Properties

Label 8400.2.a.dk.1.2
Level $8400$
Weight $2$
Character 8400.1
Self dual yes
Analytic conductor $67.074$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.0743376979\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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 840)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 8400.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -0.387873 q^{11} +2.96239 q^{13} -3.35026 q^{17} +2.96239 q^{19} +1.00000 q^{21} -0.962389 q^{23} +1.00000 q^{27} +1.22425 q^{29} -2.96239 q^{31} -0.387873 q^{33} -5.92478 q^{37} +2.96239 q^{39} +1.03761 q^{41} +10.7005 q^{43} +3.22425 q^{47} +1.00000 q^{49} -3.35026 q^{51} +5.66291 q^{53} +2.96239 q^{57} -3.22425 q^{59} +14.6253 q^{61} +1.00000 q^{63} -0.962389 q^{69} -5.53690 q^{71} -6.18664 q^{73} -0.387873 q^{77} +13.9248 q^{79} +1.00000 q^{81} +3.22425 q^{83} +1.22425 q^{87} +3.73813 q^{89} +2.96239 q^{91} -2.96239 q^{93} +7.73813 q^{97} -0.387873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 2 q^{11} - 2 q^{13} - 2 q^{19} + 3 q^{21} + 8 q^{23} + 3 q^{27} + 2 q^{29} + 2 q^{31} - 2 q^{33} + 4 q^{37} - 2 q^{39} + 14 q^{41} + 12 q^{43} + 8 q^{47} + 3 q^{49} - 14 q^{53} - 2 q^{57} - 8 q^{59} + 2 q^{61} + 3 q^{63} + 8 q^{69} + 6 q^{71} - 6 q^{73} - 2 q^{77} + 20 q^{79} + 3 q^{81} + 8 q^{83} + 2 q^{87} + 2 q^{89} - 2 q^{91} + 2 q^{93} + 14 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.387873 −0.116948 −0.0584741 0.998289i \(-0.518623\pi\)
−0.0584741 + 0.998289i \(0.518623\pi\)
\(12\) 0 0
\(13\) 2.96239 0.821619 0.410809 0.911721i \(-0.365246\pi\)
0.410809 + 0.911721i \(0.365246\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.35026 −0.812558 −0.406279 0.913749i \(-0.633174\pi\)
−0.406279 + 0.913749i \(0.633174\pi\)
\(18\) 0 0
\(19\) 2.96239 0.679619 0.339809 0.940494i \(-0.389637\pi\)
0.339809 + 0.940494i \(0.389637\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −0.962389 −0.200672 −0.100336 0.994954i \(-0.531992\pi\)
−0.100336 + 0.994954i \(0.531992\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.22425 0.227338 0.113669 0.993519i \(-0.463740\pi\)
0.113669 + 0.993519i \(0.463740\pi\)
\(30\) 0 0
\(31\) −2.96239 −0.532061 −0.266030 0.963965i \(-0.585712\pi\)
−0.266030 + 0.963965i \(0.585712\pi\)
\(32\) 0 0
\(33\) −0.387873 −0.0675200
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.92478 −0.974027 −0.487014 0.873394i \(-0.661914\pi\)
−0.487014 + 0.873394i \(0.661914\pi\)
\(38\) 0 0
\(39\) 2.96239 0.474362
\(40\) 0 0
\(41\) 1.03761 0.162048 0.0810238 0.996712i \(-0.474181\pi\)
0.0810238 + 0.996712i \(0.474181\pi\)
\(42\) 0 0
\(43\) 10.7005 1.63181 0.815907 0.578183i \(-0.196238\pi\)
0.815907 + 0.578183i \(0.196238\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.22425 0.470306 0.235153 0.971958i \(-0.424441\pi\)
0.235153 + 0.971958i \(0.424441\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.35026 −0.469130
\(52\) 0 0
\(53\) 5.66291 0.777861 0.388930 0.921267i \(-0.372845\pi\)
0.388930 + 0.921267i \(0.372845\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.96239 0.392378
\(58\) 0 0
\(59\) −3.22425 −0.419762 −0.209881 0.977727i \(-0.567308\pi\)
−0.209881 + 0.977727i \(0.567308\pi\)
\(60\) 0 0
\(61\) 14.6253 1.87258 0.936289 0.351231i \(-0.114237\pi\)
0.936289 + 0.351231i \(0.114237\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −0.962389 −0.115858
\(70\) 0 0
\(71\) −5.53690 −0.657110 −0.328555 0.944485i \(-0.606562\pi\)
−0.328555 + 0.944485i \(0.606562\pi\)
\(72\) 0 0
\(73\) −6.18664 −0.724092 −0.362046 0.932160i \(-0.617922\pi\)
−0.362046 + 0.932160i \(0.617922\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.387873 −0.0442022
\(78\) 0 0
\(79\) 13.9248 1.56666 0.783330 0.621606i \(-0.213520\pi\)
0.783330 + 0.621606i \(0.213520\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.22425 0.353908 0.176954 0.984219i \(-0.443376\pi\)
0.176954 + 0.984219i \(0.443376\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.22425 0.131254
\(88\) 0 0
\(89\) 3.73813 0.396242 0.198121 0.980178i \(-0.436516\pi\)
0.198121 + 0.980178i \(0.436516\pi\)
\(90\) 0 0
\(91\) 2.96239 0.310543
\(92\) 0 0
\(93\) −2.96239 −0.307185
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.73813 0.785689 0.392844 0.919605i \(-0.371491\pi\)
0.392844 + 0.919605i \(0.371491\pi\)
\(98\) 0 0
\(99\) −0.387873 −0.0389827
\(100\) 0 0
\(101\) −7.58769 −0.755003 −0.377502 0.926009i \(-0.623217\pi\)
−0.377502 + 0.926009i \(0.623217\pi\)
\(102\) 0 0
\(103\) 14.7005 1.44849 0.724243 0.689545i \(-0.242189\pi\)
0.724243 + 0.689545i \(0.242189\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.4387 1.58919 0.794593 0.607143i \(-0.207685\pi\)
0.794593 + 0.607143i \(0.207685\pi\)
\(108\) 0 0
\(109\) −2.77575 −0.265868 −0.132934 0.991125i \(-0.542440\pi\)
−0.132934 + 0.991125i \(0.542440\pi\)
\(110\) 0 0
\(111\) −5.92478 −0.562355
\(112\) 0 0
\(113\) −4.26187 −0.400923 −0.200461 0.979702i \(-0.564244\pi\)
−0.200461 + 0.979702i \(0.564244\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.96239 0.273873
\(118\) 0 0
\(119\) −3.35026 −0.307118
\(120\) 0 0
\(121\) −10.8496 −0.986323
\(122\) 0 0
\(123\) 1.03761 0.0935583
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −14.5501 −1.29111 −0.645555 0.763714i \(-0.723374\pi\)
−0.645555 + 0.763714i \(0.723374\pi\)
\(128\) 0 0
\(129\) 10.7005 0.942129
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 2.96239 0.256872
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.4387 −0.891835 −0.445917 0.895074i \(-0.647122\pi\)
−0.445917 + 0.895074i \(0.647122\pi\)
\(138\) 0 0
\(139\) −5.66291 −0.480322 −0.240161 0.970733i \(-0.577200\pi\)
−0.240161 + 0.970733i \(0.577200\pi\)
\(140\) 0 0
\(141\) 3.22425 0.271531
\(142\) 0 0
\(143\) −1.14903 −0.0960868
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 20.5501 1.68353 0.841764 0.539846i \(-0.181517\pi\)
0.841764 + 0.539846i \(0.181517\pi\)
\(150\) 0 0
\(151\) 22.7005 1.84734 0.923671 0.383186i \(-0.125173\pi\)
0.923671 + 0.383186i \(0.125173\pi\)
\(152\) 0 0
\(153\) −3.35026 −0.270853
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.18664 0.174513 0.0872565 0.996186i \(-0.472190\pi\)
0.0872565 + 0.996186i \(0.472190\pi\)
\(158\) 0 0
\(159\) 5.66291 0.449098
\(160\) 0 0
\(161\) −0.962389 −0.0758468
\(162\) 0 0
\(163\) 7.22425 0.565847 0.282924 0.959142i \(-0.408696\pi\)
0.282924 + 0.959142i \(0.408696\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.8496 −1.22648 −0.613238 0.789898i \(-0.710133\pi\)
−0.613238 + 0.789898i \(0.710133\pi\)
\(168\) 0 0
\(169\) −4.22425 −0.324943
\(170\) 0 0
\(171\) 2.96239 0.226540
\(172\) 0 0
\(173\) −23.9756 −1.82283 −0.911414 0.411490i \(-0.865008\pi\)
−0.911414 + 0.411490i \(0.865008\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −3.22425 −0.242350
\(178\) 0 0
\(179\) −12.2374 −0.914668 −0.457334 0.889295i \(-0.651196\pi\)
−0.457334 + 0.889295i \(0.651196\pi\)
\(180\) 0 0
\(181\) 8.70052 0.646705 0.323352 0.946279i \(-0.395190\pi\)
0.323352 + 0.946279i \(0.395190\pi\)
\(182\) 0 0
\(183\) 14.6253 1.08113
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.29948 0.0950271
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 13.5369 0.979496 0.489748 0.871864i \(-0.337089\pi\)
0.489748 + 0.871864i \(0.337089\pi\)
\(192\) 0 0
\(193\) 1.92478 0.138548 0.0692742 0.997598i \(-0.477932\pi\)
0.0692742 + 0.997598i \(0.477932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −20.1114 −1.43288 −0.716440 0.697649i \(-0.754229\pi\)
−0.716440 + 0.697649i \(0.754229\pi\)
\(198\) 0 0
\(199\) 8.26187 0.585668 0.292834 0.956163i \(-0.405402\pi\)
0.292834 + 0.956163i \(0.405402\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.22425 0.0859258
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.962389 −0.0668906
\(208\) 0 0
\(209\) −1.14903 −0.0794801
\(210\) 0 0
\(211\) 18.1768 1.25134 0.625671 0.780087i \(-0.284825\pi\)
0.625671 + 0.780087i \(0.284825\pi\)
\(212\) 0 0
\(213\) −5.53690 −0.379382
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.96239 −0.201100
\(218\) 0 0
\(219\) −6.18664 −0.418055
\(220\) 0 0
\(221\) −9.92478 −0.667613
\(222\) 0 0
\(223\) 21.4010 1.43312 0.716560 0.697525i \(-0.245716\pi\)
0.716560 + 0.697525i \(0.245716\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.5501 1.23121 0.615606 0.788054i \(-0.288911\pi\)
0.615606 + 0.788054i \(0.288911\pi\)
\(228\) 0 0
\(229\) −28.5501 −1.88664 −0.943321 0.331881i \(-0.892317\pi\)
−0.943321 + 0.331881i \(0.892317\pi\)
\(230\) 0 0
\(231\) −0.387873 −0.0255202
\(232\) 0 0
\(233\) −18.9624 −1.24227 −0.621134 0.783705i \(-0.713328\pi\)
−0.621134 + 0.783705i \(0.713328\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 13.9248 0.904511
\(238\) 0 0
\(239\) 18.1622 1.17482 0.587408 0.809291i \(-0.300149\pi\)
0.587408 + 0.809291i \(0.300149\pi\)
\(240\) 0 0
\(241\) 14.3733 0.925865 0.462932 0.886394i \(-0.346797\pi\)
0.462932 + 0.886394i \(0.346797\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.77575 0.558387
\(248\) 0 0
\(249\) 3.22425 0.204329
\(250\) 0 0
\(251\) 8.62530 0.544424 0.272212 0.962237i \(-0.412245\pi\)
0.272212 + 0.962237i \(0.412245\pi\)
\(252\) 0 0
\(253\) 0.373285 0.0234682
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.8251 1.73568 0.867842 0.496841i \(-0.165507\pi\)
0.867842 + 0.496841i \(0.165507\pi\)
\(258\) 0 0
\(259\) −5.92478 −0.368148
\(260\) 0 0
\(261\) 1.22425 0.0757794
\(262\) 0 0
\(263\) −0.589104 −0.0363257 −0.0181629 0.999835i \(-0.505782\pi\)
−0.0181629 + 0.999835i \(0.505782\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 3.73813 0.228770
\(268\) 0 0
\(269\) 15.7381 0.959571 0.479786 0.877386i \(-0.340714\pi\)
0.479786 + 0.877386i \(0.340714\pi\)
\(270\) 0 0
\(271\) 7.11283 0.432074 0.216037 0.976385i \(-0.430687\pi\)
0.216037 + 0.976385i \(0.430687\pi\)
\(272\) 0 0
\(273\) 2.96239 0.179292
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −5.92478 −0.355985 −0.177993 0.984032i \(-0.556960\pi\)
−0.177993 + 0.984032i \(0.556960\pi\)
\(278\) 0 0
\(279\) −2.96239 −0.177354
\(280\) 0 0
\(281\) 29.3258 1.74943 0.874716 0.484636i \(-0.161048\pi\)
0.874716 + 0.484636i \(0.161048\pi\)
\(282\) 0 0
\(283\) −16.1016 −0.957139 −0.478570 0.878050i \(-0.658845\pi\)
−0.478570 + 0.878050i \(0.658845\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.03761 0.0612483
\(288\) 0 0
\(289\) −5.77575 −0.339750
\(290\) 0 0
\(291\) 7.73813 0.453617
\(292\) 0 0
\(293\) −9.27504 −0.541854 −0.270927 0.962600i \(-0.587330\pi\)
−0.270927 + 0.962600i \(0.587330\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.387873 −0.0225067
\(298\) 0 0
\(299\) −2.85097 −0.164876
\(300\) 0 0
\(301\) 10.7005 0.616768
\(302\) 0 0
\(303\) −7.58769 −0.435901
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 10.7005 0.610711 0.305356 0.952238i \(-0.401225\pi\)
0.305356 + 0.952238i \(0.401225\pi\)
\(308\) 0 0
\(309\) 14.7005 0.836284
\(310\) 0 0
\(311\) −12.1016 −0.686217 −0.343109 0.939296i \(-0.611480\pi\)
−0.343109 + 0.939296i \(0.611480\pi\)
\(312\) 0 0
\(313\) −28.3634 −1.60320 −0.801598 0.597863i \(-0.796017\pi\)
−0.801598 + 0.597863i \(0.796017\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.5125 0.983598 0.491799 0.870709i \(-0.336339\pi\)
0.491799 + 0.870709i \(0.336339\pi\)
\(318\) 0 0
\(319\) −0.474855 −0.0265868
\(320\) 0 0
\(321\) 16.4387 0.917516
\(322\) 0 0
\(323\) −9.92478 −0.552229
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.77575 −0.153499
\(328\) 0 0
\(329\) 3.22425 0.177759
\(330\) 0 0
\(331\) 19.2243 1.05666 0.528330 0.849039i \(-0.322818\pi\)
0.528330 + 0.849039i \(0.322818\pi\)
\(332\) 0 0
\(333\) −5.92478 −0.324676
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −16.0000 −0.871576 −0.435788 0.900049i \(-0.643530\pi\)
−0.435788 + 0.900049i \(0.643530\pi\)
\(338\) 0 0
\(339\) −4.26187 −0.231473
\(340\) 0 0
\(341\) 1.14903 0.0622235
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.11283 −0.0597401 −0.0298700 0.999554i \(-0.509509\pi\)
−0.0298700 + 0.999554i \(0.509509\pi\)
\(348\) 0 0
\(349\) −32.0263 −1.71433 −0.857166 0.515041i \(-0.827777\pi\)
−0.857166 + 0.515041i \(0.827777\pi\)
\(350\) 0 0
\(351\) 2.96239 0.158121
\(352\) 0 0
\(353\) −6.20123 −0.330058 −0.165029 0.986289i \(-0.552772\pi\)
−0.165029 + 0.986289i \(0.552772\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −3.35026 −0.177315
\(358\) 0 0
\(359\) 1.68735 0.0890549 0.0445275 0.999008i \(-0.485822\pi\)
0.0445275 + 0.999008i \(0.485822\pi\)
\(360\) 0 0
\(361\) −10.2243 −0.538119
\(362\) 0 0
\(363\) −10.8496 −0.569454
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.84955 0.200945 0.100473 0.994940i \(-0.467965\pi\)
0.100473 + 0.994940i \(0.467965\pi\)
\(368\) 0 0
\(369\) 1.03761 0.0540159
\(370\) 0 0
\(371\) 5.66291 0.294004
\(372\) 0 0
\(373\) 24.8773 1.28810 0.644049 0.764984i \(-0.277253\pi\)
0.644049 + 0.764984i \(0.277253\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.62672 0.186785
\(378\) 0 0
\(379\) −2.70052 −0.138717 −0.0693583 0.997592i \(-0.522095\pi\)
−0.0693583 + 0.997592i \(0.522095\pi\)
\(380\) 0 0
\(381\) −14.5501 −0.745423
\(382\) 0 0
\(383\) −0.523730 −0.0267614 −0.0133807 0.999910i \(-0.504259\pi\)
−0.0133807 + 0.999910i \(0.504259\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.7005 0.543938
\(388\) 0 0
\(389\) 6.77575 0.343544 0.171772 0.985137i \(-0.445051\pi\)
0.171772 + 0.985137i \(0.445051\pi\)
\(390\) 0 0
\(391\) 3.22425 0.163058
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −0.635150 −0.0318773 −0.0159386 0.999873i \(-0.505074\pi\)
−0.0159386 + 0.999873i \(0.505074\pi\)
\(398\) 0 0
\(399\) 2.96239 0.148305
\(400\) 0 0
\(401\) 10.3733 0.518017 0.259009 0.965875i \(-0.416604\pi\)
0.259009 + 0.965875i \(0.416604\pi\)
\(402\) 0 0
\(403\) −8.77575 −0.437151
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.29806 0.113911
\(408\) 0 0
\(409\) 15.7743 0.779991 0.389995 0.920817i \(-0.372477\pi\)
0.389995 + 0.920817i \(0.372477\pi\)
\(410\) 0 0
\(411\) −10.4387 −0.514901
\(412\) 0 0
\(413\) −3.22425 −0.158655
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5.66291 −0.277314
\(418\) 0 0
\(419\) −25.1490 −1.22861 −0.614305 0.789068i \(-0.710564\pi\)
−0.614305 + 0.789068i \(0.710564\pi\)
\(420\) 0 0
\(421\) 20.1768 0.983357 0.491678 0.870777i \(-0.336384\pi\)
0.491678 + 0.870777i \(0.336384\pi\)
\(422\) 0 0
\(423\) 3.22425 0.156769
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 14.6253 0.707768
\(428\) 0 0
\(429\) −1.14903 −0.0554757
\(430\) 0 0
\(431\) 38.3127 1.84546 0.922728 0.385452i \(-0.125955\pi\)
0.922728 + 0.385452i \(0.125955\pi\)
\(432\) 0 0
\(433\) 12.8872 0.619318 0.309659 0.950848i \(-0.399785\pi\)
0.309659 + 0.950848i \(0.399785\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.85097 −0.136380
\(438\) 0 0
\(439\) 29.6629 1.41573 0.707867 0.706346i \(-0.249658\pi\)
0.707867 + 0.706346i \(0.249658\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −24.8119 −1.17885 −0.589425 0.807823i \(-0.700646\pi\)
−0.589425 + 0.807823i \(0.700646\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 20.5501 0.971985
\(448\) 0 0
\(449\) −36.6516 −1.72970 −0.864849 0.502032i \(-0.832586\pi\)
−0.864849 + 0.502032i \(0.832586\pi\)
\(450\) 0 0
\(451\) −0.402462 −0.0189512
\(452\) 0 0
\(453\) 22.7005 1.06656
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −15.4763 −0.723949 −0.361975 0.932188i \(-0.617897\pi\)
−0.361975 + 0.932188i \(0.617897\pi\)
\(458\) 0 0
\(459\) −3.35026 −0.156377
\(460\) 0 0
\(461\) 41.2605 1.92169 0.960845 0.277085i \(-0.0893684\pi\)
0.960845 + 0.277085i \(0.0893684\pi\)
\(462\) 0 0
\(463\) 15.5975 0.724879 0.362440 0.932007i \(-0.381944\pi\)
0.362440 + 0.932007i \(0.381944\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18.5501 −0.858395 −0.429198 0.903211i \(-0.641204\pi\)
−0.429198 + 0.903211i \(0.641204\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.18664 0.100755
\(472\) 0 0
\(473\) −4.15045 −0.190838
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 5.66291 0.259287
\(478\) 0 0
\(479\) 41.5026 1.89630 0.948151 0.317819i \(-0.102950\pi\)
0.948151 + 0.317819i \(0.102950\pi\)
\(480\) 0 0
\(481\) −17.5515 −0.800279
\(482\) 0 0
\(483\) −0.962389 −0.0437902
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −13.6728 −0.619572 −0.309786 0.950806i \(-0.600257\pi\)
−0.309786 + 0.950806i \(0.600257\pi\)
\(488\) 0 0
\(489\) 7.22425 0.326692
\(490\) 0 0
\(491\) −23.3404 −1.05334 −0.526669 0.850070i \(-0.676559\pi\)
−0.526669 + 0.850070i \(0.676559\pi\)
\(492\) 0 0
\(493\) −4.10157 −0.184725
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −5.53690 −0.248364
\(498\) 0 0
\(499\) 6.85097 0.306691 0.153346 0.988173i \(-0.450995\pi\)
0.153346 + 0.988173i \(0.450995\pi\)
\(500\) 0 0
\(501\) −15.8496 −0.708106
\(502\) 0 0
\(503\) 3.32582 0.148291 0.0741456 0.997247i \(-0.476377\pi\)
0.0741456 + 0.997247i \(0.476377\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −4.22425 −0.187606
\(508\) 0 0
\(509\) −6.18664 −0.274218 −0.137109 0.990556i \(-0.543781\pi\)
−0.137109 + 0.990556i \(0.543781\pi\)
\(510\) 0 0
\(511\) −6.18664 −0.273681
\(512\) 0 0
\(513\) 2.96239 0.130793
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.25060 −0.0550014
\(518\) 0 0
\(519\) −23.9756 −1.05241
\(520\) 0 0
\(521\) 34.4387 1.50879 0.754393 0.656424i \(-0.227932\pi\)
0.754393 + 0.656424i \(0.227932\pi\)
\(522\) 0 0
\(523\) 6.59895 0.288552 0.144276 0.989537i \(-0.453915\pi\)
0.144276 + 0.989537i \(0.453915\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.92478 0.432330
\(528\) 0 0
\(529\) −22.0738 −0.959731
\(530\) 0 0
\(531\) −3.22425 −0.139921
\(532\) 0 0
\(533\) 3.07381 0.133141
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −12.2374 −0.528084
\(538\) 0 0
\(539\) −0.387873 −0.0167069
\(540\) 0 0
\(541\) −9.07381 −0.390113 −0.195057 0.980792i \(-0.562489\pi\)
−0.195057 + 0.980792i \(0.562489\pi\)
\(542\) 0 0
\(543\) 8.70052 0.373375
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −19.8496 −0.848706 −0.424353 0.905497i \(-0.639498\pi\)
−0.424353 + 0.905497i \(0.639498\pi\)
\(548\) 0 0
\(549\) 14.6253 0.624193
\(550\) 0 0
\(551\) 3.62672 0.154503
\(552\) 0 0
\(553\) 13.9248 0.592142
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −0.785595 −0.0332867 −0.0166434 0.999861i \(-0.505298\pi\)
−0.0166434 + 0.999861i \(0.505298\pi\)
\(558\) 0 0
\(559\) 31.6991 1.34073
\(560\) 0 0
\(561\) 1.29948 0.0548639
\(562\) 0 0
\(563\) −32.2228 −1.35803 −0.679015 0.734124i \(-0.737593\pi\)
−0.679015 + 0.734124i \(0.737593\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 16.0752 0.673908 0.336954 0.941521i \(-0.390603\pi\)
0.336954 + 0.941521i \(0.390603\pi\)
\(570\) 0 0
\(571\) 39.8496 1.66765 0.833826 0.552027i \(-0.186146\pi\)
0.833826 + 0.552027i \(0.186146\pi\)
\(572\) 0 0
\(573\) 13.5369 0.565512
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.1378 1.25465 0.627326 0.778757i \(-0.284149\pi\)
0.627326 + 0.778757i \(0.284149\pi\)
\(578\) 0 0
\(579\) 1.92478 0.0799910
\(580\) 0 0
\(581\) 3.22425 0.133765
\(582\) 0 0
\(583\) −2.19649 −0.0909694
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −33.7743 −1.39402 −0.697008 0.717063i \(-0.745486\pi\)
−0.697008 + 0.717063i \(0.745486\pi\)
\(588\) 0 0
\(589\) −8.77575 −0.361598
\(590\) 0 0
\(591\) −20.1114 −0.827273
\(592\) 0 0
\(593\) −7.19982 −0.295661 −0.147831 0.989013i \(-0.547229\pi\)
−0.147831 + 0.989013i \(0.547229\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.26187 0.338136
\(598\) 0 0
\(599\) 29.5369 1.20685 0.603423 0.797422i \(-0.293803\pi\)
0.603423 + 0.797422i \(0.293803\pi\)
\(600\) 0 0
\(601\) 8.59895 0.350759 0.175379 0.984501i \(-0.443885\pi\)
0.175379 + 0.984501i \(0.443885\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 37.4010 1.51806 0.759031 0.651055i \(-0.225673\pi\)
0.759031 + 0.651055i \(0.225673\pi\)
\(608\) 0 0
\(609\) 1.22425 0.0496093
\(610\) 0 0
\(611\) 9.55149 0.386412
\(612\) 0 0
\(613\) 14.5990 0.589646 0.294823 0.955552i \(-0.404739\pi\)
0.294823 + 0.955552i \(0.404739\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22.8119 −0.918374 −0.459187 0.888340i \(-0.651859\pi\)
−0.459187 + 0.888340i \(0.651859\pi\)
\(618\) 0 0
\(619\) −0.261865 −0.0105252 −0.00526262 0.999986i \(-0.501675\pi\)
−0.00526262 + 0.999986i \(0.501675\pi\)
\(620\) 0 0
\(621\) −0.962389 −0.0386193
\(622\) 0 0
\(623\) 3.73813 0.149765
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −1.14903 −0.0458879
\(628\) 0 0
\(629\) 19.8496 0.791454
\(630\) 0 0
\(631\) 5.92478 0.235862 0.117931 0.993022i \(-0.462374\pi\)
0.117931 + 0.993022i \(0.462374\pi\)
\(632\) 0 0
\(633\) 18.1768 0.722463
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.96239 0.117374
\(638\) 0 0
\(639\) −5.53690 −0.219037
\(640\) 0 0
\(641\) 6.87732 0.271638 0.135819 0.990734i \(-0.456633\pi\)
0.135819 + 0.990734i \(0.456633\pi\)
\(642\) 0 0
\(643\) −10.7005 −0.421987 −0.210994 0.977487i \(-0.567670\pi\)
−0.210994 + 0.977487i \(0.567670\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −45.2506 −1.77898 −0.889492 0.456950i \(-0.848942\pi\)
−0.889492 + 0.456950i \(0.848942\pi\)
\(648\) 0 0
\(649\) 1.25060 0.0490904
\(650\) 0 0
\(651\) −2.96239 −0.116105
\(652\) 0 0
\(653\) −36.0625 −1.41124 −0.705618 0.708592i \(-0.749331\pi\)
−0.705618 + 0.708592i \(0.749331\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −6.18664 −0.241364
\(658\) 0 0
\(659\) −43.4617 −1.69303 −0.846513 0.532367i \(-0.821302\pi\)
−0.846513 + 0.532367i \(0.821302\pi\)
\(660\) 0 0
\(661\) −22.4749 −0.874171 −0.437085 0.899420i \(-0.643989\pi\)
−0.437085 + 0.899420i \(0.643989\pi\)
\(662\) 0 0
\(663\) −9.92478 −0.385446
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.17821 −0.0456204
\(668\) 0 0
\(669\) 21.4010 0.827412
\(670\) 0 0
\(671\) −5.67276 −0.218995
\(672\) 0 0
\(673\) −47.5487 −1.83287 −0.916433 0.400188i \(-0.868945\pi\)
−0.916433 + 0.400188i \(0.868945\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.67750 −0.218204 −0.109102 0.994031i \(-0.534798\pi\)
−0.109102 + 0.994031i \(0.534798\pi\)
\(678\) 0 0
\(679\) 7.73813 0.296962
\(680\) 0 0
\(681\) 18.5501 0.710841
\(682\) 0 0
\(683\) 12.2882 0.470195 0.235098 0.971972i \(-0.424459\pi\)
0.235098 + 0.971972i \(0.424459\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −28.5501 −1.08925
\(688\) 0 0
\(689\) 16.7757 0.639105
\(690\) 0 0
\(691\) 1.81336 0.0689834 0.0344917 0.999405i \(-0.489019\pi\)
0.0344917 + 0.999405i \(0.489019\pi\)
\(692\) 0 0
\(693\) −0.387873 −0.0147341
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −3.47627 −0.131673
\(698\) 0 0
\(699\) −18.9624 −0.717223
\(700\) 0 0
\(701\) 29.5778 1.11714 0.558570 0.829458i \(-0.311350\pi\)
0.558570 + 0.829458i \(0.311350\pi\)
\(702\) 0 0
\(703\) −17.5515 −0.661967
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.58769 −0.285364
\(708\) 0 0
\(709\) 20.8021 0.781239 0.390620 0.920552i \(-0.372261\pi\)
0.390620 + 0.920552i \(0.372261\pi\)
\(710\) 0 0
\(711\) 13.9248 0.522220
\(712\) 0 0
\(713\) 2.85097 0.106770
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 18.1622 0.678280
\(718\) 0 0
\(719\) −19.3258 −0.720732 −0.360366 0.932811i \(-0.617348\pi\)
−0.360366 + 0.932811i \(0.617348\pi\)
\(720\) 0 0
\(721\) 14.7005 0.547476
\(722\) 0 0
\(723\) 14.3733 0.534548
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21.6531 −0.803068 −0.401534 0.915844i \(-0.631523\pi\)
−0.401534 + 0.915844i \(0.631523\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −35.8496 −1.32594
\(732\) 0 0
\(733\) 48.0625 1.77523 0.887615 0.460586i \(-0.152361\pi\)
0.887615 + 0.460586i \(0.152361\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 30.2981 1.11453 0.557266 0.830334i \(-0.311850\pi\)
0.557266 + 0.830334i \(0.311850\pi\)
\(740\) 0 0
\(741\) 8.77575 0.322385
\(742\) 0 0
\(743\) −40.9624 −1.50276 −0.751382 0.659867i \(-0.770612\pi\)
−0.751382 + 0.659867i \(0.770612\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.22425 0.117969
\(748\) 0 0
\(749\) 16.4387 0.600656
\(750\) 0 0
\(751\) −9.52232 −0.347474 −0.173737 0.984792i \(-0.555584\pi\)
−0.173737 + 0.984792i \(0.555584\pi\)
\(752\) 0 0
\(753\) 8.62530 0.314323
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −31.3258 −1.13856 −0.569278 0.822145i \(-0.692777\pi\)
−0.569278 + 0.822145i \(0.692777\pi\)
\(758\) 0 0
\(759\) 0.373285 0.0135494
\(760\) 0 0
\(761\) −54.2393 −1.96617 −0.983087 0.183138i \(-0.941375\pi\)
−0.983087 + 0.183138i \(0.941375\pi\)
\(762\) 0 0
\(763\) −2.77575 −0.100489
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −9.55149 −0.344884
\(768\) 0 0
\(769\) 35.1002 1.26574 0.632872 0.774256i \(-0.281876\pi\)
0.632872 + 0.774256i \(0.281876\pi\)
\(770\) 0 0
\(771\) 27.8251 1.00210
\(772\) 0 0
\(773\) −19.8740 −0.714818 −0.357409 0.933948i \(-0.616340\pi\)
−0.357409 + 0.933948i \(0.616340\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −5.92478 −0.212550
\(778\) 0 0
\(779\) 3.07381 0.110131
\(780\) 0 0
\(781\) 2.14762 0.0768478
\(782\) 0 0
\(783\) 1.22425 0.0437513
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 13.2506 0.472333 0.236166 0.971713i \(-0.424109\pi\)
0.236166 + 0.971713i \(0.424109\pi\)
\(788\) 0 0
\(789\) −0.589104 −0.0209727
\(790\) 0 0
\(791\) −4.26187 −0.151534
\(792\) 0 0
\(793\) 43.3258 1.53855
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.7235 0.556957 0.278478 0.960443i \(-0.410170\pi\)
0.278478 + 0.960443i \(0.410170\pi\)
\(798\) 0 0
\(799\) −10.8021 −0.382151
\(800\) 0 0
\(801\) 3.73813 0.132081
\(802\) 0 0
\(803\) 2.39963 0.0846812
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15.7381 0.554009
\(808\) 0 0
\(809\) −18.5040 −0.650567 −0.325284 0.945617i \(-0.605460\pi\)
−0.325284 + 0.945617i \(0.605460\pi\)
\(810\) 0 0
\(811\) −37.6629 −1.32252 −0.661262 0.750155i \(-0.729979\pi\)
−0.661262 + 0.750155i \(0.729979\pi\)
\(812\) 0 0
\(813\) 7.11283 0.249458
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 31.6991 1.10901
\(818\) 0 0
\(819\) 2.96239 0.103514
\(820\) 0 0
\(821\) −17.9511 −0.626499 −0.313249 0.949671i \(-0.601418\pi\)
−0.313249 + 0.949671i \(0.601418\pi\)
\(822\) 0 0
\(823\) −32.1016 −1.11899 −0.559495 0.828834i \(-0.689005\pi\)
−0.559495 + 0.828834i \(0.689005\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −25.8594 −0.899220 −0.449610 0.893225i \(-0.648437\pi\)
−0.449610 + 0.893225i \(0.648437\pi\)
\(828\) 0 0
\(829\) 11.8035 0.409953 0.204976 0.978767i \(-0.434288\pi\)
0.204976 + 0.978767i \(0.434288\pi\)
\(830\) 0 0
\(831\) −5.92478 −0.205528
\(832\) 0 0
\(833\) −3.35026 −0.116080
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.96239 −0.102395
\(838\) 0 0
\(839\) −19.5778 −0.675902 −0.337951 0.941164i \(-0.609734\pi\)
−0.337951 + 0.941164i \(0.609734\pi\)
\(840\) 0 0
\(841\) −27.5012 −0.948317
\(842\) 0 0
\(843\) 29.3258 1.01004
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −10.8496 −0.372795
\(848\) 0 0
\(849\) −16.1016 −0.552604
\(850\) 0 0
\(851\) 5.70194 0.195460
\(852\) 0 0
\(853\) 40.6155 1.39065 0.695323 0.718697i \(-0.255261\pi\)
0.695323 + 0.718697i \(0.255261\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −29.1246 −0.994877 −0.497439 0.867499i \(-0.665726\pi\)
−0.497439 + 0.867499i \(0.665726\pi\)
\(858\) 0 0
\(859\) −35.7090 −1.21837 −0.609187 0.793027i \(-0.708504\pi\)
−0.609187 + 0.793027i \(0.708504\pi\)
\(860\) 0 0
\(861\) 1.03761 0.0353617
\(862\) 0 0
\(863\) 21.1852 0.721154 0.360577 0.932730i \(-0.382580\pi\)
0.360577 + 0.932730i \(0.382580\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −5.77575 −0.196155
\(868\) 0 0
\(869\) −5.40105 −0.183218
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 7.73813 0.261896
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 26.4485 0.893103 0.446551 0.894758i \(-0.352652\pi\)
0.446551 + 0.894758i \(0.352652\pi\)
\(878\) 0 0
\(879\) −9.27504 −0.312839
\(880\) 0 0
\(881\) −26.0362 −0.877182 −0.438591 0.898687i \(-0.644522\pi\)
−0.438591 + 0.898687i \(0.644522\pi\)
\(882\) 0 0
\(883\) 8.87732 0.298745 0.149373 0.988781i \(-0.452275\pi\)
0.149373 + 0.988781i \(0.452275\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.17679 0.0730896 0.0365448 0.999332i \(-0.488365\pi\)
0.0365448 + 0.999332i \(0.488365\pi\)
\(888\) 0 0
\(889\) −14.5501 −0.487994
\(890\) 0 0
\(891\) −0.387873 −0.0129942
\(892\) 0 0
\(893\) 9.55149 0.319629
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.85097 −0.0951911
\(898\) 0 0
\(899\) −3.62672 −0.120958
\(900\) 0 0
\(901\) −18.9722 −0.632057
\(902\) 0 0
\(903\) 10.7005 0.356091
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −44.7269 −1.48513 −0.742566 0.669773i \(-0.766391\pi\)
−0.742566 + 0.669773i \(0.766391\pi\)
\(908\) 0 0
\(909\) −7.58769 −0.251668
\(910\) 0 0
\(911\) −26.4631 −0.876761 −0.438381 0.898789i \(-0.644448\pi\)
−0.438381 + 0.898789i \(0.644448\pi\)
\(912\) 0 0
\(913\) −1.25060 −0.0413889
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) 59.2769 1.95537 0.977683 0.210085i \(-0.0673742\pi\)
0.977683 + 0.210085i \(0.0673742\pi\)
\(920\) 0 0
\(921\) 10.7005 0.352594
\(922\) 0 0
\(923\) −16.4025 −0.539894
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 14.7005 0.482829
\(928\) 0 0
\(929\) 25.3620 0.832101 0.416050 0.909342i \(-0.363414\pi\)
0.416050 + 0.909342i \(0.363414\pi\)
\(930\) 0 0
\(931\) 2.96239 0.0970884
\(932\) 0 0
\(933\) −12.1016 −0.396188
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.85940 0.126081 0.0630406 0.998011i \(-0.479920\pi\)
0.0630406 + 0.998011i \(0.479920\pi\)
\(938\) 0 0
\(939\) −28.3634 −0.925606
\(940\) 0 0
\(941\) 27.4861 0.896022 0.448011 0.894028i \(-0.352133\pi\)
0.448011 + 0.894028i \(0.352133\pi\)
\(942\) 0 0
\(943\) −0.998585 −0.0325184
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13.8397 −0.449730 −0.224865 0.974390i \(-0.572194\pi\)
−0.224865 + 0.974390i \(0.572194\pi\)
\(948\) 0 0
\(949\) −18.3272 −0.594927
\(950\) 0 0
\(951\) 17.5125 0.567881
\(952\) 0 0
\(953\) 39.2868 1.27262 0.636312 0.771432i \(-0.280459\pi\)
0.636312 + 0.771432i \(0.280459\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −0.474855 −0.0153499
\(958\) 0 0
\(959\) −10.4387 −0.337082
\(960\) 0 0
\(961\) −22.2243 −0.716911
\(962\) 0 0
\(963\) 16.4387 0.529728
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −8.67418 −0.278943 −0.139471 0.990226i \(-0.544540\pi\)
−0.139471 + 0.990226i \(0.544540\pi\)
\(968\) 0 0
\(969\) −9.92478 −0.318830
\(970\) 0 0
\(971\) 25.9248 0.831966 0.415983 0.909372i \(-0.363438\pi\)
0.415983 + 0.909372i \(0.363438\pi\)
\(972\) 0 0
\(973\) −5.66291 −0.181545
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −7.85940 −0.251445 −0.125722 0.992065i \(-0.540125\pi\)
−0.125722 + 0.992065i \(0.540125\pi\)
\(978\) 0 0
\(979\) −1.44992 −0.0463397
\(980\) 0 0
\(981\) −2.77575 −0.0886228
\(982\) 0 0
\(983\) −18.8510 −0.601253 −0.300626 0.953742i \(-0.597196\pi\)
−0.300626 + 0.953742i \(0.597196\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 3.22425 0.102629
\(988\) 0 0
\(989\) −10.2981 −0.327459
\(990\) 0 0
\(991\) 43.0738 1.36828 0.684142 0.729349i \(-0.260177\pi\)
0.684142 + 0.729349i \(0.260177\pi\)
\(992\) 0 0
\(993\) 19.2243 0.610063
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 24.6155 0.779579 0.389790 0.920904i \(-0.372548\pi\)
0.389790 + 0.920904i \(0.372548\pi\)
\(998\) 0 0
\(999\) −5.92478 −0.187452
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 8400.2.a.dk.1.2 3
4.3 odd 2 4200.2.a.bo.1.2 3
5.2 odd 4 1680.2.t.i.1009.3 6
5.3 odd 4 1680.2.t.i.1009.6 6
5.4 even 2 8400.2.a.dh.1.2 3
15.2 even 4 5040.2.t.ba.1009.2 6
15.8 even 4 5040.2.t.ba.1009.1 6
20.3 even 4 840.2.t.e.169.3 6
20.7 even 4 840.2.t.e.169.6 yes 6
20.19 odd 2 4200.2.a.bq.1.2 3
60.23 odd 4 2520.2.t.j.1009.1 6
60.47 odd 4 2520.2.t.j.1009.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.e.169.3 6 20.3 even 4
840.2.t.e.169.6 yes 6 20.7 even 4
1680.2.t.i.1009.3 6 5.2 odd 4
1680.2.t.i.1009.6 6 5.3 odd 4
2520.2.t.j.1009.1 6 60.23 odd 4
2520.2.t.j.1009.2 6 60.47 odd 4
4200.2.a.bo.1.2 3 4.3 odd 2
4200.2.a.bq.1.2 3 20.19 odd 2
5040.2.t.ba.1009.1 6 15.8 even 4
5040.2.t.ba.1009.2 6 15.2 even 4
8400.2.a.dh.1.2 3 5.4 even 2
8400.2.a.dk.1.2 3 1.1 even 1 trivial