Properties

Label 7800.2.a.bm.1.3
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.76156 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.76156 q^{7} +1.00000 q^{9} +4.49084 q^{11} +1.00000 q^{13} +3.62620 q^{17} -6.14931 q^{19} -2.76156 q^{21} +5.52311 q^{23} -1.00000 q^{27} +6.25240 q^{29} +8.49084 q^{31} -4.49084 q^{33} -0.896916 q^{37} -1.00000 q^{39} +2.89692 q^{41} +0.270718 q^{43} +3.38776 q^{47} +0.626198 q^{49} -3.62620 q^{51} -3.27072 q^{53} +6.14931 q^{57} -9.53707 q^{59} +11.4200 q^{61} +2.76156 q^{63} -12.4340 q^{67} -5.52311 q^{69} +8.89692 q^{71} +3.25240 q^{73} +12.4017 q^{77} +13.6724 q^{79} +1.00000 q^{81} -3.03228 q^{83} -6.25240 q^{87} -2.20617 q^{89} +2.76156 q^{91} -8.49084 q^{93} -17.7572 q^{97} +4.49084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + 2 q^{11} + 3 q^{13} + 2 q^{17} + 3 q^{19} - 2 q^{21} + 4 q^{23} - 3 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} + q^{37} - 3 q^{39} + 5 q^{41} + 6 q^{43} - 5 q^{47} - 7 q^{49} - 2 q^{51} - 15 q^{53} - 3 q^{57} + 8 q^{59} + 18 q^{61} + 2 q^{63} + 3 q^{67} - 4 q^{69} + 23 q^{71} - 8 q^{73} - 2 q^{77} + 7 q^{79} + 3 q^{81} - 8 q^{83} - q^{87} - 14 q^{89} + 2 q^{91} - 14 q^{93} + 2 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.76156 1.04377 0.521885 0.853016i \(-0.325229\pi\)
0.521885 + 0.853016i \(0.325229\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.49084 1.35404 0.677019 0.735965i \(-0.263271\pi\)
0.677019 + 0.735965i \(0.263271\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.62620 0.879482 0.439741 0.898125i \(-0.355070\pi\)
0.439741 + 0.898125i \(0.355070\pi\)
\(18\) 0 0
\(19\) −6.14931 −1.41075 −0.705375 0.708835i \(-0.749221\pi\)
−0.705375 + 0.708835i \(0.749221\pi\)
\(20\) 0 0
\(21\) −2.76156 −0.602621
\(22\) 0 0
\(23\) 5.52311 1.15165 0.575824 0.817573i \(-0.304681\pi\)
0.575824 + 0.817573i \(0.304681\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.25240 1.16104 0.580520 0.814246i \(-0.302849\pi\)
0.580520 + 0.814246i \(0.302849\pi\)
\(30\) 0 0
\(31\) 8.49084 1.52500 0.762500 0.646988i \(-0.223972\pi\)
0.762500 + 0.646988i \(0.223972\pi\)
\(32\) 0 0
\(33\) −4.49084 −0.781755
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.896916 −0.147452 −0.0737261 0.997279i \(-0.523489\pi\)
−0.0737261 + 0.997279i \(0.523489\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 2.89692 0.452422 0.226211 0.974078i \(-0.427366\pi\)
0.226211 + 0.974078i \(0.427366\pi\)
\(42\) 0 0
\(43\) 0.270718 0.0412841 0.0206421 0.999787i \(-0.493429\pi\)
0.0206421 + 0.999787i \(0.493429\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.38776 0.494155 0.247077 0.968996i \(-0.420530\pi\)
0.247077 + 0.968996i \(0.420530\pi\)
\(48\) 0 0
\(49\) 0.626198 0.0894569
\(50\) 0 0
\(51\) −3.62620 −0.507769
\(52\) 0 0
\(53\) −3.27072 −0.449268 −0.224634 0.974443i \(-0.572119\pi\)
−0.224634 + 0.974443i \(0.572119\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.14931 0.814496
\(58\) 0 0
\(59\) −9.53707 −1.24162 −0.620810 0.783961i \(-0.713196\pi\)
−0.620810 + 0.783961i \(0.713196\pi\)
\(60\) 0 0
\(61\) 11.4200 1.46219 0.731093 0.682278i \(-0.239011\pi\)
0.731093 + 0.682278i \(0.239011\pi\)
\(62\) 0 0
\(63\) 2.76156 0.347924
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.4340 −1.51905 −0.759526 0.650476i \(-0.774569\pi\)
−0.759526 + 0.650476i \(0.774569\pi\)
\(68\) 0 0
\(69\) −5.52311 −0.664905
\(70\) 0 0
\(71\) 8.89692 1.05587 0.527935 0.849285i \(-0.322967\pi\)
0.527935 + 0.849285i \(0.322967\pi\)
\(72\) 0 0
\(73\) 3.25240 0.380664 0.190332 0.981720i \(-0.439044\pi\)
0.190332 + 0.981720i \(0.439044\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.4017 1.41331
\(78\) 0 0
\(79\) 13.6724 1.53827 0.769134 0.639087i \(-0.220688\pi\)
0.769134 + 0.639087i \(0.220688\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.03228 −0.332835 −0.166418 0.986055i \(-0.553220\pi\)
−0.166418 + 0.986055i \(0.553220\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.25240 −0.670327
\(88\) 0 0
\(89\) −2.20617 −0.233853 −0.116927 0.993141i \(-0.537304\pi\)
−0.116927 + 0.993141i \(0.537304\pi\)
\(90\) 0 0
\(91\) 2.76156 0.289490
\(92\) 0 0
\(93\) −8.49084 −0.880459
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −17.7572 −1.80297 −0.901485 0.432811i \(-0.857522\pi\)
−0.901485 + 0.432811i \(0.857522\pi\)
\(98\) 0 0
\(99\) 4.49084 0.451346
\(100\) 0 0
\(101\) 5.83237 0.580342 0.290171 0.956975i \(-0.406288\pi\)
0.290171 + 0.956975i \(0.406288\pi\)
\(102\) 0 0
\(103\) −3.45856 −0.340782 −0.170391 0.985376i \(-0.554503\pi\)
−0.170391 + 0.985376i \(0.554503\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.832365 −0.0804678 −0.0402339 0.999190i \(-0.512810\pi\)
−0.0402339 + 0.999190i \(0.512810\pi\)
\(108\) 0 0
\(109\) −12.1493 −1.16369 −0.581847 0.813299i \(-0.697670\pi\)
−0.581847 + 0.813299i \(0.697670\pi\)
\(110\) 0 0
\(111\) 0.896916 0.0851315
\(112\) 0 0
\(113\) 4.50479 0.423775 0.211888 0.977294i \(-0.432039\pi\)
0.211888 + 0.977294i \(0.432039\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 10.0140 0.917978
\(120\) 0 0
\(121\) 9.16763 0.833421
\(122\) 0 0
\(123\) −2.89692 −0.261206
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.62620 0.587980 0.293990 0.955808i \(-0.405017\pi\)
0.293990 + 0.955808i \(0.405017\pi\)
\(128\) 0 0
\(129\) −0.270718 −0.0238354
\(130\) 0 0
\(131\) 1.10308 0.0963769 0.0481884 0.998838i \(-0.484655\pi\)
0.0481884 + 0.998838i \(0.484655\pi\)
\(132\) 0 0
\(133\) −16.9817 −1.47250
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −20.4479 −1.74699 −0.873493 0.486837i \(-0.838150\pi\)
−0.873493 + 0.486837i \(0.838150\pi\)
\(138\) 0 0
\(139\) 16.5693 1.40539 0.702697 0.711490i \(-0.251979\pi\)
0.702697 + 0.711490i \(0.251979\pi\)
\(140\) 0 0
\(141\) −3.38776 −0.285300
\(142\) 0 0
\(143\) 4.49084 0.375543
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −0.626198 −0.0516479
\(148\) 0 0
\(149\) 3.79383 0.310803 0.155401 0.987851i \(-0.450333\pi\)
0.155401 + 0.987851i \(0.450333\pi\)
\(150\) 0 0
\(151\) −5.80779 −0.472631 −0.236315 0.971676i \(-0.575940\pi\)
−0.236315 + 0.971676i \(0.575940\pi\)
\(152\) 0 0
\(153\) 3.62620 0.293161
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 13.6541 1.08972 0.544858 0.838528i \(-0.316584\pi\)
0.544858 + 0.838528i \(0.316584\pi\)
\(158\) 0 0
\(159\) 3.27072 0.259385
\(160\) 0 0
\(161\) 15.2524 1.20206
\(162\) 0 0
\(163\) 7.25240 0.568052 0.284026 0.958817i \(-0.408330\pi\)
0.284026 + 0.958817i \(0.408330\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.14931 −0.166319 −0.0831594 0.996536i \(-0.526501\pi\)
−0.0831594 + 0.996536i \(0.526501\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −6.14931 −0.470250
\(172\) 0 0
\(173\) −25.5693 −1.94400 −0.972001 0.234978i \(-0.924498\pi\)
−0.972001 + 0.234978i \(0.924498\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 9.53707 0.716850
\(178\) 0 0
\(179\) −21.5510 −1.61080 −0.805399 0.592732i \(-0.798049\pi\)
−0.805399 + 0.592732i \(0.798049\pi\)
\(180\) 0 0
\(181\) −0.607876 −0.0451831 −0.0225915 0.999745i \(-0.507192\pi\)
−0.0225915 + 0.999745i \(0.507192\pi\)
\(182\) 0 0
\(183\) −11.4200 −0.844193
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 16.2847 1.19085
\(188\) 0 0
\(189\) −2.76156 −0.200874
\(190\) 0 0
\(191\) −21.2803 −1.53979 −0.769894 0.638171i \(-0.779691\pi\)
−0.769894 + 0.638171i \(0.779691\pi\)
\(192\) 0 0
\(193\) 7.45856 0.536879 0.268440 0.963297i \(-0.413492\pi\)
0.268440 + 0.963297i \(0.413492\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −7.45856 −0.531401 −0.265700 0.964056i \(-0.585603\pi\)
−0.265700 + 0.964056i \(0.585603\pi\)
\(198\) 0 0
\(199\) 19.3372 1.37077 0.685387 0.728179i \(-0.259633\pi\)
0.685387 + 0.728179i \(0.259633\pi\)
\(200\) 0 0
\(201\) 12.4340 0.877026
\(202\) 0 0
\(203\) 17.2663 1.21186
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 5.52311 0.383883
\(208\) 0 0
\(209\) −27.6156 −1.91021
\(210\) 0 0
\(211\) −8.02791 −0.552664 −0.276332 0.961062i \(-0.589119\pi\)
−0.276332 + 0.961062i \(0.589119\pi\)
\(212\) 0 0
\(213\) −8.89692 −0.609607
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 23.4479 1.59175
\(218\) 0 0
\(219\) −3.25240 −0.219777
\(220\) 0 0
\(221\) 3.62620 0.243924
\(222\) 0 0
\(223\) −16.5048 −1.10524 −0.552621 0.833432i \(-0.686372\pi\)
−0.552621 + 0.833432i \(0.686372\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.69701 0.179007 0.0895033 0.995987i \(-0.471472\pi\)
0.0895033 + 0.995987i \(0.471472\pi\)
\(228\) 0 0
\(229\) −10.5616 −0.697933 −0.348967 0.937135i \(-0.613467\pi\)
−0.348967 + 0.937135i \(0.613467\pi\)
\(230\) 0 0
\(231\) −12.4017 −0.815973
\(232\) 0 0
\(233\) −25.2158 −1.65194 −0.825969 0.563715i \(-0.809372\pi\)
−0.825969 + 0.563715i \(0.809372\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −13.6724 −0.888120
\(238\) 0 0
\(239\) −8.51875 −0.551032 −0.275516 0.961297i \(-0.588849\pi\)
−0.275516 + 0.961297i \(0.588849\pi\)
\(240\) 0 0
\(241\) 7.45856 0.480448 0.240224 0.970717i \(-0.422779\pi\)
0.240224 + 0.970717i \(0.422779\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.14931 −0.391271
\(248\) 0 0
\(249\) 3.03228 0.192163
\(250\) 0 0
\(251\) −9.64452 −0.608757 −0.304378 0.952551i \(-0.598449\pi\)
−0.304378 + 0.952551i \(0.598449\pi\)
\(252\) 0 0
\(253\) 24.8034 1.55938
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.3555 0.833092 0.416546 0.909115i \(-0.363240\pi\)
0.416546 + 0.909115i \(0.363240\pi\)
\(258\) 0 0
\(259\) −2.47689 −0.153906
\(260\) 0 0
\(261\) 6.25240 0.387014
\(262\) 0 0
\(263\) −9.25240 −0.570527 −0.285264 0.958449i \(-0.592081\pi\)
−0.285264 + 0.958449i \(0.592081\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.20617 0.135015
\(268\) 0 0
\(269\) 25.2986 1.54248 0.771242 0.636542i \(-0.219636\pi\)
0.771242 + 0.636542i \(0.219636\pi\)
\(270\) 0 0
\(271\) 31.3309 1.90322 0.951608 0.307314i \(-0.0994300\pi\)
0.951608 + 0.307314i \(0.0994300\pi\)
\(272\) 0 0
\(273\) −2.76156 −0.167137
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −23.0096 −1.38251 −0.691256 0.722610i \(-0.742942\pi\)
−0.691256 + 0.722610i \(0.742942\pi\)
\(278\) 0 0
\(279\) 8.49084 0.508333
\(280\) 0 0
\(281\) −9.19554 −0.548560 −0.274280 0.961650i \(-0.588440\pi\)
−0.274280 + 0.961650i \(0.588440\pi\)
\(282\) 0 0
\(283\) 9.93545 0.590601 0.295301 0.955404i \(-0.404580\pi\)
0.295301 + 0.955404i \(0.404580\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) −3.85069 −0.226511
\(290\) 0 0
\(291\) 17.7572 1.04094
\(292\) 0 0
\(293\) −23.7938 −1.39005 −0.695025 0.718985i \(-0.744607\pi\)
−0.695025 + 0.718985i \(0.744607\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −4.49084 −0.260585
\(298\) 0 0
\(299\) 5.52311 0.319410
\(300\) 0 0
\(301\) 0.747604 0.0430912
\(302\) 0 0
\(303\) −5.83237 −0.335061
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 25.0664 1.43062 0.715309 0.698809i \(-0.246286\pi\)
0.715309 + 0.698809i \(0.246286\pi\)
\(308\) 0 0
\(309\) 3.45856 0.196751
\(310\) 0 0
\(311\) 18.3632 1.04128 0.520640 0.853776i \(-0.325693\pi\)
0.520640 + 0.853776i \(0.325693\pi\)
\(312\) 0 0
\(313\) 10.7293 0.606455 0.303227 0.952918i \(-0.401936\pi\)
0.303227 + 0.952918i \(0.401936\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.20617 0.460904 0.230452 0.973084i \(-0.425979\pi\)
0.230452 + 0.973084i \(0.425979\pi\)
\(318\) 0 0
\(319\) 28.0785 1.57209
\(320\) 0 0
\(321\) 0.832365 0.0464581
\(322\) 0 0
\(323\) −22.2986 −1.24073
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 12.1493 0.671859
\(328\) 0 0
\(329\) 9.35548 0.515784
\(330\) 0 0
\(331\) 30.7110 1.68803 0.844014 0.536322i \(-0.180187\pi\)
0.844014 + 0.536322i \(0.180187\pi\)
\(332\) 0 0
\(333\) −0.896916 −0.0491507
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8.13099 −0.442923 −0.221462 0.975169i \(-0.571083\pi\)
−0.221462 + 0.975169i \(0.571083\pi\)
\(338\) 0 0
\(339\) −4.50479 −0.244667
\(340\) 0 0
\(341\) 38.1310 2.06491
\(342\) 0 0
\(343\) −17.6016 −0.950398
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.9248 0.801206 0.400603 0.916252i \(-0.368801\pi\)
0.400603 + 0.916252i \(0.368801\pi\)
\(348\) 0 0
\(349\) −16.5048 −0.883481 −0.441741 0.897143i \(-0.645639\pi\)
−0.441741 + 0.897143i \(0.645639\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −23.0664 −1.22770 −0.613851 0.789422i \(-0.710381\pi\)
−0.613851 + 0.789422i \(0.710381\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −10.0140 −0.529995
\(358\) 0 0
\(359\) 32.6681 1.72415 0.862077 0.506777i \(-0.169163\pi\)
0.862077 + 0.506777i \(0.169163\pi\)
\(360\) 0 0
\(361\) 18.8140 0.990213
\(362\) 0 0
\(363\) −9.16763 −0.481176
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 32.1493 1.67818 0.839090 0.543992i \(-0.183088\pi\)
0.839090 + 0.543992i \(0.183088\pi\)
\(368\) 0 0
\(369\) 2.89692 0.150807
\(370\) 0 0
\(371\) −9.03228 −0.468932
\(372\) 0 0
\(373\) −6.81404 −0.352818 −0.176409 0.984317i \(-0.556448\pi\)
−0.176409 + 0.984317i \(0.556448\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.25240 0.322015
\(378\) 0 0
\(379\) 1.65078 0.0847948 0.0423974 0.999101i \(-0.486500\pi\)
0.0423974 + 0.999101i \(0.486500\pi\)
\(380\) 0 0
\(381\) −6.62620 −0.339470
\(382\) 0 0
\(383\) −10.0202 −0.512009 −0.256004 0.966676i \(-0.582406\pi\)
−0.256004 + 0.966676i \(0.582406\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.270718 0.0137614
\(388\) 0 0
\(389\) 19.4017 0.983706 0.491853 0.870678i \(-0.336320\pi\)
0.491853 + 0.870678i \(0.336320\pi\)
\(390\) 0 0
\(391\) 20.0279 1.01285
\(392\) 0 0
\(393\) −1.10308 −0.0556432
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.56165 0.128565 0.0642827 0.997932i \(-0.479524\pi\)
0.0642827 + 0.997932i \(0.479524\pi\)
\(398\) 0 0
\(399\) 16.9817 0.850147
\(400\) 0 0
\(401\) −3.55102 −0.177330 −0.0886648 0.996062i \(-0.528260\pi\)
−0.0886648 + 0.996062i \(0.528260\pi\)
\(402\) 0 0
\(403\) 8.49084 0.422959
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.02791 −0.199656
\(408\) 0 0
\(409\) 28.0558 1.38727 0.693635 0.720326i \(-0.256008\pi\)
0.693635 + 0.720326i \(0.256008\pi\)
\(410\) 0 0
\(411\) 20.4479 1.00862
\(412\) 0 0
\(413\) −26.3372 −1.29597
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −16.5693 −0.811404
\(418\) 0 0
\(419\) 36.3188 1.77429 0.887146 0.461490i \(-0.152685\pi\)
0.887146 + 0.461490i \(0.152685\pi\)
\(420\) 0 0
\(421\) 10.2062 0.497418 0.248709 0.968578i \(-0.419994\pi\)
0.248709 + 0.968578i \(0.419994\pi\)
\(422\) 0 0
\(423\) 3.38776 0.164718
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 31.5371 1.52619
\(428\) 0 0
\(429\) −4.49084 −0.216820
\(430\) 0 0
\(431\) −11.8140 −0.569062 −0.284531 0.958667i \(-0.591838\pi\)
−0.284531 + 0.958667i \(0.591838\pi\)
\(432\) 0 0
\(433\) 0.598291 0.0287521 0.0143760 0.999897i \(-0.495424\pi\)
0.0143760 + 0.999897i \(0.495424\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −33.9634 −1.62469
\(438\) 0 0
\(439\) −13.6724 −0.652549 −0.326275 0.945275i \(-0.605793\pi\)
−0.326275 + 0.945275i \(0.605793\pi\)
\(440\) 0 0
\(441\) 0.626198 0.0298190
\(442\) 0 0
\(443\) 32.1772 1.52879 0.764393 0.644751i \(-0.223039\pi\)
0.764393 + 0.644751i \(0.223039\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −3.79383 −0.179442
\(448\) 0 0
\(449\) −9.81404 −0.463153 −0.231577 0.972817i \(-0.574388\pi\)
−0.231577 + 0.972817i \(0.574388\pi\)
\(450\) 0 0
\(451\) 13.0096 0.612597
\(452\) 0 0
\(453\) 5.80779 0.272874
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.5877 −0.916272 −0.458136 0.888882i \(-0.651483\pi\)
−0.458136 + 0.888882i \(0.651483\pi\)
\(458\) 0 0
\(459\) −3.62620 −0.169256
\(460\) 0 0
\(461\) 11.7938 0.549294 0.274647 0.961545i \(-0.411439\pi\)
0.274647 + 0.961545i \(0.411439\pi\)
\(462\) 0 0
\(463\) 38.8819 1.80700 0.903498 0.428592i \(-0.140990\pi\)
0.903498 + 0.428592i \(0.140990\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.4017 1.17545 0.587725 0.809060i \(-0.300024\pi\)
0.587725 + 0.809060i \(0.300024\pi\)
\(468\) 0 0
\(469\) −34.3372 −1.58554
\(470\) 0 0
\(471\) −13.6541 −0.629148
\(472\) 0 0
\(473\) 1.21575 0.0559003
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −3.27072 −0.149756
\(478\) 0 0
\(479\) 34.0987 1.55801 0.779005 0.627018i \(-0.215725\pi\)
0.779005 + 0.627018i \(0.215725\pi\)
\(480\) 0 0
\(481\) −0.896916 −0.0408959
\(482\) 0 0
\(483\) −15.2524 −0.694008
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 16.7616 0.759539 0.379769 0.925081i \(-0.376003\pi\)
0.379769 + 0.925081i \(0.376003\pi\)
\(488\) 0 0
\(489\) −7.25240 −0.327965
\(490\) 0 0
\(491\) 1.65222 0.0745635 0.0372817 0.999305i \(-0.488130\pi\)
0.0372817 + 0.999305i \(0.488130\pi\)
\(492\) 0 0
\(493\) 22.6724 1.02111
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.5693 1.10209
\(498\) 0 0
\(499\) −22.7326 −1.01765 −0.508826 0.860870i \(-0.669920\pi\)
−0.508826 + 0.860870i \(0.669920\pi\)
\(500\) 0 0
\(501\) 2.14931 0.0960242
\(502\) 0 0
\(503\) 6.09246 0.271649 0.135825 0.990733i \(-0.456632\pi\)
0.135825 + 0.990733i \(0.456632\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 7.87090 0.348871 0.174436 0.984669i \(-0.444190\pi\)
0.174436 + 0.984669i \(0.444190\pi\)
\(510\) 0 0
\(511\) 8.98168 0.397326
\(512\) 0 0
\(513\) 6.14931 0.271499
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 15.2139 0.669105
\(518\) 0 0
\(519\) 25.5693 1.12237
\(520\) 0 0
\(521\) 19.7938 0.867184 0.433592 0.901109i \(-0.357246\pi\)
0.433592 + 0.901109i \(0.357246\pi\)
\(522\) 0 0
\(523\) 27.1878 1.18884 0.594421 0.804154i \(-0.297381\pi\)
0.594421 + 0.804154i \(0.297381\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 30.7895 1.34121
\(528\) 0 0
\(529\) 7.50479 0.326295
\(530\) 0 0
\(531\) −9.53707 −0.413873
\(532\) 0 0
\(533\) 2.89692 0.125479
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 21.5510 0.929995
\(538\) 0 0
\(539\) 2.81215 0.121128
\(540\) 0 0
\(541\) −38.0558 −1.63615 −0.818074 0.575114i \(-0.804958\pi\)
−0.818074 + 0.575114i \(0.804958\pi\)
\(542\) 0 0
\(543\) 0.607876 0.0260865
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 10.3353 0.441904 0.220952 0.975285i \(-0.429084\pi\)
0.220952 + 0.975285i \(0.429084\pi\)
\(548\) 0 0
\(549\) 11.4200 0.487395
\(550\) 0 0
\(551\) −38.4479 −1.63794
\(552\) 0 0
\(553\) 37.7572 1.60560
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.25240 0.0530657 0.0265329 0.999648i \(-0.491553\pi\)
0.0265329 + 0.999648i \(0.491553\pi\)
\(558\) 0 0
\(559\) 0.270718 0.0114502
\(560\) 0 0
\(561\) −16.2847 −0.687539
\(562\) 0 0
\(563\) −45.1868 −1.90440 −0.952198 0.305480i \(-0.901183\pi\)
−0.952198 + 0.305480i \(0.901183\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.76156 0.115975
\(568\) 0 0
\(569\) −15.8603 −0.664897 −0.332449 0.943121i \(-0.607875\pi\)
−0.332449 + 0.943121i \(0.607875\pi\)
\(570\) 0 0
\(571\) −29.5789 −1.23784 −0.618920 0.785454i \(-0.712429\pi\)
−0.618920 + 0.785454i \(0.712429\pi\)
\(572\) 0 0
\(573\) 21.2803 0.888997
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 26.8401 1.11737 0.558683 0.829381i \(-0.311307\pi\)
0.558683 + 0.829381i \(0.311307\pi\)
\(578\) 0 0
\(579\) −7.45856 −0.309967
\(580\) 0 0
\(581\) −8.37380 −0.347404
\(582\) 0 0
\(583\) −14.6883 −0.608326
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −23.4359 −0.967302 −0.483651 0.875261i \(-0.660690\pi\)
−0.483651 + 0.875261i \(0.660690\pi\)
\(588\) 0 0
\(589\) −52.2128 −2.15139
\(590\) 0 0
\(591\) 7.45856 0.306804
\(592\) 0 0
\(593\) −43.2880 −1.77763 −0.888813 0.458271i \(-0.848469\pi\)
−0.888813 + 0.458271i \(0.848469\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −19.3372 −0.791417
\(598\) 0 0
\(599\) 3.58767 0.146588 0.0732940 0.997310i \(-0.476649\pi\)
0.0732940 + 0.997310i \(0.476649\pi\)
\(600\) 0 0
\(601\) 31.0646 1.26715 0.633575 0.773681i \(-0.281587\pi\)
0.633575 + 0.773681i \(0.281587\pi\)
\(602\) 0 0
\(603\) −12.4340 −0.506351
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −30.2697 −1.22861 −0.614304 0.789069i \(-0.710563\pi\)
−0.614304 + 0.789069i \(0.710563\pi\)
\(608\) 0 0
\(609\) −17.2663 −0.699668
\(610\) 0 0
\(611\) 3.38776 0.137054
\(612\) 0 0
\(613\) −46.0558 −1.86018 −0.930088 0.367336i \(-0.880270\pi\)
−0.930088 + 0.367336i \(0.880270\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.43835 −0.0579059 −0.0289530 0.999581i \(-0.509217\pi\)
−0.0289530 + 0.999581i \(0.509217\pi\)
\(618\) 0 0
\(619\) −36.2620 −1.45749 −0.728746 0.684784i \(-0.759897\pi\)
−0.728746 + 0.684784i \(0.759897\pi\)
\(620\) 0 0
\(621\) −5.52311 −0.221635
\(622\) 0 0
\(623\) −6.09246 −0.244089
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 27.6156 1.10286
\(628\) 0 0
\(629\) −3.25240 −0.129682
\(630\) 0 0
\(631\) 34.8401 1.38696 0.693480 0.720475i \(-0.256076\pi\)
0.693480 + 0.720475i \(0.256076\pi\)
\(632\) 0 0
\(633\) 8.02791 0.319081
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.626198 0.0248109
\(638\) 0 0
\(639\) 8.89692 0.351957
\(640\) 0 0
\(641\) −29.8882 −1.18051 −0.590256 0.807216i \(-0.700973\pi\)
−0.590256 + 0.807216i \(0.700973\pi\)
\(642\) 0 0
\(643\) 18.8603 0.743777 0.371888 0.928278i \(-0.378710\pi\)
0.371888 + 0.928278i \(0.378710\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.50479 −0.255730 −0.127865 0.991792i \(-0.540812\pi\)
−0.127865 + 0.991792i \(0.540812\pi\)
\(648\) 0 0
\(649\) −42.8294 −1.68120
\(650\) 0 0
\(651\) −23.4479 −0.918997
\(652\) 0 0
\(653\) 28.1589 1.10194 0.550971 0.834524i \(-0.314257\pi\)
0.550971 + 0.834524i \(0.314257\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.25240 0.126888
\(658\) 0 0
\(659\) 30.2139 1.17697 0.588483 0.808510i \(-0.299726\pi\)
0.588483 + 0.808510i \(0.299726\pi\)
\(660\) 0 0
\(661\) 36.0356 1.40162 0.700811 0.713347i \(-0.252822\pi\)
0.700811 + 0.713347i \(0.252822\pi\)
\(662\) 0 0
\(663\) −3.62620 −0.140830
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 34.5327 1.33711
\(668\) 0 0
\(669\) 16.5048 0.638112
\(670\) 0 0
\(671\) 51.2855 1.97986
\(672\) 0 0
\(673\) 1.68305 0.0648769 0.0324385 0.999474i \(-0.489673\pi\)
0.0324385 + 0.999474i \(0.489673\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.50479 0.173133 0.0865666 0.996246i \(-0.472410\pi\)
0.0865666 + 0.996246i \(0.472410\pi\)
\(678\) 0 0
\(679\) −49.0375 −1.88189
\(680\) 0 0
\(681\) −2.69701 −0.103350
\(682\) 0 0
\(683\) 35.5650 1.36086 0.680428 0.732815i \(-0.261794\pi\)
0.680428 + 0.732815i \(0.261794\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 10.5616 0.402952
\(688\) 0 0
\(689\) −3.27072 −0.124604
\(690\) 0 0
\(691\) 6.29237 0.239373 0.119686 0.992812i \(-0.461811\pi\)
0.119686 + 0.992812i \(0.461811\pi\)
\(692\) 0 0
\(693\) 12.4017 0.471102
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 10.5048 0.397897
\(698\) 0 0
\(699\) 25.2158 0.953747
\(700\) 0 0
\(701\) −36.9065 −1.39394 −0.696970 0.717101i \(-0.745469\pi\)
−0.696970 + 0.717101i \(0.745469\pi\)
\(702\) 0 0
\(703\) 5.51542 0.208018
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16.1064 0.605744
\(708\) 0 0
\(709\) −5.25240 −0.197258 −0.0986289 0.995124i \(-0.531446\pi\)
−0.0986289 + 0.995124i \(0.531446\pi\)
\(710\) 0 0
\(711\) 13.6724 0.512756
\(712\) 0 0
\(713\) 46.8959 1.75626
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 8.51875 0.318138
\(718\) 0 0
\(719\) −22.6339 −0.844102 −0.422051 0.906572i \(-0.638690\pi\)
−0.422051 + 0.906572i \(0.638690\pi\)
\(720\) 0 0
\(721\) −9.55102 −0.355699
\(722\) 0 0
\(723\) −7.45856 −0.277387
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −34.5606 −1.28178 −0.640891 0.767632i \(-0.721435\pi\)
−0.640891 + 0.767632i \(0.721435\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.981678 0.0363087
\(732\) 0 0
\(733\) 11.4942 0.424547 0.212273 0.977210i \(-0.431913\pi\)
0.212273 + 0.977210i \(0.431913\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −55.8390 −2.05686
\(738\) 0 0
\(739\) 20.4619 0.752703 0.376351 0.926477i \(-0.377179\pi\)
0.376351 + 0.926477i \(0.377179\pi\)
\(740\) 0 0
\(741\) 6.14931 0.225901
\(742\) 0 0
\(743\) 34.2836 1.25774 0.628872 0.777509i \(-0.283517\pi\)
0.628872 + 0.777509i \(0.283517\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −3.03228 −0.110945
\(748\) 0 0
\(749\) −2.29862 −0.0839899
\(750\) 0 0
\(751\) −23.6358 −0.862482 −0.431241 0.902237i \(-0.641924\pi\)
−0.431241 + 0.902237i \(0.641924\pi\)
\(752\) 0 0
\(753\) 9.64452 0.351466
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −7.54913 −0.274378 −0.137189 0.990545i \(-0.543807\pi\)
−0.137189 + 0.990545i \(0.543807\pi\)
\(758\) 0 0
\(759\) −24.8034 −0.900307
\(760\) 0 0
\(761\) 48.2418 1.74876 0.874381 0.485239i \(-0.161268\pi\)
0.874381 + 0.485239i \(0.161268\pi\)
\(762\) 0 0
\(763\) −33.5510 −1.21463
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −9.53707 −0.344364
\(768\) 0 0
\(769\) 24.0558 0.867475 0.433737 0.901039i \(-0.357195\pi\)
0.433737 + 0.901039i \(0.357195\pi\)
\(770\) 0 0
\(771\) −13.3555 −0.480986
\(772\) 0 0
\(773\) −37.8863 −1.36268 −0.681338 0.731969i \(-0.738601\pi\)
−0.681338 + 0.731969i \(0.738601\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.47689 0.0888578
\(778\) 0 0
\(779\) −17.8140 −0.638254
\(780\) 0 0
\(781\) 39.9546 1.42969
\(782\) 0 0
\(783\) −6.25240 −0.223442
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 14.3213 0.510500 0.255250 0.966875i \(-0.417842\pi\)
0.255250 + 0.966875i \(0.417842\pi\)
\(788\) 0 0
\(789\) 9.25240 0.329394
\(790\) 0 0
\(791\) 12.4402 0.442324
\(792\) 0 0
\(793\) 11.4200 0.405537
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7.97398 −0.282453 −0.141226 0.989977i \(-0.545105\pi\)
−0.141226 + 0.989977i \(0.545105\pi\)
\(798\) 0 0
\(799\) 12.2847 0.434600
\(800\) 0 0
\(801\) −2.20617 −0.0779511
\(802\) 0 0
\(803\) 14.6060 0.515434
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −25.2986 −0.890554
\(808\) 0 0
\(809\) 7.42192 0.260941 0.130470 0.991452i \(-0.458351\pi\)
0.130470 + 0.991452i \(0.458351\pi\)
\(810\) 0 0
\(811\) −54.6391 −1.91864 −0.959319 0.282323i \(-0.908895\pi\)
−0.959319 + 0.282323i \(0.908895\pi\)
\(812\) 0 0
\(813\) −31.3309 −1.09882
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.66473 −0.0582416
\(818\) 0 0
\(819\) 2.76156 0.0964966
\(820\) 0 0
\(821\) −2.12910 −0.0743062 −0.0371531 0.999310i \(-0.511829\pi\)
−0.0371531 + 0.999310i \(0.511829\pi\)
\(822\) 0 0
\(823\) 36.3188 1.26600 0.632998 0.774154i \(-0.281824\pi\)
0.632998 + 0.774154i \(0.281824\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22.0419 −0.766471 −0.383235 0.923651i \(-0.625190\pi\)
−0.383235 + 0.923651i \(0.625190\pi\)
\(828\) 0 0
\(829\) 48.9344 1.69956 0.849781 0.527136i \(-0.176734\pi\)
0.849781 + 0.527136i \(0.176734\pi\)
\(830\) 0 0
\(831\) 23.0096 0.798194
\(832\) 0 0
\(833\) 2.27072 0.0786757
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −8.49084 −0.293486
\(838\) 0 0
\(839\) 10.2986 0.355548 0.177774 0.984071i \(-0.443110\pi\)
0.177774 + 0.984071i \(0.443110\pi\)
\(840\) 0 0
\(841\) 10.0925 0.348016
\(842\) 0 0
\(843\) 9.19554 0.316711
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 25.3169 0.869901
\(848\) 0 0
\(849\) −9.93545 −0.340984
\(850\) 0 0
\(851\) −4.95377 −0.169813
\(852\) 0 0
\(853\) −24.0356 −0.822963 −0.411482 0.911418i \(-0.634989\pi\)
−0.411482 + 0.911418i \(0.634989\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15.0096 0.512718 0.256359 0.966582i \(-0.417477\pi\)
0.256359 + 0.966582i \(0.417477\pi\)
\(858\) 0 0
\(859\) −24.2707 −0.828106 −0.414053 0.910253i \(-0.635887\pi\)
−0.414053 + 0.910253i \(0.635887\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) −42.4985 −1.44667 −0.723333 0.690499i \(-0.757391\pi\)
−0.723333 + 0.690499i \(0.757391\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 3.85069 0.130776
\(868\) 0 0
\(869\) 61.4007 2.08287
\(870\) 0 0
\(871\) −12.4340 −0.421309
\(872\) 0 0
\(873\) −17.7572 −0.600990
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 4.56165 0.154036 0.0770179 0.997030i \(-0.475460\pi\)
0.0770179 + 0.997030i \(0.475460\pi\)
\(878\) 0 0
\(879\) 23.7938 0.802546
\(880\) 0 0
\(881\) 6.10308 0.205618 0.102809 0.994701i \(-0.467217\pi\)
0.102809 + 0.994701i \(0.467217\pi\)
\(882\) 0 0
\(883\) −13.0096 −0.437807 −0.218904 0.975746i \(-0.570248\pi\)
−0.218904 + 0.975746i \(0.570248\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 11.4307 0.383804 0.191902 0.981414i \(-0.438534\pi\)
0.191902 + 0.981414i \(0.438534\pi\)
\(888\) 0 0
\(889\) 18.2986 0.613716
\(890\) 0 0
\(891\) 4.49084 0.150449
\(892\) 0 0
\(893\) −20.8324 −0.697129
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −5.52311 −0.184411
\(898\) 0 0
\(899\) 53.0881 1.77059
\(900\) 0 0
\(901\) −11.8603 −0.395123
\(902\) 0 0
\(903\) −0.747604 −0.0248787
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 19.8863 0.660313 0.330157 0.943926i \(-0.392898\pi\)
0.330157 + 0.943926i \(0.392898\pi\)
\(908\) 0 0
\(909\) 5.83237 0.193447
\(910\) 0 0
\(911\) −57.8217 −1.91572 −0.957860 0.287236i \(-0.907264\pi\)
−0.957860 + 0.287236i \(0.907264\pi\)
\(912\) 0 0
\(913\) −13.6175 −0.450672
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.04623 0.100595
\(918\) 0 0
\(919\) −8.17722 −0.269742 −0.134871 0.990863i \(-0.543062\pi\)
−0.134871 + 0.990863i \(0.543062\pi\)
\(920\) 0 0
\(921\) −25.0664 −0.825967
\(922\) 0 0
\(923\) 8.89692 0.292846
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −3.45856 −0.113594
\(928\) 0 0
\(929\) −14.7312 −0.483314 −0.241657 0.970362i \(-0.577691\pi\)
−0.241657 + 0.970362i \(0.577691\pi\)
\(930\) 0 0
\(931\) −3.85069 −0.126201
\(932\) 0 0
\(933\) −18.3632 −0.601183
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 18.9431 0.618846 0.309423 0.950925i \(-0.399864\pi\)
0.309423 + 0.950925i \(0.399864\pi\)
\(938\) 0 0
\(939\) −10.7293 −0.350137
\(940\) 0 0
\(941\) 12.3353 0.402118 0.201059 0.979579i \(-0.435562\pi\)
0.201059 + 0.979579i \(0.435562\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18.1710 −0.590477 −0.295238 0.955424i \(-0.595399\pi\)
−0.295238 + 0.955424i \(0.595399\pi\)
\(948\) 0 0
\(949\) 3.25240 0.105577
\(950\) 0 0
\(951\) −8.20617 −0.266103
\(952\) 0 0
\(953\) 37.4113 1.21187 0.605935 0.795514i \(-0.292799\pi\)
0.605935 + 0.795514i \(0.292799\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −28.0785 −0.907649
\(958\) 0 0
\(959\) −56.4681 −1.82345
\(960\) 0 0
\(961\) 41.0943 1.32562
\(962\) 0 0
\(963\) −0.832365 −0.0268226
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 38.4821 1.23750 0.618750 0.785588i \(-0.287639\pi\)
0.618750 + 0.785588i \(0.287639\pi\)
\(968\) 0 0
\(969\) 22.2986 0.716335
\(970\) 0 0
\(971\) −44.5896 −1.43095 −0.715473 0.698640i \(-0.753789\pi\)
−0.715473 + 0.698640i \(0.753789\pi\)
\(972\) 0 0
\(973\) 45.7572 1.46691
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −33.2524 −1.06384 −0.531919 0.846795i \(-0.678529\pi\)
−0.531919 + 0.846795i \(0.678529\pi\)
\(978\) 0 0
\(979\) −9.90754 −0.316646
\(980\) 0 0
\(981\) −12.1493 −0.387898
\(982\) 0 0
\(983\) −10.9186 −0.348248 −0.174124 0.984724i \(-0.555709\pi\)
−0.174124 + 0.984724i \(0.555709\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −9.35548 −0.297788
\(988\) 0 0
\(989\) 1.49521 0.0475448
\(990\) 0 0
\(991\) 17.3738 0.551897 0.275949 0.961172i \(-0.411008\pi\)
0.275949 + 0.961172i \(0.411008\pi\)
\(992\) 0 0
\(993\) −30.7110 −0.974583
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −12.0752 −0.382425 −0.191212 0.981549i \(-0.561242\pi\)
−0.191212 + 0.981549i \(0.561242\pi\)
\(998\) 0 0
\(999\) 0.896916 0.0283772
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 7800.2.a.bm.1.3 3
5.4 even 2 7800.2.a.bn.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.bm.1.3 3 1.1 even 1 trivial
7800.2.a.bn.1.1 yes 3 5.4 even 2