Properties

Label 1170.2.b.c.181.2
Level $1170$
Weight $2$
Character 1170.181
Analytic conductor $9.342$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 181.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1170.181
Dual form 1170.2.b.c.181.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{8} +1.00000 q^{10} +6.00000i q^{11} +(2.00000 - 3.00000i) q^{13} +1.00000 q^{16} +6.00000 q^{17} -6.00000i q^{19} +1.00000i q^{20} -6.00000 q^{22} -6.00000 q^{23} -1.00000 q^{25} +(3.00000 + 2.00000i) q^{26} +6.00000 q^{29} +1.00000i q^{32} +6.00000i q^{34} +6.00000i q^{37} +6.00000 q^{38} -1.00000 q^{40} +12.0000i q^{41} +8.00000 q^{43} -6.00000i q^{44} -6.00000i q^{46} +7.00000 q^{49} -1.00000i q^{50} +(-2.00000 + 3.00000i) q^{52} +12.0000 q^{53} +6.00000 q^{55} +6.00000i q^{58} +6.00000i q^{59} +10.0000 q^{61} -1.00000 q^{64} +(-3.00000 - 2.00000i) q^{65} -6.00000 q^{68} -12.0000i q^{71} +6.00000i q^{73} -6.00000 q^{74} +6.00000i q^{76} -8.00000 q^{79} -1.00000i q^{80} -12.0000 q^{82} -6.00000i q^{85} +8.00000i q^{86} +6.00000 q^{88} +6.00000 q^{92} -6.00000 q^{95} +6.00000i q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + O(q^{10}) \) \( 2 q - 2 q^{4} + 2 q^{10} + 4 q^{13} + 2 q^{16} + 12 q^{17} - 12 q^{22} - 12 q^{23} - 2 q^{25} + 6 q^{26} + 12 q^{29} + 12 q^{38} - 2 q^{40} + 16 q^{43} + 14 q^{49} - 4 q^{52} + 24 q^{53} + 12 q^{55} + 20 q^{61} - 2 q^{64} - 6 q^{65} - 12 q^{68} - 12 q^{74} - 16 q^{79} - 24 q^{82} + 12 q^{88} + 12 q^{92} - 12 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 6.00000i 1.80907i 0.426401 + 0.904534i \(0.359781\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) 2.00000 3.00000i 0.554700 0.832050i
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 3.00000 + 2.00000i 0.588348 + 0.392232i
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 6.00000i 1.02899i
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 12.0000i 1.87409i 0.349215 + 0.937043i \(0.386448\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 6.00000i 0.904534i
\(45\) 0 0
\(46\) 6.00000i 0.884652i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 1.00000i 0.141421i
\(51\) 0 0
\(52\) −2.00000 + 3.00000i −0.277350 + 0.416025i
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −3.00000 2.00000i −0.372104 0.248069i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000i 1.42414i −0.702109 0.712069i \(-0.747758\pi\)
0.702109 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 6.00000i 0.688247i
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 0 0
\(82\) −12.0000 −1.32518
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 6.00000i 0.650791i
\(86\) 8.00000i 0.862662i
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 7.00000i 0.707107i
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −3.00000 2.00000i −0.294174 0.196116i
\(105\) 0 0
\(106\) 12.0000i 1.16554i
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 6.00000i 0.574696i 0.957826 + 0.287348i \(0.0927736\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 6.00000i 0.572078i
\(111\) 0 0
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 6.00000i 0.559503i
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) −25.0000 −2.27273
\(122\) 10.0000i 0.905357i
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 2.00000 3.00000i 0.175412 0.263117i
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 6.00000i 0.514496i
\(137\) 18.0000i 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 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
\(142\) 12.0000 1.00702
\(143\) 18.0000 + 12.0000i 1.50524 + 1.00349i
\(144\) 0 0
\(145\) 6.00000i 0.498273i
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) 6.00000i 0.493197i
\(149\) 18.0000i 1.47462i −0.675556 0.737309i \(-0.736096\pi\)
0.675556 0.737309i \(-0.263904\pi\)
\(150\) 0 0
\(151\) 12.0000i 0.976546i −0.872691 0.488273i \(-0.837627\pi\)
0.872691 0.488273i \(-0.162373\pi\)
\(152\) −6.00000 −0.486664
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0000i 0.939913i −0.882690 0.469956i \(-0.844270\pi\)
0.882690 0.469956i \(-0.155730\pi\)
\(164\) 12.0000i 0.937043i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −5.00000 12.0000i −0.384615 0.923077i
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.00000i 0.452267i
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6.00000i 0.442326i
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 36.0000i 2.63258i
\(188\) 0 0
\(189\) 0 0
\(190\) 6.00000i 0.435286i
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 6.00000i 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 0 0
\(202\) 18.0000i 1.26648i
\(203\) 0 0
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 14.0000i 0.975426i
\(207\) 0 0
\(208\) 2.00000 3.00000i 0.138675 0.208013i
\(209\) 36.0000 2.49017
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −12.0000 −0.824163
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000i 0.545595i
\(216\) 0 0
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 0 0
\(220\) −6.00000 −0.404520
\(221\) 12.0000 18.0000i 0.807207 1.21081i
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.0000i 1.19734i
\(227\) 24.0000i 1.59294i 0.604681 + 0.796468i \(0.293301\pi\)
−0.604681 + 0.796468i \(0.706699\pi\)
\(228\) 0 0
\(229\) 18.0000i 1.18947i −0.803921 0.594737i \(-0.797256\pi\)
0.803921 0.594737i \(-0.202744\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.00000i 0.390567i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 24.0000i 1.54598i 0.634421 + 0.772988i \(0.281239\pi\)
−0.634421 + 0.772988i \(0.718761\pi\)
\(242\) 25.0000i 1.60706i
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 7.00000i 0.447214i
\(246\) 0 0
\(247\) −18.0000 12.0000i −1.14531 0.763542i
\(248\) 0 0
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 36.0000i 2.26330i
\(254\) 2.00000i 0.125491i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 3.00000 + 2.00000i 0.186052 + 0.124035i
\(261\) 0 0
\(262\) 12.0000i 0.741362i
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) 12.0000i 0.737154i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) 12.0000i 0.728948i −0.931214 0.364474i \(-0.881249\pi\)
0.931214 0.364474i \(-0.118751\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 6.00000i 0.361814i
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 4.00000i 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 12.0000i 0.712069i
\(285\) 0 0
\(286\) −12.0000 + 18.0000i −0.709575 + 1.06436i
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) 6.00000i 0.351123i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) −12.0000 + 18.0000i −0.693978 + 1.04097i
\(300\) 0 0
\(301\) 0 0
\(302\) 12.0000 0.690522
\(303\) 0 0
\(304\) 6.00000i 0.344124i
\(305\) 10.0000i 0.572598i
\(306\) 0 0
\(307\) 12.0000i 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 4.00000i 0.225733i
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 6.00000i 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 0 0
\(319\) 36.0000i 2.01561i
\(320\) 1.00000i 0.0559017i
\(321\) 0 0
\(322\) 0 0
\(323\) 36.0000i 2.00309i
\(324\) 0 0
\(325\) −2.00000 + 3.00000i −0.110940 + 0.166410i
\(326\) 12.0000 0.664619
\(327\) 0 0
\(328\) 12.0000 0.662589
\(329\) 0 0
\(330\) 0 0
\(331\) 18.0000i 0.989369i 0.869072 + 0.494685i \(0.164716\pi\)
−0.869072 + 0.494685i \(0.835284\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 12.0000 5.00000i 0.652714 0.271964i
\(339\) 0 0
\(340\) 6.00000i 0.325396i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 8.00000i 0.431331i
\(345\) 0 0
\(346\) 12.0000i 0.645124i
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 6.00000i 0.321173i −0.987022 0.160586i \(-0.948662\pi\)
0.987022 0.160586i \(-0.0513385\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6.00000 −0.319801
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 12.0000i 0.634220i
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 2.00000i 0.105118i
\(363\) 0 0
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 26.0000 1.35719 0.678594 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) 6.00000i 0.311925i
\(371\) 0 0
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) −36.0000 −1.86152
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 18.0000i 0.618031 0.927047i
\(378\) 0 0
\(379\) 6.00000i 0.308199i −0.988055 0.154100i \(-0.950752\pi\)
0.988055 0.154100i \(-0.0492477\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) 12.0000i 0.613973i
\(383\) 24.0000i 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) 0 0
\(388\) 6.00000i 0.304604i
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 7.00000i 0.353553i
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 8.00000i 0.402524i
\(396\) 0 0
\(397\) 18.0000i 0.903394i 0.892171 + 0.451697i \(0.149181\pi\)
−0.892171 + 0.451697i \(0.850819\pi\)
\(398\) 20.0000i 1.00251i
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 24.0000i 1.19850i −0.800561 0.599251i \(-0.795465\pi\)
0.800561 0.599251i \(-0.204535\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) −36.0000 −1.78445
\(408\) 0 0
\(409\) 12.0000i 0.593362i 0.954977 + 0.296681i \(0.0958798\pi\)
−0.954977 + 0.296681i \(0.904120\pi\)
\(410\) 12.0000i 0.592638i
\(411\) 0 0
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 3.00000 + 2.00000i 0.147087 + 0.0980581i
\(417\) 0 0
\(418\) 36.0000i 1.76082i
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) 6.00000i 0.292422i 0.989253 + 0.146211i \(0.0467079\pi\)
−0.989253 + 0.146211i \(0.953292\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 0 0
\(424\) 12.0000i 0.582772i
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 12.0000i 0.578020i 0.957326 + 0.289010i \(0.0933260\pi\)
−0.957326 + 0.289010i \(0.906674\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.00000i 0.287348i
\(437\) 36.0000i 1.72211i
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 6.00000i 0.286039i
\(441\) 0 0
\(442\) 18.0000 + 12.0000i 0.856173 + 0.570782i
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) −72.0000 −3.39035
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000i 0.280668i 0.990104 + 0.140334i \(0.0448177\pi\)
−0.990104 + 0.140334i \(0.955182\pi\)
\(458\) 18.0000 0.841085
\(459\) 0 0
\(460\) 6.00000i 0.279751i
\(461\) 6.00000i 0.279448i −0.990190 0.139724i \(-0.955378\pi\)
0.990190 0.139724i \(-0.0446215\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 6.00000i 0.277945i
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) 48.0000i 2.20704i
\(474\) 0 0
\(475\) 6.00000i 0.275299i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.0000i 1.09659i 0.836286 + 0.548294i \(0.184723\pi\)
−0.836286 + 0.548294i \(0.815277\pi\)
\(480\) 0 0
\(481\) 18.0000 + 12.0000i 0.820729 + 0.547153i
\(482\) −24.0000 −1.09317
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 6.00000 0.272446
\(486\) 0 0
\(487\) 24.0000i 1.08754i −0.839233 0.543772i \(-0.816996\pi\)
0.839233 0.543772i \(-0.183004\pi\)
\(488\) 10.0000i 0.452679i
\(489\) 0 0
\(490\) 7.00000 0.316228
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 12.0000 18.0000i 0.539906 0.809858i
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 6.00000i 0.268597i 0.990941 + 0.134298i \(0.0428781\pi\)
−0.990941 + 0.134298i \(0.957122\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 0 0
\(502\) 12.0000i 0.535586i
\(503\) −18.0000 −0.802580 −0.401290 0.915951i \(-0.631438\pi\)
−0.401290 + 0.915951i \(0.631438\pi\)
\(504\) 0 0
\(505\) 18.0000i 0.800989i
\(506\) 36.0000 1.60040
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 6.00000i 0.265945i 0.991120 + 0.132973i \(0.0424523\pi\)
−0.991120 + 0.132973i \(0.957548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 18.0000i 0.793946i
\(515\) 14.0000i 0.616914i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −2.00000 + 3.00000i −0.0877058 + 0.131559i
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 6.00000i 0.261612i
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) 0 0
\(533\) 36.0000 + 24.0000i 1.55933 + 1.03956i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 30.0000i 1.29339i
\(539\) 42.0000i 1.80907i
\(540\) 0 0
\(541\) 18.0000i 0.773880i −0.922105 0.386940i \(-0.873532\pi\)
0.922105 0.386940i \(-0.126468\pi\)
\(542\) 12.0000 0.515444
\(543\) 0 0
\(544\) 6.00000i 0.257248i
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 18.0000i 0.768922i
\(549\) 0 0
\(550\) 6.00000 0.255841
\(551\) 36.0000i 1.53365i
\(552\) 0 0
\(553\) 0 0
\(554\) 8.00000i 0.339887i
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 0 0
\(559\) 16.0000 24.0000i 0.676728 1.01509i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 18.0000i 0.757266i
\(566\) 4.00000i 0.168133i
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −18.0000 12.0000i −0.752618 0.501745i
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) 30.0000i 1.24892i −0.781058 0.624458i \(-0.785320\pi\)
0.781058 0.624458i \(-0.214680\pi\)
\(578\) 19.0000i 0.790296i
\(579\) 0 0
\(580\) 6.00000i 0.249136i
\(581\) 0 0
\(582\) 0 0
\(583\) 72.0000i 2.98194i
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 6.00000i 0.247016i
\(591\) 0 0
\(592\) 6.00000i 0.246598i
\(593\) 30.0000i 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000i 0.737309i
\(597\) 0 0
\(598\) −18.0000 12.0000i −0.736075 0.490716i
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 12.0000i 0.488273i
\(605\) 25.0000i 1.01639i
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) 0 0
\(613\) 6.00000i 0.242338i 0.992632 + 0.121169i \(0.0386643\pi\)
−0.992632 + 0.121169i \(0.961336\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 0 0
\(619\) 30.0000i 1.20580i 0.797816 + 0.602901i \(0.205989\pi\)
−0.797816 + 0.602901i \(0.794011\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 12.0000i 0.481156i
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 10.0000i 0.399680i
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) 36.0000i 1.43541i
\(630\) 0 0
\(631\) 12.0000i 0.477712i 0.971055 + 0.238856i \(0.0767725\pi\)
−0.971055 + 0.238856i \(0.923228\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 0 0
\(634\) 6.00000 0.238290
\(635\) 2.00000i 0.0793676i
\(636\) 0 0
\(637\) 14.0000 21.0000i 0.554700 0.832050i
\(638\) −36.0000 −1.42525
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 36.0000 1.41640
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) −3.00000 2.00000i −0.117670 0.0784465i
\(651\) 0 0
\(652\) 12.0000i 0.469956i
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 12.0000i 0.468879i
\(656\) 12.0000i 0.468521i
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 42.0000i 1.63361i −0.576913 0.816805i \(-0.695743\pi\)
0.576913 0.816805i \(-0.304257\pi\)
\(662\) −18.0000 −0.699590
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −36.0000 −1.39393
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 60.0000i 2.31627i
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 22.0000i 0.847408i
\(675\) 0 0
\(676\) 5.00000 + 12.0000i 0.192308 + 0.461538i
\(677\) −24.0000 −0.922395 −0.461197 0.887298i \(-0.652580\pi\)
−0.461197 + 0.887298i \(0.652580\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000i 0.918334i −0.888350 0.459167i \(-0.848148\pi\)
0.888350 0.459167i \(-0.151852\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 24.0000 36.0000i 0.914327 1.37149i
\(690\) 0 0
\(691\) 42.0000i 1.59776i 0.601494 + 0.798878i \(0.294573\pi\)
−0.601494 + 0.798878i \(0.705427\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) 0 0
\(695\) 4.00000i 0.151729i
\(696\) 0 0
\(697\) 72.0000i 2.72719i
\(698\) 6.00000 0.227103
\(699\) 0 0
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 36.0000 1.35777
\(704\) 6.00000i 0.226134i
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) 30.0000i 1.12667i 0.826227 + 0.563337i \(0.190483\pi\)
−0.826227 + 0.563337i \(0.809517\pi\)
\(710\) 12.0000i 0.450352i
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 12.0000 18.0000i 0.448775 0.673162i
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 17.0000i 0.632674i
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −26.0000 −0.964287 −0.482143 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 6.00000i 0.222070i
\(731\) 48.0000 1.77534
\(732\) 0 0
\(733\) 30.0000i 1.10808i −0.832492 0.554038i \(-0.813086\pi\)
0.832492 0.554038i \(-0.186914\pi\)
\(734\) 26.0000i 0.959678i
\(735\) 0 0
\(736\) 6.00000i 0.221163i
\(737\) 0 0
\(738\) 0 0
\(739\) 42.0000i 1.54499i 0.635018 + 0.772497i \(0.280993\pi\)
−0.635018 + 0.772497i \(0.719007\pi\)
\(740\) −6.00000 −0.220564
\(741\) 0 0
\(742\) 0 0
\(743\) 48.0000i 1.76095i −0.474093 0.880475i \(-0.657224\pi\)
0.474093 0.880475i \(-0.342776\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 4.00000i 0.146450i
\(747\) 0 0
\(748\) 36.0000i 1.31629i
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 18.0000 + 12.0000i 0.655521 + 0.437014i
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 16.0000 0.581530 0.290765 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) 6.00000 0.217930
\(759\) 0 0
\(760\) 6.00000i 0.217643i
\(761\) 48.0000i 1.74000i −0.493053 0.869999i \(-0.664119\pi\)
0.493053 0.869999i \(-0.335881\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 18.0000 + 12.0000i 0.649942 + 0.433295i
\(768\) 0 0
\(769\) 24.0000i 0.865462i −0.901523 0.432731i \(-0.857550\pi\)
0.901523 0.432731i \(-0.142450\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6.00000i 0.215945i
\(773\) 6.00000i 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 30.0000i 1.07555i
\(779\) 72.0000 2.57967
\(780\) 0 0
\(781\) 72.0000 2.57636
\(782\) 36.0000i 1.28736i
\(783\) 0 0
\(784\) 7.00000 0.250000
\(785\) 4.00000i 0.142766i
\(786\) 0 0
\(787\) 12.0000i 0.427754i 0.976861 + 0.213877i \(0.0686091\pi\)
−0.976861 + 0.213877i \(0.931391\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) 0 0
\(793\) 20.0000 30.0000i 0.710221 1.06533i
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000i 0.0353553i
\(801\) 0 0
\(802\) 24.0000 0.847469
\(803\) −36.0000 −1.27041
\(804\) 0 0