Properties

Label 3234.2.a.bb.1.1
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.23607 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.23607 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.23607 q^{10} -1.00000 q^{11} -1.00000 q^{12} +3.23607 q^{15} +1.00000 q^{16} -0.763932 q^{17} +1.00000 q^{18} +5.70820 q^{19} -3.23607 q^{20} -1.00000 q^{22} +6.47214 q^{23} -1.00000 q^{24} +5.47214 q^{25} -1.00000 q^{27} -4.47214 q^{29} +3.23607 q^{30} -7.23607 q^{31} +1.00000 q^{32} +1.00000 q^{33} -0.763932 q^{34} +1.00000 q^{36} +6.94427 q^{37} +5.70820 q^{38} -3.23607 q^{40} -0.763932 q^{41} -2.47214 q^{43} -1.00000 q^{44} -3.23607 q^{45} +6.47214 q^{46} -9.70820 q^{47} -1.00000 q^{48} +5.47214 q^{50} +0.763932 q^{51} -6.00000 q^{53} -1.00000 q^{54} +3.23607 q^{55} -5.70820 q^{57} -4.47214 q^{58} -10.4721 q^{59} +3.23607 q^{60} -7.23607 q^{62} +1.00000 q^{64} +1.00000 q^{66} -11.4164 q^{67} -0.763932 q^{68} -6.47214 q^{69} +6.47214 q^{71} +1.00000 q^{72} -2.29180 q^{73} +6.94427 q^{74} -5.47214 q^{75} +5.70820 q^{76} -15.4164 q^{79} -3.23607 q^{80} +1.00000 q^{81} -0.763932 q^{82} -3.23607 q^{83} +2.47214 q^{85} -2.47214 q^{86} +4.47214 q^{87} -1.00000 q^{88} +2.47214 q^{89} -3.23607 q^{90} +6.47214 q^{92} +7.23607 q^{93} -9.70820 q^{94} -18.4721 q^{95} -1.00000 q^{96} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 2 q^{12} + 2 q^{15} + 2 q^{16} - 6 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{27} + 2 q^{30} - 10 q^{31} + 2 q^{32} + 2 q^{33} - 6 q^{34} + 2 q^{36} - 4 q^{37} - 2 q^{38} - 2 q^{40} - 6 q^{41} + 4 q^{43} - 2 q^{44} - 2 q^{45} + 4 q^{46} - 6 q^{47} - 2 q^{48} + 2 q^{50} + 6 q^{51} - 12 q^{53} - 2 q^{54} + 2 q^{55} + 2 q^{57} - 12 q^{59} + 2 q^{60} - 10 q^{62} + 2 q^{64} + 2 q^{66} + 4 q^{67} - 6 q^{68} - 4 q^{69} + 4 q^{71} + 2 q^{72} - 18 q^{73} - 4 q^{74} - 2 q^{75} - 2 q^{76} - 4 q^{79} - 2 q^{80} + 2 q^{81} - 6 q^{82} - 2 q^{83} - 4 q^{85} + 4 q^{86} - 2 q^{88} - 4 q^{89} - 2 q^{90} + 4 q^{92} + 10 q^{93} - 6 q^{94} - 28 q^{95} - 2 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −3.23607 −1.44721 −0.723607 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.23607 −1.02333
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 3.23607 0.835549
\(16\) 1.00000 0.250000
\(17\) −0.763932 −0.185281 −0.0926404 0.995700i \(-0.529531\pi\)
−0.0926404 + 0.995700i \(0.529531\pi\)
\(18\) 1.00000 0.235702
\(19\) 5.70820 1.30955 0.654776 0.755823i \(-0.272763\pi\)
0.654776 + 0.755823i \(0.272763\pi\)
\(20\) −3.23607 −0.723607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 6.47214 1.34953 0.674767 0.738031i \(-0.264244\pi\)
0.674767 + 0.738031i \(0.264244\pi\)
\(24\) −1.00000 −0.204124
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 3.23607 0.590822
\(31\) −7.23607 −1.29964 −0.649818 0.760090i \(-0.725155\pi\)
−0.649818 + 0.760090i \(0.725155\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −0.763932 −0.131013
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.94427 1.14163 0.570816 0.821078i \(-0.306627\pi\)
0.570816 + 0.821078i \(0.306627\pi\)
\(38\) 5.70820 0.925993
\(39\) 0 0
\(40\) −3.23607 −0.511667
\(41\) −0.763932 −0.119306 −0.0596531 0.998219i \(-0.518999\pi\)
−0.0596531 + 0.998219i \(0.518999\pi\)
\(42\) 0 0
\(43\) −2.47214 −0.376997 −0.188499 0.982073i \(-0.560362\pi\)
−0.188499 + 0.982073i \(0.560362\pi\)
\(44\) −1.00000 −0.150756
\(45\) −3.23607 −0.482405
\(46\) 6.47214 0.954264
\(47\) −9.70820 −1.41609 −0.708044 0.706169i \(-0.750422\pi\)
−0.708044 + 0.706169i \(0.750422\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 5.47214 0.773877
\(51\) 0.763932 0.106972
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 3.23607 0.436351
\(56\) 0 0
\(57\) −5.70820 −0.756070
\(58\) −4.47214 −0.587220
\(59\) −10.4721 −1.36336 −0.681678 0.731652i \(-0.738749\pi\)
−0.681678 + 0.731652i \(0.738749\pi\)
\(60\) 3.23607 0.417775
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −7.23607 −0.918982
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) −11.4164 −1.39474 −0.697368 0.716713i \(-0.745646\pi\)
−0.697368 + 0.716713i \(0.745646\pi\)
\(68\) −0.763932 −0.0926404
\(69\) −6.47214 −0.779154
\(70\) 0 0
\(71\) 6.47214 0.768101 0.384051 0.923312i \(-0.374529\pi\)
0.384051 + 0.923312i \(0.374529\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.29180 −0.268234 −0.134117 0.990965i \(-0.542820\pi\)
−0.134117 + 0.990965i \(0.542820\pi\)
\(74\) 6.94427 0.807255
\(75\) −5.47214 −0.631868
\(76\) 5.70820 0.654776
\(77\) 0 0
\(78\) 0 0
\(79\) −15.4164 −1.73448 −0.867241 0.497889i \(-0.834109\pi\)
−0.867241 + 0.497889i \(0.834109\pi\)
\(80\) −3.23607 −0.361803
\(81\) 1.00000 0.111111
\(82\) −0.763932 −0.0843622
\(83\) −3.23607 −0.355205 −0.177602 0.984102i \(-0.556834\pi\)
−0.177602 + 0.984102i \(0.556834\pi\)
\(84\) 0 0
\(85\) 2.47214 0.268141
\(86\) −2.47214 −0.266577
\(87\) 4.47214 0.479463
\(88\) −1.00000 −0.106600
\(89\) 2.47214 0.262046 0.131023 0.991379i \(-0.458174\pi\)
0.131023 + 0.991379i \(0.458174\pi\)
\(90\) −3.23607 −0.341112
\(91\) 0 0
\(92\) 6.47214 0.674767
\(93\) 7.23607 0.750345
\(94\) −9.70820 −1.00132
\(95\) −18.4721 −1.89520
\(96\) −1.00000 −0.102062
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 5.47214 0.547214
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0.763932 0.0756405
\(103\) 15.2361 1.50125 0.750627 0.660726i \(-0.229751\pi\)
0.750627 + 0.660726i \(0.229751\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −16.9443 −1.63806 −0.819032 0.573747i \(-0.805489\pi\)
−0.819032 + 0.573747i \(0.805489\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 3.23607 0.308547
\(111\) −6.94427 −0.659121
\(112\) 0 0
\(113\) −13.4164 −1.26211 −0.631055 0.775738i \(-0.717378\pi\)
−0.631055 + 0.775738i \(0.717378\pi\)
\(114\) −5.70820 −0.534622
\(115\) −20.9443 −1.95306
\(116\) −4.47214 −0.415227
\(117\) 0 0
\(118\) −10.4721 −0.964038
\(119\) 0 0
\(120\) 3.23607 0.295411
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0.763932 0.0688814
\(124\) −7.23607 −0.649818
\(125\) −1.52786 −0.136656
\(126\) 0 0
\(127\) 2.47214 0.219367 0.109683 0.993967i \(-0.465016\pi\)
0.109683 + 0.993967i \(0.465016\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.47214 0.217659
\(130\) 0 0
\(131\) 11.2361 0.981700 0.490850 0.871244i \(-0.336686\pi\)
0.490850 + 0.871244i \(0.336686\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) −11.4164 −0.986227
\(135\) 3.23607 0.278516
\(136\) −0.763932 −0.0655066
\(137\) −7.52786 −0.643149 −0.321574 0.946884i \(-0.604212\pi\)
−0.321574 + 0.946884i \(0.604212\pi\)
\(138\) −6.47214 −0.550945
\(139\) 12.1803 1.03312 0.516561 0.856250i \(-0.327212\pi\)
0.516561 + 0.856250i \(0.327212\pi\)
\(140\) 0 0
\(141\) 9.70820 0.817578
\(142\) 6.47214 0.543130
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 14.4721 1.20185
\(146\) −2.29180 −0.189670
\(147\) 0 0
\(148\) 6.94427 0.570816
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −5.47214 −0.446798
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 5.70820 0.462996
\(153\) −0.763932 −0.0617602
\(154\) 0 0
\(155\) 23.4164 1.88085
\(156\) 0 0
\(157\) −19.2361 −1.53521 −0.767603 0.640926i \(-0.778551\pi\)
−0.767603 + 0.640926i \(0.778551\pi\)
\(158\) −15.4164 −1.22646
\(159\) 6.00000 0.475831
\(160\) −3.23607 −0.255834
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −0.763932 −0.0596531
\(165\) −3.23607 −0.251928
\(166\) −3.23607 −0.251168
\(167\) 9.52786 0.737288 0.368644 0.929571i \(-0.379822\pi\)
0.368644 + 0.929571i \(0.379822\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 2.47214 0.189604
\(171\) 5.70820 0.436517
\(172\) −2.47214 −0.188499
\(173\) −7.41641 −0.563859 −0.281930 0.959435i \(-0.590974\pi\)
−0.281930 + 0.959435i \(0.590974\pi\)
\(174\) 4.47214 0.339032
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 10.4721 0.787134
\(178\) 2.47214 0.185294
\(179\) −14.4721 −1.08170 −0.540849 0.841120i \(-0.681897\pi\)
−0.540849 + 0.841120i \(0.681897\pi\)
\(180\) −3.23607 −0.241202
\(181\) −1.70820 −0.126970 −0.0634849 0.997983i \(-0.520221\pi\)
−0.0634849 + 0.997983i \(0.520221\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6.47214 0.477132
\(185\) −22.4721 −1.65218
\(186\) 7.23607 0.530574
\(187\) 0.763932 0.0558642
\(188\) −9.70820 −0.708044
\(189\) 0 0
\(190\) −18.4721 −1.34011
\(191\) −1.52786 −0.110552 −0.0552762 0.998471i \(-0.517604\pi\)
−0.0552762 + 0.998471i \(0.517604\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 22.9443 1.65156 0.825782 0.563989i \(-0.190734\pi\)
0.825782 + 0.563989i \(0.190734\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.3607 1.59313 0.796566 0.604551i \(-0.206648\pi\)
0.796566 + 0.604551i \(0.206648\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 17.1246 1.21393 0.606966 0.794728i \(-0.292386\pi\)
0.606966 + 0.794728i \(0.292386\pi\)
\(200\) 5.47214 0.386938
\(201\) 11.4164 0.805251
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) 0.763932 0.0534859
\(205\) 2.47214 0.172661
\(206\) 15.2361 1.06155
\(207\) 6.47214 0.449845
\(208\) 0 0
\(209\) −5.70820 −0.394845
\(210\) 0 0
\(211\) −23.4164 −1.61205 −0.806026 0.591880i \(-0.798386\pi\)
−0.806026 + 0.591880i \(0.798386\pi\)
\(212\) −6.00000 −0.412082
\(213\) −6.47214 −0.443463
\(214\) −16.9443 −1.15829
\(215\) 8.00000 0.545595
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 6.00000 0.406371
\(219\) 2.29180 0.154865
\(220\) 3.23607 0.218176
\(221\) 0 0
\(222\) −6.94427 −0.466069
\(223\) −2.29180 −0.153470 −0.0767350 0.997052i \(-0.524450\pi\)
−0.0767350 + 0.997052i \(0.524450\pi\)
\(224\) 0 0
\(225\) 5.47214 0.364809
\(226\) −13.4164 −0.892446
\(227\) −22.6525 −1.50350 −0.751749 0.659450i \(-0.770789\pi\)
−0.751749 + 0.659450i \(0.770789\pi\)
\(228\) −5.70820 −0.378035
\(229\) 6.29180 0.415774 0.207887 0.978153i \(-0.433341\pi\)
0.207887 + 0.978153i \(0.433341\pi\)
\(230\) −20.9443 −1.38102
\(231\) 0 0
\(232\) −4.47214 −0.293610
\(233\) −26.9443 −1.76518 −0.882589 0.470145i \(-0.844201\pi\)
−0.882589 + 0.470145i \(0.844201\pi\)
\(234\) 0 0
\(235\) 31.4164 2.04938
\(236\) −10.4721 −0.681678
\(237\) 15.4164 1.00140
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 3.23607 0.208887
\(241\) −4.18034 −0.269279 −0.134640 0.990895i \(-0.542988\pi\)
−0.134640 + 0.990895i \(0.542988\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0.763932 0.0487065
\(247\) 0 0
\(248\) −7.23607 −0.459491
\(249\) 3.23607 0.205077
\(250\) −1.52786 −0.0966306
\(251\) 2.47214 0.156040 0.0780199 0.996952i \(-0.475140\pi\)
0.0780199 + 0.996952i \(0.475140\pi\)
\(252\) 0 0
\(253\) −6.47214 −0.406900
\(254\) 2.47214 0.155116
\(255\) −2.47214 −0.154811
\(256\) 1.00000 0.0625000
\(257\) 13.5279 0.843845 0.421922 0.906632i \(-0.361355\pi\)
0.421922 + 0.906632i \(0.361355\pi\)
\(258\) 2.47214 0.153908
\(259\) 0 0
\(260\) 0 0
\(261\) −4.47214 −0.276818
\(262\) 11.2361 0.694167
\(263\) −18.4721 −1.13904 −0.569520 0.821977i \(-0.692871\pi\)
−0.569520 + 0.821977i \(0.692871\pi\)
\(264\) 1.00000 0.0615457
\(265\) 19.4164 1.19274
\(266\) 0 0
\(267\) −2.47214 −0.151292
\(268\) −11.4164 −0.697368
\(269\) −14.2918 −0.871386 −0.435693 0.900095i \(-0.643497\pi\)
−0.435693 + 0.900095i \(0.643497\pi\)
\(270\) 3.23607 0.196941
\(271\) 6.47214 0.393154 0.196577 0.980488i \(-0.437017\pi\)
0.196577 + 0.980488i \(0.437017\pi\)
\(272\) −0.763932 −0.0463202
\(273\) 0 0
\(274\) −7.52786 −0.454775
\(275\) −5.47214 −0.329982
\(276\) −6.47214 −0.389577
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 12.1803 0.730528
\(279\) −7.23607 −0.433212
\(280\) 0 0
\(281\) −15.8885 −0.947831 −0.473916 0.880570i \(-0.657160\pi\)
−0.473916 + 0.880570i \(0.657160\pi\)
\(282\) 9.70820 0.578115
\(283\) 10.2918 0.611784 0.305892 0.952066i \(-0.401045\pi\)
0.305892 + 0.952066i \(0.401045\pi\)
\(284\) 6.47214 0.384051
\(285\) 18.4721 1.09419
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −16.4164 −0.965671
\(290\) 14.4721 0.849833
\(291\) 0 0
\(292\) −2.29180 −0.134117
\(293\) −31.4164 −1.83537 −0.917683 0.397313i \(-0.869943\pi\)
−0.917683 + 0.397313i \(0.869943\pi\)
\(294\) 0 0
\(295\) 33.8885 1.97307
\(296\) 6.94427 0.403628
\(297\) 1.00000 0.0580259
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −5.47214 −0.315934
\(301\) 0 0
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) 5.70820 0.327388
\(305\) 0 0
\(306\) −0.763932 −0.0436711
\(307\) −23.5967 −1.34674 −0.673369 0.739307i \(-0.735153\pi\)
−0.673369 + 0.739307i \(0.735153\pi\)
\(308\) 0 0
\(309\) −15.2361 −0.866750
\(310\) 23.4164 1.32996
\(311\) 22.6525 1.28450 0.642252 0.766494i \(-0.278000\pi\)
0.642252 + 0.766494i \(0.278000\pi\)
\(312\) 0 0
\(313\) 14.4721 0.818013 0.409007 0.912531i \(-0.365875\pi\)
0.409007 + 0.912531i \(0.365875\pi\)
\(314\) −19.2361 −1.08555
\(315\) 0 0
\(316\) −15.4164 −0.867241
\(317\) 3.88854 0.218402 0.109201 0.994020i \(-0.465171\pi\)
0.109201 + 0.994020i \(0.465171\pi\)
\(318\) 6.00000 0.336463
\(319\) 4.47214 0.250392
\(320\) −3.23607 −0.180902
\(321\) 16.9443 0.945737
\(322\) 0 0
\(323\) −4.36068 −0.242635
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) −6.00000 −0.331801
\(328\) −0.763932 −0.0421811
\(329\) 0 0
\(330\) −3.23607 −0.178140
\(331\) 19.4164 1.06722 0.533611 0.845730i \(-0.320835\pi\)
0.533611 + 0.845730i \(0.320835\pi\)
\(332\) −3.23607 −0.177602
\(333\) 6.94427 0.380544
\(334\) 9.52786 0.521342
\(335\) 36.9443 2.01848
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −13.0000 −0.707107
\(339\) 13.4164 0.728679
\(340\) 2.47214 0.134070
\(341\) 7.23607 0.391855
\(342\) 5.70820 0.308664
\(343\) 0 0
\(344\) −2.47214 −0.133289
\(345\) 20.9443 1.12760
\(346\) −7.41641 −0.398709
\(347\) 26.8328 1.44046 0.720231 0.693735i \(-0.244036\pi\)
0.720231 + 0.693735i \(0.244036\pi\)
\(348\) 4.47214 0.239732
\(349\) −27.4164 −1.46757 −0.733783 0.679384i \(-0.762247\pi\)
−0.733783 + 0.679384i \(0.762247\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −20.3607 −1.08369 −0.541845 0.840479i \(-0.682274\pi\)
−0.541845 + 0.840479i \(0.682274\pi\)
\(354\) 10.4721 0.556588
\(355\) −20.9443 −1.11161
\(356\) 2.47214 0.131023
\(357\) 0 0
\(358\) −14.4721 −0.764876
\(359\) 10.4721 0.552698 0.276349 0.961057i \(-0.410875\pi\)
0.276349 + 0.961057i \(0.410875\pi\)
\(360\) −3.23607 −0.170556
\(361\) 13.5836 0.714926
\(362\) −1.70820 −0.0897812
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 7.41641 0.388193
\(366\) 0 0
\(367\) 21.7082 1.13316 0.566580 0.824007i \(-0.308266\pi\)
0.566580 + 0.824007i \(0.308266\pi\)
\(368\) 6.47214 0.337383
\(369\) −0.763932 −0.0397687
\(370\) −22.4721 −1.16827
\(371\) 0 0
\(372\) 7.23607 0.375173
\(373\) 25.4164 1.31601 0.658006 0.753013i \(-0.271400\pi\)
0.658006 + 0.753013i \(0.271400\pi\)
\(374\) 0.763932 0.0395020
\(375\) 1.52786 0.0788986
\(376\) −9.70820 −0.500662
\(377\) 0 0
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −18.4721 −0.947601
\(381\) −2.47214 −0.126651
\(382\) −1.52786 −0.0781723
\(383\) 30.6525 1.56627 0.783134 0.621853i \(-0.213620\pi\)
0.783134 + 0.621853i \(0.213620\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 22.9443 1.16783
\(387\) −2.47214 −0.125666
\(388\) 0 0
\(389\) 10.9443 0.554897 0.277448 0.960741i \(-0.410511\pi\)
0.277448 + 0.960741i \(0.410511\pi\)
\(390\) 0 0
\(391\) −4.94427 −0.250043
\(392\) 0 0
\(393\) −11.2361 −0.566785
\(394\) 22.3607 1.12651
\(395\) 49.8885 2.51017
\(396\) −1.00000 −0.0502519
\(397\) −24.1803 −1.21358 −0.606788 0.794864i \(-0.707542\pi\)
−0.606788 + 0.794864i \(0.707542\pi\)
\(398\) 17.1246 0.858379
\(399\) 0 0
\(400\) 5.47214 0.273607
\(401\) 0.472136 0.0235773 0.0117887 0.999931i \(-0.496247\pi\)
0.0117887 + 0.999931i \(0.496247\pi\)
\(402\) 11.4164 0.569399
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) −3.23607 −0.160802
\(406\) 0 0
\(407\) −6.94427 −0.344215
\(408\) 0.763932 0.0378203
\(409\) 7.23607 0.357801 0.178900 0.983867i \(-0.442746\pi\)
0.178900 + 0.983867i \(0.442746\pi\)
\(410\) 2.47214 0.122090
\(411\) 7.52786 0.371322
\(412\) 15.2361 0.750627
\(413\) 0 0
\(414\) 6.47214 0.318088
\(415\) 10.4721 0.514057
\(416\) 0 0
\(417\) −12.1803 −0.596474
\(418\) −5.70820 −0.279197
\(419\) 32.9443 1.60943 0.804717 0.593659i \(-0.202317\pi\)
0.804717 + 0.593659i \(0.202317\pi\)
\(420\) 0 0
\(421\) −22.9443 −1.11824 −0.559118 0.829088i \(-0.688860\pi\)
−0.559118 + 0.829088i \(0.688860\pi\)
\(422\) −23.4164 −1.13989
\(423\) −9.70820 −0.472029
\(424\) −6.00000 −0.291386
\(425\) −4.18034 −0.202776
\(426\) −6.47214 −0.313576
\(427\) 0 0
\(428\) −16.9443 −0.819032
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) −13.5279 −0.651614 −0.325807 0.945436i \(-0.605636\pi\)
−0.325807 + 0.945436i \(0.605636\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −25.8885 −1.24412 −0.622062 0.782968i \(-0.713705\pi\)
−0.622062 + 0.782968i \(0.713705\pi\)
\(434\) 0 0
\(435\) −14.4721 −0.693886
\(436\) 6.00000 0.287348
\(437\) 36.9443 1.76728
\(438\) 2.29180 0.109506
\(439\) 4.58359 0.218763 0.109381 0.994000i \(-0.465113\pi\)
0.109381 + 0.994000i \(0.465113\pi\)
\(440\) 3.23607 0.154273
\(441\) 0 0
\(442\) 0 0
\(443\) −19.4164 −0.922501 −0.461251 0.887270i \(-0.652599\pi\)
−0.461251 + 0.887270i \(0.652599\pi\)
\(444\) −6.94427 −0.329561
\(445\) −8.00000 −0.379236
\(446\) −2.29180 −0.108520
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 28.4721 1.34368 0.671842 0.740695i \(-0.265504\pi\)
0.671842 + 0.740695i \(0.265504\pi\)
\(450\) 5.47214 0.257959
\(451\) 0.763932 0.0359722
\(452\) −13.4164 −0.631055
\(453\) 0 0
\(454\) −22.6525 −1.06313
\(455\) 0 0
\(456\) −5.70820 −0.267311
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 6.29180 0.293996
\(459\) 0.763932 0.0356573
\(460\) −20.9443 −0.976532
\(461\) 26.8328 1.24973 0.624864 0.780733i \(-0.285154\pi\)
0.624864 + 0.780733i \(0.285154\pi\)
\(462\) 0 0
\(463\) 33.8885 1.57493 0.787467 0.616357i \(-0.211392\pi\)
0.787467 + 0.616357i \(0.211392\pi\)
\(464\) −4.47214 −0.207614
\(465\) −23.4164 −1.08591
\(466\) −26.9443 −1.24817
\(467\) 5.88854 0.272489 0.136245 0.990675i \(-0.456497\pi\)
0.136245 + 0.990675i \(0.456497\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 31.4164 1.44913
\(471\) 19.2361 0.886351
\(472\) −10.4721 −0.482019
\(473\) 2.47214 0.113669
\(474\) 15.4164 0.708099
\(475\) 31.2361 1.43321
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) −35.7771 −1.63470 −0.817348 0.576144i \(-0.804557\pi\)
−0.817348 + 0.576144i \(0.804557\pi\)
\(480\) 3.23607 0.147706
\(481\) 0 0
\(482\) −4.18034 −0.190409
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 19.0557 0.863497 0.431749 0.901994i \(-0.357897\pi\)
0.431749 + 0.901994i \(0.357897\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 5.88854 0.265746 0.132873 0.991133i \(-0.457580\pi\)
0.132873 + 0.991133i \(0.457580\pi\)
\(492\) 0.763932 0.0344407
\(493\) 3.41641 0.153867
\(494\) 0 0
\(495\) 3.23607 0.145450
\(496\) −7.23607 −0.324909
\(497\) 0 0
\(498\) 3.23607 0.145012
\(499\) −19.4164 −0.869198 −0.434599 0.900624i \(-0.643110\pi\)
−0.434599 + 0.900624i \(0.643110\pi\)
\(500\) −1.52786 −0.0683282
\(501\) −9.52786 −0.425674
\(502\) 2.47214 0.110337
\(503\) 21.3050 0.949941 0.474970 0.880002i \(-0.342459\pi\)
0.474970 + 0.880002i \(0.342459\pi\)
\(504\) 0 0
\(505\) 38.8328 1.72804
\(506\) −6.47214 −0.287722
\(507\) 13.0000 0.577350
\(508\) 2.47214 0.109683
\(509\) −11.2361 −0.498030 −0.249015 0.968500i \(-0.580107\pi\)
−0.249015 + 0.968500i \(0.580107\pi\)
\(510\) −2.47214 −0.109468
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −5.70820 −0.252023
\(514\) 13.5279 0.596689
\(515\) −49.3050 −2.17264
\(516\) 2.47214 0.108830
\(517\) 9.70820 0.426966
\(518\) 0 0
\(519\) 7.41641 0.325544
\(520\) 0 0
\(521\) 41.3050 1.80960 0.904801 0.425834i \(-0.140019\pi\)
0.904801 + 0.425834i \(0.140019\pi\)
\(522\) −4.47214 −0.195740
\(523\) −37.7082 −1.64886 −0.824432 0.565961i \(-0.808505\pi\)
−0.824432 + 0.565961i \(0.808505\pi\)
\(524\) 11.2361 0.490850
\(525\) 0 0
\(526\) −18.4721 −0.805423
\(527\) 5.52786 0.240798
\(528\) 1.00000 0.0435194
\(529\) 18.8885 0.821241
\(530\) 19.4164 0.843395
\(531\) −10.4721 −0.454452
\(532\) 0 0
\(533\) 0 0
\(534\) −2.47214 −0.106980
\(535\) 54.8328 2.37063
\(536\) −11.4164 −0.493114
\(537\) 14.4721 0.624519
\(538\) −14.2918 −0.616163
\(539\) 0 0
\(540\) 3.23607 0.139258
\(541\) −5.41641 −0.232870 −0.116435 0.993198i \(-0.537147\pi\)
−0.116435 + 0.993198i \(0.537147\pi\)
\(542\) 6.47214 0.278002
\(543\) 1.70820 0.0733060
\(544\) −0.763932 −0.0327533
\(545\) −19.4164 −0.831708
\(546\) 0 0
\(547\) 29.5279 1.26252 0.631260 0.775571i \(-0.282538\pi\)
0.631260 + 0.775571i \(0.282538\pi\)
\(548\) −7.52786 −0.321574
\(549\) 0 0
\(550\) −5.47214 −0.233333
\(551\) −25.5279 −1.08752
\(552\) −6.47214 −0.275472
\(553\) 0 0
\(554\) 26.0000 1.10463
\(555\) 22.4721 0.953889
\(556\) 12.1803 0.516561
\(557\) −37.4164 −1.58538 −0.792692 0.609622i \(-0.791321\pi\)
−0.792692 + 0.609622i \(0.791321\pi\)
\(558\) −7.23607 −0.306327
\(559\) 0 0
\(560\) 0 0
\(561\) −0.763932 −0.0322532
\(562\) −15.8885 −0.670218
\(563\) 11.2361 0.473544 0.236772 0.971565i \(-0.423911\pi\)
0.236772 + 0.971565i \(0.423911\pi\)
\(564\) 9.70820 0.408789
\(565\) 43.4164 1.82654
\(566\) 10.2918 0.432596
\(567\) 0 0
\(568\) 6.47214 0.271565
\(569\) −17.0557 −0.715013 −0.357507 0.933911i \(-0.616373\pi\)
−0.357507 + 0.933911i \(0.616373\pi\)
\(570\) 18.4721 0.773713
\(571\) −26.4721 −1.10782 −0.553912 0.832575i \(-0.686866\pi\)
−0.553912 + 0.832575i \(0.686866\pi\)
\(572\) 0 0
\(573\) 1.52786 0.0638274
\(574\) 0 0
\(575\) 35.4164 1.47697
\(576\) 1.00000 0.0416667
\(577\) −12.5836 −0.523862 −0.261931 0.965087i \(-0.584359\pi\)
−0.261931 + 0.965087i \(0.584359\pi\)
\(578\) −16.4164 −0.682833
\(579\) −22.9443 −0.953531
\(580\) 14.4721 0.600923
\(581\) 0 0
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) −2.29180 −0.0948352
\(585\) 0 0
\(586\) −31.4164 −1.29780
\(587\) 26.4721 1.09262 0.546311 0.837582i \(-0.316032\pi\)
0.546311 + 0.837582i \(0.316032\pi\)
\(588\) 0 0
\(589\) −41.3050 −1.70194
\(590\) 33.8885 1.39517
\(591\) −22.3607 −0.919795
\(592\) 6.94427 0.285408
\(593\) 23.2361 0.954191 0.477095 0.878851i \(-0.341690\pi\)
0.477095 + 0.878851i \(0.341690\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −17.1246 −0.700864
\(598\) 0 0
\(599\) −37.3050 −1.52424 −0.762120 0.647436i \(-0.775841\pi\)
−0.762120 + 0.647436i \(0.775841\pi\)
\(600\) −5.47214 −0.223399
\(601\) −13.7082 −0.559169 −0.279585 0.960121i \(-0.590197\pi\)
−0.279585 + 0.960121i \(0.590197\pi\)
\(602\) 0 0
\(603\) −11.4164 −0.464912
\(604\) 0 0
\(605\) −3.23607 −0.131565
\(606\) 12.0000 0.487467
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 5.70820 0.231498
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −0.763932 −0.0308801
\(613\) 3.52786 0.142489 0.0712445 0.997459i \(-0.477303\pi\)
0.0712445 + 0.997459i \(0.477303\pi\)
\(614\) −23.5967 −0.952287
\(615\) −2.47214 −0.0996861
\(616\) 0 0
\(617\) −11.8885 −0.478615 −0.239307 0.970944i \(-0.576920\pi\)
−0.239307 + 0.970944i \(0.576920\pi\)
\(618\) −15.2361 −0.612885
\(619\) −34.8328 −1.40005 −0.700025 0.714119i \(-0.746828\pi\)
−0.700025 + 0.714119i \(0.746828\pi\)
\(620\) 23.4164 0.940426
\(621\) −6.47214 −0.259718
\(622\) 22.6525 0.908282
\(623\) 0 0
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 14.4721 0.578423
\(627\) 5.70820 0.227964
\(628\) −19.2361 −0.767603
\(629\) −5.30495 −0.211522
\(630\) 0 0
\(631\) −25.8885 −1.03061 −0.515303 0.857008i \(-0.672321\pi\)
−0.515303 + 0.857008i \(0.672321\pi\)
\(632\) −15.4164 −0.613232
\(633\) 23.4164 0.930719
\(634\) 3.88854 0.154434
\(635\) −8.00000 −0.317470
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 4.47214 0.177054
\(639\) 6.47214 0.256034
\(640\) −3.23607 −0.127917
\(641\) −0.111456 −0.00440225 −0.00220113 0.999998i \(-0.500701\pi\)
−0.00220113 + 0.999998i \(0.500701\pi\)
\(642\) 16.9443 0.668737
\(643\) 16.5836 0.653993 0.326997 0.945026i \(-0.393963\pi\)
0.326997 + 0.945026i \(0.393963\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) −4.36068 −0.171569
\(647\) −31.0132 −1.21925 −0.609626 0.792689i \(-0.708681\pi\)
−0.609626 + 0.792689i \(0.708681\pi\)
\(648\) 1.00000 0.0392837
\(649\) 10.4721 0.411067
\(650\) 0 0
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) 2.94427 0.115218 0.0576091 0.998339i \(-0.481652\pi\)
0.0576091 + 0.998339i \(0.481652\pi\)
\(654\) −6.00000 −0.234619
\(655\) −36.3607 −1.42073
\(656\) −0.763932 −0.0298265
\(657\) −2.29180 −0.0894115
\(658\) 0 0
\(659\) −21.8885 −0.852657 −0.426328 0.904569i \(-0.640193\pi\)
−0.426328 + 0.904569i \(0.640193\pi\)
\(660\) −3.23607 −0.125964
\(661\) −43.5967 −1.69572 −0.847858 0.530223i \(-0.822108\pi\)
−0.847858 + 0.530223i \(0.822108\pi\)
\(662\) 19.4164 0.754640
\(663\) 0 0
\(664\) −3.23607 −0.125584
\(665\) 0 0
\(666\) 6.94427 0.269085
\(667\) −28.9443 −1.12073
\(668\) 9.52786 0.368644
\(669\) 2.29180 0.0886060
\(670\) 36.9443 1.42728
\(671\) 0 0
\(672\) 0 0
\(673\) 17.0557 0.657450 0.328725 0.944426i \(-0.393381\pi\)
0.328725 + 0.944426i \(0.393381\pi\)
\(674\) −18.0000 −0.693334
\(675\) −5.47214 −0.210623
\(676\) −13.0000 −0.500000
\(677\) −37.5279 −1.44231 −0.721156 0.692772i \(-0.756389\pi\)
−0.721156 + 0.692772i \(0.756389\pi\)
\(678\) 13.4164 0.515254
\(679\) 0 0
\(680\) 2.47214 0.0948021
\(681\) 22.6525 0.868045
\(682\) 7.23607 0.277083
\(683\) 40.3607 1.54436 0.772179 0.635405i \(-0.219167\pi\)
0.772179 + 0.635405i \(0.219167\pi\)
\(684\) 5.70820 0.218259
\(685\) 24.3607 0.930774
\(686\) 0 0
\(687\) −6.29180 −0.240047
\(688\) −2.47214 −0.0942493
\(689\) 0 0
\(690\) 20.9443 0.797335
\(691\) −26.4721 −1.00705 −0.503524 0.863981i \(-0.667963\pi\)
−0.503524 + 0.863981i \(0.667963\pi\)
\(692\) −7.41641 −0.281930
\(693\) 0 0
\(694\) 26.8328 1.01856
\(695\) −39.4164 −1.49515
\(696\) 4.47214 0.169516
\(697\) 0.583592 0.0221051
\(698\) −27.4164 −1.03773
\(699\) 26.9443 1.01913
\(700\) 0 0
\(701\) −43.8885 −1.65765 −0.828824 0.559510i \(-0.810989\pi\)
−0.828824 + 0.559510i \(0.810989\pi\)
\(702\) 0 0
\(703\) 39.6393 1.49503
\(704\) −1.00000 −0.0376889
\(705\) −31.4164 −1.18321
\(706\) −20.3607 −0.766284
\(707\) 0 0
\(708\) 10.4721 0.393567
\(709\) 22.9443 0.861690 0.430845 0.902426i \(-0.358216\pi\)
0.430845 + 0.902426i \(0.358216\pi\)
\(710\) −20.9443 −0.786025
\(711\) −15.4164 −0.578160
\(712\) 2.47214 0.0926472
\(713\) −46.8328 −1.75390
\(714\) 0 0
\(715\) 0 0
\(716\) −14.4721 −0.540849
\(717\) 0 0
\(718\) 10.4721 0.390817
\(719\) 33.7082 1.25710 0.628552 0.777768i \(-0.283648\pi\)
0.628552 + 0.777768i \(0.283648\pi\)
\(720\) −3.23607 −0.120601
\(721\) 0 0
\(722\) 13.5836 0.505529
\(723\) 4.18034 0.155469
\(724\) −1.70820 −0.0634849
\(725\) −24.4721 −0.908872
\(726\) −1.00000 −0.0371135
\(727\) 40.7639 1.51185 0.755925 0.654658i \(-0.227187\pi\)
0.755925 + 0.654658i \(0.227187\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 7.41641 0.274494
\(731\) 1.88854 0.0698503
\(732\) 0 0
\(733\) 30.8328 1.13884 0.569418 0.822048i \(-0.307169\pi\)
0.569418 + 0.822048i \(0.307169\pi\)
\(734\) 21.7082 0.801264
\(735\) 0 0
\(736\) 6.47214 0.238566
\(737\) 11.4164 0.420529
\(738\) −0.763932 −0.0281207
\(739\) 16.5836 0.610037 0.305019 0.952346i \(-0.401337\pi\)
0.305019 + 0.952346i \(0.401337\pi\)
\(740\) −22.4721 −0.826092
\(741\) 0 0
\(742\) 0 0
\(743\) 9.88854 0.362775 0.181388 0.983412i \(-0.441941\pi\)
0.181388 + 0.983412i \(0.441941\pi\)
\(744\) 7.23607 0.265287
\(745\) 19.4164 0.711362
\(746\) 25.4164 0.930561
\(747\) −3.23607 −0.118402
\(748\) 0.763932 0.0279321
\(749\) 0 0
\(750\) 1.52786 0.0557897
\(751\) 38.8328 1.41703 0.708515 0.705696i \(-0.249366\pi\)
0.708515 + 0.705696i \(0.249366\pi\)
\(752\) −9.70820 −0.354022
\(753\) −2.47214 −0.0900896
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 5.05573 0.183754 0.0918768 0.995770i \(-0.470713\pi\)
0.0918768 + 0.995770i \(0.470713\pi\)
\(758\) −4.00000 −0.145287
\(759\) 6.47214 0.234924
\(760\) −18.4721 −0.670055
\(761\) −17.1246 −0.620767 −0.310383 0.950611i \(-0.600457\pi\)
−0.310383 + 0.950611i \(0.600457\pi\)
\(762\) −2.47214 −0.0895560
\(763\) 0 0
\(764\) −1.52786 −0.0552762
\(765\) 2.47214 0.0893803
\(766\) 30.6525 1.10752
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 46.0689 1.66129 0.830643 0.556805i \(-0.187973\pi\)
0.830643 + 0.556805i \(0.187973\pi\)
\(770\) 0 0
\(771\) −13.5279 −0.487194
\(772\) 22.9443 0.825782
\(773\) 43.9574 1.58104 0.790519 0.612437i \(-0.209811\pi\)
0.790519 + 0.612437i \(0.209811\pi\)
\(774\) −2.47214 −0.0888591
\(775\) −39.5967 −1.42236
\(776\) 0 0
\(777\) 0 0
\(778\) 10.9443 0.392371
\(779\) −4.36068 −0.156238
\(780\) 0 0
\(781\) −6.47214 −0.231591
\(782\) −4.94427 −0.176807
\(783\) 4.47214 0.159821
\(784\) 0 0
\(785\) 62.2492 2.22177
\(786\) −11.2361 −0.400777
\(787\) 44.5410 1.58772 0.793858 0.608103i \(-0.208069\pi\)
0.793858 + 0.608103i \(0.208069\pi\)
\(788\) 22.3607 0.796566
\(789\) 18.4721 0.657625
\(790\) 49.8885 1.77495
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) 0 0
\(794\) −24.1803 −0.858128
\(795\) −19.4164 −0.688629
\(796\) 17.1246 0.606966
\(797\) 53.1246 1.88177 0.940885 0.338726i \(-0.109996\pi\)
0.940885 + 0.338726i \(0.109996\pi\)
\(798\) 0 0
\(799\) 7.41641 0.262374
\(800\) 5.47214 0.193469
\(801\) 2.47214 0.0873486
\(802\) 0.472136 0.0166717
\(803\) 2.29180 0.0808757
\(804\) 11.4164 0.402626
\(805\) 0 0
\(806\) 0 0
\(807\) 14.2918 0.503095
\(808\) −12.0000 −0.422159
\(809\) −8.83282 −0.310545 −0.155273 0.987872i \(-0.549626\pi\)
−0.155273 + 0.987872i \(0.549626\pi\)
\(810\) −3.23607 −0.113704
\(811\) −31.5967 −1.10951 −0.554756 0.832013i \(-0.687188\pi\)
−0.554756 + 0.832013i \(0.687188\pi\)
\(812\) 0 0
\(813\) −6.47214 −0.226988
\(814\) −6.94427 −0.243397
\(815\) 38.8328 1.36025
\(816\) 0.763932 0.0267430
\(817\) −14.1115 −0.493697
\(818\) 7.23607 0.253003
\(819\) 0 0
\(820\) 2.47214 0.0863307
\(821\) −41.7771 −1.45803 −0.729015 0.684498i \(-0.760022\pi\)
−0.729015 + 0.684498i \(0.760022\pi\)
\(822\) 7.52786 0.262564
\(823\) 22.8328 0.795902 0.397951 0.917407i \(-0.369721\pi\)
0.397951 + 0.917407i \(0.369721\pi\)
\(824\) 15.2361 0.530774
\(825\) 5.47214 0.190515
\(826\) 0 0
\(827\) −52.7214 −1.83330 −0.916651 0.399689i \(-0.869118\pi\)
−0.916651 + 0.399689i \(0.869118\pi\)
\(828\) 6.47214 0.224922
\(829\) −37.4853 −1.30192 −0.650959 0.759113i \(-0.725633\pi\)
−0.650959 + 0.759113i \(0.725633\pi\)
\(830\) 10.4721 0.363493
\(831\) −26.0000 −0.901930
\(832\) 0 0
\(833\) 0 0
\(834\) −12.1803 −0.421771
\(835\) −30.8328 −1.06701
\(836\) −5.70820 −0.197422
\(837\) 7.23607 0.250115
\(838\) 32.9443 1.13804
\(839\) 20.7639 0.716851 0.358425 0.933558i \(-0.383314\pi\)
0.358425 + 0.933558i \(0.383314\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) −22.9443 −0.790712
\(843\) 15.8885 0.547231
\(844\) −23.4164 −0.806026
\(845\) 42.0689 1.44721
\(846\) −9.70820 −0.333775
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −10.2918 −0.353214
\(850\) −4.18034 −0.143384
\(851\) 44.9443 1.54067
\(852\) −6.47214 −0.221732
\(853\) 1.88854 0.0646625 0.0323313 0.999477i \(-0.489707\pi\)
0.0323313 + 0.999477i \(0.489707\pi\)
\(854\) 0 0
\(855\) −18.4721 −0.631734
\(856\) −16.9443 −0.579143
\(857\) 6.87539 0.234859 0.117429 0.993081i \(-0.462535\pi\)
0.117429 + 0.993081i \(0.462535\pi\)
\(858\) 0 0
\(859\) −16.5836 −0.565825 −0.282912 0.959146i \(-0.591301\pi\)
−0.282912 + 0.959146i \(0.591301\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −13.5279 −0.460761
\(863\) 0.360680 0.0122777 0.00613884 0.999981i \(-0.498046\pi\)
0.00613884 + 0.999981i \(0.498046\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 24.0000 0.816024
\(866\) −25.8885 −0.879729
\(867\) 16.4164 0.557530
\(868\) 0 0
\(869\) 15.4164 0.522966
\(870\) −14.4721 −0.490651
\(871\) 0 0
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) 36.9443 1.24966
\(875\) 0 0
\(876\) 2.29180 0.0774326
\(877\) 30.0000 1.01303 0.506514 0.862232i \(-0.330934\pi\)
0.506514 + 0.862232i \(0.330934\pi\)
\(878\) 4.58359 0.154689
\(879\) 31.4164 1.05965
\(880\) 3.23607 0.109088
\(881\) −26.8328 −0.904021 −0.452010 0.892013i \(-0.649293\pi\)
−0.452010 + 0.892013i \(0.649293\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) −33.8885 −1.13915
\(886\) −19.4164 −0.652307
\(887\) −19.0557 −0.639829 −0.319914 0.947446i \(-0.603654\pi\)
−0.319914 + 0.947446i \(0.603654\pi\)
\(888\) −6.94427 −0.233035
\(889\) 0 0
\(890\) −8.00000 −0.268161
\(891\) −1.00000 −0.0335013
\(892\) −2.29180 −0.0767350
\(893\) −55.4164 −1.85444
\(894\) 6.00000 0.200670
\(895\) 46.8328 1.56545
\(896\) 0 0
\(897\) 0 0
\(898\) 28.4721 0.950127
\(899\) 32.3607 1.07929
\(900\) 5.47214 0.182405
\(901\) 4.58359 0.152702
\(902\) 0.763932 0.0254362
\(903\) 0 0
\(904\) −13.4164 −0.446223
\(905\) 5.52786 0.183752
\(906\) 0 0
\(907\) 29.3050 0.973055 0.486527 0.873665i \(-0.338263\pi\)
0.486527 + 0.873665i \(0.338263\pi\)
\(908\) −22.6525 −0.751749
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −4.58359 −0.151861 −0.0759306 0.997113i \(-0.524193\pi\)
−0.0759306 + 0.997113i \(0.524193\pi\)
\(912\) −5.70820 −0.189018
\(913\) 3.23607 0.107098
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) 6.29180 0.207887
\(917\) 0 0
\(918\) 0.763932 0.0252135
\(919\) 53.6656 1.77027 0.885133 0.465338i \(-0.154067\pi\)
0.885133 + 0.465338i \(0.154067\pi\)
\(920\) −20.9443 −0.690512
\(921\) 23.5967 0.777539
\(922\) 26.8328 0.883692
\(923\) 0 0
\(924\) 0 0
\(925\) 38.0000 1.24943
\(926\) 33.8885 1.11365
\(927\) 15.2361 0.500418
\(928\) −4.47214 −0.146805
\(929\) −36.3607 −1.19296 −0.596478 0.802630i \(-0.703434\pi\)
−0.596478 + 0.802630i \(0.703434\pi\)
\(930\) −23.4164 −0.767854
\(931\) 0 0
\(932\) −26.9443 −0.882589
\(933\) −22.6525 −0.741609
\(934\) 5.88854 0.192679
\(935\) −2.47214 −0.0808475
\(936\) 0 0
\(937\) 47.5967 1.55492 0.777459 0.628934i \(-0.216508\pi\)
0.777459 + 0.628934i \(0.216508\pi\)
\(938\) 0 0
\(939\) −14.4721 −0.472280
\(940\) 31.4164 1.02469
\(941\) −52.3607 −1.70691 −0.853455 0.521167i \(-0.825497\pi\)
−0.853455 + 0.521167i \(0.825497\pi\)
\(942\) 19.2361 0.626745
\(943\) −4.94427 −0.161008
\(944\) −10.4721 −0.340839
\(945\) 0 0
\(946\) 2.47214 0.0803761
\(947\) −2.11146 −0.0686131 −0.0343066 0.999411i \(-0.510922\pi\)
−0.0343066 + 0.999411i \(0.510922\pi\)
\(948\) 15.4164 0.500702
\(949\) 0 0
\(950\) 31.2361 1.01343
\(951\) −3.88854 −0.126095
\(952\) 0 0
\(953\) −13.0557 −0.422917 −0.211458 0.977387i \(-0.567821\pi\)
−0.211458 + 0.977387i \(0.567821\pi\)
\(954\) −6.00000 −0.194257
\(955\) 4.94427 0.159993
\(956\) 0 0
\(957\) −4.47214 −0.144564
\(958\) −35.7771 −1.15591
\(959\) 0 0
\(960\) 3.23607 0.104444
\(961\) 21.3607 0.689054
\(962\) 0 0
\(963\) −16.9443 −0.546022
\(964\) −4.18034 −0.134640
\(965\) −74.2492 −2.39017
\(966\) 0 0
\(967\) 1.16718 0.0375341 0.0187671 0.999824i \(-0.494026\pi\)
0.0187671 + 0.999824i \(0.494026\pi\)
\(968\) 1.00000 0.0321412
\(969\) 4.36068 0.140085
\(970\) 0 0
\(971\) −56.9443 −1.82743 −0.913714 0.406357i \(-0.866799\pi\)
−0.913714 + 0.406357i \(0.866799\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 19.0557 0.610585
\(975\) 0 0
\(976\) 0 0
\(977\) 23.8885 0.764262 0.382131 0.924108i \(-0.375190\pi\)
0.382131 + 0.924108i \(0.375190\pi\)
\(978\) 12.0000 0.383718
\(979\) −2.47214 −0.0790098
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 5.88854 0.187911
\(983\) 38.2918 1.22132 0.610659 0.791893i \(-0.290904\pi\)
0.610659 + 0.791893i \(0.290904\pi\)
\(984\) 0.763932 0.0243533
\(985\) −72.3607 −2.30560
\(986\) 3.41641 0.108801
\(987\) 0 0
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 3.23607 0.102849
\(991\) 22.8328 0.725308 0.362654 0.931924i \(-0.381871\pi\)
0.362654 + 0.931924i \(0.381871\pi\)
\(992\) −7.23607 −0.229745
\(993\) −19.4164 −0.616161
\(994\) 0 0
\(995\) −55.4164 −1.75682
\(996\) 3.23607 0.102539
\(997\) −26.2492 −0.831321 −0.415661 0.909520i \(-0.636450\pi\)
−0.415661 + 0.909520i \(0.636450\pi\)
\(998\) −19.4164 −0.614616
\(999\) −6.94427 −0.219707
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 3234.2.a.bb.1.1 2
3.2 odd 2 9702.2.a.cw.1.2 2
7.6 odd 2 3234.2.a.be.1.2 yes 2
21.20 even 2 9702.2.a.ck.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bb.1.1 2 1.1 even 1 trivial
3234.2.a.be.1.2 yes 2 7.6 odd 2
9702.2.a.ck.1.1 2 21.20 even 2
9702.2.a.cw.1.2 2 3.2 odd 2