Properties

Label 418.2.a.g.1.3
Level $418$
Weight $2$
Character 418.1
Self dual yes
Analytic conductor $3.338$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.33774680449\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.66908\) of defining polynomial
Character \(\chi\) \(=\) 418.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.66908 q^{3} +1.00000 q^{4} -4.12398 q^{5} -2.66908 q^{6} -4.21417 q^{7} -1.00000 q^{8} +4.12398 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.66908 q^{3} +1.00000 q^{4} -4.12398 q^{5} -2.66908 q^{6} -4.21417 q^{7} -1.00000 q^{8} +4.12398 q^{9} +4.12398 q^{10} -1.00000 q^{11} +2.66908 q^{12} -2.21417 q^{13} +4.21417 q^{14} -11.0072 q^{15} +1.00000 q^{16} -3.45490 q^{17} -4.12398 q^{18} -1.00000 q^{19} -4.12398 q^{20} -11.2480 q^{21} +1.00000 q^{22} -5.45490 q^{23} -2.66908 q^{24} +12.0072 q^{25} +2.21417 q^{26} +3.00000 q^{27} -4.21417 q^{28} +5.57889 q^{29} +11.0072 q^{30} +7.00724 q^{31} -1.00000 q^{32} -2.66908 q^{33} +3.45490 q^{34} +17.3792 q^{35} +4.12398 q^{36} -2.90981 q^{37} +1.00000 q^{38} -5.90981 q^{39} +4.12398 q^{40} -11.9170 q^{41} +11.2480 q^{42} +1.46214 q^{43} -1.00000 q^{44} -17.0072 q^{45} +5.45490 q^{46} +7.58612 q^{47} +2.66908 q^{48} +10.7593 q^{49} -12.0072 q^{50} -9.22141 q^{51} -2.21417 q^{52} -13.2214 q^{53} -3.00000 q^{54} +4.12398 q^{55} +4.21417 q^{56} -2.66908 q^{57} -5.57889 q^{58} +4.79306 q^{59} -11.0072 q^{60} +8.90981 q^{61} -7.00724 q^{62} -17.3792 q^{63} +1.00000 q^{64} +9.13122 q^{65} +2.66908 q^{66} +1.30437 q^{67} -3.45490 q^{68} -14.5596 q^{69} -17.3792 q^{70} -6.80030 q^{71} -4.12398 q^{72} -1.45490 q^{73} +2.90981 q^{74} +32.0483 q^{75} -1.00000 q^{76} +4.21417 q^{77} +5.90981 q^{78} -9.15777 q^{79} -4.12398 q^{80} -4.36471 q^{81} +11.9170 q^{82} -13.2142 q^{83} -11.2480 q^{84} +14.2480 q^{85} -1.46214 q^{86} +14.8905 q^{87} +1.00000 q^{88} +8.24797 q^{89} +17.0072 q^{90} +9.33092 q^{91} -5.45490 q^{92} +18.7029 q^{93} -7.58612 q^{94} +4.12398 q^{95} -2.66908 q^{96} +2.18038 q^{97} -10.7593 q^{98} -4.12398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{11} + 6 q^{14} - 9 q^{15} + 3 q^{16} - 9 q^{17} - 3 q^{18} - 3 q^{19} - 3 q^{20} - 15 q^{21} + 3 q^{22} - 15 q^{23} + 12 q^{25} + 9 q^{27} - 6 q^{28} + 6 q^{29} + 9 q^{30} - 3 q^{31} - 3 q^{32} + 9 q^{34} + 3 q^{36} - 6 q^{37} + 3 q^{38} - 15 q^{39} + 3 q^{40} - 9 q^{41} + 15 q^{42} - 21 q^{43} - 3 q^{44} - 27 q^{45} + 15 q^{46} - 12 q^{47} + 27 q^{49} - 12 q^{50} + 3 q^{51} - 9 q^{53} - 9 q^{54} + 3 q^{55} + 6 q^{56} - 6 q^{58} - 3 q^{59} - 9 q^{60} + 24 q^{61} + 3 q^{62} + 3 q^{64} - 6 q^{65} - 9 q^{68} + 3 q^{69} + 21 q^{71} - 3 q^{72} - 3 q^{73} + 6 q^{74} + 36 q^{75} - 3 q^{76} + 6 q^{77} + 15 q^{78} - 6 q^{79} - 3 q^{80} - 9 q^{81} + 9 q^{82} - 33 q^{83} - 15 q^{84} + 24 q^{85} + 21 q^{86} + 6 q^{87} + 3 q^{88} + 6 q^{89} + 27 q^{90} + 36 q^{91} - 15 q^{92} + 36 q^{93} + 12 q^{94} + 3 q^{95} + 12 q^{97} - 27 q^{98} - 3 q^{99}+O(q^{100}) \) 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
\(3\) 2.66908 1.54099 0.770497 0.637444i \(-0.220008\pi\)
0.770497 + 0.637444i \(0.220008\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.12398 −1.84430 −0.922151 0.386831i \(-0.873570\pi\)
−0.922151 + 0.386831i \(0.873570\pi\)
\(6\) −2.66908 −1.08965
\(7\) −4.21417 −1.59281 −0.796404 0.604765i \(-0.793267\pi\)
−0.796404 + 0.604765i \(0.793267\pi\)
\(8\) −1.00000 −0.353553
\(9\) 4.12398 1.37466
\(10\) 4.12398 1.30412
\(11\) −1.00000 −0.301511
\(12\) 2.66908 0.770497
\(13\) −2.21417 −0.614102 −0.307051 0.951693i \(-0.599342\pi\)
−0.307051 + 0.951693i \(0.599342\pi\)
\(14\) 4.21417 1.12629
\(15\) −11.0072 −2.84206
\(16\) 1.00000 0.250000
\(17\) −3.45490 −0.837937 −0.418969 0.908001i \(-0.637608\pi\)
−0.418969 + 0.908001i \(0.637608\pi\)
\(18\) −4.12398 −0.972032
\(19\) −1.00000 −0.229416
\(20\) −4.12398 −0.922151
\(21\) −11.2480 −2.45451
\(22\) 1.00000 0.213201
\(23\) −5.45490 −1.13743 −0.568713 0.822536i \(-0.692559\pi\)
−0.568713 + 0.822536i \(0.692559\pi\)
\(24\) −2.66908 −0.544823
\(25\) 12.0072 2.40145
\(26\) 2.21417 0.434235
\(27\) 3.00000 0.577350
\(28\) −4.21417 −0.796404
\(29\) 5.57889 1.03597 0.517987 0.855389i \(-0.326682\pi\)
0.517987 + 0.855389i \(0.326682\pi\)
\(30\) 11.0072 2.00964
\(31\) 7.00724 1.25854 0.629268 0.777188i \(-0.283355\pi\)
0.629268 + 0.777188i \(0.283355\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.66908 −0.464627
\(34\) 3.45490 0.592511
\(35\) 17.3792 2.93762
\(36\) 4.12398 0.687331
\(37\) −2.90981 −0.478370 −0.239185 0.970974i \(-0.576880\pi\)
−0.239185 + 0.970974i \(0.576880\pi\)
\(38\) 1.00000 0.162221
\(39\) −5.90981 −0.946327
\(40\) 4.12398 0.652059
\(41\) −11.9170 −1.86113 −0.930565 0.366127i \(-0.880684\pi\)
−0.930565 + 0.366127i \(0.880684\pi\)
\(42\) 11.2480 1.73560
\(43\) 1.46214 0.222974 0.111487 0.993766i \(-0.464439\pi\)
0.111487 + 0.993766i \(0.464439\pi\)
\(44\) −1.00000 −0.150756
\(45\) −17.0072 −2.53529
\(46\) 5.45490 0.804282
\(47\) 7.58612 1.10655 0.553275 0.832999i \(-0.313378\pi\)
0.553275 + 0.832999i \(0.313378\pi\)
\(48\) 2.66908 0.385248
\(49\) 10.7593 1.53704
\(50\) −12.0072 −1.69808
\(51\) −9.22141 −1.29126
\(52\) −2.21417 −0.307051
\(53\) −13.2214 −1.81610 −0.908050 0.418861i \(-0.862429\pi\)
−0.908050 + 0.418861i \(0.862429\pi\)
\(54\) −3.00000 −0.408248
\(55\) 4.12398 0.556078
\(56\) 4.21417 0.563143
\(57\) −2.66908 −0.353528
\(58\) −5.57889 −0.732544
\(59\) 4.79306 0.624004 0.312002 0.950082i \(-0.399001\pi\)
0.312002 + 0.950082i \(0.399001\pi\)
\(60\) −11.0072 −1.42103
\(61\) 8.90981 1.14078 0.570392 0.821373i \(-0.306791\pi\)
0.570392 + 0.821373i \(0.306791\pi\)
\(62\) −7.00724 −0.889920
\(63\) −17.3792 −2.18957
\(64\) 1.00000 0.125000
\(65\) 9.13122 1.13259
\(66\) 2.66908 0.328541
\(67\) 1.30437 0.159354 0.0796769 0.996821i \(-0.474611\pi\)
0.0796769 + 0.996821i \(0.474611\pi\)
\(68\) −3.45490 −0.418969
\(69\) −14.5596 −1.75277
\(70\) −17.3792 −2.07721
\(71\) −6.80030 −0.807047 −0.403524 0.914969i \(-0.632215\pi\)
−0.403524 + 0.914969i \(0.632215\pi\)
\(72\) −4.12398 −0.486016
\(73\) −1.45490 −0.170284 −0.0851418 0.996369i \(-0.527134\pi\)
−0.0851418 + 0.996369i \(0.527134\pi\)
\(74\) 2.90981 0.338258
\(75\) 32.0483 3.70061
\(76\) −1.00000 −0.114708
\(77\) 4.21417 0.480250
\(78\) 5.90981 0.669154
\(79\) −9.15777 −1.03033 −0.515165 0.857091i \(-0.672269\pi\)
−0.515165 + 0.857091i \(0.672269\pi\)
\(80\) −4.12398 −0.461075
\(81\) −4.36471 −0.484968
\(82\) 11.9170 1.31602
\(83\) −13.2142 −1.45044 −0.725222 0.688515i \(-0.758263\pi\)
−0.725222 + 0.688515i \(0.758263\pi\)
\(84\) −11.2480 −1.22725
\(85\) 14.2480 1.54541
\(86\) −1.46214 −0.157667
\(87\) 14.8905 1.59643
\(88\) 1.00000 0.106600
\(89\) 8.24797 0.874283 0.437141 0.899393i \(-0.355991\pi\)
0.437141 + 0.899393i \(0.355991\pi\)
\(90\) 17.0072 1.79272
\(91\) 9.33092 0.978146
\(92\) −5.45490 −0.568713
\(93\) 18.7029 1.93940
\(94\) −7.58612 −0.782449
\(95\) 4.12398 0.423112
\(96\) −2.66908 −0.272412
\(97\) 2.18038 0.221384 0.110692 0.993855i \(-0.464693\pi\)
0.110692 + 0.993855i \(0.464693\pi\)
\(98\) −10.7593 −1.08685
\(99\) −4.12398 −0.414476
\(100\) 12.0072 1.20072
\(101\) −6.24797 −0.621696 −0.310848 0.950460i \(-0.600613\pi\)
−0.310848 + 0.950460i \(0.600613\pi\)
\(102\) 9.22141 0.913056
\(103\) 0.605441 0.0596559 0.0298280 0.999555i \(-0.490504\pi\)
0.0298280 + 0.999555i \(0.490504\pi\)
\(104\) 2.21417 0.217118
\(105\) 46.3864 4.52685
\(106\) 13.2214 1.28418
\(107\) 3.88325 0.375408 0.187704 0.982226i \(-0.439895\pi\)
0.187704 + 0.982226i \(0.439895\pi\)
\(108\) 3.00000 0.288675
\(109\) −4.97345 −0.476370 −0.238185 0.971220i \(-0.576552\pi\)
−0.238185 + 0.971220i \(0.576552\pi\)
\(110\) −4.12398 −0.393206
\(111\) −7.76651 −0.737164
\(112\) −4.21417 −0.398202
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 2.66908 0.249982
\(115\) 22.4959 2.09776
\(116\) 5.57889 0.517987
\(117\) −9.13122 −0.844182
\(118\) −4.79306 −0.441237
\(119\) 14.5596 1.33467
\(120\) 11.0072 1.00482
\(121\) 1.00000 0.0909091
\(122\) −8.90981 −0.806656
\(123\) −31.8075 −2.86799
\(124\) 7.00724 0.629268
\(125\) −28.8977 −2.58469
\(126\) 17.3792 1.54826
\(127\) −2.48146 −0.220194 −0.110097 0.993921i \(-0.535116\pi\)
−0.110097 + 0.993921i \(0.535116\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 3.90257 0.343602
\(130\) −9.13122 −0.800861
\(131\) 11.0072 0.961707 0.480853 0.876801i \(-0.340327\pi\)
0.480853 + 0.876801i \(0.340327\pi\)
\(132\) −2.66908 −0.232314
\(133\) 4.21417 0.365415
\(134\) −1.30437 −0.112680
\(135\) −12.3719 −1.06481
\(136\) 3.45490 0.296256
\(137\) 6.94360 0.593232 0.296616 0.954997i \(-0.404142\pi\)
0.296616 + 0.954997i \(0.404142\pi\)
\(138\) 14.5596 1.23939
\(139\) −4.37195 −0.370824 −0.185412 0.982661i \(-0.559362\pi\)
−0.185412 + 0.982661i \(0.559362\pi\)
\(140\) 17.3792 1.46881
\(141\) 20.2480 1.70519
\(142\) 6.80030 0.570668
\(143\) 2.21417 0.185159
\(144\) 4.12398 0.343665
\(145\) −23.0072 −1.91065
\(146\) 1.45490 0.120409
\(147\) 28.7173 2.36857
\(148\) −2.90981 −0.239185
\(149\) 11.7665 0.963950 0.481975 0.876185i \(-0.339920\pi\)
0.481975 + 0.876185i \(0.339920\pi\)
\(150\) −32.0483 −2.61673
\(151\) −19.5861 −1.59390 −0.796948 0.604048i \(-0.793554\pi\)
−0.796948 + 0.604048i \(0.793554\pi\)
\(152\) 1.00000 0.0811107
\(153\) −14.2480 −1.15188
\(154\) −4.21417 −0.339588
\(155\) −28.8977 −2.32112
\(156\) −5.90981 −0.473163
\(157\) 20.6199 1.64565 0.822824 0.568296i \(-0.192397\pi\)
0.822824 + 0.568296i \(0.192397\pi\)
\(158\) 9.15777 0.728553
\(159\) −35.2890 −2.79860
\(160\) 4.12398 0.326029
\(161\) 22.9879 1.81170
\(162\) 4.36471 0.342924
\(163\) −17.8341 −1.39687 −0.698437 0.715672i \(-0.746121\pi\)
−0.698437 + 0.715672i \(0.746121\pi\)
\(164\) −11.9170 −0.930565
\(165\) 11.0072 0.856912
\(166\) 13.2142 1.02562
\(167\) 4.18038 0.323488 0.161744 0.986833i \(-0.448288\pi\)
0.161744 + 0.986833i \(0.448288\pi\)
\(168\) 11.2480 0.867799
\(169\) −8.09743 −0.622879
\(170\) −14.2480 −1.09277
\(171\) −4.12398 −0.315369
\(172\) 1.46214 0.111487
\(173\) 23.7101 1.80265 0.901323 0.433147i \(-0.142597\pi\)
0.901323 + 0.433147i \(0.142597\pi\)
\(174\) −14.8905 −1.12885
\(175\) −50.6006 −3.82505
\(176\) −1.00000 −0.0753778
\(177\) 12.7931 0.961585
\(178\) −8.24797 −0.618211
\(179\) −10.8269 −0.809237 −0.404619 0.914486i \(-0.632596\pi\)
−0.404619 + 0.914486i \(0.632596\pi\)
\(180\) −17.0072 −1.26764
\(181\) −17.8341 −1.32560 −0.662799 0.748798i \(-0.730632\pi\)
−0.662799 + 0.748798i \(0.730632\pi\)
\(182\) −9.33092 −0.691654
\(183\) 23.7810 1.75794
\(184\) 5.45490 0.402141
\(185\) 12.0000 0.882258
\(186\) −18.7029 −1.37136
\(187\) 3.45490 0.252648
\(188\) 7.58612 0.553275
\(189\) −12.6425 −0.919608
\(190\) −4.12398 −0.299185
\(191\) 0.612679 0.0443319 0.0221659 0.999754i \(-0.492944\pi\)
0.0221659 + 0.999754i \(0.492944\pi\)
\(192\) 2.66908 0.192624
\(193\) 11.6425 0.838047 0.419024 0.907975i \(-0.362372\pi\)
0.419024 + 0.907975i \(0.362372\pi\)
\(194\) −2.18038 −0.156542
\(195\) 24.3719 1.74531
\(196\) 10.7593 0.768519
\(197\) −12.9098 −0.919786 −0.459893 0.887974i \(-0.652112\pi\)
−0.459893 + 0.887974i \(0.652112\pi\)
\(198\) 4.12398 0.293079
\(199\) −18.3647 −1.30184 −0.650920 0.759146i \(-0.725617\pi\)
−0.650920 + 0.759146i \(0.725617\pi\)
\(200\) −12.0072 −0.849040
\(201\) 3.48146 0.245563
\(202\) 6.24797 0.439605
\(203\) −23.5104 −1.65011
\(204\) −9.22141 −0.645628
\(205\) 49.1457 3.43248
\(206\) −0.605441 −0.0421831
\(207\) −22.4959 −1.56358
\(208\) −2.21417 −0.153525
\(209\) 1.00000 0.0691714
\(210\) −46.3864 −3.20097
\(211\) 2.13122 0.146719 0.0733596 0.997306i \(-0.476628\pi\)
0.0733596 + 0.997306i \(0.476628\pi\)
\(212\) −13.2214 −0.908050
\(213\) −18.1505 −1.24365
\(214\) −3.88325 −0.265454
\(215\) −6.02985 −0.411232
\(216\) −3.00000 −0.204124
\(217\) −29.5297 −2.00461
\(218\) 4.97345 0.336844
\(219\) −3.88325 −0.262406
\(220\) 4.12398 0.278039
\(221\) 7.64976 0.514579
\(222\) 7.76651 0.521254
\(223\) 6.90981 0.462715 0.231357 0.972869i \(-0.425683\pi\)
0.231357 + 0.972869i \(0.425683\pi\)
\(224\) 4.21417 0.281571
\(225\) 49.5176 3.30118
\(226\) 8.00000 0.532152
\(227\) −2.72548 −0.180896 −0.0904482 0.995901i \(-0.528830\pi\)
−0.0904482 + 0.995901i \(0.528830\pi\)
\(228\) −2.66908 −0.176764
\(229\) 3.48870 0.230539 0.115270 0.993334i \(-0.463227\pi\)
0.115270 + 0.993334i \(0.463227\pi\)
\(230\) −22.4959 −1.48334
\(231\) 11.2480 0.740062
\(232\) −5.57889 −0.366272
\(233\) −17.0902 −1.11962 −0.559808 0.828622i \(-0.689125\pi\)
−0.559808 + 0.828622i \(0.689125\pi\)
\(234\) 9.13122 0.596927
\(235\) −31.2850 −2.04081
\(236\) 4.79306 0.312002
\(237\) −24.4428 −1.58773
\(238\) −14.5596 −0.943757
\(239\) 6.06035 0.392011 0.196006 0.980603i \(-0.437203\pi\)
0.196006 + 0.980603i \(0.437203\pi\)
\(240\) −11.0072 −0.710514
\(241\) 4.85341 0.312635 0.156318 0.987707i \(-0.450038\pi\)
0.156318 + 0.987707i \(0.450038\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −20.6498 −1.32468
\(244\) 8.90981 0.570392
\(245\) −44.3711 −2.83476
\(246\) 31.8075 2.02797
\(247\) 2.21417 0.140885
\(248\) −7.00724 −0.444960
\(249\) −35.2697 −2.23513
\(250\) 28.8977 1.82765
\(251\) −1.58612 −0.100115 −0.0500576 0.998746i \(-0.515941\pi\)
−0.0500576 + 0.998746i \(0.515941\pi\)
\(252\) −17.3792 −1.09479
\(253\) 5.45490 0.342947
\(254\) 2.48146 0.155701
\(255\) 38.0289 2.38147
\(256\) 1.00000 0.0625000
\(257\) 19.5185 1.21753 0.608767 0.793349i \(-0.291665\pi\)
0.608767 + 0.793349i \(0.291665\pi\)
\(258\) −3.90257 −0.242963
\(259\) 12.2624 0.761951
\(260\) 9.13122 0.566294
\(261\) 23.0072 1.42411
\(262\) −11.0072 −0.680029
\(263\) −13.0483 −0.804591 −0.402295 0.915510i \(-0.631787\pi\)
−0.402295 + 0.915510i \(0.631787\pi\)
\(264\) 2.66908 0.164270
\(265\) 54.5249 3.34944
\(266\) −4.21417 −0.258388
\(267\) 22.0145 1.34726
\(268\) 1.30437 0.0796769
\(269\) 23.6006 1.43895 0.719477 0.694516i \(-0.244382\pi\)
0.719477 + 0.694516i \(0.244382\pi\)
\(270\) 12.3719 0.752933
\(271\) −16.4090 −0.996778 −0.498389 0.866954i \(-0.666075\pi\)
−0.498389 + 0.866954i \(0.666075\pi\)
\(272\) −3.45490 −0.209484
\(273\) 24.9050 1.50732
\(274\) −6.94360 −0.419478
\(275\) −12.0072 −0.724064
\(276\) −14.5596 −0.876383
\(277\) −11.5861 −0.696143 −0.348071 0.937468i \(-0.613163\pi\)
−0.348071 + 0.937468i \(0.613163\pi\)
\(278\) 4.37195 0.262212
\(279\) 28.8977 1.73006
\(280\) −17.3792 −1.03861
\(281\) 7.03379 0.419601 0.209800 0.977744i \(-0.432719\pi\)
0.209800 + 0.977744i \(0.432719\pi\)
\(282\) −20.2480 −1.20575
\(283\) 8.33092 0.495222 0.247611 0.968860i \(-0.420355\pi\)
0.247611 + 0.968860i \(0.420355\pi\)
\(284\) −6.80030 −0.403524
\(285\) 11.0072 0.652012
\(286\) −2.21417 −0.130927
\(287\) 50.2205 2.96442
\(288\) −4.12398 −0.243008
\(289\) −5.06364 −0.297861
\(290\) 23.0072 1.35103
\(291\) 5.81962 0.341152
\(292\) −1.45490 −0.0851418
\(293\) 10.2142 0.596718 0.298359 0.954454i \(-0.403561\pi\)
0.298359 + 0.954454i \(0.403561\pi\)
\(294\) −28.7173 −1.67483
\(295\) −19.7665 −1.15085
\(296\) 2.90981 0.169129
\(297\) −3.00000 −0.174078
\(298\) −11.7665 −0.681616
\(299\) 12.0781 0.698495
\(300\) 32.0483 1.85031
\(301\) −6.16172 −0.355156
\(302\) 19.5861 1.12705
\(303\) −16.6763 −0.958029
\(304\) −1.00000 −0.0573539
\(305\) −36.7439 −2.10395
\(306\) 14.2480 0.814502
\(307\) −1.93242 −0.110289 −0.0551444 0.998478i \(-0.517562\pi\)
−0.0551444 + 0.998478i \(0.517562\pi\)
\(308\) 4.21417 0.240125
\(309\) 1.61597 0.0919294
\(310\) 28.8977 1.64128
\(311\) −27.7173 −1.57171 −0.785853 0.618413i \(-0.787776\pi\)
−0.785853 + 0.618413i \(0.787776\pi\)
\(312\) 5.90981 0.334577
\(313\) 11.5258 0.651476 0.325738 0.945460i \(-0.394387\pi\)
0.325738 + 0.945460i \(0.394387\pi\)
\(314\) −20.6199 −1.16365
\(315\) 71.6715 4.03823
\(316\) −9.15777 −0.515165
\(317\) −33.8977 −1.90389 −0.951943 0.306275i \(-0.900917\pi\)
−0.951943 + 0.306275i \(0.900917\pi\)
\(318\) 35.2890 1.97891
\(319\) −5.57889 −0.312358
\(320\) −4.12398 −0.230538
\(321\) 10.3647 0.578502
\(322\) −22.9879 −1.28107
\(323\) 3.45490 0.192236
\(324\) −4.36471 −0.242484
\(325\) −26.5861 −1.47473
\(326\) 17.8341 0.987739
\(327\) −13.2745 −0.734083
\(328\) 11.9170 0.658009
\(329\) −31.9693 −1.76252
\(330\) −11.0072 −0.605928
\(331\) −16.2697 −0.894262 −0.447131 0.894468i \(-0.647554\pi\)
−0.447131 + 0.894468i \(0.647554\pi\)
\(332\) −13.2142 −0.725222
\(333\) −12.0000 −0.657596
\(334\) −4.18038 −0.228740
\(335\) −5.37919 −0.293896
\(336\) −11.2480 −0.613627
\(337\) −6.82685 −0.371882 −0.185941 0.982561i \(-0.559533\pi\)
−0.185941 + 0.982561i \(0.559533\pi\)
\(338\) 8.09743 0.440442
\(339\) −21.3526 −1.15972
\(340\) 14.2480 0.772704
\(341\) −7.00724 −0.379463
\(342\) 4.12398 0.222999
\(343\) −15.8422 −0.855400
\(344\) −1.46214 −0.0788334
\(345\) 60.0434 3.23263
\(346\) −23.7101 −1.27466
\(347\) −5.09019 −0.273256 −0.136628 0.990622i \(-0.543626\pi\)
−0.136628 + 0.990622i \(0.543626\pi\)
\(348\) 14.8905 0.798214
\(349\) 30.9919 1.65896 0.829478 0.558539i \(-0.188638\pi\)
0.829478 + 0.558539i \(0.188638\pi\)
\(350\) 50.6006 2.70472
\(351\) −6.64252 −0.354552
\(352\) 1.00000 0.0533002
\(353\) −17.8301 −0.949003 −0.474501 0.880255i \(-0.657372\pi\)
−0.474501 + 0.880255i \(0.657372\pi\)
\(354\) −12.7931 −0.679944
\(355\) 28.0443 1.48844
\(356\) 8.24797 0.437141
\(357\) 38.8606 2.05672
\(358\) 10.8269 0.572217
\(359\) 24.4090 1.28826 0.644130 0.764916i \(-0.277220\pi\)
0.644130 + 0.764916i \(0.277220\pi\)
\(360\) 17.0072 0.896360
\(361\) 1.00000 0.0526316
\(362\) 17.8341 0.937339
\(363\) 2.66908 0.140090
\(364\) 9.33092 0.489073
\(365\) 6.00000 0.314054
\(366\) −23.7810 −1.24305
\(367\) −26.6232 −1.38972 −0.694860 0.719145i \(-0.744534\pi\)
−0.694860 + 0.719145i \(0.744534\pi\)
\(368\) −5.45490 −0.284357
\(369\) −49.1457 −2.55842
\(370\) −12.0000 −0.623850
\(371\) 55.7173 2.89270
\(372\) 18.7029 0.969699
\(373\) −12.6160 −0.653230 −0.326615 0.945157i \(-0.605908\pi\)
−0.326615 + 0.945157i \(0.605908\pi\)
\(374\) −3.45490 −0.178649
\(375\) −77.1303 −3.98299
\(376\) −7.58612 −0.391225
\(377\) −12.3526 −0.636193
\(378\) 12.6425 0.650261
\(379\) −22.8905 −1.17581 −0.587903 0.808932i \(-0.700046\pi\)
−0.587903 + 0.808932i \(0.700046\pi\)
\(380\) 4.12398 0.211556
\(381\) −6.62321 −0.339317
\(382\) −0.612679 −0.0313474
\(383\) −0.0298464 −0.00152508 −0.000762539 1.00000i \(-0.500243\pi\)
−0.000762539 1.00000i \(0.500243\pi\)
\(384\) −2.66908 −0.136206
\(385\) −17.3792 −0.885725
\(386\) −11.6425 −0.592589
\(387\) 6.02985 0.306514
\(388\) 2.18038 0.110692
\(389\) 9.22865 0.467911 0.233956 0.972247i \(-0.424833\pi\)
0.233956 + 0.972247i \(0.424833\pi\)
\(390\) −24.3719 −1.23412
\(391\) 18.8462 0.953092
\(392\) −10.7593 −0.543425
\(393\) 29.3792 1.48198
\(394\) 12.9098 0.650387
\(395\) 37.7665 1.90024
\(396\) −4.12398 −0.207238
\(397\) 17.2697 0.866740 0.433370 0.901216i \(-0.357324\pi\)
0.433370 + 0.901216i \(0.357324\pi\)
\(398\) 18.3647 0.920540
\(399\) 11.2480 0.563103
\(400\) 12.0072 0.600362
\(401\) −13.2850 −0.663424 −0.331712 0.943381i \(-0.607626\pi\)
−0.331712 + 0.943381i \(0.607626\pi\)
\(402\) −3.48146 −0.173639
\(403\) −15.5152 −0.772870
\(404\) −6.24797 −0.310848
\(405\) 18.0000 0.894427
\(406\) 23.5104 1.16680
\(407\) 2.90981 0.144234
\(408\) 9.22141 0.456528
\(409\) 1.94360 0.0961048 0.0480524 0.998845i \(-0.484699\pi\)
0.0480524 + 0.998845i \(0.484699\pi\)
\(410\) −49.1457 −2.42713
\(411\) 18.5330 0.914166
\(412\) 0.605441 0.0298280
\(413\) −20.1988 −0.993918
\(414\) 22.4959 1.10561
\(415\) 54.4950 2.67506
\(416\) 2.21417 0.108559
\(417\) −11.6691 −0.571437
\(418\) −1.00000 −0.0489116
\(419\) −2.49593 −0.121934 −0.0609671 0.998140i \(-0.519418\pi\)
−0.0609671 + 0.998140i \(0.519418\pi\)
\(420\) 46.3864 2.26343
\(421\) 15.6353 0.762017 0.381009 0.924571i \(-0.375577\pi\)
0.381009 + 0.924571i \(0.375577\pi\)
\(422\) −2.13122 −0.103746
\(423\) 31.2850 1.52113
\(424\) 13.2214 0.642089
\(425\) −41.4839 −2.01226
\(426\) 18.1505 0.879396
\(427\) −37.5475 −1.81705
\(428\) 3.88325 0.187704
\(429\) 5.90981 0.285328
\(430\) 6.02985 0.290785
\(431\) 0.262441 0.0126413 0.00632067 0.999980i \(-0.497988\pi\)
0.00632067 + 0.999980i \(0.497988\pi\)
\(432\) 3.00000 0.144338
\(433\) −32.2769 −1.55113 −0.775565 0.631268i \(-0.782535\pi\)
−0.775565 + 0.631268i \(0.782535\pi\)
\(434\) 29.5297 1.41747
\(435\) −61.4081 −2.94429
\(436\) −4.97345 −0.238185
\(437\) 5.45490 0.260943
\(438\) 3.88325 0.185549
\(439\) −17.0902 −0.815670 −0.407835 0.913056i \(-0.633716\pi\)
−0.407835 + 0.913056i \(0.633716\pi\)
\(440\) −4.12398 −0.196603
\(441\) 44.3711 2.11291
\(442\) −7.64976 −0.363862
\(443\) 16.2624 0.772652 0.386326 0.922362i \(-0.373744\pi\)
0.386326 + 0.922362i \(0.373744\pi\)
\(444\) −7.76651 −0.368582
\(445\) −34.0145 −1.61244
\(446\) −6.90981 −0.327189
\(447\) 31.4057 1.48544
\(448\) −4.21417 −0.199101
\(449\) −27.3382 −1.29017 −0.645084 0.764112i \(-0.723178\pi\)
−0.645084 + 0.764112i \(0.723178\pi\)
\(450\) −49.5176 −2.33428
\(451\) 11.9170 0.561152
\(452\) −8.00000 −0.376288
\(453\) −52.2769 −2.45618
\(454\) 2.72548 0.127913
\(455\) −38.4806 −1.80400
\(456\) 2.66908 0.124991
\(457\) 10.4920 0.490794 0.245397 0.969423i \(-0.421082\pi\)
0.245397 + 0.969423i \(0.421082\pi\)
\(458\) −3.48870 −0.163016
\(459\) −10.3647 −0.483783
\(460\) 22.4959 1.04888
\(461\) 6.18038 0.287849 0.143925 0.989589i \(-0.454028\pi\)
0.143925 + 0.989589i \(0.454028\pi\)
\(462\) −11.2480 −0.523303
\(463\) −11.1433 −0.517873 −0.258937 0.965894i \(-0.583372\pi\)
−0.258937 + 0.965894i \(0.583372\pi\)
\(464\) 5.57889 0.258993
\(465\) −77.1303 −3.57683
\(466\) 17.0902 0.791688
\(467\) −34.5104 −1.59695 −0.798476 0.602027i \(-0.794360\pi\)
−0.798476 + 0.602027i \(0.794360\pi\)
\(468\) −9.13122 −0.422091
\(469\) −5.49683 −0.253820
\(470\) 31.2850 1.44307
\(471\) 55.0362 2.53593
\(472\) −4.79306 −0.220619
\(473\) −1.46214 −0.0672293
\(474\) 24.4428 1.12270
\(475\) −12.0072 −0.550930
\(476\) 14.5596 0.667337
\(477\) −54.5249 −2.49652
\(478\) −6.06035 −0.277194
\(479\) 41.6980 1.90523 0.952616 0.304176i \(-0.0983812\pi\)
0.952616 + 0.304176i \(0.0983812\pi\)
\(480\) 11.0072 0.502409
\(481\) 6.44282 0.293768
\(482\) −4.85341 −0.221067
\(483\) 61.3566 2.79182
\(484\) 1.00000 0.0454545
\(485\) −8.99187 −0.408300
\(486\) 20.6498 0.936692
\(487\) 9.10137 0.412423 0.206211 0.978507i \(-0.433887\pi\)
0.206211 + 0.978507i \(0.433887\pi\)
\(488\) −8.90981 −0.403328
\(489\) −47.6006 −2.15257
\(490\) 44.3711 2.00448
\(491\) −30.8413 −1.39185 −0.695925 0.718115i \(-0.745005\pi\)
−0.695925 + 0.718115i \(0.745005\pi\)
\(492\) −31.8075 −1.43399
\(493\) −19.2745 −0.868081
\(494\) −2.21417 −0.0996204
\(495\) 17.0072 0.764418
\(496\) 7.00724 0.314634
\(497\) 28.6577 1.28547
\(498\) 35.2697 1.58047
\(499\) 21.7665 0.974403 0.487201 0.873290i \(-0.338018\pi\)
0.487201 + 0.873290i \(0.338018\pi\)
\(500\) −28.8977 −1.29235
\(501\) 11.1578 0.498493
\(502\) 1.58612 0.0707922
\(503\) 38.0893 1.69832 0.849159 0.528138i \(-0.177109\pi\)
0.849159 + 0.528138i \(0.177109\pi\)
\(504\) 17.3792 0.774131
\(505\) 25.7665 1.14659
\(506\) −5.45490 −0.242500
\(507\) −21.6127 −0.959853
\(508\) −2.48146 −0.110097
\(509\) 14.9388 0.662149 0.331074 0.943605i \(-0.392589\pi\)
0.331074 + 0.943605i \(0.392589\pi\)
\(510\) −38.0289 −1.68395
\(511\) 6.13122 0.271229
\(512\) −1.00000 −0.0441942
\(513\) −3.00000 −0.132453
\(514\) −19.5185 −0.860926
\(515\) −2.49683 −0.110023
\(516\) 3.90257 0.171801
\(517\) −7.58612 −0.333637
\(518\) −12.2624 −0.538781
\(519\) 63.2842 2.77787
\(520\) −9.13122 −0.400431
\(521\) 7.22536 0.316549 0.158274 0.987395i \(-0.449407\pi\)
0.158274 + 0.987395i \(0.449407\pi\)
\(522\) −23.0072 −1.00700
\(523\) 4.90586 0.214518 0.107259 0.994231i \(-0.465793\pi\)
0.107259 + 0.994231i \(0.465793\pi\)
\(524\) 11.0072 0.480853
\(525\) −135.057 −5.89437
\(526\) 13.0483 0.568931
\(527\) −24.2093 −1.05458
\(528\) −2.66908 −0.116157
\(529\) 6.75598 0.293738
\(530\) −54.5249 −2.36841
\(531\) 19.7665 0.857793
\(532\) 4.21417 0.182708
\(533\) 26.3864 1.14292
\(534\) −22.0145 −0.952659
\(535\) −16.0145 −0.692366
\(536\) −1.30437 −0.0563401
\(537\) −28.8977 −1.24703
\(538\) −23.6006 −1.01749
\(539\) −10.7593 −0.463435
\(540\) −12.3719 −0.532404
\(541\) 30.8712 1.32726 0.663628 0.748063i \(-0.269016\pi\)
0.663628 + 0.748063i \(0.269016\pi\)
\(542\) 16.4090 0.704828
\(543\) −47.6006 −2.04274
\(544\) 3.45490 0.148128
\(545\) 20.5104 0.878569
\(546\) −24.9050 −1.06583
\(547\) 2.54115 0.108652 0.0543259 0.998523i \(-0.482699\pi\)
0.0543259 + 0.998523i \(0.482699\pi\)
\(548\) 6.94360 0.296616
\(549\) 36.7439 1.56819
\(550\) 12.0072 0.511990
\(551\) −5.57889 −0.237669
\(552\) 14.5596 0.619696
\(553\) 38.5925 1.64112
\(554\) 11.5861 0.492247
\(555\) 32.0289 1.35955
\(556\) −4.37195 −0.185412
\(557\) 23.9017 1.01275 0.506373 0.862314i \(-0.330986\pi\)
0.506373 + 0.862314i \(0.330986\pi\)
\(558\) −28.8977 −1.22334
\(559\) −3.23744 −0.136929
\(560\) 17.3792 0.734405
\(561\) 9.22141 0.389328
\(562\) −7.03379 −0.296703
\(563\) −10.6232 −0.447715 −0.223857 0.974622i \(-0.571865\pi\)
−0.223857 + 0.974622i \(0.571865\pi\)
\(564\) 20.2480 0.852593
\(565\) 32.9919 1.38798
\(566\) −8.33092 −0.350175
\(567\) 18.3937 0.772461
\(568\) 6.80030 0.285334
\(569\) 29.3373 1.22988 0.614941 0.788573i \(-0.289180\pi\)
0.614941 + 0.788573i \(0.289180\pi\)
\(570\) −11.0072 −0.461042
\(571\) 19.8639 0.831280 0.415640 0.909529i \(-0.363558\pi\)
0.415640 + 0.909529i \(0.363558\pi\)
\(572\) 2.21417 0.0925793
\(573\) 1.63529 0.0683151
\(574\) −50.2205 −2.09616
\(575\) −65.4983 −2.73147
\(576\) 4.12398 0.171833
\(577\) −3.17709 −0.132264 −0.0661320 0.997811i \(-0.521066\pi\)
−0.0661320 + 0.997811i \(0.521066\pi\)
\(578\) 5.06364 0.210620
\(579\) 31.0748 1.29143
\(580\) −23.0072 −0.955324
\(581\) 55.6868 2.31028
\(582\) −5.81962 −0.241231
\(583\) 13.2214 0.547575
\(584\) 1.45490 0.0602044
\(585\) 37.6570 1.55693
\(586\) −10.2142 −0.421944
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 28.7173 1.18428
\(589\) −7.00724 −0.288728
\(590\) 19.7665 0.813774
\(591\) −34.4573 −1.41738
\(592\) −2.90981 −0.119592
\(593\) −3.45885 −0.142038 −0.0710190 0.997475i \(-0.522625\pi\)
−0.0710190 + 0.997475i \(0.522625\pi\)
\(594\) 3.00000 0.123091
\(595\) −60.0434 −2.46154
\(596\) 11.7665 0.481975
\(597\) −49.0169 −2.00613
\(598\) −12.0781 −0.493911
\(599\) 40.3864 1.65014 0.825072 0.565027i \(-0.191134\pi\)
0.825072 + 0.565027i \(0.191134\pi\)
\(600\) −32.0483 −1.30836
\(601\) 18.3309 0.747734 0.373867 0.927482i \(-0.378032\pi\)
0.373867 + 0.927482i \(0.378032\pi\)
\(602\) 6.16172 0.251133
\(603\) 5.37919 0.219057
\(604\) −19.5861 −0.796948
\(605\) −4.12398 −0.167664
\(606\) 16.6763 0.677429
\(607\) 12.4283 0.504451 0.252226 0.967668i \(-0.418838\pi\)
0.252226 + 0.967668i \(0.418838\pi\)
\(608\) 1.00000 0.0405554
\(609\) −62.7511 −2.54280
\(610\) 36.7439 1.48772
\(611\) −16.7970 −0.679534
\(612\) −14.2480 −0.575940
\(613\) −0.0531082 −0.00214502 −0.00107251 0.999999i \(-0.500341\pi\)
−0.00107251 + 0.999999i \(0.500341\pi\)
\(614\) 1.93242 0.0779860
\(615\) 131.174 5.28944
\(616\) −4.21417 −0.169794
\(617\) −17.4887 −0.704068 −0.352034 0.935987i \(-0.614510\pi\)
−0.352034 + 0.935987i \(0.614510\pi\)
\(618\) −1.61597 −0.0650039
\(619\) 25.2995 1.01687 0.508437 0.861099i \(-0.330224\pi\)
0.508437 + 0.861099i \(0.330224\pi\)
\(620\) −28.8977 −1.16056
\(621\) −16.3647 −0.656693
\(622\) 27.7173 1.11136
\(623\) −34.7584 −1.39256
\(624\) −5.90981 −0.236582
\(625\) 59.1376 2.36550
\(626\) −11.5258 −0.460663
\(627\) 2.66908 0.106593
\(628\) 20.6199 0.822824
\(629\) 10.0531 0.400844
\(630\) −71.6715 −2.85546
\(631\) 9.02261 0.359184 0.179592 0.983741i \(-0.442522\pi\)
0.179592 + 0.983741i \(0.442522\pi\)
\(632\) 9.15777 0.364277
\(633\) 5.68840 0.226093
\(634\) 33.8977 1.34625
\(635\) 10.2335 0.406104
\(636\) −35.2890 −1.39930
\(637\) −23.8229 −0.943898
\(638\) 5.57889 0.220870
\(639\) −28.0443 −1.10942
\(640\) 4.12398 0.163015
\(641\) 30.4573 1.20299 0.601495 0.798876i \(-0.294572\pi\)
0.601495 + 0.798876i \(0.294572\pi\)
\(642\) −10.3647 −0.409063
\(643\) −36.7584 −1.44961 −0.724804 0.688955i \(-0.758070\pi\)
−0.724804 + 0.688955i \(0.758070\pi\)
\(644\) 22.9879 0.905851
\(645\) −16.0941 −0.633706
\(646\) −3.45490 −0.135931
\(647\) −41.0700 −1.61463 −0.807314 0.590122i \(-0.799079\pi\)
−0.807314 + 0.590122i \(0.799079\pi\)
\(648\) 4.36471 0.171462
\(649\) −4.79306 −0.188144
\(650\) 26.5861 1.04279
\(651\) −78.8172 −3.08909
\(652\) −17.8341 −0.698437
\(653\) 28.4806 1.11453 0.557265 0.830335i \(-0.311851\pi\)
0.557265 + 0.830335i \(0.311851\pi\)
\(654\) 13.2745 0.519075
\(655\) −45.3937 −1.77368
\(656\) −11.9170 −0.465282
\(657\) −6.00000 −0.234082
\(658\) 31.9693 1.24629
\(659\) −10.5740 −0.411906 −0.205953 0.978562i \(-0.566029\pi\)
−0.205953 + 0.978562i \(0.566029\pi\)
\(660\) 11.0072 0.428456
\(661\) 7.70287 0.299607 0.149803 0.988716i \(-0.452136\pi\)
0.149803 + 0.988716i \(0.452136\pi\)
\(662\) 16.2697 0.632339
\(663\) 20.4178 0.792962
\(664\) 13.2142 0.512809
\(665\) −17.3792 −0.673936
\(666\) 12.0000 0.464991
\(667\) −30.4323 −1.17834
\(668\) 4.18038 0.161744
\(669\) 18.4428 0.713041
\(670\) 5.37919 0.207816
\(671\) −8.90981 −0.343959
\(672\) 11.2480 0.433900
\(673\) −31.5176 −1.21492 −0.607458 0.794352i \(-0.707811\pi\)
−0.607458 + 0.794352i \(0.707811\pi\)
\(674\) 6.82685 0.262961
\(675\) 36.0217 1.38648
\(676\) −8.09743 −0.311440
\(677\) 11.9928 0.460919 0.230460 0.973082i \(-0.425977\pi\)
0.230460 + 0.973082i \(0.425977\pi\)
\(678\) 21.3526 0.820043
\(679\) −9.18852 −0.352623
\(680\) −14.2480 −0.546385
\(681\) −7.27452 −0.278760
\(682\) 7.00724 0.268321
\(683\) 39.1722 1.49888 0.749442 0.662070i \(-0.230322\pi\)
0.749442 + 0.662070i \(0.230322\pi\)
\(684\) −4.12398 −0.157684
\(685\) −28.6353 −1.09410
\(686\) 15.8422 0.604859
\(687\) 9.31160 0.355260
\(688\) 1.46214 0.0557436
\(689\) 29.2745 1.11527
\(690\) −60.0434 −2.28581
\(691\) −49.1722 −1.87060 −0.935300 0.353855i \(-0.884871\pi\)
−0.935300 + 0.353855i \(0.884871\pi\)
\(692\) 23.7101 0.901323
\(693\) 17.3792 0.660181
\(694\) 5.09019 0.193221
\(695\) 18.0298 0.683911
\(696\) −14.8905 −0.564423
\(697\) 41.1722 1.55951
\(698\) −30.9919 −1.17306
\(699\) −45.6151 −1.72532
\(700\) −50.6006 −1.91252
\(701\) 37.8486 1.42952 0.714760 0.699370i \(-0.246536\pi\)
0.714760 + 0.699370i \(0.246536\pi\)
\(702\) 6.64252 0.250706
\(703\) 2.90981 0.109745
\(704\) −1.00000 −0.0376889
\(705\) −83.5023 −3.14488
\(706\) 17.8301 0.671046
\(707\) 26.3300 0.990242
\(708\) 12.7931 0.480793
\(709\) 27.7511 1.04222 0.521108 0.853491i \(-0.325519\pi\)
0.521108 + 0.853491i \(0.325519\pi\)
\(710\) −28.0443 −1.05248
\(711\) −37.7665 −1.41635
\(712\) −8.24797 −0.309106
\(713\) −38.2238 −1.43149
\(714\) −38.8606 −1.45432
\(715\) −9.13122 −0.341488
\(716\) −10.8269 −0.404619
\(717\) 16.1755 0.604087
\(718\) −24.4090 −0.910937
\(719\) −17.0410 −0.635523 −0.317762 0.948171i \(-0.602931\pi\)
−0.317762 + 0.948171i \(0.602931\pi\)
\(720\) −17.0072 −0.633822
\(721\) −2.55144 −0.0950204
\(722\) −1.00000 −0.0372161
\(723\) 12.9541 0.481769
\(724\) −17.8341 −0.662799
\(725\) 66.9870 2.48784
\(726\) −2.66908 −0.0990588
\(727\) 13.5225 0.501521 0.250761 0.968049i \(-0.419319\pi\)
0.250761 + 0.968049i \(0.419319\pi\)
\(728\) −9.33092 −0.345827
\(729\) −42.0217 −1.55636
\(730\) −6.00000 −0.222070
\(731\) −5.05156 −0.186839
\(732\) 23.7810 0.878970
\(733\) −25.6682 −0.948076 −0.474038 0.880504i \(-0.657204\pi\)
−0.474038 + 0.880504i \(0.657204\pi\)
\(734\) 26.6232 0.982681
\(735\) −118.430 −4.36835
\(736\) 5.45490 0.201070
\(737\) −1.30437 −0.0480470
\(738\) 49.1457 1.80908
\(739\) 21.3493 0.785348 0.392674 0.919678i \(-0.371550\pi\)
0.392674 + 0.919678i \(0.371550\pi\)
\(740\) 12.0000 0.441129
\(741\) 5.90981 0.217102
\(742\) −55.7173 −2.04545
\(743\) −7.55718 −0.277246 −0.138623 0.990345i \(-0.544268\pi\)
−0.138623 + 0.990345i \(0.544268\pi\)
\(744\) −18.7029 −0.685680
\(745\) −48.5249 −1.77781
\(746\) 12.6160 0.461904
\(747\) −54.4950 −1.99387
\(748\) 3.45490 0.126324
\(749\) −16.3647 −0.597954
\(750\) 77.1303 2.81640
\(751\) −53.5861 −1.95539 −0.977693 0.210040i \(-0.932640\pi\)
−0.977693 + 0.210040i \(0.932640\pi\)
\(752\) 7.58612 0.276637
\(753\) −4.23349 −0.154277
\(754\) 12.3526 0.449856
\(755\) 80.7728 2.93962
\(756\) −12.6425 −0.459804
\(757\) −13.9436 −0.506789 −0.253394 0.967363i \(-0.581547\pi\)
−0.253394 + 0.967363i \(0.581547\pi\)
\(758\) 22.8905 0.831420
\(759\) 14.5596 0.528479
\(760\) −4.12398 −0.149593
\(761\) −45.7994 −1.66023 −0.830114 0.557594i \(-0.811724\pi\)
−0.830114 + 0.557594i \(0.811724\pi\)
\(762\) 6.62321 0.239934
\(763\) 20.9590 0.758766
\(764\) 0.612679 0.0221659
\(765\) 58.7584 2.12441
\(766\) 0.0298464 0.00107839
\(767\) −10.6127 −0.383202
\(768\) 2.66908 0.0963121
\(769\) −40.8751 −1.47399 −0.736997 0.675896i \(-0.763757\pi\)
−0.736997 + 0.675896i \(0.763757\pi\)
\(770\) 17.3792 0.626302
\(771\) 52.0965 1.87621
\(772\) 11.6425 0.419024
\(773\) 37.4983 1.34872 0.674361 0.738402i \(-0.264419\pi\)
0.674361 + 0.738402i \(0.264419\pi\)
\(774\) −6.02985 −0.216738
\(775\) 84.1376 3.02231
\(776\) −2.18038 −0.0782712
\(777\) 32.7294 1.17416
\(778\) −9.22865 −0.330863
\(779\) 11.9170 0.426972
\(780\) 24.3719 0.872656
\(781\) 6.80030 0.243334
\(782\) −18.8462 −0.673938
\(783\) 16.7367 0.598119
\(784\) 10.7593 0.384260
\(785\) −85.0362 −3.03507
\(786\) −29.3792 −1.04792
\(787\) 8.80754 0.313955 0.156977 0.987602i \(-0.449825\pi\)
0.156977 + 0.987602i \(0.449825\pi\)
\(788\) −12.9098 −0.459893
\(789\) −34.8269 −1.23987
\(790\) −37.7665 −1.34367
\(791\) 33.7134 1.19871
\(792\) 4.12398 0.146539
\(793\) −19.7279 −0.700557
\(794\) −17.2697 −0.612878
\(795\) 145.531 5.16146
\(796\) −18.3647 −0.650920
\(797\) −26.1312 −0.925615 −0.462808 0.886459i \(-0.653158\pi\)
−0.462808 + 0.886459i \(0.653158\pi\)
\(798\) −11.2480 −0.398174
\(799\) −26.2093 −0.927220
\(800\) −12.0072 −0.424520
\(801\) 34.0145 1.20184
\(802\) 13.2850 0.469111
\(803\) 1.45490 0.0513425
\(804\) 3.48146 0.122782
\(805\) −94.8018 −3.34132
\(806\) 15.5152 0.546501
\(807\) 62.9919 2.21742
\(808\) 6.24797 0.219803
\(809\) −6.01053 −0.211319 −0.105659 0.994402i \(-0.533695\pi\)
−0.105659 + 0.994402i \(0.533695\pi\)
\(810\) −18.0000 −0.632456
\(811\) −40.2809 −1.41445 −0.707226 0.706987i \(-0.750054\pi\)
−0.707226 + 0.706987i \(0.750054\pi\)
\(812\) −23.5104 −0.825054
\(813\) −43.7970 −1.53603
\(814\) −2.90981 −0.101989
\(815\) 73.5475 2.57626
\(816\) −9.22141 −0.322814
\(817\) −1.46214 −0.0511539
\(818\) −1.94360 −0.0679564
\(819\) 38.4806 1.34462
\(820\) 49.1457 1.71624
\(821\) 50.1496 1.75023 0.875117 0.483911i \(-0.160784\pi\)
0.875117 + 0.483911i \(0.160784\pi\)
\(822\) −18.5330 −0.646413
\(823\) −42.7400 −1.48982 −0.744911 0.667164i \(-0.767508\pi\)
−0.744911 + 0.667164i \(0.767508\pi\)
\(824\) −0.605441 −0.0210915
\(825\) −32.0483 −1.11578
\(826\) 20.1988 0.702806
\(827\) 49.0555 1.70583 0.852913 0.522052i \(-0.174833\pi\)
0.852913 + 0.522052i \(0.174833\pi\)
\(828\) −22.4959 −0.781788
\(829\) 49.9653 1.73537 0.867683 0.497117i \(-0.165608\pi\)
0.867683 + 0.497117i \(0.165608\pi\)
\(830\) −54.4950 −1.89155
\(831\) −30.9243 −1.07275
\(832\) −2.21417 −0.0767627
\(833\) −37.1722 −1.28794
\(834\) 11.6691 0.404067
\(835\) −17.2398 −0.596609
\(836\) 1.00000 0.0345857
\(837\) 21.0217 0.726617
\(838\) 2.49593 0.0862206
\(839\) −52.0257 −1.79613 −0.898063 0.439868i \(-0.855025\pi\)
−0.898063 + 0.439868i \(0.855025\pi\)
\(840\) −46.3864 −1.60048
\(841\) 2.12398 0.0732408
\(842\) −15.6353 −0.538828
\(843\) 18.7737 0.646602
\(844\) 2.13122 0.0733596
\(845\) 33.3937 1.14878
\(846\) −31.2850 −1.07560
\(847\) −4.21417 −0.144801
\(848\) −13.2214 −0.454025
\(849\) 22.2359 0.763134
\(850\) 41.4839 1.42288
\(851\) 15.8727 0.544110
\(852\) −18.1505 −0.621827
\(853\) 16.2335 0.555824 0.277912 0.960607i \(-0.410358\pi\)
0.277912 + 0.960607i \(0.410358\pi\)
\(854\) 37.5475 1.28485
\(855\) 17.0072 0.581635
\(856\) −3.88325 −0.132727
\(857\) 9.60150 0.327981 0.163990 0.986462i \(-0.447563\pi\)
0.163990 + 0.986462i \(0.447563\pi\)
\(858\) −5.90981 −0.201758
\(859\) 19.4057 0.662115 0.331058 0.943611i \(-0.392595\pi\)
0.331058 + 0.943611i \(0.392595\pi\)
\(860\) −6.02985 −0.205616
\(861\) 134.043 4.56816
\(862\) −0.262441 −0.00893877
\(863\) 1.54839 0.0527077 0.0263539 0.999653i \(-0.491610\pi\)
0.0263539 + 0.999653i \(0.491610\pi\)
\(864\) −3.00000 −0.102062
\(865\) −97.7801 −3.32462
\(866\) 32.2769 1.09681
\(867\) −13.5152 −0.459002
\(868\) −29.5297 −1.00230
\(869\) 9.15777 0.310656
\(870\) 61.4081 2.08193
\(871\) −2.88810 −0.0978594
\(872\) 4.97345 0.168422
\(873\) 8.99187 0.304329
\(874\) −5.45490 −0.184515
\(875\) 121.780 4.11692
\(876\) −3.88325 −0.131203
\(877\) −12.1200 −0.409265 −0.204632 0.978839i \(-0.565600\pi\)
−0.204632 + 0.978839i \(0.565600\pi\)
\(878\) 17.0902 0.576766
\(879\) 27.2624 0.919539
\(880\) 4.12398 0.139019
\(881\) 53.6836 1.80864 0.904322 0.426850i \(-0.140377\pi\)
0.904322 + 0.426850i \(0.140377\pi\)
\(882\) −44.3711 −1.49405
\(883\) 34.4283 1.15861 0.579303 0.815112i \(-0.303325\pi\)
0.579303 + 0.815112i \(0.303325\pi\)
\(884\) 7.64976 0.257289
\(885\) −52.7584 −1.77345
\(886\) −16.2624 −0.546347
\(887\) −27.5330 −0.924468 −0.462234 0.886758i \(-0.652952\pi\)
−0.462234 + 0.886758i \(0.652952\pi\)
\(888\) 7.76651 0.260627
\(889\) 10.4573 0.350727
\(890\) 34.0145 1.14017
\(891\) 4.36471 0.146223
\(892\) 6.90981 0.231357
\(893\) −7.58612 −0.253860
\(894\) −31.4057 −1.05037
\(895\) 44.6498 1.49248
\(896\) 4.21417 0.140786
\(897\) 32.2374 1.07638
\(898\) 27.3382 0.912286
\(899\) 39.0926 1.30381
\(900\) 49.5176 1.65059
\(901\) 45.6787 1.52178
\(902\) −11.9170 −0.396794
\(903\) −16.4461 −0.547292
\(904\) 8.00000 0.266076
\(905\) 73.5475 2.44480
\(906\) 52.2769 1.73678
\(907\) 15.5677 0.516917 0.258459 0.966022i \(-0.416785\pi\)
0.258459 + 0.966022i \(0.416785\pi\)
\(908\) −2.72548 −0.0904482
\(909\) −25.7665 −0.854621
\(910\) 38.4806 1.27562
\(911\) −0.623208 −0.0206478 −0.0103239 0.999947i \(-0.503286\pi\)
−0.0103239 + 0.999947i \(0.503286\pi\)
\(912\) −2.66908 −0.0883820
\(913\) 13.2142 0.437325
\(914\) −10.4920 −0.347044
\(915\) −98.0724 −3.24217
\(916\) 3.48870 0.115270
\(917\) −46.3864 −1.53181
\(918\) 10.3647 0.342086
\(919\) −44.4887 −1.46755 −0.733773 0.679394i \(-0.762243\pi\)
−0.733773 + 0.679394i \(0.762243\pi\)
\(920\) −22.4959 −0.741669
\(921\) −5.15777 −0.169954
\(922\) −6.18038 −0.203540
\(923\) 15.0571 0.495609
\(924\) 11.2480 0.370031
\(925\) −34.9388 −1.14878
\(926\) 11.1433 0.366192
\(927\) 2.49683 0.0820067
\(928\) −5.57889 −0.183136
\(929\) 44.1224 1.44761 0.723805 0.690005i \(-0.242391\pi\)
0.723805 + 0.690005i \(0.242391\pi\)
\(930\) 77.1303 2.52920
\(931\) −10.7593 −0.352621
\(932\) −17.0902 −0.559808
\(933\) −73.9798 −2.42199
\(934\) 34.5104 1.12922
\(935\) −14.2480 −0.465958
\(936\) 9.13122 0.298463
\(937\) 24.8672 0.812377 0.406188 0.913789i \(-0.366858\pi\)
0.406188 + 0.913789i \(0.366858\pi\)
\(938\) 5.49683 0.179478
\(939\) 30.7632 1.00392
\(940\) −31.2850 −1.02041
\(941\) 29.8301 0.972435 0.486217 0.873838i \(-0.338376\pi\)
0.486217 + 0.873838i \(0.338376\pi\)
\(942\) −55.0362 −1.79318
\(943\) 65.0063 2.11690
\(944\) 4.79306 0.156001
\(945\) 52.1376 1.69603
\(946\) 1.46214 0.0475383
\(947\) 9.07572 0.294921 0.147461 0.989068i \(-0.452890\pi\)
0.147461 + 0.989068i \(0.452890\pi\)
\(948\) −24.4428 −0.793866
\(949\) 3.22141 0.104571
\(950\) 12.0072 0.389566
\(951\) −90.4757 −2.93388
\(952\) −14.5596 −0.471878
\(953\) 1.27058 0.0411580 0.0205790 0.999788i \(-0.493449\pi\)
0.0205790 + 0.999788i \(0.493449\pi\)
\(954\) 54.5249 1.76531
\(955\) −2.52668 −0.0817613
\(956\) 6.06035 0.196006
\(957\) −14.8905 −0.481341
\(958\) −41.6980 −1.34720
\(959\) −29.2615 −0.944905
\(960\) −11.0072 −0.355257
\(961\) 18.1014 0.583915
\(962\) −6.44282 −0.207725
\(963\) 16.0145 0.516059
\(964\) 4.85341 0.156318
\(965\) −48.0136 −1.54561
\(966\) −61.3566 −1.97412
\(967\) 6.67632 0.214696 0.107348 0.994222i \(-0.465764\pi\)
0.107348 + 0.994222i \(0.465764\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 9.22141 0.296234
\(970\) 8.99187 0.288711
\(971\) −0.191566 −0.00614764 −0.00307382 0.999995i \(-0.500978\pi\)
−0.00307382 + 0.999995i \(0.500978\pi\)
\(972\) −20.6498 −0.662341
\(973\) 18.4242 0.590651
\(974\) −9.10137 −0.291627
\(975\) −70.9605 −2.27255
\(976\) 8.90981 0.285196
\(977\) 16.4959 0.527752 0.263876 0.964557i \(-0.414999\pi\)
0.263876 + 0.964557i \(0.414999\pi\)
\(978\) 47.6006 1.52210
\(979\) −8.24797 −0.263606
\(980\) −44.3711 −1.41738
\(981\) −20.5104 −0.654847
\(982\) 30.8413 0.984186
\(983\) −35.0072 −1.11656 −0.558279 0.829653i \(-0.688538\pi\)
−0.558279 + 0.829653i \(0.688538\pi\)
\(984\) 31.8075 1.01399
\(985\) 53.2398 1.69636
\(986\) 19.2745 0.613826
\(987\) −85.3285 −2.71604
\(988\) 2.21417 0.0704423
\(989\) −7.97584 −0.253617
\(990\) −17.0072 −0.540525
\(991\) −11.3759 −0.361367 −0.180684 0.983541i \(-0.557831\pi\)
−0.180684 + 0.983541i \(0.557831\pi\)
\(992\) −7.00724 −0.222480
\(993\) −43.4251 −1.37805
\(994\) −28.6577 −0.908966
\(995\) 75.7358 2.40099
\(996\) −35.2697 −1.11756
\(997\) 8.89533 0.281718 0.140859 0.990030i \(-0.455014\pi\)
0.140859 + 0.990030i \(0.455014\pi\)
\(998\) −21.7665 −0.689007
\(999\) −8.72942 −0.276187
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 418.2.a.g.1.3 3
3.2 odd 2 3762.2.a.bg.1.3 3
4.3 odd 2 3344.2.a.q.1.1 3
11.10 odd 2 4598.2.a.bo.1.3 3
19.18 odd 2 7942.2.a.bi.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.3 3 1.1 even 1 trivial
3344.2.a.q.1.1 3 4.3 odd 2
3762.2.a.bg.1.3 3 3.2 odd 2
4598.2.a.bo.1.3 3 11.10 odd 2
7942.2.a.bi.1.1 3 19.18 odd 2