Properties

Label 1875.2.b.a.1249.1
Level $1875$
Weight $2$
Character 1875.1249
Analytic conductor $14.972$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1875.1249
Dual form 1875.2.b.a.1249.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.61803i q^{2} +1.00000i q^{3} -0.618034 q^{4} +1.61803 q^{6} +2.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.61803i q^{2} +1.00000i q^{3} -0.618034 q^{4} +1.61803 q^{6} +2.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} -3.00000 q^{11} -0.618034i q^{12} -1.00000i q^{13} +3.23607 q^{14} -4.85410 q^{16} +4.23607i q^{17} +1.61803i q^{18} +6.70820 q^{19} -2.00000 q^{21} +4.85410i q^{22} +5.38197i q^{23} +2.23607 q^{24} -1.61803 q^{26} -1.00000i q^{27} -1.23607i q^{28} +3.61803 q^{29} +8.70820 q^{31} +3.38197i q^{32} -3.00000i q^{33} +6.85410 q^{34} +0.618034 q^{36} +2.00000i q^{37} -10.8541i q^{38} +1.00000 q^{39} -9.38197 q^{41} +3.23607i q^{42} -7.38197i q^{43} +1.85410 q^{44} +8.70820 q^{46} +4.76393i q^{47} -4.85410i q^{48} +3.00000 q^{49} -4.23607 q^{51} +0.618034i q^{52} +11.2361i q^{53} -1.61803 q^{54} +4.47214 q^{56} +6.70820i q^{57} -5.85410i q^{58} -3.94427 q^{59} +8.70820 q^{61} -14.0902i q^{62} -2.00000i q^{63} -4.23607 q^{64} -4.85410 q^{66} +13.1803i q^{67} -2.61803i q^{68} -5.38197 q^{69} +10.0902 q^{71} +2.23607i q^{72} +15.7082i q^{73} +3.23607 q^{74} -4.14590 q^{76} -6.00000i q^{77} -1.61803i q^{78} +9.14590 q^{79} +1.00000 q^{81} +15.1803i q^{82} +9.00000i q^{83} +1.23607 q^{84} -11.9443 q^{86} +3.61803i q^{87} +6.70820i q^{88} +11.1803 q^{89} +2.00000 q^{91} -3.32624i q^{92} +8.70820i q^{93} +7.70820 q^{94} -3.38197 q^{96} -3.85410i q^{97} -4.85410i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{6} - 4 q^{9} - 12 q^{11} + 4 q^{14} - 6 q^{16} - 8 q^{21} - 2 q^{26} + 10 q^{29} + 8 q^{31} + 14 q^{34} - 2 q^{36} + 4 q^{39} - 42 q^{41} - 6 q^{44} + 8 q^{46} + 12 q^{49} - 8 q^{51} - 2 q^{54} + 20 q^{59} + 8 q^{61} - 8 q^{64} - 6 q^{66} - 26 q^{69} + 18 q^{71} + 4 q^{74} - 30 q^{76} + 50 q^{79} + 4 q^{81} - 4 q^{84} - 12 q^{86} + 8 q^{91} + 4 q^{94} - 18 q^{96} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.61803i − 1.14412i −0.820211 0.572061i \(-0.806144\pi\)
0.820211 0.572061i \(-0.193856\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −0.618034 −0.309017
\(5\) 0 0
\(6\) 1.61803 0.660560
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 2.23607i − 0.790569i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) − 0.618034i − 0.178411i
\(13\) − 1.00000i − 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) 3.23607 0.864876
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 4.23607i 1.02740i 0.857971 + 0.513699i \(0.171725\pi\)
−0.857971 + 0.513699i \(0.828275\pi\)
\(18\) 1.61803i 0.381374i
\(19\) 6.70820 1.53897 0.769484 0.638666i \(-0.220514\pi\)
0.769484 + 0.638666i \(0.220514\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 4.85410i 1.03490i
\(23\) 5.38197i 1.12222i 0.827742 + 0.561109i \(0.189625\pi\)
−0.827742 + 0.561109i \(0.810375\pi\)
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) −1.61803 −0.317323
\(27\) − 1.00000i − 0.192450i
\(28\) − 1.23607i − 0.233595i
\(29\) 3.61803 0.671852 0.335926 0.941888i \(-0.390951\pi\)
0.335926 + 0.941888i \(0.390951\pi\)
\(30\) 0 0
\(31\) 8.70820 1.56404 0.782020 0.623254i \(-0.214190\pi\)
0.782020 + 0.623254i \(0.214190\pi\)
\(32\) 3.38197i 0.597853i
\(33\) − 3.00000i − 0.522233i
\(34\) 6.85410 1.17547
\(35\) 0 0
\(36\) 0.618034 0.103006
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) − 10.8541i − 1.76077i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −9.38197 −1.46522 −0.732608 0.680650i \(-0.761697\pi\)
−0.732608 + 0.680650i \(0.761697\pi\)
\(42\) 3.23607i 0.499336i
\(43\) − 7.38197i − 1.12574i −0.826546 0.562870i \(-0.809697\pi\)
0.826546 0.562870i \(-0.190303\pi\)
\(44\) 1.85410 0.279516
\(45\) 0 0
\(46\) 8.70820 1.28395
\(47\) 4.76393i 0.694891i 0.937700 + 0.347445i \(0.112951\pi\)
−0.937700 + 0.347445i \(0.887049\pi\)
\(48\) − 4.85410i − 0.700629i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −4.23607 −0.593168
\(52\) 0.618034i 0.0857059i
\(53\) 11.2361i 1.54339i 0.635991 + 0.771696i \(0.280591\pi\)
−0.635991 + 0.771696i \(0.719409\pi\)
\(54\) −1.61803 −0.220187
\(55\) 0 0
\(56\) 4.47214 0.597614
\(57\) 6.70820i 0.888523i
\(58\) − 5.85410i − 0.768681i
\(59\) −3.94427 −0.513500 −0.256750 0.966478i \(-0.582652\pi\)
−0.256750 + 0.966478i \(0.582652\pi\)
\(60\) 0 0
\(61\) 8.70820 1.11497 0.557486 0.830187i \(-0.311766\pi\)
0.557486 + 0.830187i \(0.311766\pi\)
\(62\) − 14.0902i − 1.78945i
\(63\) − 2.00000i − 0.251976i
\(64\) −4.23607 −0.529508
\(65\) 0 0
\(66\) −4.85410 −0.597499
\(67\) 13.1803i 1.61023i 0.593115 + 0.805117i \(0.297898\pi\)
−0.593115 + 0.805117i \(0.702102\pi\)
\(68\) − 2.61803i − 0.317483i
\(69\) −5.38197 −0.647913
\(70\) 0 0
\(71\) 10.0902 1.19748 0.598741 0.800942i \(-0.295668\pi\)
0.598741 + 0.800942i \(0.295668\pi\)
\(72\) 2.23607i 0.263523i
\(73\) 15.7082i 1.83851i 0.393667 + 0.919253i \(0.371206\pi\)
−0.393667 + 0.919253i \(0.628794\pi\)
\(74\) 3.23607 0.376185
\(75\) 0 0
\(76\) −4.14590 −0.475567
\(77\) − 6.00000i − 0.683763i
\(78\) − 1.61803i − 0.183206i
\(79\) 9.14590 1.02899 0.514497 0.857492i \(-0.327979\pi\)
0.514497 + 0.857492i \(0.327979\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 15.1803i 1.67639i
\(83\) 9.00000i 0.987878i 0.869496 + 0.493939i \(0.164443\pi\)
−0.869496 + 0.493939i \(0.835557\pi\)
\(84\) 1.23607 0.134866
\(85\) 0 0
\(86\) −11.9443 −1.28798
\(87\) 3.61803i 0.387894i
\(88\) 6.70820i 0.715097i
\(89\) 11.1803 1.18511 0.592557 0.805529i \(-0.298119\pi\)
0.592557 + 0.805529i \(0.298119\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) − 3.32624i − 0.346784i
\(93\) 8.70820i 0.902999i
\(94\) 7.70820 0.795041
\(95\) 0 0
\(96\) −3.38197 −0.345170
\(97\) − 3.85410i − 0.391325i −0.980671 0.195662i \(-0.937314\pi\)
0.980671 0.195662i \(-0.0626857\pi\)
\(98\) − 4.85410i − 0.490338i
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) −9.38197 −0.933541 −0.466770 0.884379i \(-0.654583\pi\)
−0.466770 + 0.884379i \(0.654583\pi\)
\(102\) 6.85410i 0.678657i
\(103\) − 14.4164i − 1.42049i −0.703954 0.710245i \(-0.748584\pi\)
0.703954 0.710245i \(-0.251416\pi\)
\(104\) −2.23607 −0.219265
\(105\) 0 0
\(106\) 18.1803 1.76583
\(107\) 1.14590i 0.110778i 0.998465 + 0.0553891i \(0.0176399\pi\)
−0.998465 + 0.0553891i \(0.982360\pi\)
\(108\) 0.618034i 0.0594703i
\(109\) 4.14590 0.397105 0.198553 0.980090i \(-0.436376\pi\)
0.198553 + 0.980090i \(0.436376\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) − 9.70820i − 0.917339i
\(113\) − 3.76393i − 0.354081i −0.984204 0.177040i \(-0.943348\pi\)
0.984204 0.177040i \(-0.0566523\pi\)
\(114\) 10.8541 1.01658
\(115\) 0 0
\(116\) −2.23607 −0.207614
\(117\) 1.00000i 0.0924500i
\(118\) 6.38197i 0.587508i
\(119\) −8.47214 −0.776639
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) − 14.0902i − 1.27566i
\(123\) − 9.38197i − 0.845943i
\(124\) −5.38197 −0.483315
\(125\) 0 0
\(126\) −3.23607 −0.288292
\(127\) − 13.6525i − 1.21146i −0.795670 0.605731i \(-0.792881\pi\)
0.795670 0.605731i \(-0.207119\pi\)
\(128\) 13.6180i 1.20368i
\(129\) 7.38197 0.649946
\(130\) 0 0
\(131\) −14.1803 −1.23894 −0.619471 0.785020i \(-0.712653\pi\)
−0.619471 + 0.785020i \(0.712653\pi\)
\(132\) 1.85410i 0.161379i
\(133\) 13.4164i 1.16335i
\(134\) 21.3262 1.84231
\(135\) 0 0
\(136\) 9.47214 0.812229
\(137\) − 0.437694i − 0.0373947i −0.999825 0.0186974i \(-0.994048\pi\)
0.999825 0.0186974i \(-0.00595190\pi\)
\(138\) 8.70820i 0.741292i
\(139\) −13.4164 −1.13796 −0.568982 0.822350i \(-0.692663\pi\)
−0.568982 + 0.822350i \(0.692663\pi\)
\(140\) 0 0
\(141\) −4.76393 −0.401195
\(142\) − 16.3262i − 1.37007i
\(143\) 3.00000i 0.250873i
\(144\) 4.85410 0.404508
\(145\) 0 0
\(146\) 25.4164 2.10348
\(147\) 3.00000i 0.247436i
\(148\) − 1.23607i − 0.101604i
\(149\) 13.0902 1.07239 0.536194 0.844095i \(-0.319861\pi\)
0.536194 + 0.844095i \(0.319861\pi\)
\(150\) 0 0
\(151\) −6.61803 −0.538568 −0.269284 0.963061i \(-0.586787\pi\)
−0.269284 + 0.963061i \(0.586787\pi\)
\(152\) − 15.0000i − 1.21666i
\(153\) − 4.23607i − 0.342466i
\(154\) −9.70820 −0.782309
\(155\) 0 0
\(156\) −0.618034 −0.0494823
\(157\) 2.85410i 0.227782i 0.993493 + 0.113891i \(0.0363315\pi\)
−0.993493 + 0.113891i \(0.963669\pi\)
\(158\) − 14.7984i − 1.17730i
\(159\) −11.2361 −0.891078
\(160\) 0 0
\(161\) −10.7639 −0.848317
\(162\) − 1.61803i − 0.127125i
\(163\) 18.2705i 1.43106i 0.698584 + 0.715528i \(0.253814\pi\)
−0.698584 + 0.715528i \(0.746186\pi\)
\(164\) 5.79837 0.452777
\(165\) 0 0
\(166\) 14.5623 1.13025
\(167\) − 17.7984i − 1.37728i −0.725104 0.688640i \(-0.758208\pi\)
0.725104 0.688640i \(-0.241792\pi\)
\(168\) 4.47214i 0.345033i
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −6.70820 −0.512989
\(172\) 4.56231i 0.347873i
\(173\) 0.909830i 0.0691731i 0.999402 + 0.0345865i \(0.0110114\pi\)
−0.999402 + 0.0345865i \(0.988989\pi\)
\(174\) 5.85410 0.443798
\(175\) 0 0
\(176\) 14.5623 1.09768
\(177\) − 3.94427i − 0.296470i
\(178\) − 18.0902i − 1.35592i
\(179\) 15.6525 1.16992 0.584960 0.811062i \(-0.301110\pi\)
0.584960 + 0.811062i \(0.301110\pi\)
\(180\) 0 0
\(181\) −12.4721 −0.927047 −0.463523 0.886085i \(-0.653415\pi\)
−0.463523 + 0.886085i \(0.653415\pi\)
\(182\) − 3.23607i − 0.239873i
\(183\) 8.70820i 0.643729i
\(184\) 12.0344 0.887191
\(185\) 0 0
\(186\) 14.0902 1.03314
\(187\) − 12.7082i − 0.929316i
\(188\) − 2.94427i − 0.214733i
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 17.3262 1.25368 0.626841 0.779147i \(-0.284347\pi\)
0.626841 + 0.779147i \(0.284347\pi\)
\(192\) − 4.23607i − 0.305712i
\(193\) − 11.0000i − 0.791797i −0.918294 0.395899i \(-0.870433\pi\)
0.918294 0.395899i \(-0.129567\pi\)
\(194\) −6.23607 −0.447724
\(195\) 0 0
\(196\) −1.85410 −0.132436
\(197\) 0.0901699i 0.00642434i 0.999995 + 0.00321217i \(0.00102247\pi\)
−0.999995 + 0.00321217i \(0.998978\pi\)
\(198\) − 4.85410i − 0.344966i
\(199\) −11.7082 −0.829973 −0.414986 0.909828i \(-0.636214\pi\)
−0.414986 + 0.909828i \(0.636214\pi\)
\(200\) 0 0
\(201\) −13.1803 −0.929669
\(202\) 15.1803i 1.06808i
\(203\) 7.23607i 0.507872i
\(204\) 2.61803 0.183299
\(205\) 0 0
\(206\) −23.3262 −1.62522
\(207\) − 5.38197i − 0.374072i
\(208\) 4.85410i 0.336571i
\(209\) −20.1246 −1.39205
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) − 6.94427i − 0.476935i
\(213\) 10.0902i 0.691367i
\(214\) 1.85410 0.126744
\(215\) 0 0
\(216\) −2.23607 −0.152145
\(217\) 17.4164i 1.18230i
\(218\) − 6.70820i − 0.454337i
\(219\) −15.7082 −1.06146
\(220\) 0 0
\(221\) 4.23607 0.284949
\(222\) 3.23607i 0.217191i
\(223\) − 10.1459i − 0.679420i −0.940530 0.339710i \(-0.889671\pi\)
0.940530 0.339710i \(-0.110329\pi\)
\(224\) −6.76393 −0.451934
\(225\) 0 0
\(226\) −6.09017 −0.405112
\(227\) − 5.76393i − 0.382566i −0.981535 0.191283i \(-0.938735\pi\)
0.981535 0.191283i \(-0.0612648\pi\)
\(228\) − 4.14590i − 0.274569i
\(229\) 16.1803 1.06923 0.534613 0.845097i \(-0.320457\pi\)
0.534613 + 0.845097i \(0.320457\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) − 8.09017i − 0.531146i
\(233\) 10.1803i 0.666936i 0.942761 + 0.333468i \(0.108219\pi\)
−0.942761 + 0.333468i \(0.891781\pi\)
\(234\) 1.61803 0.105774
\(235\) 0 0
\(236\) 2.43769 0.158680
\(237\) 9.14590i 0.594090i
\(238\) 13.7082i 0.888571i
\(239\) −21.3820 −1.38308 −0.691542 0.722336i \(-0.743068\pi\)
−0.691542 + 0.722336i \(0.743068\pi\)
\(240\) 0 0
\(241\) 7.32624 0.471924 0.235962 0.971762i \(-0.424176\pi\)
0.235962 + 0.971762i \(0.424176\pi\)
\(242\) 3.23607i 0.208022i
\(243\) 1.00000i 0.0641500i
\(244\) −5.38197 −0.344545
\(245\) 0 0
\(246\) −15.1803 −0.967863
\(247\) − 6.70820i − 0.426833i
\(248\) − 19.4721i − 1.23648i
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) −18.9787 −1.19793 −0.598963 0.800777i \(-0.704420\pi\)
−0.598963 + 0.800777i \(0.704420\pi\)
\(252\) 1.23607i 0.0778650i
\(253\) − 16.1459i − 1.01508i
\(254\) −22.0902 −1.38606
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) − 31.2148i − 1.94712i −0.228422 0.973562i \(-0.573356\pi\)
0.228422 0.973562i \(-0.426644\pi\)
\(258\) − 11.9443i − 0.743618i
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −3.61803 −0.223951
\(262\) 22.9443i 1.41750i
\(263\) − 12.5066i − 0.771189i −0.922668 0.385594i \(-0.873996\pi\)
0.922668 0.385594i \(-0.126004\pi\)
\(264\) −6.70820 −0.412861
\(265\) 0 0
\(266\) 21.7082 1.33102
\(267\) 11.1803i 0.684226i
\(268\) − 8.14590i − 0.497590i
\(269\) 20.5279 1.25161 0.625803 0.779981i \(-0.284771\pi\)
0.625803 + 0.779981i \(0.284771\pi\)
\(270\) 0 0
\(271\) −11.4164 −0.693497 −0.346749 0.937958i \(-0.612714\pi\)
−0.346749 + 0.937958i \(0.612714\pi\)
\(272\) − 20.5623i − 1.24677i
\(273\) 2.00000i 0.121046i
\(274\) −0.708204 −0.0427842
\(275\) 0 0
\(276\) 3.32624 0.200216
\(277\) 13.0557i 0.784443i 0.919871 + 0.392221i \(0.128293\pi\)
−0.919871 + 0.392221i \(0.871707\pi\)
\(278\) 21.7082i 1.30197i
\(279\) −8.70820 −0.521347
\(280\) 0 0
\(281\) −14.1803 −0.845928 −0.422964 0.906146i \(-0.639010\pi\)
−0.422964 + 0.906146i \(0.639010\pi\)
\(282\) 7.70820i 0.459017i
\(283\) 2.29180i 0.136233i 0.997677 + 0.0681166i \(0.0216990\pi\)
−0.997677 + 0.0681166i \(0.978301\pi\)
\(284\) −6.23607 −0.370043
\(285\) 0 0
\(286\) 4.85410 0.287029
\(287\) − 18.7639i − 1.10760i
\(288\) − 3.38197i − 0.199284i
\(289\) −0.944272 −0.0555454
\(290\) 0 0
\(291\) 3.85410 0.225931
\(292\) − 9.70820i − 0.568130i
\(293\) − 6.32624i − 0.369583i −0.982778 0.184791i \(-0.940839\pi\)
0.982778 0.184791i \(-0.0591609\pi\)
\(294\) 4.85410 0.283097
\(295\) 0 0
\(296\) 4.47214 0.259938
\(297\) 3.00000i 0.174078i
\(298\) − 21.1803i − 1.22694i
\(299\) 5.38197 0.311247
\(300\) 0 0
\(301\) 14.7639 0.850979
\(302\) 10.7082i 0.616188i
\(303\) − 9.38197i − 0.538980i
\(304\) −32.5623 −1.86758
\(305\) 0 0
\(306\) −6.85410 −0.391823
\(307\) − 8.85410i − 0.505330i −0.967554 0.252665i \(-0.918693\pi\)
0.967554 0.252665i \(-0.0813071\pi\)
\(308\) 3.70820i 0.211295i
\(309\) 14.4164 0.820121
\(310\) 0 0
\(311\) −13.5279 −0.767095 −0.383547 0.923521i \(-0.625298\pi\)
−0.383547 + 0.923521i \(0.625298\pi\)
\(312\) − 2.23607i − 0.126592i
\(313\) 2.29180i 0.129540i 0.997900 + 0.0647700i \(0.0206314\pi\)
−0.997900 + 0.0647700i \(0.979369\pi\)
\(314\) 4.61803 0.260611
\(315\) 0 0
\(316\) −5.65248 −0.317977
\(317\) − 20.5623i − 1.15489i −0.816428 0.577447i \(-0.804049\pi\)
0.816428 0.577447i \(-0.195951\pi\)
\(318\) 18.1803i 1.01950i
\(319\) −10.8541 −0.607713
\(320\) 0 0
\(321\) −1.14590 −0.0639578
\(322\) 17.4164i 0.970578i
\(323\) 28.4164i 1.58113i
\(324\) −0.618034 −0.0343352
\(325\) 0 0
\(326\) 29.5623 1.63730
\(327\) 4.14590i 0.229269i
\(328\) 20.9787i 1.15836i
\(329\) −9.52786 −0.525288
\(330\) 0 0
\(331\) −30.6869 −1.68671 −0.843353 0.537360i \(-0.819422\pi\)
−0.843353 + 0.537360i \(0.819422\pi\)
\(332\) − 5.56231i − 0.305271i
\(333\) − 2.00000i − 0.109599i
\(334\) −28.7984 −1.57578
\(335\) 0 0
\(336\) 9.70820 0.529626
\(337\) 33.1803i 1.80745i 0.428115 + 0.903724i \(0.359178\pi\)
−0.428115 + 0.903724i \(0.640822\pi\)
\(338\) − 19.4164i − 1.05611i
\(339\) 3.76393 0.204429
\(340\) 0 0
\(341\) −26.1246 −1.41473
\(342\) 10.8541i 0.586923i
\(343\) 20.0000i 1.07990i
\(344\) −16.5066 −0.889975
\(345\) 0 0
\(346\) 1.47214 0.0791425
\(347\) − 12.2705i − 0.658715i −0.944205 0.329358i \(-0.893168\pi\)
0.944205 0.329358i \(-0.106832\pi\)
\(348\) − 2.23607i − 0.119866i
\(349\) −7.23607 −0.387338 −0.193669 0.981067i \(-0.562039\pi\)
−0.193669 + 0.981067i \(0.562039\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) − 10.1459i − 0.540778i
\(353\) − 12.3820i − 0.659026i −0.944151 0.329513i \(-0.893116\pi\)
0.944151 0.329513i \(-0.106884\pi\)
\(354\) −6.38197 −0.339198
\(355\) 0 0
\(356\) −6.90983 −0.366220
\(357\) − 8.47214i − 0.448393i
\(358\) − 25.3262i − 1.33853i
\(359\) −23.9443 −1.26373 −0.631865 0.775078i \(-0.717710\pi\)
−0.631865 + 0.775078i \(0.717710\pi\)
\(360\) 0 0
\(361\) 26.0000 1.36842
\(362\) 20.1803i 1.06066i
\(363\) − 2.00000i − 0.104973i
\(364\) −1.23607 −0.0647876
\(365\) 0 0
\(366\) 14.0902 0.736505
\(367\) 12.5279i 0.653949i 0.945033 + 0.326975i \(0.106029\pi\)
−0.945033 + 0.326975i \(0.893971\pi\)
\(368\) − 26.1246i − 1.36184i
\(369\) 9.38197 0.488406
\(370\) 0 0
\(371\) −22.4721 −1.16670
\(372\) − 5.38197i − 0.279042i
\(373\) 17.4164i 0.901787i 0.892578 + 0.450894i \(0.148895\pi\)
−0.892578 + 0.450894i \(0.851105\pi\)
\(374\) −20.5623 −1.06325
\(375\) 0 0
\(376\) 10.6525 0.549359
\(377\) − 3.61803i − 0.186338i
\(378\) − 3.23607i − 0.166445i
\(379\) 13.6180 0.699511 0.349756 0.936841i \(-0.386265\pi\)
0.349756 + 0.936841i \(0.386265\pi\)
\(380\) 0 0
\(381\) 13.6525 0.699438
\(382\) − 28.0344i − 1.43437i
\(383\) 5.05573i 0.258336i 0.991623 + 0.129168i \(0.0412306\pi\)
−0.991623 + 0.129168i \(0.958769\pi\)
\(384\) −13.6180 −0.694942
\(385\) 0 0
\(386\) −17.7984 −0.905913
\(387\) 7.38197i 0.375246i
\(388\) 2.38197i 0.120926i
\(389\) −0.652476 −0.0330818 −0.0165409 0.999863i \(-0.505265\pi\)
−0.0165409 + 0.999863i \(0.505265\pi\)
\(390\) 0 0
\(391\) −22.7984 −1.15296
\(392\) − 6.70820i − 0.338815i
\(393\) − 14.1803i − 0.715304i
\(394\) 0.145898 0.00735024
\(395\) 0 0
\(396\) −1.85410 −0.0931721
\(397\) 2.52786i 0.126870i 0.997986 + 0.0634349i \(0.0202055\pi\)
−0.997986 + 0.0634349i \(0.979794\pi\)
\(398\) 18.9443i 0.949591i
\(399\) −13.4164 −0.671660
\(400\) 0 0
\(401\) 36.2705 1.81126 0.905631 0.424066i \(-0.139397\pi\)
0.905631 + 0.424066i \(0.139397\pi\)
\(402\) 21.3262i 1.06366i
\(403\) − 8.70820i − 0.433787i
\(404\) 5.79837 0.288480
\(405\) 0 0
\(406\) 11.7082 0.581068
\(407\) − 6.00000i − 0.297409i
\(408\) 9.47214i 0.468941i
\(409\) 5.12461 0.253396 0.126698 0.991941i \(-0.459562\pi\)
0.126698 + 0.991941i \(0.459562\pi\)
\(410\) 0 0
\(411\) 0.437694 0.0215899
\(412\) 8.90983i 0.438956i
\(413\) − 7.88854i − 0.388170i
\(414\) −8.70820 −0.427985
\(415\) 0 0
\(416\) 3.38197 0.165815
\(417\) − 13.4164i − 0.657004i
\(418\) 32.5623i 1.59267i
\(419\) −0.326238 −0.0159378 −0.00796888 0.999968i \(-0.502537\pi\)
−0.00796888 + 0.999968i \(0.502537\pi\)
\(420\) 0 0
\(421\) −30.3607 −1.47969 −0.739844 0.672778i \(-0.765101\pi\)
−0.739844 + 0.672778i \(0.765101\pi\)
\(422\) 4.85410i 0.236294i
\(423\) − 4.76393i − 0.231630i
\(424\) 25.1246 1.22016
\(425\) 0 0
\(426\) 16.3262 0.791009
\(427\) 17.4164i 0.842839i
\(428\) − 0.708204i − 0.0342323i
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) 29.7639 1.43368 0.716839 0.697239i \(-0.245588\pi\)
0.716839 + 0.697239i \(0.245588\pi\)
\(432\) 4.85410i 0.233543i
\(433\) 3.47214i 0.166860i 0.996514 + 0.0834301i \(0.0265875\pi\)
−0.996514 + 0.0834301i \(0.973412\pi\)
\(434\) 28.1803 1.35270
\(435\) 0 0
\(436\) −2.56231 −0.122712
\(437\) 36.1033i 1.72706i
\(438\) 25.4164i 1.21444i
\(439\) 32.0344 1.52892 0.764460 0.644671i \(-0.223006\pi\)
0.764460 + 0.644671i \(0.223006\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) − 6.85410i − 0.326016i
\(443\) − 19.4164i − 0.922501i −0.887270 0.461251i \(-0.847401\pi\)
0.887270 0.461251i \(-0.152599\pi\)
\(444\) 1.23607 0.0586612
\(445\) 0 0
\(446\) −16.4164 −0.777339
\(447\) 13.0902i 0.619144i
\(448\) − 8.47214i − 0.400271i
\(449\) −16.5066 −0.778994 −0.389497 0.921028i \(-0.627351\pi\)
−0.389497 + 0.921028i \(0.627351\pi\)
\(450\) 0 0
\(451\) 28.1459 1.32534
\(452\) 2.32624i 0.109417i
\(453\) − 6.61803i − 0.310942i
\(454\) −9.32624 −0.437702
\(455\) 0 0
\(456\) 15.0000 0.702439
\(457\) 9.88854i 0.462567i 0.972886 + 0.231283i \(0.0742924\pi\)
−0.972886 + 0.231283i \(0.925708\pi\)
\(458\) − 26.1803i − 1.22333i
\(459\) 4.23607 0.197723
\(460\) 0 0
\(461\) −19.1803 −0.893317 −0.446659 0.894704i \(-0.647386\pi\)
−0.446659 + 0.894704i \(0.647386\pi\)
\(462\) − 9.70820i − 0.451667i
\(463\) − 33.6869i − 1.56556i −0.622296 0.782782i \(-0.713800\pi\)
0.622296 0.782782i \(-0.286200\pi\)
\(464\) −17.5623 −0.815310
\(465\) 0 0
\(466\) 16.4721 0.763057
\(467\) 10.4164i 0.482014i 0.970523 + 0.241007i \(0.0774777\pi\)
−0.970523 + 0.241007i \(0.922522\pi\)
\(468\) − 0.618034i − 0.0285686i
\(469\) −26.3607 −1.21722
\(470\) 0 0
\(471\) −2.85410 −0.131510
\(472\) 8.81966i 0.405958i
\(473\) 22.1459i 1.01827i
\(474\) 14.7984 0.679712
\(475\) 0 0
\(476\) 5.23607 0.239995
\(477\) − 11.2361i − 0.514464i
\(478\) 34.5967i 1.58242i
\(479\) −4.79837 −0.219243 −0.109622 0.993973i \(-0.534964\pi\)
−0.109622 + 0.993973i \(0.534964\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) − 11.8541i − 0.539940i
\(483\) − 10.7639i − 0.489776i
\(484\) 1.23607 0.0561849
\(485\) 0 0
\(486\) 1.61803 0.0733955
\(487\) − 16.6180i − 0.753035i −0.926410 0.376518i \(-0.877121\pi\)
0.926410 0.376518i \(-0.122879\pi\)
\(488\) − 19.4721i − 0.881462i
\(489\) −18.2705 −0.826221
\(490\) 0 0
\(491\) 22.3262 1.00757 0.503785 0.863829i \(-0.331941\pi\)
0.503785 + 0.863829i \(0.331941\pi\)
\(492\) 5.79837i 0.261411i
\(493\) 15.3262i 0.690259i
\(494\) −10.8541 −0.488349
\(495\) 0 0
\(496\) −42.2705 −1.89800
\(497\) 20.1803i 0.905212i
\(498\) 14.5623i 0.652553i
\(499\) 15.0000 0.671492 0.335746 0.941953i \(-0.391012\pi\)
0.335746 + 0.941953i \(0.391012\pi\)
\(500\) 0 0
\(501\) 17.7984 0.795173
\(502\) 30.7082i 1.37057i
\(503\) − 3.96556i − 0.176815i −0.996084 0.0884077i \(-0.971822\pi\)
0.996084 0.0884077i \(-0.0281778\pi\)
\(504\) −4.47214 −0.199205
\(505\) 0 0
\(506\) −26.1246 −1.16138
\(507\) 12.0000i 0.532939i
\(508\) 8.43769i 0.374362i
\(509\) −32.8885 −1.45776 −0.728880 0.684642i \(-0.759959\pi\)
−0.728880 + 0.684642i \(0.759959\pi\)
\(510\) 0 0
\(511\) −31.4164 −1.38978
\(512\) 5.29180i 0.233867i
\(513\) − 6.70820i − 0.296174i
\(514\) −50.5066 −2.22775
\(515\) 0 0
\(516\) −4.56231 −0.200844
\(517\) − 14.2918i − 0.628552i
\(518\) 6.47214i 0.284369i
\(519\) −0.909830 −0.0399371
\(520\) 0 0
\(521\) 40.0902 1.75638 0.878191 0.478310i \(-0.158750\pi\)
0.878191 + 0.478310i \(0.158750\pi\)
\(522\) 5.85410i 0.256227i
\(523\) 1.56231i 0.0683149i 0.999416 + 0.0341574i \(0.0108748\pi\)
−0.999416 + 0.0341574i \(0.989125\pi\)
\(524\) 8.76393 0.382854
\(525\) 0 0
\(526\) −20.2361 −0.882334
\(527\) 36.8885i 1.60689i
\(528\) 14.5623i 0.633743i
\(529\) −5.96556 −0.259372
\(530\) 0 0
\(531\) 3.94427 0.171167
\(532\) − 8.29180i − 0.359495i
\(533\) 9.38197i 0.406378i
\(534\) 18.0902 0.782838
\(535\) 0 0
\(536\) 29.4721 1.27300
\(537\) 15.6525i 0.675454i
\(538\) − 33.2148i − 1.43199i
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) −26.2918 −1.13037 −0.565186 0.824963i \(-0.691196\pi\)
−0.565186 + 0.824963i \(0.691196\pi\)
\(542\) 18.4721i 0.793446i
\(543\) − 12.4721i − 0.535231i
\(544\) −14.3262 −0.614232
\(545\) 0 0
\(546\) 3.23607 0.138491
\(547\) − 24.7082i − 1.05645i −0.849106 0.528223i \(-0.822858\pi\)
0.849106 0.528223i \(-0.177142\pi\)
\(548\) 0.270510i 0.0115556i
\(549\) −8.70820 −0.371657
\(550\) 0 0
\(551\) 24.2705 1.03396
\(552\) 12.0344i 0.512220i
\(553\) 18.2918i 0.777846i
\(554\) 21.1246 0.897499
\(555\) 0 0
\(556\) 8.29180 0.351650
\(557\) 37.6525i 1.59539i 0.603063 + 0.797693i \(0.293947\pi\)
−0.603063 + 0.797693i \(0.706053\pi\)
\(558\) 14.0902i 0.596484i
\(559\) −7.38197 −0.312224
\(560\) 0 0
\(561\) 12.7082 0.536541
\(562\) 22.9443i 0.967846i
\(563\) 9.00000i 0.379305i 0.981851 + 0.189652i \(0.0607361\pi\)
−0.981851 + 0.189652i \(0.939264\pi\)
\(564\) 2.94427 0.123976
\(565\) 0 0
\(566\) 3.70820 0.155867
\(567\) 2.00000i 0.0839921i
\(568\) − 22.5623i − 0.946693i
\(569\) 10.8541 0.455028 0.227514 0.973775i \(-0.426940\pi\)
0.227514 + 0.973775i \(0.426940\pi\)
\(570\) 0 0
\(571\) −38.1246 −1.59547 −0.797733 0.603011i \(-0.793967\pi\)
−0.797733 + 0.603011i \(0.793967\pi\)
\(572\) − 1.85410i − 0.0775239i
\(573\) 17.3262i 0.723814i
\(574\) −30.3607 −1.26723
\(575\) 0 0
\(576\) 4.23607 0.176503
\(577\) − 3.72949i − 0.155261i −0.996982 0.0776304i \(-0.975265\pi\)
0.996982 0.0776304i \(-0.0247354\pi\)
\(578\) 1.52786i 0.0635508i
\(579\) 11.0000 0.457144
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) − 6.23607i − 0.258493i
\(583\) − 33.7082i − 1.39605i
\(584\) 35.1246 1.45347
\(585\) 0 0
\(586\) −10.2361 −0.422848
\(587\) − 39.3050i − 1.62229i −0.584846 0.811144i \(-0.698845\pi\)
0.584846 0.811144i \(-0.301155\pi\)
\(588\) − 1.85410i − 0.0764619i
\(589\) 58.4164 2.40701
\(590\) 0 0
\(591\) −0.0901699 −0.00370910
\(592\) − 9.70820i − 0.399005i
\(593\) 17.6180i 0.723486i 0.932278 + 0.361743i \(0.117818\pi\)
−0.932278 + 0.361743i \(0.882182\pi\)
\(594\) 4.85410 0.199166
\(595\) 0 0
\(596\) −8.09017 −0.331386
\(597\) − 11.7082i − 0.479185i
\(598\) − 8.70820i − 0.356105i
\(599\) −39.2705 −1.60455 −0.802275 0.596955i \(-0.796377\pi\)
−0.802275 + 0.596955i \(0.796377\pi\)
\(600\) 0 0
\(601\) −24.7082 −1.00787 −0.503934 0.863742i \(-0.668115\pi\)
−0.503934 + 0.863742i \(0.668115\pi\)
\(602\) − 23.8885i − 0.973624i
\(603\) − 13.1803i − 0.536745i
\(604\) 4.09017 0.166427
\(605\) 0 0
\(606\) −15.1803 −0.616659
\(607\) 22.8541i 0.927619i 0.885935 + 0.463810i \(0.153518\pi\)
−0.885935 + 0.463810i \(0.846482\pi\)
\(608\) 22.6869i 0.920076i
\(609\) −7.23607 −0.293220
\(610\) 0 0
\(611\) 4.76393 0.192728
\(612\) 2.61803i 0.105828i
\(613\) − 5.87539i − 0.237305i −0.992936 0.118652i \(-0.962143\pi\)
0.992936 0.118652i \(-0.0378574\pi\)
\(614\) −14.3262 −0.578160
\(615\) 0 0
\(616\) −13.4164 −0.540562
\(617\) − 25.2361i − 1.01597i −0.861367 0.507983i \(-0.830391\pi\)
0.861367 0.507983i \(-0.169609\pi\)
\(618\) − 23.3262i − 0.938319i
\(619\) −34.2705 −1.37745 −0.688724 0.725024i \(-0.741829\pi\)
−0.688724 + 0.725024i \(0.741829\pi\)
\(620\) 0 0
\(621\) 5.38197 0.215971
\(622\) 21.8885i 0.877651i
\(623\) 22.3607i 0.895862i
\(624\) −4.85410 −0.194320
\(625\) 0 0
\(626\) 3.70820 0.148210
\(627\) − 20.1246i − 0.803700i
\(628\) − 1.76393i − 0.0703886i
\(629\) −8.47214 −0.337806
\(630\) 0 0
\(631\) −10.7639 −0.428505 −0.214253 0.976778i \(-0.568732\pi\)
−0.214253 + 0.976778i \(0.568732\pi\)
\(632\) − 20.4508i − 0.813491i
\(633\) − 3.00000i − 0.119239i
\(634\) −33.2705 −1.32134
\(635\) 0 0
\(636\) 6.94427 0.275358
\(637\) − 3.00000i − 0.118864i
\(638\) 17.5623i 0.695298i
\(639\) −10.0902 −0.399161
\(640\) 0 0
\(641\) −23.3262 −0.921331 −0.460666 0.887574i \(-0.652389\pi\)
−0.460666 + 0.887574i \(0.652389\pi\)
\(642\) 1.85410i 0.0731756i
\(643\) 5.90983i 0.233061i 0.993187 + 0.116530i \(0.0371773\pi\)
−0.993187 + 0.116530i \(0.962823\pi\)
\(644\) 6.65248 0.262144
\(645\) 0 0
\(646\) 45.9787 1.80901
\(647\) 19.0344i 0.748321i 0.927364 + 0.374161i \(0.122069\pi\)
−0.927364 + 0.374161i \(0.877931\pi\)
\(648\) − 2.23607i − 0.0878410i
\(649\) 11.8328 0.464479
\(650\) 0 0
\(651\) −17.4164 −0.682603
\(652\) − 11.2918i − 0.442221i
\(653\) 29.6525i 1.16039i 0.814477 + 0.580196i \(0.197024\pi\)
−0.814477 + 0.580196i \(0.802976\pi\)
\(654\) 6.70820 0.262312
\(655\) 0 0
\(656\) 45.5410 1.77808
\(657\) − 15.7082i − 0.612835i
\(658\) 15.4164i 0.600994i
\(659\) 2.23607 0.0871048 0.0435524 0.999051i \(-0.486132\pi\)
0.0435524 + 0.999051i \(0.486132\pi\)
\(660\) 0 0
\(661\) 4.88854 0.190142 0.0950712 0.995470i \(-0.469692\pi\)
0.0950712 + 0.995470i \(0.469692\pi\)
\(662\) 49.6525i 1.92980i
\(663\) 4.23607i 0.164515i
\(664\) 20.1246 0.780986
\(665\) 0 0
\(666\) −3.23607 −0.125395
\(667\) 19.4721i 0.753964i
\(668\) 11.0000i 0.425603i
\(669\) 10.1459 0.392263
\(670\) 0 0
\(671\) −26.1246 −1.00853
\(672\) − 6.76393i − 0.260924i
\(673\) − 46.7771i − 1.80312i −0.432650 0.901562i \(-0.642421\pi\)
0.432650 0.901562i \(-0.357579\pi\)
\(674\) 53.6869 2.06794
\(675\) 0 0
\(676\) −7.41641 −0.285246
\(677\) 44.8885i 1.72521i 0.505881 + 0.862603i \(0.331168\pi\)
−0.505881 + 0.862603i \(0.668832\pi\)
\(678\) − 6.09017i − 0.233892i
\(679\) 7.70820 0.295814
\(680\) 0 0
\(681\) 5.76393 0.220874
\(682\) 42.2705i 1.61862i
\(683\) − 0.596748i − 0.0228339i −0.999935 0.0114170i \(-0.996366\pi\)
0.999935 0.0114170i \(-0.00363421\pi\)
\(684\) 4.14590 0.158522
\(685\) 0 0
\(686\) 32.3607 1.23554
\(687\) 16.1803i 0.617318i
\(688\) 35.8328i 1.36611i
\(689\) 11.2361 0.428060
\(690\) 0 0
\(691\) 15.0902 0.574057 0.287029 0.957922i \(-0.407333\pi\)
0.287029 + 0.957922i \(0.407333\pi\)
\(692\) − 0.562306i − 0.0213757i
\(693\) 6.00000i 0.227921i
\(694\) −19.8541 −0.753651
\(695\) 0 0
\(696\) 8.09017 0.306657
\(697\) − 39.7426i − 1.50536i
\(698\) 11.7082i 0.443162i
\(699\) −10.1803 −0.385056
\(700\) 0 0
\(701\) −48.6525 −1.83758 −0.918789 0.394748i \(-0.870832\pi\)
−0.918789 + 0.394748i \(0.870832\pi\)
\(702\) 1.61803i 0.0610688i
\(703\) 13.4164i 0.506009i
\(704\) 12.7082 0.478958
\(705\) 0 0
\(706\) −20.0344 −0.754006
\(707\) − 18.7639i − 0.705690i
\(708\) 2.43769i 0.0916142i
\(709\) 5.20163 0.195351 0.0976756 0.995218i \(-0.468859\pi\)
0.0976756 + 0.995218i \(0.468859\pi\)
\(710\) 0 0
\(711\) −9.14590 −0.342998
\(712\) − 25.0000i − 0.936915i
\(713\) 46.8673i 1.75519i
\(714\) −13.7082 −0.513017
\(715\) 0 0
\(716\) −9.67376 −0.361525
\(717\) − 21.3820i − 0.798524i
\(718\) 38.7426i 1.44586i
\(719\) 35.1246 1.30993 0.654963 0.755661i \(-0.272684\pi\)
0.654963 + 0.755661i \(0.272684\pi\)
\(720\) 0 0
\(721\) 28.8328 1.07379
\(722\) − 42.0689i − 1.56564i
\(723\) 7.32624i 0.272466i
\(724\) 7.70820 0.286473
\(725\) 0 0
\(726\) −3.23607 −0.120102
\(727\) 14.4377i 0.535464i 0.963493 + 0.267732i \(0.0862743\pi\)
−0.963493 + 0.267732i \(0.913726\pi\)
\(728\) − 4.47214i − 0.165748i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 31.2705 1.15658
\(732\) − 5.38197i − 0.198923i
\(733\) − 10.1459i − 0.374747i −0.982289 0.187374i \(-0.940002\pi\)
0.982289 0.187374i \(-0.0599975\pi\)
\(734\) 20.2705 0.748198
\(735\) 0 0
\(736\) −18.2016 −0.670921
\(737\) − 39.5410i − 1.45651i
\(738\) − 15.1803i − 0.558796i
\(739\) 1.70820 0.0628373 0.0314186 0.999506i \(-0.489997\pi\)
0.0314186 + 0.999506i \(0.489997\pi\)
\(740\) 0 0
\(741\) 6.70820 0.246432
\(742\) 36.3607i 1.33484i
\(743\) − 16.5279i − 0.606349i −0.952935 0.303174i \(-0.901954\pi\)
0.952935 0.303174i \(-0.0980464\pi\)
\(744\) 19.4721 0.713883
\(745\) 0 0
\(746\) 28.1803 1.03176
\(747\) − 9.00000i − 0.329293i
\(748\) 7.85410i 0.287174i
\(749\) −2.29180 −0.0837404
\(750\) 0 0
\(751\) 15.2918 0.558006 0.279003 0.960290i \(-0.409996\pi\)
0.279003 + 0.960290i \(0.409996\pi\)
\(752\) − 23.1246i − 0.843268i
\(753\) − 18.9787i − 0.691623i
\(754\) −5.85410 −0.213194
\(755\) 0 0
\(756\) −1.23607 −0.0449554
\(757\) − 32.2705i − 1.17289i −0.809988 0.586446i \(-0.800527\pi\)
0.809988 0.586446i \(-0.199473\pi\)
\(758\) − 22.0344i − 0.800327i
\(759\) 16.1459 0.586059
\(760\) 0 0
\(761\) 3.18034 0.115287 0.0576436 0.998337i \(-0.481641\pi\)
0.0576436 + 0.998337i \(0.481641\pi\)
\(762\) − 22.0902i − 0.800242i
\(763\) 8.29180i 0.300183i
\(764\) −10.7082 −0.387409
\(765\) 0 0
\(766\) 8.18034 0.295568
\(767\) 3.94427i 0.142419i
\(768\) 13.5623i 0.489388i
\(769\) 36.3050 1.30919 0.654595 0.755980i \(-0.272839\pi\)
0.654595 + 0.755980i \(0.272839\pi\)
\(770\) 0 0
\(771\) 31.2148 1.12417
\(772\) 6.79837i 0.244679i
\(773\) − 41.7771i − 1.50262i −0.659951 0.751309i \(-0.729423\pi\)
0.659951 0.751309i \(-0.270577\pi\)
\(774\) 11.9443 0.429328
\(775\) 0 0
\(776\) −8.61803 −0.309369
\(777\) − 4.00000i − 0.143499i
\(778\) 1.05573i 0.0378497i
\(779\) −62.9361 −2.25492
\(780\) 0 0
\(781\) −30.2705 −1.08316
\(782\) 36.8885i 1.31913i
\(783\) − 3.61803i − 0.129298i
\(784\) −14.5623 −0.520082
\(785\) 0 0
\(786\) −22.9443 −0.818395
\(787\) − 17.1459i − 0.611185i −0.952162 0.305593i \(-0.901145\pi\)
0.952162 0.305593i \(-0.0988546\pi\)
\(788\) − 0.0557281i − 0.00198523i
\(789\) 12.5066 0.445246
\(790\) 0 0
\(791\) 7.52786 0.267660
\(792\) − 6.70820i − 0.238366i
\(793\) − 8.70820i − 0.309237i
\(794\) 4.09017 0.145155
\(795\) 0 0
\(796\) 7.23607 0.256476
\(797\) 30.0132i 1.06312i 0.847020 + 0.531560i \(0.178394\pi\)
−0.847020 + 0.531560i \(0.821606\pi\)
\(798\) 21.7082i 0.768462i
\(799\) −20.1803 −0.713929
\(800\) 0 0
\(801\) −11.1803 −0.395038
\(802\) − 58.6869i − 2.07231i
\(803\) − 47.1246i − 1.66299i
\(804\) 8.14590 0.287284
\(805\) 0 0
\(806\) −14.0902 −0.496305
\(807\) 20.5279i 0.722615i
\(808\) 20.9787i 0.738029i
\(809\) 39.9230 1.40362 0.701809 0.712365i \(-0.252376\pi\)
0.701809 + 0.712365i \(0.252376\pi\)
\(810\) 0 0
\(811\) −33.7771 −1.18607 −0.593037 0.805175i \(-0.702071\pi\)
−0.593037 + 0.805175i \(0.702071\pi\)
\(812\) − 4.47214i − 0.156941i
\(813\) − 11.4164i − 0.400391i
\(814\) −9.70820 −0.340272
\(815\) 0 0
\(816\) 20.5623 0.719825
\(817\) − 49.5197i − 1.73248i
\(818\) − 8.29180i − 0.289916i
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 5.94427 0.207457 0.103728 0.994606i \(-0.466923\pi\)
0.103728 + 0.994606i \(0.466923\pi\)
\(822\) − 0.708204i − 0.0247014i
\(823\) − 28.5623i − 0.995619i −0.867286 0.497810i \(-0.834138\pi\)
0.867286 0.497810i \(-0.165862\pi\)
\(824\) −32.2361 −1.12300
\(825\) 0 0
\(826\) −12.7639 −0.444114
\(827\) − 48.9787i − 1.70316i −0.524227 0.851578i \(-0.675646\pi\)
0.524227 0.851578i \(-0.324354\pi\)
\(828\) 3.32624i 0.115595i
\(829\) 50.1246 1.74090 0.870450 0.492257i \(-0.163828\pi\)
0.870450 + 0.492257i \(0.163828\pi\)
\(830\) 0 0
\(831\) −13.0557 −0.452898
\(832\) 4.23607i 0.146859i
\(833\) 12.7082i 0.440313i
\(834\) −21.7082 −0.751694
\(835\) 0 0
\(836\) 12.4377 0.430167
\(837\) − 8.70820i − 0.301000i
\(838\) 0.527864i 0.0182348i
\(839\) 3.21478 0.110987 0.0554933 0.998459i \(-0.482327\pi\)
0.0554933 + 0.998459i \(0.482327\pi\)
\(840\) 0 0
\(841\) −15.9098 −0.548615
\(842\) 49.1246i 1.69295i
\(843\) − 14.1803i − 0.488397i
\(844\) 1.85410 0.0638208
\(845\) 0 0
\(846\) −7.70820 −0.265014
\(847\) − 4.00000i − 0.137442i
\(848\) − 54.5410i − 1.87295i
\(849\) −2.29180 −0.0786542
\(850\) 0 0
\(851\) −10.7639 −0.368983
\(852\) − 6.23607i − 0.213644i
\(853\) − 20.3951i − 0.698316i −0.937064 0.349158i \(-0.886468\pi\)
0.937064 0.349158i \(-0.113532\pi\)
\(854\) 28.1803 0.964311
\(855\) 0 0
\(856\) 2.56231 0.0875778
\(857\) − 9.05573i − 0.309338i −0.987966 0.154669i \(-0.950569\pi\)
0.987966 0.154669i \(-0.0494311\pi\)
\(858\) 4.85410i 0.165716i
\(859\) −15.1246 −0.516045 −0.258023 0.966139i \(-0.583071\pi\)
−0.258023 + 0.966139i \(0.583071\pi\)
\(860\) 0 0
\(861\) 18.7639 0.639473
\(862\) − 48.1591i − 1.64030i
\(863\) 13.0689i 0.444870i 0.974948 + 0.222435i \(0.0714005\pi\)
−0.974948 + 0.222435i \(0.928599\pi\)
\(864\) 3.38197 0.115057
\(865\) 0 0
\(866\) 5.61803 0.190909
\(867\) − 0.944272i − 0.0320692i
\(868\) − 10.7639i − 0.365352i
\(869\) −27.4377 −0.930760
\(870\) 0 0
\(871\) 13.1803 0.446599
\(872\) − 9.27051i − 0.313939i
\(873\) 3.85410i 0.130442i
\(874\) 58.4164 1.97596
\(875\) 0 0
\(876\) 9.70820 0.328010
\(877\) − 43.1246i − 1.45621i −0.685463 0.728107i \(-0.740400\pi\)
0.685463 0.728107i \(-0.259600\pi\)
\(878\) − 51.8328i − 1.74927i
\(879\) 6.32624 0.213379
\(880\) 0 0
\(881\) 15.0902 0.508401 0.254200 0.967152i \(-0.418188\pi\)
0.254200 + 0.967152i \(0.418188\pi\)
\(882\) 4.85410i 0.163446i
\(883\) 29.2016i 0.982713i 0.870959 + 0.491356i \(0.163499\pi\)
−0.870959 + 0.491356i \(0.836501\pi\)
\(884\) −2.61803 −0.0880540
\(885\) 0 0
\(886\) −31.4164 −1.05545
\(887\) − 42.9230i − 1.44121i −0.693344 0.720606i \(-0.743864\pi\)
0.693344 0.720606i \(-0.256136\pi\)
\(888\) 4.47214i 0.150075i
\(889\) 27.3050 0.915779
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 6.27051i 0.209952i
\(893\) 31.9574i 1.06941i
\(894\) 21.1803 0.708377
\(895\) 0 0
\(896\) −27.2361 −0.909893
\(897\) 5.38197i 0.179699i
\(898\) 26.7082i 0.891264i
\(899\) 31.5066 1.05080
\(900\) 0 0
\(901\) −47.5967 −1.58568
\(902\) − 45.5410i − 1.51635i
\(903\) 14.7639i 0.491313i
\(904\) −8.41641 −0.279926
\(905\) 0 0
\(906\) −10.7082 −0.355756
\(907\) 17.0000i 0.564476i 0.959344 + 0.282238i \(0.0910767\pi\)
−0.959344 + 0.282238i \(0.908923\pi\)
\(908\) 3.56231i 0.118219i
\(909\) 9.38197 0.311180
\(910\) 0 0
\(911\) 14.8885 0.493279 0.246640 0.969107i \(-0.420674\pi\)
0.246640 + 0.969107i \(0.420674\pi\)
\(912\) − 32.5623i − 1.07825i
\(913\) − 27.0000i − 0.893570i
\(914\) 16.0000 0.529233
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) − 28.3607i − 0.936552i
\(918\) − 6.85410i − 0.226219i
\(919\) 5.00000 0.164935 0.0824674 0.996594i \(-0.473720\pi\)
0.0824674 + 0.996594i \(0.473720\pi\)
\(920\) 0 0
\(921\) 8.85410 0.291753
\(922\) 31.0344i 1.02206i
\(923\) − 10.0902i − 0.332122i
\(924\) −3.70820 −0.121991
\(925\) 0 0
\(926\) −54.5066 −1.79120
\(927\) 14.4164i 0.473497i
\(928\) 12.2361i 0.401669i
\(929\) 29.5967 0.971038 0.485519 0.874226i \(-0.338631\pi\)
0.485519 + 0.874226i \(0.338631\pi\)
\(930\) 0 0
\(931\) 20.1246 0.659558
\(932\) − 6.29180i − 0.206095i
\(933\) − 13.5279i − 0.442882i
\(934\) 16.8541 0.551483
\(935\) 0 0
\(936\) 2.23607 0.0730882
\(937\) 10.4164i 0.340289i 0.985419 + 0.170145i \(0.0544235\pi\)
−0.985419 + 0.170145i \(0.945577\pi\)
\(938\) 42.6525i 1.39265i
\(939\) −2.29180 −0.0747899
\(940\) 0 0
\(941\) 57.9787 1.89005 0.945026 0.326995i \(-0.106036\pi\)
0.945026 + 0.326995i \(0.106036\pi\)
\(942\) 4.61803i 0.150464i
\(943\) − 50.4934i − 1.64429i
\(944\) 19.1459 0.623146
\(945\) 0 0
\(946\) 35.8328 1.16503
\(947\) 23.8328i 0.774462i 0.921983 + 0.387231i \(0.126568\pi\)
−0.921983 + 0.387231i \(0.873432\pi\)
\(948\) − 5.65248i − 0.183584i
\(949\) 15.7082 0.509910
\(950\) 0 0
\(951\) 20.5623 0.666778
\(952\) 18.9443i 0.613987i
\(953\) − 42.0557i − 1.36232i −0.732135 0.681159i \(-0.761476\pi\)
0.732135 0.681159i \(-0.238524\pi\)
\(954\) −18.1803 −0.588610
\(955\) 0 0
\(956\) 13.2148 0.427397
\(957\) − 10.8541i − 0.350863i
\(958\) 7.76393i 0.250841i
\(959\) 0.875388 0.0282678
\(960\) 0 0
\(961\) 44.8328 1.44622
\(962\) − 3.23607i − 0.104335i
\(963\) − 1.14590i − 0.0369260i
\(964\) −4.52786 −0.145833
\(965\) 0 0
\(966\) −17.4164 −0.560364
\(967\) 35.4164i 1.13891i 0.822021 + 0.569457i \(0.192847\pi\)
−0.822021 + 0.569457i \(0.807153\pi\)
\(968\) 4.47214i 0.143740i
\(969\) −28.4164 −0.912867
\(970\) 0 0
\(971\) 29.8885 0.959169 0.479585 0.877496i \(-0.340787\pi\)
0.479585 + 0.877496i \(0.340787\pi\)
\(972\) − 0.618034i − 0.0198234i
\(973\) − 26.8328i − 0.860221i
\(974\) −26.8885 −0.861565
\(975\) 0 0
\(976\) −42.2705 −1.35305
\(977\) 37.6525i 1.20461i 0.798266 + 0.602305i \(0.205751\pi\)
−0.798266 + 0.602305i \(0.794249\pi\)
\(978\) 29.5623i 0.945298i
\(979\) −33.5410 −1.07198
\(980\) 0 0
\(981\) −4.14590 −0.132368
\(982\) − 36.1246i − 1.15278i
\(983\) − 24.6180i − 0.785193i −0.919711 0.392597i \(-0.871577\pi\)
0.919711 0.392597i \(-0.128423\pi\)
\(984\) −20.9787 −0.668777
\(985\) 0 0
\(986\) 24.7984 0.789741
\(987\) − 9.52786i − 0.303275i
\(988\) 4.14590i 0.131899i
\(989\) 39.7295 1.26332
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 29.4508i 0.935065i
\(993\) − 30.6869i − 0.973820i
\(994\) 32.6525 1.03567
\(995\) 0 0
\(996\) 5.56231 0.176248
\(997\) 2.93112i 0.0928294i 0.998922 + 0.0464147i \(0.0147796\pi\)
−0.998922 + 0.0464147i \(0.985220\pi\)
\(998\) − 24.2705i − 0.768270i
\(999\) 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.b.a.1249.1 4
5.2 odd 4 1875.2.a.c.1.2 yes 2
5.3 odd 4 1875.2.a.b.1.1 2
5.4 even 2 inner 1875.2.b.a.1249.4 4
15.2 even 4 5625.2.a.b.1.1 2
15.8 even 4 5625.2.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.b.1.1 2 5.3 odd 4
1875.2.a.c.1.2 yes 2 5.2 odd 4
1875.2.b.a.1249.1 4 1.1 even 1 trivial
1875.2.b.a.1249.4 4 5.4 even 2 inner
5625.2.a.b.1.1 2 15.2 even 4
5625.2.a.g.1.2 2 15.8 even 4