Properties

Label 2166.2.a.u.1.1
Level $2166$
Weight $2$
Character 2166.1
Self dual yes
Analytic conductor $17.296$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.2955970778\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.41147 q^{5} +1.00000 q^{6} -2.87939 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.41147 q^{5} +1.00000 q^{6} -2.87939 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.41147 q^{10} -0.347296 q^{11} +1.00000 q^{12} +1.65270 q^{13} -2.87939 q^{14} -3.41147 q^{15} +1.00000 q^{16} +6.94356 q^{17} +1.00000 q^{18} -3.41147 q^{20} -2.87939 q^{21} -0.347296 q^{22} +6.80066 q^{23} +1.00000 q^{24} +6.63816 q^{25} +1.65270 q^{26} +1.00000 q^{27} -2.87939 q^{28} -6.35504 q^{29} -3.41147 q^{30} -1.59627 q^{31} +1.00000 q^{32} -0.347296 q^{33} +6.94356 q^{34} +9.82295 q^{35} +1.00000 q^{36} +11.2121 q^{37} +1.65270 q^{39} -3.41147 q^{40} +3.49020 q^{41} -2.87939 q^{42} +2.28312 q^{43} -0.347296 q^{44} -3.41147 q^{45} +6.80066 q^{46} +5.59627 q^{47} +1.00000 q^{48} +1.29086 q^{49} +6.63816 q^{50} +6.94356 q^{51} +1.65270 q^{52} -1.98040 q^{53} +1.00000 q^{54} +1.18479 q^{55} -2.87939 q^{56} -6.35504 q^{58} -0.445622 q^{59} -3.41147 q^{60} -12.5321 q^{61} -1.59627 q^{62} -2.87939 q^{63} +1.00000 q^{64} -5.63816 q^{65} -0.347296 q^{66} +1.07873 q^{67} +6.94356 q^{68} +6.80066 q^{69} +9.82295 q^{70} +16.6236 q^{71} +1.00000 q^{72} +12.4192 q^{73} +11.2121 q^{74} +6.63816 q^{75} +1.00000 q^{77} +1.65270 q^{78} +10.9240 q^{79} -3.41147 q^{80} +1.00000 q^{81} +3.49020 q^{82} +11.7169 q^{83} -2.87939 q^{84} -23.6878 q^{85} +2.28312 q^{86} -6.35504 q^{87} -0.347296 q^{88} -1.79292 q^{89} -3.41147 q^{90} -4.75877 q^{91} +6.80066 q^{92} -1.59627 q^{93} +5.59627 q^{94} +1.00000 q^{96} -3.65270 q^{97} +1.29086 q^{98} -0.347296 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + 3q^{12} + 6q^{13} - 3q^{14} + 3q^{16} + 6q^{17} + 3q^{18} - 3q^{21} + 6q^{23} + 3q^{24} + 3q^{25} + 6q^{26} + 3q^{27} - 3q^{28} + 6q^{29} + 9q^{31} + 3q^{32} + 6q^{34} + 9q^{35} + 3q^{36} + 9q^{37} + 6q^{39} + 9q^{41} - 3q^{42} + 15q^{43} + 6q^{46} + 3q^{47} + 3q^{48} - 12q^{49} + 3q^{50} + 6q^{51} + 6q^{52} - 3q^{53} + 3q^{54} - 3q^{56} + 6q^{58} - 12q^{59} - 33q^{61} + 9q^{62} - 3q^{63} + 3q^{64} + 12q^{67} + 6q^{68} + 6q^{69} + 9q^{70} + 15q^{71} + 3q^{72} + 3q^{73} + 9q^{74} + 3q^{75} + 3q^{77} + 6q^{78} + 15q^{79} + 3q^{81} + 9q^{82} + 27q^{83} - 3q^{84} - 27q^{85} + 15q^{86} + 6q^{87} - 15q^{89} - 3q^{91} + 6q^{92} + 9q^{93} + 3q^{94} + 3q^{96} - 12q^{97} - 12q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −3.41147 −1.52566 −0.762829 0.646601i \(-0.776190\pi\)
−0.762829 + 0.646601i \(0.776190\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.87939 −1.08831 −0.544153 0.838986i \(-0.683149\pi\)
−0.544153 + 0.838986i \(0.683149\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.41147 −1.07880
\(11\) −0.347296 −0.104714 −0.0523569 0.998628i \(-0.516673\pi\)
−0.0523569 + 0.998628i \(0.516673\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.65270 0.458378 0.229189 0.973382i \(-0.426393\pi\)
0.229189 + 0.973382i \(0.426393\pi\)
\(14\) −2.87939 −0.769548
\(15\) −3.41147 −0.880839
\(16\) 1.00000 0.250000
\(17\) 6.94356 1.68406 0.842031 0.539430i \(-0.181360\pi\)
0.842031 + 0.539430i \(0.181360\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0
\(20\) −3.41147 −0.762829
\(21\) −2.87939 −0.628333
\(22\) −0.347296 −0.0740438
\(23\) 6.80066 1.41804 0.709018 0.705191i \(-0.249139\pi\)
0.709018 + 0.705191i \(0.249139\pi\)
\(24\) 1.00000 0.204124
\(25\) 6.63816 1.32763
\(26\) 1.65270 0.324122
\(27\) 1.00000 0.192450
\(28\) −2.87939 −0.544153
\(29\) −6.35504 −1.18010 −0.590050 0.807366i \(-0.700892\pi\)
−0.590050 + 0.807366i \(0.700892\pi\)
\(30\) −3.41147 −0.622847
\(31\) −1.59627 −0.286698 −0.143349 0.989672i \(-0.545787\pi\)
−0.143349 + 0.989672i \(0.545787\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.347296 −0.0604565
\(34\) 6.94356 1.19081
\(35\) 9.82295 1.66038
\(36\) 1.00000 0.166667
\(37\) 11.2121 1.84326 0.921632 0.388066i \(-0.126857\pi\)
0.921632 + 0.388066i \(0.126857\pi\)
\(38\) 0 0
\(39\) 1.65270 0.264644
\(40\) −3.41147 −0.539401
\(41\) 3.49020 0.545078 0.272539 0.962145i \(-0.412137\pi\)
0.272539 + 0.962145i \(0.412137\pi\)
\(42\) −2.87939 −0.444299
\(43\) 2.28312 0.348172 0.174086 0.984730i \(-0.444303\pi\)
0.174086 + 0.984730i \(0.444303\pi\)
\(44\) −0.347296 −0.0523569
\(45\) −3.41147 −0.508553
\(46\) 6.80066 1.00270
\(47\) 5.59627 0.816299 0.408150 0.912915i \(-0.366174\pi\)
0.408150 + 0.912915i \(0.366174\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.29086 0.184408
\(50\) 6.63816 0.938777
\(51\) 6.94356 0.972293
\(52\) 1.65270 0.229189
\(53\) −1.98040 −0.272029 −0.136014 0.990707i \(-0.543429\pi\)
−0.136014 + 0.990707i \(0.543429\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.18479 0.159757
\(56\) −2.87939 −0.384774
\(57\) 0 0
\(58\) −6.35504 −0.834457
\(59\) −0.445622 −0.0580151 −0.0290075 0.999579i \(-0.509235\pi\)
−0.0290075 + 0.999579i \(0.509235\pi\)
\(60\) −3.41147 −0.440419
\(61\) −12.5321 −1.60457 −0.802285 0.596941i \(-0.796382\pi\)
−0.802285 + 0.596941i \(0.796382\pi\)
\(62\) −1.59627 −0.202726
\(63\) −2.87939 −0.362768
\(64\) 1.00000 0.125000
\(65\) −5.63816 −0.699327
\(66\) −0.347296 −0.0427492
\(67\) 1.07873 0.131787 0.0658937 0.997827i \(-0.479010\pi\)
0.0658937 + 0.997827i \(0.479010\pi\)
\(68\) 6.94356 0.842031
\(69\) 6.80066 0.818703
\(70\) 9.82295 1.17407
\(71\) 16.6236 1.97286 0.986430 0.164185i \(-0.0524993\pi\)
0.986430 + 0.164185i \(0.0524993\pi\)
\(72\) 1.00000 0.117851
\(73\) 12.4192 1.45356 0.726780 0.686871i \(-0.241016\pi\)
0.726780 + 0.686871i \(0.241016\pi\)
\(74\) 11.2121 1.30338
\(75\) 6.63816 0.766508
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 1.65270 0.187132
\(79\) 10.9240 1.22904 0.614521 0.788901i \(-0.289349\pi\)
0.614521 + 0.788901i \(0.289349\pi\)
\(80\) −3.41147 −0.381414
\(81\) 1.00000 0.111111
\(82\) 3.49020 0.385428
\(83\) 11.7169 1.28609 0.643047 0.765826i \(-0.277670\pi\)
0.643047 + 0.765826i \(0.277670\pi\)
\(84\) −2.87939 −0.314167
\(85\) −23.6878 −2.56930
\(86\) 2.28312 0.246195
\(87\) −6.35504 −0.681331
\(88\) −0.347296 −0.0370219
\(89\) −1.79292 −0.190049 −0.0950245 0.995475i \(-0.530293\pi\)
−0.0950245 + 0.995475i \(0.530293\pi\)
\(90\) −3.41147 −0.359601
\(91\) −4.75877 −0.498855
\(92\) 6.80066 0.709018
\(93\) −1.59627 −0.165525
\(94\) 5.59627 0.577211
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −3.65270 −0.370876 −0.185438 0.982656i \(-0.559370\pi\)
−0.185438 + 0.982656i \(0.559370\pi\)
\(98\) 1.29086 0.130396
\(99\) −0.347296 −0.0349046
\(100\) 6.63816 0.663816
\(101\) −9.90673 −0.985756 −0.492878 0.870098i \(-0.664055\pi\)
−0.492878 + 0.870098i \(0.664055\pi\)
\(102\) 6.94356 0.687515
\(103\) −7.84524 −0.773014 −0.386507 0.922286i \(-0.626319\pi\)
−0.386507 + 0.922286i \(0.626319\pi\)
\(104\) 1.65270 0.162061
\(105\) 9.82295 0.958622
\(106\) −1.98040 −0.192353
\(107\) 0.0273411 0.00264317 0.00132158 0.999999i \(-0.499579\pi\)
0.00132158 + 0.999999i \(0.499579\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.7733 1.03190 0.515948 0.856620i \(-0.327440\pi\)
0.515948 + 0.856620i \(0.327440\pi\)
\(110\) 1.18479 0.112966
\(111\) 11.2121 1.06421
\(112\) −2.87939 −0.272076
\(113\) 11.6604 1.09692 0.548461 0.836176i \(-0.315214\pi\)
0.548461 + 0.836176i \(0.315214\pi\)
\(114\) 0 0
\(115\) −23.2003 −2.16344
\(116\) −6.35504 −0.590050
\(117\) 1.65270 0.152793
\(118\) −0.445622 −0.0410229
\(119\) −19.9932 −1.83277
\(120\) −3.41147 −0.311424
\(121\) −10.8794 −0.989035
\(122\) −12.5321 −1.13460
\(123\) 3.49020 0.314701
\(124\) −1.59627 −0.143349
\(125\) −5.58853 −0.499853
\(126\) −2.87939 −0.256516
\(127\) −16.2121 −1.43859 −0.719297 0.694703i \(-0.755536\pi\)
−0.719297 + 0.694703i \(0.755536\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.28312 0.201017
\(130\) −5.63816 −0.494499
\(131\) −2.80066 −0.244695 −0.122347 0.992487i \(-0.539042\pi\)
−0.122347 + 0.992487i \(0.539042\pi\)
\(132\) −0.347296 −0.0302283
\(133\) 0 0
\(134\) 1.07873 0.0931877
\(135\) −3.41147 −0.293613
\(136\) 6.94356 0.595406
\(137\) 16.0351 1.36997 0.684985 0.728557i \(-0.259809\pi\)
0.684985 + 0.728557i \(0.259809\pi\)
\(138\) 6.80066 0.578911
\(139\) −2.07873 −0.176315 −0.0881576 0.996107i \(-0.528098\pi\)
−0.0881576 + 0.996107i \(0.528098\pi\)
\(140\) 9.82295 0.830191
\(141\) 5.59627 0.471291
\(142\) 16.6236 1.39502
\(143\) −0.573978 −0.0479984
\(144\) 1.00000 0.0833333
\(145\) 21.6800 1.80043
\(146\) 12.4192 1.02782
\(147\) 1.29086 0.106468
\(148\) 11.2121 0.921632
\(149\) −1.66725 −0.136587 −0.0682933 0.997665i \(-0.521755\pi\)
−0.0682933 + 0.997665i \(0.521755\pi\)
\(150\) 6.63816 0.542003
\(151\) −20.0523 −1.63183 −0.815917 0.578169i \(-0.803768\pi\)
−0.815917 + 0.578169i \(0.803768\pi\)
\(152\) 0 0
\(153\) 6.94356 0.561354
\(154\) 1.00000 0.0805823
\(155\) 5.44562 0.437403
\(156\) 1.65270 0.132322
\(157\) −4.09833 −0.327082 −0.163541 0.986537i \(-0.552292\pi\)
−0.163541 + 0.986537i \(0.552292\pi\)
\(158\) 10.9240 0.869064
\(159\) −1.98040 −0.157056
\(160\) −3.41147 −0.269701
\(161\) −19.5817 −1.54326
\(162\) 1.00000 0.0785674
\(163\) −8.13516 −0.637195 −0.318598 0.947890i \(-0.603212\pi\)
−0.318598 + 0.947890i \(0.603212\pi\)
\(164\) 3.49020 0.272539
\(165\) 1.18479 0.0922360
\(166\) 11.7169 0.909406
\(167\) 13.3969 1.03669 0.518343 0.855173i \(-0.326549\pi\)
0.518343 + 0.855173i \(0.326549\pi\)
\(168\) −2.87939 −0.222149
\(169\) −10.2686 −0.789890
\(170\) −23.6878 −1.81677
\(171\) 0 0
\(172\) 2.28312 0.174086
\(173\) 1.02229 0.0777232 0.0388616 0.999245i \(-0.487627\pi\)
0.0388616 + 0.999245i \(0.487627\pi\)
\(174\) −6.35504 −0.481774
\(175\) −19.1138 −1.44487
\(176\) −0.347296 −0.0261784
\(177\) −0.445622 −0.0334950
\(178\) −1.79292 −0.134385
\(179\) −2.03508 −0.152109 −0.0760546 0.997104i \(-0.524232\pi\)
−0.0760546 + 0.997104i \(0.524232\pi\)
\(180\) −3.41147 −0.254276
\(181\) −22.3037 −1.65782 −0.828909 0.559384i \(-0.811038\pi\)
−0.828909 + 0.559384i \(0.811038\pi\)
\(182\) −4.75877 −0.352744
\(183\) −12.5321 −0.926399
\(184\) 6.80066 0.501351
\(185\) −38.2499 −2.81219
\(186\) −1.59627 −0.117044
\(187\) −2.41147 −0.176344
\(188\) 5.59627 0.408150
\(189\) −2.87939 −0.209444
\(190\) 0 0
\(191\) 10.7861 0.780456 0.390228 0.920718i \(-0.372396\pi\)
0.390228 + 0.920718i \(0.372396\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.86484 0.206216 0.103108 0.994670i \(-0.467121\pi\)
0.103108 + 0.994670i \(0.467121\pi\)
\(194\) −3.65270 −0.262249
\(195\) −5.63816 −0.403757
\(196\) 1.29086 0.0922042
\(197\) −8.33544 −0.593875 −0.296938 0.954897i \(-0.595965\pi\)
−0.296938 + 0.954897i \(0.595965\pi\)
\(198\) −0.347296 −0.0246813
\(199\) −3.03415 −0.215085 −0.107543 0.994200i \(-0.534298\pi\)
−0.107543 + 0.994200i \(0.534298\pi\)
\(200\) 6.63816 0.469388
\(201\) 1.07873 0.0760874
\(202\) −9.90673 −0.697035
\(203\) 18.2986 1.28431
\(204\) 6.94356 0.486147
\(205\) −11.9067 −0.831602
\(206\) −7.84524 −0.546604
\(207\) 6.80066 0.472679
\(208\) 1.65270 0.114594
\(209\) 0 0
\(210\) 9.82295 0.677848
\(211\) 16.6382 1.14542 0.572709 0.819759i \(-0.305893\pi\)
0.572709 + 0.819759i \(0.305893\pi\)
\(212\) −1.98040 −0.136014
\(213\) 16.6236 1.13903
\(214\) 0.0273411 0.00186900
\(215\) −7.78880 −0.531192
\(216\) 1.00000 0.0680414
\(217\) 4.59627 0.312015
\(218\) 10.7733 0.729661
\(219\) 12.4192 0.839213
\(220\) 1.18479 0.0798787
\(221\) 11.4757 0.771936
\(222\) 11.2121 0.752509
\(223\) 8.17024 0.547120 0.273560 0.961855i \(-0.411799\pi\)
0.273560 + 0.961855i \(0.411799\pi\)
\(224\) −2.87939 −0.192387
\(225\) 6.63816 0.442544
\(226\) 11.6604 0.775641
\(227\) −1.80066 −0.119514 −0.0597570 0.998213i \(-0.519033\pi\)
−0.0597570 + 0.998213i \(0.519033\pi\)
\(228\) 0 0
\(229\) −18.7392 −1.23832 −0.619160 0.785265i \(-0.712527\pi\)
−0.619160 + 0.785265i \(0.712527\pi\)
\(230\) −23.2003 −1.52978
\(231\) 1.00000 0.0657952
\(232\) −6.35504 −0.417229
\(233\) −4.09833 −0.268490 −0.134245 0.990948i \(-0.542861\pi\)
−0.134245 + 0.990948i \(0.542861\pi\)
\(234\) 1.65270 0.108041
\(235\) −19.0915 −1.24539
\(236\) −0.445622 −0.0290075
\(237\) 10.9240 0.709588
\(238\) −19.9932 −1.29597
\(239\) −28.2276 −1.82589 −0.912946 0.408080i \(-0.866199\pi\)
−0.912946 + 0.408080i \(0.866199\pi\)
\(240\) −3.41147 −0.220210
\(241\) −6.56212 −0.422703 −0.211352 0.977410i \(-0.567787\pi\)
−0.211352 + 0.977410i \(0.567787\pi\)
\(242\) −10.8794 −0.699353
\(243\) 1.00000 0.0641500
\(244\) −12.5321 −0.802285
\(245\) −4.40373 −0.281344
\(246\) 3.49020 0.222527
\(247\) 0 0
\(248\) −1.59627 −0.101363
\(249\) 11.7169 0.742527
\(250\) −5.58853 −0.353449
\(251\) 8.89393 0.561380 0.280690 0.959798i \(-0.409437\pi\)
0.280690 + 0.959798i \(0.409437\pi\)
\(252\) −2.87939 −0.181384
\(253\) −2.36184 −0.148488
\(254\) −16.2121 −1.01724
\(255\) −23.6878 −1.48339
\(256\) 1.00000 0.0625000
\(257\) 13.4662 0.839996 0.419998 0.907525i \(-0.362031\pi\)
0.419998 + 0.907525i \(0.362031\pi\)
\(258\) 2.28312 0.142141
\(259\) −32.2841 −2.00603
\(260\) −5.63816 −0.349664
\(261\) −6.35504 −0.393367
\(262\) −2.80066 −0.173025
\(263\) −16.7520 −1.03297 −0.516485 0.856296i \(-0.672760\pi\)
−0.516485 + 0.856296i \(0.672760\pi\)
\(264\) −0.347296 −0.0213746
\(265\) 6.75608 0.415023
\(266\) 0 0
\(267\) −1.79292 −0.109725
\(268\) 1.07873 0.0658937
\(269\) −3.58946 −0.218853 −0.109427 0.993995i \(-0.534901\pi\)
−0.109427 + 0.993995i \(0.534901\pi\)
\(270\) −3.41147 −0.207616
\(271\) 23.9590 1.45541 0.727704 0.685891i \(-0.240587\pi\)
0.727704 + 0.685891i \(0.240587\pi\)
\(272\) 6.94356 0.421015
\(273\) −4.75877 −0.288014
\(274\) 16.0351 0.968715
\(275\) −2.30541 −0.139021
\(276\) 6.80066 0.409352
\(277\) −18.7219 −1.12489 −0.562446 0.826834i \(-0.690140\pi\)
−0.562446 + 0.826834i \(0.690140\pi\)
\(278\) −2.07873 −0.124674
\(279\) −1.59627 −0.0955660
\(280\) 9.82295 0.587033
\(281\) 10.6382 0.634619 0.317310 0.948322i \(-0.397221\pi\)
0.317310 + 0.948322i \(0.397221\pi\)
\(282\) 5.59627 0.333253
\(283\) −14.0128 −0.832974 −0.416487 0.909142i \(-0.636739\pi\)
−0.416487 + 0.909142i \(0.636739\pi\)
\(284\) 16.6236 0.986430
\(285\) 0 0
\(286\) −0.573978 −0.0339400
\(287\) −10.0496 −0.593211
\(288\) 1.00000 0.0589256
\(289\) 31.2131 1.83606
\(290\) 21.6800 1.27310
\(291\) −3.65270 −0.214125
\(292\) 12.4192 0.726780
\(293\) 11.5202 0.673019 0.336509 0.941680i \(-0.390754\pi\)
0.336509 + 0.941680i \(0.390754\pi\)
\(294\) 1.29086 0.0752844
\(295\) 1.52023 0.0885112
\(296\) 11.2121 0.651692
\(297\) −0.347296 −0.0201522
\(298\) −1.66725 −0.0965813
\(299\) 11.2395 0.649996
\(300\) 6.63816 0.383254
\(301\) −6.57398 −0.378918
\(302\) −20.0523 −1.15388
\(303\) −9.90673 −0.569127
\(304\) 0 0
\(305\) 42.7529 2.44802
\(306\) 6.94356 0.396937
\(307\) −4.29767 −0.245281 −0.122640 0.992451i \(-0.539136\pi\)
−0.122640 + 0.992451i \(0.539136\pi\)
\(308\) 1.00000 0.0569803
\(309\) −7.84524 −0.446300
\(310\) 5.44562 0.309291
\(311\) −22.9932 −1.30382 −0.651912 0.758294i \(-0.726033\pi\)
−0.651912 + 0.758294i \(0.726033\pi\)
\(312\) 1.65270 0.0935659
\(313\) −9.20977 −0.520567 −0.260283 0.965532i \(-0.583816\pi\)
−0.260283 + 0.965532i \(0.583816\pi\)
\(314\) −4.09833 −0.231282
\(315\) 9.82295 0.553460
\(316\) 10.9240 0.614521
\(317\) −3.35504 −0.188438 −0.0942188 0.995552i \(-0.530035\pi\)
−0.0942188 + 0.995552i \(0.530035\pi\)
\(318\) −1.98040 −0.111055
\(319\) 2.20708 0.123573
\(320\) −3.41147 −0.190707
\(321\) 0.0273411 0.00152603
\(322\) −19.5817 −1.09125
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 10.9709 0.608556
\(326\) −8.13516 −0.450565
\(327\) 10.7733 0.595766
\(328\) 3.49020 0.192714
\(329\) −16.1138 −0.888383
\(330\) 1.18479 0.0652207
\(331\) 22.0743 1.21331 0.606656 0.794964i \(-0.292510\pi\)
0.606656 + 0.794964i \(0.292510\pi\)
\(332\) 11.7169 0.643047
\(333\) 11.2121 0.614421
\(334\) 13.3969 0.733047
\(335\) −3.68004 −0.201062
\(336\) −2.87939 −0.157083
\(337\) 13.1165 0.714501 0.357251 0.934009i \(-0.383714\pi\)
0.357251 + 0.934009i \(0.383714\pi\)
\(338\) −10.2686 −0.558537
\(339\) 11.6604 0.633308
\(340\) −23.6878 −1.28465
\(341\) 0.554378 0.0300212
\(342\) 0 0
\(343\) 16.4388 0.887613
\(344\) 2.28312 0.123098
\(345\) −23.2003 −1.24906
\(346\) 1.02229 0.0549586
\(347\) 8.75877 0.470195 0.235098 0.971972i \(-0.424459\pi\)
0.235098 + 0.971972i \(0.424459\pi\)
\(348\) −6.35504 −0.340666
\(349\) 3.26352 0.174692 0.0873461 0.996178i \(-0.472161\pi\)
0.0873461 + 0.996178i \(0.472161\pi\)
\(350\) −19.1138 −1.02168
\(351\) 1.65270 0.0882148
\(352\) −0.347296 −0.0185110
\(353\) 29.0838 1.54797 0.773987 0.633202i \(-0.218260\pi\)
0.773987 + 0.633202i \(0.218260\pi\)
\(354\) −0.445622 −0.0236846
\(355\) −56.7110 −3.00991
\(356\) −1.79292 −0.0950245
\(357\) −19.9932 −1.05815
\(358\) −2.03508 −0.107557
\(359\) 6.61081 0.348905 0.174453 0.984666i \(-0.444184\pi\)
0.174453 + 0.984666i \(0.444184\pi\)
\(360\) −3.41147 −0.179800
\(361\) 0 0
\(362\) −22.3037 −1.17225
\(363\) −10.8794 −0.571020
\(364\) −4.75877 −0.249427
\(365\) −42.3678 −2.21763
\(366\) −12.5321 −0.655063
\(367\) −5.79561 −0.302528 −0.151264 0.988493i \(-0.548334\pi\)
−0.151264 + 0.988493i \(0.548334\pi\)
\(368\) 6.80066 0.354509
\(369\) 3.49020 0.181693
\(370\) −38.2499 −1.98852
\(371\) 5.70233 0.296050
\(372\) −1.59627 −0.0827626
\(373\) 25.8898 1.34052 0.670262 0.742125i \(-0.266182\pi\)
0.670262 + 0.742125i \(0.266182\pi\)
\(374\) −2.41147 −0.124694
\(375\) −5.58853 −0.288590
\(376\) 5.59627 0.288605
\(377\) −10.5030 −0.540932
\(378\) −2.87939 −0.148100
\(379\) 19.1557 0.983962 0.491981 0.870606i \(-0.336273\pi\)
0.491981 + 0.870606i \(0.336273\pi\)
\(380\) 0 0
\(381\) −16.2121 −0.830573
\(382\) 10.7861 0.551865
\(383\) 9.69459 0.495371 0.247685 0.968841i \(-0.420330\pi\)
0.247685 + 0.968841i \(0.420330\pi\)
\(384\) 1.00000 0.0510310
\(385\) −3.41147 −0.173865
\(386\) 2.86484 0.145816
\(387\) 2.28312 0.116057
\(388\) −3.65270 −0.185438
\(389\) −0.817896 −0.0414690 −0.0207345 0.999785i \(-0.506600\pi\)
−0.0207345 + 0.999785i \(0.506600\pi\)
\(390\) −5.63816 −0.285499
\(391\) 47.2208 2.38806
\(392\) 1.29086 0.0651982
\(393\) −2.80066 −0.141275
\(394\) −8.33544 −0.419933
\(395\) −37.2668 −1.87510
\(396\) −0.347296 −0.0174523
\(397\) −3.69728 −0.185561 −0.0927806 0.995687i \(-0.529576\pi\)
−0.0927806 + 0.995687i \(0.529576\pi\)
\(398\) −3.03415 −0.152088
\(399\) 0 0
\(400\) 6.63816 0.331908
\(401\) 21.7401 1.08565 0.542824 0.839846i \(-0.317355\pi\)
0.542824 + 0.839846i \(0.317355\pi\)
\(402\) 1.07873 0.0538019
\(403\) −2.63816 −0.131416
\(404\) −9.90673 −0.492878
\(405\) −3.41147 −0.169518
\(406\) 18.2986 0.908144
\(407\) −3.89393 −0.193015
\(408\) 6.94356 0.343758
\(409\) 4.05468 0.200491 0.100246 0.994963i \(-0.468037\pi\)
0.100246 + 0.994963i \(0.468037\pi\)
\(410\) −11.9067 −0.588031
\(411\) 16.0351 0.790952
\(412\) −7.84524 −0.386507
\(413\) 1.28312 0.0631381
\(414\) 6.80066 0.334234
\(415\) −39.9718 −1.96214
\(416\) 1.65270 0.0810305
\(417\) −2.07873 −0.101796
\(418\) 0 0
\(419\) 23.4989 1.14800 0.573998 0.818857i \(-0.305392\pi\)
0.573998 + 0.818857i \(0.305392\pi\)
\(420\) 9.82295 0.479311
\(421\) 23.0729 1.12450 0.562251 0.826967i \(-0.309935\pi\)
0.562251 + 0.826967i \(0.309935\pi\)
\(422\) 16.6382 0.809933
\(423\) 5.59627 0.272100
\(424\) −1.98040 −0.0961767
\(425\) 46.0925 2.23581
\(426\) 16.6236 0.805416
\(427\) 36.0847 1.74626
\(428\) 0.0273411 0.00132158
\(429\) −0.573978 −0.0277119
\(430\) −7.78880 −0.375609
\(431\) 3.51485 0.169304 0.0846522 0.996411i \(-0.473022\pi\)
0.0846522 + 0.996411i \(0.473022\pi\)
\(432\) 1.00000 0.0481125
\(433\) 34.1147 1.63945 0.819725 0.572757i \(-0.194126\pi\)
0.819725 + 0.572757i \(0.194126\pi\)
\(434\) 4.59627 0.220628
\(435\) 21.6800 1.03948
\(436\) 10.7733 0.515948
\(437\) 0 0
\(438\) 12.4192 0.593413
\(439\) −9.70140 −0.463023 −0.231511 0.972832i \(-0.574367\pi\)
−0.231511 + 0.972832i \(0.574367\pi\)
\(440\) 1.18479 0.0564828
\(441\) 1.29086 0.0614695
\(442\) 11.4757 0.545841
\(443\) 15.1821 0.721324 0.360662 0.932697i \(-0.382551\pi\)
0.360662 + 0.932697i \(0.382551\pi\)
\(444\) 11.2121 0.532104
\(445\) 6.11650 0.289950
\(446\) 8.17024 0.386872
\(447\) −1.66725 −0.0788583
\(448\) −2.87939 −0.136038
\(449\) −34.8384 −1.64413 −0.822064 0.569396i \(-0.807177\pi\)
−0.822064 + 0.569396i \(0.807177\pi\)
\(450\) 6.63816 0.312926
\(451\) −1.21213 −0.0570771
\(452\) 11.6604 0.548461
\(453\) −20.0523 −0.942140
\(454\) −1.80066 −0.0845091
\(455\) 16.2344 0.761081
\(456\) 0 0
\(457\) −31.8749 −1.49105 −0.745523 0.666479i \(-0.767800\pi\)
−0.745523 + 0.666479i \(0.767800\pi\)
\(458\) −18.7392 −0.875624
\(459\) 6.94356 0.324098
\(460\) −23.2003 −1.08172
\(461\) −10.6895 −0.497862 −0.248931 0.968521i \(-0.580079\pi\)
−0.248931 + 0.968521i \(0.580079\pi\)
\(462\) 1.00000 0.0465242
\(463\) 15.3182 0.711897 0.355949 0.934506i \(-0.384158\pi\)
0.355949 + 0.934506i \(0.384158\pi\)
\(464\) −6.35504 −0.295025
\(465\) 5.44562 0.252535
\(466\) −4.09833 −0.189851
\(467\) 28.5125 1.31940 0.659700 0.751529i \(-0.270683\pi\)
0.659700 + 0.751529i \(0.270683\pi\)
\(468\) 1.65270 0.0763963
\(469\) −3.10607 −0.143425
\(470\) −19.0915 −0.880626
\(471\) −4.09833 −0.188841
\(472\) −0.445622 −0.0205114
\(473\) −0.792919 −0.0364584
\(474\) 10.9240 0.501754
\(475\) 0 0
\(476\) −19.9932 −0.916386
\(477\) −1.98040 −0.0906763
\(478\) −28.2276 −1.29110
\(479\) 8.05138 0.367877 0.183939 0.982938i \(-0.441115\pi\)
0.183939 + 0.982938i \(0.441115\pi\)
\(480\) −3.41147 −0.155712
\(481\) 18.5303 0.844911
\(482\) −6.56212 −0.298896
\(483\) −19.5817 −0.890999
\(484\) −10.8794 −0.494518
\(485\) 12.4611 0.565830
\(486\) 1.00000 0.0453609
\(487\) −6.11556 −0.277123 −0.138561 0.990354i \(-0.544248\pi\)
−0.138561 + 0.990354i \(0.544248\pi\)
\(488\) −12.5321 −0.567301
\(489\) −8.13516 −0.367885
\(490\) −4.40373 −0.198940
\(491\) −22.6851 −1.02376 −0.511882 0.859056i \(-0.671052\pi\)
−0.511882 + 0.859056i \(0.671052\pi\)
\(492\) 3.49020 0.157350
\(493\) −44.1266 −1.98736
\(494\) 0 0
\(495\) 1.18479 0.0532525
\(496\) −1.59627 −0.0716745
\(497\) −47.8658 −2.14707
\(498\) 11.7169 0.525046
\(499\) 24.8557 1.11269 0.556346 0.830951i \(-0.312203\pi\)
0.556346 + 0.830951i \(0.312203\pi\)
\(500\) −5.58853 −0.249926
\(501\) 13.3969 0.598531
\(502\) 8.89393 0.396956
\(503\) 25.0232 1.11573 0.557865 0.829932i \(-0.311621\pi\)
0.557865 + 0.829932i \(0.311621\pi\)
\(504\) −2.87939 −0.128258
\(505\) 33.7965 1.50393
\(506\) −2.36184 −0.104997
\(507\) −10.2686 −0.456043
\(508\) −16.2121 −0.719297
\(509\) −0.899310 −0.0398612 −0.0199306 0.999801i \(-0.506345\pi\)
−0.0199306 + 0.999801i \(0.506345\pi\)
\(510\) −23.6878 −1.04891
\(511\) −35.7597 −1.58192
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 13.4662 0.593967
\(515\) 26.7638 1.17935
\(516\) 2.28312 0.100509
\(517\) −1.94356 −0.0854778
\(518\) −32.2841 −1.41848
\(519\) 1.02229 0.0448735
\(520\) −5.63816 −0.247249
\(521\) 6.63547 0.290705 0.145353 0.989380i \(-0.453568\pi\)
0.145353 + 0.989380i \(0.453568\pi\)
\(522\) −6.35504 −0.278152
\(523\) −37.9522 −1.65954 −0.829768 0.558109i \(-0.811527\pi\)
−0.829768 + 0.558109i \(0.811527\pi\)
\(524\) −2.80066 −0.122347
\(525\) −19.1138 −0.834195
\(526\) −16.7520 −0.730420
\(527\) −11.0838 −0.482817
\(528\) −0.347296 −0.0151141
\(529\) 23.2490 1.01082
\(530\) 6.75608 0.293465
\(531\) −0.445622 −0.0193384
\(532\) 0 0
\(533\) 5.76827 0.249851
\(534\) −1.79292 −0.0775872
\(535\) −0.0932736 −0.00403257
\(536\) 1.07873 0.0465939
\(537\) −2.03508 −0.0878203
\(538\) −3.58946 −0.154753
\(539\) −0.448311 −0.0193101
\(540\) −3.41147 −0.146806
\(541\) 15.1307 0.650520 0.325260 0.945625i \(-0.394548\pi\)
0.325260 + 0.945625i \(0.394548\pi\)
\(542\) 23.9590 1.02913
\(543\) −22.3037 −0.957141
\(544\) 6.94356 0.297703
\(545\) −36.7529 −1.57432
\(546\) −4.75877 −0.203657
\(547\) −32.5868 −1.39331 −0.696655 0.717406i \(-0.745329\pi\)
−0.696655 + 0.717406i \(0.745329\pi\)
\(548\) 16.0351 0.684985
\(549\) −12.5321 −0.534857
\(550\) −2.30541 −0.0983029
\(551\) 0 0
\(552\) 6.80066 0.289455
\(553\) −31.4543 −1.33757
\(554\) −18.7219 −0.795419
\(555\) −38.2499 −1.62362
\(556\) −2.07873 −0.0881576
\(557\) −19.9436 −0.845036 −0.422518 0.906355i \(-0.638854\pi\)
−0.422518 + 0.906355i \(0.638854\pi\)
\(558\) −1.59627 −0.0675754
\(559\) 3.77332 0.159594
\(560\) 9.82295 0.415095
\(561\) −2.41147 −0.101813
\(562\) 10.6382 0.448744
\(563\) −3.96822 −0.167240 −0.0836202 0.996498i \(-0.526648\pi\)
−0.0836202 + 0.996498i \(0.526648\pi\)
\(564\) 5.59627 0.235645
\(565\) −39.7793 −1.67353
\(566\) −14.0128 −0.589002
\(567\) −2.87939 −0.120923
\(568\) 16.6236 0.697511
\(569\) −14.8135 −0.621012 −0.310506 0.950571i \(-0.600499\pi\)
−0.310506 + 0.950571i \(0.600499\pi\)
\(570\) 0 0
\(571\) 21.0615 0.881396 0.440698 0.897655i \(-0.354731\pi\)
0.440698 + 0.897655i \(0.354731\pi\)
\(572\) −0.573978 −0.0239992
\(573\) 10.7861 0.450596
\(574\) −10.0496 −0.419463
\(575\) 45.1438 1.88263
\(576\) 1.00000 0.0416667
\(577\) −44.2422 −1.84183 −0.920913 0.389769i \(-0.872555\pi\)
−0.920913 + 0.389769i \(0.872555\pi\)
\(578\) 31.2131 1.29829
\(579\) 2.86484 0.119059
\(580\) 21.6800 0.900215
\(581\) −33.7374 −1.39966
\(582\) −3.65270 −0.151409
\(583\) 0.687786 0.0284852
\(584\) 12.4192 0.513911
\(585\) −5.63816 −0.233109
\(586\) 11.5202 0.475896
\(587\) 10.0024 0.412842 0.206421 0.978463i \(-0.433818\pi\)
0.206421 + 0.978463i \(0.433818\pi\)
\(588\) 1.29086 0.0532341
\(589\) 0 0
\(590\) 1.52023 0.0625869
\(591\) −8.33544 −0.342874
\(592\) 11.2121 0.460816
\(593\) −11.4911 −0.471884 −0.235942 0.971767i \(-0.575818\pi\)
−0.235942 + 0.971767i \(0.575818\pi\)
\(594\) −0.347296 −0.0142497
\(595\) 68.2063 2.79618
\(596\) −1.66725 −0.0682933
\(597\) −3.03415 −0.124179
\(598\) 11.2395 0.459616
\(599\) −6.41653 −0.262172 −0.131086 0.991371i \(-0.541846\pi\)
−0.131086 + 0.991371i \(0.541846\pi\)
\(600\) 6.63816 0.271002
\(601\) −31.5158 −1.28556 −0.642778 0.766053i \(-0.722218\pi\)
−0.642778 + 0.766053i \(0.722218\pi\)
\(602\) −6.57398 −0.267935
\(603\) 1.07873 0.0439291
\(604\) −20.0523 −0.815917
\(605\) 37.1147 1.50893
\(606\) −9.90673 −0.402433
\(607\) −0.748341 −0.0303742 −0.0151871 0.999885i \(-0.504834\pi\)
−0.0151871 + 0.999885i \(0.504834\pi\)
\(608\) 0 0
\(609\) 18.2986 0.741497
\(610\) 42.7529 1.73101
\(611\) 9.24897 0.374173
\(612\) 6.94356 0.280677
\(613\) −30.6878 −1.23947 −0.619734 0.784812i \(-0.712760\pi\)
−0.619734 + 0.784812i \(0.712760\pi\)
\(614\) −4.29767 −0.173440
\(615\) −11.9067 −0.480126
\(616\) 1.00000 0.0402911
\(617\) −27.2918 −1.09873 −0.549363 0.835584i \(-0.685130\pi\)
−0.549363 + 0.835584i \(0.685130\pi\)
\(618\) −7.84524 −0.315582
\(619\) −13.2371 −0.532044 −0.266022 0.963967i \(-0.585709\pi\)
−0.266022 + 0.963967i \(0.585709\pi\)
\(620\) 5.44562 0.218701
\(621\) 6.80066 0.272901
\(622\) −22.9932 −0.921943
\(623\) 5.16250 0.206831
\(624\) 1.65270 0.0661611
\(625\) −14.1257 −0.565027
\(626\) −9.20977 −0.368096
\(627\) 0 0
\(628\) −4.09833 −0.163541
\(629\) 77.8522 3.10417
\(630\) 9.82295 0.391356
\(631\) −12.2395 −0.487246 −0.243623 0.969870i \(-0.578336\pi\)
−0.243623 + 0.969870i \(0.578336\pi\)
\(632\) 10.9240 0.434532
\(633\) 16.6382 0.661307
\(634\) −3.35504 −0.133246
\(635\) 55.3073 2.19480
\(636\) −1.98040 −0.0785280
\(637\) 2.13341 0.0845287
\(638\) 2.20708 0.0873792
\(639\) 16.6236 0.657620
\(640\) −3.41147 −0.134850
\(641\) −8.95367 −0.353649 −0.176824 0.984242i \(-0.556582\pi\)
−0.176824 + 0.984242i \(0.556582\pi\)
\(642\) 0.0273411 0.00107907
\(643\) −31.3696 −1.23710 −0.618548 0.785747i \(-0.712279\pi\)
−0.618548 + 0.785747i \(0.712279\pi\)
\(644\) −19.5817 −0.771628
\(645\) −7.78880 −0.306684
\(646\) 0 0
\(647\) −5.54933 −0.218166 −0.109083 0.994033i \(-0.534792\pi\)
−0.109083 + 0.994033i \(0.534792\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0.154763 0.00607498
\(650\) 10.9709 0.430314
\(651\) 4.59627 0.180142
\(652\) −8.13516 −0.318598
\(653\) 38.9103 1.52268 0.761340 0.648353i \(-0.224542\pi\)
0.761340 + 0.648353i \(0.224542\pi\)
\(654\) 10.7733 0.421270
\(655\) 9.55438 0.373320
\(656\) 3.49020 0.136269
\(657\) 12.4192 0.484520
\(658\) −16.1138 −0.628182
\(659\) 39.5158 1.53932 0.769658 0.638456i \(-0.220427\pi\)
0.769658 + 0.638456i \(0.220427\pi\)
\(660\) 1.18479 0.0461180
\(661\) 0.0915189 0.00355967 0.00177984 0.999998i \(-0.499433\pi\)
0.00177984 + 0.999998i \(0.499433\pi\)
\(662\) 22.0743 0.857941
\(663\) 11.4757 0.445677
\(664\) 11.7169 0.454703
\(665\) 0 0
\(666\) 11.2121 0.434461
\(667\) −43.2184 −1.67342
\(668\) 13.3969 0.518343
\(669\) 8.17024 0.315880
\(670\) −3.68004 −0.142173
\(671\) 4.35235 0.168021
\(672\) −2.87939 −0.111075
\(673\) 39.8563 1.53635 0.768173 0.640242i \(-0.221166\pi\)
0.768173 + 0.640242i \(0.221166\pi\)
\(674\) 13.1165 0.505229
\(675\) 6.63816 0.255503
\(676\) −10.2686 −0.394945
\(677\) −29.3824 −1.12926 −0.564628 0.825345i \(-0.690980\pi\)
−0.564628 + 0.825345i \(0.690980\pi\)
\(678\) 11.6604 0.447817
\(679\) 10.5175 0.403626
\(680\) −23.6878 −0.908385
\(681\) −1.80066 −0.0690014
\(682\) 0.554378 0.0212282
\(683\) 21.0933 0.807112 0.403556 0.914955i \(-0.367774\pi\)
0.403556 + 0.914955i \(0.367774\pi\)
\(684\) 0 0
\(685\) −54.7033 −2.09010
\(686\) 16.4388 0.627637
\(687\) −18.7392 −0.714944
\(688\) 2.28312 0.0870431
\(689\) −3.27301 −0.124692
\(690\) −23.2003 −0.883219
\(691\) −16.1634 −0.614886 −0.307443 0.951566i \(-0.599473\pi\)
−0.307443 + 0.951566i \(0.599473\pi\)
\(692\) 1.02229 0.0388616
\(693\) 1.00000 0.0379869
\(694\) 8.75877 0.332478
\(695\) 7.09152 0.268997
\(696\) −6.35504 −0.240887
\(697\) 24.2344 0.917944
\(698\) 3.26352 0.123526
\(699\) −4.09833 −0.155013
\(700\) −19.1138 −0.722434
\(701\) −20.0729 −0.758141 −0.379071 0.925368i \(-0.623756\pi\)
−0.379071 + 0.925368i \(0.623756\pi\)
\(702\) 1.65270 0.0623773
\(703\) 0 0
\(704\) −0.347296 −0.0130892
\(705\) −19.0915 −0.719028
\(706\) 29.0838 1.09458
\(707\) 28.5253 1.07280
\(708\) −0.445622 −0.0167475
\(709\) −38.9009 −1.46095 −0.730476 0.682938i \(-0.760702\pi\)
−0.730476 + 0.682938i \(0.760702\pi\)
\(710\) −56.7110 −2.12833
\(711\) 10.9240 0.409681
\(712\) −1.79292 −0.0671925
\(713\) −10.8557 −0.406548
\(714\) −19.9932 −0.748226
\(715\) 1.95811 0.0732292
\(716\) −2.03508 −0.0760546
\(717\) −28.2276 −1.05418
\(718\) 6.61081 0.246713
\(719\) 3.06181 0.114186 0.0570932 0.998369i \(-0.481817\pi\)
0.0570932 + 0.998369i \(0.481817\pi\)
\(720\) −3.41147 −0.127138
\(721\) 22.5895 0.841275
\(722\) 0 0
\(723\) −6.56212 −0.244048
\(724\) −22.3037 −0.828909
\(725\) −42.1857 −1.56674
\(726\) −10.8794 −0.403772
\(727\) −17.4766 −0.648171 −0.324085 0.946028i \(-0.605056\pi\)
−0.324085 + 0.946028i \(0.605056\pi\)
\(728\) −4.75877 −0.176372
\(729\) 1.00000 0.0370370
\(730\) −42.3678 −1.56810
\(731\) 15.8530 0.586344
\(732\) −12.5321 −0.463199
\(733\) −41.4175 −1.52979 −0.764894 0.644156i \(-0.777209\pi\)
−0.764894 + 0.644156i \(0.777209\pi\)
\(734\) −5.79561 −0.213920
\(735\) −4.40373 −0.162434
\(736\) 6.80066 0.250676
\(737\) −0.374638 −0.0137999
\(738\) 3.49020 0.128476
\(739\) 26.4534 0.973103 0.486551 0.873652i \(-0.338255\pi\)
0.486551 + 0.873652i \(0.338255\pi\)
\(740\) −38.2499 −1.40609
\(741\) 0 0
\(742\) 5.70233 0.209339
\(743\) 15.0760 0.553086 0.276543 0.961002i \(-0.410811\pi\)
0.276543 + 0.961002i \(0.410811\pi\)
\(744\) −1.59627 −0.0585220
\(745\) 5.68779 0.208384
\(746\) 25.8898 0.947893
\(747\) 11.7169 0.428698
\(748\) −2.41147 −0.0881722
\(749\) −0.0787257 −0.00287657
\(750\) −5.58853 −0.204064
\(751\) 5.30541 0.193597 0.0967985 0.995304i \(-0.469140\pi\)
0.0967985 + 0.995304i \(0.469140\pi\)
\(752\) 5.59627 0.204075
\(753\) 8.89393 0.324113
\(754\) −10.5030 −0.382496
\(755\) 68.4080 2.48962
\(756\) −2.87939 −0.104722
\(757\) 8.13846 0.295797 0.147899 0.989003i \(-0.452749\pi\)
0.147899 + 0.989003i \(0.452749\pi\)
\(758\) 19.1557 0.695766
\(759\) −2.36184 −0.0857295
\(760\) 0 0
\(761\) −46.2113 −1.67516 −0.837579 0.546316i \(-0.816030\pi\)
−0.837579 + 0.546316i \(0.816030\pi\)
\(762\) −16.2121 −0.587304
\(763\) −31.0205 −1.12302
\(764\) 10.7861 0.390228
\(765\) −23.6878 −0.856434
\(766\) 9.69459 0.350280
\(767\) −0.736482 −0.0265928
\(768\) 1.00000 0.0360844
\(769\) −26.8435 −0.968001 −0.484000 0.875068i \(-0.660817\pi\)
−0.484000 + 0.875068i \(0.660817\pi\)
\(770\) −3.41147 −0.122941
\(771\) 13.4662 0.484972
\(772\) 2.86484 0.103108
\(773\) 21.0256 0.756238 0.378119 0.925757i \(-0.376571\pi\)
0.378119 + 0.925757i \(0.376571\pi\)
\(774\) 2.28312 0.0820650
\(775\) −10.5963 −0.380629
\(776\) −3.65270 −0.131124
\(777\) −32.2841 −1.15818
\(778\) −0.817896 −0.0293230
\(779\) 0 0
\(780\) −5.63816 −0.201878
\(781\) −5.77332 −0.206586
\(782\) 47.2208 1.68861
\(783\) −6.35504 −0.227110
\(784\) 1.29086 0.0461021
\(785\) 13.9813 0.499015
\(786\) −2.80066 −0.0998962
\(787\) −45.5313 −1.62301 −0.811507 0.584342i \(-0.801353\pi\)
−0.811507 + 0.584342i \(0.801353\pi\)
\(788\) −8.33544 −0.296938
\(789\) −16.7520 −0.596386
\(790\) −37.2668 −1.32589
\(791\) −33.5749 −1.19379
\(792\) −0.347296 −0.0123406
\(793\) −20.7118 −0.735499
\(794\) −3.69728 −0.131212
\(795\) 6.75608 0.239614
\(796\) −3.03415 −0.107543
\(797\) −36.1780 −1.28149 −0.640745 0.767754i \(-0.721374\pi\)
−0.640745 + 0.767754i \(0.721374\pi\)
\(798\) 0 0
\(799\) 38.8580 1.37470
\(800\) 6.63816 0.234694
\(801\) −1.79292 −0.0633497
\(802\) 21.7401 0.767670
\(803\) −4.31315 −0.152208
\(804\) 1.07873 0.0380437
\(805\) 66.8025 2.35448
\(806\) −2.63816 −0.0929251
\(807\) −3.58946 −0.126355
\(808\) −9.90673 −0.348517
\(809\) 2.37876 0.0836326 0.0418163 0.999125i \(-0.486686\pi\)
0.0418163 + 0.999125i \(0.486686\pi\)
\(810\) −3.41147 −0.119867
\(811\) 37.0610 1.30139 0.650694 0.759340i \(-0.274478\pi\)
0.650694 + 0.759340i \(0.274478\pi\)
\(812\) 18.2986 0.642155
\(813\) 23.9590 0.840280
\(814\) −3.89393 −0.136482
\(815\) 27.7529 0.972142
\(816\) 6.94356 0.243073
\(817\) 0 0
\(818\) 4.05468 0.141769
\(819\) −4.75877 −0.166285
\(820\) −11.9067 −0.415801
\(821\) −25.9605 −0.906027 −0.453013 0.891504i \(-0.649651\pi\)
−0.453013 + 0.891504i \(0.649651\pi\)
\(822\) 16.0351 0.559288
\(823\) −45.0110 −1.56899 −0.784493 0.620138i \(-0.787077\pi\)
−0.784493 + 0.620138i \(0.787077\pi\)
\(824\) −7.84524 −0.273302
\(825\) −2.30541 −0.0802640
\(826\) 1.28312 0.0446454
\(827\) 16.8429 0.585684 0.292842 0.956161i \(-0.405399\pi\)
0.292842 + 0.956161i \(0.405399\pi\)
\(828\) 6.80066 0.236339
\(829\) 13.0000 0.451509 0.225754 0.974184i \(-0.427515\pi\)
0.225754 + 0.974184i \(0.427515\pi\)
\(830\) −39.9718 −1.38744
\(831\) −18.7219 −0.649457
\(832\) 1.65270 0.0572972
\(833\) 8.96316 0.310555
\(834\) −2.07873 −0.0719804
\(835\) −45.7033 −1.58163
\(836\) 0 0
\(837\) −1.59627 −0.0551750
\(838\) 23.4989 0.811755
\(839\) −1.34049 −0.0462788 −0.0231394 0.999732i \(-0.507366\pi\)
−0.0231394 + 0.999732i \(0.507366\pi\)
\(840\) 9.82295 0.338924
\(841\) 11.3865 0.392638
\(842\) 23.0729 0.795143
\(843\) 10.6382 0.366398
\(844\) 16.6382 0.572709
\(845\) 35.0310 1.20510
\(846\) 5.59627 0.192404
\(847\) 31.3259 1.07637
\(848\) −1.98040 −0.0680072
\(849\) −14.0128 −0.480918
\(850\) 46.0925 1.58096
\(851\) 76.2499 2.61381
\(852\) 16.6236 0.569515
\(853\) −7.94532 −0.272042 −0.136021 0.990706i \(-0.543432\pi\)
−0.136021 + 0.990706i \(0.543432\pi\)
\(854\) 36.0847 1.23479
\(855\) 0 0
\(856\) 0.0273411 0.000934501 0
\(857\) −47.8759 −1.63541 −0.817704 0.575638i \(-0.804754\pi\)
−0.817704 + 0.575638i \(0.804754\pi\)
\(858\) −0.573978 −0.0195953
\(859\) 1.63722 0.0558613 0.0279306 0.999610i \(-0.491108\pi\)
0.0279306 + 0.999610i \(0.491108\pi\)
\(860\) −7.78880 −0.265596
\(861\) −10.0496 −0.342490
\(862\) 3.51485 0.119716
\(863\) −34.9231 −1.18880 −0.594399 0.804170i \(-0.702610\pi\)
−0.594399 + 0.804170i \(0.702610\pi\)
\(864\) 1.00000 0.0340207
\(865\) −3.48751 −0.118579
\(866\) 34.1147 1.15927
\(867\) 31.2131 1.06005
\(868\) 4.59627 0.156007
\(869\) −3.79385 −0.128698
\(870\) 21.6800 0.735022
\(871\) 1.78281 0.0604083
\(872\) 10.7733 0.364831
\(873\) −3.65270 −0.123625
\(874\) 0 0
\(875\) 16.0915 0.543993
\(876\) 12.4192 0.419606
\(877\) 6.37195 0.215165 0.107583 0.994196i \(-0.465689\pi\)
0.107583 + 0.994196i \(0.465689\pi\)
\(878\) −9.70140 −0.327406
\(879\) 11.5202 0.388568
\(880\) 1.18479 0.0399393
\(881\) 5.27395 0.177684 0.0888419 0.996046i \(-0.471683\pi\)
0.0888419 + 0.996046i \(0.471683\pi\)
\(882\) 1.29086 0.0434655
\(883\) −3.25133 −0.109416 −0.0547081 0.998502i \(-0.517423\pi\)
−0.0547081 + 0.998502i \(0.517423\pi\)
\(884\) 11.4757 0.385968
\(885\) 1.52023 0.0511019
\(886\) 15.1821 0.510053
\(887\) 22.3705 0.751129 0.375564 0.926796i \(-0.377449\pi\)
0.375564 + 0.926796i \(0.377449\pi\)
\(888\) 11.2121 0.376255
\(889\) 46.6810 1.56563
\(890\) 6.11650 0.205025
\(891\) −0.347296 −0.0116349
\(892\) 8.17024 0.273560
\(893\) 0 0
\(894\) −1.66725 −0.0557612
\(895\) 6.94263 0.232067
\(896\) −2.87939 −0.0961935
\(897\) 11.2395 0.375275
\(898\) −34.8384 −1.16257
\(899\) 10.1443 0.338332
\(900\) 6.63816 0.221272
\(901\) −13.7510 −0.458113
\(902\) −1.21213 −0.0403596
\(903\) −6.57398 −0.218768
\(904\) 11.6604 0.387821
\(905\) 76.0883 2.52926
\(906\) −20.0523 −0.666194
\(907\) −3.15745 −0.104841 −0.0524207 0.998625i \(-0.516694\pi\)
−0.0524207 + 0.998625i \(0.516694\pi\)
\(908\) −1.80066 −0.0597570
\(909\) −9.90673 −0.328585
\(910\) 16.2344 0.538166
\(911\) 44.3387 1.46901 0.734504 0.678604i \(-0.237415\pi\)
0.734504 + 0.678604i \(0.237415\pi\)
\(912\) 0 0
\(913\) −4.06923 −0.134672
\(914\) −31.8749 −1.05433
\(915\) 42.7529 1.41337
\(916\) −18.7392 −0.619160
\(917\) 8.06418 0.266303
\(918\) 6.94356 0.229172
\(919\) 14.2044 0.468560 0.234280 0.972169i \(-0.424727\pi\)
0.234280 + 0.972169i \(0.424727\pi\)
\(920\) −23.2003 −0.764890
\(921\) −4.29767 −0.141613
\(922\) −10.6895 −0.352041
\(923\) 27.4739 0.904314
\(924\) 1.00000 0.0328976
\(925\) 74.4279 2.44717
\(926\) 15.3182 0.503387
\(927\) −7.84524 −0.257671
\(928\) −6.35504 −0.208614
\(929\) −15.4266 −0.506131 −0.253066 0.967449i \(-0.581439\pi\)
−0.253066 + 0.967449i \(0.581439\pi\)
\(930\) 5.44562 0.178569
\(931\) 0 0
\(932\) −4.09833 −0.134245
\(933\) −22.9932 −0.752763
\(934\) 28.5125 0.932957
\(935\) 8.22668 0.269041
\(936\) 1.65270 0.0540203
\(937\) −9.69696 −0.316786 −0.158393 0.987376i \(-0.550631\pi\)
−0.158393 + 0.987376i \(0.550631\pi\)
\(938\) −3.10607 −0.101417
\(939\) −9.20977 −0.300549
\(940\) −19.0915 −0.622697
\(941\) 17.9855 0.586309 0.293154 0.956065i \(-0.405295\pi\)
0.293154 + 0.956065i \(0.405295\pi\)
\(942\) −4.09833 −0.133531
\(943\) 23.7357 0.772939
\(944\) −0.445622 −0.0145038
\(945\) 9.82295 0.319541
\(946\) −0.792919 −0.0257800
\(947\) 2.19396 0.0712942 0.0356471 0.999364i \(-0.488651\pi\)
0.0356471 + 0.999364i \(0.488651\pi\)
\(948\) 10.9240 0.354794
\(949\) 20.5253 0.666279
\(950\) 0 0
\(951\) −3.35504 −0.108795
\(952\) −19.9932 −0.647983
\(953\) 11.3108 0.366392 0.183196 0.983076i \(-0.441356\pi\)
0.183196 + 0.983076i \(0.441356\pi\)
\(954\) −1.98040 −0.0641178
\(955\) −36.7965 −1.19071
\(956\) −28.2276 −0.912946
\(957\) 2.20708 0.0713448
\(958\) 8.05138 0.260128
\(959\) −46.1712 −1.49095
\(960\) −3.41147 −0.110105
\(961\) −28.4519 −0.917804
\(962\) 18.5303 0.597442
\(963\) 0.0273411 0.000881056 0
\(964\) −6.56212 −0.211352
\(965\) −9.77332 −0.314614
\(966\) −19.5817 −0.630031
\(967\) 46.7844 1.50448 0.752242 0.658887i \(-0.228973\pi\)
0.752242 + 0.658887i \(0.228973\pi\)
\(968\) −10.8794 −0.349677
\(969\) 0 0
\(970\) 12.4611 0.400102
\(971\) −60.3715 −1.93741 −0.968706 0.248211i \(-0.920157\pi\)
−0.968706 + 0.248211i \(0.920157\pi\)
\(972\) 1.00000 0.0320750
\(973\) 5.98545 0.191885
\(974\) −6.11556 −0.195955
\(975\) 10.9709 0.351350
\(976\) −12.5321 −0.401142
\(977\) −22.7956 −0.729296 −0.364648 0.931145i \(-0.618811\pi\)
−0.364648 + 0.931145i \(0.618811\pi\)
\(978\) −8.13516 −0.260134
\(979\) 0.622674 0.0199008
\(980\) −4.40373 −0.140672
\(981\) 10.7733 0.343966
\(982\) −22.6851 −0.723911
\(983\) 30.7929 0.982142 0.491071 0.871120i \(-0.336606\pi\)
0.491071 + 0.871120i \(0.336606\pi\)
\(984\) 3.49020 0.111264
\(985\) 28.4361 0.906050
\(986\) −44.1266 −1.40528
\(987\) −16.1138 −0.512908
\(988\) 0 0
\(989\) 15.5267 0.493721
\(990\) 1.18479 0.0376552
\(991\) 38.1789 1.21279 0.606397 0.795162i \(-0.292614\pi\)
0.606397 + 0.795162i \(0.292614\pi\)
\(992\) −1.59627 −0.0506815
\(993\) 22.0743 0.700506
\(994\) −47.8658 −1.51821
\(995\) 10.3509 0.328146
\(996\) 11.7169 0.371264
\(997\) 30.9489 0.980163 0.490081 0.871677i \(-0.336967\pi\)
0.490081 + 0.871677i \(0.336967\pi\)
\(998\) 24.8557 0.786792
\(999\) 11.2121 0.354736
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 2166.2.a.u.1.1 3
3.2 odd 2 6498.2.a.bn.1.3 3
19.3 odd 18 114.2.i.d.85.1 yes 6
19.13 odd 18 114.2.i.d.55.1 6
19.18 odd 2 2166.2.a.o.1.1 3
57.32 even 18 342.2.u.a.55.1 6
57.41 even 18 342.2.u.a.199.1 6
57.56 even 2 6498.2.a.bs.1.3 3
76.3 even 18 912.2.bo.f.769.1 6
76.51 even 18 912.2.bo.f.625.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.55.1 6 19.13 odd 18
114.2.i.d.85.1 yes 6 19.3 odd 18
342.2.u.a.55.1 6 57.32 even 18
342.2.u.a.199.1 6 57.41 even 18
912.2.bo.f.625.1 6 76.51 even 18
912.2.bo.f.769.1 6 76.3 even 18
2166.2.a.o.1.1 3 19.18 odd 2
2166.2.a.u.1.1 3 1.1 even 1 trivial
6498.2.a.bn.1.3 3 3.2 odd 2
6498.2.a.bs.1.3 3 57.56 even 2