Properties

Label 3822.2.a.bk.1.2
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.64575 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.64575 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.64575 q^{10} -4.64575 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.64575 q^{15} +1.00000 q^{16} -1.35425 q^{17} -1.00000 q^{18} +5.00000 q^{19} +1.64575 q^{20} +4.64575 q^{22} -7.64575 q^{23} -1.00000 q^{24} -2.29150 q^{25} -1.00000 q^{26} +1.00000 q^{27} -6.29150 q^{29} -1.64575 q^{30} -7.29150 q^{31} -1.00000 q^{32} -4.64575 q^{33} +1.35425 q^{34} +1.00000 q^{36} +0.354249 q^{37} -5.00000 q^{38} +1.00000 q^{39} -1.64575 q^{40} -7.64575 q^{41} +5.29150 q^{43} -4.64575 q^{44} +1.64575 q^{45} +7.64575 q^{46} +3.00000 q^{47} +1.00000 q^{48} +2.29150 q^{50} -1.35425 q^{51} +1.00000 q^{52} -3.00000 q^{53} -1.00000 q^{54} -7.64575 q^{55} +5.00000 q^{57} +6.29150 q^{58} +7.93725 q^{59} +1.64575 q^{60} +3.93725 q^{61} +7.29150 q^{62} +1.00000 q^{64} +1.64575 q^{65} +4.64575 q^{66} -13.5830 q^{67} -1.35425 q^{68} -7.64575 q^{69} +5.70850 q^{71} -1.00000 q^{72} -8.35425 q^{73} -0.354249 q^{74} -2.29150 q^{75} +5.00000 q^{76} -1.00000 q^{78} -10.0000 q^{79} +1.64575 q^{80} +1.00000 q^{81} +7.64575 q^{82} -2.70850 q^{83} -2.22876 q^{85} -5.29150 q^{86} -6.29150 q^{87} +4.64575 q^{88} +1.06275 q^{89} -1.64575 q^{90} -7.64575 q^{92} -7.29150 q^{93} -3.00000 q^{94} +8.22876 q^{95} -1.00000 q^{96} -14.9373 q^{97} -4.64575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{10} - 4 q^{11} + 2 q^{12} + 2 q^{13} - 2 q^{15} + 2 q^{16} - 8 q^{17} - 2 q^{18} + 10 q^{19} - 2 q^{20} + 4 q^{22} - 10 q^{23} - 2 q^{24} + 6 q^{25} - 2 q^{26} + 2 q^{27} - 2 q^{29} + 2 q^{30} - 4 q^{31} - 2 q^{32} - 4 q^{33} + 8 q^{34} + 2 q^{36} + 6 q^{37} - 10 q^{38} + 2 q^{39} + 2 q^{40} - 10 q^{41} - 4 q^{44} - 2 q^{45} + 10 q^{46} + 6 q^{47} + 2 q^{48} - 6 q^{50} - 8 q^{51} + 2 q^{52} - 6 q^{53} - 2 q^{54} - 10 q^{55} + 10 q^{57} + 2 q^{58} - 2 q^{60} - 8 q^{61} + 4 q^{62} + 2 q^{64} - 2 q^{65} + 4 q^{66} - 6 q^{67} - 8 q^{68} - 10 q^{69} + 22 q^{71} - 2 q^{72} - 22 q^{73} - 6 q^{74} + 6 q^{75} + 10 q^{76} - 2 q^{78} - 20 q^{79} - 2 q^{80} + 2 q^{81} + 10 q^{82} - 16 q^{83} + 22 q^{85} - 2 q^{87} + 4 q^{88} + 18 q^{89} + 2 q^{90} - 10 q^{92} - 4 q^{93} - 6 q^{94} - 10 q^{95} - 2 q^{96} - 14 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(5\) 1.64575 0.736002 0.368001 0.929825i \(-0.380042\pi\)
0.368001 + 0.929825i \(0.380042\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.64575 −0.520432
\(11\) −4.64575 −1.40075 −0.700373 0.713777i \(-0.746983\pi\)
−0.700373 + 0.713777i \(0.746983\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.64575 0.424931
\(16\) 1.00000 0.250000
\(17\) −1.35425 −0.328454 −0.164227 0.986423i \(-0.552513\pi\)
−0.164227 + 0.986423i \(0.552513\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 1.64575 0.368001
\(21\) 0 0
\(22\) 4.64575 0.990478
\(23\) −7.64575 −1.59425 −0.797125 0.603815i \(-0.793647\pi\)
−0.797125 + 0.603815i \(0.793647\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.29150 −0.458301
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.29150 −1.16830 −0.584151 0.811645i \(-0.698573\pi\)
−0.584151 + 0.811645i \(0.698573\pi\)
\(30\) −1.64575 −0.300472
\(31\) −7.29150 −1.30959 −0.654796 0.755805i \(-0.727246\pi\)
−0.654796 + 0.755805i \(0.727246\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.64575 −0.808721
\(34\) 1.35425 0.232252
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0.354249 0.0582381 0.0291191 0.999576i \(-0.490730\pi\)
0.0291191 + 0.999576i \(0.490730\pi\)
\(38\) −5.00000 −0.811107
\(39\) 1.00000 0.160128
\(40\) −1.64575 −0.260216
\(41\) −7.64575 −1.19407 −0.597033 0.802217i \(-0.703654\pi\)
−0.597033 + 0.802217i \(0.703654\pi\)
\(42\) 0 0
\(43\) 5.29150 0.806947 0.403473 0.914991i \(-0.367803\pi\)
0.403473 + 0.914991i \(0.367803\pi\)
\(44\) −4.64575 −0.700373
\(45\) 1.64575 0.245334
\(46\) 7.64575 1.12730
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 2.29150 0.324067
\(51\) −1.35425 −0.189633
\(52\) 1.00000 0.138675
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) −1.00000 −0.136083
\(55\) −7.64575 −1.03095
\(56\) 0 0
\(57\) 5.00000 0.662266
\(58\) 6.29150 0.826115
\(59\) 7.93725 1.03334 0.516671 0.856184i \(-0.327171\pi\)
0.516671 + 0.856184i \(0.327171\pi\)
\(60\) 1.64575 0.212466
\(61\) 3.93725 0.504114 0.252057 0.967712i \(-0.418893\pi\)
0.252057 + 0.967712i \(0.418893\pi\)
\(62\) 7.29150 0.926022
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.64575 0.204130
\(66\) 4.64575 0.571852
\(67\) −13.5830 −1.65943 −0.829714 0.558189i \(-0.811497\pi\)
−0.829714 + 0.558189i \(0.811497\pi\)
\(68\) −1.35425 −0.164227
\(69\) −7.64575 −0.920440
\(70\) 0 0
\(71\) 5.70850 0.677474 0.338737 0.940881i \(-0.390000\pi\)
0.338737 + 0.940881i \(0.390000\pi\)
\(72\) −1.00000 −0.117851
\(73\) −8.35425 −0.977791 −0.488895 0.872342i \(-0.662600\pi\)
−0.488895 + 0.872342i \(0.662600\pi\)
\(74\) −0.354249 −0.0411806
\(75\) −2.29150 −0.264600
\(76\) 5.00000 0.573539
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 1.64575 0.184001
\(81\) 1.00000 0.111111
\(82\) 7.64575 0.844332
\(83\) −2.70850 −0.297296 −0.148648 0.988890i \(-0.547492\pi\)
−0.148648 + 0.988890i \(0.547492\pi\)
\(84\) 0 0
\(85\) −2.22876 −0.241743
\(86\) −5.29150 −0.570597
\(87\) −6.29150 −0.674520
\(88\) 4.64575 0.495239
\(89\) 1.06275 0.112651 0.0563254 0.998412i \(-0.482062\pi\)
0.0563254 + 0.998412i \(0.482062\pi\)
\(90\) −1.64575 −0.173477
\(91\) 0 0
\(92\) −7.64575 −0.797125
\(93\) −7.29150 −0.756094
\(94\) −3.00000 −0.309426
\(95\) 8.22876 0.844253
\(96\) −1.00000 −0.102062
\(97\) −14.9373 −1.51665 −0.758324 0.651878i \(-0.773982\pi\)
−0.758324 + 0.651878i \(0.773982\pi\)
\(98\) 0 0
\(99\) −4.64575 −0.466916
\(100\) −2.29150 −0.229150
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 1.35425 0.134091
\(103\) 5.29150 0.521387 0.260694 0.965422i \(-0.416049\pi\)
0.260694 + 0.965422i \(0.416049\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 7.64575 0.728994
\(111\) 0.354249 0.0336238
\(112\) 0 0
\(113\) −11.2288 −1.05631 −0.528156 0.849147i \(-0.677116\pi\)
−0.528156 + 0.849147i \(0.677116\pi\)
\(114\) −5.00000 −0.468293
\(115\) −12.5830 −1.17337
\(116\) −6.29150 −0.584151
\(117\) 1.00000 0.0924500
\(118\) −7.93725 −0.730683
\(119\) 0 0
\(120\) −1.64575 −0.150236
\(121\) 10.5830 0.962091
\(122\) −3.93725 −0.356462
\(123\) −7.64575 −0.689394
\(124\) −7.29150 −0.654796
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 16.2288 1.44007 0.720035 0.693938i \(-0.244126\pi\)
0.720035 + 0.693938i \(0.244126\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.29150 0.465891
\(130\) −1.64575 −0.144342
\(131\) −4.93725 −0.431370 −0.215685 0.976463i \(-0.569198\pi\)
−0.215685 + 0.976463i \(0.569198\pi\)
\(132\) −4.64575 −0.404361
\(133\) 0 0
\(134\) 13.5830 1.17339
\(135\) 1.64575 0.141644
\(136\) 1.35425 0.116126
\(137\) 17.5203 1.49686 0.748428 0.663216i \(-0.230809\pi\)
0.748428 + 0.663216i \(0.230809\pi\)
\(138\) 7.64575 0.650850
\(139\) 4.22876 0.358678 0.179339 0.983787i \(-0.442604\pi\)
0.179339 + 0.983787i \(0.442604\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) −5.70850 −0.479046
\(143\) −4.64575 −0.388497
\(144\) 1.00000 0.0833333
\(145\) −10.3542 −0.859874
\(146\) 8.35425 0.691403
\(147\) 0 0
\(148\) 0.354249 0.0291191
\(149\) −22.9373 −1.87909 −0.939547 0.342421i \(-0.888753\pi\)
−0.939547 + 0.342421i \(0.888753\pi\)
\(150\) 2.29150 0.187100
\(151\) 21.9373 1.78523 0.892614 0.450821i \(-0.148869\pi\)
0.892614 + 0.450821i \(0.148869\pi\)
\(152\) −5.00000 −0.405554
\(153\) −1.35425 −0.109485
\(154\) 0 0
\(155\) −12.0000 −0.963863
\(156\) 1.00000 0.0800641
\(157\) −2.64575 −0.211154 −0.105577 0.994411i \(-0.533669\pi\)
−0.105577 + 0.994411i \(0.533669\pi\)
\(158\) 10.0000 0.795557
\(159\) −3.00000 −0.237915
\(160\) −1.64575 −0.130108
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 23.5830 1.84716 0.923582 0.383402i \(-0.125247\pi\)
0.923582 + 0.383402i \(0.125247\pi\)
\(164\) −7.64575 −0.597033
\(165\) −7.64575 −0.595221
\(166\) 2.70850 0.210220
\(167\) 24.8745 1.92485 0.962424 0.271553i \(-0.0875371\pi\)
0.962424 + 0.271553i \(0.0875371\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 2.22876 0.170938
\(171\) 5.00000 0.382360
\(172\) 5.29150 0.403473
\(173\) −11.7085 −0.890181 −0.445090 0.895486i \(-0.646828\pi\)
−0.445090 + 0.895486i \(0.646828\pi\)
\(174\) 6.29150 0.476958
\(175\) 0 0
\(176\) −4.64575 −0.350187
\(177\) 7.93725 0.596601
\(178\) −1.06275 −0.0796562
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 1.64575 0.122667
\(181\) 9.35425 0.695296 0.347648 0.937625i \(-0.386980\pi\)
0.347648 + 0.937625i \(0.386980\pi\)
\(182\) 0 0
\(183\) 3.93725 0.291050
\(184\) 7.64575 0.563652
\(185\) 0.583005 0.0428634
\(186\) 7.29150 0.534639
\(187\) 6.29150 0.460080
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) −8.22876 −0.596977
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 3.06275 0.220461 0.110231 0.993906i \(-0.464841\pi\)
0.110231 + 0.993906i \(0.464841\pi\)
\(194\) 14.9373 1.07243
\(195\) 1.64575 0.117855
\(196\) 0 0
\(197\) 20.8118 1.48278 0.741388 0.671076i \(-0.234168\pi\)
0.741388 + 0.671076i \(0.234168\pi\)
\(198\) 4.64575 0.330159
\(199\) 4.22876 0.299769 0.149884 0.988704i \(-0.452110\pi\)
0.149884 + 0.988704i \(0.452110\pi\)
\(200\) 2.29150 0.162034
\(201\) −13.5830 −0.958071
\(202\) 0 0
\(203\) 0 0
\(204\) −1.35425 −0.0948164
\(205\) −12.5830 −0.878835
\(206\) −5.29150 −0.368676
\(207\) −7.64575 −0.531416
\(208\) 1.00000 0.0693375
\(209\) −23.2288 −1.60677
\(210\) 0 0
\(211\) −11.6458 −0.801727 −0.400863 0.916138i \(-0.631290\pi\)
−0.400863 + 0.916138i \(0.631290\pi\)
\(212\) −3.00000 −0.206041
\(213\) 5.70850 0.391140
\(214\) 12.0000 0.820303
\(215\) 8.70850 0.593915
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) −8.35425 −0.564528
\(220\) −7.64575 −0.515476
\(221\) −1.35425 −0.0910966
\(222\) −0.354249 −0.0237756
\(223\) 16.5203 1.10628 0.553139 0.833089i \(-0.313430\pi\)
0.553139 + 0.833089i \(0.313430\pi\)
\(224\) 0 0
\(225\) −2.29150 −0.152767
\(226\) 11.2288 0.746926
\(227\) −15.2915 −1.01493 −0.507466 0.861672i \(-0.669418\pi\)
−0.507466 + 0.861672i \(0.669418\pi\)
\(228\) 5.00000 0.331133
\(229\) −26.4575 −1.74836 −0.874181 0.485601i \(-0.838601\pi\)
−0.874181 + 0.485601i \(0.838601\pi\)
\(230\) 12.5830 0.829699
\(231\) 0 0
\(232\) 6.29150 0.413057
\(233\) −4.64575 −0.304353 −0.152177 0.988353i \(-0.548628\pi\)
−0.152177 + 0.988353i \(0.548628\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 4.93725 0.322071
\(236\) 7.93725 0.516671
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) 3.00000 0.194054 0.0970269 0.995282i \(-0.469067\pi\)
0.0970269 + 0.995282i \(0.469067\pi\)
\(240\) 1.64575 0.106233
\(241\) 17.8745 1.15140 0.575699 0.817662i \(-0.304730\pi\)
0.575699 + 0.817662i \(0.304730\pi\)
\(242\) −10.5830 −0.680301
\(243\) 1.00000 0.0641500
\(244\) 3.93725 0.252057
\(245\) 0 0
\(246\) 7.64575 0.487475
\(247\) 5.00000 0.318142
\(248\) 7.29150 0.463011
\(249\) −2.70850 −0.171644
\(250\) 12.0000 0.758947
\(251\) −2.70850 −0.170959 −0.0854794 0.996340i \(-0.527242\pi\)
−0.0854794 + 0.996340i \(0.527242\pi\)
\(252\) 0 0
\(253\) 35.5203 2.23314
\(254\) −16.2288 −1.01828
\(255\) −2.22876 −0.139570
\(256\) 1.00000 0.0625000
\(257\) 15.8745 0.990225 0.495112 0.868829i \(-0.335127\pi\)
0.495112 + 0.868829i \(0.335127\pi\)
\(258\) −5.29150 −0.329435
\(259\) 0 0
\(260\) 1.64575 0.102065
\(261\) −6.29150 −0.389434
\(262\) 4.93725 0.305025
\(263\) 1.64575 0.101481 0.0507407 0.998712i \(-0.483842\pi\)
0.0507407 + 0.998712i \(0.483842\pi\)
\(264\) 4.64575 0.285926
\(265\) −4.93725 −0.303293
\(266\) 0 0
\(267\) 1.06275 0.0650390
\(268\) −13.5830 −0.829714
\(269\) −24.8745 −1.51663 −0.758313 0.651891i \(-0.773976\pi\)
−0.758313 + 0.651891i \(0.773976\pi\)
\(270\) −1.64575 −0.100157
\(271\) −17.3542 −1.05420 −0.527098 0.849805i \(-0.676720\pi\)
−0.527098 + 0.849805i \(0.676720\pi\)
\(272\) −1.35425 −0.0821134
\(273\) 0 0
\(274\) −17.5203 −1.05844
\(275\) 10.6458 0.641963
\(276\) −7.64575 −0.460220
\(277\) −18.5203 −1.11277 −0.556387 0.830923i \(-0.687813\pi\)
−0.556387 + 0.830923i \(0.687813\pi\)
\(278\) −4.22876 −0.253624
\(279\) −7.29150 −0.436531
\(280\) 0 0
\(281\) 18.5830 1.10857 0.554285 0.832327i \(-0.312992\pi\)
0.554285 + 0.832327i \(0.312992\pi\)
\(282\) −3.00000 −0.178647
\(283\) −26.9373 −1.60125 −0.800627 0.599163i \(-0.795500\pi\)
−0.800627 + 0.599163i \(0.795500\pi\)
\(284\) 5.70850 0.338737
\(285\) 8.22876 0.487429
\(286\) 4.64575 0.274709
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −15.1660 −0.892118
\(290\) 10.3542 0.608022
\(291\) −14.9373 −0.875637
\(292\) −8.35425 −0.488895
\(293\) −29.5203 −1.72459 −0.862296 0.506405i \(-0.830974\pi\)
−0.862296 + 0.506405i \(0.830974\pi\)
\(294\) 0 0
\(295\) 13.0627 0.760542
\(296\) −0.354249 −0.0205903
\(297\) −4.64575 −0.269574
\(298\) 22.9373 1.32872
\(299\) −7.64575 −0.442165
\(300\) −2.29150 −0.132300
\(301\) 0 0
\(302\) −21.9373 −1.26235
\(303\) 0 0
\(304\) 5.00000 0.286770
\(305\) 6.47974 0.371029
\(306\) 1.35425 0.0774172
\(307\) 11.5830 0.661077 0.330539 0.943793i \(-0.392770\pi\)
0.330539 + 0.943793i \(0.392770\pi\)
\(308\) 0 0
\(309\) 5.29150 0.301023
\(310\) 12.0000 0.681554
\(311\) −16.9373 −0.960424 −0.480212 0.877153i \(-0.659440\pi\)
−0.480212 + 0.877153i \(0.659440\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −29.1660 −1.64856 −0.824280 0.566182i \(-0.808420\pi\)
−0.824280 + 0.566182i \(0.808420\pi\)
\(314\) 2.64575 0.149308
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −2.70850 −0.152124 −0.0760622 0.997103i \(-0.524235\pi\)
−0.0760622 + 0.997103i \(0.524235\pi\)
\(318\) 3.00000 0.168232
\(319\) 29.2288 1.63650
\(320\) 1.64575 0.0920003
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −6.77124 −0.376762
\(324\) 1.00000 0.0555556
\(325\) −2.29150 −0.127110
\(326\) −23.5830 −1.30614
\(327\) −4.00000 −0.221201
\(328\) 7.64575 0.422166
\(329\) 0 0
\(330\) 7.64575 0.420885
\(331\) 32.5830 1.79092 0.895462 0.445138i \(-0.146845\pi\)
0.895462 + 0.445138i \(0.146845\pi\)
\(332\) −2.70850 −0.148648
\(333\) 0.354249 0.0194127
\(334\) −24.8745 −1.36107
\(335\) −22.3542 −1.22134
\(336\) 0 0
\(337\) 5.58301 0.304126 0.152063 0.988371i \(-0.451408\pi\)
0.152063 + 0.988371i \(0.451408\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −11.2288 −0.609862
\(340\) −2.22876 −0.120871
\(341\) 33.8745 1.83441
\(342\) −5.00000 −0.270369
\(343\) 0 0
\(344\) −5.29150 −0.285299
\(345\) −12.5830 −0.677446
\(346\) 11.7085 0.629453
\(347\) −26.2288 −1.40803 −0.704017 0.710183i \(-0.748612\pi\)
−0.704017 + 0.710183i \(0.748612\pi\)
\(348\) −6.29150 −0.337260
\(349\) 3.06275 0.163945 0.0819725 0.996635i \(-0.473878\pi\)
0.0819725 + 0.996635i \(0.473878\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 4.64575 0.247619
\(353\) 0.583005 0.0310302 0.0155151 0.999880i \(-0.495061\pi\)
0.0155151 + 0.999880i \(0.495061\pi\)
\(354\) −7.93725 −0.421860
\(355\) 9.39477 0.498622
\(356\) 1.06275 0.0563254
\(357\) 0 0
\(358\) 6.00000 0.317110
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) −1.64575 −0.0867387
\(361\) 6.00000 0.315789
\(362\) −9.35425 −0.491648
\(363\) 10.5830 0.555464
\(364\) 0 0
\(365\) −13.7490 −0.719656
\(366\) −3.93725 −0.205804
\(367\) 20.5830 1.07442 0.537212 0.843447i \(-0.319477\pi\)
0.537212 + 0.843447i \(0.319477\pi\)
\(368\) −7.64575 −0.398562
\(369\) −7.64575 −0.398022
\(370\) −0.583005 −0.0303090
\(371\) 0 0
\(372\) −7.29150 −0.378047
\(373\) 9.35425 0.484345 0.242172 0.970233i \(-0.422140\pi\)
0.242172 + 0.970233i \(0.422140\pi\)
\(374\) −6.29150 −0.325326
\(375\) −12.0000 −0.619677
\(376\) −3.00000 −0.154713
\(377\) −6.29150 −0.324029
\(378\) 0 0
\(379\) 33.1660 1.70362 0.851812 0.523848i \(-0.175504\pi\)
0.851812 + 0.523848i \(0.175504\pi\)
\(380\) 8.22876 0.422126
\(381\) 16.2288 0.831424
\(382\) 0 0
\(383\) −37.7490 −1.92888 −0.964442 0.264293i \(-0.914861\pi\)
−0.964442 + 0.264293i \(0.914861\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −3.06275 −0.155890
\(387\) 5.29150 0.268982
\(388\) −14.9373 −0.758324
\(389\) 18.2915 0.927416 0.463708 0.885988i \(-0.346519\pi\)
0.463708 + 0.885988i \(0.346519\pi\)
\(390\) −1.64575 −0.0833359
\(391\) 10.3542 0.523637
\(392\) 0 0
\(393\) −4.93725 −0.249052
\(394\) −20.8118 −1.04848
\(395\) −16.4575 −0.828067
\(396\) −4.64575 −0.233458
\(397\) 0.937254 0.0470394 0.0235197 0.999723i \(-0.492513\pi\)
0.0235197 + 0.999723i \(0.492513\pi\)
\(398\) −4.22876 −0.211968
\(399\) 0 0
\(400\) −2.29150 −0.114575
\(401\) 8.22876 0.410924 0.205462 0.978665i \(-0.434130\pi\)
0.205462 + 0.978665i \(0.434130\pi\)
\(402\) 13.5830 0.677459
\(403\) −7.29150 −0.363216
\(404\) 0 0
\(405\) 1.64575 0.0817780
\(406\) 0 0
\(407\) −1.64575 −0.0815769
\(408\) 1.35425 0.0670453
\(409\) −11.0627 −0.547018 −0.273509 0.961870i \(-0.588184\pi\)
−0.273509 + 0.961870i \(0.588184\pi\)
\(410\) 12.5830 0.621430
\(411\) 17.5203 0.864211
\(412\) 5.29150 0.260694
\(413\) 0 0
\(414\) 7.64575 0.375768
\(415\) −4.45751 −0.218811
\(416\) −1.00000 −0.0490290
\(417\) 4.22876 0.207083
\(418\) 23.2288 1.13616
\(419\) −22.4575 −1.09712 −0.548561 0.836111i \(-0.684824\pi\)
−0.548561 + 0.836111i \(0.684824\pi\)
\(420\) 0 0
\(421\) −29.6458 −1.44485 −0.722423 0.691452i \(-0.756972\pi\)
−0.722423 + 0.691452i \(0.756972\pi\)
\(422\) 11.6458 0.566906
\(423\) 3.00000 0.145865
\(424\) 3.00000 0.145693
\(425\) 3.10326 0.150530
\(426\) −5.70850 −0.276578
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) −4.64575 −0.224299
\(430\) −8.70850 −0.419961
\(431\) −0.583005 −0.0280824 −0.0140412 0.999901i \(-0.504470\pi\)
−0.0140412 + 0.999901i \(0.504470\pi\)
\(432\) 1.00000 0.0481125
\(433\) −12.4170 −0.596723 −0.298361 0.954453i \(-0.596440\pi\)
−0.298361 + 0.954453i \(0.596440\pi\)
\(434\) 0 0
\(435\) −10.3542 −0.496448
\(436\) −4.00000 −0.191565
\(437\) −38.2288 −1.82873
\(438\) 8.35425 0.399181
\(439\) 18.3542 0.876000 0.438000 0.898975i \(-0.355687\pi\)
0.438000 + 0.898975i \(0.355687\pi\)
\(440\) 7.64575 0.364497
\(441\) 0 0
\(442\) 1.35425 0.0644150
\(443\) 34.9373 1.65992 0.829960 0.557824i \(-0.188363\pi\)
0.829960 + 0.557824i \(0.188363\pi\)
\(444\) 0.354249 0.0168119
\(445\) 1.74902 0.0829113
\(446\) −16.5203 −0.782257
\(447\) −22.9373 −1.08489
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 2.29150 0.108022
\(451\) 35.5203 1.67258
\(452\) −11.2288 −0.528156
\(453\) 21.9373 1.03070
\(454\) 15.2915 0.717666
\(455\) 0 0
\(456\) −5.00000 −0.234146
\(457\) 5.77124 0.269967 0.134984 0.990848i \(-0.456902\pi\)
0.134984 + 0.990848i \(0.456902\pi\)
\(458\) 26.4575 1.23628
\(459\) −1.35425 −0.0632109
\(460\) −12.5830 −0.586686
\(461\) 16.4575 0.766503 0.383251 0.923644i \(-0.374804\pi\)
0.383251 + 0.923644i \(0.374804\pi\)
\(462\) 0 0
\(463\) −17.1660 −0.797772 −0.398886 0.917000i \(-0.630603\pi\)
−0.398886 + 0.917000i \(0.630603\pi\)
\(464\) −6.29150 −0.292076
\(465\) −12.0000 −0.556487
\(466\) 4.64575 0.215210
\(467\) −0.479741 −0.0221998 −0.0110999 0.999938i \(-0.503533\pi\)
−0.0110999 + 0.999938i \(0.503533\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) −4.93725 −0.227739
\(471\) −2.64575 −0.121910
\(472\) −7.93725 −0.365342
\(473\) −24.5830 −1.13033
\(474\) 10.0000 0.459315
\(475\) −11.4575 −0.525707
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) −3.00000 −0.137217
\(479\) 28.7490 1.31358 0.656788 0.754075i \(-0.271915\pi\)
0.656788 + 0.754075i \(0.271915\pi\)
\(480\) −1.64575 −0.0751179
\(481\) 0.354249 0.0161523
\(482\) −17.8745 −0.814162
\(483\) 0 0
\(484\) 10.5830 0.481046
\(485\) −24.5830 −1.11626
\(486\) −1.00000 −0.0453609
\(487\) −8.06275 −0.365358 −0.182679 0.983173i \(-0.558477\pi\)
−0.182679 + 0.983173i \(0.558477\pi\)
\(488\) −3.93725 −0.178231
\(489\) 23.5830 1.06646
\(490\) 0 0
\(491\) −31.1660 −1.40650 −0.703251 0.710941i \(-0.748269\pi\)
−0.703251 + 0.710941i \(0.748269\pi\)
\(492\) −7.64575 −0.344697
\(493\) 8.52026 0.383733
\(494\) −5.00000 −0.224961
\(495\) −7.64575 −0.343651
\(496\) −7.29150 −0.327398
\(497\) 0 0
\(498\) 2.70850 0.121371
\(499\) −18.7085 −0.837507 −0.418754 0.908100i \(-0.637533\pi\)
−0.418754 + 0.908100i \(0.637533\pi\)
\(500\) −12.0000 −0.536656
\(501\) 24.8745 1.11131
\(502\) 2.70850 0.120886
\(503\) −16.4575 −0.733804 −0.366902 0.930260i \(-0.619582\pi\)
−0.366902 + 0.930260i \(0.619582\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −35.5203 −1.57907
\(507\) 1.00000 0.0444116
\(508\) 16.2288 0.720035
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 2.22876 0.0986910
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 5.00000 0.220755
\(514\) −15.8745 −0.700195
\(515\) 8.70850 0.383742
\(516\) 5.29150 0.232945
\(517\) −13.9373 −0.612960
\(518\) 0 0
\(519\) −11.7085 −0.513946
\(520\) −1.64575 −0.0721710
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 6.29150 0.275372
\(523\) −37.2915 −1.63064 −0.815322 0.579008i \(-0.803440\pi\)
−0.815322 + 0.579008i \(0.803440\pi\)
\(524\) −4.93725 −0.215685
\(525\) 0 0
\(526\) −1.64575 −0.0717582
\(527\) 9.87451 0.430140
\(528\) −4.64575 −0.202180
\(529\) 35.4575 1.54163
\(530\) 4.93725 0.214461
\(531\) 7.93725 0.344447
\(532\) 0 0
\(533\) −7.64575 −0.331174
\(534\) −1.06275 −0.0459895
\(535\) −19.7490 −0.853824
\(536\) 13.5830 0.586696
\(537\) −6.00000 −0.258919
\(538\) 24.8745 1.07242
\(539\) 0 0
\(540\) 1.64575 0.0708219
\(541\) 36.3542 1.56299 0.781496 0.623911i \(-0.214457\pi\)
0.781496 + 0.623911i \(0.214457\pi\)
\(542\) 17.3542 0.745429
\(543\) 9.35425 0.401429
\(544\) 1.35425 0.0580629
\(545\) −6.58301 −0.281985
\(546\) 0 0
\(547\) −5.06275 −0.216467 −0.108234 0.994125i \(-0.534519\pi\)
−0.108234 + 0.994125i \(0.534519\pi\)
\(548\) 17.5203 0.748428
\(549\) 3.93725 0.168038
\(550\) −10.6458 −0.453936
\(551\) −31.4575 −1.34014
\(552\) 7.64575 0.325425
\(553\) 0 0
\(554\) 18.5203 0.786850
\(555\) 0.583005 0.0247472
\(556\) 4.22876 0.179339
\(557\) −31.1660 −1.32055 −0.660273 0.751026i \(-0.729559\pi\)
−0.660273 + 0.751026i \(0.729559\pi\)
\(558\) 7.29150 0.308674
\(559\) 5.29150 0.223807
\(560\) 0 0
\(561\) 6.29150 0.265627
\(562\) −18.5830 −0.783877
\(563\) 47.0405 1.98252 0.991261 0.131917i \(-0.0421133\pi\)
0.991261 + 0.131917i \(0.0421133\pi\)
\(564\) 3.00000 0.126323
\(565\) −18.4797 −0.777449
\(566\) 26.9373 1.13226
\(567\) 0 0
\(568\) −5.70850 −0.239523
\(569\) −6.77124 −0.283865 −0.141933 0.989876i \(-0.545332\pi\)
−0.141933 + 0.989876i \(0.545332\pi\)
\(570\) −8.22876 −0.344665
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) −4.64575 −0.194249
\(573\) 0 0
\(574\) 0 0
\(575\) 17.5203 0.730645
\(576\) 1.00000 0.0416667
\(577\) 8.58301 0.357315 0.178658 0.983911i \(-0.442825\pi\)
0.178658 + 0.983911i \(0.442825\pi\)
\(578\) 15.1660 0.630823
\(579\) 3.06275 0.127283
\(580\) −10.3542 −0.429937
\(581\) 0 0
\(582\) 14.9373 0.619169
\(583\) 13.9373 0.577222
\(584\) 8.35425 0.345701
\(585\) 1.64575 0.0680434
\(586\) 29.5203 1.21947
\(587\) 16.0627 0.662980 0.331490 0.943459i \(-0.392449\pi\)
0.331490 + 0.943459i \(0.392449\pi\)
\(588\) 0 0
\(589\) −36.4575 −1.50221
\(590\) −13.0627 −0.537785
\(591\) 20.8118 0.856081
\(592\) 0.354249 0.0145595
\(593\) 6.47974 0.266091 0.133046 0.991110i \(-0.457524\pi\)
0.133046 + 0.991110i \(0.457524\pi\)
\(594\) 4.64575 0.190617
\(595\) 0 0
\(596\) −22.9373 −0.939547
\(597\) 4.22876 0.173071
\(598\) 7.64575 0.312658
\(599\) 14.8118 0.605192 0.302596 0.953119i \(-0.402147\pi\)
0.302596 + 0.953119i \(0.402147\pi\)
\(600\) 2.29150 0.0935502
\(601\) −16.8745 −0.688326 −0.344163 0.938910i \(-0.611837\pi\)
−0.344163 + 0.938910i \(0.611837\pi\)
\(602\) 0 0
\(603\) −13.5830 −0.553143
\(604\) 21.9373 0.892614
\(605\) 17.4170 0.708102
\(606\) 0 0
\(607\) −30.8118 −1.25061 −0.625305 0.780380i \(-0.715026\pi\)
−0.625305 + 0.780380i \(0.715026\pi\)
\(608\) −5.00000 −0.202777
\(609\) 0 0
\(610\) −6.47974 −0.262357
\(611\) 3.00000 0.121367
\(612\) −1.35425 −0.0547423
\(613\) 10.1255 0.408965 0.204482 0.978870i \(-0.434449\pi\)
0.204482 + 0.978870i \(0.434449\pi\)
\(614\) −11.5830 −0.467452
\(615\) −12.5830 −0.507396
\(616\) 0 0
\(617\) −21.2915 −0.857164 −0.428582 0.903503i \(-0.640987\pi\)
−0.428582 + 0.903503i \(0.640987\pi\)
\(618\) −5.29150 −0.212855
\(619\) 21.7490 0.874167 0.437083 0.899421i \(-0.356011\pi\)
0.437083 + 0.899421i \(0.356011\pi\)
\(620\) −12.0000 −0.481932
\(621\) −7.64575 −0.306813
\(622\) 16.9373 0.679122
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) −8.29150 −0.331660
\(626\) 29.1660 1.16571
\(627\) −23.2288 −0.927667
\(628\) −2.64575 −0.105577
\(629\) −0.479741 −0.0191285
\(630\) 0 0
\(631\) −38.4575 −1.53097 −0.765485 0.643454i \(-0.777501\pi\)
−0.765485 + 0.643454i \(0.777501\pi\)
\(632\) 10.0000 0.397779
\(633\) −11.6458 −0.462877
\(634\) 2.70850 0.107568
\(635\) 26.7085 1.05989
\(636\) −3.00000 −0.118958
\(637\) 0 0
\(638\) −29.2288 −1.15718
\(639\) 5.70850 0.225825
\(640\) −1.64575 −0.0650540
\(641\) −42.5830 −1.68193 −0.840964 0.541091i \(-0.818011\pi\)
−0.840964 + 0.541091i \(0.818011\pi\)
\(642\) 12.0000 0.473602
\(643\) 20.8745 0.823210 0.411605 0.911362i \(-0.364968\pi\)
0.411605 + 0.911362i \(0.364968\pi\)
\(644\) 0 0
\(645\) 8.70850 0.342897
\(646\) 6.77124 0.266411
\(647\) −37.7490 −1.48407 −0.742033 0.670363i \(-0.766138\pi\)
−0.742033 + 0.670363i \(0.766138\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −36.8745 −1.44745
\(650\) 2.29150 0.0898801
\(651\) 0 0
\(652\) 23.5830 0.923582
\(653\) 12.0000 0.469596 0.234798 0.972044i \(-0.424557\pi\)
0.234798 + 0.972044i \(0.424557\pi\)
\(654\) 4.00000 0.156412
\(655\) −8.12549 −0.317489
\(656\) −7.64575 −0.298516
\(657\) −8.35425 −0.325930
\(658\) 0 0
\(659\) −1.06275 −0.0413987 −0.0206994 0.999786i \(-0.506589\pi\)
−0.0206994 + 0.999786i \(0.506589\pi\)
\(660\) −7.64575 −0.297610
\(661\) −21.5203 −0.837041 −0.418521 0.908207i \(-0.637451\pi\)
−0.418521 + 0.908207i \(0.637451\pi\)
\(662\) −32.5830 −1.26637
\(663\) −1.35425 −0.0525947
\(664\) 2.70850 0.105110
\(665\) 0 0
\(666\) −0.354249 −0.0137269
\(667\) 48.1033 1.86257
\(668\) 24.8745 0.962424
\(669\) 16.5203 0.638710
\(670\) 22.3542 0.863620
\(671\) −18.2915 −0.706136
\(672\) 0 0
\(673\) 14.5830 0.562134 0.281067 0.959688i \(-0.409312\pi\)
0.281067 + 0.959688i \(0.409312\pi\)
\(674\) −5.58301 −0.215049
\(675\) −2.29150 −0.0882000
\(676\) 1.00000 0.0384615
\(677\) −22.1660 −0.851909 −0.425954 0.904745i \(-0.640062\pi\)
−0.425954 + 0.904745i \(0.640062\pi\)
\(678\) 11.2288 0.431238
\(679\) 0 0
\(680\) 2.22876 0.0854689
\(681\) −15.2915 −0.585972
\(682\) −33.8745 −1.29712
\(683\) 8.70850 0.333221 0.166611 0.986023i \(-0.446718\pi\)
0.166611 + 0.986023i \(0.446718\pi\)
\(684\) 5.00000 0.191180
\(685\) 28.8340 1.10169
\(686\) 0 0
\(687\) −26.4575 −1.00942
\(688\) 5.29150 0.201737
\(689\) −3.00000 −0.114291
\(690\) 12.5830 0.479027
\(691\) 34.0405 1.29496 0.647481 0.762081i \(-0.275822\pi\)
0.647481 + 0.762081i \(0.275822\pi\)
\(692\) −11.7085 −0.445090
\(693\) 0 0
\(694\) 26.2288 0.995630
\(695\) 6.95948 0.263988
\(696\) 6.29150 0.238479
\(697\) 10.3542 0.392195
\(698\) −3.06275 −0.115927
\(699\) −4.64575 −0.175718
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 1.77124 0.0668037
\(704\) −4.64575 −0.175093
\(705\) 4.93725 0.185948
\(706\) −0.583005 −0.0219417
\(707\) 0 0
\(708\) 7.93725 0.298300
\(709\) 43.5203 1.63444 0.817219 0.576328i \(-0.195515\pi\)
0.817219 + 0.576328i \(0.195515\pi\)
\(710\) −9.39477 −0.352579
\(711\) −10.0000 −0.375029
\(712\) −1.06275 −0.0398281
\(713\) 55.7490 2.08782
\(714\) 0 0
\(715\) −7.64575 −0.285935
\(716\) −6.00000 −0.224231
\(717\) 3.00000 0.112037
\(718\) 6.00000 0.223918
\(719\) 23.4170 0.873307 0.436653 0.899630i \(-0.356164\pi\)
0.436653 + 0.899630i \(0.356164\pi\)
\(720\) 1.64575 0.0613335
\(721\) 0 0
\(722\) −6.00000 −0.223297
\(723\) 17.8745 0.664760
\(724\) 9.35425 0.347648
\(725\) 14.4170 0.535434
\(726\) −10.5830 −0.392772
\(727\) 36.4575 1.35213 0.676067 0.736840i \(-0.263683\pi\)
0.676067 + 0.736840i \(0.263683\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 13.7490 0.508874
\(731\) −7.16601 −0.265044
\(732\) 3.93725 0.145525
\(733\) −12.2288 −0.451679 −0.225840 0.974164i \(-0.572513\pi\)
−0.225840 + 0.974164i \(0.572513\pi\)
\(734\) −20.5830 −0.759733
\(735\) 0 0
\(736\) 7.64575 0.281826
\(737\) 63.1033 2.32444
\(738\) 7.64575 0.281444
\(739\) −6.70850 −0.246776 −0.123388 0.992359i \(-0.539376\pi\)
−0.123388 + 0.992359i \(0.539376\pi\)
\(740\) 0.583005 0.0214317
\(741\) 5.00000 0.183680
\(742\) 0 0
\(743\) 1.45751 0.0534710 0.0267355 0.999643i \(-0.491489\pi\)
0.0267355 + 0.999643i \(0.491489\pi\)
\(744\) 7.29150 0.267319
\(745\) −37.7490 −1.38302
\(746\) −9.35425 −0.342483
\(747\) −2.70850 −0.0990987
\(748\) 6.29150 0.230040
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 9.16601 0.334472 0.167236 0.985917i \(-0.446516\pi\)
0.167236 + 0.985917i \(0.446516\pi\)
\(752\) 3.00000 0.109399
\(753\) −2.70850 −0.0987031
\(754\) 6.29150 0.229123
\(755\) 36.1033 1.31393
\(756\) 0 0
\(757\) 21.3542 0.776133 0.388067 0.921631i \(-0.373143\pi\)
0.388067 + 0.921631i \(0.373143\pi\)
\(758\) −33.1660 −1.20464
\(759\) 35.5203 1.28930
\(760\) −8.22876 −0.298488
\(761\) −3.29150 −0.119317 −0.0596584 0.998219i \(-0.519001\pi\)
−0.0596584 + 0.998219i \(0.519001\pi\)
\(762\) −16.2288 −0.587906
\(763\) 0 0
\(764\) 0 0
\(765\) −2.22876 −0.0805809
\(766\) 37.7490 1.36393
\(767\) 7.93725 0.286598
\(768\) 1.00000 0.0360844
\(769\) 0.354249 0.0127745 0.00638727 0.999980i \(-0.497967\pi\)
0.00638727 + 0.999980i \(0.497967\pi\)
\(770\) 0 0
\(771\) 15.8745 0.571706
\(772\) 3.06275 0.110231
\(773\) −44.3320 −1.59451 −0.797256 0.603641i \(-0.793716\pi\)
−0.797256 + 0.603641i \(0.793716\pi\)
\(774\) −5.29150 −0.190199
\(775\) 16.7085 0.600187
\(776\) 14.9373 0.536216
\(777\) 0 0
\(778\) −18.2915 −0.655782
\(779\) −38.2288 −1.36969
\(780\) 1.64575 0.0589273
\(781\) −26.5203 −0.948969
\(782\) −10.3542 −0.370267
\(783\) −6.29150 −0.224840
\(784\) 0 0
\(785\) −4.35425 −0.155410
\(786\) 4.93725 0.176106
\(787\) −42.6235 −1.51936 −0.759682 0.650294i \(-0.774646\pi\)
−0.759682 + 0.650294i \(0.774646\pi\)
\(788\) 20.8118 0.741388
\(789\) 1.64575 0.0585903
\(790\) 16.4575 0.585532
\(791\) 0 0
\(792\) 4.64575 0.165080
\(793\) 3.93725 0.139816
\(794\) −0.937254 −0.0332619
\(795\) −4.93725 −0.175106
\(796\) 4.22876 0.149884
\(797\) −14.1255 −0.500351 −0.250175 0.968201i \(-0.580488\pi\)
−0.250175 + 0.968201i \(0.580488\pi\)
\(798\) 0 0
\(799\) −4.06275 −0.143730
\(800\) 2.29150 0.0810169
\(801\) 1.06275 0.0375503
\(802\) −8.22876 −0.290567
\(803\) 38.8118 1.36964
\(804\) −13.5830 −0.479036
\(805\) 0 0
\(806\) 7.29150 0.256832
\(807\) −24.8745 −0.875624
\(808\) 0 0
\(809\) −23.8118 −0.837177 −0.418588 0.908176i \(-0.637475\pi\)
−0.418588 + 0.908176i \(0.637475\pi\)
\(810\) −1.64575 −0.0578258
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 0 0
\(813\) −17.3542 −0.608640
\(814\) 1.64575 0.0576836
\(815\) 38.8118 1.35952
\(816\) −1.35425 −0.0474082
\(817\) 26.4575 0.925631
\(818\) 11.0627 0.386800
\(819\) 0 0
\(820\) −12.5830 −0.439418
\(821\) −32.2288 −1.12479 −0.562396 0.826868i \(-0.690120\pi\)
−0.562396 + 0.826868i \(0.690120\pi\)
\(822\) −17.5203 −0.611089
\(823\) −21.5203 −0.750149 −0.375075 0.926995i \(-0.622383\pi\)
−0.375075 + 0.926995i \(0.622383\pi\)
\(824\) −5.29150 −0.184338
\(825\) 10.6458 0.370637
\(826\) 0 0
\(827\) 40.6458 1.41339 0.706696 0.707518i \(-0.250185\pi\)
0.706696 + 0.707518i \(0.250185\pi\)
\(828\) −7.64575 −0.265708
\(829\) 20.7712 0.721415 0.360708 0.932679i \(-0.382535\pi\)
0.360708 + 0.932679i \(0.382535\pi\)
\(830\) 4.45751 0.154723
\(831\) −18.5203 −0.642461
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −4.22876 −0.146430
\(835\) 40.9373 1.41669
\(836\) −23.2288 −0.803383
\(837\) −7.29150 −0.252031
\(838\) 22.4575 0.775782
\(839\) 21.0000 0.725001 0.362500 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(840\) 0 0
\(841\) 10.5830 0.364931
\(842\) 29.6458 1.02166
\(843\) 18.5830 0.640033
\(844\) −11.6458 −0.400863
\(845\) 1.64575 0.0566156
\(846\) −3.00000 −0.103142
\(847\) 0 0
\(848\) −3.00000 −0.103020
\(849\) −26.9373 −0.924484
\(850\) −3.10326 −0.106441
\(851\) −2.70850 −0.0928461
\(852\) 5.70850 0.195570
\(853\) 38.1033 1.30463 0.652315 0.757948i \(-0.273798\pi\)
0.652315 + 0.757948i \(0.273798\pi\)
\(854\) 0 0
\(855\) 8.22876 0.281418
\(856\) 12.0000 0.410152
\(857\) −1.93725 −0.0661753 −0.0330877 0.999452i \(-0.510534\pi\)
−0.0330877 + 0.999452i \(0.510534\pi\)
\(858\) 4.64575 0.158603
\(859\) −8.45751 −0.288567 −0.144283 0.989536i \(-0.546088\pi\)
−0.144283 + 0.989536i \(0.546088\pi\)
\(860\) 8.70850 0.296957
\(861\) 0 0
\(862\) 0.583005 0.0198572
\(863\) −31.7490 −1.08075 −0.540375 0.841425i \(-0.681717\pi\)
−0.540375 + 0.841425i \(0.681717\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −19.2693 −0.655175
\(866\) 12.4170 0.421947
\(867\) −15.1660 −0.515065
\(868\) 0 0
\(869\) 46.4575 1.57596
\(870\) 10.3542 0.351042
\(871\) −13.5830 −0.460243
\(872\) 4.00000 0.135457
\(873\) −14.9373 −0.505549
\(874\) 38.2288 1.29311
\(875\) 0 0
\(876\) −8.35425 −0.282264
\(877\) −7.18824 −0.242730 −0.121365 0.992608i \(-0.538727\pi\)
−0.121365 + 0.992608i \(0.538727\pi\)
\(878\) −18.3542 −0.619426
\(879\) −29.5203 −0.995694
\(880\) −7.64575 −0.257738
\(881\) −21.2915 −0.717329 −0.358664 0.933467i \(-0.616768\pi\)
−0.358664 + 0.933467i \(0.616768\pi\)
\(882\) 0 0
\(883\) −44.9373 −1.51226 −0.756130 0.654422i \(-0.772912\pi\)
−0.756130 + 0.654422i \(0.772912\pi\)
\(884\) −1.35425 −0.0455483
\(885\) 13.0627 0.439099
\(886\) −34.9373 −1.17374
\(887\) −0.583005 −0.0195754 −0.00978770 0.999952i \(-0.503116\pi\)
−0.00978770 + 0.999952i \(0.503116\pi\)
\(888\) −0.354249 −0.0118878
\(889\) 0 0
\(890\) −1.74902 −0.0586271
\(891\) −4.64575 −0.155639
\(892\) 16.5203 0.553139
\(893\) 15.0000 0.501956
\(894\) 22.9373 0.767137
\(895\) −9.87451 −0.330068
\(896\) 0 0
\(897\) −7.64575 −0.255284
\(898\) −12.0000 −0.400445
\(899\) 45.8745 1.53000
\(900\) −2.29150 −0.0763834
\(901\) 4.06275 0.135350
\(902\) −35.5203 −1.18270
\(903\) 0 0
\(904\) 11.2288 0.373463
\(905\) 15.3948 0.511739
\(906\) −21.9373 −0.728816
\(907\) −30.2288 −1.00373 −0.501865 0.864946i \(-0.667352\pi\)
−0.501865 + 0.864946i \(0.667352\pi\)
\(908\) −15.2915 −0.507466
\(909\) 0 0
\(910\) 0 0
\(911\) 25.7490 0.853103 0.426551 0.904463i \(-0.359728\pi\)
0.426551 + 0.904463i \(0.359728\pi\)
\(912\) 5.00000 0.165567
\(913\) 12.5830 0.416437
\(914\) −5.77124 −0.190896
\(915\) 6.47974 0.214214
\(916\) −26.4575 −0.874181
\(917\) 0 0
\(918\) 1.35425 0.0446969
\(919\) 16.1255 0.531931 0.265965 0.963983i \(-0.414309\pi\)
0.265965 + 0.963983i \(0.414309\pi\)
\(920\) 12.5830 0.414849
\(921\) 11.5830 0.381673
\(922\) −16.4575 −0.541999
\(923\) 5.70850 0.187897
\(924\) 0 0
\(925\) −0.811762 −0.0266906
\(926\) 17.1660 0.564110
\(927\) 5.29150 0.173796
\(928\) 6.29150 0.206529
\(929\) 49.7490 1.63221 0.816106 0.577902i \(-0.196128\pi\)
0.816106 + 0.577902i \(0.196128\pi\)
\(930\) 12.0000 0.393496
\(931\) 0 0
\(932\) −4.64575 −0.152177
\(933\) −16.9373 −0.554501
\(934\) 0.479741 0.0156976
\(935\) 10.3542 0.338620
\(936\) −1.00000 −0.0326860
\(937\) 21.4575 0.700986 0.350493 0.936565i \(-0.386014\pi\)
0.350493 + 0.936565i \(0.386014\pi\)
\(938\) 0 0
\(939\) −29.1660 −0.951797
\(940\) 4.93725 0.161035
\(941\) 25.1660 0.820388 0.410194 0.911998i \(-0.365461\pi\)
0.410194 + 0.911998i \(0.365461\pi\)
\(942\) 2.64575 0.0862032
\(943\) 58.4575 1.90364
\(944\) 7.93725 0.258336
\(945\) 0 0
\(946\) 24.5830 0.799262
\(947\) 26.5203 0.861793 0.430896 0.902401i \(-0.358197\pi\)
0.430896 + 0.902401i \(0.358197\pi\)
\(948\) −10.0000 −0.324785
\(949\) −8.35425 −0.271190
\(950\) 11.4575 0.371731
\(951\) −2.70850 −0.0878290
\(952\) 0 0
\(953\) 33.1033 1.07232 0.536160 0.844116i \(-0.319874\pi\)
0.536160 + 0.844116i \(0.319874\pi\)
\(954\) 3.00000 0.0971286
\(955\) 0 0
\(956\) 3.00000 0.0970269
\(957\) 29.2288 0.944832
\(958\) −28.7490 −0.928839
\(959\) 0 0
\(960\) 1.64575 0.0531164
\(961\) 22.1660 0.715033
\(962\) −0.354249 −0.0114214
\(963\) −12.0000 −0.386695
\(964\) 17.8745 0.575699
\(965\) 5.04052 0.162260
\(966\) 0 0
\(967\) −49.1033 −1.57905 −0.789527 0.613715i \(-0.789674\pi\)
−0.789527 + 0.613715i \(0.789674\pi\)
\(968\) −10.5830 −0.340151
\(969\) −6.77124 −0.217524
\(970\) 24.5830 0.789313
\(971\) 1.06275 0.0341051 0.0170526 0.999855i \(-0.494572\pi\)
0.0170526 + 0.999855i \(0.494572\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 8.06275 0.258347
\(975\) −2.29150 −0.0733868
\(976\) 3.93725 0.126028
\(977\) 51.8745 1.65961 0.829806 0.558052i \(-0.188451\pi\)
0.829806 + 0.558052i \(0.188451\pi\)
\(978\) −23.5830 −0.754101
\(979\) −4.93725 −0.157795
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 31.1660 0.994547
\(983\) 44.0405 1.40467 0.702337 0.711844i \(-0.252140\pi\)
0.702337 + 0.711844i \(0.252140\pi\)
\(984\) 7.64575 0.243738
\(985\) 34.2510 1.09133
\(986\) −8.52026 −0.271340
\(987\) 0 0
\(988\) 5.00000 0.159071
\(989\) −40.4575 −1.28647
\(990\) 7.64575 0.242998
\(991\) 26.5830 0.844437 0.422218 0.906494i \(-0.361252\pi\)
0.422218 + 0.906494i \(0.361252\pi\)
\(992\) 7.29150 0.231505
\(993\) 32.5830 1.03399
\(994\) 0 0
\(995\) 6.95948 0.220630
\(996\) −2.70850 −0.0858220
\(997\) 25.2288 0.799003 0.399501 0.916733i \(-0.369183\pi\)
0.399501 + 0.916733i \(0.369183\pi\)
\(998\) 18.7085 0.592207
\(999\) 0.354249 0.0112079
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 3822.2.a.bk.1.2 2
7.2 even 3 546.2.i.j.235.1 yes 4
7.4 even 3 546.2.i.j.79.1 4
7.6 odd 2 3822.2.a.bi.1.1 2
21.2 odd 6 1638.2.j.k.235.2 4
21.11 odd 6 1638.2.j.k.1171.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.j.79.1 4 7.4 even 3
546.2.i.j.235.1 yes 4 7.2 even 3
1638.2.j.k.235.2 4 21.2 odd 6
1638.2.j.k.1171.2 4 21.11 odd 6
3822.2.a.bi.1.1 2 7.6 odd 2
3822.2.a.bk.1.2 2 1.1 even 1 trivial