Properties

Label 5265.2.a.ba.1.3
Level $5265$
Weight $2$
Character 5265.1
Self dual yes
Analytic conductor $42.041$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5265.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.0412366642\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 31x^{5} - x^{4} - 70x^{3} + 66x^{2} - 19x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 585)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.63404\) of defining polynomial
Character \(\chi\) \(=\) 5265.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.63404 q^{2} +0.670078 q^{4} +1.00000 q^{5} +2.13012 q^{7} +2.17314 q^{8} +O(q^{10})\) \(q-1.63404 q^{2} +0.670078 q^{4} +1.00000 q^{5} +2.13012 q^{7} +2.17314 q^{8} -1.63404 q^{10} +0.0526457 q^{11} -1.00000 q^{13} -3.48069 q^{14} -4.89115 q^{16} +2.48047 q^{17} +2.13105 q^{19} +0.670078 q^{20} -0.0860250 q^{22} -4.93083 q^{23} +1.00000 q^{25} +1.63404 q^{26} +1.42734 q^{28} +2.48694 q^{29} -8.16059 q^{31} +3.64604 q^{32} -4.05318 q^{34} +2.13012 q^{35} -1.11963 q^{37} -3.48221 q^{38} +2.17314 q^{40} +5.47417 q^{41} -9.47722 q^{43} +0.0352767 q^{44} +8.05716 q^{46} -9.76560 q^{47} -2.46260 q^{49} -1.63404 q^{50} -0.670078 q^{52} +3.64091 q^{53} +0.0526457 q^{55} +4.62905 q^{56} -4.06374 q^{58} -7.49087 q^{59} -5.78404 q^{61} +13.3347 q^{62} +3.82454 q^{64} -1.00000 q^{65} -6.23591 q^{67} +1.66211 q^{68} -3.48069 q^{70} -2.50050 q^{71} -1.10245 q^{73} +1.82952 q^{74} +1.42797 q^{76} +0.112141 q^{77} -15.6156 q^{79} -4.89115 q^{80} -8.94500 q^{82} -0.489152 q^{83} +2.48047 q^{85} +15.4861 q^{86} +0.114407 q^{88} +4.64901 q^{89} -2.13012 q^{91} -3.30404 q^{92} +15.9574 q^{94} +2.13105 q^{95} +7.34097 q^{97} +4.02399 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8} - 3 q^{10} - 6 q^{11} - 8 q^{13} - 10 q^{14} + 11 q^{16} + 2 q^{17} - 10 q^{19} + 9 q^{20} + 3 q^{22} - 6 q^{23} + 8 q^{25} + 3 q^{26} - 34 q^{28} - 14 q^{29} - 31 q^{31} - q^{32} - 7 q^{34} - 11 q^{35} + q^{37} - 9 q^{38} + 6 q^{40} + 12 q^{41} + 15 q^{43} - 16 q^{44} - 32 q^{46} + 18 q^{47} + 17 q^{49} - 3 q^{50} - 9 q^{52} - 2 q^{53} - 6 q^{55} - 16 q^{56} - 42 q^{58} - 24 q^{59} - 9 q^{61} + 20 q^{62} - 30 q^{64} - 8 q^{65} - 18 q^{67} + 14 q^{68} - 10 q^{70} - 10 q^{71} + 6 q^{73} + 37 q^{74} - 53 q^{76} + 34 q^{77} - 3 q^{79} + 11 q^{80} - 34 q^{82} + 10 q^{83} + 2 q^{85} - 60 q^{86} - 14 q^{88} + 13 q^{89} + 11 q^{91} - 5 q^{92} + 17 q^{94} - 10 q^{95} - 34 q^{97} + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.63404 −1.15544 −0.577719 0.816235i \(-0.696057\pi\)
−0.577719 + 0.816235i \(0.696057\pi\)
\(3\) 0 0
\(4\) 0.670078 0.335039
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.13012 0.805108 0.402554 0.915396i \(-0.368123\pi\)
0.402554 + 0.915396i \(0.368123\pi\)
\(8\) 2.17314 0.768322
\(9\) 0 0
\(10\) −1.63404 −0.516728
\(11\) 0.0526457 0.0158733 0.00793664 0.999969i \(-0.497474\pi\)
0.00793664 + 0.999969i \(0.497474\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −3.48069 −0.930253
\(15\) 0 0
\(16\) −4.89115 −1.22279
\(17\) 2.48047 0.601603 0.300801 0.953687i \(-0.402746\pi\)
0.300801 + 0.953687i \(0.402746\pi\)
\(18\) 0 0
\(19\) 2.13105 0.488895 0.244448 0.969662i \(-0.421393\pi\)
0.244448 + 0.969662i \(0.421393\pi\)
\(20\) 0.670078 0.149834
\(21\) 0 0
\(22\) −0.0860250 −0.0183406
\(23\) −4.93083 −1.02815 −0.514075 0.857745i \(-0.671865\pi\)
−0.514075 + 0.857745i \(0.671865\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.63404 0.320461
\(27\) 0 0
\(28\) 1.42734 0.269743
\(29\) 2.48694 0.461812 0.230906 0.972976i \(-0.425831\pi\)
0.230906 + 0.972976i \(0.425831\pi\)
\(30\) 0 0
\(31\) −8.16059 −1.46569 −0.732843 0.680398i \(-0.761807\pi\)
−0.732843 + 0.680398i \(0.761807\pi\)
\(32\) 3.64604 0.644535
\(33\) 0 0
\(34\) −4.05318 −0.695115
\(35\) 2.13012 0.360055
\(36\) 0 0
\(37\) −1.11963 −0.184066 −0.0920331 0.995756i \(-0.529337\pi\)
−0.0920331 + 0.995756i \(0.529337\pi\)
\(38\) −3.48221 −0.564889
\(39\) 0 0
\(40\) 2.17314 0.343604
\(41\) 5.47417 0.854922 0.427461 0.904034i \(-0.359408\pi\)
0.427461 + 0.904034i \(0.359408\pi\)
\(42\) 0 0
\(43\) −9.47722 −1.44526 −0.722632 0.691233i \(-0.757068\pi\)
−0.722632 + 0.691233i \(0.757068\pi\)
\(44\) 0.0352767 0.00531816
\(45\) 0 0
\(46\) 8.05716 1.18796
\(47\) −9.76560 −1.42446 −0.712230 0.701946i \(-0.752315\pi\)
−0.712230 + 0.701946i \(0.752315\pi\)
\(48\) 0 0
\(49\) −2.46260 −0.351801
\(50\) −1.63404 −0.231088
\(51\) 0 0
\(52\) −0.670078 −0.0929231
\(53\) 3.64091 0.500118 0.250059 0.968231i \(-0.419550\pi\)
0.250059 + 0.968231i \(0.419550\pi\)
\(54\) 0 0
\(55\) 0.0526457 0.00709875
\(56\) 4.62905 0.618582
\(57\) 0 0
\(58\) −4.06374 −0.533596
\(59\) −7.49087 −0.975228 −0.487614 0.873059i \(-0.662133\pi\)
−0.487614 + 0.873059i \(0.662133\pi\)
\(60\) 0 0
\(61\) −5.78404 −0.740570 −0.370285 0.928918i \(-0.620740\pi\)
−0.370285 + 0.928918i \(0.620740\pi\)
\(62\) 13.3347 1.69351
\(63\) 0 0
\(64\) 3.82454 0.478067
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −6.23591 −0.761837 −0.380919 0.924609i \(-0.624392\pi\)
−0.380919 + 0.924609i \(0.624392\pi\)
\(68\) 1.66211 0.201560
\(69\) 0 0
\(70\) −3.48069 −0.416022
\(71\) −2.50050 −0.296755 −0.148377 0.988931i \(-0.547405\pi\)
−0.148377 + 0.988931i \(0.547405\pi\)
\(72\) 0 0
\(73\) −1.10245 −0.129032 −0.0645159 0.997917i \(-0.520550\pi\)
−0.0645159 + 0.997917i \(0.520550\pi\)
\(74\) 1.82952 0.212677
\(75\) 0 0
\(76\) 1.42797 0.163799
\(77\) 0.112141 0.0127797
\(78\) 0 0
\(79\) −15.6156 −1.75689 −0.878445 0.477844i \(-0.841418\pi\)
−0.878445 + 0.477844i \(0.841418\pi\)
\(80\) −4.89115 −0.546847
\(81\) 0 0
\(82\) −8.94500 −0.987810
\(83\) −0.489152 −0.0536914 −0.0268457 0.999640i \(-0.508546\pi\)
−0.0268457 + 0.999640i \(0.508546\pi\)
\(84\) 0 0
\(85\) 2.48047 0.269045
\(86\) 15.4861 1.66991
\(87\) 0 0
\(88\) 0.114407 0.0121958
\(89\) 4.64901 0.492794 0.246397 0.969169i \(-0.420753\pi\)
0.246397 + 0.969169i \(0.420753\pi\)
\(90\) 0 0
\(91\) −2.13012 −0.223297
\(92\) −3.30404 −0.344470
\(93\) 0 0
\(94\) 15.9574 1.64588
\(95\) 2.13105 0.218641
\(96\) 0 0
\(97\) 7.34097 0.745363 0.372682 0.927959i \(-0.378438\pi\)
0.372682 + 0.927959i \(0.378438\pi\)
\(98\) 4.02399 0.406484
\(99\) 0 0
\(100\) 0.670078 0.0670078
\(101\) −0.552859 −0.0550115 −0.0275058 0.999622i \(-0.508756\pi\)
−0.0275058 + 0.999622i \(0.508756\pi\)
\(102\) 0 0
\(103\) −7.83795 −0.772296 −0.386148 0.922437i \(-0.626195\pi\)
−0.386148 + 0.922437i \(0.626195\pi\)
\(104\) −2.17314 −0.213094
\(105\) 0 0
\(106\) −5.94939 −0.577855
\(107\) −19.4173 −1.87714 −0.938572 0.345084i \(-0.887851\pi\)
−0.938572 + 0.345084i \(0.887851\pi\)
\(108\) 0 0
\(109\) −2.13502 −0.204498 −0.102249 0.994759i \(-0.532604\pi\)
−0.102249 + 0.994759i \(0.532604\pi\)
\(110\) −0.0860250 −0.00820217
\(111\) 0 0
\(112\) −10.4187 −0.984477
\(113\) 6.23529 0.586567 0.293283 0.956026i \(-0.405252\pi\)
0.293283 + 0.956026i \(0.405252\pi\)
\(114\) 0 0
\(115\) −4.93083 −0.459802
\(116\) 1.66644 0.154725
\(117\) 0 0
\(118\) 12.2404 1.12682
\(119\) 5.28369 0.484355
\(120\) 0 0
\(121\) −10.9972 −0.999748
\(122\) 9.45133 0.855683
\(123\) 0 0
\(124\) −5.46823 −0.491062
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −10.6036 −0.940918 −0.470459 0.882422i \(-0.655912\pi\)
−0.470459 + 0.882422i \(0.655912\pi\)
\(128\) −13.5415 −1.19691
\(129\) 0 0
\(130\) 1.63404 0.143315
\(131\) 16.1923 1.41473 0.707365 0.706849i \(-0.249884\pi\)
0.707365 + 0.706849i \(0.249884\pi\)
\(132\) 0 0
\(133\) 4.53938 0.393614
\(134\) 10.1897 0.880256
\(135\) 0 0
\(136\) 5.39042 0.462224
\(137\) 20.7686 1.77438 0.887189 0.461406i \(-0.152655\pi\)
0.887189 + 0.461406i \(0.152655\pi\)
\(138\) 0 0
\(139\) −2.50075 −0.212111 −0.106055 0.994360i \(-0.533822\pi\)
−0.106055 + 0.994360i \(0.533822\pi\)
\(140\) 1.42734 0.120633
\(141\) 0 0
\(142\) 4.08591 0.342882
\(143\) −0.0526457 −0.00440245
\(144\) 0 0
\(145\) 2.48694 0.206529
\(146\) 1.80144 0.149088
\(147\) 0 0
\(148\) −0.750240 −0.0616693
\(149\) 1.89025 0.154856 0.0774279 0.996998i \(-0.475329\pi\)
0.0774279 + 0.996998i \(0.475329\pi\)
\(150\) 0 0
\(151\) −4.88685 −0.397686 −0.198843 0.980031i \(-0.563718\pi\)
−0.198843 + 0.980031i \(0.563718\pi\)
\(152\) 4.63107 0.375629
\(153\) 0 0
\(154\) −0.183243 −0.0147662
\(155\) −8.16059 −0.655475
\(156\) 0 0
\(157\) −17.9998 −1.43654 −0.718269 0.695765i \(-0.755065\pi\)
−0.718269 + 0.695765i \(0.755065\pi\)
\(158\) 25.5164 2.02998
\(159\) 0 0
\(160\) 3.64604 0.288245
\(161\) −10.5032 −0.827772
\(162\) 0 0
\(163\) 9.41748 0.737634 0.368817 0.929502i \(-0.379763\pi\)
0.368817 + 0.929502i \(0.379763\pi\)
\(164\) 3.66812 0.286432
\(165\) 0 0
\(166\) 0.799292 0.0620371
\(167\) 15.4905 1.19869 0.599344 0.800492i \(-0.295428\pi\)
0.599344 + 0.800492i \(0.295428\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −4.05318 −0.310865
\(171\) 0 0
\(172\) −6.35048 −0.484219
\(173\) 9.67663 0.735700 0.367850 0.929885i \(-0.380094\pi\)
0.367850 + 0.929885i \(0.380094\pi\)
\(174\) 0 0
\(175\) 2.13012 0.161022
\(176\) −0.257498 −0.0194096
\(177\) 0 0
\(178\) −7.59665 −0.569393
\(179\) 3.31630 0.247872 0.123936 0.992290i \(-0.460448\pi\)
0.123936 + 0.992290i \(0.460448\pi\)
\(180\) 0 0
\(181\) −22.6189 −1.68125 −0.840623 0.541621i \(-0.817811\pi\)
−0.840623 + 0.541621i \(0.817811\pi\)
\(182\) 3.48069 0.258006
\(183\) 0 0
\(184\) −10.7154 −0.789950
\(185\) −1.11963 −0.0823169
\(186\) 0 0
\(187\) 0.130586 0.00954940
\(188\) −6.54371 −0.477249
\(189\) 0 0
\(190\) −3.48221 −0.252626
\(191\) −5.88211 −0.425614 −0.212807 0.977094i \(-0.568261\pi\)
−0.212807 + 0.977094i \(0.568261\pi\)
\(192\) 0 0
\(193\) −0.459539 −0.0330783 −0.0165392 0.999863i \(-0.505265\pi\)
−0.0165392 + 0.999863i \(0.505265\pi\)
\(194\) −11.9954 −0.861221
\(195\) 0 0
\(196\) −1.65014 −0.117867
\(197\) 2.31119 0.164665 0.0823326 0.996605i \(-0.473763\pi\)
0.0823326 + 0.996605i \(0.473763\pi\)
\(198\) 0 0
\(199\) 24.3734 1.72779 0.863893 0.503675i \(-0.168019\pi\)
0.863893 + 0.503675i \(0.168019\pi\)
\(200\) 2.17314 0.153664
\(201\) 0 0
\(202\) 0.903392 0.0635624
\(203\) 5.29746 0.371809
\(204\) 0 0
\(205\) 5.47417 0.382333
\(206\) 12.8075 0.892341
\(207\) 0 0
\(208\) 4.89115 0.339140
\(209\) 0.112190 0.00776037
\(210\) 0 0
\(211\) 27.3938 1.88587 0.942934 0.332980i \(-0.108054\pi\)
0.942934 + 0.332980i \(0.108054\pi\)
\(212\) 2.43969 0.167559
\(213\) 0 0
\(214\) 31.7286 2.16893
\(215\) −9.47722 −0.646341
\(216\) 0 0
\(217\) −17.3830 −1.18004
\(218\) 3.48870 0.236285
\(219\) 0 0
\(220\) 0.0352767 0.00237836
\(221\) −2.48047 −0.166855
\(222\) 0 0
\(223\) 25.3359 1.69661 0.848307 0.529504i \(-0.177622\pi\)
0.848307 + 0.529504i \(0.177622\pi\)
\(224\) 7.76649 0.518920
\(225\) 0 0
\(226\) −10.1887 −0.677742
\(227\) 21.2417 1.40986 0.704931 0.709276i \(-0.250978\pi\)
0.704931 + 0.709276i \(0.250978\pi\)
\(228\) 0 0
\(229\) 12.2337 0.808427 0.404214 0.914665i \(-0.367545\pi\)
0.404214 + 0.914665i \(0.367545\pi\)
\(230\) 8.05716 0.531274
\(231\) 0 0
\(232\) 5.40446 0.354820
\(233\) −11.4048 −0.747154 −0.373577 0.927599i \(-0.621869\pi\)
−0.373577 + 0.927599i \(0.621869\pi\)
\(234\) 0 0
\(235\) −9.76560 −0.637038
\(236\) −5.01947 −0.326739
\(237\) 0 0
\(238\) −8.63375 −0.559643
\(239\) −14.8571 −0.961025 −0.480512 0.876988i \(-0.659549\pi\)
−0.480512 + 0.876988i \(0.659549\pi\)
\(240\) 0 0
\(241\) −18.5908 −1.19754 −0.598768 0.800922i \(-0.704343\pi\)
−0.598768 + 0.800922i \(0.704343\pi\)
\(242\) 17.9699 1.15515
\(243\) 0 0
\(244\) −3.87575 −0.248120
\(245\) −2.46260 −0.157330
\(246\) 0 0
\(247\) −2.13105 −0.135595
\(248\) −17.7341 −1.12612
\(249\) 0 0
\(250\) −1.63404 −0.103346
\(251\) −30.0723 −1.89815 −0.949073 0.315058i \(-0.897976\pi\)
−0.949073 + 0.315058i \(0.897976\pi\)
\(252\) 0 0
\(253\) −0.259587 −0.0163201
\(254\) 17.3267 1.08717
\(255\) 0 0
\(256\) 14.4783 0.904892
\(257\) −3.04064 −0.189670 −0.0948348 0.995493i \(-0.530232\pi\)
−0.0948348 + 0.995493i \(0.530232\pi\)
\(258\) 0 0
\(259\) −2.38494 −0.148193
\(260\) −0.670078 −0.0415565
\(261\) 0 0
\(262\) −26.4588 −1.63463
\(263\) 4.80102 0.296044 0.148022 0.988984i \(-0.452709\pi\)
0.148022 + 0.988984i \(0.452709\pi\)
\(264\) 0 0
\(265\) 3.64091 0.223659
\(266\) −7.41751 −0.454797
\(267\) 0 0
\(268\) −4.17854 −0.255245
\(269\) 16.8723 1.02873 0.514363 0.857573i \(-0.328029\pi\)
0.514363 + 0.857573i \(0.328029\pi\)
\(270\) 0 0
\(271\) −2.34099 −0.142205 −0.0711026 0.997469i \(-0.522652\pi\)
−0.0711026 + 0.997469i \(0.522652\pi\)
\(272\) −12.1324 −0.735632
\(273\) 0 0
\(274\) −33.9366 −2.05019
\(275\) 0.0526457 0.00317466
\(276\) 0 0
\(277\) −3.77233 −0.226657 −0.113329 0.993558i \(-0.536151\pi\)
−0.113329 + 0.993558i \(0.536151\pi\)
\(278\) 4.08632 0.245081
\(279\) 0 0
\(280\) 4.62905 0.276638
\(281\) −10.9560 −0.653577 −0.326789 0.945097i \(-0.605967\pi\)
−0.326789 + 0.945097i \(0.605967\pi\)
\(282\) 0 0
\(283\) 17.5697 1.04441 0.522204 0.852821i \(-0.325110\pi\)
0.522204 + 0.852821i \(0.325110\pi\)
\(284\) −1.67553 −0.0994243
\(285\) 0 0
\(286\) 0.0860250 0.00508677
\(287\) 11.6606 0.688305
\(288\) 0 0
\(289\) −10.8473 −0.638074
\(290\) −4.06374 −0.238631
\(291\) 0 0
\(292\) −0.738726 −0.0432307
\(293\) −24.5445 −1.43391 −0.716953 0.697122i \(-0.754464\pi\)
−0.716953 + 0.697122i \(0.754464\pi\)
\(294\) 0 0
\(295\) −7.49087 −0.436135
\(296\) −2.43312 −0.141422
\(297\) 0 0
\(298\) −3.08875 −0.178926
\(299\) 4.93083 0.285157
\(300\) 0 0
\(301\) −20.1876 −1.16359
\(302\) 7.98530 0.459502
\(303\) 0 0
\(304\) −10.4233 −0.597815
\(305\) −5.78404 −0.331193
\(306\) 0 0
\(307\) −11.0689 −0.631735 −0.315868 0.948803i \(-0.602296\pi\)
−0.315868 + 0.948803i \(0.602296\pi\)
\(308\) 0.0751435 0.00428170
\(309\) 0 0
\(310\) 13.3347 0.757361
\(311\) −14.7933 −0.838853 −0.419426 0.907789i \(-0.637769\pi\)
−0.419426 + 0.907789i \(0.637769\pi\)
\(312\) 0 0
\(313\) 27.0525 1.52910 0.764548 0.644567i \(-0.222962\pi\)
0.764548 + 0.644567i \(0.222962\pi\)
\(314\) 29.4123 1.65983
\(315\) 0 0
\(316\) −10.4636 −0.588626
\(317\) −0.577998 −0.0324636 −0.0162318 0.999868i \(-0.505167\pi\)
−0.0162318 + 0.999868i \(0.505167\pi\)
\(318\) 0 0
\(319\) 0.130926 0.00733047
\(320\) 3.82454 0.213798
\(321\) 0 0
\(322\) 17.1627 0.956440
\(323\) 5.28600 0.294121
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) −15.3885 −0.852291
\(327\) 0 0
\(328\) 11.8962 0.656855
\(329\) −20.8019 −1.14684
\(330\) 0 0
\(331\) −24.1140 −1.32542 −0.662712 0.748874i \(-0.730595\pi\)
−0.662712 + 0.748874i \(0.730595\pi\)
\(332\) −0.327770 −0.0179887
\(333\) 0 0
\(334\) −25.3120 −1.38501
\(335\) −6.23591 −0.340704
\(336\) 0 0
\(337\) 6.13491 0.334190 0.167095 0.985941i \(-0.446561\pi\)
0.167095 + 0.985941i \(0.446561\pi\)
\(338\) −1.63404 −0.0888799
\(339\) 0 0
\(340\) 1.66211 0.0901405
\(341\) −0.429620 −0.0232652
\(342\) 0 0
\(343\) −20.1564 −1.08835
\(344\) −20.5954 −1.11043
\(345\) 0 0
\(346\) −15.8120 −0.850057
\(347\) 3.72399 0.199914 0.0999572 0.994992i \(-0.468129\pi\)
0.0999572 + 0.994992i \(0.468129\pi\)
\(348\) 0 0
\(349\) −3.34865 −0.179249 −0.0896246 0.995976i \(-0.528567\pi\)
−0.0896246 + 0.995976i \(0.528567\pi\)
\(350\) −3.48069 −0.186051
\(351\) 0 0
\(352\) 0.191948 0.0102309
\(353\) 35.5626 1.89281 0.946403 0.322988i \(-0.104687\pi\)
0.946403 + 0.322988i \(0.104687\pi\)
\(354\) 0 0
\(355\) −2.50050 −0.132713
\(356\) 3.11520 0.165105
\(357\) 0 0
\(358\) −5.41896 −0.286401
\(359\) −9.06902 −0.478645 −0.239322 0.970940i \(-0.576925\pi\)
−0.239322 + 0.970940i \(0.576925\pi\)
\(360\) 0 0
\(361\) −14.4586 −0.760981
\(362\) 36.9600 1.94258
\(363\) 0 0
\(364\) −1.42734 −0.0748131
\(365\) −1.10245 −0.0577048
\(366\) 0 0
\(367\) −24.6186 −1.28508 −0.642541 0.766252i \(-0.722120\pi\)
−0.642541 + 0.766252i \(0.722120\pi\)
\(368\) 24.1174 1.25721
\(369\) 0 0
\(370\) 1.82952 0.0951121
\(371\) 7.75557 0.402649
\(372\) 0 0
\(373\) 26.9758 1.39675 0.698376 0.715731i \(-0.253906\pi\)
0.698376 + 0.715731i \(0.253906\pi\)
\(374\) −0.213383 −0.0110338
\(375\) 0 0
\(376\) −21.2220 −1.09444
\(377\) −2.48694 −0.128084
\(378\) 0 0
\(379\) 14.2429 0.731611 0.365805 0.930691i \(-0.380794\pi\)
0.365805 + 0.930691i \(0.380794\pi\)
\(380\) 1.42797 0.0732531
\(381\) 0 0
\(382\) 9.61158 0.491771
\(383\) −25.9736 −1.32719 −0.663595 0.748092i \(-0.730970\pi\)
−0.663595 + 0.748092i \(0.730970\pi\)
\(384\) 0 0
\(385\) 0.112141 0.00571526
\(386\) 0.750903 0.0382200
\(387\) 0 0
\(388\) 4.91902 0.249726
\(389\) −25.7906 −1.30764 −0.653819 0.756651i \(-0.726834\pi\)
−0.653819 + 0.756651i \(0.726834\pi\)
\(390\) 0 0
\(391\) −12.2308 −0.618537
\(392\) −5.35159 −0.270296
\(393\) 0 0
\(394\) −3.77657 −0.190261
\(395\) −15.6156 −0.785705
\(396\) 0 0
\(397\) −19.2890 −0.968086 −0.484043 0.875044i \(-0.660832\pi\)
−0.484043 + 0.875044i \(0.660832\pi\)
\(398\) −39.8271 −1.99635
\(399\) 0 0
\(400\) −4.89115 −0.244558
\(401\) 16.4631 0.822128 0.411064 0.911607i \(-0.365157\pi\)
0.411064 + 0.911607i \(0.365157\pi\)
\(402\) 0 0
\(403\) 8.16059 0.406508
\(404\) −0.370458 −0.0184310
\(405\) 0 0
\(406\) −8.65625 −0.429602
\(407\) −0.0589438 −0.00292173
\(408\) 0 0
\(409\) −19.7526 −0.976704 −0.488352 0.872647i \(-0.662402\pi\)
−0.488352 + 0.872647i \(0.662402\pi\)
\(410\) −8.94500 −0.441762
\(411\) 0 0
\(412\) −5.25204 −0.258749
\(413\) −15.9564 −0.785164
\(414\) 0 0
\(415\) −0.489152 −0.0240115
\(416\) −3.64604 −0.178762
\(417\) 0 0
\(418\) −0.183323 −0.00896664
\(419\) 17.4583 0.852894 0.426447 0.904513i \(-0.359765\pi\)
0.426447 + 0.904513i \(0.359765\pi\)
\(420\) 0 0
\(421\) −7.41596 −0.361432 −0.180716 0.983535i \(-0.557841\pi\)
−0.180716 + 0.983535i \(0.557841\pi\)
\(422\) −44.7625 −2.17900
\(423\) 0 0
\(424\) 7.91222 0.384251
\(425\) 2.48047 0.120321
\(426\) 0 0
\(427\) −12.3207 −0.596239
\(428\) −13.0111 −0.628916
\(429\) 0 0
\(430\) 15.4861 0.746808
\(431\) −33.6097 −1.61892 −0.809461 0.587174i \(-0.800240\pi\)
−0.809461 + 0.587174i \(0.800240\pi\)
\(432\) 0 0
\(433\) 30.2826 1.45529 0.727644 0.685955i \(-0.240615\pi\)
0.727644 + 0.685955i \(0.240615\pi\)
\(434\) 28.4045 1.36346
\(435\) 0 0
\(436\) −1.43063 −0.0685147
\(437\) −10.5078 −0.502658
\(438\) 0 0
\(439\) −28.7370 −1.37154 −0.685772 0.727817i \(-0.740535\pi\)
−0.685772 + 0.727817i \(0.740535\pi\)
\(440\) 0.114407 0.00545412
\(441\) 0 0
\(442\) 4.05318 0.192790
\(443\) 22.4218 1.06529 0.532646 0.846338i \(-0.321198\pi\)
0.532646 + 0.846338i \(0.321198\pi\)
\(444\) 0 0
\(445\) 4.64901 0.220384
\(446\) −41.3997 −1.96033
\(447\) 0 0
\(448\) 8.14671 0.384896
\(449\) −16.0149 −0.755791 −0.377895 0.925848i \(-0.623352\pi\)
−0.377895 + 0.925848i \(0.623352\pi\)
\(450\) 0 0
\(451\) 0.288192 0.0135704
\(452\) 4.17813 0.196523
\(453\) 0 0
\(454\) −34.7097 −1.62901
\(455\) −2.13012 −0.0998614
\(456\) 0 0
\(457\) 1.06556 0.0498448 0.0249224 0.999689i \(-0.492066\pi\)
0.0249224 + 0.999689i \(0.492066\pi\)
\(458\) −19.9904 −0.934089
\(459\) 0 0
\(460\) −3.30404 −0.154052
\(461\) −24.6387 −1.14754 −0.573769 0.819017i \(-0.694519\pi\)
−0.573769 + 0.819017i \(0.694519\pi\)
\(462\) 0 0
\(463\) 7.12328 0.331047 0.165524 0.986206i \(-0.447069\pi\)
0.165524 + 0.986206i \(0.447069\pi\)
\(464\) −12.1640 −0.564698
\(465\) 0 0
\(466\) 18.6359 0.863290
\(467\) −20.4662 −0.947063 −0.473531 0.880777i \(-0.657021\pi\)
−0.473531 + 0.880777i \(0.657021\pi\)
\(468\) 0 0
\(469\) −13.2832 −0.613361
\(470\) 15.9574 0.736058
\(471\) 0 0
\(472\) −16.2787 −0.749289
\(473\) −0.498935 −0.0229411
\(474\) 0 0
\(475\) 2.13105 0.0977791
\(476\) 3.54048 0.162278
\(477\) 0 0
\(478\) 24.2770 1.11041
\(479\) 14.4699 0.661145 0.330573 0.943781i \(-0.392758\pi\)
0.330573 + 0.943781i \(0.392758\pi\)
\(480\) 0 0
\(481\) 1.11963 0.0510508
\(482\) 30.3780 1.38368
\(483\) 0 0
\(484\) −7.36900 −0.334954
\(485\) 7.34097 0.333336
\(486\) 0 0
\(487\) 29.8595 1.35306 0.676531 0.736414i \(-0.263483\pi\)
0.676531 + 0.736414i \(0.263483\pi\)
\(488\) −12.5695 −0.568996
\(489\) 0 0
\(490\) 4.02399 0.181785
\(491\) −22.0470 −0.994965 −0.497483 0.867474i \(-0.665742\pi\)
−0.497483 + 0.867474i \(0.665742\pi\)
\(492\) 0 0
\(493\) 6.16877 0.277827
\(494\) 3.48221 0.156672
\(495\) 0 0
\(496\) 39.9147 1.79222
\(497\) −5.32635 −0.238920
\(498\) 0 0
\(499\) 21.1723 0.947802 0.473901 0.880578i \(-0.342845\pi\)
0.473901 + 0.880578i \(0.342845\pi\)
\(500\) 0.670078 0.0299668
\(501\) 0 0
\(502\) 49.1392 2.19319
\(503\) 40.3143 1.79752 0.898762 0.438436i \(-0.144467\pi\)
0.898762 + 0.438436i \(0.144467\pi\)
\(504\) 0 0
\(505\) −0.552859 −0.0246019
\(506\) 0.424175 0.0188569
\(507\) 0 0
\(508\) −7.10524 −0.315244
\(509\) −19.0167 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(510\) 0 0
\(511\) −2.34834 −0.103885
\(512\) 3.42501 0.151366
\(513\) 0 0
\(514\) 4.96851 0.219152
\(515\) −7.83795 −0.345381
\(516\) 0 0
\(517\) −0.514117 −0.0226108
\(518\) 3.89709 0.171228
\(519\) 0 0
\(520\) −2.17314 −0.0952986
\(521\) 33.4407 1.46507 0.732533 0.680732i \(-0.238338\pi\)
0.732533 + 0.680732i \(0.238338\pi\)
\(522\) 0 0
\(523\) −6.40250 −0.279962 −0.139981 0.990154i \(-0.544704\pi\)
−0.139981 + 0.990154i \(0.544704\pi\)
\(524\) 10.8501 0.473989
\(525\) 0 0
\(526\) −7.84504 −0.342060
\(527\) −20.2421 −0.881761
\(528\) 0 0
\(529\) 1.31310 0.0570915
\(530\) −5.94939 −0.258425
\(531\) 0 0
\(532\) 3.04173 0.131876
\(533\) −5.47417 −0.237113
\(534\) 0 0
\(535\) −19.4173 −0.839484
\(536\) −13.5515 −0.585336
\(537\) 0 0
\(538\) −27.5700 −1.18863
\(539\) −0.129646 −0.00558423
\(540\) 0 0
\(541\) −17.4745 −0.751286 −0.375643 0.926764i \(-0.622578\pi\)
−0.375643 + 0.926764i \(0.622578\pi\)
\(542\) 3.82527 0.164309
\(543\) 0 0
\(544\) 9.04389 0.387754
\(545\) −2.13502 −0.0914542
\(546\) 0 0
\(547\) −10.0747 −0.430761 −0.215381 0.976530i \(-0.569099\pi\)
−0.215381 + 0.976530i \(0.569099\pi\)
\(548\) 13.9166 0.594486
\(549\) 0 0
\(550\) −0.0860250 −0.00366812
\(551\) 5.29977 0.225778
\(552\) 0 0
\(553\) −33.2630 −1.41449
\(554\) 6.16413 0.261889
\(555\) 0 0
\(556\) −1.67570 −0.0710654
\(557\) −21.2315 −0.899607 −0.449804 0.893127i \(-0.648506\pi\)
−0.449804 + 0.893127i \(0.648506\pi\)
\(558\) 0 0
\(559\) 9.47722 0.400844
\(560\) −10.4187 −0.440271
\(561\) 0 0
\(562\) 17.9024 0.755169
\(563\) −12.3449 −0.520276 −0.260138 0.965571i \(-0.583768\pi\)
−0.260138 + 0.965571i \(0.583768\pi\)
\(564\) 0 0
\(565\) 6.23529 0.262321
\(566\) −28.7095 −1.20675
\(567\) 0 0
\(568\) −5.43394 −0.228003
\(569\) −7.75747 −0.325210 −0.162605 0.986691i \(-0.551990\pi\)
−0.162605 + 0.986691i \(0.551990\pi\)
\(570\) 0 0
\(571\) −10.2139 −0.427440 −0.213720 0.976895i \(-0.568558\pi\)
−0.213720 + 0.976895i \(0.568558\pi\)
\(572\) −0.0352767 −0.00147499
\(573\) 0 0
\(574\) −19.0539 −0.795294
\(575\) −4.93083 −0.205630
\(576\) 0 0
\(577\) −44.5539 −1.85480 −0.927402 0.374065i \(-0.877964\pi\)
−0.927402 + 0.374065i \(0.877964\pi\)
\(578\) 17.7248 0.737256
\(579\) 0 0
\(580\) 1.66644 0.0691952
\(581\) −1.04195 −0.0432274
\(582\) 0 0
\(583\) 0.191678 0.00793851
\(584\) −2.39578 −0.0991380
\(585\) 0 0
\(586\) 40.1066 1.65679
\(587\) −20.9069 −0.862921 −0.431461 0.902132i \(-0.642002\pi\)
−0.431461 + 0.902132i \(0.642002\pi\)
\(588\) 0 0
\(589\) −17.3906 −0.716567
\(590\) 12.2404 0.503928
\(591\) 0 0
\(592\) 5.47628 0.225074
\(593\) 7.48843 0.307513 0.153756 0.988109i \(-0.450863\pi\)
0.153756 + 0.988109i \(0.450863\pi\)
\(594\) 0 0
\(595\) 5.28369 0.216610
\(596\) 1.26662 0.0518827
\(597\) 0 0
\(598\) −8.05716 −0.329482
\(599\) 27.9314 1.14125 0.570623 0.821212i \(-0.306702\pi\)
0.570623 + 0.821212i \(0.306702\pi\)
\(600\) 0 0
\(601\) 32.7780 1.33704 0.668521 0.743693i \(-0.266927\pi\)
0.668521 + 0.743693i \(0.266927\pi\)
\(602\) 32.9873 1.34446
\(603\) 0 0
\(604\) −3.27457 −0.133240
\(605\) −10.9972 −0.447101
\(606\) 0 0
\(607\) 39.1704 1.58988 0.794940 0.606689i \(-0.207503\pi\)
0.794940 + 0.606689i \(0.207503\pi\)
\(608\) 7.76988 0.315110
\(609\) 0 0
\(610\) 9.45133 0.382673
\(611\) 9.76560 0.395074
\(612\) 0 0
\(613\) −10.2019 −0.412051 −0.206026 0.978547i \(-0.566053\pi\)
−0.206026 + 0.978547i \(0.566053\pi\)
\(614\) 18.0870 0.729932
\(615\) 0 0
\(616\) 0.243699 0.00981893
\(617\) 9.68947 0.390084 0.195042 0.980795i \(-0.437516\pi\)
0.195042 + 0.980795i \(0.437516\pi\)
\(618\) 0 0
\(619\) −19.8750 −0.798842 −0.399421 0.916768i \(-0.630789\pi\)
−0.399421 + 0.916768i \(0.630789\pi\)
\(620\) −5.46823 −0.219610
\(621\) 0 0
\(622\) 24.1728 0.969243
\(623\) 9.90292 0.396752
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −44.2047 −1.76678
\(627\) 0 0
\(628\) −12.0612 −0.481296
\(629\) −2.77721 −0.110735
\(630\) 0 0
\(631\) −38.9270 −1.54966 −0.774830 0.632170i \(-0.782165\pi\)
−0.774830 + 0.632170i \(0.782165\pi\)
\(632\) −33.9349 −1.34986
\(633\) 0 0
\(634\) 0.944470 0.0375097
\(635\) −10.6036 −0.420791
\(636\) 0 0
\(637\) 2.46260 0.0975719
\(638\) −0.213939 −0.00846991
\(639\) 0 0
\(640\) −13.5415 −0.535275
\(641\) 7.89412 0.311799 0.155899 0.987773i \(-0.450172\pi\)
0.155899 + 0.987773i \(0.450172\pi\)
\(642\) 0 0
\(643\) 35.7751 1.41083 0.705415 0.708794i \(-0.250760\pi\)
0.705415 + 0.708794i \(0.250760\pi\)
\(644\) −7.03799 −0.277336
\(645\) 0 0
\(646\) −8.63752 −0.339839
\(647\) 1.47889 0.0581412 0.0290706 0.999577i \(-0.490745\pi\)
0.0290706 + 0.999577i \(0.490745\pi\)
\(648\) 0 0
\(649\) −0.394362 −0.0154801
\(650\) 1.63404 0.0640922
\(651\) 0 0
\(652\) 6.31044 0.247136
\(653\) 6.95046 0.271992 0.135996 0.990709i \(-0.456577\pi\)
0.135996 + 0.990709i \(0.456577\pi\)
\(654\) 0 0
\(655\) 16.1923 0.632686
\(656\) −26.7750 −1.04539
\(657\) 0 0
\(658\) 33.9910 1.32511
\(659\) −30.9546 −1.20582 −0.602910 0.797810i \(-0.705992\pi\)
−0.602910 + 0.797810i \(0.705992\pi\)
\(660\) 0 0
\(661\) 3.57678 0.139121 0.0695603 0.997578i \(-0.477840\pi\)
0.0695603 + 0.997578i \(0.477840\pi\)
\(662\) 39.4032 1.53145
\(663\) 0 0
\(664\) −1.06300 −0.0412522
\(665\) 4.53938 0.176029
\(666\) 0 0
\(667\) −12.2627 −0.474812
\(668\) 10.3798 0.401607
\(669\) 0 0
\(670\) 10.1897 0.393663
\(671\) −0.304505 −0.0117553
\(672\) 0 0
\(673\) −28.0993 −1.08315 −0.541575 0.840653i \(-0.682172\pi\)
−0.541575 + 0.840653i \(0.682172\pi\)
\(674\) −10.0247 −0.386136
\(675\) 0 0
\(676\) 0.670078 0.0257722
\(677\) −44.2934 −1.70233 −0.851167 0.524895i \(-0.824105\pi\)
−0.851167 + 0.524895i \(0.824105\pi\)
\(678\) 0 0
\(679\) 15.6371 0.600098
\(680\) 5.39042 0.206713
\(681\) 0 0
\(682\) 0.702016 0.0268816
\(683\) 1.69041 0.0646816 0.0323408 0.999477i \(-0.489704\pi\)
0.0323408 + 0.999477i \(0.489704\pi\)
\(684\) 0 0
\(685\) 20.7686 0.793526
\(686\) 32.9364 1.25752
\(687\) 0 0
\(688\) 46.3545 1.76725
\(689\) −3.64091 −0.138708
\(690\) 0 0
\(691\) −10.8144 −0.411398 −0.205699 0.978615i \(-0.565947\pi\)
−0.205699 + 0.978615i \(0.565947\pi\)
\(692\) 6.48409 0.246488
\(693\) 0 0
\(694\) −6.08514 −0.230989
\(695\) −2.50075 −0.0948589
\(696\) 0 0
\(697\) 13.5785 0.514323
\(698\) 5.47182 0.207111
\(699\) 0 0
\(700\) 1.42734 0.0539485
\(701\) 32.1755 1.21525 0.607626 0.794224i \(-0.292122\pi\)
0.607626 + 0.794224i \(0.292122\pi\)
\(702\) 0 0
\(703\) −2.38598 −0.0899891
\(704\) 0.201346 0.00758850
\(705\) 0 0
\(706\) −58.1106 −2.18702
\(707\) −1.17765 −0.0442902
\(708\) 0 0
\(709\) 31.9120 1.19848 0.599240 0.800570i \(-0.295470\pi\)
0.599240 + 0.800570i \(0.295470\pi\)
\(710\) 4.08591 0.153341
\(711\) 0 0
\(712\) 10.1030 0.378624
\(713\) 40.2385 1.50694
\(714\) 0 0
\(715\) −0.0526457 −0.00196884
\(716\) 2.22218 0.0830467
\(717\) 0 0
\(718\) 14.8191 0.553045
\(719\) −20.6891 −0.771572 −0.385786 0.922588i \(-0.626070\pi\)
−0.385786 + 0.922588i \(0.626070\pi\)
\(720\) 0 0
\(721\) −16.6958 −0.621782
\(722\) 23.6260 0.879267
\(723\) 0 0
\(724\) −15.1564 −0.563283
\(725\) 2.48694 0.0923625
\(726\) 0 0
\(727\) −24.6791 −0.915299 −0.457649 0.889133i \(-0.651308\pi\)
−0.457649 + 0.889133i \(0.651308\pi\)
\(728\) −4.62905 −0.171564
\(729\) 0 0
\(730\) 1.80144 0.0666744
\(731\) −23.5080 −0.869474
\(732\) 0 0
\(733\) −21.7459 −0.803203 −0.401602 0.915814i \(-0.631546\pi\)
−0.401602 + 0.915814i \(0.631546\pi\)
\(734\) 40.2277 1.48483
\(735\) 0 0
\(736\) −17.9780 −0.662678
\(737\) −0.328294 −0.0120929
\(738\) 0 0
\(739\) 43.0460 1.58347 0.791736 0.610864i \(-0.209178\pi\)
0.791736 + 0.610864i \(0.209178\pi\)
\(740\) −0.750240 −0.0275794
\(741\) 0 0
\(742\) −12.6729 −0.465236
\(743\) −7.89816 −0.289755 −0.144878 0.989450i \(-0.546279\pi\)
−0.144878 + 0.989450i \(0.546279\pi\)
\(744\) 0 0
\(745\) 1.89025 0.0692536
\(746\) −44.0794 −1.61386
\(747\) 0 0
\(748\) 0.0875029 0.00319942
\(749\) −41.3612 −1.51130
\(750\) 0 0
\(751\) 38.6450 1.41018 0.705089 0.709119i \(-0.250907\pi\)
0.705089 + 0.709119i \(0.250907\pi\)
\(752\) 47.7650 1.74181
\(753\) 0 0
\(754\) 4.06374 0.147993
\(755\) −4.88685 −0.177851
\(756\) 0 0
\(757\) −15.3437 −0.557678 −0.278839 0.960338i \(-0.589950\pi\)
−0.278839 + 0.960338i \(0.589950\pi\)
\(758\) −23.2735 −0.845331
\(759\) 0 0
\(760\) 4.63107 0.167986
\(761\) −48.7170 −1.76599 −0.882995 0.469382i \(-0.844477\pi\)
−0.882995 + 0.469382i \(0.844477\pi\)
\(762\) 0 0
\(763\) −4.54784 −0.164643
\(764\) −3.94147 −0.142597
\(765\) 0 0
\(766\) 42.4418 1.53349
\(767\) 7.49087 0.270480
\(768\) 0 0
\(769\) 6.37109 0.229747 0.114874 0.993380i \(-0.463354\pi\)
0.114874 + 0.993380i \(0.463354\pi\)
\(770\) −0.183243 −0.00660363
\(771\) 0 0
\(772\) −0.307927 −0.0110825
\(773\) −29.8732 −1.07446 −0.537232 0.843435i \(-0.680530\pi\)
−0.537232 + 0.843435i \(0.680530\pi\)
\(774\) 0 0
\(775\) −8.16059 −0.293137
\(776\) 15.9530 0.572679
\(777\) 0 0
\(778\) 42.1429 1.51090
\(779\) 11.6657 0.417967
\(780\) 0 0
\(781\) −0.131641 −0.00471047
\(782\) 19.9856 0.714682
\(783\) 0 0
\(784\) 12.0450 0.430178
\(785\) −17.9998 −0.642439
\(786\) 0 0
\(787\) −23.6556 −0.843232 −0.421616 0.906774i \(-0.638537\pi\)
−0.421616 + 0.906774i \(0.638537\pi\)
\(788\) 1.54868 0.0551693
\(789\) 0 0
\(790\) 25.5164 0.907834
\(791\) 13.2819 0.472250
\(792\) 0 0
\(793\) 5.78404 0.205397
\(794\) 31.5189 1.11856
\(795\) 0 0
\(796\) 16.3321 0.578876
\(797\) 30.0006 1.06268 0.531339 0.847160i \(-0.321689\pi\)
0.531339 + 0.847160i \(0.321689\pi\)
\(798\) 0 0
\(799\) −24.2233 −0.856959
\(800\) 3.64604 0.128907
\(801\) 0 0
\(802\) −26.9013 −0.949918
\(803\) −0.0580392 −0.00204816
\(804\) 0 0
\(805\) −10.5032 −0.370191
\(806\) −13.3347 −0.469695
\(807\) 0 0
\(808\) −1.20144 −0.0422665
\(809\) −37.0787 −1.30362 −0.651808 0.758384i \(-0.725989\pi\)
−0.651808 + 0.758384i \(0.725989\pi\)
\(810\) 0 0
\(811\) −42.7582 −1.50144 −0.750722 0.660618i \(-0.770294\pi\)
−0.750722 + 0.660618i \(0.770294\pi\)
\(812\) 3.54971 0.124570
\(813\) 0 0
\(814\) 0.0963163 0.00337588
\(815\) 9.41748 0.329880
\(816\) 0 0
\(817\) −20.1964 −0.706583
\(818\) 32.2765 1.12852
\(819\) 0 0
\(820\) 3.66812 0.128096
\(821\) −28.5669 −0.996992 −0.498496 0.866892i \(-0.666114\pi\)
−0.498496 + 0.866892i \(0.666114\pi\)
\(822\) 0 0
\(823\) 32.0041 1.11559 0.557797 0.829977i \(-0.311647\pi\)
0.557797 + 0.829977i \(0.311647\pi\)
\(824\) −17.0330 −0.593372
\(825\) 0 0
\(826\) 26.0734 0.907209
\(827\) −31.4986 −1.09531 −0.547657 0.836703i \(-0.684480\pi\)
−0.547657 + 0.836703i \(0.684480\pi\)
\(828\) 0 0
\(829\) 32.3851 1.12478 0.562390 0.826872i \(-0.309882\pi\)
0.562390 + 0.826872i \(0.309882\pi\)
\(830\) 0.799292 0.0277438
\(831\) 0 0
\(832\) −3.82454 −0.132592
\(833\) −6.10842 −0.211644
\(834\) 0 0
\(835\) 15.4905 0.536070
\(836\) 0.0751763 0.00260003
\(837\) 0 0
\(838\) −28.5275 −0.985467
\(839\) −46.0763 −1.59073 −0.795365 0.606131i \(-0.792721\pi\)
−0.795365 + 0.606131i \(0.792721\pi\)
\(840\) 0 0
\(841\) −22.8152 −0.786729
\(842\) 12.1180 0.417612
\(843\) 0 0
\(844\) 18.3560 0.631839
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −23.4254 −0.804905
\(848\) −17.8083 −0.611538
\(849\) 0 0
\(850\) −4.05318 −0.139023
\(851\) 5.52071 0.189248
\(852\) 0 0
\(853\) 48.7573 1.66942 0.834709 0.550692i \(-0.185636\pi\)
0.834709 + 0.550692i \(0.185636\pi\)
\(854\) 20.1324 0.688918
\(855\) 0 0
\(856\) −42.1966 −1.44225
\(857\) −31.9479 −1.09132 −0.545660 0.838007i \(-0.683721\pi\)
−0.545660 + 0.838007i \(0.683721\pi\)
\(858\) 0 0
\(859\) −7.33499 −0.250267 −0.125133 0.992140i \(-0.539936\pi\)
−0.125133 + 0.992140i \(0.539936\pi\)
\(860\) −6.35048 −0.216549
\(861\) 0 0
\(862\) 54.9195 1.87056
\(863\) −28.4742 −0.969273 −0.484637 0.874716i \(-0.661048\pi\)
−0.484637 + 0.874716i \(0.661048\pi\)
\(864\) 0 0
\(865\) 9.67663 0.329015
\(866\) −49.4829 −1.68150
\(867\) 0 0
\(868\) −11.6480 −0.395358
\(869\) −0.822093 −0.0278876
\(870\) 0 0
\(871\) 6.23591 0.211296
\(872\) −4.63970 −0.157120
\(873\) 0 0
\(874\) 17.1702 0.580790
\(875\) 2.13012 0.0720111
\(876\) 0 0
\(877\) 45.8529 1.54834 0.774171 0.632977i \(-0.218167\pi\)
0.774171 + 0.632977i \(0.218167\pi\)
\(878\) 46.9574 1.58473
\(879\) 0 0
\(880\) −0.257498 −0.00868026
\(881\) 37.4454 1.26157 0.630784 0.775959i \(-0.282733\pi\)
0.630784 + 0.775959i \(0.282733\pi\)
\(882\) 0 0
\(883\) 50.8068 1.70978 0.854892 0.518805i \(-0.173623\pi\)
0.854892 + 0.518805i \(0.173623\pi\)
\(884\) −1.66211 −0.0559028
\(885\) 0 0
\(886\) −36.6381 −1.23088
\(887\) 45.2968 1.52092 0.760459 0.649386i \(-0.224974\pi\)
0.760459 + 0.649386i \(0.224974\pi\)
\(888\) 0 0
\(889\) −22.5869 −0.757541
\(890\) −7.59665 −0.254640
\(891\) 0 0
\(892\) 16.9770 0.568432
\(893\) −20.8109 −0.696412
\(894\) 0 0
\(895\) 3.31630 0.110852
\(896\) −28.8450 −0.963644
\(897\) 0 0
\(898\) 26.1690 0.873270
\(899\) −20.2949 −0.676872
\(900\) 0 0
\(901\) 9.03118 0.300872
\(902\) −0.470916 −0.0156798
\(903\) 0 0
\(904\) 13.5502 0.450672
\(905\) −22.6189 −0.751876
\(906\) 0 0
\(907\) 14.3690 0.477116 0.238558 0.971128i \(-0.423325\pi\)
0.238558 + 0.971128i \(0.423325\pi\)
\(908\) 14.2336 0.472359
\(909\) 0 0
\(910\) 3.48069 0.115384
\(911\) −53.9063 −1.78600 −0.892998 0.450060i \(-0.851403\pi\)
−0.892998 + 0.450060i \(0.851403\pi\)
\(912\) 0 0
\(913\) −0.0257517 −0.000852258 0
\(914\) −1.74117 −0.0575926
\(915\) 0 0
\(916\) 8.19755 0.270855
\(917\) 34.4915 1.13901
\(918\) 0 0
\(919\) −15.6573 −0.516487 −0.258244 0.966080i \(-0.583144\pi\)
−0.258244 + 0.966080i \(0.583144\pi\)
\(920\) −10.7154 −0.353276
\(921\) 0 0
\(922\) 40.2605 1.32591
\(923\) 2.50050 0.0823049
\(924\) 0 0
\(925\) −1.11963 −0.0368132
\(926\) −11.6397 −0.382505
\(927\) 0 0
\(928\) 9.06746 0.297654
\(929\) 23.1439 0.759327 0.379664 0.925125i \(-0.376040\pi\)
0.379664 + 0.925125i \(0.376040\pi\)
\(930\) 0 0
\(931\) −5.24792 −0.171994
\(932\) −7.64211 −0.250326
\(933\) 0 0
\(934\) 33.4425 1.09427
\(935\) 0.130586 0.00427062
\(936\) 0 0
\(937\) 14.6354 0.478118 0.239059 0.971005i \(-0.423161\pi\)
0.239059 + 0.971005i \(0.423161\pi\)
\(938\) 21.7052 0.708702
\(939\) 0 0
\(940\) −6.54371 −0.213432
\(941\) 22.0393 0.718462 0.359231 0.933249i \(-0.383039\pi\)
0.359231 + 0.933249i \(0.383039\pi\)
\(942\) 0 0
\(943\) −26.9922 −0.878987
\(944\) 36.6390 1.19250
\(945\) 0 0
\(946\) 0.815279 0.0265070
\(947\) 50.3906 1.63747 0.818737 0.574168i \(-0.194674\pi\)
0.818737 + 0.574168i \(0.194674\pi\)
\(948\) 0 0
\(949\) 1.10245 0.0357870
\(950\) −3.48221 −0.112978
\(951\) 0 0
\(952\) 11.4822 0.372141
\(953\) 7.28058 0.235841 0.117921 0.993023i \(-0.462377\pi\)
0.117921 + 0.993023i \(0.462377\pi\)
\(954\) 0 0
\(955\) −5.88211 −0.190340
\(956\) −9.95540 −0.321981
\(957\) 0 0
\(958\) −23.6443 −0.763913
\(959\) 44.2395 1.42857
\(960\) 0 0
\(961\) 35.5953 1.14824
\(962\) −1.82952 −0.0589861
\(963\) 0 0
\(964\) −12.4573 −0.401221
\(965\) −0.459539 −0.0147931
\(966\) 0 0
\(967\) 1.64327 0.0528441 0.0264220 0.999651i \(-0.491589\pi\)
0.0264220 + 0.999651i \(0.491589\pi\)
\(968\) −23.8985 −0.768128
\(969\) 0 0
\(970\) −11.9954 −0.385150
\(971\) −59.5135 −1.90988 −0.954939 0.296803i \(-0.904080\pi\)
−0.954939 + 0.296803i \(0.904080\pi\)
\(972\) 0 0
\(973\) −5.32689 −0.170772
\(974\) −48.7915 −1.56338
\(975\) 0 0
\(976\) 28.2906 0.905560
\(977\) 15.3273 0.490363 0.245181 0.969477i \(-0.421153\pi\)
0.245181 + 0.969477i \(0.421153\pi\)
\(978\) 0 0
\(979\) 0.244750 0.00782225
\(980\) −1.65014 −0.0527117
\(981\) 0 0
\(982\) 36.0255 1.14962
\(983\) 3.26954 0.104282 0.0521411 0.998640i \(-0.483395\pi\)
0.0521411 + 0.998640i \(0.483395\pi\)
\(984\) 0 0
\(985\) 2.31119 0.0736405
\(986\) −10.0800 −0.321013
\(987\) 0 0
\(988\) −1.42797 −0.0454297
\(989\) 46.7306 1.48595
\(990\) 0 0
\(991\) −39.2309 −1.24621 −0.623106 0.782138i \(-0.714129\pi\)
−0.623106 + 0.782138i \(0.714129\pi\)
\(992\) −29.7538 −0.944685
\(993\) 0 0
\(994\) 8.70346 0.276057
\(995\) 24.3734 0.772690
\(996\) 0 0
\(997\) −32.8254 −1.03959 −0.519795 0.854291i \(-0.673992\pi\)
−0.519795 + 0.854291i \(0.673992\pi\)
\(998\) −34.5963 −1.09513
\(999\) 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 5265.2.a.ba.1.3 8
3.2 odd 2 5265.2.a.bf.1.6 8
9.2 odd 6 585.2.i.e.391.3 yes 16
9.4 even 3 1755.2.i.f.586.6 16
9.5 odd 6 585.2.i.e.196.3 16
9.7 even 3 1755.2.i.f.1171.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.i.e.196.3 16 9.5 odd 6
585.2.i.e.391.3 yes 16 9.2 odd 6
1755.2.i.f.586.6 16 9.4 even 3
1755.2.i.f.1171.6 16 9.7 even 3
5265.2.a.ba.1.3 8 1.1 even 1 trivial
5265.2.a.bf.1.6 8 3.2 odd 2