Properties

Label 1875.2.a.b.1.2
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 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +2.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +2.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.61803 q^{12} +1.00000 q^{13} +1.23607 q^{14} +1.85410 q^{16} -0.236068 q^{17} +0.618034 q^{18} +6.70820 q^{19} -2.00000 q^{21} -1.85410 q^{22} -7.61803 q^{23} +2.23607 q^{24} +0.618034 q^{26} -1.00000 q^{27} -3.23607 q^{28} -1.38197 q^{29} -4.70820 q^{31} +5.61803 q^{32} +3.00000 q^{33} -0.145898 q^{34} -1.61803 q^{36} +2.00000 q^{37} +4.14590 q^{38} -1.00000 q^{39} -11.6180 q^{41} -1.23607 q^{42} +9.61803 q^{43} +4.85410 q^{44} -4.70820 q^{46} +9.23607 q^{47} -1.85410 q^{48} -3.00000 q^{49} +0.236068 q^{51} -1.61803 q^{52} -6.76393 q^{53} -0.618034 q^{54} -4.47214 q^{56} -6.70820 q^{57} -0.854102 q^{58} -13.9443 q^{59} -4.70820 q^{61} -2.90983 q^{62} +2.00000 q^{63} -0.236068 q^{64} +1.85410 q^{66} -9.18034 q^{67} +0.381966 q^{68} +7.61803 q^{69} -1.09017 q^{71} -2.23607 q^{72} -2.29180 q^{73} +1.23607 q^{74} -10.8541 q^{76} -6.00000 q^{77} -0.618034 q^{78} -15.8541 q^{79} +1.00000 q^{81} -7.18034 q^{82} -9.00000 q^{83} +3.23607 q^{84} +5.94427 q^{86} +1.38197 q^{87} +6.70820 q^{88} +11.1803 q^{89} +2.00000 q^{91} +12.3262 q^{92} +4.70820 q^{93} +5.70820 q^{94} -5.61803 q^{96} +2.85410 q^{97} -1.85410 q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) −0.618034 −0.252311
\(7\) 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 \(0.649385\pi\)
\(12\) 1.61803 0.467086
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 1.23607 0.330353
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) −0.236068 −0.0572549 −0.0286274 0.999590i \(-0.509114\pi\)
−0.0286274 + 0.999590i \(0.509114\pi\)
\(18\) 0.618034 0.145672
\(19\) 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
\(22\) −1.85410 −0.395296
\(23\) −7.61803 −1.58847 −0.794235 0.607611i \(-0.792128\pi\)
−0.794235 + 0.607611i \(0.792128\pi\)
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) 0.618034 0.121206
\(27\) −1.00000 −0.192450
\(28\) −3.23607 −0.611559
\(29\) −1.38197 −0.256625 −0.128312 0.991734i \(-0.540956\pi\)
−0.128312 + 0.991734i \(0.540956\pi\)
\(30\) 0 0
\(31\) −4.70820 −0.845618 −0.422809 0.906219i \(-0.638956\pi\)
−0.422809 + 0.906219i \(0.638956\pi\)
\(32\) 5.61803 0.993137
\(33\) 3.00000 0.522233
\(34\) −0.145898 −0.0250213
\(35\) 0 0
\(36\) −1.61803 −0.269672
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.14590 0.672553
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −11.6180 −1.81443 −0.907216 0.420665i \(-0.861797\pi\)
−0.907216 + 0.420665i \(0.861797\pi\)
\(42\) −1.23607 −0.190729
\(43\) 9.61803 1.46674 0.733368 0.679832i \(-0.237947\pi\)
0.733368 + 0.679832i \(0.237947\pi\)
\(44\) 4.85410 0.731783
\(45\) 0 0
\(46\) −4.70820 −0.694187
\(47\) 9.23607 1.34722 0.673609 0.739087i \(-0.264743\pi\)
0.673609 + 0.739087i \(0.264743\pi\)
\(48\) −1.85410 −0.267617
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0.236068 0.0330561
\(52\) −1.61803 −0.224381
\(53\) −6.76393 −0.929098 −0.464549 0.885548i \(-0.653783\pi\)
−0.464549 + 0.885548i \(0.653783\pi\)
\(54\) −0.618034 −0.0841038
\(55\) 0 0
\(56\) −4.47214 −0.597614
\(57\) −6.70820 −0.888523
\(58\) −0.854102 −0.112149
\(59\) −13.9443 −1.81539 −0.907695 0.419631i \(-0.862159\pi\)
−0.907695 + 0.419631i \(0.862159\pi\)
\(60\) 0 0
\(61\) −4.70820 −0.602824 −0.301412 0.953494i \(-0.597458\pi\)
−0.301412 + 0.953494i \(0.597458\pi\)
\(62\) −2.90983 −0.369549
\(63\) 2.00000 0.251976
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) 1.85410 0.228224
\(67\) −9.18034 −1.12156 −0.560779 0.827966i \(-0.689498\pi\)
−0.560779 + 0.827966i \(0.689498\pi\)
\(68\) 0.381966 0.0463202
\(69\) 7.61803 0.917104
\(70\) 0 0
\(71\) −1.09017 −0.129379 −0.0646897 0.997905i \(-0.520606\pi\)
−0.0646897 + 0.997905i \(0.520606\pi\)
\(72\) −2.23607 −0.263523
\(73\) −2.29180 −0.268234 −0.134117 0.990965i \(-0.542820\pi\)
−0.134117 + 0.990965i \(0.542820\pi\)
\(74\) 1.23607 0.143690
\(75\) 0 0
\(76\) −10.8541 −1.24505
\(77\) −6.00000 −0.683763
\(78\) −0.618034 −0.0699786
\(79\) −15.8541 −1.78373 −0.891863 0.452306i \(-0.850602\pi\)
−0.891863 + 0.452306i \(0.850602\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −7.18034 −0.792936
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 3.23607 0.353084
\(85\) 0 0
\(86\) 5.94427 0.640987
\(87\) 1.38197 0.148162
\(88\) 6.70820 0.715097
\(89\) 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
\(92\) 12.3262 1.28510
\(93\) 4.70820 0.488218
\(94\) 5.70820 0.588756
\(95\) 0 0
\(96\) −5.61803 −0.573388
\(97\) 2.85410 0.289790 0.144895 0.989447i \(-0.453716\pi\)
0.144895 + 0.989447i \(0.453716\pi\)
\(98\) −1.85410 −0.187293
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −11.6180 −1.15604 −0.578019 0.816023i \(-0.696174\pi\)
−0.578019 + 0.816023i \(0.696174\pi\)
\(102\) 0.145898 0.0144461
\(103\) −12.4164 −1.22343 −0.611713 0.791080i \(-0.709519\pi\)
−0.611713 + 0.791080i \(0.709519\pi\)
\(104\) −2.23607 −0.219265
\(105\) 0 0
\(106\) −4.18034 −0.406031
\(107\) 7.85410 0.759285 0.379642 0.925133i \(-0.376047\pi\)
0.379642 + 0.925133i \(0.376047\pi\)
\(108\) 1.61803 0.155695
\(109\) −10.8541 −1.03963 −0.519817 0.854278i \(-0.674000\pi\)
−0.519817 + 0.854278i \(0.674000\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 3.70820 0.350392
\(113\) 8.23607 0.774784 0.387392 0.921915i \(-0.373376\pi\)
0.387392 + 0.921915i \(0.373376\pi\)
\(114\) −4.14590 −0.388299
\(115\) 0 0
\(116\) 2.23607 0.207614
\(117\) 1.00000 0.0924500
\(118\) −8.61803 −0.793354
\(119\) −0.472136 −0.0432806
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −2.90983 −0.263444
\(123\) 11.6180 1.04756
\(124\) 7.61803 0.684120
\(125\) 0 0
\(126\) 1.23607 0.110118
\(127\) 17.6525 1.56640 0.783202 0.621767i \(-0.213585\pi\)
0.783202 + 0.621767i \(0.213585\pi\)
\(128\) −11.3820 −1.00603
\(129\) −9.61803 −0.846821
\(130\) 0 0
\(131\) 8.18034 0.714720 0.357360 0.933967i \(-0.383677\pi\)
0.357360 + 0.933967i \(0.383677\pi\)
\(132\) −4.85410 −0.422495
\(133\) 13.4164 1.16335
\(134\) −5.67376 −0.490138
\(135\) 0 0
\(136\) 0.527864 0.0452640
\(137\) −20.5623 −1.75676 −0.878378 0.477966i \(-0.841374\pi\)
−0.878378 + 0.477966i \(0.841374\pi\)
\(138\) 4.70820 0.400789
\(139\) −13.4164 −1.13796 −0.568982 0.822350i \(-0.692663\pi\)
−0.568982 + 0.822350i \(0.692663\pi\)
\(140\) 0 0
\(141\) −9.23607 −0.777817
\(142\) −0.673762 −0.0565409
\(143\) −3.00000 −0.250873
\(144\) 1.85410 0.154508
\(145\) 0 0
\(146\) −1.41641 −0.117223
\(147\) 3.00000 0.247436
\(148\) −3.23607 −0.266003
\(149\) −1.90983 −0.156459 −0.0782297 0.996935i \(-0.524927\pi\)
−0.0782297 + 0.996935i \(0.524927\pi\)
\(150\) 0 0
\(151\) −4.38197 −0.356599 −0.178300 0.983976i \(-0.557060\pi\)
−0.178300 + 0.983976i \(0.557060\pi\)
\(152\) −15.0000 −1.21666
\(153\) −0.236068 −0.0190850
\(154\) −3.70820 −0.298816
\(155\) 0 0
\(156\) 1.61803 0.129546
\(157\) −3.85410 −0.307591 −0.153795 0.988103i \(-0.549150\pi\)
−0.153795 + 0.988103i \(0.549150\pi\)
\(158\) −9.79837 −0.779517
\(159\) 6.76393 0.536415
\(160\) 0 0
\(161\) −15.2361 −1.20077
\(162\) 0.618034 0.0485573
\(163\) 15.2705 1.19608 0.598039 0.801467i \(-0.295947\pi\)
0.598039 + 0.801467i \(0.295947\pi\)
\(164\) 18.7984 1.46791
\(165\) 0 0
\(166\) −5.56231 −0.431719
\(167\) 6.79837 0.526074 0.263037 0.964786i \(-0.415276\pi\)
0.263037 + 0.964786i \(0.415276\pi\)
\(168\) 4.47214 0.345033
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 6.70820 0.512989
\(172\) −15.5623 −1.18661
\(173\) −12.0902 −0.919199 −0.459599 0.888126i \(-0.652007\pi\)
−0.459599 + 0.888126i \(0.652007\pi\)
\(174\) 0.854102 0.0647493
\(175\) 0 0
\(176\) −5.56231 −0.419275
\(177\) 13.9443 1.04812
\(178\) 6.90983 0.517914
\(179\) 15.6525 1.16992 0.584960 0.811062i \(-0.301110\pi\)
0.584960 + 0.811062i \(0.301110\pi\)
\(180\) 0 0
\(181\) −3.52786 −0.262224 −0.131112 0.991368i \(-0.541855\pi\)
−0.131112 + 0.991368i \(0.541855\pi\)
\(182\) 1.23607 0.0916235
\(183\) 4.70820 0.348040
\(184\) 17.0344 1.25580
\(185\) 0 0
\(186\) 2.90983 0.213359
\(187\) 0.708204 0.0517890
\(188\) −14.9443 −1.08992
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 1.67376 0.121109 0.0605546 0.998165i \(-0.480713\pi\)
0.0605546 + 0.998165i \(0.480713\pi\)
\(192\) 0.236068 0.0170367
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 1.76393 0.126643
\(195\) 0 0
\(196\) 4.85410 0.346722
\(197\) −11.0902 −0.790142 −0.395071 0.918651i \(-0.629280\pi\)
−0.395071 + 0.918651i \(0.629280\pi\)
\(198\) −1.85410 −0.131765
\(199\) −1.70820 −0.121091 −0.0605457 0.998165i \(-0.519284\pi\)
−0.0605457 + 0.998165i \(0.519284\pi\)
\(200\) 0 0
\(201\) 9.18034 0.647531
\(202\) −7.18034 −0.505207
\(203\) −2.76393 −0.193990
\(204\) −0.381966 −0.0267430
\(205\) 0 0
\(206\) −7.67376 −0.534656
\(207\) −7.61803 −0.529490
\(208\) 1.85410 0.128559
\(209\) −20.1246 −1.39205
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) 10.9443 0.751656
\(213\) 1.09017 0.0746972
\(214\) 4.85410 0.331820
\(215\) 0 0
\(216\) 2.23607 0.152145
\(217\) −9.41641 −0.639227
\(218\) −6.70820 −0.454337
\(219\) 2.29180 0.154865
\(220\) 0 0
\(221\) −0.236068 −0.0158797
\(222\) −1.23607 −0.0829595
\(223\) 16.8541 1.12863 0.564317 0.825558i \(-0.309140\pi\)
0.564317 + 0.825558i \(0.309140\pi\)
\(224\) 11.2361 0.750741
\(225\) 0 0
\(226\) 5.09017 0.338593
\(227\) −10.2361 −0.679392 −0.339696 0.940535i \(-0.610324\pi\)
−0.339696 + 0.940535i \(0.610324\pi\)
\(228\) 10.8541 0.718830
\(229\) 6.18034 0.408408 0.204204 0.978928i \(-0.434539\pi\)
0.204204 + 0.978928i \(0.434539\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 3.09017 0.202880
\(233\) 12.1803 0.797961 0.398980 0.916959i \(-0.369364\pi\)
0.398980 + 0.916959i \(0.369364\pi\)
\(234\) 0.618034 0.0404021
\(235\) 0 0
\(236\) 22.5623 1.46868
\(237\) 15.8541 1.02983
\(238\) −0.291796 −0.0189143
\(239\) 23.6180 1.52772 0.763862 0.645380i \(-0.223301\pi\)
0.763862 + 0.645380i \(0.223301\pi\)
\(240\) 0 0
\(241\) −8.32624 −0.536340 −0.268170 0.963372i \(-0.586419\pi\)
−0.268170 + 0.963372i \(0.586419\pi\)
\(242\) −1.23607 −0.0794575
\(243\) −1.00000 −0.0641500
\(244\) 7.61803 0.487695
\(245\) 0 0
\(246\) 7.18034 0.457802
\(247\) 6.70820 0.426833
\(248\) 10.5279 0.668520
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) 27.9787 1.76600 0.883000 0.469372i \(-0.155520\pi\)
0.883000 + 0.469372i \(0.155520\pi\)
\(252\) −3.23607 −0.203853
\(253\) 22.8541 1.43683
\(254\) 10.9098 0.684544
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) 20.2148 1.26096 0.630482 0.776204i \(-0.282857\pi\)
0.630482 + 0.776204i \(0.282857\pi\)
\(258\) −5.94427 −0.370074
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −1.38197 −0.0855415
\(262\) 5.05573 0.312344
\(263\) −25.5066 −1.57280 −0.786401 0.617716i \(-0.788058\pi\)
−0.786401 + 0.617716i \(0.788058\pi\)
\(264\) −6.70820 −0.412861
\(265\) 0 0
\(266\) 8.29180 0.508403
\(267\) −11.1803 −0.684226
\(268\) 14.8541 0.907359
\(269\) −29.4721 −1.79695 −0.898474 0.439027i \(-0.855323\pi\)
−0.898474 + 0.439027i \(0.855323\pi\)
\(270\) 0 0
\(271\) 15.4164 0.936480 0.468240 0.883601i \(-0.344888\pi\)
0.468240 + 0.883601i \(0.344888\pi\)
\(272\) −0.437694 −0.0265391
\(273\) −2.00000 −0.121046
\(274\) −12.7082 −0.767731
\(275\) 0 0
\(276\) −12.3262 −0.741952
\(277\) 30.9443 1.85926 0.929631 0.368493i \(-0.120126\pi\)
0.929631 + 0.368493i \(0.120126\pi\)
\(278\) −8.29180 −0.497309
\(279\) −4.70820 −0.281873
\(280\) 0 0
\(281\) 8.18034 0.487998 0.243999 0.969775i \(-0.421541\pi\)
0.243999 + 0.969775i \(0.421541\pi\)
\(282\) −5.70820 −0.339919
\(283\) −15.7082 −0.933756 −0.466878 0.884322i \(-0.654621\pi\)
−0.466878 + 0.884322i \(0.654621\pi\)
\(284\) 1.76393 0.104670
\(285\) 0 0
\(286\) −1.85410 −0.109635
\(287\) −23.2361 −1.37158
\(288\) 5.61803 0.331046
\(289\) −16.9443 −0.996722
\(290\) 0 0
\(291\) −2.85410 −0.167310
\(292\) 3.70820 0.217006
\(293\) −9.32624 −0.544845 −0.272422 0.962178i \(-0.587825\pi\)
−0.272422 + 0.962178i \(0.587825\pi\)
\(294\) 1.85410 0.108133
\(295\) 0 0
\(296\) −4.47214 −0.259938
\(297\) 3.00000 0.174078
\(298\) −1.18034 −0.0683753
\(299\) −7.61803 −0.440562
\(300\) 0 0
\(301\) 19.2361 1.10875
\(302\) −2.70820 −0.155840
\(303\) 11.6180 0.667439
\(304\) 12.4377 0.713351
\(305\) 0 0
\(306\) −0.145898 −0.00834044
\(307\) −2.14590 −0.122473 −0.0612364 0.998123i \(-0.519504\pi\)
−0.0612364 + 0.998123i \(0.519504\pi\)
\(308\) 9.70820 0.553176
\(309\) 12.4164 0.706345
\(310\) 0 0
\(311\) −22.4721 −1.27428 −0.637139 0.770749i \(-0.719882\pi\)
−0.637139 + 0.770749i \(0.719882\pi\)
\(312\) 2.23607 0.126592
\(313\) −15.7082 −0.887880 −0.443940 0.896056i \(-0.646420\pi\)
−0.443940 + 0.896056i \(0.646420\pi\)
\(314\) −2.38197 −0.134422
\(315\) 0 0
\(316\) 25.6525 1.44306
\(317\) −0.437694 −0.0245833 −0.0122917 0.999924i \(-0.503913\pi\)
−0.0122917 + 0.999924i \(0.503913\pi\)
\(318\) 4.18034 0.234422
\(319\) 4.14590 0.232126
\(320\) 0 0
\(321\) −7.85410 −0.438373
\(322\) −9.41641 −0.524756
\(323\) −1.58359 −0.0881134
\(324\) −1.61803 −0.0898908
\(325\) 0 0
\(326\) 9.43769 0.522706
\(327\) 10.8541 0.600233
\(328\) 25.9787 1.43443
\(329\) 18.4721 1.01840
\(330\) 0 0
\(331\) 29.6869 1.63174 0.815870 0.578235i \(-0.196258\pi\)
0.815870 + 0.578235i \(0.196258\pi\)
\(332\) 14.5623 0.799210
\(333\) 2.00000 0.109599
\(334\) 4.20163 0.229903
\(335\) 0 0
\(336\) −3.70820 −0.202299
\(337\) 10.8197 0.589384 0.294692 0.955592i \(-0.404783\pi\)
0.294692 + 0.955592i \(0.404783\pi\)
\(338\) −7.41641 −0.403399
\(339\) −8.23607 −0.447322
\(340\) 0 0
\(341\) 14.1246 0.764891
\(342\) 4.14590 0.224184
\(343\) −20.0000 −1.07990
\(344\) −21.5066 −1.15956
\(345\) 0 0
\(346\) −7.47214 −0.401705
\(347\) 21.2705 1.14186 0.570930 0.820998i \(-0.306583\pi\)
0.570930 + 0.820998i \(0.306583\pi\)
\(348\) −2.23607 −0.119866
\(349\) 2.76393 0.147950 0.0739749 0.997260i \(-0.476432\pi\)
0.0739749 + 0.997260i \(0.476432\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −16.8541 −0.898327
\(353\) 14.6180 0.778039 0.389020 0.921229i \(-0.372814\pi\)
0.389020 + 0.921229i \(0.372814\pi\)
\(354\) 8.61803 0.458043
\(355\) 0 0
\(356\) −18.0902 −0.958777
\(357\) 0.472136 0.0249881
\(358\) 9.67376 0.511274
\(359\) 6.05573 0.319609 0.159805 0.987149i \(-0.448914\pi\)
0.159805 + 0.987149i \(0.448914\pi\)
\(360\) 0 0
\(361\) 26.0000 1.36842
\(362\) −2.18034 −0.114596
\(363\) 2.00000 0.104973
\(364\) −3.23607 −0.169616
\(365\) 0 0
\(366\) 2.90983 0.152099
\(367\) 21.4721 1.12084 0.560418 0.828210i \(-0.310640\pi\)
0.560418 + 0.828210i \(0.310640\pi\)
\(368\) −14.1246 −0.736296
\(369\) −11.6180 −0.604811
\(370\) 0 0
\(371\) −13.5279 −0.702332
\(372\) −7.61803 −0.394977
\(373\) 9.41641 0.487563 0.243782 0.969830i \(-0.421612\pi\)
0.243782 + 0.969830i \(0.421612\pi\)
\(374\) 0.437694 0.0226326
\(375\) 0 0
\(376\) −20.6525 −1.06507
\(377\) −1.38197 −0.0711749
\(378\) −1.23607 −0.0635765
\(379\) −11.3820 −0.584652 −0.292326 0.956319i \(-0.594429\pi\)
−0.292326 + 0.956319i \(0.594429\pi\)
\(380\) 0 0
\(381\) −17.6525 −0.904364
\(382\) 1.03444 0.0529266
\(383\) −22.9443 −1.17240 −0.586199 0.810167i \(-0.699376\pi\)
−0.586199 + 0.810167i \(0.699376\pi\)
\(384\) 11.3820 0.580834
\(385\) 0 0
\(386\) 6.79837 0.346028
\(387\) 9.61803 0.488912
\(388\) −4.61803 −0.234445
\(389\) −30.6525 −1.55414 −0.777071 0.629413i \(-0.783295\pi\)
−0.777071 + 0.629413i \(0.783295\pi\)
\(390\) 0 0
\(391\) 1.79837 0.0909477
\(392\) 6.70820 0.338815
\(393\) −8.18034 −0.412644
\(394\) −6.85410 −0.345305
\(395\) 0 0
\(396\) 4.85410 0.243928
\(397\) 11.4721 0.575770 0.287885 0.957665i \(-0.407048\pi\)
0.287885 + 0.957665i \(0.407048\pi\)
\(398\) −1.05573 −0.0529189
\(399\) −13.4164 −0.671660
\(400\) 0 0
\(401\) 2.72949 0.136304 0.0681521 0.997675i \(-0.478290\pi\)
0.0681521 + 0.997675i \(0.478290\pi\)
\(402\) 5.67376 0.282982
\(403\) −4.70820 −0.234532
\(404\) 18.7984 0.935254
\(405\) 0 0
\(406\) −1.70820 −0.0847767
\(407\) −6.00000 −0.297409
\(408\) −0.527864 −0.0261332
\(409\) 35.1246 1.73680 0.868400 0.495864i \(-0.165149\pi\)
0.868400 + 0.495864i \(0.165149\pi\)
\(410\) 0 0
\(411\) 20.5623 1.01426
\(412\) 20.0902 0.989772
\(413\) −27.8885 −1.37231
\(414\) −4.70820 −0.231396
\(415\) 0 0
\(416\) 5.61803 0.275447
\(417\) 13.4164 0.657004
\(418\) −12.4377 −0.608348
\(419\) −15.3262 −0.748736 −0.374368 0.927280i \(-0.622140\pi\)
−0.374368 + 0.927280i \(0.622140\pi\)
\(420\) 0 0
\(421\) 14.3607 0.699897 0.349948 0.936769i \(-0.386199\pi\)
0.349948 + 0.936769i \(0.386199\pi\)
\(422\) −1.85410 −0.0902563
\(423\) 9.23607 0.449073
\(424\) 15.1246 0.734516
\(425\) 0 0
\(426\) 0.673762 0.0326439
\(427\) −9.41641 −0.455692
\(428\) −12.7082 −0.614274
\(429\) 3.00000 0.144841
\(430\) 0 0
\(431\) 34.2361 1.64909 0.824547 0.565794i \(-0.191430\pi\)
0.824547 + 0.565794i \(0.191430\pi\)
\(432\) −1.85410 −0.0892055
\(433\) 5.47214 0.262974 0.131487 0.991318i \(-0.458025\pi\)
0.131487 + 0.991318i \(0.458025\pi\)
\(434\) −5.81966 −0.279353
\(435\) 0 0
\(436\) 17.5623 0.841082
\(437\) −51.1033 −2.44460
\(438\) 1.41641 0.0676786
\(439\) −2.96556 −0.141538 −0.0707692 0.997493i \(-0.522545\pi\)
−0.0707692 + 0.997493i \(0.522545\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −0.145898 −0.00693966
\(443\) −7.41641 −0.352364 −0.176182 0.984358i \(-0.556375\pi\)
−0.176182 + 0.984358i \(0.556375\pi\)
\(444\) 3.23607 0.153577
\(445\) 0 0
\(446\) 10.4164 0.493231
\(447\) 1.90983 0.0903319
\(448\) −0.472136 −0.0223063
\(449\) −21.5066 −1.01496 −0.507479 0.861664i \(-0.669423\pi\)
−0.507479 + 0.861664i \(0.669423\pi\)
\(450\) 0 0
\(451\) 34.8541 1.64122
\(452\) −13.3262 −0.626814
\(453\) 4.38197 0.205883
\(454\) −6.32624 −0.296905
\(455\) 0 0
\(456\) 15.0000 0.702439
\(457\) −25.8885 −1.21102 −0.605508 0.795840i \(-0.707030\pi\)
−0.605508 + 0.795840i \(0.707030\pi\)
\(458\) 3.81966 0.178481
\(459\) 0.236068 0.0110187
\(460\) 0 0
\(461\) 3.18034 0.148123 0.0740616 0.997254i \(-0.476404\pi\)
0.0740616 + 0.997254i \(0.476404\pi\)
\(462\) 3.70820 0.172521
\(463\) −26.6869 −1.24025 −0.620123 0.784505i \(-0.712917\pi\)
−0.620123 + 0.784505i \(0.712917\pi\)
\(464\) −2.56231 −0.118952
\(465\) 0 0
\(466\) 7.52786 0.348722
\(467\) −16.4164 −0.759661 −0.379830 0.925056i \(-0.624018\pi\)
−0.379830 + 0.925056i \(0.624018\pi\)
\(468\) −1.61803 −0.0747936
\(469\) −18.3607 −0.847817
\(470\) 0 0
\(471\) 3.85410 0.177588
\(472\) 31.1803 1.43519
\(473\) −28.8541 −1.32671
\(474\) 9.79837 0.450054
\(475\) 0 0
\(476\) 0.763932 0.0350148
\(477\) −6.76393 −0.309699
\(478\) 14.5967 0.667640
\(479\) −19.7984 −0.904611 −0.452305 0.891863i \(-0.649398\pi\)
−0.452305 + 0.891863i \(0.649398\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) −5.14590 −0.234389
\(483\) 15.2361 0.693265
\(484\) 3.23607 0.147094
\(485\) 0 0
\(486\) −0.618034 −0.0280346
\(487\) −14.3820 −0.651709 −0.325855 0.945420i \(-0.605652\pi\)
−0.325855 + 0.945420i \(0.605652\pi\)
\(488\) 10.5279 0.476574
\(489\) −15.2705 −0.690556
\(490\) 0 0
\(491\) 6.67376 0.301183 0.150591 0.988596i \(-0.451882\pi\)
0.150591 + 0.988596i \(0.451882\pi\)
\(492\) −18.7984 −0.847496
\(493\) 0.326238 0.0146930
\(494\) 4.14590 0.186533
\(495\) 0 0
\(496\) −8.72949 −0.391966
\(497\) −2.18034 −0.0978016
\(498\) 5.56231 0.249253
\(499\) −15.0000 −0.671492 −0.335746 0.941953i \(-0.608988\pi\)
−0.335746 + 0.941953i \(0.608988\pi\)
\(500\) 0 0
\(501\) −6.79837 −0.303729
\(502\) 17.2918 0.771771
\(503\) 33.0344 1.47293 0.736466 0.676474i \(-0.236493\pi\)
0.736466 + 0.676474i \(0.236493\pi\)
\(504\) −4.47214 −0.199205
\(505\) 0 0
\(506\) 14.1246 0.627916
\(507\) 12.0000 0.532939
\(508\) −28.5623 −1.26725
\(509\) −2.88854 −0.128032 −0.0640162 0.997949i \(-0.520391\pi\)
−0.0640162 + 0.997949i \(0.520391\pi\)
\(510\) 0 0
\(511\) −4.58359 −0.202766
\(512\) 18.7082 0.826794
\(513\) −6.70820 −0.296174
\(514\) 12.4934 0.551061
\(515\) 0 0
\(516\) 15.5623 0.685092
\(517\) −27.7082 −1.21861
\(518\) 2.47214 0.108619
\(519\) 12.0902 0.530700
\(520\) 0 0
\(521\) 28.9098 1.26656 0.633281 0.773922i \(-0.281708\pi\)
0.633281 + 0.773922i \(0.281708\pi\)
\(522\) −0.854102 −0.0373830
\(523\) 18.5623 0.811673 0.405836 0.913946i \(-0.366980\pi\)
0.405836 + 0.913946i \(0.366980\pi\)
\(524\) −13.2361 −0.578220
\(525\) 0 0
\(526\) −15.7639 −0.687340
\(527\) 1.11146 0.0484158
\(528\) 5.56231 0.242068
\(529\) 35.0344 1.52324
\(530\) 0 0
\(531\) −13.9443 −0.605130
\(532\) −21.7082 −0.941170
\(533\) −11.6180 −0.503233
\(534\) −6.90983 −0.299018
\(535\) 0 0
\(536\) 20.5279 0.886669
\(537\) −15.6525 −0.675454
\(538\) −18.2148 −0.785295
\(539\) 9.00000 0.387657
\(540\) 0 0
\(541\) −39.7082 −1.70719 −0.853595 0.520938i \(-0.825582\pi\)
−0.853595 + 0.520938i \(0.825582\pi\)
\(542\) 9.52786 0.409257
\(543\) 3.52786 0.151395
\(544\) −1.32624 −0.0568620
\(545\) 0 0
\(546\) −1.23607 −0.0528988
\(547\) −11.2918 −0.482802 −0.241401 0.970425i \(-0.577607\pi\)
−0.241401 + 0.970425i \(0.577607\pi\)
\(548\) 33.2705 1.42125
\(549\) −4.70820 −0.200941
\(550\) 0 0
\(551\) −9.27051 −0.394937
\(552\) −17.0344 −0.725034
\(553\) −31.7082 −1.34837
\(554\) 19.1246 0.812527
\(555\) 0 0
\(556\) 21.7082 0.920633
\(557\) 6.34752 0.268953 0.134477 0.990917i \(-0.457065\pi\)
0.134477 + 0.990917i \(0.457065\pi\)
\(558\) −2.90983 −0.123183
\(559\) 9.61803 0.406799
\(560\) 0 0
\(561\) −0.708204 −0.0299004
\(562\) 5.05573 0.213263
\(563\) −9.00000 −0.379305 −0.189652 0.981851i \(-0.560736\pi\)
−0.189652 + 0.981851i \(0.560736\pi\)
\(564\) 14.9443 0.629267
\(565\) 0 0
\(566\) −9.70820 −0.408066
\(567\) 2.00000 0.0839921
\(568\) 2.43769 0.102283
\(569\) −4.14590 −0.173805 −0.0869025 0.996217i \(-0.527697\pi\)
−0.0869025 + 0.996217i \(0.527697\pi\)
\(570\) 0 0
\(571\) 2.12461 0.0889122 0.0444561 0.999011i \(-0.485845\pi\)
0.0444561 + 0.999011i \(0.485845\pi\)
\(572\) 4.85410 0.202960
\(573\) −1.67376 −0.0699224
\(574\) −14.3607 −0.599403
\(575\) 0 0
\(576\) −0.236068 −0.00983617
\(577\) −37.2705 −1.55159 −0.775796 0.630984i \(-0.782651\pi\)
−0.775796 + 0.630984i \(0.782651\pi\)
\(578\) −10.4721 −0.435583
\(579\) −11.0000 −0.457144
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) −1.76393 −0.0731173
\(583\) 20.2918 0.840400
\(584\) 5.12461 0.212058
\(585\) 0 0
\(586\) −5.76393 −0.238106
\(587\) 23.3050 0.961898 0.480949 0.876748i \(-0.340292\pi\)
0.480949 + 0.876748i \(0.340292\pi\)
\(588\) −4.85410 −0.200180
\(589\) −31.5836 −1.30138
\(590\) 0 0
\(591\) 11.0902 0.456189
\(592\) 3.70820 0.152406
\(593\) −15.3820 −0.631662 −0.315831 0.948816i \(-0.602283\pi\)
−0.315831 + 0.948816i \(0.602283\pi\)
\(594\) 1.85410 0.0760747
\(595\) 0 0
\(596\) 3.09017 0.126578
\(597\) 1.70820 0.0699121
\(598\) −4.70820 −0.192533
\(599\) 5.72949 0.234101 0.117050 0.993126i \(-0.462656\pi\)
0.117050 + 0.993126i \(0.462656\pi\)
\(600\) 0 0
\(601\) −11.2918 −0.460602 −0.230301 0.973119i \(-0.573971\pi\)
−0.230301 + 0.973119i \(0.573971\pi\)
\(602\) 11.8885 0.484541
\(603\) −9.18034 −0.373852
\(604\) 7.09017 0.288495
\(605\) 0 0
\(606\) 7.18034 0.291681
\(607\) 16.1459 0.655342 0.327671 0.944792i \(-0.393736\pi\)
0.327671 + 0.944792i \(0.393736\pi\)
\(608\) 37.6869 1.52841
\(609\) 2.76393 0.112000
\(610\) 0 0
\(611\) 9.23607 0.373651
\(612\) 0.381966 0.0154401
\(613\) 46.1246 1.86296 0.931478 0.363798i \(-0.118520\pi\)
0.931478 + 0.363798i \(0.118520\pi\)
\(614\) −1.32624 −0.0535226
\(615\) 0 0
\(616\) 13.4164 0.540562
\(617\) −20.7639 −0.835924 −0.417962 0.908464i \(-0.637256\pi\)
−0.417962 + 0.908464i \(0.637256\pi\)
\(618\) 7.67376 0.308684
\(619\) 0.729490 0.0293207 0.0146603 0.999893i \(-0.495333\pi\)
0.0146603 + 0.999893i \(0.495333\pi\)
\(620\) 0 0
\(621\) 7.61803 0.305701
\(622\) −13.8885 −0.556880
\(623\) 22.3607 0.895862
\(624\) −1.85410 −0.0742235
\(625\) 0 0
\(626\) −9.70820 −0.388018
\(627\) 20.1246 0.803700
\(628\) 6.23607 0.248846
\(629\) −0.472136 −0.0188253
\(630\) 0 0
\(631\) −15.2361 −0.606538 −0.303269 0.952905i \(-0.598078\pi\)
−0.303269 + 0.952905i \(0.598078\pi\)
\(632\) 35.4508 1.41016
\(633\) 3.00000 0.119239
\(634\) −0.270510 −0.0107433
\(635\) 0 0
\(636\) −10.9443 −0.433969
\(637\) −3.00000 −0.118864
\(638\) 2.56231 0.101443
\(639\) −1.09017 −0.0431265
\(640\) 0 0
\(641\) −7.67376 −0.303095 −0.151548 0.988450i \(-0.548426\pi\)
−0.151548 + 0.988450i \(0.548426\pi\)
\(642\) −4.85410 −0.191576
\(643\) −17.0902 −0.673971 −0.336985 0.941510i \(-0.609407\pi\)
−0.336985 + 0.941510i \(0.609407\pi\)
\(644\) 24.6525 0.971444
\(645\) 0 0
\(646\) −0.978714 −0.0385070
\(647\) −10.0344 −0.394495 −0.197247 0.980354i \(-0.563200\pi\)
−0.197247 + 0.980354i \(0.563200\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 41.8328 1.64208
\(650\) 0 0
\(651\) 9.41641 0.369058
\(652\) −24.7082 −0.967648
\(653\) 1.65248 0.0646664 0.0323332 0.999477i \(-0.489706\pi\)
0.0323332 + 0.999477i \(0.489706\pi\)
\(654\) 6.70820 0.262312
\(655\) 0 0
\(656\) −21.5410 −0.841036
\(657\) −2.29180 −0.0894115
\(658\) 11.4164 0.445058
\(659\) 2.23607 0.0871048 0.0435524 0.999051i \(-0.486132\pi\)
0.0435524 + 0.999051i \(0.486132\pi\)
\(660\) 0 0
\(661\) −30.8885 −1.20143 −0.600713 0.799465i \(-0.705116\pi\)
−0.600713 + 0.799465i \(0.705116\pi\)
\(662\) 18.3475 0.713097
\(663\) 0.236068 0.00916812
\(664\) 20.1246 0.780986
\(665\) 0 0
\(666\) 1.23607 0.0478967
\(667\) 10.5279 0.407641
\(668\) −11.0000 −0.425603
\(669\) −16.8541 −0.651617
\(670\) 0 0
\(671\) 14.1246 0.545275
\(672\) −11.2361 −0.433441
\(673\) −24.7771 −0.955087 −0.477543 0.878608i \(-0.658473\pi\)
−0.477543 + 0.878608i \(0.658473\pi\)
\(674\) 6.68692 0.257570
\(675\) 0 0
\(676\) 19.4164 0.746785
\(677\) 9.11146 0.350182 0.175091 0.984552i \(-0.443978\pi\)
0.175091 + 0.984552i \(0.443978\pi\)
\(678\) −5.09017 −0.195487
\(679\) 5.70820 0.219061
\(680\) 0 0
\(681\) 10.2361 0.392247
\(682\) 8.72949 0.334269
\(683\) −48.5967 −1.85950 −0.929751 0.368188i \(-0.879978\pi\)
−0.929751 + 0.368188i \(0.879978\pi\)
\(684\) −10.8541 −0.415017
\(685\) 0 0
\(686\) −12.3607 −0.471933
\(687\) −6.18034 −0.235795
\(688\) 17.8328 0.679870
\(689\) −6.76393 −0.257685
\(690\) 0 0
\(691\) 3.90983 0.148737 0.0743685 0.997231i \(-0.476306\pi\)
0.0743685 + 0.997231i \(0.476306\pi\)
\(692\) 19.5623 0.743647
\(693\) −6.00000 −0.227921
\(694\) 13.1459 0.499011
\(695\) 0 0
\(696\) −3.09017 −0.117133
\(697\) 2.74265 0.103885
\(698\) 1.70820 0.0646565
\(699\) −12.1803 −0.460703
\(700\) 0 0
\(701\) −17.3475 −0.655207 −0.327603 0.944815i \(-0.606241\pi\)
−0.327603 + 0.944815i \(0.606241\pi\)
\(702\) −0.618034 −0.0233262
\(703\) 13.4164 0.506009
\(704\) 0.708204 0.0266914
\(705\) 0 0
\(706\) 9.03444 0.340016
\(707\) −23.2361 −0.873882
\(708\) −22.5623 −0.847943
\(709\) −29.7984 −1.11910 −0.559551 0.828796i \(-0.689026\pi\)
−0.559551 + 0.828796i \(0.689026\pi\)
\(710\) 0 0
\(711\) −15.8541 −0.594575
\(712\) −25.0000 −0.936915
\(713\) 35.8673 1.34324
\(714\) 0.291796 0.0109202
\(715\) 0 0
\(716\) −25.3262 −0.946486
\(717\) −23.6180 −0.882032
\(718\) 3.74265 0.139674
\(719\) 5.12461 0.191116 0.0955579 0.995424i \(-0.469536\pi\)
0.0955579 + 0.995424i \(0.469536\pi\)
\(720\) 0 0
\(721\) −24.8328 −0.924822
\(722\) 16.0689 0.598022
\(723\) 8.32624 0.309656
\(724\) 5.70820 0.212144
\(725\) 0 0
\(726\) 1.23607 0.0458748
\(727\) 34.5623 1.28184 0.640922 0.767606i \(-0.278552\pi\)
0.640922 + 0.767606i \(0.278552\pi\)
\(728\) −4.47214 −0.165748
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −2.27051 −0.0839778
\(732\) −7.61803 −0.281571
\(733\) 16.8541 0.622520 0.311260 0.950325i \(-0.399249\pi\)
0.311260 + 0.950325i \(0.399249\pi\)
\(734\) 13.2705 0.489823
\(735\) 0 0
\(736\) −42.7984 −1.57757
\(737\) 27.5410 1.01449
\(738\) −7.18034 −0.264312
\(739\) 11.7082 0.430693 0.215347 0.976538i \(-0.430912\pi\)
0.215347 + 0.976538i \(0.430912\pi\)
\(740\) 0 0
\(741\) −6.70820 −0.246432
\(742\) −8.36068 −0.306930
\(743\) 25.4721 0.934482 0.467241 0.884130i \(-0.345248\pi\)
0.467241 + 0.884130i \(0.345248\pi\)
\(744\) −10.5279 −0.385970
\(745\) 0 0
\(746\) 5.81966 0.213073
\(747\) −9.00000 −0.329293
\(748\) −1.14590 −0.0418982
\(749\) 15.7082 0.573965
\(750\) 0 0
\(751\) 28.7082 1.04758 0.523789 0.851848i \(-0.324518\pi\)
0.523789 + 0.851848i \(0.324518\pi\)
\(752\) 17.1246 0.624470
\(753\) −27.9787 −1.01960
\(754\) −0.854102 −0.0311046
\(755\) 0 0
\(756\) 3.23607 0.117695
\(757\) 1.27051 0.0461775 0.0230887 0.999733i \(-0.492650\pi\)
0.0230887 + 0.999733i \(0.492650\pi\)
\(758\) −7.03444 −0.255502
\(759\) −22.8541 −0.829551
\(760\) 0 0
\(761\) −19.1803 −0.695287 −0.347643 0.937627i \(-0.613018\pi\)
−0.347643 + 0.937627i \(0.613018\pi\)
\(762\) −10.9098 −0.395221
\(763\) −21.7082 −0.785890
\(764\) −2.70820 −0.0979794
\(765\) 0 0
\(766\) −14.1803 −0.512357
\(767\) −13.9443 −0.503498
\(768\) 6.56231 0.236797
\(769\) 26.3050 0.948581 0.474290 0.880368i \(-0.342705\pi\)
0.474290 + 0.880368i \(0.342705\pi\)
\(770\) 0 0
\(771\) −20.2148 −0.728018
\(772\) −17.7984 −0.640577
\(773\) −29.7771 −1.07101 −0.535504 0.844533i \(-0.679878\pi\)
−0.535504 + 0.844533i \(0.679878\pi\)
\(774\) 5.94427 0.213662
\(775\) 0 0
\(776\) −6.38197 −0.229099
\(777\) −4.00000 −0.143499
\(778\) −18.9443 −0.679185
\(779\) −77.9361 −2.79235
\(780\) 0 0
\(781\) 3.27051 0.117028
\(782\) 1.11146 0.0397456
\(783\) 1.38197 0.0493874
\(784\) −5.56231 −0.198654
\(785\) 0 0
\(786\) −5.05573 −0.180332
\(787\) −23.8541 −0.850307 −0.425153 0.905121i \(-0.639780\pi\)
−0.425153 + 0.905121i \(0.639780\pi\)
\(788\) 17.9443 0.639238
\(789\) 25.5066 0.908058
\(790\) 0 0
\(791\) 16.4721 0.585682
\(792\) 6.70820 0.238366
\(793\) −4.70820 −0.167193
\(794\) 7.09017 0.251621
\(795\) 0 0
\(796\) 2.76393 0.0979650
\(797\) −46.0132 −1.62987 −0.814935 0.579553i \(-0.803227\pi\)
−0.814935 + 0.579553i \(0.803227\pi\)
\(798\) −8.29180 −0.293526
\(799\) −2.18034 −0.0771349
\(800\) 0 0
\(801\) 11.1803 0.395038
\(802\) 1.68692 0.0595671
\(803\) 6.87539 0.242627
\(804\) −14.8541 −0.523864
\(805\) 0 0
\(806\) −2.90983 −0.102494
\(807\) 29.4721 1.03747
\(808\) 25.9787 0.913928
\(809\) 24.9230 0.876246 0.438123 0.898915i \(-0.355644\pi\)
0.438123 + 0.898915i \(0.355644\pi\)
\(810\) 0 0
\(811\) 37.7771 1.32653 0.663266 0.748383i \(-0.269170\pi\)
0.663266 + 0.748383i \(0.269170\pi\)
\(812\) 4.47214 0.156941
\(813\) −15.4164 −0.540677
\(814\) −3.70820 −0.129972
\(815\) 0 0
\(816\) 0.437694 0.0153224
\(817\) 64.5197 2.25726
\(818\) 21.7082 0.759010
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −11.9443 −0.416858 −0.208429 0.978038i \(-0.566835\pi\)
−0.208429 + 0.978038i \(0.566835\pi\)
\(822\) 12.7082 0.443250
\(823\) 8.43769 0.294120 0.147060 0.989128i \(-0.453019\pi\)
0.147060 + 0.989128i \(0.453019\pi\)
\(824\) 27.7639 0.967202
\(825\) 0 0
\(826\) −17.2361 −0.599720
\(827\) −2.02129 −0.0702870 −0.0351435 0.999382i \(-0.511189\pi\)
−0.0351435 + 0.999382i \(0.511189\pi\)
\(828\) 12.3262 0.428366
\(829\) −9.87539 −0.342986 −0.171493 0.985185i \(-0.554859\pi\)
−0.171493 + 0.985185i \(0.554859\pi\)
\(830\) 0 0
\(831\) −30.9443 −1.07344
\(832\) −0.236068 −0.00818418
\(833\) 0.708204 0.0245378
\(834\) 8.29180 0.287121
\(835\) 0 0
\(836\) 32.5623 1.12619
\(837\) 4.70820 0.162739
\(838\) −9.47214 −0.327210
\(839\) 48.2148 1.66456 0.832280 0.554356i \(-0.187035\pi\)
0.832280 + 0.554356i \(0.187035\pi\)
\(840\) 0 0
\(841\) −27.0902 −0.934144
\(842\) 8.87539 0.305866
\(843\) −8.18034 −0.281746
\(844\) 4.85410 0.167085
\(845\) 0 0
\(846\) 5.70820 0.196252
\(847\) −4.00000 −0.137442
\(848\) −12.5410 −0.430660
\(849\) 15.7082 0.539104
\(850\) 0 0
\(851\) −15.2361 −0.522286
\(852\) −1.76393 −0.0604313
\(853\) −53.3951 −1.82821 −0.914107 0.405473i \(-0.867107\pi\)
−0.914107 + 0.405473i \(0.867107\pi\)
\(854\) −5.81966 −0.199145
\(855\) 0 0
\(856\) −17.5623 −0.600267
\(857\) −26.9443 −0.920399 −0.460199 0.887816i \(-0.652222\pi\)
−0.460199 + 0.887816i \(0.652222\pi\)
\(858\) 1.85410 0.0632980
\(859\) −25.1246 −0.857241 −0.428620 0.903485i \(-0.641000\pi\)
−0.428620 + 0.903485i \(0.641000\pi\)
\(860\) 0 0
\(861\) 23.2361 0.791883
\(862\) 21.1591 0.720680
\(863\) 45.0689 1.53416 0.767081 0.641550i \(-0.221708\pi\)
0.767081 + 0.641550i \(0.221708\pi\)
\(864\) −5.61803 −0.191129
\(865\) 0 0
\(866\) 3.38197 0.114924
\(867\) 16.9443 0.575458
\(868\) 15.2361 0.517146
\(869\) 47.5623 1.61344
\(870\) 0 0
\(871\) −9.18034 −0.311064
\(872\) 24.2705 0.821903
\(873\) 2.85410 0.0965967
\(874\) −31.5836 −1.06833
\(875\) 0 0
\(876\) −3.70820 −0.125289
\(877\) −2.87539 −0.0970950 −0.0485475 0.998821i \(-0.515459\pi\)
−0.0485475 + 0.998821i \(0.515459\pi\)
\(878\) −1.83282 −0.0618545
\(879\) 9.32624 0.314566
\(880\) 0 0
\(881\) 3.90983 0.131726 0.0658628 0.997829i \(-0.479020\pi\)
0.0658628 + 0.997829i \(0.479020\pi\)
\(882\) −1.85410 −0.0624309
\(883\) −53.7984 −1.81046 −0.905230 0.424923i \(-0.860301\pi\)
−0.905230 + 0.424923i \(0.860301\pi\)
\(884\) 0.381966 0.0128469
\(885\) 0 0
\(886\) −4.58359 −0.153989
\(887\) 21.9230 0.736102 0.368051 0.929806i \(-0.380025\pi\)
0.368051 + 0.929806i \(0.380025\pi\)
\(888\) 4.47214 0.150075
\(889\) 35.3050 1.18409
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) −27.2705 −0.913084
\(893\) 61.9574 2.07333
\(894\) 1.18034 0.0394765
\(895\) 0 0
\(896\) −22.7639 −0.760490
\(897\) 7.61803 0.254359
\(898\) −13.2918 −0.443553
\(899\) 6.50658 0.217007
\(900\) 0 0
\(901\) 1.59675 0.0531954
\(902\) 21.5410 0.717238
\(903\) −19.2361 −0.640136
\(904\) −18.4164 −0.612521
\(905\) 0 0
\(906\) 2.70820 0.0899740
\(907\) 17.0000 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) 16.5623 0.549639
\(909\) −11.6180 −0.385346
\(910\) 0 0
\(911\) −20.8885 −0.692068 −0.346034 0.938222i \(-0.612472\pi\)
−0.346034 + 0.938222i \(0.612472\pi\)
\(912\) −12.4377 −0.411853
\(913\) 27.0000 0.893570
\(914\) −16.0000 −0.529233
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 16.3607 0.540277
\(918\) 0.145898 0.00481535
\(919\) −5.00000 −0.164935 −0.0824674 0.996594i \(-0.526280\pi\)
−0.0824674 + 0.996594i \(0.526280\pi\)
\(920\) 0 0
\(921\) 2.14590 0.0707097
\(922\) 1.96556 0.0647322
\(923\) −1.09017 −0.0358834
\(924\) −9.70820 −0.319376
\(925\) 0 0
\(926\) −16.4934 −0.542007
\(927\) −12.4164 −0.407808
\(928\) −7.76393 −0.254864
\(929\) 19.5967 0.642948 0.321474 0.946918i \(-0.395822\pi\)
0.321474 + 0.946918i \(0.395822\pi\)
\(930\) 0 0
\(931\) −20.1246 −0.659558
\(932\) −19.7082 −0.645564
\(933\) 22.4721 0.735705
\(934\) −10.1459 −0.331984
\(935\) 0 0
\(936\) −2.23607 −0.0730882
\(937\) −16.4164 −0.536301 −0.268150 0.963377i \(-0.586412\pi\)
−0.268150 + 0.963377i \(0.586412\pi\)
\(938\) −11.3475 −0.370510
\(939\) 15.7082 0.512618
\(940\) 0 0
\(941\) 11.0213 0.359284 0.179642 0.983732i \(-0.442506\pi\)
0.179642 + 0.983732i \(0.442506\pi\)
\(942\) 2.38197 0.0776086
\(943\) 88.5066 2.88217
\(944\) −25.8541 −0.841479
\(945\) 0 0
\(946\) −17.8328 −0.579795
\(947\) −29.8328 −0.969436 −0.484718 0.874670i \(-0.661078\pi\)
−0.484718 + 0.874670i \(0.661078\pi\)
\(948\) −25.6525 −0.833154
\(949\) −2.29180 −0.0743948
\(950\) 0 0
\(951\) 0.437694 0.0141932
\(952\) 1.05573 0.0342163
\(953\) 59.9443 1.94179 0.970893 0.239515i \(-0.0769884\pi\)
0.970893 + 0.239515i \(0.0769884\pi\)
\(954\) −4.18034 −0.135344
\(955\) 0 0
\(956\) −38.2148 −1.23595
\(957\) −4.14590 −0.134018
\(958\) −12.2361 −0.395329
\(959\) −41.1246 −1.32798
\(960\) 0 0
\(961\) −8.83282 −0.284930
\(962\) 1.23607 0.0398524
\(963\) 7.85410 0.253095
\(964\) 13.4721 0.433908
\(965\) 0 0
\(966\) 9.41641 0.302968
\(967\) 8.58359 0.276030 0.138015 0.990430i \(-0.455928\pi\)
0.138015 + 0.990430i \(0.455928\pi\)
\(968\) 4.47214 0.143740
\(969\) 1.58359 0.0508723
\(970\) 0 0
\(971\) −5.88854 −0.188972 −0.0944862 0.995526i \(-0.530121\pi\)
−0.0944862 + 0.995526i \(0.530121\pi\)
\(972\) 1.61803 0.0518985
\(973\) −26.8328 −0.860221
\(974\) −8.88854 −0.284807
\(975\) 0 0
\(976\) −8.72949 −0.279424
\(977\) 6.34752 0.203075 0.101538 0.994832i \(-0.467624\pi\)
0.101538 + 0.994832i \(0.467624\pi\)
\(978\) −9.43769 −0.301784
\(979\) −33.5410 −1.07198
\(980\) 0 0
\(981\) −10.8541 −0.346545
\(982\) 4.12461 0.131622
\(983\) 22.3820 0.713874 0.356937 0.934128i \(-0.383821\pi\)
0.356937 + 0.934128i \(0.383821\pi\)
\(984\) −25.9787 −0.828171
\(985\) 0 0
\(986\) 0.201626 0.00642108
\(987\) −18.4721 −0.587975
\(988\) −10.8541 −0.345315
\(989\) −73.2705 −2.32987
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −26.4508 −0.839815
\(993\) −29.6869 −0.942086
\(994\) −1.34752 −0.0427409
\(995\) 0 0
\(996\) −14.5623 −0.461424
\(997\) 61.0689 1.93407 0.967035 0.254642i \(-0.0819575\pi\)
0.967035 + 0.254642i \(0.0819575\pi\)
\(998\) −9.27051 −0.293453
\(999\) −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.1.2 2
3.2 odd 2 5625.2.a.g.1.1 2
5.2 odd 4 1875.2.b.a.1249.3 4
5.3 odd 4 1875.2.b.a.1249.2 4
5.4 even 2 1875.2.a.c.1.1 yes 2
15.14 odd 2 5625.2.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.b.1.2 2 1.1 even 1 trivial
1875.2.a.c.1.1 yes 2 5.4 even 2
1875.2.b.a.1249.2 4 5.3 odd 4
1875.2.b.a.1249.3 4 5.2 odd 4
5625.2.a.b.1.2 2 15.14 odd 2
5625.2.a.g.1.1 2 3.2 odd 2