Properties

Label 8470.2.a.bp.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
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 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.61803 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.61803 q^{9} +1.00000 q^{10} -0.618034 q^{12} -0.763932 q^{13} -1.00000 q^{14} +0.618034 q^{15} +1.00000 q^{16} +0.381966 q^{17} +2.61803 q^{18} +1.14590 q^{19} -1.00000 q^{20} -0.618034 q^{21} +8.47214 q^{23} +0.618034 q^{24} +1.00000 q^{25} +0.763932 q^{26} +3.47214 q^{27} +1.00000 q^{28} -4.47214 q^{29} -0.618034 q^{30} -6.47214 q^{31} -1.00000 q^{32} -0.381966 q^{34} -1.00000 q^{35} -2.61803 q^{36} -6.76393 q^{37} -1.14590 q^{38} +0.472136 q^{39} +1.00000 q^{40} -3.85410 q^{41} +0.618034 q^{42} +9.09017 q^{43} +2.61803 q^{45} -8.47214 q^{46} +9.23607 q^{47} -0.618034 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.236068 q^{51} -0.763932 q^{52} -7.23607 q^{53} -3.47214 q^{54} -1.00000 q^{56} -0.708204 q^{57} +4.47214 q^{58} +10.8541 q^{59} +0.618034 q^{60} +0.291796 q^{61} +6.47214 q^{62} -2.61803 q^{63} +1.00000 q^{64} +0.763932 q^{65} -5.61803 q^{67} +0.381966 q^{68} -5.23607 q^{69} +1.00000 q^{70} -6.94427 q^{71} +2.61803 q^{72} -3.85410 q^{73} +6.76393 q^{74} -0.618034 q^{75} +1.14590 q^{76} -0.472136 q^{78} +4.00000 q^{79} -1.00000 q^{80} +5.70820 q^{81} +3.85410 q^{82} -2.14590 q^{83} -0.618034 q^{84} -0.381966 q^{85} -9.09017 q^{86} +2.76393 q^{87} -8.14590 q^{89} -2.61803 q^{90} -0.763932 q^{91} +8.47214 q^{92} +4.00000 q^{93} -9.23607 q^{94} -1.14590 q^{95} +0.618034 q^{96} -10.6180 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + q^{3} + 2q^{4} - 2q^{5} - q^{6} + 2q^{7} - 2q^{8} - 3q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + q^{3} + 2q^{4} - 2q^{5} - q^{6} + 2q^{7} - 2q^{8} - 3q^{9} + 2q^{10} + q^{12} - 6q^{13} - 2q^{14} - q^{15} + 2q^{16} + 3q^{17} + 3q^{18} + 9q^{19} - 2q^{20} + q^{21} + 8q^{23} - q^{24} + 2q^{25} + 6q^{26} - 2q^{27} + 2q^{28} + q^{30} - 4q^{31} - 2q^{32} - 3q^{34} - 2q^{35} - 3q^{36} - 18q^{37} - 9q^{38} - 8q^{39} + 2q^{40} - q^{41} - q^{42} + 7q^{43} + 3q^{45} - 8q^{46} + 14q^{47} + q^{48} + 2q^{49} - 2q^{50} + 4q^{51} - 6q^{52} - 10q^{53} + 2q^{54} - 2q^{56} + 12q^{57} + 15q^{59} - q^{60} + 14q^{61} + 4q^{62} - 3q^{63} + 2q^{64} + 6q^{65} - 9q^{67} + 3q^{68} - 6q^{69} + 2q^{70} + 4q^{71} + 3q^{72} - q^{73} + 18q^{74} + q^{75} + 9q^{76} + 8q^{78} + 8q^{79} - 2q^{80} - 2q^{81} + q^{82} - 11q^{83} + q^{84} - 3q^{85} - 7q^{86} + 10q^{87} - 23q^{89} - 3q^{90} - 6q^{91} + 8q^{92} + 8q^{93} - 14q^{94} - 9q^{95} - q^{96} - 19q^{97} - 2q^{98} + O(q^{100}) \)

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 \(-0.557096\pi\)
−0.178411 + 0.983956i \(0.557096\pi\)
\(4\) 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
\(13\) −0.763932 −0.211877 −0.105938 0.994373i \(-0.533785\pi\)
−0.105938 + 0.994373i \(0.533785\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0.618034 0.159576
\(16\) 1.00000 0.250000
\(17\) 0.381966 0.0926404 0.0463202 0.998927i \(-0.485251\pi\)
0.0463202 + 0.998927i \(0.485251\pi\)
\(18\) 2.61803 0.617077
\(19\) 1.14590 0.262887 0.131444 0.991324i \(-0.458039\pi\)
0.131444 + 0.991324i \(0.458039\pi\)
\(20\) −1.00000 −0.223607
\(21\) −0.618034 −0.134866
\(22\) 0 0
\(23\) 8.47214 1.76656 0.883281 0.468844i \(-0.155329\pi\)
0.883281 + 0.468844i \(0.155329\pi\)
\(24\) 0.618034 0.126156
\(25\) 1.00000 0.200000
\(26\) 0.763932 0.149819
\(27\) 3.47214 0.668213
\(28\) 1.00000 0.188982
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) −0.618034 −0.112837
\(31\) −6.47214 −1.16243 −0.581215 0.813750i \(-0.697422\pi\)
−0.581215 + 0.813750i \(0.697422\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −0.381966 −0.0655066
\(35\) −1.00000 −0.169031
\(36\) −2.61803 −0.436339
\(37\) −6.76393 −1.11198 −0.555992 0.831188i \(-0.687661\pi\)
−0.555992 + 0.831188i \(0.687661\pi\)
\(38\) −1.14590 −0.185889
\(39\) 0.472136 0.0756023
\(40\) 1.00000 0.158114
\(41\) −3.85410 −0.601910 −0.300955 0.953638i \(-0.597305\pi\)
−0.300955 + 0.953638i \(0.597305\pi\)
\(42\) 0.618034 0.0953647
\(43\) 9.09017 1.38624 0.693119 0.720823i \(-0.256236\pi\)
0.693119 + 0.720823i \(0.256236\pi\)
\(44\) 0 0
\(45\) 2.61803 0.390273
\(46\) −8.47214 −1.24915
\(47\) 9.23607 1.34722 0.673609 0.739087i \(-0.264743\pi\)
0.673609 + 0.739087i \(0.264743\pi\)
\(48\) −0.618034 −0.0892055
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −0.236068 −0.0330561
\(52\) −0.763932 −0.105938
\(53\) −7.23607 −0.993950 −0.496975 0.867765i \(-0.665556\pi\)
−0.496975 + 0.867765i \(0.665556\pi\)
\(54\) −3.47214 −0.472498
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −0.708204 −0.0938039
\(58\) 4.47214 0.587220
\(59\) 10.8541 1.41308 0.706542 0.707671i \(-0.250254\pi\)
0.706542 + 0.707671i \(0.250254\pi\)
\(60\) 0.618034 0.0797878
\(61\) 0.291796 0.0373607 0.0186803 0.999826i \(-0.494054\pi\)
0.0186803 + 0.999826i \(0.494054\pi\)
\(62\) 6.47214 0.821962
\(63\) −2.61803 −0.329841
\(64\) 1.00000 0.125000
\(65\) 0.763932 0.0947541
\(66\) 0 0
\(67\) −5.61803 −0.686352 −0.343176 0.939271i \(-0.611503\pi\)
−0.343176 + 0.939271i \(0.611503\pi\)
\(68\) 0.381966 0.0463202
\(69\) −5.23607 −0.630349
\(70\) 1.00000 0.119523
\(71\) −6.94427 −0.824133 −0.412067 0.911154i \(-0.635193\pi\)
−0.412067 + 0.911154i \(0.635193\pi\)
\(72\) 2.61803 0.308538
\(73\) −3.85410 −0.451089 −0.225544 0.974233i \(-0.572416\pi\)
−0.225544 + 0.974233i \(0.572416\pi\)
\(74\) 6.76393 0.786291
\(75\) −0.618034 −0.0713644
\(76\) 1.14590 0.131444
\(77\) 0 0
\(78\) −0.472136 −0.0534589
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.00000 −0.111803
\(81\) 5.70820 0.634245
\(82\) 3.85410 0.425614
\(83\) −2.14590 −0.235543 −0.117771 0.993041i \(-0.537575\pi\)
−0.117771 + 0.993041i \(0.537575\pi\)
\(84\) −0.618034 −0.0674330
\(85\) −0.381966 −0.0414300
\(86\) −9.09017 −0.980218
\(87\) 2.76393 0.296325
\(88\) 0 0
\(89\) −8.14590 −0.863463 −0.431732 0.902002i \(-0.642097\pi\)
−0.431732 + 0.902002i \(0.642097\pi\)
\(90\) −2.61803 −0.275965
\(91\) −0.763932 −0.0800818
\(92\) 8.47214 0.883281
\(93\) 4.00000 0.414781
\(94\) −9.23607 −0.952628
\(95\) −1.14590 −0.117567
\(96\) 0.618034 0.0630778
\(97\) −10.6180 −1.07810 −0.539049 0.842274i \(-0.681216\pi\)
−0.539049 + 0.842274i \(0.681216\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 2.94427 0.292966 0.146483 0.989213i \(-0.453205\pi\)
0.146483 + 0.989213i \(0.453205\pi\)
\(102\) 0.236068 0.0233742
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0.763932 0.0749097
\(105\) 0.618034 0.0603139
\(106\) 7.23607 0.702829
\(107\) −3.85410 −0.372590 −0.186295 0.982494i \(-0.559648\pi\)
−0.186295 + 0.982494i \(0.559648\pi\)
\(108\) 3.47214 0.334106
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 4.18034 0.396780
\(112\) 1.00000 0.0944911
\(113\) −11.3262 −1.06548 −0.532741 0.846278i \(-0.678838\pi\)
−0.532741 + 0.846278i \(0.678838\pi\)
\(114\) 0.708204 0.0663294
\(115\) −8.47214 −0.790031
\(116\) −4.47214 −0.415227
\(117\) 2.00000 0.184900
\(118\) −10.8541 −0.999201
\(119\) 0.381966 0.0350148
\(120\) −0.618034 −0.0564185
\(121\) 0 0
\(122\) −0.291796 −0.0264180
\(123\) 2.38197 0.214775
\(124\) −6.47214 −0.581215
\(125\) −1.00000 −0.0894427
\(126\) 2.61803 0.233233
\(127\) 3.52786 0.313047 0.156524 0.987674i \(-0.449971\pi\)
0.156524 + 0.987674i \(0.449971\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.61803 −0.494640
\(130\) −0.763932 −0.0670013
\(131\) −6.09017 −0.532101 −0.266050 0.963959i \(-0.585719\pi\)
−0.266050 + 0.963959i \(0.585719\pi\)
\(132\) 0 0
\(133\) 1.14590 0.0993620
\(134\) 5.61803 0.485324
\(135\) −3.47214 −0.298834
\(136\) −0.381966 −0.0327533
\(137\) −1.85410 −0.158407 −0.0792033 0.996858i \(-0.525238\pi\)
−0.0792033 + 0.996858i \(0.525238\pi\)
\(138\) 5.23607 0.445724
\(139\) 12.9443 1.09792 0.548959 0.835849i \(-0.315024\pi\)
0.548959 + 0.835849i \(0.315024\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −5.70820 −0.480717
\(142\) 6.94427 0.582750
\(143\) 0 0
\(144\) −2.61803 −0.218169
\(145\) 4.47214 0.371391
\(146\) 3.85410 0.318968
\(147\) −0.618034 −0.0509746
\(148\) −6.76393 −0.555992
\(149\) −1.70820 −0.139942 −0.0699708 0.997549i \(-0.522291\pi\)
−0.0699708 + 0.997549i \(0.522291\pi\)
\(150\) 0.618034 0.0504623
\(151\) 9.52786 0.775367 0.387683 0.921793i \(-0.373275\pi\)
0.387683 + 0.921793i \(0.373275\pi\)
\(152\) −1.14590 −0.0929446
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) 6.47214 0.519854
\(156\) 0.472136 0.0378011
\(157\) 2.94427 0.234978 0.117489 0.993074i \(-0.462515\pi\)
0.117489 + 0.993074i \(0.462515\pi\)
\(158\) −4.00000 −0.318223
\(159\) 4.47214 0.354663
\(160\) 1.00000 0.0790569
\(161\) 8.47214 0.667698
\(162\) −5.70820 −0.448479
\(163\) −1.79837 −0.140860 −0.0704298 0.997517i \(-0.522437\pi\)
−0.0704298 + 0.997517i \(0.522437\pi\)
\(164\) −3.85410 −0.300955
\(165\) 0 0
\(166\) 2.14590 0.166554
\(167\) 21.7082 1.67983 0.839916 0.542717i \(-0.182604\pi\)
0.839916 + 0.542717i \(0.182604\pi\)
\(168\) 0.618034 0.0476824
\(169\) −12.4164 −0.955108
\(170\) 0.381966 0.0292955
\(171\) −3.00000 −0.229416
\(172\) 9.09017 0.693119
\(173\) −15.8885 −1.20798 −0.603992 0.796991i \(-0.706424\pi\)
−0.603992 + 0.796991i \(0.706424\pi\)
\(174\) −2.76393 −0.209533
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −6.70820 −0.504219
\(178\) 8.14590 0.610561
\(179\) 8.85410 0.661787 0.330893 0.943668i \(-0.392650\pi\)
0.330893 + 0.943668i \(0.392650\pi\)
\(180\) 2.61803 0.195137
\(181\) 3.05573 0.227130 0.113565 0.993531i \(-0.463773\pi\)
0.113565 + 0.993531i \(0.463773\pi\)
\(182\) 0.763932 0.0566264
\(183\) −0.180340 −0.0133311
\(184\) −8.47214 −0.624574
\(185\) 6.76393 0.497294
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) 9.23607 0.673609
\(189\) 3.47214 0.252561
\(190\) 1.14590 0.0831322
\(191\) 3.23607 0.234154 0.117077 0.993123i \(-0.462648\pi\)
0.117077 + 0.993123i \(0.462648\pi\)
\(192\) −0.618034 −0.0446028
\(193\) 5.41641 0.389882 0.194941 0.980815i \(-0.437549\pi\)
0.194941 + 0.980815i \(0.437549\pi\)
\(194\) 10.6180 0.762330
\(195\) −0.472136 −0.0338104
\(196\) 1.00000 0.0714286
\(197\) 20.4721 1.45858 0.729290 0.684205i \(-0.239851\pi\)
0.729290 + 0.684205i \(0.239851\pi\)
\(198\) 0 0
\(199\) 22.9443 1.62648 0.813238 0.581931i \(-0.197703\pi\)
0.813238 + 0.581931i \(0.197703\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 3.47214 0.244906
\(202\) −2.94427 −0.207158
\(203\) −4.47214 −0.313882
\(204\) −0.236068 −0.0165281
\(205\) 3.85410 0.269182
\(206\) 6.00000 0.418040
\(207\) −22.1803 −1.54164
\(208\) −0.763932 −0.0529692
\(209\) 0 0
\(210\) −0.618034 −0.0426484
\(211\) 26.3262 1.81237 0.906186 0.422878i \(-0.138980\pi\)
0.906186 + 0.422878i \(0.138980\pi\)
\(212\) −7.23607 −0.496975
\(213\) 4.29180 0.294069
\(214\) 3.85410 0.263461
\(215\) −9.09017 −0.619944
\(216\) −3.47214 −0.236249
\(217\) −6.47214 −0.439357
\(218\) −4.00000 −0.270914
\(219\) 2.38197 0.160958
\(220\) 0 0
\(221\) −0.291796 −0.0196283
\(222\) −4.18034 −0.280566
\(223\) −27.1246 −1.81640 −0.908199 0.418538i \(-0.862543\pi\)
−0.908199 + 0.418538i \(0.862543\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.61803 −0.174536
\(226\) 11.3262 0.753410
\(227\) 29.3262 1.94645 0.973225 0.229853i \(-0.0738245\pi\)
0.973225 + 0.229853i \(0.0738245\pi\)
\(228\) −0.708204 −0.0469020
\(229\) −22.3607 −1.47764 −0.738818 0.673905i \(-0.764616\pi\)
−0.738818 + 0.673905i \(0.764616\pi\)
\(230\) 8.47214 0.558636
\(231\) 0 0
\(232\) 4.47214 0.293610
\(233\) 16.3262 1.06957 0.534784 0.844989i \(-0.320393\pi\)
0.534784 + 0.844989i \(0.320393\pi\)
\(234\) −2.00000 −0.130744
\(235\) −9.23607 −0.602495
\(236\) 10.8541 0.706542
\(237\) −2.47214 −0.160582
\(238\) −0.381966 −0.0247592
\(239\) 19.7082 1.27482 0.637409 0.770526i \(-0.280006\pi\)
0.637409 + 0.770526i \(0.280006\pi\)
\(240\) 0.618034 0.0398939
\(241\) 15.9787 1.02928 0.514640 0.857407i \(-0.327926\pi\)
0.514640 + 0.857407i \(0.327926\pi\)
\(242\) 0 0
\(243\) −13.9443 −0.894525
\(244\) 0.291796 0.0186803
\(245\) −1.00000 −0.0638877
\(246\) −2.38197 −0.151869
\(247\) −0.875388 −0.0556996
\(248\) 6.47214 0.410981
\(249\) 1.32624 0.0840469
\(250\) 1.00000 0.0632456
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) −2.61803 −0.164921
\(253\) 0 0
\(254\) −3.52786 −0.221358
\(255\) 0.236068 0.0147832
\(256\) 1.00000 0.0625000
\(257\) 8.14590 0.508127 0.254064 0.967188i \(-0.418233\pi\)
0.254064 + 0.967188i \(0.418233\pi\)
\(258\) 5.61803 0.349764
\(259\) −6.76393 −0.420290
\(260\) 0.763932 0.0473771
\(261\) 11.7082 0.724720
\(262\) 6.09017 0.376252
\(263\) −18.3607 −1.13217 −0.566084 0.824348i \(-0.691542\pi\)
−0.566084 + 0.824348i \(0.691542\pi\)
\(264\) 0 0
\(265\) 7.23607 0.444508
\(266\) −1.14590 −0.0702595
\(267\) 5.03444 0.308103
\(268\) −5.61803 −0.343176
\(269\) 12.6525 0.771435 0.385718 0.922617i \(-0.373954\pi\)
0.385718 + 0.922617i \(0.373954\pi\)
\(270\) 3.47214 0.211307
\(271\) 16.1803 0.982886 0.491443 0.870910i \(-0.336470\pi\)
0.491443 + 0.870910i \(0.336470\pi\)
\(272\) 0.381966 0.0231601
\(273\) 0.472136 0.0285750
\(274\) 1.85410 0.112010
\(275\) 0 0
\(276\) −5.23607 −0.315174
\(277\) −12.0000 −0.721010 −0.360505 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) −12.9443 −0.776346
\(279\) 16.9443 1.01443
\(280\) 1.00000 0.0597614
\(281\) −4.32624 −0.258082 −0.129041 0.991639i \(-0.541190\pi\)
−0.129041 + 0.991639i \(0.541190\pi\)
\(282\) 5.70820 0.339919
\(283\) −28.3607 −1.68587 −0.842934 0.538017i \(-0.819173\pi\)
−0.842934 + 0.538017i \(0.819173\pi\)
\(284\) −6.94427 −0.412067
\(285\) 0.708204 0.0419504
\(286\) 0 0
\(287\) −3.85410 −0.227500
\(288\) 2.61803 0.154269
\(289\) −16.8541 −0.991418
\(290\) −4.47214 −0.262613
\(291\) 6.56231 0.384689
\(292\) −3.85410 −0.225544
\(293\) 9.52786 0.556624 0.278312 0.960491i \(-0.410225\pi\)
0.278312 + 0.960491i \(0.410225\pi\)
\(294\) 0.618034 0.0360445
\(295\) −10.8541 −0.631950
\(296\) 6.76393 0.393146
\(297\) 0 0
\(298\) 1.70820 0.0989536
\(299\) −6.47214 −0.374293
\(300\) −0.618034 −0.0356822
\(301\) 9.09017 0.523949
\(302\) −9.52786 −0.548267
\(303\) −1.81966 −0.104537
\(304\) 1.14590 0.0657218
\(305\) −0.291796 −0.0167082
\(306\) 1.00000 0.0571662
\(307\) −1.56231 −0.0891655 −0.0445828 0.999006i \(-0.514196\pi\)
−0.0445828 + 0.999006i \(0.514196\pi\)
\(308\) 0 0
\(309\) 3.70820 0.210952
\(310\) −6.47214 −0.367593
\(311\) 20.1803 1.14432 0.572161 0.820141i \(-0.306105\pi\)
0.572161 + 0.820141i \(0.306105\pi\)
\(312\) −0.472136 −0.0267294
\(313\) 2.27051 0.128337 0.0641684 0.997939i \(-0.479561\pi\)
0.0641684 + 0.997939i \(0.479561\pi\)
\(314\) −2.94427 −0.166155
\(315\) 2.61803 0.147510
\(316\) 4.00000 0.225018
\(317\) −14.3607 −0.806576 −0.403288 0.915073i \(-0.632133\pi\)
−0.403288 + 0.915073i \(0.632133\pi\)
\(318\) −4.47214 −0.250785
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 2.38197 0.132948
\(322\) −8.47214 −0.472134
\(323\) 0.437694 0.0243540
\(324\) 5.70820 0.317122
\(325\) −0.763932 −0.0423753
\(326\) 1.79837 0.0996027
\(327\) −2.47214 −0.136709
\(328\) 3.85410 0.212807
\(329\) 9.23607 0.509201
\(330\) 0 0
\(331\) −30.0344 −1.65084 −0.825421 0.564517i \(-0.809062\pi\)
−0.825421 + 0.564517i \(0.809062\pi\)
\(332\) −2.14590 −0.117771
\(333\) 17.7082 0.970404
\(334\) −21.7082 −1.18782
\(335\) 5.61803 0.306946
\(336\) −0.618034 −0.0337165
\(337\) −25.7984 −1.40533 −0.702663 0.711522i \(-0.748006\pi\)
−0.702663 + 0.711522i \(0.748006\pi\)
\(338\) 12.4164 0.675364
\(339\) 7.00000 0.380188
\(340\) −0.381966 −0.0207150
\(341\) 0 0
\(342\) 3.00000 0.162221
\(343\) 1.00000 0.0539949
\(344\) −9.09017 −0.490109
\(345\) 5.23607 0.281900
\(346\) 15.8885 0.854173
\(347\) 3.90983 0.209891 0.104945 0.994478i \(-0.466533\pi\)
0.104945 + 0.994478i \(0.466533\pi\)
\(348\) 2.76393 0.148162
\(349\) 7.41641 0.396991 0.198496 0.980102i \(-0.436394\pi\)
0.198496 + 0.980102i \(0.436394\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.65248 −0.141579
\(352\) 0 0
\(353\) 8.56231 0.455726 0.227863 0.973693i \(-0.426826\pi\)
0.227863 + 0.973693i \(0.426826\pi\)
\(354\) 6.70820 0.356537
\(355\) 6.94427 0.368564
\(356\) −8.14590 −0.431732
\(357\) −0.236068 −0.0124940
\(358\) −8.85410 −0.467954
\(359\) −19.5967 −1.03428 −0.517138 0.855902i \(-0.673003\pi\)
−0.517138 + 0.855902i \(0.673003\pi\)
\(360\) −2.61803 −0.137983
\(361\) −17.6869 −0.930890
\(362\) −3.05573 −0.160606
\(363\) 0 0
\(364\) −0.763932 −0.0400409
\(365\) 3.85410 0.201733
\(366\) 0.180340 0.00942652
\(367\) −19.8885 −1.03817 −0.519087 0.854722i \(-0.673728\pi\)
−0.519087 + 0.854722i \(0.673728\pi\)
\(368\) 8.47214 0.441641
\(369\) 10.0902 0.525273
\(370\) −6.76393 −0.351640
\(371\) −7.23607 −0.375678
\(372\) 4.00000 0.207390
\(373\) −16.6525 −0.862233 −0.431116 0.902296i \(-0.641880\pi\)
−0.431116 + 0.902296i \(0.641880\pi\)
\(374\) 0 0
\(375\) 0.618034 0.0319151
\(376\) −9.23607 −0.476314
\(377\) 3.41641 0.175954
\(378\) −3.47214 −0.178587
\(379\) 13.3820 0.687385 0.343693 0.939082i \(-0.388322\pi\)
0.343693 + 0.939082i \(0.388322\pi\)
\(380\) −1.14590 −0.0587833
\(381\) −2.18034 −0.111702
\(382\) −3.23607 −0.165572
\(383\) 2.18034 0.111410 0.0557051 0.998447i \(-0.482259\pi\)
0.0557051 + 0.998447i \(0.482259\pi\)
\(384\) 0.618034 0.0315389
\(385\) 0 0
\(386\) −5.41641 −0.275688
\(387\) −23.7984 −1.20974
\(388\) −10.6180 −0.539049
\(389\) 12.2918 0.623219 0.311609 0.950210i \(-0.399132\pi\)
0.311609 + 0.950210i \(0.399132\pi\)
\(390\) 0.472136 0.0239075
\(391\) 3.23607 0.163655
\(392\) −1.00000 −0.0505076
\(393\) 3.76393 0.189865
\(394\) −20.4721 −1.03137
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 35.8885 1.80119 0.900597 0.434655i \(-0.143130\pi\)
0.900597 + 0.434655i \(0.143130\pi\)
\(398\) −22.9443 −1.15009
\(399\) −0.708204 −0.0354545
\(400\) 1.00000 0.0500000
\(401\) −17.5623 −0.877020 −0.438510 0.898726i \(-0.644494\pi\)
−0.438510 + 0.898726i \(0.644494\pi\)
\(402\) −3.47214 −0.173174
\(403\) 4.94427 0.246292
\(404\) 2.94427 0.146483
\(405\) −5.70820 −0.283643
\(406\) 4.47214 0.221948
\(407\) 0 0
\(408\) 0.236068 0.0116871
\(409\) −5.05573 −0.249990 −0.124995 0.992157i \(-0.539891\pi\)
−0.124995 + 0.992157i \(0.539891\pi\)
\(410\) −3.85410 −0.190341
\(411\) 1.14590 0.0565230
\(412\) −6.00000 −0.295599
\(413\) 10.8541 0.534095
\(414\) 22.1803 1.09010
\(415\) 2.14590 0.105338
\(416\) 0.763932 0.0374548
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 2.43769 0.119089 0.0595446 0.998226i \(-0.481035\pi\)
0.0595446 + 0.998226i \(0.481035\pi\)
\(420\) 0.618034 0.0301570
\(421\) 16.8328 0.820381 0.410191 0.912000i \(-0.365462\pi\)
0.410191 + 0.912000i \(0.365462\pi\)
\(422\) −26.3262 −1.28154
\(423\) −24.1803 −1.17569
\(424\) 7.23607 0.351415
\(425\) 0.381966 0.0185281
\(426\) −4.29180 −0.207938
\(427\) 0.291796 0.0141210
\(428\) −3.85410 −0.186295
\(429\) 0 0
\(430\) 9.09017 0.438367
\(431\) 30.6525 1.47648 0.738239 0.674539i \(-0.235658\pi\)
0.738239 + 0.674539i \(0.235658\pi\)
\(432\) 3.47214 0.167053
\(433\) 15.7984 0.759221 0.379611 0.925146i \(-0.376058\pi\)
0.379611 + 0.925146i \(0.376058\pi\)
\(434\) 6.47214 0.310672
\(435\) −2.76393 −0.132520
\(436\) 4.00000 0.191565
\(437\) 9.70820 0.464406
\(438\) −2.38197 −0.113815
\(439\) −36.5410 −1.74401 −0.872004 0.489499i \(-0.837180\pi\)
−0.872004 + 0.489499i \(0.837180\pi\)
\(440\) 0 0
\(441\) −2.61803 −0.124668
\(442\) 0.291796 0.0138793
\(443\) −8.43769 −0.400887 −0.200443 0.979705i \(-0.564238\pi\)
−0.200443 + 0.979705i \(0.564238\pi\)
\(444\) 4.18034 0.198390
\(445\) 8.14590 0.386153
\(446\) 27.1246 1.28439
\(447\) 1.05573 0.0499342
\(448\) 1.00000 0.0472456
\(449\) 9.03444 0.426362 0.213181 0.977013i \(-0.431618\pi\)
0.213181 + 0.977013i \(0.431618\pi\)
\(450\) 2.61803 0.123415
\(451\) 0 0
\(452\) −11.3262 −0.532741
\(453\) −5.88854 −0.276668
\(454\) −29.3262 −1.37635
\(455\) 0.763932 0.0358137
\(456\) 0.708204 0.0331647
\(457\) 20.9787 0.981343 0.490671 0.871345i \(-0.336752\pi\)
0.490671 + 0.871345i \(0.336752\pi\)
\(458\) 22.3607 1.04485
\(459\) 1.32624 0.0619035
\(460\) −8.47214 −0.395015
\(461\) 40.9443 1.90696 0.953482 0.301449i \(-0.0974702\pi\)
0.953482 + 0.301449i \(0.0974702\pi\)
\(462\) 0 0
\(463\) 6.94427 0.322728 0.161364 0.986895i \(-0.448411\pi\)
0.161364 + 0.986895i \(0.448411\pi\)
\(464\) −4.47214 −0.207614
\(465\) −4.00000 −0.185496
\(466\) −16.3262 −0.756298
\(467\) −4.94427 −0.228794 −0.114397 0.993435i \(-0.536494\pi\)
−0.114397 + 0.993435i \(0.536494\pi\)
\(468\) 2.00000 0.0924500
\(469\) −5.61803 −0.259417
\(470\) 9.23607 0.426028
\(471\) −1.81966 −0.0838455
\(472\) −10.8541 −0.499601
\(473\) 0 0
\(474\) 2.47214 0.113549
\(475\) 1.14590 0.0525774
\(476\) 0.381966 0.0175074
\(477\) 18.9443 0.867399
\(478\) −19.7082 −0.901432
\(479\) 35.7082 1.63155 0.815775 0.578370i \(-0.196311\pi\)
0.815775 + 0.578370i \(0.196311\pi\)
\(480\) −0.618034 −0.0282093
\(481\) 5.16718 0.235603
\(482\) −15.9787 −0.727810
\(483\) −5.23607 −0.238249
\(484\) 0 0
\(485\) 10.6180 0.482140
\(486\) 13.9443 0.632525
\(487\) 15.8197 0.716857 0.358429 0.933557i \(-0.383313\pi\)
0.358429 + 0.933557i \(0.383313\pi\)
\(488\) −0.291796 −0.0132090
\(489\) 1.11146 0.0502618
\(490\) 1.00000 0.0451754
\(491\) 32.2148 1.45383 0.726916 0.686726i \(-0.240953\pi\)
0.726916 + 0.686726i \(0.240953\pi\)
\(492\) 2.38197 0.107387
\(493\) −1.70820 −0.0769336
\(494\) 0.875388 0.0393856
\(495\) 0 0
\(496\) −6.47214 −0.290607
\(497\) −6.94427 −0.311493
\(498\) −1.32624 −0.0594301
\(499\) 26.9787 1.20773 0.603867 0.797085i \(-0.293626\pi\)
0.603867 + 0.797085i \(0.293626\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −13.4164 −0.599401
\(502\) −20.0000 −0.892644
\(503\) 10.6525 0.474970 0.237485 0.971391i \(-0.423677\pi\)
0.237485 + 0.971391i \(0.423677\pi\)
\(504\) 2.61803 0.116617
\(505\) −2.94427 −0.131018
\(506\) 0 0
\(507\) 7.67376 0.340804
\(508\) 3.52786 0.156524
\(509\) −27.1246 −1.20228 −0.601139 0.799145i \(-0.705286\pi\)
−0.601139 + 0.799145i \(0.705286\pi\)
\(510\) −0.236068 −0.0104533
\(511\) −3.85410 −0.170495
\(512\) −1.00000 −0.0441942
\(513\) 3.97871 0.175665
\(514\) −8.14590 −0.359300
\(515\) 6.00000 0.264392
\(516\) −5.61803 −0.247320
\(517\) 0 0
\(518\) 6.76393 0.297190
\(519\) 9.81966 0.431035
\(520\) −0.763932 −0.0335006
\(521\) −25.3262 −1.10956 −0.554781 0.831996i \(-0.687198\pi\)
−0.554781 + 0.831996i \(0.687198\pi\)
\(522\) −11.7082 −0.512454
\(523\) 3.79837 0.166091 0.0830456 0.996546i \(-0.473535\pi\)
0.0830456 + 0.996546i \(0.473535\pi\)
\(524\) −6.09017 −0.266050
\(525\) −0.618034 −0.0269732
\(526\) 18.3607 0.800564
\(527\) −2.47214 −0.107688
\(528\) 0 0
\(529\) 48.7771 2.12074
\(530\) −7.23607 −0.314315
\(531\) −28.4164 −1.23317
\(532\) 1.14590 0.0496810
\(533\) 2.94427 0.127531
\(534\) −5.03444 −0.217862
\(535\) 3.85410 0.166627
\(536\) 5.61803 0.242662
\(537\) −5.47214 −0.236140
\(538\) −12.6525 −0.545487
\(539\) 0 0
\(540\) −3.47214 −0.149417
\(541\) 33.0132 1.41935 0.709673 0.704531i \(-0.248843\pi\)
0.709673 + 0.704531i \(0.248843\pi\)
\(542\) −16.1803 −0.695005
\(543\) −1.88854 −0.0810452
\(544\) −0.381966 −0.0163767
\(545\) −4.00000 −0.171341
\(546\) −0.472136 −0.0202056
\(547\) −28.8541 −1.23371 −0.616856 0.787076i \(-0.711594\pi\)
−0.616856 + 0.787076i \(0.711594\pi\)
\(548\) −1.85410 −0.0792033
\(549\) −0.763932 −0.0326038
\(550\) 0 0
\(551\) −5.12461 −0.218316
\(552\) 5.23607 0.222862
\(553\) 4.00000 0.170097
\(554\) 12.0000 0.509831
\(555\) −4.18034 −0.177446
\(556\) 12.9443 0.548959
\(557\) 23.2361 0.984544 0.492272 0.870441i \(-0.336167\pi\)
0.492272 + 0.870441i \(0.336167\pi\)
\(558\) −16.9443 −0.717308
\(559\) −6.94427 −0.293711
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 4.32624 0.182491
\(563\) 27.3820 1.15401 0.577006 0.816740i \(-0.304221\pi\)
0.577006 + 0.816740i \(0.304221\pi\)
\(564\) −5.70820 −0.240359
\(565\) 11.3262 0.476498
\(566\) 28.3607 1.19209
\(567\) 5.70820 0.239722
\(568\) 6.94427 0.291375
\(569\) 28.6738 1.20207 0.601033 0.799224i \(-0.294756\pi\)
0.601033 + 0.799224i \(0.294756\pi\)
\(570\) −0.708204 −0.0296634
\(571\) 18.8328 0.788129 0.394064 0.919083i \(-0.371069\pi\)
0.394064 + 0.919083i \(0.371069\pi\)
\(572\) 0 0
\(573\) −2.00000 −0.0835512
\(574\) 3.85410 0.160867
\(575\) 8.47214 0.353312
\(576\) −2.61803 −0.109085
\(577\) 47.3951 1.97308 0.986542 0.163506i \(-0.0522803\pi\)
0.986542 + 0.163506i \(0.0522803\pi\)
\(578\) 16.8541 0.701038
\(579\) −3.34752 −0.139118
\(580\) 4.47214 0.185695
\(581\) −2.14590 −0.0890269
\(582\) −6.56231 −0.272016
\(583\) 0 0
\(584\) 3.85410 0.159484
\(585\) −2.00000 −0.0826898
\(586\) −9.52786 −0.393592
\(587\) 43.2148 1.78366 0.891832 0.452366i \(-0.149420\pi\)
0.891832 + 0.452366i \(0.149420\pi\)
\(588\) −0.618034 −0.0254873
\(589\) −7.41641 −0.305588
\(590\) 10.8541 0.446856
\(591\) −12.6525 −0.520453
\(592\) −6.76393 −0.277996
\(593\) −4.96556 −0.203911 −0.101956 0.994789i \(-0.532510\pi\)
−0.101956 + 0.994789i \(0.532510\pi\)
\(594\) 0 0
\(595\) −0.381966 −0.0156591
\(596\) −1.70820 −0.0699708
\(597\) −14.1803 −0.580363
\(598\) 6.47214 0.264665
\(599\) 22.0689 0.901710 0.450855 0.892597i \(-0.351119\pi\)
0.450855 + 0.892597i \(0.351119\pi\)
\(600\) 0.618034 0.0252311
\(601\) −17.6738 −0.720928 −0.360464 0.932773i \(-0.617382\pi\)
−0.360464 + 0.932773i \(0.617382\pi\)
\(602\) −9.09017 −0.370488
\(603\) 14.7082 0.598964
\(604\) 9.52786 0.387683
\(605\) 0 0
\(606\) 1.81966 0.0739186
\(607\) 46.8328 1.90089 0.950443 0.310900i \(-0.100630\pi\)
0.950443 + 0.310900i \(0.100630\pi\)
\(608\) −1.14590 −0.0464723
\(609\) 2.76393 0.112000
\(610\) 0.291796 0.0118145
\(611\) −7.05573 −0.285444
\(612\) −1.00000 −0.0404226
\(613\) −1.52786 −0.0617098 −0.0308549 0.999524i \(-0.509823\pi\)
−0.0308549 + 0.999524i \(0.509823\pi\)
\(614\) 1.56231 0.0630495
\(615\) −2.38197 −0.0960501
\(616\) 0 0
\(617\) 14.5623 0.586256 0.293128 0.956073i \(-0.405304\pi\)
0.293128 + 0.956073i \(0.405304\pi\)
\(618\) −3.70820 −0.149166
\(619\) 9.72949 0.391061 0.195531 0.980698i \(-0.437357\pi\)
0.195531 + 0.980698i \(0.437357\pi\)
\(620\) 6.47214 0.259927
\(621\) 29.4164 1.18044
\(622\) −20.1803 −0.809158
\(623\) −8.14590 −0.326359
\(624\) 0.472136 0.0189006
\(625\) 1.00000 0.0400000
\(626\) −2.27051 −0.0907478
\(627\) 0 0
\(628\) 2.94427 0.117489
\(629\) −2.58359 −0.103015
\(630\) −2.61803 −0.104305
\(631\) −15.1246 −0.602101 −0.301051 0.953608i \(-0.597337\pi\)
−0.301051 + 0.953608i \(0.597337\pi\)
\(632\) −4.00000 −0.159111
\(633\) −16.2705 −0.646695
\(634\) 14.3607 0.570335
\(635\) −3.52786 −0.139999
\(636\) 4.47214 0.177332
\(637\) −0.763932 −0.0302681
\(638\) 0 0
\(639\) 18.1803 0.719203
\(640\) 1.00000 0.0395285
\(641\) −34.1459 −1.34868 −0.674341 0.738420i \(-0.735572\pi\)
−0.674341 + 0.738420i \(0.735572\pi\)
\(642\) −2.38197 −0.0940087
\(643\) −38.7984 −1.53006 −0.765029 0.643996i \(-0.777276\pi\)
−0.765029 + 0.643996i \(0.777276\pi\)
\(644\) 8.47214 0.333849
\(645\) 5.61803 0.221210
\(646\) −0.437694 −0.0172208
\(647\) 24.2918 0.955009 0.477505 0.878629i \(-0.341541\pi\)
0.477505 + 0.878629i \(0.341541\pi\)
\(648\) −5.70820 −0.224239
\(649\) 0 0
\(650\) 0.763932 0.0299639
\(651\) 4.00000 0.156772
\(652\) −1.79837 −0.0704298
\(653\) 13.3475 0.522329 0.261164 0.965294i \(-0.415893\pi\)
0.261164 + 0.965294i \(0.415893\pi\)
\(654\) 2.47214 0.0966682
\(655\) 6.09017 0.237963
\(656\) −3.85410 −0.150477
\(657\) 10.0902 0.393655
\(658\) −9.23607 −0.360059
\(659\) 25.9787 1.01199 0.505994 0.862537i \(-0.331126\pi\)
0.505994 + 0.862537i \(0.331126\pi\)
\(660\) 0 0
\(661\) 24.1803 0.940506 0.470253 0.882532i \(-0.344163\pi\)
0.470253 + 0.882532i \(0.344163\pi\)
\(662\) 30.0344 1.16732
\(663\) 0.180340 0.00700382
\(664\) 2.14590 0.0832770
\(665\) −1.14590 −0.0444360
\(666\) −17.7082 −0.686179
\(667\) −37.8885 −1.46705
\(668\) 21.7082 0.839916
\(669\) 16.7639 0.648131
\(670\) −5.61803 −0.217044
\(671\) 0 0
\(672\) 0.618034 0.0238412
\(673\) −40.4508 −1.55927 −0.779633 0.626237i \(-0.784594\pi\)
−0.779633 + 0.626237i \(0.784594\pi\)
\(674\) 25.7984 0.993716
\(675\) 3.47214 0.133643
\(676\) −12.4164 −0.477554
\(677\) −15.4164 −0.592501 −0.296250 0.955110i \(-0.595736\pi\)
−0.296250 + 0.955110i \(0.595736\pi\)
\(678\) −7.00000 −0.268833
\(679\) −10.6180 −0.407483
\(680\) 0.381966 0.0146477
\(681\) −18.1246 −0.694537
\(682\) 0 0
\(683\) 30.4721 1.16598 0.582992 0.812478i \(-0.301882\pi\)
0.582992 + 0.812478i \(0.301882\pi\)
\(684\) −3.00000 −0.114708
\(685\) 1.85410 0.0708416
\(686\) −1.00000 −0.0381802
\(687\) 13.8197 0.527253
\(688\) 9.09017 0.346559
\(689\) 5.52786 0.210595
\(690\) −5.23607 −0.199334
\(691\) 12.5623 0.477893 0.238946 0.971033i \(-0.423198\pi\)
0.238946 + 0.971033i \(0.423198\pi\)
\(692\) −15.8885 −0.603992
\(693\) 0 0
\(694\) −3.90983 −0.148415
\(695\) −12.9443 −0.491004
\(696\) −2.76393 −0.104767
\(697\) −1.47214 −0.0557611
\(698\) −7.41641 −0.280715
\(699\) −10.0902 −0.381645
\(700\) 1.00000 0.0377964
\(701\) 18.8328 0.711306 0.355653 0.934618i \(-0.384259\pi\)
0.355653 + 0.934618i \(0.384259\pi\)
\(702\) 2.65248 0.100111
\(703\) −7.75078 −0.292326
\(704\) 0 0
\(705\) 5.70820 0.214983
\(706\) −8.56231 −0.322247
\(707\) 2.94427 0.110731
\(708\) −6.70820 −0.252110
\(709\) −37.7771 −1.41875 −0.709374 0.704832i \(-0.751022\pi\)
−0.709374 + 0.704832i \(0.751022\pi\)
\(710\) −6.94427 −0.260614
\(711\) −10.4721 −0.392736
\(712\) 8.14590 0.305280
\(713\) −54.8328 −2.05351
\(714\) 0.236068 0.00883462
\(715\) 0 0
\(716\) 8.85410 0.330893
\(717\) −12.1803 −0.454883
\(718\) 19.5967 0.731344
\(719\) −34.0689 −1.27055 −0.635277 0.772284i \(-0.719114\pi\)
−0.635277 + 0.772284i \(0.719114\pi\)
\(720\) 2.61803 0.0975684
\(721\) −6.00000 −0.223452
\(722\) 17.6869 0.658239
\(723\) −9.87539 −0.367270
\(724\) 3.05573 0.113565
\(725\) −4.47214 −0.166091
\(726\) 0 0
\(727\) −30.2492 −1.12188 −0.560941 0.827856i \(-0.689560\pi\)
−0.560941 + 0.827856i \(0.689560\pi\)
\(728\) 0.763932 0.0283132
\(729\) −8.50658 −0.315058
\(730\) −3.85410 −0.142647
\(731\) 3.47214 0.128422
\(732\) −0.180340 −0.00666555
\(733\) 13.5967 0.502207 0.251104 0.967960i \(-0.419206\pi\)
0.251104 + 0.967960i \(0.419206\pi\)
\(734\) 19.8885 0.734100
\(735\) 0.618034 0.0227965
\(736\) −8.47214 −0.312287
\(737\) 0 0
\(738\) −10.0902 −0.371424
\(739\) 16.8541 0.619988 0.309994 0.950738i \(-0.399673\pi\)
0.309994 + 0.950738i \(0.399673\pi\)
\(740\) 6.76393 0.248647
\(741\) 0.541020 0.0198749
\(742\) 7.23607 0.265644
\(743\) 40.2492 1.47660 0.738300 0.674472i \(-0.235629\pi\)
0.738300 + 0.674472i \(0.235629\pi\)
\(744\) −4.00000 −0.146647
\(745\) 1.70820 0.0625837
\(746\) 16.6525 0.609690
\(747\) 5.61803 0.205553
\(748\) 0 0
\(749\) −3.85410 −0.140826
\(750\) −0.618034 −0.0225674
\(751\) 17.3050 0.631467 0.315733 0.948848i \(-0.397750\pi\)
0.315733 + 0.948848i \(0.397750\pi\)
\(752\) 9.23607 0.336805
\(753\) −12.3607 −0.450448
\(754\) −3.41641 −0.124418
\(755\) −9.52786 −0.346754
\(756\) 3.47214 0.126280
\(757\) 13.3475 0.485124 0.242562 0.970136i \(-0.422012\pi\)
0.242562 + 0.970136i \(0.422012\pi\)
\(758\) −13.3820 −0.486055
\(759\) 0 0
\(760\) 1.14590 0.0415661
\(761\) 50.8115 1.84192 0.920958 0.389661i \(-0.127408\pi\)
0.920958 + 0.389661i \(0.127408\pi\)
\(762\) 2.18034 0.0789854
\(763\) 4.00000 0.144810
\(764\) 3.23607 0.117077
\(765\) 1.00000 0.0361551
\(766\) −2.18034 −0.0787789
\(767\) −8.29180 −0.299399
\(768\) −0.618034 −0.0223014
\(769\) −31.3050 −1.12889 −0.564443 0.825472i \(-0.690909\pi\)
−0.564443 + 0.825472i \(0.690909\pi\)
\(770\) 0 0
\(771\) −5.03444 −0.181311
\(772\) 5.41641 0.194941
\(773\) 39.2361 1.41122 0.705612 0.708599i \(-0.250672\pi\)
0.705612 + 0.708599i \(0.250672\pi\)
\(774\) 23.7984 0.855415
\(775\) −6.47214 −0.232486
\(776\) 10.6180 0.381165
\(777\) 4.18034 0.149969
\(778\) −12.2918 −0.440682
\(779\) −4.41641 −0.158234
\(780\) −0.472136 −0.0169052
\(781\) 0 0
\(782\) −3.23607 −0.115722
\(783\) −15.5279 −0.554921
\(784\) 1.00000 0.0357143
\(785\) −2.94427 −0.105086
\(786\) −3.76393 −0.134255
\(787\) 9.72949 0.346819 0.173409 0.984850i \(-0.444522\pi\)
0.173409 + 0.984850i \(0.444522\pi\)
\(788\) 20.4721 0.729290
\(789\) 11.3475 0.403983
\(790\) 4.00000 0.142314
\(791\) −11.3262 −0.402715
\(792\) 0 0
\(793\) −0.222912 −0.00791585
\(794\) −35.8885 −1.27364
\(795\) −4.47214 −0.158610
\(796\) 22.9443 0.813238
\(797\) 14.1803 0.502293 0.251147 0.967949i \(-0.419192\pi\)
0.251147 + 0.967949i \(0.419192\pi\)
\(798\) 0.708204 0.0250701
\(799\) 3.52786 0.124807
\(800\) −1.00000 −0.0353553
\(801\) 21.3262 0.753526
\(802\) 17.5623 0.620147
\(803\) 0 0
\(804\) 3.47214 0.122453
\(805\) −8.47214 −0.298604
\(806\) −4.94427 −0.174155
\(807\) −7.81966 −0.275265
\(808\) −2.94427 −0.103579
\(809\) −42.1591 −1.48223 −0.741117 0.671376i \(-0.765703\pi\)
−0.741117 + 0.671376i \(0.765703\pi\)
\(810\) 5.70820 0.200566
\(811\) −24.6869 −0.866875 −0.433437 0.901184i \(-0.642699\pi\)
−0.433437 + 0.901184i \(0.642699\pi\)
\(812\) −4.47214 −0.156941
\(813\) −10.0000 −0.350715
\(814\) 0 0
\(815\) 1.79837 0.0629943
\(816\) −0.236068 −0.00826403
\(817\) 10.4164 0.364424
\(818\) 5.05573 0.176769
\(819\) 2.00000 0.0698857
\(820\) 3.85410 0.134591
\(821\) −35.8885 −1.25252 −0.626259 0.779615i \(-0.715415\pi\)
−0.626259 + 0.779615i \(0.715415\pi\)
\(822\) −1.14590 −0.0399678
\(823\) −26.1803 −0.912589 −0.456295 0.889829i \(-0.650824\pi\)
−0.456295 + 0.889829i \(0.650824\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) −10.8541 −0.377663
\(827\) 36.6738 1.27527 0.637636 0.770338i \(-0.279912\pi\)
0.637636 + 0.770338i \(0.279912\pi\)
\(828\) −22.1803 −0.770820
\(829\) 42.4721 1.47512 0.737559 0.675283i \(-0.235978\pi\)
0.737559 + 0.675283i \(0.235978\pi\)
\(830\) −2.14590 −0.0744852
\(831\) 7.41641 0.257272
\(832\) −0.763932 −0.0264846
\(833\) 0.381966 0.0132343
\(834\) 8.00000 0.277017
\(835\) −21.7082 −0.751243
\(836\) 0 0
\(837\) −22.4721 −0.776751
\(838\) −2.43769 −0.0842087
\(839\) 49.7082 1.71612 0.858059 0.513551i \(-0.171670\pi\)
0.858059 + 0.513551i \(0.171670\pi\)
\(840\) −0.618034 −0.0213242
\(841\) −9.00000 −0.310345
\(842\) −16.8328 −0.580097
\(843\) 2.67376 0.0920893
\(844\) 26.3262 0.906186
\(845\) 12.4164 0.427137
\(846\) 24.1803 0.831337
\(847\) 0 0
\(848\) −7.23607 −0.248488
\(849\) 17.5279 0.601555
\(850\) −0.381966 −0.0131013
\(851\) −57.3050 −1.96439
\(852\) 4.29180 0.147035
\(853\) 13.4164 0.459369 0.229685 0.973265i \(-0.426231\pi\)
0.229685 + 0.973265i \(0.426231\pi\)
\(854\) −0.291796 −0.00998506
\(855\) 3.00000 0.102598
\(856\) 3.85410 0.131730
\(857\) 4.43769 0.151589 0.0757944 0.997123i \(-0.475851\pi\)
0.0757944 + 0.997123i \(0.475851\pi\)
\(858\) 0 0
\(859\) −1.32624 −0.0452507 −0.0226253 0.999744i \(-0.507202\pi\)
−0.0226253 + 0.999744i \(0.507202\pi\)
\(860\) −9.09017 −0.309972
\(861\) 2.38197 0.0811772
\(862\) −30.6525 −1.04403
\(863\) −49.8885 −1.69823 −0.849113 0.528211i \(-0.822863\pi\)
−0.849113 + 0.528211i \(0.822863\pi\)
\(864\) −3.47214 −0.118124
\(865\) 15.8885 0.540227
\(866\) −15.7984 −0.536851
\(867\) 10.4164 0.353760
\(868\) −6.47214 −0.219679
\(869\) 0 0
\(870\) 2.76393 0.0937061
\(871\) 4.29180 0.145422
\(872\) −4.00000 −0.135457
\(873\) 27.7984 0.940832
\(874\) −9.70820 −0.328385
\(875\) −1.00000 −0.0338062
\(876\) 2.38197 0.0804792
\(877\) −7.23607 −0.244345 −0.122172 0.992509i \(-0.538986\pi\)
−0.122172 + 0.992509i \(0.538986\pi\)
\(878\) 36.5410 1.23320
\(879\) −5.88854 −0.198616
\(880\) 0 0
\(881\) 30.3262 1.02172 0.510858 0.859665i \(-0.329328\pi\)
0.510858 + 0.859665i \(0.329328\pi\)
\(882\) 2.61803 0.0881538
\(883\) 8.21478 0.276449 0.138225 0.990401i \(-0.455860\pi\)
0.138225 + 0.990401i \(0.455860\pi\)
\(884\) −0.291796 −0.00981416
\(885\) 6.70820 0.225494
\(886\) 8.43769 0.283470
\(887\) −40.5410 −1.36123 −0.680617 0.732639i \(-0.738288\pi\)
−0.680617 + 0.732639i \(0.738288\pi\)
\(888\) −4.18034 −0.140283
\(889\) 3.52786 0.118321
\(890\) −8.14590 −0.273051
\(891\) 0 0
\(892\) −27.1246 −0.908199
\(893\) 10.5836 0.354166
\(894\) −1.05573 −0.0353088
\(895\) −8.85410 −0.295960
\(896\) −1.00000 −0.0334077
\(897\) 4.00000 0.133556
\(898\) −9.03444 −0.301483
\(899\) 28.9443 0.965346
\(900\) −2.61803 −0.0872678
\(901\) −2.76393 −0.0920799
\(902\) 0 0
\(903\) −5.61803 −0.186956
\(904\) 11.3262 0.376705
\(905\) −3.05573 −0.101576
\(906\) 5.88854 0.195634
\(907\) 19.1459 0.635729 0.317865 0.948136i \(-0.397034\pi\)
0.317865 + 0.948136i \(0.397034\pi\)
\(908\) 29.3262 0.973225
\(909\) −7.70820 −0.255665
\(910\) −0.763932 −0.0253241
\(911\) 18.4721 0.612009 0.306005 0.952030i \(-0.401008\pi\)
0.306005 + 0.952030i \(0.401008\pi\)
\(912\) −0.708204 −0.0234510
\(913\) 0 0
\(914\) −20.9787 −0.693914
\(915\) 0.180340 0.00596185
\(916\) −22.3607 −0.738818
\(917\) −6.09017 −0.201115
\(918\) −1.32624 −0.0437724
\(919\) 24.9443 0.822836 0.411418 0.911447i \(-0.365034\pi\)
0.411418 + 0.911447i \(0.365034\pi\)
\(920\) 8.47214 0.279318
\(921\) 0.965558 0.0318162
\(922\) −40.9443 −1.34843
\(923\) 5.30495 0.174615
\(924\) 0 0
\(925\) −6.76393 −0.222397
\(926\) −6.94427 −0.228203
\(927\) 15.7082 0.515925
\(928\) 4.47214 0.146805
\(929\) 22.0902 0.724755 0.362377 0.932031i \(-0.381965\pi\)
0.362377 + 0.932031i \(0.381965\pi\)
\(930\) 4.00000 0.131165
\(931\) 1.14590 0.0375553
\(932\) 16.3262 0.534784
\(933\) −12.4721 −0.408319
\(934\) 4.94427 0.161782
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 9.50658 0.310566 0.155283 0.987870i \(-0.450371\pi\)
0.155283 + 0.987870i \(0.450371\pi\)
\(938\) 5.61803 0.183435
\(939\) −1.40325 −0.0457934
\(940\) −9.23607 −0.301247
\(941\) 20.3607 0.663739 0.331870 0.943325i \(-0.392321\pi\)
0.331870 + 0.943325i \(0.392321\pi\)
\(942\) 1.81966 0.0592877
\(943\) −32.6525 −1.06331
\(944\) 10.8541 0.353271
\(945\) −3.47214 −0.112949
\(946\) 0 0
\(947\) 24.6312 0.800406 0.400203 0.916426i \(-0.368940\pi\)
0.400203 + 0.916426i \(0.368940\pi\)
\(948\) −2.47214 −0.0802912
\(949\) 2.94427 0.0955751
\(950\) −1.14590 −0.0371778
\(951\) 8.87539 0.287804
\(952\) −0.381966 −0.0123796
\(953\) −5.85410 −0.189633 −0.0948165 0.995495i \(-0.530226\pi\)
−0.0948165 + 0.995495i \(0.530226\pi\)
\(954\) −18.9443 −0.613343
\(955\) −3.23607 −0.104717
\(956\) 19.7082 0.637409
\(957\) 0 0
\(958\) −35.7082 −1.15368
\(959\) −1.85410 −0.0598721
\(960\) 0.618034 0.0199470
\(961\) 10.8885 0.351243
\(962\) −5.16718 −0.166597
\(963\) 10.0902 0.325151
\(964\) 15.9787 0.514640
\(965\) −5.41641 −0.174360
\(966\) 5.23607 0.168468
\(967\) 48.6525 1.56456 0.782279 0.622928i \(-0.214057\pi\)
0.782279 + 0.622928i \(0.214057\pi\)
\(968\) 0 0
\(969\) −0.270510 −0.00869003
\(970\) −10.6180 −0.340925
\(971\) 18.8328 0.604374 0.302187 0.953249i \(-0.402283\pi\)
0.302187 + 0.953249i \(0.402283\pi\)
\(972\) −13.9443 −0.447263
\(973\) 12.9443 0.414974
\(974\) −15.8197 −0.506895
\(975\) 0.472136 0.0151205
\(976\) 0.291796 0.00934016
\(977\) 61.1935 1.95775 0.978877 0.204452i \(-0.0655411\pi\)
0.978877 + 0.204452i \(0.0655411\pi\)
\(978\) −1.11146 −0.0355404
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −10.4721 −0.334350
\(982\) −32.2148 −1.02801
\(983\) 8.94427 0.285278 0.142639 0.989775i \(-0.454441\pi\)
0.142639 + 0.989775i \(0.454441\pi\)
\(984\) −2.38197 −0.0759343
\(985\) −20.4721 −0.652296
\(986\) 1.70820 0.0544003
\(987\) −5.70820 −0.181694
\(988\) −0.875388 −0.0278498
\(989\) 77.0132 2.44888
\(990\) 0 0
\(991\) −10.3607 −0.329118 −0.164559 0.986367i \(-0.552620\pi\)
−0.164559 + 0.986367i \(0.552620\pi\)
\(992\) 6.47214 0.205491
\(993\) 18.5623 0.589057
\(994\) 6.94427 0.220259
\(995\) −22.9443 −0.727382
\(996\) 1.32624 0.0420235
\(997\) 4.83282 0.153057 0.0765284 0.997067i \(-0.475616\pi\)
0.0765284 + 0.997067i \(0.475616\pi\)
\(998\) −26.9787 −0.853996
\(999\) −23.4853 −0.743042
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.bp.1.1 2
11.3 even 5 770.2.n.d.141.1 yes 4
11.4 even 5 770.2.n.d.71.1 4
11.10 odd 2 8470.2.a.cb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.d.71.1 4 11.4 even 5
770.2.n.d.141.1 yes 4 11.3 even 5
8470.2.a.bp.1.1 2 1.1 even 1 trivial
8470.2.a.cb.1.1 2 11.10 odd 2