Properties

Label 7350.2.a.dg.1.1
Level 7350
Weight 2
Character 7350.1
Self dual yes
Analytic conductor 58.690
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.6900454856\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 7350.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +0.585786 q^{11} +1.00000 q^{12} +3.00000 q^{13} +1.00000 q^{16} +0.171573 q^{17} -1.00000 q^{18} -1.41421 q^{19} -0.585786 q^{22} +2.41421 q^{23} -1.00000 q^{24} -3.00000 q^{26} +1.00000 q^{27} +1.00000 q^{29} +0.414214 q^{31} -1.00000 q^{32} +0.585786 q^{33} -0.171573 q^{34} +1.00000 q^{36} -7.07107 q^{37} +1.41421 q^{38} +3.00000 q^{39} -0.171573 q^{41} +4.41421 q^{43} +0.585786 q^{44} -2.41421 q^{46} +9.65685 q^{47} +1.00000 q^{48} +0.171573 q^{51} +3.00000 q^{52} +1.34315 q^{53} -1.00000 q^{54} -1.41421 q^{57} -1.00000 q^{58} -1.24264 q^{59} -5.82843 q^{61} -0.414214 q^{62} +1.00000 q^{64} -0.585786 q^{66} +7.31371 q^{67} +0.171573 q^{68} +2.41421 q^{69} +13.3137 q^{71} -1.00000 q^{72} +0.928932 q^{73} +7.07107 q^{74} -1.41421 q^{76} -3.00000 q^{78} -13.0711 q^{79} +1.00000 q^{81} +0.171573 q^{82} +14.0711 q^{83} -4.41421 q^{86} +1.00000 q^{87} -0.585786 q^{88} +4.00000 q^{89} +2.41421 q^{92} +0.414214 q^{93} -9.65685 q^{94} -1.00000 q^{96} -11.6569 q^{97} +0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} + 2q^{9} + 4q^{11} + 2q^{12} + 6q^{13} + 2q^{16} + 6q^{17} - 2q^{18} - 4q^{22} + 2q^{23} - 2q^{24} - 6q^{26} + 2q^{27} + 2q^{29} - 2q^{31} - 2q^{32} + 4q^{33} - 6q^{34} + 2q^{36} + 6q^{39} - 6q^{41} + 6q^{43} + 4q^{44} - 2q^{46} + 8q^{47} + 2q^{48} + 6q^{51} + 6q^{52} + 14q^{53} - 2q^{54} - 2q^{58} + 6q^{59} - 6q^{61} + 2q^{62} + 2q^{64} - 4q^{66} - 8q^{67} + 6q^{68} + 2q^{69} + 4q^{71} - 2q^{72} + 16q^{73} - 6q^{78} - 12q^{79} + 2q^{81} + 6q^{82} + 14q^{83} - 6q^{86} + 2q^{87} - 4q^{88} + 8q^{89} + 2q^{92} - 2q^{93} - 8q^{94} - 2q^{96} - 12q^{97} + 4q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.585786 0.176621 0.0883106 0.996093i \(-0.471853\pi\)
0.0883106 + 0.996093i \(0.471853\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.171573 0.0416125 0.0208063 0.999784i \(-0.493377\pi\)
0.0208063 + 0.999784i \(0.493377\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.41421 −0.324443 −0.162221 0.986754i \(-0.551866\pi\)
−0.162221 + 0.986754i \(0.551866\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.585786 −0.124890
\(23\) 2.41421 0.503398 0.251699 0.967806i \(-0.419011\pi\)
0.251699 + 0.967806i \(0.419011\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 0.414214 0.0743950 0.0371975 0.999308i \(-0.488157\pi\)
0.0371975 + 0.999308i \(0.488157\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.585786 0.101972
\(34\) −0.171573 −0.0294245
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.07107 −1.16248 −0.581238 0.813733i \(-0.697432\pi\)
−0.581238 + 0.813733i \(0.697432\pi\)
\(38\) 1.41421 0.229416
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) −0.171573 −0.0267952 −0.0133976 0.999910i \(-0.504265\pi\)
−0.0133976 + 0.999910i \(0.504265\pi\)
\(42\) 0 0
\(43\) 4.41421 0.673161 0.336581 0.941655i \(-0.390730\pi\)
0.336581 + 0.941655i \(0.390730\pi\)
\(44\) 0.585786 0.0883106
\(45\) 0 0
\(46\) −2.41421 −0.355956
\(47\) 9.65685 1.40860 0.704298 0.709904i \(-0.251262\pi\)
0.704298 + 0.709904i \(0.251262\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 0 0
\(51\) 0.171573 0.0240250
\(52\) 3.00000 0.416025
\(53\) 1.34315 0.184495 0.0922476 0.995736i \(-0.470595\pi\)
0.0922476 + 0.995736i \(0.470595\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −1.41421 −0.187317
\(58\) −1.00000 −0.131306
\(59\) −1.24264 −0.161778 −0.0808890 0.996723i \(-0.525776\pi\)
−0.0808890 + 0.996723i \(0.525776\pi\)
\(60\) 0 0
\(61\) −5.82843 −0.746254 −0.373127 0.927780i \(-0.621714\pi\)
−0.373127 + 0.927780i \(0.621714\pi\)
\(62\) −0.414214 −0.0526052
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −0.585786 −0.0721053
\(67\) 7.31371 0.893512 0.446756 0.894656i \(-0.352579\pi\)
0.446756 + 0.894656i \(0.352579\pi\)
\(68\) 0.171573 0.0208063
\(69\) 2.41421 0.290637
\(70\) 0 0
\(71\) 13.3137 1.58005 0.790023 0.613077i \(-0.210068\pi\)
0.790023 + 0.613077i \(0.210068\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0.928932 0.108723 0.0543616 0.998521i \(-0.482688\pi\)
0.0543616 + 0.998521i \(0.482688\pi\)
\(74\) 7.07107 0.821995
\(75\) 0 0
\(76\) −1.41421 −0.162221
\(77\) 0 0
\(78\) −3.00000 −0.339683
\(79\) −13.0711 −1.47061 −0.735305 0.677736i \(-0.762961\pi\)
−0.735305 + 0.677736i \(0.762961\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.171573 0.0189471
\(83\) 14.0711 1.54450 0.772250 0.635319i \(-0.219131\pi\)
0.772250 + 0.635319i \(0.219131\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.41421 −0.475997
\(87\) 1.00000 0.107211
\(88\) −0.585786 −0.0624450
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.41421 0.251699
\(93\) 0.414214 0.0429519
\(94\) −9.65685 −0.996028
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −11.6569 −1.18357 −0.591787 0.806094i \(-0.701577\pi\)
−0.591787 + 0.806094i \(0.701577\pi\)
\(98\) 0 0
\(99\) 0.585786 0.0588738
\(100\) 0 0
\(101\) −5.65685 −0.562878 −0.281439 0.959579i \(-0.590812\pi\)
−0.281439 + 0.959579i \(0.590812\pi\)
\(102\) −0.171573 −0.0169882
\(103\) 14.0711 1.38646 0.693232 0.720715i \(-0.256186\pi\)
0.693232 + 0.720715i \(0.256186\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) −1.34315 −0.130458
\(107\) −10.4853 −1.01365 −0.506825 0.862049i \(-0.669181\pi\)
−0.506825 + 0.862049i \(0.669181\pi\)
\(108\) 1.00000 0.0962250
\(109\) 11.3137 1.08366 0.541828 0.840489i \(-0.317732\pi\)
0.541828 + 0.840489i \(0.317732\pi\)
\(110\) 0 0
\(111\) −7.07107 −0.671156
\(112\) 0 0
\(113\) 9.41421 0.885615 0.442807 0.896617i \(-0.353983\pi\)
0.442807 + 0.896617i \(0.353983\pi\)
\(114\) 1.41421 0.132453
\(115\) 0 0
\(116\) 1.00000 0.0928477
\(117\) 3.00000 0.277350
\(118\) 1.24264 0.114394
\(119\) 0 0
\(120\) 0 0
\(121\) −10.6569 −0.968805
\(122\) 5.82843 0.527681
\(123\) −0.171573 −0.0154702
\(124\) 0.414214 0.0371975
\(125\) 0 0
\(126\) 0 0
\(127\) −8.82843 −0.783396 −0.391698 0.920094i \(-0.628112\pi\)
−0.391698 + 0.920094i \(0.628112\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.41421 0.388650
\(130\) 0 0
\(131\) −13.1716 −1.15081 −0.575403 0.817870i \(-0.695155\pi\)
−0.575403 + 0.817870i \(0.695155\pi\)
\(132\) 0.585786 0.0509862
\(133\) 0 0
\(134\) −7.31371 −0.631808
\(135\) 0 0
\(136\) −0.171573 −0.0147123
\(137\) 19.2132 1.64149 0.820747 0.571291i \(-0.193557\pi\)
0.820747 + 0.571291i \(0.193557\pi\)
\(138\) −2.41421 −0.205512
\(139\) 15.5563 1.31947 0.659736 0.751497i \(-0.270668\pi\)
0.659736 + 0.751497i \(0.270668\pi\)
\(140\) 0 0
\(141\) 9.65685 0.813254
\(142\) −13.3137 −1.11726
\(143\) 1.75736 0.146958
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −0.928932 −0.0768790
\(147\) 0 0
\(148\) −7.07107 −0.581238
\(149\) −6.65685 −0.545351 −0.272675 0.962106i \(-0.587908\pi\)
−0.272675 + 0.962106i \(0.587908\pi\)
\(150\) 0 0
\(151\) −11.8995 −0.968367 −0.484184 0.874966i \(-0.660883\pi\)
−0.484184 + 0.874966i \(0.660883\pi\)
\(152\) 1.41421 0.114708
\(153\) 0.171573 0.0138708
\(154\) 0 0
\(155\) 0 0
\(156\) 3.00000 0.240192
\(157\) 19.7990 1.58013 0.790066 0.613022i \(-0.210046\pi\)
0.790066 + 0.613022i \(0.210046\pi\)
\(158\) 13.0711 1.03988
\(159\) 1.34315 0.106518
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −3.24264 −0.253983 −0.126992 0.991904i \(-0.540532\pi\)
−0.126992 + 0.991904i \(0.540532\pi\)
\(164\) −0.171573 −0.0133976
\(165\) 0 0
\(166\) −14.0711 −1.09213
\(167\) −3.17157 −0.245424 −0.122712 0.992442i \(-0.539159\pi\)
−0.122712 + 0.992442i \(0.539159\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −1.41421 −0.108148
\(172\) 4.41421 0.336581
\(173\) 5.75736 0.437724 0.218862 0.975756i \(-0.429766\pi\)
0.218862 + 0.975756i \(0.429766\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) 0.585786 0.0441553
\(177\) −1.24264 −0.0934026
\(178\) −4.00000 −0.299813
\(179\) 9.07107 0.678003 0.339002 0.940786i \(-0.389911\pi\)
0.339002 + 0.940786i \(0.389911\pi\)
\(180\) 0 0
\(181\) −0.686292 −0.0510116 −0.0255058 0.999675i \(-0.508120\pi\)
−0.0255058 + 0.999675i \(0.508120\pi\)
\(182\) 0 0
\(183\) −5.82843 −0.430850
\(184\) −2.41421 −0.177978
\(185\) 0 0
\(186\) −0.414214 −0.0303716
\(187\) 0.100505 0.00734966
\(188\) 9.65685 0.704298
\(189\) 0 0
\(190\) 0 0
\(191\) 1.92893 0.139573 0.0697863 0.997562i \(-0.477768\pi\)
0.0697863 + 0.997562i \(0.477768\pi\)
\(192\) 1.00000 0.0721688
\(193\) −1.17157 −0.0843317 −0.0421658 0.999111i \(-0.513426\pi\)
−0.0421658 + 0.999111i \(0.513426\pi\)
\(194\) 11.6569 0.836913
\(195\) 0 0
\(196\) 0 0
\(197\) −5.14214 −0.366362 −0.183181 0.983079i \(-0.558639\pi\)
−0.183181 + 0.983079i \(0.558639\pi\)
\(198\) −0.585786 −0.0416300
\(199\) 8.34315 0.591430 0.295715 0.955276i \(-0.404442\pi\)
0.295715 + 0.955276i \(0.404442\pi\)
\(200\) 0 0
\(201\) 7.31371 0.515869
\(202\) 5.65685 0.398015
\(203\) 0 0
\(204\) 0.171573 0.0120125
\(205\) 0 0
\(206\) −14.0711 −0.980378
\(207\) 2.41421 0.167799
\(208\) 3.00000 0.208013
\(209\) −0.828427 −0.0573035
\(210\) 0 0
\(211\) 4.55635 0.313672 0.156836 0.987625i \(-0.449871\pi\)
0.156836 + 0.987625i \(0.449871\pi\)
\(212\) 1.34315 0.0922476
\(213\) 13.3137 0.912240
\(214\) 10.4853 0.716759
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −11.3137 −0.766261
\(219\) 0.928932 0.0627714
\(220\) 0 0
\(221\) 0.514719 0.0346237
\(222\) 7.07107 0.474579
\(223\) −6.55635 −0.439046 −0.219523 0.975607i \(-0.570450\pi\)
−0.219523 + 0.975607i \(0.570450\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −9.41421 −0.626224
\(227\) −11.3848 −0.755634 −0.377817 0.925880i \(-0.623325\pi\)
−0.377817 + 0.925880i \(0.623325\pi\)
\(228\) −1.41421 −0.0936586
\(229\) 6.68629 0.441843 0.220921 0.975292i \(-0.429094\pi\)
0.220921 + 0.975292i \(0.429094\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 5.75736 0.377177 0.188589 0.982056i \(-0.439609\pi\)
0.188589 + 0.982056i \(0.439609\pi\)
\(234\) −3.00000 −0.196116
\(235\) 0 0
\(236\) −1.24264 −0.0808890
\(237\) −13.0711 −0.849057
\(238\) 0 0
\(239\) 11.3137 0.731823 0.365911 0.930650i \(-0.380757\pi\)
0.365911 + 0.930650i \(0.380757\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 10.6569 0.685049
\(243\) 1.00000 0.0641500
\(244\) −5.82843 −0.373127
\(245\) 0 0
\(246\) 0.171573 0.0109391
\(247\) −4.24264 −0.269953
\(248\) −0.414214 −0.0263026
\(249\) 14.0711 0.891718
\(250\) 0 0
\(251\) −16.0711 −1.01440 −0.507198 0.861829i \(-0.669319\pi\)
−0.507198 + 0.861829i \(0.669319\pi\)
\(252\) 0 0
\(253\) 1.41421 0.0889108
\(254\) 8.82843 0.553945
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 24.6569 1.53805 0.769026 0.639217i \(-0.220742\pi\)
0.769026 + 0.639217i \(0.220742\pi\)
\(258\) −4.41421 −0.274817
\(259\) 0 0
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 13.1716 0.813742
\(263\) 20.5563 1.26756 0.633779 0.773514i \(-0.281503\pi\)
0.633779 + 0.773514i \(0.281503\pi\)
\(264\) −0.585786 −0.0360527
\(265\) 0 0
\(266\) 0 0
\(267\) 4.00000 0.244796
\(268\) 7.31371 0.446756
\(269\) 31.5563 1.92402 0.962012 0.273006i \(-0.0880179\pi\)
0.962012 + 0.273006i \(0.0880179\pi\)
\(270\) 0 0
\(271\) −28.4853 −1.73036 −0.865179 0.501463i \(-0.832795\pi\)
−0.865179 + 0.501463i \(0.832795\pi\)
\(272\) 0.171573 0.0104031
\(273\) 0 0
\(274\) −19.2132 −1.16071
\(275\) 0 0
\(276\) 2.41421 0.145319
\(277\) 24.0416 1.44452 0.722261 0.691621i \(-0.243103\pi\)
0.722261 + 0.691621i \(0.243103\pi\)
\(278\) −15.5563 −0.933008
\(279\) 0.414214 0.0247983
\(280\) 0 0
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) −9.65685 −0.575057
\(283\) 8.24264 0.489974 0.244987 0.969526i \(-0.421216\pi\)
0.244987 + 0.969526i \(0.421216\pi\)
\(284\) 13.3137 0.790023
\(285\) 0 0
\(286\) −1.75736 −0.103915
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −16.9706 −0.998268
\(290\) 0 0
\(291\) −11.6569 −0.683337
\(292\) 0.928932 0.0543616
\(293\) 11.3137 0.660954 0.330477 0.943814i \(-0.392790\pi\)
0.330477 + 0.943814i \(0.392790\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7.07107 0.410997
\(297\) 0.585786 0.0339908
\(298\) 6.65685 0.385621
\(299\) 7.24264 0.418853
\(300\) 0 0
\(301\) 0 0
\(302\) 11.8995 0.684739
\(303\) −5.65685 −0.324978
\(304\) −1.41421 −0.0811107
\(305\) 0 0
\(306\) −0.171573 −0.00980817
\(307\) 11.4142 0.651444 0.325722 0.945466i \(-0.394393\pi\)
0.325722 + 0.945466i \(0.394393\pi\)
\(308\) 0 0
\(309\) 14.0711 0.800475
\(310\) 0 0
\(311\) −8.58579 −0.486855 −0.243428 0.969919i \(-0.578272\pi\)
−0.243428 + 0.969919i \(0.578272\pi\)
\(312\) −3.00000 −0.169842
\(313\) 32.0416 1.81110 0.905550 0.424240i \(-0.139459\pi\)
0.905550 + 0.424240i \(0.139459\pi\)
\(314\) −19.7990 −1.11732
\(315\) 0 0
\(316\) −13.0711 −0.735305
\(317\) −13.9706 −0.784665 −0.392332 0.919823i \(-0.628332\pi\)
−0.392332 + 0.919823i \(0.628332\pi\)
\(318\) −1.34315 −0.0753199
\(319\) 0.585786 0.0327977
\(320\) 0 0
\(321\) −10.4853 −0.585231
\(322\) 0 0
\(323\) −0.242641 −0.0135009
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 3.24264 0.179593
\(327\) 11.3137 0.625650
\(328\) 0.171573 0.00947353
\(329\) 0 0
\(330\) 0 0
\(331\) −9.92893 −0.545743 −0.272872 0.962050i \(-0.587973\pi\)
−0.272872 + 0.962050i \(0.587973\pi\)
\(332\) 14.0711 0.772250
\(333\) −7.07107 −0.387492
\(334\) 3.17157 0.173541
\(335\) 0 0
\(336\) 0 0
\(337\) −34.7990 −1.89562 −0.947811 0.318833i \(-0.896709\pi\)
−0.947811 + 0.318833i \(0.896709\pi\)
\(338\) 4.00000 0.217571
\(339\) 9.41421 0.511310
\(340\) 0 0
\(341\) 0.242641 0.0131397
\(342\) 1.41421 0.0764719
\(343\) 0 0
\(344\) −4.41421 −0.237998
\(345\) 0 0
\(346\) −5.75736 −0.309518
\(347\) 29.7990 1.59969 0.799847 0.600204i \(-0.204914\pi\)
0.799847 + 0.600204i \(0.204914\pi\)
\(348\) 1.00000 0.0536056
\(349\) 13.3431 0.714242 0.357121 0.934058i \(-0.383758\pi\)
0.357121 + 0.934058i \(0.383758\pi\)
\(350\) 0 0
\(351\) 3.00000 0.160128
\(352\) −0.585786 −0.0312225
\(353\) 3.51472 0.187070 0.0935348 0.995616i \(-0.470183\pi\)
0.0935348 + 0.995616i \(0.470183\pi\)
\(354\) 1.24264 0.0660456
\(355\) 0 0
\(356\) 4.00000 0.212000
\(357\) 0 0
\(358\) −9.07107 −0.479421
\(359\) −19.8701 −1.04870 −0.524351 0.851502i \(-0.675692\pi\)
−0.524351 + 0.851502i \(0.675692\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0.686292 0.0360707
\(363\) −10.6569 −0.559340
\(364\) 0 0
\(365\) 0 0
\(366\) 5.82843 0.304657
\(367\) 9.24264 0.482462 0.241231 0.970468i \(-0.422449\pi\)
0.241231 + 0.970468i \(0.422449\pi\)
\(368\) 2.41421 0.125850
\(369\) −0.171573 −0.00893173
\(370\) 0 0
\(371\) 0 0
\(372\) 0.414214 0.0214760
\(373\) 13.6569 0.707125 0.353563 0.935411i \(-0.384970\pi\)
0.353563 + 0.935411i \(0.384970\pi\)
\(374\) −0.100505 −0.00519699
\(375\) 0 0
\(376\) −9.65685 −0.498014
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) 8.21320 0.421884 0.210942 0.977499i \(-0.432347\pi\)
0.210942 + 0.977499i \(0.432347\pi\)
\(380\) 0 0
\(381\) −8.82843 −0.452294
\(382\) −1.92893 −0.0986928
\(383\) −3.55635 −0.181721 −0.0908605 0.995864i \(-0.528962\pi\)
−0.0908605 + 0.995864i \(0.528962\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 1.17157 0.0596315
\(387\) 4.41421 0.224387
\(388\) −11.6569 −0.591787
\(389\) −21.6569 −1.09805 −0.549023 0.835807i \(-0.685000\pi\)
−0.549023 + 0.835807i \(0.685000\pi\)
\(390\) 0 0
\(391\) 0.414214 0.0209477
\(392\) 0 0
\(393\) −13.1716 −0.664418
\(394\) 5.14214 0.259057
\(395\) 0 0
\(396\) 0.585786 0.0294369
\(397\) 7.68629 0.385764 0.192882 0.981222i \(-0.438217\pi\)
0.192882 + 0.981222i \(0.438217\pi\)
\(398\) −8.34315 −0.418204
\(399\) 0 0
\(400\) 0 0
\(401\) 31.8995 1.59298 0.796492 0.604649i \(-0.206686\pi\)
0.796492 + 0.604649i \(0.206686\pi\)
\(402\) −7.31371 −0.364775
\(403\) 1.24264 0.0619003
\(404\) −5.65685 −0.281439
\(405\) 0 0
\(406\) 0 0
\(407\) −4.14214 −0.205318
\(408\) −0.171573 −0.00849412
\(409\) 25.4142 1.25665 0.628326 0.777950i \(-0.283740\pi\)
0.628326 + 0.777950i \(0.283740\pi\)
\(410\) 0 0
\(411\) 19.2132 0.947717
\(412\) 14.0711 0.693232
\(413\) 0 0
\(414\) −2.41421 −0.118652
\(415\) 0 0
\(416\) −3.00000 −0.147087
\(417\) 15.5563 0.761798
\(418\) 0.828427 0.0405197
\(419\) −6.55635 −0.320299 −0.160149 0.987093i \(-0.551198\pi\)
−0.160149 + 0.987093i \(0.551198\pi\)
\(420\) 0 0
\(421\) −24.3431 −1.18641 −0.593206 0.805051i \(-0.702138\pi\)
−0.593206 + 0.805051i \(0.702138\pi\)
\(422\) −4.55635 −0.221800
\(423\) 9.65685 0.469532
\(424\) −1.34315 −0.0652289
\(425\) 0 0
\(426\) −13.3137 −0.645051
\(427\) 0 0
\(428\) −10.4853 −0.506825
\(429\) 1.75736 0.0848461
\(430\) 0 0
\(431\) −15.0416 −0.724530 −0.362265 0.932075i \(-0.617996\pi\)
−0.362265 + 0.932075i \(0.617996\pi\)
\(432\) 1.00000 0.0481125
\(433\) −16.6274 −0.799063 −0.399531 0.916720i \(-0.630827\pi\)
−0.399531 + 0.916720i \(0.630827\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 11.3137 0.541828
\(437\) −3.41421 −0.163324
\(438\) −0.928932 −0.0443861
\(439\) 4.41421 0.210679 0.105339 0.994436i \(-0.466407\pi\)
0.105339 + 0.994436i \(0.466407\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −0.514719 −0.0244827
\(443\) −25.6569 −1.21899 −0.609497 0.792788i \(-0.708629\pi\)
−0.609497 + 0.792788i \(0.708629\pi\)
\(444\) −7.07107 −0.335578
\(445\) 0 0
\(446\) 6.55635 0.310452
\(447\) −6.65685 −0.314858
\(448\) 0 0
\(449\) −10.9289 −0.515768 −0.257884 0.966176i \(-0.583025\pi\)
−0.257884 + 0.966176i \(0.583025\pi\)
\(450\) 0 0
\(451\) −0.100505 −0.00473260
\(452\) 9.41421 0.442807
\(453\) −11.8995 −0.559087
\(454\) 11.3848 0.534314
\(455\) 0 0
\(456\) 1.41421 0.0662266
\(457\) 1.82843 0.0855302 0.0427651 0.999085i \(-0.486383\pi\)
0.0427651 + 0.999085i \(0.486383\pi\)
\(458\) −6.68629 −0.312430
\(459\) 0.171573 0.00800834
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) 10.1005 0.469410 0.234705 0.972067i \(-0.424588\pi\)
0.234705 + 0.972067i \(0.424588\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −5.75736 −0.266705
\(467\) 7.24264 0.335149 0.167575 0.985859i \(-0.446406\pi\)
0.167575 + 0.985859i \(0.446406\pi\)
\(468\) 3.00000 0.138675
\(469\) 0 0
\(470\) 0 0
\(471\) 19.7990 0.912289
\(472\) 1.24264 0.0571972
\(473\) 2.58579 0.118895
\(474\) 13.0711 0.600374
\(475\) 0 0
\(476\) 0 0
\(477\) 1.34315 0.0614984
\(478\) −11.3137 −0.517477
\(479\) 11.7574 0.537207 0.268604 0.963251i \(-0.413438\pi\)
0.268604 + 0.963251i \(0.413438\pi\)
\(480\) 0 0
\(481\) −21.2132 −0.967239
\(482\) 6.00000 0.273293
\(483\) 0 0
\(484\) −10.6569 −0.484402
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 22.4853 1.01891 0.509453 0.860499i \(-0.329848\pi\)
0.509453 + 0.860499i \(0.329848\pi\)
\(488\) 5.82843 0.263840
\(489\) −3.24264 −0.146637
\(490\) 0 0
\(491\) 28.1421 1.27004 0.635018 0.772497i \(-0.280993\pi\)
0.635018 + 0.772497i \(0.280993\pi\)
\(492\) −0.171573 −0.00773510
\(493\) 0.171573 0.00772725
\(494\) 4.24264 0.190885
\(495\) 0 0
\(496\) 0.414214 0.0185987
\(497\) 0 0
\(498\) −14.0711 −0.630540
\(499\) −11.9289 −0.534012 −0.267006 0.963695i \(-0.586034\pi\)
−0.267006 + 0.963695i \(0.586034\pi\)
\(500\) 0 0
\(501\) −3.17157 −0.141695
\(502\) 16.0711 0.717287
\(503\) 28.8701 1.28725 0.643626 0.765340i \(-0.277429\pi\)
0.643626 + 0.765340i \(0.277429\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.41421 −0.0628695
\(507\) −4.00000 −0.177646
\(508\) −8.82843 −0.391698
\(509\) 42.0416 1.86346 0.931731 0.363149i \(-0.118298\pi\)
0.931731 + 0.363149i \(0.118298\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −1.41421 −0.0624391
\(514\) −24.6569 −1.08757
\(515\) 0 0
\(516\) 4.41421 0.194325
\(517\) 5.65685 0.248788
\(518\) 0 0
\(519\) 5.75736 0.252720
\(520\) 0 0
\(521\) 0.656854 0.0287773 0.0143887 0.999896i \(-0.495420\pi\)
0.0143887 + 0.999896i \(0.495420\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 32.2843 1.41169 0.705846 0.708365i \(-0.250567\pi\)
0.705846 + 0.708365i \(0.250567\pi\)
\(524\) −13.1716 −0.575403
\(525\) 0 0
\(526\) −20.5563 −0.896299
\(527\) 0.0710678 0.00309576
\(528\) 0.585786 0.0254931
\(529\) −17.1716 −0.746590
\(530\) 0 0
\(531\) −1.24264 −0.0539260
\(532\) 0 0
\(533\) −0.514719 −0.0222949
\(534\) −4.00000 −0.173097
\(535\) 0 0
\(536\) −7.31371 −0.315904
\(537\) 9.07107 0.391445
\(538\) −31.5563 −1.36049
\(539\) 0 0
\(540\) 0 0
\(541\) −25.7574 −1.10740 −0.553698 0.832718i \(-0.686784\pi\)
−0.553698 + 0.832718i \(0.686784\pi\)
\(542\) 28.4853 1.22355
\(543\) −0.686292 −0.0294516
\(544\) −0.171573 −0.00735613
\(545\) 0 0
\(546\) 0 0
\(547\) −19.8701 −0.849582 −0.424791 0.905291i \(-0.639652\pi\)
−0.424791 + 0.905291i \(0.639652\pi\)
\(548\) 19.2132 0.820747
\(549\) −5.82843 −0.248751
\(550\) 0 0
\(551\) −1.41421 −0.0602475
\(552\) −2.41421 −0.102756
\(553\) 0 0
\(554\) −24.0416 −1.02143
\(555\) 0 0
\(556\) 15.5563 0.659736
\(557\) 3.31371 0.140406 0.0702032 0.997533i \(-0.477635\pi\)
0.0702032 + 0.997533i \(0.477635\pi\)
\(558\) −0.414214 −0.0175351
\(559\) 13.2426 0.560104
\(560\) 0 0
\(561\) 0.100505 0.00424333
\(562\) −20.0000 −0.843649
\(563\) −9.38478 −0.395521 −0.197761 0.980250i \(-0.563367\pi\)
−0.197761 + 0.980250i \(0.563367\pi\)
\(564\) 9.65685 0.406627
\(565\) 0 0
\(566\) −8.24264 −0.346464
\(567\) 0 0
\(568\) −13.3137 −0.558631
\(569\) −19.7574 −0.828272 −0.414136 0.910215i \(-0.635916\pi\)
−0.414136 + 0.910215i \(0.635916\pi\)
\(570\) 0 0
\(571\) −46.0122 −1.92555 −0.962775 0.270303i \(-0.912876\pi\)
−0.962775 + 0.270303i \(0.912876\pi\)
\(572\) 1.75736 0.0734789
\(573\) 1.92893 0.0805823
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 8.34315 0.347330 0.173665 0.984805i \(-0.444439\pi\)
0.173665 + 0.984805i \(0.444439\pi\)
\(578\) 16.9706 0.705882
\(579\) −1.17157 −0.0486889
\(580\) 0 0
\(581\) 0 0
\(582\) 11.6569 0.483192
\(583\) 0.786797 0.0325858
\(584\) −0.928932 −0.0384395
\(585\) 0 0
\(586\) −11.3137 −0.467365
\(587\) 23.5269 0.971060 0.485530 0.874220i \(-0.338627\pi\)
0.485530 + 0.874220i \(0.338627\pi\)
\(588\) 0 0
\(589\) −0.585786 −0.0241369
\(590\) 0 0
\(591\) −5.14214 −0.211519
\(592\) −7.07107 −0.290619
\(593\) −26.0000 −1.06769 −0.533846 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) −0.585786 −0.0240351
\(595\) 0 0
\(596\) −6.65685 −0.272675
\(597\) 8.34315 0.341462
\(598\) −7.24264 −0.296174
\(599\) −27.5858 −1.12712 −0.563562 0.826074i \(-0.690570\pi\)
−0.563562 + 0.826074i \(0.690570\pi\)
\(600\) 0 0
\(601\) 26.7279 1.09025 0.545127 0.838353i \(-0.316481\pi\)
0.545127 + 0.838353i \(0.316481\pi\)
\(602\) 0 0
\(603\) 7.31371 0.297837
\(604\) −11.8995 −0.484184
\(605\) 0 0
\(606\) 5.65685 0.229794
\(607\) −27.9411 −1.13410 −0.567048 0.823685i \(-0.691914\pi\)
−0.567048 + 0.823685i \(0.691914\pi\)
\(608\) 1.41421 0.0573539
\(609\) 0 0
\(610\) 0 0
\(611\) 28.9706 1.17202
\(612\) 0.171573 0.00693542
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) −11.4142 −0.460640
\(615\) 0 0
\(616\) 0 0
\(617\) −14.3848 −0.579109 −0.289555 0.957161i \(-0.593507\pi\)
−0.289555 + 0.957161i \(0.593507\pi\)
\(618\) −14.0711 −0.566021
\(619\) 16.1421 0.648807 0.324404 0.945919i \(-0.394836\pi\)
0.324404 + 0.945919i \(0.394836\pi\)
\(620\) 0 0
\(621\) 2.41421 0.0968791
\(622\) 8.58579 0.344259
\(623\) 0 0
\(624\) 3.00000 0.120096
\(625\) 0 0
\(626\) −32.0416 −1.28064
\(627\) −0.828427 −0.0330842
\(628\) 19.7990 0.790066
\(629\) −1.21320 −0.0483736
\(630\) 0 0
\(631\) −35.1716 −1.40016 −0.700079 0.714065i \(-0.746852\pi\)
−0.700079 + 0.714065i \(0.746852\pi\)
\(632\) 13.0711 0.519939
\(633\) 4.55635 0.181099
\(634\) 13.9706 0.554842
\(635\) 0 0
\(636\) 1.34315 0.0532592
\(637\) 0 0
\(638\) −0.585786 −0.0231915
\(639\) 13.3137 0.526682
\(640\) 0 0
\(641\) 40.2426 1.58949 0.794744 0.606944i \(-0.207605\pi\)
0.794744 + 0.606944i \(0.207605\pi\)
\(642\) 10.4853 0.413821
\(643\) −22.2426 −0.877164 −0.438582 0.898691i \(-0.644519\pi\)
−0.438582 + 0.898691i \(0.644519\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0.242641 0.00954657
\(647\) −48.8701 −1.92128 −0.960640 0.277796i \(-0.910396\pi\)
−0.960640 + 0.277796i \(0.910396\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −0.727922 −0.0285734
\(650\) 0 0
\(651\) 0 0
\(652\) −3.24264 −0.126992
\(653\) −35.3137 −1.38193 −0.690966 0.722887i \(-0.742815\pi\)
−0.690966 + 0.722887i \(0.742815\pi\)
\(654\) −11.3137 −0.442401
\(655\) 0 0
\(656\) −0.171573 −0.00669880
\(657\) 0.928932 0.0362411
\(658\) 0 0
\(659\) 8.82843 0.343907 0.171953 0.985105i \(-0.444992\pi\)
0.171953 + 0.985105i \(0.444992\pi\)
\(660\) 0 0
\(661\) 45.6569 1.77585 0.887923 0.459992i \(-0.152148\pi\)
0.887923 + 0.459992i \(0.152148\pi\)
\(662\) 9.92893 0.385899
\(663\) 0.514719 0.0199900
\(664\) −14.0711 −0.546063
\(665\) 0 0
\(666\) 7.07107 0.273998
\(667\) 2.41421 0.0934787
\(668\) −3.17157 −0.122712
\(669\) −6.55635 −0.253483
\(670\) 0 0
\(671\) −3.41421 −0.131804
\(672\) 0 0
\(673\) 38.7990 1.49559 0.747796 0.663929i \(-0.231112\pi\)
0.747796 + 0.663929i \(0.231112\pi\)
\(674\) 34.7990 1.34041
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) 32.3848 1.24465 0.622324 0.782760i \(-0.286189\pi\)
0.622324 + 0.782760i \(0.286189\pi\)
\(678\) −9.41421 −0.361551
\(679\) 0 0
\(680\) 0 0
\(681\) −11.3848 −0.436266
\(682\) −0.242641 −0.00929119
\(683\) 10.9289 0.418184 0.209092 0.977896i \(-0.432949\pi\)
0.209092 + 0.977896i \(0.432949\pi\)
\(684\) −1.41421 −0.0540738
\(685\) 0 0
\(686\) 0 0
\(687\) 6.68629 0.255098
\(688\) 4.41421 0.168290
\(689\) 4.02944 0.153509
\(690\) 0 0
\(691\) −29.1127 −1.10750 −0.553750 0.832683i \(-0.686804\pi\)
−0.553750 + 0.832683i \(0.686804\pi\)
\(692\) 5.75736 0.218862
\(693\) 0 0
\(694\) −29.7990 −1.13115
\(695\) 0 0
\(696\) −1.00000 −0.0379049
\(697\) −0.0294373 −0.00111502
\(698\) −13.3431 −0.505046
\(699\) 5.75736 0.217763
\(700\) 0 0
\(701\) 13.6863 0.516924 0.258462 0.966021i \(-0.416784\pi\)
0.258462 + 0.966021i \(0.416784\pi\)
\(702\) −3.00000 −0.113228
\(703\) 10.0000 0.377157
\(704\) 0.585786 0.0220777
\(705\) 0 0
\(706\) −3.51472 −0.132278
\(707\) 0 0
\(708\) −1.24264 −0.0467013
\(709\) 0.100505 0.00377455 0.00188727 0.999998i \(-0.499399\pi\)
0.00188727 + 0.999998i \(0.499399\pi\)
\(710\) 0 0
\(711\) −13.0711 −0.490203
\(712\) −4.00000 −0.149906
\(713\) 1.00000 0.0374503
\(714\) 0 0
\(715\) 0 0
\(716\) 9.07107 0.339002
\(717\) 11.3137 0.422518
\(718\) 19.8701 0.741544
\(719\) 33.0711 1.23334 0.616671 0.787221i \(-0.288481\pi\)
0.616671 + 0.787221i \(0.288481\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 17.0000 0.632674
\(723\) −6.00000 −0.223142
\(724\) −0.686292 −0.0255058
\(725\) 0 0
\(726\) 10.6569 0.395513
\(727\) 15.8701 0.588588 0.294294 0.955715i \(-0.404916\pi\)
0.294294 + 0.955715i \(0.404916\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.757359 0.0280119
\(732\) −5.82843 −0.215425
\(733\) −17.4853 −0.645834 −0.322917 0.946427i \(-0.604663\pi\)
−0.322917 + 0.946427i \(0.604663\pi\)
\(734\) −9.24264 −0.341152
\(735\) 0 0
\(736\) −2.41421 −0.0889891
\(737\) 4.28427 0.157813
\(738\) 0.171573 0.00631568
\(739\) 0.129942 0.00478001 0.00239000 0.999997i \(-0.499239\pi\)
0.00239000 + 0.999997i \(0.499239\pi\)
\(740\) 0 0
\(741\) −4.24264 −0.155857
\(742\) 0 0
\(743\) −5.44365 −0.199708 −0.0998541 0.995002i \(-0.531838\pi\)
−0.0998541 + 0.995002i \(0.531838\pi\)
\(744\) −0.414214 −0.0151858
\(745\) 0 0
\(746\) −13.6569 −0.500013
\(747\) 14.0711 0.514833
\(748\) 0.100505 0.00367483
\(749\) 0 0
\(750\) 0 0
\(751\) 20.9706 0.765227 0.382613 0.923909i \(-0.375024\pi\)
0.382613 + 0.923909i \(0.375024\pi\)
\(752\) 9.65685 0.352149
\(753\) −16.0711 −0.585662
\(754\) −3.00000 −0.109254
\(755\) 0 0
\(756\) 0 0
\(757\) −35.7574 −1.29962 −0.649812 0.760095i \(-0.725152\pi\)
−0.649812 + 0.760095i \(0.725152\pi\)
\(758\) −8.21320 −0.298317
\(759\) 1.41421 0.0513327
\(760\) 0 0
\(761\) 48.9706 1.77518 0.887591 0.460633i \(-0.152378\pi\)
0.887591 + 0.460633i \(0.152378\pi\)
\(762\) 8.82843 0.319820
\(763\) 0 0
\(764\) 1.92893 0.0697863
\(765\) 0 0
\(766\) 3.55635 0.128496
\(767\) −3.72792 −0.134607
\(768\) 1.00000 0.0360844
\(769\) 7.31371 0.263739 0.131870 0.991267i \(-0.457902\pi\)
0.131870 + 0.991267i \(0.457902\pi\)
\(770\) 0 0
\(771\) 24.6569 0.887995
\(772\) −1.17157 −0.0421658
\(773\) 3.31371 0.119186 0.0595929 0.998223i \(-0.481020\pi\)
0.0595929 + 0.998223i \(0.481020\pi\)
\(774\) −4.41421 −0.158666
\(775\) 0 0
\(776\) 11.6569 0.418457
\(777\) 0 0
\(778\) 21.6569 0.776436
\(779\) 0.242641 0.00869350
\(780\) 0 0
\(781\) 7.79899 0.279070
\(782\) −0.414214 −0.0148122
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 0 0
\(786\) 13.1716 0.469814
\(787\) 7.45584 0.265772 0.132886 0.991131i \(-0.457576\pi\)
0.132886 + 0.991131i \(0.457576\pi\)
\(788\) −5.14214 −0.183181
\(789\) 20.5563 0.731825
\(790\) 0 0
\(791\) 0 0
\(792\) −0.585786 −0.0208150
\(793\) −17.4853 −0.620921
\(794\) −7.68629 −0.272776
\(795\) 0 0
\(796\) 8.34315 0.295715
\(797\) 7.75736 0.274780 0.137390 0.990517i \(-0.456129\pi\)
0.137390 + 0.990517i \(0.456129\pi\)
\(798\) 0 0
\(799\) 1.65685 0.0586153
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) −31.8995 −1.12641
\(803\) 0.544156 0.0192028
\(804\) 7.31371 0.257935
\(805\) 0 0
\(806\) −1.24264 −0.0437702
\(807\) 31.5563 1.11084
\(808\) 5.65685 0.199007
\(809\) 10.6274 0.373640 0.186820 0.982394i \(-0.440182\pi\)
0.186820 + 0.982394i \(0.440182\pi\)
\(810\) 0 0
\(811\) 42.2426 1.48334 0.741670 0.670765i \(-0.234034\pi\)
0.741670 + 0.670765i \(0.234034\pi\)
\(812\) 0 0
\(813\) −28.4853 −0.999022
\(814\) 4.14214 0.145182
\(815\) 0 0
\(816\) 0.171573 0.00600625
\(817\) −6.24264 −0.218402
\(818\) −25.4142 −0.888587
\(819\) 0 0
\(820\) 0 0
\(821\) 18.8284 0.657117 0.328558 0.944484i \(-0.393437\pi\)
0.328558 + 0.944484i \(0.393437\pi\)
\(822\) −19.2132 −0.670137
\(823\) 31.4142 1.09503 0.547515 0.836796i \(-0.315574\pi\)
0.547515 + 0.836796i \(0.315574\pi\)
\(824\) −14.0711 −0.490189
\(825\) 0 0
\(826\) 0 0
\(827\) 7.89949 0.274692 0.137346 0.990523i \(-0.456143\pi\)
0.137346 + 0.990523i \(0.456143\pi\)
\(828\) 2.41421 0.0838997
\(829\) 38.7990 1.34754 0.673772 0.738939i \(-0.264673\pi\)
0.673772 + 0.738939i \(0.264673\pi\)
\(830\) 0 0
\(831\) 24.0416 0.833995
\(832\) 3.00000 0.104006
\(833\) 0 0
\(834\) −15.5563 −0.538672
\(835\) 0 0
\(836\) −0.828427 −0.0286518
\(837\) 0.414214 0.0143173
\(838\) 6.55635 0.226485
\(839\) 14.5858 0.503557 0.251779 0.967785i \(-0.418985\pi\)
0.251779 + 0.967785i \(0.418985\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 24.3431 0.838920
\(843\) 20.0000 0.688837
\(844\) 4.55635 0.156836
\(845\) 0 0
\(846\) −9.65685 −0.332009
\(847\) 0 0
\(848\) 1.34315 0.0461238
\(849\) 8.24264 0.282887
\(850\) 0 0
\(851\) −17.0711 −0.585189
\(852\) 13.3137 0.456120
\(853\) −8.31371 −0.284656 −0.142328 0.989820i \(-0.545459\pi\)
−0.142328 + 0.989820i \(0.545459\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 10.4853 0.358380
\(857\) −21.4558 −0.732918 −0.366459 0.930434i \(-0.619430\pi\)
−0.366459 + 0.930434i \(0.619430\pi\)
\(858\) −1.75736 −0.0599953
\(859\) −45.9411 −1.56749 −0.783745 0.621082i \(-0.786693\pi\)
−0.783745 + 0.621082i \(0.786693\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 15.0416 0.512320
\(863\) −8.97056 −0.305362 −0.152681 0.988276i \(-0.548791\pi\)
−0.152681 + 0.988276i \(0.548791\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 16.6274 0.565023
\(867\) −16.9706 −0.576351
\(868\) 0 0
\(869\) −7.65685 −0.259741
\(870\) 0 0
\(871\) 21.9411 0.743447
\(872\) −11.3137 −0.383131
\(873\) −11.6569 −0.394525
\(874\) 3.41421 0.115487
\(875\) 0 0
\(876\) 0.928932 0.0313857
\(877\) −37.6569 −1.27158 −0.635791 0.771861i \(-0.719326\pi\)
−0.635791 + 0.771861i \(0.719326\pi\)
\(878\) −4.41421 −0.148972
\(879\) 11.3137 0.381602
\(880\) 0 0
\(881\) −3.62742 −0.122211 −0.0611054 0.998131i \(-0.519463\pi\)
−0.0611054 + 0.998131i \(0.519463\pi\)
\(882\) 0 0
\(883\) 33.5269 1.12827 0.564135 0.825682i \(-0.309210\pi\)
0.564135 + 0.825682i \(0.309210\pi\)
\(884\) 0.514719 0.0173119
\(885\) 0 0
\(886\) 25.6569 0.861959
\(887\) −34.2426 −1.14976 −0.574878 0.818239i \(-0.694950\pi\)
−0.574878 + 0.818239i \(0.694950\pi\)
\(888\) 7.07107 0.237289
\(889\) 0 0
\(890\) 0 0
\(891\) 0.585786 0.0196246
\(892\) −6.55635 −0.219523
\(893\) −13.6569 −0.457009
\(894\) 6.65685 0.222639
\(895\) 0 0
\(896\) 0 0
\(897\) 7.24264 0.241825
\(898\) 10.9289 0.364703
\(899\) 0.414214 0.0138148
\(900\) 0 0
\(901\) 0.230447 0.00767732
\(902\) 0.100505 0.00334645
\(903\) 0 0
\(904\) −9.41421 −0.313112
\(905\) 0 0
\(906\) 11.8995 0.395334
\(907\) −18.6985 −0.620873 −0.310436 0.950594i \(-0.600475\pi\)
−0.310436 + 0.950594i \(0.600475\pi\)
\(908\) −11.3848 −0.377817
\(909\) −5.65685 −0.187626
\(910\) 0 0
\(911\) 7.92893 0.262697 0.131349 0.991336i \(-0.458069\pi\)
0.131349 + 0.991336i \(0.458069\pi\)
\(912\) −1.41421 −0.0468293
\(913\) 8.24264 0.272792
\(914\) −1.82843 −0.0604790
\(915\) 0 0
\(916\) 6.68629 0.220921
\(917\) 0 0
\(918\) −0.171573 −0.00566275
\(919\) 35.5563 1.17290 0.586448 0.809987i \(-0.300526\pi\)
0.586448 + 0.809987i \(0.300526\pi\)
\(920\) 0 0
\(921\) 11.4142 0.376111
\(922\) −20.0000 −0.658665
\(923\) 39.9411 1.31468
\(924\) 0 0
\(925\) 0 0
\(926\) −10.1005 −0.331923
\(927\) 14.0711 0.462155
\(928\) −1.00000 −0.0328266
\(929\) 2.65685 0.0871686 0.0435843 0.999050i \(-0.486122\pi\)
0.0435843 + 0.999050i \(0.486122\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 5.75736 0.188589
\(933\) −8.58579 −0.281086
\(934\) −7.24264 −0.236986
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) −8.24264 −0.269275 −0.134638 0.990895i \(-0.542987\pi\)
−0.134638 + 0.990895i \(0.542987\pi\)
\(938\) 0 0
\(939\) 32.0416 1.04564
\(940\) 0 0
\(941\) 28.6863 0.935146 0.467573 0.883954i \(-0.345128\pi\)
0.467573 + 0.883954i \(0.345128\pi\)
\(942\) −19.7990 −0.645086
\(943\) −0.414214 −0.0134886
\(944\) −1.24264 −0.0404445
\(945\) 0 0
\(946\) −2.58579 −0.0840712
\(947\) 17.2132 0.559354 0.279677 0.960094i \(-0.409773\pi\)
0.279677 + 0.960094i \(0.409773\pi\)
\(948\) −13.0711 −0.424529
\(949\) 2.78680 0.0904632
\(950\) 0 0
\(951\) −13.9706 −0.453027
\(952\) 0 0
\(953\) 47.5980 1.54185 0.770925 0.636926i \(-0.219794\pi\)
0.770925 + 0.636926i \(0.219794\pi\)
\(954\) −1.34315 −0.0434859
\(955\) 0 0
\(956\) 11.3137 0.365911
\(957\) 0.585786 0.0189358
\(958\) −11.7574 −0.379863
\(959\) 0 0
\(960\) 0 0
\(961\) −30.8284 −0.994465
\(962\) 21.2132 0.683941
\(963\) −10.4853 −0.337883
\(964\) −6.00000 −0.193247
\(965\) 0 0
\(966\) 0 0
\(967\) 4.04163 0.129970 0.0649850 0.997886i \(-0.479300\pi\)
0.0649850 + 0.997886i \(0.479300\pi\)
\(968\) 10.6569 0.342524
\(969\) −0.242641 −0.00779474
\(970\) 0 0
\(971\) −11.0294 −0.353951 −0.176976 0.984215i \(-0.556631\pi\)
−0.176976 + 0.984215i \(0.556631\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −22.4853 −0.720475
\(975\) 0 0
\(976\) −5.82843 −0.186563
\(977\) −39.5563 −1.26552 −0.632760 0.774348i \(-0.718078\pi\)
−0.632760 + 0.774348i \(0.718078\pi\)
\(978\) 3.24264 0.103688
\(979\) 2.34315 0.0748873
\(980\) 0 0
\(981\) 11.3137 0.361219
\(982\) −28.1421 −0.898052
\(983\) 51.3137 1.63665 0.818327 0.574754i \(-0.194902\pi\)
0.818327 + 0.574754i \(0.194902\pi\)
\(984\) 0.171573 0.00546954
\(985\) 0 0
\(986\) −0.171573 −0.00546399
\(987\) 0 0
\(988\) −4.24264 −0.134976
\(989\) 10.6569 0.338868
\(990\) 0 0
\(991\) −29.6569 −0.942081 −0.471041 0.882112i \(-0.656121\pi\)
−0.471041 + 0.882112i \(0.656121\pi\)
\(992\) −0.414214 −0.0131513
\(993\) −9.92893 −0.315085
\(994\) 0 0
\(995\) 0 0
\(996\) 14.0711 0.445859
\(997\) −2.68629 −0.0850757 −0.0425379 0.999095i \(-0.513544\pi\)
−0.0425379 + 0.999095i \(0.513544\pi\)
\(998\) 11.9289 0.377604
\(999\) −7.07107 −0.223719
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 7350.2.a.dg.1.1 yes 2
5.4 even 2 7350.2.a.dj.1.1 yes 2
7.6 odd 2 7350.2.a.dc.1.1 2
35.34 odd 2 7350.2.a.dm.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7350.2.a.dc.1.1 2 7.6 odd 2
7350.2.a.dg.1.1 yes 2 1.1 even 1 trivial
7350.2.a.dj.1.1 yes 2 5.4 even 2
7350.2.a.dm.1.1 yes 2 35.34 odd 2