Properties

Label 1152.3.m.f.415.8
Level $1152$
Weight $3$
Character 1152.415
Analytic conductor $31.390$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.m (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 415.8
Root \(1.80398 + 0.863518i\) of defining polynomial
Character \(\chi\) \(=\) 1152.415
Dual form 1152.3.m.f.991.8

$q$-expansion

\(f(q)\) \(=\) \(q+(6.49473 + 6.49473i) q^{5} +3.94273 q^{7} +O(q^{10})\) \(q+(6.49473 + 6.49473i) q^{5} +3.94273 q^{7} +(4.31091 - 4.31091i) q^{11} +(-4.06281 + 4.06281i) q^{13} +14.5538 q^{17} +(-4.94805 - 4.94805i) q^{19} +43.6717 q^{23} +59.3629i q^{25} +(25.0979 - 25.0979i) q^{29} -32.5024i q^{31} +(25.6069 + 25.6069i) q^{35} +(-4.14345 - 4.14345i) q^{37} +55.3348i q^{41} +(16.1189 - 16.1189i) q^{43} -7.92420i q^{47} -33.4549 q^{49} +(-31.5748 - 31.5748i) q^{53} +55.9964 q^{55} +(-49.7172 + 49.7172i) q^{59} +(-44.4711 + 44.4711i) q^{61} -52.7736 q^{65} +(1.64068 + 1.64068i) q^{67} -24.1145 q^{71} -10.7741i q^{73} +(16.9967 - 16.9967i) q^{77} +72.0517i q^{79} +(42.0499 + 42.0499i) q^{83} +(94.5229 + 94.5229i) q^{85} -28.9853i q^{89} +(-16.0185 + 16.0185i) q^{91} -64.2724i q^{95} -54.2698 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 32q^{11} + 32q^{19} + 128q^{23} + 32q^{29} + 96q^{35} + 96q^{37} - 160q^{43} + 112q^{49} - 160q^{53} - 256q^{55} - 128q^{59} + 32q^{61} + 32q^{65} - 320q^{67} - 512q^{71} + 224q^{77} - 160q^{83} - 160q^{85} + 480q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 6.49473 + 6.49473i 1.29895 + 1.29895i 0.929089 + 0.369856i \(0.120593\pi\)
0.369856 + 0.929089i \(0.379407\pi\)
\(6\) 0 0
\(7\) 3.94273 0.563247 0.281623 0.959525i \(-0.409127\pi\)
0.281623 + 0.959525i \(0.409127\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.31091 4.31091i 0.391901 0.391901i −0.483464 0.875364i \(-0.660621\pi\)
0.875364 + 0.483464i \(0.160621\pi\)
\(12\) 0 0
\(13\) −4.06281 + 4.06281i −0.312524 + 0.312524i −0.845887 0.533363i \(-0.820928\pi\)
0.533363 + 0.845887i \(0.320928\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 14.5538 0.856106 0.428053 0.903754i \(-0.359200\pi\)
0.428053 + 0.903754i \(0.359200\pi\)
\(18\) 0 0
\(19\) −4.94805 4.94805i −0.260423 0.260423i 0.564803 0.825226i \(-0.308952\pi\)
−0.825226 + 0.564803i \(0.808952\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 43.6717 1.89877 0.949385 0.314115i \(-0.101708\pi\)
0.949385 + 0.314115i \(0.101708\pi\)
\(24\) 0 0
\(25\) 59.3629i 2.37452i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 25.0979 25.0979i 0.865445 0.865445i −0.126519 0.991964i \(-0.540381\pi\)
0.991964 + 0.126519i \(0.0403806\pi\)
\(30\) 0 0
\(31\) 32.5024i 1.04846i −0.851576 0.524232i \(-0.824352\pi\)
0.851576 0.524232i \(-0.175648\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 25.6069 + 25.6069i 0.731627 + 0.731627i
\(36\) 0 0
\(37\) −4.14345 4.14345i −0.111985 0.111985i 0.648894 0.760879i \(-0.275232\pi\)
−0.760879 + 0.648894i \(0.775232\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 55.3348i 1.34963i 0.737987 + 0.674814i \(0.235776\pi\)
−0.737987 + 0.674814i \(0.764224\pi\)
\(42\) 0 0
\(43\) 16.1189 16.1189i 0.374858 0.374858i −0.494385 0.869243i \(-0.664607\pi\)
0.869243 + 0.494385i \(0.164607\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.92420i 0.168600i −0.996440 0.0843001i \(-0.973135\pi\)
0.996440 0.0843001i \(-0.0268654\pi\)
\(48\) 0 0
\(49\) −33.4549 −0.682753
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −31.5748 31.5748i −0.595750 0.595750i 0.343429 0.939179i \(-0.388412\pi\)
−0.939179 + 0.343429i \(0.888412\pi\)
\(54\) 0 0
\(55\) 55.9964 1.01812
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −49.7172 + 49.7172i −0.842665 + 0.842665i −0.989205 0.146540i \(-0.953186\pi\)
0.146540 + 0.989205i \(0.453186\pi\)
\(60\) 0 0
\(61\) −44.4711 + 44.4711i −0.729035 + 0.729035i −0.970427 0.241393i \(-0.922396\pi\)
0.241393 + 0.970427i \(0.422396\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −52.7736 −0.811902
\(66\) 0 0
\(67\) 1.64068 + 1.64068i 0.0244878 + 0.0244878i 0.719245 0.694757i \(-0.244488\pi\)
−0.694757 + 0.719245i \(0.744488\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −24.1145 −0.339641 −0.169821 0.985475i \(-0.554319\pi\)
−0.169821 + 0.985475i \(0.554319\pi\)
\(72\) 0 0
\(73\) 10.7741i 0.147591i −0.997273 0.0737955i \(-0.976489\pi\)
0.997273 0.0737955i \(-0.0235112\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.9967 16.9967i 0.220737 0.220737i
\(78\) 0 0
\(79\) 72.0517i 0.912047i 0.889968 + 0.456024i \(0.150727\pi\)
−0.889968 + 0.456024i \(0.849273\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 42.0499 + 42.0499i 0.506625 + 0.506625i 0.913489 0.406864i \(-0.133378\pi\)
−0.406864 + 0.913489i \(0.633378\pi\)
\(84\) 0 0
\(85\) 94.5229 + 94.5229i 1.11203 + 1.11203i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 28.9853i 0.325677i −0.986653 0.162839i \(-0.947935\pi\)
0.986653 0.162839i \(-0.0520650\pi\)
\(90\) 0 0
\(91\) −16.0185 + 16.0185i −0.176028 + 0.176028i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 64.2724i 0.676552i
\(96\) 0 0
\(97\) −54.2698 −0.559483 −0.279741 0.960075i \(-0.590249\pi\)
−0.279741 + 0.960075i \(0.590249\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 57.0829 + 57.0829i 0.565177 + 0.565177i 0.930773 0.365597i \(-0.119135\pi\)
−0.365597 + 0.930773i \(0.619135\pi\)
\(102\) 0 0
\(103\) 39.3048 0.381600 0.190800 0.981629i \(-0.438892\pi\)
0.190800 + 0.981629i \(0.438892\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 25.6981 25.6981i 0.240169 0.240169i −0.576751 0.816920i \(-0.695680\pi\)
0.816920 + 0.576751i \(0.195680\pi\)
\(108\) 0 0
\(109\) 9.66133 9.66133i 0.0886360 0.0886360i −0.661399 0.750035i \(-0.730037\pi\)
0.750035 + 0.661399i \(0.230037\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −64.2927 −0.568962 −0.284481 0.958682i \(-0.591821\pi\)
−0.284481 + 0.958682i \(0.591821\pi\)
\(114\) 0 0
\(115\) 283.636 + 283.636i 2.46640 + 2.46640i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 57.3816 0.482199
\(120\) 0 0
\(121\) 83.8321i 0.692827i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −223.178 + 223.178i −1.78542 + 1.78542i
\(126\) 0 0
\(127\) 129.668i 1.02101i −0.859875 0.510504i \(-0.829459\pi\)
0.859875 0.510504i \(-0.170541\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −118.504 118.504i −0.904613 0.904613i 0.0912183 0.995831i \(-0.470924\pi\)
−0.995831 + 0.0912183i \(0.970924\pi\)
\(132\) 0 0
\(133\) −19.5088 19.5088i −0.146683 0.146683i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 157.472i 1.14943i 0.818353 + 0.574716i \(0.194888\pi\)
−0.818353 + 0.574716i \(0.805112\pi\)
\(138\) 0 0
\(139\) 118.943 118.943i 0.855703 0.855703i −0.135125 0.990829i \(-0.543144\pi\)
0.990829 + 0.135125i \(0.0431437\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 35.0288i 0.244957i
\(144\) 0 0
\(145\) 326.008 2.24833
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 99.5402 + 99.5402i 0.668055 + 0.668055i 0.957266 0.289210i \(-0.0933927\pi\)
−0.289210 + 0.957266i \(0.593393\pi\)
\(150\) 0 0
\(151\) 273.705 1.81262 0.906308 0.422618i \(-0.138889\pi\)
0.906308 + 0.422618i \(0.138889\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 211.094 211.094i 1.36190 1.36190i
\(156\) 0 0
\(157\) 75.8792 75.8792i 0.483307 0.483307i −0.422879 0.906186i \(-0.638981\pi\)
0.906186 + 0.422879i \(0.138981\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 172.186 1.06948
\(162\) 0 0
\(163\) −177.242 177.242i −1.08737 1.08737i −0.995798 0.0915766i \(-0.970809\pi\)
−0.0915766 0.995798i \(-0.529191\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −61.6774 −0.369326 −0.184663 0.982802i \(-0.559119\pi\)
−0.184663 + 0.982802i \(0.559119\pi\)
\(168\) 0 0
\(169\) 135.987i 0.804658i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 69.7012 69.7012i 0.402897 0.402897i −0.476355 0.879253i \(-0.658042\pi\)
0.879253 + 0.476355i \(0.158042\pi\)
\(174\) 0 0
\(175\) 234.052i 1.33744i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 43.6228 + 43.6228i 0.243703 + 0.243703i 0.818380 0.574677i \(-0.194872\pi\)
−0.574677 + 0.818380i \(0.694872\pi\)
\(180\) 0 0
\(181\) 44.7291 + 44.7291i 0.247122 + 0.247122i 0.819788 0.572666i \(-0.194091\pi\)
−0.572666 + 0.819788i \(0.694091\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 53.8211i 0.290925i
\(186\) 0 0
\(187\) 62.7401 62.7401i 0.335509 0.335509i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 171.759i 0.899263i 0.893214 + 0.449632i \(0.148445\pi\)
−0.893214 + 0.449632i \(0.851555\pi\)
\(192\) 0 0
\(193\) −215.384 −1.11598 −0.557989 0.829848i \(-0.688427\pi\)
−0.557989 + 0.829848i \(0.688427\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.3354 18.3354i −0.0930731 0.0930731i 0.659037 0.752110i \(-0.270964\pi\)
−0.752110 + 0.659037i \(0.770964\pi\)
\(198\) 0 0
\(199\) −227.112 −1.14127 −0.570634 0.821205i \(-0.693302\pi\)
−0.570634 + 0.821205i \(0.693302\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 98.9542 98.9542i 0.487459 0.487459i
\(204\) 0 0
\(205\) −359.384 + 359.384i −1.75309 + 1.75309i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −42.6612 −0.204120
\(210\) 0 0
\(211\) −190.206 190.206i −0.901451 0.901451i 0.0941112 0.995562i \(-0.469999\pi\)
−0.995562 + 0.0941112i \(0.969999\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 209.375 0.973839
\(216\) 0 0
\(217\) 128.148i 0.590544i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −59.1293 + 59.1293i −0.267553 + 0.267553i
\(222\) 0 0
\(223\) 154.401i 0.692379i 0.938165 + 0.346190i \(0.112525\pi\)
−0.938165 + 0.346190i \(0.887475\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −36.8204 36.8204i −0.162204 0.162204i 0.621338 0.783543i \(-0.286589\pi\)
−0.783543 + 0.621338i \(0.786589\pi\)
\(228\) 0 0
\(229\) −17.9692 17.9692i −0.0784683 0.0784683i 0.666783 0.745252i \(-0.267671\pi\)
−0.745252 + 0.666783i \(0.767671\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 167.669i 0.719608i −0.933028 0.359804i \(-0.882844\pi\)
0.933028 0.359804i \(-0.117156\pi\)
\(234\) 0 0
\(235\) 51.4655 51.4655i 0.219002 0.219002i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 29.7509i 0.124481i −0.998061 0.0622403i \(-0.980175\pi\)
0.998061 0.0622403i \(-0.0198245\pi\)
\(240\) 0 0
\(241\) −107.373 −0.445531 −0.222766 0.974872i \(-0.571508\pi\)
−0.222766 + 0.974872i \(0.571508\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −217.280 217.280i −0.886859 0.886859i
\(246\) 0 0
\(247\) 40.2059 0.162777
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −342.946 + 342.946i −1.36632 + 1.36632i −0.500697 + 0.865623i \(0.666923\pi\)
−0.865623 + 0.500697i \(0.833077\pi\)
\(252\) 0 0
\(253\) 188.265 188.265i 0.744130 0.744130i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 393.565 1.53138 0.765691 0.643209i \(-0.222397\pi\)
0.765691 + 0.643209i \(0.222397\pi\)
\(258\) 0 0
\(259\) −16.3365 16.3365i −0.0630753 0.0630753i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −413.800 −1.57338 −0.786692 0.617346i \(-0.788208\pi\)
−0.786692 + 0.617346i \(0.788208\pi\)
\(264\) 0 0
\(265\) 410.139i 1.54769i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 165.389 165.389i 0.614830 0.614830i −0.329371 0.944201i \(-0.606837\pi\)
0.944201 + 0.329371i \(0.106837\pi\)
\(270\) 0 0
\(271\) 309.821i 1.14325i −0.820514 0.571626i \(-0.806313\pi\)
0.820514 0.571626i \(-0.193687\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 255.908 + 255.908i 0.930575 + 0.930575i
\(276\) 0 0
\(277\) −157.397 157.397i −0.568221 0.568221i 0.363409 0.931630i \(-0.381613\pi\)
−0.931630 + 0.363409i \(0.881613\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 411.141i 1.46313i −0.681769 0.731567i \(-0.738789\pi\)
0.681769 0.731567i \(-0.261211\pi\)
\(282\) 0 0
\(283\) −343.521 + 343.521i −1.21385 + 1.21385i −0.244106 + 0.969748i \(0.578495\pi\)
−0.969748 + 0.244106i \(0.921505\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 218.170i 0.760174i
\(288\) 0 0
\(289\) −77.1870 −0.267083
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 35.1386 + 35.1386i 0.119927 + 0.119927i 0.764523 0.644596i \(-0.222975\pi\)
−0.644596 + 0.764523i \(0.722975\pi\)
\(294\) 0 0
\(295\) −645.799 −2.18915
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −177.430 + 177.430i −0.593410 + 0.593410i
\(300\) 0 0
\(301\) 63.5524 63.5524i 0.211137 0.211137i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −577.655 −1.89395
\(306\) 0 0
\(307\) 16.4432 + 16.4432i 0.0535609 + 0.0535609i 0.733380 0.679819i \(-0.237942\pi\)
−0.679819 + 0.733380i \(0.737942\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −39.8016 −0.127980 −0.0639898 0.997951i \(-0.520382\pi\)
−0.0639898 + 0.997951i \(0.520382\pi\)
\(312\) 0 0
\(313\) 431.885i 1.37982i 0.723894 + 0.689911i \(0.242351\pi\)
−0.723894 + 0.689911i \(0.757649\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 255.063 255.063i 0.804615 0.804615i −0.179198 0.983813i \(-0.557350\pi\)
0.983813 + 0.179198i \(0.0573503\pi\)
\(318\) 0 0
\(319\) 216.390i 0.678337i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −72.0128 72.0128i −0.222950 0.222950i
\(324\) 0 0
\(325\) −241.180 241.180i −0.742092 0.742092i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 31.2430i 0.0949635i
\(330\) 0 0
\(331\) −205.897 + 205.897i −0.622045 + 0.622045i −0.946054 0.324009i \(-0.894969\pi\)
0.324009 + 0.946054i \(0.394969\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 21.3115i 0.0636165i
\(336\) 0 0
\(337\) 45.7312 0.135701 0.0678504 0.997696i \(-0.478386\pi\)
0.0678504 + 0.997696i \(0.478386\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −140.115 140.115i −0.410894 0.410894i
\(342\) 0 0
\(343\) −325.097 −0.947805
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 296.512 296.512i 0.854500 0.854500i −0.136183 0.990684i \(-0.543484\pi\)
0.990684 + 0.136183i \(0.0434836\pi\)
\(348\) 0 0
\(349\) 198.107 198.107i 0.567641 0.567641i −0.363826 0.931467i \(-0.618530\pi\)
0.931467 + 0.363826i \(0.118530\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −85.4490 −0.242065 −0.121033 0.992649i \(-0.538621\pi\)
−0.121033 + 0.992649i \(0.538621\pi\)
\(354\) 0 0
\(355\) −156.617 156.617i −0.441176 0.441176i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 302.214 0.841823 0.420911 0.907102i \(-0.361710\pi\)
0.420911 + 0.907102i \(0.361710\pi\)
\(360\) 0 0
\(361\) 312.034i 0.864359i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 69.9751 69.9751i 0.191713 0.191713i
\(366\) 0 0
\(367\) 372.554i 1.01513i −0.861612 0.507567i \(-0.830545\pi\)
0.861612 0.507567i \(-0.169455\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −124.491 124.491i −0.335554 0.335554i
\(372\) 0 0
\(373\) 407.130 + 407.130i 1.09150 + 1.09150i 0.995369 + 0.0961318i \(0.0306470\pi\)
0.0961318 + 0.995369i \(0.469353\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 203.936i 0.540944i
\(378\) 0 0
\(379\) 117.854 117.854i 0.310961 0.310961i −0.534321 0.845282i \(-0.679433\pi\)
0.845282 + 0.534321i \(0.179433\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 407.983i 1.06523i −0.846357 0.532615i \(-0.821209\pi\)
0.846357 0.532615i \(-0.178791\pi\)
\(384\) 0 0
\(385\) 220.778 0.573450
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 458.508 + 458.508i 1.17868 + 1.17868i 0.980080 + 0.198605i \(0.0636411\pi\)
0.198605 + 0.980080i \(0.436359\pi\)
\(390\) 0 0
\(391\) 635.589 1.62555
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −467.956 + 467.956i −1.18470 + 1.18470i
\(396\) 0 0
\(397\) 259.865 259.865i 0.654573 0.654573i −0.299518 0.954091i \(-0.596826\pi\)
0.954091 + 0.299518i \(0.0968259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 499.197 1.24488 0.622441 0.782667i \(-0.286141\pi\)
0.622441 + 0.782667i \(0.286141\pi\)
\(402\) 0 0
\(403\) 132.051 + 132.051i 0.327670 + 0.327670i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −35.7241 −0.0877741
\(408\) 0 0
\(409\) 494.949i 1.21014i −0.796171 0.605072i \(-0.793144\pi\)
0.796171 0.605072i \(-0.206856\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −196.021 + 196.021i −0.474628 + 0.474628i
\(414\) 0 0
\(415\) 546.205i 1.31616i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −560.555 560.555i −1.33784 1.33784i −0.898148 0.439693i \(-0.855087\pi\)
−0.439693 0.898148i \(-0.644913\pi\)
\(420\) 0 0
\(421\) −397.946 397.946i −0.945239 0.945239i 0.0533373 0.998577i \(-0.483014\pi\)
−0.998577 + 0.0533373i \(0.983014\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 863.956i 2.03284i
\(426\) 0 0
\(427\) −175.337 + 175.337i −0.410626 + 0.410626i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 662.874i 1.53799i −0.639255 0.768995i \(-0.720757\pi\)
0.639255 0.768995i \(-0.279243\pi\)
\(432\) 0 0
\(433\) −338.800 −0.782448 −0.391224 0.920296i \(-0.627948\pi\)
−0.391224 + 0.920296i \(0.627948\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −216.090 216.090i −0.494484 0.494484i
\(438\) 0 0
\(439\) −234.566 −0.534319 −0.267160 0.963652i \(-0.586085\pi\)
−0.267160 + 0.963652i \(0.586085\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 421.096 421.096i 0.950555 0.950555i −0.0482792 0.998834i \(-0.515374\pi\)
0.998834 + 0.0482792i \(0.0153737\pi\)
\(444\) 0 0
\(445\) 188.251 188.251i 0.423037 0.423037i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −492.636 −1.09718 −0.548592 0.836090i \(-0.684836\pi\)
−0.548592 + 0.836090i \(0.684836\pi\)
\(450\) 0 0
\(451\) 238.543 + 238.543i 0.528921 + 0.528921i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −208.072 −0.457301
\(456\) 0 0
\(457\) 516.831i 1.13092i −0.824775 0.565461i \(-0.808698\pi\)
0.824775 0.565461i \(-0.191302\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −27.5260 + 27.5260i −0.0597093 + 0.0597093i −0.736331 0.676622i \(-0.763443\pi\)
0.676622 + 0.736331i \(0.263443\pi\)
\(462\) 0 0
\(463\) 122.111i 0.263740i 0.991267 + 0.131870i \(0.0420981\pi\)
−0.991267 + 0.131870i \(0.957902\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 267.964 + 267.964i 0.573798 + 0.573798i 0.933188 0.359390i \(-0.117015\pi\)
−0.359390 + 0.933188i \(0.617015\pi\)
\(468\) 0 0
\(469\) 6.46875 + 6.46875i 0.0137926 + 0.0137926i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 138.974i 0.293814i
\(474\) 0 0
\(475\) 293.730 293.730i 0.618380 0.618380i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 419.084i 0.874915i −0.899239 0.437457i \(-0.855879\pi\)
0.899239 0.437457i \(-0.144121\pi\)
\(480\) 0 0
\(481\) 33.6681 0.0699960
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −352.468 352.468i −0.726738 0.726738i
\(486\) 0 0
\(487\) −57.2378 −0.117531 −0.0587657 0.998272i \(-0.518716\pi\)
−0.0587657 + 0.998272i \(0.518716\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −301.955 + 301.955i −0.614979 + 0.614979i −0.944239 0.329260i \(-0.893201\pi\)
0.329260 + 0.944239i \(0.393201\pi\)
\(492\) 0 0
\(493\) 365.270 365.270i 0.740912 0.740912i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −95.0771 −0.191302
\(498\) 0 0
\(499\) −619.990 619.990i −1.24247 1.24247i −0.958975 0.283491i \(-0.908507\pi\)
−0.283491 0.958975i \(-0.591493\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −222.446 −0.442239 −0.221120 0.975247i \(-0.570971\pi\)
−0.221120 + 0.975247i \(0.570971\pi\)
\(504\) 0 0
\(505\) 741.475i 1.46827i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 489.873 489.873i 0.962421 0.962421i −0.0368976 0.999319i \(-0.511748\pi\)
0.999319 + 0.0368976i \(0.0117475\pi\)
\(510\) 0 0
\(511\) 42.4795i 0.0831302i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 255.274 + 255.274i 0.495678 + 0.495678i
\(516\) 0 0
\(517\) −34.1605 34.1605i −0.0660745 0.0660745i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 197.152i 0.378412i −0.981937 0.189206i \(-0.939409\pi\)
0.981937 0.189206i \(-0.0605913\pi\)
\(522\) 0 0
\(523\) −621.874 + 621.874i −1.18905 + 1.18905i −0.211721 + 0.977330i \(0.567907\pi\)
−0.977330 + 0.211721i \(0.932093\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 473.033i 0.897596i
\(528\) 0 0
\(529\) 1378.22 2.60533
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −224.815 224.815i −0.421791 0.421791i
\(534\) 0 0
\(535\) 333.804 0.623933
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −144.221 + 144.221i −0.267572 + 0.267572i
\(540\) 0 0
\(541\) −423.563 + 423.563i −0.782925 + 0.782925i −0.980323 0.197398i \(-0.936751\pi\)
0.197398 + 0.980323i \(0.436751\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 125.495 0.230267
\(546\) 0 0
\(547\) 14.5553 + 14.5553i 0.0266093 + 0.0266093i 0.720286 0.693677i \(-0.244010\pi\)
−0.693677 + 0.720286i \(0.744010\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −248.371 −0.450764
\(552\) 0 0
\(553\) 284.080i 0.513708i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −351.991 + 351.991i −0.631941 + 0.631941i −0.948554 0.316614i \(-0.897454\pi\)
0.316614 + 0.948554i \(0.397454\pi\)
\(558\) 0 0
\(559\) 130.976i 0.234304i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −150.902 150.902i −0.268031 0.268031i 0.560275 0.828307i \(-0.310695\pi\)
−0.828307 + 0.560275i \(0.810695\pi\)
\(564\) 0 0
\(565\) −417.563 417.563i −0.739050 0.739050i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 113.300i 0.199121i −0.995032 0.0995603i \(-0.968256\pi\)
0.995032 0.0995603i \(-0.0317436\pi\)
\(570\) 0 0
\(571\) 207.486 207.486i 0.363373 0.363373i −0.501680 0.865053i \(-0.667285\pi\)
0.865053 + 0.501680i \(0.167285\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2592.48i 4.50866i
\(576\) 0 0
\(577\) −484.715 −0.840061 −0.420031 0.907510i \(-0.637981\pi\)
−0.420031 + 0.907510i \(0.637981\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 165.791 + 165.791i 0.285355 + 0.285355i
\(582\) 0 0
\(583\) −272.232 −0.466950
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −540.404 + 540.404i −0.920619 + 0.920619i −0.997073 0.0764537i \(-0.975640\pi\)
0.0764537 + 0.997073i \(0.475640\pi\)
\(588\) 0 0
\(589\) −160.823 + 160.823i −0.273045 + 0.273045i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −411.176 −0.693383 −0.346692 0.937979i \(-0.612695\pi\)
−0.346692 + 0.937979i \(0.612695\pi\)
\(594\) 0 0
\(595\) 372.678 + 372.678i 0.626350 + 0.626350i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 552.839 0.922936 0.461468 0.887157i \(-0.347323\pi\)
0.461468 + 0.887157i \(0.347323\pi\)
\(600\) 0 0
\(601\) 881.159i 1.46615i −0.680145 0.733077i \(-0.738083\pi\)
0.680145 0.733077i \(-0.261917\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −544.467 + 544.467i −0.899945 + 0.899945i
\(606\) 0 0
\(607\) 1175.08i 1.93588i 0.251186 + 0.967939i \(0.419180\pi\)
−0.251186 + 0.967939i \(0.580820\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.1945 + 32.1945i 0.0526915 + 0.0526915i
\(612\) 0 0
\(613\) 496.928 + 496.928i 0.810649 + 0.810649i 0.984731 0.174082i \(-0.0556959\pi\)
−0.174082 + 0.984731i \(0.555696\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 623.301i 1.01021i −0.863057 0.505106i \(-0.831453\pi\)
0.863057 0.505106i \(-0.168547\pi\)
\(618\) 0 0
\(619\) −7.45302 + 7.45302i −0.0120404 + 0.0120404i −0.713101 0.701061i \(-0.752710\pi\)
0.701061 + 0.713101i \(0.252710\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 114.281i 0.183437i
\(624\) 0 0
\(625\) −1414.88 −2.26381
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −60.3029 60.3029i −0.0958711 0.0958711i
\(630\) 0 0
\(631\) 147.833 0.234284 0.117142 0.993115i \(-0.462627\pi\)
0.117142 + 0.993115i \(0.462627\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 842.158 842.158i 1.32623 1.32623i
\(636\) 0 0
\(637\) 135.921 135.921i 0.213376 0.213376i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 782.691 1.22105 0.610523 0.791998i \(-0.290959\pi\)
0.610523 + 0.791998i \(0.290959\pi\)
\(642\) 0 0
\(643\) −126.760 126.760i −0.197138 0.197138i 0.601634 0.798772i \(-0.294517\pi\)
−0.798772 + 0.601634i \(0.794517\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1226.09 −1.89504 −0.947520 0.319697i \(-0.896419\pi\)
−0.947520 + 0.319697i \(0.896419\pi\)
\(648\) 0 0
\(649\) 428.653i 0.660482i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 326.300 326.300i 0.499694 0.499694i −0.411649 0.911343i \(-0.635047\pi\)
0.911343 + 0.411649i \(0.135047\pi\)
\(654\) 0 0
\(655\) 1539.31i 2.35008i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −574.901 574.901i −0.872384 0.872384i 0.120347 0.992732i \(-0.461599\pi\)
−0.992732 + 0.120347i \(0.961599\pi\)
\(660\) 0 0
\(661\) −52.8795 52.8795i −0.0799993 0.0799993i 0.665975 0.745974i \(-0.268016\pi\)
−0.745974 + 0.665975i \(0.768016\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 253.409i 0.381065i
\(666\) 0 0
\(667\) 1096.07 1096.07i 1.64328 1.64328i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 383.422i 0.571419i
\(672\) 0 0
\(673\) 342.318 0.508645 0.254322 0.967119i \(-0.418148\pi\)
0.254322 + 0.967119i \(0.418148\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 107.154 + 107.154i 0.158278 + 0.158278i 0.781803 0.623525i \(-0.214300\pi\)
−0.623525 + 0.781803i \(0.714300\pi\)
\(678\) 0 0
\(679\) −213.971 −0.315127
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 724.233 724.233i 1.06037 1.06037i 0.0623142 0.998057i \(-0.480152\pi\)
0.998057 0.0623142i \(-0.0198481\pi\)
\(684\) 0 0
\(685\) −1022.74 + 1022.74i −1.49305 + 1.49305i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 256.564 0.372372
\(690\) 0 0
\(691\) −162.528 162.528i −0.235207 0.235207i 0.579655 0.814862i \(-0.303187\pi\)
−0.814862 + 0.579655i \(0.803187\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1545.00 2.22302
\(696\) 0 0
\(697\) 805.331i 1.15543i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −301.659 + 301.659i −0.430327 + 0.430327i −0.888740 0.458412i \(-0.848418\pi\)
0.458412 + 0.888740i \(0.348418\pi\)
\(702\) 0 0
\(703\) 41.0040i 0.0583271i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 225.062 + 225.062i 0.318334 + 0.318334i
\(708\) 0 0
\(709\) 629.100 + 629.100i 0.887306 + 0.887306i 0.994264 0.106958i \(-0.0341109\pi\)
−0.106958 + 0.994264i \(0.534111\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1419.43i 1.99079i
\(714\) 0 0
\(715\) −227.502 + 227.502i −0.318185 + 0.318185i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 145.542i 0.202422i 0.994865 + 0.101211i \(0.0322718\pi\)
−0.994865 + 0.101211i \(0.967728\pi\)
\(720\) 0 0
\(721\) 154.968 0.214935
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1489.88 + 1489.88i 2.05501 + 2.05501i
\(726\) 0 0
\(727\) −938.214 −1.29053 −0.645264 0.763960i \(-0.723253\pi\)
−0.645264 + 0.763960i \(0.723253\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 234.591 234.591i 0.320918 0.320918i
\(732\) 0 0
\(733\) −692.101 + 692.101i −0.944203 + 0.944203i −0.998524 0.0543203i \(-0.982701\pi\)
0.0543203 + 0.998524i \(0.482701\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 14.1456 0.0191935
\(738\) 0 0
\(739\) 440.389 + 440.389i 0.595926 + 0.595926i 0.939226 0.343300i \(-0.111545\pi\)
−0.343300 + 0.939226i \(0.611545\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1010.54 1.36008 0.680039 0.733176i \(-0.261963\pi\)
0.680039 + 0.733176i \(0.261963\pi\)
\(744\) 0 0
\(745\) 1292.97i 1.73553i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 101.321 101.321i 0.135275 0.135275i
\(750\) 0 0
\(751\) 776.971i 1.03458i −0.855810 0.517291i \(-0.826940\pi\)
0.855810 0.517291i \(-0.173060\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1777.64 + 1777.64i 2.35449 + 2.35449i
\(756\) 0 0
\(757\) −375.481 375.481i −0.496012 0.496012i 0.414182 0.910194i \(-0.364068\pi\)
−0.910194 + 0.414182i \(0.864068\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1502.22i 1.97400i 0.160711 + 0.987001i \(0.448621\pi\)
−0.160711 + 0.987001i \(0.551379\pi\)
\(762\) 0 0
\(763\) 38.0920 38.0920i 0.0499239 0.0499239i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 403.983i 0.526705i
\(768\) 0 0
\(769\) −293.930 −0.382223 −0.191112 0.981568i \(-0.561209\pi\)
−0.191112 + 0.981568i \(0.561209\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −748.271 748.271i −0.968009 0.968009i 0.0314945 0.999504i \(-0.489973\pi\)
−0.999504 + 0.0314945i \(0.989973\pi\)
\(774\) 0 0
\(775\) 1929.44 2.48960
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 273.799 273.799i 0.351475 0.351475i
\(780\) 0 0
\(781\) −103.956 + 103.956i −0.133106 + 0.133106i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 985.629 1.25558
\(786\) 0 0
\(787\) 735.839 + 735.839i 0.934992 + 0.934992i 0.998012 0.0630203i \(-0.0200733\pi\)
−0.0630203 + 0.998012i \(0.520073\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −253.488 −0.320466
\(792\) 0 0
\(793\) 361.355i 0.455681i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 212.043 212.043i 0.266051 0.266051i −0.561455 0.827507i \(-0.689758\pi\)
0.827507 + 0.561455i \(0.189758\pi\)
\(798\) 0 0
\(799\) 115.327i 0.144340i
\(800\) 0