Properties

Label 1089.3.b.h.485.1
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.7
Defining polynomial: \(x^{8} + 12 x^{6} + 23 x^{4} + 12 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 485.1
Root \(0.319918i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.h.485.8

$q$-expansion

\(f(q)\) \(=\) \(q-2.80588i q^{2} -3.87298 q^{4} -2.03151i q^{5} +0.504017 q^{7} -0.356394i q^{8} +O(q^{10})\) \(q-2.80588i q^{2} -3.87298 q^{4} -2.03151i q^{5} +0.504017 q^{7} -0.356394i q^{8} -5.70017 q^{10} -22.0767 q^{13} -1.41421i q^{14} -16.4919 q^{16} +11.9816i q^{17} +5.92017 q^{19} +7.86799i q^{20} -42.7628i q^{23} +20.8730 q^{25} +61.9445i q^{26} -1.95205 q^{28} +36.4765i q^{29} -3.49193 q^{31} +44.8489i q^{32} +33.6190 q^{34} -1.02391i q^{35} -32.8569 q^{37} -16.6113i q^{38} -0.724016 q^{40} +27.2102i q^{41} -58.0737 q^{43} -119.988 q^{46} -11.8298i q^{47} -48.7460 q^{49} -58.5672i q^{50} +85.5025 q^{52} +61.7877i q^{53} -0.179629i q^{56} +102.349 q^{58} +59.9359i q^{59} -49.5055 q^{61} +9.79796i q^{62} +59.8730 q^{64} +44.8489i q^{65} +108.333 q^{67} -46.4045i q^{68} -2.87298 q^{70} +51.4049i q^{71} -123.204 q^{73} +92.1925i q^{74} -22.9287 q^{76} -9.12820 q^{79} +33.5035i q^{80} +76.3488 q^{82} -88.2722i q^{83} +24.3407 q^{85} +162.948i q^{86} -4.52349i q^{89} -11.1270 q^{91} +165.620i q^{92} -33.1930 q^{94} -12.0269i q^{95} +71.1109 q^{97} +136.776i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q - 8 q^{16} + 136 q^{25} + 96 q^{31} + 176 q^{34} + 16 q^{37} - 328 q^{49} + 416 q^{58} + 448 q^{64} + 216 q^{67} + 8 q^{70} + 208 q^{82} - 120 q^{91} + 352 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.80588i − 1.40294i −0.712698 0.701471i \(-0.752527\pi\)
0.712698 0.701471i \(-0.247473\pi\)
\(3\) 0 0
\(4\) −3.87298 −0.968246
\(5\) − 2.03151i − 0.406301i −0.979148 0.203151i \(-0.934882\pi\)
0.979148 0.203151i \(-0.0651181\pi\)
\(6\) 0 0
\(7\) 0.504017 0.0720025 0.0360012 0.999352i \(-0.488538\pi\)
0.0360012 + 0.999352i \(0.488538\pi\)
\(8\) − 0.356394i − 0.0445492i
\(9\) 0 0
\(10\) −5.70017 −0.570017
\(11\) 0 0
\(12\) 0 0
\(13\) −22.0767 −1.69820 −0.849102 0.528228i \(-0.822857\pi\)
−0.849102 + 0.528228i \(0.822857\pi\)
\(14\) − 1.41421i − 0.101015i
\(15\) 0 0
\(16\) −16.4919 −1.03075
\(17\) 11.9816i 0.704799i 0.935850 + 0.352400i \(0.114634\pi\)
−0.935850 + 0.352400i \(0.885366\pi\)
\(18\) 0 0
\(19\) 5.92017 0.311588 0.155794 0.987790i \(-0.450206\pi\)
0.155794 + 0.987790i \(0.450206\pi\)
\(20\) 7.86799i 0.393399i
\(21\) 0 0
\(22\) 0 0
\(23\) − 42.7628i − 1.85925i −0.368502 0.929627i \(-0.620129\pi\)
0.368502 0.929627i \(-0.379871\pi\)
\(24\) 0 0
\(25\) 20.8730 0.834919
\(26\) 61.9445i 2.38248i
\(27\) 0 0
\(28\) −1.95205 −0.0697161
\(29\) 36.4765i 1.25781i 0.777482 + 0.628905i \(0.216497\pi\)
−0.777482 + 0.628905i \(0.783503\pi\)
\(30\) 0 0
\(31\) −3.49193 −0.112643 −0.0563215 0.998413i \(-0.517937\pi\)
−0.0563215 + 0.998413i \(0.517937\pi\)
\(32\) 44.8489i 1.40153i
\(33\) 0 0
\(34\) 33.6190 0.988793
\(35\) − 1.02391i − 0.0292547i
\(36\) 0 0
\(37\) −32.8569 −0.888023 −0.444011 0.896021i \(-0.646445\pi\)
−0.444011 + 0.896021i \(0.646445\pi\)
\(38\) − 16.6113i − 0.437140i
\(39\) 0 0
\(40\) −0.724016 −0.0181004
\(41\) 27.2102i 0.663665i 0.943338 + 0.331832i \(0.107667\pi\)
−0.943338 + 0.331832i \(0.892333\pi\)
\(42\) 0 0
\(43\) −58.0737 −1.35055 −0.675276 0.737565i \(-0.735976\pi\)
−0.675276 + 0.737565i \(0.735976\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −119.988 −2.60843
\(47\) − 11.8298i − 0.251697i −0.992049 0.125849i \(-0.959835\pi\)
0.992049 0.125849i \(-0.0401654\pi\)
\(48\) 0 0
\(49\) −48.7460 −0.994816
\(50\) − 58.5672i − 1.17134i
\(51\) 0 0
\(52\) 85.5025 1.64428
\(53\) 61.7877i 1.16581i 0.812542 + 0.582903i \(0.198083\pi\)
−0.812542 + 0.582903i \(0.801917\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 0.179629i − 0.00320765i
\(57\) 0 0
\(58\) 102.349 1.76463
\(59\) 59.9359i 1.01586i 0.861398 + 0.507931i \(0.169590\pi\)
−0.861398 + 0.507931i \(0.830410\pi\)
\(60\) 0 0
\(61\) −49.5055 −0.811565 −0.405782 0.913970i \(-0.633001\pi\)
−0.405782 + 0.913970i \(0.633001\pi\)
\(62\) 9.79796i 0.158032i
\(63\) 0 0
\(64\) 59.8730 0.935515
\(65\) 44.8489i 0.689983i
\(66\) 0 0
\(67\) 108.333 1.61691 0.808453 0.588561i \(-0.200305\pi\)
0.808453 + 0.588561i \(0.200305\pi\)
\(68\) − 46.4045i − 0.682419i
\(69\) 0 0
\(70\) −2.87298 −0.0410426
\(71\) 51.4049i 0.724013i 0.932176 + 0.362007i \(0.117908\pi\)
−0.932176 + 0.362007i \(0.882092\pi\)
\(72\) 0 0
\(73\) −123.204 −1.68772 −0.843861 0.536561i \(-0.819723\pi\)
−0.843861 + 0.536561i \(0.819723\pi\)
\(74\) 92.1925i 1.24584i
\(75\) 0 0
\(76\) −22.9287 −0.301694
\(77\) 0 0
\(78\) 0 0
\(79\) −9.12820 −0.115547 −0.0577734 0.998330i \(-0.518400\pi\)
−0.0577734 + 0.998330i \(0.518400\pi\)
\(80\) 33.5035i 0.418793i
\(81\) 0 0
\(82\) 76.3488 0.931083
\(83\) − 88.2722i − 1.06352i −0.846895 0.531760i \(-0.821531\pi\)
0.846895 0.531760i \(-0.178469\pi\)
\(84\) 0 0
\(85\) 24.3407 0.286361
\(86\) 162.948i 1.89475i
\(87\) 0 0
\(88\) 0 0
\(89\) − 4.52349i − 0.0508257i −0.999677 0.0254129i \(-0.991910\pi\)
0.999677 0.0254129i \(-0.00809004\pi\)
\(90\) 0 0
\(91\) −11.1270 −0.122275
\(92\) 165.620i 1.80022i
\(93\) 0 0
\(94\) −33.1930 −0.353117
\(95\) − 12.0269i − 0.126599i
\(96\) 0 0
\(97\) 71.1109 0.733102 0.366551 0.930398i \(-0.380539\pi\)
0.366551 + 0.930398i \(0.380539\pi\)
\(98\) 136.776i 1.39567i
\(99\) 0 0
\(100\) −80.8407 −0.808407
\(101\) 56.7909i 0.562287i 0.959666 + 0.281143i \(0.0907136\pi\)
−0.959666 + 0.281143i \(0.909286\pi\)
\(102\) 0 0
\(103\) 168.619 1.63708 0.818539 0.574452i \(-0.194785\pi\)
0.818539 + 0.574452i \(0.194785\pi\)
\(104\) 7.86799i 0.0756537i
\(105\) 0 0
\(106\) 173.369 1.63556
\(107\) 138.818i 1.29736i 0.761061 + 0.648680i \(0.224679\pi\)
−0.761061 + 0.648680i \(0.775321\pi\)
\(108\) 0 0
\(109\) 130.288 1.19530 0.597651 0.801756i \(-0.296101\pi\)
0.597651 + 0.801756i \(0.296101\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −8.31222 −0.0742162
\(113\) − 8.56369i − 0.0757848i −0.999282 0.0378924i \(-0.987936\pi\)
0.999282 0.0378924i \(-0.0120644\pi\)
\(114\) 0 0
\(115\) −86.8730 −0.755417
\(116\) − 141.273i − 1.21787i
\(117\) 0 0
\(118\) 168.173 1.42520
\(119\) 6.03893i 0.0507473i
\(120\) 0 0
\(121\) 0 0
\(122\) 138.907i 1.13858i
\(123\) 0 0
\(124\) 13.5242 0.109066
\(125\) − 93.1912i − 0.745530i
\(126\) 0 0
\(127\) −146.756 −1.15556 −0.577781 0.816192i \(-0.696081\pi\)
−0.577781 + 0.816192i \(0.696081\pi\)
\(128\) 11.3989i 0.0890536i
\(129\) 0 0
\(130\) 125.841 0.968006
\(131\) − 182.026i − 1.38951i −0.719246 0.694756i \(-0.755512\pi\)
0.719246 0.694756i \(-0.244488\pi\)
\(132\) 0 0
\(133\) 2.98387 0.0224351
\(134\) − 303.969i − 2.26842i
\(135\) 0 0
\(136\) 4.27017 0.0313983
\(137\) − 156.416i − 1.14172i −0.821046 0.570861i \(-0.806609\pi\)
0.821046 0.570861i \(-0.193391\pi\)
\(138\) 0 0
\(139\) −34.0090 −0.244669 −0.122334 0.992489i \(-0.539038\pi\)
−0.122334 + 0.992489i \(0.539038\pi\)
\(140\) 3.96560i 0.0283257i
\(141\) 0 0
\(142\) 144.236 1.01575
\(143\) 0 0
\(144\) 0 0
\(145\) 74.1022 0.511050
\(146\) 345.695i 2.36778i
\(147\) 0 0
\(148\) 127.254 0.859825
\(149\) 134.377i 0.901859i 0.892559 + 0.450930i \(0.148907\pi\)
−0.892559 + 0.450930i \(0.851093\pi\)
\(150\) 0 0
\(151\) −163.581 −1.08332 −0.541659 0.840598i \(-0.682204\pi\)
−0.541659 + 0.840598i \(0.682204\pi\)
\(152\) − 2.10991i − 0.0138810i
\(153\) 0 0
\(154\) 0 0
\(155\) 7.09388i 0.0457670i
\(156\) 0 0
\(157\) −175.173 −1.11575 −0.557877 0.829924i \(-0.688384\pi\)
−0.557877 + 0.829924i \(0.688384\pi\)
\(158\) 25.6127i 0.162105i
\(159\) 0 0
\(160\) 91.1108 0.569442
\(161\) − 21.5532i − 0.133871i
\(162\) 0 0
\(163\) −144.268 −0.885080 −0.442540 0.896749i \(-0.645923\pi\)
−0.442540 + 0.896749i \(0.645923\pi\)
\(164\) − 105.385i − 0.642590i
\(165\) 0 0
\(166\) −247.681 −1.49206
\(167\) − 260.059i − 1.55724i −0.627496 0.778620i \(-0.715920\pi\)
0.627496 0.778620i \(-0.284080\pi\)
\(168\) 0 0
\(169\) 318.379 1.88390
\(170\) − 68.2971i − 0.401748i
\(171\) 0 0
\(172\) 224.919 1.30767
\(173\) 269.404i 1.55725i 0.627489 + 0.778625i \(0.284083\pi\)
−0.627489 + 0.778625i \(0.715917\pi\)
\(174\) 0 0
\(175\) 10.5203 0.0601162
\(176\) 0 0
\(177\) 0 0
\(178\) −12.6924 −0.0713056
\(179\) − 34.3460i − 0.191877i −0.995387 0.0959386i \(-0.969415\pi\)
0.995387 0.0959386i \(-0.0305852\pi\)
\(180\) 0 0
\(181\) −243.190 −1.34359 −0.671794 0.740738i \(-0.734476\pi\)
−0.671794 + 0.740738i \(0.734476\pi\)
\(182\) 31.2211i 0.171545i
\(183\) 0 0
\(184\) −15.2404 −0.0828284
\(185\) 66.7489i 0.360805i
\(186\) 0 0
\(187\) 0 0
\(188\) 45.8165i 0.243705i
\(189\) 0 0
\(190\) −33.7460 −0.177610
\(191\) − 137.156i − 0.718094i −0.933320 0.359047i \(-0.883102\pi\)
0.933320 0.359047i \(-0.116898\pi\)
\(192\) 0 0
\(193\) 133.624 0.692352 0.346176 0.938170i \(-0.387480\pi\)
0.346176 + 0.938170i \(0.387480\pi\)
\(194\) − 199.529i − 1.02850i
\(195\) 0 0
\(196\) 188.792 0.963226
\(197\) − 166.299i − 0.844160i −0.906559 0.422080i \(-0.861300\pi\)
0.906559 0.422080i \(-0.138700\pi\)
\(198\) 0 0
\(199\) −309.728 −1.55642 −0.778211 0.628004i \(-0.783872\pi\)
−0.778211 + 0.628004i \(0.783872\pi\)
\(200\) − 7.43901i − 0.0371950i
\(201\) 0 0
\(202\) 159.349 0.788855
\(203\) 18.3848i 0.0905654i
\(204\) 0 0
\(205\) 55.2778 0.269648
\(206\) − 473.125i − 2.29672i
\(207\) 0 0
\(208\) 364.087 1.75042
\(209\) 0 0
\(210\) 0 0
\(211\) 207.470 0.983271 0.491635 0.870801i \(-0.336399\pi\)
0.491635 + 0.870801i \(0.336399\pi\)
\(212\) − 239.303i − 1.12879i
\(213\) 0 0
\(214\) 389.506 1.82012
\(215\) 117.977i 0.548731i
\(216\) 0 0
\(217\) −1.75999 −0.00811057
\(218\) − 365.573i − 1.67694i
\(219\) 0 0
\(220\) 0 0
\(221\) − 264.514i − 1.19689i
\(222\) 0 0
\(223\) −288.730 −1.29475 −0.647376 0.762171i \(-0.724134\pi\)
−0.647376 + 0.762171i \(0.724134\pi\)
\(224\) 22.6046i 0.100913i
\(225\) 0 0
\(226\) −24.0287 −0.106322
\(227\) 400.444i 1.76407i 0.471184 + 0.882035i \(0.343827\pi\)
−0.471184 + 0.882035i \(0.656173\pi\)
\(228\) 0 0
\(229\) 12.4597 0.0544090 0.0272045 0.999630i \(-0.491339\pi\)
0.0272045 + 0.999630i \(0.491339\pi\)
\(230\) 243.755i 1.05981i
\(231\) 0 0
\(232\) 13.0000 0.0560345
\(233\) 71.3406i 0.306183i 0.988212 + 0.153091i \(0.0489229\pi\)
−0.988212 + 0.153091i \(0.951077\pi\)
\(234\) 0 0
\(235\) −24.0323 −0.102265
\(236\) − 232.131i − 0.983604i
\(237\) 0 0
\(238\) 16.9445 0.0711955
\(239\) 132.906i 0.556093i 0.960568 + 0.278046i \(0.0896869\pi\)
−0.960568 + 0.278046i \(0.910313\pi\)
\(240\) 0 0
\(241\) −130.884 −0.543087 −0.271543 0.962426i \(-0.587534\pi\)
−0.271543 + 0.962426i \(0.587534\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 191.734 0.785794
\(245\) 99.0277i 0.404195i
\(246\) 0 0
\(247\) −130.698 −0.529140
\(248\) 1.24450i 0.00501816i
\(249\) 0 0
\(250\) −261.484 −1.04594
\(251\) 166.247i 0.662339i 0.943571 + 0.331169i \(0.107443\pi\)
−0.943571 + 0.331169i \(0.892557\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 411.781i 1.62119i
\(255\) 0 0
\(256\) 271.476 1.06045
\(257\) − 389.435i − 1.51531i −0.652655 0.757655i \(-0.726345\pi\)
0.652655 0.757655i \(-0.273655\pi\)
\(258\) 0 0
\(259\) −16.5604 −0.0639398
\(260\) − 173.699i − 0.668073i
\(261\) 0 0
\(262\) −510.744 −1.94940
\(263\) − 272.114i − 1.03465i −0.855788 0.517327i \(-0.826927\pi\)
0.855788 0.517327i \(-0.173073\pi\)
\(264\) 0 0
\(265\) 125.522 0.473668
\(266\) − 8.37238i − 0.0314751i
\(267\) 0 0
\(268\) −419.571 −1.56556
\(269\) 237.350i 0.882341i 0.897423 + 0.441171i \(0.145437\pi\)
−0.897423 + 0.441171i \(0.854563\pi\)
\(270\) 0 0
\(271\) −301.401 −1.11218 −0.556091 0.831122i \(-0.687699\pi\)
−0.556091 + 0.831122i \(0.687699\pi\)
\(272\) − 197.600i − 0.726469i
\(273\) 0 0
\(274\) −438.885 −1.60177
\(275\) 0 0
\(276\) 0 0
\(277\) −15.5326 −0.0560743 −0.0280371 0.999607i \(-0.508926\pi\)
−0.0280371 + 0.999607i \(0.508926\pi\)
\(278\) 95.4252i 0.343256i
\(279\) 0 0
\(280\) −0.364917 −0.00130327
\(281\) − 211.250i − 0.751781i −0.926664 0.375890i \(-0.877337\pi\)
0.926664 0.375890i \(-0.122663\pi\)
\(282\) 0 0
\(283\) 29.6649 0.104823 0.0524114 0.998626i \(-0.483309\pi\)
0.0524114 + 0.998626i \(0.483309\pi\)
\(284\) − 199.090i − 0.701023i
\(285\) 0 0
\(286\) 0 0
\(287\) 13.7144i 0.0477855i
\(288\) 0 0
\(289\) 145.441 0.503258
\(290\) − 207.922i − 0.716973i
\(291\) 0 0
\(292\) 477.166 1.63413
\(293\) − 440.744i − 1.50425i −0.659022 0.752123i \(-0.729030\pi\)
0.659022 0.752123i \(-0.270970\pi\)
\(294\) 0 0
\(295\) 121.760 0.412746
\(296\) 11.7100i 0.0395608i
\(297\) 0 0
\(298\) 377.046 1.26526
\(299\) 944.061i 3.15739i
\(300\) 0 0
\(301\) −29.2702 −0.0972431
\(302\) 458.989i 1.51983i
\(303\) 0 0
\(304\) −97.6350 −0.321168
\(305\) 100.571i 0.329740i
\(306\) 0 0
\(307\) −114.307 −0.372336 −0.186168 0.982518i \(-0.559607\pi\)
−0.186168 + 0.982518i \(0.559607\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 19.9046 0.0642084
\(311\) − 492.303i − 1.58297i −0.611190 0.791484i \(-0.709309\pi\)
0.611190 0.791484i \(-0.290691\pi\)
\(312\) 0 0
\(313\) −202.714 −0.647648 −0.323824 0.946117i \(-0.604969\pi\)
−0.323824 + 0.946117i \(0.604969\pi\)
\(314\) 491.516i 1.56534i
\(315\) 0 0
\(316\) 35.3534 0.111878
\(317\) − 400.582i − 1.26366i −0.775105 0.631832i \(-0.782303\pi\)
0.775105 0.631832i \(-0.217697\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 121.632i − 0.380101i
\(321\) 0 0
\(322\) −60.4758 −0.187813
\(323\) 70.9330i 0.219607i
\(324\) 0 0
\(325\) −460.806 −1.41786
\(326\) 404.800i 1.24172i
\(327\) 0 0
\(328\) 9.69757 0.0295658
\(329\) − 5.96241i − 0.0181228i
\(330\) 0 0
\(331\) 433.077 1.30839 0.654194 0.756327i \(-0.273008\pi\)
0.654194 + 0.756327i \(0.273008\pi\)
\(332\) 341.877i 1.02975i
\(333\) 0 0
\(334\) −729.696 −2.18472
\(335\) − 220.078i − 0.656951i
\(336\) 0 0
\(337\) −403.120 −1.19620 −0.598100 0.801421i \(-0.704078\pi\)
−0.598100 + 0.801421i \(0.704078\pi\)
\(338\) − 893.334i − 2.64300i
\(339\) 0 0
\(340\) −94.2710 −0.277268
\(341\) 0 0
\(342\) 0 0
\(343\) −49.2656 −0.143632
\(344\) 20.6971i 0.0601661i
\(345\) 0 0
\(346\) 755.917 2.18473
\(347\) − 104.418i − 0.300916i −0.988616 0.150458i \(-0.951925\pi\)
0.988616 0.150458i \(-0.0480747\pi\)
\(348\) 0 0
\(349\) −49.2133 −0.141012 −0.0705062 0.997511i \(-0.522461\pi\)
−0.0705062 + 0.997511i \(0.522461\pi\)
\(350\) − 29.5189i − 0.0843396i
\(351\) 0 0
\(352\) 0 0
\(353\) − 234.322i − 0.663801i −0.943314 0.331901i \(-0.892310\pi\)
0.943314 0.331901i \(-0.107690\pi\)
\(354\) 0 0
\(355\) 104.429 0.294167
\(356\) 17.5194i 0.0492118i
\(357\) 0 0
\(358\) −96.3709 −0.269193
\(359\) − 227.101i − 0.632594i −0.948660 0.316297i \(-0.897560\pi\)
0.948660 0.316297i \(-0.102440\pi\)
\(360\) 0 0
\(361\) −325.952 −0.902913
\(362\) 682.361i 1.88498i
\(363\) 0 0
\(364\) 43.0948 0.118392
\(365\) 250.289i 0.685724i
\(366\) 0 0
\(367\) −509.982 −1.38960 −0.694798 0.719205i \(-0.744506\pi\)
−0.694798 + 0.719205i \(0.744506\pi\)
\(368\) 705.242i 1.91642i
\(369\) 0 0
\(370\) 187.290 0.506188
\(371\) 31.1421i 0.0839409i
\(372\) 0 0
\(373\) −416.132 −1.11564 −0.557818 0.829963i \(-0.688361\pi\)
−0.557818 + 0.829963i \(0.688361\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −4.21606 −0.0112129
\(377\) − 805.279i − 2.13602i
\(378\) 0 0
\(379\) −372.587 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(380\) 46.5798i 0.122578i
\(381\) 0 0
\(382\) −384.844 −1.00744
\(383\) 649.614i 1.69612i 0.529899 + 0.848061i \(0.322230\pi\)
−0.529899 + 0.848061i \(0.677770\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 374.933i − 0.971330i
\(387\) 0 0
\(388\) −275.411 −0.709823
\(389\) 448.470i 1.15288i 0.817140 + 0.576439i \(0.195558\pi\)
−0.817140 + 0.576439i \(0.804442\pi\)
\(390\) 0 0
\(391\) 512.367 1.31040
\(392\) 17.3728i 0.0443183i
\(393\) 0 0
\(394\) −466.617 −1.18431
\(395\) 18.5440i 0.0469468i
\(396\) 0 0
\(397\) −6.63508 −0.0167131 −0.00835653 0.999965i \(-0.502660\pi\)
−0.00835653 + 0.999965i \(0.502660\pi\)
\(398\) 869.060i 2.18357i
\(399\) 0 0
\(400\) −344.236 −0.860590
\(401\) − 16.2649i − 0.0405609i −0.999794 0.0202804i \(-0.993544\pi\)
0.999794 0.0202804i \(-0.00645590\pi\)
\(402\) 0 0
\(403\) 77.0902 0.191291
\(404\) − 219.950i − 0.544432i
\(405\) 0 0
\(406\) 51.5855 0.127058
\(407\) 0 0
\(408\) 0 0
\(409\) −80.9908 −0.198021 −0.0990107 0.995086i \(-0.531568\pi\)
−0.0990107 + 0.995086i \(0.531568\pi\)
\(410\) − 155.103i − 0.378300i
\(411\) 0 0
\(412\) −653.058 −1.58509
\(413\) 30.2087i 0.0731446i
\(414\) 0 0
\(415\) −179.325 −0.432109
\(416\) − 990.113i − 2.38008i
\(417\) 0 0
\(418\) 0 0
\(419\) − 156.122i − 0.372607i −0.982492 0.186303i \(-0.940349\pi\)
0.982492 0.186303i \(-0.0596508\pi\)
\(420\) 0 0
\(421\) −173.540 −0.412210 −0.206105 0.978530i \(-0.566079\pi\)
−0.206105 + 0.978530i \(0.566079\pi\)
\(422\) − 582.137i − 1.37947i
\(423\) 0 0
\(424\) 22.0208 0.0519358
\(425\) 250.092i 0.588451i
\(426\) 0 0
\(427\) −24.9516 −0.0584347
\(428\) − 537.638i − 1.25616i
\(429\) 0 0
\(430\) 331.030 0.769838
\(431\) − 204.134i − 0.473629i −0.971555 0.236814i \(-0.923897\pi\)
0.971555 0.236814i \(-0.0761033\pi\)
\(432\) 0 0
\(433\) −345.097 −0.796990 −0.398495 0.917170i \(-0.630467\pi\)
−0.398495 + 0.917170i \(0.630467\pi\)
\(434\) 4.93834i 0.0113787i
\(435\) 0 0
\(436\) −504.603 −1.15735
\(437\) − 253.163i − 0.579321i
\(438\) 0 0
\(439\) −688.693 −1.56878 −0.784388 0.620270i \(-0.787023\pi\)
−0.784388 + 0.620270i \(0.787023\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −742.194 −1.67917
\(443\) 702.545i 1.58588i 0.609300 + 0.792940i \(0.291451\pi\)
−0.609300 + 0.792940i \(0.708549\pi\)
\(444\) 0 0
\(445\) −9.18950 −0.0206506
\(446\) 810.142i 1.81646i
\(447\) 0 0
\(448\) 30.1770 0.0673594
\(449\) − 471.864i − 1.05092i −0.850818 0.525461i \(-0.823893\pi\)
0.850818 0.525461i \(-0.176107\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 33.1670i 0.0733784i
\(453\) 0 0
\(454\) 1123.60 2.47489
\(455\) 22.6046i 0.0496804i
\(456\) 0 0
\(457\) −170.345 −0.372746 −0.186373 0.982479i \(-0.559673\pi\)
−0.186373 + 0.982479i \(0.559673\pi\)
\(458\) − 34.9604i − 0.0763327i
\(459\) 0 0
\(460\) 336.458 0.731430
\(461\) − 279.960i − 0.607289i −0.952785 0.303645i \(-0.901796\pi\)
0.952785 0.303645i \(-0.0982036\pi\)
\(462\) 0 0
\(463\) 147.921 0.319485 0.159742 0.987159i \(-0.448934\pi\)
0.159742 + 0.987159i \(0.448934\pi\)
\(464\) − 601.568i − 1.29648i
\(465\) 0 0
\(466\) 200.173 0.429557
\(467\) − 826.636i − 1.77010i −0.465497 0.885050i \(-0.654124\pi\)
0.465497 0.885050i \(-0.345876\pi\)
\(468\) 0 0
\(469\) 54.6015 0.116421
\(470\) 67.4317i 0.143472i
\(471\) 0 0
\(472\) 21.3608 0.0452559
\(473\) 0 0
\(474\) 0 0
\(475\) 123.572 0.260151
\(476\) − 23.3887i − 0.0491359i
\(477\) 0 0
\(478\) 372.919 0.780166
\(479\) 531.483i 1.10957i 0.831994 + 0.554784i \(0.187199\pi\)
−0.831994 + 0.554784i \(0.812801\pi\)
\(480\) 0 0
\(481\) 725.370 1.50804
\(482\) 367.245i 0.761919i
\(483\) 0 0
\(484\) 0 0
\(485\) − 144.462i − 0.297860i
\(486\) 0 0
\(487\) −607.825 −1.24810 −0.624050 0.781385i \(-0.714514\pi\)
−0.624050 + 0.781385i \(0.714514\pi\)
\(488\) 17.6434i 0.0361546i
\(489\) 0 0
\(490\) 277.860 0.567062
\(491\) − 127.294i − 0.259255i −0.991563 0.129628i \(-0.958622\pi\)
0.991563 0.129628i \(-0.0413782\pi\)
\(492\) 0 0
\(493\) −437.046 −0.886504
\(494\) 366.722i 0.742353i
\(495\) 0 0
\(496\) 57.5887 0.116106
\(497\) 25.9090i 0.0521307i
\(498\) 0 0
\(499\) 380.683 0.762893 0.381446 0.924391i \(-0.375426\pi\)
0.381446 + 0.924391i \(0.375426\pi\)
\(500\) 360.928i 0.721856i
\(501\) 0 0
\(502\) 466.470 0.929223
\(503\) − 141.680i − 0.281670i −0.990033 0.140835i \(-0.955021\pi\)
0.990033 0.140835i \(-0.0449788\pi\)
\(504\) 0 0
\(505\) 115.371 0.228458
\(506\) 0 0
\(507\) 0 0
\(508\) 568.385 1.11887
\(509\) − 527.724i − 1.03679i −0.855143 0.518393i \(-0.826530\pi\)
0.855143 0.518393i \(-0.173470\pi\)
\(510\) 0 0
\(511\) −62.0968 −0.121520
\(512\) − 716.134i − 1.39870i
\(513\) 0 0
\(514\) −1092.71 −2.12589
\(515\) − 342.550i − 0.665146i
\(516\) 0 0
\(517\) 0 0
\(518\) 46.4666i 0.0897039i
\(519\) 0 0
\(520\) 15.9839 0.0307382
\(521\) − 20.4491i − 0.0392496i −0.999807 0.0196248i \(-0.993753\pi\)
0.999807 0.0196248i \(-0.00624717\pi\)
\(522\) 0 0
\(523\) 404.213 0.772873 0.386437 0.922316i \(-0.373706\pi\)
0.386437 + 0.922316i \(0.373706\pi\)
\(524\) 704.984i 1.34539i
\(525\) 0 0
\(526\) −763.520 −1.45156
\(527\) − 41.8389i − 0.0793907i
\(528\) 0 0
\(529\) −1299.66 −2.45683
\(530\) − 352.201i − 0.664529i
\(531\) 0 0
\(532\) −11.5565 −0.0217227
\(533\) − 600.711i − 1.12704i
\(534\) 0 0
\(535\) 282.009 0.527119
\(536\) − 38.6091i − 0.0720319i
\(537\) 0 0
\(538\) 665.976 1.23787
\(539\) 0 0
\(540\) 0 0
\(541\) −35.1928 −0.0650514 −0.0325257 0.999471i \(-0.510355\pi\)
−0.0325257 + 0.999471i \(0.510355\pi\)
\(542\) 845.697i 1.56033i
\(543\) 0 0
\(544\) −537.361 −0.987796
\(545\) − 264.681i − 0.485653i
\(546\) 0 0
\(547\) 368.347 0.673395 0.336698 0.941613i \(-0.390690\pi\)
0.336698 + 0.941613i \(0.390690\pi\)
\(548\) 605.797i 1.10547i
\(549\) 0 0
\(550\) 0 0
\(551\) 215.947i 0.391918i
\(552\) 0 0
\(553\) −4.60077 −0.00831965
\(554\) 43.5826i 0.0786689i
\(555\) 0 0
\(556\) 131.716 0.236900
\(557\) − 178.620i − 0.320683i −0.987062 0.160341i \(-0.948741\pi\)
0.987062 0.160341i \(-0.0512595\pi\)
\(558\) 0 0
\(559\) 1282.07 2.29351
\(560\) 16.8863i 0.0301541i
\(561\) 0 0
\(562\) −592.744 −1.05470
\(563\) − 171.436i − 0.304505i −0.988342 0.152252i \(-0.951347\pi\)
0.988342 0.152252i \(-0.0486527\pi\)
\(564\) 0 0
\(565\) −17.3972 −0.0307915
\(566\) − 83.2362i − 0.147060i
\(567\) 0 0
\(568\) 18.3204 0.0322542
\(569\) 653.652i 1.14877i 0.818584 + 0.574386i \(0.194759\pi\)
−0.818584 + 0.574386i \(0.805241\pi\)
\(570\) 0 0
\(571\) 172.238 0.301642 0.150821 0.988561i \(-0.451808\pi\)
0.150821 + 0.988561i \(0.451808\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 38.4811 0.0670402
\(575\) − 892.588i − 1.55233i
\(576\) 0 0
\(577\) 628.202 1.08874 0.544369 0.838846i \(-0.316769\pi\)
0.544369 + 0.838846i \(0.316769\pi\)
\(578\) − 408.092i − 0.706041i
\(579\) 0 0
\(580\) −286.997 −0.494822
\(581\) − 44.4907i − 0.0765761i
\(582\) 0 0
\(583\) 0 0
\(584\) 43.9091i 0.0751868i
\(585\) 0 0
\(586\) −1236.68 −2.11037
\(587\) − 122.740i − 0.209097i −0.994520 0.104549i \(-0.966660\pi\)
0.994520 0.104549i \(-0.0333398\pi\)
\(588\) 0 0
\(589\) −20.6728 −0.0350982
\(590\) − 341.645i − 0.579059i
\(591\) 0 0
\(592\) 541.873 0.915326
\(593\) − 365.467i − 0.616302i −0.951338 0.308151i \(-0.900290\pi\)
0.951338 0.308151i \(-0.0997101\pi\)
\(594\) 0 0
\(595\) 12.2681 0.0206187
\(596\) − 520.440i − 0.873222i
\(597\) 0 0
\(598\) 2648.93 4.42964
\(599\) 317.385i 0.529859i 0.964268 + 0.264929i \(0.0853486\pi\)
−0.964268 + 0.264929i \(0.914651\pi\)
\(600\) 0 0
\(601\) 296.732 0.493731 0.246866 0.969050i \(-0.420599\pi\)
0.246866 + 0.969050i \(0.420599\pi\)
\(602\) 82.1287i 0.136426i
\(603\) 0 0
\(604\) 633.546 1.04892
\(605\) 0 0
\(606\) 0 0
\(607\) 829.161 1.36600 0.682999 0.730419i \(-0.260675\pi\)
0.682999 + 0.730419i \(0.260675\pi\)
\(608\) 265.513i 0.436699i
\(609\) 0 0
\(610\) 282.190 0.462606
\(611\) 261.162i 0.427434i
\(612\) 0 0
\(613\) 824.364 1.34480 0.672401 0.740187i \(-0.265263\pi\)
0.672401 + 0.740187i \(0.265263\pi\)
\(614\) 320.733i 0.522366i
\(615\) 0 0
\(616\) 0 0
\(617\) − 147.294i − 0.238725i −0.992851 0.119363i \(-0.961915\pi\)
0.992851 0.119363i \(-0.0380851\pi\)
\(618\) 0 0
\(619\) 454.329 0.733972 0.366986 0.930227i \(-0.380390\pi\)
0.366986 + 0.930227i \(0.380390\pi\)
\(620\) − 27.4745i − 0.0443137i
\(621\) 0 0
\(622\) −1381.35 −2.22081
\(623\) − 2.27992i − 0.00365958i
\(624\) 0 0
\(625\) 332.506 0.532010
\(626\) 568.791i 0.908612i
\(627\) 0 0
\(628\) 678.444 1.08032
\(629\) − 393.677i − 0.625878i
\(630\) 0 0
\(631\) 36.0464 0.0571258 0.0285629 0.999592i \(-0.490907\pi\)
0.0285629 + 0.999592i \(0.490907\pi\)
\(632\) 3.25323i 0.00514752i
\(633\) 0 0
\(634\) −1123.99 −1.77285
\(635\) 298.136i 0.469506i
\(636\) 0 0
\(637\) 1076.15 1.68940
\(638\) 0 0
\(639\) 0 0
\(640\) 23.1568 0.0361826
\(641\) 1209.52i 1.88692i 0.331486 + 0.943460i \(0.392450\pi\)
−0.331486 + 0.943460i \(0.607550\pi\)
\(642\) 0 0
\(643\) 1057.01 1.64388 0.821938 0.569577i \(-0.192893\pi\)
0.821938 + 0.569577i \(0.192893\pi\)
\(644\) 83.4752i 0.129620i
\(645\) 0 0
\(646\) 199.030 0.308096
\(647\) 730.812i 1.12954i 0.825249 + 0.564770i \(0.191035\pi\)
−0.825249 + 0.564770i \(0.808965\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 1292.97i 1.98918i
\(651\) 0 0
\(652\) 558.748 0.856975
\(653\) − 493.064i − 0.755076i −0.925994 0.377538i \(-0.876771\pi\)
0.925994 0.377538i \(-0.123229\pi\)
\(654\) 0 0
\(655\) −369.787 −0.564560
\(656\) − 448.750i − 0.684069i
\(657\) 0 0
\(658\) −16.7298 −0.0254253
\(659\) − 887.112i − 1.34615i −0.739575 0.673074i \(-0.764973\pi\)
0.739575 0.673074i \(-0.235027\pi\)
\(660\) 0 0
\(661\) −574.603 −0.869293 −0.434647 0.900601i \(-0.643127\pi\)
−0.434647 + 0.900601i \(0.643127\pi\)
\(662\) − 1215.16i − 1.83559i
\(663\) 0 0
\(664\) −31.4597 −0.0473790
\(665\) − 6.06174i − 0.00911540i
\(666\) 0 0
\(667\) 1559.84 2.33859
\(668\) 1007.20i 1.50779i
\(669\) 0 0
\(670\) −617.514 −0.921663
\(671\) 0 0
\(672\) 0 0
\(673\) 311.362 0.462647 0.231324 0.972877i \(-0.425694\pi\)
0.231324 + 0.972877i \(0.425694\pi\)
\(674\) 1131.11i 1.67820i
\(675\) 0 0
\(676\) −1233.08 −1.82408
\(677\) − 1283.45i − 1.89580i −0.318573 0.947898i \(-0.603204\pi\)
0.318573 0.947898i \(-0.396796\pi\)
\(678\) 0 0
\(679\) 35.8411 0.0527851
\(680\) − 8.67487i − 0.0127572i
\(681\) 0 0
\(682\) 0 0
\(683\) 1285.45i 1.88206i 0.338319 + 0.941031i \(0.390142\pi\)
−0.338319 + 0.941031i \(0.609858\pi\)
\(684\) 0 0
\(685\) −317.760 −0.463883
\(686\) 138.234i 0.201507i
\(687\) 0 0
\(688\) 957.748 1.39208
\(689\) − 1364.07i − 1.97978i
\(690\) 0 0
\(691\) −171.853 −0.248702 −0.124351 0.992238i \(-0.539685\pi\)
−0.124351 + 0.992238i \(0.539685\pi\)
\(692\) − 1043.40i − 1.50780i
\(693\) 0 0
\(694\) −292.984 −0.422167
\(695\) 69.0894i 0.0994092i
\(696\) 0 0
\(697\) −326.022 −0.467750
\(698\) 138.087i 0.197832i
\(699\) 0 0
\(700\) −40.7451 −0.0582073
\(701\) 690.179i 0.984564i 0.870436 + 0.492282i \(0.163837\pi\)
−0.870436 + 0.492282i \(0.836163\pi\)
\(702\) 0 0
\(703\) −194.518 −0.276697
\(704\) 0 0
\(705\) 0 0
\(706\) −657.480 −0.931274
\(707\) 28.6236i 0.0404860i
\(708\) 0 0
\(709\) 479.613 0.676464 0.338232 0.941063i \(-0.390171\pi\)
0.338232 + 0.941063i \(0.390171\pi\)
\(710\) − 293.017i − 0.412700i
\(711\) 0 0
\(712\) −1.61215 −0.00226425
\(713\) 149.325i 0.209432i
\(714\) 0 0
\(715\) 0 0
\(716\) 133.022i 0.185784i
\(717\) 0 0
\(718\) −637.220 −0.887493
\(719\) 1025.44i 1.42620i 0.701064 + 0.713099i \(0.252709\pi\)
−0.701064 + 0.713099i \(0.747291\pi\)
\(720\) 0 0
\(721\) 84.9868 0.117874
\(722\) 914.582i 1.26673i
\(723\) 0 0
\(724\) 941.869 1.30092
\(725\) 761.373i 1.05017i
\(726\) 0 0
\(727\) −250.204 −0.344159 −0.172079 0.985083i \(-0.555049\pi\)
−0.172079 + 0.985083i \(0.555049\pi\)
\(728\) 3.96560i 0.00544725i
\(729\) 0 0
\(730\) 702.282 0.962030
\(731\) − 695.816i − 0.951869i
\(732\) 0 0
\(733\) −945.472 −1.28987 −0.644933 0.764239i \(-0.723115\pi\)
−0.644933 + 0.764239i \(0.723115\pi\)
\(734\) 1430.95i 1.94952i
\(735\) 0 0
\(736\) 1917.87 2.60580
\(737\) 0 0
\(738\) 0 0
\(739\) 654.012 0.884996 0.442498 0.896769i \(-0.354092\pi\)
0.442498 + 0.896769i \(0.354092\pi\)
\(740\) − 258.517i − 0.349348i
\(741\) 0 0
\(742\) 87.3810 0.117764
\(743\) − 1183.38i − 1.59270i −0.604837 0.796349i \(-0.706762\pi\)
0.604837 0.796349i \(-0.293238\pi\)
\(744\) 0 0
\(745\) 272.988 0.366427
\(746\) 1167.62i 1.56517i
\(747\) 0 0
\(748\) 0 0
\(749\) 69.9664i 0.0934132i
\(750\) 0 0
\(751\) 75.3851 0.100380 0.0501898 0.998740i \(-0.484017\pi\)
0.0501898 + 0.998740i \(0.484017\pi\)
\(752\) 195.096i 0.259436i
\(753\) 0 0
\(754\) −2259.52 −2.99671
\(755\) 332.316i 0.440153i
\(756\) 0 0
\(757\) 488.488 0.645294 0.322647 0.946519i \(-0.395427\pi\)
0.322647 + 0.946519i \(0.395427\pi\)
\(758\) 1045.43i 1.37920i
\(759\) 0 0
\(760\) −4.28630 −0.00563987
\(761\) − 187.728i − 0.246685i −0.992364 0.123343i \(-0.960639\pi\)
0.992364 0.123343i \(-0.0393614\pi\)
\(762\) 0 0
\(763\) 65.6673 0.0860647
\(764\) 531.203i 0.695291i
\(765\) 0 0
\(766\) 1822.74 2.37956
\(767\) − 1323.18i − 1.72514i
\(768\) 0 0
\(769\) −257.324 −0.334621 −0.167311 0.985904i \(-0.553508\pi\)
−0.167311 + 0.985904i \(0.553508\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −517.523 −0.670367
\(773\) 475.467i 0.615093i 0.951533 + 0.307546i \(0.0995079\pi\)
−0.951533 + 0.307546i \(0.900492\pi\)
\(774\) 0 0
\(775\) −72.8871 −0.0940478
\(776\) − 25.3435i − 0.0326591i
\(777\) 0 0
\(778\) 1258.35 1.61742
\(779\) 161.089i 0.206790i
\(780\) 0 0
\(781\) 0 0
\(782\) − 1437.64i − 1.83842i
\(783\) 0 0
\(784\) 803.915 1.02540
\(785\) 355.866i 0.453332i
\(786\) 0 0
\(787\) −290.097 −0.368612 −0.184306 0.982869i \(-0.559004\pi\)
−0.184306 + 0.982869i \(0.559004\pi\)
\(788\) 644.075i 0.817354i
\(789\) 0 0
\(790\) 52.0323 0.0658636
\(791\) − 4.31625i − 0.00545669i
\(792\) 0 0
\(793\) 1092.92 1.37820
\(794\) 18.6173i 0.0234474i
\(795\) 0 0
\(796\) 1199.57 1.50700
\(797\) − 792.022i − 0.993754i −0.867821 0.496877i \(-0.834480\pi\)
0.867821 0.496877i \(-0.165520\pi\)
\(798\) 0 0
\(799\) 141.740 0.177396
\(800\) 936.130i 1.17016i
\(801\) 0 0
\(802\) −45.6374 −0.0569045
\(803\) 0 0
\(804\) 0 0
\(805\) −43.7855 −0.0543919
\(806\) − 216.306i − 0.268370i
\(807\) 0 0
\(808\) 20.2399 0.0250494
\(809\) 1021.86i 1.26311i 0.775331 + 0.631556i \(0.217583\pi\)
−0.775331 + 0.631556i \(0.782417\pi\)
\(810\) 0 0
\(811\) 113.251 0.139644 0.0698221 0.997559i \(-0.477757\pi\)
0.0698221 + 0.997559i \(0.477757\pi\)
\(812\) − 71.2039i − 0.0876896i
\(813\) 0 0
\(814\) 0 0
\(815\) 293.082i 0.359609i
\(816\) 0 0
\(817\) −343.806 −0.420816
\(818\) 227.251i 0.277813i
\(819\) 0 0
\(820\) −214.090 −0.261085
\(821\) − 344.122i − 0.419150i −0.977793 0.209575i \(-0.932792\pi\)
0.977793 0.209575i \(-0.0672080\pi\)
\(822\) 0 0
\(823\) 121.006 0.147031 0.0735153 0.997294i \(-0.476578\pi\)
0.0735153 + 0.997294i \(0.476578\pi\)
\(824\) − 60.0948i − 0.0729306i
\(825\) 0 0
\(826\) 84.7621 0.102618
\(827\) − 251.924i − 0.304624i −0.988332 0.152312i \(-0.951328\pi\)
0.988332 0.152312i \(-0.0486718\pi\)
\(828\) 0 0
\(829\) −226.464 −0.273177 −0.136589 0.990628i \(-0.543614\pi\)
−0.136589 + 0.990628i \(0.543614\pi\)
\(830\) 503.166i 0.606224i
\(831\) 0 0
\(832\) −1321.80 −1.58870
\(833\) − 584.054i − 0.701146i
\(834\) 0 0
\(835\) −528.312 −0.632708
\(836\) 0 0
\(837\) 0 0
\(838\) −438.061 −0.522746
\(839\) − 580.612i − 0.692028i −0.938229 0.346014i \(-0.887535\pi\)
0.938229 0.346014i \(-0.112465\pi\)
\(840\) 0 0
\(841\) −489.534 −0.582086
\(842\) 486.934i 0.578306i
\(843\) 0 0
\(844\) −803.528 −0.952048
\(845\) − 646.789i − 0.765431i
\(846\) 0 0
\(847\) 0 0
\(848\) − 1019.00i − 1.20165i
\(849\) 0 0
\(850\) 701.728 0.825562
\(851\) 1405.05i 1.65106i
\(852\) 0 0
\(853\) 1056.85 1.23898 0.619491 0.785003i \(-0.287339\pi\)
0.619491 + 0.785003i \(0.287339\pi\)
\(854\) 70.0113i 0.0819804i
\(855\) 0 0
\(856\) 49.4738 0.0577964
\(857\) 1471.20i 1.71669i 0.513071 + 0.858346i \(0.328508\pi\)
−0.513071 + 0.858346i \(0.671492\pi\)
\(858\) 0 0
\(859\) −1109.66 −1.29180 −0.645902 0.763421i \(-0.723518\pi\)
−0.645902 + 0.763421i \(0.723518\pi\)
\(860\) − 456.924i − 0.531307i
\(861\) 0 0
\(862\) −572.776 −0.664474
\(863\) 369.201i 0.427811i 0.976854 + 0.213906i \(0.0686185\pi\)
−0.976854 + 0.213906i \(0.931382\pi\)
\(864\) 0 0
\(865\) 547.297 0.632713
\(866\) 968.301i 1.11813i
\(867\) 0 0
\(868\) 6.81643 0.00785303
\(869\) 0 0
\(870\) 0 0
\(871\) −2391.62 −2.74584
\(872\) − 46.4338i − 0.0532498i
\(873\) 0 0
\(874\) −710.347 −0.812754
\(875\) − 46.9700i − 0.0536800i
\(876\) 0 0
\(877\) 84.1500 0.0959521 0.0479761 0.998848i \(-0.484723\pi\)
0.0479761 + 0.998848i \(0.484723\pi\)
\(878\) 1932.39i 2.20090i
\(879\) 0 0
\(880\) 0 0
\(881\) − 763.999i − 0.867195i −0.901107 0.433598i \(-0.857244\pi\)
0.901107 0.433598i \(-0.142756\pi\)
\(882\) 0 0
\(883\) −3.97183 −0.00449811 −0.00224906 0.999997i \(-0.500716\pi\)
−0.00224906 + 0.999997i \(0.500716\pi\)
\(884\) 1024.46i 1.15889i
\(885\) 0 0
\(886\) 1971.26 2.22490
\(887\) 1020.39i 1.15038i 0.818021 + 0.575189i \(0.195071\pi\)
−0.818021 + 0.575189i \(0.804929\pi\)
\(888\) 0 0
\(889\) −73.9677 −0.0832033
\(890\) 25.7847i 0.0289715i
\(891\) 0 0
\(892\) 1118.25 1.25364
\(893\) − 70.0343i − 0.0784259i
\(894\) 0 0
\(895\) −69.7741 −0.0779599
\(896\) 5.74522i 0.00641208i
\(897\) 0 0
\(898\) −1324.00 −1.47438
\(899\) − 127.373i − 0.141684i
\(900\) 0 0
\(901\) −740.315 −0.821660
\(902\) 0 0
\(903\) 0 0
\(904\) −3.05205 −0.00337616
\(905\) 494.041i 0.545902i
\(906\) 0 0
\(907\) −672.333 −0.741271 −0.370635 0.928778i \(-0.620860\pi\)
−0.370635 + 0.928778i \(0.620860\pi\)
\(908\) − 1550.91i − 1.70805i
\(909\) 0 0
\(910\) 63.4259 0.0696988
\(911\) 320.354i 0.351650i 0.984421 + 0.175825i \(0.0562594\pi\)
−0.984421 + 0.175825i \(0.943741\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 477.969i 0.522941i
\(915\) 0 0
\(916\) −48.2561 −0.0526813
\(917\) − 91.7443i − 0.100048i
\(918\) 0 0
\(919\) −755.334 −0.821908 −0.410954 0.911656i \(-0.634804\pi\)
−0.410954 + 0.911656i \(0.634804\pi\)
\(920\) 30.9610i 0.0336533i
\(921\) 0 0
\(922\) −785.536 −0.851992
\(923\) − 1134.85i − 1.22952i
\(924\) 0 0
\(925\) −685.820 −0.741428
\(926\) − 415.050i − 0.448218i
\(927\) 0 0
\(928\) −1635.93 −1.76285
\(929\) − 653.730i − 0.703692i −0.936058 0.351846i \(-0.885554\pi\)
0.936058 0.351846i \(-0.114446\pi\)
\(930\) 0 0
\(931\) −288.584 −0.309972
\(932\) − 276.301i − 0.296460i
\(933\) 0 0
\(934\) −2319.45 −2.48335
\(935\) 0 0
\(936\) 0 0
\(937\) −1051.46 −1.12216 −0.561078 0.827763i \(-0.689613\pi\)
−0.561078 + 0.827763i \(0.689613\pi\)
\(938\) − 153.206i − 0.163332i
\(939\) 0 0
\(940\) 93.0766 0.0990176
\(941\) 503.345i 0.534904i 0.963571 + 0.267452i \(0.0861817\pi\)
−0.963571 + 0.267452i \(0.913818\pi\)
\(942\) 0 0
\(943\) 1163.59 1.23392
\(944\) − 988.458i − 1.04710i
\(945\) 0 0
\(946\) 0 0
\(947\) − 369.018i − 0.389671i −0.980836 0.194836i \(-0.937583\pi\)
0.980836 0.194836i \(-0.0624173\pi\)
\(948\) 0 0
\(949\) 2719.93 2.86610
\(950\) − 346.728i − 0.364976i
\(951\) 0 0
\(952\) 2.15224 0.00226075
\(953\) − 852.586i − 0.894634i −0.894375 0.447317i \(-0.852380\pi\)
0.894375 0.447317i \(-0.147620\pi\)
\(954\) 0 0
\(955\) −278.633 −0.291762
\(956\) − 514.743i − 0.538435i
\(957\) 0 0
\(958\) 1491.28 1.55666
\(959\) − 78.8364i − 0.0822068i
\(960\) 0 0
\(961\) −948.806 −0.987312
\(962\) − 2035.30i − 2.11570i
\(963\) 0 0
\(964\) 506.911 0.525841
\(965\) − 271.458i − 0.281304i
\(966\) 0 0
\(967\) −683.004 −0.706312 −0.353156 0.935564i \(-0.614892\pi\)
−0.353156 + 0.935564i \(0.614892\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −405.344 −0.417881
\(971\) − 869.351i − 0.895315i −0.894205 0.447657i \(-0.852258\pi\)
0.894205 0.447657i \(-0.147742\pi\)
\(972\) 0 0
\(973\) −17.1411 −0.0176168
\(974\) 1705.49i 1.75101i
\(975\) 0 0
\(976\) 816.441 0.836517
\(977\) − 743.912i − 0.761425i −0.924694 0.380712i \(-0.875679\pi\)
0.924694 0.380712i \(-0.124321\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) − 383.533i − 0.391360i
\(981\) 0 0
\(982\) −357.173 −0.363720
\(983\) − 352.071i − 0.358159i −0.983835 0.179080i \(-0.942688\pi\)
0.983835 0.179080i \(-0.0573120\pi\)
\(984\) 0 0
\(985\) −337.838 −0.342983
\(986\) 1226.30i 1.24371i
\(987\) 0 0
\(988\) 506.190 0.512338
\(989\) 2483.40i 2.51102i
\(990\) 0 0
\(991\) 1939.01 1.95662 0.978309 0.207152i \(-0.0664195\pi\)
0.978309 + 0.207152i \(0.0664195\pi\)
\(992\) − 156.609i − 0.157872i
\(993\) 0 0
\(994\) 72.6976 0.0731364
\(995\) 629.214i 0.632376i
\(996\) 0 0
\(997\) 115.855 0.116204 0.0581020 0.998311i \(-0.481495\pi\)
0.0581020 + 0.998311i \(0.481495\pi\)
\(998\) − 1068.15i − 1.07029i
\(999\) 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 1089.3.b.h.485.1 8
3.2 odd 2 inner 1089.3.b.h.485.8 yes 8
11.10 odd 2 inner 1089.3.b.h.485.7 yes 8
33.32 even 2 inner 1089.3.b.h.485.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1089.3.b.h.485.1 8 1.1 even 1 trivial
1089.3.b.h.485.2 yes 8 33.32 even 2 inner
1089.3.b.h.485.7 yes 8 11.10 odd 2 inner
1089.3.b.h.485.8 yes 8 3.2 odd 2 inner