Properties

Label 7728.2.a.bz.1.4
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.39605.1
Defining polynomial: \(x^{4} - 2 x^{3} - 8 x^{2} + 9 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.57685\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.57685 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.57685 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.81840 q^{11} +4.91214 q^{13} -2.57685 q^{15} -7.21699 q^{17} -2.33529 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.64014 q^{25} -1.00000 q^{27} +3.10555 q^{29} +1.21699 q^{31} +4.81840 q^{33} +2.57685 q^{35} +0.223858 q^{37} -4.91214 q^{39} +8.25925 q^{41} -7.31073 q^{43} +2.57685 q^{45} -8.11144 q^{47} +1.00000 q^{49} +7.21699 q^{51} -1.13600 q^{53} -12.4163 q^{55} +2.33529 q^{57} +9.56998 q^{59} -14.1773 q^{61} +1.00000 q^{63} +12.6578 q^{65} -6.01770 q^{67} -1.00000 q^{69} +4.75158 q^{71} -9.70010 q^{73} -1.64014 q^{75} -4.81840 q^{77} -5.87582 q^{79} +1.00000 q^{81} -4.60729 q^{83} -18.5971 q^{85} -3.10555 q^{87} +7.95440 q^{89} +4.91214 q^{91} -1.21699 q^{93} -6.01770 q^{95} -4.43497 q^{97} -4.81840 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 2q^{5} + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 2q^{5} + 4q^{7} + 4q^{9} + 2q^{13} + 2q^{15} - 10q^{17} - 4q^{19} - 4q^{21} + 4q^{23} - 4q^{27} + 11q^{29} - 14q^{31} - 2q^{35} + 13q^{37} - 2q^{39} + 7q^{41} - 12q^{43} - 2q^{45} - 15q^{47} + 4q^{49} + 10q^{51} + q^{53} - 31q^{55} + 4q^{57} - 5q^{59} + 3q^{61} + 4q^{63} + 25q^{65} - 5q^{67} - 4q^{69} - 5q^{71} - 6q^{73} - 26q^{79} + 4q^{81} - 2q^{83} + 5q^{85} - 11q^{87} + 7q^{89} + 2q^{91} + 14q^{93} - 5q^{95} - 27q^{97} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.57685 1.15240 0.576201 0.817308i \(-0.304535\pi\)
0.576201 + 0.817308i \(0.304535\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.81840 −1.45280 −0.726401 0.687271i \(-0.758809\pi\)
−0.726401 + 0.687271i \(0.758809\pi\)
\(12\) 0 0
\(13\) 4.91214 1.36238 0.681191 0.732105i \(-0.261462\pi\)
0.681191 + 0.732105i \(0.261462\pi\)
\(14\) 0 0
\(15\) −2.57685 −0.665339
\(16\) 0 0
\(17\) −7.21699 −1.75038 −0.875189 0.483782i \(-0.839263\pi\)
−0.875189 + 0.483782i \(0.839263\pi\)
\(18\) 0 0
\(19\) −2.33529 −0.535753 −0.267877 0.963453i \(-0.586322\pi\)
−0.267877 + 0.963453i \(0.586322\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.64014 0.328029
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.10555 0.576687 0.288344 0.957527i \(-0.406895\pi\)
0.288344 + 0.957527i \(0.406895\pi\)
\(30\) 0 0
\(31\) 1.21699 0.218578 0.109289 0.994010i \(-0.465143\pi\)
0.109289 + 0.994010i \(0.465143\pi\)
\(32\) 0 0
\(33\) 4.81840 0.838776
\(34\) 0 0
\(35\) 2.57685 0.435567
\(36\) 0 0
\(37\) 0.223858 0.0368020 0.0184010 0.999831i \(-0.494142\pi\)
0.0184010 + 0.999831i \(0.494142\pi\)
\(38\) 0 0
\(39\) −4.91214 −0.786572
\(40\) 0 0
\(41\) 8.25925 1.28988 0.644939 0.764234i \(-0.276883\pi\)
0.644939 + 0.764234i \(0.276883\pi\)
\(42\) 0 0
\(43\) −7.31073 −1.11488 −0.557438 0.830219i \(-0.688216\pi\)
−0.557438 + 0.830219i \(0.688216\pi\)
\(44\) 0 0
\(45\) 2.57685 0.384134
\(46\) 0 0
\(47\) −8.11144 −1.18317 −0.591587 0.806241i \(-0.701498\pi\)
−0.591587 + 0.806241i \(0.701498\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 7.21699 1.01058
\(52\) 0 0
\(53\) −1.13600 −0.156041 −0.0780207 0.996952i \(-0.524860\pi\)
−0.0780207 + 0.996952i \(0.524860\pi\)
\(54\) 0 0
\(55\) −12.4163 −1.67421
\(56\) 0 0
\(57\) 2.33529 0.309317
\(58\) 0 0
\(59\) 9.56998 1.24591 0.622953 0.782260i \(-0.285933\pi\)
0.622953 + 0.782260i \(0.285933\pi\)
\(60\) 0 0
\(61\) −14.1773 −1.81521 −0.907607 0.419821i \(-0.862093\pi\)
−0.907607 + 0.419821i \(0.862093\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 12.6578 1.57001
\(66\) 0 0
\(67\) −6.01770 −0.735179 −0.367589 0.929988i \(-0.619817\pi\)
−0.367589 + 0.929988i \(0.619817\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 4.75158 0.563909 0.281954 0.959428i \(-0.409017\pi\)
0.281954 + 0.959428i \(0.409017\pi\)
\(72\) 0 0
\(73\) −9.70010 −1.13531 −0.567655 0.823266i \(-0.692149\pi\)
−0.567655 + 0.823266i \(0.692149\pi\)
\(74\) 0 0
\(75\) −1.64014 −0.189387
\(76\) 0 0
\(77\) −4.81840 −0.549108
\(78\) 0 0
\(79\) −5.87582 −0.661081 −0.330540 0.943792i \(-0.607231\pi\)
−0.330540 + 0.943792i \(0.607231\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.60729 −0.505716 −0.252858 0.967503i \(-0.581371\pi\)
−0.252858 + 0.967503i \(0.581371\pi\)
\(84\) 0 0
\(85\) −18.5971 −2.01714
\(86\) 0 0
\(87\) −3.10555 −0.332950
\(88\) 0 0
\(89\) 7.95440 0.843165 0.421582 0.906790i \(-0.361475\pi\)
0.421582 + 0.906790i \(0.361475\pi\)
\(90\) 0 0
\(91\) 4.91214 0.514932
\(92\) 0 0
\(93\) −1.21699 −0.126196
\(94\) 0 0
\(95\) −6.01770 −0.617403
\(96\) 0 0
\(97\) −4.43497 −0.450303 −0.225151 0.974324i \(-0.572288\pi\)
−0.225151 + 0.974324i \(0.572288\pi\)
\(98\) 0 0
\(99\) −4.81840 −0.484268
\(100\) 0 0
\(101\) 3.93318 0.391366 0.195683 0.980667i \(-0.437308\pi\)
0.195683 + 0.980667i \(0.437308\pi\)
\(102\) 0 0
\(103\) 1.92396 0.189573 0.0947865 0.995498i \(-0.469783\pi\)
0.0947865 + 0.995498i \(0.469783\pi\)
\(104\) 0 0
\(105\) −2.57685 −0.251475
\(106\) 0 0
\(107\) 2.13012 0.205926 0.102963 0.994685i \(-0.467168\pi\)
0.102963 + 0.994685i \(0.467168\pi\)
\(108\) 0 0
\(109\) −13.0108 −1.24621 −0.623106 0.782138i \(-0.714129\pi\)
−0.623106 + 0.782138i \(0.714129\pi\)
\(110\) 0 0
\(111\) −0.223858 −0.0212476
\(112\) 0 0
\(113\) −11.5431 −1.08588 −0.542940 0.839771i \(-0.682689\pi\)
−0.542940 + 0.839771i \(0.682689\pi\)
\(114\) 0 0
\(115\) 2.57685 0.240292
\(116\) 0 0
\(117\) 4.91214 0.454128
\(118\) 0 0
\(119\) −7.21699 −0.661580
\(120\) 0 0
\(121\) 12.2170 1.11064
\(122\) 0 0
\(123\) −8.25925 −0.744711
\(124\) 0 0
\(125\) −8.65784 −0.774381
\(126\) 0 0
\(127\) −20.3378 −1.80469 −0.902345 0.431013i \(-0.858156\pi\)
−0.902345 + 0.431013i \(0.858156\pi\)
\(128\) 0 0
\(129\) 7.31073 0.643674
\(130\) 0 0
\(131\) 14.7693 1.29040 0.645199 0.764015i \(-0.276775\pi\)
0.645199 + 0.764015i \(0.276775\pi\)
\(132\) 0 0
\(133\) −2.33529 −0.202496
\(134\) 0 0
\(135\) −2.57685 −0.221780
\(136\) 0 0
\(137\) 15.4975 1.32404 0.662019 0.749487i \(-0.269700\pi\)
0.662019 + 0.749487i \(0.269700\pi\)
\(138\) 0 0
\(139\) −16.1291 −1.36806 −0.684028 0.729456i \(-0.739773\pi\)
−0.684028 + 0.729456i \(0.739773\pi\)
\(140\) 0 0
\(141\) 8.11144 0.683106
\(142\) 0 0
\(143\) −23.6687 −1.97927
\(144\) 0 0
\(145\) 8.00254 0.664575
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −14.2583 −1.16808 −0.584041 0.811724i \(-0.698529\pi\)
−0.584041 + 0.811724i \(0.698529\pi\)
\(150\) 0 0
\(151\) −1.76197 −0.143387 −0.0716937 0.997427i \(-0.522840\pi\)
−0.0716937 + 0.997427i \(0.522840\pi\)
\(152\) 0 0
\(153\) −7.21699 −0.583459
\(154\) 0 0
\(155\) 3.13600 0.251890
\(156\) 0 0
\(157\) −3.82428 −0.305211 −0.152605 0.988287i \(-0.548766\pi\)
−0.152605 + 0.988287i \(0.548766\pi\)
\(158\) 0 0
\(159\) 1.13600 0.0900906
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 11.3943 0.892468 0.446234 0.894916i \(-0.352765\pi\)
0.446234 + 0.894916i \(0.352765\pi\)
\(164\) 0 0
\(165\) 12.4163 0.966606
\(166\) 0 0
\(167\) −2.33529 −0.180710 −0.0903552 0.995910i \(-0.528800\pi\)
−0.0903552 + 0.995910i \(0.528800\pi\)
\(168\) 0 0
\(169\) 11.1291 0.856087
\(170\) 0 0
\(171\) −2.33529 −0.178584
\(172\) 0 0
\(173\) 13.6765 1.03980 0.519901 0.854226i \(-0.325969\pi\)
0.519901 + 0.854226i \(0.325969\pi\)
\(174\) 0 0
\(175\) 1.64014 0.123983
\(176\) 0 0
\(177\) −9.56998 −0.719324
\(178\) 0 0
\(179\) −19.2337 −1.43759 −0.718797 0.695220i \(-0.755307\pi\)
−0.718797 + 0.695220i \(0.755307\pi\)
\(180\) 0 0
\(181\) −8.28617 −0.615906 −0.307953 0.951402i \(-0.599644\pi\)
−0.307953 + 0.951402i \(0.599644\pi\)
\(182\) 0 0
\(183\) 14.1773 1.04801
\(184\) 0 0
\(185\) 0.576848 0.0424107
\(186\) 0 0
\(187\) 34.7744 2.54295
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −11.3648 −0.822328 −0.411164 0.911561i \(-0.634878\pi\)
−0.411164 + 0.911561i \(0.634878\pi\)
\(192\) 0 0
\(193\) −16.2077 −1.16666 −0.583328 0.812236i \(-0.698250\pi\)
−0.583328 + 0.812236i \(0.698250\pi\)
\(194\) 0 0
\(195\) −12.6578 −0.906447
\(196\) 0 0
\(197\) 15.8784 1.13129 0.565643 0.824650i \(-0.308628\pi\)
0.565643 + 0.824650i \(0.308628\pi\)
\(198\) 0 0
\(199\) −12.4503 −0.882575 −0.441288 0.897366i \(-0.645478\pi\)
−0.441288 + 0.897366i \(0.645478\pi\)
\(200\) 0 0
\(201\) 6.01770 0.424456
\(202\) 0 0
\(203\) 3.10555 0.217967
\(204\) 0 0
\(205\) 21.2828 1.48646
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 11.2524 0.778344
\(210\) 0 0
\(211\) 2.15958 0.148671 0.0743357 0.997233i \(-0.476316\pi\)
0.0743357 + 0.997233i \(0.476316\pi\)
\(212\) 0 0
\(213\) −4.75158 −0.325573
\(214\) 0 0
\(215\) −18.8386 −1.28478
\(216\) 0 0
\(217\) 1.21699 0.0826147
\(218\) 0 0
\(219\) 9.70010 0.655472
\(220\) 0 0
\(221\) −35.4509 −2.38468
\(222\) 0 0
\(223\) 3.89110 0.260568 0.130284 0.991477i \(-0.458411\pi\)
0.130284 + 0.991477i \(0.458411\pi\)
\(224\) 0 0
\(225\) 1.64014 0.109343
\(226\) 0 0
\(227\) 17.7423 1.17760 0.588799 0.808279i \(-0.299601\pi\)
0.588799 + 0.808279i \(0.299601\pi\)
\(228\) 0 0
\(229\) 24.4026 1.61257 0.806283 0.591530i \(-0.201476\pi\)
0.806283 + 0.591530i \(0.201476\pi\)
\(230\) 0 0
\(231\) 4.81840 0.317028
\(232\) 0 0
\(233\) −12.9839 −0.850604 −0.425302 0.905051i \(-0.639832\pi\)
−0.425302 + 0.905051i \(0.639832\pi\)
\(234\) 0 0
\(235\) −20.9019 −1.36349
\(236\) 0 0
\(237\) 5.87582 0.381675
\(238\) 0 0
\(239\) 4.58532 0.296600 0.148300 0.988942i \(-0.452620\pi\)
0.148300 + 0.988942i \(0.452620\pi\)
\(240\) 0 0
\(241\) 19.1881 1.23601 0.618007 0.786173i \(-0.287940\pi\)
0.618007 + 0.786173i \(0.287940\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.57685 0.164629
\(246\) 0 0
\(247\) −11.4713 −0.729901
\(248\) 0 0
\(249\) 4.60729 0.291975
\(250\) 0 0
\(251\) 20.4071 1.28808 0.644041 0.764991i \(-0.277257\pi\)
0.644041 + 0.764991i \(0.277257\pi\)
\(252\) 0 0
\(253\) −4.81840 −0.302930
\(254\) 0 0
\(255\) 18.5971 1.16459
\(256\) 0 0
\(257\) −3.30578 −0.206209 −0.103105 0.994671i \(-0.532878\pi\)
−0.103105 + 0.994671i \(0.532878\pi\)
\(258\) 0 0
\(259\) 0.223858 0.0139098
\(260\) 0 0
\(261\) 3.10555 0.192229
\(262\) 0 0
\(263\) 25.0682 1.54577 0.772885 0.634546i \(-0.218813\pi\)
0.772885 + 0.634546i \(0.218813\pi\)
\(264\) 0 0
\(265\) −2.92730 −0.179822
\(266\) 0 0
\(267\) −7.95440 −0.486801
\(268\) 0 0
\(269\) −10.5220 −0.641539 −0.320770 0.947157i \(-0.603942\pi\)
−0.320770 + 0.947157i \(0.603942\pi\)
\(270\) 0 0
\(271\) 12.0075 0.729403 0.364701 0.931124i \(-0.381171\pi\)
0.364701 + 0.931124i \(0.381171\pi\)
\(272\) 0 0
\(273\) −4.91214 −0.297296
\(274\) 0 0
\(275\) −7.90287 −0.476561
\(276\) 0 0
\(277\) 15.3310 0.921149 0.460574 0.887621i \(-0.347643\pi\)
0.460574 + 0.887621i \(0.347643\pi\)
\(278\) 0 0
\(279\) 1.21699 0.0728593
\(280\) 0 0
\(281\) −6.47463 −0.386244 −0.193122 0.981175i \(-0.561861\pi\)
−0.193122 + 0.981175i \(0.561861\pi\)
\(282\) 0 0
\(283\) −20.0531 −1.19203 −0.596016 0.802972i \(-0.703251\pi\)
−0.596016 + 0.802972i \(0.703251\pi\)
\(284\) 0 0
\(285\) 6.01770 0.356458
\(286\) 0 0
\(287\) 8.25925 0.487528
\(288\) 0 0
\(289\) 35.0850 2.06382
\(290\) 0 0
\(291\) 4.43497 0.259982
\(292\) 0 0
\(293\) −9.32941 −0.545030 −0.272515 0.962152i \(-0.587855\pi\)
−0.272515 + 0.962152i \(0.587855\pi\)
\(294\) 0 0
\(295\) 24.6604 1.43578
\(296\) 0 0
\(297\) 4.81840 0.279592
\(298\) 0 0
\(299\) 4.91214 0.284076
\(300\) 0 0
\(301\) −7.31073 −0.421384
\(302\) 0 0
\(303\) −3.93318 −0.225955
\(304\) 0 0
\(305\) −36.5327 −2.09185
\(306\) 0 0
\(307\) −14.7070 −0.839371 −0.419685 0.907670i \(-0.637860\pi\)
−0.419685 + 0.907670i \(0.637860\pi\)
\(308\) 0 0
\(309\) −1.92396 −0.109450
\(310\) 0 0
\(311\) −17.3192 −0.982082 −0.491041 0.871136i \(-0.663383\pi\)
−0.491041 + 0.871136i \(0.663383\pi\)
\(312\) 0 0
\(313\) −30.5303 −1.72567 −0.862836 0.505484i \(-0.831314\pi\)
−0.862836 + 0.505484i \(0.831314\pi\)
\(314\) 0 0
\(315\) 2.57685 0.145189
\(316\) 0 0
\(317\) 10.0937 0.566921 0.283460 0.958984i \(-0.408518\pi\)
0.283460 + 0.958984i \(0.408518\pi\)
\(318\) 0 0
\(319\) −14.9638 −0.837812
\(320\) 0 0
\(321\) −2.13012 −0.118892
\(322\) 0 0
\(323\) 16.8538 0.937770
\(324\) 0 0
\(325\) 8.05661 0.446901
\(326\) 0 0
\(327\) 13.0108 0.719500
\(328\) 0 0
\(329\) −8.11144 −0.447198
\(330\) 0 0
\(331\) −10.3632 −0.569613 −0.284806 0.958585i \(-0.591929\pi\)
−0.284806 + 0.958585i \(0.591929\pi\)
\(332\) 0 0
\(333\) 0.223858 0.0122673
\(334\) 0 0
\(335\) −15.5067 −0.847221
\(336\) 0 0
\(337\) −27.8798 −1.51871 −0.759354 0.650678i \(-0.774485\pi\)
−0.759354 + 0.650678i \(0.774485\pi\)
\(338\) 0 0
\(339\) 11.5431 0.626933
\(340\) 0 0
\(341\) −5.86395 −0.317551
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −2.57685 −0.138733
\(346\) 0 0
\(347\) 21.6849 1.16411 0.582054 0.813150i \(-0.302249\pi\)
0.582054 + 0.813150i \(0.302249\pi\)
\(348\) 0 0
\(349\) 29.1038 1.55789 0.778946 0.627091i \(-0.215755\pi\)
0.778946 + 0.627091i \(0.215755\pi\)
\(350\) 0 0
\(351\) −4.91214 −0.262191
\(352\) 0 0
\(353\) −34.6932 −1.84653 −0.923267 0.384158i \(-0.874492\pi\)
−0.923267 + 0.384158i \(0.874492\pi\)
\(354\) 0 0
\(355\) 12.2441 0.649849
\(356\) 0 0
\(357\) 7.21699 0.381964
\(358\) 0 0
\(359\) 13.9888 0.738301 0.369150 0.929370i \(-0.379649\pi\)
0.369150 + 0.929370i \(0.379649\pi\)
\(360\) 0 0
\(361\) −13.5464 −0.712969
\(362\) 0 0
\(363\) −12.2170 −0.641226
\(364\) 0 0
\(365\) −24.9957 −1.30833
\(366\) 0 0
\(367\) −29.1704 −1.52268 −0.761341 0.648351i \(-0.775459\pi\)
−0.761341 + 0.648351i \(0.775459\pi\)
\(368\) 0 0
\(369\) 8.25925 0.429959
\(370\) 0 0
\(371\) −1.13600 −0.0589781
\(372\) 0 0
\(373\) −16.5818 −0.858573 −0.429286 0.903168i \(-0.641235\pi\)
−0.429286 + 0.903168i \(0.641235\pi\)
\(374\) 0 0
\(375\) 8.65784 0.447089
\(376\) 0 0
\(377\) 15.2549 0.785669
\(378\) 0 0
\(379\) −36.1331 −1.85603 −0.928016 0.372540i \(-0.878487\pi\)
−0.928016 + 0.372540i \(0.878487\pi\)
\(380\) 0 0
\(381\) 20.3378 1.04194
\(382\) 0 0
\(383\) 7.05501 0.360494 0.180247 0.983621i \(-0.442310\pi\)
0.180247 + 0.983621i \(0.442310\pi\)
\(384\) 0 0
\(385\) −12.4163 −0.632792
\(386\) 0 0
\(387\) −7.31073 −0.371625
\(388\) 0 0
\(389\) −26.1832 −1.32754 −0.663771 0.747936i \(-0.731045\pi\)
−0.663771 + 0.747936i \(0.731045\pi\)
\(390\) 0 0
\(391\) −7.21699 −0.364979
\(392\) 0 0
\(393\) −14.7693 −0.745011
\(394\) 0 0
\(395\) −15.1411 −0.761830
\(396\) 0 0
\(397\) 6.19924 0.311131 0.155566 0.987826i \(-0.450280\pi\)
0.155566 + 0.987826i \(0.450280\pi\)
\(398\) 0 0
\(399\) 2.33529 0.116911
\(400\) 0 0
\(401\) −0.546403 −0.0272861 −0.0136430 0.999907i \(-0.504343\pi\)
−0.0136430 + 0.999907i \(0.504343\pi\)
\(402\) 0 0
\(403\) 5.97803 0.297787
\(404\) 0 0
\(405\) 2.57685 0.128045
\(406\) 0 0
\(407\) −1.07864 −0.0534660
\(408\) 0 0
\(409\) 30.0311 1.48494 0.742472 0.669878i \(-0.233653\pi\)
0.742472 + 0.669878i \(0.233653\pi\)
\(410\) 0 0
\(411\) −15.4975 −0.764433
\(412\) 0 0
\(413\) 9.56998 0.470908
\(414\) 0 0
\(415\) −11.8723 −0.582788
\(416\) 0 0
\(417\) 16.1291 0.789847
\(418\) 0 0
\(419\) −18.1022 −0.884351 −0.442176 0.896929i \(-0.645793\pi\)
−0.442176 + 0.896929i \(0.645793\pi\)
\(420\) 0 0
\(421\) 13.2549 0.646005 0.323003 0.946398i \(-0.395308\pi\)
0.323003 + 0.946398i \(0.395308\pi\)
\(422\) 0 0
\(423\) −8.11144 −0.394392
\(424\) 0 0
\(425\) −11.8369 −0.574174
\(426\) 0 0
\(427\) −14.1773 −0.686086
\(428\) 0 0
\(429\) 23.6687 1.14273
\(430\) 0 0
\(431\) −29.3678 −1.41460 −0.707298 0.706915i \(-0.750086\pi\)
−0.707298 + 0.706915i \(0.750086\pi\)
\(432\) 0 0
\(433\) −15.8748 −0.762896 −0.381448 0.924390i \(-0.624574\pi\)
−0.381448 + 0.924390i \(0.624574\pi\)
\(434\) 0 0
\(435\) −8.00254 −0.383692
\(436\) 0 0
\(437\) −2.33529 −0.111712
\(438\) 0 0
\(439\) 9.38603 0.447971 0.223985 0.974593i \(-0.428093\pi\)
0.223985 + 0.974593i \(0.428093\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −13.3800 −0.635701 −0.317851 0.948141i \(-0.602961\pi\)
−0.317851 + 0.948141i \(0.602961\pi\)
\(444\) 0 0
\(445\) 20.4973 0.971664
\(446\) 0 0
\(447\) 14.2583 0.674393
\(448\) 0 0
\(449\) −21.9383 −1.03533 −0.517665 0.855583i \(-0.673199\pi\)
−0.517665 + 0.855583i \(0.673199\pi\)
\(450\) 0 0
\(451\) −39.7964 −1.87394
\(452\) 0 0
\(453\) 1.76197 0.0827847
\(454\) 0 0
\(455\) 12.6578 0.593409
\(456\) 0 0
\(457\) 22.1376 1.03555 0.517777 0.855516i \(-0.326760\pi\)
0.517777 + 0.855516i \(0.326760\pi\)
\(458\) 0 0
\(459\) 7.21699 0.336860
\(460\) 0 0
\(461\) −11.1006 −0.517007 −0.258503 0.966010i \(-0.583229\pi\)
−0.258503 + 0.966010i \(0.583229\pi\)
\(462\) 0 0
\(463\) 8.71873 0.405194 0.202597 0.979262i \(-0.435062\pi\)
0.202597 + 0.979262i \(0.435062\pi\)
\(464\) 0 0
\(465\) −3.13600 −0.145428
\(466\) 0 0
\(467\) 14.3084 0.662113 0.331056 0.943611i \(-0.392595\pi\)
0.331056 + 0.943611i \(0.392595\pi\)
\(468\) 0 0
\(469\) −6.01770 −0.277871
\(470\) 0 0
\(471\) 3.82428 0.176214
\(472\) 0 0
\(473\) 35.2260 1.61969
\(474\) 0 0
\(475\) −3.83022 −0.175742
\(476\) 0 0
\(477\) −1.13600 −0.0520138
\(478\) 0 0
\(479\) −37.8738 −1.73050 −0.865250 0.501341i \(-0.832840\pi\)
−0.865250 + 0.501341i \(0.832840\pi\)
\(480\) 0 0
\(481\) 1.09962 0.0501384
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −11.4282 −0.518929
\(486\) 0 0
\(487\) 12.2172 0.553613 0.276807 0.960926i \(-0.410724\pi\)
0.276807 + 0.960926i \(0.410724\pi\)
\(488\) 0 0
\(489\) −11.3943 −0.515266
\(490\) 0 0
\(491\) 40.4964 1.82758 0.913789 0.406189i \(-0.133143\pi\)
0.913789 + 0.406189i \(0.133143\pi\)
\(492\) 0 0
\(493\) −22.4128 −1.00942
\(494\) 0 0
\(495\) −12.4163 −0.558071
\(496\) 0 0
\(497\) 4.75158 0.213137
\(498\) 0 0
\(499\) −7.00941 −0.313784 −0.156892 0.987616i \(-0.550147\pi\)
−0.156892 + 0.987616i \(0.550147\pi\)
\(500\) 0 0
\(501\) 2.33529 0.104333
\(502\) 0 0
\(503\) −31.4905 −1.40409 −0.702047 0.712131i \(-0.747730\pi\)
−0.702047 + 0.712131i \(0.747730\pi\)
\(504\) 0 0
\(505\) 10.1352 0.451010
\(506\) 0 0
\(507\) −11.1291 −0.494262
\(508\) 0 0
\(509\) −27.4356 −1.21606 −0.608031 0.793914i \(-0.708040\pi\)
−0.608031 + 0.793914i \(0.708040\pi\)
\(510\) 0 0
\(511\) −9.70010 −0.429107
\(512\) 0 0
\(513\) 2.33529 0.103106
\(514\) 0 0
\(515\) 4.95774 0.218464
\(516\) 0 0
\(517\) 39.0842 1.71892
\(518\) 0 0
\(519\) −13.6765 −0.600330
\(520\) 0 0
\(521\) −13.9579 −0.611508 −0.305754 0.952111i \(-0.598908\pi\)
−0.305754 + 0.952111i \(0.598908\pi\)
\(522\) 0 0
\(523\) 17.1046 0.747931 0.373965 0.927443i \(-0.377998\pi\)
0.373965 + 0.927443i \(0.377998\pi\)
\(524\) 0 0
\(525\) −1.64014 −0.0715817
\(526\) 0 0
\(527\) −8.78301 −0.382594
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 9.56998 0.415302
\(532\) 0 0
\(533\) 40.5706 1.75731
\(534\) 0 0
\(535\) 5.48899 0.237310
\(536\) 0 0
\(537\) 19.2337 0.829995
\(538\) 0 0
\(539\) −4.81840 −0.207543
\(540\) 0 0
\(541\) 18.9847 0.816214 0.408107 0.912934i \(-0.366189\pi\)
0.408107 + 0.912934i \(0.366189\pi\)
\(542\) 0 0
\(543\) 8.28617 0.355593
\(544\) 0 0
\(545\) −33.5269 −1.43614
\(546\) 0 0
\(547\) 10.4026 0.444781 0.222390 0.974958i \(-0.428614\pi\)
0.222390 + 0.974958i \(0.428614\pi\)
\(548\) 0 0
\(549\) −14.1773 −0.605071
\(550\) 0 0
\(551\) −7.25238 −0.308962
\(552\) 0 0
\(553\) −5.87582 −0.249865
\(554\) 0 0
\(555\) −0.576848 −0.0244858
\(556\) 0 0
\(557\) −0.972896 −0.0412229 −0.0206115 0.999788i \(-0.506561\pi\)
−0.0206115 + 0.999788i \(0.506561\pi\)
\(558\) 0 0
\(559\) −35.9113 −1.51889
\(560\) 0 0
\(561\) −34.7744 −1.46817
\(562\) 0 0
\(563\) 42.0521 1.77228 0.886142 0.463413i \(-0.153375\pi\)
0.886142 + 0.463413i \(0.153375\pi\)
\(564\) 0 0
\(565\) −29.7447 −1.25137
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −14.1586 −0.593559 −0.296779 0.954946i \(-0.595913\pi\)
−0.296779 + 0.954946i \(0.595913\pi\)
\(570\) 0 0
\(571\) 18.4433 0.771830 0.385915 0.922534i \(-0.373886\pi\)
0.385915 + 0.922534i \(0.373886\pi\)
\(572\) 0 0
\(573\) 11.3648 0.474772
\(574\) 0 0
\(575\) 1.64014 0.0683987
\(576\) 0 0
\(577\) −32.7296 −1.36255 −0.681276 0.732027i \(-0.738575\pi\)
−0.681276 + 0.732027i \(0.738575\pi\)
\(578\) 0 0
\(579\) 16.2077 0.673570
\(580\) 0 0
\(581\) −4.60729 −0.191143
\(582\) 0 0
\(583\) 5.47370 0.226697
\(584\) 0 0
\(585\) 12.6578 0.523337
\(586\) 0 0
\(587\) 32.6230 1.34650 0.673248 0.739417i \(-0.264899\pi\)
0.673248 + 0.739417i \(0.264899\pi\)
\(588\) 0 0
\(589\) −2.84203 −0.117104
\(590\) 0 0
\(591\) −15.8784 −0.653148
\(592\) 0 0
\(593\) −3.76197 −0.154486 −0.0772429 0.997012i \(-0.524612\pi\)
−0.0772429 + 0.997012i \(0.524612\pi\)
\(594\) 0 0
\(595\) −18.5971 −0.762406
\(596\) 0 0
\(597\) 12.4503 0.509555
\(598\) 0 0
\(599\) 2.51694 0.102840 0.0514198 0.998677i \(-0.483625\pi\)
0.0514198 + 0.998677i \(0.483625\pi\)
\(600\) 0 0
\(601\) −15.0330 −0.613210 −0.306605 0.951837i \(-0.599193\pi\)
−0.306605 + 0.951837i \(0.599193\pi\)
\(602\) 0 0
\(603\) −6.01770 −0.245060
\(604\) 0 0
\(605\) 31.4813 1.27990
\(606\) 0 0
\(607\) 13.7041 0.556231 0.278115 0.960548i \(-0.410290\pi\)
0.278115 + 0.960548i \(0.410290\pi\)
\(608\) 0 0
\(609\) −3.10555 −0.125843
\(610\) 0 0
\(611\) −39.8445 −1.61194
\(612\) 0 0
\(613\) −29.6429 −1.19726 −0.598632 0.801024i \(-0.704289\pi\)
−0.598632 + 0.801024i \(0.704289\pi\)
\(614\) 0 0
\(615\) −21.2828 −0.858206
\(616\) 0 0
\(617\) 21.6687 0.872348 0.436174 0.899862i \(-0.356333\pi\)
0.436174 + 0.899862i \(0.356333\pi\)
\(618\) 0 0
\(619\) −10.9815 −0.441384 −0.220692 0.975344i \(-0.570832\pi\)
−0.220692 + 0.975344i \(0.570832\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 7.95440 0.318686
\(624\) 0 0
\(625\) −30.5106 −1.22043
\(626\) 0 0
\(627\) −11.2524 −0.449377
\(628\) 0 0
\(629\) −1.61558 −0.0644174
\(630\) 0 0
\(631\) −45.8837 −1.82660 −0.913301 0.407284i \(-0.866476\pi\)
−0.913301 + 0.407284i \(0.866476\pi\)
\(632\) 0 0
\(633\) −2.15958 −0.0858355
\(634\) 0 0
\(635\) −52.4075 −2.07973
\(636\) 0 0
\(637\) 4.91214 0.194626
\(638\) 0 0
\(639\) 4.75158 0.187970
\(640\) 0 0
\(641\) −10.9121 −0.431004 −0.215502 0.976503i \(-0.569139\pi\)
−0.215502 + 0.976503i \(0.569139\pi\)
\(642\) 0 0
\(643\) 29.3526 1.15755 0.578777 0.815486i \(-0.303530\pi\)
0.578777 + 0.815486i \(0.303530\pi\)
\(644\) 0 0
\(645\) 18.8386 0.741771
\(646\) 0 0
\(647\) 45.5259 1.78981 0.894905 0.446257i \(-0.147243\pi\)
0.894905 + 0.446257i \(0.147243\pi\)
\(648\) 0 0
\(649\) −46.1120 −1.81005
\(650\) 0 0
\(651\) −1.21699 −0.0476976
\(652\) 0 0
\(653\) −12.7251 −0.497971 −0.248986 0.968507i \(-0.580097\pi\)
−0.248986 + 0.968507i \(0.580097\pi\)
\(654\) 0 0
\(655\) 38.0582 1.48706
\(656\) 0 0
\(657\) −9.70010 −0.378437
\(658\) 0 0
\(659\) 18.4552 0.718913 0.359456 0.933162i \(-0.382962\pi\)
0.359456 + 0.933162i \(0.382962\pi\)
\(660\) 0 0
\(661\) −21.3506 −0.830443 −0.415222 0.909720i \(-0.636296\pi\)
−0.415222 + 0.909720i \(0.636296\pi\)
\(662\) 0 0
\(663\) 35.4509 1.37680
\(664\) 0 0
\(665\) −6.01770 −0.233356
\(666\) 0 0
\(667\) 3.10555 0.120248
\(668\) 0 0
\(669\) −3.89110 −0.150439
\(670\) 0 0
\(671\) 68.3118 2.63715
\(672\) 0 0
\(673\) 10.1492 0.391224 0.195612 0.980681i \(-0.437331\pi\)
0.195612 + 0.980681i \(0.437331\pi\)
\(674\) 0 0
\(675\) −1.64014 −0.0631291
\(676\) 0 0
\(677\) −38.4171 −1.47649 −0.738244 0.674533i \(-0.764345\pi\)
−0.738244 + 0.674533i \(0.764345\pi\)
\(678\) 0 0
\(679\) −4.43497 −0.170198
\(680\) 0 0
\(681\) −17.7423 −0.679887
\(682\) 0 0
\(683\) 11.4418 0.437810 0.218905 0.975746i \(-0.429752\pi\)
0.218905 + 0.975746i \(0.429752\pi\)
\(684\) 0 0
\(685\) 39.9346 1.52582
\(686\) 0 0
\(687\) −24.4026 −0.931015
\(688\) 0 0
\(689\) −5.58019 −0.212588
\(690\) 0 0
\(691\) 37.3225 1.41981 0.709907 0.704295i \(-0.248737\pi\)
0.709907 + 0.704295i \(0.248737\pi\)
\(692\) 0 0
\(693\) −4.81840 −0.183036
\(694\) 0 0
\(695\) −41.5623 −1.57655
\(696\) 0 0
\(697\) −59.6069 −2.25777
\(698\) 0 0
\(699\) 12.9839 0.491097
\(700\) 0 0
\(701\) −4.18165 −0.157939 −0.0789694 0.996877i \(-0.525163\pi\)
−0.0789694 + 0.996877i \(0.525163\pi\)
\(702\) 0 0
\(703\) −0.522774 −0.0197168
\(704\) 0 0
\(705\) 20.9019 0.787212
\(706\) 0 0
\(707\) 3.93318 0.147922
\(708\) 0 0
\(709\) 27.1435 1.01940 0.509698 0.860354i \(-0.329757\pi\)
0.509698 + 0.860354i \(0.329757\pi\)
\(710\) 0 0
\(711\) −5.87582 −0.220360
\(712\) 0 0
\(713\) 1.21699 0.0455767
\(714\) 0 0
\(715\) −60.9905 −2.28092
\(716\) 0 0
\(717\) −4.58532 −0.171242
\(718\) 0 0
\(719\) 29.7369 1.10900 0.554500 0.832184i \(-0.312910\pi\)
0.554500 + 0.832184i \(0.312910\pi\)
\(720\) 0 0
\(721\) 1.92396 0.0716519
\(722\) 0 0
\(723\) −19.1881 −0.713613
\(724\) 0 0
\(725\) 5.09355 0.189170
\(726\) 0 0
\(727\) −27.2170 −1.00942 −0.504711 0.863288i \(-0.668401\pi\)
−0.504711 + 0.863288i \(0.668401\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 52.7615 1.95145
\(732\) 0 0
\(733\) 16.0775 0.593835 0.296917 0.954903i \(-0.404041\pi\)
0.296917 + 0.954903i \(0.404041\pi\)
\(734\) 0 0
\(735\) −2.57685 −0.0950485
\(736\) 0 0
\(737\) 28.9957 1.06807
\(738\) 0 0
\(739\) −9.16786 −0.337245 −0.168623 0.985681i \(-0.553932\pi\)
−0.168623 + 0.985681i \(0.553932\pi\)
\(740\) 0 0
\(741\) 11.4713 0.421408
\(742\) 0 0
\(743\) 20.4163 0.749001 0.374500 0.927227i \(-0.377814\pi\)
0.374500 + 0.927227i \(0.377814\pi\)
\(744\) 0 0
\(745\) −36.7414 −1.34610
\(746\) 0 0
\(747\) −4.60729 −0.168572
\(748\) 0 0
\(749\) 2.13012 0.0778328
\(750\) 0 0
\(751\) 12.9249 0.471636 0.235818 0.971797i \(-0.424223\pi\)
0.235818 + 0.971797i \(0.424223\pi\)
\(752\) 0 0
\(753\) −20.4071 −0.743675
\(754\) 0 0
\(755\) −4.54034 −0.165240
\(756\) 0 0
\(757\) −42.4953 −1.54452 −0.772259 0.635308i \(-0.780873\pi\)
−0.772259 + 0.635308i \(0.780873\pi\)
\(758\) 0 0
\(759\) 4.81840 0.174897
\(760\) 0 0
\(761\) 49.9167 1.80948 0.904740 0.425964i \(-0.140065\pi\)
0.904740 + 0.425964i \(0.140065\pi\)
\(762\) 0 0
\(763\) −13.0108 −0.471024
\(764\) 0 0
\(765\) −18.5971 −0.672379
\(766\) 0 0
\(767\) 47.0091 1.69740
\(768\) 0 0
\(769\) 1.94240 0.0700447 0.0350224 0.999387i \(-0.488850\pi\)
0.0350224 + 0.999387i \(0.488850\pi\)
\(770\) 0 0
\(771\) 3.30578 0.119055
\(772\) 0 0
\(773\) −29.7576 −1.07031 −0.535153 0.844755i \(-0.679746\pi\)
−0.535153 + 0.844755i \(0.679746\pi\)
\(774\) 0 0
\(775\) 1.99604 0.0716998
\(776\) 0 0
\(777\) −0.223858 −0.00803086
\(778\) 0 0
\(779\) −19.2878 −0.691056
\(780\) 0 0
\(781\) −22.8950 −0.819248
\(782\) 0 0
\(783\) −3.10555 −0.110983
\(784\) 0 0
\(785\) −9.85459 −0.351726
\(786\) 0 0
\(787\) −20.8758 −0.744141 −0.372070 0.928205i \(-0.621352\pi\)
−0.372070 + 0.928205i \(0.621352\pi\)
\(788\) 0 0
\(789\) −25.0682 −0.892451
\(790\) 0 0
\(791\) −11.5431 −0.410424
\(792\) 0 0
\(793\) −69.6408 −2.47302
\(794\) 0 0
\(795\) 2.92730 0.103821
\(796\) 0 0
\(797\) −6.09486 −0.215891 −0.107946 0.994157i \(-0.534427\pi\)
−0.107946 + 0.994157i \(0.534427\pi\)
\(798\) 0 0
\(799\) 58.5402 2.07100
\(800\) 0 0
\(801\) 7.95440 0.281055
\(802\) 0 0
\(803\) 46.7390 1.64938
\(804\) 0 0
\(805\) 2.57685 0.0908219
\(806\) 0 0
\(807\) 10.5220 0.370393
\(808\) 0 0
\(809\) −6.87817 −0.241824 −0.120912 0.992663i \(-0.538582\pi\)
−0.120912 + 0.992663i \(0.538582\pi\)
\(810\) 0 0
\(811\) 1.30293 0.0457520 0.0228760 0.999738i \(-0.492718\pi\)
0.0228760 + 0.999738i \(0.492718\pi\)
\(812\) 0 0
\(813\) −12.0075 −0.421121
\(814\) 0 0
\(815\) 29.3613 1.02848
\(816\) 0 0
\(817\) 17.0727 0.597298
\(818\) 0 0
\(819\) 4.91214 0.171644
\(820\) 0 0
\(821\) 32.1030 1.12040 0.560201 0.828357i \(-0.310724\pi\)
0.560201 + 0.828357i \(0.310724\pi\)
\(822\) 0 0
\(823\) −37.1916 −1.29642 −0.648209 0.761462i \(-0.724482\pi\)
−0.648209 + 0.761462i \(0.724482\pi\)
\(824\) 0 0
\(825\) 7.90287 0.275143
\(826\) 0 0
\(827\) 8.61082 0.299428 0.149714 0.988729i \(-0.452165\pi\)
0.149714 + 0.988729i \(0.452165\pi\)
\(828\) 0 0
\(829\) −52.6544 −1.82876 −0.914382 0.404852i \(-0.867323\pi\)
−0.914382 + 0.404852i \(0.867323\pi\)
\(830\) 0 0
\(831\) −15.3310 −0.531825
\(832\) 0 0
\(833\) −7.21699 −0.250054
\(834\) 0 0
\(835\) −6.01770 −0.208251
\(836\) 0 0
\(837\) −1.21699 −0.0420653
\(838\) 0 0
\(839\) −49.9387 −1.72408 −0.862038 0.506844i \(-0.830812\pi\)
−0.862038 + 0.506844i \(0.830812\pi\)
\(840\) 0 0
\(841\) −19.3555 −0.667432
\(842\) 0 0
\(843\) 6.47463 0.222998
\(844\) 0 0
\(845\) 28.6781 0.986556
\(846\) 0 0
\(847\) 12.2170 0.419781
\(848\) 0 0
\(849\) 20.0531 0.688220
\(850\) 0 0
\(851\) 0.223858 0.00767375
\(852\) 0 0
\(853\) −19.8513 −0.679696 −0.339848 0.940480i \(-0.610376\pi\)
−0.339848 + 0.940480i \(0.610376\pi\)
\(854\) 0 0
\(855\) −6.01770 −0.205801
\(856\) 0 0
\(857\) −47.8328 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(858\) 0 0
\(859\) −2.81742 −0.0961290 −0.0480645 0.998844i \(-0.515305\pi\)
−0.0480645 + 0.998844i \(0.515305\pi\)
\(860\) 0 0
\(861\) −8.25925 −0.281474
\(862\) 0 0
\(863\) −19.6840 −0.670050 −0.335025 0.942209i \(-0.608745\pi\)
−0.335025 + 0.942209i \(0.608745\pi\)
\(864\) 0 0
\(865\) 35.2422 1.19827
\(866\) 0 0
\(867\) −35.0850 −1.19155
\(868\) 0 0
\(869\) 28.3120 0.960420
\(870\) 0 0
\(871\) −29.5598 −1.00159
\(872\) 0 0
\(873\) −4.43497 −0.150101
\(874\) 0 0
\(875\) −8.65784 −0.292688
\(876\) 0 0
\(877\) 14.5185 0.490255 0.245127 0.969491i \(-0.421170\pi\)
0.245127 + 0.969491i \(0.421170\pi\)
\(878\) 0 0
\(879\) 9.32941 0.314673
\(880\) 0 0
\(881\) −34.5657 −1.16455 −0.582273 0.812993i \(-0.697837\pi\)
−0.582273 + 0.812993i \(0.697837\pi\)
\(882\) 0 0
\(883\) −28.9929 −0.975689 −0.487844 0.872931i \(-0.662217\pi\)
−0.487844 + 0.872931i \(0.662217\pi\)
\(884\) 0 0
\(885\) −24.6604 −0.828950
\(886\) 0 0
\(887\) −39.5048 −1.32644 −0.663221 0.748424i \(-0.730811\pi\)
−0.663221 + 0.748424i \(0.730811\pi\)
\(888\) 0 0
\(889\) −20.3378 −0.682109
\(890\) 0 0
\(891\) −4.81840 −0.161423
\(892\) 0 0
\(893\) 18.9426 0.633889
\(894\) 0 0
\(895\) −49.5623 −1.65669
\(896\) 0 0
\(897\) −4.91214 −0.164012
\(898\) 0 0
\(899\) 3.77943 0.126051
\(900\) 0 0
\(901\) 8.19849 0.273131
\(902\) 0 0
\(903\) 7.31073 0.243286
\(904\) 0 0
\(905\) −21.3522 −0.709771
\(906\) 0 0
\(907\) 24.2911 0.806573 0.403287 0.915074i \(-0.367868\pi\)
0.403287 + 0.915074i \(0.367868\pi\)
\(908\) 0 0
\(909\) 3.93318 0.130455
\(910\) 0 0
\(911\) 33.1064 1.09686 0.548431 0.836196i \(-0.315225\pi\)
0.548431 + 0.836196i \(0.315225\pi\)
\(912\) 0 0
\(913\) 22.1998 0.734706
\(914\) 0 0
\(915\) 36.5327 1.20773
\(916\) 0 0
\(917\) 14.7693 0.487724
\(918\) 0 0
\(919\) 38.3521 1.26512 0.632560 0.774511i \(-0.282004\pi\)
0.632560 + 0.774511i \(0.282004\pi\)
\(920\) 0 0
\(921\) 14.7070 0.484611
\(922\) 0 0
\(923\) 23.3404 0.768260
\(924\) 0 0
\(925\) 0.367159 0.0120721
\(926\) 0 0
\(927\) 1.92396 0.0631910
\(928\) 0 0
\(929\) 20.9805 0.688348 0.344174 0.938906i \(-0.388159\pi\)
0.344174 + 0.938906i \(0.388159\pi\)
\(930\) 0 0
\(931\) −2.33529 −0.0765362
\(932\) 0 0
\(933\) 17.3192 0.567005
\(934\) 0 0
\(935\) 89.6082 2.93050
\(936\) 0 0
\(937\) 49.5054 1.61727 0.808635 0.588310i \(-0.200207\pi\)
0.808635 + 0.588310i \(0.200207\pi\)
\(938\) 0 0
\(939\) 30.5303 0.996317
\(940\) 0 0
\(941\) 17.2093 0.561008 0.280504 0.959853i \(-0.409498\pi\)
0.280504 + 0.959853i \(0.409498\pi\)
\(942\) 0 0
\(943\) 8.25925 0.268958
\(944\) 0 0
\(945\) −2.57685 −0.0838249
\(946\) 0 0
\(947\) −57.5167 −1.86904 −0.934521 0.355908i \(-0.884172\pi\)
−0.934521 + 0.355908i \(0.884172\pi\)
\(948\) 0 0
\(949\) −47.6483 −1.54673
\(950\) 0 0
\(951\) −10.0937 −0.327312
\(952\) 0 0
\(953\) 50.4525 1.63432 0.817158 0.576414i \(-0.195548\pi\)
0.817158 + 0.576414i \(0.195548\pi\)
\(954\) 0 0
\(955\) −29.2854 −0.947652
\(956\) 0 0
\(957\) 14.9638 0.483711
\(958\) 0 0
\(959\) 15.4975 0.500439
\(960\) 0 0
\(961\) −29.5189 −0.952224
\(962\) 0 0
\(963\) 2.13012 0.0686421
\(964\) 0 0
\(965\) −41.7648 −1.34446
\(966\) 0 0
\(967\) 35.6769 1.14729 0.573646 0.819103i \(-0.305529\pi\)
0.573646 + 0.819103i \(0.305529\pi\)
\(968\) 0 0
\(969\) −16.8538 −0.541422
\(970\) 0 0
\(971\) 27.3986 0.879264 0.439632 0.898178i \(-0.355109\pi\)
0.439632 + 0.898178i \(0.355109\pi\)
\(972\) 0 0
\(973\) −16.1291 −0.517076
\(974\) 0 0
\(975\) −8.05661 −0.258018
\(976\) 0 0
\(977\) 10.1552 0.324893 0.162446 0.986717i \(-0.448062\pi\)
0.162446 + 0.986717i \(0.448062\pi\)
\(978\) 0 0
\(979\) −38.3275 −1.22495
\(980\) 0 0
\(981\) −13.0108 −0.415404
\(982\) 0 0
\(983\) 4.05153 0.129224 0.0646119 0.997910i \(-0.479419\pi\)
0.0646119 + 0.997910i \(0.479419\pi\)
\(984\) 0 0
\(985\) 40.9161 1.30370
\(986\) 0 0
\(987\) 8.11144 0.258190
\(988\) 0 0
\(989\) −7.31073 −0.232468
\(990\) 0 0
\(991\) −17.7787 −0.564758 −0.282379 0.959303i \(-0.591124\pi\)
−0.282379 + 0.959303i \(0.591124\pi\)
\(992\) 0 0
\(993\) 10.3632 0.328866
\(994\) 0 0
\(995\) −32.0824 −1.01708
\(996\) 0 0
\(997\) −11.6882 −0.370170 −0.185085 0.982723i \(-0.559256\pi\)
−0.185085 + 0.982723i \(0.559256\pi\)
\(998\) 0 0
\(999\) −0.223858 −0.00708255
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 7728.2.a.bz.1.4 4
4.3 odd 2 3864.2.a.t.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.t.1.4 4 4.3 odd 2
7728.2.a.bz.1.4 4 1.1 even 1 trivial