Properties

Label 8034.2.a.bb.1.10
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-1.34813\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.34813 q^{5} -1.00000 q^{6} +2.71387 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.34813 q^{5} -1.00000 q^{6} +2.71387 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.34813 q^{10} -2.39135 q^{11} +1.00000 q^{12} -1.00000 q^{13} -2.71387 q^{14} +1.34813 q^{15} +1.00000 q^{16} -6.50957 q^{17} -1.00000 q^{18} -0.431762 q^{19} +1.34813 q^{20} +2.71387 q^{21} +2.39135 q^{22} -4.50549 q^{23} -1.00000 q^{24} -3.18255 q^{25} +1.00000 q^{26} +1.00000 q^{27} +2.71387 q^{28} -1.56456 q^{29} -1.34813 q^{30} +10.2484 q^{31} -1.00000 q^{32} -2.39135 q^{33} +6.50957 q^{34} +3.65864 q^{35} +1.00000 q^{36} -6.36996 q^{37} +0.431762 q^{38} -1.00000 q^{39} -1.34813 q^{40} -5.06676 q^{41} -2.71387 q^{42} +5.51257 q^{43} -2.39135 q^{44} +1.34813 q^{45} +4.50549 q^{46} -1.41946 q^{47} +1.00000 q^{48} +0.365083 q^{49} +3.18255 q^{50} -6.50957 q^{51} -1.00000 q^{52} -8.06106 q^{53} -1.00000 q^{54} -3.22385 q^{55} -2.71387 q^{56} -0.431762 q^{57} +1.56456 q^{58} +6.88037 q^{59} +1.34813 q^{60} -11.9507 q^{61} -10.2484 q^{62} +2.71387 q^{63} +1.00000 q^{64} -1.34813 q^{65} +2.39135 q^{66} +8.38179 q^{67} -6.50957 q^{68} -4.50549 q^{69} -3.65864 q^{70} +12.9886 q^{71} -1.00000 q^{72} -1.57753 q^{73} +6.36996 q^{74} -3.18255 q^{75} -0.431762 q^{76} -6.48982 q^{77} +1.00000 q^{78} +11.4119 q^{79} +1.34813 q^{80} +1.00000 q^{81} +5.06676 q^{82} -5.72642 q^{83} +2.71387 q^{84} -8.77573 q^{85} -5.51257 q^{86} -1.56456 q^{87} +2.39135 q^{88} -11.4913 q^{89} -1.34813 q^{90} -2.71387 q^{91} -4.50549 q^{92} +10.2484 q^{93} +1.41946 q^{94} -0.582070 q^{95} -1.00000 q^{96} -6.79470 q^{97} -0.365083 q^{98} -2.39135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.34813 0.602901 0.301450 0.953482i \(-0.402529\pi\)
0.301450 + 0.953482i \(0.402529\pi\)
\(6\) −1.00000 −0.408248
\(7\) 2.71387 1.02575 0.512873 0.858465i \(-0.328581\pi\)
0.512873 + 0.858465i \(0.328581\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.34813 −0.426315
\(11\) −2.39135 −0.721020 −0.360510 0.932755i \(-0.617397\pi\)
−0.360510 + 0.932755i \(0.617397\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) −2.71387 −0.725312
\(15\) 1.34813 0.348085
\(16\) 1.00000 0.250000
\(17\) −6.50957 −1.57880 −0.789402 0.613877i \(-0.789609\pi\)
−0.789402 + 0.613877i \(0.789609\pi\)
\(18\) −1.00000 −0.235702
\(19\) −0.431762 −0.0990531 −0.0495265 0.998773i \(-0.515771\pi\)
−0.0495265 + 0.998773i \(0.515771\pi\)
\(20\) 1.34813 0.301450
\(21\) 2.71387 0.592215
\(22\) 2.39135 0.509838
\(23\) −4.50549 −0.939460 −0.469730 0.882810i \(-0.655649\pi\)
−0.469730 + 0.882810i \(0.655649\pi\)
\(24\) −1.00000 −0.204124
\(25\) −3.18255 −0.636511
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 2.71387 0.512873
\(29\) −1.56456 −0.290531 −0.145266 0.989393i \(-0.546404\pi\)
−0.145266 + 0.989393i \(0.546404\pi\)
\(30\) −1.34813 −0.246133
\(31\) 10.2484 1.84067 0.920336 0.391128i \(-0.127915\pi\)
0.920336 + 0.391128i \(0.127915\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.39135 −0.416281
\(34\) 6.50957 1.11638
\(35\) 3.65864 0.618423
\(36\) 1.00000 0.166667
\(37\) −6.36996 −1.04721 −0.523607 0.851960i \(-0.675414\pi\)
−0.523607 + 0.851960i \(0.675414\pi\)
\(38\) 0.431762 0.0700411
\(39\) −1.00000 −0.160128
\(40\) −1.34813 −0.213158
\(41\) −5.06676 −0.791295 −0.395647 0.918403i \(-0.629480\pi\)
−0.395647 + 0.918403i \(0.629480\pi\)
\(42\) −2.71387 −0.418759
\(43\) 5.51257 0.840660 0.420330 0.907371i \(-0.361914\pi\)
0.420330 + 0.907371i \(0.361914\pi\)
\(44\) −2.39135 −0.360510
\(45\) 1.34813 0.200967
\(46\) 4.50549 0.664298
\(47\) −1.41946 −0.207050 −0.103525 0.994627i \(-0.533012\pi\)
−0.103525 + 0.994627i \(0.533012\pi\)
\(48\) 1.00000 0.144338
\(49\) 0.365083 0.0521547
\(50\) 3.18255 0.450081
\(51\) −6.50957 −0.911522
\(52\) −1.00000 −0.138675
\(53\) −8.06106 −1.10727 −0.553636 0.832759i \(-0.686760\pi\)
−0.553636 + 0.832759i \(0.686760\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.22385 −0.434703
\(56\) −2.71387 −0.362656
\(57\) −0.431762 −0.0571883
\(58\) 1.56456 0.205436
\(59\) 6.88037 0.895747 0.447874 0.894097i \(-0.352181\pi\)
0.447874 + 0.894097i \(0.352181\pi\)
\(60\) 1.34813 0.174042
\(61\) −11.9507 −1.53013 −0.765064 0.643954i \(-0.777293\pi\)
−0.765064 + 0.643954i \(0.777293\pi\)
\(62\) −10.2484 −1.30155
\(63\) 2.71387 0.341915
\(64\) 1.00000 0.125000
\(65\) −1.34813 −0.167215
\(66\) 2.39135 0.294355
\(67\) 8.38179 1.02400 0.511999 0.858986i \(-0.328905\pi\)
0.511999 + 0.858986i \(0.328905\pi\)
\(68\) −6.50957 −0.789402
\(69\) −4.50549 −0.542397
\(70\) −3.65864 −0.437291
\(71\) 12.9886 1.54146 0.770732 0.637160i \(-0.219891\pi\)
0.770732 + 0.637160i \(0.219891\pi\)
\(72\) −1.00000 −0.117851
\(73\) −1.57753 −0.184636 −0.0923182 0.995730i \(-0.529428\pi\)
−0.0923182 + 0.995730i \(0.529428\pi\)
\(74\) 6.36996 0.740492
\(75\) −3.18255 −0.367490
\(76\) −0.431762 −0.0495265
\(77\) −6.48982 −0.739583
\(78\) 1.00000 0.113228
\(79\) 11.4119 1.28394 0.641969 0.766730i \(-0.278118\pi\)
0.641969 + 0.766730i \(0.278118\pi\)
\(80\) 1.34813 0.150725
\(81\) 1.00000 0.111111
\(82\) 5.06676 0.559530
\(83\) −5.72642 −0.628556 −0.314278 0.949331i \(-0.601762\pi\)
−0.314278 + 0.949331i \(0.601762\pi\)
\(84\) 2.71387 0.296107
\(85\) −8.77573 −0.951861
\(86\) −5.51257 −0.594436
\(87\) −1.56456 −0.167738
\(88\) 2.39135 0.254919
\(89\) −11.4913 −1.21808 −0.609039 0.793140i \(-0.708445\pi\)
−0.609039 + 0.793140i \(0.708445\pi\)
\(90\) −1.34813 −0.142105
\(91\) −2.71387 −0.284491
\(92\) −4.50549 −0.469730
\(93\) 10.2484 1.06271
\(94\) 1.41946 0.146407
\(95\) −0.582070 −0.0597192
\(96\) −1.00000 −0.102062
\(97\) −6.79470 −0.689897 −0.344949 0.938622i \(-0.612104\pi\)
−0.344949 + 0.938622i \(0.612104\pi\)
\(98\) −0.365083 −0.0368790
\(99\) −2.39135 −0.240340
\(100\) −3.18255 −0.318255
\(101\) −14.9341 −1.48599 −0.742997 0.669295i \(-0.766596\pi\)
−0.742997 + 0.669295i \(0.766596\pi\)
\(102\) 6.50957 0.644544
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 3.65864 0.357047
\(106\) 8.06106 0.782959
\(107\) 8.05560 0.778764 0.389382 0.921076i \(-0.372689\pi\)
0.389382 + 0.921076i \(0.372689\pi\)
\(108\) 1.00000 0.0962250
\(109\) −3.57732 −0.342645 −0.171322 0.985215i \(-0.554804\pi\)
−0.171322 + 0.985215i \(0.554804\pi\)
\(110\) 3.22385 0.307382
\(111\) −6.36996 −0.604610
\(112\) 2.71387 0.256436
\(113\) −10.1751 −0.957192 −0.478596 0.878035i \(-0.658854\pi\)
−0.478596 + 0.878035i \(0.658854\pi\)
\(114\) 0.431762 0.0404383
\(115\) −6.07397 −0.566401
\(116\) −1.56456 −0.145266
\(117\) −1.00000 −0.0924500
\(118\) −6.88037 −0.633389
\(119\) −17.6661 −1.61945
\(120\) −1.34813 −0.123067
\(121\) −5.28143 −0.480130
\(122\) 11.9507 1.08196
\(123\) −5.06676 −0.456854
\(124\) 10.2484 0.920336
\(125\) −11.0311 −0.986653
\(126\) −2.71387 −0.241771
\(127\) −9.58182 −0.850249 −0.425124 0.905135i \(-0.639770\pi\)
−0.425124 + 0.905135i \(0.639770\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.51257 0.485355
\(130\) 1.34813 0.118239
\(131\) −9.94744 −0.869112 −0.434556 0.900645i \(-0.643095\pi\)
−0.434556 + 0.900645i \(0.643095\pi\)
\(132\) −2.39135 −0.208140
\(133\) −1.17175 −0.101603
\(134\) −8.38179 −0.724076
\(135\) 1.34813 0.116028
\(136\) 6.50957 0.558191
\(137\) 3.56849 0.304877 0.152438 0.988313i \(-0.451287\pi\)
0.152438 + 0.988313i \(0.451287\pi\)
\(138\) 4.50549 0.383533
\(139\) −16.3086 −1.38327 −0.691637 0.722245i \(-0.743110\pi\)
−0.691637 + 0.722245i \(0.743110\pi\)
\(140\) 3.65864 0.309211
\(141\) −1.41946 −0.119540
\(142\) −12.9886 −1.08998
\(143\) 2.39135 0.199975
\(144\) 1.00000 0.0833333
\(145\) −2.10922 −0.175161
\(146\) 1.57753 0.130558
\(147\) 0.365083 0.0301115
\(148\) −6.36996 −0.523607
\(149\) −1.55123 −0.127082 −0.0635409 0.997979i \(-0.520239\pi\)
−0.0635409 + 0.997979i \(0.520239\pi\)
\(150\) 3.18255 0.259854
\(151\) −14.3699 −1.16941 −0.584703 0.811247i \(-0.698789\pi\)
−0.584703 + 0.811247i \(0.698789\pi\)
\(152\) 0.431762 0.0350206
\(153\) −6.50957 −0.526268
\(154\) 6.48982 0.522964
\(155\) 13.8162 1.10974
\(156\) −1.00000 −0.0800641
\(157\) 8.54243 0.681760 0.340880 0.940107i \(-0.389275\pi\)
0.340880 + 0.940107i \(0.389275\pi\)
\(158\) −11.4119 −0.907882
\(159\) −8.06106 −0.639283
\(160\) −1.34813 −0.106579
\(161\) −12.2273 −0.963647
\(162\) −1.00000 −0.0785674
\(163\) −0.232060 −0.0181763 −0.00908816 0.999959i \(-0.502893\pi\)
−0.00908816 + 0.999959i \(0.502893\pi\)
\(164\) −5.06676 −0.395647
\(165\) −3.22385 −0.250976
\(166\) 5.72642 0.444456
\(167\) −1.99495 −0.154374 −0.0771868 0.997017i \(-0.524594\pi\)
−0.0771868 + 0.997017i \(0.524594\pi\)
\(168\) −2.71387 −0.209380
\(169\) 1.00000 0.0769231
\(170\) 8.77573 0.673068
\(171\) −0.431762 −0.0330177
\(172\) 5.51257 0.420330
\(173\) 24.4936 1.86222 0.931108 0.364743i \(-0.118843\pi\)
0.931108 + 0.364743i \(0.118843\pi\)
\(174\) 1.56456 0.118609
\(175\) −8.63703 −0.652898
\(176\) −2.39135 −0.180255
\(177\) 6.88037 0.517160
\(178\) 11.4913 0.861312
\(179\) −24.9563 −1.86532 −0.932660 0.360758i \(-0.882518\pi\)
−0.932660 + 0.360758i \(0.882518\pi\)
\(180\) 1.34813 0.100483
\(181\) 19.7726 1.46968 0.734842 0.678238i \(-0.237256\pi\)
0.734842 + 0.678238i \(0.237256\pi\)
\(182\) 2.71387 0.201165
\(183\) −11.9507 −0.883420
\(184\) 4.50549 0.332149
\(185\) −8.58751 −0.631366
\(186\) −10.2484 −0.751451
\(187\) 15.5667 1.13835
\(188\) −1.41946 −0.103525
\(189\) 2.71387 0.197405
\(190\) 0.582070 0.0422278
\(191\) 4.73569 0.342662 0.171331 0.985214i \(-0.445193\pi\)
0.171331 + 0.985214i \(0.445193\pi\)
\(192\) 1.00000 0.0721688
\(193\) −6.52330 −0.469558 −0.234779 0.972049i \(-0.575437\pi\)
−0.234779 + 0.972049i \(0.575437\pi\)
\(194\) 6.79470 0.487831
\(195\) −1.34813 −0.0965414
\(196\) 0.365083 0.0260774
\(197\) −8.88078 −0.632729 −0.316365 0.948638i \(-0.602462\pi\)
−0.316365 + 0.948638i \(0.602462\pi\)
\(198\) 2.39135 0.169946
\(199\) −23.6494 −1.67646 −0.838231 0.545315i \(-0.816410\pi\)
−0.838231 + 0.545315i \(0.816410\pi\)
\(200\) 3.18255 0.225041
\(201\) 8.38179 0.591206
\(202\) 14.9341 1.05076
\(203\) −4.24600 −0.298011
\(204\) −6.50957 −0.455761
\(205\) −6.83063 −0.477072
\(206\) −1.00000 −0.0696733
\(207\) −4.50549 −0.313153
\(208\) −1.00000 −0.0693375
\(209\) 1.03250 0.0714192
\(210\) −3.65864 −0.252470
\(211\) −11.2030 −0.771249 −0.385625 0.922656i \(-0.626014\pi\)
−0.385625 + 0.922656i \(0.626014\pi\)
\(212\) −8.06106 −0.553636
\(213\) 12.9886 0.889964
\(214\) −8.05560 −0.550670
\(215\) 7.43165 0.506834
\(216\) −1.00000 −0.0680414
\(217\) 27.8129 1.88806
\(218\) 3.57732 0.242287
\(219\) −1.57753 −0.106600
\(220\) −3.22385 −0.217352
\(221\) 6.50957 0.437881
\(222\) 6.36996 0.427524
\(223\) 27.5336 1.84379 0.921893 0.387445i \(-0.126642\pi\)
0.921893 + 0.387445i \(0.126642\pi\)
\(224\) −2.71387 −0.181328
\(225\) −3.18255 −0.212170
\(226\) 10.1751 0.676837
\(227\) 23.3657 1.55083 0.775416 0.631450i \(-0.217540\pi\)
0.775416 + 0.631450i \(0.217540\pi\)
\(228\) −0.431762 −0.0285942
\(229\) −13.7682 −0.909831 −0.454916 0.890535i \(-0.650331\pi\)
−0.454916 + 0.890535i \(0.650331\pi\)
\(230\) 6.07397 0.400506
\(231\) −6.48982 −0.426999
\(232\) 1.56456 0.102718
\(233\) 22.0201 1.44259 0.721294 0.692629i \(-0.243548\pi\)
0.721294 + 0.692629i \(0.243548\pi\)
\(234\) 1.00000 0.0653720
\(235\) −1.91362 −0.124831
\(236\) 6.88037 0.447874
\(237\) 11.4119 0.741282
\(238\) 17.6661 1.14512
\(239\) −23.6471 −1.52960 −0.764800 0.644267i \(-0.777162\pi\)
−0.764800 + 0.644267i \(0.777162\pi\)
\(240\) 1.34813 0.0870212
\(241\) −13.2240 −0.851830 −0.425915 0.904763i \(-0.640048\pi\)
−0.425915 + 0.904763i \(0.640048\pi\)
\(242\) 5.28143 0.339503
\(243\) 1.00000 0.0641500
\(244\) −11.9507 −0.765064
\(245\) 0.492178 0.0314441
\(246\) 5.06676 0.323045
\(247\) 0.431762 0.0274724
\(248\) −10.2484 −0.650776
\(249\) −5.72642 −0.362897
\(250\) 11.0311 0.697669
\(251\) 4.85647 0.306538 0.153269 0.988185i \(-0.451020\pi\)
0.153269 + 0.988185i \(0.451020\pi\)
\(252\) 2.71387 0.170958
\(253\) 10.7742 0.677369
\(254\) 9.58182 0.601217
\(255\) −8.77573 −0.549557
\(256\) 1.00000 0.0625000
\(257\) −11.7803 −0.734834 −0.367417 0.930056i \(-0.619758\pi\)
−0.367417 + 0.930056i \(0.619758\pi\)
\(258\) −5.51257 −0.343198
\(259\) −17.2872 −1.07418
\(260\) −1.34813 −0.0836073
\(261\) −1.56456 −0.0968437
\(262\) 9.94744 0.614555
\(263\) 6.54794 0.403763 0.201882 0.979410i \(-0.435294\pi\)
0.201882 + 0.979410i \(0.435294\pi\)
\(264\) 2.39135 0.147178
\(265\) −10.8673 −0.667575
\(266\) 1.17175 0.0718444
\(267\) −11.4913 −0.703258
\(268\) 8.38179 0.511999
\(269\) 4.08408 0.249011 0.124505 0.992219i \(-0.460266\pi\)
0.124505 + 0.992219i \(0.460266\pi\)
\(270\) −1.34813 −0.0820444
\(271\) −9.81272 −0.596080 −0.298040 0.954553i \(-0.596333\pi\)
−0.298040 + 0.954553i \(0.596333\pi\)
\(272\) −6.50957 −0.394701
\(273\) −2.71387 −0.164251
\(274\) −3.56849 −0.215580
\(275\) 7.61061 0.458937
\(276\) −4.50549 −0.271199
\(277\) −10.6253 −0.638413 −0.319206 0.947685i \(-0.603416\pi\)
−0.319206 + 0.947685i \(0.603416\pi\)
\(278\) 16.3086 0.978122
\(279\) 10.2484 0.613558
\(280\) −3.65864 −0.218645
\(281\) −11.0617 −0.659884 −0.329942 0.944001i \(-0.607029\pi\)
−0.329942 + 0.944001i \(0.607029\pi\)
\(282\) 1.41946 0.0845278
\(283\) 6.36457 0.378334 0.189167 0.981945i \(-0.439421\pi\)
0.189167 + 0.981945i \(0.439421\pi\)
\(284\) 12.9886 0.770732
\(285\) −0.582070 −0.0344789
\(286\) −2.39135 −0.141404
\(287\) −13.7505 −0.811667
\(288\) −1.00000 −0.0589256
\(289\) 25.3745 1.49262
\(290\) 2.10922 0.123858
\(291\) −6.79470 −0.398312
\(292\) −1.57753 −0.0923182
\(293\) 7.74495 0.452465 0.226232 0.974073i \(-0.427359\pi\)
0.226232 + 0.974073i \(0.427359\pi\)
\(294\) −0.365083 −0.0212921
\(295\) 9.27560 0.540047
\(296\) 6.36996 0.370246
\(297\) −2.39135 −0.138760
\(298\) 1.55123 0.0898605
\(299\) 4.50549 0.260559
\(300\) −3.18255 −0.183745
\(301\) 14.9604 0.862303
\(302\) 14.3699 0.826895
\(303\) −14.9341 −0.857939
\(304\) −0.431762 −0.0247633
\(305\) −16.1110 −0.922516
\(306\) 6.50957 0.372127
\(307\) −12.8933 −0.735857 −0.367928 0.929854i \(-0.619933\pi\)
−0.367928 + 0.929854i \(0.619933\pi\)
\(308\) −6.48982 −0.369792
\(309\) 1.00000 0.0568880
\(310\) −13.8162 −0.784707
\(311\) 22.5094 1.27639 0.638197 0.769873i \(-0.279681\pi\)
0.638197 + 0.769873i \(0.279681\pi\)
\(312\) 1.00000 0.0566139
\(313\) −14.2391 −0.804843 −0.402421 0.915455i \(-0.631831\pi\)
−0.402421 + 0.915455i \(0.631831\pi\)
\(314\) −8.54243 −0.482077
\(315\) 3.65864 0.206141
\(316\) 11.4119 0.641969
\(317\) 7.36812 0.413835 0.206917 0.978358i \(-0.433657\pi\)
0.206917 + 0.978358i \(0.433657\pi\)
\(318\) 8.06106 0.452042
\(319\) 3.74141 0.209479
\(320\) 1.34813 0.0753626
\(321\) 8.05560 0.449620
\(322\) 12.2273 0.681401
\(323\) 2.81059 0.156385
\(324\) 1.00000 0.0555556
\(325\) 3.18255 0.176536
\(326\) 0.232060 0.0128526
\(327\) −3.57732 −0.197826
\(328\) 5.06676 0.279765
\(329\) −3.85224 −0.212381
\(330\) 3.22385 0.177467
\(331\) 18.1845 0.999509 0.499755 0.866167i \(-0.333424\pi\)
0.499755 + 0.866167i \(0.333424\pi\)
\(332\) −5.72642 −0.314278
\(333\) −6.36996 −0.349072
\(334\) 1.99495 0.109159
\(335\) 11.2997 0.617369
\(336\) 2.71387 0.148054
\(337\) 30.5645 1.66496 0.832478 0.554059i \(-0.186922\pi\)
0.832478 + 0.554059i \(0.186922\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −10.1751 −0.552635
\(340\) −8.77573 −0.475931
\(341\) −24.5076 −1.32716
\(342\) 0.431762 0.0233470
\(343\) −18.0063 −0.972248
\(344\) −5.51257 −0.297218
\(345\) −6.07397 −0.327012
\(346\) −24.4936 −1.31679
\(347\) −25.2946 −1.35788 −0.678941 0.734192i \(-0.737561\pi\)
−0.678941 + 0.734192i \(0.737561\pi\)
\(348\) −1.56456 −0.0838691
\(349\) 14.5811 0.780510 0.390255 0.920707i \(-0.372387\pi\)
0.390255 + 0.920707i \(0.372387\pi\)
\(350\) 8.63703 0.461669
\(351\) −1.00000 −0.0533761
\(352\) 2.39135 0.127460
\(353\) −29.6881 −1.58014 −0.790069 0.613018i \(-0.789955\pi\)
−0.790069 + 0.613018i \(0.789955\pi\)
\(354\) −6.88037 −0.365687
\(355\) 17.5103 0.929349
\(356\) −11.4913 −0.609039
\(357\) −17.6661 −0.934990
\(358\) 24.9563 1.31898
\(359\) −15.1234 −0.798185 −0.399092 0.916911i \(-0.630675\pi\)
−0.399092 + 0.916911i \(0.630675\pi\)
\(360\) −1.34813 −0.0710525
\(361\) −18.8136 −0.990188
\(362\) −19.7726 −1.03922
\(363\) −5.28143 −0.277203
\(364\) −2.71387 −0.142245
\(365\) −2.12671 −0.111317
\(366\) 11.9507 0.624673
\(367\) −10.8564 −0.566701 −0.283350 0.959016i \(-0.591446\pi\)
−0.283350 + 0.959016i \(0.591446\pi\)
\(368\) −4.50549 −0.234865
\(369\) −5.06676 −0.263765
\(370\) 8.58751 0.446443
\(371\) −21.8766 −1.13578
\(372\) 10.2484 0.531356
\(373\) 7.66103 0.396673 0.198337 0.980134i \(-0.436446\pi\)
0.198337 + 0.980134i \(0.436446\pi\)
\(374\) −15.5667 −0.804934
\(375\) −11.0311 −0.569645
\(376\) 1.41946 0.0732033
\(377\) 1.56456 0.0805788
\(378\) −2.71387 −0.139586
\(379\) 10.0157 0.514471 0.257236 0.966349i \(-0.417188\pi\)
0.257236 + 0.966349i \(0.417188\pi\)
\(380\) −0.582070 −0.0298596
\(381\) −9.58182 −0.490891
\(382\) −4.73569 −0.242299
\(383\) 17.7480 0.906883 0.453441 0.891286i \(-0.350196\pi\)
0.453441 + 0.891286i \(0.350196\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −8.74909 −0.445895
\(386\) 6.52330 0.332027
\(387\) 5.51257 0.280220
\(388\) −6.79470 −0.344949
\(389\) 17.6122 0.892973 0.446486 0.894790i \(-0.352675\pi\)
0.446486 + 0.894790i \(0.352675\pi\)
\(390\) 1.34813 0.0682651
\(391\) 29.3288 1.48322
\(392\) −0.365083 −0.0184395
\(393\) −9.94744 −0.501782
\(394\) 8.88078 0.447407
\(395\) 15.3847 0.774087
\(396\) −2.39135 −0.120170
\(397\) 15.9439 0.800199 0.400100 0.916472i \(-0.368975\pi\)
0.400100 + 0.916472i \(0.368975\pi\)
\(398\) 23.6494 1.18544
\(399\) −1.17175 −0.0586607
\(400\) −3.18255 −0.159128
\(401\) −13.5652 −0.677414 −0.338707 0.940892i \(-0.609990\pi\)
−0.338707 + 0.940892i \(0.609990\pi\)
\(402\) −8.38179 −0.418046
\(403\) −10.2484 −0.510511
\(404\) −14.9341 −0.742997
\(405\) 1.34813 0.0669890
\(406\) 4.24600 0.210726
\(407\) 15.2328 0.755062
\(408\) 6.50957 0.322272
\(409\) 23.6532 1.16958 0.584788 0.811186i \(-0.301178\pi\)
0.584788 + 0.811186i \(0.301178\pi\)
\(410\) 6.83063 0.337341
\(411\) 3.56849 0.176021
\(412\) 1.00000 0.0492665
\(413\) 18.6724 0.918809
\(414\) 4.50549 0.221433
\(415\) −7.71994 −0.378957
\(416\) 1.00000 0.0490290
\(417\) −16.3086 −0.798634
\(418\) −1.03250 −0.0505010
\(419\) 30.1554 1.47319 0.736594 0.676336i \(-0.236433\pi\)
0.736594 + 0.676336i \(0.236433\pi\)
\(420\) 3.65864 0.178523
\(421\) 13.2847 0.647456 0.323728 0.946150i \(-0.395064\pi\)
0.323728 + 0.946150i \(0.395064\pi\)
\(422\) 11.2030 0.545356
\(423\) −1.41946 −0.0690167
\(424\) 8.06106 0.391479
\(425\) 20.7171 1.00493
\(426\) −12.9886 −0.629300
\(427\) −32.4326 −1.56952
\(428\) 8.05560 0.389382
\(429\) 2.39135 0.115456
\(430\) −7.43165 −0.358386
\(431\) −8.15311 −0.392721 −0.196361 0.980532i \(-0.562912\pi\)
−0.196361 + 0.980532i \(0.562912\pi\)
\(432\) 1.00000 0.0481125
\(433\) −1.67772 −0.0806258 −0.0403129 0.999187i \(-0.512835\pi\)
−0.0403129 + 0.999187i \(0.512835\pi\)
\(434\) −27.8129 −1.33506
\(435\) −2.10922 −0.101129
\(436\) −3.57732 −0.171322
\(437\) 1.94530 0.0930564
\(438\) 1.57753 0.0753775
\(439\) 38.8832 1.85579 0.927897 0.372836i \(-0.121615\pi\)
0.927897 + 0.372836i \(0.121615\pi\)
\(440\) 3.22385 0.153691
\(441\) 0.365083 0.0173849
\(442\) −6.50957 −0.309629
\(443\) −18.8022 −0.893319 −0.446659 0.894704i \(-0.647386\pi\)
−0.446659 + 0.894704i \(0.647386\pi\)
\(444\) −6.36996 −0.302305
\(445\) −15.4918 −0.734380
\(446\) −27.5336 −1.30375
\(447\) −1.55123 −0.0733708
\(448\) 2.71387 0.128218
\(449\) −4.65776 −0.219813 −0.109907 0.993942i \(-0.535055\pi\)
−0.109907 + 0.993942i \(0.535055\pi\)
\(450\) 3.18255 0.150027
\(451\) 12.1164 0.570539
\(452\) −10.1751 −0.478596
\(453\) −14.3699 −0.675157
\(454\) −23.3657 −1.09660
\(455\) −3.65864 −0.171520
\(456\) 0.431762 0.0202191
\(457\) 16.6855 0.780514 0.390257 0.920706i \(-0.372386\pi\)
0.390257 + 0.920706i \(0.372386\pi\)
\(458\) 13.7682 0.643348
\(459\) −6.50957 −0.303841
\(460\) −6.07397 −0.283200
\(461\) −10.4652 −0.487413 −0.243706 0.969849i \(-0.578363\pi\)
−0.243706 + 0.969849i \(0.578363\pi\)
\(462\) 6.48982 0.301934
\(463\) 31.2686 1.45318 0.726588 0.687073i \(-0.241105\pi\)
0.726588 + 0.687073i \(0.241105\pi\)
\(464\) −1.56456 −0.0726328
\(465\) 13.8162 0.640710
\(466\) −22.0201 −1.02006
\(467\) −15.9687 −0.738942 −0.369471 0.929242i \(-0.620461\pi\)
−0.369471 + 0.929242i \(0.620461\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 22.7471 1.05036
\(470\) 1.91362 0.0882686
\(471\) 8.54243 0.393614
\(472\) −6.88037 −0.316694
\(473\) −13.1825 −0.606132
\(474\) −11.4119 −0.524166
\(475\) 1.37411 0.0630484
\(476\) −17.6661 −0.809725
\(477\) −8.06106 −0.369090
\(478\) 23.6471 1.08159
\(479\) 18.5527 0.847693 0.423846 0.905734i \(-0.360680\pi\)
0.423846 + 0.905734i \(0.360680\pi\)
\(480\) −1.34813 −0.0615333
\(481\) 6.36996 0.290445
\(482\) 13.2240 0.602335
\(483\) −12.2273 −0.556362
\(484\) −5.28143 −0.240065
\(485\) −9.16011 −0.415939
\(486\) −1.00000 −0.0453609
\(487\) −35.5161 −1.60939 −0.804695 0.593689i \(-0.797671\pi\)
−0.804695 + 0.593689i \(0.797671\pi\)
\(488\) 11.9507 0.540982
\(489\) −0.232060 −0.0104941
\(490\) −0.492178 −0.0222343
\(491\) 5.28868 0.238675 0.119337 0.992854i \(-0.461923\pi\)
0.119337 + 0.992854i \(0.461923\pi\)
\(492\) −5.06676 −0.228427
\(493\) 10.1846 0.458691
\(494\) −0.431762 −0.0194259
\(495\) −3.22385 −0.144901
\(496\) 10.2484 0.460168
\(497\) 35.2494 1.58115
\(498\) 5.72642 0.256607
\(499\) −10.8970 −0.487815 −0.243908 0.969798i \(-0.578429\pi\)
−0.243908 + 0.969798i \(0.578429\pi\)
\(500\) −11.0311 −0.493327
\(501\) −1.99495 −0.0891276
\(502\) −4.85647 −0.216755
\(503\) 5.95779 0.265645 0.132822 0.991140i \(-0.457596\pi\)
0.132822 + 0.991140i \(0.457596\pi\)
\(504\) −2.71387 −0.120885
\(505\) −20.1330 −0.895907
\(506\) −10.7742 −0.478972
\(507\) 1.00000 0.0444116
\(508\) −9.58182 −0.425124
\(509\) −6.66761 −0.295537 −0.147768 0.989022i \(-0.547209\pi\)
−0.147768 + 0.989022i \(0.547209\pi\)
\(510\) 8.77573 0.388596
\(511\) −4.28122 −0.189390
\(512\) −1.00000 −0.0441942
\(513\) −0.431762 −0.0190628
\(514\) 11.7803 0.519606
\(515\) 1.34813 0.0594056
\(516\) 5.51257 0.242678
\(517\) 3.39444 0.149287
\(518\) 17.2872 0.759557
\(519\) 24.4936 1.07515
\(520\) 1.34813 0.0591193
\(521\) 9.06386 0.397095 0.198547 0.980091i \(-0.436378\pi\)
0.198547 + 0.980091i \(0.436378\pi\)
\(522\) 1.56456 0.0684788
\(523\) −21.9878 −0.961460 −0.480730 0.876869i \(-0.659628\pi\)
−0.480730 + 0.876869i \(0.659628\pi\)
\(524\) −9.94744 −0.434556
\(525\) −8.63703 −0.376951
\(526\) −6.54794 −0.285504
\(527\) −66.7129 −2.90606
\(528\) −2.39135 −0.104070
\(529\) −2.70056 −0.117416
\(530\) 10.8673 0.472046
\(531\) 6.88037 0.298582
\(532\) −1.17175 −0.0508017
\(533\) 5.06676 0.219466
\(534\) 11.4913 0.497278
\(535\) 10.8600 0.469518
\(536\) −8.38179 −0.362038
\(537\) −24.9563 −1.07694
\(538\) −4.08408 −0.176077
\(539\) −0.873042 −0.0376046
\(540\) 1.34813 0.0580141
\(541\) −7.02793 −0.302154 −0.151077 0.988522i \(-0.548274\pi\)
−0.151077 + 0.988522i \(0.548274\pi\)
\(542\) 9.81272 0.421492
\(543\) 19.7726 0.848523
\(544\) 6.50957 0.279096
\(545\) −4.82268 −0.206581
\(546\) 2.71387 0.116143
\(547\) −36.3334 −1.55350 −0.776752 0.629806i \(-0.783134\pi\)
−0.776752 + 0.629806i \(0.783134\pi\)
\(548\) 3.56849 0.152438
\(549\) −11.9507 −0.510043
\(550\) −7.61061 −0.324517
\(551\) 0.675517 0.0287780
\(552\) 4.50549 0.191766
\(553\) 30.9704 1.31699
\(554\) 10.6253 0.451426
\(555\) −8.58751 −0.364519
\(556\) −16.3086 −0.691637
\(557\) −19.8696 −0.841902 −0.420951 0.907083i \(-0.638304\pi\)
−0.420951 + 0.907083i \(0.638304\pi\)
\(558\) −10.2484 −0.433851
\(559\) −5.51257 −0.233157
\(560\) 3.65864 0.154606
\(561\) 15.5667 0.657226
\(562\) 11.0617 0.466608
\(563\) 13.2306 0.557602 0.278801 0.960349i \(-0.410063\pi\)
0.278801 + 0.960349i \(0.410063\pi\)
\(564\) −1.41946 −0.0597702
\(565\) −13.7173 −0.577092
\(566\) −6.36457 −0.267523
\(567\) 2.71387 0.113972
\(568\) −12.9886 −0.544990
\(569\) −7.31748 −0.306765 −0.153382 0.988167i \(-0.549017\pi\)
−0.153382 + 0.988167i \(0.549017\pi\)
\(570\) 0.582070 0.0243802
\(571\) 24.0308 1.00566 0.502828 0.864386i \(-0.332293\pi\)
0.502828 + 0.864386i \(0.332293\pi\)
\(572\) 2.39135 0.0999875
\(573\) 4.73569 0.197836
\(574\) 13.7505 0.573935
\(575\) 14.3390 0.597976
\(576\) 1.00000 0.0416667
\(577\) 23.1047 0.961859 0.480930 0.876759i \(-0.340299\pi\)
0.480930 + 0.876759i \(0.340299\pi\)
\(578\) −25.3745 −1.05544
\(579\) −6.52330 −0.271099
\(580\) −2.10922 −0.0875807
\(581\) −15.5408 −0.644739
\(582\) 6.79470 0.281649
\(583\) 19.2768 0.798364
\(584\) 1.57753 0.0652788
\(585\) −1.34813 −0.0557382
\(586\) −7.74495 −0.319941
\(587\) 3.64806 0.150571 0.0752857 0.997162i \(-0.476013\pi\)
0.0752857 + 0.997162i \(0.476013\pi\)
\(588\) 0.365083 0.0150558
\(589\) −4.42489 −0.182324
\(590\) −9.27560 −0.381871
\(591\) −8.88078 −0.365306
\(592\) −6.36996 −0.261804
\(593\) −43.6328 −1.79179 −0.895893 0.444270i \(-0.853463\pi\)
−0.895893 + 0.444270i \(0.853463\pi\)
\(594\) 2.39135 0.0981184
\(595\) −23.8162 −0.976368
\(596\) −1.55123 −0.0635409
\(597\) −23.6494 −0.967906
\(598\) −4.50549 −0.184243
\(599\) 26.7488 1.09293 0.546463 0.837483i \(-0.315974\pi\)
0.546463 + 0.837483i \(0.315974\pi\)
\(600\) 3.18255 0.129927
\(601\) 17.5744 0.716873 0.358436 0.933554i \(-0.383310\pi\)
0.358436 + 0.933554i \(0.383310\pi\)
\(602\) −14.9604 −0.609740
\(603\) 8.38179 0.341333
\(604\) −14.3699 −0.584703
\(605\) −7.12004 −0.289471
\(606\) 14.9341 0.606655
\(607\) 45.0663 1.82919 0.914593 0.404375i \(-0.132511\pi\)
0.914593 + 0.404375i \(0.132511\pi\)
\(608\) 0.431762 0.0175103
\(609\) −4.24600 −0.172057
\(610\) 16.1110 0.652317
\(611\) 1.41946 0.0574254
\(612\) −6.50957 −0.263134
\(613\) −10.3838 −0.419399 −0.209700 0.977766i \(-0.567249\pi\)
−0.209700 + 0.977766i \(0.567249\pi\)
\(614\) 12.8933 0.520329
\(615\) −6.83063 −0.275438
\(616\) 6.48982 0.261482
\(617\) 30.9502 1.24601 0.623004 0.782218i \(-0.285912\pi\)
0.623004 + 0.782218i \(0.285912\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 12.0955 0.486159 0.243080 0.970006i \(-0.421842\pi\)
0.243080 + 0.970006i \(0.421842\pi\)
\(620\) 13.8162 0.554871
\(621\) −4.50549 −0.180799
\(622\) −22.5094 −0.902547
\(623\) −31.1860 −1.24944
\(624\) −1.00000 −0.0400320
\(625\) 1.04142 0.0416569
\(626\) 14.2391 0.569110
\(627\) 1.03250 0.0412339
\(628\) 8.54243 0.340880
\(629\) 41.4657 1.65335
\(630\) −3.65864 −0.145764
\(631\) −42.2963 −1.68379 −0.841894 0.539643i \(-0.818559\pi\)
−0.841894 + 0.539643i \(0.818559\pi\)
\(632\) −11.4119 −0.453941
\(633\) −11.2030 −0.445281
\(634\) −7.36812 −0.292625
\(635\) −12.9175 −0.512616
\(636\) −8.06106 −0.319642
\(637\) −0.365083 −0.0144651
\(638\) −3.74141 −0.148124
\(639\) 12.9886 0.513821
\(640\) −1.34813 −0.0532894
\(641\) −3.38669 −0.133766 −0.0668832 0.997761i \(-0.521305\pi\)
−0.0668832 + 0.997761i \(0.521305\pi\)
\(642\) −8.05560 −0.317929
\(643\) −26.1572 −1.03154 −0.515769 0.856728i \(-0.672494\pi\)
−0.515769 + 0.856728i \(0.672494\pi\)
\(644\) −12.2273 −0.481823
\(645\) 7.43165 0.292621
\(646\) −2.81059 −0.110581
\(647\) −35.2804 −1.38702 −0.693508 0.720449i \(-0.743936\pi\)
−0.693508 + 0.720449i \(0.743936\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −16.4534 −0.645852
\(650\) −3.18255 −0.124830
\(651\) 27.8129 1.09007
\(652\) −0.232060 −0.00908816
\(653\) 9.91311 0.387930 0.193965 0.981008i \(-0.437865\pi\)
0.193965 + 0.981008i \(0.437865\pi\)
\(654\) 3.57732 0.139884
\(655\) −13.4104 −0.523988
\(656\) −5.06676 −0.197824
\(657\) −1.57753 −0.0615454
\(658\) 3.85224 0.150176
\(659\) −43.4838 −1.69389 −0.846945 0.531680i \(-0.821561\pi\)
−0.846945 + 0.531680i \(0.821561\pi\)
\(660\) −3.22385 −0.125488
\(661\) 17.9106 0.696640 0.348320 0.937376i \(-0.386752\pi\)
0.348320 + 0.937376i \(0.386752\pi\)
\(662\) −18.1845 −0.706760
\(663\) 6.50957 0.252811
\(664\) 5.72642 0.222228
\(665\) −1.57966 −0.0612567
\(666\) 6.36996 0.246831
\(667\) 7.04910 0.272942
\(668\) −1.99495 −0.0771868
\(669\) 27.5336 1.06451
\(670\) −11.2997 −0.436546
\(671\) 28.5783 1.10325
\(672\) −2.71387 −0.104690
\(673\) −11.0858 −0.427327 −0.213663 0.976907i \(-0.568540\pi\)
−0.213663 + 0.976907i \(0.568540\pi\)
\(674\) −30.5645 −1.17730
\(675\) −3.18255 −0.122497
\(676\) 1.00000 0.0384615
\(677\) 27.8562 1.07060 0.535300 0.844662i \(-0.320198\pi\)
0.535300 + 0.844662i \(0.320198\pi\)
\(678\) 10.1751 0.390772
\(679\) −18.4399 −0.707659
\(680\) 8.77573 0.336534
\(681\) 23.3657 0.895374
\(682\) 24.5076 0.938445
\(683\) 25.9979 0.994783 0.497392 0.867526i \(-0.334291\pi\)
0.497392 + 0.867526i \(0.334291\pi\)
\(684\) −0.431762 −0.0165088
\(685\) 4.81078 0.183810
\(686\) 18.0063 0.687483
\(687\) −13.7682 −0.525291
\(688\) 5.51257 0.210165
\(689\) 8.06106 0.307102
\(690\) 6.07397 0.231232
\(691\) −26.2325 −0.997931 −0.498966 0.866622i \(-0.666287\pi\)
−0.498966 + 0.866622i \(0.666287\pi\)
\(692\) 24.4936 0.931108
\(693\) −6.48982 −0.246528
\(694\) 25.2946 0.960168
\(695\) −21.9860 −0.833977
\(696\) 1.56456 0.0593044
\(697\) 32.9824 1.24930
\(698\) −14.5811 −0.551904
\(699\) 22.0201 0.832878
\(700\) −8.63703 −0.326449
\(701\) −10.6850 −0.403568 −0.201784 0.979430i \(-0.564674\pi\)
−0.201784 + 0.979430i \(0.564674\pi\)
\(702\) 1.00000 0.0377426
\(703\) 2.75031 0.103730
\(704\) −2.39135 −0.0901275
\(705\) −1.91362 −0.0720710
\(706\) 29.6881 1.11733
\(707\) −40.5291 −1.52425
\(708\) 6.88037 0.258580
\(709\) −3.50709 −0.131712 −0.0658558 0.997829i \(-0.520978\pi\)
−0.0658558 + 0.997829i \(0.520978\pi\)
\(710\) −17.5103 −0.657149
\(711\) 11.4119 0.427980
\(712\) 11.4913 0.430656
\(713\) −46.1742 −1.72924
\(714\) 17.6661 0.661138
\(715\) 3.22385 0.120565
\(716\) −24.9563 −0.932660
\(717\) −23.6471 −0.883115
\(718\) 15.1234 0.564402
\(719\) −3.78949 −0.141324 −0.0706621 0.997500i \(-0.522511\pi\)
−0.0706621 + 0.997500i \(0.522511\pi\)
\(720\) 1.34813 0.0502417
\(721\) 2.71387 0.101070
\(722\) 18.8136 0.700169
\(723\) −13.2240 −0.491804
\(724\) 19.7726 0.734842
\(725\) 4.97929 0.184926
\(726\) 5.28143 0.196012
\(727\) 38.5644 1.43027 0.715137 0.698984i \(-0.246364\pi\)
0.715137 + 0.698984i \(0.246364\pi\)
\(728\) 2.71387 0.100583
\(729\) 1.00000 0.0370370
\(730\) 2.12671 0.0787133
\(731\) −35.8845 −1.32724
\(732\) −11.9507 −0.441710
\(733\) −23.8425 −0.880644 −0.440322 0.897840i \(-0.645136\pi\)
−0.440322 + 0.897840i \(0.645136\pi\)
\(734\) 10.8564 0.400718
\(735\) 0.492178 0.0181543
\(736\) 4.50549 0.166075
\(737\) −20.0438 −0.738323
\(738\) 5.06676 0.186510
\(739\) 33.6980 1.23960 0.619800 0.784760i \(-0.287214\pi\)
0.619800 + 0.784760i \(0.287214\pi\)
\(740\) −8.58751 −0.315683
\(741\) 0.431762 0.0158612
\(742\) 21.8766 0.803117
\(743\) 2.63822 0.0967871 0.0483935 0.998828i \(-0.484590\pi\)
0.0483935 + 0.998828i \(0.484590\pi\)
\(744\) −10.2484 −0.375726
\(745\) −2.09126 −0.0766177
\(746\) −7.66103 −0.280490
\(747\) −5.72642 −0.209519
\(748\) 15.5667 0.569174
\(749\) 21.8618 0.798814
\(750\) 11.0311 0.402800
\(751\) −15.9161 −0.580788 −0.290394 0.956907i \(-0.593786\pi\)
−0.290394 + 0.956907i \(0.593786\pi\)
\(752\) −1.41946 −0.0517625
\(753\) 4.85647 0.176980
\(754\) −1.56456 −0.0569778
\(755\) −19.3725 −0.705036
\(756\) 2.71387 0.0987024
\(757\) 13.2525 0.481670 0.240835 0.970566i \(-0.422579\pi\)
0.240835 + 0.970566i \(0.422579\pi\)
\(758\) −10.0157 −0.363786
\(759\) 10.7742 0.391079
\(760\) 0.582070 0.0211139
\(761\) 1.63852 0.0593963 0.0296982 0.999559i \(-0.490545\pi\)
0.0296982 + 0.999559i \(0.490545\pi\)
\(762\) 9.58182 0.347113
\(763\) −9.70837 −0.351467
\(764\) 4.73569 0.171331
\(765\) −8.77573 −0.317287
\(766\) −17.7480 −0.641263
\(767\) −6.88037 −0.248436
\(768\) 1.00000 0.0360844
\(769\) −49.4014 −1.78146 −0.890730 0.454533i \(-0.849806\pi\)
−0.890730 + 0.454533i \(0.849806\pi\)
\(770\) 8.74909 0.315295
\(771\) −11.7803 −0.424257
\(772\) −6.52330 −0.234779
\(773\) −30.0731 −1.08165 −0.540827 0.841134i \(-0.681889\pi\)
−0.540827 + 0.841134i \(0.681889\pi\)
\(774\) −5.51257 −0.198145
\(775\) −32.6162 −1.17161
\(776\) 6.79470 0.243915
\(777\) −17.2872 −0.620176
\(778\) −17.6122 −0.631427
\(779\) 2.18764 0.0783802
\(780\) −1.34813 −0.0482707
\(781\) −31.0603 −1.11143
\(782\) −29.3288 −1.04880
\(783\) −1.56456 −0.0559127
\(784\) 0.365083 0.0130387
\(785\) 11.5163 0.411033
\(786\) 9.94744 0.354813
\(787\) −43.4901 −1.55025 −0.775127 0.631806i \(-0.782314\pi\)
−0.775127 + 0.631806i \(0.782314\pi\)
\(788\) −8.88078 −0.316365
\(789\) 6.54794 0.233113
\(790\) −15.3847 −0.547362
\(791\) −27.6139 −0.981836
\(792\) 2.39135 0.0849730
\(793\) 11.9507 0.424381
\(794\) −15.9439 −0.565826
\(795\) −10.8673 −0.385424
\(796\) −23.6494 −0.838231
\(797\) 21.0117 0.744271 0.372136 0.928178i \(-0.378626\pi\)
0.372136 + 0.928178i \(0.378626\pi\)
\(798\) 1.17175 0.0414794
\(799\) 9.24010 0.326891
\(800\) 3.18255 0.112520
\(801\) −11.4913 −0.406026
\(802\) 13.5652 0.479004
\(803\) 3.77244 0.133126
\(804\) 8.38179 0.295603
\(805\) −16.4840 −0.580983
\(806\) 10.2484 0.360986
\(807\) 4.08408 0.143766
\(808\) 14.9341 0.525378
\(809\) −39.2784 −1.38096 −0.690478 0.723353i \(-0.742600\pi\)
−0.690478 + 0.723353i \(0.742600\pi\)
\(810\) −1.34813 −0.0473683
\(811\) 3.38932 0.119015 0.0595075 0.998228i \(-0.481047\pi\)
0.0595075 + 0.998228i \(0.481047\pi\)
\(812\) −4.24600 −0.149006
\(813\) −9.81272 −0.344147
\(814\) −15.2328 −0.533910
\(815\) −0.312846 −0.0109585
\(816\) −6.50957 −0.227881
\(817\) −2.38012 −0.0832699
\(818\) −23.6532 −0.827015
\(819\) −2.71387 −0.0948302
\(820\) −6.83063 −0.238536
\(821\) −4.00943 −0.139930 −0.0699651 0.997549i \(-0.522289\pi\)
−0.0699651 + 0.997549i \(0.522289\pi\)
\(822\) −3.56849 −0.124465
\(823\) 28.8401 1.00530 0.502652 0.864489i \(-0.332358\pi\)
0.502652 + 0.864489i \(0.332358\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 7.61061 0.264967
\(826\) −18.6724 −0.649696
\(827\) 43.1464 1.50035 0.750174 0.661240i \(-0.229970\pi\)
0.750174 + 0.661240i \(0.229970\pi\)
\(828\) −4.50549 −0.156577
\(829\) −18.0415 −0.626606 −0.313303 0.949653i \(-0.601435\pi\)
−0.313303 + 0.949653i \(0.601435\pi\)
\(830\) 7.71994 0.267963
\(831\) −10.6253 −0.368588
\(832\) −1.00000 −0.0346688
\(833\) −2.37653 −0.0823420
\(834\) 16.3086 0.564719
\(835\) −2.68944 −0.0930719
\(836\) 1.03250 0.0357096
\(837\) 10.2484 0.354238
\(838\) −30.1554 −1.04170
\(839\) −3.49785 −0.120759 −0.0603796 0.998175i \(-0.519231\pi\)
−0.0603796 + 0.998175i \(0.519231\pi\)
\(840\) −3.65864 −0.126235
\(841\) −26.5522 −0.915592
\(842\) −13.2847 −0.457821
\(843\) −11.0617 −0.380984
\(844\) −11.2030 −0.385625
\(845\) 1.34813 0.0463770
\(846\) 1.41946 0.0488022
\(847\) −14.3331 −0.492492
\(848\) −8.06106 −0.276818
\(849\) 6.36457 0.218431
\(850\) −20.7171 −0.710589
\(851\) 28.6998 0.983816
\(852\) 12.9886 0.444982
\(853\) 40.2794 1.37914 0.689571 0.724218i \(-0.257799\pi\)
0.689571 + 0.724218i \(0.257799\pi\)
\(854\) 32.4326 1.10982
\(855\) −0.582070 −0.0199064
\(856\) −8.05560 −0.275335
\(857\) 27.3567 0.934486 0.467243 0.884129i \(-0.345247\pi\)
0.467243 + 0.884129i \(0.345247\pi\)
\(858\) −2.39135 −0.0816394
\(859\) 9.68118 0.330317 0.165159 0.986267i \(-0.447186\pi\)
0.165159 + 0.986267i \(0.447186\pi\)
\(860\) 7.43165 0.253417
\(861\) −13.7505 −0.468616
\(862\) 8.15311 0.277696
\(863\) 31.4569 1.07081 0.535403 0.844597i \(-0.320160\pi\)
0.535403 + 0.844597i \(0.320160\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 33.0205 1.12273
\(866\) 1.67772 0.0570111
\(867\) 25.3745 0.861764
\(868\) 27.8129 0.944031
\(869\) −27.2899 −0.925745
\(870\) 2.10922 0.0715093
\(871\) −8.38179 −0.284006
\(872\) 3.57732 0.121143
\(873\) −6.79470 −0.229966
\(874\) −1.94530 −0.0658008
\(875\) −29.9370 −1.01206
\(876\) −1.57753 −0.0532999
\(877\) 27.4115 0.925621 0.462810 0.886457i \(-0.346841\pi\)
0.462810 + 0.886457i \(0.346841\pi\)
\(878\) −38.8832 −1.31224
\(879\) 7.74495 0.261231
\(880\) −3.22385 −0.108676
\(881\) 2.91785 0.0983048 0.0491524 0.998791i \(-0.484348\pi\)
0.0491524 + 0.998791i \(0.484348\pi\)
\(882\) −0.365083 −0.0122930
\(883\) −27.9384 −0.940203 −0.470101 0.882612i \(-0.655783\pi\)
−0.470101 + 0.882612i \(0.655783\pi\)
\(884\) 6.50957 0.218941
\(885\) 9.27560 0.311796
\(886\) 18.8022 0.631672
\(887\) −10.3031 −0.345943 −0.172972 0.984927i \(-0.555337\pi\)
−0.172972 + 0.984927i \(0.555337\pi\)
\(888\) 6.36996 0.213762
\(889\) −26.0038 −0.872139
\(890\) 15.4918 0.519285
\(891\) −2.39135 −0.0801133
\(892\) 27.5336 0.921893
\(893\) 0.612871 0.0205090
\(894\) 1.55123 0.0518810
\(895\) −33.6442 −1.12460
\(896\) −2.71387 −0.0906640
\(897\) 4.50549 0.150434
\(898\) 4.65776 0.155431
\(899\) −16.0343 −0.534772
\(900\) −3.18255 −0.106085
\(901\) 52.4740 1.74816
\(902\) −12.1164 −0.403432
\(903\) 14.9604 0.497851
\(904\) 10.1751 0.338418
\(905\) 26.6559 0.886074
\(906\) 14.3699 0.477408
\(907\) 19.1032 0.634310 0.317155 0.948374i \(-0.397272\pi\)
0.317155 + 0.948374i \(0.397272\pi\)
\(908\) 23.3657 0.775416
\(909\) −14.9341 −0.495331
\(910\) 3.65864 0.121283
\(911\) −41.7725 −1.38398 −0.691992 0.721905i \(-0.743267\pi\)
−0.691992 + 0.721905i \(0.743267\pi\)
\(912\) −0.431762 −0.0142971
\(913\) 13.6939 0.453202
\(914\) −16.6855 −0.551907
\(915\) −16.1110 −0.532615
\(916\) −13.7682 −0.454916
\(917\) −26.9960 −0.891488
\(918\) 6.50957 0.214848
\(919\) 53.3955 1.76135 0.880677 0.473717i \(-0.157088\pi\)
0.880677 + 0.473717i \(0.157088\pi\)
\(920\) 6.07397 0.200253
\(921\) −12.8933 −0.424847
\(922\) 10.4652 0.344653
\(923\) −12.9886 −0.427525
\(924\) −6.48982 −0.213499
\(925\) 20.2727 0.666563
\(926\) −31.2686 −1.02755
\(927\) 1.00000 0.0328443
\(928\) 1.56456 0.0513591
\(929\) −4.38364 −0.143823 −0.0719113 0.997411i \(-0.522910\pi\)
−0.0719113 + 0.997411i \(0.522910\pi\)
\(930\) −13.8162 −0.453051
\(931\) −0.157629 −0.00516609
\(932\) 22.0201 0.721294
\(933\) 22.5094 0.736926
\(934\) 15.9687 0.522511
\(935\) 20.9859 0.686311
\(936\) 1.00000 0.0326860
\(937\) −28.5199 −0.931704 −0.465852 0.884863i \(-0.654252\pi\)
−0.465852 + 0.884863i \(0.654252\pi\)
\(938\) −22.7471 −0.742718
\(939\) −14.2391 −0.464676
\(940\) −1.91362 −0.0624153
\(941\) −12.7299 −0.414982 −0.207491 0.978237i \(-0.566530\pi\)
−0.207491 + 0.978237i \(0.566530\pi\)
\(942\) −8.54243 −0.278327
\(943\) 22.8282 0.743389
\(944\) 6.88037 0.223937
\(945\) 3.65864 0.119016
\(946\) 13.1825 0.428600
\(947\) 53.4256 1.73610 0.868050 0.496477i \(-0.165373\pi\)
0.868050 + 0.496477i \(0.165373\pi\)
\(948\) 11.4119 0.370641
\(949\) 1.57753 0.0512089
\(950\) −1.37411 −0.0445819
\(951\) 7.36812 0.238928
\(952\) 17.6661 0.572562
\(953\) −15.7274 −0.509459 −0.254730 0.967012i \(-0.581986\pi\)
−0.254730 + 0.967012i \(0.581986\pi\)
\(954\) 8.06106 0.260986
\(955\) 6.38431 0.206591
\(956\) −23.6471 −0.764800
\(957\) 3.74141 0.120943
\(958\) −18.5527 −0.599409
\(959\) 9.68441 0.312726
\(960\) 1.34813 0.0435106
\(961\) 74.0303 2.38808
\(962\) −6.36996 −0.205376
\(963\) 8.05560 0.259588
\(964\) −13.2240 −0.425915
\(965\) −8.79424 −0.283097
\(966\) 12.2273 0.393407
\(967\) −7.42064 −0.238632 −0.119316 0.992856i \(-0.538070\pi\)
−0.119316 + 0.992856i \(0.538070\pi\)
\(968\) 5.28143 0.169752
\(969\) 2.81059 0.0902891
\(970\) 9.16011 0.294114
\(971\) −43.6841 −1.40189 −0.700945 0.713215i \(-0.747238\pi\)
−0.700945 + 0.713215i \(0.747238\pi\)
\(972\) 1.00000 0.0320750
\(973\) −44.2593 −1.41889
\(974\) 35.5161 1.13801
\(975\) 3.18255 0.101923
\(976\) −11.9507 −0.382532
\(977\) −5.79119 −0.185277 −0.0926383 0.995700i \(-0.529530\pi\)
−0.0926383 + 0.995700i \(0.529530\pi\)
\(978\) 0.232060 0.00742045
\(979\) 27.4798 0.878259
\(980\) 0.492178 0.0157221
\(981\) −3.57732 −0.114215
\(982\) −5.28868 −0.168769
\(983\) −24.9652 −0.796267 −0.398133 0.917328i \(-0.630342\pi\)
−0.398133 + 0.917328i \(0.630342\pi\)
\(984\) 5.06676 0.161522
\(985\) −11.9724 −0.381473
\(986\) −10.1846 −0.324344
\(987\) −3.85224 −0.122618
\(988\) 0.431762 0.0137362
\(989\) −24.8368 −0.789766
\(990\) 3.22385 0.102461
\(991\) 39.6768 1.26037 0.630187 0.776443i \(-0.282978\pi\)
0.630187 + 0.776443i \(0.282978\pi\)
\(992\) −10.2484 −0.325388
\(993\) 18.1845 0.577067
\(994\) −35.2494 −1.11804
\(995\) −31.8824 −1.01074
\(996\) −5.72642 −0.181449
\(997\) −12.4981 −0.395818 −0.197909 0.980220i \(-0.563415\pi\)
−0.197909 + 0.980220i \(0.563415\pi\)
\(998\) 10.8970 0.344937
\(999\) −6.36996 −0.201537
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 8034.2.a.bb.1.10 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bb.1.10 14 1.1 even 1 trivial