Properties

Label 845.2.a.l.1.3
Level $845$
Weight $2$
Character 845.1
Self dual yes
Analytic conductor $6.747$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.74735897080\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.219687\) of defining polynomial
Character \(\chi\) \(=\) 845.1

$q$-expansion

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.219687 0.155342 0.0776710 0.996979i \(-0.475252\pi\)
0.0776710 + 0.996979i \(0.475252\pi\)
\(3\) −1.60020 −0.923873 −0.461937 0.886913i \(-0.652845\pi\)
−0.461937 + 0.886913i \(0.652845\pi\)
\(4\) −1.95174 −0.975869
\(5\) 1.00000 0.447214
\(6\) −0.351542 −0.143516
\(7\) 0.332247 0.125577 0.0627887 0.998027i \(-0.480001\pi\)
0.0627887 + 0.998027i \(0.480001\pi\)
\(8\) −0.868145 −0.306936
\(9\) −0.439374 −0.146458
\(10\) 0.219687 0.0694711
\(11\) 5.37182 1.61966 0.809832 0.586662i \(-0.199558\pi\)
0.809832 + 0.586662i \(0.199558\pi\)
\(12\) 3.12316 0.901579
\(13\) 0 0
\(14\) 0.0729902 0.0195074
\(15\) −1.60020 −0.413169
\(16\) 3.71276 0.928189
\(17\) −5.06430 −1.22827 −0.614136 0.789200i \(-0.710495\pi\)
−0.614136 + 0.789200i \(0.710495\pi\)
\(18\) −0.0965246 −0.0227511
\(19\) −2.26795 −0.520303 −0.260152 0.965568i \(-0.583773\pi\)
−0.260152 + 0.965568i \(0.583773\pi\)
\(20\) −1.95174 −0.436422
\(21\) −0.531659 −0.116018
\(22\) 1.18012 0.251602
\(23\) −2.83918 −0.592010 −0.296005 0.955186i \(-0.595654\pi\)
−0.296005 + 0.955186i \(0.595654\pi\)
\(24\) 1.38920 0.283570
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.50367 1.05918
\(28\) −0.648458 −0.122547
\(29\) −2.90348 −0.539162 −0.269581 0.962978i \(-0.586885\pi\)
−0.269581 + 0.962978i \(0.586885\pi\)
\(30\) −0.351542 −0.0641825
\(31\) −5.46410 −0.981382 −0.490691 0.871334i \(-0.663256\pi\)
−0.490691 + 0.871334i \(0.663256\pi\)
\(32\) 2.55193 0.451122
\(33\) −8.59596 −1.49636
\(34\) −1.11256 −0.190802
\(35\) 0.332247 0.0561599
\(36\) 0.857542 0.142924
\(37\) −5.97201 −0.981793 −0.490896 0.871218i \(-0.663331\pi\)
−0.490896 + 0.871218i \(0.663331\pi\)
\(38\) −0.498239 −0.0808250
\(39\) 0 0
\(40\) −0.868145 −0.137266
\(41\) −3.73205 −0.582848 −0.291424 0.956594i \(-0.594129\pi\)
−0.291424 + 0.956594i \(0.594129\pi\)
\(42\) −0.116799 −0.0180224
\(43\) −5.06430 −0.772298 −0.386149 0.922436i \(-0.626195\pi\)
−0.386149 + 0.922436i \(0.626195\pi\)
\(44\) −10.4844 −1.58058
\(45\) −0.439374 −0.0654980
\(46\) −0.623730 −0.0919640
\(47\) −8.34285 −1.21693 −0.608465 0.793581i \(-0.708214\pi\)
−0.608465 + 0.793581i \(0.708214\pi\)
\(48\) −5.94114 −0.857529
\(49\) −6.88961 −0.984230
\(50\) 0.219687 0.0310684
\(51\) 8.10387 1.13477
\(52\) 0 0
\(53\) −1.56063 −0.214369 −0.107184 0.994239i \(-0.534183\pi\)
−0.107184 + 0.994239i \(0.534183\pi\)
\(54\) 1.20908 0.164536
\(55\) 5.37182 0.724336
\(56\) −0.288438 −0.0385442
\(57\) 3.62916 0.480694
\(58\) −0.637855 −0.0837545
\(59\) −2.70732 −0.352463 −0.176232 0.984349i \(-0.556391\pi\)
−0.176232 + 0.984349i \(0.556391\pi\)
\(60\) 3.12316 0.403199
\(61\) 14.1039 1.80582 0.902908 0.429835i \(-0.141428\pi\)
0.902908 + 0.429835i \(0.141428\pi\)
\(62\) −1.20039 −0.152450
\(63\) −0.145980 −0.0183918
\(64\) −6.86488 −0.858111
\(65\) 0 0
\(66\) −1.88842 −0.232448
\(67\) −10.3322 −1.26228 −0.631142 0.775667i \(-0.717414\pi\)
−0.631142 + 0.775667i \(0.717414\pi\)
\(68\) 9.88418 1.19863
\(69\) 4.54324 0.546942
\(70\) 0.0729902 0.00872400
\(71\) −12.7973 −1.51876 −0.759382 0.650645i \(-0.774498\pi\)
−0.759382 + 0.650645i \(0.774498\pi\)
\(72\) 0.381440 0.0449531
\(73\) 9.68922 1.13404 0.567019 0.823705i \(-0.308097\pi\)
0.567019 + 0.823705i \(0.308097\pi\)
\(74\) −1.31197 −0.152514
\(75\) −1.60020 −0.184775
\(76\) 4.42644 0.507748
\(77\) 1.78477 0.203393
\(78\) 0 0
\(79\) 4.51851 0.508372 0.254186 0.967155i \(-0.418192\pi\)
0.254186 + 0.967155i \(0.418192\pi\)
\(80\) 3.71276 0.415099
\(81\) −7.48883 −0.832092
\(82\) −0.819883 −0.0905409
\(83\) 4.26371 0.468003 0.234001 0.972236i \(-0.424818\pi\)
0.234001 + 0.972236i \(0.424818\pi\)
\(84\) 1.03766 0.113218
\(85\) −5.06430 −0.549300
\(86\) −1.11256 −0.119970
\(87\) 4.64613 0.498117
\(88\) −4.66351 −0.497132
\(89\) −3.22584 −0.341938 −0.170969 0.985276i \(-0.554690\pi\)
−0.170969 + 0.985276i \(0.554690\pi\)
\(90\) −0.0965246 −0.0101746
\(91\) 0 0
\(92\) 5.54133 0.577724
\(93\) 8.74363 0.906672
\(94\) −1.83281 −0.189040
\(95\) −2.26795 −0.232687
\(96\) −4.08359 −0.416780
\(97\) 2.50791 0.254640 0.127320 0.991862i \(-0.459363\pi\)
0.127320 + 0.991862i \(0.459363\pi\)
\(98\) −1.51356 −0.152892
\(99\) −2.36023 −0.237213
\(100\) −1.95174 −0.195174
\(101\) 12.4467 1.23849 0.619247 0.785196i \(-0.287438\pi\)
0.619247 + 0.785196i \(0.287438\pi\)
\(102\) 1.78031 0.176277
\(103\) −15.0247 −1.48043 −0.740215 0.672370i \(-0.765276\pi\)
−0.740215 + 0.672370i \(0.765276\pi\)
\(104\) 0 0
\(105\) −0.531659 −0.0518846
\(106\) −0.342849 −0.0333004
\(107\) −13.0643 −1.26297 −0.631487 0.775387i \(-0.717555\pi\)
−0.631487 + 0.775387i \(0.717555\pi\)
\(108\) −10.7417 −1.03362
\(109\) 11.2325 1.07587 0.537937 0.842985i \(-0.319204\pi\)
0.537937 + 0.842985i \(0.319204\pi\)
\(110\) 1.18012 0.112520
\(111\) 9.55639 0.907052
\(112\) 1.23355 0.116560
\(113\) −18.3438 −1.72564 −0.862821 0.505509i \(-0.831305\pi\)
−0.862821 + 0.505509i \(0.831305\pi\)
\(114\) 0.797279 0.0746721
\(115\) −2.83918 −0.264755
\(116\) 5.66682 0.526151
\(117\) 0 0
\(118\) −0.594763 −0.0547524
\(119\) −1.68260 −0.154243
\(120\) 1.38920 0.126816
\(121\) 17.8564 1.62331
\(122\) 3.09843 0.280519
\(123\) 5.97201 0.538478
\(124\) 10.6645 0.957700
\(125\) 1.00000 0.0894427
\(126\) −0.0320700 −0.00285702
\(127\) 3.23996 0.287500 0.143750 0.989614i \(-0.454084\pi\)
0.143750 + 0.989614i \(0.454084\pi\)
\(128\) −6.61199 −0.584423
\(129\) 8.10387 0.713506
\(130\) 0 0
\(131\) 0.175664 0.0153478 0.00767390 0.999971i \(-0.497557\pi\)
0.00767390 + 0.999971i \(0.497557\pi\)
\(132\) 16.7771 1.46025
\(133\) −0.753518 −0.0653383
\(134\) −2.26986 −0.196086
\(135\) 5.50367 0.473681
\(136\) 4.39654 0.377001
\(137\) 17.9829 1.53638 0.768190 0.640221i \(-0.221157\pi\)
0.768190 + 0.640221i \(0.221157\pi\)
\(138\) 0.998090 0.0849631
\(139\) 11.9861 1.01665 0.508325 0.861165i \(-0.330265\pi\)
0.508325 + 0.861165i \(0.330265\pi\)
\(140\) −0.648458 −0.0548047
\(141\) 13.3502 1.12429
\(142\) −2.81140 −0.235928
\(143\) 0 0
\(144\) −1.63129 −0.135941
\(145\) −2.90348 −0.241121
\(146\) 2.12859 0.176164
\(147\) 11.0247 0.909304
\(148\) 11.6558 0.958101
\(149\) 3.41041 0.279391 0.139696 0.990194i \(-0.455388\pi\)
0.139696 + 0.990194i \(0.455388\pi\)
\(150\) −0.351542 −0.0287033
\(151\) −7.96141 −0.647890 −0.323945 0.946076i \(-0.605009\pi\)
−0.323945 + 0.946076i \(0.605009\pi\)
\(152\) 1.96891 0.159700
\(153\) 2.22512 0.179890
\(154\) 0.392090 0.0315955
\(155\) −5.46410 −0.438887
\(156\) 0 0
\(157\) −16.4329 −1.31148 −0.655742 0.754985i \(-0.727644\pi\)
−0.655742 + 0.754985i \(0.727644\pi\)
\(158\) 0.992658 0.0789716
\(159\) 2.49731 0.198049
\(160\) 2.55193 0.201748
\(161\) −0.943307 −0.0743430
\(162\) −1.64520 −0.129259
\(163\) −17.8072 −1.39477 −0.697384 0.716697i \(-0.745653\pi\)
−0.697384 + 0.716697i \(0.745653\pi\)
\(164\) 7.28398 0.568784
\(165\) −8.59596 −0.669194
\(166\) 0.936681 0.0727006
\(167\) 6.29366 0.487018 0.243509 0.969899i \(-0.421702\pi\)
0.243509 + 0.969899i \(0.421702\pi\)
\(168\) 0.461557 0.0356099
\(169\) 0 0
\(170\) −1.11256 −0.0853294
\(171\) 0.996477 0.0762025
\(172\) 9.88418 0.753662
\(173\) 15.9751 1.21457 0.607283 0.794486i \(-0.292260\pi\)
0.607283 + 0.794486i \(0.292260\pi\)
\(174\) 1.02069 0.0773786
\(175\) 0.332247 0.0251155
\(176\) 19.9442 1.50335
\(177\) 4.33225 0.325632
\(178\) −0.708674 −0.0531173
\(179\) 23.6174 1.76525 0.882625 0.470079i \(-0.155774\pi\)
0.882625 + 0.470079i \(0.155774\pi\)
\(180\) 0.857542 0.0639174
\(181\) −2.62590 −0.195182 −0.0975909 0.995227i \(-0.531114\pi\)
−0.0975909 + 0.995227i \(0.531114\pi\)
\(182\) 0 0
\(183\) −22.5689 −1.66834
\(184\) 2.46482 0.181709
\(185\) −5.97201 −0.439071
\(186\) 1.92086 0.140844
\(187\) −27.2045 −1.98939
\(188\) 16.2831 1.18756
\(189\) 1.82858 0.133009
\(190\) −0.498239 −0.0361460
\(191\) −2.01582 −0.145860 −0.0729298 0.997337i \(-0.523235\pi\)
−0.0729298 + 0.997337i \(0.523235\pi\)
\(192\) 10.9852 0.792786
\(193\) −22.8211 −1.64270 −0.821348 0.570427i \(-0.806778\pi\)
−0.821348 + 0.570427i \(0.806778\pi\)
\(194\) 0.550955 0.0395563
\(195\) 0 0
\(196\) 13.4467 0.960480
\(197\) 0.643026 0.0458137 0.0229068 0.999738i \(-0.492708\pi\)
0.0229068 + 0.999738i \(0.492708\pi\)
\(198\) −0.518513 −0.0368491
\(199\) 3.06684 0.217403 0.108701 0.994074i \(-0.465331\pi\)
0.108701 + 0.994074i \(0.465331\pi\)
\(200\) −0.868145 −0.0613871
\(201\) 16.5336 1.16619
\(202\) 2.73438 0.192390
\(203\) −0.964670 −0.0677065
\(204\) −15.8166 −1.10739
\(205\) −3.73205 −0.260658
\(206\) −3.30074 −0.229973
\(207\) 1.24746 0.0867045
\(208\) 0 0
\(209\) −12.1830 −0.842716
\(210\) −0.116799 −0.00805987
\(211\) −8.20039 −0.564538 −0.282269 0.959335i \(-0.591087\pi\)
−0.282269 + 0.959335i \(0.591087\pi\)
\(212\) 3.04593 0.209196
\(213\) 20.4782 1.40314
\(214\) −2.87005 −0.196193
\(215\) −5.06430 −0.345382
\(216\) −4.77798 −0.325101
\(217\) −1.81543 −0.123239
\(218\) 2.46762 0.167129
\(219\) −15.5046 −1.04771
\(220\) −10.4844 −0.706856
\(221\) 0 0
\(222\) 2.09941 0.140903
\(223\) −10.2442 −0.686002 −0.343001 0.939335i \(-0.611443\pi\)
−0.343001 + 0.939335i \(0.611443\pi\)
\(224\) 0.847871 0.0566508
\(225\) −0.439374 −0.0292916
\(226\) −4.02990 −0.268065
\(227\) −7.04381 −0.467514 −0.233757 0.972295i \(-0.575102\pi\)
−0.233757 + 0.972295i \(0.575102\pi\)
\(228\) −7.08317 −0.469095
\(229\) −1.32899 −0.0878219 −0.0439109 0.999035i \(-0.513982\pi\)
−0.0439109 + 0.999035i \(0.513982\pi\)
\(230\) −0.623730 −0.0411275
\(231\) −2.85598 −0.187909
\(232\) 2.52064 0.165488
\(233\) −1.24746 −0.0817238 −0.0408619 0.999165i \(-0.513010\pi\)
−0.0408619 + 0.999165i \(0.513010\pi\)
\(234\) 0 0
\(235\) −8.34285 −0.544227
\(236\) 5.28398 0.343958
\(237\) −7.23050 −0.469672
\(238\) −0.369644 −0.0239605
\(239\) 9.94207 0.643099 0.321549 0.946893i \(-0.395796\pi\)
0.321549 + 0.946893i \(0.395796\pi\)
\(240\) −5.94114 −0.383499
\(241\) 22.5869 1.45495 0.727475 0.686134i \(-0.240694\pi\)
0.727475 + 0.686134i \(0.240694\pi\)
\(242\) 3.92282 0.252168
\(243\) −4.52742 −0.290434
\(244\) −27.5270 −1.76224
\(245\) −6.88961 −0.440161
\(246\) 1.31197 0.0836483
\(247\) 0 0
\(248\) 4.74363 0.301221
\(249\) −6.82277 −0.432376
\(250\) 0.219687 0.0138942
\(251\) −6.76836 −0.427215 −0.213608 0.976920i \(-0.568521\pi\)
−0.213608 + 0.976920i \(0.568521\pi\)
\(252\) 0.284915 0.0179480
\(253\) −15.2515 −0.958856
\(254\) 0.711777 0.0446609
\(255\) 8.10387 0.507484
\(256\) 12.2772 0.767325
\(257\) −10.2538 −0.639616 −0.319808 0.947482i \(-0.603618\pi\)
−0.319808 + 0.947482i \(0.603618\pi\)
\(258\) 1.78031 0.110837
\(259\) −1.98418 −0.123291
\(260\) 0 0
\(261\) 1.27571 0.0789645
\(262\) 0.0385910 0.00238416
\(263\) 18.6570 1.15044 0.575220 0.817999i \(-0.304917\pi\)
0.575220 + 0.817999i \(0.304917\pi\)
\(264\) 7.46254 0.459287
\(265\) −1.56063 −0.0958685
\(266\) −0.165538 −0.0101498
\(267\) 5.16197 0.315907
\(268\) 20.1658 1.23182
\(269\) 17.9579 1.09491 0.547456 0.836835i \(-0.315596\pi\)
0.547456 + 0.836835i \(0.315596\pi\)
\(270\) 1.20908 0.0735825
\(271\) −30.8977 −1.87690 −0.938450 0.345415i \(-0.887738\pi\)
−0.938450 + 0.345415i \(0.887738\pi\)
\(272\) −18.8025 −1.14007
\(273\) 0 0
\(274\) 3.95060 0.238665
\(275\) 5.37182 0.323933
\(276\) −8.86721 −0.533744
\(277\) 26.5045 1.59250 0.796250 0.604967i \(-0.206814\pi\)
0.796250 + 0.604967i \(0.206814\pi\)
\(278\) 2.63320 0.157929
\(279\) 2.40078 0.143731
\(280\) −0.288438 −0.0172375
\(281\) −4.97766 −0.296942 −0.148471 0.988917i \(-0.547435\pi\)
−0.148471 + 0.988917i \(0.547435\pi\)
\(282\) 2.93286 0.174649
\(283\) −12.5863 −0.748180 −0.374090 0.927392i \(-0.622045\pi\)
−0.374090 + 0.927392i \(0.622045\pi\)
\(284\) 24.9770 1.48211
\(285\) 3.62916 0.214973
\(286\) 0 0
\(287\) −1.23996 −0.0731926
\(288\) −1.12125 −0.0660704
\(289\) 8.64711 0.508653
\(290\) −0.637855 −0.0374562
\(291\) −4.01315 −0.235255
\(292\) −18.9108 −1.10667
\(293\) −16.9176 −0.988337 −0.494168 0.869366i \(-0.664527\pi\)
−0.494168 + 0.869366i \(0.664527\pi\)
\(294\) 2.42199 0.141253
\(295\) −2.70732 −0.157626
\(296\) 5.18457 0.301347
\(297\) 29.5647 1.71552
\(298\) 0.749222 0.0434012
\(299\) 0 0
\(300\) 3.12316 0.180316
\(301\) −1.68260 −0.0969832
\(302\) −1.74902 −0.100645
\(303\) −19.9172 −1.14421
\(304\) −8.42034 −0.482940
\(305\) 14.1039 0.807585
\(306\) 0.488829 0.0279445
\(307\) 4.30426 0.245657 0.122828 0.992428i \(-0.460803\pi\)
0.122828 + 0.992428i \(0.460803\pi\)
\(308\) −3.48340 −0.198485
\(309\) 24.0425 1.36773
\(310\) −1.20039 −0.0681776
\(311\) −2.22512 −0.126175 −0.0630875 0.998008i \(-0.520095\pi\)
−0.0630875 + 0.998008i \(0.520095\pi\)
\(312\) 0 0
\(313\) 7.20887 0.407469 0.203735 0.979026i \(-0.434692\pi\)
0.203735 + 0.979026i \(0.434692\pi\)
\(314\) −3.61008 −0.203729
\(315\) −0.145980 −0.00822506
\(316\) −8.81895 −0.496105
\(317\) 0.321644 0.0180653 0.00903266 0.999959i \(-0.497125\pi\)
0.00903266 + 0.999959i \(0.497125\pi\)
\(318\) 0.548626 0.0307654
\(319\) −15.5969 −0.873261
\(320\) −6.86488 −0.383759
\(321\) 20.9054 1.16683
\(322\) −0.207232 −0.0115486
\(323\) 11.4856 0.639074
\(324\) 14.6162 0.812013
\(325\) 0 0
\(326\) −3.91201 −0.216666
\(327\) −17.9741 −0.993972
\(328\) 3.23996 0.178897
\(329\) −2.77188 −0.152819
\(330\) −1.88842 −0.103954
\(331\) −16.6320 −0.914178 −0.457089 0.889421i \(-0.651108\pi\)
−0.457089 + 0.889421i \(0.651108\pi\)
\(332\) −8.32164 −0.456710
\(333\) 2.62395 0.143791
\(334\) 1.38263 0.0756543
\(335\) −10.3322 −0.564511
\(336\) −1.97392 −0.107686
\(337\) 24.2186 1.31927 0.659636 0.751586i \(-0.270711\pi\)
0.659636 + 0.751586i \(0.270711\pi\)
\(338\) 0 0
\(339\) 29.3537 1.59427
\(340\) 9.88418 0.536045
\(341\) −29.3521 −1.58951
\(342\) 0.218913 0.0118375
\(343\) −4.61478 −0.249174
\(344\) 4.39654 0.237046
\(345\) 4.54324 0.244600
\(346\) 3.50952 0.188673
\(347\) −6.27360 −0.336784 −0.168392 0.985720i \(-0.553857\pi\)
−0.168392 + 0.985720i \(0.553857\pi\)
\(348\) −9.06802 −0.486097
\(349\) −7.06994 −0.378445 −0.189223 0.981934i \(-0.560597\pi\)
−0.189223 + 0.981934i \(0.560597\pi\)
\(350\) 0.0729902 0.00390149
\(351\) 0 0
\(352\) 13.7085 0.730666
\(353\) 21.7898 1.15976 0.579878 0.814704i \(-0.303100\pi\)
0.579878 + 0.814704i \(0.303100\pi\)
\(354\) 0.951738 0.0505843
\(355\) −12.7973 −0.679212
\(356\) 6.29598 0.333687
\(357\) 2.69248 0.142501
\(358\) 5.18844 0.274217
\(359\) −23.9737 −1.26528 −0.632642 0.774444i \(-0.718029\pi\)
−0.632642 + 0.774444i \(0.718029\pi\)
\(360\) 0.381440 0.0201037
\(361\) −13.8564 −0.729285
\(362\) −0.576876 −0.0303199
\(363\) −28.5737 −1.49973
\(364\) 0 0
\(365\) 9.68922 0.507157
\(366\) −4.95810 −0.259164
\(367\) 6.39133 0.333625 0.166812 0.985989i \(-0.446653\pi\)
0.166812 + 0.985989i \(0.446653\pi\)
\(368\) −10.5412 −0.549497
\(369\) 1.63977 0.0853628
\(370\) −1.31197 −0.0682062
\(371\) −0.518513 −0.0269198
\(372\) −17.0653 −0.884793
\(373\) −20.0801 −1.03971 −0.519855 0.854255i \(-0.674014\pi\)
−0.519855 + 0.854255i \(0.674014\pi\)
\(374\) −5.97647 −0.309036
\(375\) −1.60020 −0.0826338
\(376\) 7.24280 0.373519
\(377\) 0 0
\(378\) 0.401714 0.0206619
\(379\) 5.46182 0.280555 0.140277 0.990112i \(-0.455201\pi\)
0.140277 + 0.990112i \(0.455201\pi\)
\(380\) 4.42644 0.227072
\(381\) −5.18457 −0.265614
\(382\) −0.442849 −0.0226581
\(383\) 5.66775 0.289609 0.144804 0.989460i \(-0.453745\pi\)
0.144804 + 0.989460i \(0.453745\pi\)
\(384\) 10.5805 0.539933
\(385\) 1.78477 0.0909602
\(386\) −5.01349 −0.255180
\(387\) 2.22512 0.113109
\(388\) −4.89478 −0.248495
\(389\) 10.6174 0.538325 0.269162 0.963095i \(-0.413253\pi\)
0.269162 + 0.963095i \(0.413253\pi\)
\(390\) 0 0
\(391\) 14.3784 0.727149
\(392\) 5.98118 0.302095
\(393\) −0.281096 −0.0141794
\(394\) 0.141264 0.00711679
\(395\) 4.51851 0.227351
\(396\) 4.60656 0.231488
\(397\) −28.0338 −1.40697 −0.703487 0.710708i \(-0.748375\pi\)
−0.703487 + 0.710708i \(0.748375\pi\)
\(398\) 0.673745 0.0337718
\(399\) 1.20578 0.0603643
\(400\) 3.71276 0.185638
\(401\) 22.5143 1.12431 0.562155 0.827032i \(-0.309973\pi\)
0.562155 + 0.827032i \(0.309973\pi\)
\(402\) 3.63222 0.181159
\(403\) 0 0
\(404\) −24.2927 −1.20861
\(405\) −7.48883 −0.372123
\(406\) −0.211925 −0.0105177
\(407\) −32.0805 −1.59017
\(408\) −7.03533 −0.348301
\(409\) −4.28772 −0.212014 −0.106007 0.994365i \(-0.533807\pi\)
−0.106007 + 0.994365i \(0.533807\pi\)
\(410\) −0.819883 −0.0404911
\(411\) −28.7761 −1.41942
\(412\) 29.3243 1.44471
\(413\) −0.899499 −0.0442614
\(414\) 0.274051 0.0134689
\(415\) 4.26371 0.209297
\(416\) 0 0
\(417\) −19.1802 −0.939257
\(418\) −2.67645 −0.130909
\(419\) 17.7116 0.865266 0.432633 0.901570i \(-0.357585\pi\)
0.432633 + 0.901570i \(0.357585\pi\)
\(420\) 1.03766 0.0506326
\(421\) −12.8787 −0.627672 −0.313836 0.949477i \(-0.601614\pi\)
−0.313836 + 0.949477i \(0.601614\pi\)
\(422\) −1.80152 −0.0876965
\(423\) 3.66563 0.178229
\(424\) 1.35485 0.0657973
\(425\) −5.06430 −0.245655
\(426\) 4.49880 0.217967
\(427\) 4.68596 0.226770
\(428\) 25.4981 1.23250
\(429\) 0 0
\(430\) −1.11256 −0.0536524
\(431\) −9.49845 −0.457524 −0.228762 0.973482i \(-0.573468\pi\)
−0.228762 + 0.973482i \(0.573468\pi\)
\(432\) 20.4338 0.983121
\(433\) −1.39628 −0.0671010 −0.0335505 0.999437i \(-0.510681\pi\)
−0.0335505 + 0.999437i \(0.510681\pi\)
\(434\) −0.398826 −0.0191443
\(435\) 4.64613 0.222765
\(436\) −21.9228 −1.04991
\(437\) 6.43911 0.308024
\(438\) −3.40617 −0.162753
\(439\) 4.16180 0.198632 0.0993159 0.995056i \(-0.468335\pi\)
0.0993159 + 0.995056i \(0.468335\pi\)
\(440\) −4.66351 −0.222324
\(441\) 3.02711 0.144148
\(442\) 0 0
\(443\) 9.54563 0.453526 0.226763 0.973950i \(-0.427186\pi\)
0.226763 + 0.973950i \(0.427186\pi\)
\(444\) −18.6516 −0.885164
\(445\) −3.22584 −0.152919
\(446\) −2.25052 −0.106565
\(447\) −5.45732 −0.258122
\(448\) −2.28083 −0.107759
\(449\) −21.7171 −1.02489 −0.512446 0.858720i \(-0.671260\pi\)
−0.512446 + 0.858720i \(0.671260\pi\)
\(450\) −0.0965246 −0.00455022
\(451\) −20.0479 −0.944018
\(452\) 35.8023 1.68400
\(453\) 12.7398 0.598569
\(454\) −1.54743 −0.0726246
\(455\) 0 0
\(456\) −3.15064 −0.147542
\(457\) −4.72259 −0.220914 −0.110457 0.993881i \(-0.535231\pi\)
−0.110457 + 0.993881i \(0.535231\pi\)
\(458\) −0.291961 −0.0136424
\(459\) −27.8722 −1.30096
\(460\) 5.54133 0.258366
\(461\) 1.78151 0.0829730 0.0414865 0.999139i \(-0.486791\pi\)
0.0414865 + 0.999139i \(0.486791\pi\)
\(462\) −0.627421 −0.0291902
\(463\) −6.80200 −0.316116 −0.158058 0.987430i \(-0.550523\pi\)
−0.158058 + 0.987430i \(0.550523\pi\)
\(464\) −10.7799 −0.500444
\(465\) 8.74363 0.405476
\(466\) −0.274051 −0.0126952
\(467\) 18.2374 0.843927 0.421963 0.906613i \(-0.361341\pi\)
0.421963 + 0.906613i \(0.361341\pi\)
\(468\) 0 0
\(469\) −3.43285 −0.158514
\(470\) −1.83281 −0.0845414
\(471\) 26.2958 1.21165
\(472\) 2.35035 0.108184
\(473\) −27.2045 −1.25086
\(474\) −1.58845 −0.0729598
\(475\) −2.26795 −0.104061
\(476\) 3.28398 0.150521
\(477\) 0.685698 0.0313960
\(478\) 2.18414 0.0999003
\(479\) 35.1807 1.60745 0.803724 0.595002i \(-0.202849\pi\)
0.803724 + 0.595002i \(0.202849\pi\)
\(480\) −4.08359 −0.186390
\(481\) 0 0
\(482\) 4.96204 0.226015
\(483\) 1.50948 0.0686835
\(484\) −34.8510 −1.58414
\(485\) 2.50791 0.113878
\(486\) −0.994615 −0.0451166
\(487\) 10.3040 0.466919 0.233459 0.972367i \(-0.424995\pi\)
0.233459 + 0.972367i \(0.424995\pi\)
\(488\) −12.2442 −0.554269
\(489\) 28.4950 1.28859
\(490\) −1.51356 −0.0683756
\(491\) 9.33198 0.421147 0.210573 0.977578i \(-0.432467\pi\)
0.210573 + 0.977578i \(0.432467\pi\)
\(492\) −11.6558 −0.525484
\(493\) 14.7041 0.662238
\(494\) 0 0
\(495\) −2.36023 −0.106085
\(496\) −20.2869 −0.910907
\(497\) −4.25187 −0.190722
\(498\) −1.49887 −0.0671661
\(499\) −23.9421 −1.07179 −0.535897 0.844283i \(-0.680026\pi\)
−0.535897 + 0.844283i \(0.680026\pi\)
\(500\) −1.95174 −0.0872844
\(501\) −10.0711 −0.449943
\(502\) −1.48692 −0.0663645
\(503\) 42.1443 1.87912 0.939560 0.342385i \(-0.111235\pi\)
0.939560 + 0.342385i \(0.111235\pi\)
\(504\) 0.126732 0.00564510
\(505\) 12.4467 0.553872
\(506\) −3.35056 −0.148951
\(507\) 0 0
\(508\) −6.32355 −0.280562
\(509\) 33.5602 1.48753 0.743765 0.668441i \(-0.233038\pi\)
0.743765 + 0.668441i \(0.233038\pi\)
\(510\) 1.78031 0.0788336
\(511\) 3.21921 0.142409
\(512\) 15.9211 0.703621
\(513\) −12.4820 −0.551096
\(514\) −2.25263 −0.0993593
\(515\) −15.0247 −0.662069
\(516\) −15.8166 −0.696288
\(517\) −44.8162 −1.97102
\(518\) −0.435898 −0.0191523
\(519\) −25.5633 −1.12210
\(520\) 0 0
\(521\) 12.4649 0.546098 0.273049 0.962000i \(-0.411968\pi\)
0.273049 + 0.962000i \(0.411968\pi\)
\(522\) 0.280257 0.0122665
\(523\) −5.65956 −0.247475 −0.123738 0.992315i \(-0.539488\pi\)
−0.123738 + 0.992315i \(0.539488\pi\)
\(524\) −0.342849 −0.0149774
\(525\) −0.531659 −0.0232035
\(526\) 4.09870 0.178712
\(527\) 27.6718 1.20540
\(528\) −31.9147 −1.38891
\(529\) −14.9391 −0.649525
\(530\) −0.342849 −0.0148924
\(531\) 1.18953 0.0516211
\(532\) 1.47067 0.0637616
\(533\) 0 0
\(534\) 1.13402 0.0490737
\(535\) −13.0643 −0.564819
\(536\) 8.96989 0.387440
\(537\) −37.7925 −1.63087
\(538\) 3.94511 0.170086
\(539\) −37.0097 −1.59412
\(540\) −10.7417 −0.462250
\(541\) 15.4750 0.665321 0.332660 0.943047i \(-0.392054\pi\)
0.332660 + 0.943047i \(0.392054\pi\)
\(542\) −6.78781 −0.291562
\(543\) 4.20196 0.180323
\(544\) −12.9237 −0.554101
\(545\) 11.2325 0.481146
\(546\) 0 0
\(547\) 25.1765 1.07647 0.538234 0.842795i \(-0.319092\pi\)
0.538234 + 0.842795i \(0.319092\pi\)
\(548\) −35.0979 −1.49931
\(549\) −6.19687 −0.264476
\(550\) 1.18012 0.0503204
\(551\) 6.58493 0.280528
\(552\) −3.94419 −0.167876
\(553\) 1.50126 0.0638401
\(554\) 5.82269 0.247382
\(555\) 9.55639 0.405646
\(556\) −23.3938 −0.992118
\(557\) −42.3489 −1.79438 −0.897190 0.441645i \(-0.854395\pi\)
−0.897190 + 0.441645i \(0.854395\pi\)
\(558\) 0.527420 0.0223275
\(559\) 0 0
\(560\) 1.23355 0.0521270
\(561\) 43.5325 1.83794
\(562\) −1.09353 −0.0461276
\(563\) 23.7905 1.00265 0.501326 0.865259i \(-0.332846\pi\)
0.501326 + 0.865259i \(0.332846\pi\)
\(564\) −26.0561 −1.09716
\(565\) −18.3438 −0.771731
\(566\) −2.76505 −0.116224
\(567\) −2.48814 −0.104492
\(568\) 11.1099 0.466162
\(569\) −26.7421 −1.12109 −0.560543 0.828125i \(-0.689407\pi\)
−0.560543 + 0.828125i \(0.689407\pi\)
\(570\) 0.797279 0.0333944
\(571\) −16.7159 −0.699539 −0.349769 0.936836i \(-0.613740\pi\)
−0.349769 + 0.936836i \(0.613740\pi\)
\(572\) 0 0
\(573\) 3.22571 0.134756
\(574\) −0.272403 −0.0113699
\(575\) −2.83918 −0.118402
\(576\) 3.01625 0.125677
\(577\) 20.6768 0.860786 0.430393 0.902642i \(-0.358375\pi\)
0.430393 + 0.902642i \(0.358375\pi\)
\(578\) 1.89966 0.0790153
\(579\) 36.5182 1.51764
\(580\) 5.66682 0.235302
\(581\) 1.41660 0.0587706
\(582\) −0.881636 −0.0365450
\(583\) −8.38340 −0.347205
\(584\) −8.41165 −0.348076
\(585\) 0 0
\(586\) −3.71657 −0.153530
\(587\) −20.7972 −0.858391 −0.429196 0.903212i \(-0.641203\pi\)
−0.429196 + 0.903212i \(0.641203\pi\)
\(588\) −21.5174 −0.887362
\(589\) 12.3923 0.510616
\(590\) −0.594763 −0.0244860
\(591\) −1.02897 −0.0423260
\(592\) −22.1726 −0.911289
\(593\) 21.8475 0.897169 0.448585 0.893740i \(-0.351928\pi\)
0.448585 + 0.893740i \(0.351928\pi\)
\(594\) 6.49498 0.266492
\(595\) −1.68260 −0.0689797
\(596\) −6.65622 −0.272649
\(597\) −4.90755 −0.200853
\(598\) 0 0
\(599\) −3.58040 −0.146291 −0.0731456 0.997321i \(-0.523304\pi\)
−0.0731456 + 0.997321i \(0.523304\pi\)
\(600\) 1.38920 0.0567139
\(601\) 21.3486 0.870829 0.435414 0.900230i \(-0.356602\pi\)
0.435414 + 0.900230i \(0.356602\pi\)
\(602\) −0.369644 −0.0150656
\(603\) 4.53972 0.184872
\(604\) 15.5386 0.632256
\(605\) 17.8564 0.725966
\(606\) −4.37554 −0.177744
\(607\) −3.29976 −0.133933 −0.0669665 0.997755i \(-0.521332\pi\)
−0.0669665 + 0.997755i \(0.521332\pi\)
\(608\) −5.78766 −0.234720
\(609\) 1.54366 0.0625523
\(610\) 3.09843 0.125452
\(611\) 0 0
\(612\) −4.34285 −0.175549
\(613\) −9.88635 −0.399306 −0.199653 0.979867i \(-0.563981\pi\)
−0.199653 + 0.979867i \(0.563981\pi\)
\(614\) 0.945589 0.0381609
\(615\) 5.97201 0.240815
\(616\) −1.54944 −0.0624286
\(617\) 45.7169 1.84049 0.920246 0.391339i \(-0.127988\pi\)
0.920246 + 0.391339i \(0.127988\pi\)
\(618\) 5.28182 0.212466
\(619\) −19.9143 −0.800425 −0.400212 0.916422i \(-0.631064\pi\)
−0.400212 + 0.916422i \(0.631064\pi\)
\(620\) 10.6645 0.428296
\(621\) −15.6259 −0.627046
\(622\) −0.488829 −0.0196003
\(623\) −1.07177 −0.0429397
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 1.58369 0.0632971
\(627\) 19.4952 0.778563
\(628\) 32.0726 1.27984
\(629\) 30.2440 1.20591
\(630\) −0.0320700 −0.00127770
\(631\) −14.5958 −0.581050 −0.290525 0.956867i \(-0.593830\pi\)
−0.290525 + 0.956867i \(0.593830\pi\)
\(632\) −3.92272 −0.156038
\(633\) 13.1222 0.521562
\(634\) 0.0706609 0.00280630
\(635\) 3.23996 0.128574
\(636\) −4.87409 −0.193270
\(637\) 0 0
\(638\) −3.42644 −0.135654
\(639\) 5.62281 0.222435
\(640\) −6.61199 −0.261362
\(641\) −14.1637 −0.559431 −0.279716 0.960083i \(-0.590240\pi\)
−0.279716 + 0.960083i \(0.590240\pi\)
\(642\) 4.59265 0.181257
\(643\) −16.7716 −0.661408 −0.330704 0.943735i \(-0.607286\pi\)
−0.330704 + 0.943735i \(0.607286\pi\)
\(644\) 1.84109 0.0725490
\(645\) 8.10387 0.319089
\(646\) 2.52323 0.0992751
\(647\) −2.99168 −0.117615 −0.0588075 0.998269i \(-0.518730\pi\)
−0.0588075 + 0.998269i \(0.518730\pi\)
\(648\) 6.50139 0.255399
\(649\) −14.5432 −0.570872
\(650\) 0 0
\(651\) 2.90504 0.113858
\(652\) 34.7550 1.36111
\(653\) −11.6643 −0.456461 −0.228230 0.973607i \(-0.573294\pi\)
−0.228230 + 0.973607i \(0.573294\pi\)
\(654\) −3.94868 −0.154406
\(655\) 0.175664 0.00686374
\(656\) −13.8562 −0.540993
\(657\) −4.25719 −0.166089
\(658\) −0.608946 −0.0237392
\(659\) −1.81047 −0.0705260 −0.0352630 0.999378i \(-0.511227\pi\)
−0.0352630 + 0.999378i \(0.511227\pi\)
\(660\) 16.7771 0.653046
\(661\) 12.3406 0.479992 0.239996 0.970774i \(-0.422854\pi\)
0.239996 + 0.970774i \(0.422854\pi\)
\(662\) −3.65383 −0.142010
\(663\) 0 0
\(664\) −3.70152 −0.143647
\(665\) −0.753518 −0.0292202
\(666\) 0.576446 0.0223368
\(667\) 8.24348 0.319189
\(668\) −12.2836 −0.475265
\(669\) 16.3927 0.633779
\(670\) −2.26986 −0.0876923
\(671\) 75.7634 2.92481
\(672\) −1.35676 −0.0523381
\(673\) 9.26625 0.357188 0.178594 0.983923i \(-0.442845\pi\)
0.178594 + 0.983923i \(0.442845\pi\)
\(674\) 5.32051 0.204938
\(675\) 5.50367 0.211836
\(676\) 0 0
\(677\) 13.8984 0.534158 0.267079 0.963675i \(-0.413941\pi\)
0.267079 + 0.963675i \(0.413941\pi\)
\(678\) 6.44863 0.247658
\(679\) 0.833244 0.0319770
\(680\) 4.39654 0.168600
\(681\) 11.2715 0.431924
\(682\) −6.44828 −0.246917
\(683\) 37.7512 1.44451 0.722255 0.691626i \(-0.243105\pi\)
0.722255 + 0.691626i \(0.243105\pi\)
\(684\) −1.94486 −0.0743637
\(685\) 17.9829 0.687090
\(686\) −1.01381 −0.0387073
\(687\) 2.12664 0.0811363
\(688\) −18.8025 −0.716838
\(689\) 0 0
\(690\) 0.998090 0.0379966
\(691\) 1.65291 0.0628797 0.0314399 0.999506i \(-0.489991\pi\)
0.0314399 + 0.999506i \(0.489991\pi\)
\(692\) −31.1792 −1.18526
\(693\) −0.784180 −0.0297885
\(694\) −1.37823 −0.0523168
\(695\) 11.9861 0.454660
\(696\) −4.03351 −0.152890
\(697\) 18.9002 0.715897
\(698\) −1.55317 −0.0587885
\(699\) 1.99618 0.0755025
\(700\) −0.648458 −0.0245094
\(701\) 20.4819 0.773590 0.386795 0.922166i \(-0.373582\pi\)
0.386795 + 0.922166i \(0.373582\pi\)
\(702\) 0 0
\(703\) 13.5442 0.510830
\(704\) −36.8769 −1.38985
\(705\) 13.3502 0.502797
\(706\) 4.78694 0.180159
\(707\) 4.13538 0.155527
\(708\) −8.45541 −0.317774
\(709\) −21.9417 −0.824039 −0.412020 0.911175i \(-0.635177\pi\)
−0.412020 + 0.911175i \(0.635177\pi\)
\(710\) −2.81140 −0.105510
\(711\) −1.98532 −0.0744552
\(712\) 2.80049 0.104953
\(713\) 15.5136 0.580987
\(714\) 0.591503 0.0221364
\(715\) 0 0
\(716\) −46.0950 −1.72265
\(717\) −15.9093 −0.594142
\(718\) −5.26671 −0.196552
\(719\) 38.8475 1.44877 0.724384 0.689397i \(-0.242124\pi\)
0.724384 + 0.689397i \(0.242124\pi\)
\(720\) −1.63129 −0.0607945
\(721\) −4.99191 −0.185909
\(722\) −3.04407 −0.113289
\(723\) −36.1434 −1.34419
\(724\) 5.12507 0.190472
\(725\) −2.90348 −0.107832
\(726\) −6.27728 −0.232972
\(727\) −30.6598 −1.13711 −0.568555 0.822645i \(-0.692497\pi\)
−0.568555 + 0.822645i \(0.692497\pi\)
\(728\) 0 0
\(729\) 29.7112 1.10042
\(730\) 2.12859 0.0787828
\(731\) 25.6471 0.948593
\(732\) 44.0487 1.62809
\(733\) −24.3858 −0.900709 −0.450355 0.892850i \(-0.648702\pi\)
−0.450355 + 0.892850i \(0.648702\pi\)
\(734\) 1.40409 0.0518259
\(735\) 11.0247 0.406653
\(736\) −7.24539 −0.267069
\(737\) −55.5029 −2.04448
\(738\) 0.360235 0.0132604
\(739\) −38.2788 −1.40811 −0.704054 0.710146i \(-0.748629\pi\)
−0.704054 + 0.710146i \(0.748629\pi\)
\(740\) 11.6558 0.428476
\(741\) 0 0
\(742\) −0.113910 −0.00418178
\(743\) 40.0079 1.46775 0.733874 0.679286i \(-0.237710\pi\)
0.733874 + 0.679286i \(0.237710\pi\)
\(744\) −7.59074 −0.278290
\(745\) 3.41041 0.124948
\(746\) −4.41134 −0.161511
\(747\) −1.87336 −0.0685427
\(748\) 53.0960 1.94138
\(749\) −4.34057 −0.158601
\(750\) −0.351542 −0.0128365
\(751\) 25.6020 0.934230 0.467115 0.884197i \(-0.345293\pi\)
0.467115 + 0.884197i \(0.345293\pi\)
\(752\) −30.9750 −1.12954
\(753\) 10.8307 0.394693
\(754\) 0 0
\(755\) −7.96141 −0.289745
\(756\) −3.56890 −0.129800
\(757\) −1.84848 −0.0671840 −0.0335920 0.999436i \(-0.510695\pi\)
−0.0335920 + 0.999436i \(0.510695\pi\)
\(758\) 1.19989 0.0435820
\(759\) 24.4055 0.885862
\(760\) 1.96891 0.0714198
\(761\) −26.2124 −0.950199 −0.475099 0.879932i \(-0.657588\pi\)
−0.475099 + 0.879932i \(0.657588\pi\)
\(762\) −1.13898 −0.0412610
\(763\) 3.73195 0.135106
\(764\) 3.93435 0.142340
\(765\) 2.22512 0.0804494
\(766\) 1.24513 0.0449884
\(767\) 0 0
\(768\) −19.6459 −0.708911
\(769\) 44.3495 1.59928 0.799641 0.600478i \(-0.205023\pi\)
0.799641 + 0.600478i \(0.205023\pi\)
\(770\) 0.392090 0.0141299
\(771\) 16.4081 0.590924
\(772\) 44.5408 1.60306
\(773\) 23.2638 0.836742 0.418371 0.908276i \(-0.362601\pi\)
0.418371 + 0.908276i \(0.362601\pi\)
\(774\) 0.488829 0.0175706
\(775\) −5.46410 −0.196276
\(776\) −2.17723 −0.0781580
\(777\) 3.17508 0.113905
\(778\) 2.33251 0.0836245
\(779\) 8.46410 0.303258
\(780\) 0 0
\(781\) −68.7449 −2.45989
\(782\) 3.15875 0.112957
\(783\) −15.9798 −0.571071
\(784\) −25.5794 −0.913552
\(785\) −16.4329 −0.586514
\(786\) −0.0617531 −0.00220266
\(787\) 47.9133 1.70793 0.853963 0.520334i \(-0.174192\pi\)
0.853963 + 0.520334i \(0.174192\pi\)
\(788\) −1.25502 −0.0447081
\(789\) −29.8548 −1.06286
\(790\) 0.992658 0.0353172
\(791\) −6.09467 −0.216702
\(792\) 2.04903 0.0728090
\(793\) 0 0
\(794\) −6.15865 −0.218562
\(795\) 2.49731 0.0885704
\(796\) −5.98567 −0.212156
\(797\) −20.6952 −0.733060 −0.366530 0.930406i \(-0.619454\pi\)
−0.366530 + 0.930406i \(0.619454\pi\)
\(798\) 0.264893 0.00937712
\(799\) 42.2507 1.49472
\(800\) 2.55193 0.0902245
\(801\) 1.41735 0.0500795
\(802\) 4.94609 0.174653
\(803\) 52.0487 1.83676
\(804\) −32.2693 −1.13805
\(805\) −0.943307 −0.0332472
\(806\) 0 0
\(807\) −28.7361 −1.01156
\(808\) −10.8056 −0.380138
\(809\) 15.8915 0.558714 0.279357 0.960187i \(-0.409879\pi\)
0.279357 + 0.960187i \(0.409879\pi\)
\(810\) −1.64520 −0.0578063
\(811\) 23.8796 0.838525 0.419263 0.907865i \(-0.362289\pi\)
0.419263 + 0.907865i \(0.362289\pi\)
\(812\) 1.88278 0.0660727
\(813\) 49.4423 1.73402
\(814\) −7.04768 −0.247021
\(815\) −17.8072 −0.623759
\(816\) 30.0877 1.05328
\(817\) 11.4856 0.401829
\(818\) −0.941956 −0.0329347
\(819\) 0 0
\(820\) 7.28398 0.254368
\(821\) −15.9097 −0.555251 −0.277626 0.960689i \(-0.589547\pi\)
−0.277626 + 0.960689i \(0.589547\pi\)
\(822\) −6.32174 −0.220496
\(823\) −14.8115 −0.516295 −0.258147 0.966106i \(-0.583112\pi\)
−0.258147 + 0.966106i \(0.583112\pi\)
\(824\) 13.0436 0.454397
\(825\) −8.59596 −0.299273
\(826\) −0.197608 −0.00687566
\(827\) −33.9498 −1.18055 −0.590275 0.807202i \(-0.700981\pi\)
−0.590275 + 0.807202i \(0.700981\pi\)
\(828\) −2.43472 −0.0846122
\(829\) 23.3146 0.809749 0.404875 0.914372i \(-0.367315\pi\)
0.404875 + 0.914372i \(0.367315\pi\)
\(830\) 0.936681 0.0325127
\(831\) −42.4124 −1.47127
\(832\) 0 0
\(833\) 34.8910 1.20890
\(834\) −4.21363 −0.145906
\(835\) 6.29366 0.217801
\(836\) 23.7780 0.822380
\(837\) −30.0726 −1.03946
\(838\) 3.89100 0.134412
\(839\) 14.7930 0.510710 0.255355 0.966847i \(-0.417808\pi\)
0.255355 + 0.966847i \(0.417808\pi\)
\(840\) 0.461557 0.0159252
\(841\) −20.5698 −0.709305
\(842\) −2.82929 −0.0975038
\(843\) 7.96523 0.274337
\(844\) 16.0050 0.550915
\(845\) 0 0
\(846\) 0.805291 0.0276865
\(847\) 5.93273 0.203851
\(848\) −5.79422 −0.198974
\(849\) 20.1406 0.691223
\(850\) −1.11256 −0.0381605
\(851\) 16.9556 0.581231
\(852\) −39.9681 −1.36929
\(853\) 16.3452 0.559650 0.279825 0.960051i \(-0.409724\pi\)
0.279825 + 0.960051i \(0.409724\pi\)
\(854\) 1.02944 0.0352268
\(855\) 0.996477 0.0340788
\(856\) 11.3417 0.387651
\(857\) −34.1418 −1.16626 −0.583132 0.812378i \(-0.698173\pi\)
−0.583132 + 0.812378i \(0.698173\pi\)
\(858\) 0 0
\(859\) −45.1996 −1.54219 −0.771096 0.636719i \(-0.780291\pi\)
−0.771096 + 0.636719i \(0.780291\pi\)
\(860\) 9.88418 0.337048
\(861\) 1.98418 0.0676207
\(862\) −2.08669 −0.0710728
\(863\) −4.75058 −0.161712 −0.0808559 0.996726i \(-0.525765\pi\)
−0.0808559 + 0.996726i \(0.525765\pi\)
\(864\) 14.0450 0.477821
\(865\) 15.9751 0.543170
\(866\) −0.306745 −0.0104236
\(867\) −13.8371 −0.469931
\(868\) 3.54324 0.120265
\(869\) 24.2726 0.823392
\(870\) 1.02069 0.0346048
\(871\) 0 0
\(872\) −9.75140 −0.330224
\(873\) −1.10191 −0.0372940
\(874\) 1.41459 0.0478492
\(875\) 0.332247 0.0112320
\(876\) 30.2610 1.02242
\(877\) −2.25506 −0.0761481 −0.0380741 0.999275i \(-0.512122\pi\)
−0.0380741 + 0.999275i \(0.512122\pi\)
\(878\) 0.914293 0.0308559
\(879\) 27.0715 0.913098
\(880\) 19.9442 0.672320
\(881\) −2.98304 −0.100501 −0.0502507 0.998737i \(-0.516002\pi\)
−0.0502507 + 0.998737i \(0.516002\pi\)
\(882\) 0.665017 0.0223923
\(883\) 28.2874 0.951947 0.475973 0.879460i \(-0.342096\pi\)
0.475973 + 0.879460i \(0.342096\pi\)
\(884\) 0 0
\(885\) 4.33225 0.145627
\(886\) 2.09705 0.0704517
\(887\) −27.9816 −0.939531 −0.469766 0.882791i \(-0.655662\pi\)
−0.469766 + 0.882791i \(0.655662\pi\)
\(888\) −8.29633 −0.278407
\(889\) 1.07647 0.0361035
\(890\) −0.708674 −0.0237548
\(891\) −40.2286 −1.34771
\(892\) 19.9940 0.669448
\(893\) 18.9212 0.633172
\(894\) −1.19890 −0.0400973
\(895\) 23.6174 0.789443
\(896\) −2.19681 −0.0733903
\(897\) 0 0
\(898\) −4.77095 −0.159209
\(899\) 15.8649 0.529124
\(900\) 0.857542 0.0285847
\(901\) 7.90348 0.263303
\(902\) −4.40426 −0.146646
\(903\) 2.69248 0.0896002
\(904\) 15.9251 0.529661
\(905\) −2.62590 −0.0872879
\(906\) 2.79877 0.0929829
\(907\) 16.5520 0.549600 0.274800 0.961501i \(-0.411388\pi\)
0.274800 + 0.961501i \(0.411388\pi\)
\(908\) 13.7477 0.456232
\(909\) −5.46876 −0.181387
\(910\) 0 0
\(911\) 7.04863 0.233532 0.116766 0.993159i \(-0.462747\pi\)
0.116766 + 0.993159i \(0.462747\pi\)
\(912\) 13.4742 0.446175
\(913\) 22.9039 0.758007
\(914\) −1.03749 −0.0343172
\(915\) −22.5689 −0.746106
\(916\) 2.59383 0.0857026
\(917\) 0.0583636 0.00192734
\(918\) −6.12316 −0.202094
\(919\) 16.5438 0.545728 0.272864 0.962053i \(-0.412029\pi\)
0.272864 + 0.962053i \(0.412029\pi\)
\(920\) 2.46482 0.0812626
\(921\) −6.88766 −0.226956
\(922\) 0.391374 0.0128892
\(923\) 0 0
\(924\) 5.57412 0.183375
\(925\) −5.97201 −0.196359
\(926\) −1.49431 −0.0491060
\(927\) 6.60147 0.216821
\(928\) −7.40948 −0.243228
\(929\) −33.8367 −1.11015 −0.555074 0.831801i \(-0.687310\pi\)
−0.555074 + 0.831801i \(0.687310\pi\)
\(930\) 1.92086 0.0629875
\(931\) 15.6253 0.512098
\(932\) 2.43472 0.0797517
\(933\) 3.56063 0.116570
\(934\) 4.00652 0.131097
\(935\) −27.2045 −0.889681
\(936\) 0 0
\(937\) −30.4606 −0.995104 −0.497552 0.867434i \(-0.665768\pi\)
−0.497552 + 0.867434i \(0.665768\pi\)
\(938\) −0.754153 −0.0246240
\(939\) −11.5356 −0.376450
\(940\) 16.2831 0.531095
\(941\) 38.2101 1.24561 0.622807 0.782375i \(-0.285992\pi\)
0.622807 + 0.782375i \(0.285992\pi\)
\(942\) 5.77684 0.188220
\(943\) 10.5960 0.345052
\(944\) −10.0516 −0.327153
\(945\) 1.82858 0.0594836
\(946\) −5.97647 −0.194312
\(947\) −52.4482 −1.70434 −0.852169 0.523266i \(-0.824713\pi\)
−0.852169 + 0.523266i \(0.824713\pi\)
\(948\) 14.1120 0.458338
\(949\) 0 0
\(950\) −0.498239 −0.0161650
\(951\) −0.514693 −0.0166901
\(952\) 1.46074 0.0473427
\(953\) 39.7500 1.28763 0.643814 0.765182i \(-0.277351\pi\)
0.643814 + 0.765182i \(0.277351\pi\)
\(954\) 0.150639 0.00487711
\(955\) −2.01582 −0.0652304
\(956\) −19.4043 −0.627580
\(957\) 24.9581 0.806782
\(958\) 7.72874 0.249704
\(959\) 5.97475 0.192935
\(960\) 10.9852 0.354544
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) 5.74011 0.184972
\(964\) −44.0837 −1.41984
\(965\) −22.8211 −0.734636
\(966\) 0.331612 0.0106694
\(967\) −25.7857 −0.829214 −0.414607 0.910001i \(-0.636081\pi\)
−0.414607 + 0.910001i \(0.636081\pi\)
\(968\) −15.5019 −0.498251
\(969\) −18.3792 −0.590424
\(970\) 0.550955 0.0176901
\(971\) −55.5252 −1.78189 −0.890945 0.454111i \(-0.849957\pi\)
−0.890945 + 0.454111i \(0.849957\pi\)
\(972\) 8.83634 0.283426
\(973\) 3.98235 0.127668
\(974\) 2.26365 0.0725321
\(975\) 0 0
\(976\) 52.3642 1.67614
\(977\) 40.5161 1.29622 0.648112 0.761545i \(-0.275559\pi\)
0.648112 + 0.761545i \(0.275559\pi\)
\(978\) 6.25998 0.200172
\(979\) −17.3286 −0.553824
\(980\) 13.4467 0.429540
\(981\) −4.93525 −0.157570
\(982\) 2.05011 0.0654218
\(983\) −34.8059 −1.11014 −0.555068 0.831805i \(-0.687308\pi\)
−0.555068 + 0.831805i \(0.687308\pi\)
\(984\) −5.18457 −0.165278
\(985\) 0.643026 0.0204885
\(986\) 3.23029 0.102873
\(987\) 4.43555 0.141185
\(988\) 0 0
\(989\) 14.3784 0.457208
\(990\) −0.518513 −0.0164794
\(991\) 43.8855 1.39407 0.697034 0.717038i \(-0.254503\pi\)
0.697034 + 0.717038i \(0.254503\pi\)
\(992\) −13.9440 −0.442723
\(993\) 26.6145 0.844584
\(994\) −0.934079 −0.0296272
\(995\) 3.06684 0.0972254
\(996\) 13.3163 0.421942
\(997\) −5.49137 −0.173914 −0.0869568 0.996212i \(-0.527714\pi\)
−0.0869568 + 0.996212i \(0.527714\pi\)
\(998\) −5.25976 −0.166495
\(999\) −32.8680 −1.03990
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.a.l.1.3 4
3.2 odd 2 7605.2.a.cj.1.2 4
5.4 even 2 4225.2.a.bl.1.2 4
13.2 odd 12 845.2.m.g.316.3 8
13.3 even 3 845.2.e.n.191.2 8
13.4 even 6 845.2.e.m.146.3 8
13.5 odd 4 845.2.c.g.506.4 8
13.6 odd 12 65.2.m.a.36.2 8
13.7 odd 12 845.2.m.g.361.3 8
13.8 odd 4 845.2.c.g.506.5 8
13.9 even 3 845.2.e.n.146.2 8
13.10 even 6 845.2.e.m.191.3 8
13.11 odd 12 65.2.m.a.56.2 yes 8
13.12 even 2 845.2.a.m.1.2 4
39.11 even 12 585.2.bu.c.316.3 8
39.32 even 12 585.2.bu.c.361.3 8
39.38 odd 2 7605.2.a.cf.1.3 4
52.11 even 12 1040.2.da.b.641.2 8
52.19 even 12 1040.2.da.b.881.2 8
65.19 odd 12 325.2.n.d.101.3 8
65.24 odd 12 325.2.n.d.251.3 8
65.32 even 12 325.2.m.c.49.2 8
65.37 even 12 325.2.m.b.199.3 8
65.58 even 12 325.2.m.b.49.3 8
65.63 even 12 325.2.m.c.199.2 8
65.64 even 2 4225.2.a.bi.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.2 8 13.6 odd 12
65.2.m.a.56.2 yes 8 13.11 odd 12
325.2.m.b.49.3 8 65.58 even 12
325.2.m.b.199.3 8 65.37 even 12
325.2.m.c.49.2 8 65.32 even 12
325.2.m.c.199.2 8 65.63 even 12
325.2.n.d.101.3 8 65.19 odd 12
325.2.n.d.251.3 8 65.24 odd 12
585.2.bu.c.316.3 8 39.11 even 12
585.2.bu.c.361.3 8 39.32 even 12
845.2.a.l.1.3 4 1.1 even 1 trivial
845.2.a.m.1.2 4 13.12 even 2
845.2.c.g.506.4 8 13.5 odd 4
845.2.c.g.506.5 8 13.8 odd 4
845.2.e.m.146.3 8 13.4 even 6
845.2.e.m.191.3 8 13.10 even 6
845.2.e.n.146.2 8 13.9 even 3
845.2.e.n.191.2 8 13.3 even 3
845.2.m.g.316.3 8 13.2 odd 12
845.2.m.g.361.3 8 13.7 odd 12
1040.2.da.b.641.2 8 52.11 even 12
1040.2.da.b.881.2 8 52.19 even 12
4225.2.a.bi.1.3 4 65.64 even 2
4225.2.a.bl.1.2 4 5.4 even 2
7605.2.a.cf.1.3 4 39.38 odd 2
7605.2.a.cj.1.2 4 3.2 odd 2