Properties

Label 1152.3.m.f.415.5
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 415.5
Root \(1.78012 - 0.911682i\) of defining polynomial
Character \(\chi\) \(=\) 1152.415
Dual form 1152.3.m.f.991.5

$q$-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 \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

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

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
\(5\) 1.00772 + 1.00772i 0.201544 + 0.201544i 0.800661 0.599117i \(-0.204482\pi\)
−0.599117 + 0.800661i \(0.704482\pi\)
\(6\) 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 −0.596608 0.802533i \(-0.703485\pi\)
0.802533 + 0.596608i \(0.203485\pi\)
\(12\) 0 0
\(13\) 6.88229 6.88229i 0.529407 0.529407i −0.390989 0.920395i \(-0.627867\pi\)
0.920395 + 0.390989i \(0.127867\pi\)
\(14\) 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 0.994431 0.105390i \(-0.0336092\pi\)
−0.105390 + 0.994431i \(0.533609\pi\)
\(20\) 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 −0.990095 0.140399i \(-0.955161\pi\)
0.140399 + 0.990095i \(0.455161\pi\)
\(30\) 0 0
\(31\) 41.3761i 1.33471i −0.744738 0.667357i \(-0.767426\pi\)
0.744738 0.667357i \(-0.232574\pi\)
\(32\) 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 0.790652 0.612266i \(-0.209742\pi\)
−0.612266 + 0.790652i \(0.709742\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 47.1477i 1.14994i 0.818173 + 0.574972i \(0.194987\pi\)
−0.818173 + 0.574972i \(0.805013\pi\)
\(42\) 0 0
\(43\) 48.8218 48.8218i 1.13539 1.13539i 0.146124 0.989266i \(-0.453320\pi\)
0.989266 0.146124i \(-0.0466799\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 45.6048i 0.970315i −0.874427 0.485157i \(-0.838762\pi\)
0.874427 0.485157i \(-0.161238\pi\)
\(48\) 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 0.903316 0.428975i \(-0.141125\pi\)
−0.428975 + 0.903316i \(0.641125\pi\)
\(54\) 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 −0.652248 0.758006i \(-0.726174\pi\)
0.758006 + 0.652248i \(0.226174\pi\)
\(60\) 0 0
\(61\) −35.9513 + 35.9513i −0.589366 + 0.589366i −0.937460 0.348093i \(-0.886829\pi\)
0.348093 + 0.937460i \(0.386829\pi\)
\(62\) 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 0.626841 0.779147i \(-0.284348\pi\)
−0.779147 + 0.626841i \(0.784348\pi\)
\(68\) 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 \(0.276053\pi\)
−0.646929 + 0.762550i \(0.723947\pi\)
\(74\) 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 \(0.00898702\pi\)
−0.999601 + 0.0282298i \(0.991013\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.1751 + 10.1751i 0.122592 + 0.122592i 0.765741 0.643149i \(-0.222373\pi\)
−0.643149 + 0.765741i \(0.722373\pi\)
\(84\) 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 \(0.0393279\pi\)
−0.992377 + 0.123238i \(0.960672\pi\)
\(90\) 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 0.00217389 0.999998i \(-0.499308\pi\)
−0.999998 + 0.00217389i \(0.999308\pi\)
\(102\) 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 0.0552381 0.998473i \(-0.482408\pi\)
0.998473 0.0552381i \(-0.0175918\pi\)
\(108\) 0 0
\(109\) 81.1384 81.1384i 0.744389 0.744389i −0.229030 0.973419i \(-0.573555\pi\)
0.973419 + 0.229030i \(0.0735554\pi\)
\(110\) 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 \(-0.953678\pi\)
0.989430 0.145013i \(-0.0463224\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.3686 12.3686i −0.0944170 0.0944170i 0.658321 0.752738i \(-0.271267\pi\)
−0.752738 + 0.658321i \(0.771267\pi\)
\(132\) 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 \(-0.821563\pi\)
0.846949 0.531674i \(-0.178437\pi\)
\(138\) 0 0
\(139\) −82.5709 + 82.5709i −0.594035 + 0.594035i −0.938719 0.344684i \(-0.887986\pi\)
0.344684 + 0.938719i \(0.387986\pi\)
\(140\) 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 0.916059 + 0.401043i \(0.131352\pi\)
0.401043 + 0.916059i \(0.368648\pi\)
\(150\) 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 −0.858851 0.512226i \(-0.828821\pi\)
0.512226 + 0.858851i \(0.328821\pi\)
\(158\) 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 0.308321 0.951282i \(-0.400233\pi\)
−0.951282 + 0.308321i \(0.900233\pi\)
\(164\) 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 −0.816989 0.576654i \(-0.804358\pi\)
0.576654 + 0.816989i \(0.304358\pi\)
\(174\) 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 0.839636 0.543150i \(-0.182769\pi\)
−0.543150 + 0.839636i \(0.682769\pi\)
\(180\) 0 0
\(181\) 66.6042 + 66.6042i 0.367979 + 0.367979i 0.866740 0.498761i \(-0.166211\pi\)
−0.498761 + 0.866740i \(0.666211\pi\)
\(182\) 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 \(-0.903753\pi\)
0.954633 0.297784i \(-0.0962474\pi\)
\(192\) 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 0.826442 0.563023i \(-0.190362\pi\)
−0.563023 + 0.826442i \(0.690362\pi\)
\(198\) 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 0.957498 0.288441i \(-0.0931368\pi\)
−0.288441 + 0.957498i \(0.593137\pi\)
\(212\) 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 \(0.0886457\pi\)
−0.961472 + 0.274903i \(0.911354\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −295.844 295.844i −1.30328 1.30328i −0.926168 0.377112i \(-0.876917\pi\)
−0.377112 0.926168i \(-0.623083\pi\)
\(228\) 0 0
\(229\) −73.3817 73.3817i −0.320444 0.320444i 0.528493 0.848937i \(-0.322757\pi\)
−0.848937 + 0.528493i \(0.822757\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 156.229i 0.670509i 0.942128 + 0.335255i \(0.108822\pi\)
−0.942128 + 0.335255i \(0.891178\pi\)
\(234\) 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 \(-0.991228\pi\)
0.999620 0.0275557i \(-0.00877235\pi\)
\(240\) 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 −0.759658 0.650322i \(-0.774634\pi\)
0.650322 + 0.759658i \(0.274634\pi\)
\(252\) 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 −0.969008 0.247028i \(-0.920546\pi\)
0.247028 + 0.969008i \(0.420546\pi\)
\(270\) 0 0
\(271\) 291.647i 1.07619i 0.842884 + 0.538095i \(0.180856\pi\)
−0.842884 + 0.538095i \(0.819144\pi\)
\(272\) 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 −0.994208 0.107475i \(-0.965723\pi\)
−0.107475 0.994208i \(-0.534277\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 211.861i 0.753955i 0.926222 + 0.376978i \(0.123037\pi\)
−0.926222 + 0.376978i \(0.876963\pi\)
\(282\) 0 0
\(283\) −105.325 + 105.325i −0.372175 + 0.372175i −0.868269 0.496094i \(-0.834767\pi\)
0.496094 + 0.868269i \(0.334767\pi\)
\(284\) 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 0.351562 0.936165i \(-0.385651\pi\)
−0.936165 + 0.351562i \(0.885651\pi\)
\(294\) 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 0.749884 0.661570i \(-0.230109\pi\)
−0.661570 + 0.749884i \(0.730109\pi\)
\(308\) 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 \(-0.795865\pi\)
0.801313 0.598245i \(-0.204135\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −48.5840 + 48.5840i −0.153262 + 0.153262i −0.779573 0.626311i \(-0.784564\pi\)
0.626311 + 0.779573i \(0.284564\pi\)
\(318\) 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 −0.704248 0.709954i \(-0.748716\pi\)
0.709954 + 0.704248i \(0.248716\pi\)
\(332\) 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 0.329183 0.944266i \(-0.393227\pi\)
0.944266 0.329183i \(-0.106773\pi\)
\(348\) 0 0
\(349\) −119.382 + 119.382i −0.342068 + 0.342068i −0.857144 0.515076i \(-0.827764\pi\)
0.515076 + 0.857144i \(0.327764\pi\)
\(350\) 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 \(0.0966602\pi\)
−0.954246 + 0.299021i \(0.903340\pi\)
\(368\) 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 −0.988554 0.150870i \(-0.951793\pi\)
−0.150870 0.988554i \(-0.548207\pi\)
\(374\) 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 −0.999424 0.0339473i \(-0.989192\pi\)
0.0339473 + 0.999424i \(0.489192\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 213.276i 0.556857i −0.960457 0.278428i \(-0.910187\pi\)
0.960457 0.278428i \(-0.0898135\pi\)
\(384\) 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 0.382187 0.924085i \(-0.375171\pi\)
−0.924085 + 0.382187i \(0.875171\pi\)
\(390\) 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 −0.999947 0.0102579i \(-0.996735\pi\)
0.0102579 + 0.999947i \(0.496735\pi\)
\(398\) 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 \(0.262752\pi\)
−0.678219 + 0.734860i \(0.737248\pi\)
\(410\) 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 −0.960659 0.277729i \(-0.910418\pi\)
−0.277729 0.960659i \(-0.589582\pi\)
\(420\) 0 0
\(421\) 411.213 + 411.213i 0.976754 + 0.976754i 0.999736 0.0229817i \(-0.00731596\pi\)
−0.0229817 + 0.999736i \(0.507316\pi\)
\(422\) 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 \(-0.984814\pi\)
0.998862 0.0476895i \(-0.0151858\pi\)
\(432\) 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 −0.924949 0.380091i \(-0.875893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 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 \(-0.809368\pi\)
0.825963 0.563725i \(-0.190632\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.67717 5.67717i 0.0123149 0.0123149i −0.700923 0.713237i \(-0.747228\pi\)
0.713237 + 0.700923i \(0.247228\pi\)
\(462\) 0 0
\(463\) 464.510i 1.00326i −0.865082 0.501631i \(-0.832733\pi\)
0.865082 0.501631i \(-0.167267\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 495.985 + 495.985i 1.06207 + 1.06207i 0.997942 + 0.0641248i \(0.0204256\pi\)
0.0641248 + 0.997942i \(0.479574\pi\)
\(468\) 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 \(0.129397\pi\)
−0.918505 + 0.395409i \(0.870603\pi\)
\(480\) 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 −0.586697 0.809807i \(-0.699572\pi\)
0.809807 + 0.586697i \(0.199572\pi\)
\(492\) 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 0.969112 0.246621i \(-0.0793202\pi\)
−0.246621 + 0.969112i \(0.579320\pi\)
\(500\) 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 0.428208 0.903680i \(-0.359145\pi\)
0.903680 0.428208i \(-0.140855\pi\)
\(510\) 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 \(0.0438753\pi\)
−0.990515 + 0.137402i \(0.956125\pi\)
\(522\) 0 0
\(523\) −226.187 + 226.187i −0.432481 + 0.432481i −0.889471 0.456991i \(-0.848927\pi\)
0.456991 + 0.889471i \(0.348927\pi\)
\(524\) 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 −0.547284 0.836947i \(-0.684338\pi\)
0.836947 + 0.547284i \(0.184338\pi\)
\(542\) 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 0.443880 0.896086i \(-0.353602\pi\)
−0.896086 + 0.443880i \(0.853602\pi\)
\(548\) 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 −0.209184 + 0.977876i \(0.567081\pi\)
−0.977876 + 0.209184i \(0.932919\pi\)
\(558\) 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 0.891393 0.453231i \(-0.149729\pi\)
−0.453231 + 0.891393i \(0.649729\pi\)
\(564\) 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 \(0.0686592\pi\)
−0.976827 + 0.214030i \(0.931341\pi\)
\(570\) 0 0
\(571\) −59.9229 + 59.9229i −0.104944 + 0.104944i −0.757629 0.652685i \(-0.773642\pi\)
0.652685 + 0.757629i \(0.273642\pi\)
\(572\) 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 −0.930801 0.365525i \(-0.880889\pi\)
0.365525 + 0.930801i \(0.380889\pi\)
\(588\) 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 \(-0.863858\pi\)
0.909922 0.414780i \(-0.136142\pi\)
\(602\) 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 \(-0.844800\pi\)
0.883472 0.468484i \(-0.155200\pi\)
\(608\) 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 0.831022 0.556240i \(-0.187756\pi\)
−0.556240 + 0.831022i \(0.687756\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 599.157i 0.971081i −0.874214 0.485541i \(-0.838623\pi\)
0.874214 0.485541i \(-0.161377\pi\)
\(618\) 0 0
\(619\) −126.719 + 126.719i −0.204715 + 0.204715i −0.802017 0.597301i \(-0.796240\pi\)
0.597301 + 0.802017i \(0.296240\pi\)
\(620\) 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 0.646126 0.763230i \(-0.276388\pi\)
−0.763230 + 0.646126i \(0.776388\pi\)
\(644\) 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 −0.263986 0.964527i \(-0.585037\pi\)
0.964527 + 0.263986i \(0.0850371\pi\)
\(654\) 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 0.301001 0.953624i \(-0.402679\pi\)
−0.953624 + 0.301001i \(0.902679\pi\)
\(660\) 0 0
\(661\) 513.622 + 513.622i 0.777038 + 0.777038i 0.979326 0.202288i \(-0.0648376\pi\)
−0.202288 + 0.979326i \(0.564838\pi\)
\(662\) 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 0.998037 0.0626291i \(-0.0199485\pi\)
−0.0626291 + 0.998037i \(0.519949\pi\)
\(678\) 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 −0.947833 0.318768i \(-0.896731\pi\)
0.318768 + 0.947833i \(0.396731\pi\)
\(684\) 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 0.808253 0.588836i \(-0.200414\pi\)
−0.588836 + 0.808253i \(0.700414\pi\)
\(692\) 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 −0.394533 + 0.918882i \(0.629094\pi\)
−0.918882 + 0.394533i \(0.870906\pi\)
\(702\) 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 0.360890 0.932608i \(-0.382473\pi\)
−0.932608 + 0.360890i \(0.882473\pi\)
\(710\) 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 \(-0.790184\pi\)
0.790509 0.612450i \(-0.209816\pi\)
\(720\) 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 −0.972019 0.234903i \(-0.924523\pi\)
0.234903 + 0.972019i \(0.424523\pi\)
\(734\) 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 −0.971230 0.238142i \(-0.923462\pi\)
−0.238142 0.971230i \(-0.576538\pi\)
\(740\) 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 \(-0.951104\pi\)
0.988225 0.153008i \(-0.0488961\pi\)
\(752\) 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 0.909448 0.415818i \(-0.136505\pi\)
−0.415818 + 0.909448i \(0.636505\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 384.012i 0.504615i −0.967647 0.252307i \(-0.918811\pi\)
0.967647 0.252307i \(-0.0811894\pi\)
\(762\) 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 0.705949 0.708263i \(-0.250521\pi\)
−0.708263 + 0.705949i \(0.750521\pi\)
\(774\) 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 0.609891 0.792485i \(-0.291213\pi\)
−0.792485 + 0.609891i \(0.791213\pi\)
\(788\) 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 −0.940222 0.340562i \(-0.889383\pi\)
0.340562 + 0.940222i \(0.389383\pi\)
\(798\) 0 0
\(799\) 1020.64i 1.27740i
\(800\) 0 0