Properties

Label 9702.2.a.ci.1.2
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3234)
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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{8} +2.00000 q^{10} +1.00000 q^{11} +5.41421 q^{13} +1.00000 q^{16} -2.00000 q^{17} -2.24264 q^{19} -2.00000 q^{20} -1.00000 q^{22} +4.82843 q^{23} -1.00000 q^{25} -5.41421 q^{26} -9.65685 q^{29} +6.24264 q^{31} -1.00000 q^{32} +2.00000 q^{34} +0.828427 q^{37} +2.24264 q^{38} +2.00000 q^{40} -11.6569 q^{41} -4.82843 q^{43} +1.00000 q^{44} -4.82843 q^{46} +7.89949 q^{47} +1.00000 q^{50} +5.41421 q^{52} +10.4853 q^{53} -2.00000 q^{55} +9.65685 q^{58} -6.82843 q^{59} +2.58579 q^{61} -6.24264 q^{62} +1.00000 q^{64} -10.8284 q^{65} -1.17157 q^{67} -2.00000 q^{68} -5.65685 q^{71} +11.6569 q^{73} -0.828427 q^{74} -2.24264 q^{76} -16.4853 q^{79} -2.00000 q^{80} +11.6569 q^{82} -5.07107 q^{83} +4.00000 q^{85} +4.82843 q^{86} -1.00000 q^{88} -3.75736 q^{89} +4.82843 q^{92} -7.89949 q^{94} +4.48528 q^{95} -9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} + 2 q^{11} + 8 q^{13} + 2 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{25} - 8 q^{26} - 8 q^{29} + 4 q^{31} - 2 q^{32} + 4 q^{34} - 4 q^{37} - 4 q^{38} + 4 q^{40} - 12 q^{41} - 4 q^{43} + 2 q^{44} - 4 q^{46} - 4 q^{47} + 2 q^{50} + 8 q^{52} + 4 q^{53} - 4 q^{55} + 8 q^{58} - 8 q^{59} + 8 q^{61} - 4 q^{62} + 2 q^{64} - 16 q^{65} - 8 q^{67} - 4 q^{68} + 12 q^{73} + 4 q^{74} + 4 q^{76} - 16 q^{79} - 4 q^{80} + 12 q^{82} + 4 q^{83} + 8 q^{85} + 4 q^{86} - 2 q^{88} - 16 q^{89} + 4 q^{92} + 4 q^{94} - 8 q^{95} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.41421 1.50163 0.750816 0.660511i \(-0.229660\pi\)
0.750816 + 0.660511i \(0.229660\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −2.24264 −0.514497 −0.257249 0.966345i \(-0.582816\pi\)
−0.257249 + 0.966345i \(0.582816\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 4.82843 1.00680 0.503398 0.864054i \(-0.332083\pi\)
0.503398 + 0.864054i \(0.332083\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −5.41421 −1.06181
\(27\) 0 0
\(28\) 0 0
\(29\) −9.65685 −1.79323 −0.896616 0.442808i \(-0.853982\pi\)
−0.896616 + 0.442808i \(0.853982\pi\)
\(30\) 0 0
\(31\) 6.24264 1.12121 0.560606 0.828083i \(-0.310568\pi\)
0.560606 + 0.828083i \(0.310568\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) 0.828427 0.136193 0.0680963 0.997679i \(-0.478307\pi\)
0.0680963 + 0.997679i \(0.478307\pi\)
\(38\) 2.24264 0.363804
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) −11.6569 −1.82049 −0.910247 0.414065i \(-0.864109\pi\)
−0.910247 + 0.414065i \(0.864109\pi\)
\(42\) 0 0
\(43\) −4.82843 −0.736328 −0.368164 0.929761i \(-0.620014\pi\)
−0.368164 + 0.929761i \(0.620014\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −4.82843 −0.711913
\(47\) 7.89949 1.15226 0.576130 0.817358i \(-0.304562\pi\)
0.576130 + 0.817358i \(0.304562\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 5.41421 0.750816
\(53\) 10.4853 1.44026 0.720132 0.693837i \(-0.244081\pi\)
0.720132 + 0.693837i \(0.244081\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 9.65685 1.26801
\(59\) −6.82843 −0.888985 −0.444493 0.895782i \(-0.646616\pi\)
−0.444493 + 0.895782i \(0.646616\pi\)
\(60\) 0 0
\(61\) 2.58579 0.331076 0.165538 0.986203i \(-0.447064\pi\)
0.165538 + 0.986203i \(0.447064\pi\)
\(62\) −6.24264 −0.792816
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −10.8284 −1.34310
\(66\) 0 0
\(67\) −1.17157 −0.143130 −0.0715652 0.997436i \(-0.522799\pi\)
−0.0715652 + 0.997436i \(0.522799\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) 11.6569 1.36433 0.682166 0.731198i \(-0.261038\pi\)
0.682166 + 0.731198i \(0.261038\pi\)
\(74\) −0.828427 −0.0963027
\(75\) 0 0
\(76\) −2.24264 −0.257249
\(77\) 0 0
\(78\) 0 0
\(79\) −16.4853 −1.85474 −0.927370 0.374147i \(-0.877936\pi\)
−0.927370 + 0.374147i \(0.877936\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) 11.6569 1.28728
\(83\) −5.07107 −0.556622 −0.278311 0.960491i \(-0.589775\pi\)
−0.278311 + 0.960491i \(0.589775\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 4.82843 0.520663
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −3.75736 −0.398279 −0.199140 0.979971i \(-0.563815\pi\)
−0.199140 + 0.979971i \(0.563815\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.82843 0.503398
\(93\) 0 0
\(94\) −7.89949 −0.814771
\(95\) 4.48528 0.460180
\(96\) 0 0
\(97\) −9.89949 −1.00514 −0.502571 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 15.0711 1.49963 0.749814 0.661649i \(-0.230143\pi\)
0.749814 + 0.661649i \(0.230143\pi\)
\(102\) 0 0
\(103\) 6.24264 0.615106 0.307553 0.951531i \(-0.400490\pi\)
0.307553 + 0.951531i \(0.400490\pi\)
\(104\) −5.41421 −0.530907
\(105\) 0 0
\(106\) −10.4853 −1.01842
\(107\) −12.8284 −1.24017 −0.620085 0.784534i \(-0.712902\pi\)
−0.620085 + 0.784534i \(0.712902\pi\)
\(108\) 0 0
\(109\) 4.34315 0.415998 0.207999 0.978129i \(-0.433305\pi\)
0.207999 + 0.978129i \(0.433305\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) 0 0
\(113\) −5.65685 −0.532152 −0.266076 0.963952i \(-0.585727\pi\)
−0.266076 + 0.963952i \(0.585727\pi\)
\(114\) 0 0
\(115\) −9.65685 −0.900506
\(116\) −9.65685 −0.896616
\(117\) 0 0
\(118\) 6.82843 0.628608
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.58579 −0.234106
\(123\) 0 0
\(124\) 6.24264 0.560606
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 14.1421 1.25491 0.627456 0.778652i \(-0.284096\pi\)
0.627456 + 0.778652i \(0.284096\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 10.8284 0.949716
\(131\) −17.5563 −1.53391 −0.766953 0.641704i \(-0.778228\pi\)
−0.766953 + 0.641704i \(0.778228\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.17157 0.101208
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 5.31371 0.453981 0.226990 0.973897i \(-0.427111\pi\)
0.226990 + 0.973897i \(0.427111\pi\)
\(138\) 0 0
\(139\) 0.100505 0.00852473 0.00426236 0.999991i \(-0.498643\pi\)
0.00426236 + 0.999991i \(0.498643\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 5.65685 0.474713
\(143\) 5.41421 0.452759
\(144\) 0 0
\(145\) 19.3137 1.60392
\(146\) −11.6569 −0.964728
\(147\) 0 0
\(148\) 0.828427 0.0680963
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 2.24264 0.181902
\(153\) 0 0
\(154\) 0 0
\(155\) −12.4853 −1.00284
\(156\) 0 0
\(157\) 12.8284 1.02382 0.511910 0.859039i \(-0.328938\pi\)
0.511910 + 0.859039i \(0.328938\pi\)
\(158\) 16.4853 1.31150
\(159\) 0 0
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 0 0
\(163\) −2.34315 −0.183529 −0.0917647 0.995781i \(-0.529251\pi\)
−0.0917647 + 0.995781i \(0.529251\pi\)
\(164\) −11.6569 −0.910247
\(165\) 0 0
\(166\) 5.07107 0.393591
\(167\) 5.17157 0.400188 0.200094 0.979777i \(-0.435875\pi\)
0.200094 + 0.979777i \(0.435875\pi\)
\(168\) 0 0
\(169\) 16.3137 1.25490
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −4.82843 −0.368164
\(173\) 5.89949 0.448530 0.224265 0.974528i \(-0.428002\pi\)
0.224265 + 0.974528i \(0.428002\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 3.75736 0.281626
\(179\) 20.4853 1.53114 0.765571 0.643352i \(-0.222457\pi\)
0.765571 + 0.643352i \(0.222457\pi\)
\(180\) 0 0
\(181\) −9.31371 −0.692283 −0.346141 0.938182i \(-0.612508\pi\)
−0.346141 + 0.938182i \(0.612508\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.82843 −0.355956
\(185\) −1.65685 −0.121814
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 7.89949 0.576130
\(189\) 0 0
\(190\) −4.48528 −0.325397
\(191\) −21.7990 −1.57732 −0.788660 0.614830i \(-0.789225\pi\)
−0.788660 + 0.614830i \(0.789225\pi\)
\(192\) 0 0
\(193\) −14.9706 −1.07760 −0.538802 0.842432i \(-0.681123\pi\)
−0.538802 + 0.842432i \(0.681123\pi\)
\(194\) 9.89949 0.710742
\(195\) 0 0
\(196\) 0 0
\(197\) −15.6569 −1.11550 −0.557752 0.830007i \(-0.688336\pi\)
−0.557752 + 0.830007i \(0.688336\pi\)
\(198\) 0 0
\(199\) 15.2132 1.07844 0.539218 0.842166i \(-0.318720\pi\)
0.539218 + 0.842166i \(0.318720\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −15.0711 −1.06040
\(203\) 0 0
\(204\) 0 0
\(205\) 23.3137 1.62830
\(206\) −6.24264 −0.434945
\(207\) 0 0
\(208\) 5.41421 0.375408
\(209\) −2.24264 −0.155127
\(210\) 0 0
\(211\) −23.1716 −1.59520 −0.797598 0.603189i \(-0.793897\pi\)
−0.797598 + 0.603189i \(0.793897\pi\)
\(212\) 10.4853 0.720132
\(213\) 0 0
\(214\) 12.8284 0.876933
\(215\) 9.65685 0.658592
\(216\) 0 0
\(217\) 0 0
\(218\) −4.34315 −0.294155
\(219\) 0 0
\(220\) −2.00000 −0.134840
\(221\) −10.8284 −0.728399
\(222\) 0 0
\(223\) 6.24264 0.418038 0.209019 0.977912i \(-0.432973\pi\)
0.209019 + 0.977912i \(0.432973\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 5.65685 0.376288
\(227\) 11.8995 0.789797 0.394899 0.918725i \(-0.370780\pi\)
0.394899 + 0.918725i \(0.370780\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 9.65685 0.636754
\(231\) 0 0
\(232\) 9.65685 0.634004
\(233\) 7.65685 0.501617 0.250809 0.968037i \(-0.419304\pi\)
0.250809 + 0.968037i \(0.419304\pi\)
\(234\) 0 0
\(235\) −15.7990 −1.03061
\(236\) −6.82843 −0.444493
\(237\) 0 0
\(238\) 0 0
\(239\) −0.686292 −0.0443925 −0.0221963 0.999754i \(-0.507066\pi\)
−0.0221963 + 0.999754i \(0.507066\pi\)
\(240\) 0 0
\(241\) −7.65685 −0.493221 −0.246611 0.969115i \(-0.579317\pi\)
−0.246611 + 0.969115i \(0.579317\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 2.58579 0.165538
\(245\) 0 0
\(246\) 0 0
\(247\) −12.1421 −0.772586
\(248\) −6.24264 −0.396408
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) −8.48528 −0.535586 −0.267793 0.963476i \(-0.586294\pi\)
−0.267793 + 0.963476i \(0.586294\pi\)
\(252\) 0 0
\(253\) 4.82843 0.303561
\(254\) −14.1421 −0.887357
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 29.2132 1.82227 0.911135 0.412108i \(-0.135208\pi\)
0.911135 + 0.412108i \(0.135208\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −10.8284 −0.671551
\(261\) 0 0
\(262\) 17.5563 1.08463
\(263\) −9.17157 −0.565543 −0.282772 0.959187i \(-0.591254\pi\)
−0.282772 + 0.959187i \(0.591254\pi\)
\(264\) 0 0
\(265\) −20.9706 −1.28821
\(266\) 0 0
\(267\) 0 0
\(268\) −1.17157 −0.0715652
\(269\) −5.31371 −0.323983 −0.161991 0.986792i \(-0.551792\pi\)
−0.161991 + 0.986792i \(0.551792\pi\)
\(270\) 0 0
\(271\) 22.1421 1.34504 0.672519 0.740079i \(-0.265212\pi\)
0.672519 + 0.740079i \(0.265212\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −5.31371 −0.321013
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) −0.100505 −0.00602789
\(279\) 0 0
\(280\) 0 0
\(281\) 1.31371 0.0783693 0.0391846 0.999232i \(-0.487524\pi\)
0.0391846 + 0.999232i \(0.487524\pi\)
\(282\) 0 0
\(283\) −2.24264 −0.133311 −0.0666556 0.997776i \(-0.521233\pi\)
−0.0666556 + 0.997776i \(0.521233\pi\)
\(284\) −5.65685 −0.335673
\(285\) 0 0
\(286\) −5.41421 −0.320149
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −19.3137 −1.13414
\(291\) 0 0
\(292\) 11.6569 0.682166
\(293\) 4.92893 0.287951 0.143976 0.989581i \(-0.454011\pi\)
0.143976 + 0.989581i \(0.454011\pi\)
\(294\) 0 0
\(295\) 13.6569 0.795133
\(296\) −0.828427 −0.0481513
\(297\) 0 0
\(298\) 4.00000 0.231714
\(299\) 26.1421 1.51184
\(300\) 0 0
\(301\) 0 0
\(302\) −4.00000 −0.230174
\(303\) 0 0
\(304\) −2.24264 −0.128624
\(305\) −5.17157 −0.296123
\(306\) 0 0
\(307\) −25.0711 −1.43088 −0.715441 0.698673i \(-0.753774\pi\)
−0.715441 + 0.698673i \(0.753774\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 12.4853 0.709116
\(311\) −8.10051 −0.459338 −0.229669 0.973269i \(-0.573764\pi\)
−0.229669 + 0.973269i \(0.573764\pi\)
\(312\) 0 0
\(313\) −8.24264 −0.465902 −0.232951 0.972489i \(-0.574838\pi\)
−0.232951 + 0.972489i \(0.574838\pi\)
\(314\) −12.8284 −0.723950
\(315\) 0 0
\(316\) −16.4853 −0.927370
\(317\) −12.8284 −0.720516 −0.360258 0.932853i \(-0.617311\pi\)
−0.360258 + 0.932853i \(0.617311\pi\)
\(318\) 0 0
\(319\) −9.65685 −0.540680
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) 0 0
\(323\) 4.48528 0.249568
\(324\) 0 0
\(325\) −5.41421 −0.300327
\(326\) 2.34315 0.129775
\(327\) 0 0
\(328\) 11.6569 0.643642
\(329\) 0 0
\(330\) 0 0
\(331\) −27.7990 −1.52797 −0.763985 0.645234i \(-0.776760\pi\)
−0.763985 + 0.645234i \(0.776760\pi\)
\(332\) −5.07107 −0.278311
\(333\) 0 0
\(334\) −5.17157 −0.282976
\(335\) 2.34315 0.128020
\(336\) 0 0
\(337\) 12.8284 0.698809 0.349404 0.936972i \(-0.386384\pi\)
0.349404 + 0.936972i \(0.386384\pi\)
\(338\) −16.3137 −0.887349
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) 6.24264 0.338058
\(342\) 0 0
\(343\) 0 0
\(344\) 4.82843 0.260331
\(345\) 0 0
\(346\) −5.89949 −0.317159
\(347\) −19.4558 −1.04444 −0.522222 0.852809i \(-0.674897\pi\)
−0.522222 + 0.852809i \(0.674897\pi\)
\(348\) 0 0
\(349\) −17.4142 −0.932161 −0.466081 0.884742i \(-0.654334\pi\)
−0.466081 + 0.884742i \(0.654334\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −21.2132 −1.12906 −0.564532 0.825411i \(-0.690943\pi\)
−0.564532 + 0.825411i \(0.690943\pi\)
\(354\) 0 0
\(355\) 11.3137 0.600469
\(356\) −3.75736 −0.199140
\(357\) 0 0
\(358\) −20.4853 −1.08268
\(359\) −5.17157 −0.272945 −0.136473 0.990644i \(-0.543577\pi\)
−0.136473 + 0.990644i \(0.543577\pi\)
\(360\) 0 0
\(361\) −13.9706 −0.735293
\(362\) 9.31371 0.489518
\(363\) 0 0
\(364\) 0 0
\(365\) −23.3137 −1.22030
\(366\) 0 0
\(367\) −32.5858 −1.70096 −0.850482 0.526004i \(-0.823690\pi\)
−0.850482 + 0.526004i \(0.823690\pi\)
\(368\) 4.82843 0.251699
\(369\) 0 0
\(370\) 1.65685 0.0861358
\(371\) 0 0
\(372\) 0 0
\(373\) 35.9411 1.86096 0.930480 0.366342i \(-0.119390\pi\)
0.930480 + 0.366342i \(0.119390\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) −7.89949 −0.407385
\(377\) −52.2843 −2.69278
\(378\) 0 0
\(379\) 32.2843 1.65833 0.829166 0.559003i \(-0.188816\pi\)
0.829166 + 0.559003i \(0.188816\pi\)
\(380\) 4.48528 0.230090
\(381\) 0 0
\(382\) 21.7990 1.11533
\(383\) 22.7279 1.16134 0.580671 0.814138i \(-0.302790\pi\)
0.580671 + 0.814138i \(0.302790\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.9706 0.761982
\(387\) 0 0
\(388\) −9.89949 −0.502571
\(389\) 34.2843 1.73828 0.869141 0.494565i \(-0.164673\pi\)
0.869141 + 0.494565i \(0.164673\pi\)
\(390\) 0 0
\(391\) −9.65685 −0.488368
\(392\) 0 0
\(393\) 0 0
\(394\) 15.6569 0.788781
\(395\) 32.9706 1.65893
\(396\) 0 0
\(397\) −11.1716 −0.560685 −0.280343 0.959900i \(-0.590448\pi\)
−0.280343 + 0.959900i \(0.590448\pi\)
\(398\) −15.2132 −0.762569
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −10.6863 −0.533648 −0.266824 0.963745i \(-0.585974\pi\)
−0.266824 + 0.963745i \(0.585974\pi\)
\(402\) 0 0
\(403\) 33.7990 1.68365
\(404\) 15.0711 0.749814
\(405\) 0 0
\(406\) 0 0
\(407\) 0.828427 0.0410636
\(408\) 0 0
\(409\) −1.51472 −0.0748980 −0.0374490 0.999299i \(-0.511923\pi\)
−0.0374490 + 0.999299i \(0.511923\pi\)
\(410\) −23.3137 −1.15138
\(411\) 0 0
\(412\) 6.24264 0.307553
\(413\) 0 0
\(414\) 0 0
\(415\) 10.1421 0.497858
\(416\) −5.41421 −0.265454
\(417\) 0 0
\(418\) 2.24264 0.109691
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −27.6569 −1.34791 −0.673956 0.738771i \(-0.735406\pi\)
−0.673956 + 0.738771i \(0.735406\pi\)
\(422\) 23.1716 1.12797
\(423\) 0 0
\(424\) −10.4853 −0.509210
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) −12.8284 −0.620085
\(429\) 0 0
\(430\) −9.65685 −0.465695
\(431\) −2.14214 −0.103183 −0.0515915 0.998668i \(-0.516429\pi\)
−0.0515915 + 0.998668i \(0.516429\pi\)
\(432\) 0 0
\(433\) −34.3848 −1.65243 −0.826213 0.563357i \(-0.809509\pi\)
−0.826213 + 0.563357i \(0.809509\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.34315 0.207999
\(437\) −10.8284 −0.517994
\(438\) 0 0
\(439\) −29.1716 −1.39228 −0.696142 0.717904i \(-0.745101\pi\)
−0.696142 + 0.717904i \(0.745101\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) 10.8284 0.515056
\(443\) −12.4853 −0.593194 −0.296597 0.955003i \(-0.595852\pi\)
−0.296597 + 0.955003i \(0.595852\pi\)
\(444\) 0 0
\(445\) 7.51472 0.356232
\(446\) −6.24264 −0.295598
\(447\) 0 0
\(448\) 0 0
\(449\) −16.9706 −0.800890 −0.400445 0.916321i \(-0.631145\pi\)
−0.400445 + 0.916321i \(0.631145\pi\)
\(450\) 0 0
\(451\) −11.6569 −0.548900
\(452\) −5.65685 −0.266076
\(453\) 0 0
\(454\) −11.8995 −0.558471
\(455\) 0 0
\(456\) 0 0
\(457\) −29.3137 −1.37124 −0.685619 0.727961i \(-0.740468\pi\)
−0.685619 + 0.727961i \(0.740468\pi\)
\(458\) −14.0000 −0.654177
\(459\) 0 0
\(460\) −9.65685 −0.450253
\(461\) 30.5858 1.42452 0.712261 0.701915i \(-0.247671\pi\)
0.712261 + 0.701915i \(0.247671\pi\)
\(462\) 0 0
\(463\) −3.31371 −0.154001 −0.0770005 0.997031i \(-0.524534\pi\)
−0.0770005 + 0.997031i \(0.524534\pi\)
\(464\) −9.65685 −0.448308
\(465\) 0 0
\(466\) −7.65685 −0.354697
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 15.7990 0.728753
\(471\) 0 0
\(472\) 6.82843 0.314304
\(473\) −4.82843 −0.222011
\(474\) 0 0
\(475\) 2.24264 0.102899
\(476\) 0 0
\(477\) 0 0
\(478\) 0.686292 0.0313902
\(479\) −7.31371 −0.334172 −0.167086 0.985942i \(-0.553436\pi\)
−0.167086 + 0.985942i \(0.553436\pi\)
\(480\) 0 0
\(481\) 4.48528 0.204511
\(482\) 7.65685 0.348760
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 19.7990 0.899026
\(486\) 0 0
\(487\) 0.142136 0.00644078 0.00322039 0.999995i \(-0.498975\pi\)
0.00322039 + 0.999995i \(0.498975\pi\)
\(488\) −2.58579 −0.117053
\(489\) 0 0
\(490\) 0 0
\(491\) 42.6274 1.92375 0.961874 0.273492i \(-0.0881788\pi\)
0.961874 + 0.273492i \(0.0881788\pi\)
\(492\) 0 0
\(493\) 19.3137 0.869846
\(494\) 12.1421 0.546301
\(495\) 0 0
\(496\) 6.24264 0.280303
\(497\) 0 0
\(498\) 0 0
\(499\) −6.14214 −0.274960 −0.137480 0.990505i \(-0.543900\pi\)
−0.137480 + 0.990505i \(0.543900\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 8.48528 0.378717
\(503\) 8.48528 0.378340 0.189170 0.981944i \(-0.439420\pi\)
0.189170 + 0.981944i \(0.439420\pi\)
\(504\) 0 0
\(505\) −30.1421 −1.34131
\(506\) −4.82843 −0.214650
\(507\) 0 0
\(508\) 14.1421 0.627456
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −29.2132 −1.28854
\(515\) −12.4853 −0.550167
\(516\) 0 0
\(517\) 7.89949 0.347419
\(518\) 0 0
\(519\) 0 0
\(520\) 10.8284 0.474858
\(521\) −21.2132 −0.929367 −0.464684 0.885477i \(-0.653832\pi\)
−0.464684 + 0.885477i \(0.653832\pi\)
\(522\) 0 0
\(523\) −9.07107 −0.396650 −0.198325 0.980136i \(-0.563550\pi\)
−0.198325 + 0.980136i \(0.563550\pi\)
\(524\) −17.5563 −0.766953
\(525\) 0 0
\(526\) 9.17157 0.399900
\(527\) −12.4853 −0.543867
\(528\) 0 0
\(529\) 0.313708 0.0136395
\(530\) 20.9706 0.910903
\(531\) 0 0
\(532\) 0 0
\(533\) −63.1127 −2.73371
\(534\) 0 0
\(535\) 25.6569 1.10924
\(536\) 1.17157 0.0506042
\(537\) 0 0
\(538\) 5.31371 0.229090
\(539\) 0 0
\(540\) 0 0
\(541\) −45.3137 −1.94819 −0.974094 0.226142i \(-0.927389\pi\)
−0.974094 + 0.226142i \(0.927389\pi\)
\(542\) −22.1421 −0.951086
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) −8.68629 −0.372080
\(546\) 0 0
\(547\) −42.6274 −1.82262 −0.911308 0.411724i \(-0.864927\pi\)
−0.911308 + 0.411724i \(0.864927\pi\)
\(548\) 5.31371 0.226990
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 21.6569 0.922613
\(552\) 0 0
\(553\) 0 0
\(554\) 16.0000 0.679775
\(555\) 0 0
\(556\) 0.100505 0.00426236
\(557\) −19.6569 −0.832888 −0.416444 0.909161i \(-0.636724\pi\)
−0.416444 + 0.909161i \(0.636724\pi\)
\(558\) 0 0
\(559\) −26.1421 −1.10569
\(560\) 0 0
\(561\) 0 0
\(562\) −1.31371 −0.0554154
\(563\) 21.0711 0.888040 0.444020 0.896017i \(-0.353552\pi\)
0.444020 + 0.896017i \(0.353552\pi\)
\(564\) 0 0
\(565\) 11.3137 0.475971
\(566\) 2.24264 0.0942652
\(567\) 0 0
\(568\) 5.65685 0.237356
\(569\) 10.6863 0.447993 0.223996 0.974590i \(-0.428090\pi\)
0.223996 + 0.974590i \(0.428090\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 5.41421 0.226380
\(573\) 0 0
\(574\) 0 0
\(575\) −4.82843 −0.201359
\(576\) 0 0
\(577\) −4.44365 −0.184992 −0.0924958 0.995713i \(-0.529484\pi\)
−0.0924958 + 0.995713i \(0.529484\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 19.3137 0.801958
\(581\) 0 0
\(582\) 0 0
\(583\) 10.4853 0.434256
\(584\) −11.6569 −0.482364
\(585\) 0 0
\(586\) −4.92893 −0.203612
\(587\) −11.7990 −0.486996 −0.243498 0.969901i \(-0.578295\pi\)
−0.243498 + 0.969901i \(0.578295\pi\)
\(588\) 0 0
\(589\) −14.0000 −0.576860
\(590\) −13.6569 −0.562244
\(591\) 0 0
\(592\) 0.828427 0.0340481
\(593\) 37.3137 1.53229 0.766145 0.642668i \(-0.222172\pi\)
0.766145 + 0.642668i \(0.222172\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) 0 0
\(598\) −26.1421 −1.06903
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) 15.1716 0.618861 0.309431 0.950922i \(-0.399862\pi\)
0.309431 + 0.950922i \(0.399862\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 4.00000 0.162758
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) −35.3137 −1.43334 −0.716670 0.697413i \(-0.754334\pi\)
−0.716670 + 0.697413i \(0.754334\pi\)
\(608\) 2.24264 0.0909511
\(609\) 0 0
\(610\) 5.17157 0.209391
\(611\) 42.7696 1.73027
\(612\) 0 0
\(613\) −30.3431 −1.22555 −0.612774 0.790258i \(-0.709946\pi\)
−0.612774 + 0.790258i \(0.709946\pi\)
\(614\) 25.0711 1.01179
\(615\) 0 0
\(616\) 0 0
\(617\) 8.00000 0.322068 0.161034 0.986949i \(-0.448517\pi\)
0.161034 + 0.986949i \(0.448517\pi\)
\(618\) 0 0
\(619\) −21.6569 −0.870462 −0.435231 0.900319i \(-0.643333\pi\)
−0.435231 + 0.900319i \(0.643333\pi\)
\(620\) −12.4853 −0.501421
\(621\) 0 0
\(622\) 8.10051 0.324801
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 8.24264 0.329442
\(627\) 0 0
\(628\) 12.8284 0.511910
\(629\) −1.65685 −0.0660631
\(630\) 0 0
\(631\) 36.1421 1.43880 0.719398 0.694598i \(-0.244418\pi\)
0.719398 + 0.694598i \(0.244418\pi\)
\(632\) 16.4853 0.655749
\(633\) 0 0
\(634\) 12.8284 0.509482
\(635\) −28.2843 −1.12243
\(636\) 0 0
\(637\) 0 0
\(638\) 9.65685 0.382319
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) 35.3137 1.39481 0.697404 0.716678i \(-0.254338\pi\)
0.697404 + 0.716678i \(0.254338\pi\)
\(642\) 0 0
\(643\) 37.4558 1.47711 0.738557 0.674191i \(-0.235507\pi\)
0.738557 + 0.674191i \(0.235507\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −4.48528 −0.176471
\(647\) −1.07107 −0.0421080 −0.0210540 0.999778i \(-0.506702\pi\)
−0.0210540 + 0.999778i \(0.506702\pi\)
\(648\) 0 0
\(649\) −6.82843 −0.268039
\(650\) 5.41421 0.212363
\(651\) 0 0
\(652\) −2.34315 −0.0917647
\(653\) −22.2843 −0.872051 −0.436025 0.899934i \(-0.643614\pi\)
−0.436025 + 0.899934i \(0.643614\pi\)
\(654\) 0 0
\(655\) 35.1127 1.37197
\(656\) −11.6569 −0.455124
\(657\) 0 0
\(658\) 0 0
\(659\) 49.1127 1.91316 0.956580 0.291471i \(-0.0941448\pi\)
0.956580 + 0.291471i \(0.0941448\pi\)
\(660\) 0 0
\(661\) −39.9411 −1.55353 −0.776765 0.629791i \(-0.783141\pi\)
−0.776765 + 0.629791i \(0.783141\pi\)
\(662\) 27.7990 1.08044
\(663\) 0 0
\(664\) 5.07107 0.196796
\(665\) 0 0
\(666\) 0 0
\(667\) −46.6274 −1.80542
\(668\) 5.17157 0.200094
\(669\) 0 0
\(670\) −2.34315 −0.0905236
\(671\) 2.58579 0.0998232
\(672\) 0 0
\(673\) −17.5147 −0.675143 −0.337571 0.941300i \(-0.609605\pi\)
−0.337571 + 0.941300i \(0.609605\pi\)
\(674\) −12.8284 −0.494133
\(675\) 0 0
\(676\) 16.3137 0.627450
\(677\) 36.0416 1.38519 0.692596 0.721326i \(-0.256467\pi\)
0.692596 + 0.721326i \(0.256467\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −4.00000 −0.153393
\(681\) 0 0
\(682\) −6.24264 −0.239043
\(683\) −28.7696 −1.10084 −0.550418 0.834889i \(-0.685532\pi\)
−0.550418 + 0.834889i \(0.685532\pi\)
\(684\) 0 0
\(685\) −10.6274 −0.406053
\(686\) 0 0
\(687\) 0 0
\(688\) −4.82843 −0.184082
\(689\) 56.7696 2.16275
\(690\) 0 0
\(691\) −14.1421 −0.537992 −0.268996 0.963141i \(-0.586692\pi\)
−0.268996 + 0.963141i \(0.586692\pi\)
\(692\) 5.89949 0.224265
\(693\) 0 0
\(694\) 19.4558 0.738534
\(695\) −0.201010 −0.00762475
\(696\) 0 0
\(697\) 23.3137 0.883070
\(698\) 17.4142 0.659138
\(699\) 0 0
\(700\) 0 0
\(701\) −9.31371 −0.351774 −0.175887 0.984410i \(-0.556279\pi\)
−0.175887 + 0.984410i \(0.556279\pi\)
\(702\) 0 0
\(703\) −1.85786 −0.0700707
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 21.2132 0.798369
\(707\) 0 0
\(708\) 0 0
\(709\) −34.7696 −1.30580 −0.652899 0.757445i \(-0.726447\pi\)
−0.652899 + 0.757445i \(0.726447\pi\)
\(710\) −11.3137 −0.424596
\(711\) 0 0
\(712\) 3.75736 0.140813
\(713\) 30.1421 1.12883
\(714\) 0 0
\(715\) −10.8284 −0.404960
\(716\) 20.4853 0.765571
\(717\) 0 0
\(718\) 5.17157 0.193001
\(719\) 44.1838 1.64778 0.823888 0.566752i \(-0.191800\pi\)
0.823888 + 0.566752i \(0.191800\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 13.9706 0.519931
\(723\) 0 0
\(724\) −9.31371 −0.346141
\(725\) 9.65685 0.358647
\(726\) 0 0
\(727\) 31.2132 1.15763 0.578817 0.815458i \(-0.303515\pi\)
0.578817 + 0.815458i \(0.303515\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 23.3137 0.862879
\(731\) 9.65685 0.357172
\(732\) 0 0
\(733\) 19.5563 0.722330 0.361165 0.932502i \(-0.382379\pi\)
0.361165 + 0.932502i \(0.382379\pi\)
\(734\) 32.5858 1.20276
\(735\) 0 0
\(736\) −4.82843 −0.177978
\(737\) −1.17157 −0.0431554
\(738\) 0 0
\(739\) 2.62742 0.0966511 0.0483255 0.998832i \(-0.484612\pi\)
0.0483255 + 0.998832i \(0.484612\pi\)
\(740\) −1.65685 −0.0609072
\(741\) 0 0
\(742\) 0 0
\(743\) 18.6274 0.683374 0.341687 0.939814i \(-0.389002\pi\)
0.341687 + 0.939814i \(0.389002\pi\)
\(744\) 0 0
\(745\) 8.00000 0.293097
\(746\) −35.9411 −1.31590
\(747\) 0 0
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) 0 0
\(751\) −12.1421 −0.443073 −0.221536 0.975152i \(-0.571107\pi\)
−0.221536 + 0.975152i \(0.571107\pi\)
\(752\) 7.89949 0.288065
\(753\) 0 0
\(754\) 52.2843 1.90408
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −48.1421 −1.74976 −0.874878 0.484344i \(-0.839058\pi\)
−0.874878 + 0.484344i \(0.839058\pi\)
\(758\) −32.2843 −1.17262
\(759\) 0 0
\(760\) −4.48528 −0.162698
\(761\) 1.51472 0.0549085 0.0274543 0.999623i \(-0.491260\pi\)
0.0274543 + 0.999623i \(0.491260\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −21.7990 −0.788660
\(765\) 0 0
\(766\) −22.7279 −0.821193
\(767\) −36.9706 −1.33493
\(768\) 0 0
\(769\) −15.8579 −0.571849 −0.285925 0.958252i \(-0.592301\pi\)
−0.285925 + 0.958252i \(0.592301\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14.9706 −0.538802
\(773\) −5.51472 −0.198351 −0.0991753 0.995070i \(-0.531620\pi\)
−0.0991753 + 0.995070i \(0.531620\pi\)
\(774\) 0 0
\(775\) −6.24264 −0.224242
\(776\) 9.89949 0.355371
\(777\) 0 0
\(778\) −34.2843 −1.22915
\(779\) 26.1421 0.936639
\(780\) 0 0
\(781\) −5.65685 −0.202418
\(782\) 9.65685 0.345328
\(783\) 0 0
\(784\) 0 0
\(785\) −25.6569 −0.915732
\(786\) 0 0
\(787\) −3.41421 −0.121704 −0.0608518 0.998147i \(-0.519382\pi\)
−0.0608518 + 0.998147i \(0.519382\pi\)
\(788\) −15.6569 −0.557752
\(789\) 0 0
\(790\) −32.9706 −1.17304
\(791\) 0 0
\(792\) 0 0
\(793\) 14.0000 0.497155
\(794\) 11.1716 0.396464
\(795\) 0 0
\(796\) 15.2132 0.539218
\(797\) −17.5147 −0.620403 −0.310202 0.950671i \(-0.600397\pi\)
−0.310202 + 0.950671i \(0.600397\pi\)
\(798\) 0 0
\(799\) −15.7990 −0.558928
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 10.6863 0.377346
\(803\) 11.6569 0.411361
\(804\) 0 0
\(805\) 0 0
\(806\) −33.7990 −1.19052
\(807\) 0 0
\(808\) −15.0711 −0.530198
\(809\) −26.7696 −0.941167 −0.470584 0.882355i \(-0.655957\pi\)
−0.470584 + 0.882355i \(0.655957\pi\)
\(810\) 0 0
\(811\) 36.3848 1.27764 0.638821 0.769355i \(-0.279422\pi\)
0.638821 + 0.769355i \(0.279422\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −0.828427 −0.0290364
\(815\) 4.68629 0.164154
\(816\) 0 0
\(817\) 10.8284 0.378839
\(818\) 1.51472 0.0529609
\(819\) 0 0
\(820\) 23.3137 0.814150
\(821\) −47.3137 −1.65126 −0.825630 0.564212i \(-0.809180\pi\)
−0.825630 + 0.564212i \(0.809180\pi\)
\(822\) 0 0
\(823\) −28.1421 −0.980973 −0.490487 0.871449i \(-0.663181\pi\)
−0.490487 + 0.871449i \(0.663181\pi\)
\(824\) −6.24264 −0.217473
\(825\) 0 0
\(826\) 0 0
\(827\) −39.3137 −1.36707 −0.683536 0.729917i \(-0.739559\pi\)
−0.683536 + 0.729917i \(0.739559\pi\)
\(828\) 0 0
\(829\) 50.2843 1.74644 0.873222 0.487322i \(-0.162026\pi\)
0.873222 + 0.487322i \(0.162026\pi\)
\(830\) −10.1421 −0.352039
\(831\) 0 0
\(832\) 5.41421 0.187704
\(833\) 0 0
\(834\) 0 0
\(835\) −10.3431 −0.357939
\(836\) −2.24264 −0.0775634
\(837\) 0 0
\(838\) 16.0000 0.552711
\(839\) 17.0711 0.589359 0.294679 0.955596i \(-0.404787\pi\)
0.294679 + 0.955596i \(0.404787\pi\)
\(840\) 0 0
\(841\) 64.2548 2.21568
\(842\) 27.6569 0.953118
\(843\) 0 0
\(844\) −23.1716 −0.797598
\(845\) −32.6274 −1.12242
\(846\) 0 0
\(847\) 0 0
\(848\) 10.4853 0.360066
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) 35.5563 1.21743 0.608713 0.793390i \(-0.291686\pi\)
0.608713 + 0.793390i \(0.291686\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.8284 0.438467
\(857\) −0.142136 −0.00485526 −0.00242763 0.999997i \(-0.500773\pi\)
−0.00242763 + 0.999997i \(0.500773\pi\)
\(858\) 0 0
\(859\) −30.4264 −1.03814 −0.519068 0.854733i \(-0.673721\pi\)
−0.519068 + 0.854733i \(0.673721\pi\)
\(860\) 9.65685 0.329296
\(861\) 0 0
\(862\) 2.14214 0.0729614
\(863\) 38.7696 1.31973 0.659865 0.751384i \(-0.270613\pi\)
0.659865 + 0.751384i \(0.270613\pi\)
\(864\) 0 0
\(865\) −11.7990 −0.401178
\(866\) 34.3848 1.16844
\(867\) 0 0
\(868\) 0 0
\(869\) −16.4853 −0.559225
\(870\) 0 0
\(871\) −6.34315 −0.214929
\(872\) −4.34315 −0.147077
\(873\) 0 0
\(874\) 10.8284 0.366277
\(875\) 0 0
\(876\) 0 0
\(877\) −6.68629 −0.225780 −0.112890 0.993607i \(-0.536011\pi\)
−0.112890 + 0.993607i \(0.536011\pi\)
\(878\) 29.1716 0.984493
\(879\) 0 0
\(880\) −2.00000 −0.0674200
\(881\) −37.8995 −1.27687 −0.638433 0.769677i \(-0.720417\pi\)
−0.638433 + 0.769677i \(0.720417\pi\)
\(882\) 0 0
\(883\) −10.6274 −0.357641 −0.178821 0.983882i \(-0.557228\pi\)
−0.178821 + 0.983882i \(0.557228\pi\)
\(884\) −10.8284 −0.364199
\(885\) 0 0
\(886\) 12.4853 0.419451
\(887\) −4.28427 −0.143852 −0.0719259 0.997410i \(-0.522915\pi\)
−0.0719259 + 0.997410i \(0.522915\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −7.51472 −0.251894
\(891\) 0 0
\(892\) 6.24264 0.209019
\(893\) −17.7157 −0.592834
\(894\) 0 0
\(895\) −40.9706 −1.36949
\(896\) 0 0
\(897\) 0 0
\(898\) 16.9706 0.566315
\(899\) −60.2843 −2.01059
\(900\) 0 0
\(901\) −20.9706 −0.698631
\(902\) 11.6569 0.388131
\(903\) 0 0
\(904\) 5.65685 0.188144
\(905\) 18.6274 0.619196
\(906\) 0 0
\(907\) −8.20101 −0.272310 −0.136155 0.990688i \(-0.543475\pi\)
−0.136155 + 0.990688i \(0.543475\pi\)
\(908\) 11.8995 0.394899
\(909\) 0 0
\(910\) 0 0
\(911\) 11.4558 0.379549 0.189775 0.981828i \(-0.439224\pi\)
0.189775 + 0.981828i \(0.439224\pi\)
\(912\) 0 0
\(913\) −5.07107 −0.167828
\(914\) 29.3137 0.969611
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) 0 0
\(919\) −0.970563 −0.0320159 −0.0160080 0.999872i \(-0.505096\pi\)
−0.0160080 + 0.999872i \(0.505096\pi\)
\(920\) 9.65685 0.318377
\(921\) 0 0
\(922\) −30.5858 −1.00729
\(923\) −30.6274 −1.00811
\(924\) 0 0
\(925\) −0.828427 −0.0272385
\(926\) 3.31371 0.108895
\(927\) 0 0
\(928\) 9.65685 0.317002
\(929\) −10.7868 −0.353903 −0.176952 0.984220i \(-0.556624\pi\)
−0.176952 + 0.984220i \(0.556624\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 7.65685 0.250809
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −56.9117 −1.85922 −0.929612 0.368540i \(-0.879858\pi\)
−0.929612 + 0.368540i \(0.879858\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −15.7990 −0.515306
\(941\) −27.5563 −0.898311 −0.449156 0.893454i \(-0.648275\pi\)
−0.449156 + 0.893454i \(0.648275\pi\)
\(942\) 0 0
\(943\) −56.2843 −1.83287
\(944\) −6.82843 −0.222246
\(945\) 0 0
\(946\) 4.82843 0.156986
\(947\) 52.5685 1.70825 0.854124 0.520069i \(-0.174094\pi\)
0.854124 + 0.520069i \(0.174094\pi\)
\(948\) 0 0
\(949\) 63.1127 2.04872
\(950\) −2.24264 −0.0727609
\(951\) 0 0
\(952\) 0 0
\(953\) −9.51472 −0.308212 −0.154106 0.988054i \(-0.549250\pi\)
−0.154106 + 0.988054i \(0.549250\pi\)
\(954\) 0 0
\(955\) 43.5980 1.41080
\(956\) −0.686292 −0.0221963
\(957\) 0 0
\(958\) 7.31371 0.236295
\(959\) 0 0
\(960\) 0 0
\(961\) 7.97056 0.257115
\(962\) −4.48528 −0.144611
\(963\) 0 0
\(964\) −7.65685 −0.246611
\(965\) 29.9411 0.963839
\(966\) 0 0
\(967\) −9.94113 −0.319685 −0.159843 0.987143i \(-0.551099\pi\)
−0.159843 + 0.987143i \(0.551099\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −19.7990 −0.635707
\(971\) −4.68629 −0.150390 −0.0751951 0.997169i \(-0.523958\pi\)
−0.0751951 + 0.997169i \(0.523958\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −0.142136 −0.00455432
\(975\) 0 0
\(976\) 2.58579 0.0827690
\(977\) −26.6274 −0.851887 −0.425943 0.904750i \(-0.640058\pi\)
−0.425943 + 0.904750i \(0.640058\pi\)
\(978\) 0 0
\(979\) −3.75736 −0.120086
\(980\) 0 0
\(981\) 0 0
\(982\) −42.6274 −1.36030
\(983\) −0.870058 −0.0277505 −0.0138753 0.999904i \(-0.504417\pi\)
−0.0138753 + 0.999904i \(0.504417\pi\)
\(984\) 0 0
\(985\) 31.3137 0.997738
\(986\) −19.3137 −0.615074
\(987\) 0 0
\(988\) −12.1421 −0.386293
\(989\) −23.3137 −0.741333
\(990\) 0 0
\(991\) 50.9117 1.61726 0.808632 0.588315i \(-0.200209\pi\)
0.808632 + 0.588315i \(0.200209\pi\)
\(992\) −6.24264 −0.198204
\(993\) 0 0
\(994\) 0 0
\(995\) −30.4264 −0.964582
\(996\) 0 0
\(997\) 30.5858 0.968662 0.484331 0.874885i \(-0.339063\pi\)
0.484331 + 0.874885i \(0.339063\pi\)
\(998\) 6.14214 0.194426
\(999\) 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.ci.1.2 2
3.2 odd 2 3234.2.a.bc.1.2 2
7.6 odd 2 9702.2.a.cy.1.1 2
21.20 even 2 3234.2.a.bd.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bc.1.2 2 3.2 odd 2
3234.2.a.bd.1.1 yes 2 21.20 even 2
9702.2.a.ci.1.2 2 1.1 even 1 trivial
9702.2.a.cy.1.1 2 7.6 odd 2