Properties

Label 8470.2.a.da.1.5
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.38498000.1
Defining polynomial: \(x^{6} - x^{5} - 12 x^{4} + 5 x^{3} + 38 x^{2} - x - 19\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.15804\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.15804 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.15804 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.65894 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.15804 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.15804 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.65894 q^{9} -1.00000 q^{10} +1.15804 q^{12} -3.58654 q^{13} +1.00000 q^{14} -1.15804 q^{15} +1.00000 q^{16} +4.29025 q^{17} -1.65894 q^{18} -4.89322 q^{19} -1.00000 q^{20} +1.15804 q^{21} +2.85699 q^{23} +1.15804 q^{24} +1.00000 q^{25} -3.58654 q^{26} -5.39525 q^{27} +1.00000 q^{28} -0.754826 q^{29} -1.15804 q^{30} -9.35354 q^{31} +1.00000 q^{32} +4.29025 q^{34} -1.00000 q^{35} -1.65894 q^{36} -4.24080 q^{37} -4.89322 q^{38} -4.15337 q^{39} -1.00000 q^{40} +2.02891 q^{41} +1.15804 q^{42} -0.0689189 q^{43} +1.65894 q^{45} +2.85699 q^{46} +2.23785 q^{47} +1.15804 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.96830 q^{51} -3.58654 q^{52} +13.1026 q^{53} -5.39525 q^{54} +1.00000 q^{56} -5.66656 q^{57} -0.754826 q^{58} +1.22512 q^{59} -1.15804 q^{60} -4.03036 q^{61} -9.35354 q^{62} -1.65894 q^{63} +1.00000 q^{64} +3.58654 q^{65} -14.3059 q^{67} +4.29025 q^{68} +3.30852 q^{69} -1.00000 q^{70} -5.79751 q^{71} -1.65894 q^{72} -9.06652 q^{73} -4.24080 q^{74} +1.15804 q^{75} -4.89322 q^{76} -4.15337 q^{78} -5.03269 q^{79} -1.00000 q^{80} -1.27112 q^{81} +2.02891 q^{82} -16.6258 q^{83} +1.15804 q^{84} -4.29025 q^{85} -0.0689189 q^{86} -0.874121 q^{87} +9.92243 q^{89} +1.65894 q^{90} -3.58654 q^{91} +2.85699 q^{92} -10.8318 q^{93} +2.23785 q^{94} +4.89322 q^{95} +1.15804 q^{96} -6.71733 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} - 5q^{3} + 6q^{4} - 6q^{5} - 5q^{6} + 6q^{7} + 6q^{8} + 11q^{9} + O(q^{10}) \) \( 6q + 6q^{2} - 5q^{3} + 6q^{4} - 6q^{5} - 5q^{6} + 6q^{7} + 6q^{8} + 11q^{9} - 6q^{10} - 5q^{12} + 6q^{14} + 5q^{15} + 6q^{16} - 15q^{17} + 11q^{18} - 13q^{19} - 6q^{20} - 5q^{21} - 2q^{23} - 5q^{24} + 6q^{25} - 26q^{27} + 6q^{28} - 4q^{29} + 5q^{30} - 2q^{31} + 6q^{32} - 15q^{34} - 6q^{35} + 11q^{36} - 13q^{38} - 30q^{39} - 6q^{40} - 17q^{41} - 5q^{42} + 15q^{43} - 11q^{45} - 2q^{46} - 18q^{47} - 5q^{48} + 6q^{49} + 6q^{50} + 6q^{51} + 22q^{53} - 26q^{54} + 6q^{56} - 2q^{57} - 4q^{58} - 7q^{59} + 5q^{60} - 20q^{61} - 2q^{62} + 11q^{63} + 6q^{64} - 29q^{67} - 15q^{68} + 12q^{69} - 6q^{70} - 2q^{71} + 11q^{72} + q^{73} - 5q^{75} - 13q^{76} - 30q^{78} - 12q^{79} - 6q^{80} + 22q^{81} - 17q^{82} - 35q^{83} - 5q^{84} + 15q^{85} + 15q^{86} - 29q^{89} - 11q^{90} - 2q^{92} + 8q^{93} - 18q^{94} + 13q^{95} - 5q^{96} - 19q^{97} + 6q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.15804 0.668596 0.334298 0.942467i \(-0.391501\pi\)
0.334298 + 0.942467i \(0.391501\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.15804 0.472769
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −1.65894 −0.552979
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 1.15804 0.334298
\(13\) −3.58654 −0.994727 −0.497363 0.867542i \(-0.665698\pi\)
−0.497363 + 0.867542i \(0.665698\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.15804 −0.299005
\(16\) 1.00000 0.250000
\(17\) 4.29025 1.04054 0.520270 0.854002i \(-0.325831\pi\)
0.520270 + 0.854002i \(0.325831\pi\)
\(18\) −1.65894 −0.391015
\(19\) −4.89322 −1.12258 −0.561291 0.827619i \(-0.689695\pi\)
−0.561291 + 0.827619i \(0.689695\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.15804 0.252706
\(22\) 0 0
\(23\) 2.85699 0.595724 0.297862 0.954609i \(-0.403726\pi\)
0.297862 + 0.954609i \(0.403726\pi\)
\(24\) 1.15804 0.236385
\(25\) 1.00000 0.200000
\(26\) −3.58654 −0.703378
\(27\) −5.39525 −1.03832
\(28\) 1.00000 0.188982
\(29\) −0.754826 −0.140168 −0.0700838 0.997541i \(-0.522327\pi\)
−0.0700838 + 0.997541i \(0.522327\pi\)
\(30\) −1.15804 −0.211429
\(31\) −9.35354 −1.67995 −0.839973 0.542628i \(-0.817429\pi\)
−0.839973 + 0.542628i \(0.817429\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.29025 0.735772
\(35\) −1.00000 −0.169031
\(36\) −1.65894 −0.276489
\(37\) −4.24080 −0.697183 −0.348592 0.937275i \(-0.613340\pi\)
−0.348592 + 0.937275i \(0.613340\pi\)
\(38\) −4.89322 −0.793785
\(39\) −4.15337 −0.665071
\(40\) −1.00000 −0.158114
\(41\) 2.02891 0.316862 0.158431 0.987370i \(-0.449356\pi\)
0.158431 + 0.987370i \(0.449356\pi\)
\(42\) 1.15804 0.178690
\(43\) −0.0689189 −0.0105100 −0.00525502 0.999986i \(-0.501673\pi\)
−0.00525502 + 0.999986i \(0.501673\pi\)
\(44\) 0 0
\(45\) 1.65894 0.247300
\(46\) 2.85699 0.421240
\(47\) 2.23785 0.326425 0.163212 0.986591i \(-0.447814\pi\)
0.163212 + 0.986591i \(0.447814\pi\)
\(48\) 1.15804 0.167149
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 4.96830 0.695701
\(52\) −3.58654 −0.497363
\(53\) 13.1026 1.79977 0.899887 0.436123i \(-0.143649\pi\)
0.899887 + 0.436123i \(0.143649\pi\)
\(54\) −5.39525 −0.734200
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −5.66656 −0.750554
\(58\) −0.754826 −0.0991135
\(59\) 1.22512 0.159497 0.0797486 0.996815i \(-0.474588\pi\)
0.0797486 + 0.996815i \(0.474588\pi\)
\(60\) −1.15804 −0.149503
\(61\) −4.03036 −0.516034 −0.258017 0.966140i \(-0.583069\pi\)
−0.258017 + 0.966140i \(0.583069\pi\)
\(62\) −9.35354 −1.18790
\(63\) −1.65894 −0.209006
\(64\) 1.00000 0.125000
\(65\) 3.58654 0.444855
\(66\) 0 0
\(67\) −14.3059 −1.74774 −0.873870 0.486159i \(-0.838398\pi\)
−0.873870 + 0.486159i \(0.838398\pi\)
\(68\) 4.29025 0.520270
\(69\) 3.30852 0.398299
\(70\) −1.00000 −0.119523
\(71\) −5.79751 −0.688038 −0.344019 0.938963i \(-0.611789\pi\)
−0.344019 + 0.938963i \(0.611789\pi\)
\(72\) −1.65894 −0.195508
\(73\) −9.06652 −1.06116 −0.530578 0.847636i \(-0.678025\pi\)
−0.530578 + 0.847636i \(0.678025\pi\)
\(74\) −4.24080 −0.492983
\(75\) 1.15804 0.133719
\(76\) −4.89322 −0.561291
\(77\) 0 0
\(78\) −4.15337 −0.470276
\(79\) −5.03269 −0.566222 −0.283111 0.959087i \(-0.591367\pi\)
−0.283111 + 0.959087i \(0.591367\pi\)
\(80\) −1.00000 −0.111803
\(81\) −1.27112 −0.141236
\(82\) 2.02891 0.224055
\(83\) −16.6258 −1.82492 −0.912460 0.409167i \(-0.865819\pi\)
−0.912460 + 0.409167i \(0.865819\pi\)
\(84\) 1.15804 0.126353
\(85\) −4.29025 −0.465343
\(86\) −0.0689189 −0.00743171
\(87\) −0.874121 −0.0937156
\(88\) 0 0
\(89\) 9.92243 1.05177 0.525887 0.850554i \(-0.323733\pi\)
0.525887 + 0.850554i \(0.323733\pi\)
\(90\) 1.65894 0.174867
\(91\) −3.58654 −0.375971
\(92\) 2.85699 0.297862
\(93\) −10.8318 −1.12321
\(94\) 2.23785 0.230817
\(95\) 4.89322 0.502034
\(96\) 1.15804 0.118192
\(97\) −6.71733 −0.682042 −0.341021 0.940056i \(-0.610773\pi\)
−0.341021 + 0.940056i \(0.610773\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 3.15928 0.314360 0.157180 0.987570i \(-0.449760\pi\)
0.157180 + 0.987570i \(0.449760\pi\)
\(102\) 4.96830 0.491935
\(103\) −14.4797 −1.42673 −0.713365 0.700792i \(-0.752830\pi\)
−0.713365 + 0.700792i \(0.752830\pi\)
\(104\) −3.58654 −0.351689
\(105\) −1.15804 −0.113013
\(106\) 13.1026 1.27263
\(107\) −7.11161 −0.687505 −0.343753 0.939060i \(-0.611698\pi\)
−0.343753 + 0.939060i \(0.611698\pi\)
\(108\) −5.39525 −0.519158
\(109\) −15.0710 −1.44354 −0.721770 0.692133i \(-0.756671\pi\)
−0.721770 + 0.692133i \(0.756671\pi\)
\(110\) 0 0
\(111\) −4.91103 −0.466134
\(112\) 1.00000 0.0944911
\(113\) 0.563261 0.0529871 0.0264936 0.999649i \(-0.491566\pi\)
0.0264936 + 0.999649i \(0.491566\pi\)
\(114\) −5.66656 −0.530722
\(115\) −2.85699 −0.266416
\(116\) −0.754826 −0.0700838
\(117\) 5.94984 0.550063
\(118\) 1.22512 0.112782
\(119\) 4.29025 0.393287
\(120\) −1.15804 −0.105714
\(121\) 0 0
\(122\) −4.03036 −0.364891
\(123\) 2.34956 0.211853
\(124\) −9.35354 −0.839973
\(125\) −1.00000 −0.0894427
\(126\) −1.65894 −0.147790
\(127\) 12.3097 1.09231 0.546153 0.837685i \(-0.316092\pi\)
0.546153 + 0.837685i \(0.316092\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.0798110 −0.00702697
\(130\) 3.58654 0.314560
\(131\) −14.9957 −1.31018 −0.655090 0.755551i \(-0.727369\pi\)
−0.655090 + 0.755551i \(0.727369\pi\)
\(132\) 0 0
\(133\) −4.89322 −0.424296
\(134\) −14.3059 −1.23584
\(135\) 5.39525 0.464349
\(136\) 4.29025 0.367886
\(137\) −0.423951 −0.0362206 −0.0181103 0.999836i \(-0.505765\pi\)
−0.0181103 + 0.999836i \(0.505765\pi\)
\(138\) 3.30852 0.281640
\(139\) 19.3990 1.64540 0.822700 0.568475i \(-0.192466\pi\)
0.822700 + 0.568475i \(0.192466\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 2.59153 0.218246
\(142\) −5.79751 −0.486517
\(143\) 0 0
\(144\) −1.65894 −0.138245
\(145\) 0.754826 0.0626849
\(146\) −9.06652 −0.750350
\(147\) 1.15804 0.0955138
\(148\) −4.24080 −0.348592
\(149\) 20.2199 1.65648 0.828241 0.560372i \(-0.189342\pi\)
0.828241 + 0.560372i \(0.189342\pi\)
\(150\) 1.15804 0.0945538
\(151\) −15.3843 −1.25195 −0.625977 0.779842i \(-0.715299\pi\)
−0.625977 + 0.779842i \(0.715299\pi\)
\(152\) −4.89322 −0.396892
\(153\) −7.11726 −0.575396
\(154\) 0 0
\(155\) 9.35354 0.751295
\(156\) −4.15337 −0.332535
\(157\) −4.83845 −0.386150 −0.193075 0.981184i \(-0.561846\pi\)
−0.193075 + 0.981184i \(0.561846\pi\)
\(158\) −5.03269 −0.400380
\(159\) 15.1733 1.20332
\(160\) −1.00000 −0.0790569
\(161\) 2.85699 0.225163
\(162\) −1.27112 −0.0998688
\(163\) 3.98234 0.311921 0.155961 0.987763i \(-0.450153\pi\)
0.155961 + 0.987763i \(0.450153\pi\)
\(164\) 2.02891 0.158431
\(165\) 0 0
\(166\) −16.6258 −1.29041
\(167\) −23.9536 −1.85359 −0.926794 0.375570i \(-0.877447\pi\)
−0.926794 + 0.375570i \(0.877447\pi\)
\(168\) 1.15804 0.0893450
\(169\) −0.136737 −0.0105182
\(170\) −4.29025 −0.329047
\(171\) 8.11754 0.620764
\(172\) −0.0689189 −0.00525502
\(173\) 9.32676 0.709100 0.354550 0.935037i \(-0.384634\pi\)
0.354550 + 0.935037i \(0.384634\pi\)
\(174\) −0.874121 −0.0662669
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 1.41874 0.106639
\(178\) 9.92243 0.743717
\(179\) 21.0160 1.57081 0.785405 0.618982i \(-0.212454\pi\)
0.785405 + 0.618982i \(0.212454\pi\)
\(180\) 1.65894 0.123650
\(181\) −2.21284 −0.164479 −0.0822394 0.996613i \(-0.526207\pi\)
−0.0822394 + 0.996613i \(0.526207\pi\)
\(182\) −3.58654 −0.265852
\(183\) −4.66733 −0.345019
\(184\) 2.85699 0.210620
\(185\) 4.24080 0.311790
\(186\) −10.8318 −0.794227
\(187\) 0 0
\(188\) 2.23785 0.163212
\(189\) −5.39525 −0.392447
\(190\) 4.89322 0.354991
\(191\) 19.1652 1.38674 0.693372 0.720580i \(-0.256124\pi\)
0.693372 + 0.720580i \(0.256124\pi\)
\(192\) 1.15804 0.0835746
\(193\) 8.44654 0.607996 0.303998 0.952673i \(-0.401678\pi\)
0.303998 + 0.952673i \(0.401678\pi\)
\(194\) −6.71733 −0.482276
\(195\) 4.15337 0.297429
\(196\) 1.00000 0.0714286
\(197\) −11.1821 −0.796693 −0.398347 0.917235i \(-0.630416\pi\)
−0.398347 + 0.917235i \(0.630416\pi\)
\(198\) 0 0
\(199\) −4.23045 −0.299888 −0.149944 0.988694i \(-0.547909\pi\)
−0.149944 + 0.988694i \(0.547909\pi\)
\(200\) 1.00000 0.0707107
\(201\) −16.5668 −1.16853
\(202\) 3.15928 0.222286
\(203\) −0.754826 −0.0529784
\(204\) 4.96830 0.347850
\(205\) −2.02891 −0.141705
\(206\) −14.4797 −1.00885
\(207\) −4.73957 −0.329423
\(208\) −3.58654 −0.248682
\(209\) 0 0
\(210\) −1.15804 −0.0799126
\(211\) −2.78498 −0.191726 −0.0958628 0.995395i \(-0.530561\pi\)
−0.0958628 + 0.995395i \(0.530561\pi\)
\(212\) 13.1026 0.899887
\(213\) −6.71377 −0.460020
\(214\) −7.11161 −0.486139
\(215\) 0.0689189 0.00470023
\(216\) −5.39525 −0.367100
\(217\) −9.35354 −0.634960
\(218\) −15.0710 −1.02074
\(219\) −10.4994 −0.709485
\(220\) 0 0
\(221\) −15.3872 −1.03505
\(222\) −4.91103 −0.329607
\(223\) −11.2604 −0.754052 −0.377026 0.926203i \(-0.623053\pi\)
−0.377026 + 0.926203i \(0.623053\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.65894 −0.110596
\(226\) 0.563261 0.0374676
\(227\) 4.17183 0.276894 0.138447 0.990370i \(-0.455789\pi\)
0.138447 + 0.990370i \(0.455789\pi\)
\(228\) −5.66656 −0.375277
\(229\) 14.1775 0.936879 0.468439 0.883496i \(-0.344816\pi\)
0.468439 + 0.883496i \(0.344816\pi\)
\(230\) −2.85699 −0.188384
\(231\) 0 0
\(232\) −0.754826 −0.0495568
\(233\) −9.49782 −0.622223 −0.311111 0.950373i \(-0.600701\pi\)
−0.311111 + 0.950373i \(0.600701\pi\)
\(234\) 5.94984 0.388953
\(235\) −2.23785 −0.145982
\(236\) 1.22512 0.0797486
\(237\) −5.82808 −0.378574
\(238\) 4.29025 0.278096
\(239\) 2.03180 0.131426 0.0657130 0.997839i \(-0.479068\pi\)
0.0657130 + 0.997839i \(0.479068\pi\)
\(240\) −1.15804 −0.0747514
\(241\) −18.6496 −1.20133 −0.600663 0.799502i \(-0.705097\pi\)
−0.600663 + 0.799502i \(0.705097\pi\)
\(242\) 0 0
\(243\) 14.7137 0.943886
\(244\) −4.03036 −0.258017
\(245\) −1.00000 −0.0638877
\(246\) 2.34956 0.149803
\(247\) 17.5497 1.11666
\(248\) −9.35354 −0.593951
\(249\) −19.2534 −1.22013
\(250\) −1.00000 −0.0632456
\(251\) −14.5068 −0.915663 −0.457832 0.889039i \(-0.651374\pi\)
−0.457832 + 0.889039i \(0.651374\pi\)
\(252\) −1.65894 −0.104503
\(253\) 0 0
\(254\) 12.3097 0.772377
\(255\) −4.96830 −0.311127
\(256\) 1.00000 0.0625000
\(257\) −14.8443 −0.925962 −0.462981 0.886368i \(-0.653220\pi\)
−0.462981 + 0.886368i \(0.653220\pi\)
\(258\) −0.0798110 −0.00496882
\(259\) −4.24080 −0.263510
\(260\) 3.58654 0.222428
\(261\) 1.25221 0.0775097
\(262\) −14.9957 −0.926437
\(263\) −31.6286 −1.95030 −0.975152 0.221539i \(-0.928892\pi\)
−0.975152 + 0.221539i \(0.928892\pi\)
\(264\) 0 0
\(265\) −13.1026 −0.804884
\(266\) −4.89322 −0.300022
\(267\) 11.4906 0.703213
\(268\) −14.3059 −0.873870
\(269\) 11.1022 0.676912 0.338456 0.940982i \(-0.390095\pi\)
0.338456 + 0.940982i \(0.390095\pi\)
\(270\) 5.39525 0.328344
\(271\) −14.3128 −0.869442 −0.434721 0.900565i \(-0.643153\pi\)
−0.434721 + 0.900565i \(0.643153\pi\)
\(272\) 4.29025 0.260135
\(273\) −4.15337 −0.251373
\(274\) −0.423951 −0.0256118
\(275\) 0 0
\(276\) 3.30852 0.199149
\(277\) 3.88359 0.233342 0.116671 0.993171i \(-0.462778\pi\)
0.116671 + 0.993171i \(0.462778\pi\)
\(278\) 19.3990 1.16347
\(279\) 15.5169 0.928974
\(280\) −1.00000 −0.0597614
\(281\) −23.6842 −1.41288 −0.706441 0.707772i \(-0.749700\pi\)
−0.706441 + 0.707772i \(0.749700\pi\)
\(282\) 2.59153 0.154323
\(283\) 8.95928 0.532574 0.266287 0.963894i \(-0.414203\pi\)
0.266287 + 0.963894i \(0.414203\pi\)
\(284\) −5.79751 −0.344019
\(285\) 5.66656 0.335658
\(286\) 0 0
\(287\) 2.02891 0.119763
\(288\) −1.65894 −0.0977538
\(289\) 1.40627 0.0827218
\(290\) 0.754826 0.0443249
\(291\) −7.77896 −0.456011
\(292\) −9.06652 −0.530578
\(293\) 12.7431 0.744460 0.372230 0.928141i \(-0.378593\pi\)
0.372230 + 0.928141i \(0.378593\pi\)
\(294\) 1.15804 0.0675384
\(295\) −1.22512 −0.0713293
\(296\) −4.24080 −0.246491
\(297\) 0 0
\(298\) 20.2199 1.17131
\(299\) −10.2467 −0.592583
\(300\) 1.15804 0.0668596
\(301\) −0.0689189 −0.00397242
\(302\) −15.3843 −0.885265
\(303\) 3.65858 0.210180
\(304\) −4.89322 −0.280645
\(305\) 4.03036 0.230778
\(306\) −7.11726 −0.406866
\(307\) −5.75775 −0.328612 −0.164306 0.986409i \(-0.552538\pi\)
−0.164306 + 0.986409i \(0.552538\pi\)
\(308\) 0 0
\(309\) −16.7682 −0.953907
\(310\) 9.35354 0.531246
\(311\) 12.2054 0.692106 0.346053 0.938215i \(-0.387522\pi\)
0.346053 + 0.938215i \(0.387522\pi\)
\(312\) −4.15337 −0.235138
\(313\) 20.3162 1.14834 0.574170 0.818736i \(-0.305325\pi\)
0.574170 + 0.818736i \(0.305325\pi\)
\(314\) −4.83845 −0.273050
\(315\) 1.65894 0.0934705
\(316\) −5.03269 −0.283111
\(317\) 26.5294 1.49004 0.745020 0.667042i \(-0.232440\pi\)
0.745020 + 0.667042i \(0.232440\pi\)
\(318\) 15.1733 0.850878
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −8.23555 −0.459663
\(322\) 2.85699 0.159214
\(323\) −20.9931 −1.16809
\(324\) −1.27112 −0.0706179
\(325\) −3.58654 −0.198945
\(326\) 3.98234 0.220562
\(327\) −17.4529 −0.965146
\(328\) 2.02891 0.112028
\(329\) 2.23785 0.123377
\(330\) 0 0
\(331\) 0.393891 0.0216502 0.0108251 0.999941i \(-0.496554\pi\)
0.0108251 + 0.999941i \(0.496554\pi\)
\(332\) −16.6258 −0.912460
\(333\) 7.03522 0.385527
\(334\) −23.9536 −1.31068
\(335\) 14.3059 0.781614
\(336\) 1.15804 0.0631764
\(337\) −1.24378 −0.0677532 −0.0338766 0.999426i \(-0.510785\pi\)
−0.0338766 + 0.999426i \(0.510785\pi\)
\(338\) −0.136737 −0.00743752
\(339\) 0.652280 0.0354270
\(340\) −4.29025 −0.232672
\(341\) 0 0
\(342\) 8.11754 0.438946
\(343\) 1.00000 0.0539949
\(344\) −0.0689189 −0.00371586
\(345\) −3.30852 −0.178125
\(346\) 9.32676 0.501410
\(347\) −15.7099 −0.843354 −0.421677 0.906746i \(-0.638558\pi\)
−0.421677 + 0.906746i \(0.638558\pi\)
\(348\) −0.874121 −0.0468578
\(349\) 16.4116 0.878493 0.439247 0.898366i \(-0.355245\pi\)
0.439247 + 0.898366i \(0.355245\pi\)
\(350\) 1.00000 0.0534522
\(351\) 19.3503 1.03284
\(352\) 0 0
\(353\) 22.8233 1.21476 0.607382 0.794410i \(-0.292220\pi\)
0.607382 + 0.794410i \(0.292220\pi\)
\(354\) 1.41874 0.0754053
\(355\) 5.79751 0.307700
\(356\) 9.92243 0.525887
\(357\) 4.96830 0.262950
\(358\) 21.0160 1.11073
\(359\) −10.8548 −0.572894 −0.286447 0.958096i \(-0.592474\pi\)
−0.286447 + 0.958096i \(0.592474\pi\)
\(360\) 1.65894 0.0874336
\(361\) 4.94358 0.260189
\(362\) −2.21284 −0.116304
\(363\) 0 0
\(364\) −3.58654 −0.187986
\(365\) 9.06652 0.474563
\(366\) −4.66733 −0.243965
\(367\) 21.3899 1.11654 0.558271 0.829658i \(-0.311465\pi\)
0.558271 + 0.829658i \(0.311465\pi\)
\(368\) 2.85699 0.148931
\(369\) −3.36583 −0.175218
\(370\) 4.24080 0.220469
\(371\) 13.1026 0.680251
\(372\) −10.8318 −0.561603
\(373\) 2.90463 0.150396 0.0751981 0.997169i \(-0.476041\pi\)
0.0751981 + 0.997169i \(0.476041\pi\)
\(374\) 0 0
\(375\) −1.15804 −0.0598011
\(376\) 2.23785 0.115409
\(377\) 2.70721 0.139429
\(378\) −5.39525 −0.277502
\(379\) 0.0627414 0.00322281 0.00161140 0.999999i \(-0.499487\pi\)
0.00161140 + 0.999999i \(0.499487\pi\)
\(380\) 4.89322 0.251017
\(381\) 14.2551 0.730312
\(382\) 19.1652 0.980576
\(383\) −8.43394 −0.430954 −0.215477 0.976509i \(-0.569131\pi\)
−0.215477 + 0.976509i \(0.569131\pi\)
\(384\) 1.15804 0.0590961
\(385\) 0 0
\(386\) 8.44654 0.429918
\(387\) 0.114332 0.00581182
\(388\) −6.71733 −0.341021
\(389\) −36.6681 −1.85915 −0.929573 0.368637i \(-0.879825\pi\)
−0.929573 + 0.368637i \(0.879825\pi\)
\(390\) 4.15337 0.210314
\(391\) 12.2572 0.619874
\(392\) 1.00000 0.0505076
\(393\) −17.3657 −0.875982
\(394\) −11.1821 −0.563347
\(395\) 5.03269 0.253222
\(396\) 0 0
\(397\) −2.91177 −0.146137 −0.0730687 0.997327i \(-0.523279\pi\)
−0.0730687 + 0.997327i \(0.523279\pi\)
\(398\) −4.23045 −0.212053
\(399\) −5.66656 −0.283683
\(400\) 1.00000 0.0500000
\(401\) 15.3525 0.766667 0.383334 0.923610i \(-0.374776\pi\)
0.383334 + 0.923610i \(0.374776\pi\)
\(402\) −16.5668 −0.826278
\(403\) 33.5469 1.67109
\(404\) 3.15928 0.157180
\(405\) 1.27112 0.0631626
\(406\) −0.754826 −0.0374614
\(407\) 0 0
\(408\) 4.96830 0.245967
\(409\) −14.9938 −0.741397 −0.370699 0.928753i \(-0.620882\pi\)
−0.370699 + 0.928753i \(0.620882\pi\)
\(410\) −2.02891 −0.100201
\(411\) −0.490954 −0.0242170
\(412\) −14.4797 −0.713365
\(413\) 1.22512 0.0602843
\(414\) −4.73957 −0.232937
\(415\) 16.6258 0.816129
\(416\) −3.58654 −0.175845
\(417\) 22.4649 1.10011
\(418\) 0 0
\(419\) 26.8166 1.31008 0.655038 0.755596i \(-0.272653\pi\)
0.655038 + 0.755596i \(0.272653\pi\)
\(420\) −1.15804 −0.0565067
\(421\) 39.0662 1.90397 0.951984 0.306148i \(-0.0990401\pi\)
0.951984 + 0.306148i \(0.0990401\pi\)
\(422\) −2.78498 −0.135571
\(423\) −3.71246 −0.180506
\(424\) 13.1026 0.636316
\(425\) 4.29025 0.208108
\(426\) −6.71377 −0.325283
\(427\) −4.03036 −0.195043
\(428\) −7.11161 −0.343753
\(429\) 0 0
\(430\) 0.0689189 0.00332356
\(431\) 36.9117 1.77797 0.888987 0.457932i \(-0.151410\pi\)
0.888987 + 0.457932i \(0.151410\pi\)
\(432\) −5.39525 −0.259579
\(433\) −12.4708 −0.599307 −0.299654 0.954048i \(-0.596871\pi\)
−0.299654 + 0.954048i \(0.596871\pi\)
\(434\) −9.35354 −0.448984
\(435\) 0.874121 0.0419109
\(436\) −15.0710 −0.721770
\(437\) −13.9799 −0.668749
\(438\) −10.4994 −0.501682
\(439\) −5.79416 −0.276540 −0.138270 0.990395i \(-0.544154\pi\)
−0.138270 + 0.990395i \(0.544154\pi\)
\(440\) 0 0
\(441\) −1.65894 −0.0789970
\(442\) −15.3872 −0.731893
\(443\) 11.8578 0.563380 0.281690 0.959506i \(-0.409105\pi\)
0.281690 + 0.959506i \(0.409105\pi\)
\(444\) −4.91103 −0.233067
\(445\) −9.92243 −0.470368
\(446\) −11.2604 −0.533195
\(447\) 23.4156 1.10752
\(448\) 1.00000 0.0472456
\(449\) −6.26914 −0.295859 −0.147929 0.988998i \(-0.547261\pi\)
−0.147929 + 0.988998i \(0.547261\pi\)
\(450\) −1.65894 −0.0782030
\(451\) 0 0
\(452\) 0.563261 0.0264936
\(453\) −17.8156 −0.837051
\(454\) 4.17183 0.195794
\(455\) 3.58654 0.168140
\(456\) −5.66656 −0.265361
\(457\) −30.0377 −1.40510 −0.702552 0.711632i \(-0.747956\pi\)
−0.702552 + 0.711632i \(0.747956\pi\)
\(458\) 14.1775 0.662473
\(459\) −23.1470 −1.08041
\(460\) −2.85699 −0.133208
\(461\) −37.8904 −1.76473 −0.882365 0.470566i \(-0.844050\pi\)
−0.882365 + 0.470566i \(0.844050\pi\)
\(462\) 0 0
\(463\) 2.23043 0.103657 0.0518285 0.998656i \(-0.483495\pi\)
0.0518285 + 0.998656i \(0.483495\pi\)
\(464\) −0.754826 −0.0350419
\(465\) 10.8318 0.502313
\(466\) −9.49782 −0.439978
\(467\) 40.1699 1.85884 0.929420 0.369024i \(-0.120308\pi\)
0.929420 + 0.369024i \(0.120308\pi\)
\(468\) 5.94984 0.275031
\(469\) −14.3059 −0.660584
\(470\) −2.23785 −0.103225
\(471\) −5.60313 −0.258179
\(472\) 1.22512 0.0563908
\(473\) 0 0
\(474\) −5.82808 −0.267692
\(475\) −4.89322 −0.224516
\(476\) 4.29025 0.196643
\(477\) −21.7363 −0.995237
\(478\) 2.03180 0.0929322
\(479\) −15.4649 −0.706609 −0.353304 0.935508i \(-0.614942\pi\)
−0.353304 + 0.935508i \(0.614942\pi\)
\(480\) −1.15804 −0.0528572
\(481\) 15.2098 0.693507
\(482\) −18.6496 −0.849466
\(483\) 3.30852 0.150543
\(484\) 0 0
\(485\) 6.71733 0.305018
\(486\) 14.7137 0.667428
\(487\) −17.9184 −0.811960 −0.405980 0.913882i \(-0.633070\pi\)
−0.405980 + 0.913882i \(0.633070\pi\)
\(488\) −4.03036 −0.182446
\(489\) 4.61172 0.208549
\(490\) −1.00000 −0.0451754
\(491\) 26.6413 1.20230 0.601152 0.799135i \(-0.294709\pi\)
0.601152 + 0.799135i \(0.294709\pi\)
\(492\) 2.34956 0.105926
\(493\) −3.23839 −0.145850
\(494\) 17.5497 0.789599
\(495\) 0 0
\(496\) −9.35354 −0.419987
\(497\) −5.79751 −0.260054
\(498\) −19.2534 −0.862765
\(499\) 29.4299 1.31746 0.658732 0.752378i \(-0.271093\pi\)
0.658732 + 0.752378i \(0.271093\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −27.7393 −1.23930
\(502\) −14.5068 −0.647472
\(503\) −25.1768 −1.12258 −0.561288 0.827620i \(-0.689694\pi\)
−0.561288 + 0.827620i \(0.689694\pi\)
\(504\) −1.65894 −0.0738949
\(505\) −3.15928 −0.140586
\(506\) 0 0
\(507\) −0.158348 −0.00703246
\(508\) 12.3097 0.546153
\(509\) 28.8350 1.27809 0.639043 0.769171i \(-0.279330\pi\)
0.639043 + 0.769171i \(0.279330\pi\)
\(510\) −4.96830 −0.220000
\(511\) −9.06652 −0.401079
\(512\) 1.00000 0.0441942
\(513\) 26.4001 1.16559
\(514\) −14.8443 −0.654754
\(515\) 14.4797 0.638053
\(516\) −0.0798110 −0.00351348
\(517\) 0 0
\(518\) −4.24080 −0.186330
\(519\) 10.8008 0.474102
\(520\) 3.58654 0.157280
\(521\) 8.86274 0.388284 0.194142 0.980973i \(-0.437808\pi\)
0.194142 + 0.980973i \(0.437808\pi\)
\(522\) 1.25221 0.0548077
\(523\) −23.2339 −1.01595 −0.507973 0.861373i \(-0.669605\pi\)
−0.507973 + 0.861373i \(0.669605\pi\)
\(524\) −14.9957 −0.655090
\(525\) 1.15804 0.0505411
\(526\) −31.6286 −1.37907
\(527\) −40.1291 −1.74805
\(528\) 0 0
\(529\) −14.8376 −0.645113
\(530\) −13.1026 −0.569139
\(531\) −2.03240 −0.0881985
\(532\) −4.89322 −0.212148
\(533\) −7.27675 −0.315191
\(534\) 11.4906 0.497247
\(535\) 7.11161 0.307462
\(536\) −14.3059 −0.617920
\(537\) 24.3375 1.05024
\(538\) 11.1022 0.478649
\(539\) 0 0
\(540\) 5.39525 0.232175
\(541\) −21.2530 −0.913740 −0.456870 0.889534i \(-0.651030\pi\)
−0.456870 + 0.889534i \(0.651030\pi\)
\(542\) −14.3128 −0.614789
\(543\) −2.56256 −0.109970
\(544\) 4.29025 0.183943
\(545\) 15.0710 0.645571
\(546\) −4.15337 −0.177748
\(547\) 39.8544 1.70405 0.852026 0.523500i \(-0.175374\pi\)
0.852026 + 0.523500i \(0.175374\pi\)
\(548\) −0.423951 −0.0181103
\(549\) 6.68611 0.285356
\(550\) 0 0
\(551\) 3.69353 0.157350
\(552\) 3.30852 0.140820
\(553\) −5.03269 −0.214012
\(554\) 3.88359 0.164998
\(555\) 4.91103 0.208462
\(556\) 19.3990 0.822700
\(557\) 19.7255 0.835794 0.417897 0.908494i \(-0.362767\pi\)
0.417897 + 0.908494i \(0.362767\pi\)
\(558\) 15.5169 0.656884
\(559\) 0.247180 0.0104546
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −23.6842 −0.999058
\(563\) 20.5630 0.866627 0.433314 0.901243i \(-0.357344\pi\)
0.433314 + 0.901243i \(0.357344\pi\)
\(564\) 2.59153 0.109123
\(565\) −0.563261 −0.0236966
\(566\) 8.95928 0.376587
\(567\) −1.27112 −0.0533821
\(568\) −5.79751 −0.243258
\(569\) 21.9128 0.918632 0.459316 0.888273i \(-0.348095\pi\)
0.459316 + 0.888273i \(0.348095\pi\)
\(570\) 5.66656 0.237346
\(571\) −17.3899 −0.727745 −0.363872 0.931449i \(-0.618546\pi\)
−0.363872 + 0.931449i \(0.618546\pi\)
\(572\) 0 0
\(573\) 22.1941 0.927172
\(574\) 2.02891 0.0846849
\(575\) 2.85699 0.119145
\(576\) −1.65894 −0.0691223
\(577\) 25.6014 1.06580 0.532899 0.846179i \(-0.321102\pi\)
0.532899 + 0.846179i \(0.321102\pi\)
\(578\) 1.40627 0.0584932
\(579\) 9.78146 0.406504
\(580\) 0.754826 0.0313424
\(581\) −16.6258 −0.689755
\(582\) −7.77896 −0.322448
\(583\) 0 0
\(584\) −9.06652 −0.375175
\(585\) −5.94984 −0.245996
\(586\) 12.7431 0.526413
\(587\) −38.2688 −1.57952 −0.789762 0.613413i \(-0.789796\pi\)
−0.789762 + 0.613413i \(0.789796\pi\)
\(588\) 1.15804 0.0477569
\(589\) 45.7689 1.88588
\(590\) −1.22512 −0.0504374
\(591\) −12.9494 −0.532666
\(592\) −4.24080 −0.174296
\(593\) −11.8855 −0.488078 −0.244039 0.969765i \(-0.578473\pi\)
−0.244039 + 0.969765i \(0.578473\pi\)
\(594\) 0 0
\(595\) −4.29025 −0.175883
\(596\) 20.2199 0.828241
\(597\) −4.89904 −0.200504
\(598\) −10.2467 −0.419019
\(599\) −4.65368 −0.190144 −0.0950722 0.995470i \(-0.530308\pi\)
−0.0950722 + 0.995470i \(0.530308\pi\)
\(600\) 1.15804 0.0472769
\(601\) −13.2306 −0.539688 −0.269844 0.962904i \(-0.586972\pi\)
−0.269844 + 0.962904i \(0.586972\pi\)
\(602\) −0.0689189 −0.00280892
\(603\) 23.7325 0.966464
\(604\) −15.3843 −0.625977
\(605\) 0 0
\(606\) 3.65858 0.148620
\(607\) 30.3137 1.23040 0.615198 0.788372i \(-0.289076\pi\)
0.615198 + 0.788372i \(0.289076\pi\)
\(608\) −4.89322 −0.198446
\(609\) −0.874121 −0.0354212
\(610\) 4.03036 0.163184
\(611\) −8.02615 −0.324703
\(612\) −7.11726 −0.287698
\(613\) −12.6321 −0.510206 −0.255103 0.966914i \(-0.582109\pi\)
−0.255103 + 0.966914i \(0.582109\pi\)
\(614\) −5.75775 −0.232364
\(615\) −2.34956 −0.0947434
\(616\) 0 0
\(617\) −48.4310 −1.94976 −0.974880 0.222731i \(-0.928503\pi\)
−0.974880 + 0.222731i \(0.928503\pi\)
\(618\) −16.7682 −0.674514
\(619\) −2.76873 −0.111285 −0.0556423 0.998451i \(-0.517721\pi\)
−0.0556423 + 0.998451i \(0.517721\pi\)
\(620\) 9.35354 0.375647
\(621\) −15.4142 −0.618550
\(622\) 12.2054 0.489393
\(623\) 9.92243 0.397534
\(624\) −4.15337 −0.166268
\(625\) 1.00000 0.0400000
\(626\) 20.3162 0.811999
\(627\) 0 0
\(628\) −4.83845 −0.193075
\(629\) −18.1941 −0.725446
\(630\) 1.65894 0.0660936
\(631\) 21.0875 0.839479 0.419740 0.907645i \(-0.362121\pi\)
0.419740 + 0.907645i \(0.362121\pi\)
\(632\) −5.03269 −0.200190
\(633\) −3.22512 −0.128187
\(634\) 26.5294 1.05362
\(635\) −12.3097 −0.488494
\(636\) 15.1733 0.601661
\(637\) −3.58654 −0.142104
\(638\) 0 0
\(639\) 9.61771 0.380471
\(640\) −1.00000 −0.0395285
\(641\) 10.8696 0.429324 0.214662 0.976688i \(-0.431135\pi\)
0.214662 + 0.976688i \(0.431135\pi\)
\(642\) −8.23555 −0.325031
\(643\) 3.71180 0.146379 0.0731896 0.997318i \(-0.476682\pi\)
0.0731896 + 0.997318i \(0.476682\pi\)
\(644\) 2.85699 0.112581
\(645\) 0.0798110 0.00314256
\(646\) −20.9931 −0.825964
\(647\) −25.1297 −0.987951 −0.493976 0.869476i \(-0.664457\pi\)
−0.493976 + 0.869476i \(0.664457\pi\)
\(648\) −1.27112 −0.0499344
\(649\) 0 0
\(650\) −3.58654 −0.140676
\(651\) −10.8318 −0.424532
\(652\) 3.98234 0.155961
\(653\) 27.3470 1.07017 0.535085 0.844798i \(-0.320279\pi\)
0.535085 + 0.844798i \(0.320279\pi\)
\(654\) −17.4529 −0.682461
\(655\) 14.9957 0.585930
\(656\) 2.02891 0.0792155
\(657\) 15.0408 0.586796
\(658\) 2.23785 0.0872406
\(659\) −6.74968 −0.262930 −0.131465 0.991321i \(-0.541968\pi\)
−0.131465 + 0.991321i \(0.541968\pi\)
\(660\) 0 0
\(661\) −34.5615 −1.34429 −0.672144 0.740421i \(-0.734626\pi\)
−0.672144 + 0.740421i \(0.734626\pi\)
\(662\) 0.393891 0.0153090
\(663\) −17.8190 −0.692032
\(664\) −16.6258 −0.645206
\(665\) 4.89322 0.189751
\(666\) 7.03522 0.272609
\(667\) −2.15653 −0.0835013
\(668\) −23.9536 −0.926794
\(669\) −13.0400 −0.504157
\(670\) 14.3059 0.552684
\(671\) 0 0
\(672\) 1.15804 0.0446725
\(673\) 2.98731 0.115152 0.0575761 0.998341i \(-0.481663\pi\)
0.0575761 + 0.998341i \(0.481663\pi\)
\(674\) −1.24378 −0.0479087
\(675\) −5.39525 −0.207663
\(676\) −0.136737 −0.00525912
\(677\) 5.39584 0.207379 0.103690 0.994610i \(-0.466935\pi\)
0.103690 + 0.994610i \(0.466935\pi\)
\(678\) 0.652280 0.0250507
\(679\) −6.71733 −0.257787
\(680\) −4.29025 −0.164524
\(681\) 4.83116 0.185130
\(682\) 0 0
\(683\) 2.39397 0.0916028 0.0458014 0.998951i \(-0.485416\pi\)
0.0458014 + 0.998951i \(0.485416\pi\)
\(684\) 8.11754 0.310382
\(685\) 0.423951 0.0161983
\(686\) 1.00000 0.0381802
\(687\) 16.4182 0.626394
\(688\) −0.0689189 −0.00262751
\(689\) −46.9928 −1.79028
\(690\) −3.30852 −0.125953
\(691\) −28.2727 −1.07554 −0.537771 0.843091i \(-0.680734\pi\)
−0.537771 + 0.843091i \(0.680734\pi\)
\(692\) 9.32676 0.354550
\(693\) 0 0
\(694\) −15.7099 −0.596341
\(695\) −19.3990 −0.735846
\(696\) −0.874121 −0.0331335
\(697\) 8.70452 0.329707
\(698\) 16.4116 0.621189
\(699\) −10.9989 −0.416016
\(700\) 1.00000 0.0377964
\(701\) 42.6691 1.61159 0.805795 0.592194i \(-0.201738\pi\)
0.805795 + 0.592194i \(0.201738\pi\)
\(702\) 19.3503 0.730329
\(703\) 20.7512 0.782645
\(704\) 0 0
\(705\) −2.59153 −0.0976027
\(706\) 22.8233 0.858967
\(707\) 3.15928 0.118817
\(708\) 1.41874 0.0533196
\(709\) 49.1230 1.84485 0.922427 0.386171i \(-0.126203\pi\)
0.922427 + 0.386171i \(0.126203\pi\)
\(710\) 5.79751 0.217577
\(711\) 8.34892 0.313109
\(712\) 9.92243 0.371859
\(713\) −26.7230 −1.00078
\(714\) 4.96830 0.185934
\(715\) 0 0
\(716\) 21.0160 0.785405
\(717\) 2.35291 0.0878709
\(718\) −10.8548 −0.405097
\(719\) 31.0636 1.15848 0.579239 0.815158i \(-0.303350\pi\)
0.579239 + 0.815158i \(0.303350\pi\)
\(720\) 1.65894 0.0618249
\(721\) −14.4797 −0.539254
\(722\) 4.94358 0.183981
\(723\) −21.5971 −0.803203
\(724\) −2.21284 −0.0822394
\(725\) −0.754826 −0.0280335
\(726\) 0 0
\(727\) 41.2246 1.52894 0.764468 0.644662i \(-0.223002\pi\)
0.764468 + 0.644662i \(0.223002\pi\)
\(728\) −3.58654 −0.132926
\(729\) 20.8525 0.772315
\(730\) 9.06652 0.335567
\(731\) −0.295679 −0.0109361
\(732\) −4.66733 −0.172509
\(733\) 45.0160 1.66271 0.831353 0.555745i \(-0.187567\pi\)
0.831353 + 0.555745i \(0.187567\pi\)
\(734\) 21.3899 0.789515
\(735\) −1.15804 −0.0427151
\(736\) 2.85699 0.105310
\(737\) 0 0
\(738\) −3.36583 −0.123898
\(739\) −39.4450 −1.45101 −0.725504 0.688218i \(-0.758393\pi\)
−0.725504 + 0.688218i \(0.758393\pi\)
\(740\) 4.24080 0.155895
\(741\) 20.3233 0.746596
\(742\) 13.1026 0.481010
\(743\) 37.9890 1.39368 0.696840 0.717227i \(-0.254589\pi\)
0.696840 + 0.717227i \(0.254589\pi\)
\(744\) −10.8318 −0.397113
\(745\) −20.2199 −0.740801
\(746\) 2.90463 0.106346
\(747\) 27.5811 1.00914
\(748\) 0 0
\(749\) −7.11161 −0.259852
\(750\) −1.15804 −0.0422858
\(751\) 44.7783 1.63399 0.816993 0.576648i \(-0.195640\pi\)
0.816993 + 0.576648i \(0.195640\pi\)
\(752\) 2.23785 0.0816061
\(753\) −16.7995 −0.612209
\(754\) 2.70721 0.0985909
\(755\) 15.3843 0.559890
\(756\) −5.39525 −0.196223
\(757\) −45.7050 −1.66118 −0.830588 0.556888i \(-0.811995\pi\)
−0.830588 + 0.556888i \(0.811995\pi\)
\(758\) 0.0627414 0.00227887
\(759\) 0 0
\(760\) 4.89322 0.177496
\(761\) −21.6462 −0.784675 −0.392338 0.919821i \(-0.628333\pi\)
−0.392338 + 0.919821i \(0.628333\pi\)
\(762\) 14.2551 0.516409
\(763\) −15.0710 −0.545607
\(764\) 19.1652 0.693372
\(765\) 7.11726 0.257325
\(766\) −8.43394 −0.304731
\(767\) −4.39394 −0.158656
\(768\) 1.15804 0.0417873
\(769\) −38.7211 −1.39632 −0.698159 0.715942i \(-0.745997\pi\)
−0.698159 + 0.715942i \(0.745997\pi\)
\(770\) 0 0
\(771\) −17.1903 −0.619095
\(772\) 8.44654 0.303998
\(773\) −31.6653 −1.13892 −0.569461 0.822018i \(-0.692848\pi\)
−0.569461 + 0.822018i \(0.692848\pi\)
\(774\) 0.114332 0.00410958
\(775\) −9.35354 −0.335989
\(776\) −6.71733 −0.241138
\(777\) −4.91103 −0.176182
\(778\) −36.6681 −1.31462
\(779\) −9.92788 −0.355703
\(780\) 4.15337 0.148714
\(781\) 0 0
\(782\) 12.2572 0.438317
\(783\) 4.07247 0.145538
\(784\) 1.00000 0.0357143
\(785\) 4.83845 0.172692
\(786\) −17.3657 −0.619413
\(787\) −42.4147 −1.51192 −0.755961 0.654617i \(-0.772830\pi\)
−0.755961 + 0.654617i \(0.772830\pi\)
\(788\) −11.1821 −0.398347
\(789\) −36.6273 −1.30397
\(790\) 5.03269 0.179055
\(791\) 0.563261 0.0200272
\(792\) 0 0
\(793\) 14.4550 0.513313
\(794\) −2.91177 −0.103335
\(795\) −15.1733 −0.538142
\(796\) −4.23045 −0.149944
\(797\) 13.7248 0.486159 0.243079 0.970006i \(-0.421842\pi\)
0.243079 + 0.970006i \(0.421842\pi\)
\(798\) −5.66656 −0.200594
\(799\) 9.60096 0.339658
\(800\) 1.00000 0.0353553
\(801\) −16.4607 −0.581609
\(802\) 15.3525 0.542116
\(803\) 0 0
\(804\) −16.5668 −0.584267
\(805\) −2.85699 −0.100696
\(806\) 33.5469 1.18164
\(807\) 12.8568 0.452581
\(808\) 3.15928 0.111143
\(809\) −32.0943 −1.12838 −0.564189 0.825646i \(-0.690811\pi\)
−0.564189 + 0.825646i \(0.690811\pi\)
\(810\) 1.27112 0.0446627
\(811\) 44.8968 1.57654 0.788271 0.615329i \(-0.210977\pi\)
0.788271 + 0.615329i \(0.210977\pi\)
\(812\) −0.754826 −0.0264892
\(813\) −16.5749 −0.581306
\(814\) 0 0
\(815\) −3.98234 −0.139495
\(816\) 4.96830 0.173925
\(817\) 0.337235 0.0117984
\(818\) −14.9938 −0.524247
\(819\) 5.94984 0.207904
\(820\) −2.02891 −0.0708525
\(821\) −41.2775 −1.44060 −0.720298 0.693665i \(-0.755995\pi\)
−0.720298 + 0.693665i \(0.755995\pi\)
\(822\) −0.490954 −0.0171240
\(823\) 14.7542 0.514298 0.257149 0.966372i \(-0.417217\pi\)
0.257149 + 0.966372i \(0.417217\pi\)
\(824\) −14.4797 −0.504426
\(825\) 0 0
\(826\) 1.22512 0.0426274
\(827\) 51.5767 1.79350 0.896748 0.442541i \(-0.145923\pi\)
0.896748 + 0.442541i \(0.145923\pi\)
\(828\) −4.73957 −0.164711
\(829\) 10.6814 0.370982 0.185491 0.982646i \(-0.440612\pi\)
0.185491 + 0.982646i \(0.440612\pi\)
\(830\) 16.6258 0.577090
\(831\) 4.49737 0.156012
\(832\) −3.58654 −0.124341
\(833\) 4.29025 0.148648
\(834\) 22.4649 0.777895
\(835\) 23.9536 0.828950
\(836\) 0 0
\(837\) 50.4647 1.74432
\(838\) 26.8166 0.926363
\(839\) −27.4262 −0.946857 −0.473429 0.880832i \(-0.656984\pi\)
−0.473429 + 0.880832i \(0.656984\pi\)
\(840\) −1.15804 −0.0399563
\(841\) −28.4302 −0.980353
\(842\) 39.0662 1.34631
\(843\) −27.4273 −0.944648
\(844\) −2.78498 −0.0958628
\(845\) 0.136737 0.00470390
\(846\) −3.71246 −0.127637
\(847\) 0 0
\(848\) 13.1026 0.449944
\(849\) 10.3752 0.356077
\(850\) 4.29025 0.147154
\(851\) −12.1159 −0.415329
\(852\) −6.71377 −0.230010
\(853\) 50.2244 1.71965 0.859826 0.510587i \(-0.170572\pi\)
0.859826 + 0.510587i \(0.170572\pi\)
\(854\) −4.03036 −0.137916
\(855\) −8.11754 −0.277614
\(856\) −7.11161 −0.243070
\(857\) −2.69776 −0.0921539 −0.0460769 0.998938i \(-0.514672\pi\)
−0.0460769 + 0.998938i \(0.514672\pi\)
\(858\) 0 0
\(859\) 19.0873 0.651252 0.325626 0.945499i \(-0.394425\pi\)
0.325626 + 0.945499i \(0.394425\pi\)
\(860\) 0.0689189 0.00235011
\(861\) 2.34956 0.0800728
\(862\) 36.9117 1.25722
\(863\) −21.2845 −0.724532 −0.362266 0.932075i \(-0.617997\pi\)
−0.362266 + 0.932075i \(0.617997\pi\)
\(864\) −5.39525 −0.183550
\(865\) −9.32676 −0.317119
\(866\) −12.4708 −0.423774
\(867\) 1.62852 0.0553075
\(868\) −9.35354 −0.317480
\(869\) 0 0
\(870\) 0.874121 0.0296355
\(871\) 51.3086 1.73853
\(872\) −15.0710 −0.510369
\(873\) 11.1436 0.377155
\(874\) −13.9799 −0.472877
\(875\) −1.00000 −0.0338062
\(876\) −10.4994 −0.354742
\(877\) 14.7968 0.499654 0.249827 0.968291i \(-0.419626\pi\)
0.249827 + 0.968291i \(0.419626\pi\)
\(878\) −5.79416 −0.195543
\(879\) 14.7571 0.497743
\(880\) 0 0
\(881\) −38.9767 −1.31316 −0.656579 0.754257i \(-0.727997\pi\)
−0.656579 + 0.754257i \(0.727997\pi\)
\(882\) −1.65894 −0.0558593
\(883\) −40.8954 −1.37624 −0.688119 0.725597i \(-0.741563\pi\)
−0.688119 + 0.725597i \(0.741563\pi\)
\(884\) −15.3872 −0.517526
\(885\) −1.41874 −0.0476905
\(886\) 11.8578 0.398369
\(887\) −30.9885 −1.04049 −0.520247 0.854016i \(-0.674160\pi\)
−0.520247 + 0.854016i \(0.674160\pi\)
\(888\) −4.91103 −0.164803
\(889\) 12.3097 0.412853
\(890\) −9.92243 −0.332600
\(891\) 0 0
\(892\) −11.2604 −0.377026
\(893\) −10.9503 −0.366438
\(894\) 23.4156 0.783134
\(895\) −21.0160 −0.702488
\(896\) 1.00000 0.0334077
\(897\) −11.8661 −0.396199
\(898\) −6.26914 −0.209204
\(899\) 7.06030 0.235474
\(900\) −1.65894 −0.0552979
\(901\) 56.2133 1.87274
\(902\) 0 0
\(903\) −0.0798110 −0.00265594
\(904\) 0.563261 0.0187338
\(905\) 2.21284 0.0735572
\(906\) −17.8156 −0.591885
\(907\) −7.92243 −0.263060 −0.131530 0.991312i \(-0.541989\pi\)
−0.131530 + 0.991312i \(0.541989\pi\)
\(908\) 4.17183 0.138447
\(909\) −5.24104 −0.173834
\(910\) 3.58654 0.118893
\(911\) 30.5244 1.01132 0.505659 0.862734i \(-0.331250\pi\)
0.505659 + 0.862734i \(0.331250\pi\)
\(912\) −5.66656 −0.187638
\(913\) 0 0
\(914\) −30.0377 −0.993559
\(915\) 4.66733 0.154297
\(916\) 14.1775 0.468439
\(917\) −14.9957 −0.495202
\(918\) −23.1470 −0.763964
\(919\) −5.14113 −0.169590 −0.0847951 0.996398i \(-0.527024\pi\)
−0.0847951 + 0.996398i \(0.527024\pi\)
\(920\) −2.85699 −0.0941922
\(921\) −6.66772 −0.219709
\(922\) −37.8904 −1.24785
\(923\) 20.7930 0.684410
\(924\) 0 0
\(925\) −4.24080 −0.139437
\(926\) 2.23043 0.0732966
\(927\) 24.0210 0.788952
\(928\) −0.754826 −0.0247784
\(929\) 9.36591 0.307286 0.153643 0.988126i \(-0.450899\pi\)
0.153643 + 0.988126i \(0.450899\pi\)
\(930\) 10.8318 0.355189
\(931\) −4.89322 −0.160369
\(932\) −9.49782 −0.311111
\(933\) 14.1344 0.462739
\(934\) 40.1699 1.31440
\(935\) 0 0
\(936\) 5.94984 0.194477
\(937\) −32.4445 −1.05991 −0.529957 0.848024i \(-0.677792\pi\)
−0.529957 + 0.848024i \(0.677792\pi\)
\(938\) −14.3059 −0.467103
\(939\) 23.5270 0.767776
\(940\) −2.23785 −0.0729908
\(941\) −11.9651 −0.390052 −0.195026 0.980798i \(-0.562479\pi\)
−0.195026 + 0.980798i \(0.562479\pi\)
\(942\) −5.60313 −0.182560
\(943\) 5.79657 0.188762
\(944\) 1.22512 0.0398743
\(945\) 5.39525 0.175507
\(946\) 0 0
\(947\) 5.99311 0.194750 0.0973749 0.995248i \(-0.468955\pi\)
0.0973749 + 0.995248i \(0.468955\pi\)
\(948\) −5.82808 −0.189287
\(949\) 32.5174 1.05556
\(950\) −4.89322 −0.158757
\(951\) 30.7222 0.996235
\(952\) 4.29025 0.139048
\(953\) 37.6907 1.22092 0.610461 0.792046i \(-0.290984\pi\)
0.610461 + 0.792046i \(0.290984\pi\)
\(954\) −21.7363 −0.703739
\(955\) −19.1652 −0.620171
\(956\) 2.03180 0.0657130
\(957\) 0 0
\(958\) −15.4649 −0.499648
\(959\) −0.423951 −0.0136901
\(960\) −1.15804 −0.0373757
\(961\) 56.4888 1.82222
\(962\) 15.2098 0.490383
\(963\) 11.7977 0.380176
\(964\) −18.6496 −0.600663
\(965\) −8.44654 −0.271904
\(966\) 3.30852 0.106450
\(967\) −49.5830 −1.59448 −0.797241 0.603662i \(-0.793708\pi\)
−0.797241 + 0.603662i \(0.793708\pi\)
\(968\) 0 0
\(969\) −24.3110 −0.780981
\(970\) 6.71733 0.215680
\(971\) −38.2393 −1.22716 −0.613579 0.789633i \(-0.710271\pi\)
−0.613579 + 0.789633i \(0.710271\pi\)
\(972\) 14.7137 0.471943
\(973\) 19.3990 0.621903
\(974\) −17.9184 −0.574143
\(975\) −4.15337 −0.133014
\(976\) −4.03036 −0.129009
\(977\) −7.02182 −0.224648 −0.112324 0.993672i \(-0.535829\pi\)
−0.112324 + 0.993672i \(0.535829\pi\)
\(978\) 4.61172 0.147467
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 25.0018 0.798247
\(982\) 26.6413 0.850157
\(983\) −31.0557 −0.990525 −0.495262 0.868744i \(-0.664928\pi\)
−0.495262 + 0.868744i \(0.664928\pi\)
\(984\) 2.34956 0.0749013
\(985\) 11.1821 0.356292
\(986\) −3.23839 −0.103131
\(987\) 2.59153 0.0824894
\(988\) 17.5497 0.558331
\(989\) −0.196901 −0.00626108
\(990\) 0 0
\(991\) 20.2609 0.643608 0.321804 0.946806i \(-0.395711\pi\)
0.321804 + 0.946806i \(0.395711\pi\)
\(992\) −9.35354 −0.296975
\(993\) 0.456143 0.0144753
\(994\) −5.79751 −0.183886
\(995\) 4.23045 0.134114
\(996\) −19.2534 −0.610067
\(997\) 16.0793 0.509236 0.254618 0.967042i \(-0.418050\pi\)
0.254618 + 0.967042i \(0.418050\pi\)
\(998\) 29.4299 0.931588
\(999\) 22.8802 0.723896
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 8470.2.a.da.1.5 6
11.5 even 5 770.2.n.h.421.3 12
11.9 even 5 770.2.n.h.631.3 yes 12
11.10 odd 2 8470.2.a.cu.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.h.421.3 12 11.5 even 5
770.2.n.h.631.3 yes 12 11.9 even 5
8470.2.a.cu.1.5 6 11.10 odd 2
8470.2.a.da.1.5 6 1.1 even 1 trivial