Properties

Label 8470.2.a.cy.1.5
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.13298000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 3 x^{3} + 26 x^{2} + 13 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 1.5
Root \(0.0677009\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.44508 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.44508 q^{6} -1.00000 q^{7} -1.00000 q^{8} +2.97843 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.44508 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.44508 q^{6} -1.00000 q^{7} -1.00000 q^{8} +2.97843 q^{9} -1.00000 q^{10} +2.44508 q^{12} -1.00429 q^{13} +1.00000 q^{14} +2.44508 q^{15} +1.00000 q^{16} -0.974141 q^{17} -2.97843 q^{18} -5.13082 q^{19} +1.00000 q^{20} -2.44508 q^{21} -4.97385 q^{23} -2.44508 q^{24} +1.00000 q^{25} +1.00429 q^{26} -0.0527323 q^{27} -1.00000 q^{28} -2.89329 q^{29} -2.44508 q^{30} -6.46901 q^{31} -1.00000 q^{32} +0.974141 q^{34} -1.00000 q^{35} +2.97843 q^{36} +9.79242 q^{37} +5.13082 q^{38} -2.45558 q^{39} -1.00000 q^{40} -3.10213 q^{41} +2.44508 q^{42} +9.46992 q^{43} +2.97843 q^{45} +4.97385 q^{46} +7.29309 q^{47} +2.44508 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.38186 q^{51} -1.00429 q^{52} +3.98903 q^{53} +0.0527323 q^{54} +1.00000 q^{56} -12.5453 q^{57} +2.89329 q^{58} -0.902167 q^{59} +2.44508 q^{60} -7.11156 q^{61} +6.46901 q^{62} -2.97843 q^{63} +1.00000 q^{64} -1.00429 q^{65} -1.88571 q^{67} -0.974141 q^{68} -12.1615 q^{69} +1.00000 q^{70} +2.21005 q^{71} -2.97843 q^{72} +4.41995 q^{73} -9.79242 q^{74} +2.44508 q^{75} -5.13082 q^{76} +2.45558 q^{78} -12.7528 q^{79} +1.00000 q^{80} -9.06423 q^{81} +3.10213 q^{82} -8.95284 q^{83} -2.44508 q^{84} -0.974141 q^{85} -9.46992 q^{86} -7.07434 q^{87} -5.28764 q^{89} -2.97843 q^{90} +1.00429 q^{91} -4.97385 q^{92} -15.8173 q^{93} -7.29309 q^{94} -5.13082 q^{95} -2.44508 q^{96} -9.41520 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} - 6q^{8} + 15q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} - 6q^{8} + 15q^{9} - 6q^{10} + q^{12} - 2q^{13} + 6q^{14} + q^{15} + 6q^{16} - 7q^{17} - 15q^{18} - 11q^{19} + 6q^{20} - q^{21} - 6q^{23} - q^{24} + 6q^{25} + 2q^{26} + 4q^{27} - 6q^{28} - 2q^{29} - q^{30} - 6q^{32} + 7q^{34} - 6q^{35} + 15q^{36} - 14q^{37} + 11q^{38} - 20q^{39} - 6q^{40} - 13q^{41} + q^{42} - 19q^{43} + 15q^{45} + 6q^{46} + 22q^{47} + q^{48} + 6q^{49} - 6q^{50} + 14q^{51} - 2q^{52} - 10q^{53} - 4q^{54} + 6q^{56} - 32q^{57} + 2q^{58} - 7q^{59} + q^{60} - 22q^{61} - 15q^{63} + 6q^{64} - 2q^{65} + 5q^{67} - 7q^{68} - 36q^{69} + 6q^{70} + 8q^{71} - 15q^{72} - 13q^{73} + 14q^{74} + q^{75} - 11q^{76} + 20q^{78} + 16q^{79} + 6q^{80} + 18q^{81} + 13q^{82} + 5q^{83} - q^{84} - 7q^{85} + 19q^{86} - 14q^{87} + q^{89} - 15q^{90} + 2q^{91} - 6q^{92} - 42q^{93} - 22q^{94} - 11q^{95} - q^{96} - 3q^{97} - 6q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.44508 1.41167 0.705835 0.708376i \(-0.250572\pi\)
0.705835 + 0.708376i \(0.250572\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.44508 −0.998201
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 2.97843 0.992811
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 2.44508 0.705835
\(13\) −1.00429 −0.278541 −0.139270 0.990254i \(-0.544476\pi\)
−0.139270 + 0.990254i \(0.544476\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.44508 0.631318
\(16\) 1.00000 0.250000
\(17\) −0.974141 −0.236264 −0.118132 0.992998i \(-0.537691\pi\)
−0.118132 + 0.992998i \(0.537691\pi\)
\(18\) −2.97843 −0.702023
\(19\) −5.13082 −1.17709 −0.588545 0.808464i \(-0.700299\pi\)
−0.588545 + 0.808464i \(0.700299\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.44508 −0.533561
\(22\) 0 0
\(23\) −4.97385 −1.03712 −0.518560 0.855041i \(-0.673532\pi\)
−0.518560 + 0.855041i \(0.673532\pi\)
\(24\) −2.44508 −0.499101
\(25\) 1.00000 0.200000
\(26\) 1.00429 0.196958
\(27\) −0.0527323 −0.0101483
\(28\) −1.00000 −0.188982
\(29\) −2.89329 −0.537271 −0.268635 0.963242i \(-0.586573\pi\)
−0.268635 + 0.963242i \(0.586573\pi\)
\(30\) −2.44508 −0.446409
\(31\) −6.46901 −1.16187 −0.580934 0.813950i \(-0.697313\pi\)
−0.580934 + 0.813950i \(0.697313\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0.974141 0.167064
\(35\) −1.00000 −0.169031
\(36\) 2.97843 0.496406
\(37\) 9.79242 1.60986 0.804932 0.593367i \(-0.202202\pi\)
0.804932 + 0.593367i \(0.202202\pi\)
\(38\) 5.13082 0.832329
\(39\) −2.45558 −0.393207
\(40\) −1.00000 −0.158114
\(41\) −3.10213 −0.484471 −0.242235 0.970218i \(-0.577881\pi\)
−0.242235 + 0.970218i \(0.577881\pi\)
\(42\) 2.44508 0.377285
\(43\) 9.46992 1.44415 0.722074 0.691816i \(-0.243189\pi\)
0.722074 + 0.691816i \(0.243189\pi\)
\(44\) 0 0
\(45\) 2.97843 0.443999
\(46\) 4.97385 0.733354
\(47\) 7.29309 1.06381 0.531903 0.846805i \(-0.321477\pi\)
0.531903 + 0.846805i \(0.321477\pi\)
\(48\) 2.44508 0.352917
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −2.38186 −0.333526
\(52\) −1.00429 −0.139270
\(53\) 3.98903 0.547936 0.273968 0.961739i \(-0.411664\pi\)
0.273968 + 0.961739i \(0.411664\pi\)
\(54\) 0.0527323 0.00717595
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −12.5453 −1.66166
\(58\) 2.89329 0.379908
\(59\) −0.902167 −0.117452 −0.0587260 0.998274i \(-0.518704\pi\)
−0.0587260 + 0.998274i \(0.518704\pi\)
\(60\) 2.44508 0.315659
\(61\) −7.11156 −0.910542 −0.455271 0.890353i \(-0.650458\pi\)
−0.455271 + 0.890353i \(0.650458\pi\)
\(62\) 6.46901 0.821565
\(63\) −2.97843 −0.375247
\(64\) 1.00000 0.125000
\(65\) −1.00429 −0.124567
\(66\) 0 0
\(67\) −1.88571 −0.230377 −0.115188 0.993344i \(-0.536747\pi\)
−0.115188 + 0.993344i \(0.536747\pi\)
\(68\) −0.974141 −0.118132
\(69\) −12.1615 −1.46407
\(70\) 1.00000 0.119523
\(71\) 2.21005 0.262285 0.131142 0.991364i \(-0.458136\pi\)
0.131142 + 0.991364i \(0.458136\pi\)
\(72\) −2.97843 −0.351012
\(73\) 4.41995 0.517316 0.258658 0.965969i \(-0.416720\pi\)
0.258658 + 0.965969i \(0.416720\pi\)
\(74\) −9.79242 −1.13835
\(75\) 2.44508 0.282334
\(76\) −5.13082 −0.588545
\(77\) 0 0
\(78\) 2.45558 0.278040
\(79\) −12.7528 −1.43480 −0.717399 0.696663i \(-0.754667\pi\)
−0.717399 + 0.696663i \(0.754667\pi\)
\(80\) 1.00000 0.111803
\(81\) −9.06423 −1.00714
\(82\) 3.10213 0.342573
\(83\) −8.95284 −0.982702 −0.491351 0.870962i \(-0.663497\pi\)
−0.491351 + 0.870962i \(0.663497\pi\)
\(84\) −2.44508 −0.266780
\(85\) −0.974141 −0.105660
\(86\) −9.46992 −1.02117
\(87\) −7.07434 −0.758448
\(88\) 0 0
\(89\) −5.28764 −0.560489 −0.280245 0.959929i \(-0.590416\pi\)
−0.280245 + 0.959929i \(0.590416\pi\)
\(90\) −2.97843 −0.313954
\(91\) 1.00429 0.105278
\(92\) −4.97385 −0.518560
\(93\) −15.8173 −1.64018
\(94\) −7.29309 −0.752225
\(95\) −5.13082 −0.526411
\(96\) −2.44508 −0.249550
\(97\) −9.41520 −0.955969 −0.477985 0.878368i \(-0.658633\pi\)
−0.477985 + 0.878368i \(0.658633\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.43796 −0.640601 −0.320300 0.947316i \(-0.603784\pi\)
−0.320300 + 0.947316i \(0.603784\pi\)
\(102\) 2.38186 0.235839
\(103\) 6.53919 0.644325 0.322163 0.946684i \(-0.395590\pi\)
0.322163 + 0.946684i \(0.395590\pi\)
\(104\) 1.00429 0.0984790
\(105\) −2.44508 −0.238616
\(106\) −3.98903 −0.387449
\(107\) −13.6151 −1.31622 −0.658110 0.752922i \(-0.728644\pi\)
−0.658110 + 0.752922i \(0.728644\pi\)
\(108\) −0.0527323 −0.00507417
\(109\) 4.46766 0.427925 0.213962 0.976842i \(-0.431363\pi\)
0.213962 + 0.976842i \(0.431363\pi\)
\(110\) 0 0
\(111\) 23.9433 2.27260
\(112\) −1.00000 −0.0944911
\(113\) −17.2962 −1.62709 −0.813544 0.581504i \(-0.802465\pi\)
−0.813544 + 0.581504i \(0.802465\pi\)
\(114\) 12.5453 1.17497
\(115\) −4.97385 −0.463814
\(116\) −2.89329 −0.268635
\(117\) −2.99122 −0.276538
\(118\) 0.902167 0.0830512
\(119\) 0.974141 0.0892993
\(120\) −2.44508 −0.223205
\(121\) 0 0
\(122\) 7.11156 0.643850
\(123\) −7.58496 −0.683913
\(124\) −6.46901 −0.580934
\(125\) 1.00000 0.0894427
\(126\) 2.97843 0.265340
\(127\) −20.1769 −1.79041 −0.895207 0.445651i \(-0.852972\pi\)
−0.895207 + 0.445651i \(0.852972\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 23.1547 2.03866
\(130\) 1.00429 0.0880823
\(131\) 11.0785 0.967935 0.483967 0.875086i \(-0.339195\pi\)
0.483967 + 0.875086i \(0.339195\pi\)
\(132\) 0 0
\(133\) 5.13082 0.444898
\(134\) 1.88571 0.162901
\(135\) −0.0527323 −0.00453847
\(136\) 0.974141 0.0835319
\(137\) −19.6702 −1.68054 −0.840269 0.542170i \(-0.817603\pi\)
−0.840269 + 0.542170i \(0.817603\pi\)
\(138\) 12.1615 1.03525
\(139\) 9.53624 0.808854 0.404427 0.914570i \(-0.367471\pi\)
0.404427 + 0.914570i \(0.367471\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 17.8322 1.50174
\(142\) −2.21005 −0.185463
\(143\) 0 0
\(144\) 2.97843 0.248203
\(145\) −2.89329 −0.240275
\(146\) −4.41995 −0.365797
\(147\) 2.44508 0.201667
\(148\) 9.79242 0.804932
\(149\) 9.27662 0.759970 0.379985 0.924993i \(-0.375929\pi\)
0.379985 + 0.924993i \(0.375929\pi\)
\(150\) −2.44508 −0.199640
\(151\) −0.800899 −0.0651763 −0.0325881 0.999469i \(-0.510375\pi\)
−0.0325881 + 0.999469i \(0.510375\pi\)
\(152\) 5.13082 0.416164
\(153\) −2.90141 −0.234565
\(154\) 0 0
\(155\) −6.46901 −0.519604
\(156\) −2.45558 −0.196604
\(157\) −18.0983 −1.44440 −0.722199 0.691685i \(-0.756868\pi\)
−0.722199 + 0.691685i \(0.756868\pi\)
\(158\) 12.7528 1.01456
\(159\) 9.75352 0.773504
\(160\) −1.00000 −0.0790569
\(161\) 4.97385 0.391994
\(162\) 9.06423 0.712154
\(163\) 8.78949 0.688446 0.344223 0.938888i \(-0.388142\pi\)
0.344223 + 0.938888i \(0.388142\pi\)
\(164\) −3.10213 −0.242235
\(165\) 0 0
\(166\) 8.95284 0.694875
\(167\) 11.5772 0.895871 0.447935 0.894066i \(-0.352159\pi\)
0.447935 + 0.894066i \(0.352159\pi\)
\(168\) 2.44508 0.188642
\(169\) −11.9914 −0.922415
\(170\) 0.974141 0.0747132
\(171\) −15.2818 −1.16863
\(172\) 9.46992 0.722074
\(173\) −5.23233 −0.397807 −0.198903 0.980019i \(-0.563738\pi\)
−0.198903 + 0.980019i \(0.563738\pi\)
\(174\) 7.07434 0.536304
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −2.20587 −0.165804
\(178\) 5.28764 0.396326
\(179\) 9.84737 0.736027 0.368014 0.929820i \(-0.380038\pi\)
0.368014 + 0.929820i \(0.380038\pi\)
\(180\) 2.97843 0.221999
\(181\) −20.0533 −1.49055 −0.745275 0.666757i \(-0.767682\pi\)
−0.745275 + 0.666757i \(0.767682\pi\)
\(182\) −1.00429 −0.0744431
\(183\) −17.3884 −1.28538
\(184\) 4.97385 0.366677
\(185\) 9.79242 0.719953
\(186\) 15.8173 1.15978
\(187\) 0 0
\(188\) 7.29309 0.531903
\(189\) 0.0527323 0.00383571
\(190\) 5.13082 0.372229
\(191\) −17.4033 −1.25926 −0.629629 0.776896i \(-0.716793\pi\)
−0.629629 + 0.776896i \(0.716793\pi\)
\(192\) 2.44508 0.176459
\(193\) −25.2939 −1.82069 −0.910346 0.413847i \(-0.864185\pi\)
−0.910346 + 0.413847i \(0.864185\pi\)
\(194\) 9.41520 0.675972
\(195\) −2.45558 −0.175848
\(196\) 1.00000 0.0714286
\(197\) 9.89470 0.704968 0.352484 0.935818i \(-0.385337\pi\)
0.352484 + 0.935818i \(0.385337\pi\)
\(198\) 0 0
\(199\) −1.57633 −0.111743 −0.0558717 0.998438i \(-0.517794\pi\)
−0.0558717 + 0.998438i \(0.517794\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.61073 −0.325216
\(202\) 6.43796 0.452973
\(203\) 2.89329 0.203069
\(204\) −2.38186 −0.166763
\(205\) −3.10213 −0.216662
\(206\) −6.53919 −0.455607
\(207\) −14.8143 −1.02966
\(208\) −1.00429 −0.0696352
\(209\) 0 0
\(210\) 2.44508 0.168727
\(211\) 10.2108 0.702938 0.351469 0.936199i \(-0.385682\pi\)
0.351469 + 0.936199i \(0.385682\pi\)
\(212\) 3.98903 0.273968
\(213\) 5.40375 0.370259
\(214\) 13.6151 0.930708
\(215\) 9.46992 0.645843
\(216\) 0.0527323 0.00358798
\(217\) 6.46901 0.439145
\(218\) −4.46766 −0.302588
\(219\) 10.8071 0.730279
\(220\) 0 0
\(221\) 0.978322 0.0658091
\(222\) −23.9433 −1.60697
\(223\) −9.38022 −0.628146 −0.314073 0.949399i \(-0.601694\pi\)
−0.314073 + 0.949399i \(0.601694\pi\)
\(224\) 1.00000 0.0668153
\(225\) 2.97843 0.198562
\(226\) 17.2962 1.15052
\(227\) −13.6115 −0.903430 −0.451715 0.892162i \(-0.649188\pi\)
−0.451715 + 0.892162i \(0.649188\pi\)
\(228\) −12.5453 −0.830831
\(229\) −16.2404 −1.07319 −0.536597 0.843839i \(-0.680290\pi\)
−0.536597 + 0.843839i \(0.680290\pi\)
\(230\) 4.97385 0.327966
\(231\) 0 0
\(232\) 2.89329 0.189954
\(233\) −8.03950 −0.526685 −0.263343 0.964702i \(-0.584825\pi\)
−0.263343 + 0.964702i \(0.584825\pi\)
\(234\) 2.99122 0.195542
\(235\) 7.29309 0.475749
\(236\) −0.902167 −0.0587260
\(237\) −31.1816 −2.02546
\(238\) −0.974141 −0.0631442
\(239\) 2.23140 0.144337 0.0721686 0.997392i \(-0.477008\pi\)
0.0721686 + 0.997392i \(0.477008\pi\)
\(240\) 2.44508 0.157829
\(241\) −3.19243 −0.205643 −0.102821 0.994700i \(-0.532787\pi\)
−0.102821 + 0.994700i \(0.532787\pi\)
\(242\) 0 0
\(243\) −22.0046 −1.41160
\(244\) −7.11156 −0.455271
\(245\) 1.00000 0.0638877
\(246\) 7.58496 0.483599
\(247\) 5.15284 0.327868
\(248\) 6.46901 0.410783
\(249\) −21.8904 −1.38725
\(250\) −1.00000 −0.0632456
\(251\) 18.3437 1.15785 0.578923 0.815383i \(-0.303473\pi\)
0.578923 + 0.815383i \(0.303473\pi\)
\(252\) −2.97843 −0.187624
\(253\) 0 0
\(254\) 20.1769 1.26601
\(255\) −2.38186 −0.149158
\(256\) 1.00000 0.0625000
\(257\) 17.7717 1.10857 0.554285 0.832327i \(-0.312992\pi\)
0.554285 + 0.832327i \(0.312992\pi\)
\(258\) −23.1547 −1.44155
\(259\) −9.79242 −0.608471
\(260\) −1.00429 −0.0622836
\(261\) −8.61747 −0.533408
\(262\) −11.0785 −0.684433
\(263\) 16.3390 1.00751 0.503754 0.863847i \(-0.331952\pi\)
0.503754 + 0.863847i \(0.331952\pi\)
\(264\) 0 0
\(265\) 3.98903 0.245044
\(266\) −5.13082 −0.314591
\(267\) −12.9287 −0.791226
\(268\) −1.88571 −0.115188
\(269\) 22.6073 1.37839 0.689197 0.724574i \(-0.257964\pi\)
0.689197 + 0.724574i \(0.257964\pi\)
\(270\) 0.0527323 0.00320918
\(271\) 0.691377 0.0419981 0.0209991 0.999779i \(-0.493315\pi\)
0.0209991 + 0.999779i \(0.493315\pi\)
\(272\) −0.974141 −0.0590659
\(273\) 2.45558 0.148618
\(274\) 19.6702 1.18832
\(275\) 0 0
\(276\) −12.1615 −0.732035
\(277\) 7.69613 0.462415 0.231208 0.972904i \(-0.425732\pi\)
0.231208 + 0.972904i \(0.425732\pi\)
\(278\) −9.53624 −0.571946
\(279\) −19.2675 −1.15352
\(280\) 1.00000 0.0597614
\(281\) −12.7976 −0.763443 −0.381722 0.924277i \(-0.624669\pi\)
−0.381722 + 0.924277i \(0.624669\pi\)
\(282\) −17.8322 −1.06189
\(283\) 16.7641 0.996522 0.498261 0.867027i \(-0.333972\pi\)
0.498261 + 0.867027i \(0.333972\pi\)
\(284\) 2.21005 0.131142
\(285\) −12.5453 −0.743118
\(286\) 0 0
\(287\) 3.10213 0.183113
\(288\) −2.97843 −0.175506
\(289\) −16.0511 −0.944179
\(290\) 2.89329 0.169900
\(291\) −23.0210 −1.34951
\(292\) 4.41995 0.258658
\(293\) −7.67994 −0.448667 −0.224333 0.974512i \(-0.572020\pi\)
−0.224333 + 0.974512i \(0.572020\pi\)
\(294\) −2.44508 −0.142600
\(295\) −0.902167 −0.0525262
\(296\) −9.79242 −0.569173
\(297\) 0 0
\(298\) −9.27662 −0.537380
\(299\) 4.99520 0.288880
\(300\) 2.44508 0.141167
\(301\) −9.46992 −0.545837
\(302\) 0.800899 0.0460866
\(303\) −15.7413 −0.904317
\(304\) −5.13082 −0.294273
\(305\) −7.11156 −0.407207
\(306\) 2.90141 0.165863
\(307\) 18.3456 1.04704 0.523518 0.852015i \(-0.324619\pi\)
0.523518 + 0.852015i \(0.324619\pi\)
\(308\) 0 0
\(309\) 15.9889 0.909574
\(310\) 6.46901 0.367415
\(311\) −17.8031 −1.00952 −0.504761 0.863259i \(-0.668420\pi\)
−0.504761 + 0.863259i \(0.668420\pi\)
\(312\) 2.45558 0.139020
\(313\) 26.7803 1.51371 0.756857 0.653581i \(-0.226734\pi\)
0.756857 + 0.653581i \(0.226734\pi\)
\(314\) 18.0983 1.02134
\(315\) −2.97843 −0.167816
\(316\) −12.7528 −0.717399
\(317\) −35.0441 −1.96827 −0.984137 0.177411i \(-0.943228\pi\)
−0.984137 + 0.177411i \(0.943228\pi\)
\(318\) −9.75352 −0.546950
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −33.2900 −1.85807
\(322\) −4.97385 −0.277182
\(323\) 4.99814 0.278104
\(324\) −9.06423 −0.503569
\(325\) −1.00429 −0.0557081
\(326\) −8.78949 −0.486805
\(327\) 10.9238 0.604088
\(328\) 3.10213 0.171286
\(329\) −7.29309 −0.402081
\(330\) 0 0
\(331\) 17.7187 0.973908 0.486954 0.873428i \(-0.338108\pi\)
0.486954 + 0.873428i \(0.338108\pi\)
\(332\) −8.95284 −0.491351
\(333\) 29.1661 1.59829
\(334\) −11.5772 −0.633476
\(335\) −1.88571 −0.103028
\(336\) −2.44508 −0.133390
\(337\) 33.1859 1.80775 0.903875 0.427796i \(-0.140710\pi\)
0.903875 + 0.427796i \(0.140710\pi\)
\(338\) 11.9914 0.652246
\(339\) −42.2906 −2.29691
\(340\) −0.974141 −0.0528302
\(341\) 0 0
\(342\) 15.2818 0.826345
\(343\) −1.00000 −0.0539949
\(344\) −9.46992 −0.510584
\(345\) −12.1615 −0.654752
\(346\) 5.23233 0.281292
\(347\) −32.1471 −1.72575 −0.862874 0.505420i \(-0.831338\pi\)
−0.862874 + 0.505420i \(0.831338\pi\)
\(348\) −7.07434 −0.379224
\(349\) 2.51940 0.134861 0.0674303 0.997724i \(-0.478520\pi\)
0.0674303 + 0.997724i \(0.478520\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0.0529586 0.00282672
\(352\) 0 0
\(353\) −5.22226 −0.277953 −0.138976 0.990296i \(-0.544381\pi\)
−0.138976 + 0.990296i \(0.544381\pi\)
\(354\) 2.20587 0.117241
\(355\) 2.21005 0.117297
\(356\) −5.28764 −0.280245
\(357\) 2.38186 0.126061
\(358\) −9.84737 −0.520450
\(359\) −9.49040 −0.500884 −0.250442 0.968132i \(-0.580576\pi\)
−0.250442 + 0.968132i \(0.580576\pi\)
\(360\) −2.97843 −0.156977
\(361\) 7.32530 0.385542
\(362\) 20.0533 1.05398
\(363\) 0 0
\(364\) 1.00429 0.0526392
\(365\) 4.41995 0.231351
\(366\) 17.3884 0.908904
\(367\) −19.6449 −1.02545 −0.512727 0.858552i \(-0.671365\pi\)
−0.512727 + 0.858552i \(0.671365\pi\)
\(368\) −4.97385 −0.259280
\(369\) −9.23948 −0.480988
\(370\) −9.79242 −0.509084
\(371\) −3.98903 −0.207100
\(372\) −15.8173 −0.820088
\(373\) −20.5117 −1.06206 −0.531029 0.847354i \(-0.678194\pi\)
−0.531029 + 0.847354i \(0.678194\pi\)
\(374\) 0 0
\(375\) 2.44508 0.126264
\(376\) −7.29309 −0.376113
\(377\) 2.90571 0.149652
\(378\) −0.0527323 −0.00271226
\(379\) 32.1657 1.65224 0.826120 0.563494i \(-0.190543\pi\)
0.826120 + 0.563494i \(0.190543\pi\)
\(380\) −5.13082 −0.263205
\(381\) −49.3343 −2.52747
\(382\) 17.4033 0.890430
\(383\) 6.01133 0.307165 0.153582 0.988136i \(-0.450919\pi\)
0.153582 + 0.988136i \(0.450919\pi\)
\(384\) −2.44508 −0.124775
\(385\) 0 0
\(386\) 25.2939 1.28742
\(387\) 28.2055 1.43377
\(388\) −9.41520 −0.477985
\(389\) 18.5721 0.941644 0.470822 0.882228i \(-0.343957\pi\)
0.470822 + 0.882228i \(0.343957\pi\)
\(390\) 2.45558 0.124343
\(391\) 4.84523 0.245034
\(392\) −1.00000 −0.0505076
\(393\) 27.0879 1.36640
\(394\) −9.89470 −0.498488
\(395\) −12.7528 −0.641661
\(396\) 0 0
\(397\) 5.21019 0.261492 0.130746 0.991416i \(-0.458263\pi\)
0.130746 + 0.991416i \(0.458263\pi\)
\(398\) 1.57633 0.0790145
\(399\) 12.5453 0.628050
\(400\) 1.00000 0.0500000
\(401\) −22.0522 −1.10124 −0.550618 0.834758i \(-0.685608\pi\)
−0.550618 + 0.834758i \(0.685608\pi\)
\(402\) 4.61073 0.229962
\(403\) 6.49678 0.323628
\(404\) −6.43796 −0.320300
\(405\) −9.06423 −0.450405
\(406\) −2.89329 −0.143592
\(407\) 0 0
\(408\) 2.38186 0.117919
\(409\) 35.6323 1.76190 0.880952 0.473205i \(-0.156903\pi\)
0.880952 + 0.473205i \(0.156903\pi\)
\(410\) 3.10213 0.153203
\(411\) −48.0953 −2.37236
\(412\) 6.53919 0.322163
\(413\) 0.902167 0.0443927
\(414\) 14.8143 0.728082
\(415\) −8.95284 −0.439478
\(416\) 1.00429 0.0492395
\(417\) 23.3169 1.14183
\(418\) 0 0
\(419\) 2.71308 0.132543 0.0662713 0.997802i \(-0.478890\pi\)
0.0662713 + 0.997802i \(0.478890\pi\)
\(420\) −2.44508 −0.119308
\(421\) −19.3933 −0.945169 −0.472585 0.881285i \(-0.656679\pi\)
−0.472585 + 0.881285i \(0.656679\pi\)
\(422\) −10.2108 −0.497053
\(423\) 21.7220 1.05616
\(424\) −3.98903 −0.193725
\(425\) −0.974141 −0.0472528
\(426\) −5.40375 −0.261813
\(427\) 7.11156 0.344152
\(428\) −13.6151 −0.658110
\(429\) 0 0
\(430\) −9.46992 −0.456680
\(431\) −27.1661 −1.30854 −0.654271 0.756260i \(-0.727025\pi\)
−0.654271 + 0.756260i \(0.727025\pi\)
\(432\) −0.0527323 −0.00253708
\(433\) 6.77403 0.325539 0.162770 0.986664i \(-0.447957\pi\)
0.162770 + 0.986664i \(0.447957\pi\)
\(434\) −6.46901 −0.310523
\(435\) −7.07434 −0.339188
\(436\) 4.46766 0.213962
\(437\) 25.5199 1.22078
\(438\) −10.8071 −0.516385
\(439\) 36.5942 1.74654 0.873272 0.487232i \(-0.161993\pi\)
0.873272 + 0.487232i \(0.161993\pi\)
\(440\) 0 0
\(441\) 2.97843 0.141830
\(442\) −0.978322 −0.0465340
\(443\) −13.8146 −0.656352 −0.328176 0.944617i \(-0.606434\pi\)
−0.328176 + 0.944617i \(0.606434\pi\)
\(444\) 23.9433 1.13630
\(445\) −5.28764 −0.250658
\(446\) 9.38022 0.444166
\(447\) 22.6821 1.07283
\(448\) −1.00000 −0.0472456
\(449\) 24.7466 1.16787 0.583933 0.811802i \(-0.301513\pi\)
0.583933 + 0.811802i \(0.301513\pi\)
\(450\) −2.97843 −0.140405
\(451\) 0 0
\(452\) −17.2962 −0.813544
\(453\) −1.95827 −0.0920074
\(454\) 13.6115 0.638821
\(455\) 1.00429 0.0470820
\(456\) 12.5453 0.587487
\(457\) −11.0626 −0.517485 −0.258743 0.965946i \(-0.583308\pi\)
−0.258743 + 0.965946i \(0.583308\pi\)
\(458\) 16.2404 0.758862
\(459\) 0.0513687 0.00239768
\(460\) −4.97385 −0.231907
\(461\) 17.8851 0.832990 0.416495 0.909138i \(-0.363258\pi\)
0.416495 + 0.909138i \(0.363258\pi\)
\(462\) 0 0
\(463\) −8.47374 −0.393808 −0.196904 0.980423i \(-0.563089\pi\)
−0.196904 + 0.980423i \(0.563089\pi\)
\(464\) −2.89329 −0.134318
\(465\) −15.8173 −0.733509
\(466\) 8.03950 0.372423
\(467\) 14.7272 0.681495 0.340747 0.940155i \(-0.389320\pi\)
0.340747 + 0.940155i \(0.389320\pi\)
\(468\) −2.99122 −0.138269
\(469\) 1.88571 0.0870742
\(470\) −7.29309 −0.336405
\(471\) −44.2517 −2.03901
\(472\) 0.902167 0.0415256
\(473\) 0 0
\(474\) 31.1816 1.43222
\(475\) −5.13082 −0.235418
\(476\) 0.974141 0.0446497
\(477\) 11.8811 0.543997
\(478\) −2.23140 −0.102062
\(479\) 34.1872 1.56205 0.781026 0.624498i \(-0.214696\pi\)
0.781026 + 0.624498i \(0.214696\pi\)
\(480\) −2.44508 −0.111602
\(481\) −9.83445 −0.448413
\(482\) 3.19243 0.145411
\(483\) 12.1615 0.553366
\(484\) 0 0
\(485\) −9.41520 −0.427522
\(486\) 22.0046 0.998150
\(487\) −32.1518 −1.45694 −0.728468 0.685080i \(-0.759767\pi\)
−0.728468 + 0.685080i \(0.759767\pi\)
\(488\) 7.11156 0.321925
\(489\) 21.4910 0.971858
\(490\) −1.00000 −0.0451754
\(491\) 16.6210 0.750097 0.375048 0.927005i \(-0.377626\pi\)
0.375048 + 0.927005i \(0.377626\pi\)
\(492\) −7.58496 −0.341956
\(493\) 2.81847 0.126938
\(494\) −5.15284 −0.231837
\(495\) 0 0
\(496\) −6.46901 −0.290467
\(497\) −2.21005 −0.0991342
\(498\) 21.8904 0.980934
\(499\) 26.8144 1.20038 0.600189 0.799858i \(-0.295092\pi\)
0.600189 + 0.799858i \(0.295092\pi\)
\(500\) 1.00000 0.0447214
\(501\) 28.3072 1.26467
\(502\) −18.3437 −0.818720
\(503\) 1.76608 0.0787458 0.0393729 0.999225i \(-0.487464\pi\)
0.0393729 + 0.999225i \(0.487464\pi\)
\(504\) 2.97843 0.132670
\(505\) −6.43796 −0.286485
\(506\) 0 0
\(507\) −29.3200 −1.30215
\(508\) −20.1769 −0.895207
\(509\) 26.5080 1.17495 0.587474 0.809243i \(-0.300122\pi\)
0.587474 + 0.809243i \(0.300122\pi\)
\(510\) 2.38186 0.105470
\(511\) −4.41995 −0.195527
\(512\) −1.00000 −0.0441942
\(513\) 0.270560 0.0119455
\(514\) −17.7717 −0.783878
\(515\) 6.53919 0.288151
\(516\) 23.1547 1.01933
\(517\) 0 0
\(518\) 9.79242 0.430254
\(519\) −12.7935 −0.561572
\(520\) 1.00429 0.0440412
\(521\) −15.0004 −0.657181 −0.328590 0.944473i \(-0.606574\pi\)
−0.328590 + 0.944473i \(0.606574\pi\)
\(522\) 8.61747 0.377177
\(523\) 28.8065 1.25962 0.629809 0.776750i \(-0.283133\pi\)
0.629809 + 0.776750i \(0.283133\pi\)
\(524\) 11.0785 0.483967
\(525\) −2.44508 −0.106712
\(526\) −16.3390 −0.712416
\(527\) 6.30173 0.274508
\(528\) 0 0
\(529\) 1.73918 0.0756167
\(530\) −3.98903 −0.173273
\(531\) −2.68704 −0.116608
\(532\) 5.13082 0.222449
\(533\) 3.11544 0.134945
\(534\) 12.9287 0.559481
\(535\) −13.6151 −0.588632
\(536\) 1.88571 0.0814505
\(537\) 24.0776 1.03903
\(538\) −22.6073 −0.974671
\(539\) 0 0
\(540\) −0.0527323 −0.00226924
\(541\) 26.0652 1.12063 0.560315 0.828280i \(-0.310680\pi\)
0.560315 + 0.828280i \(0.310680\pi\)
\(542\) −0.691377 −0.0296972
\(543\) −49.0320 −2.10417
\(544\) 0.974141 0.0417659
\(545\) 4.46766 0.191374
\(546\) −2.45558 −0.105089
\(547\) 23.8493 1.01972 0.509860 0.860257i \(-0.329697\pi\)
0.509860 + 0.860257i \(0.329697\pi\)
\(548\) −19.6702 −0.840269
\(549\) −21.1813 −0.903996
\(550\) 0 0
\(551\) 14.8449 0.632416
\(552\) 12.1615 0.517627
\(553\) 12.7528 0.542302
\(554\) −7.69613 −0.326977
\(555\) 23.9433 1.01634
\(556\) 9.53624 0.404427
\(557\) −28.5937 −1.21155 −0.605776 0.795635i \(-0.707137\pi\)
−0.605776 + 0.795635i \(0.707137\pi\)
\(558\) 19.2675 0.815659
\(559\) −9.51057 −0.402254
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 12.7976 0.539836
\(563\) −46.9795 −1.97995 −0.989974 0.141249i \(-0.954888\pi\)
−0.989974 + 0.141249i \(0.954888\pi\)
\(564\) 17.8322 0.750872
\(565\) −17.2962 −0.727656
\(566\) −16.7641 −0.704648
\(567\) 9.06423 0.380662
\(568\) −2.21005 −0.0927316
\(569\) 10.8453 0.454657 0.227329 0.973818i \(-0.427001\pi\)
0.227329 + 0.973818i \(0.427001\pi\)
\(570\) 12.5453 0.525464
\(571\) −12.5385 −0.524718 −0.262359 0.964970i \(-0.584500\pi\)
−0.262359 + 0.964970i \(0.584500\pi\)
\(572\) 0 0
\(573\) −42.5525 −1.77766
\(574\) −3.10213 −0.129480
\(575\) −4.97385 −0.207424
\(576\) 2.97843 0.124101
\(577\) −16.5167 −0.687601 −0.343800 0.939043i \(-0.611714\pi\)
−0.343800 + 0.939043i \(0.611714\pi\)
\(578\) 16.0511 0.667636
\(579\) −61.8456 −2.57022
\(580\) −2.89329 −0.120137
\(581\) 8.95284 0.371426
\(582\) 23.0210 0.954249
\(583\) 0 0
\(584\) −4.41995 −0.182899
\(585\) −2.99122 −0.123672
\(586\) 7.67994 0.317255
\(587\) −0.475698 −0.0196341 −0.00981707 0.999952i \(-0.503125\pi\)
−0.00981707 + 0.999952i \(0.503125\pi\)
\(588\) 2.44508 0.100834
\(589\) 33.1913 1.36762
\(590\) 0.902167 0.0371416
\(591\) 24.1934 0.995183
\(592\) 9.79242 0.402466
\(593\) −27.6428 −1.13515 −0.567576 0.823321i \(-0.692119\pi\)
−0.567576 + 0.823321i \(0.692119\pi\)
\(594\) 0 0
\(595\) 0.974141 0.0399359
\(596\) 9.27662 0.379985
\(597\) −3.85427 −0.157745
\(598\) −4.99520 −0.204269
\(599\) −4.39806 −0.179700 −0.0898500 0.995955i \(-0.528639\pi\)
−0.0898500 + 0.995955i \(0.528639\pi\)
\(600\) −2.44508 −0.0998201
\(601\) −32.1891 −1.31302 −0.656511 0.754316i \(-0.727968\pi\)
−0.656511 + 0.754316i \(0.727968\pi\)
\(602\) 9.46992 0.385965
\(603\) −5.61647 −0.228721
\(604\) −0.800899 −0.0325881
\(605\) 0 0
\(606\) 15.7413 0.639448
\(607\) −15.0219 −0.609721 −0.304861 0.952397i \(-0.598610\pi\)
−0.304861 + 0.952397i \(0.598610\pi\)
\(608\) 5.13082 0.208082
\(609\) 7.07434 0.286667
\(610\) 7.11156 0.287939
\(611\) −7.32440 −0.296314
\(612\) −2.90141 −0.117283
\(613\) 1.12624 0.0454885 0.0227443 0.999741i \(-0.492760\pi\)
0.0227443 + 0.999741i \(0.492760\pi\)
\(614\) −18.3456 −0.740366
\(615\) −7.58496 −0.305855
\(616\) 0 0
\(617\) −6.79925 −0.273728 −0.136864 0.990590i \(-0.543702\pi\)
−0.136864 + 0.990590i \(0.543702\pi\)
\(618\) −15.9889 −0.643166
\(619\) 36.5101 1.46746 0.733732 0.679439i \(-0.237777\pi\)
0.733732 + 0.679439i \(0.237777\pi\)
\(620\) −6.46901 −0.259802
\(621\) 0.262282 0.0105250
\(622\) 17.8031 0.713841
\(623\) 5.28764 0.211845
\(624\) −2.45558 −0.0983019
\(625\) 1.00000 0.0400000
\(626\) −26.7803 −1.07036
\(627\) 0 0
\(628\) −18.0983 −0.722199
\(629\) −9.53919 −0.380352
\(630\) 2.97843 0.118664
\(631\) 0.487747 0.0194169 0.00970845 0.999953i \(-0.496910\pi\)
0.00970845 + 0.999953i \(0.496910\pi\)
\(632\) 12.7528 0.507278
\(633\) 24.9662 0.992317
\(634\) 35.0441 1.39178
\(635\) −20.1769 −0.800697
\(636\) 9.75352 0.386752
\(637\) −1.00429 −0.0397915
\(638\) 0 0
\(639\) 6.58248 0.260399
\(640\) −1.00000 −0.0395285
\(641\) 29.9446 1.18274 0.591371 0.806400i \(-0.298587\pi\)
0.591371 + 0.806400i \(0.298587\pi\)
\(642\) 33.2900 1.31385
\(643\) 36.7794 1.45044 0.725220 0.688518i \(-0.241738\pi\)
0.725220 + 0.688518i \(0.241738\pi\)
\(644\) 4.97385 0.195997
\(645\) 23.1547 0.911717
\(646\) −4.99814 −0.196649
\(647\) −24.8585 −0.977287 −0.488643 0.872484i \(-0.662508\pi\)
−0.488643 + 0.872484i \(0.662508\pi\)
\(648\) 9.06423 0.356077
\(649\) 0 0
\(650\) 1.00429 0.0393916
\(651\) 15.8173 0.619928
\(652\) 8.78949 0.344223
\(653\) −25.7286 −1.00684 −0.503420 0.864042i \(-0.667925\pi\)
−0.503420 + 0.864042i \(0.667925\pi\)
\(654\) −10.9238 −0.427155
\(655\) 11.0785 0.432874
\(656\) −3.10213 −0.121118
\(657\) 13.1645 0.513597
\(658\) 7.29309 0.284314
\(659\) 45.1117 1.75730 0.878652 0.477463i \(-0.158443\pi\)
0.878652 + 0.477463i \(0.158443\pi\)
\(660\) 0 0
\(661\) −41.3111 −1.60682 −0.803409 0.595428i \(-0.796983\pi\)
−0.803409 + 0.595428i \(0.796983\pi\)
\(662\) −17.7187 −0.688657
\(663\) 2.39208 0.0929007
\(664\) 8.95284 0.347437
\(665\) 5.13082 0.198965
\(666\) −29.1661 −1.13016
\(667\) 14.3908 0.557214
\(668\) 11.5772 0.447935
\(669\) −22.9354 −0.886735
\(670\) 1.88571 0.0728515
\(671\) 0 0
\(672\) 2.44508 0.0943211
\(673\) 35.7108 1.37655 0.688275 0.725450i \(-0.258368\pi\)
0.688275 + 0.725450i \(0.258368\pi\)
\(674\) −33.1859 −1.27827
\(675\) −0.0527323 −0.00202967
\(676\) −11.9914 −0.461208
\(677\) 44.0214 1.69188 0.845940 0.533278i \(-0.179040\pi\)
0.845940 + 0.533278i \(0.179040\pi\)
\(678\) 42.2906 1.62416
\(679\) 9.41520 0.361322
\(680\) 0.974141 0.0373566
\(681\) −33.2814 −1.27534
\(682\) 0 0
\(683\) 48.6683 1.86224 0.931120 0.364712i \(-0.118833\pi\)
0.931120 + 0.364712i \(0.118833\pi\)
\(684\) −15.2818 −0.584314
\(685\) −19.6702 −0.751559
\(686\) 1.00000 0.0381802
\(687\) −39.7091 −1.51499
\(688\) 9.46992 0.361037
\(689\) −4.00616 −0.152622
\(690\) 12.1615 0.462980
\(691\) 2.67864 0.101900 0.0509501 0.998701i \(-0.483775\pi\)
0.0509501 + 0.998701i \(0.483775\pi\)
\(692\) −5.23233 −0.198903
\(693\) 0 0
\(694\) 32.1471 1.22029
\(695\) 9.53624 0.361730
\(696\) 7.07434 0.268152
\(697\) 3.02191 0.114463
\(698\) −2.51940 −0.0953608
\(699\) −19.6572 −0.743505
\(700\) −1.00000 −0.0377964
\(701\) 3.34202 0.126226 0.0631131 0.998006i \(-0.479897\pi\)
0.0631131 + 0.998006i \(0.479897\pi\)
\(702\) −0.0529586 −0.00199880
\(703\) −50.2431 −1.89496
\(704\) 0 0
\(705\) 17.8322 0.671600
\(706\) 5.22226 0.196542
\(707\) 6.43796 0.242124
\(708\) −2.20587 −0.0829018
\(709\) −14.6950 −0.551882 −0.275941 0.961175i \(-0.588989\pi\)
−0.275941 + 0.961175i \(0.588989\pi\)
\(710\) −2.21005 −0.0829416
\(711\) −37.9832 −1.42448
\(712\) 5.28764 0.198163
\(713\) 32.1759 1.20500
\(714\) −2.38186 −0.0891387
\(715\) 0 0
\(716\) 9.84737 0.368014
\(717\) 5.45596 0.203757
\(718\) 9.49040 0.354178
\(719\) −22.0502 −0.822332 −0.411166 0.911560i \(-0.634878\pi\)
−0.411166 + 0.911560i \(0.634878\pi\)
\(720\) 2.97843 0.111000
\(721\) −6.53919 −0.243532
\(722\) −7.32530 −0.272619
\(723\) −7.80576 −0.290299
\(724\) −20.0533 −0.745275
\(725\) −2.89329 −0.107454
\(726\) 0 0
\(727\) 51.5714 1.91268 0.956339 0.292259i \(-0.0944068\pi\)
0.956339 + 0.292259i \(0.0944068\pi\)
\(728\) −1.00429 −0.0372216
\(729\) −26.6104 −0.985571
\(730\) −4.41995 −0.163590
\(731\) −9.22503 −0.341200
\(732\) −17.3884 −0.642692
\(733\) 15.0644 0.556417 0.278209 0.960521i \(-0.410259\pi\)
0.278209 + 0.960521i \(0.410259\pi\)
\(734\) 19.6449 0.725106
\(735\) 2.44508 0.0901883
\(736\) 4.97385 0.183339
\(737\) 0 0
\(738\) 9.23948 0.340110
\(739\) −2.91374 −0.107184 −0.0535918 0.998563i \(-0.517067\pi\)
−0.0535918 + 0.998563i \(0.517067\pi\)
\(740\) 9.79242 0.359976
\(741\) 12.5991 0.462841
\(742\) 3.98903 0.146442
\(743\) 19.3058 0.708261 0.354131 0.935196i \(-0.384777\pi\)
0.354131 + 0.935196i \(0.384777\pi\)
\(744\) 15.8173 0.579889
\(745\) 9.27662 0.339869
\(746\) 20.5117 0.750988
\(747\) −26.6654 −0.975637
\(748\) 0 0
\(749\) 13.6151 0.497485
\(750\) −2.44508 −0.0892818
\(751\) −2.00896 −0.0733080 −0.0366540 0.999328i \(-0.511670\pi\)
−0.0366540 + 0.999328i \(0.511670\pi\)
\(752\) 7.29309 0.265952
\(753\) 44.8519 1.63449
\(754\) −2.90571 −0.105820
\(755\) −0.800899 −0.0291477
\(756\) 0.0527323 0.00191785
\(757\) −12.9220 −0.469658 −0.234829 0.972037i \(-0.575453\pi\)
−0.234829 + 0.972037i \(0.575453\pi\)
\(758\) −32.1657 −1.16831
\(759\) 0 0
\(760\) 5.13082 0.186114
\(761\) 28.5718 1.03573 0.517864 0.855463i \(-0.326727\pi\)
0.517864 + 0.855463i \(0.326727\pi\)
\(762\) 49.3343 1.78719
\(763\) −4.46766 −0.161740
\(764\) −17.4033 −0.629629
\(765\) −2.90141 −0.104901
\(766\) −6.01133 −0.217198
\(767\) 0.906039 0.0327152
\(768\) 2.44508 0.0882294
\(769\) −28.4564 −1.02616 −0.513082 0.858340i \(-0.671496\pi\)
−0.513082 + 0.858340i \(0.671496\pi\)
\(770\) 0 0
\(771\) 43.4534 1.56494
\(772\) −25.2939 −0.910346
\(773\) −17.7792 −0.639472 −0.319736 0.947507i \(-0.603594\pi\)
−0.319736 + 0.947507i \(0.603594\pi\)
\(774\) −28.2055 −1.01383
\(775\) −6.46901 −0.232374
\(776\) 9.41520 0.337986
\(777\) −23.9433 −0.858960
\(778\) −18.5721 −0.665843
\(779\) 15.9164 0.570266
\(780\) −2.45558 −0.0879239
\(781\) 0 0
\(782\) −4.84523 −0.173265
\(783\) 0.152570 0.00545240
\(784\) 1.00000 0.0357143
\(785\) −18.0983 −0.645954
\(786\) −27.0879 −0.966193
\(787\) 36.4440 1.29909 0.649544 0.760324i \(-0.274960\pi\)
0.649544 + 0.760324i \(0.274960\pi\)
\(788\) 9.89470 0.352484
\(789\) 39.9503 1.42227
\(790\) 12.7528 0.453723
\(791\) 17.2962 0.614981
\(792\) 0 0
\(793\) 7.14209 0.253623
\(794\) −5.21019 −0.184903
\(795\) 9.75352 0.345922
\(796\) −1.57633 −0.0558717
\(797\) 45.6253 1.61613 0.808066 0.589092i \(-0.200514\pi\)
0.808066 + 0.589092i \(0.200514\pi\)
\(798\) −12.5453 −0.444098
\(799\) −7.10450 −0.251339
\(800\) −1.00000 −0.0353553
\(801\) −15.7489 −0.556460
\(802\) 22.0522 0.778691
\(803\) 0 0
\(804\) −4.61073 −0.162608
\(805\) 4.97385 0.175305
\(806\) −6.49678 −0.228839
\(807\) 55.2768 1.94584
\(808\) 6.43796 0.226487
\(809\) −29.3137 −1.03062 −0.515308 0.857005i \(-0.672322\pi\)
−0.515308 + 0.857005i \(0.672322\pi\)
\(810\) 9.06423 0.318485
\(811\) −22.8889 −0.803739 −0.401870 0.915697i \(-0.631640\pi\)
−0.401870 + 0.915697i \(0.631640\pi\)
\(812\) 2.89329 0.101535
\(813\) 1.69047 0.0592875
\(814\) 0 0
\(815\) 8.78949 0.307882
\(816\) −2.38186 −0.0833816
\(817\) −48.5884 −1.69989
\(818\) −35.6323 −1.24585
\(819\) 2.99122 0.104522
\(820\) −3.10213 −0.108331
\(821\) −17.6975 −0.617648 −0.308824 0.951119i \(-0.599935\pi\)
−0.308824 + 0.951119i \(0.599935\pi\)
\(822\) 48.0953 1.67751
\(823\) −18.5622 −0.647037 −0.323518 0.946222i \(-0.604866\pi\)
−0.323518 + 0.946222i \(0.604866\pi\)
\(824\) −6.53919 −0.227803
\(825\) 0 0
\(826\) −0.902167 −0.0313904
\(827\) −26.0843 −0.907041 −0.453521 0.891246i \(-0.649832\pi\)
−0.453521 + 0.891246i \(0.649832\pi\)
\(828\) −14.8143 −0.514832
\(829\) 42.9069 1.49022 0.745109 0.666943i \(-0.232398\pi\)
0.745109 + 0.666943i \(0.232398\pi\)
\(830\) 8.95284 0.310758
\(831\) 18.8177 0.652778
\(832\) −1.00429 −0.0348176
\(833\) −0.974141 −0.0337520
\(834\) −23.3169 −0.807399
\(835\) 11.5772 0.400645
\(836\) 0 0
\(837\) 0.341126 0.0117910
\(838\) −2.71308 −0.0937217
\(839\) −16.6107 −0.573464 −0.286732 0.958011i \(-0.592569\pi\)
−0.286732 + 0.958011i \(0.592569\pi\)
\(840\) 2.44508 0.0843634
\(841\) −20.6289 −0.711340
\(842\) 19.3933 0.668336
\(843\) −31.2913 −1.07773
\(844\) 10.2108 0.351469
\(845\) −11.9914 −0.412517
\(846\) −21.7220 −0.746817
\(847\) 0 0
\(848\) 3.98903 0.136984
\(849\) 40.9896 1.40676
\(850\) 0.974141 0.0334127
\(851\) −48.7060 −1.66962
\(852\) 5.40375 0.185130
\(853\) −44.6067 −1.52730 −0.763652 0.645628i \(-0.776596\pi\)
−0.763652 + 0.645628i \(0.776596\pi\)
\(854\) −7.11156 −0.243352
\(855\) −15.2818 −0.522627
\(856\) 13.6151 0.465354
\(857\) 30.6150 1.04579 0.522895 0.852397i \(-0.324852\pi\)
0.522895 + 0.852397i \(0.324852\pi\)
\(858\) 0 0
\(859\) −11.1817 −0.381515 −0.190757 0.981637i \(-0.561094\pi\)
−0.190757 + 0.981637i \(0.561094\pi\)
\(860\) 9.46992 0.322921
\(861\) 7.58496 0.258495
\(862\) 27.1661 0.925279
\(863\) −44.5350 −1.51599 −0.757995 0.652260i \(-0.773821\pi\)
−0.757995 + 0.652260i \(0.773821\pi\)
\(864\) 0.0527323 0.00179399
\(865\) −5.23233 −0.177905
\(866\) −6.77403 −0.230191
\(867\) −39.2462 −1.33287
\(868\) 6.46901 0.219573
\(869\) 0 0
\(870\) 7.07434 0.239842
\(871\) 1.89381 0.0641693
\(872\) −4.46766 −0.151294
\(873\) −28.0426 −0.949097
\(874\) −25.5199 −0.863224
\(875\) −1.00000 −0.0338062
\(876\) 10.8071 0.365139
\(877\) −16.9159 −0.571209 −0.285604 0.958348i \(-0.592194\pi\)
−0.285604 + 0.958348i \(0.592194\pi\)
\(878\) −36.5942 −1.23499
\(879\) −18.7781 −0.633369
\(880\) 0 0
\(881\) −9.06950 −0.305559 −0.152780 0.988260i \(-0.548822\pi\)
−0.152780 + 0.988260i \(0.548822\pi\)
\(882\) −2.97843 −0.100289
\(883\) 25.8923 0.871344 0.435672 0.900105i \(-0.356511\pi\)
0.435672 + 0.900105i \(0.356511\pi\)
\(884\) 0.978322 0.0329045
\(885\) −2.20587 −0.0741496
\(886\) 13.8146 0.464111
\(887\) 25.9193 0.870284 0.435142 0.900362i \(-0.356698\pi\)
0.435142 + 0.900362i \(0.356698\pi\)
\(888\) −23.9433 −0.803484
\(889\) 20.1769 0.676713
\(890\) 5.28764 0.177242
\(891\) 0 0
\(892\) −9.38022 −0.314073
\(893\) −37.4195 −1.25220
\(894\) −22.6821 −0.758603
\(895\) 9.84737 0.329161
\(896\) 1.00000 0.0334077
\(897\) 12.2137 0.407803
\(898\) −24.7466 −0.825806
\(899\) 18.7167 0.624238
\(900\) 2.97843 0.0992811
\(901\) −3.88588 −0.129457
\(902\) 0 0
\(903\) −23.1547 −0.770541
\(904\) 17.2962 0.575262
\(905\) −20.0533 −0.666595
\(906\) 1.95827 0.0650590
\(907\) 12.0946 0.401596 0.200798 0.979633i \(-0.435647\pi\)
0.200798 + 0.979633i \(0.435647\pi\)
\(908\) −13.6115 −0.451715
\(909\) −19.1750 −0.635996
\(910\) −1.00429 −0.0332920
\(911\) 12.1158 0.401416 0.200708 0.979651i \(-0.435676\pi\)
0.200708 + 0.979651i \(0.435676\pi\)
\(912\) −12.5453 −0.415416
\(913\) 0 0
\(914\) 11.0626 0.365917
\(915\) −17.3884 −0.574841
\(916\) −16.2404 −0.536597
\(917\) −11.0785 −0.365845
\(918\) −0.0513687 −0.00169542
\(919\) 53.3173 1.75877 0.879387 0.476107i \(-0.157953\pi\)
0.879387 + 0.476107i \(0.157953\pi\)
\(920\) 4.97385 0.163983
\(921\) 44.8564 1.47807
\(922\) −17.8851 −0.589013
\(923\) −2.21954 −0.0730569
\(924\) 0 0
\(925\) 9.79242 0.321973
\(926\) 8.47374 0.278464
\(927\) 19.4765 0.639693
\(928\) 2.89329 0.0949769
\(929\) −33.2434 −1.09068 −0.545340 0.838215i \(-0.683600\pi\)
−0.545340 + 0.838215i \(0.683600\pi\)
\(930\) 15.8173 0.518669
\(931\) −5.13082 −0.168156
\(932\) −8.03950 −0.263343
\(933\) −43.5301 −1.42511
\(934\) −14.7272 −0.481889
\(935\) 0 0
\(936\) 2.99122 0.0977711
\(937\) 41.0718 1.34176 0.670879 0.741567i \(-0.265917\pi\)
0.670879 + 0.741567i \(0.265917\pi\)
\(938\) −1.88571 −0.0615708
\(939\) 65.4801 2.13686
\(940\) 7.29309 0.237874
\(941\) 13.1369 0.428251 0.214125 0.976806i \(-0.431310\pi\)
0.214125 + 0.976806i \(0.431310\pi\)
\(942\) 44.2517 1.44180
\(943\) 15.4295 0.502454
\(944\) −0.902167 −0.0293630
\(945\) 0.0527323 0.00171538
\(946\) 0 0
\(947\) −49.2785 −1.60133 −0.800667 0.599109i \(-0.795522\pi\)
−0.800667 + 0.599109i \(0.795522\pi\)
\(948\) −31.1816 −1.01273
\(949\) −4.43892 −0.144093
\(950\) 5.13082 0.166466
\(951\) −85.6858 −2.77855
\(952\) −0.974141 −0.0315721
\(953\) 42.4198 1.37411 0.687056 0.726604i \(-0.258903\pi\)
0.687056 + 0.726604i \(0.258903\pi\)
\(954\) −11.8811 −0.384664
\(955\) −17.4033 −0.563158
\(956\) 2.23140 0.0721686
\(957\) 0 0
\(958\) −34.1872 −1.10454
\(959\) 19.6702 0.635184
\(960\) 2.44508 0.0789147
\(961\) 10.8481 0.349939
\(962\) 9.83445 0.317076
\(963\) −40.5516 −1.30676
\(964\) −3.19243 −0.102821
\(965\) −25.2939 −0.814238
\(966\) −12.1615 −0.391289
\(967\) 26.3349 0.846874 0.423437 0.905926i \(-0.360823\pi\)
0.423437 + 0.905926i \(0.360823\pi\)
\(968\) 0 0
\(969\) 12.2209 0.392591
\(970\) 9.41520 0.302304
\(971\) 18.1663 0.582983 0.291491 0.956573i \(-0.405848\pi\)
0.291491 + 0.956573i \(0.405848\pi\)
\(972\) −22.0046 −0.705798
\(973\) −9.53624 −0.305718
\(974\) 32.1518 1.03021
\(975\) −2.45558 −0.0786415
\(976\) −7.11156 −0.227635
\(977\) −10.9394 −0.349983 −0.174992 0.984570i \(-0.555990\pi\)
−0.174992 + 0.984570i \(0.555990\pi\)
\(978\) −21.4910 −0.687207
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 13.3066 0.424848
\(982\) −16.6210 −0.530398
\(983\) −38.4069 −1.22499 −0.612496 0.790474i \(-0.709834\pi\)
−0.612496 + 0.790474i \(0.709834\pi\)
\(984\) 7.58496 0.241800
\(985\) 9.89470 0.315271
\(986\) −2.81847 −0.0897584
\(987\) −17.8322 −0.567606
\(988\) 5.15284 0.163934
\(989\) −47.1019 −1.49775
\(990\) 0 0
\(991\) 55.6207 1.76685 0.883425 0.468572i \(-0.155232\pi\)
0.883425 + 0.468572i \(0.155232\pi\)
\(992\) 6.46901 0.205391
\(993\) 43.3237 1.37484
\(994\) 2.21005 0.0700985
\(995\) −1.57633 −0.0499731
\(996\) −21.8904 −0.693625
\(997\) 30.2402 0.957718 0.478859 0.877892i \(-0.341051\pi\)
0.478859 + 0.877892i \(0.341051\pi\)
\(998\) −26.8144 −0.848796
\(999\) −0.516377 −0.0163374
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.cy.1.5 6
11.7 odd 10 770.2.n.i.71.1 12
11.8 odd 10 770.2.n.i.141.1 yes 12
11.10 odd 2 8470.2.a.de.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.i.71.1 12 11.7 odd 10
770.2.n.i.141.1 yes 12 11.8 odd 10
8470.2.a.cy.1.5 6 1.1 even 1 trivial
8470.2.a.de.1.5 6 11.10 odd 2