Properties

Label 7623.2.a.cp.1.3
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7674048.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 847)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.276564\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.27656 q^{2} -0.370384 q^{4} +4.09144 q^{5} +1.00000 q^{7} +3.02595 q^{8} +O(q^{10})\) \(q-1.27656 q^{2} -0.370384 q^{4} +4.09144 q^{5} +1.00000 q^{7} +3.02595 q^{8} -5.22298 q^{10} -4.39091 q^{13} -1.27656 q^{14} -3.12205 q^{16} -4.19146 q^{17} -1.24880 q^{19} -1.51540 q^{20} -4.97180 q^{23} +11.7399 q^{25} +5.60527 q^{26} -0.370384 q^{28} +1.93542 q^{29} -1.56278 q^{31} -2.06640 q^{32} +5.35067 q^{34} +4.09144 q^{35} -0.716296 q^{37} +1.59417 q^{38} +12.3805 q^{40} -4.80626 q^{41} +1.35362 q^{43} +6.34682 q^{46} +10.4662 q^{47} +1.00000 q^{49} -14.9867 q^{50} +1.62632 q^{52} +3.97180 q^{53} +3.02595 q^{56} -2.47069 q^{58} -13.7588 q^{59} -11.7271 q^{61} +1.99499 q^{62} +8.88199 q^{64} -17.9651 q^{65} +7.59274 q^{67} +1.55245 q^{68} -5.22298 q^{70} -0.218316 q^{71} +10.9714 q^{73} +0.914398 q^{74} +0.462535 q^{76} -4.56248 q^{79} -12.7737 q^{80} +6.13550 q^{82} -2.45458 q^{83} -17.1491 q^{85} -1.72798 q^{86} -4.20456 q^{89} -4.39091 q^{91} +1.84148 q^{92} -13.3608 q^{94} -5.10938 q^{95} -10.9249 q^{97} -1.27656 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 4q^{2} + 4q^{4} + 4q^{5} + 6q^{7} - 12q^{8} + O(q^{10}) \) \( 6q - 4q^{2} + 4q^{4} + 4q^{5} + 6q^{7} - 12q^{8} - 8q^{10} + 4q^{13} - 4q^{14} + 8q^{16} - 22q^{17} + 6q^{19} - 2q^{20} - 2q^{23} + 4q^{25} - 6q^{26} + 4q^{28} - 12q^{29} - 2q^{31} - 8q^{32} + 24q^{34} + 4q^{35} + 14q^{37} + 22q^{38} + 18q^{40} - 26q^{41} - 4q^{43} + 12q^{46} + 16q^{47} + 6q^{49} + 4q^{50} + 12q^{52} - 4q^{53} - 12q^{56} - 2q^{58} + 4q^{59} - 8q^{61} - 20q^{62} + 26q^{64} - 24q^{65} + 6q^{67} - 12q^{68} - 8q^{70} - 22q^{71} + 14q^{73} - 44q^{74} - 30q^{76} - 28q^{79} + 4q^{80} - 4q^{82} - 22q^{83} - 24q^{85} + 30q^{86} + 4q^{91} - 10q^{92} - 38q^{94} + 24q^{95} - 4q^{97} - 4q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.27656 −0.902667 −0.451334 0.892355i \(-0.649052\pi\)
−0.451334 + 0.892355i \(0.649052\pi\)
\(3\) 0 0
\(4\) −0.370384 −0.185192
\(5\) 4.09144 1.82975 0.914873 0.403741i \(-0.132290\pi\)
0.914873 + 0.403741i \(0.132290\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 3.02595 1.06983
\(9\) 0 0
\(10\) −5.22298 −1.65165
\(11\) 0 0
\(12\) 0 0
\(13\) −4.39091 −1.21782 −0.608909 0.793240i \(-0.708393\pi\)
−0.608909 + 0.793240i \(0.708393\pi\)
\(14\) −1.27656 −0.341176
\(15\) 0 0
\(16\) −3.12205 −0.780512
\(17\) −4.19146 −1.01658 −0.508289 0.861186i \(-0.669722\pi\)
−0.508289 + 0.861186i \(0.669722\pi\)
\(18\) 0 0
\(19\) −1.24880 −0.286494 −0.143247 0.989687i \(-0.545754\pi\)
−0.143247 + 0.989687i \(0.545754\pi\)
\(20\) −1.51540 −0.338855
\(21\) 0 0
\(22\) 0 0
\(23\) −4.97180 −1.03669 −0.518346 0.855171i \(-0.673452\pi\)
−0.518346 + 0.855171i \(0.673452\pi\)
\(24\) 0 0
\(25\) 11.7399 2.34797
\(26\) 5.60527 1.09928
\(27\) 0 0
\(28\) −0.370384 −0.0699960
\(29\) 1.93542 0.359399 0.179699 0.983722i \(-0.442488\pi\)
0.179699 + 0.983722i \(0.442488\pi\)
\(30\) 0 0
\(31\) −1.56278 −0.280684 −0.140342 0.990103i \(-0.544820\pi\)
−0.140342 + 0.990103i \(0.544820\pi\)
\(32\) −2.06640 −0.365292
\(33\) 0 0
\(34\) 5.35067 0.917632
\(35\) 4.09144 0.691579
\(36\) 0 0
\(37\) −0.716296 −0.117758 −0.0588792 0.998265i \(-0.518753\pi\)
−0.0588792 + 0.998265i \(0.518753\pi\)
\(38\) 1.59417 0.258609
\(39\) 0 0
\(40\) 12.3805 1.95752
\(41\) −4.80626 −0.750611 −0.375306 0.926901i \(-0.622462\pi\)
−0.375306 + 0.926901i \(0.622462\pi\)
\(42\) 0 0
\(43\) 1.35362 0.206424 0.103212 0.994659i \(-0.467088\pi\)
0.103212 + 0.994659i \(0.467088\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.34682 0.935788
\(47\) 10.4662 1.52665 0.763327 0.646013i \(-0.223565\pi\)
0.763327 + 0.646013i \(0.223565\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −14.9867 −2.11944
\(51\) 0 0
\(52\) 1.62632 0.225530
\(53\) 3.97180 0.545569 0.272784 0.962075i \(-0.412055\pi\)
0.272784 + 0.962075i \(0.412055\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.02595 0.404359
\(57\) 0 0
\(58\) −2.47069 −0.324417
\(59\) −13.7588 −1.79125 −0.895624 0.444811i \(-0.853271\pi\)
−0.895624 + 0.444811i \(0.853271\pi\)
\(60\) 0 0
\(61\) −11.7271 −1.50151 −0.750753 0.660583i \(-0.770309\pi\)
−0.750753 + 0.660583i \(0.770309\pi\)
\(62\) 1.99499 0.253364
\(63\) 0 0
\(64\) 8.88199 1.11025
\(65\) −17.9651 −2.22830
\(66\) 0 0
\(67\) 7.59274 0.927600 0.463800 0.885940i \(-0.346486\pi\)
0.463800 + 0.885940i \(0.346486\pi\)
\(68\) 1.55245 0.188262
\(69\) 0 0
\(70\) −5.22298 −0.624266
\(71\) −0.218316 −0.0259094 −0.0129547 0.999916i \(-0.504124\pi\)
−0.0129547 + 0.999916i \(0.504124\pi\)
\(72\) 0 0
\(73\) 10.9714 1.28411 0.642054 0.766659i \(-0.278082\pi\)
0.642054 + 0.766659i \(0.278082\pi\)
\(74\) 0.914398 0.106297
\(75\) 0 0
\(76\) 0.462535 0.0530564
\(77\) 0 0
\(78\) 0 0
\(79\) −4.56248 −0.513319 −0.256659 0.966502i \(-0.582622\pi\)
−0.256659 + 0.966502i \(0.582622\pi\)
\(80\) −12.7737 −1.42814
\(81\) 0 0
\(82\) 6.13550 0.677552
\(83\) −2.45458 −0.269425 −0.134712 0.990885i \(-0.543011\pi\)
−0.134712 + 0.990885i \(0.543011\pi\)
\(84\) 0 0
\(85\) −17.1491 −1.86008
\(86\) −1.72798 −0.186333
\(87\) 0 0
\(88\) 0 0
\(89\) −4.20456 −0.445683 −0.222841 0.974855i \(-0.571533\pi\)
−0.222841 + 0.974855i \(0.571533\pi\)
\(90\) 0 0
\(91\) −4.39091 −0.460292
\(92\) 1.84148 0.191987
\(93\) 0 0
\(94\) −13.3608 −1.37806
\(95\) −5.10938 −0.524211
\(96\) 0 0
\(97\) −10.9249 −1.10925 −0.554627 0.832099i \(-0.687139\pi\)
−0.554627 + 0.832099i \(0.687139\pi\)
\(98\) −1.27656 −0.128952
\(99\) 0 0
\(100\) −4.34826 −0.434826
\(101\) −7.36579 −0.732924 −0.366462 0.930433i \(-0.619431\pi\)
−0.366462 + 0.930433i \(0.619431\pi\)
\(102\) 0 0
\(103\) −0.153886 −0.0151629 −0.00758143 0.999971i \(-0.502413\pi\)
−0.00758143 + 0.999971i \(0.502413\pi\)
\(104\) −13.2867 −1.30286
\(105\) 0 0
\(106\) −5.07026 −0.492467
\(107\) −7.94617 −0.768185 −0.384092 0.923295i \(-0.625486\pi\)
−0.384092 + 0.923295i \(0.625486\pi\)
\(108\) 0 0
\(109\) −4.91440 −0.470714 −0.235357 0.971909i \(-0.575626\pi\)
−0.235357 + 0.971909i \(0.575626\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.12205 −0.295006
\(113\) −3.04208 −0.286175 −0.143088 0.989710i \(-0.545703\pi\)
−0.143088 + 0.989710i \(0.545703\pi\)
\(114\) 0 0
\(115\) −20.3418 −1.89688
\(116\) −0.716849 −0.0665578
\(117\) 0 0
\(118\) 17.5640 1.61690
\(119\) −4.19146 −0.384231
\(120\) 0 0
\(121\) 0 0
\(122\) 14.9704 1.35536
\(123\) 0 0
\(124\) 0.578830 0.0519804
\(125\) 27.5757 2.46645
\(126\) 0 0
\(127\) −10.7954 −0.957932 −0.478966 0.877833i \(-0.658988\pi\)
−0.478966 + 0.877833i \(0.658988\pi\)
\(128\) −7.20562 −0.636893
\(129\) 0 0
\(130\) 22.9336 2.01141
\(131\) 8.70429 0.760497 0.380249 0.924884i \(-0.375838\pi\)
0.380249 + 0.924884i \(0.375838\pi\)
\(132\) 0 0
\(133\) −1.24880 −0.108285
\(134\) −9.69261 −0.837314
\(135\) 0 0
\(136\) −12.6831 −1.08757
\(137\) −11.9977 −1.02503 −0.512516 0.858677i \(-0.671287\pi\)
−0.512516 + 0.858677i \(0.671287\pi\)
\(138\) 0 0
\(139\) 8.18967 0.694639 0.347319 0.937747i \(-0.387092\pi\)
0.347319 + 0.937747i \(0.387092\pi\)
\(140\) −1.51540 −0.128075
\(141\) 0 0
\(142\) 0.278695 0.0233875
\(143\) 0 0
\(144\) 0 0
\(145\) 7.91865 0.657608
\(146\) −14.0057 −1.15912
\(147\) 0 0
\(148\) 0.265305 0.0218079
\(149\) −12.0816 −0.989766 −0.494883 0.868960i \(-0.664789\pi\)
−0.494883 + 0.868960i \(0.664789\pi\)
\(150\) 0 0
\(151\) −6.94850 −0.565461 −0.282730 0.959199i \(-0.591240\pi\)
−0.282730 + 0.959199i \(0.591240\pi\)
\(152\) −3.77880 −0.306501
\(153\) 0 0
\(154\) 0 0
\(155\) −6.39402 −0.513580
\(156\) 0 0
\(157\) 10.4930 0.837436 0.418718 0.908116i \(-0.362480\pi\)
0.418718 + 0.908116i \(0.362480\pi\)
\(158\) 5.82429 0.463356
\(159\) 0 0
\(160\) −8.45455 −0.668391
\(161\) −4.97180 −0.391833
\(162\) 0 0
\(163\) −7.81743 −0.612309 −0.306154 0.951982i \(-0.599042\pi\)
−0.306154 + 0.951982i \(0.599042\pi\)
\(164\) 1.78016 0.139007
\(165\) 0 0
\(166\) 3.13342 0.243201
\(167\) −21.3503 −1.65214 −0.826068 0.563571i \(-0.809427\pi\)
−0.826068 + 0.563571i \(0.809427\pi\)
\(168\) 0 0
\(169\) 6.28007 0.483082
\(170\) 21.8919 1.67903
\(171\) 0 0
\(172\) −0.501358 −0.0382282
\(173\) −19.4384 −1.47787 −0.738936 0.673776i \(-0.764671\pi\)
−0.738936 + 0.673776i \(0.764671\pi\)
\(174\) 0 0
\(175\) 11.7399 0.887450
\(176\) 0 0
\(177\) 0 0
\(178\) 5.36740 0.402303
\(179\) −17.9964 −1.34511 −0.672557 0.740045i \(-0.734804\pi\)
−0.672557 + 0.740045i \(0.734804\pi\)
\(180\) 0 0
\(181\) 15.1575 1.12665 0.563326 0.826235i \(-0.309522\pi\)
0.563326 + 0.826235i \(0.309522\pi\)
\(182\) 5.60527 0.415491
\(183\) 0 0
\(184\) −15.0444 −1.10909
\(185\) −2.93068 −0.215468
\(186\) 0 0
\(187\) 0 0
\(188\) −3.87652 −0.282724
\(189\) 0 0
\(190\) 6.52245 0.473188
\(191\) 23.7574 1.71903 0.859513 0.511115i \(-0.170767\pi\)
0.859513 + 0.511115i \(0.170767\pi\)
\(192\) 0 0
\(193\) 14.0582 1.01193 0.505965 0.862554i \(-0.331137\pi\)
0.505965 + 0.862554i \(0.331137\pi\)
\(194\) 13.9463 1.00129
\(195\) 0 0
\(196\) −0.370384 −0.0264560
\(197\) −18.0665 −1.28718 −0.643591 0.765369i \(-0.722556\pi\)
−0.643591 + 0.765369i \(0.722556\pi\)
\(198\) 0 0
\(199\) 1.54374 0.109433 0.0547163 0.998502i \(-0.482575\pi\)
0.0547163 + 0.998502i \(0.482575\pi\)
\(200\) 35.5242 2.51194
\(201\) 0 0
\(202\) 9.40291 0.661586
\(203\) 1.93542 0.135840
\(204\) 0 0
\(205\) −19.6645 −1.37343
\(206\) 0.196446 0.0136870
\(207\) 0 0
\(208\) 13.7086 0.950522
\(209\) 0 0
\(210\) 0 0
\(211\) −22.0836 −1.52030 −0.760149 0.649749i \(-0.774874\pi\)
−0.760149 + 0.649749i \(0.774874\pi\)
\(212\) −1.47109 −0.101035
\(213\) 0 0
\(214\) 10.1438 0.693415
\(215\) 5.53824 0.377704
\(216\) 0 0
\(217\) −1.56278 −0.106089
\(218\) 6.27354 0.424898
\(219\) 0 0
\(220\) 0 0
\(221\) 18.4043 1.23801
\(222\) 0 0
\(223\) −8.57414 −0.574167 −0.287083 0.957906i \(-0.592686\pi\)
−0.287083 + 0.957906i \(0.592686\pi\)
\(224\) −2.06640 −0.138067
\(225\) 0 0
\(226\) 3.88342 0.258321
\(227\) 26.5709 1.76357 0.881786 0.471650i \(-0.156341\pi\)
0.881786 + 0.471650i \(0.156341\pi\)
\(228\) 0 0
\(229\) 11.4113 0.754081 0.377040 0.926197i \(-0.376942\pi\)
0.377040 + 0.926197i \(0.376942\pi\)
\(230\) 25.9676 1.71225
\(231\) 0 0
\(232\) 5.85648 0.384497
\(233\) −26.6130 −1.74348 −0.871738 0.489972i \(-0.837007\pi\)
−0.871738 + 0.489972i \(0.837007\pi\)
\(234\) 0 0
\(235\) 42.8218 2.79339
\(236\) 5.09606 0.331725
\(237\) 0 0
\(238\) 5.35067 0.346832
\(239\) 7.51280 0.485963 0.242981 0.970031i \(-0.421875\pi\)
0.242981 + 0.970031i \(0.421875\pi\)
\(240\) 0 0
\(241\) 27.4388 1.76749 0.883744 0.467971i \(-0.155015\pi\)
0.883744 + 0.467971i \(0.155015\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 4.34355 0.278067
\(245\) 4.09144 0.261392
\(246\) 0 0
\(247\) 5.48336 0.348898
\(248\) −4.72889 −0.300285
\(249\) 0 0
\(250\) −35.2022 −2.22638
\(251\) −1.92514 −0.121514 −0.0607568 0.998153i \(-0.519351\pi\)
−0.0607568 + 0.998153i \(0.519351\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 13.7810 0.864694
\(255\) 0 0
\(256\) −8.56553 −0.535346
\(257\) −10.1604 −0.633791 −0.316895 0.948460i \(-0.602640\pi\)
−0.316895 + 0.948460i \(0.602640\pi\)
\(258\) 0 0
\(259\) −0.716296 −0.0445085
\(260\) 6.65400 0.412663
\(261\) 0 0
\(262\) −11.1116 −0.686476
\(263\) 4.38774 0.270560 0.135280 0.990807i \(-0.456807\pi\)
0.135280 + 0.990807i \(0.456807\pi\)
\(264\) 0 0
\(265\) 16.2504 0.998253
\(266\) 1.59417 0.0977449
\(267\) 0 0
\(268\) −2.81223 −0.171784
\(269\) −0.625379 −0.0381300 −0.0190650 0.999818i \(-0.506069\pi\)
−0.0190650 + 0.999818i \(0.506069\pi\)
\(270\) 0 0
\(271\) −11.9375 −0.725151 −0.362575 0.931954i \(-0.618102\pi\)
−0.362575 + 0.931954i \(0.618102\pi\)
\(272\) 13.0859 0.793452
\(273\) 0 0
\(274\) 15.3158 0.925263
\(275\) 0 0
\(276\) 0 0
\(277\) 28.5571 1.71583 0.857915 0.513792i \(-0.171760\pi\)
0.857915 + 0.513792i \(0.171760\pi\)
\(278\) −10.4546 −0.627028
\(279\) 0 0
\(280\) 12.3805 0.739875
\(281\) −14.6981 −0.876814 −0.438407 0.898777i \(-0.644457\pi\)
−0.438407 + 0.898777i \(0.644457\pi\)
\(282\) 0 0
\(283\) 22.5396 1.33984 0.669921 0.742433i \(-0.266328\pi\)
0.669921 + 0.742433i \(0.266328\pi\)
\(284\) 0.0808609 0.00479821
\(285\) 0 0
\(286\) 0 0
\(287\) −4.80626 −0.283704
\(288\) 0 0
\(289\) 0.568350 0.0334323
\(290\) −10.1087 −0.593601
\(291\) 0 0
\(292\) −4.06364 −0.237807
\(293\) −16.8280 −0.983105 −0.491553 0.870848i \(-0.663570\pi\)
−0.491553 + 0.870848i \(0.663570\pi\)
\(294\) 0 0
\(295\) −56.2934 −3.27753
\(296\) −2.16747 −0.125982
\(297\) 0 0
\(298\) 15.4230 0.893429
\(299\) 21.8307 1.26250
\(300\) 0 0
\(301\) 1.35362 0.0780211
\(302\) 8.87021 0.510423
\(303\) 0 0
\(304\) 3.89881 0.223612
\(305\) −47.9808 −2.74738
\(306\) 0 0
\(307\) 0.238354 0.0136036 0.00680179 0.999977i \(-0.497835\pi\)
0.00680179 + 0.999977i \(0.497835\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8.16238 0.463592
\(311\) 0.656523 0.0372280 0.0186140 0.999827i \(-0.494075\pi\)
0.0186140 + 0.999827i \(0.494075\pi\)
\(312\) 0 0
\(313\) −7.45262 −0.421247 −0.210624 0.977567i \(-0.567549\pi\)
−0.210624 + 0.977567i \(0.567549\pi\)
\(314\) −13.3950 −0.755926
\(315\) 0 0
\(316\) 1.68987 0.0950625
\(317\) −17.4412 −0.979595 −0.489797 0.871836i \(-0.662929\pi\)
−0.489797 + 0.871836i \(0.662929\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 36.3401 2.03147
\(321\) 0 0
\(322\) 6.34682 0.353695
\(323\) 5.23429 0.291244
\(324\) 0 0
\(325\) −51.5486 −2.85940
\(326\) 9.97946 0.552711
\(327\) 0 0
\(328\) −14.5435 −0.803030
\(329\) 10.4662 0.577021
\(330\) 0 0
\(331\) −28.6146 −1.57280 −0.786399 0.617719i \(-0.788057\pi\)
−0.786399 + 0.617719i \(0.788057\pi\)
\(332\) 0.909136 0.0498953
\(333\) 0 0
\(334\) 27.2550 1.49133
\(335\) 31.0652 1.69727
\(336\) 0 0
\(337\) 1.73053 0.0942679 0.0471339 0.998889i \(-0.484991\pi\)
0.0471339 + 0.998889i \(0.484991\pi\)
\(338\) −8.01691 −0.436062
\(339\) 0 0
\(340\) 6.35176 0.344472
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 4.09597 0.220840
\(345\) 0 0
\(346\) 24.8143 1.33403
\(347\) −2.73009 −0.146559 −0.0732796 0.997311i \(-0.523347\pi\)
−0.0732796 + 0.997311i \(0.523347\pi\)
\(348\) 0 0
\(349\) 30.9063 1.65437 0.827187 0.561927i \(-0.189940\pi\)
0.827187 + 0.561927i \(0.189940\pi\)
\(350\) −14.9867 −0.801072
\(351\) 0 0
\(352\) 0 0
\(353\) 23.9543 1.27496 0.637480 0.770467i \(-0.279977\pi\)
0.637480 + 0.770467i \(0.279977\pi\)
\(354\) 0 0
\(355\) −0.893227 −0.0474076
\(356\) 1.55730 0.0825369
\(357\) 0 0
\(358\) 22.9736 1.21419
\(359\) 1.51611 0.0800170 0.0400085 0.999199i \(-0.487261\pi\)
0.0400085 + 0.999199i \(0.487261\pi\)
\(360\) 0 0
\(361\) −17.4405 −0.917921
\(362\) −19.3496 −1.01699
\(363\) 0 0
\(364\) 1.62632 0.0852425
\(365\) 44.8889 2.34959
\(366\) 0 0
\(367\) 8.72207 0.455288 0.227644 0.973744i \(-0.426898\pi\)
0.227644 + 0.973744i \(0.426898\pi\)
\(368\) 15.5222 0.809151
\(369\) 0 0
\(370\) 3.74120 0.194496
\(371\) 3.97180 0.206206
\(372\) 0 0
\(373\) −36.8111 −1.90601 −0.953004 0.302957i \(-0.902026\pi\)
−0.953004 + 0.302957i \(0.902026\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 31.6702 1.63327
\(377\) −8.49825 −0.437682
\(378\) 0 0
\(379\) 2.59838 0.133470 0.0667349 0.997771i \(-0.478742\pi\)
0.0667349 + 0.997771i \(0.478742\pi\)
\(380\) 1.89243 0.0970798
\(381\) 0 0
\(382\) −30.3278 −1.55171
\(383\) −19.2392 −0.983078 −0.491539 0.870856i \(-0.663565\pi\)
−0.491539 + 0.870856i \(0.663565\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −17.9462 −0.913435
\(387\) 0 0
\(388\) 4.04640 0.205425
\(389\) 28.3331 1.43654 0.718272 0.695762i \(-0.244933\pi\)
0.718272 + 0.695762i \(0.244933\pi\)
\(390\) 0 0
\(391\) 20.8391 1.05388
\(392\) 3.02595 0.152833
\(393\) 0 0
\(394\) 23.0630 1.16190
\(395\) −18.6671 −0.939243
\(396\) 0 0
\(397\) −10.3666 −0.520287 −0.260143 0.965570i \(-0.583770\pi\)
−0.260143 + 0.965570i \(0.583770\pi\)
\(398\) −1.97068 −0.0987812
\(399\) 0 0
\(400\) −36.6524 −1.83262
\(401\) 12.7878 0.638592 0.319296 0.947655i \(-0.396554\pi\)
0.319296 + 0.947655i \(0.396554\pi\)
\(402\) 0 0
\(403\) 6.86203 0.341822
\(404\) 2.72817 0.135732
\(405\) 0 0
\(406\) −2.47069 −0.122618
\(407\) 0 0
\(408\) 0 0
\(409\) −32.0217 −1.58337 −0.791686 0.610928i \(-0.790797\pi\)
−0.791686 + 0.610928i \(0.790797\pi\)
\(410\) 25.1030 1.23975
\(411\) 0 0
\(412\) 0.0569970 0.00280804
\(413\) −13.7588 −0.677028
\(414\) 0 0
\(415\) −10.0427 −0.492979
\(416\) 9.07338 0.444859
\(417\) 0 0
\(418\) 0 0
\(419\) 20.0934 0.981629 0.490815 0.871264i \(-0.336699\pi\)
0.490815 + 0.871264i \(0.336699\pi\)
\(420\) 0 0
\(421\) −22.4243 −1.09289 −0.546446 0.837494i \(-0.684020\pi\)
−0.546446 + 0.837494i \(0.684020\pi\)
\(422\) 28.1911 1.37232
\(423\) 0 0
\(424\) 12.0185 0.583668
\(425\) −49.2072 −2.38690
\(426\) 0 0
\(427\) −11.7271 −0.567516
\(428\) 2.94313 0.142262
\(429\) 0 0
\(430\) −7.06991 −0.340941
\(431\) −12.0856 −0.582144 −0.291072 0.956701i \(-0.594012\pi\)
−0.291072 + 0.956701i \(0.594012\pi\)
\(432\) 0 0
\(433\) −0.302453 −0.0145350 −0.00726749 0.999974i \(-0.502313\pi\)
−0.00726749 + 0.999974i \(0.502313\pi\)
\(434\) 1.99499 0.0957626
\(435\) 0 0
\(436\) 1.82022 0.0871725
\(437\) 6.20878 0.297006
\(438\) 0 0
\(439\) −3.52592 −0.168283 −0.0841416 0.996454i \(-0.526815\pi\)
−0.0841416 + 0.996454i \(0.526815\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −23.4943 −1.11751
\(443\) 21.8443 1.03786 0.518928 0.854818i \(-0.326331\pi\)
0.518928 + 0.854818i \(0.326331\pi\)
\(444\) 0 0
\(445\) −17.2027 −0.815487
\(446\) 10.9454 0.518281
\(447\) 0 0
\(448\) 8.88199 0.419634
\(449\) −9.30172 −0.438975 −0.219488 0.975615i \(-0.570439\pi\)
−0.219488 + 0.975615i \(0.570439\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 1.12674 0.0529974
\(453\) 0 0
\(454\) −33.9194 −1.59192
\(455\) −17.9651 −0.842218
\(456\) 0 0
\(457\) −11.3165 −0.529365 −0.264683 0.964336i \(-0.585267\pi\)
−0.264683 + 0.964336i \(0.585267\pi\)
\(458\) −14.5673 −0.680684
\(459\) 0 0
\(460\) 7.53429 0.351288
\(461\) 0.678821 0.0316158 0.0158079 0.999875i \(-0.494968\pi\)
0.0158079 + 0.999875i \(0.494968\pi\)
\(462\) 0 0
\(463\) −13.4936 −0.627103 −0.313551 0.949571i \(-0.601519\pi\)
−0.313551 + 0.949571i \(0.601519\pi\)
\(464\) −6.04247 −0.280515
\(465\) 0 0
\(466\) 33.9732 1.57378
\(467\) −29.6568 −1.37235 −0.686176 0.727435i \(-0.740712\pi\)
−0.686176 + 0.727435i \(0.740712\pi\)
\(468\) 0 0
\(469\) 7.59274 0.350600
\(470\) −54.6648 −2.52150
\(471\) 0 0
\(472\) −41.6335 −1.91634
\(473\) 0 0
\(474\) 0 0
\(475\) −14.6607 −0.672680
\(476\) 1.55245 0.0711565
\(477\) 0 0
\(478\) −9.59057 −0.438662
\(479\) −4.70136 −0.214811 −0.107405 0.994215i \(-0.534254\pi\)
−0.107405 + 0.994215i \(0.534254\pi\)
\(480\) 0 0
\(481\) 3.14519 0.143408
\(482\) −35.0274 −1.59545
\(483\) 0 0
\(484\) 0 0
\(485\) −44.6985 −2.02965
\(486\) 0 0
\(487\) 32.3738 1.46700 0.733498 0.679692i \(-0.237886\pi\)
0.733498 + 0.679692i \(0.237886\pi\)
\(488\) −35.4857 −1.60636
\(489\) 0 0
\(490\) −5.22298 −0.235950
\(491\) 6.85594 0.309404 0.154702 0.987961i \(-0.450558\pi\)
0.154702 + 0.987961i \(0.450558\pi\)
\(492\) 0 0
\(493\) −8.11224 −0.365357
\(494\) −6.99986 −0.314939
\(495\) 0 0
\(496\) 4.87908 0.219077
\(497\) −0.218316 −0.00979282
\(498\) 0 0
\(499\) 10.9670 0.490952 0.245476 0.969403i \(-0.421056\pi\)
0.245476 + 0.969403i \(0.421056\pi\)
\(500\) −10.2136 −0.456766
\(501\) 0 0
\(502\) 2.45756 0.109686
\(503\) 26.9334 1.20090 0.600451 0.799662i \(-0.294988\pi\)
0.600451 + 0.799662i \(0.294988\pi\)
\(504\) 0 0
\(505\) −30.1367 −1.34106
\(506\) 0 0
\(507\) 0 0
\(508\) 3.99843 0.177402
\(509\) −30.3958 −1.34727 −0.673635 0.739064i \(-0.735268\pi\)
−0.673635 + 0.739064i \(0.735268\pi\)
\(510\) 0 0
\(511\) 10.9714 0.485347
\(512\) 25.3457 1.12013
\(513\) 0 0
\(514\) 12.9705 0.572102
\(515\) −0.629616 −0.0277442
\(516\) 0 0
\(517\) 0 0
\(518\) 0.914398 0.0401763
\(519\) 0 0
\(520\) −54.3615 −2.38391
\(521\) 0.935426 0.0409818 0.0204909 0.999790i \(-0.493477\pi\)
0.0204909 + 0.999790i \(0.493477\pi\)
\(522\) 0 0
\(523\) 26.1853 1.14500 0.572502 0.819903i \(-0.305973\pi\)
0.572502 + 0.819903i \(0.305973\pi\)
\(524\) −3.22393 −0.140838
\(525\) 0 0
\(526\) −5.60124 −0.244226
\(527\) 6.55034 0.285337
\(528\) 0 0
\(529\) 1.71881 0.0747307
\(530\) −20.7446 −0.901090
\(531\) 0 0
\(532\) 0.462535 0.0200534
\(533\) 21.1038 0.914109
\(534\) 0 0
\(535\) −32.5112 −1.40558
\(536\) 22.9752 0.992378
\(537\) 0 0
\(538\) 0.798336 0.0344187
\(539\) 0 0
\(540\) 0 0
\(541\) 26.7566 1.15035 0.575177 0.818029i \(-0.304933\pi\)
0.575177 + 0.818029i \(0.304933\pi\)
\(542\) 15.2390 0.654570
\(543\) 0 0
\(544\) 8.66124 0.371348
\(545\) −20.1070 −0.861287
\(546\) 0 0
\(547\) 46.4581 1.98641 0.993203 0.116397i \(-0.0371345\pi\)
0.993203 + 0.116397i \(0.0371345\pi\)
\(548\) 4.44376 0.189828
\(549\) 0 0
\(550\) 0 0
\(551\) −2.41695 −0.102966
\(552\) 0 0
\(553\) −4.56248 −0.194016
\(554\) −36.4550 −1.54882
\(555\) 0 0
\(556\) −3.03332 −0.128642
\(557\) −35.0974 −1.48712 −0.743562 0.668667i \(-0.766865\pi\)
−0.743562 + 0.668667i \(0.766865\pi\)
\(558\) 0 0
\(559\) −5.94360 −0.251388
\(560\) −12.7737 −0.539786
\(561\) 0 0
\(562\) 18.7630 0.791471
\(563\) 8.94791 0.377109 0.188555 0.982063i \(-0.439620\pi\)
0.188555 + 0.982063i \(0.439620\pi\)
\(564\) 0 0
\(565\) −12.4465 −0.523628
\(566\) −28.7733 −1.20943
\(567\) 0 0
\(568\) −0.660614 −0.0277187
\(569\) −19.4256 −0.814365 −0.407183 0.913347i \(-0.633489\pi\)
−0.407183 + 0.913347i \(0.633489\pi\)
\(570\) 0 0
\(571\) 16.0171 0.670295 0.335148 0.942166i \(-0.391214\pi\)
0.335148 + 0.942166i \(0.391214\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 6.13550 0.256091
\(575\) −58.3683 −2.43412
\(576\) 0 0
\(577\) −32.9192 −1.37044 −0.685222 0.728335i \(-0.740295\pi\)
−0.685222 + 0.728335i \(0.740295\pi\)
\(578\) −0.725535 −0.0301783
\(579\) 0 0
\(580\) −2.93294 −0.121784
\(581\) −2.45458 −0.101833
\(582\) 0 0
\(583\) 0 0
\(584\) 33.1990 1.37378
\(585\) 0 0
\(586\) 21.4821 0.887417
\(587\) −27.4372 −1.13246 −0.566228 0.824249i \(-0.691598\pi\)
−0.566228 + 0.824249i \(0.691598\pi\)
\(588\) 0 0
\(589\) 1.95160 0.0804142
\(590\) 71.8622 2.95852
\(591\) 0 0
\(592\) 2.23631 0.0919118
\(593\) −14.6132 −0.600092 −0.300046 0.953925i \(-0.597002\pi\)
−0.300046 + 0.953925i \(0.597002\pi\)
\(594\) 0 0
\(595\) −17.1491 −0.703045
\(596\) 4.47485 0.183297
\(597\) 0 0
\(598\) −27.8683 −1.13962
\(599\) −25.2765 −1.03277 −0.516384 0.856357i \(-0.672722\pi\)
−0.516384 + 0.856357i \(0.672722\pi\)
\(600\) 0 0
\(601\) −9.30098 −0.379395 −0.189697 0.981843i \(-0.560751\pi\)
−0.189697 + 0.981843i \(0.560751\pi\)
\(602\) −1.72798 −0.0704271
\(603\) 0 0
\(604\) 2.57361 0.104719
\(605\) 0 0
\(606\) 0 0
\(607\) −2.12671 −0.0863207 −0.0431603 0.999068i \(-0.513743\pi\)
−0.0431603 + 0.999068i \(0.513743\pi\)
\(608\) 2.58052 0.104654
\(609\) 0 0
\(610\) 61.2506 2.47997
\(611\) −45.9561 −1.85919
\(612\) 0 0
\(613\) −40.9448 −1.65374 −0.826872 0.562390i \(-0.809882\pi\)
−0.826872 + 0.562390i \(0.809882\pi\)
\(614\) −0.304274 −0.0122795
\(615\) 0 0
\(616\) 0 0
\(617\) 7.53813 0.303474 0.151737 0.988421i \(-0.451513\pi\)
0.151737 + 0.988421i \(0.451513\pi\)
\(618\) 0 0
\(619\) −19.5055 −0.783994 −0.391997 0.919967i \(-0.628216\pi\)
−0.391997 + 0.919967i \(0.628216\pi\)
\(620\) 2.36824 0.0951110
\(621\) 0 0
\(622\) −0.838094 −0.0336045
\(623\) −4.20456 −0.168452
\(624\) 0 0
\(625\) 54.1250 2.16500
\(626\) 9.51375 0.380246
\(627\) 0 0
\(628\) −3.88645 −0.155086
\(629\) 3.00233 0.119711
\(630\) 0 0
\(631\) −10.6654 −0.424582 −0.212291 0.977207i \(-0.568092\pi\)
−0.212291 + 0.977207i \(0.568092\pi\)
\(632\) −13.8058 −0.549166
\(633\) 0 0
\(634\) 22.2648 0.884248
\(635\) −44.1685 −1.75277
\(636\) 0 0
\(637\) −4.39091 −0.173974
\(638\) 0 0
\(639\) 0 0
\(640\) −29.4814 −1.16535
\(641\) −0.448349 −0.0177087 −0.00885436 0.999961i \(-0.502818\pi\)
−0.00885436 + 0.999961i \(0.502818\pi\)
\(642\) 0 0
\(643\) −31.6440 −1.24792 −0.623958 0.781458i \(-0.714476\pi\)
−0.623958 + 0.781458i \(0.714476\pi\)
\(644\) 1.84148 0.0725643
\(645\) 0 0
\(646\) −6.68191 −0.262896
\(647\) 30.1242 1.18430 0.592152 0.805826i \(-0.298279\pi\)
0.592152 + 0.805826i \(0.298279\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 65.8051 2.58109
\(651\) 0 0
\(652\) 2.89545 0.113395
\(653\) −34.6110 −1.35443 −0.677217 0.735784i \(-0.736814\pi\)
−0.677217 + 0.735784i \(0.736814\pi\)
\(654\) 0 0
\(655\) 35.6131 1.39152
\(656\) 15.0054 0.585861
\(657\) 0 0
\(658\) −13.3608 −0.520858
\(659\) 20.2587 0.789167 0.394584 0.918860i \(-0.370889\pi\)
0.394584 + 0.918860i \(0.370889\pi\)
\(660\) 0 0
\(661\) 40.6737 1.58202 0.791012 0.611801i \(-0.209555\pi\)
0.791012 + 0.611801i \(0.209555\pi\)
\(662\) 36.5283 1.41971
\(663\) 0 0
\(664\) −7.42742 −0.288240
\(665\) −5.10938 −0.198133
\(666\) 0 0
\(667\) −9.62253 −0.372586
\(668\) 7.90781 0.305962
\(669\) 0 0
\(670\) −39.6567 −1.53207
\(671\) 0 0
\(672\) 0 0
\(673\) 24.7484 0.953980 0.476990 0.878909i \(-0.341728\pi\)
0.476990 + 0.878909i \(0.341728\pi\)
\(674\) −2.20913 −0.0850925
\(675\) 0 0
\(676\) −2.32604 −0.0894630
\(677\) 11.3821 0.437452 0.218726 0.975786i \(-0.429810\pi\)
0.218726 + 0.975786i \(0.429810\pi\)
\(678\) 0 0
\(679\) −10.9249 −0.419259
\(680\) −51.8923 −1.98998
\(681\) 0 0
\(682\) 0 0
\(683\) 0.212700 0.00813875 0.00406938 0.999992i \(-0.498705\pi\)
0.00406938 + 0.999992i \(0.498705\pi\)
\(684\) 0 0
\(685\) −49.0878 −1.87555
\(686\) −1.27656 −0.0487394
\(687\) 0 0
\(688\) −4.22605 −0.161117
\(689\) −17.4398 −0.664404
\(690\) 0 0
\(691\) 36.3804 1.38398 0.691988 0.721909i \(-0.256735\pi\)
0.691988 + 0.721909i \(0.256735\pi\)
\(692\) 7.19966 0.273690
\(693\) 0 0
\(694\) 3.48514 0.132294
\(695\) 33.5075 1.27101
\(696\) 0 0
\(697\) 20.1452 0.763056
\(698\) −39.4538 −1.49335
\(699\) 0 0
\(700\) −4.34826 −0.164349
\(701\) 18.7394 0.707779 0.353889 0.935287i \(-0.384859\pi\)
0.353889 + 0.935287i \(0.384859\pi\)
\(702\) 0 0
\(703\) 0.894509 0.0337371
\(704\) 0 0
\(705\) 0 0
\(706\) −30.5792 −1.15086
\(707\) −7.36579 −0.277019
\(708\) 0 0
\(709\) −12.7596 −0.479195 −0.239598 0.970872i \(-0.577015\pi\)
−0.239598 + 0.970872i \(0.577015\pi\)
\(710\) 1.14026 0.0427933
\(711\) 0 0
\(712\) −12.7228 −0.476807
\(713\) 7.76984 0.290983
\(714\) 0 0
\(715\) 0 0
\(716\) 6.66558 0.249105
\(717\) 0 0
\(718\) −1.93541 −0.0722287
\(719\) 17.1228 0.638571 0.319286 0.947658i \(-0.396557\pi\)
0.319286 + 0.947658i \(0.396557\pi\)
\(720\) 0 0
\(721\) −0.153886 −0.00573102
\(722\) 22.2639 0.828577
\(723\) 0 0
\(724\) −5.61411 −0.208647
\(725\) 22.7216 0.843858
\(726\) 0 0
\(727\) −8.65786 −0.321102 −0.160551 0.987028i \(-0.551327\pi\)
−0.160551 + 0.987028i \(0.551327\pi\)
\(728\) −13.2867 −0.492436
\(729\) 0 0
\(730\) −57.3036 −2.12090
\(731\) −5.67363 −0.209847
\(732\) 0 0
\(733\) 28.0866 1.03740 0.518702 0.854955i \(-0.326416\pi\)
0.518702 + 0.854955i \(0.326416\pi\)
\(734\) −11.1343 −0.410974
\(735\) 0 0
\(736\) 10.2737 0.378695
\(737\) 0 0
\(738\) 0 0
\(739\) −33.8965 −1.24690 −0.623451 0.781862i \(-0.714270\pi\)
−0.623451 + 0.781862i \(0.714270\pi\)
\(740\) 1.08548 0.0399030
\(741\) 0 0
\(742\) −5.07026 −0.186135
\(743\) 5.65321 0.207396 0.103698 0.994609i \(-0.466932\pi\)
0.103698 + 0.994609i \(0.466932\pi\)
\(744\) 0 0
\(745\) −49.4313 −1.81102
\(746\) 46.9918 1.72049
\(747\) 0 0
\(748\) 0 0
\(749\) −7.94617 −0.290347
\(750\) 0 0
\(751\) 44.3409 1.61802 0.809012 0.587792i \(-0.200003\pi\)
0.809012 + 0.587792i \(0.200003\pi\)
\(752\) −32.6760 −1.19157
\(753\) 0 0
\(754\) 10.8486 0.395081
\(755\) −28.4294 −1.03465
\(756\) 0 0
\(757\) −21.3781 −0.777001 −0.388500 0.921449i \(-0.627007\pi\)
−0.388500 + 0.921449i \(0.627007\pi\)
\(758\) −3.31700 −0.120479
\(759\) 0 0
\(760\) −15.4607 −0.560819
\(761\) 4.58574 0.166233 0.0831164 0.996540i \(-0.473513\pi\)
0.0831164 + 0.996540i \(0.473513\pi\)
\(762\) 0 0
\(763\) −4.91440 −0.177913
\(764\) −8.79936 −0.318350
\(765\) 0 0
\(766\) 24.5601 0.887392
\(767\) 60.4138 2.18142
\(768\) 0 0
\(769\) 12.8223 0.462385 0.231193 0.972908i \(-0.425737\pi\)
0.231193 + 0.972908i \(0.425737\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −5.20692 −0.187401
\(773\) −54.0639 −1.94454 −0.972272 0.233855i \(-0.924866\pi\)
−0.972272 + 0.233855i \(0.924866\pi\)
\(774\) 0 0
\(775\) −18.3468 −0.659038
\(776\) −33.0581 −1.18672
\(777\) 0 0
\(778\) −36.1690 −1.29672
\(779\) 6.00205 0.215046
\(780\) 0 0
\(781\) 0 0
\(782\) −26.6025 −0.951302
\(783\) 0 0
\(784\) −3.12205 −0.111502
\(785\) 42.9316 1.53229
\(786\) 0 0
\(787\) −43.8514 −1.56314 −0.781568 0.623821i \(-0.785580\pi\)
−0.781568 + 0.623821i \(0.785580\pi\)
\(788\) 6.69153 0.238376
\(789\) 0 0
\(790\) 23.8297 0.847824
\(791\) −3.04208 −0.108164
\(792\) 0 0
\(793\) 51.4928 1.82856
\(794\) 13.2337 0.469646
\(795\) 0 0
\(796\) −0.571776 −0.0202660
\(797\) 35.7697 1.26703 0.633514 0.773731i \(-0.281612\pi\)
0.633514 + 0.773731i \(0.281612\pi\)
\(798\) 0 0
\(799\) −43.8687 −1.55196
\(800\) −24.2593 −0.857694
\(801\) 0 0
\(802\) −16.3244 −0.576436
\(803\) 0 0
\(804\) 0 0
\(805\) −20.3418 −0.716955
\(806\) −8.75982 −0.308551
\(807\) 0 0
\(808\) −22.2885 −0.784107
\(809\) 30.7068 1.07959 0.539797 0.841795i \(-0.318501\pi\)
0.539797 + 0.841795i \(0.318501\pi\)
\(810\) 0 0
\(811\) 17.0984 0.600405 0.300202 0.953876i \(-0.402946\pi\)
0.300202 + 0.953876i \(0.402946\pi\)
\(812\) −0.716849 −0.0251565
\(813\) 0 0
\(814\) 0 0
\(815\) −31.9845 −1.12037
\(816\) 0 0
\(817\) −1.69039 −0.0591394
\(818\) 40.8778 1.42926
\(819\) 0 0
\(820\) 7.28342 0.254348
\(821\) 12.3822 0.432142 0.216071 0.976378i \(-0.430676\pi\)
0.216071 + 0.976378i \(0.430676\pi\)
\(822\) 0 0
\(823\) 46.1384 1.60829 0.804143 0.594436i \(-0.202625\pi\)
0.804143 + 0.594436i \(0.202625\pi\)
\(824\) −0.465652 −0.0162217
\(825\) 0 0
\(826\) 17.5640 0.611131
\(827\) −46.7103 −1.62428 −0.812138 0.583466i \(-0.801696\pi\)
−0.812138 + 0.583466i \(0.801696\pi\)
\(828\) 0 0
\(829\) 4.02555 0.139813 0.0699065 0.997554i \(-0.477730\pi\)
0.0699065 + 0.997554i \(0.477730\pi\)
\(830\) 12.8202 0.444996
\(831\) 0 0
\(832\) −39.0000 −1.35208
\(833\) −4.19146 −0.145226
\(834\) 0 0
\(835\) −87.3534 −3.02299
\(836\) 0 0
\(837\) 0 0
\(838\) −25.6506 −0.886084
\(839\) 14.3021 0.493765 0.246882 0.969045i \(-0.420594\pi\)
0.246882 + 0.969045i \(0.420594\pi\)
\(840\) 0 0
\(841\) −25.2541 −0.870833
\(842\) 28.6260 0.986518
\(843\) 0 0
\(844\) 8.17941 0.281547
\(845\) 25.6945 0.883918
\(846\) 0 0
\(847\) 0 0
\(848\) −12.4002 −0.425823
\(849\) 0 0
\(850\) 62.8161 2.15457
\(851\) 3.56128 0.122079
\(852\) 0 0
\(853\) −32.3007 −1.10595 −0.552977 0.833197i \(-0.686508\pi\)
−0.552977 + 0.833197i \(0.686508\pi\)
\(854\) 14.9704 0.512278
\(855\) 0 0
\(856\) −24.0447 −0.821830
\(857\) −16.2318 −0.554468 −0.277234 0.960802i \(-0.589418\pi\)
−0.277234 + 0.960802i \(0.589418\pi\)
\(858\) 0 0
\(859\) −28.2893 −0.965220 −0.482610 0.875835i \(-0.660311\pi\)
−0.482610 + 0.875835i \(0.660311\pi\)
\(860\) −2.05127 −0.0699479
\(861\) 0 0
\(862\) 15.4281 0.525482
\(863\) −31.9865 −1.08883 −0.544417 0.838815i \(-0.683249\pi\)
−0.544417 + 0.838815i \(0.683249\pi\)
\(864\) 0 0
\(865\) −79.5309 −2.70413
\(866\) 0.386101 0.0131203
\(867\) 0 0
\(868\) 0.578830 0.0196468
\(869\) 0 0
\(870\) 0 0
\(871\) −33.3390 −1.12965
\(872\) −14.8707 −0.503586
\(873\) 0 0
\(874\) −7.92590 −0.268098
\(875\) 27.5757 0.932229
\(876\) 0 0
\(877\) 45.7968 1.54645 0.773224 0.634133i \(-0.218643\pi\)
0.773224 + 0.634133i \(0.218643\pi\)
\(878\) 4.50107 0.151904
\(879\) 0 0
\(880\) 0 0
\(881\) −26.7142 −0.900025 −0.450012 0.893022i \(-0.648580\pi\)
−0.450012 + 0.893022i \(0.648580\pi\)
\(882\) 0 0
\(883\) −28.4869 −0.958662 −0.479331 0.877634i \(-0.659121\pi\)
−0.479331 + 0.877634i \(0.659121\pi\)
\(884\) −6.81667 −0.229269
\(885\) 0 0
\(886\) −27.8857 −0.936838
\(887\) 23.6139 0.792879 0.396439 0.918061i \(-0.370246\pi\)
0.396439 + 0.918061i \(0.370246\pi\)
\(888\) 0 0
\(889\) −10.7954 −0.362064
\(890\) 21.9604 0.736113
\(891\) 0 0
\(892\) 3.17572 0.106331
\(893\) −13.0702 −0.437377
\(894\) 0 0
\(895\) −73.6312 −2.46122
\(896\) −7.20562 −0.240723
\(897\) 0 0
\(898\) 11.8742 0.396249
\(899\) −3.02464 −0.100877
\(900\) 0 0
\(901\) −16.6477 −0.554614
\(902\) 0 0
\(903\) 0 0
\(904\) −9.20519 −0.306160
\(905\) 62.0161 2.06149
\(906\) 0 0
\(907\) 23.9434 0.795029 0.397514 0.917596i \(-0.369873\pi\)
0.397514 + 0.917596i \(0.369873\pi\)
\(908\) −9.84143 −0.326599
\(909\) 0 0
\(910\) 22.9336 0.760242
\(911\) −18.2388 −0.604280 −0.302140 0.953264i \(-0.597701\pi\)
−0.302140 + 0.953264i \(0.597701\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 14.4463 0.477841
\(915\) 0 0
\(916\) −4.22657 −0.139650
\(917\) 8.70429 0.287441
\(918\) 0 0
\(919\) 28.4024 0.936909 0.468455 0.883488i \(-0.344811\pi\)
0.468455 + 0.883488i \(0.344811\pi\)
\(920\) −61.5533 −2.02935
\(921\) 0 0
\(922\) −0.866558 −0.0285386
\(923\) 0.958607 0.0315529
\(924\) 0 0
\(925\) −8.40922 −0.276493
\(926\) 17.2255 0.566065
\(927\) 0 0
\(928\) −3.99936 −0.131285
\(929\) −3.38963 −0.111210 −0.0556051 0.998453i \(-0.517709\pi\)
−0.0556051 + 0.998453i \(0.517709\pi\)
\(930\) 0 0
\(931\) −1.24880 −0.0409277
\(932\) 9.85704 0.322878
\(933\) 0 0
\(934\) 37.8588 1.23878
\(935\) 0 0
\(936\) 0 0
\(937\) 45.3871 1.48273 0.741367 0.671100i \(-0.234178\pi\)
0.741367 + 0.671100i \(0.234178\pi\)
\(938\) −9.69261 −0.316475
\(939\) 0 0
\(940\) −15.8605 −0.517313
\(941\) −21.5279 −0.701789 −0.350894 0.936415i \(-0.614122\pi\)
−0.350894 + 0.936415i \(0.614122\pi\)
\(942\) 0 0
\(943\) 23.8958 0.778153
\(944\) 42.9558 1.39809
\(945\) 0 0
\(946\) 0 0
\(947\) 11.2673 0.366140 0.183070 0.983100i \(-0.441397\pi\)
0.183070 + 0.983100i \(0.441397\pi\)
\(948\) 0 0
\(949\) −48.1745 −1.56381
\(950\) 18.7153 0.607206
\(951\) 0 0
\(952\) −12.6831 −0.411063
\(953\) 23.8696 0.773213 0.386607 0.922245i \(-0.373647\pi\)
0.386607 + 0.922245i \(0.373647\pi\)
\(954\) 0 0
\(955\) 97.2019 3.14538
\(956\) −2.78262 −0.0899964
\(957\) 0 0
\(958\) 6.00159 0.193902
\(959\) −11.9977 −0.387426
\(960\) 0 0
\(961\) −28.5577 −0.921217
\(962\) −4.01504 −0.129450
\(963\) 0 0
\(964\) −10.1629 −0.327325
\(965\) 57.5181 1.85157
\(966\) 0 0
\(967\) 20.5165 0.659765 0.329882 0.944022i \(-0.392991\pi\)
0.329882 + 0.944022i \(0.392991\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 57.0605 1.83210
\(971\) 34.7645 1.11565 0.557824 0.829960i \(-0.311637\pi\)
0.557824 + 0.829960i \(0.311637\pi\)
\(972\) 0 0
\(973\) 8.18967 0.262549
\(974\) −41.3272 −1.32421
\(975\) 0 0
\(976\) 36.6127 1.17194
\(977\) −5.78294 −0.185013 −0.0925064 0.995712i \(-0.529488\pi\)
−0.0925064 + 0.995712i \(0.529488\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.51540 −0.0484078
\(981\) 0 0
\(982\) −8.75204 −0.279289
\(983\) 17.2321 0.549618 0.274809 0.961499i \(-0.411385\pi\)
0.274809 + 0.961499i \(0.411385\pi\)
\(984\) 0 0
\(985\) −73.9178 −2.35522
\(986\) 10.3558 0.329796
\(987\) 0 0
\(988\) −2.03095 −0.0646131
\(989\) −6.72991 −0.213999
\(990\) 0 0
\(991\) −24.0077 −0.762630 −0.381315 0.924445i \(-0.624529\pi\)
−0.381315 + 0.924445i \(0.624529\pi\)
\(992\) 3.22933 0.102531
\(993\) 0 0
\(994\) 0.278695 0.00883966
\(995\) 6.31610 0.200234
\(996\) 0 0
\(997\) 36.0335 1.14119 0.570596 0.821231i \(-0.306712\pi\)
0.570596 + 0.821231i \(0.306712\pi\)
\(998\) −14.0001 −0.443166
\(999\) 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 7623.2.a.cp.1.3 6
3.2 odd 2 847.2.a.n.1.4 yes 6
11.10 odd 2 7623.2.a.cs.1.4 6
21.20 even 2 5929.2.a.bm.1.4 6
33.2 even 10 847.2.f.z.323.4 24
33.5 odd 10 847.2.f.y.729.3 24
33.8 even 10 847.2.f.z.372.3 24
33.14 odd 10 847.2.f.y.372.4 24
33.17 even 10 847.2.f.z.729.4 24
33.20 odd 10 847.2.f.y.323.3 24
33.26 odd 10 847.2.f.y.148.4 24
33.29 even 10 847.2.f.z.148.3 24
33.32 even 2 847.2.a.m.1.3 6
231.230 odd 2 5929.2.a.bj.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.m.1.3 6 33.32 even 2
847.2.a.n.1.4 yes 6 3.2 odd 2
847.2.f.y.148.4 24 33.26 odd 10
847.2.f.y.323.3 24 33.20 odd 10
847.2.f.y.372.4 24 33.14 odd 10
847.2.f.y.729.3 24 33.5 odd 10
847.2.f.z.148.3 24 33.29 even 10
847.2.f.z.323.4 24 33.2 even 10
847.2.f.z.372.3 24 33.8 even 10
847.2.f.z.729.4 24 33.17 even 10
5929.2.a.bj.1.3 6 231.230 odd 2
5929.2.a.bm.1.4 6 21.20 even 2
7623.2.a.cp.1.3 6 1.1 even 1 trivial
7623.2.a.cs.1.4 6 11.10 odd 2