Properties

Label 6720.2.a.ct.1.1
Level $6720$
Weight $2$
Character 6720.1
Self dual yes
Analytic conductor $53.659$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3360)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 6720.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.12311 q^{11} +3.12311 q^{13} -1.00000 q^{15} +2.00000 q^{17} +1.12311 q^{19} -1.00000 q^{21} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +5.12311 q^{31} +5.12311 q^{33} +1.00000 q^{35} +2.00000 q^{37} -3.12311 q^{39} +2.00000 q^{41} -10.2462 q^{43} +1.00000 q^{45} +1.00000 q^{49} -2.00000 q^{51} -13.3693 q^{53} -5.12311 q^{55} -1.12311 q^{57} +4.00000 q^{59} +8.24621 q^{61} +1.00000 q^{63} +3.12311 q^{65} +2.24621 q^{67} -5.12311 q^{71} +15.1231 q^{73} -1.00000 q^{75} -5.12311 q^{77} -2.24621 q^{79} +1.00000 q^{81} +4.00000 q^{83} +2.00000 q^{85} -2.00000 q^{87} -0.246211 q^{89} +3.12311 q^{91} -5.12311 q^{93} +1.12311 q^{95} +4.87689 q^{97} -5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} - 6 q^{19} - 2 q^{21} + 2 q^{25} - 2 q^{27} + 4 q^{29} + 2 q^{31} + 2 q^{33} + 2 q^{35} + 4 q^{37} + 2 q^{39} + 4 q^{41} - 4 q^{43} + 2 q^{45} + 2 q^{49} - 4 q^{51} - 2 q^{53} - 2 q^{55} + 6 q^{57} + 8 q^{59} + 2 q^{63} - 2 q^{65} - 12 q^{67} - 2 q^{71} + 22 q^{73} - 2 q^{75} - 2 q^{77} + 12 q^{79} + 2 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} + 16 q^{89} - 2 q^{91} - 2 q^{93} - 6 q^{95} + 18 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(11\) −5.12311 −1.54467 −0.772337 0.635213i \(-0.780912\pi\)
−0.772337 + 0.635213i \(0.780912\pi\)
\(12\) 0 0
\(13\) 3.12311 0.866194 0.433097 0.901347i \(-0.357421\pi\)
0.433097 + 0.901347i \(0.357421\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 1.12311 0.257658 0.128829 0.991667i \(-0.458878\pi\)
0.128829 + 0.991667i \(0.458878\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 5.12311 0.920137 0.460068 0.887883i \(-0.347825\pi\)
0.460068 + 0.887883i \(0.347825\pi\)
\(32\) 0 0
\(33\) 5.12311 0.891818
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −3.12311 −0.500097
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −10.2462 −1.56253 −0.781266 0.624198i \(-0.785426\pi\)
−0.781266 + 0.624198i \(0.785426\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −13.3693 −1.83642 −0.918208 0.396098i \(-0.870364\pi\)
−0.918208 + 0.396098i \(0.870364\pi\)
\(54\) 0 0
\(55\) −5.12311 −0.690799
\(56\) 0 0
\(57\) −1.12311 −0.148759
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 8.24621 1.05582 0.527910 0.849301i \(-0.322976\pi\)
0.527910 + 0.849301i \(0.322976\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 3.12311 0.387374
\(66\) 0 0
\(67\) 2.24621 0.274418 0.137209 0.990542i \(-0.456187\pi\)
0.137209 + 0.990542i \(0.456187\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.12311 −0.608001 −0.304000 0.952672i \(-0.598322\pi\)
−0.304000 + 0.952672i \(0.598322\pi\)
\(72\) 0 0
\(73\) 15.1231 1.77003 0.885013 0.465567i \(-0.154149\pi\)
0.885013 + 0.465567i \(0.154149\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −5.12311 −0.583832
\(78\) 0 0
\(79\) −2.24621 −0.252719 −0.126359 0.991985i \(-0.540329\pi\)
−0.126359 + 0.991985i \(0.540329\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −0.246211 −0.0260983 −0.0130492 0.999915i \(-0.504154\pi\)
−0.0130492 + 0.999915i \(0.504154\pi\)
\(90\) 0 0
\(91\) 3.12311 0.327390
\(92\) 0 0
\(93\) −5.12311 −0.531241
\(94\) 0 0
\(95\) 1.12311 0.115228
\(96\) 0 0
\(97\) 4.87689 0.495174 0.247587 0.968866i \(-0.420362\pi\)
0.247587 + 0.968866i \(0.420362\pi\)
\(98\) 0 0
\(99\) −5.12311 −0.514891
\(100\) 0 0
\(101\) 0.246211 0.0244989 0.0122495 0.999925i \(-0.496101\pi\)
0.0122495 + 0.999925i \(0.496101\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −8.24621 −0.789844 −0.394922 0.918715i \(-0.629228\pi\)
−0.394922 + 0.918715i \(0.629228\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −5.36932 −0.505103 −0.252551 0.967583i \(-0.581270\pi\)
−0.252551 + 0.967583i \(0.581270\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.12311 0.288731
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 15.2462 1.38602
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 20.4924 1.81841 0.909204 0.416350i \(-0.136691\pi\)
0.909204 + 0.416350i \(0.136691\pi\)
\(128\) 0 0
\(129\) 10.2462 0.902129
\(130\) 0 0
\(131\) −1.75379 −0.153229 −0.0766146 0.997061i \(-0.524411\pi\)
−0.0766146 + 0.997061i \(0.524411\pi\)
\(132\) 0 0
\(133\) 1.12311 0.0973856
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 12.8769 1.10015 0.550074 0.835116i \(-0.314600\pi\)
0.550074 + 0.835116i \(0.314600\pi\)
\(138\) 0 0
\(139\) 9.12311 0.773812 0.386906 0.922119i \(-0.373544\pi\)
0.386906 + 0.922119i \(0.373544\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −16.0000 −1.33799
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 12.2462 1.00325 0.501624 0.865086i \(-0.332736\pi\)
0.501624 + 0.865086i \(0.332736\pi\)
\(150\) 0 0
\(151\) 20.4924 1.66765 0.833825 0.552029i \(-0.186146\pi\)
0.833825 + 0.552029i \(0.186146\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 5.12311 0.411498
\(156\) 0 0
\(157\) −7.12311 −0.568486 −0.284243 0.958752i \(-0.591742\pi\)
−0.284243 + 0.958752i \(0.591742\pi\)
\(158\) 0 0
\(159\) 13.3693 1.06026
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −10.2462 −0.802545 −0.401273 0.915959i \(-0.631432\pi\)
−0.401273 + 0.915959i \(0.631432\pi\)
\(164\) 0 0
\(165\) 5.12311 0.398833
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −3.24621 −0.249709
\(170\) 0 0
\(171\) 1.12311 0.0858860
\(172\) 0 0
\(173\) 0.246211 0.0187191 0.00935955 0.999956i \(-0.497021\pi\)
0.00935955 + 0.999956i \(0.497021\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) 2.87689 0.215029 0.107515 0.994204i \(-0.465711\pi\)
0.107515 + 0.994204i \(0.465711\pi\)
\(180\) 0 0
\(181\) −22.4924 −1.67185 −0.835924 0.548845i \(-0.815068\pi\)
−0.835924 + 0.548845i \(0.815068\pi\)
\(182\) 0 0
\(183\) −8.24621 −0.609577
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) −10.2462 −0.749277
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 5.12311 0.370695 0.185347 0.982673i \(-0.440659\pi\)
0.185347 + 0.982673i \(0.440659\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 0 0
\(195\) −3.12311 −0.223650
\(196\) 0 0
\(197\) −23.6155 −1.68254 −0.841268 0.540618i \(-0.818191\pi\)
−0.841268 + 0.540618i \(0.818191\pi\)
\(198\) 0 0
\(199\) 5.12311 0.363167 0.181584 0.983375i \(-0.441878\pi\)
0.181584 + 0.983375i \(0.441878\pi\)
\(200\) 0 0
\(201\) −2.24621 −0.158436
\(202\) 0 0
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5.75379 −0.397998
\(210\) 0 0
\(211\) −5.75379 −0.396107 −0.198054 0.980191i \(-0.563462\pi\)
−0.198054 + 0.980191i \(0.563462\pi\)
\(212\) 0 0
\(213\) 5.12311 0.351029
\(214\) 0 0
\(215\) −10.2462 −0.698786
\(216\) 0 0
\(217\) 5.12311 0.347779
\(218\) 0 0
\(219\) −15.1231 −1.02192
\(220\) 0 0
\(221\) 6.24621 0.420166
\(222\) 0 0
\(223\) −18.2462 −1.22186 −0.610928 0.791686i \(-0.709204\pi\)
−0.610928 + 0.791686i \(0.709204\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 3.75379 0.248057 0.124029 0.992279i \(-0.460419\pi\)
0.124029 + 0.992279i \(0.460419\pi\)
\(230\) 0 0
\(231\) 5.12311 0.337076
\(232\) 0 0
\(233\) 23.1231 1.51485 0.757423 0.652925i \(-0.226458\pi\)
0.757423 + 0.652925i \(0.226458\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.24621 0.145907
\(238\) 0 0
\(239\) −13.1231 −0.848863 −0.424432 0.905460i \(-0.639526\pi\)
−0.424432 + 0.905460i \(0.639526\pi\)
\(240\) 0 0
\(241\) 20.2462 1.30417 0.652087 0.758145i \(-0.273894\pi\)
0.652087 + 0.758145i \(0.273894\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 3.50758 0.223182
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −2.00000 −0.125245
\(256\) 0 0
\(257\) 20.2462 1.26292 0.631462 0.775407i \(-0.282455\pi\)
0.631462 + 0.775407i \(0.282455\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) −18.2462 −1.12511 −0.562555 0.826760i \(-0.690181\pi\)
−0.562555 + 0.826760i \(0.690181\pi\)
\(264\) 0 0
\(265\) −13.3693 −0.821271
\(266\) 0 0
\(267\) 0.246211 0.0150679
\(268\) 0 0
\(269\) 16.2462 0.990549 0.495274 0.868737i \(-0.335067\pi\)
0.495274 + 0.868737i \(0.335067\pi\)
\(270\) 0 0
\(271\) −17.6155 −1.07007 −0.535034 0.844831i \(-0.679701\pi\)
−0.535034 + 0.844831i \(0.679701\pi\)
\(272\) 0 0
\(273\) −3.12311 −0.189019
\(274\) 0 0
\(275\) −5.12311 −0.308935
\(276\) 0 0
\(277\) 14.4924 0.870765 0.435383 0.900245i \(-0.356613\pi\)
0.435383 + 0.900245i \(0.356613\pi\)
\(278\) 0 0
\(279\) 5.12311 0.306712
\(280\) 0 0
\(281\) 24.7386 1.47578 0.737892 0.674919i \(-0.235822\pi\)
0.737892 + 0.674919i \(0.235822\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) −1.12311 −0.0665270
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −4.87689 −0.285889
\(292\) 0 0
\(293\) 16.2462 0.949114 0.474557 0.880225i \(-0.342608\pi\)
0.474557 + 0.880225i \(0.342608\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 5.12311 0.297273
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −10.2462 −0.590582
\(302\) 0 0
\(303\) −0.246211 −0.0141445
\(304\) 0 0
\(305\) 8.24621 0.472177
\(306\) 0 0
\(307\) 9.75379 0.556678 0.278339 0.960483i \(-0.410216\pi\)
0.278339 + 0.960483i \(0.410216\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) 33.3693 1.88615 0.943073 0.332587i \(-0.107921\pi\)
0.943073 + 0.332587i \(0.107921\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) 15.1231 0.849398 0.424699 0.905335i \(-0.360380\pi\)
0.424699 + 0.905335i \(0.360380\pi\)
\(318\) 0 0
\(319\) −10.2462 −0.573678
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 2.24621 0.124983
\(324\) 0 0
\(325\) 3.12311 0.173239
\(326\) 0 0
\(327\) 8.24621 0.456017
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.24621 −0.123463 −0.0617315 0.998093i \(-0.519662\pi\)
−0.0617315 + 0.998093i \(0.519662\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 2.24621 0.122724
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 5.36932 0.291621
\(340\) 0 0
\(341\) −26.2462 −1.42131
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 26.2462 1.40897 0.704485 0.709719i \(-0.251178\pi\)
0.704485 + 0.709719i \(0.251178\pi\)
\(348\) 0 0
\(349\) 11.7538 0.629166 0.314583 0.949230i \(-0.398135\pi\)
0.314583 + 0.949230i \(0.398135\pi\)
\(350\) 0 0
\(351\) −3.12311 −0.166699
\(352\) 0 0
\(353\) −10.4924 −0.558455 −0.279228 0.960225i \(-0.590078\pi\)
−0.279228 + 0.960225i \(0.590078\pi\)
\(354\) 0 0
\(355\) −5.12311 −0.271906
\(356\) 0 0
\(357\) −2.00000 −0.105851
\(358\) 0 0
\(359\) 33.6155 1.77416 0.887080 0.461616i \(-0.152730\pi\)
0.887080 + 0.461616i \(0.152730\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) 0 0
\(363\) −15.2462 −0.800219
\(364\) 0 0
\(365\) 15.1231 0.791580
\(366\) 0 0
\(367\) 13.7538 0.717942 0.358971 0.933349i \(-0.383128\pi\)
0.358971 + 0.933349i \(0.383128\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) −13.3693 −0.694100
\(372\) 0 0
\(373\) 20.2462 1.04831 0.524155 0.851623i \(-0.324381\pi\)
0.524155 + 0.851623i \(0.324381\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 6.24621 0.321696
\(378\) 0 0
\(379\) −12.4924 −0.641693 −0.320846 0.947131i \(-0.603967\pi\)
−0.320846 + 0.947131i \(0.603967\pi\)
\(380\) 0 0
\(381\) −20.4924 −1.04986
\(382\) 0 0
\(383\) 4.49242 0.229552 0.114776 0.993391i \(-0.463385\pi\)
0.114776 + 0.993391i \(0.463385\pi\)
\(384\) 0 0
\(385\) −5.12311 −0.261098
\(386\) 0 0
\(387\) −10.2462 −0.520844
\(388\) 0 0
\(389\) 20.2462 1.02652 0.513262 0.858232i \(-0.328437\pi\)
0.513262 + 0.858232i \(0.328437\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 1.75379 0.0884669
\(394\) 0 0
\(395\) −2.24621 −0.113019
\(396\) 0 0
\(397\) 19.1231 0.959761 0.479881 0.877334i \(-0.340680\pi\)
0.479881 + 0.877334i \(0.340680\pi\)
\(398\) 0 0
\(399\) −1.12311 −0.0562256
\(400\) 0 0
\(401\) −38.9848 −1.94681 −0.973405 0.229090i \(-0.926425\pi\)
−0.973405 + 0.229090i \(0.926425\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −10.2462 −0.507886
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −12.8769 −0.635170
\(412\) 0 0
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) −9.12311 −0.446760
\(418\) 0 0
\(419\) 17.7538 0.867329 0.433665 0.901074i \(-0.357220\pi\)
0.433665 + 0.901074i \(0.357220\pi\)
\(420\) 0 0
\(421\) 6.49242 0.316421 0.158211 0.987405i \(-0.449427\pi\)
0.158211 + 0.987405i \(0.449427\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 8.24621 0.399062
\(428\) 0 0
\(429\) 16.0000 0.772487
\(430\) 0 0
\(431\) −33.6155 −1.61920 −0.809602 0.586980i \(-0.800317\pi\)
−0.809602 + 0.586980i \(0.800317\pi\)
\(432\) 0 0
\(433\) 4.87689 0.234369 0.117184 0.993110i \(-0.462613\pi\)
0.117184 + 0.993110i \(0.462613\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 33.6155 1.60438 0.802191 0.597068i \(-0.203668\pi\)
0.802191 + 0.597068i \(0.203668\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 22.7386 1.08035 0.540173 0.841554i \(-0.318359\pi\)
0.540173 + 0.841554i \(0.318359\pi\)
\(444\) 0 0
\(445\) −0.246211 −0.0116715
\(446\) 0 0
\(447\) −12.2462 −0.579226
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) −10.2462 −0.482475
\(452\) 0 0
\(453\) −20.4924 −0.962818
\(454\) 0 0
\(455\) 3.12311 0.146413
\(456\) 0 0
\(457\) 4.24621 0.198629 0.0993147 0.995056i \(-0.468335\pi\)
0.0993147 + 0.995056i \(0.468335\pi\)
\(458\) 0 0
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −14.4924 −0.674979 −0.337490 0.941329i \(-0.609578\pi\)
−0.337490 + 0.941329i \(0.609578\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) −5.12311 −0.237578
\(466\) 0 0
\(467\) −36.9848 −1.71145 −0.855727 0.517427i \(-0.826890\pi\)
−0.855727 + 0.517427i \(0.826890\pi\)
\(468\) 0 0
\(469\) 2.24621 0.103720
\(470\) 0 0
\(471\) 7.12311 0.328215
\(472\) 0 0
\(473\) 52.4924 2.41360
\(474\) 0 0
\(475\) 1.12311 0.0515316
\(476\) 0 0
\(477\) −13.3693 −0.612139
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 6.24621 0.284803
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.87689 0.221448
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 0 0
\(489\) 10.2462 0.463350
\(490\) 0 0
\(491\) −13.1231 −0.592237 −0.296119 0.955151i \(-0.595692\pi\)
−0.296119 + 0.955151i \(0.595692\pi\)
\(492\) 0 0
\(493\) 4.00000 0.180151
\(494\) 0 0
\(495\) −5.12311 −0.230266
\(496\) 0 0
\(497\) −5.12311 −0.229803
\(498\) 0 0
\(499\) 20.4924 0.917367 0.458683 0.888600i \(-0.348321\pi\)
0.458683 + 0.888600i \(0.348321\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) 0 0
\(503\) −3.50758 −0.156395 −0.0781976 0.996938i \(-0.524916\pi\)
−0.0781976 + 0.996938i \(0.524916\pi\)
\(504\) 0 0
\(505\) 0.246211 0.0109563
\(506\) 0 0
\(507\) 3.24621 0.144169
\(508\) 0 0
\(509\) 11.7538 0.520978 0.260489 0.965477i \(-0.416116\pi\)
0.260489 + 0.965477i \(0.416116\pi\)
\(510\) 0 0
\(511\) 15.1231 0.669007
\(512\) 0 0
\(513\) −1.12311 −0.0495863
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −0.246211 −0.0108075
\(520\) 0 0
\(521\) 22.4924 0.985411 0.492705 0.870196i \(-0.336008\pi\)
0.492705 + 0.870196i \(0.336008\pi\)
\(522\) 0 0
\(523\) 34.7386 1.51901 0.759507 0.650499i \(-0.225440\pi\)
0.759507 + 0.650499i \(0.225440\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 10.2462 0.446332
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 6.24621 0.270553
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 0 0
\(537\) −2.87689 −0.124147
\(538\) 0 0
\(539\) −5.12311 −0.220668
\(540\) 0 0
\(541\) 14.4924 0.623078 0.311539 0.950233i \(-0.399156\pi\)
0.311539 + 0.950233i \(0.399156\pi\)
\(542\) 0 0
\(543\) 22.4924 0.965242
\(544\) 0 0
\(545\) −8.24621 −0.353229
\(546\) 0 0
\(547\) −21.7538 −0.930125 −0.465062 0.885278i \(-0.653968\pi\)
−0.465062 + 0.885278i \(0.653968\pi\)
\(548\) 0 0
\(549\) 8.24621 0.351940
\(550\) 0 0
\(551\) 2.24621 0.0956918
\(552\) 0 0
\(553\) −2.24621 −0.0955186
\(554\) 0 0
\(555\) −2.00000 −0.0848953
\(556\) 0 0
\(557\) 15.1231 0.640787 0.320393 0.947285i \(-0.396185\pi\)
0.320393 + 0.947285i \(0.396185\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) 10.2462 0.432595
\(562\) 0 0
\(563\) 16.4924 0.695073 0.347536 0.937667i \(-0.387018\pi\)
0.347536 + 0.937667i \(0.387018\pi\)
\(564\) 0 0
\(565\) −5.36932 −0.225889
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 3.50758 0.146788 0.0733938 0.997303i \(-0.476617\pi\)
0.0733938 + 0.997303i \(0.476617\pi\)
\(572\) 0 0
\(573\) −5.12311 −0.214021
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −23.6155 −0.983127 −0.491564 0.870842i \(-0.663574\pi\)
−0.491564 + 0.870842i \(0.663574\pi\)
\(578\) 0 0
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 68.4924 2.83667
\(584\) 0 0
\(585\) 3.12311 0.129125
\(586\) 0 0
\(587\) 32.4924 1.34111 0.670553 0.741862i \(-0.266057\pi\)
0.670553 + 0.741862i \(0.266057\pi\)
\(588\) 0 0
\(589\) 5.75379 0.237081
\(590\) 0 0
\(591\) 23.6155 0.971413
\(592\) 0 0
\(593\) 30.4924 1.25217 0.626087 0.779753i \(-0.284656\pi\)
0.626087 + 0.779753i \(0.284656\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 0 0
\(597\) −5.12311 −0.209675
\(598\) 0 0
\(599\) 5.12311 0.209324 0.104662 0.994508i \(-0.466624\pi\)
0.104662 + 0.994508i \(0.466624\pi\)
\(600\) 0 0
\(601\) −16.2462 −0.662697 −0.331348 0.943508i \(-0.607504\pi\)
−0.331348 + 0.943508i \(0.607504\pi\)
\(602\) 0 0
\(603\) 2.24621 0.0914728
\(604\) 0 0
\(605\) 15.2462 0.619847
\(606\) 0 0
\(607\) 13.7538 0.558249 0.279125 0.960255i \(-0.409956\pi\)
0.279125 + 0.960255i \(0.409956\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −22.9848 −0.928349 −0.464175 0.885744i \(-0.653649\pi\)
−0.464175 + 0.885744i \(0.653649\pi\)
\(614\) 0 0
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) −13.3693 −0.538228 −0.269114 0.963108i \(-0.586731\pi\)
−0.269114 + 0.963108i \(0.586731\pi\)
\(618\) 0 0
\(619\) 17.1231 0.688236 0.344118 0.938926i \(-0.388178\pi\)
0.344118 + 0.938926i \(0.388178\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −0.246211 −0.00986425
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 5.75379 0.229784
\(628\) 0 0
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −2.24621 −0.0894203 −0.0447101 0.999000i \(-0.514236\pi\)
−0.0447101 + 0.999000i \(0.514236\pi\)
\(632\) 0 0
\(633\) 5.75379 0.228693
\(634\) 0 0
\(635\) 20.4924 0.813217
\(636\) 0 0
\(637\) 3.12311 0.123742
\(638\) 0 0
\(639\) −5.12311 −0.202667
\(640\) 0 0
\(641\) −8.24621 −0.325706 −0.162853 0.986650i \(-0.552070\pi\)
−0.162853 + 0.986650i \(0.552070\pi\)
\(642\) 0 0
\(643\) 22.2462 0.877305 0.438652 0.898657i \(-0.355456\pi\)
0.438652 + 0.898657i \(0.355456\pi\)
\(644\) 0 0
\(645\) 10.2462 0.403444
\(646\) 0 0
\(647\) −40.9848 −1.61128 −0.805640 0.592405i \(-0.798179\pi\)
−0.805640 + 0.592405i \(0.798179\pi\)
\(648\) 0 0
\(649\) −20.4924 −0.804398
\(650\) 0 0
\(651\) −5.12311 −0.200790
\(652\) 0 0
\(653\) −7.61553 −0.298019 −0.149009 0.988836i \(-0.547608\pi\)
−0.149009 + 0.988836i \(0.547608\pi\)
\(654\) 0 0
\(655\) −1.75379 −0.0685262
\(656\) 0 0
\(657\) 15.1231 0.590009
\(658\) 0 0
\(659\) 9.61553 0.374568 0.187284 0.982306i \(-0.440032\pi\)
0.187284 + 0.982306i \(0.440032\pi\)
\(660\) 0 0
\(661\) −31.7538 −1.23508 −0.617540 0.786540i \(-0.711870\pi\)
−0.617540 + 0.786540i \(0.711870\pi\)
\(662\) 0 0
\(663\) −6.24621 −0.242583
\(664\) 0 0
\(665\) 1.12311 0.0435522
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 18.2462 0.705439
\(670\) 0 0
\(671\) −42.2462 −1.63090
\(672\) 0 0
\(673\) 37.2311 1.43515 0.717576 0.696480i \(-0.245252\pi\)
0.717576 + 0.696480i \(0.245252\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −7.75379 −0.298002 −0.149001 0.988837i \(-0.547606\pi\)
−0.149001 + 0.988837i \(0.547606\pi\)
\(678\) 0 0
\(679\) 4.87689 0.187158
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 13.7538 0.526274 0.263137 0.964758i \(-0.415243\pi\)
0.263137 + 0.964758i \(0.415243\pi\)
\(684\) 0 0
\(685\) 12.8769 0.492001
\(686\) 0 0
\(687\) −3.75379 −0.143216
\(688\) 0 0
\(689\) −41.7538 −1.59069
\(690\) 0 0
\(691\) −20.6307 −0.784828 −0.392414 0.919789i \(-0.628360\pi\)
−0.392414 + 0.919789i \(0.628360\pi\)
\(692\) 0 0
\(693\) −5.12311 −0.194611
\(694\) 0 0
\(695\) 9.12311 0.346059
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) −23.1231 −0.874596
\(700\) 0 0
\(701\) 38.4924 1.45384 0.726919 0.686723i \(-0.240951\pi\)
0.726919 + 0.686723i \(0.240951\pi\)
\(702\) 0 0
\(703\) 2.24621 0.0847175
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.246211 0.00925973
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) −2.24621 −0.0842395
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 0 0
\(717\) 13.1231 0.490091
\(718\) 0 0
\(719\) −43.2311 −1.61225 −0.806123 0.591748i \(-0.798438\pi\)
−0.806123 + 0.591748i \(0.798438\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −20.2462 −0.752965
\(724\) 0 0
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) 30.7386 1.14003 0.570016 0.821633i \(-0.306937\pi\)
0.570016 + 0.821633i \(0.306937\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −20.4924 −0.757940
\(732\) 0 0
\(733\) 47.6155 1.75872 0.879360 0.476158i \(-0.157971\pi\)
0.879360 + 0.476158i \(0.157971\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) −11.5076 −0.423887
\(738\) 0 0
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) −3.50758 −0.128854
\(742\) 0 0
\(743\) 14.7386 0.540708 0.270354 0.962761i \(-0.412859\pi\)
0.270354 + 0.962761i \(0.412859\pi\)
\(744\) 0 0
\(745\) 12.2462 0.448666
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) 10.2462 0.373890 0.186945 0.982370i \(-0.440141\pi\)
0.186945 + 0.982370i \(0.440141\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 20.4924 0.745796
\(756\) 0 0
\(757\) −35.7538 −1.29949 −0.649747 0.760151i \(-0.725125\pi\)
−0.649747 + 0.760151i \(0.725125\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −28.7386 −1.04177 −0.520887 0.853625i \(-0.674399\pi\)
−0.520887 + 0.853625i \(0.674399\pi\)
\(762\) 0 0
\(763\) −8.24621 −0.298533
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) 12.4924 0.451075
\(768\) 0 0
\(769\) −32.2462 −1.16283 −0.581414 0.813608i \(-0.697500\pi\)
−0.581414 + 0.813608i \(0.697500\pi\)
\(770\) 0 0
\(771\) −20.2462 −0.729149
\(772\) 0 0
\(773\) 16.2462 0.584336 0.292168 0.956367i \(-0.405623\pi\)
0.292168 + 0.956367i \(0.405623\pi\)
\(774\) 0 0
\(775\) 5.12311 0.184027
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) 0 0
\(779\) 2.24621 0.0804789
\(780\) 0 0
\(781\) 26.2462 0.939163
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) −7.12311 −0.254235
\(786\) 0 0
\(787\) −31.2311 −1.11327 −0.556633 0.830758i \(-0.687907\pi\)
−0.556633 + 0.830758i \(0.687907\pi\)
\(788\) 0 0
\(789\) 18.2462 0.649582
\(790\) 0 0
\(791\) −5.36932 −0.190911
\(792\) 0 0
\(793\) 25.7538 0.914544
\(794\) 0 0
\(795\) 13.3693 0.474161
\(796\) 0 0
\(797\) 36.7386 1.30135 0.650675 0.759357i \(-0.274486\pi\)
0.650675 + 0.759357i \(0.274486\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −0.246211 −0.00869945
\(802\) 0 0
\(803\) −77.4773 −2.73411
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −16.2462 −0.571894
\(808\) 0 0
\(809\) 20.2462 0.711819 0.355909 0.934520i \(-0.384171\pi\)
0.355909 + 0.934520i \(0.384171\pi\)
\(810\) 0 0
\(811\) 53.6155 1.88270 0.941348 0.337438i \(-0.109560\pi\)
0.941348 + 0.337438i \(0.109560\pi\)
\(812\) 0 0
\(813\) 17.6155 0.617804
\(814\) 0 0
\(815\) −10.2462 −0.358909
\(816\) 0 0
\(817\) −11.5076 −0.402599
\(818\) 0 0
\(819\) 3.12311 0.109130
\(820\) 0 0
\(821\) −54.9848 −1.91898 −0.959492 0.281735i \(-0.909090\pi\)
−0.959492 + 0.281735i \(0.909090\pi\)
\(822\) 0 0
\(823\) −12.4924 −0.435458 −0.217729 0.976009i \(-0.569865\pi\)
−0.217729 + 0.976009i \(0.569865\pi\)
\(824\) 0 0
\(825\) 5.12311 0.178364
\(826\) 0 0
\(827\) −48.9848 −1.70337 −0.851685 0.524054i \(-0.824419\pi\)
−0.851685 + 0.524054i \(0.824419\pi\)
\(828\) 0 0
\(829\) −6.49242 −0.225491 −0.112746 0.993624i \(-0.535965\pi\)
−0.112746 + 0.993624i \(0.535965\pi\)
\(830\) 0 0
\(831\) −14.4924 −0.502737
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) −5.12311 −0.177080
\(838\) 0 0
\(839\) −38.7386 −1.33741 −0.668703 0.743530i \(-0.733150\pi\)
−0.668703 + 0.743530i \(0.733150\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −24.7386 −0.852044
\(844\) 0 0
\(845\) −3.24621 −0.111673
\(846\) 0 0
\(847\) 15.2462 0.523866
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −0.384472 −0.0131641 −0.00658203 0.999978i \(-0.502095\pi\)
−0.00658203 + 0.999978i \(0.502095\pi\)
\(854\) 0 0
\(855\) 1.12311 0.0384094
\(856\) 0 0
\(857\) −48.2462 −1.64806 −0.824030 0.566547i \(-0.808279\pi\)
−0.824030 + 0.566547i \(0.808279\pi\)
\(858\) 0 0
\(859\) −21.6155 −0.737512 −0.368756 0.929526i \(-0.620216\pi\)
−0.368756 + 0.929526i \(0.620216\pi\)
\(860\) 0 0
\(861\) −2.00000 −0.0681598
\(862\) 0 0
\(863\) 40.0000 1.36162 0.680808 0.732462i \(-0.261629\pi\)
0.680808 + 0.732462i \(0.261629\pi\)
\(864\) 0 0
\(865\) 0.246211 0.00837143
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 11.5076 0.390368
\(870\) 0 0
\(871\) 7.01515 0.237699
\(872\) 0 0
\(873\) 4.87689 0.165058
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −10.4924 −0.354304 −0.177152 0.984184i \(-0.556688\pi\)
−0.177152 + 0.984184i \(0.556688\pi\)
\(878\) 0 0
\(879\) −16.2462 −0.547971
\(880\) 0 0
\(881\) −40.2462 −1.35593 −0.677965 0.735095i \(-0.737138\pi\)
−0.677965 + 0.735095i \(0.737138\pi\)
\(882\) 0 0
\(883\) −18.2462 −0.614034 −0.307017 0.951704i \(-0.599331\pi\)
−0.307017 + 0.951704i \(0.599331\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 20.4924 0.687294
\(890\) 0 0
\(891\) −5.12311 −0.171630
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 2.87689 0.0961640
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 10.2462 0.341730
\(900\) 0 0
\(901\) −26.7386 −0.890793
\(902\) 0 0
\(903\) 10.2462 0.340973
\(904\) 0 0
\(905\) −22.4924 −0.747673
\(906\) 0 0
\(907\) 38.7386 1.28630 0.643148 0.765742i \(-0.277628\pi\)
0.643148 + 0.765742i \(0.277628\pi\)
\(908\) 0 0
\(909\) 0.246211 0.00816631
\(910\) 0 0
\(911\) 1.61553 0.0535248 0.0267624 0.999642i \(-0.491480\pi\)
0.0267624 + 0.999642i \(0.491480\pi\)
\(912\) 0 0
\(913\) −20.4924 −0.678200
\(914\) 0 0
\(915\) −8.24621 −0.272611
\(916\) 0 0
\(917\) −1.75379 −0.0579152
\(918\) 0 0
\(919\) −18.2462 −0.601887 −0.300943 0.953642i \(-0.597302\pi\)
−0.300943 + 0.953642i \(0.597302\pi\)
\(920\) 0 0
\(921\) −9.75379 −0.321398
\(922\) 0 0
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −11.7538 −0.385629 −0.192815 0.981235i \(-0.561762\pi\)
−0.192815 + 0.981235i \(0.561762\pi\)
\(930\) 0 0
\(931\) 1.12311 0.0368083
\(932\) 0 0
\(933\) 16.0000 0.523816
\(934\) 0 0
\(935\) −10.2462 −0.335087
\(936\) 0 0
\(937\) 40.1080 1.31027 0.655135 0.755512i \(-0.272612\pi\)
0.655135 + 0.755512i \(0.272612\pi\)
\(938\) 0 0
\(939\) −33.3693 −1.08897
\(940\) 0 0
\(941\) −42.9848 −1.40127 −0.700633 0.713522i \(-0.747099\pi\)
−0.700633 + 0.713522i \(0.747099\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) 51.2311 1.66479 0.832393 0.554186i \(-0.186970\pi\)
0.832393 + 0.554186i \(0.186970\pi\)
\(948\) 0 0
\(949\) 47.2311 1.53318
\(950\) 0 0
\(951\) −15.1231 −0.490400
\(952\) 0 0
\(953\) 2.63068 0.0852162 0.0426081 0.999092i \(-0.486433\pi\)
0.0426081 + 0.999092i \(0.486433\pi\)
\(954\) 0 0
\(955\) 5.12311 0.165780
\(956\) 0 0
\(957\) 10.2462 0.331213
\(958\) 0 0
\(959\) 12.8769 0.415817
\(960\) 0 0
\(961\) −4.75379 −0.153348
\(962\) 0 0
\(963\) 8.00000 0.257796
\(964\) 0 0
\(965\) 18.0000 0.579441
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 0 0
\(969\) −2.24621 −0.0721587
\(970\) 0 0
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) 0 0
\(973\) 9.12311 0.292473
\(974\) 0 0
\(975\) −3.12311 −0.100019
\(976\) 0 0
\(977\) 10.6307 0.340106 0.170053 0.985435i \(-0.445606\pi\)
0.170053 + 0.985435i \(0.445606\pi\)
\(978\) 0 0
\(979\) 1.26137 0.0403134
\(980\) 0 0
\(981\) −8.24621 −0.263281
\(982\) 0 0
\(983\) −11.5076 −0.367035 −0.183517 0.983016i \(-0.558748\pi\)
−0.183517 + 0.983016i \(0.558748\pi\)
\(984\) 0 0
\(985\) −23.6155 −0.752453
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −9.26137 −0.294197 −0.147098 0.989122i \(-0.546993\pi\)
−0.147098 + 0.989122i \(0.546993\pi\)
\(992\) 0 0
\(993\) 2.24621 0.0712814
\(994\) 0 0
\(995\) 5.12311 0.162413
\(996\) 0 0
\(997\) −37.8617 −1.19909 −0.599547 0.800340i \(-0.704652\pi\)
−0.599547 + 0.800340i \(0.704652\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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.ct.1.1 2
4.3 odd 2 6720.2.a.cw.1.2 2
8.3 odd 2 3360.2.a.ba.1.1 2
8.5 even 2 3360.2.a.bg.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.a.ba.1.1 2 8.3 odd 2
3360.2.a.bg.1.2 yes 2 8.5 even 2
6720.2.a.ct.1.1 2 1.1 even 1 trivial
6720.2.a.cw.1.2 2 4.3 odd 2