Properties

Label 8470.2.a.cd.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.73205 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.73205 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} -1.00000 q^{10} +2.73205 q^{12} -4.00000 q^{13} -1.00000 q^{14} -2.73205 q^{15} +1.00000 q^{16} -0.535898 q^{17} +4.46410 q^{18} +0.732051 q^{19} -1.00000 q^{20} -2.73205 q^{21} +8.19615 q^{23} +2.73205 q^{24} +1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} +7.66025 q^{29} -2.73205 q^{30} +0.535898 q^{31} +1.00000 q^{32} -0.535898 q^{34} +1.00000 q^{35} +4.46410 q^{36} -2.73205 q^{37} +0.732051 q^{38} -10.9282 q^{39} -1.00000 q^{40} -2.19615 q^{41} -2.73205 q^{42} +4.92820 q^{43} -4.46410 q^{45} +8.19615 q^{46} +6.00000 q^{47} +2.73205 q^{48} +1.00000 q^{49} +1.00000 q^{50} -1.46410 q^{51} -4.00000 q^{52} +5.26795 q^{53} +4.00000 q^{54} -1.00000 q^{56} +2.00000 q^{57} +7.66025 q^{58} +8.39230 q^{59} -2.73205 q^{60} +8.00000 q^{61} +0.535898 q^{62} -4.46410 q^{63} +1.00000 q^{64} +4.00000 q^{65} +15.8564 q^{67} -0.535898 q^{68} +22.3923 q^{69} +1.00000 q^{70} +1.07180 q^{71} +4.46410 q^{72} +6.00000 q^{73} -2.73205 q^{74} +2.73205 q^{75} +0.732051 q^{76} -10.9282 q^{78} -7.66025 q^{79} -1.00000 q^{80} -2.46410 q^{81} -2.19615 q^{82} -14.9282 q^{83} -2.73205 q^{84} +0.535898 q^{85} +4.92820 q^{86} +20.9282 q^{87} -8.92820 q^{89} -4.46410 q^{90} +4.00000 q^{91} +8.19615 q^{92} +1.46410 q^{93} +6.00000 q^{94} -0.732051 q^{95} +2.73205 q^{96} +2.73205 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{12} - 8 q^{13} - 2 q^{14} - 2 q^{15} + 2 q^{16} - 8 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{21} + 6 q^{23} + 2 q^{24} + 2 q^{25} - 8 q^{26} + 8 q^{27} - 2 q^{28} - 2 q^{29} - 2 q^{30} + 8 q^{31} + 2 q^{32} - 8 q^{34} + 2 q^{35} + 2 q^{36} - 2 q^{37} - 2 q^{38} - 8 q^{39} - 2 q^{40} + 6 q^{41} - 2 q^{42} - 4 q^{43} - 2 q^{45} + 6 q^{46} + 12 q^{47} + 2 q^{48} + 2 q^{49} + 2 q^{50} + 4 q^{51} - 8 q^{52} + 14 q^{53} + 8 q^{54} - 2 q^{56} + 4 q^{57} - 2 q^{58} - 4 q^{59} - 2 q^{60} + 16 q^{61} + 8 q^{62} - 2 q^{63} + 2 q^{64} + 8 q^{65} + 4 q^{67} - 8 q^{68} + 24 q^{69} + 2 q^{70} + 16 q^{71} + 2 q^{72} + 12 q^{73} - 2 q^{74} + 2 q^{75} - 2 q^{76} - 8 q^{78} + 2 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} - 16 q^{83} - 2 q^{84} + 8 q^{85} - 4 q^{86} + 28 q^{87} - 4 q^{89} - 2 q^{90} + 8 q^{91} + 6 q^{92} - 4 q^{93} + 12 q^{94} + 2 q^{95} + 2 q^{96} + 2 q^{97} + 2 q^{98} + O(q^{100}) \)

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
\(3\) 2.73205 1.57735 0.788675 0.614810i \(-0.210767\pi\)
0.788675 + 0.614810i \(0.210767\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.73205 1.11536
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 4.46410 1.48803
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 2.73205 0.788675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.73205 −0.705412
\(16\) 1.00000 0.250000
\(17\) −0.535898 −0.129974 −0.0649872 0.997886i \(-0.520701\pi\)
−0.0649872 + 0.997886i \(0.520701\pi\)
\(18\) 4.46410 1.05220
\(19\) 0.732051 0.167944 0.0839720 0.996468i \(-0.473239\pi\)
0.0839720 + 0.996468i \(0.473239\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.73205 −0.596182
\(22\) 0 0
\(23\) 8.19615 1.70902 0.854508 0.519438i \(-0.173859\pi\)
0.854508 + 0.519438i \(0.173859\pi\)
\(24\) 2.73205 0.557678
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 4.00000 0.769800
\(28\) −1.00000 −0.188982
\(29\) 7.66025 1.42247 0.711237 0.702953i \(-0.248135\pi\)
0.711237 + 0.702953i \(0.248135\pi\)
\(30\) −2.73205 −0.498802
\(31\) 0.535898 0.0962502 0.0481251 0.998841i \(-0.484675\pi\)
0.0481251 + 0.998841i \(0.484675\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.535898 −0.0919058
\(35\) 1.00000 0.169031
\(36\) 4.46410 0.744017
\(37\) −2.73205 −0.449146 −0.224573 0.974457i \(-0.572099\pi\)
−0.224573 + 0.974457i \(0.572099\pi\)
\(38\) 0.732051 0.118754
\(39\) −10.9282 −1.74991
\(40\) −1.00000 −0.158114
\(41\) −2.19615 −0.342981 −0.171491 0.985186i \(-0.554858\pi\)
−0.171491 + 0.985186i \(0.554858\pi\)
\(42\) −2.73205 −0.421565
\(43\) 4.92820 0.751544 0.375772 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(44\) 0 0
\(45\) −4.46410 −0.665469
\(46\) 8.19615 1.20846
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 2.73205 0.394338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −1.46410 −0.205015
\(52\) −4.00000 −0.554700
\(53\) 5.26795 0.723608 0.361804 0.932254i \(-0.382161\pi\)
0.361804 + 0.932254i \(0.382161\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 2.00000 0.264906
\(58\) 7.66025 1.00584
\(59\) 8.39230 1.09259 0.546293 0.837594i \(-0.316039\pi\)
0.546293 + 0.837594i \(0.316039\pi\)
\(60\) −2.73205 −0.352706
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0.535898 0.0680592
\(63\) −4.46410 −0.562424
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 15.8564 1.93717 0.968584 0.248686i \(-0.0799987\pi\)
0.968584 + 0.248686i \(0.0799987\pi\)
\(68\) −0.535898 −0.0649872
\(69\) 22.3923 2.69572
\(70\) 1.00000 0.119523
\(71\) 1.07180 0.127199 0.0635994 0.997976i \(-0.479742\pi\)
0.0635994 + 0.997976i \(0.479742\pi\)
\(72\) 4.46410 0.526099
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −2.73205 −0.317594
\(75\) 2.73205 0.315470
\(76\) 0.732051 0.0839720
\(77\) 0 0
\(78\) −10.9282 −1.23738
\(79\) −7.66025 −0.861846 −0.430923 0.902389i \(-0.641812\pi\)
−0.430923 + 0.902389i \(0.641812\pi\)
\(80\) −1.00000 −0.111803
\(81\) −2.46410 −0.273789
\(82\) −2.19615 −0.242524
\(83\) −14.9282 −1.63858 −0.819292 0.573377i \(-0.805633\pi\)
−0.819292 + 0.573377i \(0.805633\pi\)
\(84\) −2.73205 −0.298091
\(85\) 0.535898 0.0581263
\(86\) 4.92820 0.531422
\(87\) 20.9282 2.24374
\(88\) 0 0
\(89\) −8.92820 −0.946388 −0.473194 0.880958i \(-0.656899\pi\)
−0.473194 + 0.880958i \(0.656899\pi\)
\(90\) −4.46410 −0.470558
\(91\) 4.00000 0.419314
\(92\) 8.19615 0.854508
\(93\) 1.46410 0.151820
\(94\) 6.00000 0.618853
\(95\) −0.732051 −0.0751068
\(96\) 2.73205 0.278839
\(97\) 2.73205 0.277398 0.138699 0.990335i \(-0.455708\pi\)
0.138699 + 0.990335i \(0.455708\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) −1.46410 −0.144968
\(103\) 2.39230 0.235721 0.117860 0.993030i \(-0.462396\pi\)
0.117860 + 0.993030i \(0.462396\pi\)
\(104\) −4.00000 −0.392232
\(105\) 2.73205 0.266621
\(106\) 5.26795 0.511668
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 4.00000 0.384900
\(109\) 16.7321 1.60264 0.801320 0.598236i \(-0.204132\pi\)
0.801320 + 0.598236i \(0.204132\pi\)
\(110\) 0 0
\(111\) −7.46410 −0.708461
\(112\) −1.00000 −0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 2.00000 0.187317
\(115\) −8.19615 −0.764295
\(116\) 7.66025 0.711237
\(117\) −17.8564 −1.65083
\(118\) 8.39230 0.772574
\(119\) 0.535898 0.0491257
\(120\) −2.73205 −0.249401
\(121\) 0 0
\(122\) 8.00000 0.724286
\(123\) −6.00000 −0.541002
\(124\) 0.535898 0.0481251
\(125\) −1.00000 −0.0894427
\(126\) −4.46410 −0.397694
\(127\) −8.39230 −0.744697 −0.372348 0.928093i \(-0.621447\pi\)
−0.372348 + 0.928093i \(0.621447\pi\)
\(128\) 1.00000 0.0883883
\(129\) 13.4641 1.18545
\(130\) 4.00000 0.350823
\(131\) −10.1962 −0.890842 −0.445421 0.895321i \(-0.646946\pi\)
−0.445421 + 0.895321i \(0.646946\pi\)
\(132\) 0 0
\(133\) −0.732051 −0.0634769
\(134\) 15.8564 1.36978
\(135\) −4.00000 −0.344265
\(136\) −0.535898 −0.0459529
\(137\) −7.85641 −0.671218 −0.335609 0.942001i \(-0.608942\pi\)
−0.335609 + 0.942001i \(0.608942\pi\)
\(138\) 22.3923 1.90616
\(139\) 3.66025 0.310459 0.155229 0.987878i \(-0.450388\pi\)
0.155229 + 0.987878i \(0.450388\pi\)
\(140\) 1.00000 0.0845154
\(141\) 16.3923 1.38048
\(142\) 1.07180 0.0899432
\(143\) 0 0
\(144\) 4.46410 0.372008
\(145\) −7.66025 −0.636149
\(146\) 6.00000 0.496564
\(147\) 2.73205 0.225336
\(148\) −2.73205 −0.224573
\(149\) −1.80385 −0.147777 −0.0738885 0.997267i \(-0.523541\pi\)
−0.0738885 + 0.997267i \(0.523541\pi\)
\(150\) 2.73205 0.223071
\(151\) 1.80385 0.146795 0.0733975 0.997303i \(-0.476616\pi\)
0.0733975 + 0.997303i \(0.476616\pi\)
\(152\) 0.732051 0.0593772
\(153\) −2.39230 −0.193406
\(154\) 0 0
\(155\) −0.535898 −0.0430444
\(156\) −10.9282 −0.874957
\(157\) −16.5359 −1.31971 −0.659854 0.751394i \(-0.729382\pi\)
−0.659854 + 0.751394i \(0.729382\pi\)
\(158\) −7.66025 −0.609417
\(159\) 14.3923 1.14138
\(160\) −1.00000 −0.0790569
\(161\) −8.19615 −0.645947
\(162\) −2.46410 −0.193598
\(163\) 12.9282 1.01262 0.506308 0.862353i \(-0.331010\pi\)
0.506308 + 0.862353i \(0.331010\pi\)
\(164\) −2.19615 −0.171491
\(165\) 0 0
\(166\) −14.9282 −1.15865
\(167\) −5.07180 −0.392467 −0.196234 0.980557i \(-0.562871\pi\)
−0.196234 + 0.980557i \(0.562871\pi\)
\(168\) −2.73205 −0.210782
\(169\) 3.00000 0.230769
\(170\) 0.535898 0.0411015
\(171\) 3.26795 0.249906
\(172\) 4.92820 0.375772
\(173\) −4.92820 −0.374684 −0.187342 0.982295i \(-0.559987\pi\)
−0.187342 + 0.982295i \(0.559987\pi\)
\(174\) 20.9282 1.58656
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 22.9282 1.72339
\(178\) −8.92820 −0.669197
\(179\) 6.39230 0.477783 0.238892 0.971046i \(-0.423216\pi\)
0.238892 + 0.971046i \(0.423216\pi\)
\(180\) −4.46410 −0.332734
\(181\) −4.92820 −0.366310 −0.183155 0.983084i \(-0.558631\pi\)
−0.183155 + 0.983084i \(0.558631\pi\)
\(182\) 4.00000 0.296500
\(183\) 21.8564 1.61567
\(184\) 8.19615 0.604228
\(185\) 2.73205 0.200864
\(186\) 1.46410 0.107353
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) −4.00000 −0.290957
\(190\) −0.732051 −0.0531085
\(191\) −20.3923 −1.47554 −0.737768 0.675055i \(-0.764120\pi\)
−0.737768 + 0.675055i \(0.764120\pi\)
\(192\) 2.73205 0.197169
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 2.73205 0.196150
\(195\) 10.9282 0.782585
\(196\) 1.00000 0.0714286
\(197\) 3.46410 0.246807 0.123404 0.992357i \(-0.460619\pi\)
0.123404 + 0.992357i \(0.460619\pi\)
\(198\) 0 0
\(199\) −1.85641 −0.131597 −0.0657986 0.997833i \(-0.520959\pi\)
−0.0657986 + 0.997833i \(0.520959\pi\)
\(200\) 1.00000 0.0707107
\(201\) 43.3205 3.05559
\(202\) −8.00000 −0.562878
\(203\) −7.66025 −0.537644
\(204\) −1.46410 −0.102508
\(205\) 2.19615 0.153386
\(206\) 2.39230 0.166680
\(207\) 36.5885 2.54307
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) 2.73205 0.188529
\(211\) 13.8564 0.953914 0.476957 0.878927i \(-0.341740\pi\)
0.476957 + 0.878927i \(0.341740\pi\)
\(212\) 5.26795 0.361804
\(213\) 2.92820 0.200637
\(214\) 2.00000 0.136717
\(215\) −4.92820 −0.336101
\(216\) 4.00000 0.272166
\(217\) −0.535898 −0.0363792
\(218\) 16.7321 1.13324
\(219\) 16.3923 1.10769
\(220\) 0 0
\(221\) 2.14359 0.144194
\(222\) −7.46410 −0.500958
\(223\) −11.4641 −0.767693 −0.383847 0.923397i \(-0.625401\pi\)
−0.383847 + 0.923397i \(0.625401\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 4.46410 0.297607
\(226\) 18.0000 1.19734
\(227\) 23.3205 1.54784 0.773918 0.633286i \(-0.218294\pi\)
0.773918 + 0.633286i \(0.218294\pi\)
\(228\) 2.00000 0.132453
\(229\) 23.3205 1.54106 0.770531 0.637402i \(-0.219991\pi\)
0.770531 + 0.637402i \(0.219991\pi\)
\(230\) −8.19615 −0.540438
\(231\) 0 0
\(232\) 7.66025 0.502920
\(233\) −23.8564 −1.56289 −0.781443 0.623977i \(-0.785516\pi\)
−0.781443 + 0.623977i \(0.785516\pi\)
\(234\) −17.8564 −1.16731
\(235\) −6.00000 −0.391397
\(236\) 8.39230 0.546293
\(237\) −20.9282 −1.35943
\(238\) 0.535898 0.0347371
\(239\) −20.0526 −1.29709 −0.648546 0.761175i \(-0.724623\pi\)
−0.648546 + 0.761175i \(0.724623\pi\)
\(240\) −2.73205 −0.176353
\(241\) 22.9808 1.48032 0.740161 0.672430i \(-0.234749\pi\)
0.740161 + 0.672430i \(0.234749\pi\)
\(242\) 0 0
\(243\) −18.7321 −1.20166
\(244\) 8.00000 0.512148
\(245\) −1.00000 −0.0638877
\(246\) −6.00000 −0.382546
\(247\) −2.92820 −0.186317
\(248\) 0.535898 0.0340296
\(249\) −40.7846 −2.58462
\(250\) −1.00000 −0.0632456
\(251\) −24.7846 −1.56439 −0.782195 0.623033i \(-0.785900\pi\)
−0.782195 + 0.623033i \(0.785900\pi\)
\(252\) −4.46410 −0.281212
\(253\) 0 0
\(254\) −8.39230 −0.526580
\(255\) 1.46410 0.0916856
\(256\) 1.00000 0.0625000
\(257\) −7.80385 −0.486791 −0.243395 0.969927i \(-0.578261\pi\)
−0.243395 + 0.969927i \(0.578261\pi\)
\(258\) 13.4641 0.838238
\(259\) 2.73205 0.169761
\(260\) 4.00000 0.248069
\(261\) 34.1962 2.11669
\(262\) −10.1962 −0.629920
\(263\) −28.3923 −1.75074 −0.875372 0.483449i \(-0.839384\pi\)
−0.875372 + 0.483449i \(0.839384\pi\)
\(264\) 0 0
\(265\) −5.26795 −0.323608
\(266\) −0.732051 −0.0448849
\(267\) −24.3923 −1.49278
\(268\) 15.8564 0.968584
\(269\) −4.39230 −0.267804 −0.133902 0.990995i \(-0.542751\pi\)
−0.133902 + 0.990995i \(0.542751\pi\)
\(270\) −4.00000 −0.243432
\(271\) −32.3923 −1.96769 −0.983846 0.179016i \(-0.942709\pi\)
−0.983846 + 0.179016i \(0.942709\pi\)
\(272\) −0.535898 −0.0324936
\(273\) 10.9282 0.661405
\(274\) −7.85641 −0.474623
\(275\) 0 0
\(276\) 22.3923 1.34786
\(277\) 17.3205 1.04069 0.520344 0.853957i \(-0.325804\pi\)
0.520344 + 0.853957i \(0.325804\pi\)
\(278\) 3.66025 0.219527
\(279\) 2.39230 0.143224
\(280\) 1.00000 0.0597614
\(281\) 23.3205 1.39118 0.695592 0.718437i \(-0.255142\pi\)
0.695592 + 0.718437i \(0.255142\pi\)
\(282\) 16.3923 0.976148
\(283\) 9.46410 0.562582 0.281291 0.959622i \(-0.409237\pi\)
0.281291 + 0.959622i \(0.409237\pi\)
\(284\) 1.07180 0.0635994
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 2.19615 0.129635
\(288\) 4.46410 0.263050
\(289\) −16.7128 −0.983107
\(290\) −7.66025 −0.449826
\(291\) 7.46410 0.437553
\(292\) 6.00000 0.351123
\(293\) −27.7128 −1.61900 −0.809500 0.587120i \(-0.800262\pi\)
−0.809500 + 0.587120i \(0.800262\pi\)
\(294\) 2.73205 0.159336
\(295\) −8.39230 −0.488619
\(296\) −2.73205 −0.158797
\(297\) 0 0
\(298\) −1.80385 −0.104494
\(299\) −32.7846 −1.89598
\(300\) 2.73205 0.157735
\(301\) −4.92820 −0.284057
\(302\) 1.80385 0.103800
\(303\) −21.8564 −1.25562
\(304\) 0.732051 0.0419860
\(305\) −8.00000 −0.458079
\(306\) −2.39230 −0.136759
\(307\) −13.4641 −0.768437 −0.384218 0.923242i \(-0.625529\pi\)
−0.384218 + 0.923242i \(0.625529\pi\)
\(308\) 0 0
\(309\) 6.53590 0.371814
\(310\) −0.535898 −0.0304370
\(311\) 17.3205 0.982156 0.491078 0.871116i \(-0.336603\pi\)
0.491078 + 0.871116i \(0.336603\pi\)
\(312\) −10.9282 −0.618688
\(313\) −2.73205 −0.154425 −0.0772123 0.997015i \(-0.524602\pi\)
−0.0772123 + 0.997015i \(0.524602\pi\)
\(314\) −16.5359 −0.933175
\(315\) 4.46410 0.251524
\(316\) −7.66025 −0.430923
\(317\) −4.58846 −0.257713 −0.128857 0.991663i \(-0.541131\pi\)
−0.128857 + 0.991663i \(0.541131\pi\)
\(318\) 14.3923 0.807080
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 5.46410 0.304976
\(322\) −8.19615 −0.456754
\(323\) −0.392305 −0.0218284
\(324\) −2.46410 −0.136895
\(325\) −4.00000 −0.221880
\(326\) 12.9282 0.716027
\(327\) 45.7128 2.52792
\(328\) −2.19615 −0.121262
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 5.60770 0.308227 0.154113 0.988053i \(-0.450748\pi\)
0.154113 + 0.988053i \(0.450748\pi\)
\(332\) −14.9282 −0.819292
\(333\) −12.1962 −0.668345
\(334\) −5.07180 −0.277516
\(335\) −15.8564 −0.866328
\(336\) −2.73205 −0.149046
\(337\) −27.8564 −1.51744 −0.758718 0.651420i \(-0.774174\pi\)
−0.758718 + 0.651420i \(0.774174\pi\)
\(338\) 3.00000 0.163178
\(339\) 49.1769 2.67092
\(340\) 0.535898 0.0290632
\(341\) 0 0
\(342\) 3.26795 0.176710
\(343\) −1.00000 −0.0539949
\(344\) 4.92820 0.265711
\(345\) −22.3923 −1.20556
\(346\) −4.92820 −0.264942
\(347\) −4.92820 −0.264560 −0.132280 0.991212i \(-0.542230\pi\)
−0.132280 + 0.991212i \(0.542230\pi\)
\(348\) 20.9282 1.12187
\(349\) −5.46410 −0.292487 −0.146243 0.989249i \(-0.546718\pi\)
−0.146243 + 0.989249i \(0.546718\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −16.0000 −0.854017
\(352\) 0 0
\(353\) 24.9808 1.32959 0.664796 0.747025i \(-0.268519\pi\)
0.664796 + 0.747025i \(0.268519\pi\)
\(354\) 22.9282 1.21862
\(355\) −1.07180 −0.0568851
\(356\) −8.92820 −0.473194
\(357\) 1.46410 0.0774885
\(358\) 6.39230 0.337844
\(359\) −24.4449 −1.29015 −0.645075 0.764119i \(-0.723174\pi\)
−0.645075 + 0.764119i \(0.723174\pi\)
\(360\) −4.46410 −0.235279
\(361\) −18.4641 −0.971795
\(362\) −4.92820 −0.259021
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −6.00000 −0.314054
\(366\) 21.8564 1.14245
\(367\) −14.3923 −0.751272 −0.375636 0.926767i \(-0.622576\pi\)
−0.375636 + 0.926767i \(0.622576\pi\)
\(368\) 8.19615 0.427254
\(369\) −9.80385 −0.510368
\(370\) 2.73205 0.142033
\(371\) −5.26795 −0.273498
\(372\) 1.46410 0.0759101
\(373\) −33.3205 −1.72527 −0.862635 0.505826i \(-0.831188\pi\)
−0.862635 + 0.505826i \(0.831188\pi\)
\(374\) 0 0
\(375\) −2.73205 −0.141082
\(376\) 6.00000 0.309426
\(377\) −30.6410 −1.57809
\(378\) −4.00000 −0.205738
\(379\) 29.8564 1.53362 0.766810 0.641874i \(-0.221843\pi\)
0.766810 + 0.641874i \(0.221843\pi\)
\(380\) −0.732051 −0.0375534
\(381\) −22.9282 −1.17465
\(382\) −20.3923 −1.04336
\(383\) 3.46410 0.177007 0.0885037 0.996076i \(-0.471792\pi\)
0.0885037 + 0.996076i \(0.471792\pi\)
\(384\) 2.73205 0.139419
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) 22.0000 1.11832
\(388\) 2.73205 0.138699
\(389\) 26.3923 1.33814 0.669071 0.743198i \(-0.266692\pi\)
0.669071 + 0.743198i \(0.266692\pi\)
\(390\) 10.9282 0.553371
\(391\) −4.39230 −0.222128
\(392\) 1.00000 0.0505076
\(393\) −27.8564 −1.40517
\(394\) 3.46410 0.174519
\(395\) 7.66025 0.385429
\(396\) 0 0
\(397\) 7.46410 0.374613 0.187306 0.982302i \(-0.440024\pi\)
0.187306 + 0.982302i \(0.440024\pi\)
\(398\) −1.85641 −0.0930532
\(399\) −2.00000 −0.100125
\(400\) 1.00000 0.0500000
\(401\) 20.3923 1.01834 0.509172 0.860665i \(-0.329952\pi\)
0.509172 + 0.860665i \(0.329952\pi\)
\(402\) 43.3205 2.16063
\(403\) −2.14359 −0.106780
\(404\) −8.00000 −0.398015
\(405\) 2.46410 0.122442
\(406\) −7.66025 −0.380172
\(407\) 0 0
\(408\) −1.46410 −0.0724838
\(409\) −30.1962 −1.49310 −0.746552 0.665327i \(-0.768292\pi\)
−0.746552 + 0.665327i \(0.768292\pi\)
\(410\) 2.19615 0.108460
\(411\) −21.4641 −1.05875
\(412\) 2.39230 0.117860
\(413\) −8.39230 −0.412958
\(414\) 36.5885 1.79822
\(415\) 14.9282 0.732797
\(416\) −4.00000 −0.196116
\(417\) 10.0000 0.489702
\(418\) 0 0
\(419\) 37.8564 1.84941 0.924703 0.380689i \(-0.124313\pi\)
0.924703 + 0.380689i \(0.124313\pi\)
\(420\) 2.73205 0.133310
\(421\) 24.2487 1.18181 0.590905 0.806741i \(-0.298771\pi\)
0.590905 + 0.806741i \(0.298771\pi\)
\(422\) 13.8564 0.674519
\(423\) 26.7846 1.30231
\(424\) 5.26795 0.255834
\(425\) −0.535898 −0.0259949
\(426\) 2.92820 0.141872
\(427\) −8.00000 −0.387147
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) −4.92820 −0.237659
\(431\) 16.7321 0.805955 0.402977 0.915210i \(-0.367975\pi\)
0.402977 + 0.915210i \(0.367975\pi\)
\(432\) 4.00000 0.192450
\(433\) −29.2679 −1.40653 −0.703264 0.710929i \(-0.748275\pi\)
−0.703264 + 0.710929i \(0.748275\pi\)
\(434\) −0.535898 −0.0257239
\(435\) −20.9282 −1.00343
\(436\) 16.7321 0.801320
\(437\) 6.00000 0.287019
\(438\) 16.3923 0.783255
\(439\) 29.4641 1.40624 0.703122 0.711069i \(-0.251789\pi\)
0.703122 + 0.711069i \(0.251789\pi\)
\(440\) 0 0
\(441\) 4.46410 0.212576
\(442\) 2.14359 0.101960
\(443\) −28.9282 −1.37442 −0.687210 0.726459i \(-0.741165\pi\)
−0.687210 + 0.726459i \(0.741165\pi\)
\(444\) −7.46410 −0.354231
\(445\) 8.92820 0.423237
\(446\) −11.4641 −0.542841
\(447\) −4.92820 −0.233096
\(448\) −1.00000 −0.0472456
\(449\) −10.2487 −0.483667 −0.241833 0.970318i \(-0.577749\pi\)
−0.241833 + 0.970318i \(0.577749\pi\)
\(450\) 4.46410 0.210440
\(451\) 0 0
\(452\) 18.0000 0.846649
\(453\) 4.92820 0.231547
\(454\) 23.3205 1.09449
\(455\) −4.00000 −0.187523
\(456\) 2.00000 0.0936586
\(457\) 15.8564 0.741731 0.370866 0.928687i \(-0.379061\pi\)
0.370866 + 0.928687i \(0.379061\pi\)
\(458\) 23.3205 1.08970
\(459\) −2.14359 −0.100054
\(460\) −8.19615 −0.382148
\(461\) 4.67949 0.217946 0.108973 0.994045i \(-0.465244\pi\)
0.108973 + 0.994045i \(0.465244\pi\)
\(462\) 0 0
\(463\) 7.41154 0.344444 0.172222 0.985058i \(-0.444905\pi\)
0.172222 + 0.985058i \(0.444905\pi\)
\(464\) 7.66025 0.355618
\(465\) −1.46410 −0.0678961
\(466\) −23.8564 −1.10513
\(467\) 40.5885 1.87821 0.939105 0.343631i \(-0.111657\pi\)
0.939105 + 0.343631i \(0.111657\pi\)
\(468\) −17.8564 −0.825413
\(469\) −15.8564 −0.732181
\(470\) −6.00000 −0.276759
\(471\) −45.1769 −2.08164
\(472\) 8.39230 0.386287
\(473\) 0 0
\(474\) −20.9282 −0.961264
\(475\) 0.732051 0.0335888
\(476\) 0.535898 0.0245629
\(477\) 23.5167 1.07675
\(478\) −20.0526 −0.917183
\(479\) 19.7128 0.900701 0.450351 0.892852i \(-0.351299\pi\)
0.450351 + 0.892852i \(0.351299\pi\)
\(480\) −2.73205 −0.124700
\(481\) 10.9282 0.498283
\(482\) 22.9808 1.04675
\(483\) −22.3923 −1.01889
\(484\) 0 0
\(485\) −2.73205 −0.124056
\(486\) −18.7321 −0.849703
\(487\) −30.0526 −1.36181 −0.680906 0.732371i \(-0.738414\pi\)
−0.680906 + 0.732371i \(0.738414\pi\)
\(488\) 8.00000 0.362143
\(489\) 35.3205 1.59725
\(490\) −1.00000 −0.0451754
\(491\) 9.07180 0.409404 0.204702 0.978824i \(-0.434377\pi\)
0.204702 + 0.978824i \(0.434377\pi\)
\(492\) −6.00000 −0.270501
\(493\) −4.10512 −0.184885
\(494\) −2.92820 −0.131746
\(495\) 0 0
\(496\) 0.535898 0.0240625
\(497\) −1.07180 −0.0480767
\(498\) −40.7846 −1.82760
\(499\) −22.3923 −1.00242 −0.501209 0.865326i \(-0.667111\pi\)
−0.501209 + 0.865326i \(0.667111\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −13.8564 −0.619059
\(502\) −24.7846 −1.10619
\(503\) 10.1436 0.452280 0.226140 0.974095i \(-0.427389\pi\)
0.226140 + 0.974095i \(0.427389\pi\)
\(504\) −4.46410 −0.198847
\(505\) 8.00000 0.355995
\(506\) 0 0
\(507\) 8.19615 0.364004
\(508\) −8.39230 −0.372348
\(509\) −0.928203 −0.0411419 −0.0205709 0.999788i \(-0.506548\pi\)
−0.0205709 + 0.999788i \(0.506548\pi\)
\(510\) 1.46410 0.0648315
\(511\) −6.00000 −0.265424
\(512\) 1.00000 0.0441942
\(513\) 2.92820 0.129283
\(514\) −7.80385 −0.344213
\(515\) −2.39230 −0.105418
\(516\) 13.4641 0.592724
\(517\) 0 0
\(518\) 2.73205 0.120039
\(519\) −13.4641 −0.591008
\(520\) 4.00000 0.175412
\(521\) −24.2487 −1.06236 −0.531178 0.847260i \(-0.678250\pi\)
−0.531178 + 0.847260i \(0.678250\pi\)
\(522\) 34.1962 1.49672
\(523\) 14.5359 0.635610 0.317805 0.948156i \(-0.397054\pi\)
0.317805 + 0.948156i \(0.397054\pi\)
\(524\) −10.1962 −0.445421
\(525\) −2.73205 −0.119236
\(526\) −28.3923 −1.23796
\(527\) −0.287187 −0.0125101
\(528\) 0 0
\(529\) 44.1769 1.92074
\(530\) −5.26795 −0.228825
\(531\) 37.4641 1.62580
\(532\) −0.732051 −0.0317384
\(533\) 8.78461 0.380504
\(534\) −24.3923 −1.05556
\(535\) −2.00000 −0.0864675
\(536\) 15.8564 0.684892
\(537\) 17.4641 0.753632
\(538\) −4.39230 −0.189366
\(539\) 0 0
\(540\) −4.00000 −0.172133
\(541\) 12.7321 0.547394 0.273697 0.961816i \(-0.411754\pi\)
0.273697 + 0.961816i \(0.411754\pi\)
\(542\) −32.3923 −1.39137
\(543\) −13.4641 −0.577800
\(544\) −0.535898 −0.0229765
\(545\) −16.7321 −0.716722
\(546\) 10.9282 0.467684
\(547\) −23.7128 −1.01389 −0.506943 0.861979i \(-0.669225\pi\)
−0.506943 + 0.861979i \(0.669225\pi\)
\(548\) −7.85641 −0.335609
\(549\) 35.7128 1.52419
\(550\) 0 0
\(551\) 5.60770 0.238896
\(552\) 22.3923 0.953080
\(553\) 7.66025 0.325747
\(554\) 17.3205 0.735878
\(555\) 7.46410 0.316833
\(556\) 3.66025 0.155229
\(557\) −3.07180 −0.130156 −0.0650781 0.997880i \(-0.520730\pi\)
−0.0650781 + 0.997880i \(0.520730\pi\)
\(558\) 2.39230 0.101274
\(559\) −19.7128 −0.833763
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 23.3205 0.983716
\(563\) −39.7128 −1.67370 −0.836848 0.547436i \(-0.815604\pi\)
−0.836848 + 0.547436i \(0.815604\pi\)
\(564\) 16.3923 0.690241
\(565\) −18.0000 −0.757266
\(566\) 9.46410 0.397806
\(567\) 2.46410 0.103483
\(568\) 1.07180 0.0449716
\(569\) 33.8564 1.41933 0.709667 0.704537i \(-0.248845\pi\)
0.709667 + 0.704537i \(0.248845\pi\)
\(570\) −2.00000 −0.0837708
\(571\) −19.3205 −0.808538 −0.404269 0.914640i \(-0.632474\pi\)
−0.404269 + 0.914640i \(0.632474\pi\)
\(572\) 0 0
\(573\) −55.7128 −2.32744
\(574\) 2.19615 0.0916656
\(575\) 8.19615 0.341803
\(576\) 4.46410 0.186004
\(577\) 1.26795 0.0527854 0.0263927 0.999652i \(-0.491598\pi\)
0.0263927 + 0.999652i \(0.491598\pi\)
\(578\) −16.7128 −0.695161
\(579\) 43.7128 1.81664
\(580\) −7.66025 −0.318075
\(581\) 14.9282 0.619326
\(582\) 7.46410 0.309397
\(583\) 0 0
\(584\) 6.00000 0.248282
\(585\) 17.8564 0.738272
\(586\) −27.7128 −1.14481
\(587\) 26.7321 1.10335 0.551675 0.834059i \(-0.313989\pi\)
0.551675 + 0.834059i \(0.313989\pi\)
\(588\) 2.73205 0.112668
\(589\) 0.392305 0.0161646
\(590\) −8.39230 −0.345506
\(591\) 9.46410 0.389301
\(592\) −2.73205 −0.112287
\(593\) 28.6410 1.17615 0.588073 0.808808i \(-0.299887\pi\)
0.588073 + 0.808808i \(0.299887\pi\)
\(594\) 0 0
\(595\) −0.535898 −0.0219697
\(596\) −1.80385 −0.0738885
\(597\) −5.07180 −0.207575
\(598\) −32.7846 −1.34066
\(599\) −46.9282 −1.91743 −0.958717 0.284361i \(-0.908218\pi\)
−0.958717 + 0.284361i \(0.908218\pi\)
\(600\) 2.73205 0.111536
\(601\) −2.87564 −0.117300 −0.0586500 0.998279i \(-0.518680\pi\)
−0.0586500 + 0.998279i \(0.518680\pi\)
\(602\) −4.92820 −0.200859
\(603\) 70.7846 2.88257
\(604\) 1.80385 0.0733975
\(605\) 0 0
\(606\) −21.8564 −0.887856
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 0.732051 0.0296886
\(609\) −20.9282 −0.848054
\(610\) −8.00000 −0.323911
\(611\) −24.0000 −0.970936
\(612\) −2.39230 −0.0967032
\(613\) −41.7128 −1.68476 −0.842382 0.538880i \(-0.818847\pi\)
−0.842382 + 0.538880i \(0.818847\pi\)
\(614\) −13.4641 −0.543367
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) 38.3923 1.54562 0.772808 0.634640i \(-0.218852\pi\)
0.772808 + 0.634640i \(0.218852\pi\)
\(618\) 6.53590 0.262912
\(619\) 5.07180 0.203853 0.101926 0.994792i \(-0.467499\pi\)
0.101926 + 0.994792i \(0.467499\pi\)
\(620\) −0.535898 −0.0215222
\(621\) 32.7846 1.31560
\(622\) 17.3205 0.694489
\(623\) 8.92820 0.357701
\(624\) −10.9282 −0.437478
\(625\) 1.00000 0.0400000
\(626\) −2.73205 −0.109195
\(627\) 0 0
\(628\) −16.5359 −0.659854
\(629\) 1.46410 0.0583776
\(630\) 4.46410 0.177854
\(631\) −25.8564 −1.02933 −0.514664 0.857392i \(-0.672083\pi\)
−0.514664 + 0.857392i \(0.672083\pi\)
\(632\) −7.66025 −0.304709
\(633\) 37.8564 1.50466
\(634\) −4.58846 −0.182231
\(635\) 8.39230 0.333038
\(636\) 14.3923 0.570692
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) 4.78461 0.189276
\(640\) −1.00000 −0.0395285
\(641\) −23.8564 −0.942271 −0.471136 0.882061i \(-0.656156\pi\)
−0.471136 + 0.882061i \(0.656156\pi\)
\(642\) 5.46410 0.215651
\(643\) 4.19615 0.165480 0.0827400 0.996571i \(-0.473633\pi\)
0.0827400 + 0.996571i \(0.473633\pi\)
\(644\) −8.19615 −0.322974
\(645\) −13.4641 −0.530148
\(646\) −0.392305 −0.0154350
\(647\) −5.60770 −0.220461 −0.110231 0.993906i \(-0.535159\pi\)
−0.110231 + 0.993906i \(0.535159\pi\)
\(648\) −2.46410 −0.0967991
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) −1.46410 −0.0573827
\(652\) 12.9282 0.506308
\(653\) −34.7321 −1.35917 −0.679585 0.733597i \(-0.737840\pi\)
−0.679585 + 0.733597i \(0.737840\pi\)
\(654\) 45.7128 1.78751
\(655\) 10.1962 0.398397
\(656\) −2.19615 −0.0857453
\(657\) 26.7846 1.04497
\(658\) −6.00000 −0.233904
\(659\) −40.3923 −1.57346 −0.786730 0.617297i \(-0.788228\pi\)
−0.786730 + 0.617297i \(0.788228\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 5.60770 0.217949
\(663\) 5.85641 0.227444
\(664\) −14.9282 −0.579327
\(665\) 0.732051 0.0283877
\(666\) −12.1962 −0.472591
\(667\) 62.7846 2.43103
\(668\) −5.07180 −0.196234
\(669\) −31.3205 −1.21092
\(670\) −15.8564 −0.612586
\(671\) 0 0
\(672\) −2.73205 −0.105391
\(673\) 43.8564 1.69054 0.845270 0.534339i \(-0.179440\pi\)
0.845270 + 0.534339i \(0.179440\pi\)
\(674\) −27.8564 −1.07299
\(675\) 4.00000 0.153960
\(676\) 3.00000 0.115385
\(677\) 12.9282 0.496871 0.248436 0.968648i \(-0.420084\pi\)
0.248436 + 0.968648i \(0.420084\pi\)
\(678\) 49.1769 1.88863
\(679\) −2.73205 −0.104846
\(680\) 0.535898 0.0205508
\(681\) 63.7128 2.44148
\(682\) 0 0
\(683\) 1.60770 0.0615167 0.0307584 0.999527i \(-0.490208\pi\)
0.0307584 + 0.999527i \(0.490208\pi\)
\(684\) 3.26795 0.124953
\(685\) 7.85641 0.300178
\(686\) −1.00000 −0.0381802
\(687\) 63.7128 2.43080
\(688\) 4.92820 0.187886
\(689\) −21.0718 −0.802772
\(690\) −22.3923 −0.852460
\(691\) −14.9282 −0.567896 −0.283948 0.958840i \(-0.591644\pi\)
−0.283948 + 0.958840i \(0.591644\pi\)
\(692\) −4.92820 −0.187342
\(693\) 0 0
\(694\) −4.92820 −0.187072
\(695\) −3.66025 −0.138841
\(696\) 20.9282 0.793281
\(697\) 1.17691 0.0445788
\(698\) −5.46410 −0.206819
\(699\) −65.1769 −2.46522
\(700\) −1.00000 −0.0377964
\(701\) −31.2679 −1.18097 −0.590487 0.807047i \(-0.701064\pi\)
−0.590487 + 0.807047i \(0.701064\pi\)
\(702\) −16.0000 −0.603881
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) −16.3923 −0.617370
\(706\) 24.9808 0.940163
\(707\) 8.00000 0.300871
\(708\) 22.9282 0.861695
\(709\) −3.85641 −0.144830 −0.0724152 0.997375i \(-0.523071\pi\)
−0.0724152 + 0.997375i \(0.523071\pi\)
\(710\) −1.07180 −0.0402238
\(711\) −34.1962 −1.28246
\(712\) −8.92820 −0.334599
\(713\) 4.39230 0.164493
\(714\) 1.46410 0.0547926
\(715\) 0 0
\(716\) 6.39230 0.238892
\(717\) −54.7846 −2.04597
\(718\) −24.4449 −0.912274
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) −4.46410 −0.166367
\(721\) −2.39230 −0.0890941
\(722\) −18.4641 −0.687163
\(723\) 62.7846 2.33498
\(724\) −4.92820 −0.183155
\(725\) 7.66025 0.284495
\(726\) 0 0
\(727\) −45.7128 −1.69539 −0.847697 0.530480i \(-0.822012\pi\)
−0.847697 + 0.530480i \(0.822012\pi\)
\(728\) 4.00000 0.148250
\(729\) −43.7846 −1.62165
\(730\) −6.00000 −0.222070
\(731\) −2.64102 −0.0976815
\(732\) 21.8564 0.807836
\(733\) −51.8564 −1.91536 −0.957680 0.287835i \(-0.907065\pi\)
−0.957680 + 0.287835i \(0.907065\pi\)
\(734\) −14.3923 −0.531230
\(735\) −2.73205 −0.100773
\(736\) 8.19615 0.302114
\(737\) 0 0
\(738\) −9.80385 −0.360885
\(739\) 9.85641 0.362574 0.181287 0.983430i \(-0.441974\pi\)
0.181287 + 0.983430i \(0.441974\pi\)
\(740\) 2.73205 0.100432
\(741\) −8.00000 −0.293887
\(742\) −5.26795 −0.193392
\(743\) 3.60770 0.132353 0.0661767 0.997808i \(-0.478920\pi\)
0.0661767 + 0.997808i \(0.478920\pi\)
\(744\) 1.46410 0.0536766
\(745\) 1.80385 0.0660879
\(746\) −33.3205 −1.21995
\(747\) −66.6410 −2.43827
\(748\) 0 0
\(749\) −2.00000 −0.0730784
\(750\) −2.73205 −0.0997604
\(751\) 22.1436 0.808031 0.404016 0.914752i \(-0.367614\pi\)
0.404016 + 0.914752i \(0.367614\pi\)
\(752\) 6.00000 0.218797
\(753\) −67.7128 −2.46759
\(754\) −30.6410 −1.11588
\(755\) −1.80385 −0.0656487
\(756\) −4.00000 −0.145479
\(757\) −10.3397 −0.375804 −0.187902 0.982188i \(-0.560169\pi\)
−0.187902 + 0.982188i \(0.560169\pi\)
\(758\) 29.8564 1.08443
\(759\) 0 0
\(760\) −0.732051 −0.0265543
\(761\) 34.5885 1.25383 0.626915 0.779087i \(-0.284317\pi\)
0.626915 + 0.779087i \(0.284317\pi\)
\(762\) −22.9282 −0.830601
\(763\) −16.7321 −0.605741
\(764\) −20.3923 −0.737768
\(765\) 2.39230 0.0864940
\(766\) 3.46410 0.125163
\(767\) −33.5692 −1.21211
\(768\) 2.73205 0.0985844
\(769\) −3.26795 −0.117845 −0.0589226 0.998263i \(-0.518767\pi\)
−0.0589226 + 0.998263i \(0.518767\pi\)
\(770\) 0 0
\(771\) −21.3205 −0.767839
\(772\) 16.0000 0.575853
\(773\) −45.7128 −1.64418 −0.822088 0.569361i \(-0.807191\pi\)
−0.822088 + 0.569361i \(0.807191\pi\)
\(774\) 22.0000 0.790774
\(775\) 0.535898 0.0192500
\(776\) 2.73205 0.0980749
\(777\) 7.46410 0.267773
\(778\) 26.3923 0.946210
\(779\) −1.60770 −0.0576017
\(780\) 10.9282 0.391292
\(781\) 0 0
\(782\) −4.39230 −0.157069
\(783\) 30.6410 1.09502
\(784\) 1.00000 0.0357143
\(785\) 16.5359 0.590192
\(786\) −27.8564 −0.993605
\(787\) −4.78461 −0.170553 −0.0852765 0.996357i \(-0.527177\pi\)
−0.0852765 + 0.996357i \(0.527177\pi\)
\(788\) 3.46410 0.123404
\(789\) −77.5692 −2.76154
\(790\) 7.66025 0.272540
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −32.0000 −1.13635
\(794\) 7.46410 0.264891
\(795\) −14.3923 −0.510442
\(796\) −1.85641 −0.0657986
\(797\) 18.3923 0.651489 0.325744 0.945458i \(-0.394385\pi\)
0.325744 + 0.945458i \(0.394385\pi\)
\(798\) −2.00000 −0.0707992
\(799\) −3.21539 −0.113752
\(800\) 1.00000 0.0353553
\(801\) −39.8564 −1.40826
\(802\) 20.3923 0.720077
\(803\) 0 0
\(804\) 43.3205 1.52780
\(805\) 8.19615 0.288876
\(806\) −2.14359 −0.0755049
\(807\) −12.0000 −0.422420
\(808\) −8.00000 −0.281439
\(809\) 44.1051 1.55065 0.775327 0.631560i \(-0.217585\pi\)
0.775327 + 0.631560i \(0.217585\pi\)
\(810\) 2.46410 0.0865797
\(811\) −3.26795 −0.114753 −0.0573766 0.998353i \(-0.518274\pi\)
−0.0573766 + 0.998353i \(0.518274\pi\)
\(812\) −7.66025 −0.268822
\(813\) −88.4974 −3.10374
\(814\) 0 0
\(815\) −12.9282 −0.452855
\(816\) −1.46410 −0.0512538
\(817\) 3.60770 0.126217
\(818\) −30.1962 −1.05578
\(819\) 17.8564 0.623953
\(820\) 2.19615 0.0766930
\(821\) 23.3731 0.815726 0.407863 0.913043i \(-0.366274\pi\)
0.407863 + 0.913043i \(0.366274\pi\)
\(822\) −21.4641 −0.748647
\(823\) 1.94744 0.0678835 0.0339418 0.999424i \(-0.489194\pi\)
0.0339418 + 0.999424i \(0.489194\pi\)
\(824\) 2.39230 0.0833399
\(825\) 0 0
\(826\) −8.39230 −0.292006
\(827\) 20.7846 0.722752 0.361376 0.932420i \(-0.382307\pi\)
0.361376 + 0.932420i \(0.382307\pi\)
\(828\) 36.5885 1.27154
\(829\) 3.60770 0.125300 0.0626502 0.998036i \(-0.480045\pi\)
0.0626502 + 0.998036i \(0.480045\pi\)
\(830\) 14.9282 0.518165
\(831\) 47.3205 1.64153
\(832\) −4.00000 −0.138675
\(833\) −0.535898 −0.0185678
\(834\) 10.0000 0.346272
\(835\) 5.07180 0.175517
\(836\) 0 0
\(837\) 2.14359 0.0740934
\(838\) 37.8564 1.30773
\(839\) −28.2487 −0.975254 −0.487627 0.873052i \(-0.662137\pi\)
−0.487627 + 0.873052i \(0.662137\pi\)
\(840\) 2.73205 0.0942647
\(841\) 29.6795 1.02343
\(842\) 24.2487 0.835666
\(843\) 63.7128 2.19439
\(844\) 13.8564 0.476957
\(845\) −3.00000 −0.103203
\(846\) 26.7846 0.920874
\(847\) 0 0
\(848\) 5.26795 0.180902
\(849\) 25.8564 0.887390
\(850\) −0.535898 −0.0183812
\(851\) −22.3923 −0.767598
\(852\) 2.92820 0.100319
\(853\) 24.6410 0.843692 0.421846 0.906667i \(-0.361382\pi\)
0.421846 + 0.906667i \(0.361382\pi\)
\(854\) −8.00000 −0.273754
\(855\) −3.26795 −0.111762
\(856\) 2.00000 0.0683586
\(857\) 24.9282 0.851531 0.425766 0.904833i \(-0.360005\pi\)
0.425766 + 0.904833i \(0.360005\pi\)
\(858\) 0 0
\(859\) 13.0718 0.446004 0.223002 0.974818i \(-0.428414\pi\)
0.223002 + 0.974818i \(0.428414\pi\)
\(860\) −4.92820 −0.168050
\(861\) 6.00000 0.204479
\(862\) 16.7321 0.569896
\(863\) 5.66025 0.192677 0.0963386 0.995349i \(-0.469287\pi\)
0.0963386 + 0.995349i \(0.469287\pi\)
\(864\) 4.00000 0.136083
\(865\) 4.92820 0.167564
\(866\) −29.2679 −0.994565
\(867\) −45.6603 −1.55070
\(868\) −0.535898 −0.0181896
\(869\) 0 0
\(870\) −20.9282 −0.709533
\(871\) −63.4256 −2.14910
\(872\) 16.7321 0.566619
\(873\) 12.1962 0.412777
\(874\) 6.00000 0.202953
\(875\) 1.00000 0.0338062
\(876\) 16.3923 0.553845
\(877\) 38.3923 1.29642 0.648208 0.761463i \(-0.275519\pi\)
0.648208 + 0.761463i \(0.275519\pi\)
\(878\) 29.4641 0.994365
\(879\) −75.7128 −2.55373
\(880\) 0 0
\(881\) 2.28719 0.0770573 0.0385286 0.999257i \(-0.487733\pi\)
0.0385286 + 0.999257i \(0.487733\pi\)
\(882\) 4.46410 0.150314
\(883\) −57.3205 −1.92899 −0.964494 0.264104i \(-0.914924\pi\)
−0.964494 + 0.264104i \(0.914924\pi\)
\(884\) 2.14359 0.0720969
\(885\) −22.9282 −0.770723
\(886\) −28.9282 −0.971862
\(887\) 46.6410 1.56605 0.783026 0.621989i \(-0.213675\pi\)
0.783026 + 0.621989i \(0.213675\pi\)
\(888\) −7.46410 −0.250479
\(889\) 8.39230 0.281469
\(890\) 8.92820 0.299274
\(891\) 0 0
\(892\) −11.4641 −0.383847
\(893\) 4.39230 0.146983
\(894\) −4.92820 −0.164824
\(895\) −6.39230 −0.213671
\(896\) −1.00000 −0.0334077
\(897\) −89.5692 −2.99063
\(898\) −10.2487 −0.342004
\(899\) 4.10512 0.136913
\(900\) 4.46410 0.148803
\(901\) −2.82309 −0.0940506
\(902\) 0 0
\(903\) −13.4641 −0.448057
\(904\) 18.0000 0.598671
\(905\) 4.92820 0.163819
\(906\) 4.92820 0.163729
\(907\) −9.32051 −0.309482 −0.154741 0.987955i \(-0.549454\pi\)
−0.154741 + 0.987955i \(0.549454\pi\)
\(908\) 23.3205 0.773918
\(909\) −35.7128 −1.18452
\(910\) −4.00000 −0.132599
\(911\) 43.0333 1.42576 0.712879 0.701287i \(-0.247391\pi\)
0.712879 + 0.701287i \(0.247391\pi\)
\(912\) 2.00000 0.0662266
\(913\) 0 0
\(914\) 15.8564 0.524483
\(915\) −21.8564 −0.722551
\(916\) 23.3205 0.770531
\(917\) 10.1962 0.336707
\(918\) −2.14359 −0.0707491
\(919\) 35.3731 1.16685 0.583425 0.812167i \(-0.301712\pi\)
0.583425 + 0.812167i \(0.301712\pi\)
\(920\) −8.19615 −0.270219
\(921\) −36.7846 −1.21209
\(922\) 4.67949 0.154111
\(923\) −4.28719 −0.141114
\(924\) 0 0
\(925\) −2.73205 −0.0898293
\(926\) 7.41154 0.243558
\(927\) 10.6795 0.350761
\(928\) 7.66025 0.251460
\(929\) −36.2487 −1.18928 −0.594641 0.803991i \(-0.702706\pi\)
−0.594641 + 0.803991i \(0.702706\pi\)
\(930\) −1.46410 −0.0480098
\(931\) 0.732051 0.0239920
\(932\) −23.8564 −0.781443
\(933\) 47.3205 1.54920
\(934\) 40.5885 1.32809
\(935\) 0 0
\(936\) −17.8564 −0.583655
\(937\) 6.67949 0.218209 0.109105 0.994030i \(-0.465202\pi\)
0.109105 + 0.994030i \(0.465202\pi\)
\(938\) −15.8564 −0.517730
\(939\) −7.46410 −0.243582
\(940\) −6.00000 −0.195698
\(941\) 6.92820 0.225853 0.112926 0.993603i \(-0.463978\pi\)
0.112926 + 0.993603i \(0.463978\pi\)
\(942\) −45.1769 −1.47194
\(943\) −18.0000 −0.586161
\(944\) 8.39230 0.273146
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) 2.39230 0.0777395 0.0388697 0.999244i \(-0.487624\pi\)
0.0388697 + 0.999244i \(0.487624\pi\)
\(948\) −20.9282 −0.679716
\(949\) −24.0000 −0.779073
\(950\) 0.732051 0.0237509
\(951\) −12.5359 −0.406504
\(952\) 0.535898 0.0173686
\(953\) −22.6410 −0.733414 −0.366707 0.930336i \(-0.619515\pi\)
−0.366707 + 0.930336i \(0.619515\pi\)
\(954\) 23.5167 0.761380
\(955\) 20.3923 0.659879
\(956\) −20.0526 −0.648546
\(957\) 0 0
\(958\) 19.7128 0.636892
\(959\) 7.85641 0.253697
\(960\) −2.73205 −0.0881766
\(961\) −30.7128 −0.990736
\(962\) 10.9282 0.352339
\(963\) 8.92820 0.287707
\(964\) 22.9808 0.740161
\(965\) −16.0000 −0.515058
\(966\) −22.3923 −0.720461
\(967\) −36.3923 −1.17030 −0.585149 0.810926i \(-0.698964\pi\)
−0.585149 + 0.810926i \(0.698964\pi\)
\(968\) 0 0
\(969\) −1.07180 −0.0344311
\(970\) −2.73205 −0.0877209
\(971\) 3.60770 0.115776 0.0578882 0.998323i \(-0.481563\pi\)
0.0578882 + 0.998323i \(0.481563\pi\)
\(972\) −18.7321 −0.600831
\(973\) −3.66025 −0.117342
\(974\) −30.0526 −0.962946
\(975\) −10.9282 −0.349983
\(976\) 8.00000 0.256074
\(977\) 31.8564 1.01918 0.509588 0.860418i \(-0.329798\pi\)
0.509588 + 0.860418i \(0.329798\pi\)
\(978\) 35.3205 1.12943
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 74.6936 2.38478
\(982\) 9.07180 0.289493
\(983\) 27.4641 0.875969 0.437984 0.898983i \(-0.355693\pi\)
0.437984 + 0.898983i \(0.355693\pi\)
\(984\) −6.00000 −0.191273
\(985\) −3.46410 −0.110375
\(986\) −4.10512 −0.130734
\(987\) −16.3923 −0.521773
\(988\) −2.92820 −0.0931586
\(989\) 40.3923 1.28440
\(990\) 0 0
\(991\) −3.60770 −0.114602 −0.0573011 0.998357i \(-0.518250\pi\)
−0.0573011 + 0.998357i \(0.518250\pi\)
\(992\) 0.535898 0.0170148
\(993\) 15.3205 0.486182
\(994\) −1.07180 −0.0339953
\(995\) 1.85641 0.0588520
\(996\) −40.7846 −1.29231
\(997\) −23.8564 −0.755540 −0.377770 0.925899i \(-0.623309\pi\)
−0.377770 + 0.925899i \(0.623309\pi\)
\(998\) −22.3923 −0.708816
\(999\) −10.9282 −0.345753
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.cd.1.2 yes 2
11.10 odd 2 8470.2.a.bs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.bs.1.2 2 11.10 odd 2
8470.2.a.cd.1.2 yes 2 1.1 even 1 trivial