Properties

Label 1323.2.a.be.1.4
Level $1323$
Weight $2$
Character 1323.1
Self dual yes
Analytic conductor $10.564$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \(x^{4} - 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.28825\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.28825 q^{2} +3.23607 q^{4} +4.23607 q^{5} +2.82843 q^{8} +O(q^{10})\) \(q+2.28825 q^{2} +3.23607 q^{4} +4.23607 q^{5} +2.82843 q^{8} +9.69316 q^{10} -3.70246 q^{11} -2.28825 q^{13} +5.00000 q^{17} +5.45052 q^{19} +13.7082 q^{20} -8.47214 q^{22} +0.333851 q^{23} +12.9443 q^{25} -5.23607 q^{26} -7.19859 q^{29} -3.36861 q^{31} -5.65685 q^{32} +11.4412 q^{34} -4.70820 q^{37} +12.4721 q^{38} +11.9814 q^{40} -4.70820 q^{41} +2.23607 q^{43} -11.9814 q^{44} +0.763932 q^{46} -1.47214 q^{47} +29.6197 q^{50} -7.40492 q^{52} +3.16228 q^{53} -15.6839 q^{55} -16.4721 q^{58} -3.94427 q^{59} +3.70246 q^{61} -7.70820 q^{62} -12.9443 q^{64} -9.69316 q^{65} -11.2361 q^{67} +16.1803 q^{68} +13.0618 q^{71} +0.746512 q^{73} -10.7735 q^{74} +17.6383 q^{76} -11.4721 q^{79} -10.7735 q^{82} -0.236068 q^{83} +21.1803 q^{85} +5.11667 q^{86} -10.4721 q^{88} -1.70820 q^{89} +1.08036 q^{92} -3.36861 q^{94} +23.0888 q^{95} +1.95440 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + 8q^{5} + O(q^{10}) \) \( 4q + 4q^{4} + 8q^{5} + 20q^{17} + 28q^{20} - 16q^{22} + 16q^{25} - 12q^{26} + 8q^{37} + 32q^{38} + 8q^{41} + 12q^{46} + 12q^{47} - 48q^{58} + 20q^{59} - 4q^{62} - 16q^{64} - 36q^{67} + 20q^{68} - 28q^{79} + 8q^{83} + 40q^{85} - 24q^{88} + 20q^{89} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.28825 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(3\) 0 0
\(4\) 3.23607 1.61803
\(5\) 4.23607 1.89443 0.947214 0.320603i \(-0.103886\pi\)
0.947214 + 0.320603i \(0.103886\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.82843 1.00000
\(9\) 0 0
\(10\) 9.69316 3.06525
\(11\) −3.70246 −1.11633 −0.558167 0.829729i \(-0.688495\pi\)
−0.558167 + 0.829729i \(0.688495\pi\)
\(12\) 0 0
\(13\) −2.28825 −0.634645 −0.317323 0.948318i \(-0.602784\pi\)
−0.317323 + 0.948318i \(0.602784\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) 5.45052 1.25044 0.625218 0.780450i \(-0.285010\pi\)
0.625218 + 0.780450i \(0.285010\pi\)
\(20\) 13.7082 3.06525
\(21\) 0 0
\(22\) −8.47214 −1.80627
\(23\) 0.333851 0.0696126 0.0348063 0.999394i \(-0.488919\pi\)
0.0348063 + 0.999394i \(0.488919\pi\)
\(24\) 0 0
\(25\) 12.9443 2.58885
\(26\) −5.23607 −1.02688
\(27\) 0 0
\(28\) 0 0
\(29\) −7.19859 −1.33674 −0.668372 0.743827i \(-0.733009\pi\)
−0.668372 + 0.743827i \(0.733009\pi\)
\(30\) 0 0
\(31\) −3.36861 −0.605020 −0.302510 0.953146i \(-0.597825\pi\)
−0.302510 + 0.953146i \(0.597825\pi\)
\(32\) −5.65685 −1.00000
\(33\) 0 0
\(34\) 11.4412 1.96215
\(35\) 0 0
\(36\) 0 0
\(37\) −4.70820 −0.774024 −0.387012 0.922075i \(-0.626493\pi\)
−0.387012 + 0.922075i \(0.626493\pi\)
\(38\) 12.4721 2.02325
\(39\) 0 0
\(40\) 11.9814 1.89443
\(41\) −4.70820 −0.735298 −0.367649 0.929965i \(-0.619837\pi\)
−0.367649 + 0.929965i \(0.619837\pi\)
\(42\) 0 0
\(43\) 2.23607 0.340997 0.170499 0.985358i \(-0.445462\pi\)
0.170499 + 0.985358i \(0.445462\pi\)
\(44\) −11.9814 −1.80627
\(45\) 0 0
\(46\) 0.763932 0.112636
\(47\) −1.47214 −0.214733 −0.107367 0.994220i \(-0.534242\pi\)
−0.107367 + 0.994220i \(0.534242\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 29.6197 4.18885
\(51\) 0 0
\(52\) −7.40492 −1.02688
\(53\) 3.16228 0.434372 0.217186 0.976130i \(-0.430312\pi\)
0.217186 + 0.976130i \(0.430312\pi\)
\(54\) 0 0
\(55\) −15.6839 −2.11481
\(56\) 0 0
\(57\) 0 0
\(58\) −16.4721 −2.16290
\(59\) −3.94427 −0.513500 −0.256750 0.966478i \(-0.582652\pi\)
−0.256750 + 0.966478i \(0.582652\pi\)
\(60\) 0 0
\(61\) 3.70246 0.474051 0.237026 0.971503i \(-0.423827\pi\)
0.237026 + 0.971503i \(0.423827\pi\)
\(62\) −7.70820 −0.978943
\(63\) 0 0
\(64\) −12.9443 −1.61803
\(65\) −9.69316 −1.20229
\(66\) 0 0
\(67\) −11.2361 −1.37270 −0.686352 0.727269i \(-0.740789\pi\)
−0.686352 + 0.727269i \(0.740789\pi\)
\(68\) 16.1803 1.96215
\(69\) 0 0
\(70\) 0 0
\(71\) 13.0618 1.55015 0.775074 0.631871i \(-0.217713\pi\)
0.775074 + 0.631871i \(0.217713\pi\)
\(72\) 0 0
\(73\) 0.746512 0.0873727 0.0436863 0.999045i \(-0.486090\pi\)
0.0436863 + 0.999045i \(0.486090\pi\)
\(74\) −10.7735 −1.25240
\(75\) 0 0
\(76\) 17.6383 2.02325
\(77\) 0 0
\(78\) 0 0
\(79\) −11.4721 −1.29072 −0.645358 0.763880i \(-0.723292\pi\)
−0.645358 + 0.763880i \(0.723292\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −10.7735 −1.18974
\(83\) −0.236068 −0.0259118 −0.0129559 0.999916i \(-0.504124\pi\)
−0.0129559 + 0.999916i \(0.504124\pi\)
\(84\) 0 0
\(85\) 21.1803 2.29733
\(86\) 5.11667 0.551745
\(87\) 0 0
\(88\) −10.4721 −1.11633
\(89\) −1.70820 −0.181069 −0.0905346 0.995893i \(-0.528858\pi\)
−0.0905346 + 0.995893i \(0.528858\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.08036 0.112636
\(93\) 0 0
\(94\) −3.36861 −0.347445
\(95\) 23.0888 2.36886
\(96\) 0 0
\(97\) 1.95440 0.198439 0.0992194 0.995066i \(-0.468365\pi\)
0.0992194 + 0.995066i \(0.468365\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 41.8885 4.18885
\(101\) 9.23607 0.919023 0.459512 0.888172i \(-0.348024\pi\)
0.459512 + 0.888172i \(0.348024\pi\)
\(102\) 0 0
\(103\) −8.40647 −0.828314 −0.414157 0.910205i \(-0.635924\pi\)
−0.414157 + 0.910205i \(0.635924\pi\)
\(104\) −6.47214 −0.634645
\(105\) 0 0
\(106\) 7.23607 0.702829
\(107\) −9.89949 −0.957020 −0.478510 0.878082i \(-0.658823\pi\)
−0.478510 + 0.878082i \(0.658823\pi\)
\(108\) 0 0
\(109\) 9.76393 0.935215 0.467608 0.883936i \(-0.345116\pi\)
0.467608 + 0.883936i \(0.345116\pi\)
\(110\) −35.8885 −3.42184
\(111\) 0 0
\(112\) 0 0
\(113\) 1.20788 0.113628 0.0568140 0.998385i \(-0.481906\pi\)
0.0568140 + 0.998385i \(0.481906\pi\)
\(114\) 0 0
\(115\) 1.41421 0.131876
\(116\) −23.2951 −2.16290
\(117\) 0 0
\(118\) −9.02546 −0.830861
\(119\) 0 0
\(120\) 0 0
\(121\) 2.70820 0.246200
\(122\) 8.47214 0.767031
\(123\) 0 0
\(124\) −10.9010 −0.978943
\(125\) 33.6525 3.00997
\(126\) 0 0
\(127\) 5.47214 0.485574 0.242787 0.970080i \(-0.421938\pi\)
0.242787 + 0.970080i \(0.421938\pi\)
\(128\) −18.3060 −1.61803
\(129\) 0 0
\(130\) −22.1803 −1.94534
\(131\) 10.9443 0.956205 0.478103 0.878304i \(-0.341325\pi\)
0.478103 + 0.878304i \(0.341325\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −25.7109 −2.22108
\(135\) 0 0
\(136\) 14.1421 1.21268
\(137\) 4.44897 0.380101 0.190051 0.981774i \(-0.439135\pi\)
0.190051 + 0.981774i \(0.439135\pi\)
\(138\) 0 0
\(139\) −19.1800 −1.62683 −0.813413 0.581687i \(-0.802393\pi\)
−0.813413 + 0.581687i \(0.802393\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 29.8885 2.50819
\(143\) 8.47214 0.708476
\(144\) 0 0
\(145\) −30.4937 −2.53236
\(146\) 1.70820 0.141372
\(147\) 0 0
\(148\) −15.2361 −1.25240
\(149\) 22.0084 1.80300 0.901500 0.432779i \(-0.142467\pi\)
0.901500 + 0.432779i \(0.142467\pi\)
\(150\) 0 0
\(151\) −16.4164 −1.33595 −0.667974 0.744184i \(-0.732838\pi\)
−0.667974 + 0.744184i \(0.732838\pi\)
\(152\) 15.4164 1.25044
\(153\) 0 0
\(154\) 0 0
\(155\) −14.2697 −1.14617
\(156\) 0 0
\(157\) −20.6730 −1.64989 −0.824943 0.565215i \(-0.808793\pi\)
−0.824943 + 0.565215i \(0.808793\pi\)
\(158\) −26.2511 −2.08842
\(159\) 0 0
\(160\) −23.9628 −1.89443
\(161\) 0 0
\(162\) 0 0
\(163\) −5.18034 −0.405756 −0.202878 0.979204i \(-0.565029\pi\)
−0.202878 + 0.979204i \(0.565029\pi\)
\(164\) −15.2361 −1.18974
\(165\) 0 0
\(166\) −0.540182 −0.0419262
\(167\) −3.76393 −0.291262 −0.145631 0.989339i \(-0.546521\pi\)
−0.145631 + 0.989339i \(0.546521\pi\)
\(168\) 0 0
\(169\) −7.76393 −0.597226
\(170\) 48.4658 3.71716
\(171\) 0 0
\(172\) 7.23607 0.551745
\(173\) 18.4721 1.40441 0.702205 0.711975i \(-0.252199\pi\)
0.702205 + 0.711975i \(0.252199\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −3.90879 −0.292976
\(179\) 14.6823 1.09741 0.548704 0.836017i \(-0.315121\pi\)
0.548704 + 0.836017i \(0.315121\pi\)
\(180\) 0 0
\(181\) 22.8825 1.70084 0.850420 0.526105i \(-0.176348\pi\)
0.850420 + 0.526105i \(0.176348\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.944272 0.0696126
\(185\) −19.9443 −1.46633
\(186\) 0 0
\(187\) −18.5123 −1.35375
\(188\) −4.76393 −0.347445
\(189\) 0 0
\(190\) 52.8328 3.83290
\(191\) −4.24264 −0.306987 −0.153493 0.988150i \(-0.549052\pi\)
−0.153493 + 0.988150i \(0.549052\pi\)
\(192\) 0 0
\(193\) 17.1803 1.23667 0.618334 0.785915i \(-0.287808\pi\)
0.618334 + 0.785915i \(0.287808\pi\)
\(194\) 4.47214 0.321081
\(195\) 0 0
\(196\) 0 0
\(197\) 0.461370 0.0328713 0.0164356 0.999865i \(-0.494768\pi\)
0.0164356 + 0.999865i \(0.494768\pi\)
\(198\) 0 0
\(199\) 13.4744 0.955177 0.477589 0.878584i \(-0.341511\pi\)
0.477589 + 0.878584i \(0.341511\pi\)
\(200\) 36.6119 2.58885
\(201\) 0 0
\(202\) 21.1344 1.48701
\(203\) 0 0
\(204\) 0 0
\(205\) −19.9443 −1.39297
\(206\) −19.2361 −1.34024
\(207\) 0 0
\(208\) 0 0
\(209\) −20.1803 −1.39590
\(210\) 0 0
\(211\) −17.4164 −1.19899 −0.599497 0.800377i \(-0.704633\pi\)
−0.599497 + 0.800377i \(0.704633\pi\)
\(212\) 10.2333 0.702829
\(213\) 0 0
\(214\) −22.6525 −1.54849
\(215\) 9.47214 0.645994
\(216\) 0 0
\(217\) 0 0
\(218\) 22.3423 1.51321
\(219\) 0 0
\(220\) −50.7541 −3.42184
\(221\) −11.4412 −0.769620
\(222\) 0 0
\(223\) −19.5138 −1.30674 −0.653372 0.757037i \(-0.726646\pi\)
−0.653372 + 0.757037i \(0.726646\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.76393 0.183854
\(227\) 12.1803 0.808438 0.404219 0.914662i \(-0.367543\pi\)
0.404219 + 0.914662i \(0.367543\pi\)
\(228\) 0 0
\(229\) 10.6460 0.703508 0.351754 0.936092i \(-0.385585\pi\)
0.351754 + 0.936092i \(0.385585\pi\)
\(230\) 3.23607 0.213380
\(231\) 0 0
\(232\) −20.3607 −1.33674
\(233\) −0.127520 −0.00835408 −0.00417704 0.999991i \(-0.501330\pi\)
−0.00417704 + 0.999991i \(0.501330\pi\)
\(234\) 0 0
\(235\) −6.23607 −0.406796
\(236\) −12.7639 −0.830861
\(237\) 0 0
\(238\) 0 0
\(239\) −25.5834 −1.65485 −0.827425 0.561576i \(-0.810195\pi\)
−0.827425 + 0.561576i \(0.810195\pi\)
\(240\) 0 0
\(241\) 6.86474 0.442197 0.221098 0.975252i \(-0.429036\pi\)
0.221098 + 0.975252i \(0.429036\pi\)
\(242\) 6.19704 0.398361
\(243\) 0 0
\(244\) 11.9814 0.767031
\(245\) 0 0
\(246\) 0 0
\(247\) −12.4721 −0.793583
\(248\) −9.52786 −0.605020
\(249\) 0 0
\(250\) 77.0051 4.87023
\(251\) −9.29180 −0.586493 −0.293246 0.956037i \(-0.594736\pi\)
−0.293246 + 0.956037i \(0.594736\pi\)
\(252\) 0 0
\(253\) −1.23607 −0.0777109
\(254\) 12.5216 0.785675
\(255\) 0 0
\(256\) −16.0000 −1.00000
\(257\) −5.41641 −0.337866 −0.168933 0.985628i \(-0.554032\pi\)
−0.168933 + 0.985628i \(0.554032\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −31.3677 −1.94534
\(261\) 0 0
\(262\) 25.0432 1.54717
\(263\) −20.1815 −1.24445 −0.622224 0.782839i \(-0.713771\pi\)
−0.622224 + 0.782839i \(0.713771\pi\)
\(264\) 0 0
\(265\) 13.3956 0.822887
\(266\) 0 0
\(267\) 0 0
\(268\) −36.3607 −2.22108
\(269\) 7.00000 0.426798 0.213399 0.976965i \(-0.431547\pi\)
0.213399 + 0.976965i \(0.431547\pi\)
\(270\) 0 0
\(271\) 5.57804 0.338842 0.169421 0.985544i \(-0.445810\pi\)
0.169421 + 0.985544i \(0.445810\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 10.1803 0.615017
\(275\) −47.9256 −2.89002
\(276\) 0 0
\(277\) −13.7639 −0.826995 −0.413497 0.910505i \(-0.635693\pi\)
−0.413497 + 0.910505i \(0.635693\pi\)
\(278\) −43.8885 −2.63226
\(279\) 0 0
\(280\) 0 0
\(281\) −22.2936 −1.32992 −0.664961 0.746878i \(-0.731552\pi\)
−0.664961 + 0.746878i \(0.731552\pi\)
\(282\) 0 0
\(283\) 16.3516 0.972000 0.486000 0.873959i \(-0.338456\pi\)
0.486000 + 0.873959i \(0.338456\pi\)
\(284\) 42.2688 2.50819
\(285\) 0 0
\(286\) 19.3863 1.14634
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) −69.7771 −4.09745
\(291\) 0 0
\(292\) 2.41577 0.141372
\(293\) 7.47214 0.436527 0.218263 0.975890i \(-0.429961\pi\)
0.218263 + 0.975890i \(0.429961\pi\)
\(294\) 0 0
\(295\) −16.7082 −0.972789
\(296\) −13.3168 −0.774024
\(297\) 0 0
\(298\) 50.3607 2.91732
\(299\) −0.763932 −0.0441793
\(300\) 0 0
\(301\) 0 0
\(302\) −37.5648 −2.16161
\(303\) 0 0
\(304\) 0 0
\(305\) 15.6839 0.898056
\(306\) 0 0
\(307\) 2.74962 0.156929 0.0784644 0.996917i \(-0.474998\pi\)
0.0784644 + 0.996917i \(0.474998\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −32.6525 −1.85454
\(311\) 27.0689 1.53494 0.767468 0.641088i \(-0.221516\pi\)
0.767468 + 0.641088i \(0.221516\pi\)
\(312\) 0 0
\(313\) −28.9520 −1.63646 −0.818231 0.574889i \(-0.805045\pi\)
−0.818231 + 0.574889i \(0.805045\pi\)
\(314\) −47.3050 −2.66957
\(315\) 0 0
\(316\) −37.1246 −2.08842
\(317\) −2.08191 −0.116932 −0.0584660 0.998289i \(-0.518621\pi\)
−0.0584660 + 0.998289i \(0.518621\pi\)
\(318\) 0 0
\(319\) 26.6525 1.49225
\(320\) −54.8328 −3.06525
\(321\) 0 0
\(322\) 0 0
\(323\) 27.2526 1.51638
\(324\) 0 0
\(325\) −29.6197 −1.64300
\(326\) −11.8539 −0.656526
\(327\) 0 0
\(328\) −13.3168 −0.735298
\(329\) 0 0
\(330\) 0 0
\(331\) 18.4164 1.01226 0.506129 0.862458i \(-0.331076\pi\)
0.506129 + 0.862458i \(0.331076\pi\)
\(332\) −0.763932 −0.0419262
\(333\) 0 0
\(334\) −8.61280 −0.471271
\(335\) −47.5967 −2.60049
\(336\) 0 0
\(337\) 3.76393 0.205034 0.102517 0.994731i \(-0.467310\pi\)
0.102517 + 0.994731i \(0.467310\pi\)
\(338\) −17.7658 −0.966331
\(339\) 0 0
\(340\) 68.5410 3.71716
\(341\) 12.4721 0.675404
\(342\) 0 0
\(343\) 0 0
\(344\) 6.32456 0.340997
\(345\) 0 0
\(346\) 42.2688 2.27238
\(347\) 23.1676 1.24370 0.621851 0.783136i \(-0.286381\pi\)
0.621851 + 0.783136i \(0.286381\pi\)
\(348\) 0 0
\(349\) 10.9010 0.583520 0.291760 0.956492i \(-0.405759\pi\)
0.291760 + 0.956492i \(0.405759\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 20.9443 1.11633
\(353\) 30.1246 1.60337 0.801686 0.597746i \(-0.203937\pi\)
0.801686 + 0.597746i \(0.203937\pi\)
\(354\) 0 0
\(355\) 55.3306 2.93664
\(356\) −5.52786 −0.292976
\(357\) 0 0
\(358\) 33.5967 1.77564
\(359\) −4.65530 −0.245697 −0.122849 0.992425i \(-0.539203\pi\)
−0.122849 + 0.992425i \(0.539203\pi\)
\(360\) 0 0
\(361\) 10.7082 0.563590
\(362\) 52.3607 2.75202
\(363\) 0 0
\(364\) 0 0
\(365\) 3.16228 0.165521
\(366\) 0 0
\(367\) 18.6398 0.972990 0.486495 0.873683i \(-0.338275\pi\)
0.486495 + 0.873683i \(0.338275\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −45.6374 −2.37258
\(371\) 0 0
\(372\) 0 0
\(373\) −13.6525 −0.706898 −0.353449 0.935454i \(-0.614991\pi\)
−0.353449 + 0.935454i \(0.614991\pi\)
\(374\) −42.3607 −2.19042
\(375\) 0 0
\(376\) −4.16383 −0.214733
\(377\) 16.4721 0.848358
\(378\) 0 0
\(379\) −30.8885 −1.58664 −0.793319 0.608806i \(-0.791649\pi\)
−0.793319 + 0.608806i \(0.791649\pi\)
\(380\) 74.7169 3.83290
\(381\) 0 0
\(382\) −9.70820 −0.496715
\(383\) −11.1803 −0.571289 −0.285644 0.958336i \(-0.592208\pi\)
−0.285644 + 0.958336i \(0.592208\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 39.3128 2.00097
\(387\) 0 0
\(388\) 6.32456 0.321081
\(389\) 23.0888 1.17065 0.585324 0.810800i \(-0.300967\pi\)
0.585324 + 0.810800i \(0.300967\pi\)
\(390\) 0 0
\(391\) 1.66925 0.0844177
\(392\) 0 0
\(393\) 0 0
\(394\) 1.05573 0.0531868
\(395\) −48.5967 −2.44517
\(396\) 0 0
\(397\) 27.7928 1.39488 0.697440 0.716643i \(-0.254322\pi\)
0.697440 + 0.716643i \(0.254322\pi\)
\(398\) 30.8328 1.54551
\(399\) 0 0
\(400\) 0 0
\(401\) 23.7078 1.18391 0.591955 0.805971i \(-0.298356\pi\)
0.591955 + 0.805971i \(0.298356\pi\)
\(402\) 0 0
\(403\) 7.70820 0.383973
\(404\) 29.8885 1.48701
\(405\) 0 0
\(406\) 0 0
\(407\) 17.4319 0.864069
\(408\) 0 0
\(409\) 32.0354 1.58405 0.792025 0.610488i \(-0.209027\pi\)
0.792025 + 0.610488i \(0.209027\pi\)
\(410\) −45.6374 −2.25387
\(411\) 0 0
\(412\) −27.2039 −1.34024
\(413\) 0 0
\(414\) 0 0
\(415\) −1.00000 −0.0490881
\(416\) 12.9443 0.634645
\(417\) 0 0
\(418\) −46.1776 −2.25862
\(419\) 15.6525 0.764673 0.382337 0.924023i \(-0.375119\pi\)
0.382337 + 0.924023i \(0.375119\pi\)
\(420\) 0 0
\(421\) −4.47214 −0.217959 −0.108979 0.994044i \(-0.534758\pi\)
−0.108979 + 0.994044i \(0.534758\pi\)
\(422\) −39.8530 −1.94001
\(423\) 0 0
\(424\) 8.94427 0.434372
\(425\) 64.7214 3.13945
\(426\) 0 0
\(427\) 0 0
\(428\) −32.0354 −1.54849
\(429\) 0 0
\(430\) 21.6746 1.04524
\(431\) 20.0540 0.965969 0.482984 0.875629i \(-0.339553\pi\)
0.482984 + 0.875629i \(0.339553\pi\)
\(432\) 0 0
\(433\) −16.2728 −0.782019 −0.391009 0.920387i \(-0.627874\pi\)
−0.391009 + 0.920387i \(0.627874\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 31.5967 1.51321
\(437\) 1.81966 0.0870461
\(438\) 0 0
\(439\) −0.412662 −0.0196953 −0.00984764 0.999952i \(-0.503135\pi\)
−0.00984764 + 0.999952i \(0.503135\pi\)
\(440\) −44.3607 −2.11481
\(441\) 0 0
\(442\) −26.1803 −1.24527
\(443\) 21.5958 1.02605 0.513023 0.858375i \(-0.328526\pi\)
0.513023 + 0.858375i \(0.328526\pi\)
\(444\) 0 0
\(445\) −7.23607 −0.343023
\(446\) −44.6525 −2.11436
\(447\) 0 0
\(448\) 0 0
\(449\) −34.7363 −1.63931 −0.819655 0.572858i \(-0.805835\pi\)
−0.819655 + 0.572858i \(0.805835\pi\)
\(450\) 0 0
\(451\) 17.4319 0.820838
\(452\) 3.90879 0.183854
\(453\) 0 0
\(454\) 27.8716 1.30808
\(455\) 0 0
\(456\) 0 0
\(457\) 32.8328 1.53585 0.767927 0.640537i \(-0.221288\pi\)
0.767927 + 0.640537i \(0.221288\pi\)
\(458\) 24.3607 1.13830
\(459\) 0 0
\(460\) 4.57649 0.213380
\(461\) 22.5279 1.04923 0.524614 0.851340i \(-0.324210\pi\)
0.524614 + 0.851340i \(0.324210\pi\)
\(462\) 0 0
\(463\) 37.1803 1.72792 0.863958 0.503563i \(-0.167978\pi\)
0.863958 + 0.503563i \(0.167978\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −0.291796 −0.0135172
\(467\) −37.5967 −1.73977 −0.869885 0.493255i \(-0.835807\pi\)
−0.869885 + 0.493255i \(0.835807\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −14.2697 −0.658210
\(471\) 0 0
\(472\) −11.1561 −0.513500
\(473\) −8.27895 −0.380667
\(474\) 0 0
\(475\) 70.5531 3.23720
\(476\) 0 0
\(477\) 0 0
\(478\) −58.5410 −2.67760
\(479\) −31.0689 −1.41957 −0.709787 0.704417i \(-0.751209\pi\)
−0.709787 + 0.704417i \(0.751209\pi\)
\(480\) 0 0
\(481\) 10.7735 0.491231
\(482\) 15.7082 0.715489
\(483\) 0 0
\(484\) 8.76393 0.398361
\(485\) 8.27895 0.375928
\(486\) 0 0
\(487\) −25.8885 −1.17312 −0.586561 0.809905i \(-0.699519\pi\)
−0.586561 + 0.809905i \(0.699519\pi\)
\(488\) 10.4721 0.474051
\(489\) 0 0
\(490\) 0 0
\(491\) −1.46292 −0.0660207 −0.0330104 0.999455i \(-0.510509\pi\)
−0.0330104 + 0.999455i \(0.510509\pi\)
\(492\) 0 0
\(493\) −35.9929 −1.62104
\(494\) −28.5393 −1.28404
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 36.8885 1.65136 0.825679 0.564140i \(-0.190792\pi\)
0.825679 + 0.564140i \(0.190792\pi\)
\(500\) 108.902 4.87023
\(501\) 0 0
\(502\) −21.2619 −0.948966
\(503\) 35.8328 1.59771 0.798853 0.601526i \(-0.205440\pi\)
0.798853 + 0.601526i \(0.205440\pi\)
\(504\) 0 0
\(505\) 39.1246 1.74102
\(506\) −2.82843 −0.125739
\(507\) 0 0
\(508\) 17.7082 0.785675
\(509\) −17.3607 −0.769499 −0.384749 0.923021i \(-0.625712\pi\)
−0.384749 + 0.923021i \(0.625712\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) −12.3941 −0.546679
\(515\) −35.6104 −1.56918
\(516\) 0 0
\(517\) 5.45052 0.239714
\(518\) 0 0
\(519\) 0 0
\(520\) −27.4164 −1.20229
\(521\) −8.52786 −0.373613 −0.186806 0.982397i \(-0.559814\pi\)
−0.186806 + 0.982397i \(0.559814\pi\)
\(522\) 0 0
\(523\) −4.29135 −0.187648 −0.0938238 0.995589i \(-0.529909\pi\)
−0.0938238 + 0.995589i \(0.529909\pi\)
\(524\) 35.4164 1.54717
\(525\) 0 0
\(526\) −46.1803 −2.01356
\(527\) −16.8430 −0.733694
\(528\) 0 0
\(529\) −22.8885 −0.995154
\(530\) 30.6525 1.33146
\(531\) 0 0
\(532\) 0 0
\(533\) 10.7735 0.466653
\(534\) 0 0
\(535\) −41.9349 −1.81301
\(536\) −31.7804 −1.37270
\(537\) 0 0
\(538\) 16.0177 0.690573
\(539\) 0 0
\(540\) 0 0
\(541\) 28.3050 1.21692 0.608462 0.793583i \(-0.291787\pi\)
0.608462 + 0.793583i \(0.291787\pi\)
\(542\) 12.7639 0.548258
\(543\) 0 0
\(544\) −28.2843 −1.21268
\(545\) 41.3607 1.77170
\(546\) 0 0
\(547\) 18.1246 0.774952 0.387476 0.921880i \(-0.373347\pi\)
0.387476 + 0.921880i \(0.373347\pi\)
\(548\) 14.3972 0.615017
\(549\) 0 0
\(550\) −109.666 −4.67616
\(551\) −39.2361 −1.67151
\(552\) 0 0
\(553\) 0 0
\(554\) −31.4953 −1.33811
\(555\) 0 0
\(556\) −62.0678 −2.63226
\(557\) 6.32456 0.267980 0.133990 0.990983i \(-0.457221\pi\)
0.133990 + 0.990983i \(0.457221\pi\)
\(558\) 0 0
\(559\) −5.11667 −0.216412
\(560\) 0 0
\(561\) 0 0
\(562\) −51.0132 −2.15186
\(563\) 34.1803 1.44053 0.720265 0.693699i \(-0.244020\pi\)
0.720265 + 0.693699i \(0.244020\pi\)
\(564\) 0 0
\(565\) 5.11667 0.215260
\(566\) 37.4164 1.57273
\(567\) 0 0
\(568\) 36.9443 1.55015
\(569\) 25.3770 1.06386 0.531930 0.846788i \(-0.321467\pi\)
0.531930 + 0.846788i \(0.321467\pi\)
\(570\) 0 0
\(571\) −19.0689 −0.798008 −0.399004 0.916949i \(-0.630644\pi\)
−0.399004 + 0.916949i \(0.630644\pi\)
\(572\) 27.4164 1.14634
\(573\) 0 0
\(574\) 0 0
\(575\) 4.32145 0.180217
\(576\) 0 0
\(577\) 38.3600 1.59695 0.798474 0.602030i \(-0.205641\pi\)
0.798474 + 0.602030i \(0.205641\pi\)
\(578\) 18.3060 0.761428
\(579\) 0 0
\(580\) −98.6797 −4.09745
\(581\) 0 0
\(582\) 0 0
\(583\) −11.7082 −0.484904
\(584\) 2.11146 0.0873727
\(585\) 0 0
\(586\) 17.0981 0.706315
\(587\) −18.4721 −0.762427 −0.381213 0.924487i \(-0.624494\pi\)
−0.381213 + 0.924487i \(0.624494\pi\)
\(588\) 0 0
\(589\) −18.3607 −0.756539
\(590\) −38.2325 −1.57401
\(591\) 0 0
\(592\) 0 0
\(593\) 11.4721 0.471104 0.235552 0.971862i \(-0.424310\pi\)
0.235552 + 0.971862i \(0.424310\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 71.2208 2.91732
\(597\) 0 0
\(598\) −1.74806 −0.0714837
\(599\) 32.2418 1.31736 0.658681 0.752422i \(-0.271114\pi\)
0.658681 + 0.752422i \(0.271114\pi\)
\(600\) 0 0
\(601\) −25.3283 −1.03316 −0.516582 0.856238i \(-0.672796\pi\)
−0.516582 + 0.856238i \(0.672796\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −53.1246 −2.16161
\(605\) 11.4721 0.466409
\(606\) 0 0
\(607\) 26.6637 1.08225 0.541124 0.840943i \(-0.317999\pi\)
0.541124 + 0.840943i \(0.317999\pi\)
\(608\) −30.8328 −1.25044
\(609\) 0 0
\(610\) 35.8885 1.45308
\(611\) 3.36861 0.136279
\(612\) 0 0
\(613\) −31.8885 −1.28797 −0.643983 0.765040i \(-0.722719\pi\)
−0.643983 + 0.765040i \(0.722719\pi\)
\(614\) 6.29180 0.253916
\(615\) 0 0
\(616\) 0 0
\(617\) 10.5672 0.425419 0.212710 0.977115i \(-0.431771\pi\)
0.212710 + 0.977115i \(0.431771\pi\)
\(618\) 0 0
\(619\) −38.5663 −1.55011 −0.775056 0.631893i \(-0.782278\pi\)
−0.775056 + 0.631893i \(0.782278\pi\)
\(620\) −46.1776 −1.85454
\(621\) 0 0
\(622\) 61.9403 2.48358
\(623\) 0 0
\(624\) 0 0
\(625\) 77.8328 3.11331
\(626\) −66.2492 −2.64785
\(627\) 0 0
\(628\) −66.8993 −2.66957
\(629\) −23.5410 −0.938642
\(630\) 0 0
\(631\) −26.2361 −1.04444 −0.522221 0.852810i \(-0.674896\pi\)
−0.522221 + 0.852810i \(0.674896\pi\)
\(632\) −32.4481 −1.29072
\(633\) 0 0
\(634\) −4.76393 −0.189200
\(635\) 23.1803 0.919884
\(636\) 0 0
\(637\) 0 0
\(638\) 60.9874 2.41451
\(639\) 0 0
\(640\) −77.5453 −3.06525
\(641\) −34.8152 −1.37512 −0.687558 0.726129i \(-0.741317\pi\)
−0.687558 + 0.726129i \(0.741317\pi\)
\(642\) 0 0
\(643\) −33.8623 −1.33540 −0.667700 0.744431i \(-0.732721\pi\)
−0.667700 + 0.744431i \(0.732721\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 62.3607 2.45355
\(647\) −4.36068 −0.171436 −0.0857180 0.996319i \(-0.527318\pi\)
−0.0857180 + 0.996319i \(0.527318\pi\)
\(648\) 0 0
\(649\) 14.6035 0.573238
\(650\) −67.7771 −2.65844
\(651\) 0 0
\(652\) −16.7639 −0.656526
\(653\) −18.4335 −0.721358 −0.360679 0.932690i \(-0.617455\pi\)
−0.360679 + 0.932690i \(0.617455\pi\)
\(654\) 0 0
\(655\) 46.3607 1.81146
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 38.5176 1.50043 0.750217 0.661192i \(-0.229949\pi\)
0.750217 + 0.661192i \(0.229949\pi\)
\(660\) 0 0
\(661\) −29.3345 −1.14098 −0.570491 0.821304i \(-0.693247\pi\)
−0.570491 + 0.821304i \(0.693247\pi\)
\(662\) 42.1413 1.63787
\(663\) 0 0
\(664\) −0.667701 −0.0259118
\(665\) 0 0
\(666\) 0 0
\(667\) −2.40325 −0.0930543
\(668\) −12.1803 −0.471271
\(669\) 0 0
\(670\) −108.913 −4.20768
\(671\) −13.7082 −0.529199
\(672\) 0 0
\(673\) −41.2361 −1.58953 −0.794767 0.606915i \(-0.792407\pi\)
−0.794767 + 0.606915i \(0.792407\pi\)
\(674\) 8.61280 0.331753
\(675\) 0 0
\(676\) −25.1246 −0.966331
\(677\) 12.1115 0.465481 0.232741 0.972539i \(-0.425231\pi\)
0.232741 + 0.972539i \(0.425231\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 59.9070 2.29733
\(681\) 0 0
\(682\) 28.5393 1.09283
\(683\) 38.9002 1.48847 0.744237 0.667916i \(-0.232813\pi\)
0.744237 + 0.667916i \(0.232813\pi\)
\(684\) 0 0
\(685\) 18.8461 0.720074
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −7.23607 −0.275672
\(690\) 0 0
\(691\) 0.795221 0.0302516 0.0151258 0.999886i \(-0.495185\pi\)
0.0151258 + 0.999886i \(0.495185\pi\)
\(692\) 59.7771 2.27238
\(693\) 0 0
\(694\) 53.0132 2.01235
\(695\) −81.2478 −3.08190
\(696\) 0 0
\(697\) −23.5410 −0.891680
\(698\) 24.9443 0.944155
\(699\) 0 0
\(700\) 0 0
\(701\) 9.43812 0.356473 0.178237 0.983988i \(-0.442961\pi\)
0.178237 + 0.983988i \(0.442961\pi\)
\(702\) 0 0
\(703\) −25.6622 −0.967867
\(704\) 47.9256 1.80627
\(705\) 0 0
\(706\) 68.9325 2.59431
\(707\) 0 0
\(708\) 0 0
\(709\) −21.4721 −0.806403 −0.403201 0.915111i \(-0.632103\pi\)
−0.403201 + 0.915111i \(0.632103\pi\)
\(710\) 126.610 4.75159
\(711\) 0 0
\(712\) −4.83153 −0.181069
\(713\) −1.12461 −0.0421170
\(714\) 0 0
\(715\) 35.8885 1.34216
\(716\) 47.5130 1.77564
\(717\) 0 0
\(718\) −10.6525 −0.397547
\(719\) 0.416408 0.0155294 0.00776470 0.999970i \(-0.497528\pi\)
0.00776470 + 0.999970i \(0.497528\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 24.5030 0.911907
\(723\) 0 0
\(724\) 74.0492 2.75202
\(725\) −93.1805 −3.46064
\(726\) 0 0
\(727\) 35.4829 1.31599 0.657993 0.753024i \(-0.271406\pi\)
0.657993 + 0.753024i \(0.271406\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 7.23607 0.267819
\(731\) 11.1803 0.413520
\(732\) 0 0
\(733\) 6.32456 0.233603 0.116801 0.993155i \(-0.462736\pi\)
0.116801 + 0.993155i \(0.462736\pi\)
\(734\) 42.6525 1.57433
\(735\) 0 0
\(736\) −1.88854 −0.0696126
\(737\) 41.6011 1.53240
\(738\) 0 0
\(739\) −3.63932 −0.133875 −0.0669373 0.997757i \(-0.521323\pi\)
−0.0669373 + 0.997757i \(0.521323\pi\)
\(740\) −64.5410 −2.37258
\(741\) 0 0
\(742\) 0 0
\(743\) 25.9172 0.950810 0.475405 0.879767i \(-0.342301\pi\)
0.475405 + 0.879767i \(0.342301\pi\)
\(744\) 0 0
\(745\) 93.2292 3.41565
\(746\) −31.2402 −1.14379
\(747\) 0 0
\(748\) −59.9070 −2.19042
\(749\) 0 0
\(750\) 0 0
\(751\) 44.6525 1.62939 0.814696 0.579888i \(-0.196904\pi\)
0.814696 + 0.579888i \(0.196904\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 37.6923 1.37267
\(755\) −69.5410 −2.53086
\(756\) 0 0
\(757\) −35.3607 −1.28521 −0.642603 0.766199i \(-0.722145\pi\)
−0.642603 + 0.766199i \(0.722145\pi\)
\(758\) −70.6806 −2.56723
\(759\) 0 0
\(760\) 65.3050 2.36886
\(761\) −35.4721 −1.28586 −0.642932 0.765923i \(-0.722282\pi\)
−0.642932 + 0.765923i \(0.722282\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −13.7295 −0.496715
\(765\) 0 0
\(766\) −25.5834 −0.924365
\(767\) 9.02546 0.325891
\(768\) 0 0
\(769\) −28.6969 −1.03484 −0.517419 0.855732i \(-0.673107\pi\)
−0.517419 + 0.855732i \(0.673107\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 55.5967 2.00097
\(773\) −0.596748 −0.0214635 −0.0107318 0.999942i \(-0.503416\pi\)
−0.0107318 + 0.999942i \(0.503416\pi\)
\(774\) 0 0
\(775\) −43.6042 −1.56631
\(776\) 5.52786 0.198439
\(777\) 0 0
\(778\) 52.8328 1.89415
\(779\) −25.6622 −0.919443
\(780\) 0 0
\(781\) −48.3607 −1.73048
\(782\) 3.81966 0.136591
\(783\) 0 0
\(784\) 0 0
\(785\) −87.5723 −3.12559
\(786\) 0 0
\(787\) −3.41732 −0.121814 −0.0609071 0.998143i \(-0.519399\pi\)
−0.0609071 + 0.998143i \(0.519399\pi\)
\(788\) 1.49302 0.0531868
\(789\) 0 0
\(790\) −111.201 −3.95636
\(791\) 0 0
\(792\) 0 0
\(793\) −8.47214 −0.300854
\(794\) 63.5967 2.25696
\(795\) 0 0
\(796\) 43.6042 1.54551
\(797\) −16.0689 −0.569189 −0.284595 0.958648i \(-0.591859\pi\)
−0.284595 + 0.958648i \(0.591859\pi\)
\(798\) 0 0
\(799\) −7.36068 −0.260402
\(800\) −73.2239 −2.58885
\(801\) 0 0
\(802\) 54.2492 1.91561
\(803\) −2.76393 −0.0975370
\(804\) 0 0
\(805\) 0 0
\(806\) 17.6383 0.621281
\(807\) 0 0
\(808\) 26.1235 0.919023
\(809\) −10.5672 −0.371523 −0.185761 0.982595i \(-0.559475\pi\)
−0.185761 + 0.982595i \(0.559475\pi\)
\(810\) 0 0
\(811\) 37.9774 1.33357 0.666784 0.745251i \(-0.267670\pi\)
0.666784 + 0.745251i \(0.267670\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 39.8885 1.39809
\(815\) −21.9443 −0.768674
\(816\) 0 0
\(817\) 12.1877 0.426395
\(818\) 73.3050 2.56305
\(819\) 0 0
\(820\) −64.5410 −2.25387
\(821\) −12.2364 −0.427055 −0.213528 0.976937i \(-0.568495\pi\)
−0.213528 + 0.976937i \(0.568495\pi\)
\(822\) 0 0
\(823\) −9.58359 −0.334063 −0.167032 0.985952i \(-0.553418\pi\)
−0.167032 + 0.985952i \(0.553418\pi\)
\(824\) −23.7771 −0.828314
\(825\) 0 0
\(826\) 0 0
\(827\) −39.9805 −1.39026 −0.695130 0.718884i \(-0.744653\pi\)
−0.695130 + 0.718884i \(0.744653\pi\)
\(828\) 0 0
\(829\) −22.8337 −0.793049 −0.396524 0.918024i \(-0.629784\pi\)
−0.396524 + 0.918024i \(0.629784\pi\)
\(830\) −2.28825 −0.0794262
\(831\) 0 0
\(832\) 29.6197 1.02688
\(833\) 0 0
\(834\) 0 0
\(835\) −15.9443 −0.551774
\(836\) −65.3050 −2.25862
\(837\) 0 0
\(838\) 35.8167 1.23727
\(839\) −24.8197 −0.856870 −0.428435 0.903573i \(-0.640935\pi\)
−0.428435 + 0.903573i \(0.640935\pi\)
\(840\) 0 0
\(841\) 22.8197 0.786885
\(842\) −10.2333 −0.352664
\(843\) 0 0
\(844\) −56.3607 −1.94001
\(845\) −32.8885 −1.13140
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 148.098 5.07973
\(851\) −1.57184 −0.0538819
\(852\) 0 0
\(853\) −21.3894 −0.732360 −0.366180 0.930544i \(-0.619335\pi\)
−0.366180 + 0.930544i \(0.619335\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −28.0000 −0.957020
\(857\) −3.18034 −0.108638 −0.0543192 0.998524i \(-0.517299\pi\)
−0.0543192 + 0.998524i \(0.517299\pi\)
\(858\) 0 0
\(859\) −47.5918 −1.62381 −0.811905 0.583789i \(-0.801570\pi\)
−0.811905 + 0.583789i \(0.801570\pi\)
\(860\) 30.6525 1.04524
\(861\) 0 0
\(862\) 45.8885 1.56297
\(863\) 28.8732 0.982854 0.491427 0.870919i \(-0.336475\pi\)
0.491427 + 0.870919i \(0.336475\pi\)
\(864\) 0 0
\(865\) 78.2492 2.66055
\(866\) −37.2361 −1.26533
\(867\) 0 0
\(868\) 0 0
\(869\) 42.4751 1.44087
\(870\) 0 0
\(871\) 25.7109 0.871180
\(872\) 27.6166 0.935215
\(873\) 0 0
\(874\) 4.16383 0.140844
\(875\) 0 0
\(876\) 0 0
\(877\) −50.0132 −1.68882 −0.844412 0.535694i \(-0.820050\pi\)
−0.844412 + 0.535694i \(0.820050\pi\)
\(878\) −0.944272 −0.0318676
\(879\) 0 0
\(880\) 0 0
\(881\) 16.8328 0.567112 0.283556 0.958956i \(-0.408486\pi\)
0.283556 + 0.958956i \(0.408486\pi\)
\(882\) 0 0
\(883\) 16.8885 0.568345 0.284172 0.958773i \(-0.408281\pi\)
0.284172 + 0.958773i \(0.408281\pi\)
\(884\) −37.0246 −1.24527
\(885\) 0 0
\(886\) 49.4164 1.66018
\(887\) 52.1935 1.75249 0.876243 0.481869i \(-0.160042\pi\)
0.876243 + 0.481869i \(0.160042\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −16.5579 −0.555022
\(891\) 0 0
\(892\) −63.1481 −2.11436
\(893\) −8.02391 −0.268510
\(894\) 0 0
\(895\) 62.1953 2.07896
\(896\) 0 0
\(897\) 0 0
\(898\) −79.4853 −2.65246
\(899\) 24.2492 0.808757
\(900\) 0 0
\(901\) 15.8114 0.526754
\(902\) 39.8885 1.32814
\(903\) 0 0
\(904\) 3.41641 0.113628
\(905\) 96.9316 3.22212
\(906\) 0 0
\(907\) 47.0000 1.56061 0.780305 0.625400i \(-0.215064\pi\)
0.780305 + 0.625400i \(0.215064\pi\)
\(908\) 39.4164 1.30808
\(909\) 0 0
\(910\) 0 0
\(911\) 22.8337 0.756516 0.378258 0.925700i \(-0.376523\pi\)
0.378258 + 0.925700i \(0.376523\pi\)
\(912\) 0 0
\(913\) 0.874032 0.0289262
\(914\) 75.1295 2.48506
\(915\) 0 0
\(916\) 34.4512 1.13830
\(917\) 0 0
\(918\) 0 0
\(919\) 1.94427 0.0641356 0.0320678 0.999486i \(-0.489791\pi\)
0.0320678 + 0.999486i \(0.489791\pi\)
\(920\) 4.00000 0.131876
\(921\) 0 0
\(922\) 51.5493 1.69769
\(923\) −29.8885 −0.983793
\(924\) 0 0
\(925\) −60.9443 −2.00384
\(926\) 85.0777 2.79583
\(927\) 0 0
\(928\) 40.7214 1.33674
\(929\) −17.8328 −0.585076 −0.292538 0.956254i \(-0.594500\pi\)
−0.292538 + 0.956254i \(0.594500\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.412662 −0.0135172
\(933\) 0 0
\(934\) −86.0306 −2.81501
\(935\) −78.4193 −2.56459
\(936\) 0 0
\(937\) 17.8446 0.582958 0.291479 0.956577i \(-0.405853\pi\)
0.291479 + 0.956577i \(0.405853\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −20.1803 −0.658210
\(941\) 20.7082 0.675068 0.337534 0.941313i \(-0.390407\pi\)
0.337534 + 0.941313i \(0.390407\pi\)
\(942\) 0 0
\(943\) −1.57184 −0.0511860
\(944\) 0 0
\(945\) 0 0
\(946\) −18.9443 −0.615931
\(947\) 40.7758 1.32503 0.662517 0.749047i \(-0.269488\pi\)
0.662517 + 0.749047i \(0.269488\pi\)
\(948\) 0 0
\(949\) −1.70820 −0.0554506
\(950\) 161.443 5.23789
\(951\) 0 0
\(952\) 0 0
\(953\) −14.8886 −0.482291 −0.241145 0.970489i \(-0.577523\pi\)
−0.241145 + 0.970489i \(0.577523\pi\)
\(954\) 0 0
\(955\) −17.9721 −0.581564
\(956\) −82.7895 −2.67760
\(957\) 0 0
\(958\) −71.0932 −2.29692
\(959\) 0 0
\(960\) 0 0
\(961\) −19.6525 −0.633951
\(962\) 24.6525 0.794828
\(963\) 0 0
\(964\) 22.2148 0.715489
\(965\) 72.7771 2.34278
\(966\) 0 0
\(967\) 5.52786 0.177764 0.0888821 0.996042i \(-0.471671\pi\)
0.0888821 + 0.996042i \(0.471671\pi\)
\(968\) 7.65996 0.246200
\(969\) 0 0
\(970\) 18.9443 0.608264
\(971\) −20.5279 −0.658771 −0.329385 0.944196i \(-0.606841\pi\)
−0.329385 + 0.944196i \(0.606841\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −59.2393 −1.89815
\(975\) 0 0
\(976\) 0 0
\(977\) −4.83153 −0.154574 −0.0772872 0.997009i \(-0.524626\pi\)
−0.0772872 + 0.997009i \(0.524626\pi\)
\(978\) 0 0
\(979\) 6.32456 0.202134
\(980\) 0 0
\(981\) 0 0
\(982\) −3.34752 −0.106824
\(983\) −26.0557 −0.831049 −0.415524 0.909582i \(-0.636402\pi\)
−0.415524 + 0.909582i \(0.636402\pi\)
\(984\) 0 0
\(985\) 1.95440 0.0622722
\(986\) −82.3607 −2.62290
\(987\) 0 0
\(988\) −40.3607 −1.28404
\(989\) 0.746512 0.0237377
\(990\) 0 0
\(991\) 0.124612 0.00395842 0.00197921 0.999998i \(-0.499370\pi\)
0.00197921 + 0.999998i \(0.499370\pi\)
\(992\) 19.0557 0.605020
\(993\) 0 0
\(994\) 0 0
\(995\) 57.0786 1.80951
\(996\) 0 0
\(997\) 61.4488 1.94610 0.973051 0.230589i \(-0.0740653\pi\)
0.973051 + 0.230589i \(0.0740653\pi\)
\(998\) 84.4100 2.67195
\(999\) 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 1323.2.a.be.1.4 yes 4
3.2 odd 2 1323.2.a.bb.1.1 4
7.6 odd 2 1323.2.a.bb.1.4 yes 4
21.20 even 2 inner 1323.2.a.be.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.2.a.bb.1.1 4 3.2 odd 2
1323.2.a.bb.1.4 yes 4 7.6 odd 2
1323.2.a.be.1.1 yes 4 21.20 even 2 inner
1323.2.a.be.1.4 yes 4 1.1 even 1 trivial