Properties

Label 475.2.a.j.1.3
Level $475$
Weight $2$
Character 475.1
Self dual yes
Analytic conductor $3.793$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 475.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.906968 q^{2} +3.21789 q^{3} -1.17741 q^{4} -2.91852 q^{6} +2.59637 q^{7} +2.88181 q^{8} +7.35482 q^{9} +O(q^{10})\) \(q-0.906968 q^{2} +3.21789 q^{3} -1.17741 q^{4} -2.91852 q^{6} +2.59637 q^{7} +2.88181 q^{8} +7.35482 q^{9} +0.741113 q^{11} -3.78878 q^{12} -3.78878 q^{13} -2.35482 q^{14} -0.258887 q^{16} -3.16725 q^{17} -6.67058 q^{18} -1.00000 q^{19} +8.35482 q^{21} -0.672165 q^{22} +0.570885 q^{23} +9.27334 q^{24} +3.43630 q^{26} +14.0133 q^{27} -3.05699 q^{28} +6.00000 q^{29} +5.83705 q^{31} -5.52881 q^{32} +2.38482 q^{33} +2.87259 q^{34} -8.65964 q^{36} -1.40396 q^{37} +0.906968 q^{38} -12.1919 q^{39} -3.83705 q^{41} -7.57755 q^{42} -2.59637 q^{43} -0.872594 q^{44} -0.517774 q^{46} +5.08247 q^{47} -0.833070 q^{48} -0.258887 q^{49} -10.1919 q^{51} +4.46094 q^{52} -0.160905 q^{53} -12.7096 q^{54} +7.48223 q^{56} -3.21789 q^{57} -5.44181 q^{58} +8.35482 q^{59} -8.57816 q^{61} -5.29401 q^{62} +19.0958 q^{63} +5.53223 q^{64} -2.16295 q^{66} -14.8464 q^{67} +3.72915 q^{68} +1.83705 q^{69} +3.64518 q^{71} +21.1952 q^{72} -10.8461 q^{73} +1.27334 q^{74} +1.17741 q^{76} +1.92420 q^{77} +11.0576 q^{78} +1.83705 q^{79} +23.0289 q^{81} +3.48008 q^{82} -4.19876 q^{83} -9.83705 q^{84} +2.35482 q^{86} +19.3073 q^{87} +2.13574 q^{88} -16.9015 q^{89} -9.83705 q^{91} -0.672165 q^{92} +18.7830 q^{93} -4.60963 q^{94} -17.7911 q^{96} -3.78878 q^{97} +0.234802 q^{98} +5.45075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51} - 16 q^{54} + 40 q^{56} + 20 q^{59} - 14 q^{61} - 12 q^{64} - 48 q^{66} - 24 q^{69} + 52 q^{71} - 40 q^{74} - 8 q^{76} - 24 q^{79} + 38 q^{81} - 24 q^{84} - 16 q^{86} + 24 q^{89} - 24 q^{91} - 48 q^{94} - 64 q^{96} - 30 q^{99}+O(q^{100}) \) 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 \(0.603905\pi\)
\(3\) 3.21789 1.85785 0.928925 0.370268i \(-0.120734\pi\)
0.928925 + 0.370268i \(0.120734\pi\)
\(4\) −1.17741 −0.588705
\(5\) 0 0
\(6\) −2.91852 −1.19148
\(7\) 2.59637 0.981334 0.490667 0.871347i \(-0.336753\pi\)
0.490667 + 0.871347i \(0.336753\pi\)
\(8\) 2.88181 1.01887
\(9\) 7.35482 2.45161
\(10\) 0 0
\(11\) 0.741113 0.223454 0.111727 0.993739i \(-0.464362\pi\)
0.111727 + 0.993739i \(0.464362\pi\)
\(12\) −3.78878 −1.09373
\(13\) −3.78878 −1.05082 −0.525409 0.850850i \(-0.676088\pi\)
−0.525409 + 0.850850i \(0.676088\pi\)
\(14\) −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 \(0.625483\pi\)
\(18\) −6.67058 −1.57227
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 8.35482 1.82317
\(22\) −0.672165 −0.143306
\(23\) 0.570885 0.119038 0.0595189 0.998227i \(-0.481043\pi\)
0.0595189 + 0.998227i \(0.481043\pi\)
\(24\) 9.27334 1.89291
\(25\) 0 0
\(26\) 3.43630 0.673913
\(27\) 14.0133 2.69687
\(28\) −3.05699 −0.577716
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 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
\(33\) 2.38482 0.415144
\(34\) 2.87259 0.492646
\(35\) 0 0
\(36\) −8.65964 −1.44327
\(37\) −1.40396 −0.230809 −0.115404 0.993319i \(-0.536816\pi\)
−0.115404 + 0.993319i \(0.536816\pi\)
\(38\) 0.906968 0.147130
\(39\) −12.1919 −1.95226
\(40\) 0 0
\(41\) −3.83705 −0.599246 −0.299623 0.954058i \(-0.596861\pi\)
−0.299623 + 0.954058i \(0.596861\pi\)
\(42\) −7.57755 −1.16924
\(43\) −2.59637 −0.395942 −0.197971 0.980208i \(-0.563435\pi\)
−0.197971 + 0.980208i \(0.563435\pi\)
\(44\) −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 \(0.379126\pi\)
\(48\) −0.833070 −0.120243
\(49\) −0.258887 −0.0369839
\(50\) 0 0
\(51\) −10.1919 −1.42715
\(52\) 4.46094 0.618621
\(53\) −0.160905 −0.0221020 −0.0110510 0.999939i \(-0.503518\pi\)
−0.0110510 + 0.999939i \(0.503518\pi\)
\(54\) −12.7096 −1.72956
\(55\) 0 0
\(56\) 7.48223 0.999854
\(57\) −3.21789 −0.426220
\(58\) −5.44181 −0.714544
\(59\) 8.35482 1.08770 0.543852 0.839181i \(-0.316965\pi\)
0.543852 + 0.839181i \(0.316965\pi\)
\(60\) 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
\(63\) 19.0958 2.40584
\(64\) 5.53223 0.691529
\(65\) 0 0
\(66\) −2.16295 −0.266241
\(67\) −14.8464 −1.81378 −0.906888 0.421371i \(-0.861549\pi\)
−0.906888 + 0.421371i \(0.861549\pi\)
\(68\) 3.72915 0.452226
\(69\) 1.83705 0.221154
\(70\) 0 0
\(71\) 3.64518 0.432603 0.216302 0.976327i \(-0.430601\pi\)
0.216302 + 0.976327i \(0.430601\pi\)
\(72\) 21.1952 2.49788
\(73\) −10.8461 −1.26944 −0.634719 0.772743i \(-0.718884\pi\)
−0.634719 + 0.772743i \(0.718884\pi\)
\(74\) 1.27334 0.148023
\(75\) 0 0
\(76\) 1.17741 0.135058
\(77\) 1.92420 0.219283
\(78\) 11.0576 1.25203
\(79\) 1.83705 0.206684 0.103342 0.994646i \(-0.467046\pi\)
0.103342 + 0.994646i \(0.467046\pi\)
\(80\) 0 0
\(81\) 23.0289 2.55877
\(82\) 3.48008 0.384310
\(83\) −4.19876 −0.460873 −0.230437 0.973087i \(-0.574015\pi\)
−0.230437 + 0.973087i \(0.574015\pi\)
\(84\) −9.83705 −1.07331
\(85\) 0 0
\(86\) 2.35482 0.253927
\(87\) 19.3073 2.06996
\(88\) 2.13574 0.227671
\(89\) −16.9015 −1.79156 −0.895778 0.444502i \(-0.853381\pi\)
−0.895778 + 0.444502i \(0.853381\pi\)
\(90\) 0 0
\(91\) −9.83705 −1.03120
\(92\) −0.672165 −0.0700781
\(93\) 18.7830 1.94770
\(94\) −4.60963 −0.475447
\(95\) 0 0
\(96\) −17.7911 −1.81580
\(97\) −3.78878 −0.384692 −0.192346 0.981327i \(-0.561610\pi\)
−0.192346 + 0.981327i \(0.561610\pi\)
\(98\) 0.234802 0.0237186
\(99\) 5.45075 0.547821
\(100\) 0 0
\(101\) 8.35482 0.831336 0.415668 0.909517i \(-0.363548\pi\)
0.415668 + 0.909517i \(0.363548\pi\)
\(102\) 9.24369 0.915262
\(103\) 2.07612 0.204566 0.102283 0.994755i \(-0.467385\pi\)
0.102283 + 0.994755i \(0.467385\pi\)
\(104\) −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 \(0.588914\pi\)
\(108\) −16.4994 −1.58766
\(109\) 1.64518 0.157580 0.0787899 0.996891i \(-0.474894\pi\)
0.0787899 + 0.996891i \(0.474894\pi\)
\(110\) 0 0
\(111\) −4.51777 −0.428808
\(112\) −0.672165 −0.0635137
\(113\) −3.89006 −0.365946 −0.182973 0.983118i \(-0.558572\pi\)
−0.182973 + 0.983118i \(0.558572\pi\)
\(114\) 2.91852 0.273345
\(115\) 0 0
\(116\) −7.06446 −0.655918
\(117\) −27.8658 −2.57619
\(118\) −7.57755 −0.697570
\(119\) −8.22334 −0.753832
\(120\) 0 0
\(121\) −10.4508 −0.950068
\(122\) 7.78011 0.704378
\(123\) −12.3472 −1.11331
\(124\) −6.87259 −0.617177
\(125\) 0 0
\(126\) −17.3193 −1.54292
\(127\) −14.4233 −1.27986 −0.639931 0.768432i \(-0.721037\pi\)
−0.639931 + 0.768432i \(0.721037\pi\)
\(128\) 6.04007 0.533872
\(129\) −8.35482 −0.735601
\(130\) 0 0
\(131\) 9.96853 0.870954 0.435477 0.900200i \(-0.356580\pi\)
0.435477 + 0.900200i \(0.356580\pi\)
\(132\) −2.80791 −0.244397
\(133\) −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 \(0.636060\pi\)
\(138\) −1.66614 −0.141831
\(139\) −13.4508 −1.14088 −0.570439 0.821340i \(-0.693227\pi\)
−0.570439 + 0.821340i \(0.693227\pi\)
\(140\) 0 0
\(141\) 16.3548 1.37732
\(142\) −3.30606 −0.277438
\(143\) −2.80791 −0.234809
\(144\) −1.90407 −0.158672
\(145\) 0 0
\(146\) 9.83705 0.814120
\(147\) −0.833070 −0.0687105
\(148\) 1.65303 0.135878
\(149\) 15.0959 1.23671 0.618353 0.785900i \(-0.287800\pi\)
0.618353 + 0.785900i \(0.287800\pi\)
\(150\) 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
\(153\) −23.2946 −1.88325
\(154\) −1.74519 −0.140631
\(155\) 0 0
\(156\) 14.3548 1.14931
\(157\) −7.57755 −0.604754 −0.302377 0.953188i \(-0.597780\pi\)
−0.302377 + 0.953188i \(0.597780\pi\)
\(158\) −1.66614 −0.132551
\(159\) −0.517774 −0.0410622
\(160\) 0 0
\(161\) 1.48223 0.116816
\(162\) −20.8865 −1.64100
\(163\) 19.6757 1.54112 0.770559 0.637369i \(-0.219977\pi\)
0.770559 + 0.637369i \(0.219977\pi\)
\(164\) 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 \(0.362840\pi\)
\(168\) 24.0770 1.85758
\(169\) 1.35482 0.104217
\(170\) 0 0
\(171\) −7.35482 −0.562437
\(172\) 3.05699 0.233093
\(173\) −20.3895 −1.55018 −0.775092 0.631848i \(-0.782297\pi\)
−0.775092 + 0.631848i \(0.782297\pi\)
\(174\) −17.5111 −1.32752
\(175\) 0 0
\(176\) −0.191865 −0.0144623
\(177\) 26.8849 2.02079
\(178\) 15.3291 1.14897
\(179\) 25.0645 1.87341 0.936703 0.350126i \(-0.113861\pi\)
0.936703 + 0.350126i \(0.113861\pi\)
\(180\) 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
\(183\) −27.6036 −2.04051
\(184\) 1.64518 0.121284
\(185\) 0 0
\(186\) −17.0355 −1.24911
\(187\) −2.34729 −0.171651
\(188\) −5.98414 −0.436439
\(189\) 36.3837 2.64653
\(190\) 0 0
\(191\) 11.4508 0.828547 0.414274 0.910152i \(-0.364036\pi\)
0.414274 + 0.910152i \(0.364036\pi\)
\(192\) 17.8021 1.28476
\(193\) −3.78878 −0.272722 −0.136361 0.990659i \(-0.543541\pi\)
−0.136361 + 0.990659i \(0.543541\pi\)
\(194\) 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 \(0.525922\pi\)
\(198\) −4.94366 −0.351330
\(199\) −19.4508 −1.37883 −0.689414 0.724368i \(-0.742132\pi\)
−0.689414 + 0.724368i \(0.742132\pi\)
\(200\) 0 0
\(201\) −47.7741 −3.36972
\(202\) −7.57755 −0.533155
\(203\) 15.5782 1.09337
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) −1.88297 −0.131193
\(207\) 4.19876 0.291834
\(208\) 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
\(213\) 11.7298 0.803712
\(214\) 5.17334 0.353642
\(215\) 0 0
\(216\) 40.3837 2.74776
\(217\) 15.1551 1.02880
\(218\) −1.49213 −0.101059
\(219\) −34.9015 −2.35843
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 4.09748 0.275005
\(223\) 4.03785 0.270394 0.135197 0.990819i \(-0.456833\pi\)
0.135197 + 0.990819i \(0.456833\pi\)
\(224\) −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 \(0.378569\pi\)
\(228\) 3.78878 0.250918
\(229\) 16.1315 1.06600 0.532999 0.846116i \(-0.321065\pi\)
0.532999 + 0.846116i \(0.321065\pi\)
\(230\) 0 0
\(231\) 6.19186 0.407395
\(232\) 17.2908 1.13520
\(233\) 2.12676 0.139329 0.0696644 0.997570i \(-0.477807\pi\)
0.0696644 + 0.997570i \(0.477807\pi\)
\(234\) 25.2733 1.65217
\(235\) 0 0
\(236\) −9.83705 −0.640337
\(237\) 5.91141 0.383987
\(238\) 7.45830 0.483450
\(239\) −14.4152 −0.932442 −0.466221 0.884668i \(-0.654385\pi\)
−0.466221 + 0.884668i \(0.654385\pi\)
\(240\) 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
\(243\) 32.0645 2.05694
\(244\) 10.1000 0.646587
\(245\) 0 0
\(246\) 11.1985 0.713990
\(247\) 3.78878 0.241074
\(248\) 16.8212 1.06815
\(249\) −13.5111 −0.856233
\(250\) 0 0
\(251\) 12.9330 0.816322 0.408161 0.912910i \(-0.366170\pi\)
0.408161 + 0.912910i \(0.366170\pi\)
\(252\) −22.4836 −1.41633
\(253\) 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 \(0.388059\pi\)
\(258\) 7.57755 0.471758
\(259\) −3.64518 −0.226501
\(260\) 0 0
\(261\) 44.1289 2.73151
\(262\) −9.04113 −0.558563
\(263\) −17.8527 −1.10085 −0.550424 0.834885i \(-0.685534\pi\)
−0.550424 + 0.834885i \(0.685534\pi\)
\(264\) 6.87259 0.422979
\(265\) 0 0
\(266\) 2.35482 0.144383
\(267\) −54.3872 −3.32844
\(268\) 17.4803 1.06778
\(269\) 24.9934 1.52387 0.761936 0.647652i \(-0.224249\pi\)
0.761936 + 0.647652i \(0.224249\pi\)
\(270\) 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
\(273\) −31.6545 −1.91582
\(274\) 8.80150 0.531718
\(275\) 0 0
\(276\) −2.16295 −0.130195
\(277\) 21.2315 1.27568 0.637840 0.770169i \(-0.279828\pi\)
0.637840 + 0.770169i \(0.279828\pi\)
\(278\) 12.1994 0.731671
\(279\) 42.9304 2.57018
\(280\) 0 0
\(281\) 3.83705 0.228899 0.114449 0.993429i \(-0.463490\pi\)
0.114449 + 0.993429i \(0.463490\pi\)
\(282\) −14.8333 −0.883310
\(283\) −0.211545 −0.0125751 −0.00628753 0.999980i \(-0.502001\pi\)
−0.00628753 + 0.999980i \(0.502001\pi\)
\(284\) −4.29187 −0.254676
\(285\) 0 0
\(286\) 2.54668 0.150589
\(287\) −9.96237 −0.588060
\(288\) −40.6634 −2.39612
\(289\) −6.96853 −0.409913
\(290\) 0 0
\(291\) −12.1919 −0.714700
\(292\) 12.7703 0.747324
\(293\) 14.9942 0.875970 0.437985 0.898982i \(-0.355692\pi\)
0.437985 + 0.898982i \(0.355692\pi\)
\(294\) 0.755568 0.0440656
\(295\) 0 0
\(296\) −4.04593 −0.235165
\(297\) 10.3855 0.602626
\(298\) −13.6915 −0.793129
\(299\) −2.16295 −0.125087
\(300\) 0 0
\(301\) −6.74111 −0.388551
\(302\) −12.8716 −0.740675
\(303\) 26.8849 1.54450
\(304\) 0.258887 0.0148482
\(305\) 0 0
\(306\) 21.1274 1.20777
\(307\) −1.65303 −0.0943434 −0.0471717 0.998887i \(-0.515021\pi\)
−0.0471717 + 0.998887i \(0.515021\pi\)
\(308\) −2.26557 −0.129093
\(309\) 6.68073 0.380053
\(310\) 0 0
\(311\) −0.741113 −0.0420247 −0.0210123 0.999779i \(-0.506689\pi\)
−0.0210123 + 0.999779i \(0.506689\pi\)
\(312\) −35.1346 −1.98911
\(313\) 26.8849 1.51962 0.759812 0.650143i \(-0.225291\pi\)
0.759812 + 0.650143i \(0.225291\pi\)
\(314\) 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 \(0.426389\pi\)
\(318\) 0.469604 0.0263341
\(319\) 4.44668 0.248966
\(320\) 0 0
\(321\) −18.3548 −1.02447
\(322\) −1.34433 −0.0749166
\(323\) 3.16725 0.176231
\(324\) −27.1145 −1.50636
\(325\) 0 0
\(326\) −17.8452 −0.988354
\(327\) 5.29401 0.292759
\(328\) −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
\(333\) −10.3258 −0.565852
\(334\) −9.79112 −0.535746
\(335\) 0 0
\(336\) −2.16295 −0.117999
\(337\) 9.90275 0.539437 0.269718 0.962939i \(-0.413069\pi\)
0.269718 + 0.962939i \(0.413069\pi\)
\(338\) −1.22878 −0.0668367
\(339\) −12.5178 −0.679872
\(340\) 0 0
\(341\) 4.32591 0.234261
\(342\) 6.67058 0.360704
\(343\) −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 \(0.306504\pi\)
\(348\) −22.7327 −1.21860
\(349\) −16.4152 −0.878686 −0.439343 0.898319i \(-0.644789\pi\)
−0.439343 + 0.898319i \(0.644789\pi\)
\(350\) 0 0
\(351\) −53.0934 −2.83391
\(352\) −4.09748 −0.218396
\(353\) −23.8744 −1.27071 −0.635354 0.772221i \(-0.719146\pi\)
−0.635354 + 0.772221i \(0.719146\pi\)
\(354\) −24.3837 −1.29598
\(355\) 0 0
\(356\) 19.9000 1.05470
\(357\) −26.4618 −1.40051
\(358\) −22.7327 −1.20146
\(359\) 2.22334 0.117343 0.0586717 0.998277i \(-0.481313\pi\)
0.0586717 + 0.998277i \(0.481313\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 17.6127 0.925701
\(363\) −33.6294 −1.76508
\(364\) 11.5822 0.607074
\(365\) 0 0
\(366\) 25.0355 1.30863
\(367\) 4.52057 0.235972 0.117986 0.993015i \(-0.462356\pi\)
0.117986 + 0.993015i \(0.462356\pi\)
\(368\) −0.147795 −0.00770433
\(369\) −28.2208 −1.46911
\(370\) 0 0
\(371\) −0.417768 −0.0216894
\(372\) −22.1153 −1.14662
\(373\) −15.5186 −0.803521 −0.401760 0.915745i \(-0.631602\pi\)
−0.401760 + 0.915745i \(0.631602\pi\)
\(374\) 2.12892 0.110084
\(375\) 0 0
\(376\) 14.6467 0.755345
\(377\) −22.7327 −1.17079
\(378\) −32.9989 −1.69728
\(379\) −18.9015 −0.970905 −0.485453 0.874263i \(-0.661345\pi\)
−0.485453 + 0.874263i \(0.661345\pi\)
\(380\) 0 0
\(381\) −46.4126 −2.37779
\(382\) −10.3855 −0.531366
\(383\) 13.7046 0.700274 0.350137 0.936699i \(-0.386135\pi\)
0.350137 + 0.936699i \(0.386135\pi\)
\(384\) 19.4363 0.991854
\(385\) 0 0
\(386\) 3.43630 0.174903
\(387\) −19.0958 −0.970694
\(388\) 4.46094 0.226470
\(389\) 12.7411 0.646000 0.323000 0.946399i \(-0.395309\pi\)
0.323000 + 0.946399i \(0.395309\pi\)
\(390\) 0 0
\(391\) −1.80814 −0.0914413
\(392\) −0.746063 −0.0376819
\(393\) 32.0776 1.61810
\(394\) 2.07110 0.104340
\(395\) 0 0
\(396\) −6.41777 −0.322505
\(397\) 38.6522 1.93990 0.969950 0.243306i \(-0.0782319\pi\)
0.969950 + 0.243306i \(0.0782319\pi\)
\(398\) 17.6412 0.884274
\(399\) −8.35482 −0.418264
\(400\) 0 0
\(401\) 31.8660 1.59131 0.795655 0.605750i \(-0.207127\pi\)
0.795655 + 0.605750i \(0.207127\pi\)
\(402\) 43.3296 2.16108
\(403\) −22.1153 −1.10164
\(404\) −9.83705 −0.489411
\(405\) 0 0
\(406\) −14.1289 −0.701206
\(407\) −1.04049 −0.0515751
\(408\) −29.3710 −1.45408
\(409\) 11.0645 0.547102 0.273551 0.961857i \(-0.411802\pi\)
0.273551 + 0.961857i \(0.411802\pi\)
\(410\) 0 0
\(411\) −31.2274 −1.54033
\(412\) −2.44444 −0.120429
\(413\) 21.6922 1.06740
\(414\) −3.80814 −0.187160
\(415\) 0 0
\(416\) 20.9474 1.02703
\(417\) −43.2830 −2.11958
\(418\) 0.672165 0.0328767
\(419\) −25.7452 −1.25773 −0.628867 0.777513i \(-0.716481\pi\)
−0.628867 + 0.777513i \(0.716481\pi\)
\(420\) 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
\(423\) 37.3806 1.81751
\(424\) −0.463697 −0.0225191
\(425\) 0 0
\(426\) −10.6385 −0.515439
\(427\) −22.2720 −1.07782
\(428\) 6.71593 0.324627
\(429\) −9.03555 −0.436240
\(430\) 0 0
\(431\) 1.74519 0.0840627 0.0420314 0.999116i \(-0.486617\pi\)
0.0420314 + 0.999116i \(0.486617\pi\)
\(432\) −3.62787 −0.174546
\(433\) 18.5208 0.890052 0.445026 0.895518i \(-0.353194\pi\)
0.445026 + 0.895518i \(0.353194\pi\)
\(434\) −13.7452 −0.659790
\(435\) 0 0
\(436\) −1.93705 −0.0927679
\(437\) −0.570885 −0.0273091
\(438\) 31.6545 1.51251
\(439\) 29.4482 1.40549 0.702743 0.711444i \(-0.251959\pi\)
0.702743 + 0.711444i \(0.251959\pi\)
\(440\) 0 0
\(441\) −1.90407 −0.0906699
\(442\) −10.8836 −0.517681
\(443\) −11.7388 −0.557726 −0.278863 0.960331i \(-0.589958\pi\)
−0.278863 + 0.960331i \(0.589958\pi\)
\(444\) 5.31927 0.252441
\(445\) 0 0
\(446\) −3.66220 −0.173410
\(447\) 48.5771 2.29762
\(448\) 14.3637 0.678620
\(449\) −7.06446 −0.333392 −0.166696 0.986008i \(-0.553310\pi\)
−0.166696 + 0.986008i \(0.553310\pi\)
\(450\) 0 0
\(451\) −2.84368 −0.133904
\(452\) 4.58019 0.215434
\(453\) 45.6679 2.14566
\(454\) −10.1748 −0.477529
\(455\) 0 0
\(456\) −9.27334 −0.434264
\(457\) 34.5000 1.61384 0.806920 0.590660i \(-0.201133\pi\)
0.806920 + 0.590660i \(0.201133\pi\)
\(458\) −14.6307 −0.683649
\(459\) −44.3837 −2.07166
\(460\) 0 0
\(461\) 8.03147 0.374063 0.187032 0.982354i \(-0.440113\pi\)
0.187032 + 0.982354i \(0.440113\pi\)
\(462\) −5.61582 −0.261272
\(463\) 25.3290 1.17714 0.588570 0.808447i \(-0.299691\pi\)
0.588570 + 0.808447i \(0.299691\pi\)
\(464\) −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 \(0.713612\pi\)
\(468\) 32.8094 1.51662
\(469\) −38.5467 −1.77992
\(470\) 0 0
\(471\) −24.3837 −1.12354
\(472\) 24.0770 1.10823
\(473\) −1.92420 −0.0884748
\(474\) −5.36146 −0.246260
\(475\) 0 0
\(476\) 9.68224 0.443785
\(477\) −1.18343 −0.0541854
\(478\) 13.0741 0.597996
\(479\) −28.9015 −1.32054 −0.660272 0.751027i \(-0.729559\pi\)
−0.660272 + 0.751027i \(0.729559\pi\)
\(480\) 0 0
\(481\) 5.31927 0.242538
\(482\) 0.147795 0.00673187
\(483\) 4.76964 0.217026
\(484\) 12.3048 0.559310
\(485\) 0 0
\(486\) −29.0815 −1.31916
\(487\) 17.7294 0.803395 0.401697 0.915773i \(-0.368420\pi\)
0.401697 + 0.915773i \(0.368420\pi\)
\(488\) −24.7206 −1.11905
\(489\) 63.3141 2.86316
\(490\) 0 0
\(491\) −35.1645 −1.58695 −0.793475 0.608603i \(-0.791730\pi\)
−0.793475 + 0.608603i \(0.791730\pi\)
\(492\) 14.5377 0.655410
\(493\) −19.0035 −0.855875
\(494\) −3.43630 −0.154606
\(495\) 0 0
\(496\) −1.51114 −0.0678520
\(497\) 9.46422 0.424528
\(498\) 12.2542 0.549122
\(499\) 21.4508 0.960268 0.480134 0.877195i \(-0.340588\pi\)
0.480134 + 0.877195i \(0.340588\pi\)
\(500\) 0 0
\(501\) 34.7385 1.55200
\(502\) −11.7298 −0.523526
\(503\) 5.34053 0.238122 0.119061 0.992887i \(-0.462012\pi\)
0.119061 + 0.992887i \(0.462012\pi\)
\(504\) 55.0304 2.45125
\(505\) 0 0
\(506\) −0.383729 −0.0170588
\(507\) 4.35966 0.193619
\(508\) 16.9821 0.753461
\(509\) 36.1919 1.60418 0.802088 0.597206i \(-0.203722\pi\)
0.802088 + 0.597206i \(0.203722\pi\)
\(510\) 0 0
\(511\) −28.1604 −1.24574
\(512\) 2.92346 0.129200
\(513\) −14.0133 −0.618704
\(514\) −10.0170 −0.441832
\(515\) 0 0
\(516\) 9.83705 0.433052
\(517\) 3.76668 0.165658
\(518\) 3.30606 0.145260
\(519\) −65.6111 −2.88001
\(520\) 0 0
\(521\) −2.77259 −0.121469 −0.0607346 0.998154i \(-0.519344\pi\)
−0.0607346 + 0.998154i \(0.519344\pi\)
\(522\) −40.0235 −1.75178
\(523\) 20.5373 0.898033 0.449016 0.893524i \(-0.351774\pi\)
0.449016 + 0.893524i \(0.351774\pi\)
\(524\) −11.7370 −0.512735
\(525\) 0 0
\(526\) 16.1919 0.705999
\(527\) −18.4874 −0.805323
\(528\) −0.617399 −0.0268688
\(529\) −22.6741 −0.985830
\(530\) 0 0
\(531\) 61.4482 2.66662
\(532\) 3.05699 0.132537
\(533\) 14.5377 0.629698
\(534\) 49.3274 2.13461
\(535\) 0 0
\(536\) −42.7845 −1.84801
\(537\) 80.6547 3.48051
\(538\) −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
\(543\) −62.4891 −2.68166
\(544\) 17.5111 0.750784
\(545\) 0 0
\(546\) 28.7096 1.22866
\(547\) −43.0756 −1.84178 −0.920890 0.389822i \(-0.872537\pi\)
−0.920890 + 0.389822i \(0.872537\pi\)
\(548\) 11.4260 0.488092
\(549\) −63.0908 −2.69265
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 5.29401 0.225328
\(553\) 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 \(0.827488\pi\)
\(558\) −38.9365 −1.64831
\(559\) 9.83705 0.416063
\(560\) 0 0
\(561\) −7.55332 −0.318902
\(562\) −3.48008 −0.146798
\(563\) −19.7173 −0.830986 −0.415493 0.909596i \(-0.636391\pi\)
−0.415493 + 0.909596i \(0.636391\pi\)
\(564\) −19.2563 −0.810837
\(565\) 0 0
\(566\) 0.191865 0.00806467
\(567\) 59.7915 2.51101
\(568\) 10.5047 0.440768
\(569\) 18.6807 0.783137 0.391568 0.920149i \(-0.371933\pi\)
0.391568 + 0.920149i \(0.371933\pi\)
\(570\) 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
\(573\) 36.8473 1.53932
\(574\) 9.03555 0.377137
\(575\) 0 0
\(576\) 40.6885 1.69536
\(577\) 0.156779 0.00652679 0.00326339 0.999995i \(-0.498961\pi\)
0.00326339 + 0.999995i \(0.498961\pi\)
\(578\) 6.32023 0.262887
\(579\) −12.1919 −0.506677
\(580\) 0 0
\(581\) −10.9015 −0.452271
\(582\) 11.0576 0.458353
\(583\) −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 \(0.722225\pi\)
\(588\) 0.980865 0.0404502
\(589\) −5.83705 −0.240511
\(590\) 0 0
\(591\) −7.34818 −0.302264
\(592\) 0.363466 0.0149384
\(593\) −28.8728 −1.18567 −0.592833 0.805326i \(-0.701991\pi\)
−0.592833 + 0.805326i \(0.701991\pi\)
\(594\) −9.41928 −0.386478
\(595\) 0 0
\(596\) −17.7741 −0.728055
\(597\) −62.5904 −2.56165
\(598\) 1.96173 0.0802211
\(599\) 25.3274 1.03485 0.517425 0.855728i \(-0.326891\pi\)
0.517425 + 0.855728i \(0.326891\pi\)
\(600\) 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
\(603\) −109.193 −4.44667
\(604\) −16.7096 −0.679906
\(605\) 0 0
\(606\) −24.3837 −0.990521
\(607\) 2.49921 0.101440 0.0507199 0.998713i \(-0.483848\pi\)
0.0507199 + 0.998713i \(0.483848\pi\)
\(608\) 5.52881 0.224223
\(609\) 50.1289 2.03133
\(610\) 0 0
\(611\) −19.2563 −0.779027
\(612\) 27.4272 1.10868
\(613\) 0.883711 0.0356927 0.0178464 0.999841i \(-0.494319\pi\)
0.0178464 + 0.999841i \(0.494319\pi\)
\(614\) 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 \(0.298350\pi\)
\(618\) −6.05921 −0.243737
\(619\) 30.3208 1.21870 0.609348 0.792903i \(-0.291431\pi\)
0.609348 + 0.792903i \(0.291431\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 0.672165 0.0269514
\(623\) −43.8825 −1.75811
\(624\) 3.15632 0.126354
\(625\) 0 0
\(626\) −24.3837 −0.974570
\(627\) −2.38482 −0.0952405
\(628\) 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
\(633\) 36.1286 1.43598
\(634\) −7.40226 −0.293981
\(635\) 0 0
\(636\) 0.609632 0.0241735
\(637\) 0.980865 0.0388633
\(638\) −4.03299 −0.159668
\(639\) 26.8096 1.06057
\(640\) 0 0
\(641\) −32.6675 −1.29029 −0.645143 0.764062i \(-0.723202\pi\)
−0.645143 + 0.764062i \(0.723202\pi\)
\(642\) 16.6472 0.657014
\(643\) −31.8661 −1.25668 −0.628338 0.777941i \(-0.716264\pi\)
−0.628338 + 0.777941i \(0.716264\pi\)
\(644\) −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 \(0.637237\pi\)
\(648\) 66.3649 2.60706
\(649\) 6.19186 0.243052
\(650\) 0 0
\(651\) 48.7675 1.91135
\(652\) −23.1663 −0.907263
\(653\) −12.8340 −0.502234 −0.251117 0.967957i \(-0.580798\pi\)
−0.251117 + 0.967957i \(0.580798\pi\)
\(654\) −4.80150 −0.187753
\(655\) 0 0
\(656\) 0.993361 0.0387842
\(657\) −79.7710 −3.11216
\(658\) −11.9683 −0.466573
\(659\) 20.3548 0.792911 0.396456 0.918054i \(-0.370240\pi\)
0.396456 + 0.918054i \(0.370240\pi\)
\(660\) 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
\(663\) 38.6147 1.49967
\(664\) −12.1000 −0.469571
\(665\) 0 0
\(666\) 9.36520 0.362894
\(667\) 3.42531 0.132629
\(668\) −12.7107 −0.491790
\(669\) 12.9934 0.502352
\(670\) 0 0
\(671\) −6.35738 −0.245424
\(672\) −46.1922 −1.78190
\(673\) −21.2094 −0.817564 −0.408782 0.912632i \(-0.634046\pi\)
−0.408782 + 0.912632i \(0.634046\pi\)
\(674\) −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 \(0.569456\pi\)
\(678\) 11.3532 0.436018
\(679\) −9.83705 −0.377511
\(680\) 0 0
\(681\) 36.1000 1.38336
\(682\) −3.92346 −0.150237
\(683\) 12.3603 0.472954 0.236477 0.971637i \(-0.424007\pi\)
0.236477 + 0.971637i \(0.424007\pi\)
\(684\) 8.65964 0.331109
\(685\) 0 0
\(686\) 17.0934 0.652628
\(687\) 51.9093 1.98046
\(688\) 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
\(693\) 14.1521 0.537595
\(694\) −19.2985 −0.732561
\(695\) 0 0
\(696\) 55.6401 2.10903
\(697\) 12.1529 0.460323
\(698\) 14.8881 0.563521
\(699\) 6.84368 0.258852
\(700\) 0 0
\(701\) −16.0289 −0.605404 −0.302702 0.953085i \(-0.597889\pi\)
−0.302702 + 0.953085i \(0.597889\pi\)
\(702\) 48.1540 1.81745
\(703\) 1.40396 0.0529512
\(704\) 4.10001 0.154525
\(705\) 0 0
\(706\) 21.6533 0.814934
\(707\) 21.6922 0.815818
\(708\) −31.6545 −1.18965
\(709\) 31.4193 1.17998 0.589988 0.807412i \(-0.299133\pi\)
0.589988 + 0.807412i \(0.299133\pi\)
\(710\) 0 0
\(711\) 13.5111 0.506707
\(712\) −48.7069 −1.82537
\(713\) 3.33228 0.124795
\(714\) 24.0000 0.898177
\(715\) 0 0
\(716\) −29.5111 −1.10288
\(717\) −46.3865 −1.73234
\(718\) −2.01650 −0.0752550
\(719\) 11.2589 0.419886 0.209943 0.977714i \(-0.432672\pi\)
0.209943 + 0.977714i \(0.432672\pi\)
\(720\) 0 0
\(721\) 5.39037 0.200748
\(722\) −0.906968 −0.0337538
\(723\) −0.524371 −0.0195016
\(724\) 22.8644 0.849750
\(725\) 0 0
\(726\) 30.5008 1.13199
\(727\) −48.9829 −1.81668 −0.908338 0.418237i \(-0.862648\pi\)
−0.908338 + 0.418237i \(0.862648\pi\)
\(728\) −28.3485 −1.05066
\(729\) 34.0934 1.26272
\(730\) 0 0
\(731\) 8.22334 0.304151
\(732\) 32.5007 1.20126
\(733\) 35.9260 1.32696 0.663479 0.748195i \(-0.269079\pi\)
0.663479 + 0.748195i \(0.269079\pi\)
\(734\) −4.10001 −0.151334
\(735\) 0 0
\(736\) −3.15632 −0.116343
\(737\) −11.0029 −0.405296
\(738\) 25.5953 0.942177
\(739\) 14.3523 0.527956 0.263978 0.964529i \(-0.414965\pi\)
0.263978 + 0.964529i \(0.414965\pi\)
\(740\) 0 0
\(741\) 12.1919 0.447879
\(742\) 0.378902 0.0139099
\(743\) −12.5629 −0.460887 −0.230443 0.973086i \(-0.574018\pi\)
−0.230443 + 0.973086i \(0.574018\pi\)
\(744\) 54.1289 1.98446
\(745\) 0 0
\(746\) 14.0748 0.515316
\(747\) −30.8811 −1.12988
\(748\) 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
\(753\) 41.6169 1.51660
\(754\) 20.6178 0.750855
\(755\) 0 0
\(756\) −42.8386 −1.55802
\(757\) 15.7350 0.571897 0.285949 0.958245i \(-0.407691\pi\)
0.285949 + 0.958245i \(0.407691\pi\)
\(758\) 17.1431 0.622664
\(759\) 1.36146 0.0494178
\(760\) 0 0
\(761\) 16.9619 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(762\) 42.0948 1.52493
\(763\) 4.27149 0.154638
\(764\) −13.4822 −0.487770
\(765\) 0 0
\(766\) −12.4297 −0.449102
\(767\) −31.6545 −1.14298
\(768\) −53.2323 −1.92086
\(769\) 41.9974 1.51447 0.757233 0.653145i \(-0.226551\pi\)
0.757233 + 0.653145i \(0.226551\pi\)
\(770\) 0 0
\(771\) 35.5400 1.27994
\(772\) 4.46094 0.160553
\(773\) 40.9579 1.47315 0.736576 0.676355i \(-0.236441\pi\)
0.736576 + 0.676355i \(0.236441\pi\)
\(774\) 17.3193 0.622528
\(775\) 0 0
\(776\) −10.9185 −0.391952
\(777\) −11.7298 −0.420804
\(778\) −11.5558 −0.414295
\(779\) 3.83705 0.137476
\(780\) 0 0
\(781\) 2.70149 0.0966669
\(782\) 1.63992 0.0586434
\(783\) 84.0800 3.00477
\(784\) 0.0670225 0.00239366
\(785\) 0 0
\(786\) −29.0934 −1.03773
\(787\) 28.5379 1.01727 0.508634 0.860983i \(-0.330151\pi\)
0.508634 + 0.860983i \(0.330151\pi\)
\(788\) 2.68866 0.0957796
\(789\) −57.4482 −2.04521
\(790\) 0 0
\(791\) −10.1000 −0.359115
\(792\) 15.7080 0.558160
\(793\) 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 \(0.299364\pi\)
\(798\) 7.57755 0.268242
\(799\) −16.0974 −0.569487
\(800\) 0 0
\(801\) −124.308 −4.39219
\(802\) −28.9014 −1.02054
\(803\) −8.03817 −0.283661
\(804\) 56.2497 1.98377
\(805\) 0 0
\(806\) 20.0578 0.706507
\(807\) 80.4259 2.83113
\(808\) 24.0770 0.847025
\(809\) −34.4234 −1.21026 −0.605130 0.796126i \(-0.706879\pi\)
−0.605130 + 0.796126i \(0.706879\pi\)
\(810\) 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
\(813\) −76.7980 −2.69342
\(814\) 0.943690 0.0330763
\(815\) 0 0
\(816\) 2.63854 0.0923674
\(817\) 2.59637 0.0908353
\(818\) −10.0351 −0.350869
\(819\) −72.3497 −2.52810
\(820\) 0 0
\(821\) −31.3167 −1.09296 −0.546480 0.837472i \(-0.684033\pi\)
−0.546480 + 0.837472i \(0.684033\pi\)
\(822\) 28.3223 0.987852
\(823\) −13.4800 −0.469882 −0.234941 0.972010i \(-0.575490\pi\)
−0.234941 + 0.972010i \(0.575490\pi\)
\(824\) 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 \(0.410335\pi\)
\(828\) −4.94366 −0.171804
\(829\) −48.4837 −1.68391 −0.841955 0.539548i \(-0.818595\pi\)
−0.841955 + 0.539548i \(0.818595\pi\)
\(830\) 0 0
\(831\) 68.3208 2.37002
\(832\) −20.9604 −0.726670
\(833\) 0.819960 0.0284099
\(834\) 39.2563 1.35934
\(835\) 0 0
\(836\) 0.872594 0.0301793
\(837\) 81.7965 2.82730
\(838\) 23.3501 0.806614
\(839\) −18.9934 −0.655724 −0.327862 0.944726i \(-0.606328\pi\)
−0.327862 + 0.944726i \(0.606328\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −24.8946 −0.857925
\(843\) 12.3472 0.425260
\(844\) −13.2193 −0.455026
\(845\) 0 0
\(846\) −33.9030 −1.16561
\(847\) −27.1340 −0.932334
\(848\) 0.0416562 0.00143048
\(849\) −0.680729 −0.0233626
\(850\) 0 0
\(851\) −0.801497 −0.0274750
\(852\) −13.8108 −0.473149
\(853\) −44.1210 −1.51067 −0.755337 0.655337i \(-0.772527\pi\)
−0.755337 + 0.655337i \(0.772527\pi\)
\(854\) 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 \(0.686760\pi\)
\(858\) 8.19495 0.279771
\(859\) 6.29444 0.214763 0.107382 0.994218i \(-0.465753\pi\)
0.107382 + 0.994218i \(0.465753\pi\)
\(860\) 0 0
\(861\) −32.0578 −1.09253
\(862\) −1.58283 −0.0539113
\(863\) 17.4338 0.593453 0.296726 0.954963i \(-0.404105\pi\)
0.296726 + 0.954963i \(0.404105\pi\)
\(864\) −77.4771 −2.63582
\(865\) 0 0
\(866\) −16.7978 −0.570811
\(867\) −22.4240 −0.761557
\(868\) −17.8438 −0.605657
\(869\) 1.36146 0.0461843
\(870\) 0 0
\(871\) 56.2497 1.90595
\(872\) 4.74109 0.160554
\(873\) −27.8658 −0.943113
\(874\) 0.517774 0.0175140
\(875\) 0 0
\(876\) 41.0934 1.38842
\(877\) −23.2904 −0.786462 −0.393231 0.919440i \(-0.628643\pi\)
−0.393231 + 0.919440i \(0.628643\pi\)
\(878\) −26.7086 −0.901370
\(879\) 48.2497 1.62742
\(880\) 0 0
\(881\) 28.9619 0.975751 0.487875 0.872913i \(-0.337772\pi\)
0.487875 + 0.872913i \(0.337772\pi\)
\(882\) 1.72693 0.0581487
\(883\) 37.3627 1.25735 0.628677 0.777667i \(-0.283597\pi\)
0.628677 + 0.777667i \(0.283597\pi\)
\(884\) −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 \(0.373675\pi\)
\(888\) −13.0194 −0.436901
\(889\) −37.4482 −1.25597
\(890\) 0 0
\(891\) 17.0670 0.571767
\(892\) −4.75420 −0.159183
\(893\) −5.08247 −0.170078
\(894\) −44.0578 −1.47351
\(895\) 0 0
\(896\) 15.6822 0.523907
\(897\) −6.96015 −0.232393
\(898\) 6.40723 0.213812
\(899\) 35.0223 1.16806
\(900\) 0 0
\(901\) 0.509626 0.0169781
\(902\) 2.57913 0.0858756
\(903\) −21.6922 −0.721870
\(904\) −11.2104 −0.372852
\(905\) 0 0
\(906\) −41.4193 −1.37606
\(907\) 35.2693 1.17110 0.585549 0.810637i \(-0.300879\pi\)
0.585549 + 0.810637i \(0.300879\pi\)
\(908\) −13.2088 −0.438350
\(909\) 61.4482 2.03811
\(910\) 0 0
\(911\) −4.97260 −0.164750 −0.0823748 0.996601i \(-0.526250\pi\)
−0.0823748 + 0.996601i \(0.526250\pi\)
\(912\) 0.833070 0.0275857
\(913\) −3.11175 −0.102984
\(914\) −31.2904 −1.03499
\(915\) 0 0
\(916\) −18.9934 −0.627558
\(917\) 25.8819 0.854697
\(918\) 40.2546 1.32860
\(919\) −48.7096 −1.60678 −0.803391 0.595451i \(-0.796973\pi\)
−0.803391 + 0.595451i \(0.796973\pi\)
\(920\) 0 0
\(921\) −5.31927 −0.175276
\(922\) −7.28429 −0.239895
\(923\) −13.8108 −0.454587
\(924\) −7.29036 −0.239835
\(925\) 0 0
\(926\) −22.9726 −0.754926
\(927\) 15.2695 0.501516
\(928\) −33.1729 −1.08895
\(929\) 32.5126 1.06671 0.533353 0.845893i \(-0.320932\pi\)
0.533353 + 0.845893i \(0.320932\pi\)
\(930\) 0 0
\(931\) 0.258887 0.00848468
\(932\) −2.50407 −0.0820235
\(933\) −2.38482 −0.0780755
\(934\) 24.3756 0.797593
\(935\) 0 0
\(936\) −80.3038 −2.62481
\(937\) 0.385560 0.0125957 0.00629785 0.999980i \(-0.497995\pi\)
0.00629785 + 0.999980i \(0.497995\pi\)
\(938\) 34.9606 1.14150
\(939\) 86.5126 2.82323
\(940\) 0 0
\(941\) 45.3482 1.47831 0.739154 0.673536i \(-0.235225\pi\)
0.739154 + 0.673536i \(0.235225\pi\)
\(942\) 22.1153 0.720554
\(943\) −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 \(0.549916\pi\)
\(948\) −6.96015 −0.226055
\(949\) 41.0934 1.33395
\(950\) 0 0
\(951\) 26.2630 0.851635
\(952\) −23.6981 −0.768059
\(953\) 59.8421 1.93848 0.969238 0.246126i \(-0.0791576\pi\)
0.969238 + 0.246126i \(0.0791576\pi\)
\(954\) 1.07333 0.0347503
\(955\) 0 0
\(956\) 16.9726 0.548933
\(957\) 14.3089 0.462542
\(958\) 26.2127 0.846895
\(959\) −25.1959 −0.813619
\(960\) 0 0
\(961\) 3.07110 0.0990676
\(962\) −4.82441 −0.155545
\(963\) −41.9518 −1.35188
\(964\) 0.191865 0.00617954
\(965\) 0 0
\(966\) −4.32591 −0.139184
\(967\) 0.368324 0.0118445 0.00592225 0.999982i \(-0.498115\pi\)
0.00592225 + 0.999982i \(0.498115\pi\)
\(968\) −30.1171 −0.967999
\(969\) 10.1919 0.327410
\(970\) 0 0
\(971\) −45.8370 −1.47098 −0.735490 0.677535i \(-0.763048\pi\)
−0.735490 + 0.677535i \(0.763048\pi\)
\(972\) −37.7531 −1.21093
\(973\) −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 \(0.653743\pi\)
\(978\) −57.4239 −1.83621
\(979\) −12.5259 −0.400330
\(980\) 0 0
\(981\) 12.1000 0.386323
\(982\) 31.8930 1.01775
\(983\) 11.0160 0.351355 0.175677 0.984448i \(-0.443788\pi\)
0.175677 + 0.984448i \(0.443788\pi\)
\(984\) −35.5822 −1.13432
\(985\) 0 0
\(986\) 17.2356 0.548892
\(987\) 42.4631 1.35161
\(988\) −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
\(993\) 25.7431 0.816933
\(994\) −8.58374 −0.272260
\(995\) 0 0
\(996\) 15.9081 0.504069
\(997\) −20.3854 −0.645611 −0.322805 0.946465i \(-0.604626\pi\)
−0.322805 + 0.946465i \(0.604626\pi\)
\(998\) −19.4551 −0.615842
\(999\) −19.6741 −0.622461
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 475.2.a.j.1.3 6
3.2 odd 2 4275.2.a.br.1.4 6
4.3 odd 2 7600.2.a.ck.1.1 6
5.2 odd 4 95.2.b.b.39.3 6
5.3 odd 4 95.2.b.b.39.4 yes 6
5.4 even 2 inner 475.2.a.j.1.4 6
15.2 even 4 855.2.c.d.514.4 6
15.8 even 4 855.2.c.d.514.3 6
15.14 odd 2 4275.2.a.br.1.3 6
19.18 odd 2 9025.2.a.bx.1.4 6
20.3 even 4 1520.2.d.h.609.1 6
20.7 even 4 1520.2.d.h.609.6 6
20.19 odd 2 7600.2.a.ck.1.6 6
95.18 even 4 1805.2.b.e.1084.3 6
95.37 even 4 1805.2.b.e.1084.4 6
95.94 odd 2 9025.2.a.bx.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.b.39.3 6 5.2 odd 4
95.2.b.b.39.4 yes 6 5.3 odd 4
475.2.a.j.1.3 6 1.1 even 1 trivial
475.2.a.j.1.4 6 5.4 even 2 inner
855.2.c.d.514.3 6 15.8 even 4
855.2.c.d.514.4 6 15.2 even 4
1520.2.d.h.609.1 6 20.3 even 4
1520.2.d.h.609.6 6 20.7 even 4
1805.2.b.e.1084.3 6 95.18 even 4
1805.2.b.e.1084.4 6 95.37 even 4
4275.2.a.br.1.3 6 15.14 odd 2
4275.2.a.br.1.4 6 3.2 odd 2
7600.2.a.ck.1.1 6 4.3 odd 2
7600.2.a.ck.1.6 6 20.19 odd 2
9025.2.a.bx.1.3 6 95.94 odd 2
9025.2.a.bx.1.4 6 19.18 odd 2