Properties

Label 8470.2.a.bv.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: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.23607 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.23607 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.47214 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.23607 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.23607 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.47214 q^{9} +1.00000 q^{10} +1.23607 q^{12} -4.47214 q^{13} -1.00000 q^{14} +1.23607 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.47214 q^{18} -0.763932 q^{19} +1.00000 q^{20} -1.23607 q^{21} -2.76393 q^{23} +1.23607 q^{24} +1.00000 q^{25} -4.47214 q^{26} -5.52786 q^{27} -1.00000 q^{28} +3.23607 q^{29} +1.23607 q^{30} -8.94427 q^{31} +1.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} -1.47214 q^{36} +1.23607 q^{37} -0.763932 q^{38} -5.52786 q^{39} +1.00000 q^{40} +5.70820 q^{41} -1.23607 q^{42} -8.00000 q^{43} -1.47214 q^{45} -2.76393 q^{46} -8.47214 q^{47} +1.23607 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.47214 q^{51} -4.47214 q^{52} -5.23607 q^{53} -5.52786 q^{54} -1.00000 q^{56} -0.944272 q^{57} +3.23607 q^{58} -8.00000 q^{59} +1.23607 q^{60} +5.52786 q^{61} -8.94427 q^{62} +1.47214 q^{63} +1.00000 q^{64} -4.47214 q^{65} -6.00000 q^{67} +2.00000 q^{68} -3.41641 q^{69} -1.00000 q^{70} -8.00000 q^{71} -1.47214 q^{72} +0.472136 q^{73} +1.23607 q^{74} +1.23607 q^{75} -0.763932 q^{76} -5.52786 q^{78} +0.763932 q^{79} +1.00000 q^{80} -2.41641 q^{81} +5.70820 q^{82} +8.00000 q^{83} -1.23607 q^{84} +2.00000 q^{85} -8.00000 q^{86} +4.00000 q^{87} +2.94427 q^{89} -1.47214 q^{90} +4.47214 q^{91} -2.76393 q^{92} -11.0557 q^{93} -8.47214 q^{94} -0.763932 q^{95} +1.23607 q^{96} -11.7082 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 6q^{9} + 2q^{10} - 2q^{12} - 2q^{14} - 2q^{15} + 2q^{16} + 4q^{17} + 6q^{18} - 6q^{19} + 2q^{20} + 2q^{21} - 10q^{23} - 2q^{24} + 2q^{25} - 20q^{27} - 2q^{28} + 2q^{29} - 2q^{30} + 2q^{32} + 4q^{34} - 2q^{35} + 6q^{36} - 2q^{37} - 6q^{38} - 20q^{39} + 2q^{40} - 2q^{41} + 2q^{42} - 16q^{43} + 6q^{45} - 10q^{46} - 8q^{47} - 2q^{48} + 2q^{49} + 2q^{50} - 4q^{51} - 6q^{53} - 20q^{54} - 2q^{56} + 16q^{57} + 2q^{58} - 16q^{59} - 2q^{60} + 20q^{61} - 6q^{63} + 2q^{64} - 12q^{67} + 4q^{68} + 20q^{69} - 2q^{70} - 16q^{71} + 6q^{72} - 8q^{73} - 2q^{74} - 2q^{75} - 6q^{76} - 20q^{78} + 6q^{79} + 2q^{80} + 22q^{81} - 2q^{82} + 16q^{83} + 2q^{84} + 4q^{85} - 16q^{86} + 8q^{87} - 12q^{89} + 6q^{90} - 10q^{92} - 40q^{93} - 8q^{94} - 6q^{95} - 2q^{96} - 10q^{97} + 2q^{98} + O(q^{100}) \)

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
\(3\) 1.23607 0.713644 0.356822 0.934172i \(-0.383860\pi\)
0.356822 + 0.934172i \(0.383860\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.23607 0.504623
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −1.47214 −0.490712
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.23607 0.356822
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.23607 0.319151
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.47214 −0.346986
\(19\) −0.763932 −0.175258 −0.0876290 0.996153i \(-0.527929\pi\)
−0.0876290 + 0.996153i \(0.527929\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.23607 −0.269732
\(22\) 0 0
\(23\) −2.76393 −0.576320 −0.288160 0.957582i \(-0.593043\pi\)
−0.288160 + 0.957582i \(0.593043\pi\)
\(24\) 1.23607 0.252311
\(25\) 1.00000 0.200000
\(26\) −4.47214 −0.877058
\(27\) −5.52786 −1.06384
\(28\) −1.00000 −0.188982
\(29\) 3.23607 0.600923 0.300461 0.953794i \(-0.402859\pi\)
0.300461 + 0.953794i \(0.402859\pi\)
\(30\) 1.23607 0.225674
\(31\) −8.94427 −1.60644 −0.803219 0.595683i \(-0.796881\pi\)
−0.803219 + 0.595683i \(0.796881\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) −1.47214 −0.245356
\(37\) 1.23607 0.203208 0.101604 0.994825i \(-0.467602\pi\)
0.101604 + 0.994825i \(0.467602\pi\)
\(38\) −0.763932 −0.123926
\(39\) −5.52786 −0.885167
\(40\) 1.00000 0.158114
\(41\) 5.70820 0.891472 0.445736 0.895165i \(-0.352942\pi\)
0.445736 + 0.895165i \(0.352942\pi\)
\(42\) −1.23607 −0.190729
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) −1.47214 −0.219453
\(46\) −2.76393 −0.407520
\(47\) −8.47214 −1.23579 −0.617894 0.786261i \(-0.712014\pi\)
−0.617894 + 0.786261i \(0.712014\pi\)
\(48\) 1.23607 0.178411
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 2.47214 0.346168
\(52\) −4.47214 −0.620174
\(53\) −5.23607 −0.719229 −0.359615 0.933101i \(-0.617092\pi\)
−0.359615 + 0.933101i \(0.617092\pi\)
\(54\) −5.52786 −0.752247
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −0.944272 −0.125072
\(58\) 3.23607 0.424917
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 1.23607 0.159576
\(61\) 5.52786 0.707770 0.353885 0.935289i \(-0.384860\pi\)
0.353885 + 0.935289i \(0.384860\pi\)
\(62\) −8.94427 −1.13592
\(63\) 1.47214 0.185472
\(64\) 1.00000 0.125000
\(65\) −4.47214 −0.554700
\(66\) 0 0
\(67\) −6.00000 −0.733017 −0.366508 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) 2.00000 0.242536
\(69\) −3.41641 −0.411287
\(70\) −1.00000 −0.119523
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.47214 −0.173493
\(73\) 0.472136 0.0552593 0.0276297 0.999618i \(-0.491204\pi\)
0.0276297 + 0.999618i \(0.491204\pi\)
\(74\) 1.23607 0.143690
\(75\) 1.23607 0.142729
\(76\) −0.763932 −0.0876290
\(77\) 0 0
\(78\) −5.52786 −0.625907
\(79\) 0.763932 0.0859491 0.0429745 0.999076i \(-0.486317\pi\)
0.0429745 + 0.999076i \(0.486317\pi\)
\(80\) 1.00000 0.111803
\(81\) −2.41641 −0.268490
\(82\) 5.70820 0.630366
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −1.23607 −0.134866
\(85\) 2.00000 0.216930
\(86\) −8.00000 −0.862662
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) 2.94427 0.312092 0.156046 0.987750i \(-0.450125\pi\)
0.156046 + 0.987750i \(0.450125\pi\)
\(90\) −1.47214 −0.155177
\(91\) 4.47214 0.468807
\(92\) −2.76393 −0.288160
\(93\) −11.0557 −1.14643
\(94\) −8.47214 −0.873834
\(95\) −0.763932 −0.0783778
\(96\) 1.23607 0.126156
\(97\) −11.7082 −1.18879 −0.594394 0.804174i \(-0.702608\pi\)
−0.594394 + 0.804174i \(0.702608\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 19.4164 1.93200 0.966002 0.258533i \(-0.0832391\pi\)
0.966002 + 0.258533i \(0.0832391\pi\)
\(102\) 2.47214 0.244778
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −4.47214 −0.438529
\(105\) −1.23607 −0.120628
\(106\) −5.23607 −0.508572
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) −5.52786 −0.531919
\(109\) 2.29180 0.219514 0.109757 0.993958i \(-0.464993\pi\)
0.109757 + 0.993958i \(0.464993\pi\)
\(110\) 0 0
\(111\) 1.52786 0.145018
\(112\) −1.00000 −0.0944911
\(113\) −6.94427 −0.653262 −0.326631 0.945152i \(-0.605913\pi\)
−0.326631 + 0.945152i \(0.605913\pi\)
\(114\) −0.944272 −0.0884392
\(115\) −2.76393 −0.257738
\(116\) 3.23607 0.300461
\(117\) 6.58359 0.608653
\(118\) −8.00000 −0.736460
\(119\) −2.00000 −0.183340
\(120\) 1.23607 0.112837
\(121\) 0 0
\(122\) 5.52786 0.500469
\(123\) 7.05573 0.636194
\(124\) −8.94427 −0.803219
\(125\) 1.00000 0.0894427
\(126\) 1.47214 0.131148
\(127\) −14.4721 −1.28419 −0.642097 0.766623i \(-0.721935\pi\)
−0.642097 + 0.766623i \(0.721935\pi\)
\(128\) 1.00000 0.0883883
\(129\) −9.88854 −0.870638
\(130\) −4.47214 −0.392232
\(131\) −16.1803 −1.41368 −0.706841 0.707372i \(-0.749881\pi\)
−0.706841 + 0.707372i \(0.749881\pi\)
\(132\) 0 0
\(133\) 0.763932 0.0662413
\(134\) −6.00000 −0.518321
\(135\) −5.52786 −0.475763
\(136\) 2.00000 0.171499
\(137\) 15.8885 1.35745 0.678725 0.734393i \(-0.262533\pi\)
0.678725 + 0.734393i \(0.262533\pi\)
\(138\) −3.41641 −0.290824
\(139\) −8.18034 −0.693847 −0.346924 0.937893i \(-0.612774\pi\)
−0.346924 + 0.937893i \(0.612774\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −10.4721 −0.881913
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) −1.47214 −0.122678
\(145\) 3.23607 0.268741
\(146\) 0.472136 0.0390742
\(147\) 1.23607 0.101949
\(148\) 1.23607 0.101604
\(149\) −5.70820 −0.467634 −0.233817 0.972281i \(-0.575122\pi\)
−0.233817 + 0.972281i \(0.575122\pi\)
\(150\) 1.23607 0.100925
\(151\) −6.29180 −0.512019 −0.256010 0.966674i \(-0.582408\pi\)
−0.256010 + 0.966674i \(0.582408\pi\)
\(152\) −0.763932 −0.0619631
\(153\) −2.94427 −0.238030
\(154\) 0 0
\(155\) −8.94427 −0.718421
\(156\) −5.52786 −0.442583
\(157\) −16.4721 −1.31462 −0.657310 0.753620i \(-0.728306\pi\)
−0.657310 + 0.753620i \(0.728306\pi\)
\(158\) 0.763932 0.0607752
\(159\) −6.47214 −0.513274
\(160\) 1.00000 0.0790569
\(161\) 2.76393 0.217828
\(162\) −2.41641 −0.189851
\(163\) −2.94427 −0.230613 −0.115307 0.993330i \(-0.536785\pi\)
−0.115307 + 0.993330i \(0.536785\pi\)
\(164\) 5.70820 0.445736
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 12.9443 1.00166 0.500829 0.865546i \(-0.333029\pi\)
0.500829 + 0.865546i \(0.333029\pi\)
\(168\) −1.23607 −0.0953647
\(169\) 7.00000 0.538462
\(170\) 2.00000 0.153393
\(171\) 1.12461 0.0860012
\(172\) −8.00000 −0.609994
\(173\) 3.52786 0.268219 0.134109 0.990967i \(-0.457183\pi\)
0.134109 + 0.990967i \(0.457183\pi\)
\(174\) 4.00000 0.303239
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −9.88854 −0.743268
\(178\) 2.94427 0.220683
\(179\) 9.52786 0.712146 0.356073 0.934458i \(-0.384115\pi\)
0.356073 + 0.934458i \(0.384115\pi\)
\(180\) −1.47214 −0.109727
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 4.47214 0.331497
\(183\) 6.83282 0.505096
\(184\) −2.76393 −0.203760
\(185\) 1.23607 0.0908775
\(186\) −11.0557 −0.810645
\(187\) 0 0
\(188\) −8.47214 −0.617894
\(189\) 5.52786 0.402093
\(190\) −0.763932 −0.0554215
\(191\) −2.47214 −0.178877 −0.0894387 0.995992i \(-0.528507\pi\)
−0.0894387 + 0.995992i \(0.528507\pi\)
\(192\) 1.23607 0.0892055
\(193\) −9.05573 −0.651846 −0.325923 0.945396i \(-0.605675\pi\)
−0.325923 + 0.945396i \(0.605675\pi\)
\(194\) −11.7082 −0.840600
\(195\) −5.52786 −0.395859
\(196\) 1.00000 0.0714286
\(197\) 5.41641 0.385903 0.192952 0.981208i \(-0.438194\pi\)
0.192952 + 0.981208i \(0.438194\pi\)
\(198\) 0 0
\(199\) −14.4721 −1.02590 −0.512951 0.858418i \(-0.671448\pi\)
−0.512951 + 0.858418i \(0.671448\pi\)
\(200\) 1.00000 0.0707107
\(201\) −7.41641 −0.523113
\(202\) 19.4164 1.36613
\(203\) −3.23607 −0.227127
\(204\) 2.47214 0.173084
\(205\) 5.70820 0.398678
\(206\) 14.0000 0.975426
\(207\) 4.06888 0.282807
\(208\) −4.47214 −0.310087
\(209\) 0 0
\(210\) −1.23607 −0.0852968
\(211\) 21.8885 1.50687 0.753435 0.657523i \(-0.228396\pi\)
0.753435 + 0.657523i \(0.228396\pi\)
\(212\) −5.23607 −0.359615
\(213\) −9.88854 −0.677552
\(214\) −16.0000 −1.09374
\(215\) −8.00000 −0.545595
\(216\) −5.52786 −0.376124
\(217\) 8.94427 0.607177
\(218\) 2.29180 0.155220
\(219\) 0.583592 0.0394355
\(220\) 0 0
\(221\) −8.94427 −0.601657
\(222\) 1.52786 0.102544
\(223\) −18.9443 −1.26860 −0.634301 0.773086i \(-0.718712\pi\)
−0.634301 + 0.773086i \(0.718712\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.47214 −0.0981424
\(226\) −6.94427 −0.461926
\(227\) 1.52786 0.101408 0.0507039 0.998714i \(-0.483854\pi\)
0.0507039 + 0.998714i \(0.483854\pi\)
\(228\) −0.944272 −0.0625359
\(229\) −11.5279 −0.761783 −0.380891 0.924620i \(-0.624383\pi\)
−0.380891 + 0.924620i \(0.624383\pi\)
\(230\) −2.76393 −0.182248
\(231\) 0 0
\(232\) 3.23607 0.212458
\(233\) −2.94427 −0.192886 −0.0964428 0.995339i \(-0.530746\pi\)
−0.0964428 + 0.995339i \(0.530746\pi\)
\(234\) 6.58359 0.430383
\(235\) −8.47214 −0.552661
\(236\) −8.00000 −0.520756
\(237\) 0.944272 0.0613371
\(238\) −2.00000 −0.129641
\(239\) 7.23607 0.468062 0.234031 0.972229i \(-0.424808\pi\)
0.234031 + 0.972229i \(0.424808\pi\)
\(240\) 1.23607 0.0797878
\(241\) −17.1246 −1.10309 −0.551547 0.834144i \(-0.685962\pi\)
−0.551547 + 0.834144i \(0.685962\pi\)
\(242\) 0 0
\(243\) 13.5967 0.872232
\(244\) 5.52786 0.353885
\(245\) 1.00000 0.0638877
\(246\) 7.05573 0.449857
\(247\) 3.41641 0.217381
\(248\) −8.94427 −0.567962
\(249\) 9.88854 0.626661
\(250\) 1.00000 0.0632456
\(251\) −5.52786 −0.348916 −0.174458 0.984665i \(-0.555817\pi\)
−0.174458 + 0.984665i \(0.555817\pi\)
\(252\) 1.47214 0.0927358
\(253\) 0 0
\(254\) −14.4721 −0.908063
\(255\) 2.47214 0.154811
\(256\) 1.00000 0.0625000
\(257\) 15.1246 0.943447 0.471724 0.881746i \(-0.343632\pi\)
0.471724 + 0.881746i \(0.343632\pi\)
\(258\) −9.88854 −0.615634
\(259\) −1.23607 −0.0768055
\(260\) −4.47214 −0.277350
\(261\) −4.76393 −0.294880
\(262\) −16.1803 −0.999625
\(263\) 16.3607 1.00884 0.504421 0.863458i \(-0.331706\pi\)
0.504421 + 0.863458i \(0.331706\pi\)
\(264\) 0 0
\(265\) −5.23607 −0.321649
\(266\) 0.763932 0.0468397
\(267\) 3.63932 0.222723
\(268\) −6.00000 −0.366508
\(269\) −12.4721 −0.760440 −0.380220 0.924896i \(-0.624152\pi\)
−0.380220 + 0.924896i \(0.624152\pi\)
\(270\) −5.52786 −0.336415
\(271\) 6.47214 0.393154 0.196577 0.980488i \(-0.437017\pi\)
0.196577 + 0.980488i \(0.437017\pi\)
\(272\) 2.00000 0.121268
\(273\) 5.52786 0.334562
\(274\) 15.8885 0.959862
\(275\) 0 0
\(276\) −3.41641 −0.205644
\(277\) −11.5279 −0.692642 −0.346321 0.938116i \(-0.612569\pi\)
−0.346321 + 0.938116i \(0.612569\pi\)
\(278\) −8.18034 −0.490624
\(279\) 13.1672 0.788299
\(280\) −1.00000 −0.0597614
\(281\) 28.3607 1.69186 0.845928 0.533297i \(-0.179047\pi\)
0.845928 + 0.533297i \(0.179047\pi\)
\(282\) −10.4721 −0.623607
\(283\) −19.4164 −1.15419 −0.577093 0.816679i \(-0.695813\pi\)
−0.577093 + 0.816679i \(0.695813\pi\)
\(284\) −8.00000 −0.474713
\(285\) −0.944272 −0.0559338
\(286\) 0 0
\(287\) −5.70820 −0.336945
\(288\) −1.47214 −0.0867464
\(289\) −13.0000 −0.764706
\(290\) 3.23607 0.190028
\(291\) −14.4721 −0.848372
\(292\) 0.472136 0.0276297
\(293\) 0.472136 0.0275825 0.0137912 0.999905i \(-0.495610\pi\)
0.0137912 + 0.999905i \(0.495610\pi\)
\(294\) 1.23607 0.0720889
\(295\) −8.00000 −0.465778
\(296\) 1.23607 0.0718450
\(297\) 0 0
\(298\) −5.70820 −0.330667
\(299\) 12.3607 0.714837
\(300\) 1.23607 0.0713644
\(301\) 8.00000 0.461112
\(302\) −6.29180 −0.362052
\(303\) 24.0000 1.37876
\(304\) −0.763932 −0.0438145
\(305\) 5.52786 0.316525
\(306\) −2.94427 −0.168313
\(307\) 9.52786 0.543784 0.271892 0.962328i \(-0.412351\pi\)
0.271892 + 0.962328i \(0.412351\pi\)
\(308\) 0 0
\(309\) 17.3050 0.984444
\(310\) −8.94427 −0.508001
\(311\) 30.8328 1.74837 0.874184 0.485594i \(-0.161397\pi\)
0.874184 + 0.485594i \(0.161397\pi\)
\(312\) −5.52786 −0.312954
\(313\) −20.6525 −1.16735 −0.583673 0.811988i \(-0.698385\pi\)
−0.583673 + 0.811988i \(0.698385\pi\)
\(314\) −16.4721 −0.929576
\(315\) 1.47214 0.0829455
\(316\) 0.763932 0.0429745
\(317\) 2.18034 0.122460 0.0612300 0.998124i \(-0.480498\pi\)
0.0612300 + 0.998124i \(0.480498\pi\)
\(318\) −6.47214 −0.362939
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −19.7771 −1.10385
\(322\) 2.76393 0.154028
\(323\) −1.52786 −0.0850126
\(324\) −2.41641 −0.134245
\(325\) −4.47214 −0.248069
\(326\) −2.94427 −0.163068
\(327\) 2.83282 0.156655
\(328\) 5.70820 0.315183
\(329\) 8.47214 0.467084
\(330\) 0 0
\(331\) −15.4164 −0.847362 −0.423681 0.905811i \(-0.639262\pi\)
−0.423681 + 0.905811i \(0.639262\pi\)
\(332\) 8.00000 0.439057
\(333\) −1.81966 −0.0997168
\(334\) 12.9443 0.708279
\(335\) −6.00000 −0.327815
\(336\) −1.23607 −0.0674330
\(337\) 28.8328 1.57062 0.785312 0.619100i \(-0.212503\pi\)
0.785312 + 0.619100i \(0.212503\pi\)
\(338\) 7.00000 0.380750
\(339\) −8.58359 −0.466197
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) 1.12461 0.0608120
\(343\) −1.00000 −0.0539949
\(344\) −8.00000 −0.431331
\(345\) −3.41641 −0.183933
\(346\) 3.52786 0.189659
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 4.00000 0.214423
\(349\) −24.9443 −1.33524 −0.667618 0.744504i \(-0.732686\pi\)
−0.667618 + 0.744504i \(0.732686\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 24.7214 1.31953
\(352\) 0 0
\(353\) −18.7639 −0.998703 −0.499352 0.866399i \(-0.666428\pi\)
−0.499352 + 0.866399i \(0.666428\pi\)
\(354\) −9.88854 −0.525570
\(355\) −8.00000 −0.424596
\(356\) 2.94427 0.156046
\(357\) −2.47214 −0.130839
\(358\) 9.52786 0.503563
\(359\) −12.1803 −0.642854 −0.321427 0.946934i \(-0.604162\pi\)
−0.321427 + 0.946934i \(0.604162\pi\)
\(360\) −1.47214 −0.0775884
\(361\) −18.4164 −0.969285
\(362\) 14.0000 0.735824
\(363\) 0 0
\(364\) 4.47214 0.234404
\(365\) 0.472136 0.0247127
\(366\) 6.83282 0.357157
\(367\) 28.8328 1.50506 0.752530 0.658558i \(-0.228833\pi\)
0.752530 + 0.658558i \(0.228833\pi\)
\(368\) −2.76393 −0.144080
\(369\) −8.40325 −0.437456
\(370\) 1.23607 0.0642601
\(371\) 5.23607 0.271843
\(372\) −11.0557 −0.573213
\(373\) −38.3607 −1.98624 −0.993120 0.117098i \(-0.962641\pi\)
−0.993120 + 0.117098i \(0.962641\pi\)
\(374\) 0 0
\(375\) 1.23607 0.0638303
\(376\) −8.47214 −0.436917
\(377\) −14.4721 −0.745353
\(378\) 5.52786 0.284323
\(379\) −4.94427 −0.253970 −0.126985 0.991905i \(-0.540530\pi\)
−0.126985 + 0.991905i \(0.540530\pi\)
\(380\) −0.763932 −0.0391889
\(381\) −17.8885 −0.916458
\(382\) −2.47214 −0.126485
\(383\) −3.88854 −0.198695 −0.0993477 0.995053i \(-0.531676\pi\)
−0.0993477 + 0.995053i \(0.531676\pi\)
\(384\) 1.23607 0.0630778
\(385\) 0 0
\(386\) −9.05573 −0.460924
\(387\) 11.7771 0.598663
\(388\) −11.7082 −0.594394
\(389\) 38.3607 1.94496 0.972482 0.232979i \(-0.0748472\pi\)
0.972482 + 0.232979i \(0.0748472\pi\)
\(390\) −5.52786 −0.279914
\(391\) −5.52786 −0.279556
\(392\) 1.00000 0.0505076
\(393\) −20.0000 −1.00887
\(394\) 5.41641 0.272875
\(395\) 0.763932 0.0384376
\(396\) 0 0
\(397\) 25.4164 1.27561 0.637806 0.770197i \(-0.279842\pi\)
0.637806 + 0.770197i \(0.279842\pi\)
\(398\) −14.4721 −0.725423
\(399\) 0.944272 0.0472727
\(400\) 1.00000 0.0500000
\(401\) −3.52786 −0.176173 −0.0880866 0.996113i \(-0.528075\pi\)
−0.0880866 + 0.996113i \(0.528075\pi\)
\(402\) −7.41641 −0.369897
\(403\) 40.0000 1.99254
\(404\) 19.4164 0.966002
\(405\) −2.41641 −0.120072
\(406\) −3.23607 −0.160603
\(407\) 0 0
\(408\) 2.47214 0.122389
\(409\) 4.76393 0.235561 0.117781 0.993040i \(-0.462422\pi\)
0.117781 + 0.993040i \(0.462422\pi\)
\(410\) 5.70820 0.281908
\(411\) 19.6393 0.968736
\(412\) 14.0000 0.689730
\(413\) 8.00000 0.393654
\(414\) 4.06888 0.199975
\(415\) 8.00000 0.392705
\(416\) −4.47214 −0.219265
\(417\) −10.1115 −0.495160
\(418\) 0 0
\(419\) 23.4164 1.14397 0.571983 0.820265i \(-0.306174\pi\)
0.571983 + 0.820265i \(0.306174\pi\)
\(420\) −1.23607 −0.0603139
\(421\) 31.3050 1.52571 0.762855 0.646570i \(-0.223797\pi\)
0.762855 + 0.646570i \(0.223797\pi\)
\(422\) 21.8885 1.06552
\(423\) 12.4721 0.606416
\(424\) −5.23607 −0.254286
\(425\) 2.00000 0.0970143
\(426\) −9.88854 −0.479102
\(427\) −5.52786 −0.267512
\(428\) −16.0000 −0.773389
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 2.65248 0.127765 0.0638826 0.997957i \(-0.479652\pi\)
0.0638826 + 0.997957i \(0.479652\pi\)
\(432\) −5.52786 −0.265959
\(433\) −5.23607 −0.251629 −0.125815 0.992054i \(-0.540154\pi\)
−0.125815 + 0.992054i \(0.540154\pi\)
\(434\) 8.94427 0.429339
\(435\) 4.00000 0.191785
\(436\) 2.29180 0.109757
\(437\) 2.11146 0.101005
\(438\) 0.583592 0.0278851
\(439\) 19.4164 0.926695 0.463347 0.886177i \(-0.346648\pi\)
0.463347 + 0.886177i \(0.346648\pi\)
\(440\) 0 0
\(441\) −1.47214 −0.0701017
\(442\) −8.94427 −0.425436
\(443\) −2.94427 −0.139887 −0.0699433 0.997551i \(-0.522282\pi\)
−0.0699433 + 0.997551i \(0.522282\pi\)
\(444\) 1.52786 0.0725092
\(445\) 2.94427 0.139572
\(446\) −18.9443 −0.897037
\(447\) −7.05573 −0.333724
\(448\) −1.00000 −0.0472456
\(449\) −18.3607 −0.866494 −0.433247 0.901275i \(-0.642632\pi\)
−0.433247 + 0.901275i \(0.642632\pi\)
\(450\) −1.47214 −0.0693972
\(451\) 0 0
\(452\) −6.94427 −0.326631
\(453\) −7.77709 −0.365399
\(454\) 1.52786 0.0717062
\(455\) 4.47214 0.209657
\(456\) −0.944272 −0.0442196
\(457\) 13.0557 0.610721 0.305361 0.952237i \(-0.401223\pi\)
0.305361 + 0.952237i \(0.401223\pi\)
\(458\) −11.5279 −0.538662
\(459\) −11.0557 −0.516037
\(460\) −2.76393 −0.128869
\(461\) 20.9443 0.975472 0.487736 0.872991i \(-0.337823\pi\)
0.487736 + 0.872991i \(0.337823\pi\)
\(462\) 0 0
\(463\) −15.7082 −0.730022 −0.365011 0.931003i \(-0.618935\pi\)
−0.365011 + 0.931003i \(0.618935\pi\)
\(464\) 3.23607 0.150231
\(465\) −11.0557 −0.512697
\(466\) −2.94427 −0.136391
\(467\) 13.8197 0.639498 0.319749 0.947502i \(-0.396401\pi\)
0.319749 + 0.947502i \(0.396401\pi\)
\(468\) 6.58359 0.304327
\(469\) 6.00000 0.277054
\(470\) −8.47214 −0.390790
\(471\) −20.3607 −0.938171
\(472\) −8.00000 −0.368230
\(473\) 0 0
\(474\) 0.944272 0.0433718
\(475\) −0.763932 −0.0350516
\(476\) −2.00000 −0.0916698
\(477\) 7.70820 0.352934
\(478\) 7.23607 0.330970
\(479\) −5.88854 −0.269054 −0.134527 0.990910i \(-0.542952\pi\)
−0.134527 + 0.990910i \(0.542952\pi\)
\(480\) 1.23607 0.0564185
\(481\) −5.52786 −0.252049
\(482\) −17.1246 −0.780005
\(483\) 3.41641 0.155452
\(484\) 0 0
\(485\) −11.7082 −0.531642
\(486\) 13.5967 0.616761
\(487\) 24.0689 1.09067 0.545333 0.838220i \(-0.316403\pi\)
0.545333 + 0.838220i \(0.316403\pi\)
\(488\) 5.52786 0.250235
\(489\) −3.63932 −0.164576
\(490\) 1.00000 0.0451754
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 7.05573 0.318097
\(493\) 6.47214 0.291490
\(494\) 3.41641 0.153711
\(495\) 0 0
\(496\) −8.94427 −0.401610
\(497\) 8.00000 0.358849
\(498\) 9.88854 0.443116
\(499\) 33.3050 1.49093 0.745467 0.666542i \(-0.232226\pi\)
0.745467 + 0.666542i \(0.232226\pi\)
\(500\) 1.00000 0.0447214
\(501\) 16.0000 0.714827
\(502\) −5.52786 −0.246721
\(503\) 14.8328 0.661363 0.330681 0.943742i \(-0.392721\pi\)
0.330681 + 0.943742i \(0.392721\pi\)
\(504\) 1.47214 0.0655741
\(505\) 19.4164 0.864019
\(506\) 0 0
\(507\) 8.65248 0.384270
\(508\) −14.4721 −0.642097
\(509\) 6.94427 0.307799 0.153900 0.988086i \(-0.450817\pi\)
0.153900 + 0.988086i \(0.450817\pi\)
\(510\) 2.47214 0.109468
\(511\) −0.472136 −0.0208861
\(512\) 1.00000 0.0441942
\(513\) 4.22291 0.186446
\(514\) 15.1246 0.667118
\(515\) 14.0000 0.616914
\(516\) −9.88854 −0.435319
\(517\) 0 0
\(518\) −1.23607 −0.0543097
\(519\) 4.36068 0.191413
\(520\) −4.47214 −0.196116
\(521\) 8.47214 0.371171 0.185586 0.982628i \(-0.440582\pi\)
0.185586 + 0.982628i \(0.440582\pi\)
\(522\) −4.76393 −0.208512
\(523\) −30.4721 −1.33245 −0.666227 0.745749i \(-0.732092\pi\)
−0.666227 + 0.745749i \(0.732092\pi\)
\(524\) −16.1803 −0.706841
\(525\) −1.23607 −0.0539464
\(526\) 16.3607 0.713360
\(527\) −17.8885 −0.779237
\(528\) 0 0
\(529\) −15.3607 −0.667856
\(530\) −5.23607 −0.227440
\(531\) 11.7771 0.511082
\(532\) 0.763932 0.0331207
\(533\) −25.5279 −1.10573
\(534\) 3.63932 0.157489
\(535\) −16.0000 −0.691740
\(536\) −6.00000 −0.259161
\(537\) 11.7771 0.508219
\(538\) −12.4721 −0.537712
\(539\) 0 0
\(540\) −5.52786 −0.237881
\(541\) −15.2361 −0.655050 −0.327525 0.944843i \(-0.606215\pi\)
−0.327525 + 0.944843i \(0.606215\pi\)
\(542\) 6.47214 0.278002
\(543\) 17.3050 0.742627
\(544\) 2.00000 0.0857493
\(545\) 2.29180 0.0981698
\(546\) 5.52786 0.236571
\(547\) −7.05573 −0.301681 −0.150841 0.988558i \(-0.548198\pi\)
−0.150841 + 0.988558i \(0.548198\pi\)
\(548\) 15.8885 0.678725
\(549\) −8.13777 −0.347311
\(550\) 0 0
\(551\) −2.47214 −0.105317
\(552\) −3.41641 −0.145412
\(553\) −0.763932 −0.0324857
\(554\) −11.5279 −0.489772
\(555\) 1.52786 0.0648542
\(556\) −8.18034 −0.346924
\(557\) 24.8328 1.05220 0.526100 0.850423i \(-0.323654\pi\)
0.526100 + 0.850423i \(0.323654\pi\)
\(558\) 13.1672 0.557411
\(559\) 35.7771 1.51321
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 28.3607 1.19632
\(563\) 14.8328 0.625129 0.312564 0.949897i \(-0.398812\pi\)
0.312564 + 0.949897i \(0.398812\pi\)
\(564\) −10.4721 −0.440956
\(565\) −6.94427 −0.292148
\(566\) −19.4164 −0.816132
\(567\) 2.41641 0.101480
\(568\) −8.00000 −0.335673
\(569\) −46.8328 −1.96333 −0.981667 0.190605i \(-0.938955\pi\)
−0.981667 + 0.190605i \(0.938955\pi\)
\(570\) −0.944272 −0.0395512
\(571\) 27.4164 1.14734 0.573670 0.819086i \(-0.305519\pi\)
0.573670 + 0.819086i \(0.305519\pi\)
\(572\) 0 0
\(573\) −3.05573 −0.127655
\(574\) −5.70820 −0.238256
\(575\) −2.76393 −0.115264
\(576\) −1.47214 −0.0613390
\(577\) −11.7082 −0.487419 −0.243709 0.969848i \(-0.578364\pi\)
−0.243709 + 0.969848i \(0.578364\pi\)
\(578\) −13.0000 −0.540729
\(579\) −11.1935 −0.465186
\(580\) 3.23607 0.134370
\(581\) −8.00000 −0.331896
\(582\) −14.4721 −0.599889
\(583\) 0 0
\(584\) 0.472136 0.0195371
\(585\) 6.58359 0.272198
\(586\) 0.472136 0.0195038
\(587\) −0.291796 −0.0120437 −0.00602186 0.999982i \(-0.501917\pi\)
−0.00602186 + 0.999982i \(0.501917\pi\)
\(588\) 1.23607 0.0509746
\(589\) 6.83282 0.281541
\(590\) −8.00000 −0.329355
\(591\) 6.69505 0.275397
\(592\) 1.23607 0.0508021
\(593\) −6.36068 −0.261202 −0.130601 0.991435i \(-0.541691\pi\)
−0.130601 + 0.991435i \(0.541691\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) −5.70820 −0.233817
\(597\) −17.8885 −0.732129
\(598\) 12.3607 0.505466
\(599\) −17.8885 −0.730906 −0.365453 0.930830i \(-0.619086\pi\)
−0.365453 + 0.930830i \(0.619086\pi\)
\(600\) 1.23607 0.0504623
\(601\) 18.2918 0.746138 0.373069 0.927804i \(-0.378305\pi\)
0.373069 + 0.927804i \(0.378305\pi\)
\(602\) 8.00000 0.326056
\(603\) 8.83282 0.359700
\(604\) −6.29180 −0.256010
\(605\) 0 0
\(606\) 24.0000 0.974933
\(607\) 34.8328 1.41382 0.706910 0.707303i \(-0.250088\pi\)
0.706910 + 0.707303i \(0.250088\pi\)
\(608\) −0.763932 −0.0309815
\(609\) −4.00000 −0.162088
\(610\) 5.52786 0.223817
\(611\) 37.8885 1.53281
\(612\) −2.94427 −0.119015
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 9.52786 0.384513
\(615\) 7.05573 0.284514
\(616\) 0 0
\(617\) −40.4721 −1.62935 −0.814673 0.579920i \(-0.803084\pi\)
−0.814673 + 0.579920i \(0.803084\pi\)
\(618\) 17.3050 0.696107
\(619\) 10.4721 0.420911 0.210455 0.977603i \(-0.432505\pi\)
0.210455 + 0.977603i \(0.432505\pi\)
\(620\) −8.94427 −0.359211
\(621\) 15.2786 0.613111
\(622\) 30.8328 1.23628
\(623\) −2.94427 −0.117960
\(624\) −5.52786 −0.221292
\(625\) 1.00000 0.0400000
\(626\) −20.6525 −0.825439
\(627\) 0 0
\(628\) −16.4721 −0.657310
\(629\) 2.47214 0.0985705
\(630\) 1.47214 0.0586513
\(631\) −38.8328 −1.54591 −0.772955 0.634461i \(-0.781222\pi\)
−0.772955 + 0.634461i \(0.781222\pi\)
\(632\) 0.763932 0.0303876
\(633\) 27.0557 1.07537
\(634\) 2.18034 0.0865924
\(635\) −14.4721 −0.574309
\(636\) −6.47214 −0.256637
\(637\) −4.47214 −0.177192
\(638\) 0 0
\(639\) 11.7771 0.465894
\(640\) 1.00000 0.0395285
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −19.7771 −0.780539
\(643\) −25.8197 −1.01823 −0.509114 0.860699i \(-0.670027\pi\)
−0.509114 + 0.860699i \(0.670027\pi\)
\(644\) 2.76393 0.108914
\(645\) −9.88854 −0.389361
\(646\) −1.52786 −0.0601130
\(647\) −30.0000 −1.17942 −0.589711 0.807614i \(-0.700758\pi\)
−0.589711 + 0.807614i \(0.700758\pi\)
\(648\) −2.41641 −0.0949255
\(649\) 0 0
\(650\) −4.47214 −0.175412
\(651\) 11.0557 0.433308
\(652\) −2.94427 −0.115307
\(653\) 33.5967 1.31474 0.657371 0.753567i \(-0.271668\pi\)
0.657371 + 0.753567i \(0.271668\pi\)
\(654\) 2.83282 0.110772
\(655\) −16.1803 −0.632218
\(656\) 5.70820 0.222868
\(657\) −0.695048 −0.0271164
\(658\) 8.47214 0.330278
\(659\) −12.3607 −0.481504 −0.240752 0.970587i \(-0.577394\pi\)
−0.240752 + 0.970587i \(0.577394\pi\)
\(660\) 0 0
\(661\) 14.9443 0.581265 0.290632 0.956835i \(-0.406134\pi\)
0.290632 + 0.956835i \(0.406134\pi\)
\(662\) −15.4164 −0.599176
\(663\) −11.0557 −0.429369
\(664\) 8.00000 0.310460
\(665\) 0.763932 0.0296240
\(666\) −1.81966 −0.0705104
\(667\) −8.94427 −0.346324
\(668\) 12.9443 0.500829
\(669\) −23.4164 −0.905331
\(670\) −6.00000 −0.231800
\(671\) 0 0
\(672\) −1.23607 −0.0476824
\(673\) −11.8885 −0.458270 −0.229135 0.973395i \(-0.573590\pi\)
−0.229135 + 0.973395i \(0.573590\pi\)
\(674\) 28.8328 1.11060
\(675\) −5.52786 −0.212768
\(676\) 7.00000 0.269231
\(677\) 23.5279 0.904249 0.452125 0.891955i \(-0.350666\pi\)
0.452125 + 0.891955i \(0.350666\pi\)
\(678\) −8.58359 −0.329651
\(679\) 11.7082 0.449320
\(680\) 2.00000 0.0766965
\(681\) 1.88854 0.0723692
\(682\) 0 0
\(683\) −48.4721 −1.85473 −0.927367 0.374152i \(-0.877934\pi\)
−0.927367 + 0.374152i \(0.877934\pi\)
\(684\) 1.12461 0.0430006
\(685\) 15.8885 0.607070
\(686\) −1.00000 −0.0381802
\(687\) −14.2492 −0.543642
\(688\) −8.00000 −0.304997
\(689\) 23.4164 0.892094
\(690\) −3.41641 −0.130060
\(691\) −36.3607 −1.38323 −0.691613 0.722269i \(-0.743099\pi\)
−0.691613 + 0.722269i \(0.743099\pi\)
\(692\) 3.52786 0.134109
\(693\) 0 0
\(694\) 0 0
\(695\) −8.18034 −0.310298
\(696\) 4.00000 0.151620
\(697\) 11.4164 0.432427
\(698\) −24.9443 −0.944155
\(699\) −3.63932 −0.137652
\(700\) −1.00000 −0.0377964
\(701\) 9.70820 0.366674 0.183337 0.983050i \(-0.441310\pi\)
0.183337 + 0.983050i \(0.441310\pi\)
\(702\) 24.7214 0.933048
\(703\) −0.944272 −0.0356139
\(704\) 0 0
\(705\) −10.4721 −0.394403
\(706\) −18.7639 −0.706190
\(707\) −19.4164 −0.730229
\(708\) −9.88854 −0.371634
\(709\) 14.9443 0.561244 0.280622 0.959818i \(-0.409459\pi\)
0.280622 + 0.959818i \(0.409459\pi\)
\(710\) −8.00000 −0.300235
\(711\) −1.12461 −0.0421762
\(712\) 2.94427 0.110341
\(713\) 24.7214 0.925822
\(714\) −2.47214 −0.0925174
\(715\) 0 0
\(716\) 9.52786 0.356073
\(717\) 8.94427 0.334030
\(718\) −12.1803 −0.454566
\(719\) 12.3607 0.460976 0.230488 0.973075i \(-0.425968\pi\)
0.230488 + 0.973075i \(0.425968\pi\)
\(720\) −1.47214 −0.0548633
\(721\) −14.0000 −0.521387
\(722\) −18.4164 −0.685388
\(723\) −21.1672 −0.787216
\(724\) 14.0000 0.520306
\(725\) 3.23607 0.120185
\(726\) 0 0
\(727\) 10.3607 0.384256 0.192128 0.981370i \(-0.438461\pi\)
0.192128 + 0.981370i \(0.438461\pi\)
\(728\) 4.47214 0.165748
\(729\) 24.0557 0.890953
\(730\) 0.472136 0.0174745
\(731\) −16.0000 −0.591781
\(732\) 6.83282 0.252548
\(733\) 36.2492 1.33890 0.669448 0.742859i \(-0.266531\pi\)
0.669448 + 0.742859i \(0.266531\pi\)
\(734\) 28.8328 1.06424
\(735\) 1.23607 0.0455931
\(736\) −2.76393 −0.101880
\(737\) 0 0
\(738\) −8.40325 −0.309328
\(739\) −44.7214 −1.64510 −0.822551 0.568691i \(-0.807450\pi\)
−0.822551 + 0.568691i \(0.807450\pi\)
\(740\) 1.23607 0.0454388
\(741\) 4.22291 0.155133
\(742\) 5.23607 0.192222
\(743\) 38.4721 1.41141 0.705703 0.708508i \(-0.250631\pi\)
0.705703 + 0.708508i \(0.250631\pi\)
\(744\) −11.0557 −0.405323
\(745\) −5.70820 −0.209132
\(746\) −38.3607 −1.40448
\(747\) −11.7771 −0.430901
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) 1.23607 0.0451348
\(751\) 6.11146 0.223010 0.111505 0.993764i \(-0.464433\pi\)
0.111505 + 0.993764i \(0.464433\pi\)
\(752\) −8.47214 −0.308947
\(753\) −6.83282 −0.249002
\(754\) −14.4721 −0.527044
\(755\) −6.29180 −0.228982
\(756\) 5.52786 0.201046
\(757\) −43.1246 −1.56739 −0.783695 0.621145i \(-0.786668\pi\)
−0.783695 + 0.621145i \(0.786668\pi\)
\(758\) −4.94427 −0.179584
\(759\) 0 0
\(760\) −0.763932 −0.0277107
\(761\) −18.0689 −0.654997 −0.327498 0.944852i \(-0.606206\pi\)
−0.327498 + 0.944852i \(0.606206\pi\)
\(762\) −17.8885 −0.648034
\(763\) −2.29180 −0.0829686
\(764\) −2.47214 −0.0894387
\(765\) −2.94427 −0.106450
\(766\) −3.88854 −0.140499
\(767\) 35.7771 1.29184
\(768\) 1.23607 0.0446028
\(769\) 35.0132 1.26261 0.631303 0.775536i \(-0.282520\pi\)
0.631303 + 0.775536i \(0.282520\pi\)
\(770\) 0 0
\(771\) 18.6950 0.673286
\(772\) −9.05573 −0.325923
\(773\) 29.7771 1.07101 0.535504 0.844533i \(-0.320122\pi\)
0.535504 + 0.844533i \(0.320122\pi\)
\(774\) 11.7771 0.423319
\(775\) −8.94427 −0.321288
\(776\) −11.7082 −0.420300
\(777\) −1.52786 −0.0548118
\(778\) 38.3607 1.37530
\(779\) −4.36068 −0.156238
\(780\) −5.52786 −0.197929
\(781\) 0 0
\(782\) −5.52786 −0.197676
\(783\) −17.8885 −0.639284
\(784\) 1.00000 0.0357143
\(785\) −16.4721 −0.587916
\(786\) −20.0000 −0.713376
\(787\) −46.8328 −1.66941 −0.834705 0.550698i \(-0.814362\pi\)
−0.834705 + 0.550698i \(0.814362\pi\)
\(788\) 5.41641 0.192952
\(789\) 20.2229 0.719955
\(790\) 0.763932 0.0271795
\(791\) 6.94427 0.246910
\(792\) 0 0
\(793\) −24.7214 −0.877881
\(794\) 25.4164 0.901995
\(795\) −6.47214 −0.229543
\(796\) −14.4721 −0.512951
\(797\) −6.58359 −0.233203 −0.116601 0.993179i \(-0.537200\pi\)
−0.116601 + 0.993179i \(0.537200\pi\)
\(798\) 0.944272 0.0334269
\(799\) −16.9443 −0.599445
\(800\) 1.00000 0.0353553
\(801\) −4.33437 −0.153147
\(802\) −3.52786 −0.124573
\(803\) 0 0
\(804\) −7.41641 −0.261557
\(805\) 2.76393 0.0974158
\(806\) 40.0000 1.40894
\(807\) −15.4164 −0.542683
\(808\) 19.4164 0.683067
\(809\) −20.3607 −0.715843 −0.357922 0.933752i \(-0.616515\pi\)
−0.357922 + 0.933752i \(0.616515\pi\)
\(810\) −2.41641 −0.0849039
\(811\) −4.18034 −0.146792 −0.0733958 0.997303i \(-0.523384\pi\)
−0.0733958 + 0.997303i \(0.523384\pi\)
\(812\) −3.23607 −0.113564
\(813\) 8.00000 0.280572
\(814\) 0 0
\(815\) −2.94427 −0.103133
\(816\) 2.47214 0.0865421
\(817\) 6.11146 0.213813
\(818\) 4.76393 0.166567
\(819\) −6.58359 −0.230049
\(820\) 5.70820 0.199339
\(821\) −22.2918 −0.777989 −0.388995 0.921240i \(-0.627178\pi\)
−0.388995 + 0.921240i \(0.627178\pi\)
\(822\) 19.6393 0.685000
\(823\) 37.0132 1.29020 0.645099 0.764099i \(-0.276816\pi\)
0.645099 + 0.764099i \(0.276816\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) −39.7771 −1.38318 −0.691592 0.722288i \(-0.743091\pi\)
−0.691592 + 0.722288i \(0.743091\pi\)
\(828\) 4.06888 0.141403
\(829\) −7.52786 −0.261454 −0.130727 0.991418i \(-0.541731\pi\)
−0.130727 + 0.991418i \(0.541731\pi\)
\(830\) 8.00000 0.277684
\(831\) −14.2492 −0.494300
\(832\) −4.47214 −0.155043
\(833\) 2.00000 0.0692959
\(834\) −10.1115 −0.350131
\(835\) 12.9443 0.447955
\(836\) 0 0
\(837\) 49.4427 1.70899
\(838\) 23.4164 0.808906
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) −1.23607 −0.0426484
\(841\) −18.5279 −0.638892
\(842\) 31.3050 1.07884
\(843\) 35.0557 1.20738
\(844\) 21.8885 0.753435
\(845\) 7.00000 0.240807
\(846\) 12.4721 0.428801
\(847\) 0 0
\(848\) −5.23607 −0.179807
\(849\) −24.0000 −0.823678
\(850\) 2.00000 0.0685994
\(851\) −3.41641 −0.117113
\(852\) −9.88854 −0.338776
\(853\) 22.5836 0.773247 0.386624 0.922238i \(-0.373641\pi\)
0.386624 + 0.922238i \(0.373641\pi\)
\(854\) −5.52786 −0.189160
\(855\) 1.12461 0.0384609
\(856\) −16.0000 −0.546869
\(857\) 8.47214 0.289403 0.144701 0.989475i \(-0.453778\pi\)
0.144701 + 0.989475i \(0.453778\pi\)
\(858\) 0 0
\(859\) 3.63932 0.124172 0.0620860 0.998071i \(-0.480225\pi\)
0.0620860 + 0.998071i \(0.480225\pi\)
\(860\) −8.00000 −0.272798
\(861\) −7.05573 −0.240459
\(862\) 2.65248 0.0903437
\(863\) −14.7639 −0.502570 −0.251285 0.967913i \(-0.580853\pi\)
−0.251285 + 0.967913i \(0.580853\pi\)
\(864\) −5.52786 −0.188062
\(865\) 3.52786 0.119951
\(866\) −5.23607 −0.177929
\(867\) −16.0689 −0.545728
\(868\) 8.94427 0.303588
\(869\) 0 0
\(870\) 4.00000 0.135613
\(871\) 26.8328 0.909195
\(872\) 2.29180 0.0776100
\(873\) 17.2361 0.583353
\(874\) 2.11146 0.0714211
\(875\) −1.00000 −0.0338062
\(876\) 0.583592 0.0197178
\(877\) −27.3050 −0.922023 −0.461011 0.887394i \(-0.652513\pi\)
−0.461011 + 0.887394i \(0.652513\pi\)
\(878\) 19.4164 0.655272
\(879\) 0.583592 0.0196841
\(880\) 0 0
\(881\) 21.0557 0.709386 0.354693 0.934983i \(-0.384585\pi\)
0.354693 + 0.934983i \(0.384585\pi\)
\(882\) −1.47214 −0.0495694
\(883\) −33.4164 −1.12455 −0.562276 0.826950i \(-0.690074\pi\)
−0.562276 + 0.826950i \(0.690074\pi\)
\(884\) −8.94427 −0.300828
\(885\) −9.88854 −0.332400
\(886\) −2.94427 −0.0989147
\(887\) 41.8885 1.40648 0.703240 0.710953i \(-0.251736\pi\)
0.703240 + 0.710953i \(0.251736\pi\)
\(888\) 1.52786 0.0512718
\(889\) 14.4721 0.485380
\(890\) 2.94427 0.0986922
\(891\) 0 0
\(892\) −18.9443 −0.634301
\(893\) 6.47214 0.216582
\(894\) −7.05573 −0.235979
\(895\) 9.52786 0.318481
\(896\) −1.00000 −0.0334077
\(897\) 15.2786 0.510139
\(898\) −18.3607 −0.612704
\(899\) −28.9443 −0.965346
\(900\) −1.47214 −0.0490712
\(901\) −10.4721 −0.348877
\(902\) 0 0
\(903\) 9.88854 0.329070
\(904\) −6.94427 −0.230963
\(905\) 14.0000 0.465376
\(906\) −7.77709 −0.258376
\(907\) 24.4721 0.812584 0.406292 0.913743i \(-0.366822\pi\)
0.406292 + 0.913743i \(0.366822\pi\)
\(908\) 1.52786 0.0507039
\(909\) −28.5836 −0.948058
\(910\) 4.47214 0.148250
\(911\) −23.4164 −0.775820 −0.387910 0.921697i \(-0.626803\pi\)
−0.387910 + 0.921697i \(0.626803\pi\)
\(912\) −0.944272 −0.0312680
\(913\) 0 0
\(914\) 13.0557 0.431845
\(915\) 6.83282 0.225886
\(916\) −11.5279 −0.380891
\(917\) 16.1803 0.534322
\(918\) −11.0557 −0.364893
\(919\) 15.8197 0.521842 0.260921 0.965360i \(-0.415974\pi\)
0.260921 + 0.965360i \(0.415974\pi\)
\(920\) −2.76393 −0.0911241
\(921\) 11.7771 0.388068
\(922\) 20.9443 0.689763
\(923\) 35.7771 1.17762
\(924\) 0 0
\(925\) 1.23607 0.0406417
\(926\) −15.7082 −0.516204
\(927\) −20.6099 −0.676918
\(928\) 3.23607 0.106229
\(929\) −14.3607 −0.471159 −0.235579 0.971855i \(-0.575699\pi\)
−0.235579 + 0.971855i \(0.575699\pi\)
\(930\) −11.0557 −0.362532
\(931\) −0.763932 −0.0250369
\(932\) −2.94427 −0.0964428
\(933\) 38.1115 1.24771
\(934\) 13.8197 0.452193
\(935\) 0 0
\(936\) 6.58359 0.215191
\(937\) −13.0557 −0.426512 −0.213256 0.976996i \(-0.568407\pi\)
−0.213256 + 0.976996i \(0.568407\pi\)
\(938\) 6.00000 0.195907
\(939\) −25.5279 −0.833070
\(940\) −8.47214 −0.276331
\(941\) −26.2492 −0.855700 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(942\) −20.3607 −0.663387
\(943\) −15.7771 −0.513773
\(944\) −8.00000 −0.260378
\(945\) 5.52786 0.179821
\(946\) 0 0
\(947\) 11.5279 0.374605 0.187303 0.982302i \(-0.440025\pi\)
0.187303 + 0.982302i \(0.440025\pi\)
\(948\) 0.944272 0.0306685
\(949\) −2.11146 −0.0685408
\(950\) −0.763932 −0.0247852
\(951\) 2.69505 0.0873929
\(952\) −2.00000 −0.0648204
\(953\) 31.8885 1.03297 0.516486 0.856296i \(-0.327240\pi\)
0.516486 + 0.856296i \(0.327240\pi\)
\(954\) 7.70820 0.249562
\(955\) −2.47214 −0.0799964
\(956\) 7.23607 0.234031
\(957\) 0 0
\(958\) −5.88854 −0.190250
\(959\) −15.8885 −0.513068
\(960\) 1.23607 0.0398939
\(961\) 49.0000 1.58065
\(962\) −5.52786 −0.178225
\(963\) 23.5542 0.759023
\(964\) −17.1246 −0.551547
\(965\) −9.05573 −0.291514
\(966\) 3.41641 0.109921
\(967\) 29.3050 0.942384 0.471192 0.882031i \(-0.343824\pi\)
0.471192 + 0.882031i \(0.343824\pi\)
\(968\) 0 0
\(969\) −1.88854 −0.0606688
\(970\) −11.7082 −0.375928
\(971\) 30.8328 0.989472 0.494736 0.869043i \(-0.335265\pi\)
0.494736 + 0.869043i \(0.335265\pi\)
\(972\) 13.5967 0.436116
\(973\) 8.18034 0.262250
\(974\) 24.0689 0.771217
\(975\) −5.52786 −0.177033
\(976\) 5.52786 0.176943
\(977\) 52.8328 1.69027 0.845136 0.534552i \(-0.179520\pi\)
0.845136 + 0.534552i \(0.179520\pi\)
\(978\) −3.63932 −0.116373
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −3.37384 −0.107718
\(982\) −8.00000 −0.255290
\(983\) −30.9443 −0.986969 −0.493484 0.869755i \(-0.664277\pi\)
−0.493484 + 0.869755i \(0.664277\pi\)
\(984\) 7.05573 0.224928
\(985\) 5.41641 0.172581
\(986\) 6.47214 0.206115
\(987\) 10.4721 0.333332
\(988\) 3.41641 0.108690
\(989\) 22.1115 0.703103
\(990\) 0 0
\(991\) −25.3050 −0.803838 −0.401919 0.915675i \(-0.631657\pi\)
−0.401919 + 0.915675i \(0.631657\pi\)
\(992\) −8.94427 −0.283981
\(993\) −19.0557 −0.604715
\(994\) 8.00000 0.253745
\(995\) −14.4721 −0.458798
\(996\) 9.88854 0.313331
\(997\) 40.4721 1.28177 0.640883 0.767639i \(-0.278568\pi\)
0.640883 + 0.767639i \(0.278568\pi\)
\(998\) 33.3050 1.05425
\(999\) −6.83282 −0.216181
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.bv.1.2 yes 2
11.10 odd 2 8470.2.a.bj.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.bj.1.2 2 11.10 odd 2
8470.2.a.bv.1.2 yes 2 1.1 even 1 trivial