Properties

Label 8470.2.a.bx.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: \( 1 \)
Twist minimal: no (minimal twist has level 770)
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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} -1.00000 q^{10} +0.618034 q^{12} +5.23607 q^{13} -1.00000 q^{14} -0.618034 q^{15} +1.00000 q^{16} +1.61803 q^{17} -2.61803 q^{18} +0.381966 q^{19} -1.00000 q^{20} -0.618034 q^{21} -8.47214 q^{23} +0.618034 q^{24} +1.00000 q^{25} +5.23607 q^{26} -3.47214 q^{27} -1.00000 q^{28} -4.00000 q^{29} -0.618034 q^{30} -4.47214 q^{31} +1.00000 q^{32} +1.61803 q^{34} +1.00000 q^{35} -2.61803 q^{36} +9.70820 q^{37} +0.381966 q^{38} +3.23607 q^{39} -1.00000 q^{40} -8.61803 q^{41} -0.618034 q^{42} +4.61803 q^{43} +2.61803 q^{45} -8.47214 q^{46} -6.76393 q^{47} +0.618034 q^{48} +1.00000 q^{49} +1.00000 q^{50} +1.00000 q^{51} +5.23607 q^{52} -6.47214 q^{53} -3.47214 q^{54} -1.00000 q^{56} +0.236068 q^{57} -4.00000 q^{58} -3.61803 q^{59} -0.618034 q^{60} +8.00000 q^{61} -4.47214 q^{62} +2.61803 q^{63} +1.00000 q^{64} -5.23607 q^{65} -6.09017 q^{67} +1.61803 q^{68} -5.23607 q^{69} +1.00000 q^{70} +2.47214 q^{71} -2.61803 q^{72} -4.32624 q^{73} +9.70820 q^{74} +0.618034 q^{75} +0.381966 q^{76} +3.23607 q^{78} -1.70820 q^{79} -1.00000 q^{80} +5.70820 q^{81} -8.61803 q^{82} -11.0902 q^{83} -0.618034 q^{84} -1.61803 q^{85} +4.61803 q^{86} -2.47214 q^{87} +9.85410 q^{89} +2.61803 q^{90} -5.23607 q^{91} -8.47214 q^{92} -2.76393 q^{93} -6.76393 q^{94} -0.381966 q^{95} +0.618034 q^{96} -10.7984 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - q^{3} + 2q^{4} - 2q^{5} - q^{6} - 2q^{7} + 2q^{8} - 3q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - q^{3} + 2q^{4} - 2q^{5} - q^{6} - 2q^{7} + 2q^{8} - 3q^{9} - 2q^{10} - q^{12} + 6q^{13} - 2q^{14} + q^{15} + 2q^{16} + q^{17} - 3q^{18} + 3q^{19} - 2q^{20} + q^{21} - 8q^{23} - q^{24} + 2q^{25} + 6q^{26} + 2q^{27} - 2q^{28} - 8q^{29} + q^{30} + 2q^{32} + q^{34} + 2q^{35} - 3q^{36} + 6q^{37} + 3q^{38} + 2q^{39} - 2q^{40} - 15q^{41} + q^{42} + 7q^{43} + 3q^{45} - 8q^{46} - 18q^{47} - q^{48} + 2q^{49} + 2q^{50} + 2q^{51} + 6q^{52} - 4q^{53} + 2q^{54} - 2q^{56} - 4q^{57} - 8q^{58} - 5q^{59} + q^{60} + 16q^{61} + 3q^{63} + 2q^{64} - 6q^{65} - q^{67} + q^{68} - 6q^{69} + 2q^{70} - 4q^{71} - 3q^{72} + 7q^{73} + 6q^{74} - q^{75} + 3q^{76} + 2q^{78} + 10q^{79} - 2q^{80} - 2q^{81} - 15q^{82} - 11q^{83} + q^{84} - q^{85} + 7q^{86} + 4q^{87} + 13q^{89} + 3q^{90} - 6q^{91} - 8q^{92} - 10q^{93} - 18q^{94} - 3q^{95} - q^{96} + 3q^{97} + 2q^{98} + O(q^{100}) \)

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
\(3\) 0.618034 0.356822 0.178411 0.983956i \(-0.442904\pi\)
0.178411 + 0.983956i \(0.442904\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0.618034 0.252311
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.61803 −0.872678
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 0.618034 0.178411
\(13\) 5.23607 1.45222 0.726112 0.687576i \(-0.241325\pi\)
0.726112 + 0.687576i \(0.241325\pi\)
\(14\) −1.00000 −0.267261
\(15\) −0.618034 −0.159576
\(16\) 1.00000 0.250000
\(17\) 1.61803 0.392431 0.196215 0.980561i \(-0.437135\pi\)
0.196215 + 0.980561i \(0.437135\pi\)
\(18\) −2.61803 −0.617077
\(19\) 0.381966 0.0876290 0.0438145 0.999040i \(-0.486049\pi\)
0.0438145 + 0.999040i \(0.486049\pi\)
\(20\) −1.00000 −0.223607
\(21\) −0.618034 −0.134866
\(22\) 0 0
\(23\) −8.47214 −1.76656 −0.883281 0.468844i \(-0.844671\pi\)
−0.883281 + 0.468844i \(0.844671\pi\)
\(24\) 0.618034 0.126156
\(25\) 1.00000 0.200000
\(26\) 5.23607 1.02688
\(27\) −3.47214 −0.668213
\(28\) −1.00000 −0.188982
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) −0.618034 −0.112837
\(31\) −4.47214 −0.803219 −0.401610 0.915811i \(-0.631549\pi\)
−0.401610 + 0.915811i \(0.631549\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.61803 0.277491
\(35\) 1.00000 0.169031
\(36\) −2.61803 −0.436339
\(37\) 9.70820 1.59602 0.798009 0.602645i \(-0.205886\pi\)
0.798009 + 0.602645i \(0.205886\pi\)
\(38\) 0.381966 0.0619631
\(39\) 3.23607 0.518186
\(40\) −1.00000 −0.158114
\(41\) −8.61803 −1.34591 −0.672955 0.739683i \(-0.734975\pi\)
−0.672955 + 0.739683i \(0.734975\pi\)
\(42\) −0.618034 −0.0953647
\(43\) 4.61803 0.704244 0.352122 0.935954i \(-0.385460\pi\)
0.352122 + 0.935954i \(0.385460\pi\)
\(44\) 0 0
\(45\) 2.61803 0.390273
\(46\) −8.47214 −1.24915
\(47\) −6.76393 −0.986621 −0.493310 0.869853i \(-0.664213\pi\)
−0.493310 + 0.869853i \(0.664213\pi\)
\(48\) 0.618034 0.0892055
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 1.00000 0.140028
\(52\) 5.23607 0.726112
\(53\) −6.47214 −0.889016 −0.444508 0.895775i \(-0.646622\pi\)
−0.444508 + 0.895775i \(0.646622\pi\)
\(54\) −3.47214 −0.472498
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0.236068 0.0312680
\(58\) −4.00000 −0.525226
\(59\) −3.61803 −0.471028 −0.235514 0.971871i \(-0.575677\pi\)
−0.235514 + 0.971871i \(0.575677\pi\)
\(60\) −0.618034 −0.0797878
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −4.47214 −0.567962
\(63\) 2.61803 0.329841
\(64\) 1.00000 0.125000
\(65\) −5.23607 −0.649454
\(66\) 0 0
\(67\) −6.09017 −0.744033 −0.372016 0.928226i \(-0.621333\pi\)
−0.372016 + 0.928226i \(0.621333\pi\)
\(68\) 1.61803 0.196215
\(69\) −5.23607 −0.630349
\(70\) 1.00000 0.119523
\(71\) 2.47214 0.293389 0.146694 0.989182i \(-0.453137\pi\)
0.146694 + 0.989182i \(0.453137\pi\)
\(72\) −2.61803 −0.308538
\(73\) −4.32624 −0.506348 −0.253174 0.967421i \(-0.581474\pi\)
−0.253174 + 0.967421i \(0.581474\pi\)
\(74\) 9.70820 1.12856
\(75\) 0.618034 0.0713644
\(76\) 0.381966 0.0438145
\(77\) 0 0
\(78\) 3.23607 0.366413
\(79\) −1.70820 −0.192188 −0.0960940 0.995372i \(-0.530635\pi\)
−0.0960940 + 0.995372i \(0.530635\pi\)
\(80\) −1.00000 −0.111803
\(81\) 5.70820 0.634245
\(82\) −8.61803 −0.951703
\(83\) −11.0902 −1.21730 −0.608652 0.793437i \(-0.708289\pi\)
−0.608652 + 0.793437i \(0.708289\pi\)
\(84\) −0.618034 −0.0674330
\(85\) −1.61803 −0.175500
\(86\) 4.61803 0.497975
\(87\) −2.47214 −0.265041
\(88\) 0 0
\(89\) 9.85410 1.04453 0.522266 0.852782i \(-0.325087\pi\)
0.522266 + 0.852782i \(0.325087\pi\)
\(90\) 2.61803 0.275965
\(91\) −5.23607 −0.548889
\(92\) −8.47214 −0.883281
\(93\) −2.76393 −0.286606
\(94\) −6.76393 −0.697646
\(95\) −0.381966 −0.0391889
\(96\) 0.618034 0.0630778
\(97\) −10.7984 −1.09641 −0.548204 0.836344i \(-0.684688\pi\)
−0.548204 + 0.836344i \(0.684688\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −17.2361 −1.71505 −0.857526 0.514440i \(-0.828000\pi\)
−0.857526 + 0.514440i \(0.828000\pi\)
\(102\) 1.00000 0.0990148
\(103\) 0.291796 0.0287515 0.0143758 0.999897i \(-0.495424\pi\)
0.0143758 + 0.999897i \(0.495424\pi\)
\(104\) 5.23607 0.513439
\(105\) 0.618034 0.0603139
\(106\) −6.47214 −0.628629
\(107\) 3.09017 0.298738 0.149369 0.988782i \(-0.452276\pi\)
0.149369 + 0.988782i \(0.452276\pi\)
\(108\) −3.47214 −0.334106
\(109\) 12.1803 1.16666 0.583332 0.812233i \(-0.301748\pi\)
0.583332 + 0.812233i \(0.301748\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) −1.00000 −0.0944911
\(113\) 13.7984 1.29804 0.649021 0.760771i \(-0.275179\pi\)
0.649021 + 0.760771i \(0.275179\pi\)
\(114\) 0.236068 0.0221098
\(115\) 8.47214 0.790031
\(116\) −4.00000 −0.371391
\(117\) −13.7082 −1.26732
\(118\) −3.61803 −0.333067
\(119\) −1.61803 −0.148325
\(120\) −0.618034 −0.0564185
\(121\) 0 0
\(122\) 8.00000 0.724286
\(123\) −5.32624 −0.480251
\(124\) −4.47214 −0.401610
\(125\) −1.00000 −0.0894427
\(126\) 2.61803 0.233233
\(127\) −14.1803 −1.25830 −0.629151 0.777283i \(-0.716597\pi\)
−0.629151 + 0.777283i \(0.716597\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.85410 0.251290
\(130\) −5.23607 −0.459234
\(131\) −14.8541 −1.29781 −0.648904 0.760870i \(-0.724773\pi\)
−0.648904 + 0.760870i \(0.724773\pi\)
\(132\) 0 0
\(133\) −0.381966 −0.0331207
\(134\) −6.09017 −0.526111
\(135\) 3.47214 0.298834
\(136\) 1.61803 0.138745
\(137\) −6.14590 −0.525080 −0.262540 0.964921i \(-0.584560\pi\)
−0.262540 + 0.964921i \(0.584560\pi\)
\(138\) −5.23607 −0.445724
\(139\) 10.4721 0.888235 0.444117 0.895969i \(-0.353517\pi\)
0.444117 + 0.895969i \(0.353517\pi\)
\(140\) 1.00000 0.0845154
\(141\) −4.18034 −0.352048
\(142\) 2.47214 0.207457
\(143\) 0 0
\(144\) −2.61803 −0.218169
\(145\) 4.00000 0.332182
\(146\) −4.32624 −0.358042
\(147\) 0.618034 0.0509746
\(148\) 9.70820 0.798009
\(149\) −5.70820 −0.467634 −0.233817 0.972281i \(-0.575122\pi\)
−0.233817 + 0.972281i \(0.575122\pi\)
\(150\) 0.618034 0.0504623
\(151\) 2.18034 0.177434 0.0887168 0.996057i \(-0.471723\pi\)
0.0887168 + 0.996057i \(0.471723\pi\)
\(152\) 0.381966 0.0309815
\(153\) −4.23607 −0.342466
\(154\) 0 0
\(155\) 4.47214 0.359211
\(156\) 3.23607 0.259093
\(157\) 16.1803 1.29133 0.645666 0.763620i \(-0.276580\pi\)
0.645666 + 0.763620i \(0.276580\pi\)
\(158\) −1.70820 −0.135897
\(159\) −4.00000 −0.317221
\(160\) −1.00000 −0.0790569
\(161\) 8.47214 0.667698
\(162\) 5.70820 0.448479
\(163\) −20.3820 −1.59644 −0.798219 0.602367i \(-0.794224\pi\)
−0.798219 + 0.602367i \(0.794224\pi\)
\(164\) −8.61803 −0.672955
\(165\) 0 0
\(166\) −11.0902 −0.860764
\(167\) −10.0000 −0.773823 −0.386912 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) −0.618034 −0.0476824
\(169\) 14.4164 1.10895
\(170\) −1.61803 −0.124098
\(171\) −1.00000 −0.0764719
\(172\) 4.61803 0.352122
\(173\) −9.23607 −0.702205 −0.351103 0.936337i \(-0.614193\pi\)
−0.351103 + 0.936337i \(0.614193\pi\)
\(174\) −2.47214 −0.187412
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −2.23607 −0.168073
\(178\) 9.85410 0.738596
\(179\) −14.8541 −1.11025 −0.555124 0.831768i \(-0.687329\pi\)
−0.555124 + 0.831768i \(0.687329\pi\)
\(180\) 2.61803 0.195137
\(181\) 17.1246 1.27286 0.636431 0.771333i \(-0.280410\pi\)
0.636431 + 0.771333i \(0.280410\pi\)
\(182\) −5.23607 −0.388123
\(183\) 4.94427 0.365491
\(184\) −8.47214 −0.624574
\(185\) −9.70820 −0.713761
\(186\) −2.76393 −0.202661
\(187\) 0 0
\(188\) −6.76393 −0.493310
\(189\) 3.47214 0.252561
\(190\) −0.381966 −0.0277107
\(191\) −2.00000 −0.144715 −0.0723575 0.997379i \(-0.523052\pi\)
−0.0723575 + 0.997379i \(0.523052\pi\)
\(192\) 0.618034 0.0446028
\(193\) 12.4721 0.897764 0.448882 0.893591i \(-0.351822\pi\)
0.448882 + 0.893591i \(0.351822\pi\)
\(194\) −10.7984 −0.775278
\(195\) −3.23607 −0.231740
\(196\) 1.00000 0.0714286
\(197\) −21.4164 −1.52586 −0.762928 0.646484i \(-0.776239\pi\)
−0.762928 + 0.646484i \(0.776239\pi\)
\(198\) 0 0
\(199\) −1.23607 −0.0876225 −0.0438113 0.999040i \(-0.513950\pi\)
−0.0438113 + 0.999040i \(0.513950\pi\)
\(200\) 1.00000 0.0707107
\(201\) −3.76393 −0.265487
\(202\) −17.2361 −1.21273
\(203\) 4.00000 0.280745
\(204\) 1.00000 0.0700140
\(205\) 8.61803 0.601910
\(206\) 0.291796 0.0203304
\(207\) 22.1803 1.54164
\(208\) 5.23607 0.363056
\(209\) 0 0
\(210\) 0.618034 0.0426484
\(211\) −16.9098 −1.16412 −0.582060 0.813146i \(-0.697753\pi\)
−0.582060 + 0.813146i \(0.697753\pi\)
\(212\) −6.47214 −0.444508
\(213\) 1.52786 0.104688
\(214\) 3.09017 0.211240
\(215\) −4.61803 −0.314947
\(216\) −3.47214 −0.236249
\(217\) 4.47214 0.303588
\(218\) 12.1803 0.824957
\(219\) −2.67376 −0.180676
\(220\) 0 0
\(221\) 8.47214 0.569898
\(222\) 6.00000 0.402694
\(223\) −5.05573 −0.338557 −0.169278 0.985568i \(-0.554144\pi\)
−0.169278 + 0.985568i \(0.554144\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.61803 −0.174536
\(226\) 13.7984 0.917854
\(227\) −8.56231 −0.568300 −0.284150 0.958780i \(-0.591711\pi\)
−0.284150 + 0.958780i \(0.591711\pi\)
\(228\) 0.236068 0.0156340
\(229\) −5.70820 −0.377209 −0.188604 0.982053i \(-0.560396\pi\)
−0.188604 + 0.982053i \(0.560396\pi\)
\(230\) 8.47214 0.558636
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) −3.67376 −0.240676 −0.120338 0.992733i \(-0.538398\pi\)
−0.120338 + 0.992733i \(0.538398\pi\)
\(234\) −13.7082 −0.896133
\(235\) 6.76393 0.441230
\(236\) −3.61803 −0.235514
\(237\) −1.05573 −0.0685769
\(238\) −1.61803 −0.104882
\(239\) −17.7082 −1.14545 −0.572724 0.819748i \(-0.694113\pi\)
−0.572724 + 0.819748i \(0.694113\pi\)
\(240\) −0.618034 −0.0398939
\(241\) 7.14590 0.460308 0.230154 0.973154i \(-0.426077\pi\)
0.230154 + 0.973154i \(0.426077\pi\)
\(242\) 0 0
\(243\) 13.9443 0.894525
\(244\) 8.00000 0.512148
\(245\) −1.00000 −0.0638877
\(246\) −5.32624 −0.339589
\(247\) 2.00000 0.127257
\(248\) −4.47214 −0.283981
\(249\) −6.85410 −0.434361
\(250\) −1.00000 −0.0632456
\(251\) −17.8885 −1.12911 −0.564557 0.825394i \(-0.690953\pi\)
−0.564557 + 0.825394i \(0.690953\pi\)
\(252\) 2.61803 0.164921
\(253\) 0 0
\(254\) −14.1803 −0.889754
\(255\) −1.00000 −0.0626224
\(256\) 1.00000 0.0625000
\(257\) −25.5623 −1.59453 −0.797266 0.603628i \(-0.793721\pi\)
−0.797266 + 0.603628i \(0.793721\pi\)
\(258\) 2.85410 0.177689
\(259\) −9.70820 −0.603238
\(260\) −5.23607 −0.324727
\(261\) 10.4721 0.648209
\(262\) −14.8541 −0.917689
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 6.47214 0.397580
\(266\) −0.381966 −0.0234198
\(267\) 6.09017 0.372712
\(268\) −6.09017 −0.372016
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 3.47214 0.211307
\(271\) −19.7082 −1.19719 −0.598594 0.801053i \(-0.704274\pi\)
−0.598594 + 0.801053i \(0.704274\pi\)
\(272\) 1.61803 0.0981077
\(273\) −3.23607 −0.195856
\(274\) −6.14590 −0.371287
\(275\) 0 0
\(276\) −5.23607 −0.315174
\(277\) −23.1246 −1.38942 −0.694712 0.719288i \(-0.744468\pi\)
−0.694712 + 0.719288i \(0.744468\pi\)
\(278\) 10.4721 0.628077
\(279\) 11.7082 0.700952
\(280\) 1.00000 0.0597614
\(281\) 6.79837 0.405557 0.202778 0.979225i \(-0.435003\pi\)
0.202778 + 0.979225i \(0.435003\pi\)
\(282\) −4.18034 −0.248936
\(283\) 25.8885 1.53891 0.769457 0.638699i \(-0.220527\pi\)
0.769457 + 0.638699i \(0.220527\pi\)
\(284\) 2.47214 0.146694
\(285\) −0.236068 −0.0139835
\(286\) 0 0
\(287\) 8.61803 0.508706
\(288\) −2.61803 −0.154269
\(289\) −14.3820 −0.845998
\(290\) 4.00000 0.234888
\(291\) −6.67376 −0.391223
\(292\) −4.32624 −0.253174
\(293\) −2.29180 −0.133888 −0.0669441 0.997757i \(-0.521325\pi\)
−0.0669441 + 0.997757i \(0.521325\pi\)
\(294\) 0.618034 0.0360445
\(295\) 3.61803 0.210650
\(296\) 9.70820 0.564278
\(297\) 0 0
\(298\) −5.70820 −0.330667
\(299\) −44.3607 −2.56544
\(300\) 0.618034 0.0356822
\(301\) −4.61803 −0.266179
\(302\) 2.18034 0.125464
\(303\) −10.6525 −0.611969
\(304\) 0.381966 0.0219073
\(305\) −8.00000 −0.458079
\(306\) −4.23607 −0.242160
\(307\) −34.5066 −1.96939 −0.984697 0.174274i \(-0.944242\pi\)
−0.984697 + 0.174274i \(0.944242\pi\)
\(308\) 0 0
\(309\) 0.180340 0.0102592
\(310\) 4.47214 0.254000
\(311\) 3.23607 0.183501 0.0917503 0.995782i \(-0.470754\pi\)
0.0917503 + 0.995782i \(0.470754\pi\)
\(312\) 3.23607 0.183206
\(313\) 3.32624 0.188010 0.0940050 0.995572i \(-0.470033\pi\)
0.0940050 + 0.995572i \(0.470033\pi\)
\(314\) 16.1803 0.913109
\(315\) −2.61803 −0.147510
\(316\) −1.70820 −0.0960940
\(317\) −17.8885 −1.00472 −0.502360 0.864658i \(-0.667535\pi\)
−0.502360 + 0.864658i \(0.667535\pi\)
\(318\) −4.00000 −0.224309
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 1.90983 0.106596
\(322\) 8.47214 0.472134
\(323\) 0.618034 0.0343883
\(324\) 5.70820 0.317122
\(325\) 5.23607 0.290445
\(326\) −20.3820 −1.12885
\(327\) 7.52786 0.416292
\(328\) −8.61803 −0.475851
\(329\) 6.76393 0.372908
\(330\) 0 0
\(331\) 12.3262 0.677511 0.338756 0.940874i \(-0.389994\pi\)
0.338756 + 0.940874i \(0.389994\pi\)
\(332\) −11.0902 −0.608652
\(333\) −25.4164 −1.39281
\(334\) −10.0000 −0.547176
\(335\) 6.09017 0.332742
\(336\) −0.618034 −0.0337165
\(337\) −32.2705 −1.75789 −0.878943 0.476926i \(-0.841751\pi\)
−0.878943 + 0.476926i \(0.841751\pi\)
\(338\) 14.4164 0.784149
\(339\) 8.52786 0.463170
\(340\) −1.61803 −0.0877502
\(341\) 0 0
\(342\) −1.00000 −0.0540738
\(343\) −1.00000 −0.0539949
\(344\) 4.61803 0.248988
\(345\) 5.23607 0.281900
\(346\) −9.23607 −0.496534
\(347\) 27.0344 1.45128 0.725642 0.688072i \(-0.241543\pi\)
0.725642 + 0.688072i \(0.241543\pi\)
\(348\) −2.47214 −0.132520
\(349\) −10.4721 −0.560561 −0.280280 0.959918i \(-0.590427\pi\)
−0.280280 + 0.959918i \(0.590427\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −18.1803 −0.970395
\(352\) 0 0
\(353\) 17.0344 0.906652 0.453326 0.891345i \(-0.350237\pi\)
0.453326 + 0.891345i \(0.350237\pi\)
\(354\) −2.23607 −0.118846
\(355\) −2.47214 −0.131207
\(356\) 9.85410 0.522266
\(357\) −1.00000 −0.0529256
\(358\) −14.8541 −0.785064
\(359\) −25.7082 −1.35683 −0.678414 0.734680i \(-0.737332\pi\)
−0.678414 + 0.734680i \(0.737332\pi\)
\(360\) 2.61803 0.137983
\(361\) −18.8541 −0.992321
\(362\) 17.1246 0.900050
\(363\) 0 0
\(364\) −5.23607 −0.274445
\(365\) 4.32624 0.226446
\(366\) 4.94427 0.258441
\(367\) 11.0557 0.577104 0.288552 0.957464i \(-0.406826\pi\)
0.288552 + 0.957464i \(0.406826\pi\)
\(368\) −8.47214 −0.441641
\(369\) 22.5623 1.17455
\(370\) −9.70820 −0.504705
\(371\) 6.47214 0.336017
\(372\) −2.76393 −0.143303
\(373\) 4.94427 0.256005 0.128002 0.991774i \(-0.459143\pi\)
0.128002 + 0.991774i \(0.459143\pi\)
\(374\) 0 0
\(375\) −0.618034 −0.0319151
\(376\) −6.76393 −0.348823
\(377\) −20.9443 −1.07868
\(378\) 3.47214 0.178587
\(379\) −10.1459 −0.521160 −0.260580 0.965452i \(-0.583914\pi\)
−0.260580 + 0.965452i \(0.583914\pi\)
\(380\) −0.381966 −0.0195944
\(381\) −8.76393 −0.448990
\(382\) −2.00000 −0.102329
\(383\) 17.7082 0.904847 0.452423 0.891803i \(-0.350560\pi\)
0.452423 + 0.891803i \(0.350560\pi\)
\(384\) 0.618034 0.0315389
\(385\) 0 0
\(386\) 12.4721 0.634815
\(387\) −12.0902 −0.614578
\(388\) −10.7984 −0.548204
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) −3.23607 −0.163865
\(391\) −13.7082 −0.693254
\(392\) 1.00000 0.0505076
\(393\) −9.18034 −0.463087
\(394\) −21.4164 −1.07894
\(395\) 1.70820 0.0859491
\(396\) 0 0
\(397\) −34.5410 −1.73356 −0.866782 0.498687i \(-0.833816\pi\)
−0.866782 + 0.498687i \(0.833816\pi\)
\(398\) −1.23607 −0.0619585
\(399\) −0.236068 −0.0118182
\(400\) 1.00000 0.0500000
\(401\) −10.5066 −0.524673 −0.262337 0.964976i \(-0.584493\pi\)
−0.262337 + 0.964976i \(0.584493\pi\)
\(402\) −3.76393 −0.187728
\(403\) −23.4164 −1.16645
\(404\) −17.2361 −0.857526
\(405\) −5.70820 −0.283643
\(406\) 4.00000 0.198517
\(407\) 0 0
\(408\) 1.00000 0.0495074
\(409\) 31.3050 1.54793 0.773965 0.633228i \(-0.218271\pi\)
0.773965 + 0.633228i \(0.218271\pi\)
\(410\) 8.61803 0.425614
\(411\) −3.79837 −0.187360
\(412\) 0.291796 0.0143758
\(413\) 3.61803 0.178032
\(414\) 22.1803 1.09010
\(415\) 11.0902 0.544395
\(416\) 5.23607 0.256719
\(417\) 6.47214 0.316942
\(418\) 0 0
\(419\) 27.7426 1.35532 0.677658 0.735377i \(-0.262995\pi\)
0.677658 + 0.735377i \(0.262995\pi\)
\(420\) 0.618034 0.0301570
\(421\) −0.472136 −0.0230105 −0.0115052 0.999934i \(-0.503662\pi\)
−0.0115052 + 0.999934i \(0.503662\pi\)
\(422\) −16.9098 −0.823158
\(423\) 17.7082 0.861002
\(424\) −6.47214 −0.314315
\(425\) 1.61803 0.0784862
\(426\) 1.52786 0.0740253
\(427\) −8.00000 −0.387147
\(428\) 3.09017 0.149369
\(429\) 0 0
\(430\) −4.61803 −0.222701
\(431\) −6.29180 −0.303065 −0.151533 0.988452i \(-0.548421\pi\)
−0.151533 + 0.988452i \(0.548421\pi\)
\(432\) −3.47214 −0.167053
\(433\) 25.7984 1.23979 0.619895 0.784684i \(-0.287175\pi\)
0.619895 + 0.784684i \(0.287175\pi\)
\(434\) 4.47214 0.214669
\(435\) 2.47214 0.118530
\(436\) 12.1803 0.583332
\(437\) −3.23607 −0.154802
\(438\) −2.67376 −0.127757
\(439\) 20.3607 0.971762 0.485881 0.874025i \(-0.338499\pi\)
0.485881 + 0.874025i \(0.338499\pi\)
\(440\) 0 0
\(441\) −2.61803 −0.124668
\(442\) 8.47214 0.402978
\(443\) −28.6180 −1.35968 −0.679842 0.733359i \(-0.737952\pi\)
−0.679842 + 0.733359i \(0.737952\pi\)
\(444\) 6.00000 0.284747
\(445\) −9.85410 −0.467129
\(446\) −5.05573 −0.239396
\(447\) −3.52786 −0.166862
\(448\) −1.00000 −0.0472456
\(449\) −14.7426 −0.695748 −0.347874 0.937541i \(-0.613096\pi\)
−0.347874 + 0.937541i \(0.613096\pi\)
\(450\) −2.61803 −0.123415
\(451\) 0 0
\(452\) 13.7984 0.649021
\(453\) 1.34752 0.0633122
\(454\) −8.56231 −0.401849
\(455\) 5.23607 0.245471
\(456\) 0.236068 0.0110549
\(457\) 25.5623 1.19575 0.597877 0.801588i \(-0.296011\pi\)
0.597877 + 0.801588i \(0.296011\pi\)
\(458\) −5.70820 −0.266727
\(459\) −5.61803 −0.262227
\(460\) 8.47214 0.395015
\(461\) 1.12461 0.0523784 0.0261892 0.999657i \(-0.491663\pi\)
0.0261892 + 0.999657i \(0.491663\pi\)
\(462\) 0 0
\(463\) 0.652476 0.0303231 0.0151616 0.999885i \(-0.495174\pi\)
0.0151616 + 0.999885i \(0.495174\pi\)
\(464\) −4.00000 −0.185695
\(465\) 2.76393 0.128174
\(466\) −3.67376 −0.170184
\(467\) 16.9443 0.784087 0.392044 0.919947i \(-0.371768\pi\)
0.392044 + 0.919947i \(0.371768\pi\)
\(468\) −13.7082 −0.633662
\(469\) 6.09017 0.281218
\(470\) 6.76393 0.311997
\(471\) 10.0000 0.460776
\(472\) −3.61803 −0.166534
\(473\) 0 0
\(474\) −1.05573 −0.0484912
\(475\) 0.381966 0.0175258
\(476\) −1.61803 −0.0741625
\(477\) 16.9443 0.775825
\(478\) −17.7082 −0.809954
\(479\) 17.1246 0.782443 0.391222 0.920296i \(-0.372053\pi\)
0.391222 + 0.920296i \(0.372053\pi\)
\(480\) −0.618034 −0.0282093
\(481\) 50.8328 2.31778
\(482\) 7.14590 0.325487
\(483\) 5.23607 0.238249
\(484\) 0 0
\(485\) 10.7984 0.490329
\(486\) 13.9443 0.632525
\(487\) 37.7771 1.71184 0.855922 0.517106i \(-0.172991\pi\)
0.855922 + 0.517106i \(0.172991\pi\)
\(488\) 8.00000 0.362143
\(489\) −12.5967 −0.569645
\(490\) −1.00000 −0.0451754
\(491\) 37.9230 1.71144 0.855720 0.517439i \(-0.173115\pi\)
0.855720 + 0.517439i \(0.173115\pi\)
\(492\) −5.32624 −0.240125
\(493\) −6.47214 −0.291490
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) −4.47214 −0.200805
\(497\) −2.47214 −0.110890
\(498\) −6.85410 −0.307140
\(499\) 0.618034 0.0276670 0.0138335 0.999904i \(-0.495597\pi\)
0.0138335 + 0.999904i \(0.495597\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −6.18034 −0.276117
\(502\) −17.8885 −0.798405
\(503\) 25.7082 1.14627 0.573136 0.819460i \(-0.305727\pi\)
0.573136 + 0.819460i \(0.305727\pi\)
\(504\) 2.61803 0.116617
\(505\) 17.2361 0.766995
\(506\) 0 0
\(507\) 8.90983 0.395699
\(508\) −14.1803 −0.629151
\(509\) 28.1803 1.24907 0.624536 0.780996i \(-0.285288\pi\)
0.624536 + 0.780996i \(0.285288\pi\)
\(510\) −1.00000 −0.0442807
\(511\) 4.32624 0.191381
\(512\) 1.00000 0.0441942
\(513\) −1.32624 −0.0585548
\(514\) −25.5623 −1.12750
\(515\) −0.291796 −0.0128581
\(516\) 2.85410 0.125645
\(517\) 0 0
\(518\) −9.70820 −0.426554
\(519\) −5.70820 −0.250562
\(520\) −5.23607 −0.229617
\(521\) 17.4377 0.763959 0.381980 0.924171i \(-0.375242\pi\)
0.381980 + 0.924171i \(0.375242\pi\)
\(522\) 10.4721 0.458353
\(523\) −28.9230 −1.26471 −0.632357 0.774677i \(-0.717912\pi\)
−0.632357 + 0.774677i \(0.717912\pi\)
\(524\) −14.8541 −0.648904
\(525\) −0.618034 −0.0269732
\(526\) 12.0000 0.523225
\(527\) −7.23607 −0.315208
\(528\) 0 0
\(529\) 48.7771 2.12074
\(530\) 6.47214 0.281132
\(531\) 9.47214 0.411056
\(532\) −0.381966 −0.0165603
\(533\) −45.1246 −1.95456
\(534\) 6.09017 0.263547
\(535\) −3.09017 −0.133600
\(536\) −6.09017 −0.263055
\(537\) −9.18034 −0.396161
\(538\) 14.0000 0.603583
\(539\) 0 0
\(540\) 3.47214 0.149417
\(541\) −40.1803 −1.72749 −0.863744 0.503931i \(-0.831887\pi\)
−0.863744 + 0.503931i \(0.831887\pi\)
\(542\) −19.7082 −0.846540
\(543\) 10.5836 0.454185
\(544\) 1.61803 0.0693726
\(545\) −12.1803 −0.521748
\(546\) −3.23607 −0.138491
\(547\) −38.0902 −1.62862 −0.814309 0.580432i \(-0.802884\pi\)
−0.814309 + 0.580432i \(0.802884\pi\)
\(548\) −6.14590 −0.262540
\(549\) −20.9443 −0.893880
\(550\) 0 0
\(551\) −1.52786 −0.0650892
\(552\) −5.23607 −0.222862
\(553\) 1.70820 0.0726402
\(554\) −23.1246 −0.982471
\(555\) −6.00000 −0.254686
\(556\) 10.4721 0.444117
\(557\) 40.3607 1.71014 0.855068 0.518515i \(-0.173515\pi\)
0.855068 + 0.518515i \(0.173515\pi\)
\(558\) 11.7082 0.495648
\(559\) 24.1803 1.02272
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 6.79837 0.286772
\(563\) 16.6869 0.703270 0.351635 0.936137i \(-0.385626\pi\)
0.351635 + 0.936137i \(0.385626\pi\)
\(564\) −4.18034 −0.176024
\(565\) −13.7984 −0.580502
\(566\) 25.8885 1.08818
\(567\) −5.70820 −0.239722
\(568\) 2.47214 0.103729
\(569\) −36.4508 −1.52810 −0.764050 0.645158i \(-0.776792\pi\)
−0.764050 + 0.645158i \(0.776792\pi\)
\(570\) −0.236068 −0.00988780
\(571\) 26.8328 1.12292 0.561459 0.827504i \(-0.310240\pi\)
0.561459 + 0.827504i \(0.310240\pi\)
\(572\) 0 0
\(573\) −1.23607 −0.0516375
\(574\) 8.61803 0.359710
\(575\) −8.47214 −0.353312
\(576\) −2.61803 −0.109085
\(577\) 8.67376 0.361093 0.180547 0.983566i \(-0.442213\pi\)
0.180547 + 0.983566i \(0.442213\pi\)
\(578\) −14.3820 −0.598211
\(579\) 7.70820 0.320342
\(580\) 4.00000 0.166091
\(581\) 11.0902 0.460098
\(582\) −6.67376 −0.276636
\(583\) 0 0
\(584\) −4.32624 −0.179021
\(585\) 13.7082 0.566764
\(586\) −2.29180 −0.0946732
\(587\) −8.38197 −0.345961 −0.172980 0.984925i \(-0.555340\pi\)
−0.172980 + 0.984925i \(0.555340\pi\)
\(588\) 0.618034 0.0254873
\(589\) −1.70820 −0.0703853
\(590\) 3.61803 0.148952
\(591\) −13.2361 −0.544459
\(592\) 9.70820 0.399005
\(593\) 10.3820 0.426336 0.213168 0.977016i \(-0.431622\pi\)
0.213168 + 0.977016i \(0.431622\pi\)
\(594\) 0 0
\(595\) 1.61803 0.0663329
\(596\) −5.70820 −0.233817
\(597\) −0.763932 −0.0312657
\(598\) −44.3607 −1.81404
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0.618034 0.0252311
\(601\) 13.5623 0.553218 0.276609 0.960983i \(-0.410789\pi\)
0.276609 + 0.960983i \(0.410789\pi\)
\(602\) −4.61803 −0.188217
\(603\) 15.9443 0.649301
\(604\) 2.18034 0.0887168
\(605\) 0 0
\(606\) −10.6525 −0.432727
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) 0.381966 0.0154908
\(609\) 2.47214 0.100176
\(610\) −8.00000 −0.323911
\(611\) −35.4164 −1.43279
\(612\) −4.23607 −0.171233
\(613\) 10.0689 0.406678 0.203339 0.979108i \(-0.434821\pi\)
0.203339 + 0.979108i \(0.434821\pi\)
\(614\) −34.5066 −1.39257
\(615\) 5.32624 0.214775
\(616\) 0 0
\(617\) 37.7984 1.52171 0.760853 0.648925i \(-0.224781\pi\)
0.760853 + 0.648925i \(0.224781\pi\)
\(618\) 0.180340 0.00725433
\(619\) 40.5623 1.63034 0.815168 0.579224i \(-0.196644\pi\)
0.815168 + 0.579224i \(0.196644\pi\)
\(620\) 4.47214 0.179605
\(621\) 29.4164 1.18044
\(622\) 3.23607 0.129755
\(623\) −9.85410 −0.394796
\(624\) 3.23607 0.129546
\(625\) 1.00000 0.0400000
\(626\) 3.32624 0.132943
\(627\) 0 0
\(628\) 16.1803 0.645666
\(629\) 15.7082 0.626327
\(630\) −2.61803 −0.104305
\(631\) −25.2361 −1.00463 −0.502316 0.864684i \(-0.667519\pi\)
−0.502316 + 0.864684i \(0.667519\pi\)
\(632\) −1.70820 −0.0679487
\(633\) −10.4508 −0.415384
\(634\) −17.8885 −0.710445
\(635\) 14.1803 0.562730
\(636\) −4.00000 −0.158610
\(637\) 5.23607 0.207461
\(638\) 0 0
\(639\) −6.47214 −0.256034
\(640\) −1.00000 −0.0395285
\(641\) 40.3262 1.59279 0.796395 0.604776i \(-0.206738\pi\)
0.796395 + 0.604776i \(0.206738\pi\)
\(642\) 1.90983 0.0753750
\(643\) −13.5623 −0.534845 −0.267423 0.963579i \(-0.586172\pi\)
−0.267423 + 0.963579i \(0.586172\pi\)
\(644\) 8.47214 0.333849
\(645\) −2.85410 −0.112380
\(646\) 0.618034 0.0243162
\(647\) 27.8885 1.09641 0.548206 0.836343i \(-0.315311\pi\)
0.548206 + 0.836343i \(0.315311\pi\)
\(648\) 5.70820 0.224239
\(649\) 0 0
\(650\) 5.23607 0.205375
\(651\) 2.76393 0.108327
\(652\) −20.3820 −0.798219
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 7.52786 0.294363
\(655\) 14.8541 0.580398
\(656\) −8.61803 −0.336478
\(657\) 11.3262 0.441879
\(658\) 6.76393 0.263686
\(659\) −9.90983 −0.386032 −0.193016 0.981196i \(-0.561827\pi\)
−0.193016 + 0.981196i \(0.561827\pi\)
\(660\) 0 0
\(661\) −33.7771 −1.31378 −0.656888 0.753988i \(-0.728128\pi\)
−0.656888 + 0.753988i \(0.728128\pi\)
\(662\) 12.3262 0.479073
\(663\) 5.23607 0.203352
\(664\) −11.0902 −0.430382
\(665\) 0.381966 0.0148120
\(666\) −25.4164 −0.984866
\(667\) 33.8885 1.31217
\(668\) −10.0000 −0.386912
\(669\) −3.12461 −0.120804
\(670\) 6.09017 0.235284
\(671\) 0 0
\(672\) −0.618034 −0.0238412
\(673\) −7.14590 −0.275454 −0.137727 0.990470i \(-0.543980\pi\)
−0.137727 + 0.990470i \(0.543980\pi\)
\(674\) −32.2705 −1.24301
\(675\) −3.47214 −0.133643
\(676\) 14.4164 0.554477
\(677\) −6.94427 −0.266890 −0.133445 0.991056i \(-0.542604\pi\)
−0.133445 + 0.991056i \(0.542604\pi\)
\(678\) 8.52786 0.327511
\(679\) 10.7984 0.414404
\(680\) −1.61803 −0.0620488
\(681\) −5.29180 −0.202782
\(682\) 0 0
\(683\) 17.3050 0.662156 0.331078 0.943603i \(-0.392588\pi\)
0.331078 + 0.943603i \(0.392588\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 6.14590 0.234823
\(686\) −1.00000 −0.0381802
\(687\) −3.52786 −0.134596
\(688\) 4.61803 0.176061
\(689\) −33.8885 −1.29105
\(690\) 5.23607 0.199334
\(691\) −27.7984 −1.05750 −0.528750 0.848778i \(-0.677339\pi\)
−0.528750 + 0.848778i \(0.677339\pi\)
\(692\) −9.23607 −0.351103
\(693\) 0 0
\(694\) 27.0344 1.02621
\(695\) −10.4721 −0.397231
\(696\) −2.47214 −0.0937061
\(697\) −13.9443 −0.528177
\(698\) −10.4721 −0.396376
\(699\) −2.27051 −0.0858786
\(700\) −1.00000 −0.0377964
\(701\) −18.1803 −0.686662 −0.343331 0.939214i \(-0.611555\pi\)
−0.343331 + 0.939214i \(0.611555\pi\)
\(702\) −18.1803 −0.686173
\(703\) 3.70820 0.139858
\(704\) 0 0
\(705\) 4.18034 0.157441
\(706\) 17.0344 0.641100
\(707\) 17.2361 0.648229
\(708\) −2.23607 −0.0840366
\(709\) 37.2361 1.39843 0.699215 0.714912i \(-0.253533\pi\)
0.699215 + 0.714912i \(0.253533\pi\)
\(710\) −2.47214 −0.0927776
\(711\) 4.47214 0.167718
\(712\) 9.85410 0.369298
\(713\) 37.8885 1.41894
\(714\) −1.00000 −0.0374241
\(715\) 0 0
\(716\) −14.8541 −0.555124
\(717\) −10.9443 −0.408721
\(718\) −25.7082 −0.959422
\(719\) 50.3607 1.87814 0.939068 0.343731i \(-0.111691\pi\)
0.939068 + 0.343731i \(0.111691\pi\)
\(720\) 2.61803 0.0975684
\(721\) −0.291796 −0.0108671
\(722\) −18.8541 −0.701677
\(723\) 4.41641 0.164248
\(724\) 17.1246 0.636431
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) 37.3050 1.38356 0.691782 0.722106i \(-0.256826\pi\)
0.691782 + 0.722106i \(0.256826\pi\)
\(728\) −5.23607 −0.194062
\(729\) −8.50658 −0.315058
\(730\) 4.32624 0.160121
\(731\) 7.47214 0.276367
\(732\) 4.94427 0.182746
\(733\) 14.7639 0.545318 0.272659 0.962111i \(-0.412097\pi\)
0.272659 + 0.962111i \(0.412097\pi\)
\(734\) 11.0557 0.408074
\(735\) −0.618034 −0.0227965
\(736\) −8.47214 −0.312287
\(737\) 0 0
\(738\) 22.5623 0.830530
\(739\) −19.9787 −0.734929 −0.367464 0.930038i \(-0.619774\pi\)
−0.367464 + 0.930038i \(0.619774\pi\)
\(740\) −9.70820 −0.356881
\(741\) 1.23607 0.0454081
\(742\) 6.47214 0.237600
\(743\) 22.0000 0.807102 0.403551 0.914957i \(-0.367776\pi\)
0.403551 + 0.914957i \(0.367776\pi\)
\(744\) −2.76393 −0.101331
\(745\) 5.70820 0.209132
\(746\) 4.94427 0.181023
\(747\) 29.0344 1.06231
\(748\) 0 0
\(749\) −3.09017 −0.112912
\(750\) −0.618034 −0.0225674
\(751\) −22.5836 −0.824087 −0.412043 0.911164i \(-0.635185\pi\)
−0.412043 + 0.911164i \(0.635185\pi\)
\(752\) −6.76393 −0.246655
\(753\) −11.0557 −0.402893
\(754\) −20.9443 −0.762745
\(755\) −2.18034 −0.0793507
\(756\) 3.47214 0.126280
\(757\) −32.0689 −1.16556 −0.582782 0.812629i \(-0.698036\pi\)
−0.582782 + 0.812629i \(0.698036\pi\)
\(758\) −10.1459 −0.368516
\(759\) 0 0
\(760\) −0.381966 −0.0138554
\(761\) −40.2705 −1.45980 −0.729902 0.683551i \(-0.760435\pi\)
−0.729902 + 0.683551i \(0.760435\pi\)
\(762\) −8.76393 −0.317484
\(763\) −12.1803 −0.440958
\(764\) −2.00000 −0.0723575
\(765\) 4.23607 0.153155
\(766\) 17.7082 0.639823
\(767\) −18.9443 −0.684038
\(768\) 0.618034 0.0223014
\(769\) 1.41641 0.0510770 0.0255385 0.999674i \(-0.491870\pi\)
0.0255385 + 0.999674i \(0.491870\pi\)
\(770\) 0 0
\(771\) −15.7984 −0.568965
\(772\) 12.4721 0.448882
\(773\) −8.36068 −0.300713 −0.150356 0.988632i \(-0.548042\pi\)
−0.150356 + 0.988632i \(0.548042\pi\)
\(774\) −12.0902 −0.434572
\(775\) −4.47214 −0.160644
\(776\) −10.7984 −0.387639
\(777\) −6.00000 −0.215249
\(778\) 8.00000 0.286814
\(779\) −3.29180 −0.117941
\(780\) −3.23607 −0.115870
\(781\) 0 0
\(782\) −13.7082 −0.490204
\(783\) 13.8885 0.496336
\(784\) 1.00000 0.0357143
\(785\) −16.1803 −0.577501
\(786\) −9.18034 −0.327452
\(787\) 15.0344 0.535920 0.267960 0.963430i \(-0.413650\pi\)
0.267960 + 0.963430i \(0.413650\pi\)
\(788\) −21.4164 −0.762928
\(789\) 7.41641 0.264031
\(790\) 1.70820 0.0607752
\(791\) −13.7984 −0.490614
\(792\) 0 0
\(793\) 41.8885 1.48751
\(794\) −34.5410 −1.22581
\(795\) 4.00000 0.141865
\(796\) −1.23607 −0.0438113
\(797\) 25.5279 0.904243 0.452122 0.891956i \(-0.350667\pi\)
0.452122 + 0.891956i \(0.350667\pi\)
\(798\) −0.236068 −0.00835672
\(799\) −10.9443 −0.387181
\(800\) 1.00000 0.0353553
\(801\) −25.7984 −0.911541
\(802\) −10.5066 −0.371000
\(803\) 0 0
\(804\) −3.76393 −0.132744
\(805\) −8.47214 −0.298604
\(806\) −23.4164 −0.824808
\(807\) 8.65248 0.304582
\(808\) −17.2361 −0.606363
\(809\) −46.5623 −1.63704 −0.818522 0.574476i \(-0.805206\pi\)
−0.818522 + 0.574476i \(0.805206\pi\)
\(810\) −5.70820 −0.200566
\(811\) −25.2016 −0.884949 −0.442474 0.896781i \(-0.645899\pi\)
−0.442474 + 0.896781i \(0.645899\pi\)
\(812\) 4.00000 0.140372
\(813\) −12.1803 −0.427183
\(814\) 0 0
\(815\) 20.3820 0.713949
\(816\) 1.00000 0.0350070
\(817\) 1.76393 0.0617122
\(818\) 31.3050 1.09455
\(819\) 13.7082 0.479003
\(820\) 8.61803 0.300955
\(821\) 35.4853 1.23845 0.619223 0.785215i \(-0.287448\pi\)
0.619223 + 0.785215i \(0.287448\pi\)
\(822\) −3.79837 −0.132484
\(823\) −33.4853 −1.16722 −0.583612 0.812033i \(-0.698361\pi\)
−0.583612 + 0.812033i \(0.698361\pi\)
\(824\) 0.291796 0.0101652
\(825\) 0 0
\(826\) 3.61803 0.125888
\(827\) 16.7426 0.582199 0.291099 0.956693i \(-0.405979\pi\)
0.291099 + 0.956693i \(0.405979\pi\)
\(828\) 22.1803 0.770820
\(829\) −14.9443 −0.519036 −0.259518 0.965738i \(-0.583564\pi\)
−0.259518 + 0.965738i \(0.583564\pi\)
\(830\) 11.0902 0.384945
\(831\) −14.2918 −0.495777
\(832\) 5.23607 0.181528
\(833\) 1.61803 0.0560616
\(834\) 6.47214 0.224112
\(835\) 10.0000 0.346064
\(836\) 0 0
\(837\) 15.5279 0.536721
\(838\) 27.7426 0.958354
\(839\) 20.6525 0.713003 0.356501 0.934295i \(-0.383970\pi\)
0.356501 + 0.934295i \(0.383970\pi\)
\(840\) 0.618034 0.0213242
\(841\) −13.0000 −0.448276
\(842\) −0.472136 −0.0162709
\(843\) 4.20163 0.144712
\(844\) −16.9098 −0.582060
\(845\) −14.4164 −0.495940
\(846\) 17.7082 0.608821
\(847\) 0 0
\(848\) −6.47214 −0.222254
\(849\) 16.0000 0.549119
\(850\) 1.61803 0.0554981
\(851\) −82.2492 −2.81947
\(852\) 1.52786 0.0523438
\(853\) 49.7082 1.70198 0.850988 0.525185i \(-0.176004\pi\)
0.850988 + 0.525185i \(0.176004\pi\)
\(854\) −8.00000 −0.273754
\(855\) 1.00000 0.0341993
\(856\) 3.09017 0.105620
\(857\) 12.6869 0.433377 0.216688 0.976241i \(-0.430474\pi\)
0.216688 + 0.976241i \(0.430474\pi\)
\(858\) 0 0
\(859\) 37.1459 1.26740 0.633701 0.773578i \(-0.281535\pi\)
0.633701 + 0.773578i \(0.281535\pi\)
\(860\) −4.61803 −0.157474
\(861\) 5.32624 0.181518
\(862\) −6.29180 −0.214299
\(863\) −40.2492 −1.37010 −0.685050 0.728496i \(-0.740220\pi\)
−0.685050 + 0.728496i \(0.740220\pi\)
\(864\) −3.47214 −0.118124
\(865\) 9.23607 0.314036
\(866\) 25.7984 0.876664
\(867\) −8.88854 −0.301871
\(868\) 4.47214 0.151794
\(869\) 0 0
\(870\) 2.47214 0.0838133
\(871\) −31.8885 −1.08050
\(872\) 12.1803 0.412478
\(873\) 28.2705 0.956812
\(874\) −3.23607 −0.109462
\(875\) 1.00000 0.0338062
\(876\) −2.67376 −0.0903380
\(877\) −19.1246 −0.645792 −0.322896 0.946434i \(-0.604656\pi\)
−0.322896 + 0.946434i \(0.604656\pi\)
\(878\) 20.3607 0.687140
\(879\) −1.41641 −0.0477743
\(880\) 0 0
\(881\) −57.3394 −1.93181 −0.965907 0.258891i \(-0.916643\pi\)
−0.965907 + 0.258891i \(0.916643\pi\)
\(882\) −2.61803 −0.0881538
\(883\) 36.6180 1.23229 0.616147 0.787631i \(-0.288693\pi\)
0.616147 + 0.787631i \(0.288693\pi\)
\(884\) 8.47214 0.284949
\(885\) 2.23607 0.0751646
\(886\) −28.6180 −0.961442
\(887\) 22.0000 0.738688 0.369344 0.929293i \(-0.379582\pi\)
0.369344 + 0.929293i \(0.379582\pi\)
\(888\) 6.00000 0.201347
\(889\) 14.1803 0.475593
\(890\) −9.85410 −0.330310
\(891\) 0 0
\(892\) −5.05573 −0.169278
\(893\) −2.58359 −0.0864566
\(894\) −3.52786 −0.117989
\(895\) 14.8541 0.496518
\(896\) −1.00000 −0.0334077
\(897\) −27.4164 −0.915407
\(898\) −14.7426 −0.491968
\(899\) 17.8885 0.596616
\(900\) −2.61803 −0.0872678
\(901\) −10.4721 −0.348877
\(902\) 0 0
\(903\) −2.85410 −0.0949786
\(904\) 13.7984 0.458927
\(905\) −17.1246 −0.569241
\(906\) 1.34752 0.0447685
\(907\) −3.79837 −0.126123 −0.0630615 0.998010i \(-0.520086\pi\)
−0.0630615 + 0.998010i \(0.520086\pi\)
\(908\) −8.56231 −0.284150
\(909\) 45.1246 1.49669
\(910\) 5.23607 0.173574
\(911\) 32.0689 1.06249 0.531245 0.847218i \(-0.321724\pi\)
0.531245 + 0.847218i \(0.321724\pi\)
\(912\) 0.236068 0.00781699
\(913\) 0 0
\(914\) 25.5623 0.845526
\(915\) −4.94427 −0.163453
\(916\) −5.70820 −0.188604
\(917\) 14.8541 0.490526
\(918\) −5.61803 −0.185423
\(919\) 16.7639 0.552991 0.276496 0.961015i \(-0.410827\pi\)
0.276496 + 0.961015i \(0.410827\pi\)
\(920\) 8.47214 0.279318
\(921\) −21.3262 −0.702723
\(922\) 1.12461 0.0370371
\(923\) 12.9443 0.426066
\(924\) 0 0
\(925\) 9.70820 0.319204
\(926\) 0.652476 0.0214417
\(927\) −0.763932 −0.0250908
\(928\) −4.00000 −0.131306
\(929\) 35.3262 1.15902 0.579508 0.814966i \(-0.303245\pi\)
0.579508 + 0.814966i \(0.303245\pi\)
\(930\) 2.76393 0.0906329
\(931\) 0.381966 0.0125184
\(932\) −3.67376 −0.120338
\(933\) 2.00000 0.0654771
\(934\) 16.9443 0.554434
\(935\) 0 0
\(936\) −13.7082 −0.448067
\(937\) −45.3951 −1.48299 −0.741497 0.670956i \(-0.765884\pi\)
−0.741497 + 0.670956i \(0.765884\pi\)
\(938\) 6.09017 0.198851
\(939\) 2.05573 0.0670862
\(940\) 6.76393 0.220615
\(941\) 17.7771 0.579516 0.289758 0.957100i \(-0.406425\pi\)
0.289758 + 0.957100i \(0.406425\pi\)
\(942\) 10.0000 0.325818
\(943\) 73.0132 2.37764
\(944\) −3.61803 −0.117757
\(945\) −3.47214 −0.112949
\(946\) 0 0
\(947\) −40.3394 −1.31085 −0.655427 0.755258i \(-0.727511\pi\)
−0.655427 + 0.755258i \(0.727511\pi\)
\(948\) −1.05573 −0.0342885
\(949\) −22.6525 −0.735330
\(950\) 0.381966 0.0123926
\(951\) −11.0557 −0.358507
\(952\) −1.61803 −0.0524408
\(953\) −33.8541 −1.09664 −0.548321 0.836268i \(-0.684733\pi\)
−0.548321 + 0.836268i \(0.684733\pi\)
\(954\) 16.9443 0.548591
\(955\) 2.00000 0.0647185
\(956\) −17.7082 −0.572724
\(957\) 0 0
\(958\) 17.1246 0.553271
\(959\) 6.14590 0.198461
\(960\) −0.618034 −0.0199470
\(961\) −11.0000 −0.354839
\(962\) 50.8328 1.63892
\(963\) −8.09017 −0.260702
\(964\) 7.14590 0.230154
\(965\) −12.4721 −0.401492
\(966\) 5.23607 0.168468
\(967\) 26.0000 0.836104 0.418052 0.908423i \(-0.362713\pi\)
0.418052 + 0.908423i \(0.362713\pi\)
\(968\) 0 0
\(969\) 0.381966 0.0122705
\(970\) 10.7984 0.346715
\(971\) 41.3050 1.32554 0.662769 0.748823i \(-0.269381\pi\)
0.662769 + 0.748823i \(0.269381\pi\)
\(972\) 13.9443 0.447263
\(973\) −10.4721 −0.335721
\(974\) 37.7771 1.21046
\(975\) 3.23607 0.103637
\(976\) 8.00000 0.256074
\(977\) 40.4721 1.29482 0.647409 0.762143i \(-0.275853\pi\)
0.647409 + 0.762143i \(0.275853\pi\)
\(978\) −12.5967 −0.402800
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −31.8885 −1.01812
\(982\) 37.9230 1.21017
\(983\) 26.7639 0.853637 0.426818 0.904337i \(-0.359634\pi\)
0.426818 + 0.904337i \(0.359634\pi\)
\(984\) −5.32624 −0.169794
\(985\) 21.4164 0.682383
\(986\) −6.47214 −0.206115
\(987\) 4.18034 0.133062
\(988\) 2.00000 0.0636285
\(989\) −39.1246 −1.24409
\(990\) 0 0
\(991\) −56.7639 −1.80317 −0.901583 0.432606i \(-0.857594\pi\)
−0.901583 + 0.432606i \(0.857594\pi\)
\(992\) −4.47214 −0.141990
\(993\) 7.61803 0.241751
\(994\) −2.47214 −0.0784114
\(995\) 1.23607 0.0391860
\(996\) −6.85410 −0.217181
\(997\) −53.1246 −1.68247 −0.841237 0.540667i \(-0.818172\pi\)
−0.841237 + 0.540667i \(0.818172\pi\)
\(998\) 0.618034 0.0195635
\(999\) −33.7082 −1.06648
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.bx.1.2 2
11.7 odd 10 770.2.n.c.71.1 4
11.8 odd 10 770.2.n.c.141.1 yes 4
11.10 odd 2 8470.2.a.bk.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.c.71.1 4 11.7 odd 10
770.2.n.c.141.1 yes 4 11.8 odd 10
8470.2.a.bk.1.2 2 11.10 odd 2
8470.2.a.bx.1.2 2 1.1 even 1 trivial