Properties

Label 165.2.a.a.1.2
Level $165$
Weight $2$
Character 165.1
Self dual yes
Analytic conductor $1.318$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 165.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.31753163335\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
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 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} -0.414214 q^{6} -4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} -0.414214 q^{6} -4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} -0.414214 q^{10} -1.00000 q^{11} +1.82843 q^{12} +5.65685 q^{13} -2.00000 q^{14} +1.00000 q^{15} +3.00000 q^{16} -6.82843 q^{17} +0.414214 q^{18} -1.17157 q^{19} +1.82843 q^{20} +4.82843 q^{21} -0.414214 q^{22} -4.00000 q^{23} +1.58579 q^{24} +1.00000 q^{25} +2.34315 q^{26} -1.00000 q^{27} +8.82843 q^{28} +0.828427 q^{29} +0.414214 q^{30} +4.41421 q^{32} +1.00000 q^{33} -2.82843 q^{34} +4.82843 q^{35} -1.82843 q^{36} +0.343146 q^{37} -0.485281 q^{38} -5.65685 q^{39} +1.58579 q^{40} -0.828427 q^{41} +2.00000 q^{42} -3.17157 q^{43} +1.82843 q^{44} -1.00000 q^{45} -1.65685 q^{46} -4.00000 q^{47} -3.00000 q^{48} +16.3137 q^{49} +0.414214 q^{50} +6.82843 q^{51} -10.3431 q^{52} -13.3137 q^{53} -0.414214 q^{54} +1.00000 q^{55} +7.65685 q^{56} +1.17157 q^{57} +0.343146 q^{58} -4.00000 q^{59} -1.82843 q^{60} -0.343146 q^{61} -4.82843 q^{63} -4.17157 q^{64} -5.65685 q^{65} +0.414214 q^{66} +5.65685 q^{67} +12.4853 q^{68} +4.00000 q^{69} +2.00000 q^{70} +13.6569 q^{71} -1.58579 q^{72} -11.3137 q^{73} +0.142136 q^{74} -1.00000 q^{75} +2.14214 q^{76} +4.82843 q^{77} -2.34315 q^{78} -8.48528 q^{79} -3.00000 q^{80} +1.00000 q^{81} -0.343146 q^{82} -10.0000 q^{83} -8.82843 q^{84} +6.82843 q^{85} -1.31371 q^{86} -0.828427 q^{87} +1.58579 q^{88} -7.65685 q^{89} -0.414214 q^{90} -27.3137 q^{91} +7.31371 q^{92} -1.65685 q^{94} +1.17157 q^{95} -4.41421 q^{96} +0.343146 q^{97} +6.75736 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} - 6q^{8} + 2q^{9} + 2q^{10} - 2q^{11} - 2q^{12} - 4q^{14} + 2q^{15} + 6q^{16} - 8q^{17} - 2q^{18} - 8q^{19} - 2q^{20} + 4q^{21} + 2q^{22} - 8q^{23} + 6q^{24} + 2q^{25} + 16q^{26} - 2q^{27} + 12q^{28} - 4q^{29} - 2q^{30} + 6q^{32} + 2q^{33} + 4q^{35} + 2q^{36} + 12q^{37} + 16q^{38} + 6q^{40} + 4q^{41} + 4q^{42} - 12q^{43} - 2q^{44} - 2q^{45} + 8q^{46} - 8q^{47} - 6q^{48} + 10q^{49} - 2q^{50} + 8q^{51} - 32q^{52} - 4q^{53} + 2q^{54} + 2q^{55} + 4q^{56} + 8q^{57} + 12q^{58} - 8q^{59} + 2q^{60} - 12q^{61} - 4q^{63} - 14q^{64} - 2q^{66} + 8q^{68} + 8q^{69} + 4q^{70} + 16q^{71} - 6q^{72} - 28q^{74} - 2q^{75} - 24q^{76} + 4q^{77} - 16q^{78} - 6q^{80} + 2q^{81} - 12q^{82} - 20q^{83} - 12q^{84} + 8q^{85} + 20q^{86} + 4q^{87} + 6q^{88} - 4q^{89} + 2q^{90} - 32q^{91} - 8q^{92} + 8q^{94} + 8q^{95} - 6q^{96} + 12q^{97} + 22q^{98} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.82843 −0.914214
\(5\) −1.00000 −0.447214
\(6\) −0.414214 −0.169102
\(7\) −4.82843 −1.82497 −0.912487 0.409106i \(-0.865841\pi\)
−0.912487 + 0.409106i \(0.865841\pi\)
\(8\) −1.58579 −0.560660
\(9\) 1.00000 0.333333
\(10\) −0.414214 −0.130986
\(11\) −1.00000 −0.301511
\(12\) 1.82843 0.527821
\(13\) 5.65685 1.56893 0.784465 0.620174i \(-0.212938\pi\)
0.784465 + 0.620174i \(0.212938\pi\)
\(14\) −2.00000 −0.534522
\(15\) 1.00000 0.258199
\(16\) 3.00000 0.750000
\(17\) −6.82843 −1.65614 −0.828068 0.560627i \(-0.810560\pi\)
−0.828068 + 0.560627i \(0.810560\pi\)
\(18\) 0.414214 0.0976311
\(19\) −1.17157 −0.268777 −0.134389 0.990929i \(-0.542907\pi\)
−0.134389 + 0.990929i \(0.542907\pi\)
\(20\) 1.82843 0.408849
\(21\) 4.82843 1.05365
\(22\) −0.414214 −0.0883106
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.58579 0.323697
\(25\) 1.00000 0.200000
\(26\) 2.34315 0.459529
\(27\) −1.00000 −0.192450
\(28\) 8.82843 1.66842
\(29\) 0.828427 0.153835 0.0769175 0.997037i \(-0.475492\pi\)
0.0769175 + 0.997037i \(0.475492\pi\)
\(30\) 0.414214 0.0756247
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 4.41421 0.780330
\(33\) 1.00000 0.174078
\(34\) −2.82843 −0.485071
\(35\) 4.82843 0.816153
\(36\) −1.82843 −0.304738
\(37\) 0.343146 0.0564128 0.0282064 0.999602i \(-0.491020\pi\)
0.0282064 + 0.999602i \(0.491020\pi\)
\(38\) −0.485281 −0.0787230
\(39\) −5.65685 −0.905822
\(40\) 1.58579 0.250735
\(41\) −0.828427 −0.129379 −0.0646893 0.997905i \(-0.520606\pi\)
−0.0646893 + 0.997905i \(0.520606\pi\)
\(42\) 2.00000 0.308607
\(43\) −3.17157 −0.483660 −0.241830 0.970319i \(-0.577748\pi\)
−0.241830 + 0.970319i \(0.577748\pi\)
\(44\) 1.82843 0.275646
\(45\) −1.00000 −0.149071
\(46\) −1.65685 −0.244290
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −3.00000 −0.433013
\(49\) 16.3137 2.33053
\(50\) 0.414214 0.0585786
\(51\) 6.82843 0.956171
\(52\) −10.3431 −1.43434
\(53\) −13.3137 −1.82878 −0.914389 0.404836i \(-0.867329\pi\)
−0.914389 + 0.404836i \(0.867329\pi\)
\(54\) −0.414214 −0.0563673
\(55\) 1.00000 0.134840
\(56\) 7.65685 1.02319
\(57\) 1.17157 0.155179
\(58\) 0.343146 0.0450572
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) −1.82843 −0.236049
\(61\) −0.343146 −0.0439353 −0.0219677 0.999759i \(-0.506993\pi\)
−0.0219677 + 0.999759i \(0.506993\pi\)
\(62\) 0 0
\(63\) −4.82843 −0.608325
\(64\) −4.17157 −0.521447
\(65\) −5.65685 −0.701646
\(66\) 0.414214 0.0509862
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) 12.4853 1.51406
\(69\) 4.00000 0.481543
\(70\) 2.00000 0.239046
\(71\) 13.6569 1.62077 0.810385 0.585897i \(-0.199258\pi\)
0.810385 + 0.585897i \(0.199258\pi\)
\(72\) −1.58579 −0.186887
\(73\) −11.3137 −1.32417 −0.662085 0.749429i \(-0.730328\pi\)
−0.662085 + 0.749429i \(0.730328\pi\)
\(74\) 0.142136 0.0165229
\(75\) −1.00000 −0.115470
\(76\) 2.14214 0.245720
\(77\) 4.82843 0.550250
\(78\) −2.34315 −0.265309
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −0.343146 −0.0378941
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) −8.82843 −0.963260
\(85\) 6.82843 0.740647
\(86\) −1.31371 −0.141661
\(87\) −0.828427 −0.0888167
\(88\) 1.58579 0.169045
\(89\) −7.65685 −0.811625 −0.405812 0.913956i \(-0.633011\pi\)
−0.405812 + 0.913956i \(0.633011\pi\)
\(90\) −0.414214 −0.0436619
\(91\) −27.3137 −2.86325
\(92\) 7.31371 0.762507
\(93\) 0 0
\(94\) −1.65685 −0.170891
\(95\) 1.17157 0.120201
\(96\) −4.41421 −0.450524
\(97\) 0.343146 0.0348412 0.0174206 0.999848i \(-0.494455\pi\)
0.0174206 + 0.999848i \(0.494455\pi\)
\(98\) 6.75736 0.682596
\(99\) −1.00000 −0.100504
\(100\) −1.82843 −0.182843
\(101\) 4.82843 0.480446 0.240223 0.970718i \(-0.422779\pi\)
0.240223 + 0.970718i \(0.422779\pi\)
\(102\) 2.82843 0.280056
\(103\) 19.3137 1.90304 0.951518 0.307593i \(-0.0995234\pi\)
0.951518 + 0.307593i \(0.0995234\pi\)
\(104\) −8.97056 −0.879636
\(105\) −4.82843 −0.471206
\(106\) −5.51472 −0.535637
\(107\) −5.31371 −0.513696 −0.256848 0.966452i \(-0.582684\pi\)
−0.256848 + 0.966452i \(0.582684\pi\)
\(108\) 1.82843 0.175940
\(109\) −5.31371 −0.508961 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(110\) 0.414214 0.0394937
\(111\) −0.343146 −0.0325700
\(112\) −14.4853 −1.36873
\(113\) 14.9706 1.40831 0.704156 0.710045i \(-0.251326\pi\)
0.704156 + 0.710045i \(0.251326\pi\)
\(114\) 0.485281 0.0454508
\(115\) 4.00000 0.373002
\(116\) −1.51472 −0.140638
\(117\) 5.65685 0.522976
\(118\) −1.65685 −0.152526
\(119\) 32.9706 3.02241
\(120\) −1.58579 −0.144762
\(121\) 1.00000 0.0909091
\(122\) −0.142136 −0.0128684
\(123\) 0.828427 0.0746968
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) −2.00000 −0.178174
\(127\) 2.48528 0.220533 0.110267 0.993902i \(-0.464830\pi\)
0.110267 + 0.993902i \(0.464830\pi\)
\(128\) −10.5563 −0.933058
\(129\) 3.17157 0.279241
\(130\) −2.34315 −0.205507
\(131\) −19.3137 −1.68745 −0.843723 0.536778i \(-0.819641\pi\)
−0.843723 + 0.536778i \(0.819641\pi\)
\(132\) −1.82843 −0.159144
\(133\) 5.65685 0.490511
\(134\) 2.34315 0.202417
\(135\) 1.00000 0.0860663
\(136\) 10.8284 0.928530
\(137\) 9.31371 0.795724 0.397862 0.917445i \(-0.369752\pi\)
0.397862 + 0.917445i \(0.369752\pi\)
\(138\) 1.65685 0.141041
\(139\) −16.4853 −1.39826 −0.699132 0.714993i \(-0.746430\pi\)
−0.699132 + 0.714993i \(0.746430\pi\)
\(140\) −8.82843 −0.746138
\(141\) 4.00000 0.336861
\(142\) 5.65685 0.474713
\(143\) −5.65685 −0.473050
\(144\) 3.00000 0.250000
\(145\) −0.828427 −0.0687971
\(146\) −4.68629 −0.387840
\(147\) −16.3137 −1.34553
\(148\) −0.627417 −0.0515734
\(149\) 18.4853 1.51437 0.757187 0.653199i \(-0.226573\pi\)
0.757187 + 0.653199i \(0.226573\pi\)
\(150\) −0.414214 −0.0338204
\(151\) −0.485281 −0.0394916 −0.0197458 0.999805i \(-0.506286\pi\)
−0.0197458 + 0.999805i \(0.506286\pi\)
\(152\) 1.85786 0.150693
\(153\) −6.82843 −0.552046
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 10.3431 0.828114
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −3.51472 −0.279616
\(159\) 13.3137 1.05585
\(160\) −4.41421 −0.348974
\(161\) 19.3137 1.52213
\(162\) 0.414214 0.0325437
\(163\) 15.3137 1.19946 0.599731 0.800202i \(-0.295274\pi\)
0.599731 + 0.800202i \(0.295274\pi\)
\(164\) 1.51472 0.118280
\(165\) −1.00000 −0.0778499
\(166\) −4.14214 −0.321492
\(167\) 9.31371 0.720716 0.360358 0.932814i \(-0.382654\pi\)
0.360358 + 0.932814i \(0.382654\pi\)
\(168\) −7.65685 −0.590739
\(169\) 19.0000 1.46154
\(170\) 2.82843 0.216930
\(171\) −1.17157 −0.0895924
\(172\) 5.79899 0.442169
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) −0.343146 −0.0260138
\(175\) −4.82843 −0.364995
\(176\) −3.00000 −0.226134
\(177\) 4.00000 0.300658
\(178\) −3.17157 −0.237719
\(179\) −6.34315 −0.474109 −0.237054 0.971496i \(-0.576182\pi\)
−0.237054 + 0.971496i \(0.576182\pi\)
\(180\) 1.82843 0.136283
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −11.3137 −0.838628
\(183\) 0.343146 0.0253661
\(184\) 6.34315 0.467623
\(185\) −0.343146 −0.0252286
\(186\) 0 0
\(187\) 6.82843 0.499344
\(188\) 7.31371 0.533407
\(189\) 4.82843 0.351216
\(190\) 0.485281 0.0352060
\(191\) −5.65685 −0.409316 −0.204658 0.978834i \(-0.565608\pi\)
−0.204658 + 0.978834i \(0.565608\pi\)
\(192\) 4.17157 0.301057
\(193\) 2.34315 0.168663 0.0843317 0.996438i \(-0.473124\pi\)
0.0843317 + 0.996438i \(0.473124\pi\)
\(194\) 0.142136 0.0102047
\(195\) 5.65685 0.405096
\(196\) −29.8284 −2.13060
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) −0.414214 −0.0294369
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) −1.58579 −0.112132
\(201\) −5.65685 −0.399004
\(202\) 2.00000 0.140720
\(203\) −4.00000 −0.280745
\(204\) −12.4853 −0.874145
\(205\) 0.828427 0.0578599
\(206\) 8.00000 0.557386
\(207\) −4.00000 −0.278019
\(208\) 16.9706 1.17670
\(209\) 1.17157 0.0810394
\(210\) −2.00000 −0.138013
\(211\) 6.82843 0.470088 0.235044 0.971985i \(-0.424477\pi\)
0.235044 + 0.971985i \(0.424477\pi\)
\(212\) 24.3431 1.67189
\(213\) −13.6569 −0.935752
\(214\) −2.20101 −0.150458
\(215\) 3.17157 0.216299
\(216\) 1.58579 0.107899
\(217\) 0 0
\(218\) −2.20101 −0.149071
\(219\) 11.3137 0.764510
\(220\) −1.82843 −0.123273
\(221\) −38.6274 −2.59836
\(222\) −0.142136 −0.00953952
\(223\) −17.6569 −1.18239 −0.591195 0.806529i \(-0.701344\pi\)
−0.591195 + 0.806529i \(0.701344\pi\)
\(224\) −21.3137 −1.42408
\(225\) 1.00000 0.0666667
\(226\) 6.20101 0.412485
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) −2.14214 −0.141866
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 1.65685 0.109250
\(231\) −4.82843 −0.317687
\(232\) −1.31371 −0.0862492
\(233\) −13.1716 −0.862898 −0.431449 0.902137i \(-0.641998\pi\)
−0.431449 + 0.902137i \(0.641998\pi\)
\(234\) 2.34315 0.153176
\(235\) 4.00000 0.260931
\(236\) 7.31371 0.476082
\(237\) 8.48528 0.551178
\(238\) 13.6569 0.885242
\(239\) 6.34315 0.410304 0.205152 0.978730i \(-0.434231\pi\)
0.205152 + 0.978730i \(0.434231\pi\)
\(240\) 3.00000 0.193649
\(241\) −23.6569 −1.52387 −0.761936 0.647652i \(-0.775751\pi\)
−0.761936 + 0.647652i \(0.775751\pi\)
\(242\) 0.414214 0.0266267
\(243\) −1.00000 −0.0641500
\(244\) 0.627417 0.0401663
\(245\) −16.3137 −1.04224
\(246\) 0.343146 0.0218782
\(247\) −6.62742 −0.421692
\(248\) 0 0
\(249\) 10.0000 0.633724
\(250\) −0.414214 −0.0261972
\(251\) −12.9706 −0.818695 −0.409347 0.912379i \(-0.634244\pi\)
−0.409347 + 0.912379i \(0.634244\pi\)
\(252\) 8.82843 0.556139
\(253\) 4.00000 0.251478
\(254\) 1.02944 0.0645926
\(255\) −6.82843 −0.427613
\(256\) 3.97056 0.248160
\(257\) −27.6569 −1.72519 −0.862594 0.505898i \(-0.831161\pi\)
−0.862594 + 0.505898i \(0.831161\pi\)
\(258\) 1.31371 0.0817879
\(259\) −1.65685 −0.102952
\(260\) 10.3431 0.641455
\(261\) 0.828427 0.0512784
\(262\) −8.00000 −0.494242
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) −1.58579 −0.0975984
\(265\) 13.3137 0.817855
\(266\) 2.34315 0.143667
\(267\) 7.65685 0.468592
\(268\) −10.3431 −0.631808
\(269\) −24.6274 −1.50156 −0.750780 0.660552i \(-0.770322\pi\)
−0.750780 + 0.660552i \(0.770322\pi\)
\(270\) 0.414214 0.0252082
\(271\) 27.7990 1.68867 0.844334 0.535817i \(-0.179996\pi\)
0.844334 + 0.535817i \(0.179996\pi\)
\(272\) −20.4853 −1.24210
\(273\) 27.3137 1.65310
\(274\) 3.85786 0.233062
\(275\) −1.00000 −0.0603023
\(276\) −7.31371 −0.440234
\(277\) −13.6569 −0.820561 −0.410280 0.911959i \(-0.634569\pi\)
−0.410280 + 0.911959i \(0.634569\pi\)
\(278\) −6.82843 −0.409542
\(279\) 0 0
\(280\) −7.65685 −0.457585
\(281\) −16.8284 −1.00390 −0.501950 0.864897i \(-0.667384\pi\)
−0.501950 + 0.864897i \(0.667384\pi\)
\(282\) 1.65685 0.0986642
\(283\) −3.17157 −0.188530 −0.0942652 0.995547i \(-0.530050\pi\)
−0.0942652 + 0.995547i \(0.530050\pi\)
\(284\) −24.9706 −1.48173
\(285\) −1.17157 −0.0693980
\(286\) −2.34315 −0.138553
\(287\) 4.00000 0.236113
\(288\) 4.41421 0.260110
\(289\) 29.6274 1.74279
\(290\) −0.343146 −0.0201502
\(291\) −0.343146 −0.0201156
\(292\) 20.6863 1.21057
\(293\) −1.17157 −0.0684440 −0.0342220 0.999414i \(-0.510895\pi\)
−0.0342220 + 0.999414i \(0.510895\pi\)
\(294\) −6.75736 −0.394097
\(295\) 4.00000 0.232889
\(296\) −0.544156 −0.0316284
\(297\) 1.00000 0.0580259
\(298\) 7.65685 0.443550
\(299\) −22.6274 −1.30858
\(300\) 1.82843 0.105564
\(301\) 15.3137 0.882667
\(302\) −0.201010 −0.0115668
\(303\) −4.82843 −0.277386
\(304\) −3.51472 −0.201583
\(305\) 0.343146 0.0196485
\(306\) −2.82843 −0.161690
\(307\) −8.82843 −0.503865 −0.251932 0.967745i \(-0.581066\pi\)
−0.251932 + 0.967745i \(0.581066\pi\)
\(308\) −8.82843 −0.503046
\(309\) −19.3137 −1.09872
\(310\) 0 0
\(311\) 19.3137 1.09518 0.547590 0.836747i \(-0.315545\pi\)
0.547590 + 0.836747i \(0.315545\pi\)
\(312\) 8.97056 0.507858
\(313\) 4.34315 0.245489 0.122745 0.992438i \(-0.460830\pi\)
0.122745 + 0.992438i \(0.460830\pi\)
\(314\) 7.45584 0.420758
\(315\) 4.82843 0.272051
\(316\) 15.5147 0.872771
\(317\) 30.2843 1.70093 0.850467 0.526028i \(-0.176319\pi\)
0.850467 + 0.526028i \(0.176319\pi\)
\(318\) 5.51472 0.309250
\(319\) −0.828427 −0.0463830
\(320\) 4.17157 0.233198
\(321\) 5.31371 0.296582
\(322\) 8.00000 0.445823
\(323\) 8.00000 0.445132
\(324\) −1.82843 −0.101579
\(325\) 5.65685 0.313786
\(326\) 6.34315 0.351314
\(327\) 5.31371 0.293849
\(328\) 1.31371 0.0725374
\(329\) 19.3137 1.06480
\(330\) −0.414214 −0.0228017
\(331\) 17.6569 0.970508 0.485254 0.874373i \(-0.338727\pi\)
0.485254 + 0.874373i \(0.338727\pi\)
\(332\) 18.2843 1.00348
\(333\) 0.343146 0.0188043
\(334\) 3.85786 0.211093
\(335\) −5.65685 −0.309067
\(336\) 14.4853 0.790237
\(337\) −19.3137 −1.05208 −0.526042 0.850458i \(-0.676325\pi\)
−0.526042 + 0.850458i \(0.676325\pi\)
\(338\) 7.87006 0.428075
\(339\) −14.9706 −0.813089
\(340\) −12.4853 −0.677109
\(341\) 0 0
\(342\) −0.485281 −0.0262410
\(343\) −44.9706 −2.42818
\(344\) 5.02944 0.271169
\(345\) −4.00000 −0.215353
\(346\) −1.17157 −0.0629841
\(347\) −6.68629 −0.358939 −0.179469 0.983764i \(-0.557438\pi\)
−0.179469 + 0.983764i \(0.557438\pi\)
\(348\) 1.51472 0.0811974
\(349\) −22.9706 −1.22959 −0.614793 0.788688i \(-0.710760\pi\)
−0.614793 + 0.788688i \(0.710760\pi\)
\(350\) −2.00000 −0.106904
\(351\) −5.65685 −0.301941
\(352\) −4.41421 −0.235278
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 1.65685 0.0880608
\(355\) −13.6569 −0.724831
\(356\) 14.0000 0.741999
\(357\) −32.9706 −1.74499
\(358\) −2.62742 −0.138863
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 1.58579 0.0835783
\(361\) −17.6274 −0.927759
\(362\) −5.79899 −0.304788
\(363\) −1.00000 −0.0524864
\(364\) 49.9411 2.61763
\(365\) 11.3137 0.592187
\(366\) 0.142136 0.00742955
\(367\) −1.65685 −0.0864871 −0.0432435 0.999065i \(-0.513769\pi\)
−0.0432435 + 0.999065i \(0.513769\pi\)
\(368\) −12.0000 −0.625543
\(369\) −0.828427 −0.0431262
\(370\) −0.142136 −0.00738928
\(371\) 64.2843 3.33747
\(372\) 0 0
\(373\) −34.6274 −1.79294 −0.896470 0.443105i \(-0.853877\pi\)
−0.896470 + 0.443105i \(0.853877\pi\)
\(374\) 2.82843 0.146254
\(375\) 1.00000 0.0516398
\(376\) 6.34315 0.327123
\(377\) 4.68629 0.241356
\(378\) 2.00000 0.102869
\(379\) −0.686292 −0.0352524 −0.0176262 0.999845i \(-0.505611\pi\)
−0.0176262 + 0.999845i \(0.505611\pi\)
\(380\) −2.14214 −0.109889
\(381\) −2.48528 −0.127325
\(382\) −2.34315 −0.119886
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 10.5563 0.538701
\(385\) −4.82843 −0.246079
\(386\) 0.970563 0.0494003
\(387\) −3.17157 −0.161220
\(388\) −0.627417 −0.0318523
\(389\) −12.3431 −0.625822 −0.312911 0.949782i \(-0.601304\pi\)
−0.312911 + 0.949782i \(0.601304\pi\)
\(390\) 2.34315 0.118650
\(391\) 27.3137 1.38131
\(392\) −25.8701 −1.30664
\(393\) 19.3137 0.974248
\(394\) 3.51472 0.177069
\(395\) 8.48528 0.426941
\(396\) 1.82843 0.0918819
\(397\) 18.9706 0.952105 0.476053 0.879417i \(-0.342067\pi\)
0.476053 + 0.879417i \(0.342067\pi\)
\(398\) −4.28427 −0.214751
\(399\) −5.65685 −0.283197
\(400\) 3.00000 0.150000
\(401\) −29.3137 −1.46386 −0.731928 0.681382i \(-0.761379\pi\)
−0.731928 + 0.681382i \(0.761379\pi\)
\(402\) −2.34315 −0.116865
\(403\) 0 0
\(404\) −8.82843 −0.439231
\(405\) −1.00000 −0.0496904
\(406\) −1.65685 −0.0822283
\(407\) −0.343146 −0.0170091
\(408\) −10.8284 −0.536087
\(409\) −8.34315 −0.412542 −0.206271 0.978495i \(-0.566133\pi\)
−0.206271 + 0.978495i \(0.566133\pi\)
\(410\) 0.343146 0.0169468
\(411\) −9.31371 −0.459411
\(412\) −35.3137 −1.73978
\(413\) 19.3137 0.950365
\(414\) −1.65685 −0.0814299
\(415\) 10.0000 0.490881
\(416\) 24.9706 1.22428
\(417\) 16.4853 0.807288
\(418\) 0.485281 0.0237359
\(419\) −3.02944 −0.147998 −0.0739988 0.997258i \(-0.523576\pi\)
−0.0739988 + 0.997258i \(0.523576\pi\)
\(420\) 8.82843 0.430783
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 2.82843 0.137686
\(423\) −4.00000 −0.194487
\(424\) 21.1127 1.02532
\(425\) −6.82843 −0.331227
\(426\) −5.65685 −0.274075
\(427\) 1.65685 0.0801808
\(428\) 9.71573 0.469627
\(429\) 5.65685 0.273115
\(430\) 1.31371 0.0633526
\(431\) 10.3431 0.498212 0.249106 0.968476i \(-0.419863\pi\)
0.249106 + 0.968476i \(0.419863\pi\)
\(432\) −3.00000 −0.144338
\(433\) −4.34315 −0.208718 −0.104359 0.994540i \(-0.533279\pi\)
−0.104359 + 0.994540i \(0.533279\pi\)
\(434\) 0 0
\(435\) 0.828427 0.0397200
\(436\) 9.71573 0.465299
\(437\) 4.68629 0.224176
\(438\) 4.68629 0.223920
\(439\) −3.51472 −0.167748 −0.0838742 0.996476i \(-0.526729\pi\)
−0.0838742 + 0.996476i \(0.526729\pi\)
\(440\) −1.58579 −0.0755994
\(441\) 16.3137 0.776843
\(442\) −16.0000 −0.761042
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0.627417 0.0297759
\(445\) 7.65685 0.362970
\(446\) −7.31371 −0.346314
\(447\) −18.4853 −0.874324
\(448\) 20.1421 0.951626
\(449\) −2.97056 −0.140190 −0.0700948 0.997540i \(-0.522330\pi\)
−0.0700948 + 0.997540i \(0.522330\pi\)
\(450\) 0.414214 0.0195262
\(451\) 0.828427 0.0390091
\(452\) −27.3726 −1.28750
\(453\) 0.485281 0.0228005
\(454\) −5.79899 −0.272160
\(455\) 27.3137 1.28049
\(456\) −1.85786 −0.0870025
\(457\) −0.686292 −0.0321034 −0.0160517 0.999871i \(-0.505110\pi\)
−0.0160517 + 0.999871i \(0.505110\pi\)
\(458\) −0.828427 −0.0387099
\(459\) 6.82843 0.318724
\(460\) −7.31371 −0.341003
\(461\) −28.1421 −1.31071 −0.655355 0.755321i \(-0.727481\pi\)
−0.655355 + 0.755321i \(0.727481\pi\)
\(462\) −2.00000 −0.0930484
\(463\) −28.9706 −1.34638 −0.673188 0.739471i \(-0.735076\pi\)
−0.673188 + 0.739471i \(0.735076\pi\)
\(464\) 2.48528 0.115376
\(465\) 0 0
\(466\) −5.45584 −0.252737
\(467\) 22.6274 1.04707 0.523536 0.852004i \(-0.324613\pi\)
0.523536 + 0.852004i \(0.324613\pi\)
\(468\) −10.3431 −0.478112
\(469\) −27.3137 −1.26123
\(470\) 1.65685 0.0764250
\(471\) −18.0000 −0.829396
\(472\) 6.34315 0.291967
\(473\) 3.17157 0.145829
\(474\) 3.51472 0.161436
\(475\) −1.17157 −0.0537555
\(476\) −60.2843 −2.76313
\(477\) −13.3137 −0.609593
\(478\) 2.62742 0.120175
\(479\) 3.02944 0.138419 0.0692093 0.997602i \(-0.477952\pi\)
0.0692093 + 0.997602i \(0.477952\pi\)
\(480\) 4.41421 0.201480
\(481\) 1.94113 0.0885077
\(482\) −9.79899 −0.446332
\(483\) −19.3137 −0.878804
\(484\) −1.82843 −0.0831103
\(485\) −0.343146 −0.0155814
\(486\) −0.414214 −0.0187891
\(487\) 20.9706 0.950267 0.475133 0.879914i \(-0.342400\pi\)
0.475133 + 0.879914i \(0.342400\pi\)
\(488\) 0.544156 0.0246328
\(489\) −15.3137 −0.692510
\(490\) −6.75736 −0.305266
\(491\) 25.6569 1.15788 0.578939 0.815371i \(-0.303467\pi\)
0.578939 + 0.815371i \(0.303467\pi\)
\(492\) −1.51472 −0.0682888
\(493\) −5.65685 −0.254772
\(494\) −2.74517 −0.123511
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) −65.9411 −2.95786
\(498\) 4.14214 0.185614
\(499\) −33.6569 −1.50669 −0.753344 0.657627i \(-0.771560\pi\)
−0.753344 + 0.657627i \(0.771560\pi\)
\(500\) 1.82843 0.0817697
\(501\) −9.31371 −0.416106
\(502\) −5.37258 −0.239790
\(503\) −5.31371 −0.236927 −0.118463 0.992958i \(-0.537797\pi\)
−0.118463 + 0.992958i \(0.537797\pi\)
\(504\) 7.65685 0.341063
\(505\) −4.82843 −0.214862
\(506\) 1.65685 0.0736562
\(507\) −19.0000 −0.843820
\(508\) −4.54416 −0.201614
\(509\) 41.3137 1.83120 0.915599 0.402093i \(-0.131717\pi\)
0.915599 + 0.402093i \(0.131717\pi\)
\(510\) −2.82843 −0.125245
\(511\) 54.6274 2.41657
\(512\) 22.7574 1.00574
\(513\) 1.17157 0.0517262
\(514\) −11.4558 −0.505296
\(515\) −19.3137 −0.851064
\(516\) −5.79899 −0.255286
\(517\) 4.00000 0.175920
\(518\) −0.686292 −0.0301539
\(519\) 2.82843 0.124154
\(520\) 8.97056 0.393385
\(521\) 12.6274 0.553217 0.276609 0.960983i \(-0.410789\pi\)
0.276609 + 0.960983i \(0.410789\pi\)
\(522\) 0.343146 0.0150191
\(523\) −26.4853 −1.15812 −0.579060 0.815285i \(-0.696580\pi\)
−0.579060 + 0.815285i \(0.696580\pi\)
\(524\) 35.3137 1.54269
\(525\) 4.82843 0.210730
\(526\) 7.45584 0.325090
\(527\) 0 0
\(528\) 3.00000 0.130558
\(529\) −7.00000 −0.304348
\(530\) 5.51472 0.239544
\(531\) −4.00000 −0.173585
\(532\) −10.3431 −0.448432
\(533\) −4.68629 −0.202986
\(534\) 3.17157 0.137247
\(535\) 5.31371 0.229732
\(536\) −8.97056 −0.387469
\(537\) 6.34315 0.273727
\(538\) −10.2010 −0.439797
\(539\) −16.3137 −0.702681
\(540\) −1.82843 −0.0786830
\(541\) −5.31371 −0.228454 −0.114227 0.993455i \(-0.536439\pi\)
−0.114227 + 0.993455i \(0.536439\pi\)
\(542\) 11.5147 0.494600
\(543\) 14.0000 0.600798
\(544\) −30.1421 −1.29233
\(545\) 5.31371 0.227614
\(546\) 11.3137 0.484182
\(547\) 20.1421 0.861216 0.430608 0.902539i \(-0.358299\pi\)
0.430608 + 0.902539i \(0.358299\pi\)
\(548\) −17.0294 −0.727462
\(549\) −0.343146 −0.0146451
\(550\) −0.414214 −0.0176621
\(551\) −0.970563 −0.0413474
\(552\) −6.34315 −0.269982
\(553\) 40.9706 1.74225
\(554\) −5.65685 −0.240337
\(555\) 0.343146 0.0145657
\(556\) 30.1421 1.27831
\(557\) 10.8284 0.458815 0.229408 0.973330i \(-0.426321\pi\)
0.229408 + 0.973330i \(0.426321\pi\)
\(558\) 0 0
\(559\) −17.9411 −0.758829
\(560\) 14.4853 0.612115
\(561\) −6.82843 −0.288296
\(562\) −6.97056 −0.294035
\(563\) 20.3431 0.857361 0.428681 0.903456i \(-0.358979\pi\)
0.428681 + 0.903456i \(0.358979\pi\)
\(564\) −7.31371 −0.307963
\(565\) −14.9706 −0.629816
\(566\) −1.31371 −0.0552193
\(567\) −4.82843 −0.202775
\(568\) −21.6569 −0.908701
\(569\) 15.4558 0.647943 0.323971 0.946067i \(-0.394982\pi\)
0.323971 + 0.946067i \(0.394982\pi\)
\(570\) −0.485281 −0.0203262
\(571\) 0.485281 0.0203084 0.0101542 0.999948i \(-0.496768\pi\)
0.0101542 + 0.999948i \(0.496768\pi\)
\(572\) 10.3431 0.432469
\(573\) 5.65685 0.236318
\(574\) 1.65685 0.0691558
\(575\) −4.00000 −0.166812
\(576\) −4.17157 −0.173816
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 12.2721 0.510451
\(579\) −2.34315 −0.0973778
\(580\) 1.51472 0.0628953
\(581\) 48.2843 2.00317
\(582\) −0.142136 −0.00589171
\(583\) 13.3137 0.551397
\(584\) 17.9411 0.742409
\(585\) −5.65685 −0.233882
\(586\) −0.485281 −0.0200468
\(587\) −30.6274 −1.26413 −0.632064 0.774916i \(-0.717792\pi\)
−0.632064 + 0.774916i \(0.717792\pi\)
\(588\) 29.8284 1.23010
\(589\) 0 0
\(590\) 1.65685 0.0682116
\(591\) −8.48528 −0.349038
\(592\) 1.02944 0.0423096
\(593\) −17.1716 −0.705152 −0.352576 0.935783i \(-0.614694\pi\)
−0.352576 + 0.935783i \(0.614694\pi\)
\(594\) 0.414214 0.0169954
\(595\) −32.9706 −1.35166
\(596\) −33.7990 −1.38446
\(597\) 10.3431 0.423317
\(598\) −9.37258 −0.383273
\(599\) 4.68629 0.191477 0.0957383 0.995407i \(-0.469479\pi\)
0.0957383 + 0.995407i \(0.469479\pi\)
\(600\) 1.58579 0.0647395
\(601\) 17.3137 0.706241 0.353120 0.935578i \(-0.385121\pi\)
0.353120 + 0.935578i \(0.385121\pi\)
\(602\) 6.34315 0.258527
\(603\) 5.65685 0.230365
\(604\) 0.887302 0.0361038
\(605\) −1.00000 −0.0406558
\(606\) −2.00000 −0.0812444
\(607\) 18.4853 0.750294 0.375147 0.926965i \(-0.377592\pi\)
0.375147 + 0.926965i \(0.377592\pi\)
\(608\) −5.17157 −0.209735
\(609\) 4.00000 0.162088
\(610\) 0.142136 0.00575490
\(611\) −22.6274 −0.915407
\(612\) 12.4853 0.504688
\(613\) 21.9411 0.886194 0.443097 0.896474i \(-0.353880\pi\)
0.443097 + 0.896474i \(0.353880\pi\)
\(614\) −3.65685 −0.147579
\(615\) −0.828427 −0.0334054
\(616\) −7.65685 −0.308503
\(617\) 11.6569 0.469287 0.234644 0.972081i \(-0.424608\pi\)
0.234644 + 0.972081i \(0.424608\pi\)
\(618\) −8.00000 −0.321807
\(619\) −25.6569 −1.03124 −0.515618 0.856819i \(-0.672438\pi\)
−0.515618 + 0.856819i \(0.672438\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 8.00000 0.320771
\(623\) 36.9706 1.48119
\(624\) −16.9706 −0.679366
\(625\) 1.00000 0.0400000
\(626\) 1.79899 0.0719021
\(627\) −1.17157 −0.0467881
\(628\) −32.9117 −1.31332
\(629\) −2.34315 −0.0934273
\(630\) 2.00000 0.0796819
\(631\) 34.3431 1.36718 0.683590 0.729867i \(-0.260418\pi\)
0.683590 + 0.729867i \(0.260418\pi\)
\(632\) 13.4558 0.535245
\(633\) −6.82843 −0.271406
\(634\) 12.5442 0.498192
\(635\) −2.48528 −0.0986254
\(636\) −24.3431 −0.965269
\(637\) 92.2843 3.65644
\(638\) −0.343146 −0.0135853
\(639\) 13.6569 0.540257
\(640\) 10.5563 0.417276
\(641\) −26.9706 −1.06527 −0.532637 0.846344i \(-0.678799\pi\)
−0.532637 + 0.846344i \(0.678799\pi\)
\(642\) 2.20101 0.0868669
\(643\) −29.9411 −1.18076 −0.590381 0.807124i \(-0.701023\pi\)
−0.590381 + 0.807124i \(0.701023\pi\)
\(644\) −35.3137 −1.39156
\(645\) −3.17157 −0.124881
\(646\) 3.31371 0.130376
\(647\) 27.3137 1.07381 0.536906 0.843642i \(-0.319593\pi\)
0.536906 + 0.843642i \(0.319593\pi\)
\(648\) −1.58579 −0.0622956
\(649\) 4.00000 0.157014
\(650\) 2.34315 0.0919057
\(651\) 0 0
\(652\) −28.0000 −1.09656
\(653\) −26.9706 −1.05544 −0.527720 0.849418i \(-0.676953\pi\)
−0.527720 + 0.849418i \(0.676953\pi\)
\(654\) 2.20101 0.0860663
\(655\) 19.3137 0.754649
\(656\) −2.48528 −0.0970339
\(657\) −11.3137 −0.441390
\(658\) 8.00000 0.311872
\(659\) 7.31371 0.284902 0.142451 0.989802i \(-0.454502\pi\)
0.142451 + 0.989802i \(0.454502\pi\)
\(660\) 1.82843 0.0711714
\(661\) −13.3137 −0.517843 −0.258922 0.965898i \(-0.583367\pi\)
−0.258922 + 0.965898i \(0.583367\pi\)
\(662\) 7.31371 0.284255
\(663\) 38.6274 1.50016
\(664\) 15.8579 0.615404
\(665\) −5.65685 −0.219363
\(666\) 0.142136 0.00550764
\(667\) −3.31371 −0.128307
\(668\) −17.0294 −0.658889
\(669\) 17.6569 0.682653
\(670\) −2.34315 −0.0905236
\(671\) 0.343146 0.0132470
\(672\) 21.3137 0.822194
\(673\) 29.6569 1.14319 0.571594 0.820537i \(-0.306325\pi\)
0.571594 + 0.820537i \(0.306325\pi\)
\(674\) −8.00000 −0.308148
\(675\) −1.00000 −0.0384900
\(676\) −34.7401 −1.33616
\(677\) −21.4558 −0.824615 −0.412308 0.911045i \(-0.635277\pi\)
−0.412308 + 0.911045i \(0.635277\pi\)
\(678\) −6.20101 −0.238148
\(679\) −1.65685 −0.0635842
\(680\) −10.8284 −0.415251
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 2.14214 0.0819066
\(685\) −9.31371 −0.355859
\(686\) −18.6274 −0.711198
\(687\) 2.00000 0.0763048
\(688\) −9.51472 −0.362745
\(689\) −75.3137 −2.86922
\(690\) −1.65685 −0.0630754
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 5.17157 0.196594
\(693\) 4.82843 0.183417
\(694\) −2.76955 −0.105131
\(695\) 16.4853 0.625322
\(696\) 1.31371 0.0497960
\(697\) 5.65685 0.214269
\(698\) −9.51472 −0.360137
\(699\) 13.1716 0.498195
\(700\) 8.82843 0.333683
\(701\) 7.85786 0.296787 0.148394 0.988928i \(-0.452590\pi\)
0.148394 + 0.988928i \(0.452590\pi\)
\(702\) −2.34315 −0.0884363
\(703\) −0.402020 −0.0151625
\(704\) 4.17157 0.157222
\(705\) −4.00000 −0.150649
\(706\) 10.7696 0.405317
\(707\) −23.3137 −0.876802
\(708\) −7.31371 −0.274866
\(709\) 29.3137 1.10090 0.550450 0.834868i \(-0.314456\pi\)
0.550450 + 0.834868i \(0.314456\pi\)
\(710\) −5.65685 −0.212298
\(711\) −8.48528 −0.318223
\(712\) 12.1421 0.455046
\(713\) 0 0
\(714\) −13.6569 −0.511095
\(715\) 5.65685 0.211554
\(716\) 11.5980 0.433437
\(717\) −6.34315 −0.236889
\(718\) 4.97056 0.185500
\(719\) 31.5980 1.17841 0.589203 0.807985i \(-0.299442\pi\)
0.589203 + 0.807985i \(0.299442\pi\)
\(720\) −3.00000 −0.111803
\(721\) −93.2548 −3.47299
\(722\) −7.30152 −0.271734
\(723\) 23.6569 0.879808
\(724\) 25.5980 0.951341
\(725\) 0.828427 0.0307670
\(726\) −0.414214 −0.0153729
\(727\) −33.9411 −1.25881 −0.629403 0.777079i \(-0.716701\pi\)
−0.629403 + 0.777079i \(0.716701\pi\)
\(728\) 43.3137 1.60531
\(729\) 1.00000 0.0370370
\(730\) 4.68629 0.173447
\(731\) 21.6569 0.801008
\(732\) −0.627417 −0.0231900
\(733\) −17.6569 −0.652171 −0.326085 0.945340i \(-0.605730\pi\)
−0.326085 + 0.945340i \(0.605730\pi\)
\(734\) −0.686292 −0.0253315
\(735\) 16.3137 0.601740
\(736\) −17.6569 −0.650840
\(737\) −5.65685 −0.208373
\(738\) −0.343146 −0.0126314
\(739\) 47.1127 1.73307 0.866534 0.499118i \(-0.166342\pi\)
0.866534 + 0.499118i \(0.166342\pi\)
\(740\) 0.627417 0.0230643
\(741\) 6.62742 0.243464
\(742\) 26.6274 0.977523
\(743\) 47.6569 1.74836 0.874180 0.485602i \(-0.161399\pi\)
0.874180 + 0.485602i \(0.161399\pi\)
\(744\) 0 0
\(745\) −18.4853 −0.677248
\(746\) −14.3431 −0.525140
\(747\) −10.0000 −0.365881
\(748\) −12.4853 −0.456507
\(749\) 25.6569 0.937481
\(750\) 0.414214 0.0151249
\(751\) −36.2843 −1.32403 −0.662016 0.749490i \(-0.730299\pi\)
−0.662016 + 0.749490i \(0.730299\pi\)
\(752\) −12.0000 −0.437595
\(753\) 12.9706 0.472674
\(754\) 1.94113 0.0706916
\(755\) 0.485281 0.0176612
\(756\) −8.82843 −0.321087
\(757\) 8.62742 0.313569 0.156784 0.987633i \(-0.449887\pi\)
0.156784 + 0.987633i \(0.449887\pi\)
\(758\) −0.284271 −0.0103252
\(759\) −4.00000 −0.145191
\(760\) −1.85786 −0.0673918
\(761\) −23.1716 −0.839969 −0.419984 0.907531i \(-0.637964\pi\)
−0.419984 + 0.907531i \(0.637964\pi\)
\(762\) −1.02944 −0.0372926
\(763\) 25.6569 0.928840
\(764\) 10.3431 0.374202
\(765\) 6.82843 0.246882
\(766\) −3.31371 −0.119729
\(767\) −22.6274 −0.817029
\(768\) −3.97056 −0.143275
\(769\) 33.3137 1.20132 0.600662 0.799503i \(-0.294904\pi\)
0.600662 + 0.799503i \(0.294904\pi\)
\(770\) −2.00000 −0.0720750
\(771\) 27.6569 0.996037
\(772\) −4.28427 −0.154194
\(773\) −7.65685 −0.275398 −0.137699 0.990474i \(-0.543971\pi\)
−0.137699 + 0.990474i \(0.543971\pi\)
\(774\) −1.31371 −0.0472203
\(775\) 0 0
\(776\) −0.544156 −0.0195341
\(777\) 1.65685 0.0594393
\(778\) −5.11270 −0.183299
\(779\) 0.970563 0.0347740
\(780\) −10.3431 −0.370344
\(781\) −13.6569 −0.488681
\(782\) 11.3137 0.404577
\(783\) −0.828427 −0.0296056
\(784\) 48.9411 1.74790
\(785\) −18.0000 −0.642448
\(786\) 8.00000 0.285351
\(787\) 8.14214 0.290236 0.145118 0.989414i \(-0.453644\pi\)
0.145118 + 0.989414i \(0.453644\pi\)
\(788\) −15.5147 −0.552689
\(789\) −18.0000 −0.640817
\(790\) 3.51472 0.125048
\(791\) −72.2843 −2.57013
\(792\) 1.58579 0.0563485
\(793\) −1.94113 −0.0689314
\(794\) 7.85786 0.278865
\(795\) −13.3137 −0.472189
\(796\) 18.9117 0.670307
\(797\) 1.02944 0.0364645 0.0182323 0.999834i \(-0.494196\pi\)
0.0182323 + 0.999834i \(0.494196\pi\)
\(798\) −2.34315 −0.0829465
\(799\) 27.3137 0.966290
\(800\) 4.41421 0.156066
\(801\) −7.65685 −0.270542
\(802\) −12.1421 −0.428754
\(803\) 11.3137 0.399252
\(804\) 10.3431 0.364775
\(805\) −19.3137 −0.680719
\(806\) 0 0
\(807\) 24.6274 0.866926
\(808\) −7.65685 −0.269367
\(809\) −56.4264 −1.98385 −0.991923 0.126838i \(-0.959517\pi\)
−0.991923 + 0.126838i \(0.959517\pi\)
\(810\) −0.414214 −0.0145540
\(811\) −16.4853 −0.578877 −0.289438 0.957197i \(-0.593468\pi\)
−0.289438 + 0.957197i \(0.593468\pi\)
\(812\) 7.31371 0.256661
\(813\) −27.7990 −0.974953
\(814\) −0.142136 −0.00498185
\(815\) −15.3137 −0.536416
\(816\) 20.4853 0.717128
\(817\) 3.71573 0.129997
\(818\) −3.45584 −0.120831
\(819\) −27.3137 −0.954418
\(820\) −1.51472 −0.0528963
\(821\) −7.17157 −0.250290 −0.125145 0.992138i \(-0.539940\pi\)
−0.125145 + 0.992138i \(0.539940\pi\)
\(822\) −3.85786 −0.134558
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −30.6274 −1.06696
\(825\) 1.00000 0.0348155
\(826\) 8.00000 0.278356
\(827\) 18.6863 0.649786 0.324893 0.945751i \(-0.394672\pi\)
0.324893 + 0.945751i \(0.394672\pi\)
\(828\) 7.31371 0.254169
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 4.14214 0.143776
\(831\) 13.6569 0.473751
\(832\) −23.5980 −0.818113
\(833\) −111.397 −3.85968
\(834\) 6.82843 0.236449
\(835\) −9.31371 −0.322314
\(836\) −2.14214 −0.0740873
\(837\) 0 0
\(838\) −1.25483 −0.0433475
\(839\) 22.6274 0.781185 0.390593 0.920564i \(-0.372270\pi\)
0.390593 + 0.920564i \(0.372270\pi\)
\(840\) 7.65685 0.264187
\(841\) −28.3137 −0.976335
\(842\) −2.48528 −0.0856485
\(843\) 16.8284 0.579602
\(844\) −12.4853 −0.429761
\(845\) −19.0000 −0.653620
\(846\) −1.65685 −0.0569638
\(847\) −4.82843 −0.165907
\(848\) −39.9411 −1.37158
\(849\) 3.17157 0.108848
\(850\) −2.82843 −0.0970143
\(851\) −1.37258 −0.0470515
\(852\) 24.9706 0.855477
\(853\) −31.3137 −1.07216 −0.536080 0.844167i \(-0.680096\pi\)
−0.536080 + 0.844167i \(0.680096\pi\)
\(854\) 0.686292 0.0234844
\(855\) 1.17157 0.0400669
\(856\) 8.42641 0.288009
\(857\) −11.5147 −0.393335 −0.196668 0.980470i \(-0.563012\pi\)
−0.196668 + 0.980470i \(0.563012\pi\)
\(858\) 2.34315 0.0799937
\(859\) −19.0294 −0.649276 −0.324638 0.945838i \(-0.605242\pi\)
−0.324638 + 0.945838i \(0.605242\pi\)
\(860\) −5.79899 −0.197744
\(861\) −4.00000 −0.136320
\(862\) 4.28427 0.145923
\(863\) −43.3137 −1.47442 −0.737208 0.675666i \(-0.763856\pi\)
−0.737208 + 0.675666i \(0.763856\pi\)
\(864\) −4.41421 −0.150175
\(865\) 2.82843 0.0961694
\(866\) −1.79899 −0.0611322
\(867\) −29.6274 −1.00620
\(868\) 0 0
\(869\) 8.48528 0.287843
\(870\) 0.343146 0.0116337
\(871\) 32.0000 1.08428
\(872\) 8.42641 0.285354
\(873\) 0.343146 0.0116137
\(874\) 1.94113 0.0656595
\(875\) 4.82843 0.163231
\(876\) −20.6863 −0.698925
\(877\) −42.6274 −1.43943 −0.719713 0.694272i \(-0.755727\pi\)
−0.719713 + 0.694272i \(0.755727\pi\)
\(878\) −1.45584 −0.0491324
\(879\) 1.17157 0.0395162
\(880\) 3.00000 0.101130
\(881\) −13.0294 −0.438973 −0.219486 0.975616i \(-0.570438\pi\)
−0.219486 + 0.975616i \(0.570438\pi\)
\(882\) 6.75736 0.227532
\(883\) −50.6274 −1.70375 −0.851874 0.523747i \(-0.824534\pi\)
−0.851874 + 0.523747i \(0.824534\pi\)
\(884\) 70.6274 2.37546
\(885\) −4.00000 −0.134459
\(886\) 4.97056 0.166989
\(887\) −4.34315 −0.145829 −0.0729143 0.997338i \(-0.523230\pi\)
−0.0729143 + 0.997338i \(0.523230\pi\)
\(888\) 0.544156 0.0182607
\(889\) −12.0000 −0.402467
\(890\) 3.17157 0.106311
\(891\) −1.00000 −0.0335013
\(892\) 32.2843 1.08096
\(893\) 4.68629 0.156821
\(894\) −7.65685 −0.256084
\(895\) 6.34315 0.212028
\(896\) 50.9706 1.70281
\(897\) 22.6274 0.755507
\(898\) −1.23045 −0.0410606
\(899\) 0 0
\(900\) −1.82843 −0.0609476
\(901\) 90.9117 3.02871
\(902\) 0.343146 0.0114255
\(903\) −15.3137 −0.509608
\(904\) −23.7401 −0.789584
\(905\) 14.0000 0.465376
\(906\) 0.201010 0.00667811
\(907\) 7.02944 0.233409 0.116704 0.993167i \(-0.462767\pi\)
0.116704 + 0.993167i \(0.462767\pi\)
\(908\) 25.5980 0.849499
\(909\) 4.82843 0.160149
\(910\) 11.3137 0.375046
\(911\) −15.0294 −0.497947 −0.248974 0.968510i \(-0.580093\pi\)
−0.248974 + 0.968510i \(0.580093\pi\)
\(912\) 3.51472 0.116384
\(913\) 10.0000 0.330952
\(914\) −0.284271 −0.00940286
\(915\) −0.343146 −0.0113440
\(916\) 3.65685 0.120826
\(917\) 93.2548 3.07955
\(918\) 2.82843 0.0933520
\(919\) 28.4853 0.939643 0.469821 0.882762i \(-0.344318\pi\)
0.469821 + 0.882762i \(0.344318\pi\)
\(920\) −6.34315 −0.209127
\(921\) 8.82843 0.290907
\(922\) −11.6569 −0.383898
\(923\) 77.2548 2.54287
\(924\) 8.82843 0.290434
\(925\) 0.343146 0.0112826
\(926\) −12.0000 −0.394344
\(927\) 19.3137 0.634345
\(928\) 3.65685 0.120042
\(929\) 33.5980 1.10231 0.551157 0.834402i \(-0.314187\pi\)
0.551157 + 0.834402i \(0.314187\pi\)
\(930\) 0 0
\(931\) −19.1127 −0.626393
\(932\) 24.0833 0.788873
\(933\) −19.3137 −0.632302
\(934\) 9.37258 0.306680
\(935\) −6.82843 −0.223313
\(936\) −8.97056 −0.293212
\(937\) 44.9706 1.46912 0.734562 0.678541i \(-0.237388\pi\)
0.734562 + 0.678541i \(0.237388\pi\)
\(938\) −11.3137 −0.369406
\(939\) −4.34315 −0.141733
\(940\) −7.31371 −0.238547
\(941\) 38.7696 1.26385 0.631926 0.775029i \(-0.282265\pi\)
0.631926 + 0.775029i \(0.282265\pi\)
\(942\) −7.45584 −0.242925
\(943\) 3.31371 0.107909
\(944\) −12.0000 −0.390567
\(945\) −4.82843 −0.157069
\(946\) 1.31371 0.0427123
\(947\) −38.6274 −1.25522 −0.627611 0.778527i \(-0.715967\pi\)
−0.627611 + 0.778527i \(0.715967\pi\)
\(948\) −15.5147 −0.503895
\(949\) −64.0000 −2.07753
\(950\) −0.485281 −0.0157446
\(951\) −30.2843 −0.982035
\(952\) −52.2843 −1.69454
\(953\) 27.7990 0.900498 0.450249 0.892903i \(-0.351335\pi\)
0.450249 + 0.892903i \(0.351335\pi\)
\(954\) −5.51472 −0.178546
\(955\) 5.65685 0.183052
\(956\) −11.5980 −0.375105
\(957\) 0.828427 0.0267792
\(958\) 1.25483 0.0405418
\(959\) −44.9706 −1.45218
\(960\) −4.17157 −0.134637
\(961\) −31.0000 −1.00000
\(962\) 0.804041 0.0259233
\(963\) −5.31371 −0.171232
\(964\) 43.2548 1.39314
\(965\) −2.34315 −0.0754285
\(966\) −8.00000 −0.257396
\(967\) −39.4558 −1.26881 −0.634407 0.772999i \(-0.718756\pi\)
−0.634407 + 0.772999i \(0.718756\pi\)
\(968\) −1.58579 −0.0509691
\(969\) −8.00000 −0.256997
\(970\) −0.142136 −0.00456370
\(971\) 10.6274 0.341050 0.170525 0.985353i \(-0.445454\pi\)
0.170525 + 0.985353i \(0.445454\pi\)
\(972\) 1.82843 0.0586468
\(973\) 79.5980 2.55179
\(974\) 8.68629 0.278327
\(975\) −5.65685 −0.181164
\(976\) −1.02944 −0.0329515
\(977\) 25.3137 0.809857 0.404929 0.914348i \(-0.367296\pi\)
0.404929 + 0.914348i \(0.367296\pi\)
\(978\) −6.34315 −0.202831
\(979\) 7.65685 0.244714
\(980\) 29.8284 0.952834
\(981\) −5.31371 −0.169654
\(982\) 10.6274 0.339135
\(983\) 14.6274 0.466542 0.233271 0.972412i \(-0.425057\pi\)
0.233271 + 0.972412i \(0.425057\pi\)
\(984\) −1.31371 −0.0418795
\(985\) −8.48528 −0.270364
\(986\) −2.34315 −0.0746210
\(987\) −19.3137 −0.614762
\(988\) 12.1177 0.385517
\(989\) 12.6863 0.403401
\(990\) 0.414214 0.0131646
\(991\) 14.6274 0.464655 0.232328 0.972638i \(-0.425366\pi\)
0.232328 + 0.972638i \(0.425366\pi\)
\(992\) 0 0
\(993\) −17.6569 −0.560323
\(994\) −27.3137 −0.866338
\(995\) 10.3431 0.327900
\(996\) −18.2843 −0.579359
\(997\) −16.6863 −0.528460 −0.264230 0.964460i \(-0.585118\pi\)
−0.264230 + 0.964460i \(0.585118\pi\)
\(998\) −13.9411 −0.441299
\(999\) −0.343146 −0.0108567
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 165.2.a.a.1.2 2
3.2 odd 2 495.2.a.d.1.1 2
4.3 odd 2 2640.2.a.bb.1.2 2
5.2 odd 4 825.2.c.e.199.3 4
5.3 odd 4 825.2.c.e.199.2 4
5.4 even 2 825.2.a.g.1.1 2
7.6 odd 2 8085.2.a.ba.1.2 2
11.10 odd 2 1815.2.a.k.1.1 2
12.11 even 2 7920.2.a.cg.1.2 2
15.2 even 4 2475.2.c.m.199.2 4
15.8 even 4 2475.2.c.m.199.3 4
15.14 odd 2 2475.2.a.m.1.2 2
33.32 even 2 5445.2.a.m.1.2 2
55.54 odd 2 9075.2.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.2 2 1.1 even 1 trivial
495.2.a.d.1.1 2 3.2 odd 2
825.2.a.g.1.1 2 5.4 even 2
825.2.c.e.199.2 4 5.3 odd 4
825.2.c.e.199.3 4 5.2 odd 4
1815.2.a.k.1.1 2 11.10 odd 2
2475.2.a.m.1.2 2 15.14 odd 2
2475.2.c.m.199.2 4 15.2 even 4
2475.2.c.m.199.3 4 15.8 even 4
2640.2.a.bb.1.2 2 4.3 odd 2
5445.2.a.m.1.2 2 33.32 even 2
7920.2.a.cg.1.2 2 12.11 even 2
8085.2.a.ba.1.2 2 7.6 odd 2
9075.2.a.v.1.2 2 55.54 odd 2