Properties

Label 165.2.a.b.1.1
Level $165$
Weight $2$
Character 165.1
Self dual yes
Analytic conductor $1.318$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} +2.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} +2.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} +1.73205 q^{10} -1.00000 q^{11} +1.00000 q^{12} +5.46410 q^{13} -3.46410 q^{14} -1.00000 q^{15} -5.00000 q^{16} -1.73205 q^{18} +5.46410 q^{19} -1.00000 q^{20} +2.00000 q^{21} +1.73205 q^{22} +6.92820 q^{23} +1.73205 q^{24} +1.00000 q^{25} -9.46410 q^{26} +1.00000 q^{27} +2.00000 q^{28} -3.46410 q^{29} +1.73205 q^{30} -10.9282 q^{31} +5.19615 q^{32} -1.00000 q^{33} -2.00000 q^{35} +1.00000 q^{36} -4.92820 q^{37} -9.46410 q^{38} +5.46410 q^{39} -1.73205 q^{40} +3.46410 q^{41} -3.46410 q^{42} -4.92820 q^{43} -1.00000 q^{44} -1.00000 q^{45} -12.0000 q^{46} -6.92820 q^{47} -5.00000 q^{48} -3.00000 q^{49} -1.73205 q^{50} +5.46410 q^{52} +0.928203 q^{53} -1.73205 q^{54} +1.00000 q^{55} +3.46410 q^{56} +5.46410 q^{57} +6.00000 q^{58} -6.92820 q^{59} -1.00000 q^{60} +2.00000 q^{61} +18.9282 q^{62} +2.00000 q^{63} +1.00000 q^{64} -5.46410 q^{65} +1.73205 q^{66} +8.00000 q^{67} +6.92820 q^{69} +3.46410 q^{70} +13.8564 q^{71} +1.73205 q^{72} -8.39230 q^{73} +8.53590 q^{74} +1.00000 q^{75} +5.46410 q^{76} -2.00000 q^{77} -9.46410 q^{78} -6.53590 q^{79} +5.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +8.53590 q^{83} +2.00000 q^{84} +8.53590 q^{86} -3.46410 q^{87} -1.73205 q^{88} +0.928203 q^{89} +1.73205 q^{90} +10.9282 q^{91} +6.92820 q^{92} -10.9282 q^{93} +12.0000 q^{94} -5.46410 q^{95} +5.19615 q^{96} -10.0000 q^{97} +5.19615 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{4} - 2q^{5} + 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{4} - 2q^{5} + 4q^{7} + 2q^{9} - 2q^{11} + 2q^{12} + 4q^{13} - 2q^{15} - 10q^{16} + 4q^{19} - 2q^{20} + 4q^{21} + 2q^{25} - 12q^{26} + 2q^{27} + 4q^{28} - 8q^{31} - 2q^{33} - 4q^{35} + 2q^{36} + 4q^{37} - 12q^{38} + 4q^{39} + 4q^{43} - 2q^{44} - 2q^{45} - 24q^{46} - 10q^{48} - 6q^{49} + 4q^{52} - 12q^{53} + 2q^{55} + 4q^{57} + 12q^{58} - 2q^{60} + 4q^{61} + 24q^{62} + 4q^{63} + 2q^{64} - 4q^{65} + 16q^{67} + 4q^{73} + 24q^{74} + 2q^{75} + 4q^{76} - 4q^{77} - 12q^{78} - 20q^{79} + 10q^{80} + 2q^{81} - 12q^{82} + 24q^{83} + 4q^{84} + 24q^{86} - 12q^{89} + 8q^{91} - 8q^{93} + 24q^{94} - 4q^{95} - 20q^{97} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.73205 −0.707107
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.73205 0.612372
\(9\) 1.00000 0.333333
\(10\) 1.73205 0.547723
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 5.46410 1.51547 0.757735 0.652563i \(-0.226306\pi\)
0.757735 + 0.652563i \(0.226306\pi\)
\(14\) −3.46410 −0.925820
\(15\) −1.00000 −0.258199
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.73205 −0.408248
\(19\) 5.46410 1.25355 0.626775 0.779200i \(-0.284374\pi\)
0.626775 + 0.779200i \(0.284374\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.00000 0.436436
\(22\) 1.73205 0.369274
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 1.73205 0.353553
\(25\) 1.00000 0.200000
\(26\) −9.46410 −1.85606
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 1.73205 0.316228
\(31\) −10.9282 −1.96276 −0.981382 0.192068i \(-0.938481\pi\)
−0.981382 + 0.192068i \(0.938481\pi\)
\(32\) 5.19615 0.918559
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −4.92820 −0.810192 −0.405096 0.914274i \(-0.632762\pi\)
−0.405096 + 0.914274i \(0.632762\pi\)
\(38\) −9.46410 −1.53528
\(39\) 5.46410 0.874957
\(40\) −1.73205 −0.273861
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) −3.46410 −0.534522
\(43\) −4.92820 −0.751544 −0.375772 0.926712i \(-0.622622\pi\)
−0.375772 + 0.926712i \(0.622622\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) −12.0000 −1.76930
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) −5.00000 −0.721688
\(49\) −3.00000 −0.428571
\(50\) −1.73205 −0.244949
\(51\) 0 0
\(52\) 5.46410 0.757735
\(53\) 0.928203 0.127499 0.0637493 0.997966i \(-0.479694\pi\)
0.0637493 + 0.997966i \(0.479694\pi\)
\(54\) −1.73205 −0.235702
\(55\) 1.00000 0.134840
\(56\) 3.46410 0.462910
\(57\) 5.46410 0.723738
\(58\) 6.00000 0.787839
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 18.9282 2.40388
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −5.46410 −0.677738
\(66\) 1.73205 0.213201
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 6.92820 0.834058
\(70\) 3.46410 0.414039
\(71\) 13.8564 1.64445 0.822226 0.569160i \(-0.192732\pi\)
0.822226 + 0.569160i \(0.192732\pi\)
\(72\) 1.73205 0.204124
\(73\) −8.39230 −0.982245 −0.491122 0.871091i \(-0.663413\pi\)
−0.491122 + 0.871091i \(0.663413\pi\)
\(74\) 8.53590 0.992278
\(75\) 1.00000 0.115470
\(76\) 5.46410 0.626775
\(77\) −2.00000 −0.227921
\(78\) −9.46410 −1.07160
\(79\) −6.53590 −0.735346 −0.367673 0.929955i \(-0.619845\pi\)
−0.367673 + 0.929955i \(0.619845\pi\)
\(80\) 5.00000 0.559017
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 8.53590 0.936937 0.468468 0.883480i \(-0.344806\pi\)
0.468468 + 0.883480i \(0.344806\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 8.53590 0.920450
\(87\) −3.46410 −0.371391
\(88\) −1.73205 −0.184637
\(89\) 0.928203 0.0983893 0.0491947 0.998789i \(-0.484335\pi\)
0.0491947 + 0.998789i \(0.484335\pi\)
\(90\) 1.73205 0.182574
\(91\) 10.9282 1.14559
\(92\) 6.92820 0.722315
\(93\) −10.9282 −1.13320
\(94\) 12.0000 1.23771
\(95\) −5.46410 −0.560605
\(96\) 5.19615 0.530330
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 5.19615 0.524891
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 9.46410 0.928032
\(105\) −2.00000 −0.195180
\(106\) −1.60770 −0.156153
\(107\) 8.53590 0.825196 0.412598 0.910913i \(-0.364621\pi\)
0.412598 + 0.910913i \(0.364621\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −1.73205 −0.165145
\(111\) −4.92820 −0.467764
\(112\) −10.0000 −0.944911
\(113\) −12.9282 −1.21618 −0.608092 0.793867i \(-0.708065\pi\)
−0.608092 + 0.793867i \(0.708065\pi\)
\(114\) −9.46410 −0.886394
\(115\) −6.92820 −0.646058
\(116\) −3.46410 −0.321634
\(117\) 5.46410 0.505156
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) −1.73205 −0.158114
\(121\) 1.00000 0.0909091
\(122\) −3.46410 −0.313625
\(123\) 3.46410 0.312348
\(124\) −10.9282 −0.981382
\(125\) −1.00000 −0.0894427
\(126\) −3.46410 −0.308607
\(127\) 8.92820 0.792250 0.396125 0.918197i \(-0.370355\pi\)
0.396125 + 0.918197i \(0.370355\pi\)
\(128\) −12.1244 −1.07165
\(129\) −4.92820 −0.433904
\(130\) 9.46410 0.830057
\(131\) 18.9282 1.65376 0.826882 0.562375i \(-0.190112\pi\)
0.826882 + 0.562375i \(0.190112\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 10.9282 0.947595
\(134\) −13.8564 −1.19701
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) −12.0000 −1.02151
\(139\) 12.3923 1.05110 0.525551 0.850762i \(-0.323859\pi\)
0.525551 + 0.850762i \(0.323859\pi\)
\(140\) −2.00000 −0.169031
\(141\) −6.92820 −0.583460
\(142\) −24.0000 −2.01404
\(143\) −5.46410 −0.456931
\(144\) −5.00000 −0.416667
\(145\) 3.46410 0.287678
\(146\) 14.5359 1.20300
\(147\) −3.00000 −0.247436
\(148\) −4.92820 −0.405096
\(149\) −15.4641 −1.26687 −0.633434 0.773796i \(-0.718355\pi\)
−0.633434 + 0.773796i \(0.718355\pi\)
\(150\) −1.73205 −0.141421
\(151\) −20.3923 −1.65950 −0.829751 0.558134i \(-0.811518\pi\)
−0.829751 + 0.558134i \(0.811518\pi\)
\(152\) 9.46410 0.767640
\(153\) 0 0
\(154\) 3.46410 0.279145
\(155\) 10.9282 0.877774
\(156\) 5.46410 0.437478
\(157\) −3.07180 −0.245156 −0.122578 0.992459i \(-0.539116\pi\)
−0.122578 + 0.992459i \(0.539116\pi\)
\(158\) 11.3205 0.900611
\(159\) 0.928203 0.0736113
\(160\) −5.19615 −0.410792
\(161\) 13.8564 1.09204
\(162\) −1.73205 −0.136083
\(163\) 9.85641 0.772013 0.386007 0.922496i \(-0.373854\pi\)
0.386007 + 0.922496i \(0.373854\pi\)
\(164\) 3.46410 0.270501
\(165\) 1.00000 0.0778499
\(166\) −14.7846 −1.14751
\(167\) −10.3923 −0.804181 −0.402090 0.915600i \(-0.631716\pi\)
−0.402090 + 0.915600i \(0.631716\pi\)
\(168\) 3.46410 0.267261
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) 5.46410 0.417850
\(172\) −4.92820 −0.375772
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 6.00000 0.454859
\(175\) 2.00000 0.151186
\(176\) 5.00000 0.376889
\(177\) −6.92820 −0.520756
\(178\) −1.60770 −0.120502
\(179\) 6.92820 0.517838 0.258919 0.965899i \(-0.416634\pi\)
0.258919 + 0.965899i \(0.416634\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 15.8564 1.17860 0.589299 0.807915i \(-0.299404\pi\)
0.589299 + 0.807915i \(0.299404\pi\)
\(182\) −18.9282 −1.40305
\(183\) 2.00000 0.147844
\(184\) 12.0000 0.884652
\(185\) 4.92820 0.362329
\(186\) 18.9282 1.38788
\(187\) 0 0
\(188\) −6.92820 −0.505291
\(189\) 2.00000 0.145479
\(190\) 9.46410 0.686598
\(191\) −18.9282 −1.36960 −0.684798 0.728733i \(-0.740110\pi\)
−0.684798 + 0.728733i \(0.740110\pi\)
\(192\) 1.00000 0.0721688
\(193\) 24.3923 1.75580 0.877898 0.478847i \(-0.158945\pi\)
0.877898 + 0.478847i \(0.158945\pi\)
\(194\) 17.3205 1.24354
\(195\) −5.46410 −0.391292
\(196\) −3.00000 −0.214286
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 1.73205 0.123091
\(199\) −24.7846 −1.75693 −0.878467 0.477803i \(-0.841433\pi\)
−0.878467 + 0.477803i \(0.841433\pi\)
\(200\) 1.73205 0.122474
\(201\) 8.00000 0.564276
\(202\) 18.0000 1.26648
\(203\) −6.92820 −0.486265
\(204\) 0 0
\(205\) −3.46410 −0.241943
\(206\) −13.8564 −0.965422
\(207\) 6.92820 0.481543
\(208\) −27.3205 −1.89434
\(209\) −5.46410 −0.377960
\(210\) 3.46410 0.239046
\(211\) −8.39230 −0.577750 −0.288875 0.957367i \(-0.593281\pi\)
−0.288875 + 0.957367i \(0.593281\pi\)
\(212\) 0.928203 0.0637493
\(213\) 13.8564 0.949425
\(214\) −14.7846 −1.01066
\(215\) 4.92820 0.336101
\(216\) 1.73205 0.117851
\(217\) −21.8564 −1.48371
\(218\) 17.3205 1.17309
\(219\) −8.39230 −0.567099
\(220\) 1.00000 0.0674200
\(221\) 0 0
\(222\) 8.53590 0.572892
\(223\) 9.85641 0.660034 0.330017 0.943975i \(-0.392946\pi\)
0.330017 + 0.943975i \(0.392946\pi\)
\(224\) 10.3923 0.694365
\(225\) 1.00000 0.0666667
\(226\) 22.3923 1.48951
\(227\) 15.4641 1.02639 0.513194 0.858272i \(-0.328462\pi\)
0.513194 + 0.858272i \(0.328462\pi\)
\(228\) 5.46410 0.361869
\(229\) −23.8564 −1.57648 −0.788238 0.615371i \(-0.789006\pi\)
−0.788238 + 0.615371i \(0.789006\pi\)
\(230\) 12.0000 0.791257
\(231\) −2.00000 −0.131590
\(232\) −6.00000 −0.393919
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) −9.46410 −0.618688
\(235\) 6.92820 0.451946
\(236\) −6.92820 −0.450988
\(237\) −6.53590 −0.424552
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 5.00000 0.322749
\(241\) 0.143594 0.00924967 0.00462484 0.999989i \(-0.498528\pi\)
0.00462484 + 0.999989i \(0.498528\pi\)
\(242\) −1.73205 −0.111340
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 3.00000 0.191663
\(246\) −6.00000 −0.382546
\(247\) 29.8564 1.89972
\(248\) −18.9282 −1.20194
\(249\) 8.53590 0.540941
\(250\) 1.73205 0.109545
\(251\) −1.85641 −0.117175 −0.0585877 0.998282i \(-0.518660\pi\)
−0.0585877 + 0.998282i \(0.518660\pi\)
\(252\) 2.00000 0.125988
\(253\) −6.92820 −0.435572
\(254\) −15.4641 −0.970304
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −19.8564 −1.23861 −0.619304 0.785151i \(-0.712585\pi\)
−0.619304 + 0.785151i \(0.712585\pi\)
\(258\) 8.53590 0.531422
\(259\) −9.85641 −0.612447
\(260\) −5.46410 −0.338869
\(261\) −3.46410 −0.214423
\(262\) −32.7846 −2.02544
\(263\) 20.5359 1.26630 0.633149 0.774030i \(-0.281762\pi\)
0.633149 + 0.774030i \(0.281762\pi\)
\(264\) −1.73205 −0.106600
\(265\) −0.928203 −0.0570191
\(266\) −18.9282 −1.16056
\(267\) 0.928203 0.0568051
\(268\) 8.00000 0.488678
\(269\) 19.8564 1.21067 0.605333 0.795972i \(-0.293040\pi\)
0.605333 + 0.795972i \(0.293040\pi\)
\(270\) 1.73205 0.105409
\(271\) −11.6077 −0.705117 −0.352559 0.935790i \(-0.614688\pi\)
−0.352559 + 0.935790i \(0.614688\pi\)
\(272\) 0 0
\(273\) 10.9282 0.661405
\(274\) 31.1769 1.88347
\(275\) −1.00000 −0.0603023
\(276\) 6.92820 0.417029
\(277\) 29.4641 1.77033 0.885163 0.465281i \(-0.154047\pi\)
0.885163 + 0.465281i \(0.154047\pi\)
\(278\) −21.4641 −1.28733
\(279\) −10.9282 −0.654254
\(280\) −3.46410 −0.207020
\(281\) 3.46410 0.206651 0.103325 0.994648i \(-0.467052\pi\)
0.103325 + 0.994648i \(0.467052\pi\)
\(282\) 12.0000 0.714590
\(283\) −4.92820 −0.292951 −0.146476 0.989214i \(-0.546793\pi\)
−0.146476 + 0.989214i \(0.546793\pi\)
\(284\) 13.8564 0.822226
\(285\) −5.46410 −0.323665
\(286\) 9.46410 0.559624
\(287\) 6.92820 0.408959
\(288\) 5.19615 0.306186
\(289\) −17.0000 −1.00000
\(290\) −6.00000 −0.352332
\(291\) −10.0000 −0.586210
\(292\) −8.39230 −0.491122
\(293\) −13.8564 −0.809500 −0.404750 0.914427i \(-0.632641\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) 5.19615 0.303046
\(295\) 6.92820 0.403376
\(296\) −8.53590 −0.496139
\(297\) −1.00000 −0.0580259
\(298\) 26.7846 1.55159
\(299\) 37.8564 2.18929
\(300\) 1.00000 0.0577350
\(301\) −9.85641 −0.568114
\(302\) 35.3205 2.03247
\(303\) −10.3923 −0.597022
\(304\) −27.3205 −1.56694
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 14.0000 0.799022 0.399511 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) −2.00000 −0.113961
\(309\) 8.00000 0.455104
\(310\) −18.9282 −1.07505
\(311\) 5.07180 0.287595 0.143798 0.989607i \(-0.454069\pi\)
0.143798 + 0.989607i \(0.454069\pi\)
\(312\) 9.46410 0.535799
\(313\) 20.9282 1.18293 0.591466 0.806330i \(-0.298549\pi\)
0.591466 + 0.806330i \(0.298549\pi\)
\(314\) 5.32051 0.300254
\(315\) −2.00000 −0.112687
\(316\) −6.53590 −0.367673
\(317\) −24.9282 −1.40011 −0.700054 0.714090i \(-0.746841\pi\)
−0.700054 + 0.714090i \(0.746841\pi\)
\(318\) −1.60770 −0.0901551
\(319\) 3.46410 0.193952
\(320\) −1.00000 −0.0559017
\(321\) 8.53590 0.476427
\(322\) −24.0000 −1.33747
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 5.46410 0.303094
\(326\) −17.0718 −0.945519
\(327\) −10.0000 −0.553001
\(328\) 6.00000 0.331295
\(329\) −13.8564 −0.763928
\(330\) −1.73205 −0.0953463
\(331\) 9.85641 0.541757 0.270879 0.962614i \(-0.412686\pi\)
0.270879 + 0.962614i \(0.412686\pi\)
\(332\) 8.53590 0.468468
\(333\) −4.92820 −0.270064
\(334\) 18.0000 0.984916
\(335\) −8.00000 −0.437087
\(336\) −10.0000 −0.545545
\(337\) 33.1769 1.80726 0.903631 0.428312i \(-0.140892\pi\)
0.903631 + 0.428312i \(0.140892\pi\)
\(338\) −29.1962 −1.58806
\(339\) −12.9282 −0.702164
\(340\) 0 0
\(341\) 10.9282 0.591795
\(342\) −9.46410 −0.511760
\(343\) −20.0000 −1.07990
\(344\) −8.53590 −0.460225
\(345\) −6.92820 −0.373002
\(346\) 20.7846 1.11739
\(347\) 22.3923 1.20208 0.601041 0.799218i \(-0.294753\pi\)
0.601041 + 0.799218i \(0.294753\pi\)
\(348\) −3.46410 −0.185695
\(349\) −8.14359 −0.435917 −0.217958 0.975958i \(-0.569940\pi\)
−0.217958 + 0.975958i \(0.569940\pi\)
\(350\) −3.46410 −0.185164
\(351\) 5.46410 0.291652
\(352\) −5.19615 −0.276956
\(353\) 12.9282 0.688099 0.344049 0.938952i \(-0.388201\pi\)
0.344049 + 0.938952i \(0.388201\pi\)
\(354\) 12.0000 0.637793
\(355\) −13.8564 −0.735422
\(356\) 0.928203 0.0491947
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −20.7846 −1.09697 −0.548485 0.836160i \(-0.684795\pi\)
−0.548485 + 0.836160i \(0.684795\pi\)
\(360\) −1.73205 −0.0912871
\(361\) 10.8564 0.571390
\(362\) −27.4641 −1.44348
\(363\) 1.00000 0.0524864
\(364\) 10.9282 0.572793
\(365\) 8.39230 0.439273
\(366\) −3.46410 −0.181071
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) −34.6410 −1.80579
\(369\) 3.46410 0.180334
\(370\) −8.53590 −0.443760
\(371\) 1.85641 0.0963798
\(372\) −10.9282 −0.566601
\(373\) −20.3923 −1.05587 −0.527937 0.849284i \(-0.677034\pi\)
−0.527937 + 0.849284i \(0.677034\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −12.0000 −0.618853
\(377\) −18.9282 −0.974852
\(378\) −3.46410 −0.178174
\(379\) −17.8564 −0.917222 −0.458611 0.888637i \(-0.651653\pi\)
−0.458611 + 0.888637i \(0.651653\pi\)
\(380\) −5.46410 −0.280302
\(381\) 8.92820 0.457406
\(382\) 32.7846 1.67741
\(383\) −13.8564 −0.708029 −0.354015 0.935240i \(-0.615184\pi\)
−0.354015 + 0.935240i \(0.615184\pi\)
\(384\) −12.1244 −0.618718
\(385\) 2.00000 0.101929
\(386\) −42.2487 −2.15040
\(387\) −4.92820 −0.250515
\(388\) −10.0000 −0.507673
\(389\) 11.0718 0.561362 0.280681 0.959801i \(-0.409440\pi\)
0.280681 + 0.959801i \(0.409440\pi\)
\(390\) 9.46410 0.479233
\(391\) 0 0
\(392\) −5.19615 −0.262445
\(393\) 18.9282 0.954802
\(394\) 20.7846 1.04711
\(395\) 6.53590 0.328857
\(396\) −1.00000 −0.0502519
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 42.9282 2.15180
\(399\) 10.9282 0.547094
\(400\) −5.00000 −0.250000
\(401\) −7.85641 −0.392330 −0.196165 0.980571i \(-0.562849\pi\)
−0.196165 + 0.980571i \(0.562849\pi\)
\(402\) −13.8564 −0.691095
\(403\) −59.7128 −2.97451
\(404\) −10.3923 −0.517036
\(405\) −1.00000 −0.0496904
\(406\) 12.0000 0.595550
\(407\) 4.92820 0.244282
\(408\) 0 0
\(409\) −6.78461 −0.335477 −0.167739 0.985831i \(-0.553646\pi\)
−0.167739 + 0.985831i \(0.553646\pi\)
\(410\) 6.00000 0.296319
\(411\) −18.0000 −0.887875
\(412\) 8.00000 0.394132
\(413\) −13.8564 −0.681829
\(414\) −12.0000 −0.589768
\(415\) −8.53590 −0.419011
\(416\) 28.3923 1.39205
\(417\) 12.3923 0.606854
\(418\) 9.46410 0.462904
\(419\) 30.9282 1.51094 0.755471 0.655182i \(-0.227408\pi\)
0.755471 + 0.655182i \(0.227408\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 14.5359 0.707596
\(423\) −6.92820 −0.336861
\(424\) 1.60770 0.0780766
\(425\) 0 0
\(426\) −24.0000 −1.16280
\(427\) 4.00000 0.193574
\(428\) 8.53590 0.412598
\(429\) −5.46410 −0.263809
\(430\) −8.53590 −0.411638
\(431\) 8.78461 0.423140 0.211570 0.977363i \(-0.432142\pi\)
0.211570 + 0.977363i \(0.432142\pi\)
\(432\) −5.00000 −0.240563
\(433\) 0.143594 0.00690067 0.00345033 0.999994i \(-0.498902\pi\)
0.00345033 + 0.999994i \(0.498902\pi\)
\(434\) 37.8564 1.81717
\(435\) 3.46410 0.166091
\(436\) −10.0000 −0.478913
\(437\) 37.8564 1.81092
\(438\) 14.5359 0.694552
\(439\) 33.1769 1.58345 0.791724 0.610879i \(-0.209184\pi\)
0.791724 + 0.610879i \(0.209184\pi\)
\(440\) 1.73205 0.0825723
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −4.92820 −0.233882
\(445\) −0.928203 −0.0440011
\(446\) −17.0718 −0.808373
\(447\) −15.4641 −0.731427
\(448\) 2.00000 0.0944911
\(449\) −26.7846 −1.26404 −0.632022 0.774950i \(-0.717775\pi\)
−0.632022 + 0.774950i \(0.717775\pi\)
\(450\) −1.73205 −0.0816497
\(451\) −3.46410 −0.163118
\(452\) −12.9282 −0.608092
\(453\) −20.3923 −0.958114
\(454\) −26.7846 −1.25706
\(455\) −10.9282 −0.512322
\(456\) 9.46410 0.443197
\(457\) 12.3923 0.579688 0.289844 0.957074i \(-0.406397\pi\)
0.289844 + 0.957074i \(0.406397\pi\)
\(458\) 41.3205 1.93078
\(459\) 0 0
\(460\) −6.92820 −0.323029
\(461\) 36.2487 1.68827 0.844135 0.536130i \(-0.180114\pi\)
0.844135 + 0.536130i \(0.180114\pi\)
\(462\) 3.46410 0.161165
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) 17.3205 0.804084
\(465\) 10.9282 0.506783
\(466\) −20.7846 −0.962828
\(467\) 5.07180 0.234695 0.117347 0.993091i \(-0.462561\pi\)
0.117347 + 0.993091i \(0.462561\pi\)
\(468\) 5.46410 0.252578
\(469\) 16.0000 0.738811
\(470\) −12.0000 −0.553519
\(471\) −3.07180 −0.141541
\(472\) −12.0000 −0.552345
\(473\) 4.92820 0.226599
\(474\) 11.3205 0.519968
\(475\) 5.46410 0.250710
\(476\) 0 0
\(477\) 0.928203 0.0424995
\(478\) 20.7846 0.950666
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) −5.19615 −0.237171
\(481\) −26.9282 −1.22782
\(482\) −0.248711 −0.0113285
\(483\) 13.8564 0.630488
\(484\) 1.00000 0.0454545
\(485\) 10.0000 0.454077
\(486\) −1.73205 −0.0785674
\(487\) −31.7128 −1.43704 −0.718522 0.695504i \(-0.755181\pi\)
−0.718522 + 0.695504i \(0.755181\pi\)
\(488\) 3.46410 0.156813
\(489\) 9.85641 0.445722
\(490\) −5.19615 −0.234738
\(491\) −30.9282 −1.39577 −0.697885 0.716210i \(-0.745875\pi\)
−0.697885 + 0.716210i \(0.745875\pi\)
\(492\) 3.46410 0.156174
\(493\) 0 0
\(494\) −51.7128 −2.32667
\(495\) 1.00000 0.0449467
\(496\) 54.6410 2.45345
\(497\) 27.7128 1.24309
\(498\) −14.7846 −0.662514
\(499\) 28.7846 1.28858 0.644288 0.764783i \(-0.277154\pi\)
0.644288 + 0.764783i \(0.277154\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −10.3923 −0.464294
\(502\) 3.21539 0.143510
\(503\) 31.1769 1.39011 0.695055 0.718957i \(-0.255380\pi\)
0.695055 + 0.718957i \(0.255380\pi\)
\(504\) 3.46410 0.154303
\(505\) 10.3923 0.462451
\(506\) 12.0000 0.533465
\(507\) 16.8564 0.748619
\(508\) 8.92820 0.396125
\(509\) 19.8564 0.880120 0.440060 0.897968i \(-0.354957\pi\)
0.440060 + 0.897968i \(0.354957\pi\)
\(510\) 0 0
\(511\) −16.7846 −0.742507
\(512\) −8.66025 −0.382733
\(513\) 5.46410 0.241246
\(514\) 34.3923 1.51698
\(515\) −8.00000 −0.352522
\(516\) −4.92820 −0.216952
\(517\) 6.92820 0.304702
\(518\) 17.0718 0.750092
\(519\) −12.0000 −0.526742
\(520\) −9.46410 −0.415028
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 6.00000 0.262613
\(523\) −22.0000 −0.961993 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) 18.9282 0.826882
\(525\) 2.00000 0.0872872
\(526\) −35.5692 −1.55089
\(527\) 0 0
\(528\) 5.00000 0.217597
\(529\) 25.0000 1.08696
\(530\) 1.60770 0.0698338
\(531\) −6.92820 −0.300658
\(532\) 10.9282 0.473798
\(533\) 18.9282 0.819871
\(534\) −1.60770 −0.0695718
\(535\) −8.53590 −0.369039
\(536\) 13.8564 0.598506
\(537\) 6.92820 0.298974
\(538\) −34.3923 −1.48276
\(539\) 3.00000 0.129219
\(540\) −1.00000 −0.0430331
\(541\) 27.8564 1.19764 0.598820 0.800883i \(-0.295636\pi\)
0.598820 + 0.800883i \(0.295636\pi\)
\(542\) 20.1051 0.863589
\(543\) 15.8564 0.680464
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) −18.9282 −0.810052
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) −18.0000 −0.768922
\(549\) 2.00000 0.0853579
\(550\) 1.73205 0.0738549
\(551\) −18.9282 −0.806369
\(552\) 12.0000 0.510754
\(553\) −13.0718 −0.555869
\(554\) −51.0333 −2.16820
\(555\) 4.92820 0.209191
\(556\) 12.3923 0.525551
\(557\) 3.21539 0.136240 0.0681202 0.997677i \(-0.478300\pi\)
0.0681202 + 0.997677i \(0.478300\pi\)
\(558\) 18.9282 0.801295
\(559\) −26.9282 −1.13894
\(560\) 10.0000 0.422577
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 10.3923 0.437983 0.218992 0.975727i \(-0.429723\pi\)
0.218992 + 0.975727i \(0.429723\pi\)
\(564\) −6.92820 −0.291730
\(565\) 12.9282 0.543894
\(566\) 8.53590 0.358791
\(567\) 2.00000 0.0839921
\(568\) 24.0000 1.00702
\(569\) 5.32051 0.223047 0.111524 0.993762i \(-0.464427\pi\)
0.111524 + 0.993762i \(0.464427\pi\)
\(570\) 9.46410 0.396408
\(571\) 3.60770 0.150977 0.0754887 0.997147i \(-0.475948\pi\)
0.0754887 + 0.997147i \(0.475948\pi\)
\(572\) −5.46410 −0.228466
\(573\) −18.9282 −0.790737
\(574\) −12.0000 −0.500870
\(575\) 6.92820 0.288926
\(576\) 1.00000 0.0416667
\(577\) −18.7846 −0.782014 −0.391007 0.920388i \(-0.627873\pi\)
−0.391007 + 0.920388i \(0.627873\pi\)
\(578\) 29.4449 1.22474
\(579\) 24.3923 1.01371
\(580\) 3.46410 0.143839
\(581\) 17.0718 0.708257
\(582\) 17.3205 0.717958
\(583\) −0.928203 −0.0384422
\(584\) −14.5359 −0.601500
\(585\) −5.46410 −0.225913
\(586\) 24.0000 0.991431
\(587\) −18.9282 −0.781251 −0.390625 0.920550i \(-0.627741\pi\)
−0.390625 + 0.920550i \(0.627741\pi\)
\(588\) −3.00000 −0.123718
\(589\) −59.7128 −2.46042
\(590\) −12.0000 −0.494032
\(591\) −12.0000 −0.493614
\(592\) 24.6410 1.01274
\(593\) 8.78461 0.360741 0.180370 0.983599i \(-0.442270\pi\)
0.180370 + 0.983599i \(0.442270\pi\)
\(594\) 1.73205 0.0710669
\(595\) 0 0
\(596\) −15.4641 −0.633434
\(597\) −24.7846 −1.01437
\(598\) −65.5692 −2.68132
\(599\) −37.8564 −1.54677 −0.773385 0.633936i \(-0.781438\pi\)
−0.773385 + 0.633936i \(0.781438\pi\)
\(600\) 1.73205 0.0707107
\(601\) −32.6410 −1.33145 −0.665727 0.746195i \(-0.731879\pi\)
−0.665727 + 0.746195i \(0.731879\pi\)
\(602\) 17.0718 0.695794
\(603\) 8.00000 0.325785
\(604\) −20.3923 −0.829751
\(605\) −1.00000 −0.0406558
\(606\) 18.0000 0.731200
\(607\) −18.7846 −0.762444 −0.381222 0.924484i \(-0.624497\pi\)
−0.381222 + 0.924484i \(0.624497\pi\)
\(608\) 28.3923 1.15146
\(609\) −6.92820 −0.280745
\(610\) 3.46410 0.140257
\(611\) −37.8564 −1.53151
\(612\) 0 0
\(613\) −20.3923 −0.823637 −0.411819 0.911266i \(-0.635106\pi\)
−0.411819 + 0.911266i \(0.635106\pi\)
\(614\) −24.2487 −0.978598
\(615\) −3.46410 −0.139686
\(616\) −3.46410 −0.139573
\(617\) −36.9282 −1.48667 −0.743337 0.668917i \(-0.766758\pi\)
−0.743337 + 0.668917i \(0.766758\pi\)
\(618\) −13.8564 −0.557386
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 10.9282 0.438887
\(621\) 6.92820 0.278019
\(622\) −8.78461 −0.352231
\(623\) 1.85641 0.0743754
\(624\) −27.3205 −1.09370
\(625\) 1.00000 0.0400000
\(626\) −36.2487 −1.44879
\(627\) −5.46410 −0.218215
\(628\) −3.07180 −0.122578
\(629\) 0 0
\(630\) 3.46410 0.138013
\(631\) −21.0718 −0.838855 −0.419427 0.907789i \(-0.637769\pi\)
−0.419427 + 0.907789i \(0.637769\pi\)
\(632\) −11.3205 −0.450306
\(633\) −8.39230 −0.333564
\(634\) 43.1769 1.71477
\(635\) −8.92820 −0.354305
\(636\) 0.928203 0.0368057
\(637\) −16.3923 −0.649487
\(638\) −6.00000 −0.237542
\(639\) 13.8564 0.548151
\(640\) 12.1244 0.479257
\(641\) −12.9282 −0.510633 −0.255317 0.966857i \(-0.582180\pi\)
−0.255317 + 0.966857i \(0.582180\pi\)
\(642\) −14.7846 −0.583502
\(643\) 37.5692 1.48159 0.740793 0.671734i \(-0.234450\pi\)
0.740793 + 0.671734i \(0.234450\pi\)
\(644\) 13.8564 0.546019
\(645\) 4.92820 0.194048
\(646\) 0 0
\(647\) 27.7128 1.08950 0.544752 0.838597i \(-0.316624\pi\)
0.544752 + 0.838597i \(0.316624\pi\)
\(648\) 1.73205 0.0680414
\(649\) 6.92820 0.271956
\(650\) −9.46410 −0.371213
\(651\) −21.8564 −0.856620
\(652\) 9.85641 0.386007
\(653\) 19.8564 0.777041 0.388521 0.921440i \(-0.372986\pi\)
0.388521 + 0.921440i \(0.372986\pi\)
\(654\) 17.3205 0.677285
\(655\) −18.9282 −0.739586
\(656\) −17.3205 −0.676252
\(657\) −8.39230 −0.327415
\(658\) 24.0000 0.935617
\(659\) −15.7128 −0.612084 −0.306042 0.952018i \(-0.599005\pi\)
−0.306042 + 0.952018i \(0.599005\pi\)
\(660\) 1.00000 0.0389249
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −17.0718 −0.663514
\(663\) 0 0
\(664\) 14.7846 0.573754
\(665\) −10.9282 −0.423778
\(666\) 8.53590 0.330759
\(667\) −24.0000 −0.929284
\(668\) −10.3923 −0.402090
\(669\) 9.85641 0.381071
\(670\) 13.8564 0.535320
\(671\) −2.00000 −0.0772091
\(672\) 10.3923 0.400892
\(673\) −3.32051 −0.127996 −0.0639981 0.997950i \(-0.520385\pi\)
−0.0639981 + 0.997950i \(0.520385\pi\)
\(674\) −57.4641 −2.21343
\(675\) 1.00000 0.0384900
\(676\) 16.8564 0.648323
\(677\) 8.78461 0.337620 0.168810 0.985649i \(-0.446008\pi\)
0.168810 + 0.985649i \(0.446008\pi\)
\(678\) 22.3923 0.859971
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 15.4641 0.592586
\(682\) −18.9282 −0.724798
\(683\) −32.7846 −1.25447 −0.627234 0.778831i \(-0.715813\pi\)
−0.627234 + 0.778831i \(0.715813\pi\)
\(684\) 5.46410 0.208925
\(685\) 18.0000 0.687745
\(686\) 34.6410 1.32260
\(687\) −23.8564 −0.910179
\(688\) 24.6410 0.939430
\(689\) 5.07180 0.193220
\(690\) 12.0000 0.456832
\(691\) 47.7128 1.81508 0.907540 0.419965i \(-0.137958\pi\)
0.907540 + 0.419965i \(0.137958\pi\)
\(692\) −12.0000 −0.456172
\(693\) −2.00000 −0.0759737
\(694\) −38.7846 −1.47224
\(695\) −12.3923 −0.470067
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) 14.1051 0.533887
\(699\) 12.0000 0.453882
\(700\) 2.00000 0.0755929
\(701\) 39.4641 1.49054 0.745269 0.666764i \(-0.232321\pi\)
0.745269 + 0.666764i \(0.232321\pi\)
\(702\) −9.46410 −0.357199
\(703\) −26.9282 −1.01562
\(704\) −1.00000 −0.0376889
\(705\) 6.92820 0.260931
\(706\) −22.3923 −0.842746
\(707\) −20.7846 −0.781686
\(708\) −6.92820 −0.260378
\(709\) −11.8564 −0.445277 −0.222638 0.974901i \(-0.571467\pi\)
−0.222638 + 0.974901i \(0.571467\pi\)
\(710\) 24.0000 0.900704
\(711\) −6.53590 −0.245115
\(712\) 1.60770 0.0602509
\(713\) −75.7128 −2.83547
\(714\) 0 0
\(715\) 5.46410 0.204346
\(716\) 6.92820 0.258919
\(717\) −12.0000 −0.448148
\(718\) 36.0000 1.34351
\(719\) −5.07180 −0.189146 −0.0945731 0.995518i \(-0.530149\pi\)
−0.0945731 + 0.995518i \(0.530149\pi\)
\(720\) 5.00000 0.186339
\(721\) 16.0000 0.595871
\(722\) −18.8038 −0.699807
\(723\) 0.143594 0.00534030
\(724\) 15.8564 0.589299
\(725\) −3.46410 −0.128654
\(726\) −1.73205 −0.0642824
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 18.9282 0.701526
\(729\) 1.00000 0.0370370
\(730\) −14.5359 −0.537998
\(731\) 0 0
\(732\) 2.00000 0.0739221
\(733\) 53.9615 1.99311 0.996557 0.0829082i \(-0.0264208\pi\)
0.996557 + 0.0829082i \(0.0264208\pi\)
\(734\) −34.6410 −1.27862
\(735\) 3.00000 0.110657
\(736\) 36.0000 1.32698
\(737\) −8.00000 −0.294684
\(738\) −6.00000 −0.220863
\(739\) 17.4641 0.642427 0.321214 0.947007i \(-0.395909\pi\)
0.321214 + 0.947007i \(0.395909\pi\)
\(740\) 4.92820 0.181164
\(741\) 29.8564 1.09680
\(742\) −3.21539 −0.118041
\(743\) 25.6077 0.939455 0.469728 0.882811i \(-0.344352\pi\)
0.469728 + 0.882811i \(0.344352\pi\)
\(744\) −18.9282 −0.693942
\(745\) 15.4641 0.566561
\(746\) 35.3205 1.29318
\(747\) 8.53590 0.312312
\(748\) 0 0
\(749\) 17.0718 0.623790
\(750\) 1.73205 0.0632456
\(751\) 26.9282 0.982624 0.491312 0.870984i \(-0.336517\pi\)
0.491312 + 0.870984i \(0.336517\pi\)
\(752\) 34.6410 1.26323
\(753\) −1.85641 −0.0676512
\(754\) 32.7846 1.19395
\(755\) 20.3923 0.742152
\(756\) 2.00000 0.0727393
\(757\) 34.7846 1.26427 0.632134 0.774859i \(-0.282179\pi\)
0.632134 + 0.774859i \(0.282179\pi\)
\(758\) 30.9282 1.12336
\(759\) −6.92820 −0.251478
\(760\) −9.46410 −0.343299
\(761\) −32.5359 −1.17943 −0.589713 0.807613i \(-0.700759\pi\)
−0.589713 + 0.807613i \(0.700759\pi\)
\(762\) −15.4641 −0.560205
\(763\) −20.0000 −0.724049
\(764\) −18.9282 −0.684798
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −37.8564 −1.36692
\(768\) 19.0000 0.685603
\(769\) 50.4974 1.82098 0.910492 0.413527i \(-0.135703\pi\)
0.910492 + 0.413527i \(0.135703\pi\)
\(770\) −3.46410 −0.124838
\(771\) −19.8564 −0.715111
\(772\) 24.3923 0.877898
\(773\) −4.14359 −0.149035 −0.0745174 0.997220i \(-0.523742\pi\)
−0.0745174 + 0.997220i \(0.523742\pi\)
\(774\) 8.53590 0.306817
\(775\) −10.9282 −0.392553
\(776\) −17.3205 −0.621770
\(777\) −9.85641 −0.353597
\(778\) −19.1769 −0.687526
\(779\) 18.9282 0.678173
\(780\) −5.46410 −0.195646
\(781\) −13.8564 −0.495821
\(782\) 0 0
\(783\) −3.46410 −0.123797
\(784\) 15.0000 0.535714
\(785\) 3.07180 0.109637
\(786\) −32.7846 −1.16939
\(787\) 22.7846 0.812184 0.406092 0.913832i \(-0.366891\pi\)
0.406092 + 0.913832i \(0.366891\pi\)
\(788\) −12.0000 −0.427482
\(789\) 20.5359 0.731097
\(790\) −11.3205 −0.402766
\(791\) −25.8564 −0.919348
\(792\) −1.73205 −0.0615457
\(793\) 10.9282 0.388072
\(794\) −3.46410 −0.122936
\(795\) −0.928203 −0.0329200
\(796\) −24.7846 −0.878467
\(797\) −52.6410 −1.86464 −0.932320 0.361634i \(-0.882219\pi\)
−0.932320 + 0.361634i \(0.882219\pi\)
\(798\) −18.9282 −0.670051
\(799\) 0 0
\(800\) 5.19615 0.183712
\(801\) 0.928203 0.0327964
\(802\) 13.6077 0.480504
\(803\) 8.39230 0.296158
\(804\) 8.00000 0.282138
\(805\) −13.8564 −0.488374
\(806\) 103.426 3.64301
\(807\) 19.8564 0.698979
\(808\) −18.0000 −0.633238
\(809\) −15.4641 −0.543689 −0.271844 0.962341i \(-0.587634\pi\)
−0.271844 + 0.962341i \(0.587634\pi\)
\(810\) 1.73205 0.0608581
\(811\) 12.3923 0.435153 0.217576 0.976043i \(-0.430185\pi\)
0.217576 + 0.976043i \(0.430185\pi\)
\(812\) −6.92820 −0.243132
\(813\) −11.6077 −0.407100
\(814\) −8.53590 −0.299183
\(815\) −9.85641 −0.345255
\(816\) 0 0
\(817\) −26.9282 −0.942099
\(818\) 11.7513 0.410874
\(819\) 10.9282 0.381862
\(820\) −3.46410 −0.120972
\(821\) 20.5359 0.716708 0.358354 0.933586i \(-0.383338\pi\)
0.358354 + 0.933586i \(0.383338\pi\)
\(822\) 31.1769 1.08742
\(823\) −33.5692 −1.17015 −0.585075 0.810979i \(-0.698935\pi\)
−0.585075 + 0.810979i \(0.698935\pi\)
\(824\) 13.8564 0.482711
\(825\) −1.00000 −0.0348155
\(826\) 24.0000 0.835067
\(827\) 22.3923 0.778657 0.389328 0.921099i \(-0.372707\pi\)
0.389328 + 0.921099i \(0.372707\pi\)
\(828\) 6.92820 0.240772
\(829\) 29.7128 1.03197 0.515984 0.856598i \(-0.327426\pi\)
0.515984 + 0.856598i \(0.327426\pi\)
\(830\) 14.7846 0.513181
\(831\) 29.4641 1.02210
\(832\) 5.46410 0.189434
\(833\) 0 0
\(834\) −21.4641 −0.743241
\(835\) 10.3923 0.359641
\(836\) −5.46410 −0.188980
\(837\) −10.9282 −0.377734
\(838\) −53.5692 −1.85052
\(839\) 56.7846 1.96042 0.980211 0.197954i \(-0.0634298\pi\)
0.980211 + 0.197954i \(0.0634298\pi\)
\(840\) −3.46410 −0.119523
\(841\) −17.0000 −0.586207
\(842\) −3.46410 −0.119381
\(843\) 3.46410 0.119310
\(844\) −8.39230 −0.288875
\(845\) −16.8564 −0.579878
\(846\) 12.0000 0.412568
\(847\) 2.00000 0.0687208
\(848\) −4.64102 −0.159373
\(849\) −4.92820 −0.169135
\(850\) 0 0
\(851\) −34.1436 −1.17043
\(852\) 13.8564 0.474713
\(853\) 3.60770 0.123525 0.0617626 0.998091i \(-0.480328\pi\)
0.0617626 + 0.998091i \(0.480328\pi\)
\(854\) −6.92820 −0.237078
\(855\) −5.46410 −0.186868
\(856\) 14.7846 0.505328
\(857\) −37.8564 −1.29315 −0.646575 0.762850i \(-0.723799\pi\)
−0.646575 + 0.762850i \(0.723799\pi\)
\(858\) 9.46410 0.323099
\(859\) −7.71281 −0.263158 −0.131579 0.991306i \(-0.542005\pi\)
−0.131579 + 0.991306i \(0.542005\pi\)
\(860\) 4.92820 0.168050
\(861\) 6.92820 0.236113
\(862\) −15.2154 −0.518238
\(863\) −37.8564 −1.28865 −0.644324 0.764753i \(-0.722861\pi\)
−0.644324 + 0.764753i \(0.722861\pi\)
\(864\) 5.19615 0.176777
\(865\) 12.0000 0.408012
\(866\) −0.248711 −0.00845155
\(867\) −17.0000 −0.577350
\(868\) −21.8564 −0.741855
\(869\) 6.53590 0.221715
\(870\) −6.00000 −0.203419
\(871\) 43.7128 1.48115
\(872\) −17.3205 −0.586546
\(873\) −10.0000 −0.338449
\(874\) −65.5692 −2.21791
\(875\) −2.00000 −0.0676123
\(876\) −8.39230 −0.283550
\(877\) −34.2487 −1.15650 −0.578248 0.815861i \(-0.696264\pi\)
−0.578248 + 0.815861i \(0.696264\pi\)
\(878\) −57.4641 −1.93932
\(879\) −13.8564 −0.467365
\(880\) −5.00000 −0.168550
\(881\) −0.928203 −0.0312720 −0.0156360 0.999878i \(-0.504977\pi\)
−0.0156360 + 0.999878i \(0.504977\pi\)
\(882\) 5.19615 0.174964
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 6.92820 0.232889
\(886\) −20.7846 −0.698273
\(887\) 12.2487 0.411271 0.205636 0.978629i \(-0.434074\pi\)
0.205636 + 0.978629i \(0.434074\pi\)
\(888\) −8.53590 −0.286446
\(889\) 17.8564 0.598885
\(890\) 1.60770 0.0538901
\(891\) −1.00000 −0.0335013
\(892\) 9.85641 0.330017
\(893\) −37.8564 −1.26682
\(894\) 26.7846 0.895811
\(895\) −6.92820 −0.231584
\(896\) −24.2487 −0.810093
\(897\) 37.8564 1.26399
\(898\) 46.3923 1.54813
\(899\) 37.8564 1.26258
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 6.00000 0.199778
\(903\) −9.85641 −0.328001
\(904\) −22.3923 −0.744757
\(905\) −15.8564 −0.527085
\(906\) 35.3205 1.17345
\(907\) 18.1436 0.602448 0.301224 0.953553i \(-0.402605\pi\)
0.301224 + 0.953553i \(0.402605\pi\)
\(908\) 15.4641 0.513194
\(909\) −10.3923 −0.344691
\(910\) 18.9282 0.627464
\(911\) 18.9282 0.627119 0.313560 0.949568i \(-0.398478\pi\)
0.313560 + 0.949568i \(0.398478\pi\)
\(912\) −27.3205 −0.904672
\(913\) −8.53590 −0.282497
\(914\) −21.4641 −0.709969
\(915\) −2.00000 −0.0661180
\(916\) −23.8564 −0.788238
\(917\) 37.8564 1.25013
\(918\) 0 0
\(919\) −32.3923 −1.06852 −0.534262 0.845319i \(-0.679410\pi\)
−0.534262 + 0.845319i \(0.679410\pi\)
\(920\) −12.0000 −0.395628
\(921\) 14.0000 0.461316
\(922\) −62.7846 −2.06770
\(923\) 75.7128 2.49212
\(924\) −2.00000 −0.0657952
\(925\) −4.92820 −0.162038
\(926\) 48.4974 1.59372
\(927\) 8.00000 0.262754
\(928\) −18.0000 −0.590879
\(929\) 2.78461 0.0913601 0.0456800 0.998956i \(-0.485455\pi\)
0.0456800 + 0.998956i \(0.485455\pi\)
\(930\) −18.9282 −0.620680
\(931\) −16.3923 −0.537236
\(932\) 12.0000 0.393073
\(933\) 5.07180 0.166043
\(934\) −8.78461 −0.287441
\(935\) 0 0
\(936\) 9.46410 0.309344
\(937\) −20.3923 −0.666188 −0.333094 0.942894i \(-0.608093\pi\)
−0.333094 + 0.942894i \(0.608093\pi\)
\(938\) −27.7128 −0.904855
\(939\) 20.9282 0.682966
\(940\) 6.92820 0.225973
\(941\) 27.4641 0.895304 0.447652 0.894208i \(-0.352260\pi\)
0.447652 + 0.894208i \(0.352260\pi\)
\(942\) 5.32051 0.173352
\(943\) 24.0000 0.781548
\(944\) 34.6410 1.12747
\(945\) −2.00000 −0.0650600
\(946\) −8.53590 −0.277526
\(947\) −18.9282 −0.615084 −0.307542 0.951535i \(-0.599506\pi\)
−0.307542 + 0.951535i \(0.599506\pi\)
\(948\) −6.53590 −0.212276
\(949\) −45.8564 −1.48856
\(950\) −9.46410 −0.307056
\(951\) −24.9282 −0.808352
\(952\) 0 0
\(953\) −3.21539 −0.104157 −0.0520784 0.998643i \(-0.516585\pi\)
−0.0520784 + 0.998643i \(0.516585\pi\)
\(954\) −1.60770 −0.0520511
\(955\) 18.9282 0.612502
\(956\) −12.0000 −0.388108
\(957\) 3.46410 0.111979
\(958\) −20.7846 −0.671520
\(959\) −36.0000 −1.16250
\(960\) −1.00000 −0.0322749
\(961\) 88.4256 2.85244
\(962\) 46.6410 1.50377
\(963\) 8.53590 0.275065
\(964\) 0.143594 0.00462484
\(965\) −24.3923 −0.785216
\(966\) −24.0000 −0.772187
\(967\) 22.7846 0.732704 0.366352 0.930476i \(-0.380607\pi\)
0.366352 + 0.930476i \(0.380607\pi\)
\(968\) 1.73205 0.0556702
\(969\) 0 0
\(970\) −17.3205 −0.556128
\(971\) 25.8564 0.829772 0.414886 0.909873i \(-0.363822\pi\)
0.414886 + 0.909873i \(0.363822\pi\)
\(972\) 1.00000 0.0320750
\(973\) 24.7846 0.794558
\(974\) 54.9282 1.76001
\(975\) 5.46410 0.174991
\(976\) −10.0000 −0.320092
\(977\) 47.5692 1.52187 0.760937 0.648826i \(-0.224740\pi\)
0.760937 + 0.648826i \(0.224740\pi\)
\(978\) −17.0718 −0.545896
\(979\) −0.928203 −0.0296655
\(980\) 3.00000 0.0958315
\(981\) −10.0000 −0.319275
\(982\) 53.5692 1.70946
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 6.00000 0.191273
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) −13.8564 −0.441054
\(988\) 29.8564 0.949859
\(989\) −34.1436 −1.08570
\(990\) −1.73205 −0.0550482
\(991\) −7.21539 −0.229204 −0.114602 0.993411i \(-0.536559\pi\)
−0.114602 + 0.993411i \(0.536559\pi\)
\(992\) −56.7846 −1.80291
\(993\) 9.85641 0.312784
\(994\) −48.0000 −1.52247
\(995\) 24.7846 0.785725
\(996\) 8.53590 0.270470
\(997\) 27.6077 0.874344 0.437172 0.899378i \(-0.355980\pi\)
0.437172 + 0.899378i \(0.355980\pi\)
\(998\) −49.8564 −1.57818
\(999\) −4.92820 −0.155921
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.b.1.1 2
3.2 odd 2 495.2.a.c.1.2 2
4.3 odd 2 2640.2.a.x.1.2 2
5.2 odd 4 825.2.c.c.199.1 4
5.3 odd 4 825.2.c.c.199.4 4
5.4 even 2 825.2.a.e.1.2 2
7.6 odd 2 8085.2.a.bd.1.1 2
11.10 odd 2 1815.2.a.i.1.2 2
12.11 even 2 7920.2.a.bz.1.2 2
15.2 even 4 2475.2.c.n.199.4 4
15.8 even 4 2475.2.c.n.199.1 4
15.14 odd 2 2475.2.a.r.1.1 2
33.32 even 2 5445.2.a.s.1.1 2
55.54 odd 2 9075.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.1 2 1.1 even 1 trivial
495.2.a.c.1.2 2 3.2 odd 2
825.2.a.e.1.2 2 5.4 even 2
825.2.c.c.199.1 4 5.2 odd 4
825.2.c.c.199.4 4 5.3 odd 4
1815.2.a.i.1.2 2 11.10 odd 2
2475.2.a.r.1.1 2 15.14 odd 2
2475.2.c.n.199.1 4 15.8 even 4
2475.2.c.n.199.4 4 15.2 even 4
2640.2.a.x.1.2 2 4.3 odd 2
5445.2.a.s.1.1 2 33.32 even 2
7920.2.a.bz.1.2 2 12.11 even 2
8085.2.a.bd.1.1 2 7.6 odd 2
9075.2.a.bh.1.1 2 55.54 odd 2