Properties

Label 117.2.b.a.64.2
Level $117$
Weight $2$
Character 117.64
Analytic conductor $0.934$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 64.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 117.64
Dual form 117.2.b.a.64.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205i q^{2} -1.00000 q^{4} +3.46410i q^{7} +1.73205i q^{8} +O(q^{10})\) \(q+1.73205i q^{2} -1.00000 q^{4} +3.46410i q^{7} +1.73205i q^{8} -3.46410i q^{11} +(-1.00000 - 3.46410i) q^{13} -6.00000 q^{14} -5.00000 q^{16} +6.00000 q^{17} -3.46410i q^{19} +6.00000 q^{22} +5.00000 q^{25} +(6.00000 - 1.73205i) q^{26} -3.46410i q^{28} -6.00000 q^{29} +3.46410i q^{31} -5.19615i q^{32} +10.3923i q^{34} -6.92820i q^{37} +6.00000 q^{38} -6.92820i q^{41} -4.00000 q^{43} +3.46410i q^{44} +3.46410i q^{47} -5.00000 q^{49} +8.66025i q^{50} +(1.00000 + 3.46410i) q^{52} -6.00000 q^{53} -6.00000 q^{56} -10.3923i q^{58} +10.3923i q^{59} -2.00000 q^{61} -6.00000 q^{62} -1.00000 q^{64} +10.3923i q^{67} -6.00000 q^{68} +3.46410i q^{71} +12.0000 q^{74} +3.46410i q^{76} +12.0000 q^{77} -8.00000 q^{79} +12.0000 q^{82} -3.46410i q^{83} -6.92820i q^{86} +6.00000 q^{88} -6.92820i q^{89} +(12.0000 - 3.46410i) q^{91} -6.00000 q^{94} +13.8564i q^{97} -8.66025i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{13} - 12 q^{14} - 10 q^{16} + 12 q^{17} + 12 q^{22} + 10 q^{25} + 12 q^{26} - 12 q^{29} + 12 q^{38} - 8 q^{43} - 10 q^{49} + 2 q^{52} - 12 q^{53} - 12 q^{56} - 4 q^{61} - 12 q^{62} - 2 q^{64} - 12 q^{68} + 24 q^{74} + 24 q^{77} - 16 q^{79} + 24 q^{82} + 12 q^{88} + 24 q^{91} - 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205i 1.22474i 0.790569 + 0.612372i \(0.209785\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 3.46410i 1.30931i 0.755929 + 0.654654i \(0.227186\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.73205i 0.612372i
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410i 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) −1.00000 3.46410i −0.277350 0.960769i
\(14\) −6.00000 −1.60357
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i −0.917663 0.397360i \(-0.869927\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 6.00000 1.73205i 1.17670 0.339683i
\(27\) 0 0
\(28\) 3.46410i 0.654654i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 5.19615i 0.918559i
\(33\) 0 0
\(34\) 10.3923i 1.78227i
\(35\) 0 0
\(36\) 0 0
\(37\) 6.92820i 1.13899i −0.821995 0.569495i \(-0.807139\pi\)
0.821995 0.569495i \(-0.192861\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820i 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.46410i 0.522233i
\(45\) 0 0
\(46\) 0 0
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 8.66025i 1.22474i
\(51\) 0 0
\(52\) 1.00000 + 3.46410i 0.138675 + 0.480384i
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) 10.3923i 1.36458i
\(59\) 10.3923i 1.35296i 0.736460 + 0.676481i \(0.236496\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 10.3923i 1.26962i 0.772667 + 0.634811i \(0.218922\pi\)
−0.772667 + 0.634811i \(0.781078\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 3.46410i 0.411113i 0.978645 + 0.205557i \(0.0659005\pi\)
−0.978645 + 0.205557i \(0.934100\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 12.0000 1.39497
\(75\) 0 0
\(76\) 3.46410i 0.397360i
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 12.0000 1.32518
\(83\) 3.46410i 0.380235i −0.981761 0.190117i \(-0.939113\pi\)
0.981761 0.190117i \(-0.0608868\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.92820i 0.747087i
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) 6.92820i 0.734388i −0.930144 0.367194i \(-0.880318\pi\)
0.930144 0.367194i \(-0.119682\pi\)
\(90\) 0 0
\(91\) 12.0000 3.46410i 1.25794 0.363137i
\(92\) 0 0
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) 13.8564i 1.40690i 0.710742 + 0.703452i \(0.248359\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 8.66025i 0.874818i
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 6.00000 1.73205i 0.588348 0.169842i
\(105\) 0 0
\(106\) 10.3923i 1.00939i
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 6.92820i 0.663602i −0.943349 0.331801i \(-0.892344\pi\)
0.943349 0.331801i \(-0.107656\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 17.3205i 1.63663i
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) −18.0000 −1.65703
\(119\) 20.7846i 1.90532i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 3.46410i 0.313625i
\(123\) 0 0
\(124\) 3.46410i 0.311086i
\(125\) 0 0
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 12.1244i 1.07165i
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) −18.0000 −1.55496
\(135\) 0 0
\(136\) 10.3923i 0.891133i
\(137\) 20.7846i 1.77575i −0.460086 0.887875i \(-0.652181\pi\)
0.460086 0.887875i \(-0.347819\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) −12.0000 + 3.46410i −1.00349 + 0.289683i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 6.92820i 0.569495i
\(149\) 13.8564i 1.13516i 0.823318 + 0.567581i \(0.192120\pi\)
−0.823318 + 0.567581i \(0.807880\pi\)
\(150\) 0 0
\(151\) 10.3923i 0.845714i −0.906196 0.422857i \(-0.861027\pi\)
0.906196 0.422857i \(-0.138973\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 20.7846i 1.67487i
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 13.8564i 1.10236i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.46410i 0.271329i −0.990755 0.135665i \(-0.956683\pi\)
0.990755 0.135665i \(-0.0433170\pi\)
\(164\) 6.92820i 0.541002i
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 17.3205i 1.34030i 0.742225 + 0.670151i \(0.233770\pi\)
−0.742225 + 0.670151i \(0.766230\pi\)
\(168\) 0 0
\(169\) −11.0000 + 6.92820i −0.846154 + 0.532939i
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 17.3205i 1.30931i
\(176\) 17.3205i 1.30558i
\(177\) 0 0
\(178\) 12.0000 0.899438
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 6.00000 + 20.7846i 0.444750 + 1.54066i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 20.7846i 1.51992i
\(188\) 3.46410i 0.252646i
\(189\) 0 0
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) −24.0000 −1.72310
\(195\) 0 0
\(196\) 5.00000 0.357143
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 8.66025i 0.612372i
\(201\) 0 0
\(202\) 10.3923i 0.731200i
\(203\) 20.7846i 1.45879i
\(204\) 0 0
\(205\) 0 0
\(206\) 13.8564i 0.965422i
\(207\) 0 0
\(208\) 5.00000 + 17.3205i 0.346688 + 1.20096i
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 20.7846i 1.42081i
\(215\) 0 0
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 12.0000 0.812743
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 20.7846i −0.403604 1.39812i
\(222\) 0 0
\(223\) 3.46410i 0.231973i 0.993251 + 0.115987i \(0.0370030\pi\)
−0.993251 + 0.115987i \(0.962997\pi\)
\(224\) 18.0000 1.20268
\(225\) 0 0
\(226\) 10.3923i 0.691286i
\(227\) 17.3205i 1.14960i −0.818293 0.574801i \(-0.805079\pi\)
0.818293 0.574801i \(-0.194921\pi\)
\(228\) 0 0
\(229\) 6.92820i 0.457829i 0.973447 + 0.228914i \(0.0735176\pi\)
−0.973447 + 0.228914i \(0.926482\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 10.3923i 0.682288i
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 10.3923i 0.676481i
\(237\) 0 0
\(238\) −36.0000 −2.33353
\(239\) 10.3923i 0.672222i −0.941822 0.336111i \(-0.890888\pi\)
0.941822 0.336111i \(-0.109112\pi\)
\(240\) 0 0
\(241\) 13.8564i 0.892570i −0.894891 0.446285i \(-0.852747\pi\)
0.894891 0.446285i \(-0.147253\pi\)
\(242\) 1.73205i 0.111340i
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) −12.0000 + 3.46410i −0.763542 + 0.220416i
\(248\) −6.00000 −0.381000
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 13.8564i 0.869428i
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) 0 0
\(262\) 20.7846i 1.28408i
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 20.7846i 1.27439i
\(267\) 0 0
\(268\) 10.3923i 0.634811i
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 10.3923i 0.631288i −0.948878 0.315644i \(-0.897780\pi\)
0.948878 0.315644i \(-0.102220\pi\)
\(272\) −30.0000 −1.81902
\(273\) 0 0
\(274\) 36.0000 2.17484
\(275\) 17.3205i 1.04447i
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 6.92820i 0.415526i
\(279\) 0 0
\(280\) 0 0
\(281\) 6.92820i 0.413302i 0.978415 + 0.206651i \(0.0662565\pi\)
−0.978415 + 0.206651i \(0.933744\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 3.46410i 0.205557i
\(285\) 0 0
\(286\) −6.00000 20.7846i −0.354787 1.22902i
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 27.7128i 1.61900i 0.587120 + 0.809500i \(0.300262\pi\)
−0.587120 + 0.809500i \(0.699738\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 12.0000 0.697486
\(297\) 0 0
\(298\) −24.0000 −1.39028
\(299\) 0 0
\(300\) 0 0
\(301\) 13.8564i 0.798670i
\(302\) 18.0000 1.03578
\(303\) 0 0
\(304\) 17.3205i 0.993399i
\(305\) 0 0
\(306\) 0 0
\(307\) 10.3923i 0.593120i 0.955014 + 0.296560i \(0.0958395\pi\)
−0.955014 + 0.296560i \(0.904160\pi\)
\(308\) −12.0000 −0.683763
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 24.2487i 1.36843i
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 13.8564i 0.778253i 0.921184 + 0.389127i \(0.127223\pi\)
−0.921184 + 0.389127i \(0.872777\pi\)
\(318\) 0 0
\(319\) 20.7846i 1.16371i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 20.7846i 1.15649i
\(324\) 0 0
\(325\) −5.00000 17.3205i −0.277350 0.960769i
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) 12.0000 0.662589
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 3.46410i 0.190404i −0.995458 0.0952021i \(-0.969650\pi\)
0.995458 0.0952021i \(-0.0303497\pi\)
\(332\) 3.46410i 0.190117i
\(333\) 0 0
\(334\) −30.0000 −1.64153
\(335\) 0 0
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −12.0000 19.0526i −0.652714 1.03632i
\(339\) 0 0
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) 6.92820i 0.374088i
\(344\) 6.92820i 0.373544i
\(345\) 0 0
\(346\) 31.1769i 1.67608i
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 0 0
\(349\) 6.92820i 0.370858i 0.982658 + 0.185429i \(0.0593675\pi\)
−0.982658 + 0.185429i \(0.940632\pi\)
\(350\) −30.0000 −1.60357
\(351\) 0 0
\(352\) −18.0000 −0.959403
\(353\) 34.6410i 1.84376i 0.387481 + 0.921878i \(0.373345\pi\)
−0.387481 + 0.921878i \(0.626655\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.92820i 0.367194i
\(357\) 0 0
\(358\) 20.7846i 1.09850i
\(359\) 17.3205i 0.914141i 0.889430 + 0.457071i \(0.151101\pi\)
−0.889430 + 0.457071i \(0.848899\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 17.3205i 0.910346i
\(363\) 0 0
\(364\) −12.0000 + 3.46410i −0.628971 + 0.181568i
\(365\) 0 0
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 20.7846i 1.07908i
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 36.0000 1.86152
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) 6.00000 + 20.7846i 0.309016 + 1.07046i
\(378\) 0 0
\(379\) 17.3205i 0.889695i −0.895606 0.444847i \(-0.853258\pi\)
0.895606 0.444847i \(-0.146742\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 41.5692i 2.12687i
\(383\) 3.46410i 0.177007i 0.996076 + 0.0885037i \(0.0282085\pi\)
−0.996076 + 0.0885037i \(0.971792\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 13.8564i 0.703452i
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 8.66025i 0.437409i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 34.6410i 1.73858i 0.494300 + 0.869291i \(0.335424\pi\)
−0.494300 + 0.869291i \(0.664576\pi\)
\(398\) 27.7128i 1.38912i
\(399\) 0 0
\(400\) −25.0000 −1.25000
\(401\) 6.92820i 0.345978i 0.984924 + 0.172989i \(0.0553425\pi\)
−0.984924 + 0.172989i \(0.944657\pi\)
\(402\) 0 0
\(403\) 12.0000 3.46410i 0.597763 0.172559i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 36.0000 1.78665
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 27.7128i 1.37031i −0.728397 0.685155i \(-0.759734\pi\)
0.728397 0.685155i \(-0.240266\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) −36.0000 −1.77144
\(414\) 0 0
\(415\) 0 0
\(416\) −18.0000 + 5.19615i −0.882523 + 0.254762i
\(417\) 0 0
\(418\) 20.7846i 1.01661i
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 34.6410i 1.68830i 0.536107 + 0.844150i \(0.319894\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 34.6410i 1.68630i
\(423\) 0 0
\(424\) 10.3923i 0.504695i
\(425\) 30.0000 1.45521
\(426\) 0 0
\(427\) 6.92820i 0.335279i
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 24.2487i 1.16802i −0.811747 0.584010i \(-0.801483\pi\)
0.811747 0.584010i \(-0.198517\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 20.7846i 0.997693i
\(435\) 0 0
\(436\) 6.92820i 0.331801i
\(437\) 0 0
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 36.0000 10.3923i 1.71235 0.494312i
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −6.00000 −0.284108
\(447\) 0 0
\(448\) 3.46410i 0.163663i
\(449\) 6.92820i 0.326962i 0.986546 + 0.163481i \(0.0522723\pi\)
−0.986546 + 0.163481i \(0.947728\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 30.0000 1.40797
\(455\) 0 0
\(456\) 0 0
\(457\) 27.7128i 1.29635i 0.761491 + 0.648175i \(0.224468\pi\)
−0.761491 + 0.648175i \(0.775532\pi\)
\(458\) −12.0000 −0.560723
\(459\) 0 0
\(460\) 0 0
\(461\) 13.8564i 0.645357i −0.946509 0.322679i \(-0.895417\pi\)
0.946509 0.322679i \(-0.104583\pi\)
\(462\) 0 0
\(463\) 17.3205i 0.804952i 0.915430 + 0.402476i \(0.131850\pi\)
−0.915430 + 0.402476i \(0.868150\pi\)
\(464\) 30.0000 1.39272
\(465\) 0 0
\(466\) 10.3923i 0.481414i
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −36.0000 −1.66233
\(470\) 0 0
\(471\) 0 0
\(472\) −18.0000 −0.828517
\(473\) 13.8564i 0.637118i
\(474\) 0 0
\(475\) 17.3205i 0.794719i
\(476\) 20.7846i 0.952661i
\(477\) 0 0
\(478\) 18.0000 0.823301
\(479\) 10.3923i 0.474837i −0.971408 0.237418i \(-0.923699\pi\)
0.971408 0.237418i \(-0.0763012\pi\)
\(480\) 0 0
\(481\) −24.0000 + 6.92820i −1.09431 + 0.315899i
\(482\) 24.0000 1.09317
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) 38.1051i 1.72671i −0.504599 0.863354i \(-0.668360\pi\)
0.504599 0.863354i \(-0.331640\pi\)
\(488\) 3.46410i 0.156813i
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) −6.00000 20.7846i −0.269953 0.935144i
\(495\) 0 0
\(496\) 17.3205i 0.777714i
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 10.3923i 0.465223i 0.972570 + 0.232612i \(0.0747271\pi\)
−0.972570 + 0.232612i \(0.925273\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 20.7846i 0.927663i
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 41.5692i 1.84252i −0.388943 0.921262i \(-0.627160\pi\)
0.388943 0.921262i \(-0.372840\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 8.66025i 0.382733i
\(513\) 0 0
\(514\) 31.1769i 1.37515i
\(515\) 0 0
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 41.5692i 1.82645i
\(519\) 0 0
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 41.5692i 1.81250i
\(527\) 20.7846i 0.905392i
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −12.0000 −0.520266
\(533\) −24.0000 + 6.92820i −1.03956 + 0.300094i
\(534\) 0 0
\(535\) 0 0
\(536\) −18.0000 −0.777482
\(537\) 0 0
\(538\) 10.3923i 0.448044i
\(539\) 17.3205i 0.746047i
\(540\) 0 0
\(541\) 6.92820i 0.297867i −0.988847 0.148933i \(-0.952416\pi\)
0.988847 0.148933i \(-0.0475840\pi\)
\(542\) 18.0000 0.773166
\(543\) 0 0
\(544\) 31.1769i 1.33670i
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 20.7846i 0.887875i
\(549\) 0 0
\(550\) 30.0000 1.27920
\(551\) 20.7846i 0.885454i
\(552\) 0 0
\(553\) 27.7128i 1.17847i
\(554\) 17.3205i 0.735878i
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 13.8564i 0.587115i 0.955941 + 0.293557i \(0.0948392\pi\)
−0.955941 + 0.293557i \(0.905161\pi\)
\(558\) 0 0
\(559\) 4.00000 + 13.8564i 0.169182 + 0.586064i
\(560\) 0 0
\(561\) 0 0
\(562\) −12.0000 −0.506189
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6.92820i 0.291214i
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 12.0000 3.46410i 0.501745 0.144841i
\(573\) 0 0
\(574\) 41.5692i 1.73507i
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 32.9090i 1.36883i
\(579\) 0 0
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 20.7846i 0.860811i
\(584\) 0 0
\(585\) 0 0
\(586\) −48.0000 −1.98286
\(587\) 10.3923i 0.428936i 0.976731 + 0.214468i \(0.0688018\pi\)
−0.976731 + 0.214468i \(0.931198\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 0 0
\(592\) 34.6410i 1.42374i
\(593\) 6.92820i 0.284507i 0.989830 + 0.142254i \(0.0454349\pi\)
−0.989830 + 0.142254i \(0.954565\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 13.8564i 0.567581i
\(597\) 0 0
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 24.0000 0.978167
\(603\) 0 0
\(604\) 10.3923i 0.422857i
\(605\) 0 0
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −18.0000 −0.729996
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 3.46410i 0.485468 0.140143i
\(612\) 0 0
\(613\) 20.7846i 0.839482i 0.907644 + 0.419741i \(0.137879\pi\)
−0.907644 + 0.419741i \(0.862121\pi\)
\(614\) −18.0000 −0.726421
\(615\) 0 0
\(616\) 20.7846i 0.837436i
\(617\) 6.92820i 0.278919i −0.990228 0.139459i \(-0.955464\pi\)
0.990228 0.139459i \(-0.0445365\pi\)
\(618\) 0 0
\(619\) 31.1769i 1.25311i −0.779379 0.626553i \(-0.784465\pi\)
0.779379 0.626553i \(-0.215535\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 17.3205i 0.692267i
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 41.5692i 1.65747i
\(630\) 0 0
\(631\) 38.1051i 1.51694i −0.651707 0.758470i \(-0.725947\pi\)
0.651707 0.758470i \(-0.274053\pi\)
\(632\) 13.8564i 0.551178i
\(633\) 0 0
\(634\) −24.0000 −0.953162
\(635\) 0 0
\(636\) 0 0
\(637\) 5.00000 + 17.3205i 0.198107 + 0.686264i
\(638\) −36.0000 −1.42525
\(639\) 0 0
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) 10.3923i 0.409832i 0.978780 + 0.204916i \(0.0656922\pi\)
−0.978780 + 0.204916i \(0.934308\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 36.0000 1.41640
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 30.0000 8.66025i 1.17670 0.339683i
\(651\) 0 0
\(652\) 3.46410i 0.135665i
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 34.6410i 1.35250i
\(657\) 0 0
\(658\) 20.7846i 0.810268i
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 20.7846i 0.808428i 0.914665 + 0.404214i \(0.132455\pi\)
−0.914665 + 0.404214i \(0.867545\pi\)
\(662\) 6.00000 0.233197
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 17.3205i 0.670151i
\(669\) 0 0
\(670\) 0 0
\(671\) 6.92820i 0.267460i
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 24.2487i 0.934025i
\(675\) 0 0
\(676\) 11.0000 6.92820i 0.423077 0.266469i
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) −48.0000 −1.84207
\(680\) 0 0
\(681\) 0 0
\(682\) 20.7846i 0.795884i
\(683\) 31.1769i 1.19295i −0.802631 0.596476i \(-0.796567\pi\)
0.802631 0.596476i \(-0.203433\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −12.0000 −0.458162
\(687\) 0 0
\(688\) 20.0000 0.762493
\(689\) 6.00000 + 20.7846i 0.228582 + 0.791831i
\(690\) 0 0
\(691\) 45.0333i 1.71315i −0.516024 0.856574i \(-0.672588\pi\)
0.516024 0.856574i \(-0.327412\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 62.3538i 2.36692i
\(695\) 0 0
\(696\) 0 0
\(697\) 41.5692i 1.57455i
\(698\) −12.0000 −0.454207
\(699\) 0 0
\(700\) 17.3205i 0.654654i
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 3.46410i 0.130558i
\(705\) 0 0
\(706\) −60.0000 −2.25813
\(707\) 20.7846i 0.781686i
\(708\) 0 0
\(709\) 6.92820i 0.260194i 0.991501 + 0.130097i \(0.0415289\pi\)
−0.991501 + 0.130097i \(0.958471\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 12.0000 0.449719
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −30.0000 −1.11959
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 27.7128i 1.03208i
\(722\) 12.1244i 0.451222i
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) −30.0000 −1.11417
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 6.00000 + 20.7846i 0.222375 + 0.770329i
\(729\) 0 0
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 34.6410i 1.27950i −0.768585 0.639748i \(-0.779039\pi\)
0.768585 0.639748i \(-0.220961\pi\)
\(734\) 27.7128i 1.02290i
\(735\) 0 0
\(736\) 0 0
\(737\) 36.0000 1.32608
\(738\) 0 0
\(739\) 38.1051i 1.40172i 0.713299 + 0.700860i \(0.247200\pi\)
−0.713299 + 0.700860i \(0.752800\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 36.0000 1.32160
\(743\) 3.46410i 0.127086i 0.997979 + 0.0635428i \(0.0202399\pi\)
−0.997979 + 0.0635428i \(0.979760\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 38.1051i 1.39513i
\(747\) 0 0
\(748\) 20.7846i 0.759961i
\(749\) 41.5692i 1.51891i
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 17.3205i 0.631614i
\(753\) 0 0
\(754\) −36.0000 + 10.3923i −1.31104 + 0.378465i
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 30.0000 1.08965
\(759\) 0 0
\(760\) 0 0
\(761\) 48.4974i 1.75803i −0.476794 0.879015i \(-0.658201\pi\)
0.476794 0.879015i \(-0.341799\pi\)
\(762\) 0 0
\(763\) 24.0000 0.868858
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) 36.0000 10.3923i 1.29988 0.375244i
\(768\) 0 0
\(769\) 27.7128i 0.999350i −0.866213 0.499675i \(-0.833453\pi\)
0.866213 0.499675i \(-0.166547\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 13.8564i 0.498380i −0.968455 0.249190i \(-0.919836\pi\)
0.968455 0.249190i \(-0.0801644\pi\)
\(774\) 0 0
\(775\) 17.3205i 0.622171i
\(776\) −24.0000 −0.861550
\(777\) 0 0
\(778\) 31.1769i 1.11775i
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) 0 0
\(784\) 25.0000 0.892857
\(785\) 0 0
\(786\) 0 0
\(787\) 10.3923i 0.370446i 0.982697 + 0.185223i \(0.0593007\pi\)
−0.982697 + 0.185223i \(0.940699\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 20.7846i 0.739016i
\(792\) 0 0
\(793\) 2.00000 + 6.92820i 0.0710221 + 0.246028i
\(794\) −60.0000 −2.12932
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 20.7846i 0.735307i
\(800\) 25.9808i 0.918559i
\(801\) 0 0