Properties

Label 2850.2.a.bj.1.2
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 2850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.44949 q^{11} +1.00000 q^{12} -2.44949 q^{13} +4.44949 q^{14} +1.00000 q^{16} +4.44949 q^{17} +1.00000 q^{18} +1.00000 q^{19} +4.44949 q^{21} +3.44949 q^{22} -1.00000 q^{23} +1.00000 q^{24} -2.44949 q^{26} +1.00000 q^{27} +4.44949 q^{28} -4.34847 q^{29} -3.00000 q^{31} +1.00000 q^{32} +3.44949 q^{33} +4.44949 q^{34} +1.00000 q^{36} -7.79796 q^{37} +1.00000 q^{38} -2.44949 q^{39} -0.898979 q^{41} +4.44949 q^{42} -2.44949 q^{43} +3.44949 q^{44} -1.00000 q^{46} -7.79796 q^{47} +1.00000 q^{48} +12.7980 q^{49} +4.44949 q^{51} -2.44949 q^{52} -7.44949 q^{53} +1.00000 q^{54} +4.44949 q^{56} +1.00000 q^{57} -4.34847 q^{58} +6.44949 q^{59} -9.44949 q^{61} -3.00000 q^{62} +4.44949 q^{63} +1.00000 q^{64} +3.44949 q^{66} +15.2474 q^{67} +4.44949 q^{68} -1.00000 q^{69} -1.55051 q^{71} +1.00000 q^{72} -1.00000 q^{73} -7.79796 q^{74} +1.00000 q^{76} +15.3485 q^{77} -2.44949 q^{78} -5.00000 q^{79} +1.00000 q^{81} -0.898979 q^{82} +8.34847 q^{83} +4.44949 q^{84} -2.44949 q^{86} -4.34847 q^{87} +3.44949 q^{88} +2.10102 q^{89} -10.8990 q^{91} -1.00000 q^{92} -3.00000 q^{93} -7.79796 q^{94} +1.00000 q^{96} +1.55051 q^{97} +12.7980 q^{98} +3.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + 2q^{11} + 2q^{12} + 4q^{14} + 2q^{16} + 4q^{17} + 2q^{18} + 2q^{19} + 4q^{21} + 2q^{22} - 2q^{23} + 2q^{24} + 2q^{27} + 4q^{28} + 6q^{29} - 6q^{31} + 2q^{32} + 2q^{33} + 4q^{34} + 2q^{36} + 4q^{37} + 2q^{38} + 8q^{41} + 4q^{42} + 2q^{44} - 2q^{46} + 4q^{47} + 2q^{48} + 6q^{49} + 4q^{51} - 10q^{53} + 2q^{54} + 4q^{56} + 2q^{57} + 6q^{58} + 8q^{59} - 14q^{61} - 6q^{62} + 4q^{63} + 2q^{64} + 2q^{66} + 6q^{67} + 4q^{68} - 2q^{69} - 8q^{71} + 2q^{72} - 2q^{73} + 4q^{74} + 2q^{76} + 16q^{77} - 10q^{79} + 2q^{81} + 8q^{82} + 2q^{83} + 4q^{84} + 6q^{87} + 2q^{88} + 14q^{89} - 12q^{91} - 2q^{92} - 6q^{93} + 4q^{94} + 2q^{96} + 8q^{97} + 6q^{98} + 2q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 4.44949 1.68175 0.840875 0.541230i \(-0.182041\pi\)
0.840875 + 0.541230i \(0.182041\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.44949 1.04006 0.520030 0.854148i \(-0.325921\pi\)
0.520030 + 0.854148i \(0.325921\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.44949 −0.679366 −0.339683 0.940540i \(-0.610320\pi\)
−0.339683 + 0.940540i \(0.610320\pi\)
\(14\) 4.44949 1.18918
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.44949 1.07916 0.539580 0.841934i \(-0.318583\pi\)
0.539580 + 0.841934i \(0.318583\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.44949 0.970958
\(22\) 3.44949 0.735434
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.44949 −0.480384
\(27\) 1.00000 0.192450
\(28\) 4.44949 0.840875
\(29\) −4.34847 −0.807490 −0.403745 0.914871i \(-0.632292\pi\)
−0.403745 + 0.914871i \(0.632292\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.44949 0.600479
\(34\) 4.44949 0.763081
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.79796 −1.28198 −0.640988 0.767551i \(-0.721475\pi\)
−0.640988 + 0.767551i \(0.721475\pi\)
\(38\) 1.00000 0.162221
\(39\) −2.44949 −0.392232
\(40\) 0 0
\(41\) −0.898979 −0.140397 −0.0701985 0.997533i \(-0.522363\pi\)
−0.0701985 + 0.997533i \(0.522363\pi\)
\(42\) 4.44949 0.686571
\(43\) −2.44949 −0.373544 −0.186772 0.982403i \(-0.559803\pi\)
−0.186772 + 0.982403i \(0.559803\pi\)
\(44\) 3.44949 0.520030
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −7.79796 −1.13745 −0.568725 0.822528i \(-0.692563\pi\)
−0.568725 + 0.822528i \(0.692563\pi\)
\(48\) 1.00000 0.144338
\(49\) 12.7980 1.82828
\(50\) 0 0
\(51\) 4.44949 0.623053
\(52\) −2.44949 −0.339683
\(53\) −7.44949 −1.02327 −0.511633 0.859204i \(-0.670959\pi\)
−0.511633 + 0.859204i \(0.670959\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 4.44949 0.594588
\(57\) 1.00000 0.132453
\(58\) −4.34847 −0.570982
\(59\) 6.44949 0.839652 0.419826 0.907605i \(-0.362091\pi\)
0.419826 + 0.907605i \(0.362091\pi\)
\(60\) 0 0
\(61\) −9.44949 −1.20988 −0.604942 0.796270i \(-0.706804\pi\)
−0.604942 + 0.796270i \(0.706804\pi\)
\(62\) −3.00000 −0.381000
\(63\) 4.44949 0.560583
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.44949 0.424603
\(67\) 15.2474 1.86277 0.931386 0.364033i \(-0.118600\pi\)
0.931386 + 0.364033i \(0.118600\pi\)
\(68\) 4.44949 0.539580
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −1.55051 −0.184012 −0.0920059 0.995758i \(-0.529328\pi\)
−0.0920059 + 0.995758i \(0.529328\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) −7.79796 −0.906494
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 15.3485 1.74912
\(78\) −2.44949 −0.277350
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −0.898979 −0.0992757
\(83\) 8.34847 0.916364 0.458182 0.888859i \(-0.348501\pi\)
0.458182 + 0.888859i \(0.348501\pi\)
\(84\) 4.44949 0.485479
\(85\) 0 0
\(86\) −2.44949 −0.264135
\(87\) −4.34847 −0.466205
\(88\) 3.44949 0.367717
\(89\) 2.10102 0.222708 0.111354 0.993781i \(-0.464481\pi\)
0.111354 + 0.993781i \(0.464481\pi\)
\(90\) 0 0
\(91\) −10.8990 −1.14252
\(92\) −1.00000 −0.104257
\(93\) −3.00000 −0.311086
\(94\) −7.79796 −0.804298
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 1.55051 0.157430 0.0787152 0.996897i \(-0.474918\pi\)
0.0787152 + 0.996897i \(0.474918\pi\)
\(98\) 12.7980 1.29279
\(99\) 3.44949 0.346687
\(100\) 0 0
\(101\) −1.55051 −0.154282 −0.0771408 0.997020i \(-0.524579\pi\)
−0.0771408 + 0.997020i \(0.524579\pi\)
\(102\) 4.44949 0.440565
\(103\) 1.89898 0.187112 0.0935560 0.995614i \(-0.470177\pi\)
0.0935560 + 0.995614i \(0.470177\pi\)
\(104\) −2.44949 −0.240192
\(105\) 0 0
\(106\) −7.44949 −0.723558
\(107\) 17.3485 1.67714 0.838570 0.544794i \(-0.183392\pi\)
0.838570 + 0.544794i \(0.183392\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.89898 −0.277672 −0.138836 0.990315i \(-0.544336\pi\)
−0.138836 + 0.990315i \(0.544336\pi\)
\(110\) 0 0
\(111\) −7.79796 −0.740150
\(112\) 4.44949 0.420437
\(113\) −16.7980 −1.58022 −0.790110 0.612966i \(-0.789976\pi\)
−0.790110 + 0.612966i \(0.789976\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −4.34847 −0.403745
\(117\) −2.44949 −0.226455
\(118\) 6.44949 0.593724
\(119\) 19.7980 1.81488
\(120\) 0 0
\(121\) 0.898979 0.0817254
\(122\) −9.44949 −0.855517
\(123\) −0.898979 −0.0810583
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) 4.44949 0.396392
\(127\) 0.101021 0.00896412 0.00448206 0.999990i \(-0.498573\pi\)
0.00448206 + 0.999990i \(0.498573\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.44949 −0.215666
\(130\) 0 0
\(131\) 9.24745 0.807953 0.403977 0.914769i \(-0.367628\pi\)
0.403977 + 0.914769i \(0.367628\pi\)
\(132\) 3.44949 0.300240
\(133\) 4.44949 0.385820
\(134\) 15.2474 1.31718
\(135\) 0 0
\(136\) 4.44949 0.381541
\(137\) −4.89898 −0.418548 −0.209274 0.977857i \(-0.567110\pi\)
−0.209274 + 0.977857i \(0.567110\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −22.2474 −1.88700 −0.943502 0.331367i \(-0.892490\pi\)
−0.943502 + 0.331367i \(0.892490\pi\)
\(140\) 0 0
\(141\) −7.79796 −0.656707
\(142\) −1.55051 −0.130116
\(143\) −8.44949 −0.706582
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −1.00000 −0.0827606
\(147\) 12.7980 1.05556
\(148\) −7.79796 −0.640988
\(149\) 5.79796 0.474987 0.237494 0.971389i \(-0.423674\pi\)
0.237494 + 0.971389i \(0.423674\pi\)
\(150\) 0 0
\(151\) −23.7980 −1.93665 −0.968325 0.249692i \(-0.919671\pi\)
−0.968325 + 0.249692i \(0.919671\pi\)
\(152\) 1.00000 0.0811107
\(153\) 4.44949 0.359720
\(154\) 15.3485 1.23681
\(155\) 0 0
\(156\) −2.44949 −0.196116
\(157\) −4.89898 −0.390981 −0.195491 0.980706i \(-0.562630\pi\)
−0.195491 + 0.980706i \(0.562630\pi\)
\(158\) −5.00000 −0.397779
\(159\) −7.44949 −0.590783
\(160\) 0 0
\(161\) −4.44949 −0.350669
\(162\) 1.00000 0.0785674
\(163\) 19.7980 1.55070 0.775348 0.631534i \(-0.217575\pi\)
0.775348 + 0.631534i \(0.217575\pi\)
\(164\) −0.898979 −0.0701985
\(165\) 0 0
\(166\) 8.34847 0.647967
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 4.44949 0.343286
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) −2.44949 −0.186772
\(173\) −1.65153 −0.125564 −0.0627818 0.998027i \(-0.519997\pi\)
−0.0627818 + 0.998027i \(0.519997\pi\)
\(174\) −4.34847 −0.329657
\(175\) 0 0
\(176\) 3.44949 0.260015
\(177\) 6.44949 0.484773
\(178\) 2.10102 0.157478
\(179\) 25.7980 1.92823 0.964115 0.265485i \(-0.0855321\pi\)
0.964115 + 0.265485i \(0.0855321\pi\)
\(180\) 0 0
\(181\) −14.4495 −1.07402 −0.537011 0.843575i \(-0.680447\pi\)
−0.537011 + 0.843575i \(0.680447\pi\)
\(182\) −10.8990 −0.807886
\(183\) −9.44949 −0.698526
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −3.00000 −0.219971
\(187\) 15.3485 1.12239
\(188\) −7.79796 −0.568725
\(189\) 4.44949 0.323653
\(190\) 0 0
\(191\) −0.101021 −0.00730959 −0.00365479 0.999993i \(-0.501163\pi\)
−0.00365479 + 0.999993i \(0.501163\pi\)
\(192\) 1.00000 0.0721688
\(193\) −15.3485 −1.10481 −0.552403 0.833577i \(-0.686289\pi\)
−0.552403 + 0.833577i \(0.686289\pi\)
\(194\) 1.55051 0.111320
\(195\) 0 0
\(196\) 12.7980 0.914140
\(197\) 27.3485 1.94850 0.974249 0.225475i \(-0.0723935\pi\)
0.974249 + 0.225475i \(0.0723935\pi\)
\(198\) 3.44949 0.245145
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) 15.2474 1.07547
\(202\) −1.55051 −0.109094
\(203\) −19.3485 −1.35800
\(204\) 4.44949 0.311527
\(205\) 0 0
\(206\) 1.89898 0.132308
\(207\) −1.00000 −0.0695048
\(208\) −2.44949 −0.169842
\(209\) 3.44949 0.238606
\(210\) 0 0
\(211\) −3.65153 −0.251382 −0.125691 0.992069i \(-0.540115\pi\)
−0.125691 + 0.992069i \(0.540115\pi\)
\(212\) −7.44949 −0.511633
\(213\) −1.55051 −0.106239
\(214\) 17.3485 1.18592
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −13.3485 −0.906153
\(218\) −2.89898 −0.196344
\(219\) −1.00000 −0.0675737
\(220\) 0 0
\(221\) −10.8990 −0.733145
\(222\) −7.79796 −0.523365
\(223\) 16.1010 1.07820 0.539102 0.842240i \(-0.318764\pi\)
0.539102 + 0.842240i \(0.318764\pi\)
\(224\) 4.44949 0.297294
\(225\) 0 0
\(226\) −16.7980 −1.11738
\(227\) 29.5959 1.96435 0.982175 0.187969i \(-0.0601903\pi\)
0.982175 + 0.187969i \(0.0601903\pi\)
\(228\) 1.00000 0.0662266
\(229\) 7.24745 0.478925 0.239462 0.970906i \(-0.423029\pi\)
0.239462 + 0.970906i \(0.423029\pi\)
\(230\) 0 0
\(231\) 15.3485 1.00986
\(232\) −4.34847 −0.285491
\(233\) 6.24745 0.409284 0.204642 0.978837i \(-0.434397\pi\)
0.204642 + 0.978837i \(0.434397\pi\)
\(234\) −2.44949 −0.160128
\(235\) 0 0
\(236\) 6.44949 0.419826
\(237\) −5.00000 −0.324785
\(238\) 19.7980 1.28331
\(239\) −8.69694 −0.562558 −0.281279 0.959626i \(-0.590759\pi\)
−0.281279 + 0.959626i \(0.590759\pi\)
\(240\) 0 0
\(241\) −4.44949 −0.286617 −0.143308 0.989678i \(-0.545774\pi\)
−0.143308 + 0.989678i \(0.545774\pi\)
\(242\) 0.898979 0.0577886
\(243\) 1.00000 0.0641500
\(244\) −9.44949 −0.604942
\(245\) 0 0
\(246\) −0.898979 −0.0573168
\(247\) −2.44949 −0.155857
\(248\) −3.00000 −0.190500
\(249\) 8.34847 0.529063
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 4.44949 0.280292
\(253\) −3.44949 −0.216868
\(254\) 0.101021 0.00633859
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −21.4949 −1.34081 −0.670407 0.741993i \(-0.733881\pi\)
−0.670407 + 0.741993i \(0.733881\pi\)
\(258\) −2.44949 −0.152499
\(259\) −34.6969 −2.15596
\(260\) 0 0
\(261\) −4.34847 −0.269163
\(262\) 9.24745 0.571309
\(263\) −6.79796 −0.419180 −0.209590 0.977789i \(-0.567213\pi\)
−0.209590 + 0.977789i \(0.567213\pi\)
\(264\) 3.44949 0.212301
\(265\) 0 0
\(266\) 4.44949 0.272816
\(267\) 2.10102 0.128580
\(268\) 15.2474 0.931386
\(269\) −2.89898 −0.176754 −0.0883769 0.996087i \(-0.528168\pi\)
−0.0883769 + 0.996087i \(0.528168\pi\)
\(270\) 0 0
\(271\) 22.9444 1.39377 0.696886 0.717182i \(-0.254568\pi\)
0.696886 + 0.717182i \(0.254568\pi\)
\(272\) 4.44949 0.269790
\(273\) −10.8990 −0.659636
\(274\) −4.89898 −0.295958
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 21.0454 1.26450 0.632248 0.774766i \(-0.282132\pi\)
0.632248 + 0.774766i \(0.282132\pi\)
\(278\) −22.2474 −1.33431
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) 7.00000 0.417585 0.208792 0.977960i \(-0.433047\pi\)
0.208792 + 0.977960i \(0.433047\pi\)
\(282\) −7.79796 −0.464362
\(283\) 29.7980 1.77130 0.885652 0.464349i \(-0.153712\pi\)
0.885652 + 0.464349i \(0.153712\pi\)
\(284\) −1.55051 −0.0920059
\(285\) 0 0
\(286\) −8.44949 −0.499629
\(287\) −4.00000 −0.236113
\(288\) 1.00000 0.0589256
\(289\) 2.79796 0.164586
\(290\) 0 0
\(291\) 1.55051 0.0908925
\(292\) −1.00000 −0.0585206
\(293\) −23.2474 −1.35813 −0.679065 0.734078i \(-0.737615\pi\)
−0.679065 + 0.734078i \(0.737615\pi\)
\(294\) 12.7980 0.746392
\(295\) 0 0
\(296\) −7.79796 −0.453247
\(297\) 3.44949 0.200160
\(298\) 5.79796 0.335867
\(299\) 2.44949 0.141658
\(300\) 0 0
\(301\) −10.8990 −0.628207
\(302\) −23.7980 −1.36942
\(303\) −1.55051 −0.0890745
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 4.44949 0.254360
\(307\) −16.3485 −0.933056 −0.466528 0.884506i \(-0.654495\pi\)
−0.466528 + 0.884506i \(0.654495\pi\)
\(308\) 15.3485 0.874560
\(309\) 1.89898 0.108029
\(310\) 0 0
\(311\) −23.7980 −1.34946 −0.674729 0.738065i \(-0.735740\pi\)
−0.674729 + 0.738065i \(0.735740\pi\)
\(312\) −2.44949 −0.138675
\(313\) 31.8990 1.80304 0.901518 0.432741i \(-0.142453\pi\)
0.901518 + 0.432741i \(0.142453\pi\)
\(314\) −4.89898 −0.276465
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) 15.2474 0.856382 0.428191 0.903688i \(-0.359151\pi\)
0.428191 + 0.903688i \(0.359151\pi\)
\(318\) −7.44949 −0.417747
\(319\) −15.0000 −0.839839
\(320\) 0 0
\(321\) 17.3485 0.968297
\(322\) −4.44949 −0.247960
\(323\) 4.44949 0.247576
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 19.7980 1.09651
\(327\) −2.89898 −0.160314
\(328\) −0.898979 −0.0496378
\(329\) −34.6969 −1.91290
\(330\) 0 0
\(331\) −22.3485 −1.22838 −0.614191 0.789157i \(-0.710518\pi\)
−0.614191 + 0.789157i \(0.710518\pi\)
\(332\) 8.34847 0.458182
\(333\) −7.79796 −0.427326
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 4.44949 0.242740
\(337\) −24.8990 −1.35633 −0.678167 0.734908i \(-0.737225\pi\)
−0.678167 + 0.734908i \(0.737225\pi\)
\(338\) −7.00000 −0.380750
\(339\) −16.7980 −0.912340
\(340\) 0 0
\(341\) −10.3485 −0.560401
\(342\) 1.00000 0.0540738
\(343\) 25.7980 1.39296
\(344\) −2.44949 −0.132068
\(345\) 0 0
\(346\) −1.65153 −0.0887868
\(347\) −23.5959 −1.26670 −0.633348 0.773867i \(-0.718320\pi\)
−0.633348 + 0.773867i \(0.718320\pi\)
\(348\) −4.34847 −0.233102
\(349\) −25.9444 −1.38877 −0.694386 0.719603i \(-0.744324\pi\)
−0.694386 + 0.719603i \(0.744324\pi\)
\(350\) 0 0
\(351\) −2.44949 −0.130744
\(352\) 3.44949 0.183858
\(353\) −18.2474 −0.971214 −0.485607 0.874177i \(-0.661401\pi\)
−0.485607 + 0.874177i \(0.661401\pi\)
\(354\) 6.44949 0.342787
\(355\) 0 0
\(356\) 2.10102 0.111354
\(357\) 19.7980 1.04782
\(358\) 25.7980 1.36346
\(359\) −22.8990 −1.20856 −0.604281 0.796771i \(-0.706540\pi\)
−0.604281 + 0.796771i \(0.706540\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −14.4495 −0.759448
\(363\) 0.898979 0.0471842
\(364\) −10.8990 −0.571262
\(365\) 0 0
\(366\) −9.44949 −0.493933
\(367\) −5.55051 −0.289734 −0.144867 0.989451i \(-0.546275\pi\)
−0.144867 + 0.989451i \(0.546275\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −0.898979 −0.0467990
\(370\) 0 0
\(371\) −33.1464 −1.72088
\(372\) −3.00000 −0.155543
\(373\) −22.4495 −1.16239 −0.581195 0.813764i \(-0.697415\pi\)
−0.581195 + 0.813764i \(0.697415\pi\)
\(374\) 15.3485 0.793650
\(375\) 0 0
\(376\) −7.79796 −0.402149
\(377\) 10.6515 0.548582
\(378\) 4.44949 0.228857
\(379\) −1.30306 −0.0669338 −0.0334669 0.999440i \(-0.510655\pi\)
−0.0334669 + 0.999440i \(0.510655\pi\)
\(380\) 0 0
\(381\) 0.101021 0.00517544
\(382\) −0.101021 −0.00516866
\(383\) 16.2474 0.830206 0.415103 0.909774i \(-0.363746\pi\)
0.415103 + 0.909774i \(0.363746\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −15.3485 −0.781217
\(387\) −2.44949 −0.124515
\(388\) 1.55051 0.0787152
\(389\) 21.5959 1.09496 0.547478 0.836820i \(-0.315588\pi\)
0.547478 + 0.836820i \(0.315588\pi\)
\(390\) 0 0
\(391\) −4.44949 −0.225020
\(392\) 12.7980 0.646395
\(393\) 9.24745 0.466472
\(394\) 27.3485 1.37780
\(395\) 0 0
\(396\) 3.44949 0.173343
\(397\) 23.9444 1.20173 0.600867 0.799349i \(-0.294822\pi\)
0.600867 + 0.799349i \(0.294822\pi\)
\(398\) −10.0000 −0.501255
\(399\) 4.44949 0.222753
\(400\) 0 0
\(401\) 11.2020 0.559403 0.279702 0.960087i \(-0.409764\pi\)
0.279702 + 0.960087i \(0.409764\pi\)
\(402\) 15.2474 0.760474
\(403\) 7.34847 0.366053
\(404\) −1.55051 −0.0771408
\(405\) 0 0
\(406\) −19.3485 −0.960248
\(407\) −26.8990 −1.33333
\(408\) 4.44949 0.220283
\(409\) −22.8990 −1.13228 −0.566141 0.824309i \(-0.691564\pi\)
−0.566141 + 0.824309i \(0.691564\pi\)
\(410\) 0 0
\(411\) −4.89898 −0.241649
\(412\) 1.89898 0.0935560
\(413\) 28.6969 1.41208
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −2.44949 −0.120096
\(417\) −22.2474 −1.08946
\(418\) 3.44949 0.168720
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −3.65153 −0.177754
\(423\) −7.79796 −0.379150
\(424\) −7.44949 −0.361779
\(425\) 0 0
\(426\) −1.55051 −0.0751225
\(427\) −42.0454 −2.03472
\(428\) 17.3485 0.838570
\(429\) −8.44949 −0.407945
\(430\) 0 0
\(431\) −5.10102 −0.245708 −0.122854 0.992425i \(-0.539205\pi\)
−0.122854 + 0.992425i \(0.539205\pi\)
\(432\) 1.00000 0.0481125
\(433\) 4.65153 0.223538 0.111769 0.993734i \(-0.464348\pi\)
0.111769 + 0.993734i \(0.464348\pi\)
\(434\) −13.3485 −0.640747
\(435\) 0 0
\(436\) −2.89898 −0.138836
\(437\) −1.00000 −0.0478365
\(438\) −1.00000 −0.0477818
\(439\) −12.1010 −0.577550 −0.288775 0.957397i \(-0.593248\pi\)
−0.288775 + 0.957397i \(0.593248\pi\)
\(440\) 0 0
\(441\) 12.7980 0.609427
\(442\) −10.8990 −0.518412
\(443\) 21.2474 1.00950 0.504748 0.863267i \(-0.331585\pi\)
0.504748 + 0.863267i \(0.331585\pi\)
\(444\) −7.79796 −0.370075
\(445\) 0 0
\(446\) 16.1010 0.762405
\(447\) 5.79796 0.274234
\(448\) 4.44949 0.210219
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) −3.10102 −0.146021
\(452\) −16.7980 −0.790110
\(453\) −23.7980 −1.11813
\(454\) 29.5959 1.38901
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) −24.8990 −1.16473 −0.582363 0.812929i \(-0.697872\pi\)
−0.582363 + 0.812929i \(0.697872\pi\)
\(458\) 7.24745 0.338651
\(459\) 4.44949 0.207684
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 15.3485 0.714075
\(463\) −14.6969 −0.683025 −0.341512 0.939877i \(-0.610939\pi\)
−0.341512 + 0.939877i \(0.610939\pi\)
\(464\) −4.34847 −0.201873
\(465\) 0 0
\(466\) 6.24745 0.289407
\(467\) −6.34847 −0.293772 −0.146886 0.989153i \(-0.546925\pi\)
−0.146886 + 0.989153i \(0.546925\pi\)
\(468\) −2.44949 −0.113228
\(469\) 67.8434 3.13272
\(470\) 0 0
\(471\) −4.89898 −0.225733
\(472\) 6.44949 0.296862
\(473\) −8.44949 −0.388508
\(474\) −5.00000 −0.229658
\(475\) 0 0
\(476\) 19.7980 0.907438
\(477\) −7.44949 −0.341089
\(478\) −8.69694 −0.397789
\(479\) 16.5959 0.758287 0.379143 0.925338i \(-0.376219\pi\)
0.379143 + 0.925338i \(0.376219\pi\)
\(480\) 0 0
\(481\) 19.1010 0.870932
\(482\) −4.44949 −0.202669
\(483\) −4.44949 −0.202459
\(484\) 0.898979 0.0408627
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) −9.44949 −0.427758
\(489\) 19.7980 0.895295
\(490\) 0 0
\(491\) 43.5959 1.96746 0.983728 0.179664i \(-0.0575009\pi\)
0.983728 + 0.179664i \(0.0575009\pi\)
\(492\) −0.898979 −0.0405291
\(493\) −19.3485 −0.871411
\(494\) −2.44949 −0.110208
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) −6.89898 −0.309462
\(498\) 8.34847 0.374104
\(499\) −23.8434 −1.06738 −0.533688 0.845682i \(-0.679194\pi\)
−0.533688 + 0.845682i \(0.679194\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) −18.0000 −0.803379
\(503\) 29.7980 1.32863 0.664313 0.747455i \(-0.268724\pi\)
0.664313 + 0.747455i \(0.268724\pi\)
\(504\) 4.44949 0.198196
\(505\) 0 0
\(506\) −3.44949 −0.153349
\(507\) −7.00000 −0.310881
\(508\) 0.101021 0.00448206
\(509\) −30.1464 −1.33622 −0.668108 0.744064i \(-0.732896\pi\)
−0.668108 + 0.744064i \(0.732896\pi\)
\(510\) 0 0
\(511\) −4.44949 −0.196834
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −21.4949 −0.948099
\(515\) 0 0
\(516\) −2.44949 −0.107833
\(517\) −26.8990 −1.18302
\(518\) −34.6969 −1.52450
\(519\) −1.65153 −0.0724942
\(520\) 0 0
\(521\) 15.6969 0.687695 0.343848 0.939025i \(-0.388270\pi\)
0.343848 + 0.939025i \(0.388270\pi\)
\(522\) −4.34847 −0.190327
\(523\) −33.3939 −1.46021 −0.730106 0.683334i \(-0.760529\pi\)
−0.730106 + 0.683334i \(0.760529\pi\)
\(524\) 9.24745 0.403977
\(525\) 0 0
\(526\) −6.79796 −0.296405
\(527\) −13.3485 −0.581468
\(528\) 3.44949 0.150120
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 6.44949 0.279884
\(532\) 4.44949 0.192910
\(533\) 2.20204 0.0953810
\(534\) 2.10102 0.0909200
\(535\) 0 0
\(536\) 15.2474 0.658589
\(537\) 25.7980 1.11326
\(538\) −2.89898 −0.124984
\(539\) 44.1464 1.90152
\(540\) 0 0
\(541\) 22.1464 0.952149 0.476075 0.879405i \(-0.342059\pi\)
0.476075 + 0.879405i \(0.342059\pi\)
\(542\) 22.9444 0.985546
\(543\) −14.4495 −0.620087
\(544\) 4.44949 0.190770
\(545\) 0 0
\(546\) −10.8990 −0.466433
\(547\) −46.6413 −1.99424 −0.997120 0.0758461i \(-0.975834\pi\)
−0.997120 + 0.0758461i \(0.975834\pi\)
\(548\) −4.89898 −0.209274
\(549\) −9.44949 −0.403294
\(550\) 0 0
\(551\) −4.34847 −0.185251
\(552\) −1.00000 −0.0425628
\(553\) −22.2474 −0.946058
\(554\) 21.0454 0.894134
\(555\) 0 0
\(556\) −22.2474 −0.943502
\(557\) 27.3485 1.15879 0.579396 0.815046i \(-0.303289\pi\)
0.579396 + 0.815046i \(0.303289\pi\)
\(558\) −3.00000 −0.127000
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 15.3485 0.648013
\(562\) 7.00000 0.295277
\(563\) 6.24745 0.263299 0.131649 0.991296i \(-0.457973\pi\)
0.131649 + 0.991296i \(0.457973\pi\)
\(564\) −7.79796 −0.328353
\(565\) 0 0
\(566\) 29.7980 1.25250
\(567\) 4.44949 0.186861
\(568\) −1.55051 −0.0650580
\(569\) −33.1918 −1.39147 −0.695737 0.718297i \(-0.744922\pi\)
−0.695737 + 0.718297i \(0.744922\pi\)
\(570\) 0 0
\(571\) −10.2474 −0.428842 −0.214421 0.976741i \(-0.568787\pi\)
−0.214421 + 0.976741i \(0.568787\pi\)
\(572\) −8.44949 −0.353291
\(573\) −0.101021 −0.00422019
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 0.101021 0.00420554 0.00210277 0.999998i \(-0.499331\pi\)
0.00210277 + 0.999998i \(0.499331\pi\)
\(578\) 2.79796 0.116380
\(579\) −15.3485 −0.637861
\(580\) 0 0
\(581\) 37.1464 1.54109
\(582\) 1.55051 0.0642707
\(583\) −25.6969 −1.06426
\(584\) −1.00000 −0.0413803
\(585\) 0 0
\(586\) −23.2474 −0.960343
\(587\) −33.4495 −1.38061 −0.690304 0.723519i \(-0.742523\pi\)
−0.690304 + 0.723519i \(0.742523\pi\)
\(588\) 12.7980 0.527779
\(589\) −3.00000 −0.123613
\(590\) 0 0
\(591\) 27.3485 1.12497
\(592\) −7.79796 −0.320494
\(593\) 8.49490 0.348844 0.174422 0.984671i \(-0.444194\pi\)
0.174422 + 0.984671i \(0.444194\pi\)
\(594\) 3.44949 0.141534
\(595\) 0 0
\(596\) 5.79796 0.237494
\(597\) −10.0000 −0.409273
\(598\) 2.44949 0.100167
\(599\) 16.4495 0.672108 0.336054 0.941843i \(-0.390908\pi\)
0.336054 + 0.941843i \(0.390908\pi\)
\(600\) 0 0
\(601\) 17.1464 0.699417 0.349709 0.936858i \(-0.386281\pi\)
0.349709 + 0.936858i \(0.386281\pi\)
\(602\) −10.8990 −0.444209
\(603\) 15.2474 0.620924
\(604\) −23.7980 −0.968325
\(605\) 0 0
\(606\) −1.55051 −0.0629852
\(607\) 46.1918 1.87487 0.937434 0.348162i \(-0.113194\pi\)
0.937434 + 0.348162i \(0.113194\pi\)
\(608\) 1.00000 0.0405554
\(609\) −19.3485 −0.784040
\(610\) 0 0
\(611\) 19.1010 0.772745
\(612\) 4.44949 0.179860
\(613\) 37.1918 1.50216 0.751082 0.660209i \(-0.229532\pi\)
0.751082 + 0.660209i \(0.229532\pi\)
\(614\) −16.3485 −0.659771
\(615\) 0 0
\(616\) 15.3485 0.618407
\(617\) 28.2929 1.13903 0.569514 0.821982i \(-0.307132\pi\)
0.569514 + 0.821982i \(0.307132\pi\)
\(618\) 1.89898 0.0763882
\(619\) 45.1464 1.81459 0.907294 0.420497i \(-0.138144\pi\)
0.907294 + 0.420497i \(0.138144\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −23.7980 −0.954211
\(623\) 9.34847 0.374539
\(624\) −2.44949 −0.0980581
\(625\) 0 0
\(626\) 31.8990 1.27494
\(627\) 3.44949 0.137759
\(628\) −4.89898 −0.195491
\(629\) −34.6969 −1.38346
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) −5.00000 −0.198889
\(633\) −3.65153 −0.145135
\(634\) 15.2474 0.605554
\(635\) 0 0
\(636\) −7.44949 −0.295391
\(637\) −31.3485 −1.24207
\(638\) −15.0000 −0.593856
\(639\) −1.55051 −0.0613372
\(640\) 0 0
\(641\) 37.7980 1.49293 0.746465 0.665425i \(-0.231750\pi\)
0.746465 + 0.665425i \(0.231750\pi\)
\(642\) 17.3485 0.684689
\(643\) −0.202041 −0.00796772 −0.00398386 0.999992i \(-0.501268\pi\)
−0.00398386 + 0.999992i \(0.501268\pi\)
\(644\) −4.44949 −0.175334
\(645\) 0 0
\(646\) 4.44949 0.175063
\(647\) 25.8990 1.01819 0.509097 0.860709i \(-0.329979\pi\)
0.509097 + 0.860709i \(0.329979\pi\)
\(648\) 1.00000 0.0392837
\(649\) 22.2474 0.873289
\(650\) 0 0
\(651\) −13.3485 −0.523168
\(652\) 19.7980 0.775348
\(653\) −14.6969 −0.575136 −0.287568 0.957760i \(-0.592847\pi\)
−0.287568 + 0.957760i \(0.592847\pi\)
\(654\) −2.89898 −0.113359
\(655\) 0 0
\(656\) −0.898979 −0.0350993
\(657\) −1.00000 −0.0390137
\(658\) −34.6969 −1.35263
\(659\) −18.0454 −0.702949 −0.351475 0.936197i \(-0.614320\pi\)
−0.351475 + 0.936197i \(0.614320\pi\)
\(660\) 0 0
\(661\) 23.5959 0.917775 0.458887 0.888494i \(-0.348248\pi\)
0.458887 + 0.888494i \(0.348248\pi\)
\(662\) −22.3485 −0.868598
\(663\) −10.8990 −0.423281
\(664\) 8.34847 0.323983
\(665\) 0 0
\(666\) −7.79796 −0.302165
\(667\) 4.34847 0.168373
\(668\) 18.0000 0.696441
\(669\) 16.1010 0.622501
\(670\) 0 0
\(671\) −32.5959 −1.25835
\(672\) 4.44949 0.171643
\(673\) −45.3485 −1.74806 −0.874028 0.485876i \(-0.838501\pi\)
−0.874028 + 0.485876i \(0.838501\pi\)
\(674\) −24.8990 −0.959073
\(675\) 0 0
\(676\) −7.00000 −0.269231
\(677\) 22.3485 0.858921 0.429461 0.903086i \(-0.358704\pi\)
0.429461 + 0.903086i \(0.358704\pi\)
\(678\) −16.7980 −0.645122
\(679\) 6.89898 0.264759
\(680\) 0 0
\(681\) 29.5959 1.13412
\(682\) −10.3485 −0.396263
\(683\) −35.6413 −1.36378 −0.681889 0.731456i \(-0.738841\pi\)
−0.681889 + 0.731456i \(0.738841\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 25.7980 0.984971
\(687\) 7.24745 0.276507
\(688\) −2.44949 −0.0933859
\(689\) 18.2474 0.695172
\(690\) 0 0
\(691\) 9.75255 0.371005 0.185502 0.982644i \(-0.440609\pi\)
0.185502 + 0.982644i \(0.440609\pi\)
\(692\) −1.65153 −0.0627818
\(693\) 15.3485 0.583040
\(694\) −23.5959 −0.895689
\(695\) 0 0
\(696\) −4.34847 −0.164828
\(697\) −4.00000 −0.151511
\(698\) −25.9444 −0.982010
\(699\) 6.24745 0.236300
\(700\) 0 0
\(701\) −32.4949 −1.22732 −0.613658 0.789572i \(-0.710302\pi\)
−0.613658 + 0.789572i \(0.710302\pi\)
\(702\) −2.44949 −0.0924500
\(703\) −7.79796 −0.294106
\(704\) 3.44949 0.130008
\(705\) 0 0
\(706\) −18.2474 −0.686752
\(707\) −6.89898 −0.259463
\(708\) 6.44949 0.242387
\(709\) −12.7526 −0.478932 −0.239466 0.970905i \(-0.576972\pi\)
−0.239466 + 0.970905i \(0.576972\pi\)
\(710\) 0 0
\(711\) −5.00000 −0.187515
\(712\) 2.10102 0.0787391
\(713\) 3.00000 0.112351
\(714\) 19.7980 0.740920
\(715\) 0 0
\(716\) 25.7980 0.964115
\(717\) −8.69694 −0.324793
\(718\) −22.8990 −0.854582
\(719\) −19.2020 −0.716115 −0.358058 0.933699i \(-0.616561\pi\)
−0.358058 + 0.933699i \(0.616561\pi\)
\(720\) 0 0
\(721\) 8.44949 0.314675
\(722\) 1.00000 0.0372161
\(723\) −4.44949 −0.165478
\(724\) −14.4495 −0.537011
\(725\) 0 0
\(726\) 0.898979 0.0333643
\(727\) 22.4949 0.834290 0.417145 0.908840i \(-0.363031\pi\)
0.417145 + 0.908840i \(0.363031\pi\)
\(728\) −10.8990 −0.403943
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −10.8990 −0.403113
\(732\) −9.44949 −0.349263
\(733\) 4.14643 0.153152 0.0765759 0.997064i \(-0.475601\pi\)
0.0765759 + 0.997064i \(0.475601\pi\)
\(734\) −5.55051 −0.204873
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 52.5959 1.93740
\(738\) −0.898979 −0.0330919
\(739\) 32.8990 1.21021 0.605104 0.796146i \(-0.293131\pi\)
0.605104 + 0.796146i \(0.293131\pi\)
\(740\) 0 0
\(741\) −2.44949 −0.0899843
\(742\) −33.1464 −1.21684
\(743\) −53.3939 −1.95883 −0.979416 0.201854i \(-0.935303\pi\)
−0.979416 + 0.201854i \(0.935303\pi\)
\(744\) −3.00000 −0.109985
\(745\) 0 0
\(746\) −22.4495 −0.821934
\(747\) 8.34847 0.305455
\(748\) 15.3485 0.561196
\(749\) 77.1918 2.82053
\(750\) 0 0
\(751\) −23.7980 −0.868400 −0.434200 0.900817i \(-0.642969\pi\)
−0.434200 + 0.900817i \(0.642969\pi\)
\(752\) −7.79796 −0.284362
\(753\) −18.0000 −0.655956
\(754\) 10.6515 0.387906
\(755\) 0 0
\(756\) 4.44949 0.161826
\(757\) 48.1464 1.74991 0.874956 0.484203i \(-0.160890\pi\)
0.874956 + 0.484203i \(0.160890\pi\)
\(758\) −1.30306 −0.0473293
\(759\) −3.44949 −0.125209
\(760\) 0 0
\(761\) 51.3485 1.86138 0.930690 0.365808i \(-0.119207\pi\)
0.930690 + 0.365808i \(0.119207\pi\)
\(762\) 0.101021 0.00365959
\(763\) −12.8990 −0.466974
\(764\) −0.101021 −0.00365479
\(765\) 0 0
\(766\) 16.2474 0.587044
\(767\) −15.7980 −0.570431
\(768\) 1.00000 0.0360844
\(769\) −7.89898 −0.284844 −0.142422 0.989806i \(-0.545489\pi\)
−0.142422 + 0.989806i \(0.545489\pi\)
\(770\) 0 0
\(771\) −21.4949 −0.774120
\(772\) −15.3485 −0.552403
\(773\) 18.4949 0.665215 0.332608 0.943065i \(-0.392072\pi\)
0.332608 + 0.943065i \(0.392072\pi\)
\(774\) −2.44949 −0.0880451
\(775\) 0 0
\(776\) 1.55051 0.0556601
\(777\) −34.6969 −1.24475
\(778\) 21.5959 0.774251
\(779\) −0.898979 −0.0322093
\(780\) 0 0
\(781\) −5.34847 −0.191383
\(782\) −4.44949 −0.159113
\(783\) −4.34847 −0.155402
\(784\) 12.7980 0.457070
\(785\) 0 0
\(786\) 9.24745 0.329846
\(787\) −33.4495 −1.19235 −0.596173 0.802856i \(-0.703313\pi\)
−0.596173 + 0.802856i \(0.703313\pi\)
\(788\) 27.3485 0.974249
\(789\) −6.79796 −0.242014
\(790\) 0 0
\(791\) −74.7423 −2.65753
\(792\) 3.44949 0.122572
\(793\) 23.1464 0.821954
\(794\) 23.9444 0.849755
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 3.50510 0.124157 0.0620786 0.998071i \(-0.480227\pi\)
0.0620786 + 0.998071i \(0.480227\pi\)
\(798\) 4.44949 0.157510
\(799\) −34.6969 −1.22749
\(800\) 0 0
\(801\) 2.10102 0.0742359
\(802\) 11.2020 0.395558
\(803\) −3.44949 −0.121730
\(804\) 15.2474 0.537736
\(805\) 0 0
\(806\) 7.34847 0.258839
\(807\) −2.89898 −0.102049
\(808\) −1.55051 −0.0545468
\(809\) −3.55051 −0.124829 −0.0624146 0.998050i \(-0.519880\pi\)
−0.0624146 + 0.998050i \(0.519880\pi\)
\(810\) 0 0
\(811\) −19.4495 −0.682964 −0.341482 0.939888i \(-0.610929\pi\)
−0.341482 + 0.939888i \(0.610929\pi\)
\(812\) −19.3485 −0.678998
\(813\) 22.9444 0.804695
\(814\) −26.8990 −0.942809
\(815\) 0 0
\(816\) 4.44949 0.155763
\(817\) −2.44949 −0.0856968
\(818\) −22.8990 −0.800644
\(819\) −10.8990 −0.380841
\(820\) 0 0
\(821\) −23.1464 −0.807816 −0.403908 0.914800i \(-0.632348\pi\)
−0.403908 + 0.914800i \(0.632348\pi\)
\(822\) −4.89898 −0.170872
\(823\) −31.7980 −1.10841 −0.554204 0.832381i \(-0.686977\pi\)
−0.554204 + 0.832381i \(0.686977\pi\)
\(824\) 1.89898 0.0661541
\(825\) 0 0
\(826\) 28.6969 0.998494
\(827\) 0.247449 0.00860463 0.00430232 0.999991i \(-0.498631\pi\)
0.00430232 + 0.999991i \(0.498631\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −39.6413 −1.37680 −0.688400 0.725331i \(-0.741687\pi\)
−0.688400 + 0.725331i \(0.741687\pi\)
\(830\) 0 0
\(831\) 21.0454 0.730057
\(832\) −2.44949 −0.0849208
\(833\) 56.9444 1.97301
\(834\) −22.2474 −0.770366
\(835\) 0 0
\(836\) 3.44949 0.119303
\(837\) −3.00000 −0.103695
\(838\) 0 0
\(839\) 10.6515 0.367732 0.183866 0.982951i \(-0.441139\pi\)
0.183866 + 0.982951i \(0.441139\pi\)
\(840\) 0 0
\(841\) −10.0908 −0.347959
\(842\) 22.0000 0.758170
\(843\) 7.00000 0.241093
\(844\) −3.65153 −0.125691
\(845\) 0 0
\(846\) −7.79796 −0.268099
\(847\) 4.00000 0.137442
\(848\) −7.44949 −0.255817
\(849\) 29.7980 1.02266
\(850\) 0 0
\(851\) 7.79796 0.267311
\(852\) −1.55051 −0.0531196
\(853\) 1.10102 0.0376982 0.0188491 0.999822i \(-0.494000\pi\)
0.0188491 + 0.999822i \(0.494000\pi\)
\(854\) −42.0454 −1.43876
\(855\) 0 0
\(856\) 17.3485 0.592958
\(857\) −36.4949 −1.24664 −0.623321 0.781966i \(-0.714217\pi\)
−0.623321 + 0.781966i \(0.714217\pi\)
\(858\) −8.44949 −0.288461
\(859\) −4.20204 −0.143372 −0.0716859 0.997427i \(-0.522838\pi\)
−0.0716859 + 0.997427i \(0.522838\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) −5.10102 −0.173741
\(863\) −17.3031 −0.589003 −0.294502 0.955651i \(-0.595154\pi\)
−0.294502 + 0.955651i \(0.595154\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 4.65153 0.158065
\(867\) 2.79796 0.0950237
\(868\) −13.3485 −0.453077
\(869\) −17.2474 −0.585080
\(870\) 0 0
\(871\) −37.3485 −1.26550
\(872\) −2.89898 −0.0981718
\(873\) 1.55051 0.0524768
\(874\) −1.00000 −0.0338255
\(875\) 0 0
\(876\) −1.00000 −0.0337869
\(877\) −9.10102 −0.307320 −0.153660 0.988124i \(-0.549106\pi\)
−0.153660 + 0.988124i \(0.549106\pi\)
\(878\) −12.1010 −0.408390
\(879\) −23.2474 −0.784117
\(880\) 0 0
\(881\) 50.6969 1.70802 0.854012 0.520254i \(-0.174163\pi\)
0.854012 + 0.520254i \(0.174163\pi\)
\(882\) 12.7980 0.430930
\(883\) 34.6515 1.16612 0.583058 0.812430i \(-0.301856\pi\)
0.583058 + 0.812430i \(0.301856\pi\)
\(884\) −10.8990 −0.366572
\(885\) 0 0
\(886\) 21.2474 0.713822
\(887\) −4.89898 −0.164492 −0.0822458 0.996612i \(-0.526209\pi\)
−0.0822458 + 0.996612i \(0.526209\pi\)
\(888\) −7.79796 −0.261682
\(889\) 0.449490 0.0150754
\(890\) 0 0
\(891\) 3.44949 0.115562
\(892\) 16.1010 0.539102
\(893\) −7.79796 −0.260949
\(894\) 5.79796 0.193913
\(895\) 0 0
\(896\) 4.44949 0.148647
\(897\) 2.44949 0.0817861
\(898\) −15.0000 −0.500556
\(899\) 13.0454 0.435089
\(900\) 0 0
\(901\) −33.1464 −1.10427
\(902\) −3.10102 −0.103253
\(903\) −10.8990 −0.362695
\(904\) −16.7980 −0.558692
\(905\) 0 0
\(906\) −23.7980 −0.790634
\(907\) −22.0000 −0.730498 −0.365249 0.930910i \(-0.619016\pi\)
−0.365249 + 0.930910i \(0.619016\pi\)
\(908\) 29.5959 0.982175
\(909\) −1.55051 −0.0514272
\(910\) 0 0
\(911\) 6.49490 0.215186 0.107593 0.994195i \(-0.465686\pi\)
0.107593 + 0.994195i \(0.465686\pi\)
\(912\) 1.00000 0.0331133
\(913\) 28.7980 0.953073
\(914\) −24.8990 −0.823585
\(915\) 0 0
\(916\) 7.24745 0.239462
\(917\) 41.1464 1.35877
\(918\) 4.44949 0.146855
\(919\) 23.8434 0.786520 0.393260 0.919427i \(-0.371347\pi\)
0.393260 + 0.919427i \(0.371347\pi\)
\(920\) 0 0
\(921\) −16.3485 −0.538700
\(922\) 12.0000 0.395199
\(923\) 3.79796 0.125011
\(924\) 15.3485 0.504928
\(925\) 0 0
\(926\) −14.6969 −0.482971
\(927\) 1.89898 0.0623707
\(928\) −4.34847 −0.142745
\(929\) −31.5959 −1.03663 −0.518314 0.855190i \(-0.673440\pi\)
−0.518314 + 0.855190i \(0.673440\pi\)
\(930\) 0 0
\(931\) 12.7980 0.419436
\(932\) 6.24745 0.204642
\(933\) −23.7980 −0.779110
\(934\) −6.34847 −0.207728
\(935\) 0 0
\(936\) −2.44949 −0.0800641
\(937\) −19.3939 −0.633570 −0.316785 0.948497i \(-0.602603\pi\)
−0.316785 + 0.948497i \(0.602603\pi\)
\(938\) 67.8434 2.21516
\(939\) 31.8990 1.04098
\(940\) 0 0
\(941\) 36.6413 1.19447 0.597237 0.802065i \(-0.296265\pi\)
0.597237 + 0.802065i \(0.296265\pi\)
\(942\) −4.89898 −0.159617
\(943\) 0.898979 0.0292748
\(944\) 6.44949 0.209913
\(945\) 0 0
\(946\) −8.44949 −0.274717
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) −5.00000 −0.162392
\(949\) 2.44949 0.0795138
\(950\) 0 0
\(951\) 15.2474 0.494432
\(952\) 19.7980 0.641656
\(953\) −15.4949 −0.501929 −0.250964 0.967996i \(-0.580748\pi\)
−0.250964 + 0.967996i \(0.580748\pi\)
\(954\) −7.44949 −0.241186
\(955\) 0 0
\(956\) −8.69694 −0.281279
\(957\) −15.0000 −0.484881
\(958\) 16.5959 0.536190
\(959\) −21.7980 −0.703893
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 19.1010 0.615842
\(963\) 17.3485 0.559047
\(964\) −4.44949 −0.143308
\(965\) 0 0
\(966\) −4.44949 −0.143160
\(967\) −34.8990 −1.12228 −0.561138 0.827722i \(-0.689636\pi\)
−0.561138 + 0.827722i \(0.689636\pi\)
\(968\) 0.898979 0.0288943
\(969\) 4.44949 0.142938
\(970\) 0 0
\(971\) 41.6413 1.33633 0.668167 0.744011i \(-0.267079\pi\)
0.668167 + 0.744011i \(0.267079\pi\)
\(972\) 1.00000 0.0320750
\(973\) −98.9898 −3.17347
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) −9.44949 −0.302471
\(977\) 19.5959 0.626929 0.313464 0.949600i \(-0.398510\pi\)
0.313464 + 0.949600i \(0.398510\pi\)
\(978\) 19.7980 0.633069
\(979\) 7.24745 0.231629
\(980\) 0 0
\(981\) −2.89898 −0.0925573
\(982\) 43.5959 1.39120
\(983\) 50.4495 1.60909 0.804544 0.593892i \(-0.202410\pi\)
0.804544 + 0.593892i \(0.202410\pi\)
\(984\) −0.898979 −0.0286584
\(985\) 0 0
\(986\) −19.3485 −0.616181
\(987\) −34.6969 −1.10442
\(988\) −2.44949 −0.0779287
\(989\) 2.44949 0.0778892
\(990\) 0 0
\(991\) 44.1010 1.40092 0.700458 0.713694i \(-0.252979\pi\)
0.700458 + 0.713694i \(0.252979\pi\)
\(992\) −3.00000 −0.0952501
\(993\) −22.3485 −0.709207
\(994\) −6.89898 −0.218822
\(995\) 0 0
\(996\) 8.34847 0.264531
\(997\) 39.4495 1.24938 0.624689 0.780874i \(-0.285226\pi\)
0.624689 + 0.780874i \(0.285226\pi\)
\(998\) −23.8434 −0.754749
\(999\) −7.79796 −0.246717
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 2850.2.a.bj.1.2 yes 2
3.2 odd 2 8550.2.a.bu.1.2 2
5.2 odd 4 2850.2.d.w.799.4 4
5.3 odd 4 2850.2.d.w.799.1 4
5.4 even 2 2850.2.a.bc.1.1 2
15.14 odd 2 8550.2.a.bv.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bc.1.1 2 5.4 even 2
2850.2.a.bj.1.2 yes 2 1.1 even 1 trivial
2850.2.d.w.799.1 4 5.3 odd 4
2850.2.d.w.799.4 4 5.2 odd 4
8550.2.a.bu.1.2 2 3.2 odd 2
8550.2.a.bv.1.1 2 15.14 odd 2