Properties

Label 483.2.a.d.1.1
Level $483$
Weight $2$
Character 483.1
Self dual yes
Analytic conductor $3.857$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.85677441763\)
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 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 483.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -0.618034 q^{5} +1.61803 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -0.618034 q^{5} +1.61803 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.23607 q^{11} -0.618034 q^{12} -2.38197 q^{13} +1.61803 q^{14} +0.618034 q^{15} -4.85410 q^{16} +6.70820 q^{17} -1.61803 q^{18} -3.47214 q^{19} -0.381966 q^{20} +1.00000 q^{21} -3.61803 q^{22} -1.00000 q^{23} -2.23607 q^{24} -4.61803 q^{25} +3.85410 q^{26} -1.00000 q^{27} -0.618034 q^{28} -8.23607 q^{29} -1.00000 q^{30} +6.70820 q^{31} +3.38197 q^{32} -2.23607 q^{33} -10.8541 q^{34} +0.618034 q^{35} +0.618034 q^{36} -11.0000 q^{37} +5.61803 q^{38} +2.38197 q^{39} -1.38197 q^{40} -1.47214 q^{41} -1.61803 q^{42} -1.61803 q^{43} +1.38197 q^{44} -0.618034 q^{45} +1.61803 q^{46} -7.23607 q^{47} +4.85410 q^{48} +1.00000 q^{49} +7.47214 q^{50} -6.70820 q^{51} -1.47214 q^{52} -13.0902 q^{53} +1.61803 q^{54} -1.38197 q^{55} -2.23607 q^{56} +3.47214 q^{57} +13.3262 q^{58} +9.38197 q^{59} +0.381966 q^{60} -4.85410 q^{61} -10.8541 q^{62} -1.00000 q^{63} +4.23607 q^{64} +1.47214 q^{65} +3.61803 q^{66} -5.09017 q^{67} +4.14590 q^{68} +1.00000 q^{69} -1.00000 q^{70} +4.38197 q^{71} +2.23607 q^{72} -12.7082 q^{73} +17.7984 q^{74} +4.61803 q^{75} -2.14590 q^{76} -2.23607 q^{77} -3.85410 q^{78} -9.47214 q^{79} +3.00000 q^{80} +1.00000 q^{81} +2.38197 q^{82} -9.18034 q^{83} +0.618034 q^{84} -4.14590 q^{85} +2.61803 q^{86} +8.23607 q^{87} +5.00000 q^{88} +11.6180 q^{89} +1.00000 q^{90} +2.38197 q^{91} -0.618034 q^{92} -6.70820 q^{93} +11.7082 q^{94} +2.14590 q^{95} -3.38197 q^{96} +10.4164 q^{97} -1.61803 q^{98} +2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{5} + q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{5} + q^{6} - 2 q^{7} + 2 q^{9} + 2 q^{10} + q^{12} - 7 q^{13} + q^{14} - q^{15} - 3 q^{16} - q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} - 5 q^{22} - 2 q^{23} - 7 q^{25} + q^{26} - 2 q^{27} + q^{28} - 12 q^{29} - 2 q^{30} + 9 q^{32} - 15 q^{34} - q^{35} - q^{36} - 22 q^{37} + 9 q^{38} + 7 q^{39} - 5 q^{40} + 6 q^{41} - q^{42} - q^{43} + 5 q^{44} + q^{45} + q^{46} - 10 q^{47} + 3 q^{48} + 2 q^{49} + 6 q^{50} + 6 q^{52} - 15 q^{53} + q^{54} - 5 q^{55} - 2 q^{57} + 11 q^{58} + 21 q^{59} + 3 q^{60} - 3 q^{61} - 15 q^{62} - 2 q^{63} + 4 q^{64} - 6 q^{65} + 5 q^{66} + q^{67} + 15 q^{68} + 2 q^{69} - 2 q^{70} + 11 q^{71} - 12 q^{73} + 11 q^{74} + 7 q^{75} - 11 q^{76} - q^{78} - 10 q^{79} + 6 q^{80} + 2 q^{81} + 7 q^{82} + 4 q^{83} - q^{84} - 15 q^{85} + 3 q^{86} + 12 q^{87} + 10 q^{88} + 21 q^{89} + 2 q^{90} + 7 q^{91} + q^{92} + 10 q^{94} + 11 q^{95} - 9 q^{96} - 6 q^{97} - q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.14412 −0.572061 0.820211i \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.618034 0.309017
\(5\) −0.618034 −0.276393 −0.138197 0.990405i \(-0.544131\pi\)
−0.138197 + 0.990405i \(0.544131\pi\)
\(6\) 1.61803 0.660560
\(7\) −1.00000 −0.377964
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 2.23607 0.674200 0.337100 0.941469i \(-0.390554\pi\)
0.337100 + 0.941469i \(0.390554\pi\)
\(12\) −0.618034 −0.178411
\(13\) −2.38197 −0.660639 −0.330319 0.943869i \(-0.607156\pi\)
−0.330319 + 0.943869i \(0.607156\pi\)
\(14\) 1.61803 0.432438
\(15\) 0.618034 0.159576
\(16\) −4.85410 −1.21353
\(17\) 6.70820 1.62698 0.813489 0.581580i \(-0.197565\pi\)
0.813489 + 0.581580i \(0.197565\pi\)
\(18\) −1.61803 −0.381374
\(19\) −3.47214 −0.796563 −0.398281 0.917263i \(-0.630393\pi\)
−0.398281 + 0.917263i \(0.630393\pi\)
\(20\) −0.381966 −0.0854102
\(21\) 1.00000 0.218218
\(22\) −3.61803 −0.771367
\(23\) −1.00000 −0.208514
\(24\) −2.23607 −0.456435
\(25\) −4.61803 −0.923607
\(26\) 3.85410 0.755852
\(27\) −1.00000 −0.192450
\(28\) −0.618034 −0.116797
\(29\) −8.23607 −1.52940 −0.764700 0.644387i \(-0.777113\pi\)
−0.764700 + 0.644387i \(0.777113\pi\)
\(30\) −1.00000 −0.182574
\(31\) 6.70820 1.20483 0.602414 0.798183i \(-0.294205\pi\)
0.602414 + 0.798183i \(0.294205\pi\)
\(32\) 3.38197 0.597853
\(33\) −2.23607 −0.389249
\(34\) −10.8541 −1.86146
\(35\) 0.618034 0.104467
\(36\) 0.618034 0.103006
\(37\) −11.0000 −1.80839 −0.904194 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 5.61803 0.911365
\(39\) 2.38197 0.381420
\(40\) −1.38197 −0.218508
\(41\) −1.47214 −0.229909 −0.114955 0.993371i \(-0.536672\pi\)
−0.114955 + 0.993371i \(0.536672\pi\)
\(42\) −1.61803 −0.249668
\(43\) −1.61803 −0.246748 −0.123374 0.992360i \(-0.539371\pi\)
−0.123374 + 0.992360i \(0.539371\pi\)
\(44\) 1.38197 0.208339
\(45\) −0.618034 −0.0921311
\(46\) 1.61803 0.238566
\(47\) −7.23607 −1.05549 −0.527744 0.849403i \(-0.676962\pi\)
−0.527744 + 0.849403i \(0.676962\pi\)
\(48\) 4.85410 0.700629
\(49\) 1.00000 0.142857
\(50\) 7.47214 1.05672
\(51\) −6.70820 −0.939336
\(52\) −1.47214 −0.204149
\(53\) −13.0902 −1.79807 −0.899037 0.437874i \(-0.855732\pi\)
−0.899037 + 0.437874i \(0.855732\pi\)
\(54\) 1.61803 0.220187
\(55\) −1.38197 −0.186344
\(56\) −2.23607 −0.298807
\(57\) 3.47214 0.459896
\(58\) 13.3262 1.74982
\(59\) 9.38197 1.22143 0.610714 0.791851i \(-0.290883\pi\)
0.610714 + 0.791851i \(0.290883\pi\)
\(60\) 0.381966 0.0493116
\(61\) −4.85410 −0.621504 −0.310752 0.950491i \(-0.600581\pi\)
−0.310752 + 0.950491i \(0.600581\pi\)
\(62\) −10.8541 −1.37847
\(63\) −1.00000 −0.125988
\(64\) 4.23607 0.529508
\(65\) 1.47214 0.182596
\(66\) 3.61803 0.445349
\(67\) −5.09017 −0.621863 −0.310932 0.950432i \(-0.600641\pi\)
−0.310932 + 0.950432i \(0.600641\pi\)
\(68\) 4.14590 0.502764
\(69\) 1.00000 0.120386
\(70\) −1.00000 −0.119523
\(71\) 4.38197 0.520044 0.260022 0.965603i \(-0.416270\pi\)
0.260022 + 0.965603i \(0.416270\pi\)
\(72\) 2.23607 0.263523
\(73\) −12.7082 −1.48738 −0.743691 0.668523i \(-0.766927\pi\)
−0.743691 + 0.668523i \(0.766927\pi\)
\(74\) 17.7984 2.06902
\(75\) 4.61803 0.533245
\(76\) −2.14590 −0.246151
\(77\) −2.23607 −0.254824
\(78\) −3.85410 −0.436391
\(79\) −9.47214 −1.06570 −0.532849 0.846210i \(-0.678879\pi\)
−0.532849 + 0.846210i \(0.678879\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) 2.38197 0.263044
\(83\) −9.18034 −1.00767 −0.503837 0.863799i \(-0.668079\pi\)
−0.503837 + 0.863799i \(0.668079\pi\)
\(84\) 0.618034 0.0674330
\(85\) −4.14590 −0.449686
\(86\) 2.61803 0.282310
\(87\) 8.23607 0.882999
\(88\) 5.00000 0.533002
\(89\) 11.6180 1.23151 0.615755 0.787938i \(-0.288851\pi\)
0.615755 + 0.787938i \(0.288851\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.38197 0.249698
\(92\) −0.618034 −0.0644345
\(93\) −6.70820 −0.695608
\(94\) 11.7082 1.20761
\(95\) 2.14590 0.220164
\(96\) −3.38197 −0.345170
\(97\) 10.4164 1.05763 0.528813 0.848738i \(-0.322637\pi\)
0.528813 + 0.848738i \(0.322637\pi\)
\(98\) −1.61803 −0.163446
\(99\) 2.23607 0.224733
\(100\) −2.85410 −0.285410
\(101\) 4.14590 0.412532 0.206266 0.978496i \(-0.433869\pi\)
0.206266 + 0.978496i \(0.433869\pi\)
\(102\) 10.8541 1.07472
\(103\) −7.41641 −0.730760 −0.365380 0.930858i \(-0.619061\pi\)
−0.365380 + 0.930858i \(0.619061\pi\)
\(104\) −5.32624 −0.522281
\(105\) −0.618034 −0.0603139
\(106\) 21.1803 2.05722
\(107\) 3.32624 0.321560 0.160780 0.986990i \(-0.448599\pi\)
0.160780 + 0.986990i \(0.448599\pi\)
\(108\) −0.618034 −0.0594703
\(109\) 19.2705 1.84578 0.922890 0.385064i \(-0.125820\pi\)
0.922890 + 0.385064i \(0.125820\pi\)
\(110\) 2.23607 0.213201
\(111\) 11.0000 1.04407
\(112\) 4.85410 0.458670
\(113\) −0.909830 −0.0855896 −0.0427948 0.999084i \(-0.513626\pi\)
−0.0427948 + 0.999084i \(0.513626\pi\)
\(114\) −5.61803 −0.526177
\(115\) 0.618034 0.0576320
\(116\) −5.09017 −0.472610
\(117\) −2.38197 −0.220213
\(118\) −15.1803 −1.39746
\(119\) −6.70820 −0.614940
\(120\) 1.38197 0.126156
\(121\) −6.00000 −0.545455
\(122\) 7.85410 0.711077
\(123\) 1.47214 0.132738
\(124\) 4.14590 0.372313
\(125\) 5.94427 0.531672
\(126\) 1.61803 0.144146
\(127\) 22.2705 1.97619 0.988094 0.153851i \(-0.0491675\pi\)
0.988094 + 0.153851i \(0.0491675\pi\)
\(128\) −13.6180 −1.20368
\(129\) 1.61803 0.142460
\(130\) −2.38197 −0.208912
\(131\) 15.1803 1.32631 0.663156 0.748481i \(-0.269216\pi\)
0.663156 + 0.748481i \(0.269216\pi\)
\(132\) −1.38197 −0.120285
\(133\) 3.47214 0.301072
\(134\) 8.23607 0.711488
\(135\) 0.618034 0.0531919
\(136\) 15.0000 1.28624
\(137\) −16.4164 −1.40255 −0.701274 0.712892i \(-0.747385\pi\)
−0.701274 + 0.712892i \(0.747385\pi\)
\(138\) −1.61803 −0.137736
\(139\) −11.3820 −0.965406 −0.482703 0.875784i \(-0.660345\pi\)
−0.482703 + 0.875784i \(0.660345\pi\)
\(140\) 0.381966 0.0322820
\(141\) 7.23607 0.609387
\(142\) −7.09017 −0.594994
\(143\) −5.32624 −0.445402
\(144\) −4.85410 −0.404508
\(145\) 5.09017 0.422716
\(146\) 20.5623 1.70175
\(147\) −1.00000 −0.0824786
\(148\) −6.79837 −0.558823
\(149\) −19.2361 −1.57588 −0.787940 0.615752i \(-0.788852\pi\)
−0.787940 + 0.615752i \(0.788852\pi\)
\(150\) −7.47214 −0.610097
\(151\) −15.2361 −1.23989 −0.619947 0.784644i \(-0.712846\pi\)
−0.619947 + 0.784644i \(0.712846\pi\)
\(152\) −7.76393 −0.629738
\(153\) 6.70820 0.542326
\(154\) 3.61803 0.291549
\(155\) −4.14590 −0.333007
\(156\) 1.47214 0.117865
\(157\) −0.291796 −0.0232879 −0.0116439 0.999932i \(-0.503706\pi\)
−0.0116439 + 0.999932i \(0.503706\pi\)
\(158\) 15.3262 1.21929
\(159\) 13.0902 1.03812
\(160\) −2.09017 −0.165242
\(161\) 1.00000 0.0788110
\(162\) −1.61803 −0.127125
\(163\) −0.618034 −0.0484082 −0.0242041 0.999707i \(-0.507705\pi\)
−0.0242041 + 0.999707i \(0.507705\pi\)
\(164\) −0.909830 −0.0710458
\(165\) 1.38197 0.107586
\(166\) 14.8541 1.15290
\(167\) −7.18034 −0.555631 −0.277816 0.960634i \(-0.589610\pi\)
−0.277816 + 0.960634i \(0.589610\pi\)
\(168\) 2.23607 0.172516
\(169\) −7.32624 −0.563557
\(170\) 6.70820 0.514496
\(171\) −3.47214 −0.265521
\(172\) −1.00000 −0.0762493
\(173\) −3.47214 −0.263982 −0.131991 0.991251i \(-0.542137\pi\)
−0.131991 + 0.991251i \(0.542137\pi\)
\(174\) −13.3262 −1.01026
\(175\) 4.61803 0.349091
\(176\) −10.8541 −0.818159
\(177\) −9.38197 −0.705192
\(178\) −18.7984 −1.40900
\(179\) 1.85410 0.138582 0.0692910 0.997596i \(-0.477926\pi\)
0.0692910 + 0.997596i \(0.477926\pi\)
\(180\) −0.381966 −0.0284701
\(181\) −1.94427 −0.144517 −0.0722583 0.997386i \(-0.523021\pi\)
−0.0722583 + 0.997386i \(0.523021\pi\)
\(182\) −3.85410 −0.285685
\(183\) 4.85410 0.358826
\(184\) −2.23607 −0.164845
\(185\) 6.79837 0.499826
\(186\) 10.8541 0.795861
\(187\) 15.0000 1.09691
\(188\) −4.47214 −0.326164
\(189\) 1.00000 0.0727393
\(190\) −3.47214 −0.251895
\(191\) 16.1803 1.17077 0.585384 0.810756i \(-0.300944\pi\)
0.585384 + 0.810756i \(0.300944\pi\)
\(192\) −4.23607 −0.305712
\(193\) 8.29180 0.596857 0.298428 0.954432i \(-0.403538\pi\)
0.298428 + 0.954432i \(0.403538\pi\)
\(194\) −16.8541 −1.21005
\(195\) −1.47214 −0.105422
\(196\) 0.618034 0.0441453
\(197\) −25.5066 −1.81727 −0.908634 0.417593i \(-0.862874\pi\)
−0.908634 + 0.417593i \(0.862874\pi\)
\(198\) −3.61803 −0.257122
\(199\) −6.90983 −0.489825 −0.244912 0.969545i \(-0.578759\pi\)
−0.244912 + 0.969545i \(0.578759\pi\)
\(200\) −10.3262 −0.730175
\(201\) 5.09017 0.359033
\(202\) −6.70820 −0.471988
\(203\) 8.23607 0.578059
\(204\) −4.14590 −0.290271
\(205\) 0.909830 0.0635453
\(206\) 12.0000 0.836080
\(207\) −1.00000 −0.0695048
\(208\) 11.5623 0.801702
\(209\) −7.76393 −0.537042
\(210\) 1.00000 0.0690066
\(211\) 10.4164 0.717095 0.358548 0.933511i \(-0.383272\pi\)
0.358548 + 0.933511i \(0.383272\pi\)
\(212\) −8.09017 −0.555635
\(213\) −4.38197 −0.300247
\(214\) −5.38197 −0.367904
\(215\) 1.00000 0.0681994
\(216\) −2.23607 −0.152145
\(217\) −6.70820 −0.455383
\(218\) −31.1803 −2.11180
\(219\) 12.7082 0.858741
\(220\) −0.854102 −0.0575835
\(221\) −15.9787 −1.07484
\(222\) −17.7984 −1.19455
\(223\) −17.8541 −1.19560 −0.597800 0.801646i \(-0.703958\pi\)
−0.597800 + 0.801646i \(0.703958\pi\)
\(224\) −3.38197 −0.225967
\(225\) −4.61803 −0.307869
\(226\) 1.47214 0.0979250
\(227\) 24.3262 1.61459 0.807295 0.590149i \(-0.200931\pi\)
0.807295 + 0.590149i \(0.200931\pi\)
\(228\) 2.14590 0.142116
\(229\) −16.3262 −1.07887 −0.539434 0.842028i \(-0.681362\pi\)
−0.539434 + 0.842028i \(0.681362\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 2.23607 0.147122
\(232\) −18.4164 −1.20910
\(233\) −11.0902 −0.726541 −0.363271 0.931684i \(-0.618340\pi\)
−0.363271 + 0.931684i \(0.618340\pi\)
\(234\) 3.85410 0.251951
\(235\) 4.47214 0.291730
\(236\) 5.79837 0.377442
\(237\) 9.47214 0.615281
\(238\) 10.8541 0.703567
\(239\) −4.79837 −0.310381 −0.155191 0.987885i \(-0.549599\pi\)
−0.155191 + 0.987885i \(0.549599\pi\)
\(240\) −3.00000 −0.193649
\(241\) −11.0000 −0.708572 −0.354286 0.935137i \(-0.615276\pi\)
−0.354286 + 0.935137i \(0.615276\pi\)
\(242\) 9.70820 0.624067
\(243\) −1.00000 −0.0641500
\(244\) −3.00000 −0.192055
\(245\) −0.618034 −0.0394847
\(246\) −2.38197 −0.151869
\(247\) 8.27051 0.526240
\(248\) 15.0000 0.952501
\(249\) 9.18034 0.581780
\(250\) −9.61803 −0.608298
\(251\) 23.1246 1.45961 0.729806 0.683654i \(-0.239610\pi\)
0.729806 + 0.683654i \(0.239610\pi\)
\(252\) −0.618034 −0.0389325
\(253\) −2.23607 −0.140580
\(254\) −36.0344 −2.26100
\(255\) 4.14590 0.259626
\(256\) 13.5623 0.847644
\(257\) −3.23607 −0.201860 −0.100930 0.994894i \(-0.532182\pi\)
−0.100930 + 0.994894i \(0.532182\pi\)
\(258\) −2.61803 −0.162992
\(259\) 11.0000 0.683507
\(260\) 0.909830 0.0564253
\(261\) −8.23607 −0.509800
\(262\) −24.5623 −1.51746
\(263\) −0.0557281 −0.00343634 −0.00171817 0.999999i \(-0.500547\pi\)
−0.00171817 + 0.999999i \(0.500547\pi\)
\(264\) −5.00000 −0.307729
\(265\) 8.09017 0.496975
\(266\) −5.61803 −0.344464
\(267\) −11.6180 −0.711012
\(268\) −3.14590 −0.192166
\(269\) −32.0902 −1.95657 −0.978286 0.207259i \(-0.933546\pi\)
−0.978286 + 0.207259i \(0.933546\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 21.9443 1.33302 0.666510 0.745496i \(-0.267787\pi\)
0.666510 + 0.745496i \(0.267787\pi\)
\(272\) −32.5623 −1.97438
\(273\) −2.38197 −0.144163
\(274\) 26.5623 1.60469
\(275\) −10.3262 −0.622696
\(276\) 0.618034 0.0372013
\(277\) 4.27051 0.256590 0.128295 0.991736i \(-0.459050\pi\)
0.128295 + 0.991736i \(0.459050\pi\)
\(278\) 18.4164 1.10454
\(279\) 6.70820 0.401610
\(280\) 1.38197 0.0825883
\(281\) −3.70820 −0.221213 −0.110606 0.993864i \(-0.535279\pi\)
−0.110606 + 0.993864i \(0.535279\pi\)
\(282\) −11.7082 −0.697213
\(283\) −26.2148 −1.55831 −0.779154 0.626833i \(-0.784351\pi\)
−0.779154 + 0.626833i \(0.784351\pi\)
\(284\) 2.70820 0.160702
\(285\) −2.14590 −0.127112
\(286\) 8.61803 0.509595
\(287\) 1.47214 0.0868974
\(288\) 3.38197 0.199284
\(289\) 28.0000 1.64706
\(290\) −8.23607 −0.483639
\(291\) −10.4164 −0.610621
\(292\) −7.85410 −0.459627
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 1.61803 0.0943657
\(295\) −5.79837 −0.337594
\(296\) −24.5967 −1.42966
\(297\) −2.23607 −0.129750
\(298\) 31.1246 1.80300
\(299\) 2.38197 0.137753
\(300\) 2.85410 0.164782
\(301\) 1.61803 0.0932619
\(302\) 24.6525 1.41859
\(303\) −4.14590 −0.238176
\(304\) 16.8541 0.966649
\(305\) 3.00000 0.171780
\(306\) −10.8541 −0.620488
\(307\) 16.1246 0.920280 0.460140 0.887846i \(-0.347799\pi\)
0.460140 + 0.887846i \(0.347799\pi\)
\(308\) −1.38197 −0.0787448
\(309\) 7.41641 0.421905
\(310\) 6.70820 0.381000
\(311\) 15.3262 0.869071 0.434536 0.900655i \(-0.356913\pi\)
0.434536 + 0.900655i \(0.356913\pi\)
\(312\) 5.32624 0.301539
\(313\) −2.47214 −0.139733 −0.0698667 0.997556i \(-0.522257\pi\)
−0.0698667 + 0.997556i \(0.522257\pi\)
\(314\) 0.472136 0.0266442
\(315\) 0.618034 0.0348223
\(316\) −5.85410 −0.329319
\(317\) 14.5066 0.814771 0.407385 0.913256i \(-0.366441\pi\)
0.407385 + 0.913256i \(0.366441\pi\)
\(318\) −21.1803 −1.18773
\(319\) −18.4164 −1.03112
\(320\) −2.61803 −0.146353
\(321\) −3.32624 −0.185652
\(322\) −1.61803 −0.0901695
\(323\) −23.2918 −1.29599
\(324\) 0.618034 0.0343352
\(325\) 11.0000 0.610170
\(326\) 1.00000 0.0553849
\(327\) −19.2705 −1.06566
\(328\) −3.29180 −0.181759
\(329\) 7.23607 0.398937
\(330\) −2.23607 −0.123091
\(331\) −13.4164 −0.737432 −0.368716 0.929542i \(-0.620203\pi\)
−0.368716 + 0.929542i \(0.620203\pi\)
\(332\) −5.67376 −0.311388
\(333\) −11.0000 −0.602796
\(334\) 11.6180 0.635711
\(335\) 3.14590 0.171879
\(336\) −4.85410 −0.264813
\(337\) −7.50658 −0.408909 −0.204455 0.978876i \(-0.565542\pi\)
−0.204455 + 0.978876i \(0.565542\pi\)
\(338\) 11.8541 0.644778
\(339\) 0.909830 0.0494152
\(340\) −2.56231 −0.138961
\(341\) 15.0000 0.812296
\(342\) 5.61803 0.303788
\(343\) −1.00000 −0.0539949
\(344\) −3.61803 −0.195071
\(345\) −0.618034 −0.0332738
\(346\) 5.61803 0.302027
\(347\) 5.18034 0.278095 0.139048 0.990286i \(-0.455596\pi\)
0.139048 + 0.990286i \(0.455596\pi\)
\(348\) 5.09017 0.272862
\(349\) −11.8541 −0.634536 −0.317268 0.948336i \(-0.602765\pi\)
−0.317268 + 0.948336i \(0.602765\pi\)
\(350\) −7.47214 −0.399402
\(351\) 2.38197 0.127140
\(352\) 7.56231 0.403072
\(353\) 30.3050 1.61297 0.806485 0.591255i \(-0.201367\pi\)
0.806485 + 0.591255i \(0.201367\pi\)
\(354\) 15.1803 0.806826
\(355\) −2.70820 −0.143737
\(356\) 7.18034 0.380557
\(357\) 6.70820 0.355036
\(358\) −3.00000 −0.158555
\(359\) 23.0902 1.21865 0.609326 0.792920i \(-0.291440\pi\)
0.609326 + 0.792920i \(0.291440\pi\)
\(360\) −1.38197 −0.0728360
\(361\) −6.94427 −0.365488
\(362\) 3.14590 0.165345
\(363\) 6.00000 0.314918
\(364\) 1.47214 0.0771609
\(365\) 7.85410 0.411102
\(366\) −7.85410 −0.410540
\(367\) 10.8541 0.566580 0.283290 0.959034i \(-0.408574\pi\)
0.283290 + 0.959034i \(0.408574\pi\)
\(368\) 4.85410 0.253038
\(369\) −1.47214 −0.0766363
\(370\) −11.0000 −0.571863
\(371\) 13.0902 0.679608
\(372\) −4.14590 −0.214955
\(373\) 24.4164 1.26423 0.632117 0.774873i \(-0.282186\pi\)
0.632117 + 0.774873i \(0.282186\pi\)
\(374\) −24.2705 −1.25500
\(375\) −5.94427 −0.306961
\(376\) −16.1803 −0.834437
\(377\) 19.6180 1.01038
\(378\) −1.61803 −0.0832227
\(379\) 7.41641 0.380955 0.190478 0.981692i \(-0.438996\pi\)
0.190478 + 0.981692i \(0.438996\pi\)
\(380\) 1.32624 0.0680346
\(381\) −22.2705 −1.14095
\(382\) −26.1803 −1.33950
\(383\) −0.708204 −0.0361875 −0.0180938 0.999836i \(-0.505760\pi\)
−0.0180938 + 0.999836i \(0.505760\pi\)
\(384\) 13.6180 0.694942
\(385\) 1.38197 0.0704315
\(386\) −13.4164 −0.682877
\(387\) −1.61803 −0.0822493
\(388\) 6.43769 0.326824
\(389\) 5.29180 0.268305 0.134152 0.990961i \(-0.457169\pi\)
0.134152 + 0.990961i \(0.457169\pi\)
\(390\) 2.38197 0.120616
\(391\) −6.70820 −0.339248
\(392\) 2.23607 0.112938
\(393\) −15.1803 −0.765747
\(394\) 41.2705 2.07918
\(395\) 5.85410 0.294552
\(396\) 1.38197 0.0694464
\(397\) 22.0689 1.10761 0.553803 0.832648i \(-0.313176\pi\)
0.553803 + 0.832648i \(0.313176\pi\)
\(398\) 11.1803 0.560420
\(399\) −3.47214 −0.173824
\(400\) 22.4164 1.12082
\(401\) 33.1803 1.65695 0.828474 0.560028i \(-0.189210\pi\)
0.828474 + 0.560028i \(0.189210\pi\)
\(402\) −8.23607 −0.410778
\(403\) −15.9787 −0.795956
\(404\) 2.56231 0.127479
\(405\) −0.618034 −0.0307104
\(406\) −13.3262 −0.661370
\(407\) −24.5967 −1.21922
\(408\) −15.0000 −0.742611
\(409\) 6.41641 0.317271 0.158635 0.987337i \(-0.449291\pi\)
0.158635 + 0.987337i \(0.449291\pi\)
\(410\) −1.47214 −0.0727036
\(411\) 16.4164 0.809762
\(412\) −4.58359 −0.225817
\(413\) −9.38197 −0.461656
\(414\) 1.61803 0.0795220
\(415\) 5.67376 0.278514
\(416\) −8.05573 −0.394965
\(417\) 11.3820 0.557377
\(418\) 12.5623 0.614442
\(419\) 15.6738 0.765713 0.382857 0.923808i \(-0.374940\pi\)
0.382857 + 0.923808i \(0.374940\pi\)
\(420\) −0.381966 −0.0186380
\(421\) −10.2705 −0.500554 −0.250277 0.968174i \(-0.580522\pi\)
−0.250277 + 0.968174i \(0.580522\pi\)
\(422\) −16.8541 −0.820445
\(423\) −7.23607 −0.351830
\(424\) −29.2705 −1.42150
\(425\) −30.9787 −1.50269
\(426\) 7.09017 0.343520
\(427\) 4.85410 0.234906
\(428\) 2.05573 0.0993674
\(429\) 5.32624 0.257153
\(430\) −1.61803 −0.0780285
\(431\) 0.673762 0.0324540 0.0162270 0.999868i \(-0.494835\pi\)
0.0162270 + 0.999868i \(0.494835\pi\)
\(432\) 4.85410 0.233543
\(433\) 19.1246 0.919070 0.459535 0.888160i \(-0.348016\pi\)
0.459535 + 0.888160i \(0.348016\pi\)
\(434\) 10.8541 0.521014
\(435\) −5.09017 −0.244055
\(436\) 11.9098 0.570377
\(437\) 3.47214 0.166095
\(438\) −20.5623 −0.982505
\(439\) 23.6525 1.12887 0.564436 0.825477i \(-0.309094\pi\)
0.564436 + 0.825477i \(0.309094\pi\)
\(440\) −3.09017 −0.147318
\(441\) 1.00000 0.0476190
\(442\) 25.8541 1.22975
\(443\) −9.52786 −0.452682 −0.226341 0.974048i \(-0.572676\pi\)
−0.226341 + 0.974048i \(0.572676\pi\)
\(444\) 6.79837 0.322637
\(445\) −7.18034 −0.340381
\(446\) 28.8885 1.36791
\(447\) 19.2361 0.909835
\(448\) −4.23607 −0.200135
\(449\) −18.4377 −0.870129 −0.435064 0.900399i \(-0.643274\pi\)
−0.435064 + 0.900399i \(0.643274\pi\)
\(450\) 7.47214 0.352240
\(451\) −3.29180 −0.155005
\(452\) −0.562306 −0.0264486
\(453\) 15.2361 0.715853
\(454\) −39.3607 −1.84729
\(455\) −1.47214 −0.0690148
\(456\) 7.76393 0.363579
\(457\) −14.3262 −0.670153 −0.335077 0.942191i \(-0.608762\pi\)
−0.335077 + 0.942191i \(0.608762\pi\)
\(458\) 26.4164 1.23436
\(459\) −6.70820 −0.313112
\(460\) 0.381966 0.0178093
\(461\) 25.4508 1.18536 0.592682 0.805436i \(-0.298069\pi\)
0.592682 + 0.805436i \(0.298069\pi\)
\(462\) −3.61803 −0.168326
\(463\) 17.2918 0.803618 0.401809 0.915724i \(-0.368382\pi\)
0.401809 + 0.915724i \(0.368382\pi\)
\(464\) 39.9787 1.85597
\(465\) 4.14590 0.192261
\(466\) 17.9443 0.831252
\(467\) −15.1803 −0.702462 −0.351231 0.936289i \(-0.614237\pi\)
−0.351231 + 0.936289i \(0.614237\pi\)
\(468\) −1.47214 −0.0680495
\(469\) 5.09017 0.235042
\(470\) −7.23607 −0.333775
\(471\) 0.291796 0.0134453
\(472\) 20.9787 0.965624
\(473\) −3.61803 −0.166357
\(474\) −15.3262 −0.703957
\(475\) 16.0344 0.735711
\(476\) −4.14590 −0.190027
\(477\) −13.0902 −0.599358
\(478\) 7.76393 0.355114
\(479\) 21.0000 0.959514 0.479757 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(480\) 2.09017 0.0954028
\(481\) 26.2016 1.19469
\(482\) 17.7984 0.810694
\(483\) −1.00000 −0.0455016
\(484\) −3.70820 −0.168555
\(485\) −6.43769 −0.292321
\(486\) 1.61803 0.0733955
\(487\) −29.2918 −1.32734 −0.663669 0.748026i \(-0.731002\pi\)
−0.663669 + 0.748026i \(0.731002\pi\)
\(488\) −10.8541 −0.491342
\(489\) 0.618034 0.0279485
\(490\) 1.00000 0.0451754
\(491\) 9.79837 0.442194 0.221097 0.975252i \(-0.429036\pi\)
0.221097 + 0.975252i \(0.429036\pi\)
\(492\) 0.909830 0.0410183
\(493\) −55.2492 −2.48830
\(494\) −13.3820 −0.602083
\(495\) −1.38197 −0.0621148
\(496\) −32.5623 −1.46209
\(497\) −4.38197 −0.196558
\(498\) −14.8541 −0.665628
\(499\) −38.3951 −1.71880 −0.859401 0.511302i \(-0.829163\pi\)
−0.859401 + 0.511302i \(0.829163\pi\)
\(500\) 3.67376 0.164296
\(501\) 7.18034 0.320794
\(502\) −37.4164 −1.66998
\(503\) 36.3262 1.61971 0.809853 0.586632i \(-0.199547\pi\)
0.809853 + 0.586632i \(0.199547\pi\)
\(504\) −2.23607 −0.0996024
\(505\) −2.56231 −0.114021
\(506\) 3.61803 0.160841
\(507\) 7.32624 0.325370
\(508\) 13.7639 0.610676
\(509\) −15.5967 −0.691314 −0.345657 0.938361i \(-0.612344\pi\)
−0.345657 + 0.938361i \(0.612344\pi\)
\(510\) −6.70820 −0.297044
\(511\) 12.7082 0.562178
\(512\) 5.29180 0.233867
\(513\) 3.47214 0.153299
\(514\) 5.23607 0.230953
\(515\) 4.58359 0.201977
\(516\) 1.00000 0.0440225
\(517\) −16.1803 −0.711611
\(518\) −17.7984 −0.782016
\(519\) 3.47214 0.152410
\(520\) 3.29180 0.144355
\(521\) 3.52786 0.154559 0.0772793 0.997009i \(-0.475377\pi\)
0.0772793 + 0.997009i \(0.475377\pi\)
\(522\) 13.3262 0.583274
\(523\) 21.4164 0.936474 0.468237 0.883603i \(-0.344889\pi\)
0.468237 + 0.883603i \(0.344889\pi\)
\(524\) 9.38197 0.409853
\(525\) −4.61803 −0.201548
\(526\) 0.0901699 0.00393160
\(527\) 45.0000 1.96023
\(528\) 10.8541 0.472364
\(529\) 1.00000 0.0434783
\(530\) −13.0902 −0.568601
\(531\) 9.38197 0.407143
\(532\) 2.14590 0.0930365
\(533\) 3.50658 0.151887
\(534\) 18.7984 0.813485
\(535\) −2.05573 −0.0888769
\(536\) −11.3820 −0.491626
\(537\) −1.85410 −0.0800104
\(538\) 51.9230 2.23856
\(539\) 2.23607 0.0963143
\(540\) 0.381966 0.0164372
\(541\) −5.12461 −0.220324 −0.110162 0.993914i \(-0.535137\pi\)
−0.110162 + 0.993914i \(0.535137\pi\)
\(542\) −35.5066 −1.52514
\(543\) 1.94427 0.0834367
\(544\) 22.6869 0.972694
\(545\) −11.9098 −0.510161
\(546\) 3.85410 0.164940
\(547\) 12.7984 0.547219 0.273609 0.961841i \(-0.411782\pi\)
0.273609 + 0.961841i \(0.411782\pi\)
\(548\) −10.1459 −0.433411
\(549\) −4.85410 −0.207168
\(550\) 16.7082 0.712440
\(551\) 28.5967 1.21826
\(552\) 2.23607 0.0951734
\(553\) 9.47214 0.402796
\(554\) −6.90983 −0.293571
\(555\) −6.79837 −0.288575
\(556\) −7.03444 −0.298327
\(557\) −5.81966 −0.246587 −0.123293 0.992370i \(-0.539346\pi\)
−0.123293 + 0.992370i \(0.539346\pi\)
\(558\) −10.8541 −0.459491
\(559\) 3.85410 0.163011
\(560\) −3.00000 −0.126773
\(561\) −15.0000 −0.633300
\(562\) 6.00000 0.253095
\(563\) −23.5066 −0.990684 −0.495342 0.868698i \(-0.664957\pi\)
−0.495342 + 0.868698i \(0.664957\pi\)
\(564\) 4.47214 0.188311
\(565\) 0.562306 0.0236564
\(566\) 42.4164 1.78289
\(567\) −1.00000 −0.0419961
\(568\) 9.79837 0.411131
\(569\) −16.3607 −0.685875 −0.342938 0.939358i \(-0.611422\pi\)
−0.342938 + 0.939358i \(0.611422\pi\)
\(570\) 3.47214 0.145432
\(571\) −44.4164 −1.85877 −0.929384 0.369113i \(-0.879661\pi\)
−0.929384 + 0.369113i \(0.879661\pi\)
\(572\) −3.29180 −0.137637
\(573\) −16.1803 −0.675943
\(574\) −2.38197 −0.0994213
\(575\) 4.61803 0.192585
\(576\) 4.23607 0.176503
\(577\) −36.8328 −1.53337 −0.766685 0.642023i \(-0.778095\pi\)
−0.766685 + 0.642023i \(0.778095\pi\)
\(578\) −45.3050 −1.88444
\(579\) −8.29180 −0.344595
\(580\) 3.14590 0.130626
\(581\) 9.18034 0.380865
\(582\) 16.8541 0.698625
\(583\) −29.2705 −1.21226
\(584\) −28.4164 −1.17588
\(585\) 1.47214 0.0608653
\(586\) −19.4164 −0.802084
\(587\) −31.0344 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(588\) −0.618034 −0.0254873
\(589\) −23.2918 −0.959722
\(590\) 9.38197 0.386249
\(591\) 25.5066 1.04920
\(592\) 53.3951 2.19453
\(593\) −32.0689 −1.31691 −0.658456 0.752620i \(-0.728790\pi\)
−0.658456 + 0.752620i \(0.728790\pi\)
\(594\) 3.61803 0.148450
\(595\) 4.14590 0.169965
\(596\) −11.8885 −0.486974
\(597\) 6.90983 0.282801
\(598\) −3.85410 −0.157606
\(599\) −15.5066 −0.633582 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(600\) 10.3262 0.421567
\(601\) 0.909830 0.0371127 0.0185564 0.999828i \(-0.494093\pi\)
0.0185564 + 0.999828i \(0.494093\pi\)
\(602\) −2.61803 −0.106703
\(603\) −5.09017 −0.207288
\(604\) −9.41641 −0.383148
\(605\) 3.70820 0.150760
\(606\) 6.70820 0.272502
\(607\) −33.7984 −1.37183 −0.685917 0.727680i \(-0.740599\pi\)
−0.685917 + 0.727680i \(0.740599\pi\)
\(608\) −11.7426 −0.476227
\(609\) −8.23607 −0.333742
\(610\) −4.85410 −0.196537
\(611\) 17.2361 0.697297
\(612\) 4.14590 0.167588
\(613\) −11.6525 −0.470639 −0.235320 0.971918i \(-0.575614\pi\)
−0.235320 + 0.971918i \(0.575614\pi\)
\(614\) −26.0902 −1.05291
\(615\) −0.909830 −0.0366879
\(616\) −5.00000 −0.201456
\(617\) −2.67376 −0.107642 −0.0538208 0.998551i \(-0.517140\pi\)
−0.0538208 + 0.998551i \(0.517140\pi\)
\(618\) −12.0000 −0.482711
\(619\) −19.6180 −0.788515 −0.394258 0.919000i \(-0.628998\pi\)
−0.394258 + 0.919000i \(0.628998\pi\)
\(620\) −2.56231 −0.102905
\(621\) 1.00000 0.0401286
\(622\) −24.7984 −0.994324
\(623\) −11.6180 −0.465467
\(624\) −11.5623 −0.462863
\(625\) 19.4164 0.776656
\(626\) 4.00000 0.159872
\(627\) 7.76393 0.310062
\(628\) −0.180340 −0.00719634
\(629\) −73.7902 −2.94221
\(630\) −1.00000 −0.0398410
\(631\) 16.1246 0.641911 0.320955 0.947094i \(-0.395996\pi\)
0.320955 + 0.947094i \(0.395996\pi\)
\(632\) −21.1803 −0.842509
\(633\) −10.4164 −0.414015
\(634\) −23.4721 −0.932198
\(635\) −13.7639 −0.546205
\(636\) 8.09017 0.320796
\(637\) −2.38197 −0.0943769
\(638\) 29.7984 1.17973
\(639\) 4.38197 0.173348
\(640\) 8.41641 0.332688
\(641\) 17.7984 0.702994 0.351497 0.936189i \(-0.385673\pi\)
0.351497 + 0.936189i \(0.385673\pi\)
\(642\) 5.38197 0.212409
\(643\) −11.9787 −0.472394 −0.236197 0.971705i \(-0.575901\pi\)
−0.236197 + 0.971705i \(0.575901\pi\)
\(644\) 0.618034 0.0243540
\(645\) −1.00000 −0.0393750
\(646\) 37.6869 1.48277
\(647\) −24.0344 −0.944891 −0.472446 0.881360i \(-0.656629\pi\)
−0.472446 + 0.881360i \(0.656629\pi\)
\(648\) 2.23607 0.0878410
\(649\) 20.9787 0.823487
\(650\) −17.7984 −0.698110
\(651\) 6.70820 0.262915
\(652\) −0.381966 −0.0149589
\(653\) 27.7426 1.08565 0.542827 0.839845i \(-0.317354\pi\)
0.542827 + 0.839845i \(0.317354\pi\)
\(654\) 31.1803 1.21925
\(655\) −9.38197 −0.366584
\(656\) 7.14590 0.279000
\(657\) −12.7082 −0.495794
\(658\) −11.7082 −0.456433
\(659\) 9.65248 0.376007 0.188004 0.982168i \(-0.439798\pi\)
0.188004 + 0.982168i \(0.439798\pi\)
\(660\) 0.854102 0.0332459
\(661\) 14.4164 0.560733 0.280367 0.959893i \(-0.409544\pi\)
0.280367 + 0.959893i \(0.409544\pi\)
\(662\) 21.7082 0.843713
\(663\) 15.9787 0.620562
\(664\) −20.5279 −0.796636
\(665\) −2.14590 −0.0832144
\(666\) 17.7984 0.689673
\(667\) 8.23607 0.318902
\(668\) −4.43769 −0.171700
\(669\) 17.8541 0.690279
\(670\) −5.09017 −0.196650
\(671\) −10.8541 −0.419018
\(672\) 3.38197 0.130462
\(673\) 12.4164 0.478617 0.239309 0.970944i \(-0.423079\pi\)
0.239309 + 0.970944i \(0.423079\pi\)
\(674\) 12.1459 0.467843
\(675\) 4.61803 0.177748
\(676\) −4.52786 −0.174149
\(677\) −1.25735 −0.0483240 −0.0241620 0.999708i \(-0.507692\pi\)
−0.0241620 + 0.999708i \(0.507692\pi\)
\(678\) −1.47214 −0.0565370
\(679\) −10.4164 −0.399745
\(680\) −9.27051 −0.355508
\(681\) −24.3262 −0.932183
\(682\) −24.2705 −0.929366
\(683\) 26.0132 0.995366 0.497683 0.867359i \(-0.334184\pi\)
0.497683 + 0.867359i \(0.334184\pi\)
\(684\) −2.14590 −0.0820505
\(685\) 10.1459 0.387655
\(686\) 1.61803 0.0617768
\(687\) 16.3262 0.622885
\(688\) 7.85410 0.299435
\(689\) 31.1803 1.18788
\(690\) 1.00000 0.0380693
\(691\) −7.72949 −0.294044 −0.147022 0.989133i \(-0.546969\pi\)
−0.147022 + 0.989133i \(0.546969\pi\)
\(692\) −2.14590 −0.0815748
\(693\) −2.23607 −0.0849412
\(694\) −8.38197 −0.318175
\(695\) 7.03444 0.266832
\(696\) 18.4164 0.698072
\(697\) −9.87539 −0.374057
\(698\) 19.1803 0.725987
\(699\) 11.0902 0.419469
\(700\) 2.85410 0.107875
\(701\) 37.4508 1.41450 0.707250 0.706964i \(-0.249936\pi\)
0.707250 + 0.706964i \(0.249936\pi\)
\(702\) −3.85410 −0.145464
\(703\) 38.1935 1.44049
\(704\) 9.47214 0.356995
\(705\) −4.47214 −0.168430
\(706\) −49.0344 −1.84544
\(707\) −4.14590 −0.155923
\(708\) −5.79837 −0.217916
\(709\) −2.56231 −0.0962294 −0.0481147 0.998842i \(-0.515321\pi\)
−0.0481147 + 0.998842i \(0.515321\pi\)
\(710\) 4.38197 0.164452
\(711\) −9.47214 −0.355233
\(712\) 25.9787 0.973593
\(713\) −6.70820 −0.251224
\(714\) −10.8541 −0.406205
\(715\) 3.29180 0.123106
\(716\) 1.14590 0.0428242
\(717\) 4.79837 0.179199
\(718\) −37.3607 −1.39429
\(719\) −3.00000 −0.111881 −0.0559406 0.998434i \(-0.517816\pi\)
−0.0559406 + 0.998434i \(0.517816\pi\)
\(720\) 3.00000 0.111803
\(721\) 7.41641 0.276201
\(722\) 11.2361 0.418163
\(723\) 11.0000 0.409094
\(724\) −1.20163 −0.0446581
\(725\) 38.0344 1.41256
\(726\) −9.70820 −0.360305
\(727\) 18.8885 0.700537 0.350269 0.936649i \(-0.386090\pi\)
0.350269 + 0.936649i \(0.386090\pi\)
\(728\) 5.32624 0.197404
\(729\) 1.00000 0.0370370
\(730\) −12.7082 −0.470352
\(731\) −10.8541 −0.401453
\(732\) 3.00000 0.110883
\(733\) 25.5967 0.945437 0.472719 0.881213i \(-0.343273\pi\)
0.472719 + 0.881213i \(0.343273\pi\)
\(734\) −17.5623 −0.648237
\(735\) 0.618034 0.0227965
\(736\) −3.38197 −0.124661
\(737\) −11.3820 −0.419260
\(738\) 2.38197 0.0876814
\(739\) 45.2492 1.66452 0.832260 0.554386i \(-0.187047\pi\)
0.832260 + 0.554386i \(0.187047\pi\)
\(740\) 4.20163 0.154455
\(741\) −8.27051 −0.303825
\(742\) −21.1803 −0.777555
\(743\) 11.6738 0.428269 0.214134 0.976804i \(-0.431307\pi\)
0.214134 + 0.976804i \(0.431307\pi\)
\(744\) −15.0000 −0.549927
\(745\) 11.8885 0.435563
\(746\) −39.5066 −1.44644
\(747\) −9.18034 −0.335891
\(748\) 9.27051 0.338963
\(749\) −3.32624 −0.121538
\(750\) 9.61803 0.351201
\(751\) 32.9787 1.20341 0.601705 0.798718i \(-0.294488\pi\)
0.601705 + 0.798718i \(0.294488\pi\)
\(752\) 35.1246 1.28086
\(753\) −23.1246 −0.842708
\(754\) −31.7426 −1.15600
\(755\) 9.41641 0.342698
\(756\) 0.618034 0.0224777
\(757\) −17.0000 −0.617876 −0.308938 0.951082i \(-0.599973\pi\)
−0.308938 + 0.951082i \(0.599973\pi\)
\(758\) −12.0000 −0.435860
\(759\) 2.23607 0.0811641
\(760\) 4.79837 0.174055
\(761\) −14.4721 −0.524615 −0.262307 0.964984i \(-0.584483\pi\)
−0.262307 + 0.964984i \(0.584483\pi\)
\(762\) 36.0344 1.30539
\(763\) −19.2705 −0.697639
\(764\) 10.0000 0.361787
\(765\) −4.14590 −0.149895
\(766\) 1.14590 0.0414030
\(767\) −22.3475 −0.806922
\(768\) −13.5623 −0.489388
\(769\) 33.2361 1.19852 0.599262 0.800553i \(-0.295461\pi\)
0.599262 + 0.800553i \(0.295461\pi\)
\(770\) −2.23607 −0.0805823
\(771\) 3.23607 0.116544
\(772\) 5.12461 0.184439
\(773\) −19.9443 −0.717346 −0.358673 0.933463i \(-0.616771\pi\)
−0.358673 + 0.933463i \(0.616771\pi\)
\(774\) 2.61803 0.0941033
\(775\) −30.9787 −1.11279
\(776\) 23.2918 0.836127
\(777\) −11.0000 −0.394623
\(778\) −8.56231 −0.306974
\(779\) 5.11146 0.183137
\(780\) −0.909830 −0.0325771
\(781\) 9.79837 0.350613
\(782\) 10.8541 0.388142
\(783\) 8.23607 0.294333
\(784\) −4.85410 −0.173361
\(785\) 0.180340 0.00643661
\(786\) 24.5623 0.876108
\(787\) −48.5755 −1.73153 −0.865764 0.500452i \(-0.833167\pi\)
−0.865764 + 0.500452i \(0.833167\pi\)
\(788\) −15.7639 −0.561567
\(789\) 0.0557281 0.00198397
\(790\) −9.47214 −0.337003
\(791\) 0.909830 0.0323498
\(792\) 5.00000 0.177667
\(793\) 11.5623 0.410590
\(794\) −35.7082 −1.26724
\(795\) −8.09017 −0.286929
\(796\) −4.27051 −0.151364
\(797\) 44.1803 1.56495 0.782474 0.622683i \(-0.213957\pi\)
0.782474 + 0.622683i \(0.213957\pi\)
\(798\) 5.61803 0.198876
\(799\) −48.5410 −1.71726
\(800\) −15.6180 −0.552181
\(801\) 11.6180 0.410503
\(802\) −53.6869 −1.89575
\(803\) −28.4164 −1.00279
\(804\) 3.14590 0.110947
\(805\) −0.618034 −0.0217828
\(806\) 25.8541 0.910672
\(807\) 32.0902 1.12963
\(808\) 9.27051 0.326135
\(809\) −9.43769 −0.331812 −0.165906 0.986142i \(-0.553055\pi\)
−0.165906 + 0.986142i \(0.553055\pi\)
\(810\) 1.00000 0.0351364
\(811\) 8.52786 0.299454 0.149727 0.988727i \(-0.452161\pi\)
0.149727 + 0.988727i \(0.452161\pi\)
\(812\) 5.09017 0.178630
\(813\) −21.9443 −0.769619
\(814\) 39.7984 1.39493
\(815\) 0.381966 0.0133797
\(816\) 32.5623 1.13991
\(817\) 5.61803 0.196550
\(818\) −10.3820 −0.362997
\(819\) 2.38197 0.0832326
\(820\) 0.562306 0.0196366
\(821\) −29.1246 −1.01646 −0.508228 0.861223i \(-0.669699\pi\)
−0.508228 + 0.861223i \(0.669699\pi\)
\(822\) −26.5623 −0.926467
\(823\) −39.8541 −1.38923 −0.694613 0.719383i \(-0.744425\pi\)
−0.694613 + 0.719383i \(0.744425\pi\)
\(824\) −16.5836 −0.577717
\(825\) 10.3262 0.359513
\(826\) 15.1803 0.528192
\(827\) −28.2148 −0.981124 −0.490562 0.871406i \(-0.663208\pi\)
−0.490562 + 0.871406i \(0.663208\pi\)
\(828\) −0.618034 −0.0214782
\(829\) −30.4164 −1.05641 −0.528203 0.849118i \(-0.677134\pi\)
−0.528203 + 0.849118i \(0.677134\pi\)
\(830\) −9.18034 −0.318654
\(831\) −4.27051 −0.148142
\(832\) −10.0902 −0.349814
\(833\) 6.70820 0.232425
\(834\) −18.4164 −0.637708
\(835\) 4.43769 0.153573
\(836\) −4.79837 −0.165955
\(837\) −6.70820 −0.231869
\(838\) −25.3607 −0.876070
\(839\) 16.7426 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(840\) −1.38197 −0.0476824
\(841\) 38.8328 1.33906
\(842\) 16.6180 0.572695
\(843\) 3.70820 0.127717
\(844\) 6.43769 0.221595
\(845\) 4.52786 0.155763
\(846\) 11.7082 0.402536
\(847\) 6.00000 0.206162
\(848\) 63.5410 2.18201
\(849\) 26.2148 0.899689
\(850\) 50.1246 1.71926
\(851\) 11.0000 0.377075
\(852\) −2.70820 −0.0927815
\(853\) −47.1246 −1.61352 −0.806758 0.590882i \(-0.798780\pi\)
−0.806758 + 0.590882i \(0.798780\pi\)
\(854\) −7.85410 −0.268762
\(855\) 2.14590 0.0733882
\(856\) 7.43769 0.254215
\(857\) 39.7082 1.35641 0.678203 0.734874i \(-0.262759\pi\)
0.678203 + 0.734874i \(0.262759\pi\)
\(858\) −8.61803 −0.294215
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0.618034 0.0210748
\(861\) −1.47214 −0.0501703
\(862\) −1.09017 −0.0371313
\(863\) −8.06888 −0.274668 −0.137334 0.990525i \(-0.543853\pi\)
−0.137334 + 0.990525i \(0.543853\pi\)
\(864\) −3.38197 −0.115057
\(865\) 2.14590 0.0729627
\(866\) −30.9443 −1.05153
\(867\) −28.0000 −0.950930
\(868\) −4.14590 −0.140721
\(869\) −21.1803 −0.718494
\(870\) 8.23607 0.279229
\(871\) 12.1246 0.410827
\(872\) 43.0902 1.45922
\(873\) 10.4164 0.352542
\(874\) −5.61803 −0.190033
\(875\) −5.94427 −0.200953
\(876\) 7.85410 0.265366
\(877\) 16.5836 0.559988 0.279994 0.960002i \(-0.409667\pi\)
0.279994 + 0.960002i \(0.409667\pi\)
\(878\) −38.2705 −1.29157
\(879\) −12.0000 −0.404750
\(880\) 6.70820 0.226134
\(881\) 1.81966 0.0613059 0.0306530 0.999530i \(-0.490241\pi\)
0.0306530 + 0.999530i \(0.490241\pi\)
\(882\) −1.61803 −0.0544820
\(883\) −3.90983 −0.131576 −0.0657881 0.997834i \(-0.520956\pi\)
−0.0657881 + 0.997834i \(0.520956\pi\)
\(884\) −9.87539 −0.332145
\(885\) 5.79837 0.194910
\(886\) 15.4164 0.517924
\(887\) 9.79837 0.328997 0.164499 0.986377i \(-0.447399\pi\)
0.164499 + 0.986377i \(0.447399\pi\)
\(888\) 24.5967 0.825413
\(889\) −22.2705 −0.746929
\(890\) 11.6180 0.389437
\(891\) 2.23607 0.0749111
\(892\) −11.0344 −0.369460
\(893\) 25.1246 0.840763
\(894\) −31.1246 −1.04096
\(895\) −1.14590 −0.0383031
\(896\) 13.6180 0.454947
\(897\) −2.38197 −0.0795315
\(898\) 29.8328 0.995534
\(899\) −55.2492 −1.84266
\(900\) −2.85410 −0.0951367
\(901\) −87.8115 −2.92543
\(902\) 5.32624 0.177344
\(903\) −1.61803 −0.0538448
\(904\) −2.03444 −0.0676645
\(905\) 1.20163 0.0399434
\(906\) −24.6525 −0.819024
\(907\) 48.9787 1.62631 0.813156 0.582046i \(-0.197748\pi\)
0.813156 + 0.582046i \(0.197748\pi\)
\(908\) 15.0344 0.498935
\(909\) 4.14590 0.137511
\(910\) 2.38197 0.0789614
\(911\) −7.59675 −0.251691 −0.125846 0.992050i \(-0.540164\pi\)
−0.125846 + 0.992050i \(0.540164\pi\)
\(912\) −16.8541 −0.558095
\(913\) −20.5279 −0.679373
\(914\) 23.1803 0.766737
\(915\) −3.00000 −0.0991769
\(916\) −10.0902 −0.333389
\(917\) −15.1803 −0.501299
\(918\) 10.8541 0.358239
\(919\) −50.1246 −1.65346 −0.826729 0.562600i \(-0.809801\pi\)
−0.826729 + 0.562600i \(0.809801\pi\)
\(920\) 1.38197 0.0455621
\(921\) −16.1246 −0.531324
\(922\) −41.1803 −1.35620
\(923\) −10.4377 −0.343561
\(924\) 1.38197 0.0454633
\(925\) 50.7984 1.67024
\(926\) −27.9787 −0.919438
\(927\) −7.41641 −0.243587
\(928\) −27.8541 −0.914356
\(929\) 28.9098 0.948501 0.474250 0.880390i \(-0.342719\pi\)
0.474250 + 0.880390i \(0.342719\pi\)
\(930\) −6.70820 −0.219971
\(931\) −3.47214 −0.113795
\(932\) −6.85410 −0.224514
\(933\) −15.3262 −0.501759
\(934\) 24.5623 0.803703
\(935\) −9.27051 −0.303178
\(936\) −5.32624 −0.174094
\(937\) 2.70820 0.0884732 0.0442366 0.999021i \(-0.485914\pi\)
0.0442366 + 0.999021i \(0.485914\pi\)
\(938\) −8.23607 −0.268917
\(939\) 2.47214 0.0806751
\(940\) 2.76393 0.0901495
\(941\) −15.0557 −0.490803 −0.245401 0.969422i \(-0.578920\pi\)
−0.245401 + 0.969422i \(0.578920\pi\)
\(942\) −0.472136 −0.0153830
\(943\) 1.47214 0.0479393
\(944\) −45.5410 −1.48223
\(945\) −0.618034 −0.0201046
\(946\) 5.85410 0.190333
\(947\) −18.9443 −0.615606 −0.307803 0.951450i \(-0.599594\pi\)
−0.307803 + 0.951450i \(0.599594\pi\)
\(948\) 5.85410 0.190132
\(949\) 30.2705 0.982622
\(950\) −25.9443 −0.841743
\(951\) −14.5066 −0.470408
\(952\) −15.0000 −0.486153
\(953\) 12.1591 0.393870 0.196935 0.980417i \(-0.436901\pi\)
0.196935 + 0.980417i \(0.436901\pi\)
\(954\) 21.1803 0.685739
\(955\) −10.0000 −0.323592
\(956\) −2.96556 −0.0959130
\(957\) 18.4164 0.595318
\(958\) −33.9787 −1.09780
\(959\) 16.4164 0.530113
\(960\) 2.61803 0.0844967
\(961\) 14.0000 0.451613
\(962\) −42.3951 −1.36687
\(963\) 3.32624 0.107187
\(964\) −6.79837 −0.218961
\(965\) −5.12461 −0.164967
\(966\) 1.61803 0.0520594
\(967\) 50.3607 1.61949 0.809745 0.586782i \(-0.199605\pi\)
0.809745 + 0.586782i \(0.199605\pi\)
\(968\) −13.4164 −0.431220
\(969\) 23.2918 0.748240
\(970\) 10.4164 0.334451
\(971\) −19.6869 −0.631783 −0.315892 0.948795i \(-0.602304\pi\)
−0.315892 + 0.948795i \(0.602304\pi\)
\(972\) −0.618034 −0.0198234
\(973\) 11.3820 0.364889
\(974\) 47.3951 1.51864
\(975\) −11.0000 −0.352282
\(976\) 23.5623 0.754211
\(977\) 49.7984 1.59319 0.796596 0.604513i \(-0.206632\pi\)
0.796596 + 0.604513i \(0.206632\pi\)
\(978\) −1.00000 −0.0319765
\(979\) 25.9787 0.830283
\(980\) −0.381966 −0.0122015
\(981\) 19.2705 0.615260
\(982\) −15.8541 −0.505925
\(983\) −7.40325 −0.236127 −0.118064 0.993006i \(-0.537669\pi\)
−0.118064 + 0.993006i \(0.537669\pi\)
\(984\) 3.29180 0.104939
\(985\) 15.7639 0.502281
\(986\) 89.3951 2.84692
\(987\) −7.23607 −0.230327
\(988\) 5.11146 0.162617
\(989\) 1.61803 0.0514505
\(990\) 2.23607 0.0710669
\(991\) −35.5755 −1.13009 −0.565046 0.825059i \(-0.691142\pi\)
−0.565046 + 0.825059i \(0.691142\pi\)
\(992\) 22.6869 0.720310
\(993\) 13.4164 0.425757
\(994\) 7.09017 0.224887
\(995\) 4.27051 0.135384
\(996\) 5.67376 0.179780
\(997\) 57.7214 1.82805 0.914027 0.405654i \(-0.132956\pi\)
0.914027 + 0.405654i \(0.132956\pi\)
\(998\) 62.1246 1.96652
\(999\) 11.0000 0.348025
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 483.2.a.d.1.1 2
3.2 odd 2 1449.2.a.h.1.2 2
4.3 odd 2 7728.2.a.bn.1.1 2
7.6 odd 2 3381.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.d.1.1 2 1.1 even 1 trivial
1449.2.a.h.1.2 2 3.2 odd 2
3381.2.a.r.1.1 2 7.6 odd 2
7728.2.a.bn.1.1 2 4.3 odd 2