Properties

Label 4225.2.a.bq.1.4
Level $4225$
Weight $2$
Character 4225.1
Self dual yes
Analytic conductor $33.737$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Coefficient field: 6.6.199374400.1
Defining polynomial: \(x^{6} - 8 x^{4} + 10 x^{2} - 1\)
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 1.4
Root \(0.330837\) of defining polynomial
Character \(\chi\) \(=\) 4225.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.330837 q^{2} +2.69180 q^{3} -1.89055 q^{4} +0.890547 q^{6} +3.35348 q^{7} -1.28714 q^{8} +4.24581 q^{9} +O(q^{10})\) \(q+0.330837 q^{2} +2.69180 q^{3} -1.89055 q^{4} +0.890547 q^{6} +3.35348 q^{7} -1.28714 q^{8} +4.24581 q^{9} +3.24581 q^{11} -5.08898 q^{12} +1.10945 q^{14} +3.35526 q^{16} +1.94881 q^{17} +1.40467 q^{18} +1.24581 q^{19} +9.02690 q^{21} +1.07383 q^{22} -2.69180 q^{23} -3.46472 q^{24} +3.35348 q^{27} -6.33991 q^{28} +3.00000 q^{29} -3.78109 q^{31} +3.68431 q^{32} +8.73709 q^{33} +0.644737 q^{34} -8.02690 q^{36} +1.94881 q^{37} +0.412160 q^{38} -2.78109 q^{41} +2.98643 q^{42} +8.73709 q^{43} -6.13636 q^{44} -0.890547 q^{46} -6.86960 q^{47} +9.03171 q^{48} +4.24581 q^{49} +5.24581 q^{51} +12.8336 q^{53} +1.10945 q^{54} -4.31638 q^{56} +3.35348 q^{57} +0.992510 q^{58} +2.53528 q^{59} -7.49162 q^{61} -1.25092 q^{62} +14.2382 q^{63} -5.49162 q^{64} +2.89055 q^{66} -4.01515 q^{67} -3.68431 q^{68} -7.24581 q^{69} -5.24581 q^{71} -5.46493 q^{72} -5.46493 q^{73} +0.644737 q^{74} -2.35526 q^{76} +10.8848 q^{77} +13.7811 q^{79} -3.71053 q^{81} -0.920088 q^{82} +8.61955 q^{83} -17.0658 q^{84} +2.89055 q^{86} +8.07541 q^{87} -4.17780 q^{88} -10.3164 q^{89} +5.08898 q^{92} -10.1780 q^{93} -2.27271 q^{94} +9.91745 q^{96} -5.26607 q^{97} +1.40467 q^{98} +13.7811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{4} - 10q^{6} + 6q^{9} + O(q^{10}) \) \( 6q + 4q^{4} - 10q^{6} + 6q^{9} + 22q^{14} + 16q^{16} - 12q^{19} + 4q^{21} - 32q^{24} + 18q^{29} + 8q^{31} + 8q^{34} + 2q^{36} + 14q^{41} - 2q^{44} + 10q^{46} + 6q^{49} + 12q^{51} + 22q^{54} + 16q^{56} + 4q^{59} - 6q^{61} + 6q^{64} + 2q^{66} - 24q^{69} - 12q^{71} + 8q^{74} - 10q^{76} + 52q^{79} - 14q^{81} - 90q^{84} + 2q^{86} - 20q^{89} + 56q^{94} - 6q^{96} + 52q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.330837 0.233937 0.116968 0.993136i \(-0.462682\pi\)
0.116968 + 0.993136i \(0.462682\pi\)
\(3\) 2.69180 1.55411 0.777057 0.629430i \(-0.216712\pi\)
0.777057 + 0.629430i \(0.216712\pi\)
\(4\) −1.89055 −0.945274
\(5\) 0 0
\(6\) 0.890547 0.363564
\(7\) 3.35348 1.26750 0.633748 0.773540i \(-0.281516\pi\)
0.633748 + 0.773540i \(0.281516\pi\)
\(8\) −1.28714 −0.455071
\(9\) 4.24581 1.41527
\(10\) 0 0
\(11\) 3.24581 0.978649 0.489324 0.872102i \(-0.337243\pi\)
0.489324 + 0.872102i \(0.337243\pi\)
\(12\) −5.08898 −1.46906
\(13\) 0 0
\(14\) 1.10945 0.296514
\(15\) 0 0
\(16\) 3.35526 0.838816
\(17\) 1.94881 0.472655 0.236328 0.971673i \(-0.424056\pi\)
0.236328 + 0.971673i \(0.424056\pi\)
\(18\) 1.40467 0.331084
\(19\) 1.24581 0.285808 0.142904 0.989737i \(-0.454356\pi\)
0.142904 + 0.989737i \(0.454356\pi\)
\(20\) 0 0
\(21\) 9.02690 1.96983
\(22\) 1.07383 0.228942
\(23\) −2.69180 −0.561280 −0.280640 0.959813i \(-0.590547\pi\)
−0.280640 + 0.959813i \(0.590547\pi\)
\(24\) −3.46472 −0.707232
\(25\) 0 0
\(26\) 0 0
\(27\) 3.35348 0.645377
\(28\) −6.33991 −1.19813
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −3.78109 −0.679105 −0.339552 0.940587i \(-0.610276\pi\)
−0.339552 + 0.940587i \(0.610276\pi\)
\(32\) 3.68431 0.651301
\(33\) 8.73709 1.52093
\(34\) 0.644737 0.110571
\(35\) 0 0
\(36\) −8.02690 −1.33782
\(37\) 1.94881 0.320382 0.160191 0.987086i \(-0.448789\pi\)
0.160191 + 0.987086i \(0.448789\pi\)
\(38\) 0.412160 0.0668611
\(39\) 0 0
\(40\) 0 0
\(41\) −2.78109 −0.434334 −0.217167 0.976134i \(-0.569682\pi\)
−0.217167 + 0.976134i \(0.569682\pi\)
\(42\) 2.98643 0.460816
\(43\) 8.73709 1.33239 0.666197 0.745776i \(-0.267921\pi\)
0.666197 + 0.745776i \(0.267921\pi\)
\(44\) −6.13636 −0.925091
\(45\) 0 0
\(46\) −0.890547 −0.131304
\(47\) −6.86960 −1.00203 −0.501017 0.865437i \(-0.667041\pi\)
−0.501017 + 0.865437i \(0.667041\pi\)
\(48\) 9.03171 1.30362
\(49\) 4.24581 0.606544
\(50\) 0 0
\(51\) 5.24581 0.734560
\(52\) 0 0
\(53\) 12.8336 1.76282 0.881412 0.472347i \(-0.156593\pi\)
0.881412 + 0.472347i \(0.156593\pi\)
\(54\) 1.10945 0.150977
\(55\) 0 0
\(56\) −4.31638 −0.576800
\(57\) 3.35348 0.444179
\(58\) 0.992510 0.130323
\(59\) 2.53528 0.330066 0.165033 0.986288i \(-0.447227\pi\)
0.165033 + 0.986288i \(0.447227\pi\)
\(60\) 0 0
\(61\) −7.49162 −0.959204 −0.479602 0.877486i \(-0.659219\pi\)
−0.479602 + 0.877486i \(0.659219\pi\)
\(62\) −1.25092 −0.158868
\(63\) 14.2382 1.79385
\(64\) −5.49162 −0.686453
\(65\) 0 0
\(66\) 2.89055 0.355802
\(67\) −4.01515 −0.490529 −0.245264 0.969456i \(-0.578875\pi\)
−0.245264 + 0.969456i \(0.578875\pi\)
\(68\) −3.68431 −0.446789
\(69\) −7.24581 −0.872293
\(70\) 0 0
\(71\) −5.24581 −0.622563 −0.311282 0.950318i \(-0.600758\pi\)
−0.311282 + 0.950318i \(0.600758\pi\)
\(72\) −5.46493 −0.644048
\(73\) −5.46493 −0.639622 −0.319811 0.947481i \(-0.603619\pi\)
−0.319811 + 0.947481i \(0.603619\pi\)
\(74\) 0.644737 0.0749491
\(75\) 0 0
\(76\) −2.35526 −0.270167
\(77\) 10.8848 1.24043
\(78\) 0 0
\(79\) 13.7811 1.55049 0.775247 0.631658i \(-0.217625\pi\)
0.775247 + 0.631658i \(0.217625\pi\)
\(80\) 0 0
\(81\) −3.71053 −0.412281
\(82\) −0.920088 −0.101607
\(83\) 8.61955 0.946119 0.473059 0.881031i \(-0.343150\pi\)
0.473059 + 0.881031i \(0.343150\pi\)
\(84\) −17.0658 −1.86203
\(85\) 0 0
\(86\) 2.89055 0.311696
\(87\) 8.07541 0.865775
\(88\) −4.17780 −0.445355
\(89\) −10.3164 −1.09353 −0.546767 0.837285i \(-0.684142\pi\)
−0.546767 + 0.837285i \(0.684142\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.08898 0.530563
\(93\) −10.1780 −1.05541
\(94\) −2.27271 −0.234413
\(95\) 0 0
\(96\) 9.91745 1.01220
\(97\) −5.26607 −0.534689 −0.267344 0.963601i \(-0.586146\pi\)
−0.267344 + 0.963601i \(0.586146\pi\)
\(98\) 1.40467 0.141893
\(99\) 13.7811 1.38505
\(100\) 0 0
\(101\) 5.71053 0.568219 0.284109 0.958792i \(-0.408302\pi\)
0.284109 + 0.958792i \(0.408302\pi\)
\(102\) 1.73551 0.171841
\(103\) 7.36863 0.726052 0.363026 0.931779i \(-0.381744\pi\)
0.363026 + 0.931779i \(0.381744\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 4.24581 0.412390
\(107\) −8.57444 −0.828922 −0.414461 0.910067i \(-0.636030\pi\)
−0.414461 + 0.910067i \(0.636030\pi\)
\(108\) −6.33991 −0.610058
\(109\) 8.49162 0.813350 0.406675 0.913573i \(-0.366688\pi\)
0.406675 + 0.913573i \(0.366688\pi\)
\(110\) 0 0
\(111\) 5.24581 0.497910
\(112\) 11.2518 1.06320
\(113\) 7.33242 0.689776 0.344888 0.938644i \(-0.387917\pi\)
0.344888 + 0.938644i \(0.387917\pi\)
\(114\) 1.10945 0.103910
\(115\) 0 0
\(116\) −5.67164 −0.526599
\(117\) 0 0
\(118\) 0.838765 0.0772145
\(119\) 6.53528 0.599089
\(120\) 0 0
\(121\) −0.464716 −0.0422469
\(122\) −2.47850 −0.224393
\(123\) −7.48616 −0.675004
\(124\) 7.14834 0.641940
\(125\) 0 0
\(126\) 4.71053 0.419647
\(127\) 9.16369 0.813146 0.406573 0.913618i \(-0.366724\pi\)
0.406573 + 0.913618i \(0.366724\pi\)
\(128\) −9.18546 −0.811887
\(129\) 23.5185 2.07069
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) −16.5179 −1.43770
\(133\) 4.17780 0.362261
\(134\) −1.32836 −0.114753
\(135\) 0 0
\(136\) −2.50838 −0.215092
\(137\) 16.8487 1.43948 0.719741 0.694242i \(-0.244260\pi\)
0.719741 + 0.694242i \(0.244260\pi\)
\(138\) −2.39718 −0.204061
\(139\) −1.02690 −0.0871009 −0.0435505 0.999051i \(-0.513867\pi\)
−0.0435505 + 0.999051i \(0.513867\pi\)
\(140\) 0 0
\(141\) −18.4916 −1.55728
\(142\) −1.73551 −0.145640
\(143\) 0 0
\(144\) 14.2458 1.18715
\(145\) 0 0
\(146\) −1.80800 −0.149631
\(147\) 11.4289 0.942639
\(148\) −3.68431 −0.302849
\(149\) −15.8517 −1.29862 −0.649309 0.760524i \(-0.724942\pi\)
−0.649309 + 0.760524i \(0.724942\pi\)
\(150\) 0 0
\(151\) −14.5454 −1.18369 −0.591845 0.806052i \(-0.701600\pi\)
−0.591845 + 0.806052i \(0.701600\pi\)
\(152\) −1.60353 −0.130063
\(153\) 8.27427 0.668935
\(154\) 3.60107 0.290183
\(155\) 0 0
\(156\) 0 0
\(157\) −10.9210 −0.871588 −0.435794 0.900047i \(-0.643532\pi\)
−0.435794 + 0.900047i \(0.643532\pi\)
\(158\) 4.55929 0.362718
\(159\) 34.5454 2.73963
\(160\) 0 0
\(161\) −9.02690 −0.711420
\(162\) −1.22758 −0.0964476
\(163\) −4.17780 −0.327230 −0.163615 0.986524i \(-0.552316\pi\)
−0.163615 + 0.986524i \(0.552316\pi\)
\(164\) 5.25779 0.410564
\(165\) 0 0
\(166\) 2.85166 0.221332
\(167\) −3.35348 −0.259500 −0.129750 0.991547i \(-0.541417\pi\)
−0.129750 + 0.991547i \(0.541417\pi\)
\(168\) −11.6188 −0.896413
\(169\) 0 0
\(170\) 0 0
\(171\) 5.28947 0.404496
\(172\) −16.5179 −1.25948
\(173\) −8.73709 −0.664268 −0.332134 0.943232i \(-0.607769\pi\)
−0.332134 + 0.943232i \(0.607769\pi\)
\(174\) 2.67164 0.202537
\(175\) 0 0
\(176\) 10.8905 0.820906
\(177\) 6.82449 0.512960
\(178\) −3.41303 −0.255818
\(179\) −18.0101 −1.34614 −0.673071 0.739578i \(-0.735025\pi\)
−0.673071 + 0.739578i \(0.735025\pi\)
\(180\) 0 0
\(181\) 1.04366 0.0775749 0.0387875 0.999247i \(-0.487650\pi\)
0.0387875 + 0.999247i \(0.487650\pi\)
\(182\) 0 0
\(183\) −20.1660 −1.49071
\(184\) 3.46472 0.255422
\(185\) 0 0
\(186\) −3.36724 −0.246898
\(187\) 6.32546 0.462564
\(188\) 12.9873 0.947197
\(189\) 11.2458 0.818012
\(190\) 0 0
\(191\) 25.5185 1.84646 0.923228 0.384253i \(-0.125541\pi\)
0.923228 + 0.384253i \(0.125541\pi\)
\(192\) −14.7824 −1.06683
\(193\) −19.8207 −1.42672 −0.713362 0.700795i \(-0.752829\pi\)
−0.713362 + 0.700795i \(0.752829\pi\)
\(194\) −1.74221 −0.125083
\(195\) 0 0
\(196\) −8.02690 −0.573350
\(197\) 21.6520 1.54264 0.771319 0.636448i \(-0.219597\pi\)
0.771319 + 0.636448i \(0.219597\pi\)
\(198\) 4.55929 0.324015
\(199\) 18.2291 1.29222 0.646112 0.763243i \(-0.276394\pi\)
0.646112 + 0.763243i \(0.276394\pi\)
\(200\) 0 0
\(201\) −10.8080 −0.762337
\(202\) 1.88925 0.132927
\(203\) 10.0604 0.706104
\(204\) −9.91745 −0.694361
\(205\) 0 0
\(206\) 2.43781 0.169850
\(207\) −11.4289 −0.794363
\(208\) 0 0
\(209\) 4.04366 0.279706
\(210\) 0 0
\(211\) 19.2996 1.32864 0.664320 0.747448i \(-0.268721\pi\)
0.664320 + 0.747448i \(0.268721\pi\)
\(212\) −24.2624 −1.66635
\(213\) −14.1207 −0.967534
\(214\) −2.83674 −0.193915
\(215\) 0 0
\(216\) −4.31638 −0.293692
\(217\) −12.6798 −0.860762
\(218\) 2.80934 0.190272
\(219\) −14.7105 −0.994045
\(220\) 0 0
\(221\) 0 0
\(222\) 1.73551 0.116480
\(223\) 12.3707 0.828406 0.414203 0.910184i \(-0.364060\pi\)
0.414203 + 0.910184i \(0.364060\pi\)
\(224\) 12.3553 0.825521
\(225\) 0 0
\(226\) 2.42583 0.161364
\(227\) −6.16282 −0.409040 −0.204520 0.978862i \(-0.565563\pi\)
−0.204520 + 0.978862i \(0.565563\pi\)
\(228\) −6.33991 −0.419871
\(229\) 26.9832 1.78310 0.891551 0.452920i \(-0.149618\pi\)
0.891551 + 0.452920i \(0.149618\pi\)
\(230\) 0 0
\(231\) 29.2996 1.92777
\(232\) −3.86141 −0.253514
\(233\) −0.824319 −0.0540029 −0.0270015 0.999635i \(-0.508596\pi\)
−0.0270015 + 0.999635i \(0.508596\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.79307 −0.312003
\(237\) 37.0960 2.40964
\(238\) 2.16211 0.140149
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) −22.6938 −1.46183 −0.730917 0.682466i \(-0.760907\pi\)
−0.730917 + 0.682466i \(0.760907\pi\)
\(242\) −0.153745 −0.00988310
\(243\) −20.0484 −1.28611
\(244\) 14.1633 0.906710
\(245\) 0 0
\(246\) −2.47670 −0.157908
\(247\) 0 0
\(248\) 4.86678 0.309041
\(249\) 23.2021 1.47038
\(250\) 0 0
\(251\) 19.0269 1.20097 0.600484 0.799637i \(-0.294975\pi\)
0.600484 + 0.799637i \(0.294975\pi\)
\(252\) −26.9180 −1.69568
\(253\) −8.73709 −0.549296
\(254\) 3.03168 0.190225
\(255\) 0 0
\(256\) 7.94436 0.496522
\(257\) 2.11145 0.131709 0.0658544 0.997829i \(-0.479023\pi\)
0.0658544 + 0.997829i \(0.479023\pi\)
\(258\) 7.78079 0.484411
\(259\) 6.53528 0.406083
\(260\) 0 0
\(261\) 12.7374 0.788427
\(262\) 3.30837 0.204391
\(263\) −29.9173 −1.84478 −0.922391 0.386257i \(-0.873768\pi\)
−0.922391 + 0.386257i \(0.873768\pi\)
\(264\) −11.2458 −0.692132
\(265\) 0 0
\(266\) 1.38217 0.0847461
\(267\) −27.7697 −1.69948
\(268\) 7.59083 0.463684
\(269\) 18.5891 1.13340 0.566699 0.823925i \(-0.308220\pi\)
0.566699 + 0.823925i \(0.308220\pi\)
\(270\) 0 0
\(271\) −5.82476 −0.353829 −0.176914 0.984226i \(-0.556612\pi\)
−0.176914 + 0.984226i \(0.556612\pi\)
\(272\) 6.53876 0.396471
\(273\) 0 0
\(274\) 5.57417 0.336748
\(275\) 0 0
\(276\) 13.6985 0.824556
\(277\) 13.5403 0.813560 0.406780 0.913526i \(-0.366651\pi\)
0.406780 + 0.913526i \(0.366651\pi\)
\(278\) −0.339738 −0.0203761
\(279\) −16.0538 −0.961116
\(280\) 0 0
\(281\) 0.464716 0.0277226 0.0138613 0.999904i \(-0.495588\pi\)
0.0138613 + 0.999904i \(0.495588\pi\)
\(282\) −6.11770 −0.364304
\(283\) 10.0604 0.598031 0.299015 0.954248i \(-0.403342\pi\)
0.299015 + 0.954248i \(0.403342\pi\)
\(284\) 9.91745 0.588492
\(285\) 0 0
\(286\) 0 0
\(287\) −9.32634 −0.550516
\(288\) 15.6429 0.921767
\(289\) −13.2021 −0.776597
\(290\) 0 0
\(291\) −14.1752 −0.830967
\(292\) 10.3317 0.604618
\(293\) −13.4501 −0.785764 −0.392882 0.919589i \(-0.628522\pi\)
−0.392882 + 0.919589i \(0.628522\pi\)
\(294\) 3.78109 0.220518
\(295\) 0 0
\(296\) −2.50838 −0.145797
\(297\) 10.8848 0.631597
\(298\) −5.24431 −0.303795
\(299\) 0 0
\(300\) 0 0
\(301\) 29.2996 1.68880
\(302\) −4.81216 −0.276909
\(303\) 15.3716 0.883076
\(304\) 4.18002 0.239741
\(305\) 0 0
\(306\) 2.73743 0.156488
\(307\) −24.6077 −1.40444 −0.702219 0.711961i \(-0.747807\pi\)
−0.702219 + 0.711961i \(0.747807\pi\)
\(308\) −20.5781 −1.17255
\(309\) 19.8349 1.12837
\(310\) 0 0
\(311\) 2.43781 0.138236 0.0691178 0.997609i \(-0.477982\pi\)
0.0691178 + 0.997609i \(0.477982\pi\)
\(312\) 0 0
\(313\) −19.2965 −1.09071 −0.545353 0.838207i \(-0.683604\pi\)
−0.545353 + 0.838207i \(0.683604\pi\)
\(314\) −3.61305 −0.203896
\(315\) 0 0
\(316\) −26.0538 −1.46564
\(317\) 28.8217 1.61879 0.809395 0.587265i \(-0.199795\pi\)
0.809395 + 0.587265i \(0.199795\pi\)
\(318\) 11.4289 0.640900
\(319\) 9.73743 0.545191
\(320\) 0 0
\(321\) −23.0807 −1.28824
\(322\) −2.98643 −0.166427
\(323\) 2.42785 0.135089
\(324\) 7.01492 0.389718
\(325\) 0 0
\(326\) −1.38217 −0.0765512
\(327\) 22.8578 1.26404
\(328\) 3.57964 0.197653
\(329\) −23.0371 −1.27007
\(330\) 0 0
\(331\) 2.97310 0.163416 0.0817081 0.996656i \(-0.473962\pi\)
0.0817081 + 0.996656i \(0.473962\pi\)
\(332\) −16.2957 −0.894341
\(333\) 8.27427 0.453427
\(334\) −1.10945 −0.0607066
\(335\) 0 0
\(336\) 30.2876 1.65233
\(337\) 1.90370 0.103701 0.0518505 0.998655i \(-0.483488\pi\)
0.0518505 + 0.998655i \(0.483488\pi\)
\(338\) 0 0
\(339\) 19.7374 1.07199
\(340\) 0 0
\(341\) −12.2727 −0.664605
\(342\) 1.74995 0.0946265
\(343\) −9.23611 −0.498703
\(344\) −11.2458 −0.606333
\(345\) 0 0
\(346\) −2.89055 −0.155397
\(347\) −12.6347 −0.678266 −0.339133 0.940738i \(-0.610134\pi\)
−0.339133 + 0.940738i \(0.610134\pi\)
\(348\) −15.2669 −0.818394
\(349\) −8.97310 −0.480319 −0.240159 0.970733i \(-0.577200\pi\)
−0.240159 + 0.970733i \(0.577200\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 11.9586 0.637395
\(353\) 34.2505 1.82297 0.911484 0.411336i \(-0.134938\pi\)
0.911484 + 0.411336i \(0.134938\pi\)
\(354\) 2.25779 0.120000
\(355\) 0 0
\(356\) 19.5036 1.03369
\(357\) 17.5917 0.931052
\(358\) −5.95841 −0.314912
\(359\) −22.4043 −1.18245 −0.591227 0.806505i \(-0.701356\pi\)
−0.591227 + 0.806505i \(0.701356\pi\)
\(360\) 0 0
\(361\) −17.4480 −0.918314
\(362\) 0.345282 0.0181476
\(363\) −1.25092 −0.0656565
\(364\) 0 0
\(365\) 0 0
\(366\) −6.67164 −0.348732
\(367\) −13.1951 −0.688777 −0.344388 0.938827i \(-0.611914\pi\)
−0.344388 + 0.938827i \(0.611914\pi\)
\(368\) −9.03171 −0.470811
\(369\) −11.8080 −0.614700
\(370\) 0 0
\(371\) 43.0371 2.23437
\(372\) 19.2419 0.997647
\(373\) −15.2614 −0.790206 −0.395103 0.918637i \(-0.629291\pi\)
−0.395103 + 0.918637i \(0.629291\pi\)
\(374\) 2.09269 0.108211
\(375\) 0 0
\(376\) 8.84210 0.455997
\(377\) 0 0
\(378\) 3.72052 0.191363
\(379\) −18.2291 −0.936363 −0.468182 0.883632i \(-0.655091\pi\)
−0.468182 + 0.883632i \(0.655091\pi\)
\(380\) 0 0
\(381\) 24.6669 1.26372
\(382\) 8.44246 0.431954
\(383\) −1.44088 −0.0736255 −0.0368128 0.999322i \(-0.511721\pi\)
−0.0368128 + 0.999322i \(0.511721\pi\)
\(384\) −24.7255 −1.26177
\(385\) 0 0
\(386\) −6.55741 −0.333763
\(387\) 37.0960 1.88570
\(388\) 9.95576 0.505427
\(389\) −18.7912 −0.952754 −0.476377 0.879241i \(-0.658050\pi\)
−0.476377 + 0.879241i \(0.658050\pi\)
\(390\) 0 0
\(391\) −5.24581 −0.265292
\(392\) −5.46493 −0.276021
\(393\) 26.9180 1.35784
\(394\) 7.16326 0.360880
\(395\) 0 0
\(396\) −26.0538 −1.30925
\(397\) 17.0927 0.857857 0.428928 0.903338i \(-0.358891\pi\)
0.428928 + 0.903338i \(0.358891\pi\)
\(398\) 6.03084 0.302298
\(399\) 11.2458 0.562995
\(400\) 0 0
\(401\) −22.2021 −1.10872 −0.554361 0.832276i \(-0.687037\pi\)
−0.554361 + 0.832276i \(0.687037\pi\)
\(402\) −3.57568 −0.178339
\(403\) 0 0
\(404\) −10.7960 −0.537122
\(405\) 0 0
\(406\) 3.32836 0.165184
\(407\) 6.32546 0.313541
\(408\) −6.75207 −0.334277
\(409\) 9.63276 0.476309 0.238155 0.971227i \(-0.423458\pi\)
0.238155 + 0.971227i \(0.423458\pi\)
\(410\) 0 0
\(411\) 45.3534 2.23712
\(412\) −13.9307 −0.686318
\(413\) 8.50202 0.418357
\(414\) −3.78109 −0.185831
\(415\) 0 0
\(416\) 0 0
\(417\) −2.76423 −0.135365
\(418\) 1.33779 0.0654335
\(419\) −1.95634 −0.0955733 −0.0477866 0.998858i \(-0.515217\pi\)
−0.0477866 + 0.998858i \(0.515217\pi\)
\(420\) 0 0
\(421\) 12.0807 0.588778 0.294389 0.955686i \(-0.404884\pi\)
0.294389 + 0.955686i \(0.404884\pi\)
\(422\) 6.38502 0.310818
\(423\) −29.1670 −1.41815
\(424\) −16.5185 −0.802210
\(425\) 0 0
\(426\) −4.67164 −0.226342
\(427\) −25.1230 −1.21579
\(428\) 16.2104 0.783558
\(429\) 0 0
\(430\) 0 0
\(431\) −24.5891 −1.18441 −0.592207 0.805786i \(-0.701743\pi\)
−0.592207 + 0.805786i \(0.701743\pi\)
\(432\) 11.2518 0.541352
\(433\) −36.0728 −1.73355 −0.866775 0.498700i \(-0.833811\pi\)
−0.866775 + 0.498700i \(0.833811\pi\)
\(434\) −4.19495 −0.201364
\(435\) 0 0
\(436\) −16.0538 −0.768838
\(437\) −3.35348 −0.160419
\(438\) −4.86678 −0.232544
\(439\) 2.53528 0.121003 0.0605013 0.998168i \(-0.480730\pi\)
0.0605013 + 0.998168i \(0.480730\pi\)
\(440\) 0 0
\(441\) 18.0269 0.858424
\(442\) 0 0
\(443\) −19.3579 −0.919721 −0.459860 0.887991i \(-0.652101\pi\)
−0.459860 + 0.887991i \(0.652101\pi\)
\(444\) −9.91745 −0.470661
\(445\) 0 0
\(446\) 4.09269 0.193795
\(447\) −42.6696 −2.01820
\(448\) −18.4160 −0.870075
\(449\) 24.8080 1.17076 0.585381 0.810758i \(-0.300945\pi\)
0.585381 + 0.810758i \(0.300945\pi\)
\(450\) 0 0
\(451\) −9.02690 −0.425060
\(452\) −13.8623 −0.652027
\(453\) −39.1534 −1.83959
\(454\) −2.03888 −0.0956896
\(455\) 0 0
\(456\) −4.31638 −0.202133
\(457\) −7.56748 −0.353992 −0.176996 0.984212i \(-0.556638\pi\)
−0.176996 + 0.984212i \(0.556638\pi\)
\(458\) 8.92704 0.417133
\(459\) 6.53528 0.305041
\(460\) 0 0
\(461\) 12.3433 0.574884 0.287442 0.957798i \(-0.407195\pi\)
0.287442 + 0.957798i \(0.407195\pi\)
\(462\) 9.69338 0.450977
\(463\) −22.8578 −1.06229 −0.531146 0.847281i \(-0.678238\pi\)
−0.531146 + 0.847281i \(0.678238\pi\)
\(464\) 10.0658 0.467293
\(465\) 0 0
\(466\) −0.272715 −0.0126333
\(467\) −15.2976 −0.707889 −0.353945 0.935266i \(-0.615160\pi\)
−0.353945 + 0.935266i \(0.615160\pi\)
\(468\) 0 0
\(469\) −13.4647 −0.621743
\(470\) 0 0
\(471\) −29.3971 −1.35455
\(472\) −3.26325 −0.150203
\(473\) 28.3589 1.30394
\(474\) 12.2727 0.563704
\(475\) 0 0
\(476\) −12.3553 −0.566303
\(477\) 54.4889 2.49487
\(478\) −1.32335 −0.0605284
\(479\) −24.2829 −1.10951 −0.554756 0.832013i \(-0.687188\pi\)
−0.554756 + 0.832013i \(0.687188\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −7.50793 −0.341977
\(483\) −24.2987 −1.10563
\(484\) 0.878567 0.0399349
\(485\) 0 0
\(486\) −6.63276 −0.300868
\(487\) −36.8883 −1.67157 −0.835783 0.549060i \(-0.814986\pi\)
−0.835783 + 0.549060i \(0.814986\pi\)
\(488\) 9.64273 0.436506
\(489\) −11.2458 −0.508553
\(490\) 0 0
\(491\) −35.3534 −1.59548 −0.797739 0.603003i \(-0.793971\pi\)
−0.797739 + 0.603003i \(0.793971\pi\)
\(492\) 14.1529 0.638064
\(493\) 5.84642 0.263310
\(494\) 0 0
\(495\) 0 0
\(496\) −12.6866 −0.569644
\(497\) −17.5917 −0.789096
\(498\) 7.67612 0.343975
\(499\) −16.2189 −0.726058 −0.363029 0.931778i \(-0.618257\pi\)
−0.363029 + 0.931778i \(0.618257\pi\)
\(500\) 0 0
\(501\) −9.02690 −0.403292
\(502\) 6.29480 0.280950
\(503\) 20.2384 0.902386 0.451193 0.892427i \(-0.350999\pi\)
0.451193 + 0.892427i \(0.350999\pi\)
\(504\) −18.3265 −0.816328
\(505\) 0 0
\(506\) −2.89055 −0.128500
\(507\) 0 0
\(508\) −17.3244 −0.768646
\(509\) 20.0371 0.888127 0.444063 0.895995i \(-0.353537\pi\)
0.444063 + 0.895995i \(0.353537\pi\)
\(510\) 0 0
\(511\) −18.3265 −0.810718
\(512\) 20.9992 0.928042
\(513\) 4.17780 0.184454
\(514\) 0.698546 0.0308116
\(515\) 0 0
\(516\) −44.4629 −1.95737
\(517\) −22.2974 −0.980639
\(518\) 2.16211 0.0949977
\(519\) −23.5185 −1.03235
\(520\) 0 0
\(521\) 16.0269 0.702151 0.351076 0.936347i \(-0.385816\pi\)
0.351076 + 0.936347i \(0.385816\pi\)
\(522\) 4.21401 0.184442
\(523\) −11.7380 −0.513265 −0.256633 0.966509i \(-0.582613\pi\)
−0.256633 + 0.966509i \(0.582613\pi\)
\(524\) −18.9055 −0.825889
\(525\) 0 0
\(526\) −9.89775 −0.431562
\(527\) −7.36863 −0.320982
\(528\) 29.3152 1.27578
\(529\) −15.7542 −0.684965
\(530\) 0 0
\(531\) 10.7643 0.467132
\(532\) −7.89832 −0.342436
\(533\) 0 0
\(534\) −9.18722 −0.397570
\(535\) 0 0
\(536\) 5.16804 0.223225
\(537\) −48.4798 −2.09206
\(538\) 6.14995 0.265143
\(539\) 13.7811 0.593594
\(540\) 0 0
\(541\) 21.8080 0.937599 0.468800 0.883305i \(-0.344687\pi\)
0.468800 + 0.883305i \(0.344687\pi\)
\(542\) −1.92704 −0.0827736
\(543\) 2.80934 0.120560
\(544\) 7.18002 0.307841
\(545\) 0 0
\(546\) 0 0
\(547\) −6.30924 −0.269764 −0.134882 0.990862i \(-0.543065\pi\)
−0.134882 + 0.990862i \(0.543065\pi\)
\(548\) −31.8533 −1.36070
\(549\) −31.8080 −1.35753
\(550\) 0 0
\(551\) 3.73743 0.159220
\(552\) 9.32634 0.396955
\(553\) 46.2146 1.96524
\(554\) 4.47964 0.190322
\(555\) 0 0
\(556\) 1.94141 0.0823342
\(557\) −35.8378 −1.51849 −0.759247 0.650802i \(-0.774433\pi\)
−0.759247 + 0.650802i \(0.774433\pi\)
\(558\) −5.31119 −0.224840
\(559\) 0 0
\(560\) 0 0
\(561\) 17.0269 0.718876
\(562\) 0.153745 0.00648534
\(563\) 5.00212 0.210814 0.105407 0.994429i \(-0.466385\pi\)
0.105407 + 0.994429i \(0.466385\pi\)
\(564\) 34.9593 1.47205
\(565\) 0 0
\(566\) 3.32836 0.139901
\(567\) −12.4432 −0.522564
\(568\) 6.75207 0.283310
\(569\) −13.1680 −0.552033 −0.276017 0.961153i \(-0.589014\pi\)
−0.276017 + 0.961153i \(0.589014\pi\)
\(570\) 0 0
\(571\) 19.8349 0.830065 0.415032 0.909807i \(-0.363770\pi\)
0.415032 + 0.909807i \(0.363770\pi\)
\(572\) 0 0
\(573\) 68.6909 2.86960
\(574\) −3.08549 −0.128786
\(575\) 0 0
\(576\) −23.3164 −0.971516
\(577\) −10.9210 −0.454646 −0.227323 0.973819i \(-0.572997\pi\)
−0.227323 + 0.973819i \(0.572997\pi\)
\(578\) −4.36775 −0.181675
\(579\) −53.3534 −2.21729
\(580\) 0 0
\(581\) 28.9055 1.19920
\(582\) −4.68969 −0.194394
\(583\) 41.6553 1.72519
\(584\) 7.03411 0.291073
\(585\) 0 0
\(586\) −4.44979 −0.183819
\(587\) −40.4495 −1.66953 −0.834764 0.550607i \(-0.814396\pi\)
−0.834764 + 0.550607i \(0.814396\pi\)
\(588\) −21.6069 −0.891052
\(589\) −4.71053 −0.194094
\(590\) 0 0
\(591\) 58.2829 2.39744
\(592\) 6.53876 0.268742
\(593\) −1.47709 −0.0606569 −0.0303284 0.999540i \(-0.509655\pi\)
−0.0303284 + 0.999540i \(0.509655\pi\)
\(594\) 3.60107 0.147754
\(595\) 0 0
\(596\) 29.9683 1.22755
\(597\) 49.0690 2.00826
\(598\) 0 0
\(599\) 2.27271 0.0928606 0.0464303 0.998922i \(-0.485215\pi\)
0.0464303 + 0.998922i \(0.485215\pi\)
\(600\) 0 0
\(601\) 7.40429 0.302027 0.151014 0.988532i \(-0.451746\pi\)
0.151014 + 0.988532i \(0.451746\pi\)
\(602\) 9.69338 0.395073
\(603\) −17.0476 −0.694231
\(604\) 27.4988 1.11891
\(605\) 0 0
\(606\) 5.08549 0.206584
\(607\) −10.6932 −0.434024 −0.217012 0.976169i \(-0.569631\pi\)
−0.217012 + 0.976169i \(0.569631\pi\)
\(608\) 4.58996 0.186147
\(609\) 27.0807 1.09737
\(610\) 0 0
\(611\) 0 0
\(612\) −15.6429 −0.632327
\(613\) −6.08149 −0.245629 −0.122815 0.992430i \(-0.539192\pi\)
−0.122815 + 0.992430i \(0.539192\pi\)
\(614\) −8.14114 −0.328550
\(615\) 0 0
\(616\) −14.0101 −0.564485
\(617\) −31.8388 −1.28178 −0.640892 0.767631i \(-0.721435\pi\)
−0.640892 + 0.767631i \(0.721435\pi\)
\(618\) 6.56211 0.263967
\(619\) 26.4043 1.06128 0.530639 0.847598i \(-0.321952\pi\)
0.530639 + 0.847598i \(0.321952\pi\)
\(620\) 0 0
\(621\) −9.02690 −0.362237
\(622\) 0.806517 0.0323384
\(623\) −34.5957 −1.38605
\(624\) 0 0
\(625\) 0 0
\(626\) −6.38400 −0.255156
\(627\) 10.8848 0.434695
\(628\) 20.6466 0.823889
\(629\) 3.79785 0.151430
\(630\) 0 0
\(631\) −35.1680 −1.40002 −0.700009 0.714134i \(-0.746821\pi\)
−0.700009 + 0.714134i \(0.746821\pi\)
\(632\) −17.7381 −0.705585
\(633\) 51.9508 2.06486
\(634\) 9.53528 0.378695
\(635\) 0 0
\(636\) −65.3098 −2.58970
\(637\) 0 0
\(638\) 3.22150 0.127540
\(639\) −22.2727 −0.881095
\(640\) 0 0
\(641\) 5.52514 0.218230 0.109115 0.994029i \(-0.465198\pi\)
0.109115 + 0.994029i \(0.465198\pi\)
\(642\) −7.63594 −0.301367
\(643\) 32.1552 1.26808 0.634039 0.773301i \(-0.281396\pi\)
0.634039 + 0.773301i \(0.281396\pi\)
\(644\) 17.0658 0.672486
\(645\) 0 0
\(646\) 0.803220 0.0316023
\(647\) −13.7843 −0.541917 −0.270959 0.962591i \(-0.587341\pi\)
−0.270959 + 0.962591i \(0.587341\pi\)
\(648\) 4.77595 0.187617
\(649\) 8.22905 0.323019
\(650\) 0 0
\(651\) −34.1316 −1.33772
\(652\) 7.89832 0.309322
\(653\) 8.50202 0.332710 0.166355 0.986066i \(-0.446800\pi\)
0.166355 + 0.986066i \(0.446800\pi\)
\(654\) 7.56219 0.295705
\(655\) 0 0
\(656\) −9.33130 −0.364326
\(657\) −23.2031 −0.905238
\(658\) −7.62150 −0.297117
\(659\) 4.04366 0.157519 0.0787594 0.996894i \(-0.474904\pi\)
0.0787594 + 0.996894i \(0.474904\pi\)
\(660\) 0 0
\(661\) −31.2727 −1.21637 −0.608184 0.793796i \(-0.708102\pi\)
−0.608184 + 0.793796i \(0.708102\pi\)
\(662\) 0.983609 0.0382290
\(663\) 0 0
\(664\) −11.0945 −0.430551
\(665\) 0 0
\(666\) 2.73743 0.106073
\(667\) −8.07541 −0.312681
\(668\) 6.33991 0.245298
\(669\) 33.2996 1.28744
\(670\) 0 0
\(671\) −24.3164 −0.938723
\(672\) 33.2579 1.28295
\(673\) −32.0739 −1.23636 −0.618179 0.786037i \(-0.712129\pi\)
−0.618179 + 0.786037i \(0.712129\pi\)
\(674\) 0.629812 0.0242595
\(675\) 0 0
\(676\) 0 0
\(677\) 14.2382 0.547220 0.273610 0.961841i \(-0.411782\pi\)
0.273610 + 0.961841i \(0.411782\pi\)
\(678\) 6.52986 0.250778
\(679\) −17.6597 −0.677716
\(680\) 0 0
\(681\) −16.5891 −0.635695
\(682\) −4.06026 −0.155475
\(683\) 25.7847 0.986622 0.493311 0.869853i \(-0.335786\pi\)
0.493311 + 0.869853i \(0.335786\pi\)
\(684\) −10.0000 −0.382360
\(685\) 0 0
\(686\) −3.05564 −0.116665
\(687\) 72.6336 2.77114
\(688\) 29.3152 1.11763
\(689\) 0 0
\(690\) 0 0
\(691\) 0.0436636 0.00166104 0.000830522 1.00000i \(-0.499736\pi\)
0.000830522 1.00000i \(0.499736\pi\)
\(692\) 16.5179 0.627915
\(693\) 46.2146 1.75555
\(694\) −4.18002 −0.158671
\(695\) 0 0
\(696\) −10.3941 −0.393989
\(697\) −5.41982 −0.205290
\(698\) −2.96863 −0.112364
\(699\) −2.21891 −0.0839267
\(700\) 0 0
\(701\) −14.5454 −0.549373 −0.274687 0.961534i \(-0.588574\pi\)
−0.274687 + 0.961534i \(0.588574\pi\)
\(702\) 0 0
\(703\) 2.42785 0.0915679
\(704\) −17.8248 −0.671796
\(705\) 0 0
\(706\) 11.3313 0.426459
\(707\) 19.1501 0.720214
\(708\) −12.9020 −0.484888
\(709\) −19.6328 −0.737324 −0.368662 0.929564i \(-0.620184\pi\)
−0.368662 + 0.929564i \(0.620184\pi\)
\(710\) 0 0
\(711\) 58.5119 2.19437
\(712\) 13.2786 0.497636
\(713\) 10.1780 0.381168
\(714\) 5.81998 0.217807
\(715\) 0 0
\(716\) 34.0490 1.27247
\(717\) −10.7672 −0.402109
\(718\) −7.41216 −0.276619
\(719\) 47.4312 1.76889 0.884443 0.466649i \(-0.154539\pi\)
0.884443 + 0.466649i \(0.154539\pi\)
\(720\) 0 0
\(721\) 24.7105 0.920268
\(722\) −5.77242 −0.214827
\(723\) −61.0872 −2.27186
\(724\) −1.97310 −0.0733295
\(725\) 0 0
\(726\) −0.413851 −0.0153595
\(727\) 34.0951 1.26452 0.632259 0.774757i \(-0.282128\pi\)
0.632259 + 0.774757i \(0.282128\pi\)
\(728\) 0 0
\(729\) −42.8349 −1.58648
\(730\) 0 0
\(731\) 17.0269 0.629763
\(732\) 38.1247 1.40913
\(733\) 14.3920 0.531580 0.265790 0.964031i \(-0.414367\pi\)
0.265790 + 0.964031i \(0.414367\pi\)
\(734\) −4.36541 −0.161130
\(735\) 0 0
\(736\) −9.91745 −0.365562
\(737\) −13.0324 −0.480055
\(738\) −3.90652 −0.143801
\(739\) 34.4480 1.26719 0.633594 0.773665i \(-0.281579\pi\)
0.633594 + 0.773665i \(0.281579\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 14.2382 0.522702
\(743\) −40.7134 −1.49363 −0.746816 0.665031i \(-0.768418\pi\)
−0.746816 + 0.665031i \(0.768418\pi\)
\(744\) 13.1004 0.480285
\(745\) 0 0
\(746\) −5.04903 −0.184858
\(747\) 36.5970 1.33901
\(748\) −11.9586 −0.437249
\(749\) −28.7542 −1.05066
\(750\) 0 0
\(751\) 32.5018 1.18601 0.593003 0.805200i \(-0.297942\pi\)
0.593003 + 0.805200i \(0.297942\pi\)
\(752\) −23.0493 −0.840522
\(753\) 51.2167 1.86644
\(754\) 0 0
\(755\) 0 0
\(756\) −21.2607 −0.773245
\(757\) −13.0324 −0.473671 −0.236836 0.971550i \(-0.576110\pi\)
−0.236836 + 0.971550i \(0.576110\pi\)
\(758\) −6.03084 −0.219050
\(759\) −23.5185 −0.853668
\(760\) 0 0
\(761\) 3.98985 0.144632 0.0723161 0.997382i \(-0.476961\pi\)
0.0723161 + 0.997382i \(0.476961\pi\)
\(762\) 8.16070 0.295631
\(763\) 28.4765 1.03092
\(764\) −48.2440 −1.74541
\(765\) 0 0
\(766\) −0.476696 −0.0172237
\(767\) 0 0
\(768\) 21.3847 0.771652
\(769\) −6.66686 −0.240413 −0.120207 0.992749i \(-0.538356\pi\)
−0.120207 + 0.992749i \(0.538356\pi\)
\(770\) 0 0
\(771\) 5.68362 0.204691
\(772\) 37.4720 1.34865
\(773\) −48.3349 −1.73849 −0.869244 0.494384i \(-0.835394\pi\)
−0.869244 + 0.494384i \(0.835394\pi\)
\(774\) 12.2727 0.441134
\(775\) 0 0
\(776\) 6.77815 0.243321
\(777\) 17.5917 0.631099
\(778\) −6.21683 −0.222884
\(779\) −3.46472 −0.124136
\(780\) 0 0
\(781\) −17.0269 −0.609271
\(782\) −1.73551 −0.0620616
\(783\) 10.0604 0.359531
\(784\) 14.2458 0.508779
\(785\) 0 0
\(786\) 8.90547 0.317648
\(787\) 27.9612 0.996709 0.498355 0.866973i \(-0.333938\pi\)
0.498355 + 0.866973i \(0.333938\pi\)
\(788\) −40.9341 −1.45822
\(789\) −80.5316 −2.86700
\(790\) 0 0
\(791\) 24.5891 0.874288
\(792\) −17.7381 −0.630297
\(793\) 0 0
\(794\) 5.65488 0.200684
\(795\) 0 0
\(796\) −34.4629 −1.22150
\(797\) 37.2424 1.31919 0.659597 0.751619i \(-0.270727\pi\)
0.659597 + 0.751619i \(0.270727\pi\)
\(798\) 3.72052 0.131705
\(799\) −13.3875 −0.473617
\(800\) 0 0
\(801\) −43.8014 −1.54765
\(802\) −7.34528 −0.259371
\(803\) −17.7381 −0.625965
\(804\) 20.4330 0.720617
\(805\) 0 0
\(806\) 0 0
\(807\) 50.0382 1.76143
\(808\) −7.35022 −0.258580
\(809\) 14.5287 0.510801 0.255400 0.966835i \(-0.417793\pi\)
0.255400 + 0.966835i \(0.417793\pi\)
\(810\) 0 0
\(811\) −44.0538 −1.54694 −0.773469 0.633834i \(-0.781480\pi\)
−0.773469 + 0.633834i \(0.781480\pi\)
\(812\) −19.0197 −0.667461
\(813\) −15.6791 −0.549890
\(814\) 2.09269 0.0733489
\(815\) 0 0
\(816\) 17.6011 0.616161
\(817\) 10.8848 0.380809
\(818\) 3.18687 0.111426
\(819\) 0 0
\(820\) 0 0
\(821\) −27.0269 −0.943245 −0.471623 0.881800i \(-0.656332\pi\)
−0.471623 + 0.881800i \(0.656332\pi\)
\(822\) 15.0046 0.523345
\(823\) −39.5963 −1.38024 −0.690120 0.723695i \(-0.742442\pi\)
−0.690120 + 0.723695i \(0.742442\pi\)
\(824\) −9.48442 −0.330405
\(825\) 0 0
\(826\) 2.81278 0.0978691
\(827\) 26.5639 0.923716 0.461858 0.886954i \(-0.347183\pi\)
0.461858 + 0.886954i \(0.347183\pi\)
\(828\) 21.6069 0.750890
\(829\) 13.9832 0.485658 0.242829 0.970069i \(-0.421925\pi\)
0.242829 + 0.970069i \(0.421925\pi\)
\(830\) 0 0
\(831\) 36.4480 1.26437
\(832\) 0 0
\(833\) 8.27427 0.286686
\(834\) −0.914507 −0.0316668
\(835\) 0 0
\(836\) −7.64474 −0.264399
\(837\) −12.6798 −0.438278
\(838\) −0.647228 −0.0223581
\(839\) −14.3941 −0.496941 −0.248471 0.968639i \(-0.579928\pi\)
−0.248471 + 0.968639i \(0.579928\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 3.99674 0.137737
\(843\) 1.25092 0.0430841
\(844\) −36.4868 −1.25593
\(845\) 0 0
\(846\) −9.64952 −0.331757
\(847\) −1.55841 −0.0535477
\(848\) 43.0600 1.47869
\(849\) 27.0807 0.929408
\(850\) 0 0
\(851\) −5.24581 −0.179824
\(852\) 26.6958 0.914584
\(853\) 27.2633 0.933478 0.466739 0.884395i \(-0.345429\pi\)
0.466739 + 0.884395i \(0.345429\pi\)
\(854\) −8.31160 −0.284417
\(855\) 0 0
\(856\) 11.0365 0.377219
\(857\) −50.6201 −1.72915 −0.864575 0.502503i \(-0.832413\pi\)
−0.864575 + 0.502503i \(0.832413\pi\)
\(858\) 0 0
\(859\) 1.27992 0.0436702 0.0218351 0.999762i \(-0.493049\pi\)
0.0218351 + 0.999762i \(0.493049\pi\)
\(860\) 0 0
\(861\) −25.1047 −0.855565
\(862\) −8.13497 −0.277078
\(863\) −8.38448 −0.285411 −0.142706 0.989765i \(-0.545580\pi\)
−0.142706 + 0.989765i \(0.545580\pi\)
\(864\) 12.3553 0.420335
\(865\) 0 0
\(866\) −11.9342 −0.405541
\(867\) −35.5376 −1.20692
\(868\) 23.9718 0.813655
\(869\) 44.7308 1.51739
\(870\) 0 0
\(871\) 0 0
\(872\) −10.9299 −0.370132
\(873\) −22.3588 −0.756729
\(874\) −1.10945 −0.0375278
\(875\) 0 0
\(876\) 27.8109 0.939645
\(877\) 55.8862 1.88714 0.943572 0.331169i \(-0.107443\pi\)
0.943572 + 0.331169i \(0.107443\pi\)
\(878\) 0.838765 0.0283069
\(879\) −36.2051 −1.22117
\(880\) 0 0
\(881\) 25.1949 0.848839 0.424420 0.905466i \(-0.360478\pi\)
0.424420 + 0.905466i \(0.360478\pi\)
\(882\) 5.96396 0.200817
\(883\) −30.7868 −1.03606 −0.518029 0.855363i \(-0.673334\pi\)
−0.518029 + 0.855363i \(0.673334\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −6.40429 −0.215156
\(887\) −12.4721 −0.418771 −0.209385 0.977833i \(-0.567146\pi\)
−0.209385 + 0.977833i \(0.567146\pi\)
\(888\) −6.75207 −0.226585
\(889\) 30.7302 1.03066
\(890\) 0 0
\(891\) −12.0437 −0.403478
\(892\) −23.3875 −0.783071
\(893\) −8.55822 −0.286390
\(894\) −14.1167 −0.472132
\(895\) 0 0
\(896\) −30.8032 −1.02906
\(897\) 0 0
\(898\) 8.20739 0.273884
\(899\) −11.3433 −0.378320
\(900\) 0 0
\(901\) 25.0101 0.833209
\(902\) −2.98643 −0.0994372
\(903\) 78.8688 2.62459
\(904\) −9.43781 −0.313897
\(905\) 0 0
\(906\) −12.9534 −0.430348
\(907\) −38.8911 −1.29136 −0.645678 0.763609i \(-0.723425\pi\)
−0.645678 + 0.763609i \(0.723425\pi\)
\(908\) 11.6511 0.386655
\(909\) 24.2458 0.804183
\(910\) 0 0
\(911\) 0.165096 0.00546989 0.00273494 0.999996i \(-0.499129\pi\)
0.00273494 + 0.999996i \(0.499129\pi\)
\(912\) 11.2518 0.372584
\(913\) 27.9774 0.925918
\(914\) −2.50360 −0.0828117
\(915\) 0 0
\(916\) −51.0131 −1.68552
\(917\) 33.5348 1.10742
\(918\) 2.16211 0.0713603
\(919\) 0.895326 0.0295341 0.0147670 0.999891i \(-0.495299\pi\)
0.0147670 + 0.999891i \(0.495299\pi\)
\(920\) 0 0
\(921\) −66.2392 −2.18266
\(922\) 4.08361 0.134486
\(923\) 0 0
\(924\) −55.3923 −1.82227
\(925\) 0 0
\(926\) −7.56219 −0.248509
\(927\) 31.2858 1.02756
\(928\) 11.0529 0.362831
\(929\) 12.2895 0.403205 0.201602 0.979467i \(-0.435385\pi\)
0.201602 + 0.979467i \(0.435385\pi\)
\(930\) 0 0
\(931\) 5.28947 0.173356
\(932\) 1.55841 0.0510476
\(933\) 6.56211 0.214834
\(934\) −5.06101 −0.165601
\(935\) 0 0
\(936\) 0 0
\(937\) −5.77242 −0.188577 −0.0942884 0.995545i \(-0.530058\pi\)
−0.0942884 + 0.995545i \(0.530058\pi\)
\(938\) −4.45462 −0.145448
\(939\) −51.9425 −1.69508
\(940\) 0 0
\(941\) 55.8887 1.82192 0.910960 0.412495i \(-0.135342\pi\)
0.910960 + 0.412495i \(0.135342\pi\)
\(942\) −9.72563 −0.316878
\(943\) 7.48616 0.243783
\(944\) 8.50655 0.276864
\(945\) 0 0
\(946\) 9.38217 0.305041
\(947\) 1.89638 0.0616239 0.0308120 0.999525i \(-0.490191\pi\)
0.0308120 + 0.999525i \(0.490191\pi\)
\(948\) −70.1318 −2.27777
\(949\) 0 0
\(950\) 0 0
\(951\) 77.5825 2.51578
\(952\) −8.41179 −0.272628
\(953\) 33.6523 1.09011 0.545053 0.838402i \(-0.316510\pi\)
0.545053 + 0.838402i \(0.316510\pi\)
\(954\) 18.0269 0.583643
\(955\) 0 0
\(956\) 7.56219 0.244579
\(957\) 26.2113 0.847290
\(958\) −8.03366 −0.259556
\(959\) 56.5018 1.82454
\(960\) 0 0
\(961\) −16.7033 −0.538817
\(962\) 0 0
\(963\) −36.4054 −1.17315
\(964\) 42.9036 1.38183
\(965\) 0 0
\(966\) −8.03888 −0.258647
\(967\) 23.0493 0.741216 0.370608 0.928789i \(-0.379149\pi\)
0.370608 + 0.928789i \(0.379149\pi\)
\(968\) 0.598152 0.0192253
\(969\) 6.53528 0.209944
\(970\) 0 0
\(971\) 14.9193 0.478783 0.239391 0.970923i \(-0.423052\pi\)
0.239391 + 0.970923i \(0.423052\pi\)
\(972\) 37.9025 1.21572
\(973\) −3.44370 −0.110400
\(974\) −12.2040 −0.391041
\(975\) 0 0
\(976\) −25.1364 −0.804595
\(977\) 23.3641 0.747485 0.373742 0.927533i \(-0.378074\pi\)
0.373742 + 0.927533i \(0.378074\pi\)
\(978\) −3.72052 −0.118969
\(979\) −33.4850 −1.07019
\(980\) 0 0
\(981\) 36.0538 1.15111
\(982\) −11.6962 −0.373241
\(983\) −5.31119 −0.169401 −0.0847003 0.996406i \(-0.526993\pi\)
−0.0847003 + 0.996406i \(0.526993\pi\)
\(984\) 9.63570 0.307175
\(985\) 0 0
\(986\) 1.93421 0.0615978
\(987\) −62.0112 −1.97384
\(988\) 0 0
\(989\) −23.5185 −0.747846
\(990\) 0 0
\(991\) −24.0879 −0.765178 −0.382589 0.923919i \(-0.624967\pi\)
−0.382589 + 0.923919i \(0.624967\pi\)
\(992\) −13.9307 −0.442301
\(993\) 8.00299 0.253967
\(994\) −5.81998 −0.184599
\(995\) 0 0
\(996\) −43.8648 −1.38991
\(997\) 20.4099 0.646389 0.323195 0.946332i \(-0.395243\pi\)
0.323195 + 0.946332i \(0.395243\pi\)
\(998\) −5.36581 −0.169852
\(999\) 6.53528 0.206767
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.bq.1.4 6
5.2 odd 4 845.2.b.e.339.4 6
5.3 odd 4 845.2.b.e.339.3 6
5.4 even 2 inner 4225.2.a.bq.1.3 6
13.4 even 6 325.2.e.e.276.4 12
13.10 even 6 325.2.e.e.126.4 12
13.12 even 2 4225.2.a.br.1.3 6
65.2 even 12 845.2.l.f.654.6 24
65.3 odd 12 845.2.n.e.529.4 12
65.4 even 6 325.2.e.e.276.3 12
65.7 even 12 845.2.l.f.699.7 24
65.8 even 4 845.2.d.d.844.8 12
65.12 odd 4 845.2.b.d.339.3 6
65.17 odd 12 65.2.n.a.29.3 yes 12
65.18 even 4 845.2.d.d.844.6 12
65.22 odd 12 845.2.n.e.484.4 12
65.23 odd 12 65.2.n.a.9.3 12
65.28 even 12 845.2.l.f.654.7 24
65.32 even 12 845.2.l.f.699.5 24
65.33 even 12 845.2.l.f.699.6 24
65.37 even 12 845.2.l.f.654.8 24
65.38 odd 4 845.2.b.d.339.4 6
65.42 odd 12 845.2.n.e.529.3 12
65.43 odd 12 65.2.n.a.29.4 yes 12
65.47 even 4 845.2.d.d.844.5 12
65.48 odd 12 845.2.n.e.484.3 12
65.49 even 6 325.2.e.e.126.3 12
65.57 even 4 845.2.d.d.844.7 12
65.58 even 12 845.2.l.f.699.8 24
65.62 odd 12 65.2.n.a.9.4 yes 12
65.63 even 12 845.2.l.f.654.5 24
65.64 even 2 4225.2.a.br.1.4 6
195.17 even 12 585.2.bs.a.289.4 12
195.23 even 12 585.2.bs.a.334.4 12
195.62 even 12 585.2.bs.a.334.3 12
195.173 even 12 585.2.bs.a.289.3 12
260.23 even 12 1040.2.dh.a.529.6 12
260.43 even 12 1040.2.dh.a.289.1 12
260.127 even 12 1040.2.dh.a.529.1 12
260.147 even 12 1040.2.dh.a.289.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.n.a.9.3 12 65.23 odd 12
65.2.n.a.9.4 yes 12 65.62 odd 12
65.2.n.a.29.3 yes 12 65.17 odd 12
65.2.n.a.29.4 yes 12 65.43 odd 12
325.2.e.e.126.3 12 65.49 even 6
325.2.e.e.126.4 12 13.10 even 6
325.2.e.e.276.3 12 65.4 even 6
325.2.e.e.276.4 12 13.4 even 6
585.2.bs.a.289.3 12 195.173 even 12
585.2.bs.a.289.4 12 195.17 even 12
585.2.bs.a.334.3 12 195.62 even 12
585.2.bs.a.334.4 12 195.23 even 12
845.2.b.d.339.3 6 65.12 odd 4
845.2.b.d.339.4 6 65.38 odd 4
845.2.b.e.339.3 6 5.3 odd 4
845.2.b.e.339.4 6 5.2 odd 4
845.2.d.d.844.5 12 65.47 even 4
845.2.d.d.844.6 12 65.18 even 4
845.2.d.d.844.7 12 65.57 even 4
845.2.d.d.844.8 12 65.8 even 4
845.2.l.f.654.5 24 65.63 even 12
845.2.l.f.654.6 24 65.2 even 12
845.2.l.f.654.7 24 65.28 even 12
845.2.l.f.654.8 24 65.37 even 12
845.2.l.f.699.5 24 65.32 even 12
845.2.l.f.699.6 24 65.33 even 12
845.2.l.f.699.7 24 65.7 even 12
845.2.l.f.699.8 24 65.58 even 12
845.2.n.e.484.3 12 65.48 odd 12
845.2.n.e.484.4 12 65.22 odd 12
845.2.n.e.529.3 12 65.42 odd 12
845.2.n.e.529.4 12 65.3 odd 12
1040.2.dh.a.289.1 12 260.43 even 12
1040.2.dh.a.289.6 12 260.147 even 12
1040.2.dh.a.529.1 12 260.127 even 12
1040.2.dh.a.529.6 12 260.23 even 12
4225.2.a.bq.1.3 6 5.4 even 2 inner
4225.2.a.bq.1.4 6 1.1 even 1 trivial
4225.2.a.br.1.3 6 13.12 even 2
4225.2.a.br.1.4 6 65.64 even 2