Properties

Label 1875.2.b.a.1249.2
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 1249.2
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1875.1249
Dual form 1875.2.b.a.1249.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034i q^{2} -1.00000i q^{3} +1.61803 q^{4} -0.618034 q^{6} -2.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-0.618034i q^{2} -1.00000i q^{3} +1.61803 q^{4} -0.618034 q^{6} -2.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.61803i q^{12} +1.00000i q^{13} -1.23607 q^{14} +1.85410 q^{16} +0.236068i q^{17} +0.618034i q^{18} -6.70820 q^{19} -2.00000 q^{21} +1.85410i q^{22} -7.61803i q^{23} -2.23607 q^{24} +0.618034 q^{26} +1.00000i q^{27} -3.23607i q^{28} +1.38197 q^{29} -4.70820 q^{31} -5.61803i q^{32} +3.00000i q^{33} +0.145898 q^{34} -1.61803 q^{36} -2.00000i q^{37} +4.14590i q^{38} +1.00000 q^{39} -11.6180 q^{41} +1.23607i q^{42} +9.61803i q^{43} -4.85410 q^{44} -4.70820 q^{46} -9.23607i q^{47} -1.85410i q^{48} +3.00000 q^{49} +0.236068 q^{51} +1.61803i q^{52} -6.76393i q^{53} +0.618034 q^{54} -4.47214 q^{56} +6.70820i q^{57} -0.854102i q^{58} +13.9443 q^{59} -4.70820 q^{61} +2.90983i q^{62} +2.00000i q^{63} +0.236068 q^{64} +1.85410 q^{66} +9.18034i q^{67} +0.381966i q^{68} -7.61803 q^{69} -1.09017 q^{71} +2.23607i q^{72} -2.29180i q^{73} -1.23607 q^{74} -10.8541 q^{76} +6.00000i q^{77} -0.618034i q^{78} +15.8541 q^{79} +1.00000 q^{81} +7.18034i q^{82} -9.00000i q^{83} -3.23607 q^{84} +5.94427 q^{86} -1.38197i q^{87} +6.70820i q^{88} -11.1803 q^{89} +2.00000 q^{91} -12.3262i q^{92} +4.70820i q^{93} -5.70820 q^{94} -5.61803 q^{96} -2.85410i q^{97} -1.85410i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{6} - 4 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.618034i − 0.437016i −0.975835 0.218508i \(-0.929881\pi\)
0.975835 0.218508i \(-0.0701190\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 1.61803 0.809017
\(5\) 0 0
\(6\) −0.618034 −0.252311
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) − 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 \(0.649385\pi\)
\(12\) − 1.61803i − 0.467086i
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) −1.23607 −0.330353
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 0.236068i 0.0572549i 0.999590 + 0.0286274i \(0.00911364\pi\)
−0.999590 + 0.0286274i \(0.990886\pi\)
\(18\) 0.618034i 0.145672i
\(19\) −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
\(22\) 1.85410i 0.395296i
\(23\) − 7.61803i − 1.58847i −0.607611 0.794235i \(-0.707872\pi\)
0.607611 0.794235i \(-0.292128\pi\)
\(24\) −2.23607 −0.456435
\(25\) 0 0
\(26\) 0.618034 0.121206
\(27\) 1.00000i 0.192450i
\(28\) − 3.23607i − 0.611559i
\(29\) 1.38197 0.256625 0.128312 0.991734i \(-0.459044\pi\)
0.128312 + 0.991734i \(0.459044\pi\)
\(30\) 0 0
\(31\) −4.70820 −0.845618 −0.422809 0.906219i \(-0.638956\pi\)
−0.422809 + 0.906219i \(0.638956\pi\)
\(32\) − 5.61803i − 0.993137i
\(33\) 3.00000i 0.522233i
\(34\) 0.145898 0.0250213
\(35\) 0 0
\(36\) −1.61803 −0.269672
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 4.14590i 0.672553i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −11.6180 −1.81443 −0.907216 0.420665i \(-0.861797\pi\)
−0.907216 + 0.420665i \(0.861797\pi\)
\(42\) 1.23607i 0.190729i
\(43\) 9.61803i 1.46674i 0.679832 + 0.733368i \(0.262053\pi\)
−0.679832 + 0.733368i \(0.737947\pi\)
\(44\) −4.85410 −0.731783
\(45\) 0 0
\(46\) −4.70820 −0.694187
\(47\) − 9.23607i − 1.34722i −0.739087 0.673609i \(-0.764743\pi\)
0.739087 0.673609i \(-0.235257\pi\)
\(48\) − 1.85410i − 0.267617i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0.236068 0.0330561
\(52\) 1.61803i 0.224381i
\(53\) − 6.76393i − 0.929098i −0.885548 0.464549i \(-0.846217\pi\)
0.885548 0.464549i \(-0.153783\pi\)
\(54\) 0.618034 0.0841038
\(55\) 0 0
\(56\) −4.47214 −0.597614
\(57\) 6.70820i 0.888523i
\(58\) − 0.854102i − 0.112149i
\(59\) 13.9443 1.81539 0.907695 0.419631i \(-0.137841\pi\)
0.907695 + 0.419631i \(0.137841\pi\)
\(60\) 0 0
\(61\) −4.70820 −0.602824 −0.301412 0.953494i \(-0.597458\pi\)
−0.301412 + 0.953494i \(0.597458\pi\)
\(62\) 2.90983i 0.369549i
\(63\) 2.00000i 0.251976i
\(64\) 0.236068 0.0295085
\(65\) 0 0
\(66\) 1.85410 0.228224
\(67\) 9.18034i 1.12156i 0.827966 + 0.560779i \(0.189498\pi\)
−0.827966 + 0.560779i \(0.810502\pi\)
\(68\) 0.381966i 0.0463202i
\(69\) −7.61803 −0.917104
\(70\) 0 0
\(71\) −1.09017 −0.129379 −0.0646897 0.997905i \(-0.520606\pi\)
−0.0646897 + 0.997905i \(0.520606\pi\)
\(72\) 2.23607i 0.263523i
\(73\) − 2.29180i − 0.268234i −0.990965 0.134117i \(-0.957180\pi\)
0.990965 0.134117i \(-0.0428199\pi\)
\(74\) −1.23607 −0.143690
\(75\) 0 0
\(76\) −10.8541 −1.24505
\(77\) 6.00000i 0.683763i
\(78\) − 0.618034i − 0.0699786i
\(79\) 15.8541 1.78373 0.891863 0.452306i \(-0.149398\pi\)
0.891863 + 0.452306i \(0.149398\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 7.18034i 0.792936i
\(83\) − 9.00000i − 0.987878i −0.869496 0.493939i \(-0.835557\pi\)
0.869496 0.493939i \(-0.164443\pi\)
\(84\) −3.23607 −0.353084
\(85\) 0 0
\(86\) 5.94427 0.640987
\(87\) − 1.38197i − 0.148162i
\(88\) 6.70820i 0.715097i
\(89\) −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
\(92\) − 12.3262i − 1.28510i
\(93\) 4.70820i 0.488218i
\(94\) −5.70820 −0.588756
\(95\) 0 0
\(96\) −5.61803 −0.573388
\(97\) − 2.85410i − 0.289790i −0.989447 0.144895i \(-0.953716\pi\)
0.989447 0.144895i \(-0.0462845\pi\)
\(98\) − 1.85410i − 0.187293i
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) −11.6180 −1.15604 −0.578019 0.816023i \(-0.696174\pi\)
−0.578019 + 0.816023i \(0.696174\pi\)
\(102\) − 0.145898i − 0.0144461i
\(103\) − 12.4164i − 1.22343i −0.791080 0.611713i \(-0.790481\pi\)
0.791080 0.611713i \(-0.209519\pi\)
\(104\) 2.23607 0.219265
\(105\) 0 0
\(106\) −4.18034 −0.406031
\(107\) − 7.85410i − 0.759285i −0.925133 0.379642i \(-0.876047\pi\)
0.925133 0.379642i \(-0.123953\pi\)
\(108\) 1.61803i 0.155695i
\(109\) 10.8541 1.03963 0.519817 0.854278i \(-0.326000\pi\)
0.519817 + 0.854278i \(0.326000\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) − 3.70820i − 0.350392i
\(113\) 8.23607i 0.774784i 0.921915 + 0.387392i \(0.126624\pi\)
−0.921915 + 0.387392i \(0.873376\pi\)
\(114\) 4.14590 0.388299
\(115\) 0 0
\(116\) 2.23607 0.207614
\(117\) − 1.00000i − 0.0924500i
\(118\) − 8.61803i − 0.793354i
\(119\) 0.472136 0.0432806
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 2.90983i 0.263444i
\(123\) 11.6180i 1.04756i
\(124\) −7.61803 −0.684120
\(125\) 0 0
\(126\) 1.23607 0.110118
\(127\) − 17.6525i − 1.56640i −0.621767 0.783202i \(-0.713585\pi\)
0.621767 0.783202i \(-0.286415\pi\)
\(128\) − 11.3820i − 1.00603i
\(129\) 9.61803 0.846821
\(130\) 0 0
\(131\) 8.18034 0.714720 0.357360 0.933967i \(-0.383677\pi\)
0.357360 + 0.933967i \(0.383677\pi\)
\(132\) 4.85410i 0.422495i
\(133\) 13.4164i 1.16335i
\(134\) 5.67376 0.490138
\(135\) 0 0
\(136\) 0.527864 0.0452640
\(137\) 20.5623i 1.75676i 0.477966 + 0.878378i \(0.341374\pi\)
−0.477966 + 0.878378i \(0.658626\pi\)
\(138\) 4.70820i 0.400789i
\(139\) 13.4164 1.13796 0.568982 0.822350i \(-0.307337\pi\)
0.568982 + 0.822350i \(0.307337\pi\)
\(140\) 0 0
\(141\) −9.23607 −0.777817
\(142\) 0.673762i 0.0565409i
\(143\) − 3.00000i − 0.250873i
\(144\) −1.85410 −0.154508
\(145\) 0 0
\(146\) −1.41641 −0.117223
\(147\) − 3.00000i − 0.247436i
\(148\) − 3.23607i − 0.266003i
\(149\) 1.90983 0.156459 0.0782297 0.996935i \(-0.475073\pi\)
0.0782297 + 0.996935i \(0.475073\pi\)
\(150\) 0 0
\(151\) −4.38197 −0.356599 −0.178300 0.983976i \(-0.557060\pi\)
−0.178300 + 0.983976i \(0.557060\pi\)
\(152\) 15.0000i 1.21666i
\(153\) − 0.236068i − 0.0190850i
\(154\) 3.70820 0.298816
\(155\) 0 0
\(156\) 1.61803 0.129546
\(157\) 3.85410i 0.307591i 0.988103 + 0.153795i \(0.0491497\pi\)
−0.988103 + 0.153795i \(0.950850\pi\)
\(158\) − 9.79837i − 0.779517i
\(159\) −6.76393 −0.536415
\(160\) 0 0
\(161\) −15.2361 −1.20077
\(162\) − 0.618034i − 0.0485573i
\(163\) 15.2705i 1.19608i 0.801467 + 0.598039i \(0.204053\pi\)
−0.801467 + 0.598039i \(0.795947\pi\)
\(164\) −18.7984 −1.46791
\(165\) 0 0
\(166\) −5.56231 −0.431719
\(167\) − 6.79837i − 0.526074i −0.964786 0.263037i \(-0.915276\pi\)
0.964786 0.263037i \(-0.0847241\pi\)
\(168\) 4.47214i 0.345033i
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 6.70820 0.512989
\(172\) 15.5623i 1.18661i
\(173\) − 12.0902i − 0.919199i −0.888126 0.459599i \(-0.847993\pi\)
0.888126 0.459599i \(-0.152007\pi\)
\(174\) −0.854102 −0.0647493
\(175\) 0 0
\(176\) −5.56231 −0.419275
\(177\) − 13.9443i − 1.04812i
\(178\) 6.90983i 0.517914i
\(179\) −15.6525 −1.16992 −0.584960 0.811062i \(-0.698890\pi\)
−0.584960 + 0.811062i \(0.698890\pi\)
\(180\) 0 0
\(181\) −3.52786 −0.262224 −0.131112 0.991368i \(-0.541855\pi\)
−0.131112 + 0.991368i \(0.541855\pi\)
\(182\) − 1.23607i − 0.0916235i
\(183\) 4.70820i 0.348040i
\(184\) −17.0344 −1.25580
\(185\) 0 0
\(186\) 2.90983 0.213359
\(187\) − 0.708204i − 0.0517890i
\(188\) − 14.9443i − 1.08992i
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 1.67376 0.121109 0.0605546 0.998165i \(-0.480713\pi\)
0.0605546 + 0.998165i \(0.480713\pi\)
\(192\) − 0.236068i − 0.0170367i
\(193\) 11.0000i 0.791797i 0.918294 + 0.395899i \(0.129567\pi\)
−0.918294 + 0.395899i \(0.870433\pi\)
\(194\) −1.76393 −0.126643
\(195\) 0 0
\(196\) 4.85410 0.346722
\(197\) 11.0902i 0.790142i 0.918651 + 0.395071i \(0.129280\pi\)
−0.918651 + 0.395071i \(0.870720\pi\)
\(198\) − 1.85410i − 0.131765i
\(199\) 1.70820 0.121091 0.0605457 0.998165i \(-0.480716\pi\)
0.0605457 + 0.998165i \(0.480716\pi\)
\(200\) 0 0
\(201\) 9.18034 0.647531
\(202\) 7.18034i 0.505207i
\(203\) − 2.76393i − 0.193990i
\(204\) 0.381966 0.0267430
\(205\) 0 0
\(206\) −7.67376 −0.534656
\(207\) 7.61803i 0.529490i
\(208\) 1.85410i 0.128559i
\(209\) 20.1246 1.39205
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) − 10.9443i − 0.751656i
\(213\) 1.09017i 0.0746972i
\(214\) −4.85410 −0.331820
\(215\) 0 0
\(216\) 2.23607 0.152145
\(217\) 9.41641i 0.639227i
\(218\) − 6.70820i − 0.454337i
\(219\) −2.29180 −0.154865
\(220\) 0 0
\(221\) −0.236068 −0.0158797
\(222\) 1.23607i 0.0829595i
\(223\) 16.8541i 1.12863i 0.825558 + 0.564317i \(0.190860\pi\)
−0.825558 + 0.564317i \(0.809140\pi\)
\(224\) −11.2361 −0.750741
\(225\) 0 0
\(226\) 5.09017 0.338593
\(227\) 10.2361i 0.679392i 0.940535 + 0.339696i \(0.110324\pi\)
−0.940535 + 0.339696i \(0.889676\pi\)
\(228\) 10.8541i 0.718830i
\(229\) −6.18034 −0.408408 −0.204204 0.978928i \(-0.565461\pi\)
−0.204204 + 0.978928i \(0.565461\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) − 3.09017i − 0.202880i
\(233\) 12.1803i 0.797961i 0.916959 + 0.398980i \(0.130636\pi\)
−0.916959 + 0.398980i \(0.869364\pi\)
\(234\) −0.618034 −0.0404021
\(235\) 0 0
\(236\) 22.5623 1.46868
\(237\) − 15.8541i − 1.02983i
\(238\) − 0.291796i − 0.0189143i
\(239\) −23.6180 −1.52772 −0.763862 0.645380i \(-0.776699\pi\)
−0.763862 + 0.645380i \(0.776699\pi\)
\(240\) 0 0
\(241\) −8.32624 −0.536340 −0.268170 0.963372i \(-0.586419\pi\)
−0.268170 + 0.963372i \(0.586419\pi\)
\(242\) 1.23607i 0.0794575i
\(243\) − 1.00000i − 0.0641500i
\(244\) −7.61803 −0.487695
\(245\) 0 0
\(246\) 7.18034 0.457802
\(247\) − 6.70820i − 0.426833i
\(248\) 10.5279i 0.668520i
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) 27.9787 1.76600 0.883000 0.469372i \(-0.155520\pi\)
0.883000 + 0.469372i \(0.155520\pi\)
\(252\) 3.23607i 0.203853i
\(253\) 22.8541i 1.43683i
\(254\) −10.9098 −0.684544
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) − 20.2148i − 1.26096i −0.776204 0.630482i \(-0.782857\pi\)
0.776204 0.630482i \(-0.217143\pi\)
\(258\) − 5.94427i − 0.370074i
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −1.38197 −0.0855415
\(262\) − 5.05573i − 0.312344i
\(263\) − 25.5066i − 1.57280i −0.617716 0.786401i \(-0.711942\pi\)
0.617716 0.786401i \(-0.288058\pi\)
\(264\) 6.70820 0.412861
\(265\) 0 0
\(266\) 8.29180 0.508403
\(267\) 11.1803i 0.684226i
\(268\) 14.8541i 0.907359i
\(269\) 29.4721 1.79695 0.898474 0.439027i \(-0.144677\pi\)
0.898474 + 0.439027i \(0.144677\pi\)
\(270\) 0 0
\(271\) 15.4164 0.936480 0.468240 0.883601i \(-0.344888\pi\)
0.468240 + 0.883601i \(0.344888\pi\)
\(272\) 0.437694i 0.0265391i
\(273\) − 2.00000i − 0.121046i
\(274\) 12.7082 0.767731
\(275\) 0 0
\(276\) −12.3262 −0.741952
\(277\) − 30.9443i − 1.85926i −0.368493 0.929631i \(-0.620126\pi\)
0.368493 0.929631i \(-0.379874\pi\)
\(278\) − 8.29180i − 0.497309i
\(279\) 4.70820 0.281873
\(280\) 0 0
\(281\) 8.18034 0.487998 0.243999 0.969775i \(-0.421541\pi\)
0.243999 + 0.969775i \(0.421541\pi\)
\(282\) 5.70820i 0.339919i
\(283\) − 15.7082i − 0.933756i −0.884322 0.466878i \(-0.845379\pi\)
0.884322 0.466878i \(-0.154621\pi\)
\(284\) −1.76393 −0.104670
\(285\) 0 0
\(286\) −1.85410 −0.109635
\(287\) 23.2361i 1.37158i
\(288\) 5.61803i 0.331046i
\(289\) 16.9443 0.996722
\(290\) 0 0
\(291\) −2.85410 −0.167310
\(292\) − 3.70820i − 0.217006i
\(293\) − 9.32624i − 0.544845i −0.962178 0.272422i \(-0.912175\pi\)
0.962178 0.272422i \(-0.0878248\pi\)
\(294\) −1.85410 −0.108133
\(295\) 0 0
\(296\) −4.47214 −0.259938
\(297\) − 3.00000i − 0.174078i
\(298\) − 1.18034i − 0.0683753i
\(299\) 7.61803 0.440562
\(300\) 0 0
\(301\) 19.2361 1.10875
\(302\) 2.70820i 0.155840i
\(303\) 11.6180i 0.667439i
\(304\) −12.4377 −0.713351
\(305\) 0 0
\(306\) −0.145898 −0.00834044
\(307\) 2.14590i 0.122473i 0.998123 + 0.0612364i \(0.0195044\pi\)
−0.998123 + 0.0612364i \(0.980496\pi\)
\(308\) 9.70820i 0.553176i
\(309\) −12.4164 −0.706345
\(310\) 0 0
\(311\) −22.4721 −1.27428 −0.637139 0.770749i \(-0.719882\pi\)
−0.637139 + 0.770749i \(0.719882\pi\)
\(312\) − 2.23607i − 0.126592i
\(313\) − 15.7082i − 0.887880i −0.896056 0.443940i \(-0.853580\pi\)
0.896056 0.443940i \(-0.146420\pi\)
\(314\) 2.38197 0.134422
\(315\) 0 0
\(316\) 25.6525 1.44306
\(317\) 0.437694i 0.0245833i 0.999924 + 0.0122917i \(0.00391266\pi\)
−0.999924 + 0.0122917i \(0.996087\pi\)
\(318\) 4.18034i 0.234422i
\(319\) −4.14590 −0.232126
\(320\) 0 0
\(321\) −7.85410 −0.438373
\(322\) 9.41641i 0.524756i
\(323\) − 1.58359i − 0.0881134i
\(324\) 1.61803 0.0898908
\(325\) 0 0
\(326\) 9.43769 0.522706
\(327\) − 10.8541i − 0.600233i
\(328\) 25.9787i 1.43443i
\(329\) −18.4721 −1.01840
\(330\) 0 0
\(331\) 29.6869 1.63174 0.815870 0.578235i \(-0.196258\pi\)
0.815870 + 0.578235i \(0.196258\pi\)
\(332\) − 14.5623i − 0.799210i
\(333\) 2.00000i 0.109599i
\(334\) −4.20163 −0.229903
\(335\) 0 0
\(336\) −3.70820 −0.202299
\(337\) − 10.8197i − 0.589384i −0.955592 0.294692i \(-0.904783\pi\)
0.955592 0.294692i \(-0.0952171\pi\)
\(338\) − 7.41641i − 0.403399i
\(339\) 8.23607 0.447322
\(340\) 0 0
\(341\) 14.1246 0.764891
\(342\) − 4.14590i − 0.224184i
\(343\) − 20.0000i − 1.07990i
\(344\) 21.5066 1.15956
\(345\) 0 0
\(346\) −7.47214 −0.401705
\(347\) − 21.2705i − 1.14186i −0.820998 0.570930i \(-0.806583\pi\)
0.820998 0.570930i \(-0.193417\pi\)
\(348\) − 2.23607i − 0.119866i
\(349\) −2.76393 −0.147950 −0.0739749 0.997260i \(-0.523568\pi\)
−0.0739749 + 0.997260i \(0.523568\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 16.8541i 0.898327i
\(353\) 14.6180i 0.778039i 0.921229 + 0.389020i \(0.127186\pi\)
−0.921229 + 0.389020i \(0.872814\pi\)
\(354\) −8.61803 −0.458043
\(355\) 0 0
\(356\) −18.0902 −0.958777
\(357\) − 0.472136i − 0.0249881i
\(358\) 9.67376i 0.511274i
\(359\) −6.05573 −0.319609 −0.159805 0.987149i \(-0.551086\pi\)
−0.159805 + 0.987149i \(0.551086\pi\)
\(360\) 0 0
\(361\) 26.0000 1.36842
\(362\) 2.18034i 0.114596i
\(363\) 2.00000i 0.104973i
\(364\) 3.23607 0.169616
\(365\) 0 0
\(366\) 2.90983 0.152099
\(367\) − 21.4721i − 1.12084i −0.828210 0.560418i \(-0.810640\pi\)
0.828210 0.560418i \(-0.189360\pi\)
\(368\) − 14.1246i − 0.736296i
\(369\) 11.6180 0.604811
\(370\) 0 0
\(371\) −13.5279 −0.702332
\(372\) 7.61803i 0.394977i
\(373\) 9.41641i 0.487563i 0.969830 + 0.243782i \(0.0783880\pi\)
−0.969830 + 0.243782i \(0.921612\pi\)
\(374\) −0.437694 −0.0226326
\(375\) 0 0
\(376\) −20.6525 −1.06507
\(377\) 1.38197i 0.0711749i
\(378\) − 1.23607i − 0.0635765i
\(379\) 11.3820 0.584652 0.292326 0.956319i \(-0.405571\pi\)
0.292326 + 0.956319i \(0.405571\pi\)
\(380\) 0 0
\(381\) −17.6525 −0.904364
\(382\) − 1.03444i − 0.0529266i
\(383\) − 22.9443i − 1.17240i −0.810167 0.586199i \(-0.800624\pi\)
0.810167 0.586199i \(-0.199376\pi\)
\(384\) −11.3820 −0.580834
\(385\) 0 0
\(386\) 6.79837 0.346028
\(387\) − 9.61803i − 0.488912i
\(388\) − 4.61803i − 0.234445i
\(389\) 30.6525 1.55414 0.777071 0.629413i \(-0.216705\pi\)
0.777071 + 0.629413i \(0.216705\pi\)
\(390\) 0 0
\(391\) 1.79837 0.0909477
\(392\) − 6.70820i − 0.338815i
\(393\) − 8.18034i − 0.412644i
\(394\) 6.85410 0.345305
\(395\) 0 0
\(396\) 4.85410 0.243928
\(397\) − 11.4721i − 0.575770i −0.957665 0.287885i \(-0.907048\pi\)
0.957665 0.287885i \(-0.0929521\pi\)
\(398\) − 1.05573i − 0.0529189i
\(399\) 13.4164 0.671660
\(400\) 0 0
\(401\) 2.72949 0.136304 0.0681521 0.997675i \(-0.478290\pi\)
0.0681521 + 0.997675i \(0.478290\pi\)
\(402\) − 5.67376i − 0.282982i
\(403\) − 4.70820i − 0.234532i
\(404\) −18.7984 −0.935254
\(405\) 0 0
\(406\) −1.70820 −0.0847767
\(407\) 6.00000i 0.297409i
\(408\) − 0.527864i − 0.0261332i
\(409\) −35.1246 −1.73680 −0.868400 0.495864i \(-0.834851\pi\)
−0.868400 + 0.495864i \(0.834851\pi\)
\(410\) 0 0
\(411\) 20.5623 1.01426
\(412\) − 20.0902i − 0.989772i
\(413\) − 27.8885i − 1.37231i
\(414\) 4.70820 0.231396
\(415\) 0 0
\(416\) 5.61803 0.275447
\(417\) − 13.4164i − 0.657004i
\(418\) − 12.4377i − 0.608348i
\(419\) 15.3262 0.748736 0.374368 0.927280i \(-0.377860\pi\)
0.374368 + 0.927280i \(0.377860\pi\)
\(420\) 0 0
\(421\) 14.3607 0.699897 0.349948 0.936769i \(-0.386199\pi\)
0.349948 + 0.936769i \(0.386199\pi\)
\(422\) 1.85410i 0.0902563i
\(423\) 9.23607i 0.449073i
\(424\) −15.1246 −0.734516
\(425\) 0 0
\(426\) 0.673762 0.0326439
\(427\) 9.41641i 0.455692i
\(428\) − 12.7082i − 0.614274i
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) 34.2361 1.64909 0.824547 0.565794i \(-0.191430\pi\)
0.824547 + 0.565794i \(0.191430\pi\)
\(432\) 1.85410i 0.0892055i
\(433\) 5.47214i 0.262974i 0.991318 + 0.131487i \(0.0419752\pi\)
−0.991318 + 0.131487i \(0.958025\pi\)
\(434\) 5.81966 0.279353
\(435\) 0 0
\(436\) 17.5623 0.841082
\(437\) 51.1033i 2.44460i
\(438\) 1.41641i 0.0676786i
\(439\) 2.96556 0.141538 0.0707692 0.997493i \(-0.477455\pi\)
0.0707692 + 0.997493i \(0.477455\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0.145898i 0.00693966i
\(443\) − 7.41641i − 0.352364i −0.984358 0.176182i \(-0.943625\pi\)
0.984358 0.176182i \(-0.0563748\pi\)
\(444\) −3.23607 −0.153577
\(445\) 0 0
\(446\) 10.4164 0.493231
\(447\) − 1.90983i − 0.0903319i
\(448\) − 0.472136i − 0.0223063i
\(449\) 21.5066 1.01496 0.507479 0.861664i \(-0.330577\pi\)
0.507479 + 0.861664i \(0.330577\pi\)
\(450\) 0 0
\(451\) 34.8541 1.64122
\(452\) 13.3262i 0.626814i
\(453\) 4.38197i 0.205883i
\(454\) 6.32624 0.296905
\(455\) 0 0
\(456\) 15.0000 0.702439
\(457\) 25.8885i 1.21102i 0.795840 + 0.605508i \(0.207030\pi\)
−0.795840 + 0.605508i \(0.792970\pi\)
\(458\) 3.81966i 0.178481i
\(459\) −0.236068 −0.0110187
\(460\) 0 0
\(461\) 3.18034 0.148123 0.0740616 0.997254i \(-0.476404\pi\)
0.0740616 + 0.997254i \(0.476404\pi\)
\(462\) − 3.70820i − 0.172521i
\(463\) − 26.6869i − 1.24025i −0.784505 0.620123i \(-0.787083\pi\)
0.784505 0.620123i \(-0.212917\pi\)
\(464\) 2.56231 0.118952
\(465\) 0 0
\(466\) 7.52786 0.348722
\(467\) 16.4164i 0.759661i 0.925056 + 0.379830i \(0.124018\pi\)
−0.925056 + 0.379830i \(0.875982\pi\)
\(468\) − 1.61803i − 0.0747936i
\(469\) 18.3607 0.847817
\(470\) 0 0
\(471\) 3.85410 0.177588
\(472\) − 31.1803i − 1.43519i
\(473\) − 28.8541i − 1.32671i
\(474\) −9.79837 −0.450054
\(475\) 0 0
\(476\) 0.763932 0.0350148
\(477\) 6.76393i 0.309699i
\(478\) 14.5967i 0.667640i
\(479\) 19.7984 0.904611 0.452305 0.891863i \(-0.350602\pi\)
0.452305 + 0.891863i \(0.350602\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 5.14590i 0.234389i
\(483\) 15.2361i 0.693265i
\(484\) −3.23607 −0.147094
\(485\) 0 0
\(486\) −0.618034 −0.0280346
\(487\) 14.3820i 0.651709i 0.945420 + 0.325855i \(0.105652\pi\)
−0.945420 + 0.325855i \(0.894348\pi\)
\(488\) 10.5279i 0.476574i
\(489\) 15.2705 0.690556
\(490\) 0 0
\(491\) 6.67376 0.301183 0.150591 0.988596i \(-0.451882\pi\)
0.150591 + 0.988596i \(0.451882\pi\)
\(492\) 18.7984i 0.847496i
\(493\) 0.326238i 0.0146930i
\(494\) −4.14590 −0.186533
\(495\) 0 0
\(496\) −8.72949 −0.391966
\(497\) 2.18034i 0.0978016i
\(498\) 5.56231i 0.249253i
\(499\) 15.0000 0.671492 0.335746 0.941953i \(-0.391012\pi\)
0.335746 + 0.941953i \(0.391012\pi\)
\(500\) 0 0
\(501\) −6.79837 −0.303729
\(502\) − 17.2918i − 0.771771i
\(503\) 33.0344i 1.47293i 0.676474 + 0.736466i \(0.263507\pi\)
−0.676474 + 0.736466i \(0.736493\pi\)
\(504\) 4.47214 0.199205
\(505\) 0 0
\(506\) 14.1246 0.627916
\(507\) − 12.0000i − 0.532939i
\(508\) − 28.5623i − 1.26725i
\(509\) 2.88854 0.128032 0.0640162 0.997949i \(-0.479609\pi\)
0.0640162 + 0.997949i \(0.479609\pi\)
\(510\) 0 0
\(511\) −4.58359 −0.202766
\(512\) − 18.7082i − 0.826794i
\(513\) − 6.70820i − 0.296174i
\(514\) −12.4934 −0.551061
\(515\) 0 0
\(516\) 15.5623 0.685092
\(517\) 27.7082i 1.21861i
\(518\) 2.47214i 0.108619i
\(519\) −12.0902 −0.530700
\(520\) 0 0
\(521\) 28.9098 1.26656 0.633281 0.773922i \(-0.281708\pi\)
0.633281 + 0.773922i \(0.281708\pi\)
\(522\) 0.854102i 0.0373830i
\(523\) 18.5623i 0.811673i 0.913946 + 0.405836i \(0.133020\pi\)
−0.913946 + 0.405836i \(0.866980\pi\)
\(524\) 13.2361 0.578220
\(525\) 0 0
\(526\) −15.7639 −0.687340
\(527\) − 1.11146i − 0.0484158i
\(528\) 5.56231i 0.242068i
\(529\) −35.0344 −1.52324
\(530\) 0 0
\(531\) −13.9443 −0.605130
\(532\) 21.7082i 0.941170i
\(533\) − 11.6180i − 0.503233i
\(534\) 6.90983 0.299018
\(535\) 0 0
\(536\) 20.5279 0.886669
\(537\) 15.6525i 0.675454i
\(538\) − 18.2148i − 0.785295i
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) −39.7082 −1.70719 −0.853595 0.520938i \(-0.825582\pi\)
−0.853595 + 0.520938i \(0.825582\pi\)
\(542\) − 9.52786i − 0.409257i
\(543\) 3.52786i 0.151395i
\(544\) 1.32624 0.0568620
\(545\) 0 0
\(546\) −1.23607 −0.0528988
\(547\) 11.2918i 0.482802i 0.970425 + 0.241401i \(0.0776070\pi\)
−0.970425 + 0.241401i \(0.922393\pi\)
\(548\) 33.2705i 1.42125i
\(549\) 4.70820 0.200941
\(550\) 0 0
\(551\) −9.27051 −0.394937
\(552\) 17.0344i 0.725034i
\(553\) − 31.7082i − 1.34837i
\(554\) −19.1246 −0.812527
\(555\) 0 0
\(556\) 21.7082 0.920633
\(557\) − 6.34752i − 0.268953i −0.990917 0.134477i \(-0.957065\pi\)
0.990917 0.134477i \(-0.0429353\pi\)
\(558\) − 2.90983i − 0.123183i
\(559\) −9.61803 −0.406799
\(560\) 0 0
\(561\) −0.708204 −0.0299004
\(562\) − 5.05573i − 0.213263i
\(563\) − 9.00000i − 0.379305i −0.981851 0.189652i \(-0.939264\pi\)
0.981851 0.189652i \(-0.0607361\pi\)
\(564\) −14.9443 −0.629267
\(565\) 0 0
\(566\) −9.70820 −0.408066
\(567\) − 2.00000i − 0.0839921i
\(568\) 2.43769i 0.102283i
\(569\) 4.14590 0.173805 0.0869025 0.996217i \(-0.472303\pi\)
0.0869025 + 0.996217i \(0.472303\pi\)
\(570\) 0 0
\(571\) 2.12461 0.0889122 0.0444561 0.999011i \(-0.485845\pi\)
0.0444561 + 0.999011i \(0.485845\pi\)
\(572\) − 4.85410i − 0.202960i
\(573\) − 1.67376i − 0.0699224i
\(574\) 14.3607 0.599403
\(575\) 0 0
\(576\) −0.236068 −0.00983617
\(577\) 37.2705i 1.55159i 0.630984 + 0.775796i \(0.282651\pi\)
−0.630984 + 0.775796i \(0.717349\pi\)
\(578\) − 10.4721i − 0.435583i
\(579\) 11.0000 0.457144
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) 1.76393i 0.0731173i
\(583\) 20.2918i 0.840400i
\(584\) −5.12461 −0.212058
\(585\) 0 0
\(586\) −5.76393 −0.238106
\(587\) − 23.3050i − 0.961898i −0.876748 0.480949i \(-0.840292\pi\)
0.876748 0.480949i \(-0.159708\pi\)
\(588\) − 4.85410i − 0.200180i
\(589\) 31.5836 1.30138
\(590\) 0 0
\(591\) 11.0902 0.456189
\(592\) − 3.70820i − 0.152406i
\(593\) − 15.3820i − 0.631662i −0.948816 0.315831i \(-0.897717\pi\)
0.948816 0.315831i \(-0.102283\pi\)
\(594\) −1.85410 −0.0760747
\(595\) 0 0
\(596\) 3.09017 0.126578
\(597\) − 1.70820i − 0.0699121i
\(598\) − 4.70820i − 0.192533i
\(599\) −5.72949 −0.234101 −0.117050 0.993126i \(-0.537344\pi\)
−0.117050 + 0.993126i \(0.537344\pi\)
\(600\) 0 0
\(601\) −11.2918 −0.460602 −0.230301 0.973119i \(-0.573971\pi\)
−0.230301 + 0.973119i \(0.573971\pi\)
\(602\) − 11.8885i − 0.484541i
\(603\) − 9.18034i − 0.373852i
\(604\) −7.09017 −0.288495
\(605\) 0 0
\(606\) 7.18034 0.291681
\(607\) − 16.1459i − 0.655342i −0.944792 0.327671i \(-0.893736\pi\)
0.944792 0.327671i \(-0.106264\pi\)
\(608\) 37.6869i 1.52841i
\(609\) −2.76393 −0.112000
\(610\) 0 0
\(611\) 9.23607 0.373651
\(612\) − 0.381966i − 0.0154401i
\(613\) 46.1246i 1.86296i 0.363798 + 0.931478i \(0.381480\pi\)
−0.363798 + 0.931478i \(0.618520\pi\)
\(614\) 1.32624 0.0535226
\(615\) 0 0
\(616\) 13.4164 0.540562
\(617\) 20.7639i 0.835924i 0.908464 + 0.417962i \(0.137256\pi\)
−0.908464 + 0.417962i \(0.862744\pi\)
\(618\) 7.67376i 0.308684i
\(619\) −0.729490 −0.0293207 −0.0146603 0.999893i \(-0.504667\pi\)
−0.0146603 + 0.999893i \(0.504667\pi\)
\(620\) 0 0
\(621\) 7.61803 0.305701
\(622\) 13.8885i 0.556880i
\(623\) 22.3607i 0.895862i
\(624\) 1.85410 0.0742235
\(625\) 0 0
\(626\) −9.70820 −0.388018
\(627\) − 20.1246i − 0.803700i
\(628\) 6.23607i 0.248846i
\(629\) 0.472136 0.0188253
\(630\) 0 0
\(631\) −15.2361 −0.606538 −0.303269 0.952905i \(-0.598078\pi\)
−0.303269 + 0.952905i \(0.598078\pi\)
\(632\) − 35.4508i − 1.41016i
\(633\) 3.00000i 0.119239i
\(634\) 0.270510 0.0107433
\(635\) 0 0
\(636\) −10.9443 −0.433969
\(637\) 3.00000i 0.118864i
\(638\) 2.56231i 0.101443i
\(639\) 1.09017 0.0431265
\(640\) 0 0
\(641\) −7.67376 −0.303095 −0.151548 0.988450i \(-0.548426\pi\)
−0.151548 + 0.988450i \(0.548426\pi\)
\(642\) 4.85410i 0.191576i
\(643\) − 17.0902i − 0.673971i −0.941510 0.336985i \(-0.890593\pi\)
0.941510 0.336985i \(-0.109407\pi\)
\(644\) −24.6525 −0.971444
\(645\) 0 0
\(646\) −0.978714 −0.0385070
\(647\) 10.0344i 0.394495i 0.980354 + 0.197247i \(0.0632002\pi\)
−0.980354 + 0.197247i \(0.936800\pi\)
\(648\) − 2.23607i − 0.0878410i
\(649\) −41.8328 −1.64208
\(650\) 0 0
\(651\) 9.41641 0.369058
\(652\) 24.7082i 0.967648i
\(653\) 1.65248i 0.0646664i 0.999477 + 0.0323332i \(0.0102938\pi\)
−0.999477 + 0.0323332i \(0.989706\pi\)
\(654\) −6.70820 −0.262312
\(655\) 0 0
\(656\) −21.5410 −0.841036
\(657\) 2.29180i 0.0894115i
\(658\) 11.4164i 0.445058i
\(659\) −2.23607 −0.0871048 −0.0435524 0.999051i \(-0.513868\pi\)
−0.0435524 + 0.999051i \(0.513868\pi\)
\(660\) 0 0
\(661\) −30.8885 −1.20143 −0.600713 0.799465i \(-0.705116\pi\)
−0.600713 + 0.799465i \(0.705116\pi\)
\(662\) − 18.3475i − 0.713097i
\(663\) 0.236068i 0.00916812i
\(664\) −20.1246 −0.780986
\(665\) 0 0
\(666\) 1.23607 0.0478967
\(667\) − 10.5279i − 0.407641i
\(668\) − 11.0000i − 0.425603i
\(669\) 16.8541 0.651617
\(670\) 0 0
\(671\) 14.1246 0.545275
\(672\) 11.2361i 0.433441i
\(673\) − 24.7771i − 0.955087i −0.878608 0.477543i \(-0.841527\pi\)
0.878608 0.477543i \(-0.158473\pi\)
\(674\) −6.68692 −0.257570
\(675\) 0 0
\(676\) 19.4164 0.746785
\(677\) − 9.11146i − 0.350182i −0.984552 0.175091i \(-0.943978\pi\)
0.984552 0.175091i \(-0.0560219\pi\)
\(678\) − 5.09017i − 0.195487i
\(679\) −5.70820 −0.219061
\(680\) 0 0
\(681\) 10.2361 0.392247
\(682\) − 8.72949i − 0.334269i
\(683\) − 48.5967i − 1.85950i −0.368188 0.929751i \(-0.620022\pi\)
0.368188 0.929751i \(-0.379978\pi\)
\(684\) 10.8541 0.415017
\(685\) 0 0
\(686\) −12.3607 −0.471933
\(687\) 6.18034i 0.235795i
\(688\) 17.8328i 0.679870i
\(689\) 6.76393 0.257685
\(690\) 0 0
\(691\) 3.90983 0.148737 0.0743685 0.997231i \(-0.476306\pi\)
0.0743685 + 0.997231i \(0.476306\pi\)
\(692\) − 19.5623i − 0.743647i
\(693\) − 6.00000i − 0.227921i
\(694\) −13.1459 −0.499011
\(695\) 0 0
\(696\) −3.09017 −0.117133
\(697\) − 2.74265i − 0.103885i
\(698\) 1.70820i 0.0646565i
\(699\) 12.1803 0.460703
\(700\) 0 0
\(701\) −17.3475 −0.655207 −0.327603 0.944815i \(-0.606241\pi\)
−0.327603 + 0.944815i \(0.606241\pi\)
\(702\) 0.618034i 0.0233262i
\(703\) 13.4164i 0.506009i
\(704\) −0.708204 −0.0266914
\(705\) 0 0
\(706\) 9.03444 0.340016
\(707\) 23.2361i 0.873882i
\(708\) − 22.5623i − 0.847943i
\(709\) 29.7984 1.11910 0.559551 0.828796i \(-0.310974\pi\)
0.559551 + 0.828796i \(0.310974\pi\)
\(710\) 0 0
\(711\) −15.8541 −0.594575
\(712\) 25.0000i 0.936915i
\(713\) 35.8673i 1.34324i
\(714\) −0.291796 −0.0109202
\(715\) 0 0
\(716\) −25.3262 −0.946486
\(717\) 23.6180i 0.882032i
\(718\) 3.74265i 0.139674i
\(719\) −5.12461 −0.191116 −0.0955579 0.995424i \(-0.530464\pi\)
−0.0955579 + 0.995424i \(0.530464\pi\)
\(720\) 0 0
\(721\) −24.8328 −0.924822
\(722\) − 16.0689i − 0.598022i
\(723\) 8.32624i 0.309656i
\(724\) −5.70820 −0.212144
\(725\) 0 0
\(726\) 1.23607 0.0458748
\(727\) − 34.5623i − 1.28184i −0.767606 0.640922i \(-0.778552\pi\)
0.767606 0.640922i \(-0.221448\pi\)
\(728\) − 4.47214i − 0.165748i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −2.27051 −0.0839778
\(732\) 7.61803i 0.281571i
\(733\) 16.8541i 0.622520i 0.950325 + 0.311260i \(0.100751\pi\)
−0.950325 + 0.311260i \(0.899249\pi\)
\(734\) −13.2705 −0.489823
\(735\) 0 0
\(736\) −42.7984 −1.57757
\(737\) − 27.5410i − 1.01449i
\(738\) − 7.18034i − 0.264312i
\(739\) −11.7082 −0.430693 −0.215347 0.976538i \(-0.569088\pi\)
−0.215347 + 0.976538i \(0.569088\pi\)
\(740\) 0 0
\(741\) −6.70820 −0.246432
\(742\) 8.36068i 0.306930i
\(743\) 25.4721i 0.934482i 0.884130 + 0.467241i \(0.154752\pi\)
−0.884130 + 0.467241i \(0.845248\pi\)
\(744\) 10.5279 0.385970
\(745\) 0 0
\(746\) 5.81966 0.213073
\(747\) 9.00000i 0.329293i
\(748\) − 1.14590i − 0.0418982i
\(749\) −15.7082 −0.573965
\(750\) 0 0
\(751\) 28.7082 1.04758 0.523789 0.851848i \(-0.324518\pi\)
0.523789 + 0.851848i \(0.324518\pi\)
\(752\) − 17.1246i − 0.624470i
\(753\) − 27.9787i − 1.01960i
\(754\) 0.854102 0.0311046
\(755\) 0 0
\(756\) 3.23607 0.117695
\(757\) − 1.27051i − 0.0461775i −0.999733 0.0230887i \(-0.992650\pi\)
0.999733 0.0230887i \(-0.00735002\pi\)
\(758\) − 7.03444i − 0.255502i
\(759\) 22.8541 0.829551
\(760\) 0 0
\(761\) −19.1803 −0.695287 −0.347643 0.937627i \(-0.613018\pi\)
−0.347643 + 0.937627i \(0.613018\pi\)
\(762\) 10.9098i 0.395221i
\(763\) − 21.7082i − 0.785890i
\(764\) 2.70820 0.0979794
\(765\) 0 0
\(766\) −14.1803 −0.512357
\(767\) 13.9443i 0.503498i
\(768\) 6.56231i 0.236797i
\(769\) −26.3050 −0.948581 −0.474290 0.880368i \(-0.657295\pi\)
−0.474290 + 0.880368i \(0.657295\pi\)
\(770\) 0 0
\(771\) −20.2148 −0.728018
\(772\) 17.7984i 0.640577i
\(773\) − 29.7771i − 1.07101i −0.844533 0.535504i \(-0.820122\pi\)
0.844533 0.535504i \(-0.179878\pi\)
\(774\) −5.94427 −0.213662
\(775\) 0 0
\(776\) −6.38197 −0.229099
\(777\) 4.00000i 0.143499i
\(778\) − 18.9443i − 0.679185i
\(779\) 77.9361 2.79235
\(780\) 0 0
\(781\) 3.27051 0.117028
\(782\) − 1.11146i − 0.0397456i
\(783\) 1.38197i 0.0493874i
\(784\) 5.56231 0.198654
\(785\) 0 0
\(786\) −5.05573 −0.180332
\(787\) 23.8541i 0.850307i 0.905121 + 0.425153i \(0.139780\pi\)
−0.905121 + 0.425153i \(0.860220\pi\)
\(788\) 17.9443i 0.639238i
\(789\) −25.5066 −0.908058
\(790\) 0 0
\(791\) 16.4721 0.585682
\(792\) − 6.70820i − 0.238366i
\(793\) − 4.70820i − 0.167193i
\(794\) −7.09017 −0.251621
\(795\) 0 0
\(796\) 2.76393 0.0979650
\(797\) 46.0132i 1.62987i 0.579553 + 0.814935i \(0.303227\pi\)
−0.579553 + 0.814935i \(0.696773\pi\)
\(798\) − 8.29180i − 0.293526i
\(799\) 2.18034 0.0771349
\(800\) 0 0
\(801\) 11.1803 0.395038
\(802\) − 1.68692i − 0.0595671i
\(803\) 6.87539i 0.242627i
\(804\) 14.8541 0.523864
\(805\) 0 0
\(806\) −2.90983 −0.102494
\(807\) − 29.4721i − 1.03747i
\(808\) 25.9787i 0.913928i
\(809\) −24.9230 −0.876246 −0.438123 0.898915i \(-0.644356\pi\)
−0.438123 + 0.898915i \(0.644356\pi\)
\(810\) 0 0
\(811\) 37.7771 1.32653 0.663266 0.748383i \(-0.269170\pi\)
0.663266 + 0.748383i \(0.269170\pi\)
\(812\) − 4.47214i − 0.156941i
\(813\) − 15.4164i − 0.540677i
\(814\) 3.70820 0.129972
\(815\) 0 0
\(816\) 0.437694 0.0153224
\(817\) − 64.5197i − 2.25726i
\(818\) 21.7082i 0.759010i
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −11.9443 −0.416858 −0.208429 0.978038i \(-0.566835\pi\)
−0.208429 + 0.978038i \(0.566835\pi\)
\(822\) − 12.7082i − 0.443250i
\(823\) 8.43769i 0.294120i 0.989128 + 0.147060i \(0.0469810\pi\)
−0.989128 + 0.147060i \(0.953019\pi\)
\(824\) −27.7639 −0.967202
\(825\) 0 0
\(826\) −17.2361 −0.599720
\(827\) 2.02129i 0.0702870i 0.999382 + 0.0351435i \(0.0111888\pi\)
−0.999382 + 0.0351435i \(0.988811\pi\)
\(828\) 12.3262i 0.428366i
\(829\) 9.87539 0.342986 0.171493 0.985185i \(-0.445141\pi\)
0.171493 + 0.985185i \(0.445141\pi\)
\(830\) 0 0
\(831\) −30.9443 −1.07344
\(832\) 0.236068i 0.00818418i
\(833\) 0.708204i 0.0245378i
\(834\) −8.29180 −0.287121
\(835\) 0 0
\(836\) 32.5623 1.12619
\(837\) − 4.70820i − 0.162739i
\(838\) − 9.47214i − 0.327210i
\(839\) −48.2148 −1.66456 −0.832280 0.554356i \(-0.812965\pi\)
−0.832280 + 0.554356i \(0.812965\pi\)
\(840\) 0 0
\(841\) −27.0902 −0.934144
\(842\) − 8.87539i − 0.305866i
\(843\) − 8.18034i − 0.281746i
\(844\) −4.85410 −0.167085
\(845\) 0 0
\(846\) 5.70820 0.196252
\(847\) 4.00000i 0.137442i
\(848\) − 12.5410i − 0.430660i
\(849\) −15.7082 −0.539104
\(850\) 0 0
\(851\) −15.2361 −0.522286
\(852\) 1.76393i 0.0604313i
\(853\) − 53.3951i − 1.82821i −0.405473 0.914107i \(-0.632893\pi\)
0.405473 0.914107i \(-0.367107\pi\)
\(854\) 5.81966 0.199145
\(855\) 0 0
\(856\) −17.5623 −0.600267
\(857\) 26.9443i 0.920399i 0.887816 + 0.460199i \(0.152222\pi\)
−0.887816 + 0.460199i \(0.847778\pi\)
\(858\) 1.85410i 0.0632980i
\(859\) 25.1246 0.857241 0.428620 0.903485i \(-0.359000\pi\)
0.428620 + 0.903485i \(0.359000\pi\)
\(860\) 0 0
\(861\) 23.2361 0.791883
\(862\) − 21.1591i − 0.720680i
\(863\) 45.0689i 1.53416i 0.641550 + 0.767081i \(0.278292\pi\)
−0.641550 + 0.767081i \(0.721708\pi\)
\(864\) 5.61803 0.191129
\(865\) 0 0
\(866\) 3.38197 0.114924
\(867\) − 16.9443i − 0.575458i
\(868\) 15.2361i 0.517146i
\(869\) −47.5623 −1.61344
\(870\) 0 0
\(871\) −9.18034 −0.311064
\(872\) − 24.2705i − 0.821903i
\(873\) 2.85410i 0.0965967i
\(874\) 31.5836 1.06833
\(875\) 0 0
\(876\) −3.70820 −0.125289
\(877\) 2.87539i 0.0970950i 0.998821 + 0.0485475i \(0.0154592\pi\)
−0.998821 + 0.0485475i \(0.984541\pi\)
\(878\) − 1.83282i − 0.0618545i
\(879\) −9.32624 −0.314566
\(880\) 0 0
\(881\) 3.90983 0.131726 0.0658628 0.997829i \(-0.479020\pi\)
0.0658628 + 0.997829i \(0.479020\pi\)
\(882\) 1.85410i 0.0624309i
\(883\) − 53.7984i − 1.81046i −0.424923 0.905230i \(-0.639699\pi\)
0.424923 0.905230i \(-0.360301\pi\)
\(884\) −0.381966 −0.0128469
\(885\) 0 0
\(886\) −4.58359 −0.153989
\(887\) − 21.9230i − 0.736102i −0.929806 0.368051i \(-0.880025\pi\)
0.929806 0.368051i \(-0.119975\pi\)
\(888\) 4.47214i 0.150075i
\(889\) −35.3050 −1.18409
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 27.2705i 0.913084i
\(893\) 61.9574i 2.07333i
\(894\) −1.18034 −0.0394765
\(895\) 0 0
\(896\) −22.7639 −0.760490
\(897\) − 7.61803i − 0.254359i
\(898\) − 13.2918i − 0.443553i
\(899\) −6.50658 −0.217007
\(900\) 0 0
\(901\) 1.59675 0.0531954
\(902\) − 21.5410i − 0.717238i
\(903\) − 19.2361i − 0.640136i
\(904\) 18.4164 0.612521
\(905\) 0 0
\(906\) 2.70820 0.0899740
\(907\) − 17.0000i − 0.564476i −0.959344 0.282238i \(-0.908923\pi\)
0.959344 0.282238i \(-0.0910767\pi\)
\(908\) 16.5623i 0.549639i
\(909\) 11.6180 0.385346
\(910\) 0 0
\(911\) −20.8885 −0.692068 −0.346034 0.938222i \(-0.612472\pi\)
−0.346034 + 0.938222i \(0.612472\pi\)
\(912\) 12.4377i 0.411853i
\(913\) 27.0000i 0.893570i
\(914\) 16.0000 0.529233
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) − 16.3607i − 0.540277i
\(918\) 0.145898i 0.00481535i
\(919\) 5.00000 0.164935 0.0824674 0.996594i \(-0.473720\pi\)
0.0824674 + 0.996594i \(0.473720\pi\)
\(920\) 0 0
\(921\) 2.14590 0.0707097
\(922\) − 1.96556i − 0.0647322i
\(923\) − 1.09017i − 0.0358834i
\(924\) 9.70820 0.319376
\(925\) 0 0
\(926\) −16.4934 −0.542007
\(927\) 12.4164i 0.407808i
\(928\) − 7.76393i − 0.254864i
\(929\) −19.5967 −0.642948 −0.321474 0.946918i \(-0.604178\pi\)
−0.321474 + 0.946918i \(0.604178\pi\)
\(930\) 0 0
\(931\) −20.1246 −0.659558
\(932\) 19.7082i 0.645564i
\(933\) 22.4721i 0.735705i
\(934\) 10.1459 0.331984
\(935\) 0 0
\(936\) −2.23607 −0.0730882
\(937\) 16.4164i 0.536301i 0.963377 + 0.268150i \(0.0864124\pi\)
−0.963377 + 0.268150i \(0.913588\pi\)
\(938\) − 11.3475i − 0.370510i
\(939\) −15.7082 −0.512618
\(940\) 0 0
\(941\) 11.0213 0.359284 0.179642 0.983732i \(-0.442506\pi\)
0.179642 + 0.983732i \(0.442506\pi\)
\(942\) − 2.38197i − 0.0776086i
\(943\) 88.5066i 2.88217i
\(944\) 25.8541 0.841479
\(945\) 0 0
\(946\) −17.8328 −0.579795
\(947\) 29.8328i 0.969436i 0.874670 + 0.484718i \(0.161078\pi\)
−0.874670 + 0.484718i \(0.838922\pi\)
\(948\) − 25.6525i − 0.833154i
\(949\) 2.29180 0.0743948
\(950\) 0 0
\(951\) 0.437694 0.0141932
\(952\) − 1.05573i − 0.0342163i
\(953\) 59.9443i 1.94179i 0.239515 + 0.970893i \(0.423012\pi\)
−0.239515 + 0.970893i \(0.576988\pi\)
\(954\) 4.18034 0.135344
\(955\) 0 0
\(956\) −38.2148 −1.23595
\(957\) 4.14590i 0.134018i
\(958\) − 12.2361i − 0.395329i
\(959\) 41.1246 1.32798
\(960\) 0 0
\(961\) −8.83282 −0.284930
\(962\) − 1.23607i − 0.0398524i
\(963\) 7.85410i 0.253095i
\(964\) −13.4721 −0.433908
\(965\) 0 0
\(966\) 9.41641 0.302968
\(967\) − 8.58359i − 0.276030i −0.990430 0.138015i \(-0.955928\pi\)
0.990430 0.138015i \(-0.0440722\pi\)
\(968\) 4.47214i 0.143740i
\(969\) −1.58359 −0.0508723
\(970\) 0 0
\(971\) −5.88854 −0.188972 −0.0944862 0.995526i \(-0.530121\pi\)
−0.0944862 + 0.995526i \(0.530121\pi\)
\(972\) − 1.61803i − 0.0518985i
\(973\) − 26.8328i − 0.860221i
\(974\) 8.88854 0.284807
\(975\) 0 0
\(976\) −8.72949 −0.279424
\(977\) − 6.34752i − 0.203075i −0.994832 0.101538i \(-0.967624\pi\)
0.994832 0.101538i \(-0.0323762\pi\)
\(978\) − 9.43769i − 0.301784i
\(979\) 33.5410 1.07198
\(980\) 0 0
\(981\) −10.8541 −0.346545
\(982\) − 4.12461i − 0.131622i
\(983\) 22.3820i 0.713874i 0.934128 + 0.356937i \(0.116179\pi\)
−0.934128 + 0.356937i \(0.883821\pi\)
\(984\) 25.9787 0.828171
\(985\) 0 0
\(986\) 0.201626 0.00642108
\(987\) 18.4721i 0.587975i
\(988\) − 10.8541i − 0.345315i
\(989\) 73.2705 2.32987
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 26.4508i 0.839815i
\(993\) − 29.6869i − 0.942086i
\(994\) 1.34752 0.0427409
\(995\) 0 0
\(996\) −14.5623 −0.461424
\(997\) − 61.0689i − 1.93407i −0.254642 0.967035i \(-0.581958\pi\)
0.254642 0.967035i \(-0.418042\pi\)
\(998\) − 9.27051i − 0.293453i
\(999\) 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.1249.2 4
5.2 odd 4 1875.2.a.b.1.2 2
5.3 odd 4 1875.2.a.c.1.1 yes 2
5.4 even 2 inner 1875.2.b.a.1249.3 4
15.2 even 4 5625.2.a.g.1.1 2
15.8 even 4 5625.2.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.b.1.2 2 5.2 odd 4
1875.2.a.c.1.1 yes 2 5.3 odd 4
1875.2.b.a.1249.2 4 1.1 even 1 trivial
1875.2.b.a.1249.3 4 5.4 even 2 inner
5625.2.a.b.1.2 2 15.8 even 4
5625.2.a.g.1.1 2 15.2 even 4