Properties

Label 5445.2.a.bf.1.4
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Defining polynomial: \(x^{4} - x^{3} - 3 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.09529\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.09529 q^{2} -0.800331 q^{4} -1.00000 q^{5} +0.705037 q^{7} -3.06719 q^{8} +O(q^{10})\) \(q+1.09529 q^{2} -0.800331 q^{4} -1.00000 q^{5} +0.705037 q^{7} -3.06719 q^{8} -1.09529 q^{10} +4.71333 q^{13} +0.772223 q^{14} -1.75881 q^{16} -7.78051 q^{17} +1.19967 q^{19} +0.800331 q^{20} +6.89318 q^{23} +1.00000 q^{25} +5.16248 q^{26} -0.564263 q^{28} +1.32741 q^{29} -7.68126 q^{31} +4.20796 q^{32} -8.52195 q^{34} -0.705037 q^{35} +8.43763 q^{37} +1.31399 q^{38} +3.06719 q^{40} -0.232901 q^{41} -7.32892 q^{43} +7.55006 q^{46} -8.32228 q^{47} -6.50292 q^{49} +1.09529 q^{50} -3.77222 q^{52} -6.82332 q^{53} -2.16248 q^{56} +1.45390 q^{58} +3.54011 q^{59} +10.8719 q^{61} -8.41324 q^{62} +8.12657 q^{64} -4.71333 q^{65} -2.04036 q^{67} +6.22699 q^{68} -0.772223 q^{70} -0.670527 q^{71} +5.00433 q^{73} +9.24168 q^{74} -0.960132 q^{76} -2.28027 q^{79} +1.75881 q^{80} -0.255095 q^{82} -2.10999 q^{83} +7.78051 q^{85} -8.02732 q^{86} +3.34722 q^{89} +3.32307 q^{91} -5.51683 q^{92} -9.11534 q^{94} -1.19967 q^{95} +3.32228 q^{97} -7.12261 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{2} + q^{4} - 4q^{5} + 6q^{7} - 3q^{8} + O(q^{10}) \) \( 4q - 3q^{2} + q^{4} - 4q^{5} + 6q^{7} - 3q^{8} + 3q^{10} + 7q^{13} - 3q^{14} - q^{16} - 10q^{17} + 9q^{19} - q^{20} + 3q^{23} + 4q^{25} + 4q^{26} - 7q^{28} - 15q^{29} - 13q^{31} + 6q^{32} - 3q^{34} - 6q^{35} - 3q^{37} - 15q^{38} + 3q^{40} - 22q^{41} + q^{46} + 2q^{47} - 12q^{49} - 3q^{50} - 9q^{52} - 10q^{53} + 8q^{56} + 39q^{58} + 21q^{59} + 11q^{61} - 10q^{62} - 3q^{64} - 7q^{65} + q^{67} - 3q^{68} + 3q^{70} + 13q^{71} + q^{73} + 11q^{74} + 19q^{76} - 4q^{79} + q^{80} + 25q^{82} - 3q^{83} + 10q^{85} + 10q^{89} + 12q^{91} + 24q^{92} - 35q^{94} - 9q^{95} - 22q^{97} + 11q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.09529 0.774490 0.387245 0.921977i \(-0.373427\pi\)
0.387245 + 0.921977i \(0.373427\pi\)
\(3\) 0 0
\(4\) −0.800331 −0.400166
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.705037 0.266479 0.133239 0.991084i \(-0.457462\pi\)
0.133239 + 0.991084i \(0.457462\pi\)
\(8\) −3.06719 −1.08441
\(9\) 0 0
\(10\) −1.09529 −0.346362
\(11\) 0 0
\(12\) 0 0
\(13\) 4.71333 1.30724 0.653621 0.756822i \(-0.273249\pi\)
0.653621 + 0.756822i \(0.273249\pi\)
\(14\) 0.772223 0.206385
\(15\) 0 0
\(16\) −1.75881 −0.439702
\(17\) −7.78051 −1.88705 −0.943526 0.331299i \(-0.892513\pi\)
−0.943526 + 0.331299i \(0.892513\pi\)
\(18\) 0 0
\(19\) 1.19967 0.275223 0.137611 0.990486i \(-0.456057\pi\)
0.137611 + 0.990486i \(0.456057\pi\)
\(20\) 0.800331 0.178959
\(21\) 0 0
\(22\) 0 0
\(23\) 6.89318 1.43733 0.718664 0.695358i \(-0.244754\pi\)
0.718664 + 0.695358i \(0.244754\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 5.16248 1.01245
\(27\) 0 0
\(28\) −0.564263 −0.106636
\(29\) 1.32741 0.246493 0.123246 0.992376i \(-0.460669\pi\)
0.123246 + 0.992376i \(0.460669\pi\)
\(30\) 0 0
\(31\) −7.68126 −1.37960 −0.689798 0.724002i \(-0.742301\pi\)
−0.689798 + 0.724002i \(0.742301\pi\)
\(32\) 4.20796 0.743869
\(33\) 0 0
\(34\) −8.52195 −1.46150
\(35\) −0.705037 −0.119173
\(36\) 0 0
\(37\) 8.43763 1.38714 0.693569 0.720391i \(-0.256037\pi\)
0.693569 + 0.720391i \(0.256037\pi\)
\(38\) 1.31399 0.213157
\(39\) 0 0
\(40\) 3.06719 0.484965
\(41\) −0.232901 −0.0363730 −0.0181865 0.999835i \(-0.505789\pi\)
−0.0181865 + 0.999835i \(0.505789\pi\)
\(42\) 0 0
\(43\) −7.32892 −1.11765 −0.558825 0.829286i \(-0.688748\pi\)
−0.558825 + 0.829286i \(0.688748\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 7.55006 1.11320
\(47\) −8.32228 −1.21393 −0.606965 0.794729i \(-0.707613\pi\)
−0.606965 + 0.794729i \(0.707613\pi\)
\(48\) 0 0
\(49\) −6.50292 −0.928989
\(50\) 1.09529 0.154898
\(51\) 0 0
\(52\) −3.77222 −0.523113
\(53\) −6.82332 −0.937254 −0.468627 0.883396i \(-0.655251\pi\)
−0.468627 + 0.883396i \(0.655251\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.16248 −0.288974
\(57\) 0 0
\(58\) 1.45390 0.190906
\(59\) 3.54011 0.460883 0.230442 0.973086i \(-0.425983\pi\)
0.230442 + 0.973086i \(0.425983\pi\)
\(60\) 0 0
\(61\) 10.8719 1.39200 0.695999 0.718043i \(-0.254962\pi\)
0.695999 + 0.718043i \(0.254962\pi\)
\(62\) −8.41324 −1.06848
\(63\) 0 0
\(64\) 8.12657 1.01582
\(65\) −4.71333 −0.584616
\(66\) 0 0
\(67\) −2.04036 −0.249269 −0.124635 0.992203i \(-0.539776\pi\)
−0.124635 + 0.992203i \(0.539776\pi\)
\(68\) 6.22699 0.755133
\(69\) 0 0
\(70\) −0.772223 −0.0922983
\(71\) −0.670527 −0.0795769 −0.0397884 0.999208i \(-0.512668\pi\)
−0.0397884 + 0.999208i \(0.512668\pi\)
\(72\) 0 0
\(73\) 5.00433 0.585713 0.292856 0.956156i \(-0.405394\pi\)
0.292856 + 0.956156i \(0.405394\pi\)
\(74\) 9.24168 1.07432
\(75\) 0 0
\(76\) −0.960132 −0.110135
\(77\) 0 0
\(78\) 0 0
\(79\) −2.28027 −0.256550 −0.128275 0.991739i \(-0.540944\pi\)
−0.128275 + 0.991739i \(0.540944\pi\)
\(80\) 1.75881 0.196641
\(81\) 0 0
\(82\) −0.255095 −0.0281706
\(83\) −2.10999 −0.231601 −0.115801 0.993272i \(-0.536943\pi\)
−0.115801 + 0.993272i \(0.536943\pi\)
\(84\) 0 0
\(85\) 7.78051 0.843915
\(86\) −8.02732 −0.865608
\(87\) 0 0
\(88\) 0 0
\(89\) 3.34722 0.354805 0.177402 0.984138i \(-0.443231\pi\)
0.177402 + 0.984138i \(0.443231\pi\)
\(90\) 0 0
\(91\) 3.32307 0.348353
\(92\) −5.51683 −0.575169
\(93\) 0 0
\(94\) −9.11534 −0.940176
\(95\) −1.19967 −0.123083
\(96\) 0 0
\(97\) 3.32228 0.337327 0.168663 0.985674i \(-0.446055\pi\)
0.168663 + 0.985674i \(0.446055\pi\)
\(98\) −7.12261 −0.719492
\(99\) 0 0
\(100\) −0.800331 −0.0800331
\(101\) −18.9218 −1.88279 −0.941394 0.337310i \(-0.890483\pi\)
−0.941394 + 0.337310i \(0.890483\pi\)
\(102\) 0 0
\(103\) −18.0964 −1.78309 −0.891545 0.452932i \(-0.850378\pi\)
−0.891545 + 0.452932i \(0.850378\pi\)
\(104\) −14.4567 −1.41759
\(105\) 0 0
\(106\) −7.47354 −0.725894
\(107\) −6.11945 −0.591589 −0.295795 0.955252i \(-0.595584\pi\)
−0.295795 + 0.955252i \(0.595584\pi\)
\(108\) 0 0
\(109\) −9.03128 −0.865039 −0.432520 0.901625i \(-0.642375\pi\)
−0.432520 + 0.901625i \(0.642375\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.24002 −0.117171
\(113\) −3.91684 −0.368466 −0.184233 0.982883i \(-0.558980\pi\)
−0.184233 + 0.982883i \(0.558980\pi\)
\(114\) 0 0
\(115\) −6.89318 −0.642792
\(116\) −1.06236 −0.0986380
\(117\) 0 0
\(118\) 3.87746 0.356949
\(119\) −5.48555 −0.502860
\(120\) 0 0
\(121\) 0 0
\(122\) 11.9079 1.07809
\(123\) 0 0
\(124\) 6.14755 0.552067
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −10.8675 −0.964336 −0.482168 0.876079i \(-0.660151\pi\)
−0.482168 + 0.876079i \(0.660151\pi\)
\(128\) 0.485063 0.0428739
\(129\) 0 0
\(130\) −5.16248 −0.452779
\(131\) −11.7094 −1.02305 −0.511526 0.859268i \(-0.670920\pi\)
−0.511526 + 0.859268i \(0.670920\pi\)
\(132\) 0 0
\(133\) 0.845811 0.0733411
\(134\) −2.23479 −0.193056
\(135\) 0 0
\(136\) 23.8643 2.04635
\(137\) −20.4302 −1.74547 −0.872735 0.488194i \(-0.837656\pi\)
−0.872735 + 0.488194i \(0.837656\pi\)
\(138\) 0 0
\(139\) 8.17100 0.693055 0.346528 0.938040i \(-0.387361\pi\)
0.346528 + 0.938040i \(0.387361\pi\)
\(140\) 0.564263 0.0476889
\(141\) 0 0
\(142\) −0.734424 −0.0616315
\(143\) 0 0
\(144\) 0 0
\(145\) −1.32741 −0.110235
\(146\) 5.48122 0.453629
\(147\) 0 0
\(148\) −6.75289 −0.555084
\(149\) −9.60217 −0.786641 −0.393320 0.919401i \(-0.628674\pi\)
−0.393320 + 0.919401i \(0.628674\pi\)
\(150\) 0 0
\(151\) −6.90776 −0.562146 −0.281073 0.959686i \(-0.590690\pi\)
−0.281073 + 0.959686i \(0.590690\pi\)
\(152\) −3.67961 −0.298456
\(153\) 0 0
\(154\) 0 0
\(155\) 7.68126 0.616974
\(156\) 0 0
\(157\) 11.3519 0.905980 0.452990 0.891516i \(-0.350357\pi\)
0.452990 + 0.891516i \(0.350357\pi\)
\(158\) −2.49757 −0.198696
\(159\) 0 0
\(160\) −4.20796 −0.332668
\(161\) 4.85995 0.383018
\(162\) 0 0
\(163\) −1.79253 −0.140402 −0.0702008 0.997533i \(-0.522364\pi\)
−0.0702008 + 0.997533i \(0.522364\pi\)
\(164\) 0.186398 0.0145552
\(165\) 0 0
\(166\) −2.31106 −0.179373
\(167\) 6.02450 0.466189 0.233095 0.972454i \(-0.425115\pi\)
0.233095 + 0.972454i \(0.425115\pi\)
\(168\) 0 0
\(169\) 9.21546 0.708882
\(170\) 8.52195 0.653604
\(171\) 0 0
\(172\) 5.86556 0.447245
\(173\) 4.18674 0.318312 0.159156 0.987253i \(-0.449123\pi\)
0.159156 + 0.987253i \(0.449123\pi\)
\(174\) 0 0
\(175\) 0.705037 0.0532958
\(176\) 0 0
\(177\) 0 0
\(178\) 3.66619 0.274793
\(179\) −4.85166 −0.362630 −0.181315 0.983425i \(-0.558035\pi\)
−0.181315 + 0.983425i \(0.558035\pi\)
\(180\) 0 0
\(181\) −23.0877 −1.71610 −0.858049 0.513569i \(-0.828323\pi\)
−0.858049 + 0.513569i \(0.828323\pi\)
\(182\) 3.63974 0.269795
\(183\) 0 0
\(184\) −21.1427 −1.55866
\(185\) −8.43763 −0.620347
\(186\) 0 0
\(187\) 0 0
\(188\) 6.66058 0.485773
\(189\) 0 0
\(190\) −1.31399 −0.0953269
\(191\) −6.22571 −0.450476 −0.225238 0.974304i \(-0.572316\pi\)
−0.225238 + 0.974304i \(0.572316\pi\)
\(192\) 0 0
\(193\) −19.0868 −1.37390 −0.686950 0.726705i \(-0.741051\pi\)
−0.686950 + 0.726705i \(0.741051\pi\)
\(194\) 3.63887 0.261256
\(195\) 0 0
\(196\) 5.20449 0.371749
\(197\) 6.80056 0.484520 0.242260 0.970211i \(-0.422111\pi\)
0.242260 + 0.970211i \(0.422111\pi\)
\(198\) 0 0
\(199\) 21.2972 1.50972 0.754860 0.655886i \(-0.227705\pi\)
0.754860 + 0.655886i \(0.227705\pi\)
\(200\) −3.06719 −0.216883
\(201\) 0 0
\(202\) −20.7249 −1.45820
\(203\) 0.935870 0.0656852
\(204\) 0 0
\(205\) 0.232901 0.0162665
\(206\) −19.8209 −1.38099
\(207\) 0 0
\(208\) −8.28984 −0.574797
\(209\) 0 0
\(210\) 0 0
\(211\) 8.89073 0.612063 0.306032 0.952021i \(-0.400999\pi\)
0.306032 + 0.952021i \(0.400999\pi\)
\(212\) 5.46091 0.375057
\(213\) 0 0
\(214\) −6.70259 −0.458180
\(215\) 7.32892 0.499828
\(216\) 0 0
\(217\) −5.41558 −0.367633
\(218\) −9.89190 −0.669964
\(219\) 0 0
\(220\) 0 0
\(221\) −36.6721 −2.46683
\(222\) 0 0
\(223\) 5.41720 0.362762 0.181381 0.983413i \(-0.441943\pi\)
0.181381 + 0.983413i \(0.441943\pi\)
\(224\) 2.96677 0.198226
\(225\) 0 0
\(226\) −4.29009 −0.285373
\(227\) −8.43842 −0.560077 −0.280039 0.959989i \(-0.590347\pi\)
−0.280039 + 0.959989i \(0.590347\pi\)
\(228\) 0 0
\(229\) −11.2053 −0.740466 −0.370233 0.928939i \(-0.620722\pi\)
−0.370233 + 0.928939i \(0.620722\pi\)
\(230\) −7.55006 −0.497836
\(231\) 0 0
\(232\) −4.07140 −0.267300
\(233\) 5.83979 0.382577 0.191289 0.981534i \(-0.438733\pi\)
0.191289 + 0.981534i \(0.438733\pi\)
\(234\) 0 0
\(235\) 8.32228 0.542886
\(236\) −2.83326 −0.184430
\(237\) 0 0
\(238\) −6.00829 −0.389460
\(239\) −16.5261 −1.06899 −0.534494 0.845173i \(-0.679498\pi\)
−0.534494 + 0.845173i \(0.679498\pi\)
\(240\) 0 0
\(241\) 3.29180 0.212043 0.106022 0.994364i \(-0.466189\pi\)
0.106022 + 0.994364i \(0.466189\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −8.70108 −0.557030
\(245\) 6.50292 0.415456
\(246\) 0 0
\(247\) 5.65443 0.359783
\(248\) 23.5599 1.49605
\(249\) 0 0
\(250\) −1.09529 −0.0692725
\(251\) 7.39934 0.467042 0.233521 0.972352i \(-0.424975\pi\)
0.233521 + 0.972352i \(0.424975\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −11.9031 −0.746869
\(255\) 0 0
\(256\) −15.7219 −0.982616
\(257\) 18.6991 1.16642 0.583210 0.812322i \(-0.301797\pi\)
0.583210 + 0.812322i \(0.301797\pi\)
\(258\) 0 0
\(259\) 5.94884 0.369643
\(260\) 3.77222 0.233943
\(261\) 0 0
\(262\) −12.8252 −0.792344
\(263\) 3.99020 0.246046 0.123023 0.992404i \(-0.460741\pi\)
0.123023 + 0.992404i \(0.460741\pi\)
\(264\) 0 0
\(265\) 6.82332 0.419153
\(266\) 0.926412 0.0568020
\(267\) 0 0
\(268\) 1.63296 0.0997489
\(269\) 12.7150 0.775246 0.387623 0.921818i \(-0.373296\pi\)
0.387623 + 0.921818i \(0.373296\pi\)
\(270\) 0 0
\(271\) −23.8280 −1.44745 −0.723724 0.690090i \(-0.757571\pi\)
−0.723724 + 0.690090i \(0.757571\pi\)
\(272\) 13.6844 0.829740
\(273\) 0 0
\(274\) −22.3771 −1.35185
\(275\) 0 0
\(276\) 0 0
\(277\) −7.41252 −0.445375 −0.222688 0.974890i \(-0.571483\pi\)
−0.222688 + 0.974890i \(0.571483\pi\)
\(278\) 8.94965 0.536764
\(279\) 0 0
\(280\) 2.16248 0.129233
\(281\) −29.0815 −1.73485 −0.867427 0.497564i \(-0.834228\pi\)
−0.867427 + 0.497564i \(0.834228\pi\)
\(282\) 0 0
\(283\) 21.6729 1.28832 0.644160 0.764891i \(-0.277207\pi\)
0.644160 + 0.764891i \(0.277207\pi\)
\(284\) 0.536643 0.0318439
\(285\) 0 0
\(286\) 0 0
\(287\) −0.164204 −0.00969265
\(288\) 0 0
\(289\) 43.5364 2.56096
\(290\) −1.45390 −0.0853759
\(291\) 0 0
\(292\) −4.00512 −0.234382
\(293\) −8.41220 −0.491446 −0.245723 0.969340i \(-0.579025\pi\)
−0.245723 + 0.969340i \(0.579025\pi\)
\(294\) 0 0
\(295\) −3.54011 −0.206113
\(296\) −25.8798 −1.50423
\(297\) 0 0
\(298\) −10.5172 −0.609245
\(299\) 32.4898 1.87893
\(300\) 0 0
\(301\) −5.16716 −0.297830
\(302\) −7.56603 −0.435376
\(303\) 0 0
\(304\) −2.10999 −0.121016
\(305\) −10.8719 −0.622520
\(306\) 0 0
\(307\) −23.7431 −1.35509 −0.677545 0.735481i \(-0.736956\pi\)
−0.677545 + 0.735481i \(0.736956\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8.41324 0.477840
\(311\) −23.0471 −1.30688 −0.653440 0.756979i \(-0.726675\pi\)
−0.653440 + 0.756979i \(0.726675\pi\)
\(312\) 0 0
\(313\) −17.3638 −0.981460 −0.490730 0.871312i \(-0.663270\pi\)
−0.490730 + 0.871312i \(0.663270\pi\)
\(314\) 12.4337 0.701673
\(315\) 0 0
\(316\) 1.82497 0.102663
\(317\) 6.15095 0.345472 0.172736 0.984968i \(-0.444739\pi\)
0.172736 + 0.984968i \(0.444739\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −8.12657 −0.454289
\(321\) 0 0
\(322\) 5.32307 0.296643
\(323\) −9.33404 −0.519360
\(324\) 0 0
\(325\) 4.71333 0.261448
\(326\) −1.96335 −0.108740
\(327\) 0 0
\(328\) 0.714351 0.0394434
\(329\) −5.86752 −0.323487
\(330\) 0 0
\(331\) 19.4191 1.06737 0.533685 0.845683i \(-0.320807\pi\)
0.533685 + 0.845683i \(0.320807\pi\)
\(332\) 1.68869 0.0926788
\(333\) 0 0
\(334\) 6.59859 0.361059
\(335\) 2.04036 0.111477
\(336\) 0 0
\(337\) −31.6868 −1.72609 −0.863045 0.505127i \(-0.831446\pi\)
−0.863045 + 0.505127i \(0.831446\pi\)
\(338\) 10.0936 0.549022
\(339\) 0 0
\(340\) −6.22699 −0.337706
\(341\) 0 0
\(342\) 0 0
\(343\) −9.52006 −0.514035
\(344\) 22.4791 1.21199
\(345\) 0 0
\(346\) 4.58571 0.246529
\(347\) −27.9434 −1.50008 −0.750041 0.661392i \(-0.769966\pi\)
−0.750041 + 0.661392i \(0.769966\pi\)
\(348\) 0 0
\(349\) 0.683331 0.0365779 0.0182889 0.999833i \(-0.494178\pi\)
0.0182889 + 0.999833i \(0.494178\pi\)
\(350\) 0.772223 0.0412771
\(351\) 0 0
\(352\) 0 0
\(353\) −1.55900 −0.0829769 −0.0414885 0.999139i \(-0.513210\pi\)
−0.0414885 + 0.999139i \(0.513210\pi\)
\(354\) 0 0
\(355\) 0.670527 0.0355879
\(356\) −2.67889 −0.141981
\(357\) 0 0
\(358\) −5.31399 −0.280853
\(359\) 15.8404 0.836022 0.418011 0.908442i \(-0.362727\pi\)
0.418011 + 0.908442i \(0.362727\pi\)
\(360\) 0 0
\(361\) −17.5608 −0.924252
\(362\) −25.2878 −1.32910
\(363\) 0 0
\(364\) −2.65956 −0.139399
\(365\) −5.00433 −0.261939
\(366\) 0 0
\(367\) 15.7361 0.821419 0.410710 0.911766i \(-0.365281\pi\)
0.410710 + 0.911766i \(0.365281\pi\)
\(368\) −12.1238 −0.631996
\(369\) 0 0
\(370\) −9.24168 −0.480452
\(371\) −4.81069 −0.249759
\(372\) 0 0
\(373\) 5.27703 0.273234 0.136617 0.990624i \(-0.456377\pi\)
0.136617 + 0.990624i \(0.456377\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 25.5260 1.31640
\(377\) 6.25650 0.322226
\(378\) 0 0
\(379\) 33.5093 1.72125 0.860627 0.509235i \(-0.170072\pi\)
0.860627 + 0.509235i \(0.170072\pi\)
\(380\) 0.960132 0.0492537
\(381\) 0 0
\(382\) −6.81898 −0.348889
\(383\) 20.2323 1.03382 0.516910 0.856039i \(-0.327082\pi\)
0.516910 + 0.856039i \(0.327082\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −20.9057 −1.06407
\(387\) 0 0
\(388\) −2.65892 −0.134986
\(389\) 27.5729 1.39800 0.699000 0.715122i \(-0.253629\pi\)
0.699000 + 0.715122i \(0.253629\pi\)
\(390\) 0 0
\(391\) −53.6325 −2.71231
\(392\) 19.9457 1.00741
\(393\) 0 0
\(394\) 7.44862 0.375256
\(395\) 2.28027 0.114733
\(396\) 0 0
\(397\) 11.7601 0.590222 0.295111 0.955463i \(-0.404643\pi\)
0.295111 + 0.955463i \(0.404643\pi\)
\(398\) 23.3267 1.16926
\(399\) 0 0
\(400\) −1.75881 −0.0879404
\(401\) 7.72406 0.385721 0.192861 0.981226i \(-0.438223\pi\)
0.192861 + 0.981226i \(0.438223\pi\)
\(402\) 0 0
\(403\) −36.2043 −1.80347
\(404\) 15.1437 0.753427
\(405\) 0 0
\(406\) 1.02505 0.0508725
\(407\) 0 0
\(408\) 0 0
\(409\) −16.7409 −0.827783 −0.413892 0.910326i \(-0.635831\pi\)
−0.413892 + 0.910326i \(0.635831\pi\)
\(410\) 0.255095 0.0125983
\(411\) 0 0
\(412\) 14.4831 0.713531
\(413\) 2.49591 0.122816
\(414\) 0 0
\(415\) 2.10999 0.103575
\(416\) 19.8335 0.972417
\(417\) 0 0
\(418\) 0 0
\(419\) 38.0968 1.86115 0.930576 0.366100i \(-0.119307\pi\)
0.930576 + 0.366100i \(0.119307\pi\)
\(420\) 0 0
\(421\) −22.6633 −1.10454 −0.552272 0.833664i \(-0.686239\pi\)
−0.552272 + 0.833664i \(0.686239\pi\)
\(422\) 9.73797 0.474037
\(423\) 0 0
\(424\) 20.9284 1.01637
\(425\) −7.78051 −0.377410
\(426\) 0 0
\(427\) 7.66506 0.370938
\(428\) 4.89758 0.236734
\(429\) 0 0
\(430\) 8.02732 0.387112
\(431\) −33.9766 −1.63660 −0.818299 0.574793i \(-0.805082\pi\)
−0.818299 + 0.574793i \(0.805082\pi\)
\(432\) 0 0
\(433\) −36.6753 −1.76250 −0.881251 0.472649i \(-0.843298\pi\)
−0.881251 + 0.472649i \(0.843298\pi\)
\(434\) −5.93165 −0.284728
\(435\) 0 0
\(436\) 7.22801 0.346159
\(437\) 8.26953 0.395585
\(438\) 0 0
\(439\) −1.05012 −0.0501193 −0.0250596 0.999686i \(-0.507978\pi\)
−0.0250596 + 0.999686i \(0.507978\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −40.1667 −1.91054
\(443\) 30.5206 1.45008 0.725039 0.688708i \(-0.241822\pi\)
0.725039 + 0.688708i \(0.241822\pi\)
\(444\) 0 0
\(445\) −3.34722 −0.158674
\(446\) 5.93342 0.280956
\(447\) 0 0
\(448\) 5.72953 0.270695
\(449\) 36.5695 1.72582 0.862910 0.505357i \(-0.168639\pi\)
0.862910 + 0.505357i \(0.168639\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 3.13477 0.147447
\(453\) 0 0
\(454\) −9.24255 −0.433774
\(455\) −3.32307 −0.155788
\(456\) 0 0
\(457\) 2.38409 0.111523 0.0557615 0.998444i \(-0.482241\pi\)
0.0557615 + 0.998444i \(0.482241\pi\)
\(458\) −12.2731 −0.573483
\(459\) 0 0
\(460\) 5.51683 0.257223
\(461\) 28.5962 1.33186 0.665929 0.746015i \(-0.268035\pi\)
0.665929 + 0.746015i \(0.268035\pi\)
\(462\) 0 0
\(463\) −30.5806 −1.42120 −0.710600 0.703596i \(-0.751577\pi\)
−0.710600 + 0.703596i \(0.751577\pi\)
\(464\) −2.33465 −0.108383
\(465\) 0 0
\(466\) 6.39628 0.296302
\(467\) 3.74219 0.173168 0.0865840 0.996245i \(-0.472405\pi\)
0.0865840 + 0.996245i \(0.472405\pi\)
\(468\) 0 0
\(469\) −1.43853 −0.0664250
\(470\) 9.11534 0.420459
\(471\) 0 0
\(472\) −10.8582 −0.499788
\(473\) 0 0
\(474\) 0 0
\(475\) 1.19967 0.0550446
\(476\) 4.39026 0.201227
\(477\) 0 0
\(478\) −18.1010 −0.827920
\(479\) −0.944951 −0.0431759 −0.0215880 0.999767i \(-0.506872\pi\)
−0.0215880 + 0.999767i \(0.506872\pi\)
\(480\) 0 0
\(481\) 39.7693 1.81332
\(482\) 3.60548 0.164225
\(483\) 0 0
\(484\) 0 0
\(485\) −3.32228 −0.150857
\(486\) 0 0
\(487\) −13.3873 −0.606638 −0.303319 0.952889i \(-0.598095\pi\)
−0.303319 + 0.952889i \(0.598095\pi\)
\(488\) −33.3460 −1.50950
\(489\) 0 0
\(490\) 7.12261 0.321767
\(491\) −2.78887 −0.125860 −0.0629300 0.998018i \(-0.520045\pi\)
−0.0629300 + 0.998018i \(0.520045\pi\)
\(492\) 0 0
\(493\) −10.3279 −0.465145
\(494\) 6.19327 0.278648
\(495\) 0 0
\(496\) 13.5099 0.606611
\(497\) −0.472746 −0.0212056
\(498\) 0 0
\(499\) 17.7790 0.795899 0.397950 0.917407i \(-0.369722\pi\)
0.397950 + 0.917407i \(0.369722\pi\)
\(500\) 0.800331 0.0357919
\(501\) 0 0
\(502\) 8.10445 0.361719
\(503\) 25.7838 1.14964 0.574822 0.818279i \(-0.305071\pi\)
0.574822 + 0.818279i \(0.305071\pi\)
\(504\) 0 0
\(505\) 18.9218 0.842008
\(506\) 0 0
\(507\) 0 0
\(508\) 8.69761 0.385894
\(509\) 28.2301 1.25128 0.625639 0.780113i \(-0.284838\pi\)
0.625639 + 0.780113i \(0.284838\pi\)
\(510\) 0 0
\(511\) 3.52824 0.156080
\(512\) −18.1902 −0.803900
\(513\) 0 0
\(514\) 20.4810 0.903380
\(515\) 18.0964 0.797422
\(516\) 0 0
\(517\) 0 0
\(518\) 6.51573 0.286285
\(519\) 0 0
\(520\) 14.4567 0.633966
\(521\) −11.6955 −0.512388 −0.256194 0.966625i \(-0.582469\pi\)
−0.256194 + 0.966625i \(0.582469\pi\)
\(522\) 0 0
\(523\) 19.6871 0.860857 0.430429 0.902625i \(-0.358362\pi\)
0.430429 + 0.902625i \(0.358362\pi\)
\(524\) 9.37137 0.409390
\(525\) 0 0
\(526\) 4.37044 0.190560
\(527\) 59.7642 2.60337
\(528\) 0 0
\(529\) 24.5159 1.06591
\(530\) 7.47354 0.324630
\(531\) 0 0
\(532\) −0.676929 −0.0293486
\(533\) −1.09774 −0.0475484
\(534\) 0 0
\(535\) 6.11945 0.264567
\(536\) 6.25815 0.270311
\(537\) 0 0
\(538\) 13.9266 0.600420
\(539\) 0 0
\(540\) 0 0
\(541\) 5.45092 0.234353 0.117177 0.993111i \(-0.462616\pi\)
0.117177 + 0.993111i \(0.462616\pi\)
\(542\) −26.0987 −1.12103
\(543\) 0 0
\(544\) −32.7401 −1.40372
\(545\) 9.03128 0.386857
\(546\) 0 0
\(547\) −12.7892 −0.546827 −0.273414 0.961897i \(-0.588153\pi\)
−0.273414 + 0.961897i \(0.588153\pi\)
\(548\) 16.3509 0.698477
\(549\) 0 0
\(550\) 0 0
\(551\) 1.59245 0.0678405
\(552\) 0 0
\(553\) −1.60767 −0.0683653
\(554\) −8.11889 −0.344939
\(555\) 0 0
\(556\) −6.53951 −0.277337
\(557\) −0.354468 −0.0150193 −0.00750964 0.999972i \(-0.502390\pi\)
−0.00750964 + 0.999972i \(0.502390\pi\)
\(558\) 0 0
\(559\) −34.5436 −1.46104
\(560\) 1.24002 0.0524006
\(561\) 0 0
\(562\) −31.8528 −1.34363
\(563\) 35.0818 1.47852 0.739261 0.673419i \(-0.235175\pi\)
0.739261 + 0.673419i \(0.235175\pi\)
\(564\) 0 0
\(565\) 3.91684 0.164783
\(566\) 23.7382 0.997791
\(567\) 0 0
\(568\) 2.05663 0.0862943
\(569\) −16.3179 −0.684084 −0.342042 0.939685i \(-0.611119\pi\)
−0.342042 + 0.939685i \(0.611119\pi\)
\(570\) 0 0
\(571\) 14.4160 0.603291 0.301645 0.953420i \(-0.402464\pi\)
0.301645 + 0.953420i \(0.402464\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −0.179852 −0.00750686
\(575\) 6.89318 0.287465
\(576\) 0 0
\(577\) 8.90863 0.370871 0.185435 0.982656i \(-0.440630\pi\)
0.185435 + 0.982656i \(0.440630\pi\)
\(578\) 47.6852 1.98344
\(579\) 0 0
\(580\) 1.06236 0.0441122
\(581\) −1.48762 −0.0617168
\(582\) 0 0
\(583\) 0 0
\(584\) −15.3492 −0.635155
\(585\) 0 0
\(586\) −9.21383 −0.380620
\(587\) 13.8014 0.569644 0.284822 0.958580i \(-0.408066\pi\)
0.284822 + 0.958580i \(0.408066\pi\)
\(588\) 0 0
\(589\) −9.21497 −0.379696
\(590\) −3.87746 −0.159633
\(591\) 0 0
\(592\) −14.8402 −0.609927
\(593\) −16.4676 −0.676242 −0.338121 0.941103i \(-0.609791\pi\)
−0.338121 + 0.941103i \(0.609791\pi\)
\(594\) 0 0
\(595\) 5.48555 0.224886
\(596\) 7.68492 0.314787
\(597\) 0 0
\(598\) 35.5859 1.45522
\(599\) −37.8599 −1.54692 −0.773458 0.633848i \(-0.781475\pi\)
−0.773458 + 0.633848i \(0.781475\pi\)
\(600\) 0 0
\(601\) −1.56441 −0.0638135 −0.0319067 0.999491i \(-0.510158\pi\)
−0.0319067 + 0.999491i \(0.510158\pi\)
\(602\) −5.65956 −0.230666
\(603\) 0 0
\(604\) 5.52850 0.224951
\(605\) 0 0
\(606\) 0 0
\(607\) −18.8523 −0.765191 −0.382595 0.923916i \(-0.624970\pi\)
−0.382595 + 0.923916i \(0.624970\pi\)
\(608\) 5.04816 0.204730
\(609\) 0 0
\(610\) −11.9079 −0.482136
\(611\) −39.2256 −1.58690
\(612\) 0 0
\(613\) −32.6700 −1.31953 −0.659765 0.751472i \(-0.729344\pi\)
−0.659765 + 0.751472i \(0.729344\pi\)
\(614\) −26.0057 −1.04950
\(615\) 0 0
\(616\) 0 0
\(617\) 33.6559 1.35493 0.677467 0.735553i \(-0.263078\pi\)
0.677467 + 0.735553i \(0.263078\pi\)
\(618\) 0 0
\(619\) −8.94001 −0.359329 −0.179665 0.983728i \(-0.557501\pi\)
−0.179665 + 0.983728i \(0.557501\pi\)
\(620\) −6.14755 −0.246892
\(621\) 0 0
\(622\) −25.2433 −1.01216
\(623\) 2.35992 0.0945480
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −19.0185 −0.760131
\(627\) 0 0
\(628\) −9.08528 −0.362542
\(629\) −65.6491 −2.61760
\(630\) 0 0
\(631\) −19.2577 −0.766638 −0.383319 0.923616i \(-0.625219\pi\)
−0.383319 + 0.923616i \(0.625219\pi\)
\(632\) 6.99401 0.278207
\(633\) 0 0
\(634\) 6.73710 0.267565
\(635\) 10.8675 0.431264
\(636\) 0 0
\(637\) −30.6504 −1.21441
\(638\) 0 0
\(639\) 0 0
\(640\) −0.485063 −0.0191738
\(641\) 14.8928 0.588229 0.294114 0.955770i \(-0.404975\pi\)
0.294114 + 0.955770i \(0.404975\pi\)
\(642\) 0 0
\(643\) 35.3193 1.39286 0.696429 0.717626i \(-0.254771\pi\)
0.696429 + 0.717626i \(0.254771\pi\)
\(644\) −3.88957 −0.153270
\(645\) 0 0
\(646\) −10.2235 −0.402239
\(647\) −31.1428 −1.22435 −0.612175 0.790722i \(-0.709705\pi\)
−0.612175 + 0.790722i \(0.709705\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 5.16248 0.202489
\(651\) 0 0
\(652\) 1.43462 0.0561839
\(653\) −5.05494 −0.197815 −0.0989075 0.995097i \(-0.531535\pi\)
−0.0989075 + 0.995097i \(0.531535\pi\)
\(654\) 0 0
\(655\) 11.7094 0.457523
\(656\) 0.409628 0.0159933
\(657\) 0 0
\(658\) −6.42666 −0.250537
\(659\) −14.4486 −0.562837 −0.281419 0.959585i \(-0.590805\pi\)
−0.281419 + 0.959585i \(0.590805\pi\)
\(660\) 0 0
\(661\) 14.8696 0.578361 0.289181 0.957275i \(-0.406617\pi\)
0.289181 + 0.957275i \(0.406617\pi\)
\(662\) 21.2696 0.826667
\(663\) 0 0
\(664\) 6.47172 0.251152
\(665\) −0.845811 −0.0327991
\(666\) 0 0
\(667\) 9.15004 0.354291
\(668\) −4.82159 −0.186553
\(669\) 0 0
\(670\) 2.23479 0.0863375
\(671\) 0 0
\(672\) 0 0
\(673\) 41.6153 1.60415 0.802075 0.597223i \(-0.203729\pi\)
0.802075 + 0.597223i \(0.203729\pi\)
\(674\) −34.7064 −1.33684
\(675\) 0 0
\(676\) −7.37542 −0.283670
\(677\) 44.2366 1.70015 0.850076 0.526660i \(-0.176556\pi\)
0.850076 + 0.526660i \(0.176556\pi\)
\(678\) 0 0
\(679\) 2.34233 0.0898904
\(680\) −23.8643 −0.915153
\(681\) 0 0
\(682\) 0 0
\(683\) −24.5318 −0.938683 −0.469342 0.883017i \(-0.655509\pi\)
−0.469342 + 0.883017i \(0.655509\pi\)
\(684\) 0 0
\(685\) 20.4302 0.780598
\(686\) −10.4273 −0.398115
\(687\) 0 0
\(688\) 12.8902 0.491433
\(689\) −32.1605 −1.22522
\(690\) 0 0
\(691\) 23.7670 0.904139 0.452069 0.891983i \(-0.350686\pi\)
0.452069 + 0.891983i \(0.350686\pi\)
\(692\) −3.35078 −0.127378
\(693\) 0 0
\(694\) −30.6063 −1.16180
\(695\) −8.17100 −0.309944
\(696\) 0 0
\(697\) 1.81209 0.0686378
\(698\) 0.748449 0.0283292
\(699\) 0 0
\(700\) −0.564263 −0.0213271
\(701\) −32.6939 −1.23483 −0.617416 0.786637i \(-0.711820\pi\)
−0.617416 + 0.786637i \(0.711820\pi\)
\(702\) 0 0
\(703\) 10.1224 0.381772
\(704\) 0 0
\(705\) 0 0
\(706\) −1.70756 −0.0642648
\(707\) −13.3406 −0.501723
\(708\) 0 0
\(709\) 23.8264 0.894821 0.447410 0.894329i \(-0.352346\pi\)
0.447410 + 0.894329i \(0.352346\pi\)
\(710\) 0.734424 0.0275624
\(711\) 0 0
\(712\) −10.2666 −0.384755
\(713\) −52.9483 −1.98293
\(714\) 0 0
\(715\) 0 0
\(716\) 3.88293 0.145112
\(717\) 0 0
\(718\) 17.3499 0.647491
\(719\) 2.84680 0.106168 0.0530838 0.998590i \(-0.483095\pi\)
0.0530838 + 0.998590i \(0.483095\pi\)
\(720\) 0 0
\(721\) −12.7586 −0.475156
\(722\) −19.2342 −0.715824
\(723\) 0 0
\(724\) 18.4778 0.686723
\(725\) 1.32741 0.0492986
\(726\) 0 0
\(727\) −13.5192 −0.501399 −0.250700 0.968065i \(-0.580661\pi\)
−0.250700 + 0.968065i \(0.580661\pi\)
\(728\) −10.1925 −0.377758
\(729\) 0 0
\(730\) −5.48122 −0.202869
\(731\) 57.0227 2.10906
\(732\) 0 0
\(733\) −32.8352 −1.21280 −0.606399 0.795161i \(-0.707386\pi\)
−0.606399 + 0.795161i \(0.707386\pi\)
\(734\) 17.2357 0.636181
\(735\) 0 0
\(736\) 29.0062 1.06918
\(737\) 0 0
\(738\) 0 0
\(739\) −50.8927 −1.87212 −0.936058 0.351844i \(-0.885555\pi\)
−0.936058 + 0.351844i \(0.885555\pi\)
\(740\) 6.75289 0.248241
\(741\) 0 0
\(742\) −5.26912 −0.193435
\(743\) −15.4855 −0.568107 −0.284054 0.958808i \(-0.591679\pi\)
−0.284054 + 0.958808i \(0.591679\pi\)
\(744\) 0 0
\(745\) 9.60217 0.351796
\(746\) 5.77990 0.211617
\(747\) 0 0
\(748\) 0 0
\(749\) −4.31444 −0.157646
\(750\) 0 0
\(751\) 11.0575 0.403494 0.201747 0.979438i \(-0.435338\pi\)
0.201747 + 0.979438i \(0.435338\pi\)
\(752\) 14.6373 0.533767
\(753\) 0 0
\(754\) 6.85270 0.249561
\(755\) 6.90776 0.251399
\(756\) 0 0
\(757\) 9.00282 0.327213 0.163607 0.986526i \(-0.447687\pi\)
0.163607 + 0.986526i \(0.447687\pi\)
\(758\) 36.7025 1.33309
\(759\) 0 0
\(760\) 3.67961 0.133473
\(761\) 8.29644 0.300746 0.150373 0.988629i \(-0.451953\pi\)
0.150373 + 0.988629i \(0.451953\pi\)
\(762\) 0 0
\(763\) −6.36738 −0.230515
\(764\) 4.98263 0.180265
\(765\) 0 0
\(766\) 22.1603 0.800684
\(767\) 16.6857 0.602486
\(768\) 0 0
\(769\) −2.89088 −0.104248 −0.0521239 0.998641i \(-0.516599\pi\)
−0.0521239 + 0.998641i \(0.516599\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 15.2758 0.549787
\(773\) 27.8477 1.00161 0.500807 0.865559i \(-0.333037\pi\)
0.500807 + 0.865559i \(0.333037\pi\)
\(774\) 0 0
\(775\) −7.68126 −0.275919
\(776\) −10.1901 −0.365802
\(777\) 0 0
\(778\) 30.2004 1.08274
\(779\) −0.279404 −0.0100107
\(780\) 0 0
\(781\) 0 0
\(782\) −58.7433 −2.10066
\(783\) 0 0
\(784\) 11.4374 0.408478
\(785\) −11.3519 −0.405167
\(786\) 0 0
\(787\) 24.8356 0.885292 0.442646 0.896696i \(-0.354040\pi\)
0.442646 + 0.896696i \(0.354040\pi\)
\(788\) −5.44270 −0.193888
\(789\) 0 0
\(790\) 2.49757 0.0888594
\(791\) −2.76152 −0.0981883
\(792\) 0 0
\(793\) 51.2426 1.81968
\(794\) 12.8808 0.457121
\(795\) 0 0
\(796\) −17.0448 −0.604138
\(797\) 2.81107 0.0995731 0.0497866 0.998760i \(-0.484146\pi\)
0.0497866 + 0.998760i \(0.484146\pi\)
\(798\) 0 0
\(799\) 64.7516 2.29075
\(800\) 4.20796 0.148774
\(801\) 0 0
\(802\) 8.46012 0.298737
\(803\) 0 0
\(804\) 0 0
\(805\) −4.85995 −0.171291
\(806\) −39.6544 −1.39677
\(807\) 0 0
\(808\) 58.0366 2.04172
\(809\) 11.6766 0.410526 0.205263 0.978707i \(-0.434195\pi\)
0.205263 + 0.978707i \(0.434195\pi\)
\(810\) 0 0
\(811\) −24.0690 −0.845175 −0.422588 0.906322i \(-0.638878\pi\)
−0.422588 + 0.906322i \(0.638878\pi\)
\(812\) −0.749006 −0.0262849
\(813\) 0 0
\(814\) 0 0
\(815\) 1.79253 0.0627895
\(816\) 0 0
\(817\) −8.79227 −0.307603
\(818\) −18.3362 −0.641110
\(819\) 0 0
\(820\) −0.186398 −0.00650930
\(821\) 4.85561 0.169462 0.0847310 0.996404i \(-0.472997\pi\)
0.0847310 + 0.996404i \(0.472997\pi\)
\(822\) 0 0
\(823\) 26.0601 0.908397 0.454198 0.890901i \(-0.349926\pi\)
0.454198 + 0.890901i \(0.349926\pi\)
\(824\) 55.5050 1.93361
\(825\) 0 0
\(826\) 2.73376 0.0951195
\(827\) −16.2843 −0.566260 −0.283130 0.959081i \(-0.591373\pi\)
−0.283130 + 0.959081i \(0.591373\pi\)
\(828\) 0 0
\(829\) −10.0141 −0.347805 −0.173903 0.984763i \(-0.555638\pi\)
−0.173903 + 0.984763i \(0.555638\pi\)
\(830\) 2.31106 0.0802179
\(831\) 0 0
\(832\) 38.3032 1.32792
\(833\) 50.5961 1.75305
\(834\) 0 0
\(835\) −6.02450 −0.208486
\(836\) 0 0
\(837\) 0 0
\(838\) 41.7272 1.44144
\(839\) −11.7532 −0.405766 −0.202883 0.979203i \(-0.565031\pi\)
−0.202883 + 0.979203i \(0.565031\pi\)
\(840\) 0 0
\(841\) −27.2380 −0.939241
\(842\) −24.8230 −0.855458
\(843\) 0 0
\(844\) −7.11553 −0.244927
\(845\) −9.21546 −0.317021
\(846\) 0 0
\(847\) 0 0
\(848\) 12.0009 0.412113
\(849\) 0 0
\(850\) −8.52195 −0.292300
\(851\) 58.1621 1.99377
\(852\) 0 0
\(853\) 45.6353 1.56252 0.781262 0.624203i \(-0.214576\pi\)
0.781262 + 0.624203i \(0.214576\pi\)
\(854\) 8.39549 0.287288
\(855\) 0 0
\(856\) 18.7695 0.641527
\(857\) −8.41558 −0.287471 −0.143735 0.989616i \(-0.545911\pi\)
−0.143735 + 0.989616i \(0.545911\pi\)
\(858\) 0 0
\(859\) −45.3009 −1.54565 −0.772823 0.634622i \(-0.781156\pi\)
−0.772823 + 0.634622i \(0.781156\pi\)
\(860\) −5.86556 −0.200014
\(861\) 0 0
\(862\) −37.2144 −1.26753
\(863\) 12.0590 0.410493 0.205247 0.978710i \(-0.434200\pi\)
0.205247 + 0.978710i \(0.434200\pi\)
\(864\) 0 0
\(865\) −4.18674 −0.142354
\(866\) −40.1702 −1.36504
\(867\) 0 0
\(868\) 4.33425 0.147114
\(869\) 0 0
\(870\) 0 0
\(871\) −9.61687 −0.325855
\(872\) 27.7006 0.938061
\(873\) 0 0
\(874\) 9.05757 0.306377
\(875\) −0.705037 −0.0238346
\(876\) 0 0
\(877\) 21.1439 0.713979 0.356989 0.934108i \(-0.383803\pi\)
0.356989 + 0.934108i \(0.383803\pi\)
\(878\) −1.15019 −0.0388169
\(879\) 0 0
\(880\) 0 0
\(881\) 13.1669 0.443605 0.221803 0.975092i \(-0.428806\pi\)
0.221803 + 0.975092i \(0.428806\pi\)
\(882\) 0 0
\(883\) −34.7697 −1.17009 −0.585047 0.810999i \(-0.698924\pi\)
−0.585047 + 0.810999i \(0.698924\pi\)
\(884\) 29.3498 0.987142
\(885\) 0 0
\(886\) 33.4290 1.12307
\(887\) −20.9565 −0.703651 −0.351826 0.936066i \(-0.614439\pi\)
−0.351826 + 0.936066i \(0.614439\pi\)
\(888\) 0 0
\(889\) −7.66200 −0.256975
\(890\) −3.66619 −0.122891
\(891\) 0 0
\(892\) −4.33555 −0.145165
\(893\) −9.98398 −0.334101
\(894\) 0 0
\(895\) 4.85166 0.162173
\(896\) 0.341987 0.0114250
\(897\) 0 0
\(898\) 40.0543 1.33663
\(899\) −10.1961 −0.340061
\(900\) 0 0
\(901\) 53.0889 1.76865
\(902\) 0 0
\(903\) 0 0
\(904\) 12.0137 0.399569
\(905\) 23.0877 0.767462
\(906\) 0 0
\(907\) 26.2408 0.871312 0.435656 0.900113i \(-0.356516\pi\)
0.435656 + 0.900113i \(0.356516\pi\)
\(908\) 6.75353 0.224124
\(909\) 0 0
\(910\) −3.63974 −0.120656
\(911\) 39.6599 1.31399 0.656995 0.753895i \(-0.271827\pi\)
0.656995 + 0.753895i \(0.271827\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 2.61128 0.0863734
\(915\) 0 0
\(916\) 8.96793 0.296309
\(917\) −8.25554 −0.272622
\(918\) 0 0
\(919\) −14.9758 −0.494005 −0.247003 0.969015i \(-0.579446\pi\)
−0.247003 + 0.969015i \(0.579446\pi\)
\(920\) 21.1427 0.697053
\(921\) 0 0
\(922\) 31.3213 1.03151
\(923\) −3.16041 −0.104026
\(924\) 0 0
\(925\) 8.43763 0.277427
\(926\) −33.4947 −1.10071
\(927\) 0 0
\(928\) 5.58567 0.183359
\(929\) 49.6461 1.62884 0.814418 0.580279i \(-0.197057\pi\)
0.814418 + 0.580279i \(0.197057\pi\)
\(930\) 0 0
\(931\) −7.80135 −0.255679
\(932\) −4.67376 −0.153094
\(933\) 0 0
\(934\) 4.09880 0.134117
\(935\) 0 0
\(936\) 0 0
\(937\) 54.2914 1.77362 0.886811 0.462132i \(-0.152915\pi\)
0.886811 + 0.462132i \(0.152915\pi\)
\(938\) −1.57561 −0.0514455
\(939\) 0 0
\(940\) −6.66058 −0.217244
\(941\) 6.66066 0.217131 0.108566 0.994089i \(-0.465374\pi\)
0.108566 + 0.994089i \(0.465374\pi\)
\(942\) 0 0
\(943\) −1.60543 −0.0522800
\(944\) −6.22638 −0.202651
\(945\) 0 0
\(946\) 0 0
\(947\) 42.6250 1.38513 0.692563 0.721358i \(-0.256482\pi\)
0.692563 + 0.721358i \(0.256482\pi\)
\(948\) 0 0
\(949\) 23.5871 0.765669
\(950\) 1.31399 0.0426315
\(951\) 0 0
\(952\) 16.8252 0.545308
\(953\) −16.7539 −0.542713 −0.271357 0.962479i \(-0.587472\pi\)
−0.271357 + 0.962479i \(0.587472\pi\)
\(954\) 0 0
\(955\) 6.22571 0.201459
\(956\) 13.2264 0.427772
\(957\) 0 0
\(958\) −1.03500 −0.0334393
\(959\) −14.4040 −0.465131
\(960\) 0 0
\(961\) 28.0018 0.903284
\(962\) 43.5591 1.40440
\(963\) 0 0
\(964\) −2.63453 −0.0848524
\(965\) 19.0868 0.614427
\(966\) 0 0
\(967\) 50.1233 1.61186 0.805928 0.592014i \(-0.201667\pi\)
0.805928 + 0.592014i \(0.201667\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −3.63887 −0.116837
\(971\) −40.1230 −1.28761 −0.643804 0.765190i \(-0.722645\pi\)
−0.643804 + 0.765190i \(0.722645\pi\)
\(972\) 0 0
\(973\) 5.76086 0.184685
\(974\) −14.6631 −0.469835
\(975\) 0 0
\(976\) −19.1215 −0.612064
\(977\) 24.1682 0.773209 0.386604 0.922246i \(-0.373648\pi\)
0.386604 + 0.922246i \(0.373648\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −5.20449 −0.166251
\(981\) 0 0
\(982\) −3.05464 −0.0974774
\(983\) −23.3021 −0.743220 −0.371610 0.928389i \(-0.621194\pi\)
−0.371610 + 0.928389i \(0.621194\pi\)
\(984\) 0 0
\(985\) −6.80056 −0.216684
\(986\) −11.3121 −0.360250
\(987\) 0 0
\(988\) −4.52542 −0.143973
\(989\) −50.5195 −1.60643
\(990\) 0 0
\(991\) −54.0689 −1.71756 −0.858778 0.512348i \(-0.828776\pi\)
−0.858778 + 0.512348i \(0.828776\pi\)
\(992\) −32.3224 −1.02624
\(993\) 0 0
\(994\) −0.517796 −0.0164235
\(995\) −21.2972 −0.675168
\(996\) 0 0
\(997\) −50.7109 −1.60603 −0.803015 0.595959i \(-0.796772\pi\)
−0.803015 + 0.595959i \(0.796772\pi\)
\(998\) 19.4733 0.616416
\(999\) 0 0
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 5445.2.a.bf.1.4 4
3.2 odd 2 1815.2.a.w.1.1 4
11.7 odd 10 495.2.n.a.181.1 8
11.8 odd 10 495.2.n.a.361.1 8
11.10 odd 2 5445.2.a.bt.1.1 4
15.14 odd 2 9075.2.a.cm.1.4 4
33.8 even 10 165.2.m.d.31.2 yes 8
33.29 even 10 165.2.m.d.16.2 8
33.32 even 2 1815.2.a.p.1.4 4
165.8 odd 20 825.2.bx.f.724.2 16
165.29 even 10 825.2.n.g.676.1 8
165.62 odd 20 825.2.bx.f.49.2 16
165.74 even 10 825.2.n.g.526.1 8
165.107 odd 20 825.2.bx.f.724.3 16
165.128 odd 20 825.2.bx.f.49.3 16
165.164 even 2 9075.2.a.di.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.d.16.2 8 33.29 even 10
165.2.m.d.31.2 yes 8 33.8 even 10
495.2.n.a.181.1 8 11.7 odd 10
495.2.n.a.361.1 8 11.8 odd 10
825.2.n.g.526.1 8 165.74 even 10
825.2.n.g.676.1 8 165.29 even 10
825.2.bx.f.49.2 16 165.62 odd 20
825.2.bx.f.49.3 16 165.128 odd 20
825.2.bx.f.724.2 16 165.8 odd 20
825.2.bx.f.724.3 16 165.107 odd 20
1815.2.a.p.1.4 4 33.32 even 2
1815.2.a.w.1.1 4 3.2 odd 2
5445.2.a.bf.1.4 4 1.1 even 1 trivial
5445.2.a.bt.1.1 4 11.10 odd 2
9075.2.a.cm.1.4 4 15.14 odd 2
9075.2.a.di.1.1 4 165.164 even 2