Properties

Label 7168.2.a.bb.1.6
Level $7168$
Weight $2$
Character 7168.1
Self dual yes
Analytic conductor $57.237$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7168 = 2^{10} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7168.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.2367681689\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.9433055232.1
Defining polynomial: \(x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 9 x^{4} - 76 x^{3} - 4 x^{2} + 48 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1792)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.177920\) of defining polynomial
Character \(\chi\) \(=\) 7168.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.67251 q^{3} +2.65618 q^{5} +1.00000 q^{7} -0.202696 q^{9} +O(q^{10})\) \(q+1.67251 q^{3} +2.65618 q^{5} +1.00000 q^{7} -0.202696 q^{9} -0.826484 q^{11} -5.57281 q^{13} +4.44250 q^{15} -1.74896 q^{17} -5.93216 q^{19} +1.67251 q^{21} +3.04150 q^{23} +2.05530 q^{25} -5.35655 q^{27} -6.26943 q^{29} -7.90794 q^{31} -1.38231 q^{33} +2.65618 q^{35} +8.30514 q^{37} -9.32061 q^{39} +1.38922 q^{41} -2.45934 q^{43} -0.538396 q^{45} -1.80017 q^{47} +1.00000 q^{49} -2.92516 q^{51} -13.7698 q^{53} -2.19529 q^{55} -9.92162 q^{57} -6.70340 q^{59} +4.38770 q^{61} -0.202696 q^{63} -14.8024 q^{65} +6.80390 q^{67} +5.08696 q^{69} +1.11625 q^{71} +11.2521 q^{73} +3.43752 q^{75} -0.826484 q^{77} -7.61158 q^{79} -8.35083 q^{81} +15.8207 q^{83} -4.64556 q^{85} -10.4857 q^{87} -0.428825 q^{89} -5.57281 q^{91} -13.2261 q^{93} -15.7569 q^{95} -19.2163 q^{97} +0.167525 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + 8q^{7} + 12q^{9} + O(q^{10}) \) \( 8q - 8q^{5} + 8q^{7} + 12q^{9} - 12q^{11} - 20q^{13} + 4q^{17} - 4q^{19} + 8q^{23} + 12q^{25} + 12q^{27} - 8q^{29} - 4q^{31} + 8q^{33} - 8q^{35} - 8q^{37} - 16q^{39} - 12q^{41} + 4q^{43} - 52q^{45} + 20q^{47} + 8q^{49} - 32q^{51} - 40q^{53} + 24q^{55} - 4q^{57} - 4q^{59} + 8q^{61} + 12q^{63} + 36q^{65} - 28q^{67} - 4q^{69} - 16q^{71} + 16q^{73} + 28q^{75} - 12q^{77} + 20q^{81} + 8q^{83} - 16q^{85} + 20q^{87} + 16q^{89} - 20q^{91} - 16q^{93} - 40q^{95} - 36q^{97} + 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.67251 0.965627 0.482813 0.875723i \(-0.339615\pi\)
0.482813 + 0.875723i \(0.339615\pi\)
\(4\) 0 0
\(5\) 2.65618 1.18788 0.593940 0.804509i \(-0.297572\pi\)
0.593940 + 0.804509i \(0.297572\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −0.202696 −0.0675652
\(10\) 0 0
\(11\) −0.826484 −0.249194 −0.124597 0.992207i \(-0.539764\pi\)
−0.124597 + 0.992207i \(0.539764\pi\)
\(12\) 0 0
\(13\) −5.57281 −1.54562 −0.772810 0.634637i \(-0.781150\pi\)
−0.772810 + 0.634637i \(0.781150\pi\)
\(14\) 0 0
\(15\) 4.44250 1.14705
\(16\) 0 0
\(17\) −1.74896 −0.424185 −0.212093 0.977250i \(-0.568028\pi\)
−0.212093 + 0.977250i \(0.568028\pi\)
\(18\) 0 0
\(19\) −5.93216 −1.36093 −0.680465 0.732781i \(-0.738222\pi\)
−0.680465 + 0.732781i \(0.738222\pi\)
\(20\) 0 0
\(21\) 1.67251 0.364973
\(22\) 0 0
\(23\) 3.04150 0.634198 0.317099 0.948393i \(-0.397291\pi\)
0.317099 + 0.948393i \(0.397291\pi\)
\(24\) 0 0
\(25\) 2.05530 0.411060
\(26\) 0 0
\(27\) −5.35655 −1.03087
\(28\) 0 0
\(29\) −6.26943 −1.16420 −0.582102 0.813116i \(-0.697770\pi\)
−0.582102 + 0.813116i \(0.697770\pi\)
\(30\) 0 0
\(31\) −7.90794 −1.42031 −0.710154 0.704046i \(-0.751375\pi\)
−0.710154 + 0.704046i \(0.751375\pi\)
\(32\) 0 0
\(33\) −1.38231 −0.240629
\(34\) 0 0
\(35\) 2.65618 0.448977
\(36\) 0 0
\(37\) 8.30514 1.36536 0.682678 0.730719i \(-0.260815\pi\)
0.682678 + 0.730719i \(0.260815\pi\)
\(38\) 0 0
\(39\) −9.32061 −1.49249
\(40\) 0 0
\(41\) 1.38922 0.216960 0.108480 0.994099i \(-0.465402\pi\)
0.108480 + 0.994099i \(0.465402\pi\)
\(42\) 0 0
\(43\) −2.45934 −0.375046 −0.187523 0.982260i \(-0.560046\pi\)
−0.187523 + 0.982260i \(0.560046\pi\)
\(44\) 0 0
\(45\) −0.538396 −0.0802594
\(46\) 0 0
\(47\) −1.80017 −0.262582 −0.131291 0.991344i \(-0.541912\pi\)
−0.131291 + 0.991344i \(0.541912\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.92516 −0.409604
\(52\) 0 0
\(53\) −13.7698 −1.89143 −0.945717 0.324992i \(-0.894639\pi\)
−0.945717 + 0.324992i \(0.894639\pi\)
\(54\) 0 0
\(55\) −2.19529 −0.296013
\(56\) 0 0
\(57\) −9.92162 −1.31415
\(58\) 0 0
\(59\) −6.70340 −0.872708 −0.436354 0.899775i \(-0.643731\pi\)
−0.436354 + 0.899775i \(0.643731\pi\)
\(60\) 0 0
\(61\) 4.38770 0.561787 0.280893 0.959739i \(-0.409369\pi\)
0.280893 + 0.959739i \(0.409369\pi\)
\(62\) 0 0
\(63\) −0.202696 −0.0255372
\(64\) 0 0
\(65\) −14.8024 −1.83601
\(66\) 0 0
\(67\) 6.80390 0.831228 0.415614 0.909541i \(-0.363567\pi\)
0.415614 + 0.909541i \(0.363567\pi\)
\(68\) 0 0
\(69\) 5.08696 0.612398
\(70\) 0 0
\(71\) 1.11625 0.132475 0.0662375 0.997804i \(-0.478900\pi\)
0.0662375 + 0.997804i \(0.478900\pi\)
\(72\) 0 0
\(73\) 11.2521 1.31696 0.658478 0.752600i \(-0.271201\pi\)
0.658478 + 0.752600i \(0.271201\pi\)
\(74\) 0 0
\(75\) 3.43752 0.396931
\(76\) 0 0
\(77\) −0.826484 −0.0941866
\(78\) 0 0
\(79\) −7.61158 −0.856370 −0.428185 0.903691i \(-0.640847\pi\)
−0.428185 + 0.903691i \(0.640847\pi\)
\(80\) 0 0
\(81\) −8.35083 −0.927870
\(82\) 0 0
\(83\) 15.8207 1.73655 0.868273 0.496086i \(-0.165230\pi\)
0.868273 + 0.496086i \(0.165230\pi\)
\(84\) 0 0
\(85\) −4.64556 −0.503881
\(86\) 0 0
\(87\) −10.4857 −1.12419
\(88\) 0 0
\(89\) −0.428825 −0.0454554 −0.0227277 0.999742i \(-0.507235\pi\)
−0.0227277 + 0.999742i \(0.507235\pi\)
\(90\) 0 0
\(91\) −5.57281 −0.584190
\(92\) 0 0
\(93\) −13.2261 −1.37149
\(94\) 0 0
\(95\) −15.7569 −1.61662
\(96\) 0 0
\(97\) −19.2163 −1.95111 −0.975557 0.219745i \(-0.929478\pi\)
−0.975557 + 0.219745i \(0.929478\pi\)
\(98\) 0 0
\(99\) 0.167525 0.0168369
\(100\) 0 0
\(101\) −6.89495 −0.686073 −0.343037 0.939322i \(-0.611456\pi\)
−0.343037 + 0.939322i \(0.611456\pi\)
\(102\) 0 0
\(103\) −6.09849 −0.600902 −0.300451 0.953797i \(-0.597137\pi\)
−0.300451 + 0.953797i \(0.597137\pi\)
\(104\) 0 0
\(105\) 4.44250 0.433544
\(106\) 0 0
\(107\) 7.35376 0.710915 0.355457 0.934692i \(-0.384325\pi\)
0.355457 + 0.934692i \(0.384325\pi\)
\(108\) 0 0
\(109\) −10.8902 −1.04309 −0.521547 0.853222i \(-0.674645\pi\)
−0.521547 + 0.853222i \(0.674645\pi\)
\(110\) 0 0
\(111\) 13.8905 1.31842
\(112\) 0 0
\(113\) 14.2646 1.34190 0.670952 0.741500i \(-0.265885\pi\)
0.670952 + 0.741500i \(0.265885\pi\)
\(114\) 0 0
\(115\) 8.07879 0.753351
\(116\) 0 0
\(117\) 1.12958 0.104430
\(118\) 0 0
\(119\) −1.74896 −0.160327
\(120\) 0 0
\(121\) −10.3169 −0.937902
\(122\) 0 0
\(123\) 2.32349 0.209502
\(124\) 0 0
\(125\) −7.82165 −0.699590
\(126\) 0 0
\(127\) 8.98310 0.797121 0.398561 0.917142i \(-0.369510\pi\)
0.398561 + 0.917142i \(0.369510\pi\)
\(128\) 0 0
\(129\) −4.11328 −0.362154
\(130\) 0 0
\(131\) 18.0378 1.57597 0.787985 0.615694i \(-0.211124\pi\)
0.787985 + 0.615694i \(0.211124\pi\)
\(132\) 0 0
\(133\) −5.93216 −0.514383
\(134\) 0 0
\(135\) −14.2280 −1.22455
\(136\) 0 0
\(137\) −11.7927 −1.00751 −0.503757 0.863845i \(-0.668049\pi\)
−0.503757 + 0.863845i \(0.668049\pi\)
\(138\) 0 0
\(139\) 0.741071 0.0628568 0.0314284 0.999506i \(-0.489994\pi\)
0.0314284 + 0.999506i \(0.489994\pi\)
\(140\) 0 0
\(141\) −3.01081 −0.253556
\(142\) 0 0
\(143\) 4.60584 0.385160
\(144\) 0 0
\(145\) −16.6527 −1.38294
\(146\) 0 0
\(147\) 1.67251 0.137947
\(148\) 0 0
\(149\) −2.81229 −0.230392 −0.115196 0.993343i \(-0.536750\pi\)
−0.115196 + 0.993343i \(0.536750\pi\)
\(150\) 0 0
\(151\) −15.1887 −1.23604 −0.618020 0.786162i \(-0.712065\pi\)
−0.618020 + 0.786162i \(0.712065\pi\)
\(152\) 0 0
\(153\) 0.354507 0.0286602
\(154\) 0 0
\(155\) −21.0049 −1.68716
\(156\) 0 0
\(157\) 11.1953 0.893484 0.446742 0.894663i \(-0.352584\pi\)
0.446742 + 0.894663i \(0.352584\pi\)
\(158\) 0 0
\(159\) −23.0303 −1.82642
\(160\) 0 0
\(161\) 3.04150 0.239704
\(162\) 0 0
\(163\) −8.94026 −0.700255 −0.350128 0.936702i \(-0.613862\pi\)
−0.350128 + 0.936702i \(0.613862\pi\)
\(164\) 0 0
\(165\) −3.67166 −0.285838
\(166\) 0 0
\(167\) 22.5263 1.74314 0.871570 0.490271i \(-0.163102\pi\)
0.871570 + 0.490271i \(0.163102\pi\)
\(168\) 0 0
\(169\) 18.0563 1.38894
\(170\) 0 0
\(171\) 1.20242 0.0919515
\(172\) 0 0
\(173\) −0.0149780 −0.00113876 −0.000569379 1.00000i \(-0.500181\pi\)
−0.000569379 1.00000i \(0.500181\pi\)
\(174\) 0 0
\(175\) 2.05530 0.155366
\(176\) 0 0
\(177\) −11.2115 −0.842710
\(178\) 0 0
\(179\) −20.3038 −1.51757 −0.758787 0.651338i \(-0.774208\pi\)
−0.758787 + 0.651338i \(0.774208\pi\)
\(180\) 0 0
\(181\) −12.2375 −0.909605 −0.454803 0.890592i \(-0.650290\pi\)
−0.454803 + 0.890592i \(0.650290\pi\)
\(182\) 0 0
\(183\) 7.33848 0.542476
\(184\) 0 0
\(185\) 22.0600 1.62188
\(186\) 0 0
\(187\) 1.44549 0.105705
\(188\) 0 0
\(189\) −5.35655 −0.389632
\(190\) 0 0
\(191\) 17.2085 1.24517 0.622583 0.782554i \(-0.286083\pi\)
0.622583 + 0.782554i \(0.286083\pi\)
\(192\) 0 0
\(193\) 7.00982 0.504578 0.252289 0.967652i \(-0.418817\pi\)
0.252289 + 0.967652i \(0.418817\pi\)
\(194\) 0 0
\(195\) −24.7572 −1.77290
\(196\) 0 0
\(197\) −21.8036 −1.55345 −0.776723 0.629843i \(-0.783119\pi\)
−0.776723 + 0.629843i \(0.783119\pi\)
\(198\) 0 0
\(199\) 15.3483 1.08801 0.544005 0.839082i \(-0.316907\pi\)
0.544005 + 0.839082i \(0.316907\pi\)
\(200\) 0 0
\(201\) 11.3796 0.802656
\(202\) 0 0
\(203\) −6.26943 −0.440028
\(204\) 0 0
\(205\) 3.69002 0.257722
\(206\) 0 0
\(207\) −0.616500 −0.0428497
\(208\) 0 0
\(209\) 4.90283 0.339136
\(210\) 0 0
\(211\) −6.19572 −0.426531 −0.213266 0.976994i \(-0.568410\pi\)
−0.213266 + 0.976994i \(0.568410\pi\)
\(212\) 0 0
\(213\) 1.86695 0.127921
\(214\) 0 0
\(215\) −6.53246 −0.445510
\(216\) 0 0
\(217\) −7.90794 −0.536826
\(218\) 0 0
\(219\) 18.8193 1.27169
\(220\) 0 0
\(221\) 9.74663 0.655629
\(222\) 0 0
\(223\) −0.528935 −0.0354201 −0.0177101 0.999843i \(-0.505638\pi\)
−0.0177101 + 0.999843i \(0.505638\pi\)
\(224\) 0 0
\(225\) −0.416601 −0.0277734
\(226\) 0 0
\(227\) 25.2769 1.67769 0.838845 0.544371i \(-0.183232\pi\)
0.838845 + 0.544371i \(0.183232\pi\)
\(228\) 0 0
\(229\) −12.3496 −0.816084 −0.408042 0.912963i \(-0.633788\pi\)
−0.408042 + 0.912963i \(0.633788\pi\)
\(230\) 0 0
\(231\) −1.38231 −0.0909491
\(232\) 0 0
\(233\) 26.9485 1.76545 0.882727 0.469885i \(-0.155705\pi\)
0.882727 + 0.469885i \(0.155705\pi\)
\(234\) 0 0
\(235\) −4.78158 −0.311916
\(236\) 0 0
\(237\) −12.7305 −0.826934
\(238\) 0 0
\(239\) −19.8050 −1.28108 −0.640539 0.767926i \(-0.721289\pi\)
−0.640539 + 0.767926i \(0.721289\pi\)
\(240\) 0 0
\(241\) −3.73993 −0.240910 −0.120455 0.992719i \(-0.538435\pi\)
−0.120455 + 0.992719i \(0.538435\pi\)
\(242\) 0 0
\(243\) 2.10278 0.134894
\(244\) 0 0
\(245\) 2.65618 0.169697
\(246\) 0 0
\(247\) 33.0588 2.10348
\(248\) 0 0
\(249\) 26.4603 1.67686
\(250\) 0 0
\(251\) 14.3509 0.905818 0.452909 0.891557i \(-0.350386\pi\)
0.452909 + 0.891557i \(0.350386\pi\)
\(252\) 0 0
\(253\) −2.51376 −0.158039
\(254\) 0 0
\(255\) −7.76976 −0.486561
\(256\) 0 0
\(257\) 12.6111 0.786657 0.393328 0.919398i \(-0.371324\pi\)
0.393328 + 0.919398i \(0.371324\pi\)
\(258\) 0 0
\(259\) 8.30514 0.516056
\(260\) 0 0
\(261\) 1.27079 0.0786597
\(262\) 0 0
\(263\) −7.01176 −0.432364 −0.216182 0.976353i \(-0.569360\pi\)
−0.216182 + 0.976353i \(0.569360\pi\)
\(264\) 0 0
\(265\) −36.5752 −2.24680
\(266\) 0 0
\(267\) −0.717217 −0.0438929
\(268\) 0 0
\(269\) −5.54805 −0.338271 −0.169135 0.985593i \(-0.554098\pi\)
−0.169135 + 0.985593i \(0.554098\pi\)
\(270\) 0 0
\(271\) −23.5746 −1.43205 −0.716026 0.698073i \(-0.754041\pi\)
−0.716026 + 0.698073i \(0.754041\pi\)
\(272\) 0 0
\(273\) −9.32061 −0.564109
\(274\) 0 0
\(275\) −1.69868 −0.102434
\(276\) 0 0
\(277\) −11.1677 −0.671003 −0.335502 0.942040i \(-0.608906\pi\)
−0.335502 + 0.942040i \(0.608906\pi\)
\(278\) 0 0
\(279\) 1.60290 0.0959634
\(280\) 0 0
\(281\) 33.1753 1.97907 0.989536 0.144288i \(-0.0460891\pi\)
0.989536 + 0.144288i \(0.0460891\pi\)
\(282\) 0 0
\(283\) −2.91674 −0.173382 −0.0866912 0.996235i \(-0.527629\pi\)
−0.0866912 + 0.996235i \(0.527629\pi\)
\(284\) 0 0
\(285\) −26.3536 −1.56105
\(286\) 0 0
\(287\) 1.38922 0.0820030
\(288\) 0 0
\(289\) −13.9411 −0.820067
\(290\) 0 0
\(291\) −32.1395 −1.88405
\(292\) 0 0
\(293\) 7.53003 0.439909 0.219954 0.975510i \(-0.429409\pi\)
0.219954 + 0.975510i \(0.429409\pi\)
\(294\) 0 0
\(295\) −17.8054 −1.03667
\(296\) 0 0
\(297\) 4.42711 0.256887
\(298\) 0 0
\(299\) −16.9497 −0.980229
\(300\) 0 0
\(301\) −2.45934 −0.141754
\(302\) 0 0
\(303\) −11.5319 −0.662491
\(304\) 0 0
\(305\) 11.6545 0.667336
\(306\) 0 0
\(307\) 14.0353 0.801037 0.400518 0.916289i \(-0.368830\pi\)
0.400518 + 0.916289i \(0.368830\pi\)
\(308\) 0 0
\(309\) −10.1998 −0.580247
\(310\) 0 0
\(311\) 9.78126 0.554644 0.277322 0.960777i \(-0.410553\pi\)
0.277322 + 0.960777i \(0.410553\pi\)
\(312\) 0 0
\(313\) 5.68720 0.321460 0.160730 0.986998i \(-0.448615\pi\)
0.160730 + 0.986998i \(0.448615\pi\)
\(314\) 0 0
\(315\) −0.538396 −0.0303352
\(316\) 0 0
\(317\) −30.9235 −1.73683 −0.868417 0.495834i \(-0.834863\pi\)
−0.868417 + 0.495834i \(0.834863\pi\)
\(318\) 0 0
\(319\) 5.18159 0.290113
\(320\) 0 0
\(321\) 12.2993 0.686478
\(322\) 0 0
\(323\) 10.3751 0.577286
\(324\) 0 0
\(325\) −11.4538 −0.635343
\(326\) 0 0
\(327\) −18.2141 −1.00724
\(328\) 0 0
\(329\) −1.80017 −0.0992466
\(330\) 0 0
\(331\) 12.3877 0.680890 0.340445 0.940265i \(-0.389422\pi\)
0.340445 + 0.940265i \(0.389422\pi\)
\(332\) 0 0
\(333\) −1.68342 −0.0922506
\(334\) 0 0
\(335\) 18.0724 0.987400
\(336\) 0 0
\(337\) −19.1758 −1.04457 −0.522286 0.852771i \(-0.674921\pi\)
−0.522286 + 0.852771i \(0.674921\pi\)
\(338\) 0 0
\(339\) 23.8578 1.29578
\(340\) 0 0
\(341\) 6.53579 0.353933
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 13.5119 0.727456
\(346\) 0 0
\(347\) 9.00913 0.483635 0.241818 0.970322i \(-0.422256\pi\)
0.241818 + 0.970322i \(0.422256\pi\)
\(348\) 0 0
\(349\) 19.9165 1.06610 0.533052 0.846082i \(-0.321045\pi\)
0.533052 + 0.846082i \(0.321045\pi\)
\(350\) 0 0
\(351\) 29.8511 1.59333
\(352\) 0 0
\(353\) 12.8957 0.686371 0.343185 0.939268i \(-0.388494\pi\)
0.343185 + 0.939268i \(0.388494\pi\)
\(354\) 0 0
\(355\) 2.96498 0.157365
\(356\) 0 0
\(357\) −2.92516 −0.154816
\(358\) 0 0
\(359\) 33.5832 1.77245 0.886227 0.463252i \(-0.153318\pi\)
0.886227 + 0.463252i \(0.153318\pi\)
\(360\) 0 0
\(361\) 16.1905 0.852130
\(362\) 0 0
\(363\) −17.2552 −0.905663
\(364\) 0 0
\(365\) 29.8875 1.56439
\(366\) 0 0
\(367\) −7.74580 −0.404328 −0.202164 0.979352i \(-0.564797\pi\)
−0.202164 + 0.979352i \(0.564797\pi\)
\(368\) 0 0
\(369\) −0.281589 −0.0146589
\(370\) 0 0
\(371\) −13.7698 −0.714895
\(372\) 0 0
\(373\) 2.61073 0.135178 0.0675892 0.997713i \(-0.478469\pi\)
0.0675892 + 0.997713i \(0.478469\pi\)
\(374\) 0 0
\(375\) −13.0818 −0.675543
\(376\) 0 0
\(377\) 34.9384 1.79942
\(378\) 0 0
\(379\) 34.5705 1.77577 0.887884 0.460068i \(-0.152175\pi\)
0.887884 + 0.460068i \(0.152175\pi\)
\(380\) 0 0
\(381\) 15.0244 0.769722
\(382\) 0 0
\(383\) 22.2480 1.13682 0.568411 0.822745i \(-0.307559\pi\)
0.568411 + 0.822745i \(0.307559\pi\)
\(384\) 0 0
\(385\) −2.19529 −0.111882
\(386\) 0 0
\(387\) 0.498498 0.0253401
\(388\) 0 0
\(389\) −9.80992 −0.497383 −0.248691 0.968583i \(-0.580000\pi\)
−0.248691 + 0.968583i \(0.580000\pi\)
\(390\) 0 0
\(391\) −5.31947 −0.269017
\(392\) 0 0
\(393\) 30.1685 1.52180
\(394\) 0 0
\(395\) −20.2177 −1.01727
\(396\) 0 0
\(397\) −23.1991 −1.16433 −0.582164 0.813072i \(-0.697794\pi\)
−0.582164 + 0.813072i \(0.697794\pi\)
\(398\) 0 0
\(399\) −9.92162 −0.496702
\(400\) 0 0
\(401\) 22.8150 1.13932 0.569662 0.821879i \(-0.307074\pi\)
0.569662 + 0.821879i \(0.307074\pi\)
\(402\) 0 0
\(403\) 44.0695 2.19526
\(404\) 0 0
\(405\) −22.1813 −1.10220
\(406\) 0 0
\(407\) −6.86407 −0.340239
\(408\) 0 0
\(409\) 13.9196 0.688281 0.344140 0.938918i \(-0.388170\pi\)
0.344140 + 0.938918i \(0.388170\pi\)
\(410\) 0 0
\(411\) −19.7234 −0.972883
\(412\) 0 0
\(413\) −6.70340 −0.329853
\(414\) 0 0
\(415\) 42.0226 2.06281
\(416\) 0 0
\(417\) 1.23945 0.0606962
\(418\) 0 0
\(419\) −35.6151 −1.73991 −0.869955 0.493131i \(-0.835852\pi\)
−0.869955 + 0.493131i \(0.835852\pi\)
\(420\) 0 0
\(421\) −22.7369 −1.10813 −0.554064 0.832474i \(-0.686924\pi\)
−0.554064 + 0.832474i \(0.686924\pi\)
\(422\) 0 0
\(423\) 0.364887 0.0177414
\(424\) 0 0
\(425\) −3.59464 −0.174366
\(426\) 0 0
\(427\) 4.38770 0.212335
\(428\) 0 0
\(429\) 7.70334 0.371921
\(430\) 0 0
\(431\) 0.695976 0.0335240 0.0167620 0.999860i \(-0.494664\pi\)
0.0167620 + 0.999860i \(0.494664\pi\)
\(432\) 0 0
\(433\) −26.4982 −1.27342 −0.636711 0.771103i \(-0.719705\pi\)
−0.636711 + 0.771103i \(0.719705\pi\)
\(434\) 0 0
\(435\) −27.8520 −1.33540
\(436\) 0 0
\(437\) −18.0427 −0.863099
\(438\) 0 0
\(439\) −5.34131 −0.254927 −0.127463 0.991843i \(-0.540684\pi\)
−0.127463 + 0.991843i \(0.540684\pi\)
\(440\) 0 0
\(441\) −0.202696 −0.00965217
\(442\) 0 0
\(443\) 25.5034 1.21171 0.605853 0.795577i \(-0.292832\pi\)
0.605853 + 0.795577i \(0.292832\pi\)
\(444\) 0 0
\(445\) −1.13904 −0.0539956
\(446\) 0 0
\(447\) −4.70360 −0.222473
\(448\) 0 0
\(449\) 5.57561 0.263129 0.131565 0.991308i \(-0.458000\pi\)
0.131565 + 0.991308i \(0.458000\pi\)
\(450\) 0 0
\(451\) −1.14817 −0.0540651
\(452\) 0 0
\(453\) −25.4034 −1.19355
\(454\) 0 0
\(455\) −14.8024 −0.693948
\(456\) 0 0
\(457\) −11.8678 −0.555151 −0.277576 0.960704i \(-0.589531\pi\)
−0.277576 + 0.960704i \(0.589531\pi\)
\(458\) 0 0
\(459\) 9.36840 0.437279
\(460\) 0 0
\(461\) −35.6344 −1.65966 −0.829830 0.558017i \(-0.811562\pi\)
−0.829830 + 0.558017i \(0.811562\pi\)
\(462\) 0 0
\(463\) 31.1785 1.44899 0.724494 0.689281i \(-0.242074\pi\)
0.724494 + 0.689281i \(0.242074\pi\)
\(464\) 0 0
\(465\) −35.1310 −1.62916
\(466\) 0 0
\(467\) −6.87676 −0.318219 −0.159109 0.987261i \(-0.550862\pi\)
−0.159109 + 0.987261i \(0.550862\pi\)
\(468\) 0 0
\(469\) 6.80390 0.314175
\(470\) 0 0
\(471\) 18.7243 0.862772
\(472\) 0 0
\(473\) 2.03261 0.0934594
\(474\) 0 0
\(475\) −12.1924 −0.559424
\(476\) 0 0
\(477\) 2.79109 0.127795
\(478\) 0 0
\(479\) −13.2313 −0.604555 −0.302277 0.953220i \(-0.597747\pi\)
−0.302277 + 0.953220i \(0.597747\pi\)
\(480\) 0 0
\(481\) −46.2830 −2.11032
\(482\) 0 0
\(483\) 5.08696 0.231465
\(484\) 0 0
\(485\) −51.0419 −2.31769
\(486\) 0 0
\(487\) 27.6451 1.25272 0.626360 0.779534i \(-0.284544\pi\)
0.626360 + 0.779534i \(0.284544\pi\)
\(488\) 0 0
\(489\) −14.9527 −0.676185
\(490\) 0 0
\(491\) −7.46915 −0.337078 −0.168539 0.985695i \(-0.553905\pi\)
−0.168539 + 0.985695i \(0.553905\pi\)
\(492\) 0 0
\(493\) 10.9650 0.493838
\(494\) 0 0
\(495\) 0.444976 0.0200002
\(496\) 0 0
\(497\) 1.11625 0.0500709
\(498\) 0 0
\(499\) −40.7824 −1.82567 −0.912836 0.408327i \(-0.866112\pi\)
−0.912836 + 0.408327i \(0.866112\pi\)
\(500\) 0 0
\(501\) 37.6756 1.68322
\(502\) 0 0
\(503\) −16.8595 −0.751729 −0.375864 0.926675i \(-0.622654\pi\)
−0.375864 + 0.926675i \(0.622654\pi\)
\(504\) 0 0
\(505\) −18.3142 −0.814973
\(506\) 0 0
\(507\) 30.1993 1.34120
\(508\) 0 0
\(509\) −30.2500 −1.34081 −0.670403 0.741997i \(-0.733879\pi\)
−0.670403 + 0.741997i \(0.733879\pi\)
\(510\) 0 0
\(511\) 11.2521 0.497762
\(512\) 0 0
\(513\) 31.7759 1.40294
\(514\) 0 0
\(515\) −16.1987 −0.713799
\(516\) 0 0
\(517\) 1.48781 0.0654339
\(518\) 0 0
\(519\) −0.0250509 −0.00109961
\(520\) 0 0
\(521\) −29.9153 −1.31061 −0.655307 0.755363i \(-0.727461\pi\)
−0.655307 + 0.755363i \(0.727461\pi\)
\(522\) 0 0
\(523\) 17.8490 0.780484 0.390242 0.920712i \(-0.372391\pi\)
0.390242 + 0.920712i \(0.372391\pi\)
\(524\) 0 0
\(525\) 3.43752 0.150026
\(526\) 0 0
\(527\) 13.8307 0.602473
\(528\) 0 0
\(529\) −13.7492 −0.597793
\(530\) 0 0
\(531\) 1.35875 0.0589647
\(532\) 0 0
\(533\) −7.74186 −0.335337
\(534\) 0 0
\(535\) 19.5329 0.844482
\(536\) 0 0
\(537\) −33.9583 −1.46541
\(538\) 0 0
\(539\) −0.826484 −0.0355992
\(540\) 0 0
\(541\) 9.87716 0.424652 0.212326 0.977199i \(-0.431896\pi\)
0.212326 + 0.977199i \(0.431896\pi\)
\(542\) 0 0
\(543\) −20.4674 −0.878339
\(544\) 0 0
\(545\) −28.9264 −1.23907
\(546\) 0 0
\(547\) −26.9678 −1.15306 −0.576530 0.817076i \(-0.695594\pi\)
−0.576530 + 0.817076i \(0.695594\pi\)
\(548\) 0 0
\(549\) −0.889367 −0.0379572
\(550\) 0 0
\(551\) 37.1912 1.58440
\(552\) 0 0
\(553\) −7.61158 −0.323677
\(554\) 0 0
\(555\) 36.8956 1.56613
\(556\) 0 0
\(557\) −15.1355 −0.641310 −0.320655 0.947196i \(-0.603903\pi\)
−0.320655 + 0.947196i \(0.603903\pi\)
\(558\) 0 0
\(559\) 13.7055 0.579679
\(560\) 0 0
\(561\) 2.41760 0.102071
\(562\) 0 0
\(563\) 9.23929 0.389390 0.194695 0.980864i \(-0.437628\pi\)
0.194695 + 0.980864i \(0.437628\pi\)
\(564\) 0 0
\(565\) 37.8895 1.59402
\(566\) 0 0
\(567\) −8.35083 −0.350702
\(568\) 0 0
\(569\) −36.7286 −1.53974 −0.769871 0.638199i \(-0.779680\pi\)
−0.769871 + 0.638199i \(0.779680\pi\)
\(570\) 0 0
\(571\) −33.6452 −1.40801 −0.704005 0.710195i \(-0.748607\pi\)
−0.704005 + 0.710195i \(0.748607\pi\)
\(572\) 0 0
\(573\) 28.7815 1.20236
\(574\) 0 0
\(575\) 6.25121 0.260694
\(576\) 0 0
\(577\) −5.87217 −0.244462 −0.122231 0.992502i \(-0.539005\pi\)
−0.122231 + 0.992502i \(0.539005\pi\)
\(578\) 0 0
\(579\) 11.7240 0.487234
\(580\) 0 0
\(581\) 15.8207 0.656353
\(582\) 0 0
\(583\) 11.3806 0.471335
\(584\) 0 0
\(585\) 3.00038 0.124051
\(586\) 0 0
\(587\) 15.5578 0.642138 0.321069 0.947056i \(-0.395958\pi\)
0.321069 + 0.947056i \(0.395958\pi\)
\(588\) 0 0
\(589\) 46.9111 1.93294
\(590\) 0 0
\(591\) −36.4669 −1.50005
\(592\) 0 0
\(593\) −26.8167 −1.10123 −0.550616 0.834759i \(-0.685607\pi\)
−0.550616 + 0.834759i \(0.685607\pi\)
\(594\) 0 0
\(595\) −4.64556 −0.190449
\(596\) 0 0
\(597\) 25.6702 1.05061
\(598\) 0 0
\(599\) −44.9307 −1.83582 −0.917909 0.396791i \(-0.870124\pi\)
−0.917909 + 0.396791i \(0.870124\pi\)
\(600\) 0 0
\(601\) 40.2092 1.64017 0.820083 0.572244i \(-0.193927\pi\)
0.820083 + 0.572244i \(0.193927\pi\)
\(602\) 0 0
\(603\) −1.37912 −0.0561621
\(604\) 0 0
\(605\) −27.4036 −1.11412
\(606\) 0 0
\(607\) 20.7271 0.841286 0.420643 0.907226i \(-0.361805\pi\)
0.420643 + 0.907226i \(0.361805\pi\)
\(608\) 0 0
\(609\) −10.4857 −0.424903
\(610\) 0 0
\(611\) 10.0320 0.405852
\(612\) 0 0
\(613\) −7.68740 −0.310491 −0.155246 0.987876i \(-0.549617\pi\)
−0.155246 + 0.987876i \(0.549617\pi\)
\(614\) 0 0
\(615\) 6.17161 0.248863
\(616\) 0 0
\(617\) 24.8297 0.999607 0.499803 0.866139i \(-0.333406\pi\)
0.499803 + 0.866139i \(0.333406\pi\)
\(618\) 0 0
\(619\) 18.6927 0.751324 0.375662 0.926757i \(-0.377415\pi\)
0.375662 + 0.926757i \(0.377415\pi\)
\(620\) 0 0
\(621\) −16.2920 −0.653775
\(622\) 0 0
\(623\) −0.428825 −0.0171805
\(624\) 0 0
\(625\) −31.0522 −1.24209
\(626\) 0 0
\(627\) 8.20006 0.327479
\(628\) 0 0
\(629\) −14.5254 −0.579164
\(630\) 0 0
\(631\) 30.4138 1.21075 0.605377 0.795939i \(-0.293022\pi\)
0.605377 + 0.795939i \(0.293022\pi\)
\(632\) 0 0
\(633\) −10.3624 −0.411870
\(634\) 0 0
\(635\) 23.8608 0.946885
\(636\) 0 0
\(637\) −5.57281 −0.220803
\(638\) 0 0
\(639\) −0.226260 −0.00895071
\(640\) 0 0
\(641\) −41.7486 −1.64897 −0.824485 0.565884i \(-0.808535\pi\)
−0.824485 + 0.565884i \(0.808535\pi\)
\(642\) 0 0
\(643\) −21.7983 −0.859640 −0.429820 0.902915i \(-0.641423\pi\)
−0.429820 + 0.902915i \(0.641423\pi\)
\(644\) 0 0
\(645\) −10.9256 −0.430196
\(646\) 0 0
\(647\) 8.85923 0.348292 0.174146 0.984720i \(-0.444283\pi\)
0.174146 + 0.984720i \(0.444283\pi\)
\(648\) 0 0
\(649\) 5.54026 0.217474
\(650\) 0 0
\(651\) −13.2261 −0.518373
\(652\) 0 0
\(653\) 3.43172 0.134293 0.0671467 0.997743i \(-0.478610\pi\)
0.0671467 + 0.997743i \(0.478610\pi\)
\(654\) 0 0
\(655\) 47.9117 1.87206
\(656\) 0 0
\(657\) −2.28075 −0.0889804
\(658\) 0 0
\(659\) −32.9576 −1.28384 −0.641922 0.766770i \(-0.721863\pi\)
−0.641922 + 0.766770i \(0.721863\pi\)
\(660\) 0 0
\(661\) −32.8710 −1.27853 −0.639266 0.768985i \(-0.720762\pi\)
−0.639266 + 0.768985i \(0.720762\pi\)
\(662\) 0 0
\(663\) 16.3014 0.633093
\(664\) 0 0
\(665\) −15.7569 −0.611026
\(666\) 0 0
\(667\) −19.0685 −0.738335
\(668\) 0 0
\(669\) −0.884651 −0.0342026
\(670\) 0 0
\(671\) −3.62636 −0.139994
\(672\) 0 0
\(673\) 46.7430 1.80181 0.900906 0.434014i \(-0.142903\pi\)
0.900906 + 0.434014i \(0.142903\pi\)
\(674\) 0 0
\(675\) −11.0093 −0.423750
\(676\) 0 0
\(677\) 42.5577 1.63562 0.817812 0.575485i \(-0.195187\pi\)
0.817812 + 0.575485i \(0.195187\pi\)
\(678\) 0 0
\(679\) −19.2163 −0.737452
\(680\) 0 0
\(681\) 42.2760 1.62002
\(682\) 0 0
\(683\) −36.0188 −1.37822 −0.689110 0.724657i \(-0.741998\pi\)
−0.689110 + 0.724657i \(0.741998\pi\)
\(684\) 0 0
\(685\) −31.3234 −1.19681
\(686\) 0 0
\(687\) −20.6549 −0.788033
\(688\) 0 0
\(689\) 76.7368 2.92344
\(690\) 0 0
\(691\) 21.9821 0.836240 0.418120 0.908392i \(-0.362689\pi\)
0.418120 + 0.908392i \(0.362689\pi\)
\(692\) 0 0
\(693\) 0.167525 0.00636374
\(694\) 0 0
\(695\) 1.96842 0.0746664
\(696\) 0 0
\(697\) −2.42969 −0.0920311
\(698\) 0 0
\(699\) 45.0718 1.70477
\(700\) 0 0
\(701\) 7.91840 0.299074 0.149537 0.988756i \(-0.452222\pi\)
0.149537 + 0.988756i \(0.452222\pi\)
\(702\) 0 0
\(703\) −49.2674 −1.85815
\(704\) 0 0
\(705\) −7.99726 −0.301194
\(706\) 0 0
\(707\) −6.89495 −0.259311
\(708\) 0 0
\(709\) −0.969949 −0.0364272 −0.0182136 0.999834i \(-0.505798\pi\)
−0.0182136 + 0.999834i \(0.505798\pi\)
\(710\) 0 0
\(711\) 1.54283 0.0578608
\(712\) 0 0
\(713\) −24.0520 −0.900756
\(714\) 0 0
\(715\) 12.2340 0.457524
\(716\) 0 0
\(717\) −33.1241 −1.23704
\(718\) 0 0
\(719\) −10.0440 −0.374579 −0.187289 0.982305i \(-0.559970\pi\)
−0.187289 + 0.982305i \(0.559970\pi\)
\(720\) 0 0
\(721\) −6.09849 −0.227119
\(722\) 0 0
\(723\) −6.25509 −0.232629
\(724\) 0 0
\(725\) −12.8856 −0.478558
\(726\) 0 0
\(727\) −5.53235 −0.205184 −0.102592 0.994724i \(-0.532714\pi\)
−0.102592 + 0.994724i \(0.532714\pi\)
\(728\) 0 0
\(729\) 28.5694 1.05813
\(730\) 0 0
\(731\) 4.30129 0.159089
\(732\) 0 0
\(733\) −5.63576 −0.208162 −0.104081 0.994569i \(-0.533190\pi\)
−0.104081 + 0.994569i \(0.533190\pi\)
\(734\) 0 0
\(735\) 4.44250 0.163864
\(736\) 0 0
\(737\) −5.62331 −0.207137
\(738\) 0 0
\(739\) 3.10659 0.114278 0.0571388 0.998366i \(-0.481802\pi\)
0.0571388 + 0.998366i \(0.481802\pi\)
\(740\) 0 0
\(741\) 55.2913 2.03118
\(742\) 0 0
\(743\) −22.5404 −0.826927 −0.413464 0.910521i \(-0.635681\pi\)
−0.413464 + 0.910521i \(0.635681\pi\)
\(744\) 0 0
\(745\) −7.46996 −0.273678
\(746\) 0 0
\(747\) −3.20678 −0.117330
\(748\) 0 0
\(749\) 7.35376 0.268701
\(750\) 0 0
\(751\) 14.8752 0.542805 0.271403 0.962466i \(-0.412513\pi\)
0.271403 + 0.962466i \(0.412513\pi\)
\(752\) 0 0
\(753\) 24.0020 0.874682
\(754\) 0 0
\(755\) −40.3440 −1.46827
\(756\) 0 0
\(757\) −1.38945 −0.0505003 −0.0252502 0.999681i \(-0.508038\pi\)
−0.0252502 + 0.999681i \(0.508038\pi\)
\(758\) 0 0
\(759\) −4.20429 −0.152606
\(760\) 0 0
\(761\) −19.9186 −0.722049 −0.361024 0.932556i \(-0.617573\pi\)
−0.361024 + 0.932556i \(0.617573\pi\)
\(762\) 0 0
\(763\) −10.8902 −0.394253
\(764\) 0 0
\(765\) 0.941634 0.0340448
\(766\) 0 0
\(767\) 37.3568 1.34888
\(768\) 0 0
\(769\) −9.78665 −0.352915 −0.176458 0.984308i \(-0.556464\pi\)
−0.176458 + 0.984308i \(0.556464\pi\)
\(770\) 0 0
\(771\) 21.0922 0.759617
\(772\) 0 0
\(773\) 27.8576 1.00197 0.500985 0.865456i \(-0.332971\pi\)
0.500985 + 0.865456i \(0.332971\pi\)
\(774\) 0 0
\(775\) −16.2532 −0.583832
\(776\) 0 0
\(777\) 13.8905 0.498318
\(778\) 0 0
\(779\) −8.24107 −0.295267
\(780\) 0 0
\(781\) −0.922567 −0.0330121
\(782\) 0 0
\(783\) 33.5825 1.20014
\(784\) 0 0
\(785\) 29.7368 1.06135
\(786\) 0 0
\(787\) 43.7329 1.55891 0.779454 0.626459i \(-0.215496\pi\)
0.779454 + 0.626459i \(0.215496\pi\)
\(788\) 0 0
\(789\) −11.7273 −0.417502
\(790\) 0 0
\(791\) 14.2646 0.507192
\(792\) 0 0
\(793\) −24.4518 −0.868309
\(794\) 0 0
\(795\) −61.1726 −2.16957
\(796\) 0 0
\(797\) −45.9626 −1.62808 −0.814040 0.580809i \(-0.802736\pi\)
−0.814040 + 0.580809i \(0.802736\pi\)
\(798\) 0 0
\(799\) 3.14843 0.111383
\(800\) 0 0
\(801\) 0.0869210 0.00307120
\(802\) 0 0
\(803\) −9.29966 −0.328178
\(804\) 0 0
\(805\) 8.07879 0.284740
\(806\) 0 0
\(807\) −9.27920 −0.326643
\(808\) 0 0
\(809\) −20.8417 −0.732756 −0.366378 0.930466i \(-0.619402\pi\)
−0.366378 + 0.930466i \(0.619402\pi\)
\(810\) 0 0
\(811\) −2.73444 −0.0960193 −0.0480096 0.998847i \(-0.515288\pi\)
−0.0480096 + 0.998847i \(0.515288\pi\)
\(812\) 0 0
\(813\) −39.4288 −1.38283
\(814\) 0 0
\(815\) −23.7469 −0.831819
\(816\) 0 0
\(817\) 14.5892 0.510411
\(818\) 0 0
\(819\) 1.12958 0.0394709
\(820\) 0 0
\(821\) 9.56765 0.333913 0.166957 0.985964i \(-0.446606\pi\)
0.166957 + 0.985964i \(0.446606\pi\)
\(822\) 0 0
\(823\) 35.9651 1.25366 0.626832 0.779154i \(-0.284351\pi\)
0.626832 + 0.779154i \(0.284351\pi\)
\(824\) 0 0
\(825\) −2.84106 −0.0989130
\(826\) 0 0
\(827\) −32.7916 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(828\) 0 0
\(829\) −26.0624 −0.905185 −0.452593 0.891717i \(-0.649501\pi\)
−0.452593 + 0.891717i \(0.649501\pi\)
\(830\) 0 0
\(831\) −18.6782 −0.647938
\(832\) 0 0
\(833\) −1.74896 −0.0605979
\(834\) 0 0
\(835\) 59.8340 2.07064
\(836\) 0 0
\(837\) 42.3593 1.46415
\(838\) 0 0
\(839\) 34.0165 1.17438 0.587191 0.809449i \(-0.300234\pi\)
0.587191 + 0.809449i \(0.300234\pi\)
\(840\) 0 0
\(841\) 10.3058 0.355371
\(842\) 0 0
\(843\) 55.4861 1.91104
\(844\) 0 0
\(845\) 47.9607 1.64990
\(846\) 0 0
\(847\) −10.3169 −0.354494
\(848\) 0 0
\(849\) −4.87829 −0.167423
\(850\) 0 0
\(851\) 25.2601 0.865906
\(852\) 0 0
\(853\) 15.5570 0.532663 0.266331 0.963882i \(-0.414188\pi\)
0.266331 + 0.963882i \(0.414188\pi\)
\(854\) 0 0
\(855\) 3.19385 0.109227
\(856\) 0 0
\(857\) −0.996249 −0.0340312 −0.0170156 0.999855i \(-0.505416\pi\)
−0.0170156 + 0.999855i \(0.505416\pi\)
\(858\) 0 0
\(859\) −33.3053 −1.13636 −0.568181 0.822903i \(-0.692353\pi\)
−0.568181 + 0.822903i \(0.692353\pi\)
\(860\) 0 0
\(861\) 2.32349 0.0791843
\(862\) 0 0
\(863\) −45.1680 −1.53754 −0.768768 0.639528i \(-0.779130\pi\)
−0.768768 + 0.639528i \(0.779130\pi\)
\(864\) 0 0
\(865\) −0.0397843 −0.00135271
\(866\) 0 0
\(867\) −23.3168 −0.791879
\(868\) 0 0
\(869\) 6.29086 0.213403
\(870\) 0 0
\(871\) −37.9168 −1.28476
\(872\) 0 0
\(873\) 3.89505 0.131827
\(874\) 0 0
\(875\) −7.82165 −0.264420
\(876\) 0 0
\(877\) −6.46529 −0.218317 −0.109159 0.994024i \(-0.534816\pi\)
−0.109159 + 0.994024i \(0.534816\pi\)
\(878\) 0 0
\(879\) 12.5941 0.424788
\(880\) 0 0
\(881\) −29.5811 −0.996613 −0.498306 0.867001i \(-0.666045\pi\)
−0.498306 + 0.867001i \(0.666045\pi\)
\(882\) 0 0
\(883\) −29.6304 −0.997143 −0.498572 0.866849i \(-0.666142\pi\)
−0.498572 + 0.866849i \(0.666142\pi\)
\(884\) 0 0
\(885\) −29.7799 −1.00104
\(886\) 0 0
\(887\) 33.7805 1.13424 0.567120 0.823635i \(-0.308058\pi\)
0.567120 + 0.823635i \(0.308058\pi\)
\(888\) 0 0
\(889\) 8.98310 0.301284
\(890\) 0 0
\(891\) 6.90183 0.231220
\(892\) 0 0
\(893\) 10.6789 0.357355
\(894\) 0 0
\(895\) −53.9305 −1.80270
\(896\) 0 0
\(897\) −28.3487 −0.946535
\(898\) 0 0
\(899\) 49.5783 1.65353
\(900\) 0 0
\(901\) 24.0829 0.802318
\(902\) 0 0
\(903\) −4.11328 −0.136881
\(904\) 0 0
\(905\) −32.5050 −1.08050
\(906\) 0 0
\(907\) −53.6507 −1.78144 −0.890721 0.454551i \(-0.849800\pi\)
−0.890721 + 0.454551i \(0.849800\pi\)
\(908\) 0 0
\(909\) 1.39758 0.0463547
\(910\) 0 0
\(911\) −16.3562 −0.541906 −0.270953 0.962593i \(-0.587339\pi\)
−0.270953 + 0.962593i \(0.587339\pi\)
\(912\) 0 0
\(913\) −13.0756 −0.432738
\(914\) 0 0
\(915\) 19.4924 0.644397
\(916\) 0 0
\(917\) 18.0378 0.595661
\(918\) 0 0
\(919\) 6.74372 0.222455 0.111227 0.993795i \(-0.464522\pi\)
0.111227 + 0.993795i \(0.464522\pi\)
\(920\) 0 0
\(921\) 23.4742 0.773503
\(922\) 0 0
\(923\) −6.22068 −0.204756
\(924\) 0 0
\(925\) 17.0696 0.561244
\(926\) 0 0
\(927\) 1.23614 0.0406000
\(928\) 0 0
\(929\) −3.61776 −0.118695 −0.0593475 0.998237i \(-0.518902\pi\)
−0.0593475 + 0.998237i \(0.518902\pi\)
\(930\) 0 0
\(931\) −5.93216 −0.194419
\(932\) 0 0
\(933\) 16.3593 0.535579
\(934\) 0 0
\(935\) 3.83948 0.125564
\(936\) 0 0
\(937\) −2.79455 −0.0912940 −0.0456470 0.998958i \(-0.514535\pi\)
−0.0456470 + 0.998958i \(0.514535\pi\)
\(938\) 0 0
\(939\) 9.51193 0.310410
\(940\) 0 0
\(941\) −24.5879 −0.801541 −0.400771 0.916178i \(-0.631258\pi\)
−0.400771 + 0.916178i \(0.631258\pi\)
\(942\) 0 0
\(943\) 4.22532 0.137595
\(944\) 0 0
\(945\) −14.2280 −0.462836
\(946\) 0 0
\(947\) −42.2713 −1.37363 −0.686816 0.726832i \(-0.740992\pi\)
−0.686816 + 0.726832i \(0.740992\pi\)
\(948\) 0 0
\(949\) −62.7057 −2.03551
\(950\) 0 0
\(951\) −51.7199 −1.67713
\(952\) 0 0
\(953\) −26.9944 −0.874434 −0.437217 0.899356i \(-0.644036\pi\)
−0.437217 + 0.899356i \(0.644036\pi\)
\(954\) 0 0
\(955\) 45.7090 1.47911
\(956\) 0 0
\(957\) 8.66628 0.280141
\(958\) 0 0
\(959\) −11.7927 −0.380805
\(960\) 0 0
\(961\) 31.5355 1.01727
\(962\) 0 0
\(963\) −1.49058 −0.0480331
\(964\) 0 0
\(965\) 18.6194 0.599379
\(966\) 0 0
\(967\) 3.64431 0.117193 0.0585966 0.998282i \(-0.481337\pi\)
0.0585966 + 0.998282i \(0.481337\pi\)
\(968\) 0 0
\(969\) 17.3525 0.557443
\(970\) 0 0
\(971\) 14.6356 0.469680 0.234840 0.972034i \(-0.424543\pi\)
0.234840 + 0.972034i \(0.424543\pi\)
\(972\) 0 0
\(973\) 0.741071 0.0237576
\(974\) 0 0
\(975\) −19.1567 −0.613504
\(976\) 0 0
\(977\) 11.7791 0.376846 0.188423 0.982088i \(-0.439662\pi\)
0.188423 + 0.982088i \(0.439662\pi\)
\(978\) 0 0
\(979\) 0.354418 0.0113272
\(980\) 0 0
\(981\) 2.20740 0.0704769
\(982\) 0 0
\(983\) 3.39640 0.108328 0.0541642 0.998532i \(-0.482751\pi\)
0.0541642 + 0.998532i \(0.482751\pi\)
\(984\) 0 0
\(985\) −57.9145 −1.84531
\(986\) 0 0
\(987\) −3.01081 −0.0958352
\(988\) 0 0
\(989\) −7.48010 −0.237853
\(990\) 0 0
\(991\) 24.0326 0.763421 0.381710 0.924282i \(-0.375335\pi\)
0.381710 + 0.924282i \(0.375335\pi\)
\(992\) 0 0
\(993\) 20.7186 0.657485
\(994\) 0 0
\(995\) 40.7678 1.29243
\(996\) 0 0
\(997\) −2.41840 −0.0765913 −0.0382957 0.999266i \(-0.512193\pi\)
−0.0382957 + 0.999266i \(0.512193\pi\)
\(998\) 0 0
\(999\) −44.4869 −1.40750
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 7168.2.a.bb.1.6 8
4.3 odd 2 7168.2.a.ba.1.3 8
8.3 odd 2 7168.2.a.be.1.6 8
8.5 even 2 7168.2.a.bf.1.3 8
32.3 odd 8 1792.2.m.g.1345.3 yes 16
32.5 even 8 1792.2.m.h.449.3 yes 16
32.11 odd 8 1792.2.m.g.449.3 yes 16
32.13 even 8 1792.2.m.h.1345.3 yes 16
32.19 odd 8 1792.2.m.f.1345.6 yes 16
32.21 even 8 1792.2.m.e.449.6 16
32.27 odd 8 1792.2.m.f.449.6 yes 16
32.29 even 8 1792.2.m.e.1345.6 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.e.449.6 16 32.21 even 8
1792.2.m.e.1345.6 yes 16 32.29 even 8
1792.2.m.f.449.6 yes 16 32.27 odd 8
1792.2.m.f.1345.6 yes 16 32.19 odd 8
1792.2.m.g.449.3 yes 16 32.11 odd 8
1792.2.m.g.1345.3 yes 16 32.3 odd 8
1792.2.m.h.449.3 yes 16 32.5 even 8
1792.2.m.h.1345.3 yes 16 32.13 even 8
7168.2.a.ba.1.3 8 4.3 odd 2
7168.2.a.bb.1.6 8 1.1 even 1 trivial
7168.2.a.be.1.6 8 8.3 odd 2
7168.2.a.bf.1.3 8 8.5 even 2