Properties

Label 6720.2.a.cp.1.2
Level $6720$
Weight $2$
Character 6720.1
Self dual yes
Analytic conductor $53.659$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6720.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.6594701583\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3360)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 6720.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.47214 q^{13} +1.00000 q^{15} -4.47214 q^{17} -1.00000 q^{21} +6.47214 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.47214 q^{29} -6.47214 q^{31} -1.00000 q^{35} -8.47214 q^{37} -4.47214 q^{39} +10.9443 q^{41} -10.4721 q^{43} -1.00000 q^{45} -2.47214 q^{47} +1.00000 q^{49} +4.47214 q^{51} -2.00000 q^{53} +4.00000 q^{59} -12.4721 q^{61} +1.00000 q^{63} -4.47214 q^{65} -2.47214 q^{67} -6.47214 q^{69} +15.4164 q^{71} +8.47214 q^{73} -1.00000 q^{75} +12.9443 q^{79} +1.00000 q^{81} +16.9443 q^{83} +4.47214 q^{85} +8.47214 q^{87} -6.94427 q^{89} +4.47214 q^{91} +6.47214 q^{93} -4.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{15} - 2 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 8 q^{29} - 4 q^{31} - 2 q^{35} - 8 q^{37} + 4 q^{41} - 12 q^{43} - 2 q^{45} + 4 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{59} - 16 q^{61} + 2 q^{63} + 4 q^{67} - 4 q^{69} + 4 q^{71} + 8 q^{73} - 2 q^{75} + 8 q^{79} + 2 q^{81} + 16 q^{83} + 8 q^{87} + 4 q^{89} + 4 q^{93}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 6.47214 1.34953 0.674767 0.738031i \(-0.264244\pi\)
0.674767 + 0.738031i \(0.264244\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.47214 −1.57324 −0.786618 0.617440i \(-0.788170\pi\)
−0.786618 + 0.617440i \(0.788170\pi\)
\(30\) 0 0
\(31\) −6.47214 −1.16243 −0.581215 0.813750i \(-0.697422\pi\)
−0.581215 + 0.813750i \(0.697422\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −8.47214 −1.39281 −0.696405 0.717649i \(-0.745218\pi\)
−0.696405 + 0.717649i \(0.745218\pi\)
\(38\) 0 0
\(39\) −4.47214 −0.716115
\(40\) 0 0
\(41\) 10.9443 1.70921 0.854604 0.519280i \(-0.173800\pi\)
0.854604 + 0.519280i \(0.173800\pi\)
\(42\) 0 0
\(43\) −10.4721 −1.59699 −0.798493 0.602004i \(-0.794369\pi\)
−0.798493 + 0.602004i \(0.794369\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −2.47214 −0.360598 −0.180299 0.983612i \(-0.557707\pi\)
−0.180299 + 0.983612i \(0.557707\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.47214 0.626224
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −12.4721 −1.59689 −0.798447 0.602066i \(-0.794345\pi\)
−0.798447 + 0.602066i \(0.794345\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −4.47214 −0.554700
\(66\) 0 0
\(67\) −2.47214 −0.302019 −0.151010 0.988532i \(-0.548252\pi\)
−0.151010 + 0.988532i \(0.548252\pi\)
\(68\) 0 0
\(69\) −6.47214 −0.779154
\(70\) 0 0
\(71\) 15.4164 1.82959 0.914796 0.403917i \(-0.132352\pi\)
0.914796 + 0.403917i \(0.132352\pi\)
\(72\) 0 0
\(73\) 8.47214 0.991589 0.495794 0.868440i \(-0.334877\pi\)
0.495794 + 0.868440i \(0.334877\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 12.9443 1.45634 0.728172 0.685394i \(-0.240370\pi\)
0.728172 + 0.685394i \(0.240370\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.9443 1.85988 0.929938 0.367717i \(-0.119860\pi\)
0.929938 + 0.367717i \(0.119860\pi\)
\(84\) 0 0
\(85\) 4.47214 0.485071
\(86\) 0 0
\(87\) 8.47214 0.908308
\(88\) 0 0
\(89\) −6.94427 −0.736091 −0.368046 0.929808i \(-0.619973\pi\)
−0.368046 + 0.929808i \(0.619973\pi\)
\(90\) 0 0
\(91\) 4.47214 0.468807
\(92\) 0 0
\(93\) 6.47214 0.671129
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.47214 −0.454077 −0.227038 0.973886i \(-0.572904\pi\)
−0.227038 + 0.973886i \(0.572904\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.94427 0.690981 0.345490 0.938422i \(-0.387713\pi\)
0.345490 + 0.938422i \(0.387713\pi\)
\(102\) 0 0
\(103\) −17.8885 −1.76261 −0.881305 0.472547i \(-0.843335\pi\)
−0.881305 + 0.472547i \(0.843335\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 6.47214 0.625685 0.312842 0.949805i \(-0.398719\pi\)
0.312842 + 0.949805i \(0.398719\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 8.47214 0.804140
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −6.47214 −0.603530
\(116\) 0 0
\(117\) 4.47214 0.413449
\(118\) 0 0
\(119\) −4.47214 −0.409960
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −10.9443 −0.986812
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 10.4721 0.922020
\(130\) 0 0
\(131\) −0.944272 −0.0825014 −0.0412507 0.999149i \(-0.513134\pi\)
−0.0412507 + 0.999149i \(0.513134\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 10.9443 0.935032 0.467516 0.883985i \(-0.345149\pi\)
0.467516 + 0.883985i \(0.345149\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 2.47214 0.208191
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 8.47214 0.703573
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −3.52786 −0.289014 −0.144507 0.989504i \(-0.546160\pi\)
−0.144507 + 0.989504i \(0.546160\pi\)
\(150\) 0 0
\(151\) −20.9443 −1.70442 −0.852210 0.523199i \(-0.824738\pi\)
−0.852210 + 0.523199i \(0.824738\pi\)
\(152\) 0 0
\(153\) −4.47214 −0.361551
\(154\) 0 0
\(155\) 6.47214 0.519854
\(156\) 0 0
\(157\) −16.4721 −1.31462 −0.657310 0.753620i \(-0.728306\pi\)
−0.657310 + 0.753620i \(0.728306\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 6.47214 0.510076
\(162\) 0 0
\(163\) 2.47214 0.193633 0.0968163 0.995302i \(-0.469134\pi\)
0.0968163 + 0.995302i \(0.469134\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −10.4721 −0.810358 −0.405179 0.914237i \(-0.632791\pi\)
−0.405179 + 0.914237i \(0.632791\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) 13.4164 0.997234 0.498617 0.866822i \(-0.333841\pi\)
0.498617 + 0.866822i \(0.333841\pi\)
\(182\) 0 0
\(183\) 12.4721 0.921967
\(184\) 0 0
\(185\) 8.47214 0.622884
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −23.4164 −1.69435 −0.847176 0.531313i \(-0.821699\pi\)
−0.847176 + 0.531313i \(0.821699\pi\)
\(192\) 0 0
\(193\) −18.9443 −1.36364 −0.681819 0.731521i \(-0.738811\pi\)
−0.681819 + 0.731521i \(0.738811\pi\)
\(194\) 0 0
\(195\) 4.47214 0.320256
\(196\) 0 0
\(197\) −14.9443 −1.06474 −0.532368 0.846513i \(-0.678698\pi\)
−0.532368 + 0.846513i \(0.678698\pi\)
\(198\) 0 0
\(199\) 11.4164 0.809288 0.404644 0.914474i \(-0.367395\pi\)
0.404644 + 0.914474i \(0.367395\pi\)
\(200\) 0 0
\(201\) 2.47214 0.174371
\(202\) 0 0
\(203\) −8.47214 −0.594627
\(204\) 0 0
\(205\) −10.9443 −0.764381
\(206\) 0 0
\(207\) 6.47214 0.449845
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) −15.4164 −1.05631
\(214\) 0 0
\(215\) 10.4721 0.714194
\(216\) 0 0
\(217\) −6.47214 −0.439357
\(218\) 0 0
\(219\) −8.47214 −0.572494
\(220\) 0 0
\(221\) −20.0000 −1.34535
\(222\) 0 0
\(223\) 25.8885 1.73363 0.866813 0.498634i \(-0.166165\pi\)
0.866813 + 0.498634i \(0.166165\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −0.944272 −0.0626735 −0.0313368 0.999509i \(-0.509976\pi\)
−0.0313368 + 0.999509i \(0.509976\pi\)
\(228\) 0 0
\(229\) −20.4721 −1.35284 −0.676418 0.736518i \(-0.736469\pi\)
−0.676418 + 0.736518i \(0.736469\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.94427 0.192886 0.0964428 0.995339i \(-0.469254\pi\)
0.0964428 + 0.995339i \(0.469254\pi\)
\(234\) 0 0
\(235\) 2.47214 0.161264
\(236\) 0 0
\(237\) −12.9443 −0.840821
\(238\) 0 0
\(239\) −20.3607 −1.31702 −0.658511 0.752571i \(-0.728814\pi\)
−0.658511 + 0.752571i \(0.728814\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −16.9443 −1.07380
\(250\) 0 0
\(251\) 7.05573 0.445354 0.222677 0.974892i \(-0.428521\pi\)
0.222677 + 0.974892i \(0.428521\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −4.47214 −0.280056
\(256\) 0 0
\(257\) 8.47214 0.528477 0.264239 0.964457i \(-0.414879\pi\)
0.264239 + 0.964457i \(0.414879\pi\)
\(258\) 0 0
\(259\) −8.47214 −0.526433
\(260\) 0 0
\(261\) −8.47214 −0.524412
\(262\) 0 0
\(263\) −9.52786 −0.587513 −0.293757 0.955880i \(-0.594906\pi\)
−0.293757 + 0.955880i \(0.594906\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 6.94427 0.424983
\(268\) 0 0
\(269\) −22.0000 −1.34136 −0.670682 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) 0 0
\(271\) 24.3607 1.47981 0.739903 0.672714i \(-0.234871\pi\)
0.739903 + 0.672714i \(0.234871\pi\)
\(272\) 0 0
\(273\) −4.47214 −0.270666
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.47214 0.268705 0.134352 0.990934i \(-0.457105\pi\)
0.134352 + 0.990934i \(0.457105\pi\)
\(278\) 0 0
\(279\) −6.47214 −0.387477
\(280\) 0 0
\(281\) −31.8885 −1.90231 −0.951156 0.308712i \(-0.900102\pi\)
−0.951156 + 0.308712i \(0.900102\pi\)
\(282\) 0 0
\(283\) 5.88854 0.350038 0.175019 0.984565i \(-0.444001\pi\)
0.175019 + 0.984565i \(0.444001\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.9443 0.646020
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 4.47214 0.262161
\(292\) 0 0
\(293\) 5.05573 0.295359 0.147679 0.989035i \(-0.452820\pi\)
0.147679 + 0.989035i \(0.452820\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 28.9443 1.67389
\(300\) 0 0
\(301\) −10.4721 −0.603604
\(302\) 0 0
\(303\) −6.94427 −0.398938
\(304\) 0 0
\(305\) 12.4721 0.714152
\(306\) 0 0
\(307\) 24.9443 1.42364 0.711822 0.702360i \(-0.247870\pi\)
0.711822 + 0.702360i \(0.247870\pi\)
\(308\) 0 0
\(309\) 17.8885 1.01764
\(310\) 0 0
\(311\) −33.8885 −1.92164 −0.960822 0.277168i \(-0.910604\pi\)
−0.960822 + 0.277168i \(0.910604\pi\)
\(312\) 0 0
\(313\) 0.472136 0.0266867 0.0133434 0.999911i \(-0.495753\pi\)
0.0133434 + 0.999911i \(0.495753\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) 15.8885 0.892390 0.446195 0.894936i \(-0.352779\pi\)
0.446195 + 0.894936i \(0.352779\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −6.47214 −0.361239
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 4.47214 0.248069
\(326\) 0 0
\(327\) 14.0000 0.774202
\(328\) 0 0
\(329\) −2.47214 −0.136293
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) −8.47214 −0.464270
\(334\) 0 0
\(335\) 2.47214 0.135067
\(336\) 0 0
\(337\) 6.94427 0.378279 0.189139 0.981950i \(-0.439430\pi\)
0.189139 + 0.981950i \(0.439430\pi\)
\(338\) 0 0
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 6.47214 0.348448
\(346\) 0 0
\(347\) 9.52786 0.511483 0.255741 0.966745i \(-0.417680\pi\)
0.255741 + 0.966745i \(0.417680\pi\)
\(348\) 0 0
\(349\) −15.5279 −0.831188 −0.415594 0.909550i \(-0.636426\pi\)
−0.415594 + 0.909550i \(0.636426\pi\)
\(350\) 0 0
\(351\) −4.47214 −0.238705
\(352\) 0 0
\(353\) −17.4164 −0.926982 −0.463491 0.886102i \(-0.653403\pi\)
−0.463491 + 0.886102i \(0.653403\pi\)
\(354\) 0 0
\(355\) −15.4164 −0.818218
\(356\) 0 0
\(357\) 4.47214 0.236691
\(358\) 0 0
\(359\) 15.4164 0.813647 0.406823 0.913507i \(-0.366636\pi\)
0.406823 + 0.913507i \(0.366636\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) −8.47214 −0.443452
\(366\) 0 0
\(367\) −20.9443 −1.09328 −0.546641 0.837367i \(-0.684094\pi\)
−0.546641 + 0.837367i \(0.684094\pi\)
\(368\) 0 0
\(369\) 10.9443 0.569736
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) 2.58359 0.133773 0.0668867 0.997761i \(-0.478693\pi\)
0.0668867 + 0.997761i \(0.478693\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −37.8885 −1.95136
\(378\) 0 0
\(379\) −19.0557 −0.978827 −0.489414 0.872052i \(-0.662789\pi\)
−0.489414 + 0.872052i \(0.662789\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 23.4164 1.19652 0.598261 0.801301i \(-0.295858\pi\)
0.598261 + 0.801301i \(0.295858\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −10.4721 −0.532329
\(388\) 0 0
\(389\) −24.4721 −1.24079 −0.620393 0.784291i \(-0.713027\pi\)
−0.620393 + 0.784291i \(0.713027\pi\)
\(390\) 0 0
\(391\) −28.9443 −1.46377
\(392\) 0 0
\(393\) 0.944272 0.0476322
\(394\) 0 0
\(395\) −12.9443 −0.651297
\(396\) 0 0
\(397\) 1.41641 0.0710875 0.0355437 0.999368i \(-0.488684\pi\)
0.0355437 + 0.999368i \(0.488684\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −26.9443 −1.34553 −0.672766 0.739855i \(-0.734894\pi\)
−0.672766 + 0.739855i \(0.734894\pi\)
\(402\) 0 0
\(403\) −28.9443 −1.44182
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 5.05573 0.249990 0.124995 0.992157i \(-0.460109\pi\)
0.124995 + 0.992157i \(0.460109\pi\)
\(410\) 0 0
\(411\) −10.9443 −0.539841
\(412\) 0 0
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −16.9443 −0.831762
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15.0557 0.735520 0.367760 0.929921i \(-0.380125\pi\)
0.367760 + 0.929921i \(0.380125\pi\)
\(420\) 0 0
\(421\) −23.8885 −1.16426 −0.582128 0.813097i \(-0.697780\pi\)
−0.582128 + 0.813097i \(0.697780\pi\)
\(422\) 0 0
\(423\) −2.47214 −0.120199
\(424\) 0 0
\(425\) −4.47214 −0.216930
\(426\) 0 0
\(427\) −12.4721 −0.603569
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −34.4721 −1.66046 −0.830232 0.557418i \(-0.811792\pi\)
−0.830232 + 0.557418i \(0.811792\pi\)
\(432\) 0 0
\(433\) 18.3607 0.882358 0.441179 0.897419i \(-0.354560\pi\)
0.441179 + 0.897419i \(0.354560\pi\)
\(434\) 0 0
\(435\) −8.47214 −0.406208
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 30.4721 1.45436 0.727178 0.686449i \(-0.240832\pi\)
0.727178 + 0.686449i \(0.240832\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 1.52786 0.0725910 0.0362955 0.999341i \(-0.488444\pi\)
0.0362955 + 0.999341i \(0.488444\pi\)
\(444\) 0 0
\(445\) 6.94427 0.329190
\(446\) 0 0
\(447\) 3.52786 0.166862
\(448\) 0 0
\(449\) 32.8328 1.54948 0.774738 0.632282i \(-0.217882\pi\)
0.774738 + 0.632282i \(0.217882\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 20.9443 0.984048
\(454\) 0 0
\(455\) −4.47214 −0.209657
\(456\) 0 0
\(457\) −34.9443 −1.63462 −0.817312 0.576195i \(-0.804537\pi\)
−0.817312 + 0.576195i \(0.804537\pi\)
\(458\) 0 0
\(459\) 4.47214 0.208741
\(460\) 0 0
\(461\) −28.8328 −1.34288 −0.671439 0.741060i \(-0.734323\pi\)
−0.671439 + 0.741060i \(0.734323\pi\)
\(462\) 0 0
\(463\) 3.05573 0.142012 0.0710059 0.997476i \(-0.477379\pi\)
0.0710059 + 0.997476i \(0.477379\pi\)
\(464\) 0 0
\(465\) −6.47214 −0.300138
\(466\) 0 0
\(467\) 29.8885 1.38308 0.691538 0.722340i \(-0.256933\pi\)
0.691538 + 0.722340i \(0.256933\pi\)
\(468\) 0 0
\(469\) −2.47214 −0.114153
\(470\) 0 0
\(471\) 16.4721 0.758996
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 9.88854 0.451819 0.225910 0.974148i \(-0.427465\pi\)
0.225910 + 0.974148i \(0.427465\pi\)
\(480\) 0 0
\(481\) −37.8885 −1.72757
\(482\) 0 0
\(483\) −6.47214 −0.294492
\(484\) 0 0
\(485\) 4.47214 0.203069
\(486\) 0 0
\(487\) −19.0557 −0.863497 −0.431749 0.901994i \(-0.642103\pi\)
−0.431749 + 0.901994i \(0.642103\pi\)
\(488\) 0 0
\(489\) −2.47214 −0.111794
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 37.8885 1.70641
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15.4164 0.691520
\(498\) 0 0
\(499\) −11.0557 −0.494922 −0.247461 0.968898i \(-0.579596\pi\)
−0.247461 + 0.968898i \(0.579596\pi\)
\(500\) 0 0
\(501\) 10.4721 0.467861
\(502\) 0 0
\(503\) −21.5279 −0.959880 −0.479940 0.877301i \(-0.659342\pi\)
−0.479940 + 0.877301i \(0.659342\pi\)
\(504\) 0 0
\(505\) −6.94427 −0.309016
\(506\) 0 0
\(507\) −7.00000 −0.310881
\(508\) 0 0
\(509\) −22.0000 −0.975133 −0.487566 0.873086i \(-0.662115\pi\)
−0.487566 + 0.873086i \(0.662115\pi\)
\(510\) 0 0
\(511\) 8.47214 0.374785
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 17.8885 0.788263
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) −24.9443 −1.09074 −0.545368 0.838196i \(-0.683610\pi\)
−0.545368 + 0.838196i \(0.683610\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 28.9443 1.26083
\(528\) 0 0
\(529\) 18.8885 0.821241
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 48.9443 2.12001
\(534\) 0 0
\(535\) −6.47214 −0.279815
\(536\) 0 0
\(537\) −16.0000 −0.690451
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 14.9443 0.642504 0.321252 0.946994i \(-0.395896\pi\)
0.321252 + 0.946994i \(0.395896\pi\)
\(542\) 0 0
\(543\) −13.4164 −0.575753
\(544\) 0 0
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 15.4164 0.659158 0.329579 0.944128i \(-0.393093\pi\)
0.329579 + 0.944128i \(0.393093\pi\)
\(548\) 0 0
\(549\) −12.4721 −0.532298
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 12.9443 0.550446
\(554\) 0 0
\(555\) −8.47214 −0.359622
\(556\) 0 0
\(557\) −10.0000 −0.423714 −0.211857 0.977301i \(-0.567951\pi\)
−0.211857 + 0.977301i \(0.567951\pi\)
\(558\) 0 0
\(559\) −46.8328 −1.98082
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 32.8328 1.37642 0.688212 0.725510i \(-0.258396\pi\)
0.688212 + 0.725510i \(0.258396\pi\)
\(570\) 0 0
\(571\) 4.94427 0.206911 0.103456 0.994634i \(-0.467010\pi\)
0.103456 + 0.994634i \(0.467010\pi\)
\(572\) 0 0
\(573\) 23.4164 0.978234
\(574\) 0 0
\(575\) 6.47214 0.269907
\(576\) 0 0
\(577\) 42.3607 1.76350 0.881749 0.471719i \(-0.156366\pi\)
0.881749 + 0.471719i \(0.156366\pi\)
\(578\) 0 0
\(579\) 18.9443 0.787297
\(580\) 0 0
\(581\) 16.9443 0.702967
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −4.47214 −0.184900
\(586\) 0 0
\(587\) −5.88854 −0.243046 −0.121523 0.992589i \(-0.538778\pi\)
−0.121523 + 0.992589i \(0.538778\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 14.9443 0.614725
\(592\) 0 0
\(593\) −12.4721 −0.512169 −0.256085 0.966654i \(-0.582433\pi\)
−0.256085 + 0.966654i \(0.582433\pi\)
\(594\) 0 0
\(595\) 4.47214 0.183340
\(596\) 0 0
\(597\) −11.4164 −0.467242
\(598\) 0 0
\(599\) 2.47214 0.101009 0.0505044 0.998724i \(-0.483917\pi\)
0.0505044 + 0.998724i \(0.483917\pi\)
\(600\) 0 0
\(601\) −9.05573 −0.369391 −0.184695 0.982796i \(-0.559130\pi\)
−0.184695 + 0.982796i \(0.559130\pi\)
\(602\) 0 0
\(603\) −2.47214 −0.100673
\(604\) 0 0
\(605\) 11.0000 0.447214
\(606\) 0 0
\(607\) −9.88854 −0.401364 −0.200682 0.979656i \(-0.564316\pi\)
−0.200682 + 0.979656i \(0.564316\pi\)
\(608\) 0 0
\(609\) 8.47214 0.343308
\(610\) 0 0
\(611\) −11.0557 −0.447267
\(612\) 0 0
\(613\) 23.5279 0.950281 0.475141 0.879910i \(-0.342397\pi\)
0.475141 + 0.879910i \(0.342397\pi\)
\(614\) 0 0
\(615\) 10.9443 0.441316
\(616\) 0 0
\(617\) −38.9443 −1.56784 −0.783919 0.620864i \(-0.786782\pi\)
−0.783919 + 0.620864i \(0.786782\pi\)
\(618\) 0 0
\(619\) −44.9443 −1.80646 −0.903231 0.429154i \(-0.858812\pi\)
−0.903231 + 0.429154i \(0.858812\pi\)
\(620\) 0 0
\(621\) −6.47214 −0.259718
\(622\) 0 0
\(623\) −6.94427 −0.278216
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 37.8885 1.51072
\(630\) 0 0
\(631\) 33.8885 1.34908 0.674541 0.738238i \(-0.264342\pi\)
0.674541 + 0.738238i \(0.264342\pi\)
\(632\) 0 0
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.47214 0.177192
\(638\) 0 0
\(639\) 15.4164 0.609864
\(640\) 0 0
\(641\) 6.94427 0.274282 0.137141 0.990552i \(-0.456209\pi\)
0.137141 + 0.990552i \(0.456209\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) −10.4721 −0.412340
\(646\) 0 0
\(647\) −31.4164 −1.23511 −0.617553 0.786529i \(-0.711876\pi\)
−0.617553 + 0.786529i \(0.711876\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 6.47214 0.253663
\(652\) 0 0
\(653\) 39.8885 1.56096 0.780480 0.625181i \(-0.214975\pi\)
0.780480 + 0.625181i \(0.214975\pi\)
\(654\) 0 0
\(655\) 0.944272 0.0368958
\(656\) 0 0
\(657\) 8.47214 0.330530
\(658\) 0 0
\(659\) −22.8328 −0.889440 −0.444720 0.895670i \(-0.646697\pi\)
−0.444720 + 0.895670i \(0.646697\pi\)
\(660\) 0 0
\(661\) 1.63932 0.0637622 0.0318811 0.999492i \(-0.489850\pi\)
0.0318811 + 0.999492i \(0.489850\pi\)
\(662\) 0 0
\(663\) 20.0000 0.776736
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −54.8328 −2.12314
\(668\) 0 0
\(669\) −25.8885 −1.00091
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −25.7771 −0.993634 −0.496817 0.867855i \(-0.665498\pi\)
−0.496817 + 0.867855i \(0.665498\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −23.8885 −0.918111 −0.459056 0.888408i \(-0.651812\pi\)
−0.459056 + 0.888408i \(0.651812\pi\)
\(678\) 0 0
\(679\) −4.47214 −0.171625
\(680\) 0 0
\(681\) 0.944272 0.0361846
\(682\) 0 0
\(683\) 32.3607 1.23825 0.619123 0.785294i \(-0.287488\pi\)
0.619123 + 0.785294i \(0.287488\pi\)
\(684\) 0 0
\(685\) −10.9443 −0.418159
\(686\) 0 0
\(687\) 20.4721 0.781061
\(688\) 0 0
\(689\) −8.94427 −0.340750
\(690\) 0 0
\(691\) 9.88854 0.376178 0.188089 0.982152i \(-0.439771\pi\)
0.188089 + 0.982152i \(0.439771\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −48.9443 −1.85390
\(698\) 0 0
\(699\) −2.94427 −0.111363
\(700\) 0 0
\(701\) 17.4164 0.657809 0.328904 0.944363i \(-0.393321\pi\)
0.328904 + 0.944363i \(0.393321\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −2.47214 −0.0931060
\(706\) 0 0
\(707\) 6.94427 0.261166
\(708\) 0 0
\(709\) 40.8328 1.53351 0.766754 0.641941i \(-0.221870\pi\)
0.766754 + 0.641941i \(0.221870\pi\)
\(710\) 0 0
\(711\) 12.9443 0.485448
\(712\) 0 0
\(713\) −41.8885 −1.56874
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 20.3607 0.760384
\(718\) 0 0
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 0 0
\(721\) −17.8885 −0.666204
\(722\) 0 0
\(723\) 6.00000 0.223142
\(724\) 0 0
\(725\) −8.47214 −0.314647
\(726\) 0 0
\(727\) −11.0557 −0.410034 −0.205017 0.978758i \(-0.565725\pi\)
−0.205017 + 0.978758i \(0.565725\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 46.8328 1.73217
\(732\) 0 0
\(733\) 2.58359 0.0954272 0.0477136 0.998861i \(-0.484807\pi\)
0.0477136 + 0.998861i \(0.484807\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −19.4164 −0.712319 −0.356159 0.934425i \(-0.615914\pi\)
−0.356159 + 0.934425i \(0.615914\pi\)
\(744\) 0 0
\(745\) 3.52786 0.129251
\(746\) 0 0
\(747\) 16.9443 0.619958
\(748\) 0 0
\(749\) 6.47214 0.236487
\(750\) 0 0
\(751\) 28.9443 1.05619 0.528096 0.849185i \(-0.322906\pi\)
0.528096 + 0.849185i \(0.322906\pi\)
\(752\) 0 0
\(753\) −7.05573 −0.257125
\(754\) 0 0
\(755\) 20.9443 0.762240
\(756\) 0 0
\(757\) 20.4721 0.744072 0.372036 0.928218i \(-0.378660\pi\)
0.372036 + 0.928218i \(0.378660\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −0.111456 −0.00404028 −0.00202014 0.999998i \(-0.500643\pi\)
−0.00202014 + 0.999998i \(0.500643\pi\)
\(762\) 0 0
\(763\) −14.0000 −0.506834
\(764\) 0 0
\(765\) 4.47214 0.161690
\(766\) 0 0
\(767\) 17.8885 0.645918
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) −8.47214 −0.305117
\(772\) 0 0
\(773\) 21.0557 0.757322 0.378661 0.925535i \(-0.376385\pi\)
0.378661 + 0.925535i \(0.376385\pi\)
\(774\) 0 0
\(775\) −6.47214 −0.232486
\(776\) 0 0
\(777\) 8.47214 0.303936
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 8.47214 0.302769
\(784\) 0 0
\(785\) 16.4721 0.587916
\(786\) 0 0
\(787\) −23.7771 −0.847562 −0.423781 0.905765i \(-0.639297\pi\)
−0.423781 + 0.905765i \(0.639297\pi\)
\(788\) 0 0
\(789\) 9.52786 0.339201
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) −55.7771 −1.98070
\(794\) 0 0
\(795\) −2.00000 −0.0709327
\(796\) 0 0
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 11.0557 0.391124
\(800\) 0 0
\(801\) −6.94427 −0.245364
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −6.47214 −0.228113
\(806\) 0 0
\(807\) 22.0000 0.774437
\(808\) 0 0
\(809\) −12.8328 −0.451178 −0.225589 0.974223i \(-0.572431\pi\)
−0.225589 + 0.974223i \(0.572431\pi\)
\(810\) 0 0
\(811\) −27.7771 −0.975385 −0.487693 0.873015i \(-0.662161\pi\)
−0.487693 + 0.873015i \(0.662161\pi\)
\(812\) 0 0
\(813\) −24.3607 −0.854366
\(814\) 0 0
\(815\) −2.47214 −0.0865951
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 4.47214 0.156269
\(820\) 0 0
\(821\) −31.3050 −1.09255 −0.546275 0.837606i \(-0.683955\pi\)
−0.546275 + 0.837606i \(0.683955\pi\)
\(822\) 0 0
\(823\) −3.05573 −0.106516 −0.0532580 0.998581i \(-0.516961\pi\)
−0.0532580 + 0.998581i \(0.516961\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.4164 0.396987 0.198494 0.980102i \(-0.436395\pi\)
0.198494 + 0.980102i \(0.436395\pi\)
\(828\) 0 0
\(829\) −33.4164 −1.16060 −0.580300 0.814403i \(-0.697065\pi\)
−0.580300 + 0.814403i \(0.697065\pi\)
\(830\) 0 0
\(831\) −4.47214 −0.155137
\(832\) 0 0
\(833\) −4.47214 −0.154950
\(834\) 0 0
\(835\) 10.4721 0.362403
\(836\) 0 0
\(837\) 6.47214 0.223710
\(838\) 0 0
\(839\) −49.8885 −1.72234 −0.861172 0.508314i \(-0.830269\pi\)
−0.861172 + 0.508314i \(0.830269\pi\)
\(840\) 0 0
\(841\) 42.7771 1.47507
\(842\) 0 0
\(843\) 31.8885 1.09830
\(844\) 0 0
\(845\) −7.00000 −0.240807
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 0 0
\(849\) −5.88854 −0.202094
\(850\) 0 0
\(851\) −54.8328 −1.87964
\(852\) 0 0
\(853\) 31.5279 1.07949 0.539747 0.841827i \(-0.318520\pi\)
0.539747 + 0.841827i \(0.318520\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.3607 0.627189 0.313594 0.949557i \(-0.398467\pi\)
0.313594 + 0.949557i \(0.398467\pi\)
\(858\) 0 0
\(859\) 4.94427 0.168696 0.0843482 0.996436i \(-0.473119\pi\)
0.0843482 + 0.996436i \(0.473119\pi\)
\(860\) 0 0
\(861\) −10.9443 −0.372980
\(862\) 0 0
\(863\) −48.3607 −1.64622 −0.823108 0.567884i \(-0.807762\pi\)
−0.823108 + 0.567884i \(0.807762\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 0 0
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −11.0557 −0.374609
\(872\) 0 0
\(873\) −4.47214 −0.151359
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −47.3050 −1.59737 −0.798687 0.601746i \(-0.794472\pi\)
−0.798687 + 0.601746i \(0.794472\pi\)
\(878\) 0 0
\(879\) −5.05573 −0.170525
\(880\) 0 0
\(881\) −32.8328 −1.10617 −0.553083 0.833126i \(-0.686549\pi\)
−0.553083 + 0.833126i \(0.686549\pi\)
\(882\) 0 0
\(883\) −25.3050 −0.851579 −0.425790 0.904822i \(-0.640004\pi\)
−0.425790 + 0.904822i \(0.640004\pi\)
\(884\) 0 0
\(885\) 4.00000 0.134459
\(886\) 0 0
\(887\) 24.5836 0.825436 0.412718 0.910859i \(-0.364579\pi\)
0.412718 + 0.910859i \(0.364579\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) 0 0
\(897\) −28.9443 −0.966421
\(898\) 0 0
\(899\) 54.8328 1.82878
\(900\) 0 0
\(901\) 8.94427 0.297977
\(902\) 0 0
\(903\) 10.4721 0.348491
\(904\) 0 0
\(905\) −13.4164 −0.445976
\(906\) 0 0
\(907\) 36.3607 1.20734 0.603668 0.797236i \(-0.293705\pi\)
0.603668 + 0.797236i \(0.293705\pi\)
\(908\) 0 0
\(909\) 6.94427 0.230327
\(910\) 0 0
\(911\) 0.583592 0.0193353 0.00966764 0.999953i \(-0.496923\pi\)
0.00966764 + 0.999953i \(0.496923\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −12.4721 −0.412316
\(916\) 0 0
\(917\) −0.944272 −0.0311826
\(918\) 0 0
\(919\) −56.7214 −1.87107 −0.935533 0.353241i \(-0.885080\pi\)
−0.935533 + 0.353241i \(0.885080\pi\)
\(920\) 0 0
\(921\) −24.9443 −0.821942
\(922\) 0 0
\(923\) 68.9443 2.26933
\(924\) 0 0
\(925\) −8.47214 −0.278562
\(926\) 0 0
\(927\) −17.8885 −0.587537
\(928\) 0 0
\(929\) 4.11146 0.134893 0.0674463 0.997723i \(-0.478515\pi\)
0.0674463 + 0.997723i \(0.478515\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 33.8885 1.10946
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 31.3050 1.02269 0.511344 0.859376i \(-0.329148\pi\)
0.511344 + 0.859376i \(0.329148\pi\)
\(938\) 0 0
\(939\) −0.472136 −0.0154076
\(940\) 0 0
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 0 0
\(943\) 70.8328 2.30663
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −56.3607 −1.83148 −0.915738 0.401776i \(-0.868393\pi\)
−0.915738 + 0.401776i \(0.868393\pi\)
\(948\) 0 0
\(949\) 37.8885 1.22991
\(950\) 0 0
\(951\) −15.8885 −0.515221
\(952\) 0 0
\(953\) 41.7771 1.35329 0.676646 0.736308i \(-0.263433\pi\)
0.676646 + 0.736308i \(0.263433\pi\)
\(954\) 0 0
\(955\) 23.4164 0.757737
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 10.9443 0.353409
\(960\) 0 0
\(961\) 10.8885 0.351243
\(962\) 0 0
\(963\) 6.47214 0.208562
\(964\) 0 0
\(965\) 18.9443 0.609838
\(966\) 0 0
\(967\) −3.05573 −0.0982656 −0.0491328 0.998792i \(-0.515646\pi\)
−0.0491328 + 0.998792i \(0.515646\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −4.47214 −0.143223
\(976\) 0 0
\(977\) −24.8328 −0.794472 −0.397236 0.917716i \(-0.630031\pi\)
−0.397236 + 0.917716i \(0.630031\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 0 0
\(983\) 30.2492 0.964800 0.482400 0.875951i \(-0.339765\pi\)
0.482400 + 0.875951i \(0.339765\pi\)
\(984\) 0 0
\(985\) 14.9443 0.476164
\(986\) 0 0
\(987\) 2.47214 0.0786890
\(988\) 0 0
\(989\) −67.7771 −2.15519
\(990\) 0 0
\(991\) −36.9443 −1.17357 −0.586787 0.809742i \(-0.699607\pi\)
−0.586787 + 0.809742i \(0.699607\pi\)
\(992\) 0 0
\(993\) 32.0000 1.01549
\(994\) 0 0
\(995\) −11.4164 −0.361924
\(996\) 0 0
\(997\) 17.4164 0.551583 0.275792 0.961217i \(-0.411060\pi\)
0.275792 + 0.961217i \(0.411060\pi\)
\(998\) 0 0
\(999\) 8.47214 0.268047
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 6720.2.a.cp.1.2 2
4.3 odd 2 6720.2.a.cv.1.2 2
8.3 odd 2 3360.2.a.bd.1.1 2
8.5 even 2 3360.2.a.bh.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.a.bd.1.1 2 8.3 odd 2
3360.2.a.bh.1.1 yes 2 8.5 even 2
6720.2.a.cp.1.2 2 1.1 even 1 trivial
6720.2.a.cv.1.2 2 4.3 odd 2