Properties

Label 1764.2.f.b.881.1
Level $1764$
Weight $2$
Character 1764.881
Analytic conductor $14.086$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.1
Root \(0.923880 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 1764.881
Dual form 1764.2.f.b.881.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.84776 q^{5} +O(q^{10})\) \(q-1.84776 q^{5} -2.00000i q^{11} +4.46088i q^{13} +2.29610 q^{17} -1.53073i q^{19} -8.82843i q^{23} -1.58579 q^{25} +1.17157i q^{29} +5.86030i q^{31} +8.24264 q^{37} +11.8519 q^{41} +1.17157 q^{43} -8.02509 q^{47} -3.75736i q^{53} +3.69552i q^{55} +9.81845 q^{59} -12.3003i q^{61} -8.24264i q^{65} +12.4853 q^{67} -13.3137i q^{71} -2.74444i q^{73} +11.3137 q^{79} +10.4525 q^{83} -4.24264 q^{85} -14.4650 q^{89} +2.82843i q^{95} +2.74444i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{67}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.84776 −0.826343 −0.413171 0.910653i \(-0.635579\pi\)
−0.413171 + 0.910653i \(0.635579\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 4.46088i 1.23723i 0.785695 + 0.618613i \(0.212305\pi\)
−0.785695 + 0.618613i \(0.787695\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.29610 0.556886 0.278443 0.960453i \(-0.410182\pi\)
0.278443 + 0.960453i \(0.410182\pi\)
\(18\) 0 0
\(19\) − 1.53073i − 0.351174i −0.984464 0.175587i \(-0.943818\pi\)
0.984464 0.175587i \(-0.0561824\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 8.82843i − 1.84085i −0.390914 0.920427i \(-0.627841\pi\)
0.390914 0.920427i \(-0.372159\pi\)
\(24\) 0 0
\(25\) −1.58579 −0.317157
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.17157i 0.217556i 0.994066 + 0.108778i \(0.0346937\pi\)
−0.994066 + 0.108778i \(0.965306\pi\)
\(30\) 0 0
\(31\) 5.86030i 1.05254i 0.850317 + 0.526271i \(0.176410\pi\)
−0.850317 + 0.526271i \(0.823590\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.24264 1.35508 0.677541 0.735485i \(-0.263046\pi\)
0.677541 + 0.735485i \(0.263046\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.8519 1.85096 0.925480 0.378798i \(-0.123662\pi\)
0.925480 + 0.378798i \(0.123662\pi\)
\(42\) 0 0
\(43\) 1.17157 0.178663 0.0893316 0.996002i \(-0.471527\pi\)
0.0893316 + 0.996002i \(0.471527\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.02509 −1.17058 −0.585290 0.810824i \(-0.699019\pi\)
−0.585290 + 0.810824i \(0.699019\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 3.75736i − 0.516113i −0.966130 0.258056i \(-0.916918\pi\)
0.966130 0.258056i \(-0.0830821\pi\)
\(54\) 0 0
\(55\) 3.69552i 0.498304i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.81845 1.27825 0.639127 0.769101i \(-0.279296\pi\)
0.639127 + 0.769101i \(0.279296\pi\)
\(60\) 0 0
\(61\) − 12.3003i − 1.57489i −0.616387 0.787444i \(-0.711404\pi\)
0.616387 0.787444i \(-0.288596\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 8.24264i − 1.02237i
\(66\) 0 0
\(67\) 12.4853 1.52532 0.762660 0.646800i \(-0.223893\pi\)
0.762660 + 0.646800i \(0.223893\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 13.3137i − 1.58005i −0.613077 0.790023i \(-0.710068\pi\)
0.613077 0.790023i \(-0.289932\pi\)
\(72\) 0 0
\(73\) − 2.74444i − 0.321213i −0.987019 0.160606i \(-0.948655\pi\)
0.987019 0.160606i \(-0.0513450\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.4525 1.14731 0.573656 0.819097i \(-0.305525\pi\)
0.573656 + 0.819097i \(0.305525\pi\)
\(84\) 0 0
\(85\) −4.24264 −0.460179
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.4650 −1.53329 −0.766646 0.642070i \(-0.778076\pi\)
−0.766646 + 0.642070i \(0.778076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.82843i 0.290191i
\(96\) 0 0
\(97\) 2.74444i 0.278656i 0.990246 + 0.139328i \(0.0444942\pi\)
−0.990246 + 0.139328i \(0.955506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.74444 0.273082 0.136541 0.990634i \(-0.456401\pi\)
0.136541 + 0.990634i \(0.456401\pi\)
\(102\) 0 0
\(103\) 0.634051i 0.0624749i 0.999512 + 0.0312374i \(0.00994480\pi\)
−0.999512 + 0.0312374i \(0.990055\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.8284i − 1.24017i −0.784534 0.620085i \(-0.787098\pi\)
0.784534 0.620085i \(-0.212902\pi\)
\(108\) 0 0
\(109\) 3.07107 0.294155 0.147077 0.989125i \(-0.453013\pi\)
0.147077 + 0.989125i \(0.453013\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.4142i 1.26190i 0.775822 + 0.630952i \(0.217335\pi\)
−0.775822 + 0.630952i \(0.782665\pi\)
\(114\) 0 0
\(115\) 16.3128i 1.52118i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.1689 1.08842
\(126\) 0 0
\(127\) 6.82843 0.605925 0.302962 0.953002i \(-0.402024\pi\)
0.302962 + 0.953002i \(0.402024\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 11.3492 0.991583 0.495792 0.868442i \(-0.334878\pi\)
0.495792 + 0.868442i \(0.334878\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 5.17157i − 0.441837i −0.975292 0.220919i \(-0.929094\pi\)
0.975292 0.220919i \(-0.0709055\pi\)
\(138\) 0 0
\(139\) 6.49435i 0.550844i 0.961323 + 0.275422i \(0.0888176\pi\)
−0.961323 + 0.275422i \(0.911182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.92177 0.746076
\(144\) 0 0
\(145\) − 2.16478i − 0.179776i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 7.07107i − 0.579284i −0.957135 0.289642i \(-0.906464\pi\)
0.957135 0.289642i \(-0.0935363\pi\)
\(150\) 0 0
\(151\) −18.1421 −1.47639 −0.738193 0.674590i \(-0.764321\pi\)
−0.738193 + 0.674590i \(0.764321\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 10.8284i − 0.869760i
\(156\) 0 0
\(157\) − 11.2179i − 0.895284i −0.894213 0.447642i \(-0.852264\pi\)
0.894213 0.447642i \(-0.147736\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.82843 −0.221540 −0.110770 0.993846i \(-0.535332\pi\)
−0.110770 + 0.993846i \(0.535332\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.3128 −1.26232 −0.631161 0.775652i \(-0.717421\pi\)
−0.631161 + 0.775652i \(0.717421\pi\)
\(168\) 0 0
\(169\) −6.89949 −0.530730
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −12.1146 −0.921052 −0.460526 0.887646i \(-0.652339\pi\)
−0.460526 + 0.887646i \(0.652339\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 13.3137i − 0.995113i −0.867431 0.497557i \(-0.834231\pi\)
0.867431 0.497557i \(-0.165769\pi\)
\(180\) 0 0
\(181\) 14.4650i 1.07518i 0.843207 + 0.537589i \(0.180665\pi\)
−0.843207 + 0.537589i \(0.819335\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −15.2304 −1.11976
\(186\) 0 0
\(187\) − 4.59220i − 0.335815i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.31371i 0.384486i 0.981347 + 0.192243i \(0.0615763\pi\)
−0.981347 + 0.192243i \(0.938424\pi\)
\(192\) 0 0
\(193\) 1.65685 0.119263 0.0596315 0.998220i \(-0.481007\pi\)
0.0596315 + 0.998220i \(0.481007\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.0711i 0.788781i 0.918943 + 0.394390i \(0.129044\pi\)
−0.918943 + 0.394390i \(0.870956\pi\)
\(198\) 0 0
\(199\) 3.06147i 0.217022i 0.994095 + 0.108511i \(0.0346082\pi\)
−0.994095 + 0.108511i \(0.965392\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −21.8995 −1.52953
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.06147 −0.211766
\(210\) 0 0
\(211\) 26.6274 1.83311 0.916553 0.399912i \(-0.130959\pi\)
0.916553 + 0.399912i \(0.130959\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.16478 −0.147637
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 10.2426i 0.688995i
\(222\) 0 0
\(223\) − 21.2764i − 1.42477i −0.701786 0.712387i \(-0.747614\pi\)
0.701786 0.712387i \(-0.252386\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.86030 0.388962 0.194481 0.980906i \(-0.437698\pi\)
0.194481 + 0.980906i \(0.437698\pi\)
\(228\) 0 0
\(229\) 13.1969i 0.872079i 0.899928 + 0.436039i \(0.143619\pi\)
−0.899928 + 0.436039i \(0.856381\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.8284i 1.36452i 0.731112 + 0.682258i \(0.239002\pi\)
−0.731112 + 0.682258i \(0.760998\pi\)
\(234\) 0 0
\(235\) 14.8284 0.967300
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 3.17157i − 0.205152i −0.994725 0.102576i \(-0.967292\pi\)
0.994725 0.102576i \(-0.0327085\pi\)
\(240\) 0 0
\(241\) 6.25425i 0.402872i 0.979502 + 0.201436i \(0.0645608\pi\)
−0.979502 + 0.201436i \(0.935439\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.82843 0.434482
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −19.7457 −1.24634 −0.623169 0.782088i \(-0.714155\pi\)
−0.623169 + 0.782088i \(0.714155\pi\)
\(252\) 0 0
\(253\) −17.6569 −1.11008
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.1200 0.818405 0.409202 0.912444i \(-0.365807\pi\)
0.409202 + 0.912444i \(0.365807\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.34315i 0.514460i 0.966350 + 0.257230i \(0.0828099\pi\)
−0.966350 + 0.257230i \(0.917190\pi\)
\(264\) 0 0
\(265\) 6.94269i 0.426486i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −24.7318 −1.50793 −0.753964 0.656916i \(-0.771860\pi\)
−0.753964 + 0.656916i \(0.771860\pi\)
\(270\) 0 0
\(271\) 23.7038i 1.43991i 0.694023 + 0.719953i \(0.255837\pi\)
−0.694023 + 0.719953i \(0.744163\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.17157i 0.191253i
\(276\) 0 0
\(277\) −20.9706 −1.26000 −0.630000 0.776596i \(-0.716945\pi\)
−0.630000 + 0.776596i \(0.716945\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.48528i 0.386879i 0.981112 + 0.193440i \(0.0619644\pi\)
−0.981112 + 0.193440i \(0.938036\pi\)
\(282\) 0 0
\(283\) − 31.9916i − 1.90170i −0.309650 0.950850i \(-0.600212\pi\)
0.309650 0.950850i \(-0.399788\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −11.7279 −0.689878
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.34211 0.487351 0.243676 0.969857i \(-0.421647\pi\)
0.243676 + 0.969857i \(0.421647\pi\)
\(294\) 0 0
\(295\) −18.1421 −1.05628
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 39.3826 2.27755
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 22.7279i 1.30140i
\(306\) 0 0
\(307\) − 28.5587i − 1.62993i −0.579510 0.814965i \(-0.696756\pi\)
0.579510 0.814965i \(-0.303244\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.9105 1.24243 0.621215 0.783641i \(-0.286640\pi\)
0.621215 + 0.783641i \(0.286640\pi\)
\(312\) 0 0
\(313\) − 22.6758i − 1.28171i −0.767660 0.640857i \(-0.778579\pi\)
0.767660 0.640857i \(-0.221421\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.2426i 1.36160i 0.732468 + 0.680801i \(0.238368\pi\)
−0.732468 + 0.680801i \(0.761632\pi\)
\(318\) 0 0
\(319\) 2.34315 0.131191
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 3.51472i − 0.195564i
\(324\) 0 0
\(325\) − 7.07401i − 0.392396i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.68629 −0.257582 −0.128791 0.991672i \(-0.541110\pi\)
−0.128791 + 0.991672i \(0.541110\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −23.0698 −1.26044
\(336\) 0 0
\(337\) −12.9289 −0.704284 −0.352142 0.935947i \(-0.614547\pi\)
−0.352142 + 0.935947i \(0.614547\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 11.7206 0.634706
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 9.79899i − 0.526037i −0.964791 0.263019i \(-0.915282\pi\)
0.964791 0.263019i \(-0.0847181\pi\)
\(348\) 0 0
\(349\) − 30.0669i − 1.60944i −0.593652 0.804722i \(-0.702315\pi\)
0.593652 0.804722i \(-0.297685\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −32.5712 −1.73359 −0.866796 0.498664i \(-0.833824\pi\)
−0.866796 + 0.498664i \(0.833824\pi\)
\(354\) 0 0
\(355\) 24.6005i 1.30566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 31.4558i 1.66018i 0.557632 + 0.830088i \(0.311710\pi\)
−0.557632 + 0.830088i \(0.688290\pi\)
\(360\) 0 0
\(361\) 16.6569 0.876677
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.07107i 0.265432i
\(366\) 0 0
\(367\) 27.9246i 1.45765i 0.684698 + 0.728827i \(0.259934\pi\)
−0.684698 + 0.728827i \(0.740066\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −14.6274 −0.757379 −0.378689 0.925524i \(-0.623625\pi\)
−0.378689 + 0.925524i \(0.623625\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.22625 −0.269166
\(378\) 0 0
\(379\) 20.2843 1.04193 0.520967 0.853577i \(-0.325572\pi\)
0.520967 + 0.853577i \(0.325572\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −6.12293 −0.312867 −0.156434 0.987688i \(-0.550000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.82843i 0.346215i 0.984903 + 0.173107i \(0.0553808\pi\)
−0.984903 + 0.173107i \(0.944619\pi\)
\(390\) 0 0
\(391\) − 20.2710i − 1.02515i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −20.9050 −1.05185
\(396\) 0 0
\(397\) 7.97069i 0.400038i 0.979792 + 0.200019i \(0.0641003\pi\)
−0.979792 + 0.200019i \(0.935900\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.4853i 1.12286i 0.827524 + 0.561431i \(0.189749\pi\)
−0.827524 + 0.561431i \(0.810251\pi\)
\(402\) 0 0
\(403\) −26.1421 −1.30223
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 16.4853i − 0.817145i
\(408\) 0 0
\(409\) − 28.6131i − 1.41483i −0.706801 0.707413i \(-0.749862\pi\)
0.706801 0.707413i \(-0.250138\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −19.3137 −0.948073
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.5194 −0.709321 −0.354661 0.934995i \(-0.615404\pi\)
−0.354661 + 0.934995i \(0.615404\pi\)
\(420\) 0 0
\(421\) −7.31371 −0.356448 −0.178224 0.983990i \(-0.557035\pi\)
−0.178224 + 0.983990i \(0.557035\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.64113 −0.176621
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.6863i 0.514741i 0.966313 + 0.257370i \(0.0828560\pi\)
−0.966313 + 0.257370i \(0.917144\pi\)
\(432\) 0 0
\(433\) 19.6913i 0.946303i 0.880981 + 0.473152i \(0.156884\pi\)
−0.880981 + 0.473152i \(0.843116\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −13.5140 −0.646461
\(438\) 0 0
\(439\) − 18.7402i − 0.894422i −0.894428 0.447211i \(-0.852417\pi\)
0.894428 0.447211i \(-0.147583\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.68629i 0.317675i 0.987305 + 0.158838i \(0.0507746\pi\)
−0.987305 + 0.158838i \(0.949225\pi\)
\(444\) 0 0
\(445\) 26.7279 1.26703
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 19.5563i 0.922921i 0.887161 + 0.461461i \(0.152674\pi\)
−0.887161 + 0.461461i \(0.847326\pi\)
\(450\) 0 0
\(451\) − 23.7038i − 1.11617i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −28.1647 −1.31176 −0.655881 0.754864i \(-0.727703\pi\)
−0.655881 + 0.754864i \(0.727703\pi\)
\(462\) 0 0
\(463\) 2.82843 0.131448 0.0657241 0.997838i \(-0.479064\pi\)
0.0657241 + 0.997838i \(0.479064\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 37.2178 1.72224 0.861118 0.508406i \(-0.169765\pi\)
0.861118 + 0.508406i \(0.169765\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 2.34315i − 0.107738i
\(474\) 0 0
\(475\) 2.42742i 0.111378i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.9719 1.14100 0.570499 0.821299i \(-0.306750\pi\)
0.570499 + 0.821299i \(0.306750\pi\)
\(480\) 0 0
\(481\) 36.7695i 1.67654i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 5.07107i − 0.230265i
\(486\) 0 0
\(487\) −19.7990 −0.897178 −0.448589 0.893738i \(-0.648073\pi\)
−0.448589 + 0.893738i \(0.648073\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 17.5147i − 0.790428i −0.918589 0.395214i \(-0.870670\pi\)
0.918589 0.395214i \(-0.129330\pi\)
\(492\) 0 0
\(493\) 2.69005i 0.121154i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2.14214 −0.0958952 −0.0479476 0.998850i \(-0.515268\pi\)
−0.0479476 + 0.998850i \(0.515268\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.2459 −0.546016 −0.273008 0.962012i \(-0.588019\pi\)
−0.273008 + 0.962012i \(0.588019\pi\)
\(504\) 0 0
\(505\) −5.07107 −0.225660
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.35757 −0.237470 −0.118735 0.992926i \(-0.537884\pi\)
−0.118735 + 0.992926i \(0.537884\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 1.17157i − 0.0516257i
\(516\) 0 0
\(517\) 16.0502i 0.705886i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.9958 0.700788 0.350394 0.936602i \(-0.386048\pi\)
0.350394 + 0.936602i \(0.386048\pi\)
\(522\) 0 0
\(523\) 29.1927i 1.27651i 0.769826 + 0.638254i \(0.220343\pi\)
−0.769826 + 0.638254i \(0.779657\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 13.4558i 0.586146i
\(528\) 0 0
\(529\) −54.9411 −2.38874
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 52.8701i 2.29006i
\(534\) 0 0
\(535\) 23.7038i 1.02481i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 19.6569 0.845114 0.422557 0.906336i \(-0.361133\pi\)
0.422557 + 0.906336i \(0.361133\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.67459 −0.243073
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.79337 0.0764000
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 8.72792i − 0.369814i −0.982756 0.184907i \(-0.940802\pi\)
0.982756 0.184907i \(-0.0591984\pi\)
\(558\) 0 0
\(559\) 5.22625i 0.221047i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −42.8155 −1.80446 −0.902229 0.431258i \(-0.858070\pi\)
−0.902229 + 0.431258i \(0.858070\pi\)
\(564\) 0 0
\(565\) − 24.7862i − 1.04276i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 16.4853i − 0.691099i −0.938401 0.345549i \(-0.887693\pi\)
0.938401 0.345549i \(-0.112307\pi\)
\(570\) 0 0
\(571\) 42.6274 1.78390 0.891951 0.452132i \(-0.149336\pi\)
0.891951 + 0.452132i \(0.149336\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.0000i 0.583840i
\(576\) 0 0
\(577\) − 14.7277i − 0.613121i −0.951851 0.306561i \(-0.900822\pi\)
0.951851 0.306561i \(-0.0991782\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −7.51472 −0.311228
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.9414 0.657971 0.328986 0.944335i \(-0.393293\pi\)
0.328986 + 0.944335i \(0.393293\pi\)
\(588\) 0 0
\(589\) 8.97056 0.369626
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4.08947 −0.167934 −0.0839671 0.996469i \(-0.526759\pi\)
−0.0839671 + 0.996469i \(0.526759\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 13.3137i 0.543983i 0.962300 + 0.271992i \(0.0876823\pi\)
−0.962300 + 0.271992i \(0.912318\pi\)
\(600\) 0 0
\(601\) − 42.0501i − 1.71526i −0.514267 0.857630i \(-0.671936\pi\)
0.514267 0.857630i \(-0.328064\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −12.9343 −0.525855
\(606\) 0 0
\(607\) − 0.896683i − 0.0363952i −0.999834 0.0181976i \(-0.994207\pi\)
0.999834 0.0181976i \(-0.00579280\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 35.7990i − 1.44827i
\(612\) 0 0
\(613\) −2.78680 −0.112558 −0.0562788 0.998415i \(-0.517924\pi\)
−0.0562788 + 0.998415i \(0.517924\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 33.4558i − 1.34688i −0.739241 0.673441i \(-0.764816\pi\)
0.739241 0.673441i \(-0.235184\pi\)
\(618\) 0 0
\(619\) 25.2346i 1.01426i 0.861869 + 0.507132i \(0.169294\pi\)
−0.861869 + 0.507132i \(0.830706\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −14.5563 −0.582254
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 18.9259 0.754626
\(630\) 0 0
\(631\) 4.48528 0.178556 0.0892781 0.996007i \(-0.471544\pi\)
0.0892781 + 0.996007i \(0.471544\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −12.6173 −0.500702
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 27.4558i − 1.08444i −0.840236 0.542220i \(-0.817584\pi\)
0.840236 0.542220i \(-0.182416\pi\)
\(642\) 0 0
\(643\) 47.1451i 1.85922i 0.368546 + 0.929610i \(0.379856\pi\)
−0.368546 + 0.929610i \(0.620144\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 36.3211 1.42793 0.713965 0.700181i \(-0.246897\pi\)
0.713965 + 0.700181i \(0.246897\pi\)
\(648\) 0 0
\(649\) − 19.6369i − 0.770816i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.1421i 1.33609i 0.744123 + 0.668043i \(0.232868\pi\)
−0.744123 + 0.668043i \(0.767132\pi\)
\(654\) 0 0
\(655\) −20.9706 −0.819388
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 30.9706i − 1.20644i −0.797574 0.603221i \(-0.793884\pi\)
0.797574 0.603221i \(-0.206116\pi\)
\(660\) 0 0
\(661\) − 1.02800i − 0.0399845i −0.999800 0.0199923i \(-0.993636\pi\)
0.999800 0.0199923i \(-0.00636416\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10.3431 0.400488
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −24.6005 −0.949693
\(672\) 0 0
\(673\) −28.0416 −1.08093 −0.540463 0.841368i \(-0.681751\pi\)
−0.540463 + 0.841368i \(0.681751\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −10.0586 −0.386582 −0.193291 0.981142i \(-0.561916\pi\)
−0.193291 + 0.981142i \(0.561916\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 13.0294i − 0.498558i −0.968432 0.249279i \(-0.919806\pi\)
0.968432 0.249279i \(-0.0801935\pi\)
\(684\) 0 0
\(685\) 9.55582i 0.365109i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 16.7611 0.638549
\(690\) 0 0
\(691\) − 32.2542i − 1.22701i −0.789692 0.613504i \(-0.789760\pi\)
0.789692 0.613504i \(-0.210240\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 12.0000i − 0.455186i
\(696\) 0 0
\(697\) 27.2132 1.03077
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 35.4558i − 1.33915i −0.742745 0.669574i \(-0.766477\pi\)
0.742745 0.669574i \(-0.233523\pi\)
\(702\) 0 0
\(703\) − 12.6173i − 0.475870i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 22.8701 0.858903 0.429452 0.903090i \(-0.358707\pi\)
0.429452 + 0.903090i \(0.358707\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 51.7373 1.93758
\(714\) 0 0
\(715\) −16.4853 −0.616515
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −26.5027 −0.988383 −0.494192 0.869353i \(-0.664536\pi\)
−0.494192 + 0.869353i \(0.664536\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 1.85786i − 0.0689994i
\(726\) 0 0
\(727\) − 25.3434i − 0.939933i −0.882684 0.469967i \(-0.844266\pi\)
0.882684 0.469967i \(-0.155734\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.69005 0.0994951
\(732\) 0 0
\(733\) − 1.58513i − 0.0585480i −0.999571 0.0292740i \(-0.990680\pi\)
0.999571 0.0292740i \(-0.00931953\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 24.9706i − 0.919803i
\(738\) 0 0
\(739\) 53.4558 1.96641 0.983203 0.182518i \(-0.0584248\pi\)
0.983203 + 0.182518i \(0.0584248\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 32.3431i 1.18655i 0.804998 + 0.593277i \(0.202166\pi\)
−0.804998 + 0.593277i \(0.797834\pi\)
\(744\) 0 0
\(745\) 13.0656i 0.478688i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 20.2843 0.740184 0.370092 0.928995i \(-0.379326\pi\)
0.370092 + 0.928995i \(0.379326\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 33.5223 1.22000
\(756\) 0 0
\(757\) 37.8995 1.37748 0.688740 0.725008i \(-0.258164\pi\)
0.688740 + 0.725008i \(0.258164\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −14.5738 −0.528301 −0.264151 0.964481i \(-0.585092\pi\)
−0.264151 + 0.964481i \(0.585092\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 43.7990i 1.58149i
\(768\) 0 0
\(769\) 13.5684i 0.489288i 0.969613 + 0.244644i \(0.0786710\pi\)
−0.969613 + 0.244644i \(0.921329\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18.2375 0.655957 0.327978 0.944685i \(-0.393633\pi\)
0.327978 + 0.944685i \(0.393633\pi\)
\(774\) 0 0
\(775\) − 9.29319i − 0.333821i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 18.1421i − 0.650009i
\(780\) 0 0
\(781\) −26.6274 −0.952804
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 20.7279i 0.739811i
\(786\) 0 0
\(787\) − 3.58673i − 0.127853i −0.997955 0.0639266i \(-0.979638\pi\)
0.997955 0.0639266i \(-0.0203624\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 54.8701 1.94849
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 30.4064 1.07705 0.538526 0.842609i \(-0.318982\pi\)
0.538526 + 0.842609i \(0.318982\pi\)
\(798\) 0 0
\(799\) −18.4264 −0.651879
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −5.48888 −0.193699
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 21.2132i 0.745817i 0.927868 + 0.372908i \(0.121639\pi\)
−0.927868 + 0.372908i \(0.878361\pi\)
\(810\) 0 0
\(811\) − 13.1426i − 0.461497i −0.973013 0.230749i \(-0.925882\pi\)
0.973013 0.230749i \(-0.0741175\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.22625 0.183068
\(816\) 0 0
\(817\) − 1.79337i − 0.0627419i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 36.9289i − 1.28883i −0.764677 0.644414i \(-0.777101\pi\)
0.764677 0.644414i \(-0.222899\pi\)
\(822\) 0 0
\(823\) 16.2843 0.567634 0.283817 0.958878i \(-0.408399\pi\)
0.283817 + 0.958878i \(0.408399\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1.51472i − 0.0526719i −0.999653 0.0263360i \(-0.991616\pi\)
0.999653 0.0263360i \(-0.00838397\pi\)
\(828\) 0 0
\(829\) 6.99709i 0.243019i 0.992590 + 0.121509i \(0.0387735\pi\)
−0.992590 + 0.121509i \(0.961227\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 30.1421 1.04311
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 16.6842 0.576003 0.288002 0.957630i \(-0.407009\pi\)
0.288002 + 0.957630i \(0.407009\pi\)
\(840\) 0 0
\(841\) 27.6274 0.952670
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.7486 0.438565
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 72.7696i − 2.49451i
\(852\) 0 0
\(853\) 23.7264i 0.812376i 0.913790 + 0.406188i \(0.133142\pi\)
−0.913790 + 0.406188i \(0.866858\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 40.2249 1.37406 0.687028 0.726631i \(-0.258915\pi\)
0.687028 + 0.726631i \(0.258915\pi\)
\(858\) 0 0
\(859\) 33.2597i 1.13481i 0.823441 + 0.567403i \(0.192052\pi\)
−0.823441 + 0.567403i \(0.807948\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 37.1127i − 1.26333i −0.775241 0.631665i \(-0.782372\pi\)
0.775241 0.631665i \(-0.217628\pi\)
\(864\) 0 0
\(865\) 22.3848 0.761105
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 22.6274i − 0.767583i
\(870\) 0 0
\(871\) 55.6954i 1.88717i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 15.0711 0.508914 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −16.8155 −0.566530 −0.283265 0.959042i \(-0.591418\pi\)
−0.283265 + 0.959042i \(0.591418\pi\)
\(882\) 0 0
\(883\) −50.1421 −1.68742 −0.843709 0.536801i \(-0.819632\pi\)
−0.843709 + 0.536801i \(0.819632\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −48.0417 −1.61308 −0.806542 0.591177i \(-0.798663\pi\)
−0.806542 + 0.591177i \(0.798663\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 12.2843i 0.411077i
\(894\) 0 0
\(895\) 24.6005i 0.822305i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −6.86577 −0.228986
\(900\) 0 0
\(901\) − 8.62727i − 0.287416i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 26.7279i − 0.888466i
\(906\) 0 0
\(907\) −15.5147 −0.515158 −0.257579 0.966257i \(-0.582925\pi\)
−0.257579 + 0.966257i \(0.582925\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 14.2843i − 0.473259i −0.971600 0.236630i \(-0.923957\pi\)
0.971600 0.236630i \(-0.0760428\pi\)
\(912\) 0 0
\(913\) − 20.9050i − 0.691855i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −32.4853 −1.07159 −0.535795 0.844348i \(-0.679988\pi\)
−0.535795 + 0.844348i \(0.679988\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 59.3909 1.95488
\(924\) 0 0
\(925\) −13.0711 −0.429774
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.58513 −0.0520063 −0.0260032 0.999662i \(-0.508278\pi\)
−0.0260032 + 0.999662i \(0.508278\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 8.48528i 0.277498i
\(936\) 0 0
\(937\) 45.1116i 1.47373i 0.676039 + 0.736866i \(0.263695\pi\)
−0.676039 + 0.736866i \(0.736305\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 42.3127 1.37936 0.689678 0.724116i \(-0.257752\pi\)
0.689678 + 0.724116i \(0.257752\pi\)
\(942\) 0 0
\(943\) − 104.634i − 3.40735i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.17157i − 0.103062i −0.998671 0.0515311i \(-0.983590\pi\)
0.998671 0.0515311i \(-0.0164101\pi\)
\(948\) 0 0
\(949\) 12.2426 0.397413
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 5.12994i 0.166175i 0.996542 + 0.0830876i \(0.0264781\pi\)
−0.996542 + 0.0830876i \(0.973522\pi\)
\(954\) 0 0
\(955\) − 9.81845i − 0.317718i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −3.34315 −0.107843
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −3.06147 −0.0985521
\(966\) 0 0
\(967\) 32.4853 1.04466 0.522328 0.852745i \(-0.325064\pi\)
0.522328 + 0.852745i \(0.325064\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.79337 0.0575519 0.0287759 0.999586i \(-0.490839\pi\)
0.0287759 + 0.999586i \(0.490839\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 16.4853i − 0.527411i −0.964603 0.263705i \(-0.915055\pi\)
0.964603 0.263705i \(-0.0849447\pi\)
\(978\) 0 0
\(979\) 28.9301i 0.924610i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −27.3994 −0.873904 −0.436952 0.899485i \(-0.643942\pi\)
−0.436952 + 0.899485i \(0.643942\pi\)
\(984\) 0 0
\(985\) − 20.4567i − 0.651804i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 10.3431i − 0.328893i
\(990\) 0 0
\(991\) −38.1421 −1.21162 −0.605812 0.795608i \(-0.707152\pi\)
−0.605812 + 0.795608i \(0.707152\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 5.65685i − 0.179334i
\(996\) 0 0
\(997\) 41.0446i 1.29990i 0.759978 + 0.649949i \(0.225210\pi\)
−0.759978 + 0.649949i \(0.774790\pi\)
\(998\) 0 0
\(999\) 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 1764.2.f.b.881.1 8
3.2 odd 2 inner 1764.2.f.b.881.8 yes 8
4.3 odd 2 7056.2.k.e.881.2 8
7.2 even 3 1764.2.t.c.521.8 16
7.3 odd 6 1764.2.t.c.1097.1 16
7.4 even 3 1764.2.t.c.1097.7 16
7.5 odd 6 1764.2.t.c.521.2 16
7.6 odd 2 inner 1764.2.f.b.881.7 yes 8
12.11 even 2 7056.2.k.e.881.7 8
21.2 odd 6 1764.2.t.c.521.1 16
21.5 even 6 1764.2.t.c.521.7 16
21.11 odd 6 1764.2.t.c.1097.2 16
21.17 even 6 1764.2.t.c.1097.8 16
21.20 even 2 inner 1764.2.f.b.881.2 yes 8
28.27 even 2 7056.2.k.e.881.8 8
84.83 odd 2 7056.2.k.e.881.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.2.f.b.881.1 8 1.1 even 1 trivial
1764.2.f.b.881.2 yes 8 21.20 even 2 inner
1764.2.f.b.881.7 yes 8 7.6 odd 2 inner
1764.2.f.b.881.8 yes 8 3.2 odd 2 inner
1764.2.t.c.521.1 16 21.2 odd 6
1764.2.t.c.521.2 16 7.5 odd 6
1764.2.t.c.521.7 16 21.5 even 6
1764.2.t.c.521.8 16 7.2 even 3
1764.2.t.c.1097.1 16 7.3 odd 6
1764.2.t.c.1097.2 16 21.11 odd 6
1764.2.t.c.1097.7 16 7.4 even 3
1764.2.t.c.1097.8 16 21.17 even 6
7056.2.k.e.881.1 8 84.83 odd 2
7056.2.k.e.881.2 8 4.3 odd 2
7056.2.k.e.881.7 8 12.11 even 2
7056.2.k.e.881.8 8 28.27 even 2