Properties

Label 4864.2.a.bp.1.5
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 13 x^{6} + 24 x^{5} + 48 x^{4} - 68 x^{3} - 62 x^{2} + 32 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.932340\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.579017 q^{3} -2.10882 q^{5} -2.73436 q^{7} -2.66474 q^{9} +O(q^{10})\) \(q+0.579017 q^{3} -2.10882 q^{5} -2.73436 q^{7} -2.66474 q^{9} -4.66474 q^{11} -4.47791 q^{13} -1.22104 q^{15} -6.85237 q^{17} -1.00000 q^{19} -1.58324 q^{21} -1.20416 q^{23} -0.552894 q^{25} -3.27998 q^{27} +9.57484 q^{29} +5.35842 q^{31} -2.70096 q^{33} +5.76625 q^{35} +1.09693 q^{37} -2.59279 q^{39} -7.33797 q^{41} -7.64408 q^{43} +5.61945 q^{45} -7.56486 q^{47} +0.476702 q^{49} -3.96764 q^{51} +3.11949 q^{53} +9.83708 q^{55} -0.579017 q^{57} +10.2442 q^{59} -0.722061 q^{61} +7.28634 q^{63} +9.44309 q^{65} -6.13698 q^{67} -0.697232 q^{69} +4.62247 q^{71} +6.19270 q^{73} -0.320135 q^{75} +12.7551 q^{77} +3.26723 q^{79} +6.09505 q^{81} -8.97934 q^{83} +14.4504 q^{85} +5.54400 q^{87} -0.620707 q^{89} +12.2442 q^{91} +3.10262 q^{93} +2.10882 q^{95} -1.67284 q^{97} +12.4303 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{5} - 4q^{7} + 12q^{9} + O(q^{10}) \) \( 8q + 8q^{5} - 4q^{7} + 12q^{9} - 4q^{11} + 8q^{13} - 4q^{17} - 8q^{19} + 16q^{21} + 12q^{25} + 28q^{29} - 8q^{31} + 12q^{35} + 4q^{37} + 4q^{39} - 8q^{41} + 4q^{43} + 24q^{45} - 12q^{47} + 12q^{49} - 12q^{51} + 32q^{53} + 8q^{55} - 12q^{59} + 8q^{61} + 16q^{63} + 8q^{65} + 4q^{67} + 28q^{69} + 24q^{71} + 24q^{77} + 24q^{79} - 8q^{81} - 40q^{83} + 24q^{85} - 24q^{87} + 8q^{89} + 4q^{91} + 32q^{93} - 8q^{95} + 16q^{97} + 76q^{99} + O(q^{100}) \)

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
\(3\) 0.579017 0.334296 0.167148 0.985932i \(-0.446544\pi\)
0.167148 + 0.985932i \(0.446544\pi\)
\(4\) 0 0
\(5\) −2.10882 −0.943091 −0.471546 0.881842i \(-0.656304\pi\)
−0.471546 + 0.881842i \(0.656304\pi\)
\(6\) 0 0
\(7\) −2.73436 −1.03349 −0.516745 0.856140i \(-0.672856\pi\)
−0.516745 + 0.856140i \(0.672856\pi\)
\(8\) 0 0
\(9\) −2.66474 −0.888246
\(10\) 0 0
\(11\) −4.66474 −1.40647 −0.703236 0.710957i \(-0.748262\pi\)
−0.703236 + 0.710957i \(0.748262\pi\)
\(12\) 0 0
\(13\) −4.47791 −1.24195 −0.620974 0.783831i \(-0.713263\pi\)
−0.620974 + 0.783831i \(0.713263\pi\)
\(14\) 0 0
\(15\) −1.22104 −0.315271
\(16\) 0 0
\(17\) −6.85237 −1.66194 −0.830972 0.556314i \(-0.812215\pi\)
−0.830972 + 0.556314i \(0.812215\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.58324 −0.345491
\(22\) 0 0
\(23\) −1.20416 −0.251085 −0.125543 0.992088i \(-0.540067\pi\)
−0.125543 + 0.992088i \(0.540067\pi\)
\(24\) 0 0
\(25\) −0.552894 −0.110579
\(26\) 0 0
\(27\) −3.27998 −0.631233
\(28\) 0 0
\(29\) 9.57484 1.77800 0.889002 0.457904i \(-0.151400\pi\)
0.889002 + 0.457904i \(0.151400\pi\)
\(30\) 0 0
\(31\) 5.35842 0.962400 0.481200 0.876611i \(-0.340201\pi\)
0.481200 + 0.876611i \(0.340201\pi\)
\(32\) 0 0
\(33\) −2.70096 −0.470178
\(34\) 0 0
\(35\) 5.76625 0.974675
\(36\) 0 0
\(37\) 1.09693 0.180335 0.0901674 0.995927i \(-0.471260\pi\)
0.0901674 + 0.995927i \(0.471260\pi\)
\(38\) 0 0
\(39\) −2.59279 −0.415178
\(40\) 0 0
\(41\) −7.33797 −1.14600 −0.573000 0.819556i \(-0.694220\pi\)
−0.573000 + 0.819556i \(0.694220\pi\)
\(42\) 0 0
\(43\) −7.64408 −1.16571 −0.582856 0.812576i \(-0.698065\pi\)
−0.582856 + 0.812576i \(0.698065\pi\)
\(44\) 0 0
\(45\) 5.61945 0.837697
\(46\) 0 0
\(47\) −7.56486 −1.10345 −0.551724 0.834027i \(-0.686030\pi\)
−0.551724 + 0.834027i \(0.686030\pi\)
\(48\) 0 0
\(49\) 0.476702 0.0681003
\(50\) 0 0
\(51\) −3.96764 −0.555581
\(52\) 0 0
\(53\) 3.11949 0.428495 0.214248 0.976779i \(-0.431270\pi\)
0.214248 + 0.976779i \(0.431270\pi\)
\(54\) 0 0
\(55\) 9.83708 1.32643
\(56\) 0 0
\(57\) −0.579017 −0.0766927
\(58\) 0 0
\(59\) 10.2442 1.33368 0.666840 0.745201i \(-0.267646\pi\)
0.666840 + 0.745201i \(0.267646\pi\)
\(60\) 0 0
\(61\) −0.722061 −0.0924504 −0.0462252 0.998931i \(-0.514719\pi\)
−0.0462252 + 0.998931i \(0.514719\pi\)
\(62\) 0 0
\(63\) 7.28634 0.917993
\(64\) 0 0
\(65\) 9.44309 1.17127
\(66\) 0 0
\(67\) −6.13698 −0.749752 −0.374876 0.927075i \(-0.622315\pi\)
−0.374876 + 0.927075i \(0.622315\pi\)
\(68\) 0 0
\(69\) −0.697232 −0.0839368
\(70\) 0 0
\(71\) 4.62247 0.548587 0.274293 0.961646i \(-0.411556\pi\)
0.274293 + 0.961646i \(0.411556\pi\)
\(72\) 0 0
\(73\) 6.19270 0.724801 0.362400 0.932022i \(-0.381957\pi\)
0.362400 + 0.932022i \(0.381957\pi\)
\(74\) 0 0
\(75\) −0.320135 −0.0369660
\(76\) 0 0
\(77\) 12.7551 1.45357
\(78\) 0 0
\(79\) 3.26723 0.367592 0.183796 0.982964i \(-0.441161\pi\)
0.183796 + 0.982964i \(0.441161\pi\)
\(80\) 0 0
\(81\) 6.09505 0.677228
\(82\) 0 0
\(83\) −8.97934 −0.985611 −0.492805 0.870140i \(-0.664029\pi\)
−0.492805 + 0.870140i \(0.664029\pi\)
\(84\) 0 0
\(85\) 14.4504 1.56736
\(86\) 0 0
\(87\) 5.54400 0.594379
\(88\) 0 0
\(89\) −0.620707 −0.0657948 −0.0328974 0.999459i \(-0.510473\pi\)
−0.0328974 + 0.999459i \(0.510473\pi\)
\(90\) 0 0
\(91\) 12.2442 1.28354
\(92\) 0 0
\(93\) 3.10262 0.321726
\(94\) 0 0
\(95\) 2.10882 0.216360
\(96\) 0 0
\(97\) −1.67284 −0.169851 −0.0849257 0.996387i \(-0.527065\pi\)
−0.0849257 + 0.996387i \(0.527065\pi\)
\(98\) 0 0
\(99\) 12.4303 1.24929
\(100\) 0 0
\(101\) 7.30571 0.726946 0.363473 0.931605i \(-0.381591\pi\)
0.363473 + 0.931605i \(0.381591\pi\)
\(102\) 0 0
\(103\) −8.77332 −0.864461 −0.432231 0.901763i \(-0.642273\pi\)
−0.432231 + 0.901763i \(0.642273\pi\)
\(104\) 0 0
\(105\) 3.33876 0.325830
\(106\) 0 0
\(107\) 9.91355 0.958379 0.479190 0.877711i \(-0.340931\pi\)
0.479190 + 0.877711i \(0.340931\pi\)
\(108\) 0 0
\(109\) −2.84589 −0.272587 −0.136293 0.990669i \(-0.543519\pi\)
−0.136293 + 0.990669i \(0.543519\pi\)
\(110\) 0 0
\(111\) 0.635143 0.0602851
\(112\) 0 0
\(113\) −1.58487 −0.149092 −0.0745462 0.997218i \(-0.523751\pi\)
−0.0745462 + 0.997218i \(0.523751\pi\)
\(114\) 0 0
\(115\) 2.53936 0.236797
\(116\) 0 0
\(117\) 11.9325 1.10316
\(118\) 0 0
\(119\) 18.7368 1.71760
\(120\) 0 0
\(121\) 10.7598 0.978163
\(122\) 0 0
\(123\) −4.24881 −0.383103
\(124\) 0 0
\(125\) 11.7100 1.04738
\(126\) 0 0
\(127\) −14.2933 −1.26833 −0.634163 0.773199i \(-0.718655\pi\)
−0.634163 + 0.773199i \(0.718655\pi\)
\(128\) 0 0
\(129\) −4.42605 −0.389692
\(130\) 0 0
\(131\) −0.609006 −0.0532091 −0.0266046 0.999646i \(-0.508469\pi\)
−0.0266046 + 0.999646i \(0.508469\pi\)
\(132\) 0 0
\(133\) 2.73436 0.237099
\(134\) 0 0
\(135\) 6.91688 0.595310
\(136\) 0 0
\(137\) −18.1589 −1.55142 −0.775710 0.631089i \(-0.782608\pi\)
−0.775710 + 0.631089i \(0.782608\pi\)
\(138\) 0 0
\(139\) −1.93421 −0.164058 −0.0820289 0.996630i \(-0.526140\pi\)
−0.0820289 + 0.996630i \(0.526140\pi\)
\(140\) 0 0
\(141\) −4.38018 −0.368878
\(142\) 0 0
\(143\) 20.8883 1.74677
\(144\) 0 0
\(145\) −20.1916 −1.67682
\(146\) 0 0
\(147\) 0.276019 0.0227656
\(148\) 0 0
\(149\) 15.5775 1.27616 0.638080 0.769970i \(-0.279729\pi\)
0.638080 + 0.769970i \(0.279729\pi\)
\(150\) 0 0
\(151\) −19.9746 −1.62551 −0.812756 0.582605i \(-0.802034\pi\)
−0.812756 + 0.582605i \(0.802034\pi\)
\(152\) 0 0
\(153\) 18.2598 1.47622
\(154\) 0 0
\(155\) −11.2999 −0.907631
\(156\) 0 0
\(157\) −20.8872 −1.66698 −0.833489 0.552536i \(-0.813660\pi\)
−0.833489 + 0.552536i \(0.813660\pi\)
\(158\) 0 0
\(159\) 1.80624 0.143244
\(160\) 0 0
\(161\) 3.29261 0.259494
\(162\) 0 0
\(163\) 2.54835 0.199602 0.0998010 0.995007i \(-0.468179\pi\)
0.0998010 + 0.995007i \(0.468179\pi\)
\(164\) 0 0
\(165\) 5.69584 0.443420
\(166\) 0 0
\(167\) −18.6644 −1.44429 −0.722146 0.691740i \(-0.756844\pi\)
−0.722146 + 0.691740i \(0.756844\pi\)
\(168\) 0 0
\(169\) 7.05167 0.542436
\(170\) 0 0
\(171\) 2.66474 0.203778
\(172\) 0 0
\(173\) −8.42370 −0.640442 −0.320221 0.947343i \(-0.603757\pi\)
−0.320221 + 0.947343i \(0.603757\pi\)
\(174\) 0 0
\(175\) 1.51181 0.114282
\(176\) 0 0
\(177\) 5.93157 0.445844
\(178\) 0 0
\(179\) −16.6809 −1.24679 −0.623394 0.781908i \(-0.714247\pi\)
−0.623394 + 0.781908i \(0.714247\pi\)
\(180\) 0 0
\(181\) 4.65888 0.346292 0.173146 0.984896i \(-0.444607\pi\)
0.173146 + 0.984896i \(0.444607\pi\)
\(182\) 0 0
\(183\) −0.418086 −0.0309058
\(184\) 0 0
\(185\) −2.31323 −0.170072
\(186\) 0 0
\(187\) 31.9645 2.33748
\(188\) 0 0
\(189\) 8.96864 0.652372
\(190\) 0 0
\(191\) −20.0609 −1.45155 −0.725777 0.687930i \(-0.758520\pi\)
−0.725777 + 0.687930i \(0.758520\pi\)
\(192\) 0 0
\(193\) 1.85117 0.133250 0.0666251 0.997778i \(-0.478777\pi\)
0.0666251 + 0.997778i \(0.478777\pi\)
\(194\) 0 0
\(195\) 5.46771 0.391551
\(196\) 0 0
\(197\) 0.224883 0.0160222 0.00801112 0.999968i \(-0.497450\pi\)
0.00801112 + 0.999968i \(0.497450\pi\)
\(198\) 0 0
\(199\) −1.77685 −0.125957 −0.0629787 0.998015i \(-0.520060\pi\)
−0.0629787 + 0.998015i \(0.520060\pi\)
\(200\) 0 0
\(201\) −3.55342 −0.250639
\(202\) 0 0
\(203\) −26.1810 −1.83755
\(204\) 0 0
\(205\) 15.4744 1.08078
\(206\) 0 0
\(207\) 3.20878 0.223026
\(208\) 0 0
\(209\) 4.66474 0.322667
\(210\) 0 0
\(211\) 17.0194 1.17166 0.585831 0.810433i \(-0.300768\pi\)
0.585831 + 0.810433i \(0.300768\pi\)
\(212\) 0 0
\(213\) 2.67649 0.183390
\(214\) 0 0
\(215\) 16.1200 1.09937
\(216\) 0 0
\(217\) −14.6518 −0.994630
\(218\) 0 0
\(219\) 3.58568 0.242298
\(220\) 0 0
\(221\) 30.6843 2.06405
\(222\) 0 0
\(223\) −13.6634 −0.914971 −0.457486 0.889217i \(-0.651250\pi\)
−0.457486 + 0.889217i \(0.651250\pi\)
\(224\) 0 0
\(225\) 1.47332 0.0982212
\(226\) 0 0
\(227\) −5.46152 −0.362494 −0.181247 0.983438i \(-0.558013\pi\)
−0.181247 + 0.983438i \(0.558013\pi\)
\(228\) 0 0
\(229\) 3.22791 0.213306 0.106653 0.994296i \(-0.465987\pi\)
0.106653 + 0.994296i \(0.465987\pi\)
\(230\) 0 0
\(231\) 7.38540 0.485923
\(232\) 0 0
\(233\) 17.8344 1.16837 0.584185 0.811621i \(-0.301414\pi\)
0.584185 + 0.811621i \(0.301414\pi\)
\(234\) 0 0
\(235\) 15.9529 1.04065
\(236\) 0 0
\(237\) 1.89178 0.122884
\(238\) 0 0
\(239\) −1.82556 −0.118086 −0.0590428 0.998255i \(-0.518805\pi\)
−0.0590428 + 0.998255i \(0.518805\pi\)
\(240\) 0 0
\(241\) 6.47608 0.417161 0.208581 0.978005i \(-0.433116\pi\)
0.208581 + 0.978005i \(0.433116\pi\)
\(242\) 0 0
\(243\) 13.3691 0.857627
\(244\) 0 0
\(245\) −1.00528 −0.0642248
\(246\) 0 0
\(247\) 4.47791 0.284923
\(248\) 0 0
\(249\) −5.19919 −0.329485
\(250\) 0 0
\(251\) −3.10899 −0.196238 −0.0981189 0.995175i \(-0.531283\pi\)
−0.0981189 + 0.995175i \(0.531283\pi\)
\(252\) 0 0
\(253\) 5.61711 0.353145
\(254\) 0 0
\(255\) 8.36702 0.523963
\(256\) 0 0
\(257\) 5.75606 0.359053 0.179526 0.983753i \(-0.442543\pi\)
0.179526 + 0.983753i \(0.442543\pi\)
\(258\) 0 0
\(259\) −2.99941 −0.186374
\(260\) 0 0
\(261\) −25.5145 −1.57931
\(262\) 0 0
\(263\) −4.58595 −0.282782 −0.141391 0.989954i \(-0.545157\pi\)
−0.141391 + 0.989954i \(0.545157\pi\)
\(264\) 0 0
\(265\) −6.57844 −0.404110
\(266\) 0 0
\(267\) −0.359400 −0.0219949
\(268\) 0 0
\(269\) 11.3913 0.694542 0.347271 0.937765i \(-0.387108\pi\)
0.347271 + 0.937765i \(0.387108\pi\)
\(270\) 0 0
\(271\) −25.1252 −1.52625 −0.763124 0.646252i \(-0.776336\pi\)
−0.763124 + 0.646252i \(0.776336\pi\)
\(272\) 0 0
\(273\) 7.08960 0.429082
\(274\) 0 0
\(275\) 2.57910 0.155526
\(276\) 0 0
\(277\) 18.2998 1.09953 0.549763 0.835321i \(-0.314718\pi\)
0.549763 + 0.835321i \(0.314718\pi\)
\(278\) 0 0
\(279\) −14.2788 −0.854848
\(280\) 0 0
\(281\) 18.0708 1.07802 0.539008 0.842301i \(-0.318799\pi\)
0.539008 + 0.842301i \(0.318799\pi\)
\(282\) 0 0
\(283\) −15.3972 −0.915269 −0.457634 0.889140i \(-0.651303\pi\)
−0.457634 + 0.889140i \(0.651303\pi\)
\(284\) 0 0
\(285\) 1.22104 0.0723282
\(286\) 0 0
\(287\) 20.0646 1.18438
\(288\) 0 0
\(289\) 29.9550 1.76206
\(290\) 0 0
\(291\) −0.968605 −0.0567806
\(292\) 0 0
\(293\) 11.4407 0.668374 0.334187 0.942507i \(-0.391538\pi\)
0.334187 + 0.942507i \(0.391538\pi\)
\(294\) 0 0
\(295\) −21.6031 −1.25778
\(296\) 0 0
\(297\) 15.3003 0.887811
\(298\) 0 0
\(299\) 5.39214 0.311835
\(300\) 0 0
\(301\) 20.9016 1.20475
\(302\) 0 0
\(303\) 4.23013 0.243015
\(304\) 0 0
\(305\) 1.52269 0.0871892
\(306\) 0 0
\(307\) −21.3453 −1.21824 −0.609122 0.793077i \(-0.708478\pi\)
−0.609122 + 0.793077i \(0.708478\pi\)
\(308\) 0 0
\(309\) −5.07991 −0.288986
\(310\) 0 0
\(311\) −21.7295 −1.23216 −0.616082 0.787682i \(-0.711281\pi\)
−0.616082 + 0.787682i \(0.711281\pi\)
\(312\) 0 0
\(313\) −34.1009 −1.92750 −0.963748 0.266814i \(-0.914029\pi\)
−0.963748 + 0.266814i \(0.914029\pi\)
\(314\) 0 0
\(315\) −15.3656 −0.865751
\(316\) 0 0
\(317\) 22.2281 1.24845 0.624226 0.781244i \(-0.285414\pi\)
0.624226 + 0.781244i \(0.285414\pi\)
\(318\) 0 0
\(319\) −44.6641 −2.50071
\(320\) 0 0
\(321\) 5.74012 0.320382
\(322\) 0 0
\(323\) 6.85237 0.381276
\(324\) 0 0
\(325\) 2.47581 0.137333
\(326\) 0 0
\(327\) −1.64782 −0.0911245
\(328\) 0 0
\(329\) 20.6850 1.14040
\(330\) 0 0
\(331\) −24.1969 −1.32998 −0.664991 0.746852i \(-0.731564\pi\)
−0.664991 + 0.746852i \(0.731564\pi\)
\(332\) 0 0
\(333\) −2.92304 −0.160182
\(334\) 0 0
\(335\) 12.9418 0.707084
\(336\) 0 0
\(337\) −34.2735 −1.86700 −0.933498 0.358583i \(-0.883260\pi\)
−0.933498 + 0.358583i \(0.883260\pi\)
\(338\) 0 0
\(339\) −0.917670 −0.0498410
\(340\) 0 0
\(341\) −24.9956 −1.35359
\(342\) 0 0
\(343\) 17.8370 0.963108
\(344\) 0 0
\(345\) 1.47033 0.0791601
\(346\) 0 0
\(347\) −26.9233 −1.44532 −0.722658 0.691206i \(-0.757080\pi\)
−0.722658 + 0.691206i \(0.757080\pi\)
\(348\) 0 0
\(349\) −8.65769 −0.463436 −0.231718 0.972783i \(-0.574435\pi\)
−0.231718 + 0.972783i \(0.574435\pi\)
\(350\) 0 0
\(351\) 14.6875 0.783959
\(352\) 0 0
\(353\) −9.11265 −0.485018 −0.242509 0.970149i \(-0.577970\pi\)
−0.242509 + 0.970149i \(0.577970\pi\)
\(354\) 0 0
\(355\) −9.74795 −0.517367
\(356\) 0 0
\(357\) 10.8489 0.574187
\(358\) 0 0
\(359\) 4.29978 0.226934 0.113467 0.993542i \(-0.463804\pi\)
0.113467 + 0.993542i \(0.463804\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 6.23010 0.326996
\(364\) 0 0
\(365\) −13.0593 −0.683553
\(366\) 0 0
\(367\) 21.2873 1.11119 0.555594 0.831454i \(-0.312491\pi\)
0.555594 + 0.831454i \(0.312491\pi\)
\(368\) 0 0
\(369\) 19.5538 1.01793
\(370\) 0 0
\(371\) −8.52980 −0.442845
\(372\) 0 0
\(373\) −11.1076 −0.575129 −0.287565 0.957761i \(-0.592846\pi\)
−0.287565 + 0.957761i \(0.592846\pi\)
\(374\) 0 0
\(375\) 6.78031 0.350134
\(376\) 0 0
\(377\) −42.8753 −2.20819
\(378\) 0 0
\(379\) 20.6908 1.06281 0.531407 0.847117i \(-0.321664\pi\)
0.531407 + 0.847117i \(0.321664\pi\)
\(380\) 0 0
\(381\) −8.27608 −0.423996
\(382\) 0 0
\(383\) 12.1869 0.622720 0.311360 0.950292i \(-0.399215\pi\)
0.311360 + 0.950292i \(0.399215\pi\)
\(384\) 0 0
\(385\) −26.8981 −1.37085
\(386\) 0 0
\(387\) 20.3695 1.03544
\(388\) 0 0
\(389\) 26.1769 1.32722 0.663611 0.748078i \(-0.269023\pi\)
0.663611 + 0.748078i \(0.269023\pi\)
\(390\) 0 0
\(391\) 8.25137 0.417290
\(392\) 0 0
\(393\) −0.352625 −0.0177876
\(394\) 0 0
\(395\) −6.88999 −0.346673
\(396\) 0 0
\(397\) 14.8451 0.745055 0.372528 0.928021i \(-0.378491\pi\)
0.372528 + 0.928021i \(0.378491\pi\)
\(398\) 0 0
\(399\) 1.58324 0.0792611
\(400\) 0 0
\(401\) 34.4111 1.71841 0.859205 0.511632i \(-0.170959\pi\)
0.859205 + 0.511632i \(0.170959\pi\)
\(402\) 0 0
\(403\) −23.9945 −1.19525
\(404\) 0 0
\(405\) −12.8533 −0.638688
\(406\) 0 0
\(407\) −5.11691 −0.253636
\(408\) 0 0
\(409\) −1.43283 −0.0708487 −0.0354243 0.999372i \(-0.511278\pi\)
−0.0354243 + 0.999372i \(0.511278\pi\)
\(410\) 0 0
\(411\) −10.5143 −0.518633
\(412\) 0 0
\(413\) −28.0113 −1.37834
\(414\) 0 0
\(415\) 18.9358 0.929521
\(416\) 0 0
\(417\) −1.11994 −0.0548438
\(418\) 0 0
\(419\) 33.6886 1.64579 0.822897 0.568190i \(-0.192356\pi\)
0.822897 + 0.568190i \(0.192356\pi\)
\(420\) 0 0
\(421\) 0.237430 0.0115716 0.00578582 0.999983i \(-0.498158\pi\)
0.00578582 + 0.999983i \(0.498158\pi\)
\(422\) 0 0
\(423\) 20.1584 0.980134
\(424\) 0 0
\(425\) 3.78863 0.183776
\(426\) 0 0
\(427\) 1.97437 0.0955465
\(428\) 0 0
\(429\) 12.0947 0.583936
\(430\) 0 0
\(431\) −13.0743 −0.629766 −0.314883 0.949130i \(-0.601965\pi\)
−0.314883 + 0.949130i \(0.601965\pi\)
\(432\) 0 0
\(433\) −9.67718 −0.465056 −0.232528 0.972590i \(-0.574700\pi\)
−0.232528 + 0.972590i \(0.574700\pi\)
\(434\) 0 0
\(435\) −11.6913 −0.560554
\(436\) 0 0
\(437\) 1.20416 0.0576030
\(438\) 0 0
\(439\) 17.7983 0.849468 0.424734 0.905318i \(-0.360368\pi\)
0.424734 + 0.905318i \(0.360368\pi\)
\(440\) 0 0
\(441\) −1.27029 −0.0604898
\(442\) 0 0
\(443\) 3.30468 0.157010 0.0785050 0.996914i \(-0.474985\pi\)
0.0785050 + 0.996914i \(0.474985\pi\)
\(444\) 0 0
\(445\) 1.30896 0.0620505
\(446\) 0 0
\(447\) 9.01966 0.426615
\(448\) 0 0
\(449\) −29.6543 −1.39947 −0.699735 0.714402i \(-0.746699\pi\)
−0.699735 + 0.714402i \(0.746699\pi\)
\(450\) 0 0
\(451\) 34.2297 1.61182
\(452\) 0 0
\(453\) −11.5656 −0.543402
\(454\) 0 0
\(455\) −25.8208 −1.21050
\(456\) 0 0
\(457\) 20.2310 0.946368 0.473184 0.880964i \(-0.343105\pi\)
0.473184 + 0.880964i \(0.343105\pi\)
\(458\) 0 0
\(459\) 22.4756 1.04907
\(460\) 0 0
\(461\) 41.2429 1.92087 0.960436 0.278502i \(-0.0898377\pi\)
0.960436 + 0.278502i \(0.0898377\pi\)
\(462\) 0 0
\(463\) 16.5232 0.767896 0.383948 0.923355i \(-0.374564\pi\)
0.383948 + 0.923355i \(0.374564\pi\)
\(464\) 0 0
\(465\) −6.54285 −0.303417
\(466\) 0 0
\(467\) −24.9967 −1.15671 −0.578355 0.815786i \(-0.696305\pi\)
−0.578355 + 0.815786i \(0.696305\pi\)
\(468\) 0 0
\(469\) 16.7807 0.774860
\(470\) 0 0
\(471\) −12.0940 −0.557264
\(472\) 0 0
\(473\) 35.6576 1.63954
\(474\) 0 0
\(475\) 0.552894 0.0253685
\(476\) 0 0
\(477\) −8.31263 −0.380609
\(478\) 0 0
\(479\) −10.0298 −0.458273 −0.229137 0.973394i \(-0.573590\pi\)
−0.229137 + 0.973394i \(0.573590\pi\)
\(480\) 0 0
\(481\) −4.91197 −0.223966
\(482\) 0 0
\(483\) 1.90648 0.0867478
\(484\) 0 0
\(485\) 3.52772 0.160185
\(486\) 0 0
\(487\) 41.6248 1.88620 0.943101 0.332508i \(-0.107895\pi\)
0.943101 + 0.332508i \(0.107895\pi\)
\(488\) 0 0
\(489\) 1.47554 0.0667261
\(490\) 0 0
\(491\) −34.6939 −1.56572 −0.782858 0.622200i \(-0.786239\pi\)
−0.782858 + 0.622200i \(0.786239\pi\)
\(492\) 0 0
\(493\) −65.6104 −2.95494
\(494\) 0 0
\(495\) −26.2132 −1.17820
\(496\) 0 0
\(497\) −12.6395 −0.566959
\(498\) 0 0
\(499\) −14.1270 −0.632413 −0.316207 0.948690i \(-0.602409\pi\)
−0.316207 + 0.948690i \(0.602409\pi\)
\(500\) 0 0
\(501\) −10.8070 −0.482821
\(502\) 0 0
\(503\) 3.01157 0.134279 0.0671395 0.997744i \(-0.478613\pi\)
0.0671395 + 0.997744i \(0.478613\pi\)
\(504\) 0 0
\(505\) −15.4064 −0.685576
\(506\) 0 0
\(507\) 4.08304 0.181334
\(508\) 0 0
\(509\) 2.38122 0.105546 0.0527730 0.998607i \(-0.483194\pi\)
0.0527730 + 0.998607i \(0.483194\pi\)
\(510\) 0 0
\(511\) −16.9330 −0.749074
\(512\) 0 0
\(513\) 3.27998 0.144815
\(514\) 0 0
\(515\) 18.5013 0.815266
\(516\) 0 0
\(517\) 35.2881 1.55197
\(518\) 0 0
\(519\) −4.87747 −0.214097
\(520\) 0 0
\(521\) −17.9133 −0.784798 −0.392399 0.919795i \(-0.628355\pi\)
−0.392399 + 0.919795i \(0.628355\pi\)
\(522\) 0 0
\(523\) 18.9602 0.829071 0.414536 0.910033i \(-0.363944\pi\)
0.414536 + 0.910033i \(0.363944\pi\)
\(524\) 0 0
\(525\) 0.875363 0.0382040
\(526\) 0 0
\(527\) −36.7178 −1.59945
\(528\) 0 0
\(529\) −21.5500 −0.936956
\(530\) 0 0
\(531\) −27.2981 −1.18464
\(532\) 0 0
\(533\) 32.8588 1.42327
\(534\) 0 0
\(535\) −20.9059 −0.903839
\(536\) 0 0
\(537\) −9.65852 −0.416796
\(538\) 0 0
\(539\) −2.22369 −0.0957811
\(540\) 0 0
\(541\) 34.2027 1.47049 0.735245 0.677802i \(-0.237067\pi\)
0.735245 + 0.677802i \(0.237067\pi\)
\(542\) 0 0
\(543\) 2.69757 0.115764
\(544\) 0 0
\(545\) 6.00145 0.257074
\(546\) 0 0
\(547\) −0.645113 −0.0275830 −0.0137915 0.999905i \(-0.504390\pi\)
−0.0137915 + 0.999905i \(0.504390\pi\)
\(548\) 0 0
\(549\) 1.92410 0.0821187
\(550\) 0 0
\(551\) −9.57484 −0.407902
\(552\) 0 0
\(553\) −8.93377 −0.379903
\(554\) 0 0
\(555\) −1.33940 −0.0568544
\(556\) 0 0
\(557\) −10.2596 −0.434714 −0.217357 0.976092i \(-0.569744\pi\)
−0.217357 + 0.976092i \(0.569744\pi\)
\(558\) 0 0
\(559\) 34.2295 1.44775
\(560\) 0 0
\(561\) 18.5080 0.781409
\(562\) 0 0
\(563\) −12.9293 −0.544906 −0.272453 0.962169i \(-0.587835\pi\)
−0.272453 + 0.962169i \(0.587835\pi\)
\(564\) 0 0
\(565\) 3.34221 0.140608
\(566\) 0 0
\(567\) −16.6660 −0.699908
\(568\) 0 0
\(569\) −6.32637 −0.265215 −0.132608 0.991169i \(-0.542335\pi\)
−0.132608 + 0.991169i \(0.542335\pi\)
\(570\) 0 0
\(571\) 31.1997 1.30566 0.652832 0.757502i \(-0.273581\pi\)
0.652832 + 0.757502i \(0.273581\pi\)
\(572\) 0 0
\(573\) −11.6156 −0.485248
\(574\) 0 0
\(575\) 0.665774 0.0277647
\(576\) 0 0
\(577\) 5.48996 0.228550 0.114275 0.993449i \(-0.463545\pi\)
0.114275 + 0.993449i \(0.463545\pi\)
\(578\) 0 0
\(579\) 1.07186 0.0445450
\(580\) 0 0
\(581\) 24.5527 1.01862
\(582\) 0 0
\(583\) −14.5516 −0.602666
\(584\) 0 0
\(585\) −25.1634 −1.04038
\(586\) 0 0
\(587\) 17.4576 0.720552 0.360276 0.932846i \(-0.382683\pi\)
0.360276 + 0.932846i \(0.382683\pi\)
\(588\) 0 0
\(589\) −5.35842 −0.220790
\(590\) 0 0
\(591\) 0.130211 0.00535617
\(592\) 0 0
\(593\) −16.7657 −0.688486 −0.344243 0.938881i \(-0.611864\pi\)
−0.344243 + 0.938881i \(0.611864\pi\)
\(594\) 0 0
\(595\) −39.5125 −1.61985
\(596\) 0 0
\(597\) −1.02883 −0.0421070
\(598\) 0 0
\(599\) −0.357982 −0.0146267 −0.00731337 0.999973i \(-0.502328\pi\)
−0.00731337 + 0.999973i \(0.502328\pi\)
\(600\) 0 0
\(601\) −22.8949 −0.933903 −0.466952 0.884283i \(-0.654648\pi\)
−0.466952 + 0.884283i \(0.654648\pi\)
\(602\) 0 0
\(603\) 16.3535 0.665964
\(604\) 0 0
\(605\) −22.6904 −0.922497
\(606\) 0 0
\(607\) −1.33917 −0.0543551 −0.0271775 0.999631i \(-0.508652\pi\)
−0.0271775 + 0.999631i \(0.508652\pi\)
\(608\) 0 0
\(609\) −15.1593 −0.614284
\(610\) 0 0
\(611\) 33.8747 1.37043
\(612\) 0 0
\(613\) −29.8667 −1.20631 −0.603153 0.797625i \(-0.706089\pi\)
−0.603153 + 0.797625i \(0.706089\pi\)
\(614\) 0 0
\(615\) 8.95997 0.361301
\(616\) 0 0
\(617\) 36.4935 1.46917 0.734587 0.678514i \(-0.237376\pi\)
0.734587 + 0.678514i \(0.237376\pi\)
\(618\) 0 0
\(619\) 8.88669 0.357186 0.178593 0.983923i \(-0.442845\pi\)
0.178593 + 0.983923i \(0.442845\pi\)
\(620\) 0 0
\(621\) 3.94963 0.158493
\(622\) 0 0
\(623\) 1.69723 0.0679982
\(624\) 0 0
\(625\) −21.9298 −0.877194
\(626\) 0 0
\(627\) 2.70096 0.107866
\(628\) 0 0
\(629\) −7.51659 −0.299706
\(630\) 0 0
\(631\) −28.1724 −1.12153 −0.560763 0.827977i \(-0.689492\pi\)
−0.560763 + 0.827977i \(0.689492\pi\)
\(632\) 0 0
\(633\) 9.85452 0.391682
\(634\) 0 0
\(635\) 30.1420 1.19615
\(636\) 0 0
\(637\) −2.13463 −0.0845771
\(638\) 0 0
\(639\) −12.3177 −0.487280
\(640\) 0 0
\(641\) 3.58435 0.141573 0.0707867 0.997491i \(-0.477449\pi\)
0.0707867 + 0.997491i \(0.477449\pi\)
\(642\) 0 0
\(643\) 19.8837 0.784137 0.392069 0.919936i \(-0.371760\pi\)
0.392069 + 0.919936i \(0.371760\pi\)
\(644\) 0 0
\(645\) 9.33373 0.367515
\(646\) 0 0
\(647\) 35.7712 1.40631 0.703156 0.711036i \(-0.251774\pi\)
0.703156 + 0.711036i \(0.251774\pi\)
\(648\) 0 0
\(649\) −47.7865 −1.87578
\(650\) 0 0
\(651\) −8.48365 −0.332501
\(652\) 0 0
\(653\) 8.57378 0.335518 0.167759 0.985828i \(-0.446347\pi\)
0.167759 + 0.985828i \(0.446347\pi\)
\(654\) 0 0
\(655\) 1.28428 0.0501810
\(656\) 0 0
\(657\) −16.5019 −0.643802
\(658\) 0 0
\(659\) −22.9993 −0.895927 −0.447964 0.894052i \(-0.647851\pi\)
−0.447964 + 0.894052i \(0.647851\pi\)
\(660\) 0 0
\(661\) −28.4720 −1.10743 −0.553716 0.832706i \(-0.686791\pi\)
−0.553716 + 0.832706i \(0.686791\pi\)
\(662\) 0 0
\(663\) 17.7667 0.690003
\(664\) 0 0
\(665\) −5.76625 −0.223606
\(666\) 0 0
\(667\) −11.5297 −0.446431
\(668\) 0 0
\(669\) −7.91136 −0.305871
\(670\) 0 0
\(671\) 3.36822 0.130029
\(672\) 0 0
\(673\) 6.63685 0.255832 0.127916 0.991785i \(-0.459171\pi\)
0.127916 + 0.991785i \(0.459171\pi\)
\(674\) 0 0
\(675\) 1.81348 0.0698009
\(676\) 0 0
\(677\) 29.0823 1.11772 0.558862 0.829261i \(-0.311238\pi\)
0.558862 + 0.829261i \(0.311238\pi\)
\(678\) 0 0
\(679\) 4.57415 0.175540
\(680\) 0 0
\(681\) −3.16231 −0.121180
\(682\) 0 0
\(683\) −4.63939 −0.177521 −0.0887607 0.996053i \(-0.528291\pi\)
−0.0887607 + 0.996053i \(0.528291\pi\)
\(684\) 0 0
\(685\) 38.2938 1.46313
\(686\) 0 0
\(687\) 1.86902 0.0713074
\(688\) 0 0
\(689\) −13.9688 −0.532169
\(690\) 0 0
\(691\) 5.11245 0.194487 0.0972434 0.995261i \(-0.468997\pi\)
0.0972434 + 0.995261i \(0.468997\pi\)
\(692\) 0 0
\(693\) −33.9889 −1.29113
\(694\) 0 0
\(695\) 4.07890 0.154721
\(696\) 0 0
\(697\) 50.2825 1.90459
\(698\) 0 0
\(699\) 10.3264 0.390581
\(700\) 0 0
\(701\) −5.27938 −0.199399 −0.0996997 0.995018i \(-0.531788\pi\)
−0.0996997 + 0.995018i \(0.531788\pi\)
\(702\) 0 0
\(703\) −1.09693 −0.0413716
\(704\) 0 0
\(705\) 9.23700 0.347886
\(706\) 0 0
\(707\) −19.9764 −0.751290
\(708\) 0 0
\(709\) −28.1441 −1.05697 −0.528486 0.848942i \(-0.677240\pi\)
−0.528486 + 0.848942i \(0.677240\pi\)
\(710\) 0 0
\(711\) −8.70632 −0.326512
\(712\) 0 0
\(713\) −6.45241 −0.241645
\(714\) 0 0
\(715\) −44.0495 −1.64736
\(716\) 0 0
\(717\) −1.05703 −0.0394755
\(718\) 0 0
\(719\) −9.51160 −0.354723 −0.177361 0.984146i \(-0.556756\pi\)
−0.177361 + 0.984146i \(0.556756\pi\)
\(720\) 0 0
\(721\) 23.9894 0.893412
\(722\) 0 0
\(723\) 3.74976 0.139455
\(724\) 0 0
\(725\) −5.29387 −0.196609
\(726\) 0 0
\(727\) 7.06196 0.261914 0.130957 0.991388i \(-0.458195\pi\)
0.130957 + 0.991388i \(0.458195\pi\)
\(728\) 0 0
\(729\) −10.5442 −0.390527
\(730\) 0 0
\(731\) 52.3801 1.93735
\(732\) 0 0
\(733\) −7.62443 −0.281615 −0.140807 0.990037i \(-0.544970\pi\)
−0.140807 + 0.990037i \(0.544970\pi\)
\(734\) 0 0
\(735\) −0.582073 −0.0214701
\(736\) 0 0
\(737\) 28.6274 1.05450
\(738\) 0 0
\(739\) 45.0131 1.65583 0.827917 0.560851i \(-0.189526\pi\)
0.827917 + 0.560851i \(0.189526\pi\)
\(740\) 0 0
\(741\) 2.59279 0.0952484
\(742\) 0 0
\(743\) 15.3137 0.561805 0.280903 0.959736i \(-0.409366\pi\)
0.280903 + 0.959736i \(0.409366\pi\)
\(744\) 0 0
\(745\) −32.8501 −1.20354
\(746\) 0 0
\(747\) 23.9276 0.875465
\(748\) 0 0
\(749\) −27.1072 −0.990475
\(750\) 0 0
\(751\) −4.09580 −0.149458 −0.0747290 0.997204i \(-0.523809\pi\)
−0.0747290 + 0.997204i \(0.523809\pi\)
\(752\) 0 0
\(753\) −1.80016 −0.0656015
\(754\) 0 0
\(755\) 42.1228 1.53301
\(756\) 0 0
\(757\) −9.51720 −0.345909 −0.172954 0.984930i \(-0.555331\pi\)
−0.172954 + 0.984930i \(0.555331\pi\)
\(758\) 0 0
\(759\) 3.25240 0.118055
\(760\) 0 0
\(761\) 38.4841 1.39505 0.697523 0.716562i \(-0.254285\pi\)
0.697523 + 0.716562i \(0.254285\pi\)
\(762\) 0 0
\(763\) 7.78167 0.281715
\(764\) 0 0
\(765\) −38.5065 −1.39221
\(766\) 0 0
\(767\) −45.8726 −1.65636
\(768\) 0 0
\(769\) 15.0210 0.541670 0.270835 0.962626i \(-0.412700\pi\)
0.270835 + 0.962626i \(0.412700\pi\)
\(770\) 0 0
\(771\) 3.33286 0.120030
\(772\) 0 0
\(773\) −38.4471 −1.38285 −0.691423 0.722451i \(-0.743016\pi\)
−0.691423 + 0.722451i \(0.743016\pi\)
\(774\) 0 0
\(775\) −2.96263 −0.106421
\(776\) 0 0
\(777\) −1.73671 −0.0623040
\(778\) 0 0
\(779\) 7.33797 0.262910
\(780\) 0 0
\(781\) −21.5626 −0.771572
\(782\) 0 0
\(783\) −31.4053 −1.12233
\(784\) 0 0
\(785\) 44.0472 1.57211
\(786\) 0 0
\(787\) −30.8457 −1.09953 −0.549765 0.835319i \(-0.685283\pi\)
−0.549765 + 0.835319i \(0.685283\pi\)
\(788\) 0 0
\(789\) −2.65535 −0.0945328
\(790\) 0 0
\(791\) 4.33361 0.154085
\(792\) 0 0
\(793\) 3.23332 0.114819
\(794\) 0 0
\(795\) −3.80903 −0.135092
\(796\) 0 0
\(797\) 34.1285 1.20889 0.604446 0.796646i \(-0.293394\pi\)
0.604446 + 0.796646i \(0.293394\pi\)
\(798\) 0 0
\(799\) 51.8372 1.83387
\(800\) 0 0
\(801\) 1.65402 0.0584420
\(802\) 0 0
\(803\) −28.8873 −1.01941
\(804\) 0 0
\(805\) −6.94351 −0.244727
\(806\) 0 0
\(807\) 6.59578 0.232182
\(808\) 0 0
\(809\) 25.9170 0.911192 0.455596 0.890187i \(-0.349426\pi\)
0.455596 + 0.890187i \(0.349426\pi\)
\(810\) 0 0
\(811\) −33.7874 −1.18644 −0.593219 0.805041i \(-0.702143\pi\)
−0.593219 + 0.805041i \(0.702143\pi\)
\(812\) 0 0
\(813\) −14.5479 −0.510218
\(814\) 0 0
\(815\) −5.37400 −0.188243
\(816\) 0 0
\(817\) 7.64408 0.267432
\(818\) 0 0
\(819\) −32.6276 −1.14010
\(820\) 0 0
\(821\) −27.6475 −0.964903 −0.482451 0.875923i \(-0.660254\pi\)
−0.482451 + 0.875923i \(0.660254\pi\)
\(822\) 0 0
\(823\) −12.4983 −0.435665 −0.217832 0.975986i \(-0.569899\pi\)
−0.217832 + 0.975986i \(0.569899\pi\)
\(824\) 0 0
\(825\) 1.49335 0.0519916
\(826\) 0 0
\(827\) −9.03084 −0.314033 −0.157017 0.987596i \(-0.550188\pi\)
−0.157017 + 0.987596i \(0.550188\pi\)
\(828\) 0 0
\(829\) −55.7684 −1.93692 −0.968458 0.249175i \(-0.919841\pi\)
−0.968458 + 0.249175i \(0.919841\pi\)
\(830\) 0 0
\(831\) 10.5959 0.367567
\(832\) 0 0
\(833\) −3.26654 −0.113179
\(834\) 0 0
\(835\) 39.3597 1.36210
\(836\) 0 0
\(837\) −17.5755 −0.607498
\(838\) 0 0
\(839\) −14.3128 −0.494134 −0.247067 0.968998i \(-0.579467\pi\)
−0.247067 + 0.968998i \(0.579467\pi\)
\(840\) 0 0
\(841\) 62.6776 2.16130
\(842\) 0 0
\(843\) 10.4633 0.360376
\(844\) 0 0
\(845\) −14.8707 −0.511567
\(846\) 0 0
\(847\) −29.4211 −1.01092
\(848\) 0 0
\(849\) −8.91525 −0.305971
\(850\) 0 0
\(851\) −1.32089 −0.0452794
\(852\) 0 0
\(853\) −38.0325 −1.30221 −0.651105 0.758988i \(-0.725694\pi\)
−0.651105 + 0.758988i \(0.725694\pi\)
\(854\) 0 0
\(855\) −5.61945 −0.192181
\(856\) 0 0
\(857\) 23.1869 0.792048 0.396024 0.918240i \(-0.370390\pi\)
0.396024 + 0.918240i \(0.370390\pi\)
\(858\) 0 0
\(859\) −46.5870 −1.58953 −0.794764 0.606919i \(-0.792405\pi\)
−0.794764 + 0.606919i \(0.792405\pi\)
\(860\) 0 0
\(861\) 11.6178 0.395933
\(862\) 0 0
\(863\) 54.3149 1.84890 0.924451 0.381301i \(-0.124524\pi\)
0.924451 + 0.381301i \(0.124524\pi\)
\(864\) 0 0
\(865\) 17.7640 0.603995
\(866\) 0 0
\(867\) 17.3444 0.589048
\(868\) 0 0
\(869\) −15.2408 −0.517008
\(870\) 0 0
\(871\) 27.4808 0.931153
\(872\) 0 0
\(873\) 4.45769 0.150870
\(874\) 0 0
\(875\) −32.0194 −1.08245
\(876\) 0 0
\(877\) −1.80324 −0.0608910 −0.0304455 0.999536i \(-0.509693\pi\)
−0.0304455 + 0.999536i \(0.509693\pi\)
\(878\) 0 0
\(879\) 6.62438 0.223435
\(880\) 0 0
\(881\) 12.6372 0.425759 0.212880 0.977078i \(-0.431716\pi\)
0.212880 + 0.977078i \(0.431716\pi\)
\(882\) 0 0
\(883\) 39.9276 1.34367 0.671835 0.740700i \(-0.265506\pi\)
0.671835 + 0.740700i \(0.265506\pi\)
\(884\) 0 0
\(885\) −12.5086 −0.420471
\(886\) 0 0
\(887\) −14.3642 −0.482304 −0.241152 0.970487i \(-0.577525\pi\)
−0.241152 + 0.970487i \(0.577525\pi\)
\(888\) 0 0
\(889\) 39.0830 1.31080
\(890\) 0 0
\(891\) −28.4318 −0.952502
\(892\) 0 0
\(893\) 7.56486 0.253148
\(894\) 0 0
\(895\) 35.1769 1.17584
\(896\) 0 0
\(897\) 3.12214 0.104245
\(898\) 0 0
\(899\) 51.3060 1.71115
\(900\) 0 0
\(901\) −21.3759 −0.712135
\(902\) 0 0
\(903\) 12.1024 0.402743
\(904\) 0 0
\(905\) −9.82473 −0.326585
\(906\) 0 0
\(907\) 3.32787 0.110500 0.0552501 0.998473i \(-0.482404\pi\)
0.0552501 + 0.998473i \(0.482404\pi\)
\(908\) 0 0
\(909\) −19.4678 −0.645707
\(910\) 0 0
\(911\) 35.8252 1.18694 0.593471 0.804855i \(-0.297757\pi\)
0.593471 + 0.804855i \(0.297757\pi\)
\(912\) 0 0
\(913\) 41.8863 1.38623
\(914\) 0 0
\(915\) 0.881666 0.0291470
\(916\) 0 0
\(917\) 1.66524 0.0549910
\(918\) 0 0
\(919\) −48.3186 −1.59388 −0.796942 0.604055i \(-0.793551\pi\)
−0.796942 + 0.604055i \(0.793551\pi\)
\(920\) 0 0
\(921\) −12.3593 −0.407254
\(922\) 0 0
\(923\) −20.6990 −0.681316
\(924\) 0 0
\(925\) −0.606487 −0.0199412
\(926\) 0 0
\(927\) 23.3786 0.767855
\(928\) 0 0
\(929\) −23.2127 −0.761585 −0.380792 0.924661i \(-0.624349\pi\)
−0.380792 + 0.924661i \(0.624349\pi\)
\(930\) 0 0
\(931\) −0.476702 −0.0156233
\(932\) 0 0
\(933\) −12.5817 −0.411908
\(934\) 0 0
\(935\) −67.4073 −2.20445
\(936\) 0 0
\(937\) −7.10467 −0.232099 −0.116050 0.993243i \(-0.537023\pi\)
−0.116050 + 0.993243i \(0.537023\pi\)
\(938\) 0 0
\(939\) −19.7450 −0.644354
\(940\) 0 0
\(941\) −55.5352 −1.81040 −0.905199 0.424989i \(-0.860278\pi\)
−0.905199 + 0.424989i \(0.860278\pi\)
\(942\) 0 0
\(943\) 8.83612 0.287744
\(944\) 0 0
\(945\) −18.9132 −0.615247
\(946\) 0 0
\(947\) −15.5988 −0.506894 −0.253447 0.967349i \(-0.581564\pi\)
−0.253447 + 0.967349i \(0.581564\pi\)
\(948\) 0 0
\(949\) −27.7304 −0.900165
\(950\) 0 0
\(951\) 12.8704 0.417352
\(952\) 0 0
\(953\) −21.1243 −0.684282 −0.342141 0.939649i \(-0.611152\pi\)
−0.342141 + 0.939649i \(0.611152\pi\)
\(954\) 0 0
\(955\) 42.3047 1.36895
\(956\) 0 0
\(957\) −25.8613 −0.835977
\(958\) 0 0
\(959\) 49.6529 1.60338
\(960\) 0 0
\(961\) −2.28738 −0.0737863
\(962\) 0 0
\(963\) −26.4170 −0.851277
\(964\) 0 0
\(965\) −3.90378 −0.125667
\(966\) 0 0
\(967\) 46.1207 1.48314 0.741570 0.670875i \(-0.234081\pi\)
0.741570 + 0.670875i \(0.234081\pi\)
\(968\) 0 0
\(969\) 3.96764 0.127459
\(970\) 0 0
\(971\) −22.3253 −0.716454 −0.358227 0.933635i \(-0.616619\pi\)
−0.358227 + 0.933635i \(0.616619\pi\)
\(972\) 0 0
\(973\) 5.28882 0.169552
\(974\) 0 0
\(975\) 1.43354 0.0459099
\(976\) 0 0
\(977\) 14.2650 0.456379 0.228190 0.973617i \(-0.426719\pi\)
0.228190 + 0.973617i \(0.426719\pi\)
\(978\) 0 0
\(979\) 2.89544 0.0925385
\(980\) 0 0
\(981\) 7.58355 0.242124
\(982\) 0 0
\(983\) 12.7242 0.405838 0.202919 0.979196i \(-0.434957\pi\)
0.202919 + 0.979196i \(0.434957\pi\)
\(984\) 0 0
\(985\) −0.474237 −0.0151104
\(986\) 0 0
\(987\) 11.9770 0.381232
\(988\) 0 0
\(989\) 9.20472 0.292693
\(990\) 0 0
\(991\) 10.8169 0.343611 0.171806 0.985131i \(-0.445040\pi\)
0.171806 + 0.985131i \(0.445040\pi\)
\(992\) 0 0
\(993\) −14.0104 −0.444607
\(994\) 0 0
\(995\) 3.74705 0.118789
\(996\) 0 0
\(997\) 5.39134 0.170746 0.0853728 0.996349i \(-0.472792\pi\)
0.0853728 + 0.996349i \(0.472792\pi\)
\(998\) 0 0
\(999\) −3.59792 −0.113833
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 4864.2.a.bp.1.5 8
4.3 odd 2 4864.2.a.bq.1.4 8
8.3 odd 2 4864.2.a.bo.1.5 8
8.5 even 2 4864.2.a.bn.1.4 8
16.3 odd 4 152.2.c.b.77.16 yes 16
16.5 even 4 608.2.c.b.305.8 16
16.11 odd 4 152.2.c.b.77.15 16
16.13 even 4 608.2.c.b.305.9 16
48.5 odd 4 5472.2.g.b.2737.12 16
48.11 even 4 1368.2.g.b.685.2 16
48.29 odd 4 5472.2.g.b.2737.5 16
48.35 even 4 1368.2.g.b.685.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.c.b.77.15 16 16.11 odd 4
152.2.c.b.77.16 yes 16 16.3 odd 4
608.2.c.b.305.8 16 16.5 even 4
608.2.c.b.305.9 16 16.13 even 4
1368.2.g.b.685.1 16 48.35 even 4
1368.2.g.b.685.2 16 48.11 even 4
4864.2.a.bn.1.4 8 8.5 even 2
4864.2.a.bo.1.5 8 8.3 odd 2
4864.2.a.bp.1.5 8 1.1 even 1 trivial
4864.2.a.bq.1.4 8 4.3 odd 2
5472.2.g.b.2737.5 16 48.29 odd 4
5472.2.g.b.2737.12 16 48.5 odd 4