Properties

Label 1850.2.a.be.1.2
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1791440.1
Defining polynomial: \(x^{5} - 9 x^{3} + 13 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.09441\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.09441 q^{3} +1.00000 q^{4} -1.09441 q^{6} -3.20984 q^{7} +1.00000 q^{8} -1.80226 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.09441 q^{3} +1.00000 q^{4} -1.09441 q^{6} -3.20984 q^{7} +1.00000 q^{8} -1.80226 q^{9} +3.82327 q^{11} -1.09441 q^{12} +0.147332 q^{13} -3.20984 q^{14} +1.00000 q^{16} +0.978989 q^{17} -1.80226 q^{18} +2.67594 q^{19} +3.51289 q^{21} +3.82327 q^{22} -2.33616 q^{23} -1.09441 q^{24} +0.147332 q^{26} +5.25565 q^{27} -3.20984 q^{28} +6.30425 q^{29} -3.62372 q^{31} +1.00000 q^{32} -4.18424 q^{33} +0.978989 q^{34} -1.80226 q^{36} +1.00000 q^{37} +2.67594 q^{38} -0.161242 q^{39} +11.8265 q^{41} +3.51289 q^{42} -4.53390 q^{43} +3.82327 q^{44} -2.33616 q^{46} +6.23085 q^{47} -1.09441 q^{48} +3.30305 q^{49} -1.07142 q^{51} +0.147332 q^{52} +11.2978 q^{53} +5.25565 q^{54} -3.20984 q^{56} -2.92858 q^{57} +6.30425 q^{58} +6.92858 q^{59} +10.4885 q^{61} -3.62372 q^{62} +5.78496 q^{63} +1.00000 q^{64} -4.18424 q^{66} +2.80936 q^{67} +0.978989 q^{68} +2.55672 q^{69} -12.3189 q^{71} -1.80226 q^{72} -13.9966 q^{73} +1.00000 q^{74} +2.67594 q^{76} -12.2721 q^{77} -0.161242 q^{78} +15.6057 q^{79} -0.345071 q^{81} +11.8265 q^{82} +13.5371 q^{83} +3.51289 q^{84} -4.53390 q^{86} -6.89945 q^{87} +3.82327 q^{88} -6.46929 q^{89} -0.472913 q^{91} -2.33616 q^{92} +3.96585 q^{93} +6.23085 q^{94} -1.09441 q^{96} -3.07063 q^{97} +3.30305 q^{98} -6.89054 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9} + O(q^{10}) \) \( 5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9} + 3 q^{11} - 6 q^{13} - q^{14} + 5 q^{16} + 9 q^{17} + 3 q^{18} + 4 q^{19} + 16 q^{21} + 3 q^{22} + 6 q^{23} - 6 q^{26} - q^{28} + 11 q^{29} + 23 q^{31} + 5 q^{32} + 20 q^{33} + 9 q^{34} + 3 q^{36} + 5 q^{37} + 4 q^{38} + 20 q^{39} - 7 q^{41} + 16 q^{42} - 17 q^{43} + 3 q^{44} + 6 q^{46} + 12 q^{47} + 30 q^{49} - 20 q^{51} - 6 q^{52} - 7 q^{53} - q^{56} + 11 q^{58} + 20 q^{59} - 9 q^{61} + 23 q^{62} - 33 q^{63} + 5 q^{64} + 20 q^{66} + 12 q^{67} + 9 q^{68} + 16 q^{69} + 6 q^{71} + 3 q^{72} - 6 q^{73} + 5 q^{74} + 4 q^{76} - q^{77} + 20 q^{78} + 20 q^{79} - 7 q^{81} - 7 q^{82} + 12 q^{83} + 16 q^{84} - 17 q^{86} - 34 q^{87} + 3 q^{88} + 12 q^{89} + 16 q^{91} + 6 q^{92} - 4 q^{93} + 12 q^{94} + 3 q^{97} + 30 q^{98} - 11 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.09441 −0.631859 −0.315930 0.948783i \(-0.602316\pi\)
−0.315930 + 0.948783i \(0.602316\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.09441 −0.446792
\(7\) −3.20984 −1.21320 −0.606602 0.795006i \(-0.707468\pi\)
−0.606602 + 0.795006i \(0.707468\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.80226 −0.600754
\(10\) 0 0
\(11\) 3.82327 1.15276 0.576380 0.817182i \(-0.304465\pi\)
0.576380 + 0.817182i \(0.304465\pi\)
\(12\) −1.09441 −0.315930
\(13\) 0.147332 0.0408626 0.0204313 0.999791i \(-0.493496\pi\)
0.0204313 + 0.999791i \(0.493496\pi\)
\(14\) −3.20984 −0.857865
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.978989 0.237440 0.118720 0.992928i \(-0.462121\pi\)
0.118720 + 0.992928i \(0.462121\pi\)
\(18\) −1.80226 −0.424797
\(19\) 2.67594 0.613903 0.306951 0.951725i \(-0.400691\pi\)
0.306951 + 0.951725i \(0.400691\pi\)
\(20\) 0 0
\(21\) 3.51289 0.766574
\(22\) 3.82327 0.815124
\(23\) −2.33616 −0.487122 −0.243561 0.969886i \(-0.578316\pi\)
−0.243561 + 0.969886i \(0.578316\pi\)
\(24\) −1.09441 −0.223396
\(25\) 0 0
\(26\) 0.147332 0.0288942
\(27\) 5.25565 1.01145
\(28\) −3.20984 −0.606602
\(29\) 6.30425 1.17067 0.585335 0.810792i \(-0.300963\pi\)
0.585335 + 0.810792i \(0.300963\pi\)
\(30\) 0 0
\(31\) −3.62372 −0.650839 −0.325420 0.945570i \(-0.605506\pi\)
−0.325420 + 0.945570i \(0.605506\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.18424 −0.728382
\(34\) 0.978989 0.167895
\(35\) 0 0
\(36\) −1.80226 −0.300377
\(37\) 1.00000 0.164399
\(38\) 2.67594 0.434095
\(39\) −0.161242 −0.0258194
\(40\) 0 0
\(41\) 11.8265 1.84698 0.923491 0.383620i \(-0.125323\pi\)
0.923491 + 0.383620i \(0.125323\pi\)
\(42\) 3.51289 0.542050
\(43\) −4.53390 −0.691413 −0.345706 0.938343i \(-0.612361\pi\)
−0.345706 + 0.938343i \(0.612361\pi\)
\(44\) 3.82327 0.576380
\(45\) 0 0
\(46\) −2.33616 −0.344448
\(47\) 6.23085 0.908863 0.454431 0.890782i \(-0.349843\pi\)
0.454431 + 0.890782i \(0.349843\pi\)
\(48\) −1.09441 −0.157965
\(49\) 3.30305 0.471864
\(50\) 0 0
\(51\) −1.07142 −0.150028
\(52\) 0.147332 0.0204313
\(53\) 11.2978 1.55188 0.775939 0.630807i \(-0.217276\pi\)
0.775939 + 0.630807i \(0.217276\pi\)
\(54\) 5.25565 0.715204
\(55\) 0 0
\(56\) −3.20984 −0.428932
\(57\) −2.92858 −0.387900
\(58\) 6.30425 0.827788
\(59\) 6.92858 0.902025 0.451012 0.892518i \(-0.351063\pi\)
0.451012 + 0.892518i \(0.351063\pi\)
\(60\) 0 0
\(61\) 10.4885 1.34291 0.671457 0.741044i \(-0.265669\pi\)
0.671457 + 0.741044i \(0.265669\pi\)
\(62\) −3.62372 −0.460213
\(63\) 5.78496 0.728837
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.18424 −0.515044
\(67\) 2.80936 0.343218 0.171609 0.985165i \(-0.445103\pi\)
0.171609 + 0.985165i \(0.445103\pi\)
\(68\) 0.978989 0.118720
\(69\) 2.55672 0.307793
\(70\) 0 0
\(71\) −12.3189 −1.46198 −0.730990 0.682388i \(-0.760941\pi\)
−0.730990 + 0.682388i \(0.760941\pi\)
\(72\) −1.80226 −0.212399
\(73\) −13.9966 −1.63818 −0.819090 0.573665i \(-0.805521\pi\)
−0.819090 + 0.573665i \(0.805521\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.67594 0.306951
\(77\) −12.2721 −1.39853
\(78\) −0.161242 −0.0182571
\(79\) 15.6057 1.75578 0.877890 0.478861i \(-0.158950\pi\)
0.877890 + 0.478861i \(0.158950\pi\)
\(80\) 0 0
\(81\) −0.345071 −0.0383412
\(82\) 11.8265 1.30601
\(83\) 13.5371 1.48589 0.742944 0.669354i \(-0.233429\pi\)
0.742944 + 0.669354i \(0.233429\pi\)
\(84\) 3.51289 0.383287
\(85\) 0 0
\(86\) −4.53390 −0.488903
\(87\) −6.89945 −0.739699
\(88\) 3.82327 0.407562
\(89\) −6.46929 −0.685743 −0.342872 0.939382i \(-0.611400\pi\)
−0.342872 + 0.939382i \(0.611400\pi\)
\(90\) 0 0
\(91\) −0.472913 −0.0495747
\(92\) −2.33616 −0.243561
\(93\) 3.96585 0.411239
\(94\) 6.23085 0.642663
\(95\) 0 0
\(96\) −1.09441 −0.111698
\(97\) −3.07063 −0.311775 −0.155887 0.987775i \(-0.549824\pi\)
−0.155887 + 0.987775i \(0.549824\pi\)
\(98\) 3.30305 0.333658
\(99\) −6.89054 −0.692525
\(100\) 0 0
\(101\) −6.57513 −0.654250 −0.327125 0.944981i \(-0.606080\pi\)
−0.327125 + 0.944981i \(0.606080\pi\)
\(102\) −1.07142 −0.106086
\(103\) −2.52180 −0.248480 −0.124240 0.992252i \(-0.539649\pi\)
−0.124240 + 0.992252i \(0.539649\pi\)
\(104\) 0.147332 0.0144471
\(105\) 0 0
\(106\) 11.2978 1.09734
\(107\) 5.70291 0.551321 0.275661 0.961255i \(-0.411103\pi\)
0.275661 + 0.961255i \(0.411103\pi\)
\(108\) 5.25565 0.505726
\(109\) 11.9318 1.14286 0.571428 0.820652i \(-0.306390\pi\)
0.571428 + 0.820652i \(0.306390\pi\)
\(110\) 0 0
\(111\) −1.09441 −0.103877
\(112\) −3.20984 −0.303301
\(113\) −2.18044 −0.205119 −0.102559 0.994727i \(-0.532703\pi\)
−0.102559 + 0.994727i \(0.532703\pi\)
\(114\) −2.92858 −0.274287
\(115\) 0 0
\(116\) 6.30425 0.585335
\(117\) −0.265531 −0.0245484
\(118\) 6.92858 0.637828
\(119\) −3.14239 −0.288063
\(120\) 0 0
\(121\) 3.61741 0.328856
\(122\) 10.4885 0.949583
\(123\) −12.9430 −1.16703
\(124\) −3.62372 −0.325420
\(125\) 0 0
\(126\) 5.78496 0.515365
\(127\) −5.39390 −0.478631 −0.239316 0.970942i \(-0.576923\pi\)
−0.239316 + 0.970942i \(0.576923\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.96195 0.436876
\(130\) 0 0
\(131\) −8.97060 −0.783765 −0.391883 0.920015i \(-0.628176\pi\)
−0.391883 + 0.920015i \(0.628176\pi\)
\(132\) −4.18424 −0.364191
\(133\) −8.58933 −0.744789
\(134\) 2.80936 0.242692
\(135\) 0 0
\(136\) 0.978989 0.0839476
\(137\) −2.70352 −0.230978 −0.115489 0.993309i \(-0.536843\pi\)
−0.115489 + 0.993309i \(0.536843\pi\)
\(138\) 2.55672 0.217642
\(139\) 4.30358 0.365025 0.182512 0.983204i \(-0.441577\pi\)
0.182512 + 0.983204i \(0.441577\pi\)
\(140\) 0 0
\(141\) −6.81912 −0.574273
\(142\) −12.3189 −1.03378
\(143\) 0.563292 0.0471048
\(144\) −1.80226 −0.150188
\(145\) 0 0
\(146\) −13.9966 −1.15837
\(147\) −3.61490 −0.298152
\(148\) 1.00000 0.0821995
\(149\) 21.6949 1.77732 0.888659 0.458568i \(-0.151638\pi\)
0.888659 + 0.458568i \(0.151638\pi\)
\(150\) 0 0
\(151\) 2.06744 0.168246 0.0841230 0.996455i \(-0.473191\pi\)
0.0841230 + 0.996455i \(0.473191\pi\)
\(152\) 2.67594 0.217047
\(153\) −1.76439 −0.142643
\(154\) −12.2721 −0.988912
\(155\) 0 0
\(156\) −0.161242 −0.0129097
\(157\) 18.5439 1.47996 0.739982 0.672627i \(-0.234834\pi\)
0.739982 + 0.672627i \(0.234834\pi\)
\(158\) 15.6057 1.24152
\(159\) −12.3645 −0.980569
\(160\) 0 0
\(161\) 7.49868 0.590979
\(162\) −0.345071 −0.0271113
\(163\) −21.6240 −1.69372 −0.846860 0.531817i \(-0.821510\pi\)
−0.846860 + 0.531817i \(0.821510\pi\)
\(164\) 11.8265 0.923491
\(165\) 0 0
\(166\) 13.5371 1.05068
\(167\) −11.7729 −0.911012 −0.455506 0.890233i \(-0.650542\pi\)
−0.455506 + 0.890233i \(0.650542\pi\)
\(168\) 3.51289 0.271025
\(169\) −12.9783 −0.998330
\(170\) 0 0
\(171\) −4.82274 −0.368804
\(172\) −4.53390 −0.345706
\(173\) 14.0158 1.06560 0.532801 0.846240i \(-0.321139\pi\)
0.532801 + 0.846240i \(0.321139\pi\)
\(174\) −6.89945 −0.523046
\(175\) 0 0
\(176\) 3.82327 0.288190
\(177\) −7.58273 −0.569953
\(178\) −6.46929 −0.484894
\(179\) −12.4160 −0.928019 −0.464009 0.885830i \(-0.653590\pi\)
−0.464009 + 0.885830i \(0.653590\pi\)
\(180\) 0 0
\(181\) 8.19748 0.609314 0.304657 0.952462i \(-0.401458\pi\)
0.304657 + 0.952462i \(0.401458\pi\)
\(182\) −0.472913 −0.0350546
\(183\) −11.4787 −0.848532
\(184\) −2.33616 −0.172224
\(185\) 0 0
\(186\) 3.96585 0.290790
\(187\) 3.74294 0.273711
\(188\) 6.23085 0.454431
\(189\) −16.8698 −1.22710
\(190\) 0 0
\(191\) 8.00137 0.578959 0.289479 0.957184i \(-0.406518\pi\)
0.289479 + 0.957184i \(0.406518\pi\)
\(192\) −1.09441 −0.0789824
\(193\) −14.0420 −1.01077 −0.505383 0.862895i \(-0.668649\pi\)
−0.505383 + 0.862895i \(0.668649\pi\)
\(194\) −3.07063 −0.220458
\(195\) 0 0
\(196\) 3.30305 0.235932
\(197\) 3.36608 0.239823 0.119912 0.992785i \(-0.461739\pi\)
0.119912 + 0.992785i \(0.461739\pi\)
\(198\) −6.89054 −0.489689
\(199\) −14.8890 −1.05545 −0.527725 0.849415i \(-0.676955\pi\)
−0.527725 + 0.849415i \(0.676955\pi\)
\(200\) 0 0
\(201\) −3.07460 −0.216866
\(202\) −6.57513 −0.462624
\(203\) −20.2356 −1.42026
\(204\) −1.07142 −0.0750142
\(205\) 0 0
\(206\) −2.52180 −0.175702
\(207\) 4.21037 0.292641
\(208\) 0.147332 0.0102157
\(209\) 10.2308 0.707683
\(210\) 0 0
\(211\) −9.80544 −0.675035 −0.337517 0.941319i \(-0.609587\pi\)
−0.337517 + 0.941319i \(0.609587\pi\)
\(212\) 11.2978 0.775939
\(213\) 13.4819 0.923766
\(214\) 5.70291 0.389843
\(215\) 0 0
\(216\) 5.25565 0.357602
\(217\) 11.6316 0.789601
\(218\) 11.9318 0.808121
\(219\) 15.3181 1.03510
\(220\) 0 0
\(221\) 0.144237 0.00970241
\(222\) −1.09441 −0.0734522
\(223\) −15.4447 −1.03425 −0.517125 0.855910i \(-0.672998\pi\)
−0.517125 + 0.855910i \(0.672998\pi\)
\(224\) −3.20984 −0.214466
\(225\) 0 0
\(226\) −2.18044 −0.145041
\(227\) 6.46090 0.428825 0.214413 0.976743i \(-0.431216\pi\)
0.214413 + 0.976743i \(0.431216\pi\)
\(228\) −2.92858 −0.193950
\(229\) 7.97458 0.526975 0.263488 0.964663i \(-0.415127\pi\)
0.263488 + 0.964663i \(0.415127\pi\)
\(230\) 0 0
\(231\) 13.4307 0.883676
\(232\) 6.30425 0.413894
\(233\) 25.2907 1.65685 0.828423 0.560103i \(-0.189238\pi\)
0.828423 + 0.560103i \(0.189238\pi\)
\(234\) −0.265531 −0.0173583
\(235\) 0 0
\(236\) 6.92858 0.451012
\(237\) −17.0791 −1.10941
\(238\) −3.14239 −0.203691
\(239\) 5.11481 0.330850 0.165425 0.986222i \(-0.447100\pi\)
0.165425 + 0.986222i \(0.447100\pi\)
\(240\) 0 0
\(241\) 16.4415 1.05909 0.529544 0.848282i \(-0.322363\pi\)
0.529544 + 0.848282i \(0.322363\pi\)
\(242\) 3.61741 0.232536
\(243\) −15.3893 −0.987225
\(244\) 10.4885 0.671457
\(245\) 0 0
\(246\) −12.9430 −0.825217
\(247\) 0.394252 0.0250857
\(248\) −3.62372 −0.230107
\(249\) −14.8152 −0.938872
\(250\) 0 0
\(251\) 6.35586 0.401178 0.200589 0.979675i \(-0.435714\pi\)
0.200589 + 0.979675i \(0.435714\pi\)
\(252\) 5.78496 0.364418
\(253\) −8.93177 −0.561535
\(254\) −5.39390 −0.338444
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −5.45772 −0.340443 −0.170222 0.985406i \(-0.554448\pi\)
−0.170222 + 0.985406i \(0.554448\pi\)
\(258\) 4.96195 0.308918
\(259\) −3.20984 −0.199450
\(260\) 0 0
\(261\) −11.3619 −0.703284
\(262\) −8.97060 −0.554206
\(263\) −14.8882 −0.918044 −0.459022 0.888425i \(-0.651800\pi\)
−0.459022 + 0.888425i \(0.651800\pi\)
\(264\) −4.18424 −0.257522
\(265\) 0 0
\(266\) −8.58933 −0.526646
\(267\) 7.08007 0.433293
\(268\) 2.80936 0.171609
\(269\) 15.5573 0.948546 0.474273 0.880378i \(-0.342711\pi\)
0.474273 + 0.880378i \(0.342711\pi\)
\(270\) 0 0
\(271\) 0.462920 0.0281204 0.0140602 0.999901i \(-0.495524\pi\)
0.0140602 + 0.999901i \(0.495524\pi\)
\(272\) 0.978989 0.0593599
\(273\) 0.517562 0.0313242
\(274\) −2.70352 −0.163326
\(275\) 0 0
\(276\) 2.55672 0.153896
\(277\) 9.12112 0.548035 0.274018 0.961725i \(-0.411647\pi\)
0.274018 + 0.961725i \(0.411647\pi\)
\(278\) 4.30358 0.258111
\(279\) 6.53089 0.390994
\(280\) 0 0
\(281\) −24.8734 −1.48382 −0.741912 0.670497i \(-0.766081\pi\)
−0.741912 + 0.670497i \(0.766081\pi\)
\(282\) −6.81912 −0.406073
\(283\) 19.7259 1.17258 0.586292 0.810100i \(-0.300587\pi\)
0.586292 + 0.810100i \(0.300587\pi\)
\(284\) −12.3189 −0.730990
\(285\) 0 0
\(286\) 0.563292 0.0333081
\(287\) −37.9610 −2.24077
\(288\) −1.80226 −0.106199
\(289\) −16.0416 −0.943622
\(290\) 0 0
\(291\) 3.36053 0.196998
\(292\) −13.9966 −0.819090
\(293\) −2.65528 −0.155123 −0.0775615 0.996988i \(-0.524713\pi\)
−0.0775615 + 0.996988i \(0.524713\pi\)
\(294\) −3.61490 −0.210825
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 20.0938 1.16596
\(298\) 21.6949 1.25675
\(299\) −0.344191 −0.0199051
\(300\) 0 0
\(301\) 14.5531 0.838825
\(302\) 2.06744 0.118968
\(303\) 7.19590 0.413394
\(304\) 2.67594 0.153476
\(305\) 0 0
\(306\) −1.76439 −0.100864
\(307\) −25.3545 −1.44706 −0.723528 0.690295i \(-0.757481\pi\)
−0.723528 + 0.690295i \(0.757481\pi\)
\(308\) −12.2721 −0.699267
\(309\) 2.75989 0.157005
\(310\) 0 0
\(311\) 11.1234 0.630748 0.315374 0.948967i \(-0.397870\pi\)
0.315374 + 0.948967i \(0.397870\pi\)
\(312\) −0.161242 −0.00912855
\(313\) 3.49109 0.197328 0.0986640 0.995121i \(-0.468543\pi\)
0.0986640 + 0.995121i \(0.468543\pi\)
\(314\) 18.5439 1.04649
\(315\) 0 0
\(316\) 15.6057 0.877890
\(317\) 18.3656 1.03152 0.515759 0.856734i \(-0.327510\pi\)
0.515759 + 0.856734i \(0.327510\pi\)
\(318\) −12.3645 −0.693367
\(319\) 24.1029 1.34950
\(320\) 0 0
\(321\) −6.24134 −0.348357
\(322\) 7.49868 0.417885
\(323\) 2.61972 0.145765
\(324\) −0.345071 −0.0191706
\(325\) 0 0
\(326\) −21.6240 −1.19764
\(327\) −13.0583 −0.722124
\(328\) 11.8265 0.653007
\(329\) −20.0000 −1.10264
\(330\) 0 0
\(331\) −21.9365 −1.20574 −0.602870 0.797839i \(-0.705976\pi\)
−0.602870 + 0.797839i \(0.705976\pi\)
\(332\) 13.5371 0.742944
\(333\) −1.80226 −0.0987633
\(334\) −11.7729 −0.644183
\(335\) 0 0
\(336\) 3.51289 0.191644
\(337\) −8.65472 −0.471453 −0.235726 0.971819i \(-0.575747\pi\)
−0.235726 + 0.971819i \(0.575747\pi\)
\(338\) −12.9783 −0.705926
\(339\) 2.38630 0.129606
\(340\) 0 0
\(341\) −13.8545 −0.750262
\(342\) −4.82274 −0.260784
\(343\) 11.8666 0.640737
\(344\) −4.53390 −0.244451
\(345\) 0 0
\(346\) 14.0158 0.753495
\(347\) −21.3265 −1.14486 −0.572432 0.819952i \(-0.694000\pi\)
−0.572432 + 0.819952i \(0.694000\pi\)
\(348\) −6.89945 −0.369849
\(349\) −1.16305 −0.0622569 −0.0311285 0.999515i \(-0.509910\pi\)
−0.0311285 + 0.999515i \(0.509910\pi\)
\(350\) 0 0
\(351\) 0.774328 0.0413306
\(352\) 3.82327 0.203781
\(353\) −0.470079 −0.0250198 −0.0125099 0.999922i \(-0.503982\pi\)
−0.0125099 + 0.999922i \(0.503982\pi\)
\(354\) −7.58273 −0.403017
\(355\) 0 0
\(356\) −6.46929 −0.342872
\(357\) 3.43908 0.182015
\(358\) −12.4160 −0.656208
\(359\) −33.3086 −1.75796 −0.878981 0.476856i \(-0.841776\pi\)
−0.878981 + 0.476856i \(0.841776\pi\)
\(360\) 0 0
\(361\) −11.8393 −0.623123
\(362\) 8.19748 0.430850
\(363\) −3.95894 −0.207790
\(364\) −0.472913 −0.0247874
\(365\) 0 0
\(366\) −11.4787 −0.600003
\(367\) −11.1884 −0.584029 −0.292014 0.956414i \(-0.594326\pi\)
−0.292014 + 0.956414i \(0.594326\pi\)
\(368\) −2.33616 −0.121781
\(369\) −21.3144 −1.10958
\(370\) 0 0
\(371\) −36.2642 −1.88275
\(372\) 3.96585 0.205620
\(373\) 23.1634 1.19936 0.599678 0.800242i \(-0.295295\pi\)
0.599678 + 0.800242i \(0.295295\pi\)
\(374\) 3.74294 0.193543
\(375\) 0 0
\(376\) 6.23085 0.321331
\(377\) 0.928820 0.0478366
\(378\) −16.8698 −0.867688
\(379\) −26.7721 −1.37519 −0.687595 0.726095i \(-0.741333\pi\)
−0.687595 + 0.726095i \(0.741333\pi\)
\(380\) 0 0
\(381\) 5.90315 0.302428
\(382\) 8.00137 0.409386
\(383\) 22.3189 1.14044 0.570220 0.821492i \(-0.306858\pi\)
0.570220 + 0.821492i \(0.306858\pi\)
\(384\) −1.09441 −0.0558490
\(385\) 0 0
\(386\) −14.0420 −0.714720
\(387\) 8.17127 0.415369
\(388\) −3.07063 −0.155887
\(389\) −24.3011 −1.23211 −0.616057 0.787701i \(-0.711271\pi\)
−0.616057 + 0.787701i \(0.711271\pi\)
\(390\) 0 0
\(391\) −2.28707 −0.115662
\(392\) 3.30305 0.166829
\(393\) 9.81754 0.495229
\(394\) 3.36608 0.169581
\(395\) 0 0
\(396\) −6.89054 −0.346262
\(397\) 28.1888 1.41476 0.707378 0.706835i \(-0.249878\pi\)
0.707378 + 0.706835i \(0.249878\pi\)
\(398\) −14.8890 −0.746316
\(399\) 9.40027 0.470602
\(400\) 0 0
\(401\) −15.9528 −0.796644 −0.398322 0.917246i \(-0.630407\pi\)
−0.398322 + 0.917246i \(0.630407\pi\)
\(402\) −3.07460 −0.153347
\(403\) −0.533891 −0.0265950
\(404\) −6.57513 −0.327125
\(405\) 0 0
\(406\) −20.2356 −1.00428
\(407\) 3.82327 0.189513
\(408\) −1.07142 −0.0530431
\(409\) 31.6014 1.56259 0.781294 0.624164i \(-0.214560\pi\)
0.781294 + 0.624164i \(0.214560\pi\)
\(410\) 0 0
\(411\) 2.95877 0.145945
\(412\) −2.52180 −0.124240
\(413\) −22.2396 −1.09434
\(414\) 4.21037 0.206928
\(415\) 0 0
\(416\) 0.147332 0.00722356
\(417\) −4.70989 −0.230644
\(418\) 10.2308 0.500407
\(419\) −18.4266 −0.900197 −0.450098 0.892979i \(-0.648611\pi\)
−0.450098 + 0.892979i \(0.648611\pi\)
\(420\) 0 0
\(421\) −19.1570 −0.933655 −0.466828 0.884348i \(-0.654603\pi\)
−0.466828 + 0.884348i \(0.654603\pi\)
\(422\) −9.80544 −0.477322
\(423\) −11.2296 −0.546003
\(424\) 11.2978 0.548672
\(425\) 0 0
\(426\) 13.4819 0.653201
\(427\) −33.6663 −1.62923
\(428\) 5.70291 0.275661
\(429\) −0.616473 −0.0297636
\(430\) 0 0
\(431\) −10.3947 −0.500694 −0.250347 0.968156i \(-0.580545\pi\)
−0.250347 + 0.968156i \(0.580545\pi\)
\(432\) 5.25565 0.252863
\(433\) 3.36622 0.161770 0.0808852 0.996723i \(-0.474225\pi\)
0.0808852 + 0.996723i \(0.474225\pi\)
\(434\) 11.6316 0.558332
\(435\) 0 0
\(436\) 11.9318 0.571428
\(437\) −6.25142 −0.299046
\(438\) 15.3181 0.731926
\(439\) −32.5045 −1.55135 −0.775677 0.631130i \(-0.782591\pi\)
−0.775677 + 0.631130i \(0.782591\pi\)
\(440\) 0 0
\(441\) −5.95296 −0.283474
\(442\) 0.144237 0.00686064
\(443\) −23.5479 −1.11879 −0.559397 0.828900i \(-0.688967\pi\)
−0.559397 + 0.828900i \(0.688967\pi\)
\(444\) −1.09441 −0.0519385
\(445\) 0 0
\(446\) −15.4447 −0.731325
\(447\) −23.7432 −1.12302
\(448\) −3.20984 −0.151651
\(449\) 12.5629 0.592878 0.296439 0.955052i \(-0.404201\pi\)
0.296439 + 0.955052i \(0.404201\pi\)
\(450\) 0 0
\(451\) 45.2158 2.12913
\(452\) −2.18044 −0.102559
\(453\) −2.26263 −0.106308
\(454\) 6.46090 0.303225
\(455\) 0 0
\(456\) −2.92858 −0.137143
\(457\) 6.53390 0.305643 0.152821 0.988254i \(-0.451164\pi\)
0.152821 + 0.988254i \(0.451164\pi\)
\(458\) 7.97458 0.372628
\(459\) 5.14523 0.240159
\(460\) 0 0
\(461\) 24.7616 1.15326 0.576631 0.817005i \(-0.304367\pi\)
0.576631 + 0.817005i \(0.304367\pi\)
\(462\) 13.4307 0.624853
\(463\) 25.9239 1.20479 0.602393 0.798200i \(-0.294214\pi\)
0.602393 + 0.798200i \(0.294214\pi\)
\(464\) 6.30425 0.292667
\(465\) 0 0
\(466\) 25.2907 1.17157
\(467\) 28.8182 1.33355 0.666773 0.745261i \(-0.267675\pi\)
0.666773 + 0.745261i \(0.267675\pi\)
\(468\) −0.265531 −0.0122742
\(469\) −9.01759 −0.416394
\(470\) 0 0
\(471\) −20.2947 −0.935129
\(472\) 6.92858 0.318914
\(473\) −17.3343 −0.797033
\(474\) −17.0791 −0.784469
\(475\) 0 0
\(476\) −3.14239 −0.144031
\(477\) −20.3617 −0.932297
\(478\) 5.11481 0.233946
\(479\) −41.7873 −1.90931 −0.954655 0.297714i \(-0.903776\pi\)
−0.954655 + 0.297714i \(0.903776\pi\)
\(480\) 0 0
\(481\) 0.147332 0.00671778
\(482\) 16.4415 0.748888
\(483\) −8.20665 −0.373416
\(484\) 3.61741 0.164428
\(485\) 0 0
\(486\) −15.3893 −0.698073
\(487\) 9.34463 0.423446 0.211723 0.977330i \(-0.432093\pi\)
0.211723 + 0.977330i \(0.432093\pi\)
\(488\) 10.4885 0.474791
\(489\) 23.6655 1.07019
\(490\) 0 0
\(491\) −10.7558 −0.485404 −0.242702 0.970101i \(-0.578034\pi\)
−0.242702 + 0.970101i \(0.578034\pi\)
\(492\) −12.9430 −0.583516
\(493\) 6.17179 0.277963
\(494\) 0.394252 0.0177383
\(495\) 0 0
\(496\) −3.62372 −0.162710
\(497\) 39.5415 1.77368
\(498\) −14.8152 −0.663883
\(499\) 29.0702 1.30136 0.650680 0.759352i \(-0.274484\pi\)
0.650680 + 0.759352i \(0.274484\pi\)
\(500\) 0 0
\(501\) 12.8844 0.575631
\(502\) 6.35586 0.283676
\(503\) 36.2683 1.61712 0.808561 0.588412i \(-0.200247\pi\)
0.808561 + 0.588412i \(0.200247\pi\)
\(504\) 5.78496 0.257683
\(505\) 0 0
\(506\) −8.93177 −0.397065
\(507\) 14.2036 0.630804
\(508\) −5.39390 −0.239316
\(509\) −22.2170 −0.984751 −0.492375 0.870383i \(-0.663871\pi\)
−0.492375 + 0.870383i \(0.663871\pi\)
\(510\) 0 0
\(511\) 44.9268 1.98745
\(512\) 1.00000 0.0441942
\(513\) 14.0638 0.620933
\(514\) −5.45772 −0.240730
\(515\) 0 0
\(516\) 4.96195 0.218438
\(517\) 23.8222 1.04770
\(518\) −3.20984 −0.141032
\(519\) −15.3391 −0.673311
\(520\) 0 0
\(521\) 41.2732 1.80821 0.904107 0.427306i \(-0.140537\pi\)
0.904107 + 0.427306i \(0.140537\pi\)
\(522\) −11.3619 −0.497297
\(523\) 11.0354 0.482545 0.241273 0.970457i \(-0.422435\pi\)
0.241273 + 0.970457i \(0.422435\pi\)
\(524\) −8.97060 −0.391883
\(525\) 0 0
\(526\) −14.8882 −0.649155
\(527\) −3.54758 −0.154535
\(528\) −4.18424 −0.182096
\(529\) −17.5424 −0.762712
\(530\) 0 0
\(531\) −12.4871 −0.541895
\(532\) −8.58933 −0.372395
\(533\) 1.74242 0.0754726
\(534\) 7.08007 0.306385
\(535\) 0 0
\(536\) 2.80936 0.121346
\(537\) 13.5883 0.586377
\(538\) 15.5573 0.670723
\(539\) 12.6285 0.543946
\(540\) 0 0
\(541\) −32.2338 −1.38584 −0.692920 0.721014i \(-0.743676\pi\)
−0.692920 + 0.721014i \(0.743676\pi\)
\(542\) 0.462920 0.0198841
\(543\) −8.97142 −0.385001
\(544\) 0.978989 0.0419738
\(545\) 0 0
\(546\) 0.517562 0.0221496
\(547\) 15.0234 0.642355 0.321177 0.947019i \(-0.395921\pi\)
0.321177 + 0.947019i \(0.395921\pi\)
\(548\) −2.70352 −0.115489
\(549\) −18.9030 −0.806760
\(550\) 0 0
\(551\) 16.8698 0.718677
\(552\) 2.55672 0.108821
\(553\) −50.0918 −2.13012
\(554\) 9.12112 0.387519
\(555\) 0 0
\(556\) 4.30358 0.182512
\(557\) 39.1275 1.65788 0.828942 0.559334i \(-0.188943\pi\)
0.828942 + 0.559334i \(0.188943\pi\)
\(558\) 6.53089 0.276475
\(559\) −0.667989 −0.0282529
\(560\) 0 0
\(561\) −4.09632 −0.172947
\(562\) −24.8734 −1.04922
\(563\) −27.1970 −1.14622 −0.573109 0.819479i \(-0.694263\pi\)
−0.573109 + 0.819479i \(0.694263\pi\)
\(564\) −6.81912 −0.287137
\(565\) 0 0
\(566\) 19.7259 0.829142
\(567\) 1.10762 0.0465157
\(568\) −12.3189 −0.516888
\(569\) −33.4499 −1.40229 −0.701146 0.713018i \(-0.747328\pi\)
−0.701146 + 0.713018i \(0.747328\pi\)
\(570\) 0 0
\(571\) 28.9145 1.21003 0.605017 0.796213i \(-0.293166\pi\)
0.605017 + 0.796213i \(0.293166\pi\)
\(572\) 0.563292 0.0235524
\(573\) −8.75680 −0.365821
\(574\) −37.9610 −1.58446
\(575\) 0 0
\(576\) −1.80226 −0.0750942
\(577\) 33.5613 1.39717 0.698587 0.715525i \(-0.253812\pi\)
0.698587 + 0.715525i \(0.253812\pi\)
\(578\) −16.0416 −0.667242
\(579\) 15.3678 0.638663
\(580\) 0 0
\(581\) −43.4518 −1.80268
\(582\) 3.36053 0.139299
\(583\) 43.1948 1.78894
\(584\) −13.9966 −0.579184
\(585\) 0 0
\(586\) −2.65528 −0.109689
\(587\) −4.83581 −0.199595 −0.0997976 0.995008i \(-0.531820\pi\)
−0.0997976 + 0.995008i \(0.531820\pi\)
\(588\) −3.61490 −0.149076
\(589\) −9.69686 −0.399552
\(590\) 0 0
\(591\) −3.68388 −0.151535
\(592\) 1.00000 0.0410997
\(593\) 43.1680 1.77270 0.886348 0.463020i \(-0.153234\pi\)
0.886348 + 0.463020i \(0.153234\pi\)
\(594\) 20.0938 0.824459
\(595\) 0 0
\(596\) 21.6949 0.888659
\(597\) 16.2947 0.666896
\(598\) −0.344191 −0.0140750
\(599\) −38.7714 −1.58416 −0.792078 0.610420i \(-0.791001\pi\)
−0.792078 + 0.610420i \(0.791001\pi\)
\(600\) 0 0
\(601\) 1.56064 0.0636597 0.0318299 0.999493i \(-0.489867\pi\)
0.0318299 + 0.999493i \(0.489867\pi\)
\(602\) 14.5531 0.593139
\(603\) −5.06320 −0.206190
\(604\) 2.06744 0.0841230
\(605\) 0 0
\(606\) 7.19590 0.292314
\(607\) 21.0494 0.854370 0.427185 0.904164i \(-0.359505\pi\)
0.427185 + 0.904164i \(0.359505\pi\)
\(608\) 2.67594 0.108524
\(609\) 22.1461 0.897405
\(610\) 0 0
\(611\) 0.918005 0.0371385
\(612\) −1.76439 −0.0713214
\(613\) −5.28786 −0.213574 −0.106787 0.994282i \(-0.534056\pi\)
−0.106787 + 0.994282i \(0.534056\pi\)
\(614\) −25.3545 −1.02322
\(615\) 0 0
\(616\) −12.2721 −0.494456
\(617\) −34.8545 −1.40319 −0.701595 0.712576i \(-0.747528\pi\)
−0.701595 + 0.712576i \(0.747528\pi\)
\(618\) 2.75989 0.111019
\(619\) −7.30122 −0.293461 −0.146730 0.989177i \(-0.546875\pi\)
−0.146730 + 0.989177i \(0.546875\pi\)
\(620\) 0 0
\(621\) −12.2780 −0.492701
\(622\) 11.1234 0.446006
\(623\) 20.7654 0.831946
\(624\) −0.161242 −0.00645486
\(625\) 0 0
\(626\) 3.49109 0.139532
\(627\) −11.1968 −0.447156
\(628\) 18.5439 0.739982
\(629\) 0.978989 0.0390348
\(630\) 0 0
\(631\) 24.2752 0.966381 0.483190 0.875515i \(-0.339478\pi\)
0.483190 + 0.875515i \(0.339478\pi\)
\(632\) 15.6057 0.620762
\(633\) 10.7312 0.426527
\(634\) 18.3656 0.729393
\(635\) 0 0
\(636\) −12.3645 −0.490285
\(637\) 0.486646 0.0192816
\(638\) 24.1029 0.954241
\(639\) 22.2018 0.878290
\(640\) 0 0
\(641\) −8.22087 −0.324705 −0.162353 0.986733i \(-0.551908\pi\)
−0.162353 + 0.986733i \(0.551908\pi\)
\(642\) −6.24134 −0.246326
\(643\) 29.7758 1.17424 0.587121 0.809499i \(-0.300261\pi\)
0.587121 + 0.809499i \(0.300261\pi\)
\(644\) 7.49868 0.295489
\(645\) 0 0
\(646\) 2.61972 0.103071
\(647\) −35.6356 −1.40098 −0.700490 0.713662i \(-0.747035\pi\)
−0.700490 + 0.713662i \(0.747035\pi\)
\(648\) −0.345071 −0.0135557
\(649\) 26.4899 1.03982
\(650\) 0 0
\(651\) −12.7297 −0.498917
\(652\) −21.6240 −0.846860
\(653\) 37.6128 1.47190 0.735951 0.677035i \(-0.236736\pi\)
0.735951 + 0.677035i \(0.236736\pi\)
\(654\) −13.0583 −0.510619
\(655\) 0 0
\(656\) 11.8265 0.461746
\(657\) 25.2256 0.984142
\(658\) −20.0000 −0.779681
\(659\) 45.0553 1.75510 0.877552 0.479481i \(-0.159175\pi\)
0.877552 + 0.479481i \(0.159175\pi\)
\(660\) 0 0
\(661\) 7.32365 0.284857 0.142429 0.989805i \(-0.454509\pi\)
0.142429 + 0.989805i \(0.454509\pi\)
\(662\) −21.9365 −0.852587
\(663\) −0.157854 −0.00613056
\(664\) 13.5371 0.525341
\(665\) 0 0
\(666\) −1.80226 −0.0698362
\(667\) −14.7277 −0.570260
\(668\) −11.7729 −0.455506
\(669\) 16.9028 0.653501
\(670\) 0 0
\(671\) 40.1003 1.54806
\(672\) 3.51289 0.135512
\(673\) −8.58807 −0.331046 −0.165523 0.986206i \(-0.552931\pi\)
−0.165523 + 0.986206i \(0.552931\pi\)
\(674\) −8.65472 −0.333368
\(675\) 0 0
\(676\) −12.9783 −0.499165
\(677\) −9.13523 −0.351096 −0.175548 0.984471i \(-0.556170\pi\)
−0.175548 + 0.984471i \(0.556170\pi\)
\(678\) 2.38630 0.0916454
\(679\) 9.85621 0.378247
\(680\) 0 0
\(681\) −7.07089 −0.270957
\(682\) −13.8545 −0.530515
\(683\) −40.7414 −1.55893 −0.779463 0.626449i \(-0.784508\pi\)
−0.779463 + 0.626449i \(0.784508\pi\)
\(684\) −4.82274 −0.184402
\(685\) 0 0
\(686\) 11.8666 0.453069
\(687\) −8.72748 −0.332974
\(688\) −4.53390 −0.172853
\(689\) 1.66454 0.0634139
\(690\) 0 0
\(691\) −28.7543 −1.09387 −0.546933 0.837176i \(-0.684205\pi\)
−0.546933 + 0.837176i \(0.684205\pi\)
\(692\) 14.0158 0.532801
\(693\) 22.1175 0.840174
\(694\) −21.3265 −0.809541
\(695\) 0 0
\(696\) −6.89945 −0.261523
\(697\) 11.5780 0.438547
\(698\) −1.16305 −0.0440223
\(699\) −27.6784 −1.04689
\(700\) 0 0
\(701\) −7.87020 −0.297253 −0.148627 0.988893i \(-0.547485\pi\)
−0.148627 + 0.988893i \(0.547485\pi\)
\(702\) 0.774328 0.0292251
\(703\) 2.67594 0.100925
\(704\) 3.82327 0.144095
\(705\) 0 0
\(706\) −0.470079 −0.0176917
\(707\) 21.1051 0.793738
\(708\) −7.58273 −0.284976
\(709\) 33.0458 1.24106 0.620530 0.784182i \(-0.286917\pi\)
0.620530 + 0.784182i \(0.286917\pi\)
\(710\) 0 0
\(711\) −28.1256 −1.05479
\(712\) −6.46929 −0.242447
\(713\) 8.46558 0.317039
\(714\) 3.43908 0.128704
\(715\) 0 0
\(716\) −12.4160 −0.464009
\(717\) −5.59771 −0.209050
\(718\) −33.3086 −1.24307
\(719\) 4.94138 0.184282 0.0921412 0.995746i \(-0.470629\pi\)
0.0921412 + 0.995746i \(0.470629\pi\)
\(720\) 0 0
\(721\) 8.09456 0.301457
\(722\) −11.8393 −0.440615
\(723\) −17.9937 −0.669195
\(724\) 8.19748 0.304657
\(725\) 0 0
\(726\) −3.95894 −0.146930
\(727\) 42.5178 1.57690 0.788448 0.615101i \(-0.210885\pi\)
0.788448 + 0.615101i \(0.210885\pi\)
\(728\) −0.472913 −0.0175273
\(729\) 17.8775 0.662129
\(730\) 0 0
\(731\) −4.43863 −0.164169
\(732\) −11.4787 −0.424266
\(733\) 10.2114 0.377167 0.188584 0.982057i \(-0.439610\pi\)
0.188584 + 0.982057i \(0.439610\pi\)
\(734\) −11.1884 −0.412971
\(735\) 0 0
\(736\) −2.33616 −0.0861119
\(737\) 10.7410 0.395648
\(738\) −21.3144 −0.784593
\(739\) 25.5528 0.939975 0.469988 0.882673i \(-0.344258\pi\)
0.469988 + 0.882673i \(0.344258\pi\)
\(740\) 0 0
\(741\) −0.431475 −0.0158506
\(742\) −36.2642 −1.33130
\(743\) 44.2852 1.62467 0.812334 0.583193i \(-0.198197\pi\)
0.812334 + 0.583193i \(0.198197\pi\)
\(744\) 3.96585 0.145395
\(745\) 0 0
\(746\) 23.1634 0.848072
\(747\) −24.3974 −0.892652
\(748\) 3.74294 0.136855
\(749\) −18.3054 −0.668865
\(750\) 0 0
\(751\) 41.2897 1.50668 0.753342 0.657629i \(-0.228440\pi\)
0.753342 + 0.657629i \(0.228440\pi\)
\(752\) 6.23085 0.227216
\(753\) −6.95593 −0.253488
\(754\) 0.928820 0.0338256
\(755\) 0 0
\(756\) −16.8698 −0.613548
\(757\) 6.57802 0.239082 0.119541 0.992829i \(-0.461858\pi\)
0.119541 + 0.992829i \(0.461858\pi\)
\(758\) −26.7721 −0.972406
\(759\) 9.77504 0.354811
\(760\) 0 0
\(761\) 20.3166 0.736475 0.368237 0.929732i \(-0.379961\pi\)
0.368237 + 0.929732i \(0.379961\pi\)
\(762\) 5.90315 0.213849
\(763\) −38.2990 −1.38652
\(764\) 8.00137 0.289479
\(765\) 0 0
\(766\) 22.3189 0.806413
\(767\) 1.02080 0.0368591
\(768\) −1.09441 −0.0394912
\(769\) 25.2435 0.910303 0.455151 0.890414i \(-0.349585\pi\)
0.455151 + 0.890414i \(0.349585\pi\)
\(770\) 0 0
\(771\) 5.97300 0.215112
\(772\) −14.0420 −0.505383
\(773\) −48.9781 −1.76162 −0.880810 0.473470i \(-0.843001\pi\)
−0.880810 + 0.473470i \(0.843001\pi\)
\(774\) 8.17127 0.293710
\(775\) 0 0
\(776\) −3.07063 −0.110229
\(777\) 3.51289 0.126024
\(778\) −24.3011 −0.871237
\(779\) 31.6469 1.13387
\(780\) 0 0
\(781\) −47.0984 −1.68531
\(782\) −2.28707 −0.0817855
\(783\) 33.1330 1.18408
\(784\) 3.30305 0.117966
\(785\) 0 0
\(786\) 9.81754 0.350180
\(787\) −3.78073 −0.134768 −0.0673842 0.997727i \(-0.521465\pi\)
−0.0673842 + 0.997727i \(0.521465\pi\)
\(788\) 3.36608 0.119912
\(789\) 16.2938 0.580075
\(790\) 0 0
\(791\) 6.99886 0.248851
\(792\) −6.89054 −0.244845
\(793\) 1.54529 0.0548750
\(794\) 28.1888 1.00038
\(795\) 0 0
\(796\) −14.8890 −0.527725
\(797\) −29.5487 −1.04667 −0.523334 0.852128i \(-0.675312\pi\)
−0.523334 + 0.852128i \(0.675312\pi\)
\(798\) 9.40027 0.332766
\(799\) 6.09993 0.215800
\(800\) 0 0
\(801\) 11.6593 0.411963
\(802\) −15.9528 −0.563312
\(803\) −53.5129 −1.88843
\(804\) −3.07460 −0.108433
\(805\) 0 0
\(806\) −0.533891 −0.0188055
\(807\) −17.0261 −0.599348
\(808\) −6.57513 −0.231312
\(809\) 47.4231 1.66731 0.833654 0.552287i \(-0.186245\pi\)
0.833654 + 0.552287i \(0.186245\pi\)
\(810\) 0 0
\(811\) 25.6962 0.902317 0.451159 0.892444i \(-0.351011\pi\)
0.451159 + 0.892444i \(0.351011\pi\)
\(812\) −20.2356 −0.710131
\(813\) −0.506626 −0.0177681
\(814\) 3.82327 0.134006
\(815\) 0 0
\(816\) −1.07142 −0.0375071
\(817\) −12.1324 −0.424460
\(818\) 31.6014 1.10492
\(819\) 0.852312 0.0297822
\(820\) 0 0
\(821\) 0.286013 0.00998192 0.00499096 0.999988i \(-0.498411\pi\)
0.00499096 + 0.999988i \(0.498411\pi\)
\(822\) 2.95877 0.103199
\(823\) 44.6093 1.55498 0.777491 0.628893i \(-0.216492\pi\)
0.777491 + 0.628893i \(0.216492\pi\)
\(824\) −2.52180 −0.0878510
\(825\) 0 0
\(826\) −22.2396 −0.773815
\(827\) −55.1277 −1.91698 −0.958488 0.285131i \(-0.907963\pi\)
−0.958488 + 0.285131i \(0.907963\pi\)
\(828\) 4.21037 0.146320
\(829\) −17.2029 −0.597482 −0.298741 0.954334i \(-0.596567\pi\)
−0.298741 + 0.954334i \(0.596567\pi\)
\(830\) 0 0
\(831\) −9.98227 −0.346281
\(832\) 0.147332 0.00510783
\(833\) 3.23365 0.112039
\(834\) −4.70989 −0.163090
\(835\) 0 0
\(836\) 10.2308 0.353841
\(837\) −19.0450 −0.658292
\(838\) −18.4266 −0.636535
\(839\) 27.3527 0.944320 0.472160 0.881513i \(-0.343474\pi\)
0.472160 + 0.881513i \(0.343474\pi\)
\(840\) 0 0
\(841\) 10.7436 0.370467
\(842\) −19.1570 −0.660194
\(843\) 27.2218 0.937568
\(844\) −9.80544 −0.337517
\(845\) 0 0
\(846\) −11.2296 −0.386082
\(847\) −11.6113 −0.398969
\(848\) 11.2978 0.387970
\(849\) −21.5883 −0.740908
\(850\) 0 0
\(851\) −2.33616 −0.0800824
\(852\) 13.4819 0.461883
\(853\) −51.8565 −1.77553 −0.887766 0.460294i \(-0.847744\pi\)
−0.887766 + 0.460294i \(0.847744\pi\)
\(854\) −33.6663 −1.15204
\(855\) 0 0
\(856\) 5.70291 0.194921
\(857\) −4.63553 −0.158347 −0.0791733 0.996861i \(-0.525228\pi\)
−0.0791733 + 0.996861i \(0.525228\pi\)
\(858\) −0.616473 −0.0210461
\(859\) 43.1006 1.47057 0.735287 0.677755i \(-0.237047\pi\)
0.735287 + 0.677755i \(0.237047\pi\)
\(860\) 0 0
\(861\) 41.5450 1.41585
\(862\) −10.3947 −0.354044
\(863\) 11.2463 0.382829 0.191414 0.981509i \(-0.438693\pi\)
0.191414 + 0.981509i \(0.438693\pi\)
\(864\) 5.25565 0.178801
\(865\) 0 0
\(866\) 3.36622 0.114389
\(867\) 17.5561 0.596237
\(868\) 11.6316 0.394801
\(869\) 59.6649 2.02399
\(870\) 0 0
\(871\) 0.413910 0.0140248
\(872\) 11.9318 0.404061
\(873\) 5.53407 0.187300
\(874\) −6.25142 −0.211457
\(875\) 0 0
\(876\) 15.3181 0.517550
\(877\) −17.9444 −0.605939 −0.302969 0.953000i \(-0.597978\pi\)
−0.302969 + 0.953000i \(0.597978\pi\)
\(878\) −32.5045 −1.09697
\(879\) 2.90597 0.0980160
\(880\) 0 0
\(881\) −16.6495 −0.560935 −0.280468 0.959864i \(-0.590489\pi\)
−0.280468 + 0.959864i \(0.590489\pi\)
\(882\) −5.95296 −0.200446
\(883\) 28.7758 0.968383 0.484191 0.874962i \(-0.339114\pi\)
0.484191 + 0.874962i \(0.339114\pi\)
\(884\) 0.144237 0.00485121
\(885\) 0 0
\(886\) −23.5479 −0.791107
\(887\) −23.6957 −0.795625 −0.397812 0.917467i \(-0.630230\pi\)
−0.397812 + 0.917467i \(0.630230\pi\)
\(888\) −1.09441 −0.0367261
\(889\) 17.3135 0.580678
\(890\) 0 0
\(891\) −1.31930 −0.0441982
\(892\) −15.4447 −0.517125
\(893\) 16.6734 0.557953
\(894\) −23.7432 −0.794092
\(895\) 0 0
\(896\) −3.20984 −0.107233
\(897\) 0.376687 0.0125772
\(898\) 12.5629 0.419228
\(899\) −22.8448 −0.761918
\(900\) 0 0
\(901\) 11.0605 0.368478
\(902\) 45.2158 1.50552
\(903\) −15.9271 −0.530019
\(904\) −2.18044 −0.0725204
\(905\) 0 0
\(906\) −2.26263 −0.0751710
\(907\) −5.56548 −0.184799 −0.0923994 0.995722i \(-0.529454\pi\)
−0.0923994 + 0.995722i \(0.529454\pi\)
\(908\) 6.46090 0.214413
\(909\) 11.8501 0.393043
\(910\) 0 0
\(911\) −17.2514 −0.571565 −0.285782 0.958295i \(-0.592253\pi\)
−0.285782 + 0.958295i \(0.592253\pi\)
\(912\) −2.92858 −0.0969751
\(913\) 51.7559 1.71287
\(914\) 6.53390 0.216122
\(915\) 0 0
\(916\) 7.97458 0.263488
\(917\) 28.7942 0.950867
\(918\) 5.14523 0.169818
\(919\) −8.34007 −0.275114 −0.137557 0.990494i \(-0.543925\pi\)
−0.137557 + 0.990494i \(0.543925\pi\)
\(920\) 0 0
\(921\) 27.7483 0.914336
\(922\) 24.7616 0.815480
\(923\) −1.81497 −0.0597403
\(924\) 13.4307 0.441838
\(925\) 0 0
\(926\) 25.9239 0.851913
\(927\) 4.54494 0.149275
\(928\) 6.30425 0.206947
\(929\) −48.2529 −1.58313 −0.791564 0.611087i \(-0.790733\pi\)
−0.791564 + 0.611087i \(0.790733\pi\)
\(930\) 0 0
\(931\) 8.83876 0.289679
\(932\) 25.2907 0.828423
\(933\) −12.1736 −0.398544
\(934\) 28.8182 0.942959
\(935\) 0 0
\(936\) −0.265531 −0.00867916
\(937\) −30.2000 −0.986592 −0.493296 0.869862i \(-0.664208\pi\)
−0.493296 + 0.869862i \(0.664208\pi\)
\(938\) −9.01759 −0.294435
\(939\) −3.82069 −0.124684
\(940\) 0 0
\(941\) 14.9206 0.486399 0.243199 0.969976i \(-0.421803\pi\)
0.243199 + 0.969976i \(0.421803\pi\)
\(942\) −20.2947 −0.661236
\(943\) −27.6285 −0.899707
\(944\) 6.92858 0.225506
\(945\) 0 0
\(946\) −17.3343 −0.563587
\(947\) −48.1091 −1.56334 −0.781668 0.623694i \(-0.785631\pi\)
−0.781668 + 0.623694i \(0.785631\pi\)
\(948\) −17.0791 −0.554703
\(949\) −2.06215 −0.0669403
\(950\) 0 0
\(951\) −20.0996 −0.651774
\(952\) −3.14239 −0.101846
\(953\) −8.67149 −0.280897 −0.140449 0.990088i \(-0.544854\pi\)
−0.140449 + 0.990088i \(0.544854\pi\)
\(954\) −20.3617 −0.659234
\(955\) 0 0
\(956\) 5.11481 0.165425
\(957\) −26.3785 −0.852695
\(958\) −41.7873 −1.35009
\(959\) 8.67787 0.280223
\(960\) 0 0
\(961\) −17.8686 −0.576408
\(962\) 0.147332 0.00475018
\(963\) −10.2781 −0.331208
\(964\) 16.4415 0.529544
\(965\) 0 0
\(966\) −8.20665 −0.264045
\(967\) −18.5944 −0.597955 −0.298978 0.954260i \(-0.596646\pi\)
−0.298978 + 0.954260i \(0.596646\pi\)
\(968\) 3.61741 0.116268
\(969\) −2.86705 −0.0921029
\(970\) 0 0
\(971\) −32.6679 −1.04836 −0.524182 0.851606i \(-0.675629\pi\)
−0.524182 + 0.851606i \(0.675629\pi\)
\(972\) −15.3893 −0.493612
\(973\) −13.8138 −0.442850
\(974\) 9.34463 0.299421
\(975\) 0 0
\(976\) 10.4885 0.335728
\(977\) 28.1856 0.901737 0.450868 0.892590i \(-0.351114\pi\)
0.450868 + 0.892590i \(0.351114\pi\)
\(978\) 23.6655 0.756740
\(979\) −24.7338 −0.790497
\(980\) 0 0
\(981\) −21.5042 −0.686575
\(982\) −10.7558 −0.343232
\(983\) −25.3845 −0.809639 −0.404819 0.914397i \(-0.632666\pi\)
−0.404819 + 0.914397i \(0.632666\pi\)
\(984\) −12.9430 −0.412608
\(985\) 0 0
\(986\) 6.17179 0.196550
\(987\) 21.8883 0.696711
\(988\) 0.394252 0.0125428
\(989\) 10.5919 0.336803
\(990\) 0 0
\(991\) −21.1083 −0.670527 −0.335264 0.942124i \(-0.608825\pi\)
−0.335264 + 0.942124i \(0.608825\pi\)
\(992\) −3.62372 −0.115053
\(993\) 24.0076 0.761858
\(994\) 39.5415 1.25418
\(995\) 0 0
\(996\) −14.8152 −0.469436
\(997\) 39.0978 1.23824 0.619120 0.785296i \(-0.287489\pi\)
0.619120 + 0.785296i \(0.287489\pi\)
\(998\) 29.0702 0.920201
\(999\) 5.25565 0.166282
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 1850.2.a.be.1.2 5
5.2 odd 4 370.2.b.d.149.9 yes 10
5.3 odd 4 370.2.b.d.149.2 10
5.4 even 2 1850.2.a.bd.1.4 5
15.2 even 4 3330.2.d.p.1999.2 10
15.8 even 4 3330.2.d.p.1999.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.2 10 5.3 odd 4
370.2.b.d.149.9 yes 10 5.2 odd 4
1850.2.a.bd.1.4 5 5.4 even 2
1850.2.a.be.1.2 5 1.1 even 1 trivial
3330.2.d.p.1999.2 10 15.2 even 4
3330.2.d.p.1999.7 10 15.8 even 4