Properties

Label 4140.2.a.q.1.1
Level $4140$
Weight $2$
Character 4140.1
Self dual yes
Analytic conductor $33.058$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.0580664368\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 4140.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} -3.44949 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} -3.44949 q^{7} +2.44949 q^{11} +4.44949 q^{13} -5.44949 q^{17} -1.55051 q^{19} -1.00000 q^{23} +1.00000 q^{25} -1.89898 q^{29} -7.00000 q^{31} -3.44949 q^{35} +6.34847 q^{37} -7.89898 q^{41} -8.89898 q^{43} +2.44949 q^{47} +4.89898 q^{49} +4.34847 q^{53} +2.44949 q^{55} +1.89898 q^{59} -5.34847 q^{61} +4.44949 q^{65} +1.44949 q^{67} -3.00000 q^{71} +9.34847 q^{73} -8.44949 q^{77} -4.00000 q^{79} +10.3485 q^{83} -5.44949 q^{85} +16.8990 q^{89} -15.3485 q^{91} -1.55051 q^{95} -14.8990 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{7} + 4 q^{13} - 6 q^{17} - 8 q^{19} - 2 q^{23} + 2 q^{25} + 6 q^{29} - 14 q^{31} - 2 q^{35} - 2 q^{37} - 6 q^{41} - 8 q^{43} - 6 q^{53} - 6 q^{59} + 4 q^{61} + 4 q^{65} - 2 q^{67} - 6 q^{71} + 4 q^{73} - 12 q^{77} - 8 q^{79} + 6 q^{83} - 6 q^{85} + 24 q^{89} - 16 q^{91} - 8 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.44949 −1.30378 −0.651892 0.758312i \(-0.726025\pi\)
−0.651892 + 0.758312i \(0.726025\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.44949 0.738549 0.369274 0.929320i \(-0.379606\pi\)
0.369274 + 0.929320i \(0.379606\pi\)
\(12\) 0 0
\(13\) 4.44949 1.23407 0.617033 0.786937i \(-0.288334\pi\)
0.617033 + 0.786937i \(0.288334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.44949 −1.32170 −0.660848 0.750520i \(-0.729803\pi\)
−0.660848 + 0.750520i \(0.729803\pi\)
\(18\) 0 0
\(19\) −1.55051 −0.355711 −0.177856 0.984057i \(-0.556916\pi\)
−0.177856 + 0.984057i \(0.556916\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.89898 −0.352632 −0.176316 0.984334i \(-0.556418\pi\)
−0.176316 + 0.984334i \(0.556418\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.44949 −0.583070
\(36\) 0 0
\(37\) 6.34847 1.04368 0.521841 0.853043i \(-0.325245\pi\)
0.521841 + 0.853043i \(0.325245\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.89898 −1.23361 −0.616807 0.787115i \(-0.711574\pi\)
−0.616807 + 0.787115i \(0.711574\pi\)
\(42\) 0 0
\(43\) −8.89898 −1.35708 −0.678541 0.734563i \(-0.737387\pi\)
−0.678541 + 0.734563i \(0.737387\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.44949 0.357295 0.178647 0.983913i \(-0.442828\pi\)
0.178647 + 0.983913i \(0.442828\pi\)
\(48\) 0 0
\(49\) 4.89898 0.699854
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.34847 0.597308 0.298654 0.954361i \(-0.403462\pi\)
0.298654 + 0.954361i \(0.403462\pi\)
\(54\) 0 0
\(55\) 2.44949 0.330289
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.89898 0.247226 0.123613 0.992330i \(-0.460552\pi\)
0.123613 + 0.992330i \(0.460552\pi\)
\(60\) 0 0
\(61\) −5.34847 −0.684801 −0.342401 0.939554i \(-0.611240\pi\)
−0.342401 + 0.939554i \(0.611240\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.44949 0.551891
\(66\) 0 0
\(67\) 1.44949 0.177083 0.0885417 0.996072i \(-0.471779\pi\)
0.0885417 + 0.996072i \(0.471779\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) 9.34847 1.09416 0.547078 0.837082i \(-0.315740\pi\)
0.547078 + 0.837082i \(0.315740\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.44949 −0.962909
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.3485 1.13589 0.567946 0.823066i \(-0.307738\pi\)
0.567946 + 0.823066i \(0.307738\pi\)
\(84\) 0 0
\(85\) −5.44949 −0.591080
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.8990 1.79129 0.895644 0.444771i \(-0.146715\pi\)
0.895644 + 0.444771i \(0.146715\pi\)
\(90\) 0 0
\(91\) −15.3485 −1.60896
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.55051 −0.159079
\(96\) 0 0
\(97\) −14.8990 −1.51276 −0.756381 0.654131i \(-0.773034\pi\)
−0.756381 + 0.654131i \(0.773034\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −18.7980 −1.87047 −0.935233 0.354032i \(-0.884810\pi\)
−0.935233 + 0.354032i \(0.884810\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.1464 −1.36759 −0.683793 0.729676i \(-0.739671\pi\)
−0.683793 + 0.729676i \(0.739671\pi\)
\(108\) 0 0
\(109\) −11.3485 −1.08699 −0.543493 0.839414i \(-0.682899\pi\)
−0.543493 + 0.839414i \(0.682899\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.3485 1.53793 0.768967 0.639288i \(-0.220771\pi\)
0.768967 + 0.639288i \(0.220771\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 18.7980 1.72321
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 3.34847 0.297129 0.148564 0.988903i \(-0.452535\pi\)
0.148564 + 0.988903i \(0.452535\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −10.8990 −0.952248 −0.476124 0.879378i \(-0.657959\pi\)
−0.476124 + 0.879378i \(0.657959\pi\)
\(132\) 0 0
\(133\) 5.34847 0.463771
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.8990 1.44378 0.721889 0.692009i \(-0.243274\pi\)
0.721889 + 0.692009i \(0.243274\pi\)
\(138\) 0 0
\(139\) −21.6969 −1.84031 −0.920155 0.391554i \(-0.871938\pi\)
−0.920155 + 0.391554i \(0.871938\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.8990 0.911418
\(144\) 0 0
\(145\) −1.89898 −0.157702
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.2474 −1.49489 −0.747445 0.664324i \(-0.768719\pi\)
−0.747445 + 0.664324i \(0.768719\pi\)
\(150\) 0 0
\(151\) −19.7980 −1.61114 −0.805568 0.592504i \(-0.798139\pi\)
−0.805568 + 0.592504i \(0.798139\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −7.00000 −0.562254
\(156\) 0 0
\(157\) −14.3485 −1.14513 −0.572566 0.819858i \(-0.694052\pi\)
−0.572566 + 0.819858i \(0.694052\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.44949 0.271858
\(162\) 0 0
\(163\) −24.6969 −1.93441 −0.967207 0.253990i \(-0.918257\pi\)
−0.967207 + 0.253990i \(0.918257\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.247449 −0.0191482 −0.00957408 0.999954i \(-0.503048\pi\)
−0.00957408 + 0.999954i \(0.503048\pi\)
\(168\) 0 0
\(169\) 6.79796 0.522920
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.79796 0.744925 0.372463 0.928047i \(-0.378514\pi\)
0.372463 + 0.928047i \(0.378514\pi\)
\(174\) 0 0
\(175\) −3.44949 −0.260757
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0.898979 0.0668206 0.0334103 0.999442i \(-0.489363\pi\)
0.0334103 + 0.999442i \(0.489363\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.34847 0.466749
\(186\) 0 0
\(187\) −13.3485 −0.976137
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −15.5505 −1.12520 −0.562598 0.826731i \(-0.690198\pi\)
−0.562598 + 0.826731i \(0.690198\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.7980 1.12556 0.562779 0.826607i \(-0.309732\pi\)
0.562779 + 0.826607i \(0.309732\pi\)
\(198\) 0 0
\(199\) 11.7980 0.836335 0.418168 0.908370i \(-0.362672\pi\)
0.418168 + 0.908370i \(0.362672\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.55051 0.459756
\(204\) 0 0
\(205\) −7.89898 −0.551689
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.79796 −0.262710
\(210\) 0 0
\(211\) 13.6969 0.942936 0.471468 0.881883i \(-0.343724\pi\)
0.471468 + 0.881883i \(0.343724\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.89898 −0.606905
\(216\) 0 0
\(217\) 24.1464 1.63917
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −24.2474 −1.63106
\(222\) 0 0
\(223\) −17.1010 −1.14517 −0.572585 0.819846i \(-0.694059\pi\)
−0.572585 + 0.819846i \(0.694059\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −1.79796 −0.118812 −0.0594062 0.998234i \(-0.518921\pi\)
−0.0594062 + 0.998234i \(0.518921\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.4949 −1.60471 −0.802357 0.596844i \(-0.796421\pi\)
−0.802357 + 0.596844i \(0.796421\pi\)
\(234\) 0 0
\(235\) 2.44949 0.159787
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −19.8990 −1.28716 −0.643579 0.765380i \(-0.722551\pi\)
−0.643579 + 0.765380i \(0.722551\pi\)
\(240\) 0 0
\(241\) 3.34847 0.215694 0.107847 0.994168i \(-0.465604\pi\)
0.107847 + 0.994168i \(0.465604\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.89898 0.312984
\(246\) 0 0
\(247\) −6.89898 −0.438972
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.10102 0.0694958 0.0347479 0.999396i \(-0.488937\pi\)
0.0347479 + 0.999396i \(0.488937\pi\)
\(252\) 0 0
\(253\) −2.44949 −0.153998
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.3485 1.20692 0.603462 0.797392i \(-0.293787\pi\)
0.603462 + 0.797392i \(0.293787\pi\)
\(258\) 0 0
\(259\) −21.8990 −1.36074
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −21.2474 −1.31017 −0.655087 0.755554i \(-0.727368\pi\)
−0.655087 + 0.755554i \(0.727368\pi\)
\(264\) 0 0
\(265\) 4.34847 0.267124
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.69694 0.347348 0.173674 0.984803i \(-0.444436\pi\)
0.173674 + 0.984803i \(0.444436\pi\)
\(270\) 0 0
\(271\) 25.6969 1.56098 0.780489 0.625170i \(-0.214970\pi\)
0.780489 + 0.625170i \(0.214970\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.44949 0.147710
\(276\) 0 0
\(277\) −14.8990 −0.895193 −0.447596 0.894236i \(-0.647720\pi\)
−0.447596 + 0.894236i \(0.647720\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.34847 0.438373 0.219186 0.975683i \(-0.429660\pi\)
0.219186 + 0.975683i \(0.429660\pi\)
\(282\) 0 0
\(283\) 8.55051 0.508275 0.254138 0.967168i \(-0.418208\pi\)
0.254138 + 0.967168i \(0.418208\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 27.2474 1.60837
\(288\) 0 0
\(289\) 12.6969 0.746879
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.3485 1.30561 0.652806 0.757525i \(-0.273592\pi\)
0.652806 + 0.757525i \(0.273592\pi\)
\(294\) 0 0
\(295\) 1.89898 0.110563
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.44949 −0.257321
\(300\) 0 0
\(301\) 30.6969 1.76934
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5.34847 −0.306252
\(306\) 0 0
\(307\) −12.4495 −0.710530 −0.355265 0.934766i \(-0.615609\pi\)
−0.355265 + 0.934766i \(0.615609\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.69694 0.493158 0.246579 0.969123i \(-0.420693\pi\)
0.246579 + 0.969123i \(0.420693\pi\)
\(312\) 0 0
\(313\) 3.65153 0.206397 0.103198 0.994661i \(-0.467092\pi\)
0.103198 + 0.994661i \(0.467092\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.2474 1.36187 0.680936 0.732343i \(-0.261573\pi\)
0.680936 + 0.732343i \(0.261573\pi\)
\(318\) 0 0
\(319\) −4.65153 −0.260436
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.44949 0.470142
\(324\) 0 0
\(325\) 4.44949 0.246813
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.44949 −0.465835
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.44949 0.0791941
\(336\) 0 0
\(337\) 15.1010 0.822605 0.411303 0.911499i \(-0.365074\pi\)
0.411303 + 0.911499i \(0.365074\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −17.1464 −0.928531
\(342\) 0 0
\(343\) 7.24745 0.391325
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.20204 −0.118212 −0.0591059 0.998252i \(-0.518825\pi\)
−0.0591059 + 0.998252i \(0.518825\pi\)
\(348\) 0 0
\(349\) −3.69694 −0.197893 −0.0989463 0.995093i \(-0.531547\pi\)
−0.0989463 + 0.995093i \(0.531547\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −14.4495 −0.769069 −0.384534 0.923111i \(-0.625638\pi\)
−0.384534 + 0.923111i \(0.625638\pi\)
\(354\) 0 0
\(355\) −3.00000 −0.159223
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −11.1464 −0.588286 −0.294143 0.955761i \(-0.595034\pi\)
−0.294143 + 0.955761i \(0.595034\pi\)
\(360\) 0 0
\(361\) −16.5959 −0.873469
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.34847 0.489321
\(366\) 0 0
\(367\) 5.24745 0.273915 0.136957 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −15.0000 −0.778761
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.44949 −0.435171
\(378\) 0 0
\(379\) 25.3939 1.30440 0.652198 0.758049i \(-0.273847\pi\)
0.652198 + 0.758049i \(0.273847\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10.3485 0.528782 0.264391 0.964416i \(-0.414829\pi\)
0.264391 + 0.964416i \(0.414829\pi\)
\(384\) 0 0
\(385\) −8.44949 −0.430626
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 22.8990 1.16102 0.580512 0.814252i \(-0.302852\pi\)
0.580512 + 0.814252i \(0.302852\pi\)
\(390\) 0 0
\(391\) 5.44949 0.275593
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −1.79796 −0.0902370 −0.0451185 0.998982i \(-0.514367\pi\)
−0.0451185 + 0.998982i \(0.514367\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8.69694 0.434304 0.217152 0.976138i \(-0.430323\pi\)
0.217152 + 0.976138i \(0.430323\pi\)
\(402\) 0 0
\(403\) −31.1464 −1.55151
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.5505 0.770810
\(408\) 0 0
\(409\) −20.5959 −1.01840 −0.509201 0.860647i \(-0.670059\pi\)
−0.509201 + 0.860647i \(0.670059\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.55051 −0.322330
\(414\) 0 0
\(415\) 10.3485 0.507986
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 17.1464 0.837658 0.418829 0.908065i \(-0.362441\pi\)
0.418829 + 0.908065i \(0.362441\pi\)
\(420\) 0 0
\(421\) −20.6515 −1.00649 −0.503247 0.864143i \(-0.667861\pi\)
−0.503247 + 0.864143i \(0.667861\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.44949 −0.264339
\(426\) 0 0
\(427\) 18.4495 0.892833
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 19.5959 0.943902 0.471951 0.881625i \(-0.343550\pi\)
0.471951 + 0.881625i \(0.343550\pi\)
\(432\) 0 0
\(433\) −26.8434 −1.29001 −0.645005 0.764178i \(-0.723145\pi\)
−0.645005 + 0.764178i \(0.723145\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.55051 0.0741710
\(438\) 0 0
\(439\) 6.89898 0.329270 0.164635 0.986355i \(-0.447355\pi\)
0.164635 + 0.986355i \(0.447355\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −38.9444 −1.85030 −0.925152 0.379597i \(-0.876063\pi\)
−0.925152 + 0.379597i \(0.876063\pi\)
\(444\) 0 0
\(445\) 16.8990 0.801088
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 26.3939 1.24560 0.622802 0.782379i \(-0.285994\pi\)
0.622802 + 0.782379i \(0.285994\pi\)
\(450\) 0 0
\(451\) −19.3485 −0.911084
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −15.3485 −0.719547
\(456\) 0 0
\(457\) −26.3485 −1.23253 −0.616265 0.787539i \(-0.711355\pi\)
−0.616265 + 0.787539i \(0.711355\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −9.30306 −0.433287 −0.216643 0.976251i \(-0.569511\pi\)
−0.216643 + 0.976251i \(0.569511\pi\)
\(462\) 0 0
\(463\) 24.0454 1.11748 0.558742 0.829341i \(-0.311284\pi\)
0.558742 + 0.829341i \(0.311284\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17.9444 −0.830367 −0.415184 0.909738i \(-0.636283\pi\)
−0.415184 + 0.909738i \(0.636283\pi\)
\(468\) 0 0
\(469\) −5.00000 −0.230879
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −21.7980 −1.00227
\(474\) 0 0
\(475\) −1.55051 −0.0711423
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.6515 0.486681 0.243340 0.969941i \(-0.421757\pi\)
0.243340 + 0.969941i \(0.421757\pi\)
\(480\) 0 0
\(481\) 28.2474 1.28797
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −14.8990 −0.676528
\(486\) 0 0
\(487\) 1.14643 0.0519496 0.0259748 0.999663i \(-0.491731\pi\)
0.0259748 + 0.999663i \(0.491731\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −38.3939 −1.73269 −0.866346 0.499445i \(-0.833537\pi\)
−0.866346 + 0.499445i \(0.833537\pi\)
\(492\) 0 0
\(493\) 10.3485 0.466072
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10.3485 0.464192
\(498\) 0 0
\(499\) −4.79796 −0.214786 −0.107393 0.994217i \(-0.534250\pi\)
−0.107393 + 0.994217i \(0.534250\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.1464 0.630758 0.315379 0.948966i \(-0.397868\pi\)
0.315379 + 0.948966i \(0.397868\pi\)
\(504\) 0 0
\(505\) −18.7980 −0.836498
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −23.3939 −1.03692 −0.518458 0.855103i \(-0.673494\pi\)
−0.518458 + 0.855103i \(0.673494\pi\)
\(510\) 0 0
\(511\) −32.2474 −1.42654
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −31.3485 −1.37340 −0.686701 0.726940i \(-0.740942\pi\)
−0.686701 + 0.726940i \(0.740942\pi\)
\(522\) 0 0
\(523\) −12.2020 −0.533558 −0.266779 0.963758i \(-0.585959\pi\)
−0.266779 + 0.963758i \(0.585959\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 38.1464 1.66168
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −35.1464 −1.52236
\(534\) 0 0
\(535\) −14.1464 −0.611603
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) 15.1010 0.649244 0.324622 0.945844i \(-0.394763\pi\)
0.324622 + 0.945844i \(0.394763\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −11.3485 −0.486115
\(546\) 0 0
\(547\) −33.3939 −1.42782 −0.713910 0.700238i \(-0.753077\pi\)
−0.713910 + 0.700238i \(0.753077\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.94439 0.125435
\(552\) 0 0
\(553\) 13.7980 0.586749
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.5505 1.04024 0.520119 0.854094i \(-0.325887\pi\)
0.520119 + 0.854094i \(0.325887\pi\)
\(558\) 0 0
\(559\) −39.5959 −1.67473
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 38.1464 1.60768 0.803840 0.594845i \(-0.202787\pi\)
0.803840 + 0.594845i \(0.202787\pi\)
\(564\) 0 0
\(565\) 16.3485 0.687785
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −11.3939 −0.477656 −0.238828 0.971062i \(-0.576763\pi\)
−0.238828 + 0.971062i \(0.576763\pi\)
\(570\) 0 0
\(571\) −32.0454 −1.34106 −0.670529 0.741883i \(-0.733933\pi\)
−0.670529 + 0.741883i \(0.733933\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −26.2929 −1.09459 −0.547293 0.836941i \(-0.684342\pi\)
−0.547293 + 0.836941i \(0.684342\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −35.6969 −1.48096
\(582\) 0 0
\(583\) 10.6515 0.441141
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.59592 −0.0658706 −0.0329353 0.999457i \(-0.510486\pi\)
−0.0329353 + 0.999457i \(0.510486\pi\)
\(588\) 0 0
\(589\) 10.8536 0.447214
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 41.1464 1.68968 0.844841 0.535018i \(-0.179695\pi\)
0.844841 + 0.535018i \(0.179695\pi\)
\(594\) 0 0
\(595\) 18.7980 0.770641
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) 14.7980 0.603621 0.301811 0.953368i \(-0.402409\pi\)
0.301811 + 0.953368i \(0.402409\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) 24.6515 1.00057 0.500287 0.865859i \(-0.333228\pi\)
0.500287 + 0.865859i \(0.333228\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 10.8990 0.440926
\(612\) 0 0
\(613\) 22.6969 0.916721 0.458360 0.888766i \(-0.348437\pi\)
0.458360 + 0.888766i \(0.348437\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 16.3485 0.658165 0.329082 0.944301i \(-0.393261\pi\)
0.329082 + 0.944301i \(0.393261\pi\)
\(618\) 0 0
\(619\) −1.79796 −0.0722661 −0.0361330 0.999347i \(-0.511504\pi\)
−0.0361330 + 0.999347i \(0.511504\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −58.2929 −2.33545
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −34.5959 −1.37943
\(630\) 0 0
\(631\) −22.2474 −0.885657 −0.442828 0.896606i \(-0.646025\pi\)
−0.442828 + 0.896606i \(0.646025\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.34847 0.132880
\(636\) 0 0
\(637\) 21.7980 0.863667
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −26.4495 −1.04469 −0.522346 0.852734i \(-0.674943\pi\)
−0.522346 + 0.852734i \(0.674943\pi\)
\(642\) 0 0
\(643\) 24.3485 0.960210 0.480105 0.877211i \(-0.340599\pi\)
0.480105 + 0.877211i \(0.340599\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −20.4495 −0.803952 −0.401976 0.915650i \(-0.631677\pi\)
−0.401976 + 0.915650i \(0.631677\pi\)
\(648\) 0 0
\(649\) 4.65153 0.182589
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −27.5505 −1.07813 −0.539067 0.842263i \(-0.681223\pi\)
−0.539067 + 0.842263i \(0.681223\pi\)
\(654\) 0 0
\(655\) −10.8990 −0.425858
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −14.4495 −0.562872 −0.281436 0.959580i \(-0.590811\pi\)
−0.281436 + 0.959580i \(0.590811\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.34847 0.207405
\(666\) 0 0
\(667\) 1.89898 0.0735288
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −13.1010 −0.505759
\(672\) 0 0
\(673\) 18.0454 0.695599 0.347800 0.937569i \(-0.386929\pi\)
0.347800 + 0.937569i \(0.386929\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 39.2474 1.50840 0.754201 0.656644i \(-0.228024\pi\)
0.754201 + 0.656644i \(0.228024\pi\)
\(678\) 0 0
\(679\) 51.3939 1.97232
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 19.3485 0.740349 0.370174 0.928962i \(-0.379298\pi\)
0.370174 + 0.928962i \(0.379298\pi\)
\(684\) 0 0
\(685\) 16.8990 0.645677
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 19.3485 0.737118
\(690\) 0 0
\(691\) 43.3939 1.65078 0.825390 0.564562i \(-0.190955\pi\)
0.825390 + 0.564562i \(0.190955\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −21.6969 −0.823012
\(696\) 0 0
\(697\) 43.0454 1.63046
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 16.0454 0.606027 0.303013 0.952986i \(-0.402007\pi\)
0.303013 + 0.952986i \(0.402007\pi\)
\(702\) 0 0
\(703\) −9.84337 −0.371250
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 64.8434 2.43869
\(708\) 0 0
\(709\) 52.9444 1.98837 0.994184 0.107694i \(-0.0343466\pi\)
0.994184 + 0.107694i \(0.0343466\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.00000 0.262152
\(714\) 0 0
\(715\) 10.8990 0.407599
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 49.2929 1.83831 0.919157 0.393892i \(-0.128872\pi\)
0.919157 + 0.393892i \(0.128872\pi\)
\(720\) 0 0
\(721\) 13.7980 0.513863
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.89898 −0.0705263
\(726\) 0 0
\(727\) −17.6515 −0.654659 −0.327330 0.944910i \(-0.606149\pi\)
−0.327330 + 0.944910i \(0.606149\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 48.4949 1.79365
\(732\) 0 0
\(733\) 21.6515 0.799718 0.399859 0.916577i \(-0.369059\pi\)
0.399859 + 0.916577i \(0.369059\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.55051 0.130785
\(738\) 0 0
\(739\) −45.6969 −1.68099 −0.840495 0.541820i \(-0.817735\pi\)
−0.840495 + 0.541820i \(0.817735\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 14.2020 0.521022 0.260511 0.965471i \(-0.416109\pi\)
0.260511 + 0.965471i \(0.416109\pi\)
\(744\) 0 0
\(745\) −18.2474 −0.668535
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 48.7980 1.78304
\(750\) 0 0
\(751\) 27.3485 0.997960 0.498980 0.866614i \(-0.333708\pi\)
0.498980 + 0.866614i \(0.333708\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −19.7980 −0.720521
\(756\) 0 0
\(757\) 10.7526 0.390808 0.195404 0.980723i \(-0.437398\pi\)
0.195404 + 0.980723i \(0.437398\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.49490 0.126690 0.0633450 0.997992i \(-0.479823\pi\)
0.0633450 + 0.997992i \(0.479823\pi\)
\(762\) 0 0
\(763\) 39.1464 1.41720
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.44949 0.305093
\(768\) 0 0
\(769\) 31.1464 1.12317 0.561584 0.827419i \(-0.310192\pi\)
0.561584 + 0.827419i \(0.310192\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.6969 −0.744417 −0.372209 0.928149i \(-0.621399\pi\)
−0.372209 + 0.928149i \(0.621399\pi\)
\(774\) 0 0
\(775\) −7.00000 −0.251447
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.2474 0.438810
\(780\) 0 0
\(781\) −7.34847 −0.262949
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −14.3485 −0.512119
\(786\) 0 0
\(787\) −13.8536 −0.493827 −0.246913 0.969038i \(-0.579416\pi\)
−0.246913 + 0.969038i \(0.579416\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −56.3939 −2.00514
\(792\) 0 0
\(793\) −23.7980 −0.845090
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.85357 0.136501 0.0682503 0.997668i \(-0.478258\pi\)
0.0682503 + 0.997668i \(0.478258\pi\)
\(798\) 0 0
\(799\) −13.3485 −0.472235
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 22.8990 0.808087
\(804\) 0 0
\(805\) 3.44949 0.121579
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.7980 0.660901 0.330451 0.943823i \(-0.392799\pi\)
0.330451 + 0.943823i \(0.392799\pi\)
\(810\) 0 0
\(811\) 2.79796 0.0982496 0.0491248 0.998793i \(-0.484357\pi\)
0.0491248 + 0.998793i \(0.484357\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −24.6969 −0.865096
\(816\) 0 0
\(817\) 13.7980 0.482729
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.5959 0.474501 0.237250 0.971449i \(-0.423754\pi\)
0.237250 + 0.971449i \(0.423754\pi\)
\(822\) 0 0
\(823\) 12.8990 0.449630 0.224815 0.974401i \(-0.427822\pi\)
0.224815 + 0.974401i \(0.427822\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −20.7526 −0.721637 −0.360818 0.932636i \(-0.617503\pi\)
−0.360818 + 0.932636i \(0.617503\pi\)
\(828\) 0 0
\(829\) −24.3939 −0.847234 −0.423617 0.905841i \(-0.639240\pi\)
−0.423617 + 0.905841i \(0.639240\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −26.6969 −0.924994
\(834\) 0 0
\(835\) −0.247449 −0.00856332
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 25.5959 0.883669 0.441835 0.897097i \(-0.354328\pi\)
0.441835 + 0.897097i \(0.354328\pi\)
\(840\) 0 0
\(841\) −25.3939 −0.875651
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.79796 0.233857
\(846\) 0 0
\(847\) 17.2474 0.592629
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −6.34847 −0.217623
\(852\) 0 0
\(853\) 23.7980 0.814827 0.407413 0.913244i \(-0.366431\pi\)
0.407413 + 0.913244i \(0.366431\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −16.2929 −0.556553 −0.278277 0.960501i \(-0.589763\pi\)
−0.278277 + 0.960501i \(0.589763\pi\)
\(858\) 0 0
\(859\) 17.0000 0.580033 0.290016 0.957022i \(-0.406339\pi\)
0.290016 + 0.957022i \(0.406339\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25.5959 −0.871295 −0.435648 0.900117i \(-0.643481\pi\)
−0.435648 + 0.900117i \(0.643481\pi\)
\(864\) 0 0
\(865\) 9.79796 0.333141
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −9.79796 −0.332373
\(870\) 0 0
\(871\) 6.44949 0.218533
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.44949 −0.116614
\(876\) 0 0
\(877\) 23.7980 0.803600 0.401800 0.915727i \(-0.368385\pi\)
0.401800 + 0.915727i \(0.368385\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 35.1464 1.18411 0.592057 0.805896i \(-0.298316\pi\)
0.592057 + 0.805896i \(0.298316\pi\)
\(882\) 0 0
\(883\) −37.5505 −1.26368 −0.631838 0.775101i \(-0.717699\pi\)
−0.631838 + 0.775101i \(0.717699\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 37.5959 1.26235 0.631174 0.775642i \(-0.282574\pi\)
0.631174 + 0.775642i \(0.282574\pi\)
\(888\) 0 0
\(889\) −11.5505 −0.387392
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.79796 −0.127094
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13.2929 0.443342
\(900\) 0 0
\(901\) −23.6969 −0.789459
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.898979 0.0298831
\(906\) 0 0
\(907\) −2.34847 −0.0779796 −0.0389898 0.999240i \(-0.512414\pi\)
−0.0389898 + 0.999240i \(0.512414\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −24.4949 −0.811552 −0.405776 0.913973i \(-0.632999\pi\)
−0.405776 + 0.913973i \(0.632999\pi\)
\(912\) 0 0
\(913\) 25.3485 0.838912
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 37.5959 1.24153
\(918\) 0 0
\(919\) 36.8990 1.21719 0.608593 0.793483i \(-0.291734\pi\)
0.608593 + 0.793483i \(0.291734\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −13.3485 −0.439370
\(924\) 0 0
\(925\) 6.34847 0.208736
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 27.0000 0.885841 0.442921 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(930\) 0 0
\(931\) −7.59592 −0.248946
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −13.3485 −0.436542
\(936\) 0 0
\(937\) 28.6969 0.937488 0.468744 0.883334i \(-0.344707\pi\)
0.468744 + 0.883334i \(0.344707\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −32.4495 −1.05782 −0.528912 0.848677i \(-0.677400\pi\)
−0.528912 + 0.848677i \(0.677400\pi\)
\(942\) 0 0
\(943\) 7.89898 0.257226
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36.4949 −1.18592 −0.592962 0.805230i \(-0.702042\pi\)
−0.592962 + 0.805230i \(0.702042\pi\)
\(948\) 0 0
\(949\) 41.5959 1.35026
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 32.2020 1.04313 0.521563 0.853213i \(-0.325349\pi\)
0.521563 + 0.853213i \(0.325349\pi\)
\(954\) 0 0
\(955\) −15.5505 −0.503203
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −58.2929 −1.88237
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −16.0000 −0.515058
\(966\) 0 0
\(967\) −8.65153 −0.278214 −0.139107 0.990277i \(-0.544423\pi\)
−0.139107 + 0.990277i \(0.544423\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.4949 0.400980 0.200490 0.979696i \(-0.435747\pi\)
0.200490 + 0.979696i \(0.435747\pi\)
\(972\) 0 0
\(973\) 74.8434 2.39937
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −40.3485 −1.29086 −0.645431 0.763819i \(-0.723322\pi\)
−0.645431 + 0.763819i \(0.723322\pi\)
\(978\) 0 0
\(979\) 41.3939 1.32295
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 16.3485 0.521435 0.260718 0.965415i \(-0.416041\pi\)
0.260718 + 0.965415i \(0.416041\pi\)
\(984\) 0 0
\(985\) 15.7980 0.503365
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.89898 0.282971
\(990\) 0 0
\(991\) 30.1010 0.956190 0.478095 0.878308i \(-0.341327\pi\)
0.478095 + 0.878308i \(0.341327\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 11.7980 0.374020
\(996\) 0 0
\(997\) 32.0000 1.01345 0.506725 0.862108i \(-0.330856\pi\)
0.506725 + 0.862108i \(0.330856\pi\)
\(998\) 0 0
\(999\) 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 4140.2.a.q.1.1 yes 2
3.2 odd 2 4140.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.a.l.1.1 2 3.2 odd 2
4140.2.a.q.1.1 yes 2 1.1 even 1 trivial