Properties

Label 1089.3.c.a.604.2
Level $1089$
Weight $3$
Character 1089.604
Analytic conductor $29.673$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 121)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 604.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 1089.604
Dual form 1089.3.c.a.604.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} +2.00000 q^{4} +7.00000 q^{5} -7.07107i q^{7} +8.48528i q^{8} +O(q^{10})\) \(q+1.41421i q^{2} +2.00000 q^{4} +7.00000 q^{5} -7.07107i q^{7} +8.48528i q^{8} +9.89949i q^{10} -16.9706i q^{13} +10.0000 q^{14} -4.00000 q^{16} -4.24264i q^{17} +16.9706i q^{19} +14.0000 q^{20} +9.00000 q^{23} +24.0000 q^{25} +24.0000 q^{26} -14.1421i q^{28} +22.6274i q^{29} +49.0000 q^{31} +28.2843i q^{32} +6.00000 q^{34} -49.4975i q^{35} +17.0000 q^{37} -24.0000 q^{38} +59.3970i q^{40} -16.9706i q^{41} -46.6690i q^{43} +12.7279i q^{46} -32.0000 q^{47} -1.00000 q^{49} +33.9411i q^{50} -33.9411i q^{52} -16.0000 q^{53} +60.0000 q^{56} -32.0000 q^{58} +71.0000 q^{59} -11.3137i q^{61} +69.2965i q^{62} -56.0000 q^{64} -118.794i q^{65} -31.0000 q^{67} -8.48528i q^{68} +70.0000 q^{70} +73.0000 q^{71} +39.5980i q^{73} +24.0416i q^{74} +33.9411i q^{76} -156.978i q^{79} -28.0000 q^{80} +24.0000 q^{82} -35.3553i q^{83} -29.6985i q^{85} +66.0000 q^{86} +9.00000 q^{89} -120.000 q^{91} +18.0000 q^{92} -45.2548i q^{94} +118.794i q^{95} -17.0000 q^{97} -1.41421i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} + 14 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} + 14 q^{5} + 20 q^{14} - 8 q^{16} + 28 q^{20} + 18 q^{23} + 48 q^{25} + 48 q^{26} + 98 q^{31} + 12 q^{34} + 34 q^{37} - 48 q^{38} - 64 q^{47} - 2 q^{49} - 32 q^{53} + 120 q^{56} - 64 q^{58} + 142 q^{59} - 112 q^{64} - 62 q^{67} + 140 q^{70} + 146 q^{71} - 56 q^{80} + 48 q^{82} + 132 q^{86} + 18 q^{89} - 240 q^{91} + 36 q^{92} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) 7.00000 1.40000 0.700000 0.714143i \(-0.253183\pi\)
0.700000 + 0.714143i \(0.253183\pi\)
\(6\) 0 0
\(7\) − 7.07107i − 1.01015i −0.863075 0.505076i \(-0.831464\pi\)
0.863075 0.505076i \(-0.168536\pi\)
\(8\) 8.48528i 1.06066i
\(9\) 0 0
\(10\) 9.89949i 0.989949i
\(11\) 0 0
\(12\) 0 0
\(13\) − 16.9706i − 1.30543i −0.757604 0.652714i \(-0.773630\pi\)
0.757604 0.652714i \(-0.226370\pi\)
\(14\) 10.0000 0.714286
\(15\) 0 0
\(16\) −4.00000 −0.250000
\(17\) − 4.24264i − 0.249567i −0.992184 0.124784i \(-0.960176\pi\)
0.992184 0.124784i \(-0.0398236\pi\)
\(18\) 0 0
\(19\) 16.9706i 0.893188i 0.894737 + 0.446594i \(0.147363\pi\)
−0.894737 + 0.446594i \(0.852637\pi\)
\(20\) 14.0000 0.700000
\(21\) 0 0
\(22\) 0 0
\(23\) 9.00000 0.391304 0.195652 0.980673i \(-0.437318\pi\)
0.195652 + 0.980673i \(0.437318\pi\)
\(24\) 0 0
\(25\) 24.0000 0.960000
\(26\) 24.0000 0.923077
\(27\) 0 0
\(28\) − 14.1421i − 0.505076i
\(29\) 22.6274i 0.780256i 0.920761 + 0.390128i \(0.127569\pi\)
−0.920761 + 0.390128i \(0.872431\pi\)
\(30\) 0 0
\(31\) 49.0000 1.58065 0.790323 0.612691i \(-0.209913\pi\)
0.790323 + 0.612691i \(0.209913\pi\)
\(32\) 28.2843i 0.883883i
\(33\) 0 0
\(34\) 6.00000 0.176471
\(35\) − 49.4975i − 1.41421i
\(36\) 0 0
\(37\) 17.0000 0.459459 0.229730 0.973254i \(-0.426216\pi\)
0.229730 + 0.973254i \(0.426216\pi\)
\(38\) −24.0000 −0.631579
\(39\) 0 0
\(40\) 59.3970i 1.48492i
\(41\) − 16.9706i − 0.413916i −0.978350 0.206958i \(-0.933644\pi\)
0.978350 0.206958i \(-0.0663564\pi\)
\(42\) 0 0
\(43\) − 46.6690i − 1.08533i −0.839950 0.542663i \(-0.817416\pi\)
0.839950 0.542663i \(-0.182584\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 12.7279i 0.276694i
\(47\) −32.0000 −0.680851 −0.340426 0.940271i \(-0.610571\pi\)
−0.340426 + 0.940271i \(0.610571\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.0204082
\(50\) 33.9411i 0.678823i
\(51\) 0 0
\(52\) − 33.9411i − 0.652714i
\(53\) −16.0000 −0.301887 −0.150943 0.988542i \(-0.548231\pi\)
−0.150943 + 0.988542i \(0.548231\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 60.0000 1.07143
\(57\) 0 0
\(58\) −32.0000 −0.551724
\(59\) 71.0000 1.20339 0.601695 0.798726i \(-0.294492\pi\)
0.601695 + 0.798726i \(0.294492\pi\)
\(60\) 0 0
\(61\) − 11.3137i − 0.185471i −0.995691 0.0927353i \(-0.970439\pi\)
0.995691 0.0927353i \(-0.0295610\pi\)
\(62\) 69.2965i 1.11768i
\(63\) 0 0
\(64\) −56.0000 −0.875000
\(65\) − 118.794i − 1.82760i
\(66\) 0 0
\(67\) −31.0000 −0.462687 −0.231343 0.972872i \(-0.574312\pi\)
−0.231343 + 0.972872i \(0.574312\pi\)
\(68\) − 8.48528i − 0.124784i
\(69\) 0 0
\(70\) 70.0000 1.00000
\(71\) 73.0000 1.02817 0.514085 0.857740i \(-0.328132\pi\)
0.514085 + 0.857740i \(0.328132\pi\)
\(72\) 0 0
\(73\) 39.5980i 0.542438i 0.962518 + 0.271219i \(0.0874268\pi\)
−0.962518 + 0.271219i \(0.912573\pi\)
\(74\) 24.0416i 0.324887i
\(75\) 0 0
\(76\) 33.9411i 0.446594i
\(77\) 0 0
\(78\) 0 0
\(79\) − 156.978i − 1.98706i −0.113572 0.993530i \(-0.536229\pi\)
0.113572 0.993530i \(-0.463771\pi\)
\(80\) −28.0000 −0.350000
\(81\) 0 0
\(82\) 24.0000 0.292683
\(83\) − 35.3553i − 0.425968i −0.977056 0.212984i \(-0.931682\pi\)
0.977056 0.212984i \(-0.0683182\pi\)
\(84\) 0 0
\(85\) − 29.6985i − 0.349394i
\(86\) 66.0000 0.767442
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000 0.101124 0.0505618 0.998721i \(-0.483899\pi\)
0.0505618 + 0.998721i \(0.483899\pi\)
\(90\) 0 0
\(91\) −120.000 −1.31868
\(92\) 18.0000 0.195652
\(93\) 0 0
\(94\) − 45.2548i − 0.481434i
\(95\) 118.794i 1.25046i
\(96\) 0 0
\(97\) −17.0000 −0.175258 −0.0876289 0.996153i \(-0.527929\pi\)
−0.0876289 + 0.996153i \(0.527929\pi\)
\(98\) − 1.41421i − 0.0144308i
\(99\) 0 0
\(100\) 48.0000 0.480000
\(101\) − 154.149i − 1.52623i −0.646262 0.763115i \(-0.723669\pi\)
0.646262 0.763115i \(-0.276331\pi\)
\(102\) 0 0
\(103\) −16.0000 −0.155340 −0.0776699 0.996979i \(-0.524748\pi\)
−0.0776699 + 0.996979i \(0.524748\pi\)
\(104\) 144.000 1.38462
\(105\) 0 0
\(106\) − 22.6274i − 0.213466i
\(107\) 185.262i 1.73142i 0.500546 + 0.865710i \(0.333133\pi\)
−0.500546 + 0.865710i \(0.666867\pi\)
\(108\) 0 0
\(109\) − 46.6690i − 0.428156i −0.976817 0.214078i \(-0.931325\pi\)
0.976817 0.214078i \(-0.0686747\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 28.2843i 0.252538i
\(113\) −65.0000 −0.575221 −0.287611 0.957747i \(-0.592861\pi\)
−0.287611 + 0.957747i \(0.592861\pi\)
\(114\) 0 0
\(115\) 63.0000 0.547826
\(116\) 45.2548i 0.390128i
\(117\) 0 0
\(118\) 100.409i 0.850925i
\(119\) −30.0000 −0.252101
\(120\) 0 0
\(121\) 0 0
\(122\) 16.0000 0.131148
\(123\) 0 0
\(124\) 98.0000 0.790323
\(125\) −7.00000 −0.0560000
\(126\) 0 0
\(127\) 175.362i 1.38081i 0.723425 + 0.690403i \(0.242567\pi\)
−0.723425 + 0.690403i \(0.757433\pi\)
\(128\) 33.9411i 0.265165i
\(129\) 0 0
\(130\) 168.000 1.29231
\(131\) 140.007i 1.06876i 0.845245 + 0.534378i \(0.179454\pi\)
−0.845245 + 0.534378i \(0.820546\pi\)
\(132\) 0 0
\(133\) 120.000 0.902256
\(134\) − 43.8406i − 0.327169i
\(135\) 0 0
\(136\) 36.0000 0.264706
\(137\) −257.000 −1.87591 −0.937956 0.346754i \(-0.887284\pi\)
−0.937956 + 0.346754i \(0.887284\pi\)
\(138\) 0 0
\(139\) 86.2670i 0.620626i 0.950634 + 0.310313i \(0.100434\pi\)
−0.950634 + 0.310313i \(0.899566\pi\)
\(140\) − 98.9949i − 0.707107i
\(141\) 0 0
\(142\) 103.238i 0.727025i
\(143\) 0 0
\(144\) 0 0
\(145\) 158.392i 1.09236i
\(146\) −56.0000 −0.383562
\(147\) 0 0
\(148\) 34.0000 0.229730
\(149\) 275.772i 1.85082i 0.378972 + 0.925408i \(0.376278\pi\)
−0.378972 + 0.925408i \(0.623722\pi\)
\(150\) 0 0
\(151\) 156.978i 1.03959i 0.854292 + 0.519794i \(0.173991\pi\)
−0.854292 + 0.519794i \(0.826009\pi\)
\(152\) −144.000 −0.947368
\(153\) 0 0
\(154\) 0 0
\(155\) 343.000 2.21290
\(156\) 0 0
\(157\) 175.000 1.11465 0.557325 0.830295i \(-0.311828\pi\)
0.557325 + 0.830295i \(0.311828\pi\)
\(158\) 222.000 1.40506
\(159\) 0 0
\(160\) 197.990i 1.23744i
\(161\) − 63.6396i − 0.395277i
\(162\) 0 0
\(163\) −160.000 −0.981595 −0.490798 0.871274i \(-0.663295\pi\)
−0.490798 + 0.871274i \(0.663295\pi\)
\(164\) − 33.9411i − 0.206958i
\(165\) 0 0
\(166\) 50.0000 0.301205
\(167\) 16.9706i 0.101620i 0.998708 + 0.0508101i \(0.0161803\pi\)
−0.998708 + 0.0508101i \(0.983820\pi\)
\(168\) 0 0
\(169\) −119.000 −0.704142
\(170\) 42.0000 0.247059
\(171\) 0 0
\(172\) − 93.3381i − 0.542663i
\(173\) 123.037i 0.711194i 0.934639 + 0.355597i \(0.115722\pi\)
−0.934639 + 0.355597i \(0.884278\pi\)
\(174\) 0 0
\(175\) − 169.706i − 0.969746i
\(176\) 0 0
\(177\) 0 0
\(178\) 12.7279i 0.0715052i
\(179\) 199.000 1.11173 0.555866 0.831272i \(-0.312387\pi\)
0.555866 + 0.831272i \(0.312387\pi\)
\(180\) 0 0
\(181\) −73.0000 −0.403315 −0.201657 0.979456i \(-0.564633\pi\)
−0.201657 + 0.979456i \(0.564633\pi\)
\(182\) − 169.706i − 0.932449i
\(183\) 0 0
\(184\) 76.3675i 0.415041i
\(185\) 119.000 0.643243
\(186\) 0 0
\(187\) 0 0
\(188\) −64.0000 −0.340426
\(189\) 0 0
\(190\) −168.000 −0.884211
\(191\) −215.000 −1.12565 −0.562827 0.826575i \(-0.690286\pi\)
−0.562827 + 0.826575i \(0.690286\pi\)
\(192\) 0 0
\(193\) − 135.765i − 0.703443i −0.936105 0.351722i \(-0.885596\pi\)
0.936105 0.351722i \(-0.114404\pi\)
\(194\) − 24.0416i − 0.123926i
\(195\) 0 0
\(196\) −2.00000 −0.0102041
\(197\) − 202.233i − 1.02656i −0.858221 0.513281i \(-0.828430\pi\)
0.858221 0.513281i \(-0.171570\pi\)
\(198\) 0 0
\(199\) 200.000 1.00503 0.502513 0.864570i \(-0.332409\pi\)
0.502513 + 0.864570i \(0.332409\pi\)
\(200\) 203.647i 1.01823i
\(201\) 0 0
\(202\) 218.000 1.07921
\(203\) 160.000 0.788177
\(204\) 0 0
\(205\) − 118.794i − 0.579483i
\(206\) − 22.6274i − 0.109842i
\(207\) 0 0
\(208\) 67.8823i 0.326357i
\(209\) 0 0
\(210\) 0 0
\(211\) − 79.1960i − 0.375336i −0.982232 0.187668i \(-0.939907\pi\)
0.982232 0.187668i \(-0.0600930\pi\)
\(212\) −32.0000 −0.150943
\(213\) 0 0
\(214\) −262.000 −1.22430
\(215\) − 326.683i − 1.51946i
\(216\) 0 0
\(217\) − 346.482i − 1.59669i
\(218\) 66.0000 0.302752
\(219\) 0 0
\(220\) 0 0
\(221\) −72.0000 −0.325792
\(222\) 0 0
\(223\) −111.000 −0.497758 −0.248879 0.968535i \(-0.580062\pi\)
−0.248879 + 0.968535i \(0.580062\pi\)
\(224\) 200.000 0.892857
\(225\) 0 0
\(226\) − 91.9239i − 0.406743i
\(227\) 131.522i 0.579391i 0.957119 + 0.289696i \(0.0935541\pi\)
−0.957119 + 0.289696i \(0.906446\pi\)
\(228\) 0 0
\(229\) −303.000 −1.32314 −0.661572 0.749882i \(-0.730110\pi\)
−0.661572 + 0.749882i \(0.730110\pi\)
\(230\) 89.0955i 0.387372i
\(231\) 0 0
\(232\) −192.000 −0.827586
\(233\) 79.1960i 0.339897i 0.985453 + 0.169948i \(0.0543601\pi\)
−0.985453 + 0.169948i \(0.945640\pi\)
\(234\) 0 0
\(235\) −224.000 −0.953191
\(236\) 142.000 0.601695
\(237\) 0 0
\(238\) − 42.4264i − 0.178262i
\(239\) − 281.428i − 1.17753i −0.808306 0.588763i \(-0.799615\pi\)
0.808306 0.588763i \(-0.200385\pi\)
\(240\) 0 0
\(241\) 373.352i 1.54918i 0.632464 + 0.774590i \(0.282044\pi\)
−0.632464 + 0.774590i \(0.717956\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) − 22.6274i − 0.0927353i
\(245\) −7.00000 −0.0285714
\(246\) 0 0
\(247\) 288.000 1.16599
\(248\) 415.779i 1.67653i
\(249\) 0 0
\(250\) − 9.89949i − 0.0395980i
\(251\) −225.000 −0.896414 −0.448207 0.893930i \(-0.647937\pi\)
−0.448207 + 0.893930i \(0.647937\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −248.000 −0.976378
\(255\) 0 0
\(256\) −272.000 −1.06250
\(257\) −424.000 −1.64981 −0.824903 0.565275i \(-0.808770\pi\)
−0.824903 + 0.565275i \(0.808770\pi\)
\(258\) 0 0
\(259\) − 120.208i − 0.464124i
\(260\) − 237.588i − 0.913800i
\(261\) 0 0
\(262\) −198.000 −0.755725
\(263\) 140.007i 0.532347i 0.963925 + 0.266173i \(0.0857593\pi\)
−0.963925 + 0.266173i \(0.914241\pi\)
\(264\) 0 0
\(265\) −112.000 −0.422642
\(266\) 169.706i 0.637991i
\(267\) 0 0
\(268\) −62.0000 −0.231343
\(269\) −136.000 −0.505576 −0.252788 0.967522i \(-0.581348\pi\)
−0.252788 + 0.967522i \(0.581348\pi\)
\(270\) 0 0
\(271\) 288.500i 1.06457i 0.846564 + 0.532287i \(0.178667\pi\)
−0.846564 + 0.532287i \(0.821333\pi\)
\(272\) 16.9706i 0.0623918i
\(273\) 0 0
\(274\) − 363.453i − 1.32647i
\(275\) 0 0
\(276\) 0 0
\(277\) 123.037i 0.444175i 0.975027 + 0.222088i \(0.0712871\pi\)
−0.975027 + 0.222088i \(0.928713\pi\)
\(278\) −122.000 −0.438849
\(279\) 0 0
\(280\) 420.000 1.50000
\(281\) − 284.257i − 1.01159i −0.862654 0.505795i \(-0.831199\pi\)
0.862654 0.505795i \(-0.168801\pi\)
\(282\) 0 0
\(283\) 79.1960i 0.279844i 0.990163 + 0.139922i \(0.0446852\pi\)
−0.990163 + 0.139922i \(0.955315\pi\)
\(284\) 146.000 0.514085
\(285\) 0 0
\(286\) 0 0
\(287\) −120.000 −0.418118
\(288\) 0 0
\(289\) 271.000 0.937716
\(290\) −224.000 −0.772414
\(291\) 0 0
\(292\) 79.1960i 0.271219i
\(293\) − 164.049i − 0.559893i −0.960016 0.279947i \(-0.909683\pi\)
0.960016 0.279947i \(-0.0903168\pi\)
\(294\) 0 0
\(295\) 497.000 1.68475
\(296\) 144.250i 0.487330i
\(297\) 0 0
\(298\) −390.000 −1.30872
\(299\) − 152.735i − 0.510820i
\(300\) 0 0
\(301\) −330.000 −1.09635
\(302\) −222.000 −0.735099
\(303\) 0 0
\(304\) − 67.8823i − 0.223297i
\(305\) − 79.1960i − 0.259659i
\(306\) 0 0
\(307\) − 186.676i − 0.608066i −0.952662 0.304033i \(-0.901667\pi\)
0.952662 0.304033i \(-0.0983333\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 485.075i 1.56476i
\(311\) −142.000 −0.456592 −0.228296 0.973592i \(-0.573315\pi\)
−0.228296 + 0.973592i \(0.573315\pi\)
\(312\) 0 0
\(313\) −447.000 −1.42812 −0.714058 0.700087i \(-0.753145\pi\)
−0.714058 + 0.700087i \(0.753145\pi\)
\(314\) 247.487i 0.788176i
\(315\) 0 0
\(316\) − 313.955i − 0.993530i
\(317\) −423.000 −1.33438 −0.667192 0.744885i \(-0.732504\pi\)
−0.667192 + 0.744885i \(0.732504\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −392.000 −1.22500
\(321\) 0 0
\(322\) 90.0000 0.279503
\(323\) 72.0000 0.222910
\(324\) 0 0
\(325\) − 407.294i − 1.25321i
\(326\) − 226.274i − 0.694093i
\(327\) 0 0
\(328\) 144.000 0.439024
\(329\) 226.274i 0.687763i
\(330\) 0 0
\(331\) 145.000 0.438066 0.219033 0.975717i \(-0.429710\pi\)
0.219033 + 0.975717i \(0.429710\pi\)
\(332\) − 70.7107i − 0.212984i
\(333\) 0 0
\(334\) −24.0000 −0.0718563
\(335\) −217.000 −0.647761
\(336\) 0 0
\(337\) − 255.973i − 0.759563i −0.925076 0.379781i \(-0.875999\pi\)
0.925076 0.379781i \(-0.124001\pi\)
\(338\) − 168.291i − 0.497904i
\(339\) 0 0
\(340\) − 59.3970i − 0.174697i
\(341\) 0 0
\(342\) 0 0
\(343\) − 339.411i − 0.989537i
\(344\) 396.000 1.15116
\(345\) 0 0
\(346\) −174.000 −0.502890
\(347\) − 548.715i − 1.58131i −0.612261 0.790655i \(-0.709740\pi\)
0.612261 0.790655i \(-0.290260\pi\)
\(348\) 0 0
\(349\) 436.992i 1.25213i 0.779772 + 0.626063i \(0.215335\pi\)
−0.779772 + 0.626063i \(0.784665\pi\)
\(350\) 240.000 0.685714
\(351\) 0 0
\(352\) 0 0
\(353\) −585.000 −1.65722 −0.828612 0.559823i \(-0.810869\pi\)
−0.828612 + 0.559823i \(0.810869\pi\)
\(354\) 0 0
\(355\) 511.000 1.43944
\(356\) 18.0000 0.0505618
\(357\) 0 0
\(358\) 281.428i 0.786113i
\(359\) − 412.950i − 1.15028i −0.818055 0.575140i \(-0.804948\pi\)
0.818055 0.575140i \(-0.195052\pi\)
\(360\) 0 0
\(361\) 73.0000 0.202216
\(362\) − 103.238i − 0.285187i
\(363\) 0 0
\(364\) −240.000 −0.659341
\(365\) 277.186i 0.759413i
\(366\) 0 0
\(367\) 545.000 1.48501 0.742507 0.669839i \(-0.233637\pi\)
0.742507 + 0.669839i \(0.233637\pi\)
\(368\) −36.0000 −0.0978261
\(369\) 0 0
\(370\) 168.291i 0.454842i
\(371\) 113.137i 0.304952i
\(372\) 0 0
\(373\) 233.345i 0.625590i 0.949821 + 0.312795i \(0.101265\pi\)
−0.949821 + 0.312795i \(0.898735\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) − 271.529i − 0.722152i
\(377\) 384.000 1.01857
\(378\) 0 0
\(379\) −447.000 −1.17942 −0.589710 0.807615i \(-0.700758\pi\)
−0.589710 + 0.807615i \(0.700758\pi\)
\(380\) 237.588i 0.625231i
\(381\) 0 0
\(382\) − 304.056i − 0.795958i
\(383\) 545.000 1.42298 0.711488 0.702698i \(-0.248021\pi\)
0.711488 + 0.702698i \(0.248021\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 192.000 0.497409
\(387\) 0 0
\(388\) −34.0000 −0.0876289
\(389\) −215.000 −0.552699 −0.276350 0.961057i \(-0.589125\pi\)
−0.276350 + 0.961057i \(0.589125\pi\)
\(390\) 0 0
\(391\) − 38.1838i − 0.0976567i
\(392\) − 8.48528i − 0.0216461i
\(393\) 0 0
\(394\) 286.000 0.725888
\(395\) − 1098.84i − 2.78188i
\(396\) 0 0
\(397\) −592.000 −1.49118 −0.745592 0.666403i \(-0.767833\pi\)
−0.745592 + 0.666403i \(0.767833\pi\)
\(398\) 282.843i 0.710660i
\(399\) 0 0
\(400\) −96.0000 −0.240000
\(401\) −488.000 −1.21696 −0.608479 0.793570i \(-0.708220\pi\)
−0.608479 + 0.793570i \(0.708220\pi\)
\(402\) 0 0
\(403\) − 831.558i − 2.06342i
\(404\) − 308.299i − 0.763115i
\(405\) 0 0
\(406\) 226.274i 0.557326i
\(407\) 0 0
\(408\) 0 0
\(409\) − 156.978i − 0.383809i −0.981414 0.191904i \(-0.938534\pi\)
0.981414 0.191904i \(-0.0614663\pi\)
\(410\) 168.000 0.409756
\(411\) 0 0
\(412\) −32.0000 −0.0776699
\(413\) − 502.046i − 1.21561i
\(414\) 0 0
\(415\) − 247.487i − 0.596355i
\(416\) 480.000 1.15385
\(417\) 0 0
\(418\) 0 0
\(419\) 328.000 0.782816 0.391408 0.920217i \(-0.371988\pi\)
0.391408 + 0.920217i \(0.371988\pi\)
\(420\) 0 0
\(421\) 208.000 0.494062 0.247031 0.969008i \(-0.420545\pi\)
0.247031 + 0.969008i \(0.420545\pi\)
\(422\) 112.000 0.265403
\(423\) 0 0
\(424\) − 135.765i − 0.320199i
\(425\) − 101.823i − 0.239584i
\(426\) 0 0
\(427\) −80.0000 −0.187354
\(428\) 370.524i 0.865710i
\(429\) 0 0
\(430\) 462.000 1.07442
\(431\) 561.443i 1.30265i 0.758798 + 0.651326i \(0.225787\pi\)
−0.758798 + 0.651326i \(0.774213\pi\)
\(432\) 0 0
\(433\) 39.0000 0.0900693 0.0450346 0.998985i \(-0.485660\pi\)
0.0450346 + 0.998985i \(0.485660\pi\)
\(434\) 490.000 1.12903
\(435\) 0 0
\(436\) − 93.3381i − 0.214078i
\(437\) 152.735i 0.349508i
\(438\) 0 0
\(439\) 248.902i 0.566974i 0.958976 + 0.283487i \(0.0914913\pi\)
−0.958976 + 0.283487i \(0.908509\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 101.823i − 0.230370i
\(443\) −175.000 −0.395034 −0.197517 0.980299i \(-0.563288\pi\)
−0.197517 + 0.980299i \(0.563288\pi\)
\(444\) 0 0
\(445\) 63.0000 0.141573
\(446\) − 156.978i − 0.351968i
\(447\) 0 0
\(448\) 395.980i 0.883883i
\(449\) −313.000 −0.697105 −0.348552 0.937289i \(-0.613327\pi\)
−0.348552 + 0.937289i \(0.613327\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −130.000 −0.287611
\(453\) 0 0
\(454\) −186.000 −0.409692
\(455\) −840.000 −1.84615
\(456\) 0 0
\(457\) − 11.3137i − 0.0247565i −0.999923 0.0123782i \(-0.996060\pi\)
0.999923 0.0123782i \(-0.00394022\pi\)
\(458\) − 428.507i − 0.935604i
\(459\) 0 0
\(460\) 126.000 0.273913
\(461\) 871.156i 1.88971i 0.327491 + 0.944854i \(0.393797\pi\)
−0.327491 + 0.944854i \(0.606203\pi\)
\(462\) 0 0
\(463\) 321.000 0.693305 0.346652 0.937994i \(-0.387318\pi\)
0.346652 + 0.937994i \(0.387318\pi\)
\(464\) − 90.5097i − 0.195064i
\(465\) 0 0
\(466\) −112.000 −0.240343
\(467\) 161.000 0.344754 0.172377 0.985031i \(-0.444855\pi\)
0.172377 + 0.985031i \(0.444855\pi\)
\(468\) 0 0
\(469\) 219.203i 0.467384i
\(470\) − 316.784i − 0.674008i
\(471\) 0 0
\(472\) 602.455i 1.27639i
\(473\) 0 0
\(474\) 0 0
\(475\) 407.294i 0.857460i
\(476\) −60.0000 −0.126050
\(477\) 0 0
\(478\) 398.000 0.832636
\(479\) 446.891i 0.932968i 0.884530 + 0.466484i \(0.154479\pi\)
−0.884530 + 0.466484i \(0.845521\pi\)
\(480\) 0 0
\(481\) − 288.500i − 0.599791i
\(482\) −528.000 −1.09544
\(483\) 0 0
\(484\) 0 0
\(485\) −119.000 −0.245361
\(486\) 0 0
\(487\) −727.000 −1.49281 −0.746407 0.665490i \(-0.768223\pi\)
−0.746407 + 0.665490i \(0.768223\pi\)
\(488\) 96.0000 0.196721
\(489\) 0 0
\(490\) − 9.89949i − 0.0202031i
\(491\) 520.431i 1.05994i 0.848016 + 0.529970i \(0.177797\pi\)
−0.848016 + 0.529970i \(0.822203\pi\)
\(492\) 0 0
\(493\) 96.0000 0.194726
\(494\) 407.294i 0.824481i
\(495\) 0 0
\(496\) −196.000 −0.395161
\(497\) − 516.188i − 1.03861i
\(498\) 0 0
\(499\) −368.000 −0.737475 −0.368737 0.929534i \(-0.620210\pi\)
−0.368737 + 0.929534i \(0.620210\pi\)
\(500\) −14.0000 −0.0280000
\(501\) 0 0
\(502\) − 318.198i − 0.633861i
\(503\) 465.276i 0.925003i 0.886619 + 0.462501i \(0.153048\pi\)
−0.886619 + 0.462501i \(0.846952\pi\)
\(504\) 0 0
\(505\) − 1079.04i − 2.13672i
\(506\) 0 0
\(507\) 0 0
\(508\) 350.725i 0.690403i
\(509\) 111.000 0.218075 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(510\) 0 0
\(511\) 280.000 0.547945
\(512\) − 248.902i − 0.486136i
\(513\) 0 0
\(514\) − 599.627i − 1.16659i
\(515\) −112.000 −0.217476
\(516\) 0 0
\(517\) 0 0
\(518\) 170.000 0.328185
\(519\) 0 0
\(520\) 1008.00 1.93846
\(521\) 511.000 0.980806 0.490403 0.871496i \(-0.336850\pi\)
0.490403 + 0.871496i \(0.336850\pi\)
\(522\) 0 0
\(523\) − 695.793i − 1.33039i −0.746670 0.665194i \(-0.768349\pi\)
0.746670 0.665194i \(-0.231651\pi\)
\(524\) 280.014i 0.534378i
\(525\) 0 0
\(526\) −198.000 −0.376426
\(527\) − 207.889i − 0.394477i
\(528\) 0 0
\(529\) −448.000 −0.846881
\(530\) − 158.392i − 0.298853i
\(531\) 0 0
\(532\) 240.000 0.451128
\(533\) −288.000 −0.540338
\(534\) 0 0
\(535\) 1296.83i 2.42399i
\(536\) − 263.044i − 0.490753i
\(537\) 0 0
\(538\) − 192.333i − 0.357496i
\(539\) 0 0
\(540\) 0 0
\(541\) 231.931i 0.428708i 0.976756 + 0.214354i \(0.0687646\pi\)
−0.976756 + 0.214354i \(0.931235\pi\)
\(542\) −408.000 −0.752768
\(543\) 0 0
\(544\) 120.000 0.220588
\(545\) − 326.683i − 0.599419i
\(546\) 0 0
\(547\) 701.450i 1.28236i 0.767391 + 0.641179i \(0.221554\pi\)
−0.767391 + 0.641179i \(0.778446\pi\)
\(548\) −514.000 −0.937956
\(549\) 0 0
\(550\) 0 0
\(551\) −384.000 −0.696915
\(552\) 0 0
\(553\) −1110.00 −2.00723
\(554\) −174.000 −0.314079
\(555\) 0 0
\(556\) 172.534i 0.310313i
\(557\) − 210.718i − 0.378308i −0.981947 0.189154i \(-0.939425\pi\)
0.981947 0.189154i \(-0.0605746\pi\)
\(558\) 0 0
\(559\) −792.000 −1.41682
\(560\) 197.990i 0.353553i
\(561\) 0 0
\(562\) 402.000 0.715302
\(563\) − 543.058i − 0.964579i −0.876012 0.482290i \(-0.839805\pi\)
0.876012 0.482290i \(-0.160195\pi\)
\(564\) 0 0
\(565\) −455.000 −0.805310
\(566\) −112.000 −0.197880
\(567\) 0 0
\(568\) 619.426i 1.09054i
\(569\) − 79.1960i − 0.139184i −0.997576 0.0695922i \(-0.977830\pi\)
0.997576 0.0695922i \(-0.0221698\pi\)
\(570\) 0 0
\(571\) − 124.451i − 0.217952i −0.994044 0.108976i \(-0.965243\pi\)
0.994044 0.108976i \(-0.0347572\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 169.706i − 0.295654i
\(575\) 216.000 0.375652
\(576\) 0 0
\(577\) 103.000 0.178510 0.0892548 0.996009i \(-0.471551\pi\)
0.0892548 + 0.996009i \(0.471551\pi\)
\(578\) 383.252i 0.663066i
\(579\) 0 0
\(580\) 316.784i 0.546179i
\(581\) −250.000 −0.430293
\(582\) 0 0
\(583\) 0 0
\(584\) −336.000 −0.575342
\(585\) 0 0
\(586\) 232.000 0.395904
\(587\) 632.000 1.07666 0.538330 0.842734i \(-0.319055\pi\)
0.538330 + 0.842734i \(0.319055\pi\)
\(588\) 0 0
\(589\) 831.558i 1.41181i
\(590\) 702.864i 1.19130i
\(591\) 0 0
\(592\) −68.0000 −0.114865
\(593\) − 497.803i − 0.839466i −0.907648 0.419733i \(-0.862124\pi\)
0.907648 0.419733i \(-0.137876\pi\)
\(594\) 0 0
\(595\) −210.000 −0.352941
\(596\) 551.543i 0.925408i
\(597\) 0 0
\(598\) 216.000 0.361204
\(599\) −312.000 −0.520868 −0.260434 0.965492i \(-0.583866\pi\)
−0.260434 + 0.965492i \(0.583866\pi\)
\(600\) 0 0
\(601\) − 411.536i − 0.684752i −0.939563 0.342376i \(-0.888768\pi\)
0.939563 0.342376i \(-0.111232\pi\)
\(602\) − 466.690i − 0.775233i
\(603\) 0 0
\(604\) 313.955i 0.519794i
\(605\) 0 0
\(606\) 0 0
\(607\) 123.037i 0.202696i 0.994851 + 0.101348i \(0.0323156\pi\)
−0.994851 + 0.101348i \(0.967684\pi\)
\(608\) −480.000 −0.789474
\(609\) 0 0
\(610\) 112.000 0.183607
\(611\) 543.058i 0.888802i
\(612\) 0 0
\(613\) − 543.058i − 0.885902i −0.896546 0.442951i \(-0.853932\pi\)
0.896546 0.442951i \(-0.146068\pi\)
\(614\) 264.000 0.429967
\(615\) 0 0
\(616\) 0 0
\(617\) 1120.00 1.81524 0.907618 0.419798i \(-0.137899\pi\)
0.907618 + 0.419798i \(0.137899\pi\)
\(618\) 0 0
\(619\) 703.000 1.13570 0.567851 0.823131i \(-0.307775\pi\)
0.567851 + 0.823131i \(0.307775\pi\)
\(620\) 686.000 1.10645
\(621\) 0 0
\(622\) − 200.818i − 0.322859i
\(623\) − 63.6396i − 0.102150i
\(624\) 0 0
\(625\) −649.000 −1.03840
\(626\) − 632.153i − 1.00983i
\(627\) 0 0
\(628\) 350.000 0.557325
\(629\) − 72.1249i − 0.114666i
\(630\) 0 0
\(631\) 391.000 0.619651 0.309826 0.950793i \(-0.399729\pi\)
0.309826 + 0.950793i \(0.399729\pi\)
\(632\) 1332.00 2.10759
\(633\) 0 0
\(634\) − 598.212i − 0.943553i
\(635\) 1227.54i 1.93313i
\(636\) 0 0
\(637\) 16.9706i 0.0266414i
\(638\) 0 0
\(639\) 0 0
\(640\) 237.588i 0.371231i
\(641\) 23.0000 0.0358814 0.0179407 0.999839i \(-0.494289\pi\)
0.0179407 + 0.999839i \(0.494289\pi\)
\(642\) 0 0
\(643\) −447.000 −0.695179 −0.347589 0.937647i \(-0.613000\pi\)
−0.347589 + 0.937647i \(0.613000\pi\)
\(644\) − 127.279i − 0.197639i
\(645\) 0 0
\(646\) 101.823i 0.157621i
\(647\) 479.000 0.740340 0.370170 0.928964i \(-0.379299\pi\)
0.370170 + 0.928964i \(0.379299\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 576.000 0.886154
\(651\) 0 0
\(652\) −320.000 −0.490798
\(653\) −721.000 −1.10413 −0.552067 0.833800i \(-0.686161\pi\)
−0.552067 + 0.833800i \(0.686161\pi\)
\(654\) 0 0
\(655\) 980.050i 1.49626i
\(656\) 67.8823i 0.103479i
\(657\) 0 0
\(658\) −320.000 −0.486322
\(659\) − 700.036i − 1.06227i −0.847287 0.531135i \(-0.821766\pi\)
0.847287 0.531135i \(-0.178234\pi\)
\(660\) 0 0
\(661\) 607.000 0.918306 0.459153 0.888357i \(-0.348153\pi\)
0.459153 + 0.888357i \(0.348153\pi\)
\(662\) 205.061i 0.309760i
\(663\) 0 0
\(664\) 300.000 0.451807
\(665\) 840.000 1.26316
\(666\) 0 0
\(667\) 203.647i 0.305317i
\(668\) 33.9411i 0.0508101i
\(669\) 0 0
\(670\) − 306.884i − 0.458036i
\(671\) 0 0
\(672\) 0 0
\(673\) − 997.021i − 1.48146i −0.671805 0.740729i \(-0.734481\pi\)
0.671805 0.740729i \(-0.265519\pi\)
\(674\) 362.000 0.537092
\(675\) 0 0
\(676\) −238.000 −0.352071
\(677\) 773.575i 1.14265i 0.820724 + 0.571326i \(0.193571\pi\)
−0.820724 + 0.571326i \(0.806429\pi\)
\(678\) 0 0
\(679\) 120.208i 0.177037i
\(680\) 252.000 0.370588
\(681\) 0 0
\(682\) 0 0
\(683\) 218.000 0.319180 0.159590 0.987183i \(-0.448983\pi\)
0.159590 + 0.987183i \(0.448983\pi\)
\(684\) 0 0
\(685\) −1799.00 −2.62628
\(686\) 480.000 0.699708
\(687\) 0 0
\(688\) 186.676i 0.271332i
\(689\) 271.529i 0.394091i
\(690\) 0 0
\(691\) 863.000 1.24891 0.624457 0.781059i \(-0.285320\pi\)
0.624457 + 0.781059i \(0.285320\pi\)
\(692\) 246.073i 0.355597i
\(693\) 0 0
\(694\) 776.000 1.11816
\(695\) 603.869i 0.868877i
\(696\) 0 0
\(697\) −72.0000 −0.103300
\(698\) −618.000 −0.885387
\(699\) 0 0
\(700\) − 339.411i − 0.484873i
\(701\) 403.051i 0.574966i 0.957786 + 0.287483i \(0.0928184\pi\)
−0.957786 + 0.287483i \(0.907182\pi\)
\(702\) 0 0
\(703\) 288.500i 0.410383i
\(704\) 0 0
\(705\) 0 0
\(706\) − 827.315i − 1.17183i
\(707\) −1090.00 −1.54173
\(708\) 0 0
\(709\) −623.000 −0.878702 −0.439351 0.898315i \(-0.644792\pi\)
−0.439351 + 0.898315i \(0.644792\pi\)
\(710\) 722.663i 1.01784i
\(711\) 0 0
\(712\) 76.3675i 0.107258i
\(713\) 441.000 0.618513
\(714\) 0 0
\(715\) 0 0
\(716\) 398.000 0.555866
\(717\) 0 0
\(718\) 584.000 0.813370
\(719\) −281.000 −0.390821 −0.195410 0.980722i \(-0.562604\pi\)
−0.195410 + 0.980722i \(0.562604\pi\)
\(720\) 0 0
\(721\) 113.137i 0.156917i
\(722\) 103.238i 0.142988i
\(723\) 0 0
\(724\) −146.000 −0.201657
\(725\) 543.058i 0.749046i
\(726\) 0 0
\(727\) 1223.00 1.68226 0.841128 0.540836i \(-0.181892\pi\)
0.841128 + 0.540836i \(0.181892\pi\)
\(728\) − 1018.23i − 1.39867i
\(729\) 0 0
\(730\) −392.000 −0.536986
\(731\) −198.000 −0.270862
\(732\) 0 0
\(733\) − 893.783i − 1.21935i −0.792652 0.609675i \(-0.791300\pi\)
0.792652 0.609675i \(-0.208700\pi\)
\(734\) 770.746i 1.05006i
\(735\) 0 0
\(736\) 254.558i 0.345867i
\(737\) 0 0
\(738\) 0 0
\(739\) 963.079i 1.30322i 0.758554 + 0.651610i \(0.225906\pi\)
−0.758554 + 0.651610i \(0.774094\pi\)
\(740\) 238.000 0.321622
\(741\) 0 0
\(742\) −160.000 −0.215633
\(743\) 11.3137i 0.0152271i 0.999971 + 0.00761353i \(0.00242349\pi\)
−0.999971 + 0.00761353i \(0.997577\pi\)
\(744\) 0 0
\(745\) 1930.40i 2.59114i
\(746\) −330.000 −0.442359
\(747\) 0 0
\(748\) 0 0
\(749\) 1310.00 1.74900
\(750\) 0 0
\(751\) 527.000 0.701731 0.350866 0.936426i \(-0.385887\pi\)
0.350866 + 0.936426i \(0.385887\pi\)
\(752\) 128.000 0.170213
\(753\) 0 0
\(754\) 543.058i 0.720236i
\(755\) 1098.84i 1.45542i
\(756\) 0 0
\(757\) 38.0000 0.0501982 0.0250991 0.999685i \(-0.492010\pi\)
0.0250991 + 0.999685i \(0.492010\pi\)
\(758\) − 632.153i − 0.833976i
\(759\) 0 0
\(760\) −1008.00 −1.32632
\(761\) − 1227.54i − 1.61306i −0.591194 0.806529i \(-0.701343\pi\)
0.591194 0.806529i \(-0.298657\pi\)
\(762\) 0 0
\(763\) −330.000 −0.432503
\(764\) −430.000 −0.562827
\(765\) 0 0
\(766\) 770.746i 1.00620i
\(767\) − 1204.91i − 1.57094i
\(768\) 0 0
\(769\) 311.127i 0.404586i 0.979325 + 0.202293i \(0.0648394\pi\)
−0.979325 + 0.202293i \(0.935161\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 271.529i − 0.351722i
\(773\) −1176.00 −1.52135 −0.760673 0.649136i \(-0.775131\pi\)
−0.760673 + 0.649136i \(0.775131\pi\)
\(774\) 0 0
\(775\) 1176.00 1.51742
\(776\) − 144.250i − 0.185889i
\(777\) 0 0
\(778\) − 304.056i − 0.390817i
\(779\) 288.000 0.369705
\(780\) 0 0
\(781\) 0 0
\(782\) 54.0000 0.0690537
\(783\) 0 0
\(784\) 4.00000 0.00510204
\(785\) 1225.00 1.56051
\(786\) 0 0
\(787\) − 1255.82i − 1.59571i −0.602851 0.797854i \(-0.705969\pi\)
0.602851 0.797854i \(-0.294031\pi\)
\(788\) − 404.465i − 0.513281i
\(789\) 0 0
\(790\) 1554.00 1.96709
\(791\) 459.619i 0.581061i
\(792\) 0 0
\(793\) −192.000 −0.242119
\(794\) − 837.214i − 1.05443i
\(795\) 0 0
\(796\) 400.000 0.502513
\(797\) −191.000 −0.239649 −0.119824 0.992795i \(-0.538233\pi\)
−0.119824 + 0.992795i \(0.538233\pi\)
\(798\) 0 0
\(799\) 135.765i 0.169918i
\(800\) 678.823i 0.848528i
\(801\) 0 0
\(802\) − 690.136i − 0.860519i
\(803\) 0 0
\(804\) 0 0
\(805\) − 445.477i − 0.553388i
\(806\) 1176.00 1.45906
\(807\) 0 0
\(808\) 1308.00 1.61881
\(809\) − 548.715i − 0.678263i −0.940739 0.339132i \(-0.889867\pi\)
0.940739 0.339132i \(-0.110133\pi\)
\(810\) 0 0
\(811\) 405.879i 0.500468i 0.968185 + 0.250234i \(0.0805075\pi\)
−0.968185 + 0.250234i \(0.919492\pi\)
\(812\) 320.000 0.394089
\(813\) 0 0
\(814\) 0 0
\(815\) −1120.00 −1.37423
\(816\) 0 0
\(817\) 792.000 0.969400
\(818\) 222.000 0.271394
\(819\) 0 0
\(820\) − 237.588i − 0.289741i
\(821\) − 55.1543i − 0.0671795i −0.999436 0.0335897i \(-0.989306\pi\)
0.999436 0.0335897i \(-0.0106940\pi\)
\(822\) 0 0
\(823\) 687.000 0.834751 0.417375 0.908734i \(-0.362950\pi\)
0.417375 + 0.908734i \(0.362950\pi\)
\(824\) − 135.765i − 0.164763i
\(825\) 0 0
\(826\) 710.000 0.859564
\(827\) 16.9706i 0.0205206i 0.999947 + 0.0102603i \(0.00326602\pi\)
−0.999947 + 0.0102603i \(0.996734\pi\)
\(828\) 0 0
\(829\) 985.000 1.18818 0.594089 0.804399i \(-0.297513\pi\)
0.594089 + 0.804399i \(0.297513\pi\)
\(830\) 350.000 0.421687
\(831\) 0 0
\(832\) 950.352i 1.14225i
\(833\) 4.24264i 0.00509321i
\(834\) 0 0
\(835\) 118.794i 0.142268i
\(836\) 0 0
\(837\) 0 0
\(838\) 463.862i 0.553535i
\(839\) −87.0000 −0.103695 −0.0518474 0.998655i \(-0.516511\pi\)
−0.0518474 + 0.998655i \(0.516511\pi\)
\(840\) 0 0
\(841\) 329.000 0.391201
\(842\) 294.156i 0.349354i
\(843\) 0 0
\(844\) − 158.392i − 0.187668i
\(845\) −833.000 −0.985799
\(846\) 0 0
\(847\) 0 0
\(848\) 64.0000 0.0754717
\(849\) 0 0
\(850\) 144.000 0.169412
\(851\) 153.000 0.179788
\(852\) 0 0
\(853\) 533.159i 0.625039i 0.949911 + 0.312520i \(0.101173\pi\)
−0.949911 + 0.312520i \(0.898827\pi\)
\(854\) − 113.137i − 0.132479i
\(855\) 0 0
\(856\) −1572.00 −1.83645
\(857\) 1368.96i 1.59738i 0.601740 + 0.798692i \(0.294475\pi\)
−0.601740 + 0.798692i \(0.705525\pi\)
\(858\) 0 0
\(859\) −977.000 −1.13737 −0.568685 0.822556i \(-0.692547\pi\)
−0.568685 + 0.822556i \(0.692547\pi\)
\(860\) − 653.367i − 0.759729i
\(861\) 0 0
\(862\) −794.000 −0.921114
\(863\) 1272.00 1.47393 0.736964 0.675932i \(-0.236259\pi\)
0.736964 + 0.675932i \(0.236259\pi\)
\(864\) 0 0
\(865\) 861.256i 0.995672i
\(866\) 55.1543i 0.0636886i
\(867\) 0 0
\(868\) − 692.965i − 0.798346i
\(869\) 0 0
\(870\) 0 0
\(871\) 526.087i 0.604004i
\(872\) 396.000 0.454128
\(873\) 0 0
\(874\) −216.000 −0.247140
\(875\) 49.4975i 0.0565685i
\(876\) 0 0
\(877\) − 605.283i − 0.690175i −0.938571 0.345087i \(-0.887849\pi\)
0.938571 0.345087i \(-0.112151\pi\)
\(878\) −352.000 −0.400911
\(879\) 0 0
\(880\) 0 0
\(881\) 295.000 0.334847 0.167423 0.985885i \(-0.446455\pi\)
0.167423 + 0.985885i \(0.446455\pi\)
\(882\) 0 0
\(883\) −584.000 −0.661382 −0.330691 0.943739i \(-0.607282\pi\)
−0.330691 + 0.943739i \(0.607282\pi\)
\(884\) −144.000 −0.162896
\(885\) 0 0
\(886\) − 247.487i − 0.279331i
\(887\) 971.565i 1.09534i 0.836695 + 0.547669i \(0.184485\pi\)
−0.836695 + 0.547669i \(0.815515\pi\)
\(888\) 0 0
\(889\) 1240.00 1.39483
\(890\) 89.0955i 0.100107i
\(891\) 0 0
\(892\) −222.000 −0.248879
\(893\) − 543.058i − 0.608128i
\(894\) 0 0
\(895\) 1393.00 1.55642
\(896\) 240.000 0.267857
\(897\) 0 0
\(898\) − 442.649i − 0.492927i
\(899\) 1108.74i 1.23331i
\(900\) 0 0
\(901\) 67.8823i 0.0753410i
\(902\) 0 0
\(903\) 0 0
\(904\) − 551.543i − 0.610114i
\(905\) −511.000 −0.564641
\(906\) 0 0
\(907\) −1536.00 −1.69350 −0.846748 0.531995i \(-0.821443\pi\)
−0.846748 + 0.531995i \(0.821443\pi\)
\(908\) 263.044i 0.289696i
\(909\) 0 0
\(910\) − 1187.94i − 1.30543i
\(911\) −830.000 −0.911087 −0.455543 0.890214i \(-0.650555\pi\)
−0.455543 + 0.890214i \(0.650555\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 16.0000 0.0175055
\(915\) 0 0
\(916\) −606.000 −0.661572
\(917\) 990.000 1.07961
\(918\) 0 0
\(919\) 564.271i 0.614006i 0.951709 + 0.307003i \(0.0993261\pi\)
−0.951709 + 0.307003i \(0.900674\pi\)
\(920\) 534.573i 0.581057i
\(921\) 0 0
\(922\) −1232.00 −1.33623
\(923\) − 1238.85i − 1.34220i
\(924\) 0 0
\(925\) 408.000 0.441081
\(926\) 453.963i 0.490240i
\(927\) 0 0
\(928\) −640.000 −0.689655
\(929\) −1544.00 −1.66200 −0.831001 0.556271i \(-0.812232\pi\)
−0.831001 + 0.556271i \(0.812232\pi\)
\(930\) 0 0
\(931\) − 16.9706i − 0.0182283i
\(932\) 158.392i 0.169948i
\(933\) 0 0
\(934\) 227.688i 0.243778i
\(935\) 0 0
\(936\) 0 0
\(937\) − 888.126i − 0.947840i −0.880568 0.473920i \(-0.842839\pi\)
0.880568 0.473920i \(-0.157161\pi\)
\(938\) −310.000 −0.330490
\(939\) 0 0
\(940\) −448.000 −0.476596
\(941\) − 813.173i − 0.864158i −0.901836 0.432079i \(-0.857780\pi\)
0.901836 0.432079i \(-0.142220\pi\)
\(942\) 0 0
\(943\) − 152.735i − 0.161967i
\(944\) −284.000 −0.300847
\(945\) 0 0
\(946\) 0 0
\(947\) −145.000 −0.153115 −0.0765576 0.997065i \(-0.524393\pi\)
−0.0765576 + 0.997065i \(0.524393\pi\)
\(948\) 0 0
\(949\) 672.000 0.708114
\(950\) −576.000 −0.606316
\(951\) 0 0
\(952\) − 254.558i − 0.267393i
\(953\) 629.325i 0.660362i 0.943918 + 0.330181i \(0.107110\pi\)
−0.943918 + 0.330181i \(0.892890\pi\)
\(954\) 0 0
\(955\) −1505.00 −1.57592
\(956\) − 562.857i − 0.588763i
\(957\) 0 0
\(958\) −632.000 −0.659708
\(959\) 1817.26i 1.89496i
\(960\) 0 0
\(961\) 1440.00 1.49844
\(962\) 408.000 0.424116
\(963\) 0 0
\(964\) 746.705i 0.774590i
\(965\) − 950.352i − 0.984820i
\(966\) 0 0
\(967\) 373.352i 0.386093i 0.981190 + 0.193047i \(0.0618369\pi\)
−0.981190 + 0.193047i \(0.938163\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) − 168.291i − 0.173496i
\(971\) 1695.00 1.74562 0.872812 0.488057i \(-0.162294\pi\)
0.872812 + 0.488057i \(0.162294\pi\)
\(972\) 0 0
\(973\) 610.000 0.626927
\(974\) − 1028.13i − 1.05558i
\(975\) 0 0
\(976\) 45.2548i 0.0463677i
\(977\) 369.000 0.377687 0.188843 0.982007i \(-0.439526\pi\)
0.188843 + 0.982007i \(0.439526\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −14.0000 −0.0142857
\(981\) 0 0
\(982\) −736.000 −0.749491
\(983\) −457.000 −0.464903 −0.232452 0.972608i \(-0.574675\pi\)
−0.232452 + 0.972608i \(0.574675\pi\)
\(984\) 0 0
\(985\) − 1415.63i − 1.43719i
\(986\) 135.765i 0.137692i
\(987\) 0 0
\(988\) 576.000 0.582996
\(989\) − 420.021i − 0.424693i
\(990\) 0 0
\(991\) −1032.00 −1.04137 −0.520686 0.853748i \(-0.674324\pi\)
−0.520686 + 0.853748i \(0.674324\pi\)
\(992\) 1385.93i 1.39711i
\(993\) 0 0
\(994\) 730.000 0.734406
\(995\) 1400.00 1.40704
\(996\) 0 0
\(997\) 708.521i 0.710653i 0.934742 + 0.355326i \(0.115630\pi\)
−0.934742 + 0.355326i \(0.884370\pi\)
\(998\) − 520.431i − 0.521474i
\(999\) 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.c.a.604.2 2
3.2 odd 2 121.3.b.a.120.1 2
11.10 odd 2 inner 1089.3.c.a.604.1 2
33.2 even 10 121.3.d.e.40.2 8
33.5 odd 10 121.3.d.e.118.2 8
33.8 even 10 121.3.d.e.112.2 8
33.14 odd 10 121.3.d.e.112.1 8
33.17 even 10 121.3.d.e.118.1 8
33.20 odd 10 121.3.d.e.40.1 8
33.26 odd 10 121.3.d.e.94.2 8
33.29 even 10 121.3.d.e.94.1 8
33.32 even 2 121.3.b.a.120.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.3.b.a.120.1 2 3.2 odd 2
121.3.b.a.120.2 yes 2 33.32 even 2
121.3.d.e.40.1 8 33.20 odd 10
121.3.d.e.40.2 8 33.2 even 10
121.3.d.e.94.1 8 33.29 even 10
121.3.d.e.94.2 8 33.26 odd 10
121.3.d.e.112.1 8 33.14 odd 10
121.3.d.e.112.2 8 33.8 even 10
121.3.d.e.118.1 8 33.17 even 10
121.3.d.e.118.2 8 33.5 odd 10
1089.3.c.a.604.1 2 11.10 odd 2 inner
1089.3.c.a.604.2 2 1.1 even 1 trivial