Properties

Label 2352.4.a.co.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
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: \(1\)
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 1.1
Root \(3.96955\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -14.6418 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -14.6418 q^{5} +9.00000 q^{9} +11.4490 q^{11} -20.8071 q^{13} -43.9254 q^{15} +18.8331 q^{17} -91.0178 q^{19} -28.9336 q^{23} +89.3825 q^{25} +27.0000 q^{27} +281.235 q^{29} +276.010 q^{31} +34.3469 q^{33} +250.567 q^{37} -62.4214 q^{39} +12.1422 q^{41} -65.2121 q^{43} -131.776 q^{45} -199.886 q^{47} +56.4992 q^{51} -64.1260 q^{53} -167.633 q^{55} -273.054 q^{57} +69.0791 q^{59} -656.210 q^{61} +304.654 q^{65} +419.714 q^{67} -86.8007 q^{69} -1022.46 q^{71} -103.280 q^{73} +268.148 q^{75} -260.616 q^{79} +81.0000 q^{81} -879.313 q^{83} -275.750 q^{85} +843.705 q^{87} -1480.70 q^{89} +828.029 q^{93} +1332.67 q^{95} -177.673 q^{97} +103.041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{3} - 8q^{5} + 36q^{9} + O(q^{10}) \) \( 4q + 12q^{3} - 8q^{5} + 36q^{9} - 40q^{11} - 48q^{13} - 24q^{15} - 72q^{17} + 32q^{19} - 8q^{23} + 164q^{25} + 108q^{27} + 144q^{29} + 48q^{31} - 120q^{33} + 48q^{37} - 144q^{39} - 72q^{41} - 512q^{43} - 72q^{45} + 160q^{47} - 216q^{51} + 536q^{53} - 336q^{55} + 96q^{57} + 240q^{59} - 896q^{61} - 136q^{65} - 1088q^{67} - 24q^{69} - 1288q^{71} - 1488q^{73} + 492q^{75} - 416q^{79} + 324q^{81} - 112q^{83} - 1512q^{85} + 432q^{87} - 3160q^{89} + 144q^{93} + 240q^{95} - 2384q^{97} - 360q^{99} + O(q^{100}) \)

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
\(5\) −14.6418 −1.30960 −0.654802 0.755801i \(-0.727248\pi\)
−0.654802 + 0.755801i \(0.727248\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 11.4490 0.313817 0.156909 0.987613i \(-0.449847\pi\)
0.156909 + 0.987613i \(0.449847\pi\)
\(12\) 0 0
\(13\) −20.8071 −0.443913 −0.221956 0.975057i \(-0.571244\pi\)
−0.221956 + 0.975057i \(0.571244\pi\)
\(14\) 0 0
\(15\) −43.9254 −0.756100
\(16\) 0 0
\(17\) 18.8331 0.268688 0.134344 0.990935i \(-0.457107\pi\)
0.134344 + 0.990935i \(0.457107\pi\)
\(18\) 0 0
\(19\) −91.0178 −1.09900 −0.549498 0.835495i \(-0.685181\pi\)
−0.549498 + 0.835495i \(0.685181\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −28.9336 −0.262307 −0.131154 0.991362i \(-0.541868\pi\)
−0.131154 + 0.991362i \(0.541868\pi\)
\(24\) 0 0
\(25\) 89.3825 0.715060
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 281.235 1.80083 0.900414 0.435034i \(-0.143264\pi\)
0.900414 + 0.435034i \(0.143264\pi\)
\(30\) 0 0
\(31\) 276.010 1.59912 0.799561 0.600585i \(-0.205065\pi\)
0.799561 + 0.600585i \(0.205065\pi\)
\(32\) 0 0
\(33\) 34.3469 0.181182
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 250.567 1.11332 0.556662 0.830739i \(-0.312082\pi\)
0.556662 + 0.830739i \(0.312082\pi\)
\(38\) 0 0
\(39\) −62.4214 −0.256293
\(40\) 0 0
\(41\) 12.1422 0.0462512 0.0231256 0.999733i \(-0.492638\pi\)
0.0231256 + 0.999733i \(0.492638\pi\)
\(42\) 0 0
\(43\) −65.2121 −0.231273 −0.115637 0.993292i \(-0.536891\pi\)
−0.115637 + 0.993292i \(0.536891\pi\)
\(44\) 0 0
\(45\) −131.776 −0.436534
\(46\) 0 0
\(47\) −199.886 −0.620350 −0.310175 0.950680i \(-0.600388\pi\)
−0.310175 + 0.950680i \(0.600388\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 56.4992 0.155127
\(52\) 0 0
\(53\) −64.1260 −0.166196 −0.0830980 0.996541i \(-0.526481\pi\)
−0.0830980 + 0.996541i \(0.526481\pi\)
\(54\) 0 0
\(55\) −167.633 −0.410976
\(56\) 0 0
\(57\) −273.054 −0.634506
\(58\) 0 0
\(59\) 69.0791 0.152429 0.0762147 0.997091i \(-0.475717\pi\)
0.0762147 + 0.997091i \(0.475717\pi\)
\(60\) 0 0
\(61\) −656.210 −1.37736 −0.688681 0.725064i \(-0.741810\pi\)
−0.688681 + 0.725064i \(0.741810\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 304.654 0.581349
\(66\) 0 0
\(67\) 419.714 0.765317 0.382658 0.923890i \(-0.375009\pi\)
0.382658 + 0.923890i \(0.375009\pi\)
\(68\) 0 0
\(69\) −86.8007 −0.151443
\(70\) 0 0
\(71\) −1022.46 −1.70907 −0.854536 0.519393i \(-0.826158\pi\)
−0.854536 + 0.519393i \(0.826158\pi\)
\(72\) 0 0
\(73\) −103.280 −0.165590 −0.0827948 0.996567i \(-0.526385\pi\)
−0.0827948 + 0.996567i \(0.526385\pi\)
\(74\) 0 0
\(75\) 268.148 0.412840
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −260.616 −0.371159 −0.185579 0.982629i \(-0.559416\pi\)
−0.185579 + 0.982629i \(0.559416\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −879.313 −1.16286 −0.581428 0.813598i \(-0.697506\pi\)
−0.581428 + 0.813598i \(0.697506\pi\)
\(84\) 0 0
\(85\) −275.750 −0.351874
\(86\) 0 0
\(87\) 843.705 1.03971
\(88\) 0 0
\(89\) −1480.70 −1.76353 −0.881763 0.471692i \(-0.843643\pi\)
−0.881763 + 0.471692i \(0.843643\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 828.029 0.923254
\(94\) 0 0
\(95\) 1332.67 1.43925
\(96\) 0 0
\(97\) −177.673 −0.185979 −0.0929895 0.995667i \(-0.529642\pi\)
−0.0929895 + 0.995667i \(0.529642\pi\)
\(98\) 0 0
\(99\) 103.041 0.104606
\(100\) 0 0
\(101\) 872.534 0.859608 0.429804 0.902922i \(-0.358583\pi\)
0.429804 + 0.902922i \(0.358583\pi\)
\(102\) 0 0
\(103\) 942.292 0.901425 0.450713 0.892669i \(-0.351170\pi\)
0.450713 + 0.892669i \(0.351170\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1162.43 1.05024 0.525121 0.851028i \(-0.324020\pi\)
0.525121 + 0.851028i \(0.324020\pi\)
\(108\) 0 0
\(109\) −35.2015 −0.0309330 −0.0154665 0.999880i \(-0.504923\pi\)
−0.0154665 + 0.999880i \(0.504923\pi\)
\(110\) 0 0
\(111\) 751.701 0.642777
\(112\) 0 0
\(113\) −420.632 −0.350175 −0.175087 0.984553i \(-0.556021\pi\)
−0.175087 + 0.984553i \(0.556021\pi\)
\(114\) 0 0
\(115\) 423.640 0.343519
\(116\) 0 0
\(117\) −187.264 −0.147971
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1199.92 −0.901519
\(122\) 0 0
\(123\) 36.4267 0.0267032
\(124\) 0 0
\(125\) 521.504 0.373158
\(126\) 0 0
\(127\) 722.609 0.504891 0.252445 0.967611i \(-0.418765\pi\)
0.252445 + 0.967611i \(0.418765\pi\)
\(128\) 0 0
\(129\) −195.636 −0.133526
\(130\) 0 0
\(131\) 799.224 0.533042 0.266521 0.963829i \(-0.414126\pi\)
0.266521 + 0.963829i \(0.414126\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −395.329 −0.252033
\(136\) 0 0
\(137\) −2951.13 −1.84038 −0.920191 0.391470i \(-0.871967\pi\)
−0.920191 + 0.391470i \(0.871967\pi\)
\(138\) 0 0
\(139\) −102.191 −0.0623579 −0.0311789 0.999514i \(-0.509926\pi\)
−0.0311789 + 0.999514i \(0.509926\pi\)
\(140\) 0 0
\(141\) −599.659 −0.358159
\(142\) 0 0
\(143\) −238.220 −0.139307
\(144\) 0 0
\(145\) −4117.79 −2.35837
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1798.70 0.988963 0.494482 0.869188i \(-0.335358\pi\)
0.494482 + 0.869188i \(0.335358\pi\)
\(150\) 0 0
\(151\) 3196.81 1.72286 0.861432 0.507873i \(-0.169568\pi\)
0.861432 + 0.507873i \(0.169568\pi\)
\(152\) 0 0
\(153\) 169.498 0.0895626
\(154\) 0 0
\(155\) −4041.28 −2.09422
\(156\) 0 0
\(157\) −3476.27 −1.76711 −0.883556 0.468326i \(-0.844857\pi\)
−0.883556 + 0.468326i \(0.844857\pi\)
\(158\) 0 0
\(159\) −192.378 −0.0959533
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3789.94 −1.82117 −0.910587 0.413318i \(-0.864370\pi\)
−0.910587 + 0.413318i \(0.864370\pi\)
\(164\) 0 0
\(165\) −502.900 −0.237277
\(166\) 0 0
\(167\) 2711.44 1.25639 0.628195 0.778056i \(-0.283794\pi\)
0.628195 + 0.778056i \(0.283794\pi\)
\(168\) 0 0
\(169\) −1764.06 −0.802942
\(170\) 0 0
\(171\) −819.161 −0.366332
\(172\) 0 0
\(173\) 2504.48 1.10065 0.550324 0.834951i \(-0.314504\pi\)
0.550324 + 0.834951i \(0.314504\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 207.237 0.0880052
\(178\) 0 0
\(179\) 2089.31 0.872413 0.436207 0.899847i \(-0.356322\pi\)
0.436207 + 0.899847i \(0.356322\pi\)
\(180\) 0 0
\(181\) −4081.29 −1.67602 −0.838010 0.545655i \(-0.816281\pi\)
−0.838010 + 0.545655i \(0.816281\pi\)
\(182\) 0 0
\(183\) −1968.63 −0.795220
\(184\) 0 0
\(185\) −3668.75 −1.45801
\(186\) 0 0
\(187\) 215.619 0.0843189
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2056.23 −0.778970 −0.389485 0.921033i \(-0.627347\pi\)
−0.389485 + 0.921033i \(0.627347\pi\)
\(192\) 0 0
\(193\) −1489.48 −0.555520 −0.277760 0.960650i \(-0.589592\pi\)
−0.277760 + 0.960650i \(0.589592\pi\)
\(194\) 0 0
\(195\) 913.963 0.335642
\(196\) 0 0
\(197\) 1591.66 0.575640 0.287820 0.957684i \(-0.407069\pi\)
0.287820 + 0.957684i \(0.407069\pi\)
\(198\) 0 0
\(199\) −4303.32 −1.53294 −0.766468 0.642282i \(-0.777988\pi\)
−0.766468 + 0.642282i \(0.777988\pi\)
\(200\) 0 0
\(201\) 1259.14 0.441856
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −177.784 −0.0605707
\(206\) 0 0
\(207\) −260.402 −0.0874358
\(208\) 0 0
\(209\) −1042.06 −0.344884
\(210\) 0 0
\(211\) −2209.01 −0.720733 −0.360367 0.932811i \(-0.617348\pi\)
−0.360367 + 0.932811i \(0.617348\pi\)
\(212\) 0 0
\(213\) −3067.39 −0.986733
\(214\) 0 0
\(215\) 954.824 0.302876
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −309.841 −0.0956033
\(220\) 0 0
\(221\) −391.863 −0.119274
\(222\) 0 0
\(223\) −4731.91 −1.42095 −0.710476 0.703721i \(-0.751520\pi\)
−0.710476 + 0.703721i \(0.751520\pi\)
\(224\) 0 0
\(225\) 804.443 0.238353
\(226\) 0 0
\(227\) 622.308 0.181956 0.0909780 0.995853i \(-0.471001\pi\)
0.0909780 + 0.995853i \(0.471001\pi\)
\(228\) 0 0
\(229\) −530.619 −0.153119 −0.0765595 0.997065i \(-0.524394\pi\)
−0.0765595 + 0.997065i \(0.524394\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3189.28 0.896723 0.448362 0.893852i \(-0.352008\pi\)
0.448362 + 0.893852i \(0.352008\pi\)
\(234\) 0 0
\(235\) 2926.70 0.812412
\(236\) 0 0
\(237\) −781.847 −0.214289
\(238\) 0 0
\(239\) −1515.36 −0.410127 −0.205064 0.978749i \(-0.565740\pi\)
−0.205064 + 0.978749i \(0.565740\pi\)
\(240\) 0 0
\(241\) −1235.38 −0.330200 −0.165100 0.986277i \(-0.552795\pi\)
−0.165100 + 0.986277i \(0.552795\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1893.82 0.487858
\(248\) 0 0
\(249\) −2637.94 −0.671376
\(250\) 0 0
\(251\) 2165.31 0.544514 0.272257 0.962225i \(-0.412230\pi\)
0.272257 + 0.962225i \(0.412230\pi\)
\(252\) 0 0
\(253\) −331.259 −0.0823166
\(254\) 0 0
\(255\) −827.251 −0.203155
\(256\) 0 0
\(257\) −3947.43 −0.958108 −0.479054 0.877786i \(-0.659020\pi\)
−0.479054 + 0.877786i \(0.659020\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2531.11 0.600276
\(262\) 0 0
\(263\) 997.979 0.233985 0.116992 0.993133i \(-0.462675\pi\)
0.116992 + 0.993133i \(0.462675\pi\)
\(264\) 0 0
\(265\) 938.921 0.217651
\(266\) 0 0
\(267\) −4442.10 −1.01817
\(268\) 0 0
\(269\) −6029.26 −1.36658 −0.683291 0.730146i \(-0.739452\pi\)
−0.683291 + 0.730146i \(0.739452\pi\)
\(270\) 0 0
\(271\) 3119.65 0.699280 0.349640 0.936884i \(-0.386304\pi\)
0.349640 + 0.936884i \(0.386304\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1023.34 0.224398
\(276\) 0 0
\(277\) −2003.80 −0.434645 −0.217323 0.976100i \(-0.569732\pi\)
−0.217323 + 0.976100i \(0.569732\pi\)
\(278\) 0 0
\(279\) 2484.09 0.533041
\(280\) 0 0
\(281\) −8915.35 −1.89269 −0.946344 0.323162i \(-0.895254\pi\)
−0.946344 + 0.323162i \(0.895254\pi\)
\(282\) 0 0
\(283\) −3239.09 −0.680366 −0.340183 0.940359i \(-0.610489\pi\)
−0.340183 + 0.940359i \(0.610489\pi\)
\(284\) 0 0
\(285\) 3998.00 0.830951
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4558.32 −0.927807
\(290\) 0 0
\(291\) −533.019 −0.107375
\(292\) 0 0
\(293\) −5997.02 −1.19573 −0.597867 0.801595i \(-0.703985\pi\)
−0.597867 + 0.801595i \(0.703985\pi\)
\(294\) 0 0
\(295\) −1011.44 −0.199622
\(296\) 0 0
\(297\) 309.122 0.0603942
\(298\) 0 0
\(299\) 602.025 0.116442
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 2617.60 0.496295
\(304\) 0 0
\(305\) 9608.10 1.80380
\(306\) 0 0
\(307\) −9676.32 −1.79888 −0.899441 0.437042i \(-0.856026\pi\)
−0.899441 + 0.437042i \(0.856026\pi\)
\(308\) 0 0
\(309\) 2826.88 0.520438
\(310\) 0 0
\(311\) 1458.55 0.265938 0.132969 0.991120i \(-0.457549\pi\)
0.132969 + 0.991120i \(0.457549\pi\)
\(312\) 0 0
\(313\) −10115.7 −1.82674 −0.913371 0.407128i \(-0.866530\pi\)
−0.913371 + 0.407128i \(0.866530\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8516.05 1.50886 0.754431 0.656379i \(-0.227913\pi\)
0.754431 + 0.656379i \(0.227913\pi\)
\(318\) 0 0
\(319\) 3219.85 0.565131
\(320\) 0 0
\(321\) 3487.28 0.606357
\(322\) 0 0
\(323\) −1714.15 −0.295287
\(324\) 0 0
\(325\) −1859.80 −0.317424
\(326\) 0 0
\(327\) −105.605 −0.0178592
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2800.56 0.465053 0.232527 0.972590i \(-0.425301\pi\)
0.232527 + 0.972590i \(0.425301\pi\)
\(332\) 0 0
\(333\) 2255.10 0.371108
\(334\) 0 0
\(335\) −6145.37 −1.00226
\(336\) 0 0
\(337\) −6972.73 −1.12709 −0.563544 0.826086i \(-0.690563\pi\)
−0.563544 + 0.826086i \(0.690563\pi\)
\(338\) 0 0
\(339\) −1261.90 −0.202173
\(340\) 0 0
\(341\) 3160.02 0.501832
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1270.92 0.198331
\(346\) 0 0
\(347\) 704.074 0.108924 0.0544621 0.998516i \(-0.482656\pi\)
0.0544621 + 0.998516i \(0.482656\pi\)
\(348\) 0 0
\(349\) 5350.53 0.820652 0.410326 0.911939i \(-0.365415\pi\)
0.410326 + 0.911939i \(0.365415\pi\)
\(350\) 0 0
\(351\) −561.793 −0.0854310
\(352\) 0 0
\(353\) −1230.18 −0.185485 −0.0927423 0.995690i \(-0.529563\pi\)
−0.0927423 + 0.995690i \(0.529563\pi\)
\(354\) 0 0
\(355\) 14970.7 2.23820
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5402.57 0.794253 0.397126 0.917764i \(-0.370007\pi\)
0.397126 + 0.917764i \(0.370007\pi\)
\(360\) 0 0
\(361\) 1425.25 0.207792
\(362\) 0 0
\(363\) −3599.76 −0.520492
\(364\) 0 0
\(365\) 1512.21 0.216857
\(366\) 0 0
\(367\) −13258.7 −1.88583 −0.942913 0.333040i \(-0.891926\pi\)
−0.942913 + 0.333040i \(0.891926\pi\)
\(368\) 0 0
\(369\) 109.280 0.0154171
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6424.77 −0.891856 −0.445928 0.895069i \(-0.647126\pi\)
−0.445928 + 0.895069i \(0.647126\pi\)
\(374\) 0 0
\(375\) 1564.51 0.215443
\(376\) 0 0
\(377\) −5851.70 −0.799410
\(378\) 0 0
\(379\) 11167.5 1.51355 0.756773 0.653678i \(-0.226775\pi\)
0.756773 + 0.653678i \(0.226775\pi\)
\(380\) 0 0
\(381\) 2167.83 0.291499
\(382\) 0 0
\(383\) −8758.23 −1.16847 −0.584236 0.811584i \(-0.698606\pi\)
−0.584236 + 0.811584i \(0.698606\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −586.909 −0.0770911
\(388\) 0 0
\(389\) 12054.8 1.57122 0.785610 0.618723i \(-0.212350\pi\)
0.785610 + 0.618723i \(0.212350\pi\)
\(390\) 0 0
\(391\) −544.908 −0.0704788
\(392\) 0 0
\(393\) 2397.67 0.307752
\(394\) 0 0
\(395\) 3815.88 0.486071
\(396\) 0 0
\(397\) 4650.10 0.587863 0.293932 0.955826i \(-0.405036\pi\)
0.293932 + 0.955826i \(0.405036\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7378.14 0.918819 0.459410 0.888225i \(-0.348061\pi\)
0.459410 + 0.888225i \(0.348061\pi\)
\(402\) 0 0
\(403\) −5742.97 −0.709871
\(404\) 0 0
\(405\) −1185.99 −0.145511
\(406\) 0 0
\(407\) 2868.73 0.349380
\(408\) 0 0
\(409\) 3727.49 0.450642 0.225321 0.974285i \(-0.427657\pi\)
0.225321 + 0.974285i \(0.427657\pi\)
\(410\) 0 0
\(411\) −8853.40 −1.06254
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 12874.7 1.52288
\(416\) 0 0
\(417\) −306.574 −0.0360023
\(418\) 0 0
\(419\) −11146.7 −1.29964 −0.649822 0.760087i \(-0.725156\pi\)
−0.649822 + 0.760087i \(0.725156\pi\)
\(420\) 0 0
\(421\) −11777.7 −1.36344 −0.681719 0.731614i \(-0.738768\pi\)
−0.681719 + 0.731614i \(0.738768\pi\)
\(422\) 0 0
\(423\) −1798.98 −0.206783
\(424\) 0 0
\(425\) 1683.35 0.192128
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −714.660 −0.0804292
\(430\) 0 0
\(431\) 5574.28 0.622978 0.311489 0.950250i \(-0.399172\pi\)
0.311489 + 0.950250i \(0.399172\pi\)
\(432\) 0 0
\(433\) −240.867 −0.0267328 −0.0133664 0.999911i \(-0.504255\pi\)
−0.0133664 + 0.999911i \(0.504255\pi\)
\(434\) 0 0
\(435\) −12353.4 −1.36161
\(436\) 0 0
\(437\) 2633.47 0.288275
\(438\) 0 0
\(439\) −9433.79 −1.02563 −0.512813 0.858500i \(-0.671397\pi\)
−0.512813 + 0.858500i \(0.671397\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9053.72 0.971005 0.485502 0.874235i \(-0.338637\pi\)
0.485502 + 0.874235i \(0.338637\pi\)
\(444\) 0 0
\(445\) 21680.1 2.30952
\(446\) 0 0
\(447\) 5396.11 0.570978
\(448\) 0 0
\(449\) 2726.05 0.286526 0.143263 0.989685i \(-0.454241\pi\)
0.143263 + 0.989685i \(0.454241\pi\)
\(450\) 0 0
\(451\) 139.016 0.0145144
\(452\) 0 0
\(453\) 9590.42 0.994696
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10628.5 1.08792 0.543960 0.839111i \(-0.316924\pi\)
0.543960 + 0.839111i \(0.316924\pi\)
\(458\) 0 0
\(459\) 508.493 0.0517090
\(460\) 0 0
\(461\) −11324.0 −1.14406 −0.572032 0.820232i \(-0.693845\pi\)
−0.572032 + 0.820232i \(0.693845\pi\)
\(462\) 0 0
\(463\) −11711.0 −1.17550 −0.587751 0.809042i \(-0.699986\pi\)
−0.587751 + 0.809042i \(0.699986\pi\)
\(464\) 0 0
\(465\) −12123.8 −1.20910
\(466\) 0 0
\(467\) −15480.6 −1.53396 −0.766980 0.641671i \(-0.778241\pi\)
−0.766980 + 0.641671i \(0.778241\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −10428.8 −1.02024
\(472\) 0 0
\(473\) −746.611 −0.0725776
\(474\) 0 0
\(475\) −8135.41 −0.785848
\(476\) 0 0
\(477\) −577.134 −0.0553987
\(478\) 0 0
\(479\) 4747.39 0.452847 0.226423 0.974029i \(-0.427297\pi\)
0.226423 + 0.974029i \(0.427297\pi\)
\(480\) 0 0
\(481\) −5213.58 −0.494218
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2601.45 0.243559
\(486\) 0 0
\(487\) 7970.85 0.741671 0.370835 0.928699i \(-0.379071\pi\)
0.370835 + 0.928699i \(0.379071\pi\)
\(488\) 0 0
\(489\) −11369.8 −1.05145
\(490\) 0 0
\(491\) −13019.1 −1.19662 −0.598311 0.801264i \(-0.704161\pi\)
−0.598311 + 0.801264i \(0.704161\pi\)
\(492\) 0 0
\(493\) 5296.52 0.483860
\(494\) 0 0
\(495\) −1508.70 −0.136992
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 15490.8 1.38971 0.694854 0.719151i \(-0.255469\pi\)
0.694854 + 0.719151i \(0.255469\pi\)
\(500\) 0 0
\(501\) 8134.31 0.725377
\(502\) 0 0
\(503\) 8646.86 0.766490 0.383245 0.923647i \(-0.374806\pi\)
0.383245 + 0.923647i \(0.374806\pi\)
\(504\) 0 0
\(505\) −12775.5 −1.12574
\(506\) 0 0
\(507\) −5292.19 −0.463579
\(508\) 0 0
\(509\) 10252.0 0.892753 0.446376 0.894845i \(-0.352714\pi\)
0.446376 + 0.894845i \(0.352714\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2457.48 −0.211502
\(514\) 0 0
\(515\) −13796.9 −1.18051
\(516\) 0 0
\(517\) −2288.49 −0.194676
\(518\) 0 0
\(519\) 7513.45 0.635460
\(520\) 0 0
\(521\) 10025.3 0.843023 0.421512 0.906823i \(-0.361500\pi\)
0.421512 + 0.906823i \(0.361500\pi\)
\(522\) 0 0
\(523\) −186.782 −0.0156164 −0.00780822 0.999970i \(-0.502485\pi\)
−0.00780822 + 0.999970i \(0.502485\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5198.11 0.429665
\(528\) 0 0
\(529\) −11329.8 −0.931195
\(530\) 0 0
\(531\) 621.712 0.0508098
\(532\) 0 0
\(533\) −252.646 −0.0205315
\(534\) 0 0
\(535\) −17020.0 −1.37540
\(536\) 0 0
\(537\) 6267.92 0.503688
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −9729.13 −0.773176 −0.386588 0.922253i \(-0.626346\pi\)
−0.386588 + 0.922253i \(0.626346\pi\)
\(542\) 0 0
\(543\) −12243.9 −0.967651
\(544\) 0 0
\(545\) 515.414 0.0405099
\(546\) 0 0
\(547\) 19387.1 1.51541 0.757707 0.652595i \(-0.226320\pi\)
0.757707 + 0.652595i \(0.226320\pi\)
\(548\) 0 0
\(549\) −5905.89 −0.459121
\(550\) 0 0
\(551\) −25597.4 −1.97910
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −11006.3 −0.841783
\(556\) 0 0
\(557\) 13182.0 1.00276 0.501381 0.865227i \(-0.332826\pi\)
0.501381 + 0.865227i \(0.332826\pi\)
\(558\) 0 0
\(559\) 1356.88 0.102665
\(560\) 0 0
\(561\) 646.857 0.0486815
\(562\) 0 0
\(563\) −14211.0 −1.06381 −0.531903 0.846805i \(-0.678523\pi\)
−0.531903 + 0.846805i \(0.678523\pi\)
\(564\) 0 0
\(565\) 6158.81 0.458590
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −855.367 −0.0630208 −0.0315104 0.999503i \(-0.510032\pi\)
−0.0315104 + 0.999503i \(0.510032\pi\)
\(570\) 0 0
\(571\) −16827.3 −1.23327 −0.616637 0.787247i \(-0.711505\pi\)
−0.616637 + 0.787247i \(0.711505\pi\)
\(572\) 0 0
\(573\) −6168.68 −0.449738
\(574\) 0 0
\(575\) −2586.16 −0.187566
\(576\) 0 0
\(577\) −3961.45 −0.285818 −0.142909 0.989736i \(-0.545646\pi\)
−0.142909 + 0.989736i \(0.545646\pi\)
\(578\) 0 0
\(579\) −4468.45 −0.320730
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −734.176 −0.0521552
\(584\) 0 0
\(585\) 2741.89 0.193783
\(586\) 0 0
\(587\) −130.006 −0.00914124 −0.00457062 0.999990i \(-0.501455\pi\)
−0.00457062 + 0.999990i \(0.501455\pi\)
\(588\) 0 0
\(589\) −25121.8 −1.75743
\(590\) 0 0
\(591\) 4774.98 0.332346
\(592\) 0 0
\(593\) 11730.9 0.812361 0.406180 0.913793i \(-0.366860\pi\)
0.406180 + 0.913793i \(0.366860\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −12910.0 −0.885041
\(598\) 0 0
\(599\) 2349.76 0.160281 0.0801407 0.996784i \(-0.474463\pi\)
0.0801407 + 0.996784i \(0.474463\pi\)
\(600\) 0 0
\(601\) 21900.5 1.48642 0.743210 0.669059i \(-0.233302\pi\)
0.743210 + 0.669059i \(0.233302\pi\)
\(602\) 0 0
\(603\) 3777.42 0.255106
\(604\) 0 0
\(605\) 17569.0 1.18063
\(606\) 0 0
\(607\) 23463.6 1.56896 0.784480 0.620154i \(-0.212930\pi\)
0.784480 + 0.620154i \(0.212930\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4159.07 0.275381
\(612\) 0 0
\(613\) −9012.69 −0.593832 −0.296916 0.954904i \(-0.595958\pi\)
−0.296916 + 0.954904i \(0.595958\pi\)
\(614\) 0 0
\(615\) −533.353 −0.0349705
\(616\) 0 0
\(617\) 18605.8 1.21401 0.607004 0.794699i \(-0.292371\pi\)
0.607004 + 0.794699i \(0.292371\pi\)
\(618\) 0 0
\(619\) −7917.28 −0.514091 −0.257045 0.966399i \(-0.582749\pi\)
−0.257045 + 0.966399i \(0.582749\pi\)
\(620\) 0 0
\(621\) −781.207 −0.0504811
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −18808.6 −1.20375
\(626\) 0 0
\(627\) −3126.18 −0.199119
\(628\) 0 0
\(629\) 4718.95 0.299136
\(630\) 0 0
\(631\) 10620.3 0.670025 0.335013 0.942214i \(-0.391259\pi\)
0.335013 + 0.942214i \(0.391259\pi\)
\(632\) 0 0
\(633\) −6627.04 −0.416116
\(634\) 0 0
\(635\) −10580.3 −0.661207
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −9202.17 −0.569690
\(640\) 0 0
\(641\) −6334.25 −0.390309 −0.195154 0.980773i \(-0.562521\pi\)
−0.195154 + 0.980773i \(0.562521\pi\)
\(642\) 0 0
\(643\) 25268.3 1.54974 0.774872 0.632118i \(-0.217814\pi\)
0.774872 + 0.632118i \(0.217814\pi\)
\(644\) 0 0
\(645\) 2864.47 0.174866
\(646\) 0 0
\(647\) −2288.39 −0.139051 −0.0695256 0.997580i \(-0.522149\pi\)
−0.0695256 + 0.997580i \(0.522149\pi\)
\(648\) 0 0
\(649\) 790.884 0.0478350
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −12721.2 −0.762357 −0.381179 0.924501i \(-0.624482\pi\)
−0.381179 + 0.924501i \(0.624482\pi\)
\(654\) 0 0
\(655\) −11702.1 −0.698073
\(656\) 0 0
\(657\) −929.523 −0.0551966
\(658\) 0 0
\(659\) −15931.0 −0.941707 −0.470853 0.882211i \(-0.656054\pi\)
−0.470853 + 0.882211i \(0.656054\pi\)
\(660\) 0 0
\(661\) 3790.79 0.223063 0.111531 0.993761i \(-0.464424\pi\)
0.111531 + 0.993761i \(0.464424\pi\)
\(662\) 0 0
\(663\) −1175.59 −0.0688628
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8137.13 −0.472370
\(668\) 0 0
\(669\) −14195.7 −0.820387
\(670\) 0 0
\(671\) −7512.92 −0.432240
\(672\) 0 0
\(673\) −27903.0 −1.59819 −0.799096 0.601203i \(-0.794688\pi\)
−0.799096 + 0.601203i \(0.794688\pi\)
\(674\) 0 0
\(675\) 2413.33 0.137613
\(676\) 0 0
\(677\) −12845.9 −0.729257 −0.364629 0.931153i \(-0.618804\pi\)
−0.364629 + 0.931153i \(0.618804\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1866.92 0.105052
\(682\) 0 0
\(683\) 27871.5 1.56146 0.780728 0.624870i \(-0.214848\pi\)
0.780728 + 0.624870i \(0.214848\pi\)
\(684\) 0 0
\(685\) 43209.9 2.41017
\(686\) 0 0
\(687\) −1591.86 −0.0884033
\(688\) 0 0
\(689\) 1334.28 0.0737765
\(690\) 0 0
\(691\) 12700.3 0.699190 0.349595 0.936901i \(-0.386319\pi\)
0.349595 + 0.936901i \(0.386319\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1496.26 0.0816641
\(696\) 0 0
\(697\) 228.676 0.0124271
\(698\) 0 0
\(699\) 9567.83 0.517723
\(700\) 0 0
\(701\) −18491.7 −0.996320 −0.498160 0.867085i \(-0.665991\pi\)
−0.498160 + 0.867085i \(0.665991\pi\)
\(702\) 0 0
\(703\) −22806.1 −1.22354
\(704\) 0 0
\(705\) 8780.10 0.469046
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −10589.8 −0.560944 −0.280472 0.959862i \(-0.590491\pi\)
−0.280472 + 0.959862i \(0.590491\pi\)
\(710\) 0 0
\(711\) −2345.54 −0.123720
\(712\) 0 0
\(713\) −7985.95 −0.419462
\(714\) 0 0
\(715\) 3487.97 0.182437
\(716\) 0 0
\(717\) −4546.07 −0.236787
\(718\) 0 0
\(719\) 4494.41 0.233120 0.116560 0.993184i \(-0.462813\pi\)
0.116560 + 0.993184i \(0.462813\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −3706.15 −0.190641
\(724\) 0 0
\(725\) 25137.5 1.28770
\(726\) 0 0
\(727\) −3251.31 −0.165866 −0.0829328 0.996555i \(-0.526429\pi\)
−0.0829328 + 0.996555i \(0.526429\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −1228.15 −0.0621403
\(732\) 0 0
\(733\) −17052.9 −0.859293 −0.429647 0.902997i \(-0.641362\pi\)
−0.429647 + 0.902997i \(0.641362\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4805.29 0.240170
\(738\) 0 0
\(739\) −33680.4 −1.67653 −0.838265 0.545264i \(-0.816429\pi\)
−0.838265 + 0.545264i \(0.816429\pi\)
\(740\) 0 0
\(741\) 5681.47 0.281665
\(742\) 0 0
\(743\) 18689.5 0.922813 0.461407 0.887189i \(-0.347345\pi\)
0.461407 + 0.887189i \(0.347345\pi\)
\(744\) 0 0
\(745\) −26336.3 −1.29515
\(746\) 0 0
\(747\) −7913.81 −0.387619
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 25028.2 1.21610 0.608052 0.793897i \(-0.291951\pi\)
0.608052 + 0.793897i \(0.291951\pi\)
\(752\) 0 0
\(753\) 6495.93 0.314375
\(754\) 0 0
\(755\) −46807.0 −2.25627
\(756\) 0 0
\(757\) −18630.2 −0.894488 −0.447244 0.894412i \(-0.647594\pi\)
−0.447244 + 0.894412i \(0.647594\pi\)
\(758\) 0 0
\(759\) −993.778 −0.0475255
\(760\) 0 0
\(761\) −28611.1 −1.36288 −0.681440 0.731874i \(-0.738646\pi\)
−0.681440 + 0.731874i \(0.738646\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2481.75 −0.117291
\(766\) 0 0
\(767\) −1437.34 −0.0676654
\(768\) 0 0
\(769\) −15072.5 −0.706798 −0.353399 0.935473i \(-0.614974\pi\)
−0.353399 + 0.935473i \(0.614974\pi\)
\(770\) 0 0
\(771\) −11842.3 −0.553164
\(772\) 0 0
\(773\) 32904.3 1.53103 0.765515 0.643418i \(-0.222484\pi\)
0.765515 + 0.643418i \(0.222484\pi\)
\(774\) 0 0
\(775\) 24670.4 1.14347
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1105.16 −0.0508299
\(780\) 0 0
\(781\) −11706.1 −0.536336
\(782\) 0 0
\(783\) 7593.34 0.346570
\(784\) 0 0
\(785\) 50898.9 2.31422
\(786\) 0 0
\(787\) 31033.3 1.40562 0.702808 0.711380i \(-0.251929\pi\)
0.702808 + 0.711380i \(0.251929\pi\)
\(788\) 0 0
\(789\) 2993.94 0.135091
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 13653.9 0.611428
\(794\) 0 0
\(795\) 2816.76 0.125661
\(796\) 0 0
\(797\) −5906.64 −0.262514 −0.131257 0.991348i \(-0.541901\pi\)
−0.131257 + 0.991348i \(0.541901\pi\)
\(798\) 0 0
\(799\) −3764.48 −0.166680
\(800\) 0 0
\(801\) −13326.3 −0.587842
\(802\) 0 0
\(803\) −1182.45 −0.0519649
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −18087.8 −0.788997
\(808\) 0 0
\(809\) −9652.69 −0.419494 −0.209747 0.977756i \(-0.567264\pi\)
−0.209747 + 0.977756i \(0.567264\pi\)
\(810\) 0 0
\(811\) 29469.2 1.27596 0.637980 0.770053i \(-0.279770\pi\)
0.637980 + 0.770053i \(0.279770\pi\)
\(812\) 0 0
\(813\) 9358.94 0.403730
\(814\) 0 0
\(815\) 55491.6 2.38501
\(816\) 0 0
\(817\) 5935.47 0.254169
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −667.350 −0.0283687 −0.0141843 0.999899i \(-0.504515\pi\)
−0.0141843 + 0.999899i \(0.504515\pi\)
\(822\) 0 0
\(823\) 18030.5 0.763673 0.381836 0.924230i \(-0.375292\pi\)
0.381836 + 0.924230i \(0.375292\pi\)
\(824\) 0 0
\(825\) 3070.01 0.129556
\(826\) 0 0
\(827\) 2362.88 0.0993535 0.0496767 0.998765i \(-0.484181\pi\)
0.0496767 + 0.998765i \(0.484181\pi\)
\(828\) 0 0
\(829\) 26128.9 1.09468 0.547342 0.836909i \(-0.315640\pi\)
0.547342 + 0.836909i \(0.315640\pi\)
\(830\) 0 0
\(831\) −6011.40 −0.250942
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −39700.3 −1.64537
\(836\) 0 0
\(837\) 7452.26 0.307751
\(838\) 0 0
\(839\) 39017.9 1.60554 0.802769 0.596290i \(-0.203359\pi\)
0.802769 + 0.596290i \(0.203359\pi\)
\(840\) 0 0
\(841\) 54704.1 2.24298
\(842\) 0 0
\(843\) −26746.0 −1.09274
\(844\) 0 0
\(845\) 25829.1 1.05153
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −9717.26 −0.392810
\(850\) 0 0
\(851\) −7249.80 −0.292033
\(852\) 0 0
\(853\) 24487.4 0.982922 0.491461 0.870900i \(-0.336463\pi\)
0.491461 + 0.870900i \(0.336463\pi\)
\(854\) 0 0
\(855\) 11994.0 0.479750
\(856\) 0 0
\(857\) −2351.90 −0.0937450 −0.0468725 0.998901i \(-0.514925\pi\)
−0.0468725 + 0.998901i \(0.514925\pi\)
\(858\) 0 0
\(859\) 5413.01 0.215005 0.107503 0.994205i \(-0.465715\pi\)
0.107503 + 0.994205i \(0.465715\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −32772.7 −1.29270 −0.646348 0.763043i \(-0.723705\pi\)
−0.646348 + 0.763043i \(0.723705\pi\)
\(864\) 0 0
\(865\) −36670.1 −1.44141
\(866\) 0 0
\(867\) −13674.9 −0.535670
\(868\) 0 0
\(869\) −2983.78 −0.116476
\(870\) 0 0
\(871\) −8733.05 −0.339734
\(872\) 0 0
\(873\) −1599.06 −0.0619930
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −40592.0 −1.56294 −0.781468 0.623946i \(-0.785529\pi\)
−0.781468 + 0.623946i \(0.785529\pi\)
\(878\) 0 0
\(879\) −17991.1 −0.690357
\(880\) 0 0
\(881\) −21588.5 −0.825578 −0.412789 0.910827i \(-0.635445\pi\)
−0.412789 + 0.910827i \(0.635445\pi\)
\(882\) 0 0
\(883\) −11223.7 −0.427757 −0.213878 0.976860i \(-0.568610\pi\)
−0.213878 + 0.976860i \(0.568610\pi\)
\(884\) 0 0
\(885\) −3034.33 −0.115252
\(886\) 0 0
\(887\) −36557.9 −1.38387 −0.691936 0.721958i \(-0.743242\pi\)
−0.691936 + 0.721958i \(0.743242\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 927.365 0.0348686
\(892\) 0 0
\(893\) 18193.2 0.681762
\(894\) 0 0
\(895\) −30591.2 −1.14252
\(896\) 0 0
\(897\) 1806.08 0.0672276
\(898\) 0 0
\(899\) 77623.5 2.87974
\(900\) 0 0
\(901\) −1207.69 −0.0446548
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 59757.4 2.19492
\(906\) 0 0
\(907\) 19252.0 0.704798 0.352399 0.935850i \(-0.385366\pi\)
0.352399 + 0.935850i \(0.385366\pi\)
\(908\) 0 0
\(909\) 7852.81 0.286536
\(910\) 0 0
\(911\) −4910.11 −0.178572 −0.0892861 0.996006i \(-0.528459\pi\)
−0.0892861 + 0.996006i \(0.528459\pi\)
\(912\) 0 0
\(913\) −10067.2 −0.364925
\(914\) 0 0
\(915\) 28824.3 1.04142
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1908.48 0.0685039 0.0342520 0.999413i \(-0.489095\pi\)
0.0342520 + 0.999413i \(0.489095\pi\)
\(920\) 0 0
\(921\) −29029.0 −1.03859
\(922\) 0 0
\(923\) 21274.5 0.758678
\(924\) 0 0
\(925\) 22396.3 0.796093
\(926\) 0 0
\(927\) 8480.63 0.300475
\(928\) 0 0
\(929\) −30463.0 −1.07584 −0.537921 0.842995i \(-0.680790\pi\)
−0.537921 + 0.842995i \(0.680790\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 4375.65 0.153540
\(934\) 0 0
\(935\) −3157.05 −0.110424
\(936\) 0 0
\(937\) −468.146 −0.0163219 −0.00816097 0.999967i \(-0.502598\pi\)
−0.00816097 + 0.999967i \(0.502598\pi\)
\(938\) 0 0
\(939\) −30347.0 −1.05467
\(940\) 0 0
\(941\) 8443.67 0.292514 0.146257 0.989247i \(-0.453277\pi\)
0.146257 + 0.989247i \(0.453277\pi\)
\(942\) 0 0
\(943\) −351.319 −0.0121320
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7269.71 0.249455 0.124727 0.992191i \(-0.460194\pi\)
0.124727 + 0.992191i \(0.460194\pi\)
\(948\) 0 0
\(949\) 2148.97 0.0735074
\(950\) 0 0
\(951\) 25548.2 0.871142
\(952\) 0 0
\(953\) 25806.9 0.877195 0.438598 0.898684i \(-0.355475\pi\)
0.438598 + 0.898684i \(0.355475\pi\)
\(954\) 0 0
\(955\) 30106.9 1.02014
\(956\) 0 0
\(957\) 9659.54 0.326279
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 46390.3 1.55719
\(962\) 0 0
\(963\) 10461.8 0.350081
\(964\) 0 0
\(965\) 21808.7 0.727511
\(966\) 0 0
\(967\) −7155.61 −0.237962 −0.118981 0.992897i \(-0.537963\pi\)
−0.118981 + 0.992897i \(0.537963\pi\)
\(968\) 0 0
\(969\) −5142.44 −0.170484
\(970\) 0 0
\(971\) 29970.2 0.990515 0.495257 0.868746i \(-0.335074\pi\)
0.495257 + 0.868746i \(0.335074\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −5579.39 −0.183265
\(976\) 0 0
\(977\) −46223.3 −1.51363 −0.756813 0.653631i \(-0.773245\pi\)
−0.756813 + 0.653631i \(0.773245\pi\)
\(978\) 0 0
\(979\) −16952.5 −0.553425
\(980\) 0 0
\(981\) −316.814 −0.0103110
\(982\) 0 0
\(983\) 41135.8 1.33472 0.667360 0.744736i \(-0.267424\pi\)
0.667360 + 0.744736i \(0.267424\pi\)
\(984\) 0 0
\(985\) −23304.8 −0.753860
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1886.82 0.0606647
\(990\) 0 0
\(991\) 4228.11 0.135530 0.0677651 0.997701i \(-0.478413\pi\)
0.0677651 + 0.997701i \(0.478413\pi\)
\(992\) 0 0
\(993\) 8401.67 0.268499
\(994\) 0 0
\(995\) 63008.4 2.00754
\(996\) 0 0
\(997\) 38767.3 1.23147 0.615734 0.787954i \(-0.288860\pi\)
0.615734 + 0.787954i \(0.288860\pi\)
\(998\) 0 0
\(999\) 6765.31 0.214259
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.co.1.1 4
4.3 odd 2 1176.4.a.z.1.1 4
7.6 odd 2 2352.4.a.cn.1.4 4
28.27 even 2 1176.4.a.be.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.z.1.1 4 4.3 odd 2
1176.4.a.be.1.4 yes 4 28.27 even 2
2352.4.a.cn.1.4 4 7.6 odd 2
2352.4.a.co.1.1 4 1.1 even 1 trivial