Properties

Label 9702.2.a.cm.1.2
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.24264 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.24264 q^{5} -1.00000 q^{8} -4.24264 q^{10} -1.00000 q^{11} -5.65685 q^{13} +1.00000 q^{16} -7.07107 q^{17} -1.41421 q^{19} +4.24264 q^{20} +1.00000 q^{22} +8.00000 q^{23} +13.0000 q^{25} +5.65685 q^{26} +8.00000 q^{29} -4.24264 q^{31} -1.00000 q^{32} +7.07107 q^{34} +2.00000 q^{37} +1.41421 q^{38} -4.24264 q^{40} +1.41421 q^{41} +8.00000 q^{43} -1.00000 q^{44} -8.00000 q^{46} +9.89949 q^{47} -13.0000 q^{50} -5.65685 q^{52} +2.00000 q^{53} -4.24264 q^{55} -8.00000 q^{58} -8.48528 q^{59} +4.24264 q^{62} +1.00000 q^{64} -24.0000 q^{65} -2.00000 q^{67} -7.07107 q^{68} -12.0000 q^{71} -7.07107 q^{73} -2.00000 q^{74} -1.41421 q^{76} +14.0000 q^{79} +4.24264 q^{80} -1.41421 q^{82} +12.7279 q^{83} -30.0000 q^{85} -8.00000 q^{86} +1.00000 q^{88} +2.82843 q^{89} +8.00000 q^{92} -9.89949 q^{94} -6.00000 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{11} + 2 q^{16} + 2 q^{22} + 16 q^{23} + 26 q^{25} + 16 q^{29} - 2 q^{32} + 4 q^{37} + 16 q^{43} - 2 q^{44} - 16 q^{46} - 26 q^{50} + 4 q^{53} - 16 q^{58} + 2 q^{64} - 48 q^{65} - 4 q^{67} - 24 q^{71} - 4 q^{74} + 28 q^{79} - 60 q^{85} - 16 q^{86} + 2 q^{88} + 16 q^{92} - 12 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 4.24264 1.89737 0.948683 0.316228i \(-0.102416\pi\)
0.948683 + 0.316228i \(0.102416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −4.24264 −1.34164
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.07107 −1.71499 −0.857493 0.514496i \(-0.827979\pi\)
−0.857493 + 0.514496i \(0.827979\pi\)
\(18\) 0 0
\(19\) −1.41421 −0.324443 −0.162221 0.986754i \(-0.551866\pi\)
−0.162221 + 0.986754i \(0.551866\pi\)
\(20\) 4.24264 0.948683
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 13.0000 2.60000
\(26\) 5.65685 1.10940
\(27\) 0 0
\(28\) 0 0
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) −4.24264 −0.762001 −0.381000 0.924575i \(-0.624420\pi\)
−0.381000 + 0.924575i \(0.624420\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.07107 1.21268
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 1.41421 0.229416
\(39\) 0 0
\(40\) −4.24264 −0.670820
\(41\) 1.41421 0.220863 0.110432 0.993884i \(-0.464777\pi\)
0.110432 + 0.993884i \(0.464777\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 9.89949 1.44399 0.721995 0.691898i \(-0.243225\pi\)
0.721995 + 0.691898i \(0.243225\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −13.0000 −1.83848
\(51\) 0 0
\(52\) −5.65685 −0.784465
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −4.24264 −0.572078
\(56\) 0 0
\(57\) 0 0
\(58\) −8.00000 −1.05045
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 4.24264 0.538816
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −24.0000 −2.97683
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −7.07107 −0.857493
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −7.07107 −0.827606 −0.413803 0.910366i \(-0.635800\pi\)
−0.413803 + 0.910366i \(0.635800\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −1.41421 −0.162221
\(77\) 0 0
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 4.24264 0.474342
\(81\) 0 0
\(82\) −1.41421 −0.156174
\(83\) 12.7279 1.39707 0.698535 0.715575i \(-0.253835\pi\)
0.698535 + 0.715575i \(0.253835\pi\)
\(84\) 0 0
\(85\) −30.0000 −3.25396
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 2.82843 0.299813 0.149906 0.988700i \(-0.452103\pi\)
0.149906 + 0.988700i \(0.452103\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) −9.89949 −1.02105
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 13.0000 1.30000
\(101\) −5.65685 −0.562878 −0.281439 0.959579i \(-0.590812\pi\)
−0.281439 + 0.959579i \(0.590812\pi\)
\(102\) 0 0
\(103\) 7.07107 0.696733 0.348367 0.937358i \(-0.386736\pi\)
0.348367 + 0.937358i \(0.386736\pi\)
\(104\) 5.65685 0.554700
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 4.24264 0.404520
\(111\) 0 0
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 33.9411 3.16503
\(116\) 8.00000 0.742781
\(117\) 0 0
\(118\) 8.48528 0.781133
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) −4.24264 −0.381000
\(125\) 33.9411 3.03579
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 24.0000 2.10494
\(131\) 9.89949 0.864923 0.432461 0.901652i \(-0.357645\pi\)
0.432461 + 0.901652i \(0.357645\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 7.07107 0.606339
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) 0 0
\(139\) −12.7279 −1.07957 −0.539784 0.841803i \(-0.681494\pi\)
−0.539784 + 0.841803i \(0.681494\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 5.65685 0.473050
\(144\) 0 0
\(145\) 33.9411 2.81866
\(146\) 7.07107 0.585206
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 1.41421 0.114708
\(153\) 0 0
\(154\) 0 0
\(155\) −18.0000 −1.44579
\(156\) 0 0
\(157\) 7.07107 0.564333 0.282166 0.959366i \(-0.408947\pi\)
0.282166 + 0.959366i \(0.408947\pi\)
\(158\) −14.0000 −1.11378
\(159\) 0 0
\(160\) −4.24264 −0.335410
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 1.41421 0.110432
\(165\) 0 0
\(166\) −12.7279 −0.987878
\(167\) −14.1421 −1.09435 −0.547176 0.837018i \(-0.684297\pi\)
−0.547176 + 0.837018i \(0.684297\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 30.0000 2.30089
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −19.7990 −1.50529 −0.752645 0.658427i \(-0.771222\pi\)
−0.752645 + 0.658427i \(0.771222\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −2.82843 −0.212000
\(179\) 14.0000 1.04641 0.523205 0.852207i \(-0.324736\pi\)
0.523205 + 0.852207i \(0.324736\pi\)
\(180\) 0 0
\(181\) −1.41421 −0.105118 −0.0525588 0.998618i \(-0.516738\pi\)
−0.0525588 + 0.998618i \(0.516738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −8.00000 −0.589768
\(185\) 8.48528 0.623850
\(186\) 0 0
\(187\) 7.07107 0.517088
\(188\) 9.89949 0.721995
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) 15.5563 1.10276 0.551380 0.834254i \(-0.314101\pi\)
0.551380 + 0.834254i \(0.314101\pi\)
\(200\) −13.0000 −0.919239
\(201\) 0 0
\(202\) 5.65685 0.398015
\(203\) 0 0
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) −7.07107 −0.492665
\(207\) 0 0
\(208\) −5.65685 −0.392232
\(209\) 1.41421 0.0978232
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 33.9411 2.31477
\(216\) 0 0
\(217\) 0 0
\(218\) −8.00000 −0.541828
\(219\) 0 0
\(220\) −4.24264 −0.286039
\(221\) 40.0000 2.69069
\(222\) 0 0
\(223\) 24.0416 1.60995 0.804973 0.593311i \(-0.202180\pi\)
0.804973 + 0.593311i \(0.202180\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) 1.41421 0.0938647 0.0469323 0.998898i \(-0.485055\pi\)
0.0469323 + 0.998898i \(0.485055\pi\)
\(228\) 0 0
\(229\) 18.3848 1.21490 0.607450 0.794358i \(-0.292192\pi\)
0.607450 + 0.794358i \(0.292192\pi\)
\(230\) −33.9411 −2.23801
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) 42.0000 2.73978
\(236\) −8.48528 −0.552345
\(237\) 0 0
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 24.0416 1.54866 0.774329 0.632783i \(-0.218088\pi\)
0.774329 + 0.632783i \(0.218088\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 4.24264 0.269408
\(249\) 0 0
\(250\) −33.9411 −2.14663
\(251\) −19.7990 −1.24970 −0.624851 0.780744i \(-0.714840\pi\)
−0.624851 + 0.780744i \(0.714840\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) −6.00000 −0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.82843 0.176432 0.0882162 0.996101i \(-0.471883\pi\)
0.0882162 + 0.996101i \(0.471883\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −24.0000 −1.48842
\(261\) 0 0
\(262\) −9.89949 −0.611593
\(263\) −22.0000 −1.35658 −0.678289 0.734795i \(-0.737278\pi\)
−0.678289 + 0.734795i \(0.737278\pi\)
\(264\) 0 0
\(265\) 8.48528 0.521247
\(266\) 0 0
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) −15.5563 −0.948487 −0.474244 0.880394i \(-0.657278\pi\)
−0.474244 + 0.880394i \(0.657278\pi\)
\(270\) 0 0
\(271\) −25.4558 −1.54633 −0.773166 0.634203i \(-0.781328\pi\)
−0.773166 + 0.634203i \(0.781328\pi\)
\(272\) −7.07107 −0.428746
\(273\) 0 0
\(274\) −22.0000 −1.32907
\(275\) −13.0000 −0.783929
\(276\) 0 0
\(277\) 4.00000 0.240337 0.120168 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) 12.7279 0.763370
\(279\) 0 0
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) 0 0
\(283\) 9.89949 0.588464 0.294232 0.955734i \(-0.404936\pi\)
0.294232 + 0.955734i \(0.404936\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) −5.65685 −0.334497
\(287\) 0 0
\(288\) 0 0
\(289\) 33.0000 1.94118
\(290\) −33.9411 −1.99309
\(291\) 0 0
\(292\) −7.07107 −0.413803
\(293\) −14.1421 −0.826192 −0.413096 0.910687i \(-0.635553\pi\)
−0.413096 + 0.910687i \(0.635553\pi\)
\(294\) 0 0
\(295\) −36.0000 −2.09600
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) −45.2548 −2.61715
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) −1.41421 −0.0811107
\(305\) 0 0
\(306\) 0 0
\(307\) 1.41421 0.0807134 0.0403567 0.999185i \(-0.487151\pi\)
0.0403567 + 0.999185i \(0.487151\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 18.0000 1.02233
\(311\) 26.8701 1.52366 0.761831 0.647776i \(-0.224301\pi\)
0.761831 + 0.647776i \(0.224301\pi\)
\(312\) 0 0
\(313\) 19.7990 1.11911 0.559553 0.828795i \(-0.310973\pi\)
0.559553 + 0.828795i \(0.310973\pi\)
\(314\) −7.07107 −0.399043
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 4.24264 0.237171
\(321\) 0 0
\(322\) 0 0
\(323\) 10.0000 0.556415
\(324\) 0 0
\(325\) −73.5391 −4.07922
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −1.41421 −0.0780869
\(329\) 0 0
\(330\) 0 0
\(331\) 26.0000 1.42909 0.714545 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) 12.7279 0.698535
\(333\) 0 0
\(334\) 14.1421 0.773823
\(335\) −8.48528 −0.463600
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −19.0000 −1.03346
\(339\) 0 0
\(340\) −30.0000 −1.62698
\(341\) 4.24264 0.229752
\(342\) 0 0
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 19.7990 1.06440
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 0 0
\(349\) 25.4558 1.36262 0.681310 0.731995i \(-0.261411\pi\)
0.681310 + 0.731995i \(0.261411\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 2.82843 0.150542 0.0752710 0.997163i \(-0.476018\pi\)
0.0752710 + 0.997163i \(0.476018\pi\)
\(354\) 0 0
\(355\) −50.9117 −2.70211
\(356\) 2.82843 0.149906
\(357\) 0 0
\(358\) −14.0000 −0.739923
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 1.41421 0.0743294
\(363\) 0 0
\(364\) 0 0
\(365\) −30.0000 −1.57027
\(366\) 0 0
\(367\) 21.2132 1.10732 0.553660 0.832743i \(-0.313231\pi\)
0.553660 + 0.832743i \(0.313231\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) −8.48528 −0.441129
\(371\) 0 0
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) −7.07107 −0.365636
\(375\) 0 0
\(376\) −9.89949 −0.510527
\(377\) −45.2548 −2.33074
\(378\) 0 0
\(379\) −36.0000 −1.84920 −0.924598 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) −6.00000 −0.307794
\(381\) 0 0
\(382\) 20.0000 1.02329
\(383\) −18.3848 −0.939418 −0.469709 0.882821i \(-0.655641\pi\)
−0.469709 + 0.882821i \(0.655641\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) 0 0
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −56.5685 −2.86079
\(392\) 0 0
\(393\) 0 0
\(394\) −8.00000 −0.403034
\(395\) 59.3970 2.98859
\(396\) 0 0
\(397\) 26.8701 1.34857 0.674285 0.738471i \(-0.264452\pi\)
0.674285 + 0.738471i \(0.264452\pi\)
\(398\) −15.5563 −0.779769
\(399\) 0 0
\(400\) 13.0000 0.650000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 24.0000 1.19553
\(404\) −5.65685 −0.281439
\(405\) 0 0
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) 4.24264 0.209785 0.104893 0.994484i \(-0.466550\pi\)
0.104893 + 0.994484i \(0.466550\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 7.07107 0.348367
\(413\) 0 0
\(414\) 0 0
\(415\) 54.0000 2.65076
\(416\) 5.65685 0.277350
\(417\) 0 0
\(418\) −1.41421 −0.0691714
\(419\) −16.9706 −0.829066 −0.414533 0.910034i \(-0.636055\pi\)
−0.414533 + 0.910034i \(0.636055\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) −91.9239 −4.45896
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −33.9411 −1.63679
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) 33.9411 1.63111 0.815553 0.578682i \(-0.196433\pi\)
0.815553 + 0.578682i \(0.196433\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) −11.3137 −0.541208
\(438\) 0 0
\(439\) −8.48528 −0.404980 −0.202490 0.979284i \(-0.564903\pi\)
−0.202490 + 0.979284i \(0.564903\pi\)
\(440\) 4.24264 0.202260
\(441\) 0 0
\(442\) −40.0000 −1.90261
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) −24.0416 −1.13840
\(447\) 0 0
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −1.41421 −0.0665927
\(452\) −10.0000 −0.470360
\(453\) 0 0
\(454\) −1.41421 −0.0663723
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −18.3848 −0.859064
\(459\) 0 0
\(460\) 33.9411 1.58251
\(461\) −5.65685 −0.263466 −0.131733 0.991285i \(-0.542054\pi\)
−0.131733 + 0.991285i \(0.542054\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 8.00000 0.371391
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) 16.9706 0.785304 0.392652 0.919687i \(-0.371558\pi\)
0.392652 + 0.919687i \(0.371558\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −42.0000 −1.93732
\(471\) 0 0
\(472\) 8.48528 0.390567
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −18.3848 −0.843551
\(476\) 0 0
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) −33.9411 −1.55081 −0.775405 0.631464i \(-0.782454\pi\)
−0.775405 + 0.631464i \(0.782454\pi\)
\(480\) 0 0
\(481\) −11.3137 −0.515861
\(482\) −24.0416 −1.09507
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) −56.5685 −2.54772
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) −4.24264 −0.190500
\(497\) 0 0
\(498\) 0 0
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 33.9411 1.51789
\(501\) 0 0
\(502\) 19.7990 0.883672
\(503\) 19.7990 0.882793 0.441397 0.897312i \(-0.354483\pi\)
0.441397 + 0.897312i \(0.354483\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 8.00000 0.355643
\(507\) 0 0
\(508\) 6.00000 0.266207
\(509\) 12.7279 0.564155 0.282078 0.959392i \(-0.408976\pi\)
0.282078 + 0.959392i \(0.408976\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −2.82843 −0.124757
\(515\) 30.0000 1.32196
\(516\) 0 0
\(517\) −9.89949 −0.435379
\(518\) 0 0
\(519\) 0 0
\(520\) 24.0000 1.05247
\(521\) 19.7990 0.867409 0.433705 0.901055i \(-0.357206\pi\)
0.433705 + 0.901055i \(0.357206\pi\)
\(522\) 0 0
\(523\) 21.2132 0.927589 0.463794 0.885943i \(-0.346488\pi\)
0.463794 + 0.885943i \(0.346488\pi\)
\(524\) 9.89949 0.432461
\(525\) 0 0
\(526\) 22.0000 0.959246
\(527\) 30.0000 1.30682
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −8.48528 −0.368577
\(531\) 0 0
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) −50.9117 −2.20110
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) 15.5563 0.670682
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 25.4558 1.09342
\(543\) 0 0
\(544\) 7.07107 0.303170
\(545\) 33.9411 1.45388
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 22.0000 0.939793
\(549\) 0 0
\(550\) 13.0000 0.554322
\(551\) −11.3137 −0.481980
\(552\) 0 0
\(553\) 0 0
\(554\) −4.00000 −0.169944
\(555\) 0 0
\(556\) −12.7279 −0.539784
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −45.2548 −1.91408
\(560\) 0 0
\(561\) 0 0
\(562\) −14.0000 −0.590554
\(563\) 1.41421 0.0596020 0.0298010 0.999556i \(-0.490513\pi\)
0.0298010 + 0.999556i \(0.490513\pi\)
\(564\) 0 0
\(565\) −42.4264 −1.78489
\(566\) −9.89949 −0.416107
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 5.65685 0.236525
\(573\) 0 0
\(574\) 0 0
\(575\) 104.000 4.33710
\(576\) 0 0
\(577\) 19.7990 0.824243 0.412121 0.911129i \(-0.364788\pi\)
0.412121 + 0.911129i \(0.364788\pi\)
\(578\) −33.0000 −1.37262
\(579\) 0 0
\(580\) 33.9411 1.40933
\(581\) 0 0
\(582\) 0 0
\(583\) −2.00000 −0.0828315
\(584\) 7.07107 0.292603
\(585\) 0 0
\(586\) 14.1421 0.584206
\(587\) 14.1421 0.583708 0.291854 0.956463i \(-0.405728\pi\)
0.291854 + 0.956463i \(0.405728\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) 36.0000 1.48210
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) 12.7279 0.522673 0.261337 0.965248i \(-0.415837\pi\)
0.261337 + 0.965248i \(0.415837\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) 45.2548 1.85061
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −12.7279 −0.519183 −0.259591 0.965719i \(-0.583588\pi\)
−0.259591 + 0.965719i \(0.583588\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) 4.24264 0.172488
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 1.41421 0.0573539
\(609\) 0 0
\(610\) 0 0
\(611\) −56.0000 −2.26552
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −1.41421 −0.0570730
\(615\) 0 0
\(616\) 0 0
\(617\) 20.0000 0.805170 0.402585 0.915383i \(-0.368112\pi\)
0.402585 + 0.915383i \(0.368112\pi\)
\(618\) 0 0
\(619\) 45.2548 1.81895 0.909473 0.415764i \(-0.136486\pi\)
0.909473 + 0.415764i \(0.136486\pi\)
\(620\) −18.0000 −0.722897
\(621\) 0 0
\(622\) −26.8701 −1.07739
\(623\) 0 0
\(624\) 0 0
\(625\) 79.0000 3.16000
\(626\) −19.7990 −0.791327
\(627\) 0 0
\(628\) 7.07107 0.282166
\(629\) −14.1421 −0.563884
\(630\) 0 0
\(631\) −24.0000 −0.955425 −0.477712 0.878516i \(-0.658534\pi\)
−0.477712 + 0.878516i \(0.658534\pi\)
\(632\) −14.0000 −0.556890
\(633\) 0 0
\(634\) −22.0000 −0.873732
\(635\) 25.4558 1.01018
\(636\) 0 0
\(637\) 0 0
\(638\) 8.00000 0.316723
\(639\) 0 0
\(640\) −4.24264 −0.167705
\(641\) −44.0000 −1.73790 −0.868948 0.494904i \(-0.835203\pi\)
−0.868948 + 0.494904i \(0.835203\pi\)
\(642\) 0 0
\(643\) −8.48528 −0.334627 −0.167313 0.985904i \(-0.553509\pi\)
−0.167313 + 0.985904i \(0.553509\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −10.0000 −0.393445
\(647\) 26.8701 1.05637 0.528185 0.849129i \(-0.322873\pi\)
0.528185 + 0.849129i \(0.322873\pi\)
\(648\) 0 0
\(649\) 8.48528 0.333076
\(650\) 73.5391 2.88444
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 0 0
\(655\) 42.0000 1.64108
\(656\) 1.41421 0.0552158
\(657\) 0 0
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) −4.24264 −0.165020 −0.0825098 0.996590i \(-0.526294\pi\)
−0.0825098 + 0.996590i \(0.526294\pi\)
\(662\) −26.0000 −1.01052
\(663\) 0 0
\(664\) −12.7279 −0.493939
\(665\) 0 0
\(666\) 0 0
\(667\) 64.0000 2.47809
\(668\) −14.1421 −0.547176
\(669\) 0 0
\(670\) 8.48528 0.327815
\(671\) 0 0
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 19.0000 0.730769
\(677\) 19.7990 0.760937 0.380468 0.924794i \(-0.375763\pi\)
0.380468 + 0.924794i \(0.375763\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 30.0000 1.15045
\(681\) 0 0
\(682\) −4.24264 −0.162459
\(683\) −38.0000 −1.45403 −0.727015 0.686622i \(-0.759093\pi\)
−0.727015 + 0.686622i \(0.759093\pi\)
\(684\) 0 0
\(685\) 93.3381 3.56627
\(686\) 0 0
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −11.3137 −0.431018
\(690\) 0 0
\(691\) −8.48528 −0.322795 −0.161398 0.986889i \(-0.551600\pi\)
−0.161398 + 0.986889i \(0.551600\pi\)
\(692\) −19.7990 −0.752645
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) −54.0000 −2.04834
\(696\) 0 0
\(697\) −10.0000 −0.378777
\(698\) −25.4558 −0.963518
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −2.82843 −0.106676
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −2.82843 −0.106449
\(707\) 0 0
\(708\) 0 0
\(709\) 2.00000 0.0751116 0.0375558 0.999295i \(-0.488043\pi\)
0.0375558 + 0.999295i \(0.488043\pi\)
\(710\) 50.9117 1.91068
\(711\) 0 0
\(712\) −2.82843 −0.106000
\(713\) −33.9411 −1.27111
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 14.0000 0.523205
\(717\) 0 0
\(718\) 10.0000 0.373197
\(719\) −24.0416 −0.896602 −0.448301 0.893883i \(-0.647971\pi\)
−0.448301 + 0.893883i \(0.647971\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) −1.41421 −0.0525588
\(725\) 104.000 3.86246
\(726\) 0 0
\(727\) −49.4975 −1.83576 −0.917880 0.396858i \(-0.870100\pi\)
−0.917880 + 0.396858i \(0.870100\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 30.0000 1.11035
\(731\) −56.5685 −2.09226
\(732\) 0 0
\(733\) 28.2843 1.04470 0.522352 0.852730i \(-0.325055\pi\)
0.522352 + 0.852730i \(0.325055\pi\)
\(734\) −21.2132 −0.782994
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) 2.00000 0.0736709
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 8.48528 0.311925
\(741\) 0 0
\(742\) 0 0
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 0 0
\(745\) 76.3675 2.79789
\(746\) −26.0000 −0.951928
\(747\) 0 0
\(748\) 7.07107 0.258544
\(749\) 0 0
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 9.89949 0.360997
\(753\) 0 0
\(754\) 45.2548 1.64808
\(755\) −33.9411 −1.23524
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 36.0000 1.30758
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) 21.2132 0.768978 0.384489 0.923130i \(-0.374378\pi\)
0.384489 + 0.923130i \(0.374378\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) 18.3848 0.664269
\(767\) 48.0000 1.73318
\(768\) 0 0
\(769\) 7.07107 0.254989 0.127495 0.991839i \(-0.459306\pi\)
0.127495 + 0.991839i \(0.459306\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 22.0000 0.791797
\(773\) −7.07107 −0.254329 −0.127164 0.991882i \(-0.540588\pi\)
−0.127164 + 0.991882i \(0.540588\pi\)
\(774\) 0 0
\(775\) −55.1543 −1.98120
\(776\) 0 0
\(777\) 0 0
\(778\) −2.00000 −0.0717035
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 56.5685 2.02289
\(783\) 0 0
\(784\) 0 0
\(785\) 30.0000 1.07075
\(786\) 0 0
\(787\) −41.0122 −1.46193 −0.730963 0.682417i \(-0.760929\pi\)
−0.730963 + 0.682417i \(0.760929\pi\)
\(788\) 8.00000 0.284988
\(789\) 0 0
\(790\) −59.3970 −2.11325
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −26.8701 −0.953583
\(795\) 0 0
\(796\) 15.5563 0.551380
\(797\) 32.5269 1.15216 0.576081 0.817392i \(-0.304581\pi\)
0.576081 + 0.817392i \(0.304581\pi\)
\(798\) 0 0
\(799\) −70.0000 −2.47642
\(800\) −13.0000 −0.459619
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 7.07107 0.249533
\(804\) 0 0
\(805\) 0 0
\(806\) −24.0000 −0.845364
\(807\) 0 0
\(808\) 5.65685 0.199007
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 9.89949 0.347618 0.173809 0.984779i \(-0.444392\pi\)
0.173809 + 0.984779i \(0.444392\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 2.00000 0.0701000
\(815\) −16.9706 −0.594453
\(816\) 0 0
\(817\) −11.3137 −0.395817
\(818\) −4.24264 −0.148340
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −7.07107 −0.246332
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 9.89949 0.343824 0.171912 0.985112i \(-0.445006\pi\)
0.171912 + 0.985112i \(0.445006\pi\)
\(830\) −54.0000 −1.87437
\(831\) 0 0
\(832\) −5.65685 −0.196116
\(833\) 0 0
\(834\) 0 0
\(835\) −60.0000 −2.07639
\(836\) 1.41421 0.0489116
\(837\) 0 0
\(838\) 16.9706 0.586238
\(839\) −18.3848 −0.634713 −0.317356 0.948306i \(-0.602795\pi\)
−0.317356 + 0.948306i \(0.602795\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 10.0000 0.344623
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 80.6102 2.77307
\(846\) 0 0
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) 91.9239 3.15296
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −4.24264 −0.144926 −0.0724629 0.997371i \(-0.523086\pi\)
−0.0724629 + 0.997371i \(0.523086\pi\)
\(858\) 0 0
\(859\) −25.4558 −0.868542 −0.434271 0.900782i \(-0.642994\pi\)
−0.434271 + 0.900782i \(0.642994\pi\)
\(860\) 33.9411 1.15738
\(861\) 0 0
\(862\) −30.0000 −1.02180
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 0 0
\(865\) −84.0000 −2.85609
\(866\) −33.9411 −1.15337
\(867\) 0 0
\(868\) 0 0
\(869\) −14.0000 −0.474917
\(870\) 0 0
\(871\) 11.3137 0.383350
\(872\) −8.00000 −0.270914
\(873\) 0 0
\(874\) 11.3137 0.382692
\(875\) 0 0
\(876\) 0 0
\(877\) −28.0000 −0.945493 −0.472746 0.881199i \(-0.656737\pi\)
−0.472746 + 0.881199i \(0.656737\pi\)
\(878\) 8.48528 0.286364
\(879\) 0 0
\(880\) −4.24264 −0.143019
\(881\) −45.2548 −1.52467 −0.762337 0.647180i \(-0.775948\pi\)
−0.762337 + 0.647180i \(0.775948\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 40.0000 1.34535
\(885\) 0 0
\(886\) −6.00000 −0.201574
\(887\) 5.65685 0.189939 0.0949693 0.995480i \(-0.469725\pi\)
0.0949693 + 0.995480i \(0.469725\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −12.0000 −0.402241
\(891\) 0 0
\(892\) 24.0416 0.804973
\(893\) −14.0000 −0.468492
\(894\) 0 0
\(895\) 59.3970 1.98542
\(896\) 0 0
\(897\) 0 0
\(898\) 2.00000 0.0667409
\(899\) −33.9411 −1.13200
\(900\) 0 0
\(901\) −14.1421 −0.471143
\(902\) 1.41421 0.0470882
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) −26.0000 −0.863316 −0.431658 0.902037i \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) 1.41421 0.0469323
\(909\) 0 0
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) −12.7279 −0.421233
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 18.3848 0.607450
\(917\) 0 0
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) −33.9411 −1.11901
\(921\) 0 0
\(922\) 5.65685 0.186299
\(923\) 67.8823 2.23437
\(924\) 0 0
\(925\) 26.0000 0.854875
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) −8.00000 −0.262613
\(929\) 2.82843 0.0927977 0.0463988 0.998923i \(-0.485225\pi\)
0.0463988 + 0.998923i \(0.485225\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −14.0000 −0.458585
\(933\) 0 0
\(934\) −16.9706 −0.555294
\(935\) 30.0000 0.981105
\(936\) 0 0
\(937\) 38.1838 1.24741 0.623705 0.781660i \(-0.285627\pi\)
0.623705 + 0.781660i \(0.285627\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 42.0000 1.36989
\(941\) −19.7990 −0.645429 −0.322714 0.946496i \(-0.604595\pi\)
−0.322714 + 0.946496i \(0.604595\pi\)
\(942\) 0 0
\(943\) 11.3137 0.368425
\(944\) −8.48528 −0.276172
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) 40.0000 1.29845
\(950\) 18.3848 0.596481
\(951\) 0 0
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) −84.8528 −2.74577
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 33.9411 1.09659
\(959\) 0 0
\(960\) 0 0
\(961\) −13.0000 −0.419355
\(962\) 11.3137 0.364769
\(963\) 0 0
\(964\) 24.0416 0.774329
\(965\) 93.3381 3.00466
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 0 0
\(971\) 11.3137 0.363074 0.181537 0.983384i \(-0.441893\pi\)
0.181537 + 0.983384i \(0.441893\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) −2.82843 −0.0903969
\(980\) 0 0
\(981\) 0 0
\(982\) −20.0000 −0.638226
\(983\) −57.9828 −1.84936 −0.924681 0.380742i \(-0.875669\pi\)
−0.924681 + 0.380742i \(0.875669\pi\)
\(984\) 0 0
\(985\) 33.9411 1.08145
\(986\) 56.5685 1.80151
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 64.0000 2.03508
\(990\) 0 0
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) 4.24264 0.134704
\(993\) 0 0
\(994\) 0 0
\(995\) 66.0000 2.09234
\(996\) 0 0
\(997\) −59.3970 −1.88112 −0.940560 0.339626i \(-0.889699\pi\)
−0.940560 + 0.339626i \(0.889699\pi\)
\(998\) 6.00000 0.189927
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9702.2.a.cm.1.2 yes 2
3.2 odd 2 9702.2.a.do.1.1 yes 2
7.6 odd 2 inner 9702.2.a.cm.1.1 2
21.20 even 2 9702.2.a.do.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9702.2.a.cm.1.1 2 7.6 odd 2 inner
9702.2.a.cm.1.2 yes 2 1.1 even 1 trivial
9702.2.a.do.1.1 yes 2 3.2 odd 2
9702.2.a.do.1.2 yes 2 21.20 even 2