21} +1.00000 q^{23} +13.3726 q^{25} -1.00000 q^{27} +1.00000 q^{29} -2.61274 q^{31} -4.51446 q^{33} -7.29183 q^{35} -0.322680 q^{37} +1.48859 q^{39} -7.70149 q^{41} +4.70934 q^{43} -4.28632 q^{45} +13.0439 q^{47} -4.10597 q^{49} +1.55072 q^{51} +7.24304 q^{53} -19.3504 q^{55} +1.31263 q^{57} +9.70167 q^{59} -4.23430 q^{61} +1.70119 q^{63} +6.38057 q^{65} -3.71674 q^{67} -1.00000 q^{69} +11.9916 q^{71} -10.7328 q^{73} -13.3726 q^{75} +7.67994 q^{77} -17.6430 q^{79} +1.00000 q^{81} +11.1598 q^{83} +6.64689 q^{85} -1.00000 q^{87} -17.7972 q^{89} -2.53236 q^{91} +2.61274 q^{93} +5.62635 q^{95} -0.650043 q^{97} +4.51446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −4.28632 −1.91690 −0.958451 0.285257i \(-0.907921\pi\)
−0.958451 + 0.285257i \(0.907921\pi\)
\(6\) 0 0
\(7\) 1.70119 0.642988 0.321494 0.946912i \(-0.395815\pi\)
0.321494 + 0.946912i \(0.395815\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.51446 1.36116 0.680581 0.732673i \(-0.261728\pi\)
0.680581 + 0.732673i \(0.261728\pi\)
\(12\) 0 0
\(13\) −1.48859 −0.412860 −0.206430 0.978461i \(-0.566185\pi\)
−0.206430 + 0.978461i \(0.566185\pi\)
\(14\) 0 0
\(15\) 4.28632 1.10672
\(16\) 0 0
\(17\) −1.55072 −0.376105 −0.188052 0.982159i \(-0.560217\pi\)
−0.188052 + 0.982159i \(0.560217\pi\)
\(18\) 0 0
\(19\) −1.31263 −0.301138 −0.150569 0.988600i \(-0.548111\pi\)
−0.150569 + 0.988600i \(0.548111\pi\)
\(20\) 0 0
\(21\) −1.70119 −0.371229
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 13.3726 2.67451
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −2.61274 −0.469262 −0.234631 0.972084i \(-0.575388\pi\)
−0.234631 + 0.972084i \(0.575388\pi\)
\(32\) 0 0
\(33\) −4.51446 −0.785867
\(34\) 0 0
\(35\) −7.29183 −1.23254
\(36\) 0 0
\(37\) −0.322680 −0.0530483 −0.0265241 0.999648i \(-0.508444\pi\)
−0.0265241 + 0.999648i \(0.508444\pi\)
\(38\) 0 0
\(39\) 1.48859 0.238365
\(40\) 0 0
\(41\) −7.70149 −1.20277 −0.601385 0.798959i \(-0.705384\pi\)
−0.601385 + 0.798959i \(0.705384\pi\)
\(42\) 0 0
\(43\) 4.70934 0.718168 0.359084 0.933305i \(-0.383089\pi\)
0.359084 + 0.933305i \(0.383089\pi\)
\(44\) 0 0
\(45\) −4.28632 −0.638967
\(46\) 0 0
\(47\) 13.0439 1.90264 0.951322 0.308199i \(-0.0997263\pi\)
0.951322 + 0.308199i \(0.0997263\pi\)
\(48\) 0 0
\(49\) −4.10597 −0.586567
\(50\) 0 0
\(51\) 1.55072 0.217144
\(52\) 0 0
\(53\) 7.24304 0.994908 0.497454 0.867490i \(-0.334268\pi\)
0.497454 + 0.867490i \(0.334268\pi\)
\(54\) 0 0
\(55\) −19.3504 −2.60921
\(56\) 0 0
\(57\) 1.31263 0.173862
\(58\) 0 0
\(59\) 9.70167 1.26305 0.631525 0.775356i \(-0.282429\pi\)
0.631525 + 0.775356i \(0.282429\pi\)
\(60\) 0 0
\(61\) −4.23430 −0.542146 −0.271073 0.962559i \(-0.587378\pi\)
−0.271073 + 0.962559i \(0.587378\pi\)
\(62\) 0 0
\(63\) 1.70119 0.214329
\(64\) 0 0
\(65\) 6.38057 0.791412
\(66\) 0 0
\(67\) −3.71674 −0.454072 −0.227036 0.973886i \(-0.572903\pi\)
−0.227036 + 0.973886i \(0.572903\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 11.9916 1.42314 0.711570 0.702616i \(-0.247985\pi\)
0.711570 + 0.702616i \(0.247985\pi\)
\(72\) 0 0
\(73\) −10.7328 −1.25618 −0.628092 0.778139i \(-0.716164\pi\)
−0.628092 + 0.778139i \(0.716164\pi\)
\(74\) 0 0
\(75\) −13.3726 −1.54413
\(76\) 0 0
\(77\) 7.67994 0.875210
\(78\) 0 0
\(79\) −17.6430 −1.98500 −0.992498 0.122262i \(-0.960985\pi\)
−0.992498 + 0.122262i \(0.960985\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.1598 1.22495 0.612473 0.790491i \(-0.290175\pi\)
0.612473 + 0.790491i \(0.290175\pi\)
\(84\) 0 0
\(85\) 6.64689 0.720956
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −17.7972 −1.88650 −0.943251 0.332080i \(-0.892250\pi\)
−0.943251 + 0.332080i \(0.892250\pi\)
\(90\) 0 0
\(91\) −2.53236 −0.265464
\(92\) 0 0
\(93\) 2.61274 0.270929
\(94\) 0 0
\(95\) 5.62635 0.577252
\(96\) 0 0
\(97\) −0.650043 −0.0660019 −0.0330009 0.999455i \(-0.510506\pi\)
−0.0330009 + 0.999455i \(0.510506\pi\)
\(98\) 0 0
\(99\) 4.51446 0.453721
\(100\) 0 0
\(101\) −16.9798 −1.68956 −0.844778 0.535117i \(-0.820267\pi\)
−0.844778 + 0.535117i \(0.820267\pi\)
\(102\) 0 0
\(103\) −19.0401 −1.87607 −0.938037 0.346535i \(-0.887358\pi\)
−0.938037 + 0.346535i \(0.887358\pi\)
\(104\) 0 0
\(105\) 7.29183 0.711610
\(106\) 0 0
\(107\) 10.3417 0.999766 0.499883 0.866093i \(-0.333376\pi\)
0.499883 + 0.866093i \(0.333376\pi\)
\(108\) 0 0
\(109\) −5.48832 −0.525685 −0.262843 0.964839i \(-0.584660\pi\)
−0.262843 + 0.964839i \(0.584660\pi\)
\(110\) 0 0
\(111\) 0.322680 0.0306274
\(112\) 0 0
\(113\) 5.27014 0.495773 0.247886 0.968789i \(-0.420264\pi\)
0.247886 + 0.968789i \(0.420264\pi\)
\(114\) 0 0
\(115\) −4.28632 −0.399702
\(116\) 0 0
\(117\) −1.48859 −0.137620
\(118\) 0 0
\(119\) −2.63806 −0.241831
\(120\) 0 0
\(121\) 9.38037 0.852761
\(122\) 0 0
\(123\) 7.70149 0.694420
\(124\) 0 0
\(125\) −35.8876 −3.20988
\(126\) 0 0
\(127\) −7.46610 −0.662509 −0.331255 0.943541i \(-0.607472\pi\)
−0.331255 + 0.943541i \(0.607472\pi\)
\(128\) 0 0
\(129\) −4.70934 −0.414634
\(130\) 0 0
\(131\) 11.3089 0.988059 0.494030 0.869445i \(-0.335523\pi\)
0.494030 + 0.869445i \(0.335523\pi\)
\(132\) 0 0
\(133\) −2.23303 −0.193628
\(134\) 0 0
\(135\) 4.28632 0.368908
\(136\) 0 0
\(137\) −9.30951 −0.795365 −0.397683 0.917523i \(-0.630186\pi\)
−0.397683 + 0.917523i \(0.630186\pi\)
\(138\) 0 0
\(139\) 18.9397 1.60645 0.803224 0.595677i \(-0.203116\pi\)
0.803224 + 0.595677i \(0.203116\pi\)
\(140\) 0 0
\(141\) −13.0439 −1.09849
\(142\) 0 0
\(143\) −6.72017 −0.561969
\(144\) 0 0
\(145\) −4.28632 −0.355960
\(146\) 0 0
\(147\) 4.10597 0.338654
\(148\) 0 0
\(149\) 0.200164 0.0163981 0.00819906 0.999966i \(-0.497390\pi\)
0.00819906 + 0.999966i \(0.497390\pi\)
\(150\) 0 0
\(151\) 16.1714 1.31601 0.658004 0.753015i \(-0.271401\pi\)
0.658004 + 0.753015i \(0.271401\pi\)
\(152\) 0 0
\(153\) −1.55072 −0.125368
\(154\) 0 0
\(155\) 11.1991 0.899530
\(156\) 0 0
\(157\) −6.72841 −0.536986 −0.268493 0.963282i \(-0.586526\pi\)
−0.268493 + 0.963282i \(0.586526\pi\)
\(158\) 0 0
\(159\) −7.24304 −0.574410
\(160\) 0 0
\(161\) 1.70119 0.134072
\(162\) 0 0
\(163\) −2.55383 −0.200032 −0.100016 0.994986i \(-0.531889\pi\)
−0.100016 + 0.994986i \(0.531889\pi\)
\(164\) 0 0
\(165\) 19.3504 1.50643
\(166\) 0 0
\(167\) −0.272414 −0.0210800 −0.0105400 0.999944i \(-0.503355\pi\)
−0.0105400 + 0.999944i \(0.503355\pi\)
\(168\) 0 0
\(169\) −10.7841 −0.829547
\(170\) 0 0
\(171\) −1.31263 −0.100379
\(172\) 0 0
\(173\) 17.0168 1.29376 0.646881 0.762591i \(-0.276073\pi\)
0.646881 + 0.762591i \(0.276073\pi\)
\(174\) 0 0
\(175\) 22.7492 1.71968
\(176\) 0 0
\(177\) −9.70167 −0.729222
\(178\) 0 0
\(179\) −12.3164 −0.920569 −0.460285 0.887771i \(-0.652253\pi\)
−0.460285 + 0.887771i \(0.652253\pi\)
\(180\) 0 0
\(181\) −4.16472 −0.309561 −0.154781 0.987949i \(-0.549467\pi\)
−0.154781 + 0.987949i \(0.549467\pi\)
\(182\) 0 0
\(183\) 4.23430 0.313008
\(184\) 0 0
\(185\) 1.38311 0.101688
\(186\) 0 0
\(187\) −7.00067 −0.511939
\(188\) 0 0
\(189\) −1.70119 −0.123743
\(190\) 0 0
\(191\) −18.8725 −1.36557 −0.682784 0.730621i \(-0.739231\pi\)
−0.682784 + 0.730621i \(0.739231\pi\)
\(192\) 0 0
\(193\) −7.72554 −0.556097 −0.278048 0.960567i \(-0.589688\pi\)
−0.278048 + 0.960567i \(0.589688\pi\)
\(194\) 0 0
\(195\) −6.38057 −0.456922
\(196\) 0 0
\(197\) 20.0387 1.42770 0.713851 0.700298i \(-0.246949\pi\)
0.713851 + 0.700298i \(0.246949\pi\)
\(198\) 0 0
\(199\) 26.1404 1.85304 0.926520 0.376244i \(-0.122785\pi\)
0.926520 + 0.376244i \(0.122785\pi\)
\(200\) 0 0
\(201\) 3.71674 0.262158
\(202\) 0 0
\(203\) 1.70119 0.119400
\(204\) 0 0
\(205\) 33.0111 2.30559
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −5.92581 −0.409897
\(210\) 0 0
\(211\) 5.36990 0.369679 0.184840 0.982769i \(-0.440823\pi\)
0.184840 + 0.982769i \(0.440823\pi\)
\(212\) 0 0
\(213\) −11.9916 −0.821650
\(214\) 0 0
\(215\) −20.1858 −1.37666
\(216\) 0 0
\(217\) −4.44476 −0.301730
\(218\) 0 0
\(219\) 10.7328 0.725258
\(220\) 0 0
\(221\) 2.30838 0.155279
\(222\) 0 0
\(223\) 21.2451 1.42268 0.711339 0.702849i \(-0.248089\pi\)
0.711339 + 0.702849i \(0.248089\pi\)
\(224\) 0 0
\(225\) 13.3726 0.891505
\(226\) 0 0
\(227\) 0.120251 0.00798134 0.00399067 0.999992i \(-0.498730\pi\)
0.00399067 + 0.999992i \(0.498730\pi\)
\(228\) 0 0
\(229\) 25.2154 1.66628 0.833141 0.553060i \(-0.186540\pi\)
0.833141 + 0.553060i \(0.186540\pi\)
\(230\) 0 0
\(231\) −7.67994 −0.505303
\(232\) 0 0
\(233\) 15.5036 1.01567 0.507837 0.861453i \(-0.330445\pi\)
0.507837 + 0.861453i \(0.330445\pi\)
\(234\) 0 0
\(235\) −55.9102 −3.64718
\(236\) 0 0
\(237\) 17.6430 1.14604
\(238\) 0 0
\(239\) −4.11436 −0.266136 −0.133068 0.991107i \(-0.542483\pi\)
−0.133068 + 0.991107i \(0.542483\pi\)
\(240\) 0 0
\(241\) 8.29590 0.534386 0.267193 0.963643i \(-0.413904\pi\)
0.267193 + 0.963643i \(0.413904\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 17.5995 1.12439
\(246\) 0 0
\(247\) 1.95396 0.124328
\(248\) 0 0
\(249\) −11.1598 −0.707223
\(250\) 0 0
\(251\) 24.4531 1.54347 0.771733 0.635946i \(-0.219390\pi\)
0.771733 + 0.635946i \(0.219390\pi\)
\(252\) 0 0
\(253\) 4.51446 0.283822
\(254\) 0 0
\(255\) −6.64689 −0.416244
\(256\) 0 0
\(257\) −6.87202 −0.428665 −0.214332 0.976761i \(-0.568758\pi\)
−0.214332 + 0.976761i \(0.568758\pi\)
\(258\) 0 0
\(259\) −0.548939 −0.0341094
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 14.2428 0.878246 0.439123 0.898427i \(-0.355289\pi\)
0.439123 + 0.898427i \(0.355289\pi\)
\(264\) 0 0
\(265\) −31.0460 −1.90714
\(266\) 0 0
\(267\) 17.7972 1.08917
\(268\) 0 0
\(269\) −23.4638 −1.43061 −0.715306 0.698811i \(-0.753713\pi\)
−0.715306 + 0.698811i \(0.753713\pi\)
\(270\) 0 0
\(271\) 21.3651 1.29784 0.648918 0.760859i \(-0.275222\pi\)
0.648918 + 0.760859i \(0.275222\pi\)
\(272\) 0 0
\(273\) 2.53236 0.153266
\(274\) 0 0
\(275\) 60.3700 3.64045
\(276\) 0 0
\(277\) −20.2990 −1.21965 −0.609824 0.792537i \(-0.708760\pi\)
−0.609824 + 0.792537i \(0.708760\pi\)
\(278\) 0 0
\(279\) −2.61274 −0.156421
\(280\) 0 0
\(281\) 13.9991 0.835117 0.417558 0.908650i \(-0.362886\pi\)
0.417558 + 0.908650i \(0.362886\pi\)
\(282\) 0 0
\(283\) 23.4813 1.39582 0.697909 0.716186i \(-0.254114\pi\)
0.697909 + 0.716186i \(0.254114\pi\)
\(284\) 0 0
\(285\) −5.62635 −0.333276
\(286\) 0 0
\(287\) −13.1017 −0.773367
\(288\) 0 0
\(289\) −14.5953 −0.858545
\(290\) 0 0
\(291\) 0.650043 0.0381062
\(292\) 0 0
\(293\) 6.87158 0.401442 0.200721 0.979648i \(-0.435672\pi\)
0.200721 + 0.979648i \(0.435672\pi\)
\(294\) 0 0
\(295\) −41.5845 −2.42114
\(296\) 0 0
\(297\) −4.51446 −0.261956
\(298\) 0 0
\(299\) −1.48859 −0.0860872
\(300\) 0 0
\(301\) 8.01147 0.461773
\(302\) 0 0
\(303\) 16.9798 0.975465
\(304\) 0 0
\(305\) 18.1496 1.03924
\(306\) 0 0
\(307\) 5.69689 0.325139 0.162569 0.986697i \(-0.448022\pi\)
0.162569 + 0.986697i \(0.448022\pi\)
\(308\) 0 0
\(309\) 19.0401 1.08315
\(310\) 0 0
\(311\) −8.45195 −0.479266 −0.239633 0.970864i \(-0.577027\pi\)
−0.239633 + 0.970864i \(0.577027\pi\)
\(312\) 0 0
\(313\) −29.2225 −1.65175 −0.825877 0.563850i \(-0.809319\pi\)
−0.825877 + 0.563850i \(0.809319\pi\)
\(314\) 0 0
\(315\) −7.29183 −0.410848
\(316\) 0 0
\(317\) −3.03917 −0.170697 −0.0853485 0.996351i \(-0.527200\pi\)
−0.0853485 + 0.996351i \(0.527200\pi\)
\(318\) 0 0
\(319\) 4.51446 0.252761
\(320\) 0 0
\(321\) −10.3417 −0.577215
\(322\) 0 0
\(323\) 2.03552 0.113259
\(324\) 0 0
\(325\) −19.9062 −1.10420
\(326\) 0 0
\(327\) 5.48832 0.303505
\(328\) 0 0
\(329\) 22.1900 1.22338
\(330\) 0 0
\(331\) 33.0230 1.81511 0.907555 0.419933i \(-0.137946\pi\)
0.907555 + 0.419933i \(0.137946\pi\)
\(332\) 0 0
\(333\) −0.322680 −0.0176828
\(334\) 0 0
\(335\) 15.9311 0.870411
\(336\) 0 0
\(337\) 33.0649 1.80116 0.900580 0.434690i \(-0.143142\pi\)
0.900580 + 0.434690i \(0.143142\pi\)
\(338\) 0 0
\(339\) −5.27014 −0.286235
\(340\) 0 0
\(341\) −11.7951 −0.638742
\(342\) 0 0
\(343\) −18.8933 −1.02014
\(344\) 0 0
\(345\) 4.28632 0.230768
\(346\) 0 0
\(347\) 26.7918 1.43826 0.719130 0.694875i \(-0.244540\pi\)
0.719130 + 0.694875i \(0.244540\pi\)
\(348\) 0 0
\(349\) −12.6622 −0.677790 −0.338895 0.940824i \(-0.610053\pi\)
−0.338895 + 0.940824i \(0.610053\pi\)
\(350\) 0 0
\(351\) 1.48859 0.0794549
\(352\) 0 0
\(353\) 2.30737 0.122809 0.0614046 0.998113i \(-0.480442\pi\)
0.0614046 + 0.998113i \(0.480442\pi\)
\(354\) 0 0
\(355\) −51.3998 −2.72802
\(356\) 0 0
\(357\) 2.63806 0.139621
\(358\) 0 0
\(359\) 30.5132 1.61043 0.805213 0.592986i \(-0.202051\pi\)
0.805213 + 0.592986i \(0.202051\pi\)
\(360\) 0 0
\(361\) −17.2770 −0.909316
\(362\) 0 0
\(363\) −9.38037 −0.492342
\(364\) 0 0
\(365\) 46.0044 2.40798
\(366\) 0 0
\(367\) −8.40674 −0.438828 −0.219414 0.975632i \(-0.570415\pi\)
−0.219414 + 0.975632i \(0.570415\pi\)
\(368\) 0 0
\(369\) −7.70149 −0.400923
\(370\) 0 0
\(371\) 12.3218 0.639714
\(372\) 0 0
\(373\) 13.9437 0.721977 0.360988 0.932570i \(-0.382439\pi\)
0.360988 + 0.932570i \(0.382439\pi\)
\(374\) 0 0
\(375\) 35.8876 1.85323
\(376\) 0 0
\(377\) −1.48859 −0.0766661
\(378\) 0 0
\(379\) 29.7324 1.52725 0.763624 0.645661i \(-0.223418\pi\)
0.763624 + 0.645661i \(0.223418\pi\)
\(380\) 0 0
\(381\) 7.46610 0.382500
\(382\) 0 0
\(383\) 5.87971 0.300439 0.150219 0.988653i \(-0.452002\pi\)
0.150219 + 0.988653i \(0.452002\pi\)
\(384\) 0 0
\(385\) −32.9187 −1.67769
\(386\) 0 0
\(387\) 4.70934 0.239389
\(388\) 0 0
\(389\) −3.17532 −0.160995 −0.0804976 0.996755i \(-0.525651\pi\)
−0.0804976 + 0.996755i \(0.525651\pi\)
\(390\) 0 0
\(391\) −1.55072 −0.0784233
\(392\) 0 0
\(393\) −11.3089 −0.570456
\(394\) 0 0
\(395\) 75.6237 3.80504
\(396\) 0 0
\(397\) 4.94465 0.248165 0.124082 0.992272i \(-0.460401\pi\)
0.124082 + 0.992272i \(0.460401\pi\)
\(398\) 0 0
\(399\) 2.23303 0.111791
\(400\) 0 0
\(401\) 32.4870 1.62232 0.811161 0.584823i \(-0.198836\pi\)
0.811161 + 0.584823i \(0.198836\pi\)
\(402\) 0 0
\(403\) 3.88929 0.193739
\(404\) 0 0
\(405\) −4.28632 −0.212989
\(406\) 0 0
\(407\) −1.45673 −0.0722073
\(408\) 0 0
\(409\) −6.19666 −0.306405 −0.153202 0.988195i \(-0.548959\pi\)
−0.153202 + 0.988195i \(0.548959\pi\)
\(410\) 0 0
\(411\) 9.30951 0.459204
\(412\) 0 0
\(413\) 16.5043 0.812125
\(414\) 0 0
\(415\) −47.8345 −2.34810
\(416\) 0 0
\(417\) −18.9397 −0.927483
\(418\) 0 0
\(419\) 28.8722 1.41050 0.705250 0.708959i \(-0.250835\pi\)
0.705250 + 0.708959i \(0.250835\pi\)
\(420\) 0 0
\(421\) 36.9336 1.80003 0.900016 0.435857i \(-0.143554\pi\)
0.900016 + 0.435857i \(0.143554\pi\)
\(422\) 0 0
\(423\) 13.0439 0.634215
\(424\) 0 0
\(425\) −20.7371 −1.00590
\(426\) 0 0
\(427\) −7.20332 −0.348593
\(428\) 0 0
\(429\) 6.72017 0.324453
\(430\) 0 0
\(431\) −18.7749 −0.904354 −0.452177 0.891928i \(-0.649353\pi\)
−0.452177 + 0.891928i \(0.649353\pi\)
\(432\) 0 0
\(433\) 25.2249 1.21223 0.606116 0.795377i \(-0.292727\pi\)
0.606116 + 0.795377i \(0.292727\pi\)
\(434\) 0 0
\(435\) 4.28632 0.205514
\(436\) 0 0
\(437\) −1.31263 −0.0627916
\(438\) 0 0
\(439\) 24.5690 1.17261 0.586306 0.810090i \(-0.300582\pi\)
0.586306 + 0.810090i \(0.300582\pi\)
\(440\) 0 0
\(441\) −4.10597 −0.195522
\(442\) 0 0
\(443\) −23.7431 −1.12807 −0.564034 0.825752i \(-0.690751\pi\)
−0.564034 + 0.825752i \(0.690751\pi\)
\(444\) 0 0
\(445\) 76.2847 3.61624
\(446\) 0 0
\(447\) −0.200164 −0.00946745
\(448\) 0 0
\(449\) 27.8149 1.31266 0.656332 0.754472i \(-0.272107\pi\)
0.656332 + 0.754472i \(0.272107\pi\)
\(450\) 0 0
\(451\) −34.7681 −1.63716
\(452\) 0 0
\(453\) −16.1714 −0.759797
\(454\) 0 0
\(455\) 10.8545 0.508868
\(456\) 0 0
\(457\) 41.9187 1.96088 0.980438 0.196829i \(-0.0630644\pi\)
0.980438 + 0.196829i \(0.0630644\pi\)
\(458\) 0 0
\(459\) 1.55072 0.0723814
\(460\) 0 0
\(461\) 4.48238 0.208765 0.104383 0.994537i \(-0.466713\pi\)
0.104383 + 0.994537i \(0.466713\pi\)
\(462\) 0 0
\(463\) 21.4327 0.996064 0.498032 0.867159i \(-0.334056\pi\)
0.498032 + 0.867159i \(0.334056\pi\)
\(464\) 0 0
\(465\) −11.1991 −0.519344
\(466\) 0 0
\(467\) −23.7109 −1.09721 −0.548605 0.836081i \(-0.684841\pi\)
−0.548605 + 0.836081i \(0.684841\pi\)
\(468\) 0 0
\(469\) −6.32286 −0.291962
\(470\) 0 0
\(471\) 6.72841 0.310029
\(472\) 0 0
\(473\) 21.2601 0.977543
\(474\) 0 0
\(475\) −17.5532 −0.805397
\(476\) 0 0
\(477\) 7.24304 0.331636
\(478\) 0 0
\(479\) −7.90312 −0.361103 −0.180551 0.983566i \(-0.557788\pi\)
−0.180551 + 0.983566i \(0.557788\pi\)
\(480\) 0 0
\(481\) 0.480337 0.0219015
\(482\) 0 0
\(483\) −1.70119 −0.0774066
\(484\) 0 0
\(485\) 2.78629 0.126519
\(486\) 0 0
\(487\) −40.5725 −1.83852 −0.919259 0.393654i \(-0.871211\pi\)
−0.919259 + 0.393654i \(0.871211\pi\)
\(488\) 0 0
\(489\) 2.55383 0.115488
\(490\) 0 0
\(491\) 32.5970 1.47108 0.735541 0.677480i \(-0.236928\pi\)
0.735541 + 0.677480i \(0.236928\pi\)
\(492\) 0 0
\(493\) −1.55072 −0.0698409
\(494\) 0 0
\(495\) −19.3504 −0.869738
\(496\) 0 0
\(497\) 20.3999 0.915061
\(498\) 0 0
\(499\) 16.3779 0.733177 0.366589 0.930383i \(-0.380526\pi\)
0.366589 + 0.930383i \(0.380526\pi\)
\(500\) 0 0
\(501\) 0.272414 0.0121706
\(502\) 0 0
\(503\) −38.6217 −1.72206 −0.861028 0.508557i \(-0.830179\pi\)
−0.861028 + 0.508557i \(0.830179\pi\)
\(504\) 0 0
\(505\) 72.7810 3.23871
\(506\) 0 0
\(507\) 10.7841 0.478939
\(508\) 0 0
\(509\) 14.7904 0.655571 0.327786 0.944752i \(-0.393698\pi\)
0.327786 + 0.944752i \(0.393698\pi\)
\(510\) 0 0
\(511\) −18.2585 −0.807711
\(512\) 0 0
\(513\) 1.31263 0.0579540
\(514\) 0 0
\(515\) 81.6119 3.59625
\(516\) 0 0
\(517\) 58.8860 2.58981
\(518\) 0 0
\(519\) −17.0168 −0.746954
\(520\) 0 0
\(521\) 17.4435 0.764212 0.382106 0.924119i \(-0.375199\pi\)
0.382106 + 0.924119i \(0.375199\pi\)
\(522\) 0 0
\(523\) 2.98273 0.130426 0.0652129 0.997871i \(-0.479227\pi\)
0.0652129 + 0.997871i \(0.479227\pi\)
\(524\) 0 0
\(525\) −22.7492 −0.992858
\(526\) 0 0
\(527\) 4.05163 0.176492
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 9.70167 0.421016
\(532\) 0 0
\(533\) 11.4643 0.496575
\(534\) 0 0
\(535\) −44.3277 −1.91645
\(536\) 0 0
\(537\) 12.3164 0.531491
\(538\) 0 0
\(539\) −18.5362 −0.798412
\(540\) 0 0
\(541\) −3.89722 −0.167554 −0.0837772 0.996485i \(-0.526698\pi\)
−0.0837772 + 0.996485i \(0.526698\pi\)
\(542\) 0 0
\(543\) 4.16472 0.178725
\(544\) 0 0
\(545\) 23.5247 1.00769
\(546\) 0 0
\(547\) −25.3710 −1.08479 −0.542393 0.840125i \(-0.682482\pi\)
−0.542393 + 0.840125i \(0.682482\pi\)
\(548\) 0 0
\(549\) −4.23430 −0.180715
\(550\) 0 0
\(551\) −1.31263 −0.0559199
\(552\) 0 0
\(553\) −30.0141 −1.27633
\(554\) 0 0
\(555\) −1.38311 −0.0587098
\(556\) 0 0
\(557\) −19.5331 −0.827646 −0.413823 0.910357i \(-0.635807\pi\)
−0.413823 + 0.910357i \(0.635807\pi\)
\(558\) 0 0
\(559\) −7.01027 −0.296503
\(560\) 0 0
\(561\) 7.00067 0.295568
\(562\) 0 0
\(563\) 27.8534 1.17388 0.586939 0.809631i \(-0.300333\pi\)
0.586939 + 0.809631i \(0.300333\pi\)
\(564\) 0 0
\(565\) −22.5895 −0.950348
\(566\) 0 0
\(567\) 1.70119 0.0714431
\(568\) 0 0
\(569\) 18.4536 0.773617 0.386808 0.922160i \(-0.373578\pi\)
0.386808 + 0.922160i \(0.373578\pi\)
\(570\) 0 0
\(571\) 22.9586 0.960789 0.480394 0.877053i \(-0.340494\pi\)
0.480394 + 0.877053i \(0.340494\pi\)
\(572\) 0 0
\(573\) 18.8725 0.788411
\(574\) 0 0
\(575\) 13.3726 0.557675
\(576\) 0 0
\(577\) −16.2346 −0.675854 −0.337927 0.941172i \(-0.609726\pi\)
−0.337927 + 0.941172i \(0.609726\pi\)
\(578\) 0 0
\(579\) 7.72554 0.321063
\(580\) 0 0
\(581\) 18.9849 0.787626
\(582\) 0 0
\(583\) 32.6984 1.35423
\(584\) 0 0
\(585\) 6.38057 0.263804
\(586\) 0 0
\(587\) 18.1911 0.750828 0.375414 0.926857i \(-0.377500\pi\)
0.375414 + 0.926857i \(0.377500\pi\)
\(588\) 0 0
\(589\) 3.42956 0.141313
\(590\) 0 0
\(591\) −20.0387 −0.824284
\(592\) 0 0
\(593\) −43.8123 −1.79915 −0.899577 0.436762i \(-0.856125\pi\)
−0.899577 + 0.436762i \(0.856125\pi\)
\(594\) 0 0
\(595\) 11.3076 0.463566
\(596\) 0 0
\(597\) −26.1404 −1.06985
\(598\) 0 0
\(599\) −30.0723 −1.22872 −0.614359 0.789026i \(-0.710586\pi\)
−0.614359 + 0.789026i \(0.710586\pi\)
\(600\) 0 0
\(601\) −4.69148 −0.191370 −0.0956848 0.995412i \(-0.530504\pi\)
−0.0956848 + 0.995412i \(0.530504\pi\)
\(602\) 0 0
\(603\) −3.71674 −0.151357
\(604\) 0 0
\(605\) −40.2073 −1.63466
\(606\) 0 0
\(607\) −2.04644 −0.0830626 −0.0415313 0.999137i \(-0.513224\pi\)
−0.0415313 + 0.999137i \(0.513224\pi\)
\(608\) 0 0
\(609\) −1.70119 −0.0689355
\(610\) 0 0
\(611\) −19.4169 −0.785525
\(612\) 0 0
\(613\) −24.0994 −0.973368 −0.486684 0.873578i \(-0.661794\pi\)
−0.486684 + 0.873578i \(0.661794\pi\)
\(614\) 0 0
\(615\) −33.0111 −1.33113
\(616\) 0 0
\(617\) −35.8902 −1.44488 −0.722442 0.691431i \(-0.756980\pi\)
−0.722442 + 0.691431i \(0.756980\pi\)
\(618\) 0 0
\(619\) −20.8832 −0.839368 −0.419684 0.907670i \(-0.637859\pi\)
−0.419684 + 0.907670i \(0.637859\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −30.2764 −1.21300
\(624\) 0 0
\(625\) 86.9628 3.47851
\(626\) 0 0
\(627\) 5.92581 0.236654
\(628\) 0 0
\(629\) 0.500387 0.0199517
\(630\) 0 0
\(631\) −28.5474 −1.13645 −0.568226 0.822872i \(-0.692370\pi\)
−0.568226 + 0.822872i \(0.692370\pi\)
\(632\) 0 0
\(633\) −5.36990 −0.213434
\(634\) 0 0
\(635\) 32.0021 1.26997
\(636\) 0 0
\(637\) 6.11209 0.242170
\(638\) 0 0
\(639\) 11.9916 0.474380
\(640\) 0 0
\(641\) 26.2128 1.03534 0.517672 0.855579i \(-0.326799\pi\)
0.517672 + 0.855579i \(0.326799\pi\)
\(642\) 0 0
\(643\) −35.7568 −1.41011 −0.705055 0.709153i \(-0.749078\pi\)
−0.705055 + 0.709153i \(0.749078\pi\)
\(644\) 0 0
\(645\) 20.1858 0.794814
\(646\) 0 0
\(647\) −26.4374 −1.03936 −0.519681 0.854360i \(-0.673949\pi\)
−0.519681 + 0.854360i \(0.673949\pi\)
\(648\) 0 0
\(649\) 43.7978 1.71921
\(650\) 0 0
\(651\) 4.44476 0.174204
\(652\) 0 0
\(653\) −13.1035 −0.512780 −0.256390 0.966573i \(-0.582533\pi\)
−0.256390 + 0.966573i \(0.582533\pi\)
\(654\) 0 0
\(655\) −48.4734 −1.89401
\(656\) 0 0
\(657\) −10.7328 −0.418728
\(658\) 0 0
\(659\) −8.19408 −0.319196 −0.159598 0.987182i \(-0.551020\pi\)
−0.159598 + 0.987182i \(0.551020\pi\)
\(660\) 0 0
\(661\) −10.1152 −0.393434 −0.196717 0.980460i \(-0.563028\pi\)
−0.196717 + 0.980460i \(0.563028\pi\)
\(662\) 0 0
\(663\) −2.30838 −0.0896501
\(664\) 0 0
\(665\) 9.57147 0.371166
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) −21.2451 −0.821383
\(670\) 0 0
\(671\) −19.1156 −0.737948
\(672\) 0 0
\(673\) 31.4242 1.21132 0.605658 0.795725i \(-0.292910\pi\)
0.605658 + 0.795725i \(0.292910\pi\)
\(674\) 0 0
\(675\) −13.3726 −0.514711
\(676\) 0 0
\(677\) −36.4259 −1.39996 −0.699980 0.714162i \(-0.746808\pi\)
−0.699980 + 0.714162i \(0.746808\pi\)
\(678\) 0 0
\(679\) −1.10584 −0.0424384
\(680\) 0 0
\(681\) −0.120251 −0.00460803
\(682\) 0 0
\(683\) −29.6658 −1.13513 −0.567566 0.823328i \(-0.692115\pi\)
−0.567566 + 0.823328i \(0.692115\pi\)
\(684\) 0 0
\(685\) 39.9036 1.52464
\(686\) 0 0
\(687\) −25.2154 −0.962029
\(688\) 0 0
\(689\) −10.7819 −0.410758
\(690\) 0 0
\(691\) 31.1234 1.18399 0.591996 0.805941i \(-0.298340\pi\)
0.591996 + 0.805941i \(0.298340\pi\)
\(692\) 0 0
\(693\) 7.67994 0.291737
\(694\) 0 0
\(695\) −81.1818 −3.07940
\(696\) 0 0
\(697\) 11.9429 0.452368
\(698\) 0 0
\(699\) −15.5036 −0.586400
\(700\) 0 0
\(701\) 21.0718 0.795870 0.397935 0.917414i \(-0.369727\pi\)
0.397935 + 0.917414i \(0.369727\pi\)
\(702\) 0 0
\(703\) 0.423559 0.0159748
\(704\) 0 0
\(705\) 55.9102 2.10570
\(706\) 0 0
\(707\) −28.8858 −1.08636
\(708\) 0 0
\(709\) −4.39042 −0.164886 −0.0824429 0.996596i \(-0.526272\pi\)
−0.0824429 + 0.996596i \(0.526272\pi\)
\(710\) 0 0
\(711\) −17.6430 −0.661665
\(712\) 0 0
\(713\) −2.61274 −0.0978479
\(714\) 0 0
\(715\) 28.8048 1.07724
\(716\) 0 0
\(717\) 4.11436 0.153653
\(718\) 0 0
\(719\) 30.6811 1.14421 0.572106 0.820180i \(-0.306127\pi\)
0.572106 + 0.820180i \(0.306127\pi\)
\(720\) 0 0
\(721\) −32.3907 −1.20629
\(722\) 0 0
\(723\) −8.29590 −0.308528
\(724\) 0 0
\(725\) 13.3726 0.496645
\(726\) 0 0
\(727\) −19.9250 −0.738976 −0.369488 0.929236i \(-0.620467\pi\)
−0.369488 + 0.929236i \(0.620467\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −7.30287 −0.270106
\(732\) 0 0
\(733\) −20.5753 −0.759965 −0.379982 0.924994i \(-0.624070\pi\)
−0.379982 + 0.924994i \(0.624070\pi\)
\(734\) 0 0
\(735\) −17.5995 −0.649167
\(736\) 0 0
\(737\) −16.7791 −0.618065
\(738\) 0 0
\(739\) −19.8600 −0.730560 −0.365280 0.930898i \(-0.619027\pi\)
−0.365280 + 0.930898i \(0.619027\pi\)
\(740\) 0 0
\(741\) −1.95396 −0.0717806
\(742\) 0 0
\(743\) 13.3852 0.491056 0.245528 0.969389i \(-0.421039\pi\)
0.245528 + 0.969389i \(0.421039\pi\)
\(744\) 0 0
\(745\) −0.857970 −0.0314336
\(746\) 0 0
\(747\) 11.1598 0.408315
\(748\) 0 0
\(749\) 17.5931 0.642837
\(750\) 0 0
\(751\) −26.0496 −0.950565 −0.475282 0.879833i \(-0.657654\pi\)
−0.475282 + 0.879833i \(0.657654\pi\)
\(752\) 0 0
\(753\) −24.4531 −0.891121
\(754\) 0 0
\(755\) −69.3157 −2.52266
\(756\) 0 0
\(757\) −28.7077 −1.04340 −0.521700 0.853129i \(-0.674702\pi\)
−0.521700 + 0.853129i \(0.674702\pi\)
\(758\) 0 0
\(759\) −4.51446 −0.163865
\(760\) 0 0
\(761\) 10.9034 0.395247 0.197624 0.980278i \(-0.436678\pi\)
0.197624 + 0.980278i \(0.436678\pi\)
\(762\) 0 0
\(763\) −9.33665 −0.338009
\(764\) 0 0
\(765\) 6.64689 0.240319
\(766\) 0 0
\(767\) −14.4418 −0.521462
\(768\) 0 0
\(769\) −35.0411 −1.26361 −0.631807 0.775126i \(-0.717686\pi\)
−0.631807 + 0.775126i \(0.717686\pi\)
\(770\) 0 0
\(771\) 6.87202 0.247490
\(772\) 0 0
\(773\) −3.19141 −0.114787 −0.0573936 0.998352i \(-0.518279\pi\)
−0.0573936 + 0.998352i \(0.518279\pi\)
\(774\) 0 0
\(775\) −34.9391 −1.25505
\(776\) 0 0
\(777\) 0.548939 0.0196931
\(778\) 0 0
\(779\) 10.1092 0.362200
\(780\) 0 0
\(781\) 54.1356 1.93712
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 28.8402 1.02935
\(786\) 0 0
\(787\) 5.29740 0.188832 0.0944160 0.995533i \(-0.469902\pi\)
0.0944160 + 0.995533i \(0.469902\pi\)
\(788\) 0 0
\(789\) −14.2428 −0.507056
\(790\) 0 0
\(791\) 8.96549 0.318776
\(792\) 0 0
\(793\) 6.30312 0.223830
\(794\) 0 0
\(795\) 31.0460 1.10109
\(796\) 0 0
\(797\) 10.2239 0.362150 0.181075 0.983469i \(-0.442042\pi\)
0.181075 + 0.983469i \(0.442042\pi\)
\(798\) 0 0
\(799\) −20.2274 −0.715594
\(800\) 0 0
\(801\) −17.7972 −0.628834
\(802\) 0 0
\(803\) −48.4530 −1.70987
\(804\) 0 0
\(805\) −7.29183 −0.257003
\(806\) 0 0
\(807\) 23.4638 0.825964
\(808\) 0 0
\(809\) 44.0830 1.54988 0.774939 0.632036i \(-0.217781\pi\)
0.774939 + 0.632036i \(0.217781\pi\)
\(810\) 0 0
\(811\) 29.5966 1.03928 0.519638 0.854387i \(-0.326067\pi\)
0.519638 + 0.854387i \(0.326067\pi\)
\(812\) 0 0
\(813\) −21.3651 −0.749306
\(814\) 0 0
\(815\) 10.9466 0.383441
\(816\) 0 0
\(817\) −6.18162 −0.216267
\(818\) 0 0
\(819\) −2.53236 −0.0884879
\(820\) 0 0
\(821\) 0.314106 0.0109624 0.00548119 0.999985i \(-0.498255\pi\)
0.00548119 + 0.999985i \(0.498255\pi\)
\(822\) 0 0
\(823\) 1.68347 0.0586819 0.0293410 0.999569i \(-0.490659\pi\)
0.0293410 + 0.999569i \(0.490659\pi\)
\(824\) 0 0
\(825\) −60.3700 −2.10181
\(826\) 0 0
\(827\) −31.8277 −1.10676 −0.553379 0.832930i \(-0.686662\pi\)
−0.553379 + 0.832930i \(0.686662\pi\)
\(828\) 0 0
\(829\) 42.0120 1.45914 0.729568 0.683908i \(-0.239721\pi\)
0.729568 + 0.683908i \(0.239721\pi\)
\(830\) 0 0
\(831\) 20.2990 0.704164
\(832\) 0 0
\(833\) 6.36721 0.220611
\(834\) 0 0
\(835\) 1.16766 0.0404084
\(836\) 0 0
\(837\) 2.61274 0.0903095
\(838\) 0 0
\(839\) 30.5412 1.05440 0.527200 0.849742i \(-0.323242\pi\)
0.527200 + 0.849742i \(0.323242\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −13.9991 −0.482155
\(844\) 0 0
\(845\) 46.2242 1.59016
\(846\) 0 0
\(847\) 15.9577 0.548315
\(848\) 0 0
\(849\) −23.4813 −0.805876
\(850\) 0 0
\(851\) −0.322680 −0.0110613
\(852\) 0 0
\(853\) −10.3151 −0.353181 −0.176591 0.984284i \(-0.556507\pi\)
−0.176591 + 0.984284i \(0.556507\pi\)
\(854\) 0 0
\(855\) 5.62635 0.192417
\(856\) 0 0
\(857\) 28.7351 0.981571 0.490786 0.871280i \(-0.336710\pi\)
0.490786 + 0.871280i \(0.336710\pi\)
\(858\) 0 0
\(859\) 22.6931 0.774278 0.387139 0.922021i \(-0.373463\pi\)
0.387139 + 0.922021i \(0.373463\pi\)
\(860\) 0 0
\(861\) 13.1017 0.446503
\(862\) 0 0
\(863\) 26.0771 0.887676 0.443838 0.896107i \(-0.353617\pi\)
0.443838 + 0.896107i \(0.353617\pi\)
\(864\) 0 0
\(865\) −72.9395 −2.48002
\(866\) 0 0
\(867\) 14.5953 0.495681
\(868\) 0 0
\(869\) −79.6488 −2.70190
\(870\) 0 0
\(871\) 5.53268 0.187468
\(872\) 0 0
\(873\) −0.650043 −0.0220006
\(874\) 0 0
\(875\) −61.0514 −2.06391
\(876\) 0 0
\(877\) −9.89634 −0.334176 −0.167088 0.985942i \(-0.553436\pi\)
−0.167088 + 0.985942i \(0.553436\pi\)
\(878\) 0 0
\(879\) −6.87158 −0.231773
\(880\) 0 0
\(881\) 43.1270 1.45298 0.726492 0.687175i \(-0.241149\pi\)
0.726492 + 0.687175i \(0.241149\pi\)
\(882\) 0 0
\(883\) −43.1136 −1.45089 −0.725445 0.688280i \(-0.758366\pi\)
−0.725445 + 0.688280i \(0.758366\pi\)
\(884\) 0 0
\(885\) 41.5845 1.39785
\(886\) 0 0
\(887\) −20.2874 −0.681185 −0.340593 0.940211i \(-0.610628\pi\)
−0.340593 + 0.940211i \(0.610628\pi\)
\(888\) 0 0
\(889\) −12.7012 −0.425985
\(890\) 0 0
\(891\) 4.51446 0.151240
\(892\) 0 0
\(893\) −17.1218 −0.572958
\(894\) 0 0
\(895\) 52.7920 1.76464
\(896\) 0 0
\(897\) 1.48859 0.0497025
\(898\) 0 0
\(899\) −2.61274 −0.0871398
\(900\) 0 0
\(901\) −11.2319 −0.374190
\(902\) 0 0
\(903\) −8.01147 −0.266605
\(904\) 0 0
\(905\) 17.8513 0.593398
\(906\) 0 0
\(907\) 3.56974 0.118531 0.0592656 0.998242i \(-0.481124\pi\)
0.0592656 + 0.998242i \(0.481124\pi\)
\(908\) 0 0
\(909\) −16.9798 −0.563185
\(910\) 0 0
\(911\) 16.8315 0.557651 0.278826 0.960342i \(-0.410055\pi\)
0.278826 + 0.960342i \(0.410055\pi\)
\(912\) 0 0
\(913\) 50.3805 1.66735
\(914\) 0 0
\(915\) −18.1496 −0.600006
\(916\) 0 0
\(917\) 19.2385 0.635310
\(918\) 0 0
\(919\) 7.69555 0.253853 0.126926 0.991912i \(-0.459489\pi\)
0.126926 + 0.991912i \(0.459489\pi\)
\(920\) 0 0
\(921\) −5.69689 −0.187719
\(922\) 0 0
\(923\) −17.8505 −0.587557
\(924\) 0 0
\(925\) −4.31506 −0.141878
\(926\) 0 0
\(927\) −19.0401 −0.625358
\(928\) 0 0
\(929\) 35.4882 1.16433 0.582165 0.813071i \(-0.302206\pi\)
0.582165 + 0.813071i \(0.302206\pi\)
\(930\) 0 0
\(931\) 5.38961 0.176637
\(932\) 0 0
\(933\) 8.45195 0.276704
\(934\) 0 0
\(935\) 30.0071 0.981338
\(936\) 0 0
\(937\) 31.7653 1.03773 0.518864 0.854857i \(-0.326355\pi\)
0.518864 + 0.854857i \(0.326355\pi\)
\(938\) 0 0
\(939\) 29.2225 0.953640
\(940\) 0 0
\(941\) 48.4539 1.57955 0.789777 0.613394i \(-0.210196\pi\)
0.789777 + 0.613394i \(0.210196\pi\)
\(942\) 0 0
\(943\) −7.70149 −0.250795
\(944\) 0 0
\(945\) 7.29183 0.237203
\(946\) 0 0
\(947\) 40.5567 1.31792 0.658958 0.752180i \(-0.270997\pi\)
0.658958 + 0.752180i \(0.270997\pi\)
\(948\) 0 0
\(949\) 15.9768 0.518628
\(950\) 0 0
\(951\) 3.03917 0.0985520
\(952\) 0 0
\(953\) −18.7158 −0.606263 −0.303131 0.952949i \(-0.598032\pi\)
−0.303131 + 0.952949i \(0.598032\pi\)
\(954\) 0 0
\(955\) 80.8937 2.61766
\(956\) 0 0
\(957\) −4.51446 −0.145932
\(958\) 0 0
\(959\) −15.8372 −0.511410
\(960\) 0 0
\(961\) −24.1736 −0.779793
\(962\) 0 0
\(963\) 10.3417 0.333255
\(964\) 0 0
\(965\) 33.1142 1.06598
\(966\) 0 0
\(967\) −22.1539 −0.712421 −0.356210 0.934406i \(-0.615931\pi\)
−0.356210 + 0.934406i \(0.615931\pi\)
\(968\) 0 0
\(969\) −2.03552 −0.0653903
\(970\) 0 0
\(971\) −12.2222 −0.392228 −0.196114 0.980581i \(-0.562832\pi\)
−0.196114 + 0.980581i \(0.562832\pi\)
\(972\) 0 0
\(973\) 32.2200 1.03293
\(974\) 0 0
\(975\) 19.9062 0.637510
\(976\) 0 0
\(977\) −29.0812 −0.930391 −0.465196 0.885208i \(-0.654016\pi\)
−0.465196 + 0.885208i \(0.654016\pi\)
\(978\) 0 0
\(979\) −80.3449 −2.56783
\(980\) 0 0
\(981\) −5.48832 −0.175228
\(982\) 0 0
\(983\) 39.2793 1.25281 0.626407 0.779496i \(-0.284525\pi\)
0.626407 + 0.779496i \(0.284525\pi\)
\(984\) 0 0
\(985\) −85.8926 −2.73676
\(986\) 0 0
\(987\) −22.1900 −0.706317
\(988\) 0 0
\(989\) 4.70934 0.149748
\(990\) 0 0
\(991\) 23.4921 0.746250 0.373125 0.927781i \(-0.378286\pi\)
0.373125 + 0.927781i \(0.378286\pi\)
\(992\) 0 0
\(993\) −33.0230 −1.04795
\(994\) 0 0
\(995\) −112.046 −3.55210
\(996\) 0 0
\(997\) 46.0294 1.45777 0.728883 0.684638i \(-0.240040\pi\)
0.728883 + 0.684638i \(0.240040\pi\)
\(998\) 0 0
\(999\) 0.322680 0.0102091
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 8004.2.a.i.1.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.1 16 1.1 even 1 trivial