Properties

Label 2352.4.a.bq.1.2
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{1345}) \)
Defining polynomial: \(x^{2} - x - 336\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-17.8371\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +20.8371 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +20.8371 q^{5} +9.00000 q^{9} -15.1629 q^{11} +2.16288 q^{13} -62.5114 q^{15} -119.348 q^{17} +33.5114 q^{19} -0.651517 q^{23} +309.186 q^{25} -27.0000 q^{27} -163.208 q^{29} +223.326 q^{31} +45.4886 q^{33} +168.534 q^{37} -6.48864 q^{39} -323.023 q^{41} -221.557 q^{43} +187.534 q^{45} -508.045 q^{47} +358.045 q^{51} -176.511 q^{53} -315.951 q^{55} -100.534 q^{57} -454.928 q^{59} +38.6515 q^{61} +45.0682 q^{65} -141.792 q^{67} +1.95455 q^{69} -602.742 q^{71} -1102.30 q^{73} -927.557 q^{75} +116.303 q^{79} +81.0000 q^{81} +568.928 q^{83} -2486.88 q^{85} +489.625 q^{87} -383.159 q^{89} -669.977 q^{93} +698.280 q^{95} +334.701 q^{97} -136.466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} + 5q^{5} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} + 5q^{5} + 18q^{9} - 67q^{11} + 41q^{13} - 15q^{15} - 92q^{17} - 43q^{19} - 148q^{23} + 435q^{25} - 54q^{27} + 77q^{29} + 520q^{31} + 201q^{33} + 7q^{37} - 123q^{39} - 426q^{41} + 107q^{43} + 45q^{45} - 576q^{47} + 276q^{51} - 243q^{53} + 505q^{55} + 129q^{57} + 7q^{59} + 224q^{61} - 570q^{65} - 687q^{67} + 444q^{69} - 472q^{71} - 921q^{73} - 1305q^{75} + 526q^{79} + 162q^{81} + 221q^{83} - 2920q^{85} - 231q^{87} + 774q^{89} - 1560q^{93} + 1910q^{95} + 1953q^{97} - 603q^{99} + O(q^{100}) \)

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
\(5\) 20.8371 1.86373 0.931864 0.362807i \(-0.118182\pi\)
0.931864 + 0.362807i \(0.118182\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −15.1629 −0.415616 −0.207808 0.978170i \(-0.566633\pi\)
−0.207808 + 0.978170i \(0.566633\pi\)
\(12\) 0 0
\(13\) 2.16288 0.0461442 0.0230721 0.999734i \(-0.492655\pi\)
0.0230721 + 0.999734i \(0.492655\pi\)
\(14\) 0 0
\(15\) −62.5114 −1.07602
\(16\) 0 0
\(17\) −119.348 −1.70272 −0.851361 0.524581i \(-0.824222\pi\)
−0.851361 + 0.524581i \(0.824222\pi\)
\(18\) 0 0
\(19\) 33.5114 0.404633 0.202317 0.979320i \(-0.435153\pi\)
0.202317 + 0.979320i \(0.435153\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.651517 −0.00590655 −0.00295327 0.999996i \(-0.500940\pi\)
−0.00295327 + 0.999996i \(0.500940\pi\)
\(24\) 0 0
\(25\) 309.186 2.47348
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −163.208 −1.04507 −0.522535 0.852618i \(-0.675014\pi\)
−0.522535 + 0.852618i \(0.675014\pi\)
\(30\) 0 0
\(31\) 223.326 1.29389 0.646943 0.762538i \(-0.276047\pi\)
0.646943 + 0.762538i \(0.276047\pi\)
\(32\) 0 0
\(33\) 45.4886 0.239956
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 168.534 0.748833 0.374417 0.927261i \(-0.377843\pi\)
0.374417 + 0.927261i \(0.377843\pi\)
\(38\) 0 0
\(39\) −6.48864 −0.0266414
\(40\) 0 0
\(41\) −323.023 −1.23043 −0.615216 0.788359i \(-0.710931\pi\)
−0.615216 + 0.788359i \(0.710931\pi\)
\(42\) 0 0
\(43\) −221.557 −0.785746 −0.392873 0.919593i \(-0.628519\pi\)
−0.392873 + 0.919593i \(0.628519\pi\)
\(44\) 0 0
\(45\) 187.534 0.621243
\(46\) 0 0
\(47\) −508.045 −1.57672 −0.788362 0.615211i \(-0.789071\pi\)
−0.788362 + 0.615211i \(0.789071\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 358.045 0.983066
\(52\) 0 0
\(53\) −176.511 −0.457466 −0.228733 0.973489i \(-0.573458\pi\)
−0.228733 + 0.973489i \(0.573458\pi\)
\(54\) 0 0
\(55\) −315.951 −0.774596
\(56\) 0 0
\(57\) −100.534 −0.233615
\(58\) 0 0
\(59\) −454.928 −1.00384 −0.501920 0.864914i \(-0.667373\pi\)
−0.501920 + 0.864914i \(0.667373\pi\)
\(60\) 0 0
\(61\) 38.6515 0.0811282 0.0405641 0.999177i \(-0.487085\pi\)
0.0405641 + 0.999177i \(0.487085\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 45.0682 0.0860003
\(66\) 0 0
\(67\) −141.792 −0.258546 −0.129273 0.991609i \(-0.541264\pi\)
−0.129273 + 0.991609i \(0.541264\pi\)
\(68\) 0 0
\(69\) 1.95455 0.00341015
\(70\) 0 0
\(71\) −602.742 −1.00750 −0.503749 0.863850i \(-0.668046\pi\)
−0.503749 + 0.863850i \(0.668046\pi\)
\(72\) 0 0
\(73\) −1102.30 −1.76732 −0.883660 0.468129i \(-0.844928\pi\)
−0.883660 + 0.468129i \(0.844928\pi\)
\(74\) 0 0
\(75\) −927.557 −1.42807
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 116.303 0.165634 0.0828172 0.996565i \(-0.473608\pi\)
0.0828172 + 0.996565i \(0.473608\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 568.928 0.752385 0.376193 0.926542i \(-0.377233\pi\)
0.376193 + 0.926542i \(0.377233\pi\)
\(84\) 0 0
\(85\) −2486.88 −3.17341
\(86\) 0 0
\(87\) 489.625 0.603371
\(88\) 0 0
\(89\) −383.159 −0.456346 −0.228173 0.973621i \(-0.573275\pi\)
−0.228173 + 0.973621i \(0.573275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −669.977 −0.747026
\(94\) 0 0
\(95\) 698.280 0.754127
\(96\) 0 0
\(97\) 334.701 0.350348 0.175174 0.984538i \(-0.443951\pi\)
0.175174 + 0.984538i \(0.443951\pi\)
\(98\) 0 0
\(99\) −136.466 −0.138539
\(100\) 0 0
\(101\) −14.7424 −0.0145240 −0.00726201 0.999974i \(-0.502312\pi\)
−0.00726201 + 0.999974i \(0.502312\pi\)
\(102\) 0 0
\(103\) 841.420 0.804928 0.402464 0.915436i \(-0.368154\pi\)
0.402464 + 0.915436i \(0.368154\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 715.670 0.646603 0.323301 0.946296i \(-0.395207\pi\)
0.323301 + 0.946296i \(0.395207\pi\)
\(108\) 0 0
\(109\) 600.019 0.527260 0.263630 0.964624i \(-0.415080\pi\)
0.263630 + 0.964624i \(0.415080\pi\)
\(110\) 0 0
\(111\) −505.602 −0.432339
\(112\) 0 0
\(113\) 622.644 0.518349 0.259174 0.965831i \(-0.416550\pi\)
0.259174 + 0.965831i \(0.416550\pi\)
\(114\) 0 0
\(115\) −13.5757 −0.0110082
\(116\) 0 0
\(117\) 19.4659 0.0153814
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1101.09 −0.827263
\(122\) 0 0
\(123\) 969.068 0.710390
\(124\) 0 0
\(125\) 3837.90 2.74618
\(126\) 0 0
\(127\) 180.076 0.125820 0.0629100 0.998019i \(-0.479962\pi\)
0.0629100 + 0.998019i \(0.479962\pi\)
\(128\) 0 0
\(129\) 664.670 0.453651
\(130\) 0 0
\(131\) −217.860 −0.145302 −0.0726508 0.997357i \(-0.523146\pi\)
−0.0726508 + 0.997357i \(0.523146\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −562.602 −0.358675
\(136\) 0 0
\(137\) −2601.86 −1.62257 −0.811283 0.584654i \(-0.801230\pi\)
−0.811283 + 0.584654i \(0.801230\pi\)
\(138\) 0 0
\(139\) 2651.55 1.61800 0.808998 0.587811i \(-0.200010\pi\)
0.808998 + 0.587811i \(0.200010\pi\)
\(140\) 0 0
\(141\) 1524.14 0.910322
\(142\) 0 0
\(143\) −32.7955 −0.0191783
\(144\) 0 0
\(145\) −3400.79 −1.94773
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 581.023 0.319458 0.159729 0.987161i \(-0.448938\pi\)
0.159729 + 0.987161i \(0.448938\pi\)
\(150\) 0 0
\(151\) 615.390 0.331654 0.165827 0.986155i \(-0.446971\pi\)
0.165827 + 0.986155i \(0.446971\pi\)
\(152\) 0 0
\(153\) −1074.14 −0.567574
\(154\) 0 0
\(155\) 4653.47 2.41145
\(156\) 0 0
\(157\) −306.932 −0.156024 −0.0780122 0.996952i \(-0.524857\pi\)
−0.0780122 + 0.996952i \(0.524857\pi\)
\(158\) 0 0
\(159\) 529.534 0.264118
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3514.50 −1.68882 −0.844408 0.535701i \(-0.820047\pi\)
−0.844408 + 0.535701i \(0.820047\pi\)
\(164\) 0 0
\(165\) 947.852 0.447213
\(166\) 0 0
\(167\) −1123.30 −0.520502 −0.260251 0.965541i \(-0.583805\pi\)
−0.260251 + 0.965541i \(0.583805\pi\)
\(168\) 0 0
\(169\) −2192.32 −0.997871
\(170\) 0 0
\(171\) 301.602 0.134878
\(172\) 0 0
\(173\) 1530.60 0.672655 0.336327 0.941745i \(-0.390815\pi\)
0.336327 + 0.941745i \(0.390815\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1364.78 0.579568
\(178\) 0 0
\(179\) −3413.43 −1.42532 −0.712659 0.701511i \(-0.752509\pi\)
−0.712659 + 0.701511i \(0.752509\pi\)
\(180\) 0 0
\(181\) 1286.71 0.528399 0.264200 0.964468i \(-0.414892\pi\)
0.264200 + 0.964468i \(0.414892\pi\)
\(182\) 0 0
\(183\) −115.955 −0.0468394
\(184\) 0 0
\(185\) 3511.77 1.39562
\(186\) 0 0
\(187\) 1809.67 0.707679
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1055.30 −0.399783 −0.199891 0.979818i \(-0.564059\pi\)
−0.199891 + 0.979818i \(0.564059\pi\)
\(192\) 0 0
\(193\) −4770.84 −1.77934 −0.889670 0.456604i \(-0.849066\pi\)
−0.889670 + 0.456604i \(0.849066\pi\)
\(194\) 0 0
\(195\) −135.205 −0.0496523
\(196\) 0 0
\(197\) 1622.31 0.586725 0.293363 0.956001i \(-0.405226\pi\)
0.293363 + 0.956001i \(0.405226\pi\)
\(198\) 0 0
\(199\) 3550.14 1.26464 0.632318 0.774709i \(-0.282104\pi\)
0.632318 + 0.774709i \(0.282104\pi\)
\(200\) 0 0
\(201\) 425.375 0.149272
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −6730.86 −2.29319
\(206\) 0 0
\(207\) −5.86365 −0.00196885
\(208\) 0 0
\(209\) −508.129 −0.168172
\(210\) 0 0
\(211\) −4653.39 −1.51826 −0.759129 0.650941i \(-0.774375\pi\)
−0.759129 + 0.650941i \(0.774375\pi\)
\(212\) 0 0
\(213\) 1808.23 0.581679
\(214\) 0 0
\(215\) −4616.61 −1.46442
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3306.90 1.02036
\(220\) 0 0
\(221\) −258.136 −0.0785707
\(222\) 0 0
\(223\) 4649.53 1.39621 0.698107 0.715993i \(-0.254026\pi\)
0.698107 + 0.715993i \(0.254026\pi\)
\(224\) 0 0
\(225\) 2782.67 0.824495
\(226\) 0 0
\(227\) −4151.72 −1.21392 −0.606958 0.794734i \(-0.707611\pi\)
−0.606958 + 0.794734i \(0.707611\pi\)
\(228\) 0 0
\(229\) 4263.63 1.23034 0.615172 0.788393i \(-0.289087\pi\)
0.615172 + 0.788393i \(0.289087\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3049.90 0.857535 0.428768 0.903415i \(-0.358948\pi\)
0.428768 + 0.903415i \(0.358948\pi\)
\(234\) 0 0
\(235\) −10586.2 −2.93859
\(236\) 0 0
\(237\) −348.909 −0.0956290
\(238\) 0 0
\(239\) −3987.20 −1.07912 −0.539562 0.841946i \(-0.681410\pi\)
−0.539562 + 0.841946i \(0.681410\pi\)
\(240\) 0 0
\(241\) −624.648 −0.166959 −0.0834795 0.996509i \(-0.526603\pi\)
−0.0834795 + 0.996509i \(0.526603\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 72.4810 0.0186715
\(248\) 0 0
\(249\) −1706.78 −0.434390
\(250\) 0 0
\(251\) 1328.78 0.334152 0.167076 0.985944i \(-0.446568\pi\)
0.167076 + 0.985944i \(0.446568\pi\)
\(252\) 0 0
\(253\) 9.87887 0.00245486
\(254\) 0 0
\(255\) 7460.64 1.83217
\(256\) 0 0
\(257\) −3226.18 −0.783049 −0.391525 0.920168i \(-0.628052\pi\)
−0.391525 + 0.920168i \(0.628052\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1468.87 −0.348357
\(262\) 0 0
\(263\) −3250.61 −0.762135 −0.381067 0.924547i \(-0.624443\pi\)
−0.381067 + 0.924547i \(0.624443\pi\)
\(264\) 0 0
\(265\) −3677.99 −0.852593
\(266\) 0 0
\(267\) 1149.48 0.263471
\(268\) 0 0
\(269\) 2826.04 0.640546 0.320273 0.947325i \(-0.396225\pi\)
0.320273 + 0.947325i \(0.396225\pi\)
\(270\) 0 0
\(271\) 2396.77 0.537245 0.268622 0.963246i \(-0.413432\pi\)
0.268622 + 0.963246i \(0.413432\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4688.14 −1.02802
\(276\) 0 0
\(277\) 1820.47 0.394878 0.197439 0.980315i \(-0.436738\pi\)
0.197439 + 0.980315i \(0.436738\pi\)
\(278\) 0 0
\(279\) 2009.93 0.431296
\(280\) 0 0
\(281\) 3083.81 0.654679 0.327339 0.944907i \(-0.393848\pi\)
0.327339 + 0.944907i \(0.393848\pi\)
\(282\) 0 0
\(283\) −2554.77 −0.536626 −0.268313 0.963332i \(-0.586466\pi\)
−0.268313 + 0.963332i \(0.586466\pi\)
\(284\) 0 0
\(285\) −2094.84 −0.435395
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 9331.06 1.89926
\(290\) 0 0
\(291\) −1004.10 −0.202273
\(292\) 0 0
\(293\) −1846.47 −0.368163 −0.184081 0.982911i \(-0.558931\pi\)
−0.184081 + 0.982911i \(0.558931\pi\)
\(294\) 0 0
\(295\) −9479.39 −1.87089
\(296\) 0 0
\(297\) 409.398 0.0799854
\(298\) 0 0
\(299\) −1.40915 −0.000272553 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 44.2272 0.00838544
\(304\) 0 0
\(305\) 805.386 0.151201
\(306\) 0 0
\(307\) −7041.50 −1.30905 −0.654527 0.756039i \(-0.727132\pi\)
−0.654527 + 0.756039i \(0.727132\pi\)
\(308\) 0 0
\(309\) −2524.26 −0.464726
\(310\) 0 0
\(311\) 2685.99 0.489738 0.244869 0.969556i \(-0.421255\pi\)
0.244869 + 0.969556i \(0.421255\pi\)
\(312\) 0 0
\(313\) 2219.19 0.400754 0.200377 0.979719i \(-0.435783\pi\)
0.200377 + 0.979719i \(0.435783\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2221.26 0.393560 0.196780 0.980448i \(-0.436952\pi\)
0.196780 + 0.980448i \(0.436952\pi\)
\(318\) 0 0
\(319\) 2474.71 0.434348
\(320\) 0 0
\(321\) −2147.01 −0.373316
\(322\) 0 0
\(323\) −3999.53 −0.688978
\(324\) 0 0
\(325\) 668.731 0.114137
\(326\) 0 0
\(327\) −1800.06 −0.304414
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4154.06 −0.689812 −0.344906 0.938637i \(-0.612089\pi\)
−0.344906 + 0.938637i \(0.612089\pi\)
\(332\) 0 0
\(333\) 1516.81 0.249611
\(334\) 0 0
\(335\) −2954.53 −0.481860
\(336\) 0 0
\(337\) −254.167 −0.0410841 −0.0205420 0.999789i \(-0.506539\pi\)
−0.0205420 + 0.999789i \(0.506539\pi\)
\(338\) 0 0
\(339\) −1867.93 −0.299269
\(340\) 0 0
\(341\) −3386.26 −0.537761
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 40.7272 0.00635559
\(346\) 0 0
\(347\) 6224.64 0.962986 0.481493 0.876450i \(-0.340095\pi\)
0.481493 + 0.876450i \(0.340095\pi\)
\(348\) 0 0
\(349\) 9732.21 1.49270 0.746352 0.665552i \(-0.231804\pi\)
0.746352 + 0.665552i \(0.231804\pi\)
\(350\) 0 0
\(351\) −58.3977 −0.00888046
\(352\) 0 0
\(353\) 1425.61 0.214951 0.107476 0.994208i \(-0.465723\pi\)
0.107476 + 0.994208i \(0.465723\pi\)
\(354\) 0 0
\(355\) −12559.4 −1.87770
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5766.49 0.847754 0.423877 0.905720i \(-0.360669\pi\)
0.423877 + 0.905720i \(0.360669\pi\)
\(360\) 0 0
\(361\) −5735.99 −0.836272
\(362\) 0 0
\(363\) 3303.26 0.477621
\(364\) 0 0
\(365\) −22968.7 −3.29381
\(366\) 0 0
\(367\) −11545.3 −1.64213 −0.821065 0.570834i \(-0.806620\pi\)
−0.821065 + 0.570834i \(0.806620\pi\)
\(368\) 0 0
\(369\) −2907.20 −0.410144
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6479.57 −0.899463 −0.449731 0.893164i \(-0.648480\pi\)
−0.449731 + 0.893164i \(0.648480\pi\)
\(374\) 0 0
\(375\) −11513.7 −1.58551
\(376\) 0 0
\(377\) −353.000 −0.0482239
\(378\) 0 0
\(379\) −611.996 −0.0829449 −0.0414725 0.999140i \(-0.513205\pi\)
−0.0414725 + 0.999140i \(0.513205\pi\)
\(380\) 0 0
\(381\) −540.227 −0.0726422
\(382\) 0 0
\(383\) −4360.81 −0.581794 −0.290897 0.956754i \(-0.593954\pi\)
−0.290897 + 0.956754i \(0.593954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1994.01 −0.261915
\(388\) 0 0
\(389\) −13146.9 −1.71356 −0.856781 0.515681i \(-0.827539\pi\)
−0.856781 + 0.515681i \(0.827539\pi\)
\(390\) 0 0
\(391\) 77.7575 0.0100572
\(392\) 0 0
\(393\) 653.580 0.0838899
\(394\) 0 0
\(395\) 2423.42 0.308697
\(396\) 0 0
\(397\) −8478.04 −1.07179 −0.535895 0.844285i \(-0.680026\pi\)
−0.535895 + 0.844285i \(0.680026\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2803.00 0.349065 0.174533 0.984651i \(-0.444159\pi\)
0.174533 + 0.984651i \(0.444159\pi\)
\(402\) 0 0
\(403\) 483.027 0.0597054
\(404\) 0 0
\(405\) 1687.81 0.207081
\(406\) 0 0
\(407\) −2555.46 −0.311227
\(408\) 0 0
\(409\) −6385.39 −0.771973 −0.385987 0.922504i \(-0.626139\pi\)
−0.385987 + 0.922504i \(0.626139\pi\)
\(410\) 0 0
\(411\) 7805.57 0.936789
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 11854.8 1.40224
\(416\) 0 0
\(417\) −7954.65 −0.934151
\(418\) 0 0
\(419\) −4831.66 −0.563346 −0.281673 0.959510i \(-0.590889\pi\)
−0.281673 + 0.959510i \(0.590889\pi\)
\(420\) 0 0
\(421\) 7475.37 0.865385 0.432693 0.901542i \(-0.357564\pi\)
0.432693 + 0.901542i \(0.357564\pi\)
\(422\) 0 0
\(423\) −4572.41 −0.525575
\(424\) 0 0
\(425\) −36900.8 −4.21165
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 98.3864 0.0110726
\(430\) 0 0
\(431\) −6991.93 −0.781414 −0.390707 0.920515i \(-0.627769\pi\)
−0.390707 + 0.920515i \(0.627769\pi\)
\(432\) 0 0
\(433\) −7699.26 −0.854510 −0.427255 0.904131i \(-0.640519\pi\)
−0.427255 + 0.904131i \(0.640519\pi\)
\(434\) 0 0
\(435\) 10202.4 1.12452
\(436\) 0 0
\(437\) −21.8332 −0.00238999
\(438\) 0 0
\(439\) −9412.32 −1.02329 −0.511646 0.859196i \(-0.670964\pi\)
−0.511646 + 0.859196i \(0.670964\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6258.18 0.671185 0.335593 0.942007i \(-0.391063\pi\)
0.335593 + 0.942007i \(0.391063\pi\)
\(444\) 0 0
\(445\) −7983.93 −0.850505
\(446\) 0 0
\(447\) −1743.07 −0.184439
\(448\) 0 0
\(449\) −11633.8 −1.22279 −0.611396 0.791325i \(-0.709392\pi\)
−0.611396 + 0.791325i \(0.709392\pi\)
\(450\) 0 0
\(451\) 4897.95 0.511387
\(452\) 0 0
\(453\) −1846.17 −0.191480
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −13104.6 −1.34138 −0.670688 0.741740i \(-0.734001\pi\)
−0.670688 + 0.741740i \(0.734001\pi\)
\(458\) 0 0
\(459\) 3222.41 0.327689
\(460\) 0 0
\(461\) −2594.63 −0.262134 −0.131067 0.991373i \(-0.541840\pi\)
−0.131067 + 0.991373i \(0.541840\pi\)
\(462\) 0 0
\(463\) 14136.2 1.41893 0.709465 0.704741i \(-0.248937\pi\)
0.709465 + 0.704741i \(0.248937\pi\)
\(464\) 0 0
\(465\) −13960.4 −1.39225
\(466\) 0 0
\(467\) −15590.2 −1.54482 −0.772409 0.635125i \(-0.780948\pi\)
−0.772409 + 0.635125i \(0.780948\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 920.795 0.0900807
\(472\) 0 0
\(473\) 3359.44 0.326569
\(474\) 0 0
\(475\) 10361.2 1.00085
\(476\) 0 0
\(477\) −1588.60 −0.152489
\(478\) 0 0
\(479\) 8453.51 0.806369 0.403184 0.915119i \(-0.367903\pi\)
0.403184 + 0.915119i \(0.367903\pi\)
\(480\) 0 0
\(481\) 364.519 0.0345543
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6974.20 0.652953
\(486\) 0 0
\(487\) 4011.07 0.373221 0.186611 0.982434i \(-0.440250\pi\)
0.186611 + 0.982434i \(0.440250\pi\)
\(488\) 0 0
\(489\) 10543.5 0.975038
\(490\) 0 0
\(491\) −13927.9 −1.28016 −0.640079 0.768309i \(-0.721098\pi\)
−0.640079 + 0.768309i \(0.721098\pi\)
\(492\) 0 0
\(493\) 19478.7 1.77946
\(494\) 0 0
\(495\) −2843.56 −0.258199
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 3947.55 0.354141 0.177071 0.984198i \(-0.443338\pi\)
0.177071 + 0.984198i \(0.443338\pi\)
\(500\) 0 0
\(501\) 3369.91 0.300512
\(502\) 0 0
\(503\) 13725.3 1.21666 0.608331 0.793684i \(-0.291839\pi\)
0.608331 + 0.793684i \(0.291839\pi\)
\(504\) 0 0
\(505\) −307.190 −0.0270688
\(506\) 0 0
\(507\) 6576.97 0.576121
\(508\) 0 0
\(509\) 7830.10 0.681853 0.340926 0.940090i \(-0.389259\pi\)
0.340926 + 0.940090i \(0.389259\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −904.807 −0.0778717
\(514\) 0 0
\(515\) 17532.8 1.50017
\(516\) 0 0
\(517\) 7703.43 0.655312
\(518\) 0 0
\(519\) −4591.80 −0.388357
\(520\) 0 0
\(521\) −5907.39 −0.496751 −0.248376 0.968664i \(-0.579897\pi\)
−0.248376 + 0.968664i \(0.579897\pi\)
\(522\) 0 0
\(523\) −7908.06 −0.661176 −0.330588 0.943775i \(-0.607247\pi\)
−0.330588 + 0.943775i \(0.607247\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −26653.6 −2.20313
\(528\) 0 0
\(529\) −12166.6 −0.999965
\(530\) 0 0
\(531\) −4094.35 −0.334613
\(532\) 0 0
\(533\) −698.659 −0.0567773
\(534\) 0 0
\(535\) 14912.5 1.20509
\(536\) 0 0
\(537\) 10240.3 0.822908
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3941.04 −0.313195 −0.156598 0.987662i \(-0.550053\pi\)
−0.156598 + 0.987662i \(0.550053\pi\)
\(542\) 0 0
\(543\) −3860.12 −0.305071
\(544\) 0 0
\(545\) 12502.7 0.982670
\(546\) 0 0
\(547\) 1828.71 0.142943 0.0714717 0.997443i \(-0.477230\pi\)
0.0714717 + 0.997443i \(0.477230\pi\)
\(548\) 0 0
\(549\) 347.864 0.0270427
\(550\) 0 0
\(551\) −5469.33 −0.422870
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −10535.3 −0.805763
\(556\) 0 0
\(557\) 22532.0 1.71402 0.857011 0.515298i \(-0.172319\pi\)
0.857011 + 0.515298i \(0.172319\pi\)
\(558\) 0 0
\(559\) −479.201 −0.0362577
\(560\) 0 0
\(561\) −5429.00 −0.408579
\(562\) 0 0
\(563\) −23355.7 −1.74836 −0.874179 0.485604i \(-0.838600\pi\)
−0.874179 + 0.485604i \(0.838600\pi\)
\(564\) 0 0
\(565\) 12974.1 0.966062
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −20887.6 −1.53894 −0.769468 0.638686i \(-0.779478\pi\)
−0.769468 + 0.638686i \(0.779478\pi\)
\(570\) 0 0
\(571\) −23745.3 −1.74029 −0.870147 0.492792i \(-0.835976\pi\)
−0.870147 + 0.492792i \(0.835976\pi\)
\(572\) 0 0
\(573\) 3165.89 0.230815
\(574\) 0 0
\(575\) −201.440 −0.0146098
\(576\) 0 0
\(577\) 2454.39 0.177084 0.0885422 0.996072i \(-0.471779\pi\)
0.0885422 + 0.996072i \(0.471779\pi\)
\(578\) 0 0
\(579\) 14312.5 1.02730
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2676.42 0.190130
\(584\) 0 0
\(585\) 405.614 0.0286668
\(586\) 0 0
\(587\) −18567.5 −1.30556 −0.652780 0.757547i \(-0.726397\pi\)
−0.652780 + 0.757547i \(0.726397\pi\)
\(588\) 0 0
\(589\) 7483.95 0.523550
\(590\) 0 0
\(591\) −4866.93 −0.338746
\(592\) 0 0
\(593\) 17112.9 1.18507 0.592533 0.805546i \(-0.298128\pi\)
0.592533 + 0.805546i \(0.298128\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −10650.4 −0.730138
\(598\) 0 0
\(599\) −23264.8 −1.58694 −0.793469 0.608611i \(-0.791727\pi\)
−0.793469 + 0.608611i \(0.791727\pi\)
\(600\) 0 0
\(601\) 25322.3 1.71867 0.859334 0.511416i \(-0.170879\pi\)
0.859334 + 0.511416i \(0.170879\pi\)
\(602\) 0 0
\(603\) −1276.13 −0.0861821
\(604\) 0 0
\(605\) −22943.5 −1.54179
\(606\) 0 0
\(607\) 21734.4 1.45333 0.726665 0.686992i \(-0.241069\pi\)
0.726665 + 0.686992i \(0.241069\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1098.84 −0.0727567
\(612\) 0 0
\(613\) −13572.4 −0.894262 −0.447131 0.894468i \(-0.647554\pi\)
−0.447131 + 0.894468i \(0.647554\pi\)
\(614\) 0 0
\(615\) 20192.6 1.32397
\(616\) 0 0
\(617\) −8497.12 −0.554427 −0.277213 0.960808i \(-0.589411\pi\)
−0.277213 + 0.960808i \(0.589411\pi\)
\(618\) 0 0
\(619\) 22982.9 1.49235 0.746173 0.665752i \(-0.231889\pi\)
0.746173 + 0.665752i \(0.231889\pi\)
\(620\) 0 0
\(621\) 17.5910 0.00113672
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 41322.5 2.64464
\(626\) 0 0
\(627\) 1524.39 0.0970943
\(628\) 0 0
\(629\) −20114.3 −1.27505
\(630\) 0 0
\(631\) 15717.9 0.991635 0.495817 0.868427i \(-0.334869\pi\)
0.495817 + 0.868427i \(0.334869\pi\)
\(632\) 0 0
\(633\) 13960.2 0.876566
\(634\) 0 0
\(635\) 3752.26 0.234494
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −5424.68 −0.335833
\(640\) 0 0
\(641\) 29107.4 1.79356 0.896780 0.442478i \(-0.145900\pi\)
0.896780 + 0.442478i \(0.145900\pi\)
\(642\) 0 0
\(643\) 3112.26 0.190880 0.0954398 0.995435i \(-0.469574\pi\)
0.0954398 + 0.995435i \(0.469574\pi\)
\(644\) 0 0
\(645\) 13849.8 0.845482
\(646\) 0 0
\(647\) 7857.59 0.477456 0.238728 0.971087i \(-0.423270\pi\)
0.238728 + 0.971087i \(0.423270\pi\)
\(648\) 0 0
\(649\) 6898.02 0.417213
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19522.0 −1.16992 −0.584958 0.811063i \(-0.698889\pi\)
−0.584958 + 0.811063i \(0.698889\pi\)
\(654\) 0 0
\(655\) −4539.57 −0.270803
\(656\) 0 0
\(657\) −9920.69 −0.589107
\(658\) 0 0
\(659\) −664.061 −0.0392536 −0.0196268 0.999807i \(-0.506248\pi\)
−0.0196268 + 0.999807i \(0.506248\pi\)
\(660\) 0 0
\(661\) 15921.6 0.936883 0.468442 0.883494i \(-0.344816\pi\)
0.468442 + 0.883494i \(0.344816\pi\)
\(662\) 0 0
\(663\) 774.409 0.0453628
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 106.333 0.00617276
\(668\) 0 0
\(669\) −13948.6 −0.806105
\(670\) 0 0
\(671\) −586.068 −0.0337182
\(672\) 0 0
\(673\) 24631.0 1.41078 0.705391 0.708819i \(-0.250771\pi\)
0.705391 + 0.708819i \(0.250771\pi\)
\(674\) 0 0
\(675\) −8348.01 −0.476022
\(676\) 0 0
\(677\) 17092.8 0.970353 0.485177 0.874416i \(-0.338755\pi\)
0.485177 + 0.874416i \(0.338755\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 12455.1 0.700855
\(682\) 0 0
\(683\) 19163.6 1.07361 0.536804 0.843707i \(-0.319632\pi\)
0.536804 + 0.843707i \(0.319632\pi\)
\(684\) 0 0
\(685\) −54215.2 −3.02402
\(686\) 0 0
\(687\) −12790.9 −0.710339
\(688\) 0 0
\(689\) −381.773 −0.0211094
\(690\) 0 0
\(691\) −8095.87 −0.445704 −0.222852 0.974852i \(-0.571537\pi\)
−0.222852 + 0.974852i \(0.571537\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 55250.7 3.01551
\(696\) 0 0
\(697\) 38552.3 2.09508
\(698\) 0 0
\(699\) −9149.70 −0.495098
\(700\) 0 0
\(701\) 12354.7 0.665664 0.332832 0.942986i \(-0.391996\pi\)
0.332832 + 0.942986i \(0.391996\pi\)
\(702\) 0 0
\(703\) 5647.81 0.303003
\(704\) 0 0
\(705\) 31758.6 1.69659
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3828.82 0.202813 0.101406 0.994845i \(-0.467666\pi\)
0.101406 + 0.994845i \(0.467666\pi\)
\(710\) 0 0
\(711\) 1046.73 0.0552114
\(712\) 0 0
\(713\) −145.500 −0.00764241
\(714\) 0 0
\(715\) −683.363 −0.0357431
\(716\) 0 0
\(717\) 11961.6 0.623033
\(718\) 0 0
\(719\) −1223.00 −0.0634356 −0.0317178 0.999497i \(-0.510098\pi\)
−0.0317178 + 0.999497i \(0.510098\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1873.94 0.0963938
\(724\) 0 0
\(725\) −50461.7 −2.58496
\(726\) 0 0
\(727\) −6368.21 −0.324875 −0.162437 0.986719i \(-0.551936\pi\)
−0.162437 + 0.986719i \(0.551936\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 26442.5 1.33791
\(732\) 0 0
\(733\) −25154.0 −1.26751 −0.633753 0.773535i \(-0.718487\pi\)
−0.633753 + 0.773535i \(0.718487\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2149.97 0.107456
\(738\) 0 0
\(739\) 10739.1 0.534566 0.267283 0.963618i \(-0.413874\pi\)
0.267283 + 0.963618i \(0.413874\pi\)
\(740\) 0 0
\(741\) −217.443 −0.0107800
\(742\) 0 0
\(743\) −28166.3 −1.39074 −0.695370 0.718652i \(-0.744760\pi\)
−0.695370 + 0.718652i \(0.744760\pi\)
\(744\) 0 0
\(745\) 12106.8 0.595383
\(746\) 0 0
\(747\) 5120.35 0.250795
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −28657.0 −1.39242 −0.696211 0.717837i \(-0.745132\pi\)
−0.696211 + 0.717837i \(0.745132\pi\)
\(752\) 0 0
\(753\) −3986.35 −0.192923
\(754\) 0 0
\(755\) 12823.0 0.618113
\(756\) 0 0
\(757\) −23604.1 −1.13330 −0.566648 0.823960i \(-0.691760\pi\)
−0.566648 + 0.823960i \(0.691760\pi\)
\(758\) 0 0
\(759\) −29.6366 −0.00141731
\(760\) 0 0
\(761\) 4630.97 0.220595 0.110297 0.993899i \(-0.464820\pi\)
0.110297 + 0.993899i \(0.464820\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −22381.9 −1.05780
\(766\) 0 0
\(767\) −983.954 −0.0463214
\(768\) 0 0
\(769\) 33276.8 1.56046 0.780228 0.625495i \(-0.215103\pi\)
0.780228 + 0.625495i \(0.215103\pi\)
\(770\) 0 0
\(771\) 9678.55 0.452094
\(772\) 0 0
\(773\) −22938.8 −1.06734 −0.533668 0.845694i \(-0.679187\pi\)
−0.533668 + 0.845694i \(0.679187\pi\)
\(774\) 0 0
\(775\) 69049.1 3.20041
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −10824.9 −0.497873
\(780\) 0 0
\(781\) 9139.31 0.418733
\(782\) 0 0
\(783\) 4406.62 0.201124
\(784\) 0 0
\(785\) −6395.58 −0.290787
\(786\) 0 0
\(787\) 13514.5 0.612120 0.306060 0.952012i \(-0.400989\pi\)
0.306060 + 0.952012i \(0.400989\pi\)
\(788\) 0 0
\(789\) 9751.84 0.440019
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 83.5986 0.00374360
\(794\) 0 0
\(795\) 11034.0 0.492245
\(796\) 0 0
\(797\) 10473.4 0.465480 0.232740 0.972539i \(-0.425231\pi\)
0.232740 + 0.972539i \(0.425231\pi\)
\(798\) 0 0
\(799\) 60634.5 2.68472
\(800\) 0 0
\(801\) −3448.43 −0.152115
\(802\) 0 0
\(803\) 16714.0 0.734527
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −8478.12 −0.369819
\(808\) 0 0
\(809\) 23568.0 1.02423 0.512117 0.858916i \(-0.328861\pi\)
0.512117 + 0.858916i \(0.328861\pi\)
\(810\) 0 0
\(811\) 6704.22 0.290280 0.145140 0.989411i \(-0.453637\pi\)
0.145140 + 0.989411i \(0.453637\pi\)
\(812\) 0 0
\(813\) −7190.31 −0.310178
\(814\) 0 0
\(815\) −73232.1 −3.14749
\(816\) 0 0
\(817\) −7424.67 −0.317939
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 24539.4 1.04316 0.521579 0.853203i \(-0.325343\pi\)
0.521579 + 0.853203i \(0.325343\pi\)
\(822\) 0 0
\(823\) 31117.0 1.31795 0.658973 0.752167i \(-0.270991\pi\)
0.658973 + 0.752167i \(0.270991\pi\)
\(824\) 0 0
\(825\) 14064.4 0.593528
\(826\) 0 0
\(827\) −31244.9 −1.31377 −0.656887 0.753989i \(-0.728127\pi\)
−0.656887 + 0.753989i \(0.728127\pi\)
\(828\) 0 0
\(829\) 4231.50 0.177281 0.0886405 0.996064i \(-0.471748\pi\)
0.0886405 + 0.996064i \(0.471748\pi\)
\(830\) 0 0
\(831\) −5461.40 −0.227983
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −23406.4 −0.970074
\(836\) 0 0
\(837\) −6029.80 −0.249009
\(838\) 0 0
\(839\) 38670.4 1.59124 0.795621 0.605795i \(-0.207145\pi\)
0.795621 + 0.605795i \(0.207145\pi\)
\(840\) 0 0
\(841\) 2247.96 0.0921710
\(842\) 0 0
\(843\) −9251.43 −0.377979
\(844\) 0 0
\(845\) −45681.7 −1.85976
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7664.31 0.309821
\(850\) 0 0
\(851\) −109.803 −0.00442302
\(852\) 0 0
\(853\) −19944.4 −0.800565 −0.400282 0.916392i \(-0.631088\pi\)
−0.400282 + 0.916392i \(0.631088\pi\)
\(854\) 0 0
\(855\) 6284.52 0.251376
\(856\) 0 0
\(857\) 13882.2 0.553334 0.276667 0.960966i \(-0.410770\pi\)
0.276667 + 0.960966i \(0.410770\pi\)
\(858\) 0 0
\(859\) 4157.16 0.165123 0.0825614 0.996586i \(-0.473690\pi\)
0.0825614 + 0.996586i \(0.473690\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16237.9 0.640493 0.320246 0.947334i \(-0.396234\pi\)
0.320246 + 0.947334i \(0.396234\pi\)
\(864\) 0 0
\(865\) 31893.3 1.25365
\(866\) 0 0
\(867\) −27993.2 −1.09654
\(868\) 0 0
\(869\) −1763.49 −0.0688403
\(870\) 0 0
\(871\) −306.678 −0.0119304
\(872\) 0 0
\(873\) 3012.31 0.116783
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −16489.4 −0.634900 −0.317450 0.948275i \(-0.602827\pi\)
−0.317450 + 0.948275i \(0.602827\pi\)
\(878\) 0 0
\(879\) 5539.40 0.212559
\(880\) 0 0
\(881\) 45411.7 1.73662 0.868309 0.496023i \(-0.165207\pi\)
0.868309 + 0.496023i \(0.165207\pi\)
\(882\) 0 0
\(883\) 2206.85 0.0841070 0.0420535 0.999115i \(-0.486610\pi\)
0.0420535 + 0.999115i \(0.486610\pi\)
\(884\) 0 0
\(885\) 28438.2 1.08016
\(886\) 0 0
\(887\) −28146.2 −1.06545 −0.532727 0.846287i \(-0.678833\pi\)
−0.532727 + 0.846287i \(0.678833\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1228.19 −0.0461796
\(892\) 0 0
\(893\) −17025.3 −0.637995
\(894\) 0 0
\(895\) −71126.1 −2.65641
\(896\) 0 0
\(897\) 4.22746 0.000157359 0
\(898\) 0 0
\(899\) −36448.6 −1.35220
\(900\) 0 0
\(901\) 21066.4 0.778937
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 26811.3 0.984793
\(906\) 0 0
\(907\) 5042.25 0.184592 0.0922960 0.995732i \(-0.470579\pi\)
0.0922960 + 0.995732i \(0.470579\pi\)
\(908\) 0 0
\(909\) −132.682 −0.00484134
\(910\) 0 0
\(911\) −29647.3 −1.07822 −0.539110 0.842235i \(-0.681239\pi\)
−0.539110 + 0.842235i \(0.681239\pi\)
\(912\) 0 0
\(913\) −8626.59 −0.312704
\(914\) 0 0
\(915\) −2416.16 −0.0872959
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 11891.3 0.426830 0.213415 0.976962i \(-0.431541\pi\)
0.213415 + 0.976962i \(0.431541\pi\)
\(920\) 0 0
\(921\) 21124.5 0.755782
\(922\) 0 0
\(923\) −1303.66 −0.0464902
\(924\) 0 0
\(925\) 52108.3 1.85223
\(926\) 0 0
\(927\) 7572.78 0.268309
\(928\) 0 0
\(929\) −39188.5 −1.38400 −0.691999 0.721898i \(-0.743270\pi\)
−0.691999 + 0.721898i \(0.743270\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −8057.98 −0.282751
\(934\) 0 0
\(935\) 37708.2 1.31892
\(936\) 0 0
\(937\) 9716.23 0.338757 0.169379 0.985551i \(-0.445824\pi\)
0.169379 + 0.985551i \(0.445824\pi\)
\(938\) 0 0
\(939\) −6657.57 −0.231375
\(940\) 0 0
\(941\) −6995.87 −0.242358 −0.121179 0.992631i \(-0.538667\pi\)
−0.121179 + 0.992631i \(0.538667\pi\)
\(942\) 0 0
\(943\) 210.455 0.00726760
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −14979.2 −0.514002 −0.257001 0.966411i \(-0.582734\pi\)
−0.257001 + 0.966411i \(0.582734\pi\)
\(948\) 0 0
\(949\) −2384.14 −0.0815516
\(950\) 0 0
\(951\) −6663.78 −0.227222
\(952\) 0 0
\(953\) 29393.3 0.999100 0.499550 0.866285i \(-0.333499\pi\)
0.499550 + 0.866285i \(0.333499\pi\)
\(954\) 0 0
\(955\) −21989.3 −0.745087
\(956\) 0 0
\(957\) −7424.12 −0.250771
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 20083.4 0.674143
\(962\) 0 0
\(963\) 6441.03 0.215534
\(964\) 0 0
\(965\) −99410.6 −3.31621
\(966\) 0 0
\(967\) 7133.95 0.237241 0.118621 0.992940i \(-0.462153\pi\)
0.118621 + 0.992940i \(0.462153\pi\)
\(968\) 0 0
\(969\) 11998.6 0.397781
\(970\) 0 0
\(971\) 9688.13 0.320192 0.160096 0.987101i \(-0.448820\pi\)
0.160096 + 0.987101i \(0.448820\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −2006.19 −0.0658970
\(976\) 0 0
\(977\) 21305.7 0.697676 0.348838 0.937183i \(-0.386576\pi\)
0.348838 + 0.937183i \(0.386576\pi\)
\(978\) 0 0
\(979\) 5809.79 0.189665
\(980\) 0 0
\(981\) 5400.17 0.175753
\(982\) 0 0
\(983\) 37280.8 1.20964 0.604818 0.796364i \(-0.293246\pi\)
0.604818 + 0.796364i \(0.293246\pi\)
\(984\) 0 0
\(985\) 33804.3 1.09350
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 144.348 0.00464105
\(990\) 0 0
\(991\) 51397.1 1.64751 0.823755 0.566946i \(-0.191875\pi\)
0.823755 + 0.566946i \(0.191875\pi\)
\(992\) 0 0
\(993\) 12462.2 0.398263
\(994\) 0 0
\(995\) 73974.6 2.35694
\(996\) 0 0
\(997\) −34373.8 −1.09191 −0.545953 0.837816i \(-0.683832\pi\)
−0.545953 + 0.837816i \(0.683832\pi\)
\(998\) 0 0
\(999\) −4550.42 −0.144113
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.bq.1.2 2
4.3 odd 2 294.4.a.n.1.2 2
7.2 even 3 336.4.q.j.193.1 4
7.4 even 3 336.4.q.j.289.1 4
7.6 odd 2 2352.4.a.ca.1.1 2
12.11 even 2 882.4.a.v.1.1 2
28.3 even 6 294.4.e.l.79.2 4
28.11 odd 6 42.4.e.c.37.1 yes 4
28.19 even 6 294.4.e.l.67.2 4
28.23 odd 6 42.4.e.c.25.1 4
28.27 even 2 294.4.a.m.1.1 2
84.11 even 6 126.4.g.g.37.2 4
84.23 even 6 126.4.g.g.109.2 4
84.47 odd 6 882.4.g.bf.361.1 4
84.59 odd 6 882.4.g.bf.667.1 4
84.83 odd 2 882.4.a.z.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.4.e.c.25.1 4 28.23 odd 6
42.4.e.c.37.1 yes 4 28.11 odd 6
126.4.g.g.37.2 4 84.11 even 6
126.4.g.g.109.2 4 84.23 even 6
294.4.a.m.1.1 2 28.27 even 2
294.4.a.n.1.2 2 4.3 odd 2
294.4.e.l.67.2 4 28.19 even 6
294.4.e.l.79.2 4 28.3 even 6
336.4.q.j.193.1 4 7.2 even 3
336.4.q.j.289.1 4 7.4 even 3
882.4.a.v.1.1 2 12.11 even 2
882.4.a.z.1.2 2 84.83 odd 2
882.4.g.bf.361.1 4 84.47 odd 6
882.4.g.bf.667.1 4 84.59 odd 6
2352.4.a.bq.1.2 2 1.1 even 1 trivial
2352.4.a.ca.1.1 2 7.6 odd 2