Properties

Label 1176.4.a.be.1.3
Level $1176$
Weight $4$
Character 1176.1
Self dual yes
Analytic conductor $69.386$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1176.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.3862461668\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-4.92368\) of defining polynomial
Character \(\chi\) \(=\) 1176.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +14.5121 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +14.5121 q^{5} +9.00000 q^{9} +10.1116 q^{11} -0.623659 q^{13} +43.5362 q^{15} -16.8120 q^{17} +75.7927 q^{19} +46.7115 q^{23} +85.5999 q^{25} +27.0000 q^{27} +73.5290 q^{29} -135.435 q^{31} +30.3348 q^{33} +133.220 q^{37} -1.87098 q^{39} -69.3010 q^{41} +279.261 q^{43} +130.609 q^{45} +372.176 q^{47} -50.4361 q^{51} +656.872 q^{53} +146.740 q^{55} +227.378 q^{57} -730.411 q^{59} +39.5121 q^{61} -9.05057 q^{65} +417.962 q^{67} +140.135 q^{69} -831.408 q^{71} +1192.72 q^{73} +256.800 q^{75} -1034.49 q^{79} +81.0000 q^{81} +117.618 q^{83} -243.977 q^{85} +220.587 q^{87} -678.390 q^{89} -406.305 q^{93} +1099.91 q^{95} +1659.31 q^{97} +91.0044 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{3} + 8q^{5} + 36q^{9} + O(q^{10}) \) \( 4q + 12q^{3} + 8q^{5} + 36q^{9} + 40q^{11} + 48q^{13} + 24q^{15} + 72q^{17} + 32q^{19} + 8q^{23} + 164q^{25} + 108q^{27} + 144q^{29} + 48q^{31} + 120q^{33} + 48q^{37} + 144q^{39} + 72q^{41} + 512q^{43} + 72q^{45} + 160q^{47} + 216q^{51} + 536q^{53} - 336q^{55} + 96q^{57} + 240q^{59} + 896q^{61} - 136q^{65} + 1088q^{67} + 24q^{69} + 1288q^{71} + 1488q^{73} + 492q^{75} + 416q^{79} + 324q^{81} - 112q^{83} - 1512q^{85} + 432q^{87} + 3160q^{89} + 144q^{93} - 240q^{95} + 2384q^{97} + 360q^{99} + O(q^{100}) \)

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
\(5\) 14.5121 1.29800 0.648999 0.760789i \(-0.275188\pi\)
0.648999 + 0.760789i \(0.275188\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 10.1116 0.277160 0.138580 0.990351i \(-0.455746\pi\)
0.138580 + 0.990351i \(0.455746\pi\)
\(12\) 0 0
\(13\) −0.623659 −0.0133055 −0.00665276 0.999978i \(-0.502118\pi\)
−0.00665276 + 0.999978i \(0.502118\pi\)
\(14\) 0 0
\(15\) 43.5362 0.749400
\(16\) 0 0
\(17\) −16.8120 −0.239854 −0.119927 0.992783i \(-0.538266\pi\)
−0.119927 + 0.992783i \(0.538266\pi\)
\(18\) 0 0
\(19\) 75.7927 0.915159 0.457580 0.889169i \(-0.348716\pi\)
0.457580 + 0.889169i \(0.348716\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 46.7115 0.423479 0.211740 0.977326i \(-0.432087\pi\)
0.211740 + 0.977326i \(0.432087\pi\)
\(24\) 0 0
\(25\) 85.5999 0.684799
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 73.5290 0.470827 0.235414 0.971895i \(-0.424356\pi\)
0.235414 + 0.971895i \(0.424356\pi\)
\(30\) 0 0
\(31\) −135.435 −0.784672 −0.392336 0.919822i \(-0.628333\pi\)
−0.392336 + 0.919822i \(0.628333\pi\)
\(32\) 0 0
\(33\) 30.3348 0.160018
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 133.220 0.591926 0.295963 0.955199i \(-0.404359\pi\)
0.295963 + 0.955199i \(0.404359\pi\)
\(38\) 0 0
\(39\) −1.87098 −0.00768195
\(40\) 0 0
\(41\) −69.3010 −0.263975 −0.131988 0.991251i \(-0.542136\pi\)
−0.131988 + 0.991251i \(0.542136\pi\)
\(42\) 0 0
\(43\) 279.261 0.990392 0.495196 0.868781i \(-0.335096\pi\)
0.495196 + 0.868781i \(0.335096\pi\)
\(44\) 0 0
\(45\) 130.609 0.432666
\(46\) 0 0
\(47\) 372.176 1.15505 0.577526 0.816372i \(-0.304018\pi\)
0.577526 + 0.816372i \(0.304018\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −50.4361 −0.138480
\(52\) 0 0
\(53\) 656.872 1.70242 0.851210 0.524825i \(-0.175869\pi\)
0.851210 + 0.524825i \(0.175869\pi\)
\(54\) 0 0
\(55\) 146.740 0.359753
\(56\) 0 0
\(57\) 227.378 0.528368
\(58\) 0 0
\(59\) −730.411 −1.61172 −0.805859 0.592107i \(-0.798296\pi\)
−0.805859 + 0.592107i \(0.798296\pi\)
\(60\) 0 0
\(61\) 39.5121 0.0829345 0.0414673 0.999140i \(-0.486797\pi\)
0.0414673 + 0.999140i \(0.486797\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −9.05057 −0.0172705
\(66\) 0 0
\(67\) 417.962 0.762122 0.381061 0.924550i \(-0.375559\pi\)
0.381061 + 0.924550i \(0.375559\pi\)
\(68\) 0 0
\(69\) 140.135 0.244496
\(70\) 0 0
\(71\) −831.408 −1.38972 −0.694859 0.719146i \(-0.744533\pi\)
−0.694859 + 0.719146i \(0.744533\pi\)
\(72\) 0 0
\(73\) 1192.72 1.91229 0.956145 0.292894i \(-0.0946184\pi\)
0.956145 + 0.292894i \(0.0946184\pi\)
\(74\) 0 0
\(75\) 256.800 0.395369
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1034.49 −1.47327 −0.736637 0.676288i \(-0.763587\pi\)
−0.736637 + 0.676288i \(0.763587\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 117.618 0.155545 0.0777726 0.996971i \(-0.475219\pi\)
0.0777726 + 0.996971i \(0.475219\pi\)
\(84\) 0 0
\(85\) −243.977 −0.311330
\(86\) 0 0
\(87\) 220.587 0.271832
\(88\) 0 0
\(89\) −678.390 −0.807969 −0.403984 0.914766i \(-0.632375\pi\)
−0.403984 + 0.914766i \(0.632375\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −406.305 −0.453031
\(94\) 0 0
\(95\) 1099.91 1.18788
\(96\) 0 0
\(97\) 1659.31 1.73688 0.868439 0.495795i \(-0.165123\pi\)
0.868439 + 0.495795i \(0.165123\pi\)
\(98\) 0 0
\(99\) 91.0044 0.0923867
\(100\) 0 0
\(101\) 485.839 0.478641 0.239321 0.970941i \(-0.423075\pi\)
0.239321 + 0.970941i \(0.423075\pi\)
\(102\) 0 0
\(103\) 276.792 0.264787 0.132394 0.991197i \(-0.457734\pi\)
0.132394 + 0.991197i \(0.457734\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 103.639 0.0936373 0.0468187 0.998903i \(-0.485092\pi\)
0.0468187 + 0.998903i \(0.485092\pi\)
\(108\) 0 0
\(109\) −50.1571 −0.0440751 −0.0220375 0.999757i \(-0.507015\pi\)
−0.0220375 + 0.999757i \(0.507015\pi\)
\(110\) 0 0
\(111\) 399.661 0.341749
\(112\) 0 0
\(113\) −1954.66 −1.62725 −0.813623 0.581393i \(-0.802508\pi\)
−0.813623 + 0.581393i \(0.802508\pi\)
\(114\) 0 0
\(115\) 677.880 0.549675
\(116\) 0 0
\(117\) −5.61293 −0.00443517
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1228.76 −0.923182
\(122\) 0 0
\(123\) −207.903 −0.152406
\(124\) 0 0
\(125\) −571.776 −0.409130
\(126\) 0 0
\(127\) −1123.20 −0.784783 −0.392392 0.919798i \(-0.628352\pi\)
−0.392392 + 0.919798i \(0.628352\pi\)
\(128\) 0 0
\(129\) 837.782 0.571803
\(130\) 0 0
\(131\) 1562.85 1.04234 0.521171 0.853453i \(-0.325496\pi\)
0.521171 + 0.853453i \(0.325496\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 391.826 0.249800
\(136\) 0 0
\(137\) 1283.91 0.800673 0.400337 0.916368i \(-0.368893\pi\)
0.400337 + 0.916368i \(0.368893\pi\)
\(138\) 0 0
\(139\) −674.069 −0.411322 −0.205661 0.978623i \(-0.565934\pi\)
−0.205661 + 0.978623i \(0.565934\pi\)
\(140\) 0 0
\(141\) 1116.53 0.666869
\(142\) 0 0
\(143\) −6.30619 −0.00368776
\(144\) 0 0
\(145\) 1067.06 0.611133
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1293.72 −0.711314 −0.355657 0.934617i \(-0.615743\pi\)
−0.355657 + 0.934617i \(0.615743\pi\)
\(150\) 0 0
\(151\) 2266.65 1.22157 0.610786 0.791795i \(-0.290853\pi\)
0.610786 + 0.791795i \(0.290853\pi\)
\(152\) 0 0
\(153\) −151.308 −0.0799512
\(154\) 0 0
\(155\) −1965.44 −1.01850
\(156\) 0 0
\(157\) 1868.03 0.949585 0.474792 0.880098i \(-0.342523\pi\)
0.474792 + 0.880098i \(0.342523\pi\)
\(158\) 0 0
\(159\) 1970.61 0.982892
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1511.45 −0.726293 −0.363147 0.931732i \(-0.618298\pi\)
−0.363147 + 0.931732i \(0.618298\pi\)
\(164\) 0 0
\(165\) 440.220 0.207704
\(166\) 0 0
\(167\) −2371.40 −1.09883 −0.549415 0.835550i \(-0.685149\pi\)
−0.549415 + 0.835550i \(0.685149\pi\)
\(168\) 0 0
\(169\) −2196.61 −0.999823
\(170\) 0 0
\(171\) 682.134 0.305053
\(172\) 0 0
\(173\) 4124.67 1.81267 0.906337 0.422555i \(-0.138867\pi\)
0.906337 + 0.422555i \(0.138867\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2191.23 −0.930526
\(178\) 0 0
\(179\) 3194.53 1.33391 0.666955 0.745098i \(-0.267597\pi\)
0.666955 + 0.745098i \(0.267597\pi\)
\(180\) 0 0
\(181\) 1389.99 0.570815 0.285407 0.958406i \(-0.407871\pi\)
0.285407 + 0.958406i \(0.407871\pi\)
\(182\) 0 0
\(183\) 118.536 0.0478823
\(184\) 0 0
\(185\) 1933.30 0.768319
\(186\) 0 0
\(187\) −169.996 −0.0664779
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1915.99 0.725844 0.362922 0.931820i \(-0.381779\pi\)
0.362922 + 0.931820i \(0.381779\pi\)
\(192\) 0 0
\(193\) −1092.29 −0.407381 −0.203690 0.979035i \(-0.565294\pi\)
−0.203690 + 0.979035i \(0.565294\pi\)
\(194\) 0 0
\(195\) −27.1517 −0.00997115
\(196\) 0 0
\(197\) 4727.17 1.70963 0.854815 0.518933i \(-0.173671\pi\)
0.854815 + 0.518933i \(0.173671\pi\)
\(198\) 0 0
\(199\) 634.583 0.226052 0.113026 0.993592i \(-0.463946\pi\)
0.113026 + 0.993592i \(0.463946\pi\)
\(200\) 0 0
\(201\) 1253.89 0.440011
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1005.70 −0.342640
\(206\) 0 0
\(207\) 420.404 0.141160
\(208\) 0 0
\(209\) 766.385 0.253646
\(210\) 0 0
\(211\) −1782.00 −0.581413 −0.290707 0.956812i \(-0.593890\pi\)
−0.290707 + 0.956812i \(0.593890\pi\)
\(212\) 0 0
\(213\) −2494.22 −0.802354
\(214\) 0 0
\(215\) 4052.65 1.28553
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3578.16 1.10406
\(220\) 0 0
\(221\) 10.4850 0.00319138
\(222\) 0 0
\(223\) −5362.96 −1.61045 −0.805225 0.592970i \(-0.797955\pi\)
−0.805225 + 0.592970i \(0.797955\pi\)
\(224\) 0 0
\(225\) 770.399 0.228266
\(226\) 0 0
\(227\) −734.718 −0.214824 −0.107412 0.994215i \(-0.534256\pi\)
−0.107412 + 0.994215i \(0.534256\pi\)
\(228\) 0 0
\(229\) 5160.26 1.48908 0.744541 0.667577i \(-0.232668\pi\)
0.744541 + 0.667577i \(0.232668\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4068.43 −1.14391 −0.571956 0.820284i \(-0.693815\pi\)
−0.571956 + 0.820284i \(0.693815\pi\)
\(234\) 0 0
\(235\) 5401.04 1.49925
\(236\) 0 0
\(237\) −3103.46 −0.850595
\(238\) 0 0
\(239\) −650.039 −0.175931 −0.0879655 0.996124i \(-0.528037\pi\)
−0.0879655 + 0.996124i \(0.528037\pi\)
\(240\) 0 0
\(241\) 3233.14 0.864170 0.432085 0.901833i \(-0.357778\pi\)
0.432085 + 0.901833i \(0.357778\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −47.2687 −0.0121767
\(248\) 0 0
\(249\) 352.854 0.0898041
\(250\) 0 0
\(251\) 1139.75 0.286615 0.143308 0.989678i \(-0.454226\pi\)
0.143308 + 0.989678i \(0.454226\pi\)
\(252\) 0 0
\(253\) 472.328 0.117372
\(254\) 0 0
\(255\) −731.931 −0.179746
\(256\) 0 0
\(257\) 2036.97 0.494408 0.247204 0.968963i \(-0.420488\pi\)
0.247204 + 0.968963i \(0.420488\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 661.761 0.156942
\(262\) 0 0
\(263\) −836.215 −0.196058 −0.0980289 0.995184i \(-0.531254\pi\)
−0.0980289 + 0.995184i \(0.531254\pi\)
\(264\) 0 0
\(265\) 9532.56 2.20974
\(266\) 0 0
\(267\) −2035.17 −0.466481
\(268\) 0 0
\(269\) 2682.99 0.608122 0.304061 0.952652i \(-0.401657\pi\)
0.304061 + 0.952652i \(0.401657\pi\)
\(270\) 0 0
\(271\) −6541.42 −1.46628 −0.733142 0.680076i \(-0.761947\pi\)
−0.733142 + 0.680076i \(0.761947\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 865.552 0.189799
\(276\) 0 0
\(277\) −1293.11 −0.280490 −0.140245 0.990117i \(-0.544789\pi\)
−0.140245 + 0.990117i \(0.544789\pi\)
\(278\) 0 0
\(279\) −1218.92 −0.261557
\(280\) 0 0
\(281\) 4684.69 0.994538 0.497269 0.867596i \(-0.334336\pi\)
0.497269 + 0.867596i \(0.334336\pi\)
\(282\) 0 0
\(283\) 565.240 0.118728 0.0593640 0.998236i \(-0.481093\pi\)
0.0593640 + 0.998236i \(0.481093\pi\)
\(284\) 0 0
\(285\) 3299.72 0.685820
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4630.36 −0.942470
\(290\) 0 0
\(291\) 4977.93 1.00279
\(292\) 0 0
\(293\) 809.451 0.161395 0.0806973 0.996739i \(-0.474285\pi\)
0.0806973 + 0.996739i \(0.474285\pi\)
\(294\) 0 0
\(295\) −10599.8 −2.09201
\(296\) 0 0
\(297\) 273.013 0.0533395
\(298\) 0 0
\(299\) −29.1320 −0.00563461
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1457.52 0.276344
\(304\) 0 0
\(305\) 573.402 0.107649
\(306\) 0 0
\(307\) 3667.95 0.681892 0.340946 0.940083i \(-0.389253\pi\)
0.340946 + 0.940083i \(0.389253\pi\)
\(308\) 0 0
\(309\) 830.375 0.152875
\(310\) 0 0
\(311\) −73.3047 −0.0133657 −0.00668284 0.999978i \(-0.502127\pi\)
−0.00668284 + 0.999978i \(0.502127\pi\)
\(312\) 0 0
\(313\) −8300.73 −1.49899 −0.749497 0.662008i \(-0.769705\pi\)
−0.749497 + 0.662008i \(0.769705\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1288.89 0.228364 0.114182 0.993460i \(-0.463575\pi\)
0.114182 + 0.993460i \(0.463575\pi\)
\(318\) 0 0
\(319\) 743.495 0.130494
\(320\) 0 0
\(321\) 310.918 0.0540615
\(322\) 0 0
\(323\) −1274.23 −0.219504
\(324\) 0 0
\(325\) −53.3851 −0.00911161
\(326\) 0 0
\(327\) −150.471 −0.0254468
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6517.90 −1.08234 −0.541172 0.840912i \(-0.682019\pi\)
−0.541172 + 0.840912i \(0.682019\pi\)
\(332\) 0 0
\(333\) 1198.98 0.197309
\(334\) 0 0
\(335\) 6065.49 0.989233
\(336\) 0 0
\(337\) −7477.45 −1.20867 −0.604337 0.796729i \(-0.706562\pi\)
−0.604337 + 0.796729i \(0.706562\pi\)
\(338\) 0 0
\(339\) −5863.97 −0.939491
\(340\) 0 0
\(341\) −1369.46 −0.217480
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2033.64 0.317355
\(346\) 0 0
\(347\) −1228.91 −0.190119 −0.0950596 0.995472i \(-0.530304\pi\)
−0.0950596 + 0.995472i \(0.530304\pi\)
\(348\) 0 0
\(349\) 1660.55 0.254691 0.127345 0.991858i \(-0.459354\pi\)
0.127345 + 0.991858i \(0.459354\pi\)
\(350\) 0 0
\(351\) −16.8388 −0.00256065
\(352\) 0 0
\(353\) 10483.4 1.58066 0.790330 0.612681i \(-0.209909\pi\)
0.790330 + 0.612681i \(0.209909\pi\)
\(354\) 0 0
\(355\) −12065.4 −1.80385
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5329.76 −0.783549 −0.391775 0.920061i \(-0.628139\pi\)
−0.391775 + 0.920061i \(0.628139\pi\)
\(360\) 0 0
\(361\) −1114.47 −0.162483
\(362\) 0 0
\(363\) −3686.27 −0.533000
\(364\) 0 0
\(365\) 17308.8 2.48215
\(366\) 0 0
\(367\) −6608.43 −0.939937 −0.469969 0.882683i \(-0.655735\pi\)
−0.469969 + 0.882683i \(0.655735\pi\)
\(368\) 0 0
\(369\) −623.709 −0.0879918
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6969.99 0.967540 0.483770 0.875195i \(-0.339267\pi\)
0.483770 + 0.875195i \(0.339267\pi\)
\(374\) 0 0
\(375\) −1715.33 −0.236211
\(376\) 0 0
\(377\) −45.8570 −0.00626460
\(378\) 0 0
\(379\) −7383.56 −1.00071 −0.500353 0.865821i \(-0.666797\pi\)
−0.500353 + 0.865821i \(0.666797\pi\)
\(380\) 0 0
\(381\) −3369.59 −0.453095
\(382\) 0 0
\(383\) 845.491 0.112800 0.0564002 0.998408i \(-0.482038\pi\)
0.0564002 + 0.998408i \(0.482038\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2513.35 0.330131
\(388\) 0 0
\(389\) −5897.87 −0.768725 −0.384362 0.923182i \(-0.625579\pi\)
−0.384362 + 0.923182i \(0.625579\pi\)
\(390\) 0 0
\(391\) −785.315 −0.101573
\(392\) 0 0
\(393\) 4688.55 0.601796
\(394\) 0 0
\(395\) −15012.5 −1.91231
\(396\) 0 0
\(397\) −10274.5 −1.29890 −0.649448 0.760406i \(-0.725000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4261.85 0.530740 0.265370 0.964147i \(-0.414506\pi\)
0.265370 + 0.964147i \(0.414506\pi\)
\(402\) 0 0
\(403\) 84.4652 0.0104405
\(404\) 0 0
\(405\) 1175.48 0.144222
\(406\) 0 0
\(407\) 1347.07 0.164058
\(408\) 0 0
\(409\) 6389.21 0.772436 0.386218 0.922408i \(-0.373781\pi\)
0.386218 + 0.922408i \(0.373781\pi\)
\(410\) 0 0
\(411\) 3851.74 0.462269
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1706.88 0.201898
\(416\) 0 0
\(417\) −2022.21 −0.237477
\(418\) 0 0
\(419\) −7.77360 −0.000906361 0 −0.000453180 1.00000i \(-0.500144\pi\)
−0.000453180 1.00000i \(0.500144\pi\)
\(420\) 0 0
\(421\) 8500.33 0.984040 0.492020 0.870584i \(-0.336259\pi\)
0.492020 + 0.870584i \(0.336259\pi\)
\(422\) 0 0
\(423\) 3349.58 0.385017
\(424\) 0 0
\(425\) −1439.11 −0.164252
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −18.9186 −0.00212913
\(430\) 0 0
\(431\) −7264.32 −0.811856 −0.405928 0.913905i \(-0.633052\pi\)
−0.405928 + 0.913905i \(0.633052\pi\)
\(432\) 0 0
\(433\) 1023.35 0.113578 0.0567890 0.998386i \(-0.481914\pi\)
0.0567890 + 0.998386i \(0.481914\pi\)
\(434\) 0 0
\(435\) 3201.17 0.352838
\(436\) 0 0
\(437\) 3540.39 0.387551
\(438\) 0 0
\(439\) −8081.20 −0.878576 −0.439288 0.898346i \(-0.644769\pi\)
−0.439288 + 0.898346i \(0.644769\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4900.66 0.525592 0.262796 0.964851i \(-0.415355\pi\)
0.262796 + 0.964851i \(0.415355\pi\)
\(444\) 0 0
\(445\) −9844.84 −1.04874
\(446\) 0 0
\(447\) −3881.16 −0.410677
\(448\) 0 0
\(449\) 8493.30 0.892703 0.446351 0.894858i \(-0.352723\pi\)
0.446351 + 0.894858i \(0.352723\pi\)
\(450\) 0 0
\(451\) −700.744 −0.0731635
\(452\) 0 0
\(453\) 6799.96 0.705275
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −13042.6 −1.33503 −0.667514 0.744597i \(-0.732642\pi\)
−0.667514 + 0.744597i \(0.732642\pi\)
\(458\) 0 0
\(459\) −453.925 −0.0461599
\(460\) 0 0
\(461\) −8900.01 −0.899164 −0.449582 0.893239i \(-0.648427\pi\)
−0.449582 + 0.893239i \(0.648427\pi\)
\(462\) 0 0
\(463\) −17336.4 −1.74015 −0.870076 0.492917i \(-0.835930\pi\)
−0.870076 + 0.492917i \(0.835930\pi\)
\(464\) 0 0
\(465\) −5896.32 −0.588033
\(466\) 0 0
\(467\) 3071.51 0.304352 0.152176 0.988353i \(-0.451372\pi\)
0.152176 + 0.988353i \(0.451372\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5604.08 0.548243
\(472\) 0 0
\(473\) 2823.77 0.274497
\(474\) 0 0
\(475\) 6487.85 0.626701
\(476\) 0 0
\(477\) 5911.84 0.567473
\(478\) 0 0
\(479\) −16272.9 −1.55225 −0.776125 0.630580i \(-0.782817\pi\)
−0.776125 + 0.630580i \(0.782817\pi\)
\(480\) 0 0
\(481\) −83.0839 −0.00787589
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 24080.0 2.25447
\(486\) 0 0
\(487\) −10851.9 −1.00975 −0.504874 0.863193i \(-0.668461\pi\)
−0.504874 + 0.863193i \(0.668461\pi\)
\(488\) 0 0
\(489\) −4534.35 −0.419326
\(490\) 0 0
\(491\) 744.455 0.0684252 0.0342126 0.999415i \(-0.489108\pi\)
0.0342126 + 0.999415i \(0.489108\pi\)
\(492\) 0 0
\(493\) −1236.17 −0.112930
\(494\) 0 0
\(495\) 1320.66 0.119918
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 13272.2 1.19067 0.595334 0.803478i \(-0.297020\pi\)
0.595334 + 0.803478i \(0.297020\pi\)
\(500\) 0 0
\(501\) −7114.21 −0.634410
\(502\) 0 0
\(503\) −19485.3 −1.72725 −0.863626 0.504133i \(-0.831812\pi\)
−0.863626 + 0.504133i \(0.831812\pi\)
\(504\) 0 0
\(505\) 7050.52 0.621275
\(506\) 0 0
\(507\) −6589.83 −0.577248
\(508\) 0 0
\(509\) 1348.71 0.117447 0.0587235 0.998274i \(-0.481297\pi\)
0.0587235 + 0.998274i \(0.481297\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2046.40 0.176123
\(514\) 0 0
\(515\) 4016.82 0.343693
\(516\) 0 0
\(517\) 3763.29 0.320134
\(518\) 0 0
\(519\) 12374.0 1.04655
\(520\) 0 0
\(521\) 3349.98 0.281699 0.140849 0.990031i \(-0.455017\pi\)
0.140849 + 0.990031i \(0.455017\pi\)
\(522\) 0 0
\(523\) −7588.42 −0.634452 −0.317226 0.948350i \(-0.602751\pi\)
−0.317226 + 0.948350i \(0.602751\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2276.94 0.188207
\(528\) 0 0
\(529\) −9985.03 −0.820665
\(530\) 0 0
\(531\) −6573.70 −0.537239
\(532\) 0 0
\(533\) 43.2202 0.00351233
\(534\) 0 0
\(535\) 1504.02 0.121541
\(536\) 0 0
\(537\) 9583.58 0.770134
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −13690.4 −1.08797 −0.543987 0.839093i \(-0.683086\pi\)
−0.543987 + 0.839093i \(0.683086\pi\)
\(542\) 0 0
\(543\) 4169.98 0.329560
\(544\) 0 0
\(545\) −727.884 −0.0572094
\(546\) 0 0
\(547\) 15250.2 1.19205 0.596024 0.802967i \(-0.296746\pi\)
0.596024 + 0.802967i \(0.296746\pi\)
\(548\) 0 0
\(549\) 355.609 0.0276448
\(550\) 0 0
\(551\) 5572.96 0.430882
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 5799.90 0.443589
\(556\) 0 0
\(557\) −14231.6 −1.08261 −0.541305 0.840826i \(-0.682070\pi\)
−0.541305 + 0.840826i \(0.682070\pi\)
\(558\) 0 0
\(559\) −174.163 −0.0131777
\(560\) 0 0
\(561\) −509.989 −0.0383810
\(562\) 0 0
\(563\) −23001.7 −1.72186 −0.860928 0.508727i \(-0.830116\pi\)
−0.860928 + 0.508727i \(0.830116\pi\)
\(564\) 0 0
\(565\) −28366.1 −2.11216
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11996.5 0.883866 0.441933 0.897048i \(-0.354293\pi\)
0.441933 + 0.897048i \(0.354293\pi\)
\(570\) 0 0
\(571\) −25436.0 −1.86421 −0.932106 0.362186i \(-0.882030\pi\)
−0.932106 + 0.362186i \(0.882030\pi\)
\(572\) 0 0
\(573\) 5747.97 0.419066
\(574\) 0 0
\(575\) 3998.50 0.289998
\(576\) 0 0
\(577\) 1605.24 0.115818 0.0579090 0.998322i \(-0.481557\pi\)
0.0579090 + 0.998322i \(0.481557\pi\)
\(578\) 0 0
\(579\) −3276.86 −0.235201
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6642.02 0.471843
\(584\) 0 0
\(585\) −81.4552 −0.00575685
\(586\) 0 0
\(587\) −20510.8 −1.44220 −0.721101 0.692829i \(-0.756364\pi\)
−0.721101 + 0.692829i \(0.756364\pi\)
\(588\) 0 0
\(589\) −10265.0 −0.718100
\(590\) 0 0
\(591\) 14181.5 0.987055
\(592\) 0 0
\(593\) 899.970 0.0623227 0.0311613 0.999514i \(-0.490079\pi\)
0.0311613 + 0.999514i \(0.490079\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1903.75 0.130511
\(598\) 0 0
\(599\) −17326.2 −1.18185 −0.590926 0.806726i \(-0.701238\pi\)
−0.590926 + 0.806726i \(0.701238\pi\)
\(600\) 0 0
\(601\) −10193.7 −0.691860 −0.345930 0.938260i \(-0.612437\pi\)
−0.345930 + 0.938260i \(0.612437\pi\)
\(602\) 0 0
\(603\) 3761.66 0.254041
\(604\) 0 0
\(605\) −17831.8 −1.19829
\(606\) 0 0
\(607\) −10674.8 −0.713803 −0.356901 0.934142i \(-0.616167\pi\)
−0.356901 + 0.934142i \(0.616167\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −232.111 −0.0153686
\(612\) 0 0
\(613\) −20442.8 −1.34694 −0.673471 0.739214i \(-0.735197\pi\)
−0.673471 + 0.739214i \(0.735197\pi\)
\(614\) 0 0
\(615\) −3017.10 −0.197823
\(616\) 0 0
\(617\) −12084.9 −0.788523 −0.394261 0.918998i \(-0.629000\pi\)
−0.394261 + 0.918998i \(0.629000\pi\)
\(618\) 0 0
\(619\) 23057.3 1.49718 0.748589 0.663035i \(-0.230732\pi\)
0.748589 + 0.663035i \(0.230732\pi\)
\(620\) 0 0
\(621\) 1261.21 0.0814986
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −18997.6 −1.21585
\(626\) 0 0
\(627\) 2299.15 0.146442
\(628\) 0 0
\(629\) −2239.70 −0.141976
\(630\) 0 0
\(631\) −19201.2 −1.21139 −0.605694 0.795697i \(-0.707105\pi\)
−0.605694 + 0.795697i \(0.707105\pi\)
\(632\) 0 0
\(633\) −5346.01 −0.335679
\(634\) 0 0
\(635\) −16299.9 −1.01865
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −7482.67 −0.463239
\(640\) 0 0
\(641\) 10581.2 0.651998 0.325999 0.945370i \(-0.394299\pi\)
0.325999 + 0.945370i \(0.394299\pi\)
\(642\) 0 0
\(643\) 29823.2 1.82910 0.914550 0.404474i \(-0.132545\pi\)
0.914550 + 0.404474i \(0.132545\pi\)
\(644\) 0 0
\(645\) 12157.9 0.742199
\(646\) 0 0
\(647\) 143.283 0.00870639 0.00435320 0.999991i \(-0.498614\pi\)
0.00435320 + 0.999991i \(0.498614\pi\)
\(648\) 0 0
\(649\) −7385.62 −0.446704
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7516.26 0.450435 0.225217 0.974309i \(-0.427691\pi\)
0.225217 + 0.974309i \(0.427691\pi\)
\(654\) 0 0
\(655\) 22680.2 1.35296
\(656\) 0 0
\(657\) 10734.5 0.637430
\(658\) 0 0
\(659\) −4791.03 −0.283205 −0.141603 0.989924i \(-0.545226\pi\)
−0.141603 + 0.989924i \(0.545226\pi\)
\(660\) 0 0
\(661\) 4711.63 0.277249 0.138624 0.990345i \(-0.455732\pi\)
0.138624 + 0.990345i \(0.455732\pi\)
\(662\) 0 0
\(663\) 31.4549 0.00184254
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3434.65 0.199386
\(668\) 0 0
\(669\) −16088.9 −0.929793
\(670\) 0 0
\(671\) 399.530 0.0229861
\(672\) 0 0
\(673\) −10953.3 −0.627367 −0.313683 0.949528i \(-0.601563\pi\)
−0.313683 + 0.949528i \(0.601563\pi\)
\(674\) 0 0
\(675\) 2311.20 0.131790
\(676\) 0 0
\(677\) −27537.8 −1.56331 −0.781656 0.623710i \(-0.785625\pi\)
−0.781656 + 0.623710i \(0.785625\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −2204.15 −0.124028
\(682\) 0 0
\(683\) 3280.93 0.183808 0.0919042 0.995768i \(-0.470705\pi\)
0.0919042 + 0.995768i \(0.470705\pi\)
\(684\) 0 0
\(685\) 18632.2 1.03927
\(686\) 0 0
\(687\) 15480.8 0.859722
\(688\) 0 0
\(689\) −409.664 −0.0226516
\(690\) 0 0
\(691\) −20420.4 −1.12421 −0.562106 0.827065i \(-0.690009\pi\)
−0.562106 + 0.827065i \(0.690009\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −9782.13 −0.533895
\(696\) 0 0
\(697\) 1165.09 0.0633155
\(698\) 0 0
\(699\) −12205.3 −0.660438
\(700\) 0 0
\(701\) −19238.0 −1.03653 −0.518267 0.855219i \(-0.673423\pi\)
−0.518267 + 0.855219i \(0.673423\pi\)
\(702\) 0 0
\(703\) 10097.1 0.541707
\(704\) 0 0
\(705\) 16203.1 0.865595
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −34685.9 −1.83731 −0.918657 0.395057i \(-0.870725\pi\)
−0.918657 + 0.395057i \(0.870725\pi\)
\(710\) 0 0
\(711\) −9310.37 −0.491091
\(712\) 0 0
\(713\) −6326.37 −0.332293
\(714\) 0 0
\(715\) −91.5157 −0.00478671
\(716\) 0 0
\(717\) −1950.12 −0.101574
\(718\) 0 0
\(719\) 21896.6 1.13575 0.567877 0.823114i \(-0.307765\pi\)
0.567877 + 0.823114i \(0.307765\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 9699.42 0.498929
\(724\) 0 0
\(725\) 6294.07 0.322422
\(726\) 0 0
\(727\) −22962.5 −1.17143 −0.585717 0.810516i \(-0.699187\pi\)
−0.585717 + 0.810516i \(0.699187\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −4694.94 −0.237549
\(732\) 0 0
\(733\) −22990.5 −1.15849 −0.579245 0.815153i \(-0.696653\pi\)
−0.579245 + 0.815153i \(0.696653\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4226.26 0.211230
\(738\) 0 0
\(739\) 5010.14 0.249392 0.124696 0.992195i \(-0.460204\pi\)
0.124696 + 0.992195i \(0.460204\pi\)
\(740\) 0 0
\(741\) −141.806 −0.00703021
\(742\) 0 0
\(743\) 14389.0 0.710472 0.355236 0.934777i \(-0.384401\pi\)
0.355236 + 0.934777i \(0.384401\pi\)
\(744\) 0 0
\(745\) −18774.6 −0.923284
\(746\) 0 0
\(747\) 1058.56 0.0518484
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 35586.5 1.72912 0.864560 0.502529i \(-0.167597\pi\)
0.864560 + 0.502529i \(0.167597\pi\)
\(752\) 0 0
\(753\) 3419.25 0.165477
\(754\) 0 0
\(755\) 32893.8 1.58560
\(756\) 0 0
\(757\) −12140.6 −0.582905 −0.291453 0.956585i \(-0.594139\pi\)
−0.291453 + 0.956585i \(0.594139\pi\)
\(758\) 0 0
\(759\) 1416.98 0.0677645
\(760\) 0 0
\(761\) 22998.0 1.09550 0.547750 0.836642i \(-0.315484\pi\)
0.547750 + 0.836642i \(0.315484\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2195.79 −0.103777
\(766\) 0 0
\(767\) 455.527 0.0214448
\(768\) 0 0
\(769\) 24004.6 1.12566 0.562828 0.826574i \(-0.309713\pi\)
0.562828 + 0.826574i \(0.309713\pi\)
\(770\) 0 0
\(771\) 6110.92 0.285447
\(772\) 0 0
\(773\) 2777.97 0.129258 0.0646292 0.997909i \(-0.479414\pi\)
0.0646292 + 0.997909i \(0.479414\pi\)
\(774\) 0 0
\(775\) −11593.2 −0.537343
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5252.50 −0.241580
\(780\) 0 0
\(781\) −8406.87 −0.385175
\(782\) 0 0
\(783\) 1985.28 0.0906107
\(784\) 0 0
\(785\) 27108.9 1.23256
\(786\) 0 0
\(787\) 25824.5 1.16969 0.584843 0.811147i \(-0.301156\pi\)
0.584843 + 0.811147i \(0.301156\pi\)
\(788\) 0 0
\(789\) −2508.64 −0.113194
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −24.6421 −0.00110349
\(794\) 0 0
\(795\) 28597.7 1.27579
\(796\) 0 0
\(797\) −5363.77 −0.238387 −0.119193 0.992871i \(-0.538031\pi\)
−0.119193 + 0.992871i \(0.538031\pi\)
\(798\) 0 0
\(799\) −6257.03 −0.277043
\(800\) 0 0
\(801\) −6105.51 −0.269323
\(802\) 0 0
\(803\) 12060.3 0.530010
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8048.98 0.351100
\(808\) 0 0
\(809\) 15270.5 0.663638 0.331819 0.943343i \(-0.392338\pi\)
0.331819 + 0.943343i \(0.392338\pi\)
\(810\) 0 0
\(811\) 2993.47 0.129612 0.0648058 0.997898i \(-0.479357\pi\)
0.0648058 + 0.997898i \(0.479357\pi\)
\(812\) 0 0
\(813\) −19624.3 −0.846559
\(814\) 0 0
\(815\) −21934.2 −0.942728
\(816\) 0 0
\(817\) 21165.9 0.906367
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 16002.9 0.680272 0.340136 0.940376i \(-0.389527\pi\)
0.340136 + 0.940376i \(0.389527\pi\)
\(822\) 0 0
\(823\) −11836.7 −0.501340 −0.250670 0.968073i \(-0.580651\pi\)
−0.250670 + 0.968073i \(0.580651\pi\)
\(824\) 0 0
\(825\) 2596.66 0.109581
\(826\) 0 0
\(827\) −5675.18 −0.238628 −0.119314 0.992857i \(-0.538069\pi\)
−0.119314 + 0.992857i \(0.538069\pi\)
\(828\) 0 0
\(829\) −28989.5 −1.21453 −0.607265 0.794499i \(-0.707734\pi\)
−0.607265 + 0.794499i \(0.707734\pi\)
\(830\) 0 0
\(831\) −3879.34 −0.161941
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −34413.9 −1.42628
\(836\) 0 0
\(837\) −3656.75 −0.151010
\(838\) 0 0
\(839\) 10021.9 0.412390 0.206195 0.978511i \(-0.433892\pi\)
0.206195 + 0.978511i \(0.433892\pi\)
\(840\) 0 0
\(841\) −18982.5 −0.778322
\(842\) 0 0
\(843\) 14054.1 0.574197
\(844\) 0 0
\(845\) −31877.4 −1.29777
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 1695.72 0.0685477
\(850\) 0 0
\(851\) 6222.92 0.250668
\(852\) 0 0
\(853\) −19446.0 −0.780562 −0.390281 0.920696i \(-0.627622\pi\)
−0.390281 + 0.920696i \(0.627622\pi\)
\(854\) 0 0
\(855\) 9899.17 0.395958
\(856\) 0 0
\(857\) 39813.3 1.58693 0.793464 0.608617i \(-0.208275\pi\)
0.793464 + 0.608617i \(0.208275\pi\)
\(858\) 0 0
\(859\) 46968.9 1.86561 0.932804 0.360384i \(-0.117354\pi\)
0.932804 + 0.360384i \(0.117354\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −13859.8 −0.546689 −0.273344 0.961916i \(-0.588130\pi\)
−0.273344 + 0.961916i \(0.588130\pi\)
\(864\) 0 0
\(865\) 59857.4 2.35285
\(866\) 0 0
\(867\) −13891.1 −0.544135
\(868\) 0 0
\(869\) −10460.3 −0.408333
\(870\) 0 0
\(871\) −260.665 −0.0101404
\(872\) 0 0
\(873\) 14933.8 0.578960
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 27382.5 1.05432 0.527162 0.849765i \(-0.323256\pi\)
0.527162 + 0.849765i \(0.323256\pi\)
\(878\) 0 0
\(879\) 2428.35 0.0931812
\(880\) 0 0
\(881\) −17720.1 −0.677647 −0.338823 0.940850i \(-0.610029\pi\)
−0.338823 + 0.940850i \(0.610029\pi\)
\(882\) 0 0
\(883\) 45478.3 1.73326 0.866629 0.498953i \(-0.166282\pi\)
0.866629 + 0.498953i \(0.166282\pi\)
\(884\) 0 0
\(885\) −31799.3 −1.20782
\(886\) 0 0
\(887\) 35483.6 1.34321 0.671603 0.740911i \(-0.265606\pi\)
0.671603 + 0.740911i \(0.265606\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 819.039 0.0307956
\(892\) 0 0
\(893\) 28208.2 1.05706
\(894\) 0 0
\(895\) 46359.1 1.73141
\(896\) 0 0
\(897\) −87.3961 −0.00325315
\(898\) 0 0
\(899\) −9958.40 −0.369445
\(900\) 0 0
\(901\) −11043.3 −0.408332
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20171.7 0.740917
\(906\) 0 0
\(907\) −4269.33 −0.156296 −0.0781481 0.996942i \(-0.524901\pi\)
−0.0781481 + 0.996942i \(0.524901\pi\)
\(908\) 0 0
\(909\) 4372.55 0.159547
\(910\) 0 0
\(911\) 45428.0 1.65214 0.826069 0.563569i \(-0.190572\pi\)
0.826069 + 0.563569i \(0.190572\pi\)
\(912\) 0 0
\(913\) 1189.31 0.0431110
\(914\) 0 0
\(915\) 1720.21 0.0621511
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −28035.2 −1.00631 −0.503153 0.864198i \(-0.667827\pi\)
−0.503153 + 0.864198i \(0.667827\pi\)
\(920\) 0 0
\(921\) 11003.8 0.393690
\(922\) 0 0
\(923\) 518.515 0.0184909
\(924\) 0 0
\(925\) 11403.6 0.405351
\(926\) 0 0
\(927\) 2491.12 0.0882624
\(928\) 0 0
\(929\) 45234.1 1.59751 0.798754 0.601658i \(-0.205493\pi\)
0.798754 + 0.601658i \(0.205493\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −219.914 −0.00771668
\(934\) 0 0
\(935\) −2467.00 −0.0862882
\(936\) 0 0
\(937\) −26573.2 −0.926476 −0.463238 0.886234i \(-0.653312\pi\)
−0.463238 + 0.886234i \(0.653312\pi\)
\(938\) 0 0
\(939\) −24902.2 −0.865445
\(940\) 0 0
\(941\) 37470.6 1.29809 0.649047 0.760748i \(-0.275168\pi\)
0.649047 + 0.760748i \(0.275168\pi\)
\(942\) 0 0
\(943\) −3237.15 −0.111788
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −46525.9 −1.59650 −0.798252 0.602323i \(-0.794242\pi\)
−0.798252 + 0.602323i \(0.794242\pi\)
\(948\) 0 0
\(949\) −743.849 −0.0254440
\(950\) 0 0
\(951\) 3866.68 0.131846
\(952\) 0 0
\(953\) 7066.68 0.240202 0.120101 0.992762i \(-0.461678\pi\)
0.120101 + 0.992762i \(0.461678\pi\)
\(954\) 0 0
\(955\) 27805.0 0.942144
\(956\) 0 0
\(957\) 2230.49 0.0753410
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −11448.4 −0.384289
\(962\) 0 0
\(963\) 932.754 0.0312124
\(964\) 0 0
\(965\) −15851.3 −0.528779
\(966\) 0 0
\(967\) −31681.8 −1.05359 −0.526793 0.849994i \(-0.676606\pi\)
−0.526793 + 0.849994i \(0.676606\pi\)
\(968\) 0 0
\(969\) −3822.68 −0.126731
\(970\) 0 0
\(971\) 53411.5 1.76525 0.882625 0.470078i \(-0.155774\pi\)
0.882625 + 0.470078i \(0.155774\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −160.155 −0.00526059
\(976\) 0 0
\(977\) 56743.2 1.85811 0.929056 0.369939i \(-0.120622\pi\)
0.929056 + 0.369939i \(0.120622\pi\)
\(978\) 0 0
\(979\) −6859.61 −0.223937
\(980\) 0 0
\(981\) −451.414 −0.0146917
\(982\) 0 0
\(983\) 40797.4 1.32374 0.661869 0.749620i \(-0.269764\pi\)
0.661869 + 0.749620i \(0.269764\pi\)
\(984\) 0 0
\(985\) 68601.0 2.21910
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 13044.7 0.419411
\(990\) 0 0
\(991\) 37456.6 1.20066 0.600328 0.799754i \(-0.295037\pi\)
0.600328 + 0.799754i \(0.295037\pi\)
\(992\) 0 0
\(993\) −19553.7 −0.624892
\(994\) 0 0
\(995\) 9209.10 0.293415
\(996\) 0 0
\(997\) −9679.23 −0.307467 −0.153733 0.988112i \(-0.549130\pi\)
−0.153733 + 0.988112i \(0.549130\pi\)
\(998\) 0 0
\(999\) 3596.95 0.113916
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 1176.4.a.be.1.3 yes 4
4.3 odd 2 2352.4.a.cn.1.3 4
7.6 odd 2 1176.4.a.z.1.2 4
28.27 even 2 2352.4.a.co.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.z.1.2 4 7.6 odd 2
1176.4.a.be.1.3 yes 4 1.1 even 1 trivial
2352.4.a.cn.1.3 4 4.3 odd 2
2352.4.a.co.1.2 4 28.27 even 2