Properties

Label 693.2.a.l.1.2
Level $693$
Weight $2$
Character 693.1
Self dual yes
Analytic conductor $5.534$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.53363286007\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 693.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.46260 q^{2} +0.139194 q^{4} -2.39821 q^{5} -1.00000 q^{7} +2.72161 q^{8} +O(q^{10})\) \(q-1.46260 q^{2} +0.139194 q^{4} -2.39821 q^{5} -1.00000 q^{7} +2.72161 q^{8} +3.50761 q^{10} +1.00000 q^{11} +5.04502 q^{13} +1.46260 q^{14} -4.25901 q^{16} +6.36842 q^{17} -5.32340 q^{19} -0.333816 q^{20} -1.46260 q^{22} -4.92520 q^{23} +0.751399 q^{25} -7.37883 q^{26} -0.139194 q^{28} -5.04502 q^{29} -7.57201 q^{31} +0.786003 q^{32} -9.31444 q^{34} +2.39821 q^{35} +4.24860 q^{37} +7.78600 q^{38} -6.52699 q^{40} +0.646809 q^{41} -10.5180 q^{43} +0.139194 q^{44} +7.20359 q^{46} -0.526989 q^{47} +1.00000 q^{49} -1.09899 q^{50} +0.702237 q^{52} -3.72161 q^{53} -2.39821 q^{55} -2.72161 q^{56} +7.37883 q^{58} -7.97021 q^{59} -2.00000 q^{61} +11.0748 q^{62} +7.36842 q^{64} -12.0990 q^{65} +8.76663 q^{67} +0.886447 q^{68} -3.50761 q^{70} +11.4432 q^{71} -13.0450 q^{73} -6.21400 q^{74} -0.740987 q^{76} -1.00000 q^{77} +11.4432 q^{79} +10.2140 q^{80} -0.946021 q^{82} -13.1648 q^{83} -15.2728 q^{85} +15.3836 q^{86} +2.72161 q^{88} -11.8504 q^{89} -5.04502 q^{91} -0.685559 q^{92} +0.770774 q^{94} +12.7666 q^{95} -1.87122 q^{97} -1.46260 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} + O(q^{10}) \) \( 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} - 11 q^{10} + 3 q^{11} - 4 q^{13} + 2 q^{14} - 4 q^{16} - 8 q^{17} - 8 q^{19} + 3 q^{20} - 2 q^{22} - 10 q^{23} + 15 q^{25} + q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} - 8 q^{32} - 4 q^{34} + 4 q^{35} + 13 q^{38} - 18 q^{40} - 14 q^{41} - 14 q^{43} + 6 q^{44} + 28 q^{46} + 3 q^{49} + 19 q^{50} - 29 q^{52} - 4 q^{55} + 3 q^{56} - q^{58} - 6 q^{61} + 38 q^{62} - 5 q^{64} - 14 q^{65} - 4 q^{67} - 42 q^{68} + 11 q^{70} + 12 q^{71} - 20 q^{73} - 29 q^{74} - 11 q^{76} - 3 q^{77} + 12 q^{79} + 41 q^{80} - 6 q^{82} - 6 q^{83} - 6 q^{85} - 24 q^{86} - 3 q^{88} - 26 q^{89} + 4 q^{91} - 26 q^{92} + 35 q^{94} + 8 q^{95} - 4 q^{97} - 2 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.46260 −1.03421 −0.517107 0.855921i \(-0.672991\pi\)
−0.517107 + 0.855921i \(0.672991\pi\)
\(3\) 0 0
\(4\) 0.139194 0.0695971
\(5\) −2.39821 −1.07251 −0.536255 0.844056i \(-0.680162\pi\)
−0.536255 + 0.844056i \(0.680162\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.72161 0.962235
\(9\) 0 0
\(10\) 3.50761 1.10921
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.04502 1.39924 0.699618 0.714517i \(-0.253354\pi\)
0.699618 + 0.714517i \(0.253354\pi\)
\(14\) 1.46260 0.390896
\(15\) 0 0
\(16\) −4.25901 −1.06475
\(17\) 6.36842 1.54457 0.772284 0.635277i \(-0.219114\pi\)
0.772284 + 0.635277i \(0.219114\pi\)
\(18\) 0 0
\(19\) −5.32340 −1.22127 −0.610636 0.791911i \(-0.709086\pi\)
−0.610636 + 0.791911i \(0.709086\pi\)
\(20\) −0.333816 −0.0746436
\(21\) 0 0
\(22\) −1.46260 −0.311827
\(23\) −4.92520 −1.02697 −0.513487 0.858097i \(-0.671647\pi\)
−0.513487 + 0.858097i \(0.671647\pi\)
\(24\) 0 0
\(25\) 0.751399 0.150280
\(26\) −7.37883 −1.44711
\(27\) 0 0
\(28\) −0.139194 −0.0263052
\(29\) −5.04502 −0.936836 −0.468418 0.883507i \(-0.655176\pi\)
−0.468418 + 0.883507i \(0.655176\pi\)
\(30\) 0 0
\(31\) −7.57201 −1.35997 −0.679986 0.733225i \(-0.738014\pi\)
−0.679986 + 0.733225i \(0.738014\pi\)
\(32\) 0.786003 0.138947
\(33\) 0 0
\(34\) −9.31444 −1.59741
\(35\) 2.39821 0.405371
\(36\) 0 0
\(37\) 4.24860 0.698466 0.349233 0.937036i \(-0.386442\pi\)
0.349233 + 0.937036i \(0.386442\pi\)
\(38\) 7.78600 1.26306
\(39\) 0 0
\(40\) −6.52699 −1.03201
\(41\) 0.646809 0.101015 0.0505073 0.998724i \(-0.483916\pi\)
0.0505073 + 0.998724i \(0.483916\pi\)
\(42\) 0 0
\(43\) −10.5180 −1.60398 −0.801992 0.597335i \(-0.796226\pi\)
−0.801992 + 0.597335i \(0.796226\pi\)
\(44\) 0.139194 0.0209843
\(45\) 0 0
\(46\) 7.20359 1.06211
\(47\) −0.526989 −0.0768693 −0.0384347 0.999261i \(-0.512237\pi\)
−0.0384347 + 0.999261i \(0.512237\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.09899 −0.155421
\(51\) 0 0
\(52\) 0.702237 0.0973827
\(53\) −3.72161 −0.511203 −0.255601 0.966782i \(-0.582273\pi\)
−0.255601 + 0.966782i \(0.582273\pi\)
\(54\) 0 0
\(55\) −2.39821 −0.323374
\(56\) −2.72161 −0.363691
\(57\) 0 0
\(58\) 7.37883 0.968888
\(59\) −7.97021 −1.03763 −0.518817 0.854886i \(-0.673627\pi\)
−0.518817 + 0.854886i \(0.673627\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 11.0748 1.40650
\(63\) 0 0
\(64\) 7.36842 0.921053
\(65\) −12.0990 −1.50070
\(66\) 0 0
\(67\) 8.76663 1.07101 0.535507 0.844531i \(-0.320121\pi\)
0.535507 + 0.844531i \(0.320121\pi\)
\(68\) 0.886447 0.107497
\(69\) 0 0
\(70\) −3.50761 −0.419240
\(71\) 11.4432 1.35806 0.679030 0.734110i \(-0.262400\pi\)
0.679030 + 0.734110i \(0.262400\pi\)
\(72\) 0 0
\(73\) −13.0450 −1.52680 −0.763402 0.645924i \(-0.776472\pi\)
−0.763402 + 0.645924i \(0.776472\pi\)
\(74\) −6.21400 −0.722363
\(75\) 0 0
\(76\) −0.740987 −0.0849970
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 11.4432 1.28746 0.643732 0.765251i \(-0.277385\pi\)
0.643732 + 0.765251i \(0.277385\pi\)
\(80\) 10.2140 1.14196
\(81\) 0 0
\(82\) −0.946021 −0.104471
\(83\) −13.1648 −1.44503 −0.722514 0.691356i \(-0.757014\pi\)
−0.722514 + 0.691356i \(0.757014\pi\)
\(84\) 0 0
\(85\) −15.2728 −1.65657
\(86\) 15.3836 1.65886
\(87\) 0 0
\(88\) 2.72161 0.290125
\(89\) −11.8504 −1.25614 −0.628070 0.778157i \(-0.716155\pi\)
−0.628070 + 0.778157i \(0.716155\pi\)
\(90\) 0 0
\(91\) −5.04502 −0.528861
\(92\) −0.685559 −0.0714744
\(93\) 0 0
\(94\) 0.770774 0.0794993
\(95\) 12.7666 1.30983
\(96\) 0 0
\(97\) −1.87122 −0.189993 −0.0949967 0.995478i \(-0.530284\pi\)
−0.0949967 + 0.995478i \(0.530284\pi\)
\(98\) −1.46260 −0.147745
\(99\) 0 0
\(100\) 0.104590 0.0104590
\(101\) −4.51803 −0.449560 −0.224780 0.974409i \(-0.572166\pi\)
−0.224780 + 0.974409i \(0.572166\pi\)
\(102\) 0 0
\(103\) −10.6468 −1.04906 −0.524531 0.851392i \(-0.675759\pi\)
−0.524531 + 0.851392i \(0.675759\pi\)
\(104\) 13.7306 1.34639
\(105\) 0 0
\(106\) 5.44322 0.528693
\(107\) −15.9702 −1.54390 −0.771949 0.635684i \(-0.780718\pi\)
−0.771949 + 0.635684i \(0.780718\pi\)
\(108\) 0 0
\(109\) 12.7756 1.22368 0.611840 0.790982i \(-0.290430\pi\)
0.611840 + 0.790982i \(0.290430\pi\)
\(110\) 3.50761 0.334438
\(111\) 0 0
\(112\) 4.25901 0.402439
\(113\) −18.7368 −1.76261 −0.881307 0.472544i \(-0.843336\pi\)
−0.881307 + 0.472544i \(0.843336\pi\)
\(114\) 0 0
\(115\) 11.8116 1.10144
\(116\) −0.702237 −0.0652010
\(117\) 0 0
\(118\) 11.6572 1.07313
\(119\) −6.36842 −0.583792
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.92520 0.264835
\(123\) 0 0
\(124\) −1.05398 −0.0946501
\(125\) 10.1890 0.911334
\(126\) 0 0
\(127\) −2.27839 −0.202174 −0.101087 0.994878i \(-0.532232\pi\)
−0.101087 + 0.994878i \(0.532232\pi\)
\(128\) −12.3490 −1.09151
\(129\) 0 0
\(130\) 17.6960 1.55204
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 5.32340 0.461598
\(134\) −12.8221 −1.10766
\(135\) 0 0
\(136\) 17.3324 1.48624
\(137\) 4.77559 0.408006 0.204003 0.978970i \(-0.434605\pi\)
0.204003 + 0.978970i \(0.434605\pi\)
\(138\) 0 0
\(139\) −15.4432 −1.30988 −0.654939 0.755682i \(-0.727306\pi\)
−0.654939 + 0.755682i \(0.727306\pi\)
\(140\) 0.333816 0.0282126
\(141\) 0 0
\(142\) −16.7368 −1.40452
\(143\) 5.04502 0.421885
\(144\) 0 0
\(145\) 12.0990 1.00477
\(146\) 19.0796 1.57904
\(147\) 0 0
\(148\) 0.591380 0.0486112
\(149\) 9.84143 0.806241 0.403121 0.915147i \(-0.367925\pi\)
0.403121 + 0.915147i \(0.367925\pi\)
\(150\) 0 0
\(151\) −4.12878 −0.335996 −0.167998 0.985787i \(-0.553730\pi\)
−0.167998 + 0.985787i \(0.553730\pi\)
\(152\) −14.4882 −1.17515
\(153\) 0 0
\(154\) 1.46260 0.117860
\(155\) 18.1592 1.45859
\(156\) 0 0
\(157\) −0.946021 −0.0755007 −0.0377504 0.999287i \(-0.512019\pi\)
−0.0377504 + 0.999287i \(0.512019\pi\)
\(158\) −16.7368 −1.33151
\(159\) 0 0
\(160\) −1.88500 −0.149022
\(161\) 4.92520 0.388160
\(162\) 0 0
\(163\) 8.76663 0.686655 0.343328 0.939216i \(-0.388446\pi\)
0.343328 + 0.939216i \(0.388446\pi\)
\(164\) 0.0900320 0.00703032
\(165\) 0 0
\(166\) 19.2549 1.49447
\(167\) 24.3684 1.88568 0.942842 0.333239i \(-0.108142\pi\)
0.942842 + 0.333239i \(0.108142\pi\)
\(168\) 0 0
\(169\) 12.4522 0.957860
\(170\) 22.3380 1.71324
\(171\) 0 0
\(172\) −1.46405 −0.111633
\(173\) −12.3476 −0.938770 −0.469385 0.882994i \(-0.655524\pi\)
−0.469385 + 0.882994i \(0.655524\pi\)
\(174\) 0 0
\(175\) −0.751399 −0.0568004
\(176\) −4.25901 −0.321035
\(177\) 0 0
\(178\) 17.3324 1.29912
\(179\) 5.59283 0.418028 0.209014 0.977913i \(-0.432975\pi\)
0.209014 + 0.977913i \(0.432975\pi\)
\(180\) 0 0
\(181\) 13.5720 1.00880 0.504400 0.863470i \(-0.331714\pi\)
0.504400 + 0.863470i \(0.331714\pi\)
\(182\) 7.37883 0.546955
\(183\) 0 0
\(184\) −13.4045 −0.988191
\(185\) −10.1890 −0.749112
\(186\) 0 0
\(187\) 6.36842 0.465705
\(188\) −0.0733538 −0.00534988
\(189\) 0 0
\(190\) −18.6724 −1.35464
\(191\) 9.42240 0.681781 0.340890 0.940103i \(-0.389271\pi\)
0.340890 + 0.940103i \(0.389271\pi\)
\(192\) 0 0
\(193\) −10.1288 −0.729086 −0.364543 0.931187i \(-0.618775\pi\)
−0.364543 + 0.931187i \(0.618775\pi\)
\(194\) 2.73684 0.196494
\(195\) 0 0
\(196\) 0.139194 0.00994244
\(197\) 2.25756 0.160845 0.0804224 0.996761i \(-0.474373\pi\)
0.0804224 + 0.996761i \(0.474373\pi\)
\(198\) 0 0
\(199\) 3.07480 0.217967 0.108984 0.994044i \(-0.465240\pi\)
0.108984 + 0.994044i \(0.465240\pi\)
\(200\) 2.04502 0.144604
\(201\) 0 0
\(202\) 6.60806 0.464941
\(203\) 5.04502 0.354091
\(204\) 0 0
\(205\) −1.55118 −0.108339
\(206\) 15.5720 1.08495
\(207\) 0 0
\(208\) −21.4868 −1.48984
\(209\) −5.32340 −0.368228
\(210\) 0 0
\(211\) −14.6468 −1.00833 −0.504164 0.863608i \(-0.668199\pi\)
−0.504164 + 0.863608i \(0.668199\pi\)
\(212\) −0.518027 −0.0355782
\(213\) 0 0
\(214\) 23.3580 1.59672
\(215\) 25.2244 1.72029
\(216\) 0 0
\(217\) 7.57201 0.514021
\(218\) −18.6856 −1.26555
\(219\) 0 0
\(220\) −0.333816 −0.0225059
\(221\) 32.1288 2.16122
\(222\) 0 0
\(223\) −1.90997 −0.127901 −0.0639505 0.997953i \(-0.520370\pi\)
−0.0639505 + 0.997953i \(0.520370\pi\)
\(224\) −0.786003 −0.0525170
\(225\) 0 0
\(226\) 27.4045 1.82292
\(227\) 3.20359 0.212629 0.106315 0.994333i \(-0.466095\pi\)
0.106315 + 0.994333i \(0.466095\pi\)
\(228\) 0 0
\(229\) −18.3088 −1.20988 −0.604941 0.796270i \(-0.706803\pi\)
−0.604941 + 0.796270i \(0.706803\pi\)
\(230\) −17.2757 −1.13913
\(231\) 0 0
\(232\) −13.7306 −0.901456
\(233\) 16.5872 1.08667 0.543333 0.839517i \(-0.317162\pi\)
0.543333 + 0.839517i \(0.317162\pi\)
\(234\) 0 0
\(235\) 1.26383 0.0824432
\(236\) −1.10941 −0.0722162
\(237\) 0 0
\(238\) 9.31444 0.603766
\(239\) −2.91623 −0.188635 −0.0943177 0.995542i \(-0.530067\pi\)
−0.0943177 + 0.995542i \(0.530067\pi\)
\(240\) 0 0
\(241\) −6.09899 −0.392871 −0.196435 0.980517i \(-0.562937\pi\)
−0.196435 + 0.980517i \(0.562937\pi\)
\(242\) −1.46260 −0.0940194
\(243\) 0 0
\(244\) −0.278388 −0.0178220
\(245\) −2.39821 −0.153216
\(246\) 0 0
\(247\) −26.8567 −1.70885
\(248\) −20.6081 −1.30861
\(249\) 0 0
\(250\) −14.9025 −0.942514
\(251\) −1.62262 −0.102419 −0.0512093 0.998688i \(-0.516308\pi\)
−0.0512093 + 0.998688i \(0.516308\pi\)
\(252\) 0 0
\(253\) −4.92520 −0.309644
\(254\) 3.33237 0.209091
\(255\) 0 0
\(256\) 3.32485 0.207803
\(257\) 6.89541 0.430124 0.215062 0.976600i \(-0.431005\pi\)
0.215062 + 0.976600i \(0.431005\pi\)
\(258\) 0 0
\(259\) −4.24860 −0.263995
\(260\) −1.68411 −0.104444
\(261\) 0 0
\(262\) 5.85039 0.361439
\(263\) −5.08377 −0.313478 −0.156739 0.987640i \(-0.550098\pi\)
−0.156739 + 0.987640i \(0.550098\pi\)
\(264\) 0 0
\(265\) 8.92520 0.548270
\(266\) −7.78600 −0.477390
\(267\) 0 0
\(268\) 1.22026 0.0745394
\(269\) 0.886447 0.0540476 0.0270238 0.999635i \(-0.491397\pi\)
0.0270238 + 0.999635i \(0.491397\pi\)
\(270\) 0 0
\(271\) −25.3234 −1.53829 −0.769144 0.639076i \(-0.779317\pi\)
−0.769144 + 0.639076i \(0.779317\pi\)
\(272\) −27.1232 −1.64458
\(273\) 0 0
\(274\) −6.98477 −0.421965
\(275\) 0.751399 0.0453111
\(276\) 0 0
\(277\) −24.8269 −1.49170 −0.745851 0.666113i \(-0.767957\pi\)
−0.745851 + 0.666113i \(0.767957\pi\)
\(278\) 22.5872 1.35469
\(279\) 0 0
\(280\) 6.52699 0.390062
\(281\) −1.90101 −0.113404 −0.0567022 0.998391i \(-0.518059\pi\)
−0.0567022 + 0.998391i \(0.518059\pi\)
\(282\) 0 0
\(283\) −22.3178 −1.32666 −0.663328 0.748329i \(-0.730857\pi\)
−0.663328 + 0.748329i \(0.730857\pi\)
\(284\) 1.59283 0.0945171
\(285\) 0 0
\(286\) −7.37883 −0.436320
\(287\) −0.646809 −0.0381799
\(288\) 0 0
\(289\) 23.5568 1.38569
\(290\) −17.6960 −1.03914
\(291\) 0 0
\(292\) −1.81579 −0.106261
\(293\) −12.0900 −0.706307 −0.353154 0.935565i \(-0.614891\pi\)
−0.353154 + 0.935565i \(0.614891\pi\)
\(294\) 0 0
\(295\) 19.1142 1.11287
\(296\) 11.5630 0.672088
\(297\) 0 0
\(298\) −14.3941 −0.833826
\(299\) −24.8477 −1.43698
\(300\) 0 0
\(301\) 10.5180 0.606249
\(302\) 6.03875 0.347491
\(303\) 0 0
\(304\) 22.6724 1.30035
\(305\) 4.79641 0.274642
\(306\) 0 0
\(307\) −13.5928 −0.775784 −0.387892 0.921705i \(-0.626797\pi\)
−0.387892 + 0.921705i \(0.626797\pi\)
\(308\) −0.139194 −0.00793132
\(309\) 0 0
\(310\) −26.5597 −1.50849
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 14.9252 0.843622 0.421811 0.906684i \(-0.361395\pi\)
0.421811 + 0.906684i \(0.361395\pi\)
\(314\) 1.38365 0.0780838
\(315\) 0 0
\(316\) 1.59283 0.0896037
\(317\) −3.97918 −0.223493 −0.111746 0.993737i \(-0.535644\pi\)
−0.111746 + 0.993737i \(0.535644\pi\)
\(318\) 0 0
\(319\) −5.04502 −0.282467
\(320\) −17.6710 −0.987839
\(321\) 0 0
\(322\) −7.20359 −0.401440
\(323\) −33.9017 −1.88634
\(324\) 0 0
\(325\) 3.79082 0.210277
\(326\) −12.8221 −0.710148
\(327\) 0 0
\(328\) 1.76036 0.0971997
\(329\) 0.526989 0.0290539
\(330\) 0 0
\(331\) 23.4432 1.28856 0.644278 0.764791i \(-0.277158\pi\)
0.644278 + 0.764791i \(0.277158\pi\)
\(332\) −1.83247 −0.100570
\(333\) 0 0
\(334\) −35.6412 −1.95020
\(335\) −21.0242 −1.14867
\(336\) 0 0
\(337\) 11.1648 0.608187 0.304094 0.952642i \(-0.401646\pi\)
0.304094 + 0.952642i \(0.401646\pi\)
\(338\) −18.2125 −0.990632
\(339\) 0 0
\(340\) −2.12588 −0.115292
\(341\) −7.57201 −0.410047
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −28.6260 −1.54341
\(345\) 0 0
\(346\) 18.0596 0.970889
\(347\) 22.5872 1.21255 0.606273 0.795256i \(-0.292664\pi\)
0.606273 + 0.795256i \(0.292664\pi\)
\(348\) 0 0
\(349\) 27.9315 1.49514 0.747568 0.664185i \(-0.231221\pi\)
0.747568 + 0.664185i \(0.231221\pi\)
\(350\) 1.09899 0.0587437
\(351\) 0 0
\(352\) 0.786003 0.0418941
\(353\) 16.5478 0.880751 0.440376 0.897814i \(-0.354845\pi\)
0.440376 + 0.897814i \(0.354845\pi\)
\(354\) 0 0
\(355\) −27.4432 −1.45654
\(356\) −1.64951 −0.0874236
\(357\) 0 0
\(358\) −8.18006 −0.432330
\(359\) −22.0305 −1.16272 −0.581362 0.813645i \(-0.697480\pi\)
−0.581362 + 0.813645i \(0.697480\pi\)
\(360\) 0 0
\(361\) 9.33863 0.491507
\(362\) −19.8504 −1.04331
\(363\) 0 0
\(364\) −0.702237 −0.0368072
\(365\) 31.2847 1.63751
\(366\) 0 0
\(367\) −19.3836 −1.01182 −0.505909 0.862587i \(-0.668843\pi\)
−0.505909 + 0.862587i \(0.668843\pi\)
\(368\) 20.9765 1.09347
\(369\) 0 0
\(370\) 14.9025 0.774742
\(371\) 3.72161 0.193216
\(372\) 0 0
\(373\) 29.2549 1.51476 0.757380 0.652975i \(-0.226479\pi\)
0.757380 + 0.652975i \(0.226479\pi\)
\(374\) −9.31444 −0.481638
\(375\) 0 0
\(376\) −1.43426 −0.0739663
\(377\) −25.4522 −1.31085
\(378\) 0 0
\(379\) 12.5270 0.643468 0.321734 0.946830i \(-0.395734\pi\)
0.321734 + 0.946830i \(0.395734\pi\)
\(380\) 1.77704 0.0911602
\(381\) 0 0
\(382\) −13.7812 −0.705107
\(383\) 17.5928 0.898952 0.449476 0.893293i \(-0.351611\pi\)
0.449476 + 0.893293i \(0.351611\pi\)
\(384\) 0 0
\(385\) 2.39821 0.122224
\(386\) 14.8143 0.754030
\(387\) 0 0
\(388\) −0.260463 −0.0132230
\(389\) −20.0900 −1.01861 −0.509303 0.860588i \(-0.670097\pi\)
−0.509303 + 0.860588i \(0.670097\pi\)
\(390\) 0 0
\(391\) −31.3657 −1.58623
\(392\) 2.72161 0.137462
\(393\) 0 0
\(394\) −3.30191 −0.166348
\(395\) −27.4432 −1.38082
\(396\) 0 0
\(397\) −35.1053 −1.76188 −0.880941 0.473226i \(-0.843090\pi\)
−0.880941 + 0.473226i \(0.843090\pi\)
\(398\) −4.49720 −0.225424
\(399\) 0 0
\(400\) −3.20022 −0.160011
\(401\) −9.57201 −0.478003 −0.239002 0.971019i \(-0.576820\pi\)
−0.239002 + 0.971019i \(0.576820\pi\)
\(402\) 0 0
\(403\) −38.2009 −1.90292
\(404\) −0.628883 −0.0312881
\(405\) 0 0
\(406\) −7.37883 −0.366205
\(407\) 4.24860 0.210595
\(408\) 0 0
\(409\) 38.1801 1.88788 0.943941 0.330113i \(-0.107087\pi\)
0.943941 + 0.330113i \(0.107087\pi\)
\(410\) 2.26875 0.112046
\(411\) 0 0
\(412\) −1.48197 −0.0730116
\(413\) 7.97021 0.392189
\(414\) 0 0
\(415\) 31.5720 1.54981
\(416\) 3.96540 0.194420
\(417\) 0 0
\(418\) 7.78600 0.380826
\(419\) −7.17380 −0.350463 −0.175231 0.984527i \(-0.556067\pi\)
−0.175231 + 0.984527i \(0.556067\pi\)
\(420\) 0 0
\(421\) 15.1530 0.738511 0.369255 0.929328i \(-0.379613\pi\)
0.369255 + 0.929328i \(0.379613\pi\)
\(422\) 21.4224 1.04283
\(423\) 0 0
\(424\) −10.1288 −0.491897
\(425\) 4.78522 0.232117
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) −2.22296 −0.107451
\(429\) 0 0
\(430\) −36.8932 −1.77915
\(431\) −5.56304 −0.267962 −0.133981 0.990984i \(-0.542776\pi\)
−0.133981 + 0.990984i \(0.542776\pi\)
\(432\) 0 0
\(433\) −25.6412 −1.23224 −0.616119 0.787653i \(-0.711296\pi\)
−0.616119 + 0.787653i \(0.711296\pi\)
\(434\) −11.0748 −0.531608
\(435\) 0 0
\(436\) 1.77829 0.0851645
\(437\) 26.2188 1.25422
\(438\) 0 0
\(439\) 23.6710 1.12976 0.564878 0.825175i \(-0.308923\pi\)
0.564878 + 0.825175i \(0.308923\pi\)
\(440\) −6.52699 −0.311162
\(441\) 0 0
\(442\) −46.9915 −2.23516
\(443\) 18.0305 0.856653 0.428326 0.903624i \(-0.359103\pi\)
0.428326 + 0.903624i \(0.359103\pi\)
\(444\) 0 0
\(445\) 28.4197 1.34722
\(446\) 2.79352 0.132277
\(447\) 0 0
\(448\) −7.36842 −0.348125
\(449\) 34.9765 1.65064 0.825321 0.564664i \(-0.190994\pi\)
0.825321 + 0.564664i \(0.190994\pi\)
\(450\) 0 0
\(451\) 0.646809 0.0304570
\(452\) −2.60806 −0.122673
\(453\) 0 0
\(454\) −4.68556 −0.219904
\(455\) 12.0990 0.567210
\(456\) 0 0
\(457\) 4.53595 0.212183 0.106091 0.994356i \(-0.466166\pi\)
0.106091 + 0.994356i \(0.466166\pi\)
\(458\) 26.7785 1.25128
\(459\) 0 0
\(460\) 1.64411 0.0766571
\(461\) 2.79641 0.130242 0.0651210 0.997877i \(-0.479257\pi\)
0.0651210 + 0.997877i \(0.479257\pi\)
\(462\) 0 0
\(463\) 38.3595 1.78272 0.891358 0.453301i \(-0.149754\pi\)
0.891358 + 0.453301i \(0.149754\pi\)
\(464\) 21.4868 0.997499
\(465\) 0 0
\(466\) −24.2605 −1.12384
\(467\) 20.4674 0.947119 0.473560 0.880762i \(-0.342969\pi\)
0.473560 + 0.880762i \(0.342969\pi\)
\(468\) 0 0
\(469\) −8.76663 −0.404805
\(470\) −1.84848 −0.0852638
\(471\) 0 0
\(472\) −21.6918 −0.998447
\(473\) −10.5180 −0.483619
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) −0.886447 −0.0406302
\(477\) 0 0
\(478\) 4.26528 0.195089
\(479\) 11.6137 0.530641 0.265321 0.964160i \(-0.414522\pi\)
0.265321 + 0.964160i \(0.414522\pi\)
\(480\) 0 0
\(481\) 21.4343 0.977318
\(482\) 8.92038 0.406312
\(483\) 0 0
\(484\) 0.139194 0.00632701
\(485\) 4.48757 0.203770
\(486\) 0 0
\(487\) 32.4793 1.47178 0.735888 0.677103i \(-0.236765\pi\)
0.735888 + 0.677103i \(0.236765\pi\)
\(488\) −5.44322 −0.246403
\(489\) 0 0
\(490\) 3.50761 0.158458
\(491\) −26.6766 −1.20390 −0.601949 0.798535i \(-0.705609\pi\)
−0.601949 + 0.798535i \(0.705609\pi\)
\(492\) 0 0
\(493\) −32.1288 −1.44701
\(494\) 39.2805 1.76731
\(495\) 0 0
\(496\) 32.2493 1.44804
\(497\) −11.4432 −0.513299
\(498\) 0 0
\(499\) −41.2459 −1.84642 −0.923210 0.384296i \(-0.874444\pi\)
−0.923210 + 0.384296i \(0.874444\pi\)
\(500\) 1.41825 0.0634262
\(501\) 0 0
\(502\) 2.37324 0.105923
\(503\) −30.5180 −1.36073 −0.680366 0.732873i \(-0.738179\pi\)
−0.680366 + 0.732873i \(0.738179\pi\)
\(504\) 0 0
\(505\) 10.8352 0.482159
\(506\) 7.20359 0.320238
\(507\) 0 0
\(508\) −0.317138 −0.0140707
\(509\) 18.9944 0.841912 0.420956 0.907081i \(-0.361695\pi\)
0.420956 + 0.907081i \(0.361695\pi\)
\(510\) 0 0
\(511\) 13.0450 0.577078
\(512\) 19.8352 0.876599
\(513\) 0 0
\(514\) −10.0852 −0.444840
\(515\) 25.5333 1.12513
\(516\) 0 0
\(517\) −0.526989 −0.0231770
\(518\) 6.21400 0.273027
\(519\) 0 0
\(520\) −32.9288 −1.44402
\(521\) −25.2430 −1.10592 −0.552958 0.833209i \(-0.686501\pi\)
−0.552958 + 0.833209i \(0.686501\pi\)
\(522\) 0 0
\(523\) −2.93416 −0.128302 −0.0641509 0.997940i \(-0.520434\pi\)
−0.0641509 + 0.997940i \(0.520434\pi\)
\(524\) −0.556777 −0.0243229
\(525\) 0 0
\(526\) 7.43551 0.324204
\(527\) −48.2217 −2.10057
\(528\) 0 0
\(529\) 1.25756 0.0546767
\(530\) −13.0540 −0.567029
\(531\) 0 0
\(532\) 0.740987 0.0321258
\(533\) 3.26316 0.141343
\(534\) 0 0
\(535\) 38.2999 1.65585
\(536\) 23.8594 1.03057
\(537\) 0 0
\(538\) −1.29652 −0.0558968
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 8.90437 0.382829 0.191414 0.981509i \(-0.438693\pi\)
0.191414 + 0.981509i \(0.438693\pi\)
\(542\) 37.0380 1.59092
\(543\) 0 0
\(544\) 5.00560 0.214613
\(545\) −30.6385 −1.31241
\(546\) 0 0
\(547\) −29.4737 −1.26020 −0.630102 0.776513i \(-0.716987\pi\)
−0.630102 + 0.776513i \(0.716987\pi\)
\(548\) 0.664734 0.0283960
\(549\) 0 0
\(550\) −1.09899 −0.0468613
\(551\) 26.8567 1.14413
\(552\) 0 0
\(553\) −11.4432 −0.486615
\(554\) 36.3117 1.54274
\(555\) 0 0
\(556\) −2.14961 −0.0911636
\(557\) −14.8954 −0.631139 −0.315569 0.948903i \(-0.602196\pi\)
−0.315569 + 0.948903i \(0.602196\pi\)
\(558\) 0 0
\(559\) −53.0636 −2.24435
\(560\) −10.2140 −0.431620
\(561\) 0 0
\(562\) 2.78041 0.117284
\(563\) 7.81164 0.329222 0.164611 0.986359i \(-0.447363\pi\)
0.164611 + 0.986359i \(0.447363\pi\)
\(564\) 0 0
\(565\) 44.9348 1.89042
\(566\) 32.6420 1.37205
\(567\) 0 0
\(568\) 31.1440 1.30677
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) 25.5512 1.06928 0.534642 0.845079i \(-0.320447\pi\)
0.534642 + 0.845079i \(0.320447\pi\)
\(572\) 0.702237 0.0293620
\(573\) 0 0
\(574\) 0.946021 0.0394862
\(575\) −3.70079 −0.154334
\(576\) 0 0
\(577\) −29.5124 −1.22862 −0.614309 0.789065i \(-0.710565\pi\)
−0.614309 + 0.789065i \(0.710565\pi\)
\(578\) −34.4541 −1.43310
\(579\) 0 0
\(580\) 1.68411 0.0699288
\(581\) 13.1648 0.546169
\(582\) 0 0
\(583\) −3.72161 −0.154133
\(584\) −35.5035 −1.46914
\(585\) 0 0
\(586\) 17.6829 0.730472
\(587\) −31.3955 −1.29583 −0.647916 0.761712i \(-0.724359\pi\)
−0.647916 + 0.761712i \(0.724359\pi\)
\(588\) 0 0
\(589\) 40.3088 1.66090
\(590\) −27.9564 −1.15095
\(591\) 0 0
\(592\) −18.0948 −0.743694
\(593\) 7.90997 0.324823 0.162412 0.986723i \(-0.448073\pi\)
0.162412 + 0.986723i \(0.448073\pi\)
\(594\) 0 0
\(595\) 15.2728 0.626123
\(596\) 1.36987 0.0561120
\(597\) 0 0
\(598\) 36.3422 1.48614
\(599\) −27.4432 −1.12130 −0.560650 0.828053i \(-0.689449\pi\)
−0.560650 + 0.828053i \(0.689449\pi\)
\(600\) 0 0
\(601\) 31.9910 1.30494 0.652471 0.757814i \(-0.273732\pi\)
0.652471 + 0.757814i \(0.273732\pi\)
\(602\) −15.3836 −0.626991
\(603\) 0 0
\(604\) −0.574702 −0.0233843
\(605\) −2.39821 −0.0975010
\(606\) 0 0
\(607\) 7.41344 0.300902 0.150451 0.988617i \(-0.451927\pi\)
0.150451 + 0.988617i \(0.451927\pi\)
\(608\) −4.18421 −0.169692
\(609\) 0 0
\(610\) −7.01523 −0.284038
\(611\) −2.65867 −0.107558
\(612\) 0 0
\(613\) −33.9917 −1.37291 −0.686456 0.727171i \(-0.740835\pi\)
−0.686456 + 0.727171i \(0.740835\pi\)
\(614\) 19.8809 0.802326
\(615\) 0 0
\(616\) −2.72161 −0.109657
\(617\) 44.0305 1.77260 0.886300 0.463112i \(-0.153267\pi\)
0.886300 + 0.463112i \(0.153267\pi\)
\(618\) 0 0
\(619\) 40.0096 1.60812 0.804061 0.594546i \(-0.202668\pi\)
0.804061 + 0.594546i \(0.202668\pi\)
\(620\) 2.52766 0.101513
\(621\) 0 0
\(622\) 11.7008 0.469159
\(623\) 11.8504 0.474776
\(624\) 0 0
\(625\) −28.1924 −1.12770
\(626\) −21.8296 −0.872485
\(627\) 0 0
\(628\) −0.131681 −0.00525463
\(629\) 27.0569 1.07883
\(630\) 0 0
\(631\) 28.5568 1.13683 0.568414 0.822743i \(-0.307557\pi\)
0.568414 + 0.822743i \(0.307557\pi\)
\(632\) 31.1440 1.23884
\(633\) 0 0
\(634\) 5.81994 0.231139
\(635\) 5.46405 0.216834
\(636\) 0 0
\(637\) 5.04502 0.199891
\(638\) 7.37883 0.292131
\(639\) 0 0
\(640\) 29.6156 1.17066
\(641\) 31.1053 1.22858 0.614292 0.789079i \(-0.289442\pi\)
0.614292 + 0.789079i \(0.289442\pi\)
\(642\) 0 0
\(643\) −5.48197 −0.216188 −0.108094 0.994141i \(-0.534475\pi\)
−0.108094 + 0.994141i \(0.534475\pi\)
\(644\) 0.685559 0.0270148
\(645\) 0 0
\(646\) 49.5845 1.95088
\(647\) 9.26383 0.364199 0.182099 0.983280i \(-0.441711\pi\)
0.182099 + 0.983280i \(0.441711\pi\)
\(648\) 0 0
\(649\) −7.97021 −0.312858
\(650\) −5.54445 −0.217471
\(651\) 0 0
\(652\) 1.22026 0.0477892
\(653\) 29.9821 1.17329 0.586645 0.809844i \(-0.300449\pi\)
0.586645 + 0.809844i \(0.300449\pi\)
\(654\) 0 0
\(655\) 9.59283 0.374823
\(656\) −2.75477 −0.107556
\(657\) 0 0
\(658\) −0.770774 −0.0300479
\(659\) −23.9702 −0.933747 −0.466873 0.884324i \(-0.654620\pi\)
−0.466873 + 0.884324i \(0.654620\pi\)
\(660\) 0 0
\(661\) −40.4585 −1.57365 −0.786826 0.617175i \(-0.788277\pi\)
−0.786826 + 0.617175i \(0.788277\pi\)
\(662\) −34.2880 −1.33264
\(663\) 0 0
\(664\) −35.8296 −1.39046
\(665\) −12.7666 −0.495069
\(666\) 0 0
\(667\) 24.8477 0.962107
\(668\) 3.39194 0.131238
\(669\) 0 0
\(670\) 30.7499 1.18797
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 21.8712 0.843073 0.421537 0.906811i \(-0.361491\pi\)
0.421537 + 0.906811i \(0.361491\pi\)
\(674\) −16.3297 −0.628995
\(675\) 0 0
\(676\) 1.73327 0.0666643
\(677\) 1.26316 0.0485472 0.0242736 0.999705i \(-0.492273\pi\)
0.0242736 + 0.999705i \(0.492273\pi\)
\(678\) 0 0
\(679\) 1.87122 0.0718108
\(680\) −41.5666 −1.59401
\(681\) 0 0
\(682\) 11.0748 0.424076
\(683\) −37.6441 −1.44041 −0.720206 0.693760i \(-0.755953\pi\)
−0.720206 + 0.693760i \(0.755953\pi\)
\(684\) 0 0
\(685\) −11.4529 −0.437591
\(686\) 1.46260 0.0558423
\(687\) 0 0
\(688\) 44.7964 1.70785
\(689\) −18.7756 −0.715293
\(690\) 0 0
\(691\) 14.3892 0.547393 0.273696 0.961816i \(-0.411754\pi\)
0.273696 + 0.961816i \(0.411754\pi\)
\(692\) −1.71871 −0.0653357
\(693\) 0 0
\(694\) −33.0361 −1.25403
\(695\) 37.0361 1.40486
\(696\) 0 0
\(697\) 4.11915 0.156024
\(698\) −40.8525 −1.54629
\(699\) 0 0
\(700\) −0.104590 −0.00395314
\(701\) 39.2936 1.48410 0.742050 0.670345i \(-0.233854\pi\)
0.742050 + 0.670345i \(0.233854\pi\)
\(702\) 0 0
\(703\) −22.6170 −0.853017
\(704\) 7.36842 0.277708
\(705\) 0 0
\(706\) −24.2028 −0.910885
\(707\) 4.51803 0.169918
\(708\) 0 0
\(709\) −49.2430 −1.84936 −0.924680 0.380745i \(-0.875667\pi\)
−0.924680 + 0.380745i \(0.875667\pi\)
\(710\) 40.1384 1.50637
\(711\) 0 0
\(712\) −32.2522 −1.20870
\(713\) 37.2936 1.39666
\(714\) 0 0
\(715\) −12.0990 −0.452477
\(716\) 0.778489 0.0290935
\(717\) 0 0
\(718\) 32.2217 1.20250
\(719\) 7.41344 0.276475 0.138237 0.990399i \(-0.455856\pi\)
0.138237 + 0.990399i \(0.455856\pi\)
\(720\) 0 0
\(721\) 10.6468 0.396508
\(722\) −13.6587 −0.508323
\(723\) 0 0
\(724\) 1.88914 0.0702095
\(725\) −3.79082 −0.140787
\(726\) 0 0
\(727\) 18.9557 0.703026 0.351513 0.936183i \(-0.385667\pi\)
0.351513 + 0.936183i \(0.385667\pi\)
\(728\) −13.7306 −0.508889
\(729\) 0 0
\(730\) −45.7569 −1.69354
\(731\) −66.9832 −2.47746
\(732\) 0 0
\(733\) −3.59283 −0.132704 −0.0663521 0.997796i \(-0.521136\pi\)
−0.0663521 + 0.997796i \(0.521136\pi\)
\(734\) 28.3505 1.04644
\(735\) 0 0
\(736\) −3.87122 −0.142695
\(737\) 8.76663 0.322923
\(738\) 0 0
\(739\) −26.7756 −0.984956 −0.492478 0.870325i \(-0.663909\pi\)
−0.492478 + 0.870325i \(0.663909\pi\)
\(740\) −1.41825 −0.0521360
\(741\) 0 0
\(742\) −5.44322 −0.199827
\(743\) 33.8027 1.24010 0.620050 0.784562i \(-0.287112\pi\)
0.620050 + 0.784562i \(0.287112\pi\)
\(744\) 0 0
\(745\) −23.6018 −0.864703
\(746\) −42.7881 −1.56658
\(747\) 0 0
\(748\) 0.886447 0.0324117
\(749\) 15.9702 0.583539
\(750\) 0 0
\(751\) 35.3955 1.29160 0.645800 0.763506i \(-0.276524\pi\)
0.645800 + 0.763506i \(0.276524\pi\)
\(752\) 2.24445 0.0818468
\(753\) 0 0
\(754\) 37.2263 1.35570
\(755\) 9.90168 0.360359
\(756\) 0 0
\(757\) −29.3442 −1.06653 −0.533267 0.845947i \(-0.679036\pi\)
−0.533267 + 0.845947i \(0.679036\pi\)
\(758\) −18.3220 −0.665483
\(759\) 0 0
\(760\) 34.7458 1.26036
\(761\) −12.9044 −0.467783 −0.233892 0.972263i \(-0.575146\pi\)
−0.233892 + 0.972263i \(0.575146\pi\)
\(762\) 0 0
\(763\) −12.7756 −0.462507
\(764\) 1.31154 0.0474500
\(765\) 0 0
\(766\) −25.7312 −0.929708
\(767\) −40.2099 −1.45189
\(768\) 0 0
\(769\) 9.78186 0.352743 0.176371 0.984324i \(-0.443564\pi\)
0.176371 + 0.984324i \(0.443564\pi\)
\(770\) −3.50761 −0.126406
\(771\) 0 0
\(772\) −1.40987 −0.0507422
\(773\) −29.7223 −1.06904 −0.534518 0.845157i \(-0.679507\pi\)
−0.534518 + 0.845157i \(0.679507\pi\)
\(774\) 0 0
\(775\) −5.68960 −0.204376
\(776\) −5.09273 −0.182818
\(777\) 0 0
\(778\) 29.3836 1.05345
\(779\) −3.44322 −0.123366
\(780\) 0 0
\(781\) 11.4432 0.409471
\(782\) 45.8755 1.64050
\(783\) 0 0
\(784\) −4.25901 −0.152108
\(785\) 2.26875 0.0809753
\(786\) 0 0
\(787\) 17.0242 0.606847 0.303423 0.952856i \(-0.401870\pi\)
0.303423 + 0.952856i \(0.401870\pi\)
\(788\) 0.314240 0.0111943
\(789\) 0 0
\(790\) 40.1384 1.42806
\(791\) 18.7368 0.666205
\(792\) 0 0
\(793\) −10.0900 −0.358308
\(794\) 51.3449 1.82216
\(795\) 0 0
\(796\) 0.427995 0.0151699
\(797\) 36.4287 1.29037 0.645185 0.764027i \(-0.276780\pi\)
0.645185 + 0.764027i \(0.276780\pi\)
\(798\) 0 0
\(799\) −3.35609 −0.118730
\(800\) 0.590602 0.0208809
\(801\) 0 0
\(802\) 14.0000 0.494357
\(803\) −13.0450 −0.460349
\(804\) 0 0
\(805\) −11.8116 −0.416306
\(806\) 55.8726 1.96803
\(807\) 0 0
\(808\) −12.2963 −0.432583
\(809\) −44.4882 −1.56412 −0.782062 0.623201i \(-0.785832\pi\)
−0.782062 + 0.623201i \(0.785832\pi\)
\(810\) 0 0
\(811\) 7.65307 0.268736 0.134368 0.990932i \(-0.457100\pi\)
0.134368 + 0.990932i \(0.457100\pi\)
\(812\) 0.702237 0.0246437
\(813\) 0 0
\(814\) −6.21400 −0.217800
\(815\) −21.0242 −0.736445
\(816\) 0 0
\(817\) 55.9917 1.95890
\(818\) −55.8421 −1.95247
\(819\) 0 0
\(820\) −0.215915 −0.00754009
\(821\) 44.3691 1.54849 0.774246 0.632885i \(-0.218129\pi\)
0.774246 + 0.632885i \(0.218129\pi\)
\(822\) 0 0
\(823\) 6.61702 0.230655 0.115327 0.993328i \(-0.463208\pi\)
0.115327 + 0.993328i \(0.463208\pi\)
\(824\) −28.9765 −1.00944
\(825\) 0 0
\(826\) −11.6572 −0.405607
\(827\) −39.7126 −1.38094 −0.690472 0.723359i \(-0.742597\pi\)
−0.690472 + 0.723359i \(0.742597\pi\)
\(828\) 0 0
\(829\) −3.90997 −0.135799 −0.0678994 0.997692i \(-0.521630\pi\)
−0.0678994 + 0.997692i \(0.521630\pi\)
\(830\) −46.1772 −1.60283
\(831\) 0 0
\(832\) 37.1738 1.28877
\(833\) 6.36842 0.220653
\(834\) 0 0
\(835\) −58.4405 −2.02242
\(836\) −0.740987 −0.0256276
\(837\) 0 0
\(838\) 10.4924 0.362453
\(839\) 9.58097 0.330772 0.165386 0.986229i \(-0.447113\pi\)
0.165386 + 0.986229i \(0.447113\pi\)
\(840\) 0 0
\(841\) −3.54781 −0.122338
\(842\) −22.1627 −0.763778
\(843\) 0 0
\(844\) −2.03875 −0.0701767
\(845\) −29.8629 −1.02732
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 15.8504 0.544305
\(849\) 0 0
\(850\) −6.99886 −0.240059
\(851\) −20.9252 −0.717307
\(852\) 0 0
\(853\) −14.5568 −0.498415 −0.249207 0.968450i \(-0.580170\pi\)
−0.249207 + 0.968450i \(0.580170\pi\)
\(854\) −2.92520 −0.100098
\(855\) 0 0
\(856\) −43.4647 −1.48559
\(857\) −10.4793 −0.357965 −0.178983 0.983852i \(-0.557281\pi\)
−0.178983 + 0.983852i \(0.557281\pi\)
\(858\) 0 0
\(859\) 2.88645 0.0984843 0.0492421 0.998787i \(-0.484319\pi\)
0.0492421 + 0.998787i \(0.484319\pi\)
\(860\) 3.51109 0.119727
\(861\) 0 0
\(862\) 8.13650 0.277130
\(863\) 20.5485 0.699479 0.349739 0.936847i \(-0.386270\pi\)
0.349739 + 0.936847i \(0.386270\pi\)
\(864\) 0 0
\(865\) 29.6121 1.00684
\(866\) 37.5028 1.27440
\(867\) 0 0
\(868\) 1.05398 0.0357744
\(869\) 11.4432 0.388185
\(870\) 0 0
\(871\) 44.2278 1.49860
\(872\) 34.7702 1.17747
\(873\) 0 0
\(874\) −38.3476 −1.29713
\(875\) −10.1890 −0.344452
\(876\) 0 0
\(877\) 59.1149 1.99617 0.998084 0.0618724i \(-0.0197072\pi\)
0.998084 + 0.0618724i \(0.0197072\pi\)
\(878\) −34.6212 −1.16841
\(879\) 0 0
\(880\) 10.2140 0.344314
\(881\) −30.3982 −1.02414 −0.512071 0.858943i \(-0.671121\pi\)
−0.512071 + 0.858943i \(0.671121\pi\)
\(882\) 0 0
\(883\) −35.6114 −1.19842 −0.599210 0.800592i \(-0.704519\pi\)
−0.599210 + 0.800592i \(0.704519\pi\)
\(884\) 4.47214 0.150414
\(885\) 0 0
\(886\) −26.3713 −0.885962
\(887\) −22.9736 −0.771377 −0.385689 0.922629i \(-0.626036\pi\)
−0.385689 + 0.922629i \(0.626036\pi\)
\(888\) 0 0
\(889\) 2.27839 0.0764147
\(890\) −41.5666 −1.39332
\(891\) 0 0
\(892\) −0.265856 −0.00890153
\(893\) 2.80538 0.0938784
\(894\) 0 0
\(895\) −13.4128 −0.448339
\(896\) 12.3490 0.412553
\(897\) 0 0
\(898\) −51.1565 −1.70712
\(899\) 38.2009 1.27407
\(900\) 0 0
\(901\) −23.7008 −0.789588
\(902\) −0.946021 −0.0314991
\(903\) 0 0
\(904\) −50.9944 −1.69605
\(905\) −32.5485 −1.08195
\(906\) 0 0
\(907\) −57.1745 −1.89845 −0.949224 0.314602i \(-0.898129\pi\)
−0.949224 + 0.314602i \(0.898129\pi\)
\(908\) 0.445920 0.0147984
\(909\) 0 0
\(910\) −17.6960 −0.586616
\(911\) −6.82687 −0.226184 −0.113092 0.993584i \(-0.536076\pi\)
−0.113092 + 0.993584i \(0.536076\pi\)
\(912\) 0 0
\(913\) −13.1648 −0.435692
\(914\) −6.63428 −0.219442
\(915\) 0 0
\(916\) −2.54848 −0.0842043
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) 12.0692 0.398126 0.199063 0.979987i \(-0.436210\pi\)
0.199063 + 0.979987i \(0.436210\pi\)
\(920\) 32.1467 1.05985
\(921\) 0 0
\(922\) −4.09003 −0.134698
\(923\) 57.7312 1.90025
\(924\) 0 0
\(925\) 3.19239 0.104965
\(926\) −56.1045 −1.84371
\(927\) 0 0
\(928\) −3.96540 −0.130171
\(929\) −26.8954 −0.882410 −0.441205 0.897406i \(-0.645449\pi\)
−0.441205 + 0.897406i \(0.645449\pi\)
\(930\) 0 0
\(931\) −5.32340 −0.174468
\(932\) 2.30885 0.0756288
\(933\) 0 0
\(934\) −29.9356 −0.979523
\(935\) −15.2728 −0.499474
\(936\) 0 0
\(937\) −14.9944 −0.489846 −0.244923 0.969543i \(-0.578763\pi\)
−0.244923 + 0.969543i \(0.578763\pi\)
\(938\) 12.8221 0.418655
\(939\) 0 0
\(940\) 0.175918 0.00573780
\(941\) 30.1205 0.981900 0.490950 0.871188i \(-0.336650\pi\)
0.490950 + 0.871188i \(0.336650\pi\)
\(942\) 0 0
\(943\) −3.18566 −0.103739
\(944\) 33.9452 1.10482
\(945\) 0 0
\(946\) 15.3836 0.500166
\(947\) 17.3532 0.563903 0.281951 0.959429i \(-0.409018\pi\)
0.281951 + 0.959429i \(0.409018\pi\)
\(948\) 0 0
\(949\) −65.8123 −2.13636
\(950\) 5.85039 0.189812
\(951\) 0 0
\(952\) −17.3324 −0.561745
\(953\) 2.14064 0.0693422 0.0346711 0.999399i \(-0.488962\pi\)
0.0346711 + 0.999399i \(0.488962\pi\)
\(954\) 0 0
\(955\) −22.5969 −0.731217
\(956\) −0.405923 −0.0131285
\(957\) 0 0
\(958\) −16.9861 −0.548796
\(959\) −4.77559 −0.154212
\(960\) 0 0
\(961\) 26.3353 0.849525
\(962\) −31.3497 −1.01076
\(963\) 0 0
\(964\) −0.848944 −0.0273427
\(965\) 24.2909 0.781952
\(966\) 0 0
\(967\) −1.53326 −0.0493062 −0.0246531 0.999696i \(-0.507848\pi\)
−0.0246531 + 0.999696i \(0.507848\pi\)
\(968\) 2.72161 0.0874759
\(969\) 0 0
\(970\) −6.56351 −0.210742
\(971\) 26.5574 0.852269 0.426135 0.904660i \(-0.359875\pi\)
0.426135 + 0.904660i \(0.359875\pi\)
\(972\) 0 0
\(973\) 15.4432 0.495087
\(974\) −47.5041 −1.52213
\(975\) 0 0
\(976\) 8.51803 0.272655
\(977\) 55.9017 1.78845 0.894227 0.447615i \(-0.147726\pi\)
0.894227 + 0.447615i \(0.147726\pi\)
\(978\) 0 0
\(979\) −11.8504 −0.378740
\(980\) −0.333816 −0.0106634
\(981\) 0 0
\(982\) 39.0171 1.24509
\(983\) 53.0361 1.69159 0.845794 0.533510i \(-0.179127\pi\)
0.845794 + 0.533510i \(0.179127\pi\)
\(984\) 0 0
\(985\) −5.41411 −0.172508
\(986\) 46.9915 1.49651
\(987\) 0 0
\(988\) −3.73829 −0.118931
\(989\) 51.8034 1.64725
\(990\) 0 0
\(991\) 14.7362 0.468110 0.234055 0.972223i \(-0.424800\pi\)
0.234055 + 0.972223i \(0.424800\pi\)
\(992\) −5.95162 −0.188964
\(993\) 0 0
\(994\) 16.7368 0.530860
\(995\) −7.37402 −0.233772
\(996\) 0 0
\(997\) −45.0665 −1.42727 −0.713635 0.700517i \(-0.752953\pi\)
−0.713635 + 0.700517i \(0.752953\pi\)
\(998\) 60.3262 1.90959
\(999\) 0 0
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 693.2.a.l.1.2 3
3.2 odd 2 231.2.a.e.1.2 3
7.6 odd 2 4851.2.a.bi.1.2 3
11.10 odd 2 7623.2.a.cd.1.2 3
12.11 even 2 3696.2.a.bo.1.2 3
15.14 odd 2 5775.2.a.bp.1.2 3
21.20 even 2 1617.2.a.t.1.2 3
33.32 even 2 2541.2.a.bg.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.2 3 3.2 odd 2
693.2.a.l.1.2 3 1.1 even 1 trivial
1617.2.a.t.1.2 3 21.20 even 2
2541.2.a.bg.1.2 3 33.32 even 2
3696.2.a.bo.1.2 3 12.11 even 2
4851.2.a.bi.1.2 3 7.6 odd 2
5775.2.a.bp.1.2 3 15.14 odd 2
7623.2.a.cd.1.2 3 11.10 odd 2