Properties

Label 1805.2.a.c.1.2
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $2$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(2\)
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 \(\chi\) \(=\) 1805.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.381966 q^{2} -0.381966 q^{3} -1.85410 q^{4} -1.00000 q^{5} +0.145898 q^{6} -4.23607 q^{7} +1.47214 q^{8} -2.85410 q^{9} +O(q^{10})\) \(q-0.381966 q^{2} -0.381966 q^{3} -1.85410 q^{4} -1.00000 q^{5} +0.145898 q^{6} -4.23607 q^{7} +1.47214 q^{8} -2.85410 q^{9} +0.381966 q^{10} -2.38197 q^{11} +0.708204 q^{12} -5.00000 q^{13} +1.61803 q^{14} +0.381966 q^{15} +3.14590 q^{16} -6.00000 q^{17} +1.09017 q^{18} +1.85410 q^{20} +1.61803 q^{21} +0.909830 q^{22} +0.618034 q^{23} -0.562306 q^{24} +1.00000 q^{25} +1.90983 q^{26} +2.23607 q^{27} +7.85410 q^{28} -4.85410 q^{29} -0.145898 q^{30} -10.8541 q^{31} -4.14590 q^{32} +0.909830 q^{33} +2.29180 q^{34} +4.23607 q^{35} +5.29180 q^{36} -4.85410 q^{37} +1.90983 q^{39} -1.47214 q^{40} +11.1803 q^{41} -0.618034 q^{42} +3.85410 q^{43} +4.41641 q^{44} +2.85410 q^{45} -0.236068 q^{46} -5.76393 q^{47} -1.20163 q^{48} +10.9443 q^{49} -0.381966 q^{50} +2.29180 q^{51} +9.27051 q^{52} -3.38197 q^{53} -0.854102 q^{54} +2.38197 q^{55} -6.23607 q^{56} +1.85410 q^{58} +11.0902 q^{59} -0.708204 q^{60} -3.00000 q^{61} +4.14590 q^{62} +12.0902 q^{63} -4.70820 q^{64} +5.00000 q^{65} -0.347524 q^{66} -8.70820 q^{67} +11.1246 q^{68} -0.236068 q^{69} -1.61803 q^{70} -8.23607 q^{71} -4.20163 q^{72} -1.00000 q^{73} +1.85410 q^{74} -0.381966 q^{75} +10.0902 q^{77} -0.729490 q^{78} -8.00000 q^{79} -3.14590 q^{80} +7.70820 q^{81} -4.27051 q^{82} -4.47214 q^{83} -3.00000 q^{84} +6.00000 q^{85} -1.47214 q^{86} +1.85410 q^{87} -3.50658 q^{88} -6.70820 q^{89} -1.09017 q^{90} +21.1803 q^{91} -1.14590 q^{92} +4.14590 q^{93} +2.20163 q^{94} +1.58359 q^{96} +5.09017 q^{97} -4.18034 q^{98} +6.79837 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 2 q^{5} + 7 q^{6} - 4 q^{7} - 6 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 2 q^{5} + 7 q^{6} - 4 q^{7} - 6 q^{8} + q^{9} + 3 q^{10} - 7 q^{11} - 12 q^{12} - 10 q^{13} + q^{14} + 3 q^{15} + 13 q^{16} - 12 q^{17} - 9 q^{18} - 3 q^{20} + q^{21} + 13 q^{22} - q^{23} + 19 q^{24} + 2 q^{25} + 15 q^{26} + 9 q^{28} - 3 q^{29} - 7 q^{30} - 15 q^{31} - 15 q^{32} + 13 q^{33} + 18 q^{34} + 4 q^{35} + 24 q^{36} - 3 q^{37} + 15 q^{39} + 6 q^{40} + q^{42} + q^{43} - 18 q^{44} - q^{45} + 4 q^{46} - 16 q^{47} - 27 q^{48} + 4 q^{49} - 3 q^{50} + 18 q^{51} - 15 q^{52} - 9 q^{53} + 5 q^{54} + 7 q^{55} - 8 q^{56} - 3 q^{58} + 11 q^{59} + 12 q^{60} - 6 q^{61} + 15 q^{62} + 13 q^{63} + 4 q^{64} + 10 q^{65} - 32 q^{66} - 4 q^{67} - 18 q^{68} + 4 q^{69} - q^{70} - 12 q^{71} - 33 q^{72} - 2 q^{73} - 3 q^{74} - 3 q^{75} + 9 q^{77} - 35 q^{78} - 16 q^{79} - 13 q^{80} + 2 q^{81} + 25 q^{82} - 6 q^{84} + 12 q^{85} + 6 q^{86} - 3 q^{87} + 31 q^{88} + 9 q^{90} + 20 q^{91} - 9 q^{92} + 15 q^{93} + 29 q^{94} + 30 q^{96} - q^{97} + 14 q^{98} - 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.381966 −0.270091 −0.135045 0.990839i \(-0.543118\pi\)
−0.135045 + 0.990839i \(0.543118\pi\)
\(3\) −0.381966 −0.220528 −0.110264 0.993902i \(-0.535170\pi\)
−0.110264 + 0.993902i \(0.535170\pi\)
\(4\) −1.85410 −0.927051
\(5\) −1.00000 −0.447214
\(6\) 0.145898 0.0595626
\(7\) −4.23607 −1.60108 −0.800542 0.599277i \(-0.795455\pi\)
−0.800542 + 0.599277i \(0.795455\pi\)
\(8\) 1.47214 0.520479
\(9\) −2.85410 −0.951367
\(10\) 0.381966 0.120788
\(11\) −2.38197 −0.718190 −0.359095 0.933301i \(-0.616915\pi\)
−0.359095 + 0.933301i \(0.616915\pi\)
\(12\) 0.708204 0.204441
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 1.61803 0.432438
\(15\) 0.381966 0.0986232
\(16\) 3.14590 0.786475
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.09017 0.256956
\(19\) 0 0
\(20\) 1.85410 0.414590
\(21\) 1.61803 0.353084
\(22\) 0.909830 0.193976
\(23\) 0.618034 0.128869 0.0644345 0.997922i \(-0.479476\pi\)
0.0644345 + 0.997922i \(0.479476\pi\)
\(24\) −0.562306 −0.114780
\(25\) 1.00000 0.200000
\(26\) 1.90983 0.374548
\(27\) 2.23607 0.430331
\(28\) 7.85410 1.48429
\(29\) −4.85410 −0.901384 −0.450692 0.892679i \(-0.648823\pi\)
−0.450692 + 0.892679i \(0.648823\pi\)
\(30\) −0.145898 −0.0266372
\(31\) −10.8541 −1.94945 −0.974727 0.223399i \(-0.928285\pi\)
−0.974727 + 0.223399i \(0.928285\pi\)
\(32\) −4.14590 −0.732898
\(33\) 0.909830 0.158381
\(34\) 2.29180 0.393040
\(35\) 4.23607 0.716026
\(36\) 5.29180 0.881966
\(37\) −4.85410 −0.798009 −0.399005 0.916949i \(-0.630644\pi\)
−0.399005 + 0.916949i \(0.630644\pi\)
\(38\) 0 0
\(39\) 1.90983 0.305818
\(40\) −1.47214 −0.232765
\(41\) 11.1803 1.74608 0.873038 0.487652i \(-0.162147\pi\)
0.873038 + 0.487652i \(0.162147\pi\)
\(42\) −0.618034 −0.0953647
\(43\) 3.85410 0.587745 0.293873 0.955845i \(-0.405056\pi\)
0.293873 + 0.955845i \(0.405056\pi\)
\(44\) 4.41641 0.665799
\(45\) 2.85410 0.425464
\(46\) −0.236068 −0.0348063
\(47\) −5.76393 −0.840756 −0.420378 0.907349i \(-0.638103\pi\)
−0.420378 + 0.907349i \(0.638103\pi\)
\(48\) −1.20163 −0.173440
\(49\) 10.9443 1.56347
\(50\) −0.381966 −0.0540182
\(51\) 2.29180 0.320916
\(52\) 9.27051 1.28559
\(53\) −3.38197 −0.464549 −0.232274 0.972650i \(-0.574617\pi\)
−0.232274 + 0.972650i \(0.574617\pi\)
\(54\) −0.854102 −0.116229
\(55\) 2.38197 0.321184
\(56\) −6.23607 −0.833330
\(57\) 0 0
\(58\) 1.85410 0.243456
\(59\) 11.0902 1.44382 0.721909 0.691988i \(-0.243265\pi\)
0.721909 + 0.691988i \(0.243265\pi\)
\(60\) −0.708204 −0.0914287
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) 4.14590 0.526530
\(63\) 12.0902 1.52322
\(64\) −4.70820 −0.588525
\(65\) 5.00000 0.620174
\(66\) −0.347524 −0.0427773
\(67\) −8.70820 −1.06388 −0.531938 0.846783i \(-0.678536\pi\)
−0.531938 + 0.846783i \(0.678536\pi\)
\(68\) 11.1246 1.34906
\(69\) −0.236068 −0.0284192
\(70\) −1.61803 −0.193392
\(71\) −8.23607 −0.977441 −0.488721 0.872440i \(-0.662536\pi\)
−0.488721 + 0.872440i \(0.662536\pi\)
\(72\) −4.20163 −0.495166
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 1.85410 0.215535
\(75\) −0.381966 −0.0441056
\(76\) 0 0
\(77\) 10.0902 1.14988
\(78\) −0.729490 −0.0825985
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −3.14590 −0.351722
\(81\) 7.70820 0.856467
\(82\) −4.27051 −0.471599
\(83\) −4.47214 −0.490881 −0.245440 0.969412i \(-0.578933\pi\)
−0.245440 + 0.969412i \(0.578933\pi\)
\(84\) −3.00000 −0.327327
\(85\) 6.00000 0.650791
\(86\) −1.47214 −0.158745
\(87\) 1.85410 0.198781
\(88\) −3.50658 −0.373802
\(89\) −6.70820 −0.711068 −0.355534 0.934663i \(-0.615701\pi\)
−0.355534 + 0.934663i \(0.615701\pi\)
\(90\) −1.09017 −0.114914
\(91\) 21.1803 2.22030
\(92\) −1.14590 −0.119468
\(93\) 4.14590 0.429910
\(94\) 2.20163 0.227080
\(95\) 0 0
\(96\) 1.58359 0.161625
\(97\) 5.09017 0.516828 0.258414 0.966034i \(-0.416800\pi\)
0.258414 + 0.966034i \(0.416800\pi\)
\(98\) −4.18034 −0.422278
\(99\) 6.79837 0.683262
\(100\) −1.85410 −0.185410
\(101\) 5.94427 0.591477 0.295739 0.955269i \(-0.404434\pi\)
0.295739 + 0.955269i \(0.404434\pi\)
\(102\) −0.875388 −0.0866763
\(103\) −1.67376 −0.164921 −0.0824603 0.996594i \(-0.526278\pi\)
−0.0824603 + 0.996594i \(0.526278\pi\)
\(104\) −7.36068 −0.721774
\(105\) −1.61803 −0.157904
\(106\) 1.29180 0.125470
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) −4.14590 −0.398939
\(109\) −1.76393 −0.168954 −0.0844770 0.996425i \(-0.526922\pi\)
−0.0844770 + 0.996425i \(0.526922\pi\)
\(110\) −0.909830 −0.0867489
\(111\) 1.85410 0.175984
\(112\) −13.3262 −1.25921
\(113\) 14.9443 1.40584 0.702919 0.711269i \(-0.251879\pi\)
0.702919 + 0.711269i \(0.251879\pi\)
\(114\) 0 0
\(115\) −0.618034 −0.0576320
\(116\) 9.00000 0.835629
\(117\) 14.2705 1.31931
\(118\) −4.23607 −0.389962
\(119\) 25.4164 2.32992
\(120\) 0.562306 0.0513313
\(121\) −5.32624 −0.484203
\(122\) 1.14590 0.103745
\(123\) −4.27051 −0.385059
\(124\) 20.1246 1.80724
\(125\) −1.00000 −0.0894427
\(126\) −4.61803 −0.411407
\(127\) −10.7082 −0.950199 −0.475100 0.879932i \(-0.657588\pi\)
−0.475100 + 0.879932i \(0.657588\pi\)
\(128\) 10.0902 0.891853
\(129\) −1.47214 −0.129614
\(130\) −1.90983 −0.167503
\(131\) −7.85410 −0.686216 −0.343108 0.939296i \(-0.611480\pi\)
−0.343108 + 0.939296i \(0.611480\pi\)
\(132\) −1.68692 −0.146827
\(133\) 0 0
\(134\) 3.32624 0.287343
\(135\) −2.23607 −0.192450
\(136\) −8.83282 −0.757408
\(137\) 1.18034 0.100843 0.0504216 0.998728i \(-0.483943\pi\)
0.0504216 + 0.998728i \(0.483943\pi\)
\(138\) 0.0901699 0.00767578
\(139\) 8.14590 0.690926 0.345463 0.938432i \(-0.387722\pi\)
0.345463 + 0.938432i \(0.387722\pi\)
\(140\) −7.85410 −0.663793
\(141\) 2.20163 0.185410
\(142\) 3.14590 0.263998
\(143\) 11.9098 0.995950
\(144\) −8.97871 −0.748226
\(145\) 4.85410 0.403111
\(146\) 0.381966 0.0316117
\(147\) −4.18034 −0.344789
\(148\) 9.00000 0.739795
\(149\) −7.61803 −0.624094 −0.312047 0.950067i \(-0.601015\pi\)
−0.312047 + 0.950067i \(0.601015\pi\)
\(150\) 0.145898 0.0119125
\(151\) 7.38197 0.600736 0.300368 0.953823i \(-0.402891\pi\)
0.300368 + 0.953823i \(0.402891\pi\)
\(152\) 0 0
\(153\) 17.1246 1.38444
\(154\) −3.85410 −0.310572
\(155\) 10.8541 0.871822
\(156\) −3.54102 −0.283508
\(157\) −24.0344 −1.91816 −0.959079 0.283140i \(-0.908624\pi\)
−0.959079 + 0.283140i \(0.908624\pi\)
\(158\) 3.05573 0.243101
\(159\) 1.29180 0.102446
\(160\) 4.14590 0.327762
\(161\) −2.61803 −0.206330
\(162\) −2.94427 −0.231324
\(163\) 8.52786 0.667954 0.333977 0.942581i \(-0.391609\pi\)
0.333977 + 0.942581i \(0.391609\pi\)
\(164\) −20.7295 −1.61870
\(165\) −0.909830 −0.0708302
\(166\) 1.70820 0.132582
\(167\) −8.23607 −0.637326 −0.318663 0.947868i \(-0.603234\pi\)
−0.318663 + 0.947868i \(0.603234\pi\)
\(168\) 2.38197 0.183773
\(169\) 12.0000 0.923077
\(170\) −2.29180 −0.175773
\(171\) 0 0
\(172\) −7.14590 −0.544870
\(173\) −13.8885 −1.05593 −0.527963 0.849267i \(-0.677044\pi\)
−0.527963 + 0.849267i \(0.677044\pi\)
\(174\) −0.708204 −0.0536888
\(175\) −4.23607 −0.320217
\(176\) −7.49342 −0.564838
\(177\) −4.23607 −0.318402
\(178\) 2.56231 0.192053
\(179\) −10.5279 −0.786890 −0.393445 0.919348i \(-0.628717\pi\)
−0.393445 + 0.919348i \(0.628717\pi\)
\(180\) −5.29180 −0.394427
\(181\) −8.29180 −0.616324 −0.308162 0.951334i \(-0.599714\pi\)
−0.308162 + 0.951334i \(0.599714\pi\)
\(182\) −8.09017 −0.599683
\(183\) 1.14590 0.0847072
\(184\) 0.909830 0.0670736
\(185\) 4.85410 0.356881
\(186\) −1.58359 −0.116115
\(187\) 14.2918 1.04512
\(188\) 10.6869 0.779424
\(189\) −9.47214 −0.688997
\(190\) 0 0
\(191\) 6.05573 0.438177 0.219089 0.975705i \(-0.429692\pi\)
0.219089 + 0.975705i \(0.429692\pi\)
\(192\) 1.79837 0.129786
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −1.94427 −0.139591
\(195\) −1.90983 −0.136766
\(196\) −20.2918 −1.44941
\(197\) 16.5279 1.17756 0.588781 0.808293i \(-0.299608\pi\)
0.588781 + 0.808293i \(0.299608\pi\)
\(198\) −2.59675 −0.184543
\(199\) 13.4164 0.951064 0.475532 0.879698i \(-0.342256\pi\)
0.475532 + 0.879698i \(0.342256\pi\)
\(200\) 1.47214 0.104096
\(201\) 3.32624 0.234615
\(202\) −2.27051 −0.159753
\(203\) 20.5623 1.44319
\(204\) −4.24922 −0.297505
\(205\) −11.1803 −0.780869
\(206\) 0.639320 0.0445436
\(207\) −1.76393 −0.122602
\(208\) −15.7295 −1.09064
\(209\) 0 0
\(210\) 0.618034 0.0426484
\(211\) 6.90983 0.475692 0.237846 0.971303i \(-0.423559\pi\)
0.237846 + 0.971303i \(0.423559\pi\)
\(212\) 6.27051 0.430660
\(213\) 3.14590 0.215553
\(214\) 1.14590 0.0783320
\(215\) −3.85410 −0.262848
\(216\) 3.29180 0.223978
\(217\) 45.9787 3.12124
\(218\) 0.673762 0.0456329
\(219\) 0.381966 0.0258109
\(220\) −4.41641 −0.297754
\(221\) 30.0000 2.01802
\(222\) −0.708204 −0.0475315
\(223\) −10.2361 −0.685458 −0.342729 0.939434i \(-0.611351\pi\)
−0.342729 + 0.939434i \(0.611351\pi\)
\(224\) 17.5623 1.17343
\(225\) −2.85410 −0.190273
\(226\) −5.70820 −0.379704
\(227\) −17.6525 −1.17164 −0.585818 0.810443i \(-0.699227\pi\)
−0.585818 + 0.810443i \(0.699227\pi\)
\(228\) 0 0
\(229\) −21.6180 −1.42856 −0.714280 0.699860i \(-0.753246\pi\)
−0.714280 + 0.699860i \(0.753246\pi\)
\(230\) 0.236068 0.0155659
\(231\) −3.85410 −0.253581
\(232\) −7.14590 −0.469151
\(233\) −6.81966 −0.446771 −0.223385 0.974730i \(-0.571711\pi\)
−0.223385 + 0.974730i \(0.571711\pi\)
\(234\) −5.45085 −0.356333
\(235\) 5.76393 0.375997
\(236\) −20.5623 −1.33849
\(237\) 3.05573 0.198491
\(238\) −9.70820 −0.629289
\(239\) 23.5623 1.52412 0.762059 0.647507i \(-0.224188\pi\)
0.762059 + 0.647507i \(0.224188\pi\)
\(240\) 1.20163 0.0775646
\(241\) −18.1246 −1.16751 −0.583754 0.811930i \(-0.698417\pi\)
−0.583754 + 0.811930i \(0.698417\pi\)
\(242\) 2.03444 0.130779
\(243\) −9.65248 −0.619207
\(244\) 5.56231 0.356090
\(245\) −10.9443 −0.699204
\(246\) 1.63119 0.104001
\(247\) 0 0
\(248\) −15.9787 −1.01465
\(249\) 1.70820 0.108253
\(250\) 0.381966 0.0241577
\(251\) 15.1803 0.958175 0.479087 0.877767i \(-0.340968\pi\)
0.479087 + 0.877767i \(0.340968\pi\)
\(252\) −22.4164 −1.41210
\(253\) −1.47214 −0.0925524
\(254\) 4.09017 0.256640
\(255\) −2.29180 −0.143518
\(256\) 5.56231 0.347644
\(257\) −13.5279 −0.843845 −0.421922 0.906632i \(-0.638645\pi\)
−0.421922 + 0.906632i \(0.638645\pi\)
\(258\) 0.562306 0.0350076
\(259\) 20.5623 1.27768
\(260\) −9.27051 −0.574933
\(261\) 13.8541 0.857547
\(262\) 3.00000 0.185341
\(263\) −19.8885 −1.22638 −0.613190 0.789935i \(-0.710114\pi\)
−0.613190 + 0.789935i \(0.710114\pi\)
\(264\) 1.33939 0.0824340
\(265\) 3.38197 0.207753
\(266\) 0 0
\(267\) 2.56231 0.156811
\(268\) 16.1459 0.986268
\(269\) −1.67376 −0.102051 −0.0510255 0.998697i \(-0.516249\pi\)
−0.0510255 + 0.998697i \(0.516249\pi\)
\(270\) 0.854102 0.0519790
\(271\) 16.3820 0.995134 0.497567 0.867426i \(-0.334227\pi\)
0.497567 + 0.867426i \(0.334227\pi\)
\(272\) −18.8754 −1.14449
\(273\) −8.09017 −0.489639
\(274\) −0.450850 −0.0272368
\(275\) −2.38197 −0.143638
\(276\) 0.437694 0.0263461
\(277\) −25.4164 −1.52712 −0.763562 0.645735i \(-0.776551\pi\)
−0.763562 + 0.645735i \(0.776551\pi\)
\(278\) −3.11146 −0.186613
\(279\) 30.9787 1.85465
\(280\) 6.23607 0.372676
\(281\) −27.5066 −1.64090 −0.820452 0.571715i \(-0.806278\pi\)
−0.820452 + 0.571715i \(0.806278\pi\)
\(282\) −0.840946 −0.0500776
\(283\) 4.61803 0.274514 0.137257 0.990535i \(-0.456171\pi\)
0.137257 + 0.990535i \(0.456171\pi\)
\(284\) 15.2705 0.906138
\(285\) 0 0
\(286\) −4.54915 −0.268997
\(287\) −47.3607 −2.79561
\(288\) 11.8328 0.697255
\(289\) 19.0000 1.11765
\(290\) −1.85410 −0.108877
\(291\) −1.94427 −0.113975
\(292\) 1.85410 0.108503
\(293\) −22.0344 −1.28727 −0.643633 0.765334i \(-0.722574\pi\)
−0.643633 + 0.765334i \(0.722574\pi\)
\(294\) 1.59675 0.0931242
\(295\) −11.0902 −0.645695
\(296\) −7.14590 −0.415347
\(297\) −5.32624 −0.309060
\(298\) 2.90983 0.168562
\(299\) −3.09017 −0.178709
\(300\) 0.708204 0.0408882
\(301\) −16.3262 −0.941029
\(302\) −2.81966 −0.162253
\(303\) −2.27051 −0.130437
\(304\) 0 0
\(305\) 3.00000 0.171780
\(306\) −6.54102 −0.373925
\(307\) 12.2705 0.700315 0.350157 0.936691i \(-0.386128\pi\)
0.350157 + 0.936691i \(0.386128\pi\)
\(308\) −18.7082 −1.06600
\(309\) 0.639320 0.0363697
\(310\) −4.14590 −0.235471
\(311\) −7.05573 −0.400094 −0.200047 0.979786i \(-0.564109\pi\)
−0.200047 + 0.979786i \(0.564109\pi\)
\(312\) 2.81153 0.159172
\(313\) −9.79837 −0.553837 −0.276918 0.960893i \(-0.589313\pi\)
−0.276918 + 0.960893i \(0.589313\pi\)
\(314\) 9.18034 0.518077
\(315\) −12.0902 −0.681204
\(316\) 14.8328 0.834411
\(317\) 13.2361 0.743412 0.371706 0.928351i \(-0.378773\pi\)
0.371706 + 0.928351i \(0.378773\pi\)
\(318\) −0.493422 −0.0276697
\(319\) 11.5623 0.647365
\(320\) 4.70820 0.263197
\(321\) 1.14590 0.0639578
\(322\) 1.00000 0.0557278
\(323\) 0 0
\(324\) −14.2918 −0.793989
\(325\) −5.00000 −0.277350
\(326\) −3.25735 −0.180408
\(327\) 0.673762 0.0372591
\(328\) 16.4590 0.908795
\(329\) 24.4164 1.34612
\(330\) 0.347524 0.0191306
\(331\) 0.291796 0.0160386 0.00801928 0.999968i \(-0.497447\pi\)
0.00801928 + 0.999968i \(0.497447\pi\)
\(332\) 8.29180 0.455071
\(333\) 13.8541 0.759200
\(334\) 3.14590 0.172136
\(335\) 8.70820 0.475780
\(336\) 5.09017 0.277692
\(337\) −28.0000 −1.52526 −0.762629 0.646837i \(-0.776092\pi\)
−0.762629 + 0.646837i \(0.776092\pi\)
\(338\) −4.58359 −0.249315
\(339\) −5.70820 −0.310027
\(340\) −11.1246 −0.603317
\(341\) 25.8541 1.40008
\(342\) 0 0
\(343\) −16.7082 −0.902158
\(344\) 5.67376 0.305909
\(345\) 0.236068 0.0127095
\(346\) 5.30495 0.285196
\(347\) −19.8885 −1.06767 −0.533836 0.845588i \(-0.679250\pi\)
−0.533836 + 0.845588i \(0.679250\pi\)
\(348\) −3.43769 −0.184280
\(349\) 21.7426 1.16386 0.581929 0.813240i \(-0.302298\pi\)
0.581929 + 0.813240i \(0.302298\pi\)
\(350\) 1.61803 0.0864876
\(351\) −11.1803 −0.596762
\(352\) 9.87539 0.526360
\(353\) −36.1591 −1.92455 −0.962276 0.272075i \(-0.912290\pi\)
−0.962276 + 0.272075i \(0.912290\pi\)
\(354\) 1.61803 0.0859975
\(355\) 8.23607 0.437125
\(356\) 12.4377 0.659196
\(357\) −9.70820 −0.513813
\(358\) 4.02129 0.212532
\(359\) 26.3820 1.39239 0.696193 0.717854i \(-0.254876\pi\)
0.696193 + 0.717854i \(0.254876\pi\)
\(360\) 4.20163 0.221445
\(361\) 0 0
\(362\) 3.16718 0.166464
\(363\) 2.03444 0.106781
\(364\) −39.2705 −2.05833
\(365\) 1.00000 0.0523424
\(366\) −0.437694 −0.0228786
\(367\) 1.65248 0.0862585 0.0431293 0.999070i \(-0.486267\pi\)
0.0431293 + 0.999070i \(0.486267\pi\)
\(368\) 1.94427 0.101352
\(369\) −31.9098 −1.66116
\(370\) −1.85410 −0.0963902
\(371\) 14.3262 0.743781
\(372\) −7.68692 −0.398548
\(373\) 6.70820 0.347338 0.173669 0.984804i \(-0.444438\pi\)
0.173669 + 0.984804i \(0.444438\pi\)
\(374\) −5.45898 −0.282277
\(375\) 0.381966 0.0197246
\(376\) −8.48529 −0.437596
\(377\) 24.2705 1.24999
\(378\) 3.61803 0.186092
\(379\) 26.8328 1.37831 0.689155 0.724614i \(-0.257982\pi\)
0.689155 + 0.724614i \(0.257982\pi\)
\(380\) 0 0
\(381\) 4.09017 0.209546
\(382\) −2.31308 −0.118348
\(383\) −32.9230 −1.68229 −0.841143 0.540813i \(-0.818117\pi\)
−0.841143 + 0.540813i \(0.818117\pi\)
\(384\) −3.85410 −0.196679
\(385\) −10.0902 −0.514243
\(386\) 1.52786 0.0777662
\(387\) −11.0000 −0.559161
\(388\) −9.43769 −0.479126
\(389\) −24.3262 −1.23339 −0.616695 0.787202i \(-0.711529\pi\)
−0.616695 + 0.787202i \(0.711529\pi\)
\(390\) 0.729490 0.0369392
\(391\) −3.70820 −0.187532
\(392\) 16.1115 0.813751
\(393\) 3.00000 0.151330
\(394\) −6.31308 −0.318048
\(395\) 8.00000 0.402524
\(396\) −12.6049 −0.633419
\(397\) −16.8885 −0.847612 −0.423806 0.905753i \(-0.639306\pi\)
−0.423806 + 0.905753i \(0.639306\pi\)
\(398\) −5.12461 −0.256874
\(399\) 0 0
\(400\) 3.14590 0.157295
\(401\) −6.76393 −0.337775 −0.168887 0.985635i \(-0.554017\pi\)
−0.168887 + 0.985635i \(0.554017\pi\)
\(402\) −1.27051 −0.0633673
\(403\) 54.2705 2.70341
\(404\) −11.0213 −0.548329
\(405\) −7.70820 −0.383024
\(406\) −7.85410 −0.389793
\(407\) 11.5623 0.573122
\(408\) 3.37384 0.167030
\(409\) 1.70820 0.0844652 0.0422326 0.999108i \(-0.486553\pi\)
0.0422326 + 0.999108i \(0.486553\pi\)
\(410\) 4.27051 0.210905
\(411\) −0.450850 −0.0222388
\(412\) 3.10333 0.152890
\(413\) −46.9787 −2.31167
\(414\) 0.673762 0.0331136
\(415\) 4.47214 0.219529
\(416\) 20.7295 1.01635
\(417\) −3.11146 −0.152369
\(418\) 0 0
\(419\) 1.05573 0.0515757 0.0257878 0.999667i \(-0.491791\pi\)
0.0257878 + 0.999667i \(0.491791\pi\)
\(420\) 3.00000 0.146385
\(421\) −26.1246 −1.27324 −0.636618 0.771179i \(-0.719667\pi\)
−0.636618 + 0.771179i \(0.719667\pi\)
\(422\) −2.63932 −0.128480
\(423\) 16.4508 0.799868
\(424\) −4.97871 −0.241788
\(425\) −6.00000 −0.291043
\(426\) −1.20163 −0.0582190
\(427\) 12.7082 0.614993
\(428\) 5.56231 0.268864
\(429\) −4.54915 −0.219635
\(430\) 1.47214 0.0709927
\(431\) 25.1803 1.21289 0.606447 0.795124i \(-0.292594\pi\)
0.606447 + 0.795124i \(0.292594\pi\)
\(432\) 7.03444 0.338445
\(433\) 29.2148 1.40397 0.701986 0.712190i \(-0.252297\pi\)
0.701986 + 0.712190i \(0.252297\pi\)
\(434\) −17.5623 −0.843018
\(435\) −1.85410 −0.0888974
\(436\) 3.27051 0.156629
\(437\) 0 0
\(438\) −0.145898 −0.00697128
\(439\) −3.47214 −0.165716 −0.0828580 0.996561i \(-0.526405\pi\)
−0.0828580 + 0.996561i \(0.526405\pi\)
\(440\) 3.50658 0.167170
\(441\) −31.2361 −1.48743
\(442\) −11.4590 −0.545048
\(443\) −29.3607 −1.39497 −0.697484 0.716600i \(-0.745697\pi\)
−0.697484 + 0.716600i \(0.745697\pi\)
\(444\) −3.43769 −0.163146
\(445\) 6.70820 0.317999
\(446\) 3.90983 0.185136
\(447\) 2.90983 0.137630
\(448\) 19.9443 0.942278
\(449\) −41.4721 −1.95719 −0.978596 0.205793i \(-0.934023\pi\)
−0.978596 + 0.205793i \(0.934023\pi\)
\(450\) 1.09017 0.0513911
\(451\) −26.6312 −1.25401
\(452\) −27.7082 −1.30328
\(453\) −2.81966 −0.132479
\(454\) 6.74265 0.316448
\(455\) −21.1803 −0.992950
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 8.25735 0.385841
\(459\) −13.4164 −0.626224
\(460\) 1.14590 0.0534278
\(461\) −1.05573 −0.0491702 −0.0245851 0.999698i \(-0.507826\pi\)
−0.0245851 + 0.999698i \(0.507826\pi\)
\(462\) 1.47214 0.0684900
\(463\) 33.0344 1.53524 0.767620 0.640905i \(-0.221441\pi\)
0.767620 + 0.640905i \(0.221441\pi\)
\(464\) −15.2705 −0.708916
\(465\) −4.14590 −0.192261
\(466\) 2.60488 0.120669
\(467\) −24.7082 −1.14336 −0.571680 0.820477i \(-0.693708\pi\)
−0.571680 + 0.820477i \(0.693708\pi\)
\(468\) −26.4590 −1.22307
\(469\) 36.8885 1.70335
\(470\) −2.20163 −0.101553
\(471\) 9.18034 0.423008
\(472\) 16.3262 0.751476
\(473\) −9.18034 −0.422112
\(474\) −1.16718 −0.0536105
\(475\) 0 0
\(476\) −47.1246 −2.15995
\(477\) 9.65248 0.441957
\(478\) −9.00000 −0.411650
\(479\) −10.6180 −0.485150 −0.242575 0.970133i \(-0.577992\pi\)
−0.242575 + 0.970133i \(0.577992\pi\)
\(480\) −1.58359 −0.0722808
\(481\) 24.2705 1.10664
\(482\) 6.92299 0.315333
\(483\) 1.00000 0.0455016
\(484\) 9.87539 0.448881
\(485\) −5.09017 −0.231133
\(486\) 3.68692 0.167242
\(487\) 35.4164 1.60487 0.802435 0.596739i \(-0.203537\pi\)
0.802435 + 0.596739i \(0.203537\pi\)
\(488\) −4.41641 −0.199921
\(489\) −3.25735 −0.147303
\(490\) 4.18034 0.188849
\(491\) 8.67376 0.391441 0.195721 0.980660i \(-0.437295\pi\)
0.195721 + 0.980660i \(0.437295\pi\)
\(492\) 7.91796 0.356969
\(493\) 29.1246 1.31171
\(494\) 0 0
\(495\) −6.79837 −0.305564
\(496\) −34.1459 −1.53320
\(497\) 34.8885 1.56497
\(498\) −0.652476 −0.0292381
\(499\) 26.6525 1.19313 0.596564 0.802565i \(-0.296532\pi\)
0.596564 + 0.802565i \(0.296532\pi\)
\(500\) 1.85410 0.0829180
\(501\) 3.14590 0.140548
\(502\) −5.79837 −0.258794
\(503\) −2.05573 −0.0916604 −0.0458302 0.998949i \(-0.514593\pi\)
−0.0458302 + 0.998949i \(0.514593\pi\)
\(504\) 17.7984 0.792803
\(505\) −5.94427 −0.264517
\(506\) 0.562306 0.0249975
\(507\) −4.58359 −0.203564
\(508\) 19.8541 0.880883
\(509\) −10.7984 −0.478630 −0.239315 0.970942i \(-0.576923\pi\)
−0.239315 + 0.970942i \(0.576923\pi\)
\(510\) 0.875388 0.0387628
\(511\) 4.23607 0.187393
\(512\) −22.3050 −0.985749
\(513\) 0 0
\(514\) 5.16718 0.227915
\(515\) 1.67376 0.0737548
\(516\) 2.72949 0.120159
\(517\) 13.7295 0.603822
\(518\) −7.85410 −0.345089
\(519\) 5.30495 0.232862
\(520\) 7.36068 0.322787
\(521\) 0.381966 0.0167342 0.00836712 0.999965i \(-0.497337\pi\)
0.00836712 + 0.999965i \(0.497337\pi\)
\(522\) −5.29180 −0.231616
\(523\) −10.5967 −0.463363 −0.231682 0.972792i \(-0.574423\pi\)
−0.231682 + 0.972792i \(0.574423\pi\)
\(524\) 14.5623 0.636157
\(525\) 1.61803 0.0706168
\(526\) 7.59675 0.331234
\(527\) 65.1246 2.83687
\(528\) 2.86223 0.124563
\(529\) −22.6180 −0.983393
\(530\) −1.29180 −0.0561120
\(531\) −31.6525 −1.37360
\(532\) 0 0
\(533\) −55.9017 −2.42137
\(534\) −0.978714 −0.0423531
\(535\) 3.00000 0.129701
\(536\) −12.8197 −0.553725
\(537\) 4.02129 0.173531
\(538\) 0.639320 0.0275631
\(539\) −26.0689 −1.12287
\(540\) 4.14590 0.178411
\(541\) 32.5967 1.40144 0.700722 0.713435i \(-0.252861\pi\)
0.700722 + 0.713435i \(0.252861\pi\)
\(542\) −6.25735 −0.268776
\(543\) 3.16718 0.135917
\(544\) 24.8754 1.06652
\(545\) 1.76393 0.0755585
\(546\) 3.09017 0.132247
\(547\) −37.9787 −1.62385 −0.811926 0.583760i \(-0.801581\pi\)
−0.811926 + 0.583760i \(0.801581\pi\)
\(548\) −2.18847 −0.0934868
\(549\) 8.56231 0.365430
\(550\) 0.909830 0.0387953
\(551\) 0 0
\(552\) −0.347524 −0.0147916
\(553\) 33.8885 1.44109
\(554\) 9.70820 0.412462
\(555\) −1.85410 −0.0787022
\(556\) −15.1033 −0.640524
\(557\) −21.4721 −0.909804 −0.454902 0.890542i \(-0.650326\pi\)
−0.454902 + 0.890542i \(0.650326\pi\)
\(558\) −11.8328 −0.500923
\(559\) −19.2705 −0.815056
\(560\) 13.3262 0.563136
\(561\) −5.45898 −0.230478
\(562\) 10.5066 0.443193
\(563\) 30.1803 1.27195 0.635975 0.771710i \(-0.280598\pi\)
0.635975 + 0.771710i \(0.280598\pi\)
\(564\) −4.08204 −0.171885
\(565\) −14.9443 −0.628710
\(566\) −1.76393 −0.0741436
\(567\) −32.6525 −1.37128
\(568\) −12.1246 −0.508737
\(569\) 0.437694 0.0183491 0.00917455 0.999958i \(-0.497080\pi\)
0.00917455 + 0.999958i \(0.497080\pi\)
\(570\) 0 0
\(571\) 26.8541 1.12381 0.561905 0.827202i \(-0.310069\pi\)
0.561905 + 0.827202i \(0.310069\pi\)
\(572\) −22.0820 −0.923296
\(573\) −2.31308 −0.0966304
\(574\) 18.0902 0.755069
\(575\) 0.618034 0.0257738
\(576\) 13.4377 0.559904
\(577\) 19.7639 0.822783 0.411392 0.911459i \(-0.365043\pi\)
0.411392 + 0.911459i \(0.365043\pi\)
\(578\) −7.25735 −0.301866
\(579\) 1.52786 0.0634959
\(580\) −9.00000 −0.373705
\(581\) 18.9443 0.785941
\(582\) 0.742646 0.0307837
\(583\) 8.05573 0.333634
\(584\) −1.47214 −0.0609174
\(585\) −14.2705 −0.590013
\(586\) 8.41641 0.347679
\(587\) −42.2361 −1.74327 −0.871635 0.490156i \(-0.836940\pi\)
−0.871635 + 0.490156i \(0.836940\pi\)
\(588\) 7.75078 0.319637
\(589\) 0 0
\(590\) 4.23607 0.174396
\(591\) −6.31308 −0.259686
\(592\) −15.2705 −0.627614
\(593\) −15.6525 −0.642770 −0.321385 0.946949i \(-0.604148\pi\)
−0.321385 + 0.946949i \(0.604148\pi\)
\(594\) 2.03444 0.0834742
\(595\) −25.4164 −1.04197
\(596\) 14.1246 0.578567
\(597\) −5.12461 −0.209736
\(598\) 1.18034 0.0482677
\(599\) −7.36068 −0.300749 −0.150375 0.988629i \(-0.548048\pi\)
−0.150375 + 0.988629i \(0.548048\pi\)
\(600\) −0.562306 −0.0229560
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 6.23607 0.254163
\(603\) 24.8541 1.01214
\(604\) −13.6869 −0.556913
\(605\) 5.32624 0.216542
\(606\) 0.867258 0.0352299
\(607\) 16.5623 0.672243 0.336122 0.941819i \(-0.390885\pi\)
0.336122 + 0.941819i \(0.390885\pi\)
\(608\) 0 0
\(609\) −7.85410 −0.318264
\(610\) −1.14590 −0.0463961
\(611\) 28.8197 1.16592
\(612\) −31.7508 −1.28345
\(613\) 4.05573 0.163809 0.0819047 0.996640i \(-0.473900\pi\)
0.0819047 + 0.996640i \(0.473900\pi\)
\(614\) −4.68692 −0.189149
\(615\) 4.27051 0.172204
\(616\) 14.8541 0.598489
\(617\) −12.9098 −0.519730 −0.259865 0.965645i \(-0.583678\pi\)
−0.259865 + 0.965645i \(0.583678\pi\)
\(618\) −0.244199 −0.00982311
\(619\) −8.05573 −0.323787 −0.161894 0.986808i \(-0.551760\pi\)
−0.161894 + 0.986808i \(0.551760\pi\)
\(620\) −20.1246 −0.808224
\(621\) 1.38197 0.0554564
\(622\) 2.69505 0.108062
\(623\) 28.4164 1.13848
\(624\) 6.00813 0.240518
\(625\) 1.00000 0.0400000
\(626\) 3.74265 0.149586
\(627\) 0 0
\(628\) 44.5623 1.77823
\(629\) 29.1246 1.16127
\(630\) 4.61803 0.183987
\(631\) 15.0000 0.597141 0.298570 0.954388i \(-0.403490\pi\)
0.298570 + 0.954388i \(0.403490\pi\)
\(632\) −11.7771 −0.468467
\(633\) −2.63932 −0.104904
\(634\) −5.05573 −0.200789
\(635\) 10.7082 0.424942
\(636\) −2.39512 −0.0949728
\(637\) −54.7214 −2.16814
\(638\) −4.41641 −0.174847
\(639\) 23.5066 0.929906
\(640\) −10.0902 −0.398849
\(641\) −5.72949 −0.226301 −0.113151 0.993578i \(-0.536094\pi\)
−0.113151 + 0.993578i \(0.536094\pi\)
\(642\) −0.437694 −0.0172744
\(643\) 31.5967 1.24605 0.623027 0.782200i \(-0.285903\pi\)
0.623027 + 0.782200i \(0.285903\pi\)
\(644\) 4.85410 0.191278
\(645\) 1.47214 0.0579653
\(646\) 0 0
\(647\) 36.3050 1.42729 0.713647 0.700505i \(-0.247042\pi\)
0.713647 + 0.700505i \(0.247042\pi\)
\(648\) 11.3475 0.445773
\(649\) −26.4164 −1.03693
\(650\) 1.90983 0.0749097
\(651\) −17.5623 −0.688321
\(652\) −15.8115 −0.619227
\(653\) 7.85410 0.307355 0.153677 0.988121i \(-0.450888\pi\)
0.153677 + 0.988121i \(0.450888\pi\)
\(654\) −0.257354 −0.0100633
\(655\) 7.85410 0.306885
\(656\) 35.1722 1.37324
\(657\) 2.85410 0.111349
\(658\) −9.32624 −0.363575
\(659\) −42.4721 −1.65448 −0.827240 0.561849i \(-0.810090\pi\)
−0.827240 + 0.561849i \(0.810090\pi\)
\(660\) 1.68692 0.0656632
\(661\) 3.70820 0.144232 0.0721162 0.997396i \(-0.477025\pi\)
0.0721162 + 0.997396i \(0.477025\pi\)
\(662\) −0.111456 −0.00433187
\(663\) −11.4590 −0.445030
\(664\) −6.58359 −0.255493
\(665\) 0 0
\(666\) −5.29180 −0.205053
\(667\) −3.00000 −0.116160
\(668\) 15.2705 0.590834
\(669\) 3.90983 0.151163
\(670\) −3.32624 −0.128504
\(671\) 7.14590 0.275864
\(672\) −6.70820 −0.258775
\(673\) 11.1115 0.428315 0.214158 0.976799i \(-0.431299\pi\)
0.214158 + 0.976799i \(0.431299\pi\)
\(674\) 10.6950 0.411958
\(675\) 2.23607 0.0860663
\(676\) −22.2492 −0.855739
\(677\) 8.38197 0.322145 0.161073 0.986943i \(-0.448505\pi\)
0.161073 + 0.986943i \(0.448505\pi\)
\(678\) 2.18034 0.0837354
\(679\) −21.5623 −0.827485
\(680\) 8.83282 0.338723
\(681\) 6.74265 0.258379
\(682\) −9.87539 −0.378148
\(683\) −23.4721 −0.898136 −0.449068 0.893498i \(-0.648244\pi\)
−0.449068 + 0.893498i \(0.648244\pi\)
\(684\) 0 0
\(685\) −1.18034 −0.0450985
\(686\) 6.38197 0.243665
\(687\) 8.25735 0.315038
\(688\) 12.1246 0.462246
\(689\) 16.9098 0.644213
\(690\) −0.0901699 −0.00343271
\(691\) 21.4721 0.816839 0.408419 0.912794i \(-0.366080\pi\)
0.408419 + 0.912794i \(0.366080\pi\)
\(692\) 25.7508 0.978898
\(693\) −28.7984 −1.09396
\(694\) 7.59675 0.288369
\(695\) −8.14590 −0.308992
\(696\) 2.72949 0.103461
\(697\) −67.0820 −2.54091
\(698\) −8.30495 −0.314347
\(699\) 2.60488 0.0985255
\(700\) 7.85410 0.296857
\(701\) 48.5755 1.83467 0.917335 0.398116i \(-0.130336\pi\)
0.917335 + 0.398116i \(0.130336\pi\)
\(702\) 4.27051 0.161180
\(703\) 0 0
\(704\) 11.2148 0.422673
\(705\) −2.20163 −0.0829180
\(706\) 13.8115 0.519804
\(707\) −25.1803 −0.947004
\(708\) 7.85410 0.295175
\(709\) −15.7082 −0.589934 −0.294967 0.955507i \(-0.595309\pi\)
−0.294967 + 0.955507i \(0.595309\pi\)
\(710\) −3.14590 −0.118063
\(711\) 22.8328 0.856297
\(712\) −9.87539 −0.370096
\(713\) −6.70820 −0.251224
\(714\) 3.70820 0.138776
\(715\) −11.9098 −0.445402
\(716\) 19.5197 0.729487
\(717\) −9.00000 −0.336111
\(718\) −10.0770 −0.376071
\(719\) 12.2016 0.455044 0.227522 0.973773i \(-0.426938\pi\)
0.227522 + 0.973773i \(0.426938\pi\)
\(720\) 8.97871 0.334617
\(721\) 7.09017 0.264052
\(722\) 0 0
\(723\) 6.92299 0.257469
\(724\) 15.3738 0.571364
\(725\) −4.85410 −0.180277
\(726\) −0.777088 −0.0288404
\(727\) −10.5836 −0.392524 −0.196262 0.980552i \(-0.562880\pi\)
−0.196262 + 0.980552i \(0.562880\pi\)
\(728\) 31.1803 1.15562
\(729\) −19.4377 −0.719915
\(730\) −0.381966 −0.0141372
\(731\) −23.1246 −0.855295
\(732\) −2.12461 −0.0785279
\(733\) 0.236068 0.00871937 0.00435968 0.999990i \(-0.498612\pi\)
0.00435968 + 0.999990i \(0.498612\pi\)
\(734\) −0.631190 −0.0232976
\(735\) 4.18034 0.154194
\(736\) −2.56231 −0.0944478
\(737\) 20.7426 0.764065
\(738\) 12.1885 0.448664
\(739\) −21.8328 −0.803133 −0.401567 0.915830i \(-0.631534\pi\)
−0.401567 + 0.915830i \(0.631534\pi\)
\(740\) −9.00000 −0.330847
\(741\) 0 0
\(742\) −5.47214 −0.200888
\(743\) −36.2361 −1.32937 −0.664686 0.747123i \(-0.731435\pi\)
−0.664686 + 0.747123i \(0.731435\pi\)
\(744\) 6.10333 0.223759
\(745\) 7.61803 0.279103
\(746\) −2.56231 −0.0938127
\(747\) 12.7639 0.467008
\(748\) −26.4984 −0.968879
\(749\) 12.7082 0.464348
\(750\) −0.145898 −0.00532744
\(751\) 2.56231 0.0934999 0.0467499 0.998907i \(-0.485114\pi\)
0.0467499 + 0.998907i \(0.485114\pi\)
\(752\) −18.1327 −0.661233
\(753\) −5.79837 −0.211304
\(754\) −9.27051 −0.337612
\(755\) −7.38197 −0.268657
\(756\) 17.5623 0.638735
\(757\) −14.3820 −0.522721 −0.261361 0.965241i \(-0.584171\pi\)
−0.261361 + 0.965241i \(0.584171\pi\)
\(758\) −10.2492 −0.372269
\(759\) 0.562306 0.0204104
\(760\) 0 0
\(761\) 37.3607 1.35432 0.677162 0.735834i \(-0.263210\pi\)
0.677162 + 0.735834i \(0.263210\pi\)
\(762\) −1.56231 −0.0565964
\(763\) 7.47214 0.270509
\(764\) −11.2279 −0.406213
\(765\) −17.1246 −0.619142
\(766\) 12.5755 0.454370
\(767\) −55.4508 −2.00221
\(768\) −2.12461 −0.0766653
\(769\) 13.6869 0.493563 0.246781 0.969071i \(-0.420627\pi\)
0.246781 + 0.969071i \(0.420627\pi\)
\(770\) 3.85410 0.138892
\(771\) 5.16718 0.186092
\(772\) 7.41641 0.266922
\(773\) 39.7426 1.42944 0.714722 0.699409i \(-0.246553\pi\)
0.714722 + 0.699409i \(0.246553\pi\)
\(774\) 4.20163 0.151024
\(775\) −10.8541 −0.389891
\(776\) 7.49342 0.268998
\(777\) −7.85410 −0.281764
\(778\) 9.29180 0.333127
\(779\) 0 0
\(780\) 3.54102 0.126789
\(781\) 19.6180 0.701988
\(782\) 1.41641 0.0506506
\(783\) −10.8541 −0.387894
\(784\) 34.4296 1.22963
\(785\) 24.0344 0.857826
\(786\) −1.14590 −0.0408728
\(787\) 35.4853 1.26491 0.632457 0.774595i \(-0.282046\pi\)
0.632457 + 0.774595i \(0.282046\pi\)
\(788\) −30.6443 −1.09166
\(789\) 7.59675 0.270451
\(790\) −3.05573 −0.108718
\(791\) −63.3050 −2.25086
\(792\) 10.0081 0.355623
\(793\) 15.0000 0.532666
\(794\) 6.45085 0.228932
\(795\) −1.29180 −0.0458153
\(796\) −24.8754 −0.881685
\(797\) 2.18034 0.0772316 0.0386158 0.999254i \(-0.487705\pi\)
0.0386158 + 0.999254i \(0.487705\pi\)
\(798\) 0 0
\(799\) 34.5836 1.22348
\(800\) −4.14590 −0.146580
\(801\) 19.1459 0.676487
\(802\) 2.58359 0.0912298
\(803\) 2.38197 0.0840578
\(804\) −6.16718 −0.217500
\(805\) 2.61803 0.0922736
\(806\) −20.7295 −0.730165
\(807\) 0.639320 0.0225051
\(808\) 8.75078 0.307851
\(809\) 28.9098 1.01642 0.508208 0.861235i \(-0.330308\pi\)
0.508208 + 0.861235i \(0.330308\pi\)
\(810\) 2.94427 0.103451
\(811\) −40.3262 −1.41605 −0.708023 0.706190i \(-0.750413\pi\)
−0.708023 + 0.706190i \(0.750413\pi\)
\(812\) −38.1246 −1.33791
\(813\) −6.25735 −0.219455
\(814\) −4.41641 −0.154795
\(815\) −8.52786 −0.298718
\(816\) 7.20976 0.252392
\(817\) 0 0
\(818\) −0.652476 −0.0228133
\(819\) −60.4508 −2.11232
\(820\) 20.7295 0.723905
\(821\) 12.3820 0.432134 0.216067 0.976379i \(-0.430677\pi\)
0.216067 + 0.976379i \(0.430677\pi\)
\(822\) 0.172209 0.00600649
\(823\) 8.36068 0.291435 0.145717 0.989326i \(-0.453451\pi\)
0.145717 + 0.989326i \(0.453451\pi\)
\(824\) −2.46401 −0.0858377
\(825\) 0.909830 0.0316762
\(826\) 17.9443 0.624361
\(827\) 35.8328 1.24603 0.623015 0.782210i \(-0.285908\pi\)
0.623015 + 0.782210i \(0.285908\pi\)
\(828\) 3.27051 0.113658
\(829\) −43.3951 −1.50717 −0.753587 0.657348i \(-0.771678\pi\)
−0.753587 + 0.657348i \(0.771678\pi\)
\(830\) −1.70820 −0.0592926
\(831\) 9.70820 0.336774
\(832\) 23.5410 0.816138
\(833\) −65.6656 −2.27518
\(834\) 1.18847 0.0411534
\(835\) 8.23607 0.285021
\(836\) 0 0
\(837\) −24.2705 −0.838912
\(838\) −0.403252 −0.0139301
\(839\) 13.3820 0.461997 0.230998 0.972954i \(-0.425801\pi\)
0.230998 + 0.972954i \(0.425801\pi\)
\(840\) −2.38197 −0.0821856
\(841\) −5.43769 −0.187507
\(842\) 9.97871 0.343889
\(843\) 10.5066 0.361866
\(844\) −12.8115 −0.440991
\(845\) −12.0000 −0.412813
\(846\) −6.28367 −0.216037
\(847\) 22.5623 0.775250
\(848\) −10.6393 −0.365356
\(849\) −1.76393 −0.0605380
\(850\) 2.29180 0.0786080
\(851\) −3.00000 −0.102839
\(852\) −5.83282 −0.199829
\(853\) 52.9574 1.81323 0.906614 0.421961i \(-0.138658\pi\)
0.906614 + 0.421961i \(0.138658\pi\)
\(854\) −4.85410 −0.166104
\(855\) 0 0
\(856\) −4.41641 −0.150950
\(857\) 32.6180 1.11421 0.557105 0.830442i \(-0.311912\pi\)
0.557105 + 0.830442i \(0.311912\pi\)
\(858\) 1.73762 0.0593214
\(859\) 13.5967 0.463915 0.231958 0.972726i \(-0.425487\pi\)
0.231958 + 0.972726i \(0.425487\pi\)
\(860\) 7.14590 0.243673
\(861\) 18.0902 0.616511
\(862\) −9.61803 −0.327592
\(863\) 4.65248 0.158372 0.0791861 0.996860i \(-0.474768\pi\)
0.0791861 + 0.996860i \(0.474768\pi\)
\(864\) −9.27051 −0.315389
\(865\) 13.8885 0.472225
\(866\) −11.1591 −0.379200
\(867\) −7.25735 −0.246473
\(868\) −85.2492 −2.89355
\(869\) 19.0557 0.646421
\(870\) 0.708204 0.0240104
\(871\) 43.5410 1.47533
\(872\) −2.59675 −0.0879370
\(873\) −14.5279 −0.491694
\(874\) 0 0
\(875\) 4.23607 0.143205
\(876\) −0.708204 −0.0239280
\(877\) 7.81966 0.264051 0.132026 0.991246i \(-0.457852\pi\)
0.132026 + 0.991246i \(0.457852\pi\)
\(878\) 1.32624 0.0447584
\(879\) 8.41641 0.283878
\(880\) 7.49342 0.252603
\(881\) 25.0344 0.843432 0.421716 0.906728i \(-0.361428\pi\)
0.421716 + 0.906728i \(0.361428\pi\)
\(882\) 11.9311 0.401742
\(883\) 46.9230 1.57908 0.789542 0.613696i \(-0.210318\pi\)
0.789542 + 0.613696i \(0.210318\pi\)
\(884\) −55.6231 −1.87081
\(885\) 4.23607 0.142394
\(886\) 11.2148 0.376768
\(887\) −12.4721 −0.418773 −0.209387 0.977833i \(-0.567147\pi\)
−0.209387 + 0.977833i \(0.567147\pi\)
\(888\) 2.72949 0.0915957
\(889\) 45.3607 1.52135
\(890\) −2.56231 −0.0858887
\(891\) −18.3607 −0.615106
\(892\) 18.9787 0.635454
\(893\) 0 0
\(894\) −1.11146 −0.0371727
\(895\) 10.5279 0.351908
\(896\) −42.7426 −1.42793
\(897\) 1.18034 0.0394104
\(898\) 15.8409 0.528619
\(899\) 52.6869 1.75721
\(900\) 5.29180 0.176393
\(901\) 20.2918 0.676018
\(902\) 10.1722 0.338698
\(903\) 6.23607 0.207523
\(904\) 22.0000 0.731709
\(905\) 8.29180 0.275629
\(906\) 1.07701 0.0357814
\(907\) 32.2492 1.07082 0.535409 0.844593i \(-0.320158\pi\)
0.535409 + 0.844593i \(0.320158\pi\)
\(908\) 32.7295 1.08617
\(909\) −16.9656 −0.562712
\(910\) 8.09017 0.268187
\(911\) −36.1033 −1.19616 −0.598078 0.801438i \(-0.704069\pi\)
−0.598078 + 0.801438i \(0.704069\pi\)
\(912\) 0 0
\(913\) 10.6525 0.352545
\(914\) 2.29180 0.0758059
\(915\) −1.14590 −0.0378822
\(916\) 40.0820 1.32435
\(917\) 33.2705 1.09869
\(918\) 5.12461 0.169137
\(919\) 22.8885 0.755023 0.377512 0.926005i \(-0.376780\pi\)
0.377512 + 0.926005i \(0.376780\pi\)
\(920\) −0.909830 −0.0299962
\(921\) −4.68692 −0.154439
\(922\) 0.403252 0.0132804
\(923\) 41.1803 1.35547
\(924\) 7.14590 0.235083
\(925\) −4.85410 −0.159602
\(926\) −12.6180 −0.414654
\(927\) 4.77709 0.156900
\(928\) 20.1246 0.660623
\(929\) 11.6180 0.381175 0.190588 0.981670i \(-0.438961\pi\)
0.190588 + 0.981670i \(0.438961\pi\)
\(930\) 1.58359 0.0519280
\(931\) 0 0
\(932\) 12.6443 0.414179
\(933\) 2.69505 0.0882319
\(934\) 9.43769 0.308811
\(935\) −14.2918 −0.467392
\(936\) 21.0081 0.686672
\(937\) 2.56231 0.0837069 0.0418534 0.999124i \(-0.486674\pi\)
0.0418534 + 0.999124i \(0.486674\pi\)
\(938\) −14.0902 −0.460060
\(939\) 3.74265 0.122137
\(940\) −10.6869 −0.348569
\(941\) −10.5066 −0.342505 −0.171252 0.985227i \(-0.554781\pi\)
−0.171252 + 0.985227i \(0.554781\pi\)
\(942\) −3.50658 −0.114250
\(943\) 6.90983 0.225015
\(944\) 34.8885 1.13553
\(945\) 9.47214 0.308129
\(946\) 3.50658 0.114009
\(947\) 48.0000 1.55979 0.779895 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) −5.66563 −0.184011
\(949\) 5.00000 0.162307
\(950\) 0 0
\(951\) −5.05573 −0.163943
\(952\) 37.4164 1.21267
\(953\) 11.8328 0.383302 0.191651 0.981463i \(-0.438616\pi\)
0.191651 + 0.981463i \(0.438616\pi\)
\(954\) −3.68692 −0.119368
\(955\) −6.05573 −0.195959
\(956\) −43.6869 −1.41294
\(957\) −4.41641 −0.142762
\(958\) 4.05573 0.131035
\(959\) −5.00000 −0.161458
\(960\) −1.79837 −0.0580423
\(961\) 86.8115 2.80037
\(962\) −9.27051 −0.298893
\(963\) 8.56231 0.275916
\(964\) 33.6049 1.08234
\(965\) 4.00000 0.128765
\(966\) −0.381966 −0.0122896
\(967\) 28.7639 0.924986 0.462493 0.886623i \(-0.346955\pi\)
0.462493 + 0.886623i \(0.346955\pi\)
\(968\) −7.84095 −0.252018
\(969\) 0 0
\(970\) 1.94427 0.0624268
\(971\) 37.7984 1.21301 0.606504 0.795081i \(-0.292572\pi\)
0.606504 + 0.795081i \(0.292572\pi\)
\(972\) 17.8967 0.574036
\(973\) −34.5066 −1.10623
\(974\) −13.5279 −0.433461
\(975\) 1.90983 0.0611635
\(976\) −9.43769 −0.302093
\(977\) −53.6525 −1.71649 −0.858247 0.513236i \(-0.828446\pi\)
−0.858247 + 0.513236i \(0.828446\pi\)
\(978\) 1.24420 0.0397851
\(979\) 15.9787 0.510682
\(980\) 20.2918 0.648198
\(981\) 5.03444 0.160737
\(982\) −3.31308 −0.105725
\(983\) −13.8541 −0.441877 −0.220939 0.975288i \(-0.570912\pi\)
−0.220939 + 0.975288i \(0.570912\pi\)
\(984\) −6.28677 −0.200415
\(985\) −16.5279 −0.526622
\(986\) −11.1246 −0.354280
\(987\) −9.32624 −0.296857
\(988\) 0 0
\(989\) 2.38197 0.0757421
\(990\) 2.59675 0.0825301
\(991\) −18.1459 −0.576423 −0.288212 0.957567i \(-0.593061\pi\)
−0.288212 + 0.957567i \(0.593061\pi\)
\(992\) 45.0000 1.42875
\(993\) −0.111456 −0.00353695
\(994\) −13.3262 −0.422683
\(995\) −13.4164 −0.425329
\(996\) −3.16718 −0.100356
\(997\) −3.23607 −0.102487 −0.0512437 0.998686i \(-0.516319\pi\)
−0.0512437 + 0.998686i \(0.516319\pi\)
\(998\) −10.1803 −0.322253
\(999\) −10.8541 −0.343409
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 1805.2.a.c.1.2 2
5.4 even 2 9025.2.a.v.1.1 2
19.18 odd 2 1805.2.a.e.1.1 yes 2
95.94 odd 2 9025.2.a.k.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.c.1.2 2 1.1 even 1 trivial
1805.2.a.e.1.1 yes 2 19.18 odd 2
9025.2.a.k.1.2 2 95.94 odd 2
9025.2.a.v.1.1 2 5.4 even 2