Properties

Label 8470.2.a.bi.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +2.46410 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +0.535898 q^{17} +2.00000 q^{18} +7.92820 q^{19} +1.00000 q^{20} +1.00000 q^{21} +2.46410 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.46410 q^{26} +5.00000 q^{27} -1.00000 q^{28} -6.92820 q^{29} +1.00000 q^{30} -7.46410 q^{31} -1.00000 q^{32} -0.535898 q^{34} -1.00000 q^{35} -2.00000 q^{36} +4.92820 q^{37} -7.92820 q^{38} -2.46410 q^{39} -1.00000 q^{40} -12.3923 q^{41} -1.00000 q^{42} -8.39230 q^{43} -2.00000 q^{45} -2.46410 q^{46} -7.46410 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.535898 q^{51} +2.46410 q^{52} -0.535898 q^{53} -5.00000 q^{54} +1.00000 q^{56} -7.92820 q^{57} +6.92820 q^{58} -3.92820 q^{59} -1.00000 q^{60} +8.92820 q^{61} +7.46410 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.46410 q^{65} -14.3923 q^{67} +0.535898 q^{68} -2.46410 q^{69} +1.00000 q^{70} +8.00000 q^{71} +2.00000 q^{72} +7.46410 q^{73} -4.92820 q^{74} -1.00000 q^{75} +7.92820 q^{76} +2.46410 q^{78} +14.4641 q^{79} +1.00000 q^{80} +1.00000 q^{81} +12.3923 q^{82} +1.00000 q^{83} +1.00000 q^{84} +0.535898 q^{85} +8.39230 q^{86} +6.92820 q^{87} -10.5359 q^{89} +2.00000 q^{90} -2.46410 q^{91} +2.46410 q^{92} +7.46410 q^{93} +7.46410 q^{94} +7.92820 q^{95} +1.00000 q^{96} -14.9282 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - 2q^{7} - 2q^{8} - 4q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - 2q^{7} - 2q^{8} - 4q^{9} - 2q^{10} - 2q^{12} - 2q^{13} + 2q^{14} - 2q^{15} + 2q^{16} + 8q^{17} + 4q^{18} + 2q^{19} + 2q^{20} + 2q^{21} - 2q^{23} + 2q^{24} + 2q^{25} + 2q^{26} + 10q^{27} - 2q^{28} + 2q^{30} - 8q^{31} - 2q^{32} - 8q^{34} - 2q^{35} - 4q^{36} - 4q^{37} - 2q^{38} + 2q^{39} - 2q^{40} - 4q^{41} - 2q^{42} + 4q^{43} - 4q^{45} + 2q^{46} - 8q^{47} - 2q^{48} + 2q^{49} - 2q^{50} - 8q^{51} - 2q^{52} - 8q^{53} - 10q^{54} + 2q^{56} - 2q^{57} + 6q^{59} - 2q^{60} + 4q^{61} + 8q^{62} + 4q^{63} + 2q^{64} - 2q^{65} - 8q^{67} + 8q^{68} + 2q^{69} + 2q^{70} + 16q^{71} + 4q^{72} + 8q^{73} + 4q^{74} - 2q^{75} + 2q^{76} - 2q^{78} + 22q^{79} + 2q^{80} + 2q^{81} + 4q^{82} + 2q^{83} + 2q^{84} + 8q^{85} - 4q^{86} - 28q^{89} + 4q^{90} + 2q^{91} - 2q^{92} + 8q^{93} + 8q^{94} + 2q^{95} + 2q^{96} - 16q^{97} - 2q^{98} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) 2.46410 0.683419 0.341709 0.939806i \(-0.388994\pi\)
0.341709 + 0.939806i \(0.388994\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 0.535898 0.129974 0.0649872 0.997886i \(-0.479299\pi\)
0.0649872 + 0.997886i \(0.479299\pi\)
\(18\) 2.00000 0.471405
\(19\) 7.92820 1.81885 0.909427 0.415863i \(-0.136520\pi\)
0.909427 + 0.415863i \(0.136520\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 2.46410 0.513801 0.256900 0.966438i \(-0.417299\pi\)
0.256900 + 0.966438i \(0.417299\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −2.46410 −0.483250
\(27\) 5.00000 0.962250
\(28\) −1.00000 −0.188982
\(29\) −6.92820 −1.28654 −0.643268 0.765641i \(-0.722422\pi\)
−0.643268 + 0.765641i \(0.722422\pi\)
\(30\) 1.00000 0.182574
\(31\) −7.46410 −1.34059 −0.670296 0.742094i \(-0.733833\pi\)
−0.670296 + 0.742094i \(0.733833\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −0.535898 −0.0919058
\(35\) −1.00000 −0.169031
\(36\) −2.00000 −0.333333
\(37\) 4.92820 0.810192 0.405096 0.914274i \(-0.367238\pi\)
0.405096 + 0.914274i \(0.367238\pi\)
\(38\) −7.92820 −1.28612
\(39\) −2.46410 −0.394572
\(40\) −1.00000 −0.158114
\(41\) −12.3923 −1.93535 −0.967676 0.252195i \(-0.918848\pi\)
−0.967676 + 0.252195i \(0.918848\pi\)
\(42\) −1.00000 −0.154303
\(43\) −8.39230 −1.27981 −0.639907 0.768452i \(-0.721027\pi\)
−0.639907 + 0.768452i \(0.721027\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −2.46410 −0.363312
\(47\) −7.46410 −1.08875 −0.544376 0.838842i \(-0.683233\pi\)
−0.544376 + 0.838842i \(0.683233\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −0.535898 −0.0750408
\(52\) 2.46410 0.341709
\(53\) −0.535898 −0.0736113 −0.0368057 0.999322i \(-0.511718\pi\)
−0.0368057 + 0.999322i \(0.511718\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −7.92820 −1.05012
\(58\) 6.92820 0.909718
\(59\) −3.92820 −0.511409 −0.255704 0.966755i \(-0.582307\pi\)
−0.255704 + 0.966755i \(0.582307\pi\)
\(60\) −1.00000 −0.129099
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) 7.46410 0.947942
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 2.46410 0.305634
\(66\) 0 0
\(67\) −14.3923 −1.75830 −0.879150 0.476545i \(-0.841889\pi\)
−0.879150 + 0.476545i \(0.841889\pi\)
\(68\) 0.535898 0.0649872
\(69\) −2.46410 −0.296643
\(70\) 1.00000 0.119523
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 2.00000 0.235702
\(73\) 7.46410 0.873607 0.436804 0.899557i \(-0.356111\pi\)
0.436804 + 0.899557i \(0.356111\pi\)
\(74\) −4.92820 −0.572892
\(75\) −1.00000 −0.115470
\(76\) 7.92820 0.909427
\(77\) 0 0
\(78\) 2.46410 0.279005
\(79\) 14.4641 1.62734 0.813669 0.581328i \(-0.197467\pi\)
0.813669 + 0.581328i \(0.197467\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 12.3923 1.36850
\(83\) 1.00000 0.109764 0.0548821 0.998493i \(-0.482522\pi\)
0.0548821 + 0.998493i \(0.482522\pi\)
\(84\) 1.00000 0.109109
\(85\) 0.535898 0.0581263
\(86\) 8.39230 0.904966
\(87\) 6.92820 0.742781
\(88\) 0 0
\(89\) −10.5359 −1.11680 −0.558401 0.829571i \(-0.688585\pi\)
−0.558401 + 0.829571i \(0.688585\pi\)
\(90\) 2.00000 0.210819
\(91\) −2.46410 −0.258308
\(92\) 2.46410 0.256900
\(93\) 7.46410 0.773991
\(94\) 7.46410 0.769863
\(95\) 7.92820 0.813416
\(96\) 1.00000 0.102062
\(97\) −14.9282 −1.51573 −0.757865 0.652412i \(-0.773757\pi\)
−0.757865 + 0.652412i \(0.773757\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.4641 1.83725 0.918623 0.395134i \(-0.129302\pi\)
0.918623 + 0.395134i \(0.129302\pi\)
\(102\) 0.535898 0.0530618
\(103\) 5.46410 0.538394 0.269197 0.963085i \(-0.413242\pi\)
0.269197 + 0.963085i \(0.413242\pi\)
\(104\) −2.46410 −0.241625
\(105\) 1.00000 0.0975900
\(106\) 0.535898 0.0520511
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 5.00000 0.481125
\(109\) 2.39230 0.229141 0.114571 0.993415i \(-0.463451\pi\)
0.114571 + 0.993415i \(0.463451\pi\)
\(110\) 0 0
\(111\) −4.92820 −0.467764
\(112\) −1.00000 −0.0944911
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 7.92820 0.742544
\(115\) 2.46410 0.229779
\(116\) −6.92820 −0.643268
\(117\) −4.92820 −0.455613
\(118\) 3.92820 0.361620
\(119\) −0.535898 −0.0491257
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) −8.92820 −0.808322
\(123\) 12.3923 1.11738
\(124\) −7.46410 −0.670296
\(125\) 1.00000 0.0894427
\(126\) −2.00000 −0.178174
\(127\) 7.39230 0.655961 0.327980 0.944684i \(-0.393632\pi\)
0.327980 + 0.944684i \(0.393632\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.39230 0.738901
\(130\) −2.46410 −0.216116
\(131\) −16.8564 −1.47275 −0.736376 0.676573i \(-0.763464\pi\)
−0.736376 + 0.676573i \(0.763464\pi\)
\(132\) 0 0
\(133\) −7.92820 −0.687462
\(134\) 14.3923 1.24331
\(135\) 5.00000 0.430331
\(136\) −0.535898 −0.0459529
\(137\) −5.92820 −0.506481 −0.253240 0.967403i \(-0.581496\pi\)
−0.253240 + 0.967403i \(0.581496\pi\)
\(138\) 2.46410 0.209758
\(139\) 10.8564 0.920828 0.460414 0.887704i \(-0.347701\pi\)
0.460414 + 0.887704i \(0.347701\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 7.46410 0.628591
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) −6.92820 −0.575356
\(146\) −7.46410 −0.617733
\(147\) −1.00000 −0.0824786
\(148\) 4.92820 0.405096
\(149\) 16.9282 1.38681 0.693406 0.720547i \(-0.256109\pi\)
0.693406 + 0.720547i \(0.256109\pi\)
\(150\) 1.00000 0.0816497
\(151\) 12.4641 1.01431 0.507157 0.861854i \(-0.330696\pi\)
0.507157 + 0.861854i \(0.330696\pi\)
\(152\) −7.92820 −0.643062
\(153\) −1.07180 −0.0866496
\(154\) 0 0
\(155\) −7.46410 −0.599531
\(156\) −2.46410 −0.197286
\(157\) 14.3205 1.14290 0.571450 0.820637i \(-0.306381\pi\)
0.571450 + 0.820637i \(0.306381\pi\)
\(158\) −14.4641 −1.15070
\(159\) 0.535898 0.0424995
\(160\) −1.00000 −0.0790569
\(161\) −2.46410 −0.194198
\(162\) −1.00000 −0.0785674
\(163\) −22.0000 −1.72317 −0.861586 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) −12.3923 −0.967676
\(165\) 0 0
\(166\) −1.00000 −0.0776151
\(167\) −16.3923 −1.26847 −0.634237 0.773138i \(-0.718686\pi\)
−0.634237 + 0.773138i \(0.718686\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −6.92820 −0.532939
\(170\) −0.535898 −0.0411015
\(171\) −15.8564 −1.21257
\(172\) −8.39230 −0.639907
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −6.92820 −0.525226
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 3.92820 0.295262
\(178\) 10.5359 0.789699
\(179\) 12.3923 0.926244 0.463122 0.886294i \(-0.346729\pi\)
0.463122 + 0.886294i \(0.346729\pi\)
\(180\) −2.00000 −0.149071
\(181\) 3.53590 0.262821 0.131411 0.991328i \(-0.458049\pi\)
0.131411 + 0.991328i \(0.458049\pi\)
\(182\) 2.46410 0.182651
\(183\) −8.92820 −0.659992
\(184\) −2.46410 −0.181656
\(185\) 4.92820 0.362329
\(186\) −7.46410 −0.547294
\(187\) 0 0
\(188\) −7.46410 −0.544376
\(189\) −5.00000 −0.363696
\(190\) −7.92820 −0.575172
\(191\) −17.3923 −1.25846 −0.629232 0.777218i \(-0.716630\pi\)
−0.629232 + 0.777218i \(0.716630\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −19.0000 −1.36765 −0.683825 0.729646i \(-0.739685\pi\)
−0.683825 + 0.729646i \(0.739685\pi\)
\(194\) 14.9282 1.07178
\(195\) −2.46410 −0.176458
\(196\) 1.00000 0.0714286
\(197\) −4.00000 −0.284988 −0.142494 0.989796i \(-0.545512\pi\)
−0.142494 + 0.989796i \(0.545512\pi\)
\(198\) 0 0
\(199\) 1.85641 0.131597 0.0657986 0.997833i \(-0.479041\pi\)
0.0657986 + 0.997833i \(0.479041\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 14.3923 1.01515
\(202\) −18.4641 −1.29913
\(203\) 6.92820 0.486265
\(204\) −0.535898 −0.0375204
\(205\) −12.3923 −0.865516
\(206\) −5.46410 −0.380702
\(207\) −4.92820 −0.342534
\(208\) 2.46410 0.170855
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) 24.0000 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) −0.535898 −0.0368057
\(213\) −8.00000 −0.548151
\(214\) −6.00000 −0.410152
\(215\) −8.39230 −0.572350
\(216\) −5.00000 −0.340207
\(217\) 7.46410 0.506696
\(218\) −2.39230 −0.162027
\(219\) −7.46410 −0.504377
\(220\) 0 0
\(221\) 1.32051 0.0888270
\(222\) 4.92820 0.330759
\(223\) 3.46410 0.231973 0.115987 0.993251i \(-0.462997\pi\)
0.115987 + 0.993251i \(0.462997\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.00000 −0.133333
\(226\) −1.00000 −0.0665190
\(227\) 16.0000 1.06196 0.530979 0.847385i \(-0.321824\pi\)
0.530979 + 0.847385i \(0.321824\pi\)
\(228\) −7.92820 −0.525058
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) −2.46410 −0.162478
\(231\) 0 0
\(232\) 6.92820 0.454859
\(233\) −8.85641 −0.580202 −0.290101 0.956996i \(-0.593689\pi\)
−0.290101 + 0.956996i \(0.593689\pi\)
\(234\) 4.92820 0.322167
\(235\) −7.46410 −0.486904
\(236\) −3.92820 −0.255704
\(237\) −14.4641 −0.939544
\(238\) 0.535898 0.0347371
\(239\) 5.53590 0.358087 0.179044 0.983841i \(-0.442700\pi\)
0.179044 + 0.983841i \(0.442700\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 8.92820 0.571570
\(245\) 1.00000 0.0638877
\(246\) −12.3923 −0.790104
\(247\) 19.5359 1.24304
\(248\) 7.46410 0.473971
\(249\) −1.00000 −0.0633724
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) −7.39230 −0.463834
\(255\) −0.535898 −0.0335593
\(256\) 1.00000 0.0625000
\(257\) −18.3923 −1.14728 −0.573640 0.819107i \(-0.694469\pi\)
−0.573640 + 0.819107i \(0.694469\pi\)
\(258\) −8.39230 −0.522482
\(259\) −4.92820 −0.306224
\(260\) 2.46410 0.152817
\(261\) 13.8564 0.857690
\(262\) 16.8564 1.04139
\(263\) −9.53590 −0.588009 −0.294004 0.955804i \(-0.594988\pi\)
−0.294004 + 0.955804i \(0.594988\pi\)
\(264\) 0 0
\(265\) −0.535898 −0.0329200
\(266\) 7.92820 0.486109
\(267\) 10.5359 0.644787
\(268\) −14.3923 −0.879150
\(269\) 24.3205 1.48285 0.741424 0.671037i \(-0.234151\pi\)
0.741424 + 0.671037i \(0.234151\pi\)
\(270\) −5.00000 −0.304290
\(271\) −2.14359 −0.130214 −0.0651070 0.997878i \(-0.520739\pi\)
−0.0651070 + 0.997878i \(0.520739\pi\)
\(272\) 0.535898 0.0324936
\(273\) 2.46410 0.149134
\(274\) 5.92820 0.358136
\(275\) 0 0
\(276\) −2.46410 −0.148321
\(277\) 7.32051 0.439847 0.219923 0.975517i \(-0.429419\pi\)
0.219923 + 0.975517i \(0.429419\pi\)
\(278\) −10.8564 −0.651124
\(279\) 14.9282 0.893728
\(280\) 1.00000 0.0597614
\(281\) −17.0000 −1.01413 −0.507067 0.861906i \(-0.669271\pi\)
−0.507067 + 0.861906i \(0.669271\pi\)
\(282\) −7.46410 −0.444481
\(283\) −27.7846 −1.65162 −0.825812 0.563946i \(-0.809283\pi\)
−0.825812 + 0.563946i \(0.809283\pi\)
\(284\) 8.00000 0.474713
\(285\) −7.92820 −0.469626
\(286\) 0 0
\(287\) 12.3923 0.731495
\(288\) 2.00000 0.117851
\(289\) −16.7128 −0.983107
\(290\) 6.92820 0.406838
\(291\) 14.9282 0.875107
\(292\) 7.46410 0.436804
\(293\) 13.5359 0.790776 0.395388 0.918514i \(-0.370610\pi\)
0.395388 + 0.918514i \(0.370610\pi\)
\(294\) 1.00000 0.0583212
\(295\) −3.92820 −0.228709
\(296\) −4.92820 −0.286446
\(297\) 0 0
\(298\) −16.9282 −0.980624
\(299\) 6.07180 0.351141
\(300\) −1.00000 −0.0577350
\(301\) 8.39230 0.483724
\(302\) −12.4641 −0.717228
\(303\) −18.4641 −1.06073
\(304\) 7.92820 0.454714
\(305\) 8.92820 0.511227
\(306\) 1.07180 0.0612705
\(307\) 1.07180 0.0611707 0.0305853 0.999532i \(-0.490263\pi\)
0.0305853 + 0.999532i \(0.490263\pi\)
\(308\) 0 0
\(309\) −5.46410 −0.310842
\(310\) 7.46410 0.423932
\(311\) −11.3205 −0.641927 −0.320964 0.947092i \(-0.604007\pi\)
−0.320964 + 0.947092i \(0.604007\pi\)
\(312\) 2.46410 0.139502
\(313\) −26.7846 −1.51396 −0.756978 0.653441i \(-0.773325\pi\)
−0.756978 + 0.653441i \(0.773325\pi\)
\(314\) −14.3205 −0.808153
\(315\) 2.00000 0.112687
\(316\) 14.4641 0.813669
\(317\) −1.60770 −0.0902972 −0.0451486 0.998980i \(-0.514376\pi\)
−0.0451486 + 0.998980i \(0.514376\pi\)
\(318\) −0.535898 −0.0300517
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −6.00000 −0.334887
\(322\) 2.46410 0.137319
\(323\) 4.24871 0.236405
\(324\) 1.00000 0.0555556
\(325\) 2.46410 0.136684
\(326\) 22.0000 1.21847
\(327\) −2.39230 −0.132295
\(328\) 12.3923 0.684251
\(329\) 7.46410 0.411509
\(330\) 0 0
\(331\) −5.46410 −0.300334 −0.150167 0.988661i \(-0.547981\pi\)
−0.150167 + 0.988661i \(0.547981\pi\)
\(332\) 1.00000 0.0548821
\(333\) −9.85641 −0.540128
\(334\) 16.3923 0.896947
\(335\) −14.3923 −0.786336
\(336\) 1.00000 0.0545545
\(337\) 9.78461 0.533002 0.266501 0.963835i \(-0.414132\pi\)
0.266501 + 0.963835i \(0.414132\pi\)
\(338\) 6.92820 0.376845
\(339\) −1.00000 −0.0543125
\(340\) 0.535898 0.0290632
\(341\) 0 0
\(342\) 15.8564 0.857416
\(343\) −1.00000 −0.0539949
\(344\) 8.39230 0.452483
\(345\) −2.46410 −0.132663
\(346\) 18.0000 0.967686
\(347\) 12.9282 0.694022 0.347011 0.937861i \(-0.387197\pi\)
0.347011 + 0.937861i \(0.387197\pi\)
\(348\) 6.92820 0.371391
\(349\) 8.46410 0.453073 0.226536 0.974003i \(-0.427260\pi\)
0.226536 + 0.974003i \(0.427260\pi\)
\(350\) 1.00000 0.0534522
\(351\) 12.3205 0.657620
\(352\) 0 0
\(353\) −27.4641 −1.46177 −0.730883 0.682502i \(-0.760892\pi\)
−0.730883 + 0.682502i \(0.760892\pi\)
\(354\) −3.92820 −0.208782
\(355\) 8.00000 0.424596
\(356\) −10.5359 −0.558401
\(357\) 0.535898 0.0283628
\(358\) −12.3923 −0.654954
\(359\) −35.7128 −1.88485 −0.942425 0.334417i \(-0.891460\pi\)
−0.942425 + 0.334417i \(0.891460\pi\)
\(360\) 2.00000 0.105409
\(361\) 43.8564 2.30823
\(362\) −3.53590 −0.185843
\(363\) 0 0
\(364\) −2.46410 −0.129154
\(365\) 7.46410 0.390689
\(366\) 8.92820 0.466685
\(367\) −14.5359 −0.758768 −0.379384 0.925239i \(-0.623864\pi\)
−0.379384 + 0.925239i \(0.623864\pi\)
\(368\) 2.46410 0.128450
\(369\) 24.7846 1.29024
\(370\) −4.92820 −0.256205
\(371\) 0.535898 0.0278225
\(372\) 7.46410 0.386996
\(373\) −11.0718 −0.573276 −0.286638 0.958039i \(-0.592538\pi\)
−0.286638 + 0.958039i \(0.592538\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 7.46410 0.384932
\(377\) −17.0718 −0.879242
\(378\) 5.00000 0.257172
\(379\) −3.85641 −0.198090 −0.0990451 0.995083i \(-0.531579\pi\)
−0.0990451 + 0.995083i \(0.531579\pi\)
\(380\) 7.92820 0.406708
\(381\) −7.39230 −0.378719
\(382\) 17.3923 0.889868
\(383\) −14.5359 −0.742750 −0.371375 0.928483i \(-0.621114\pi\)
−0.371375 + 0.928483i \(0.621114\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 19.0000 0.967075
\(387\) 16.7846 0.853210
\(388\) −14.9282 −0.757865
\(389\) 21.3205 1.08099 0.540496 0.841346i \(-0.318236\pi\)
0.540496 + 0.841346i \(0.318236\pi\)
\(390\) 2.46410 0.124775
\(391\) 1.32051 0.0667810
\(392\) −1.00000 −0.0505076
\(393\) 16.8564 0.850293
\(394\) 4.00000 0.201517
\(395\) 14.4641 0.727768
\(396\) 0 0
\(397\) −24.9282 −1.25111 −0.625555 0.780180i \(-0.715128\pi\)
−0.625555 + 0.780180i \(0.715128\pi\)
\(398\) −1.85641 −0.0930532
\(399\) 7.92820 0.396907
\(400\) 1.00000 0.0500000
\(401\) 31.8564 1.59083 0.795417 0.606063i \(-0.207252\pi\)
0.795417 + 0.606063i \(0.207252\pi\)
\(402\) −14.3923 −0.717823
\(403\) −18.3923 −0.916186
\(404\) 18.4641 0.918623
\(405\) 1.00000 0.0496904
\(406\) −6.92820 −0.343841
\(407\) 0 0
\(408\) 0.535898 0.0265309
\(409\) −23.7128 −1.17252 −0.586262 0.810122i \(-0.699401\pi\)
−0.586262 + 0.810122i \(0.699401\pi\)
\(410\) 12.3923 0.612012
\(411\) 5.92820 0.292417
\(412\) 5.46410 0.269197
\(413\) 3.92820 0.193294
\(414\) 4.92820 0.242208
\(415\) 1.00000 0.0490881
\(416\) −2.46410 −0.120813
\(417\) −10.8564 −0.531641
\(418\) 0 0
\(419\) −16.7128 −0.816474 −0.408237 0.912876i \(-0.633856\pi\)
−0.408237 + 0.912876i \(0.633856\pi\)
\(420\) 1.00000 0.0487950
\(421\) −21.8564 −1.06522 −0.532608 0.846362i \(-0.678788\pi\)
−0.532608 + 0.846362i \(0.678788\pi\)
\(422\) −24.0000 −1.16830
\(423\) 14.9282 0.725834
\(424\) 0.535898 0.0260255
\(425\) 0.535898 0.0259949
\(426\) 8.00000 0.387601
\(427\) −8.92820 −0.432066
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) 8.39230 0.404713
\(431\) −24.3205 −1.17148 −0.585739 0.810500i \(-0.699196\pi\)
−0.585739 + 0.810500i \(0.699196\pi\)
\(432\) 5.00000 0.240563
\(433\) −26.2487 −1.26143 −0.630716 0.776014i \(-0.717239\pi\)
−0.630716 + 0.776014i \(0.717239\pi\)
\(434\) −7.46410 −0.358288
\(435\) 6.92820 0.332182
\(436\) 2.39230 0.114571
\(437\) 19.5359 0.934529
\(438\) 7.46410 0.356649
\(439\) 29.4641 1.40624 0.703122 0.711069i \(-0.251789\pi\)
0.703122 + 0.711069i \(0.251789\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) −1.32051 −0.0628102
\(443\) 39.7128 1.88681 0.943406 0.331639i \(-0.107602\pi\)
0.943406 + 0.331639i \(0.107602\pi\)
\(444\) −4.92820 −0.233882
\(445\) −10.5359 −0.499449
\(446\) −3.46410 −0.164030
\(447\) −16.9282 −0.800677
\(448\) −1.00000 −0.0472456
\(449\) −21.9282 −1.03486 −0.517428 0.855727i \(-0.673110\pi\)
−0.517428 + 0.855727i \(0.673110\pi\)
\(450\) 2.00000 0.0942809
\(451\) 0 0
\(452\) 1.00000 0.0470360
\(453\) −12.4641 −0.585615
\(454\) −16.0000 −0.750917
\(455\) −2.46410 −0.115519
\(456\) 7.92820 0.371272
\(457\) −29.6410 −1.38655 −0.693274 0.720674i \(-0.743832\pi\)
−0.693274 + 0.720674i \(0.743832\pi\)
\(458\) 18.0000 0.841085
\(459\) 2.67949 0.125068
\(460\) 2.46410 0.114889
\(461\) −39.8564 −1.85630 −0.928149 0.372209i \(-0.878600\pi\)
−0.928149 + 0.372209i \(0.878600\pi\)
\(462\) 0 0
\(463\) −1.53590 −0.0713793 −0.0356896 0.999363i \(-0.511363\pi\)
−0.0356896 + 0.999363i \(0.511363\pi\)
\(464\) −6.92820 −0.321634
\(465\) 7.46410 0.346139
\(466\) 8.85641 0.410265
\(467\) −2.07180 −0.0958713 −0.0479357 0.998850i \(-0.515264\pi\)
−0.0479357 + 0.998850i \(0.515264\pi\)
\(468\) −4.92820 −0.227806
\(469\) 14.3923 0.664575
\(470\) 7.46410 0.344293
\(471\) −14.3205 −0.659854
\(472\) 3.92820 0.180810
\(473\) 0 0
\(474\) 14.4641 0.664358
\(475\) 7.92820 0.363771
\(476\) −0.535898 −0.0245629
\(477\) 1.07180 0.0490742
\(478\) −5.53590 −0.253206
\(479\) 9.46410 0.432426 0.216213 0.976346i \(-0.430629\pi\)
0.216213 + 0.976346i \(0.430629\pi\)
\(480\) 1.00000 0.0456435
\(481\) 12.1436 0.553700
\(482\) −4.00000 −0.182195
\(483\) 2.46410 0.112121
\(484\) 0 0
\(485\) −14.9282 −0.677855
\(486\) 16.0000 0.725775
\(487\) −32.3205 −1.46458 −0.732291 0.680992i \(-0.761549\pi\)
−0.732291 + 0.680992i \(0.761549\pi\)
\(488\) −8.92820 −0.404161
\(489\) 22.0000 0.994874
\(490\) −1.00000 −0.0451754
\(491\) −16.5359 −0.746255 −0.373127 0.927780i \(-0.621715\pi\)
−0.373127 + 0.927780i \(0.621715\pi\)
\(492\) 12.3923 0.558688
\(493\) −3.71281 −0.167217
\(494\) −19.5359 −0.878962
\(495\) 0 0
\(496\) −7.46410 −0.335148
\(497\) −8.00000 −0.358849
\(498\) 1.00000 0.0448111
\(499\) 4.92820 0.220617 0.110308 0.993897i \(-0.464816\pi\)
0.110308 + 0.993897i \(0.464816\pi\)
\(500\) 1.00000 0.0447214
\(501\) 16.3923 0.732354
\(502\) 0 0
\(503\) 18.2487 0.813670 0.406835 0.913502i \(-0.366632\pi\)
0.406835 + 0.913502i \(0.366632\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 18.4641 0.821642
\(506\) 0 0
\(507\) 6.92820 0.307692
\(508\) 7.39230 0.327980
\(509\) 8.32051 0.368800 0.184400 0.982851i \(-0.440966\pi\)
0.184400 + 0.982851i \(0.440966\pi\)
\(510\) 0.535898 0.0237300
\(511\) −7.46410 −0.330192
\(512\) −1.00000 −0.0441942
\(513\) 39.6410 1.75019
\(514\) 18.3923 0.811250
\(515\) 5.46410 0.240777
\(516\) 8.39230 0.369451
\(517\) 0 0
\(518\) 4.92820 0.216533
\(519\) 18.0000 0.790112
\(520\) −2.46410 −0.108058
\(521\) 16.0000 0.700973 0.350486 0.936568i \(-0.386016\pi\)
0.350486 + 0.936568i \(0.386016\pi\)
\(522\) −13.8564 −0.606478
\(523\) −9.92820 −0.434130 −0.217065 0.976157i \(-0.569648\pi\)
−0.217065 + 0.976157i \(0.569648\pi\)
\(524\) −16.8564 −0.736376
\(525\) 1.00000 0.0436436
\(526\) 9.53590 0.415785
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) −16.9282 −0.736009
\(530\) 0.535898 0.0232779
\(531\) 7.85641 0.340939
\(532\) −7.92820 −0.343731
\(533\) −30.5359 −1.32266
\(534\) −10.5359 −0.455933
\(535\) 6.00000 0.259403
\(536\) 14.3923 0.621653
\(537\) −12.3923 −0.534767
\(538\) −24.3205 −1.04853
\(539\) 0 0
\(540\) 5.00000 0.215166
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 2.14359 0.0920752
\(543\) −3.53590 −0.151740
\(544\) −0.535898 −0.0229765
\(545\) 2.39230 0.102475
\(546\) −2.46410 −0.105454
\(547\) 41.4641 1.77288 0.886438 0.462846i \(-0.153172\pi\)
0.886438 + 0.462846i \(0.153172\pi\)
\(548\) −5.92820 −0.253240
\(549\) −17.8564 −0.762093
\(550\) 0 0
\(551\) −54.9282 −2.34002
\(552\) 2.46410 0.104879
\(553\) −14.4641 −0.615076
\(554\) −7.32051 −0.311019
\(555\) −4.92820 −0.209191
\(556\) 10.8564 0.460414
\(557\) 13.3205 0.564408 0.282204 0.959354i \(-0.408934\pi\)
0.282204 + 0.959354i \(0.408934\pi\)
\(558\) −14.9282 −0.631961
\(559\) −20.6795 −0.874649
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 17.0000 0.717102
\(563\) 12.7128 0.535781 0.267891 0.963449i \(-0.413673\pi\)
0.267891 + 0.963449i \(0.413673\pi\)
\(564\) 7.46410 0.314295
\(565\) 1.00000 0.0420703
\(566\) 27.7846 1.16787
\(567\) −1.00000 −0.0419961
\(568\) −8.00000 −0.335673
\(569\) −22.7128 −0.952171 −0.476085 0.879399i \(-0.657945\pi\)
−0.476085 + 0.879399i \(0.657945\pi\)
\(570\) 7.92820 0.332076
\(571\) 25.8564 1.08206 0.541028 0.841004i \(-0.318035\pi\)
0.541028 + 0.841004i \(0.318035\pi\)
\(572\) 0 0
\(573\) 17.3923 0.726574
\(574\) −12.3923 −0.517245
\(575\) 2.46410 0.102760
\(576\) −2.00000 −0.0833333
\(577\) 0.679492 0.0282876 0.0141438 0.999900i \(-0.495498\pi\)
0.0141438 + 0.999900i \(0.495498\pi\)
\(578\) 16.7128 0.695161
\(579\) 19.0000 0.789613
\(580\) −6.92820 −0.287678
\(581\) −1.00000 −0.0414870
\(582\) −14.9282 −0.618794
\(583\) 0 0
\(584\) −7.46410 −0.308867
\(585\) −4.92820 −0.203756
\(586\) −13.5359 −0.559163
\(587\) −1.14359 −0.0472012 −0.0236006 0.999721i \(-0.507513\pi\)
−0.0236006 + 0.999721i \(0.507513\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −59.1769 −2.43834
\(590\) 3.92820 0.161722
\(591\) 4.00000 0.164538
\(592\) 4.92820 0.202548
\(593\) −28.0000 −1.14982 −0.574911 0.818216i \(-0.694963\pi\)
−0.574911 + 0.818216i \(0.694963\pi\)
\(594\) 0 0
\(595\) −0.535898 −0.0219697
\(596\) 16.9282 0.693406
\(597\) −1.85641 −0.0759777
\(598\) −6.07180 −0.248294
\(599\) −4.60770 −0.188265 −0.0941327 0.995560i \(-0.530008\pi\)
−0.0941327 + 0.995560i \(0.530008\pi\)
\(600\) 1.00000 0.0408248
\(601\) −37.3205 −1.52234 −0.761168 0.648555i \(-0.775374\pi\)
−0.761168 + 0.648555i \(0.775374\pi\)
\(602\) −8.39230 −0.342045
\(603\) 28.7846 1.17220
\(604\) 12.4641 0.507157
\(605\) 0 0
\(606\) 18.4641 0.750053
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) −7.92820 −0.321531
\(609\) −6.92820 −0.280745
\(610\) −8.92820 −0.361492
\(611\) −18.3923 −0.744073
\(612\) −1.07180 −0.0433248
\(613\) −33.4641 −1.35160 −0.675801 0.737084i \(-0.736202\pi\)
−0.675801 + 0.737084i \(0.736202\pi\)
\(614\) −1.07180 −0.0432542
\(615\) 12.3923 0.499706
\(616\) 0 0
\(617\) −42.7846 −1.72244 −0.861222 0.508229i \(-0.830300\pi\)
−0.861222 + 0.508229i \(0.830300\pi\)
\(618\) 5.46410 0.219798
\(619\) 19.9282 0.800982 0.400491 0.916301i \(-0.368840\pi\)
0.400491 + 0.916301i \(0.368840\pi\)
\(620\) −7.46410 −0.299766
\(621\) 12.3205 0.494405
\(622\) 11.3205 0.453911
\(623\) 10.5359 0.422112
\(624\) −2.46410 −0.0986430
\(625\) 1.00000 0.0400000
\(626\) 26.7846 1.07053
\(627\) 0 0
\(628\) 14.3205 0.571450
\(629\) 2.64102 0.105304
\(630\) −2.00000 −0.0796819
\(631\) 25.0718 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(632\) −14.4641 −0.575351
\(633\) −24.0000 −0.953914
\(634\) 1.60770 0.0638497
\(635\) 7.39230 0.293355
\(636\) 0.535898 0.0212498
\(637\) 2.46410 0.0976313
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) −1.00000 −0.0395285
\(641\) −7.14359 −0.282155 −0.141077 0.989999i \(-0.545057\pi\)
−0.141077 + 0.989999i \(0.545057\pi\)
\(642\) 6.00000 0.236801
\(643\) 41.5692 1.63933 0.819665 0.572843i \(-0.194160\pi\)
0.819665 + 0.572843i \(0.194160\pi\)
\(644\) −2.46410 −0.0970992
\(645\) 8.39230 0.330447
\(646\) −4.24871 −0.167163
\(647\) 23.3205 0.916824 0.458412 0.888740i \(-0.348418\pi\)
0.458412 + 0.888740i \(0.348418\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −2.46410 −0.0966500
\(651\) −7.46410 −0.292541
\(652\) −22.0000 −0.861586
\(653\) −18.5359 −0.725366 −0.362683 0.931913i \(-0.618139\pi\)
−0.362683 + 0.931913i \(0.618139\pi\)
\(654\) 2.39230 0.0935465
\(655\) −16.8564 −0.658634
\(656\) −12.3923 −0.483838
\(657\) −14.9282 −0.582405
\(658\) −7.46410 −0.290981
\(659\) −5.32051 −0.207258 −0.103629 0.994616i \(-0.533045\pi\)
−0.103629 + 0.994616i \(0.533045\pi\)
\(660\) 0 0
\(661\) −21.3923 −0.832064 −0.416032 0.909350i \(-0.636580\pi\)
−0.416032 + 0.909350i \(0.636580\pi\)
\(662\) 5.46410 0.212368
\(663\) −1.32051 −0.0512843
\(664\) −1.00000 −0.0388075
\(665\) −7.92820 −0.307443
\(666\) 9.85641 0.381928
\(667\) −17.0718 −0.661023
\(668\) −16.3923 −0.634237
\(669\) −3.46410 −0.133930
\(670\) 14.3923 0.556023
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) −11.1436 −0.429554 −0.214777 0.976663i \(-0.568902\pi\)
−0.214777 + 0.976663i \(0.568902\pi\)
\(674\) −9.78461 −0.376889
\(675\) 5.00000 0.192450
\(676\) −6.92820 −0.266469
\(677\) 12.6077 0.484553 0.242277 0.970207i \(-0.422106\pi\)
0.242277 + 0.970207i \(0.422106\pi\)
\(678\) 1.00000 0.0384048
\(679\) 14.9282 0.572892
\(680\) −0.535898 −0.0205508
\(681\) −16.0000 −0.613121
\(682\) 0 0
\(683\) 8.78461 0.336134 0.168067 0.985776i \(-0.446248\pi\)
0.168067 + 0.985776i \(0.446248\pi\)
\(684\) −15.8564 −0.606285
\(685\) −5.92820 −0.226505
\(686\) 1.00000 0.0381802
\(687\) 18.0000 0.686743
\(688\) −8.39230 −0.319954
\(689\) −1.32051 −0.0503074
\(690\) 2.46410 0.0938067
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) −12.9282 −0.490748
\(695\) 10.8564 0.411807
\(696\) −6.92820 −0.262613
\(697\) −6.64102 −0.251546
\(698\) −8.46410 −0.320371
\(699\) 8.85641 0.334980
\(700\) −1.00000 −0.0377964
\(701\) −9.60770 −0.362878 −0.181439 0.983402i \(-0.558075\pi\)
−0.181439 + 0.983402i \(0.558075\pi\)
\(702\) −12.3205 −0.465008
\(703\) 39.0718 1.47362
\(704\) 0 0
\(705\) 7.46410 0.281114
\(706\) 27.4641 1.03363
\(707\) −18.4641 −0.694414
\(708\) 3.92820 0.147631
\(709\) −7.85641 −0.295054 −0.147527 0.989058i \(-0.547131\pi\)
−0.147527 + 0.989058i \(0.547131\pi\)
\(710\) −8.00000 −0.300235
\(711\) −28.9282 −1.08489
\(712\) 10.5359 0.394849
\(713\) −18.3923 −0.688797
\(714\) −0.535898 −0.0200555
\(715\) 0 0
\(716\) 12.3923 0.463122
\(717\) −5.53590 −0.206742
\(718\) 35.7128 1.33279
\(719\) −15.7128 −0.585989 −0.292995 0.956114i \(-0.594652\pi\)
−0.292995 + 0.956114i \(0.594652\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −5.46410 −0.203494
\(722\) −43.8564 −1.63217
\(723\) −4.00000 −0.148762
\(724\) 3.53590 0.131411
\(725\) −6.92820 −0.257307
\(726\) 0 0
\(727\) 20.7846 0.770859 0.385429 0.922737i \(-0.374053\pi\)
0.385429 + 0.922737i \(0.374053\pi\)
\(728\) 2.46410 0.0913257
\(729\) 13.0000 0.481481
\(730\) −7.46410 −0.276259
\(731\) −4.49742 −0.166343
\(732\) −8.92820 −0.329996
\(733\) −14.1769 −0.523636 −0.261818 0.965117i \(-0.584322\pi\)
−0.261818 + 0.965117i \(0.584322\pi\)
\(734\) 14.5359 0.536530
\(735\) −1.00000 −0.0368856
\(736\) −2.46410 −0.0908280
\(737\) 0 0
\(738\) −24.7846 −0.912334
\(739\) −28.2487 −1.03915 −0.519573 0.854426i \(-0.673909\pi\)
−0.519573 + 0.854426i \(0.673909\pi\)
\(740\) 4.92820 0.181164
\(741\) −19.5359 −0.717669
\(742\) −0.535898 −0.0196734
\(743\) −46.9282 −1.72163 −0.860814 0.508919i \(-0.830045\pi\)
−0.860814 + 0.508919i \(0.830045\pi\)
\(744\) −7.46410 −0.273647
\(745\) 16.9282 0.620201
\(746\) 11.0718 0.405367
\(747\) −2.00000 −0.0731762
\(748\) 0 0
\(749\) −6.00000 −0.219235
\(750\) 1.00000 0.0365148
\(751\) 36.1769 1.32011 0.660057 0.751215i \(-0.270532\pi\)
0.660057 + 0.751215i \(0.270532\pi\)
\(752\) −7.46410 −0.272188
\(753\) 0 0
\(754\) 17.0718 0.621718
\(755\) 12.4641 0.453615
\(756\) −5.00000 −0.181848
\(757\) 7.32051 0.266068 0.133034 0.991111i \(-0.457528\pi\)
0.133034 + 0.991111i \(0.457528\pi\)
\(758\) 3.85641 0.140071
\(759\) 0 0
\(760\) −7.92820 −0.287586
\(761\) 0.248711 0.00901578 0.00450789 0.999990i \(-0.498565\pi\)
0.00450789 + 0.999990i \(0.498565\pi\)
\(762\) 7.39230 0.267795
\(763\) −2.39230 −0.0866073
\(764\) −17.3923 −0.629232
\(765\) −1.07180 −0.0387509
\(766\) 14.5359 0.525203
\(767\) −9.67949 −0.349506
\(768\) −1.00000 −0.0360844
\(769\) −18.7846 −0.677390 −0.338695 0.940896i \(-0.609986\pi\)
−0.338695 + 0.940896i \(0.609986\pi\)
\(770\) 0 0
\(771\) 18.3923 0.662383
\(772\) −19.0000 −0.683825
\(773\) −4.46410 −0.160563 −0.0802813 0.996772i \(-0.525582\pi\)
−0.0802813 + 0.996772i \(0.525582\pi\)
\(774\) −16.7846 −0.603310
\(775\) −7.46410 −0.268118
\(776\) 14.9282 0.535891
\(777\) 4.92820 0.176798
\(778\) −21.3205 −0.764377
\(779\) −98.2487 −3.52013
\(780\) −2.46410 −0.0882290
\(781\) 0 0
\(782\) −1.32051 −0.0472213
\(783\) −34.6410 −1.23797
\(784\) 1.00000 0.0357143
\(785\) 14.3205 0.511121
\(786\) −16.8564 −0.601248
\(787\) 48.4974 1.72875 0.864373 0.502851i \(-0.167715\pi\)
0.864373 + 0.502851i \(0.167715\pi\)
\(788\) −4.00000 −0.142494
\(789\) 9.53590 0.339487
\(790\) −14.4641 −0.514610
\(791\) −1.00000 −0.0355559
\(792\) 0 0
\(793\) 22.0000 0.781243
\(794\) 24.9282 0.884669
\(795\) 0.535898 0.0190064
\(796\) 1.85641 0.0657986
\(797\) −0.607695 −0.0215257 −0.0107628 0.999942i \(-0.503426\pi\)
−0.0107628 + 0.999942i \(0.503426\pi\)
\(798\) −7.92820 −0.280655
\(799\) −4.00000 −0.141510
\(800\) −1.00000 −0.0353553
\(801\) 21.0718 0.744535
\(802\) −31.8564 −1.12489
\(803\) 0 0
\(804\) 14.3923 0.507577
\(805\) −2.46410 −0.0868482
\(806\) 18.3923 0.647841
\(807\) −24.3205 −0.856122
\(808\) −18.4641 −0.649565
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 19.7128 0.692210 0.346105 0.938196i \(-0.387504\pi\)
0.346105 + 0.938196i \(0.387504\pi\)
\(812\) 6.92820 0.243132
\(813\) 2.14359 0.0751791
\(814\) 0 0
\(815\) −22.0000 −0.770626
\(816\) −0.535898 −0.0187602
\(817\) −66.5359 −2.32780
\(818\) 23.7128 0.829099
\(819\) 4.92820 0.172205
\(820\) −12.3923 −0.432758
\(821\) −23.0718 −0.805211 −0.402606 0.915374i \(-0.631895\pi\)
−0.402606 + 0.915374i \(0.631895\pi\)
\(822\) −5.92820 −0.206770
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −5.46410 −0.190351
\(825\) 0 0
\(826\) −3.92820 −0.136680
\(827\) −2.39230 −0.0831886 −0.0415943 0.999135i \(-0.513244\pi\)
−0.0415943 + 0.999135i \(0.513244\pi\)
\(828\) −4.92820 −0.171267
\(829\) −0.607695 −0.0211061 −0.0105531 0.999944i \(-0.503359\pi\)
−0.0105531 + 0.999944i \(0.503359\pi\)
\(830\) −1.00000 −0.0347105
\(831\) −7.32051 −0.253946
\(832\) 2.46410 0.0854274
\(833\) 0.535898 0.0185678
\(834\) 10.8564 0.375927
\(835\) −16.3923 −0.567279
\(836\) 0 0
\(837\) −37.3205 −1.28999
\(838\) 16.7128 0.577335
\(839\) −42.2487 −1.45859 −0.729294 0.684201i \(-0.760151\pi\)
−0.729294 + 0.684201i \(0.760151\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 19.0000 0.655172
\(842\) 21.8564 0.753222
\(843\) 17.0000 0.585511
\(844\) 24.0000 0.826114
\(845\) −6.92820 −0.238337
\(846\) −14.9282 −0.513242
\(847\) 0 0
\(848\) −0.535898 −0.0184028
\(849\) 27.7846 0.953565
\(850\) −0.535898 −0.0183812
\(851\) 12.1436 0.416277
\(852\) −8.00000 −0.274075
\(853\) −16.4641 −0.563720 −0.281860 0.959456i \(-0.590951\pi\)
−0.281860 + 0.959456i \(0.590951\pi\)
\(854\) 8.92820 0.305517
\(855\) −15.8564 −0.542278
\(856\) −6.00000 −0.205076
\(857\) 6.92820 0.236663 0.118331 0.992974i \(-0.462245\pi\)
0.118331 + 0.992974i \(0.462245\pi\)
\(858\) 0 0
\(859\) 42.6410 1.45489 0.727446 0.686165i \(-0.240707\pi\)
0.727446 + 0.686165i \(0.240707\pi\)
\(860\) −8.39230 −0.286175
\(861\) −12.3923 −0.422329
\(862\) 24.3205 0.828360
\(863\) −40.7846 −1.38832 −0.694162 0.719819i \(-0.744225\pi\)
−0.694162 + 0.719819i \(0.744225\pi\)
\(864\) −5.00000 −0.170103
\(865\) −18.0000 −0.612018
\(866\) 26.2487 0.891968
\(867\) 16.7128 0.567597
\(868\) 7.46410 0.253348
\(869\) 0 0
\(870\) −6.92820 −0.234888
\(871\) −35.4641 −1.20166
\(872\) −2.39230 −0.0810137
\(873\) 29.8564 1.01049
\(874\) −19.5359 −0.660812
\(875\) −1.00000 −0.0338062
\(876\) −7.46410 −0.252189
\(877\) −12.3923 −0.418458 −0.209229 0.977867i \(-0.567095\pi\)
−0.209229 + 0.977867i \(0.567095\pi\)
\(878\) −29.4641 −0.994365
\(879\) −13.5359 −0.456555
\(880\) 0 0
\(881\) 49.7128 1.67487 0.837434 0.546539i \(-0.184055\pi\)
0.837434 + 0.546539i \(0.184055\pi\)
\(882\) 2.00000 0.0673435
\(883\) 6.78461 0.228320 0.114160 0.993462i \(-0.463582\pi\)
0.114160 + 0.993462i \(0.463582\pi\)
\(884\) 1.32051 0.0444135
\(885\) 3.92820 0.132045
\(886\) −39.7128 −1.33418
\(887\) 33.4641 1.12361 0.561807 0.827268i \(-0.310106\pi\)
0.561807 + 0.827268i \(0.310106\pi\)
\(888\) 4.92820 0.165380
\(889\) −7.39230 −0.247930
\(890\) 10.5359 0.353164
\(891\) 0 0
\(892\) 3.46410 0.115987
\(893\) −59.1769 −1.98028
\(894\) 16.9282 0.566164
\(895\) 12.3923 0.414229
\(896\) 1.00000 0.0334077
\(897\) −6.07180 −0.202731
\(898\) 21.9282 0.731754
\(899\) 51.7128 1.72472
\(900\) −2.00000 −0.0666667
\(901\) −0.287187 −0.00956759
\(902\) 0 0
\(903\) −8.39230 −0.279278
\(904\) −1.00000 −0.0332595
\(905\) 3.53590 0.117537
\(906\) 12.4641 0.414092
\(907\) 26.6410 0.884600 0.442300 0.896867i \(-0.354163\pi\)
0.442300 + 0.896867i \(0.354163\pi\)
\(908\) 16.0000 0.530979
\(909\) −36.9282 −1.22483
\(910\) 2.46410 0.0816842
\(911\) −11.3923 −0.377444 −0.188722 0.982031i \(-0.560434\pi\)
−0.188722 + 0.982031i \(0.560434\pi\)
\(912\) −7.92820 −0.262529
\(913\) 0 0
\(914\) 29.6410 0.980438
\(915\) −8.92820 −0.295157
\(916\) −18.0000 −0.594737
\(917\) 16.8564 0.556648
\(918\) −2.67949 −0.0884364
\(919\) 51.7128 1.70585 0.852924 0.522035i \(-0.174827\pi\)
0.852924 + 0.522035i \(0.174827\pi\)
\(920\) −2.46410 −0.0812390
\(921\) −1.07180 −0.0353169
\(922\) 39.8564 1.31260
\(923\) 19.7128 0.648855
\(924\) 0 0
\(925\) 4.92820 0.162038
\(926\) 1.53590 0.0504728
\(927\) −10.9282 −0.358929
\(928\) 6.92820 0.227429
\(929\) 1.85641 0.0609067 0.0304534 0.999536i \(-0.490305\pi\)
0.0304534 + 0.999536i \(0.490305\pi\)
\(930\) −7.46410 −0.244758
\(931\) 7.92820 0.259836
\(932\) −8.85641 −0.290101
\(933\) 11.3205 0.370617
\(934\) 2.07180 0.0677913
\(935\) 0 0
\(936\) 4.92820 0.161083
\(937\) −33.8564 −1.10604 −0.553020 0.833168i \(-0.686525\pi\)
−0.553020 + 0.833168i \(0.686525\pi\)
\(938\) −14.3923 −0.469925
\(939\) 26.7846 0.874083
\(940\) −7.46410 −0.243452
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 14.3205 0.466587
\(943\) −30.5359 −0.994386
\(944\) −3.92820 −0.127852
\(945\) −5.00000 −0.162650
\(946\) 0 0
\(947\) 50.3923 1.63753 0.818765 0.574129i \(-0.194659\pi\)
0.818765 + 0.574129i \(0.194659\pi\)
\(948\) −14.4641 −0.469772
\(949\) 18.3923 0.597039
\(950\) −7.92820 −0.257225
\(951\) 1.60770 0.0521331
\(952\) 0.535898 0.0173686
\(953\) 50.5692 1.63810 0.819049 0.573724i \(-0.194502\pi\)
0.819049 + 0.573724i \(0.194502\pi\)
\(954\) −1.07180 −0.0347007
\(955\) −17.3923 −0.562802
\(956\) 5.53590 0.179044
\(957\) 0 0
\(958\) −9.46410 −0.305771
\(959\) 5.92820 0.191432
\(960\) −1.00000 −0.0322749
\(961\) 24.7128 0.797188
\(962\) −12.1436 −0.391525
\(963\) −12.0000 −0.386695
\(964\) 4.00000 0.128831
\(965\) −19.0000 −0.611632
\(966\) −2.46410 −0.0792812
\(967\) 43.7128 1.40571 0.702855 0.711333i \(-0.251908\pi\)
0.702855 + 0.711333i \(0.251908\pi\)
\(968\) 0 0
\(969\) −4.24871 −0.136488
\(970\) 14.9282 0.479316
\(971\) −42.7128 −1.37072 −0.685360 0.728205i \(-0.740355\pi\)
−0.685360 + 0.728205i \(0.740355\pi\)
\(972\) −16.0000 −0.513200
\(973\) −10.8564 −0.348040
\(974\) 32.3205 1.03562
\(975\) −2.46410 −0.0789144
\(976\) 8.92820 0.285785
\(977\) −8.85641 −0.283342 −0.141671 0.989914i \(-0.545247\pi\)
−0.141671 + 0.989914i \(0.545247\pi\)
\(978\) −22.0000 −0.703482
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −4.78461 −0.152761
\(982\) 16.5359 0.527682
\(983\) −26.7846 −0.854296 −0.427148 0.904182i \(-0.640482\pi\)
−0.427148 + 0.904182i \(0.640482\pi\)
\(984\) −12.3923 −0.395052
\(985\) −4.00000 −0.127451
\(986\) 3.71281 0.118240
\(987\) −7.46410 −0.237585
\(988\) 19.5359 0.621520
\(989\) −20.6795 −0.657570
\(990\) 0 0
\(991\) −33.1051 −1.05162 −0.525809 0.850602i \(-0.676237\pi\)
−0.525809 + 0.850602i \(0.676237\pi\)
\(992\) 7.46410 0.236985
\(993\) 5.46410 0.173398
\(994\) 8.00000 0.253745
\(995\) 1.85641 0.0588520
\(996\) −1.00000 −0.0316862
\(997\) −4.32051 −0.136832 −0.0684159 0.997657i \(-0.521794\pi\)
−0.0684159 + 0.997657i \(0.521794\pi\)
\(998\) −4.92820 −0.156000
\(999\) 24.6410 0.779607
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.bi.1.2 2
11.10 odd 2 8470.2.a.bw.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.bi.1.2 2 1.1 even 1 trivial
8470.2.a.bw.1.1 yes 2 11.10 odd 2