Properties

Label 4275.2.a.br.1.3
Level $4275$
Weight $2$
Character 4275.1
Self dual yes
Analytic conductor $34.136$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.1360468641\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.66064384.1
Defining polynomial: \( x^{6} - 9x^{4} + 13x^{2} - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.285442\) of defining polynomial
Character \(\chi\) \(=\) 4275.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.906968 q^{2} -1.17741 q^{4} -2.59637 q^{7} +2.88181 q^{8} +O(q^{10})\) \(q-0.906968 q^{2} -1.17741 q^{4} -2.59637 q^{7} +2.88181 q^{8} -0.741113 q^{11} +3.78878 q^{13} +2.35482 q^{14} -0.258887 q^{16} -3.16725 q^{17} -1.00000 q^{19} +0.672165 q^{22} +0.570885 q^{23} -3.43630 q^{26} +3.05699 q^{28} -6.00000 q^{29} +5.83705 q^{31} -5.52881 q^{32} +2.87259 q^{34} +1.40396 q^{37} +0.906968 q^{38} +3.83705 q^{41} +2.59637 q^{43} +0.872594 q^{44} -0.517774 q^{46} +5.08247 q^{47} -0.258887 q^{49} -4.46094 q^{52} -0.160905 q^{53} -7.48223 q^{56} +5.44181 q^{58} -8.35482 q^{59} -8.57816 q^{61} -5.29401 q^{62} +5.53223 q^{64} +14.8464 q^{67} +3.72915 q^{68} -3.64518 q^{71} +10.8461 q^{73} -1.27334 q^{74} +1.17741 q^{76} +1.92420 q^{77} +1.83705 q^{79} -3.48008 q^{82} -4.19876 q^{83} -2.35482 q^{86} -2.13574 q^{88} +16.9015 q^{89} -9.83705 q^{91} -0.672165 q^{92} -4.60963 q^{94} +3.78878 q^{97} +0.234802 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{4} - 2 q^{11} - 16 q^{14} - 4 q^{16} - 6 q^{19} - 8 q^{26} - 36 q^{29} - 8 q^{34} - 12 q^{41} - 20 q^{44} - 8 q^{46} - 4 q^{49} - 40 q^{56} - 20 q^{59} - 14 q^{61} - 12 q^{64} - 52 q^{71} + 40 q^{74} - 8 q^{76} - 24 q^{79} + 16 q^{86} - 24 q^{89} - 24 q^{91} - 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.906968 −0.641323 −0.320661 0.947194i \(-0.603905\pi\)
−0.320661 + 0.947194i \(0.603905\pi\)
\(3\) 0 0
\(4\) −1.17741 −0.588705
\(5\) 0 0
\(6\) 0 0
\(7\) −2.59637 −0.981334 −0.490667 0.871347i \(-0.663247\pi\)
−0.490667 + 0.871347i \(0.663247\pi\)
\(8\) 2.88181 1.01887
\(9\) 0 0
\(10\) 0 0
\(11\) −0.741113 −0.223454 −0.111727 0.993739i \(-0.535638\pi\)
−0.111727 + 0.993739i \(0.535638\pi\)
\(12\) 0 0
\(13\) 3.78878 1.05082 0.525409 0.850850i \(-0.323912\pi\)
0.525409 + 0.850850i \(0.323912\pi\)
\(14\) 2.35482 0.629352
\(15\) 0 0
\(16\) −0.258887 −0.0647218
\(17\) −3.16725 −0.768171 −0.384086 0.923298i \(-0.625483\pi\)
−0.384086 + 0.923298i \(0.625483\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0.672165 0.143306
\(23\) 0.570885 0.119038 0.0595189 0.998227i \(-0.481043\pi\)
0.0595189 + 0.998227i \(0.481043\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.43630 −0.673913
\(27\) 0 0
\(28\) 3.05699 0.577716
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 5.83705 1.04836 0.524182 0.851606i \(-0.324371\pi\)
0.524182 + 0.851606i \(0.324371\pi\)
\(32\) −5.52881 −0.977365
\(33\) 0 0
\(34\) 2.87259 0.492646
\(35\) 0 0
\(36\) 0 0
\(37\) 1.40396 0.230809 0.115404 0.993319i \(-0.463184\pi\)
0.115404 + 0.993319i \(0.463184\pi\)
\(38\) 0.906968 0.147130
\(39\) 0 0
\(40\) 0 0
\(41\) 3.83705 0.599246 0.299623 0.954058i \(-0.403139\pi\)
0.299623 + 0.954058i \(0.403139\pi\)
\(42\) 0 0
\(43\) 2.59637 0.395942 0.197971 0.980208i \(-0.436565\pi\)
0.197971 + 0.980208i \(0.436565\pi\)
\(44\) 0.872594 0.131548
\(45\) 0 0
\(46\) −0.517774 −0.0763416
\(47\) 5.08247 0.741354 0.370677 0.928762i \(-0.379126\pi\)
0.370677 + 0.928762i \(0.379126\pi\)
\(48\) 0 0
\(49\) −0.258887 −0.0369839
\(50\) 0 0
\(51\) 0 0
\(52\) −4.46094 −0.618621
\(53\) −0.160905 −0.0221020 −0.0110510 0.999939i \(-0.503518\pi\)
−0.0110510 + 0.999939i \(0.503518\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7.48223 −0.999854
\(57\) 0 0
\(58\) 5.44181 0.714544
\(59\) −8.35482 −1.08770 −0.543852 0.839181i \(-0.683035\pi\)
−0.543852 + 0.839181i \(0.683035\pi\)
\(60\) 0 0
\(61\) −8.57816 −1.09832 −0.549160 0.835717i \(-0.685052\pi\)
−0.549160 + 0.835717i \(0.685052\pi\)
\(62\) −5.29401 −0.672340
\(63\) 0 0
\(64\) 5.53223 0.691529
\(65\) 0 0
\(66\) 0 0
\(67\) 14.8464 1.81378 0.906888 0.421371i \(-0.138451\pi\)
0.906888 + 0.421371i \(0.138451\pi\)
\(68\) 3.72915 0.452226
\(69\) 0 0
\(70\) 0 0
\(71\) −3.64518 −0.432603 −0.216302 0.976327i \(-0.569399\pi\)
−0.216302 + 0.976327i \(0.569399\pi\)
\(72\) 0 0
\(73\) 10.8461 1.26944 0.634719 0.772743i \(-0.281116\pi\)
0.634719 + 0.772743i \(0.281116\pi\)
\(74\) −1.27334 −0.148023
\(75\) 0 0
\(76\) 1.17741 0.135058
\(77\) 1.92420 0.219283
\(78\) 0 0
\(79\) 1.83705 0.206684 0.103342 0.994646i \(-0.467046\pi\)
0.103342 + 0.994646i \(0.467046\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.48008 −0.384310
\(83\) −4.19876 −0.460873 −0.230437 0.973087i \(-0.574015\pi\)
−0.230437 + 0.973087i \(0.574015\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.35482 −0.253927
\(87\) 0 0
\(88\) −2.13574 −0.227671
\(89\) 16.9015 1.79156 0.895778 0.444502i \(-0.146619\pi\)
0.895778 + 0.444502i \(0.146619\pi\)
\(90\) 0 0
\(91\) −9.83705 −1.03120
\(92\) −0.672165 −0.0700781
\(93\) 0 0
\(94\) −4.60963 −0.475447
\(95\) 0 0
\(96\) 0 0
\(97\) 3.78878 0.384692 0.192346 0.981327i \(-0.438390\pi\)
0.192346 + 0.981327i \(0.438390\pi\)
\(98\) 0.234802 0.0237186
\(99\) 0 0
\(100\) 0 0
\(101\) −8.35482 −0.831336 −0.415668 0.909517i \(-0.636452\pi\)
−0.415668 + 0.909517i \(0.636452\pi\)
\(102\) 0 0
\(103\) −2.07612 −0.204566 −0.102283 0.994755i \(-0.532615\pi\)
−0.102283 + 0.994755i \(0.532615\pi\)
\(104\) 10.9185 1.07065
\(105\) 0 0
\(106\) 0.145935 0.0141745
\(107\) −5.70399 −0.551426 −0.275713 0.961240i \(-0.588914\pi\)
−0.275713 + 0.961240i \(0.588914\pi\)
\(108\) 0 0
\(109\) 1.64518 0.157580 0.0787899 0.996891i \(-0.474894\pi\)
0.0787899 + 0.996891i \(0.474894\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.672165 0.0635137
\(113\) −3.89006 −0.365946 −0.182973 0.983118i \(-0.558572\pi\)
−0.182973 + 0.983118i \(0.558572\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 7.06446 0.655918
\(117\) 0 0
\(118\) 7.57755 0.697570
\(119\) 8.22334 0.753832
\(120\) 0 0
\(121\) −10.4508 −0.950068
\(122\) 7.78011 0.704378
\(123\) 0 0
\(124\) −6.87259 −0.617177
\(125\) 0 0
\(126\) 0 0
\(127\) 14.4233 1.27986 0.639931 0.768432i \(-0.278963\pi\)
0.639931 + 0.768432i \(0.278963\pi\)
\(128\) 6.04007 0.533872
\(129\) 0 0
\(130\) 0 0
\(131\) −9.96853 −0.870954 −0.435477 0.900200i \(-0.643420\pi\)
−0.435477 + 0.900200i \(0.643420\pi\)
\(132\) 0 0
\(133\) 2.59637 0.225133
\(134\) −13.4652 −1.16322
\(135\) 0 0
\(136\) −9.12741 −0.782669
\(137\) −9.70431 −0.829095 −0.414548 0.910028i \(-0.636060\pi\)
−0.414548 + 0.910028i \(0.636060\pi\)
\(138\) 0 0
\(139\) −13.4508 −1.14088 −0.570439 0.821340i \(-0.693227\pi\)
−0.570439 + 0.821340i \(0.693227\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 3.30606 0.277438
\(143\) −2.80791 −0.234809
\(144\) 0 0
\(145\) 0 0
\(146\) −9.83705 −0.814120
\(147\) 0 0
\(148\) −1.65303 −0.135878
\(149\) −15.0959 −1.23671 −0.618353 0.785900i \(-0.712200\pi\)
−0.618353 + 0.785900i \(0.712200\pi\)
\(150\) 0 0
\(151\) 14.1919 1.15492 0.577459 0.816420i \(-0.304044\pi\)
0.577459 + 0.816420i \(0.304044\pi\)
\(152\) −2.88181 −0.233745
\(153\) 0 0
\(154\) −1.74519 −0.140631
\(155\) 0 0
\(156\) 0 0
\(157\) 7.57755 0.604754 0.302377 0.953188i \(-0.402220\pi\)
0.302377 + 0.953188i \(0.402220\pi\)
\(158\) −1.66614 −0.132551
\(159\) 0 0
\(160\) 0 0
\(161\) −1.48223 −0.116816
\(162\) 0 0
\(163\) −19.6757 −1.54112 −0.770559 0.637369i \(-0.780023\pi\)
−0.770559 + 0.637369i \(0.780023\pi\)
\(164\) −4.51777 −0.352779
\(165\) 0 0
\(166\) 3.80814 0.295569
\(167\) 10.7954 0.835376 0.417688 0.908590i \(-0.362840\pi\)
0.417688 + 0.908590i \(0.362840\pi\)
\(168\) 0 0
\(169\) 1.35482 0.104217
\(170\) 0 0
\(171\) 0 0
\(172\) −3.05699 −0.233093
\(173\) −20.3895 −1.55018 −0.775092 0.631848i \(-0.782297\pi\)
−0.775092 + 0.631848i \(0.782297\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.191865 0.0144623
\(177\) 0 0
\(178\) −15.3291 −1.14897
\(179\) −25.0645 −1.87341 −0.936703 0.350126i \(-0.886139\pi\)
−0.936703 + 0.350126i \(0.886139\pi\)
\(180\) 0 0
\(181\) −19.4193 −1.44342 −0.721712 0.692194i \(-0.756644\pi\)
−0.721712 + 0.692194i \(0.756644\pi\)
\(182\) 8.92188 0.661334
\(183\) 0 0
\(184\) 1.64518 0.121284
\(185\) 0 0
\(186\) 0 0
\(187\) 2.34729 0.171651
\(188\) −5.98414 −0.436439
\(189\) 0 0
\(190\) 0 0
\(191\) −11.4508 −0.828547 −0.414274 0.910152i \(-0.635964\pi\)
−0.414274 + 0.910152i \(0.635964\pi\)
\(192\) 0 0
\(193\) 3.78878 0.272722 0.136361 0.990659i \(-0.456459\pi\)
0.136361 + 0.990659i \(0.456459\pi\)
\(194\) −3.43630 −0.246712
\(195\) 0 0
\(196\) 0.304816 0.0217726
\(197\) −2.28354 −0.162695 −0.0813477 0.996686i \(-0.525922\pi\)
−0.0813477 + 0.996686i \(0.525922\pi\)
\(198\) 0 0
\(199\) −19.4508 −1.37883 −0.689414 0.724368i \(-0.742132\pi\)
−0.689414 + 0.724368i \(0.742132\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 7.57755 0.533155
\(203\) 15.5782 1.09337
\(204\) 0 0
\(205\) 0 0
\(206\) 1.88297 0.131193
\(207\) 0 0
\(208\) −0.980865 −0.0680108
\(209\) 0.741113 0.0512639
\(210\) 0 0
\(211\) 11.2274 0.772927 0.386463 0.922305i \(-0.373697\pi\)
0.386463 + 0.922305i \(0.373697\pi\)
\(212\) 0.189451 0.0130115
\(213\) 0 0
\(214\) 5.17334 0.353642
\(215\) 0 0
\(216\) 0 0
\(217\) −15.1551 −1.02880
\(218\) −1.49213 −0.101059
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −4.03785 −0.270394 −0.135197 0.990819i \(-0.543167\pi\)
−0.135197 + 0.990819i \(0.543167\pi\)
\(224\) 14.3548 0.959122
\(225\) 0 0
\(226\) 3.52815 0.234689
\(227\) 11.2185 0.744600 0.372300 0.928112i \(-0.378569\pi\)
0.372300 + 0.928112i \(0.378569\pi\)
\(228\) 0 0
\(229\) 16.1315 1.06600 0.532999 0.846116i \(-0.321065\pi\)
0.532999 + 0.846116i \(0.321065\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −17.2908 −1.13520
\(233\) 2.12676 0.139329 0.0696644 0.997570i \(-0.477807\pi\)
0.0696644 + 0.997570i \(0.477807\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 9.83705 0.640337
\(237\) 0 0
\(238\) −7.45830 −0.483450
\(239\) 14.4152 0.932442 0.466221 0.884668i \(-0.345615\pi\)
0.466221 + 0.884668i \(0.345615\pi\)
\(240\) 0 0
\(241\) −0.162955 −0.0104968 −0.00524842 0.999986i \(-0.501671\pi\)
−0.00524842 + 0.999986i \(0.501671\pi\)
\(242\) 9.47849 0.609301
\(243\) 0 0
\(244\) 10.1000 0.646587
\(245\) 0 0
\(246\) 0 0
\(247\) −3.78878 −0.241074
\(248\) 16.8212 1.06815
\(249\) 0 0
\(250\) 0 0
\(251\) −12.9330 −0.816322 −0.408161 0.912910i \(-0.633830\pi\)
−0.408161 + 0.912910i \(0.633830\pi\)
\(252\) 0 0
\(253\) −0.423090 −0.0265995
\(254\) −13.0815 −0.820805
\(255\) 0 0
\(256\) −16.5426 −1.03391
\(257\) 11.0445 0.688938 0.344469 0.938798i \(-0.388059\pi\)
0.344469 + 0.938798i \(0.388059\pi\)
\(258\) 0 0
\(259\) −3.64518 −0.226501
\(260\) 0 0
\(261\) 0 0
\(262\) 9.04113 0.558563
\(263\) −17.8527 −1.10085 −0.550424 0.834885i \(-0.685534\pi\)
−0.550424 + 0.834885i \(0.685534\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.35482 −0.144383
\(267\) 0 0
\(268\) −17.4803 −1.06778
\(269\) −24.9934 −1.52387 −0.761936 0.647652i \(-0.775751\pi\)
−0.761936 + 0.647652i \(0.775751\pi\)
\(270\) 0 0
\(271\) −23.8660 −1.44975 −0.724877 0.688879i \(-0.758103\pi\)
−0.724877 + 0.688879i \(0.758103\pi\)
\(272\) 0.819960 0.0497174
\(273\) 0 0
\(274\) 8.80150 0.531718
\(275\) 0 0
\(276\) 0 0
\(277\) −21.2315 −1.27568 −0.637840 0.770169i \(-0.720172\pi\)
−0.637840 + 0.770169i \(0.720172\pi\)
\(278\) 12.1994 0.731671
\(279\) 0 0
\(280\) 0 0
\(281\) −3.83705 −0.228899 −0.114449 0.993429i \(-0.536510\pi\)
−0.114449 + 0.993429i \(0.536510\pi\)
\(282\) 0 0
\(283\) 0.211545 0.0125751 0.00628753 0.999980i \(-0.497999\pi\)
0.00628753 + 0.999980i \(0.497999\pi\)
\(284\) 4.29187 0.254676
\(285\) 0 0
\(286\) 2.54668 0.150589
\(287\) −9.96237 −0.588060
\(288\) 0 0
\(289\) −6.96853 −0.409913
\(290\) 0 0
\(291\) 0 0
\(292\) −12.7703 −0.747324
\(293\) 14.9942 0.875970 0.437985 0.898982i \(-0.355692\pi\)
0.437985 + 0.898982i \(0.355692\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4.04593 0.235165
\(297\) 0 0
\(298\) 13.6915 0.793129
\(299\) 2.16295 0.125087
\(300\) 0 0
\(301\) −6.74111 −0.388551
\(302\) −12.8716 −0.740675
\(303\) 0 0
\(304\) 0.258887 0.0148482
\(305\) 0 0
\(306\) 0 0
\(307\) 1.65303 0.0943434 0.0471717 0.998887i \(-0.484979\pi\)
0.0471717 + 0.998887i \(0.484979\pi\)
\(308\) −2.26557 −0.129093
\(309\) 0 0
\(310\) 0 0
\(311\) 0.741113 0.0420247 0.0210123 0.999779i \(-0.493311\pi\)
0.0210123 + 0.999779i \(0.493311\pi\)
\(312\) 0 0
\(313\) −26.8849 −1.51962 −0.759812 0.650143i \(-0.774709\pi\)
−0.759812 + 0.650143i \(0.774709\pi\)
\(314\) −6.87259 −0.387843
\(315\) 0 0
\(316\) −2.16295 −0.121676
\(317\) 8.16155 0.458398 0.229199 0.973380i \(-0.426389\pi\)
0.229199 + 0.973380i \(0.426389\pi\)
\(318\) 0 0
\(319\) 4.44668 0.248966
\(320\) 0 0
\(321\) 0 0
\(322\) 1.34433 0.0749166
\(323\) 3.16725 0.176231
\(324\) 0 0
\(325\) 0 0
\(326\) 17.8452 0.988354
\(327\) 0 0
\(328\) 11.0576 0.610555
\(329\) −13.1959 −0.727516
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 4.94366 0.271318
\(333\) 0 0
\(334\) −9.79112 −0.535746
\(335\) 0 0
\(336\) 0 0
\(337\) −9.90275 −0.539437 −0.269718 0.962939i \(-0.586931\pi\)
−0.269718 + 0.962939i \(0.586931\pi\)
\(338\) −1.22878 −0.0668367
\(339\) 0 0
\(340\) 0 0
\(341\) −4.32591 −0.234261
\(342\) 0 0
\(343\) 18.8467 1.01763
\(344\) 7.48223 0.403415
\(345\) 0 0
\(346\) 18.4926 0.994169
\(347\) 21.2781 1.14227 0.571133 0.820858i \(-0.306504\pi\)
0.571133 + 0.820858i \(0.306504\pi\)
\(348\) 0 0
\(349\) −16.4152 −0.878686 −0.439343 0.898319i \(-0.644789\pi\)
−0.439343 + 0.898319i \(0.644789\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.09748 0.218396
\(353\) −23.8744 −1.27071 −0.635354 0.772221i \(-0.719146\pi\)
−0.635354 + 0.772221i \(0.719146\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −19.9000 −1.05470
\(357\) 0 0
\(358\) 22.7327 1.20146
\(359\) −2.22334 −0.117343 −0.0586717 0.998277i \(-0.518687\pi\)
−0.0586717 + 0.998277i \(0.518687\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 17.6127 0.925701
\(363\) 0 0
\(364\) 11.5822 0.607074
\(365\) 0 0
\(366\) 0 0
\(367\) −4.52057 −0.235972 −0.117986 0.993015i \(-0.537644\pi\)
−0.117986 + 0.993015i \(0.537644\pi\)
\(368\) −0.147795 −0.00770433
\(369\) 0 0
\(370\) 0 0
\(371\) 0.417768 0.0216894
\(372\) 0 0
\(373\) 15.5186 0.803521 0.401760 0.915745i \(-0.368398\pi\)
0.401760 + 0.915745i \(0.368398\pi\)
\(374\) −2.12892 −0.110084
\(375\) 0 0
\(376\) 14.6467 0.755345
\(377\) −22.7327 −1.17079
\(378\) 0 0
\(379\) −18.9015 −0.970905 −0.485453 0.874263i \(-0.661345\pi\)
−0.485453 + 0.874263i \(0.661345\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 10.3855 0.531366
\(383\) 13.7046 0.700274 0.350137 0.936699i \(-0.386135\pi\)
0.350137 + 0.936699i \(0.386135\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.43630 −0.174903
\(387\) 0 0
\(388\) −4.46094 −0.226470
\(389\) −12.7411 −0.646000 −0.323000 0.946399i \(-0.604691\pi\)
−0.323000 + 0.946399i \(0.604691\pi\)
\(390\) 0 0
\(391\) −1.80814 −0.0914413
\(392\) −0.746063 −0.0376819
\(393\) 0 0
\(394\) 2.07110 0.104340
\(395\) 0 0
\(396\) 0 0
\(397\) −38.6522 −1.93990 −0.969950 0.243306i \(-0.921768\pi\)
−0.969950 + 0.243306i \(0.921768\pi\)
\(398\) 17.6412 0.884274
\(399\) 0 0
\(400\) 0 0
\(401\) −31.8660 −1.59131 −0.795655 0.605750i \(-0.792873\pi\)
−0.795655 + 0.605750i \(0.792873\pi\)
\(402\) 0 0
\(403\) 22.1153 1.10164
\(404\) 9.83705 0.489411
\(405\) 0 0
\(406\) −14.1289 −0.701206
\(407\) −1.04049 −0.0515751
\(408\) 0 0
\(409\) 11.0645 0.547102 0.273551 0.961857i \(-0.411802\pi\)
0.273551 + 0.961857i \(0.411802\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.44444 0.120429
\(413\) 21.6922 1.06740
\(414\) 0 0
\(415\) 0 0
\(416\) −20.9474 −1.02703
\(417\) 0 0
\(418\) −0.672165 −0.0328767
\(419\) 25.7452 1.25773 0.628867 0.777513i \(-0.283519\pi\)
0.628867 + 0.777513i \(0.283519\pi\)
\(420\) 0 0
\(421\) 27.4482 1.33774 0.668871 0.743378i \(-0.266778\pi\)
0.668871 + 0.743378i \(0.266778\pi\)
\(422\) −10.1829 −0.495696
\(423\) 0 0
\(424\) −0.463697 −0.0225191
\(425\) 0 0
\(426\) 0 0
\(427\) 22.2720 1.07782
\(428\) 6.71593 0.324627
\(429\) 0 0
\(430\) 0 0
\(431\) −1.74519 −0.0840627 −0.0420314 0.999116i \(-0.513383\pi\)
−0.0420314 + 0.999116i \(0.513383\pi\)
\(432\) 0 0
\(433\) −18.5208 −0.890052 −0.445026 0.895518i \(-0.646806\pi\)
−0.445026 + 0.895518i \(0.646806\pi\)
\(434\) 13.7452 0.659790
\(435\) 0 0
\(436\) −1.93705 −0.0927679
\(437\) −0.570885 −0.0273091
\(438\) 0 0
\(439\) 29.4482 1.40549 0.702743 0.711444i \(-0.251959\pi\)
0.702743 + 0.711444i \(0.251959\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 10.8836 0.517681
\(443\) −11.7388 −0.557726 −0.278863 0.960331i \(-0.589958\pi\)
−0.278863 + 0.960331i \(0.589958\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.66220 0.173410
\(447\) 0 0
\(448\) −14.3637 −0.678620
\(449\) 7.06446 0.333392 0.166696 0.986008i \(-0.446690\pi\)
0.166696 + 0.986008i \(0.446690\pi\)
\(450\) 0 0
\(451\) −2.84368 −0.133904
\(452\) 4.58019 0.215434
\(453\) 0 0
\(454\) −10.1748 −0.477529
\(455\) 0 0
\(456\) 0 0
\(457\) −34.5000 −1.61384 −0.806920 0.590660i \(-0.798867\pi\)
−0.806920 + 0.590660i \(0.798867\pi\)
\(458\) −14.6307 −0.683649
\(459\) 0 0
\(460\) 0 0
\(461\) −8.03147 −0.374063 −0.187032 0.982354i \(-0.559887\pi\)
−0.187032 + 0.982354i \(0.559887\pi\)
\(462\) 0 0
\(463\) −25.3290 −1.17714 −0.588570 0.808447i \(-0.700309\pi\)
−0.588570 + 0.808447i \(0.700309\pi\)
\(464\) 1.55332 0.0721112
\(465\) 0 0
\(466\) −1.92890 −0.0893547
\(467\) −26.8759 −1.24367 −0.621834 0.783149i \(-0.713612\pi\)
−0.621834 + 0.783149i \(0.713612\pi\)
\(468\) 0 0
\(469\) −38.5467 −1.77992
\(470\) 0 0
\(471\) 0 0
\(472\) −24.0770 −1.10823
\(473\) −1.92420 −0.0884748
\(474\) 0 0
\(475\) 0 0
\(476\) −9.68224 −0.443785
\(477\) 0 0
\(478\) −13.0741 −0.597996
\(479\) 28.9015 1.32054 0.660272 0.751027i \(-0.270441\pi\)
0.660272 + 0.751027i \(0.270441\pi\)
\(480\) 0 0
\(481\) 5.31927 0.242538
\(482\) 0.147795 0.00673187
\(483\) 0 0
\(484\) 12.3048 0.559310
\(485\) 0 0
\(486\) 0 0
\(487\) −17.7294 −0.803395 −0.401697 0.915773i \(-0.631580\pi\)
−0.401697 + 0.915773i \(0.631580\pi\)
\(488\) −24.7206 −1.11905
\(489\) 0 0
\(490\) 0 0
\(491\) 35.1645 1.58695 0.793475 0.608603i \(-0.208270\pi\)
0.793475 + 0.608603i \(0.208270\pi\)
\(492\) 0 0
\(493\) 19.0035 0.855875
\(494\) 3.43630 0.154606
\(495\) 0 0
\(496\) −1.51114 −0.0678520
\(497\) 9.46422 0.424528
\(498\) 0 0
\(499\) 21.4508 0.960268 0.480134 0.877195i \(-0.340588\pi\)
0.480134 + 0.877195i \(0.340588\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 11.7298 0.523526
\(503\) 5.34053 0.238122 0.119061 0.992887i \(-0.462012\pi\)
0.119061 + 0.992887i \(0.462012\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0.383729 0.0170588
\(507\) 0 0
\(508\) −16.9821 −0.753461
\(509\) −36.1919 −1.60418 −0.802088 0.597206i \(-0.796278\pi\)
−0.802088 + 0.597206i \(0.796278\pi\)
\(510\) 0 0
\(511\) −28.1604 −1.24574
\(512\) 2.92346 0.129200
\(513\) 0 0
\(514\) −10.0170 −0.441832
\(515\) 0 0
\(516\) 0 0
\(517\) −3.76668 −0.165658
\(518\) 3.30606 0.145260
\(519\) 0 0
\(520\) 0 0
\(521\) 2.77259 0.121469 0.0607346 0.998154i \(-0.480656\pi\)
0.0607346 + 0.998154i \(0.480656\pi\)
\(522\) 0 0
\(523\) −20.5373 −0.898033 −0.449016 0.893524i \(-0.648226\pi\)
−0.449016 + 0.893524i \(0.648226\pi\)
\(524\) 11.7370 0.512735
\(525\) 0 0
\(526\) 16.1919 0.705999
\(527\) −18.4874 −0.805323
\(528\) 0 0
\(529\) −22.6741 −0.985830
\(530\) 0 0
\(531\) 0 0
\(532\) −3.05699 −0.132537
\(533\) 14.5377 0.629698
\(534\) 0 0
\(535\) 0 0
\(536\) 42.7845 1.84801
\(537\) 0 0
\(538\) 22.6682 0.977294
\(539\) 0.191865 0.00826419
\(540\) 0 0
\(541\) 35.4797 1.52539 0.762695 0.646758i \(-0.223876\pi\)
0.762695 + 0.646758i \(0.223876\pi\)
\(542\) 21.6456 0.929760
\(543\) 0 0
\(544\) 17.5111 0.750784
\(545\) 0 0
\(546\) 0 0
\(547\) 43.0756 1.84178 0.920890 0.389822i \(-0.127463\pi\)
0.920890 + 0.389822i \(0.127463\pi\)
\(548\) 11.4260 0.488092
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) −4.76964 −0.202826
\(554\) 19.2563 0.818123
\(555\) 0 0
\(556\) 15.8370 0.671640
\(557\) −40.4376 −1.71340 −0.856698 0.515818i \(-0.827488\pi\)
−0.856698 + 0.515818i \(0.827488\pi\)
\(558\) 0 0
\(559\) 9.83705 0.416063
\(560\) 0 0
\(561\) 0 0
\(562\) 3.48008 0.146798
\(563\) −19.7173 −0.830986 −0.415493 0.909596i \(-0.636391\pi\)
−0.415493 + 0.909596i \(0.636391\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.191865 −0.00806467
\(567\) 0 0
\(568\) −10.5047 −0.440768
\(569\) −18.6807 −0.783137 −0.391568 0.920149i \(-0.628067\pi\)
−0.391568 + 0.920149i \(0.628067\pi\)
\(570\) 0 0
\(571\) −29.9371 −1.25283 −0.626413 0.779491i \(-0.715478\pi\)
−0.626413 + 0.779491i \(0.715478\pi\)
\(572\) 3.30606 0.138233
\(573\) 0 0
\(574\) 9.03555 0.377137
\(575\) 0 0
\(576\) 0 0
\(577\) −0.156779 −0.00652679 −0.00326339 0.999995i \(-0.501039\pi\)
−0.00326339 + 0.999995i \(0.501039\pi\)
\(578\) 6.32023 0.262887
\(579\) 0 0
\(580\) 0 0
\(581\) 10.9015 0.452271
\(582\) 0 0
\(583\) 0.119249 0.00493877
\(584\) 31.2563 1.29340
\(585\) 0 0
\(586\) −13.5993 −0.561780
\(587\) −31.1474 −1.28559 −0.642795 0.766038i \(-0.722225\pi\)
−0.642795 + 0.766038i \(0.722225\pi\)
\(588\) 0 0
\(589\) −5.83705 −0.240511
\(590\) 0 0
\(591\) 0 0
\(592\) −0.363466 −0.0149384
\(593\) −28.8728 −1.18567 −0.592833 0.805326i \(-0.701991\pi\)
−0.592833 + 0.805326i \(0.701991\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17.7741 0.728055
\(597\) 0 0
\(598\) −1.96173 −0.0802211
\(599\) −25.3274 −1.03485 −0.517425 0.855728i \(-0.673109\pi\)
−0.517425 + 0.855728i \(0.673109\pi\)
\(600\) 0 0
\(601\) −19.8370 −0.809170 −0.404585 0.914500i \(-0.632584\pi\)
−0.404585 + 0.914500i \(0.632584\pi\)
\(602\) 6.11397 0.249187
\(603\) 0 0
\(604\) −16.7096 −0.679906
\(605\) 0 0
\(606\) 0 0
\(607\) −2.49921 −0.101440 −0.0507199 0.998713i \(-0.516152\pi\)
−0.0507199 + 0.998713i \(0.516152\pi\)
\(608\) 5.52881 0.224223
\(609\) 0 0
\(610\) 0 0
\(611\) 19.2563 0.779027
\(612\) 0 0
\(613\) −0.883711 −0.0356927 −0.0178464 0.999841i \(-0.505681\pi\)
−0.0178464 + 0.999841i \(0.505681\pi\)
\(614\) −1.49925 −0.0605046
\(615\) 0 0
\(616\) 5.54517 0.223421
\(617\) 29.4085 1.18394 0.591971 0.805959i \(-0.298350\pi\)
0.591971 + 0.805959i \(0.298350\pi\)
\(618\) 0 0
\(619\) 30.3208 1.21870 0.609348 0.792903i \(-0.291431\pi\)
0.609348 + 0.792903i \(0.291431\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −0.672165 −0.0269514
\(623\) −43.8825 −1.75811
\(624\) 0 0
\(625\) 0 0
\(626\) 24.3837 0.974570
\(627\) 0 0
\(628\) −8.92188 −0.356022
\(629\) −4.44668 −0.177301
\(630\) 0 0
\(631\) 17.7767 0.707678 0.353839 0.935306i \(-0.384876\pi\)
0.353839 + 0.935306i \(0.384876\pi\)
\(632\) 5.29401 0.210584
\(633\) 0 0
\(634\) −7.40226 −0.293981
\(635\) 0 0
\(636\) 0 0
\(637\) −0.980865 −0.0388633
\(638\) −4.03299 −0.159668
\(639\) 0 0
\(640\) 0 0
\(641\) 32.6675 1.29029 0.645143 0.764062i \(-0.276798\pi\)
0.645143 + 0.764062i \(0.276798\pi\)
\(642\) 0 0
\(643\) 31.8661 1.25668 0.628338 0.777941i \(-0.283736\pi\)
0.628338 + 0.777941i \(0.283736\pi\)
\(644\) 1.74519 0.0687700
\(645\) 0 0
\(646\) −2.87259 −0.113021
\(647\) −21.2601 −0.835820 −0.417910 0.908488i \(-0.637237\pi\)
−0.417910 + 0.908488i \(0.637237\pi\)
\(648\) 0 0
\(649\) 6.19186 0.243052
\(650\) 0 0
\(651\) 0 0
\(652\) 23.1663 0.907263
\(653\) −12.8340 −0.502234 −0.251117 0.967957i \(-0.580798\pi\)
−0.251117 + 0.967957i \(0.580798\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.993361 −0.0387842
\(657\) 0 0
\(658\) 11.9683 0.466573
\(659\) −20.3548 −0.792911 −0.396456 0.918054i \(-0.629760\pi\)
−0.396456 + 0.918054i \(0.629760\pi\)
\(660\) 0 0
\(661\) −30.7385 −1.19559 −0.597795 0.801649i \(-0.703957\pi\)
−0.597795 + 0.801649i \(0.703957\pi\)
\(662\) −7.25574 −0.282002
\(663\) 0 0
\(664\) −12.1000 −0.469571
\(665\) 0 0
\(666\) 0 0
\(667\) −3.42531 −0.132629
\(668\) −12.7107 −0.491790
\(669\) 0 0
\(670\) 0 0
\(671\) 6.35738 0.245424
\(672\) 0 0
\(673\) 21.2094 0.817564 0.408782 0.912632i \(-0.365954\pi\)
0.408782 + 0.912632i \(0.365954\pi\)
\(674\) 8.98147 0.345953
\(675\) 0 0
\(676\) −1.59518 −0.0613530
\(677\) −11.2650 −0.432951 −0.216475 0.976288i \(-0.569456\pi\)
−0.216475 + 0.976288i \(0.569456\pi\)
\(678\) 0 0
\(679\) −9.83705 −0.377511
\(680\) 0 0
\(681\) 0 0
\(682\) 3.92346 0.150237
\(683\) 12.3603 0.472954 0.236477 0.971637i \(-0.424007\pi\)
0.236477 + 0.971637i \(0.424007\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −17.0934 −0.652628
\(687\) 0 0
\(688\) −0.672165 −0.0256261
\(689\) −0.609632 −0.0232251
\(690\) 0 0
\(691\) 22.7493 0.865423 0.432711 0.901533i \(-0.357557\pi\)
0.432711 + 0.901533i \(0.357557\pi\)
\(692\) 24.0068 0.912601
\(693\) 0 0
\(694\) −19.2985 −0.732561
\(695\) 0 0
\(696\) 0 0
\(697\) −12.1529 −0.460323
\(698\) 14.8881 0.563521
\(699\) 0 0
\(700\) 0 0
\(701\) 16.0289 0.605404 0.302702 0.953085i \(-0.402111\pi\)
0.302702 + 0.953085i \(0.402111\pi\)
\(702\) 0 0
\(703\) −1.40396 −0.0529512
\(704\) −4.10001 −0.154525
\(705\) 0 0
\(706\) 21.6533 0.814934
\(707\) 21.6922 0.815818
\(708\) 0 0
\(709\) 31.4193 1.17998 0.589988 0.807412i \(-0.299133\pi\)
0.589988 + 0.807412i \(0.299133\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 48.7069 1.82537
\(713\) 3.33228 0.124795
\(714\) 0 0
\(715\) 0 0
\(716\) 29.5111 1.10288
\(717\) 0 0
\(718\) 2.01650 0.0752550
\(719\) −11.2589 −0.419886 −0.209943 0.977714i \(-0.567328\pi\)
−0.209943 + 0.977714i \(0.567328\pi\)
\(720\) 0 0
\(721\) 5.39037 0.200748
\(722\) −0.906968 −0.0337538
\(723\) 0 0
\(724\) 22.8644 0.849750
\(725\) 0 0
\(726\) 0 0
\(727\) 48.9829 1.81668 0.908338 0.418237i \(-0.137352\pi\)
0.908338 + 0.418237i \(0.137352\pi\)
\(728\) −28.3485 −1.05066
\(729\) 0 0
\(730\) 0 0
\(731\) −8.22334 −0.304151
\(732\) 0 0
\(733\) −35.9260 −1.32696 −0.663479 0.748195i \(-0.730921\pi\)
−0.663479 + 0.748195i \(0.730921\pi\)
\(734\) 4.10001 0.151334
\(735\) 0 0
\(736\) −3.15632 −0.116343
\(737\) −11.0029 −0.405296
\(738\) 0 0
\(739\) 14.3523 0.527956 0.263978 0.964529i \(-0.414965\pi\)
0.263978 + 0.964529i \(0.414965\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.378902 −0.0139099
\(743\) −12.5629 −0.460887 −0.230443 0.973086i \(-0.574018\pi\)
−0.230443 + 0.973086i \(0.574018\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −14.0748 −0.515316
\(747\) 0 0
\(748\) −2.76372 −0.101052
\(749\) 14.8096 0.541133
\(750\) 0 0
\(751\) 26.4548 0.965350 0.482675 0.875799i \(-0.339665\pi\)
0.482675 + 0.875799i \(0.339665\pi\)
\(752\) −1.31578 −0.0479817
\(753\) 0 0
\(754\) 20.6178 0.750855
\(755\) 0 0
\(756\) 0 0
\(757\) −15.7350 −0.571897 −0.285949 0.958245i \(-0.592309\pi\)
−0.285949 + 0.958245i \(0.592309\pi\)
\(758\) 17.1431 0.622664
\(759\) 0 0
\(760\) 0 0
\(761\) −16.9619 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(762\) 0 0
\(763\) −4.27149 −0.154638
\(764\) 13.4822 0.487770
\(765\) 0 0
\(766\) −12.4297 −0.449102
\(767\) −31.6545 −1.14298
\(768\) 0 0
\(769\) 41.9974 1.51447 0.757233 0.653145i \(-0.226551\pi\)
0.757233 + 0.653145i \(0.226551\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.46094 −0.160553
\(773\) 40.9579 1.47315 0.736576 0.676355i \(-0.236441\pi\)
0.736576 + 0.676355i \(0.236441\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 10.9185 0.391952
\(777\) 0 0
\(778\) 11.5558 0.414295
\(779\) −3.83705 −0.137476
\(780\) 0 0
\(781\) 2.70149 0.0966669
\(782\) 1.63992 0.0586434
\(783\) 0 0
\(784\) 0.0670225 0.00239366
\(785\) 0 0
\(786\) 0 0
\(787\) −28.5379 −1.01727 −0.508634 0.860983i \(-0.669849\pi\)
−0.508634 + 0.860983i \(0.669849\pi\)
\(788\) 2.68866 0.0957796
\(789\) 0 0
\(790\) 0 0
\(791\) 10.1000 0.359115
\(792\) 0 0
\(793\) −32.5007 −1.15413
\(794\) 35.0563 1.24410
\(795\) 0 0
\(796\) 22.9015 0.811722
\(797\) 33.2790 1.17880 0.589402 0.807840i \(-0.299364\pi\)
0.589402 + 0.807840i \(0.299364\pi\)
\(798\) 0 0
\(799\) −16.0974 −0.569487
\(800\) 0 0
\(801\) 0 0
\(802\) 28.9014 1.02054
\(803\) −8.03817 −0.283661
\(804\) 0 0
\(805\) 0 0
\(806\) −20.0578 −0.706507
\(807\) 0 0
\(808\) −24.0770 −0.847025
\(809\) 34.4234 1.21026 0.605130 0.796126i \(-0.293121\pi\)
0.605130 + 0.796126i \(0.293121\pi\)
\(810\) 0 0
\(811\) −11.2563 −0.395263 −0.197631 0.980276i \(-0.563325\pi\)
−0.197631 + 0.980276i \(0.563325\pi\)
\(812\) −18.3419 −0.643675
\(813\) 0 0
\(814\) 0.943690 0.0330763
\(815\) 0 0
\(816\) 0 0
\(817\) −2.59637 −0.0908353
\(818\) −10.0351 −0.350869
\(819\) 0 0
\(820\) 0 0
\(821\) 31.3167 1.09296 0.546480 0.837472i \(-0.315967\pi\)
0.546480 + 0.837472i \(0.315967\pi\)
\(822\) 0 0
\(823\) 13.4800 0.469882 0.234941 0.972010i \(-0.424510\pi\)
0.234941 + 0.972010i \(0.424510\pi\)
\(824\) −5.98298 −0.208427
\(825\) 0 0
\(826\) −19.6741 −0.684549
\(827\) 15.9882 0.555963 0.277982 0.960586i \(-0.410335\pi\)
0.277982 + 0.960586i \(0.410335\pi\)
\(828\) 0 0
\(829\) −48.4837 −1.68391 −0.841955 0.539548i \(-0.818595\pi\)
−0.841955 + 0.539548i \(0.818595\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 20.9604 0.726670
\(833\) 0.819960 0.0284099
\(834\) 0 0
\(835\) 0 0
\(836\) −0.872594 −0.0301793
\(837\) 0 0
\(838\) −23.3501 −0.806614
\(839\) 18.9934 0.655724 0.327862 0.944726i \(-0.393672\pi\)
0.327862 + 0.944726i \(0.393672\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −24.8946 −0.857925
\(843\) 0 0
\(844\) −13.2193 −0.455026
\(845\) 0 0
\(846\) 0 0
\(847\) 27.1340 0.932334
\(848\) 0.0416562 0.00143048
\(849\) 0 0
\(850\) 0 0
\(851\) 0.801497 0.0274750
\(852\) 0 0
\(853\) 44.1210 1.51067 0.755337 0.655337i \(-0.227473\pi\)
0.755337 + 0.655337i \(0.227473\pi\)
\(854\) −20.2000 −0.691230
\(855\) 0 0
\(856\) −16.4378 −0.561833
\(857\) −32.4149 −1.10727 −0.553635 0.832759i \(-0.686760\pi\)
−0.553635 + 0.832759i \(0.686760\pi\)
\(858\) 0 0
\(859\) 6.29444 0.214763 0.107382 0.994218i \(-0.465753\pi\)
0.107382 + 0.994218i \(0.465753\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.58283 0.0539113
\(863\) 17.4338 0.593453 0.296726 0.954963i \(-0.404105\pi\)
0.296726 + 0.954963i \(0.404105\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 16.7978 0.570811
\(867\) 0 0
\(868\) 17.8438 0.605657
\(869\) −1.36146 −0.0461843
\(870\) 0 0
\(871\) 56.2497 1.90595
\(872\) 4.74109 0.160554
\(873\) 0 0
\(874\) 0.517774 0.0175140
\(875\) 0 0
\(876\) 0 0
\(877\) 23.2904 0.786462 0.393231 0.919440i \(-0.371357\pi\)
0.393231 + 0.919440i \(0.371357\pi\)
\(878\) −26.7086 −0.901370
\(879\) 0 0
\(880\) 0 0
\(881\) −28.9619 −0.975751 −0.487875 0.872913i \(-0.662228\pi\)
−0.487875 + 0.872913i \(0.662228\pi\)
\(882\) 0 0
\(883\) −37.3627 −1.25735 −0.628677 0.777667i \(-0.716403\pi\)
−0.628677 + 0.777667i \(0.716403\pi\)
\(884\) 14.1289 0.475207
\(885\) 0 0
\(886\) 10.6467 0.357683
\(887\) 23.0234 0.773050 0.386525 0.922279i \(-0.373675\pi\)
0.386525 + 0.922279i \(0.373675\pi\)
\(888\) 0 0
\(889\) −37.4482 −1.25597
\(890\) 0 0
\(891\) 0 0
\(892\) 4.75420 0.159183
\(893\) −5.08247 −0.170078
\(894\) 0 0
\(895\) 0 0
\(896\) −15.6822 −0.523907
\(897\) 0 0
\(898\) −6.40723 −0.213812
\(899\) −35.0223 −1.16806
\(900\) 0 0
\(901\) 0.509626 0.0169781
\(902\) 2.57913 0.0858756
\(903\) 0 0
\(904\) −11.2104 −0.372852
\(905\) 0 0
\(906\) 0 0
\(907\) −35.2693 −1.17110 −0.585549 0.810637i \(-0.699121\pi\)
−0.585549 + 0.810637i \(0.699121\pi\)
\(908\) −13.2088 −0.438350
\(909\) 0 0
\(910\) 0 0
\(911\) 4.97260 0.164750 0.0823748 0.996601i \(-0.473750\pi\)
0.0823748 + 0.996601i \(0.473750\pi\)
\(912\) 0 0
\(913\) 3.11175 0.102984
\(914\) 31.2904 1.03499
\(915\) 0 0
\(916\) −18.9934 −0.627558
\(917\) 25.8819 0.854697
\(918\) 0 0
\(919\) −48.7096 −1.60678 −0.803391 0.595451i \(-0.796973\pi\)
−0.803391 + 0.595451i \(0.796973\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 7.28429 0.239895
\(923\) −13.8108 −0.454587
\(924\) 0 0
\(925\) 0 0
\(926\) 22.9726 0.754926
\(927\) 0 0
\(928\) 33.1729 1.08895
\(929\) −32.5126 −1.06671 −0.533353 0.845893i \(-0.679068\pi\)
−0.533353 + 0.845893i \(0.679068\pi\)
\(930\) 0 0
\(931\) 0.258887 0.00848468
\(932\) −2.50407 −0.0820235
\(933\) 0 0
\(934\) 24.3756 0.797593
\(935\) 0 0
\(936\) 0 0
\(937\) −0.385560 −0.0125957 −0.00629785 0.999980i \(-0.502005\pi\)
−0.00629785 + 0.999980i \(0.502005\pi\)
\(938\) 34.9606 1.14150
\(939\) 0 0
\(940\) 0 0
\(941\) −45.3482 −1.47831 −0.739154 0.673536i \(-0.764775\pi\)
−0.739154 + 0.673536i \(0.764775\pi\)
\(942\) 0 0
\(943\) 2.19051 0.0713329
\(944\) 2.16295 0.0703982
\(945\) 0 0
\(946\) 1.74519 0.0567409
\(947\) −9.61202 −0.312349 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(948\) 0 0
\(949\) 41.0934 1.33395
\(950\) 0 0
\(951\) 0 0
\(952\) 23.6981 0.768059
\(953\) 59.8421 1.93848 0.969238 0.246126i \(-0.0791576\pi\)
0.969238 + 0.246126i \(0.0791576\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −16.9726 −0.548933
\(957\) 0 0
\(958\) −26.2127 −0.846895
\(959\) 25.1959 0.813619
\(960\) 0 0
\(961\) 3.07110 0.0990676
\(962\) −4.82441 −0.155545
\(963\) 0 0
\(964\) 0.191865 0.00617954
\(965\) 0 0
\(966\) 0 0
\(967\) −0.368324 −0.0118445 −0.00592225 0.999982i \(-0.501885\pi\)
−0.00592225 + 0.999982i \(0.501885\pi\)
\(968\) −30.1171 −0.967999
\(969\) 0 0
\(970\) 0 0
\(971\) 45.8370 1.47098 0.735490 0.677535i \(-0.236952\pi\)
0.735490 + 0.677535i \(0.236952\pi\)
\(972\) 0 0
\(973\) 34.9231 1.11958
\(974\) 16.0800 0.515235
\(975\) 0 0
\(976\) 2.22077 0.0710853
\(977\) −29.0337 −0.928872 −0.464436 0.885607i \(-0.653743\pi\)
−0.464436 + 0.885607i \(0.653743\pi\)
\(978\) 0 0
\(979\) −12.5259 −0.400330
\(980\) 0 0
\(981\) 0 0
\(982\) −31.8930 −1.01775
\(983\) 11.0160 0.351355 0.175677 0.984448i \(-0.443788\pi\)
0.175677 + 0.984448i \(0.443788\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −17.2356 −0.548892
\(987\) 0 0
\(988\) 4.46094 0.141921
\(989\) 1.48223 0.0471320
\(990\) 0 0
\(991\) −25.6822 −0.815823 −0.407912 0.913021i \(-0.633743\pi\)
−0.407912 + 0.913021i \(0.633743\pi\)
\(992\) −32.2719 −1.02463
\(993\) 0 0
\(994\) −8.58374 −0.272260
\(995\) 0 0
\(996\) 0 0
\(997\) 20.3854 0.645611 0.322805 0.946465i \(-0.395374\pi\)
0.322805 + 0.946465i \(0.395374\pi\)
\(998\) −19.4551 −0.615842
\(999\) 0 0
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 4275.2.a.br.1.3 6
3.2 odd 2 475.2.a.j.1.4 6
5.2 odd 4 855.2.c.d.514.3 6
5.3 odd 4 855.2.c.d.514.4 6
5.4 even 2 inner 4275.2.a.br.1.4 6
12.11 even 2 7600.2.a.ck.1.6 6
15.2 even 4 95.2.b.b.39.4 yes 6
15.8 even 4 95.2.b.b.39.3 6
15.14 odd 2 475.2.a.j.1.3 6
57.56 even 2 9025.2.a.bx.1.3 6
60.23 odd 4 1520.2.d.h.609.6 6
60.47 odd 4 1520.2.d.h.609.1 6
60.59 even 2 7600.2.a.ck.1.1 6
285.113 odd 4 1805.2.b.e.1084.4 6
285.227 odd 4 1805.2.b.e.1084.3 6
285.284 even 2 9025.2.a.bx.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.b.39.3 6 15.8 even 4
95.2.b.b.39.4 yes 6 15.2 even 4
475.2.a.j.1.3 6 15.14 odd 2
475.2.a.j.1.4 6 3.2 odd 2
855.2.c.d.514.3 6 5.2 odd 4
855.2.c.d.514.4 6 5.3 odd 4
1520.2.d.h.609.1 6 60.47 odd 4
1520.2.d.h.609.6 6 60.23 odd 4
1805.2.b.e.1084.3 6 285.227 odd 4
1805.2.b.e.1084.4 6 285.113 odd 4
4275.2.a.br.1.3 6 1.1 even 1 trivial
4275.2.a.br.1.4 6 5.4 even 2 inner
7600.2.a.ck.1.1 6 60.59 even 2
7600.2.a.ck.1.6 6 12.11 even 2
9025.2.a.bx.1.3 6 57.56 even 2
9025.2.a.bx.1.4 6 285.284 even 2