Properties

Label 1152.3.m.f.991.5
Level $1152$
Weight $3$
Character 1152.991
Analytic conductor $31.390$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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 991.5
Root \(1.78012 + 0.911682i\) of defining polynomial
Character \(\chi\) \(=\) 1152.991
Dual form 1152.3.m.f.415.5

$q$-expansion

\(f(q)\) \(=\) \(q+(1.00772 - 1.00772i) q^{5} +10.0236 q^{7} +O(q^{10})\) \(q+(1.00772 - 1.00772i) q^{5} +10.0236 q^{7} +(2.26517 + 2.26517i) q^{11} +(6.88229 + 6.88229i) q^{13} +22.3801 q^{17} +(16.8918 - 16.8918i) q^{19} -33.2007 q^{23} +22.9690i q^{25} +(-24.6412 - 24.6412i) q^{29} +41.3761i q^{31} +(10.1010 - 10.1010i) q^{35} +(6.60031 - 6.60031i) q^{37} -47.1477i q^{41} +(48.8218 + 48.8218i) q^{43} +45.6048i q^{47} +51.4717 q^{49} +(25.1401 - 25.1401i) q^{53} +4.56532 q^{55} +(6.23974 + 6.23974i) q^{59} +(-35.9513 - 35.9513i) q^{61} +13.8709 q^{65} +(-10.2045 + 10.2045i) q^{67} -11.9529 q^{71} -111.332i q^{73} +(22.7051 + 22.7051i) q^{77} -4.46031i q^{79} +(10.1751 - 10.1751i) q^{83} +(22.5530 - 22.5530i) q^{85} -21.9364i q^{89} +(68.9850 + 68.9850i) q^{91} -34.0444i q^{95} +107.309 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 32q^{11} + 32q^{19} + 128q^{23} + 32q^{29} + 96q^{35} + 96q^{37} - 160q^{43} + 112q^{49} - 160q^{53} - 256q^{55} - 128q^{59} + 32q^{61} + 32q^{65} - 320q^{67} - 512q^{71} + 224q^{77} - 160q^{83} - 160q^{85} + 480q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(5\) 1.00772 1.00772i 0.201544 0.201544i −0.599117 0.800661i \(-0.704482\pi\)
0.800661 + 0.599117i \(0.204482\pi\)
\(6\) 0 0
\(7\) 10.0236 1.43194 0.715969 0.698133i \(-0.245985\pi\)
0.715969 + 0.698133i \(0.245985\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.26517 + 2.26517i 0.205925 + 0.205925i 0.802533 0.596608i \(-0.203485\pi\)
−0.596608 + 0.802533i \(0.703485\pi\)
\(12\) 0 0
\(13\) 6.88229 + 6.88229i 0.529407 + 0.529407i 0.920395 0.390989i \(-0.127867\pi\)
−0.390989 + 0.920395i \(0.627867\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 22.3801 1.31648 0.658240 0.752809i \(-0.271301\pi\)
0.658240 + 0.752809i \(0.271301\pi\)
\(18\) 0 0
\(19\) 16.8918 16.8918i 0.889041 0.889041i −0.105390 0.994431i \(-0.533609\pi\)
0.994431 + 0.105390i \(0.0336092\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −33.2007 −1.44351 −0.721755 0.692149i \(-0.756664\pi\)
−0.721755 + 0.692149i \(0.756664\pi\)
\(24\) 0 0
\(25\) 22.9690i 0.918760i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −24.6412 24.6412i −0.849696 0.849696i 0.140399 0.990095i \(-0.455161\pi\)
−0.990095 + 0.140399i \(0.955161\pi\)
\(30\) 0 0
\(31\) 41.3761i 1.33471i 0.744738 + 0.667357i \(0.232574\pi\)
−0.744738 + 0.667357i \(0.767426\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.1010 10.1010i 0.288599 0.288599i
\(36\) 0 0
\(37\) 6.60031 6.60031i 0.178387 0.178387i −0.612266 0.790652i \(-0.709742\pi\)
0.790652 + 0.612266i \(0.209742\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 47.1477i 1.14994i −0.818173 0.574972i \(-0.805013\pi\)
0.818173 0.574972i \(-0.194987\pi\)
\(42\) 0 0
\(43\) 48.8218 + 48.8218i 1.13539 + 1.13539i 0.989266 + 0.146124i \(0.0466799\pi\)
0.146124 + 0.989266i \(0.453320\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 45.6048i 0.970315i 0.874427 + 0.485157i \(0.161238\pi\)
−0.874427 + 0.485157i \(0.838762\pi\)
\(48\) 0 0
\(49\) 51.4717 1.05044
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 25.1401 25.1401i 0.474341 0.474341i −0.428975 0.903316i \(-0.641125\pi\)
0.903316 + 0.428975i \(0.141125\pi\)
\(54\) 0 0
\(55\) 4.56532 0.0830059
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.23974 + 6.23974i 0.105758 + 0.105758i 0.758006 0.652248i \(-0.226174\pi\)
−0.652248 + 0.758006i \(0.726174\pi\)
\(60\) 0 0
\(61\) −35.9513 35.9513i −0.589366 0.589366i 0.348093 0.937460i \(-0.386829\pi\)
−0.937460 + 0.348093i \(0.886829\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.8709 0.213398
\(66\) 0 0
\(67\) −10.2045 + 10.2045i −0.152307 + 0.152307i −0.779147 0.626841i \(-0.784348\pi\)
0.626841 + 0.779147i \(0.284348\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.9529 −0.168350 −0.0841752 0.996451i \(-0.526826\pi\)
−0.0841752 + 0.996451i \(0.526826\pi\)
\(72\) 0 0
\(73\) 111.332i 1.52510i −0.646929 0.762550i \(-0.723947\pi\)
0.646929 0.762550i \(-0.276053\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 22.7051 + 22.7051i 0.294871 + 0.294871i
\(78\) 0 0
\(79\) 4.46031i 0.0564596i −0.999601 0.0282298i \(-0.991013\pi\)
0.999601 0.0282298i \(-0.00898702\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.1751 10.1751i 0.122592 0.122592i −0.643149 0.765741i \(-0.722373\pi\)
0.765741 + 0.643149i \(0.222373\pi\)
\(84\) 0 0
\(85\) 22.5530 22.5530i 0.265329 0.265329i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 21.9364i 0.246476i −0.992377 0.123238i \(-0.960672\pi\)
0.992377 0.123238i \(-0.0393279\pi\)
\(90\) 0 0
\(91\) 68.9850 + 68.9850i 0.758077 + 0.758077i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 34.0444i 0.358362i
\(96\) 0 0
\(97\) 107.309 1.10628 0.553140 0.833088i \(-0.313429\pi\)
0.553140 + 0.833088i \(0.313429\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −100.780 + 100.780i −0.997824 + 0.997824i −0.999998 0.00217389i \(-0.999308\pi\)
0.00217389 + 0.999998i \(0.499308\pi\)
\(102\) 0 0
\(103\) 58.0562 0.563653 0.281826 0.959465i \(-0.409060\pi\)
0.281826 + 0.959465i \(0.409060\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 112.747 + 112.747i 1.05371 + 1.05371i 0.998473 + 0.0552381i \(0.0175918\pi\)
0.0552381 + 0.998473i \(0.482408\pi\)
\(108\) 0 0
\(109\) 81.1384 + 81.1384i 0.744389 + 0.744389i 0.973419 0.229030i \(-0.0735554\pi\)
−0.229030 + 0.973419i \(0.573555\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 171.844 1.52074 0.760371 0.649489i \(-0.225017\pi\)
0.760371 + 0.649489i \(0.225017\pi\)
\(114\) 0 0
\(115\) −33.4571 + 33.4571i −0.290931 + 0.290931i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 224.329 1.88512
\(120\) 0 0
\(121\) 110.738i 0.915190i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 48.3394 + 48.3394i 0.386715 + 0.386715i
\(126\) 0 0
\(127\) 36.8333i 0.290026i 0.989430 + 0.145013i \(0.0463224\pi\)
−0.989430 + 0.145013i \(0.953678\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.3686 + 12.3686i −0.0944170 + 0.0944170i −0.752738 0.658321i \(-0.771267\pi\)
0.658321 + 0.752738i \(0.271267\pi\)
\(132\) 0 0
\(133\) 169.316 169.316i 1.27305 1.27305i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 145.679i 1.06335i 0.846949 + 0.531674i \(0.178437\pi\)
−0.846949 + 0.531674i \(0.821563\pi\)
\(138\) 0 0
\(139\) −82.5709 82.5709i −0.594035 0.594035i 0.344684 0.938719i \(-0.387986\pi\)
−0.938719 + 0.344684i \(0.887986\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 31.1791i 0.218036i
\(144\) 0 0
\(145\) −49.6629 −0.342503
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 196.248 196.248i 1.31710 1.31710i 0.401043 0.916059i \(-0.368648\pi\)
0.916059 0.401043i \(-0.131352\pi\)
\(150\) 0 0
\(151\) 64.5007 0.427157 0.213578 0.976926i \(-0.431488\pi\)
0.213578 + 0.976926i \(0.431488\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 41.6956 + 41.6956i 0.269004 + 0.269004i
\(156\) 0 0
\(157\) −54.4202 54.4202i −0.346625 0.346625i 0.512226 0.858851i \(-0.328821\pi\)
−0.858851 + 0.512226i \(0.828821\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −332.789 −2.06701
\(162\) 0 0
\(163\) −104.803 + 104.803i −0.642961 + 0.642961i −0.951282 0.308321i \(-0.900233\pi\)
0.308321 + 0.951282i \(0.400233\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −53.3110 −0.319228 −0.159614 0.987180i \(-0.551025\pi\)
−0.159614 + 0.987180i \(0.551025\pi\)
\(168\) 0 0
\(169\) 74.2683i 0.439457i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −41.5780 41.5780i −0.240335 0.240335i 0.576654 0.816989i \(-0.304358\pi\)
−0.816989 + 0.576654i \(0.804358\pi\)
\(174\) 0 0
\(175\) 230.231i 1.31561i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 53.0709 53.0709i 0.296486 0.296486i −0.543150 0.839636i \(-0.682769\pi\)
0.839636 + 0.543150i \(0.182769\pi\)
\(180\) 0 0
\(181\) 66.6042 66.6042i 0.367979 0.367979i −0.498761 0.866740i \(-0.666211\pi\)
0.866740 + 0.498761i \(0.166211\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 13.3025i 0.0719056i
\(186\) 0 0
\(187\) 50.6949 + 50.6949i 0.271096 + 0.271096i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 113.753i 0.595567i 0.954633 + 0.297784i \(0.0962474\pi\)
−0.954633 + 0.297784i \(0.903753\pi\)
\(192\) 0 0
\(193\) −26.5596 −0.137615 −0.0688073 0.997630i \(-0.521919\pi\)
−0.0688073 + 0.997630i \(0.521919\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 51.8935 51.8935i 0.263419 0.263419i −0.563023 0.826442i \(-0.690362\pi\)
0.826442 + 0.563023i \(0.190362\pi\)
\(198\) 0 0
\(199\) −136.741 −0.687140 −0.343570 0.939127i \(-0.611636\pi\)
−0.343570 + 0.939127i \(0.611636\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −246.992 246.992i −1.21671 1.21671i
\(204\) 0 0
\(205\) −47.5118 47.5118i −0.231765 0.231765i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 76.5255 0.366151
\(210\) 0 0
\(211\) 141.171 141.171i 0.669057 0.669057i −0.288441 0.957498i \(-0.593137\pi\)
0.957498 + 0.288441i \(0.0931368\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 98.3975 0.457663
\(216\) 0 0
\(217\) 414.736i 1.91123i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 154.027 + 154.027i 0.696953 + 0.696953i
\(222\) 0 0
\(223\) 122.607i 0.549806i −0.961472 0.274903i \(-0.911354\pi\)
0.961472 0.274903i \(-0.0886457\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −295.844 + 295.844i −1.30328 + 1.30328i −0.377112 + 0.926168i \(0.623083\pi\)
−0.926168 + 0.377112i \(0.876917\pi\)
\(228\) 0 0
\(229\) −73.3817 + 73.3817i −0.320444 + 0.320444i −0.848937 0.528493i \(-0.822757\pi\)
0.528493 + 0.848937i \(0.322757\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 156.229i 0.670509i −0.942128 0.335255i \(-0.891178\pi\)
0.942128 0.335255i \(-0.108822\pi\)
\(234\) 0 0
\(235\) 45.9569 + 45.9569i 0.195561 + 0.195561i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 13.1716i 0.0551113i 0.999620 + 0.0275557i \(0.00877235\pi\)
−0.999620 + 0.0275557i \(0.991228\pi\)
\(240\) 0 0
\(241\) −189.519 −0.786386 −0.393193 0.919456i \(-0.628630\pi\)
−0.393193 + 0.919456i \(0.628630\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 51.8692 51.8692i 0.211711 0.211711i
\(246\) 0 0
\(247\) 232.508 0.941328
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −27.4434 27.4434i −0.109336 0.109336i 0.650322 0.759658i \(-0.274634\pi\)
−0.759658 + 0.650322i \(0.774634\pi\)
\(252\) 0 0
\(253\) −75.2053 75.2053i −0.297254 0.297254i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −135.375 −0.526752 −0.263376 0.964693i \(-0.584836\pi\)
−0.263376 + 0.964693i \(0.584836\pi\)
\(258\) 0 0
\(259\) 66.1586 66.1586i 0.255438 0.255438i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −31.6123 −0.120199 −0.0600994 0.998192i \(-0.519142\pi\)
−0.0600994 + 0.998192i \(0.519142\pi\)
\(264\) 0 0
\(265\) 50.6684i 0.191201i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −194.213 194.213i −0.721981 0.721981i 0.247028 0.969008i \(-0.420546\pi\)
−0.969008 + 0.247028i \(0.920546\pi\)
\(270\) 0 0
\(271\) 291.647i 1.07619i −0.842884 0.538095i \(-0.819144\pi\)
0.842884 0.538095i \(-0.180856\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −52.0287 + 52.0287i −0.189195 + 0.189195i
\(276\) 0 0
\(277\) −305.166 + 305.166i −1.10168 + 1.10168i −0.107475 + 0.994208i \(0.534277\pi\)
−0.994208 + 0.107475i \(0.965723\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 211.861i 0.753955i −0.926222 0.376978i \(-0.876963\pi\)
0.926222 0.376978i \(-0.123037\pi\)
\(282\) 0 0
\(283\) −105.325 105.325i −0.372175 0.372175i 0.496094 0.868269i \(-0.334767\pi\)
−0.868269 + 0.496094i \(0.834767\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 472.588i 1.64665i
\(288\) 0 0
\(289\) 211.871 0.733117
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −171.289 + 171.289i −0.584603 + 0.584603i −0.936165 0.351562i \(-0.885651\pi\)
0.351562 + 0.936165i \(0.385651\pi\)
\(294\) 0 0
\(295\) 12.5758 0.0426300
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −228.497 228.497i −0.764204 0.764204i
\(300\) 0 0
\(301\) 489.368 + 489.368i 1.62581 + 1.62581i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −72.4579 −0.237567
\(306\) 0 0
\(307\) 27.1124 27.1124i 0.0883140 0.0883140i −0.661570 0.749884i \(-0.730109\pi\)
0.749884 + 0.661570i \(0.230109\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −371.124 −1.19333 −0.596663 0.802492i \(-0.703507\pi\)
−0.596663 + 0.802492i \(0.703507\pi\)
\(312\) 0 0
\(313\) 374.501i 1.19649i 0.801313 + 0.598245i \(0.204135\pi\)
−0.801313 + 0.598245i \(0.795865\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −48.5840 48.5840i −0.153262 0.153262i 0.626311 0.779573i \(-0.284564\pi\)
−0.779573 + 0.626311i \(0.784564\pi\)
\(318\) 0 0
\(319\) 111.633i 0.349947i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 378.040 378.040i 1.17040 1.17040i
\(324\) 0 0
\(325\) −158.079 + 158.079i −0.486398 + 0.486398i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 457.122i 1.38943i
\(330\) 0 0
\(331\) 1.88883 + 1.88883i 0.00570644 + 0.00570644i 0.709954 0.704248i \(-0.248716\pi\)
−0.704248 + 0.709954i \(0.748716\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 20.5667i 0.0613931i
\(336\) 0 0
\(337\) −386.980 −1.14831 −0.574154 0.818747i \(-0.694669\pi\)
−0.574154 + 0.818747i \(0.694669\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −93.7240 + 93.7240i −0.274851 + 0.274851i
\(342\) 0 0
\(343\) 24.7757 0.0722325
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 441.887 + 441.887i 1.27345 + 1.27345i 0.944266 + 0.329183i \(0.106773\pi\)
0.329183 + 0.944266i \(0.393227\pi\)
\(348\) 0 0
\(349\) −119.382 119.382i −0.342068 0.342068i 0.515076 0.857144i \(-0.327764\pi\)
−0.857144 + 0.515076i \(0.827764\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 515.642 1.46074 0.730371 0.683050i \(-0.239347\pi\)
0.730371 + 0.683050i \(0.239347\pi\)
\(354\) 0 0
\(355\) −12.0452 + 12.0452i −0.0339301 + 0.0339301i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −428.264 −1.19294 −0.596468 0.802637i \(-0.703430\pi\)
−0.596468 + 0.802637i \(0.703430\pi\)
\(360\) 0 0
\(361\) 209.664i 0.580786i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −112.192 112.192i −0.307375 0.307375i
\(366\) 0 0
\(367\) 219.482i 0.598043i −0.954246 0.299021i \(-0.903340\pi\)
0.954246 0.299021i \(-0.0966602\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 251.993 251.993i 0.679226 0.679226i
\(372\) 0 0
\(373\) −425.005 + 425.005i −1.13942 + 1.13942i −0.150870 + 0.988554i \(0.548207\pi\)
−0.988554 + 0.150870i \(0.951793\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 339.175i 0.899669i
\(378\) 0 0
\(379\) −365.916 365.916i −0.965476 0.965476i 0.0339473 0.999424i \(-0.489192\pi\)
−0.999424 + 0.0339473i \(0.989192\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 213.276i 0.556857i 0.960457 + 0.278428i \(0.0898135\pi\)
−0.960457 + 0.278428i \(0.910187\pi\)
\(384\) 0 0
\(385\) 45.7608 0.118859
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −210.798 + 210.798i −0.541898 + 0.541898i −0.924085 0.382187i \(-0.875171\pi\)
0.382187 + 0.924085i \(0.375171\pi\)
\(390\) 0 0
\(391\) −743.037 −1.90035
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.49475 4.49475i −0.0113791 0.0113791i
\(396\) 0 0
\(397\) −392.907 392.907i −0.989690 0.989690i 0.0102579 0.999947i \(-0.496735\pi\)
−0.999947 + 0.0102579i \(0.996735\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −29.3290 −0.0731396 −0.0365698 0.999331i \(-0.511643\pi\)
−0.0365698 + 0.999331i \(0.511643\pi\)
\(402\) 0 0
\(403\) −284.762 + 284.762i −0.706606 + 0.706606i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 29.9017 0.0734684
\(408\) 0 0
\(409\) 601.115i 1.46972i −0.678219 0.734860i \(-0.737248\pi\)
0.678219 0.734860i \(-0.262752\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 62.5444 + 62.5444i 0.151439 + 0.151439i
\(414\) 0 0
\(415\) 20.5073i 0.0494153i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −518.885 + 518.885i −1.23839 + 1.23839i −0.277729 + 0.960659i \(0.589582\pi\)
−0.960659 + 0.277729i \(0.910418\pi\)
\(420\) 0 0
\(421\) 411.213 411.213i 0.976754 0.976754i −0.0229817 0.999736i \(-0.507316\pi\)
0.999736 + 0.0229817i \(0.00731596\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 514.049i 1.20953i
\(426\) 0 0
\(427\) −360.360 360.360i −0.843936 0.843936i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 41.1083i 0.0953789i 0.998862 + 0.0476895i \(0.0151858\pi\)
−0.998862 + 0.0476895i \(0.984814\pi\)
\(432\) 0 0
\(433\) −351.682 −0.812199 −0.406100 0.913829i \(-0.633111\pi\)
−0.406100 + 0.913829i \(0.633111\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −560.819 + 560.819i −1.28334 + 1.28334i
\(438\) 0 0
\(439\) −775.613 −1.76677 −0.883386 0.468646i \(-0.844742\pi\)
−0.883386 + 0.468646i \(0.844742\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −241.372 241.372i −0.544858 0.544858i 0.380091 0.924949i \(-0.375893\pi\)
−0.924949 + 0.380091i \(0.875893\pi\)
\(444\) 0 0
\(445\) −22.1058 22.1058i −0.0496759 0.0496759i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −266.360 −0.593228 −0.296614 0.954997i \(-0.595858\pi\)
−0.296614 + 0.954997i \(0.595858\pi\)
\(450\) 0 0
\(451\) 106.798 106.798i 0.236802 0.236802i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 139.035 0.305572
\(456\) 0 0
\(457\) 515.244i 1.12745i 0.825963 + 0.563725i \(0.190632\pi\)
−0.825963 + 0.563725i \(0.809368\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.67717 + 5.67717i 0.0123149 + 0.0123149i 0.713237 0.700923i \(-0.247228\pi\)
−0.700923 + 0.713237i \(0.747228\pi\)
\(462\) 0 0
\(463\) 464.510i 1.00326i 0.865082 + 0.501631i \(0.167267\pi\)
−0.865082 + 0.501631i \(0.832733\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 495.985 495.985i 1.06207 1.06207i 0.0641248 0.997942i \(-0.479574\pi\)
0.997942 0.0641248i \(-0.0204256\pi\)
\(468\) 0 0
\(469\) −102.286 + 102.286i −0.218094 + 0.218094i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 221.180i 0.467610i
\(474\) 0 0
\(475\) 387.987 + 387.987i 0.816815 + 0.816815i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 378.802i 0.790818i −0.918505 0.395409i \(-0.870603\pi\)
0.918505 0.395409i \(-0.129397\pi\)
\(480\) 0 0
\(481\) 90.8504 0.188878
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 108.138 108.138i 0.222964 0.222964i
\(486\) 0 0
\(487\) 147.446 0.302764 0.151382 0.988475i \(-0.451628\pi\)
0.151382 + 0.988475i \(0.451628\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 109.547 + 109.547i 0.223110 + 0.223110i 0.809807 0.586697i \(-0.199572\pi\)
−0.586697 + 0.809807i \(0.699572\pi\)
\(492\) 0 0
\(493\) −551.473 551.473i −1.11861 1.11861i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −119.810 −0.241067
\(498\) 0 0
\(499\) 360.523 360.523i 0.722491 0.722491i −0.246621 0.969112i \(-0.579320\pi\)
0.969112 + 0.246621i \(0.0793202\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 927.420 1.84378 0.921889 0.387454i \(-0.126645\pi\)
0.921889 + 0.387454i \(0.126645\pi\)
\(504\) 0 0
\(505\) 203.117i 0.402211i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 677.931 + 677.931i 1.33189 + 1.33189i 0.903680 + 0.428208i \(0.140855\pi\)
0.428208 + 0.903680i \(0.359145\pi\)
\(510\) 0 0
\(511\) 1115.95i 2.18385i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 58.5045 58.5045i 0.113601 0.113601i
\(516\) 0 0
\(517\) −103.303 + 103.303i −0.199812 + 0.199812i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 143.173i 0.274804i −0.990515 0.137402i \(-0.956125\pi\)
0.990515 0.137402i \(-0.0438753\pi\)
\(522\) 0 0
\(523\) −226.187 226.187i −0.432481 0.432481i 0.456991 0.889471i \(-0.348927\pi\)
−0.889471 + 0.456991i \(0.848927\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 926.004i 1.75712i
\(528\) 0 0
\(529\) 573.288 1.08372
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 324.484 324.484i 0.608788 0.608788i
\(534\) 0 0
\(535\) 227.235 0.424739
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 116.592 + 116.592i 0.216312 + 0.216312i
\(540\) 0 0
\(541\) 156.708 + 156.708i 0.289663 + 0.289663i 0.836947 0.547284i \(-0.184338\pi\)
−0.547284 + 0.836947i \(0.684338\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 163.530 0.300055
\(546\) 0 0
\(547\) −247.357 + 247.357i −0.452207 + 0.452207i −0.896086 0.443880i \(-0.853602\pi\)
0.443880 + 0.896086i \(0.353602\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −832.466 −1.51083
\(552\) 0 0
\(553\) 44.7082i 0.0808466i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −661.193 661.193i −1.18706 1.18706i −0.977876 0.209184i \(-0.932919\pi\)
−0.209184 0.977876i \(-0.567081\pi\)
\(558\) 0 0
\(559\) 672.011i 1.20217i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 246.685 246.685i 0.438162 0.438162i −0.453231 0.891393i \(-0.649729\pi\)
0.891393 + 0.453231i \(0.149729\pi\)
\(564\) 0 0
\(565\) 173.171 173.171i 0.306497 0.306497i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 243.567i 0.428061i −0.976827 0.214030i \(-0.931341\pi\)
0.976827 0.214030i \(-0.0686592\pi\)
\(570\) 0 0
\(571\) −59.9229 59.9229i −0.104944 0.104944i 0.652685 0.757629i \(-0.273642\pi\)
−0.757629 + 0.652685i \(0.773642\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 762.587i 1.32624i
\(576\) 0 0
\(577\) 136.609 0.236757 0.118378 0.992969i \(-0.462230\pi\)
0.118378 + 0.992969i \(0.462230\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 101.991 101.991i 0.175543 0.175543i
\(582\) 0 0
\(583\) 113.893 0.195357
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −331.817 331.817i −0.565276 0.565276i 0.365525 0.930801i \(-0.380889\pi\)
−0.930801 + 0.365525i \(0.880889\pi\)
\(588\) 0 0
\(589\) 698.916 + 698.916i 1.18661 + 1.18661i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −131.285 −0.221391 −0.110695 0.993854i \(-0.535308\pi\)
−0.110695 + 0.993854i \(0.535308\pi\)
\(594\) 0 0
\(595\) 226.061 226.061i 0.379934 0.379934i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 136.119 0.227243 0.113621 0.993524i \(-0.463755\pi\)
0.113621 + 0.993524i \(0.463755\pi\)
\(600\) 0 0
\(601\) 498.566i 0.829561i 0.909922 + 0.414780i \(0.136142\pi\)
−0.909922 + 0.414780i \(0.863858\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −111.593 111.593i −0.184451 0.184451i
\(606\) 0 0
\(607\) 568.740i 0.936969i 0.883472 + 0.468484i \(0.155200\pi\)
−0.883472 + 0.468484i \(0.844800\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −313.865 + 313.865i −0.513691 + 0.513691i
\(612\) 0 0
\(613\) 168.441 168.441i 0.274782 0.274782i −0.556240 0.831022i \(-0.687756\pi\)
0.831022 + 0.556240i \(0.187756\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 599.157i 0.971081i 0.874214 + 0.485541i \(0.161377\pi\)
−0.874214 + 0.485541i \(0.838623\pi\)
\(618\) 0 0
\(619\) −126.719 126.719i −0.204715 0.204715i 0.597301 0.802017i \(-0.296240\pi\)
−0.802017 + 0.597301i \(0.796240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 219.881i 0.352939i
\(624\) 0 0
\(625\) −476.800 −0.762879
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 147.716 147.716i 0.234842 0.234842i
\(630\) 0 0
\(631\) 668.283 1.05909 0.529543 0.848283i \(-0.322363\pi\)
0.529543 + 0.848283i \(0.322363\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 37.1177 + 37.1177i 0.0584531 + 0.0584531i
\(636\) 0 0
\(637\) 354.243 + 354.243i 0.556112 + 0.556112i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −484.574 −0.755966 −0.377983 0.925813i \(-0.623382\pi\)
−0.377983 + 0.925813i \(0.623382\pi\)
\(642\) 0 0
\(643\) −75.2980 + 75.2980i −0.117104 + 0.117104i −0.763230 0.646126i \(-0.776388\pi\)
0.646126 + 0.763230i \(0.276388\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 582.307 0.900011 0.450006 0.893026i \(-0.351422\pi\)
0.450006 + 0.893026i \(0.351422\pi\)
\(648\) 0 0
\(649\) 28.2682i 0.0435565i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 457.453 + 457.453i 0.700541 + 0.700541i 0.964527 0.263986i \(-0.0850371\pi\)
−0.263986 + 0.964527i \(0.585037\pi\)
\(654\) 0 0
\(655\) 24.9283i 0.0380584i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −430.079 + 430.079i −0.652623 + 0.652623i −0.953624 0.301001i \(-0.902679\pi\)
0.301001 + 0.953624i \(0.402679\pi\)
\(660\) 0 0
\(661\) 513.622 513.622i 0.777038 0.777038i −0.202288 0.979326i \(-0.564838\pi\)
0.979326 + 0.202288i \(0.0648376\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 341.246i 0.513152i
\(666\) 0 0
\(667\) 818.105 + 818.105i 1.22654 + 1.22654i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 162.872i 0.242730i
\(672\) 0 0
\(673\) −1112.68 −1.65332 −0.826659 0.562703i \(-0.809761\pi\)
−0.826659 + 0.562703i \(0.809761\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 633.271 633.271i 0.935408 0.935408i −0.0626291 0.998037i \(-0.519949\pi\)
0.998037 + 0.0626291i \(0.0199485\pi\)
\(678\) 0 0
\(679\) 1075.62 1.58412
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −429.651 429.651i −0.629065 0.629065i 0.318768 0.947833i \(-0.396731\pi\)
−0.947833 + 0.318768i \(0.896731\pi\)
\(684\) 0 0
\(685\) 146.803 + 146.803i 0.214312 + 0.214312i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 346.042 0.502239
\(690\) 0 0
\(691\) 151.617 151.617i 0.219417 0.219417i −0.588836 0.808253i \(-0.700414\pi\)
0.808253 + 0.588836i \(0.200414\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −166.417 −0.239449
\(696\) 0 0
\(697\) 1055.17i 1.51388i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −920.704 920.704i −1.31341 1.31341i −0.918882 0.394533i \(-0.870906\pi\)
−0.394533 0.918882i \(-0.629094\pi\)
\(702\) 0 0
\(703\) 222.982i 0.317186i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1010.18 + 1010.18i −1.42882 + 1.42882i
\(708\) 0 0
\(709\) −405.348 + 405.348i −0.571718 + 0.571718i −0.932608 0.360890i \(-0.882473\pi\)
0.360890 + 0.932608i \(0.382473\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1373.72i 1.92667i
\(714\) 0 0
\(715\) 31.4199 + 31.4199i 0.0439439 + 0.0439439i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 880.704i 1.22490i 0.790509 + 0.612450i \(0.209816\pi\)
−0.790509 + 0.612450i \(0.790184\pi\)
\(720\) 0 0
\(721\) 581.930 0.807115
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 565.983 565.983i 0.780667 0.780667i
\(726\) 0 0
\(727\) 1000.46 1.37615 0.688077 0.725637i \(-0.258455\pi\)
0.688077 + 0.725637i \(0.258455\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1092.64 + 1092.64i 1.49472 + 1.49472i
\(732\) 0 0
\(733\) −540.306 540.306i −0.737116 0.737116i 0.234903 0.972019i \(-0.424523\pi\)
−0.972019 + 0.234903i \(0.924523\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −46.2301 −0.0627274
\(738\) 0 0
\(739\) −893.726 + 893.726i −1.20937 + 1.20937i −0.238142 + 0.971230i \(0.576538\pi\)
−0.971230 + 0.238142i \(0.923462\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1295.75 −1.74394 −0.871969 0.489561i \(-0.837157\pi\)
−0.871969 + 0.489561i \(0.837157\pi\)
\(744\) 0 0
\(745\) 395.527i 0.530909i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1130.13 + 1130.13i 1.50885 + 1.50885i
\(750\) 0 0
\(751\) 229.818i 0.306016i 0.988225 + 0.153008i \(0.0488961\pi\)
−0.988225 + 0.153008i \(0.951104\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 64.9987 64.9987i 0.0860910 0.0860910i
\(756\) 0 0
\(757\) 373.678 373.678i 0.493630 0.493630i −0.415818 0.909448i \(-0.636505\pi\)
0.909448 + 0.415818i \(0.136505\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 384.012i 0.504615i 0.967647 + 0.252307i \(0.0811894\pi\)
−0.967647 + 0.252307i \(0.918811\pi\)
\(762\) 0 0
\(763\) 813.296 + 813.296i 1.06592 + 1.06592i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 85.8874i 0.111978i
\(768\) 0 0
\(769\) 865.026 1.12487 0.562436 0.826841i \(-0.309864\pi\)
0.562436 + 0.826841i \(0.309864\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.78859 + 1.78859i −0.00231383 + 0.00231383i −0.708263 0.705949i \(-0.750521\pi\)
0.705949 + 0.708263i \(0.250521\pi\)
\(774\) 0 0
\(775\) −950.368 −1.22628
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −796.409 796.409i −1.02235 1.02235i
\(780\) 0 0
\(781\) −27.0753 27.0753i −0.0346675 0.0346675i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −109.681 −0.139721
\(786\) 0 0
\(787\) −143.702 + 143.702i −0.182595 + 0.182595i −0.792485 0.609891i \(-0.791213\pi\)
0.609891 + 0.792485i \(0.291213\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1722.49 2.17761
\(792\) 0 0
\(793\) 494.855i 0.624029i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −477.929 477.929i −0.599660 0.599660i 0.340562 0.940222i \(-0.389383\pi\)
−0.940222 + 0.340562i \(0.889383\pi\)
\(798\) 0 0
\(799\) 1020.64i 1.27740i
\(800\) 0 0