Properties

Label 845.2.c.g.506.5
Level $845$
Weight $2$
Character 845.506
Analytic conductor $6.747$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
Defining polynomial: \(x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 506.5
Root \(1.40994 - 0.109843i\) of defining polynomial
Character \(\chi\) \(=\) 845.506
Dual form 845.2.c.g.506.4

$q$-expansion

\(f(q)\) \(=\) \(q+0.219687i q^{2} -1.60020 q^{3} +1.95174 q^{4} +1.00000i q^{5} -0.351542i q^{6} -0.332247i q^{7} +0.868145i q^{8} -0.439374 q^{9} +O(q^{10})\) \(q+0.219687i q^{2} -1.60020 q^{3} +1.95174 q^{4} +1.00000i q^{5} -0.351542i q^{6} -0.332247i q^{7} +0.868145i q^{8} -0.439374 q^{9} -0.219687 q^{10} -5.37182i q^{11} -3.12316 q^{12} +0.0729902 q^{14} -1.60020i q^{15} +3.71276 q^{16} +5.06430 q^{17} -0.0965246i q^{18} -2.26795i q^{19} +1.95174i q^{20} +0.531659i q^{21} +1.18012 q^{22} +2.83918 q^{23} -1.38920i q^{24} -1.00000 q^{25} +5.50367 q^{27} -0.648458i q^{28} -2.90348 q^{29} +0.351542 q^{30} -5.46410i q^{31} +2.55193i q^{32} +8.59596i q^{33} +1.11256i q^{34} +0.332247 q^{35} -0.857542 q^{36} +5.97201i q^{37} +0.498239 q^{38} -0.868145 q^{40} -3.73205i q^{41} -0.116799 q^{42} +5.06430 q^{43} -10.4844i q^{44} -0.439374i q^{45} +0.623730i q^{46} +8.34285i q^{47} -5.94114 q^{48} +6.88961 q^{49} -0.219687i q^{50} -8.10387 q^{51} -1.56063 q^{53} +1.20908i q^{54} +5.37182 q^{55} +0.288438 q^{56} +3.62916i q^{57} -0.637855i q^{58} +2.70732i q^{59} -3.12316i q^{60} +14.1039 q^{61} +1.20039 q^{62} +0.145980i q^{63} +6.86488 q^{64} -1.88842 q^{66} -10.3322i q^{67} +9.88418 q^{68} -4.54324 q^{69} +0.0729902i q^{70} -12.7973i q^{71} -0.381440i q^{72} -9.68922i q^{73} -1.31197 q^{74} +1.60020 q^{75} -4.42644i q^{76} -1.78477 q^{77} +4.51851 q^{79} +3.71276i q^{80} -7.48883 q^{81} +0.819883 q^{82} +4.26371i q^{83} +1.03766i q^{84} +5.06430i q^{85} +1.11256i q^{86} +4.64613 q^{87} +4.66351 q^{88} +3.22584i q^{89} +0.0965246 q^{90} +5.54133 q^{92} +8.74363i q^{93} -1.83281 q^{94} +2.26795 q^{95} -4.08359i q^{96} +2.50791i q^{97} +1.51356i q^{98} +2.36023i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} - 4q^{4} + 8q^{9} + O(q^{10}) \) \( 8q - 4q^{3} - 4q^{4} + 8q^{9} + 4q^{10} + 20q^{12} + 4q^{14} + 4q^{16} + 4q^{17} + 24q^{22} + 20q^{23} - 8q^{25} - 4q^{27} + 16q^{29} - 8q^{30} - 20q^{35} - 40q^{36} - 16q^{38} - 12q^{40} - 8q^{42} + 4q^{43} - 56q^{48} - 24q^{49} - 8q^{51} - 24q^{53} - 24q^{56} + 56q^{61} - 8q^{62} - 8q^{64} + 12q^{66} + 28q^{68} + 32q^{69} - 20q^{74} + 4q^{75} - 36q^{77} - 16q^{79} - 16q^{81} - 8q^{82} - 44q^{87} + 36q^{88} + 40q^{90} + 44q^{92} - 64q^{94} + 32q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.219687i 0.155342i 0.996979 + 0.0776710i \(0.0247484\pi\)
−0.996979 + 0.0776710i \(0.975252\pi\)
\(3\) −1.60020 −0.923873 −0.461937 0.886913i \(-0.652845\pi\)
−0.461937 + 0.886913i \(0.652845\pi\)
\(4\) 1.95174 0.975869
\(5\) 1.00000i 0.447214i
\(6\) − 0.351542i − 0.143516i
\(7\) − 0.332247i − 0.125577i −0.998027 0.0627887i \(-0.980001\pi\)
0.998027 0.0627887i \(-0.0199994\pi\)
\(8\) 0.868145i 0.306936i
\(9\) −0.439374 −0.146458
\(10\) −0.219687 −0.0694711
\(11\) − 5.37182i − 1.61966i −0.586662 0.809832i \(-0.699558\pi\)
0.586662 0.809832i \(-0.300442\pi\)
\(12\) −3.12316 −0.901579
\(13\) 0 0
\(14\) 0.0729902 0.0195074
\(15\) − 1.60020i − 0.413169i
\(16\) 3.71276 0.928189
\(17\) 5.06430 1.22827 0.614136 0.789200i \(-0.289505\pi\)
0.614136 + 0.789200i \(0.289505\pi\)
\(18\) − 0.0965246i − 0.0227511i
\(19\) − 2.26795i − 0.520303i −0.965568 0.260152i \(-0.916227\pi\)
0.965568 0.260152i \(-0.0837725\pi\)
\(20\) 1.95174i 0.436422i
\(21\) 0.531659i 0.116018i
\(22\) 1.18012 0.251602
\(23\) 2.83918 0.592010 0.296005 0.955186i \(-0.404346\pi\)
0.296005 + 0.955186i \(0.404346\pi\)
\(24\) − 1.38920i − 0.283570i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.50367 1.05918
\(28\) − 0.648458i − 0.122547i
\(29\) −2.90348 −0.539162 −0.269581 0.962978i \(-0.586885\pi\)
−0.269581 + 0.962978i \(0.586885\pi\)
\(30\) 0.351542 0.0641825
\(31\) − 5.46410i − 0.981382i −0.871334 0.490691i \(-0.836744\pi\)
0.871334 0.490691i \(-0.163256\pi\)
\(32\) 2.55193i 0.451122i
\(33\) 8.59596i 1.49636i
\(34\) 1.11256i 0.190802i
\(35\) 0.332247 0.0561599
\(36\) −0.857542 −0.142924
\(37\) 5.97201i 0.981793i 0.871218 + 0.490896i \(0.163331\pi\)
−0.871218 + 0.490896i \(0.836669\pi\)
\(38\) 0.498239 0.0808250
\(39\) 0 0
\(40\) −0.868145 −0.137266
\(41\) − 3.73205i − 0.582848i −0.956594 0.291424i \(-0.905871\pi\)
0.956594 0.291424i \(-0.0941291\pi\)
\(42\) −0.116799 −0.0180224
\(43\) 5.06430 0.772298 0.386149 0.922436i \(-0.373805\pi\)
0.386149 + 0.922436i \(0.373805\pi\)
\(44\) − 10.4844i − 1.58058i
\(45\) − 0.439374i − 0.0654980i
\(46\) 0.623730i 0.0919640i
\(47\) 8.34285i 1.21693i 0.793581 + 0.608465i \(0.208214\pi\)
−0.793581 + 0.608465i \(0.791786\pi\)
\(48\) −5.94114 −0.857529
\(49\) 6.88961 0.984230
\(50\) − 0.219687i − 0.0310684i
\(51\) −8.10387 −1.13477
\(52\) 0 0
\(53\) −1.56063 −0.214369 −0.107184 0.994239i \(-0.534183\pi\)
−0.107184 + 0.994239i \(0.534183\pi\)
\(54\) 1.20908i 0.164536i
\(55\) 5.37182 0.724336
\(56\) 0.288438 0.0385442
\(57\) 3.62916i 0.480694i
\(58\) − 0.637855i − 0.0837545i
\(59\) 2.70732i 0.352463i 0.984349 + 0.176232i \(0.0563908\pi\)
−0.984349 + 0.176232i \(0.943609\pi\)
\(60\) − 3.12316i − 0.403199i
\(61\) 14.1039 1.80582 0.902908 0.429835i \(-0.141428\pi\)
0.902908 + 0.429835i \(0.141428\pi\)
\(62\) 1.20039 0.152450
\(63\) 0.145980i 0.0183918i
\(64\) 6.86488 0.858111
\(65\) 0 0
\(66\) −1.88842 −0.232448
\(67\) − 10.3322i − 1.26228i −0.775667 0.631142i \(-0.782586\pi\)
0.775667 0.631142i \(-0.217414\pi\)
\(68\) 9.88418 1.19863
\(69\) −4.54324 −0.546942
\(70\) 0.0729902i 0.00872400i
\(71\) − 12.7973i − 1.51876i −0.650645 0.759382i \(-0.725502\pi\)
0.650645 0.759382i \(-0.274498\pi\)
\(72\) − 0.381440i − 0.0449531i
\(73\) − 9.68922i − 1.13404i −0.823705 0.567019i \(-0.808097\pi\)
0.823705 0.567019i \(-0.191903\pi\)
\(74\) −1.31197 −0.152514
\(75\) 1.60020 0.184775
\(76\) − 4.42644i − 0.507748i
\(77\) −1.78477 −0.203393
\(78\) 0 0
\(79\) 4.51851 0.508372 0.254186 0.967155i \(-0.418192\pi\)
0.254186 + 0.967155i \(0.418192\pi\)
\(80\) 3.71276i 0.415099i
\(81\) −7.48883 −0.832092
\(82\) 0.819883 0.0905409
\(83\) 4.26371i 0.468003i 0.972236 + 0.234001i \(0.0751821\pi\)
−0.972236 + 0.234001i \(0.924818\pi\)
\(84\) 1.03766i 0.113218i
\(85\) 5.06430i 0.549300i
\(86\) 1.11256i 0.119970i
\(87\) 4.64613 0.498117
\(88\) 4.66351 0.497132
\(89\) 3.22584i 0.341938i 0.985276 + 0.170969i \(0.0546898\pi\)
−0.985276 + 0.170969i \(0.945310\pi\)
\(90\) 0.0965246 0.0101746
\(91\) 0 0
\(92\) 5.54133 0.577724
\(93\) 8.74363i 0.906672i
\(94\) −1.83281 −0.189040
\(95\) 2.26795 0.232687
\(96\) − 4.08359i − 0.416780i
\(97\) 2.50791i 0.254640i 0.991862 + 0.127320i \(0.0406375\pi\)
−0.991862 + 0.127320i \(0.959363\pi\)
\(98\) 1.51356i 0.152892i
\(99\) 2.36023i 0.237213i
\(100\) −1.95174 −0.195174
\(101\) −12.4467 −1.23849 −0.619247 0.785196i \(-0.712562\pi\)
−0.619247 + 0.785196i \(0.712562\pi\)
\(102\) − 1.78031i − 0.176277i
\(103\) 15.0247 1.48043 0.740215 0.672370i \(-0.234724\pi\)
0.740215 + 0.672370i \(0.234724\pi\)
\(104\) 0 0
\(105\) −0.531659 −0.0518846
\(106\) − 0.342849i − 0.0333004i
\(107\) −13.0643 −1.26297 −0.631487 0.775387i \(-0.717555\pi\)
−0.631487 + 0.775387i \(0.717555\pi\)
\(108\) 10.7417 1.03362
\(109\) 11.2325i 1.07587i 0.842985 + 0.537937i \(0.180796\pi\)
−0.842985 + 0.537937i \(0.819204\pi\)
\(110\) 1.18012i 0.112520i
\(111\) − 9.55639i − 0.907052i
\(112\) − 1.23355i − 0.116560i
\(113\) −18.3438 −1.72564 −0.862821 0.505509i \(-0.831305\pi\)
−0.862821 + 0.505509i \(0.831305\pi\)
\(114\) −0.797279 −0.0746721
\(115\) 2.83918i 0.264755i
\(116\) −5.66682 −0.526151
\(117\) 0 0
\(118\) −0.594763 −0.0547524
\(119\) − 1.68260i − 0.154243i
\(120\) 1.38920 0.126816
\(121\) −17.8564 −1.62331
\(122\) 3.09843i 0.280519i
\(123\) 5.97201i 0.538478i
\(124\) − 10.6645i − 0.957700i
\(125\) − 1.00000i − 0.0894427i
\(126\) −0.0320700 −0.00285702
\(127\) −3.23996 −0.287500 −0.143750 0.989614i \(-0.545916\pi\)
−0.143750 + 0.989614i \(0.545916\pi\)
\(128\) 6.61199i 0.584423i
\(129\) −8.10387 −0.713506
\(130\) 0 0
\(131\) 0.175664 0.0153478 0.00767390 0.999971i \(-0.497557\pi\)
0.00767390 + 0.999971i \(0.497557\pi\)
\(132\) 16.7771i 1.46025i
\(133\) −0.753518 −0.0653383
\(134\) 2.26986 0.196086
\(135\) 5.50367i 0.473681i
\(136\) 4.39654i 0.377001i
\(137\) − 17.9829i − 1.53638i −0.640221 0.768190i \(-0.721157\pi\)
0.640221 0.768190i \(-0.278843\pi\)
\(138\) − 0.998090i − 0.0849631i
\(139\) 11.9861 1.01665 0.508325 0.861165i \(-0.330265\pi\)
0.508325 + 0.861165i \(0.330265\pi\)
\(140\) 0.648458 0.0548047
\(141\) − 13.3502i − 1.12429i
\(142\) 2.81140 0.235928
\(143\) 0 0
\(144\) −1.63129 −0.135941
\(145\) − 2.90348i − 0.241121i
\(146\) 2.12859 0.176164
\(147\) −11.0247 −0.909304
\(148\) 11.6558i 0.958101i
\(149\) 3.41041i 0.279391i 0.990194 + 0.139696i \(0.0446124\pi\)
−0.990194 + 0.139696i \(0.955388\pi\)
\(150\) 0.351542i 0.0287033i
\(151\) 7.96141i 0.647890i 0.946076 + 0.323945i \(0.105009\pi\)
−0.946076 + 0.323945i \(0.894991\pi\)
\(152\) 1.96891 0.159700
\(153\) −2.22512 −0.179890
\(154\) − 0.392090i − 0.0315955i
\(155\) 5.46410 0.438887
\(156\) 0 0
\(157\) −16.4329 −1.31148 −0.655742 0.754985i \(-0.727644\pi\)
−0.655742 + 0.754985i \(0.727644\pi\)
\(158\) 0.992658i 0.0789716i
\(159\) 2.49731 0.198049
\(160\) −2.55193 −0.201748
\(161\) − 0.943307i − 0.0743430i
\(162\) − 1.64520i − 0.129259i
\(163\) 17.8072i 1.39477i 0.716697 + 0.697384i \(0.245653\pi\)
−0.716697 + 0.697384i \(0.754347\pi\)
\(164\) − 7.28398i − 0.568784i
\(165\) −8.59596 −0.669194
\(166\) −0.936681 −0.0727006
\(167\) − 6.29366i − 0.487018i −0.969899 0.243509i \(-0.921702\pi\)
0.969899 0.243509i \(-0.0782985\pi\)
\(168\) −0.461557 −0.0356099
\(169\) 0 0
\(170\) −1.11256 −0.0853294
\(171\) 0.996477i 0.0762025i
\(172\) 9.88418 0.753662
\(173\) −15.9751 −1.21457 −0.607283 0.794486i \(-0.707740\pi\)
−0.607283 + 0.794486i \(0.707740\pi\)
\(174\) 1.02069i 0.0773786i
\(175\) 0.332247i 0.0251155i
\(176\) − 19.9442i − 1.50335i
\(177\) − 4.33225i − 0.325632i
\(178\) −0.708674 −0.0531173
\(179\) −23.6174 −1.76525 −0.882625 0.470079i \(-0.844226\pi\)
−0.882625 + 0.470079i \(0.844226\pi\)
\(180\) − 0.857542i − 0.0639174i
\(181\) 2.62590 0.195182 0.0975909 0.995227i \(-0.468886\pi\)
0.0975909 + 0.995227i \(0.468886\pi\)
\(182\) 0 0
\(183\) −22.5689 −1.66834
\(184\) 2.46482i 0.181709i
\(185\) −5.97201 −0.439071
\(186\) −1.92086 −0.140844
\(187\) − 27.2045i − 1.98939i
\(188\) 16.2831i 1.18756i
\(189\) − 1.82858i − 0.133009i
\(190\) 0.498239i 0.0361460i
\(191\) −2.01582 −0.145860 −0.0729298 0.997337i \(-0.523235\pi\)
−0.0729298 + 0.997337i \(0.523235\pi\)
\(192\) −10.9852 −0.792786
\(193\) 22.8211i 1.64270i 0.570427 + 0.821348i \(0.306778\pi\)
−0.570427 + 0.821348i \(0.693222\pi\)
\(194\) −0.550955 −0.0395563
\(195\) 0 0
\(196\) 13.4467 0.960480
\(197\) 0.643026i 0.0458137i 0.999738 + 0.0229068i \(0.00729211\pi\)
−0.999738 + 0.0229068i \(0.992708\pi\)
\(198\) −0.518513 −0.0368491
\(199\) −3.06684 −0.217403 −0.108701 0.994074i \(-0.534669\pi\)
−0.108701 + 0.994074i \(0.534669\pi\)
\(200\) − 0.868145i − 0.0613871i
\(201\) 16.5336i 1.16619i
\(202\) − 2.73438i − 0.192390i
\(203\) 0.964670i 0.0677065i
\(204\) −15.8166 −1.10739
\(205\) 3.73205 0.260658
\(206\) 3.30074i 0.229973i
\(207\) −1.24746 −0.0867045
\(208\) 0 0
\(209\) −12.1830 −0.842716
\(210\) − 0.116799i − 0.00805987i
\(211\) −8.20039 −0.564538 −0.282269 0.959335i \(-0.591087\pi\)
−0.282269 + 0.959335i \(0.591087\pi\)
\(212\) −3.04593 −0.209196
\(213\) 20.4782i 1.40314i
\(214\) − 2.87005i − 0.196193i
\(215\) 5.06430i 0.345382i
\(216\) 4.77798i 0.325101i
\(217\) −1.81543 −0.123239
\(218\) −2.46762 −0.167129
\(219\) 15.5046i 1.04771i
\(220\) 10.4844 0.706856
\(221\) 0 0
\(222\) 2.09941 0.140903
\(223\) − 10.2442i − 0.686002i −0.939335 0.343001i \(-0.888557\pi\)
0.939335 0.343001i \(-0.111443\pi\)
\(224\) 0.847871 0.0566508
\(225\) 0.439374 0.0292916
\(226\) − 4.02990i − 0.268065i
\(227\) − 7.04381i − 0.467514i −0.972295 0.233757i \(-0.924898\pi\)
0.972295 0.233757i \(-0.0751020\pi\)
\(228\) 7.08317i 0.469095i
\(229\) 1.32899i 0.0878219i 0.999035 + 0.0439109i \(0.0139818\pi\)
−0.999035 + 0.0439109i \(0.986018\pi\)
\(230\) −0.623730 −0.0411275
\(231\) 2.85598 0.187909
\(232\) − 2.52064i − 0.165488i
\(233\) 1.24746 0.0817238 0.0408619 0.999165i \(-0.486990\pi\)
0.0408619 + 0.999165i \(0.486990\pi\)
\(234\) 0 0
\(235\) −8.34285 −0.544227
\(236\) 5.28398i 0.343958i
\(237\) −7.23050 −0.469672
\(238\) 0.369644 0.0239605
\(239\) 9.94207i 0.643099i 0.946893 + 0.321549i \(0.104204\pi\)
−0.946893 + 0.321549i \(0.895796\pi\)
\(240\) − 5.94114i − 0.383499i
\(241\) − 22.5869i − 1.45495i −0.686134 0.727475i \(-0.740694\pi\)
0.686134 0.727475i \(-0.259306\pi\)
\(242\) − 3.92282i − 0.252168i
\(243\) −4.52742 −0.290434
\(244\) 27.5270 1.76224
\(245\) 6.88961i 0.440161i
\(246\) −1.31197 −0.0836483
\(247\) 0 0
\(248\) 4.74363 0.301221
\(249\) − 6.82277i − 0.432376i
\(250\) 0.219687 0.0138942
\(251\) 6.76836 0.427215 0.213608 0.976920i \(-0.431479\pi\)
0.213608 + 0.976920i \(0.431479\pi\)
\(252\) 0.284915i 0.0179480i
\(253\) − 15.2515i − 0.958856i
\(254\) − 0.711777i − 0.0446609i
\(255\) − 8.10387i − 0.507484i
\(256\) 12.2772 0.767325
\(257\) 10.2538 0.639616 0.319808 0.947482i \(-0.396382\pi\)
0.319808 + 0.947482i \(0.396382\pi\)
\(258\) − 1.78031i − 0.110837i
\(259\) 1.98418 0.123291
\(260\) 0 0
\(261\) 1.27571 0.0789645
\(262\) 0.0385910i 0.00238416i
\(263\) 18.6570 1.15044 0.575220 0.817999i \(-0.304917\pi\)
0.575220 + 0.817999i \(0.304917\pi\)
\(264\) −7.46254 −0.459287
\(265\) − 1.56063i − 0.0958685i
\(266\) − 0.165538i − 0.0101498i
\(267\) − 5.16197i − 0.315907i
\(268\) − 20.1658i − 1.23182i
\(269\) 17.9579 1.09491 0.547456 0.836835i \(-0.315596\pi\)
0.547456 + 0.836835i \(0.315596\pi\)
\(270\) −1.20908 −0.0735825
\(271\) 30.8977i 1.87690i 0.345415 + 0.938450i \(0.387738\pi\)
−0.345415 + 0.938450i \(0.612262\pi\)
\(272\) 18.8025 1.14007
\(273\) 0 0
\(274\) 3.95060 0.238665
\(275\) 5.37182i 0.323933i
\(276\) −8.86721 −0.533744
\(277\) −26.5045 −1.59250 −0.796250 0.604967i \(-0.793186\pi\)
−0.796250 + 0.604967i \(0.793186\pi\)
\(278\) 2.63320i 0.157929i
\(279\) 2.40078i 0.143731i
\(280\) 0.288438i 0.0172375i
\(281\) 4.97766i 0.296942i 0.988917 + 0.148471i \(0.0474352\pi\)
−0.988917 + 0.148471i \(0.952565\pi\)
\(282\) 2.93286 0.174649
\(283\) 12.5863 0.748180 0.374090 0.927392i \(-0.377955\pi\)
0.374090 + 0.927392i \(0.377955\pi\)
\(284\) − 24.9770i − 1.48211i
\(285\) −3.62916 −0.214973
\(286\) 0 0
\(287\) −1.23996 −0.0731926
\(288\) − 1.12125i − 0.0660704i
\(289\) 8.64711 0.508653
\(290\) 0.637855 0.0374562
\(291\) − 4.01315i − 0.235255i
\(292\) − 18.9108i − 1.10667i
\(293\) 16.9176i 0.988337i 0.869366 + 0.494168i \(0.164527\pi\)
−0.869366 + 0.494168i \(0.835473\pi\)
\(294\) − 2.42199i − 0.141253i
\(295\) −2.70732 −0.157626
\(296\) −5.18457 −0.301347
\(297\) − 29.5647i − 1.71552i
\(298\) −0.749222 −0.0434012
\(299\) 0 0
\(300\) 3.12316 0.180316
\(301\) − 1.68260i − 0.0969832i
\(302\) −1.74902 −0.100645
\(303\) 19.9172 1.14421
\(304\) − 8.42034i − 0.482940i
\(305\) 14.1039i 0.807585i
\(306\) − 0.488829i − 0.0279445i
\(307\) − 4.30426i − 0.245657i −0.992428 0.122828i \(-0.960803\pi\)
0.992428 0.122828i \(-0.0391965\pi\)
\(308\) −3.48340 −0.198485
\(309\) −24.0425 −1.36773
\(310\) 1.20039i 0.0681776i
\(311\) 2.22512 0.126175 0.0630875 0.998008i \(-0.479905\pi\)
0.0630875 + 0.998008i \(0.479905\pi\)
\(312\) 0 0
\(313\) 7.20887 0.407469 0.203735 0.979026i \(-0.434692\pi\)
0.203735 + 0.979026i \(0.434692\pi\)
\(314\) − 3.61008i − 0.203729i
\(315\) −0.145980 −0.00822506
\(316\) 8.81895 0.496105
\(317\) 0.321644i 0.0180653i 0.999959 + 0.00903266i \(0.00287522\pi\)
−0.999959 + 0.00903266i \(0.997125\pi\)
\(318\) 0.548626i 0.0307654i
\(319\) 15.5969i 0.873261i
\(320\) 6.86488i 0.383759i
\(321\) 20.9054 1.16683
\(322\) 0.207232 0.0115486
\(323\) − 11.4856i − 0.639074i
\(324\) −14.6162 −0.812013
\(325\) 0 0
\(326\) −3.91201 −0.216666
\(327\) − 17.9741i − 0.993972i
\(328\) 3.23996 0.178897
\(329\) 2.77188 0.152819
\(330\) − 1.88842i − 0.103954i
\(331\) − 16.6320i − 0.914178i −0.889421 0.457089i \(-0.848892\pi\)
0.889421 0.457089i \(-0.151108\pi\)
\(332\) 8.32164i 0.456710i
\(333\) − 2.62395i − 0.143791i
\(334\) 1.38263 0.0756543
\(335\) 10.3322 0.564511
\(336\) 1.97392i 0.107686i
\(337\) −24.2186 −1.31927 −0.659636 0.751586i \(-0.729289\pi\)
−0.659636 + 0.751586i \(0.729289\pi\)
\(338\) 0 0
\(339\) 29.3537 1.59427
\(340\) 9.88418i 0.536045i
\(341\) −29.3521 −1.58951
\(342\) −0.218913 −0.0118375
\(343\) − 4.61478i − 0.249174i
\(344\) 4.39654i 0.237046i
\(345\) − 4.54324i − 0.244600i
\(346\) − 3.50952i − 0.188673i
\(347\) −6.27360 −0.336784 −0.168392 0.985720i \(-0.553857\pi\)
−0.168392 + 0.985720i \(0.553857\pi\)
\(348\) 9.06802 0.486097
\(349\) 7.06994i 0.378445i 0.981934 + 0.189223i \(0.0605968\pi\)
−0.981934 + 0.189223i \(0.939403\pi\)
\(350\) −0.0729902 −0.00390149
\(351\) 0 0
\(352\) 13.7085 0.730666
\(353\) 21.7898i 1.15976i 0.814704 + 0.579878i \(0.196900\pi\)
−0.814704 + 0.579878i \(0.803100\pi\)
\(354\) 0.951738 0.0505843
\(355\) 12.7973 0.679212
\(356\) 6.29598i 0.333687i
\(357\) 2.69248i 0.142501i
\(358\) − 5.18844i − 0.274217i
\(359\) 23.9737i 1.26528i 0.774444 + 0.632642i \(0.218029\pi\)
−0.774444 + 0.632642i \(0.781971\pi\)
\(360\) 0.381440 0.0201037
\(361\) 13.8564 0.729285
\(362\) 0.576876i 0.0303199i
\(363\) 28.5737 1.49973
\(364\) 0 0
\(365\) 9.68922 0.507157
\(366\) − 4.95810i − 0.259164i
\(367\) 6.39133 0.333625 0.166812 0.985989i \(-0.446653\pi\)
0.166812 + 0.985989i \(0.446653\pi\)
\(368\) 10.5412 0.549497
\(369\) 1.63977i 0.0853628i
\(370\) − 1.31197i − 0.0682062i
\(371\) 0.518513i 0.0269198i
\(372\) 17.0653i 0.884793i
\(373\) −20.0801 −1.03971 −0.519855 0.854255i \(-0.674014\pi\)
−0.519855 + 0.854255i \(0.674014\pi\)
\(374\) 5.97647 0.309036
\(375\) 1.60020i 0.0826338i
\(376\) −7.24280 −0.373519
\(377\) 0 0
\(378\) 0.401714 0.0206619
\(379\) 5.46182i 0.280555i 0.990112 + 0.140277i \(0.0447994\pi\)
−0.990112 + 0.140277i \(0.955201\pi\)
\(380\) 4.42644 0.227072
\(381\) 5.18457 0.265614
\(382\) − 0.442849i − 0.0226581i
\(383\) 5.66775i 0.289609i 0.989460 + 0.144804i \(0.0462553\pi\)
−0.989460 + 0.144804i \(0.953745\pi\)
\(384\) − 10.5805i − 0.539933i
\(385\) − 1.78477i − 0.0909602i
\(386\) −5.01349 −0.255180
\(387\) −2.22512 −0.113109
\(388\) 4.89478i 0.248495i
\(389\) −10.6174 −0.538325 −0.269162 0.963095i \(-0.586747\pi\)
−0.269162 + 0.963095i \(0.586747\pi\)
\(390\) 0 0
\(391\) 14.3784 0.727149
\(392\) 5.98118i 0.302095i
\(393\) −0.281096 −0.0141794
\(394\) −0.141264 −0.00711679
\(395\) 4.51851i 0.227351i
\(396\) 4.60656i 0.231488i
\(397\) 28.0338i 1.40697i 0.710708 + 0.703487i \(0.248375\pi\)
−0.710708 + 0.703487i \(0.751625\pi\)
\(398\) − 0.673745i − 0.0337718i
\(399\) 1.20578 0.0603643
\(400\) −3.71276 −0.185638
\(401\) − 22.5143i − 1.12431i −0.827032 0.562155i \(-0.809973\pi\)
0.827032 0.562155i \(-0.190027\pi\)
\(402\) −3.63222 −0.181159
\(403\) 0 0
\(404\) −24.2927 −1.20861
\(405\) − 7.48883i − 0.372123i
\(406\) −0.211925 −0.0105177
\(407\) 32.0805 1.59017
\(408\) − 7.03533i − 0.348301i
\(409\) − 4.28772i − 0.212014i −0.994365 0.106007i \(-0.966193\pi\)
0.994365 0.106007i \(-0.0338066\pi\)
\(410\) 0.819883i 0.0404911i
\(411\) 28.7761i 1.41942i
\(412\) 29.3243 1.44471
\(413\) 0.899499 0.0442614
\(414\) − 0.274051i − 0.0134689i
\(415\) −4.26371 −0.209297
\(416\) 0 0
\(417\) −19.1802 −0.939257
\(418\) − 2.67645i − 0.130909i
\(419\) 17.7116 0.865266 0.432633 0.901570i \(-0.357585\pi\)
0.432633 + 0.901570i \(0.357585\pi\)
\(420\) −1.03766 −0.0506326
\(421\) − 12.8787i − 0.627672i −0.949477 0.313836i \(-0.898386\pi\)
0.949477 0.313836i \(-0.101614\pi\)
\(422\) − 1.80152i − 0.0876965i
\(423\) − 3.66563i − 0.178229i
\(424\) − 1.35485i − 0.0657973i
\(425\) −5.06430 −0.245655
\(426\) −4.49880 −0.217967
\(427\) − 4.68596i − 0.226770i
\(428\) −25.4981 −1.23250
\(429\) 0 0
\(430\) −1.11256 −0.0536524
\(431\) − 9.49845i − 0.457524i −0.973482 0.228762i \(-0.926532\pi\)
0.973482 0.228762i \(-0.0734678\pi\)
\(432\) 20.4338 0.983121
\(433\) 1.39628 0.0671010 0.0335505 0.999437i \(-0.489319\pi\)
0.0335505 + 0.999437i \(0.489319\pi\)
\(434\) − 0.398826i − 0.0191443i
\(435\) 4.64613i 0.222765i
\(436\) 21.9228i 1.04991i
\(437\) − 6.43911i − 0.308024i
\(438\) −3.40617 −0.162753
\(439\) −4.16180 −0.198632 −0.0993159 0.995056i \(-0.531665\pi\)
−0.0993159 + 0.995056i \(0.531665\pi\)
\(440\) 4.66351i 0.222324i
\(441\) −3.02711 −0.144148
\(442\) 0 0
\(443\) 9.54563 0.453526 0.226763 0.973950i \(-0.427186\pi\)
0.226763 + 0.973950i \(0.427186\pi\)
\(444\) − 18.6516i − 0.885164i
\(445\) −3.22584 −0.152919
\(446\) 2.25052 0.106565
\(447\) − 5.45732i − 0.258122i
\(448\) − 2.28083i − 0.107759i
\(449\) 21.7171i 1.02489i 0.858720 + 0.512446i \(0.171260\pi\)
−0.858720 + 0.512446i \(0.828740\pi\)
\(450\) 0.0965246i 0.00455022i
\(451\) −20.0479 −0.944018
\(452\) −35.8023 −1.68400
\(453\) − 12.7398i − 0.598569i
\(454\) 1.54743 0.0726246
\(455\) 0 0
\(456\) −3.15064 −0.147542
\(457\) − 4.72259i − 0.220914i −0.993881 0.110457i \(-0.964769\pi\)
0.993881 0.110457i \(-0.0352314\pi\)
\(458\) −0.291961 −0.0136424
\(459\) 27.8722 1.30096
\(460\) 5.54133i 0.258366i
\(461\) 1.78151i 0.0829730i 0.999139 + 0.0414865i \(0.0132094\pi\)
−0.999139 + 0.0414865i \(0.986791\pi\)
\(462\) 0.627421i 0.0291902i
\(463\) 6.80200i 0.316116i 0.987430 + 0.158058i \(0.0505232\pi\)
−0.987430 + 0.158058i \(0.949477\pi\)
\(464\) −10.7799 −0.500444
\(465\) −8.74363 −0.405476
\(466\) 0.274051i 0.0126952i
\(467\) −18.2374 −0.843927 −0.421963 0.906613i \(-0.638659\pi\)
−0.421963 + 0.906613i \(0.638659\pi\)
\(468\) 0 0
\(469\) −3.43285 −0.158514
\(470\) − 1.83281i − 0.0845414i
\(471\) 26.2958 1.21165
\(472\) −2.35035 −0.108184
\(473\) − 27.2045i − 1.25086i
\(474\) − 1.58845i − 0.0729598i
\(475\) 2.26795i 0.104061i
\(476\) − 3.28398i − 0.150521i
\(477\) 0.685698 0.0313960
\(478\) −2.18414 −0.0999003
\(479\) − 35.1807i − 1.60745i −0.595002 0.803724i \(-0.702849\pi\)
0.595002 0.803724i \(-0.297151\pi\)
\(480\) 4.08359 0.186390
\(481\) 0 0
\(482\) 4.96204 0.226015
\(483\) 1.50948i 0.0686835i
\(484\) −34.8510 −1.58414
\(485\) −2.50791 −0.113878
\(486\) − 0.994615i − 0.0451166i
\(487\) 10.3040i 0.466919i 0.972367 + 0.233459i \(0.0750046\pi\)
−0.972367 + 0.233459i \(0.924995\pi\)
\(488\) 12.2442i 0.554269i
\(489\) − 28.4950i − 1.28859i
\(490\) −1.51356 −0.0683756
\(491\) −9.33198 −0.421147 −0.210573 0.977578i \(-0.567533\pi\)
−0.210573 + 0.977578i \(0.567533\pi\)
\(492\) 11.6558i 0.525484i
\(493\) −14.7041 −0.662238
\(494\) 0 0
\(495\) −2.36023 −0.106085
\(496\) − 20.2869i − 0.910907i
\(497\) −4.25187 −0.190722
\(498\) 1.49887 0.0671661
\(499\) − 23.9421i − 1.07179i −0.844283 0.535897i \(-0.819974\pi\)
0.844283 0.535897i \(-0.180026\pi\)
\(500\) − 1.95174i − 0.0872844i
\(501\) 10.0711i 0.449943i
\(502\) 1.48692i 0.0663645i
\(503\) 42.1443 1.87912 0.939560 0.342385i \(-0.111235\pi\)
0.939560 + 0.342385i \(0.111235\pi\)
\(504\) −0.126732 −0.00564510
\(505\) − 12.4467i − 0.553872i
\(506\) 3.35056 0.148951
\(507\) 0 0
\(508\) −6.32355 −0.280562
\(509\) 33.5602i 1.48753i 0.668441 + 0.743765i \(0.266962\pi\)
−0.668441 + 0.743765i \(0.733038\pi\)
\(510\) 1.78031 0.0788336
\(511\) −3.21921 −0.142409
\(512\) 15.9211i 0.703621i
\(513\) − 12.4820i − 0.551096i
\(514\) 2.25263i 0.0993593i
\(515\) 15.0247i 0.662069i
\(516\) −15.8166 −0.696288
\(517\) 44.8162 1.97102
\(518\) 0.435898i 0.0191523i
\(519\) 25.5633 1.12210
\(520\) 0 0
\(521\) 12.4649 0.546098 0.273049 0.962000i \(-0.411968\pi\)
0.273049 + 0.962000i \(0.411968\pi\)
\(522\) 0.280257i 0.0122665i
\(523\) −5.65956 −0.247475 −0.123738 0.992315i \(-0.539488\pi\)
−0.123738 + 0.992315i \(0.539488\pi\)
\(524\) 0.342849 0.0149774
\(525\) − 0.531659i − 0.0232035i
\(526\) 4.09870i 0.178712i
\(527\) − 27.6718i − 1.20540i
\(528\) 31.9147i 1.38891i
\(529\) −14.9391 −0.649525
\(530\) 0.342849 0.0148924
\(531\) − 1.18953i − 0.0516211i
\(532\) −1.47067 −0.0637616
\(533\) 0 0
\(534\) 1.13402 0.0490737
\(535\) − 13.0643i − 0.564819i
\(536\) 8.96989 0.387440
\(537\) 37.7925 1.63087
\(538\) 3.94511i 0.170086i
\(539\) − 37.0097i − 1.59412i
\(540\) 10.7417i 0.462250i
\(541\) − 15.4750i − 0.665321i −0.943047 0.332660i \(-0.892054\pi\)
0.943047 0.332660i \(-0.107946\pi\)
\(542\) −6.78781 −0.291562
\(543\) −4.20196 −0.180323
\(544\) 12.9237i 0.554101i
\(545\) −11.2325 −0.481146
\(546\) 0 0
\(547\) 25.1765 1.07647 0.538234 0.842795i \(-0.319092\pi\)
0.538234 + 0.842795i \(0.319092\pi\)
\(548\) − 35.0979i − 1.49931i
\(549\) −6.19687 −0.264476
\(550\) −1.18012 −0.0503204
\(551\) 6.58493i 0.280528i
\(552\) − 3.94419i − 0.167876i
\(553\) − 1.50126i − 0.0638401i
\(554\) − 5.82269i − 0.247382i
\(555\) 9.55639 0.405646
\(556\) 23.3938 0.992118
\(557\) 42.3489i 1.79438i 0.441645 + 0.897190i \(0.354395\pi\)
−0.441645 + 0.897190i \(0.645605\pi\)
\(558\) −0.527420 −0.0223275
\(559\) 0 0
\(560\) 1.23355 0.0521270
\(561\) 43.5325i 1.83794i
\(562\) −1.09353 −0.0461276
\(563\) −23.7905 −1.00265 −0.501326 0.865259i \(-0.667154\pi\)
−0.501326 + 0.865259i \(0.667154\pi\)
\(564\) − 26.0561i − 1.09716i
\(565\) − 18.3438i − 0.771731i
\(566\) 2.76505i 0.116224i
\(567\) 2.48814i 0.104492i
\(568\) 11.1099 0.466162
\(569\) 26.7421 1.12109 0.560543 0.828125i \(-0.310593\pi\)
0.560543 + 0.828125i \(0.310593\pi\)
\(570\) − 0.797279i − 0.0333944i
\(571\) 16.7159 0.699539 0.349769 0.936836i \(-0.386260\pi\)
0.349769 + 0.936836i \(0.386260\pi\)
\(572\) 0 0
\(573\) 3.22571 0.134756
\(574\) − 0.272403i − 0.0113699i
\(575\) −2.83918 −0.118402
\(576\) −3.01625 −0.125677
\(577\) 20.6768i 0.860786i 0.902642 + 0.430393i \(0.141625\pi\)
−0.902642 + 0.430393i \(0.858375\pi\)
\(578\) 1.89966i 0.0790153i
\(579\) − 36.5182i − 1.51764i
\(580\) − 5.66682i − 0.235302i
\(581\) 1.41660 0.0587706
\(582\) 0.881636 0.0365450
\(583\) 8.38340i 0.347205i
\(584\) 8.41165 0.348076
\(585\) 0 0
\(586\) −3.71657 −0.153530
\(587\) − 20.7972i − 0.858391i −0.903212 0.429196i \(-0.858797\pi\)
0.903212 0.429196i \(-0.141203\pi\)
\(588\) −21.5174 −0.887362
\(589\) −12.3923 −0.510616
\(590\) − 0.594763i − 0.0244860i
\(591\) − 1.02897i − 0.0423260i
\(592\) 22.1726i 0.911289i
\(593\) − 21.8475i − 0.897169i −0.893740 0.448585i \(-0.851928\pi\)
0.893740 0.448585i \(-0.148072\pi\)
\(594\) 6.49498 0.266492
\(595\) 1.68260 0.0689797
\(596\) 6.65622i 0.272649i
\(597\) 4.90755 0.200853
\(598\) 0 0
\(599\) −3.58040 −0.146291 −0.0731456 0.997321i \(-0.523304\pi\)
−0.0731456 + 0.997321i \(0.523304\pi\)
\(600\) 1.38920i 0.0567139i
\(601\) 21.3486 0.870829 0.435414 0.900230i \(-0.356602\pi\)
0.435414 + 0.900230i \(0.356602\pi\)
\(602\) 0.369644 0.0150656
\(603\) 4.53972i 0.184872i
\(604\) 15.5386i 0.632256i
\(605\) − 17.8564i − 0.725966i
\(606\) 4.37554i 0.177744i
\(607\) −3.29976 −0.133933 −0.0669665 0.997755i \(-0.521332\pi\)
−0.0669665 + 0.997755i \(0.521332\pi\)
\(608\) 5.78766 0.234720
\(609\) − 1.54366i − 0.0625523i
\(610\) −3.09843 −0.125452
\(611\) 0 0
\(612\) −4.34285 −0.175549
\(613\) − 9.88635i − 0.399306i −0.979867 0.199653i \(-0.936019\pi\)
0.979867 0.199653i \(-0.0639815\pi\)
\(614\) 0.945589 0.0381609
\(615\) −5.97201 −0.240815
\(616\) − 1.54944i − 0.0624286i
\(617\) 45.7169i 1.84049i 0.391339 + 0.920246i \(0.372012\pi\)
−0.391339 + 0.920246i \(0.627988\pi\)
\(618\) − 5.28182i − 0.212466i
\(619\) 19.9143i 0.800425i 0.916422 + 0.400212i \(0.131064\pi\)
−0.916422 + 0.400212i \(0.868936\pi\)
\(620\) 10.6645 0.428296
\(621\) 15.6259 0.627046
\(622\) 0.488829i 0.0196003i
\(623\) 1.07177 0.0429397
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 1.58369i 0.0632971i
\(627\) 19.4952 0.778563
\(628\) −32.0726 −1.27984
\(629\) 30.2440i 1.20591i
\(630\) − 0.0320700i − 0.00127770i
\(631\) 14.5958i 0.581050i 0.956867 + 0.290525i \(0.0938299\pi\)
−0.956867 + 0.290525i \(0.906170\pi\)
\(632\) 3.92272i 0.156038i
\(633\) 13.1222 0.521562
\(634\) −0.0706609 −0.00280630
\(635\) − 3.23996i − 0.128574i
\(636\) 4.87409 0.193270
\(637\) 0 0
\(638\) −3.42644 −0.135654
\(639\) 5.62281i 0.222435i
\(640\) −6.61199 −0.261362
\(641\) 14.1637 0.559431 0.279716 0.960083i \(-0.409760\pi\)
0.279716 + 0.960083i \(0.409760\pi\)
\(642\) 4.59265i 0.181257i
\(643\) − 16.7716i − 0.661408i −0.943735 0.330704i \(-0.892714\pi\)
0.943735 0.330704i \(-0.107286\pi\)
\(644\) − 1.84109i − 0.0725490i
\(645\) − 8.10387i − 0.319089i
\(646\) 2.52323 0.0992751
\(647\) 2.99168 0.117615 0.0588075 0.998269i \(-0.481270\pi\)
0.0588075 + 0.998269i \(0.481270\pi\)
\(648\) − 6.50139i − 0.255399i
\(649\) 14.5432 0.570872
\(650\) 0 0
\(651\) 2.90504 0.113858
\(652\) 34.7550i 1.36111i
\(653\) −11.6643 −0.456461 −0.228230 0.973607i \(-0.573294\pi\)
−0.228230 + 0.973607i \(0.573294\pi\)
\(654\) 3.94868 0.154406
\(655\) 0.175664i 0.00686374i
\(656\) − 13.8562i − 0.540993i
\(657\) 4.25719i 0.166089i
\(658\) 0.608946i 0.0237392i
\(659\) −1.81047 −0.0705260 −0.0352630 0.999378i \(-0.511227\pi\)
−0.0352630 + 0.999378i \(0.511227\pi\)
\(660\) −16.7771 −0.653046
\(661\) − 12.3406i − 0.479992i −0.970774 0.239996i \(-0.922854\pi\)
0.970774 0.239996i \(-0.0771462\pi\)
\(662\) 3.65383 0.142010
\(663\) 0 0
\(664\) −3.70152 −0.143647
\(665\) − 0.753518i − 0.0292202i
\(666\) 0.576446 0.0223368
\(667\) −8.24348 −0.319189
\(668\) − 12.2836i − 0.475265i
\(669\) 16.3927i 0.633779i
\(670\) 2.26986i 0.0876923i
\(671\) − 75.7634i − 2.92481i
\(672\) −1.35676 −0.0523381
\(673\) −9.26625 −0.357188 −0.178594 0.983923i \(-0.557155\pi\)
−0.178594 + 0.983923i \(0.557155\pi\)
\(674\) − 5.32051i − 0.204938i
\(675\) −5.50367 −0.211836
\(676\) 0 0
\(677\) 13.8984 0.534158 0.267079 0.963675i \(-0.413941\pi\)
0.267079 + 0.963675i \(0.413941\pi\)
\(678\) 6.44863i 0.247658i
\(679\) 0.833244 0.0319770
\(680\) −4.39654 −0.168600
\(681\) 11.2715i 0.431924i
\(682\) − 6.44828i − 0.246917i
\(683\) − 37.7512i − 1.44451i −0.691626 0.722255i \(-0.743105\pi\)
0.691626 0.722255i \(-0.256895\pi\)
\(684\) 1.94486i 0.0743637i
\(685\) 17.9829 0.687090
\(686\) 1.01381 0.0387073
\(687\) − 2.12664i − 0.0811363i
\(688\) 18.8025 0.716838
\(689\) 0 0
\(690\) 0.998090 0.0379966
\(691\) 1.65291i 0.0628797i 0.999506 + 0.0314399i \(0.0100093\pi\)
−0.999506 + 0.0314399i \(0.989991\pi\)
\(692\) −31.1792 −1.18526
\(693\) 0.784180 0.0297885
\(694\) − 1.37823i − 0.0523168i
\(695\) 11.9861i 0.454660i
\(696\) 4.03351i 0.152890i
\(697\) − 18.9002i − 0.715897i
\(698\) −1.55317 −0.0587885
\(699\) −1.99618 −0.0755025
\(700\) 0.648458i 0.0245094i
\(701\) −20.4819 −0.773590 −0.386795 0.922166i \(-0.626418\pi\)
−0.386795 + 0.922166i \(0.626418\pi\)
\(702\) 0 0
\(703\) 13.5442 0.510830
\(704\) − 36.8769i − 1.38985i
\(705\) 13.3502 0.502797
\(706\) −4.78694 −0.180159
\(707\) 4.13538i 0.155527i
\(708\) − 8.45541i − 0.317774i
\(709\) 21.9417i 0.824039i 0.911175 + 0.412020i \(0.135177\pi\)
−0.911175 + 0.412020i \(0.864823\pi\)
\(710\) 2.81140i 0.105510i
\(711\) −1.98532 −0.0744552
\(712\) −2.80049 −0.104953
\(713\) − 15.5136i − 0.580987i
\(714\) −0.591503 −0.0221364
\(715\) 0 0
\(716\) −46.0950 −1.72265
\(717\) − 15.9093i − 0.594142i
\(718\) −5.26671 −0.196552
\(719\) −38.8475 −1.44877 −0.724384 0.689397i \(-0.757876\pi\)
−0.724384 + 0.689397i \(0.757876\pi\)
\(720\) − 1.63129i − 0.0607945i
\(721\) − 4.99191i − 0.185909i
\(722\) 3.04407i 0.113289i
\(723\) 36.1434i 1.34419i
\(724\) 5.12507 0.190472
\(725\) 2.90348 0.107832
\(726\) 6.27728i 0.232972i
\(727\) 30.6598 1.13711 0.568555 0.822645i \(-0.307503\pi\)
0.568555 + 0.822645i \(0.307503\pi\)
\(728\) 0 0
\(729\) 29.7112 1.10042
\(730\) 2.12859i 0.0787828i
\(731\) 25.6471 0.948593
\(732\) −44.0487 −1.62809
\(733\) − 24.3858i − 0.900709i −0.892850 0.450355i \(-0.851298\pi\)
0.892850 0.450355i \(-0.148702\pi\)
\(734\) 1.40409i 0.0518259i
\(735\) − 11.0247i − 0.406653i
\(736\) 7.24539i 0.267069i
\(737\) −55.5029 −2.04448
\(738\) −0.360235 −0.0132604
\(739\) 38.2788i 1.40811i 0.710146 + 0.704054i \(0.248629\pi\)
−0.710146 + 0.704054i \(0.751371\pi\)
\(740\) −11.6558 −0.428476
\(741\) 0 0
\(742\) −0.113910 −0.00418178
\(743\) 40.0079i 1.46775i 0.679286 + 0.733874i \(0.262290\pi\)
−0.679286 + 0.733874i \(0.737710\pi\)
\(744\) −7.59074 −0.278290
\(745\) −3.41041 −0.124948
\(746\) − 4.41134i − 0.161511i
\(747\) − 1.87336i − 0.0685427i
\(748\) − 53.0960i − 1.94138i
\(749\) 4.34057i 0.158601i
\(750\) −0.351542 −0.0128365
\(751\) −25.6020 −0.934230 −0.467115 0.884197i \(-0.654707\pi\)
−0.467115 + 0.884197i \(0.654707\pi\)
\(752\) 30.9750i 1.12954i
\(753\) −10.8307 −0.394693
\(754\) 0 0
\(755\) −7.96141 −0.289745
\(756\) − 3.56890i − 0.129800i
\(757\) −1.84848 −0.0671840 −0.0335920 0.999436i \(-0.510695\pi\)
−0.0335920 + 0.999436i \(0.510695\pi\)
\(758\) −1.19989 −0.0435820
\(759\) 24.4055i 0.885862i
\(760\) 1.96891i 0.0714198i
\(761\) 26.2124i 0.950199i 0.879932 + 0.475099i \(0.157588\pi\)
−0.879932 + 0.475099i \(0.842412\pi\)
\(762\) 1.13898i 0.0412610i
\(763\) 3.73195 0.135106
\(764\) −3.93435 −0.142340
\(765\) − 2.22512i − 0.0804494i
\(766\) −1.24513 −0.0449884
\(767\) 0 0
\(768\) −19.6459 −0.708911
\(769\) 44.3495i 1.59928i 0.600478 + 0.799641i \(0.294977\pi\)
−0.600478 + 0.799641i \(0.705023\pi\)
\(770\) 0.392090 0.0141299
\(771\) −16.4081 −0.590924
\(772\) 44.5408i 1.60306i
\(773\) 23.2638i 0.836742i 0.908276 + 0.418371i \(0.137399\pi\)
−0.908276 + 0.418371i \(0.862601\pi\)
\(774\) − 0.488829i − 0.0175706i
\(775\) 5.46410i 0.196276i
\(776\) −2.17723 −0.0781580
\(777\) −3.17508 −0.113905
\(778\) − 2.33251i − 0.0836245i
\(779\) −8.46410 −0.303258
\(780\) 0 0
\(781\) −68.7449 −2.45989
\(782\) 3.15875i 0.112957i
\(783\) −15.9798 −0.571071
\(784\) 25.5794 0.913552
\(785\) − 16.4329i − 0.586514i
\(786\) − 0.0617531i − 0.00220266i
\(787\) − 47.9133i − 1.70793i −0.520334 0.853963i \(-0.674192\pi\)
0.520334 0.853963i \(-0.325808\pi\)
\(788\) 1.25502i 0.0447081i
\(789\) −29.8548 −1.06286
\(790\) −0.992658 −0.0353172
\(791\) 6.09467i 0.216702i
\(792\) −2.04903 −0.0728090
\(793\) 0 0
\(794\) −6.15865 −0.218562
\(795\) 2.49731i 0.0885704i
\(796\) −5.98567 −0.212156
\(797\) 20.6952 0.733060 0.366530 0.930406i \(-0.380546\pi\)
0.366530 + 0.930406i \(0.380546\pi\)
\(798\) 0.264893i 0.00937712i
\(799\) 42.2507i 1.49472i
\(800\) − 2.55193i − 0.0902245i
\(801\) − 1.41735i − 0.0500795i
\(802\) 4.94609 0.174653
\(803\) −52.0487 −1.83676
\(804\) 32.2693i 1.13805i
\(805\) 0.943307 0.0332472
\(806\) 0 0
\(807\) −28.7361 −1.01156
\(808\) − 10.8056i − 0.380138i
\(809\) 15.8915 0.558714 0.279357 0.960187i \(-0.409879\pi\)
0.279357 + 0.960187i \(0.409879\pi\)
\(810\) 1.64520 0.0578063
\(811\) 23.8796i 0.838525i 0.907865 + 0.419263i \(0.137711\pi\)
−0.907865 + 0.419263i \(0.862289\pi\)
\(812\) 1.88278i 0.0660727i
\(813\) − 49.4423i − 1.73402i
\(814\) 7.04768i 0.247021i
\(815\) −17.8072 −0.623759
\(816\) −30.0877 −1.05328
\(817\) − 11.4856i − 0.401829i
\(818\) 0.941956 0.0329347
\(819\) 0 0
\(820\) 7.28398 0.254368
\(821\) − 15.9097i − 0.555251i −0.960689 0.277626i \(-0.910453\pi\)
0.960689 0.277626i \(-0.0895475\pi\)
\(822\) −6.32174 −0.220496
\(823\) 14.8115 0.516295 0.258147 0.966106i \(-0.416888\pi\)
0.258147 + 0.966106i \(0.416888\pi\)
\(824\) 13.0436i 0.454397i
\(825\) − 8.59596i − 0.299273i
\(826\) 0.197608i 0.00687566i
\(827\) 33.9498i 1.18055i 0.807202 + 0.590275i \(0.200981\pi\)
−0.807202 + 0.590275i \(0.799019\pi\)
\(828\) −2.43472 −0.0846122
\(829\) −23.3146 −0.809749 −0.404875 0.914372i \(-0.632685\pi\)
−0.404875 + 0.914372i \(0.632685\pi\)
\(830\) − 0.936681i − 0.0325127i
\(831\) 42.4124 1.47127
\(832\) 0 0
\(833\) 34.8910 1.20890
\(834\) − 4.21363i − 0.145906i
\(835\) 6.29366 0.217801
\(836\) −23.7780 −0.822380
\(837\) − 30.0726i − 1.03946i
\(838\) 3.89100i 0.134412i
\(839\) − 14.7930i − 0.510710i −0.966847 0.255355i \(-0.917808\pi\)
0.966847 0.255355i \(-0.0821924\pi\)
\(840\) − 0.461557i − 0.0159252i
\(841\) −20.5698 −0.709305
\(842\) 2.82929 0.0975038
\(843\) − 7.96523i − 0.274337i
\(844\) −16.0050 −0.550915
\(845\) 0 0
\(846\) 0.805291 0.0276865
\(847\) 5.93273i 0.203851i
\(848\) −5.79422 −0.198974
\(849\) −20.1406 −0.691223
\(850\) − 1.11256i − 0.0381605i
\(851\) 16.9556i 0.581231i
\(852\) 39.9681i 1.36929i
\(853\) − 16.3452i − 0.559650i −0.960051 0.279825i \(-0.909724\pi\)
0.960051 0.279825i \(-0.0902765\pi\)
\(854\) 1.02944 0.0352268
\(855\) −0.996477 −0.0340788
\(856\) − 11.3417i − 0.387651i
\(857\) 34.1418 1.16626 0.583132 0.812378i \(-0.301827\pi\)
0.583132 + 0.812378i \(0.301827\pi\)
\(858\) 0 0
\(859\) −45.1996 −1.54219 −0.771096 0.636719i \(-0.780291\pi\)
−0.771096 + 0.636719i \(0.780291\pi\)
\(860\) 9.88418i 0.337048i
\(861\) 1.98418 0.0676207
\(862\) 2.08669 0.0710728
\(863\) − 4.75058i − 0.161712i −0.996726 0.0808559i \(-0.974235\pi\)
0.996726 0.0808559i \(-0.0257653\pi\)
\(864\) 14.0450i 0.477821i
\(865\) − 15.9751i − 0.543170i
\(866\) 0.306745i 0.0104236i
\(867\) −13.8371 −0.469931
\(868\) −3.54324 −0.120265
\(869\) − 24.2726i − 0.823392i
\(870\) −1.02069 −0.0346048
\(871\) 0 0
\(872\) −9.75140 −0.330224
\(873\) − 1.10191i − 0.0372940i
\(874\) 1.41459 0.0478492
\(875\) −0.332247 −0.0112320
\(876\) 30.2610i 1.02242i
\(877\) − 2.25506i − 0.0761481i −0.999275 0.0380741i \(-0.987878\pi\)
0.999275 0.0380741i \(-0.0121223\pi\)
\(878\) − 0.914293i − 0.0308559i
\(879\) − 27.0715i − 0.913098i
\(880\) 19.9442 0.672320
\(881\) 2.98304 0.100501 0.0502507 0.998737i \(-0.483998\pi\)
0.0502507 + 0.998737i \(0.483998\pi\)
\(882\) − 0.665017i − 0.0223923i
\(883\) −28.2874 −0.951947 −0.475973 0.879460i \(-0.657904\pi\)
−0.475973 + 0.879460i \(0.657904\pi\)
\(884\) 0 0
\(885\) 4.33225 0.145627
\(886\) 2.09705i 0.0704517i
\(887\) −27.9816 −0.939531 −0.469766 0.882791i \(-0.655662\pi\)
−0.469766 + 0.882791i \(0.655662\pi\)
\(888\) 8.29633 0.278407
\(889\) 1.07647i 0.0361035i
\(890\) − 0.708674i − 0.0237548i
\(891\) 40.2286i 1.34771i
\(892\) − 19.9940i − 0.669448i
\(893\) 18.9212 0.633172
\(894\) 1.19890 0.0400973
\(895\) − 23.6174i − 0.789443i
\(896\) 2.19681 0.0733903
\(897\) 0 0
\(898\) −4.77095 −0.159209
\(899\) 15.8649i 0.529124i
\(900\) 0.857542 0.0285847
\(901\) −7.90348 −0.263303
\(902\) − 4.40426i − 0.146646i
\(903\) 2.69248i 0.0896002i
\(904\) − 15.9251i − 0.529661i
\(905\) 2.62590i 0.0872879i
\(906\) 2.79877 0.0929829
\(907\) −16.5520 −0.549600 −0.274800 0.961501i \(-0.588612\pi\)
−0.274800 + 0.961501i \(0.588612\pi\)
\(908\) − 13.7477i − 0.456232i
\(909\) 5.46876 0.181387
\(910\) 0 0
\(911\) 7.04863 0.233532 0.116766 0.993159i \(-0.462747\pi\)
0.116766 + 0.993159i \(0.462747\pi\)
\(912\) 13.4742i 0.446175i
\(913\) 22.9039 0.758007
\(914\) 1.03749 0.0343172
\(915\) − 22.5689i − 0.746106i
\(916\) 2.59383i 0.0857026i
\(917\) − 0.0583636i − 0.00192734i
\(918\) 6.12316i 0.202094i
\(919\) 16.5438 0.545728 0.272864 0.962053i \(-0.412029\pi\)
0.272864 + 0.962053i \(0.412029\pi\)
\(920\) −2.46482 −0.0812626
\(921\) 6.88766i 0.226956i
\(922\) −0.391374 −0.0128892
\(923\) 0 0
\(924\) 5.57412 0.183375
\(925\) − 5.97201i − 0.196359i
\(926\) −1.49431 −0.0491060
\(927\) −6.60147 −0.216821
\(928\) − 7.40948i − 0.243228i
\(929\) − 33.8367i − 1.11015i −0.831801 0.555074i \(-0.812690\pi\)
0.831801 0.555074i \(-0.187310\pi\)
\(930\) − 1.92086i − 0.0629875i
\(931\) − 15.6253i − 0.512098i
\(932\) 2.43472 0.0797517
\(933\) −3.56063 −0.116570
\(934\) − 4.00652i − 0.131097i
\(935\) 27.2045 0.889681
\(936\) 0 0
\(937\) −30.4606 −0.995104 −0.497552 0.867434i \(-0.665768\pi\)
−0.497552 + 0.867434i \(0.665768\pi\)
\(938\) − 0.754153i − 0.0246240i
\(939\) −11.5356 −0.376450
\(940\) −16.2831 −0.531095
\(941\) 38.2101i 1.24561i 0.782375 + 0.622807i \(0.214008\pi\)
−0.782375 + 0.622807i \(0.785992\pi\)
\(942\) 5.77684i 0.188220i
\(943\) − 10.5960i − 0.345052i
\(944\) 10.0516i 0.327153i
\(945\) 1.82858 0.0594836
\(946\) 5.97647 0.194312
\(947\) 52.4482i 1.70434i 0.523266 + 0.852169i \(0.324713\pi\)
−0.523266 + 0.852169i \(0.675287\pi\)
\(948\) −14.1120 −0.458338
\(949\) 0 0
\(950\) −0.498239 −0.0161650
\(951\) − 0.514693i − 0.0166901i
\(952\) 1.46074 0.0473427
\(953\) −39.7500 −1.28763 −0.643814 0.765182i \(-0.722649\pi\)
−0.643814 + 0.765182i \(0.722649\pi\)
\(954\) 0.150639i 0.00487711i
\(955\) − 2.01582i − 0.0652304i
\(956\) 19.4043i 0.627580i
\(957\) − 24.9581i − 0.806782i
\(958\) 7.72874 0.249704
\(959\) −5.97475 −0.192935
\(960\) − 10.9852i − 0.354544i
\(961\) 1.14359 0.0368901
\(962\) 0 0
\(963\) 5.74011 0.184972
\(964\) − 44.0837i − 1.41984i
\(965\) −22.8211 −0.734636
\(966\) −0.331612 −0.0106694
\(967\) − 25.7857i − 0.829214i −0.910001 0.414607i \(-0.863919\pi\)
0.910001 0.414607i \(-0.136081\pi\)
\(968\) − 15.5019i − 0.498251i
\(969\) 18.3792i 0.590424i
\(970\) − 0.550955i − 0.0176901i
\(971\) −55.5252 −1.78189 −0.890945 0.454111i \(-0.849957\pi\)
−0.890945 + 0.454111i \(0.849957\pi\)
\(972\) −8.83634 −0.283426
\(973\) − 3.98235i − 0.127668i
\(974\) −2.26365 −0.0725321
\(975\) 0 0
\(976\) 52.3642 1.67614
\(977\) 40.5161i 1.29622i 0.761545 + 0.648112i \(0.224441\pi\)
−0.761545 + 0.648112i \(0.775559\pi\)
\(978\) 6.25998 0.200172
\(979\) 17.3286 0.553824
\(980\) 13.4467i 0.429540i
\(981\) − 4.93525i − 0.157570i
\(982\) − 2.05011i − 0.0654218i
\(983\) 34.8059i 1.11014i 0.831805 + 0.555068i \(0.187308\pi\)
−0.831805 + 0.555068i \(0.812692\pi\)
\(984\) −5.18457 −0.165278
\(985\) −0.643026 −0.0204885
\(986\) − 3.23029i − 0.102873i
\(987\) −4.43555 −0.141185
\(988\) 0 0
\(989\) 14.3784 0.457208
\(990\) − 0.518513i − 0.0164794i
\(991\) 43.8855 1.39407 0.697034 0.717038i \(-0.254503\pi\)
0.697034 + 0.717038i \(0.254503\pi\)
\(992\) 13.9440 0.442723
\(993\) 26.6145i 0.844584i
\(994\) − 0.934079i − 0.0296272i
\(995\) − 3.06684i − 0.0972254i
\(996\) − 13.3163i − 0.421942i
\(997\) −5.49137 −0.173914 −0.0869568 0.996212i \(-0.527714\pi\)
−0.0869568 + 0.996212i \(0.527714\pi\)
\(998\) 5.25976 0.166495
\(999\) 32.8680i 1.03990i
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 845.2.c.g.506.5 8
13.2 odd 12 845.2.e.n.191.2 8
13.3 even 3 65.2.m.a.56.2 yes 8
13.4 even 6 65.2.m.a.36.2 8
13.5 odd 4 845.2.a.l.1.3 4
13.6 odd 12 845.2.e.n.146.2 8
13.7 odd 12 845.2.e.m.146.3 8
13.8 odd 4 845.2.a.m.1.2 4
13.9 even 3 845.2.m.g.361.3 8
13.10 even 6 845.2.m.g.316.3 8
13.11 odd 12 845.2.e.m.191.3 8
13.12 even 2 inner 845.2.c.g.506.4 8
39.5 even 4 7605.2.a.cj.1.2 4
39.8 even 4 7605.2.a.cf.1.3 4
39.17 odd 6 585.2.bu.c.361.3 8
39.29 odd 6 585.2.bu.c.316.3 8
52.3 odd 6 1040.2.da.b.641.2 8
52.43 odd 6 1040.2.da.b.881.2 8
65.3 odd 12 325.2.m.c.199.2 8
65.4 even 6 325.2.n.d.101.3 8
65.17 odd 12 325.2.m.c.49.2 8
65.29 even 6 325.2.n.d.251.3 8
65.34 odd 4 4225.2.a.bi.1.3 4
65.42 odd 12 325.2.m.b.199.3 8
65.43 odd 12 325.2.m.b.49.3 8
65.44 odd 4 4225.2.a.bl.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.2 8 13.4 even 6
65.2.m.a.56.2 yes 8 13.3 even 3
325.2.m.b.49.3 8 65.43 odd 12
325.2.m.b.199.3 8 65.42 odd 12
325.2.m.c.49.2 8 65.17 odd 12
325.2.m.c.199.2 8 65.3 odd 12
325.2.n.d.101.3 8 65.4 even 6
325.2.n.d.251.3 8 65.29 even 6
585.2.bu.c.316.3 8 39.29 odd 6
585.2.bu.c.361.3 8 39.17 odd 6
845.2.a.l.1.3 4 13.5 odd 4
845.2.a.m.1.2 4 13.8 odd 4
845.2.c.g.506.4 8 13.12 even 2 inner
845.2.c.g.506.5 8 1.1 even 1 trivial
845.2.e.m.146.3 8 13.7 odd 12
845.2.e.m.191.3 8 13.11 odd 12
845.2.e.n.146.2 8 13.6 odd 12
845.2.e.n.191.2 8 13.2 odd 12
845.2.m.g.316.3 8 13.10 even 6
845.2.m.g.361.3 8 13.9 even 3
1040.2.da.b.641.2 8 52.3 odd 6
1040.2.da.b.881.2 8 52.43 odd 6
4225.2.a.bi.1.3 4 65.34 odd 4
4225.2.a.bl.1.2 4 65.44 odd 4
7605.2.a.cf.1.3 4 39.8 even 4
7605.2.a.cj.1.2 4 39.5 even 4