Properties

Label 5775.2.a.bp.1.2
Level $5775$
Weight $2$
Character 5775.1
Self dual yes
Analytic conductor $46.114$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5775.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.1136071673\)
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: 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 \(\chi\) \(=\) 5775.1

$q$-expansion

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

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 \(-0.672991\pi\)
−0.517107 + 0.855921i \(0.672991\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.139194 0.0695971
\(5\) 0 0
\(6\) 1.46260 0.597103
\(7\) 1.00000 0.377964
\(8\) 2.72161 0.962235
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −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
\(15\) 0 0
\(16\) −4.25901 −1.06475
\(17\) 6.36842 1.54457 0.772284 0.635277i \(-0.219114\pi\)
0.772284 + 0.635277i \(0.219114\pi\)
\(18\) −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 0
\(21\) −1.00000 −0.218218
\(22\) 1.46260 0.311827
\(23\) −4.92520 −1.02697 −0.513487 0.858097i \(-0.671647\pi\)
−0.513487 + 0.858097i \(0.671647\pi\)
\(24\) −2.72161 −0.555547
\(25\) 0 0
\(26\) 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 \(0.344824\pi\)
\(30\) 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
\(33\) 1.00000 0.174078
\(34\) −9.31444 −1.59741
\(35\) 0 0
\(36\) 0.139194 0.0231990
\(37\) −4.24860 −0.698466 −0.349233 0.937036i \(-0.613558\pi\)
−0.349233 + 0.937036i \(0.613558\pi\)
\(38\) 7.78600 1.26306
\(39\) 5.04502 0.807849
\(40\) 0 0
\(41\) −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 \(-0.203774\pi\)
0.801992 + 0.597335i \(0.203774\pi\)
\(44\) −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 \(0.512237\pi\)
\(48\) 4.25901 0.614736
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −6.36842 −0.891757
\(52\) −0.702237 −0.0973827
\(53\) −3.72161 −0.511203 −0.255601 0.966782i \(-0.582273\pi\)
−0.255601 + 0.966782i \(0.582273\pi\)
\(54\) 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 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
\(63\) 1.00000 0.125988
\(64\) 7.36842 0.921053
\(65\) 0 0
\(66\) −1.46260 −0.180033
\(67\) −8.76663 −1.07101 −0.535507 0.844531i \(-0.679879\pi\)
−0.535507 + 0.844531i \(0.679879\pi\)
\(68\) 0.886447 0.107497
\(69\) 4.92520 0.592924
\(70\) 0 0
\(71\) −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 0
\(76\) −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 \(0.277385\pi\)
\(80\) 0 0
\(81\) 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
\(85\) 0 0
\(86\) −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 \(0.283845\pi\)
\(90\) 0 0
\(91\) −5.04502 −0.528861
\(92\) −0.685559 −0.0714744
\(93\) 7.57201 0.785180
\(94\) 0.770774 0.0794993
\(95\) 0 0
\(96\) −0.786003 −0.0802211
\(97\) 1.87122 0.189993 0.0949967 0.995478i \(-0.469716\pi\)
0.0949967 + 0.995478i \(0.469716\pi\)
\(98\) −1.46260 −0.147745
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 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 \(-0.324241\pi\)
0.524531 + 0.851392i \(0.324241\pi\)
\(104\) −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 \(0.780718\pi\)
\(108\) −0.139194 −0.0133940
\(109\) 12.7756 1.22368 0.611840 0.790982i \(-0.290430\pi\)
0.611840 + 0.790982i \(0.290430\pi\)
\(110\) 0 0
\(111\) 4.24860 0.403259
\(112\) −4.25901 −0.402439
\(113\) −18.7368 −1.76261 −0.881307 0.472544i \(-0.843336\pi\)
−0.881307 + 0.472544i \(0.843336\pi\)
\(114\) −7.78600 −0.729226
\(115\) 0 0
\(116\) 0.702237 0.0652010
\(117\) −5.04502 −0.466412
\(118\) −11.6572 −1.07313
\(119\) 6.36842 0.583792
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.92520 0.264835
\(123\) 0.646809 0.0583208
\(124\) −1.05398 −0.0946501
\(125\) 0 0
\(126\) −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
\(130\) 0 0
\(131\) 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
\(135\) 0 0
\(136\) 17.3324 1.48624
\(137\) 4.77559 0.408006 0.204003 0.978970i \(-0.434605\pi\)
0.204003 + 0.978970i \(0.434605\pi\)
\(138\) −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 0
\(141\) 0.526989 0.0443805
\(142\) 16.7368 1.40452
\(143\) 5.04502 0.421885
\(144\) −4.25901 −0.354918
\(145\) 0 0
\(146\) −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 \(0.632075\pi\)
\(150\) 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
\(153\) 6.36842 0.514856
\(154\) 1.46260 0.117860
\(155\) 0 0
\(156\) 0.702237 0.0562239
\(157\) 0.946021 0.0755007 0.0377504 0.999287i \(-0.487981\pi\)
0.0377504 + 0.999287i \(0.487981\pi\)
\(158\) −16.7368 −1.33151
\(159\) 3.72161 0.295143
\(160\) 0 0
\(161\) −4.92520 −0.388160
\(162\) −1.46260 −0.114913
\(163\) −8.76663 −0.686655 −0.343328 0.939216i \(-0.611554\pi\)
−0.343328 + 0.939216i \(0.611554\pi\)
\(164\) −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
\(170\) 0 0
\(171\) −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 0
\(176\) 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 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
\(183\) 2.00000 0.147844
\(184\) −13.4045 −0.988191
\(185\) 0 0
\(186\) −11.0748 −0.812044
\(187\) −6.36842 −0.465705
\(188\) −0.0733538 −0.00534988
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −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
\(195\) 0 0
\(196\) 0.139194 0.00994244
\(197\) 2.25756 0.160845 0.0804224 0.996761i \(-0.474373\pi\)
0.0804224 + 0.996761i \(0.474373\pi\)
\(198\) 1.46260 0.103942
\(199\) 3.07480 0.217967 0.108984 0.994044i \(-0.465240\pi\)
0.108984 + 0.994044i \(0.465240\pi\)
\(200\) 0 0
\(201\) 8.76663 0.618350
\(202\) −6.60806 −0.464941
\(203\) 5.04502 0.354091
\(204\) −0.886447 −0.0620637
\(205\) 0 0
\(206\) −15.5720 −1.08495
\(207\) −4.92520 −0.342325
\(208\) 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
\(213\) 11.4432 0.784077
\(214\) 23.3580 1.59672
\(215\) 0 0
\(216\) −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 \(-0.479630\pi\)
0.0639505 + 0.997953i \(0.479630\pi\)
\(224\) 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 \(0.466095\pi\)
\(228\) 0.740987 0.0490730
\(229\) −18.3088 −1.20988 −0.604941 0.796270i \(-0.706803\pi\)
−0.604941 + 0.796270i \(0.706803\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 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
\(235\) 0 0
\(236\) 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 \(0.469933\pi\)
\(240\) 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
\(243\) −1.00000 −0.0641500
\(244\) −0.278388 −0.0178220
\(245\) 0 0
\(246\) −0.946021 −0.0603161
\(247\) 26.8567 1.70885
\(248\) −20.6081 −1.30861
\(249\) 13.1648 0.834288
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) 3.32485 0.207803
\(257\) 6.89541 0.430124 0.215062 0.976600i \(-0.431005\pi\)
0.215062 + 0.976600i \(0.431005\pi\)
\(258\) 15.3836 0.957744
\(259\) −4.24860 −0.263995
\(260\) 0 0
\(261\) 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 \(0.550098\pi\)
\(264\) 2.72161 0.167504
\(265\) 0 0
\(266\) 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 \(0.508603\pi\)
\(270\) 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
\(273\) 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
\(280\) 0 0
\(281\) 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 \(-0.269143\pi\)
0.663328 + 0.748329i \(0.269143\pi\)
\(284\) −1.59283 −0.0945171
\(285\) 0 0
\(286\) −7.37883 −0.436320
\(287\) −0.646809 −0.0381799
\(288\) 0.786003 0.0463157
\(289\) 23.5568 1.38569
\(290\) 0 0
\(291\) −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
\(295\) 0 0
\(296\) −11.5630 −0.672088
\(297\) 1.00000 0.0580259
\(298\) 14.3941 0.833826
\(299\) 24.8477 1.43698
\(300\) 0 0
\(301\) 10.5180 0.606249
\(302\) 6.03875 0.347491
\(303\) −4.51803 −0.259554
\(304\) 22.6724 1.30035
\(305\) 0 0
\(306\) −9.31444 −0.532471
\(307\) 13.5928 0.775784 0.387892 0.921705i \(-0.373203\pi\)
0.387892 + 0.921705i \(0.373203\pi\)
\(308\) −0.139194 −0.00793132
\(309\) −10.6468 −0.605676
\(310\) 0 0
\(311\) 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 \(-0.638605\pi\)
−0.421811 + 0.906684i \(0.638605\pi\)
\(314\) −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 \(0.535644\pi\)
\(318\) −5.44322 −0.305241
\(319\) −5.04502 −0.282467
\(320\) 0 0
\(321\) 15.9702 0.891370
\(322\) 7.20359 0.401440
\(323\) −33.9017 −1.88634
\(324\) 0.139194 0.00773301
\(325\) 0 0
\(326\) 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
\(335\) 0 0
\(336\) 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
\(340\) 0 0
\(341\) 7.57201 0.410047
\(342\) 7.78600 0.421019
\(343\) 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 \(0.292664\pi\)
\(348\) −0.702237 −0.0376438
\(349\) 27.9315 1.49514 0.747568 0.664185i \(-0.231221\pi\)
0.747568 + 0.664185i \(0.231221\pi\)
\(350\) 0 0
\(351\) 5.04502 0.269283
\(352\) −0.786003 −0.0418941
\(353\) 16.5478 0.880751 0.440376 0.897814i \(-0.354845\pi\)
0.440376 + 0.897814i \(0.354845\pi\)
\(354\) 11.6572 0.619574
\(355\) 0 0
\(356\) 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 \(0.302520\pi\)
\(360\) 0 0
\(361\) 9.33863 0.491507
\(362\) −19.8504 −1.04331
\(363\) −1.00000 −0.0524864
\(364\) −0.702237 −0.0368072
\(365\) 0 0
\(366\) −2.92520 −0.152902
\(367\) 19.3836 1.01182 0.505909 0.862587i \(-0.331157\pi\)
0.505909 + 0.862587i \(0.331157\pi\)
\(368\) 20.9765 1.09347
\(369\) −0.646809 −0.0336715
\(370\) 0 0
\(371\) −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 \(0.773521\pi\)
\(374\) 9.31444 0.481638
\(375\) 0 0
\(376\) −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 \(0.395734\pi\)
\(380\) 0 0
\(381\) −2.27839 −0.116725
\(382\) 13.7812 0.705107
\(383\) 17.5928 0.898952 0.449476 0.893293i \(-0.351611\pi\)
0.449476 + 0.893293i \(0.351611\pi\)
\(384\) 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 \(0.329903\pi\)
\(390\) 0 0
\(391\) −31.3657 −1.58623
\(392\) 2.72161 0.137462
\(393\) −4.00000 −0.201773
\(394\) −3.30191 −0.166348
\(395\) 0 0
\(396\) −0.139194 −0.00699477
\(397\) 35.1053 1.76188 0.880941 0.473226i \(-0.156910\pi\)
0.880941 + 0.473226i \(0.156910\pi\)
\(398\) −4.49720 −0.225424
\(399\) 5.32340 0.266504
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) −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 \(0.107087\pi\)
\(410\) 0 0
\(411\) −4.77559 −0.235563
\(412\) 1.48197 0.0730116
\(413\) 7.97021 0.392189
\(414\) 7.20359 0.354037
\(415\) 0 0
\(416\) −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 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
\(423\) −0.526989 −0.0256231
\(424\) −10.1288 −0.491897
\(425\) 0 0
\(426\) −16.7368 −0.810903
\(427\) −2.00000 −0.0967868
\(428\) −2.22296 −0.107451
\(429\) −5.04502 −0.243576
\(430\) 0 0
\(431\) 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 \(-0.288704\pi\)
0.616119 + 0.787653i \(0.288704\pi\)
\(434\) 11.0748 0.531608
\(435\) 0 0
\(436\) 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 \(0.308923\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 46.9915 2.23516
\(443\) 18.0305 0.856653 0.428326 0.903624i \(-0.359103\pi\)
0.428326 + 0.903624i \(0.359103\pi\)
\(444\) 0.591380 0.0280657
\(445\) 0 0
\(446\) −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 \(0.809006\pi\)
\(450\) 0 0
\(451\) 0.646809 0.0304570
\(452\) −2.60806 −0.122673
\(453\) 4.12878 0.193987
\(454\) −4.68556 −0.219904
\(455\) 0 0
\(456\) 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
\(460\) 0 0
\(461\) −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 \(-0.850246\pi\)
−0.891358 + 0.453301i \(0.850246\pi\)
\(464\) −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 \(0.342969\pi\)
\(468\) −0.702237 −0.0324609
\(469\) −8.76663 −0.404805
\(470\) 0 0
\(471\) −0.946021 −0.0435904
\(472\) 21.6918 0.998447
\(473\) −10.5180 −0.483619
\(474\) 16.7368 0.768749
\(475\) 0 0
\(476\) 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 \(0.585478\pi\)
\(480\) 0 0
\(481\) 21.4343 0.977318
\(482\) 8.92038 0.406312
\(483\) 4.92520 0.224104
\(484\) 0.139194 0.00632701
\(485\) 0 0
\(486\) 1.46260 0.0663448
\(487\) −32.4793 −1.47178 −0.735888 0.677103i \(-0.763235\pi\)
−0.735888 + 0.677103i \(0.763235\pi\)
\(488\) −5.44322 −0.246403
\(489\) 8.76663 0.396441
\(490\) 0 0
\(491\) 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
\(495\) 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 \(0.874444\pi\)
\(500\) 0 0
\(501\) −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
\(505\) 0 0
\(506\) −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 \(0.638305\pi\)
\(510\) 0 0
\(511\) 13.0450 0.577078
\(512\) 19.8352 0.876599
\(513\) 5.32340 0.235034
\(514\) −10.0852 −0.444840
\(515\) 0 0
\(516\) −1.46405 −0.0644511
\(517\) 0.526989 0.0231770
\(518\) 6.21400 0.273027
\(519\) 12.3476 0.541999
\(520\) 0 0
\(521\) 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 0
\(526\) 7.43551 0.324204
\(527\) −48.2217 −2.10057
\(528\) −4.25901 −0.185350
\(529\) 1.25756 0.0546767
\(530\) 0 0
\(531\) 7.97021 0.345878
\(532\) −0.740987 −0.0321258
\(533\) 3.26316 0.141343
\(534\) 17.3324 0.750045
\(535\) 0 0
\(536\) −23.8594 −1.03057
\(537\) 5.59283 0.241348
\(538\) 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
\(543\) −13.5720 −0.582430
\(544\) 5.00560 0.214613
\(545\) 0 0
\(546\) −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
\(555\) 0 0
\(556\) −2.14961 −0.0911636
\(557\) −14.8954 −0.631139 −0.315569 0.948903i \(-0.602196\pi\)
−0.315569 + 0.948903i \(0.602196\pi\)
\(558\) 11.0748 0.468834
\(559\) −53.0636 −2.24435
\(560\) 0 0
\(561\) 6.36842 0.268875
\(562\) −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
\(565\) 0 0
\(566\) −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 \(0.432780\pi\)
\(570\) 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
\(573\) 9.42240 0.393626
\(574\) 0.946021 0.0394862
\(575\) 0 0
\(576\) 7.36842 0.307018
\(577\) 29.5124 1.22862 0.614309 0.789065i \(-0.289435\pi\)
0.614309 + 0.789065i \(0.289435\pi\)
\(578\) −34.4541 −1.43310
\(579\) −10.1288 −0.420938
\(580\) 0 0
\(581\) −13.1648 −0.546169
\(582\) 2.73684 0.113446
\(583\) 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 \(0.724359\pi\)
\(588\) −0.139194 −0.00574027
\(589\) 40.3088 1.66090
\(590\) 0 0
\(591\) −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 \(0.448073\pi\)
\(594\) −1.46260 −0.0600111
\(595\) 0 0
\(596\) −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 \(0.310551\pi\)
\(600\) 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
\(603\) −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
\(610\) 0 0
\(611\) 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
\(615\) 0 0
\(616\) −2.72161 −0.109657
\(617\) 44.0305 1.77260 0.886300 0.463112i \(-0.153267\pi\)
0.886300 + 0.463112i \(0.153267\pi\)
\(618\) 15.5720 0.626398
\(619\) 40.0096 1.60812 0.804061 0.594546i \(-0.202668\pi\)
0.804061 + 0.594546i \(0.202668\pi\)
\(620\) 0 0
\(621\) 4.92520 0.197641
\(622\) −11.7008 −0.469159
\(623\) 11.8504 0.474776
\(624\) −21.4868 −0.860160
\(625\) 0 0
\(626\) 21.8296 0.872485
\(627\) −5.32340 −0.212596
\(628\) 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
\(633\) 14.6468 0.582158
\(634\) 5.81994 0.231139
\(635\) 0 0
\(636\) 0.518027 0.0205411
\(637\) −5.04502 −0.199891
\(638\) 7.37883 0.292131
\(639\) −11.4432 −0.452687
\(640\) 0 0
\(641\) −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 \(-0.465525\pi\)
0.108094 + 0.994141i \(0.465525\pi\)
\(644\) −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 \(0.441711\pi\)
\(648\) 2.72161 0.106915
\(649\) −7.97021 −0.312858
\(650\) 0 0
\(651\) 7.57201 0.296770
\(652\) −1.22026 −0.0477892
\(653\) 29.9821 1.17329 0.586645 0.809844i \(-0.300449\pi\)
0.586645 + 0.809844i \(0.300449\pi\)
\(654\) 18.6856 0.730663
\(655\) 0 0
\(656\) 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 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
\(665\) 0 0
\(666\) 6.21400 0.240788
\(667\) −24.8477 −0.962107
\(668\) 3.39194 0.131238
\(669\) −1.90997 −0.0738436
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) −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 0
\(676\) 1.73327 0.0666643
\(677\) 1.26316 0.0485472 0.0242736 0.999705i \(-0.492273\pi\)
0.0242736 + 0.999705i \(0.492273\pi\)
\(678\) −27.4045 −1.05246
\(679\) 1.87122 0.0718108
\(680\) 0 0
\(681\) −3.20359 −0.122762
\(682\) −11.0748 −0.424076
\(683\) −37.6441 −1.44041 −0.720206 0.693760i \(-0.755953\pi\)
−0.720206 + 0.693760i \(0.755953\pi\)
\(684\) −0.740987 −0.0283323
\(685\) 0 0
\(686\) −1.46260 −0.0558423
\(687\) 18.3088 0.698526
\(688\) −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
\(693\) −1.00000 −0.0379869
\(694\) −33.0361 −1.25403
\(695\) 0 0
\(696\) −13.7306 −0.520456
\(697\) −4.11915 −0.156024
\(698\) −40.8525 −1.54629
\(699\) −16.5872 −0.627387
\(700\) 0 0
\(701\) −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
\(705\) 0 0
\(706\) −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 \(0.875667\pi\)
\(710\) 0 0
\(711\) 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 \(0.544144\pi\)
\(720\) 0 0
\(721\) 10.6468 0.396508
\(722\) −13.6587 −0.508323
\(723\) 6.09899 0.226824
\(724\) 1.88914 0.0702095
\(725\) 0 0
\(726\) 1.46260 0.0542821
\(727\) −18.9557 −0.703026 −0.351513 0.936183i \(-0.614333\pi\)
−0.351513 + 0.936183i \(0.614333\pi\)
\(728\) −13.7306 −0.508889
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 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
\(735\) 0 0
\(736\) −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 \(0.663909\pi\)
\(740\) 0 0
\(741\) −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
\(745\) 0 0
\(746\) 42.7881 1.56658
\(747\) −13.1648 −0.481676
\(748\) −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
\(753\) −1.62262 −0.0591314
\(754\) 37.2263 1.35570
\(755\) 0 0
\(756\) −0.139194 −0.00506244
\(757\) 29.3442 1.06653 0.533267 0.845947i \(-0.320964\pi\)
0.533267 + 0.845947i \(0.320964\pi\)
\(758\) −18.3220 −0.665483
\(759\) −4.92520 −0.178773
\(760\) 0 0
\(761\) 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
\(765\) 0 0
\(766\) −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 \(0.443564\pi\)
\(770\) 0 0
\(771\) −6.89541 −0.248332
\(772\) 1.40987 0.0507422
\(773\) −29.7223 −1.06904 −0.534518 0.845157i \(-0.679507\pi\)
−0.534518 + 0.845157i \(0.679507\pi\)
\(774\) −15.3836 −0.552954
\(775\) 0 0
\(776\) 5.09273 0.182818
\(777\) 4.24860 0.152418
\(778\) −29.3836 −1.05345
\(779\) 3.44322 0.123366
\(780\) 0 0
\(781\) 11.4432 0.409471
\(782\) 45.8755 1.64050
\(783\) −5.04502 −0.180294
\(784\) −4.25901 −0.152108
\(785\) 0 0
\(786\) 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
\(790\) 0 0
\(791\) −18.7368 −0.666205
\(792\) −2.72161 −0.0967083
\(793\) 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 \(0.276780\pi\)
\(798\) −7.78600 −0.275622
\(799\) −3.35609 −0.118730
\(800\) 0 0
\(801\) 11.8504 0.418713
\(802\) −14.0000 −0.494357
\(803\) −13.0450 −0.460349
\(804\) 1.22026 0.0430354
\(805\) 0 0
\(806\) −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 \(0.214168\pi\)
\(810\) 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
\(813\) 25.3234 0.888131
\(814\) −6.21400 −0.217800
\(815\) 0 0
\(816\) 27.1232 0.949501
\(817\) −55.9917 −1.95890
\(818\) −55.8421 −1.95247
\(819\) −5.04502 −0.176287
\(820\) 0 0
\(821\) −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 \(-0.536792\pi\)
−0.115327 + 0.993328i \(0.536792\pi\)
\(824\) 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 \(0.521630\pi\)
\(830\) 0 0
\(831\) −24.8269 −0.861235
\(832\) −37.1738 −1.28877
\(833\) 6.36842 0.220653
\(834\) −22.5872 −0.782132
\(835\) 0 0
\(836\) 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 \(0.552887\pi\)
\(840\) 0 0
\(841\) −3.54781 −0.122338
\(842\) −22.1627 −0.763778
\(843\) −1.90101 −0.0654741
\(844\) −2.03875 −0.0701767
\(845\) 0 0
\(846\) 0.770774 0.0264998
\(847\) 1.00000 0.0343604
\(848\) 15.8504 0.544305
\(849\) −22.3178 −0.765945
\(850\) 0 0
\(851\) 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
\(855\) 0 0
\(856\) −43.4647 −1.48559
\(857\) −10.4793 −0.357965 −0.178983 0.983852i \(-0.557281\pi\)
−0.178983 + 0.983852i \(0.557281\pi\)
\(858\) 7.37883 0.251909
\(859\) 2.88645 0.0984843 0.0492421 0.998787i \(-0.484319\pi\)
0.0492421 + 0.998787i \(0.484319\pi\)
\(860\) 0 0
\(861\) 0.646809 0.0220432
\(862\) −8.13650 −0.277130
\(863\) 20.5485 0.699479 0.349739 0.936847i \(-0.386270\pi\)
0.349739 + 0.936847i \(0.386270\pi\)
\(864\) −0.786003 −0.0267404
\(865\) 0 0
\(866\) −37.5028 −1.27440
\(867\) −23.5568 −0.800030
\(868\) −1.05398 −0.0357744
\(869\) −11.4432 −0.388185
\(870\) 0 0
\(871\) 44.2278 1.49860
\(872\) 34.7702 1.17747
\(873\) 1.87122 0.0633311
\(874\) −38.3476 −1.29713
\(875\) 0 0
\(876\) −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 \(-0.295481\pi\)
0.599210 + 0.800592i \(0.295481\pi\)
\(884\) −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 \(0.626036\pi\)
\(888\) 11.5630 0.388030
\(889\) 2.27839 0.0764147
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0.265856 0.00890153
\(893\) 2.80538 0.0938784
\(894\) −14.3941 −0.481409
\(895\) 0 0
\(896\) −12.3490 −0.412553
\(897\) −24.8477 −0.829640
\(898\) 51.1565 1.70712
\(899\) −38.2009 −1.27407
\(900\) 0 0
\(901\) −23.7008 −0.789588
\(902\) −0.946021 −0.0314991
\(903\) −10.5180 −0.350018
\(904\) −50.9944 −1.69605
\(905\) 0 0
\(906\) −6.03875 −0.200624
\(907\) 57.1745 1.89845 0.949224 0.314602i \(-0.101871\pi\)
0.949224 + 0.314602i \(0.101871\pi\)
\(908\) 0.445920 0.0147984
\(909\) 4.51803 0.149853
\(910\) 0 0
\(911\) 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
\(915\) 0 0
\(916\) −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 \(0.436210\pi\)
\(920\) 0 0
\(921\) −13.5928 −0.447899
\(922\) 4.09003 0.134698
\(923\) 57.7312 1.90025
\(924\) 0.139194 0.00457915
\(925\) 0 0
\(926\) 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 \(0.354551\pi\)
\(930\) 0 0
\(931\) −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 0
\(941\) −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
\(945\) 0 0
\(946\) 15.3836 0.500166
\(947\) 17.3532 0.563903 0.281951 0.959429i \(-0.409018\pi\)
0.281951 + 0.959429i \(0.409018\pi\)
\(948\) −1.59283 −0.0517327
\(949\) −65.8123 −2.13636
\(950\) 0 0
\(951\) 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
\(955\) 0 0
\(956\) 0.405923 0.0131285
\(957\) 5.04502 0.163082
\(958\) 16.9861 0.548796
\(959\) 4.77559 0.154212
\(960\) 0 0
\(961\) 26.3353 0.849525
\(962\) −31.3497 −1.01076
\(963\) −15.9702 −0.514633
\(964\) −0.848944 −0.0273427
\(965\) 0 0
\(966\) −7.20359 −0.231772
\(967\) 1.53326 0.0493062 0.0246531 0.999696i \(-0.492152\pi\)
0.0246531 + 0.999696i \(0.492152\pi\)
\(968\) 2.72161 0.0874759
\(969\) 33.9017 1.08908
\(970\) 0 0
\(971\) −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
\(975\) 0 0
\(976\) 8.51803 0.272655
\(977\) 55.9017 1.78845 0.894227 0.447615i \(-0.147726\pi\)
0.894227 + 0.447615i \(0.147726\pi\)
\(978\) −12.8221 −0.410004
\(979\) −11.8504 −0.378740
\(980\) 0 0
\(981\) 12.7756 0.407893
\(982\) −39.0171 −1.24509
\(983\) 53.0361 1.69159 0.845794 0.533510i \(-0.179127\pi\)
0.845794 + 0.533510i \(0.179127\pi\)
\(984\) 1.76036 0.0561183
\(985\) 0 0
\(986\) −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
\(995\) 0 0
\(996\) 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 5775.2.a.bp.1.2 3
5.4 even 2 231.2.a.e.1.2 3
15.14 odd 2 693.2.a.l.1.2 3
20.19 odd 2 3696.2.a.bo.1.2 3
35.34 odd 2 1617.2.a.t.1.2 3
55.54 odd 2 2541.2.a.bg.1.2 3
105.104 even 2 4851.2.a.bi.1.2 3
165.164 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 5.4 even 2
693.2.a.l.1.2 3 15.14 odd 2
1617.2.a.t.1.2 3 35.34 odd 2
2541.2.a.bg.1.2 3 55.54 odd 2
3696.2.a.bo.1.2 3 20.19 odd 2
4851.2.a.bi.1.2 3 105.104 even 2
5775.2.a.bp.1.2 3 1.1 even 1 trivial
7623.2.a.cd.1.2 3 165.164 even 2