Properties

Label 1521.4.a.bd.1.8
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(5.39589\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.39589 q^{2} +21.1156 q^{4} +13.0421 q^{5} +6.42494 q^{7} +70.7705 q^{8} +O(q^{10})\) \(q+5.39589 q^{2} +21.1156 q^{4} +13.0421 q^{5} +6.42494 q^{7} +70.7705 q^{8} +70.3740 q^{10} +26.2056 q^{11} +34.6683 q^{14} +212.945 q^{16} +123.877 q^{17} -109.667 q^{19} +275.393 q^{20} +141.402 q^{22} +63.4094 q^{23} +45.0976 q^{25} +135.667 q^{28} -225.410 q^{29} -200.732 q^{31} +582.863 q^{32} +668.426 q^{34} +83.7950 q^{35} -252.509 q^{37} -591.749 q^{38} +922.999 q^{40} +227.423 q^{41} -384.032 q^{43} +553.348 q^{44} +342.150 q^{46} +34.6646 q^{47} -301.720 q^{49} +243.342 q^{50} +61.0601 q^{53} +341.777 q^{55} +454.696 q^{56} -1216.29 q^{58} -80.5562 q^{59} -26.1383 q^{61} -1083.13 q^{62} +1441.50 q^{64} +931.510 q^{67} +2615.74 q^{68} +452.149 q^{70} +427.608 q^{71} -108.518 q^{73} -1362.51 q^{74} -2315.68 q^{76} +168.369 q^{77} +384.590 q^{79} +2777.26 q^{80} +1227.15 q^{82} -85.9758 q^{83} +1615.62 q^{85} -2072.19 q^{86} +1854.58 q^{88} -495.903 q^{89} +1338.93 q^{92} +187.046 q^{94} -1430.29 q^{95} -190.857 q^{97} -1628.05 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 40 q^{4} + 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{4} + 22 q^{7} + 36 q^{10} + 204 q^{16} - 244 q^{19} + 136 q^{22} + 354 q^{25} + 452 q^{28} - 242 q^{31} + 1292 q^{34} - 1018 q^{37} + 1700 q^{40} + 74 q^{43} + 896 q^{46} + 298 q^{49} + 1300 q^{55} - 812 q^{58} + 1148 q^{61} + 3636 q^{64} + 2198 q^{67} - 2200 q^{70} + 2176 q^{73} - 6936 q^{76} + 1862 q^{79} + 5436 q^{82} + 890 q^{85} + 3528 q^{88} - 3104 q^{94} + 4370 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.39589 1.90774 0.953868 0.300228i \(-0.0970626\pi\)
0.953868 + 0.300228i \(0.0970626\pi\)
\(3\) 0 0
\(4\) 21.1156 2.63945
\(5\) 13.0421 1.16653 0.583263 0.812284i \(-0.301776\pi\)
0.583263 + 0.812284i \(0.301776\pi\)
\(6\) 0 0
\(7\) 6.42494 0.346914 0.173457 0.984841i \(-0.444506\pi\)
0.173457 + 0.984841i \(0.444506\pi\)
\(8\) 70.7705 3.12764
\(9\) 0 0
\(10\) 70.3740 2.22542
\(11\) 26.2056 0.718298 0.359149 0.933280i \(-0.383067\pi\)
0.359149 + 0.933280i \(0.383067\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 34.6683 0.661820
\(15\) 0 0
\(16\) 212.945 3.32726
\(17\) 123.877 1.76733 0.883663 0.468124i \(-0.155070\pi\)
0.883663 + 0.468124i \(0.155070\pi\)
\(18\) 0 0
\(19\) −109.667 −1.32417 −0.662086 0.749428i \(-0.730329\pi\)
−0.662086 + 0.749428i \(0.730329\pi\)
\(20\) 275.393 3.07899
\(21\) 0 0
\(22\) 141.402 1.37032
\(23\) 63.4094 0.574860 0.287430 0.957802i \(-0.407199\pi\)
0.287430 + 0.957802i \(0.407199\pi\)
\(24\) 0 0
\(25\) 45.0976 0.360781
\(26\) 0 0
\(27\) 0 0
\(28\) 135.667 0.915664
\(29\) −225.410 −1.44337 −0.721683 0.692224i \(-0.756631\pi\)
−0.721683 + 0.692224i \(0.756631\pi\)
\(30\) 0 0
\(31\) −200.732 −1.16298 −0.581491 0.813553i \(-0.697530\pi\)
−0.581491 + 0.813553i \(0.697530\pi\)
\(32\) 582.863 3.21989
\(33\) 0 0
\(34\) 668.426 3.37159
\(35\) 83.7950 0.404684
\(36\) 0 0
\(37\) −252.509 −1.12195 −0.560976 0.827832i \(-0.689574\pi\)
−0.560976 + 0.827832i \(0.689574\pi\)
\(38\) −591.749 −2.52617
\(39\) 0 0
\(40\) 922.999 3.64847
\(41\) 227.423 0.866282 0.433141 0.901326i \(-0.357405\pi\)
0.433141 + 0.901326i \(0.357405\pi\)
\(42\) 0 0
\(43\) −384.032 −1.36196 −0.680980 0.732302i \(-0.738446\pi\)
−0.680980 + 0.732302i \(0.738446\pi\)
\(44\) 553.348 1.89592
\(45\) 0 0
\(46\) 342.150 1.09668
\(47\) 34.6646 0.107582 0.0537910 0.998552i \(-0.482870\pi\)
0.0537910 + 0.998552i \(0.482870\pi\)
\(48\) 0 0
\(49\) −301.720 −0.879651
\(50\) 243.342 0.688274
\(51\) 0 0
\(52\) 0 0
\(53\) 61.0601 0.158250 0.0791251 0.996865i \(-0.474787\pi\)
0.0791251 + 0.996865i \(0.474787\pi\)
\(54\) 0 0
\(55\) 341.777 0.837913
\(56\) 454.696 1.08502
\(57\) 0 0
\(58\) −1216.29 −2.75356
\(59\) −80.5562 −0.177755 −0.0888774 0.996043i \(-0.528328\pi\)
−0.0888774 + 0.996043i \(0.528328\pi\)
\(60\) 0 0
\(61\) −26.1383 −0.0548634 −0.0274317 0.999624i \(-0.508733\pi\)
−0.0274317 + 0.999624i \(0.508733\pi\)
\(62\) −1083.13 −2.21866
\(63\) 0 0
\(64\) 1441.50 2.81544
\(65\) 0 0
\(66\) 0 0
\(67\) 931.510 1.69854 0.849269 0.527960i \(-0.177043\pi\)
0.849269 + 0.527960i \(0.177043\pi\)
\(68\) 2615.74 4.66477
\(69\) 0 0
\(70\) 452.149 0.772030
\(71\) 427.608 0.714757 0.357379 0.933960i \(-0.383671\pi\)
0.357379 + 0.933960i \(0.383671\pi\)
\(72\) 0 0
\(73\) −108.518 −0.173987 −0.0869935 0.996209i \(-0.527726\pi\)
−0.0869935 + 0.996209i \(0.527726\pi\)
\(74\) −1362.51 −2.14039
\(75\) 0 0
\(76\) −2315.68 −3.49509
\(77\) 168.369 0.249188
\(78\) 0 0
\(79\) 384.590 0.547718 0.273859 0.961770i \(-0.411700\pi\)
0.273859 + 0.961770i \(0.411700\pi\)
\(80\) 2777.26 3.88133
\(81\) 0 0
\(82\) 1227.15 1.65264
\(83\) −85.9758 −0.113700 −0.0568498 0.998383i \(-0.518106\pi\)
−0.0568498 + 0.998383i \(0.518106\pi\)
\(84\) 0 0
\(85\) 1615.62 2.06163
\(86\) −2072.19 −2.59826
\(87\) 0 0
\(88\) 1854.58 2.24658
\(89\) −495.903 −0.590625 −0.295313 0.955401i \(-0.595424\pi\)
−0.295313 + 0.955401i \(0.595424\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1338.93 1.51732
\(93\) 0 0
\(94\) 187.046 0.205238
\(95\) −1430.29 −1.54468
\(96\) 0 0
\(97\) −190.857 −0.199780 −0.0998898 0.994999i \(-0.531849\pi\)
−0.0998898 + 0.994999i \(0.531849\pi\)
\(98\) −1628.05 −1.67814
\(99\) 0 0
\(100\) 952.264 0.952264
\(101\) 850.980 0.838373 0.419186 0.907900i \(-0.362315\pi\)
0.419186 + 0.907900i \(0.362315\pi\)
\(102\) 0 0
\(103\) −1309.50 −1.25270 −0.626352 0.779540i \(-0.715453\pi\)
−0.626352 + 0.779540i \(0.715453\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 329.474 0.301899
\(107\) 962.302 0.869432 0.434716 0.900567i \(-0.356849\pi\)
0.434716 + 0.900567i \(0.356849\pi\)
\(108\) 0 0
\(109\) −891.973 −0.783812 −0.391906 0.920005i \(-0.628184\pi\)
−0.391906 + 0.920005i \(0.628184\pi\)
\(110\) 1844.19 1.59852
\(111\) 0 0
\(112\) 1368.16 1.15427
\(113\) 263.839 0.219645 0.109823 0.993951i \(-0.464972\pi\)
0.109823 + 0.993951i \(0.464972\pi\)
\(114\) 0 0
\(115\) 826.995 0.670588
\(116\) −4759.68 −3.80970
\(117\) 0 0
\(118\) −434.673 −0.339109
\(119\) 795.901 0.613110
\(120\) 0 0
\(121\) −644.267 −0.484047
\(122\) −141.039 −0.104665
\(123\) 0 0
\(124\) −4238.57 −3.06964
\(125\) −1042.10 −0.745665
\(126\) 0 0
\(127\) −1543.97 −1.07878 −0.539391 0.842056i \(-0.681345\pi\)
−0.539391 + 0.842056i \(0.681345\pi\)
\(128\) 3115.30 2.15122
\(129\) 0 0
\(130\) 0 0
\(131\) −776.790 −0.518080 −0.259040 0.965867i \(-0.583406\pi\)
−0.259040 + 0.965867i \(0.583406\pi\)
\(132\) 0 0
\(133\) −704.602 −0.459374
\(134\) 5026.33 3.24036
\(135\) 0 0
\(136\) 8766.82 5.52756
\(137\) 2434.71 1.51833 0.759166 0.650897i \(-0.225607\pi\)
0.759166 + 0.650897i \(0.225607\pi\)
\(138\) 0 0
\(139\) 1251.40 0.763615 0.381807 0.924242i \(-0.375302\pi\)
0.381807 + 0.924242i \(0.375302\pi\)
\(140\) 1769.38 1.06814
\(141\) 0 0
\(142\) 2307.33 1.36357
\(143\) 0 0
\(144\) 0 0
\(145\) −2939.83 −1.68372
\(146\) −585.550 −0.331921
\(147\) 0 0
\(148\) −5331.88 −2.96134
\(149\) 783.619 0.430850 0.215425 0.976520i \(-0.430886\pi\)
0.215425 + 0.976520i \(0.430886\pi\)
\(150\) 0 0
\(151\) −162.220 −0.0874255 −0.0437128 0.999044i \(-0.513919\pi\)
−0.0437128 + 0.999044i \(0.513919\pi\)
\(152\) −7761.16 −4.14154
\(153\) 0 0
\(154\) 908.503 0.475385
\(155\) −2617.97 −1.35665
\(156\) 0 0
\(157\) −2355.76 −1.19752 −0.598758 0.800930i \(-0.704339\pi\)
−0.598758 + 0.800930i \(0.704339\pi\)
\(158\) 2075.20 1.04490
\(159\) 0 0
\(160\) 7601.78 3.75608
\(161\) 407.402 0.199427
\(162\) 0 0
\(163\) −1574.29 −0.756488 −0.378244 0.925706i \(-0.623472\pi\)
−0.378244 + 0.925706i \(0.623472\pi\)
\(164\) 4802.19 2.28651
\(165\) 0 0
\(166\) −463.916 −0.216909
\(167\) 3574.56 1.65633 0.828166 0.560483i \(-0.189384\pi\)
0.828166 + 0.560483i \(0.189384\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 8717.70 3.93304
\(171\) 0 0
\(172\) −8109.07 −3.59483
\(173\) −1.44108 −0.000633314 0 −0.000316657 1.00000i \(-0.500101\pi\)
−0.000316657 1.00000i \(0.500101\pi\)
\(174\) 0 0
\(175\) 289.749 0.125160
\(176\) 5580.34 2.38997
\(177\) 0 0
\(178\) −2675.84 −1.12676
\(179\) 2082.28 0.869478 0.434739 0.900556i \(-0.356841\pi\)
0.434739 + 0.900556i \(0.356841\pi\)
\(180\) 0 0
\(181\) −464.500 −0.190751 −0.0953756 0.995441i \(-0.530405\pi\)
−0.0953756 + 0.995441i \(0.530405\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 4487.51 1.79796
\(185\) −3293.26 −1.30878
\(186\) 0 0
\(187\) 3246.26 1.26947
\(188\) 731.965 0.283957
\(189\) 0 0
\(190\) −7717.68 −2.94684
\(191\) 4866.28 1.84352 0.921758 0.387765i \(-0.126753\pi\)
0.921758 + 0.387765i \(0.126753\pi\)
\(192\) 0 0
\(193\) 856.339 0.319381 0.159691 0.987167i \(-0.448950\pi\)
0.159691 + 0.987167i \(0.448950\pi\)
\(194\) −1029.84 −0.381127
\(195\) 0 0
\(196\) −6371.01 −2.32180
\(197\) −1571.93 −0.568504 −0.284252 0.958750i \(-0.591745\pi\)
−0.284252 + 0.958750i \(0.591745\pi\)
\(198\) 0 0
\(199\) 3126.61 1.11377 0.556883 0.830591i \(-0.311997\pi\)
0.556883 + 0.830591i \(0.311997\pi\)
\(200\) 3191.58 1.12839
\(201\) 0 0
\(202\) 4591.79 1.59939
\(203\) −1448.25 −0.500724
\(204\) 0 0
\(205\) 2966.09 1.01054
\(206\) −7065.90 −2.38983
\(207\) 0 0
\(208\) 0 0
\(209\) −2873.88 −0.951150
\(210\) 0 0
\(211\) 123.392 0.0402590 0.0201295 0.999797i \(-0.493592\pi\)
0.0201295 + 0.999797i \(0.493592\pi\)
\(212\) 1289.32 0.417694
\(213\) 0 0
\(214\) 5192.48 1.65865
\(215\) −5008.60 −1.58876
\(216\) 0 0
\(217\) −1289.69 −0.403455
\(218\) −4812.99 −1.49531
\(219\) 0 0
\(220\) 7216.84 2.21163
\(221\) 0 0
\(222\) 0 0
\(223\) 2503.69 0.751836 0.375918 0.926653i \(-0.377327\pi\)
0.375918 + 0.926653i \(0.377327\pi\)
\(224\) 3744.86 1.11703
\(225\) 0 0
\(226\) 1423.65 0.419025
\(227\) 579.270 0.169372 0.0846861 0.996408i \(-0.473011\pi\)
0.0846861 + 0.996408i \(0.473011\pi\)
\(228\) 0 0
\(229\) 768.922 0.221886 0.110943 0.993827i \(-0.464613\pi\)
0.110943 + 0.993827i \(0.464613\pi\)
\(230\) 4462.37 1.27930
\(231\) 0 0
\(232\) −15952.4 −4.51433
\(233\) −845.695 −0.237783 −0.118891 0.992907i \(-0.537934\pi\)
−0.118891 + 0.992907i \(0.537934\pi\)
\(234\) 0 0
\(235\) 452.101 0.125497
\(236\) −1701.00 −0.469175
\(237\) 0 0
\(238\) 4294.59 1.16965
\(239\) −6552.78 −1.77349 −0.886744 0.462260i \(-0.847039\pi\)
−0.886744 + 0.462260i \(0.847039\pi\)
\(240\) 0 0
\(241\) −5176.75 −1.38367 −0.691834 0.722056i \(-0.743197\pi\)
−0.691834 + 0.722056i \(0.743197\pi\)
\(242\) −3476.39 −0.923434
\(243\) 0 0
\(244\) −551.927 −0.144809
\(245\) −3935.08 −1.02613
\(246\) 0 0
\(247\) 0 0
\(248\) −14205.9 −3.63739
\(249\) 0 0
\(250\) −5623.05 −1.42253
\(251\) −4296.82 −1.08053 −0.540264 0.841495i \(-0.681676\pi\)
−0.540264 + 0.841495i \(0.681676\pi\)
\(252\) 0 0
\(253\) 1661.68 0.412921
\(254\) −8331.10 −2.05803
\(255\) 0 0
\(256\) 5277.77 1.28852
\(257\) 1383.94 0.335906 0.167953 0.985795i \(-0.446284\pi\)
0.167953 + 0.985795i \(0.446284\pi\)
\(258\) 0 0
\(259\) −1622.35 −0.389221
\(260\) 0 0
\(261\) 0 0
\(262\) −4191.47 −0.988359
\(263\) 2478.68 0.581147 0.290574 0.956853i \(-0.406154\pi\)
0.290574 + 0.956853i \(0.406154\pi\)
\(264\) 0 0
\(265\) 796.355 0.184603
\(266\) −3801.95 −0.876364
\(267\) 0 0
\(268\) 19669.4 4.48321
\(269\) 2452.64 0.555911 0.277956 0.960594i \(-0.410343\pi\)
0.277956 + 0.960594i \(0.410343\pi\)
\(270\) 0 0
\(271\) 5980.74 1.34060 0.670302 0.742088i \(-0.266164\pi\)
0.670302 + 0.742088i \(0.266164\pi\)
\(272\) 26378.9 5.88036
\(273\) 0 0
\(274\) 13137.4 2.89658
\(275\) 1181.81 0.259148
\(276\) 0 0
\(277\) −7377.99 −1.60036 −0.800182 0.599758i \(-0.795264\pi\)
−0.800182 + 0.599758i \(0.795264\pi\)
\(278\) 6752.42 1.45677
\(279\) 0 0
\(280\) 5930.22 1.26571
\(281\) −5937.17 −1.26043 −0.630217 0.776419i \(-0.717034\pi\)
−0.630217 + 0.776419i \(0.717034\pi\)
\(282\) 0 0
\(283\) 1486.52 0.312241 0.156121 0.987738i \(-0.450101\pi\)
0.156121 + 0.987738i \(0.450101\pi\)
\(284\) 9029.22 1.88657
\(285\) 0 0
\(286\) 0 0
\(287\) 1461.18 0.300526
\(288\) 0 0
\(289\) 10432.5 2.12344
\(290\) −15863.0 −3.21210
\(291\) 0 0
\(292\) −2291.42 −0.459231
\(293\) 3580.05 0.713818 0.356909 0.934139i \(-0.383831\pi\)
0.356909 + 0.934139i \(0.383831\pi\)
\(294\) 0 0
\(295\) −1050.63 −0.207355
\(296\) −17870.2 −3.50906
\(297\) 0 0
\(298\) 4228.32 0.821947
\(299\) 0 0
\(300\) 0 0
\(301\) −2467.38 −0.472483
\(302\) −875.320 −0.166785
\(303\) 0 0
\(304\) −23352.9 −4.40587
\(305\) −340.900 −0.0639995
\(306\) 0 0
\(307\) 415.013 0.0771533 0.0385767 0.999256i \(-0.487718\pi\)
0.0385767 + 0.999256i \(0.487718\pi\)
\(308\) 3555.23 0.657720
\(309\) 0 0
\(310\) −14126.3 −2.58812
\(311\) 3009.05 0.548642 0.274321 0.961638i \(-0.411547\pi\)
0.274321 + 0.961638i \(0.411547\pi\)
\(312\) 0 0
\(313\) 3760.84 0.679154 0.339577 0.940578i \(-0.389716\pi\)
0.339577 + 0.940578i \(0.389716\pi\)
\(314\) −12711.4 −2.28454
\(315\) 0 0
\(316\) 8120.86 1.44568
\(317\) −2772.04 −0.491146 −0.245573 0.969378i \(-0.578976\pi\)
−0.245573 + 0.969378i \(0.578976\pi\)
\(318\) 0 0
\(319\) −5907.01 −1.03677
\(320\) 18800.3 3.28428
\(321\) 0 0
\(322\) 2198.29 0.380454
\(323\) −13585.2 −2.34024
\(324\) 0 0
\(325\) 0 0
\(326\) −8494.67 −1.44318
\(327\) 0 0
\(328\) 16094.9 2.70942
\(329\) 222.718 0.0373217
\(330\) 0 0
\(331\) −6558.16 −1.08903 −0.544515 0.838751i \(-0.683286\pi\)
−0.544515 + 0.838751i \(0.683286\pi\)
\(332\) −1815.43 −0.300105
\(333\) 0 0
\(334\) 19287.9 3.15984
\(335\) 12148.9 1.98139
\(336\) 0 0
\(337\) 9509.17 1.53708 0.768542 0.639799i \(-0.220983\pi\)
0.768542 + 0.639799i \(0.220983\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 34114.8 5.44158
\(341\) −5260.29 −0.835368
\(342\) 0 0
\(343\) −4142.29 −0.652077
\(344\) −27178.1 −4.25973
\(345\) 0 0
\(346\) −7.77591 −0.00120820
\(347\) 9808.78 1.51747 0.758736 0.651398i \(-0.225817\pi\)
0.758736 + 0.651398i \(0.225817\pi\)
\(348\) 0 0
\(349\) −794.952 −0.121928 −0.0609639 0.998140i \(-0.519417\pi\)
−0.0609639 + 0.998140i \(0.519417\pi\)
\(350\) 1563.46 0.238772
\(351\) 0 0
\(352\) 15274.3 2.31284
\(353\) −7115.60 −1.07288 −0.536438 0.843940i \(-0.680230\pi\)
−0.536438 + 0.843940i \(0.680230\pi\)
\(354\) 0 0
\(355\) 5576.93 0.833782
\(356\) −10471.3 −1.55893
\(357\) 0 0
\(358\) 11235.7 1.65873
\(359\) −6907.19 −1.01545 −0.507726 0.861518i \(-0.669514\pi\)
−0.507726 + 0.861518i \(0.669514\pi\)
\(360\) 0 0
\(361\) 5167.78 0.753430
\(362\) −2506.39 −0.363903
\(363\) 0 0
\(364\) 0 0
\(365\) −1415.31 −0.202960
\(366\) 0 0
\(367\) −459.533 −0.0653609 −0.0326804 0.999466i \(-0.510404\pi\)
−0.0326804 + 0.999466i \(0.510404\pi\)
\(368\) 13502.7 1.91271
\(369\) 0 0
\(370\) −17770.1 −2.49681
\(371\) 392.308 0.0548992
\(372\) 0 0
\(373\) 5359.92 0.744038 0.372019 0.928225i \(-0.378666\pi\)
0.372019 + 0.928225i \(0.378666\pi\)
\(374\) 17516.5 2.42181
\(375\) 0 0
\(376\) 2453.23 0.336478
\(377\) 0 0
\(378\) 0 0
\(379\) −8594.06 −1.16477 −0.582384 0.812914i \(-0.697880\pi\)
−0.582384 + 0.812914i \(0.697880\pi\)
\(380\) −30201.4 −4.07711
\(381\) 0 0
\(382\) 26257.9 3.51694
\(383\) 9336.49 1.24562 0.622810 0.782373i \(-0.285991\pi\)
0.622810 + 0.782373i \(0.285991\pi\)
\(384\) 0 0
\(385\) 2195.90 0.290684
\(386\) 4620.71 0.609295
\(387\) 0 0
\(388\) −4030.07 −0.527309
\(389\) −12792.1 −1.66732 −0.833659 0.552279i \(-0.813758\pi\)
−0.833659 + 0.552279i \(0.813758\pi\)
\(390\) 0 0
\(391\) 7854.95 1.01596
\(392\) −21352.9 −2.75123
\(393\) 0 0
\(394\) −8481.96 −1.08456
\(395\) 5015.88 0.638927
\(396\) 0 0
\(397\) 13384.5 1.69206 0.846030 0.533135i \(-0.178986\pi\)
0.846030 + 0.533135i \(0.178986\pi\)
\(398\) 16870.8 2.12477
\(399\) 0 0
\(400\) 9603.30 1.20041
\(401\) −6357.41 −0.791706 −0.395853 0.918314i \(-0.629551\pi\)
−0.395853 + 0.918314i \(0.629551\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 17969.0 2.21285
\(405\) 0 0
\(406\) −7814.58 −0.955249
\(407\) −6617.14 −0.805896
\(408\) 0 0
\(409\) 5301.63 0.640950 0.320475 0.947257i \(-0.396157\pi\)
0.320475 + 0.947257i \(0.396157\pi\)
\(410\) 16004.7 1.92784
\(411\) 0 0
\(412\) −27650.8 −3.30645
\(413\) −517.569 −0.0616656
\(414\) 0 0
\(415\) −1121.31 −0.132633
\(416\) 0 0
\(417\) 0 0
\(418\) −15507.1 −1.81454
\(419\) −16395.8 −1.91167 −0.955835 0.293905i \(-0.905045\pi\)
−0.955835 + 0.293905i \(0.905045\pi\)
\(420\) 0 0
\(421\) −8484.68 −0.982227 −0.491114 0.871095i \(-0.663410\pi\)
−0.491114 + 0.871095i \(0.663410\pi\)
\(422\) 665.808 0.0768034
\(423\) 0 0
\(424\) 4321.26 0.494950
\(425\) 5586.55 0.637617
\(426\) 0 0
\(427\) −167.937 −0.0190329
\(428\) 20319.6 2.29483
\(429\) 0 0
\(430\) −27025.8 −3.03094
\(431\) 1594.74 0.178227 0.0891136 0.996021i \(-0.471597\pi\)
0.0891136 + 0.996021i \(0.471597\pi\)
\(432\) 0 0
\(433\) −3387.57 −0.375973 −0.187987 0.982172i \(-0.560196\pi\)
−0.187987 + 0.982172i \(0.560196\pi\)
\(434\) −6959.02 −0.769685
\(435\) 0 0
\(436\) −18834.6 −2.06884
\(437\) −6953.90 −0.761213
\(438\) 0 0
\(439\) −7211.54 −0.784027 −0.392014 0.919959i \(-0.628221\pi\)
−0.392014 + 0.919959i \(0.628221\pi\)
\(440\) 24187.7 2.62069
\(441\) 0 0
\(442\) 0 0
\(443\) 7500.66 0.804441 0.402220 0.915543i \(-0.368239\pi\)
0.402220 + 0.915543i \(0.368239\pi\)
\(444\) 0 0
\(445\) −6467.65 −0.688979
\(446\) 13509.6 1.43430
\(447\) 0 0
\(448\) 9261.58 0.976715
\(449\) −7649.53 −0.804017 −0.402009 0.915636i \(-0.631688\pi\)
−0.402009 + 0.915636i \(0.631688\pi\)
\(450\) 0 0
\(451\) 5959.77 0.622249
\(452\) 5571.13 0.579743
\(453\) 0 0
\(454\) 3125.67 0.323117
\(455\) 0 0
\(456\) 0 0
\(457\) 9001.79 0.921414 0.460707 0.887552i \(-0.347596\pi\)
0.460707 + 0.887552i \(0.347596\pi\)
\(458\) 4149.02 0.423299
\(459\) 0 0
\(460\) 17462.5 1.76999
\(461\) −1347.83 −0.136171 −0.0680854 0.997679i \(-0.521689\pi\)
−0.0680854 + 0.997679i \(0.521689\pi\)
\(462\) 0 0
\(463\) 94.9035 0.00952600 0.00476300 0.999989i \(-0.498484\pi\)
0.00476300 + 0.999989i \(0.498484\pi\)
\(464\) −47999.9 −4.80246
\(465\) 0 0
\(466\) −4563.28 −0.453626
\(467\) −13290.9 −1.31698 −0.658489 0.752591i \(-0.728804\pi\)
−0.658489 + 0.752591i \(0.728804\pi\)
\(468\) 0 0
\(469\) 5984.90 0.589247
\(470\) 2439.49 0.239415
\(471\) 0 0
\(472\) −5701.00 −0.555953
\(473\) −10063.8 −0.978294
\(474\) 0 0
\(475\) −4945.70 −0.477736
\(476\) 16806.0 1.61828
\(477\) 0 0
\(478\) −35358.1 −3.38335
\(479\) −3801.59 −0.362628 −0.181314 0.983425i \(-0.558035\pi\)
−0.181314 + 0.983425i \(0.558035\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −27933.2 −2.63967
\(483\) 0 0
\(484\) −13604.1 −1.27762
\(485\) −2489.19 −0.233048
\(486\) 0 0
\(487\) 8747.34 0.813922 0.406961 0.913446i \(-0.366588\pi\)
0.406961 + 0.913446i \(0.366588\pi\)
\(488\) −1849.82 −0.171593
\(489\) 0 0
\(490\) −21233.2 −1.95759
\(491\) −15085.4 −1.38655 −0.693273 0.720675i \(-0.743832\pi\)
−0.693273 + 0.720675i \(0.743832\pi\)
\(492\) 0 0
\(493\) −27923.1 −2.55090
\(494\) 0 0
\(495\) 0 0
\(496\) −42744.7 −3.86955
\(497\) 2747.36 0.247959
\(498\) 0 0
\(499\) −14593.9 −1.30925 −0.654623 0.755955i \(-0.727173\pi\)
−0.654623 + 0.755955i \(0.727173\pi\)
\(500\) −22004.6 −1.96815
\(501\) 0 0
\(502\) −23185.2 −2.06136
\(503\) 18204.9 1.61375 0.806876 0.590721i \(-0.201157\pi\)
0.806876 + 0.590721i \(0.201157\pi\)
\(504\) 0 0
\(505\) 11098.6 0.977983
\(506\) 8966.25 0.787744
\(507\) 0 0
\(508\) −32601.9 −2.84739
\(509\) −7891.92 −0.687236 −0.343618 0.939109i \(-0.611653\pi\)
−0.343618 + 0.939109i \(0.611653\pi\)
\(510\) 0 0
\(511\) −697.221 −0.0603586
\(512\) 3555.88 0.306932
\(513\) 0 0
\(514\) 7467.59 0.640819
\(515\) −17078.6 −1.46131
\(516\) 0 0
\(517\) 908.406 0.0772759
\(518\) −8754.05 −0.742530
\(519\) 0 0
\(520\) 0 0
\(521\) −647.902 −0.0544820 −0.0272410 0.999629i \(-0.508672\pi\)
−0.0272410 + 0.999629i \(0.508672\pi\)
\(522\) 0 0
\(523\) 9269.99 0.775044 0.387522 0.921860i \(-0.373331\pi\)
0.387522 + 0.921860i \(0.373331\pi\)
\(524\) −16402.4 −1.36745
\(525\) 0 0
\(526\) 13374.7 1.10867
\(527\) −24866.0 −2.05537
\(528\) 0 0
\(529\) −8146.25 −0.669536
\(530\) 4297.05 0.352173
\(531\) 0 0
\(532\) −14878.1 −1.21250
\(533\) 0 0
\(534\) 0 0
\(535\) 12550.5 1.01421
\(536\) 65923.4 5.31242
\(537\) 0 0
\(538\) 13234.2 1.06053
\(539\) −7906.75 −0.631852
\(540\) 0 0
\(541\) −3844.83 −0.305549 −0.152775 0.988261i \(-0.548821\pi\)
−0.152775 + 0.988261i \(0.548821\pi\)
\(542\) 32271.4 2.55752
\(543\) 0 0
\(544\) 72203.1 5.69060
\(545\) −11633.2 −0.914337
\(546\) 0 0
\(547\) −20245.4 −1.58251 −0.791253 0.611489i \(-0.790571\pi\)
−0.791253 + 0.611489i \(0.790571\pi\)
\(548\) 51410.5 4.00757
\(549\) 0 0
\(550\) 6376.91 0.494386
\(551\) 24720.0 1.91126
\(552\) 0 0
\(553\) 2470.97 0.190011
\(554\) −39810.8 −3.05307
\(555\) 0 0
\(556\) 26424.1 2.01553
\(557\) −7223.34 −0.549485 −0.274742 0.961518i \(-0.588593\pi\)
−0.274742 + 0.961518i \(0.588593\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 17843.7 1.34649
\(561\) 0 0
\(562\) −32036.3 −2.40458
\(563\) −17290.3 −1.29431 −0.647156 0.762358i \(-0.724042\pi\)
−0.647156 + 0.762358i \(0.724042\pi\)
\(564\) 0 0
\(565\) 3441.03 0.256221
\(566\) 8021.09 0.595674
\(567\) 0 0
\(568\) 30262.0 2.23551
\(569\) 17364.0 1.27933 0.639663 0.768655i \(-0.279074\pi\)
0.639663 + 0.768655i \(0.279074\pi\)
\(570\) 0 0
\(571\) 158.149 0.0115908 0.00579540 0.999983i \(-0.498155\pi\)
0.00579540 + 0.999983i \(0.498155\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 7884.38 0.573323
\(575\) 2859.61 0.207398
\(576\) 0 0
\(577\) −9627.04 −0.694591 −0.347295 0.937756i \(-0.612900\pi\)
−0.347295 + 0.937756i \(0.612900\pi\)
\(578\) 56292.4 4.05096
\(579\) 0 0
\(580\) −62076.4 −4.44411
\(581\) −552.389 −0.0394440
\(582\) 0 0
\(583\) 1600.12 0.113671
\(584\) −7679.86 −0.544169
\(585\) 0 0
\(586\) 19317.6 1.36178
\(587\) 13321.2 0.936668 0.468334 0.883551i \(-0.344854\pi\)
0.468334 + 0.883551i \(0.344854\pi\)
\(588\) 0 0
\(589\) 22013.6 1.53999
\(590\) −5669.06 −0.395579
\(591\) 0 0
\(592\) −53770.4 −3.73303
\(593\) −16723.4 −1.15809 −0.579044 0.815296i \(-0.696574\pi\)
−0.579044 + 0.815296i \(0.696574\pi\)
\(594\) 0 0
\(595\) 10380.3 0.715209
\(596\) 16546.6 1.13721
\(597\) 0 0
\(598\) 0 0
\(599\) 17160.0 1.17051 0.585256 0.810849i \(-0.300994\pi\)
0.585256 + 0.810849i \(0.300994\pi\)
\(600\) 0 0
\(601\) 11048.6 0.749885 0.374942 0.927048i \(-0.377663\pi\)
0.374942 + 0.927048i \(0.377663\pi\)
\(602\) −13313.7 −0.901373
\(603\) 0 0
\(604\) −3425.37 −0.230756
\(605\) −8402.62 −0.564653
\(606\) 0 0
\(607\) 21119.6 1.41222 0.706110 0.708102i \(-0.250448\pi\)
0.706110 + 0.708102i \(0.250448\pi\)
\(608\) −63920.6 −4.26369
\(609\) 0 0
\(610\) −1839.46 −0.122094
\(611\) 0 0
\(612\) 0 0
\(613\) 6233.83 0.410737 0.205369 0.978685i \(-0.434161\pi\)
0.205369 + 0.978685i \(0.434161\pi\)
\(614\) 2239.37 0.147188
\(615\) 0 0
\(616\) 11915.6 0.779371
\(617\) 5598.36 0.365286 0.182643 0.983179i \(-0.441535\pi\)
0.182643 + 0.983179i \(0.441535\pi\)
\(618\) 0 0
\(619\) 15874.3 1.03076 0.515382 0.856961i \(-0.327650\pi\)
0.515382 + 0.856961i \(0.327650\pi\)
\(620\) −55280.1 −3.58081
\(621\) 0 0
\(622\) 16236.5 1.04666
\(623\) −3186.15 −0.204896
\(624\) 0 0
\(625\) −19228.4 −1.23062
\(626\) 20293.1 1.29565
\(627\) 0 0
\(628\) −49743.3 −3.16079
\(629\) −31280.0 −1.98285
\(630\) 0 0
\(631\) −11249.0 −0.709693 −0.354846 0.934925i \(-0.615467\pi\)
−0.354846 + 0.934925i \(0.615467\pi\)
\(632\) 27217.6 1.71307
\(633\) 0 0
\(634\) −14957.6 −0.936976
\(635\) −20136.7 −1.25843
\(636\) 0 0
\(637\) 0 0
\(638\) −31873.6 −1.97788
\(639\) 0 0
\(640\) 40630.2 2.50945
\(641\) 23768.3 1.46457 0.732287 0.680996i \(-0.238453\pi\)
0.732287 + 0.680996i \(0.238453\pi\)
\(642\) 0 0
\(643\) 4368.95 0.267954 0.133977 0.990984i \(-0.457225\pi\)
0.133977 + 0.990984i \(0.457225\pi\)
\(644\) 8602.54 0.526378
\(645\) 0 0
\(646\) −73304.0 −4.46456
\(647\) 21305.2 1.29458 0.647290 0.762244i \(-0.275902\pi\)
0.647290 + 0.762244i \(0.275902\pi\)
\(648\) 0 0
\(649\) −2111.02 −0.127681
\(650\) 0 0
\(651\) 0 0
\(652\) −33242.0 −1.99672
\(653\) 27958.3 1.67549 0.837744 0.546064i \(-0.183874\pi\)
0.837744 + 0.546064i \(0.183874\pi\)
\(654\) 0 0
\(655\) −10131.0 −0.604353
\(656\) 48428.6 2.88235
\(657\) 0 0
\(658\) 1201.76 0.0711999
\(659\) −9619.46 −0.568621 −0.284310 0.958732i \(-0.591765\pi\)
−0.284310 + 0.958732i \(0.591765\pi\)
\(660\) 0 0
\(661\) −4414.40 −0.259759 −0.129879 0.991530i \(-0.541459\pi\)
−0.129879 + 0.991530i \(0.541459\pi\)
\(662\) −35387.1 −2.07758
\(663\) 0 0
\(664\) −6084.55 −0.355612
\(665\) −9189.52 −0.535871
\(666\) 0 0
\(667\) −14293.1 −0.829733
\(668\) 75479.0 4.37181
\(669\) 0 0
\(670\) 65554.1 3.77996
\(671\) −684.970 −0.0394083
\(672\) 0 0
\(673\) −18001.8 −1.03108 −0.515540 0.856865i \(-0.672409\pi\)
−0.515540 + 0.856865i \(0.672409\pi\)
\(674\) 51310.4 2.93235
\(675\) 0 0
\(676\) 0 0
\(677\) 11192.0 0.635369 0.317684 0.948197i \(-0.397095\pi\)
0.317684 + 0.948197i \(0.397095\pi\)
\(678\) 0 0
\(679\) −1226.25 −0.0693064
\(680\) 114338. 6.44804
\(681\) 0 0
\(682\) −28383.9 −1.59366
\(683\) 35122.3 1.96767 0.983835 0.179076i \(-0.0573109\pi\)
0.983835 + 0.179076i \(0.0573109\pi\)
\(684\) 0 0
\(685\) 31753.9 1.77117
\(686\) −22351.3 −1.24399
\(687\) 0 0
\(688\) −81777.5 −4.53160
\(689\) 0 0
\(690\) 0 0
\(691\) −29534.6 −1.62597 −0.812987 0.582282i \(-0.802160\pi\)
−0.812987 + 0.582282i \(0.802160\pi\)
\(692\) −30.4293 −0.00167160
\(693\) 0 0
\(694\) 52927.1 2.89494
\(695\) 16321.0 0.890776
\(696\) 0 0
\(697\) 28172.5 1.53100
\(698\) −4289.47 −0.232606
\(699\) 0 0
\(700\) 6118.24 0.330354
\(701\) −8804.19 −0.474365 −0.237182 0.971465i \(-0.576224\pi\)
−0.237182 + 0.971465i \(0.576224\pi\)
\(702\) 0 0
\(703\) 27691.8 1.48566
\(704\) 37775.5 2.02232
\(705\) 0 0
\(706\) −38395.0 −2.04676
\(707\) 5467.49 0.290843
\(708\) 0 0
\(709\) −27138.2 −1.43752 −0.718758 0.695261i \(-0.755289\pi\)
−0.718758 + 0.695261i \(0.755289\pi\)
\(710\) 30092.5 1.59064
\(711\) 0 0
\(712\) −35095.3 −1.84727
\(713\) −12728.3 −0.668552
\(714\) 0 0
\(715\) 0 0
\(716\) 43968.6 2.29495
\(717\) 0 0
\(718\) −37270.4 −1.93721
\(719\) −17337.4 −0.899274 −0.449637 0.893211i \(-0.648447\pi\)
−0.449637 + 0.893211i \(0.648447\pi\)
\(720\) 0 0
\(721\) −8413.44 −0.434581
\(722\) 27884.8 1.43735
\(723\) 0 0
\(724\) −9808.20 −0.503479
\(725\) −10165.5 −0.520739
\(726\) 0 0
\(727\) 25771.3 1.31473 0.657363 0.753574i \(-0.271672\pi\)
0.657363 + 0.753574i \(0.271672\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −7636.83 −0.387194
\(731\) −47572.6 −2.40703
\(732\) 0 0
\(733\) −21784.2 −1.09771 −0.548853 0.835919i \(-0.684935\pi\)
−0.548853 + 0.835919i \(0.684935\pi\)
\(734\) −2479.59 −0.124691
\(735\) 0 0
\(736\) 36959.0 1.85099
\(737\) 24410.8 1.22006
\(738\) 0 0
\(739\) −22353.8 −1.11272 −0.556360 0.830942i \(-0.687802\pi\)
−0.556360 + 0.830942i \(0.687802\pi\)
\(740\) −69539.2 −3.45448
\(741\) 0 0
\(742\) 2116.85 0.104733
\(743\) −18382.6 −0.907662 −0.453831 0.891088i \(-0.649943\pi\)
−0.453831 + 0.891088i \(0.649943\pi\)
\(744\) 0 0
\(745\) 10220.1 0.502597
\(746\) 28921.5 1.41943
\(747\) 0 0
\(748\) 68546.9 3.35070
\(749\) 6182.74 0.301618
\(750\) 0 0
\(751\) 13782.8 0.669694 0.334847 0.942272i \(-0.391315\pi\)
0.334847 + 0.942272i \(0.391315\pi\)
\(752\) 7381.64 0.357953
\(753\) 0 0
\(754\) 0 0
\(755\) −2115.69 −0.101984
\(756\) 0 0
\(757\) 25639.5 1.23102 0.615510 0.788129i \(-0.288950\pi\)
0.615510 + 0.788129i \(0.288950\pi\)
\(758\) −46372.6 −2.22207
\(759\) 0 0
\(760\) −101222. −4.83121
\(761\) 43.4757 0.00207095 0.00103548 0.999999i \(-0.499670\pi\)
0.00103548 + 0.999999i \(0.499670\pi\)
\(762\) 0 0
\(763\) −5730.88 −0.271916
\(764\) 102755. 4.86588
\(765\) 0 0
\(766\) 50378.7 2.37631
\(767\) 0 0
\(768\) 0 0
\(769\) 20485.7 0.960641 0.480321 0.877093i \(-0.340520\pi\)
0.480321 + 0.877093i \(0.340520\pi\)
\(770\) 11848.8 0.554548
\(771\) 0 0
\(772\) 18082.1 0.842992
\(773\) 29450.3 1.37031 0.685157 0.728395i \(-0.259734\pi\)
0.685157 + 0.728395i \(0.259734\pi\)
\(774\) 0 0
\(775\) −9052.51 −0.419582
\(776\) −13507.1 −0.624839
\(777\) 0 0
\(778\) −69025.0 −3.18080
\(779\) −24940.8 −1.14711
\(780\) 0 0
\(781\) 11205.7 0.513409
\(782\) 42384.5 1.93819
\(783\) 0 0
\(784\) −64249.7 −2.92683
\(785\) −30724.1 −1.39693
\(786\) 0 0
\(787\) −9751.10 −0.441664 −0.220832 0.975312i \(-0.570877\pi\)
−0.220832 + 0.975312i \(0.570877\pi\)
\(788\) −33192.3 −1.50054
\(789\) 0 0
\(790\) 27065.1 1.21890
\(791\) 1695.15 0.0761980
\(792\) 0 0
\(793\) 0 0
\(794\) 72221.2 3.22800
\(795\) 0 0
\(796\) 66020.3 2.93973
\(797\) 37750.8 1.67779 0.838896 0.544291i \(-0.183201\pi\)
0.838896 + 0.544291i \(0.183201\pi\)
\(798\) 0 0
\(799\) 4294.14 0.190132
\(800\) 26285.7 1.16167
\(801\) 0 0
\(802\) −34303.9 −1.51037
\(803\) −2843.77 −0.124975
\(804\) 0 0
\(805\) 5313.39 0.232637
\(806\) 0 0
\(807\) 0 0
\(808\) 60224.2 2.62213
\(809\) 6999.45 0.304187 0.152094 0.988366i \(-0.451398\pi\)
0.152094 + 0.988366i \(0.451398\pi\)
\(810\) 0 0
\(811\) 35792.1 1.54973 0.774864 0.632128i \(-0.217818\pi\)
0.774864 + 0.632128i \(0.217818\pi\)
\(812\) −30580.6 −1.32164
\(813\) 0 0
\(814\) −35705.4 −1.53744
\(815\) −20532.1 −0.882463
\(816\) 0 0
\(817\) 42115.5 1.80347
\(818\) 28607.0 1.22276
\(819\) 0 0
\(820\) 62630.9 2.66727
\(821\) 9776.62 0.415599 0.207799 0.978171i \(-0.433370\pi\)
0.207799 + 0.978171i \(0.433370\pi\)
\(822\) 0 0
\(823\) 19662.0 0.832777 0.416389 0.909187i \(-0.363296\pi\)
0.416389 + 0.909187i \(0.363296\pi\)
\(824\) −92673.7 −3.91801
\(825\) 0 0
\(826\) −2792.75 −0.117642
\(827\) 5281.22 0.222063 0.111032 0.993817i \(-0.464585\pi\)
0.111032 + 0.993817i \(0.464585\pi\)
\(828\) 0 0
\(829\) 21735.5 0.910621 0.455310 0.890333i \(-0.349528\pi\)
0.455310 + 0.890333i \(0.349528\pi\)
\(830\) −6050.46 −0.253030
\(831\) 0 0
\(832\) 0 0
\(833\) −37376.1 −1.55463
\(834\) 0 0
\(835\) 46619.9 1.93215
\(836\) −60683.8 −2.51052
\(837\) 0 0
\(838\) −88470.2 −3.64696
\(839\) −28574.7 −1.17582 −0.587908 0.808928i \(-0.700048\pi\)
−0.587908 + 0.808928i \(0.700048\pi\)
\(840\) 0 0
\(841\) 26420.7 1.08330
\(842\) −45782.4 −1.87383
\(843\) 0 0
\(844\) 2605.49 0.106262
\(845\) 0 0
\(846\) 0 0
\(847\) −4139.38 −0.167923
\(848\) 13002.4 0.526540
\(849\) 0 0
\(850\) 30144.4 1.21640
\(851\) −16011.4 −0.644965
\(852\) 0 0
\(853\) −5251.39 −0.210791 −0.105395 0.994430i \(-0.533611\pi\)
−0.105395 + 0.994430i \(0.533611\pi\)
\(854\) −906.170 −0.0363097
\(855\) 0 0
\(856\) 68102.6 2.71927
\(857\) 34851.0 1.38913 0.694566 0.719429i \(-0.255596\pi\)
0.694566 + 0.719429i \(0.255596\pi\)
\(858\) 0 0
\(859\) 41697.5 1.65623 0.828114 0.560559i \(-0.189414\pi\)
0.828114 + 0.560559i \(0.189414\pi\)
\(860\) −105760. −4.19346
\(861\) 0 0
\(862\) 8605.05 0.340010
\(863\) 28648.9 1.13004 0.565018 0.825079i \(-0.308869\pi\)
0.565018 + 0.825079i \(0.308869\pi\)
\(864\) 0 0
\(865\) −18.7948 −0.000738777 0
\(866\) −18279.0 −0.717257
\(867\) 0 0
\(868\) −27232.6 −1.06490
\(869\) 10078.4 0.393425
\(870\) 0 0
\(871\) 0 0
\(872\) −63125.4 −2.45149
\(873\) 0 0
\(874\) −37522.5 −1.45219
\(875\) −6695.42 −0.258682
\(876\) 0 0
\(877\) 31516.8 1.21351 0.606754 0.794890i \(-0.292471\pi\)
0.606754 + 0.794890i \(0.292471\pi\)
\(878\) −38912.7 −1.49572
\(879\) 0 0
\(880\) 72779.7 2.78796
\(881\) −31246.1 −1.19490 −0.597451 0.801905i \(-0.703820\pi\)
−0.597451 + 0.801905i \(0.703820\pi\)
\(882\) 0 0
\(883\) −16181.8 −0.616718 −0.308359 0.951270i \(-0.599780\pi\)
−0.308359 + 0.951270i \(0.599780\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 40472.8 1.53466
\(887\) 13289.0 0.503046 0.251523 0.967851i \(-0.419069\pi\)
0.251523 + 0.967851i \(0.419069\pi\)
\(888\) 0 0
\(889\) −9919.92 −0.374245
\(890\) −34898.7 −1.31439
\(891\) 0 0
\(892\) 52867.0 1.98444
\(893\) −3801.55 −0.142457
\(894\) 0 0
\(895\) 27157.4 1.01427
\(896\) 20015.6 0.746288
\(897\) 0 0
\(898\) −41276.0 −1.53385
\(899\) 45246.9 1.67861
\(900\) 0 0
\(901\) 7563.94 0.279679
\(902\) 32158.2 1.18709
\(903\) 0 0
\(904\) 18672.0 0.686971
\(905\) −6058.07 −0.222516
\(906\) 0 0
\(907\) 47128.4 1.72533 0.862664 0.505778i \(-0.168795\pi\)
0.862664 + 0.505778i \(0.168795\pi\)
\(908\) 12231.6 0.447050
\(909\) 0 0
\(910\) 0 0
\(911\) 2884.69 0.104911 0.0524556 0.998623i \(-0.483295\pi\)
0.0524556 + 0.998623i \(0.483295\pi\)
\(912\) 0 0
\(913\) −2253.05 −0.0816703
\(914\) 48572.7 1.75781
\(915\) 0 0
\(916\) 16236.3 0.585657
\(917\) −4990.83 −0.179729
\(918\) 0 0
\(919\) −25752.2 −0.924359 −0.462179 0.886787i \(-0.652932\pi\)
−0.462179 + 0.886787i \(0.652932\pi\)
\(920\) 58526.8 2.09736
\(921\) 0 0
\(922\) −7272.75 −0.259778
\(923\) 0 0
\(924\) 0 0
\(925\) −11387.5 −0.404779
\(926\) 512.089 0.0181731
\(927\) 0 0
\(928\) −131383. −4.64748
\(929\) −51394.7 −1.81508 −0.907539 0.419969i \(-0.862041\pi\)
−0.907539 + 0.419969i \(0.862041\pi\)
\(930\) 0 0
\(931\) 33088.6 1.16481
\(932\) −17857.4 −0.627616
\(933\) 0 0
\(934\) −71716.1 −2.51244
\(935\) 42338.3 1.48087
\(936\) 0 0
\(937\) 6781.86 0.236450 0.118225 0.992987i \(-0.462280\pi\)
0.118225 + 0.992987i \(0.462280\pi\)
\(938\) 32293.9 1.12413
\(939\) 0 0
\(940\) 9546.39 0.331244
\(941\) 22038.2 0.763470 0.381735 0.924272i \(-0.375327\pi\)
0.381735 + 0.924272i \(0.375327\pi\)
\(942\) 0 0
\(943\) 14420.8 0.497991
\(944\) −17154.0 −0.591436
\(945\) 0 0
\(946\) −54303.0 −1.86633
\(947\) 45531.0 1.56236 0.781182 0.624304i \(-0.214617\pi\)
0.781182 + 0.624304i \(0.214617\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −26686.5 −0.911393
\(951\) 0 0
\(952\) 56326.3 1.91759
\(953\) −26541.2 −0.902154 −0.451077 0.892485i \(-0.648960\pi\)
−0.451077 + 0.892485i \(0.648960\pi\)
\(954\) 0 0
\(955\) 63466.7 2.15051
\(956\) −138366. −4.68104
\(957\) 0 0
\(958\) −20513.0 −0.691799
\(959\) 15642.9 0.526731
\(960\) 0 0
\(961\) 10502.1 0.352527
\(962\) 0 0
\(963\) 0 0
\(964\) −109310. −3.65213
\(965\) 11168.5 0.372566
\(966\) 0 0
\(967\) −48269.6 −1.60522 −0.802609 0.596506i \(-0.796555\pi\)
−0.802609 + 0.596506i \(0.796555\pi\)
\(968\) −45595.1 −1.51393
\(969\) 0 0
\(970\) −13431.4 −0.444594
\(971\) 34161.3 1.12903 0.564515 0.825423i \(-0.309063\pi\)
0.564515 + 0.825423i \(0.309063\pi\)
\(972\) 0 0
\(973\) 8040.18 0.264909
\(974\) 47199.7 1.55275
\(975\) 0 0
\(976\) −5566.02 −0.182545
\(977\) −24124.2 −0.789971 −0.394985 0.918687i \(-0.629250\pi\)
−0.394985 + 0.918687i \(0.629250\pi\)
\(978\) 0 0
\(979\) −12995.4 −0.424245
\(980\) −83091.7 −2.70843
\(981\) 0 0
\(982\) −81399.1 −2.64516
\(983\) 19023.4 0.617246 0.308623 0.951185i \(-0.400132\pi\)
0.308623 + 0.951185i \(0.400132\pi\)
\(984\) 0 0
\(985\) −20501.3 −0.663175
\(986\) −150670. −4.86644
\(987\) 0 0
\(988\) 0 0
\(989\) −24351.2 −0.782936
\(990\) 0 0
\(991\) −229.962 −0.00737131 −0.00368566 0.999993i \(-0.501173\pi\)
−0.00368566 + 0.999993i \(0.501173\pi\)
\(992\) −116999. −3.74468
\(993\) 0 0
\(994\) 14824.4 0.473041
\(995\) 40777.7 1.29924
\(996\) 0 0
\(997\) 26991.1 0.857389 0.428694 0.903450i \(-0.358974\pi\)
0.428694 + 0.903450i \(0.358974\pi\)
\(998\) −78747.2 −2.49769
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1521.4.a.bd.1.8 8
3.2 odd 2 inner 1521.4.a.bd.1.1 8
13.4 even 6 117.4.g.f.55.8 yes 16
13.10 even 6 117.4.g.f.100.8 yes 16
13.12 even 2 1521.4.a.bc.1.1 8
39.17 odd 6 117.4.g.f.55.1 16
39.23 odd 6 117.4.g.f.100.1 yes 16
39.38 odd 2 1521.4.a.bc.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.g.f.55.1 16 39.17 odd 6
117.4.g.f.55.8 yes 16 13.4 even 6
117.4.g.f.100.1 yes 16 39.23 odd 6
117.4.g.f.100.8 yes 16 13.10 even 6
1521.4.a.bc.1.1 8 13.12 even 2
1521.4.a.bc.1.8 8 39.38 odd 2
1521.4.a.bd.1.1 8 3.2 odd 2 inner
1521.4.a.bd.1.8 8 1.1 even 1 trivial