Properties

Label 1875.2.a.b.1.1
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.9719503790\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} +2.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} +2.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{11} -0.618034 q^{12} +1.00000 q^{13} -3.23607 q^{14} -4.85410 q^{16} +4.23607 q^{17} -1.61803 q^{18} -6.70820 q^{19} -2.00000 q^{21} +4.85410 q^{22} -5.38197 q^{23} -2.23607 q^{24} -1.61803 q^{26} -1.00000 q^{27} +1.23607 q^{28} -3.61803 q^{29} +8.70820 q^{31} +3.38197 q^{32} +3.00000 q^{33} -6.85410 q^{34} +0.618034 q^{36} +2.00000 q^{37} +10.8541 q^{38} -1.00000 q^{39} -9.38197 q^{41} +3.23607 q^{42} +7.38197 q^{43} -1.85410 q^{44} +8.70820 q^{46} +4.76393 q^{47} +4.85410 q^{48} -3.00000 q^{49} -4.23607 q^{51} +0.618034 q^{52} -11.2361 q^{53} +1.61803 q^{54} +4.47214 q^{56} +6.70820 q^{57} +5.85410 q^{58} +3.94427 q^{59} +8.70820 q^{61} -14.0902 q^{62} +2.00000 q^{63} +4.23607 q^{64} -4.85410 q^{66} +13.1803 q^{67} +2.61803 q^{68} +5.38197 q^{69} +10.0902 q^{71} +2.23607 q^{72} -15.7082 q^{73} -3.23607 q^{74} -4.14590 q^{76} -6.00000 q^{77} +1.61803 q^{78} -9.14590 q^{79} +1.00000 q^{81} +15.1803 q^{82} -9.00000 q^{83} -1.23607 q^{84} -11.9443 q^{86} +3.61803 q^{87} -6.70820 q^{88} -11.1803 q^{89} +2.00000 q^{91} -3.32624 q^{92} -8.70820 q^{93} -7.70820 q^{94} -3.38197 q^{96} -3.85410 q^{97} +4.85410 q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} + 4 q^{7} + 2 q^{9} - 6 q^{11} + q^{12} + 2 q^{13} - 2 q^{14} - 3 q^{16} + 4 q^{17} - q^{18} - 4 q^{21} + 3 q^{22} - 13 q^{23} - q^{26} - 2 q^{27} - 2 q^{28} - 5 q^{29} + 4 q^{31} + 9 q^{32} + 6 q^{33} - 7 q^{34} - q^{36} + 4 q^{37} + 15 q^{38} - 2 q^{39} - 21 q^{41} + 2 q^{42} + 17 q^{43} + 3 q^{44} + 4 q^{46} + 14 q^{47} + 3 q^{48} - 6 q^{49} - 4 q^{51} - q^{52} - 18 q^{53} + q^{54} + 5 q^{58} - 10 q^{59} + 4 q^{61} - 17 q^{62} + 4 q^{63} + 4 q^{64} - 3 q^{66} + 4 q^{67} + 3 q^{68} + 13 q^{69} + 9 q^{71} - 18 q^{73} - 2 q^{74} - 15 q^{76} - 12 q^{77} + q^{78} - 25 q^{79} + 2 q^{81} + 8 q^{82} - 18 q^{83} + 2 q^{84} - 6 q^{86} + 5 q^{87} + 4 q^{91} + 9 q^{92} - 4 q^{93} - 2 q^{94} - 9 q^{96} - q^{97} + 3 q^{98} - 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.14412 −0.572061 0.820211i \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) 1.61803 0.660560
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −0.618034 −0.178411
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −3.23607 −0.864876
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 4.23607 1.02740 0.513699 0.857971i \(-0.328275\pi\)
0.513699 + 0.857971i \(0.328275\pi\)
\(18\) −1.61803 −0.381374
\(19\) −6.70820 −1.53897 −0.769484 0.638666i \(-0.779486\pi\)
−0.769484 + 0.638666i \(0.779486\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 4.85410 1.03490
\(23\) −5.38197 −1.12222 −0.561109 0.827742i \(-0.689625\pi\)
−0.561109 + 0.827742i \(0.689625\pi\)
\(24\) −2.23607 −0.456435
\(25\) 0 0
\(26\) −1.61803 −0.317323
\(27\) −1.00000 −0.192450
\(28\) 1.23607 0.233595
\(29\) −3.61803 −0.671852 −0.335926 0.941888i \(-0.609049\pi\)
−0.335926 + 0.941888i \(0.609049\pi\)
\(30\) 0 0
\(31\) 8.70820 1.56404 0.782020 0.623254i \(-0.214190\pi\)
0.782020 + 0.623254i \(0.214190\pi\)
\(32\) 3.38197 0.597853
\(33\) 3.00000 0.522233
\(34\) −6.85410 −1.17547
\(35\) 0 0
\(36\) 0.618034 0.103006
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 10.8541 1.76077
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −9.38197 −1.46522 −0.732608 0.680650i \(-0.761697\pi\)
−0.732608 + 0.680650i \(0.761697\pi\)
\(42\) 3.23607 0.499336
\(43\) 7.38197 1.12574 0.562870 0.826546i \(-0.309697\pi\)
0.562870 + 0.826546i \(0.309697\pi\)
\(44\) −1.85410 −0.279516
\(45\) 0 0
\(46\) 8.70820 1.28395
\(47\) 4.76393 0.694891 0.347445 0.937700i \(-0.387049\pi\)
0.347445 + 0.937700i \(0.387049\pi\)
\(48\) 4.85410 0.700629
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −4.23607 −0.593168
\(52\) 0.618034 0.0857059
\(53\) −11.2361 −1.54339 −0.771696 0.635991i \(-0.780591\pi\)
−0.771696 + 0.635991i \(0.780591\pi\)
\(54\) 1.61803 0.220187
\(55\) 0 0
\(56\) 4.47214 0.597614
\(57\) 6.70820 0.888523
\(58\) 5.85410 0.768681
\(59\) 3.94427 0.513500 0.256750 0.966478i \(-0.417348\pi\)
0.256750 + 0.966478i \(0.417348\pi\)
\(60\) 0 0
\(61\) 8.70820 1.11497 0.557486 0.830187i \(-0.311766\pi\)
0.557486 + 0.830187i \(0.311766\pi\)
\(62\) −14.0902 −1.78945
\(63\) 2.00000 0.251976
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) −4.85410 −0.597499
\(67\) 13.1803 1.61023 0.805117 0.593115i \(-0.202102\pi\)
0.805117 + 0.593115i \(0.202102\pi\)
\(68\) 2.61803 0.317483
\(69\) 5.38197 0.647913
\(70\) 0 0
\(71\) 10.0902 1.19748 0.598741 0.800942i \(-0.295668\pi\)
0.598741 + 0.800942i \(0.295668\pi\)
\(72\) 2.23607 0.263523
\(73\) −15.7082 −1.83851 −0.919253 0.393667i \(-0.871206\pi\)
−0.919253 + 0.393667i \(0.871206\pi\)
\(74\) −3.23607 −0.376185
\(75\) 0 0
\(76\) −4.14590 −0.475567
\(77\) −6.00000 −0.683763
\(78\) 1.61803 0.183206
\(79\) −9.14590 −1.02899 −0.514497 0.857492i \(-0.672021\pi\)
−0.514497 + 0.857492i \(0.672021\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 15.1803 1.67639
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) −1.23607 −0.134866
\(85\) 0 0
\(86\) −11.9443 −1.28798
\(87\) 3.61803 0.387894
\(88\) −6.70820 −0.715097
\(89\) −11.1803 −1.18511 −0.592557 0.805529i \(-0.701881\pi\)
−0.592557 + 0.805529i \(0.701881\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −3.32624 −0.346784
\(93\) −8.70820 −0.902999
\(94\) −7.70820 −0.795041
\(95\) 0 0
\(96\) −3.38197 −0.345170
\(97\) −3.85410 −0.391325 −0.195662 0.980671i \(-0.562686\pi\)
−0.195662 + 0.980671i \(0.562686\pi\)
\(98\) 4.85410 0.490338
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −9.38197 −0.933541 −0.466770 0.884379i \(-0.654583\pi\)
−0.466770 + 0.884379i \(0.654583\pi\)
\(102\) 6.85410 0.678657
\(103\) 14.4164 1.42049 0.710245 0.703954i \(-0.248584\pi\)
0.710245 + 0.703954i \(0.248584\pi\)
\(104\) 2.23607 0.219265
\(105\) 0 0
\(106\) 18.1803 1.76583
\(107\) 1.14590 0.110778 0.0553891 0.998465i \(-0.482360\pi\)
0.0553891 + 0.998465i \(0.482360\pi\)
\(108\) −0.618034 −0.0594703
\(109\) −4.14590 −0.397105 −0.198553 0.980090i \(-0.563624\pi\)
−0.198553 + 0.980090i \(0.563624\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −9.70820 −0.917339
\(113\) 3.76393 0.354081 0.177040 0.984204i \(-0.443348\pi\)
0.177040 + 0.984204i \(0.443348\pi\)
\(114\) −10.8541 −1.01658
\(115\) 0 0
\(116\) −2.23607 −0.207614
\(117\) 1.00000 0.0924500
\(118\) −6.38197 −0.587508
\(119\) 8.47214 0.776639
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −14.0902 −1.27566
\(123\) 9.38197 0.845943
\(124\) 5.38197 0.483315
\(125\) 0 0
\(126\) −3.23607 −0.288292
\(127\) −13.6525 −1.21146 −0.605731 0.795670i \(-0.707119\pi\)
−0.605731 + 0.795670i \(0.707119\pi\)
\(128\) −13.6180 −1.20368
\(129\) −7.38197 −0.649946
\(130\) 0 0
\(131\) −14.1803 −1.23894 −0.619471 0.785020i \(-0.712653\pi\)
−0.619471 + 0.785020i \(0.712653\pi\)
\(132\) 1.85410 0.161379
\(133\) −13.4164 −1.16335
\(134\) −21.3262 −1.84231
\(135\) 0 0
\(136\) 9.47214 0.812229
\(137\) −0.437694 −0.0373947 −0.0186974 0.999825i \(-0.505952\pi\)
−0.0186974 + 0.999825i \(0.505952\pi\)
\(138\) −8.70820 −0.741292
\(139\) 13.4164 1.13796 0.568982 0.822350i \(-0.307337\pi\)
0.568982 + 0.822350i \(0.307337\pi\)
\(140\) 0 0
\(141\) −4.76393 −0.401195
\(142\) −16.3262 −1.37007
\(143\) −3.00000 −0.250873
\(144\) −4.85410 −0.404508
\(145\) 0 0
\(146\) 25.4164 2.10348
\(147\) 3.00000 0.247436
\(148\) 1.23607 0.101604
\(149\) −13.0902 −1.07239 −0.536194 0.844095i \(-0.680139\pi\)
−0.536194 + 0.844095i \(0.680139\pi\)
\(150\) 0 0
\(151\) −6.61803 −0.538568 −0.269284 0.963061i \(-0.586787\pi\)
−0.269284 + 0.963061i \(0.586787\pi\)
\(152\) −15.0000 −1.21666
\(153\) 4.23607 0.342466
\(154\) 9.70820 0.782309
\(155\) 0 0
\(156\) −0.618034 −0.0494823
\(157\) 2.85410 0.227782 0.113891 0.993493i \(-0.463669\pi\)
0.113891 + 0.993493i \(0.463669\pi\)
\(158\) 14.7984 1.17730
\(159\) 11.2361 0.891078
\(160\) 0 0
\(161\) −10.7639 −0.848317
\(162\) −1.61803 −0.127125
\(163\) −18.2705 −1.43106 −0.715528 0.698584i \(-0.753814\pi\)
−0.715528 + 0.698584i \(0.753814\pi\)
\(164\) −5.79837 −0.452777
\(165\) 0 0
\(166\) 14.5623 1.13025
\(167\) −17.7984 −1.37728 −0.688640 0.725104i \(-0.741792\pi\)
−0.688640 + 0.725104i \(0.741792\pi\)
\(168\) −4.47214 −0.345033
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −6.70820 −0.512989
\(172\) 4.56231 0.347873
\(173\) −0.909830 −0.0691731 −0.0345865 0.999402i \(-0.511011\pi\)
−0.0345865 + 0.999402i \(0.511011\pi\)
\(174\) −5.85410 −0.443798
\(175\) 0 0
\(176\) 14.5623 1.09768
\(177\) −3.94427 −0.296470
\(178\) 18.0902 1.35592
\(179\) −15.6525 −1.16992 −0.584960 0.811062i \(-0.698890\pi\)
−0.584960 + 0.811062i \(0.698890\pi\)
\(180\) 0 0
\(181\) −12.4721 −0.927047 −0.463523 0.886085i \(-0.653415\pi\)
−0.463523 + 0.886085i \(0.653415\pi\)
\(182\) −3.23607 −0.239873
\(183\) −8.70820 −0.643729
\(184\) −12.0344 −0.887191
\(185\) 0 0
\(186\) 14.0902 1.03314
\(187\) −12.7082 −0.929316
\(188\) 2.94427 0.214733
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 17.3262 1.25368 0.626841 0.779147i \(-0.284347\pi\)
0.626841 + 0.779147i \(0.284347\pi\)
\(192\) −4.23607 −0.305712
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 6.23607 0.447724
\(195\) 0 0
\(196\) −1.85410 −0.132436
\(197\) 0.0901699 0.00642434 0.00321217 0.999995i \(-0.498978\pi\)
0.00321217 + 0.999995i \(0.498978\pi\)
\(198\) 4.85410 0.344966
\(199\) 11.7082 0.829973 0.414986 0.909828i \(-0.363786\pi\)
0.414986 + 0.909828i \(0.363786\pi\)
\(200\) 0 0
\(201\) −13.1803 −0.929669
\(202\) 15.1803 1.06808
\(203\) −7.23607 −0.507872
\(204\) −2.61803 −0.183299
\(205\) 0 0
\(206\) −23.3262 −1.62522
\(207\) −5.38197 −0.374072
\(208\) −4.85410 −0.336571
\(209\) 20.1246 1.39205
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) −6.94427 −0.476935
\(213\) −10.0902 −0.691367
\(214\) −1.85410 −0.126744
\(215\) 0 0
\(216\) −2.23607 −0.152145
\(217\) 17.4164 1.18230
\(218\) 6.70820 0.454337
\(219\) 15.7082 1.06146
\(220\) 0 0
\(221\) 4.23607 0.284949
\(222\) 3.23607 0.217191
\(223\) 10.1459 0.679420 0.339710 0.940530i \(-0.389671\pi\)
0.339710 + 0.940530i \(0.389671\pi\)
\(224\) 6.76393 0.451934
\(225\) 0 0
\(226\) −6.09017 −0.405112
\(227\) −5.76393 −0.382566 −0.191283 0.981535i \(-0.561265\pi\)
−0.191283 + 0.981535i \(0.561265\pi\)
\(228\) 4.14590 0.274569
\(229\) −16.1803 −1.06923 −0.534613 0.845097i \(-0.679543\pi\)
−0.534613 + 0.845097i \(0.679543\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) −8.09017 −0.531146
\(233\) −10.1803 −0.666936 −0.333468 0.942761i \(-0.608219\pi\)
−0.333468 + 0.942761i \(0.608219\pi\)
\(234\) −1.61803 −0.105774
\(235\) 0 0
\(236\) 2.43769 0.158680
\(237\) 9.14590 0.594090
\(238\) −13.7082 −0.888571
\(239\) 21.3820 1.38308 0.691542 0.722336i \(-0.256932\pi\)
0.691542 + 0.722336i \(0.256932\pi\)
\(240\) 0 0
\(241\) 7.32624 0.471924 0.235962 0.971762i \(-0.424176\pi\)
0.235962 + 0.971762i \(0.424176\pi\)
\(242\) 3.23607 0.208022
\(243\) −1.00000 −0.0641500
\(244\) 5.38197 0.344545
\(245\) 0 0
\(246\) −15.1803 −0.967863
\(247\) −6.70820 −0.426833
\(248\) 19.4721 1.23648
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) −18.9787 −1.19793 −0.598963 0.800777i \(-0.704420\pi\)
−0.598963 + 0.800777i \(0.704420\pi\)
\(252\) 1.23607 0.0778650
\(253\) 16.1459 1.01508
\(254\) 22.0902 1.38606
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) −31.2148 −1.94712 −0.973562 0.228422i \(-0.926644\pi\)
−0.973562 + 0.228422i \(0.926644\pi\)
\(258\) 11.9443 0.743618
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −3.61803 −0.223951
\(262\) 22.9443 1.41750
\(263\) 12.5066 0.771189 0.385594 0.922668i \(-0.373996\pi\)
0.385594 + 0.922668i \(0.373996\pi\)
\(264\) 6.70820 0.412861
\(265\) 0 0
\(266\) 21.7082 1.33102
\(267\) 11.1803 0.684226
\(268\) 8.14590 0.497590
\(269\) −20.5279 −1.25161 −0.625803 0.779981i \(-0.715229\pi\)
−0.625803 + 0.779981i \(0.715229\pi\)
\(270\) 0 0
\(271\) −11.4164 −0.693497 −0.346749 0.937958i \(-0.612714\pi\)
−0.346749 + 0.937958i \(0.612714\pi\)
\(272\) −20.5623 −1.24677
\(273\) −2.00000 −0.121046
\(274\) 0.708204 0.0427842
\(275\) 0 0
\(276\) 3.32624 0.200216
\(277\) 13.0557 0.784443 0.392221 0.919871i \(-0.371707\pi\)
0.392221 + 0.919871i \(0.371707\pi\)
\(278\) −21.7082 −1.30197
\(279\) 8.70820 0.521347
\(280\) 0 0
\(281\) −14.1803 −0.845928 −0.422964 0.906146i \(-0.639010\pi\)
−0.422964 + 0.906146i \(0.639010\pi\)
\(282\) 7.70820 0.459017
\(283\) −2.29180 −0.136233 −0.0681166 0.997677i \(-0.521699\pi\)
−0.0681166 + 0.997677i \(0.521699\pi\)
\(284\) 6.23607 0.370043
\(285\) 0 0
\(286\) 4.85410 0.287029
\(287\) −18.7639 −1.10760
\(288\) 3.38197 0.199284
\(289\) 0.944272 0.0555454
\(290\) 0 0
\(291\) 3.85410 0.225931
\(292\) −9.70820 −0.568130
\(293\) 6.32624 0.369583 0.184791 0.982778i \(-0.440839\pi\)
0.184791 + 0.982778i \(0.440839\pi\)
\(294\) −4.85410 −0.283097
\(295\) 0 0
\(296\) 4.47214 0.259938
\(297\) 3.00000 0.174078
\(298\) 21.1803 1.22694
\(299\) −5.38197 −0.311247
\(300\) 0 0
\(301\) 14.7639 0.850979
\(302\) 10.7082 0.616188
\(303\) 9.38197 0.538980
\(304\) 32.5623 1.86758
\(305\) 0 0
\(306\) −6.85410 −0.391823
\(307\) −8.85410 −0.505330 −0.252665 0.967554i \(-0.581307\pi\)
−0.252665 + 0.967554i \(0.581307\pi\)
\(308\) −3.70820 −0.211295
\(309\) −14.4164 −0.820121
\(310\) 0 0
\(311\) −13.5279 −0.767095 −0.383547 0.923521i \(-0.625298\pi\)
−0.383547 + 0.923521i \(0.625298\pi\)
\(312\) −2.23607 −0.126592
\(313\) −2.29180 −0.129540 −0.0647700 0.997900i \(-0.520631\pi\)
−0.0647700 + 0.997900i \(0.520631\pi\)
\(314\) −4.61803 −0.260611
\(315\) 0 0
\(316\) −5.65248 −0.317977
\(317\) −20.5623 −1.15489 −0.577447 0.816428i \(-0.695951\pi\)
−0.577447 + 0.816428i \(0.695951\pi\)
\(318\) −18.1803 −1.01950
\(319\) 10.8541 0.607713
\(320\) 0 0
\(321\) −1.14590 −0.0639578
\(322\) 17.4164 0.970578
\(323\) −28.4164 −1.58113
\(324\) 0.618034 0.0343352
\(325\) 0 0
\(326\) 29.5623 1.63730
\(327\) 4.14590 0.229269
\(328\) −20.9787 −1.15836
\(329\) 9.52786 0.525288
\(330\) 0 0
\(331\) −30.6869 −1.68671 −0.843353 0.537360i \(-0.819422\pi\)
−0.843353 + 0.537360i \(0.819422\pi\)
\(332\) −5.56231 −0.305271
\(333\) 2.00000 0.109599
\(334\) 28.7984 1.57578
\(335\) 0 0
\(336\) 9.70820 0.529626
\(337\) 33.1803 1.80745 0.903724 0.428115i \(-0.140822\pi\)
0.903724 + 0.428115i \(0.140822\pi\)
\(338\) 19.4164 1.05611
\(339\) −3.76393 −0.204429
\(340\) 0 0
\(341\) −26.1246 −1.41473
\(342\) 10.8541 0.586923
\(343\) −20.0000 −1.07990
\(344\) 16.5066 0.889975
\(345\) 0 0
\(346\) 1.47214 0.0791425
\(347\) −12.2705 −0.658715 −0.329358 0.944205i \(-0.606832\pi\)
−0.329358 + 0.944205i \(0.606832\pi\)
\(348\) 2.23607 0.119866
\(349\) 7.23607 0.387338 0.193669 0.981067i \(-0.437961\pi\)
0.193669 + 0.981067i \(0.437961\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −10.1459 −0.540778
\(353\) 12.3820 0.659026 0.329513 0.944151i \(-0.393116\pi\)
0.329513 + 0.944151i \(0.393116\pi\)
\(354\) 6.38197 0.339198
\(355\) 0 0
\(356\) −6.90983 −0.366220
\(357\) −8.47214 −0.448393
\(358\) 25.3262 1.33853
\(359\) 23.9443 1.26373 0.631865 0.775078i \(-0.282290\pi\)
0.631865 + 0.775078i \(0.282290\pi\)
\(360\) 0 0
\(361\) 26.0000 1.36842
\(362\) 20.1803 1.06066
\(363\) 2.00000 0.104973
\(364\) 1.23607 0.0647876
\(365\) 0 0
\(366\) 14.0902 0.736505
\(367\) 12.5279 0.653949 0.326975 0.945033i \(-0.393971\pi\)
0.326975 + 0.945033i \(0.393971\pi\)
\(368\) 26.1246 1.36184
\(369\) −9.38197 −0.488406
\(370\) 0 0
\(371\) −22.4721 −1.16670
\(372\) −5.38197 −0.279042
\(373\) −17.4164 −0.901787 −0.450894 0.892578i \(-0.648895\pi\)
−0.450894 + 0.892578i \(0.648895\pi\)
\(374\) 20.5623 1.06325
\(375\) 0 0
\(376\) 10.6525 0.549359
\(377\) −3.61803 −0.186338
\(378\) 3.23607 0.166445
\(379\) −13.6180 −0.699511 −0.349756 0.936841i \(-0.613735\pi\)
−0.349756 + 0.936841i \(0.613735\pi\)
\(380\) 0 0
\(381\) 13.6525 0.699438
\(382\) −28.0344 −1.43437
\(383\) −5.05573 −0.258336 −0.129168 0.991623i \(-0.541231\pi\)
−0.129168 + 0.991623i \(0.541231\pi\)
\(384\) 13.6180 0.694942
\(385\) 0 0
\(386\) −17.7984 −0.905913
\(387\) 7.38197 0.375246
\(388\) −2.38197 −0.120926
\(389\) 0.652476 0.0330818 0.0165409 0.999863i \(-0.494735\pi\)
0.0165409 + 0.999863i \(0.494735\pi\)
\(390\) 0 0
\(391\) −22.7984 −1.15296
\(392\) −6.70820 −0.338815
\(393\) 14.1803 0.715304
\(394\) −0.145898 −0.00735024
\(395\) 0 0
\(396\) −1.85410 −0.0931721
\(397\) 2.52786 0.126870 0.0634349 0.997986i \(-0.479794\pi\)
0.0634349 + 0.997986i \(0.479794\pi\)
\(398\) −18.9443 −0.949591
\(399\) 13.4164 0.671660
\(400\) 0 0
\(401\) 36.2705 1.81126 0.905631 0.424066i \(-0.139397\pi\)
0.905631 + 0.424066i \(0.139397\pi\)
\(402\) 21.3262 1.06366
\(403\) 8.70820 0.433787
\(404\) −5.79837 −0.288480
\(405\) 0 0
\(406\) 11.7082 0.581068
\(407\) −6.00000 −0.297409
\(408\) −9.47214 −0.468941
\(409\) −5.12461 −0.253396 −0.126698 0.991941i \(-0.540438\pi\)
−0.126698 + 0.991941i \(0.540438\pi\)
\(410\) 0 0
\(411\) 0.437694 0.0215899
\(412\) 8.90983 0.438956
\(413\) 7.88854 0.388170
\(414\) 8.70820 0.427985
\(415\) 0 0
\(416\) 3.38197 0.165815
\(417\) −13.4164 −0.657004
\(418\) −32.5623 −1.59267
\(419\) 0.326238 0.0159378 0.00796888 0.999968i \(-0.497463\pi\)
0.00796888 + 0.999968i \(0.497463\pi\)
\(420\) 0 0
\(421\) −30.3607 −1.47969 −0.739844 0.672778i \(-0.765101\pi\)
−0.739844 + 0.672778i \(0.765101\pi\)
\(422\) 4.85410 0.236294
\(423\) 4.76393 0.231630
\(424\) −25.1246 −1.22016
\(425\) 0 0
\(426\) 16.3262 0.791009
\(427\) 17.4164 0.842839
\(428\) 0.708204 0.0342323
\(429\) 3.00000 0.144841
\(430\) 0 0
\(431\) 29.7639 1.43368 0.716839 0.697239i \(-0.245588\pi\)
0.716839 + 0.697239i \(0.245588\pi\)
\(432\) 4.85410 0.233543
\(433\) −3.47214 −0.166860 −0.0834301 0.996514i \(-0.526588\pi\)
−0.0834301 + 0.996514i \(0.526588\pi\)
\(434\) −28.1803 −1.35270
\(435\) 0 0
\(436\) −2.56231 −0.122712
\(437\) 36.1033 1.72706
\(438\) −25.4164 −1.21444
\(439\) −32.0344 −1.52892 −0.764460 0.644671i \(-0.776994\pi\)
−0.764460 + 0.644671i \(0.776994\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −6.85410 −0.326016
\(443\) 19.4164 0.922501 0.461251 0.887270i \(-0.347401\pi\)
0.461251 + 0.887270i \(0.347401\pi\)
\(444\) −1.23607 −0.0586612
\(445\) 0 0
\(446\) −16.4164 −0.777339
\(447\) 13.0902 0.619144
\(448\) 8.47214 0.400271
\(449\) 16.5066 0.778994 0.389497 0.921028i \(-0.372649\pi\)
0.389497 + 0.921028i \(0.372649\pi\)
\(450\) 0 0
\(451\) 28.1459 1.32534
\(452\) 2.32624 0.109417
\(453\) 6.61803 0.310942
\(454\) 9.32624 0.437702
\(455\) 0 0
\(456\) 15.0000 0.702439
\(457\) 9.88854 0.462567 0.231283 0.972886i \(-0.425708\pi\)
0.231283 + 0.972886i \(0.425708\pi\)
\(458\) 26.1803 1.22333
\(459\) −4.23607 −0.197723
\(460\) 0 0
\(461\) −19.1803 −0.893317 −0.446659 0.894704i \(-0.647386\pi\)
−0.446659 + 0.894704i \(0.647386\pi\)
\(462\) −9.70820 −0.451667
\(463\) 33.6869 1.56556 0.782782 0.622296i \(-0.213800\pi\)
0.782782 + 0.622296i \(0.213800\pi\)
\(464\) 17.5623 0.815310
\(465\) 0 0
\(466\) 16.4721 0.763057
\(467\) 10.4164 0.482014 0.241007 0.970523i \(-0.422522\pi\)
0.241007 + 0.970523i \(0.422522\pi\)
\(468\) 0.618034 0.0285686
\(469\) 26.3607 1.21722
\(470\) 0 0
\(471\) −2.85410 −0.131510
\(472\) 8.81966 0.405958
\(473\) −22.1459 −1.01827
\(474\) −14.7984 −0.679712
\(475\) 0 0
\(476\) 5.23607 0.239995
\(477\) −11.2361 −0.514464
\(478\) −34.5967 −1.58242
\(479\) 4.79837 0.219243 0.109622 0.993973i \(-0.465036\pi\)
0.109622 + 0.993973i \(0.465036\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) −11.8541 −0.539940
\(483\) 10.7639 0.489776
\(484\) −1.23607 −0.0561849
\(485\) 0 0
\(486\) 1.61803 0.0733955
\(487\) −16.6180 −0.753035 −0.376518 0.926410i \(-0.622879\pi\)
−0.376518 + 0.926410i \(0.622879\pi\)
\(488\) 19.4721 0.881462
\(489\) 18.2705 0.826221
\(490\) 0 0
\(491\) 22.3262 1.00757 0.503785 0.863829i \(-0.331941\pi\)
0.503785 + 0.863829i \(0.331941\pi\)
\(492\) 5.79837 0.261411
\(493\) −15.3262 −0.690259
\(494\) 10.8541 0.488349
\(495\) 0 0
\(496\) −42.2705 −1.89800
\(497\) 20.1803 0.905212
\(498\) −14.5623 −0.652553
\(499\) −15.0000 −0.671492 −0.335746 0.941953i \(-0.608988\pi\)
−0.335746 + 0.941953i \(0.608988\pi\)
\(500\) 0 0
\(501\) 17.7984 0.795173
\(502\) 30.7082 1.37057
\(503\) 3.96556 0.176815 0.0884077 0.996084i \(-0.471822\pi\)
0.0884077 + 0.996084i \(0.471822\pi\)
\(504\) 4.47214 0.199205
\(505\) 0 0
\(506\) −26.1246 −1.16138
\(507\) 12.0000 0.532939
\(508\) −8.43769 −0.374362
\(509\) 32.8885 1.45776 0.728880 0.684642i \(-0.240041\pi\)
0.728880 + 0.684642i \(0.240041\pi\)
\(510\) 0 0
\(511\) −31.4164 −1.38978
\(512\) 5.29180 0.233867
\(513\) 6.70820 0.296174
\(514\) 50.5066 2.22775
\(515\) 0 0
\(516\) −4.56231 −0.200844
\(517\) −14.2918 −0.628552
\(518\) −6.47214 −0.284369
\(519\) 0.909830 0.0399371
\(520\) 0 0
\(521\) 40.0902 1.75638 0.878191 0.478310i \(-0.158750\pi\)
0.878191 + 0.478310i \(0.158750\pi\)
\(522\) 5.85410 0.256227
\(523\) −1.56231 −0.0683149 −0.0341574 0.999416i \(-0.510875\pi\)
−0.0341574 + 0.999416i \(0.510875\pi\)
\(524\) −8.76393 −0.382854
\(525\) 0 0
\(526\) −20.2361 −0.882334
\(527\) 36.8885 1.60689
\(528\) −14.5623 −0.633743
\(529\) 5.96556 0.259372
\(530\) 0 0
\(531\) 3.94427 0.171167
\(532\) −8.29180 −0.359495
\(533\) −9.38197 −0.406378
\(534\) −18.0902 −0.782838
\(535\) 0 0
\(536\) 29.4721 1.27300
\(537\) 15.6525 0.675454
\(538\) 33.2148 1.43199
\(539\) 9.00000 0.387657
\(540\) 0 0
\(541\) −26.2918 −1.13037 −0.565186 0.824963i \(-0.691196\pi\)
−0.565186 + 0.824963i \(0.691196\pi\)
\(542\) 18.4721 0.793446
\(543\) 12.4721 0.535231
\(544\) 14.3262 0.614232
\(545\) 0 0
\(546\) 3.23607 0.138491
\(547\) −24.7082 −1.05645 −0.528223 0.849106i \(-0.677142\pi\)
−0.528223 + 0.849106i \(0.677142\pi\)
\(548\) −0.270510 −0.0115556
\(549\) 8.70820 0.371657
\(550\) 0 0
\(551\) 24.2705 1.03396
\(552\) 12.0344 0.512220
\(553\) −18.2918 −0.777846
\(554\) −21.1246 −0.897499
\(555\) 0 0
\(556\) 8.29180 0.351650
\(557\) 37.6525 1.59539 0.797693 0.603063i \(-0.206053\pi\)
0.797693 + 0.603063i \(0.206053\pi\)
\(558\) −14.0902 −0.596484
\(559\) 7.38197 0.312224
\(560\) 0 0
\(561\) 12.7082 0.536541
\(562\) 22.9443 0.967846
\(563\) −9.00000 −0.379305 −0.189652 0.981851i \(-0.560736\pi\)
−0.189652 + 0.981851i \(0.560736\pi\)
\(564\) −2.94427 −0.123976
\(565\) 0 0
\(566\) 3.70820 0.155867
\(567\) 2.00000 0.0839921
\(568\) 22.5623 0.946693
\(569\) −10.8541 −0.455028 −0.227514 0.973775i \(-0.573060\pi\)
−0.227514 + 0.973775i \(0.573060\pi\)
\(570\) 0 0
\(571\) −38.1246 −1.59547 −0.797733 0.603011i \(-0.793967\pi\)
−0.797733 + 0.603011i \(0.793967\pi\)
\(572\) −1.85410 −0.0775239
\(573\) −17.3262 −0.723814
\(574\) 30.3607 1.26723
\(575\) 0 0
\(576\) 4.23607 0.176503
\(577\) −3.72949 −0.155261 −0.0776304 0.996982i \(-0.524735\pi\)
−0.0776304 + 0.996982i \(0.524735\pi\)
\(578\) −1.52786 −0.0635508
\(579\) −11.0000 −0.457144
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) −6.23607 −0.258493
\(583\) 33.7082 1.39605
\(584\) −35.1246 −1.45347
\(585\) 0 0
\(586\) −10.2361 −0.422848
\(587\) −39.3050 −1.62229 −0.811144 0.584846i \(-0.801155\pi\)
−0.811144 + 0.584846i \(0.801155\pi\)
\(588\) 1.85410 0.0764619
\(589\) −58.4164 −2.40701
\(590\) 0 0
\(591\) −0.0901699 −0.00370910
\(592\) −9.70820 −0.399005
\(593\) −17.6180 −0.723486 −0.361743 0.932278i \(-0.617818\pi\)
−0.361743 + 0.932278i \(0.617818\pi\)
\(594\) −4.85410 −0.199166
\(595\) 0 0
\(596\) −8.09017 −0.331386
\(597\) −11.7082 −0.479185
\(598\) 8.70820 0.356105
\(599\) 39.2705 1.60455 0.802275 0.596955i \(-0.203623\pi\)
0.802275 + 0.596955i \(0.203623\pi\)
\(600\) 0 0
\(601\) −24.7082 −1.00787 −0.503934 0.863742i \(-0.668115\pi\)
−0.503934 + 0.863742i \(0.668115\pi\)
\(602\) −23.8885 −0.973624
\(603\) 13.1803 0.536745
\(604\) −4.09017 −0.166427
\(605\) 0 0
\(606\) −15.1803 −0.616659
\(607\) 22.8541 0.927619 0.463810 0.885935i \(-0.346482\pi\)
0.463810 + 0.885935i \(0.346482\pi\)
\(608\) −22.6869 −0.920076
\(609\) 7.23607 0.293220
\(610\) 0 0
\(611\) 4.76393 0.192728
\(612\) 2.61803 0.105828
\(613\) 5.87539 0.237305 0.118652 0.992936i \(-0.462143\pi\)
0.118652 + 0.992936i \(0.462143\pi\)
\(614\) 14.3262 0.578160
\(615\) 0 0
\(616\) −13.4164 −0.540562
\(617\) −25.2361 −1.01597 −0.507983 0.861367i \(-0.669609\pi\)
−0.507983 + 0.861367i \(0.669609\pi\)
\(618\) 23.3262 0.938319
\(619\) 34.2705 1.37745 0.688724 0.725024i \(-0.258171\pi\)
0.688724 + 0.725024i \(0.258171\pi\)
\(620\) 0 0
\(621\) 5.38197 0.215971
\(622\) 21.8885 0.877651
\(623\) −22.3607 −0.895862
\(624\) 4.85410 0.194320
\(625\) 0 0
\(626\) 3.70820 0.148210
\(627\) −20.1246 −0.803700
\(628\) 1.76393 0.0703886
\(629\) 8.47214 0.337806
\(630\) 0 0
\(631\) −10.7639 −0.428505 −0.214253 0.976778i \(-0.568732\pi\)
−0.214253 + 0.976778i \(0.568732\pi\)
\(632\) −20.4508 −0.813491
\(633\) 3.00000 0.119239
\(634\) 33.2705 1.32134
\(635\) 0 0
\(636\) 6.94427 0.275358
\(637\) −3.00000 −0.118864
\(638\) −17.5623 −0.695298
\(639\) 10.0902 0.399161
\(640\) 0 0
\(641\) −23.3262 −0.921331 −0.460666 0.887574i \(-0.652389\pi\)
−0.460666 + 0.887574i \(0.652389\pi\)
\(642\) 1.85410 0.0731756
\(643\) −5.90983 −0.233061 −0.116530 0.993187i \(-0.537177\pi\)
−0.116530 + 0.993187i \(0.537177\pi\)
\(644\) −6.65248 −0.262144
\(645\) 0 0
\(646\) 45.9787 1.80901
\(647\) 19.0344 0.748321 0.374161 0.927364i \(-0.377931\pi\)
0.374161 + 0.927364i \(0.377931\pi\)
\(648\) 2.23607 0.0878410
\(649\) −11.8328 −0.464479
\(650\) 0 0
\(651\) −17.4164 −0.682603
\(652\) −11.2918 −0.442221
\(653\) −29.6525 −1.16039 −0.580196 0.814477i \(-0.697024\pi\)
−0.580196 + 0.814477i \(0.697024\pi\)
\(654\) −6.70820 −0.262312
\(655\) 0 0
\(656\) 45.5410 1.77808
\(657\) −15.7082 −0.612835
\(658\) −15.4164 −0.600994
\(659\) −2.23607 −0.0871048 −0.0435524 0.999051i \(-0.513868\pi\)
−0.0435524 + 0.999051i \(0.513868\pi\)
\(660\) 0 0
\(661\) 4.88854 0.190142 0.0950712 0.995470i \(-0.469692\pi\)
0.0950712 + 0.995470i \(0.469692\pi\)
\(662\) 49.6525 1.92980
\(663\) −4.23607 −0.164515
\(664\) −20.1246 −0.780986
\(665\) 0 0
\(666\) −3.23607 −0.125395
\(667\) 19.4721 0.753964
\(668\) −11.0000 −0.425603
\(669\) −10.1459 −0.392263
\(670\) 0 0
\(671\) −26.1246 −1.00853
\(672\) −6.76393 −0.260924
\(673\) 46.7771 1.80312 0.901562 0.432650i \(-0.142421\pi\)
0.901562 + 0.432650i \(0.142421\pi\)
\(674\) −53.6869 −2.06794
\(675\) 0 0
\(676\) −7.41641 −0.285246
\(677\) 44.8885 1.72521 0.862603 0.505881i \(-0.168832\pi\)
0.862603 + 0.505881i \(0.168832\pi\)
\(678\) 6.09017 0.233892
\(679\) −7.70820 −0.295814
\(680\) 0 0
\(681\) 5.76393 0.220874
\(682\) 42.2705 1.61862
\(683\) 0.596748 0.0228339 0.0114170 0.999935i \(-0.496366\pi\)
0.0114170 + 0.999935i \(0.496366\pi\)
\(684\) −4.14590 −0.158522
\(685\) 0 0
\(686\) 32.3607 1.23554
\(687\) 16.1803 0.617318
\(688\) −35.8328 −1.36611
\(689\) −11.2361 −0.428060
\(690\) 0 0
\(691\) 15.0902 0.574057 0.287029 0.957922i \(-0.407333\pi\)
0.287029 + 0.957922i \(0.407333\pi\)
\(692\) −0.562306 −0.0213757
\(693\) −6.00000 −0.227921
\(694\) 19.8541 0.753651
\(695\) 0 0
\(696\) 8.09017 0.306657
\(697\) −39.7426 −1.50536
\(698\) −11.7082 −0.443162
\(699\) 10.1803 0.385056
\(700\) 0 0
\(701\) −48.6525 −1.83758 −0.918789 0.394748i \(-0.870832\pi\)
−0.918789 + 0.394748i \(0.870832\pi\)
\(702\) 1.61803 0.0610688
\(703\) −13.4164 −0.506009
\(704\) −12.7082 −0.478958
\(705\) 0 0
\(706\) −20.0344 −0.754006
\(707\) −18.7639 −0.705690
\(708\) −2.43769 −0.0916142
\(709\) −5.20163 −0.195351 −0.0976756 0.995218i \(-0.531141\pi\)
−0.0976756 + 0.995218i \(0.531141\pi\)
\(710\) 0 0
\(711\) −9.14590 −0.342998
\(712\) −25.0000 −0.936915
\(713\) −46.8673 −1.75519
\(714\) 13.7082 0.513017
\(715\) 0 0
\(716\) −9.67376 −0.361525
\(717\) −21.3820 −0.798524
\(718\) −38.7426 −1.44586
\(719\) −35.1246 −1.30993 −0.654963 0.755661i \(-0.727316\pi\)
−0.654963 + 0.755661i \(0.727316\pi\)
\(720\) 0 0
\(721\) 28.8328 1.07379
\(722\) −42.0689 −1.56564
\(723\) −7.32624 −0.272466
\(724\) −7.70820 −0.286473
\(725\) 0 0
\(726\) −3.23607 −0.120102
\(727\) 14.4377 0.535464 0.267732 0.963493i \(-0.413726\pi\)
0.267732 + 0.963493i \(0.413726\pi\)
\(728\) 4.47214 0.165748
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 31.2705 1.15658
\(732\) −5.38197 −0.198923
\(733\) 10.1459 0.374747 0.187374 0.982289i \(-0.440002\pi\)
0.187374 + 0.982289i \(0.440002\pi\)
\(734\) −20.2705 −0.748198
\(735\) 0 0
\(736\) −18.2016 −0.670921
\(737\) −39.5410 −1.45651
\(738\) 15.1803 0.558796
\(739\) −1.70820 −0.0628373 −0.0314186 0.999506i \(-0.510003\pi\)
−0.0314186 + 0.999506i \(0.510003\pi\)
\(740\) 0 0
\(741\) 6.70820 0.246432
\(742\) 36.3607 1.33484
\(743\) 16.5279 0.606349 0.303174 0.952935i \(-0.401954\pi\)
0.303174 + 0.952935i \(0.401954\pi\)
\(744\) −19.4721 −0.713883
\(745\) 0 0
\(746\) 28.1803 1.03176
\(747\) −9.00000 −0.329293
\(748\) −7.85410 −0.287174
\(749\) 2.29180 0.0837404
\(750\) 0 0
\(751\) 15.2918 0.558006 0.279003 0.960290i \(-0.409996\pi\)
0.279003 + 0.960290i \(0.409996\pi\)
\(752\) −23.1246 −0.843268
\(753\) 18.9787 0.691623
\(754\) 5.85410 0.213194
\(755\) 0 0
\(756\) −1.23607 −0.0449554
\(757\) −32.2705 −1.17289 −0.586446 0.809988i \(-0.699473\pi\)
−0.586446 + 0.809988i \(0.699473\pi\)
\(758\) 22.0344 0.800327
\(759\) −16.1459 −0.586059
\(760\) 0 0
\(761\) 3.18034 0.115287 0.0576436 0.998337i \(-0.481641\pi\)
0.0576436 + 0.998337i \(0.481641\pi\)
\(762\) −22.0902 −0.800242
\(763\) −8.29180 −0.300183
\(764\) 10.7082 0.387409
\(765\) 0 0
\(766\) 8.18034 0.295568
\(767\) 3.94427 0.142419
\(768\) −13.5623 −0.489388
\(769\) −36.3050 −1.30919 −0.654595 0.755980i \(-0.727161\pi\)
−0.654595 + 0.755980i \(0.727161\pi\)
\(770\) 0 0
\(771\) 31.2148 1.12417
\(772\) 6.79837 0.244679
\(773\) 41.7771 1.50262 0.751309 0.659951i \(-0.229423\pi\)
0.751309 + 0.659951i \(0.229423\pi\)
\(774\) −11.9443 −0.429328
\(775\) 0 0
\(776\) −8.61803 −0.309369
\(777\) −4.00000 −0.143499
\(778\) −1.05573 −0.0378497
\(779\) 62.9361 2.25492
\(780\) 0 0
\(781\) −30.2705 −1.08316
\(782\) 36.8885 1.31913
\(783\) 3.61803 0.129298
\(784\) 14.5623 0.520082
\(785\) 0 0
\(786\) −22.9443 −0.818395
\(787\) −17.1459 −0.611185 −0.305593 0.952162i \(-0.598855\pi\)
−0.305593 + 0.952162i \(0.598855\pi\)
\(788\) 0.0557281 0.00198523
\(789\) −12.5066 −0.445246
\(790\) 0 0
\(791\) 7.52786 0.267660
\(792\) −6.70820 −0.238366
\(793\) 8.70820 0.309237
\(794\) −4.09017 −0.145155
\(795\) 0 0
\(796\) 7.23607 0.256476
\(797\) 30.0132 1.06312 0.531560 0.847020i \(-0.321606\pi\)
0.531560 + 0.847020i \(0.321606\pi\)
\(798\) −21.7082 −0.768462
\(799\) 20.1803 0.713929
\(800\) 0 0
\(801\) −11.1803 −0.395038
\(802\) −58.6869 −2.07231
\(803\) 47.1246 1.66299
\(804\) −8.14590 −0.287284
\(805\) 0 0
\(806\) −14.0902 −0.496305
\(807\) 20.5279 0.722615
\(808\) −20.9787 −0.738029
\(809\) −39.9230 −1.40362 −0.701809 0.712365i \(-0.747624\pi\)
−0.701809 + 0.712365i \(0.747624\pi\)
\(810\) 0 0
\(811\) −33.7771 −1.18607 −0.593037 0.805175i \(-0.702071\pi\)
−0.593037 + 0.805175i \(0.702071\pi\)
\(812\) −4.47214 −0.156941
\(813\) 11.4164 0.400391
\(814\) 9.70820 0.340272
\(815\) 0 0
\(816\) 20.5623 0.719825
\(817\) −49.5197 −1.73248
\(818\) 8.29180 0.289916
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 5.94427 0.207457 0.103728 0.994606i \(-0.466923\pi\)
0.103728 + 0.994606i \(0.466923\pi\)
\(822\) −0.708204 −0.0247014
\(823\) 28.5623 0.995619 0.497810 0.867286i \(-0.334138\pi\)
0.497810 + 0.867286i \(0.334138\pi\)
\(824\) 32.2361 1.12300
\(825\) 0 0
\(826\) −12.7639 −0.444114
\(827\) −48.9787 −1.70316 −0.851578 0.524227i \(-0.824354\pi\)
−0.851578 + 0.524227i \(0.824354\pi\)
\(828\) −3.32624 −0.115595
\(829\) −50.1246 −1.74090 −0.870450 0.492257i \(-0.836172\pi\)
−0.870450 + 0.492257i \(0.836172\pi\)
\(830\) 0 0
\(831\) −13.0557 −0.452898
\(832\) 4.23607 0.146859
\(833\) −12.7082 −0.440313
\(834\) 21.7082 0.751694
\(835\) 0 0
\(836\) 12.4377 0.430167
\(837\) −8.70820 −0.301000
\(838\) −0.527864 −0.0182348
\(839\) −3.21478 −0.110987 −0.0554933 0.998459i \(-0.517673\pi\)
−0.0554933 + 0.998459i \(0.517673\pi\)
\(840\) 0 0
\(841\) −15.9098 −0.548615
\(842\) 49.1246 1.69295
\(843\) 14.1803 0.488397
\(844\) −1.85410 −0.0638208
\(845\) 0 0
\(846\) −7.70820 −0.265014
\(847\) −4.00000 −0.137442
\(848\) 54.5410 1.87295
\(849\) 2.29180 0.0786542
\(850\) 0 0
\(851\) −10.7639 −0.368983
\(852\) −6.23607 −0.213644
\(853\) 20.3951 0.698316 0.349158 0.937064i \(-0.386468\pi\)
0.349158 + 0.937064i \(0.386468\pi\)
\(854\) −28.1803 −0.964311
\(855\) 0 0
\(856\) 2.56231 0.0875778
\(857\) −9.05573 −0.309338 −0.154669 0.987966i \(-0.549431\pi\)
−0.154669 + 0.987966i \(0.549431\pi\)
\(858\) −4.85410 −0.165716
\(859\) 15.1246 0.516045 0.258023 0.966139i \(-0.416929\pi\)
0.258023 + 0.966139i \(0.416929\pi\)
\(860\) 0 0
\(861\) 18.7639 0.639473
\(862\) −48.1591 −1.64030
\(863\) −13.0689 −0.444870 −0.222435 0.974948i \(-0.571401\pi\)
−0.222435 + 0.974948i \(0.571401\pi\)
\(864\) −3.38197 −0.115057
\(865\) 0 0
\(866\) 5.61803 0.190909
\(867\) −0.944272 −0.0320692
\(868\) 10.7639 0.365352
\(869\) 27.4377 0.930760
\(870\) 0 0
\(871\) 13.1803 0.446599
\(872\) −9.27051 −0.313939
\(873\) −3.85410 −0.130442
\(874\) −58.4164 −1.97596
\(875\) 0 0
\(876\) 9.70820 0.328010
\(877\) −43.1246 −1.45621 −0.728107 0.685463i \(-0.759600\pi\)
−0.728107 + 0.685463i \(0.759600\pi\)
\(878\) 51.8328 1.74927
\(879\) −6.32624 −0.213379
\(880\) 0 0
\(881\) 15.0902 0.508401 0.254200 0.967152i \(-0.418188\pi\)
0.254200 + 0.967152i \(0.418188\pi\)
\(882\) 4.85410 0.163446
\(883\) −29.2016 −0.982713 −0.491356 0.870959i \(-0.663499\pi\)
−0.491356 + 0.870959i \(0.663499\pi\)
\(884\) 2.61803 0.0880540
\(885\) 0 0
\(886\) −31.4164 −1.05545
\(887\) −42.9230 −1.44121 −0.720606 0.693344i \(-0.756136\pi\)
−0.720606 + 0.693344i \(0.756136\pi\)
\(888\) −4.47214 −0.150075
\(889\) −27.3050 −0.915779
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 6.27051 0.209952
\(893\) −31.9574 −1.06941
\(894\) −21.1803 −0.708377
\(895\) 0 0
\(896\) −27.2361 −0.909893
\(897\) 5.38197 0.179699
\(898\) −26.7082 −0.891264
\(899\) −31.5066 −1.05080
\(900\) 0 0
\(901\) −47.5967 −1.58568
\(902\) −45.5410 −1.51635
\(903\) −14.7639 −0.491313
\(904\) 8.41641 0.279926
\(905\) 0 0
\(906\) −10.7082 −0.355756
\(907\) 17.0000 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) −3.56231 −0.118219
\(909\) −9.38197 −0.311180
\(910\) 0 0
\(911\) 14.8885 0.493279 0.246640 0.969107i \(-0.420674\pi\)
0.246640 + 0.969107i \(0.420674\pi\)
\(912\) −32.5623 −1.07825
\(913\) 27.0000 0.893570
\(914\) −16.0000 −0.529233
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −28.3607 −0.936552
\(918\) 6.85410 0.226219
\(919\) −5.00000 −0.164935 −0.0824674 0.996594i \(-0.526280\pi\)
−0.0824674 + 0.996594i \(0.526280\pi\)
\(920\) 0 0
\(921\) 8.85410 0.291753
\(922\) 31.0344 1.02206
\(923\) 10.0902 0.332122
\(924\) 3.70820 0.121991
\(925\) 0 0
\(926\) −54.5066 −1.79120
\(927\) 14.4164 0.473497
\(928\) −12.2361 −0.401669
\(929\) −29.5967 −0.971038 −0.485519 0.874226i \(-0.661369\pi\)
−0.485519 + 0.874226i \(0.661369\pi\)
\(930\) 0 0
\(931\) 20.1246 0.659558
\(932\) −6.29180 −0.206095
\(933\) 13.5279 0.442882
\(934\) −16.8541 −0.551483
\(935\) 0 0
\(936\) 2.23607 0.0730882
\(937\) 10.4164 0.340289 0.170145 0.985419i \(-0.445577\pi\)
0.170145 + 0.985419i \(0.445577\pi\)
\(938\) −42.6525 −1.39265
\(939\) 2.29180 0.0747899
\(940\) 0 0
\(941\) 57.9787 1.89005 0.945026 0.326995i \(-0.106036\pi\)
0.945026 + 0.326995i \(0.106036\pi\)
\(942\) 4.61803 0.150464
\(943\) 50.4934 1.64429
\(944\) −19.1459 −0.623146
\(945\) 0 0
\(946\) 35.8328 1.16503
\(947\) 23.8328 0.774462 0.387231 0.921983i \(-0.373432\pi\)
0.387231 + 0.921983i \(0.373432\pi\)
\(948\) 5.65248 0.183584
\(949\) −15.7082 −0.509910
\(950\) 0 0
\(951\) 20.5623 0.666778
\(952\) 18.9443 0.613987
\(953\) 42.0557 1.36232 0.681159 0.732135i \(-0.261476\pi\)
0.681159 + 0.732135i \(0.261476\pi\)
\(954\) 18.1803 0.588610
\(955\) 0 0
\(956\) 13.2148 0.427397
\(957\) −10.8541 −0.350863
\(958\) −7.76393 −0.250841
\(959\) −0.875388 −0.0282678
\(960\) 0 0
\(961\) 44.8328 1.44622
\(962\) −3.23607 −0.104335
\(963\) 1.14590 0.0369260
\(964\) 4.52786 0.145833
\(965\) 0 0
\(966\) −17.4164 −0.560364
\(967\) 35.4164 1.13891 0.569457 0.822021i \(-0.307153\pi\)
0.569457 + 0.822021i \(0.307153\pi\)
\(968\) −4.47214 −0.143740
\(969\) 28.4164 0.912867
\(970\) 0 0
\(971\) 29.8885 0.959169 0.479585 0.877496i \(-0.340787\pi\)
0.479585 + 0.877496i \(0.340787\pi\)
\(972\) −0.618034 −0.0198234
\(973\) 26.8328 0.860221
\(974\) 26.8885 0.861565
\(975\) 0 0
\(976\) −42.2705 −1.35305
\(977\) 37.6525 1.20461 0.602305 0.798266i \(-0.294249\pi\)
0.602305 + 0.798266i \(0.294249\pi\)
\(978\) −29.5623 −0.945298
\(979\) 33.5410 1.07198
\(980\) 0 0
\(981\) −4.14590 −0.132368
\(982\) −36.1246 −1.15278
\(983\) 24.6180 0.785193 0.392597 0.919711i \(-0.371577\pi\)
0.392597 + 0.919711i \(0.371577\pi\)
\(984\) 20.9787 0.668777
\(985\) 0 0
\(986\) 24.7984 0.789741
\(987\) −9.52786 −0.303275
\(988\) −4.14590 −0.131899
\(989\) −39.7295 −1.26332
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 29.4508 0.935065
\(993\) 30.6869 0.973820
\(994\) −32.6525 −1.03567
\(995\) 0 0
\(996\) 5.56231 0.176248
\(997\) 2.93112 0.0928294 0.0464147 0.998922i \(-0.485220\pi\)
0.0464147 + 0.998922i \(0.485220\pi\)
\(998\) 24.2705 0.768270
\(999\) −2.00000 −0.0632772
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 1875.2.a.b.1.1 2
3.2 odd 2 5625.2.a.g.1.2 2
5.2 odd 4 1875.2.b.a.1249.1 4
5.3 odd 4 1875.2.b.a.1249.4 4
5.4 even 2 1875.2.a.c.1.2 yes 2
15.14 odd 2 5625.2.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.b.1.1 2 1.1 even 1 trivial
1875.2.a.c.1.2 yes 2 5.4 even 2
1875.2.b.a.1249.1 4 5.2 odd 4
1875.2.b.a.1249.4 4 5.3 odd 4
5625.2.a.b.1.1 2 15.14 odd 2
5625.2.a.g.1.2 2 3.2 odd 2