Properties

Label 3381.2.a.bi.1.3
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 3 x^{9} - 13 x^{8} + 41 x^{7} + 47 x^{6} - 165 x^{5} - 45 x^{4} + 207 x^{3} + 12 x^{2} - 76 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.05351\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.05351 q^{2} -1.00000 q^{3} -0.890116 q^{4} +2.17612 q^{5} +1.05351 q^{6} +3.04477 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.05351 q^{2} -1.00000 q^{3} -0.890116 q^{4} +2.17612 q^{5} +1.05351 q^{6} +3.04477 q^{8} +1.00000 q^{9} -2.29256 q^{10} -2.62243 q^{11} +0.890116 q^{12} -3.30704 q^{13} -2.17612 q^{15} -1.42746 q^{16} -2.96912 q^{17} -1.05351 q^{18} +0.697769 q^{19} -1.93700 q^{20} +2.76276 q^{22} +1.00000 q^{23} -3.04477 q^{24} -0.264516 q^{25} +3.48400 q^{26} -1.00000 q^{27} +7.31827 q^{29} +2.29256 q^{30} +5.60209 q^{31} -4.58569 q^{32} +2.62243 q^{33} +3.12800 q^{34} -0.890116 q^{36} -4.52695 q^{37} -0.735106 q^{38} +3.30704 q^{39} +6.62577 q^{40} +2.28774 q^{41} -7.79527 q^{43} +2.33427 q^{44} +2.17612 q^{45} -1.05351 q^{46} -3.81394 q^{47} +1.42746 q^{48} +0.278671 q^{50} +2.96912 q^{51} +2.94365 q^{52} +3.13059 q^{53} +1.05351 q^{54} -5.70671 q^{55} -0.697769 q^{57} -7.70988 q^{58} +12.1628 q^{59} +1.93700 q^{60} +1.48459 q^{61} -5.90186 q^{62} +7.68599 q^{64} -7.19650 q^{65} -2.76276 q^{66} +11.2755 q^{67} +2.64286 q^{68} -1.00000 q^{69} +13.8643 q^{71} +3.04477 q^{72} +0.705829 q^{73} +4.76919 q^{74} +0.264516 q^{75} -0.621095 q^{76} -3.48400 q^{78} -8.60406 q^{79} -3.10632 q^{80} +1.00000 q^{81} -2.41016 q^{82} -11.5918 q^{83} -6.46115 q^{85} +8.21240 q^{86} -7.31827 q^{87} -7.98469 q^{88} -0.333776 q^{89} -2.29256 q^{90} -0.890116 q^{92} -5.60209 q^{93} +4.01803 q^{94} +1.51843 q^{95} +4.58569 q^{96} +0.124876 q^{97} -2.62243 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + O(q^{10}) \) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + 11q^{10} + 8q^{11} - 15q^{12} + 5q^{15} + 37q^{16} - 11q^{17} + 3q^{18} + q^{19} - 15q^{20} + 6q^{22} + 10q^{23} - 9q^{24} + 21q^{25} + q^{26} - 10q^{27} + 22q^{29} - 11q^{30} - 3q^{31} + 11q^{32} - 8q^{33} - 3q^{34} + 15q^{36} - 3q^{37} + 16q^{38} + 39q^{40} - 26q^{41} + 27q^{43} + 16q^{44} - 5q^{45} + 3q^{46} + 11q^{47} - 37q^{48} + 2q^{50} + 11q^{51} + 29q^{52} + 5q^{53} - 3q^{54} - 18q^{55} - q^{57} + 16q^{58} - 10q^{59} + 15q^{60} + 22q^{61} - 32q^{62} + 69q^{64} - 11q^{65} - 6q^{66} - 2q^{67} - 21q^{68} - 10q^{69} + 27q^{71} + 9q^{72} - 8q^{73} + 14q^{74} - 21q^{75} - 22q^{76} - q^{78} + 21q^{79} - 53q^{80} + 10q^{81} + 36q^{82} - 12q^{83} + 23q^{85} + 18q^{86} - 22q^{87} - 10q^{88} + 6q^{89} + 11q^{90} + 15q^{92} + 3q^{93} + 35q^{94} + 44q^{95} - 11q^{96} + 6q^{97} + 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.05351 −0.744944 −0.372472 0.928043i \(-0.621490\pi\)
−0.372472 + 0.928043i \(0.621490\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.890116 −0.445058
\(5\) 2.17612 0.973189 0.486594 0.873628i \(-0.338239\pi\)
0.486594 + 0.873628i \(0.338239\pi\)
\(6\) 1.05351 0.430094
\(7\) 0 0
\(8\) 3.04477 1.07649
\(9\) 1.00000 0.333333
\(10\) −2.29256 −0.724972
\(11\) −2.62243 −0.790692 −0.395346 0.918532i \(-0.629375\pi\)
−0.395346 + 0.918532i \(0.629375\pi\)
\(12\) 0.890116 0.256954
\(13\) −3.30704 −0.917208 −0.458604 0.888641i \(-0.651650\pi\)
−0.458604 + 0.888641i \(0.651650\pi\)
\(14\) 0 0
\(15\) −2.17612 −0.561871
\(16\) −1.42746 −0.356865
\(17\) −2.96912 −0.720118 −0.360059 0.932930i \(-0.617243\pi\)
−0.360059 + 0.932930i \(0.617243\pi\)
\(18\) −1.05351 −0.248315
\(19\) 0.697769 0.160079 0.0800396 0.996792i \(-0.474495\pi\)
0.0800396 + 0.996792i \(0.474495\pi\)
\(20\) −1.93700 −0.433126
\(21\) 0 0
\(22\) 2.76276 0.589022
\(23\) 1.00000 0.208514
\(24\) −3.04477 −0.621510
\(25\) −0.264516 −0.0529033
\(26\) 3.48400 0.683269
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.31827 1.35897 0.679485 0.733690i \(-0.262203\pi\)
0.679485 + 0.733690i \(0.262203\pi\)
\(30\) 2.29256 0.418563
\(31\) 5.60209 1.00617 0.503083 0.864238i \(-0.332199\pi\)
0.503083 + 0.864238i \(0.332199\pi\)
\(32\) −4.58569 −0.810643
\(33\) 2.62243 0.456506
\(34\) 3.12800 0.536448
\(35\) 0 0
\(36\) −0.890116 −0.148353
\(37\) −4.52695 −0.744226 −0.372113 0.928187i \(-0.621367\pi\)
−0.372113 + 0.928187i \(0.621367\pi\)
\(38\) −0.735106 −0.119250
\(39\) 3.30704 0.529550
\(40\) 6.62577 1.04763
\(41\) 2.28774 0.357285 0.178643 0.983914i \(-0.442829\pi\)
0.178643 + 0.983914i \(0.442829\pi\)
\(42\) 0 0
\(43\) −7.79527 −1.18877 −0.594384 0.804182i \(-0.702604\pi\)
−0.594384 + 0.804182i \(0.702604\pi\)
\(44\) 2.33427 0.351904
\(45\) 2.17612 0.324396
\(46\) −1.05351 −0.155332
\(47\) −3.81394 −0.556321 −0.278160 0.960535i \(-0.589725\pi\)
−0.278160 + 0.960535i \(0.589725\pi\)
\(48\) 1.42746 0.206036
\(49\) 0 0
\(50\) 0.278671 0.0394100
\(51\) 2.96912 0.415760
\(52\) 2.94365 0.408211
\(53\) 3.13059 0.430020 0.215010 0.976612i \(-0.431022\pi\)
0.215010 + 0.976612i \(0.431022\pi\)
\(54\) 1.05351 0.143365
\(55\) −5.70671 −0.769493
\(56\) 0 0
\(57\) −0.697769 −0.0924217
\(58\) −7.70988 −1.01236
\(59\) 12.1628 1.58347 0.791733 0.610868i \(-0.209179\pi\)
0.791733 + 0.610868i \(0.209179\pi\)
\(60\) 1.93700 0.250065
\(61\) 1.48459 0.190082 0.0950412 0.995473i \(-0.469702\pi\)
0.0950412 + 0.995473i \(0.469702\pi\)
\(62\) −5.90186 −0.749537
\(63\) 0 0
\(64\) 7.68599 0.960749
\(65\) −7.19650 −0.892617
\(66\) −2.76276 −0.340072
\(67\) 11.2755 1.37753 0.688763 0.724986i \(-0.258154\pi\)
0.688763 + 0.724986i \(0.258154\pi\)
\(68\) 2.64286 0.320494
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 13.8643 1.64539 0.822693 0.568486i \(-0.192471\pi\)
0.822693 + 0.568486i \(0.192471\pi\)
\(72\) 3.04477 0.358829
\(73\) 0.705829 0.0826111 0.0413055 0.999147i \(-0.486848\pi\)
0.0413055 + 0.999147i \(0.486848\pi\)
\(74\) 4.76919 0.554407
\(75\) 0.264516 0.0305437
\(76\) −0.621095 −0.0712445
\(77\) 0 0
\(78\) −3.48400 −0.394485
\(79\) −8.60406 −0.968033 −0.484016 0.875059i \(-0.660822\pi\)
−0.484016 + 0.875059i \(0.660822\pi\)
\(80\) −3.10632 −0.347297
\(81\) 1.00000 0.111111
\(82\) −2.41016 −0.266157
\(83\) −11.5918 −1.27237 −0.636185 0.771537i \(-0.719489\pi\)
−0.636185 + 0.771537i \(0.719489\pi\)
\(84\) 0 0
\(85\) −6.46115 −0.700811
\(86\) 8.21240 0.885566
\(87\) −7.31827 −0.784601
\(88\) −7.98469 −0.851170
\(89\) −0.333776 −0.0353802 −0.0176901 0.999844i \(-0.505631\pi\)
−0.0176901 + 0.999844i \(0.505631\pi\)
\(90\) −2.29256 −0.241657
\(91\) 0 0
\(92\) −0.890116 −0.0928010
\(93\) −5.60209 −0.580910
\(94\) 4.01803 0.414428
\(95\) 1.51843 0.155787
\(96\) 4.58569 0.468025
\(97\) 0.124876 0.0126793 0.00633963 0.999980i \(-0.497982\pi\)
0.00633963 + 0.999980i \(0.497982\pi\)
\(98\) 0 0
\(99\) −2.62243 −0.263564
\(100\) 0.235450 0.0235450
\(101\) −8.77972 −0.873615 −0.436807 0.899555i \(-0.643891\pi\)
−0.436807 + 0.899555i \(0.643891\pi\)
\(102\) −3.12800 −0.309718
\(103\) −5.46482 −0.538465 −0.269232 0.963075i \(-0.586770\pi\)
−0.269232 + 0.963075i \(0.586770\pi\)
\(104\) −10.0692 −0.987363
\(105\) 0 0
\(106\) −3.29811 −0.320341
\(107\) 10.8427 1.04820 0.524100 0.851657i \(-0.324402\pi\)
0.524100 + 0.851657i \(0.324402\pi\)
\(108\) 0.890116 0.0856515
\(109\) −1.05736 −0.101277 −0.0506384 0.998717i \(-0.516126\pi\)
−0.0506384 + 0.998717i \(0.516126\pi\)
\(110\) 6.01208 0.573229
\(111\) 4.52695 0.429679
\(112\) 0 0
\(113\) −14.0938 −1.32584 −0.662918 0.748692i \(-0.730682\pi\)
−0.662918 + 0.748692i \(0.730682\pi\)
\(114\) 0.735106 0.0688490
\(115\) 2.17612 0.202924
\(116\) −6.51411 −0.604820
\(117\) −3.30704 −0.305736
\(118\) −12.8137 −1.17959
\(119\) 0 0
\(120\) −6.62577 −0.604847
\(121\) −4.12286 −0.374806
\(122\) −1.56403 −0.141601
\(123\) −2.28774 −0.206279
\(124\) −4.98651 −0.447802
\(125\) −11.4562 −1.02467
\(126\) 0 0
\(127\) 15.7177 1.39472 0.697360 0.716721i \(-0.254358\pi\)
0.697360 + 0.716721i \(0.254358\pi\)
\(128\) 1.07410 0.0949383
\(129\) 7.79527 0.686335
\(130\) 7.58159 0.664950
\(131\) −10.0029 −0.873960 −0.436980 0.899471i \(-0.643952\pi\)
−0.436980 + 0.899471i \(0.643952\pi\)
\(132\) −2.33427 −0.203172
\(133\) 0 0
\(134\) −11.8789 −1.02618
\(135\) −2.17612 −0.187290
\(136\) −9.04028 −0.775198
\(137\) 6.06773 0.518401 0.259201 0.965823i \(-0.416541\pi\)
0.259201 + 0.965823i \(0.416541\pi\)
\(138\) 1.05351 0.0896808
\(139\) 14.1199 1.19763 0.598815 0.800887i \(-0.295638\pi\)
0.598815 + 0.800887i \(0.295638\pi\)
\(140\) 0 0
\(141\) 3.81394 0.321192
\(142\) −14.6061 −1.22572
\(143\) 8.67248 0.725229
\(144\) −1.42746 −0.118955
\(145\) 15.9254 1.32253
\(146\) −0.743598 −0.0615406
\(147\) 0 0
\(148\) 4.02951 0.331224
\(149\) 1.01590 0.0832261 0.0416131 0.999134i \(-0.486750\pi\)
0.0416131 + 0.999134i \(0.486750\pi\)
\(150\) −0.278671 −0.0227534
\(151\) 22.5385 1.83416 0.917080 0.398702i \(-0.130539\pi\)
0.917080 + 0.398702i \(0.130539\pi\)
\(152\) 2.12454 0.172323
\(153\) −2.96912 −0.240039
\(154\) 0 0
\(155\) 12.1908 0.979189
\(156\) −2.94365 −0.235681
\(157\) −11.3661 −0.907114 −0.453557 0.891227i \(-0.649845\pi\)
−0.453557 + 0.891227i \(0.649845\pi\)
\(158\) 9.06447 0.721131
\(159\) −3.13059 −0.248272
\(160\) −9.97899 −0.788909
\(161\) 0 0
\(162\) −1.05351 −0.0827716
\(163\) 14.7670 1.15664 0.578322 0.815809i \(-0.303708\pi\)
0.578322 + 0.815809i \(0.303708\pi\)
\(164\) −2.03635 −0.159013
\(165\) 5.70671 0.444267
\(166\) 12.2121 0.947845
\(167\) −8.85336 −0.685093 −0.342547 0.939501i \(-0.611290\pi\)
−0.342547 + 0.939501i \(0.611290\pi\)
\(168\) 0 0
\(169\) −2.06349 −0.158730
\(170\) 6.80689 0.522065
\(171\) 0.697769 0.0533597
\(172\) 6.93869 0.529070
\(173\) 15.7040 1.19396 0.596978 0.802258i \(-0.296368\pi\)
0.596978 + 0.802258i \(0.296368\pi\)
\(174\) 7.70988 0.584484
\(175\) 0 0
\(176\) 3.74342 0.282171
\(177\) −12.1628 −0.914214
\(178\) 0.351636 0.0263562
\(179\) −5.30752 −0.396703 −0.198351 0.980131i \(-0.563559\pi\)
−0.198351 + 0.980131i \(0.563559\pi\)
\(180\) −1.93700 −0.144375
\(181\) 1.93050 0.143493 0.0717463 0.997423i \(-0.477143\pi\)
0.0717463 + 0.997423i \(0.477143\pi\)
\(182\) 0 0
\(183\) −1.48459 −0.109744
\(184\) 3.04477 0.224463
\(185\) −9.85117 −0.724273
\(186\) 5.90186 0.432745
\(187\) 7.78631 0.569391
\(188\) 3.39485 0.247595
\(189\) 0 0
\(190\) −1.59968 −0.116053
\(191\) 10.4418 0.755540 0.377770 0.925899i \(-0.376691\pi\)
0.377770 + 0.925899i \(0.376691\pi\)
\(192\) −7.68599 −0.554689
\(193\) 24.6163 1.77192 0.885959 0.463763i \(-0.153501\pi\)
0.885959 + 0.463763i \(0.153501\pi\)
\(194\) −0.131558 −0.00944534
\(195\) 7.19650 0.515352
\(196\) 0 0
\(197\) 5.17797 0.368915 0.184458 0.982840i \(-0.440947\pi\)
0.184458 + 0.982840i \(0.440947\pi\)
\(198\) 2.76276 0.196341
\(199\) −16.6371 −1.17937 −0.589687 0.807632i \(-0.700749\pi\)
−0.589687 + 0.807632i \(0.700749\pi\)
\(200\) −0.805391 −0.0569497
\(201\) −11.2755 −0.795315
\(202\) 9.24952 0.650794
\(203\) 0 0
\(204\) −2.64286 −0.185037
\(205\) 4.97839 0.347706
\(206\) 5.75724 0.401126
\(207\) 1.00000 0.0695048
\(208\) 4.72067 0.327320
\(209\) −1.82985 −0.126573
\(210\) 0 0
\(211\) 28.5995 1.96887 0.984435 0.175751i \(-0.0562354\pi\)
0.984435 + 0.175751i \(0.0562354\pi\)
\(212\) −2.78659 −0.191384
\(213\) −13.8643 −0.949964
\(214\) −11.4229 −0.780850
\(215\) −16.9634 −1.15690
\(216\) −3.04477 −0.207170
\(217\) 0 0
\(218\) 1.11394 0.0754456
\(219\) −0.705829 −0.0476955
\(220\) 5.07964 0.342469
\(221\) 9.81900 0.660498
\(222\) −4.76919 −0.320087
\(223\) 16.5464 1.10803 0.554016 0.832506i \(-0.313095\pi\)
0.554016 + 0.832506i \(0.313095\pi\)
\(224\) 0 0
\(225\) −0.264516 −0.0176344
\(226\) 14.8480 0.987674
\(227\) −17.9650 −1.19238 −0.596189 0.802844i \(-0.703319\pi\)
−0.596189 + 0.802844i \(0.703319\pi\)
\(228\) 0.621095 0.0411330
\(229\) 28.0061 1.85070 0.925348 0.379120i \(-0.123773\pi\)
0.925348 + 0.379120i \(0.123773\pi\)
\(230\) −2.29256 −0.151167
\(231\) 0 0
\(232\) 22.2824 1.46291
\(233\) 23.2462 1.52291 0.761454 0.648219i \(-0.224486\pi\)
0.761454 + 0.648219i \(0.224486\pi\)
\(234\) 3.48400 0.227756
\(235\) −8.29958 −0.541405
\(236\) −10.8263 −0.704734
\(237\) 8.60406 0.558894
\(238\) 0 0
\(239\) 6.43482 0.416234 0.208117 0.978104i \(-0.433267\pi\)
0.208117 + 0.978104i \(0.433267\pi\)
\(240\) 3.10632 0.200512
\(241\) 23.5177 1.51491 0.757455 0.652888i \(-0.226443\pi\)
0.757455 + 0.652888i \(0.226443\pi\)
\(242\) 4.34348 0.279209
\(243\) −1.00000 −0.0641500
\(244\) −1.32146 −0.0845977
\(245\) 0 0
\(246\) 2.41016 0.153666
\(247\) −2.30755 −0.146826
\(248\) 17.0571 1.08312
\(249\) 11.5918 0.734603
\(250\) 12.0692 0.763325
\(251\) −3.12594 −0.197307 −0.0986537 0.995122i \(-0.531454\pi\)
−0.0986537 + 0.995122i \(0.531454\pi\)
\(252\) 0 0
\(253\) −2.62243 −0.164871
\(254\) −16.5587 −1.03899
\(255\) 6.46115 0.404613
\(256\) −16.5036 −1.03147
\(257\) −20.5336 −1.28085 −0.640426 0.768020i \(-0.721242\pi\)
−0.640426 + 0.768020i \(0.721242\pi\)
\(258\) −8.21240 −0.511282
\(259\) 0 0
\(260\) 6.40572 0.397266
\(261\) 7.31827 0.452990
\(262\) 10.5382 0.651052
\(263\) 19.4476 1.19919 0.599594 0.800304i \(-0.295329\pi\)
0.599594 + 0.800304i \(0.295329\pi\)
\(264\) 7.98469 0.491424
\(265\) 6.81254 0.418491
\(266\) 0 0
\(267\) 0.333776 0.0204267
\(268\) −10.0365 −0.613079
\(269\) 7.71940 0.470660 0.235330 0.971916i \(-0.424383\pi\)
0.235330 + 0.971916i \(0.424383\pi\)
\(270\) 2.29256 0.139521
\(271\) 30.8637 1.87484 0.937418 0.348205i \(-0.113209\pi\)
0.937418 + 0.348205i \(0.113209\pi\)
\(272\) 4.23831 0.256985
\(273\) 0 0
\(274\) −6.39242 −0.386180
\(275\) 0.693675 0.0418302
\(276\) 0.890116 0.0535787
\(277\) 6.40144 0.384625 0.192313 0.981334i \(-0.438401\pi\)
0.192313 + 0.981334i \(0.438401\pi\)
\(278\) −14.8754 −0.892168
\(279\) 5.60209 0.335388
\(280\) 0 0
\(281\) −4.79227 −0.285883 −0.142941 0.989731i \(-0.545656\pi\)
−0.142941 + 0.989731i \(0.545656\pi\)
\(282\) −4.01803 −0.239270
\(283\) −11.7015 −0.695579 −0.347790 0.937573i \(-0.613068\pi\)
−0.347790 + 0.937573i \(0.613068\pi\)
\(284\) −12.3408 −0.732292
\(285\) −1.51843 −0.0899438
\(286\) −9.13655 −0.540255
\(287\) 0 0
\(288\) −4.58569 −0.270214
\(289\) −8.18432 −0.481431
\(290\) −16.7776 −0.985214
\(291\) −0.124876 −0.00732038
\(292\) −0.628270 −0.0367667
\(293\) 19.9031 1.16275 0.581376 0.813635i \(-0.302515\pi\)
0.581376 + 0.813635i \(0.302515\pi\)
\(294\) 0 0
\(295\) 26.4677 1.54101
\(296\) −13.7835 −0.801150
\(297\) 2.62243 0.152169
\(298\) −1.07027 −0.0619988
\(299\) −3.30704 −0.191251
\(300\) −0.235450 −0.0135937
\(301\) 0 0
\(302\) −23.7446 −1.36635
\(303\) 8.77972 0.504382
\(304\) −0.996038 −0.0571267
\(305\) 3.23064 0.184986
\(306\) 3.12800 0.178816
\(307\) 21.7588 1.24184 0.620920 0.783874i \(-0.286759\pi\)
0.620920 + 0.783874i \(0.286759\pi\)
\(308\) 0 0
\(309\) 5.46482 0.310883
\(310\) −12.8431 −0.729441
\(311\) −8.14345 −0.461773 −0.230886 0.972981i \(-0.574163\pi\)
−0.230886 + 0.972981i \(0.574163\pi\)
\(312\) 10.0692 0.570054
\(313\) 1.05044 0.0593743 0.0296871 0.999559i \(-0.490549\pi\)
0.0296871 + 0.999559i \(0.490549\pi\)
\(314\) 11.9743 0.675749
\(315\) 0 0
\(316\) 7.65861 0.430831
\(317\) 16.3680 0.919317 0.459658 0.888096i \(-0.347972\pi\)
0.459658 + 0.888096i \(0.347972\pi\)
\(318\) 3.29811 0.184949
\(319\) −19.1917 −1.07453
\(320\) 16.7256 0.934990
\(321\) −10.8427 −0.605178
\(322\) 0 0
\(323\) −2.07176 −0.115276
\(324\) −0.890116 −0.0494509
\(325\) 0.874766 0.0485233
\(326\) −15.5572 −0.861635
\(327\) 1.05736 0.0584722
\(328\) 6.96564 0.384613
\(329\) 0 0
\(330\) −6.01208 −0.330954
\(331\) 29.2929 1.61008 0.805042 0.593218i \(-0.202143\pi\)
0.805042 + 0.593218i \(0.202143\pi\)
\(332\) 10.3181 0.566278
\(333\) −4.52695 −0.248075
\(334\) 9.32711 0.510356
\(335\) 24.5369 1.34059
\(336\) 0 0
\(337\) −4.27417 −0.232829 −0.116414 0.993201i \(-0.537140\pi\)
−0.116414 + 0.993201i \(0.537140\pi\)
\(338\) 2.17391 0.118245
\(339\) 14.0938 0.765472
\(340\) 5.75118 0.311901
\(341\) −14.6911 −0.795567
\(342\) −0.735106 −0.0397500
\(343\) 0 0
\(344\) −23.7348 −1.27969
\(345\) −2.17612 −0.117158
\(346\) −16.5444 −0.889430
\(347\) 18.4439 0.990120 0.495060 0.868859i \(-0.335146\pi\)
0.495060 + 0.868859i \(0.335146\pi\)
\(348\) 6.51411 0.349193
\(349\) −19.9398 −1.06736 −0.533678 0.845688i \(-0.679190\pi\)
−0.533678 + 0.845688i \(0.679190\pi\)
\(350\) 0 0
\(351\) 3.30704 0.176517
\(352\) 12.0256 0.640969
\(353\) −22.0855 −1.17549 −0.587746 0.809046i \(-0.699985\pi\)
−0.587746 + 0.809046i \(0.699985\pi\)
\(354\) 12.8137 0.681038
\(355\) 30.1703 1.60127
\(356\) 0.297099 0.0157462
\(357\) 0 0
\(358\) 5.59153 0.295521
\(359\) −12.0390 −0.635392 −0.317696 0.948193i \(-0.602909\pi\)
−0.317696 + 0.948193i \(0.602909\pi\)
\(360\) 6.62577 0.349209
\(361\) −18.5131 −0.974375
\(362\) −2.03380 −0.106894
\(363\) 4.12286 0.216394
\(364\) 0 0
\(365\) 1.53597 0.0803962
\(366\) 1.56403 0.0817533
\(367\) 21.6477 1.13000 0.565000 0.825091i \(-0.308876\pi\)
0.565000 + 0.825091i \(0.308876\pi\)
\(368\) −1.42746 −0.0744116
\(369\) 2.28774 0.119095
\(370\) 10.3783 0.539543
\(371\) 0 0
\(372\) 4.98651 0.258539
\(373\) 21.9427 1.13615 0.568074 0.822977i \(-0.307689\pi\)
0.568074 + 0.822977i \(0.307689\pi\)
\(374\) −8.20296 −0.424165
\(375\) 11.4562 0.591596
\(376\) −11.6126 −0.598872
\(377\) −24.2018 −1.24646
\(378\) 0 0
\(379\) 11.8759 0.610026 0.305013 0.952348i \(-0.401339\pi\)
0.305013 + 0.952348i \(0.401339\pi\)
\(380\) −1.35158 −0.0693343
\(381\) −15.7177 −0.805242
\(382\) −11.0005 −0.562835
\(383\) −4.46691 −0.228248 −0.114124 0.993466i \(-0.536406\pi\)
−0.114124 + 0.993466i \(0.536406\pi\)
\(384\) −1.07410 −0.0548126
\(385\) 0 0
\(386\) −25.9335 −1.31998
\(387\) −7.79527 −0.396256
\(388\) −0.111154 −0.00564301
\(389\) −2.71425 −0.137618 −0.0688089 0.997630i \(-0.521920\pi\)
−0.0688089 + 0.997630i \(0.521920\pi\)
\(390\) −7.58159 −0.383909
\(391\) −2.96912 −0.150155
\(392\) 0 0
\(393\) 10.0029 0.504581
\(394\) −5.45505 −0.274821
\(395\) −18.7234 −0.942079
\(396\) 2.33427 0.117301
\(397\) −7.38176 −0.370480 −0.185240 0.982693i \(-0.559306\pi\)
−0.185240 + 0.982693i \(0.559306\pi\)
\(398\) 17.5274 0.878567
\(399\) 0 0
\(400\) 0.377587 0.0188793
\(401\) 2.70191 0.134927 0.0674634 0.997722i \(-0.478509\pi\)
0.0674634 + 0.997722i \(0.478509\pi\)
\(402\) 11.8789 0.592466
\(403\) −18.5263 −0.922863
\(404\) 7.81497 0.388809
\(405\) 2.17612 0.108132
\(406\) 0 0
\(407\) 11.8716 0.588454
\(408\) 9.04028 0.447561
\(409\) 24.8095 1.22675 0.613376 0.789791i \(-0.289811\pi\)
0.613376 + 0.789791i \(0.289811\pi\)
\(410\) −5.24479 −0.259021
\(411\) −6.06773 −0.299299
\(412\) 4.86432 0.239648
\(413\) 0 0
\(414\) −1.05351 −0.0517772
\(415\) −25.2252 −1.23826
\(416\) 15.1651 0.743528
\(417\) −14.1199 −0.691452
\(418\) 1.92776 0.0942901
\(419\) −20.0785 −0.980899 −0.490450 0.871470i \(-0.663167\pi\)
−0.490450 + 0.871470i \(0.663167\pi\)
\(420\) 0 0
\(421\) −26.5979 −1.29630 −0.648151 0.761512i \(-0.724458\pi\)
−0.648151 + 0.761512i \(0.724458\pi\)
\(422\) −30.1298 −1.46670
\(423\) −3.81394 −0.185440
\(424\) 9.53193 0.462911
\(425\) 0.785381 0.0380966
\(426\) 14.6061 0.707670
\(427\) 0 0
\(428\) −9.65122 −0.466509
\(429\) −8.67248 −0.418711
\(430\) 17.8711 0.861823
\(431\) −24.0219 −1.15710 −0.578548 0.815649i \(-0.696380\pi\)
−0.578548 + 0.815649i \(0.696380\pi\)
\(432\) 1.42746 0.0686788
\(433\) −14.0909 −0.677165 −0.338582 0.940937i \(-0.609947\pi\)
−0.338582 + 0.940937i \(0.609947\pi\)
\(434\) 0 0
\(435\) −15.9254 −0.763565
\(436\) 0.941174 0.0450740
\(437\) 0.697769 0.0333788
\(438\) 0.743598 0.0355305
\(439\) 2.89952 0.138387 0.0691933 0.997603i \(-0.477957\pi\)
0.0691933 + 0.997603i \(0.477957\pi\)
\(440\) −17.3756 −0.828350
\(441\) 0 0
\(442\) −10.3444 −0.492034
\(443\) 13.9959 0.664965 0.332482 0.943109i \(-0.392114\pi\)
0.332482 + 0.943109i \(0.392114\pi\)
\(444\) −4.02951 −0.191232
\(445\) −0.726335 −0.0344316
\(446\) −17.4319 −0.825422
\(447\) −1.01590 −0.0480506
\(448\) 0 0
\(449\) 12.6648 0.597690 0.298845 0.954302i \(-0.403399\pi\)
0.298845 + 0.954302i \(0.403399\pi\)
\(450\) 0.278671 0.0131367
\(451\) −5.99944 −0.282502
\(452\) 12.5451 0.590074
\(453\) −22.5385 −1.05895
\(454\) 18.9263 0.888256
\(455\) 0 0
\(456\) −2.12454 −0.0994908
\(457\) −3.55672 −0.166376 −0.0831881 0.996534i \(-0.526510\pi\)
−0.0831881 + 0.996534i \(0.526510\pi\)
\(458\) −29.5047 −1.37866
\(459\) 2.96912 0.138587
\(460\) −1.93700 −0.0903129
\(461\) 16.1441 0.751904 0.375952 0.926639i \(-0.377316\pi\)
0.375952 + 0.926639i \(0.377316\pi\)
\(462\) 0 0
\(463\) 9.78687 0.454834 0.227417 0.973797i \(-0.426972\pi\)
0.227417 + 0.973797i \(0.426972\pi\)
\(464\) −10.4466 −0.484969
\(465\) −12.1908 −0.565335
\(466\) −24.4901 −1.13448
\(467\) −30.1576 −1.39553 −0.697763 0.716329i \(-0.745821\pi\)
−0.697763 + 0.716329i \(0.745821\pi\)
\(468\) 2.94365 0.136070
\(469\) 0 0
\(470\) 8.74370 0.403317
\(471\) 11.3661 0.523722
\(472\) 37.0330 1.70458
\(473\) 20.4425 0.939949
\(474\) −9.06447 −0.416345
\(475\) −0.184571 −0.00846871
\(476\) 0 0
\(477\) 3.13059 0.143340
\(478\) −6.77914 −0.310071
\(479\) 2.92167 0.133494 0.0667471 0.997770i \(-0.478738\pi\)
0.0667471 + 0.997770i \(0.478738\pi\)
\(480\) 9.97899 0.455477
\(481\) 14.9708 0.682610
\(482\) −24.7762 −1.12852
\(483\) 0 0
\(484\) 3.66983 0.166810
\(485\) 0.271745 0.0123393
\(486\) 1.05351 0.0477882
\(487\) 32.2382 1.46085 0.730427 0.682991i \(-0.239321\pi\)
0.730427 + 0.682991i \(0.239321\pi\)
\(488\) 4.52023 0.204621
\(489\) −14.7670 −0.667789
\(490\) 0 0
\(491\) −12.1670 −0.549091 −0.274546 0.961574i \(-0.588527\pi\)
−0.274546 + 0.961574i \(0.588527\pi\)
\(492\) 2.03635 0.0918059
\(493\) −21.7288 −0.978618
\(494\) 2.43103 0.109377
\(495\) −5.70671 −0.256498
\(496\) −7.99677 −0.359066
\(497\) 0 0
\(498\) −12.2121 −0.547238
\(499\) 31.8296 1.42489 0.712444 0.701729i \(-0.247588\pi\)
0.712444 + 0.701729i \(0.247588\pi\)
\(500\) 10.1973 0.456039
\(501\) 8.85336 0.395539
\(502\) 3.29321 0.146983
\(503\) 0.217299 0.00968890 0.00484445 0.999988i \(-0.498458\pi\)
0.00484445 + 0.999988i \(0.498458\pi\)
\(504\) 0 0
\(505\) −19.1057 −0.850192
\(506\) 2.76276 0.122820
\(507\) 2.06349 0.0916427
\(508\) −13.9906 −0.620731
\(509\) 15.1597 0.671942 0.335971 0.941872i \(-0.390936\pi\)
0.335971 + 0.941872i \(0.390936\pi\)
\(510\) −6.80689 −0.301414
\(511\) 0 0
\(512\) 15.2385 0.673452
\(513\) −0.697769 −0.0308072
\(514\) 21.6324 0.954164
\(515\) −11.8921 −0.524028
\(516\) −6.93869 −0.305459
\(517\) 10.0018 0.439878
\(518\) 0 0
\(519\) −15.7040 −0.689330
\(520\) −21.9117 −0.960891
\(521\) −26.5770 −1.16436 −0.582179 0.813060i \(-0.697800\pi\)
−0.582179 + 0.813060i \(0.697800\pi\)
\(522\) −7.70988 −0.337452
\(523\) −36.5361 −1.59761 −0.798807 0.601588i \(-0.794535\pi\)
−0.798807 + 0.601588i \(0.794535\pi\)
\(524\) 8.90377 0.388963
\(525\) 0 0
\(526\) −20.4882 −0.893328
\(527\) −16.6333 −0.724557
\(528\) −3.74342 −0.162911
\(529\) 1.00000 0.0434783
\(530\) −7.17708 −0.311752
\(531\) 12.1628 0.527822
\(532\) 0 0
\(533\) −7.56565 −0.327705
\(534\) −0.351636 −0.0152168
\(535\) 23.5949 1.02010
\(536\) 34.3314 1.48289
\(537\) 5.30752 0.229036
\(538\) −8.13247 −0.350616
\(539\) 0 0
\(540\) 1.93700 0.0833550
\(541\) 9.94639 0.427629 0.213814 0.976874i \(-0.431411\pi\)
0.213814 + 0.976874i \(0.431411\pi\)
\(542\) −32.5152 −1.39665
\(543\) −1.93050 −0.0828455
\(544\) 13.6155 0.583758
\(545\) −2.30094 −0.0985614
\(546\) 0 0
\(547\) −35.8124 −1.53123 −0.765614 0.643300i \(-0.777565\pi\)
−0.765614 + 0.643300i \(0.777565\pi\)
\(548\) −5.40099 −0.230719
\(549\) 1.48459 0.0633608
\(550\) −0.730794 −0.0311612
\(551\) 5.10646 0.217543
\(552\) −3.04477 −0.129594
\(553\) 0 0
\(554\) −6.74398 −0.286524
\(555\) 9.85117 0.418159
\(556\) −12.5683 −0.533015
\(557\) −13.5026 −0.572124 −0.286062 0.958211i \(-0.592346\pi\)
−0.286062 + 0.958211i \(0.592346\pi\)
\(558\) −5.90186 −0.249846
\(559\) 25.7793 1.09035
\(560\) 0 0
\(561\) −7.78631 −0.328738
\(562\) 5.04871 0.212967
\(563\) 37.4182 1.57699 0.788495 0.615041i \(-0.210861\pi\)
0.788495 + 0.615041i \(0.210861\pi\)
\(564\) −3.39485 −0.142949
\(565\) −30.6698 −1.29029
\(566\) 12.3276 0.518168
\(567\) 0 0
\(568\) 42.2135 1.77124
\(569\) 42.9853 1.80204 0.901019 0.433779i \(-0.142820\pi\)
0.901019 + 0.433779i \(0.142820\pi\)
\(570\) 1.59968 0.0670031
\(571\) −20.7497 −0.868349 −0.434175 0.900829i \(-0.642960\pi\)
−0.434175 + 0.900829i \(0.642960\pi\)
\(572\) −7.71951 −0.322769
\(573\) −10.4418 −0.436211
\(574\) 0 0
\(575\) −0.264516 −0.0110311
\(576\) 7.68599 0.320250
\(577\) −38.3634 −1.59709 −0.798544 0.601936i \(-0.794396\pi\)
−0.798544 + 0.601936i \(0.794396\pi\)
\(578\) 8.62227 0.358639
\(579\) −24.6163 −1.02302
\(580\) −14.1755 −0.588604
\(581\) 0 0
\(582\) 0.131558 0.00545327
\(583\) −8.20976 −0.340014
\(584\) 2.14909 0.0889298
\(585\) −7.19650 −0.297539
\(586\) −20.9681 −0.866185
\(587\) −21.7316 −0.896958 −0.448479 0.893793i \(-0.648034\pi\)
−0.448479 + 0.893793i \(0.648034\pi\)
\(588\) 0 0
\(589\) 3.90896 0.161066
\(590\) −27.8840 −1.14797
\(591\) −5.17797 −0.212993
\(592\) 6.46205 0.265589
\(593\) −0.868007 −0.0356448 −0.0178224 0.999841i \(-0.505673\pi\)
−0.0178224 + 0.999841i \(0.505673\pi\)
\(594\) −2.76276 −0.113357
\(595\) 0 0
\(596\) −0.904273 −0.0370404
\(597\) 16.6371 0.680911
\(598\) 3.48400 0.142471
\(599\) −13.0652 −0.533829 −0.266914 0.963720i \(-0.586004\pi\)
−0.266914 + 0.963720i \(0.586004\pi\)
\(600\) 0.805391 0.0328799
\(601\) −14.4244 −0.588385 −0.294192 0.955746i \(-0.595051\pi\)
−0.294192 + 0.955746i \(0.595051\pi\)
\(602\) 0 0
\(603\) 11.2755 0.459176
\(604\) −20.0619 −0.816308
\(605\) −8.97183 −0.364757
\(606\) −9.24952 −0.375736
\(607\) 28.9832 1.17639 0.588195 0.808719i \(-0.299839\pi\)
0.588195 + 0.808719i \(0.299839\pi\)
\(608\) −3.19975 −0.129767
\(609\) 0 0
\(610\) −3.40352 −0.137804
\(611\) 12.6129 0.510262
\(612\) 2.64286 0.106831
\(613\) −5.95694 −0.240598 −0.120299 0.992738i \(-0.538385\pi\)
−0.120299 + 0.992738i \(0.538385\pi\)
\(614\) −22.9231 −0.925101
\(615\) −4.97839 −0.200748
\(616\) 0 0
\(617\) −40.0363 −1.61180 −0.805901 0.592050i \(-0.798319\pi\)
−0.805901 + 0.592050i \(0.798319\pi\)
\(618\) −5.75724 −0.231590
\(619\) 10.5175 0.422734 0.211367 0.977407i \(-0.432209\pi\)
0.211367 + 0.977407i \(0.432209\pi\)
\(620\) −10.8512 −0.435796
\(621\) −1.00000 −0.0401286
\(622\) 8.57921 0.343995
\(623\) 0 0
\(624\) −4.72067 −0.188978
\(625\) −23.6074 −0.944298
\(626\) −1.10665 −0.0442305
\(627\) 1.82985 0.0730771
\(628\) 10.1172 0.403718
\(629\) 13.4411 0.535930
\(630\) 0 0
\(631\) 44.9392 1.78900 0.894501 0.447065i \(-0.147531\pi\)
0.894501 + 0.447065i \(0.147531\pi\)
\(632\) −26.1974 −1.04208
\(633\) −28.5995 −1.13673
\(634\) −17.2438 −0.684840
\(635\) 34.2035 1.35733
\(636\) 2.78659 0.110496
\(637\) 0 0
\(638\) 20.2186 0.800462
\(639\) 13.8643 0.548462
\(640\) 2.33738 0.0923929
\(641\) −32.0409 −1.26554 −0.632769 0.774341i \(-0.718082\pi\)
−0.632769 + 0.774341i \(0.718082\pi\)
\(642\) 11.4229 0.450824
\(643\) 38.5772 1.52134 0.760669 0.649140i \(-0.224871\pi\)
0.760669 + 0.649140i \(0.224871\pi\)
\(644\) 0 0
\(645\) 16.9634 0.667934
\(646\) 2.18262 0.0858740
\(647\) 30.9334 1.21612 0.608058 0.793892i \(-0.291949\pi\)
0.608058 + 0.793892i \(0.291949\pi\)
\(648\) 3.04477 0.119610
\(649\) −31.8962 −1.25203
\(650\) −0.921575 −0.0361472
\(651\) 0 0
\(652\) −13.1444 −0.514774
\(653\) 42.5576 1.66541 0.832704 0.553718i \(-0.186792\pi\)
0.832704 + 0.553718i \(0.186792\pi\)
\(654\) −1.11394 −0.0435585
\(655\) −21.7676 −0.850529
\(656\) −3.26566 −0.127503
\(657\) 0.705829 0.0275370
\(658\) 0 0
\(659\) 32.5053 1.26623 0.633113 0.774059i \(-0.281777\pi\)
0.633113 + 0.774059i \(0.281777\pi\)
\(660\) −5.07964 −0.197725
\(661\) 33.5828 1.30622 0.653109 0.757264i \(-0.273464\pi\)
0.653109 + 0.757264i \(0.273464\pi\)
\(662\) −30.8604 −1.19942
\(663\) −9.81900 −0.381338
\(664\) −35.2945 −1.36969
\(665\) 0 0
\(666\) 4.76919 0.184802
\(667\) 7.31827 0.283365
\(668\) 7.88052 0.304906
\(669\) −16.5464 −0.639723
\(670\) −25.8499 −0.998668
\(671\) −3.89324 −0.150297
\(672\) 0 0
\(673\) −1.23960 −0.0477832 −0.0238916 0.999715i \(-0.507606\pi\)
−0.0238916 + 0.999715i \(0.507606\pi\)
\(674\) 4.50288 0.173444
\(675\) 0.264516 0.0101812
\(676\) 1.83674 0.0706440
\(677\) 5.69833 0.219004 0.109502 0.993987i \(-0.465074\pi\)
0.109502 + 0.993987i \(0.465074\pi\)
\(678\) −14.8480 −0.570234
\(679\) 0 0
\(680\) −19.6727 −0.754414
\(681\) 17.9650 0.688420
\(682\) 15.4772 0.592653
\(683\) 27.2518 1.04276 0.521380 0.853325i \(-0.325418\pi\)
0.521380 + 0.853325i \(0.325418\pi\)
\(684\) −0.621095 −0.0237482
\(685\) 13.2041 0.504503
\(686\) 0 0
\(687\) −28.0061 −1.06850
\(688\) 11.1274 0.424230
\(689\) −10.3530 −0.394418
\(690\) 2.29256 0.0872763
\(691\) −45.1514 −1.71764 −0.858820 0.512277i \(-0.828802\pi\)
−0.858820 + 0.512277i \(0.828802\pi\)
\(692\) −13.9784 −0.531379
\(693\) 0 0
\(694\) −19.4308 −0.737584
\(695\) 30.7264 1.16552
\(696\) −22.2824 −0.844614
\(697\) −6.79258 −0.257287
\(698\) 21.0068 0.795120
\(699\) −23.2462 −0.879251
\(700\) 0 0
\(701\) −0.456514 −0.0172423 −0.00862116 0.999963i \(-0.502744\pi\)
−0.00862116 + 0.999963i \(0.502744\pi\)
\(702\) −3.48400 −0.131495
\(703\) −3.15876 −0.119135
\(704\) −20.1560 −0.759657
\(705\) 8.29958 0.312580
\(706\) 23.2673 0.875676
\(707\) 0 0
\(708\) 10.8263 0.406878
\(709\) −9.90247 −0.371895 −0.185948 0.982560i \(-0.559535\pi\)
−0.185948 + 0.982560i \(0.559535\pi\)
\(710\) −31.7847 −1.19286
\(711\) −8.60406 −0.322678
\(712\) −1.01627 −0.0380863
\(713\) 5.60209 0.209800
\(714\) 0 0
\(715\) 18.8723 0.705785
\(716\) 4.72431 0.176556
\(717\) −6.43482 −0.240313
\(718\) 12.6832 0.473332
\(719\) −27.2086 −1.01471 −0.507355 0.861737i \(-0.669377\pi\)
−0.507355 + 0.861737i \(0.669377\pi\)
\(720\) −3.10632 −0.115766
\(721\) 0 0
\(722\) 19.5038 0.725855
\(723\) −23.5177 −0.874633
\(724\) −1.71837 −0.0638626
\(725\) −1.93580 −0.0718939
\(726\) −4.34348 −0.161202
\(727\) −29.0386 −1.07698 −0.538491 0.842631i \(-0.681006\pi\)
−0.538491 + 0.842631i \(0.681006\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −1.61816 −0.0598907
\(731\) 23.1451 0.856052
\(732\) 1.32146 0.0488425
\(733\) −20.7986 −0.768215 −0.384108 0.923288i \(-0.625491\pi\)
−0.384108 + 0.923288i \(0.625491\pi\)
\(734\) −22.8060 −0.841786
\(735\) 0 0
\(736\) −4.58569 −0.169031
\(737\) −29.5693 −1.08920
\(738\) −2.41016 −0.0887191
\(739\) 16.8470 0.619727 0.309864 0.950781i \(-0.399717\pi\)
0.309864 + 0.950781i \(0.399717\pi\)
\(740\) 8.76869 0.322343
\(741\) 2.30755 0.0847699
\(742\) 0 0
\(743\) −38.5982 −1.41603 −0.708016 0.706197i \(-0.750409\pi\)
−0.708016 + 0.706197i \(0.750409\pi\)
\(744\) −17.0571 −0.625342
\(745\) 2.21073 0.0809947
\(746\) −23.1168 −0.846367
\(747\) −11.5918 −0.424123
\(748\) −6.93072 −0.253412
\(749\) 0 0
\(750\) −12.0692 −0.440706
\(751\) 24.0032 0.875889 0.437945 0.899002i \(-0.355707\pi\)
0.437945 + 0.899002i \(0.355707\pi\)
\(752\) 5.44426 0.198532
\(753\) 3.12594 0.113915
\(754\) 25.4969 0.928541
\(755\) 49.0465 1.78498
\(756\) 0 0
\(757\) −6.54398 −0.237845 −0.118923 0.992904i \(-0.537944\pi\)
−0.118923 + 0.992904i \(0.537944\pi\)
\(758\) −12.5114 −0.454436
\(759\) 2.62243 0.0951882
\(760\) 4.62325 0.167703
\(761\) 16.0091 0.580328 0.290164 0.956977i \(-0.406290\pi\)
0.290164 + 0.956977i \(0.406290\pi\)
\(762\) 16.5587 0.599860
\(763\) 0 0
\(764\) −9.29439 −0.336259
\(765\) −6.46115 −0.233604
\(766\) 4.70593 0.170032
\(767\) −40.2229 −1.45237
\(768\) 16.5036 0.595521
\(769\) 35.3103 1.27332 0.636661 0.771144i \(-0.280315\pi\)
0.636661 + 0.771144i \(0.280315\pi\)
\(770\) 0 0
\(771\) 20.5336 0.739500
\(772\) −21.9113 −0.788607
\(773\) −54.2901 −1.95268 −0.976339 0.216246i \(-0.930619\pi\)
−0.976339 + 0.216246i \(0.930619\pi\)
\(774\) 8.21240 0.295189
\(775\) −1.48184 −0.0532294
\(776\) 0.380219 0.0136491
\(777\) 0 0
\(778\) 2.85949 0.102518
\(779\) 1.59631 0.0571939
\(780\) −6.40572 −0.229362
\(781\) −36.3581 −1.30099
\(782\) 3.12800 0.111857
\(783\) −7.31827 −0.261534
\(784\) 0 0
\(785\) −24.7340 −0.882793
\(786\) −10.5382 −0.375885
\(787\) 51.9594 1.85215 0.926077 0.377335i \(-0.123159\pi\)
0.926077 + 0.377335i \(0.123159\pi\)
\(788\) −4.60900 −0.164189
\(789\) −19.4476 −0.692352
\(790\) 19.7253 0.701796
\(791\) 0 0
\(792\) −7.98469 −0.283723
\(793\) −4.90960 −0.174345
\(794\) 7.77677 0.275987
\(795\) −6.81254 −0.241616
\(796\) 14.8090 0.524889
\(797\) 24.3737 0.863361 0.431680 0.902027i \(-0.357921\pi\)
0.431680 + 0.902027i \(0.357921\pi\)
\(798\) 0 0
\(799\) 11.3241 0.400616
\(800\) 1.21299 0.0428857
\(801\) −0.333776 −0.0117934
\(802\) −2.84649 −0.100513
\(803\) −1.85099 −0.0653199
\(804\) 10.0365 0.353961
\(805\) 0 0
\(806\) 19.5177 0.687481
\(807\) −7.71940 −0.271736
\(808\) −26.7322 −0.940435
\(809\) 9.43223 0.331619 0.165810 0.986158i \(-0.446976\pi\)
0.165810 + 0.986158i \(0.446976\pi\)
\(810\) −2.29256 −0.0805524
\(811\) 26.7248 0.938436 0.469218 0.883082i \(-0.344536\pi\)
0.469218 + 0.883082i \(0.344536\pi\)
\(812\) 0 0
\(813\) −30.8637 −1.08244
\(814\) −12.5069 −0.438365
\(815\) 32.1348 1.12563
\(816\) −4.23831 −0.148370
\(817\) −5.43929 −0.190297
\(818\) −26.1371 −0.913861
\(819\) 0 0
\(820\) −4.43134 −0.154749
\(821\) −44.1164 −1.53967 −0.769837 0.638240i \(-0.779663\pi\)
−0.769837 + 0.638240i \(0.779663\pi\)
\(822\) 6.39242 0.222961
\(823\) 1.47720 0.0514921 0.0257461 0.999669i \(-0.491804\pi\)
0.0257461 + 0.999669i \(0.491804\pi\)
\(824\) −16.6391 −0.579650
\(825\) −0.693675 −0.0241507
\(826\) 0 0
\(827\) −39.6798 −1.37980 −0.689900 0.723904i \(-0.742346\pi\)
−0.689900 + 0.723904i \(0.742346\pi\)
\(828\) −0.890116 −0.0309337
\(829\) −3.10798 −0.107945 −0.0539724 0.998542i \(-0.517188\pi\)
−0.0539724 + 0.998542i \(0.517188\pi\)
\(830\) 26.5750 0.922432
\(831\) −6.40144 −0.222063
\(832\) −25.4179 −0.881207
\(833\) 0 0
\(834\) 14.8754 0.515093
\(835\) −19.2659 −0.666725
\(836\) 1.62878 0.0563325
\(837\) −5.60209 −0.193637
\(838\) 21.1529 0.730715
\(839\) 52.8937 1.82609 0.913046 0.407856i \(-0.133724\pi\)
0.913046 + 0.407856i \(0.133724\pi\)
\(840\) 0 0
\(841\) 24.5571 0.846798
\(842\) 28.0212 0.965673
\(843\) 4.79227 0.165055
\(844\) −25.4569 −0.876261
\(845\) −4.49039 −0.154474
\(846\) 4.01803 0.138143
\(847\) 0 0
\(848\) −4.46880 −0.153459
\(849\) 11.7015 0.401593
\(850\) −0.827407 −0.0283798
\(851\) −4.52695 −0.155182
\(852\) 12.3408 0.422789
\(853\) 14.0250 0.480205 0.240103 0.970748i \(-0.422819\pi\)
0.240103 + 0.970748i \(0.422819\pi\)
\(854\) 0 0
\(855\) 1.51843 0.0519291
\(856\) 33.0134 1.12837
\(857\) 22.1856 0.757845 0.378923 0.925428i \(-0.376295\pi\)
0.378923 + 0.925428i \(0.376295\pi\)
\(858\) 9.13655 0.311917
\(859\) 6.51983 0.222454 0.111227 0.993795i \(-0.464522\pi\)
0.111227 + 0.993795i \(0.464522\pi\)
\(860\) 15.0994 0.514886
\(861\) 0 0
\(862\) 25.3073 0.861972
\(863\) −15.8555 −0.539726 −0.269863 0.962899i \(-0.586978\pi\)
−0.269863 + 0.962899i \(0.586978\pi\)
\(864\) 4.58569 0.156008
\(865\) 34.1738 1.16194
\(866\) 14.8449 0.504450
\(867\) 8.18432 0.277954
\(868\) 0 0
\(869\) 22.5636 0.765416
\(870\) 16.7776 0.568814
\(871\) −37.2887 −1.26348
\(872\) −3.21942 −0.109023
\(873\) 0.124876 0.00422642
\(874\) −0.735106 −0.0248653
\(875\) 0 0
\(876\) 0.628270 0.0212273
\(877\) −34.2400 −1.15620 −0.578102 0.815965i \(-0.696206\pi\)
−0.578102 + 0.815965i \(0.696206\pi\)
\(878\) −3.05467 −0.103090
\(879\) −19.9031 −0.671315
\(880\) 8.14611 0.274605
\(881\) −41.1806 −1.38741 −0.693704 0.720260i \(-0.744023\pi\)
−0.693704 + 0.720260i \(0.744023\pi\)
\(882\) 0 0
\(883\) 43.6733 1.46972 0.734862 0.678216i \(-0.237247\pi\)
0.734862 + 0.678216i \(0.237247\pi\)
\(884\) −8.74005 −0.293960
\(885\) −26.4677 −0.889703
\(886\) −14.7448 −0.495362
\(887\) −12.3397 −0.414327 −0.207164 0.978306i \(-0.566423\pi\)
−0.207164 + 0.978306i \(0.566423\pi\)
\(888\) 13.7835 0.462544
\(889\) 0 0
\(890\) 0.765201 0.0256496
\(891\) −2.62243 −0.0878547
\(892\) −14.7283 −0.493139
\(893\) −2.66125 −0.0890553
\(894\) 1.07027 0.0357950
\(895\) −11.5498 −0.386067
\(896\) 0 0
\(897\) 3.30704 0.110419
\(898\) −13.3425 −0.445246
\(899\) 40.9976 1.36735
\(900\) 0.235450 0.00784834
\(901\) −9.29511 −0.309665
\(902\) 6.32047 0.210449
\(903\) 0 0
\(904\) −42.9124 −1.42725
\(905\) 4.20099 0.139645
\(906\) 23.7446 0.788861
\(907\) −23.5472 −0.781873 −0.390936 0.920418i \(-0.627849\pi\)
−0.390936 + 0.920418i \(0.627849\pi\)
\(908\) 15.9909 0.530678
\(909\) −8.77972 −0.291205
\(910\) 0 0
\(911\) 19.5967 0.649268 0.324634 0.945840i \(-0.394759\pi\)
0.324634 + 0.945840i \(0.394759\pi\)
\(912\) 0.996038 0.0329821
\(913\) 30.3988 1.00605
\(914\) 3.74704 0.123941
\(915\) −3.23064 −0.106802
\(916\) −24.9287 −0.823667
\(917\) 0 0
\(918\) −3.12800 −0.103239
\(919\) −38.2527 −1.26184 −0.630920 0.775848i \(-0.717323\pi\)
−0.630920 + 0.775848i \(0.717323\pi\)
\(920\) 6.62577 0.218445
\(921\) −21.7588 −0.716976
\(922\) −17.0079 −0.560127
\(923\) −45.8497 −1.50916
\(924\) 0 0
\(925\) 1.19745 0.0393720
\(926\) −10.3106 −0.338826
\(927\) −5.46482 −0.179488
\(928\) −33.5593 −1.10164
\(929\) −32.7683 −1.07509 −0.537546 0.843235i \(-0.680648\pi\)
−0.537546 + 0.843235i \(0.680648\pi\)
\(930\) 12.8431 0.421143
\(931\) 0 0
\(932\) −20.6918 −0.677782
\(933\) 8.14345 0.266605
\(934\) 31.7713 1.03959
\(935\) 16.9439 0.554125
\(936\) −10.0692 −0.329121
\(937\) −49.1394 −1.60531 −0.802657 0.596441i \(-0.796581\pi\)
−0.802657 + 0.596441i \(0.796581\pi\)
\(938\) 0 0
\(939\) −1.05044 −0.0342797
\(940\) 7.38759 0.240957
\(941\) 59.6840 1.94564 0.972821 0.231557i \(-0.0743820\pi\)
0.972821 + 0.231557i \(0.0743820\pi\)
\(942\) −11.9743 −0.390144
\(943\) 2.28774 0.0744991
\(944\) −17.3620 −0.565084
\(945\) 0 0
\(946\) −21.5364 −0.700210
\(947\) −2.66069 −0.0864608 −0.0432304 0.999065i \(-0.513765\pi\)
−0.0432304 + 0.999065i \(0.513765\pi\)
\(948\) −7.65861 −0.248740
\(949\) −2.33421 −0.0757715
\(950\) 0.194448 0.00630872
\(951\) −16.3680 −0.530768
\(952\) 0 0
\(953\) −23.2051 −0.751686 −0.375843 0.926683i \(-0.622647\pi\)
−0.375843 + 0.926683i \(0.622647\pi\)
\(954\) −3.29811 −0.106780
\(955\) 22.7225 0.735284
\(956\) −5.72773 −0.185248
\(957\) 19.1917 0.620378
\(958\) −3.07800 −0.0994458
\(959\) 0 0
\(960\) −16.7256 −0.539817
\(961\) 0.383420 0.0123684
\(962\) −15.7719 −0.508506
\(963\) 10.8427 0.349400
\(964\) −20.9335 −0.674223
\(965\) 53.5679 1.72441
\(966\) 0 0
\(967\) −6.58436 −0.211739 −0.105869 0.994380i \(-0.533763\pi\)
−0.105869 + 0.994380i \(0.533763\pi\)
\(968\) −12.5532 −0.403474
\(969\) 2.07176 0.0665545
\(970\) −0.286286 −0.00919210
\(971\) 41.8221 1.34213 0.671067 0.741397i \(-0.265836\pi\)
0.671067 + 0.741397i \(0.265836\pi\)
\(972\) 0.890116 0.0285505
\(973\) 0 0
\(974\) −33.9633 −1.08825
\(975\) −0.874766 −0.0280149
\(976\) −2.11920 −0.0678338
\(977\) −17.2560 −0.552067 −0.276033 0.961148i \(-0.589020\pi\)
−0.276033 + 0.961148i \(0.589020\pi\)
\(978\) 15.5572 0.497465
\(979\) 0.875303 0.0279748
\(980\) 0 0
\(981\) −1.05736 −0.0337589
\(982\) 12.8181 0.409042
\(983\) 61.1190 1.94939 0.974697 0.223531i \(-0.0717586\pi\)
0.974697 + 0.223531i \(0.0717586\pi\)
\(984\) −6.96564 −0.222056
\(985\) 11.2679 0.359024
\(986\) 22.8916 0.729016
\(987\) 0 0
\(988\) 2.05399 0.0653460
\(989\) −7.79527 −0.247875
\(990\) 6.01208 0.191076
\(991\) 27.6963 0.879802 0.439901 0.898046i \(-0.355013\pi\)
0.439901 + 0.898046i \(0.355013\pi\)
\(992\) −25.6894 −0.815641
\(993\) −29.2929 −0.929582
\(994\) 0 0
\(995\) −36.2043 −1.14775
\(996\) −10.3181 −0.326941
\(997\) 15.6408 0.495350 0.247675 0.968843i \(-0.420334\pi\)
0.247675 + 0.968843i \(0.420334\pi\)
\(998\) −33.5328 −1.06146
\(999\) 4.52695 0.143226
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 3381.2.a.bi.1.3 10
7.2 even 3 483.2.i.h.277.8 20
7.4 even 3 483.2.i.h.415.8 yes 20
7.6 odd 2 3381.2.a.bj.1.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.h.277.8 20 7.2 even 3
483.2.i.h.415.8 yes 20 7.4 even 3
3381.2.a.bi.1.3 10 1.1 even 1 trivial
3381.2.a.bj.1.3 10 7.6 odd 2