Properties

Label 1152.3.m.c.991.5
Level $1152$
Weight $3$
Character 1152.991
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 991.5
Root \(1.78012 + 0.911682i\) of defining polynomial
Character \(\chi\) \(=\) 1152.991
Dual form 1152.3.m.c.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 \(-0.754015\pi\)
−0.715969 + 0.698133i \(0.754015\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.26517 2.26517i −0.205925 0.205925i 0.596608 0.802533i \(-0.296515\pi\)
−0.802533 + 0.596608i \(0.796515\pi\)
\(12\) 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 −0.994431 0.105390i \(-0.966391\pi\)
0.105390 + 0.994431i \(0.466391\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 33.2007 1.44351 0.721755 0.692149i \(-0.243336\pi\)
0.721755 + 0.692149i \(0.243336\pi\)
\(24\) 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 \(-0.767426\pi\)
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 −0.612266 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 \(-0.953320\pi\)
−0.146124 0.989266i \(-0.546680\pi\)
\(44\) 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 −0.428975 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 0.652248 0.758006i \(-0.273826\pi\)
−0.758006 + 0.652248i \(0.773826\pi\)
\(60\) 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 −0.626841 0.779147i \(-0.715652\pi\)
0.779147 + 0.626841i \(0.215652\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.9529 0.168350 0.0841752 0.996451i \(-0.473174\pi\)
0.0841752 + 0.996451i \(0.473174\pi\)
\(72\) 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 \(0.00898702\pi\)
−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 \(-0.777627\pi\)
0.643149 + 0.765741i \(0.277627\pi\)
\(84\) 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 \(-0.590940\pi\)
−0.281826 + 0.959465i \(0.590940\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −112.747 112.747i −1.05371 1.05371i −0.998473 0.0552381i \(-0.982408\pi\)
−0.0552381 0.998473i \(-0.517592\pi\)
\(108\) 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 \(-0.953678\pi\)
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 \(-0.728733\pi\)
0.752738 + 0.658321i \(0.228733\pi\)
\(132\) 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 0.938719 0.344684i \(-0.112014\pi\)
−0.344684 + 0.938719i \(0.612014\pi\)
\(140\) 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 \(-0.568512\pi\)
−0.213578 + 0.976926i \(0.568512\pi\)
\(152\) 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 −0.308321 0.951282i \(-0.599767\pi\)
0.951282 + 0.308321i \(0.0997671\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 53.3110 0.319228 0.159614 0.987180i \(-0.448975\pi\)
0.159614 + 0.987180i \(0.448975\pi\)
\(168\) 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 −0.839636 0.543150i \(-0.817231\pi\)
0.543150 + 0.839636i \(0.317231\pi\)
\(180\) 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 \(-0.903753\pi\)
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 −0.563023 0.826442i \(-0.690362\pi\)
0.826442 + 0.563023i \(0.190362\pi\)
\(198\) 0 0
\(199\) 136.741 0.687140 0.343570 0.939127i \(-0.388364\pi\)
0.343570 + 0.939127i \(0.388364\pi\)
\(200\) 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 \(-0.906863\pi\)
0.288441 + 0.957498i \(0.406863\pi\)
\(212\) 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 0.377112 0.926168i \(-0.376917\pi\)
0.926168 0.377112i \(-0.123083\pi\)
\(228\) 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 \(-0.991228\pi\)
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 \(-0.225366\pi\)
−0.650322 + 0.759658i \(0.725366\pi\)
\(252\) 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 \(-0.480858\pi\)
0.0600994 + 0.998192i \(0.480858\pi\)
\(264\) 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 \(0.180856\pi\)
−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 −0.107475 + 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 0.868269 0.496094i \(-0.165233\pi\)
−0.496094 + 0.868269i \(0.665233\pi\)
\(284\) 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 −0.749884 0.661570i \(-0.769891\pi\)
0.661570 + 0.749884i \(0.269891\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 371.124 1.19333 0.596663 0.802492i \(-0.296493\pi\)
0.596663 + 0.802492i \(0.296493\pi\)
\(312\) 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 0.704248 0.709954i \(-0.251284\pi\)
−0.709954 + 0.704248i \(0.751284\pi\)
\(332\) 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 \(-0.893227\pi\)
−0.329183 0.944266i \(-0.606773\pi\)
\(348\) 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 \(-0.296570\pi\)
0.596468 + 0.802637i \(0.296570\pi\)
\(360\) 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 −0.150870 + 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 0.999424 0.0339473i \(-0.0108078\pi\)
−0.0339473 + 0.999424i \(0.510808\pi\)
\(380\) 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 −0.924085 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 \(-0.410418\pi\)
0.960659 0.277729i \(-0.0895819\pi\)
\(420\) 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 \(-0.984814\pi\)
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 \(-0.155258\pi\)
0.883386 + 0.468646i \(0.155258\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 241.372 + 241.372i 0.544858 + 0.544858i 0.924949 0.380091i \(-0.124107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 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 \(-0.832733\pi\)
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 −0.0641248 + 0.997942i \(0.520426\pi\)
−0.997942 + 0.0641248i \(0.979574\pi\)
\(468\) 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 \(-0.548372\pi\)
−0.151382 + 0.988475i \(0.548372\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −109.547 109.547i −0.223110 0.223110i 0.586697 0.809807i \(-0.300428\pi\)
−0.809807 + 0.586697i \(0.800428\pi\)
\(492\) 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 \(-0.920680\pi\)
0.246621 + 0.969112i \(0.420680\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −927.420 −1.84378 −0.921889 0.387454i \(-0.873355\pi\)
−0.921889 + 0.387454i \(0.873355\pi\)
\(504\) 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 0.889471 0.456991i \(-0.151073\pi\)
−0.456991 + 0.889471i \(0.651073\pi\)
\(524\) 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 −0.443880 0.896086i \(-0.646398\pi\)
0.896086 + 0.443880i \(0.146398\pi\)
\(548\) 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 −0.891393 0.453231i \(-0.850271\pi\)
0.453231 + 0.891393i \(0.350271\pi\)
\(564\) 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 0.757629 0.652685i \(-0.226358\pi\)
−0.652685 + 0.757629i \(0.726358\pi\)
\(572\) 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 \(-0.119111\pi\)
−0.365525 + 0.930801i \(0.619111\pi\)
\(588\) 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 \(-0.536245\pi\)
−0.113621 + 0.993524i \(0.536245\pi\)
\(600\) 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 \(-0.844800\pi\)
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 −0.556240 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 0.802017 0.597301i \(-0.203760\pi\)
−0.597301 + 0.802017i \(0.703760\pi\)
\(620\) 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 \(-0.677637\pi\)
−0.529543 + 0.848283i \(0.677637\pi\)
\(632\) 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 \(-0.723612\pi\)
0.763230 + 0.646126i \(0.223612\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −582.307 −0.900011 −0.450006 0.893026i \(-0.648578\pi\)
−0.450006 + 0.893026i \(0.648578\pi\)
\(648\) 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 −0.301001 0.953624i \(-0.597321\pi\)
0.953624 + 0.301001i \(0.0973207\pi\)
\(660\) 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 0.947833 0.318768i \(-0.103269\pi\)
−0.318768 + 0.947833i \(0.603269\pi\)
\(684\) 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 \(-0.799586\pi\)
0.588836 + 0.808253i \(0.299586\pi\)
\(692\) 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 \(-0.790184\pi\)
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 \(-0.741545\pi\)
−0.688077 + 0.725637i \(0.741545\pi\)
\(728\) 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 \(-0.423462\pi\)
0.971230 0.238142i \(-0.0765384\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1295.75 1.74394 0.871969 0.489561i \(-0.162843\pi\)
0.871969 + 0.489561i \(0.162843\pi\)
\(744\) 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 −0.415818 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 −0.609891 0.792485i \(-0.708787\pi\)
0.792485 + 0.609891i \(0.208787\pi\)
\(788\) 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