Properties

Label 2352.4.a.cn.1.2
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.391168.1
Defining polynomial: \(x^{4} - 40 x^{2} + 382\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 1176)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.96955\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -7.81338 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -7.81338 q^{5} +9.00000 q^{9} -42.7627 q^{11} -10.9493 q^{13} +23.4401 q^{15} +80.2889 q^{17} -112.675 q^{19} +70.1884 q^{23} -63.9511 q^{25} -27.0000 q^{27} -147.010 q^{29} +144.529 q^{31} +128.288 q^{33} -0.292803 q^{37} +32.8479 q^{39} -294.097 q^{41} -337.866 q^{43} -70.3204 q^{45} -104.524 q^{47} -240.867 q^{51} -143.050 q^{53} +334.121 q^{55} +338.024 q^{57} -520.440 q^{59} +399.902 q^{61} +85.5509 q^{65} -137.813 q^{67} -210.565 q^{69} -266.418 q^{71} +524.754 q^{73} +191.853 q^{75} -433.874 q^{79} +81.0000 q^{81} -664.921 q^{83} -627.328 q^{85} +441.029 q^{87} +803.579 q^{89} -433.588 q^{93} +880.370 q^{95} +445.813 q^{97} -384.864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{3} + 8q^{5} + 36q^{9} + O(q^{10}) \) \( 4q - 12q^{3} + 8q^{5} + 36q^{9} - 40q^{11} + 48q^{13} - 24q^{15} + 72q^{17} - 32q^{19} - 8q^{23} + 164q^{25} - 108q^{27} + 144q^{29} - 48q^{31} + 120q^{33} + 48q^{37} - 144q^{39} + 72q^{41} - 512q^{43} + 72q^{45} - 160q^{47} - 216q^{51} + 536q^{53} + 336q^{55} + 96q^{57} - 240q^{59} + 896q^{61} - 136q^{65} - 1088q^{67} + 24q^{69} - 1288q^{71} + 1488q^{73} - 492q^{75} - 416q^{79} + 324q^{81} + 112q^{83} - 1512q^{85} - 432q^{87} + 3160q^{89} + 144q^{93} + 240q^{95} + 2384q^{97} - 360q^{99} + O(q^{100}) \)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −7.81338 −0.698850 −0.349425 0.936964i \(-0.613623\pi\)
−0.349425 + 0.936964i \(0.613623\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −42.7627 −1.17213 −0.586065 0.810264i \(-0.699324\pi\)
−0.586065 + 0.810264i \(0.699324\pi\)
\(12\) 0 0
\(13\) −10.9493 −0.233599 −0.116799 0.993156i \(-0.537263\pi\)
−0.116799 + 0.993156i \(0.537263\pi\)
\(14\) 0 0
\(15\) 23.4401 0.403481
\(16\) 0 0
\(17\) 80.2889 1.14547 0.572733 0.819742i \(-0.305883\pi\)
0.572733 + 0.819742i \(0.305883\pi\)
\(18\) 0 0
\(19\) −112.675 −1.36049 −0.680246 0.732984i \(-0.738127\pi\)
−0.680246 + 0.732984i \(0.738127\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 70.1884 0.636317 0.318159 0.948037i \(-0.396936\pi\)
0.318159 + 0.948037i \(0.396936\pi\)
\(24\) 0 0
\(25\) −63.9511 −0.511609
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −147.010 −0.941344 −0.470672 0.882308i \(-0.655989\pi\)
−0.470672 + 0.882308i \(0.655989\pi\)
\(30\) 0 0
\(31\) 144.529 0.837363 0.418681 0.908133i \(-0.362492\pi\)
0.418681 + 0.908133i \(0.362492\pi\)
\(32\) 0 0
\(33\) 128.288 0.676729
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.292803 −0.00130099 −0.000650494 1.00000i \(-0.500207\pi\)
−0.000650494 1.00000i \(0.500207\pi\)
\(38\) 0 0
\(39\) 32.8479 0.134868
\(40\) 0 0
\(41\) −294.097 −1.12025 −0.560126 0.828408i \(-0.689247\pi\)
−0.560126 + 0.828408i \(0.689247\pi\)
\(42\) 0 0
\(43\) −337.866 −1.19823 −0.599117 0.800661i \(-0.704482\pi\)
−0.599117 + 0.800661i \(0.704482\pi\)
\(44\) 0 0
\(45\) −70.3204 −0.232950
\(46\) 0 0
\(47\) −104.524 −0.324391 −0.162196 0.986759i \(-0.551858\pi\)
−0.162196 + 0.986759i \(0.551858\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −240.867 −0.661335
\(52\) 0 0
\(53\) −143.050 −0.370743 −0.185372 0.982668i \(-0.559349\pi\)
−0.185372 + 0.982668i \(0.559349\pi\)
\(54\) 0 0
\(55\) 334.121 0.819143
\(56\) 0 0
\(57\) 338.024 0.785480
\(58\) 0 0
\(59\) −520.440 −1.14840 −0.574199 0.818716i \(-0.694686\pi\)
−0.574199 + 0.818716i \(0.694686\pi\)
\(60\) 0 0
\(61\) 399.902 0.839380 0.419690 0.907668i \(-0.362139\pi\)
0.419690 + 0.907668i \(0.362139\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 85.5509 0.163251
\(66\) 0 0
\(67\) −137.813 −0.251292 −0.125646 0.992075i \(-0.540100\pi\)
−0.125646 + 0.992075i \(0.540100\pi\)
\(68\) 0 0
\(69\) −210.565 −0.367378
\(70\) 0 0
\(71\) −266.418 −0.445324 −0.222662 0.974896i \(-0.571475\pi\)
−0.222662 + 0.974896i \(0.571475\pi\)
\(72\) 0 0
\(73\) 524.754 0.841340 0.420670 0.907214i \(-0.361795\pi\)
0.420670 + 0.907214i \(0.361795\pi\)
\(74\) 0 0
\(75\) 191.853 0.295377
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −433.874 −0.617907 −0.308953 0.951077i \(-0.599979\pi\)
−0.308953 + 0.951077i \(0.599979\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −664.921 −0.879332 −0.439666 0.898161i \(-0.644903\pi\)
−0.439666 + 0.898161i \(0.644903\pi\)
\(84\) 0 0
\(85\) −627.328 −0.800509
\(86\) 0 0
\(87\) 441.029 0.543485
\(88\) 0 0
\(89\) 803.579 0.957069 0.478535 0.878069i \(-0.341168\pi\)
0.478535 + 0.878069i \(0.341168\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −433.588 −0.483452
\(94\) 0 0
\(95\) 880.370 0.950780
\(96\) 0 0
\(97\) 445.813 0.466654 0.233327 0.972398i \(-0.425039\pi\)
0.233327 + 0.972398i \(0.425039\pi\)
\(98\) 0 0
\(99\) −384.864 −0.390710
\(100\) 0 0
\(101\) 1547.45 1.52453 0.762263 0.647267i \(-0.224088\pi\)
0.762263 + 0.647267i \(0.224088\pi\)
\(102\) 0 0
\(103\) −679.600 −0.650126 −0.325063 0.945692i \(-0.605386\pi\)
−0.325063 + 0.945692i \(0.605386\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1998.86 −1.80595 −0.902975 0.429693i \(-0.858622\pi\)
−0.902975 + 0.429693i \(0.858622\pi\)
\(108\) 0 0
\(109\) −1455.60 −1.27909 −0.639547 0.768752i \(-0.720878\pi\)
−0.639547 + 0.768752i \(0.720878\pi\)
\(110\) 0 0
\(111\) 0.878410 0.000751126 0
\(112\) 0 0
\(113\) 523.808 0.436068 0.218034 0.975941i \(-0.430036\pi\)
0.218034 + 0.975941i \(0.430036\pi\)
\(114\) 0 0
\(115\) −548.409 −0.444690
\(116\) 0 0
\(117\) −98.5436 −0.0778663
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 497.645 0.373888
\(122\) 0 0
\(123\) 882.292 0.646777
\(124\) 0 0
\(125\) 1476.35 1.05639
\(126\) 0 0
\(127\) −2496.08 −1.74402 −0.872011 0.489486i \(-0.837185\pi\)
−0.872011 + 0.489486i \(0.837185\pi\)
\(128\) 0 0
\(129\) 1013.60 0.691801
\(130\) 0 0
\(131\) −407.249 −0.271615 −0.135807 0.990735i \(-0.543363\pi\)
−0.135807 + 0.990735i \(0.543363\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 210.961 0.134494
\(136\) 0 0
\(137\) 478.949 0.298682 0.149341 0.988786i \(-0.452285\pi\)
0.149341 + 0.988786i \(0.452285\pi\)
\(138\) 0 0
\(139\) 123.985 0.0756569 0.0378284 0.999284i \(-0.487956\pi\)
0.0378284 + 0.999284i \(0.487956\pi\)
\(140\) 0 0
\(141\) 313.572 0.187287
\(142\) 0 0
\(143\) 468.221 0.273808
\(144\) 0 0
\(145\) 1148.64 0.657858
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1645.37 0.904658 0.452329 0.891851i \(-0.350593\pi\)
0.452329 + 0.891851i \(0.350593\pi\)
\(150\) 0 0
\(151\) 3044.79 1.64094 0.820470 0.571690i \(-0.193712\pi\)
0.820470 + 0.571690i \(0.193712\pi\)
\(152\) 0 0
\(153\) 722.600 0.381822
\(154\) 0 0
\(155\) −1129.26 −0.585191
\(156\) 0 0
\(157\) −1408.22 −0.715850 −0.357925 0.933750i \(-0.616516\pi\)
−0.357925 + 0.933750i \(0.616516\pi\)
\(158\) 0 0
\(159\) 429.149 0.214049
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2083.51 1.00119 0.500593 0.865683i \(-0.333115\pi\)
0.500593 + 0.865683i \(0.333115\pi\)
\(164\) 0 0
\(165\) −1002.36 −0.472932
\(166\) 0 0
\(167\) 184.235 0.0853685 0.0426843 0.999089i \(-0.486409\pi\)
0.0426843 + 0.999089i \(0.486409\pi\)
\(168\) 0 0
\(169\) −2077.11 −0.945432
\(170\) 0 0
\(171\) −1014.07 −0.453497
\(172\) 0 0
\(173\) 3332.98 1.46475 0.732375 0.680902i \(-0.238412\pi\)
0.732375 + 0.680902i \(0.238412\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1561.32 0.663028
\(178\) 0 0
\(179\) 3273.98 1.36709 0.683544 0.729909i \(-0.260438\pi\)
0.683544 + 0.729909i \(0.260438\pi\)
\(180\) 0 0
\(181\) −2708.05 −1.11209 −0.556043 0.831154i \(-0.687681\pi\)
−0.556043 + 0.831154i \(0.687681\pi\)
\(182\) 0 0
\(183\) −1199.71 −0.484616
\(184\) 0 0
\(185\) 2.28778 0.000909196 0
\(186\) 0 0
\(187\) −3433.37 −1.34264
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 985.187 0.373223 0.186612 0.982434i \(-0.440249\pi\)
0.186612 + 0.982434i \(0.440249\pi\)
\(192\) 0 0
\(193\) −4627.92 −1.72604 −0.863018 0.505172i \(-0.831429\pi\)
−0.863018 + 0.505172i \(0.831429\pi\)
\(194\) 0 0
\(195\) −256.653 −0.0942528
\(196\) 0 0
\(197\) −3501.81 −1.26647 −0.633233 0.773961i \(-0.718273\pi\)
−0.633233 + 0.773961i \(0.718273\pi\)
\(198\) 0 0
\(199\) −169.661 −0.0604368 −0.0302184 0.999543i \(-0.509620\pi\)
−0.0302184 + 0.999543i \(0.509620\pi\)
\(200\) 0 0
\(201\) 413.439 0.145083
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2297.90 0.782888
\(206\) 0 0
\(207\) 631.696 0.212106
\(208\) 0 0
\(209\) 4818.27 1.59467
\(210\) 0 0
\(211\) 1188.76 0.387857 0.193929 0.981016i \(-0.437877\pi\)
0.193929 + 0.981016i \(0.437877\pi\)
\(212\) 0 0
\(213\) 799.255 0.257108
\(214\) 0 0
\(215\) 2639.88 0.837386
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1574.26 −0.485748
\(220\) 0 0
\(221\) −879.106 −0.267580
\(222\) 0 0
\(223\) −3395.54 −1.01965 −0.509826 0.860278i \(-0.670290\pi\)
−0.509826 + 0.860278i \(0.670290\pi\)
\(224\) 0 0
\(225\) −575.560 −0.170536
\(226\) 0 0
\(227\) −1198.05 −0.350297 −0.175148 0.984542i \(-0.556041\pi\)
−0.175148 + 0.984542i \(0.556041\pi\)
\(228\) 0 0
\(229\) 5820.40 1.67958 0.839788 0.542914i \(-0.182679\pi\)
0.839788 + 0.542914i \(0.182679\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3488.08 0.980737 0.490368 0.871515i \(-0.336862\pi\)
0.490368 + 0.871515i \(0.336862\pi\)
\(234\) 0 0
\(235\) 816.685 0.226701
\(236\) 0 0
\(237\) 1301.62 0.356749
\(238\) 0 0
\(239\) −4688.08 −1.26882 −0.634408 0.772999i \(-0.718756\pi\)
−0.634408 + 0.772999i \(0.718756\pi\)
\(240\) 0 0
\(241\) 6419.23 1.71576 0.857881 0.513848i \(-0.171780\pi\)
0.857881 + 0.513848i \(0.171780\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1233.71 0.317809
\(248\) 0 0
\(249\) 1994.76 0.507682
\(250\) 0 0
\(251\) 5928.29 1.49080 0.745400 0.666618i \(-0.232259\pi\)
0.745400 + 0.666618i \(0.232259\pi\)
\(252\) 0 0
\(253\) −3001.44 −0.745846
\(254\) 0 0
\(255\) 1881.98 0.462174
\(256\) 0 0
\(257\) −673.329 −0.163428 −0.0817142 0.996656i \(-0.526039\pi\)
−0.0817142 + 0.996656i \(0.526039\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1323.09 −0.313781
\(262\) 0 0
\(263\) 5270.35 1.23568 0.617839 0.786304i \(-0.288008\pi\)
0.617839 + 0.786304i \(0.288008\pi\)
\(264\) 0 0
\(265\) 1117.70 0.259094
\(266\) 0 0
\(267\) −2410.74 −0.552564
\(268\) 0 0
\(269\) 4064.77 0.921313 0.460656 0.887579i \(-0.347614\pi\)
0.460656 + 0.887579i \(0.347614\pi\)
\(270\) 0 0
\(271\) −6524.19 −1.46242 −0.731211 0.682151i \(-0.761045\pi\)
−0.731211 + 0.682151i \(0.761045\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2734.72 0.599672
\(276\) 0 0
\(277\) 8345.83 1.81030 0.905149 0.425095i \(-0.139759\pi\)
0.905149 + 0.425095i \(0.139759\pi\)
\(278\) 0 0
\(279\) 1300.76 0.279121
\(280\) 0 0
\(281\) 8161.43 1.73263 0.866316 0.499496i \(-0.166481\pi\)
0.866316 + 0.499496i \(0.166481\pi\)
\(282\) 0 0
\(283\) 8669.66 1.82105 0.910526 0.413451i \(-0.135677\pi\)
0.910526 + 0.413451i \(0.135677\pi\)
\(284\) 0 0
\(285\) −2641.11 −0.548933
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1533.31 0.312093
\(290\) 0 0
\(291\) −1337.44 −0.269423
\(292\) 0 0
\(293\) 99.7900 0.0198969 0.00994846 0.999951i \(-0.496833\pi\)
0.00994846 + 0.999951i \(0.496833\pi\)
\(294\) 0 0
\(295\) 4066.39 0.802558
\(296\) 0 0
\(297\) 1154.59 0.225576
\(298\) 0 0
\(299\) −768.513 −0.148643
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −4642.35 −0.880186
\(304\) 0 0
\(305\) −3124.58 −0.586601
\(306\) 0 0
\(307\) 8631.55 1.60465 0.802327 0.596885i \(-0.203595\pi\)
0.802327 + 0.596885i \(0.203595\pi\)
\(308\) 0 0
\(309\) 2038.80 0.375351
\(310\) 0 0
\(311\) −863.980 −0.157530 −0.0787650 0.996893i \(-0.525098\pi\)
−0.0787650 + 0.996893i \(0.525098\pi\)
\(312\) 0 0
\(313\) 577.824 0.104347 0.0521734 0.998638i \(-0.483385\pi\)
0.0521734 + 0.998638i \(0.483385\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −196.879 −0.0348828 −0.0174414 0.999848i \(-0.505552\pi\)
−0.0174414 + 0.999848i \(0.505552\pi\)
\(318\) 0 0
\(319\) 6286.52 1.10338
\(320\) 0 0
\(321\) 5996.57 1.04267
\(322\) 0 0
\(323\) −9046.53 −1.55840
\(324\) 0 0
\(325\) 700.219 0.119511
\(326\) 0 0
\(327\) 4366.80 0.738485
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4645.40 0.771402 0.385701 0.922624i \(-0.373960\pi\)
0.385701 + 0.922624i \(0.373960\pi\)
\(332\) 0 0
\(333\) −2.63523 −0.000433663 0
\(334\) 0 0
\(335\) 1076.79 0.175615
\(336\) 0 0
\(337\) −7255.28 −1.17276 −0.586380 0.810036i \(-0.699447\pi\)
−0.586380 + 0.810036i \(0.699447\pi\)
\(338\) 0 0
\(339\) −1571.42 −0.251764
\(340\) 0 0
\(341\) −6180.46 −0.981498
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1645.23 0.256742
\(346\) 0 0
\(347\) −3066.25 −0.474366 −0.237183 0.971465i \(-0.576224\pi\)
−0.237183 + 0.971465i \(0.576224\pi\)
\(348\) 0 0
\(349\) 884.690 0.135692 0.0678458 0.997696i \(-0.478387\pi\)
0.0678458 + 0.997696i \(0.478387\pi\)
\(350\) 0 0
\(351\) 295.631 0.0449561
\(352\) 0 0
\(353\) 2463.28 0.371409 0.185704 0.982606i \(-0.440543\pi\)
0.185704 + 0.982606i \(0.440543\pi\)
\(354\) 0 0
\(355\) 2081.63 0.311215
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1999.23 0.293915 0.146957 0.989143i \(-0.453052\pi\)
0.146957 + 0.989143i \(0.453052\pi\)
\(360\) 0 0
\(361\) 5836.59 0.850939
\(362\) 0 0
\(363\) −1492.94 −0.215865
\(364\) 0 0
\(365\) −4100.10 −0.587971
\(366\) 0 0
\(367\) 6790.82 0.965880 0.482940 0.875653i \(-0.339569\pi\)
0.482940 + 0.875653i \(0.339569\pi\)
\(368\) 0 0
\(369\) −2646.88 −0.373417
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1222.01 −0.169633 −0.0848164 0.996397i \(-0.527030\pi\)
−0.0848164 + 0.996397i \(0.527030\pi\)
\(374\) 0 0
\(375\) −4429.04 −0.609906
\(376\) 0 0
\(377\) 1609.65 0.219897
\(378\) 0 0
\(379\) −10228.6 −1.38630 −0.693149 0.720794i \(-0.743777\pi\)
−0.693149 + 0.720794i \(0.743777\pi\)
\(380\) 0 0
\(381\) 7488.23 1.00691
\(382\) 0 0
\(383\) 5452.92 0.727496 0.363748 0.931497i \(-0.381497\pi\)
0.363748 + 0.931497i \(0.381497\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3040.79 −0.399412
\(388\) 0 0
\(389\) 14423.8 1.87999 0.939994 0.341190i \(-0.110830\pi\)
0.939994 + 0.341190i \(0.110830\pi\)
\(390\) 0 0
\(391\) 5635.35 0.728880
\(392\) 0 0
\(393\) 1221.75 0.156817
\(394\) 0 0
\(395\) 3390.02 0.431824
\(396\) 0 0
\(397\) 4462.35 0.564129 0.282064 0.959395i \(-0.408981\pi\)
0.282064 + 0.959395i \(0.408981\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12380.6 1.54179 0.770897 0.636960i \(-0.219809\pi\)
0.770897 + 0.636960i \(0.219809\pi\)
\(402\) 0 0
\(403\) −1582.49 −0.195607
\(404\) 0 0
\(405\) −632.884 −0.0776500
\(406\) 0 0
\(407\) 12.5210 0.00152493
\(408\) 0 0
\(409\) 10065.9 1.21694 0.608469 0.793578i \(-0.291784\pi\)
0.608469 + 0.793578i \(0.291784\pi\)
\(410\) 0 0
\(411\) −1436.85 −0.172444
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 5195.28 0.614521
\(416\) 0 0
\(417\) −371.956 −0.0436805
\(418\) 0 0
\(419\) 645.188 0.0752256 0.0376128 0.999292i \(-0.488025\pi\)
0.0376128 + 0.999292i \(0.488025\pi\)
\(420\) 0 0
\(421\) −16460.0 −1.90549 −0.952745 0.303770i \(-0.901755\pi\)
−0.952745 + 0.303770i \(0.901755\pi\)
\(422\) 0 0
\(423\) −940.716 −0.108130
\(424\) 0 0
\(425\) −5134.56 −0.586030
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1404.66 −0.158083
\(430\) 0 0
\(431\) −4989.28 −0.557599 −0.278800 0.960349i \(-0.589937\pi\)
−0.278800 + 0.960349i \(0.589937\pi\)
\(432\) 0 0
\(433\) −12366.3 −1.37248 −0.686242 0.727374i \(-0.740741\pi\)
−0.686242 + 0.727374i \(0.740741\pi\)
\(434\) 0 0
\(435\) −3445.92 −0.379815
\(436\) 0 0
\(437\) −7908.46 −0.865705
\(438\) 0 0
\(439\) −11733.7 −1.27567 −0.637836 0.770172i \(-0.720170\pi\)
−0.637836 + 0.770172i \(0.720170\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4632.30 0.496811 0.248406 0.968656i \(-0.420093\pi\)
0.248406 + 0.968656i \(0.420093\pi\)
\(444\) 0 0
\(445\) −6278.67 −0.668848
\(446\) 0 0
\(447\) −4936.11 −0.522304
\(448\) 0 0
\(449\) −4057.84 −0.426506 −0.213253 0.976997i \(-0.568406\pi\)
−0.213253 + 0.976997i \(0.568406\pi\)
\(450\) 0 0
\(451\) 12576.4 1.31308
\(452\) 0 0
\(453\) −9134.38 −0.947397
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17692.3 −1.81097 −0.905484 0.424381i \(-0.860492\pi\)
−0.905484 + 0.424381i \(0.860492\pi\)
\(458\) 0 0
\(459\) −2167.80 −0.220445
\(460\) 0 0
\(461\) 8120.88 0.820449 0.410225 0.911984i \(-0.365450\pi\)
0.410225 + 0.911984i \(0.365450\pi\)
\(462\) 0 0
\(463\) 19397.4 1.94703 0.973515 0.228624i \(-0.0734226\pi\)
0.973515 + 0.228624i \(0.0734226\pi\)
\(464\) 0 0
\(465\) 3387.79 0.337860
\(466\) 0 0
\(467\) 11033.3 1.09327 0.546637 0.837370i \(-0.315908\pi\)
0.546637 + 0.837370i \(0.315908\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4224.67 0.413296
\(472\) 0 0
\(473\) 14448.1 1.40449
\(474\) 0 0
\(475\) 7205.67 0.696039
\(476\) 0 0
\(477\) −1287.45 −0.123581
\(478\) 0 0
\(479\) 14170.5 1.35170 0.675852 0.737038i \(-0.263776\pi\)
0.675852 + 0.737038i \(0.263776\pi\)
\(480\) 0 0
\(481\) 3.20599 0.000303909 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3483.31 −0.326121
\(486\) 0 0
\(487\) −12929.3 −1.20304 −0.601522 0.798856i \(-0.705439\pi\)
−0.601522 + 0.798856i \(0.705439\pi\)
\(488\) 0 0
\(489\) −6250.54 −0.578035
\(490\) 0 0
\(491\) −99.6829 −0.00916217 −0.00458109 0.999990i \(-0.501458\pi\)
−0.00458109 + 0.999990i \(0.501458\pi\)
\(492\) 0 0
\(493\) −11803.2 −1.07828
\(494\) 0 0
\(495\) 3007.09 0.273048
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 6207.28 0.556865 0.278433 0.960456i \(-0.410185\pi\)
0.278433 + 0.960456i \(0.410185\pi\)
\(500\) 0 0
\(501\) −552.705 −0.0492875
\(502\) 0 0
\(503\) −4743.31 −0.420464 −0.210232 0.977651i \(-0.567422\pi\)
−0.210232 + 0.977651i \(0.567422\pi\)
\(504\) 0 0
\(505\) −12090.8 −1.06542
\(506\) 0 0
\(507\) 6231.34 0.545845
\(508\) 0 0
\(509\) −9836.46 −0.856569 −0.428284 0.903644i \(-0.640882\pi\)
−0.428284 + 0.903644i \(0.640882\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3042.22 0.261827
\(514\) 0 0
\(515\) 5309.98 0.454341
\(516\) 0 0
\(517\) 4469.72 0.380229
\(518\) 0 0
\(519\) −9998.94 −0.845674
\(520\) 0 0
\(521\) −2285.55 −0.192192 −0.0960958 0.995372i \(-0.530636\pi\)
−0.0960958 + 0.995372i \(0.530636\pi\)
\(522\) 0 0
\(523\) −7740.65 −0.647180 −0.323590 0.946197i \(-0.604890\pi\)
−0.323590 + 0.946197i \(0.604890\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11604.1 0.959171
\(528\) 0 0
\(529\) −7240.59 −0.595100
\(530\) 0 0
\(531\) −4683.96 −0.382799
\(532\) 0 0
\(533\) 3220.16 0.261689
\(534\) 0 0
\(535\) 15617.8 1.26209
\(536\) 0 0
\(537\) −9821.94 −0.789289
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −15682.4 −1.24628 −0.623141 0.782110i \(-0.714144\pi\)
−0.623141 + 0.782110i \(0.714144\pi\)
\(542\) 0 0
\(543\) 8124.14 0.642063
\(544\) 0 0
\(545\) 11373.2 0.893895
\(546\) 0 0
\(547\) 3365.32 0.263054 0.131527 0.991313i \(-0.458012\pi\)
0.131527 + 0.991313i \(0.458012\pi\)
\(548\) 0 0
\(549\) 3599.12 0.279793
\(550\) 0 0
\(551\) 16564.3 1.28069
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −6.86335 −0.000524924 0
\(556\) 0 0
\(557\) 11968.8 0.910478 0.455239 0.890369i \(-0.349554\pi\)
0.455239 + 0.890369i \(0.349554\pi\)
\(558\) 0 0
\(559\) 3699.39 0.279906
\(560\) 0 0
\(561\) 10300.1 0.775171
\(562\) 0 0
\(563\) −14981.4 −1.12148 −0.560740 0.827992i \(-0.689483\pi\)
−0.560740 + 0.827992i \(0.689483\pi\)
\(564\) 0 0
\(565\) −4092.71 −0.304746
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14947.7 1.10130 0.550652 0.834735i \(-0.314379\pi\)
0.550652 + 0.834735i \(0.314379\pi\)
\(570\) 0 0
\(571\) −8138.91 −0.596503 −0.298251 0.954487i \(-0.596403\pi\)
−0.298251 + 0.954487i \(0.596403\pi\)
\(572\) 0 0
\(573\) −2955.56 −0.215481
\(574\) 0 0
\(575\) −4488.63 −0.325545
\(576\) 0 0
\(577\) −4092.69 −0.295288 −0.147644 0.989041i \(-0.547169\pi\)
−0.147644 + 0.989041i \(0.547169\pi\)
\(578\) 0 0
\(579\) 13883.8 0.996528
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6117.19 0.434559
\(584\) 0 0
\(585\) 769.958 0.0544169
\(586\) 0 0
\(587\) 15877.4 1.11641 0.558203 0.829705i \(-0.311491\pi\)
0.558203 + 0.829705i \(0.311491\pi\)
\(588\) 0 0
\(589\) −16284.8 −1.13923
\(590\) 0 0
\(591\) 10505.4 0.731195
\(592\) 0 0
\(593\) 21660.8 1.50000 0.750001 0.661436i \(-0.230053\pi\)
0.750001 + 0.661436i \(0.230053\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 508.982 0.0348932
\(598\) 0 0
\(599\) 11674.4 0.796334 0.398167 0.917313i \(-0.369646\pi\)
0.398167 + 0.917313i \(0.369646\pi\)
\(600\) 0 0
\(601\) 5818.05 0.394881 0.197440 0.980315i \(-0.436737\pi\)
0.197440 + 0.980315i \(0.436737\pi\)
\(602\) 0 0
\(603\) −1240.32 −0.0837640
\(604\) 0 0
\(605\) −3888.29 −0.261292
\(606\) 0 0
\(607\) 19894.5 1.33030 0.665150 0.746710i \(-0.268368\pi\)
0.665150 + 0.746710i \(0.268368\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1144.46 0.0757774
\(612\) 0 0
\(613\) 23979.8 1.57999 0.789996 0.613112i \(-0.210083\pi\)
0.789996 + 0.613112i \(0.210083\pi\)
\(614\) 0 0
\(615\) −6893.69 −0.452000
\(616\) 0 0
\(617\) −16527.7 −1.07841 −0.539207 0.842173i \(-0.681276\pi\)
−0.539207 + 0.842173i \(0.681276\pi\)
\(618\) 0 0
\(619\) 2135.87 0.138688 0.0693440 0.997593i \(-0.477909\pi\)
0.0693440 + 0.997593i \(0.477909\pi\)
\(620\) 0 0
\(621\) −1895.09 −0.122459
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −3541.37 −0.226648
\(626\) 0 0
\(627\) −14454.8 −0.920685
\(628\) 0 0
\(629\) −23.5089 −0.00149024
\(630\) 0 0
\(631\) 16460.9 1.03851 0.519253 0.854620i \(-0.326210\pi\)
0.519253 + 0.854620i \(0.326210\pi\)
\(632\) 0 0
\(633\) −3566.29 −0.223930
\(634\) 0 0
\(635\) 19502.8 1.21881
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2397.76 −0.148441
\(640\) 0 0
\(641\) 2506.10 0.154423 0.0772114 0.997015i \(-0.475398\pi\)
0.0772114 + 0.997015i \(0.475398\pi\)
\(642\) 0 0
\(643\) 15408.1 0.945002 0.472501 0.881330i \(-0.343351\pi\)
0.472501 + 0.881330i \(0.343351\pi\)
\(644\) 0 0
\(645\) −7919.63 −0.483465
\(646\) 0 0
\(647\) −28067.4 −1.70548 −0.852738 0.522338i \(-0.825060\pi\)
−0.852738 + 0.522338i \(0.825060\pi\)
\(648\) 0 0
\(649\) 22255.4 1.34607
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3412.13 0.204482 0.102241 0.994760i \(-0.467399\pi\)
0.102241 + 0.994760i \(0.467399\pi\)
\(654\) 0 0
\(655\) 3181.99 0.189818
\(656\) 0 0
\(657\) 4722.79 0.280447
\(658\) 0 0
\(659\) 18182.5 1.07479 0.537397 0.843329i \(-0.319408\pi\)
0.537397 + 0.843329i \(0.319408\pi\)
\(660\) 0 0
\(661\) −23113.9 −1.36010 −0.680049 0.733167i \(-0.738042\pi\)
−0.680049 + 0.733167i \(0.738042\pi\)
\(662\) 0 0
\(663\) 2637.32 0.154487
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −10318.4 −0.598994
\(668\) 0 0
\(669\) 10186.6 0.588696
\(670\) 0 0
\(671\) −17100.9 −0.983862
\(672\) 0 0
\(673\) 13586.4 0.778183 0.389092 0.921199i \(-0.372789\pi\)
0.389092 + 0.921199i \(0.372789\pi\)
\(674\) 0 0
\(675\) 1726.68 0.0984591
\(676\) 0 0
\(677\) −24054.6 −1.36557 −0.682786 0.730619i \(-0.739232\pi\)
−0.682786 + 0.730619i \(0.739232\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 3594.15 0.202244
\(682\) 0 0
\(683\) −21915.5 −1.22778 −0.613889 0.789393i \(-0.710396\pi\)
−0.613889 + 0.789393i \(0.710396\pi\)
\(684\) 0 0
\(685\) −3742.21 −0.208734
\(686\) 0 0
\(687\) −17461.2 −0.969704
\(688\) 0 0
\(689\) 1566.29 0.0866052
\(690\) 0 0
\(691\) −19309.5 −1.06305 −0.531524 0.847043i \(-0.678381\pi\)
−0.531524 + 0.847043i \(0.678381\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −968.745 −0.0528728
\(696\) 0 0
\(697\) −23612.8 −1.28321
\(698\) 0 0
\(699\) −10464.2 −0.566229
\(700\) 0 0
\(701\) 10696.2 0.576305 0.288152 0.957585i \(-0.406959\pi\)
0.288152 + 0.957585i \(0.406959\pi\)
\(702\) 0 0
\(703\) 32.9915 0.00176998
\(704\) 0 0
\(705\) −2450.06 −0.130886
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −30425.4 −1.61164 −0.805818 0.592163i \(-0.798274\pi\)
−0.805818 + 0.592163i \(0.798274\pi\)
\(710\) 0 0
\(711\) −3904.86 −0.205969
\(712\) 0 0
\(713\) 10144.3 0.532828
\(714\) 0 0
\(715\) −3658.39 −0.191351
\(716\) 0 0
\(717\) 14064.2 0.732551
\(718\) 0 0
\(719\) 30084.9 1.56047 0.780235 0.625487i \(-0.215100\pi\)
0.780235 + 0.625487i \(0.215100\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −19257.7 −0.990596
\(724\) 0 0
\(725\) 9401.42 0.481600
\(726\) 0 0
\(727\) 3894.17 0.198661 0.0993306 0.995054i \(-0.468330\pi\)
0.0993306 + 0.995054i \(0.468330\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −27126.9 −1.37254
\(732\) 0 0
\(733\) −10760.8 −0.542235 −0.271117 0.962546i \(-0.587393\pi\)
−0.271117 + 0.962546i \(0.587393\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5893.26 0.294547
\(738\) 0 0
\(739\) 13739.3 0.683907 0.341954 0.939717i \(-0.388911\pi\)
0.341954 + 0.939717i \(0.388911\pi\)
\(740\) 0 0
\(741\) −3701.12 −0.183487
\(742\) 0 0
\(743\) −26261.8 −1.29670 −0.648352 0.761341i \(-0.724541\pi\)
−0.648352 + 0.761341i \(0.724541\pi\)
\(744\) 0 0
\(745\) −12855.9 −0.632220
\(746\) 0 0
\(747\) −5984.29 −0.293111
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −15995.8 −0.777222 −0.388611 0.921402i \(-0.627045\pi\)
−0.388611 + 0.921402i \(0.627045\pi\)
\(752\) 0 0
\(753\) −17784.9 −0.860713
\(754\) 0 0
\(755\) −23790.1 −1.14677
\(756\) 0 0
\(757\) 1837.00 0.0881993 0.0440997 0.999027i \(-0.485958\pi\)
0.0440997 + 0.999027i \(0.485958\pi\)
\(758\) 0 0
\(759\) 9004.33 0.430615
\(760\) 0 0
\(761\) 2992.29 0.142537 0.0712684 0.997457i \(-0.477295\pi\)
0.0712684 + 0.997457i \(0.477295\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −5645.95 −0.266836
\(766\) 0 0
\(767\) 5698.44 0.268264
\(768\) 0 0
\(769\) −27602.4 −1.29437 −0.647183 0.762335i \(-0.724053\pi\)
−0.647183 + 0.762335i \(0.724053\pi\)
\(770\) 0 0
\(771\) 2019.99 0.0943554
\(772\) 0 0
\(773\) −29608.1 −1.37766 −0.688830 0.724923i \(-0.741875\pi\)
−0.688830 + 0.724923i \(0.741875\pi\)
\(774\) 0 0
\(775\) −9242.81 −0.428402
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 33137.3 1.52409
\(780\) 0 0
\(781\) 11392.8 0.521978
\(782\) 0 0
\(783\) 3969.26 0.181162
\(784\) 0 0
\(785\) 11003.0 0.500272
\(786\) 0 0
\(787\) 6482.32 0.293608 0.146804 0.989166i \(-0.453101\pi\)
0.146804 + 0.989166i \(0.453101\pi\)
\(788\) 0 0
\(789\) −15811.0 −0.713420
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4378.64 −0.196078
\(794\) 0 0
\(795\) −3353.11 −0.149588
\(796\) 0 0
\(797\) −27622.2 −1.22764 −0.613821 0.789446i \(-0.710368\pi\)
−0.613821 + 0.789446i \(0.710368\pi\)
\(798\) 0 0
\(799\) −8392.12 −0.371579
\(800\) 0 0
\(801\) 7232.21 0.319023
\(802\) 0 0
\(803\) −22439.9 −0.986160
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −12194.3 −0.531920
\(808\) 0 0
\(809\) −12740.7 −0.553695 −0.276848 0.960914i \(-0.589290\pi\)
−0.276848 + 0.960914i \(0.589290\pi\)
\(810\) 0 0
\(811\) −2981.71 −0.129102 −0.0645511 0.997914i \(-0.520562\pi\)
−0.0645511 + 0.997914i \(0.520562\pi\)
\(812\) 0 0
\(813\) 19572.6 0.844330
\(814\) 0 0
\(815\) −16279.3 −0.699679
\(816\) 0 0
\(817\) 38069.0 1.63019
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 39713.6 1.68820 0.844100 0.536185i \(-0.180135\pi\)
0.844100 + 0.536185i \(0.180135\pi\)
\(822\) 0 0
\(823\) −19917.5 −0.843596 −0.421798 0.906690i \(-0.638601\pi\)
−0.421798 + 0.906690i \(0.638601\pi\)
\(824\) 0 0
\(825\) −8204.16 −0.346221
\(826\) 0 0
\(827\) 9671.42 0.406660 0.203330 0.979110i \(-0.434824\pi\)
0.203330 + 0.979110i \(0.434824\pi\)
\(828\) 0 0
\(829\) 43001.4 1.80157 0.900784 0.434267i \(-0.142993\pi\)
0.900784 + 0.434267i \(0.142993\pi\)
\(830\) 0 0
\(831\) −25037.5 −1.04518
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1439.50 −0.0596598
\(836\) 0 0
\(837\) −3902.29 −0.161151
\(838\) 0 0
\(839\) 29115.1 1.19805 0.599026 0.800730i \(-0.295555\pi\)
0.599026 + 0.800730i \(0.295555\pi\)
\(840\) 0 0
\(841\) −2777.20 −0.113871
\(842\) 0 0
\(843\) −24484.3 −1.00034
\(844\) 0 0
\(845\) 16229.3 0.660715
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −26009.0 −1.05139
\(850\) 0 0
\(851\) −20.5514 −0.000827841 0
\(852\) 0 0
\(853\) −19860.6 −0.797201 −0.398601 0.917125i \(-0.630504\pi\)
−0.398601 + 0.917125i \(0.630504\pi\)
\(854\) 0 0
\(855\) 7923.33 0.316927
\(856\) 0 0
\(857\) −26027.5 −1.03744 −0.518719 0.854945i \(-0.673591\pi\)
−0.518719 + 0.854945i \(0.673591\pi\)
\(858\) 0 0
\(859\) −18974.5 −0.753669 −0.376835 0.926281i \(-0.622988\pi\)
−0.376835 + 0.926281i \(0.622988\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 21212.2 0.836699 0.418350 0.908286i \(-0.362609\pi\)
0.418350 + 0.908286i \(0.362609\pi\)
\(864\) 0 0
\(865\) −26041.8 −1.02364
\(866\) 0 0
\(867\) −4599.93 −0.180187
\(868\) 0 0
\(869\) 18553.6 0.724267
\(870\) 0 0
\(871\) 1508.96 0.0587015
\(872\) 0 0
\(873\) 4012.32 0.155551
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13227.6 0.509310 0.254655 0.967032i \(-0.418038\pi\)
0.254655 + 0.967032i \(0.418038\pi\)
\(878\) 0 0
\(879\) −299.370 −0.0114875
\(880\) 0 0
\(881\) −7026.27 −0.268696 −0.134348 0.990934i \(-0.542894\pi\)
−0.134348 + 0.990934i \(0.542894\pi\)
\(882\) 0 0
\(883\) 29986.9 1.14285 0.571427 0.820653i \(-0.306390\pi\)
0.571427 + 0.820653i \(0.306390\pi\)
\(884\) 0 0
\(885\) −12199.2 −0.463357
\(886\) 0 0
\(887\) −32874.6 −1.24444 −0.622222 0.782841i \(-0.713770\pi\)
−0.622222 + 0.782841i \(0.713770\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −3463.78 −0.130237
\(892\) 0 0
\(893\) 11777.2 0.441332
\(894\) 0 0
\(895\) −25580.9 −0.955390
\(896\) 0 0
\(897\) 2305.54 0.0858191
\(898\) 0 0
\(899\) −21247.2 −0.788247
\(900\) 0 0
\(901\) −11485.3 −0.424674
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 21159.0 0.777181
\(906\) 0 0
\(907\) 11380.2 0.416620 0.208310 0.978063i \(-0.433204\pi\)
0.208310 + 0.978063i \(0.433204\pi\)
\(908\) 0 0
\(909\) 13927.1 0.508176
\(910\) 0 0
\(911\) 12842.6 0.467062 0.233531 0.972349i \(-0.424972\pi\)
0.233531 + 0.972349i \(0.424972\pi\)
\(912\) 0 0
\(913\) 28433.8 1.03069
\(914\) 0 0
\(915\) 9373.75 0.338674
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −12184.9 −0.437369 −0.218685 0.975796i \(-0.570177\pi\)
−0.218685 + 0.975796i \(0.570177\pi\)
\(920\) 0 0
\(921\) −25894.7 −0.926447
\(922\) 0 0
\(923\) 2917.09 0.104027
\(924\) 0 0
\(925\) 18.7251 0.000665597 0
\(926\) 0 0
\(927\) −6116.40 −0.216709
\(928\) 0 0
\(929\) 32603.6 1.15144 0.575721 0.817646i \(-0.304721\pi\)
0.575721 + 0.817646i \(0.304721\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2591.94 0.0909499
\(934\) 0 0
\(935\) 26826.2 0.938301
\(936\) 0 0
\(937\) 15788.1 0.550453 0.275227 0.961379i \(-0.411247\pi\)
0.275227 + 0.961379i \(0.411247\pi\)
\(938\) 0 0
\(939\) −1733.47 −0.0602446
\(940\) 0 0
\(941\) −45207.2 −1.56611 −0.783057 0.621950i \(-0.786341\pi\)
−0.783057 + 0.621950i \(0.786341\pi\)
\(942\) 0 0
\(943\) −20642.2 −0.712835
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36204.8 −1.24234 −0.621171 0.783675i \(-0.713343\pi\)
−0.621171 + 0.783675i \(0.713343\pi\)
\(948\) 0 0
\(949\) −5745.68 −0.196536
\(950\) 0 0
\(951\) 590.637 0.0201396
\(952\) 0 0
\(953\) 41627.8 1.41496 0.707481 0.706733i \(-0.249832\pi\)
0.707481 + 0.706733i \(0.249832\pi\)
\(954\) 0 0
\(955\) −7697.64 −0.260827
\(956\) 0 0
\(957\) −18859.6 −0.637035
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −8902.25 −0.298824
\(962\) 0 0
\(963\) −17989.7 −0.601983
\(964\) 0 0
\(965\) 36159.7 1.20624
\(966\) 0 0
\(967\) 36170.4 1.20286 0.601428 0.798927i \(-0.294599\pi\)
0.601428 + 0.798927i \(0.294599\pi\)
\(968\) 0 0
\(969\) 27139.6 0.899741
\(970\) 0 0
\(971\) 7565.19 0.250029 0.125015 0.992155i \(-0.460102\pi\)
0.125015 + 0.992155i \(0.460102\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −2100.66 −0.0689998
\(976\) 0 0
\(977\) −47949.1 −1.57014 −0.785070 0.619407i \(-0.787373\pi\)
−0.785070 + 0.619407i \(0.787373\pi\)
\(978\) 0 0
\(979\) −34363.2 −1.12181
\(980\) 0 0
\(981\) −13100.4 −0.426365
\(982\) 0 0
\(983\) −10632.9 −0.345001 −0.172501 0.985009i \(-0.555185\pi\)
−0.172501 + 0.985009i \(0.555185\pi\)
\(984\) 0 0
\(985\) 27361.0 0.885070
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −23714.3 −0.762457
\(990\) 0 0
\(991\) 3411.06 0.109340 0.0546699 0.998504i \(-0.482589\pi\)
0.0546699 + 0.998504i \(0.482589\pi\)
\(992\) 0 0
\(993\) −13936.2 −0.445369
\(994\) 0 0
\(995\) 1325.62 0.0422362
\(996\) 0 0
\(997\) 17965.3 0.570678 0.285339 0.958427i \(-0.407894\pi\)
0.285339 + 0.958427i \(0.407894\pi\)
\(998\) 0 0
\(999\) 7.90569 0.000250375 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 2352.4.a.cn.1.2 4
4.3 odd 2 1176.4.a.be.1.2 yes 4
7.6 odd 2 2352.4.a.co.1.3 4
28.27 even 2 1176.4.a.z.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.z.1.3 4 28.27 even 2
1176.4.a.be.1.2 yes 4 4.3 odd 2
2352.4.a.cn.1.2 4 1.1 even 1 trivial
2352.4.a.co.1.3 4 7.6 odd 2