Properties

Label 1560.4.a.n.1.3
Level $1560$
Weight $4$
Character 1560.1
Self dual yes
Analytic conductor $92.043$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.0429796090\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 41x^{2} - 30x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(7.24301\) of defining polynomial
Character \(\chi\) \(=\) 1560.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +5.00000 q^{5} +4.92462 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +5.00000 q^{5} +4.92462 q^{7} +9.00000 q^{9} +33.0195 q^{11} +13.0000 q^{13} +15.0000 q^{15} -119.438 q^{17} -127.793 q^{19} +14.7738 q^{21} -30.2485 q^{23} +25.0000 q^{25} +27.0000 q^{27} -309.449 q^{29} -57.9486 q^{31} +99.0584 q^{33} +24.6231 q^{35} -191.930 q^{37} +39.0000 q^{39} +78.4634 q^{41} -524.538 q^{43} +45.0000 q^{45} +183.052 q^{47} -318.748 q^{49} -358.313 q^{51} -241.053 q^{53} +165.097 q^{55} -383.380 q^{57} +103.095 q^{59} +778.610 q^{61} +44.3215 q^{63} +65.0000 q^{65} +230.431 q^{67} -90.7454 q^{69} +668.821 q^{71} +706.428 q^{73} +75.0000 q^{75} +162.608 q^{77} -283.228 q^{79} +81.0000 q^{81} -564.393 q^{83} -597.188 q^{85} -928.346 q^{87} -1561.14 q^{89} +64.0200 q^{91} -173.846 q^{93} -638.967 q^{95} +1444.89 q^{97} +297.175 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9} - 38 q^{11} + 52 q^{13} + 60 q^{15} - 50 q^{17} - 180 q^{19} - 102 q^{21} - 170 q^{23} + 100 q^{25} + 108 q^{27} - 84 q^{29} - 408 q^{31} - 114 q^{33} - 170 q^{35} - 38 q^{37} + 156 q^{39} - 90 q^{41} - 732 q^{43} + 180 q^{45} - 520 q^{47} + 230 q^{49} - 150 q^{51} - 338 q^{53} - 190 q^{55} - 540 q^{57} - 232 q^{59} + 326 q^{61} - 306 q^{63} + 260 q^{65} - 1040 q^{67} - 510 q^{69} - 702 q^{71} - 368 q^{73} + 300 q^{75} + 14 q^{77} - 678 q^{79} + 324 q^{81} - 1888 q^{83} - 250 q^{85} - 252 q^{87} - 2070 q^{89} - 442 q^{91} - 1224 q^{93} - 900 q^{95} + 1126 q^{97} - 342 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 4.92462 0.265904 0.132952 0.991122i \(-0.457554\pi\)
0.132952 + 0.991122i \(0.457554\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 33.0195 0.905067 0.452534 0.891747i \(-0.350520\pi\)
0.452534 + 0.891747i \(0.350520\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) −119.438 −1.70399 −0.851996 0.523549i \(-0.824608\pi\)
−0.851996 + 0.523549i \(0.824608\pi\)
\(18\) 0 0
\(19\) −127.793 −1.54304 −0.771521 0.636204i \(-0.780504\pi\)
−0.771521 + 0.636204i \(0.780504\pi\)
\(20\) 0 0
\(21\) 14.7738 0.153520
\(22\) 0 0
\(23\) −30.2485 −0.274228 −0.137114 0.990555i \(-0.543783\pi\)
−0.137114 + 0.990555i \(0.543783\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −309.449 −1.98149 −0.990744 0.135743i \(-0.956658\pi\)
−0.990744 + 0.135743i \(0.956658\pi\)
\(30\) 0 0
\(31\) −57.9486 −0.335738 −0.167869 0.985809i \(-0.553689\pi\)
−0.167869 + 0.985809i \(0.553689\pi\)
\(32\) 0 0
\(33\) 99.0584 0.522541
\(34\) 0 0
\(35\) 24.6231 0.118916
\(36\) 0 0
\(37\) −191.930 −0.852786 −0.426393 0.904538i \(-0.640216\pi\)
−0.426393 + 0.904538i \(0.640216\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) 78.4634 0.298876 0.149438 0.988771i \(-0.452254\pi\)
0.149438 + 0.988771i \(0.452254\pi\)
\(42\) 0 0
\(43\) −524.538 −1.86026 −0.930131 0.367227i \(-0.880307\pi\)
−0.930131 + 0.367227i \(0.880307\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 0 0
\(47\) 183.052 0.568104 0.284052 0.958809i \(-0.408321\pi\)
0.284052 + 0.958809i \(0.408321\pi\)
\(48\) 0 0
\(49\) −318.748 −0.929295
\(50\) 0 0
\(51\) −358.313 −0.983800
\(52\) 0 0
\(53\) −241.053 −0.624738 −0.312369 0.949961i \(-0.601123\pi\)
−0.312369 + 0.949961i \(0.601123\pi\)
\(54\) 0 0
\(55\) 165.097 0.404758
\(56\) 0 0
\(57\) −383.380 −0.890876
\(58\) 0 0
\(59\) 103.095 0.227489 0.113745 0.993510i \(-0.463715\pi\)
0.113745 + 0.993510i \(0.463715\pi\)
\(60\) 0 0
\(61\) 778.610 1.63427 0.817137 0.576443i \(-0.195560\pi\)
0.817137 + 0.576443i \(0.195560\pi\)
\(62\) 0 0
\(63\) 44.3215 0.0886348
\(64\) 0 0
\(65\) 65.0000 0.124035
\(66\) 0 0
\(67\) 230.431 0.420173 0.210087 0.977683i \(-0.432625\pi\)
0.210087 + 0.977683i \(0.432625\pi\)
\(68\) 0 0
\(69\) −90.7454 −0.158325
\(70\) 0 0
\(71\) 668.821 1.11795 0.558975 0.829184i \(-0.311195\pi\)
0.558975 + 0.829184i \(0.311195\pi\)
\(72\) 0 0
\(73\) 706.428 1.13262 0.566309 0.824193i \(-0.308371\pi\)
0.566309 + 0.824193i \(0.308371\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) 162.608 0.240661
\(78\) 0 0
\(79\) −283.228 −0.403362 −0.201681 0.979451i \(-0.564640\pi\)
−0.201681 + 0.979451i \(0.564640\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −564.393 −0.746388 −0.373194 0.927753i \(-0.621737\pi\)
−0.373194 + 0.927753i \(0.621737\pi\)
\(84\) 0 0
\(85\) −597.188 −0.762048
\(86\) 0 0
\(87\) −928.346 −1.14401
\(88\) 0 0
\(89\) −1561.14 −1.85933 −0.929665 0.368405i \(-0.879904\pi\)
−0.929665 + 0.368405i \(0.879904\pi\)
\(90\) 0 0
\(91\) 64.0200 0.0737486
\(92\) 0 0
\(93\) −173.846 −0.193838
\(94\) 0 0
\(95\) −638.967 −0.690069
\(96\) 0 0
\(97\) 1444.89 1.51244 0.756220 0.654318i \(-0.227044\pi\)
0.756220 + 0.654318i \(0.227044\pi\)
\(98\) 0 0
\(99\) 297.175 0.301689
\(100\) 0 0
\(101\) −693.814 −0.683535 −0.341768 0.939784i \(-0.611026\pi\)
−0.341768 + 0.939784i \(0.611026\pi\)
\(102\) 0 0
\(103\) −278.392 −0.266318 −0.133159 0.991095i \(-0.542512\pi\)
−0.133159 + 0.991095i \(0.542512\pi\)
\(104\) 0 0
\(105\) 73.8692 0.0686562
\(106\) 0 0
\(107\) 1392.53 1.25814 0.629070 0.777349i \(-0.283436\pi\)
0.629070 + 0.777349i \(0.283436\pi\)
\(108\) 0 0
\(109\) −462.126 −0.406089 −0.203044 0.979170i \(-0.565084\pi\)
−0.203044 + 0.979170i \(0.565084\pi\)
\(110\) 0 0
\(111\) −575.790 −0.492356
\(112\) 0 0
\(113\) −112.029 −0.0932635 −0.0466317 0.998912i \(-0.514849\pi\)
−0.0466317 + 0.998912i \(0.514849\pi\)
\(114\) 0 0
\(115\) −151.242 −0.122638
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) −588.184 −0.453099
\(120\) 0 0
\(121\) −240.715 −0.180853
\(122\) 0 0
\(123\) 235.390 0.172556
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1986.87 −1.38824 −0.694119 0.719860i \(-0.744206\pi\)
−0.694119 + 0.719860i \(0.744206\pi\)
\(128\) 0 0
\(129\) −1573.61 −1.07402
\(130\) 0 0
\(131\) −2941.06 −1.96154 −0.980771 0.195163i \(-0.937476\pi\)
−0.980771 + 0.195163i \(0.937476\pi\)
\(132\) 0 0
\(133\) −629.333 −0.410301
\(134\) 0 0
\(135\) 135.000 0.0860663
\(136\) 0 0
\(137\) −473.482 −0.295272 −0.147636 0.989042i \(-0.547166\pi\)
−0.147636 + 0.989042i \(0.547166\pi\)
\(138\) 0 0
\(139\) 2.57545 0.00157156 0.000785780 1.00000i \(-0.499750\pi\)
0.000785780 1.00000i \(0.499750\pi\)
\(140\) 0 0
\(141\) 549.157 0.327995
\(142\) 0 0
\(143\) 429.253 0.251021
\(144\) 0 0
\(145\) −1547.24 −0.886148
\(146\) 0 0
\(147\) −956.244 −0.536529
\(148\) 0 0
\(149\) 3276.52 1.80150 0.900749 0.434340i \(-0.143018\pi\)
0.900749 + 0.434340i \(0.143018\pi\)
\(150\) 0 0
\(151\) −1431.27 −0.771360 −0.385680 0.922633i \(-0.626033\pi\)
−0.385680 + 0.922633i \(0.626033\pi\)
\(152\) 0 0
\(153\) −1074.94 −0.567997
\(154\) 0 0
\(155\) −289.743 −0.150147
\(156\) 0 0
\(157\) 297.943 0.151455 0.0757275 0.997129i \(-0.475872\pi\)
0.0757275 + 0.997129i \(0.475872\pi\)
\(158\) 0 0
\(159\) −723.158 −0.360693
\(160\) 0 0
\(161\) −148.962 −0.0729183
\(162\) 0 0
\(163\) 1548.45 0.744074 0.372037 0.928218i \(-0.378659\pi\)
0.372037 + 0.928218i \(0.378659\pi\)
\(164\) 0 0
\(165\) 495.292 0.233687
\(166\) 0 0
\(167\) 585.292 0.271205 0.135602 0.990763i \(-0.456703\pi\)
0.135602 + 0.990763i \(0.456703\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −1150.14 −0.514347
\(172\) 0 0
\(173\) 794.258 0.349054 0.174527 0.984652i \(-0.444160\pi\)
0.174527 + 0.984652i \(0.444160\pi\)
\(174\) 0 0
\(175\) 123.115 0.0531809
\(176\) 0 0
\(177\) 309.286 0.131341
\(178\) 0 0
\(179\) 1889.39 0.788935 0.394467 0.918910i \(-0.370929\pi\)
0.394467 + 0.918910i \(0.370929\pi\)
\(180\) 0 0
\(181\) −743.010 −0.305124 −0.152562 0.988294i \(-0.548752\pi\)
−0.152562 + 0.988294i \(0.548752\pi\)
\(182\) 0 0
\(183\) 2335.83 0.943549
\(184\) 0 0
\(185\) −959.650 −0.381377
\(186\) 0 0
\(187\) −3943.76 −1.54223
\(188\) 0 0
\(189\) 132.965 0.0511733
\(190\) 0 0
\(191\) 1598.20 0.605454 0.302727 0.953077i \(-0.402103\pi\)
0.302727 + 0.953077i \(0.402103\pi\)
\(192\) 0 0
\(193\) −3079.75 −1.14863 −0.574314 0.818635i \(-0.694731\pi\)
−0.574314 + 0.818635i \(0.694731\pi\)
\(194\) 0 0
\(195\) 195.000 0.0716115
\(196\) 0 0
\(197\) 3809.78 1.37785 0.688923 0.724835i \(-0.258084\pi\)
0.688923 + 0.724835i \(0.258084\pi\)
\(198\) 0 0
\(199\) 3923.17 1.39752 0.698760 0.715357i \(-0.253736\pi\)
0.698760 + 0.715357i \(0.253736\pi\)
\(200\) 0 0
\(201\) 691.293 0.242587
\(202\) 0 0
\(203\) −1523.92 −0.526886
\(204\) 0 0
\(205\) 392.317 0.133661
\(206\) 0 0
\(207\) −272.236 −0.0914092
\(208\) 0 0
\(209\) −4219.67 −1.39656
\(210\) 0 0
\(211\) 5000.96 1.63166 0.815830 0.578292i \(-0.196280\pi\)
0.815830 + 0.578292i \(0.196280\pi\)
\(212\) 0 0
\(213\) 2006.46 0.645449
\(214\) 0 0
\(215\) −2622.69 −0.831935
\(216\) 0 0
\(217\) −285.375 −0.0892741
\(218\) 0 0
\(219\) 2119.28 0.653917
\(220\) 0 0
\(221\) −1552.69 −0.472602
\(222\) 0 0
\(223\) −3007.61 −0.903158 −0.451579 0.892231i \(-0.649139\pi\)
−0.451579 + 0.892231i \(0.649139\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −1368.56 −0.400152 −0.200076 0.979780i \(-0.564119\pi\)
−0.200076 + 0.979780i \(0.564119\pi\)
\(228\) 0 0
\(229\) −3877.06 −1.11879 −0.559396 0.828901i \(-0.688967\pi\)
−0.559396 + 0.828901i \(0.688967\pi\)
\(230\) 0 0
\(231\) 487.824 0.138946
\(232\) 0 0
\(233\) −1785.04 −0.501898 −0.250949 0.968000i \(-0.580743\pi\)
−0.250949 + 0.968000i \(0.580743\pi\)
\(234\) 0 0
\(235\) 915.261 0.254064
\(236\) 0 0
\(237\) −849.683 −0.232881
\(238\) 0 0
\(239\) −2595.79 −0.702543 −0.351272 0.936274i \(-0.614251\pi\)
−0.351272 + 0.936274i \(0.614251\pi\)
\(240\) 0 0
\(241\) 1357.57 0.362858 0.181429 0.983404i \(-0.441928\pi\)
0.181429 + 0.983404i \(0.441928\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −1593.74 −0.415593
\(246\) 0 0
\(247\) −1661.31 −0.427963
\(248\) 0 0
\(249\) −1693.18 −0.430927
\(250\) 0 0
\(251\) −4350.81 −1.09411 −0.547053 0.837098i \(-0.684250\pi\)
−0.547053 + 0.837098i \(0.684250\pi\)
\(252\) 0 0
\(253\) −998.787 −0.248195
\(254\) 0 0
\(255\) −1791.56 −0.439969
\(256\) 0 0
\(257\) 1521.29 0.369242 0.184621 0.982810i \(-0.440894\pi\)
0.184621 + 0.982810i \(0.440894\pi\)
\(258\) 0 0
\(259\) −945.181 −0.226759
\(260\) 0 0
\(261\) −2785.04 −0.660496
\(262\) 0 0
\(263\) −4402.93 −1.03230 −0.516152 0.856497i \(-0.672636\pi\)
−0.516152 + 0.856497i \(0.672636\pi\)
\(264\) 0 0
\(265\) −1205.26 −0.279391
\(266\) 0 0
\(267\) −4683.42 −1.07348
\(268\) 0 0
\(269\) 507.808 0.115099 0.0575494 0.998343i \(-0.481671\pi\)
0.0575494 + 0.998343i \(0.481671\pi\)
\(270\) 0 0
\(271\) 1851.72 0.415069 0.207535 0.978228i \(-0.433456\pi\)
0.207535 + 0.978228i \(0.433456\pi\)
\(272\) 0 0
\(273\) 192.060 0.0425788
\(274\) 0 0
\(275\) 825.486 0.181013
\(276\) 0 0
\(277\) −7984.20 −1.73186 −0.865928 0.500169i \(-0.833271\pi\)
−0.865928 + 0.500169i \(0.833271\pi\)
\(278\) 0 0
\(279\) −521.537 −0.111913
\(280\) 0 0
\(281\) 6594.79 1.40004 0.700021 0.714122i \(-0.253174\pi\)
0.700021 + 0.714122i \(0.253174\pi\)
\(282\) 0 0
\(283\) −8838.66 −1.85655 −0.928276 0.371893i \(-0.878709\pi\)
−0.928276 + 0.371893i \(0.878709\pi\)
\(284\) 0 0
\(285\) −1916.90 −0.398412
\(286\) 0 0
\(287\) 386.402 0.0794724
\(288\) 0 0
\(289\) 9352.32 1.90359
\(290\) 0 0
\(291\) 4334.68 0.873207
\(292\) 0 0
\(293\) 4720.35 0.941180 0.470590 0.882352i \(-0.344041\pi\)
0.470590 + 0.882352i \(0.344041\pi\)
\(294\) 0 0
\(295\) 515.477 0.101736
\(296\) 0 0
\(297\) 891.525 0.174180
\(298\) 0 0
\(299\) −393.230 −0.0760571
\(300\) 0 0
\(301\) −2583.15 −0.494652
\(302\) 0 0
\(303\) −2081.44 −0.394639
\(304\) 0 0
\(305\) 3893.05 0.730870
\(306\) 0 0
\(307\) −5097.82 −0.947714 −0.473857 0.880602i \(-0.657139\pi\)
−0.473857 + 0.880602i \(0.657139\pi\)
\(308\) 0 0
\(309\) −835.175 −0.153759
\(310\) 0 0
\(311\) −8865.03 −1.61637 −0.808183 0.588932i \(-0.799549\pi\)
−0.808183 + 0.588932i \(0.799549\pi\)
\(312\) 0 0
\(313\) −2904.17 −0.524452 −0.262226 0.965006i \(-0.584457\pi\)
−0.262226 + 0.965006i \(0.584457\pi\)
\(314\) 0 0
\(315\) 221.608 0.0396387
\(316\) 0 0
\(317\) 3599.19 0.637699 0.318850 0.947805i \(-0.396704\pi\)
0.318850 + 0.947805i \(0.396704\pi\)
\(318\) 0 0
\(319\) −10217.8 −1.79338
\(320\) 0 0
\(321\) 4177.59 0.726388
\(322\) 0 0
\(323\) 15263.3 2.62933
\(324\) 0 0
\(325\) 325.000 0.0554700
\(326\) 0 0
\(327\) −1386.38 −0.234455
\(328\) 0 0
\(329\) 901.462 0.151061
\(330\) 0 0
\(331\) −1313.36 −0.218092 −0.109046 0.994037i \(-0.534780\pi\)
−0.109046 + 0.994037i \(0.534780\pi\)
\(332\) 0 0
\(333\) −1727.37 −0.284262
\(334\) 0 0
\(335\) 1152.15 0.187907
\(336\) 0 0
\(337\) 532.634 0.0860963 0.0430481 0.999073i \(-0.486293\pi\)
0.0430481 + 0.999073i \(0.486293\pi\)
\(338\) 0 0
\(339\) −336.086 −0.0538457
\(340\) 0 0
\(341\) −1913.43 −0.303865
\(342\) 0 0
\(343\) −3258.86 −0.513008
\(344\) 0 0
\(345\) −453.727 −0.0708053
\(346\) 0 0
\(347\) 5191.65 0.803177 0.401588 0.915820i \(-0.368458\pi\)
0.401588 + 0.915820i \(0.368458\pi\)
\(348\) 0 0
\(349\) 3081.50 0.472633 0.236316 0.971676i \(-0.424060\pi\)
0.236316 + 0.971676i \(0.424060\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) 0 0
\(353\) 5601.39 0.844566 0.422283 0.906464i \(-0.361229\pi\)
0.422283 + 0.906464i \(0.361229\pi\)
\(354\) 0 0
\(355\) 3344.11 0.499962
\(356\) 0 0
\(357\) −1764.55 −0.261597
\(358\) 0 0
\(359\) 58.2648 0.00856573 0.00428286 0.999991i \(-0.498637\pi\)
0.00428286 + 0.999991i \(0.498637\pi\)
\(360\) 0 0
\(361\) 9472.13 1.38098
\(362\) 0 0
\(363\) −722.146 −0.104416
\(364\) 0 0
\(365\) 3532.14 0.506522
\(366\) 0 0
\(367\) 10571.6 1.50363 0.751813 0.659377i \(-0.229180\pi\)
0.751813 + 0.659377i \(0.229180\pi\)
\(368\) 0 0
\(369\) 706.170 0.0996254
\(370\) 0 0
\(371\) −1187.09 −0.166121
\(372\) 0 0
\(373\) −5244.76 −0.728052 −0.364026 0.931389i \(-0.618598\pi\)
−0.364026 + 0.931389i \(0.618598\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) 0 0
\(377\) −4022.83 −0.549566
\(378\) 0 0
\(379\) −11078.0 −1.50142 −0.750708 0.660634i \(-0.770288\pi\)
−0.750708 + 0.660634i \(0.770288\pi\)
\(380\) 0 0
\(381\) −5960.61 −0.801500
\(382\) 0 0
\(383\) −12771.9 −1.70395 −0.851974 0.523585i \(-0.824594\pi\)
−0.851974 + 0.523585i \(0.824594\pi\)
\(384\) 0 0
\(385\) 813.041 0.107627
\(386\) 0 0
\(387\) −4720.84 −0.620088
\(388\) 0 0
\(389\) −7046.01 −0.918372 −0.459186 0.888340i \(-0.651859\pi\)
−0.459186 + 0.888340i \(0.651859\pi\)
\(390\) 0 0
\(391\) 3612.80 0.467282
\(392\) 0 0
\(393\) −8823.19 −1.13250
\(394\) 0 0
\(395\) −1416.14 −0.180389
\(396\) 0 0
\(397\) 4637.68 0.586293 0.293146 0.956068i \(-0.405298\pi\)
0.293146 + 0.956068i \(0.405298\pi\)
\(398\) 0 0
\(399\) −1888.00 −0.236888
\(400\) 0 0
\(401\) 4933.11 0.614333 0.307167 0.951656i \(-0.400619\pi\)
0.307167 + 0.951656i \(0.400619\pi\)
\(402\) 0 0
\(403\) −753.332 −0.0931169
\(404\) 0 0
\(405\) 405.000 0.0496904
\(406\) 0 0
\(407\) −6337.42 −0.771829
\(408\) 0 0
\(409\) −3363.64 −0.406653 −0.203327 0.979111i \(-0.565175\pi\)
−0.203327 + 0.979111i \(0.565175\pi\)
\(410\) 0 0
\(411\) −1420.45 −0.170475
\(412\) 0 0
\(413\) 507.705 0.0604904
\(414\) 0 0
\(415\) −2821.96 −0.333795
\(416\) 0 0
\(417\) 7.72635 0.000907340 0
\(418\) 0 0
\(419\) −3927.92 −0.457974 −0.228987 0.973429i \(-0.573541\pi\)
−0.228987 + 0.973429i \(0.573541\pi\)
\(420\) 0 0
\(421\) −7755.35 −0.897797 −0.448899 0.893583i \(-0.648184\pi\)
−0.448899 + 0.893583i \(0.648184\pi\)
\(422\) 0 0
\(423\) 1647.47 0.189368
\(424\) 0 0
\(425\) −2985.94 −0.340798
\(426\) 0 0
\(427\) 3834.35 0.434561
\(428\) 0 0
\(429\) 1287.76 0.144927
\(430\) 0 0
\(431\) 937.848 0.104813 0.0524067 0.998626i \(-0.483311\pi\)
0.0524067 + 0.998626i \(0.483311\pi\)
\(432\) 0 0
\(433\) 13242.5 1.46973 0.734865 0.678214i \(-0.237246\pi\)
0.734865 + 0.678214i \(0.237246\pi\)
\(434\) 0 0
\(435\) −4641.73 −0.511618
\(436\) 0 0
\(437\) 3865.55 0.423145
\(438\) 0 0
\(439\) −17088.1 −1.85779 −0.928897 0.370337i \(-0.879242\pi\)
−0.928897 + 0.370337i \(0.879242\pi\)
\(440\) 0 0
\(441\) −2868.73 −0.309765
\(442\) 0 0
\(443\) 10963.6 1.17584 0.587920 0.808919i \(-0.299947\pi\)
0.587920 + 0.808919i \(0.299947\pi\)
\(444\) 0 0
\(445\) −7805.70 −0.831518
\(446\) 0 0
\(447\) 9829.57 1.04010
\(448\) 0 0
\(449\) 13803.9 1.45088 0.725440 0.688286i \(-0.241636\pi\)
0.725440 + 0.688286i \(0.241636\pi\)
\(450\) 0 0
\(451\) 2590.82 0.270503
\(452\) 0 0
\(453\) −4293.82 −0.445345
\(454\) 0 0
\(455\) 320.100 0.0329814
\(456\) 0 0
\(457\) 10184.9 1.04252 0.521260 0.853398i \(-0.325462\pi\)
0.521260 + 0.853398i \(0.325462\pi\)
\(458\) 0 0
\(459\) −3224.81 −0.327933
\(460\) 0 0
\(461\) 16798.2 1.69712 0.848559 0.529101i \(-0.177471\pi\)
0.848559 + 0.529101i \(0.177471\pi\)
\(462\) 0 0
\(463\) −8215.56 −0.824643 −0.412321 0.911038i \(-0.635282\pi\)
−0.412321 + 0.911038i \(0.635282\pi\)
\(464\) 0 0
\(465\) −869.229 −0.0866871
\(466\) 0 0
\(467\) 10706.2 1.06086 0.530431 0.847728i \(-0.322030\pi\)
0.530431 + 0.847728i \(0.322030\pi\)
\(468\) 0 0
\(469\) 1134.78 0.111726
\(470\) 0 0
\(471\) 893.829 0.0874426
\(472\) 0 0
\(473\) −17320.0 −1.68366
\(474\) 0 0
\(475\) −3194.83 −0.308608
\(476\) 0 0
\(477\) −2169.47 −0.208246
\(478\) 0 0
\(479\) −9145.64 −0.872390 −0.436195 0.899852i \(-0.643674\pi\)
−0.436195 + 0.899852i \(0.643674\pi\)
\(480\) 0 0
\(481\) −2495.09 −0.236520
\(482\) 0 0
\(483\) −446.886 −0.0420994
\(484\) 0 0
\(485\) 7224.46 0.676383
\(486\) 0 0
\(487\) −1996.38 −0.185759 −0.0928797 0.995677i \(-0.529607\pi\)
−0.0928797 + 0.995677i \(0.529607\pi\)
\(488\) 0 0
\(489\) 4645.35 0.429591
\(490\) 0 0
\(491\) 1733.44 0.159326 0.0796628 0.996822i \(-0.474616\pi\)
0.0796628 + 0.996822i \(0.474616\pi\)
\(492\) 0 0
\(493\) 36959.8 3.37644
\(494\) 0 0
\(495\) 1485.88 0.134919
\(496\) 0 0
\(497\) 3293.69 0.297268
\(498\) 0 0
\(499\) 1391.46 0.124830 0.0624152 0.998050i \(-0.480120\pi\)
0.0624152 + 0.998050i \(0.480120\pi\)
\(500\) 0 0
\(501\) 1755.88 0.156580
\(502\) 0 0
\(503\) −10970.4 −0.972460 −0.486230 0.873831i \(-0.661628\pi\)
−0.486230 + 0.873831i \(0.661628\pi\)
\(504\) 0 0
\(505\) −3469.07 −0.305686
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) −8556.24 −0.745086 −0.372543 0.928015i \(-0.621514\pi\)
−0.372543 + 0.928015i \(0.621514\pi\)
\(510\) 0 0
\(511\) 3478.89 0.301168
\(512\) 0 0
\(513\) −3450.42 −0.296959
\(514\) 0 0
\(515\) −1391.96 −0.119101
\(516\) 0 0
\(517\) 6044.28 0.514173
\(518\) 0 0
\(519\) 2382.77 0.201526
\(520\) 0 0
\(521\) 7973.37 0.670479 0.335240 0.942133i \(-0.391183\pi\)
0.335240 + 0.942133i \(0.391183\pi\)
\(522\) 0 0
\(523\) −19676.9 −1.64514 −0.822572 0.568660i \(-0.807462\pi\)
−0.822572 + 0.568660i \(0.807462\pi\)
\(524\) 0 0
\(525\) 369.346 0.0307040
\(526\) 0 0
\(527\) 6921.23 0.572094
\(528\) 0 0
\(529\) −11252.0 −0.924799
\(530\) 0 0
\(531\) 927.859 0.0758298
\(532\) 0 0
\(533\) 1020.02 0.0828933
\(534\) 0 0
\(535\) 6962.65 0.562657
\(536\) 0 0
\(537\) 5668.16 0.455492
\(538\) 0 0
\(539\) −10524.9 −0.841074
\(540\) 0 0
\(541\) −2186.81 −0.173786 −0.0868932 0.996218i \(-0.527694\pi\)
−0.0868932 + 0.996218i \(0.527694\pi\)
\(542\) 0 0
\(543\) −2229.03 −0.176164
\(544\) 0 0
\(545\) −2310.63 −0.181608
\(546\) 0 0
\(547\) −23647.4 −1.84843 −0.924213 0.381878i \(-0.875278\pi\)
−0.924213 + 0.381878i \(0.875278\pi\)
\(548\) 0 0
\(549\) 7007.49 0.544758
\(550\) 0 0
\(551\) 39545.5 3.05752
\(552\) 0 0
\(553\) −1394.79 −0.107256
\(554\) 0 0
\(555\) −2878.95 −0.220188
\(556\) 0 0
\(557\) 21859.1 1.66283 0.831416 0.555650i \(-0.187530\pi\)
0.831416 + 0.555650i \(0.187530\pi\)
\(558\) 0 0
\(559\) −6818.99 −0.515944
\(560\) 0 0
\(561\) −11831.3 −0.890405
\(562\) 0 0
\(563\) 4854.00 0.363360 0.181680 0.983358i \(-0.441847\pi\)
0.181680 + 0.983358i \(0.441847\pi\)
\(564\) 0 0
\(565\) −560.143 −0.0417087
\(566\) 0 0
\(567\) 398.894 0.0295449
\(568\) 0 0
\(569\) 5896.70 0.434451 0.217225 0.976121i \(-0.430299\pi\)
0.217225 + 0.976121i \(0.430299\pi\)
\(570\) 0 0
\(571\) −4297.28 −0.314949 −0.157474 0.987523i \(-0.550335\pi\)
−0.157474 + 0.987523i \(0.550335\pi\)
\(572\) 0 0
\(573\) 4794.60 0.349559
\(574\) 0 0
\(575\) −756.211 −0.0548455
\(576\) 0 0
\(577\) −5148.34 −0.371453 −0.185726 0.982602i \(-0.559464\pi\)
−0.185726 + 0.982602i \(0.559464\pi\)
\(578\) 0 0
\(579\) −9239.25 −0.663161
\(580\) 0 0
\(581\) −2779.42 −0.198468
\(582\) 0 0
\(583\) −7959.43 −0.565430
\(584\) 0 0
\(585\) 585.000 0.0413449
\(586\) 0 0
\(587\) −9766.94 −0.686754 −0.343377 0.939198i \(-0.611571\pi\)
−0.343377 + 0.939198i \(0.611571\pi\)
\(588\) 0 0
\(589\) 7405.44 0.518058
\(590\) 0 0
\(591\) 11429.3 0.795500
\(592\) 0 0
\(593\) 27428.8 1.89943 0.949717 0.313111i \(-0.101371\pi\)
0.949717 + 0.313111i \(0.101371\pi\)
\(594\) 0 0
\(595\) −2940.92 −0.202632
\(596\) 0 0
\(597\) 11769.5 0.806858
\(598\) 0 0
\(599\) 14371.6 0.980311 0.490156 0.871635i \(-0.336940\pi\)
0.490156 + 0.871635i \(0.336940\pi\)
\(600\) 0 0
\(601\) 12103.3 0.821470 0.410735 0.911755i \(-0.365272\pi\)
0.410735 + 0.911755i \(0.365272\pi\)
\(602\) 0 0
\(603\) 2073.88 0.140058
\(604\) 0 0
\(605\) −1203.58 −0.0808800
\(606\) 0 0
\(607\) −14922.6 −0.997839 −0.498920 0.866648i \(-0.666270\pi\)
−0.498920 + 0.866648i \(0.666270\pi\)
\(608\) 0 0
\(609\) −4571.75 −0.304198
\(610\) 0 0
\(611\) 2379.68 0.157564
\(612\) 0 0
\(613\) 16112.5 1.06163 0.530815 0.847488i \(-0.321886\pi\)
0.530815 + 0.847488i \(0.321886\pi\)
\(614\) 0 0
\(615\) 1176.95 0.0771695
\(616\) 0 0
\(617\) −22097.7 −1.44185 −0.720923 0.693015i \(-0.756282\pi\)
−0.720923 + 0.693015i \(0.756282\pi\)
\(618\) 0 0
\(619\) −5307.99 −0.344663 −0.172331 0.985039i \(-0.555130\pi\)
−0.172331 + 0.985039i \(0.555130\pi\)
\(620\) 0 0
\(621\) −816.708 −0.0527752
\(622\) 0 0
\(623\) −7688.01 −0.494404
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −12659.0 −0.806302
\(628\) 0 0
\(629\) 22923.6 1.45314
\(630\) 0 0
\(631\) −21024.5 −1.32642 −0.663211 0.748433i \(-0.730807\pi\)
−0.663211 + 0.748433i \(0.730807\pi\)
\(632\) 0 0
\(633\) 15002.9 0.942039
\(634\) 0 0
\(635\) −9934.35 −0.620839
\(636\) 0 0
\(637\) −4143.73 −0.257740
\(638\) 0 0
\(639\) 6019.39 0.372650
\(640\) 0 0
\(641\) 12148.9 0.748599 0.374300 0.927308i \(-0.377883\pi\)
0.374300 + 0.927308i \(0.377883\pi\)
\(642\) 0 0
\(643\) 15272.5 0.936687 0.468344 0.883546i \(-0.344851\pi\)
0.468344 + 0.883546i \(0.344851\pi\)
\(644\) 0 0
\(645\) −7868.07 −0.480318
\(646\) 0 0
\(647\) 17144.3 1.04175 0.520875 0.853633i \(-0.325606\pi\)
0.520875 + 0.853633i \(0.325606\pi\)
\(648\) 0 0
\(649\) 3404.15 0.205893
\(650\) 0 0
\(651\) −856.124 −0.0515424
\(652\) 0 0
\(653\) 3379.46 0.202524 0.101262 0.994860i \(-0.467712\pi\)
0.101262 + 0.994860i \(0.467712\pi\)
\(654\) 0 0
\(655\) −14705.3 −0.877228
\(656\) 0 0
\(657\) 6357.85 0.377539
\(658\) 0 0
\(659\) 12684.6 0.749803 0.374902 0.927065i \(-0.377677\pi\)
0.374902 + 0.927065i \(0.377677\pi\)
\(660\) 0 0
\(661\) −22946.2 −1.35023 −0.675116 0.737712i \(-0.735906\pi\)
−0.675116 + 0.737712i \(0.735906\pi\)
\(662\) 0 0
\(663\) −4658.06 −0.272857
\(664\) 0 0
\(665\) −3146.67 −0.183492
\(666\) 0 0
\(667\) 9360.34 0.543379
\(668\) 0 0
\(669\) −9022.82 −0.521438
\(670\) 0 0
\(671\) 25709.3 1.47913
\(672\) 0 0
\(673\) 16725.6 0.957986 0.478993 0.877819i \(-0.341002\pi\)
0.478993 + 0.877819i \(0.341002\pi\)
\(674\) 0 0
\(675\) 675.000 0.0384900
\(676\) 0 0
\(677\) 14220.8 0.807313 0.403657 0.914911i \(-0.367739\pi\)
0.403657 + 0.914911i \(0.367739\pi\)
\(678\) 0 0
\(679\) 7115.54 0.402164
\(680\) 0 0
\(681\) −4105.68 −0.231028
\(682\) 0 0
\(683\) 32769.8 1.83587 0.917936 0.396728i \(-0.129854\pi\)
0.917936 + 0.396728i \(0.129854\pi\)
\(684\) 0 0
\(685\) −2367.41 −0.132050
\(686\) 0 0
\(687\) −11631.2 −0.645935
\(688\) 0 0
\(689\) −3133.68 −0.173271
\(690\) 0 0
\(691\) −6824.82 −0.375729 −0.187864 0.982195i \(-0.560157\pi\)
−0.187864 + 0.982195i \(0.560157\pi\)
\(692\) 0 0
\(693\) 1463.47 0.0802204
\(694\) 0 0
\(695\) 12.8772 0.000702823 0
\(696\) 0 0
\(697\) −9371.47 −0.509282
\(698\) 0 0
\(699\) −5355.13 −0.289771
\(700\) 0 0
\(701\) −2989.72 −0.161085 −0.0805423 0.996751i \(-0.525665\pi\)
−0.0805423 + 0.996751i \(0.525665\pi\)
\(702\) 0 0
\(703\) 24527.4 1.31588
\(704\) 0 0
\(705\) 2745.78 0.146684
\(706\) 0 0
\(707\) −3416.77 −0.181755
\(708\) 0 0
\(709\) −27605.9 −1.46229 −0.731143 0.682224i \(-0.761013\pi\)
−0.731143 + 0.682224i \(0.761013\pi\)
\(710\) 0 0
\(711\) −2549.05 −0.134454
\(712\) 0 0
\(713\) 1752.85 0.0920686
\(714\) 0 0
\(715\) 2146.26 0.112260
\(716\) 0 0
\(717\) −7787.38 −0.405614
\(718\) 0 0
\(719\) −15643.3 −0.811401 −0.405701 0.914006i \(-0.632972\pi\)
−0.405701 + 0.914006i \(0.632972\pi\)
\(720\) 0 0
\(721\) −1370.97 −0.0708150
\(722\) 0 0
\(723\) 4072.71 0.209496
\(724\) 0 0
\(725\) −7736.21 −0.396298
\(726\) 0 0
\(727\) 27676.5 1.41192 0.705958 0.708253i \(-0.250517\pi\)
0.705958 + 0.708253i \(0.250517\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 62649.5 3.16987
\(732\) 0 0
\(733\) 26683.5 1.34458 0.672290 0.740288i \(-0.265311\pi\)
0.672290 + 0.740288i \(0.265311\pi\)
\(734\) 0 0
\(735\) −4781.22 −0.239943
\(736\) 0 0
\(737\) 7608.70 0.380285
\(738\) 0 0
\(739\) 21455.4 1.06800 0.533998 0.845486i \(-0.320689\pi\)
0.533998 + 0.845486i \(0.320689\pi\)
\(740\) 0 0
\(741\) −4983.94 −0.247084
\(742\) 0 0
\(743\) 9493.05 0.468730 0.234365 0.972149i \(-0.424699\pi\)
0.234365 + 0.972149i \(0.424699\pi\)
\(744\) 0 0
\(745\) 16382.6 0.805654
\(746\) 0 0
\(747\) −5079.54 −0.248796
\(748\) 0 0
\(749\) 6857.68 0.334545
\(750\) 0 0
\(751\) −21585.4 −1.04882 −0.524410 0.851466i \(-0.675714\pi\)
−0.524410 + 0.851466i \(0.675714\pi\)
\(752\) 0 0
\(753\) −13052.4 −0.631682
\(754\) 0 0
\(755\) −7156.36 −0.344963
\(756\) 0 0
\(757\) 8706.02 0.418000 0.209000 0.977916i \(-0.432979\pi\)
0.209000 + 0.977916i \(0.432979\pi\)
\(758\) 0 0
\(759\) −2996.36 −0.143295
\(760\) 0 0
\(761\) −2484.28 −0.118338 −0.0591688 0.998248i \(-0.518845\pi\)
−0.0591688 + 0.998248i \(0.518845\pi\)
\(762\) 0 0
\(763\) −2275.80 −0.107981
\(764\) 0 0
\(765\) −5374.69 −0.254016
\(766\) 0 0
\(767\) 1340.24 0.0630942
\(768\) 0 0
\(769\) −6852.69 −0.321345 −0.160673 0.987008i \(-0.551366\pi\)
−0.160673 + 0.987008i \(0.551366\pi\)
\(770\) 0 0
\(771\) 4563.86 0.213182
\(772\) 0 0
\(773\) 32276.9 1.50184 0.750919 0.660394i \(-0.229611\pi\)
0.750919 + 0.660394i \(0.229611\pi\)
\(774\) 0 0
\(775\) −1448.71 −0.0671476
\(776\) 0 0
\(777\) −2835.54 −0.130920
\(778\) 0 0
\(779\) −10027.1 −0.461178
\(780\) 0 0
\(781\) 22084.1 1.01182
\(782\) 0 0
\(783\) −8355.11 −0.381338
\(784\) 0 0
\(785\) 1489.72 0.0677328
\(786\) 0 0
\(787\) 13011.8 0.589355 0.294677 0.955597i \(-0.404788\pi\)
0.294677 + 0.955597i \(0.404788\pi\)
\(788\) 0 0
\(789\) −13208.8 −0.596001
\(790\) 0 0
\(791\) −551.698 −0.0247992
\(792\) 0 0
\(793\) 10121.9 0.453266
\(794\) 0 0
\(795\) −3615.79 −0.161307
\(796\) 0 0
\(797\) 25309.1 1.12484 0.562419 0.826852i \(-0.309871\pi\)
0.562419 + 0.826852i \(0.309871\pi\)
\(798\) 0 0
\(799\) −21863.3 −0.968045
\(800\) 0 0
\(801\) −14050.3 −0.619777
\(802\) 0 0
\(803\) 23325.9 1.02510
\(804\) 0 0
\(805\) −744.810 −0.0326101
\(806\) 0 0
\(807\) 1523.42 0.0664524
\(808\) 0 0
\(809\) −37180.8 −1.61583 −0.807914 0.589300i \(-0.799404\pi\)
−0.807914 + 0.589300i \(0.799404\pi\)
\(810\) 0 0
\(811\) −18270.7 −0.791085 −0.395543 0.918448i \(-0.629443\pi\)
−0.395543 + 0.918448i \(0.629443\pi\)
\(812\) 0 0
\(813\) 5555.15 0.239640
\(814\) 0 0
\(815\) 7742.25 0.332760
\(816\) 0 0
\(817\) 67032.5 2.87046
\(818\) 0 0
\(819\) 576.180 0.0245829
\(820\) 0 0
\(821\) −25619.8 −1.08908 −0.544542 0.838733i \(-0.683297\pi\)
−0.544542 + 0.838733i \(0.683297\pi\)
\(822\) 0 0
\(823\) 20112.4 0.851854 0.425927 0.904758i \(-0.359948\pi\)
0.425927 + 0.904758i \(0.359948\pi\)
\(824\) 0 0
\(825\) 2476.46 0.104508
\(826\) 0 0
\(827\) −3220.54 −0.135416 −0.0677080 0.997705i \(-0.521569\pi\)
−0.0677080 + 0.997705i \(0.521569\pi\)
\(828\) 0 0
\(829\) −20396.5 −0.854524 −0.427262 0.904128i \(-0.640522\pi\)
−0.427262 + 0.904128i \(0.640522\pi\)
\(830\) 0 0
\(831\) −23952.6 −0.999888
\(832\) 0 0
\(833\) 38070.5 1.58351
\(834\) 0 0
\(835\) 2926.46 0.121287
\(836\) 0 0
\(837\) −1564.61 −0.0646128
\(838\) 0 0
\(839\) 36682.5 1.50944 0.754720 0.656047i \(-0.227773\pi\)
0.754720 + 0.656047i \(0.227773\pi\)
\(840\) 0 0
\(841\) 71369.4 2.92630
\(842\) 0 0
\(843\) 19784.4 0.808315
\(844\) 0 0
\(845\) 845.000 0.0344010
\(846\) 0 0
\(847\) −1185.43 −0.0480896
\(848\) 0 0
\(849\) −26516.0 −1.07188
\(850\) 0 0
\(851\) 5805.58 0.233858
\(852\) 0 0
\(853\) −22759.0 −0.913544 −0.456772 0.889584i \(-0.650994\pi\)
−0.456772 + 0.889584i \(0.650994\pi\)
\(854\) 0 0
\(855\) −5750.70 −0.230023
\(856\) 0 0
\(857\) 4632.17 0.184635 0.0923173 0.995730i \(-0.470573\pi\)
0.0923173 + 0.995730i \(0.470573\pi\)
\(858\) 0 0
\(859\) 25367.9 1.00762 0.503808 0.863816i \(-0.331932\pi\)
0.503808 + 0.863816i \(0.331932\pi\)
\(860\) 0 0
\(861\) 1159.21 0.0458834
\(862\) 0 0
\(863\) −40737.3 −1.60685 −0.803426 0.595404i \(-0.796992\pi\)
−0.803426 + 0.595404i \(0.796992\pi\)
\(864\) 0 0
\(865\) 3971.29 0.156102
\(866\) 0 0
\(867\) 28057.0 1.09904
\(868\) 0 0
\(869\) −9352.02 −0.365070
\(870\) 0 0
\(871\) 2995.60 0.116535
\(872\) 0 0
\(873\) 13004.0 0.504146
\(874\) 0 0
\(875\) 615.577 0.0237832
\(876\) 0 0
\(877\) 804.453 0.0309743 0.0154871 0.999880i \(-0.495070\pi\)
0.0154871 + 0.999880i \(0.495070\pi\)
\(878\) 0 0
\(879\) 14161.0 0.543391
\(880\) 0 0
\(881\) 10971.4 0.419564 0.209782 0.977748i \(-0.432725\pi\)
0.209782 + 0.977748i \(0.432725\pi\)
\(882\) 0 0
\(883\) −40822.6 −1.55582 −0.777910 0.628375i \(-0.783720\pi\)
−0.777910 + 0.628375i \(0.783720\pi\)
\(884\) 0 0
\(885\) 1546.43 0.0587375
\(886\) 0 0
\(887\) −34628.9 −1.31085 −0.655425 0.755260i \(-0.727510\pi\)
−0.655425 + 0.755260i \(0.727510\pi\)
\(888\) 0 0
\(889\) −9784.58 −0.369138
\(890\) 0 0
\(891\) 2674.58 0.100563
\(892\) 0 0
\(893\) −23392.8 −0.876609
\(894\) 0 0
\(895\) 9446.93 0.352822
\(896\) 0 0
\(897\) −1179.69 −0.0439116
\(898\) 0 0
\(899\) 17932.1 0.665261
\(900\) 0 0
\(901\) 28790.7 1.06455
\(902\) 0 0
\(903\) −7749.45 −0.285587
\(904\) 0 0
\(905\) −3715.05 −0.136456
\(906\) 0 0
\(907\) 15188.6 0.556040 0.278020 0.960575i \(-0.410322\pi\)
0.278020 + 0.960575i \(0.410322\pi\)
\(908\) 0 0
\(909\) −6244.33 −0.227845
\(910\) 0 0
\(911\) −22243.2 −0.808947 −0.404474 0.914550i \(-0.632545\pi\)
−0.404474 + 0.914550i \(0.632545\pi\)
\(912\) 0 0
\(913\) −18635.9 −0.675531
\(914\) 0 0
\(915\) 11679.1 0.421968
\(916\) 0 0
\(917\) −14483.6 −0.521582
\(918\) 0 0
\(919\) −22385.9 −0.803529 −0.401765 0.915743i \(-0.631603\pi\)
−0.401765 + 0.915743i \(0.631603\pi\)
\(920\) 0 0
\(921\) −15293.5 −0.547163
\(922\) 0 0
\(923\) 8694.67 0.310064
\(924\) 0 0
\(925\) −4798.25 −0.170557
\(926\) 0 0
\(927\) −2505.52 −0.0887726
\(928\) 0 0
\(929\) 2508.24 0.0885819 0.0442910 0.999019i \(-0.485897\pi\)
0.0442910 + 0.999019i \(0.485897\pi\)
\(930\) 0 0
\(931\) 40733.9 1.43394
\(932\) 0 0
\(933\) −26595.1 −0.933209
\(934\) 0 0
\(935\) −19718.8 −0.689705
\(936\) 0 0
\(937\) −47700.9 −1.66310 −0.831548 0.555452i \(-0.812545\pi\)
−0.831548 + 0.555452i \(0.812545\pi\)
\(938\) 0 0
\(939\) −8712.51 −0.302793
\(940\) 0 0
\(941\) −39333.0 −1.36261 −0.681306 0.731999i \(-0.738588\pi\)
−0.681306 + 0.731999i \(0.738588\pi\)
\(942\) 0 0
\(943\) −2373.40 −0.0819601
\(944\) 0 0
\(945\) 664.823 0.0228854
\(946\) 0 0
\(947\) −19389.4 −0.665333 −0.332667 0.943044i \(-0.607948\pi\)
−0.332667 + 0.943044i \(0.607948\pi\)
\(948\) 0 0
\(949\) 9183.56 0.314132
\(950\) 0 0
\(951\) 10797.6 0.368176
\(952\) 0 0
\(953\) 48296.3 1.64163 0.820814 0.571195i \(-0.193520\pi\)
0.820814 + 0.571195i \(0.193520\pi\)
\(954\) 0 0
\(955\) 7991.00 0.270767
\(956\) 0 0
\(957\) −30653.5 −1.03541
\(958\) 0 0
\(959\) −2331.72 −0.0785141
\(960\) 0 0
\(961\) −26433.0 −0.887280
\(962\) 0 0
\(963\) 12532.8 0.419380
\(964\) 0 0
\(965\) −15398.7 −0.513682
\(966\) 0 0
\(967\) −38596.2 −1.28353 −0.641763 0.766903i \(-0.721797\pi\)
−0.641763 + 0.766903i \(0.721797\pi\)
\(968\) 0 0
\(969\) 45789.9 1.51804
\(970\) 0 0
\(971\) 48306.0 1.59651 0.798256 0.602319i \(-0.205756\pi\)
0.798256 + 0.602319i \(0.205756\pi\)
\(972\) 0 0
\(973\) 12.6831 0.000417884 0
\(974\) 0 0
\(975\) 975.000 0.0320256
\(976\) 0 0
\(977\) 5886.02 0.192744 0.0963718 0.995345i \(-0.469276\pi\)
0.0963718 + 0.995345i \(0.469276\pi\)
\(978\) 0 0
\(979\) −51548.0 −1.68282
\(980\) 0 0
\(981\) −4159.14 −0.135363
\(982\) 0 0
\(983\) 6014.75 0.195158 0.0975792 0.995228i \(-0.468890\pi\)
0.0975792 + 0.995228i \(0.468890\pi\)
\(984\) 0 0
\(985\) 19048.9 0.616191
\(986\) 0 0
\(987\) 2704.39 0.0872153
\(988\) 0 0
\(989\) 15866.5 0.510136
\(990\) 0 0
\(991\) 55642.6 1.78360 0.891799 0.452432i \(-0.149443\pi\)
0.891799 + 0.452432i \(0.149443\pi\)
\(992\) 0 0
\(993\) −3940.07 −0.125916
\(994\) 0 0
\(995\) 19615.9 0.624990
\(996\) 0 0
\(997\) 42674.5 1.35558 0.677791 0.735255i \(-0.262937\pi\)
0.677791 + 0.735255i \(0.262937\pi\)
\(998\) 0 0
\(999\) −5182.11 −0.164119
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 1560.4.a.n.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.4.a.n.1.3 4 1.1 even 1 trivial