Properties

Label 4225.2.a.bi.1.3
Level $4225$
Weight $2$
Character 4225.1
Self dual yes
Analytic conductor $33.737$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.7367948540\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 4225.1

$q$-expansion

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

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 \(-0.347155\pi\)
0.461937 + 0.886913i \(0.347155\pi\)
\(4\) −1.95174 −0.975869
\(5\) 0 0
\(6\) 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 0
\(11\) −5.37182 −1.61966 −0.809832 0.586662i \(-0.800442\pi\)
−0.809832 + 0.586662i \(0.800442\pi\)
\(12\) −3.12316 −0.901579
\(13\) 0 0
\(14\) 0.0729902 0.0195074
\(15\) 0 0
\(16\) 3.71276 0.928189
\(17\) 5.06430 1.22827 0.614136 0.789200i \(-0.289505\pi\)
0.614136 + 0.789200i \(0.289505\pi\)
\(18\) −0.0965246 −0.0227511
\(19\) 2.26795 0.520303 0.260152 0.965568i \(-0.416227\pi\)
0.260152 + 0.965568i \(0.416227\pi\)
\(20\) 0 0
\(21\) 0.531659 0.116018
\(22\) −1.18012 −0.251602
\(23\) 2.83918 0.592010 0.296005 0.955186i \(-0.404346\pi\)
0.296005 + 0.955186i \(0.404346\pi\)
\(24\) −1.38920 −0.283570
\(25\) 0 0
\(26\) 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 0
\(31\) 5.46410 0.981382 0.490691 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) 2.55193 0.451122
\(33\) −8.59596 −1.49636
\(34\) 1.11256 0.190802
\(35\) 0 0
\(36\) 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 0
\(41\) 3.73205 0.582848 0.291424 0.956594i \(-0.405871\pi\)
0.291424 + 0.956594i \(0.405871\pi\)
\(42\) 0.116799 0.0180224
\(43\) 5.06430 0.772298 0.386149 0.922436i \(-0.373805\pi\)
0.386149 + 0.922436i \(0.373805\pi\)
\(44\) 10.4844 1.58058
\(45\) 0 0
\(46\) 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 0
\(51\) 8.10387 1.13477
\(52\) 0 0
\(53\) 1.56063 0.214369 0.107184 0.994239i \(-0.465817\pi\)
0.107184 + 0.994239i \(0.465817\pi\)
\(54\) −1.20908 −0.164536
\(55\) 0 0
\(56\) −0.288438 −0.0385442
\(57\) 3.62916 0.480694
\(58\) −0.637855 −0.0837545
\(59\) 2.70732 0.352463 0.176232 0.984349i \(-0.443609\pi\)
0.176232 + 0.984349i \(0.443609\pi\)
\(60\) 0 0
\(61\) 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 0
\(71\) 12.7973 1.51876 0.759382 0.650645i \(-0.225502\pi\)
0.759382 + 0.650645i \(0.225502\pi\)
\(72\) 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
\(75\) 0 0
\(76\) −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 \(0.418192\pi\)
\(80\) 0 0
\(81\) −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
\(85\) 0 0
\(86\) 1.11256 0.119970
\(87\) −4.64613 −0.498117
\(88\) 4.66351 0.497132
\(89\) 3.22584 0.341938 0.170969 0.985276i \(-0.445310\pi\)
0.170969 + 0.985276i \(0.445310\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5.54133 −0.577724
\(93\) 8.74363 0.906672
\(94\) −1.83281 −0.189040
\(95\) 0 0
\(96\) 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
\(100\) 0 0
\(101\) 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 \(-0.234724\pi\)
0.740215 + 0.672370i \(0.234724\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.342849 0.0333004
\(107\) 13.0643 1.26297 0.631487 0.775387i \(-0.282445\pi\)
0.631487 + 0.775387i \(0.282445\pi\)
\(108\) 10.7417 1.03362
\(109\) −11.2325 −1.07587 −0.537937 0.842985i \(-0.680796\pi\)
−0.537937 + 0.842985i \(0.680796\pi\)
\(110\) 0 0
\(111\) −9.55639 −0.907052
\(112\) 1.23355 0.116560
\(113\) 18.3438 1.72564 0.862821 0.505509i \(-0.168695\pi\)
0.862821 + 0.505509i \(0.168695\pi\)
\(114\) 0.797279 0.0746721
\(115\) 0 0
\(116\) 5.66682 0.526151
\(117\) 0 0
\(118\) 0.594763 0.0547524
\(119\) 1.68260 0.154243
\(120\) 0 0
\(121\) 17.8564 1.62331
\(122\) 3.09843 0.280519
\(123\) 5.97201 0.538478
\(124\) −10.6645 −0.957700
\(125\) 0 0
\(126\) −0.0320700 −0.00285702
\(127\) −3.23996 −0.287500 −0.143750 0.989614i \(-0.545916\pi\)
−0.143750 + 0.989614i \(0.545916\pi\)
\(128\) −6.61199 −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
\(135\) 0 0
\(136\) −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 0
\(141\) −13.3502 −1.12429
\(142\) 2.81140 0.235928
\(143\) 0 0
\(144\) −1.63129 −0.135941
\(145\) 0 0
\(146\) 2.12859 0.176164
\(147\) −11.0247 −0.909304
\(148\) 11.6558 0.958101
\(149\) −3.41041 −0.279391 −0.139696 0.990194i \(-0.544612\pi\)
−0.139696 + 0.990194i \(0.544612\pi\)
\(150\) 0 0
\(151\) 7.96141 0.647890 0.323945 0.946076i \(-0.394991\pi\)
0.323945 + 0.946076i \(0.394991\pi\)
\(152\) −1.96891 −0.159700
\(153\) −2.22512 −0.179890
\(154\) −0.392090 −0.0315955
\(155\) 0 0
\(156\) 0 0
\(157\) 16.4329 1.31148 0.655742 0.754985i \(-0.272356\pi\)
0.655742 + 0.754985i \(0.272356\pi\)
\(158\) 0.992658 0.0789716
\(159\) 2.49731 0.198049
\(160\) 0 0
\(161\) 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
\(165\) 0 0
\(166\) 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
\(170\) 0 0
\(171\) −0.996477 −0.0762025
\(172\) −9.88418 −0.753662
\(173\) −15.9751 −1.21457 −0.607283 0.794486i \(-0.707740\pi\)
−0.607283 + 0.794486i \(0.707740\pi\)
\(174\) −1.02069 −0.0773786
\(175\) 0 0
\(176\) −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 0
\(181\) −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
\(185\) 0 0
\(186\) 1.92086 0.140844
\(187\) −27.2045 −1.98939
\(188\) 16.2831 1.18756
\(189\) −1.82858 −0.133009
\(190\) 0 0
\(191\) −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 0
\(201\) −16.5336 −1.16619
\(202\) 2.73438 0.192390
\(203\) −0.964670 −0.0677065
\(204\) −15.8166 −1.10739
\(205\) 0 0
\(206\) 3.30074 0.229973
\(207\) −1.24746 −0.0867045
\(208\) 0 0
\(209\) −12.1830 −0.842716
\(210\) 0 0
\(211\) −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
\(215\) 0 0
\(216\) 4.77798 0.325101
\(217\) 1.81543 0.123239
\(218\) −2.46762 −0.167129
\(219\) 15.5046 1.04771
\(220\) 0 0
\(221\) 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 0
\(226\) 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 \(-0.486018\pi\)
0.0439109 + 0.999035i \(0.486018\pi\)
\(230\) 0 0
\(231\) −2.85598 −0.187909
\(232\) 2.52064 0.165488
\(233\) 1.24746 0.0817238 0.0408619 0.999165i \(-0.486990\pi\)
0.0408619 + 0.999165i \(0.486990\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −5.28398 −0.343958
\(237\) 7.23050 0.469672
\(238\) 0.369644 0.0239605
\(239\) −9.94207 −0.643099 −0.321549 0.946893i \(-0.604204\pi\)
−0.321549 + 0.946893i \(0.604204\pi\)
\(240\) 0 0
\(241\) −22.5869 −1.45495 −0.727475 0.686134i \(-0.759306\pi\)
−0.727475 + 0.686134i \(0.759306\pi\)
\(242\) 3.92282 0.252168
\(243\) 4.52742 0.290434
\(244\) −27.5270 −1.76224
\(245\) 0 0
\(246\) 1.31197 0.0836483
\(247\) 0 0
\(248\) −4.74363 −0.301221
\(249\) 6.82277 0.432376
\(250\) 0 0
\(251\) −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
\(255\) 0 0
\(256\) 12.2772 0.767325
\(257\) 10.2538 0.639616 0.319808 0.947482i \(-0.396382\pi\)
0.319808 + 0.947482i \(0.396382\pi\)
\(258\) 1.78031 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 \(-0.695083\pi\)
−0.575220 + 0.817999i \(0.695083\pi\)
\(264\) 7.46254 0.459287
\(265\) 0 0
\(266\) 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 \(0.315596\pi\)
\(270\) 0 0
\(271\) 30.8977 1.87690 0.938450 0.345415i \(-0.112262\pi\)
0.938450 + 0.345415i \(0.112262\pi\)
\(272\) 18.8025 1.14007
\(273\) 0 0
\(274\) 3.95060 0.238665
\(275\) 0 0
\(276\) −8.86721 −0.533744
\(277\) −26.5045 −1.59250 −0.796250 0.604967i \(-0.793186\pi\)
−0.796250 + 0.604967i \(0.793186\pi\)
\(278\) 2.63320 0.157929
\(279\) −2.40078 −0.143731
\(280\) 0 0
\(281\) 4.97766 0.296942 0.148471 0.988917i \(-0.452565\pi\)
0.148471 + 0.988917i \(0.452565\pi\)
\(282\) −2.93286 −0.174649
\(283\) 12.5863 0.748180 0.374090 0.927392i \(-0.377955\pi\)
0.374090 + 0.927392i \(0.377955\pi\)
\(284\) −24.9770 −1.48211
\(285\) 0 0
\(286\) 0 0
\(287\) 1.23996 0.0731926
\(288\) −1.12125 −0.0660704
\(289\) 8.64711 0.508653
\(290\) 0 0
\(291\) 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
\(295\) 0 0
\(296\) 5.18457 0.301347
\(297\) 29.5647 1.71552
\(298\) −0.749222 −0.0434012
\(299\) 0 0
\(300\) 0 0
\(301\) 1.68260 0.0969832
\(302\) 1.74902 0.100645
\(303\) 19.9172 1.14421
\(304\) 8.42034 0.482940
\(305\) 0 0
\(306\) −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
\(310\) 0 0
\(311\) −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 \(-0.565308\pi\)
−0.203735 + 0.979026i \(0.565308\pi\)
\(314\) 3.61008 0.203729
\(315\) 0 0
\(316\) −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
\(320\) 0 0
\(321\) 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
\(330\) 0 0
\(331\) 16.6320 0.914178 0.457089 0.889421i \(-0.348892\pi\)
0.457089 + 0.889421i \(0.348892\pi\)
\(332\) −8.32164 −0.456710
\(333\) 2.62395 0.143791
\(334\) 1.38263 0.0756543
\(335\) 0 0
\(336\) 1.97392 0.107686
\(337\) −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
\(340\) 0 0
\(341\) −29.3521 −1.58951
\(342\) −0.218913 −0.0118375
\(343\) −4.61478 −0.249174
\(344\) −4.39654 −0.237046
\(345\) 0 0
\(346\) −3.50952 −0.188673
\(347\) 6.27360 0.336784 0.168392 0.985720i \(-0.446143\pi\)
0.168392 + 0.985720i \(0.446143\pi\)
\(348\) 9.06802 0.486097
\(349\) 7.06994 0.378445 0.189223 0.981934i \(-0.439403\pi\)
0.189223 + 0.981934i \(0.439403\pi\)
\(350\) 0 0
\(351\) 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
\(355\) 0 0
\(356\) −6.29598 −0.333687
\(357\) 2.69248 0.142501
\(358\) 5.18844 0.274217
\(359\) 23.9737 1.26528 0.632642 0.774444i \(-0.281971\pi\)
0.632642 + 0.774444i \(0.281971\pi\)
\(360\) 0 0
\(361\) −13.8564 −0.729285
\(362\) −0.576876 −0.0303199
\(363\) 28.5737 1.49973
\(364\) 0 0
\(365\) 0 0
\(366\) 4.95810 0.259164
\(367\) −6.39133 −0.333625 −0.166812 0.985989i \(-0.553347\pi\)
−0.166812 + 0.985989i \(0.553347\pi\)
\(368\) 10.5412 0.549497
\(369\) −1.63977 −0.0853628
\(370\) 0 0
\(371\) 0.518513 0.0269198
\(372\) −17.0653 −0.884793
\(373\) 20.0801 1.03971 0.519855 0.854255i \(-0.325986\pi\)
0.519855 + 0.854255i \(0.325986\pi\)
\(374\) −5.97647 −0.309036
\(375\) 0 0
\(376\) 7.24280 0.373519
\(377\) 0 0
\(378\) −0.401714 −0.0206619
\(379\) −5.46182 −0.280555 −0.140277 0.990112i \(-0.544799\pi\)
−0.140277 + 0.990112i \(0.544799\pi\)
\(380\) 0 0
\(381\) −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
\(385\) 0 0
\(386\) −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
\(395\) 0 0
\(396\) −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
\(400\) 0 0
\(401\) −22.5143 −1.12431 −0.562155 0.827032i \(-0.690027\pi\)
−0.562155 + 0.827032i \(0.690027\pi\)
\(402\) −3.63222 −0.181159
\(403\) 0 0
\(404\) −24.2927 −1.20861
\(405\) 0 0
\(406\) −0.211925 −0.0105177
\(407\) 32.0805 1.59017
\(408\) −7.03533 −0.348301
\(409\) 4.28772 0.212014 0.106007 0.994365i \(-0.466193\pi\)
0.106007 + 0.994365i \(0.466193\pi\)
\(410\) 0 0
\(411\) 28.7761 1.41942
\(412\) −29.3243 −1.44471
\(413\) 0.899499 0.0442614
\(414\) −0.274051 −0.0134689
\(415\) 0 0
\(416\) 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 \(0.357585\pi\)
\(420\) 0 0
\(421\) 12.8787 0.627672 0.313836 0.949477i \(-0.398386\pi\)
0.313836 + 0.949477i \(0.398386\pi\)
\(422\) −1.80152 −0.0876965
\(423\) 3.66563 0.178229
\(424\) −1.35485 −0.0657973
\(425\) 0 0
\(426\) 4.49880 0.217967
\(427\) 4.68596 0.226770
\(428\) −25.4981 −1.23250
\(429\) 0 0
\(430\) 0 0
\(431\) 9.49845 0.457524 0.228762 0.973482i \(-0.426532\pi\)
0.228762 + 0.973482i \(0.426532\pi\)
\(432\) −20.4338 −0.983121
\(433\) 1.39628 0.0671010 0.0335505 0.999437i \(-0.489319\pi\)
0.0335505 + 0.999437i \(0.489319\pi\)
\(434\) 0.398826 0.0191443
\(435\) 0 0
\(436\) 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 \(0.468335\pi\)
\(440\) 0 0
\(441\) 3.02711 0.144148
\(442\) 0 0
\(443\) −9.54563 −0.453526 −0.226763 0.973950i \(-0.572814\pi\)
−0.226763 + 0.973950i \(0.572814\pi\)
\(444\) 18.6516 0.885164
\(445\) 0 0
\(446\) −2.25052 −0.106565
\(447\) −5.45732 −0.258122
\(448\) −2.28083 −0.107759
\(449\) 21.7171 1.02489 0.512446 0.858720i \(-0.328740\pi\)
0.512446 + 0.858720i \(0.328740\pi\)
\(450\) 0 0
\(451\) −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
\(460\) 0 0
\(461\) −1.78151 −0.0829730 −0.0414865 0.999139i \(-0.513209\pi\)
−0.0414865 + 0.999139i \(0.513209\pi\)
\(462\) −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
\(465\) 0 0
\(466\) 0.274051 0.0126952
\(467\) −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
\(470\) 0 0
\(471\) 26.2958 1.21165
\(472\) −2.35035 −0.108184
\(473\) −27.2045 −1.25086
\(474\) 1.58845 0.0729598
\(475\) 0 0
\(476\) −3.28398 −0.150521
\(477\) −0.685698 −0.0313960
\(478\) −2.18414 −0.0999003
\(479\) −35.1807 −1.60745 −0.803724 0.595002i \(-0.797151\pi\)
−0.803724 + 0.595002i \(0.797151\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −4.96204 −0.226015
\(483\) 1.50948 0.0686835
\(484\) −34.8510 −1.58414
\(485\) 0 0
\(486\) 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
\(490\) 0 0
\(491\) 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
\(495\) 0 0
\(496\) 20.2869 0.910907
\(497\) 4.25187 0.190722
\(498\) 1.49887 0.0671661
\(499\) 23.9421 1.07179 0.535897 0.844283i \(-0.319974\pi\)
0.535897 + 0.844283i \(0.319974\pi\)
\(500\) 0 0
\(501\) 10.0711 0.449943
\(502\) −1.48692 −0.0663645
\(503\) −42.1443 −1.87912 −0.939560 0.342385i \(-0.888765\pi\)
−0.939560 + 0.342385i \(0.888765\pi\)
\(504\) 0.126732 0.00564510
\(505\) 0 0
\(506\) −3.35056 −0.148951
\(507\) 0 0
\(508\) 6.32355 0.280562
\(509\) −33.5602 −1.48753 −0.743765 0.668441i \(-0.766962\pi\)
−0.743765 + 0.668441i \(0.766962\pi\)
\(510\) 0 0
\(511\) 3.21921 0.142409
\(512\) 15.9211 0.703621
\(513\) −12.4820 −0.551096
\(514\) 2.25263 0.0993593
\(515\) 0 0
\(516\) −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 \(-0.460512\pi\)
0.123738 + 0.992315i \(0.460512\pi\)
\(524\) −0.342849 −0.0149774
\(525\) 0 0
\(526\) −4.09870 −0.178712
\(527\) 27.6718 1.20540
\(528\) −31.9147 −1.38891
\(529\) −14.9391 −0.649525
\(530\) 0 0
\(531\) −1.18953 −0.0516211
\(532\) −1.47067 −0.0637616
\(533\) 0 0
\(534\) 1.13402 0.0490737
\(535\) 0 0
\(536\) 8.96989 0.387440
\(537\) 37.7925 1.63087
\(538\) 3.94511 0.170086
\(539\) 37.0097 1.59412
\(540\) 0 0
\(541\) −15.4750 −0.665321 −0.332660 0.943047i \(-0.607946\pi\)
−0.332660 + 0.943047i \(0.607946\pi\)
\(542\) 6.78781 0.291562
\(543\) −4.20196 −0.180323
\(544\) 12.9237 0.554101
\(545\) 0 0
\(546\) 0 0
\(547\) −25.1765 −1.07647 −0.538234 0.842795i \(-0.680908\pi\)
−0.538234 + 0.842795i \(0.680908\pi\)
\(548\) −35.0979 −1.49931
\(549\) −6.19687 −0.264476
\(550\) 0 0
\(551\) −6.58493 −0.280528
\(552\) −3.94419 −0.167876
\(553\) 1.50126 0.0638401
\(554\) −5.82269 −0.247382
\(555\) 0 0
\(556\) −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
\(560\) 0 0
\(561\) −43.5325 −1.83794
\(562\) 1.09353 0.0461276
\(563\) −23.7905 −1.00265 −0.501326 0.865259i \(-0.667154\pi\)
−0.501326 + 0.865259i \(0.667154\pi\)
\(564\) 26.0561 1.09716
\(565\) 0 0
\(566\) 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 0
\(571\) −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
\(575\) 0 0
\(576\) 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
\(580\) 0 0
\(581\) 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 0
\(591\) 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
\(595\) 0 0
\(596\) 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 \(0.523304\pi\)
\(600\) 0 0
\(601\) 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
\(605\) 0 0
\(606\) 4.37554 0.177744
\(607\) 3.29976 0.133933 0.0669665 0.997755i \(-0.478668\pi\)
0.0669665 + 0.997755i \(0.478668\pi\)
\(608\) 5.78766 0.234720
\(609\) −1.54366 −0.0625523
\(610\) 0 0
\(611\) 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
\(615\) 0 0
\(616\) 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 \(-0.368936\pi\)
0.400212 + 0.916422i \(0.368936\pi\)
\(620\) 0 0
\(621\) −15.6259 −0.627046
\(622\) −0.488829 −0.0196003
\(623\) 1.07177 0.0429397
\(624\) 0 0
\(625\) 0 0
\(626\) −1.58369 −0.0632971
\(627\) −19.4952 −0.778563
\(628\) −32.0726 −1.27984
\(629\) −30.2440 −1.20591
\(630\) 0 0
\(631\) 14.5958 0.581050 0.290525 0.956867i \(-0.406170\pi\)
0.290525 + 0.956867i \(0.406170\pi\)
\(632\) −3.92272 −0.156038
\(633\) −13.1222 −0.521562
\(634\) 0.0706609 0.00280630
\(635\) 0 0
\(636\) −4.87409 −0.193270
\(637\) 0 0
\(638\) 3.42644 0.135654
\(639\) −5.62281 −0.222435
\(640\) 0 0
\(641\) −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
\(645\) 0 0
\(646\) 2.52323 0.0992751
\(647\) 2.99168 0.117615 0.0588075 0.998269i \(-0.481270\pi\)
0.0588075 + 0.998269i \(0.481270\pi\)
\(648\) 6.50139 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 \(-0.426706\pi\)
0.228230 + 0.973607i \(0.426706\pi\)
\(654\) −3.94868 −0.154406
\(655\) 0 0
\(656\) 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 \(0.511227\pi\)
\(660\) 0 0
\(661\) −12.3406 −0.479992 −0.239996 0.970774i \(-0.577146\pi\)
−0.239996 + 0.970774i \(0.577146\pi\)
\(662\) 3.65383 0.142010
\(663\) 0 0
\(664\) −3.70152 −0.143647
\(665\) 0 0
\(666\) 0.576446 0.0223368
\(667\) −8.24348 −0.319189
\(668\) −12.2836 −0.475265
\(669\) −16.3927 −0.633779
\(670\) 0 0
\(671\) −75.7634 −2.92481
\(672\) 1.35676 0.0523381
\(673\) −9.26625 −0.357188 −0.178594 0.983923i \(-0.557155\pi\)
−0.178594 + 0.983923i \(0.557155\pi\)
\(674\) −5.32051 −0.204938
\(675\) 0 0
\(676\) 0 0
\(677\) −13.8984 −0.534158 −0.267079 0.963675i \(-0.586059\pi\)
−0.267079 + 0.963675i \(0.586059\pi\)
\(678\) 6.44863 0.247658
\(679\) 0.833244 0.0319770
\(680\) 0 0
\(681\) −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
\(685\) 0 0
\(686\) −1.01381 −0.0387073
\(687\) 2.12664 0.0811363
\(688\) 18.8025 0.716838
\(689\) 0 0
\(690\) 0 0
\(691\) −1.65291 −0.0628797 −0.0314399 0.999506i \(-0.510009\pi\)
−0.0314399 + 0.999506i \(0.510009\pi\)
\(692\) 31.1792 1.18526
\(693\) 0.784180 0.0297885
\(694\) 1.37823 0.0523168
\(695\) 0 0
\(696\) 4.03351 0.152890
\(697\) 18.9002 0.715897
\(698\) 1.55317 0.0587885
\(699\) 1.99618 0.0755025
\(700\) 0 0
\(701\) 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
\(705\) 0 0
\(706\) 4.78694 0.180159
\(707\) 4.13538 0.155527
\(708\) −8.45541 −0.317774
\(709\) 21.9417 0.824039 0.412020 0.911175i \(-0.364823\pi\)
0.412020 + 0.911175i \(0.364823\pi\)
\(710\) 0 0
\(711\) −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 \(0.242124\pi\)
\(720\) 0 0
\(721\) 4.99191 0.185909
\(722\) −3.04407 −0.113289
\(723\) −36.1434 −1.34419
\(724\) 5.12507 0.190472
\(725\) 0 0
\(726\) 6.27728 0.232972
\(727\) 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
\(730\) 0 0
\(731\) 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
\(735\) 0 0
\(736\) 7.24539 0.267069
\(737\) 55.5029 2.04448
\(738\) −0.360235 −0.0132604
\(739\) 38.2788 1.40811 0.704054 0.710146i \(-0.251371\pi\)
0.704054 + 0.710146i \(0.251371\pi\)
\(740\) 0 0
\(741\) 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
\(745\) 0 0
\(746\) 4.41134 0.161511
\(747\) −1.87336 −0.0685427
\(748\) 53.0960 1.94138
\(749\) 4.34057 0.158601
\(750\) 0 0
\(751\) 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
\(755\) 0 0
\(756\) 3.56890 0.129800
\(757\) 1.84848 0.0671840 0.0335920 0.999436i \(-0.489305\pi\)
0.0335920 + 0.999436i \(0.489305\pi\)
\(758\) −1.19989 −0.0435820
\(759\) −24.4055 −0.885862
\(760\) 0 0
\(761\) 26.2124 0.950199 0.475099 0.879932i \(-0.342412\pi\)
0.475099 + 0.879932i \(0.342412\pi\)
\(762\) −1.13898 −0.0412610
\(763\) −3.73195 −0.135106
\(764\) 3.93435 0.142340
\(765\) 0 0
\(766\) 1.24513 0.0449884
\(767\) 0 0
\(768\) 19.6459 0.708911
\(769\) −44.3495 −1.59928 −0.799641 0.600478i \(-0.794977\pi\)
−0.799641 + 0.600478i \(0.794977\pi\)
\(770\) 0 0
\(771\) 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
\(775\) 0 0
\(776\) −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
\(785\) 0 0
\(786\) 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 0
\(791\) 6.09467 0.216702
\(792\) −2.04903 −0.0728090
\(793\) 0 0
\(794\) −6.15865 −0.218562
\(795\) 0 0
\(796\) −5.98567 −0.212156
\(797\) 20.6952 0.733060 0.366530 0.930406i \(-0.380546\pi\)
0.366530 + 0.930406i \(0.380546\pi\)
\(798\) 0.264893 0.00937712
\(799\) −42.2507 −1.49472
\(800\) 0 0
\(801\) −1.41735 −0.0500795
\(802\) −4.94609 −0.174653
\(803\) −52.0487 −1.83676
\(804\) 32.2693 1.13805
\(805\) 0 0
\(806\) 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 \(0.409879\pi\)
\(810\) 0 0
\(811\) −23.8796 −0.838525 −0.419263 0.907865i \(-0.637711\pi\)
−0.419263 + 0.907865i \(0.637711\pi\)
\(812\) 1.88278 0.0660727
\(813\) 49.4423 1.73402
\(814\) 7.04768 0.247021
\(815\) 0 0
\(816\) 30.0877 1.05328
\(817\) 11.4856 0.401829
\(818\) 0.941956 0.0329347
\(819\) 0 0
\(820\) 0 0
\(821\) 15.9097 0.555251 0.277626 0.960689i \(-0.410453\pi\)
0.277626 + 0.960689i \(0.410453\pi\)
\(822\) 6.32174 0.220496
\(823\) 14.8115 0.516295 0.258147 0.966106i \(-0.416888\pi\)
0.258147 + 0.966106i \(0.416888\pi\)
\(824\) −13.0436 −0.454397
\(825\) 0 0
\(826\) 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 0
\(831\) −42.4124 −1.47127
\(832\) 0 0
\(833\) −34.8910 −1.20890
\(834\) 4.21363 0.145906
\(835\) 0 0
\(836\) 23.7780 0.822380
\(837\) −30.0726 −1.03946
\(838\) 3.89100 0.134412
\(839\) −14.7930 −0.510710 −0.255355 0.966847i \(-0.582192\pi\)
−0.255355 + 0.966847i \(0.582192\pi\)
\(840\) 0 0
\(841\) −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
\(850\) 0 0
\(851\) −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 0
\(856\) −11.3417 −0.387651
\(857\) 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 \(0.780291\pi\)
\(860\) 0 0
\(861\) 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
\(865\) 0 0
\(866\) 0.306745 0.0104236
\(867\) 13.8371 0.469931
\(868\) −3.54324 −0.120265
\(869\) −24.2726 −0.823392
\(870\) 0 0
\(871\) 0 0
\(872\) 9.75140 0.330224
\(873\) −1.10191 −0.0372940
\(874\) 1.41459 0.0478492
\(875\) 0 0
\(876\) −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
\(880\) 0 0
\(881\) −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 \(-0.657904\pi\)
−0.475973 + 0.879460i \(0.657904\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −2.09705 −0.0704517
\(887\) 27.9816 0.939531 0.469766 0.882791i \(-0.344338\pi\)
0.469766 + 0.882791i \(0.344338\pi\)
\(888\) 8.29633 0.278407
\(889\) −1.07647 −0.0361035
\(890\) 0 0
\(891\) 40.2286 1.34771
\(892\) 19.9940 0.669448
\(893\) −18.9212 −0.633172
\(894\) −1.19890 −0.0400973
\(895\) 0 0
\(896\) −2.19681 −0.0733903
\(897\) 0 0
\(898\) 4.77095 0.159209
\(899\) −15.8649 −0.529124
\(900\) 0 0
\(901\) 7.90348 0.263303
\(902\) −4.40426 −0.146646
\(903\) 2.69248 0.0896002
\(904\) −15.9251 −0.529661
\(905\) 0 0
\(906\) 2.79877 0.0929829
\(907\) −16.5520 −0.549600 −0.274800 0.961501i \(-0.588612\pi\)
−0.274800 + 0.961501i \(0.588612\pi\)
\(908\) 13.7477 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
\(915\) 0 0
\(916\) −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 \(0.412029\pi\)
\(920\) 0 0
\(921\) 6.88766 0.226956
\(922\) −0.391374 −0.0128892
\(923\) 0 0
\(924\) 5.57412 0.183375
\(925\) 0 0
\(926\) −1.49431 −0.0491060
\(927\) −6.60147 −0.216821
\(928\) −7.40948 −0.243228
\(929\) 33.8367 1.11015 0.555074 0.831801i \(-0.312690\pi\)
0.555074 + 0.831801i \(0.312690\pi\)
\(930\) 0 0
\(931\) −15.6253 −0.512098
\(932\) −2.43472 −0.0797517
\(933\) −3.56063 −0.116570
\(934\) −4.00652 −0.131097
\(935\) 0 0
\(936\) 0 0
\(937\) 30.4606 0.995104 0.497552 0.867434i \(-0.334232\pi\)
0.497552 + 0.867434i \(0.334232\pi\)
\(938\) −0.754153 −0.0246240
\(939\) −11.5356 −0.376450
\(940\) 0 0
\(941\) −38.2101 −1.24561 −0.622807 0.782375i \(-0.714008\pi\)
−0.622807 + 0.782375i \(0.714008\pi\)
\(942\) 5.77684 0.188220
\(943\) 10.5960 0.345052
\(944\) 10.0516 0.327153
\(945\) 0 0
\(946\) −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 0
\(951\) 0.514693 0.0166901
\(952\) −1.46074 −0.0473427
\(953\) −39.7500 −1.28763 −0.643814 0.765182i \(-0.722649\pi\)
−0.643814 + 0.765182i \(0.722649\pi\)
\(954\) −0.150639 −0.00487711
\(955\) 0 0
\(956\) 19.4043 0.627580
\(957\) 24.9581 0.806782
\(958\) −7.72874 −0.249704
\(959\) 5.97475 0.192935
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) −5.74011 −0.184972
\(964\) 44.0837 1.41984
\(965\) 0 0
\(966\) 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 0
\(971\) −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
\(980\) 0 0
\(981\) 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 0
\(986\) −3.23029 −0.102873
\(987\) −4.43555 −0.141185
\(988\) 0 0
\(989\) 14.3784 0.457208
\(990\) 0 0
\(991\) 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
\(995\) 0 0
\(996\) −13.3163 −0.421942
\(997\) 5.49137 0.173914 0.0869568 0.996212i \(-0.472286\pi\)
0.0869568 + 0.996212i \(0.472286\pi\)
\(998\) 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 4225.2.a.bi.1.3 4
5.4 even 2 845.2.a.m.1.2 4
13.2 odd 12 325.2.n.d.251.3 8
13.7 odd 12 325.2.n.d.101.3 8
13.12 even 2 4225.2.a.bl.1.2 4
15.14 odd 2 7605.2.a.cf.1.3 4
65.2 even 12 325.2.m.c.199.2 8
65.4 even 6 845.2.e.n.146.2 8
65.7 even 12 325.2.m.b.49.3 8
65.9 even 6 845.2.e.m.146.3 8
65.19 odd 12 845.2.m.g.361.3 8
65.24 odd 12 845.2.m.g.316.3 8
65.28 even 12 325.2.m.b.199.3 8
65.29 even 6 845.2.e.m.191.3 8
65.33 even 12 325.2.m.c.49.2 8
65.34 odd 4 845.2.c.g.506.4 8
65.44 odd 4 845.2.c.g.506.5 8
65.49 even 6 845.2.e.n.191.2 8
65.54 odd 12 65.2.m.a.56.2 yes 8
65.59 odd 12 65.2.m.a.36.2 8
65.64 even 2 845.2.a.l.1.3 4
195.59 even 12 585.2.bu.c.361.3 8
195.119 even 12 585.2.bu.c.316.3 8
195.194 odd 2 7605.2.a.cj.1.2 4
260.59 even 12 1040.2.da.b.881.2 8
260.119 even 12 1040.2.da.b.641.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.2 8 65.59 odd 12
65.2.m.a.56.2 yes 8 65.54 odd 12
325.2.m.b.49.3 8 65.7 even 12
325.2.m.b.199.3 8 65.28 even 12
325.2.m.c.49.2 8 65.33 even 12
325.2.m.c.199.2 8 65.2 even 12
325.2.n.d.101.3 8 13.7 odd 12
325.2.n.d.251.3 8 13.2 odd 12
585.2.bu.c.316.3 8 195.119 even 12
585.2.bu.c.361.3 8 195.59 even 12
845.2.a.l.1.3 4 65.64 even 2
845.2.a.m.1.2 4 5.4 even 2
845.2.c.g.506.4 8 65.34 odd 4
845.2.c.g.506.5 8 65.44 odd 4
845.2.e.m.146.3 8 65.9 even 6
845.2.e.m.191.3 8 65.29 even 6
845.2.e.n.146.2 8 65.4 even 6
845.2.e.n.191.2 8 65.49 even 6
845.2.m.g.316.3 8 65.24 odd 12
845.2.m.g.361.3 8 65.19 odd 12
1040.2.da.b.641.2 8 260.119 even 12
1040.2.da.b.881.2 8 260.59 even 12
4225.2.a.bi.1.3 4 1.1 even 1 trivial
4225.2.a.bl.1.2 4 13.12 even 2
7605.2.a.cf.1.3 4 15.14 odd 2
7605.2.a.cj.1.2 4 195.194 odd 2