Properties

Label 2205.4.a.ca.1.1
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.1163891200.1
Defining polynomial: \( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.05323\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.46745 q^{2} +11.9581 q^{4} +5.00000 q^{5} -17.6824 q^{8} +O(q^{10})\) \(q-4.46745 q^{2} +11.9581 q^{4} +5.00000 q^{5} -17.6824 q^{8} -22.3372 q^{10} +56.5404 q^{11} +40.9643 q^{13} -16.6692 q^{16} -2.18896 q^{17} +16.4735 q^{19} +59.7903 q^{20} -252.591 q^{22} +155.272 q^{23} +25.0000 q^{25} -183.006 q^{26} +6.26048 q^{29} +168.680 q^{31} +215.928 q^{32} +9.77908 q^{34} -37.1738 q^{37} -73.5946 q^{38} -88.4122 q^{40} +266.804 q^{41} -14.6549 q^{43} +676.114 q^{44} -693.670 q^{46} +169.840 q^{47} -111.686 q^{50} +489.854 q^{52} +151.391 q^{53} +282.702 q^{55} -27.9683 q^{58} +234.076 q^{59} -242.411 q^{61} -753.569 q^{62} -831.294 q^{64} +204.822 q^{65} +820.771 q^{67} -26.1758 q^{68} -961.611 q^{71} +934.026 q^{73} +166.072 q^{74} +196.992 q^{76} +300.236 q^{79} -83.3460 q^{80} -1191.93 q^{82} -1087.60 q^{83} -10.9448 q^{85} +65.4702 q^{86} -999.773 q^{88} +1124.05 q^{89} +1856.76 q^{92} -758.753 q^{94} +82.3676 q^{95} +752.168 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8} + 10 q^{10} + 16 q^{11} + 168 q^{13} + 298 q^{16} + 4 q^{17} + 308 q^{19} + 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} - 56 q^{26} - 176 q^{29} + 392 q^{31} + 770 q^{32} + 812 q^{34} - 140 q^{37} - 20 q^{38} + 330 q^{40} - 656 q^{41} - 388 q^{43} + 160 q^{44} - 388 q^{46} - 628 q^{47} + 50 q^{50} + 1520 q^{52} + 676 q^{53} + 80 q^{55} - 2012 q^{58} - 996 q^{59} + 740 q^{61} + 364 q^{62} + 1426 q^{64} + 840 q^{65} + 1768 q^{67} + 2940 q^{68} + 224 q^{71} + 2640 q^{73} - 928 q^{74} - 1340 q^{76} + 1636 q^{79} + 1490 q^{80} - 1756 q^{82} - 140 q^{83} + 20 q^{85} - 1180 q^{86} - 5652 q^{88} + 1904 q^{89} + 1952 q^{92} - 3332 q^{94} + 1540 q^{95} + 516 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.46745 −1.57948 −0.789740 0.613441i \(-0.789785\pi\)
−0.789740 + 0.613441i \(0.789785\pi\)
\(3\) 0 0
\(4\) 11.9581 1.49476
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −17.6824 −0.781461
\(9\) 0 0
\(10\) −22.3372 −0.706365
\(11\) 56.5404 1.54978 0.774890 0.632096i \(-0.217805\pi\)
0.774890 + 0.632096i \(0.217805\pi\)
\(12\) 0 0
\(13\) 40.9643 0.873958 0.436979 0.899472i \(-0.356048\pi\)
0.436979 + 0.899472i \(0.356048\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −16.6692 −0.260456
\(17\) −2.18896 −0.0312295 −0.0156148 0.999878i \(-0.504971\pi\)
−0.0156148 + 0.999878i \(0.504971\pi\)
\(18\) 0 0
\(19\) 16.4735 0.198910 0.0994549 0.995042i \(-0.468290\pi\)
0.0994549 + 0.995042i \(0.468290\pi\)
\(20\) 59.7903 0.668476
\(21\) 0 0
\(22\) −252.591 −2.44785
\(23\) 155.272 1.40767 0.703837 0.710361i \(-0.251468\pi\)
0.703837 + 0.710361i \(0.251468\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −183.006 −1.38040
\(27\) 0 0
\(28\) 0 0
\(29\) 6.26048 0.0400876 0.0200438 0.999799i \(-0.493619\pi\)
0.0200438 + 0.999799i \(0.493619\pi\)
\(30\) 0 0
\(31\) 168.680 0.977286 0.488643 0.872484i \(-0.337492\pi\)
0.488643 + 0.872484i \(0.337492\pi\)
\(32\) 215.928 1.19285
\(33\) 0 0
\(34\) 9.77908 0.0493264
\(35\) 0 0
\(36\) 0 0
\(37\) −37.1738 −0.165171 −0.0825856 0.996584i \(-0.526318\pi\)
−0.0825856 + 0.996584i \(0.526318\pi\)
\(38\) −73.5946 −0.314174
\(39\) 0 0
\(40\) −88.4122 −0.349480
\(41\) 266.804 1.01629 0.508143 0.861273i \(-0.330332\pi\)
0.508143 + 0.861273i \(0.330332\pi\)
\(42\) 0 0
\(43\) −14.6549 −0.0519735 −0.0259867 0.999662i \(-0.508273\pi\)
−0.0259867 + 0.999662i \(0.508273\pi\)
\(44\) 676.114 2.31655
\(45\) 0 0
\(46\) −693.670 −2.22339
\(47\) 169.840 0.527102 0.263551 0.964646i \(-0.415106\pi\)
0.263551 + 0.964646i \(0.415106\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −111.686 −0.315896
\(51\) 0 0
\(52\) 489.854 1.30636
\(53\) 151.391 0.392361 0.196180 0.980568i \(-0.437146\pi\)
0.196180 + 0.980568i \(0.437146\pi\)
\(54\) 0 0
\(55\) 282.702 0.693083
\(56\) 0 0
\(57\) 0 0
\(58\) −27.9683 −0.0633176
\(59\) 234.076 0.516510 0.258255 0.966077i \(-0.416853\pi\)
0.258255 + 0.966077i \(0.416853\pi\)
\(60\) 0 0
\(61\) −242.411 −0.508813 −0.254407 0.967097i \(-0.581880\pi\)
−0.254407 + 0.967097i \(0.581880\pi\)
\(62\) −753.569 −1.54360
\(63\) 0 0
\(64\) −831.294 −1.62362
\(65\) 204.822 0.390846
\(66\) 0 0
\(67\) 820.771 1.49661 0.748307 0.663353i \(-0.230867\pi\)
0.748307 + 0.663353i \(0.230867\pi\)
\(68\) −26.1758 −0.0466806
\(69\) 0 0
\(70\) 0 0
\(71\) −961.611 −1.60736 −0.803678 0.595065i \(-0.797126\pi\)
−0.803678 + 0.595065i \(0.797126\pi\)
\(72\) 0 0
\(73\) 934.026 1.49753 0.748763 0.662837i \(-0.230648\pi\)
0.748763 + 0.662837i \(0.230648\pi\)
\(74\) 166.072 0.260885
\(75\) 0 0
\(76\) 196.992 0.297322
\(77\) 0 0
\(78\) 0 0
\(79\) 300.236 0.427584 0.213792 0.976879i \(-0.431419\pi\)
0.213792 + 0.976879i \(0.431419\pi\)
\(80\) −83.3460 −0.116480
\(81\) 0 0
\(82\) −1191.93 −1.60520
\(83\) −1087.60 −1.43830 −0.719151 0.694854i \(-0.755469\pi\)
−0.719151 + 0.694854i \(0.755469\pi\)
\(84\) 0 0
\(85\) −10.9448 −0.0139663
\(86\) 65.4702 0.0820910
\(87\) 0 0
\(88\) −999.773 −1.21109
\(89\) 1124.05 1.33875 0.669375 0.742925i \(-0.266562\pi\)
0.669375 + 0.742925i \(0.266562\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1856.76 2.10413
\(93\) 0 0
\(94\) −758.753 −0.832547
\(95\) 82.3676 0.0889552
\(96\) 0 0
\(97\) 752.168 0.787330 0.393665 0.919254i \(-0.371207\pi\)
0.393665 + 0.919254i \(0.371207\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 298.952 0.298952
\(101\) −559.226 −0.550941 −0.275471 0.961310i \(-0.588834\pi\)
−0.275471 + 0.961310i \(0.588834\pi\)
\(102\) 0 0
\(103\) −205.331 −0.196426 −0.0982128 0.995165i \(-0.531313\pi\)
−0.0982128 + 0.995165i \(0.531313\pi\)
\(104\) −724.349 −0.682964
\(105\) 0 0
\(106\) −676.330 −0.619726
\(107\) 1470.17 1.32829 0.664145 0.747604i \(-0.268796\pi\)
0.664145 + 0.747604i \(0.268796\pi\)
\(108\) 0 0
\(109\) −774.193 −0.680314 −0.340157 0.940369i \(-0.610480\pi\)
−0.340157 + 0.940369i \(0.610480\pi\)
\(110\) −1262.96 −1.09471
\(111\) 0 0
\(112\) 0 0
\(113\) 406.121 0.338094 0.169047 0.985608i \(-0.445931\pi\)
0.169047 + 0.985608i \(0.445931\pi\)
\(114\) 0 0
\(115\) 776.361 0.629531
\(116\) 74.8632 0.0599213
\(117\) 0 0
\(118\) −1045.72 −0.815817
\(119\) 0 0
\(120\) 0 0
\(121\) 1865.82 1.40182
\(122\) 1082.96 0.803660
\(123\) 0 0
\(124\) 2017.09 1.46081
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −451.639 −0.315563 −0.157781 0.987474i \(-0.550434\pi\)
−0.157781 + 0.987474i \(0.550434\pi\)
\(128\) 1986.33 1.37163
\(129\) 0 0
\(130\) −915.029 −0.617334
\(131\) −361.932 −0.241390 −0.120695 0.992690i \(-0.538512\pi\)
−0.120695 + 0.992690i \(0.538512\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −3666.75 −2.36387
\(135\) 0 0
\(136\) 38.7062 0.0244047
\(137\) −2512.72 −1.56698 −0.783490 0.621404i \(-0.786563\pi\)
−0.783490 + 0.621404i \(0.786563\pi\)
\(138\) 0 0
\(139\) 1165.71 0.711324 0.355662 0.934615i \(-0.384255\pi\)
0.355662 + 0.934615i \(0.384255\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4295.94 2.53879
\(143\) 2316.14 1.35444
\(144\) 0 0
\(145\) 31.3024 0.0179277
\(146\) −4172.71 −2.36531
\(147\) 0 0
\(148\) −444.526 −0.246891
\(149\) 2637.49 1.45014 0.725072 0.688673i \(-0.241806\pi\)
0.725072 + 0.688673i \(0.241806\pi\)
\(150\) 0 0
\(151\) −2582.32 −1.39170 −0.695849 0.718188i \(-0.744972\pi\)
−0.695849 + 0.718188i \(0.744972\pi\)
\(152\) −291.292 −0.155440
\(153\) 0 0
\(154\) 0 0
\(155\) 843.401 0.437055
\(156\) 0 0
\(157\) 225.788 0.114776 0.0573880 0.998352i \(-0.481723\pi\)
0.0573880 + 0.998352i \(0.481723\pi\)
\(158\) −1341.29 −0.675361
\(159\) 0 0
\(160\) 1079.64 0.533457
\(161\) 0 0
\(162\) 0 0
\(163\) 1489.01 0.715509 0.357755 0.933816i \(-0.383542\pi\)
0.357755 + 0.933816i \(0.383542\pi\)
\(164\) 3190.46 1.51910
\(165\) 0 0
\(166\) 4858.77 2.27177
\(167\) −2858.73 −1.32464 −0.662322 0.749220i \(-0.730429\pi\)
−0.662322 + 0.749220i \(0.730429\pi\)
\(168\) 0 0
\(169\) −518.925 −0.236197
\(170\) 48.8954 0.0220594
\(171\) 0 0
\(172\) −175.245 −0.0776877
\(173\) 32.0040 0.0140648 0.00703242 0.999975i \(-0.497761\pi\)
0.00703242 + 0.999975i \(0.497761\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −942.483 −0.403650
\(177\) 0 0
\(178\) −5021.62 −2.11453
\(179\) −738.444 −0.308346 −0.154173 0.988044i \(-0.549271\pi\)
−0.154173 + 0.988044i \(0.549271\pi\)
\(180\) 0 0
\(181\) −1991.91 −0.817997 −0.408998 0.912535i \(-0.634122\pi\)
−0.408998 + 0.912535i \(0.634122\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2745.59 −1.10004
\(185\) −185.869 −0.0738668
\(186\) 0 0
\(187\) −123.765 −0.0483989
\(188\) 2030.96 0.787890
\(189\) 0 0
\(190\) −367.973 −0.140503
\(191\) −2153.68 −0.815890 −0.407945 0.913006i \(-0.633755\pi\)
−0.407945 + 0.913006i \(0.633755\pi\)
\(192\) 0 0
\(193\) 1003.94 0.374431 0.187215 0.982319i \(-0.440054\pi\)
0.187215 + 0.982319i \(0.440054\pi\)
\(194\) −3360.27 −1.24357
\(195\) 0 0
\(196\) 0 0
\(197\) −1716.80 −0.620899 −0.310449 0.950590i \(-0.600479\pi\)
−0.310449 + 0.950590i \(0.600479\pi\)
\(198\) 0 0
\(199\) −5337.86 −1.90146 −0.950730 0.310019i \(-0.899665\pi\)
−0.950730 + 0.310019i \(0.899665\pi\)
\(200\) −442.061 −0.156292
\(201\) 0 0
\(202\) 2498.31 0.870201
\(203\) 0 0
\(204\) 0 0
\(205\) 1334.02 0.454497
\(206\) 917.304 0.310250
\(207\) 0 0
\(208\) −682.842 −0.227628
\(209\) 931.420 0.308266
\(210\) 0 0
\(211\) 860.589 0.280784 0.140392 0.990096i \(-0.455164\pi\)
0.140392 + 0.990096i \(0.455164\pi\)
\(212\) 1810.34 0.586485
\(213\) 0 0
\(214\) −6567.92 −2.09801
\(215\) −73.2747 −0.0232432
\(216\) 0 0
\(217\) 0 0
\(218\) 3458.66 1.07454
\(219\) 0 0
\(220\) 3380.57 1.03599
\(221\) −89.6695 −0.0272933
\(222\) 0 0
\(223\) −3661.12 −1.09940 −0.549701 0.835361i \(-0.685259\pi\)
−0.549701 + 0.835361i \(0.685259\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1814.32 −0.534013
\(227\) −6032.38 −1.76380 −0.881902 0.471434i \(-0.843737\pi\)
−0.881902 + 0.471434i \(0.843737\pi\)
\(228\) 0 0
\(229\) −3849.64 −1.11088 −0.555439 0.831557i \(-0.687450\pi\)
−0.555439 + 0.831557i \(0.687450\pi\)
\(230\) −3468.35 −0.994332
\(231\) 0 0
\(232\) −110.701 −0.0313269
\(233\) 1705.20 0.479448 0.239724 0.970841i \(-0.422943\pi\)
0.239724 + 0.970841i \(0.422943\pi\)
\(234\) 0 0
\(235\) 849.202 0.235727
\(236\) 2799.09 0.772057
\(237\) 0 0
\(238\) 0 0
\(239\) −6804.54 −1.84163 −0.920814 0.390001i \(-0.872475\pi\)
−0.920814 + 0.390001i \(0.872475\pi\)
\(240\) 0 0
\(241\) 4642.98 1.24100 0.620499 0.784207i \(-0.286930\pi\)
0.620499 + 0.784207i \(0.286930\pi\)
\(242\) −8335.44 −2.21414
\(243\) 0 0
\(244\) −2898.77 −0.760553
\(245\) 0 0
\(246\) 0 0
\(247\) 674.827 0.173839
\(248\) −2982.68 −0.763711
\(249\) 0 0
\(250\) −558.431 −0.141273
\(251\) −5912.64 −1.48686 −0.743431 0.668813i \(-0.766803\pi\)
−0.743431 + 0.668813i \(0.766803\pi\)
\(252\) 0 0
\(253\) 8779.16 2.18159
\(254\) 2017.67 0.498425
\(255\) 0 0
\(256\) −2223.49 −0.542844
\(257\) 1697.83 0.412093 0.206047 0.978542i \(-0.433940\pi\)
0.206047 + 0.978542i \(0.433940\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2449.27 0.584220
\(261\) 0 0
\(262\) 1616.91 0.381271
\(263\) 5661.78 1.32745 0.663727 0.747975i \(-0.268974\pi\)
0.663727 + 0.747975i \(0.268974\pi\)
\(264\) 0 0
\(265\) 756.954 0.175469
\(266\) 0 0
\(267\) 0 0
\(268\) 9814.83 2.23708
\(269\) −5330.01 −1.20809 −0.604046 0.796950i \(-0.706446\pi\)
−0.604046 + 0.796950i \(0.706446\pi\)
\(270\) 0 0
\(271\) −2034.94 −0.456139 −0.228069 0.973645i \(-0.573241\pi\)
−0.228069 + 0.973645i \(0.573241\pi\)
\(272\) 36.4883 0.00813392
\(273\) 0 0
\(274\) 11225.5 2.47502
\(275\) 1413.51 0.309956
\(276\) 0 0
\(277\) −867.657 −0.188204 −0.0941019 0.995563i \(-0.529998\pi\)
−0.0941019 + 0.995563i \(0.529998\pi\)
\(278\) −5207.73 −1.12352
\(279\) 0 0
\(280\) 0 0
\(281\) −2049.70 −0.435142 −0.217571 0.976045i \(-0.569813\pi\)
−0.217571 + 0.976045i \(0.569813\pi\)
\(282\) 0 0
\(283\) 6625.11 1.39160 0.695799 0.718237i \(-0.255051\pi\)
0.695799 + 0.718237i \(0.255051\pi\)
\(284\) −11499.0 −2.40261
\(285\) 0 0
\(286\) −10347.2 −2.13932
\(287\) 0 0
\(288\) 0 0
\(289\) −4908.21 −0.999025
\(290\) −139.842 −0.0283165
\(291\) 0 0
\(292\) 11169.1 2.23844
\(293\) 5670.84 1.13070 0.565348 0.824852i \(-0.308742\pi\)
0.565348 + 0.824852i \(0.308742\pi\)
\(294\) 0 0
\(295\) 1170.38 0.230990
\(296\) 657.323 0.129075
\(297\) 0 0
\(298\) −11782.8 −2.29048
\(299\) 6360.62 1.23025
\(300\) 0 0
\(301\) 0 0
\(302\) 11536.4 2.19816
\(303\) 0 0
\(304\) −274.600 −0.0518073
\(305\) −1212.06 −0.227548
\(306\) 0 0
\(307\) −272.638 −0.0506849 −0.0253424 0.999679i \(-0.508068\pi\)
−0.0253424 + 0.999679i \(0.508068\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3767.85 −0.690321
\(311\) 1220.83 0.222595 0.111298 0.993787i \(-0.464499\pi\)
0.111298 + 0.993787i \(0.464499\pi\)
\(312\) 0 0
\(313\) 775.195 0.139989 0.0699946 0.997547i \(-0.477702\pi\)
0.0699946 + 0.997547i \(0.477702\pi\)
\(314\) −1008.69 −0.181286
\(315\) 0 0
\(316\) 3590.24 0.639135
\(317\) −7393.24 −1.30992 −0.654962 0.755662i \(-0.727315\pi\)
−0.654962 + 0.755662i \(0.727315\pi\)
\(318\) 0 0
\(319\) 353.970 0.0621270
\(320\) −4156.47 −0.726105
\(321\) 0 0
\(322\) 0 0
\(323\) −36.0600 −0.00621186
\(324\) 0 0
\(325\) 1024.11 0.174792
\(326\) −6652.06 −1.13013
\(327\) 0 0
\(328\) −4717.74 −0.794188
\(329\) 0 0
\(330\) 0 0
\(331\) −3736.95 −0.620548 −0.310274 0.950647i \(-0.600421\pi\)
−0.310274 + 0.950647i \(0.600421\pi\)
\(332\) −13005.5 −2.14991
\(333\) 0 0
\(334\) 12771.2 2.09225
\(335\) 4103.85 0.669306
\(336\) 0 0
\(337\) −8230.24 −1.33036 −0.665178 0.746685i \(-0.731644\pi\)
−0.665178 + 0.746685i \(0.731644\pi\)
\(338\) 2318.27 0.373068
\(339\) 0 0
\(340\) −130.879 −0.0208762
\(341\) 9537.25 1.51458
\(342\) 0 0
\(343\) 0 0
\(344\) 259.135 0.0406152
\(345\) 0 0
\(346\) −142.976 −0.0222151
\(347\) 12730.5 1.96948 0.984739 0.174036i \(-0.0556809\pi\)
0.984739 + 0.174036i \(0.0556809\pi\)
\(348\) 0 0
\(349\) 4487.30 0.688252 0.344126 0.938924i \(-0.388175\pi\)
0.344126 + 0.938924i \(0.388175\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 12208.7 1.84865
\(353\) 12263.5 1.84906 0.924532 0.381106i \(-0.124457\pi\)
0.924532 + 0.381106i \(0.124457\pi\)
\(354\) 0 0
\(355\) −4808.05 −0.718831
\(356\) 13441.4 2.00111
\(357\) 0 0
\(358\) 3298.96 0.487026
\(359\) 11073.2 1.62791 0.813957 0.580925i \(-0.197309\pi\)
0.813957 + 0.580925i \(0.197309\pi\)
\(360\) 0 0
\(361\) −6587.62 −0.960435
\(362\) 8898.74 1.29201
\(363\) 0 0
\(364\) 0 0
\(365\) 4670.13 0.669714
\(366\) 0 0
\(367\) 5864.72 0.834157 0.417079 0.908870i \(-0.363054\pi\)
0.417079 + 0.908870i \(0.363054\pi\)
\(368\) −2588.26 −0.366637
\(369\) 0 0
\(370\) 830.359 0.116671
\(371\) 0 0
\(372\) 0 0
\(373\) −9368.17 −1.30044 −0.650222 0.759744i \(-0.725324\pi\)
−0.650222 + 0.759744i \(0.725324\pi\)
\(374\) 552.913 0.0764451
\(375\) 0 0
\(376\) −3003.19 −0.411909
\(377\) 256.456 0.0350349
\(378\) 0 0
\(379\) 5537.81 0.750549 0.375275 0.926914i \(-0.377548\pi\)
0.375275 + 0.926914i \(0.377548\pi\)
\(380\) 984.958 0.132966
\(381\) 0 0
\(382\) 9621.46 1.28868
\(383\) −5243.17 −0.699513 −0.349756 0.936841i \(-0.613736\pi\)
−0.349756 + 0.936841i \(0.613736\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4485.05 −0.591406
\(387\) 0 0
\(388\) 8994.47 1.17687
\(389\) 10890.5 1.41947 0.709733 0.704471i \(-0.248816\pi\)
0.709733 + 0.704471i \(0.248816\pi\)
\(390\) 0 0
\(391\) −339.886 −0.0439610
\(392\) 0 0
\(393\) 0 0
\(394\) 7669.71 0.980697
\(395\) 1501.18 0.191221
\(396\) 0 0
\(397\) 13307.0 1.68227 0.841135 0.540826i \(-0.181888\pi\)
0.841135 + 0.540826i \(0.181888\pi\)
\(398\) 23846.6 3.00332
\(399\) 0 0
\(400\) −416.730 −0.0520912
\(401\) −7227.02 −0.900001 −0.450000 0.893028i \(-0.648576\pi\)
−0.450000 + 0.893028i \(0.648576\pi\)
\(402\) 0 0
\(403\) 6909.87 0.854107
\(404\) −6687.26 −0.823524
\(405\) 0 0
\(406\) 0 0
\(407\) −2101.82 −0.255979
\(408\) 0 0
\(409\) 5852.64 0.707565 0.353783 0.935328i \(-0.384895\pi\)
0.353783 + 0.935328i \(0.384895\pi\)
\(410\) −5959.66 −0.717869
\(411\) 0 0
\(412\) −2455.36 −0.293609
\(413\) 0 0
\(414\) 0 0
\(415\) −5437.98 −0.643228
\(416\) 8845.35 1.04250
\(417\) 0 0
\(418\) −4161.07 −0.486901
\(419\) 8344.19 0.972889 0.486444 0.873712i \(-0.338294\pi\)
0.486444 + 0.873712i \(0.338294\pi\)
\(420\) 0 0
\(421\) −10955.3 −1.26824 −0.634122 0.773233i \(-0.718638\pi\)
−0.634122 + 0.773233i \(0.718638\pi\)
\(422\) −3844.63 −0.443493
\(423\) 0 0
\(424\) −2676.96 −0.306615
\(425\) −54.7241 −0.00624591
\(426\) 0 0
\(427\) 0 0
\(428\) 17580.4 1.98547
\(429\) 0 0
\(430\) 327.351 0.0367122
\(431\) −9417.43 −1.05249 −0.526243 0.850334i \(-0.676400\pi\)
−0.526243 + 0.850334i \(0.676400\pi\)
\(432\) 0 0
\(433\) 8783.79 0.974878 0.487439 0.873157i \(-0.337931\pi\)
0.487439 + 0.873157i \(0.337931\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9257.85 −1.01690
\(437\) 2557.88 0.280000
\(438\) 0 0
\(439\) 10559.6 1.14802 0.574011 0.818847i \(-0.305386\pi\)
0.574011 + 0.818847i \(0.305386\pi\)
\(440\) −4998.86 −0.541617
\(441\) 0 0
\(442\) 400.593 0.0431092
\(443\) 4878.14 0.523177 0.261589 0.965179i \(-0.415754\pi\)
0.261589 + 0.965179i \(0.415754\pi\)
\(444\) 0 0
\(445\) 5620.23 0.598707
\(446\) 16355.9 1.73649
\(447\) 0 0
\(448\) 0 0
\(449\) −43.7917 −0.00460280 −0.00230140 0.999997i \(-0.500733\pi\)
−0.00230140 + 0.999997i \(0.500733\pi\)
\(450\) 0 0
\(451\) 15085.2 1.57502
\(452\) 4856.42 0.505369
\(453\) 0 0
\(454\) 26949.3 2.78589
\(455\) 0 0
\(456\) 0 0
\(457\) 2869.76 0.293745 0.146873 0.989155i \(-0.453079\pi\)
0.146873 + 0.989155i \(0.453079\pi\)
\(458\) 17198.0 1.75461
\(459\) 0 0
\(460\) 9283.78 0.940997
\(461\) 17910.1 1.80945 0.904723 0.426001i \(-0.140078\pi\)
0.904723 + 0.426001i \(0.140078\pi\)
\(462\) 0 0
\(463\) −7630.72 −0.765938 −0.382969 0.923761i \(-0.625098\pi\)
−0.382969 + 0.923761i \(0.625098\pi\)
\(464\) −104.357 −0.0104411
\(465\) 0 0
\(466\) −7617.89 −0.757279
\(467\) 6246.30 0.618939 0.309469 0.950909i \(-0.399849\pi\)
0.309469 + 0.950909i \(0.399849\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −3793.76 −0.372326
\(471\) 0 0
\(472\) −4139.03 −0.403632
\(473\) −828.597 −0.0805474
\(474\) 0 0
\(475\) 411.838 0.0397820
\(476\) 0 0
\(477\) 0 0
\(478\) 30398.9 2.90882
\(479\) 4119.58 0.392961 0.196480 0.980508i \(-0.437049\pi\)
0.196480 + 0.980508i \(0.437049\pi\)
\(480\) 0 0
\(481\) −1522.80 −0.144353
\(482\) −20742.2 −1.96013
\(483\) 0 0
\(484\) 22311.6 2.09538
\(485\) 3760.84 0.352105
\(486\) 0 0
\(487\) −1421.46 −0.132264 −0.0661320 0.997811i \(-0.521066\pi\)
−0.0661320 + 0.997811i \(0.521066\pi\)
\(488\) 4286.43 0.397618
\(489\) 0 0
\(490\) 0 0
\(491\) −19241.1 −1.76851 −0.884253 0.467007i \(-0.845332\pi\)
−0.884253 + 0.467007i \(0.845332\pi\)
\(492\) 0 0
\(493\) −13.7040 −0.00125192
\(494\) −3014.75 −0.274575
\(495\) 0 0
\(496\) −2811.76 −0.254540
\(497\) 0 0
\(498\) 0 0
\(499\) 9576.48 0.859123 0.429561 0.903038i \(-0.358668\pi\)
0.429561 + 0.903038i \(0.358668\pi\)
\(500\) 1494.76 0.133695
\(501\) 0 0
\(502\) 26414.4 2.34847
\(503\) 10581.9 0.938019 0.469009 0.883193i \(-0.344611\pi\)
0.469009 + 0.883193i \(0.344611\pi\)
\(504\) 0 0
\(505\) −2796.13 −0.246388
\(506\) −39220.4 −3.44577
\(507\) 0 0
\(508\) −5400.72 −0.471690
\(509\) −7082.86 −0.616782 −0.308391 0.951260i \(-0.599791\pi\)
−0.308391 + 0.951260i \(0.599791\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −5957.37 −0.514220
\(513\) 0 0
\(514\) −7584.98 −0.650893
\(515\) −1026.65 −0.0878442
\(516\) 0 0
\(517\) 9602.85 0.816891
\(518\) 0 0
\(519\) 0 0
\(520\) −3621.75 −0.305431
\(521\) 7153.92 0.601572 0.300786 0.953692i \(-0.402751\pi\)
0.300786 + 0.953692i \(0.402751\pi\)
\(522\) 0 0
\(523\) −15781.3 −1.31944 −0.659721 0.751511i \(-0.729326\pi\)
−0.659721 + 0.751511i \(0.729326\pi\)
\(524\) −4328.00 −0.360820
\(525\) 0 0
\(526\) −25293.7 −2.09669
\(527\) −369.235 −0.0305202
\(528\) 0 0
\(529\) 11942.5 0.981547
\(530\) −3381.65 −0.277150
\(531\) 0 0
\(532\) 0 0
\(533\) 10929.4 0.888192
\(534\) 0 0
\(535\) 7350.86 0.594029
\(536\) −14513.2 −1.16955
\(537\) 0 0
\(538\) 23811.5 1.90816
\(539\) 0 0
\(540\) 0 0
\(541\) −24277.2 −1.92931 −0.964656 0.263511i \(-0.915119\pi\)
−0.964656 + 0.263511i \(0.915119\pi\)
\(542\) 9090.97 0.720463
\(543\) 0 0
\(544\) −472.659 −0.0372520
\(545\) −3870.96 −0.304246
\(546\) 0 0
\(547\) −191.079 −0.0149359 −0.00746796 0.999972i \(-0.502377\pi\)
−0.00746796 + 0.999972i \(0.502377\pi\)
\(548\) −30047.3 −2.34226
\(549\) 0 0
\(550\) −6314.78 −0.489569
\(551\) 103.132 0.00797382
\(552\) 0 0
\(553\) 0 0
\(554\) 3876.21 0.297264
\(555\) 0 0
\(556\) 13939.6 1.06326
\(557\) 15721.6 1.19595 0.597974 0.801515i \(-0.295972\pi\)
0.597974 + 0.801515i \(0.295972\pi\)
\(558\) 0 0
\(559\) −600.330 −0.0454226
\(560\) 0 0
\(561\) 0 0
\(562\) 9156.92 0.687298
\(563\) −3044.50 −0.227905 −0.113952 0.993486i \(-0.536351\pi\)
−0.113952 + 0.993486i \(0.536351\pi\)
\(564\) 0 0
\(565\) 2030.60 0.151200
\(566\) −29597.3 −2.19800
\(567\) 0 0
\(568\) 17003.6 1.25609
\(569\) −13052.2 −0.961647 −0.480823 0.876817i \(-0.659662\pi\)
−0.480823 + 0.876817i \(0.659662\pi\)
\(570\) 0 0
\(571\) −5811.05 −0.425893 −0.212946 0.977064i \(-0.568306\pi\)
−0.212946 + 0.977064i \(0.568306\pi\)
\(572\) 27696.5 2.02456
\(573\) 0 0
\(574\) 0 0
\(575\) 3881.81 0.281535
\(576\) 0 0
\(577\) −22790.3 −1.64432 −0.822162 0.569254i \(-0.807232\pi\)
−0.822162 + 0.569254i \(0.807232\pi\)
\(578\) 21927.2 1.57794
\(579\) 0 0
\(580\) 374.316 0.0267976
\(581\) 0 0
\(582\) 0 0
\(583\) 8559.70 0.608073
\(584\) −16515.9 −1.17026
\(585\) 0 0
\(586\) −25334.2 −1.78591
\(587\) 20600.3 1.44850 0.724248 0.689540i \(-0.242187\pi\)
0.724248 + 0.689540i \(0.242187\pi\)
\(588\) 0 0
\(589\) 2778.76 0.194392
\(590\) −5228.61 −0.364845
\(591\) 0 0
\(592\) 619.657 0.0430198
\(593\) −773.830 −0.0535875 −0.0267938 0.999641i \(-0.508530\pi\)
−0.0267938 + 0.999641i \(0.508530\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 31539.3 2.16762
\(597\) 0 0
\(598\) −28415.7 −1.94315
\(599\) 13641.0 0.930479 0.465239 0.885185i \(-0.345968\pi\)
0.465239 + 0.885185i \(0.345968\pi\)
\(600\) 0 0
\(601\) 14271.8 0.968650 0.484325 0.874888i \(-0.339065\pi\)
0.484325 + 0.874888i \(0.339065\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −30879.6 −2.08025
\(605\) 9329.09 0.626912
\(606\) 0 0
\(607\) 3234.90 0.216311 0.108155 0.994134i \(-0.465506\pi\)
0.108155 + 0.994134i \(0.465506\pi\)
\(608\) 3557.10 0.237269
\(609\) 0 0
\(610\) 5414.80 0.359408
\(611\) 6957.40 0.460665
\(612\) 0 0
\(613\) 8413.52 0.554354 0.277177 0.960819i \(-0.410601\pi\)
0.277177 + 0.960819i \(0.410601\pi\)
\(614\) 1217.99 0.0800558
\(615\) 0 0
\(616\) 0 0
\(617\) −7077.93 −0.461826 −0.230913 0.972974i \(-0.574171\pi\)
−0.230913 + 0.972974i \(0.574171\pi\)
\(618\) 0 0
\(619\) 5464.74 0.354841 0.177420 0.984135i \(-0.443225\pi\)
0.177420 + 0.984135i \(0.443225\pi\)
\(620\) 10085.4 0.653292
\(621\) 0 0
\(622\) −5454.01 −0.351585
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −3463.14 −0.221110
\(627\) 0 0
\(628\) 2699.98 0.171562
\(629\) 81.3721 0.00515822
\(630\) 0 0
\(631\) −1569.98 −0.0990492 −0.0495246 0.998773i \(-0.515771\pi\)
−0.0495246 + 0.998773i \(0.515771\pi\)
\(632\) −5308.90 −0.334140
\(633\) 0 0
\(634\) 33028.9 2.06900
\(635\) −2258.19 −0.141124
\(636\) 0 0
\(637\) 0 0
\(638\) −1581.34 −0.0981284
\(639\) 0 0
\(640\) 9931.67 0.613412
\(641\) 14578.3 0.898298 0.449149 0.893457i \(-0.351727\pi\)
0.449149 + 0.893457i \(0.351727\pi\)
\(642\) 0 0
\(643\) −11980.0 −0.734750 −0.367375 0.930073i \(-0.619744\pi\)
−0.367375 + 0.930073i \(0.619744\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 161.096 0.00981151
\(647\) 5651.81 0.343424 0.171712 0.985147i \(-0.445070\pi\)
0.171712 + 0.985147i \(0.445070\pi\)
\(648\) 0 0
\(649\) 13234.7 0.800476
\(650\) −4575.15 −0.276080
\(651\) 0 0
\(652\) 17805.6 1.06951
\(653\) 2370.86 0.142081 0.0710405 0.997473i \(-0.477368\pi\)
0.0710405 + 0.997473i \(0.477368\pi\)
\(654\) 0 0
\(655\) −1809.66 −0.107953
\(656\) −4447.40 −0.264698
\(657\) 0 0
\(658\) 0 0
\(659\) −5233.92 −0.309385 −0.154692 0.987963i \(-0.549439\pi\)
−0.154692 + 0.987963i \(0.549439\pi\)
\(660\) 0 0
\(661\) 10456.0 0.615265 0.307633 0.951505i \(-0.400463\pi\)
0.307633 + 0.951505i \(0.400463\pi\)
\(662\) 16694.6 0.980144
\(663\) 0 0
\(664\) 19231.3 1.12398
\(665\) 0 0
\(666\) 0 0
\(667\) 972.079 0.0564303
\(668\) −34184.9 −1.98002
\(669\) 0 0
\(670\) −18333.7 −1.05716
\(671\) −13706.0 −0.788548
\(672\) 0 0
\(673\) −17658.6 −1.01143 −0.505714 0.862701i \(-0.668771\pi\)
−0.505714 + 0.862701i \(0.668771\pi\)
\(674\) 36768.1 2.10127
\(675\) 0 0
\(676\) −6205.34 −0.353057
\(677\) 9124.03 0.517969 0.258984 0.965881i \(-0.416612\pi\)
0.258984 + 0.965881i \(0.416612\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 193.531 0.0109141
\(681\) 0 0
\(682\) −42607.1 −2.39225
\(683\) −10523.7 −0.589570 −0.294785 0.955564i \(-0.595248\pi\)
−0.294785 + 0.955564i \(0.595248\pi\)
\(684\) 0 0
\(685\) −12563.6 −0.700775
\(686\) 0 0
\(687\) 0 0
\(688\) 244.286 0.0135368
\(689\) 6201.62 0.342907
\(690\) 0 0
\(691\) 32928.1 1.81280 0.906400 0.422420i \(-0.138819\pi\)
0.906400 + 0.422420i \(0.138819\pi\)
\(692\) 382.706 0.0210235
\(693\) 0 0
\(694\) −56872.8 −3.11075
\(695\) 5828.54 0.318114
\(696\) 0 0
\(697\) −584.024 −0.0317381
\(698\) −20046.8 −1.08708
\(699\) 0 0
\(700\) 0 0
\(701\) 2582.50 0.139144 0.0695719 0.997577i \(-0.477837\pi\)
0.0695719 + 0.997577i \(0.477837\pi\)
\(702\) 0 0
\(703\) −612.383 −0.0328542
\(704\) −47001.7 −2.51625
\(705\) 0 0
\(706\) −54786.4 −2.92056
\(707\) 0 0
\(708\) 0 0
\(709\) −6866.00 −0.363693 −0.181846 0.983327i \(-0.558207\pi\)
−0.181846 + 0.983327i \(0.558207\pi\)
\(710\) 21479.7 1.13538
\(711\) 0 0
\(712\) −19875.9 −1.04618
\(713\) 26191.4 1.37570
\(714\) 0 0
\(715\) 11580.7 0.605725
\(716\) −8830.36 −0.460902
\(717\) 0 0
\(718\) −49468.9 −2.57126
\(719\) 6581.85 0.341393 0.170696 0.985324i \(-0.445398\pi\)
0.170696 + 0.985324i \(0.445398\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 29429.8 1.51699
\(723\) 0 0
\(724\) −23819.4 −1.22271
\(725\) 156.512 0.00801753
\(726\) 0 0
\(727\) 15527.2 0.792119 0.396060 0.918225i \(-0.370377\pi\)
0.396060 + 0.918225i \(0.370377\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −20863.5 −1.05780
\(731\) 32.0792 0.00162311
\(732\) 0 0
\(733\) −3391.32 −0.170889 −0.0854443 0.996343i \(-0.527231\pi\)
−0.0854443 + 0.996343i \(0.527231\pi\)
\(734\) −26200.3 −1.31754
\(735\) 0 0
\(736\) 33527.7 1.67914
\(737\) 46406.7 2.31942
\(738\) 0 0
\(739\) −14335.0 −0.713562 −0.356781 0.934188i \(-0.616126\pi\)
−0.356781 + 0.934188i \(0.616126\pi\)
\(740\) −2222.63 −0.110413
\(741\) 0 0
\(742\) 0 0
\(743\) 27668.2 1.36615 0.683073 0.730350i \(-0.260643\pi\)
0.683073 + 0.730350i \(0.260643\pi\)
\(744\) 0 0
\(745\) 13187.4 0.648524
\(746\) 41851.8 2.05403
\(747\) 0 0
\(748\) −1479.99 −0.0723446
\(749\) 0 0
\(750\) 0 0
\(751\) −16929.9 −0.822609 −0.411305 0.911498i \(-0.634927\pi\)
−0.411305 + 0.911498i \(0.634927\pi\)
\(752\) −2831.10 −0.137287
\(753\) 0 0
\(754\) −1145.70 −0.0553370
\(755\) −12911.6 −0.622386
\(756\) 0 0
\(757\) 19445.3 0.933622 0.466811 0.884357i \(-0.345403\pi\)
0.466811 + 0.884357i \(0.345403\pi\)
\(758\) −24739.9 −1.18548
\(759\) 0 0
\(760\) −1456.46 −0.0695150
\(761\) −9996.97 −0.476202 −0.238101 0.971240i \(-0.576525\pi\)
−0.238101 + 0.971240i \(0.576525\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −25753.9 −1.21956
\(765\) 0 0
\(766\) 23423.6 1.10487
\(767\) 9588.76 0.451408
\(768\) 0 0
\(769\) 8382.34 0.393075 0.196538 0.980496i \(-0.437030\pi\)
0.196538 + 0.980496i \(0.437030\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 12005.2 0.559684
\(773\) 11297.5 0.525668 0.262834 0.964841i \(-0.415343\pi\)
0.262834 + 0.964841i \(0.415343\pi\)
\(774\) 0 0
\(775\) 4217.00 0.195457
\(776\) −13300.2 −0.615268
\(777\) 0 0
\(778\) −48652.9 −2.24202
\(779\) 4395.20 0.202149
\(780\) 0 0
\(781\) −54369.9 −2.49105
\(782\) 1518.42 0.0694356
\(783\) 0 0
\(784\) 0 0
\(785\) 1128.94 0.0513294
\(786\) 0 0
\(787\) −25174.4 −1.14024 −0.570120 0.821561i \(-0.693103\pi\)
−0.570120 + 0.821561i \(0.693103\pi\)
\(788\) −20529.6 −0.928093
\(789\) 0 0
\(790\) −6706.43 −0.302030
\(791\) 0 0
\(792\) 0 0
\(793\) −9930.22 −0.444681
\(794\) −59448.4 −2.65711
\(795\) 0 0
\(796\) −63830.5 −2.84222
\(797\) 26277.4 1.16787 0.583936 0.811800i \(-0.301512\pi\)
0.583936 + 0.811800i \(0.301512\pi\)
\(798\) 0 0
\(799\) −371.775 −0.0164611
\(800\) 5398.21 0.238569
\(801\) 0 0
\(802\) 32286.3 1.42153
\(803\) 52810.2 2.32084
\(804\) 0 0
\(805\) 0 0
\(806\) −30869.5 −1.34905
\(807\) 0 0
\(808\) 9888.48 0.430539
\(809\) −15676.9 −0.681298 −0.340649 0.940191i \(-0.610647\pi\)
−0.340649 + 0.940191i \(0.610647\pi\)
\(810\) 0 0
\(811\) 26600.5 1.15175 0.575876 0.817537i \(-0.304661\pi\)
0.575876 + 0.817537i \(0.304661\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 9389.77 0.404314
\(815\) 7445.04 0.319986
\(816\) 0 0
\(817\) −241.419 −0.0103380
\(818\) −26146.3 −1.11759
\(819\) 0 0
\(820\) 15952.3 0.679363
\(821\) −35601.3 −1.51339 −0.756696 0.653767i \(-0.773188\pi\)
−0.756696 + 0.653767i \(0.773188\pi\)
\(822\) 0 0
\(823\) 41975.0 1.77783 0.888916 0.458071i \(-0.151459\pi\)
0.888916 + 0.458071i \(0.151459\pi\)
\(824\) 3630.75 0.153499
\(825\) 0 0
\(826\) 0 0
\(827\) −27719.4 −1.16554 −0.582768 0.812639i \(-0.698030\pi\)
−0.582768 + 0.812639i \(0.698030\pi\)
\(828\) 0 0
\(829\) 27222.3 1.14049 0.570246 0.821474i \(-0.306848\pi\)
0.570246 + 0.821474i \(0.306848\pi\)
\(830\) 24293.9 1.01597
\(831\) 0 0
\(832\) −34053.4 −1.41898
\(833\) 0 0
\(834\) 0 0
\(835\) −14293.7 −0.592399
\(836\) 11138.0 0.460784
\(837\) 0 0
\(838\) −37277.2 −1.53666
\(839\) 24081.1 0.990909 0.495454 0.868634i \(-0.335002\pi\)
0.495454 + 0.868634i \(0.335002\pi\)
\(840\) 0 0
\(841\) −24349.8 −0.998393
\(842\) 48942.4 2.00317
\(843\) 0 0
\(844\) 10291.0 0.419704
\(845\) −2594.62 −0.105631
\(846\) 0 0
\(847\) 0 0
\(848\) −2523.56 −0.102193
\(849\) 0 0
\(850\) 244.477 0.00986529
\(851\) −5772.06 −0.232507
\(852\) 0 0
\(853\) −18493.0 −0.742309 −0.371155 0.928571i \(-0.621038\pi\)
−0.371155 + 0.928571i \(0.621038\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −25996.2 −1.03801
\(857\) 3499.25 0.139477 0.0697387 0.997565i \(-0.477783\pi\)
0.0697387 + 0.997565i \(0.477783\pi\)
\(858\) 0 0
\(859\) −32158.0 −1.27732 −0.638660 0.769489i \(-0.720511\pi\)
−0.638660 + 0.769489i \(0.720511\pi\)
\(860\) −876.224 −0.0347430
\(861\) 0 0
\(862\) 42071.9 1.66238
\(863\) 47704.7 1.88168 0.940838 0.338857i \(-0.110040\pi\)
0.940838 + 0.338857i \(0.110040\pi\)
\(864\) 0 0
\(865\) 160.020 0.00628999
\(866\) −39241.1 −1.53980
\(867\) 0 0
\(868\) 0 0
\(869\) 16975.4 0.662661
\(870\) 0 0
\(871\) 33622.3 1.30798
\(872\) 13689.6 0.531639
\(873\) 0 0
\(874\) −11427.2 −0.442255
\(875\) 0 0
\(876\) 0 0
\(877\) −24437.3 −0.940923 −0.470461 0.882421i \(-0.655913\pi\)
−0.470461 + 0.882421i \(0.655913\pi\)
\(878\) −47174.4 −1.81328
\(879\) 0 0
\(880\) −4712.42 −0.180518
\(881\) 13975.8 0.534458 0.267229 0.963633i \(-0.413892\pi\)
0.267229 + 0.963633i \(0.413892\pi\)
\(882\) 0 0
\(883\) 3193.55 0.121712 0.0608558 0.998147i \(-0.480617\pi\)
0.0608558 + 0.998147i \(0.480617\pi\)
\(884\) −1072.27 −0.0407969
\(885\) 0 0
\(886\) −21792.8 −0.826348
\(887\) −20099.3 −0.760843 −0.380422 0.924813i \(-0.624221\pi\)
−0.380422 + 0.924813i \(0.624221\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −25108.1 −0.945646
\(891\) 0 0
\(892\) −43779.9 −1.64334
\(893\) 2797.87 0.104846
\(894\) 0 0
\(895\) −3692.22 −0.137896
\(896\) 0 0
\(897\) 0 0
\(898\) 195.637 0.00727004
\(899\) 1056.02 0.0391771
\(900\) 0 0
\(901\) −331.389 −0.0122532
\(902\) −67392.3 −2.48771
\(903\) 0 0
\(904\) −7181.20 −0.264207
\(905\) −9959.55 −0.365819
\(906\) 0 0
\(907\) 23212.6 0.849792 0.424896 0.905242i \(-0.360311\pi\)
0.424896 + 0.905242i \(0.360311\pi\)
\(908\) −72135.6 −2.63646
\(909\) 0 0
\(910\) 0 0
\(911\) −2754.63 −0.100181 −0.0500905 0.998745i \(-0.515951\pi\)
−0.0500905 + 0.998745i \(0.515951\pi\)
\(912\) 0 0
\(913\) −61493.1 −2.22905
\(914\) −12820.5 −0.463965
\(915\) 0 0
\(916\) −46034.2 −1.66049
\(917\) 0 0
\(918\) 0 0
\(919\) 16218.0 0.582137 0.291068 0.956702i \(-0.405989\pi\)
0.291068 + 0.956702i \(0.405989\pi\)
\(920\) −13728.0 −0.491954
\(921\) 0 0
\(922\) −80012.2 −2.85798
\(923\) −39391.7 −1.40476
\(924\) 0 0
\(925\) −929.344 −0.0330342
\(926\) 34089.8 1.20978
\(927\) 0 0
\(928\) 1351.81 0.0478184
\(929\) −42592.6 −1.50422 −0.752109 0.659038i \(-0.770964\pi\)
−0.752109 + 0.659038i \(0.770964\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 20390.9 0.716659
\(933\) 0 0
\(934\) −27905.0 −0.977602
\(935\) −618.825 −0.0216446
\(936\) 0 0
\(937\) −4670.79 −0.162848 −0.0814238 0.996680i \(-0.525947\pi\)
−0.0814238 + 0.996680i \(0.525947\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 10154.8 0.352355
\(941\) −11036.5 −0.382338 −0.191169 0.981557i \(-0.561228\pi\)
−0.191169 + 0.981557i \(0.561228\pi\)
\(942\) 0 0
\(943\) 41427.2 1.43060
\(944\) −3901.86 −0.134528
\(945\) 0 0
\(946\) 3701.71 0.127223
\(947\) −28219.0 −0.968314 −0.484157 0.874981i \(-0.660874\pi\)
−0.484157 + 0.874981i \(0.660874\pi\)
\(948\) 0 0
\(949\) 38261.7 1.30878
\(950\) −1839.86 −0.0628348
\(951\) 0 0
\(952\) 0 0
\(953\) −42209.7 −1.43474 −0.717370 0.696693i \(-0.754654\pi\)
−0.717370 + 0.696693i \(0.754654\pi\)
\(954\) 0 0
\(955\) −10768.4 −0.364877
\(956\) −81369.2 −2.75279
\(957\) 0 0
\(958\) −18404.0 −0.620674
\(959\) 0 0
\(960\) 0 0
\(961\) −1337.99 −0.0449126
\(962\) 6803.02 0.228002
\(963\) 0 0
\(964\) 55521.0 1.85499
\(965\) 5019.70 0.167451
\(966\) 0 0
\(967\) 11522.8 0.383193 0.191596 0.981474i \(-0.438634\pi\)
0.191596 + 0.981474i \(0.438634\pi\)
\(968\) −32992.2 −1.09547
\(969\) 0 0
\(970\) −16801.3 −0.556143
\(971\) −42660.3 −1.40992 −0.704961 0.709246i \(-0.749035\pi\)
−0.704961 + 0.709246i \(0.749035\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 6350.30 0.208908
\(975\) 0 0
\(976\) 4040.80 0.132524
\(977\) 13991.7 0.458171 0.229085 0.973406i \(-0.426427\pi\)
0.229085 + 0.973406i \(0.426427\pi\)
\(978\) 0 0
\(979\) 63554.1 2.07477
\(980\) 0 0
\(981\) 0 0
\(982\) 85958.4 2.79332
\(983\) −25963.2 −0.842420 −0.421210 0.906963i \(-0.638394\pi\)
−0.421210 + 0.906963i \(0.638394\pi\)
\(984\) 0 0
\(985\) −8584.01 −0.277674
\(986\) 61.2217 0.00197738
\(987\) 0 0
\(988\) 8069.62 0.259847
\(989\) −2275.51 −0.0731617
\(990\) 0 0
\(991\) 14482.3 0.464224 0.232112 0.972689i \(-0.425436\pi\)
0.232112 + 0.972689i \(0.425436\pi\)
\(992\) 36422.8 1.16575
\(993\) 0 0
\(994\) 0 0
\(995\) −26689.3 −0.850359
\(996\) 0 0
\(997\) −4184.45 −0.132922 −0.0664609 0.997789i \(-0.521171\pi\)
−0.0664609 + 0.997789i \(0.521171\pi\)
\(998\) −42782.4 −1.35697
\(999\) 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 2205.4.a.ca.1.1 6
3.2 odd 2 245.4.a.p.1.6 yes 6
7.6 odd 2 2205.4.a.bz.1.1 6
15.14 odd 2 1225.4.a.bi.1.1 6
21.2 odd 6 245.4.e.p.116.1 12
21.5 even 6 245.4.e.q.116.1 12
21.11 odd 6 245.4.e.p.226.1 12
21.17 even 6 245.4.e.q.226.1 12
21.20 even 2 245.4.a.o.1.6 6
105.104 even 2 1225.4.a.bj.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.o.1.6 6 21.20 even 2
245.4.a.p.1.6 yes 6 3.2 odd 2
245.4.e.p.116.1 12 21.2 odd 6
245.4.e.p.226.1 12 21.11 odd 6
245.4.e.q.116.1 12 21.5 even 6
245.4.e.q.226.1 12 21.17 even 6
1225.4.a.bi.1.1 6 15.14 odd 2
1225.4.a.bj.1.1 6 105.104 even 2
2205.4.a.bz.1.1 6 7.6 odd 2
2205.4.a.ca.1.1 6 1.1 even 1 trivial