Properties

Label 7605.2.a.cj.1.2
Level $7605$
Weight $2$
Character 7605.1
Self dual yes
Analytic conductor $60.726$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.7262307372\)
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 1.2
Root \(-0.219687\) of defining polynomial
Character \(\chi\) \(=\) 7605.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.219687 q^{2} -1.95174 q^{4} -1.00000 q^{5} +0.332247 q^{7} +0.868145 q^{8} +O(q^{10})\) \(q-0.219687 q^{2} -1.95174 q^{4} -1.00000 q^{5} +0.332247 q^{7} +0.868145 q^{8} +0.219687 q^{10} -5.37182 q^{11} -0.0729902 q^{14} +3.71276 q^{16} +5.06430 q^{17} -2.26795 q^{19} +1.95174 q^{20} +1.18012 q^{22} +2.83918 q^{23} +1.00000 q^{25} -0.648458 q^{28} +2.90348 q^{29} -5.46410 q^{31} -2.55193 q^{32} -1.11256 q^{34} -0.332247 q^{35} -5.97201 q^{37} +0.498239 q^{38} -0.868145 q^{40} +3.73205 q^{41} -5.06430 q^{43} +10.4844 q^{44} -0.623730 q^{46} +8.34285 q^{47} -6.88961 q^{49} -0.219687 q^{50} +1.56063 q^{53} +5.37182 q^{55} +0.288438 q^{56} -0.637855 q^{58} +2.70732 q^{59} +14.1039 q^{61} +1.20039 q^{62} -6.86488 q^{64} -10.3322 q^{67} -9.88418 q^{68} +0.0729902 q^{70} +12.7973 q^{71} +9.68922 q^{73} +1.31197 q^{74} +4.42644 q^{76} -1.78477 q^{77} +4.51851 q^{79} -3.71276 q^{80} -0.819883 q^{82} -4.26371 q^{83} -5.06430 q^{85} +1.11256 q^{86} -4.66351 q^{88} +3.22584 q^{89} -5.54133 q^{92} -1.83281 q^{94} +2.26795 q^{95} +2.50791 q^{97} +1.51356 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 2q^{4} - 4q^{5} - 10q^{7} + 6q^{8} + O(q^{10}) \) \( 4q + 2q^{2} + 2q^{4} - 4q^{5} - 10q^{7} + 6q^{8} - 2q^{10} - 2q^{14} + 2q^{16} + 2q^{17} - 16q^{19} - 2q^{20} + 12q^{22} + 10q^{23} + 4q^{25} - 8q^{28} - 8q^{29} - 8q^{31} + 4q^{32} + 4q^{34} + 10q^{35} + 2q^{37} - 8q^{38} - 6q^{40} + 8q^{41} - 2q^{43} + 12q^{44} - 16q^{46} + 8q^{47} + 12q^{49} + 2q^{50} + 12q^{53} - 12q^{56} - 22q^{58} + 12q^{59} + 28q^{61} - 4q^{62} + 4q^{64} - 30q^{67} - 14q^{68} + 2q^{70} + 4q^{71} + 8q^{73} + 10q^{74} - 20q^{76} - 18q^{77} - 8q^{79} - 2q^{80} + 4q^{82} - 12q^{83} - 2q^{85} - 4q^{86} - 18q^{88} - 12q^{89} - 22q^{92} - 32q^{94} + 16q^{95} - 2q^{97} - 24q^{98} + O(q^{100}) \)

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 \(-0.524748\pi\)
−0.0776710 + 0.996979i \(0.524748\pi\)
\(3\) 0 0
\(4\) −1.95174 −0.975869
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 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 0
\(10\) 0.219687 0.0694711
\(11\) −5.37182 −1.61966 −0.809832 0.586662i \(-0.800442\pi\)
−0.809832 + 0.586662i \(0.800442\pi\)
\(12\) 0 0
\(13\) 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 0
\(19\) −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 0
\(22\) 1.18012 0.251602
\(23\) 2.83918 0.592010 0.296005 0.955186i \(-0.404346\pi\)
0.296005 + 0.955186i \(0.404346\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) −0.648458 −0.122547
\(29\) 2.90348 0.539162 0.269581 0.962978i \(-0.413115\pi\)
0.269581 + 0.962978i \(0.413115\pi\)
\(30\) 0 0
\(31\) −5.46410 −0.981382 −0.490691 0.871334i \(-0.663256\pi\)
−0.490691 + 0.871334i \(0.663256\pi\)
\(32\) −2.55193 −0.451122
\(33\) 0 0
\(34\) −1.11256 −0.190802
\(35\) −0.332247 −0.0561599
\(36\) 0 0
\(37\) −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 \(-0.405871\pi\)
0.291424 + 0.956594i \(0.405871\pi\)
\(42\) 0 0
\(43\) −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 0
\(46\) −0.623730 −0.0919640
\(47\) 8.34285 1.21693 0.608465 0.793581i \(-0.291786\pi\)
0.608465 + 0.793581i \(0.291786\pi\)
\(48\) 0 0
\(49\) −6.88961 −0.984230
\(50\) −0.219687 −0.0310684
\(51\) 0 0
\(52\) 0 0
\(53\) 1.56063 0.214369 0.107184 0.994239i \(-0.465817\pi\)
0.107184 + 0.994239i \(0.465817\pi\)
\(54\) 0 0
\(55\) 5.37182 0.724336
\(56\) 0.288438 0.0385442
\(57\) 0 0
\(58\) −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 0
\(64\) −6.86488 −0.858111
\(65\) 0 0
\(66\) 0 0
\(67\) −10.3322 −1.26228 −0.631142 0.775667i \(-0.717414\pi\)
−0.631142 + 0.775667i \(0.717414\pi\)
\(68\) −9.88418 −1.19863
\(69\) 0 0
\(70\) 0.0729902 0.00872400
\(71\) 12.7973 1.51876 0.759382 0.650645i \(-0.225502\pi\)
0.759382 + 0.650645i \(0.225502\pi\)
\(72\) 0 0
\(73\) 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 \(0.418192\pi\)
\(80\) −3.71276 −0.415099
\(81\) 0 0
\(82\) −0.819883 −0.0905409
\(83\) −4.26371 −0.468003 −0.234001 0.972236i \(-0.575182\pi\)
−0.234001 + 0.972236i \(0.575182\pi\)
\(84\) 0 0
\(85\) −5.06430 −0.549300
\(86\) 1.11256 0.119970
\(87\) 0 0
\(88\) −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
\(93\) 0 0
\(94\) −1.83281 −0.189040
\(95\) 2.26795 0.232687
\(96\) 0 0
\(97\) 2.50791 0.254640 0.127320 0.991862i \(-0.459363\pi\)
0.127320 + 0.991862i \(0.459363\pi\)
\(98\) 1.51356 0.152892
\(99\) 0 0
\(100\) −1.95174 −0.195174
\(101\) −12.4467 −1.23849 −0.619247 0.785196i \(-0.712562\pi\)
−0.619247 + 0.785196i \(0.712562\pi\)
\(102\) 0 0
\(103\) −15.0247 −1.48043 −0.740215 0.672370i \(-0.765276\pi\)
−0.740215 + 0.672370i \(0.765276\pi\)
\(104\) 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 \(0.282445\pi\)
\(108\) 0 0
\(109\) 11.2325 1.07587 0.537937 0.842985i \(-0.319204\pi\)
0.537937 + 0.842985i \(0.319204\pi\)
\(110\) −1.18012 −0.112520
\(111\) 0 0
\(112\) 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 0
\(115\) −2.83918 −0.264755
\(116\) −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
\(123\) 0 0
\(124\) 10.6645 0.957700
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 3.23996 0.287500 0.143750 0.989614i \(-0.454084\pi\)
0.143750 + 0.989614i \(0.454084\pi\)
\(128\) 6.61199 0.584423
\(129\) 0 0
\(130\) 0 0
\(131\) −0.175664 −0.0153478 −0.00767390 0.999971i \(-0.502443\pi\)
−0.00767390 + 0.999971i \(0.502443\pi\)
\(132\) 0 0
\(133\) −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 \(-0.778843\pi\)
−0.768190 + 0.640221i \(0.778843\pi\)
\(138\) 0 0
\(139\) 11.9861 1.01665 0.508325 0.861165i \(-0.330265\pi\)
0.508325 + 0.861165i \(0.330265\pi\)
\(140\) 0.648458 0.0548047
\(141\) 0 0
\(142\) −2.81140 −0.235928
\(143\) 0 0
\(144\) 0 0
\(145\) −2.90348 −0.241121
\(146\) −2.12859 −0.176164
\(147\) 0 0
\(148\) 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 \(-0.605009\pi\)
−0.323945 + 0.946076i \(0.605009\pi\)
\(152\) −1.96891 −0.159700
\(153\) 0 0
\(154\) 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
\(159\) 0 0
\(160\) 2.55193 0.201748
\(161\) 0.943307 0.0743430
\(162\) 0 0
\(163\) −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 \(-0.578298\pi\)
−0.243509 + 0.969899i \(0.578298\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1.11256 0.0853294
\(171\) 0 0
\(172\) 9.88418 0.753662
\(173\) −15.9751 −1.21457 −0.607283 0.794486i \(-0.707740\pi\)
−0.607283 + 0.794486i \(0.707740\pi\)
\(174\) 0 0
\(175\) 0.332247 0.0251155
\(176\) −19.9442 −1.50335
\(177\) 0 0
\(178\) −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 0
\(181\) −2.62590 −0.195182 −0.0975909 0.995227i \(-0.531114\pi\)
−0.0975909 + 0.995227i \(0.531114\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2.46482 0.181709
\(185\) 5.97201 0.439071
\(186\) 0 0
\(187\) −27.2045 −1.98939
\(188\) −16.2831 −1.18756
\(189\) 0 0
\(190\) −0.498239 −0.0361460
\(191\) 2.01582 0.145860 0.0729298 0.997337i \(-0.476765\pi\)
0.0729298 + 0.997337i \(0.476765\pi\)
\(192\) 0 0
\(193\) −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 \(-0.507292\pi\)
−0.0229068 + 0.999738i \(0.507292\pi\)
\(198\) 0 0
\(199\) 3.06684 0.217403 0.108701 0.994074i \(-0.465331\pi\)
0.108701 + 0.994074i \(0.465331\pi\)
\(200\) 0.868145 0.0613871
\(201\) 0 0
\(202\) 2.73438 0.192390
\(203\) 0.964670 0.0677065
\(204\) 0 0
\(205\) −3.73205 −0.260658
\(206\) 3.30074 0.229973
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) −2.87005 −0.196193
\(215\) 5.06430 0.345382
\(216\) 0 0
\(217\) −1.81543 −0.123239
\(218\) −2.46762 −0.167129
\(219\) 0 0
\(220\) −10.4844 −0.706856
\(221\) 0 0
\(222\) 0 0
\(223\) −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 \(-0.424898\pi\)
0.233757 + 0.972295i \(0.424898\pi\)
\(228\) 0 0
\(229\) −1.32899 −0.0878219 −0.0439109 0.999035i \(-0.513982\pi\)
−0.0439109 + 0.999035i \(0.513982\pi\)
\(230\) 0.623730 0.0411275
\(231\) 0 0
\(232\) 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
\(235\) −8.34285 −0.544227
\(236\) −5.28398 −0.343958
\(237\) 0 0
\(238\) −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 \(-0.240694\pi\)
0.727475 + 0.686134i \(0.240694\pi\)
\(242\) −3.92282 −0.252168
\(243\) 0 0
\(244\) −27.5270 −1.76224
\(245\) 6.88961 0.440161
\(246\) 0 0
\(247\) 0 0
\(248\) −4.74363 −0.301221
\(249\) 0 0
\(250\) 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 0
\(253\) −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 \(0.396382\pi\)
\(258\) 0 0
\(259\) −1.98418 −0.123291
\(260\) 0 0
\(261\) 0 0
\(262\) 0.0385910 0.00238416
\(263\) −18.6570 −1.15044 −0.575220 0.817999i \(-0.695083\pi\)
−0.575220 + 0.817999i \(0.695083\pi\)
\(264\) 0 0
\(265\) −1.56063 −0.0958685
\(266\) 0.165538 0.0101498
\(267\) 0 0
\(268\) 20.1658 1.23182
\(269\) −17.9579 −1.09491 −0.547456 0.836835i \(-0.684404\pi\)
−0.547456 + 0.836835i \(0.684404\pi\)
\(270\) 0 0
\(271\) −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
\(276\) 0 0
\(277\) 26.5045 1.59250 0.796250 0.604967i \(-0.206814\pi\)
0.796250 + 0.604967i \(0.206814\pi\)
\(278\) −2.63320 −0.157929
\(279\) 0 0
\(280\) −0.288438 −0.0172375
\(281\) 4.97766 0.296942 0.148471 0.988917i \(-0.452565\pi\)
0.148471 + 0.988917i \(0.452565\pi\)
\(282\) 0 0
\(283\) −12.5863 −0.748180 −0.374090 0.927392i \(-0.622045\pi\)
−0.374090 + 0.927392i \(0.622045\pi\)
\(284\) −24.9770 −1.48211
\(285\) 0 0
\(286\) 0 0
\(287\) 1.23996 0.0731926
\(288\) 0 0
\(289\) 8.64711 0.508653
\(290\) 0.637855 0.0374562
\(291\) 0 0
\(292\) −18.9108 −1.10667
\(293\) 16.9176 0.988337 0.494168 0.869366i \(-0.335473\pi\)
0.494168 + 0.869366i \(0.335473\pi\)
\(294\) 0 0
\(295\) −2.70732 −0.157626
\(296\) −5.18457 −0.301347
\(297\) 0 0
\(298\) 0.749222 0.0434012
\(299\) 0 0
\(300\) 0 0
\(301\) −1.68260 −0.0969832
\(302\) 1.74902 0.100645
\(303\) 0 0
\(304\) −8.42034 −0.482940
\(305\) −14.1039 −0.807585
\(306\) 0 0
\(307\) 4.30426 0.245657 0.122828 0.992428i \(-0.460803\pi\)
0.122828 + 0.992428i \(0.460803\pi\)
\(308\) 3.48340 0.198485
\(309\) 0 0
\(310\) −1.20039 −0.0681776
\(311\) 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 \(0.434692\pi\)
\(314\) 3.61008 0.203729
\(315\) 0 0
\(316\) −8.81895 −0.496105
\(317\) −0.321644 −0.0180653 −0.00903266 0.999959i \(-0.502875\pi\)
−0.00903266 + 0.999959i \(0.502875\pi\)
\(318\) 0 0
\(319\) −15.5969 −0.873261
\(320\) 6.86488 0.383759
\(321\) 0 0
\(322\) −0.207232 −0.0115486
\(323\) −11.4856 −0.639074
\(324\) 0 0
\(325\) 0 0
\(326\) 3.91201 0.216666
\(327\) 0 0
\(328\) 3.23996 0.178897
\(329\) 2.77188 0.152819
\(330\) 0 0
\(331\) −16.6320 −0.914178 −0.457089 0.889421i \(-0.651108\pi\)
−0.457089 + 0.889421i \(0.651108\pi\)
\(332\) 8.32164 0.456710
\(333\) 0 0
\(334\) 1.38263 0.0756543
\(335\) 10.3322 0.564511
\(336\) 0 0
\(337\) 24.2186 1.31927 0.659636 0.751586i \(-0.270711\pi\)
0.659636 + 0.751586i \(0.270711\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 9.88418 0.536045
\(341\) 29.3521 1.58951
\(342\) 0 0
\(343\) −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 \(0.446143\pi\)
\(348\) 0 0
\(349\) −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 \(-0.696900\pi\)
−0.579878 + 0.814704i \(0.696900\pi\)
\(354\) 0 0
\(355\) −12.7973 −0.679212
\(356\) −6.29598 −0.333687
\(357\) 0 0
\(358\) 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
\(363\) 0 0
\(364\) 0 0
\(365\) −9.68922 −0.507157
\(366\) 0 0
\(367\) 6.39133 0.333625 0.166812 0.985989i \(-0.446653\pi\)
0.166812 + 0.985989i \(0.446653\pi\)
\(368\) 10.5412 0.549497
\(369\) 0 0
\(370\) −1.31197 −0.0682062
\(371\) 0.518513 0.0269198
\(372\) 0 0
\(373\) −20.0801 −1.03971 −0.519855 0.854255i \(-0.674014\pi\)
−0.519855 + 0.854255i \(0.674014\pi\)
\(374\) 5.97647 0.309036
\(375\) 0 0
\(376\) 7.24280 0.373519
\(377\) 0 0
\(378\) 0 0
\(379\) 5.46182 0.280555 0.140277 0.990112i \(-0.455201\pi\)
0.140277 + 0.990112i \(0.455201\pi\)
\(380\) −4.42644 −0.227072
\(381\) 0 0
\(382\) −0.442849 −0.0226581
\(383\) −5.66775 −0.289609 −0.144804 0.989460i \(-0.546255\pi\)
−0.144804 + 0.989460i \(0.546255\pi\)
\(384\) 0 0
\(385\) 1.78477 0.0909602
\(386\) 5.01349 0.255180
\(387\) 0 0
\(388\) −4.89478 −0.248495
\(389\) −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
\(392\) −5.98118 −0.302095
\(393\) 0 0
\(394\) 0.141264 0.00711679
\(395\) −4.51851 −0.227351
\(396\) 0 0
\(397\) −28.0338 −1.40697 −0.703487 0.710708i \(-0.748375\pi\)
−0.703487 + 0.710708i \(0.748375\pi\)
\(398\) −0.673745 −0.0337718
\(399\) 0 0
\(400\) 3.71276 0.185638
\(401\) −22.5143 −1.12431 −0.562155 0.827032i \(-0.690027\pi\)
−0.562155 + 0.827032i \(0.690027\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 24.2927 1.20861
\(405\) 0 0
\(406\) −0.211925 −0.0105177
\(407\) 32.0805 1.59017
\(408\) 0 0
\(409\) −4.28772 −0.212014 −0.106007 0.994365i \(-0.533807\pi\)
−0.106007 + 0.994365i \(0.533807\pi\)
\(410\) 0.819883 0.0404911
\(411\) 0 0
\(412\) 29.3243 1.44471
\(413\) 0.899499 0.0442614
\(414\) 0 0
\(415\) 4.26371 0.209297
\(416\) 0 0
\(417\) 0 0
\(418\) −2.67645 −0.130909
\(419\) −17.7116 −0.865266 −0.432633 0.901570i \(-0.642415\pi\)
−0.432633 + 0.901570i \(0.642415\pi\)
\(420\) 0 0
\(421\) −12.8787 −0.627672 −0.313836 0.949477i \(-0.601614\pi\)
−0.313836 + 0.949477i \(0.601614\pi\)
\(422\) 1.80152 0.0876965
\(423\) 0 0
\(424\) 1.35485 0.0657973
\(425\) 5.06430 0.245655
\(426\) 0 0
\(427\) 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 \(-0.426532\pi\)
0.228762 + 0.973482i \(0.426532\pi\)
\(432\) 0 0
\(433\) −1.39628 −0.0671010 −0.0335505 0.999437i \(-0.510681\pi\)
−0.0335505 + 0.999437i \(0.510681\pi\)
\(434\) 0.398826 0.0191443
\(435\) 0 0
\(436\) −21.9228 −1.04991
\(437\) −6.43911 −0.308024
\(438\) 0 0
\(439\) 4.16180 0.198632 0.0993159 0.995056i \(-0.468335\pi\)
0.0993159 + 0.995056i \(0.468335\pi\)
\(440\) 4.66351 0.222324
\(441\) 0 0
\(442\) 0 0
\(443\) −9.54563 −0.453526 −0.226763 0.973950i \(-0.572814\pi\)
−0.226763 + 0.973950i \(0.572814\pi\)
\(444\) 0 0
\(445\) −3.22584 −0.152919
\(446\) 2.25052 0.106565
\(447\) 0 0
\(448\) −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
\(453\) 0 0
\(454\) −1.54743 −0.0726246
\(455\) 0 0
\(456\) 0 0
\(457\) −4.72259 −0.220914 −0.110457 0.993881i \(-0.535231\pi\)
−0.110457 + 0.993881i \(0.535231\pi\)
\(458\) 0.291961 0.0136424
\(459\) 0 0
\(460\) 5.54133 0.258366
\(461\) −1.78151 −0.0829730 −0.0414865 0.999139i \(-0.513209\pi\)
−0.0414865 + 0.999139i \(0.513209\pi\)
\(462\) 0 0
\(463\) −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
\(470\) 1.83281 0.0845414
\(471\) 0 0
\(472\) 2.35035 0.108184
\(473\) 27.2045 1.25086
\(474\) 0 0
\(475\) −2.26795 −0.104061
\(476\) −3.28398 −0.150521
\(477\) 0 0
\(478\) 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
\(483\) 0 0
\(484\) −34.8510 −1.58414
\(485\) −2.50791 −0.113878
\(486\) 0 0
\(487\) 10.3040 0.466919 0.233459 0.972367i \(-0.424995\pi\)
0.233459 + 0.972367i \(0.424995\pi\)
\(488\) 12.2442 0.554269
\(489\) 0 0
\(490\) −1.51356 −0.0683756
\(491\) −9.33198 −0.421147 −0.210573 0.977578i \(-0.567533\pi\)
−0.210573 + 0.977578i \(0.567533\pi\)
\(492\) 0 0
\(493\) 14.7041 0.662238
\(494\) 0 0
\(495\) 0 0
\(496\) −20.2869 −0.910907
\(497\) 4.25187 0.190722
\(498\) 0 0
\(499\) −23.9421 −1.07179 −0.535897 0.844283i \(-0.680026\pi\)
−0.535897 + 0.844283i \(0.680026\pi\)
\(500\) 1.95174 0.0872844
\(501\) 0 0
\(502\) −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 0
\(505\) 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 \(-0.766962\pi\)
−0.743765 + 0.668441i \(0.766962\pi\)
\(510\) 0 0
\(511\) 3.21921 0.142409
\(512\) −15.9211 −0.703621
\(513\) 0 0
\(514\) −2.25263 −0.0993593
\(515\) 15.0247 0.662069
\(516\) 0 0
\(517\) −44.8162 −1.97102
\(518\) 0.435898 0.0191523
\(519\) 0 0
\(520\) 0 0
\(521\) −12.4649 −0.546098 −0.273049 0.962000i \(-0.588032\pi\)
−0.273049 + 0.962000i \(0.588032\pi\)
\(522\) 0 0
\(523\) −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 0
\(526\) 4.09870 0.178712
\(527\) −27.6718 −1.20540
\(528\) 0 0
\(529\) −14.9391 −0.649525
\(530\) 0.342849 0.0148924
\(531\) 0 0
\(532\) 1.47067 0.0637616
\(533\) 0 0
\(534\) 0 0
\(535\) −13.0643 −0.564819
\(536\) −8.96989 −0.387440
\(537\) 0 0
\(538\) 3.94511 0.170086
\(539\) 37.0097 1.59412
\(540\) 0 0
\(541\) 15.4750 0.665321 0.332660 0.943047i \(-0.392054\pi\)
0.332660 + 0.943047i \(0.392054\pi\)
\(542\) 6.78781 0.291562
\(543\) 0 0
\(544\) −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
\(549\) 0 0
\(550\) 1.18012 0.0503204
\(551\) −6.58493 −0.280528
\(552\) 0 0
\(553\) 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 \(-0.145605\pi\)
0.897190 + 0.441645i \(0.145605\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1.23355 −0.0521270
\(561\) 0 0
\(562\) −1.09353 −0.0461276
\(563\) −23.7905 −1.00265 −0.501326 0.865259i \(-0.667154\pi\)
−0.501326 + 0.865259i \(0.667154\pi\)
\(564\) 0 0
\(565\) −18.3438 −0.771731
\(566\) 2.76505 0.116224
\(567\) 0 0
\(568\) 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 0
\(571\) −16.7159 −0.699539 −0.349769 0.936836i \(-0.613740\pi\)
−0.349769 + 0.936836i \(0.613740\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −0.272403 −0.0113699
\(575\) 2.83918 0.118402
\(576\) 0 0
\(577\) 20.6768 0.860786 0.430393 0.902642i \(-0.358375\pi\)
0.430393 + 0.902642i \(0.358375\pi\)
\(578\) −1.89966 −0.0790153
\(579\) 0 0
\(580\) 5.66682 0.235302
\(581\) −1.41660 −0.0587706
\(582\) 0 0
\(583\) −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 \(-0.358797\pi\)
0.429196 + 0.903212i \(0.358797\pi\)
\(588\) 0 0
\(589\) 12.3923 0.510616
\(590\) 0.594763 0.0244860
\(591\) 0 0
\(592\) −22.1726 −0.911289
\(593\) −21.8475 −0.897169 −0.448585 0.893740i \(-0.648072\pi\)
−0.448585 + 0.893740i \(0.648072\pi\)
\(594\) 0 0
\(595\) −1.68260 −0.0689797
\(596\) 6.65622 0.272649
\(597\) 0 0
\(598\) 0 0
\(599\) 3.58040 0.146291 0.0731456 0.997321i \(-0.476696\pi\)
0.0731456 + 0.997321i \(0.476696\pi\)
\(600\) 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
\(603\) 0 0
\(604\) 15.5386 0.632256
\(605\) −17.8564 −0.725966
\(606\) 0 0
\(607\) −3.29976 −0.133933 −0.0669665 0.997755i \(-0.521332\pi\)
−0.0669665 + 0.997755i \(0.521332\pi\)
\(608\) 5.78766 0.234720
\(609\) 0 0
\(610\) 3.09843 0.125452
\(611\) 0 0
\(612\) 0 0
\(613\) −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 \(-0.872012\pi\)
−0.920246 + 0.391339i \(0.872012\pi\)
\(618\) 0 0
\(619\) −19.9143 −0.800425 −0.400212 0.916422i \(-0.631064\pi\)
−0.400212 + 0.916422i \(0.631064\pi\)
\(620\) −10.6645 −0.428296
\(621\) 0 0
\(622\) −0.488829 −0.0196003
\(623\) 1.07177 0.0429397
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −1.58369 −0.0632971
\(627\) 0 0
\(628\) 32.0726 1.27984
\(629\) −30.2440 −1.20591
\(630\) 0 0
\(631\) −14.5958 −0.581050 −0.290525 0.956867i \(-0.593830\pi\)
−0.290525 + 0.956867i \(0.593830\pi\)
\(632\) 3.92272 0.156038
\(633\) 0 0
\(634\) 0.0706609 0.00280630
\(635\) −3.23996 −0.128574
\(636\) 0 0
\(637\) 0 0
\(638\) 3.42644 0.135654
\(639\) 0 0
\(640\) −6.61199 −0.261362
\(641\) 14.1637 0.559431 0.279716 0.960083i \(-0.409760\pi\)
0.279716 + 0.960083i \(0.409760\pi\)
\(642\) 0 0
\(643\) −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 \(0.481270\pi\)
\(648\) 0 0
\(649\) −14.5432 −0.570872
\(650\) 0 0
\(651\) 0 0
\(652\) 34.7550 1.36111
\(653\) 11.6643 0.456461 0.228230 0.973607i \(-0.426706\pi\)
0.228230 + 0.973607i \(0.426706\pi\)
\(654\) 0 0
\(655\) 0.175664 0.00686374
\(656\) 13.8562 0.540993
\(657\) 0 0
\(658\) −0.608946 −0.0237392
\(659\) 1.81047 0.0705260 0.0352630 0.999378i \(-0.488773\pi\)
0.0352630 + 0.999378i \(0.488773\pi\)
\(660\) 0 0
\(661\) 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 0
\(667\) 8.24348 0.319189
\(668\) 12.2836 0.475265
\(669\) 0 0
\(670\) −2.26986 −0.0876923
\(671\) −75.7634 −2.92481
\(672\) 0 0
\(673\) 9.26625 0.357188 0.178594 0.983923i \(-0.442845\pi\)
0.178594 + 0.983923i \(0.442845\pi\)
\(674\) −5.32051 −0.204938
\(675\) 0 0
\(676\) 0 0
\(677\) −13.8984 −0.534158 −0.267079 0.963675i \(-0.586059\pi\)
−0.267079 + 0.963675i \(0.586059\pi\)
\(678\) 0 0
\(679\) 0.833244 0.0319770
\(680\) −4.39654 −0.168600
\(681\) 0 0
\(682\) −6.44828 −0.246917
\(683\) −37.7512 −1.44451 −0.722255 0.691626i \(-0.756895\pi\)
−0.722255 + 0.691626i \(0.756895\pi\)
\(684\) 0 0
\(685\) 17.9829 0.687090
\(686\) 1.01381 0.0387073
\(687\) 0 0
\(688\) −18.8025 −0.716838
\(689\) 0 0
\(690\) 0 0
\(691\) 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 0
\(694\) −1.37823 −0.0523168
\(695\) −11.9861 −0.454660
\(696\) 0 0
\(697\) 18.9002 0.715897
\(698\) 1.55317 0.0587885
\(699\) 0 0
\(700\) −0.648458 −0.0245094
\(701\) −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
\(704\) 36.8769 1.38985
\(705\) 0 0
\(706\) 4.78694 0.180159
\(707\) −4.13538 −0.155527
\(708\) 0 0
\(709\) −21.9417 −0.824039 −0.412020 0.911175i \(-0.635177\pi\)
−0.412020 + 0.911175i \(0.635177\pi\)
\(710\) 2.81140 0.105510
\(711\) 0 0
\(712\) 2.80049 0.104953
\(713\) −15.5136 −0.580987
\(714\) 0 0
\(715\) 0 0
\(716\) 46.0950 1.72265
\(717\) 0 0
\(718\) −5.26671 −0.196552
\(719\) −38.8475 −1.44877 −0.724384 0.689397i \(-0.757876\pi\)
−0.724384 + 0.689397i \(0.757876\pi\)
\(720\) 0 0
\(721\) −4.99191 −0.185909
\(722\) 3.04407 0.113289
\(723\) 0 0
\(724\) 5.12507 0.190472
\(725\) 2.90348 0.107832
\(726\) 0 0
\(727\) −30.6598 −1.13711 −0.568555 0.822645i \(-0.692497\pi\)
−0.568555 + 0.822645i \(0.692497\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.12859 0.0787828
\(731\) −25.6471 −0.948593
\(732\) 0 0
\(733\) −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 0
\(739\) −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 \(-0.762290\pi\)
−0.733874 + 0.679286i \(0.762290\pi\)
\(744\) 0 0
\(745\) 3.41041 0.124948
\(746\) 4.41134 0.161511
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) 0 0
\(755\) 7.96141 0.289745
\(756\) 0 0
\(757\) −1.84848 −0.0671840 −0.0335920 0.999436i \(-0.510695\pi\)
−0.0335920 + 0.999436i \(0.510695\pi\)
\(758\) −1.19989 −0.0435820
\(759\) 0 0
\(760\) 1.96891 0.0714198
\(761\) 26.2124 0.950199 0.475099 0.879932i \(-0.342412\pi\)
0.475099 + 0.879932i \(0.342412\pi\)
\(762\) 0 0
\(763\) 3.73195 0.135106
\(764\) −3.93435 −0.142340
\(765\) 0 0
\(766\) 1.24513 0.0449884
\(767\) 0 0
\(768\) 0 0
\(769\) 44.3495 1.59928 0.799641 0.600478i \(-0.205023\pi\)
0.799641 + 0.600478i \(0.205023\pi\)
\(770\) −0.392090 −0.0141299
\(771\) 0 0
\(772\) 44.5408 1.60306
\(773\) −23.2638 −0.836742 −0.418371 0.908276i \(-0.637399\pi\)
−0.418371 + 0.908276i \(0.637399\pi\)
\(774\) 0 0
\(775\) −5.46410 −0.196276
\(776\) 2.17723 0.0781580
\(777\) 0 0
\(778\) 2.33251 0.0836245
\(779\) −8.46410 −0.303258
\(780\) 0 0
\(781\) −68.7449 −2.45989
\(782\) −3.15875 −0.112957
\(783\) 0 0
\(784\) −25.5794 −0.913552
\(785\) 16.4329 0.586514
\(786\) 0 0
\(787\) 47.9133 1.70793 0.853963 0.520334i \(-0.174192\pi\)
0.853963 + 0.520334i \(0.174192\pi\)
\(788\) 1.25502 0.0447081
\(789\) 0 0
\(790\) 0.992658 0.0353172
\(791\) 6.09467 0.216702
\(792\) 0 0
\(793\) 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 0
\(799\) 42.2507 1.49472
\(800\) −2.55193 −0.0902245
\(801\) 0 0
\(802\) 4.94609 0.174653
\(803\) −52.0487 −1.83676
\(804\) 0 0
\(805\) −0.943307 −0.0332472
\(806\) 0 0
\(807\) 0 0
\(808\) −10.8056 −0.380138
\(809\) −15.8915 −0.558714 −0.279357 0.960187i \(-0.590121\pi\)
−0.279357 + 0.960187i \(0.590121\pi\)
\(810\) 0 0
\(811\) 23.8796 0.838525 0.419263 0.907865i \(-0.362289\pi\)
0.419263 + 0.907865i \(0.362289\pi\)
\(812\) −1.88278 −0.0660727
\(813\) 0 0
\(814\) −7.04768 −0.247021
\(815\) 17.8072 0.623759
\(816\) 0 0
\(817\) 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 \(-0.410453\pi\)
0.277626 + 0.960689i \(0.410453\pi\)
\(822\) 0 0
\(823\) −14.8115 −0.516295 −0.258147 0.966106i \(-0.583112\pi\)
−0.258147 + 0.966106i \(0.583112\pi\)
\(824\) −13.0436 −0.454397
\(825\) 0 0
\(826\) −0.197608 −0.00687566
\(827\) 33.9498 1.18055 0.590275 0.807202i \(-0.299019\pi\)
0.590275 + 0.807202i \(0.299019\pi\)
\(828\) 0 0
\(829\) 23.3146 0.809749 0.404875 0.914372i \(-0.367315\pi\)
0.404875 + 0.914372i \(0.367315\pi\)
\(830\) −0.936681 −0.0325127
\(831\) 0 0
\(832\) 0 0
\(833\) −34.8910 −1.20890
\(834\) 0 0
\(835\) 6.29366 0.217801
\(836\) −23.7780 −0.822380
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) 16.0050 0.550915
\(845\) 0 0
\(846\) 0 0
\(847\) 5.93273 0.203851
\(848\) 5.79422 0.198974
\(849\) 0 0
\(850\) −1.11256 −0.0381605
\(851\) −16.9556 −0.581231
\(852\) 0 0
\(853\) 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 \(0.780291\pi\)
\(860\) −9.88418 −0.337048
\(861\) 0 0
\(862\) −2.08669 −0.0710728
\(863\) 4.75058 0.161712 0.0808559 0.996726i \(-0.474235\pi\)
0.0808559 + 0.996726i \(0.474235\pi\)
\(864\) 0 0
\(865\) 15.9751 0.543170
\(866\) 0.306745 0.0104236
\(867\) 0 0
\(868\) 3.54324 0.120265
\(869\) −24.2726 −0.823392
\(870\) 0 0
\(871\) 0 0
\(872\) 9.75140 0.330224
\(873\) 0 0
\(874\) 1.41459 0.0478492
\(875\) −0.332247 −0.0112320
\(876\) 0 0
\(877\) −2.25506 −0.0761481 −0.0380741 0.999275i \(-0.512122\pi\)
−0.0380741 + 0.999275i \(0.512122\pi\)
\(878\) −0.914293 −0.0308559
\(879\) 0 0
\(880\) 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 0
\(883\) 28.2874 0.951947 0.475973 0.879460i \(-0.342096\pi\)
0.475973 + 0.879460i \(0.342096\pi\)
\(884\) 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 \(0.344338\pi\)
\(888\) 0 0
\(889\) 1.07647 0.0361035
\(890\) 0.708674 0.0237548
\(891\) 0 0
\(892\) 19.9940 0.669448
\(893\) −18.9212 −0.633172
\(894\) 0 0
\(895\) 23.6174 0.789443
\(896\) 2.19681 0.0733903
\(897\) 0 0
\(898\) −4.77095 −0.159209
\(899\) −15.8649 −0.529124
\(900\) 0 0
\(901\) 7.90348 0.263303
\(902\) 4.40426 0.146646
\(903\) 0 0
\(904\) 15.9251 0.529661
\(905\) 2.62590 0.0872879
\(906\) 0 0
\(907\) 16.5520 0.549600 0.274800 0.961501i \(-0.411388\pi\)
0.274800 + 0.961501i \(0.411388\pi\)
\(908\) −13.7477 −0.456232
\(909\) 0 0
\(910\) 0 0
\(911\) −7.04863 −0.233532 −0.116766 0.993159i \(-0.537253\pi\)
−0.116766 + 0.993159i \(0.537253\pi\)
\(912\) 0 0
\(913\) 22.9039 0.758007
\(914\) 1.03749 0.0343172
\(915\) 0 0
\(916\) 2.59383 0.0857026
\(917\) −0.0583636 −0.00192734
\(918\) 0 0
\(919\) 16.5438 0.545728 0.272864 0.962053i \(-0.412029\pi\)
0.272864 + 0.962053i \(0.412029\pi\)
\(920\) −2.46482 −0.0812626
\(921\) 0 0
\(922\) 0.391374 0.0128892
\(923\) 0 0
\(924\) 0 0
\(925\) −5.97201 −0.196359
\(926\) 1.49431 0.0491060
\(927\) 0 0
\(928\) −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
\(933\) 0 0
\(934\) 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
\(939\) 0 0
\(940\) 16.2831 0.531095
\(941\) −38.2101 −1.24561 −0.622807 0.782375i \(-0.714008\pi\)
−0.622807 + 0.782375i \(0.714008\pi\)
\(942\) 0 0
\(943\) 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 \(-0.175287\pi\)
0.852169 + 0.523266i \(0.175287\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.498239 0.0161650
\(951\) 0 0
\(952\) 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 0
\(955\) −2.01582 −0.0652304
\(956\) 19.4043 0.627580
\(957\) 0 0
\(958\) 7.72874 0.249704
\(959\) −5.97475 −0.192935
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) 0 0
\(964\) −44.0837 −1.41984
\(965\) 22.8211 0.734636
\(966\) 0 0
\(967\) −25.7857 −0.829214 −0.414607 0.910001i \(-0.636081\pi\)
−0.414607 + 0.910001i \(0.636081\pi\)
\(968\) 15.5019 0.498251
\(969\) 0 0
\(970\) 0.550955 0.0176901
\(971\) 55.5252 1.78189 0.890945 0.454111i \(-0.150043\pi\)
0.890945 + 0.454111i \(0.150043\pi\)
\(972\) 0 0
\(973\) 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 \(-0.724441\pi\)
−0.648112 + 0.761545i \(0.724441\pi\)
\(978\) 0 0
\(979\) −17.3286 −0.553824
\(980\) −13.4467 −0.429540
\(981\) 0 0
\(982\) 2.05011 0.0654218
\(983\) 34.8059 1.11014 0.555068 0.831805i \(-0.312692\pi\)
0.555068 + 0.831805i \(0.312692\pi\)
\(984\) 0 0
\(985\) 0.643026 0.0204885
\(986\) −3.23029 −0.102873
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) −0.934079 −0.0296272
\(995\) −3.06684 −0.0972254
\(996\) 0 0
\(997\) −5.49137 −0.173914 −0.0869568 0.996212i \(-0.527714\pi\)
−0.0869568 + 0.996212i \(0.527714\pi\)
\(998\) 5.25976 0.166495
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7605.2.a.cj.1.2 4
3.2 odd 2 845.2.a.l.1.3 4
13.6 odd 12 585.2.bu.c.361.3 8
13.11 odd 12 585.2.bu.c.316.3 8
13.12 even 2 7605.2.a.cf.1.3 4
15.14 odd 2 4225.2.a.bl.1.2 4
39.2 even 12 845.2.m.g.316.3 8
39.5 even 4 845.2.c.g.506.4 8
39.8 even 4 845.2.c.g.506.5 8
39.11 even 12 65.2.m.a.56.2 yes 8
39.17 odd 6 845.2.e.m.146.3 8
39.20 even 12 845.2.m.g.361.3 8
39.23 odd 6 845.2.e.m.191.3 8
39.29 odd 6 845.2.e.n.191.2 8
39.32 even 12 65.2.m.a.36.2 8
39.35 odd 6 845.2.e.n.146.2 8
39.38 odd 2 845.2.a.m.1.2 4
156.11 odd 12 1040.2.da.b.641.2 8
156.71 odd 12 1040.2.da.b.881.2 8
195.32 odd 12 325.2.m.c.49.2 8
195.89 even 12 325.2.n.d.251.3 8
195.128 odd 12 325.2.m.c.199.2 8
195.149 even 12 325.2.n.d.101.3 8
195.167 odd 12 325.2.m.b.199.3 8
195.188 odd 12 325.2.m.b.49.3 8
195.194 odd 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 39.32 even 12
65.2.m.a.56.2 yes 8 39.11 even 12
325.2.m.b.49.3 8 195.188 odd 12
325.2.m.b.199.3 8 195.167 odd 12
325.2.m.c.49.2 8 195.32 odd 12
325.2.m.c.199.2 8 195.128 odd 12
325.2.n.d.101.3 8 195.149 even 12
325.2.n.d.251.3 8 195.89 even 12
585.2.bu.c.316.3 8 13.11 odd 12
585.2.bu.c.361.3 8 13.6 odd 12
845.2.a.l.1.3 4 3.2 odd 2
845.2.a.m.1.2 4 39.38 odd 2
845.2.c.g.506.4 8 39.5 even 4
845.2.c.g.506.5 8 39.8 even 4
845.2.e.m.146.3 8 39.17 odd 6
845.2.e.m.191.3 8 39.23 odd 6
845.2.e.n.146.2 8 39.35 odd 6
845.2.e.n.191.2 8 39.29 odd 6
845.2.m.g.316.3 8 39.2 even 12
845.2.m.g.361.3 8 39.20 even 12
1040.2.da.b.641.2 8 156.11 odd 12
1040.2.da.b.881.2 8 156.71 odd 12
4225.2.a.bi.1.3 4 195.194 odd 2
4225.2.a.bl.1.2 4 15.14 odd 2
7605.2.a.cf.1.3 4 13.12 even 2
7605.2.a.cj.1.2 4 1.1 even 1 trivial