Properties

Label 231.2.a.e.1.2
Level $231$
Weight $2$
Character 231.1
Self dual yes
Analytic conductor $1.845$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 231.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.46260 q^{2} +1.00000 q^{3} +0.139194 q^{4} +2.39821 q^{5} +1.46260 q^{6} -1.00000 q^{7} -2.72161 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.46260 q^{2} +1.00000 q^{3} +0.139194 q^{4} +2.39821 q^{5} +1.46260 q^{6} -1.00000 q^{7} -2.72161 q^{8} +1.00000 q^{9} +3.50761 q^{10} -1.00000 q^{11} +0.139194 q^{12} +5.04502 q^{13} -1.46260 q^{14} +2.39821 q^{15} -4.25901 q^{16} -6.36842 q^{17} +1.46260 q^{18} -5.32340 q^{19} +0.333816 q^{20} -1.00000 q^{21} -1.46260 q^{22} +4.92520 q^{23} -2.72161 q^{24} +0.751399 q^{25} +7.37883 q^{26} +1.00000 q^{27} -0.139194 q^{28} +5.04502 q^{29} +3.50761 q^{30} -7.57201 q^{31} -0.786003 q^{32} -1.00000 q^{33} -9.31444 q^{34} -2.39821 q^{35} +0.139194 q^{36} +4.24860 q^{37} -7.78600 q^{38} +5.04502 q^{39} -6.52699 q^{40} -0.646809 q^{41} -1.46260 q^{42} -10.5180 q^{43} -0.139194 q^{44} +2.39821 q^{45} +7.20359 q^{46} +0.526989 q^{47} -4.25901 q^{48} +1.00000 q^{49} +1.09899 q^{50} -6.36842 q^{51} +0.702237 q^{52} +3.72161 q^{53} +1.46260 q^{54} -2.39821 q^{55} +2.72161 q^{56} -5.32340 q^{57} +7.37883 q^{58} +7.97021 q^{59} +0.333816 q^{60} -2.00000 q^{61} -11.0748 q^{62} -1.00000 q^{63} +7.36842 q^{64} +12.0990 q^{65} -1.46260 q^{66} +8.76663 q^{67} -0.886447 q^{68} +4.92520 q^{69} -3.50761 q^{70} -11.4432 q^{71} -2.72161 q^{72} -13.0450 q^{73} +6.21400 q^{74} +0.751399 q^{75} -0.740987 q^{76} +1.00000 q^{77} +7.37883 q^{78} +11.4432 q^{79} -10.2140 q^{80} +1.00000 q^{81} -0.946021 q^{82} +13.1648 q^{83} -0.139194 q^{84} -15.2728 q^{85} -15.3836 q^{86} +5.04502 q^{87} +2.72161 q^{88} +11.8504 q^{89} +3.50761 q^{90} -5.04502 q^{91} +0.685559 q^{92} -7.57201 q^{93} +0.770774 q^{94} -12.7666 q^{95} -0.786003 q^{96} -1.87122 q^{97} +1.46260 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 3 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q + 2 q^{2} + 3 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 11 q^{10} - 3 q^{11} + 6 q^{12} - 4 q^{13} - 2 q^{14} + 4 q^{15} - 4 q^{16} + 8 q^{17} + 2 q^{18} - 8 q^{19} - 3 q^{20} - 3 q^{21} - 2 q^{22} + 10 q^{23} + 3 q^{24} + 15 q^{25} - q^{26} + 3 q^{27} - 6 q^{28} - 4 q^{29} - 11 q^{30} - 2 q^{31} + 8 q^{32} - 3 q^{33} - 4 q^{34} - 4 q^{35} + 6 q^{36} - 13 q^{38} - 4 q^{39} - 18 q^{40} + 14 q^{41} - 2 q^{42} - 14 q^{43} - 6 q^{44} + 4 q^{45} + 28 q^{46} - 4 q^{48} + 3 q^{49} - 19 q^{50} + 8 q^{51} - 29 q^{52} + 2 q^{54} - 4 q^{55} - 3 q^{56} - 8 q^{57} - q^{58} - 3 q^{60} - 6 q^{61} - 38 q^{62} - 3 q^{63} - 5 q^{64} + 14 q^{65} - 2 q^{66} - 4 q^{67} + 42 q^{68} + 10 q^{69} + 11 q^{70} - 12 q^{71} + 3 q^{72} - 20 q^{73} + 29 q^{74} + 15 q^{75} - 11 q^{76} + 3 q^{77} - q^{78} + 12 q^{79} - 41 q^{80} + 3 q^{81} - 6 q^{82} + 6 q^{83} - 6 q^{84} - 6 q^{85} + 24 q^{86} - 4 q^{87} - 3 q^{88} + 26 q^{89} - 11 q^{90} + 4 q^{91} + 26 q^{92} - 2 q^{93} + 35 q^{94} - 8 q^{95} + 8 q^{96} - 4 q^{97} + 2 q^{98} - 3 q^{99} + O(q^{100}) \)

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 \(-0.327009\pi\)
0.517107 + 0.855921i \(0.327009\pi\)
\(3\) 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
\(11\) −1.00000 −0.301511
\(12\) 0.139194 0.0401819
\(13\) 5.04502 1.39924 0.699618 0.714517i \(-0.253354\pi\)
0.699618 + 0.714517i \(0.253354\pi\)
\(14\) −1.46260 −0.390896
\(15\) 2.39821 0.619214
\(16\) −4.25901 −1.06475
\(17\) −6.36842 −1.54457 −0.772284 0.635277i \(-0.780886\pi\)
−0.772284 + 0.635277i \(0.780886\pi\)
\(18\) 1.46260 0.344738
\(19\) −5.32340 −1.22127 −0.610636 0.791911i \(-0.709086\pi\)
−0.610636 + 0.791911i \(0.709086\pi\)
\(20\) 0.333816 0.0746436
\(21\) −1.00000 −0.218218
\(22\) −1.46260 −0.311827
\(23\) 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 \(-0.344824\pi\)
0.468418 + 0.883507i \(0.344824\pi\)
\(30\) 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
\(33\) −1.00000 −0.174078
\(34\) −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 \(-0.516084\pi\)
−0.0505073 + 0.998724i \(0.516084\pi\)
\(42\) −1.46260 −0.225684
\(43\) −10.5180 −1.60398 −0.801992 0.597335i \(-0.796226\pi\)
−0.801992 + 0.597335i \(0.796226\pi\)
\(44\) −0.139194 −0.0209843
\(45\) 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
\(55\) −2.39821 −0.323374
\(56\) 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 \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −11.0748 −1.40650
\(63\) −1.00000 −0.125988
\(64\) 7.36842 0.921053
\(65\) 12.0990 1.50070
\(66\) −1.46260 −0.180033
\(67\) 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 \(-0.776472\pi\)
−0.763402 + 0.645924i \(0.776472\pi\)
\(74\) 6.21400 0.722363
\(75\) 0.751399 0.0867641
\(76\) −0.740987 −0.0849970
\(77\) 1.00000 0.113961
\(78\) 7.37883 0.835488
\(79\) 11.4432 1.28746 0.643732 0.765251i \(-0.277385\pi\)
0.643732 + 0.765251i \(0.277385\pi\)
\(80\) −10.2140 −1.14196
\(81\) 1.00000 0.111111
\(82\) −0.946021 −0.104471
\(83\) 13.1648 1.44503 0.722514 0.691356i \(-0.242986\pi\)
0.722514 + 0.691356i \(0.242986\pi\)
\(84\) −0.139194 −0.0151873
\(85\) −15.2728 −1.65657
\(86\) −15.3836 −1.65886
\(87\) 5.04502 0.540882
\(88\) 2.72161 0.290125
\(89\) 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
\(99\) −1.00000 −0.100504
\(100\) 0.104590 0.0104590
\(101\) 4.51803 0.449560 0.224780 0.974409i \(-0.427834\pi\)
0.224780 + 0.974409i \(0.427834\pi\)
\(102\) −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 \(-0.219282\pi\)
0.771949 + 0.635684i \(0.219282\pi\)
\(108\) 0.139194 0.0133940
\(109\) 12.7756 1.22368 0.611840 0.790982i \(-0.290430\pi\)
0.611840 + 0.790982i \(0.290430\pi\)
\(110\) −3.50761 −0.334438
\(111\) 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
\(121\) 1.00000 0.0909091
\(122\) −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 \(-0.532232\pi\)
−0.101087 + 0.994878i \(0.532232\pi\)
\(128\) 12.3490 1.09151
\(129\) −10.5180 −0.926061
\(130\) 17.6960 1.55204
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −0.139194 −0.0121153
\(133\) 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 \(-0.727306\pi\)
−0.654939 + 0.755682i \(0.727306\pi\)
\(140\) −0.333816 −0.0282126
\(141\) 0.526989 0.0443805
\(142\) −16.7368 −1.40452
\(143\) −5.04502 −0.421885
\(144\) −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 \(-0.632075\pi\)
−0.403121 + 0.915147i \(0.632075\pi\)
\(150\) 1.09899 0.0897326
\(151\) −4.12878 −0.335996 −0.167998 0.985787i \(-0.553730\pi\)
−0.167998 + 0.985787i \(0.553730\pi\)
\(152\) 14.4882 1.17515
\(153\) −6.36842 −0.514856
\(154\) 1.46260 0.117860
\(155\) −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
\(165\) −2.39821 −0.186700
\(166\) 19.2549 1.49447
\(167\) −24.3684 −1.88568 −0.942842 0.333239i \(-0.891858\pi\)
−0.942842 + 0.333239i \(0.891858\pi\)
\(168\) 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 \(-0.344476\pi\)
0.469385 + 0.882994i \(0.344476\pi\)
\(174\) 7.37883 0.559388
\(175\) −0.751399 −0.0568004
\(176\) 4.25901 0.321035
\(177\) 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
\(187\) 6.36842 0.465705
\(188\) 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 \(-0.618775\pi\)
−0.364543 + 0.931187i \(0.618775\pi\)
\(194\) −2.73684 −0.196494
\(195\) 12.0990 0.866427
\(196\) 0.139194 0.00994244
\(197\) −2.25756 −0.160845 −0.0804224 0.996761i \(-0.525627\pi\)
−0.0804224 + 0.996761i \(0.525627\pi\)
\(198\) −1.46260 −0.103942
\(199\) 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
\(209\) 5.32340 0.368228
\(210\) −3.50761 −0.242048
\(211\) −14.6468 −1.00833 −0.504164 0.863608i \(-0.668199\pi\)
−0.504164 + 0.863608i \(0.668199\pi\)
\(212\) 0.518027 0.0355782
\(213\) −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
\(220\) −0.333816 −0.0225059
\(221\) −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 \(-0.533905\pi\)
−0.106315 + 0.994333i \(0.533905\pi\)
\(228\) −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
\(231\) 1.00000 0.0657952
\(232\) −13.7306 −0.901456
\(233\) −16.5872 −1.08667 −0.543333 0.839517i \(-0.682838\pi\)
−0.543333 + 0.839517i \(0.682838\pi\)
\(234\) 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 \(-0.469933\pi\)
0.0943177 + 0.995542i \(0.469933\pi\)
\(240\) −10.2140 −0.659311
\(241\) −6.09899 −0.392871 −0.196435 0.980517i \(-0.562937\pi\)
−0.196435 + 0.980517i \(0.562937\pi\)
\(242\) 1.46260 0.0940194
\(243\) 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
\(253\) −4.92520 −0.309644
\(254\) −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 \(-0.449902\pi\)
0.156739 + 0.987640i \(0.449902\pi\)
\(264\) 2.72161 0.167504
\(265\) 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 \(-0.779317\pi\)
−0.769144 + 0.639076i \(0.779317\pi\)
\(272\) 27.1232 1.64458
\(273\) −5.04502 −0.305338
\(274\) −6.98477 −0.421965
\(275\) −0.751399 −0.0453111
\(276\) 0.685559 0.0412658
\(277\) −24.8269 −1.49170 −0.745851 0.666113i \(-0.767957\pi\)
−0.745851 + 0.666113i \(0.767957\pi\)
\(278\) −22.5872 −1.35469
\(279\) −7.57201 −0.453324
\(280\) 6.52699 0.390062
\(281\) 1.90101 0.113404 0.0567022 0.998391i \(-0.481941\pi\)
0.0567022 + 0.998391i \(0.481941\pi\)
\(282\) 0.770774 0.0458989
\(283\) −22.3178 −1.32666 −0.663328 0.748329i \(-0.730857\pi\)
−0.663328 + 0.748329i \(0.730857\pi\)
\(284\) −1.59283 −0.0945171
\(285\) −12.7666 −0.756230
\(286\) −7.37883 −0.436320
\(287\) 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 \(-0.385109\pi\)
0.353154 + 0.935565i \(0.385109\pi\)
\(294\) 1.46260 0.0853005
\(295\) 19.1142 1.11287
\(296\) −11.5630 −0.672088
\(297\) −1.00000 −0.0580259
\(298\) −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 \(-0.626797\pi\)
−0.387892 + 0.921705i \(0.626797\pi\)
\(308\) 0.139194 0.00793132
\(309\) −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
\(319\) −5.04502 −0.282467
\(320\) 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
\(330\) −3.50761 −0.193088
\(331\) 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 \(-0.401646\pi\)
0.304094 + 0.952642i \(0.401646\pi\)
\(338\) 18.2125 0.990632
\(339\) 18.7368 1.01765
\(340\) −2.12588 −0.115292
\(341\) 7.57201 0.410047
\(342\) −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 \(-0.707336\pi\)
−0.606273 + 0.795256i \(0.707336\pi\)
\(348\) 0.702237 0.0376438
\(349\) 27.9315 1.49514 0.747568 0.664185i \(-0.231221\pi\)
0.747568 + 0.664185i \(0.231221\pi\)
\(350\) −1.09899 −0.0587437
\(351\) 5.04502 0.269283
\(352\) 0.786003 0.0418941
\(353\) −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 \(-0.302520\pi\)
0.581362 + 0.813645i \(0.302520\pi\)
\(360\) −6.52699 −0.344003
\(361\) 9.33863 0.491507
\(362\) 19.8504 1.04331
\(363\) 1.00000 0.0524864
\(364\) −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 \(-0.226479\pi\)
0.757380 + 0.652975i \(0.226479\pi\)
\(374\) 9.31444 0.481638
\(375\) −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
\(385\) 2.39821 0.122224
\(386\) −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
\(396\) −0.139194 −0.00699477
\(397\) −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
\(407\) −4.24860 −0.210595
\(408\) 17.3324 0.858080
\(409\) 38.1801 1.88788 0.943941 0.330113i \(-0.107087\pi\)
0.943941 + 0.330113i \(0.107087\pi\)
\(410\) −2.26875 −0.112046
\(411\) −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
\(418\) 7.78600 0.380826
\(419\) 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
\(429\) −5.04502 −0.243576
\(430\) −36.8932 −1.77915
\(431\) 5.56304 0.267962 0.133981 0.990984i \(-0.457224\pi\)
0.133981 + 0.990984i \(0.457224\pi\)
\(432\) −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 \(-0.308923\pi\)
0.564878 + 0.825175i \(0.308923\pi\)
\(440\) 6.52699 0.311162
\(441\) 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
\(451\) 0.646809 0.0304570
\(452\) 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 \(-0.466166\pi\)
0.106091 + 0.994356i \(0.466166\pi\)
\(458\) −26.7785 −1.25128
\(459\) −6.36842 −0.297252
\(460\) 1.64411 0.0766571
\(461\) −2.79641 −0.130242 −0.0651210 0.997877i \(-0.520743\pi\)
−0.0651210 + 0.997877i \(0.520743\pi\)
\(462\) 1.46260 0.0680462
\(463\) 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
\(473\) 10.5180 0.483619
\(474\) 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 \(-0.585478\pi\)
−0.265321 + 0.964160i \(0.585478\pi\)
\(480\) −1.88500 −0.0860380
\(481\) 21.4343 0.977318
\(482\) −8.92038 −0.406312
\(483\) −4.92520 −0.224104
\(484\) 0.139194 0.00632701
\(485\) −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 \(-0.294391\pi\)
0.601949 + 0.798535i \(0.294391\pi\)
\(492\) −0.0900320 −0.00405895
\(493\) −32.1288 −1.44701
\(494\) −39.2805 −1.76731
\(495\) −2.39821 −0.107791
\(496\) 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 \(-0.261821\pi\)
0.680366 + 0.732873i \(0.261821\pi\)
\(504\) 2.72161 0.121230
\(505\) 10.8352 0.482159
\(506\) −7.20359 −0.320238
\(507\) 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
\(517\) −0.526989 −0.0231770
\(518\) −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 \(-0.520434\pi\)
−0.0641509 + 0.997940i \(0.520434\pi\)
\(524\) 0.556777 0.0243229
\(525\) −0.751399 −0.0327937
\(526\) 7.43551 0.324204
\(527\) 48.2217 2.10057
\(528\) 4.25901 0.185350
\(529\) 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
\(539\) −1.00000 −0.0430730
\(540\) 0.333816 0.0143652
\(541\) 8.90437 0.382829 0.191414 0.981509i \(-0.438693\pi\)
0.191414 + 0.981509i \(0.438693\pi\)
\(542\) −37.0380 −1.59092
\(543\) 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 \(-0.716987\pi\)
−0.630102 + 0.776513i \(0.716987\pi\)
\(548\) −0.664734 −0.0283960
\(549\) −2.00000 −0.0853579
\(550\) −1.09899 −0.0468613
\(551\) −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 \(-0.397804\pi\)
0.315569 + 0.948903i \(0.397804\pi\)
\(558\) −11.0748 −0.468834
\(559\) −53.0636 −2.24435
\(560\) 10.2140 0.431620
\(561\) 6.36842 0.268875
\(562\) 2.78041 0.117284
\(563\) −7.81164 −0.329222 −0.164611 0.986359i \(-0.552637\pi\)
−0.164611 + 0.986359i \(0.552637\pi\)
\(564\) 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 \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) −18.6724 −0.782103
\(571\) 25.5512 1.06928 0.534642 0.845079i \(-0.320447\pi\)
0.534642 + 0.845079i \(0.320447\pi\)
\(572\) −0.702237 −0.0293620
\(573\) −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
\(583\) −3.72161 −0.154133
\(584\) 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 \(-0.551927\pi\)
−0.162412 + 0.986723i \(0.551927\pi\)
\(594\) −1.46260 −0.0600111
\(595\) 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 \(-0.273732\pi\)
0.652471 + 0.757814i \(0.273732\pi\)
\(602\) 15.3836 0.626991
\(603\) 8.76663 0.357005
\(604\) −0.574702 −0.0233843
\(605\) 2.39821 0.0975010
\(606\) 6.60806 0.268434
\(607\) 7.41344 0.300902 0.150451 0.988617i \(-0.451927\pi\)
0.150451 + 0.988617i \(0.451927\pi\)
\(608\) 4.18421 0.169692
\(609\) −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 \(-0.740835\pi\)
−0.686456 + 0.727171i \(0.740835\pi\)
\(614\) −19.8809 −0.802326
\(615\) −1.55118 −0.0625497
\(616\) −2.72161 −0.109657
\(617\) −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
\(627\) 5.32340 0.212596
\(628\) −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
\(638\) −7.37883 −0.292131
\(639\) −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
\(649\) −7.97021 −0.312858
\(650\) 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 \(-0.345380\pi\)
0.466873 + 0.884324i \(0.345380\pi\)
\(660\) −0.333816 −0.0129938
\(661\) −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
\(671\) 2.00000 0.0772091
\(672\) 0.786003 0.0303207
\(673\) 21.8712 0.843073 0.421537 0.906811i \(-0.361491\pi\)
0.421537 + 0.906811i \(0.361491\pi\)
\(674\) 16.3297 0.628995
\(675\) 0.751399 0.0289214
\(676\) 1.73327 0.0666643
\(677\) −1.26316 −0.0485472 −0.0242736 0.999705i \(-0.507727\pi\)
−0.0242736 + 0.999705i \(0.507727\pi\)
\(678\) 27.4045 1.05246
\(679\) 1.87122 0.0718108
\(680\) 41.5666 1.59401
\(681\) −3.20359 −0.122762
\(682\) 11.0748 0.424076
\(683\) 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
\(693\) 1.00000 0.0379869
\(694\) −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 \(-0.766146\pi\)
−0.742050 + 0.670345i \(0.766146\pi\)
\(702\) 7.37883 0.278496
\(703\) −22.6170 −0.853017
\(704\) −7.36842 −0.277708
\(705\) 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
\(715\) −12.0990 −0.452477
\(716\) −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
\(726\) 1.46260 0.0542821
\(727\) 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 \(-0.521136\pi\)
−0.0663521 + 0.997796i \(0.521136\pi\)
\(734\) −28.3505 −1.04644
\(735\) 2.39821 0.0884592
\(736\) −3.87122 −0.142695
\(737\) −8.76663 −0.322923
\(738\) −0.946021 −0.0348235
\(739\) −26.7756 −0.984956 −0.492478 0.870325i \(-0.663909\pi\)
−0.492478 + 0.870325i \(0.663909\pi\)
\(740\) 1.41825 0.0521360
\(741\) −26.8567 −0.986604
\(742\) −5.44322 −0.199827
\(743\) −33.8027 −1.24010 −0.620050 0.784562i \(-0.712888\pi\)
−0.620050 + 0.784562i \(0.712888\pi\)
\(744\) 20.6081 0.755528
\(745\) −23.6018 −0.864703
\(746\) 42.7881 1.56658
\(747\) 13.1648 0.481676
\(748\) 0.886447 0.0324117
\(749\) −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
\(759\) −4.92520 −0.178773
\(760\) 34.7458 1.26036
\(761\) 12.9044 0.467783 0.233892 0.972263i \(-0.424854\pi\)
0.233892 + 0.972263i \(0.424854\pi\)
\(762\) −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 \(-0.443564\pi\)
0.176371 + 0.984324i \(0.443564\pi\)
\(770\) 3.50761 0.126406
\(771\) −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
\(781\) 11.4432 0.409471
\(782\) −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 \(-0.401870\pi\)
0.303423 + 0.952856i \(0.401870\pi\)
\(788\) −0.314240 −0.0111943
\(789\) 5.08377 0.180987
\(790\) 40.1384 1.42806
\(791\) −18.7368 −0.666205
\(792\) 2.72161 0.0967083
\(793\) −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
\(803\) 13.0450 0.460349
\(804\) 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 \(-0.214168\pi\)
0.782062 + 0.623201i \(0.214168\pi\)
\(810\) 3.50761 0.123245
\(811\) 7.65307 0.268736 0.134368 0.990932i \(-0.457100\pi\)
0.134368 + 0.990932i \(0.457100\pi\)
\(812\) −0.702237 −0.0246437
\(813\) −25.3234 −0.888131
\(814\) −6.21400 −0.217800
\(815\) 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 \(-0.781871\pi\)
−0.774246 + 0.632885i \(0.781871\pi\)
\(822\) −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
\(825\) −0.751399 −0.0261604
\(826\) −11.6572 −0.405607
\(827\) 39.7126 1.38094 0.690472 0.723359i \(-0.257403\pi\)
0.690472 + 0.723359i \(0.257403\pi\)
\(828\) 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
\(836\) 0.740987 0.0256276
\(837\) −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
\(847\) −1.00000 −0.0343604
\(848\) −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 \(-0.580170\pi\)
−0.249207 + 0.968450i \(0.580170\pi\)
\(854\) 2.92520 0.100098
\(855\) −12.7666 −0.436609
\(856\) −43.4647 −1.48559
\(857\) 10.4793 0.357965 0.178983 0.983852i \(-0.442719\pi\)
0.178983 + 0.983852i \(0.442719\pi\)
\(858\) −7.37883 −0.251909
\(859\) 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
\(869\) −11.4432 −0.388185
\(870\) 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 \(-0.0197072\pi\)
0.998084 + 0.0618724i \(0.0197072\pi\)
\(878\) 34.6212 1.16841
\(879\) 12.0900 0.407787
\(880\) 10.2140 0.344314
\(881\) 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 \(-0.373964\pi\)
0.385689 + 0.922629i \(0.373964\pi\)
\(888\) −11.5630 −0.388030
\(889\) 2.27839 0.0764147
\(890\) 41.5666 1.39332
\(891\) −1.00000 −0.0335013
\(892\) −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
\(902\) 0.946021 0.0314991
\(903\) 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
\(913\) −13.1648 −0.435692
\(914\) 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 \(-0.436210\pi\)
0.199063 + 0.979987i \(0.436210\pi\)
\(920\) −32.1467 −1.05985
\(921\) −13.5928 −0.447899
\(922\) −4.09003 −0.134698
\(923\) −57.7312 −1.90025
\(924\) 0.139194 0.00457915
\(925\) 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
\(935\) 15.2728 0.499474
\(936\) −13.7306 −0.448798
\(937\) −14.9944 −0.489846 −0.244923 0.969543i \(-0.578763\pi\)
−0.244923 + 0.969543i \(0.578763\pi\)
\(938\) −12.8221 −0.418655
\(939\) 14.9252 0.487065
\(940\) 0.175918 0.00573780
\(941\) −30.1205 −0.981900 −0.490950 0.871188i \(-0.663350\pi\)
−0.490950 + 0.871188i \(0.663350\pi\)
\(942\) −1.38365 −0.0450817
\(943\) −3.18566 −0.103739
\(944\) −33.9452 −1.10482
\(945\) −2.39821 −0.0780137
\(946\) 15.3836 0.500166
\(947\) −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 \(-0.511038\pi\)
−0.0346711 + 0.999399i \(0.511038\pi\)
\(954\) 5.44322 0.176231
\(955\) −22.5969 −0.731217
\(956\) 0.405923 0.0131285
\(957\) −5.04502 −0.163082
\(958\) −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 \(-0.507848\pi\)
−0.0246531 + 0.999696i \(0.507848\pi\)
\(968\) −2.72161 −0.0874759
\(969\) 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
\(979\) −11.8504 −0.378740
\(980\) 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
\(990\) −3.50761 −0.111479
\(991\) 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 \(-0.752953\pi\)
−0.713635 + 0.700517i \(0.752953\pi\)
\(998\) −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 231.2.a.e.1.2 3
3.2 odd 2 693.2.a.l.1.2 3
4.3 odd 2 3696.2.a.bo.1.2 3
5.4 even 2 5775.2.a.bp.1.2 3
7.6 odd 2 1617.2.a.t.1.2 3
11.10 odd 2 2541.2.a.bg.1.2 3
21.20 even 2 4851.2.a.bi.1.2 3
33.32 even 2 7623.2.a.cd.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.2 3 1.1 even 1 trivial
693.2.a.l.1.2 3 3.2 odd 2
1617.2.a.t.1.2 3 7.6 odd 2
2541.2.a.bg.1.2 3 11.10 odd 2
3696.2.a.bo.1.2 3 4.3 odd 2
4851.2.a.bi.1.2 3 21.20 even 2
5775.2.a.bp.1.2 3 5.4 even 2
7623.2.a.cd.1.2 3 33.32 even 2