Properties

Label 1152.2.k.c.865.3
Level $1152$
Weight $2$
Character 1152.865
Analytic conductor $9.199$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 865.3
Root \(0.500000 - 0.691860i\) of defining polynomial
Character \(\chi\) \(=\) 1152.865
Dual form 1152.2.k.c.289.3

$q$-expansion

\(f(q)\) \(=\) \(q+(1.27133 + 1.27133i) q^{5} -0.158942i q^{7} +O(q^{10})\) \(q+(1.27133 + 1.27133i) q^{5} -0.158942i q^{7} +(-3.79793 - 3.79793i) q^{11} +(4.21215 - 4.21215i) q^{13} -3.05320 q^{17} +(2.15894 - 2.15894i) q^{19} -2.82843i q^{23} -1.76744i q^{25} +(2.09976 - 2.09976i) q^{29} +4.15894 q^{31} +(0.202067 - 0.202067i) q^{35} +(5.98737 + 5.98737i) q^{37} +2.60365i q^{41} +(-5.75481 - 5.75481i) q^{43} +2.82843 q^{47} +6.97474 q^{49} +(3.55710 + 3.55710i) q^{53} -9.65685i q^{55} +(4.00000 + 4.00000i) q^{59} +(-3.66949 + 3.66949i) q^{61} +10.7101 q^{65} +(-0.767438 + 0.767438i) q^{67} -0.317883i q^{71} -1.33897i q^{73} +(-0.603650 + 0.603650i) q^{77} -9.69382 q^{79} +(0.115816 - 0.115816i) q^{83} +(-3.88163 - 3.88163i) q^{85} -14.3990i q^{89} +(-0.669485 - 0.669485i) q^{91} +5.48946 q^{95} -0.571533 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{11} + 8q^{19} - 16q^{29} + 24q^{31} + 24q^{35} + 16q^{37} + 8q^{43} - 8q^{49} + 16q^{53} + 32q^{59} - 16q^{61} + 16q^{65} + 16q^{67} + 16q^{77} - 24q^{79} - 40q^{83} + 16q^{85} + 8q^{91} + 48q^{95} + O(q^{100}) \)

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
\(5\) 1.27133 + 1.27133i 0.568556 + 0.568556i 0.931724 0.363168i \(-0.118305\pi\)
−0.363168 + 0.931724i \(0.618305\pi\)
\(6\) 0 0
\(7\) 0.158942i 0.0600743i −0.999549 0.0300371i \(-0.990437\pi\)
0.999549 0.0300371i \(-0.00956256\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.79793 3.79793i −1.14512 1.14512i −0.987500 0.157620i \(-0.949618\pi\)
−0.157620 0.987500i \(-0.550382\pi\)
\(12\) 0 0
\(13\) 4.21215 4.21215i 1.16824 1.16824i 0.185617 0.982622i \(-0.440572\pi\)
0.982622 0.185617i \(-0.0594284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.05320 −0.740511 −0.370255 0.928930i \(-0.620730\pi\)
−0.370255 + 0.928930i \(0.620730\pi\)
\(18\) 0 0
\(19\) 2.15894 2.15894i 0.495295 0.495295i −0.414675 0.909970i \(-0.636105\pi\)
0.909970 + 0.414675i \(0.136105\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843i 0.589768i −0.955533 0.294884i \(-0.904719\pi\)
0.955533 0.294884i \(-0.0952810\pi\)
\(24\) 0 0
\(25\) 1.76744i 0.353488i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.09976 2.09976i 0.389915 0.389915i −0.484742 0.874657i \(-0.661087\pi\)
0.874657 + 0.484742i \(0.161087\pi\)
\(30\) 0 0
\(31\) 4.15894 0.746968 0.373484 0.927637i \(-0.378163\pi\)
0.373484 + 0.927637i \(0.378163\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.202067 0.202067i 0.0341556 0.0341556i
\(36\) 0 0
\(37\) 5.98737 + 5.98737i 0.984317 + 0.984317i 0.999879 0.0155615i \(-0.00495359\pi\)
−0.0155615 + 0.999879i \(0.504954\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.60365i 0.406622i 0.979114 + 0.203311i \(0.0651702\pi\)
−0.979114 + 0.203311i \(0.934830\pi\)
\(42\) 0 0
\(43\) −5.75481 5.75481i −0.877600 0.877600i 0.115686 0.993286i \(-0.463093\pi\)
−0.993286 + 0.115686i \(0.963093\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 6.97474 0.996391
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.55710 + 3.55710i 0.488605 + 0.488605i 0.907866 0.419261i \(-0.137711\pi\)
−0.419261 + 0.907866i \(0.637711\pi\)
\(54\) 0 0
\(55\) 9.65685i 1.30213i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 + 4.00000i 0.520756 + 0.520756i 0.917800 0.397044i \(-0.129964\pi\)
−0.397044 + 0.917800i \(0.629964\pi\)
\(60\) 0 0
\(61\) −3.66949 + 3.66949i −0.469829 + 0.469829i −0.901859 0.432030i \(-0.857798\pi\)
0.432030 + 0.901859i \(0.357798\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.7101 1.32842
\(66\) 0 0
\(67\) −0.767438 + 0.767438i −0.0937575 + 0.0937575i −0.752430 0.658672i \(-0.771118\pi\)
0.658672 + 0.752430i \(0.271118\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.317883i 0.0377258i −0.999822 0.0188629i \(-0.993995\pi\)
0.999822 0.0188629i \(-0.00600460\pi\)
\(72\) 0 0
\(73\) 1.33897i 0.156715i −0.996925 0.0783573i \(-0.975032\pi\)
0.996925 0.0783573i \(-0.0249675\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.603650 + 0.603650i −0.0687923 + 0.0687923i
\(78\) 0 0
\(79\) −9.69382 −1.09064 −0.545320 0.838228i \(-0.683592\pi\)
−0.545320 + 0.838228i \(0.683592\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.115816 0.115816i 0.0127125 0.0127125i −0.700722 0.713434i \(-0.747139\pi\)
0.713434 + 0.700722i \(0.247139\pi\)
\(84\) 0 0
\(85\) −3.88163 3.88163i −0.421022 0.421022i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.3990i 1.52629i −0.646225 0.763147i \(-0.723653\pi\)
0.646225 0.763147i \(-0.276347\pi\)
\(90\) 0 0
\(91\) −0.669485 0.669485i −0.0701811 0.0701811i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.48946 0.563206
\(96\) 0 0
\(97\) −0.571533 −0.0580304 −0.0290152 0.999579i \(-0.509237\pi\)
−0.0290152 + 0.999579i \(0.509237\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.15296 7.15296i −0.711746 0.711746i 0.255154 0.966900i \(-0.417874\pi\)
−0.966900 + 0.255154i \(0.917874\pi\)
\(102\) 0 0
\(103\) 11.3507i 1.11841i −0.829028 0.559207i \(-0.811106\pi\)
0.829028 0.559207i \(-0.188894\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.722018 0.722018i −0.0698001 0.0698001i 0.671345 0.741145i \(-0.265717\pi\)
−0.741145 + 0.671345i \(0.765717\pi\)
\(108\) 0 0
\(109\) 1.44471 1.44471i 0.138378 0.138378i −0.634525 0.772903i \(-0.718804\pi\)
0.772903 + 0.634525i \(0.218804\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.53488 0.332533 0.166267 0.986081i \(-0.446829\pi\)
0.166267 + 0.986081i \(0.446829\pi\)
\(114\) 0 0
\(115\) 3.59587 3.59587i 0.335316 0.335316i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.485281i 0.0444857i
\(120\) 0 0
\(121\) 17.8486i 1.62260i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 8.60365 8.60365i 0.769534 0.769534i
\(126\) 0 0
\(127\) −1.49791 −0.132918 −0.0664591 0.997789i \(-0.521170\pi\)
−0.0664591 + 0.997789i \(0.521170\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.4243 10.4243i 0.910775 0.910775i −0.0855585 0.996333i \(-0.527267\pi\)
0.996333 + 0.0855585i \(0.0272675\pi\)
\(132\) 0 0
\(133\) −0.343146 0.343146i −0.0297545 0.0297545i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.7954i 1.17862i 0.807907 + 0.589309i \(0.200600\pi\)
−0.807907 + 0.589309i \(0.799400\pi\)
\(138\) 0 0
\(139\) 2.42429 + 2.42429i 0.205626 + 0.205626i 0.802405 0.596779i \(-0.203553\pi\)
−0.596779 + 0.802405i \(0.703553\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −31.9949 −2.67555
\(144\) 0 0
\(145\) 5.33897 0.443377
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.92818 + 2.92818i 0.239886 + 0.239886i 0.816803 0.576917i \(-0.195744\pi\)
−0.576917 + 0.816803i \(0.695744\pi\)
\(150\) 0 0
\(151\) 22.6644i 1.84440i 0.386712 + 0.922201i \(0.373611\pi\)
−0.386712 + 0.922201i \(0.626389\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.28739 + 5.28739i 0.424693 + 0.424693i
\(156\) 0 0
\(157\) 2.78007 2.78007i 0.221874 0.221874i −0.587413 0.809287i \(-0.699854\pi\)
0.809287 + 0.587413i \(0.199854\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.449555 −0.0354299
\(162\) 0 0
\(163\) 5.43692 5.43692i 0.425853 0.425853i −0.461360 0.887213i \(-0.652638\pi\)
0.887213 + 0.461360i \(0.152638\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.95458i 0.306015i 0.988225 + 0.153007i \(0.0488958\pi\)
−0.988225 + 0.153007i \(0.951104\pi\)
\(168\) 0 0
\(169\) 22.4844i 1.72957i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −15.9814 + 15.9814i −1.21504 + 1.21504i −0.245695 + 0.969347i \(0.579016\pi\)
−0.969347 + 0.245695i \(0.920984\pi\)
\(174\) 0 0
\(175\) −0.280920 −0.0212355
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.2316 + 12.2316i −0.914235 + 0.914235i −0.996602 0.0823670i \(-0.973752\pi\)
0.0823670 + 0.996602i \(0.473752\pi\)
\(180\) 0 0
\(181\) −5.76259 5.76259i −0.428330 0.428330i 0.459729 0.888059i \(-0.347946\pi\)
−0.888059 + 0.459729i \(0.847946\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.2238i 1.11928i
\(186\) 0 0
\(187\) 11.5959 + 11.5959i 0.847974 + 0.847974i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.1674 1.16983 0.584916 0.811094i \(-0.301127\pi\)
0.584916 + 0.811094i \(0.301127\pi\)
\(192\) 0 0
\(193\) −22.1454 −1.59406 −0.797030 0.603940i \(-0.793597\pi\)
−0.797030 + 0.603940i \(0.793597\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.2993 14.2993i −1.01878 1.01878i −0.999820 0.0189608i \(-0.993964\pi\)
−0.0189608 0.999820i \(-0.506036\pi\)
\(198\) 0 0
\(199\) 25.0075i 1.77274i 0.462981 + 0.886368i \(0.346780\pi\)
−0.462981 + 0.886368i \(0.653220\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.333739 0.333739i −0.0234239 0.0234239i
\(204\) 0 0
\(205\) −3.31010 + 3.31010i −0.231187 + 0.231187i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −16.3990 −1.13434
\(210\) 0 0
\(211\) −18.4243 + 18.4243i −1.26838 + 1.26838i −0.321456 + 0.946924i \(0.604172\pi\)
−0.946924 + 0.321456i \(0.895828\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 14.6325i 0.997930i
\(216\) 0 0
\(217\) 0.661029i 0.0448736i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −12.8605 + 12.8605i −0.865094 + 0.865094i
\(222\) 0 0
\(223\) 18.3465 1.22857 0.614286 0.789083i \(-0.289444\pi\)
0.614286 + 0.789083i \(0.289444\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.115816 + 0.115816i −0.00768697 + 0.00768697i −0.710940 0.703253i \(-0.751730\pi\)
0.703253 + 0.710940i \(0.251730\pi\)
\(228\) 0 0
\(229\) −2.84791 2.84791i −0.188195 0.188195i 0.606720 0.794916i \(-0.292485\pi\)
−0.794916 + 0.606720i \(0.792485\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.7211i 0.767874i −0.923359 0.383937i \(-0.874568\pi\)
0.923359 0.383937i \(-0.125432\pi\)
\(234\) 0 0
\(235\) 3.59587 + 3.59587i 0.234568 + 0.234568i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 13.6517 0.883058 0.441529 0.897247i \(-0.354436\pi\)
0.441529 + 0.897247i \(0.354436\pi\)
\(240\) 0 0
\(241\) 2.13167 0.137313 0.0686565 0.997640i \(-0.478129\pi\)
0.0686565 + 0.997640i \(0.478129\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 8.86720 + 8.86720i 0.566504 + 0.566504i
\(246\) 0 0
\(247\) 18.1876i 1.15725i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4.43370 4.43370i −0.279853 0.279853i 0.553198 0.833050i \(-0.313407\pi\)
−0.833050 + 0.553198i \(0.813407\pi\)
\(252\) 0 0
\(253\) −10.7422 + 10.7422i −0.675355 + 0.675355i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.0853 −0.940997 −0.470498 0.882401i \(-0.655926\pi\)
−0.470498 + 0.882401i \(0.655926\pi\)
\(258\) 0 0
\(259\) 0.951642 0.951642i 0.0591322 0.0591322i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 26.1706i 1.61375i 0.590722 + 0.806875i \(0.298843\pi\)
−0.590722 + 0.806875i \(0.701157\pi\)
\(264\) 0 0
\(265\) 9.04449i 0.555599i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.59700 8.59700i 0.524168 0.524168i −0.394659 0.918828i \(-0.629137\pi\)
0.918828 + 0.394659i \(0.129137\pi\)
\(270\) 0 0
\(271\) 10.6644 0.647815 0.323907 0.946089i \(-0.395003\pi\)
0.323907 + 0.946089i \(0.395003\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.71261 + 6.71261i −0.404786 + 0.404786i
\(276\) 0 0
\(277\) 2.66170 + 2.66170i 0.159926 + 0.159926i 0.782534 0.622608i \(-0.213927\pi\)
−0.622608 + 0.782534i \(0.713927\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.4496i 0.623368i −0.950186 0.311684i \(-0.899107\pi\)
0.950186 0.311684i \(-0.100893\pi\)
\(282\) 0 0
\(283\) 12.4853 + 12.4853i 0.742173 + 0.742173i 0.972996 0.230823i \(-0.0741418\pi\)
−0.230823 + 0.972996i \(0.574142\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.413828 0.0244275
\(288\) 0 0
\(289\) −7.67794 −0.451644
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 21.7410 + 21.7410i 1.27013 + 1.27013i 0.946022 + 0.324104i \(0.105063\pi\)
0.324104 + 0.946022i \(0.394937\pi\)
\(294\) 0 0
\(295\) 10.1706i 0.592158i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11.9137 11.9137i −0.688990 0.688990i
\(300\) 0 0
\(301\) −0.914679 + 0.914679i −0.0527212 + 0.0527212i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −9.33026 −0.534249
\(306\) 0 0
\(307\) −15.0601 + 15.0601i −0.859523 + 0.859523i −0.991282 0.131759i \(-0.957938\pi\)
0.131759 + 0.991282i \(0.457938\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.77883i 0.100868i 0.998727 + 0.0504342i \(0.0160605\pi\)
−0.998727 + 0.0504342i \(0.983939\pi\)
\(312\) 0 0
\(313\) 2.70320i 0.152794i −0.997077 0.0763971i \(-0.975658\pi\)
0.997077 0.0763971i \(-0.0243417\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.6025 15.6025i 0.876325 0.876325i −0.116828 0.993152i \(-0.537272\pi\)
0.993152 + 0.116828i \(0.0372725\pi\)
\(318\) 0 0
\(319\) −15.9495 −0.892999
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.59169 + 6.59169i −0.366771 + 0.366771i
\(324\) 0 0
\(325\) −7.44471 7.44471i −0.412958 0.412958i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.449555i 0.0247848i
\(330\) 0 0
\(331\) −15.4454 15.4454i −0.848955 0.848955i 0.141048 0.990003i \(-0.454953\pi\)
−0.990003 + 0.141048i \(0.954953\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.95133 −0.106613
\(336\) 0 0
\(337\) −18.8738 −1.02812 −0.514062 0.857753i \(-0.671860\pi\)
−0.514062 + 0.857753i \(0.671860\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −15.7954 15.7954i −0.855368 0.855368i
\(342\) 0 0
\(343\) 2.22117i 0.119932i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.8337 + 19.8337i 1.06473 + 1.06473i 0.997755 + 0.0669717i \(0.0213337\pi\)
0.0669717 + 0.997755i \(0.478666\pi\)
\(348\) 0 0
\(349\) −11.9718 + 11.9718i −0.640836 + 0.640836i −0.950761 0.309925i \(-0.899696\pi\)
0.309925 + 0.950761i \(0.399696\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.6202 0.671705 0.335853 0.941915i \(-0.390976\pi\)
0.335853 + 0.941915i \(0.390976\pi\)
\(354\) 0 0
\(355\) 0.404135 0.404135i 0.0214492 0.0214492i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.0867i 1.42958i 0.699339 + 0.714790i \(0.253478\pi\)
−0.699339 + 0.714790i \(0.746522\pi\)
\(360\) 0 0
\(361\) 9.67794i 0.509365i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.70227 1.70227i 0.0891011 0.0891011i
\(366\) 0 0
\(367\) −20.4937 −1.06976 −0.534882 0.844927i \(-0.679644\pi\)
−0.534882 + 0.844927i \(0.679644\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.565371 0.565371i 0.0293526 0.0293526i
\(372\) 0 0
\(373\) −1.03372 1.03372i −0.0535239 0.0535239i 0.679838 0.733362i \(-0.262050\pi\)
−0.733362 + 0.679838i \(0.762050\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 17.6890i 0.911028i
\(378\) 0 0
\(379\) −17.6686 17.6686i −0.907573 0.907573i 0.0885032 0.996076i \(-0.471792\pi\)
−0.996076 + 0.0885032i \(0.971792\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 31.0958 1.58892 0.794460 0.607316i \(-0.207754\pi\)
0.794460 + 0.607316i \(0.207754\pi\)
\(384\) 0 0
\(385\) −1.53488 −0.0782245
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.56127 2.56127i −0.129862 0.129862i 0.639188 0.769050i \(-0.279270\pi\)
−0.769050 + 0.639188i \(0.779270\pi\)
\(390\) 0 0
\(391\) 8.63577i 0.436729i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.3240 12.3240i −0.620090 0.620090i
\(396\) 0 0
\(397\) 5.09795 5.09795i 0.255859 0.255859i −0.567509 0.823367i \(-0.692093\pi\)
0.823367 + 0.567509i \(0.192093\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.2660 0.762349 0.381174 0.924503i \(-0.375520\pi\)
0.381174 + 0.924503i \(0.375520\pi\)
\(402\) 0 0
\(403\) 17.5181 17.5181i 0.872637 0.872637i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 45.4792i 2.25432i
\(408\) 0 0
\(409\) 11.3779i 0.562603i −0.959619 0.281302i \(-0.909234\pi\)
0.959619 0.281302i \(-0.0907661\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.635767 0.635767i 0.0312840 0.0312840i
\(414\) 0 0
\(415\) 0.294481 0.0144555
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 23.3075 23.3075i 1.13865 1.13865i 0.149955 0.988693i \(-0.452087\pi\)
0.988693 0.149955i \(-0.0479130\pi\)
\(420\) 0 0
\(421\) 17.6154 + 17.6154i 0.858520 + 0.858520i 0.991164 0.132644i \(-0.0423467\pi\)
−0.132644 + 0.991164i \(0.542347\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.39635i 0.261761i
\(426\) 0 0
\(427\) 0.583234 + 0.583234i 0.0282247 + 0.0282247i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10.3211 −0.497151 −0.248576 0.968612i \(-0.579962\pi\)
−0.248576 + 0.968612i \(0.579962\pi\)
\(432\) 0 0
\(433\) −15.3137 −0.735930 −0.367965 0.929840i \(-0.619945\pi\)
−0.367965 + 0.929840i \(0.619945\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.10641 6.10641i −0.292109 0.292109i
\(438\) 0 0
\(439\) 22.5735i 1.07738i −0.842505 0.538688i \(-0.818920\pi\)
0.842505 0.538688i \(-0.181080\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 23.7117 + 23.7117i 1.12658 + 1.12658i 0.990730 + 0.135846i \(0.0433752\pi\)
0.135846 + 0.990730i \(0.456625\pi\)
\(444\) 0 0
\(445\) 18.3059 18.3059i 0.867784 0.867784i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.75506 0.0828266 0.0414133 0.999142i \(-0.486814\pi\)
0.0414133 + 0.999142i \(0.486814\pi\)
\(450\) 0 0
\(451\) 9.88849 9.88849i 0.465631 0.465631i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.70227i 0.0798039i
\(456\) 0 0
\(457\) 26.7422i 1.25095i 0.780246 + 0.625473i \(0.215094\pi\)
−0.780246 + 0.625473i \(0.784906\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.23921 9.23921i 0.430313 0.430313i −0.458422 0.888735i \(-0.651585\pi\)
0.888735 + 0.458422i \(0.151585\pi\)
\(462\) 0 0
\(463\) 29.4474 1.36854 0.684268 0.729231i \(-0.260122\pi\)
0.684268 + 0.729231i \(0.260122\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19.5897 + 19.5897i −0.906503 + 0.906503i −0.995988 0.0894848i \(-0.971478\pi\)
0.0894848 + 0.995988i \(0.471478\pi\)
\(468\) 0 0
\(469\) 0.121978 + 0.121978i 0.00563242 + 0.00563242i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 43.7127i 2.00991i
\(474\) 0 0
\(475\) −3.81580 3.81580i −0.175081 0.175081i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −35.5499 −1.62432 −0.812159 0.583436i \(-0.801708\pi\)
−0.812159 + 0.583436i \(0.801708\pi\)
\(480\) 0 0
\(481\) 50.4393 2.29984
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.726607 0.726607i −0.0329935 0.0329935i
\(486\) 0 0
\(487\) 9.86632i 0.447086i 0.974694 + 0.223543i \(0.0717623\pi\)
−0.974694 + 0.223543i \(0.928238\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.449555 0.449555i −0.0202881 0.0202881i 0.696890 0.717178i \(-0.254567\pi\)
−0.717178 + 0.696890i \(0.754567\pi\)
\(492\) 0 0
\(493\) −6.41099 + 6.41099i −0.288736 + 0.288736i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.0505249 −0.00226635
\(498\) 0 0
\(499\) −2.70645 + 2.70645i −0.121157 + 0.121157i −0.765086 0.643928i \(-0.777303\pi\)
0.643928 + 0.765086i \(0.277303\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 23.6719i 1.05548i 0.849407 + 0.527739i \(0.176960\pi\)
−0.849407 + 0.527739i \(0.823040\pi\)
\(504\) 0 0
\(505\) 18.1876i 0.809336i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −24.6052 + 24.6052i −1.09061 + 1.09061i −0.0951425 + 0.995464i \(0.530331\pi\)
−0.995464 + 0.0951425i \(0.969669\pi\)
\(510\) 0 0
\(511\) −0.212818 −0.00941453
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 14.4305 14.4305i 0.635882 0.635882i
\(516\) 0 0
\(517\) −10.7422 10.7422i −0.472440 0.472440i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.4889i 0.634770i −0.948297 0.317385i \(-0.897195\pi\)
0.948297 0.317385i \(-0.102805\pi\)
\(522\) 0 0
\(523\) 19.4979 + 19.4979i 0.852584 + 0.852584i 0.990451 0.137867i \(-0.0440245\pi\)
−0.137867 + 0.990451i \(0.544025\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.6981 −0.553138
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.9670 + 10.9670i 0.475031 + 0.475031i
\(534\) 0 0
\(535\) 1.83585i 0.0793706i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −26.4896 26.4896i −1.14099 1.14099i
\(540\) 0 0
\(541\) 10.0396 10.0396i 0.431638 0.431638i −0.457547 0.889185i \(-0.651272\pi\)
0.889185 + 0.457547i \(0.151272\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.67340 0.157351
\(546\) 0 0
\(547\) 7.19884 7.19884i 0.307800 0.307800i −0.536255 0.844056i \(-0.680162\pi\)
0.844056 + 0.536255i \(0.180162\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 9.06651i 0.386246i
\(552\) 0 0
\(553\) 1.54075i 0.0655194i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.02129 1.02129i 0.0432735 0.0432735i −0.685139 0.728412i \(-0.740259\pi\)
0.728412 + 0.685139i \(0.240259\pi\)
\(558\) 0 0
\(559\) −48.4802 −2.05049
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.70751 6.70751i 0.282688 0.282688i −0.551492 0.834180i \(-0.685941\pi\)
0.834180 + 0.551492i \(0.185941\pi\)
\(564\) 0 0
\(565\) 4.49400 + 4.49400i 0.189064 + 0.189064i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.98711i 0.376759i −0.982096 0.188380i \(-0.939676\pi\)
0.982096 0.188380i \(-0.0603235\pi\)
\(570\) 0 0
\(571\) 9.17157 + 9.17157i 0.383818 + 0.383818i 0.872476 0.488657i \(-0.162513\pi\)
−0.488657 + 0.872476i \(0.662513\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.99907 −0.208476
\(576\) 0 0
\(577\) 29.5013 1.22815 0.614077 0.789246i \(-0.289528\pi\)
0.614077 + 0.789246i \(0.289528\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −0.0184080 0.0184080i −0.000763692 0.000763692i
\(582\) 0 0
\(583\) 27.0192i 1.11902i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.82425 + 1.82425i 0.0752950 + 0.0752950i 0.743751 0.668456i \(-0.233045\pi\)
−0.668456 + 0.743751i \(0.733045\pi\)
\(588\) 0 0
\(589\) 8.97891 8.97891i 0.369970 0.369970i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 35.4338 1.45509 0.727546 0.686058i \(-0.240661\pi\)
0.727546 + 0.686058i \(0.240661\pi\)
\(594\) 0 0
\(595\) −0.616953 + 0.616953i −0.0252926 + 0.0252926i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 27.1632i 1.10986i −0.831897 0.554930i \(-0.812745\pi\)
0.831897 0.554930i \(-0.187255\pi\)
\(600\) 0 0
\(601\) 5.33897i 0.217781i −0.994054 0.108891i \(-0.965270\pi\)
0.994054 0.108891i \(-0.0347298\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −22.6914 + 22.6914i −0.922539 + 0.922539i
\(606\) 0 0
\(607\) 16.1084 0.653820 0.326910 0.945055i \(-0.393993\pi\)
0.326910 + 0.945055i \(0.393993\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 11.9137 11.9137i 0.481979 0.481979i
\(612\) 0 0
\(613\) −0.436924 0.436924i −0.0176472 0.0176472i 0.698228 0.715875i \(-0.253972\pi\)
−0.715875 + 0.698228i \(0.753972\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.80641i 0.354533i 0.984163 + 0.177266i \(0.0567254\pi\)
−0.984163 + 0.177266i \(0.943275\pi\)
\(618\) 0 0
\(619\) −1.92932 1.92932i −0.0775458 0.0775458i 0.667270 0.744816i \(-0.267463\pi\)
−0.744816 + 0.667270i \(0.767463\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.28861 −0.0916910
\(624\) 0 0
\(625\) 13.0390 0.521559
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −18.2807 18.2807i −0.728898 0.728898i
\(630\) 0 0
\(631\) 38.7864i 1.54406i −0.635586 0.772030i \(-0.719241\pi\)
0.635586 0.772030i \(-0.280759\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.90434 1.90434i −0.0755715 0.0755715i
\(636\) 0 0
\(637\) 29.3786 29.3786i 1.16402 1.16402i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −33.1091 −1.30773 −0.653865 0.756611i \(-0.726854\pi\)
−0.653865 + 0.756611i \(0.726854\pi\)
\(642\) 0 0
\(643\) 19.2897 19.2897i 0.760711 0.760711i −0.215740 0.976451i \(-0.569216\pi\)
0.976451 + 0.215740i \(0.0692164\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 41.8477i 1.64520i −0.568620 0.822601i \(-0.692522\pi\)
0.568620 0.822601i \(-0.307478\pi\)
\(648\) 0 0
\(649\) 30.3835i 1.19266i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 14.7741 14.7741i 0.578155 0.578155i −0.356240 0.934395i \(-0.615941\pi\)
0.934395 + 0.356240i \(0.115941\pi\)
\(654\) 0 0
\(655\) 26.5054 1.03565
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −2.22839 + 2.22839i −0.0868056 + 0.0868056i −0.749176 0.662371i \(-0.769550\pi\)
0.662371 + 0.749176i \(0.269550\pi\)
\(660\) 0 0
\(661\) 18.0685 + 18.0685i 0.702784 + 0.702784i 0.965007 0.262223i \(-0.0844557\pi\)
−0.262223 + 0.965007i \(0.584456\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.872503i 0.0338342i
\(666\) 0 0
\(667\) −5.93901 5.93901i −0.229959 0.229959i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 27.8729 1.07602
\(672\) 0 0
\(673\) 20.7981 0.801706 0.400853 0.916142i \(-0.368714\pi\)
0.400853 + 0.916142i \(0.368714\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 29.0213 + 29.0213i 1.11538 + 1.11538i 0.992411 + 0.122968i \(0.0392413\pi\)
0.122968 + 0.992411i \(0.460759\pi\)
\(678\) 0 0
\(679\) 0.0908404i 0.00348613i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −18.7938 18.7938i −0.719123 0.719123i 0.249303 0.968426i \(-0.419799\pi\)
−0.968426 + 0.249303i \(0.919799\pi\)
\(684\) 0 0
\(685\) −17.5385 + 17.5385i −0.670111 + 0.670111i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 29.9660 1.14161
\(690\) 0 0
\(691\) −10.4580 + 10.4580i −0.397841 + 0.397841i −0.877471 0.479630i \(-0.840771\pi\)
0.479630 + 0.877471i \(0.340771\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.16415i 0.233820i
\(696\) 0 0
\(697\) 7.94948i 0.301108i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 18.3314 18.3314i 0.692367 0.692367i −0.270385 0.962752i \(-0.587151\pi\)
0.962752 + 0.270385i \(0.0871511\pi\)
\(702\) 0 0
\(703\) 25.8528 0.975055
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.13690 + 1.13690i −0.0427577 + 0.0427577i
\(708\) 0 0
\(709\) −14.5722 14.5722i −0.547271 0.547271i 0.378380 0.925650i \(-0.376481\pi\)
−0.925650 + 0.378380i \(0.876481\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 11.7633i 0.440538i
\(714\) 0 0
\(715\) −40.6761 40.6761i −1.52120 1.52120i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −44.0949 −1.64446 −0.822230 0.569155i \(-0.807270\pi\)
−0.822230 + 0.569155i \(0.807270\pi\)
\(720\) 0 0
\(721\) −1.80409 −0.0671880
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.71119 3.71119i −0.137830 0.137830i
\(726\) 0 0
\(727\) 9.23457i 0.342491i 0.985228 + 0.171246i \(0.0547792\pi\)
−0.985228 + 0.171246i \(0.945221\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 17.5706 + 17.5706i 0.649872 + 0.649872i
\(732\) 0 0
\(733\) −18.2764 + 18.2764i −0.675053 + 0.675053i −0.958877 0.283823i \(-0.908397\pi\)
0.283823 + 0.958877i \(0.408397\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.82936 0.214727
\(738\) 0 0
\(739\) −16.9991 + 16.9991i −0.625321 + 0.625321i −0.946887 0.321566i \(-0.895791\pi\)
0.321566 + 0.946887i \(0.395791\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 17.8748i 0.655762i 0.944719 + 0.327881i \(0.106335\pi\)
−0.944719 + 0.327881i \(0.893665\pi\)
\(744\) 0 0
\(745\) 7.44538i 0.272778i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −0.114759 + 0.114759i −0.00419319 + 0.00419319i
\(750\) 0 0
\(751\) −35.0731 −1.27984 −0.639918 0.768443i \(-0.721032\pi\)
−0.639918 + 0.768443i \(0.721032\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −28.8139 + 28.8139i −1.04865 + 1.04865i
\(756\) 0 0
\(757\) 32.8071 + 32.8071i 1.19239 + 1.19239i 0.976393 + 0.216000i \(0.0693012\pi\)
0.216000 + 0.976393i \(0.430699\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 10.5531i 0.382550i 0.981536 + 0.191275i \(0.0612623\pi\)
−0.981536 + 0.191275i \(0.938738\pi\)
\(762\) 0 0
\(763\) −0.229624 0.229624i −0.00831296 0.00831296i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 33.6972 1.21673
\(768\) 0 0
\(769\) −35.2068 −1.26959 −0.634795 0.772681i \(-0.718915\pi\)
−0.634795 + 0.772681i \(0.718915\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −19.3897 19.3897i −0.697399 0.697399i 0.266450 0.963849i \(-0.414149\pi\)
−0.963849 + 0.266450i \(0.914149\pi\)
\(774\) 0 0
\(775\) 7.35067i 0.264044i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.62113 + 5.62113i 0.201398 + 0.201398i
\(780\) 0 0
\(781\) −1.20730 + 1.20730i −0.0432006 + 0.0432006i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.06877 0.252295
\(786\) 0 0
\(787\) 6.68964 6.68964i 0.238460 0.238460i −0.577752 0.816212i \(-0.696070\pi\)
0.816212 + 0.577752i \(0.196070\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0.561839i 0.0199767i
\(792\) 0 0
\(793\) 30.9128i 1.09775i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 13.5617 13.5617i 0.480380 0.480380i −0.424873 0.905253i \(-0.639681\pi\)
0.905253 + 0.424873i \(0.139681\pi\)
\(798\) 0 0
\(799\) −8.63577 −0.305511
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −5.08532 + 5.08532i −0.179457 + 0.179457i
\(804\) 0 0
\(805\) −0.571533 0.571533i −0.0201439