Properties

Label 8007.2.a.f.1.23
Level 8007
Weight 2
Character 8007.1
Self dual yes
Analytic conductor 63.936
Analytic rank 1
Dimension 48
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8007 = 3 \cdot 17 \cdot 157 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8007.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9362168984\)
Analytic rank: \(1\)
Dimension: \(48\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.23
Character \(\chi\) = 8007.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.206596 q^{2} -1.00000 q^{3} -1.95732 q^{4} -2.96369 q^{5} +0.206596 q^{6} -3.29821 q^{7} +0.817567 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.206596 q^{2} -1.00000 q^{3} -1.95732 q^{4} -2.96369 q^{5} +0.206596 q^{6} -3.29821 q^{7} +0.817567 q^{8} +1.00000 q^{9} +0.612286 q^{10} -2.02436 q^{11} +1.95732 q^{12} -4.52233 q^{13} +0.681397 q^{14} +2.96369 q^{15} +3.74573 q^{16} -1.00000 q^{17} -0.206596 q^{18} +2.84222 q^{19} +5.80088 q^{20} +3.29821 q^{21} +0.418225 q^{22} -6.15139 q^{23} -0.817567 q^{24} +3.78344 q^{25} +0.934296 q^{26} -1.00000 q^{27} +6.45564 q^{28} +3.35506 q^{29} -0.612286 q^{30} -3.97663 q^{31} -2.40899 q^{32} +2.02436 q^{33} +0.206596 q^{34} +9.77486 q^{35} -1.95732 q^{36} -6.02124 q^{37} -0.587192 q^{38} +4.52233 q^{39} -2.42301 q^{40} -5.50704 q^{41} -0.681397 q^{42} -0.805249 q^{43} +3.96231 q^{44} -2.96369 q^{45} +1.27085 q^{46} +2.71134 q^{47} -3.74573 q^{48} +3.87818 q^{49} -0.781645 q^{50} +1.00000 q^{51} +8.85163 q^{52} +7.81249 q^{53} +0.206596 q^{54} +5.99957 q^{55} -2.69651 q^{56} -2.84222 q^{57} -0.693142 q^{58} +0.000491989 q^{59} -5.80088 q^{60} +10.2945 q^{61} +0.821556 q^{62} -3.29821 q^{63} -6.99377 q^{64} +13.4028 q^{65} -0.418225 q^{66} +6.98424 q^{67} +1.95732 q^{68} +6.15139 q^{69} -2.01945 q^{70} +12.7098 q^{71} +0.817567 q^{72} -0.595084 q^{73} +1.24397 q^{74} -3.78344 q^{75} -5.56313 q^{76} +6.67676 q^{77} -0.934296 q^{78} +10.0119 q^{79} -11.1012 q^{80} +1.00000 q^{81} +1.13773 q^{82} -10.2303 q^{83} -6.45564 q^{84} +2.96369 q^{85} +0.166361 q^{86} -3.35506 q^{87} -1.65505 q^{88} +6.31716 q^{89} +0.612286 q^{90} +14.9156 q^{91} +12.0402 q^{92} +3.97663 q^{93} -0.560152 q^{94} -8.42345 q^{95} +2.40899 q^{96} +6.62727 q^{97} -0.801217 q^{98} -2.02436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48q - q^{2} - 48q^{3} + 45q^{4} + q^{5} + q^{6} - 13q^{7} - 6q^{8} + 48q^{9} + O(q^{10}) \) \( 48q - q^{2} - 48q^{3} + 45q^{4} + q^{5} + q^{6} - 13q^{7} - 6q^{8} + 48q^{9} - 20q^{10} + 5q^{11} - 45q^{12} - 8q^{13} + 4q^{14} - q^{15} + 39q^{16} - 48q^{17} - q^{18} - 6q^{19} + 6q^{20} + 13q^{21} - 35q^{22} - 8q^{23} + 6q^{24} + 13q^{25} + 17q^{26} - 48q^{27} - 38q^{28} + q^{29} + 20q^{30} - 21q^{31} - 3q^{32} - 5q^{33} + q^{34} + 19q^{35} + 45q^{36} - 58q^{37} - 14q^{38} + 8q^{39} - 54q^{40} - 3q^{41} - 4q^{42} - 33q^{43} + 2q^{44} + q^{45} - 26q^{46} + 9q^{47} - 39q^{48} + 11q^{49} + 4q^{50} + 48q^{51} - 31q^{52} - 33q^{53} + q^{54} - 21q^{55} + 6q^{57} - 55q^{58} + 77q^{59} - 6q^{60} - 29q^{61} - 46q^{62} - 13q^{63} + 24q^{64} - 49q^{65} + 35q^{66} - 44q^{67} - 45q^{68} + 8q^{69} + 4q^{70} + 22q^{71} - 6q^{72} - 63q^{73} - 16q^{74} - 13q^{75} - 46q^{76} - 30q^{77} - 17q^{78} - 46q^{79} - 14q^{80} + 48q^{81} - 75q^{82} + 11q^{83} + 38q^{84} - q^{85} + 8q^{86} - q^{87} - 116q^{88} + 10q^{89} - 20q^{90} - 67q^{91} - 64q^{92} + 21q^{93} - 16q^{94} - 8q^{95} + 3q^{96} - 96q^{97} - 46q^{98} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.206596 −0.146086 −0.0730428 0.997329i \(-0.523271\pi\)
−0.0730428 + 0.997329i \(0.523271\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.95732 −0.978659
\(5\) −2.96369 −1.32540 −0.662701 0.748884i \(-0.730590\pi\)
−0.662701 + 0.748884i \(0.730590\pi\)
\(6\) 0.206596 0.0843425
\(7\) −3.29821 −1.24661 −0.623303 0.781981i \(-0.714210\pi\)
−0.623303 + 0.781981i \(0.714210\pi\)
\(8\) 0.817567 0.289054
\(9\) 1.00000 0.333333
\(10\) 0.612286 0.193622
\(11\) −2.02436 −0.610367 −0.305184 0.952294i \(-0.598718\pi\)
−0.305184 + 0.952294i \(0.598718\pi\)
\(12\) 1.95732 0.565029
\(13\) −4.52233 −1.25427 −0.627134 0.778911i \(-0.715772\pi\)
−0.627134 + 0.778911i \(0.715772\pi\)
\(14\) 0.681397 0.182111
\(15\) 2.96369 0.765221
\(16\) 3.74573 0.936432
\(17\) −1.00000 −0.242536
\(18\) −0.206596 −0.0486952
\(19\) 2.84222 0.652050 0.326025 0.945361i \(-0.394291\pi\)
0.326025 + 0.945361i \(0.394291\pi\)
\(20\) 5.80088 1.29712
\(21\) 3.29821 0.719728
\(22\) 0.418225 0.0891658
\(23\) −6.15139 −1.28265 −0.641327 0.767268i \(-0.721616\pi\)
−0.641327 + 0.767268i \(0.721616\pi\)
\(24\) −0.817567 −0.166885
\(25\) 3.78344 0.756688
\(26\) 0.934296 0.183230
\(27\) −1.00000 −0.192450
\(28\) 6.45564 1.22000
\(29\) 3.35506 0.623018 0.311509 0.950243i \(-0.399166\pi\)
0.311509 + 0.950243i \(0.399166\pi\)
\(30\) −0.612286 −0.111788
\(31\) −3.97663 −0.714224 −0.357112 0.934062i \(-0.616238\pi\)
−0.357112 + 0.934062i \(0.616238\pi\)
\(32\) −2.40899 −0.425853
\(33\) 2.02436 0.352396
\(34\) 0.206596 0.0354310
\(35\) 9.77486 1.65225
\(36\) −1.95732 −0.326220
\(37\) −6.02124 −0.989886 −0.494943 0.868926i \(-0.664811\pi\)
−0.494943 + 0.868926i \(0.664811\pi\)
\(38\) −0.587192 −0.0952551
\(39\) 4.52233 0.724152
\(40\) −2.42301 −0.383112
\(41\) −5.50704 −0.860055 −0.430028 0.902816i \(-0.641496\pi\)
−0.430028 + 0.902816i \(0.641496\pi\)
\(42\) −0.681397 −0.105142
\(43\) −0.805249 −0.122799 −0.0613996 0.998113i \(-0.519556\pi\)
−0.0613996 + 0.998113i \(0.519556\pi\)
\(44\) 3.96231 0.597341
\(45\) −2.96369 −0.441800
\(46\) 1.27085 0.187377
\(47\) 2.71134 0.395489 0.197745 0.980254i \(-0.436638\pi\)
0.197745 + 0.980254i \(0.436638\pi\)
\(48\) −3.74573 −0.540650
\(49\) 3.87818 0.554026
\(50\) −0.781645 −0.110541
\(51\) 1.00000 0.140028
\(52\) 8.85163 1.22750
\(53\) 7.81249 1.07313 0.536564 0.843860i \(-0.319722\pi\)
0.536564 + 0.843860i \(0.319722\pi\)
\(54\) 0.206596 0.0281142
\(55\) 5.99957 0.808981
\(56\) −2.69651 −0.360336
\(57\) −2.84222 −0.376461
\(58\) −0.693142 −0.0910140
\(59\) 0.000491989 0 6.40516e−5 0 3.20258e−5 1.00000i \(-0.499990\pi\)
3.20258e−5 1.00000i \(0.499990\pi\)
\(60\) −5.80088 −0.748890
\(61\) 10.2945 1.31808 0.659040 0.752108i \(-0.270963\pi\)
0.659040 + 0.752108i \(0.270963\pi\)
\(62\) 0.821556 0.104338
\(63\) −3.29821 −0.415535
\(64\) −6.99377 −0.874222
\(65\) 13.4028 1.66241
\(66\) −0.418225 −0.0514799
\(67\) 6.98424 0.853261 0.426630 0.904426i \(-0.359701\pi\)
0.426630 + 0.904426i \(0.359701\pi\)
\(68\) 1.95732 0.237360
\(69\) 6.15139 0.740540
\(70\) −2.01945 −0.241370
\(71\) 12.7098 1.50838 0.754189 0.656657i \(-0.228030\pi\)
0.754189 + 0.656657i \(0.228030\pi\)
\(72\) 0.817567 0.0963512
\(73\) −0.595084 −0.0696493 −0.0348246 0.999393i \(-0.511087\pi\)
−0.0348246 + 0.999393i \(0.511087\pi\)
\(74\) 1.24397 0.144608
\(75\) −3.78344 −0.436874
\(76\) −5.56313 −0.638134
\(77\) 6.67676 0.760887
\(78\) −0.934296 −0.105788
\(79\) 10.0119 1.12643 0.563216 0.826310i \(-0.309564\pi\)
0.563216 + 0.826310i \(0.309564\pi\)
\(80\) −11.1012 −1.24115
\(81\) 1.00000 0.111111
\(82\) 1.13773 0.125642
\(83\) −10.2303 −1.12293 −0.561463 0.827502i \(-0.689761\pi\)
−0.561463 + 0.827502i \(0.689761\pi\)
\(84\) −6.45564 −0.704368
\(85\) 2.96369 0.321457
\(86\) 0.166361 0.0179392
\(87\) −3.35506 −0.359700
\(88\) −1.65505 −0.176429
\(89\) 6.31716 0.669618 0.334809 0.942286i \(-0.391328\pi\)
0.334809 + 0.942286i \(0.391328\pi\)
\(90\) 0.612286 0.0645407
\(91\) 14.9156 1.56358
\(92\) 12.0402 1.25528
\(93\) 3.97663 0.412357
\(94\) −0.560152 −0.0577752
\(95\) −8.42345 −0.864228
\(96\) 2.40899 0.245866
\(97\) 6.62727 0.672898 0.336449 0.941702i \(-0.390774\pi\)
0.336449 + 0.941702i \(0.390774\pi\)
\(98\) −0.801217 −0.0809352
\(99\) −2.02436 −0.203456
\(100\) −7.40540 −0.740540
\(101\) 4.37106 0.434937 0.217468 0.976067i \(-0.430220\pi\)
0.217468 + 0.976067i \(0.430220\pi\)
\(102\) −0.206596 −0.0204561
\(103\) 10.8198 1.06610 0.533052 0.846082i \(-0.321045\pi\)
0.533052 + 0.846082i \(0.321045\pi\)
\(104\) −3.69731 −0.362551
\(105\) −9.77486 −0.953929
\(106\) −1.61403 −0.156769
\(107\) 13.2838 1.28419 0.642097 0.766624i \(-0.278065\pi\)
0.642097 + 0.766624i \(0.278065\pi\)
\(108\) 1.95732 0.188343
\(109\) 7.51938 0.720226 0.360113 0.932909i \(-0.382738\pi\)
0.360113 + 0.932909i \(0.382738\pi\)
\(110\) −1.23949 −0.118181
\(111\) 6.02124 0.571511
\(112\) −12.3542 −1.16736
\(113\) 6.82917 0.642434 0.321217 0.947006i \(-0.395908\pi\)
0.321217 + 0.947006i \(0.395908\pi\)
\(114\) 0.587192 0.0549955
\(115\) 18.2308 1.70003
\(116\) −6.56691 −0.609722
\(117\) −4.52233 −0.418089
\(118\) −0.000101643 0 −9.35701e−6 0
\(119\) 3.29821 0.302346
\(120\) 2.42301 0.221190
\(121\) −6.90197 −0.627452
\(122\) −2.12681 −0.192552
\(123\) 5.50704 0.496553
\(124\) 7.78353 0.698981
\(125\) 3.60550 0.322485
\(126\) 0.681397 0.0607037
\(127\) −1.72390 −0.152971 −0.0764857 0.997071i \(-0.524370\pi\)
−0.0764857 + 0.997071i \(0.524370\pi\)
\(128\) 6.26286 0.553564
\(129\) 0.805249 0.0708982
\(130\) −2.76896 −0.242854
\(131\) 12.7234 1.11165 0.555825 0.831300i \(-0.312403\pi\)
0.555825 + 0.831300i \(0.312403\pi\)
\(132\) −3.96231 −0.344875
\(133\) −9.37423 −0.812849
\(134\) −1.44292 −0.124649
\(135\) 2.96369 0.255074
\(136\) −0.817567 −0.0701058
\(137\) −6.52930 −0.557836 −0.278918 0.960315i \(-0.589976\pi\)
−0.278918 + 0.960315i \(0.589976\pi\)
\(138\) −1.27085 −0.108182
\(139\) −16.1046 −1.36597 −0.682986 0.730431i \(-0.739319\pi\)
−0.682986 + 0.730431i \(0.739319\pi\)
\(140\) −19.1325 −1.61699
\(141\) −2.71134 −0.228336
\(142\) −2.62580 −0.220352
\(143\) 9.15482 0.765564
\(144\) 3.74573 0.312144
\(145\) −9.94333 −0.825749
\(146\) 0.122942 0.0101748
\(147\) −3.87818 −0.319867
\(148\) 11.7855 0.968761
\(149\) −11.1683 −0.914945 −0.457472 0.889224i \(-0.651245\pi\)
−0.457472 + 0.889224i \(0.651245\pi\)
\(150\) 0.781645 0.0638210
\(151\) 8.50804 0.692375 0.346187 0.938165i \(-0.387476\pi\)
0.346187 + 0.938165i \(0.387476\pi\)
\(152\) 2.32370 0.188477
\(153\) −1.00000 −0.0808452
\(154\) −1.37939 −0.111155
\(155\) 11.7855 0.946633
\(156\) −8.85163 −0.708698
\(157\) −1.00000 −0.0798087
\(158\) −2.06843 −0.164555
\(159\) −7.81249 −0.619571
\(160\) 7.13948 0.564426
\(161\) 20.2886 1.59896
\(162\) −0.206596 −0.0162317
\(163\) −21.7533 −1.70385 −0.851925 0.523664i \(-0.824565\pi\)
−0.851925 + 0.523664i \(0.824565\pi\)
\(164\) 10.7790 0.841701
\(165\) −5.99957 −0.467066
\(166\) 2.11355 0.164043
\(167\) −2.07171 −0.160314 −0.0801569 0.996782i \(-0.525542\pi\)
−0.0801569 + 0.996782i \(0.525542\pi\)
\(168\) 2.69651 0.208040
\(169\) 7.45145 0.573189
\(170\) −0.612286 −0.0469602
\(171\) 2.84222 0.217350
\(172\) 1.57613 0.120179
\(173\) −6.97136 −0.530023 −0.265011 0.964245i \(-0.585376\pi\)
−0.265011 + 0.964245i \(0.585376\pi\)
\(174\) 0.693142 0.0525469
\(175\) −12.4786 −0.943292
\(176\) −7.58270 −0.571568
\(177\) −0.000491989 0 −3.69802e−5 0
\(178\) −1.30510 −0.0978215
\(179\) −15.8036 −1.18122 −0.590609 0.806958i \(-0.701113\pi\)
−0.590609 + 0.806958i \(0.701113\pi\)
\(180\) 5.80088 0.432372
\(181\) 15.5742 1.15762 0.578812 0.815461i \(-0.303517\pi\)
0.578812 + 0.815461i \(0.303517\pi\)
\(182\) −3.08150 −0.228416
\(183\) −10.2945 −0.760994
\(184\) −5.02917 −0.370755
\(185\) 17.8451 1.31200
\(186\) −0.821556 −0.0602394
\(187\) 2.02436 0.148036
\(188\) −5.30695 −0.387049
\(189\) 3.29821 0.239909
\(190\) 1.74025 0.126251
\(191\) −9.33763 −0.675647 −0.337824 0.941209i \(-0.609691\pi\)
−0.337824 + 0.941209i \(0.609691\pi\)
\(192\) 6.99377 0.504732
\(193\) −12.2457 −0.881468 −0.440734 0.897638i \(-0.645282\pi\)
−0.440734 + 0.897638i \(0.645282\pi\)
\(194\) −1.36917 −0.0983006
\(195\) −13.4028 −0.959792
\(196\) −7.59083 −0.542202
\(197\) −20.2355 −1.44172 −0.720861 0.693080i \(-0.756253\pi\)
−0.720861 + 0.693080i \(0.756253\pi\)
\(198\) 0.418225 0.0297219
\(199\) 0.253590 0.0179765 0.00898825 0.999960i \(-0.497139\pi\)
0.00898825 + 0.999960i \(0.497139\pi\)
\(200\) 3.09322 0.218723
\(201\) −6.98424 −0.492630
\(202\) −0.903045 −0.0635380
\(203\) −11.0657 −0.776658
\(204\) −1.95732 −0.137040
\(205\) 16.3211 1.13992
\(206\) −2.23532 −0.155742
\(207\) −6.15139 −0.427551
\(208\) −16.9394 −1.17454
\(209\) −5.75367 −0.397990
\(210\) 2.01945 0.139355
\(211\) −22.3520 −1.53877 −0.769386 0.638784i \(-0.779438\pi\)
−0.769386 + 0.638784i \(0.779438\pi\)
\(212\) −15.2915 −1.05023
\(213\) −12.7098 −0.870863
\(214\) −2.74438 −0.187602
\(215\) 2.38651 0.162758
\(216\) −0.817567 −0.0556284
\(217\) 13.1157 0.890355
\(218\) −1.55347 −0.105215
\(219\) 0.595084 0.0402120
\(220\) −11.7431 −0.791717
\(221\) 4.52233 0.304205
\(222\) −1.24397 −0.0834895
\(223\) 9.37799 0.627996 0.313998 0.949424i \(-0.398331\pi\)
0.313998 + 0.949424i \(0.398331\pi\)
\(224\) 7.94534 0.530870
\(225\) 3.78344 0.252229
\(226\) −1.41088 −0.0938503
\(227\) −16.6181 −1.10298 −0.551490 0.834181i \(-0.685941\pi\)
−0.551490 + 0.834181i \(0.685941\pi\)
\(228\) 5.56313 0.368427
\(229\) 8.54414 0.564612 0.282306 0.959324i \(-0.408901\pi\)
0.282306 + 0.959324i \(0.408901\pi\)
\(230\) −3.76641 −0.248350
\(231\) −6.67676 −0.439298
\(232\) 2.74298 0.180086
\(233\) −10.6121 −0.695224 −0.347612 0.937638i \(-0.613007\pi\)
−0.347612 + 0.937638i \(0.613007\pi\)
\(234\) 0.934296 0.0610768
\(235\) −8.03555 −0.524182
\(236\) −0.000962980 0 −6.26846e−5 0
\(237\) −10.0119 −0.650346
\(238\) −0.681397 −0.0441684
\(239\) 10.1801 0.658494 0.329247 0.944244i \(-0.393205\pi\)
0.329247 + 0.944244i \(0.393205\pi\)
\(240\) 11.1012 0.716578
\(241\) −6.02177 −0.387896 −0.193948 0.981012i \(-0.562129\pi\)
−0.193948 + 0.981012i \(0.562129\pi\)
\(242\) 1.42592 0.0916617
\(243\) −1.00000 −0.0641500
\(244\) −20.1497 −1.28995
\(245\) −11.4937 −0.734306
\(246\) −1.13773 −0.0725392
\(247\) −12.8534 −0.817845
\(248\) −3.25116 −0.206449
\(249\) 10.2303 0.648322
\(250\) −0.744882 −0.0471105
\(251\) 9.91638 0.625916 0.312958 0.949767i \(-0.398680\pi\)
0.312958 + 0.949767i \(0.398680\pi\)
\(252\) 6.45564 0.406667
\(253\) 12.4526 0.782890
\(254\) 0.356151 0.0223469
\(255\) −2.96369 −0.185593
\(256\) 12.6937 0.793354
\(257\) −14.5981 −0.910604 −0.455302 0.890337i \(-0.650469\pi\)
−0.455302 + 0.890337i \(0.650469\pi\)
\(258\) −0.166361 −0.0103572
\(259\) 19.8593 1.23400
\(260\) −26.2335 −1.62693
\(261\) 3.35506 0.207673
\(262\) −2.62861 −0.162396
\(263\) 9.62897 0.593748 0.296874 0.954917i \(-0.404056\pi\)
0.296874 + 0.954917i \(0.404056\pi\)
\(264\) 1.65505 0.101861
\(265\) −23.1538 −1.42233
\(266\) 1.93668 0.118746
\(267\) −6.31716 −0.386604
\(268\) −13.6704 −0.835051
\(269\) −24.2163 −1.47650 −0.738248 0.674529i \(-0.764347\pi\)
−0.738248 + 0.674529i \(0.764347\pi\)
\(270\) −0.612286 −0.0372626
\(271\) 0.936971 0.0569169 0.0284585 0.999595i \(-0.490940\pi\)
0.0284585 + 0.999595i \(0.490940\pi\)
\(272\) −3.74573 −0.227118
\(273\) −14.9156 −0.902732
\(274\) 1.34893 0.0814917
\(275\) −7.65904 −0.461858
\(276\) −12.0402 −0.724736
\(277\) −14.1399 −0.849582 −0.424791 0.905291i \(-0.639652\pi\)
−0.424791 + 0.905291i \(0.639652\pi\)
\(278\) 3.32714 0.199549
\(279\) −3.97663 −0.238075
\(280\) 7.99160 0.477589
\(281\) 32.3428 1.92941 0.964704 0.263338i \(-0.0848236\pi\)
0.964704 + 0.263338i \(0.0848236\pi\)
\(282\) 0.560152 0.0333565
\(283\) 24.0543 1.42988 0.714940 0.699186i \(-0.246454\pi\)
0.714940 + 0.699186i \(0.246454\pi\)
\(284\) −24.8772 −1.47619
\(285\) 8.42345 0.498962
\(286\) −1.89135 −0.111838
\(287\) 18.1634 1.07215
\(288\) −2.40899 −0.141951
\(289\) 1.00000 0.0588235
\(290\) 2.05425 0.120630
\(291\) −6.62727 −0.388498
\(292\) 1.16477 0.0681629
\(293\) 27.9330 1.63186 0.815932 0.578148i \(-0.196224\pi\)
0.815932 + 0.578148i \(0.196224\pi\)
\(294\) 0.801217 0.0467279
\(295\) −0.00145810 −8.48940e−5 0
\(296\) −4.92277 −0.286130
\(297\) 2.02436 0.117465
\(298\) 2.30733 0.133660
\(299\) 27.8186 1.60879
\(300\) 7.40540 0.427551
\(301\) 2.65588 0.153082
\(302\) −1.75773 −0.101146
\(303\) −4.37106 −0.251111
\(304\) 10.6462 0.610601
\(305\) −30.5098 −1.74698
\(306\) 0.206596 0.0118103
\(307\) 28.8610 1.64718 0.823592 0.567182i \(-0.191966\pi\)
0.823592 + 0.567182i \(0.191966\pi\)
\(308\) −13.0685 −0.744649
\(309\) −10.8198 −0.615515
\(310\) −2.43484 −0.138289
\(311\) −14.5043 −0.822465 −0.411232 0.911531i \(-0.634902\pi\)
−0.411232 + 0.911531i \(0.634902\pi\)
\(312\) 3.69731 0.209319
\(313\) −4.22057 −0.238561 −0.119280 0.992861i \(-0.538059\pi\)
−0.119280 + 0.992861i \(0.538059\pi\)
\(314\) 0.206596 0.0116589
\(315\) 9.77486 0.550751
\(316\) −19.5966 −1.10239
\(317\) 5.40649 0.303659 0.151829 0.988407i \(-0.451484\pi\)
0.151829 + 0.988407i \(0.451484\pi\)
\(318\) 1.61403 0.0905104
\(319\) −6.79184 −0.380270
\(320\) 20.7274 1.15869
\(321\) −13.2838 −0.741429
\(322\) −4.19154 −0.233585
\(323\) −2.84222 −0.158145
\(324\) −1.95732 −0.108740
\(325\) −17.1100 −0.949090
\(326\) 4.49415 0.248908
\(327\) −7.51938 −0.415822
\(328\) −4.50237 −0.248602
\(329\) −8.94255 −0.493019
\(330\) 1.23949 0.0682315
\(331\) 23.6438 1.29958 0.649789 0.760114i \(-0.274857\pi\)
0.649789 + 0.760114i \(0.274857\pi\)
\(332\) 20.0240 1.09896
\(333\) −6.02124 −0.329962
\(334\) 0.428008 0.0234195
\(335\) −20.6991 −1.13091
\(336\) 12.3542 0.673977
\(337\) −14.6867 −0.800035 −0.400017 0.916508i \(-0.630996\pi\)
−0.400017 + 0.916508i \(0.630996\pi\)
\(338\) −1.53944 −0.0837346
\(339\) −6.82917 −0.370909
\(340\) −5.80088 −0.314597
\(341\) 8.05012 0.435939
\(342\) −0.587192 −0.0317517
\(343\) 10.2964 0.555954
\(344\) −0.658345 −0.0354956
\(345\) −18.2308 −0.981513
\(346\) 1.44026 0.0774286
\(347\) −0.0908107 −0.00487497 −0.00243749 0.999997i \(-0.500776\pi\)
−0.00243749 + 0.999997i \(0.500776\pi\)
\(348\) 6.56691 0.352023
\(349\) −28.7455 −1.53871 −0.769356 0.638820i \(-0.779423\pi\)
−0.769356 + 0.638820i \(0.779423\pi\)
\(350\) 2.57803 0.137801
\(351\) 4.52233 0.241384
\(352\) 4.87665 0.259927
\(353\) −3.49073 −0.185793 −0.0928963 0.995676i \(-0.529613\pi\)
−0.0928963 + 0.995676i \(0.529613\pi\)
\(354\) 0.000101643 0 5.40227e−6 0
\(355\) −37.6680 −1.99921
\(356\) −12.3647 −0.655327
\(357\) −3.29821 −0.174560
\(358\) 3.26497 0.172559
\(359\) −1.58297 −0.0835462 −0.0417731 0.999127i \(-0.513301\pi\)
−0.0417731 + 0.999127i \(0.513301\pi\)
\(360\) −2.42301 −0.127704
\(361\) −10.9218 −0.574831
\(362\) −3.21758 −0.169112
\(363\) 6.90197 0.362260
\(364\) −29.1945 −1.53021
\(365\) 1.76364 0.0923133
\(366\) 2.12681 0.111170
\(367\) 7.06116 0.368590 0.184295 0.982871i \(-0.441000\pi\)
0.184295 + 0.982871i \(0.441000\pi\)
\(368\) −23.0414 −1.20112
\(369\) −5.50704 −0.286685
\(370\) −3.68672 −0.191664
\(371\) −25.7672 −1.33777
\(372\) −7.78353 −0.403557
\(373\) −1.51149 −0.0782621 −0.0391310 0.999234i \(-0.512459\pi\)
−0.0391310 + 0.999234i \(0.512459\pi\)
\(374\) −0.418225 −0.0216259
\(375\) −3.60550 −0.186187
\(376\) 2.21670 0.114317
\(377\) −15.1727 −0.781432
\(378\) −0.681397 −0.0350473
\(379\) −32.5267 −1.67079 −0.835393 0.549654i \(-0.814760\pi\)
−0.835393 + 0.549654i \(0.814760\pi\)
\(380\) 16.4874 0.845784
\(381\) 1.72390 0.0883180
\(382\) 1.92912 0.0987023
\(383\) 12.0659 0.616541 0.308271 0.951299i \(-0.400250\pi\)
0.308271 + 0.951299i \(0.400250\pi\)
\(384\) −6.26286 −0.319600
\(385\) −19.7878 −1.00848
\(386\) 2.52992 0.128770
\(387\) −0.805249 −0.0409331
\(388\) −12.9717 −0.658537
\(389\) 39.1118 1.98304 0.991522 0.129937i \(-0.0414774\pi\)
0.991522 + 0.129937i \(0.0414774\pi\)
\(390\) 2.76896 0.140212
\(391\) 6.15139 0.311089
\(392\) 3.17067 0.160143
\(393\) −12.7234 −0.641811
\(394\) 4.18058 0.210615
\(395\) −29.6723 −1.49297
\(396\) 3.96231 0.199114
\(397\) 15.7417 0.790051 0.395026 0.918670i \(-0.370736\pi\)
0.395026 + 0.918670i \(0.370736\pi\)
\(398\) −0.0523907 −0.00262611
\(399\) 9.37423 0.469299
\(400\) 14.1718 0.708588
\(401\) 4.06914 0.203203 0.101602 0.994825i \(-0.467603\pi\)
0.101602 + 0.994825i \(0.467603\pi\)
\(402\) 1.44292 0.0719662
\(403\) 17.9836 0.895828
\(404\) −8.55556 −0.425655
\(405\) −2.96369 −0.147267
\(406\) 2.28613 0.113459
\(407\) 12.1892 0.604194
\(408\) 0.817567 0.0404756
\(409\) 32.9540 1.62947 0.814735 0.579834i \(-0.196882\pi\)
0.814735 + 0.579834i \(0.196882\pi\)
\(410\) −3.37189 −0.166526
\(411\) 6.52930 0.322067
\(412\) −21.1777 −1.04335
\(413\) −0.00162268 −7.98471e−5 0
\(414\) 1.27085 0.0624591
\(415\) 30.3196 1.48833
\(416\) 10.8942 0.534134
\(417\) 16.1046 0.788644
\(418\) 1.18869 0.0581406
\(419\) 0.836509 0.0408662 0.0204331 0.999791i \(-0.493495\pi\)
0.0204331 + 0.999791i \(0.493495\pi\)
\(420\) 19.1325 0.933571
\(421\) −26.6633 −1.29949 −0.649745 0.760152i \(-0.725125\pi\)
−0.649745 + 0.760152i \(0.725125\pi\)
\(422\) 4.61783 0.224792
\(423\) 2.71134 0.131830
\(424\) 6.38723 0.310192
\(425\) −3.78344 −0.183524
\(426\) 2.62580 0.127221
\(427\) −33.9535 −1.64313
\(428\) −26.0006 −1.25679
\(429\) −9.15482 −0.441999
\(430\) −0.493043 −0.0237766
\(431\) 15.4797 0.745633 0.372817 0.927905i \(-0.378392\pi\)
0.372817 + 0.927905i \(0.378392\pi\)
\(432\) −3.74573 −0.180217
\(433\) −15.4925 −0.744520 −0.372260 0.928129i \(-0.621417\pi\)
−0.372260 + 0.928129i \(0.621417\pi\)
\(434\) −2.70966 −0.130068
\(435\) 9.94333 0.476746
\(436\) −14.7178 −0.704855
\(437\) −17.4836 −0.836354
\(438\) −0.122942 −0.00587440
\(439\) −27.3681 −1.30621 −0.653104 0.757268i \(-0.726533\pi\)
−0.653104 + 0.757268i \(0.726533\pi\)
\(440\) 4.90505 0.233839
\(441\) 3.87818 0.184675
\(442\) −0.934296 −0.0444399
\(443\) −16.7822 −0.797344 −0.398672 0.917093i \(-0.630529\pi\)
−0.398672 + 0.917093i \(0.630529\pi\)
\(444\) −11.7855 −0.559314
\(445\) −18.7221 −0.887512
\(446\) −1.93746 −0.0917412
\(447\) 11.1683 0.528243
\(448\) 23.0669 1.08981
\(449\) 18.4042 0.868549 0.434275 0.900781i \(-0.357005\pi\)
0.434275 + 0.900781i \(0.357005\pi\)
\(450\) −0.781645 −0.0368471
\(451\) 11.1482 0.524949
\(452\) −13.3669 −0.628724
\(453\) −8.50804 −0.399743
\(454\) 3.43323 0.161130
\(455\) −44.2051 −2.07237
\(456\) −2.32370 −0.108817
\(457\) −1.54464 −0.0722551 −0.0361275 0.999347i \(-0.511502\pi\)
−0.0361275 + 0.999347i \(0.511502\pi\)
\(458\) −1.76519 −0.0824817
\(459\) 1.00000 0.0466760
\(460\) −35.6835 −1.66375
\(461\) 30.9109 1.43967 0.719833 0.694148i \(-0.244219\pi\)
0.719833 + 0.694148i \(0.244219\pi\)
\(462\) 1.37939 0.0641752
\(463\) −24.7695 −1.15113 −0.575567 0.817754i \(-0.695219\pi\)
−0.575567 + 0.817754i \(0.695219\pi\)
\(464\) 12.5671 0.583414
\(465\) −11.7855 −0.546539
\(466\) 2.19243 0.101562
\(467\) 18.4197 0.852361 0.426181 0.904638i \(-0.359859\pi\)
0.426181 + 0.904638i \(0.359859\pi\)
\(468\) 8.85163 0.409167
\(469\) −23.0355 −1.06368
\(470\) 1.66011 0.0765754
\(471\) 1.00000 0.0460776
\(472\) 0.000402234 0 1.85143e−5 0
\(473\) 1.63011 0.0749526
\(474\) 2.06843 0.0950062
\(475\) 10.7534 0.493399
\(476\) −6.45564 −0.295894
\(477\) 7.81249 0.357709
\(478\) −2.10316 −0.0961965
\(479\) 37.0653 1.69355 0.846777 0.531947i \(-0.178540\pi\)
0.846777 + 0.531947i \(0.178540\pi\)
\(480\) −7.13948 −0.325871
\(481\) 27.2300 1.24158
\(482\) 1.24407 0.0566661
\(483\) −20.2886 −0.923162
\(484\) 13.5094 0.614061
\(485\) −19.6412 −0.891860
\(486\) 0.206596 0.00937139
\(487\) −21.9334 −0.993895 −0.496948 0.867780i \(-0.665546\pi\)
−0.496948 + 0.867780i \(0.665546\pi\)
\(488\) 8.41647 0.380996
\(489\) 21.7533 0.983718
\(490\) 2.37456 0.107272
\(491\) 4.66692 0.210615 0.105308 0.994440i \(-0.466417\pi\)
0.105308 + 0.994440i \(0.466417\pi\)
\(492\) −10.7790 −0.485956
\(493\) −3.35506 −0.151104
\(494\) 2.65547 0.119475
\(495\) 5.99957 0.269660
\(496\) −14.8954 −0.668822
\(497\) −41.9197 −1.88035
\(498\) −2.11355 −0.0947105
\(499\) −0.286947 −0.0128455 −0.00642275 0.999979i \(-0.502044\pi\)
−0.00642275 + 0.999979i \(0.502044\pi\)
\(500\) −7.05710 −0.315603
\(501\) 2.07171 0.0925572
\(502\) −2.04869 −0.0914373
\(503\) 12.3518 0.550741 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(504\) −2.69651 −0.120112
\(505\) −12.9545 −0.576466
\(506\) −2.57266 −0.114369
\(507\) −7.45145 −0.330931
\(508\) 3.37422 0.149707
\(509\) 7.66157 0.339593 0.169796 0.985479i \(-0.445689\pi\)
0.169796 + 0.985479i \(0.445689\pi\)
\(510\) 0.612286 0.0271125
\(511\) 1.96271 0.0868252
\(512\) −15.1482 −0.669461
\(513\) −2.84222 −0.125487
\(514\) 3.01591 0.133026
\(515\) −32.0664 −1.41302
\(516\) −1.57613 −0.0693852
\(517\) −5.48872 −0.241393
\(518\) −4.10286 −0.180269
\(519\) 6.97136 0.306009
\(520\) 10.9577 0.480525
\(521\) −5.42090 −0.237494 −0.118747 0.992925i \(-0.537888\pi\)
−0.118747 + 0.992925i \(0.537888\pi\)
\(522\) −0.693142 −0.0303380
\(523\) 12.6758 0.554274 0.277137 0.960830i \(-0.410614\pi\)
0.277137 + 0.960830i \(0.410614\pi\)
\(524\) −24.9038 −1.08793
\(525\) 12.4786 0.544610
\(526\) −1.98931 −0.0867380
\(527\) 3.97663 0.173225
\(528\) 7.58270 0.329995
\(529\) 14.8396 0.645200
\(530\) 4.78348 0.207781
\(531\) 0.000491989 0 2.13505e−5 0
\(532\) 18.3484 0.795502
\(533\) 24.9046 1.07874
\(534\) 1.30510 0.0564773
\(535\) −39.3690 −1.70207
\(536\) 5.71008 0.246638
\(537\) 15.8036 0.681977
\(538\) 5.00300 0.215695
\(539\) −7.85083 −0.338159
\(540\) −5.80088 −0.249630
\(541\) 13.3446 0.573730 0.286865 0.957971i \(-0.407387\pi\)
0.286865 + 0.957971i \(0.407387\pi\)
\(542\) −0.193575 −0.00831474
\(543\) −15.5742 −0.668354
\(544\) 2.40899 0.103284
\(545\) −22.2851 −0.954588
\(546\) 3.08150 0.131876
\(547\) −37.6872 −1.61139 −0.805693 0.592333i \(-0.798207\pi\)
−0.805693 + 0.592333i \(0.798207\pi\)
\(548\) 12.7799 0.545931
\(549\) 10.2945 0.439360
\(550\) 1.58233 0.0674708
\(551\) 9.53580 0.406239
\(552\) 5.02917 0.214056
\(553\) −33.0215 −1.40422
\(554\) 2.92124 0.124112
\(555\) −17.8451 −0.757481
\(556\) 31.5218 1.33682
\(557\) −35.6177 −1.50917 −0.754586 0.656201i \(-0.772162\pi\)
−0.754586 + 0.656201i \(0.772162\pi\)
\(558\) 0.821556 0.0347792
\(559\) 3.64160 0.154023
\(560\) 36.6140 1.54722
\(561\) −2.02436 −0.0854685
\(562\) −6.68189 −0.281859
\(563\) 8.61111 0.362915 0.181457 0.983399i \(-0.441919\pi\)
0.181457 + 0.983399i \(0.441919\pi\)
\(564\) 5.30695 0.223463
\(565\) −20.2395 −0.851483
\(566\) −4.96953 −0.208885
\(567\) −3.29821 −0.138512
\(568\) 10.3911 0.436002
\(569\) 39.4240 1.65274 0.826370 0.563127i \(-0.190402\pi\)
0.826370 + 0.563127i \(0.190402\pi\)
\(570\) −1.74025 −0.0728912
\(571\) −33.0428 −1.38280 −0.691399 0.722473i \(-0.743005\pi\)
−0.691399 + 0.722473i \(0.743005\pi\)
\(572\) −17.9189 −0.749226
\(573\) 9.33763 0.390085
\(574\) −3.75248 −0.156626
\(575\) −23.2734 −0.970569
\(576\) −6.99377 −0.291407
\(577\) 22.7303 0.946275 0.473137 0.880989i \(-0.343121\pi\)
0.473137 + 0.880989i \(0.343121\pi\)
\(578\) −0.206596 −0.00859327
\(579\) 12.2457 0.508916
\(580\) 19.4623 0.808127
\(581\) 33.7418 1.39985
\(582\) 1.36917 0.0567539
\(583\) −15.8153 −0.655002
\(584\) −0.486521 −0.0201324
\(585\) 13.4028 0.554136
\(586\) −5.77085 −0.238392
\(587\) −13.8143 −0.570179 −0.285089 0.958501i \(-0.592023\pi\)
−0.285089 + 0.958501i \(0.592023\pi\)
\(588\) 7.59083 0.313041
\(589\) −11.3024 −0.465709
\(590\) 0.000301238 0 1.24018e−5 0
\(591\) 20.2355 0.832378
\(592\) −22.5539 −0.926961
\(593\) 28.1086 1.15428 0.577141 0.816644i \(-0.304168\pi\)
0.577141 + 0.816644i \(0.304168\pi\)
\(594\) −0.418225 −0.0171600
\(595\) −9.77486 −0.400730
\(596\) 21.8600 0.895419
\(597\) −0.253590 −0.0103787
\(598\) −5.74722 −0.235021
\(599\) 16.7476 0.684288 0.342144 0.939648i \(-0.388847\pi\)
0.342144 + 0.939648i \(0.388847\pi\)
\(600\) −3.09322 −0.126280
\(601\) −36.3144 −1.48130 −0.740649 0.671893i \(-0.765482\pi\)
−0.740649 + 0.671893i \(0.765482\pi\)
\(602\) −0.548694 −0.0223631
\(603\) 6.98424 0.284420
\(604\) −16.6529 −0.677599
\(605\) 20.4553 0.831626
\(606\) 0.903045 0.0366837
\(607\) 10.1764 0.413047 0.206523 0.978442i \(-0.433785\pi\)
0.206523 + 0.978442i \(0.433785\pi\)
\(608\) −6.84687 −0.277677
\(609\) 11.0657 0.448404
\(610\) 6.30320 0.255209
\(611\) −12.2616 −0.496049
\(612\) 1.95732 0.0791199
\(613\) −34.9243 −1.41058 −0.705290 0.708919i \(-0.749183\pi\)
−0.705290 + 0.708919i \(0.749183\pi\)
\(614\) −5.96257 −0.240630
\(615\) −16.3211 −0.658132
\(616\) 5.45870 0.219937
\(617\) −14.6332 −0.589112 −0.294556 0.955634i \(-0.595172\pi\)
−0.294556 + 0.955634i \(0.595172\pi\)
\(618\) 2.23532 0.0899179
\(619\) 26.5532 1.06726 0.533631 0.845717i \(-0.320827\pi\)
0.533631 + 0.845717i \(0.320827\pi\)
\(620\) −23.0679 −0.926431
\(621\) 6.15139 0.246847
\(622\) 2.99654 0.120150
\(623\) −20.8353 −0.834749
\(624\) 16.9394 0.678120
\(625\) −29.6028 −1.18411
\(626\) 0.871953 0.0348503
\(627\) 5.75367 0.229780
\(628\) 1.95732 0.0781055
\(629\) 6.02124 0.240083
\(630\) −2.01945 −0.0804568
\(631\) 4.91139 0.195519 0.0977596 0.995210i \(-0.468832\pi\)
0.0977596 + 0.995210i \(0.468832\pi\)
\(632\) 8.18544 0.325599
\(633\) 22.3520 0.888411
\(634\) −1.11696 −0.0443602
\(635\) 5.10910 0.202748
\(636\) 15.2915 0.606349
\(637\) −17.5384 −0.694897
\(638\) 1.40317 0.0555519
\(639\) 12.7098 0.502793
\(640\) −18.5612 −0.733694
\(641\) −10.3957 −0.410604 −0.205302 0.978699i \(-0.565818\pi\)
−0.205302 + 0.978699i \(0.565818\pi\)
\(642\) 2.74438 0.108312
\(643\) −28.3287 −1.11718 −0.558588 0.829445i \(-0.688657\pi\)
−0.558588 + 0.829445i \(0.688657\pi\)
\(644\) −39.7112 −1.56484
\(645\) −2.38651 −0.0939686
\(646\) 0.587192 0.0231027
\(647\) 3.79379 0.149149 0.0745745 0.997215i \(-0.476240\pi\)
0.0745745 + 0.997215i \(0.476240\pi\)
\(648\) 0.817567 0.0321171
\(649\) −0.000995963 0 −3.90950e−5 0
\(650\) 3.53485 0.138648
\(651\) −13.1157 −0.514047
\(652\) 42.5781 1.66749
\(653\) −1.93435 −0.0756971 −0.0378486 0.999283i \(-0.512050\pi\)
−0.0378486 + 0.999283i \(0.512050\pi\)
\(654\) 1.55347 0.0607457
\(655\) −37.7082 −1.47338
\(656\) −20.6279 −0.805384
\(657\) −0.595084 −0.0232164
\(658\) 1.84750 0.0720229
\(659\) 41.5604 1.61896 0.809481 0.587146i \(-0.199748\pi\)
0.809481 + 0.587146i \(0.199748\pi\)
\(660\) 11.7431 0.457098
\(661\) 7.16370 0.278635 0.139318 0.990248i \(-0.455509\pi\)
0.139318 + 0.990248i \(0.455509\pi\)
\(662\) −4.88471 −0.189850
\(663\) −4.52233 −0.175633
\(664\) −8.36399 −0.324586
\(665\) 27.7823 1.07735
\(666\) 1.24397 0.0482027
\(667\) −20.6383 −0.799116
\(668\) 4.05500 0.156893
\(669\) −9.37799 −0.362574
\(670\) 4.27636 0.165210
\(671\) −20.8398 −0.804513
\(672\) −7.94534 −0.306498
\(673\) −47.6487 −1.83672 −0.918362 0.395742i \(-0.870487\pi\)
−0.918362 + 0.395742i \(0.870487\pi\)
\(674\) 3.03421 0.116874
\(675\) −3.78344 −0.145625
\(676\) −14.5849 −0.560956
\(677\) 11.6955 0.449495 0.224747 0.974417i \(-0.427844\pi\)
0.224747 + 0.974417i \(0.427844\pi\)
\(678\) 1.41088 0.0541845
\(679\) −21.8581 −0.838838
\(680\) 2.42301 0.0929183
\(681\) 16.6181 0.636806
\(682\) −1.66312 −0.0636843
\(683\) 43.8188 1.67668 0.838340 0.545148i \(-0.183526\pi\)
0.838340 + 0.545148i \(0.183526\pi\)
\(684\) −5.56313 −0.212711
\(685\) 19.3508 0.739356
\(686\) −2.12720 −0.0812169
\(687\) −8.54414 −0.325979
\(688\) −3.01624 −0.114993
\(689\) −35.3307 −1.34599
\(690\) 3.76641 0.143385
\(691\) 20.9087 0.795404 0.397702 0.917515i \(-0.369808\pi\)
0.397702 + 0.917515i \(0.369808\pi\)
\(692\) 13.6452 0.518711
\(693\) 6.67676 0.253629
\(694\) 0.0187611 0.000712163 0
\(695\) 47.7289 1.81046
\(696\) −2.74298 −0.103972
\(697\) 5.50704 0.208594
\(698\) 5.93871 0.224784
\(699\) 10.6121 0.401388
\(700\) 24.4246 0.923161
\(701\) −14.9812 −0.565831 −0.282915 0.959145i \(-0.591302\pi\)
−0.282915 + 0.959145i \(0.591302\pi\)
\(702\) −0.934296 −0.0352627
\(703\) −17.1137 −0.645455
\(704\) 14.1579 0.533596
\(705\) 8.03555 0.302636
\(706\) 0.721171 0.0271416
\(707\) −14.4167 −0.542195
\(708\) 0.000962980 0 3.61910e−5 0
\(709\) 39.0216 1.46549 0.732743 0.680505i \(-0.238240\pi\)
0.732743 + 0.680505i \(0.238240\pi\)
\(710\) 7.78206 0.292055
\(711\) 10.0119 0.375477
\(712\) 5.16470 0.193555
\(713\) 24.4618 0.916101
\(714\) 0.681397 0.0255007
\(715\) −27.1320 −1.01468
\(716\) 30.9327 1.15601
\(717\) −10.1801 −0.380182
\(718\) 0.327036 0.0122049
\(719\) 0.467610 0.0174389 0.00871945 0.999962i \(-0.497224\pi\)
0.00871945 + 0.999962i \(0.497224\pi\)
\(720\) −11.1012 −0.413716
\(721\) −35.6859 −1.32901
\(722\) 2.25640 0.0839745
\(723\) 6.02177 0.223952
\(724\) −30.4837 −1.13292
\(725\) 12.6937 0.471431
\(726\) −1.42592 −0.0529209
\(727\) −17.1421 −0.635767 −0.317883 0.948130i \(-0.602972\pi\)
−0.317883 + 0.948130i \(0.602972\pi\)
\(728\) 12.1945 0.451958
\(729\) 1.00000 0.0370370
\(730\) −0.364362 −0.0134856
\(731\) 0.805249 0.0297832
\(732\) 20.1497 0.744753
\(733\) −10.4110 −0.384537 −0.192269 0.981342i \(-0.561584\pi\)
−0.192269 + 0.981342i \(0.561584\pi\)
\(734\) −1.45881 −0.0538456
\(735\) 11.4937 0.423952
\(736\) 14.8186 0.546222
\(737\) −14.1386 −0.520802
\(738\) 1.13773 0.0418806
\(739\) 5.63361 0.207236 0.103618 0.994617i \(-0.466958\pi\)
0.103618 + 0.994617i \(0.466958\pi\)
\(740\) −34.9285 −1.28400
\(741\) 12.8534 0.472183
\(742\) 5.32341 0.195429
\(743\) −28.7343 −1.05416 −0.527079 0.849816i \(-0.676713\pi\)
−0.527079 + 0.849816i \(0.676713\pi\)
\(744\) 3.25116 0.119193
\(745\) 33.0994 1.21267
\(746\) 0.312268 0.0114330
\(747\) −10.2303 −0.374309
\(748\) −3.96231 −0.144877
\(749\) −43.8127 −1.60088
\(750\) 0.744882 0.0271992
\(751\) −15.8107 −0.576941 −0.288470 0.957489i \(-0.593147\pi\)
−0.288470 + 0.957489i \(0.593147\pi\)
\(752\) 10.1559 0.370349
\(753\) −9.91638 −0.361373
\(754\) 3.13461 0.114156
\(755\) −25.2152 −0.917674
\(756\) −6.45564 −0.234789
\(757\) 26.1004 0.948637 0.474318 0.880353i \(-0.342695\pi\)
0.474318 + 0.880353i \(0.342695\pi\)
\(758\) 6.71990 0.244078
\(759\) −12.4526 −0.452001
\(760\) −6.88673 −0.249808
\(761\) −17.0426 −0.617794 −0.308897 0.951096i \(-0.599960\pi\)
−0.308897 + 0.951096i \(0.599960\pi\)
\(762\) −0.356151 −0.0129020
\(763\) −24.8005 −0.897837
\(764\) 18.2767 0.661228
\(765\) 2.96369 0.107152
\(766\) −2.49278 −0.0900678
\(767\) −0.00222494 −8.03378e−5 0
\(768\) −12.6937 −0.458043
\(769\) 41.2812 1.48864 0.744320 0.667823i \(-0.232774\pi\)
0.744320 + 0.667823i \(0.232774\pi\)
\(770\) 4.08809 0.147324
\(771\) 14.5981 0.525738
\(772\) 23.9688 0.862656
\(773\) −10.1049 −0.363447 −0.181723 0.983350i \(-0.558168\pi\)
−0.181723 + 0.983350i \(0.558168\pi\)
\(774\) 0.166361 0.00597973
\(775\) −15.0453 −0.540445
\(776\) 5.41824 0.194503
\(777\) −19.8593 −0.712449
\(778\) −8.08034 −0.289694
\(779\) −15.6522 −0.560799
\(780\) 26.2335 0.939309
\(781\) −25.7292 −0.920665
\(782\) −1.27085 −0.0454456
\(783\) −3.35506 −0.119900
\(784\) 14.5266 0.518808
\(785\) 2.96369 0.105779
\(786\) 2.62861 0.0937593
\(787\) 38.9370 1.38795 0.693977 0.719997i \(-0.255857\pi\)
0.693977 + 0.719997i \(0.255857\pi\)
\(788\) 39.6074 1.41095
\(789\) −9.62897 −0.342800
\(790\) 6.13018 0.218102
\(791\) −22.5240 −0.800862
\(792\) −1.65505 −0.0588096
\(793\) −46.5552 −1.65323
\(794\) −3.25217 −0.115415
\(795\) 23.1538 0.821180
\(796\) −0.496356 −0.0175929
\(797\) −5.96014 −0.211119 −0.105559 0.994413i \(-0.533663\pi\)
−0.105559 + 0.994413i \(0.533663\pi\)
\(798\) −1.93668 −0.0685577
\(799\) −2.71134 −0.0959202
\(800\) −9.11426 −0.322238
\(801\) 6.31716 0.223206
\(802\) −0.840668 −0.0296850
\(803\) 1.20466 0.0425116
\(804\) 13.6704 0.482117
\(805\) −60.1290 −2.11927
\(806\) −3.71535 −0.130868
\(807\) 24.2163 0.852456
\(808\) 3.57364 0.125720
\(809\) 45.2004 1.58916 0.794581 0.607159i \(-0.207691\pi\)
0.794581 + 0.607159i \(0.207691\pi\)
\(810\) 0.612286 0.0215136
\(811\) −18.7209 −0.657380 −0.328690 0.944438i \(-0.606607\pi\)
−0.328690 + 0.944438i \(0.606607\pi\)
\(812\) 21.6590 0.760083
\(813\) −0.936971 −0.0328610
\(814\) −2.51823 −0.0882640
\(815\) 64.4700 2.25828
\(816\) 3.74573 0.131127
\(817\) −2.28869 −0.0800712
\(818\) −6.80817 −0.238042
\(819\) 14.9156 0.521193
\(820\) −31.9457 −1.11559
\(821\) −7.15064 −0.249559 −0.124780 0.992184i \(-0.539822\pi\)
−0.124780 + 0.992184i \(0.539822\pi\)
\(822\) −1.34893 −0.0470493
\(823\) 13.0128 0.453598 0.226799 0.973942i \(-0.427174\pi\)
0.226799 + 0.973942i \(0.427174\pi\)
\(824\) 8.84589 0.308161
\(825\) 7.65904 0.266654
\(826\) 0.000335240 0 1.16645e−5 0
\(827\) 40.2504 1.39964 0.699821 0.714318i \(-0.253263\pi\)
0.699821 + 0.714318i \(0.253263\pi\)
\(828\) 12.0402 0.418427
\(829\) 22.0878 0.767142 0.383571 0.923511i \(-0.374694\pi\)
0.383571 + 0.923511i \(0.374694\pi\)
\(830\) −6.26390 −0.217423
\(831\) 14.1399 0.490506
\(832\) 31.6281 1.09651
\(833\) −3.87818 −0.134371
\(834\) −3.32714 −0.115210
\(835\) 6.13990 0.212480
\(836\) 11.2618 0.389496
\(837\) 3.97663 0.137452
\(838\) −0.172820 −0.00596995
\(839\) 24.3018 0.838990 0.419495 0.907758i \(-0.362207\pi\)
0.419495 + 0.907758i \(0.362207\pi\)
\(840\) −7.99160 −0.275736
\(841\) −17.7436 −0.611848
\(842\) 5.50854 0.189837
\(843\) −32.3428 −1.11394
\(844\) 43.7499 1.50593
\(845\) −22.0838 −0.759705
\(846\) −0.560152 −0.0192584
\(847\) 22.7641 0.782185
\(848\) 29.2635 1.00491
\(849\) −24.0543 −0.825541
\(850\) 0.781645 0.0268102
\(851\) 37.0390 1.26968
\(852\) 24.8772 0.852278
\(853\) −1.23958 −0.0424423 −0.0212211 0.999775i \(-0.506755\pi\)
−0.0212211 + 0.999775i \(0.506755\pi\)
\(854\) 7.01467 0.240037
\(855\) −8.42345 −0.288076
\(856\) 10.8604 0.371201
\(857\) −33.5881 −1.14735 −0.573674 0.819083i \(-0.694483\pi\)
−0.573674 + 0.819083i \(0.694483\pi\)
\(858\) 1.89135 0.0645696
\(859\) −41.5601 −1.41801 −0.709006 0.705203i \(-0.750856\pi\)
−0.709006 + 0.705203i \(0.750856\pi\)
\(860\) −4.67115 −0.159285
\(861\) −18.1634 −0.619006
\(862\) −3.19806 −0.108926
\(863\) −8.15597 −0.277633 −0.138816 0.990318i \(-0.544330\pi\)
−0.138816 + 0.990318i \(0.544330\pi\)
\(864\) 2.40899 0.0819554
\(865\) 20.6609 0.702493
\(866\) 3.20068 0.108764
\(867\) −1.00000 −0.0339618
\(868\) −25.6717 −0.871354
\(869\) −20.2678 −0.687537
\(870\) −2.05425 −0.0696458
\(871\) −31.5850 −1.07022
\(872\) 6.14759 0.208184
\(873\) 6.62727 0.224299
\(874\) 3.61204 0.122179
\(875\) −11.8917 −0.402012
\(876\) −1.16477 −0.0393539
\(877\) −14.7098 −0.496716 −0.248358 0.968668i \(-0.579891\pi\)
−0.248358 + 0.968668i \(0.579891\pi\)
\(878\) 5.65414 0.190818
\(879\) −27.9330 −0.942157
\(880\) 22.4728 0.757556
\(881\) −35.4621 −1.19475 −0.597374 0.801963i \(-0.703789\pi\)
−0.597374 + 0.801963i \(0.703789\pi\)
\(882\) −0.801217 −0.0269784
\(883\) 24.9029 0.838050 0.419025 0.907975i \(-0.362372\pi\)
0.419025 + 0.907975i \(0.362372\pi\)
\(884\) −8.85163 −0.297713
\(885\) 0.00145810 4.90136e−5 0
\(886\) 3.46713 0.116481
\(887\) 17.0693 0.573132 0.286566 0.958061i \(-0.407486\pi\)
0.286566 + 0.958061i \(0.407486\pi\)
\(888\) 4.92277 0.165197
\(889\) 5.68578 0.190695
\(890\) 3.86791 0.129653
\(891\) −2.02436 −0.0678186
\(892\) −18.3557 −0.614594
\(893\) 7.70621 0.257879
\(894\) −2.30733 −0.0771687
\(895\) 46.8370 1.56559
\(896\) −20.6562 −0.690076
\(897\) −27.8186 −0.928836
\(898\) −3.80224 −0.126882
\(899\) −13.3418 −0.444974
\(900\) −7.40540 −0.246847
\(901\) −7.81249 −0.260272
\(902\) −2.30318 −0.0766875
\(903\) −2.65588 −0.0883821
\(904\) 5.58330 0.185698
\(905\) −46.1571 −1.53432
\(906\) 1.75773 0.0583966
\(907\) −1.80384 −0.0598956 −0.0299478 0.999551i \(-0.509534\pi\)
−0.0299478 + 0.999551i \(0.509534\pi\)
\(908\) 32.5269 1.07944
\(909\) 4.37106 0.144979
\(910\) 9.13261 0.302743
\(911\) 17.2375 0.571104 0.285552 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(912\) −10.6462 −0.352530
\(913\) 20.7099 0.685398
\(914\) 0.319116 0.0105554
\(915\) 30.5098 1.00862
\(916\) −16.7236 −0.552563
\(917\) −41.9644 −1.38579
\(918\) −0.206596 −0.00681869
\(919\) 29.7998 0.983006 0.491503 0.870876i \(-0.336448\pi\)
0.491503 + 0.870876i \(0.336448\pi\)
\(920\) 14.9049 0.491400
\(921\) −28.8610 −0.951003
\(922\) −6.38608 −0.210314
\(923\) −57.4780 −1.89191
\(924\) 13.0685 0.429923
\(925\) −22.7810 −0.749035
\(926\) 5.11728 0.168164
\(927\) 10.8198 0.355368
\(928\) −8.08228 −0.265314
\(929\) 36.6222 1.20153 0.600767 0.799424i \(-0.294862\pi\)
0.600767 + 0.799424i \(0.294862\pi\)
\(930\) 2.43484 0.0798414
\(931\) 11.0226 0.361252
\(932\) 20.7713 0.680387
\(933\) 14.5043 0.474850
\(934\) −3.80544 −0.124518
\(935\) −5.99957 −0.196207
\(936\) −3.69731 −0.120850
\(937\) 26.2106 0.856262 0.428131 0.903717i \(-0.359172\pi\)
0.428131 + 0.903717i \(0.359172\pi\)
\(938\) 4.75904 0.155388
\(939\) 4.22057 0.137733
\(940\) 15.7281 0.512995
\(941\) 31.3776 1.02288 0.511440 0.859319i \(-0.329112\pi\)
0.511440 + 0.859319i \(0.329112\pi\)
\(942\) −0.206596 −0.00673127
\(943\) 33.8760 1.10315
\(944\) 0.00184286 5.99800e−5 0
\(945\) −9.77486 −0.317976
\(946\) −0.336775 −0.0109495
\(947\) 31.8858 1.03615 0.518075 0.855335i \(-0.326649\pi\)
0.518075 + 0.855335i \(0.326649\pi\)
\(948\) 19.5966 0.636467
\(949\) 2.69116 0.0873589
\(950\) −2.22161 −0.0720784
\(951\) −5.40649 −0.175318
\(952\) 2.69651 0.0873943
\(953\) −32.6370 −1.05722 −0.528608 0.848866i \(-0.677286\pi\)
−0.528608 + 0.848866i \(0.677286\pi\)
\(954\) −1.61403 −0.0522562
\(955\) 27.6738 0.895504
\(956\) −19.9256 −0.644441
\(957\) 6.79184 0.219549
\(958\) −7.65754 −0.247404
\(959\) 21.5350 0.695401
\(960\) −20.7274 −0.668972
\(961\) −15.1864 −0.489885
\(962\) −5.62562 −0.181377
\(963\) 13.2838 0.428064
\(964\) 11.7865 0.379618
\(965\) 36.2926 1.16830
\(966\) 4.19154 0.134861
\(967\) −0.913185 −0.0293661 −0.0146830 0.999892i \(-0.504674\pi\)
−0.0146830 + 0.999892i \(0.504674\pi\)
\(968\) −5.64282 −0.181367
\(969\) 2.84222 0.0913052
\(970\) 4.05779 0.130288
\(971\) −21.3000 −0.683550 −0.341775 0.939782i \(-0.611028\pi\)
−0.341775 + 0.939782i \(0.611028\pi\)
\(972\) 1.95732 0.0627810
\(973\) 53.1162 1.70283
\(974\) 4.53135 0.145194
\(975\) 17.1100 0.547958
\(976\) 38.5605 1.23429
\(977\) 16.5788 0.530404 0.265202 0.964193i \(-0.414561\pi\)
0.265202 + 0.964193i \(0.414561\pi\)
\(978\) −4.49415 −0.143707
\(979\) −12.7882 −0.408713
\(980\) 22.4968 0.718635
\(981\) 7.51938 0.240075
\(982\) −0.964168 −0.0307678
\(983\) 34.5744 1.10275 0.551376 0.834257i \(-0.314103\pi\)
0.551376 + 0.834257i \(0.314103\pi\)
\(984\) 4.50237 0.143530
\(985\) 59.9718 1.91086
\(986\) 0.693142 0.0220741
\(987\) 8.94255 0.284645
\(988\) 25.1583 0.800392
\(989\) 4.95340 0.157509
\(990\) −1.23949 −0.0393935
\(991\) 13.4427 0.427020 0.213510 0.976941i \(-0.431510\pi\)
0.213510 + 0.976941i \(0.431510\pi\)
\(992\) 9.57965 0.304154
\(993\) −23.6438 −0.750312
\(994\) 8.66044 0.274693
\(995\) −0.751561 −0.0238261
\(996\) −20.0240 −0.634486
\(997\) 30.8122 0.975831 0.487915 0.872891i \(-0.337757\pi\)
0.487915 + 0.872891i \(0.337757\pi\)
\(998\) 0.0592821 0.00187654
\(999\) 6.02124 0.190504
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 8007.2.a.f.1.23 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8007.2.a.f.1.23 48 1.1 even 1 trivial