Properties

Label 1400.2.g.j.449.1
Level $1400$
Weight $2$
Character 1400.449
Analytic conductor $11.179$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 1400.449
Dual form 1400.2.g.j.449.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.56155i q^{3} +1.00000i q^{7} -3.56155 q^{9} +O(q^{10})\) \(q-2.56155i q^{3} +1.00000i q^{7} -3.56155 q^{9} +2.12311 q^{11} -2.00000i q^{13} -2.56155i q^{17} +0.561553 q^{19} +2.56155 q^{21} -5.56155i q^{23} +1.43845i q^{27} -7.56155 q^{29} -0.876894 q^{31} -5.43845i q^{33} -11.8078i q^{37} -5.12311 q^{39} -6.56155 q^{41} -2.43845i q^{43} +8.24621i q^{47} -1.00000 q^{49} -6.56155 q^{51} -7.12311i q^{53} -1.43845i q^{57} +13.3693 q^{59} +2.87689 q^{61} -3.56155i q^{63} +16.1231i q^{67} -14.2462 q^{69} -10.6847 q^{71} -10.5616i q^{73} +2.12311i q^{77} -6.68466 q^{79} -7.00000 q^{81} +13.6847i q^{83} +19.3693i q^{87} +10.8078 q^{89} +2.00000 q^{91} +2.24621i q^{93} +6.00000i q^{97} -7.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{9} + O(q^{10}) \) \( 4q - 6q^{9} - 8q^{11} - 6q^{19} + 2q^{21} - 22q^{29} - 20q^{31} - 4q^{39} - 18q^{41} - 4q^{49} - 18q^{51} + 4q^{59} + 28q^{61} - 24q^{69} - 18q^{71} - 2q^{79} - 28q^{81} + 2q^{89} + 8q^{91} - 22q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 2.56155i − 1.47891i −0.673204 0.739457i \(-0.735083\pi\)
0.673204 0.739457i \(-0.264917\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −3.56155 −1.18718
\(10\) 0 0
\(11\) 2.12311 0.640140 0.320070 0.947394i \(-0.396293\pi\)
0.320070 + 0.947394i \(0.396293\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.56155i − 0.621268i −0.950530 0.310634i \(-0.899459\pi\)
0.950530 0.310634i \(-0.100541\pi\)
\(18\) 0 0
\(19\) 0.561553 0.128829 0.0644145 0.997923i \(-0.479482\pi\)
0.0644145 + 0.997923i \(0.479482\pi\)
\(20\) 0 0
\(21\) 2.56155 0.558977
\(22\) 0 0
\(23\) − 5.56155i − 1.15966i −0.814736 0.579832i \(-0.803118\pi\)
0.814736 0.579832i \(-0.196882\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.43845i 0.276829i
\(28\) 0 0
\(29\) −7.56155 −1.40415 −0.702073 0.712105i \(-0.747742\pi\)
−0.702073 + 0.712105i \(0.747742\pi\)
\(30\) 0 0
\(31\) −0.876894 −0.157495 −0.0787474 0.996895i \(-0.525092\pi\)
−0.0787474 + 0.996895i \(0.525092\pi\)
\(32\) 0 0
\(33\) − 5.43845i − 0.946712i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 11.8078i − 1.94118i −0.240730 0.970592i \(-0.577387\pi\)
0.240730 0.970592i \(-0.422613\pi\)
\(38\) 0 0
\(39\) −5.12311 −0.820353
\(40\) 0 0
\(41\) −6.56155 −1.02474 −0.512371 0.858764i \(-0.671233\pi\)
−0.512371 + 0.858764i \(0.671233\pi\)
\(42\) 0 0
\(43\) − 2.43845i − 0.371860i −0.982563 0.185930i \(-0.940470\pi\)
0.982563 0.185930i \(-0.0595297\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.24621i 1.20283i 0.798935 + 0.601417i \(0.205397\pi\)
−0.798935 + 0.601417i \(0.794603\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −6.56155 −0.918801
\(52\) 0 0
\(53\) − 7.12311i − 0.978434i −0.872162 0.489217i \(-0.837283\pi\)
0.872162 0.489217i \(-0.162717\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.43845i − 0.190527i
\(58\) 0 0
\(59\) 13.3693 1.74054 0.870268 0.492578i \(-0.163945\pi\)
0.870268 + 0.492578i \(0.163945\pi\)
\(60\) 0 0
\(61\) 2.87689 0.368349 0.184174 0.982894i \(-0.441039\pi\)
0.184174 + 0.982894i \(0.441039\pi\)
\(62\) 0 0
\(63\) − 3.56155i − 0.448713i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 16.1231i 1.96975i 0.173264 + 0.984875i \(0.444569\pi\)
−0.173264 + 0.984875i \(0.555431\pi\)
\(68\) 0 0
\(69\) −14.2462 −1.71504
\(70\) 0 0
\(71\) −10.6847 −1.26804 −0.634018 0.773318i \(-0.718595\pi\)
−0.634018 + 0.773318i \(0.718595\pi\)
\(72\) 0 0
\(73\) − 10.5616i − 1.23614i −0.786125 0.618068i \(-0.787916\pi\)
0.786125 0.618068i \(-0.212084\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.12311i 0.241950i
\(78\) 0 0
\(79\) −6.68466 −0.752083 −0.376041 0.926603i \(-0.622715\pi\)
−0.376041 + 0.926603i \(0.622715\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 13.6847i 1.50209i 0.660253 + 0.751043i \(0.270449\pi\)
−0.660253 + 0.751043i \(0.729551\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 19.3693i 2.07661i
\(88\) 0 0
\(89\) 10.8078 1.14562 0.572810 0.819688i \(-0.305853\pi\)
0.572810 + 0.819688i \(0.305853\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 2.24621i 0.232921i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 0 0
\(99\) −7.56155 −0.759965
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) − 7.12311i − 0.701860i −0.936402 0.350930i \(-0.885865\pi\)
0.936402 0.350930i \(-0.114135\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.8078i − 1.23817i −0.785323 0.619087i \(-0.787503\pi\)
0.785323 0.619087i \(-0.212497\pi\)
\(108\) 0 0
\(109\) 8.93087 0.855422 0.427711 0.903915i \(-0.359320\pi\)
0.427711 + 0.903915i \(0.359320\pi\)
\(110\) 0 0
\(111\) −30.2462 −2.87084
\(112\) 0 0
\(113\) − 0.753789i − 0.0709105i −0.999371 0.0354552i \(-0.988712\pi\)
0.999371 0.0354552i \(-0.0112881\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 7.12311i 0.658531i
\(118\) 0 0
\(119\) 2.56155 0.234817
\(120\) 0 0
\(121\) −6.49242 −0.590220
\(122\) 0 0
\(123\) 16.8078i 1.51551i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 9.80776i − 0.870298i −0.900358 0.435149i \(-0.856696\pi\)
0.900358 0.435149i \(-0.143304\pi\)
\(128\) 0 0
\(129\) −6.24621 −0.549948
\(130\) 0 0
\(131\) −1.75379 −0.153229 −0.0766146 0.997061i \(-0.524411\pi\)
−0.0766146 + 0.997061i \(0.524411\pi\)
\(132\) 0 0
\(133\) 0.561553i 0.0486928i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2.31534i − 0.197813i −0.995097 0.0989065i \(-0.968466\pi\)
0.995097 0.0989065i \(-0.0315345\pi\)
\(138\) 0 0
\(139\) 19.6847 1.66963 0.834815 0.550530i \(-0.185574\pi\)
0.834815 + 0.550530i \(0.185574\pi\)
\(140\) 0 0
\(141\) 21.1231 1.77889
\(142\) 0 0
\(143\) − 4.24621i − 0.355086i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.56155i 0.211273i
\(148\) 0 0
\(149\) −17.5616 −1.43870 −0.719349 0.694649i \(-0.755560\pi\)
−0.719349 + 0.694649i \(0.755560\pi\)
\(150\) 0 0
\(151\) −5.56155 −0.452593 −0.226296 0.974058i \(-0.572662\pi\)
−0.226296 + 0.974058i \(0.572662\pi\)
\(152\) 0 0
\(153\) 9.12311i 0.737559i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 24.4924i 1.95471i 0.211611 + 0.977354i \(0.432129\pi\)
−0.211611 + 0.977354i \(0.567871\pi\)
\(158\) 0 0
\(159\) −18.2462 −1.44702
\(160\) 0 0
\(161\) 5.56155 0.438312
\(162\) 0 0
\(163\) 13.9309i 1.09115i 0.838062 + 0.545575i \(0.183689\pi\)
−0.838062 + 0.545575i \(0.816311\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 19.6155i − 1.51790i −0.651152 0.758948i \(-0.725714\pi\)
0.651152 0.758948i \(-0.274286\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) − 14.4924i − 1.10184i −0.834559 0.550919i \(-0.814277\pi\)
0.834559 0.550919i \(-0.185723\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 34.2462i − 2.57410i
\(178\) 0 0
\(179\) 8.31534 0.621518 0.310759 0.950489i \(-0.399417\pi\)
0.310759 + 0.950489i \(0.399417\pi\)
\(180\) 0 0
\(181\) −2.87689 −0.213838 −0.106919 0.994268i \(-0.534099\pi\)
−0.106919 + 0.994268i \(0.534099\pi\)
\(182\) 0 0
\(183\) − 7.36932i − 0.544756i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 5.43845i − 0.397699i
\(188\) 0 0
\(189\) −1.43845 −0.104632
\(190\) 0 0
\(191\) 0.630683 0.0456346 0.0228173 0.999740i \(-0.492736\pi\)
0.0228173 + 0.999740i \(0.492736\pi\)
\(192\) 0 0
\(193\) − 20.6155i − 1.48394i −0.670434 0.741969i \(-0.733892\pi\)
0.670434 0.741969i \(-0.266108\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 3.56155i − 0.253750i −0.991919 0.126875i \(-0.959505\pi\)
0.991919 0.126875i \(-0.0404947\pi\)
\(198\) 0 0
\(199\) 13.1231 0.930272 0.465136 0.885239i \(-0.346005\pi\)
0.465136 + 0.885239i \(0.346005\pi\)
\(200\) 0 0
\(201\) 41.3002 2.91309
\(202\) 0 0
\(203\) − 7.56155i − 0.530717i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 19.8078i 1.37673i
\(208\) 0 0
\(209\) 1.19224 0.0824687
\(210\) 0 0
\(211\) 10.5616 0.727087 0.363544 0.931577i \(-0.381567\pi\)
0.363544 + 0.931577i \(0.381567\pi\)
\(212\) 0 0
\(213\) 27.3693i 1.87531i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 0.876894i − 0.0595275i
\(218\) 0 0
\(219\) −27.0540 −1.82814
\(220\) 0 0
\(221\) −5.12311 −0.344617
\(222\) 0 0
\(223\) 0.630683i 0.0422337i 0.999777 + 0.0211168i \(0.00672220\pi\)
−0.999777 + 0.0211168i \(0.993278\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 13.3693i − 0.887353i −0.896187 0.443676i \(-0.853674\pi\)
0.896187 0.443676i \(-0.146326\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 5.43845 0.357824
\(232\) 0 0
\(233\) − 3.56155i − 0.233325i −0.993172 0.116663i \(-0.962780\pi\)
0.993172 0.116663i \(-0.0372196\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 17.1231i 1.11227i
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) 25.0540 1.61387 0.806934 0.590641i \(-0.201125\pi\)
0.806934 + 0.590641i \(0.201125\pi\)
\(242\) 0 0
\(243\) 22.2462i 1.42710i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1.12311i − 0.0714615i
\(248\) 0 0
\(249\) 35.0540 2.22146
\(250\) 0 0
\(251\) −12.3153 −0.777337 −0.388669 0.921378i \(-0.627065\pi\)
−0.388669 + 0.921378i \(0.627065\pi\)
\(252\) 0 0
\(253\) − 11.8078i − 0.742348i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 4.24621i − 0.264871i −0.991192 0.132436i \(-0.957720\pi\)
0.991192 0.132436i \(-0.0422798\pi\)
\(258\) 0 0
\(259\) 11.8078 0.733699
\(260\) 0 0
\(261\) 26.9309 1.66698
\(262\) 0 0
\(263\) 10.6847i 0.658844i 0.944183 + 0.329422i \(0.106854\pi\)
−0.944183 + 0.329422i \(0.893146\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 27.6847i − 1.69427i
\(268\) 0 0
\(269\) −11.7538 −0.716641 −0.358321 0.933599i \(-0.616651\pi\)
−0.358321 + 0.933599i \(0.616651\pi\)
\(270\) 0 0
\(271\) 16.2462 0.986887 0.493444 0.869778i \(-0.335738\pi\)
0.493444 + 0.869778i \(0.335738\pi\)
\(272\) 0 0
\(273\) − 5.12311i − 0.310064i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 10.4924i − 0.630429i −0.949021 0.315214i \(-0.897924\pi\)
0.949021 0.315214i \(-0.102076\pi\)
\(278\) 0 0
\(279\) 3.12311 0.186975
\(280\) 0 0
\(281\) 11.5616 0.689704 0.344852 0.938657i \(-0.387929\pi\)
0.344852 + 0.938657i \(0.387929\pi\)
\(282\) 0 0
\(283\) 13.6847i 0.813469i 0.913547 + 0.406734i \(0.133333\pi\)
−0.913547 + 0.406734i \(0.866667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 6.56155i − 0.387316i
\(288\) 0 0
\(289\) 10.4384 0.614026
\(290\) 0 0
\(291\) 15.3693 0.900965
\(292\) 0 0
\(293\) 30.0000i 1.75262i 0.481749 + 0.876309i \(0.340002\pi\)
−0.481749 + 0.876309i \(0.659998\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.05398i 0.177210i
\(298\) 0 0
\(299\) −11.1231 −0.643266
\(300\) 0 0
\(301\) 2.43845 0.140550
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.31534i 0.246290i 0.992389 + 0.123145i \(0.0392980\pi\)
−0.992389 + 0.123145i \(0.960702\pi\)
\(308\) 0 0
\(309\) −18.2462 −1.03799
\(310\) 0 0
\(311\) 2.87689 0.163134 0.0815669 0.996668i \(-0.474008\pi\)
0.0815669 + 0.996668i \(0.474008\pi\)
\(312\) 0 0
\(313\) − 22.0000i − 1.24351i −0.783210 0.621757i \(-0.786419\pi\)
0.783210 0.621757i \(-0.213581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 20.6847i − 1.16177i −0.813987 0.580883i \(-0.802707\pi\)
0.813987 0.580883i \(-0.197293\pi\)
\(318\) 0 0
\(319\) −16.0540 −0.898850
\(320\) 0 0
\(321\) −32.8078 −1.83115
\(322\) 0 0
\(323\) − 1.43845i − 0.0800373i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 22.8769i − 1.26510i
\(328\) 0 0
\(329\) −8.24621 −0.454628
\(330\) 0 0
\(331\) 27.0000 1.48405 0.742027 0.670370i \(-0.233865\pi\)
0.742027 + 0.670370i \(0.233865\pi\)
\(332\) 0 0
\(333\) 42.0540i 2.30454i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 5.68466i − 0.309663i −0.987941 0.154832i \(-0.950517\pi\)
0.987941 0.154832i \(-0.0494835\pi\)
\(338\) 0 0
\(339\) −1.93087 −0.104870
\(340\) 0 0
\(341\) −1.86174 −0.100819
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.6155i 1.21407i 0.794677 + 0.607033i \(0.207640\pi\)
−0.794677 + 0.607033i \(0.792360\pi\)
\(348\) 0 0
\(349\) 12.8769 0.689284 0.344642 0.938734i \(-0.388000\pi\)
0.344642 + 0.938734i \(0.388000\pi\)
\(350\) 0 0
\(351\) 2.87689 0.153557
\(352\) 0 0
\(353\) 11.7538i 0.625591i 0.949820 + 0.312796i \(0.101265\pi\)
−0.949820 + 0.312796i \(0.898735\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 6.56155i − 0.347274i
\(358\) 0 0
\(359\) 14.6847 0.775027 0.387513 0.921864i \(-0.373334\pi\)
0.387513 + 0.921864i \(0.373334\pi\)
\(360\) 0 0
\(361\) −18.6847 −0.983403
\(362\) 0 0
\(363\) 16.6307i 0.872884i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 17.7538i 0.926740i 0.886165 + 0.463370i \(0.153360\pi\)
−0.886165 + 0.463370i \(0.846640\pi\)
\(368\) 0 0
\(369\) 23.3693 1.21656
\(370\) 0 0
\(371\) 7.12311 0.369813
\(372\) 0 0
\(373\) − 14.0540i − 0.727687i −0.931460 0.363844i \(-0.881464\pi\)
0.931460 0.363844i \(-0.118536\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.1231i 0.778880i
\(378\) 0 0
\(379\) 24.8617 1.27706 0.638531 0.769596i \(-0.279542\pi\)
0.638531 + 0.769596i \(0.279542\pi\)
\(380\) 0 0
\(381\) −25.1231 −1.28710
\(382\) 0 0
\(383\) 20.2462i 1.03453i 0.855824 + 0.517267i \(0.173050\pi\)
−0.855824 + 0.517267i \(0.826950\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.68466i 0.441466i
\(388\) 0 0
\(389\) −2.43845 −0.123634 −0.0618171 0.998087i \(-0.519690\pi\)
−0.0618171 + 0.998087i \(0.519690\pi\)
\(390\) 0 0
\(391\) −14.2462 −0.720462
\(392\) 0 0
\(393\) 4.49242i 0.226613i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 38.4924i 1.93188i 0.258767 + 0.965940i \(0.416684\pi\)
−0.258767 + 0.965940i \(0.583316\pi\)
\(398\) 0 0
\(399\) 1.43845 0.0720124
\(400\) 0 0
\(401\) −31.4924 −1.57266 −0.786328 0.617809i \(-0.788021\pi\)
−0.786328 + 0.617809i \(0.788021\pi\)
\(402\) 0 0
\(403\) 1.75379i 0.0873624i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 25.0691i − 1.24263i
\(408\) 0 0
\(409\) 5.19224 0.256740 0.128370 0.991726i \(-0.459026\pi\)
0.128370 + 0.991726i \(0.459026\pi\)
\(410\) 0 0
\(411\) −5.93087 −0.292548
\(412\) 0 0
\(413\) 13.3693i 0.657861i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 50.4233i − 2.46924i
\(418\) 0 0
\(419\) 18.8078 0.918819 0.459410 0.888224i \(-0.348061\pi\)
0.459410 + 0.888224i \(0.348061\pi\)
\(420\) 0 0
\(421\) 5.06913 0.247054 0.123527 0.992341i \(-0.460579\pi\)
0.123527 + 0.992341i \(0.460579\pi\)
\(422\) 0 0
\(423\) − 29.3693i − 1.42799i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.87689i 0.139223i
\(428\) 0 0
\(429\) −10.8769 −0.525141
\(430\) 0 0
\(431\) −13.7538 −0.662497 −0.331248 0.943544i \(-0.607470\pi\)
−0.331248 + 0.943544i \(0.607470\pi\)
\(432\) 0 0
\(433\) 35.5464i 1.70825i 0.520067 + 0.854125i \(0.325907\pi\)
−0.520067 + 0.854125i \(0.674093\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 3.12311i − 0.149398i
\(438\) 0 0
\(439\) 7.12311 0.339967 0.169984 0.985447i \(-0.445629\pi\)
0.169984 + 0.985447i \(0.445629\pi\)
\(440\) 0 0
\(441\) 3.56155 0.169598
\(442\) 0 0
\(443\) − 22.5616i − 1.07193i −0.844240 0.535966i \(-0.819948\pi\)
0.844240 0.535966i \(-0.180052\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 44.9848i 2.12771i
\(448\) 0 0
\(449\) −10.3693 −0.489358 −0.244679 0.969604i \(-0.578683\pi\)
−0.244679 + 0.969604i \(0.578683\pi\)
\(450\) 0 0
\(451\) −13.9309 −0.655979
\(452\) 0 0
\(453\) 14.2462i 0.669345i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 37.1080i 1.73584i 0.496707 + 0.867918i \(0.334542\pi\)
−0.496707 + 0.867918i \(0.665458\pi\)
\(458\) 0 0
\(459\) 3.68466 0.171985
\(460\) 0 0
\(461\) 6.49242 0.302382 0.151191 0.988505i \(-0.451689\pi\)
0.151191 + 0.988505i \(0.451689\pi\)
\(462\) 0 0
\(463\) 20.4924i 0.952364i 0.879347 + 0.476182i \(0.157980\pi\)
−0.879347 + 0.476182i \(0.842020\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 24.8769i − 1.15117i −0.817744 0.575583i \(-0.804775\pi\)
0.817744 0.575583i \(-0.195225\pi\)
\(468\) 0 0
\(469\) −16.1231 −0.744496
\(470\) 0 0
\(471\) 62.7386 2.89084
\(472\) 0 0
\(473\) − 5.17708i − 0.238042i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 25.3693i 1.16158i
\(478\) 0 0
\(479\) 40.7386 1.86140 0.930698 0.365789i \(-0.119201\pi\)
0.930698 + 0.365789i \(0.119201\pi\)
\(480\) 0 0
\(481\) −23.6155 −1.07678
\(482\) 0 0
\(483\) − 14.2462i − 0.648225i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 35.5616i 1.61145i 0.592291 + 0.805724i \(0.298223\pi\)
−0.592291 + 0.805724i \(0.701777\pi\)
\(488\) 0 0
\(489\) 35.6847 1.61372
\(490\) 0 0
\(491\) 20.6847 0.933486 0.466743 0.884393i \(-0.345427\pi\)
0.466743 + 0.884393i \(0.345427\pi\)
\(492\) 0 0
\(493\) 19.3693i 0.872350i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 10.6847i − 0.479272i
\(498\) 0 0
\(499\) 16.4924 0.738302 0.369151 0.929369i \(-0.379648\pi\)
0.369151 + 0.929369i \(0.379648\pi\)
\(500\) 0 0
\(501\) −50.2462 −2.24484
\(502\) 0 0
\(503\) − 19.7538i − 0.880778i −0.897807 0.440389i \(-0.854841\pi\)
0.897807 0.440389i \(-0.145159\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 23.0540i − 1.02386i
\(508\) 0 0
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) 10.5616 0.467216
\(512\) 0 0
\(513\) 0.807764i 0.0356637i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 17.5076i 0.769982i
\(518\) 0 0
\(519\) −37.1231 −1.62952
\(520\) 0 0
\(521\) −27.4384 −1.20210 −0.601050 0.799211i \(-0.705251\pi\)
−0.601050 + 0.799211i \(0.705251\pi\)
\(522\) 0 0
\(523\) − 28.3153i − 1.23814i −0.785334 0.619072i \(-0.787509\pi\)
0.785334 0.619072i \(-0.212491\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.24621i 0.0978465i
\(528\) 0 0
\(529\) −7.93087 −0.344820
\(530\) 0 0
\(531\) −47.6155 −2.06634
\(532\) 0 0
\(533\) 13.1231i 0.568425i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 21.3002i − 0.919171i
\(538\) 0 0
\(539\) −2.12311 −0.0914486
\(540\) 0 0
\(541\) 41.4233 1.78093 0.890463 0.455055i \(-0.150380\pi\)
0.890463 + 0.455055i \(0.150380\pi\)
\(542\) 0 0
\(543\) 7.36932i 0.316248i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 33.7386i 1.44256i 0.692644 + 0.721280i \(0.256446\pi\)
−0.692644 + 0.721280i \(0.743554\pi\)
\(548\) 0 0
\(549\) −10.2462 −0.437298
\(550\) 0 0
\(551\) −4.24621 −0.180895
\(552\) 0 0
\(553\) − 6.68466i − 0.284261i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.56155i 0.405136i 0.979268 + 0.202568i \(0.0649287\pi\)
−0.979268 + 0.202568i \(0.935071\pi\)
\(558\) 0 0
\(559\) −4.87689 −0.206271
\(560\) 0 0
\(561\) −13.9309 −0.588162
\(562\) 0 0
\(563\) 7.12311i 0.300203i 0.988671 + 0.150102i \(0.0479601\pi\)
−0.988671 + 0.150102i \(0.952040\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 7.00000i − 0.293972i
\(568\) 0 0
\(569\) 19.9848 0.837808 0.418904 0.908030i \(-0.362414\pi\)
0.418904 + 0.908030i \(0.362414\pi\)
\(570\) 0 0
\(571\) 27.3153 1.14311 0.571556 0.820563i \(-0.306340\pi\)
0.571556 + 0.820563i \(0.306340\pi\)
\(572\) 0 0
\(573\) − 1.61553i − 0.0674897i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 45.5464i − 1.89612i −0.318090 0.948061i \(-0.603041\pi\)
0.318090 0.948061i \(-0.396959\pi\)
\(578\) 0 0
\(579\) −52.8078 −2.19462
\(580\) 0 0
\(581\) −13.6847 −0.567735
\(582\) 0 0
\(583\) − 15.1231i − 0.626335i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.17708i 0.337504i 0.985659 + 0.168752i \(0.0539737\pi\)
−0.985659 + 0.168752i \(0.946026\pi\)
\(588\) 0 0
\(589\) −0.492423 −0.0202899
\(590\) 0 0
\(591\) −9.12311 −0.375274
\(592\) 0 0
\(593\) − 12.3153i − 0.505730i −0.967502 0.252865i \(-0.918627\pi\)
0.967502 0.252865i \(-0.0813729\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 33.6155i − 1.37579i
\(598\) 0 0
\(599\) −24.3002 −0.992879 −0.496439 0.868071i \(-0.665359\pi\)
−0.496439 + 0.868071i \(0.665359\pi\)
\(600\) 0 0
\(601\) −4.94602 −0.201753 −0.100876 0.994899i \(-0.532165\pi\)
−0.100876 + 0.994899i \(0.532165\pi\)
\(602\) 0 0
\(603\) − 57.4233i − 2.33846i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 37.3693i − 1.51677i −0.651805 0.758387i \(-0.725988\pi\)
0.651805 0.758387i \(-0.274012\pi\)
\(608\) 0 0
\(609\) −19.3693 −0.784884
\(610\) 0 0
\(611\) 16.4924 0.667212
\(612\) 0 0
\(613\) 6.68466i 0.269991i 0.990846 + 0.134995i \(0.0431020\pi\)
−0.990846 + 0.134995i \(0.956898\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 17.8078i 0.716914i 0.933546 + 0.358457i \(0.116697\pi\)
−0.933546 + 0.358457i \(0.883303\pi\)
\(618\) 0 0
\(619\) −29.3693 −1.18045 −0.590226 0.807238i \(-0.700961\pi\)
−0.590226 + 0.807238i \(0.700961\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) 10.8078i 0.433004i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 3.05398i − 0.121964i
\(628\) 0 0
\(629\) −30.2462 −1.20600
\(630\) 0 0
\(631\) 34.0540 1.35567 0.677834 0.735215i \(-0.262919\pi\)
0.677834 + 0.735215i \(0.262919\pi\)
\(632\) 0 0
\(633\) − 27.0540i − 1.07530i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) 38.0540 1.50539
\(640\) 0 0
\(641\) −16.4384 −0.649280 −0.324640 0.945838i \(-0.605243\pi\)
−0.324640 + 0.945838i \(0.605243\pi\)
\(642\) 0 0
\(643\) − 2.73863i − 0.108001i −0.998541 0.0540006i \(-0.982803\pi\)
0.998541 0.0540006i \(-0.0171973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 25.3693i − 0.997371i −0.866783 0.498685i \(-0.833816\pi\)
0.866783 0.498685i \(-0.166184\pi\)
\(648\) 0 0
\(649\) 28.3845 1.11419
\(650\) 0 0
\(651\) −2.24621 −0.0880360
\(652\) 0 0
\(653\) − 33.8617i − 1.32511i −0.749012 0.662556i \(-0.769472\pi\)
0.749012 0.662556i \(-0.230528\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 37.6155i 1.46752i
\(658\) 0 0
\(659\) −18.5616 −0.723055 −0.361528 0.932361i \(-0.617745\pi\)
−0.361528 + 0.932361i \(0.617745\pi\)
\(660\) 0 0
\(661\) −22.4924 −0.874854 −0.437427 0.899254i \(-0.644110\pi\)
−0.437427 + 0.899254i \(0.644110\pi\)
\(662\) 0 0
\(663\) 13.1231i 0.509659i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 42.0540i 1.62834i
\(668\) 0 0
\(669\) 1.61553 0.0624599
\(670\) 0 0
\(671\) 6.10795 0.235795
\(672\) 0 0
\(673\) − 46.4924i − 1.79215i −0.443901 0.896076i \(-0.646406\pi\)
0.443901 0.896076i \(-0.353594\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.6307i 0.562303i 0.959663 + 0.281151i \(0.0907163\pi\)
−0.959663 + 0.281151i \(0.909284\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) −34.2462 −1.31232
\(682\) 0 0
\(683\) − 34.1231i − 1.30568i −0.757494 0.652842i \(-0.773576\pi\)
0.757494 0.652842i \(-0.226424\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 35.8617i − 1.36821i
\(688\) 0 0
\(689\) −14.2462 −0.542737
\(690\) 0 0
\(691\) −17.4384 −0.663390 −0.331695 0.943387i \(-0.607620\pi\)
−0.331695 + 0.943387i \(0.607620\pi\)
\(692\) 0 0
\(693\) − 7.56155i − 0.287240i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 16.8078i 0.636639i
\(698\) 0 0
\(699\) −9.12311 −0.345068
\(700\) 0 0
\(701\) 44.1080 1.66593 0.832967 0.553322i \(-0.186640\pi\)
0.832967 + 0.553322i \(0.186640\pi\)
\(702\) 0 0
\(703\) − 6.63068i − 0.250081i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −37.3693 −1.40343 −0.701717 0.712456i \(-0.747583\pi\)
−0.701717 + 0.712456i \(0.747583\pi\)
\(710\) 0 0
\(711\) 23.8078 0.892861
\(712\) 0 0
\(713\) 4.87689i 0.182641i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 10.2462i − 0.382652i
\(718\) 0 0
\(719\) −38.2462 −1.42634 −0.713171 0.700990i \(-0.752742\pi\)
−0.713171 + 0.700990i \(0.752742\pi\)
\(720\) 0 0
\(721\) 7.12311 0.265278
\(722\) 0 0
\(723\) − 64.1771i − 2.38677i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 1.61553i − 0.0599166i −0.999551 0.0299583i \(-0.990463\pi\)
0.999551 0.0299583i \(-0.00953745\pi\)
\(728\) 0 0
\(729\) 35.9848 1.33277
\(730\) 0 0
\(731\) −6.24621 −0.231024
\(732\) 0 0
\(733\) − 22.2462i − 0.821683i −0.911707 0.410841i \(-0.865235\pi\)
0.911707 0.410841i \(-0.134765\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 34.2311i 1.26092i
\(738\) 0 0
\(739\) −17.1771 −0.631869 −0.315935 0.948781i \(-0.602318\pi\)
−0.315935 + 0.948781i \(0.602318\pi\)
\(740\) 0 0
\(741\) −2.87689 −0.105685
\(742\) 0 0
\(743\) 18.7386i 0.687454i 0.939070 + 0.343727i \(0.111689\pi\)
−0.939070 + 0.343727i \(0.888311\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 48.7386i − 1.78325i
\(748\) 0 0
\(749\) 12.8078 0.467986
\(750\) 0 0
\(751\) 27.3693 0.998721 0.499360 0.866394i \(-0.333568\pi\)
0.499360 + 0.866394i \(0.333568\pi\)
\(752\) 0 0
\(753\) 31.5464i 1.14961i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 13.3153i 0.483954i 0.970282 + 0.241977i \(0.0777959\pi\)
−0.970282 + 0.241977i \(0.922204\pi\)
\(758\) 0 0
\(759\) −30.2462 −1.09787
\(760\) 0 0
\(761\) 34.4233 1.24784 0.623922 0.781487i \(-0.285538\pi\)
0.623922 + 0.781487i \(0.285538\pi\)
\(762\) 0 0
\(763\) 8.93087i 0.323319i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 26.7386i − 0.965476i
\(768\) 0 0
\(769\) 33.3002 1.20084 0.600418 0.799687i \(-0.295001\pi\)
0.600418 + 0.799687i \(0.295001\pi\)
\(770\) 0 0
\(771\) −10.8769 −0.391722
\(772\) 0 0
\(773\) 11.6155i 0.417782i 0.977939 + 0.208891i \(0.0669853\pi\)
−0.977939 + 0.208891i \(0.933015\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 30.2462i − 1.08508i
\(778\) 0 0
\(779\) −3.68466 −0.132017
\(780\) 0 0
\(781\) −22.6847 −0.811721
\(782\) 0 0
\(783\) − 10.8769i − 0.388708i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 26.2462i 0.935576i 0.883841 + 0.467788i \(0.154949\pi\)
−0.883841 + 0.467788i \(0.845051\pi\)
\(788\) 0 0
\(789\) 27.3693 0.974373
\(790\) 0 0
\(791\) 0.753789 0.0268016
\(792\) 0 0
\(793\) − 5.75379i − 0.204323i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 41.6155i 1.47410i 0.675840 + 0.737049i \(0.263781\pi\)
−0.675840 + 0.737049i \(0.736219\pi\)
\(798\) 0 0
\(799\) 21.1231 0.747282
\(800\) 0 0
\(801\) −38.4924 −1.36006
\(802\) 0 0
\(803\) − 22.4233i − 0.791301i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 30.1080i 1.05985i
\(808\) 0 0
\(809\) −49.4233 −1.73763 −0.868815 0.495136i \(-0.835118\pi\)
−0.868815 + 0.495136i \(0.835118\pi\)
\(810\) 0 0
\(811\) 43.6155 1.53155 0.765774 0.643110i \(-0.222356\pi\)
0.765774 + 0.643110i \(0.222356\pi\)
\(812\) 0 0
\(813\) − 41.6155i − 1.45952i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 1.36932i − 0.0479063i
\(818\) 0 0
\(819\) −7.12311 −0.248901
\(820\) 0 0
\(821\) 45.8617 1.60059 0.800293 0.599609i \(-0.204677\pi\)
0.800293 + 0.599609i \(0.204677\pi\)
\(822\) 0 0
\(823\) − 12.4384i − 0.433577i −0.976219 0.216789i \(-0.930442\pi\)
0.976219 0.216789i \(-0.0695582\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 9.24621i − 0.321522i −0.986993 0.160761i \(-0.948605\pi\)
0.986993 0.160761i \(-0.0513949\pi\)
\(828\) 0 0
\(829\) 2.24621 0.0780141 0.0390071 0.999239i \(-0.487581\pi\)
0.0390071 + 0.999239i \(0.487581\pi\)
\(830\) 0 0
\(831\) −26.8769 −0.932349
\(832\) 0 0
\(833\) 2.56155i 0.0887525i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 1.26137i − 0.0435992i
\(838\) 0 0
\(839\) 1.12311 0.0387739 0.0193870 0.999812i \(-0.493829\pi\)
0.0193870 + 0.999812i \(0.493829\pi\)
\(840\) 0 0
\(841\) 28.1771 0.971623
\(842\) 0 0
\(843\) − 29.6155i − 1.02001i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 6.49242i − 0.223082i
\(848\) 0 0
\(849\) 35.0540 1.20305
\(850\) 0 0
\(851\) −65.6695 −2.25112
\(852\) 0 0
\(853\) − 10.8769i − 0.372418i −0.982510 0.186209i \(-0.940380\pi\)
0.982510 0.186209i \(-0.0596201\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.94602i 0.168953i 0.996425 + 0.0844765i \(0.0269218\pi\)
−0.996425 + 0.0844765i \(0.973078\pi\)
\(858\) 0 0
\(859\) −34.1771 −1.16611 −0.583053 0.812434i \(-0.698142\pi\)
−0.583053 + 0.812434i \(0.698142\pi\)
\(860\) 0 0
\(861\) −16.8078 −0.572807
\(862\) 0 0
\(863\) 23.8078i 0.810426i 0.914222 + 0.405213i \(0.132803\pi\)
−0.914222 + 0.405213i \(0.867197\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 26.7386i − 0.908092i
\(868\) 0 0
\(869\) −14.1922 −0.481439
\(870\) 0 0
\(871\) 32.2462 1.09262
\(872\) 0 0
\(873\) − 21.3693i − 0.723242i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 23.6155i 0.797440i 0.917073 + 0.398720i \(0.130545\pi\)
−0.917073 + 0.398720i \(0.869455\pi\)
\(878\) 0 0
\(879\) 76.8466 2.59197
\(880\) 0 0
\(881\) −40.7386 −1.37252 −0.686260 0.727357i \(-0.740749\pi\)
−0.686260 + 0.727357i \(0.740749\pi\)
\(882\) 0 0
\(883\) 1.49242i 0.0502240i 0.999685 + 0.0251120i \(0.00799424\pi\)
−0.999685 + 0.0251120i \(0.992006\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 49.6155i − 1.66593i −0.553328 0.832963i \(-0.686643\pi\)
0.553328 0.832963i \(-0.313357\pi\)
\(888\) 0 0
\(889\) 9.80776 0.328942
\(890\) 0 0
\(891\) −14.8617 −0.497887
\(892\) 0 0
\(893\) 4.63068i 0.154960i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 28.4924i 0.951334i
\(898\) 0 0
\(899\) 6.63068 0.221146
\(900\) 0 0
\(901\) −18.2462 −0.607869
\(902\) 0 0
\(903\) − 6.24621i − 0.207861i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −39.4233 −1.30615 −0.653076 0.757292i \(-0.726522\pi\)
−0.653076 + 0.757292i \(0.726522\pi\)
\(912\) 0 0
\(913\) 29.0540i 0.961546i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1.75379i − 0.0579152i
\(918\) 0 0
\(919\) −18.1922 −0.600106 −0.300053 0.953922i \(-0.597004\pi\)
−0.300053 + 0.953922i \(0.597004\pi\)
\(920\) 0 0
\(921\) 11.0540 0.364241
\(922\) 0 0
\(923\) 21.3693i 0.703380i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 25.3693i 0.833238i
\(928\) 0 0
\(929\) −56.3542 −1.84892 −0.924460 0.381279i \(-0.875484\pi\)
−0.924460 + 0.381279i \(0.875484\pi\)
\(930\) 0 0
\(931\) −0.561553 −0.0184042
\(932\) 0 0
\(933\) − 7.36932i − 0.241261i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 21.3002i 0.695847i 0.937523 + 0.347923i \(0.113113\pi\)
−0.937523 + 0.347923i \(0.886887\pi\)
\(938\) 0 0
\(939\) −56.3542 −1.83905
\(940\) 0 0
\(941\) −1.36932 −0.0446385 −0.0223192 0.999751i \(-0.507105\pi\)
−0.0223192 + 0.999751i \(0.507105\pi\)
\(942\) 0 0
\(943\) 36.4924i 1.18836i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16.4924i 0.535932i 0.963428 + 0.267966i \(0.0863514\pi\)
−0.963428 + 0.267966i \(0.913649\pi\)
\(948\) 0 0
\(949\) −21.1231 −0.685685
\(950\) 0 0
\(951\) −52.9848 −1.71815
\(952\) 0 0
\(953\) 27.8769i 0.903021i 0.892266 + 0.451511i \(0.149115\pi\)
−0.892266 + 0.451511i \(0.850885\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 41.1231i 1.32932i
\(958\) 0 0
\(959\) 2.31534 0.0747663
\(960\) 0 0
\(961\) −30.2311 −0.975195
\(962\) 0 0
\(963\) 45.6155i 1.46994i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 16.6307i − 0.534807i −0.963585 0.267403i \(-0.913834\pi\)
0.963585 0.267403i \(-0.0861656\pi\)
\(968\) 0 0
\(969\) −3.68466 −0.118368
\(970\) 0 0
\(971\) −2.31534 −0.0743028 −0.0371514 0.999310i \(-0.511828\pi\)
−0.0371514 + 0.999310i \(0.511828\pi\)
\(972\) 0 0
\(973\) 19.6847i 0.631061i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 33.2462i − 1.06364i −0.846857 0.531820i \(-0.821508\pi\)
0.846857 0.531820i \(-0.178492\pi\)
\(978\) 0 0
\(979\) 22.9460 0.733358
\(980\) 0 0
\(981\) −31.8078 −1.01554
\(982\) 0 0
\(983\) 62.3542i 1.98879i 0.105734 + 0.994394i \(0.466281\pi\)
−0.105734 + 0.994394i \(0.533719\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 21.1231i 0.672356i
\(988\) 0 0
\(989\) −13.5616 −0.431232
\(990\) 0 0
\(991\) −29.6695 −0.942483 −0.471241 0.882004i \(-0.656194\pi\)
−0.471241 + 0.882004i \(0.656194\pi\)
\(992\) 0 0
\(993\) − 69.1619i − 2.19479i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 12.2462i 0.387841i 0.981017 + 0.193921i \(0.0621204\pi\)
−0.981017 + 0.193921i \(0.937880\pi\)
\(998\) 0 0
\(999\) 16.9848 0.537377
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1400.2.g.j.449.1 4
4.3 odd 2 2800.2.g.v.449.4 4
5.2 odd 4 1400.2.a.o.1.1 2
5.3 odd 4 1400.2.a.q.1.2 yes 2
5.4 even 2 inner 1400.2.g.j.449.4 4
20.3 even 4 2800.2.a.bj.1.1 2
20.7 even 4 2800.2.a.bo.1.2 2
20.19 odd 2 2800.2.g.v.449.1 4
35.13 even 4 9800.2.a.bt.1.1 2
35.27 even 4 9800.2.a.bx.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.a.o.1.1 2 5.2 odd 4
1400.2.a.q.1.2 yes 2 5.3 odd 4
1400.2.g.j.449.1 4 1.1 even 1 trivial
1400.2.g.j.449.4 4 5.4 even 2 inner
2800.2.a.bj.1.1 2 20.3 even 4
2800.2.a.bo.1.2 2 20.7 even 4
2800.2.g.v.449.1 4 20.19 odd 2
2800.2.g.v.449.4 4 4.3 odd 2
9800.2.a.bt.1.1 2 35.13 even 4
9800.2.a.bx.1.2 2 35.27 even 4