Properties

Label 4600.2.e.r.4049.1
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.24681024.1
Defining polynomial: \(x^{6} + 12 x^{4} + 36 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.1
Root \(-2.66908i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.r.4049.6

$q$-expansion

\(f(q)\) \(=\) \(q-2.66908i q^{3} +3.66908i q^{7} -4.12398 q^{9} +O(q^{10})\) \(q-2.66908i q^{3} +3.66908i q^{7} -4.12398 q^{9} -1.21417 q^{11} -2.21417i q^{13} +1.21417i q^{17} +2.57889 q^{19} +9.79306 q^{21} -1.00000i q^{23} +3.00000i q^{27} -1.45490 q^{29} -6.46214 q^{31} +3.24073i q^{33} +4.00000i q^{37} -5.90981 q^{39} -10.9170 q^{41} -6.90981i q^{43} +5.45490i q^{47} -6.46214 q^{49} +3.24073 q^{51} -3.81962i q^{53} -6.88325i q^{57} +4.24797 q^{59} +6.78583 q^{61} -15.1312i q^{63} +12.8567i q^{67} -2.66908 q^{69} -6.91705 q^{71} +15.2214i q^{73} -4.45490i q^{77} -4.36471 q^{81} +15.5861i q^{83} +3.88325i q^{87} +10.9098 q^{89} +8.12398 q^{91} +17.2480i q^{93} +1.69563i q^{97} +5.00724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9} + O(q^{10}) \) \( 6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 2.66908i − 1.54099i −0.637444 0.770497i \(-0.720008\pi\)
0.637444 0.770497i \(-0.279992\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.66908i 1.38678i 0.720562 + 0.693391i \(0.243884\pi\)
−0.720562 + 0.693391i \(0.756116\pi\)
\(8\) 0 0
\(9\) −4.12398 −1.37466
\(10\) 0 0
\(11\) −1.21417 −0.366088 −0.183044 0.983105i \(-0.558595\pi\)
−0.183044 + 0.983105i \(0.558595\pi\)
\(12\) 0 0
\(13\) − 2.21417i − 0.614102i −0.951693 0.307051i \(-0.900658\pi\)
0.951693 0.307051i \(-0.0993422\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.21417i 0.294481i 0.989101 + 0.147240i \(0.0470391\pi\)
−0.989101 + 0.147240i \(0.952961\pi\)
\(18\) 0 0
\(19\) 2.57889 0.591637 0.295819 0.955244i \(-0.404408\pi\)
0.295819 + 0.955244i \(0.404408\pi\)
\(20\) 0 0
\(21\) 9.79306 2.13702
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.00000i 0.577350i
\(28\) 0 0
\(29\) −1.45490 −0.270169 −0.135084 0.990834i \(-0.543131\pi\)
−0.135084 + 0.990834i \(0.543131\pi\)
\(30\) 0 0
\(31\) −6.46214 −1.16063 −0.580317 0.814390i \(-0.697072\pi\)
−0.580317 + 0.814390i \(0.697072\pi\)
\(32\) 0 0
\(33\) 3.24073i 0.564139i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) −5.90981 −0.946327
\(40\) 0 0
\(41\) −10.9170 −1.70496 −0.852478 0.522763i \(-0.824901\pi\)
−0.852478 + 0.522763i \(0.824901\pi\)
\(42\) 0 0
\(43\) − 6.90981i − 1.05374i −0.849947 0.526868i \(-0.823366\pi\)
0.849947 0.526868i \(-0.176634\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.45490i 0.795680i 0.917455 + 0.397840i \(0.130240\pi\)
−0.917455 + 0.397840i \(0.869760\pi\)
\(48\) 0 0
\(49\) −6.46214 −0.923163
\(50\) 0 0
\(51\) 3.24073 0.453793
\(52\) 0 0
\(53\) − 3.81962i − 0.524665i −0.964978 0.262332i \(-0.915508\pi\)
0.964978 0.262332i \(-0.0844917\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.88325i − 0.911709i
\(58\) 0 0
\(59\) 4.24797 0.553038 0.276519 0.961008i \(-0.410819\pi\)
0.276519 + 0.961008i \(0.410819\pi\)
\(60\) 0 0
\(61\) 6.78583 0.868836 0.434418 0.900711i \(-0.356954\pi\)
0.434418 + 0.900711i \(0.356954\pi\)
\(62\) 0 0
\(63\) − 15.1312i − 1.90635i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.8567i 1.57070i 0.619055 + 0.785348i \(0.287516\pi\)
−0.619055 + 0.785348i \(0.712484\pi\)
\(68\) 0 0
\(69\) −2.66908 −0.321319
\(70\) 0 0
\(71\) −6.91705 −0.820902 −0.410451 0.911883i \(-0.634629\pi\)
−0.410451 + 0.911883i \(0.634629\pi\)
\(72\) 0 0
\(73\) 15.2214i 1.78153i 0.454463 + 0.890766i \(0.349831\pi\)
−0.454463 + 0.890766i \(0.650169\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.45490i − 0.507683i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −4.36471 −0.484968
\(82\) 0 0
\(83\) 15.5861i 1.71080i 0.517968 + 0.855400i \(0.326688\pi\)
−0.517968 + 0.855400i \(0.673312\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.88325i 0.416329i
\(88\) 0 0
\(89\) 10.9098 1.15644 0.578219 0.815882i \(-0.303748\pi\)
0.578219 + 0.815882i \(0.303748\pi\)
\(90\) 0 0
\(91\) 8.12398 0.851625
\(92\) 0 0
\(93\) 17.2480i 1.78853i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.69563i 0.172165i 0.996288 + 0.0860827i \(0.0274349\pi\)
−0.996288 + 0.0860827i \(0.972565\pi\)
\(98\) 0 0
\(99\) 5.00724 0.503246
\(100\) 0 0
\(101\) −2.24797 −0.223681 −0.111841 0.993726i \(-0.535675\pi\)
−0.111841 + 0.993726i \(0.535675\pi\)
\(102\) 0 0
\(103\) 2.78583i 0.274495i 0.990537 + 0.137248i \(0.0438256\pi\)
−0.990537 + 0.137248i \(0.956174\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.1578i 1.27201i 0.771685 + 0.636005i \(0.219414\pi\)
−0.771685 + 0.636005i \(0.780586\pi\)
\(108\) 0 0
\(109\) −3.42111 −0.327683 −0.163842 0.986487i \(-0.552389\pi\)
−0.163842 + 0.986487i \(0.552389\pi\)
\(110\) 0 0
\(111\) 10.6763 1.01335
\(112\) 0 0
\(113\) 10.0000i 0.940721i 0.882474 + 0.470360i \(0.155876\pi\)
−0.882474 + 0.470360i \(0.844124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.13122i 0.844182i
\(118\) 0 0
\(119\) −4.45490 −0.408380
\(120\) 0 0
\(121\) −9.52578 −0.865980
\(122\) 0 0
\(123\) 29.1385i 2.62733i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 19.9508i 1.77035i 0.465258 + 0.885175i \(0.345962\pi\)
−0.465258 + 0.885175i \(0.654038\pi\)
\(128\) 0 0
\(129\) −18.4428 −1.62380
\(130\) 0 0
\(131\) −9.70287 −0.847744 −0.423872 0.905722i \(-0.639329\pi\)
−0.423872 + 0.905722i \(0.639329\pi\)
\(132\) 0 0
\(133\) 9.46214i 0.820472i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.82685i 0.241514i 0.992682 + 0.120757i \(0.0385322\pi\)
−0.992682 + 0.120757i \(0.961468\pi\)
\(138\) 0 0
\(139\) −2.36471 −0.200572 −0.100286 0.994959i \(-0.531976\pi\)
−0.100286 + 0.994959i \(0.531976\pi\)
\(140\) 0 0
\(141\) 14.5596 1.22614
\(142\) 0 0
\(143\) 2.68840i 0.224815i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 17.2480i 1.42259i
\(148\) 0 0
\(149\) 11.9170 0.976282 0.488141 0.872765i \(-0.337675\pi\)
0.488141 + 0.872765i \(0.337675\pi\)
\(150\) 0 0
\(151\) 9.57889 0.779519 0.389759 0.920917i \(-0.372558\pi\)
0.389759 + 0.920917i \(0.372558\pi\)
\(152\) 0 0
\(153\) − 5.00724i − 0.404811i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 18.7439i − 1.49593i −0.663740 0.747963i \(-0.731032\pi\)
0.663740 0.747963i \(-0.268968\pi\)
\(158\) 0 0
\(159\) −10.1949 −0.808505
\(160\) 0 0
\(161\) 3.66908 0.289164
\(162\) 0 0
\(163\) 19.3188i 1.51317i 0.653896 + 0.756584i \(0.273133\pi\)
−0.653896 + 0.756584i \(0.726867\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 1.81962i − 0.140806i −0.997519 0.0704031i \(-0.977571\pi\)
0.997519 0.0704031i \(-0.0224285\pi\)
\(168\) 0 0
\(169\) 8.09743 0.622879
\(170\) 0 0
\(171\) −10.6353 −0.813301
\(172\) 0 0
\(173\) − 15.9581i − 1.21327i −0.794981 0.606635i \(-0.792519\pi\)
0.794981 0.606635i \(-0.207481\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 11.3382i − 0.852228i
\(178\) 0 0
\(179\) −17.9508 −1.34171 −0.670854 0.741589i \(-0.734072\pi\)
−0.670854 + 0.741589i \(0.734072\pi\)
\(180\) 0 0
\(181\) 6.33092 0.470574 0.235287 0.971926i \(-0.424397\pi\)
0.235287 + 0.971926i \(0.424397\pi\)
\(182\) 0 0
\(183\) − 18.1119i − 1.33887i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 1.47422i − 0.107806i
\(188\) 0 0
\(189\) −11.0072 −0.800659
\(190\) 0 0
\(191\) −15.1578 −1.09678 −0.548389 0.836223i \(-0.684759\pi\)
−0.548389 + 0.836223i \(0.684759\pi\)
\(192\) 0 0
\(193\) − 18.9879i − 1.36678i −0.730053 0.683390i \(-0.760505\pi\)
0.730053 0.683390i \(-0.239495\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 19.1650i − 1.36545i −0.730675 0.682725i \(-0.760795\pi\)
0.730675 0.682725i \(-0.239205\pi\)
\(198\) 0 0
\(199\) 5.76651 0.408777 0.204388 0.978890i \(-0.434479\pi\)
0.204388 + 0.978890i \(0.434479\pi\)
\(200\) 0 0
\(201\) 34.3155 2.42043
\(202\) 0 0
\(203\) − 5.33816i − 0.374665i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.12398i 0.286637i
\(208\) 0 0
\(209\) −3.13122 −0.216591
\(210\) 0 0
\(211\) −17.4057 −1.19826 −0.599130 0.800652i \(-0.704487\pi\)
−0.599130 + 0.800652i \(0.704487\pi\)
\(212\) 0 0
\(213\) 18.4621i 1.26501i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 23.7101i − 1.60955i
\(218\) 0 0
\(219\) 40.6272 2.74533
\(220\) 0 0
\(221\) 2.68840 0.180841
\(222\) 0 0
\(223\) 14.6763i 0.982799i 0.870934 + 0.491399i \(0.163514\pi\)
−0.870934 + 0.491399i \(0.836486\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 16.4283i 1.09039i 0.838310 + 0.545194i \(0.183544\pi\)
−0.838310 + 0.545194i \(0.816456\pi\)
\(228\) 0 0
\(229\) −19.3526 −1.27886 −0.639429 0.768850i \(-0.720829\pi\)
−0.639429 + 0.768850i \(0.720829\pi\)
\(230\) 0 0
\(231\) −11.8905 −0.782337
\(232\) 0 0
\(233\) 19.4549i 1.27453i 0.770643 + 0.637267i \(0.219935\pi\)
−0.770643 + 0.637267i \(0.780065\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −23.7029 −1.53321 −0.766606 0.642118i \(-0.778056\pi\)
−0.766606 + 0.642118i \(0.778056\pi\)
\(240\) 0 0
\(241\) 0.233492 0.0150405 0.00752027 0.999972i \(-0.497606\pi\)
0.00752027 + 0.999972i \(0.497606\pi\)
\(242\) 0 0
\(243\) 20.6498i 1.32468i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 5.71011i − 0.363325i
\(248\) 0 0
\(249\) 41.6006 2.63633
\(250\) 0 0
\(251\) 21.7511 1.37292 0.686460 0.727168i \(-0.259164\pi\)
0.686460 + 0.727168i \(0.259164\pi\)
\(252\) 0 0
\(253\) 1.21417i 0.0763345i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 16.3116i − 1.01749i −0.860917 0.508745i \(-0.830110\pi\)
0.860917 0.508745i \(-0.169890\pi\)
\(258\) 0 0
\(259\) −14.6763 −0.911942
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 2.53786i 0.156491i 0.996934 + 0.0782455i \(0.0249318\pi\)
−0.996934 + 0.0782455i \(0.975068\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 29.1191i − 1.78206i
\(268\) 0 0
\(269\) 2.54510 0.155177 0.0775886 0.996985i \(-0.475278\pi\)
0.0775886 + 0.996985i \(0.475278\pi\)
\(270\) 0 0
\(271\) 24.1795 1.46880 0.734400 0.678717i \(-0.237464\pi\)
0.734400 + 0.678717i \(0.237464\pi\)
\(272\) 0 0
\(273\) − 21.6836i − 1.31235i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 21.2359i − 1.27594i −0.770061 0.637970i \(-0.779774\pi\)
0.770061 0.637970i \(-0.220226\pi\)
\(278\) 0 0
\(279\) 26.6498 1.59548
\(280\) 0 0
\(281\) −13.8872 −0.828441 −0.414220 0.910177i \(-0.635946\pi\)
−0.414220 + 0.910177i \(0.635946\pi\)
\(282\) 0 0
\(283\) 29.5330i 1.75556i 0.479068 + 0.877778i \(0.340975\pi\)
−0.479068 + 0.877778i \(0.659025\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 40.0555i − 2.36440i
\(288\) 0 0
\(289\) 15.5258 0.913281
\(290\) 0 0
\(291\) 4.52578 0.265306
\(292\) 0 0
\(293\) 24.0821i 1.40689i 0.710750 + 0.703444i \(0.248356\pi\)
−0.710750 + 0.703444i \(0.751644\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 3.64252i − 0.211361i
\(298\) 0 0
\(299\) −2.21417 −0.128049
\(300\) 0 0
\(301\) 25.3526 1.46130
\(302\) 0 0
\(303\) 6.00000i 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 19.9436i 1.13824i 0.822254 + 0.569121i \(0.192716\pi\)
−0.822254 + 0.569121i \(0.807284\pi\)
\(308\) 0 0
\(309\) 7.43559 0.422996
\(310\) 0 0
\(311\) −2.97345 −0.168609 −0.0843043 0.996440i \(-0.526867\pi\)
−0.0843043 + 0.996440i \(0.526867\pi\)
\(312\) 0 0
\(313\) − 15.4887i − 0.875473i −0.899103 0.437736i \(-0.855780\pi\)
0.899103 0.437736i \(-0.144220\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.82685i 0.495765i 0.968790 + 0.247883i \(0.0797348\pi\)
−0.968790 + 0.247883i \(0.920265\pi\)
\(318\) 0 0
\(319\) 1.76651 0.0989055
\(320\) 0 0
\(321\) 35.1191 1.96016
\(322\) 0 0
\(323\) 3.13122i 0.174226i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 9.13122i 0.504958i
\(328\) 0 0
\(329\) −20.0145 −1.10343
\(330\) 0 0
\(331\) 17.4018 0.956489 0.478245 0.878227i \(-0.341273\pi\)
0.478245 + 0.878227i \(0.341273\pi\)
\(332\) 0 0
\(333\) − 16.4959i − 0.903972i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 26.9541i 1.46828i 0.678995 + 0.734142i \(0.262416\pi\)
−0.678995 + 0.734142i \(0.737584\pi\)
\(338\) 0 0
\(339\) 26.6908 1.44964
\(340\) 0 0
\(341\) 7.84617 0.424894
\(342\) 0 0
\(343\) 1.97345i 0.106556i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 10.1240i − 0.543484i −0.962370 0.271742i \(-0.912400\pi\)
0.962370 0.271742i \(-0.0875997\pi\)
\(348\) 0 0
\(349\) −7.38732 −0.395434 −0.197717 0.980259i \(-0.563353\pi\)
−0.197717 + 0.980259i \(0.563353\pi\)
\(350\) 0 0
\(351\) 6.64252 0.354552
\(352\) 0 0
\(353\) − 13.8833i − 0.738931i −0.929244 0.369466i \(-0.879541\pi\)
0.929244 0.369466i \(-0.120459\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 11.8905i 0.629312i
\(358\) 0 0
\(359\) −15.0902 −0.796430 −0.398215 0.917292i \(-0.630370\pi\)
−0.398215 + 0.917292i \(0.630370\pi\)
\(360\) 0 0
\(361\) −12.3493 −0.649965
\(362\) 0 0
\(363\) 25.4251i 1.33447i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 24.0289i − 1.25430i −0.778898 0.627150i \(-0.784221\pi\)
0.778898 0.627150i \(-0.215779\pi\)
\(368\) 0 0
\(369\) 45.0217 2.34374
\(370\) 0 0
\(371\) 14.0145 0.727595
\(372\) 0 0
\(373\) − 30.6908i − 1.58911i −0.607193 0.794554i \(-0.707704\pi\)
0.607193 0.794554i \(-0.292296\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.22141i 0.165911i
\(378\) 0 0
\(379\) −2.07087 −0.106374 −0.0531869 0.998585i \(-0.516938\pi\)
−0.0531869 + 0.998585i \(0.516938\pi\)
\(380\) 0 0
\(381\) 53.2504 2.72810
\(382\) 0 0
\(383\) 1.58612i 0.0810472i 0.999179 + 0.0405236i \(0.0129026\pi\)
−0.999179 + 0.0405236i \(0.987097\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 28.4959i 1.44853i
\(388\) 0 0
\(389\) −21.2142 −1.07560 −0.537801 0.843072i \(-0.680745\pi\)
−0.537801 + 0.843072i \(0.680745\pi\)
\(390\) 0 0
\(391\) 1.21417 0.0614035
\(392\) 0 0
\(393\) 25.8977i 1.30637i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 1.93965i − 0.0973485i −0.998815 0.0486742i \(-0.984500\pi\)
0.998815 0.0486742i \(-0.0154996\pi\)
\(398\) 0 0
\(399\) 25.2552 1.26434
\(400\) 0 0
\(401\) −5.69893 −0.284591 −0.142295 0.989824i \(-0.545448\pi\)
−0.142295 + 0.989824i \(0.545448\pi\)
\(402\) 0 0
\(403\) 14.3083i 0.712748i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 4.85670i − 0.240738i
\(408\) 0 0
\(409\) −3.53786 −0.174936 −0.0874679 0.996167i \(-0.527878\pi\)
−0.0874679 + 0.996167i \(0.527878\pi\)
\(410\) 0 0
\(411\) 7.54510 0.372172
\(412\) 0 0
\(413\) 15.5861i 0.766943i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.31160i 0.309081i
\(418\) 0 0
\(419\) −28.2624 −1.38071 −0.690355 0.723471i \(-0.742546\pi\)
−0.690355 + 0.723471i \(0.742546\pi\)
\(420\) 0 0
\(421\) 34.0974 1.66181 0.830904 0.556417i \(-0.187824\pi\)
0.830904 + 0.556417i \(0.187824\pi\)
\(422\) 0 0
\(423\) − 22.4959i − 1.09379i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 24.8977i 1.20489i
\(428\) 0 0
\(429\) 7.17554 0.346438
\(430\) 0 0
\(431\) −13.2850 −0.639918 −0.319959 0.947431i \(-0.603669\pi\)
−0.319959 + 0.947431i \(0.603669\pi\)
\(432\) 0 0
\(433\) − 11.0603i − 0.531526i −0.964038 0.265763i \(-0.914376\pi\)
0.964038 0.265763i \(-0.0856239\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 2.57889i − 0.123365i
\(438\) 0 0
\(439\) 19.1916 0.915963 0.457982 0.888962i \(-0.348573\pi\)
0.457982 + 0.888962i \(0.348573\pi\)
\(440\) 0 0
\(441\) 26.6498 1.26904
\(442\) 0 0
\(443\) − 24.2818i − 1.15366i −0.816864 0.576831i \(-0.804289\pi\)
0.816864 0.576831i \(-0.195711\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 31.8075i − 1.50444i
\(448\) 0 0
\(449\) −14.8413 −0.700406 −0.350203 0.936674i \(-0.613887\pi\)
−0.350203 + 0.936674i \(0.613887\pi\)
\(450\) 0 0
\(451\) 13.2552 0.624163
\(452\) 0 0
\(453\) − 25.5668i − 1.20123i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 15.5861i 0.729088i 0.931186 + 0.364544i \(0.118775\pi\)
−0.931186 + 0.364544i \(0.881225\pi\)
\(458\) 0 0
\(459\) −3.64252 −0.170019
\(460\) 0 0
\(461\) −14.1312 −0.658157 −0.329078 0.944303i \(-0.606738\pi\)
−0.329078 + 0.944303i \(0.606738\pi\)
\(462\) 0 0
\(463\) − 26.7584i − 1.24357i −0.783189 0.621784i \(-0.786408\pi\)
0.783189 0.621784i \(-0.213592\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.6908i 1.23510i 0.786531 + 0.617551i \(0.211875\pi\)
−0.786531 + 0.617551i \(0.788125\pi\)
\(468\) 0 0
\(469\) −47.1722 −2.17821
\(470\) 0 0
\(471\) −50.0289 −2.30521
\(472\) 0 0
\(473\) 8.38972i 0.385760i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 15.7520i 0.721236i
\(478\) 0 0
\(479\) −4.90981 −0.224335 −0.112167 0.993689i \(-0.535779\pi\)
−0.112167 + 0.993689i \(0.535779\pi\)
\(480\) 0 0
\(481\) 8.85670 0.403831
\(482\) 0 0
\(483\) − 9.79306i − 0.445600i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 16.7786i 0.760310i 0.924923 + 0.380155i \(0.124129\pi\)
−0.924923 + 0.380155i \(0.875871\pi\)
\(488\) 0 0
\(489\) 51.5635 2.33178
\(490\) 0 0
\(491\) 41.4839 1.87214 0.936070 0.351814i \(-0.114435\pi\)
0.936070 + 0.351814i \(0.114435\pi\)
\(492\) 0 0
\(493\) − 1.76651i − 0.0795595i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 25.3792i − 1.13841i
\(498\) 0 0
\(499\) 6.47751 0.289973 0.144987 0.989434i \(-0.453686\pi\)
0.144987 + 0.989434i \(0.453686\pi\)
\(500\) 0 0
\(501\) −4.85670 −0.216981
\(502\) 0 0
\(503\) − 37.3825i − 1.66680i −0.552669 0.833401i \(-0.686390\pi\)
0.552669 0.833401i \(-0.313610\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 21.6127i − 0.959853i
\(508\) 0 0
\(509\) 15.0410 0.666682 0.333341 0.942806i \(-0.391824\pi\)
0.333341 + 0.942806i \(0.391824\pi\)
\(510\) 0 0
\(511\) −55.8486 −2.47060
\(512\) 0 0
\(513\) 7.73666i 0.341582i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 6.62321i − 0.291288i
\(518\) 0 0
\(519\) −42.5934 −1.86964
\(520\) 0 0
\(521\) 15.9469 0.698646 0.349323 0.937002i \(-0.386412\pi\)
0.349323 + 0.937002i \(0.386412\pi\)
\(522\) 0 0
\(523\) − 10.8567i − 0.474730i −0.971420 0.237365i \(-0.923716\pi\)
0.971420 0.237365i \(-0.0762838\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 7.84617i − 0.341785i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −17.5185 −0.760240
\(532\) 0 0
\(533\) 24.1722i 1.04702i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 47.9122i 2.06756i
\(538\) 0 0
\(539\) 7.84617 0.337958
\(540\) 0 0
\(541\) −30.3792 −1.30610 −0.653052 0.757313i \(-0.726512\pi\)
−0.653052 + 0.757313i \(0.726512\pi\)
\(542\) 0 0
\(543\) − 16.8977i − 0.725151i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.4090i 1.21468i 0.794441 + 0.607341i \(0.207764\pi\)
−0.794441 + 0.607341i \(0.792236\pi\)
\(548\) 0 0
\(549\) −27.9846 −1.19435
\(550\) 0 0
\(551\) −3.75203 −0.159842
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 36.7584i 1.55750i 0.627333 + 0.778751i \(0.284147\pi\)
−0.627333 + 0.778751i \(0.715853\pi\)
\(558\) 0 0
\(559\) −15.2995 −0.647101
\(560\) 0 0
\(561\) −3.93481 −0.166128
\(562\) 0 0
\(563\) 3.03708i 0.127998i 0.997950 + 0.0639989i \(0.0203854\pi\)
−0.997950 + 0.0639989i \(0.979615\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 16.0145i − 0.672545i
\(568\) 0 0
\(569\) 32.2093 1.35029 0.675143 0.737687i \(-0.264082\pi\)
0.675143 + 0.737687i \(0.264082\pi\)
\(570\) 0 0
\(571\) 27.7777 1.16246 0.581230 0.813739i \(-0.302572\pi\)
0.581230 + 0.813739i \(0.302572\pi\)
\(572\) 0 0
\(573\) 40.4573i 1.69013i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 11.2890i 0.469967i 0.971999 + 0.234984i \(0.0755036\pi\)
−0.971999 + 0.234984i \(0.924496\pi\)
\(578\) 0 0
\(579\) −50.6803 −2.10620
\(580\) 0 0
\(581\) −57.1867 −2.37251
\(582\) 0 0
\(583\) 4.63768i 0.192073i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.6609i 0.976592i 0.872678 + 0.488296i \(0.162381\pi\)
−0.872678 + 0.488296i \(0.837619\pi\)
\(588\) 0 0
\(589\) −16.6651 −0.686675
\(590\) 0 0
\(591\) −51.1529 −2.10415
\(592\) 0 0
\(593\) 17.8341i 0.732358i 0.930544 + 0.366179i \(0.119334\pi\)
−0.930544 + 0.366179i \(0.880666\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 15.3913i − 0.629923i
\(598\) 0 0
\(599\) 23.8308 0.973700 0.486850 0.873486i \(-0.338146\pi\)
0.486850 + 0.873486i \(0.338146\pi\)
\(600\) 0 0
\(601\) 12.5297 0.511098 0.255549 0.966796i \(-0.417744\pi\)
0.255549 + 0.966796i \(0.417744\pi\)
\(602\) 0 0
\(603\) − 53.0208i − 2.15917i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 23.1722i − 0.940533i −0.882525 0.470266i \(-0.844158\pi\)
0.882525 0.470266i \(-0.155842\pi\)
\(608\) 0 0
\(609\) −14.2480 −0.577357
\(610\) 0 0
\(611\) 12.0781 0.488628
\(612\) 0 0
\(613\) 44.5780i 1.80049i 0.435386 + 0.900244i \(0.356612\pi\)
−0.435386 + 0.900244i \(0.643388\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 32.1385i − 1.29385i −0.762556 0.646923i \(-0.776056\pi\)
0.762556 0.646923i \(-0.223944\pi\)
\(618\) 0 0
\(619\) −31.5708 −1.26894 −0.634468 0.772949i \(-0.718781\pi\)
−0.634468 + 0.772949i \(0.718781\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) 40.0289i 1.60373i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 8.35748i 0.333765i
\(628\) 0 0
\(629\) −4.85670 −0.193649
\(630\) 0 0
\(631\) 24.0145 0.956001 0.478001 0.878360i \(-0.341362\pi\)
0.478001 + 0.878360i \(0.341362\pi\)
\(632\) 0 0
\(633\) 46.4573i 1.84651i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 14.3083i 0.566916i
\(638\) 0 0
\(639\) 28.5258 1.12846
\(640\) 0 0
\(641\) −10.4428 −0.412467 −0.206233 0.978503i \(-0.566121\pi\)
−0.206233 + 0.978503i \(0.566121\pi\)
\(642\) 0 0
\(643\) − 4.39940i − 0.173495i −0.996230 0.0867477i \(-0.972353\pi\)
0.996230 0.0867477i \(-0.0276474\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 23.6498i − 0.929768i −0.885372 0.464884i \(-0.846096\pi\)
0.885372 0.464884i \(-0.153904\pi\)
\(648\) 0 0
\(649\) −5.15777 −0.202460
\(650\) 0 0
\(651\) −63.2842 −2.48030
\(652\) 0 0
\(653\) − 40.9194i − 1.60130i −0.599131 0.800651i \(-0.704487\pi\)
0.599131 0.800651i \(-0.295513\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 62.7728i − 2.44900i
\(658\) 0 0
\(659\) −48.8567 −1.90319 −0.951593 0.307360i \(-0.900555\pi\)
−0.951593 + 0.307360i \(0.900555\pi\)
\(660\) 0 0
\(661\) −12.1529 −0.472694 −0.236347 0.971669i \(-0.575950\pi\)
−0.236347 + 0.971669i \(0.575950\pi\)
\(662\) 0 0
\(663\) − 7.17554i − 0.278675i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.45490i 0.0563341i
\(668\) 0 0
\(669\) 39.1722 1.51449
\(670\) 0 0
\(671\) −8.23918 −0.318070
\(672\) 0 0
\(673\) − 9.20694i − 0.354901i −0.984130 0.177451i \(-0.943215\pi\)
0.984130 0.177451i \(-0.0567850\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 39.8196i − 1.53039i −0.643797 0.765196i \(-0.722642\pi\)
0.643797 0.765196i \(-0.277358\pi\)
\(678\) 0 0
\(679\) −6.22141 −0.238756
\(680\) 0 0
\(681\) 43.8486 1.68028
\(682\) 0 0
\(683\) 5.60544i 0.214486i 0.994233 + 0.107243i \(0.0342023\pi\)
−0.994233 + 0.107243i \(0.965798\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 51.6537i 1.97071i
\(688\) 0 0
\(689\) −8.45730 −0.322197
\(690\) 0 0
\(691\) −2.84223 −0.108123 −0.0540617 0.998538i \(-0.517217\pi\)
−0.0540617 + 0.998538i \(0.517217\pi\)
\(692\) 0 0
\(693\) 18.3719i 0.697893i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 13.2552i − 0.502077i
\(698\) 0 0
\(699\) 51.9267 1.96405
\(700\) 0 0
\(701\) −24.4130 −0.922065 −0.461033 0.887383i \(-0.652521\pi\)
−0.461033 + 0.887383i \(0.652521\pi\)
\(702\) 0 0
\(703\) 10.3155i 0.389058i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 8.24797i − 0.310197i
\(708\) 0 0
\(709\) −0.123983 −0.00465629 −0.00232814 0.999997i \(-0.500741\pi\)
−0.00232814 + 0.999997i \(0.500741\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6.46214i 0.242009i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 63.2648i 2.36267i
\(718\) 0 0
\(719\) −47.1569 −1.75865 −0.879327 0.476218i \(-0.842007\pi\)
−0.879327 + 0.476218i \(0.842007\pi\)
\(720\) 0 0
\(721\) −10.2214 −0.380665
\(722\) 0 0
\(723\) − 0.623208i − 0.0231774i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 43.2962i 1.60577i 0.596135 + 0.802884i \(0.296702\pi\)
−0.596135 + 0.802884i \(0.703298\pi\)
\(728\) 0 0
\(729\) 42.0217 1.55636
\(730\) 0 0
\(731\) 8.38972 0.310305
\(732\) 0 0
\(733\) − 11.5861i − 0.427943i −0.976840 0.213972i \(-0.931360\pi\)
0.976840 0.213972i \(-0.0686400\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 15.6103i − 0.575012i
\(738\) 0 0
\(739\) 30.3792 1.11752 0.558758 0.829331i \(-0.311278\pi\)
0.558758 + 0.829331i \(0.311278\pi\)
\(740\) 0 0
\(741\) −15.2407 −0.559882
\(742\) 0 0
\(743\) − 35.5708i − 1.30496i −0.757804 0.652482i \(-0.773728\pi\)
0.757804 0.652482i \(-0.226272\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 64.2769i − 2.35177i
\(748\) 0 0
\(749\) −48.2769 −1.76400
\(750\) 0 0
\(751\) −24.9774 −0.911438 −0.455719 0.890124i \(-0.650618\pi\)
−0.455719 + 0.890124i \(0.650618\pi\)
\(752\) 0 0
\(753\) − 58.0555i − 2.11566i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 9.35263i 0.339927i 0.985450 + 0.169964i \(0.0543650\pi\)
−0.985450 + 0.169964i \(0.945635\pi\)
\(758\) 0 0
\(759\) 3.24073 0.117631
\(760\) 0 0
\(761\) −18.1466 −0.657813 −0.328907 0.944362i \(-0.606680\pi\)
−0.328907 + 0.944362i \(0.606680\pi\)
\(762\) 0 0
\(763\) − 12.5523i − 0.454425i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 9.40574i − 0.339622i
\(768\) 0 0
\(769\) −44.3445 −1.59910 −0.799552 0.600597i \(-0.794930\pi\)
−0.799552 + 0.600597i \(0.794930\pi\)
\(770\) 0 0
\(771\) −43.5370 −1.56795
\(772\) 0 0
\(773\) 10.8036i 0.388578i 0.980944 + 0.194289i \(0.0622400\pi\)
−0.980944 + 0.194289i \(0.937760\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 39.1722i 1.40530i
\(778\) 0 0
\(779\) −28.1538 −1.00872
\(780\) 0 0
\(781\) 8.39850 0.300522
\(782\) 0 0
\(783\) − 4.36471i − 0.155982i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 31.7810i − 1.13287i −0.824107 0.566435i \(-0.808322\pi\)
0.824107 0.566435i \(-0.191678\pi\)
\(788\) 0 0
\(789\) 6.77375 0.241152
\(790\) 0 0
\(791\) −36.6908 −1.30457
\(792\) 0 0
\(793\) − 15.0250i − 0.533554i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 35.0594i 1.24187i 0.783862 + 0.620935i \(0.213247\pi\)
−0.783862 + 0.620935i \(0.786753\pi\)
\(798\) 0 0
\(799\) −6.62321 −0.234312
\(800\) 0 0
\(801\) −44.9919 −1.58971
\(802\) 0 0
\(803\) − 18.4815i − 0.652196i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 6.79306i − 0.239127i
\(808\) 0 0
\(809\) 51.4911 1.81033 0.905165 0.425060i \(-0.139747\pi\)
0.905165 + 0.425060i \(0.139747\pi\)
\(810\) 0 0
\(811\) −19.7705 −0.694235 −0.347117 0.937822i \(-0.612839\pi\)
−0.347117 + 0.937822i \(0.612839\pi\)
\(812\) 0 0
\(813\) − 64.5370i − 2.26341i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 17.8196i − 0.623429i
\(818\) 0 0
\(819\) −33.5032 −1.17070
\(820\) 0 0
\(821\) 31.8486 1.11152 0.555761 0.831342i \(-0.312427\pi\)
0.555761 + 0.831342i \(0.312427\pi\)
\(822\) 0 0
\(823\) − 9.34210i − 0.325645i −0.986655 0.162823i \(-0.947940\pi\)
0.986655 0.162823i \(-0.0520598\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 30.9243i − 1.07534i −0.843155 0.537671i \(-0.819304\pi\)
0.843155 0.537671i \(-0.180696\pi\)
\(828\) 0 0
\(829\) 32.5701 1.13121 0.565603 0.824678i \(-0.308643\pi\)
0.565603 + 0.824678i \(0.308643\pi\)
\(830\) 0 0
\(831\) −56.6803 −1.96622
\(832\) 0 0
\(833\) − 7.84617i − 0.271854i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 19.3864i − 0.670093i
\(838\) 0 0
\(839\) −48.5635 −1.67660 −0.838299 0.545210i \(-0.816450\pi\)
−0.838299 + 0.545210i \(0.816450\pi\)
\(840\) 0 0
\(841\) −26.8833 −0.927009
\(842\) 0 0
\(843\) 37.0660i 1.27662i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 34.9508i − 1.20092i
\(848\) 0 0
\(849\) 78.8260 2.70530
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) 33.6546i 1.15231i 0.817340 + 0.576156i \(0.195448\pi\)
−0.817340 + 0.576156i \(0.804552\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.4081i 0.833766i 0.908960 + 0.416883i \(0.136878\pi\)
−0.908960 + 0.416883i \(0.863122\pi\)
\(858\) 0 0
\(859\) 11.1394 0.380070 0.190035 0.981777i \(-0.439140\pi\)
0.190035 + 0.981777i \(0.439140\pi\)
\(860\) 0 0
\(861\) −106.911 −3.64353
\(862\) 0 0
\(863\) 15.7173i 0.535025i 0.963554 + 0.267512i \(0.0862016\pi\)
−0.963554 + 0.267512i \(0.913798\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 41.4395i − 1.40736i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 28.4670 0.964567
\(872\) 0 0
\(873\) − 6.99276i − 0.236669i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 33.2818i − 1.12385i −0.827190 0.561923i \(-0.810062\pi\)
0.827190 0.561923i \(-0.189938\pi\)
\(878\) 0 0
\(879\) 64.2769 2.16801
\(880\) 0 0
\(881\) 8.99187 0.302944 0.151472 0.988462i \(-0.451599\pi\)
0.151472 + 0.988462i \(0.451599\pi\)
\(882\) 0 0
\(883\) − 58.5258i − 1.96955i −0.173836 0.984775i \(-0.555616\pi\)
0.173836 0.984775i \(-0.444384\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32.8220i 1.10206i 0.834487 + 0.551028i \(0.185764\pi\)
−0.834487 + 0.551028i \(0.814236\pi\)
\(888\) 0 0
\(889\) −73.2012 −2.45509
\(890\) 0 0
\(891\) 5.29952 0.177541
\(892\) 0 0
\(893\) 14.0676i 0.470754i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 5.90981i 0.197323i
\(898\) 0 0
\(899\) 9.40180 0.313567
\(900\) 0 0
\(901\) 4.63768 0.154504
\(902\) 0 0
\(903\) − 67.6682i − 2.25186i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 28.3300i 0.940683i 0.882484 + 0.470341i \(0.155869\pi\)
−0.882484 + 0.470341i \(0.844131\pi\)
\(908\) 0 0
\(909\) 9.27058 0.307486
\(910\) 0 0
\(911\) 15.3382 0.508176 0.254088 0.967181i \(-0.418225\pi\)
0.254088 + 0.967181i \(0.418225\pi\)
\(912\) 0 0
\(913\) − 18.9243i − 0.626302i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 35.6006i − 1.17564i
\(918\) 0 0
\(919\) 38.8036 1.28001 0.640006 0.768370i \(-0.278932\pi\)
0.640006 + 0.768370i \(0.278932\pi\)
\(920\) 0 0
\(921\) 53.2310 1.75402
\(922\) 0 0
\(923\) 15.3155i 0.504117i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 11.4887i − 0.377338i
\(928\) 0 0
\(929\) −4.01053 −0.131581 −0.0657906 0.997833i \(-0.520957\pi\)
−0.0657906 + 0.997833i \(0.520957\pi\)
\(930\) 0 0
\(931\) −16.6651 −0.546178
\(932\) 0 0
\(933\) 7.93636i 0.259825i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 43.6836i 1.42708i 0.700615 + 0.713540i \(0.252909\pi\)
−0.700615 + 0.713540i \(0.747091\pi\)
\(938\) 0 0
\(939\) −41.3406 −1.34910
\(940\) 0 0
\(941\) −24.8534 −0.810198 −0.405099 0.914273i \(-0.632763\pi\)
−0.405099 + 0.914273i \(0.632763\pi\)
\(942\) 0 0
\(943\) 10.9170i 0.355508i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 22.1690i − 0.720394i −0.932876 0.360197i \(-0.882709\pi\)
0.932876 0.360197i \(-0.117291\pi\)
\(948\) 0 0
\(949\) 33.7029 1.09404
\(950\) 0 0
\(951\) 23.5596 0.763971
\(952\) 0 0
\(953\) − 15.9050i − 0.515212i −0.966250 0.257606i \(-0.917066\pi\)
0.966250 0.257606i \(-0.0829337\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 4.71495i − 0.152413i
\(958\) 0 0
\(959\) −10.3719 −0.334928
\(960\) 0 0
\(961\) 10.7593 0.347073
\(962\) 0 0
\(963\) − 54.2624i − 1.74858i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 50.6272i 1.62806i 0.580823 + 0.814030i \(0.302731\pi\)
−0.580823 + 0.814030i \(0.697269\pi\)
\(968\) 0 0
\(969\) 8.35748 0.268481
\(970\) 0 0
\(971\) −52.4009 −1.68162 −0.840812 0.541327i \(-0.817922\pi\)
−0.840812 + 0.541327i \(0.817922\pi\)
\(972\) 0 0
\(973\) − 8.67632i − 0.278150i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10.5258i 0.336750i 0.985723 + 0.168375i \(0.0538519\pi\)
−0.985723 + 0.168375i \(0.946148\pi\)
\(978\) 0 0
\(979\) −13.2464 −0.423357
\(980\) 0 0
\(981\) 14.1086 0.450453
\(982\) 0 0
\(983\) − 9.44767i − 0.301334i −0.988585 0.150667i \(-0.951858\pi\)
0.988585 0.150667i \(-0.0481421\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 53.4202i 1.70038i
\(988\) 0 0
\(989\) −6.90981 −0.219719
\(990\) 0 0
\(991\) 33.3252 1.05861 0.529305 0.848432i \(-0.322453\pi\)
0.529305 + 0.848432i \(0.322453\pi\)
\(992\) 0 0
\(993\) − 46.4468i − 1.47394i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 54.0289i 1.71111i 0.517709 + 0.855557i \(0.326785\pi\)
−0.517709 + 0.855557i \(0.673215\pi\)
\(998\) 0 0
\(999\) −12.0000 −0.379663
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 4600.2.e.r.4049.1 6
5.2 odd 4 4600.2.a.y.1.1 3
5.3 odd 4 920.2.a.g.1.3 3
5.4 even 2 inner 4600.2.e.r.4049.6 6
15.8 even 4 8280.2.a.bo.1.3 3
20.3 even 4 1840.2.a.t.1.1 3
20.7 even 4 9200.2.a.cd.1.3 3
40.3 even 4 7360.2.a.ca.1.3 3
40.13 odd 4 7360.2.a.cb.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.g.1.3 3 5.3 odd 4
1840.2.a.t.1.1 3 20.3 even 4
4600.2.a.y.1.1 3 5.2 odd 4
4600.2.e.r.4049.1 6 1.1 even 1 trivial
4600.2.e.r.4049.6 6 5.4 even 2 inner
7360.2.a.ca.1.3 3 40.3 even 4
7360.2.a.cb.1.1 3 40.13 odd 4
8280.2.a.bo.1.3 3 15.8 even 4
9200.2.a.cd.1.3 3 20.7 even 4