Properties

Label 162.4.a.f.1.2
Level $162$
Weight $4$
Character 162.1
Self dual yes
Analytic conductor $9.558$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.55830942093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{105}) \)
Defining polynomial: \( x^{2} - x - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 18)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.62348\) of defining polynomial
Character \(\chi\) \(=\) 162.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} +10.8704 q^{5} +24.8704 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} +10.8704 q^{5} +24.8704 q^{7} -8.00000 q^{8} -21.7409 q^{10} -42.7409 q^{11} +15.1296 q^{13} -49.7409 q^{14} +16.0000 q^{16} -13.8704 q^{17} +143.352 q^{19} +43.4817 q^{20} +85.4817 q^{22} +19.1296 q^{23} -6.83384 q^{25} -30.2591 q^{26} +99.4817 q^{28} +226.093 q^{29} +59.3887 q^{31} -32.0000 q^{32} +27.7409 q^{34} +270.352 q^{35} -84.1860 q^{37} -286.704 q^{38} -86.9634 q^{40} +203.259 q^{41} +325.890 q^{43} -170.963 q^{44} -38.2591 q^{46} +10.9436 q^{47} +275.538 q^{49} +13.6677 q^{50} +60.5183 q^{52} -140.186 q^{53} -464.611 q^{55} -198.963 q^{56} -452.186 q^{58} +114.741 q^{59} -755.056 q^{61} -118.777 q^{62} +64.0000 q^{64} +164.465 q^{65} -767.445 q^{67} -55.4817 q^{68} -540.704 q^{70} +335.854 q^{71} +167.279 q^{73} +168.372 q^{74} +573.409 q^{76} -1062.98 q^{77} -25.3155 q^{79} +173.927 q^{80} -406.518 q^{82} +287.721 q^{83} -150.777 q^{85} -651.780 q^{86} +341.927 q^{88} -860.817 q^{89} +376.279 q^{91} +76.5183 q^{92} -21.8872 q^{94} +1558.30 q^{95} -402.149 q^{97} -551.076 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} - 9 q^{5} + 19 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 8 q^{4} - 9 q^{5} + 19 q^{7} - 16 q^{8} + 18 q^{10} - 24 q^{11} + 61 q^{13} - 38 q^{14} + 32 q^{16} + 3 q^{17} + 133 q^{19} - 36 q^{20} + 48 q^{22} + 69 q^{23} + 263 q^{25} - 122 q^{26} + 76 q^{28} + 237 q^{29} + 211 q^{31} - 64 q^{32} - 6 q^{34} + 387 q^{35} + 262 q^{37} - 266 q^{38} + 72 q^{40} + 468 q^{41} - 86 q^{43} - 96 q^{44} - 138 q^{46} + 483 q^{47} - 33 q^{49} - 526 q^{50} + 244 q^{52} + 150 q^{53} - 837 q^{55} - 152 q^{56} - 474 q^{58} + 168 q^{59} - 1049 q^{61} - 422 q^{62} + 128 q^{64} - 747 q^{65} - 1166 q^{67} + 12 q^{68} - 774 q^{70} - 312 q^{71} - 311 q^{73} - 524 q^{74} + 532 q^{76} - 1173 q^{77} + 349 q^{79} - 144 q^{80} - 936 q^{82} + 1221 q^{83} - 486 q^{85} + 172 q^{86} + 192 q^{88} - 492 q^{89} + 107 q^{91} + 276 q^{92} - 966 q^{94} + 1764 q^{95} - 128 q^{97} + 66 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 10.8704 0.972280 0.486140 0.873881i \(-0.338404\pi\)
0.486140 + 0.873881i \(0.338404\pi\)
\(6\) 0 0
\(7\) 24.8704 1.34288 0.671438 0.741060i \(-0.265677\pi\)
0.671438 + 0.741060i \(0.265677\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) −21.7409 −0.687506
\(11\) −42.7409 −1.17153 −0.585766 0.810480i \(-0.699206\pi\)
−0.585766 + 0.810480i \(0.699206\pi\)
\(12\) 0 0
\(13\) 15.1296 0.322784 0.161392 0.986890i \(-0.448402\pi\)
0.161392 + 0.986890i \(0.448402\pi\)
\(14\) −49.7409 −0.949557
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −13.8704 −0.197887 −0.0989433 0.995093i \(-0.531546\pi\)
−0.0989433 + 0.995093i \(0.531546\pi\)
\(18\) 0 0
\(19\) 143.352 1.73091 0.865454 0.500989i \(-0.167030\pi\)
0.865454 + 0.500989i \(0.167030\pi\)
\(20\) 43.4817 0.486140
\(21\) 0 0
\(22\) 85.4817 0.828398
\(23\) 19.1296 0.173426 0.0867129 0.996233i \(-0.472364\pi\)
0.0867129 + 0.996233i \(0.472364\pi\)
\(24\) 0 0
\(25\) −6.83384 −0.0546707
\(26\) −30.2591 −0.228243
\(27\) 0 0
\(28\) 99.4817 0.671438
\(29\) 226.093 1.44774 0.723869 0.689937i \(-0.242362\pi\)
0.723869 + 0.689937i \(0.242362\pi\)
\(30\) 0 0
\(31\) 59.3887 0.344082 0.172041 0.985090i \(-0.444964\pi\)
0.172041 + 0.985090i \(0.444964\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) 27.7409 0.139927
\(35\) 270.352 1.30565
\(36\) 0 0
\(37\) −84.1860 −0.374056 −0.187028 0.982355i \(-0.559886\pi\)
−0.187028 + 0.982355i \(0.559886\pi\)
\(38\) −286.704 −1.22394
\(39\) 0 0
\(40\) −86.9634 −0.343753
\(41\) 203.259 0.774238 0.387119 0.922030i \(-0.373470\pi\)
0.387119 + 0.922030i \(0.373470\pi\)
\(42\) 0 0
\(43\) 325.890 1.15576 0.577881 0.816121i \(-0.303880\pi\)
0.577881 + 0.816121i \(0.303880\pi\)
\(44\) −170.963 −0.585766
\(45\) 0 0
\(46\) −38.2591 −0.122631
\(47\) 10.9436 0.0339636 0.0169818 0.999856i \(-0.494594\pi\)
0.0169818 + 0.999856i \(0.494594\pi\)
\(48\) 0 0
\(49\) 275.538 0.803318
\(50\) 13.6677 0.0386580
\(51\) 0 0
\(52\) 60.5183 0.161392
\(53\) −140.186 −0.363321 −0.181661 0.983361i \(-0.558147\pi\)
−0.181661 + 0.983361i \(0.558147\pi\)
\(54\) 0 0
\(55\) −464.611 −1.13906
\(56\) −198.963 −0.474779
\(57\) 0 0
\(58\) −452.186 −1.02371
\(59\) 114.741 0.253186 0.126593 0.991955i \(-0.459596\pi\)
0.126593 + 0.991955i \(0.459596\pi\)
\(60\) 0 0
\(61\) −755.056 −1.58484 −0.792419 0.609978i \(-0.791178\pi\)
−0.792419 + 0.609978i \(0.791178\pi\)
\(62\) −118.777 −0.243302
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 164.465 0.313836
\(66\) 0 0
\(67\) −767.445 −1.39938 −0.699689 0.714447i \(-0.746678\pi\)
−0.699689 + 0.714447i \(0.746678\pi\)
\(68\) −55.4817 −0.0989433
\(69\) 0 0
\(70\) −540.704 −0.923236
\(71\) 335.854 0.561387 0.280694 0.959797i \(-0.409436\pi\)
0.280694 + 0.959797i \(0.409436\pi\)
\(72\) 0 0
\(73\) 167.279 0.268199 0.134099 0.990968i \(-0.457186\pi\)
0.134099 + 0.990968i \(0.457186\pi\)
\(74\) 168.372 0.264498
\(75\) 0 0
\(76\) 573.409 0.865454
\(77\) −1062.98 −1.57322
\(78\) 0 0
\(79\) −25.3155 −0.0360534 −0.0180267 0.999838i \(-0.505738\pi\)
−0.0180267 + 0.999838i \(0.505738\pi\)
\(80\) 173.927 0.243070
\(81\) 0 0
\(82\) −406.518 −0.547469
\(83\) 287.721 0.380500 0.190250 0.981736i \(-0.439070\pi\)
0.190250 + 0.981736i \(0.439070\pi\)
\(84\) 0 0
\(85\) −150.777 −0.192401
\(86\) −651.780 −0.817248
\(87\) 0 0
\(88\) 341.927 0.414199
\(89\) −860.817 −1.02524 −0.512620 0.858615i \(-0.671325\pi\)
−0.512620 + 0.858615i \(0.671325\pi\)
\(90\) 0 0
\(91\) 376.279 0.433459
\(92\) 76.5183 0.0867129
\(93\) 0 0
\(94\) −21.8872 −0.0240159
\(95\) 1558.30 1.68293
\(96\) 0 0
\(97\) −402.149 −0.420949 −0.210475 0.977599i \(-0.567501\pi\)
−0.210475 + 0.977599i \(0.567501\pi\)
\(98\) −551.076 −0.568032
\(99\) 0 0
\(100\) −27.3353 −0.0273353
\(101\) −1331.43 −1.31170 −0.655852 0.754890i \(-0.727690\pi\)
−0.655852 + 0.754890i \(0.727690\pi\)
\(102\) 0 0
\(103\) −518.505 −0.496017 −0.248009 0.968758i \(-0.579776\pi\)
−0.248009 + 0.968758i \(0.579776\pi\)
\(104\) −121.037 −0.114121
\(105\) 0 0
\(106\) 280.372 0.256907
\(107\) 1471.87 1.32982 0.664912 0.746922i \(-0.268469\pi\)
0.664912 + 0.746922i \(0.268469\pi\)
\(108\) 0 0
\(109\) −643.668 −0.565616 −0.282808 0.959176i \(-0.591266\pi\)
−0.282808 + 0.959176i \(0.591266\pi\)
\(110\) 929.223 0.805435
\(111\) 0 0
\(112\) 397.927 0.335719
\(113\) 1023.73 0.852249 0.426125 0.904665i \(-0.359879\pi\)
0.426125 + 0.904665i \(0.359879\pi\)
\(114\) 0 0
\(115\) 207.947 0.168618
\(116\) 904.372 0.723869
\(117\) 0 0
\(118\) −229.482 −0.179030
\(119\) −344.963 −0.265737
\(120\) 0 0
\(121\) 495.780 0.372487
\(122\) 1510.11 1.12065
\(123\) 0 0
\(124\) 237.555 0.172041
\(125\) −1433.09 −1.02544
\(126\) 0 0
\(127\) −31.4481 −0.0219730 −0.0109865 0.999940i \(-0.503497\pi\)
−0.0109865 + 0.999940i \(0.503497\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) −328.930 −0.221916
\(131\) −1937.36 −1.29212 −0.646059 0.763287i \(-0.723584\pi\)
−0.646059 + 0.763287i \(0.723584\pi\)
\(132\) 0 0
\(133\) 3565.23 2.32439
\(134\) 1534.89 0.989510
\(135\) 0 0
\(136\) 110.963 0.0699635
\(137\) 1158.59 0.722521 0.361261 0.932465i \(-0.382346\pi\)
0.361261 + 0.932465i \(0.382346\pi\)
\(138\) 0 0
\(139\) 2311.60 1.41055 0.705277 0.708931i \(-0.250822\pi\)
0.705277 + 0.708931i \(0.250822\pi\)
\(140\) 1081.41 0.652826
\(141\) 0 0
\(142\) −671.707 −0.396961
\(143\) −646.651 −0.378151
\(144\) 0 0
\(145\) 2457.73 1.40761
\(146\) −334.558 −0.189645
\(147\) 0 0
\(148\) −336.744 −0.187028
\(149\) −2950.21 −1.62209 −0.811043 0.584987i \(-0.801100\pi\)
−0.811043 + 0.584987i \(0.801100\pi\)
\(150\) 0 0
\(151\) −1726.40 −0.930412 −0.465206 0.885202i \(-0.654020\pi\)
−0.465206 + 0.885202i \(0.654020\pi\)
\(152\) −1146.82 −0.611968
\(153\) 0 0
\(154\) 2125.97 1.11244
\(155\) 645.581 0.334544
\(156\) 0 0
\(157\) 1283.39 0.652392 0.326196 0.945302i \(-0.394233\pi\)
0.326196 + 0.945302i \(0.394233\pi\)
\(158\) 50.6311 0.0254936
\(159\) 0 0
\(160\) −347.854 −0.171877
\(161\) 475.761 0.232889
\(162\) 0 0
\(163\) 1033.93 0.496831 0.248415 0.968654i \(-0.420090\pi\)
0.248415 + 0.968654i \(0.420090\pi\)
\(164\) 813.037 0.387119
\(165\) 0 0
\(166\) −575.442 −0.269054
\(167\) −282.617 −0.130956 −0.0654778 0.997854i \(-0.520857\pi\)
−0.0654778 + 0.997854i \(0.520857\pi\)
\(168\) 0 0
\(169\) −1968.10 −0.895811
\(170\) 301.555 0.136048
\(171\) 0 0
\(172\) 1303.56 0.577881
\(173\) −3532.72 −1.55253 −0.776266 0.630405i \(-0.782889\pi\)
−0.776266 + 0.630405i \(0.782889\pi\)
\(174\) 0 0
\(175\) −169.960 −0.0734160
\(176\) −683.854 −0.292883
\(177\) 0 0
\(178\) 1721.63 0.724955
\(179\) −4052.74 −1.69227 −0.846135 0.532969i \(-0.821076\pi\)
−0.846135 + 0.532969i \(0.821076\pi\)
\(180\) 0 0
\(181\) −2830.97 −1.16257 −0.581283 0.813702i \(-0.697449\pi\)
−0.581283 + 0.813702i \(0.697449\pi\)
\(182\) −752.558 −0.306502
\(183\) 0 0
\(184\) −153.037 −0.0613153
\(185\) −915.137 −0.363688
\(186\) 0 0
\(187\) 592.834 0.231831
\(188\) 43.7744 0.0169818
\(189\) 0 0
\(190\) −3116.60 −1.19001
\(191\) 4509.24 1.70826 0.854129 0.520061i \(-0.174091\pi\)
0.854129 + 0.520061i \(0.174091\pi\)
\(192\) 0 0
\(193\) −3221.12 −1.20136 −0.600678 0.799491i \(-0.705103\pi\)
−0.600678 + 0.799491i \(0.705103\pi\)
\(194\) 804.299 0.297656
\(195\) 0 0
\(196\) 1102.15 0.401659
\(197\) 3784.20 1.36859 0.684297 0.729204i \(-0.260109\pi\)
0.684297 + 0.729204i \(0.260109\pi\)
\(198\) 0 0
\(199\) −2926.27 −1.04240 −0.521200 0.853435i \(-0.674515\pi\)
−0.521200 + 0.853435i \(0.674515\pi\)
\(200\) 54.6707 0.0193290
\(201\) 0 0
\(202\) 2662.86 0.927515
\(203\) 5623.03 1.94413
\(204\) 0 0
\(205\) 2209.51 0.752776
\(206\) 1037.01 0.350737
\(207\) 0 0
\(208\) 242.073 0.0806959
\(209\) −6126.99 −2.02781
\(210\) 0 0
\(211\) 313.474 0.102277 0.0511385 0.998692i \(-0.483715\pi\)
0.0511385 + 0.998692i \(0.483715\pi\)
\(212\) −560.744 −0.181661
\(213\) 0 0
\(214\) −2943.74 −0.940327
\(215\) 3542.57 1.12373
\(216\) 0 0
\(217\) 1477.02 0.462059
\(218\) 1287.34 0.399951
\(219\) 0 0
\(220\) −1858.45 −0.569529
\(221\) −209.854 −0.0638746
\(222\) 0 0
\(223\) −710.386 −0.213323 −0.106661 0.994295i \(-0.534016\pi\)
−0.106661 + 0.994295i \(0.534016\pi\)
\(224\) −795.854 −0.237389
\(225\) 0 0
\(226\) −2047.45 −0.602631
\(227\) 32.6616 0.00954991 0.00477496 0.999989i \(-0.498480\pi\)
0.00477496 + 0.999989i \(0.498480\pi\)
\(228\) 0 0
\(229\) −5503.85 −1.58823 −0.794114 0.607768i \(-0.792065\pi\)
−0.794114 + 0.607768i \(0.792065\pi\)
\(230\) −415.893 −0.119231
\(231\) 0 0
\(232\) −1808.74 −0.511853
\(233\) −6788.81 −1.90880 −0.954399 0.298534i \(-0.903502\pi\)
−0.954399 + 0.298534i \(0.903502\pi\)
\(234\) 0 0
\(235\) 118.962 0.0330221
\(236\) 458.963 0.126593
\(237\) 0 0
\(238\) 689.927 0.187905
\(239\) −429.389 −0.116213 −0.0581064 0.998310i \(-0.518506\pi\)
−0.0581064 + 0.998310i \(0.518506\pi\)
\(240\) 0 0
\(241\) −4843.65 −1.29463 −0.647317 0.762221i \(-0.724109\pi\)
−0.647317 + 0.762221i \(0.724109\pi\)
\(242\) −991.561 −0.263388
\(243\) 0 0
\(244\) −3020.23 −0.792419
\(245\) 2995.22 0.781050
\(246\) 0 0
\(247\) 2168.86 0.558709
\(248\) −475.110 −0.121651
\(249\) 0 0
\(250\) 2866.18 0.725093
\(251\) 2400.87 0.603752 0.301876 0.953347i \(-0.402387\pi\)
0.301876 + 0.953347i \(0.402387\pi\)
\(252\) 0 0
\(253\) −817.614 −0.203174
\(254\) 62.8963 0.0155373
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 2980.99 0.723538 0.361769 0.932268i \(-0.382173\pi\)
0.361769 + 0.932268i \(0.382173\pi\)
\(258\) 0 0
\(259\) −2093.74 −0.502312
\(260\) 657.860 0.156918
\(261\) 0 0
\(262\) 3874.71 0.913666
\(263\) 5560.82 1.30378 0.651891 0.758313i \(-0.273976\pi\)
0.651891 + 0.758313i \(0.273976\pi\)
\(264\) 0 0
\(265\) −1523.88 −0.353250
\(266\) −7130.46 −1.64360
\(267\) 0 0
\(268\) −3069.78 −0.699689
\(269\) 6288.50 1.42534 0.712671 0.701499i \(-0.247485\pi\)
0.712671 + 0.701499i \(0.247485\pi\)
\(270\) 0 0
\(271\) 6854.90 1.53655 0.768275 0.640119i \(-0.221115\pi\)
0.768275 + 0.640119i \(0.221115\pi\)
\(272\) −221.927 −0.0494717
\(273\) 0 0
\(274\) −2317.19 −0.510900
\(275\) 292.084 0.0640485
\(276\) 0 0
\(277\) 898.172 0.194823 0.0974114 0.995244i \(-0.468944\pi\)
0.0974114 + 0.995244i \(0.468944\pi\)
\(278\) −4623.20 −0.997413
\(279\) 0 0
\(280\) −2162.82 −0.461618
\(281\) 238.770 0.0506897 0.0253448 0.999679i \(-0.491932\pi\)
0.0253448 + 0.999679i \(0.491932\pi\)
\(282\) 0 0
\(283\) 2071.16 0.435045 0.217523 0.976055i \(-0.430202\pi\)
0.217523 + 0.976055i \(0.430202\pi\)
\(284\) 1343.41 0.280694
\(285\) 0 0
\(286\) 1293.30 0.267393
\(287\) 5055.14 1.03971
\(288\) 0 0
\(289\) −4720.61 −0.960841
\(290\) −4915.45 −0.995329
\(291\) 0 0
\(292\) 669.116 0.134099
\(293\) −6577.76 −1.31153 −0.655763 0.754967i \(-0.727653\pi\)
−0.655763 + 0.754967i \(0.727653\pi\)
\(294\) 0 0
\(295\) 1247.28 0.246168
\(296\) 673.488 0.132249
\(297\) 0 0
\(298\) 5900.42 1.14699
\(299\) 289.422 0.0559790
\(300\) 0 0
\(301\) 8105.03 1.55205
\(302\) 3452.80 0.657901
\(303\) 0 0
\(304\) 2293.63 0.432727
\(305\) −8207.78 −1.54091
\(306\) 0 0
\(307\) −5237.30 −0.973644 −0.486822 0.873501i \(-0.661844\pi\)
−0.486822 + 0.873501i \(0.661844\pi\)
\(308\) −4251.93 −0.786612
\(309\) 0 0
\(310\) −1291.16 −0.236558
\(311\) 5704.99 1.04019 0.520097 0.854107i \(-0.325896\pi\)
0.520097 + 0.854107i \(0.325896\pi\)
\(312\) 0 0
\(313\) 5077.48 0.916920 0.458460 0.888715i \(-0.348401\pi\)
0.458460 + 0.888715i \(0.348401\pi\)
\(314\) −2566.78 −0.461311
\(315\) 0 0
\(316\) −101.262 −0.0180267
\(317\) 2868.91 0.508309 0.254155 0.967164i \(-0.418203\pi\)
0.254155 + 0.967164i \(0.418203\pi\)
\(318\) 0 0
\(319\) −9663.41 −1.69607
\(320\) 695.707 0.121535
\(321\) 0 0
\(322\) −951.521 −0.164678
\(323\) −1988.36 −0.342523
\(324\) 0 0
\(325\) −103.393 −0.0176468
\(326\) −2067.85 −0.351312
\(327\) 0 0
\(328\) −1626.07 −0.273734
\(329\) 272.172 0.0456089
\(330\) 0 0
\(331\) 2031.35 0.337320 0.168660 0.985674i \(-0.446056\pi\)
0.168660 + 0.985674i \(0.446056\pi\)
\(332\) 1150.88 0.190250
\(333\) 0 0
\(334\) 565.235 0.0925996
\(335\) −8342.46 −1.36059
\(336\) 0 0
\(337\) 9798.28 1.58382 0.791909 0.610640i \(-0.209088\pi\)
0.791909 + 0.610640i \(0.209088\pi\)
\(338\) 3936.19 0.633434
\(339\) 0 0
\(340\) −603.110 −0.0962006
\(341\) −2538.32 −0.403103
\(342\) 0 0
\(343\) −1677.81 −0.264120
\(344\) −2607.12 −0.408624
\(345\) 0 0
\(346\) 7065.45 1.09781
\(347\) −4556.56 −0.704925 −0.352463 0.935826i \(-0.614656\pi\)
−0.352463 + 0.935826i \(0.614656\pi\)
\(348\) 0 0
\(349\) 3348.89 0.513644 0.256822 0.966459i \(-0.417325\pi\)
0.256822 + 0.966459i \(0.417325\pi\)
\(350\) 339.921 0.0519129
\(351\) 0 0
\(352\) 1367.71 0.207100
\(353\) 1862.27 0.280789 0.140395 0.990096i \(-0.455163\pi\)
0.140395 + 0.990096i \(0.455163\pi\)
\(354\) 0 0
\(355\) 3650.87 0.545826
\(356\) −3443.27 −0.512620
\(357\) 0 0
\(358\) 8105.49 1.19662
\(359\) 6179.46 0.908466 0.454233 0.890883i \(-0.349913\pi\)
0.454233 + 0.890883i \(0.349913\pi\)
\(360\) 0 0
\(361\) 13690.8 1.99604
\(362\) 5661.94 0.822058
\(363\) 0 0
\(364\) 1505.12 0.216729
\(365\) 1818.39 0.260765
\(366\) 0 0
\(367\) 6873.95 0.977704 0.488852 0.872367i \(-0.337416\pi\)
0.488852 + 0.872367i \(0.337416\pi\)
\(368\) 306.073 0.0433564
\(369\) 0 0
\(370\) 1830.27 0.257166
\(371\) −3486.48 −0.487896
\(372\) 0 0
\(373\) 1270.03 0.176299 0.0881494 0.996107i \(-0.471905\pi\)
0.0881494 + 0.996107i \(0.471905\pi\)
\(374\) −1185.67 −0.163929
\(375\) 0 0
\(376\) −87.5489 −0.0120079
\(377\) 3420.69 0.467306
\(378\) 0 0
\(379\) 2490.54 0.337548 0.168774 0.985655i \(-0.446019\pi\)
0.168774 + 0.985655i \(0.446019\pi\)
\(380\) 6233.20 0.841464
\(381\) 0 0
\(382\) −9018.48 −1.20792
\(383\) 312.425 0.0416820 0.0208410 0.999783i \(-0.493366\pi\)
0.0208410 + 0.999783i \(0.493366\pi\)
\(384\) 0 0
\(385\) −11555.1 −1.52961
\(386\) 6442.25 0.849487
\(387\) 0 0
\(388\) −1608.60 −0.210475
\(389\) 9643.17 1.25688 0.628442 0.777856i \(-0.283693\pi\)
0.628442 + 0.777856i \(0.283693\pi\)
\(390\) 0 0
\(391\) −265.335 −0.0343186
\(392\) −2204.30 −0.284016
\(393\) 0 0
\(394\) −7568.40 −0.967742
\(395\) −275.191 −0.0350540
\(396\) 0 0
\(397\) −2260.32 −0.285749 −0.142874 0.989741i \(-0.545634\pi\)
−0.142874 + 0.989741i \(0.545634\pi\)
\(398\) 5852.53 0.737088
\(399\) 0 0
\(400\) −109.341 −0.0136677
\(401\) 10169.3 1.26641 0.633204 0.773985i \(-0.281739\pi\)
0.633204 + 0.773985i \(0.281739\pi\)
\(402\) 0 0
\(403\) 898.526 0.111064
\(404\) −5325.71 −0.655852
\(405\) 0 0
\(406\) −11246.1 −1.37471
\(407\) 3598.18 0.438219
\(408\) 0 0
\(409\) −949.833 −0.114832 −0.0574159 0.998350i \(-0.518286\pi\)
−0.0574159 + 0.998350i \(0.518286\pi\)
\(410\) −4419.03 −0.532293
\(411\) 0 0
\(412\) −2074.02 −0.248009
\(413\) 2853.65 0.339998
\(414\) 0 0
\(415\) 3127.65 0.369953
\(416\) −484.146 −0.0570606
\(417\) 0 0
\(418\) 12254.0 1.43388
\(419\) −11799.3 −1.37573 −0.687866 0.725837i \(-0.741453\pi\)
−0.687866 + 0.725837i \(0.741453\pi\)
\(420\) 0 0
\(421\) −6412.93 −0.742392 −0.371196 0.928555i \(-0.621052\pi\)
−0.371196 + 0.928555i \(0.621052\pi\)
\(422\) −626.948 −0.0723207
\(423\) 0 0
\(424\) 1121.49 0.128453
\(425\) 94.7882 0.0108186
\(426\) 0 0
\(427\) −18778.6 −2.12824
\(428\) 5887.48 0.664912
\(429\) 0 0
\(430\) −7085.13 −0.794594
\(431\) −12042.7 −1.34589 −0.672945 0.739693i \(-0.734971\pi\)
−0.672945 + 0.739693i \(0.734971\pi\)
\(432\) 0 0
\(433\) 7279.83 0.807959 0.403980 0.914768i \(-0.367627\pi\)
0.403980 + 0.914768i \(0.367627\pi\)
\(434\) −2954.05 −0.326725
\(435\) 0 0
\(436\) −2574.67 −0.282808
\(437\) 2742.27 0.300184
\(438\) 0 0
\(439\) −3598.70 −0.391245 −0.195623 0.980679i \(-0.562673\pi\)
−0.195623 + 0.980679i \(0.562673\pi\)
\(440\) 3716.89 0.402718
\(441\) 0 0
\(442\) 419.707 0.0451662
\(443\) −14786.4 −1.58583 −0.792913 0.609335i \(-0.791437\pi\)
−0.792913 + 0.609335i \(0.791437\pi\)
\(444\) 0 0
\(445\) −9357.45 −0.996822
\(446\) 1420.77 0.150842
\(447\) 0 0
\(448\) 1591.71 0.167860
\(449\) 114.489 0.0120336 0.00601681 0.999982i \(-0.498085\pi\)
0.00601681 + 0.999982i \(0.498085\pi\)
\(450\) 0 0
\(451\) −8687.47 −0.907044
\(452\) 4094.91 0.426125
\(453\) 0 0
\(454\) −65.3233 −0.00675281
\(455\) 4090.31 0.421444
\(456\) 0 0
\(457\) 6311.14 0.646001 0.323001 0.946399i \(-0.395308\pi\)
0.323001 + 0.946399i \(0.395308\pi\)
\(458\) 11007.7 1.12305
\(459\) 0 0
\(460\) 831.786 0.0843092
\(461\) −13744.8 −1.38863 −0.694317 0.719670i \(-0.744293\pi\)
−0.694317 + 0.719670i \(0.744293\pi\)
\(462\) 0 0
\(463\) 15648.6 1.57074 0.785369 0.619028i \(-0.212473\pi\)
0.785369 + 0.619028i \(0.212473\pi\)
\(464\) 3617.49 0.361935
\(465\) 0 0
\(466\) 13577.6 1.34972
\(467\) −7395.79 −0.732840 −0.366420 0.930450i \(-0.619417\pi\)
−0.366420 + 0.930450i \(0.619417\pi\)
\(468\) 0 0
\(469\) −19086.7 −1.87919
\(470\) −237.923 −0.0233502
\(471\) 0 0
\(472\) −917.927 −0.0895148
\(473\) −13928.8 −1.35401
\(474\) 0 0
\(475\) −979.645 −0.0946299
\(476\) −1379.85 −0.132869
\(477\) 0 0
\(478\) 858.777 0.0821748
\(479\) −6261.07 −0.597235 −0.298617 0.954373i \(-0.596525\pi\)
−0.298617 + 0.954373i \(0.596525\pi\)
\(480\) 0 0
\(481\) −1273.70 −0.120739
\(482\) 9687.30 0.915445
\(483\) 0 0
\(484\) 1983.12 0.186244
\(485\) −4371.54 −0.409281
\(486\) 0 0
\(487\) −10314.7 −0.959763 −0.479881 0.877333i \(-0.659320\pi\)
−0.479881 + 0.877333i \(0.659320\pi\)
\(488\) 6040.45 0.560325
\(489\) 0 0
\(490\) −5990.43 −0.552286
\(491\) −2760.90 −0.253763 −0.126881 0.991918i \(-0.540497\pi\)
−0.126881 + 0.991918i \(0.540497\pi\)
\(492\) 0 0
\(493\) −3136.01 −0.286488
\(494\) −4337.71 −0.395067
\(495\) 0 0
\(496\) 950.220 0.0860204
\(497\) 8352.82 0.753874
\(498\) 0 0
\(499\) −9585.99 −0.859976 −0.429988 0.902835i \(-0.641482\pi\)
−0.429988 + 0.902835i \(0.641482\pi\)
\(500\) −5732.36 −0.512718
\(501\) 0 0
\(502\) −4801.75 −0.426917
\(503\) 8829.60 0.782689 0.391344 0.920244i \(-0.372010\pi\)
0.391344 + 0.920244i \(0.372010\pi\)
\(504\) 0 0
\(505\) −14473.2 −1.27534
\(506\) 1635.23 0.143666
\(507\) 0 0
\(508\) −125.793 −0.0109865
\(509\) 4741.74 0.412916 0.206458 0.978456i \(-0.433806\pi\)
0.206458 + 0.978456i \(0.433806\pi\)
\(510\) 0 0
\(511\) 4160.30 0.360158
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −5961.99 −0.511619
\(515\) −5636.37 −0.482268
\(516\) 0 0
\(517\) −467.739 −0.0397894
\(518\) 4187.48 0.355188
\(519\) 0 0
\(520\) −1315.72 −0.110958
\(521\) −2753.22 −0.231518 −0.115759 0.993277i \(-0.536930\pi\)
−0.115759 + 0.993277i \(0.536930\pi\)
\(522\) 0 0
\(523\) 17115.3 1.43098 0.715489 0.698624i \(-0.246204\pi\)
0.715489 + 0.698624i \(0.246204\pi\)
\(524\) −7749.42 −0.646059
\(525\) 0 0
\(526\) −11121.6 −0.921913
\(527\) −823.747 −0.0680891
\(528\) 0 0
\(529\) −11801.1 −0.969924
\(530\) 3047.76 0.249786
\(531\) 0 0
\(532\) 14260.9 1.16220
\(533\) 3075.22 0.249911
\(534\) 0 0
\(535\) 15999.9 1.29296
\(536\) 6139.56 0.494755
\(537\) 0 0
\(538\) −12577.0 −1.00787
\(539\) −11776.7 −0.941113
\(540\) 0 0
\(541\) 17880.1 1.42093 0.710467 0.703731i \(-0.248484\pi\)
0.710467 + 0.703731i \(0.248484\pi\)
\(542\) −13709.8 −1.08651
\(543\) 0 0
\(544\) 443.854 0.0349817
\(545\) −6996.94 −0.549938
\(546\) 0 0
\(547\) −13069.5 −1.02159 −0.510795 0.859703i \(-0.670649\pi\)
−0.510795 + 0.859703i \(0.670649\pi\)
\(548\) 4634.38 0.361261
\(549\) 0 0
\(550\) −584.168 −0.0452891
\(551\) 32410.9 2.50590
\(552\) 0 0
\(553\) −629.608 −0.0484153
\(554\) −1796.34 −0.137761
\(555\) 0 0
\(556\) 9246.39 0.705277
\(557\) 9507.62 0.723251 0.361626 0.932323i \(-0.382222\pi\)
0.361626 + 0.932323i \(0.382222\pi\)
\(558\) 0 0
\(559\) 4930.58 0.373061
\(560\) 4325.63 0.326413
\(561\) 0 0
\(562\) −477.539 −0.0358430
\(563\) −20444.6 −1.53044 −0.765221 0.643768i \(-0.777370\pi\)
−0.765221 + 0.643768i \(0.777370\pi\)
\(564\) 0 0
\(565\) 11128.3 0.828625
\(566\) −4142.33 −0.307624
\(567\) 0 0
\(568\) −2686.83 −0.198480
\(569\) 2646.06 0.194954 0.0974770 0.995238i \(-0.468923\pi\)
0.0974770 + 0.995238i \(0.468923\pi\)
\(570\) 0 0
\(571\) 1757.03 0.128773 0.0643864 0.997925i \(-0.479491\pi\)
0.0643864 + 0.997925i \(0.479491\pi\)
\(572\) −2586.60 −0.189076
\(573\) 0 0
\(574\) −10110.3 −0.735183
\(575\) −130.728 −0.00948130
\(576\) 0 0
\(577\) −7515.43 −0.542238 −0.271119 0.962546i \(-0.587394\pi\)
−0.271119 + 0.962546i \(0.587394\pi\)
\(578\) 9441.22 0.679417
\(579\) 0 0
\(580\) 9830.91 0.703804
\(581\) 7155.75 0.510964
\(582\) 0 0
\(583\) 5991.67 0.425642
\(584\) −1338.23 −0.0948226
\(585\) 0 0
\(586\) 13155.5 0.927388
\(587\) −4952.49 −0.348230 −0.174115 0.984725i \(-0.555706\pi\)
−0.174115 + 0.984725i \(0.555706\pi\)
\(588\) 0 0
\(589\) 8513.50 0.595573
\(590\) −2494.56 −0.174067
\(591\) 0 0
\(592\) −1346.98 −0.0935141
\(593\) 17115.3 1.18523 0.592613 0.805487i \(-0.298096\pi\)
0.592613 + 0.805487i \(0.298096\pi\)
\(594\) 0 0
\(595\) −3749.90 −0.258371
\(596\) −11800.8 −0.811043
\(597\) 0 0
\(598\) −578.845 −0.0395831
\(599\) 8414.13 0.573943 0.286972 0.957939i \(-0.407351\pi\)
0.286972 + 0.957939i \(0.407351\pi\)
\(600\) 0 0
\(601\) 28094.2 1.90680 0.953399 0.301712i \(-0.0975582\pi\)
0.953399 + 0.301712i \(0.0975582\pi\)
\(602\) −16210.1 −1.09746
\(603\) 0 0
\(604\) −6905.59 −0.465206
\(605\) 5389.34 0.362162
\(606\) 0 0
\(607\) 1430.85 0.0956775 0.0478388 0.998855i \(-0.484767\pi\)
0.0478388 + 0.998855i \(0.484767\pi\)
\(608\) −4587.27 −0.305984
\(609\) 0 0
\(610\) 16415.6 1.08959
\(611\) 165.572 0.0109629
\(612\) 0 0
\(613\) 14438.1 0.951306 0.475653 0.879633i \(-0.342212\pi\)
0.475653 + 0.879633i \(0.342212\pi\)
\(614\) 10474.6 0.688470
\(615\) 0 0
\(616\) 8503.87 0.556218
\(617\) 25444.9 1.66025 0.830125 0.557577i \(-0.188269\pi\)
0.830125 + 0.557577i \(0.188269\pi\)
\(618\) 0 0
\(619\) −3479.46 −0.225931 −0.112966 0.993599i \(-0.536035\pi\)
−0.112966 + 0.993599i \(0.536035\pi\)
\(620\) 2582.32 0.167272
\(621\) 0 0
\(622\) −11410.0 −0.735528
\(623\) −21408.9 −1.37677
\(624\) 0 0
\(625\) −14724.1 −0.942340
\(626\) −10155.0 −0.648361
\(627\) 0 0
\(628\) 5133.55 0.326196
\(629\) 1167.70 0.0740208
\(630\) 0 0
\(631\) 11151.7 0.703552 0.351776 0.936084i \(-0.385578\pi\)
0.351776 + 0.936084i \(0.385578\pi\)
\(632\) 202.524 0.0127468
\(633\) 0 0
\(634\) −5737.82 −0.359429
\(635\) −341.855 −0.0213639
\(636\) 0 0
\(637\) 4168.77 0.259298
\(638\) 19326.8 1.19930
\(639\) 0 0
\(640\) −1391.41 −0.0859383
\(641\) −1092.78 −0.0673356 −0.0336678 0.999433i \(-0.510719\pi\)
−0.0336678 + 0.999433i \(0.510719\pi\)
\(642\) 0 0
\(643\) 31694.0 1.94384 0.971922 0.235305i \(-0.0756089\pi\)
0.971922 + 0.235305i \(0.0756089\pi\)
\(644\) 1903.04 0.116445
\(645\) 0 0
\(646\) 3976.71 0.242201
\(647\) −13719.9 −0.833672 −0.416836 0.908982i \(-0.636861\pi\)
−0.416836 + 0.908982i \(0.636861\pi\)
\(648\) 0 0
\(649\) −4904.12 −0.296616
\(650\) 206.786 0.0124782
\(651\) 0 0
\(652\) 4135.71 0.248415
\(653\) 13285.2 0.796155 0.398078 0.917352i \(-0.369677\pi\)
0.398078 + 0.917352i \(0.369677\pi\)
\(654\) 0 0
\(655\) −21059.9 −1.25630
\(656\) 3252.15 0.193559
\(657\) 0 0
\(658\) −544.344 −0.0322504
\(659\) −2593.98 −0.153334 −0.0766670 0.997057i \(-0.524428\pi\)
−0.0766670 + 0.997057i \(0.524428\pi\)
\(660\) 0 0
\(661\) −15875.3 −0.934159 −0.467079 0.884215i \(-0.654694\pi\)
−0.467079 + 0.884215i \(0.654694\pi\)
\(662\) −4062.70 −0.238522
\(663\) 0 0
\(664\) −2301.77 −0.134527
\(665\) 38755.6 2.25996
\(666\) 0 0
\(667\) 4325.06 0.251075
\(668\) −1130.47 −0.0654778
\(669\) 0 0
\(670\) 16684.9 0.962081
\(671\) 32271.8 1.85669
\(672\) 0 0
\(673\) 21434.7 1.22771 0.613853 0.789420i \(-0.289619\pi\)
0.613853 + 0.789420i \(0.289619\pi\)
\(674\) −19596.6 −1.11993
\(675\) 0 0
\(676\) −7872.38 −0.447905
\(677\) −20333.0 −1.15430 −0.577150 0.816638i \(-0.695835\pi\)
−0.577150 + 0.816638i \(0.695835\pi\)
\(678\) 0 0
\(679\) −10001.6 −0.565283
\(680\) 1206.22 0.0680241
\(681\) 0 0
\(682\) 5076.65 0.285037
\(683\) −27149.0 −1.52097 −0.760487 0.649353i \(-0.775040\pi\)
−0.760487 + 0.649353i \(0.775040\pi\)
\(684\) 0 0
\(685\) 12594.4 0.702493
\(686\) 3355.61 0.186761
\(687\) 0 0
\(688\) 5214.24 0.288941
\(689\) −2120.95 −0.117274
\(690\) 0 0
\(691\) 22710.2 1.25027 0.625134 0.780517i \(-0.285044\pi\)
0.625134 + 0.780517i \(0.285044\pi\)
\(692\) −14130.9 −0.776266
\(693\) 0 0
\(694\) 9113.12 0.498457
\(695\) 25128.0 1.37146
\(696\) 0 0
\(697\) −2819.29 −0.153211
\(698\) −6697.78 −0.363201
\(699\) 0 0
\(700\) −679.842 −0.0367080
\(701\) −20079.5 −1.08187 −0.540936 0.841064i \(-0.681930\pi\)
−0.540936 + 0.841064i \(0.681930\pi\)
\(702\) 0 0
\(703\) −12068.2 −0.647457
\(704\) −2735.41 −0.146441
\(705\) 0 0
\(706\) −3724.54 −0.198548
\(707\) −33113.2 −1.76146
\(708\) 0 0
\(709\) −28983.7 −1.53527 −0.767634 0.640889i \(-0.778566\pi\)
−0.767634 + 0.640889i \(0.778566\pi\)
\(710\) −7301.74 −0.385957
\(711\) 0 0
\(712\) 6886.54 0.362477
\(713\) 1136.08 0.0596726
\(714\) 0 0
\(715\) −7029.37 −0.367669
\(716\) −16211.0 −0.846135
\(717\) 0 0
\(718\) −12358.9 −0.642383
\(719\) 14496.2 0.751901 0.375951 0.926640i \(-0.377316\pi\)
0.375951 + 0.926640i \(0.377316\pi\)
\(720\) 0 0
\(721\) −12895.4 −0.666090
\(722\) −27381.7 −1.41141
\(723\) 0 0
\(724\) −11323.9 −0.581283
\(725\) −1545.08 −0.0791488
\(726\) 0 0
\(727\) 5000.50 0.255101 0.127550 0.991832i \(-0.459289\pi\)
0.127550 + 0.991832i \(0.459289\pi\)
\(728\) −3010.23 −0.153251
\(729\) 0 0
\(730\) −3636.79 −0.184388
\(731\) −4520.24 −0.228710
\(732\) 0 0
\(733\) 17515.6 0.882609 0.441305 0.897357i \(-0.354516\pi\)
0.441305 + 0.897357i \(0.354516\pi\)
\(734\) −13747.9 −0.691341
\(735\) 0 0
\(736\) −612.146 −0.0306576
\(737\) 32801.3 1.63942
\(738\) 0 0
\(739\) −20169.2 −1.00397 −0.501985 0.864876i \(-0.667397\pi\)
−0.501985 + 0.864876i \(0.667397\pi\)
\(740\) −3660.55 −0.181844
\(741\) 0 0
\(742\) 6972.97 0.344994
\(743\) −36702.6 −1.81223 −0.906116 0.423029i \(-0.860967\pi\)
−0.906116 + 0.423029i \(0.860967\pi\)
\(744\) 0 0
\(745\) −32070.1 −1.57712
\(746\) −2540.05 −0.124662
\(747\) 0 0
\(748\) 2371.34 0.115915
\(749\) 36606.0 1.78579
\(750\) 0 0
\(751\) −33320.1 −1.61900 −0.809499 0.587122i \(-0.800261\pi\)
−0.809499 + 0.587122i \(0.800261\pi\)
\(752\) 175.098 0.00849090
\(753\) 0 0
\(754\) −6841.38 −0.330436
\(755\) −18766.7 −0.904622
\(756\) 0 0
\(757\) 26515.6 1.27309 0.636543 0.771241i \(-0.280364\pi\)
0.636543 + 0.771241i \(0.280364\pi\)
\(758\) −4981.08 −0.238682
\(759\) 0 0
\(760\) −12466.4 −0.595005
\(761\) −5684.35 −0.270772 −0.135386 0.990793i \(-0.543227\pi\)
−0.135386 + 0.990793i \(0.543227\pi\)
\(762\) 0 0
\(763\) −16008.3 −0.759553
\(764\) 18037.0 0.854129
\(765\) 0 0
\(766\) −624.851 −0.0294736
\(767\) 1735.98 0.0817244
\(768\) 0 0
\(769\) −398.377 −0.0186812 −0.00934060 0.999956i \(-0.502973\pi\)
−0.00934060 + 0.999956i \(0.502973\pi\)
\(770\) 23110.2 1.08160
\(771\) 0 0
\(772\) −12884.5 −0.600678
\(773\) −8437.17 −0.392579 −0.196290 0.980546i \(-0.562889\pi\)
−0.196290 + 0.980546i \(0.562889\pi\)
\(774\) 0 0
\(775\) −405.853 −0.0188112
\(776\) 3217.20 0.148828
\(777\) 0 0
\(778\) −19286.3 −0.888751
\(779\) 29137.6 1.34013
\(780\) 0 0
\(781\) −14354.7 −0.657683
\(782\) 530.671 0.0242669
\(783\) 0 0
\(784\) 4408.61 0.200830
\(785\) 13951.0 0.634308
\(786\) 0 0
\(787\) 16277.9 0.737286 0.368643 0.929571i \(-0.379823\pi\)
0.368643 + 0.929571i \(0.379823\pi\)
\(788\) 15136.8 0.684297
\(789\) 0 0
\(790\) 550.381 0.0247870
\(791\) 25460.5 1.14447
\(792\) 0 0
\(793\) −11423.7 −0.511560
\(794\) 4520.64 0.202055
\(795\) 0 0
\(796\) −11705.1 −0.521200
\(797\) 5112.11 0.227202 0.113601 0.993526i \(-0.463761\pi\)
0.113601 + 0.993526i \(0.463761\pi\)
\(798\) 0 0
\(799\) −151.793 −0.00672094
\(800\) 218.683 0.00966450
\(801\) 0 0
\(802\) −20338.6 −0.895485
\(803\) −7149.64 −0.314204
\(804\) 0 0
\(805\) 5171.72 0.226434
\(806\) −1797.05 −0.0785341
\(807\) 0 0
\(808\) 10651.4 0.463757
\(809\) −13141.2 −0.571100 −0.285550 0.958364i \(-0.592176\pi\)
−0.285550 + 0.958364i \(0.592176\pi\)
\(810\) 0 0
\(811\) −18614.2 −0.805957 −0.402979 0.915209i \(-0.632025\pi\)
−0.402979 + 0.915209i \(0.632025\pi\)
\(812\) 22492.1 0.972067
\(813\) 0 0
\(814\) −7196.36 −0.309868
\(815\) 11239.2 0.483059
\(816\) 0 0
\(817\) 46717.1 2.00052
\(818\) 1899.67 0.0811983
\(819\) 0 0
\(820\) 8838.05 0.376388
\(821\) −11320.0 −0.481208 −0.240604 0.970623i \(-0.577345\pi\)
−0.240604 + 0.970623i \(0.577345\pi\)
\(822\) 0 0
\(823\) 10866.6 0.460249 0.230125 0.973161i \(-0.426087\pi\)
0.230125 + 0.973161i \(0.426087\pi\)
\(824\) 4148.04 0.175369
\(825\) 0 0
\(826\) −5707.31 −0.240415
\(827\) −13059.3 −0.549114 −0.274557 0.961571i \(-0.588531\pi\)
−0.274557 + 0.961571i \(0.588531\pi\)
\(828\) 0 0
\(829\) −21203.7 −0.888341 −0.444171 0.895942i \(-0.646502\pi\)
−0.444171 + 0.895942i \(0.646502\pi\)
\(830\) −6255.30 −0.261596
\(831\) 0 0
\(832\) 968.293 0.0403480
\(833\) −3821.83 −0.158966
\(834\) 0 0
\(835\) −3072.17 −0.127326
\(836\) −24508.0 −1.01391
\(837\) 0 0
\(838\) 23598.5 0.972790
\(839\) −14480.6 −0.595858 −0.297929 0.954588i \(-0.596296\pi\)
−0.297929 + 0.954588i \(0.596296\pi\)
\(840\) 0 0
\(841\) 26729.0 1.09595
\(842\) 12825.9 0.524950
\(843\) 0 0
\(844\) 1253.90 0.0511385
\(845\) −21394.0 −0.870979
\(846\) 0 0
\(847\) 12330.3 0.500204
\(848\) −2242.98 −0.0908303
\(849\) 0 0
\(850\) −189.576 −0.00764990
\(851\) −1610.44 −0.0648710
\(852\) 0 0
\(853\) −6467.23 −0.259594 −0.129797 0.991541i \(-0.541433\pi\)
−0.129797 + 0.991541i \(0.541433\pi\)
\(854\) 37557.1 1.50489
\(855\) 0 0
\(856\) −11775.0 −0.470164
\(857\) 3894.64 0.155237 0.0776186 0.996983i \(-0.475268\pi\)
0.0776186 + 0.996983i \(0.475268\pi\)
\(858\) 0 0
\(859\) −37653.8 −1.49561 −0.747805 0.663918i \(-0.768892\pi\)
−0.747805 + 0.663918i \(0.768892\pi\)
\(860\) 14170.3 0.561863
\(861\) 0 0
\(862\) 24085.5 0.951688
\(863\) 47067.5 1.85654 0.928271 0.371905i \(-0.121295\pi\)
0.928271 + 0.371905i \(0.121295\pi\)
\(864\) 0 0
\(865\) −38402.2 −1.50950
\(866\) −14559.7 −0.571313
\(867\) 0 0
\(868\) 5908.09 0.231030
\(869\) 1082.01 0.0422377
\(870\) 0 0
\(871\) −11611.1 −0.451697
\(872\) 5149.34 0.199976
\(873\) 0 0
\(874\) −5484.53 −0.212262
\(875\) −35641.6 −1.37703
\(876\) 0 0
\(877\) −7443.54 −0.286603 −0.143301 0.989679i \(-0.545772\pi\)
−0.143301 + 0.989679i \(0.545772\pi\)
\(878\) 7197.40 0.276652
\(879\) 0 0
\(880\) −7433.78 −0.284764
\(881\) −13781.9 −0.527040 −0.263520 0.964654i \(-0.584884\pi\)
−0.263520 + 0.964654i \(0.584884\pi\)
\(882\) 0 0
\(883\) 12230.3 0.466119 0.233060 0.972462i \(-0.425126\pi\)
0.233060 + 0.972462i \(0.425126\pi\)
\(884\) −839.415 −0.0319373
\(885\) 0 0
\(886\) 29572.7 1.12135
\(887\) −1674.87 −0.0634011 −0.0317005 0.999497i \(-0.510092\pi\)
−0.0317005 + 0.999497i \(0.510092\pi\)
\(888\) 0 0
\(889\) −782.128 −0.0295070
\(890\) 18714.9 0.704859
\(891\) 0 0
\(892\) −2841.54 −0.106661
\(893\) 1568.79 0.0587878
\(894\) 0 0
\(895\) −44055.1 −1.64536
\(896\) −3183.41 −0.118695
\(897\) 0 0
\(898\) −228.979 −0.00850905
\(899\) 13427.4 0.498140
\(900\) 0 0
\(901\) 1944.44 0.0718964
\(902\) 17374.9 0.641377
\(903\) 0 0
\(904\) −8189.82 −0.301316
\(905\) −30773.8 −1.13034
\(906\) 0 0
\(907\) 17088.1 0.625580 0.312790 0.949822i \(-0.398736\pi\)
0.312790 + 0.949822i \(0.398736\pi\)
\(908\) 130.647 0.00477496
\(909\) 0 0
\(910\) −8180.63 −0.298006
\(911\) 21288.4 0.774222 0.387111 0.922033i \(-0.373473\pi\)
0.387111 + 0.922033i \(0.373473\pi\)
\(912\) 0 0
\(913\) −12297.4 −0.445768
\(914\) −12622.3 −0.456792
\(915\) 0 0
\(916\) −22015.4 −0.794114
\(917\) −48182.8 −1.73516
\(918\) 0 0
\(919\) −10413.4 −0.373784 −0.186892 0.982380i \(-0.559841\pi\)
−0.186892 + 0.982380i \(0.559841\pi\)
\(920\) −1663.57 −0.0596156
\(921\) 0 0
\(922\) 27489.6 0.981912
\(923\) 5081.32 0.181207
\(924\) 0 0
\(925\) 575.313 0.0204499
\(926\) −31297.2 −1.11068
\(927\) 0 0
\(928\) −7234.98 −0.255926
\(929\) −6823.43 −0.240979 −0.120490 0.992715i \(-0.538446\pi\)
−0.120490 + 0.992715i \(0.538446\pi\)
\(930\) 0 0
\(931\) 39499.0 1.39047
\(932\) −27155.2 −0.954399
\(933\) 0 0
\(934\) 14791.6 0.518196
\(935\) 6444.36 0.225404
\(936\) 0 0
\(937\) 41049.8 1.43120 0.715602 0.698508i \(-0.246152\pi\)
0.715602 + 0.698508i \(0.246152\pi\)
\(938\) 38173.4 1.32879
\(939\) 0 0
\(940\) 475.847 0.0165111
\(941\) 2416.12 0.0837017 0.0418508 0.999124i \(-0.486675\pi\)
0.0418508 + 0.999124i \(0.486675\pi\)
\(942\) 0 0
\(943\) 3888.26 0.134273
\(944\) 1835.85 0.0632966
\(945\) 0 0
\(946\) 27857.7 0.957432
\(947\) −2394.07 −0.0821509 −0.0410755 0.999156i \(-0.513078\pi\)
−0.0410755 + 0.999156i \(0.513078\pi\)
\(948\) 0 0
\(949\) 2530.86 0.0865702
\(950\) 1959.29 0.0669134
\(951\) 0 0
\(952\) 2759.71 0.0939523
\(953\) −50651.3 −1.72168 −0.860838 0.508879i \(-0.830060\pi\)
−0.860838 + 0.508879i \(0.830060\pi\)
\(954\) 0 0
\(955\) 49017.4 1.66091
\(956\) −1717.55 −0.0581064
\(957\) 0 0
\(958\) 12522.1 0.422309
\(959\) 28814.7 0.970257
\(960\) 0 0
\(961\) −26264.0 −0.881608
\(962\) 2547.40 0.0853756
\(963\) 0 0
\(964\) −19374.6 −0.647317
\(965\) −35015.0 −1.16805
\(966\) 0 0
\(967\) 25034.9 0.832542 0.416271 0.909241i \(-0.363337\pi\)
0.416271 + 0.909241i \(0.363337\pi\)
\(968\) −3966.24 −0.131694
\(969\) 0 0
\(970\) 8743.07 0.289405
\(971\) 49553.7 1.63775 0.818875 0.573972i \(-0.194598\pi\)
0.818875 + 0.573972i \(0.194598\pi\)
\(972\) 0 0
\(973\) 57490.4 1.89420
\(974\) 20629.4 0.678655
\(975\) 0 0
\(976\) −12080.9 −0.396209
\(977\) −35176.2 −1.15188 −0.575939 0.817492i \(-0.695364\pi\)
−0.575939 + 0.817492i \(0.695364\pi\)
\(978\) 0 0
\(979\) 36792.1 1.20110
\(980\) 11980.9 0.390525
\(981\) 0 0
\(982\) 5521.79 0.179437
\(983\) 33349.3 1.08207 0.541036 0.840999i \(-0.318032\pi\)
0.541036 + 0.840999i \(0.318032\pi\)
\(984\) 0 0
\(985\) 41135.8 1.33066
\(986\) 6272.01 0.202578
\(987\) 0 0
\(988\) 8675.43 0.279354
\(989\) 6234.14 0.200439
\(990\) 0 0
\(991\) 23066.3 0.739378 0.369689 0.929156i \(-0.379464\pi\)
0.369689 + 0.929156i \(0.379464\pi\)
\(992\) −1900.44 −0.0608256
\(993\) 0 0
\(994\) −16705.6 −0.533069
\(995\) −31809.7 −1.01350
\(996\) 0 0
\(997\) −55769.9 −1.77156 −0.885782 0.464102i \(-0.846377\pi\)
−0.885782 + 0.464102i \(0.846377\pi\)
\(998\) 19172.0 0.608095
\(999\) 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 162.4.a.f.1.2 2
3.2 odd 2 162.4.a.g.1.1 2
4.3 odd 2 1296.4.a.l.1.2 2
9.2 odd 6 54.4.c.b.37.2 4
9.4 even 3 18.4.c.b.7.2 4
9.5 odd 6 54.4.c.b.19.2 4
9.7 even 3 18.4.c.b.13.2 yes 4
12.11 even 2 1296.4.a.r.1.1 2
36.7 odd 6 144.4.i.b.49.1 4
36.11 even 6 432.4.i.b.145.2 4
36.23 even 6 432.4.i.b.289.2 4
36.31 odd 6 144.4.i.b.97.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.4.c.b.7.2 4 9.4 even 3
18.4.c.b.13.2 yes 4 9.7 even 3
54.4.c.b.19.2 4 9.5 odd 6
54.4.c.b.37.2 4 9.2 odd 6
144.4.i.b.49.1 4 36.7 odd 6
144.4.i.b.97.1 4 36.31 odd 6
162.4.a.f.1.2 2 1.1 even 1 trivial
162.4.a.g.1.1 2 3.2 odd 2
432.4.i.b.145.2 4 36.11 even 6
432.4.i.b.289.2 4 36.23 even 6
1296.4.a.l.1.2 2 4.3 odd 2
1296.4.a.r.1.1 2 12.11 even 2