Properties

Label 4225.2.a.bq.1.3
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 1.3
Root \(-0.330837\) of defining polynomial
Character \(\chi\) \(=\) 4225.1

$q$-expansion

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

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 \(-0.537318\pi\)
−0.116968 + 0.993136i \(0.537318\pi\)
\(3\) −2.69180 −1.55411 −0.777057 0.629430i \(-0.783288\pi\)
−0.777057 + 0.629430i \(0.783288\pi\)
\(4\) −1.89055 −0.945274
\(5\) 0 0
\(6\) 0.890547 0.363564
\(7\) −3.35348 −1.26750 −0.633748 0.773540i \(-0.718484\pi\)
−0.633748 + 0.773540i \(0.718484\pi\)
\(8\) 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 \(-0.575944\pi\)
−0.236328 + 0.971673i \(0.575944\pi\)
\(18\) −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 \(-0.409453\pi\)
0.280640 + 0.959813i \(0.409453\pi\)
\(24\) −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 \(-0.551211\pi\)
−0.160191 + 0.987086i \(0.551211\pi\)
\(38\) −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 \(-0.732079\pi\)
−0.666197 + 0.745776i \(0.732079\pi\)
\(44\) −6.13636 −0.925091
\(45\) 0 0
\(46\) −0.890547 −0.131304
\(47\) 6.86960 1.00203 0.501017 0.865437i \(-0.332959\pi\)
0.501017 + 0.865437i \(0.332959\pi\)
\(48\) −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 \(-0.843407\pi\)
−0.881412 + 0.472347i \(0.843407\pi\)
\(54\) 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 \(-0.421125\pi\)
0.245264 + 0.969456i \(0.421125\pi\)
\(68\) 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 \(-0.396381\pi\)
0.319811 + 0.947481i \(0.396381\pi\)
\(74\) 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 \(-0.656850\pi\)
−0.473059 + 0.881031i \(0.656850\pi\)
\(84\) −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 \(-0.413854\pi\)
0.267344 + 0.963601i \(0.413854\pi\)
\(98\) −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 \(-0.618256\pi\)
−0.363026 + 0.931779i \(0.618256\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 4.24581 0.412390
\(107\) 8.57444 0.828922 0.414461 0.910067i \(-0.363970\pi\)
0.414461 + 0.910067i \(0.363970\pi\)
\(108\) 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 \(-0.612083\pi\)
−0.344888 + 0.938644i \(0.612083\pi\)
\(114\) 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 \(-0.633276\pi\)
−0.406573 + 0.913618i \(0.633276\pi\)
\(128\) 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 \(-0.755740\pi\)
−0.719741 + 0.694242i \(0.755740\pi\)
\(138\) 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 \(-0.356468\pi\)
0.435794 + 0.900047i \(0.356468\pi\)
\(158\) −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 \(-0.447684\pi\)
0.163615 + 0.986524i \(0.447684\pi\)
\(164\) 5.25779 0.410564
\(165\) 0 0
\(166\) 2.85166 0.221332
\(167\) 3.35348 0.259500 0.129750 0.991547i \(-0.458583\pi\)
0.129750 + 0.991547i \(0.458583\pi\)
\(168\) 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 \(-0.392231\pi\)
0.332134 + 0.943232i \(0.392231\pi\)
\(174\) 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 \(-0.247171\pi\)
0.713362 + 0.700795i \(0.247171\pi\)
\(194\) −1.74221 −0.125083
\(195\) 0 0
\(196\) −8.02690 −0.573350
\(197\) −21.6520 −1.54264 −0.771319 0.636448i \(-0.780403\pi\)
−0.771319 + 0.636448i \(0.780403\pi\)
\(198\) −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 \(-0.635940\pi\)
−0.414203 + 0.910184i \(0.635940\pi\)
\(224\) 12.3553 0.825521
\(225\) 0 0
\(226\) 2.42583 0.161364
\(227\) 6.16282 0.409040 0.204520 0.978862i \(-0.434437\pi\)
0.204520 + 0.978862i \(0.434437\pi\)
\(228\) 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 \(-0.491404\pi\)
0.0270015 + 0.999635i \(0.491404\pi\)
\(234\) 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 \(-0.520977\pi\)
−0.0658544 + 0.997829i \(0.520977\pi\)
\(258\) −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 \(-0.126232\pi\)
0.922391 + 0.386257i \(0.126232\pi\)
\(264\) −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 \(-0.633349\pi\)
−0.406780 + 0.913526i \(0.633349\pi\)
\(278\) 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 \(-0.596658\pi\)
−0.299015 + 0.954248i \(0.596658\pi\)
\(284\) 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 \(-0.371478\pi\)
0.392882 + 0.919589i \(0.371478\pi\)
\(294\) 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 \(-0.252193\pi\)
0.702219 + 0.711961i \(0.252193\pi\)
\(308\) 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 \(-0.316396\pi\)
0.545353 + 0.838207i \(0.316396\pi\)
\(314\) −3.61305 −0.203896
\(315\) 0 0
\(316\) −26.0538 −1.46564
\(317\) −28.8217 −1.61879 −0.809395 0.587265i \(-0.800205\pi\)
−0.809395 + 0.587265i \(0.800205\pi\)
\(318\) −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 \(-0.516512\pi\)
−0.0518505 + 0.998655i \(0.516512\pi\)
\(338\) 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 \(-0.389866\pi\)
0.339133 + 0.940738i \(0.389866\pi\)
\(348\) 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 \(-0.865062\pi\)
−0.911484 + 0.411336i \(0.865062\pi\)
\(354\) 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 \(-0.388086\pi\)
0.344388 + 0.938827i \(0.388086\pi\)
\(368\) 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 \(-0.370709\pi\)
0.395103 + 0.918637i \(0.370709\pi\)
\(374\) 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 \(-0.488279\pi\)
0.0368128 + 0.999322i \(0.488279\pi\)
\(384\) −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 \(-0.641109\pi\)
−0.428928 + 0.903338i \(0.641109\pi\)
\(398\) −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 \(-0.166189\pi\)
0.866775 + 0.498700i \(0.166189\pi\)
\(434\) −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 \(-0.347899\pi\)
0.459860 + 0.887991i \(0.347899\pi\)
\(444\) −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 \(-0.443362\pi\)
0.176996 + 0.984212i \(0.443362\pi\)
\(458\) −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 \(-0.321762\pi\)
0.531146 + 0.847281i \(0.321762\pi\)
\(464\) 10.0658 0.467293
\(465\) 0 0
\(466\) −0.272715 −0.0126333
\(467\) 15.2976 0.707889 0.353945 0.935266i \(-0.384840\pi\)
0.353945 + 0.935266i \(0.384840\pi\)
\(468\) 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 \(-0.185014\pi\)
0.835783 + 0.549060i \(0.185014\pi\)
\(488\) −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 \(-0.649001\pi\)
−0.451193 + 0.892427i \(0.649001\pi\)
\(504\) −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 \(-0.417387\pi\)
0.256633 + 0.966509i \(0.417387\pi\)
\(524\) −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 \(-0.456935\pi\)
0.134882 + 0.990862i \(0.456935\pi\)
\(548\) 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 \(-0.225567\pi\)
0.759247 + 0.650802i \(0.225567\pi\)
\(558\) 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 \(-0.533615\pi\)
−0.105407 + 0.994429i \(0.533615\pi\)
\(564\) 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 \(-0.427003\pi\)
0.227323 + 0.973819i \(0.427003\pi\)
\(578\) 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 \(-0.185604\pi\)
0.834764 + 0.550607i \(0.185604\pi\)
\(588\) 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 \(-0.490345\pi\)
0.0303284 + 0.999540i \(0.490345\pi\)
\(594\) 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 \(-0.430369\pi\)
0.217012 + 0.976169i \(0.430369\pi\)
\(608\) −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 \(-0.460808\pi\)
0.122815 + 0.992430i \(0.460808\pi\)
\(614\) −8.14114 −0.328550
\(615\) 0 0
\(616\) −14.0101 −0.564485
\(617\) 31.8388 1.28178 0.640892 0.767631i \(-0.278565\pi\)
0.640892 + 0.767631i \(0.278565\pi\)
\(618\) −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 \(-0.718604\pi\)
−0.634039 + 0.773301i \(0.718604\pi\)
\(644\) 17.0658 0.672486
\(645\) 0 0
\(646\) 0.803220 0.0316023
\(647\) 13.7843 0.541917 0.270959 0.962591i \(-0.412659\pi\)
0.270959 + 0.962591i \(0.412659\pi\)
\(648\) −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 \(-0.553200\pi\)
−0.166355 + 0.986066i \(0.553200\pi\)
\(654\) 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 \(-0.287871\pi\)
0.618179 + 0.786037i \(0.287871\pi\)
\(674\) 0.629812 0.0242595
\(675\) 0 0
\(676\) 0 0
\(677\) −14.2382 −0.547220 −0.273610 0.961841i \(-0.588218\pi\)
−0.273610 + 0.961841i \(0.588218\pi\)
\(678\) −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 \(-0.664214\pi\)
−0.493311 + 0.869853i \(0.664214\pi\)
\(684\) −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 \(-0.717872\pi\)
−0.632259 + 0.774757i \(0.717872\pi\)
\(728\) 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 \(-0.585633\pi\)
−0.265790 + 0.964031i \(0.585633\pi\)
\(734\) −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 \(-0.231582\pi\)
0.746816 + 0.665031i \(0.231582\pi\)
\(744\) 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 \(-0.423890\pi\)
0.236836 + 0.971550i \(0.423890\pi\)
\(758\) 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 \(-0.164606\pi\)
0.869244 + 0.494384i \(0.164606\pi\)
\(774\) 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 \(-0.666062\pi\)
−0.498355 + 0.866973i \(0.666062\pi\)
\(788\) 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 \(-0.729273\pi\)
−0.659597 + 0.751619i \(0.729273\pi\)
\(798\) −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 \(-0.257558\pi\)
0.690120 + 0.723695i \(0.257558\pi\)
\(824\) −9.48442 −0.330405
\(825\) 0 0
\(826\) 2.81278 0.0978691
\(827\) −26.5639 −0.923716 −0.461858 0.886954i \(-0.652817\pi\)
−0.461858 + 0.886954i \(0.652817\pi\)
\(828\) −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 \(-0.654571\pi\)
−0.466739 + 0.884395i \(0.654571\pi\)
\(854\) −8.31160 −0.284417
\(855\) 0 0
\(856\) 11.0365 0.377219
\(857\) 50.6201 1.72915 0.864575 0.502503i \(-0.167587\pi\)
0.864575 + 0.502503i \(0.167587\pi\)
\(858\) 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 \(-0.454420\pi\)
0.142706 + 0.989765i \(0.454420\pi\)
\(864\) 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 \(-0.892557\pi\)
−0.943572 + 0.331169i \(0.892557\pi\)
\(878\) −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 \(-0.326666\pi\)
0.518029 + 0.855363i \(0.326666\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −6.40429 −0.215156
\(887\) 12.4721 0.418771 0.209385 0.977833i \(-0.432854\pi\)
0.209385 + 0.977833i \(0.432854\pi\)
\(888\) 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 \(-0.276575\pi\)
0.645678 + 0.763609i \(0.276575\pi\)
\(908\) −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 \(-0.469942\pi\)
0.0942884 + 0.995545i \(0.469942\pi\)
\(938\) 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 \(-0.509809\pi\)
−0.0308120 + 0.999525i \(0.509809\pi\)
\(948\) 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 \(-0.683490\pi\)
−0.545053 + 0.838402i \(0.683490\pi\)
\(954\) 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 \(-0.620851\pi\)
−0.370608 + 0.928789i \(0.620851\pi\)
\(968\) −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 \(-0.621926\pi\)
−0.373742 + 0.927533i \(0.621926\pi\)
\(978\) 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 \(-0.473007\pi\)
0.0847003 + 0.996406i \(0.473007\pi\)
\(984\) 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 \(-0.604757\pi\)
−0.323195 + 0.946332i \(0.604757\pi\)
\(998\) 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.1.3 6
5.2 odd 4 845.2.b.e.339.3 6
5.3 odd 4 845.2.b.e.339.4 6
5.4 even 2 inner 4225.2.a.bq.1.4 6
13.4 even 6 325.2.e.e.276.3 12
13.10 even 6 325.2.e.e.126.3 12
13.12 even 2 4225.2.a.br.1.4 6
65.2 even 12 845.2.l.f.654.7 24
65.3 odd 12 845.2.n.e.529.3 12
65.4 even 6 325.2.e.e.276.4 12
65.7 even 12 845.2.l.f.699.6 24
65.8 even 4 845.2.d.d.844.5 12
65.12 odd 4 845.2.b.d.339.4 6
65.17 odd 12 65.2.n.a.29.4 yes 12
65.18 even 4 845.2.d.d.844.7 12
65.22 odd 12 845.2.n.e.484.3 12
65.23 odd 12 65.2.n.a.9.4 yes 12
65.28 even 12 845.2.l.f.654.6 24
65.32 even 12 845.2.l.f.699.8 24
65.33 even 12 845.2.l.f.699.7 24
65.37 even 12 845.2.l.f.654.5 24
65.38 odd 4 845.2.b.d.339.3 6
65.42 odd 12 845.2.n.e.529.4 12
65.43 odd 12 65.2.n.a.29.3 yes 12
65.47 even 4 845.2.d.d.844.8 12
65.48 odd 12 845.2.n.e.484.4 12
65.49 even 6 325.2.e.e.126.4 12
65.57 even 4 845.2.d.d.844.6 12
65.58 even 12 845.2.l.f.699.5 24
65.62 odd 12 65.2.n.a.9.3 12
65.63 even 12 845.2.l.f.654.8 24
65.64 even 2 4225.2.a.br.1.3 6
195.17 even 12 585.2.bs.a.289.3 12
195.23 even 12 585.2.bs.a.334.3 12
195.62 even 12 585.2.bs.a.334.4 12
195.173 even 12 585.2.bs.a.289.4 12
260.23 even 12 1040.2.dh.a.529.1 12
260.43 even 12 1040.2.dh.a.289.6 12
260.127 even 12 1040.2.dh.a.529.6 12
260.147 even 12 1040.2.dh.a.289.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.n.a.9.3 12 65.62 odd 12
65.2.n.a.9.4 yes 12 65.23 odd 12
65.2.n.a.29.3 yes 12 65.43 odd 12
65.2.n.a.29.4 yes 12 65.17 odd 12
325.2.e.e.126.3 12 13.10 even 6
325.2.e.e.126.4 12 65.49 even 6
325.2.e.e.276.3 12 13.4 even 6
325.2.e.e.276.4 12 65.4 even 6
585.2.bs.a.289.3 12 195.17 even 12
585.2.bs.a.289.4 12 195.173 even 12
585.2.bs.a.334.3 12 195.23 even 12
585.2.bs.a.334.4 12 195.62 even 12
845.2.b.d.339.3 6 65.38 odd 4
845.2.b.d.339.4 6 65.12 odd 4
845.2.b.e.339.3 6 5.2 odd 4
845.2.b.e.339.4 6 5.3 odd 4
845.2.d.d.844.5 12 65.8 even 4
845.2.d.d.844.6 12 65.57 even 4
845.2.d.d.844.7 12 65.18 even 4
845.2.d.d.844.8 12 65.47 even 4
845.2.l.f.654.5 24 65.37 even 12
845.2.l.f.654.6 24 65.28 even 12
845.2.l.f.654.7 24 65.2 even 12
845.2.l.f.654.8 24 65.63 even 12
845.2.l.f.699.5 24 65.58 even 12
845.2.l.f.699.6 24 65.7 even 12
845.2.l.f.699.7 24 65.33 even 12
845.2.l.f.699.8 24 65.32 even 12
845.2.n.e.484.3 12 65.22 odd 12
845.2.n.e.484.4 12 65.48 odd 12
845.2.n.e.529.3 12 65.3 odd 12
845.2.n.e.529.4 12 65.42 odd 12
1040.2.dh.a.289.1 12 260.147 even 12
1040.2.dh.a.289.6 12 260.43 even 12
1040.2.dh.a.529.1 12 260.23 even 12
1040.2.dh.a.529.6 12 260.127 even 12
4225.2.a.bq.1.3 6 1.1 even 1 trivial
4225.2.a.bq.1.4 6 5.4 even 2 inner
4225.2.a.br.1.3 6 65.64 even 2
4225.2.a.br.1.4 6 13.12 even 2