Properties

Label 6422.2.a.n.1.2
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.2799281781\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 6422.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} +0.753020 q^{5} +0.445042 q^{6} -1.60388 q^{7} -1.00000 q^{8} -2.80194 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} +0.753020 q^{5} +0.445042 q^{6} -1.60388 q^{7} -1.00000 q^{8} -2.80194 q^{9} -0.753020 q^{10} -3.60388 q^{11} -0.445042 q^{12} +1.60388 q^{14} -0.335126 q^{15} +1.00000 q^{16} -1.40581 q^{17} +2.80194 q^{18} -1.00000 q^{19} +0.753020 q^{20} +0.713792 q^{21} +3.60388 q^{22} -8.59179 q^{23} +0.445042 q^{24} -4.43296 q^{25} +2.58211 q^{27} -1.60388 q^{28} -1.38404 q^{29} +0.335126 q^{30} +2.93900 q^{31} -1.00000 q^{32} +1.60388 q^{33} +1.40581 q^{34} -1.20775 q^{35} -2.80194 q^{36} +1.32975 q^{37} +1.00000 q^{38} -0.753020 q^{40} +2.61596 q^{41} -0.713792 q^{42} +1.78017 q^{43} -3.60388 q^{44} -2.10992 q^{45} +8.59179 q^{46} +0.219833 q^{47} -0.445042 q^{48} -4.42758 q^{49} +4.43296 q^{50} +0.625646 q^{51} -5.50604 q^{53} -2.58211 q^{54} -2.71379 q^{55} +1.60388 q^{56} +0.445042 q^{57} +1.38404 q^{58} +1.42327 q^{59} -0.335126 q^{60} -10.4547 q^{61} -2.93900 q^{62} +4.49396 q^{63} +1.00000 q^{64} -1.60388 q^{66} -2.66487 q^{67} -1.40581 q^{68} +3.82371 q^{69} +1.20775 q^{70} -10.2959 q^{71} +2.80194 q^{72} -1.40581 q^{73} -1.32975 q^{74} +1.97285 q^{75} -1.00000 q^{76} +5.78017 q^{77} +11.2838 q^{79} +0.753020 q^{80} +7.25667 q^{81} -2.61596 q^{82} +3.10992 q^{83} +0.713792 q^{84} -1.05861 q^{85} -1.78017 q^{86} +0.615957 q^{87} +3.60388 q^{88} -2.19567 q^{89} +2.10992 q^{90} -8.59179 q^{92} -1.30798 q^{93} -0.219833 q^{94} -0.753020 q^{95} +0.445042 q^{96} +13.0858 q^{97} +4.42758 q^{98} +10.0978 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 7 q^{5} + q^{6} + 4 q^{7} - 3 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 7 q^{5} + q^{6} + 4 q^{7} - 3 q^{8} - 4 q^{9} - 7 q^{10} - 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 9 q^{17} + 4 q^{18} - 3 q^{19} + 7 q^{20} - 6 q^{21} + 2 q^{22} + 2 q^{23} + q^{24} + 6 q^{25} + 2 q^{27} + 4 q^{28} + 6 q^{29} - q^{31} - 3 q^{32} - 4 q^{33} - 9 q^{34} + 14 q^{35} - 4 q^{36} + 6 q^{37} + 3 q^{38} - 7 q^{40} + 18 q^{41} + 6 q^{42} + 4 q^{43} - 2 q^{44} - 7 q^{45} - 2 q^{46} + 2 q^{47} - q^{48} + 3 q^{49} - 6 q^{50} - 10 q^{51} - 26 q^{53} - 2 q^{54} - 4 q^{56} + q^{57} - 6 q^{58} + 7 q^{59} - 9 q^{61} + q^{62} + 4 q^{63} + 3 q^{64} + 4 q^{66} - 9 q^{67} + 9 q^{68} + 4 q^{69} - 14 q^{70} - 17 q^{71} + 4 q^{72} + 9 q^{73} - 6 q^{74} + 12 q^{75} - 3 q^{76} + 16 q^{77} + q^{79} + 7 q^{80} - 5 q^{81} - 18 q^{82} + 10 q^{83} - 6 q^{84} + 28 q^{85} - 4 q^{86} + 12 q^{87} + 2 q^{88} + 30 q^{89} + 7 q^{90} + 2 q^{92} - 9 q^{93} - 2 q^{94} - 7 q^{95} + q^{96} + 2 q^{97} - 3 q^{98} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.445042 −0.256945 −0.128473 0.991713i \(-0.541007\pi\)
−0.128473 + 0.991713i \(0.541007\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.753020 0.336761 0.168380 0.985722i \(-0.446146\pi\)
0.168380 + 0.985722i \(0.446146\pi\)
\(6\) 0.445042 0.181688
\(7\) −1.60388 −0.606208 −0.303104 0.952957i \(-0.598023\pi\)
−0.303104 + 0.952957i \(0.598023\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.80194 −0.933979
\(10\) −0.753020 −0.238126
\(11\) −3.60388 −1.08661 −0.543305 0.839536i \(-0.682827\pi\)
−0.543305 + 0.839536i \(0.682827\pi\)
\(12\) −0.445042 −0.128473
\(13\) 0 0
\(14\) 1.60388 0.428654
\(15\) −0.335126 −0.0865291
\(16\) 1.00000 0.250000
\(17\) −1.40581 −0.340960 −0.170480 0.985361i \(-0.554532\pi\)
−0.170480 + 0.985361i \(0.554532\pi\)
\(18\) 2.80194 0.660423
\(19\) −1.00000 −0.229416
\(20\) 0.753020 0.168380
\(21\) 0.713792 0.155762
\(22\) 3.60388 0.768349
\(23\) −8.59179 −1.79151 −0.895756 0.444545i \(-0.853365\pi\)
−0.895756 + 0.444545i \(0.853365\pi\)
\(24\) 0.445042 0.0908438
\(25\) −4.43296 −0.886592
\(26\) 0 0
\(27\) 2.58211 0.496926
\(28\) −1.60388 −0.303104
\(29\) −1.38404 −0.257010 −0.128505 0.991709i \(-0.541018\pi\)
−0.128505 + 0.991709i \(0.541018\pi\)
\(30\) 0.335126 0.0611853
\(31\) 2.93900 0.527860 0.263930 0.964542i \(-0.414981\pi\)
0.263930 + 0.964542i \(0.414981\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.60388 0.279199
\(34\) 1.40581 0.241095
\(35\) −1.20775 −0.204147
\(36\) −2.80194 −0.466990
\(37\) 1.32975 0.218609 0.109305 0.994008i \(-0.465138\pi\)
0.109305 + 0.994008i \(0.465138\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −0.753020 −0.119063
\(41\) 2.61596 0.408544 0.204272 0.978914i \(-0.434517\pi\)
0.204272 + 0.978914i \(0.434517\pi\)
\(42\) −0.713792 −0.110140
\(43\) 1.78017 0.271473 0.135736 0.990745i \(-0.456660\pi\)
0.135736 + 0.990745i \(0.456660\pi\)
\(44\) −3.60388 −0.543305
\(45\) −2.10992 −0.314528
\(46\) 8.59179 1.26679
\(47\) 0.219833 0.0320659 0.0160329 0.999871i \(-0.494896\pi\)
0.0160329 + 0.999871i \(0.494896\pi\)
\(48\) −0.445042 −0.0642363
\(49\) −4.42758 −0.632512
\(50\) 4.43296 0.626915
\(51\) 0.625646 0.0876079
\(52\) 0 0
\(53\) −5.50604 −0.756313 −0.378156 0.925742i \(-0.623442\pi\)
−0.378156 + 0.925742i \(0.623442\pi\)
\(54\) −2.58211 −0.351380
\(55\) −2.71379 −0.365928
\(56\) 1.60388 0.214327
\(57\) 0.445042 0.0589472
\(58\) 1.38404 0.181734
\(59\) 1.42327 0.185294 0.0926471 0.995699i \(-0.470467\pi\)
0.0926471 + 0.995699i \(0.470467\pi\)
\(60\) −0.335126 −0.0432645
\(61\) −10.4547 −1.33859 −0.669296 0.742996i \(-0.733404\pi\)
−0.669296 + 0.742996i \(0.733404\pi\)
\(62\) −2.93900 −0.373254
\(63\) 4.49396 0.566186
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.60388 −0.197423
\(67\) −2.66487 −0.325566 −0.162783 0.986662i \(-0.552047\pi\)
−0.162783 + 0.986662i \(0.552047\pi\)
\(68\) −1.40581 −0.170480
\(69\) 3.82371 0.460320
\(70\) 1.20775 0.144354
\(71\) −10.2959 −1.22190 −0.610949 0.791670i \(-0.709212\pi\)
−0.610949 + 0.791670i \(0.709212\pi\)
\(72\) 2.80194 0.330212
\(73\) −1.40581 −0.164538 −0.0822690 0.996610i \(-0.526217\pi\)
−0.0822690 + 0.996610i \(0.526217\pi\)
\(74\) −1.32975 −0.154580
\(75\) 1.97285 0.227805
\(76\) −1.00000 −0.114708
\(77\) 5.78017 0.658711
\(78\) 0 0
\(79\) 11.2838 1.26953 0.634764 0.772706i \(-0.281097\pi\)
0.634764 + 0.772706i \(0.281097\pi\)
\(80\) 0.753020 0.0841902
\(81\) 7.25667 0.806296
\(82\) −2.61596 −0.288884
\(83\) 3.10992 0.341358 0.170679 0.985327i \(-0.445404\pi\)
0.170679 + 0.985327i \(0.445404\pi\)
\(84\) 0.713792 0.0778811
\(85\) −1.05861 −0.114822
\(86\) −1.78017 −0.191960
\(87\) 0.615957 0.0660375
\(88\) 3.60388 0.384174
\(89\) −2.19567 −0.232740 −0.116370 0.993206i \(-0.537126\pi\)
−0.116370 + 0.993206i \(0.537126\pi\)
\(90\) 2.10992 0.222405
\(91\) 0 0
\(92\) −8.59179 −0.895756
\(93\) −1.30798 −0.135631
\(94\) −0.219833 −0.0226740
\(95\) −0.753020 −0.0772583
\(96\) 0.445042 0.0454219
\(97\) 13.0858 1.32866 0.664328 0.747441i \(-0.268718\pi\)
0.664328 + 0.747441i \(0.268718\pi\)
\(98\) 4.42758 0.447253
\(99\) 10.0978 1.01487
\(100\) −4.43296 −0.443296
\(101\) −7.16852 −0.713295 −0.356647 0.934239i \(-0.616080\pi\)
−0.356647 + 0.934239i \(0.616080\pi\)
\(102\) −0.625646 −0.0619482
\(103\) 13.5646 1.33656 0.668282 0.743908i \(-0.267030\pi\)
0.668282 + 0.743908i \(0.267030\pi\)
\(104\) 0 0
\(105\) 0.537500 0.0524546
\(106\) 5.50604 0.534794
\(107\) 1.18598 0.114653 0.0573265 0.998355i \(-0.481742\pi\)
0.0573265 + 0.998355i \(0.481742\pi\)
\(108\) 2.58211 0.248463
\(109\) 16.9879 1.62715 0.813574 0.581462i \(-0.197519\pi\)
0.813574 + 0.581462i \(0.197519\pi\)
\(110\) 2.71379 0.258750
\(111\) −0.591794 −0.0561706
\(112\) −1.60388 −0.151552
\(113\) 4.89008 0.460020 0.230010 0.973188i \(-0.426124\pi\)
0.230010 + 0.973188i \(0.426124\pi\)
\(114\) −0.445042 −0.0416820
\(115\) −6.46980 −0.603312
\(116\) −1.38404 −0.128505
\(117\) 0 0
\(118\) −1.42327 −0.131023
\(119\) 2.25475 0.206693
\(120\) 0.335126 0.0305926
\(121\) 1.98792 0.180720
\(122\) 10.4547 0.946527
\(123\) −1.16421 −0.104973
\(124\) 2.93900 0.263930
\(125\) −7.10321 −0.635331
\(126\) −4.49396 −0.400354
\(127\) −13.9511 −1.23796 −0.618979 0.785407i \(-0.712453\pi\)
−0.618979 + 0.785407i \(0.712453\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.792249 −0.0697536
\(130\) 0 0
\(131\) −2.39612 −0.209350 −0.104675 0.994506i \(-0.533380\pi\)
−0.104675 + 0.994506i \(0.533380\pi\)
\(132\) 1.60388 0.139599
\(133\) 1.60388 0.139074
\(134\) 2.66487 0.230210
\(135\) 1.94438 0.167345
\(136\) 1.40581 0.120547
\(137\) −3.81940 −0.326313 −0.163157 0.986600i \(-0.552168\pi\)
−0.163157 + 0.986600i \(0.552168\pi\)
\(138\) −3.82371 −0.325496
\(139\) −20.2935 −1.72127 −0.860636 0.509220i \(-0.829934\pi\)
−0.860636 + 0.509220i \(0.829934\pi\)
\(140\) −1.20775 −0.102074
\(141\) −0.0978347 −0.00823917
\(142\) 10.2959 0.864012
\(143\) 0 0
\(144\) −2.80194 −0.233495
\(145\) −1.04221 −0.0865510
\(146\) 1.40581 0.116346
\(147\) 1.97046 0.162521
\(148\) 1.32975 0.109305
\(149\) 22.7114 1.86059 0.930295 0.366812i \(-0.119551\pi\)
0.930295 + 0.366812i \(0.119551\pi\)
\(150\) −1.97285 −0.161083
\(151\) −0.472189 −0.0384262 −0.0192131 0.999815i \(-0.506116\pi\)
−0.0192131 + 0.999815i \(0.506116\pi\)
\(152\) 1.00000 0.0811107
\(153\) 3.93900 0.318449
\(154\) −5.78017 −0.465779
\(155\) 2.21313 0.177763
\(156\) 0 0
\(157\) 3.74764 0.299095 0.149547 0.988755i \(-0.452218\pi\)
0.149547 + 0.988755i \(0.452218\pi\)
\(158\) −11.2838 −0.897692
\(159\) 2.45042 0.194331
\(160\) −0.753020 −0.0595315
\(161\) 13.7802 1.08603
\(162\) −7.25667 −0.570138
\(163\) −4.59179 −0.359657 −0.179829 0.983698i \(-0.557554\pi\)
−0.179829 + 0.983698i \(0.557554\pi\)
\(164\) 2.61596 0.204272
\(165\) 1.20775 0.0940233
\(166\) −3.10992 −0.241376
\(167\) −12.0248 −0.930503 −0.465252 0.885178i \(-0.654036\pi\)
−0.465252 + 0.885178i \(0.654036\pi\)
\(168\) −0.713792 −0.0550702
\(169\) 0 0
\(170\) 1.05861 0.0811914
\(171\) 2.80194 0.214270
\(172\) 1.78017 0.135736
\(173\) 4.84654 0.368476 0.184238 0.982882i \(-0.441018\pi\)
0.184238 + 0.982882i \(0.441018\pi\)
\(174\) −0.615957 −0.0466956
\(175\) 7.10992 0.537459
\(176\) −3.60388 −0.271652
\(177\) −0.633415 −0.0476104
\(178\) 2.19567 0.164572
\(179\) 0.833397 0.0622910 0.0311455 0.999515i \(-0.490084\pi\)
0.0311455 + 0.999515i \(0.490084\pi\)
\(180\) −2.10992 −0.157264
\(181\) 24.1957 1.79845 0.899225 0.437487i \(-0.144131\pi\)
0.899225 + 0.437487i \(0.144131\pi\)
\(182\) 0 0
\(183\) 4.65279 0.343944
\(184\) 8.59179 0.633395
\(185\) 1.00133 0.0736191
\(186\) 1.30798 0.0959056
\(187\) 5.06638 0.370490
\(188\) 0.219833 0.0160329
\(189\) −4.14138 −0.301241
\(190\) 0.753020 0.0546298
\(191\) 0.811626 0.0587272 0.0293636 0.999569i \(-0.490652\pi\)
0.0293636 + 0.999569i \(0.490652\pi\)
\(192\) −0.445042 −0.0321181
\(193\) 20.9530 1.50823 0.754115 0.656742i \(-0.228066\pi\)
0.754115 + 0.656742i \(0.228066\pi\)
\(194\) −13.0858 −0.939502
\(195\) 0 0
\(196\) −4.42758 −0.316256
\(197\) 14.3153 1.01992 0.509961 0.860198i \(-0.329660\pi\)
0.509961 + 0.860198i \(0.329660\pi\)
\(198\) −10.0978 −0.717622
\(199\) −3.83446 −0.271818 −0.135909 0.990721i \(-0.543395\pi\)
−0.135909 + 0.990721i \(0.543395\pi\)
\(200\) 4.43296 0.313458
\(201\) 1.18598 0.0836526
\(202\) 7.16852 0.504375
\(203\) 2.21983 0.155802
\(204\) 0.625646 0.0438040
\(205\) 1.96987 0.137582
\(206\) −13.5646 −0.945094
\(207\) 24.0737 1.67324
\(208\) 0 0
\(209\) 3.60388 0.249285
\(210\) −0.537500 −0.0370910
\(211\) −13.0804 −0.900490 −0.450245 0.892905i \(-0.648663\pi\)
−0.450245 + 0.892905i \(0.648663\pi\)
\(212\) −5.50604 −0.378156
\(213\) 4.58211 0.313961
\(214\) −1.18598 −0.0810720
\(215\) 1.34050 0.0914215
\(216\) −2.58211 −0.175690
\(217\) −4.71379 −0.319993
\(218\) −16.9879 −1.15057
\(219\) 0.625646 0.0422772
\(220\) −2.71379 −0.182964
\(221\) 0 0
\(222\) 0.591794 0.0397186
\(223\) 26.0084 1.74165 0.870824 0.491594i \(-0.163586\pi\)
0.870824 + 0.491594i \(0.163586\pi\)
\(224\) 1.60388 0.107163
\(225\) 12.4209 0.828059
\(226\) −4.89008 −0.325284
\(227\) 11.5550 0.766930 0.383465 0.923556i \(-0.374731\pi\)
0.383465 + 0.923556i \(0.374731\pi\)
\(228\) 0.445042 0.0294736
\(229\) 4.03923 0.266920 0.133460 0.991054i \(-0.457391\pi\)
0.133460 + 0.991054i \(0.457391\pi\)
\(230\) 6.46980 0.426606
\(231\) −2.57242 −0.169253
\(232\) 1.38404 0.0908669
\(233\) −8.09352 −0.530224 −0.265112 0.964218i \(-0.585409\pi\)
−0.265112 + 0.964218i \(0.585409\pi\)
\(234\) 0 0
\(235\) 0.165538 0.0107985
\(236\) 1.42327 0.0926471
\(237\) −5.02177 −0.326199
\(238\) −2.25475 −0.146154
\(239\) −22.9879 −1.48696 −0.743482 0.668755i \(-0.766827\pi\)
−0.743482 + 0.668755i \(0.766827\pi\)
\(240\) −0.335126 −0.0216323
\(241\) 0.483206 0.0311260 0.0155630 0.999879i \(-0.495046\pi\)
0.0155630 + 0.999879i \(0.495046\pi\)
\(242\) −1.98792 −0.127788
\(243\) −10.9758 −0.704100
\(244\) −10.4547 −0.669296
\(245\) −3.33406 −0.213005
\(246\) 1.16421 0.0742273
\(247\) 0 0
\(248\) −2.93900 −0.186627
\(249\) −1.38404 −0.0877102
\(250\) 7.10321 0.449247
\(251\) 20.2198 1.27626 0.638132 0.769927i \(-0.279707\pi\)
0.638132 + 0.769927i \(0.279707\pi\)
\(252\) 4.49396 0.283093
\(253\) 30.9638 1.94667
\(254\) 13.9511 0.875369
\(255\) 0.471124 0.0295029
\(256\) 1.00000 0.0625000
\(257\) −11.6823 −0.728724 −0.364362 0.931257i \(-0.618713\pi\)
−0.364362 + 0.931257i \(0.618713\pi\)
\(258\) 0.792249 0.0493233
\(259\) −2.13275 −0.132523
\(260\) 0 0
\(261\) 3.87800 0.240042
\(262\) 2.39612 0.148033
\(263\) −11.8345 −0.729744 −0.364872 0.931058i \(-0.618887\pi\)
−0.364872 + 0.931058i \(0.618887\pi\)
\(264\) −1.60388 −0.0987117
\(265\) −4.14616 −0.254697
\(266\) −1.60388 −0.0983399
\(267\) 0.977165 0.0598015
\(268\) −2.66487 −0.162783
\(269\) −10.1414 −0.618331 −0.309165 0.951008i \(-0.600050\pi\)
−0.309165 + 0.951008i \(0.600050\pi\)
\(270\) −1.94438 −0.118331
\(271\) 5.40342 0.328234 0.164117 0.986441i \(-0.447522\pi\)
0.164117 + 0.986441i \(0.447522\pi\)
\(272\) −1.40581 −0.0852399
\(273\) 0 0
\(274\) 3.81940 0.230738
\(275\) 15.9758 0.963379
\(276\) 3.82371 0.230160
\(277\) −0.204767 −0.0123033 −0.00615163 0.999981i \(-0.501958\pi\)
−0.00615163 + 0.999981i \(0.501958\pi\)
\(278\) 20.2935 1.21712
\(279\) −8.23490 −0.493010
\(280\) 1.20775 0.0721769
\(281\) 20.5483 1.22581 0.612903 0.790158i \(-0.290002\pi\)
0.612903 + 0.790158i \(0.290002\pi\)
\(282\) 0.0978347 0.00582597
\(283\) −28.8418 −1.71446 −0.857232 0.514930i \(-0.827818\pi\)
−0.857232 + 0.514930i \(0.827818\pi\)
\(284\) −10.2959 −0.610949
\(285\) 0.335126 0.0198511
\(286\) 0 0
\(287\) −4.19567 −0.247663
\(288\) 2.80194 0.165106
\(289\) −15.0237 −0.883746
\(290\) 1.04221 0.0612008
\(291\) −5.82371 −0.341392
\(292\) −1.40581 −0.0822690
\(293\) 13.4819 0.787620 0.393810 0.919192i \(-0.371157\pi\)
0.393810 + 0.919192i \(0.371157\pi\)
\(294\) −1.97046 −0.114920
\(295\) 1.07175 0.0623998
\(296\) −1.32975 −0.0772901
\(297\) −9.30559 −0.539965
\(298\) −22.7114 −1.31564
\(299\) 0 0
\(300\) 1.97285 0.113903
\(301\) −2.85517 −0.164569
\(302\) 0.472189 0.0271714
\(303\) 3.19029 0.183278
\(304\) −1.00000 −0.0573539
\(305\) −7.87263 −0.450785
\(306\) −3.93900 −0.225178
\(307\) −21.6286 −1.23441 −0.617206 0.786802i \(-0.711735\pi\)
−0.617206 + 0.786802i \(0.711735\pi\)
\(308\) 5.78017 0.329356
\(309\) −6.03684 −0.343424
\(310\) −2.21313 −0.125697
\(311\) 22.2392 1.26107 0.630535 0.776161i \(-0.282836\pi\)
0.630535 + 0.776161i \(0.282836\pi\)
\(312\) 0 0
\(313\) 13.7952 0.779753 0.389876 0.920867i \(-0.372518\pi\)
0.389876 + 0.920867i \(0.372518\pi\)
\(314\) −3.74764 −0.211492
\(315\) 3.38404 0.190669
\(316\) 11.2838 0.634764
\(317\) −20.1172 −1.12990 −0.564948 0.825127i \(-0.691104\pi\)
−0.564948 + 0.825127i \(0.691104\pi\)
\(318\) −2.45042 −0.137413
\(319\) 4.98792 0.279270
\(320\) 0.753020 0.0420951
\(321\) −0.527811 −0.0294595
\(322\) −13.7802 −0.767939
\(323\) 1.40581 0.0782215
\(324\) 7.25667 0.403148
\(325\) 0 0
\(326\) 4.59179 0.254316
\(327\) −7.56033 −0.418087
\(328\) −2.61596 −0.144442
\(329\) −0.352584 −0.0194386
\(330\) −1.20775 −0.0664845
\(331\) −19.2131 −1.05605 −0.528025 0.849229i \(-0.677067\pi\)
−0.528025 + 0.849229i \(0.677067\pi\)
\(332\) 3.10992 0.170679
\(333\) −3.72587 −0.204177
\(334\) 12.0248 0.657965
\(335\) −2.00670 −0.109638
\(336\) 0.713792 0.0389405
\(337\) 3.35988 0.183024 0.0915122 0.995804i \(-0.470830\pi\)
0.0915122 + 0.995804i \(0.470830\pi\)
\(338\) 0 0
\(339\) −2.17629 −0.118200
\(340\) −1.05861 −0.0574110
\(341\) −10.5918 −0.573578
\(342\) −2.80194 −0.151511
\(343\) 18.3284 0.989642
\(344\) −1.78017 −0.0959802
\(345\) 2.87933 0.155018
\(346\) −4.84654 −0.260552
\(347\) −15.2513 −0.818732 −0.409366 0.912370i \(-0.634250\pi\)
−0.409366 + 0.912370i \(0.634250\pi\)
\(348\) 0.615957 0.0330188
\(349\) 8.26875 0.442616 0.221308 0.975204i \(-0.428967\pi\)
0.221308 + 0.975204i \(0.428967\pi\)
\(350\) −7.10992 −0.380041
\(351\) 0 0
\(352\) 3.60388 0.192087
\(353\) 18.0683 0.961678 0.480839 0.876809i \(-0.340332\pi\)
0.480839 + 0.876809i \(0.340332\pi\)
\(354\) 0.633415 0.0336657
\(355\) −7.75302 −0.411488
\(356\) −2.19567 −0.116370
\(357\) −1.00346 −0.0531086
\(358\) −0.833397 −0.0440464
\(359\) −36.8116 −1.94284 −0.971422 0.237360i \(-0.923718\pi\)
−0.971422 + 0.237360i \(0.923718\pi\)
\(360\) 2.10992 0.111202
\(361\) 1.00000 0.0526316
\(362\) −24.1957 −1.27170
\(363\) −0.884707 −0.0464351
\(364\) 0 0
\(365\) −1.05861 −0.0554100
\(366\) −4.65279 −0.243205
\(367\) 11.1099 0.579933 0.289966 0.957037i \(-0.406356\pi\)
0.289966 + 0.957037i \(0.406356\pi\)
\(368\) −8.59179 −0.447878
\(369\) −7.32975 −0.381571
\(370\) −1.00133 −0.0520566
\(371\) 8.83100 0.458483
\(372\) −1.30798 −0.0678155
\(373\) −11.0750 −0.573442 −0.286721 0.958014i \(-0.592565\pi\)
−0.286721 + 0.958014i \(0.592565\pi\)
\(374\) −5.06638 −0.261976
\(375\) 3.16123 0.163245
\(376\) −0.219833 −0.0113370
\(377\) 0 0
\(378\) 4.14138 0.213009
\(379\) 17.6233 0.905246 0.452623 0.891702i \(-0.350488\pi\)
0.452623 + 0.891702i \(0.350488\pi\)
\(380\) −0.753020 −0.0386291
\(381\) 6.20882 0.318087
\(382\) −0.811626 −0.0415264
\(383\) 13.2034 0.674664 0.337332 0.941386i \(-0.390475\pi\)
0.337332 + 0.941386i \(0.390475\pi\)
\(384\) 0.445042 0.0227109
\(385\) 4.35258 0.221828
\(386\) −20.9530 −1.06648
\(387\) −4.98792 −0.253550
\(388\) 13.0858 0.664328
\(389\) −26.9396 −1.36589 −0.682946 0.730469i \(-0.739301\pi\)
−0.682946 + 0.730469i \(0.739301\pi\)
\(390\) 0 0
\(391\) 12.0785 0.610834
\(392\) 4.42758 0.223627
\(393\) 1.06638 0.0537915
\(394\) −14.3153 −0.721193
\(395\) 8.49694 0.427528
\(396\) 10.0978 0.507435
\(397\) −27.5211 −1.38124 −0.690622 0.723216i \(-0.742663\pi\)
−0.690622 + 0.723216i \(0.742663\pi\)
\(398\) 3.83446 0.192204
\(399\) −0.713792 −0.0357343
\(400\) −4.43296 −0.221648
\(401\) −34.8068 −1.73817 −0.869085 0.494662i \(-0.835292\pi\)
−0.869085 + 0.494662i \(0.835292\pi\)
\(402\) −1.18598 −0.0591513
\(403\) 0 0
\(404\) −7.16852 −0.356647
\(405\) 5.46442 0.271529
\(406\) −2.21983 −0.110168
\(407\) −4.79225 −0.237543
\(408\) −0.625646 −0.0309741
\(409\) 27.0858 1.33930 0.669652 0.742675i \(-0.266443\pi\)
0.669652 + 0.742675i \(0.266443\pi\)
\(410\) −1.96987 −0.0972849
\(411\) 1.69979 0.0838445
\(412\) 13.5646 0.668282
\(413\) −2.28275 −0.112327
\(414\) −24.0737 −1.18316
\(415\) 2.34183 0.114956
\(416\) 0 0
\(417\) 9.03146 0.442272
\(418\) −3.60388 −0.176271
\(419\) 15.5690 0.760593 0.380297 0.924865i \(-0.375822\pi\)
0.380297 + 0.924865i \(0.375822\pi\)
\(420\) 0.537500 0.0262273
\(421\) 10.5724 0.515268 0.257634 0.966243i \(-0.417057\pi\)
0.257634 + 0.966243i \(0.417057\pi\)
\(422\) 13.0804 0.636743
\(423\) −0.615957 −0.0299489
\(424\) 5.50604 0.267397
\(425\) 6.23191 0.302292
\(426\) −4.58211 −0.222004
\(427\) 16.7681 0.811465
\(428\) 1.18598 0.0573265
\(429\) 0 0
\(430\) −1.34050 −0.0646448
\(431\) −8.96316 −0.431740 −0.215870 0.976422i \(-0.569259\pi\)
−0.215870 + 0.976422i \(0.569259\pi\)
\(432\) 2.58211 0.124232
\(433\) 28.6219 1.37548 0.687741 0.725956i \(-0.258602\pi\)
0.687741 + 0.725956i \(0.258602\pi\)
\(434\) 4.71379 0.226269
\(435\) 0.463828 0.0222389
\(436\) 16.9879 0.813574
\(437\) 8.59179 0.411001
\(438\) −0.625646 −0.0298945
\(439\) 9.25667 0.441797 0.220898 0.975297i \(-0.429101\pi\)
0.220898 + 0.975297i \(0.429101\pi\)
\(440\) 2.71379 0.129375
\(441\) 12.4058 0.590753
\(442\) 0 0
\(443\) 12.8418 0.610130 0.305065 0.952331i \(-0.401322\pi\)
0.305065 + 0.952331i \(0.401322\pi\)
\(444\) −0.591794 −0.0280853
\(445\) −1.65338 −0.0783779
\(446\) −26.0084 −1.23153
\(447\) −10.1075 −0.478069
\(448\) −1.60388 −0.0757760
\(449\) 15.0858 0.711941 0.355970 0.934497i \(-0.384150\pi\)
0.355970 + 0.934497i \(0.384150\pi\)
\(450\) −12.4209 −0.585526
\(451\) −9.42758 −0.443928
\(452\) 4.89008 0.230010
\(453\) 0.210144 0.00987342
\(454\) −11.5550 −0.542301
\(455\) 0 0
\(456\) −0.445042 −0.0208410
\(457\) 13.0513 0.610515 0.305257 0.952270i \(-0.401258\pi\)
0.305257 + 0.952270i \(0.401258\pi\)
\(458\) −4.03923 −0.188741
\(459\) −3.62996 −0.169432
\(460\) −6.46980 −0.301656
\(461\) 2.75973 0.128533 0.0642666 0.997933i \(-0.479529\pi\)
0.0642666 + 0.997933i \(0.479529\pi\)
\(462\) 2.57242 0.119680
\(463\) −29.3551 −1.36425 −0.682123 0.731237i \(-0.738943\pi\)
−0.682123 + 0.731237i \(0.738943\pi\)
\(464\) −1.38404 −0.0642526
\(465\) −0.984935 −0.0456752
\(466\) 8.09352 0.374925
\(467\) 14.0871 0.651872 0.325936 0.945392i \(-0.394321\pi\)
0.325936 + 0.945392i \(0.394321\pi\)
\(468\) 0 0
\(469\) 4.27413 0.197361
\(470\) −0.165538 −0.00763572
\(471\) −1.66786 −0.0768509
\(472\) −1.42327 −0.0655114
\(473\) −6.41550 −0.294985
\(474\) 5.02177 0.230658
\(475\) 4.43296 0.203398
\(476\) 2.25475 0.103346
\(477\) 15.4276 0.706381
\(478\) 22.9879 1.05144
\(479\) 2.41550 0.110367 0.0551835 0.998476i \(-0.482426\pi\)
0.0551835 + 0.998476i \(0.482426\pi\)
\(480\) 0.335126 0.0152963
\(481\) 0 0
\(482\) −0.483206 −0.0220094
\(483\) −6.13275 −0.279050
\(484\) 1.98792 0.0903599
\(485\) 9.85384 0.447440
\(486\) 10.9758 0.497874
\(487\) −5.24591 −0.237715 −0.118858 0.992911i \(-0.537923\pi\)
−0.118858 + 0.992911i \(0.537923\pi\)
\(488\) 10.4547 0.473263
\(489\) 2.04354 0.0924121
\(490\) 3.33406 0.150618
\(491\) 17.3491 0.782955 0.391478 0.920188i \(-0.371964\pi\)
0.391478 + 0.920188i \(0.371964\pi\)
\(492\) −1.16421 −0.0524867
\(493\) 1.94571 0.0876302
\(494\) 0 0
\(495\) 7.60388 0.341769
\(496\) 2.93900 0.131965
\(497\) 16.5133 0.740724
\(498\) 1.38404 0.0620204
\(499\) −20.7573 −0.929226 −0.464613 0.885514i \(-0.653807\pi\)
−0.464613 + 0.885514i \(0.653807\pi\)
\(500\) −7.10321 −0.317665
\(501\) 5.35152 0.239088
\(502\) −20.2198 −0.902455
\(503\) −2.67887 −0.119445 −0.0597226 0.998215i \(-0.519022\pi\)
−0.0597226 + 0.998215i \(0.519022\pi\)
\(504\) −4.49396 −0.200177
\(505\) −5.39804 −0.240210
\(506\) −30.9638 −1.37651
\(507\) 0 0
\(508\) −13.9511 −0.618979
\(509\) −21.5060 −0.953238 −0.476619 0.879110i \(-0.658138\pi\)
−0.476619 + 0.879110i \(0.658138\pi\)
\(510\) −0.471124 −0.0208617
\(511\) 2.25475 0.0997442
\(512\) −1.00000 −0.0441942
\(513\) −2.58211 −0.114003
\(514\) 11.6823 0.515286
\(515\) 10.2145 0.450103
\(516\) −0.792249 −0.0348768
\(517\) −0.792249 −0.0348431
\(518\) 2.13275 0.0937077
\(519\) −2.15691 −0.0946780
\(520\) 0 0
\(521\) 15.2513 0.668171 0.334086 0.942543i \(-0.391573\pi\)
0.334086 + 0.942543i \(0.391573\pi\)
\(522\) −3.87800 −0.169736
\(523\) −4.98792 −0.218106 −0.109053 0.994036i \(-0.534782\pi\)
−0.109053 + 0.994036i \(0.534782\pi\)
\(524\) −2.39612 −0.104675
\(525\) −3.16421 −0.138097
\(526\) 11.8345 0.516007
\(527\) −4.13169 −0.179979
\(528\) 1.60388 0.0697997
\(529\) 50.8189 2.20952
\(530\) 4.14616 0.180098
\(531\) −3.98792 −0.173061
\(532\) 1.60388 0.0695368
\(533\) 0 0
\(534\) −0.977165 −0.0422861
\(535\) 0.893068 0.0386107
\(536\) 2.66487 0.115105
\(537\) −0.370896 −0.0160054
\(538\) 10.1414 0.437226
\(539\) 15.9565 0.687293
\(540\) 1.94438 0.0836727
\(541\) 6.85192 0.294587 0.147294 0.989093i \(-0.452944\pi\)
0.147294 + 0.989093i \(0.452944\pi\)
\(542\) −5.40342 −0.232097
\(543\) −10.7681 −0.462103
\(544\) 1.40581 0.0602737
\(545\) 12.7922 0.547960
\(546\) 0 0
\(547\) −15.4058 −0.658705 −0.329353 0.944207i \(-0.606830\pi\)
−0.329353 + 0.944207i \(0.606830\pi\)
\(548\) −3.81940 −0.163157
\(549\) 29.2935 1.25022
\(550\) −15.9758 −0.681212
\(551\) 1.38404 0.0589622
\(552\) −3.82371 −0.162748
\(553\) −18.0978 −0.769598
\(554\) 0.204767 0.00869972
\(555\) −0.445633 −0.0189161
\(556\) −20.2935 −0.860636
\(557\) 17.5754 0.744694 0.372347 0.928094i \(-0.378553\pi\)
0.372347 + 0.928094i \(0.378553\pi\)
\(558\) 8.23490 0.348611
\(559\) 0 0
\(560\) −1.20775 −0.0510368
\(561\) −2.25475 −0.0951956
\(562\) −20.5483 −0.866776
\(563\) 37.0804 1.56275 0.781376 0.624061i \(-0.214518\pi\)
0.781376 + 0.624061i \(0.214518\pi\)
\(564\) −0.0978347 −0.00411958
\(565\) 3.68233 0.154917
\(566\) 28.8418 1.21231
\(567\) −11.6388 −0.488783
\(568\) 10.2959 0.432006
\(569\) −6.49396 −0.272241 −0.136120 0.990692i \(-0.543463\pi\)
−0.136120 + 0.990692i \(0.543463\pi\)
\(570\) −0.335126 −0.0140369
\(571\) −13.4491 −0.562827 −0.281413 0.959587i \(-0.590803\pi\)
−0.281413 + 0.959587i \(0.590803\pi\)
\(572\) 0 0
\(573\) −0.361208 −0.0150897
\(574\) 4.19567 0.175124
\(575\) 38.0871 1.58834
\(576\) −2.80194 −0.116747
\(577\) 6.91962 0.288068 0.144034 0.989573i \(-0.453993\pi\)
0.144034 + 0.989573i \(0.453993\pi\)
\(578\) 15.0237 0.624903
\(579\) −9.32496 −0.387532
\(580\) −1.04221 −0.0432755
\(581\) −4.98792 −0.206934
\(582\) 5.82371 0.241400
\(583\) 19.8431 0.821817
\(584\) 1.40581 0.0581730
\(585\) 0 0
\(586\) −13.4819 −0.556931
\(587\) 31.6340 1.30568 0.652838 0.757498i \(-0.273578\pi\)
0.652838 + 0.757498i \(0.273578\pi\)
\(588\) 1.97046 0.0812604
\(589\) −2.93900 −0.121099
\(590\) −1.07175 −0.0441234
\(591\) −6.37090 −0.262064
\(592\) 1.32975 0.0546523
\(593\) 39.3793 1.61711 0.808556 0.588419i \(-0.200249\pi\)
0.808556 + 0.588419i \(0.200249\pi\)
\(594\) 9.30559 0.381813
\(595\) 1.69787 0.0696060
\(596\) 22.7114 0.930295
\(597\) 1.70650 0.0698422
\(598\) 0 0
\(599\) −7.18300 −0.293489 −0.146745 0.989174i \(-0.546880\pi\)
−0.146745 + 0.989174i \(0.546880\pi\)
\(600\) −1.97285 −0.0805414
\(601\) 4.90084 0.199909 0.0999547 0.994992i \(-0.468130\pi\)
0.0999547 + 0.994992i \(0.468130\pi\)
\(602\) 2.85517 0.116368
\(603\) 7.46681 0.304072
\(604\) −0.472189 −0.0192131
\(605\) 1.49694 0.0608594
\(606\) −3.19029 −0.129597
\(607\) 35.2180 1.42946 0.714728 0.699403i \(-0.246551\pi\)
0.714728 + 0.699403i \(0.246551\pi\)
\(608\) 1.00000 0.0405554
\(609\) −0.987918 −0.0400325
\(610\) 7.87263 0.318753
\(611\) 0 0
\(612\) 3.93900 0.159225
\(613\) −10.5381 −0.425629 −0.212815 0.977093i \(-0.568263\pi\)
−0.212815 + 0.977093i \(0.568263\pi\)
\(614\) 21.6286 0.872860
\(615\) −0.876674 −0.0353509
\(616\) −5.78017 −0.232890
\(617\) 30.9855 1.24743 0.623715 0.781652i \(-0.285623\pi\)
0.623715 + 0.781652i \(0.285623\pi\)
\(618\) 6.03684 0.242837
\(619\) 7.95167 0.319605 0.159802 0.987149i \(-0.448914\pi\)
0.159802 + 0.987149i \(0.448914\pi\)
\(620\) 2.21313 0.0888813
\(621\) −22.1849 −0.890250
\(622\) −22.2392 −0.891711
\(623\) 3.52158 0.141089
\(624\) 0 0
\(625\) 16.8159 0.672638
\(626\) −13.7952 −0.551368
\(627\) −1.60388 −0.0640526
\(628\) 3.74764 0.149547
\(629\) −1.86938 −0.0745370
\(630\) −3.38404 −0.134823
\(631\) 27.8345 1.10807 0.554036 0.832493i \(-0.313087\pi\)
0.554036 + 0.832493i \(0.313087\pi\)
\(632\) −11.2838 −0.448846
\(633\) 5.82132 0.231377
\(634\) 20.1172 0.798957
\(635\) −10.5054 −0.416896
\(636\) 2.45042 0.0971654
\(637\) 0 0
\(638\) −4.98792 −0.197474
\(639\) 28.8485 1.14123
\(640\) −0.753020 −0.0297657
\(641\) 7.93230 0.313307 0.156653 0.987654i \(-0.449929\pi\)
0.156653 + 0.987654i \(0.449929\pi\)
\(642\) 0.527811 0.0208310
\(643\) 17.6146 0.694653 0.347327 0.937744i \(-0.387090\pi\)
0.347327 + 0.937744i \(0.387090\pi\)
\(644\) 13.7802 0.543015
\(645\) −0.596580 −0.0234903
\(646\) −1.40581 −0.0553110
\(647\) 26.6413 1.04738 0.523689 0.851910i \(-0.324555\pi\)
0.523689 + 0.851910i \(0.324555\pi\)
\(648\) −7.25667 −0.285069
\(649\) −5.12929 −0.201342
\(650\) 0 0
\(651\) 2.09783 0.0822206
\(652\) −4.59179 −0.179829
\(653\) −6.55927 −0.256684 −0.128342 0.991730i \(-0.540966\pi\)
−0.128342 + 0.991730i \(0.540966\pi\)
\(654\) 7.56033 0.295632
\(655\) −1.80433 −0.0705010
\(656\) 2.61596 0.102136
\(657\) 3.93900 0.153675
\(658\) 0.352584 0.0137452
\(659\) −39.9172 −1.55495 −0.777477 0.628911i \(-0.783501\pi\)
−0.777477 + 0.628911i \(0.783501\pi\)
\(660\) 1.20775 0.0470116
\(661\) 29.6582 1.15357 0.576785 0.816896i \(-0.304307\pi\)
0.576785 + 0.816896i \(0.304307\pi\)
\(662\) 19.2131 0.746739
\(663\) 0 0
\(664\) −3.10992 −0.120688
\(665\) 1.20775 0.0468346
\(666\) 3.72587 0.144375
\(667\) 11.8914 0.460437
\(668\) −12.0248 −0.465252
\(669\) −11.5748 −0.447508
\(670\) 2.00670 0.0775258
\(671\) 37.6775 1.45453
\(672\) −0.713792 −0.0275351
\(673\) 11.8974 0.458610 0.229305 0.973355i \(-0.426355\pi\)
0.229305 + 0.973355i \(0.426355\pi\)
\(674\) −3.35988 −0.129418
\(675\) −11.4464 −0.440571
\(676\) 0 0
\(677\) 25.3840 0.975588 0.487794 0.872959i \(-0.337802\pi\)
0.487794 + 0.872959i \(0.337802\pi\)
\(678\) 2.17629 0.0835800
\(679\) −20.9879 −0.805442
\(680\) 1.05861 0.0405957
\(681\) −5.14244 −0.197059
\(682\) 10.5918 0.405581
\(683\) −28.3655 −1.08538 −0.542688 0.839934i \(-0.682593\pi\)
−0.542688 + 0.839934i \(0.682593\pi\)
\(684\) 2.80194 0.107135
\(685\) −2.87608 −0.109889
\(686\) −18.3284 −0.699782
\(687\) −1.79763 −0.0685837
\(688\) 1.78017 0.0678682
\(689\) 0 0
\(690\) −2.87933 −0.109614
\(691\) 12.3854 0.471162 0.235581 0.971855i \(-0.424301\pi\)
0.235581 + 0.971855i \(0.424301\pi\)
\(692\) 4.84654 0.184238
\(693\) −16.1957 −0.615223
\(694\) 15.2513 0.578931
\(695\) −15.2814 −0.579657
\(696\) −0.615957 −0.0233478
\(697\) −3.67755 −0.139297
\(698\) −8.26875 −0.312977
\(699\) 3.60196 0.136239
\(700\) 7.10992 0.268730
\(701\) −8.17092 −0.308611 −0.154306 0.988023i \(-0.549314\pi\)
−0.154306 + 0.988023i \(0.549314\pi\)
\(702\) 0 0
\(703\) −1.32975 −0.0501524
\(704\) −3.60388 −0.135826
\(705\) −0.0736715 −0.00277463
\(706\) −18.0683 −0.680009
\(707\) 11.4974 0.432405
\(708\) −0.633415 −0.0238052
\(709\) −34.6679 −1.30198 −0.650989 0.759087i \(-0.725646\pi\)
−0.650989 + 0.759087i \(0.725646\pi\)
\(710\) 7.75302 0.290966
\(711\) −31.6165 −1.18571
\(712\) 2.19567 0.0822862
\(713\) −25.2513 −0.945668
\(714\) 1.00346 0.0375535
\(715\) 0 0
\(716\) 0.833397 0.0311455
\(717\) 10.2306 0.382068
\(718\) 36.8116 1.37380
\(719\) −1.14483 −0.0426951 −0.0213475 0.999772i \(-0.506796\pi\)
−0.0213475 + 0.999772i \(0.506796\pi\)
\(720\) −2.10992 −0.0786319
\(721\) −21.7560 −0.810236
\(722\) −1.00000 −0.0372161
\(723\) −0.215047 −0.00799767
\(724\) 24.1957 0.899225
\(725\) 6.13541 0.227863
\(726\) 0.884707 0.0328346
\(727\) 3.98062 0.147633 0.0738166 0.997272i \(-0.476482\pi\)
0.0738166 + 0.997272i \(0.476482\pi\)
\(728\) 0 0
\(729\) −16.8853 −0.625381
\(730\) 1.05861 0.0391808
\(731\) −2.50258 −0.0925614
\(732\) 4.65279 0.171972
\(733\) −28.6872 −1.05959 −0.529794 0.848127i \(-0.677731\pi\)
−0.529794 + 0.848127i \(0.677731\pi\)
\(734\) −11.1099 −0.410074
\(735\) 1.48380 0.0547307
\(736\) 8.59179 0.316698
\(737\) 9.60388 0.353763
\(738\) 7.32975 0.269812
\(739\) −43.3986 −1.59644 −0.798222 0.602363i \(-0.794226\pi\)
−0.798222 + 0.602363i \(0.794226\pi\)
\(740\) 1.00133 0.0368095
\(741\) 0 0
\(742\) −8.83100 −0.324196
\(743\) 5.14808 0.188865 0.0944324 0.995531i \(-0.469896\pi\)
0.0944324 + 0.995531i \(0.469896\pi\)
\(744\) 1.30798 0.0479528
\(745\) 17.1021 0.626574
\(746\) 11.0750 0.405485
\(747\) −8.71379 −0.318821
\(748\) 5.06638 0.185245
\(749\) −1.90217 −0.0695036
\(750\) −3.16123 −0.115432
\(751\) 36.9396 1.34794 0.673972 0.738756i \(-0.264587\pi\)
0.673972 + 0.738756i \(0.264587\pi\)
\(752\) 0.219833 0.00801647
\(753\) −8.99867 −0.327930
\(754\) 0 0
\(755\) −0.355568 −0.0129404
\(756\) −4.14138 −0.150620
\(757\) 35.5599 1.29245 0.646223 0.763149i \(-0.276348\pi\)
0.646223 + 0.763149i \(0.276348\pi\)
\(758\) −17.6233 −0.640105
\(759\) −13.7802 −0.500188
\(760\) 0.753020 0.0273149
\(761\) −28.2825 −1.02524 −0.512620 0.858616i \(-0.671325\pi\)
−0.512620 + 0.858616i \(0.671325\pi\)
\(762\) −6.20882 −0.224922
\(763\) −27.2465 −0.986390
\(764\) 0.811626 0.0293636
\(765\) 2.96615 0.107241
\(766\) −13.2034 −0.477060
\(767\) 0 0
\(768\) −0.445042 −0.0160591
\(769\) 36.1021 1.30188 0.650938 0.759131i \(-0.274376\pi\)
0.650938 + 0.759131i \(0.274376\pi\)
\(770\) −4.35258 −0.156856
\(771\) 5.19913 0.187242
\(772\) 20.9530 0.754115
\(773\) 2.05429 0.0738878 0.0369439 0.999317i \(-0.488238\pi\)
0.0369439 + 0.999317i \(0.488238\pi\)
\(774\) 4.98792 0.179287
\(775\) −13.0285 −0.467997
\(776\) −13.0858 −0.469751
\(777\) 0.949164 0.0340511
\(778\) 26.9396 0.965831
\(779\) −2.61596 −0.0937264
\(780\) 0 0
\(781\) 37.1051 1.32773
\(782\) −12.0785 −0.431925
\(783\) −3.57374 −0.127715
\(784\) −4.42758 −0.158128
\(785\) 2.82205 0.100723
\(786\) −1.06638 −0.0380364
\(787\) 48.8353 1.74079 0.870396 0.492353i \(-0.163863\pi\)
0.870396 + 0.492353i \(0.163863\pi\)
\(788\) 14.3153 0.509961
\(789\) 5.26683 0.187504
\(790\) −8.49694 −0.302308
\(791\) −7.84309 −0.278868
\(792\) −10.0978 −0.358811
\(793\) 0 0
\(794\) 27.5211 0.976688
\(795\) 1.84522 0.0654430
\(796\) −3.83446 −0.135909
\(797\) −34.4564 −1.22051 −0.610254 0.792206i \(-0.708933\pi\)
−0.610254 + 0.792206i \(0.708933\pi\)
\(798\) 0.713792 0.0252680
\(799\) −0.309043 −0.0109332
\(800\) 4.43296 0.156729
\(801\) 6.15213 0.217375
\(802\) 34.8068 1.22907
\(803\) 5.06638 0.178789
\(804\) 1.18598 0.0418263
\(805\) 10.3767 0.365732
\(806\) 0 0
\(807\) 4.51334 0.158877
\(808\) 7.16852 0.252188
\(809\) −3.80971 −0.133942 −0.0669711 0.997755i \(-0.521334\pi\)
−0.0669711 + 0.997755i \(0.521334\pi\)
\(810\) −5.46442 −0.192000
\(811\) −4.38703 −0.154049 −0.0770247 0.997029i \(-0.524542\pi\)
−0.0770247 + 0.997029i \(0.524542\pi\)
\(812\) 2.21983 0.0779009
\(813\) −2.40475 −0.0843382
\(814\) 4.79225 0.167968
\(815\) −3.45771 −0.121118
\(816\) 0.625646 0.0219020
\(817\) −1.78017 −0.0622802
\(818\) −27.0858 −0.947031
\(819\) 0 0
\(820\) 1.96987 0.0687908
\(821\) −9.30127 −0.324617 −0.162308 0.986740i \(-0.551894\pi\)
−0.162308 + 0.986740i \(0.551894\pi\)
\(822\) −1.69979 −0.0592870
\(823\) −34.1849 −1.19161 −0.595806 0.803129i \(-0.703167\pi\)
−0.595806 + 0.803129i \(0.703167\pi\)
\(824\) −13.5646 −0.472547
\(825\) −7.10992 −0.247536
\(826\) 2.28275 0.0794270
\(827\) 14.0170 0.487418 0.243709 0.969848i \(-0.421636\pi\)
0.243709 + 0.969848i \(0.421636\pi\)
\(828\) 24.0737 0.836618
\(829\) 10.1870 0.353811 0.176905 0.984228i \(-0.443391\pi\)
0.176905 + 0.984228i \(0.443391\pi\)
\(830\) −2.34183 −0.0812861
\(831\) 0.0911299 0.00316126
\(832\) 0 0
\(833\) 6.22436 0.215661
\(834\) −9.03146 −0.312734
\(835\) −9.05489 −0.313357
\(836\) 3.60388 0.124643
\(837\) 7.58881 0.262308
\(838\) −15.5690 −0.537821
\(839\) −32.1758 −1.11083 −0.555416 0.831572i \(-0.687441\pi\)
−0.555416 + 0.831572i \(0.687441\pi\)
\(840\) −0.537500 −0.0185455
\(841\) −27.0844 −0.933946
\(842\) −10.5724 −0.364350
\(843\) −9.14483 −0.314965
\(844\) −13.0804 −0.450245
\(845\) 0 0
\(846\) 0.615957 0.0211770
\(847\) −3.18837 −0.109554
\(848\) −5.50604 −0.189078
\(849\) 12.8358 0.440523
\(850\) −6.23191 −0.213753
\(851\) −11.4249 −0.391641
\(852\) 4.58211 0.156980
\(853\) 23.0331 0.788639 0.394319 0.918973i \(-0.370980\pi\)
0.394319 + 0.918973i \(0.370980\pi\)
\(854\) −16.7681 −0.573792
\(855\) 2.10992 0.0721576
\(856\) −1.18598 −0.0405360
\(857\) 13.9457 0.476376 0.238188 0.971219i \(-0.423447\pi\)
0.238188 + 0.971219i \(0.423447\pi\)
\(858\) 0 0
\(859\) 55.4094 1.89054 0.945272 0.326283i \(-0.105796\pi\)
0.945272 + 0.326283i \(0.105796\pi\)
\(860\) 1.34050 0.0457108
\(861\) 1.86725 0.0636357
\(862\) 8.96316 0.305287
\(863\) −24.8793 −0.846902 −0.423451 0.905919i \(-0.639181\pi\)
−0.423451 + 0.905919i \(0.639181\pi\)
\(864\) −2.58211 −0.0878450
\(865\) 3.64955 0.124088
\(866\) −28.6219 −0.972613
\(867\) 6.68617 0.227074
\(868\) −4.71379 −0.159997
\(869\) −40.6655 −1.37948
\(870\) −0.463828 −0.0157252
\(871\) 0 0
\(872\) −16.9879 −0.575284
\(873\) −36.6655 −1.24094
\(874\) −8.59179 −0.290622
\(875\) 11.3927 0.385142
\(876\) 0.625646 0.0211386
\(877\) 8.12200 0.274260 0.137130 0.990553i \(-0.456212\pi\)
0.137130 + 0.990553i \(0.456212\pi\)
\(878\) −9.25667 −0.312397
\(879\) −6.00000 −0.202375
\(880\) −2.71379 −0.0914819
\(881\) 53.2277 1.79329 0.896644 0.442753i \(-0.145998\pi\)
0.896644 + 0.442753i \(0.145998\pi\)
\(882\) −12.4058 −0.417725
\(883\) −27.2862 −0.918254 −0.459127 0.888371i \(-0.651838\pi\)
−0.459127 + 0.888371i \(0.651838\pi\)
\(884\) 0 0
\(885\) −0.476975 −0.0160333
\(886\) −12.8418 −0.431427
\(887\) −40.2204 −1.35047 −0.675235 0.737603i \(-0.735958\pi\)
−0.675235 + 0.737603i \(0.735958\pi\)
\(888\) 0.591794 0.0198593
\(889\) 22.3758 0.750460
\(890\) 1.65338 0.0554216
\(891\) −26.1521 −0.876129
\(892\) 26.0084 0.870824
\(893\) −0.219833 −0.00735642
\(894\) 10.1075 0.338046
\(895\) 0.627565 0.0209772
\(896\) 1.60388 0.0535817
\(897\) 0 0
\(898\) −15.0858 −0.503418
\(899\) −4.06770 −0.135666
\(900\) 12.4209 0.414029
\(901\) 7.74046 0.257872
\(902\) 9.42758 0.313904
\(903\) 1.27067 0.0422852
\(904\) −4.89008 −0.162642
\(905\) 18.2198 0.605648
\(906\) −0.210144 −0.00698156
\(907\) 43.0508 1.42948 0.714740 0.699390i \(-0.246545\pi\)
0.714740 + 0.699390i \(0.246545\pi\)
\(908\) 11.5550 0.383465
\(909\) 20.0858 0.666202
\(910\) 0 0
\(911\) −25.1250 −0.832428 −0.416214 0.909267i \(-0.636643\pi\)
−0.416214 + 0.909267i \(0.636643\pi\)
\(912\) 0.445042 0.0147368
\(913\) −11.2078 −0.370922
\(914\) −13.0513 −0.431699
\(915\) 3.50365 0.115827
\(916\) 4.03923 0.133460
\(917\) 3.84309 0.126910
\(918\) 3.62996 0.119806
\(919\) 12.1220 0.399868 0.199934 0.979809i \(-0.435927\pi\)
0.199934 + 0.979809i \(0.435927\pi\)
\(920\) 6.46980 0.213303
\(921\) 9.62565 0.317176
\(922\) −2.75973 −0.0908867
\(923\) 0 0
\(924\) −2.57242 −0.0846263
\(925\) −5.89472 −0.193817
\(926\) 29.3551 0.964668
\(927\) −38.0073 −1.24832
\(928\) 1.38404 0.0454334
\(929\) −29.9597 −0.982946 −0.491473 0.870893i \(-0.663541\pi\)
−0.491473 + 0.870893i \(0.663541\pi\)
\(930\) 0.984935 0.0322973
\(931\) 4.42758 0.145108
\(932\) −8.09352 −0.265112
\(933\) −9.89738 −0.324026
\(934\) −14.0871 −0.460943
\(935\) 3.81508 0.124767
\(936\) 0 0
\(937\) 38.9691 1.27307 0.636533 0.771249i \(-0.280368\pi\)
0.636533 + 0.771249i \(0.280368\pi\)
\(938\) −4.27413 −0.139555
\(939\) −6.13946 −0.200354
\(940\) 0.165538 0.00539927
\(941\) −47.4905 −1.54815 −0.774073 0.633096i \(-0.781784\pi\)
−0.774073 + 0.633096i \(0.781784\pi\)
\(942\) 1.66786 0.0543418
\(943\) −22.4758 −0.731912
\(944\) 1.42327 0.0463235
\(945\) −3.11854 −0.101446
\(946\) 6.41550 0.208586
\(947\) 30.8659 1.00301 0.501504 0.865155i \(-0.332780\pi\)
0.501504 + 0.865155i \(0.332780\pi\)
\(948\) −5.02177 −0.163100
\(949\) 0 0
\(950\) −4.43296 −0.143824
\(951\) 8.95300 0.290321
\(952\) −2.25475 −0.0730768
\(953\) 5.61729 0.181962 0.0909809 0.995853i \(-0.471000\pi\)
0.0909809 + 0.995853i \(0.471000\pi\)
\(954\) −15.4276 −0.499486
\(955\) 0.611171 0.0197770
\(956\) −22.9879 −0.743482
\(957\) −2.21983 −0.0717570
\(958\) −2.41550 −0.0780413
\(959\) 6.12584 0.197814
\(960\) −0.335126 −0.0108161
\(961\) −22.3623 −0.721364
\(962\) 0 0
\(963\) −3.32304 −0.107084
\(964\) 0.483206 0.0155630
\(965\) 15.7780 0.507913
\(966\) 6.13275 0.197318
\(967\) −7.40821 −0.238232 −0.119116 0.992880i \(-0.538006\pi\)
−0.119116 + 0.992880i \(0.538006\pi\)
\(968\) −1.98792 −0.0638941
\(969\) −0.625646 −0.0200986
\(970\) −9.85384 −0.316388
\(971\) 4.55363 0.146133 0.0730665 0.997327i \(-0.476721\pi\)
0.0730665 + 0.997327i \(0.476721\pi\)
\(972\) −10.9758 −0.352050
\(973\) 32.5483 1.04345
\(974\) 5.24591 0.168090
\(975\) 0 0
\(976\) −10.4547 −0.334648
\(977\) −58.9724 −1.88669 −0.943347 0.331808i \(-0.892341\pi\)
−0.943347 + 0.331808i \(0.892341\pi\)
\(978\) −2.04354 −0.0653452
\(979\) 7.91292 0.252898
\(980\) −3.33406 −0.106503
\(981\) −47.5991 −1.51972
\(982\) −17.3491 −0.553633
\(983\) 46.4432 1.48131 0.740655 0.671886i \(-0.234515\pi\)
0.740655 + 0.671886i \(0.234515\pi\)
\(984\) 1.16421 0.0371137
\(985\) 10.7797 0.343470
\(986\) −1.94571 −0.0619639
\(987\) 0.156915 0.00499465
\(988\) 0 0
\(989\) −15.2948 −0.486347
\(990\) −7.60388 −0.241667
\(991\) −46.0828 −1.46387 −0.731934 0.681376i \(-0.761382\pi\)
−0.731934 + 0.681376i \(0.761382\pi\)
\(992\) −2.93900 −0.0933134
\(993\) 8.55065 0.271347
\(994\) −16.5133 −0.523771
\(995\) −2.88743 −0.0915376
\(996\) −1.38404 −0.0438551
\(997\) −3.04759 −0.0965181 −0.0482591 0.998835i \(-0.515367\pi\)
−0.0482591 + 0.998835i \(0.515367\pi\)
\(998\) 20.7573 0.657062
\(999\) 3.43355 0.108633
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 6422.2.a.n.1.2 3
13.12 even 2 6422.2.a.v.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6422.2.a.n.1.2 3 1.1 even 1 trivial
6422.2.a.v.1.2 yes 3 13.12 even 2