Properties

Label 231.2.a.d.1.1
Level $231$
Weight $2$
Character 231.1
Self dual yes
Analytic conductor $1.845$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \(x^{3} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.36147\) of defining polynomial
Character \(\chi\) \(=\) 231.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.36147 q^{2} -1.00000 q^{3} +3.57653 q^{4} -3.93800 q^{5} +2.36147 q^{6} -1.00000 q^{7} -3.72294 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.36147 q^{2} -1.00000 q^{3} +3.57653 q^{4} -3.93800 q^{5} +2.36147 q^{6} -1.00000 q^{7} -3.72294 q^{8} +1.00000 q^{9} +9.29947 q^{10} +1.00000 q^{11} -3.57653 q^{12} -3.93800 q^{13} +2.36147 q^{14} +3.93800 q^{15} +1.63853 q^{16} +4.72294 q^{17} -2.36147 q^{18} +4.78493 q^{19} -14.0844 q^{20} +1.00000 q^{21} -2.36147 q^{22} +2.72294 q^{23} +3.72294 q^{24} +10.5079 q^{25} +9.29947 q^{26} -1.00000 q^{27} -3.57653 q^{28} +7.93800 q^{29} -9.29947 q^{30} +1.15307 q^{31} +3.57653 q^{32} -1.00000 q^{33} -11.1531 q^{34} +3.93800 q^{35} +3.57653 q^{36} -5.50787 q^{37} -11.2995 q^{38} +3.93800 q^{39} +14.6609 q^{40} +0.430132 q^{41} -2.36147 q^{42} +6.72294 q^{43} +3.57653 q^{44} -3.93800 q^{45} -6.43013 q^{46} -8.78493 q^{47} -1.63853 q^{48} +1.00000 q^{49} -24.8140 q^{50} -4.72294 q^{51} -14.0844 q^{52} -3.15307 q^{53} +2.36147 q^{54} -3.93800 q^{55} +3.72294 q^{56} -4.78493 q^{57} -18.7453 q^{58} -15.0911 q^{59} +14.0844 q^{60} +6.00000 q^{61} -2.72294 q^{62} -1.00000 q^{63} -11.7229 q^{64} +15.5079 q^{65} +2.36147 q^{66} +3.21507 q^{67} +16.8918 q^{68} -2.72294 q^{69} -9.29947 q^{70} +13.4459 q^{71} -3.72294 q^{72} +11.9380 q^{73} +13.0067 q^{74} -10.5079 q^{75} +17.1135 q^{76} -1.00000 q^{77} -9.29947 q^{78} -5.44588 q^{79} -6.45254 q^{80} +1.00000 q^{81} -1.01574 q^{82} -2.84693 q^{83} +3.57653 q^{84} -18.5989 q^{85} -15.8760 q^{86} -7.93800 q^{87} -3.72294 q^{88} +12.3061 q^{89} +9.29947 q^{90} +3.93800 q^{91} +9.73868 q^{92} -1.15307 q^{93} +20.7453 q^{94} -18.8431 q^{95} -3.57653 q^{96} +11.1531 q^{97} -2.36147 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 9 q^{10} + 3 q^{11} - 6 q^{12} + 12 q^{16} + 12 q^{19} - 21 q^{20} + 3 q^{21} - 6 q^{23} - 3 q^{24} + 15 q^{25} + 9 q^{26} - 3 q^{27} - 6 q^{28} + 12 q^{29} - 9 q^{30} - 6 q^{31} + 6 q^{32} - 3 q^{33} - 24 q^{34} + 6 q^{36} - 15 q^{38} + 18 q^{40} + 6 q^{41} + 6 q^{43} + 6 q^{44} - 24 q^{46} - 24 q^{47} - 12 q^{48} + 3 q^{49} - 39 q^{50} - 21 q^{52} - 3 q^{56} - 12 q^{57} - 9 q^{58} - 24 q^{59} + 21 q^{60} + 18 q^{61} + 6 q^{62} - 3 q^{63} - 21 q^{64} + 30 q^{65} + 12 q^{67} - 6 q^{68} + 6 q^{69} - 9 q^{70} + 12 q^{71} + 3 q^{72} + 24 q^{73} + 39 q^{74} - 15 q^{75} - 3 q^{76} - 3 q^{77} - 9 q^{78} + 12 q^{79} + 9 q^{80} + 3 q^{81} + 30 q^{82} - 18 q^{83} + 6 q^{84} - 18 q^{85} - 24 q^{86} - 12 q^{87} + 3 q^{88} + 18 q^{89} + 9 q^{90} - 18 q^{92} + 6 q^{93} + 15 q^{94} + 12 q^{95} - 6 q^{96} + 24 q^{97} + 3 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.36147 −1.66981 −0.834905 0.550394i \(-0.814478\pi\)
−0.834905 + 0.550394i \(0.814478\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.57653 1.78827
\(5\) −3.93800 −1.76113 −0.880564 0.473927i \(-0.842836\pi\)
−0.880564 + 0.473927i \(0.842836\pi\)
\(6\) 2.36147 0.964066
\(7\) −1.00000 −0.377964
\(8\) −3.72294 −1.31626
\(9\) 1.00000 0.333333
\(10\) 9.29947 2.94075
\(11\) 1.00000 0.301511
\(12\) −3.57653 −1.03246
\(13\) −3.93800 −1.09221 −0.546103 0.837718i \(-0.683889\pi\)
−0.546103 + 0.837718i \(0.683889\pi\)
\(14\) 2.36147 0.631129
\(15\) 3.93800 1.01679
\(16\) 1.63853 0.409633
\(17\) 4.72294 1.14548 0.572740 0.819737i \(-0.305880\pi\)
0.572740 + 0.819737i \(0.305880\pi\)
\(18\) −2.36147 −0.556604
\(19\) 4.78493 1.09774 0.548870 0.835908i \(-0.315058\pi\)
0.548870 + 0.835908i \(0.315058\pi\)
\(20\) −14.0844 −3.14937
\(21\) 1.00000 0.218218
\(22\) −2.36147 −0.503467
\(23\) 2.72294 0.567772 0.283886 0.958858i \(-0.408376\pi\)
0.283886 + 0.958858i \(0.408376\pi\)
\(24\) 3.72294 0.759941
\(25\) 10.5079 2.10157
\(26\) 9.29947 1.82378
\(27\) −1.00000 −0.192450
\(28\) −3.57653 −0.675902
\(29\) 7.93800 1.47405 0.737025 0.675865i \(-0.236230\pi\)
0.737025 + 0.675865i \(0.236230\pi\)
\(30\) −9.29947 −1.69784
\(31\) 1.15307 0.207097 0.103549 0.994624i \(-0.466980\pi\)
0.103549 + 0.994624i \(0.466980\pi\)
\(32\) 3.57653 0.632248
\(33\) −1.00000 −0.174078
\(34\) −11.1531 −1.91274
\(35\) 3.93800 0.665644
\(36\) 3.57653 0.596089
\(37\) −5.50787 −0.905489 −0.452744 0.891640i \(-0.649555\pi\)
−0.452744 + 0.891640i \(0.649555\pi\)
\(38\) −11.2995 −1.83302
\(39\) 3.93800 0.630585
\(40\) 14.6609 2.31810
\(41\) 0.430132 0.0671753 0.0335877 0.999436i \(-0.489307\pi\)
0.0335877 + 0.999436i \(0.489307\pi\)
\(42\) −2.36147 −0.364383
\(43\) 6.72294 1.02524 0.512619 0.858616i \(-0.328675\pi\)
0.512619 + 0.858616i \(0.328675\pi\)
\(44\) 3.57653 0.539183
\(45\) −3.93800 −0.587043
\(46\) −6.43013 −0.948071
\(47\) −8.78493 −1.28141 −0.640707 0.767785i \(-0.721359\pi\)
−0.640707 + 0.767785i \(0.721359\pi\)
\(48\) −1.63853 −0.236502
\(49\) 1.00000 0.142857
\(50\) −24.8140 −3.50923
\(51\) −4.72294 −0.661344
\(52\) −14.0844 −1.95316
\(53\) −3.15307 −0.433107 −0.216554 0.976271i \(-0.569482\pi\)
−0.216554 + 0.976271i \(0.569482\pi\)
\(54\) 2.36147 0.321355
\(55\) −3.93800 −0.531000
\(56\) 3.72294 0.497498
\(57\) −4.78493 −0.633780
\(58\) −18.7453 −2.46138
\(59\) −15.0911 −1.96469 −0.982345 0.187077i \(-0.940098\pi\)
−0.982345 + 0.187077i \(0.940098\pi\)
\(60\) 14.0844 1.81829
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −2.72294 −0.345813
\(63\) −1.00000 −0.125988
\(64\) −11.7229 −1.46537
\(65\) 15.5079 1.92351
\(66\) 2.36147 0.290677
\(67\) 3.21507 0.392783 0.196391 0.980526i \(-0.437078\pi\)
0.196391 + 0.980526i \(0.437078\pi\)
\(68\) 16.8918 2.04843
\(69\) −2.72294 −0.327803
\(70\) −9.29947 −1.11150
\(71\) 13.4459 1.59573 0.797866 0.602835i \(-0.205962\pi\)
0.797866 + 0.602835i \(0.205962\pi\)
\(72\) −3.72294 −0.438752
\(73\) 11.9380 1.39724 0.698619 0.715494i \(-0.253798\pi\)
0.698619 + 0.715494i \(0.253798\pi\)
\(74\) 13.0067 1.51199
\(75\) −10.5079 −1.21334
\(76\) 17.1135 1.96305
\(77\) −1.00000 −0.113961
\(78\) −9.29947 −1.05296
\(79\) −5.44588 −0.612709 −0.306354 0.951918i \(-0.599109\pi\)
−0.306354 + 0.951918i \(0.599109\pi\)
\(80\) −6.45254 −0.721416
\(81\) 1.00000 0.111111
\(82\) −1.01574 −0.112170
\(83\) −2.84693 −0.312491 −0.156246 0.987718i \(-0.549939\pi\)
−0.156246 + 0.987718i \(0.549939\pi\)
\(84\) 3.57653 0.390232
\(85\) −18.5989 −2.01734
\(86\) −15.8760 −1.71195
\(87\) −7.93800 −0.851043
\(88\) −3.72294 −0.396866
\(89\) 12.3061 1.30445 0.652224 0.758026i \(-0.273836\pi\)
0.652224 + 0.758026i \(0.273836\pi\)
\(90\) 9.29947 0.980250
\(91\) 3.93800 0.412815
\(92\) 9.73868 1.01533
\(93\) −1.15307 −0.119568
\(94\) 20.7453 2.13972
\(95\) −18.8431 −1.93326
\(96\) −3.57653 −0.365029
\(97\) 11.1531 1.13242 0.566211 0.824260i \(-0.308409\pi\)
0.566211 + 0.824260i \(0.308409\pi\)
\(98\) −2.36147 −0.238544
\(99\) 1.00000 0.100504
\(100\) 37.5818 3.75818
\(101\) 16.4750 1.63932 0.819659 0.572851i \(-0.194163\pi\)
0.819659 + 0.572851i \(0.194163\pi\)
\(102\) 11.1531 1.10432
\(103\) −1.56987 −0.154684 −0.0773418 0.997005i \(-0.524643\pi\)
−0.0773418 + 0.997005i \(0.524643\pi\)
\(104\) 14.6609 1.43762
\(105\) −3.93800 −0.384310
\(106\) 7.44588 0.723207
\(107\) −4.78493 −0.462577 −0.231289 0.972885i \(-0.574294\pi\)
−0.231289 + 0.972885i \(0.574294\pi\)
\(108\) −3.57653 −0.344152
\(109\) −15.0291 −1.43952 −0.719762 0.694221i \(-0.755749\pi\)
−0.719762 + 0.694221i \(0.755749\pi\)
\(110\) 9.29947 0.886670
\(111\) 5.50787 0.522784
\(112\) −1.63853 −0.154827
\(113\) 7.44588 0.700449 0.350225 0.936666i \(-0.386105\pi\)
0.350225 + 0.936666i \(0.386105\pi\)
\(114\) 11.2995 1.05829
\(115\) −10.7229 −0.999919
\(116\) 28.3905 2.63600
\(117\) −3.93800 −0.364069
\(118\) 35.6371 3.28066
\(119\) −4.72294 −0.432951
\(120\) −14.6609 −1.33835
\(121\) 1.00000 0.0909091
\(122\) −14.1688 −1.28278
\(123\) −0.430132 −0.0387837
\(124\) 4.12399 0.370346
\(125\) −21.6900 −1.94001
\(126\) 2.36147 0.210376
\(127\) 12.2928 1.09081 0.545405 0.838173i \(-0.316376\pi\)
0.545405 + 0.838173i \(0.316376\pi\)
\(128\) 20.5303 1.81464
\(129\) −6.72294 −0.591922
\(130\) −36.6214 −3.21191
\(131\) 7.13974 0.623802 0.311901 0.950115i \(-0.399034\pi\)
0.311901 + 0.950115i \(0.399034\pi\)
\(132\) −3.57653 −0.311297
\(133\) −4.78493 −0.414906
\(134\) −7.59228 −0.655873
\(135\) 3.93800 0.338929
\(136\) −17.5832 −1.50775
\(137\) 1.58320 0.135262 0.0676310 0.997710i \(-0.478456\pi\)
0.0676310 + 0.997710i \(0.478456\pi\)
\(138\) 6.43013 0.547369
\(139\) −17.1979 −1.45871 −0.729353 0.684138i \(-0.760179\pi\)
−0.729353 + 0.684138i \(0.760179\pi\)
\(140\) 14.0844 1.19035
\(141\) 8.78493 0.739825
\(142\) −31.7520 −2.66457
\(143\) −3.93800 −0.329312
\(144\) 1.63853 0.136544
\(145\) −31.2599 −2.59599
\(146\) −28.1912 −2.33312
\(147\) −1.00000 −0.0824786
\(148\) −19.6991 −1.61926
\(149\) 15.9380 1.30569 0.652846 0.757491i \(-0.273575\pi\)
0.652846 + 0.757491i \(0.273575\pi\)
\(150\) 24.8140 2.02606
\(151\) 6.59894 0.537014 0.268507 0.963278i \(-0.413470\pi\)
0.268507 + 0.963278i \(0.413470\pi\)
\(152\) −17.8140 −1.44491
\(153\) 4.72294 0.381827
\(154\) 2.36147 0.190293
\(155\) −4.54079 −0.364725
\(156\) 14.0844 1.12765
\(157\) 7.32188 0.584350 0.292175 0.956365i \(-0.405621\pi\)
0.292175 + 0.956365i \(0.405621\pi\)
\(158\) 12.8603 1.02311
\(159\) 3.15307 0.250055
\(160\) −14.0844 −1.11347
\(161\) −2.72294 −0.214598
\(162\) −2.36147 −0.185535
\(163\) 14.1068 1.10493 0.552466 0.833536i \(-0.313687\pi\)
0.552466 + 0.833536i \(0.313687\pi\)
\(164\) 1.53838 0.120127
\(165\) 3.93800 0.306573
\(166\) 6.72294 0.521801
\(167\) −0.416799 −0.0322528 −0.0161264 0.999870i \(-0.505133\pi\)
−0.0161264 + 0.999870i \(0.505133\pi\)
\(168\) −3.72294 −0.287231
\(169\) 2.50787 0.192913
\(170\) 43.9208 3.36857
\(171\) 4.78493 0.365913
\(172\) 24.0448 1.83340
\(173\) −8.43013 −0.640931 −0.320466 0.947260i \(-0.603839\pi\)
−0.320466 + 0.947260i \(0.603839\pi\)
\(174\) 18.7453 1.42108
\(175\) −10.5079 −0.794320
\(176\) 1.63853 0.123509
\(177\) 15.0911 1.13431
\(178\) −29.0606 −2.17818
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −14.0844 −1.04979
\(181\) 20.5989 1.53111 0.765554 0.643372i \(-0.222465\pi\)
0.765554 + 0.643372i \(0.222465\pi\)
\(182\) −9.29947 −0.689323
\(183\) −6.00000 −0.443533
\(184\) −10.1373 −0.747334
\(185\) 21.6900 1.59468
\(186\) 2.72294 0.199655
\(187\) 4.72294 0.345375
\(188\) −31.4196 −2.29151
\(189\) 1.00000 0.0727393
\(190\) 44.4974 3.22818
\(191\) −14.5989 −1.05634 −0.528171 0.849138i \(-0.677122\pi\)
−0.528171 + 0.849138i \(0.677122\pi\)
\(192\) 11.7229 0.846030
\(193\) 13.4592 0.968815 0.484408 0.874842i \(-0.339035\pi\)
0.484408 + 0.874842i \(0.339035\pi\)
\(194\) −26.3376 −1.89093
\(195\) −15.5079 −1.11054
\(196\) 3.57653 0.255467
\(197\) 11.4459 0.815485 0.407742 0.913097i \(-0.366316\pi\)
0.407742 + 0.913097i \(0.366316\pi\)
\(198\) −2.36147 −0.167822
\(199\) −13.0291 −0.923607 −0.461803 0.886982i \(-0.652797\pi\)
−0.461803 + 0.886982i \(0.652797\pi\)
\(200\) −39.1201 −2.76621
\(201\) −3.21507 −0.226773
\(202\) −38.9051 −2.73735
\(203\) −7.93800 −0.557139
\(204\) −16.8918 −1.18266
\(205\) −1.69386 −0.118304
\(206\) 3.70719 0.258292
\(207\) 2.72294 0.189257
\(208\) −6.45254 −0.447403
\(209\) 4.78493 0.330981
\(210\) 9.29947 0.641725
\(211\) 2.43013 0.167297 0.0836486 0.996495i \(-0.473343\pi\)
0.0836486 + 0.996495i \(0.473343\pi\)
\(212\) −11.2771 −0.774512
\(213\) −13.4459 −0.921296
\(214\) 11.2995 0.772416
\(215\) −26.4750 −1.80558
\(216\) 3.72294 0.253314
\(217\) −1.15307 −0.0782755
\(218\) 35.4907 2.40373
\(219\) −11.9380 −0.806696
\(220\) −14.0844 −0.949570
\(221\) −18.5989 −1.25110
\(222\) −13.0067 −0.872950
\(223\) 22.7678 1.52464 0.762321 0.647199i \(-0.224060\pi\)
0.762321 + 0.647199i \(0.224060\pi\)
\(224\) −3.57653 −0.238967
\(225\) 10.5079 0.700525
\(226\) −17.5832 −1.16962
\(227\) −15.8760 −1.05373 −0.526864 0.849950i \(-0.676632\pi\)
−0.526864 + 0.849950i \(0.676632\pi\)
\(228\) −17.1135 −1.13337
\(229\) −3.15307 −0.208361 −0.104180 0.994558i \(-0.533222\pi\)
−0.104180 + 0.994558i \(0.533222\pi\)
\(230\) 25.3219 1.66968
\(231\) 1.00000 0.0657952
\(232\) −29.5527 −1.94023
\(233\) 13.1397 0.860813 0.430406 0.902635i \(-0.358370\pi\)
0.430406 + 0.902635i \(0.358370\pi\)
\(234\) 9.29947 0.607926
\(235\) 34.5951 2.25674
\(236\) −53.9737 −3.51339
\(237\) 5.44588 0.353748
\(238\) 11.1531 0.722946
\(239\) −10.1068 −0.653756 −0.326878 0.945067i \(-0.605997\pi\)
−0.326878 + 0.945067i \(0.605997\pi\)
\(240\) 6.45254 0.416510
\(241\) 2.36814 0.152545 0.0762725 0.997087i \(-0.475698\pi\)
0.0762725 + 0.997087i \(0.475698\pi\)
\(242\) −2.36147 −0.151801
\(243\) −1.00000 −0.0641500
\(244\) 21.4592 1.37379
\(245\) −3.93800 −0.251590
\(246\) 1.01574 0.0647614
\(247\) −18.8431 −1.19896
\(248\) −4.29281 −0.272593
\(249\) 2.84693 0.180417
\(250\) 51.2203 3.23946
\(251\) 10.2308 0.645763 0.322881 0.946439i \(-0.395348\pi\)
0.322881 + 0.946439i \(0.395348\pi\)
\(252\) −3.57653 −0.225301
\(253\) 2.72294 0.171190
\(254\) −29.0291 −1.82145
\(255\) 18.5989 1.16471
\(256\) −25.0357 −1.56473
\(257\) −11.0777 −0.691010 −0.345505 0.938417i \(-0.612292\pi\)
−0.345505 + 0.938417i \(0.612292\pi\)
\(258\) 15.8760 0.988397
\(259\) 5.50787 0.342242
\(260\) 55.4644 3.43976
\(261\) 7.93800 0.491350
\(262\) −16.8603 −1.04163
\(263\) −8.78493 −0.541702 −0.270851 0.962621i \(-0.587305\pi\)
−0.270851 + 0.962621i \(0.587305\pi\)
\(264\) 3.72294 0.229131
\(265\) 12.4168 0.762758
\(266\) 11.2995 0.692815
\(267\) −12.3061 −0.753123
\(268\) 11.4988 0.702401
\(269\) 5.75201 0.350706 0.175353 0.984506i \(-0.443893\pi\)
0.175353 + 0.984506i \(0.443893\pi\)
\(270\) −9.29947 −0.565948
\(271\) 0.784934 0.0476813 0.0238407 0.999716i \(-0.492411\pi\)
0.0238407 + 0.999716i \(0.492411\pi\)
\(272\) 7.73868 0.469226
\(273\) −3.93800 −0.238339
\(274\) −3.73868 −0.225862
\(275\) 10.5079 0.633648
\(276\) −9.73868 −0.586200
\(277\) −11.5699 −0.695166 −0.347583 0.937649i \(-0.612998\pi\)
−0.347583 + 0.937649i \(0.612998\pi\)
\(278\) 40.6123 2.43576
\(279\) 1.15307 0.0690325
\(280\) −14.6609 −0.876159
\(281\) −25.2599 −1.50688 −0.753439 0.657518i \(-0.771607\pi\)
−0.753439 + 0.657518i \(0.771607\pi\)
\(282\) −20.7453 −1.23537
\(283\) −10.9671 −0.651925 −0.325963 0.945383i \(-0.605688\pi\)
−0.325963 + 0.945383i \(0.605688\pi\)
\(284\) 48.0896 2.85360
\(285\) 18.8431 1.11617
\(286\) 9.29947 0.549889
\(287\) −0.430132 −0.0253899
\(288\) 3.57653 0.210749
\(289\) 5.30614 0.312126
\(290\) 73.8192 4.33482
\(291\) −11.1531 −0.653805
\(292\) 42.6967 2.49863
\(293\) 14.7363 0.860902 0.430451 0.902614i \(-0.358355\pi\)
0.430451 + 0.902614i \(0.358355\pi\)
\(294\) 2.36147 0.137724
\(295\) 59.4287 3.46007
\(296\) 20.5055 1.19186
\(297\) −1.00000 −0.0580259
\(298\) −37.6371 −2.18026
\(299\) −10.7229 −0.620123
\(300\) −37.5818 −2.16978
\(301\) −6.72294 −0.387504
\(302\) −15.5832 −0.896712
\(303\) −16.4750 −0.946461
\(304\) 7.84026 0.449670
\(305\) −23.6280 −1.35294
\(306\) −11.1531 −0.637579
\(307\) 0.860264 0.0490979 0.0245489 0.999699i \(-0.492185\pi\)
0.0245489 + 0.999699i \(0.492185\pi\)
\(308\) −3.57653 −0.203792
\(309\) 1.56987 0.0893067
\(310\) 10.7229 0.609022
\(311\) 26.8918 1.52489 0.762446 0.647052i \(-0.223998\pi\)
0.762446 + 0.647052i \(0.223998\pi\)
\(312\) −14.6609 −0.830012
\(313\) 10.1688 0.574775 0.287388 0.957814i \(-0.407213\pi\)
0.287388 + 0.957814i \(0.407213\pi\)
\(314\) −17.2904 −0.975753
\(315\) 3.93800 0.221881
\(316\) −19.4774 −1.09569
\(317\) −27.4907 −1.54403 −0.772016 0.635604i \(-0.780751\pi\)
−0.772016 + 0.635604i \(0.780751\pi\)
\(318\) −7.44588 −0.417544
\(319\) 7.93800 0.444443
\(320\) 46.1650 2.58070
\(321\) 4.78493 0.267069
\(322\) 6.43013 0.358337
\(323\) 22.5989 1.25744
\(324\) 3.57653 0.198696
\(325\) −41.3800 −2.29535
\(326\) −33.3128 −1.84503
\(327\) 15.0291 0.831110
\(328\) −1.60135 −0.0884200
\(329\) 8.78493 0.484329
\(330\) −9.29947 −0.511919
\(331\) −17.1979 −0.945281 −0.472641 0.881255i \(-0.656699\pi\)
−0.472641 + 0.881255i \(0.656699\pi\)
\(332\) −10.1821 −0.558818
\(333\) −5.50787 −0.301830
\(334\) 0.984257 0.0538561
\(335\) −12.6609 −0.691741
\(336\) 1.63853 0.0893892
\(337\) 19.7387 1.07523 0.537617 0.843189i \(-0.319325\pi\)
0.537617 + 0.843189i \(0.319325\pi\)
\(338\) −5.92226 −0.322128
\(339\) −7.44588 −0.404404
\(340\) −66.5198 −3.60754
\(341\) 1.15307 0.0624422
\(342\) −11.2995 −0.611005
\(343\) −1.00000 −0.0539949
\(344\) −25.0291 −1.34948
\(345\) 10.7229 0.577304
\(346\) 19.9075 1.07023
\(347\) 16.6123 0.891794 0.445897 0.895084i \(-0.352885\pi\)
0.445897 + 0.895084i \(0.352885\pi\)
\(348\) −28.3905 −1.52189
\(349\) 6.95375 0.372226 0.186113 0.982528i \(-0.440411\pi\)
0.186113 + 0.982528i \(0.440411\pi\)
\(350\) 24.8140 1.32636
\(351\) 3.93800 0.210195
\(352\) 3.57653 0.190630
\(353\) −3.22840 −0.171830 −0.0859152 0.996302i \(-0.527381\pi\)
−0.0859152 + 0.996302i \(0.527381\pi\)
\(354\) −35.6371 −1.89409
\(355\) −52.9499 −2.81029
\(356\) 44.0133 2.33270
\(357\) 4.72294 0.249964
\(358\) 28.3376 1.49769
\(359\) 10.3061 0.543937 0.271969 0.962306i \(-0.412325\pi\)
0.271969 + 0.962306i \(0.412325\pi\)
\(360\) 14.6609 0.772699
\(361\) 3.89559 0.205031
\(362\) −48.6438 −2.55666
\(363\) −1.00000 −0.0524864
\(364\) 14.0844 0.738223
\(365\) −47.0119 −2.46072
\(366\) 14.1688 0.740616
\(367\) −11.8760 −0.619923 −0.309961 0.950749i \(-0.600316\pi\)
−0.309961 + 0.950749i \(0.600316\pi\)
\(368\) 4.46162 0.232578
\(369\) 0.430132 0.0223918
\(370\) −51.2203 −2.66282
\(371\) 3.15307 0.163699
\(372\) −4.12399 −0.213819
\(373\) 16.7229 0.865881 0.432940 0.901423i \(-0.357476\pi\)
0.432940 + 0.901423i \(0.357476\pi\)
\(374\) −11.1531 −0.576711
\(375\) 21.6900 1.12007
\(376\) 32.7058 1.68667
\(377\) −31.2599 −1.60997
\(378\) −2.36147 −0.121461
\(379\) −10.9671 −0.563341 −0.281671 0.959511i \(-0.590889\pi\)
−0.281671 + 0.959511i \(0.590889\pi\)
\(380\) −67.3930 −3.45719
\(381\) −12.2928 −0.629780
\(382\) 34.4750 1.76389
\(383\) 15.7520 0.804890 0.402445 0.915444i \(-0.368160\pi\)
0.402445 + 0.915444i \(0.368160\pi\)
\(384\) −20.5303 −1.04768
\(385\) 3.93800 0.200699
\(386\) −31.7835 −1.61774
\(387\) 6.72294 0.341746
\(388\) 39.8893 2.02507
\(389\) 36.7678 1.86420 0.932100 0.362202i \(-0.117975\pi\)
0.932100 + 0.362202i \(0.117975\pi\)
\(390\) 36.6214 1.85439
\(391\) 12.8603 0.650371
\(392\) −3.72294 −0.188037
\(393\) −7.13974 −0.360152
\(394\) −27.0291 −1.36171
\(395\) 21.4459 1.07906
\(396\) 3.57653 0.179728
\(397\) 11.8627 0.595371 0.297685 0.954664i \(-0.403785\pi\)
0.297685 + 0.954664i \(0.403785\pi\)
\(398\) 30.7678 1.54225
\(399\) 4.78493 0.239546
\(400\) 17.2175 0.860874
\(401\) 3.40106 0.169841 0.0849203 0.996388i \(-0.472936\pi\)
0.0849203 + 0.996388i \(0.472936\pi\)
\(402\) 7.59228 0.378668
\(403\) −4.54079 −0.226193
\(404\) 58.9232 2.93154
\(405\) −3.93800 −0.195681
\(406\) 18.7453 0.930316
\(407\) −5.50787 −0.273015
\(408\) 17.5832 0.870498
\(409\) −28.0315 −1.38607 −0.693034 0.720905i \(-0.743726\pi\)
−0.693034 + 0.720905i \(0.743726\pi\)
\(410\) 4.00000 0.197546
\(411\) −1.58320 −0.0780936
\(412\) −5.61469 −0.276616
\(413\) 15.0911 0.742583
\(414\) −6.43013 −0.316024
\(415\) 11.2112 0.550337
\(416\) −14.0844 −0.690545
\(417\) 17.1979 0.842184
\(418\) −11.2995 −0.552675
\(419\) 28.2890 1.38201 0.691003 0.722852i \(-0.257169\pi\)
0.691003 + 0.722852i \(0.257169\pi\)
\(420\) −14.0844 −0.687249
\(421\) −14.4921 −0.706303 −0.353152 0.935566i \(-0.614890\pi\)
−0.353152 + 0.935566i \(0.614890\pi\)
\(422\) −5.73868 −0.279355
\(423\) −8.78493 −0.427138
\(424\) 11.7387 0.570081
\(425\) 49.6280 2.40731
\(426\) 31.7520 1.53839
\(427\) −6.00000 −0.290360
\(428\) −17.1135 −0.827211
\(429\) 3.93800 0.190129
\(430\) 62.5198 3.01497
\(431\) −24.5369 −1.18190 −0.590952 0.806707i \(-0.701248\pi\)
−0.590952 + 0.806707i \(0.701248\pi\)
\(432\) −1.63853 −0.0788339
\(433\) −7.32188 −0.351867 −0.175934 0.984402i \(-0.556294\pi\)
−0.175934 + 0.984402i \(0.556294\pi\)
\(434\) 2.72294 0.130705
\(435\) 31.2599 1.49880
\(436\) −53.7520 −2.57425
\(437\) 13.0291 0.623265
\(438\) 28.1912 1.34703
\(439\) 16.5369 0.789265 0.394633 0.918839i \(-0.370872\pi\)
0.394633 + 0.918839i \(0.370872\pi\)
\(440\) 14.6609 0.698933
\(441\) 1.00000 0.0476190
\(442\) 43.9208 2.08910
\(443\) 25.1979 1.19719 0.598594 0.801053i \(-0.295726\pi\)
0.598594 + 0.801053i \(0.295726\pi\)
\(444\) 19.6991 0.934878
\(445\) −48.4616 −2.29730
\(446\) −53.7653 −2.54586
\(447\) −15.9380 −0.753842
\(448\) 11.7229 0.553857
\(449\) −36.5198 −1.72347 −0.861737 0.507355i \(-0.830623\pi\)
−0.861737 + 0.507355i \(0.830623\pi\)
\(450\) −24.8140 −1.16974
\(451\) 0.430132 0.0202541
\(452\) 26.6304 1.25259
\(453\) −6.59894 −0.310045
\(454\) 37.4907 1.75953
\(455\) −15.5079 −0.727020
\(456\) 17.8140 0.834217
\(457\) 3.15307 0.147494 0.0737472 0.997277i \(-0.476504\pi\)
0.0737472 + 0.997277i \(0.476504\pi\)
\(458\) 7.44588 0.347923
\(459\) −4.72294 −0.220448
\(460\) −38.3510 −1.78812
\(461\) −24.7678 −1.15355 −0.576775 0.816903i \(-0.695689\pi\)
−0.576775 + 0.816903i \(0.695689\pi\)
\(462\) −2.36147 −0.109865
\(463\) −35.4287 −1.64651 −0.823256 0.567671i \(-0.807845\pi\)
−0.823256 + 0.567671i \(0.807845\pi\)
\(464\) 13.0067 0.603819
\(465\) 4.54079 0.210574
\(466\) −31.0291 −1.43739
\(467\) −23.9247 −1.10710 −0.553551 0.832815i \(-0.686728\pi\)
−0.553551 + 0.832815i \(0.686728\pi\)
\(468\) −14.0844 −0.651052
\(469\) −3.21507 −0.148458
\(470\) −81.6953 −3.76832
\(471\) −7.32188 −0.337375
\(472\) 56.1831 2.58604
\(473\) 6.72294 0.309121
\(474\) −12.8603 −0.590691
\(475\) 50.2795 2.30698
\(476\) −16.8918 −0.774232
\(477\) −3.15307 −0.144369
\(478\) 23.8669 1.09165
\(479\) 4.54079 0.207474 0.103737 0.994605i \(-0.466920\pi\)
0.103737 + 0.994605i \(0.466920\pi\)
\(480\) 14.0844 0.642862
\(481\) 21.6900 0.988980
\(482\) −5.59228 −0.254721
\(483\) 2.72294 0.123898
\(484\) 3.57653 0.162570
\(485\) −43.9208 −1.99434
\(486\) 2.36147 0.107118
\(487\) 27.1397 1.22982 0.614909 0.788598i \(-0.289193\pi\)
0.614909 + 0.788598i \(0.289193\pi\)
\(488\) −22.3376 −1.01118
\(489\) −14.1068 −0.637932
\(490\) 9.29947 0.420107
\(491\) 3.46305 0.156285 0.0781427 0.996942i \(-0.475101\pi\)
0.0781427 + 0.996942i \(0.475101\pi\)
\(492\) −1.53838 −0.0693556
\(493\) 37.4907 1.68850
\(494\) 44.4974 2.00203
\(495\) −3.93800 −0.177000
\(496\) 1.88934 0.0848339
\(497\) −13.4459 −0.603130
\(498\) −6.72294 −0.301262
\(499\) −18.9671 −0.849083 −0.424542 0.905408i \(-0.639565\pi\)
−0.424542 + 0.905408i \(0.639565\pi\)
\(500\) −77.5751 −3.46926
\(501\) 0.416799 0.0186212
\(502\) −24.1597 −1.07830
\(503\) 5.86267 0.261404 0.130702 0.991422i \(-0.458277\pi\)
0.130702 + 0.991422i \(0.458277\pi\)
\(504\) 3.72294 0.165833
\(505\) −64.8784 −2.88705
\(506\) −6.43013 −0.285854
\(507\) −2.50787 −0.111378
\(508\) 43.9656 1.95066
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) −43.9208 −1.94485
\(511\) −11.9380 −0.528106
\(512\) 18.0606 0.798172
\(513\) −4.78493 −0.211260
\(514\) 26.1597 1.15386
\(515\) 6.18215 0.272418
\(516\) −24.0448 −1.05851
\(517\) −8.78493 −0.386361
\(518\) −13.0067 −0.571480
\(519\) 8.43013 0.370042
\(520\) −57.7348 −2.53184
\(521\) −2.24414 −0.0983177 −0.0491588 0.998791i \(-0.515654\pi\)
−0.0491588 + 0.998791i \(0.515654\pi\)
\(522\) −18.7453 −0.820462
\(523\) −17.9828 −0.786334 −0.393167 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(524\) 25.5355 1.11552
\(525\) 10.5079 0.458601
\(526\) 20.7453 0.904540
\(527\) 5.44588 0.237226
\(528\) −1.63853 −0.0713079
\(529\) −15.5856 −0.677635
\(530\) −29.3219 −1.27366
\(531\) −15.0911 −0.654897
\(532\) −17.1135 −0.741964
\(533\) −1.69386 −0.0733693
\(534\) 29.0606 1.25757
\(535\) 18.8431 0.814658
\(536\) −11.9695 −0.517003
\(537\) 12.0000 0.517838
\(538\) −13.5832 −0.585613
\(539\) 1.00000 0.0430730
\(540\) 14.0844 0.606096
\(541\) 28.7678 1.23682 0.618411 0.785855i \(-0.287777\pi\)
0.618411 + 0.785855i \(0.287777\pi\)
\(542\) −1.85360 −0.0796188
\(543\) −20.5989 −0.883985
\(544\) 16.8918 0.724228
\(545\) 59.1846 2.53519
\(546\) 9.29947 0.397981
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 5.66237 0.241885
\(549\) 6.00000 0.256074
\(550\) −24.8140 −1.05807
\(551\) 37.9828 1.61812
\(552\) 10.1373 0.431473
\(553\) 5.44588 0.231582
\(554\) 27.3219 1.16080
\(555\) −21.6900 −0.920690
\(556\) −61.5088 −2.60856
\(557\) −13.5079 −0.572347 −0.286173 0.958178i \(-0.592383\pi\)
−0.286173 + 0.958178i \(0.592383\pi\)
\(558\) −2.72294 −0.115271
\(559\) −26.4750 −1.11977
\(560\) 6.45254 0.272670
\(561\) −4.72294 −0.199403
\(562\) 59.6504 2.51620
\(563\) 24.7811 1.04440 0.522199 0.852824i \(-0.325112\pi\)
0.522199 + 0.852824i \(0.325112\pi\)
\(564\) 31.4196 1.32300
\(565\) −29.3219 −1.23358
\(566\) 25.8984 1.08859
\(567\) −1.00000 −0.0419961
\(568\) −50.0582 −2.10039
\(569\) 1.75201 0.0734482 0.0367241 0.999325i \(-0.488308\pi\)
0.0367241 + 0.999325i \(0.488308\pi\)
\(570\) −44.4974 −1.86379
\(571\) −3.16640 −0.132510 −0.0662549 0.997803i \(-0.521105\pi\)
−0.0662549 + 0.997803i \(0.521105\pi\)
\(572\) −14.0844 −0.588899
\(573\) 14.5989 0.609880
\(574\) 1.01574 0.0423963
\(575\) 28.6123 1.19321
\(576\) −11.7229 −0.488456
\(577\) 20.4750 0.852383 0.426192 0.904633i \(-0.359855\pi\)
0.426192 + 0.904633i \(0.359855\pi\)
\(578\) −12.5303 −0.521191
\(579\) −13.4592 −0.559346
\(580\) −111.802 −4.64233
\(581\) 2.84693 0.118111
\(582\) 26.3376 1.09173
\(583\) −3.15307 −0.130587
\(584\) −44.4444 −1.83912
\(585\) 15.5079 0.641172
\(586\) −34.7992 −1.43754
\(587\) −16.6609 −0.687671 −0.343835 0.939030i \(-0.611726\pi\)
−0.343835 + 0.939030i \(0.611726\pi\)
\(588\) −3.57653 −0.147494
\(589\) 5.51736 0.227339
\(590\) −140.339 −5.77767
\(591\) −11.4459 −0.470820
\(592\) −9.02482 −0.370918
\(593\) 27.9075 1.14602 0.573012 0.819547i \(-0.305775\pi\)
0.573012 + 0.819547i \(0.305775\pi\)
\(594\) 2.36147 0.0968922
\(595\) 18.5989 0.762482
\(596\) 57.0028 2.33493
\(597\) 13.0291 0.533245
\(598\) 25.3219 1.03549
\(599\) −24.3376 −0.994408 −0.497204 0.867634i \(-0.665640\pi\)
−0.497204 + 0.867634i \(0.665640\pi\)
\(600\) 39.1201 1.59707
\(601\) −9.50787 −0.387834 −0.193917 0.981018i \(-0.562119\pi\)
−0.193917 + 0.981018i \(0.562119\pi\)
\(602\) 15.8760 0.647058
\(603\) 3.21507 0.130928
\(604\) 23.6014 0.960325
\(605\) −3.93800 −0.160103
\(606\) 38.9051 1.58041
\(607\) 26.8431 1.08953 0.544764 0.838590i \(-0.316619\pi\)
0.544764 + 0.838590i \(0.316619\pi\)
\(608\) 17.1135 0.694043
\(609\) 7.93800 0.321664
\(610\) 55.7968 2.25915
\(611\) 34.5951 1.39957
\(612\) 16.8918 0.682809
\(613\) −23.0291 −0.930136 −0.465068 0.885275i \(-0.653970\pi\)
−0.465068 + 0.885275i \(0.653970\pi\)
\(614\) −2.03149 −0.0819841
\(615\) 1.69386 0.0683031
\(616\) 3.72294 0.150001
\(617\) 31.4459 1.26596 0.632982 0.774167i \(-0.281831\pi\)
0.632982 + 0.774167i \(0.281831\pi\)
\(618\) −3.70719 −0.149125
\(619\) 2.26132 0.0908901 0.0454450 0.998967i \(-0.485529\pi\)
0.0454450 + 0.998967i \(0.485529\pi\)
\(620\) −16.2403 −0.652226
\(621\) −2.72294 −0.109268
\(622\) −63.5040 −2.54628
\(623\) −12.3061 −0.493035
\(624\) 6.45254 0.258308
\(625\) 32.8760 1.31504
\(626\) −24.0133 −0.959766
\(627\) −4.78493 −0.191092
\(628\) 26.1870 1.04497
\(629\) −26.0133 −1.03722
\(630\) −9.29947 −0.370500
\(631\) 10.3061 0.410281 0.205140 0.978733i \(-0.434235\pi\)
0.205140 + 0.978733i \(0.434235\pi\)
\(632\) 20.2747 0.806482
\(633\) −2.43013 −0.0965891
\(634\) 64.9184 2.57824
\(635\) −48.4091 −1.92106
\(636\) 11.2771 0.447165
\(637\) −3.93800 −0.156029
\(638\) −18.7453 −0.742135
\(639\) 13.4459 0.531911
\(640\) −80.8483 −3.19581
\(641\) 4.13733 0.163415 0.0817073 0.996656i \(-0.473963\pi\)
0.0817073 + 0.996656i \(0.473963\pi\)
\(642\) −11.2995 −0.445955
\(643\) −8.44347 −0.332978 −0.166489 0.986043i \(-0.553243\pi\)
−0.166489 + 0.986043i \(0.553243\pi\)
\(644\) −9.73868 −0.383758
\(645\) 26.4750 1.04245
\(646\) −53.3667 −2.09968
\(647\) −25.2732 −0.993593 −0.496796 0.867867i \(-0.665490\pi\)
−0.496796 + 0.867867i \(0.665490\pi\)
\(648\) −3.72294 −0.146251
\(649\) −15.0911 −0.592376
\(650\) 97.7177 3.83280
\(651\) 1.15307 0.0451924
\(652\) 50.4535 1.97591
\(653\) 47.9075 1.87477 0.937383 0.348302i \(-0.113241\pi\)
0.937383 + 0.348302i \(0.113241\pi\)
\(654\) −35.4907 −1.38780
\(655\) −28.1163 −1.09859
\(656\) 0.704785 0.0275172
\(657\) 11.9380 0.465746
\(658\) −20.7453 −0.808738
\(659\) −28.2890 −1.10198 −0.550991 0.834511i \(-0.685750\pi\)
−0.550991 + 0.834511i \(0.685750\pi\)
\(660\) 14.0844 0.548235
\(661\) 19.7653 0.768783 0.384391 0.923170i \(-0.374411\pi\)
0.384391 + 0.923170i \(0.374411\pi\)
\(662\) 40.6123 1.57844
\(663\) 18.5989 0.722323
\(664\) 10.5989 0.411319
\(665\) 18.8431 0.730704
\(666\) 13.0067 0.503998
\(667\) 21.6147 0.836924
\(668\) −1.49069 −0.0576767
\(669\) −22.7678 −0.880252
\(670\) 29.8984 1.15508
\(671\) 6.00000 0.231627
\(672\) 3.57653 0.137968
\(673\) −51.5174 −1.98585 −0.992924 0.118750i \(-0.962111\pi\)
−0.992924 + 0.118750i \(0.962111\pi\)
\(674\) −46.6123 −1.79544
\(675\) −10.5079 −0.404448
\(676\) 8.96949 0.344980
\(677\) 46.3376 1.78090 0.890450 0.455081i \(-0.150390\pi\)
0.890450 + 0.455081i \(0.150390\pi\)
\(678\) 17.5832 0.675279
\(679\) −11.1531 −0.428016
\(680\) 69.2427 2.65534
\(681\) 15.8760 0.608370
\(682\) −2.72294 −0.104267
\(683\) −39.2112 −1.50038 −0.750188 0.661225i \(-0.770037\pi\)
−0.750188 + 0.661225i \(0.770037\pi\)
\(684\) 17.1135 0.654350
\(685\) −6.23465 −0.238214
\(686\) 2.36147 0.0901613
\(687\) 3.15307 0.120297
\(688\) 11.0157 0.419971
\(689\) 12.4168 0.473042
\(690\) −25.3219 −0.963988
\(691\) 7.62802 0.290184 0.145092 0.989418i \(-0.453652\pi\)
0.145092 + 0.989418i \(0.453652\pi\)
\(692\) −30.1507 −1.14616
\(693\) −1.00000 −0.0379869
\(694\) −39.2294 −1.48913
\(695\) 67.7253 2.56897
\(696\) 29.5527 1.12019
\(697\) 2.03149 0.0769480
\(698\) −16.4211 −0.621546
\(699\) −13.1397 −0.496990
\(700\) −37.5818 −1.42046
\(701\) 2.61228 0.0986644 0.0493322 0.998782i \(-0.484291\pi\)
0.0493322 + 0.998782i \(0.484291\pi\)
\(702\) −9.29947 −0.350986
\(703\) −26.3548 −0.993990
\(704\) −11.7229 −0.441825
\(705\) −34.5951 −1.30293
\(706\) 7.62376 0.286924
\(707\) −16.4750 −0.619604
\(708\) 53.9737 2.02846
\(709\) 36.1516 1.35770 0.678852 0.734276i \(-0.262478\pi\)
0.678852 + 0.734276i \(0.262478\pi\)
\(710\) 125.040 4.69265
\(711\) −5.44588 −0.204236
\(712\) −45.8150 −1.71699
\(713\) 3.13974 0.117584
\(714\) −11.1531 −0.417393
\(715\) 15.5079 0.579962
\(716\) −42.9184 −1.60394
\(717\) 10.1068 0.377446
\(718\) −24.3376 −0.908272
\(719\) 19.6767 0.733816 0.366908 0.930257i \(-0.380416\pi\)
0.366908 + 0.930257i \(0.380416\pi\)
\(720\) −6.45254 −0.240472
\(721\) 1.56987 0.0584649
\(722\) −9.19932 −0.342363
\(723\) −2.36814 −0.0880719
\(724\) 73.6728 2.73803
\(725\) 83.4115 3.09783
\(726\) 2.36147 0.0876423
\(727\) 2.47495 0.0917909 0.0458954 0.998946i \(-0.485386\pi\)
0.0458954 + 0.998946i \(0.485386\pi\)
\(728\) −14.6609 −0.543371
\(729\) 1.00000 0.0370370
\(730\) 111.017 4.10893
\(731\) 31.7520 1.17439
\(732\) −21.4592 −0.793155
\(733\) −40.0315 −1.47860 −0.739298 0.673378i \(-0.764843\pi\)
−0.739298 + 0.673378i \(0.764843\pi\)
\(734\) 28.0448 1.03515
\(735\) 3.93800 0.145255
\(736\) 9.73868 0.358973
\(737\) 3.21507 0.118428
\(738\) −1.01574 −0.0373900
\(739\) 25.2771 0.929832 0.464916 0.885355i \(-0.346085\pi\)
0.464916 + 0.885355i \(0.346085\pi\)
\(740\) 77.5751 2.85172
\(741\) 18.8431 0.692218
\(742\) −7.44588 −0.273347
\(743\) 2.10682 0.0772916 0.0386458 0.999253i \(-0.487696\pi\)
0.0386458 + 0.999253i \(0.487696\pi\)
\(744\) 4.29281 0.157382
\(745\) −62.7639 −2.29949
\(746\) −39.4907 −1.44586
\(747\) −2.84693 −0.104164
\(748\) 16.8918 0.617624
\(749\) 4.78493 0.174838
\(750\) −51.2203 −1.87030
\(751\) −45.9828 −1.67794 −0.838969 0.544180i \(-0.816841\pi\)
−0.838969 + 0.544180i \(0.816841\pi\)
\(752\) −14.3944 −0.524909
\(753\) −10.2308 −0.372831
\(754\) 73.8192 2.68834
\(755\) −25.9867 −0.945752
\(756\) 3.57653 0.130077
\(757\) −29.2599 −1.06347 −0.531734 0.846911i \(-0.678460\pi\)
−0.531734 + 0.846911i \(0.678460\pi\)
\(758\) 25.8984 0.940673
\(759\) −2.72294 −0.0988364
\(760\) 70.1516 2.54467
\(761\) 21.8760 0.793005 0.396502 0.918034i \(-0.370224\pi\)
0.396502 + 0.918034i \(0.370224\pi\)
\(762\) 29.0291 1.05161
\(763\) 15.0291 0.544089
\(764\) −52.2136 −1.88902
\(765\) −18.5989 −0.672446
\(766\) −37.1979 −1.34401
\(767\) 59.4287 2.14585
\(768\) 25.0357 0.903400
\(769\) −16.2756 −0.586914 −0.293457 0.955972i \(-0.594806\pi\)
−0.293457 + 0.955972i \(0.594806\pi\)
\(770\) −9.29947 −0.335130
\(771\) 11.0777 0.398955
\(772\) 48.1373 1.73250
\(773\) −30.4921 −1.09673 −0.548363 0.836241i \(-0.684749\pi\)
−0.548363 + 0.836241i \(0.684749\pi\)
\(774\) −15.8760 −0.570651
\(775\) 12.1163 0.435231
\(776\) −41.5222 −1.49056
\(777\) −5.50787 −0.197594
\(778\) −86.8259 −3.11286
\(779\) 2.05815 0.0737410
\(780\) −55.4644 −1.98595
\(781\) 13.4459 0.481131
\(782\) −30.3691 −1.08600
\(783\) −7.93800 −0.283681
\(784\) 1.63853 0.0585190
\(785\) −28.8336 −1.02912
\(786\) 16.8603 0.601386
\(787\) 35.2151 1.25528 0.627641 0.778503i \(-0.284021\pi\)
0.627641 + 0.778503i \(0.284021\pi\)
\(788\) 40.9366 1.45830
\(789\) 8.78493 0.312752
\(790\) −50.6438 −1.80182
\(791\) −7.44588 −0.264745
\(792\) −3.72294 −0.132289
\(793\) −23.6280 −0.839056
\(794\) −28.0133 −0.994156
\(795\) −12.4168 −0.440378
\(796\) −46.5989 −1.65166
\(797\) −14.9804 −0.530633 −0.265317 0.964161i \(-0.585477\pi\)
−0.265317 + 0.964161i \(0.585477\pi\)
\(798\) −11.2995 −0.399997
\(799\) −41.4907 −1.46784
\(800\) 37.5818 1.32872
\(801\) 12.3061 0.434816
\(802\) −8.03149 −0.283602
\(803\) 11.9380 0.421283
\(804\) −11.4988 −0.405531
\(805\) 10.7229 0.377934
\(806\) 10.7229 0.377699
\(807\) −5.75201 −0.202480
\(808\) −61.3352 −2.15777
\(809\) −25.2599 −0.888090 −0.444045 0.896004i \(-0.646457\pi\)
−0.444045 + 0.896004i \(0.646457\pi\)
\(810\) 9.29947 0.326750
\(811\) 29.5212 1.03663 0.518315 0.855190i \(-0.326560\pi\)
0.518315 + 0.855190i \(0.326560\pi\)
\(812\) −28.3905 −0.996313
\(813\) −0.784934 −0.0275288
\(814\) 13.0067 0.455883
\(815\) −55.5527 −1.94593
\(816\) −7.73868 −0.270908
\(817\) 32.1688 1.12544
\(818\) 66.1955 2.31447
\(819\) 3.93800 0.137605
\(820\) −6.05815 −0.211560
\(821\) 23.7873 0.830184 0.415092 0.909779i \(-0.363749\pi\)
0.415092 + 0.909779i \(0.363749\pi\)
\(822\) 3.73868 0.130401
\(823\) 17.7692 0.619395 0.309698 0.950835i \(-0.399772\pi\)
0.309698 + 0.950835i \(0.399772\pi\)
\(824\) 5.84452 0.203604
\(825\) −10.5079 −0.365837
\(826\) −35.6371 −1.23997
\(827\) 8.90893 0.309794 0.154897 0.987931i \(-0.450495\pi\)
0.154897 + 0.987931i \(0.450495\pi\)
\(828\) 9.73868 0.338443
\(829\) −32.7678 −1.13807 −0.569036 0.822313i \(-0.692683\pi\)
−0.569036 + 0.822313i \(0.692683\pi\)
\(830\) −26.4750 −0.918959
\(831\) 11.5699 0.401354
\(832\) 46.1650 1.60048
\(833\) 4.72294 0.163640
\(834\) −40.6123 −1.40629
\(835\) 1.64135 0.0568014
\(836\) 17.1135 0.591882
\(837\) −1.15307 −0.0398559
\(838\) −66.8035 −2.30769
\(839\) 6.96708 0.240530 0.120265 0.992742i \(-0.461626\pi\)
0.120265 + 0.992742i \(0.461626\pi\)
\(840\) 14.6609 0.505851
\(841\) 34.0119 1.17282
\(842\) 34.2227 1.17939
\(843\) 25.2599 0.869997
\(844\) 8.69145 0.299172
\(845\) −9.87601 −0.339745
\(846\) 20.7453 0.713240
\(847\) −1.00000 −0.0343604
\(848\) −5.16640 −0.177415
\(849\) 10.9671 0.376389
\(850\) −117.195 −4.01976
\(851\) −14.9976 −0.514111
\(852\) −48.0896 −1.64752
\(853\) 11.1979 0.383408 0.191704 0.981453i \(-0.438599\pi\)
0.191704 + 0.981453i \(0.438599\pi\)
\(854\) 14.1688 0.484847
\(855\) −18.8431 −0.644420
\(856\) 17.8140 0.608870
\(857\) −44.0315 −1.50409 −0.752043 0.659114i \(-0.770932\pi\)
−0.752043 + 0.659114i \(0.770932\pi\)
\(858\) −9.29947 −0.317479
\(859\) 7.13974 0.243605 0.121802 0.992554i \(-0.461133\pi\)
0.121802 + 0.992554i \(0.461133\pi\)
\(860\) −94.6886 −3.22885
\(861\) 0.430132 0.0146589
\(862\) 57.9432 1.97355
\(863\) −5.27706 −0.179633 −0.0898166 0.995958i \(-0.528628\pi\)
−0.0898166 + 0.995958i \(0.528628\pi\)
\(864\) −3.57653 −0.121676
\(865\) 33.1979 1.12876
\(866\) 17.2904 0.587552
\(867\) −5.30614 −0.180206
\(868\) −4.12399 −0.139977
\(869\) −5.44588 −0.184739
\(870\) −73.8192 −2.50271
\(871\) −12.6609 −0.429000
\(872\) 55.9523 1.89478
\(873\) 11.1531 0.377474
\(874\) −30.7678 −1.04073
\(875\) 21.6900 0.733256
\(876\) −42.6967 −1.44259
\(877\) −32.4301 −1.09509 −0.547544 0.836777i \(-0.684437\pi\)
−0.547544 + 0.836777i \(0.684437\pi\)
\(878\) −39.0515 −1.31792
\(879\) −14.7363 −0.497042
\(880\) −6.45254 −0.217515
\(881\) 39.5660 1.33301 0.666507 0.745499i \(-0.267789\pi\)
0.666507 + 0.745499i \(0.267789\pi\)
\(882\) −2.36147 −0.0795148
\(883\) −24.4130 −0.821561 −0.410781 0.911734i \(-0.634744\pi\)
−0.410781 + 0.911734i \(0.634744\pi\)
\(884\) −66.5198 −2.23730
\(885\) −59.4287 −1.99767
\(886\) −59.5040 −1.99908
\(887\) 36.2928 1.21859 0.609297 0.792942i \(-0.291452\pi\)
0.609297 + 0.792942i \(0.291452\pi\)
\(888\) −20.5055 −0.688118
\(889\) −12.2928 −0.412287
\(890\) 114.441 3.83606
\(891\) 1.00000 0.0335013
\(892\) 81.4297 2.72647
\(893\) −42.0353 −1.40666
\(894\) 37.6371 1.25877
\(895\) 47.2560 1.57960
\(896\) −20.5303 −0.685869
\(897\) 10.7229 0.358028
\(898\) 86.2403 2.87788
\(899\) 9.15307 0.305272
\(900\) 37.5818 1.25273
\(901\) −14.8918 −0.496116
\(902\) −1.01574 −0.0338205
\(903\) 6.72294 0.223725
\(904\) −27.7205 −0.921971
\(905\) −81.1187 −2.69648
\(906\) 15.5832 0.517717
\(907\) −30.3958 −1.00928 −0.504638 0.863331i \(-0.668374\pi\)
−0.504638 + 0.863331i \(0.668374\pi\)
\(908\) −56.7811 −1.88435
\(909\) 16.4750 0.546440
\(910\) 36.6214 1.21399
\(911\) −56.8259 −1.88273 −0.941363 0.337395i \(-0.890454\pi\)
−0.941363 + 0.337395i \(0.890454\pi\)
\(912\) −7.84026 −0.259617
\(913\) −2.84693 −0.0942196
\(914\) −7.44588 −0.246288
\(915\) 23.6280 0.781118
\(916\) −11.2771 −0.372605
\(917\) −7.13974 −0.235775
\(918\) 11.1531 0.368106
\(919\) −40.1688 −1.32505 −0.662523 0.749041i \(-0.730514\pi\)
−0.662523 + 0.749041i \(0.730514\pi\)
\(920\) 39.9208 1.31615
\(921\) −0.860264 −0.0283467
\(922\) 58.4883 1.92621
\(923\) −52.9499 −1.74287
\(924\) 3.57653 0.117659
\(925\) −57.8760 −1.90295
\(926\) 83.6638 2.74936
\(927\) −1.56987 −0.0515612
\(928\) 28.3905 0.931965
\(929\) −27.3180 −0.896276 −0.448138 0.893964i \(-0.647913\pi\)
−0.448138 + 0.893964i \(0.647913\pi\)
\(930\) −10.7229 −0.351619
\(931\) 4.78493 0.156820
\(932\) 46.9947 1.53936
\(933\) −26.8918 −0.880396
\(934\) 56.4974 1.84865
\(935\) −18.5989 −0.608251
\(936\) 14.6609 0.479208
\(937\) −5.75201 −0.187910 −0.0939551 0.995576i \(-0.529951\pi\)
−0.0939551 + 0.995576i \(0.529951\pi\)
\(938\) 7.59228 0.247897
\(939\) −10.1688 −0.331847
\(940\) 123.731 4.03565
\(941\) −27.5699 −0.898752 −0.449376 0.893343i \(-0.648354\pi\)
−0.449376 + 0.893343i \(0.648354\pi\)
\(942\) 17.2904 0.563352
\(943\) 1.17122 0.0381402
\(944\) −24.7272 −0.804802
\(945\) −3.93800 −0.128103
\(946\) −15.8760 −0.516174
\(947\) 37.0739 1.20474 0.602370 0.798217i \(-0.294223\pi\)
0.602370 + 0.798217i \(0.294223\pi\)
\(948\) 19.4774 0.632595
\(949\) −47.0119 −1.52607
\(950\) −118.733 −3.85222
\(951\) 27.4907 0.891447
\(952\) 17.5832 0.569875
\(953\) −12.0620 −0.390726 −0.195363 0.980731i \(-0.562589\pi\)
−0.195363 + 0.980731i \(0.562589\pi\)
\(954\) 7.44588 0.241069
\(955\) 57.4907 1.86036
\(956\) −36.1474 −1.16909
\(957\) −7.93800 −0.256599
\(958\) −10.7229 −0.346442
\(959\) −1.58320 −0.0511242
\(960\) −46.1650 −1.48997
\(961\) −29.6704 −0.957111
\(962\) −51.2203 −1.65141
\(963\) −4.78493 −0.154192
\(964\) 8.46972 0.272791
\(965\) −53.0024 −1.70621
\(966\) −6.43013 −0.206886
\(967\) 6.43013 0.206779 0.103390 0.994641i \(-0.467031\pi\)
0.103390 + 0.994641i \(0.467031\pi\)
\(968\) −3.72294 −0.119660
\(969\) −22.5989 −0.725983
\(970\) 103.718 3.33017
\(971\) −35.3047 −1.13298 −0.566491 0.824068i \(-0.691699\pi\)
−0.566491 + 0.824068i \(0.691699\pi\)
\(972\) −3.57653 −0.114717
\(973\) 17.1979 0.551339
\(974\) −64.0896 −2.05356
\(975\) 41.3800 1.32522
\(976\) 9.83119 0.314689
\(977\) 0.261319 0.00836035 0.00418017 0.999991i \(-0.498669\pi\)
0.00418017 + 0.999991i \(0.498669\pi\)
\(978\) 33.3128 1.06523
\(979\) 12.3061 0.393306
\(980\) −14.0844 −0.449910
\(981\) −15.0291 −0.479841
\(982\) −8.17789 −0.260967
\(983\) −16.3376 −0.521089 −0.260545 0.965462i \(-0.583902\pi\)
−0.260545 + 0.965462i \(0.583902\pi\)
\(984\) 1.60135 0.0510493
\(985\) −45.0739 −1.43617
\(986\) −88.5331 −2.81947
\(987\) −8.78493 −0.279628
\(988\) −67.3930 −2.14406
\(989\) 18.3061 0.582101
\(990\) 9.29947 0.295557
\(991\) −36.2623 −1.15191 −0.575955 0.817481i \(-0.695370\pi\)
−0.575955 + 0.817481i \(0.695370\pi\)
\(992\) 4.12399 0.130937
\(993\) 17.1979 0.545759
\(994\) 31.7520 1.00711
\(995\) 51.3085 1.62659
\(996\) 10.1821 0.322634
\(997\) −3.47254 −0.109977 −0.0549883 0.998487i \(-0.517512\pi\)
−0.0549883 + 0.998487i \(0.517512\pi\)
\(998\) 44.7902 1.41781
\(999\) 5.50787 0.174261
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 231.2.a.d.1.1 3
3.2 odd 2 693.2.a.m.1.3 3
4.3 odd 2 3696.2.a.bp.1.1 3
5.4 even 2 5775.2.a.bw.1.3 3
7.6 odd 2 1617.2.a.s.1.1 3
11.10 odd 2 2541.2.a.bi.1.3 3
21.20 even 2 4851.2.a.bp.1.3 3
33.32 even 2 7623.2.a.cb.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.d.1.1 3 1.1 even 1 trivial
693.2.a.m.1.3 3 3.2 odd 2
1617.2.a.s.1.1 3 7.6 odd 2
2541.2.a.bi.1.3 3 11.10 odd 2
3696.2.a.bp.1.1 3 4.3 odd 2
4851.2.a.bp.1.3 3 21.20 even 2
5775.2.a.bw.1.3 3 5.4 even 2
7623.2.a.cb.1.1 3 33.32 even 2