Properties

Label 2541.2.a.bg.1.2
Level $2541$
Weight $2$
Character 2541.1
Self dual yes
Analytic conductor $20.290$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2541.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
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 \(\chi\) \(=\) 2541.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.46260 q^{2} +1.00000 q^{3} +0.139194 q^{4} +2.39821 q^{5} -1.46260 q^{6} +1.00000 q^{7} +2.72161 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.46260 q^{2} +1.00000 q^{3} +0.139194 q^{4} +2.39821 q^{5} -1.46260 q^{6} +1.00000 q^{7} +2.72161 q^{8} +1.00000 q^{9} -3.50761 q^{10} +0.139194 q^{12} -5.04502 q^{13} -1.46260 q^{14} +2.39821 q^{15} -4.25901 q^{16} +6.36842 q^{17} -1.46260 q^{18} +5.32340 q^{19} +0.333816 q^{20} +1.00000 q^{21} +4.92520 q^{23} +2.72161 q^{24} +0.751399 q^{25} +7.37883 q^{26} +1.00000 q^{27} +0.139194 q^{28} -5.04502 q^{29} -3.50761 q^{30} -7.57201 q^{31} +0.786003 q^{32} -9.31444 q^{34} +2.39821 q^{35} +0.139194 q^{36} +4.24860 q^{37} -7.78600 q^{38} -5.04502 q^{39} +6.52699 q^{40} +0.646809 q^{41} -1.46260 q^{42} +10.5180 q^{43} +2.39821 q^{45} -7.20359 q^{46} +0.526989 q^{47} -4.25901 q^{48} +1.00000 q^{49} -1.09899 q^{50} +6.36842 q^{51} -0.702237 q^{52} +3.72161 q^{53} -1.46260 q^{54} +2.72161 q^{56} +5.32340 q^{57} +7.37883 q^{58} +7.97021 q^{59} +0.333816 q^{60} +2.00000 q^{61} +11.0748 q^{62} +1.00000 q^{63} +7.36842 q^{64} -12.0990 q^{65} +8.76663 q^{67} +0.886447 q^{68} +4.92520 q^{69} -3.50761 q^{70} -11.4432 q^{71} +2.72161 q^{72} +13.0450 q^{73} -6.21400 q^{74} +0.751399 q^{75} +0.740987 q^{76} +7.37883 q^{78} -11.4432 q^{79} -10.2140 q^{80} +1.00000 q^{81} -0.946021 q^{82} -13.1648 q^{83} +0.139194 q^{84} +15.2728 q^{85} -15.3836 q^{86} -5.04502 q^{87} +11.8504 q^{89} -3.50761 q^{90} -5.04502 q^{91} +0.685559 q^{92} -7.57201 q^{93} -0.770774 q^{94} +12.7666 q^{95} +0.786003 q^{96} -1.87122 q^{97} -1.46260 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 3 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 3 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 11 q^{10} + 6 q^{12} + 4 q^{13} - 2 q^{14} + 4 q^{15} - 4 q^{16} - 8 q^{17} - 2 q^{18} + 8 q^{19} - 3 q^{20} + 3 q^{21} + 10 q^{23} - 3 q^{24} + 15 q^{25} - q^{26} + 3 q^{27} + 6 q^{28} + 4 q^{29} + 11 q^{30} - 2 q^{31} - 8 q^{32} - 4 q^{34} + 4 q^{35} + 6 q^{36} - 13 q^{38} + 4 q^{39} + 18 q^{40} - 14 q^{41} - 2 q^{42} + 14 q^{43} + 4 q^{45} - 28 q^{46} - 4 q^{48} + 3 q^{49} + 19 q^{50} - 8 q^{51} + 29 q^{52} - 2 q^{54} - 3 q^{56} + 8 q^{57} - q^{58} - 3 q^{60} + 6 q^{61} + 38 q^{62} + 3 q^{63} - 5 q^{64} - 14 q^{65} - 4 q^{67} - 42 q^{68} + 10 q^{69} + 11 q^{70} - 12 q^{71} - 3 q^{72} + 20 q^{73} - 29 q^{74} + 15 q^{75} + 11 q^{76} - q^{78} - 12 q^{79} - 41 q^{80} + 3 q^{81} - 6 q^{82} - 6 q^{83} + 6 q^{84} + 6 q^{85} + 24 q^{86} + 4 q^{87} + 26 q^{89} + 11 q^{90} + 4 q^{91} + 26 q^{92} - 2 q^{93} - 35 q^{94} + 8 q^{95} - 8 q^{96} - 4 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.46260 −1.03421 −0.517107 0.855921i \(-0.672991\pi\)
−0.517107 + 0.855921i \(0.672991\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.139194 0.0695971
\(5\) 2.39821 1.07251 0.536255 0.844056i \(-0.319838\pi\)
0.536255 + 0.844056i \(0.319838\pi\)
\(6\) −1.46260 −0.597103
\(7\) 1.00000 0.377964
\(8\) 2.72161 0.962235
\(9\) 1.00000 0.333333
\(10\) −3.50761 −1.10921
\(11\) 0 0
\(12\) 0.139194 0.0401819
\(13\) −5.04502 −1.39924 −0.699618 0.714517i \(-0.746646\pi\)
−0.699618 + 0.714517i \(0.746646\pi\)
\(14\) −1.46260 −0.390896
\(15\) 2.39821 0.619214
\(16\) −4.25901 −1.06475
\(17\) 6.36842 1.54457 0.772284 0.635277i \(-0.219114\pi\)
0.772284 + 0.635277i \(0.219114\pi\)
\(18\) −1.46260 −0.344738
\(19\) 5.32340 1.22127 0.610636 0.791911i \(-0.290914\pi\)
0.610636 + 0.791911i \(0.290914\pi\)
\(20\) 0.333816 0.0746436
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 4.92520 1.02697 0.513487 0.858097i \(-0.328353\pi\)
0.513487 + 0.858097i \(0.328353\pi\)
\(24\) 2.72161 0.555547
\(25\) 0.751399 0.150280
\(26\) 7.37883 1.44711
\(27\) 1.00000 0.192450
\(28\) 0.139194 0.0263052
\(29\) −5.04502 −0.936836 −0.468418 0.883507i \(-0.655176\pi\)
−0.468418 + 0.883507i \(0.655176\pi\)
\(30\) −3.50761 −0.640400
\(31\) −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
\(36\) 0.139194 0.0231990
\(37\) 4.24860 0.698466 0.349233 0.937036i \(-0.386442\pi\)
0.349233 + 0.937036i \(0.386442\pi\)
\(38\) −7.78600 −1.26306
\(39\) −5.04502 −0.807849
\(40\) 6.52699 1.03201
\(41\) 0.646809 0.101015 0.0505073 0.998724i \(-0.483916\pi\)
0.0505073 + 0.998724i \(0.483916\pi\)
\(42\) −1.46260 −0.225684
\(43\) 10.5180 1.60398 0.801992 0.597335i \(-0.203774\pi\)
0.801992 + 0.597335i \(0.203774\pi\)
\(44\) 0 0
\(45\) 2.39821 0.357504
\(46\) −7.20359 −1.06211
\(47\) 0.526989 0.0768693 0.0384347 0.999261i \(-0.487763\pi\)
0.0384347 + 0.999261i \(0.487763\pi\)
\(48\) −4.25901 −0.614736
\(49\) 1.00000 0.142857
\(50\) −1.09899 −0.155421
\(51\) 6.36842 0.891757
\(52\) −0.702237 −0.0973827
\(53\) 3.72161 0.511203 0.255601 0.966782i \(-0.417727\pi\)
0.255601 + 0.966782i \(0.417727\pi\)
\(54\) −1.46260 −0.199034
\(55\) 0 0
\(56\) 2.72161 0.363691
\(57\) 5.32340 0.705102
\(58\) 7.37883 0.968888
\(59\) 7.97021 1.03763 0.518817 0.854886i \(-0.326373\pi\)
0.518817 + 0.854886i \(0.326373\pi\)
\(60\) 0.333816 0.0430955
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 11.0748 1.40650
\(63\) 1.00000 0.125988
\(64\) 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
\(69\) 4.92520 0.592924
\(70\) −3.50761 −0.419240
\(71\) −11.4432 −1.35806 −0.679030 0.734110i \(-0.737600\pi\)
−0.679030 + 0.734110i \(0.737600\pi\)
\(72\) 2.72161 0.320745
\(73\) 13.0450 1.52680 0.763402 0.645924i \(-0.223528\pi\)
0.763402 + 0.645924i \(0.223528\pi\)
\(74\) −6.21400 −0.722363
\(75\) 0.751399 0.0867641
\(76\) 0.740987 0.0849970
\(77\) 0 0
\(78\) 7.37883 0.835488
\(79\) −11.4432 −1.28746 −0.643732 0.765251i \(-0.722615\pi\)
−0.643732 + 0.765251i \(0.722615\pi\)
\(80\) −10.2140 −1.14196
\(81\) 1.00000 0.111111
\(82\) −0.946021 −0.104471
\(83\) −13.1648 −1.44503 −0.722514 0.691356i \(-0.757014\pi\)
−0.722514 + 0.691356i \(0.757014\pi\)
\(84\) 0.139194 0.0151873
\(85\) 15.2728 1.65657
\(86\) −15.3836 −1.65886
\(87\) −5.04502 −0.540882
\(88\) 0 0
\(89\) 11.8504 1.25614 0.628070 0.778157i \(-0.283845\pi\)
0.628070 + 0.778157i \(0.283845\pi\)
\(90\) −3.50761 −0.369735
\(91\) −5.04502 −0.528861
\(92\) 0.685559 0.0714744
\(93\) −7.57201 −0.785180
\(94\) −0.770774 −0.0794993
\(95\) 12.7666 1.30983
\(96\) 0.786003 0.0802211
\(97\) −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 \(0.572166\pi\)
\(102\) −9.31444 −0.922267
\(103\) −10.6468 −1.04906 −0.524531 0.851392i \(-0.675759\pi\)
−0.524531 + 0.851392i \(0.675759\pi\)
\(104\) −13.7306 −1.34639
\(105\) 2.39821 0.234041
\(106\) −5.44322 −0.528693
\(107\) −15.9702 −1.54390 −0.771949 0.635684i \(-0.780718\pi\)
−0.771949 + 0.635684i \(0.780718\pi\)
\(108\) 0.139194 0.0133940
\(109\) −12.7756 −1.22368 −0.611840 0.790982i \(-0.709570\pi\)
−0.611840 + 0.790982i \(0.709570\pi\)
\(110\) 0 0
\(111\) 4.24860 0.403259
\(112\) −4.25901 −0.402439
\(113\) 18.7368 1.76261 0.881307 0.472544i \(-0.156664\pi\)
0.881307 + 0.472544i \(0.156664\pi\)
\(114\) −7.78600 −0.729226
\(115\) 11.8116 1.10144
\(116\) −0.702237 −0.0652010
\(117\) −5.04502 −0.466412
\(118\) −11.6572 −1.07313
\(119\) 6.36842 0.583792
\(120\) 6.52699 0.595830
\(121\) 0 0
\(122\) −2.92520 −0.264835
\(123\) 0.646809 0.0583208
\(124\) −1.05398 −0.0946501
\(125\) −10.1890 −0.911334
\(126\) −1.46260 −0.130299
\(127\) 2.27839 0.202174 0.101087 0.994878i \(-0.467768\pi\)
0.101087 + 0.994878i \(0.467768\pi\)
\(128\) −12.3490 −1.09151
\(129\) 10.5180 0.926061
\(130\) 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
\(135\) 2.39821 0.206405
\(136\) 17.3324 1.48624
\(137\) −4.77559 −0.408006 −0.204003 0.978970i \(-0.565395\pi\)
−0.204003 + 0.978970i \(0.565395\pi\)
\(138\) −7.20359 −0.613210
\(139\) 15.4432 1.30988 0.654939 0.755682i \(-0.272694\pi\)
0.654939 + 0.755682i \(0.272694\pi\)
\(140\) 0.333816 0.0282126
\(141\) 0.526989 0.0443805
\(142\) 16.7368 1.40452
\(143\) 0 0
\(144\) −4.25901 −0.354918
\(145\) −12.0990 −1.00477
\(146\) −19.0796 −1.57904
\(147\) 1.00000 0.0824786
\(148\) 0.591380 0.0486112
\(149\) 9.84143 0.806241 0.403121 0.915147i \(-0.367925\pi\)
0.403121 + 0.915147i \(0.367925\pi\)
\(150\) −1.09899 −0.0897326
\(151\) 4.12878 0.335996 0.167998 0.985787i \(-0.446270\pi\)
0.167998 + 0.985787i \(0.446270\pi\)
\(152\) 14.4882 1.17515
\(153\) 6.36842 0.514856
\(154\) 0 0
\(155\) −18.1592 −1.45859
\(156\) −0.702237 −0.0562239
\(157\) −0.946021 −0.0755007 −0.0377504 0.999287i \(-0.512019\pi\)
−0.0377504 + 0.999287i \(0.512019\pi\)
\(158\) 16.7368 1.33151
\(159\) 3.72161 0.295143
\(160\) 1.88500 0.149022
\(161\) 4.92520 0.388160
\(162\) −1.46260 −0.114913
\(163\) 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 \(0.108142\pi\)
\(168\) 2.72161 0.209977
\(169\) 12.4522 0.957860
\(170\) −22.3380 −1.71324
\(171\) 5.32340 0.407091
\(172\) 1.46405 0.111633
\(173\) −12.3476 −0.938770 −0.469385 0.882994i \(-0.655524\pi\)
−0.469385 + 0.882994i \(0.655524\pi\)
\(174\) 7.37883 0.559388
\(175\) 0.751399 0.0568004
\(176\) 0 0
\(177\) 7.97021 0.599078
\(178\) −17.3324 −1.29912
\(179\) −5.59283 −0.418028 −0.209014 0.977913i \(-0.567025\pi\)
−0.209014 + 0.977913i \(0.567025\pi\)
\(180\) 0.333816 0.0248812
\(181\) 13.5720 1.00880 0.504400 0.863470i \(-0.331714\pi\)
0.504400 + 0.863470i \(0.331714\pi\)
\(182\) 7.37883 0.546955
\(183\) 2.00000 0.147844
\(184\) 13.4045 0.988191
\(185\) 10.1890 0.749112
\(186\) 11.0748 0.812044
\(187\) 0 0
\(188\) 0.0733538 0.00534988
\(189\) 1.00000 0.0727393
\(190\) −18.6724 −1.35464
\(191\) −9.42240 −0.681781 −0.340890 0.940103i \(-0.610729\pi\)
−0.340890 + 0.940103i \(0.610729\pi\)
\(192\) 7.36842 0.531770
\(193\) 10.1288 0.729086 0.364543 0.931187i \(-0.381225\pi\)
0.364543 + 0.931187i \(0.381225\pi\)
\(194\) 2.73684 0.196494
\(195\) −12.0990 −0.866427
\(196\) 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
\(201\) 8.76663 0.618350
\(202\) 6.60806 0.464941
\(203\) −5.04502 −0.354091
\(204\) 0.886447 0.0620637
\(205\) 1.55118 0.108339
\(206\) 15.5720 1.08495
\(207\) 4.92520 0.342325
\(208\) 21.4868 1.48984
\(209\) 0 0
\(210\) −3.50761 −0.242048
\(211\) 14.6468 1.00833 0.504164 0.863608i \(-0.331801\pi\)
0.504164 + 0.863608i \(0.331801\pi\)
\(212\) 0.518027 0.0355782
\(213\) −11.4432 −0.784077
\(214\) 23.3580 1.59672
\(215\) 25.2244 1.72029
\(216\) 2.72161 0.185182
\(217\) −7.57201 −0.514021
\(218\) 18.6856 1.26555
\(219\) 13.0450 0.881500
\(220\) 0 0
\(221\) −32.1288 −2.16122
\(222\) −6.21400 −0.417056
\(223\) −1.90997 −0.127901 −0.0639505 0.997953i \(-0.520370\pi\)
−0.0639505 + 0.997953i \(0.520370\pi\)
\(224\) 0.786003 0.0525170
\(225\) 0.751399 0.0500933
\(226\) −27.4045 −1.82292
\(227\) 3.20359 0.212629 0.106315 0.994333i \(-0.466095\pi\)
0.106315 + 0.994333i \(0.466095\pi\)
\(228\) 0.740987 0.0490730
\(229\) −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 \(0.317162\pi\)
\(234\) 7.37883 0.482369
\(235\) 1.26383 0.0824432
\(236\) 1.10941 0.0722162
\(237\) −11.4432 −0.743317
\(238\) −9.31444 −0.603766
\(239\) −2.91623 −0.188635 −0.0943177 0.995542i \(-0.530067\pi\)
−0.0943177 + 0.995542i \(0.530067\pi\)
\(240\) −10.2140 −0.659311
\(241\) 6.09899 0.392871 0.196435 0.980517i \(-0.437063\pi\)
0.196435 + 0.980517i \(0.437063\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0.278388 0.0178220
\(245\) 2.39821 0.153216
\(246\) −0.946021 −0.0603161
\(247\) −26.8567 −1.70885
\(248\) −20.6081 −1.30861
\(249\) −13.1648 −0.834288
\(250\) 14.9025 0.942514
\(251\) 1.62262 0.102419 0.0512093 0.998688i \(-0.483692\pi\)
0.0512093 + 0.998688i \(0.483692\pi\)
\(252\) 0.139194 0.00876841
\(253\) 0 0
\(254\) −3.33237 −0.209091
\(255\) 15.2728 0.956419
\(256\) 3.32485 0.207803
\(257\) −6.89541 −0.430124 −0.215062 0.976600i \(-0.568995\pi\)
−0.215062 + 0.976600i \(0.568995\pi\)
\(258\) −15.3836 −0.957744
\(259\) 4.24860 0.263995
\(260\) −1.68411 −0.104444
\(261\) −5.04502 −0.312279
\(262\) 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
\(267\) 11.8504 0.725232
\(268\) 1.22026 0.0745394
\(269\) −0.886447 −0.0540476 −0.0270238 0.999635i \(-0.508603\pi\)
−0.0270238 + 0.999635i \(0.508603\pi\)
\(270\) −3.50761 −0.213467
\(271\) 25.3234 1.53829 0.769144 0.639076i \(-0.220683\pi\)
0.769144 + 0.639076i \(0.220683\pi\)
\(272\) −27.1232 −1.64458
\(273\) −5.04502 −0.305338
\(274\) 6.98477 0.421965
\(275\) 0 0
\(276\) 0.685559 0.0412658
\(277\) 24.8269 1.49170 0.745851 0.666113i \(-0.232043\pi\)
0.745851 + 0.666113i \(0.232043\pi\)
\(278\) −22.5872 −1.35469
\(279\) −7.57201 −0.453324
\(280\) 6.52699 0.390062
\(281\) −1.90101 −0.113404 −0.0567022 0.998391i \(-0.518059\pi\)
−0.0567022 + 0.998391i \(0.518059\pi\)
\(282\) −0.770774 −0.0458989
\(283\) 22.3178 1.32666 0.663328 0.748329i \(-0.269143\pi\)
0.663328 + 0.748329i \(0.269143\pi\)
\(284\) −1.59283 −0.0945171
\(285\) 12.7666 0.756230
\(286\) 0 0
\(287\) 0.646809 0.0381799
\(288\) 0.786003 0.0463157
\(289\) 23.5568 1.38569
\(290\) 17.6960 1.03914
\(291\) −1.87122 −0.109693
\(292\) 1.81579 0.106261
\(293\) −12.0900 −0.706307 −0.353154 0.935565i \(-0.614891\pi\)
−0.353154 + 0.935565i \(0.614891\pi\)
\(294\) −1.46260 −0.0853005
\(295\) 19.1142 1.11287
\(296\) 11.5630 0.672088
\(297\) 0 0
\(298\) −14.3941 −0.833826
\(299\) −24.8477 −1.43698
\(300\) 0.104590 0.00603853
\(301\) 10.5180 0.606249
\(302\) −6.03875 −0.347491
\(303\) −4.51803 −0.259554
\(304\) −22.6724 −1.30035
\(305\) 4.79641 0.274642
\(306\) −9.31444 −0.532471
\(307\) 13.5928 0.775784 0.387892 0.921705i \(-0.373203\pi\)
0.387892 + 0.921705i \(0.373203\pi\)
\(308\) 0 0
\(309\) −10.6468 −0.605676
\(310\) 26.5597 1.50849
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) −13.7306 −0.777341
\(313\) 14.9252 0.843622 0.421811 0.906684i \(-0.361395\pi\)
0.421811 + 0.906684i \(0.361395\pi\)
\(314\) 1.38365 0.0780838
\(315\) 2.39821 0.135124
\(316\) −1.59283 −0.0896037
\(317\) 3.97918 0.223493 0.111746 0.993737i \(-0.464356\pi\)
0.111746 + 0.993737i \(0.464356\pi\)
\(318\) −5.44322 −0.305241
\(319\) 0 0
\(320\) 17.6710 0.987839
\(321\) −15.9702 −0.891370
\(322\) −7.20359 −0.401440
\(323\) 33.9017 1.88634
\(324\) 0.139194 0.00773301
\(325\) −3.79082 −0.210277
\(326\) −12.8221 −0.710148
\(327\) −12.7756 −0.706492
\(328\) 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
\(333\) 4.24860 0.232822
\(334\) −35.6412 −1.95020
\(335\) 21.0242 1.14867
\(336\) −4.25901 −0.232348
\(337\) −11.1648 −0.608187 −0.304094 0.952642i \(-0.598354\pi\)
−0.304094 + 0.952642i \(0.598354\pi\)
\(338\) −18.2125 −0.990632
\(339\) 18.7368 1.01765
\(340\) 2.12588 0.115292
\(341\) 0 0
\(342\) −7.78600 −0.421019
\(343\) 1.00000 0.0539949
\(344\) 28.6260 1.54341
\(345\) 11.8116 0.635918
\(346\) 18.0596 0.970889
\(347\) 22.5872 1.21255 0.606273 0.795256i \(-0.292664\pi\)
0.606273 + 0.795256i \(0.292664\pi\)
\(348\) −0.702237 −0.0376438
\(349\) −27.9315 −1.49514 −0.747568 0.664185i \(-0.768779\pi\)
−0.747568 + 0.664185i \(0.768779\pi\)
\(350\) −1.09899 −0.0587437
\(351\) −5.04502 −0.269283
\(352\) 0 0
\(353\) −16.5478 −0.880751 −0.440376 0.897814i \(-0.645155\pi\)
−0.440376 + 0.897814i \(0.645155\pi\)
\(354\) −11.6572 −0.619574
\(355\) −27.4432 −1.45654
\(356\) 1.64951 0.0874236
\(357\) 6.36842 0.337053
\(358\) 8.18006 0.432330
\(359\) −22.0305 −1.16272 −0.581362 0.813645i \(-0.697480\pi\)
−0.581362 + 0.813645i \(0.697480\pi\)
\(360\) 6.52699 0.344003
\(361\) 9.33863 0.491507
\(362\) −19.8504 −1.04331
\(363\) 0 0
\(364\) −0.702237 −0.0368072
\(365\) 31.2847 1.63751
\(366\) −2.92520 −0.152902
\(367\) −19.3836 −1.01182 −0.505909 0.862587i \(-0.668843\pi\)
−0.505909 + 0.862587i \(0.668843\pi\)
\(368\) −20.9765 −1.09347
\(369\) 0.646809 0.0336715
\(370\) −14.9025 −0.774742
\(371\) 3.72161 0.193216
\(372\) −1.05398 −0.0546463
\(373\) −29.2549 −1.51476 −0.757380 0.652975i \(-0.773521\pi\)
−0.757380 + 0.652975i \(0.773521\pi\)
\(374\) 0 0
\(375\) −10.1890 −0.526159
\(376\) 1.43426 0.0739663
\(377\) 25.4522 1.31085
\(378\) −1.46260 −0.0752279
\(379\) 12.5270 0.643468 0.321734 0.946830i \(-0.395734\pi\)
0.321734 + 0.946830i \(0.395734\pi\)
\(380\) 1.77704 0.0911602
\(381\) 2.27839 0.116725
\(382\) 13.7812 0.705107
\(383\) −17.5928 −0.898952 −0.449476 0.893293i \(-0.648389\pi\)
−0.449476 + 0.893293i \(0.648389\pi\)
\(384\) −12.3490 −0.630185
\(385\) 0 0
\(386\) −14.8143 −0.754030
\(387\) 10.5180 0.534661
\(388\) −0.260463 −0.0132230
\(389\) 20.0900 1.01861 0.509303 0.860588i \(-0.329903\pi\)
0.509303 + 0.860588i \(0.329903\pi\)
\(390\) 17.6960 0.896070
\(391\) 31.3657 1.58623
\(392\) 2.72161 0.137462
\(393\) −4.00000 −0.201773
\(394\) −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
\(399\) 5.32340 0.266504
\(400\) −3.20022 −0.160011
\(401\) 9.57201 0.478003 0.239002 0.971019i \(-0.423180\pi\)
0.239002 + 0.971019i \(0.423180\pi\)
\(402\) −12.8221 −0.639506
\(403\) 38.2009 1.90292
\(404\) −0.628883 −0.0312881
\(405\) 2.39821 0.119168
\(406\) 7.37883 0.366205
\(407\) 0 0
\(408\) 17.3324 0.858080
\(409\) −38.1801 −1.88788 −0.943941 0.330113i \(-0.892913\pi\)
−0.943941 + 0.330113i \(0.892913\pi\)
\(410\) −2.26875 −0.112046
\(411\) −4.77559 −0.235563
\(412\) −1.48197 −0.0730116
\(413\) 7.97021 0.392189
\(414\) −7.20359 −0.354037
\(415\) −31.5720 −1.54981
\(416\) −3.96540 −0.194420
\(417\) 15.4432 0.756258
\(418\) 0 0
\(419\) 7.17380 0.350463 0.175231 0.984527i \(-0.443933\pi\)
0.175231 + 0.984527i \(0.443933\pi\)
\(420\) 0.333816 0.0162886
\(421\) 15.1530 0.738511 0.369255 0.929328i \(-0.379613\pi\)
0.369255 + 0.929328i \(0.379613\pi\)
\(422\) −21.4224 −1.04283
\(423\) 0.526989 0.0256231
\(424\) 10.1288 0.491897
\(425\) 4.78522 0.232117
\(426\) 16.7368 0.810903
\(427\) 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 \(0.542776\pi\)
\(432\) −4.25901 −0.204912
\(433\) −25.6412 −1.23224 −0.616119 0.787653i \(-0.711296\pi\)
−0.616119 + 0.787653i \(0.711296\pi\)
\(434\) 11.0748 0.531608
\(435\) −12.0990 −0.580102
\(436\) −1.77829 −0.0851645
\(437\) 26.2188 1.25422
\(438\) −19.0796 −0.911659
\(439\) −23.6710 −1.12976 −0.564878 0.825175i \(-0.691077\pi\)
−0.564878 + 0.825175i \(0.691077\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 46.9915 2.23516
\(443\) −18.0305 −0.856653 −0.428326 0.903624i \(-0.640897\pi\)
−0.428326 + 0.903624i \(0.640897\pi\)
\(444\) 0.591380 0.0280657
\(445\) 28.4197 1.34722
\(446\) 2.79352 0.132277
\(447\) 9.84143 0.465484
\(448\) 7.36842 0.348125
\(449\) −34.9765 −1.65064 −0.825321 0.564664i \(-0.809006\pi\)
−0.825321 + 0.564664i \(0.809006\pi\)
\(450\) −1.09899 −0.0518071
\(451\) 0 0
\(452\) 2.60806 0.122673
\(453\) 4.12878 0.193987
\(454\) −4.68556 −0.219904
\(455\) −12.0990 −0.567210
\(456\) 14.4882 0.678474
\(457\) −4.53595 −0.212183 −0.106091 0.994356i \(-0.533834\pi\)
−0.106091 + 0.994356i \(0.533834\pi\)
\(458\) 26.7785 1.25128
\(459\) 6.36842 0.297252
\(460\) 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
\(465\) −18.1592 −0.842115
\(466\) −24.2605 −1.12384
\(467\) −20.4674 −0.947119 −0.473560 0.880762i \(-0.657031\pi\)
−0.473560 + 0.880762i \(0.657031\pi\)
\(468\) −0.702237 −0.0324609
\(469\) 8.76663 0.404805
\(470\) −1.84848 −0.0852638
\(471\) −0.946021 −0.0435904
\(472\) 21.6918 0.998447
\(473\) 0 0
\(474\) 16.7368 0.768749
\(475\) 4.00000 0.183533
\(476\) 0.886447 0.0406302
\(477\) 3.72161 0.170401
\(478\) 4.26528 0.195089
\(479\) 11.6137 0.530641 0.265321 0.964160i \(-0.414522\pi\)
0.265321 + 0.964160i \(0.414522\pi\)
\(480\) 1.88500 0.0860380
\(481\) −21.4343 −0.977318
\(482\) −8.92038 −0.406312
\(483\) 4.92520 0.224104
\(484\) 0 0
\(485\) −4.48757 −0.203770
\(486\) −1.46260 −0.0663448
\(487\) 32.4793 1.47178 0.735888 0.677103i \(-0.236765\pi\)
0.735888 + 0.677103i \(0.236765\pi\)
\(488\) 5.44322 0.246403
\(489\) 8.76663 0.396441
\(490\) −3.50761 −0.158458
\(491\) −26.6766 −1.20390 −0.601949 0.798535i \(-0.705609\pi\)
−0.601949 + 0.798535i \(0.705609\pi\)
\(492\) 0.0900320 0.00405895
\(493\) −32.1288 −1.44701
\(494\) 39.2805 1.76731
\(495\) 0 0
\(496\) 32.2493 1.44804
\(497\) −11.4432 −0.513299
\(498\) 19.2549 0.862831
\(499\) −41.2459 −1.84642 −0.923210 0.384296i \(-0.874444\pi\)
−0.923210 + 0.384296i \(0.874444\pi\)
\(500\) −1.41825 −0.0634262
\(501\) 24.3684 1.08870
\(502\) −2.37324 −0.105923
\(503\) −30.5180 −1.36073 −0.680366 0.732873i \(-0.738179\pi\)
−0.680366 + 0.732873i \(0.738179\pi\)
\(504\) 2.72161 0.121230
\(505\) −10.8352 −0.482159
\(506\) 0 0
\(507\) 12.4522 0.553021
\(508\) 0.317138 0.0140707
\(509\) −18.9944 −0.841912 −0.420956 0.907081i \(-0.638305\pi\)
−0.420956 + 0.907081i \(0.638305\pi\)
\(510\) −22.3380 −0.989142
\(511\) 13.0450 0.577078
\(512\) 19.8352 0.876599
\(513\) 5.32340 0.235034
\(514\) 10.0852 0.444840
\(515\) −25.5333 −1.12513
\(516\) 1.46405 0.0644511
\(517\) 0 0
\(518\) −6.21400 −0.273027
\(519\) −12.3476 −0.541999
\(520\) −32.9288 −1.44402
\(521\) 25.2430 1.10592 0.552958 0.833209i \(-0.313499\pi\)
0.552958 + 0.833209i \(0.313499\pi\)
\(522\) 7.37883 0.322963
\(523\) 2.93416 0.128302 0.0641509 0.997940i \(-0.479566\pi\)
0.0641509 + 0.997940i \(0.479566\pi\)
\(524\) −0.556777 −0.0243229
\(525\) 0.751399 0.0327937
\(526\) 7.43551 0.324204
\(527\) −48.2217 −2.10057
\(528\) 0 0
\(529\) 1.25756 0.0546767
\(530\) −13.0540 −0.567029
\(531\) 7.97021 0.345878
\(532\) 0.740987 0.0321258
\(533\) −3.26316 −0.141343
\(534\) −17.3324 −0.750045
\(535\) −38.2999 −1.65585
\(536\) 23.8594 1.03057
\(537\) −5.59283 −0.241348
\(538\) 1.29652 0.0558968
\(539\) 0 0
\(540\) 0.333816 0.0143652
\(541\) −8.90437 −0.382829 −0.191414 0.981509i \(-0.561307\pi\)
−0.191414 + 0.981509i \(0.561307\pi\)
\(542\) −37.0380 −1.59092
\(543\) 13.5720 0.582430
\(544\) 5.00560 0.214613
\(545\) −30.6385 −1.31241
\(546\) 7.37883 0.315785
\(547\) 29.4737 1.26020 0.630102 0.776513i \(-0.283013\pi\)
0.630102 + 0.776513i \(0.283013\pi\)
\(548\) −0.664734 −0.0283960
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −26.8567 −1.14413
\(552\) 13.4045 0.570532
\(553\) −11.4432 −0.486615
\(554\) −36.3117 −1.54274
\(555\) 10.1890 0.432500
\(556\) 2.14961 0.0911636
\(557\) −14.8954 −0.631139 −0.315569 0.948903i \(-0.602196\pi\)
−0.315569 + 0.948903i \(0.602196\pi\)
\(558\) 11.0748 0.468834
\(559\) −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 \(0.447363\pi\)
\(564\) 0.0733538 0.00308875
\(565\) 44.9348 1.89042
\(566\) −32.6420 −1.37205
\(567\) 1.00000 0.0419961
\(568\) −31.1440 −1.30677
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) −18.6724 −0.782103
\(571\) −25.5512 −1.06928 −0.534642 0.845079i \(-0.679553\pi\)
−0.534642 + 0.845079i \(0.679553\pi\)
\(572\) 0 0
\(573\) −9.42240 −0.393626
\(574\) −0.946021 −0.0394862
\(575\) 3.70079 0.154334
\(576\) 7.36842 0.307018
\(577\) −29.5124 −1.22862 −0.614309 0.789065i \(-0.710565\pi\)
−0.614309 + 0.789065i \(0.710565\pi\)
\(578\) −34.4541 −1.43310
\(579\) 10.1288 0.420938
\(580\) −1.68411 −0.0699288
\(581\) −13.1648 −0.546169
\(582\) 2.73684 0.113446
\(583\) 0 0
\(584\) 35.5035 1.46914
\(585\) −12.0990 −0.500232
\(586\) 17.6829 0.730472
\(587\) 31.3955 1.29583 0.647916 0.761712i \(-0.275641\pi\)
0.647916 + 0.761712i \(0.275641\pi\)
\(588\) 0.139194 0.00574027
\(589\) −40.3088 −1.66090
\(590\) −27.9564 −1.15095
\(591\) 2.25756 0.0928638
\(592\) −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
\(597\) 3.07480 0.125843
\(598\) 36.3422 1.48614
\(599\) 27.4432 1.12130 0.560650 0.828053i \(-0.310551\pi\)
0.560650 + 0.828053i \(0.310551\pi\)
\(600\) 2.04502 0.0834874
\(601\) −31.9910 −1.30494 −0.652471 0.757814i \(-0.726268\pi\)
−0.652471 + 0.757814i \(0.726268\pi\)
\(602\) −15.3836 −0.626991
\(603\) 8.76663 0.357005
\(604\) 0.574702 0.0233843
\(605\) 0 0
\(606\) 6.60806 0.268434
\(607\) −7.41344 −0.300902 −0.150451 0.988617i \(-0.548073\pi\)
−0.150451 + 0.988617i \(0.548073\pi\)
\(608\) 4.18421 0.169692
\(609\) −5.04502 −0.204434
\(610\) −7.01523 −0.284038
\(611\) −2.65867 −0.107558
\(612\) 0.886447 0.0358325
\(613\) 33.9917 1.37291 0.686456 0.727171i \(-0.259165\pi\)
0.686456 + 0.727171i \(0.259165\pi\)
\(614\) −19.8809 −0.802326
\(615\) 1.55118 0.0625497
\(616\) 0 0
\(617\) −44.0305 −1.77260 −0.886300 0.463112i \(-0.846733\pi\)
−0.886300 + 0.463112i \(0.846733\pi\)
\(618\) 15.5720 0.626398
\(619\) 40.0096 1.60812 0.804061 0.594546i \(-0.202668\pi\)
0.804061 + 0.594546i \(0.202668\pi\)
\(620\) −2.52766 −0.101513
\(621\) 4.92520 0.197641
\(622\) −11.7008 −0.469159
\(623\) 11.8504 0.474776
\(624\) 21.4868 0.860160
\(625\) −28.1924 −1.12770
\(626\) −21.8296 −0.872485
\(627\) 0 0
\(628\) −0.131681 −0.00525463
\(629\) 27.0569 1.07883
\(630\) −3.50761 −0.139747
\(631\) 28.5568 1.13683 0.568414 0.822743i \(-0.307557\pi\)
0.568414 + 0.822743i \(0.307557\pi\)
\(632\) −31.1440 −1.23884
\(633\) 14.6468 0.582158
\(634\) −5.81994 −0.231139
\(635\) 5.46405 0.216834
\(636\) 0.518027 0.0205411
\(637\) −5.04502 −0.199891
\(638\) 0 0
\(639\) −11.4432 −0.452687
\(640\) −29.6156 −1.17066
\(641\) −31.1053 −1.22858 −0.614292 0.789079i \(-0.710558\pi\)
−0.614292 + 0.789079i \(0.710558\pi\)
\(642\) 23.3580 0.921867
\(643\) −5.48197 −0.216188 −0.108094 0.994141i \(-0.534475\pi\)
−0.108094 + 0.994141i \(0.534475\pi\)
\(644\) 0.685559 0.0270148
\(645\) 25.2244 0.993210
\(646\) −49.5845 −1.95088
\(647\) −9.26383 −0.364199 −0.182099 0.983280i \(-0.558289\pi\)
−0.182099 + 0.983280i \(0.558289\pi\)
\(648\) 2.72161 0.106915
\(649\) 0 0
\(650\) 5.54445 0.217471
\(651\) −7.57201 −0.296770
\(652\) 1.22026 0.0477892
\(653\) −29.9821 −1.17329 −0.586645 0.809844i \(-0.699551\pi\)
−0.586645 + 0.809844i \(0.699551\pi\)
\(654\) 18.6856 0.730663
\(655\) −9.59283 −0.374823
\(656\) −2.75477 −0.107556
\(657\) 13.0450 0.508935
\(658\) −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
\(663\) −32.1288 −1.24778
\(664\) −35.8296 −1.39046
\(665\) 12.7666 0.495069
\(666\) −6.21400 −0.240788
\(667\) −24.8477 −0.962107
\(668\) 3.39194 0.131238
\(669\) −1.90997 −0.0738436
\(670\) −30.7499 −1.18797
\(671\) 0 0
\(672\) 0.786003 0.0303207
\(673\) −21.8712 −0.843073 −0.421537 0.906811i \(-0.638509\pi\)
−0.421537 + 0.906811i \(0.638509\pi\)
\(674\) 16.3297 0.628995
\(675\) 0.751399 0.0289214
\(676\) 1.73327 0.0666643
\(677\) 1.26316 0.0485472 0.0242736 0.999705i \(-0.492273\pi\)
0.0242736 + 0.999705i \(0.492273\pi\)
\(678\) −27.4045 −1.05246
\(679\) −1.87122 −0.0718108
\(680\) 41.5666 1.59401
\(681\) 3.20359 0.122762
\(682\) 0 0
\(683\) 37.6441 1.44041 0.720206 0.693760i \(-0.244047\pi\)
0.720206 + 0.693760i \(0.244047\pi\)
\(684\) 0.740987 0.0283323
\(685\) −11.4529 −0.437591
\(686\) −1.46260 −0.0558423
\(687\) −18.3088 −0.698526
\(688\) −44.7964 −1.70785
\(689\) −18.7756 −0.715293
\(690\) −17.2757 −0.657674
\(691\) 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
\(696\) −13.7306 −0.520456
\(697\) 4.11915 0.156024
\(698\) 40.8525 1.54629
\(699\) 16.5872 0.627387
\(700\) 0.104590 0.00395314
\(701\) 39.2936 1.48410 0.742050 0.670345i \(-0.233854\pi\)
0.742050 + 0.670345i \(0.233854\pi\)
\(702\) 7.37883 0.278496
\(703\) 22.6170 0.853017
\(704\) 0 0
\(705\) 1.26383 0.0475986
\(706\) 24.2028 0.910885
\(707\) −4.51803 −0.169918
\(708\) 1.10941 0.0416941
\(709\) −49.2430 −1.84936 −0.924680 0.380745i \(-0.875667\pi\)
−0.924680 + 0.380745i \(0.875667\pi\)
\(710\) 40.1384 1.50637
\(711\) −11.4432 −0.429154
\(712\) 32.2522 1.20870
\(713\) −37.2936 −1.39666
\(714\) −9.31444 −0.348584
\(715\) 0 0
\(716\) −0.778489 −0.0290935
\(717\) −2.91623 −0.108909
\(718\) 32.2217 1.20250
\(719\) −7.41344 −0.276475 −0.138237 0.990399i \(-0.544144\pi\)
−0.138237 + 0.990399i \(0.544144\pi\)
\(720\) −10.2140 −0.380653
\(721\) −10.6468 −0.396508
\(722\) −13.6587 −0.508323
\(723\) 6.09899 0.226824
\(724\) 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
\(729\) 1.00000 0.0370370
\(730\) −45.7569 −1.69354
\(731\) 66.9832 2.47746
\(732\) 0.278388 0.0102895
\(733\) 3.59283 0.132704 0.0663521 0.997796i \(-0.478864\pi\)
0.0663521 + 0.997796i \(0.478864\pi\)
\(734\) 28.3505 1.04644
\(735\) 2.39821 0.0884592
\(736\) 3.87122 0.142695
\(737\) 0 0
\(738\) −0.946021 −0.0348235
\(739\) 26.7756 0.984956 0.492478 0.870325i \(-0.336091\pi\)
0.492478 + 0.870325i \(0.336091\pi\)
\(740\) 1.41825 0.0521360
\(741\) −26.8567 −0.986604
\(742\) −5.44322 −0.199827
\(743\) 33.8027 1.24010 0.620050 0.784562i \(-0.287112\pi\)
0.620050 + 0.784562i \(0.287112\pi\)
\(744\) −20.6081 −0.755528
\(745\) 23.6018 0.864703
\(746\) 42.7881 1.56658
\(747\) −13.1648 −0.481676
\(748\) 0 0
\(749\) −15.9702 −0.583539
\(750\) 14.9025 0.544161
\(751\) 35.3955 1.29160 0.645800 0.763506i \(-0.276524\pi\)
0.645800 + 0.763506i \(0.276524\pi\)
\(752\) −2.24445 −0.0818468
\(753\) 1.62262 0.0591314
\(754\) −37.2263 −1.35570
\(755\) 9.90168 0.360359
\(756\) 0.139194 0.00506244
\(757\) −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 \(0.575146\pi\)
\(762\) −3.33237 −0.120719
\(763\) −12.7756 −0.462507
\(764\) −1.31154 −0.0474500
\(765\) 15.2728 0.552189
\(766\) 25.7312 0.929708
\(767\) −40.2099 −1.45189
\(768\) 3.32485 0.119975
\(769\) −9.78186 −0.352743 −0.176371 0.984324i \(-0.556436\pi\)
−0.176371 + 0.984324i \(0.556436\pi\)
\(770\) 0 0
\(771\) −6.89541 −0.248332
\(772\) 1.40987 0.0507422
\(773\) 29.7223 1.06904 0.534518 0.845157i \(-0.320493\pi\)
0.534518 + 0.845157i \(0.320493\pi\)
\(774\) −15.3836 −0.552954
\(775\) −5.68960 −0.204376
\(776\) −5.09273 −0.182818
\(777\) 4.24860 0.152418
\(778\) −29.3836 −1.05345
\(779\) 3.44322 0.123366
\(780\) −1.68411 −0.0603008
\(781\) 0 0
\(782\) −45.8755 −1.64050
\(783\) −5.04502 −0.180294
\(784\) −4.25901 −0.152108
\(785\) −2.26875 −0.0809753
\(786\) 5.85039 0.208677
\(787\) −17.0242 −0.606847 −0.303423 0.952856i \(-0.598130\pi\)
−0.303423 + 0.952856i \(0.598130\pi\)
\(788\) 0.314240 0.0111943
\(789\) −5.08377 −0.180987
\(790\) 40.1384 1.42806
\(791\) 18.7368 0.666205
\(792\) 0 0
\(793\) −10.0900 −0.358308
\(794\) 51.3449 1.82216
\(795\) 8.92520 0.316544
\(796\) 0.427995 0.0151699
\(797\) −36.4287 −1.29037 −0.645185 0.764027i \(-0.723220\pi\)
−0.645185 + 0.764027i \(0.723220\pi\)
\(798\) −7.78600 −0.275622
\(799\) 3.35609 0.118730
\(800\) 0.590602 0.0208809
\(801\) 11.8504 0.418713
\(802\) −14.0000 −0.494357
\(803\) 0 0
\(804\) 1.22026 0.0430354
\(805\) 11.8116 0.416306
\(806\) −55.8726 −1.96803
\(807\) −0.886447 −0.0312044
\(808\) −12.2963 −0.432583
\(809\) −44.4882 −1.56412 −0.782062 0.623201i \(-0.785832\pi\)
−0.782062 + 0.623201i \(0.785832\pi\)
\(810\) −3.50761 −0.123245
\(811\) −7.65307 −0.268736 −0.134368 0.990932i \(-0.542900\pi\)
−0.134368 + 0.990932i \(0.542900\pi\)
\(812\) −0.702237 −0.0246437
\(813\) 25.3234 0.888131
\(814\) 0 0
\(815\) 21.0242 0.736445
\(816\) −27.1232 −0.949501
\(817\) 55.9917 1.95890
\(818\) 55.8421 1.95247
\(819\) −5.04502 −0.176287
\(820\) 0.215915 0.00754009
\(821\) 44.3691 1.54849 0.774246 0.632885i \(-0.218129\pi\)
0.774246 + 0.632885i \(0.218129\pi\)
\(822\) 6.98477 0.243622
\(823\) 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 \(0.742597\pi\)
\(828\) 0.685559 0.0238248
\(829\) −3.90997 −0.135799 −0.0678994 0.997692i \(-0.521630\pi\)
−0.0678994 + 0.997692i \(0.521630\pi\)
\(830\) 46.1772 1.60283
\(831\) 24.8269 0.861235
\(832\) −37.1738 −1.28877
\(833\) 6.36842 0.220653
\(834\) −22.5872 −0.782132
\(835\) 58.4405 2.02242
\(836\) 0 0
\(837\) −7.57201 −0.261727
\(838\) −10.4924 −0.362453
\(839\) −9.58097 −0.330772 −0.165386 0.986229i \(-0.552887\pi\)
−0.165386 + 0.986229i \(0.552887\pi\)
\(840\) 6.52699 0.225203
\(841\) −3.54781 −0.122338
\(842\) −22.1627 −0.763778
\(843\) −1.90101 −0.0654741
\(844\) 2.03875 0.0701767
\(845\) 29.8629 1.02732
\(846\) −0.770774 −0.0264998
\(847\) 0 0
\(848\) −15.8504 −0.544305
\(849\) 22.3178 0.765945
\(850\) −6.99886 −0.240059
\(851\) 20.9252 0.717307
\(852\) −1.59283 −0.0545694
\(853\) 14.5568 0.498415 0.249207 0.968450i \(-0.419830\pi\)
0.249207 + 0.968450i \(0.419830\pi\)
\(854\) −2.92520 −0.100098
\(855\) 12.7666 0.436609
\(856\) −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
\(861\) 0.646809 0.0220432
\(862\) 8.13650 0.277130
\(863\) −20.5485 −0.699479 −0.349739 0.936847i \(-0.613730\pi\)
−0.349739 + 0.936847i \(0.613730\pi\)
\(864\) 0.786003 0.0267404
\(865\) −29.6121 −1.00684
\(866\) 37.5028 1.27440
\(867\) 23.5568 0.800030
\(868\) −1.05398 −0.0357744
\(869\) 0 0
\(870\) 17.6960 0.599950
\(871\) −44.2278 −1.49860
\(872\) −34.7702 −1.17747
\(873\) −1.87122 −0.0633311
\(874\) −38.3476 −1.29713
\(875\) −10.1890 −0.344452
\(876\) 1.81579 0.0613499
\(877\) −59.1149 −1.99617 −0.998084 0.0618724i \(-0.980293\pi\)
−0.998084 + 0.0618724i \(0.980293\pi\)
\(878\) 34.6212 1.16841
\(879\) −12.0900 −0.407787
\(880\) 0 0
\(881\) 30.3982 1.02414 0.512071 0.858943i \(-0.328879\pi\)
0.512071 + 0.858943i \(0.328879\pi\)
\(882\) −1.46260 −0.0492483
\(883\) −35.6114 −1.19842 −0.599210 0.800592i \(-0.704519\pi\)
−0.599210 + 0.800592i \(0.704519\pi\)
\(884\) −4.47214 −0.150414
\(885\) 19.1142 0.642518
\(886\) 26.3713 0.885962
\(887\) −22.9736 −0.771377 −0.385689 0.922629i \(-0.626036\pi\)
−0.385689 + 0.922629i \(0.626036\pi\)
\(888\) 11.5630 0.388030
\(889\) 2.27839 0.0764147
\(890\) −41.5666 −1.39332
\(891\) 0 0
\(892\) −0.265856 −0.00890153
\(893\) 2.80538 0.0938784
\(894\) −14.3941 −0.481409
\(895\) −13.4128 −0.448339
\(896\) −12.3490 −0.412553
\(897\) −24.8477 −0.829640
\(898\) 51.1565 1.70712
\(899\) 38.2009 1.27407
\(900\) 0.104590 0.00348634
\(901\) 23.7008 0.789588
\(902\) 0 0
\(903\) 10.5180 0.350018
\(904\) 50.9944 1.69605
\(905\) 32.5485 1.08195
\(906\) −6.03875 −0.200624
\(907\) −57.1745 −1.89845 −0.949224 0.314602i \(-0.898129\pi\)
−0.949224 + 0.314602i \(0.898129\pi\)
\(908\) 0.445920 0.0147984
\(909\) −4.51803 −0.149853
\(910\) 17.6960 0.586616
\(911\) 6.82687 0.226184 0.113092 0.993584i \(-0.463924\pi\)
0.113092 + 0.993584i \(0.463924\pi\)
\(912\) −22.6724 −0.750760
\(913\) 0 0
\(914\) 6.63428 0.219442
\(915\) 4.79641 0.158565
\(916\) −2.54848 −0.0842043
\(917\) −4.00000 −0.132092
\(918\) −9.31444 −0.307422
\(919\) −12.0692 −0.398126 −0.199063 0.979987i \(-0.563790\pi\)
−0.199063 + 0.979987i \(0.563790\pi\)
\(920\) 32.1467 1.05985
\(921\) 13.5928 0.447899
\(922\) −4.09003 −0.134698
\(923\) 57.7312 1.90025
\(924\) 0 0
\(925\) 3.19239 0.104965
\(926\) −56.1045 −1.84371
\(927\) −10.6468 −0.349687
\(928\) −3.96540 −0.130171
\(929\) 26.8954 0.882410 0.441205 0.897406i \(-0.354551\pi\)
0.441205 + 0.897406i \(0.354551\pi\)
\(930\) 26.5597 0.870926
\(931\) 5.32340 0.174468
\(932\) 2.30885 0.0756288
\(933\) 8.00000 0.261908
\(934\) 29.9356 0.979523
\(935\) 0 0
\(936\) −13.7306 −0.448798
\(937\) 14.9944 0.489846 0.244923 0.969543i \(-0.421237\pi\)
0.244923 + 0.969543i \(0.421237\pi\)
\(938\) −12.8221 −0.418655
\(939\) 14.9252 0.487065
\(940\) 0.175918 0.00573780
\(941\) 30.1205 0.981900 0.490950 0.871188i \(-0.336650\pi\)
0.490950 + 0.871188i \(0.336650\pi\)
\(942\) 1.38365 0.0450817
\(943\) 3.18566 0.103739
\(944\) −33.9452 −1.10482
\(945\) 2.39821 0.0780137
\(946\) 0 0
\(947\) −17.3532 −0.563903 −0.281951 0.959429i \(-0.590982\pi\)
−0.281951 + 0.959429i \(0.590982\pi\)
\(948\) −1.59283 −0.0517327
\(949\) −65.8123 −2.13636
\(950\) −5.85039 −0.189812
\(951\) 3.97918 0.129034
\(952\) 17.3324 0.561745
\(953\) 2.14064 0.0693422 0.0346711 0.999399i \(-0.488962\pi\)
0.0346711 + 0.999399i \(0.488962\pi\)
\(954\) −5.44322 −0.176231
\(955\) −22.5969 −0.731217
\(956\) −0.405923 −0.0131285
\(957\) 0 0
\(958\) −16.9861 −0.548796
\(959\) −4.77559 −0.154212
\(960\) 17.6710 0.570329
\(961\) 26.3353 0.849525
\(962\) 31.3497 1.01076
\(963\) −15.9702 −0.514633
\(964\) 0.848944 0.0273427
\(965\) 24.2909 0.781952
\(966\) −7.20359 −0.231772
\(967\) 1.53326 0.0493062 0.0246531 0.999696i \(-0.492152\pi\)
0.0246531 + 0.999696i \(0.492152\pi\)
\(968\) 0 0
\(969\) 33.9017 1.08908
\(970\) 6.56351 0.210742
\(971\) −26.5574 −0.852269 −0.426135 0.904660i \(-0.640125\pi\)
−0.426135 + 0.904660i \(0.640125\pi\)
\(972\) 0.139194 0.00446465
\(973\) 15.4432 0.495087
\(974\) −47.5041 −1.52213
\(975\) −3.79082 −0.121403
\(976\) −8.51803 −0.272655
\(977\) −55.9017 −1.78845 −0.894227 0.447615i \(-0.852274\pi\)
−0.894227 + 0.447615i \(0.852274\pi\)
\(978\) −12.8221 −0.410004
\(979\) 0 0
\(980\) 0.333816 0.0106634
\(981\) −12.7756 −0.407893
\(982\) 39.0171 1.24509
\(983\) −53.0361 −1.69159 −0.845794 0.533510i \(-0.820873\pi\)
−0.845794 + 0.533510i \(0.820873\pi\)
\(984\) 1.76036 0.0561183
\(985\) 5.41411 0.172508
\(986\) 46.9915 1.49651
\(987\) 0.526989 0.0167743
\(988\) −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
\(993\) 23.4432 0.743948
\(994\) 16.7368 0.530860
\(995\) 7.37402 0.233772
\(996\) −1.83247 −0.0580640
\(997\) 45.0665 1.42727 0.713635 0.700517i \(-0.247047\pi\)
0.713635 + 0.700517i \(0.247047\pi\)
\(998\) 60.3262 1.90959
\(999\) 4.24860 0.134420
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 2541.2.a.bg.1.2 3
3.2 odd 2 7623.2.a.cd.1.2 3
11.10 odd 2 231.2.a.e.1.2 3
33.32 even 2 693.2.a.l.1.2 3
44.43 even 2 3696.2.a.bo.1.2 3
55.54 odd 2 5775.2.a.bp.1.2 3
77.76 even 2 1617.2.a.t.1.2 3
231.230 odd 2 4851.2.a.bi.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.2 3 11.10 odd 2
693.2.a.l.1.2 3 33.32 even 2
1617.2.a.t.1.2 3 77.76 even 2
2541.2.a.bg.1.2 3 1.1 even 1 trivial
3696.2.a.bo.1.2 3 44.43 even 2
4851.2.a.bi.1.2 3 231.230 odd 2
5775.2.a.bp.1.2 3 55.54 odd 2
7623.2.a.cd.1.2 3 3.2 odd 2