Properties

Label 4598.2.a.cb.1.3
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 16 x^{6} - 4 x^{5} + 75 x^{4} + 32 x^{3} - 90 x^{2} - 28 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.55131\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.103249 q^{3} +1.00000 q^{4} -3.91798 q^{5} -0.103249 q^{6} -4.45623 q^{7} +1.00000 q^{8} -2.98934 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.103249 q^{3} +1.00000 q^{4} -3.91798 q^{5} -0.103249 q^{6} -4.45623 q^{7} +1.00000 q^{8} -2.98934 q^{9} -3.91798 q^{10} -0.103249 q^{12} -4.23103 q^{13} -4.45623 q^{14} +0.404529 q^{15} +1.00000 q^{16} -6.60160 q^{17} -2.98934 q^{18} -1.00000 q^{19} -3.91798 q^{20} +0.460102 q^{21} -1.59524 q^{23} -0.103249 q^{24} +10.3506 q^{25} -4.23103 q^{26} +0.618395 q^{27} -4.45623 q^{28} -6.26090 q^{29} +0.404529 q^{30} -6.39604 q^{31} +1.00000 q^{32} -6.60160 q^{34} +17.4594 q^{35} -2.98934 q^{36} +8.97366 q^{37} -1.00000 q^{38} +0.436851 q^{39} -3.91798 q^{40} +1.91418 q^{41} +0.460102 q^{42} +9.24310 q^{43} +11.7122 q^{45} -1.59524 q^{46} -4.59409 q^{47} -0.103249 q^{48} +12.8580 q^{49} +10.3506 q^{50} +0.681611 q^{51} -4.23103 q^{52} -4.30161 q^{53} +0.618395 q^{54} -4.45623 q^{56} +0.103249 q^{57} -6.26090 q^{58} -10.9528 q^{59} +0.404529 q^{60} -0.325652 q^{61} -6.39604 q^{62} +13.3212 q^{63} +1.00000 q^{64} +16.5771 q^{65} +8.88645 q^{67} -6.60160 q^{68} +0.164708 q^{69} +17.4594 q^{70} -2.66009 q^{71} -2.98934 q^{72} -6.64661 q^{73} +8.97366 q^{74} -1.06869 q^{75} -1.00000 q^{76} +0.436851 q^{78} -2.13991 q^{79} -3.91798 q^{80} +8.90417 q^{81} +1.91418 q^{82} +12.4605 q^{83} +0.460102 q^{84} +25.8650 q^{85} +9.24310 q^{86} +0.646434 q^{87} +8.88476 q^{89} +11.7122 q^{90} +18.8544 q^{91} -1.59524 q^{92} +0.660387 q^{93} -4.59409 q^{94} +3.91798 q^{95} -0.103249 q^{96} -11.0716 q^{97} +12.8580 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 8 q^{6} + 4 q^{7} + 8 q^{8} + 22 q^{9} + O(q^{10}) \) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 8 q^{6} + 4 q^{7} + 8 q^{8} + 22 q^{9} + 8 q^{12} - 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} - 4 q^{17} + 22 q^{18} - 8 q^{19} - 20 q^{21} + 14 q^{23} + 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} + 4 q^{28} - 2 q^{29} + 4 q^{30} + 8 q^{32} - 4 q^{34} + 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} + 16 q^{39} + 8 q^{41} - 20 q^{42} + 8 q^{43} + 16 q^{45} + 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} + 36 q^{50} + 18 q^{51} - 12 q^{52} + 36 q^{53} + 32 q^{54} + 4 q^{56} - 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} + 12 q^{61} + 24 q^{63} + 8 q^{64} + 16 q^{65} + 16 q^{67} - 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} + 22 q^{72} - 20 q^{73} + 24 q^{74} + 40 q^{75} - 8 q^{76} + 16 q^{78} - 12 q^{79} + 40 q^{81} + 8 q^{82} + 20 q^{83} - 20 q^{84} + 12 q^{85} + 8 q^{86} - 36 q^{87} + 8 q^{89} + 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} - 16 q^{94} + 8 q^{96} + 4 q^{97} + 34 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −0.103249 −0.0596110 −0.0298055 0.999556i \(-0.509489\pi\)
−0.0298055 + 0.999556i \(0.509489\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.91798 −1.75217 −0.876087 0.482153i \(-0.839855\pi\)
−0.876087 + 0.482153i \(0.839855\pi\)
\(6\) −0.103249 −0.0421513
\(7\) −4.45623 −1.68430 −0.842148 0.539247i \(-0.818709\pi\)
−0.842148 + 0.539247i \(0.818709\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.98934 −0.996447
\(10\) −3.91798 −1.23897
\(11\) 0 0
\(12\) −0.103249 −0.0298055
\(13\) −4.23103 −1.17348 −0.586739 0.809776i \(-0.699588\pi\)
−0.586739 + 0.809776i \(0.699588\pi\)
\(14\) −4.45623 −1.19098
\(15\) 0.404529 0.104449
\(16\) 1.00000 0.250000
\(17\) −6.60160 −1.60112 −0.800562 0.599250i \(-0.795466\pi\)
−0.800562 + 0.599250i \(0.795466\pi\)
\(18\) −2.98934 −0.704594
\(19\) −1.00000 −0.229416
\(20\) −3.91798 −0.876087
\(21\) 0.460102 0.100403
\(22\) 0 0
\(23\) −1.59524 −0.332631 −0.166315 0.986073i \(-0.553187\pi\)
−0.166315 + 0.986073i \(0.553187\pi\)
\(24\) −0.103249 −0.0210757
\(25\) 10.3506 2.07011
\(26\) −4.23103 −0.829774
\(27\) 0.618395 0.119010
\(28\) −4.45623 −0.842148
\(29\) −6.26090 −1.16262 −0.581310 0.813682i \(-0.697460\pi\)
−0.581310 + 0.813682i \(0.697460\pi\)
\(30\) 0.404529 0.0738565
\(31\) −6.39604 −1.14876 −0.574382 0.818588i \(-0.694757\pi\)
−0.574382 + 0.818588i \(0.694757\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.60160 −1.13217
\(35\) 17.4594 2.95118
\(36\) −2.98934 −0.498223
\(37\) 8.97366 1.47526 0.737630 0.675205i \(-0.235945\pi\)
0.737630 + 0.675205i \(0.235945\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0.436851 0.0699521
\(40\) −3.91798 −0.619487
\(41\) 1.91418 0.298945 0.149472 0.988766i \(-0.452242\pi\)
0.149472 + 0.988766i \(0.452242\pi\)
\(42\) 0.460102 0.0709953
\(43\) 9.24310 1.40956 0.704780 0.709426i \(-0.251046\pi\)
0.704780 + 0.709426i \(0.251046\pi\)
\(44\) 0 0
\(45\) 11.7122 1.74595
\(46\) −1.59524 −0.235206
\(47\) −4.59409 −0.670117 −0.335058 0.942197i \(-0.608756\pi\)
−0.335058 + 0.942197i \(0.608756\pi\)
\(48\) −0.103249 −0.0149027
\(49\) 12.8580 1.83685
\(50\) 10.3506 1.46379
\(51\) 0.681611 0.0954446
\(52\) −4.23103 −0.586739
\(53\) −4.30161 −0.590872 −0.295436 0.955363i \(-0.595465\pi\)
−0.295436 + 0.955363i \(0.595465\pi\)
\(54\) 0.618395 0.0841529
\(55\) 0 0
\(56\) −4.45623 −0.595489
\(57\) 0.103249 0.0136757
\(58\) −6.26090 −0.822097
\(59\) −10.9528 −1.42593 −0.712967 0.701198i \(-0.752649\pi\)
−0.712967 + 0.701198i \(0.752649\pi\)
\(60\) 0.404529 0.0522244
\(61\) −0.325652 −0.0416955 −0.0208477 0.999783i \(-0.506637\pi\)
−0.0208477 + 0.999783i \(0.506637\pi\)
\(62\) −6.39604 −0.812298
\(63\) 13.3212 1.67831
\(64\) 1.00000 0.125000
\(65\) 16.5771 2.05614
\(66\) 0 0
\(67\) 8.88645 1.08565 0.542826 0.839845i \(-0.317354\pi\)
0.542826 + 0.839845i \(0.317354\pi\)
\(68\) −6.60160 −0.800562
\(69\) 0.164708 0.0198285
\(70\) 17.4594 2.08680
\(71\) −2.66009 −0.315695 −0.157848 0.987463i \(-0.550455\pi\)
−0.157848 + 0.987463i \(0.550455\pi\)
\(72\) −2.98934 −0.352297
\(73\) −6.64661 −0.777927 −0.388964 0.921253i \(-0.627167\pi\)
−0.388964 + 0.921253i \(0.627167\pi\)
\(74\) 8.97366 1.04317
\(75\) −1.06869 −0.123402
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 0.436851 0.0494636
\(79\) −2.13991 −0.240758 −0.120379 0.992728i \(-0.538411\pi\)
−0.120379 + 0.992728i \(0.538411\pi\)
\(80\) −3.91798 −0.438044
\(81\) 8.90417 0.989352
\(82\) 1.91418 0.211386
\(83\) 12.4605 1.36772 0.683858 0.729615i \(-0.260301\pi\)
0.683858 + 0.729615i \(0.260301\pi\)
\(84\) 0.460102 0.0502013
\(85\) 25.8650 2.80545
\(86\) 9.24310 0.996709
\(87\) 0.646434 0.0693050
\(88\) 0 0
\(89\) 8.88476 0.941783 0.470891 0.882191i \(-0.343932\pi\)
0.470891 + 0.882191i \(0.343932\pi\)
\(90\) 11.7122 1.23457
\(91\) 18.8544 1.97648
\(92\) −1.59524 −0.166315
\(93\) 0.660387 0.0684789
\(94\) −4.59409 −0.473844
\(95\) 3.91798 0.401976
\(96\) −0.103249 −0.0105378
\(97\) −11.0716 −1.12415 −0.562073 0.827088i \(-0.689996\pi\)
−0.562073 + 0.827088i \(0.689996\pi\)
\(98\) 12.8580 1.29885
\(99\) 0 0
\(100\) 10.3506 1.03506
\(101\) −14.2396 −1.41689 −0.708445 0.705766i \(-0.750603\pi\)
−0.708445 + 0.705766i \(0.750603\pi\)
\(102\) 0.681611 0.0674895
\(103\) −16.9912 −1.67419 −0.837096 0.547056i \(-0.815749\pi\)
−0.837096 + 0.547056i \(0.815749\pi\)
\(104\) −4.23103 −0.414887
\(105\) −1.80267 −0.175923
\(106\) −4.30161 −0.417810
\(107\) −13.8759 −1.34144 −0.670718 0.741712i \(-0.734014\pi\)
−0.670718 + 0.741712i \(0.734014\pi\)
\(108\) 0.618395 0.0595051
\(109\) −6.11130 −0.585357 −0.292678 0.956211i \(-0.594547\pi\)
−0.292678 + 0.956211i \(0.594547\pi\)
\(110\) 0 0
\(111\) −0.926523 −0.0879417
\(112\) −4.45623 −0.421074
\(113\) −8.61809 −0.810722 −0.405361 0.914157i \(-0.632854\pi\)
−0.405361 + 0.914157i \(0.632854\pi\)
\(114\) 0.103249 0.00967018
\(115\) 6.25013 0.582827
\(116\) −6.26090 −0.581310
\(117\) 12.6480 1.16931
\(118\) −10.9528 −1.00829
\(119\) 29.4182 2.69677
\(120\) 0.404529 0.0369282
\(121\) 0 0
\(122\) −0.325652 −0.0294832
\(123\) −0.197638 −0.0178204
\(124\) −6.39604 −0.574382
\(125\) −20.9634 −1.87503
\(126\) 13.3212 1.18674
\(127\) −6.81088 −0.604367 −0.302184 0.953250i \(-0.597716\pi\)
−0.302184 + 0.953250i \(0.597716\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.954343 −0.0840252
\(130\) 16.5771 1.45391
\(131\) −18.3631 −1.60439 −0.802194 0.597064i \(-0.796334\pi\)
−0.802194 + 0.597064i \(0.796334\pi\)
\(132\) 0 0
\(133\) 4.45623 0.386404
\(134\) 8.88645 0.767672
\(135\) −2.42286 −0.208527
\(136\) −6.60160 −0.566083
\(137\) 12.4294 1.06192 0.530958 0.847398i \(-0.321832\pi\)
0.530958 + 0.847398i \(0.321832\pi\)
\(138\) 0.164708 0.0140208
\(139\) −0.572791 −0.0485835 −0.0242917 0.999705i \(-0.507733\pi\)
−0.0242917 + 0.999705i \(0.507733\pi\)
\(140\) 17.4594 1.47559
\(141\) 0.474336 0.0399463
\(142\) −2.66009 −0.223230
\(143\) 0 0
\(144\) −2.98934 −0.249112
\(145\) 24.5301 2.03711
\(146\) −6.64661 −0.550078
\(147\) −1.32758 −0.109497
\(148\) 8.97366 0.737630
\(149\) 12.9477 1.06072 0.530360 0.847773i \(-0.322057\pi\)
0.530360 + 0.847773i \(0.322057\pi\)
\(150\) −1.06869 −0.0872581
\(151\) −13.1700 −1.07176 −0.535879 0.844294i \(-0.680020\pi\)
−0.535879 + 0.844294i \(0.680020\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 19.7344 1.59543
\(154\) 0 0
\(155\) 25.0596 2.01283
\(156\) 0.436851 0.0349761
\(157\) 16.3314 1.30339 0.651694 0.758482i \(-0.274058\pi\)
0.651694 + 0.758482i \(0.274058\pi\)
\(158\) −2.13991 −0.170242
\(159\) 0.444138 0.0352225
\(160\) −3.91798 −0.309744
\(161\) 7.10876 0.560249
\(162\) 8.90417 0.699578
\(163\) 3.96608 0.310648 0.155324 0.987864i \(-0.450358\pi\)
0.155324 + 0.987864i \(0.450358\pi\)
\(164\) 1.91418 0.149472
\(165\) 0 0
\(166\) 12.4605 0.967121
\(167\) 13.6008 1.05247 0.526233 0.850341i \(-0.323604\pi\)
0.526233 + 0.850341i \(0.323604\pi\)
\(168\) 0.460102 0.0354977
\(169\) 4.90163 0.377049
\(170\) 25.8650 1.98375
\(171\) 2.98934 0.228601
\(172\) 9.24310 0.704780
\(173\) −14.4150 −1.09595 −0.547976 0.836494i \(-0.684602\pi\)
−0.547976 + 0.836494i \(0.684602\pi\)
\(174\) 0.646434 0.0490060
\(175\) −46.1245 −3.48669
\(176\) 0 0
\(177\) 1.13087 0.0850013
\(178\) 8.88476 0.665941
\(179\) −5.80484 −0.433874 −0.216937 0.976186i \(-0.569607\pi\)
−0.216937 + 0.976186i \(0.569607\pi\)
\(180\) 11.7122 0.872974
\(181\) −0.148577 −0.0110436 −0.00552182 0.999985i \(-0.501758\pi\)
−0.00552182 + 0.999985i \(0.501758\pi\)
\(182\) 18.8544 1.39758
\(183\) 0.0336234 0.00248551
\(184\) −1.59524 −0.117603
\(185\) −35.1586 −2.58491
\(186\) 0.660387 0.0484219
\(187\) 0 0
\(188\) −4.59409 −0.335058
\(189\) −2.75571 −0.200448
\(190\) 3.91798 0.284240
\(191\) −6.74447 −0.488013 −0.244006 0.969774i \(-0.578462\pi\)
−0.244006 + 0.969774i \(0.578462\pi\)
\(192\) −0.103249 −0.00745137
\(193\) 8.15211 0.586802 0.293401 0.955989i \(-0.405213\pi\)
0.293401 + 0.955989i \(0.405213\pi\)
\(194\) −11.0716 −0.794891
\(195\) −1.71157 −0.122568
\(196\) 12.8580 0.918426
\(197\) 3.77994 0.269309 0.134655 0.990893i \(-0.457008\pi\)
0.134655 + 0.990893i \(0.457008\pi\)
\(198\) 0 0
\(199\) −0.977228 −0.0692739 −0.0346369 0.999400i \(-0.511027\pi\)
−0.0346369 + 0.999400i \(0.511027\pi\)
\(200\) 10.3506 0.731896
\(201\) −0.917519 −0.0647168
\(202\) −14.2396 −1.00189
\(203\) 27.9000 1.95820
\(204\) 0.681611 0.0477223
\(205\) −7.49973 −0.523804
\(206\) −16.9912 −1.18383
\(207\) 4.76872 0.331449
\(208\) −4.23103 −0.293369
\(209\) 0 0
\(210\) −1.80267 −0.124396
\(211\) −11.5402 −0.794463 −0.397231 0.917718i \(-0.630029\pi\)
−0.397231 + 0.917718i \(0.630029\pi\)
\(212\) −4.30161 −0.295436
\(213\) 0.274653 0.0188189
\(214\) −13.8759 −0.948539
\(215\) −36.2143 −2.46979
\(216\) 0.618395 0.0420764
\(217\) 28.5022 1.93486
\(218\) −6.11130 −0.413910
\(219\) 0.686258 0.0463730
\(220\) 0 0
\(221\) 27.9316 1.87888
\(222\) −0.926523 −0.0621842
\(223\) −16.0725 −1.07629 −0.538146 0.842852i \(-0.680875\pi\)
−0.538146 + 0.842852i \(0.680875\pi\)
\(224\) −4.45623 −0.297744
\(225\) −30.9414 −2.06276
\(226\) −8.61809 −0.573267
\(227\) 3.14962 0.209047 0.104524 0.994522i \(-0.466668\pi\)
0.104524 + 0.994522i \(0.466668\pi\)
\(228\) 0.103249 0.00683785
\(229\) 22.0432 1.45665 0.728327 0.685229i \(-0.240298\pi\)
0.728327 + 0.685229i \(0.240298\pi\)
\(230\) 6.25013 0.412121
\(231\) 0 0
\(232\) −6.26090 −0.411048
\(233\) 10.0334 0.657308 0.328654 0.944450i \(-0.393405\pi\)
0.328654 + 0.944450i \(0.393405\pi\)
\(234\) 12.6480 0.826825
\(235\) 17.9996 1.17416
\(236\) −10.9528 −0.712967
\(237\) 0.220944 0.0143518
\(238\) 29.4182 1.90690
\(239\) 11.4321 0.739484 0.369742 0.929134i \(-0.379446\pi\)
0.369742 + 0.929134i \(0.379446\pi\)
\(240\) 0.404529 0.0261122
\(241\) −14.0982 −0.908147 −0.454073 0.890964i \(-0.650030\pi\)
−0.454073 + 0.890964i \(0.650030\pi\)
\(242\) 0 0
\(243\) −2.77453 −0.177986
\(244\) −0.325652 −0.0208477
\(245\) −50.3773 −3.21849
\(246\) −0.197638 −0.0126009
\(247\) 4.23103 0.269214
\(248\) −6.39604 −0.406149
\(249\) −1.28654 −0.0815309
\(250\) −20.9634 −1.32584
\(251\) 18.6997 1.18032 0.590158 0.807288i \(-0.299065\pi\)
0.590158 + 0.807288i \(0.299065\pi\)
\(252\) 13.3212 0.839155
\(253\) 0 0
\(254\) −6.81088 −0.427352
\(255\) −2.67054 −0.167236
\(256\) 1.00000 0.0625000
\(257\) 15.4448 0.963419 0.481709 0.876331i \(-0.340016\pi\)
0.481709 + 0.876331i \(0.340016\pi\)
\(258\) −0.954343 −0.0594148
\(259\) −39.9887 −2.48477
\(260\) 16.5771 1.02807
\(261\) 18.7160 1.15849
\(262\) −18.3631 −1.13447
\(263\) −17.4417 −1.07550 −0.537751 0.843104i \(-0.680726\pi\)
−0.537751 + 0.843104i \(0.680726\pi\)
\(264\) 0 0
\(265\) 16.8536 1.03531
\(266\) 4.45623 0.273229
\(267\) −0.917345 −0.0561406
\(268\) 8.88645 0.542826
\(269\) 21.7774 1.32779 0.663894 0.747826i \(-0.268902\pi\)
0.663894 + 0.747826i \(0.268902\pi\)
\(270\) −2.42286 −0.147451
\(271\) 27.5183 1.67162 0.835808 0.549022i \(-0.185000\pi\)
0.835808 + 0.549022i \(0.185000\pi\)
\(272\) −6.60160 −0.400281
\(273\) −1.94671 −0.117820
\(274\) 12.4294 0.750887
\(275\) 0 0
\(276\) 0.164708 0.00991423
\(277\) −5.10903 −0.306972 −0.153486 0.988151i \(-0.549050\pi\)
−0.153486 + 0.988151i \(0.549050\pi\)
\(278\) −0.572791 −0.0343537
\(279\) 19.1199 1.14468
\(280\) 17.4594 1.04340
\(281\) −27.4547 −1.63781 −0.818905 0.573929i \(-0.805419\pi\)
−0.818905 + 0.573929i \(0.805419\pi\)
\(282\) 0.474336 0.0282463
\(283\) −6.79437 −0.403883 −0.201942 0.979398i \(-0.564725\pi\)
−0.201942 + 0.979398i \(0.564725\pi\)
\(284\) −2.66009 −0.157848
\(285\) −0.404529 −0.0239622
\(286\) 0 0
\(287\) −8.53003 −0.503512
\(288\) −2.98934 −0.176149
\(289\) 26.5812 1.56360
\(290\) 24.5301 1.44046
\(291\) 1.14313 0.0670115
\(292\) −6.64661 −0.388964
\(293\) −28.6595 −1.67430 −0.837152 0.546970i \(-0.815781\pi\)
−0.837152 + 0.546970i \(0.815781\pi\)
\(294\) −1.32758 −0.0774258
\(295\) 42.9129 2.49848
\(296\) 8.97366 0.521583
\(297\) 0 0
\(298\) 12.9477 0.750042
\(299\) 6.74952 0.390335
\(300\) −1.06869 −0.0617008
\(301\) −41.1894 −2.37412
\(302\) −13.1700 −0.757848
\(303\) 1.47022 0.0844622
\(304\) −1.00000 −0.0573539
\(305\) 1.27590 0.0730578
\(306\) 19.7344 1.12814
\(307\) −14.1122 −0.805427 −0.402713 0.915326i \(-0.631933\pi\)
−0.402713 + 0.915326i \(0.631933\pi\)
\(308\) 0 0
\(309\) 1.75433 0.0998003
\(310\) 25.0596 1.42329
\(311\) 11.2126 0.635809 0.317905 0.948123i \(-0.397021\pi\)
0.317905 + 0.948123i \(0.397021\pi\)
\(312\) 0.436851 0.0247318
\(313\) −11.6375 −0.657792 −0.328896 0.944366i \(-0.606677\pi\)
−0.328896 + 0.944366i \(0.606677\pi\)
\(314\) 16.3314 0.921635
\(315\) −52.1921 −2.94069
\(316\) −2.13991 −0.120379
\(317\) −3.69201 −0.207364 −0.103682 0.994611i \(-0.533062\pi\)
−0.103682 + 0.994611i \(0.533062\pi\)
\(318\) 0.444138 0.0249060
\(319\) 0 0
\(320\) −3.91798 −0.219022
\(321\) 1.43268 0.0799644
\(322\) 7.10876 0.396156
\(323\) 6.60160 0.367323
\(324\) 8.90417 0.494676
\(325\) −43.7936 −2.42923
\(326\) 3.96608 0.219661
\(327\) 0.630987 0.0348937
\(328\) 1.91418 0.105693
\(329\) 20.4723 1.12868
\(330\) 0 0
\(331\) 13.9429 0.766373 0.383187 0.923671i \(-0.374827\pi\)
0.383187 + 0.923671i \(0.374827\pi\)
\(332\) 12.4605 0.683858
\(333\) −26.8253 −1.47002
\(334\) 13.6008 0.744205
\(335\) −34.8169 −1.90225
\(336\) 0.460102 0.0251006
\(337\) −18.9545 −1.03252 −0.516258 0.856433i \(-0.672676\pi\)
−0.516258 + 0.856433i \(0.672676\pi\)
\(338\) 4.90163 0.266614
\(339\) 0.889812 0.0483279
\(340\) 25.8650 1.40272
\(341\) 0 0
\(342\) 2.98934 0.161645
\(343\) −26.1044 −1.40951
\(344\) 9.24310 0.498354
\(345\) −0.645321 −0.0347429
\(346\) −14.4150 −0.774956
\(347\) 36.6760 1.96887 0.984436 0.175745i \(-0.0562334\pi\)
0.984436 + 0.175745i \(0.0562334\pi\)
\(348\) 0.646434 0.0346525
\(349\) 6.99845 0.374618 0.187309 0.982301i \(-0.440023\pi\)
0.187309 + 0.982301i \(0.440023\pi\)
\(350\) −46.1245 −2.46546
\(351\) −2.61645 −0.139656
\(352\) 0 0
\(353\) −11.2690 −0.599786 −0.299893 0.953973i \(-0.596951\pi\)
−0.299893 + 0.953973i \(0.596951\pi\)
\(354\) 1.13087 0.0601050
\(355\) 10.4222 0.553153
\(356\) 8.88476 0.470891
\(357\) −3.03741 −0.160757
\(358\) −5.80484 −0.306795
\(359\) −5.70457 −0.301076 −0.150538 0.988604i \(-0.548101\pi\)
−0.150538 + 0.988604i \(0.548101\pi\)
\(360\) 11.7122 0.617286
\(361\) 1.00000 0.0526316
\(362\) −0.148577 −0.00780903
\(363\) 0 0
\(364\) 18.8544 0.988241
\(365\) 26.0413 1.36306
\(366\) 0.0336234 0.00175752
\(367\) −18.7306 −0.977732 −0.488866 0.872359i \(-0.662589\pi\)
−0.488866 + 0.872359i \(0.662589\pi\)
\(368\) −1.59524 −0.0831577
\(369\) −5.72214 −0.297883
\(370\) −35.1586 −1.82781
\(371\) 19.1690 0.995204
\(372\) 0.660387 0.0342395
\(373\) −11.8435 −0.613233 −0.306616 0.951833i \(-0.599197\pi\)
−0.306616 + 0.951833i \(0.599197\pi\)
\(374\) 0 0
\(375\) 2.16446 0.111772
\(376\) −4.59409 −0.236922
\(377\) 26.4901 1.36431
\(378\) −2.75571 −0.141738
\(379\) 14.4703 0.743288 0.371644 0.928375i \(-0.378794\pi\)
0.371644 + 0.928375i \(0.378794\pi\)
\(380\) 3.91798 0.200988
\(381\) 0.703218 0.0360269
\(382\) −6.74447 −0.345077
\(383\) −23.9927 −1.22597 −0.612984 0.790095i \(-0.710031\pi\)
−0.612984 + 0.790095i \(0.710031\pi\)
\(384\) −0.103249 −0.00526892
\(385\) 0 0
\(386\) 8.15211 0.414932
\(387\) −27.6308 −1.40455
\(388\) −11.0716 −0.562073
\(389\) −27.2728 −1.38278 −0.691392 0.722480i \(-0.743002\pi\)
−0.691392 + 0.722480i \(0.743002\pi\)
\(390\) −1.71157 −0.0866689
\(391\) 10.5312 0.532583
\(392\) 12.8580 0.649425
\(393\) 1.89597 0.0956391
\(394\) 3.77994 0.190430
\(395\) 8.38411 0.421850
\(396\) 0 0
\(397\) 20.4056 1.02413 0.512063 0.858948i \(-0.328881\pi\)
0.512063 + 0.858948i \(0.328881\pi\)
\(398\) −0.977228 −0.0489840
\(399\) −0.460102 −0.0230339
\(400\) 10.3506 0.517529
\(401\) −13.8056 −0.689417 −0.344708 0.938710i \(-0.612022\pi\)
−0.344708 + 0.938710i \(0.612022\pi\)
\(402\) −0.917519 −0.0457617
\(403\) 27.0619 1.34805
\(404\) −14.2396 −0.708445
\(405\) −34.8864 −1.73352
\(406\) 27.9000 1.38465
\(407\) 0 0
\(408\) 0.681611 0.0337448
\(409\) −21.2103 −1.04878 −0.524392 0.851477i \(-0.675707\pi\)
−0.524392 + 0.851477i \(0.675707\pi\)
\(410\) −7.49973 −0.370385
\(411\) −1.28333 −0.0633018
\(412\) −16.9912 −0.837096
\(413\) 48.8082 2.40169
\(414\) 4.76872 0.234370
\(415\) −48.8199 −2.39648
\(416\) −4.23103 −0.207443
\(417\) 0.0591402 0.00289611
\(418\) 0 0
\(419\) −12.1697 −0.594527 −0.297264 0.954795i \(-0.596074\pi\)
−0.297264 + 0.954795i \(0.596074\pi\)
\(420\) −1.80267 −0.0879614
\(421\) −17.3500 −0.845585 −0.422793 0.906226i \(-0.638950\pi\)
−0.422793 + 0.906226i \(0.638950\pi\)
\(422\) −11.5402 −0.561770
\(423\) 13.7333 0.667736
\(424\) −4.30161 −0.208905
\(425\) −68.3304 −3.31451
\(426\) 0.274653 0.0133070
\(427\) 1.45118 0.0702276
\(428\) −13.8759 −0.670718
\(429\) 0 0
\(430\) −36.2143 −1.74641
\(431\) 16.6638 0.802668 0.401334 0.915932i \(-0.368547\pi\)
0.401334 + 0.915932i \(0.368547\pi\)
\(432\) 0.618395 0.0297525
\(433\) 34.2652 1.64668 0.823341 0.567547i \(-0.192108\pi\)
0.823341 + 0.567547i \(0.192108\pi\)
\(434\) 28.5022 1.36815
\(435\) −2.53271 −0.121434
\(436\) −6.11130 −0.292678
\(437\) 1.59524 0.0763108
\(438\) 0.686258 0.0327907
\(439\) 23.9778 1.14440 0.572198 0.820115i \(-0.306091\pi\)
0.572198 + 0.820115i \(0.306091\pi\)
\(440\) 0 0
\(441\) −38.4368 −1.83033
\(442\) 27.9316 1.32857
\(443\) 9.23108 0.438582 0.219291 0.975660i \(-0.429626\pi\)
0.219291 + 0.975660i \(0.429626\pi\)
\(444\) −0.926523 −0.0439708
\(445\) −34.8103 −1.65017
\(446\) −16.0725 −0.761053
\(447\) −1.33684 −0.0632305
\(448\) −4.45623 −0.210537
\(449\) 4.61870 0.217970 0.108985 0.994043i \(-0.465240\pi\)
0.108985 + 0.994043i \(0.465240\pi\)
\(450\) −30.9414 −1.45859
\(451\) 0 0
\(452\) −8.61809 −0.405361
\(453\) 1.35979 0.0638886
\(454\) 3.14962 0.147819
\(455\) −73.8713 −3.46314
\(456\) 0.103249 0.00483509
\(457\) −6.40059 −0.299407 −0.149703 0.988731i \(-0.547832\pi\)
−0.149703 + 0.988731i \(0.547832\pi\)
\(458\) 22.0432 1.03001
\(459\) −4.08240 −0.190550
\(460\) 6.25013 0.291414
\(461\) −27.9210 −1.30041 −0.650206 0.759758i \(-0.725318\pi\)
−0.650206 + 0.759758i \(0.725318\pi\)
\(462\) 0 0
\(463\) −8.30846 −0.386127 −0.193063 0.981186i \(-0.561842\pi\)
−0.193063 + 0.981186i \(0.561842\pi\)
\(464\) −6.26090 −0.290655
\(465\) −2.58738 −0.119987
\(466\) 10.0334 0.464787
\(467\) 14.7637 0.683182 0.341591 0.939849i \(-0.389034\pi\)
0.341591 + 0.939849i \(0.389034\pi\)
\(468\) 12.6480 0.584654
\(469\) −39.6000 −1.82856
\(470\) 17.9996 0.830258
\(471\) −1.68621 −0.0776963
\(472\) −10.9528 −0.504144
\(473\) 0 0
\(474\) 0.220944 0.0101483
\(475\) −10.3506 −0.474917
\(476\) 29.4182 1.34838
\(477\) 12.8590 0.588773
\(478\) 11.4321 0.522894
\(479\) −36.4821 −1.66691 −0.833456 0.552587i \(-0.813641\pi\)
−0.833456 + 0.552587i \(0.813641\pi\)
\(480\) 0.404529 0.0184641
\(481\) −37.9678 −1.73118
\(482\) −14.0982 −0.642157
\(483\) −0.733974 −0.0333970
\(484\) 0 0
\(485\) 43.3781 1.96970
\(486\) −2.77453 −0.125855
\(487\) 24.8266 1.12500 0.562500 0.826797i \(-0.309840\pi\)
0.562500 + 0.826797i \(0.309840\pi\)
\(488\) −0.325652 −0.0147416
\(489\) −0.409495 −0.0185180
\(490\) −50.3773 −2.27581
\(491\) −15.2207 −0.686902 −0.343451 0.939171i \(-0.611596\pi\)
−0.343451 + 0.939171i \(0.611596\pi\)
\(492\) −0.197638 −0.00891020
\(493\) 41.3320 1.86150
\(494\) 4.23103 0.190363
\(495\) 0 0
\(496\) −6.39604 −0.287191
\(497\) 11.8540 0.531724
\(498\) −1.28654 −0.0576510
\(499\) −39.0356 −1.74747 −0.873737 0.486399i \(-0.838310\pi\)
−0.873737 + 0.486399i \(0.838310\pi\)
\(500\) −20.9634 −0.937514
\(501\) −1.40428 −0.0627385
\(502\) 18.6997 0.834610
\(503\) −23.9042 −1.06584 −0.532918 0.846167i \(-0.678905\pi\)
−0.532918 + 0.846167i \(0.678905\pi\)
\(504\) 13.3212 0.593372
\(505\) 55.7903 2.48264
\(506\) 0 0
\(507\) −0.506090 −0.0224762
\(508\) −6.81088 −0.302184
\(509\) 39.8560 1.76659 0.883294 0.468819i \(-0.155320\pi\)
0.883294 + 0.468819i \(0.155320\pi\)
\(510\) −2.67054 −0.118253
\(511\) 29.6188 1.31026
\(512\) 1.00000 0.0441942
\(513\) −0.618395 −0.0273028
\(514\) 15.4448 0.681240
\(515\) 66.5712 2.93348
\(516\) −0.954343 −0.0420126
\(517\) 0 0
\(518\) −39.9887 −1.75700
\(519\) 1.48834 0.0653308
\(520\) 16.5771 0.726954
\(521\) −34.7437 −1.52215 −0.761075 0.648664i \(-0.775328\pi\)
−0.761075 + 0.648664i \(0.775328\pi\)
\(522\) 18.7160 0.819176
\(523\) 3.65461 0.159805 0.0799024 0.996803i \(-0.474539\pi\)
0.0799024 + 0.996803i \(0.474539\pi\)
\(524\) −18.3631 −0.802194
\(525\) 4.76232 0.207845
\(526\) −17.4417 −0.760495
\(527\) 42.2241 1.83931
\(528\) 0 0
\(529\) −20.4552 −0.889357
\(530\) 16.8536 0.732075
\(531\) 32.7416 1.42087
\(532\) 4.45623 0.193202
\(533\) −8.09896 −0.350805
\(534\) −0.917345 −0.0396974
\(535\) 54.3656 2.35043
\(536\) 8.88645 0.383836
\(537\) 0.599345 0.0258637
\(538\) 21.7774 0.938888
\(539\) 0 0
\(540\) −2.42286 −0.104263
\(541\) −23.4833 −1.00963 −0.504814 0.863228i \(-0.668439\pi\)
−0.504814 + 0.863228i \(0.668439\pi\)
\(542\) 27.5183 1.18201
\(543\) 0.0153405 0.000658322 0
\(544\) −6.60160 −0.283041
\(545\) 23.9440 1.02565
\(546\) −1.94671 −0.0833114
\(547\) −4.29735 −0.183742 −0.0918708 0.995771i \(-0.529285\pi\)
−0.0918708 + 0.995771i \(0.529285\pi\)
\(548\) 12.4294 0.530958
\(549\) 0.973485 0.0415473
\(550\) 0 0
\(551\) 6.26090 0.266723
\(552\) 0.164708 0.00701042
\(553\) 9.53591 0.405508
\(554\) −5.10903 −0.217062
\(555\) 3.63010 0.154089
\(556\) −0.572791 −0.0242917
\(557\) −14.9520 −0.633535 −0.316767 0.948503i \(-0.602597\pi\)
−0.316767 + 0.948503i \(0.602597\pi\)
\(558\) 19.1199 0.809412
\(559\) −39.1078 −1.65409
\(560\) 17.4594 0.737795
\(561\) 0 0
\(562\) −27.4547 −1.15811
\(563\) 36.2456 1.52757 0.763786 0.645470i \(-0.223339\pi\)
0.763786 + 0.645470i \(0.223339\pi\)
\(564\) 0.474336 0.0199732
\(565\) 33.7655 1.42053
\(566\) −6.79437 −0.285589
\(567\) −39.6790 −1.66636
\(568\) −2.66009 −0.111615
\(569\) 1.48123 0.0620964 0.0310482 0.999518i \(-0.490115\pi\)
0.0310482 + 0.999518i \(0.490115\pi\)
\(570\) −0.404529 −0.0169438
\(571\) −36.9110 −1.54468 −0.772338 0.635211i \(-0.780913\pi\)
−0.772338 + 0.635211i \(0.780913\pi\)
\(572\) 0 0
\(573\) 0.696362 0.0290909
\(574\) −8.53003 −0.356037
\(575\) −16.5117 −0.688584
\(576\) −2.98934 −0.124556
\(577\) 37.8575 1.57603 0.788014 0.615657i \(-0.211109\pi\)
0.788014 + 0.615657i \(0.211109\pi\)
\(578\) 26.5812 1.10563
\(579\) −0.841700 −0.0349798
\(580\) 24.5301 1.01856
\(581\) −55.5267 −2.30364
\(582\) 1.14313 0.0473843
\(583\) 0 0
\(584\) −6.64661 −0.275039
\(585\) −49.5546 −2.04883
\(586\) −28.6595 −1.18391
\(587\) −2.40075 −0.0990896 −0.0495448 0.998772i \(-0.515777\pi\)
−0.0495448 + 0.998772i \(0.515777\pi\)
\(588\) −1.32758 −0.0547483
\(589\) 6.39604 0.263544
\(590\) 42.9129 1.76670
\(591\) −0.390276 −0.0160538
\(592\) 8.97366 0.368815
\(593\) −6.20332 −0.254740 −0.127370 0.991855i \(-0.540654\pi\)
−0.127370 + 0.991855i \(0.540654\pi\)
\(594\) 0 0
\(595\) −115.260 −4.72520
\(596\) 12.9477 0.530360
\(597\) 0.100898 0.00412948
\(598\) 6.74952 0.276008
\(599\) 18.8744 0.771187 0.385593 0.922669i \(-0.373997\pi\)
0.385593 + 0.922669i \(0.373997\pi\)
\(600\) −1.06869 −0.0436290
\(601\) 40.5992 1.65608 0.828039 0.560671i \(-0.189457\pi\)
0.828039 + 0.560671i \(0.189457\pi\)
\(602\) −41.1894 −1.67875
\(603\) −26.5646 −1.08179
\(604\) −13.1700 −0.535879
\(605\) 0 0
\(606\) 1.47022 0.0597238
\(607\) 2.96039 0.120158 0.0600792 0.998194i \(-0.480865\pi\)
0.0600792 + 0.998194i \(0.480865\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −2.88066 −0.116730
\(610\) 1.27590 0.0516596
\(611\) 19.4377 0.786367
\(612\) 19.7344 0.797717
\(613\) −0.763124 −0.0308223 −0.0154112 0.999881i \(-0.504906\pi\)
−0.0154112 + 0.999881i \(0.504906\pi\)
\(614\) −14.1122 −0.569523
\(615\) 0.774341 0.0312245
\(616\) 0 0
\(617\) 1.06644 0.0429334 0.0214667 0.999770i \(-0.493166\pi\)
0.0214667 + 0.999770i \(0.493166\pi\)
\(618\) 1.75433 0.0705694
\(619\) 12.5798 0.505626 0.252813 0.967515i \(-0.418644\pi\)
0.252813 + 0.967515i \(0.418644\pi\)
\(620\) 25.0596 1.00642
\(621\) −0.986489 −0.0395865
\(622\) 11.2126 0.449585
\(623\) −39.5925 −1.58624
\(624\) 0.436851 0.0174880
\(625\) 30.3815 1.21526
\(626\) −11.6375 −0.465129
\(627\) 0 0
\(628\) 16.3314 0.651694
\(629\) −59.2405 −2.36207
\(630\) −52.1921 −2.07938
\(631\) −22.9529 −0.913741 −0.456871 0.889533i \(-0.651030\pi\)
−0.456871 + 0.889533i \(0.651030\pi\)
\(632\) −2.13991 −0.0851209
\(633\) 1.19152 0.0473587
\(634\) −3.69201 −0.146628
\(635\) 26.6849 1.05896
\(636\) 0.444138 0.0176112
\(637\) −54.4025 −2.15550
\(638\) 0 0
\(639\) 7.95192 0.314573
\(640\) −3.91798 −0.154872
\(641\) −2.91555 −0.115157 −0.0575787 0.998341i \(-0.518338\pi\)
−0.0575787 + 0.998341i \(0.518338\pi\)
\(642\) 1.43268 0.0565434
\(643\) 18.3286 0.722811 0.361406 0.932409i \(-0.382297\pi\)
0.361406 + 0.932409i \(0.382297\pi\)
\(644\) 7.10876 0.280124
\(645\) 3.73910 0.147227
\(646\) 6.60160 0.259737
\(647\) 5.12335 0.201420 0.100710 0.994916i \(-0.467889\pi\)
0.100710 + 0.994916i \(0.467889\pi\)
\(648\) 8.90417 0.349789
\(649\) 0 0
\(650\) −43.7936 −1.71773
\(651\) −2.94283 −0.115339
\(652\) 3.96608 0.155324
\(653\) 47.7078 1.86695 0.933475 0.358643i \(-0.116761\pi\)
0.933475 + 0.358643i \(0.116761\pi\)
\(654\) 0.630987 0.0246736
\(655\) 71.9461 2.81117
\(656\) 1.91418 0.0747362
\(657\) 19.8690 0.775163
\(658\) 20.4723 0.798094
\(659\) 18.6754 0.727491 0.363746 0.931498i \(-0.381498\pi\)
0.363746 + 0.931498i \(0.381498\pi\)
\(660\) 0 0
\(661\) −37.8786 −1.47331 −0.736654 0.676270i \(-0.763595\pi\)
−0.736654 + 0.676270i \(0.763595\pi\)
\(662\) 13.9429 0.541908
\(663\) −2.88392 −0.112002
\(664\) 12.4605 0.483560
\(665\) −17.4594 −0.677047
\(666\) −26.8253 −1.03946
\(667\) 9.98766 0.386724
\(668\) 13.6008 0.526233
\(669\) 1.65947 0.0641588
\(670\) −34.8169 −1.34510
\(671\) 0 0
\(672\) 0.460102 0.0177488
\(673\) 19.1798 0.739328 0.369664 0.929166i \(-0.379473\pi\)
0.369664 + 0.929166i \(0.379473\pi\)
\(674\) −18.9545 −0.730100
\(675\) 6.40074 0.246365
\(676\) 4.90163 0.188524
\(677\) 0.144722 0.00556210 0.00278105 0.999996i \(-0.499115\pi\)
0.00278105 + 0.999996i \(0.499115\pi\)
\(678\) 0.889812 0.0341730
\(679\) 49.3374 1.89339
\(680\) 25.8650 0.991876
\(681\) −0.325196 −0.0124615
\(682\) 0 0
\(683\) −8.12408 −0.310859 −0.155430 0.987847i \(-0.549676\pi\)
−0.155430 + 0.987847i \(0.549676\pi\)
\(684\) 2.98934 0.114300
\(685\) −48.6981 −1.86066
\(686\) −26.1044 −0.996672
\(687\) −2.27594 −0.0868326
\(688\) 9.24310 0.352390
\(689\) 18.2003 0.693375
\(690\) −0.645321 −0.0245669
\(691\) −38.0620 −1.44795 −0.723973 0.689828i \(-0.757686\pi\)
−0.723973 + 0.689828i \(0.757686\pi\)
\(692\) −14.4150 −0.547976
\(693\) 0 0
\(694\) 36.6760 1.39220
\(695\) 2.24418 0.0851267
\(696\) 0.646434 0.0245030
\(697\) −12.6367 −0.478648
\(698\) 6.99845 0.264895
\(699\) −1.03594 −0.0391828
\(700\) −46.1245 −1.74334
\(701\) −17.8921 −0.675774 −0.337887 0.941187i \(-0.609712\pi\)
−0.337887 + 0.941187i \(0.609712\pi\)
\(702\) −2.61645 −0.0987515
\(703\) −8.97366 −0.338448
\(704\) 0 0
\(705\) −1.85844 −0.0699929
\(706\) −11.2690 −0.424113
\(707\) 63.4548 2.38646
\(708\) 1.13087 0.0425007
\(709\) 16.5211 0.620462 0.310231 0.950661i \(-0.399594\pi\)
0.310231 + 0.950661i \(0.399594\pi\)
\(710\) 10.4222 0.391138
\(711\) 6.39691 0.239903
\(712\) 8.88476 0.332971
\(713\) 10.2032 0.382114
\(714\) −3.03741 −0.113672
\(715\) 0 0
\(716\) −5.80484 −0.216937
\(717\) −1.18036 −0.0440814
\(718\) −5.70457 −0.212893
\(719\) 44.2488 1.65020 0.825101 0.564986i \(-0.191118\pi\)
0.825101 + 0.564986i \(0.191118\pi\)
\(720\) 11.7122 0.436487
\(721\) 75.7166 2.81984
\(722\) 1.00000 0.0372161
\(723\) 1.45563 0.0541355
\(724\) −0.148577 −0.00552182
\(725\) −64.8039 −2.40676
\(726\) 0 0
\(727\) 38.4604 1.42642 0.713209 0.700952i \(-0.247241\pi\)
0.713209 + 0.700952i \(0.247241\pi\)
\(728\) 18.8544 0.698792
\(729\) −26.4260 −0.978742
\(730\) 26.0413 0.963832
\(731\) −61.0193 −2.25688
\(732\) 0.0336234 0.00124275
\(733\) 46.9096 1.73265 0.866323 0.499484i \(-0.166477\pi\)
0.866323 + 0.499484i \(0.166477\pi\)
\(734\) −18.7306 −0.691361
\(735\) 5.20142 0.191857
\(736\) −1.59524 −0.0588014
\(737\) 0 0
\(738\) −5.72214 −0.210635
\(739\) −10.9340 −0.402215 −0.201108 0.979569i \(-0.564454\pi\)
−0.201108 + 0.979569i \(0.564454\pi\)
\(740\) −35.1586 −1.29246
\(741\) −0.436851 −0.0160481
\(742\) 19.1690 0.703715
\(743\) −31.3320 −1.14946 −0.574729 0.818344i \(-0.694893\pi\)
−0.574729 + 0.818344i \(0.694893\pi\)
\(744\) 0.660387 0.0242110
\(745\) −50.7290 −1.85857
\(746\) −11.8435 −0.433621
\(747\) −37.2486 −1.36286
\(748\) 0 0
\(749\) 61.8343 2.25938
\(750\) 2.16446 0.0790349
\(751\) 39.4660 1.44013 0.720067 0.693905i \(-0.244111\pi\)
0.720067 + 0.693905i \(0.244111\pi\)
\(752\) −4.59409 −0.167529
\(753\) −1.93073 −0.0703598
\(754\) 26.4901 0.964712
\(755\) 51.5998 1.87791
\(756\) −2.75571 −0.100224
\(757\) −0.203127 −0.00738276 −0.00369138 0.999993i \(-0.501175\pi\)
−0.00369138 + 0.999993i \(0.501175\pi\)
\(758\) 14.4703 0.525584
\(759\) 0 0
\(760\) 3.91798 0.142120
\(761\) 12.4379 0.450873 0.225437 0.974258i \(-0.427619\pi\)
0.225437 + 0.974258i \(0.427619\pi\)
\(762\) 0.703218 0.0254749
\(763\) 27.2334 0.985914
\(764\) −6.74447 −0.244006
\(765\) −77.3191 −2.79548
\(766\) −23.9927 −0.866890
\(767\) 46.3417 1.67330
\(768\) −0.103249 −0.00372569
\(769\) 24.4144 0.880406 0.440203 0.897898i \(-0.354906\pi\)
0.440203 + 0.897898i \(0.354906\pi\)
\(770\) 0 0
\(771\) −1.59466 −0.0574303
\(772\) 8.15211 0.293401
\(773\) −27.8291 −1.00094 −0.500471 0.865753i \(-0.666840\pi\)
−0.500471 + 0.865753i \(0.666840\pi\)
\(774\) −27.6308 −0.993167
\(775\) −66.2027 −2.37807
\(776\) −11.0716 −0.397446
\(777\) 4.12880 0.148120
\(778\) −27.2728 −0.977776
\(779\) −1.91418 −0.0685827
\(780\) −1.71157 −0.0612842
\(781\) 0 0
\(782\) 10.5312 0.376593
\(783\) −3.87171 −0.138364
\(784\) 12.8580 0.459213
\(785\) −63.9862 −2.28376
\(786\) 1.89597 0.0676271
\(787\) −22.9480 −0.818006 −0.409003 0.912533i \(-0.634124\pi\)
−0.409003 + 0.912533i \(0.634124\pi\)
\(788\) 3.77994 0.134655
\(789\) 1.80084 0.0641118
\(790\) 8.38411 0.298293
\(791\) 38.4042 1.36550
\(792\) 0 0
\(793\) 1.37785 0.0489287
\(794\) 20.4056 0.724166
\(795\) −1.74013 −0.0617159
\(796\) −0.977228 −0.0346369
\(797\) −23.6863 −0.839013 −0.419507 0.907752i \(-0.637797\pi\)
−0.419507 + 0.907752i \(0.637797\pi\)
\(798\) −0.460102 −0.0162874
\(799\) 30.3284 1.07294
\(800\) 10.3506 0.365948
\(801\) −26.5596 −0.938436
\(802\) −13.8056 −0.487491
\(803\) 0 0
\(804\) −0.917519 −0.0323584
\(805\) −27.8520 −0.981654
\(806\) 27.0619 0.953214
\(807\) −2.24850 −0.0791508
\(808\) −14.2396 −0.500946
\(809\) −20.9439 −0.736349 −0.368175 0.929757i \(-0.620017\pi\)
−0.368175 + 0.929757i \(0.620017\pi\)
\(810\) −34.8864 −1.22578
\(811\) −19.7226 −0.692555 −0.346277 0.938132i \(-0.612554\pi\)
−0.346277 + 0.938132i \(0.612554\pi\)
\(812\) 27.9000 0.979099
\(813\) −2.84124 −0.0996467
\(814\) 0 0
\(815\) −15.5390 −0.544309
\(816\) 0.681611 0.0238611
\(817\) −9.24310 −0.323375
\(818\) −21.2103 −0.741602
\(819\) −56.3623 −1.96946
\(820\) −7.49973 −0.261902
\(821\) −35.4686 −1.23786 −0.618931 0.785445i \(-0.712434\pi\)
−0.618931 + 0.785445i \(0.712434\pi\)
\(822\) −1.28333 −0.0447611
\(823\) 18.8740 0.657906 0.328953 0.944346i \(-0.393304\pi\)
0.328953 + 0.944346i \(0.393304\pi\)
\(824\) −16.9912 −0.591916
\(825\) 0 0
\(826\) 48.8082 1.69825
\(827\) −38.4943 −1.33858 −0.669290 0.743002i \(-0.733401\pi\)
−0.669290 + 0.743002i \(0.733401\pi\)
\(828\) 4.76872 0.165724
\(829\) 20.0947 0.697917 0.348959 0.937138i \(-0.386535\pi\)
0.348959 + 0.937138i \(0.386535\pi\)
\(830\) −48.8199 −1.69456
\(831\) 0.527504 0.0182989
\(832\) −4.23103 −0.146685
\(833\) −84.8832 −2.94103
\(834\) 0.0591402 0.00204786
\(835\) −53.2879 −1.84410
\(836\) 0 0
\(837\) −3.95528 −0.136714
\(838\) −12.1697 −0.420394
\(839\) 18.8166 0.649621 0.324811 0.945779i \(-0.394699\pi\)
0.324811 + 0.945779i \(0.394699\pi\)
\(840\) −1.80267 −0.0621981
\(841\) 10.1989 0.351687
\(842\) −17.3500 −0.597919
\(843\) 2.83468 0.0976315
\(844\) −11.5402 −0.397231
\(845\) −19.2045 −0.660655
\(846\) 13.7333 0.472160
\(847\) 0 0
\(848\) −4.30161 −0.147718
\(849\) 0.701513 0.0240759
\(850\) −68.3304 −2.34371
\(851\) −14.3152 −0.490717
\(852\) 0.274653 0.00940945
\(853\) 50.7466 1.73753 0.868765 0.495225i \(-0.164914\pi\)
0.868765 + 0.495225i \(0.164914\pi\)
\(854\) 1.45118 0.0496584
\(855\) −11.7122 −0.400548
\(856\) −13.8759 −0.474270
\(857\) −44.1640 −1.50861 −0.754307 0.656522i \(-0.772027\pi\)
−0.754307 + 0.656522i \(0.772027\pi\)
\(858\) 0 0
\(859\) 23.1245 0.788999 0.394499 0.918896i \(-0.370918\pi\)
0.394499 + 0.918896i \(0.370918\pi\)
\(860\) −36.2143 −1.23490
\(861\) 0.880719 0.0300148
\(862\) 16.6638 0.567572
\(863\) 47.1399 1.60466 0.802331 0.596879i \(-0.203593\pi\)
0.802331 + 0.596879i \(0.203593\pi\)
\(864\) 0.618395 0.0210382
\(865\) 56.4777 1.92030
\(866\) 34.2652 1.16438
\(867\) −2.74448 −0.0932076
\(868\) 28.5022 0.967429
\(869\) 0 0
\(870\) −2.53271 −0.0858671
\(871\) −37.5988 −1.27399
\(872\) −6.11130 −0.206955
\(873\) 33.0966 1.12015
\(874\) 1.59524 0.0539599
\(875\) 93.4179 3.15810
\(876\) 0.686258 0.0231865
\(877\) 28.8821 0.975280 0.487640 0.873045i \(-0.337858\pi\)
0.487640 + 0.873045i \(0.337858\pi\)
\(878\) 23.9778 0.809210
\(879\) 2.95907 0.0998069
\(880\) 0 0
\(881\) −6.89494 −0.232296 −0.116148 0.993232i \(-0.537055\pi\)
−0.116148 + 0.993232i \(0.537055\pi\)
\(882\) −38.4368 −1.29424
\(883\) 14.2757 0.480416 0.240208 0.970721i \(-0.422784\pi\)
0.240208 + 0.970721i \(0.422784\pi\)
\(884\) 27.9316 0.939441
\(885\) −4.43072 −0.148937
\(886\) 9.23108 0.310124
\(887\) 4.05611 0.136191 0.0680955 0.997679i \(-0.478308\pi\)
0.0680955 + 0.997679i \(0.478308\pi\)
\(888\) −0.926523 −0.0310921
\(889\) 30.3508 1.01793
\(890\) −34.8103 −1.16684
\(891\) 0 0
\(892\) −16.0725 −0.538146
\(893\) 4.59409 0.153735
\(894\) −1.33684 −0.0447107
\(895\) 22.7432 0.760223
\(896\) −4.45623 −0.148872
\(897\) −0.696883 −0.0232682
\(898\) 4.61870 0.154128
\(899\) 40.0450 1.33558
\(900\) −30.9414 −1.03138
\(901\) 28.3975 0.946059
\(902\) 0 0
\(903\) 4.25277 0.141523
\(904\) −8.61809 −0.286634
\(905\) 0.582122 0.0193504
\(906\) 1.35979 0.0451761
\(907\) 0.368732 0.0122435 0.00612176 0.999981i \(-0.498051\pi\)
0.00612176 + 0.999981i \(0.498051\pi\)
\(908\) 3.14962 0.104524
\(909\) 42.5669 1.41186
\(910\) −73.8713 −2.44881
\(911\) 4.21077 0.139509 0.0697545 0.997564i \(-0.477778\pi\)
0.0697545 + 0.997564i \(0.477778\pi\)
\(912\) 0.103249 0.00341892
\(913\) 0 0
\(914\) −6.40059 −0.211713
\(915\) −0.131736 −0.00435505
\(916\) 22.0432 0.728327
\(917\) 81.8299 2.70226
\(918\) −4.08240 −0.134739
\(919\) 17.5496 0.578907 0.289453 0.957192i \(-0.406526\pi\)
0.289453 + 0.957192i \(0.406526\pi\)
\(920\) 6.25013 0.206061
\(921\) 1.45708 0.0480123
\(922\) −27.9210 −0.919531
\(923\) 11.2549 0.370461
\(924\) 0 0
\(925\) 92.8825 3.05396
\(926\) −8.30846 −0.273033
\(927\) 50.7925 1.66824
\(928\) −6.26090 −0.205524
\(929\) −22.3125 −0.732048 −0.366024 0.930605i \(-0.619281\pi\)
−0.366024 + 0.930605i \(0.619281\pi\)
\(930\) −2.58738 −0.0848436
\(931\) −12.8580 −0.421403
\(932\) 10.0334 0.328654
\(933\) −1.15769 −0.0379012
\(934\) 14.7637 0.483082
\(935\) 0 0
\(936\) 12.6480 0.413413
\(937\) 46.2027 1.50938 0.754689 0.656083i \(-0.227788\pi\)
0.754689 + 0.656083i \(0.227788\pi\)
\(938\) −39.6000 −1.29299
\(939\) 1.20157 0.0392116
\(940\) 17.9996 0.587081
\(941\) −7.66776 −0.249962 −0.124981 0.992159i \(-0.539887\pi\)
−0.124981 + 0.992159i \(0.539887\pi\)
\(942\) −1.68621 −0.0549396
\(943\) −3.05358 −0.0994383
\(944\) −10.9528 −0.356483
\(945\) 10.7968 0.351220
\(946\) 0 0
\(947\) −27.3140 −0.887585 −0.443793 0.896130i \(-0.646367\pi\)
−0.443793 + 0.896130i \(0.646367\pi\)
\(948\) 0.220944 0.00717592
\(949\) 28.1220 0.912880
\(950\) −10.3506 −0.335817
\(951\) 0.381197 0.0123612
\(952\) 29.4182 0.953451
\(953\) 3.09031 0.100105 0.0500525 0.998747i \(-0.484061\pi\)
0.0500525 + 0.998747i \(0.484061\pi\)
\(954\) 12.8590 0.416325
\(955\) 26.4247 0.855084
\(956\) 11.4321 0.369742
\(957\) 0 0
\(958\) −36.4821 −1.17868
\(959\) −55.3882 −1.78858
\(960\) 0.404529 0.0130561
\(961\) 9.90937 0.319657
\(962\) −37.9678 −1.22413
\(963\) 41.4799 1.33667
\(964\) −14.0982 −0.454073
\(965\) −31.9398 −1.02818
\(966\) −0.733974 −0.0236152
\(967\) 33.9900 1.09305 0.546523 0.837444i \(-0.315951\pi\)
0.546523 + 0.837444i \(0.315951\pi\)
\(968\) 0 0
\(969\) −0.681611 −0.0218965
\(970\) 43.3781 1.39279
\(971\) −36.7362 −1.17892 −0.589461 0.807797i \(-0.700660\pi\)
−0.589461 + 0.807797i \(0.700660\pi\)
\(972\) −2.77453 −0.0889932
\(973\) 2.55249 0.0818289
\(974\) 24.8266 0.795495
\(975\) 4.52166 0.144809
\(976\) −0.325652 −0.0104239
\(977\) −7.86284 −0.251554 −0.125777 0.992059i \(-0.540142\pi\)
−0.125777 + 0.992059i \(0.540142\pi\)
\(978\) −0.409495 −0.0130942
\(979\) 0 0
\(980\) −50.3773 −1.60924
\(981\) 18.2688 0.583277
\(982\) −15.2207 −0.485713
\(983\) 3.75322 0.119709 0.0598546 0.998207i \(-0.480936\pi\)
0.0598546 + 0.998207i \(0.480936\pi\)
\(984\) −0.197638 −0.00630046
\(985\) −14.8097 −0.471877
\(986\) 41.3320 1.31628
\(987\) −2.11375 −0.0672814
\(988\) 4.23103 0.134607
\(989\) −14.7450 −0.468863
\(990\) 0 0
\(991\) 44.5118 1.41396 0.706982 0.707232i \(-0.250056\pi\)
0.706982 + 0.707232i \(0.250056\pi\)
\(992\) −6.39604 −0.203075
\(993\) −1.43960 −0.0456843
\(994\) 11.8540 0.375986
\(995\) 3.82876 0.121380
\(996\) −1.28654 −0.0407654
\(997\) 42.1484 1.33485 0.667426 0.744676i \(-0.267396\pi\)
0.667426 + 0.744676i \(0.267396\pi\)
\(998\) −39.0356 −1.23565
\(999\) 5.54926 0.175571
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 4598.2.a.cb.1.3 yes 8
11.10 odd 2 4598.2.a.by.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.by.1.3 8 11.10 odd 2
4598.2.a.cb.1.3 yes 8 1.1 even 1 trivial