Properties

Label 6900.2.f.r.6349.6
Level $6900$
Weight $2$
Character 6900.6349
Analytic conductor $55.097$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6900.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(55.0967773947\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.158155776.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 6349.6
Root \(-1.42234 - 1.42234i\) of defining polynomial
Character \(\chi\) \(=\) 6900.6349
Dual form 6900.2.f.r.6349.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +3.73549i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +3.73549i q^{7} -1.00000 q^{9} +4.84469 q^{11} -2.84469i q^{13} -0.890804i q^{17} -6.84469 q^{19} -3.73549 q^{21} +1.00000i q^{23} -1.00000i q^{27} +0.890804 q^{29} +7.73549 q^{31} +4.84469i q^{33} +1.95388i q^{37} +2.84469 q^{39} +12.3618 q^{41} -3.47098i q^{43} +6.62629i q^{47} -6.95388 q^{49} +0.890804 q^{51} +12.3618i q^{53} -6.84469i q^{57} -0.890804 q^{59} +8.62629 q^{61} -3.73549i q^{63} -7.73549i q^{67} -1.00000 q^{69} -12.3618 q^{71} +16.5341i q^{73} +18.0973i q^{77} -13.4710 q^{79} +1.00000 q^{81} -4.58018i q^{83} +0.890804i q^{87} +15.1604 q^{89} +10.6263 q^{91} +7.73549i q^{93} +17.2526i q^{97} -4.84469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 8 q^{11} - 20 q^{19} + 4 q^{21} + 20 q^{31} - 4 q^{39} + 16 q^{41} - 26 q^{49} + 20 q^{61} - 6 q^{69} - 16 q^{71} - 28 q^{79} + 6 q^{81} - 4 q^{89} + 32 q^{91} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1201\) \(3451\) \(4601\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.73549i 1.41188i 0.708270 + 0.705941i \(0.249476\pi\)
−0.708270 + 0.705941i \(0.750524\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.84469 1.46073 0.730364 0.683058i \(-0.239351\pi\)
0.730364 + 0.683058i \(0.239351\pi\)
\(12\) 0 0
\(13\) − 2.84469i − 0.788974i −0.918902 0.394487i \(-0.870922\pi\)
0.918902 0.394487i \(-0.129078\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.890804i − 0.216052i −0.994148 0.108026i \(-0.965547\pi\)
0.994148 0.108026i \(-0.0344529\pi\)
\(18\) 0 0
\(19\) −6.84469 −1.57028 −0.785139 0.619319i \(-0.787409\pi\)
−0.785139 + 0.619319i \(0.787409\pi\)
\(20\) 0 0
\(21\) −3.73549 −0.815151
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 0.890804 0.165418 0.0827091 0.996574i \(-0.473643\pi\)
0.0827091 + 0.996574i \(0.473643\pi\)
\(30\) 0 0
\(31\) 7.73549 1.38933 0.694667 0.719331i \(-0.255552\pi\)
0.694667 + 0.719331i \(0.255552\pi\)
\(32\) 0 0
\(33\) 4.84469i 0.843352i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.95388i 0.321216i 0.987018 + 0.160608i \(0.0513455\pi\)
−0.987018 + 0.160608i \(0.948654\pi\)
\(38\) 0 0
\(39\) 2.84469 0.455514
\(40\) 0 0
\(41\) 12.3618 1.93059 0.965293 0.261169i \(-0.0841080\pi\)
0.965293 + 0.261169i \(0.0841080\pi\)
\(42\) 0 0
\(43\) − 3.47098i − 0.529319i −0.964342 0.264660i \(-0.914740\pi\)
0.964342 0.264660i \(-0.0852596\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.62629i 0.966544i 0.875470 + 0.483272i \(0.160552\pi\)
−0.875470 + 0.483272i \(0.839448\pi\)
\(48\) 0 0
\(49\) −6.95388 −0.993412
\(50\) 0 0
\(51\) 0.890804 0.124737
\(52\) 0 0
\(53\) 12.3618i 1.69802i 0.528376 + 0.849011i \(0.322801\pi\)
−0.528376 + 0.849011i \(0.677199\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.84469i − 0.906601i
\(58\) 0 0
\(59\) −0.890804 −0.115973 −0.0579864 0.998317i \(-0.518468\pi\)
−0.0579864 + 0.998317i \(0.518468\pi\)
\(60\) 0 0
\(61\) 8.62629 1.10448 0.552242 0.833684i \(-0.313773\pi\)
0.552242 + 0.833684i \(0.313773\pi\)
\(62\) 0 0
\(63\) − 3.73549i − 0.470627i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 7.73549i − 0.945040i −0.881320 0.472520i \(-0.843344\pi\)
0.881320 0.472520i \(-0.156656\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −12.3618 −1.46707 −0.733537 0.679650i \(-0.762132\pi\)
−0.733537 + 0.679650i \(0.762132\pi\)
\(72\) 0 0
\(73\) 16.5341i 1.93516i 0.252555 + 0.967582i \(0.418729\pi\)
−0.252555 + 0.967582i \(0.581271\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 18.0973i 2.06238i
\(78\) 0 0
\(79\) −13.4710 −1.51560 −0.757802 0.652485i \(-0.773727\pi\)
−0.757802 + 0.652485i \(0.773727\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 4.58018i − 0.502740i −0.967891 0.251370i \(-0.919119\pi\)
0.967891 0.251370i \(-0.0808810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.890804i 0.0955042i
\(88\) 0 0
\(89\) 15.1604 1.60699 0.803497 0.595309i \(-0.202970\pi\)
0.803497 + 0.595309i \(0.202970\pi\)
\(90\) 0 0
\(91\) 10.6263 1.11394
\(92\) 0 0
\(93\) 7.73549i 0.802133i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.2526i 1.75173i 0.482552 + 0.875867i \(0.339710\pi\)
−0.482552 + 0.875867i \(0.660290\pi\)
\(98\) 0 0
\(99\) −4.84469 −0.486909
\(100\) 0 0
\(101\) −2.67241 −0.265915 −0.132957 0.991122i \(-0.542447\pi\)
−0.132957 + 0.991122i \(0.542447\pi\)
\(102\) 0 0
\(103\) 6.21839i 0.612716i 0.951916 + 0.306358i \(0.0991105\pi\)
−0.951916 + 0.306358i \(0.900889\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.7986i 1.43063i 0.698801 + 0.715316i \(0.253717\pi\)
−0.698801 + 0.715316i \(0.746283\pi\)
\(108\) 0 0
\(109\) 12.5341 1.20054 0.600272 0.799796i \(-0.295059\pi\)
0.600272 + 0.799796i \(0.295059\pi\)
\(110\) 0 0
\(111\) −1.95388 −0.185454
\(112\) 0 0
\(113\) 12.3618i 1.16290i 0.813583 + 0.581449i \(0.197514\pi\)
−0.813583 + 0.581449i \(0.802486\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.84469i 0.262991i
\(118\) 0 0
\(119\) 3.32759 0.305040
\(120\) 0 0
\(121\) 12.4710 1.13373
\(122\) 0 0
\(123\) 12.3618i 1.11462i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 11.7866i − 1.04590i −0.852365 0.522948i \(-0.824832\pi\)
0.852365 0.522948i \(-0.175168\pi\)
\(128\) 0 0
\(129\) 3.47098 0.305603
\(130\) 0 0
\(131\) −17.4710 −1.52645 −0.763223 0.646135i \(-0.776384\pi\)
−0.763223 + 0.646135i \(0.776384\pi\)
\(132\) 0 0
\(133\) − 25.5683i − 2.21705i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 15.1604i − 1.29524i −0.761965 0.647618i \(-0.775765\pi\)
0.761965 0.647618i \(-0.224235\pi\)
\(138\) 0 0
\(139\) 5.51710 0.467954 0.233977 0.972242i \(-0.424826\pi\)
0.233977 + 0.972242i \(0.424826\pi\)
\(140\) 0 0
\(141\) −6.62629 −0.558035
\(142\) 0 0
\(143\) − 13.7816i − 1.15248i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 6.95388i − 0.573547i
\(148\) 0 0
\(149\) −1.15531 −0.0946470 −0.0473235 0.998880i \(-0.515069\pi\)
−0.0473235 + 0.998880i \(0.515069\pi\)
\(150\) 0 0
\(151\) −15.4710 −1.25901 −0.629505 0.776996i \(-0.716742\pi\)
−0.629505 + 0.776996i \(0.716742\pi\)
\(152\) 0 0
\(153\) 0.890804i 0.0720172i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 18.6774i − 1.49062i −0.666717 0.745311i \(-0.732301\pi\)
0.666717 0.745311i \(-0.267699\pi\)
\(158\) 0 0
\(159\) −12.3618 −0.980353
\(160\) 0 0
\(161\) −3.73549 −0.294398
\(162\) 0 0
\(163\) 0.0922364i 0.00722451i 0.999993 + 0.00361226i \(0.00114982\pi\)
−0.999993 + 0.00361226i \(0.998850\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.37371i 0.415830i 0.978147 + 0.207915i \(0.0666677\pi\)
−0.978147 + 0.207915i \(0.933332\pi\)
\(168\) 0 0
\(169\) 4.90776 0.377520
\(170\) 0 0
\(171\) 6.84469 0.523426
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 0.890804i − 0.0669569i
\(178\) 0 0
\(179\) 9.56322 0.714788 0.357394 0.933954i \(-0.383665\pi\)
0.357394 + 0.933954i \(0.383665\pi\)
\(180\) 0 0
\(181\) −7.16035 −0.532225 −0.266112 0.963942i \(-0.585739\pi\)
−0.266112 + 0.963942i \(0.585739\pi\)
\(182\) 0 0
\(183\) 8.62629i 0.637674i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 4.31566i − 0.315593i
\(188\) 0 0
\(189\) 3.73549 0.271717
\(190\) 0 0
\(191\) 14.0050 1.01337 0.506684 0.862132i \(-0.330871\pi\)
0.506684 + 0.862132i \(0.330871\pi\)
\(192\) 0 0
\(193\) 26.7236i 1.92360i 0.273745 + 0.961802i \(0.411737\pi\)
−0.273745 + 0.961802i \(0.588263\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 9.16035i − 0.652648i −0.945258 0.326324i \(-0.894190\pi\)
0.945258 0.326324i \(-0.105810\pi\)
\(198\) 0 0
\(199\) 0.310629 0.0220199 0.0110099 0.999939i \(-0.496495\pi\)
0.0110099 + 0.999939i \(0.496495\pi\)
\(200\) 0 0
\(201\) 7.73549 0.545619
\(202\) 0 0
\(203\) 3.32759i 0.233551i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.00000i − 0.0695048i
\(208\) 0 0
\(209\) −33.1604 −2.29375
\(210\) 0 0
\(211\) 5.95388 0.409882 0.204941 0.978774i \(-0.434300\pi\)
0.204941 + 0.978774i \(0.434300\pi\)
\(212\) 0 0
\(213\) − 12.3618i − 0.847015i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 28.8958i 1.96158i
\(218\) 0 0
\(219\) −16.5341 −1.11727
\(220\) 0 0
\(221\) −2.53406 −0.170459
\(222\) 0 0
\(223\) − 11.9078i − 0.797403i −0.917081 0.398701i \(-0.869461\pi\)
0.917081 0.398701i \(-0.130539\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 7.37874i − 0.489744i −0.969555 0.244872i \(-0.921254\pi\)
0.969555 0.244872i \(-0.0787460\pi\)
\(228\) 0 0
\(229\) −23.1604 −1.53048 −0.765240 0.643746i \(-0.777379\pi\)
−0.765240 + 0.643746i \(0.777379\pi\)
\(230\) 0 0
\(231\) −18.0973 −1.19071
\(232\) 0 0
\(233\) − 22.9420i − 1.50298i −0.659746 0.751489i \(-0.729336\pi\)
0.659746 0.751489i \(-0.270664\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 13.4710i − 0.875034i
\(238\) 0 0
\(239\) −0.361783 −0.0234018 −0.0117009 0.999932i \(-0.503725\pi\)
−0.0117009 + 0.999932i \(0.503725\pi\)
\(240\) 0 0
\(241\) 28.5341 1.83804 0.919020 0.394211i \(-0.128982\pi\)
0.919020 + 0.394211i \(0.128982\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 19.4710i 1.23891i
\(248\) 0 0
\(249\) 4.58018 0.290257
\(250\) 0 0
\(251\) −30.8497 −1.94722 −0.973609 0.228224i \(-0.926708\pi\)
−0.973609 + 0.228224i \(0.926708\pi\)
\(252\) 0 0
\(253\) 4.84469i 0.304583i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.7524i 0.795476i 0.917499 + 0.397738i \(0.130205\pi\)
−0.917499 + 0.397738i \(0.869795\pi\)
\(258\) 0 0
\(259\) −7.29870 −0.453519
\(260\) 0 0
\(261\) −0.890804 −0.0551394
\(262\) 0 0
\(263\) 6.36178i 0.392284i 0.980575 + 0.196142i \(0.0628414\pi\)
−0.980575 + 0.196142i \(0.937159\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 15.1604i 0.927798i
\(268\) 0 0
\(269\) −10.5802 −0.645085 −0.322542 0.946555i \(-0.604537\pi\)
−0.322542 + 0.946555i \(0.604537\pi\)
\(270\) 0 0
\(271\) −4.26451 −0.259051 −0.129525 0.991576i \(-0.541345\pi\)
−0.129525 + 0.991576i \(0.541345\pi\)
\(272\) 0 0
\(273\) 10.6263i 0.643133i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 29.3787i 1.76520i 0.470127 + 0.882599i \(0.344208\pi\)
−0.470127 + 0.882599i \(0.655792\pi\)
\(278\) 0 0
\(279\) −7.73549 −0.463112
\(280\) 0 0
\(281\) −18.7524 −1.11868 −0.559339 0.828939i \(-0.688945\pi\)
−0.559339 + 0.828939i \(0.688945\pi\)
\(282\) 0 0
\(283\) − 12.8958i − 0.766578i −0.923628 0.383289i \(-0.874791\pi\)
0.923628 0.383289i \(-0.125209\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 46.1773i 2.72576i
\(288\) 0 0
\(289\) 16.2065 0.953322
\(290\) 0 0
\(291\) −17.2526 −1.01136
\(292\) 0 0
\(293\) 8.79857i 0.514018i 0.966409 + 0.257009i \(0.0827370\pi\)
−0.966409 + 0.257009i \(0.917263\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 4.84469i − 0.281117i
\(298\) 0 0
\(299\) 2.84469 0.164512
\(300\) 0 0
\(301\) 12.9658 0.747337
\(302\) 0 0
\(303\) − 2.67241i − 0.153526i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 10.0050i − 0.571018i −0.958376 0.285509i \(-0.907837\pi\)
0.958376 0.285509i \(-0.0921626\pi\)
\(308\) 0 0
\(309\) −6.21839 −0.353752
\(310\) 0 0
\(311\) −18.5290 −1.05068 −0.525342 0.850891i \(-0.676063\pi\)
−0.525342 + 0.850891i \(0.676063\pi\)
\(312\) 0 0
\(313\) 2.39067i 0.135128i 0.997715 + 0.0675642i \(0.0215228\pi\)
−0.997715 + 0.0675642i \(0.978477\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.3157i 0.916379i 0.888855 + 0.458190i \(0.151502\pi\)
−0.888855 + 0.458190i \(0.848498\pi\)
\(318\) 0 0
\(319\) 4.31566 0.241631
\(320\) 0 0
\(321\) −14.7986 −0.825975
\(322\) 0 0
\(323\) 6.09727i 0.339261i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 12.5341i 0.693135i
\(328\) 0 0
\(329\) −24.7524 −1.36465
\(330\) 0 0
\(331\) −24.8958 −1.36840 −0.684200 0.729295i \(-0.739848\pi\)
−0.684200 + 0.729295i \(0.739848\pi\)
\(332\) 0 0
\(333\) − 1.95388i − 0.107072i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 27.4710i 1.49644i 0.663451 + 0.748220i \(0.269091\pi\)
−0.663451 + 0.748220i \(0.730909\pi\)
\(338\) 0 0
\(339\) −12.3618 −0.671400
\(340\) 0 0
\(341\) 37.4760 2.02944
\(342\) 0 0
\(343\) 0.172274i 0.00930193i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.25259i 0.0672424i 0.999435 + 0.0336212i \(0.0107040\pi\)
−0.999435 + 0.0336212i \(0.989296\pi\)
\(348\) 0 0
\(349\) −9.64325 −0.516192 −0.258096 0.966119i \(-0.583095\pi\)
−0.258096 + 0.966119i \(0.583095\pi\)
\(350\) 0 0
\(351\) −2.84469 −0.151838
\(352\) 0 0
\(353\) − 12.7524i − 0.678744i −0.940652 0.339372i \(-0.889785\pi\)
0.940652 0.339372i \(-0.110215\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.32759i 0.176115i
\(358\) 0 0
\(359\) 6.62629 0.349722 0.174861 0.984593i \(-0.444052\pi\)
0.174861 + 0.984593i \(0.444052\pi\)
\(360\) 0 0
\(361\) 27.8497 1.46577
\(362\) 0 0
\(363\) 12.4710i 0.654557i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 9.60933i 0.501603i 0.968039 + 0.250802i \(0.0806942\pi\)
−0.968039 + 0.250802i \(0.919306\pi\)
\(368\) 0 0
\(369\) −12.3618 −0.643529
\(370\) 0 0
\(371\) −46.1773 −2.39741
\(372\) 0 0
\(373\) 29.8839i 1.54733i 0.633595 + 0.773665i \(0.281579\pi\)
−0.633595 + 0.773665i \(0.718421\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 2.53406i − 0.130511i
\(378\) 0 0
\(379\) −2.52902 −0.129907 −0.0649535 0.997888i \(-0.520690\pi\)
−0.0649535 + 0.997888i \(0.520690\pi\)
\(380\) 0 0
\(381\) 11.7866 0.603848
\(382\) 0 0
\(383\) 25.2115i 1.28825i 0.764921 + 0.644124i \(0.222778\pi\)
−0.764921 + 0.644124i \(0.777222\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.47098i 0.176440i
\(388\) 0 0
\(389\) −35.0681 −1.77802 −0.889012 0.457884i \(-0.848608\pi\)
−0.889012 + 0.457884i \(0.848608\pi\)
\(390\) 0 0
\(391\) 0.890804 0.0450499
\(392\) 0 0
\(393\) − 17.4710i − 0.881294i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 8.84972i − 0.444155i −0.975029 0.222077i \(-0.928716\pi\)
0.975029 0.222077i \(-0.0712838\pi\)
\(398\) 0 0
\(399\) 25.5683 1.28001
\(400\) 0 0
\(401\) −23.0681 −1.15197 −0.575983 0.817461i \(-0.695381\pi\)
−0.575983 + 0.817461i \(0.695381\pi\)
\(402\) 0 0
\(403\) − 22.0050i − 1.09615i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.46594i 0.469209i
\(408\) 0 0
\(409\) −5.29870 −0.262004 −0.131002 0.991382i \(-0.541819\pi\)
−0.131002 + 0.991382i \(0.541819\pi\)
\(410\) 0 0
\(411\) 15.1604 0.747805
\(412\) 0 0
\(413\) − 3.32759i − 0.163740i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.51710i 0.270173i
\(418\) 0 0
\(419\) −2.53406 −0.123797 −0.0618984 0.998082i \(-0.519715\pi\)
−0.0618984 + 0.998082i \(0.519715\pi\)
\(420\) 0 0
\(421\) 22.4079 1.09209 0.546047 0.837754i \(-0.316132\pi\)
0.546047 + 0.837754i \(0.316132\pi\)
\(422\) 0 0
\(423\) − 6.62629i − 0.322181i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 32.2234i 1.55940i
\(428\) 0 0
\(429\) 13.7816 0.665382
\(430\) 0 0
\(431\) 31.5733 1.52083 0.760416 0.649436i \(-0.224995\pi\)
0.760416 + 0.649436i \(0.224995\pi\)
\(432\) 0 0
\(433\) 7.20647i 0.346321i 0.984894 + 0.173160i \(0.0553979\pi\)
−0.984894 + 0.173160i \(0.944602\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 6.84469i − 0.327426i
\(438\) 0 0
\(439\) 4.72357 0.225443 0.112722 0.993627i \(-0.464043\pi\)
0.112722 + 0.993627i \(0.464043\pi\)
\(440\) 0 0
\(441\) 6.95388 0.331137
\(442\) 0 0
\(443\) − 7.87888i − 0.374337i −0.982328 0.187168i \(-0.940069\pi\)
0.982328 0.187168i \(-0.0599310\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 1.15531i − 0.0546445i
\(448\) 0 0
\(449\) 4.70633 0.222105 0.111053 0.993815i \(-0.464578\pi\)
0.111053 + 0.993815i \(0.464578\pi\)
\(450\) 0 0
\(451\) 59.8890 2.82006
\(452\) 0 0
\(453\) − 15.4710i − 0.726890i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.04612i 0.282825i 0.989951 + 0.141413i \(0.0451645\pi\)
−0.989951 + 0.141413i \(0.954836\pi\)
\(458\) 0 0
\(459\) −0.890804 −0.0415792
\(460\) 0 0
\(461\) −19.2526 −0.896682 −0.448341 0.893863i \(-0.647985\pi\)
−0.448341 + 0.893863i \(0.647985\pi\)
\(462\) 0 0
\(463\) − 26.4418i − 1.22886i −0.788973 0.614428i \(-0.789387\pi\)
0.788973 0.614428i \(-0.210613\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 9.20143i − 0.425792i −0.977075 0.212896i \(-0.931711\pi\)
0.977075 0.212896i \(-0.0682895\pi\)
\(468\) 0 0
\(469\) 28.8958 1.33429
\(470\) 0 0
\(471\) 18.6774 0.860611
\(472\) 0 0
\(473\) − 16.8158i − 0.773191i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 12.3618i − 0.566007i
\(478\) 0 0
\(479\) 12.7524 0.582674 0.291337 0.956620i \(-0.405900\pi\)
0.291337 + 0.956620i \(0.405900\pi\)
\(480\) 0 0
\(481\) 5.55818 0.253431
\(482\) 0 0
\(483\) − 3.73549i − 0.169971i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 30.5341i 1.38363i 0.722075 + 0.691815i \(0.243189\pi\)
−0.722075 + 0.691815i \(0.756811\pi\)
\(488\) 0 0
\(489\) −0.0922364 −0.00417107
\(490\) 0 0
\(491\) −2.67241 −0.120604 −0.0603021 0.998180i \(-0.519206\pi\)
−0.0603021 + 0.998180i \(0.519206\pi\)
\(492\) 0 0
\(493\) − 0.793532i − 0.0357389i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 46.1773i − 2.07134i
\(498\) 0 0
\(499\) 20.8036 0.931297 0.465649 0.884970i \(-0.345821\pi\)
0.465649 + 0.884970i \(0.345821\pi\)
\(500\) 0 0
\(501\) −5.37371 −0.240080
\(502\) 0 0
\(503\) 3.32759i 0.148370i 0.997245 + 0.0741849i \(0.0236355\pi\)
−0.997245 + 0.0741849i \(0.976364\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4.90776i 0.217961i
\(508\) 0 0
\(509\) −16.2184 −0.718868 −0.359434 0.933171i \(-0.617030\pi\)
−0.359434 + 0.933171i \(0.617030\pi\)
\(510\) 0 0
\(511\) −61.7628 −2.73223
\(512\) 0 0
\(513\) 6.84469i 0.302200i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 32.1023i 1.41186i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.18923 0.139723 0.0698614 0.997557i \(-0.477744\pi\)
0.0698614 + 0.997557i \(0.477744\pi\)
\(522\) 0 0
\(523\) 23.8155i 1.04138i 0.853746 + 0.520690i \(0.174325\pi\)
−0.853746 + 0.520690i \(0.825675\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 6.89080i − 0.300168i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 0.890804 0.0386576
\(532\) 0 0
\(533\) − 35.1654i − 1.52318i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 9.56322i 0.412683i
\(538\) 0 0
\(539\) −33.6894 −1.45110
\(540\) 0 0
\(541\) 15.7816 0.678504 0.339252 0.940695i \(-0.389826\pi\)
0.339252 + 0.940695i \(0.389826\pi\)
\(542\) 0 0
\(543\) − 7.16035i − 0.307280i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000i 1.19719i 0.801050 + 0.598597i \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) 0 0
\(549\) −8.62629 −0.368161
\(550\) 0 0
\(551\) −6.09727 −0.259753
\(552\) 0 0
\(553\) − 50.3207i − 2.13985i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 31.7405i − 1.34489i −0.740147 0.672445i \(-0.765244\pi\)
0.740147 0.672445i \(-0.234756\pi\)
\(558\) 0 0
\(559\) −9.87384 −0.417619
\(560\) 0 0
\(561\) 4.31566 0.182208
\(562\) 0 0
\(563\) 21.9250i 0.924029i 0.886872 + 0.462014i \(0.152873\pi\)
−0.886872 + 0.462014i \(0.847127\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.73549i 0.156876i
\(568\) 0 0
\(569\) 14.6313 0.613377 0.306689 0.951810i \(-0.400779\pi\)
0.306689 + 0.951810i \(0.400779\pi\)
\(570\) 0 0
\(571\) 1.24755 0.0522084 0.0261042 0.999659i \(-0.491690\pi\)
0.0261042 + 0.999659i \(0.491690\pi\)
\(572\) 0 0
\(573\) 14.0050i 0.585069i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 18.0922i − 0.753190i −0.926378 0.376595i \(-0.877095\pi\)
0.926378 0.376595i \(-0.122905\pi\)
\(578\) 0 0
\(579\) −26.7236 −1.11059
\(580\) 0 0
\(581\) 17.1092 0.709809
\(582\) 0 0
\(583\) 59.8890i 2.48035i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 12.5290i − 0.517128i −0.965994 0.258564i \(-0.916751\pi\)
0.965994 0.258564i \(-0.0832493\pi\)
\(588\) 0 0
\(589\) −52.9470 −2.18164
\(590\) 0 0
\(591\) 9.16035 0.376806
\(592\) 0 0
\(593\) − 24.9470i − 1.02445i −0.858851 0.512225i \(-0.828821\pi\)
0.858851 0.512225i \(-0.171179\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.310629i 0.0127132i
\(598\) 0 0
\(599\) 34.5391 1.41123 0.705615 0.708596i \(-0.250671\pi\)
0.705615 + 0.708596i \(0.250671\pi\)
\(600\) 0 0
\(601\) 37.9300 1.54720 0.773599 0.633675i \(-0.218454\pi\)
0.773599 + 0.633675i \(0.218454\pi\)
\(602\) 0 0
\(603\) 7.73549i 0.315013i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 26.0973i − 1.05926i −0.848230 0.529628i \(-0.822332\pi\)
0.848230 0.529628i \(-0.177668\pi\)
\(608\) 0 0
\(609\) −3.32759 −0.134841
\(610\) 0 0
\(611\) 18.8497 0.762578
\(612\) 0 0
\(613\) 2.84972i 0.115099i 0.998343 + 0.0575496i \(0.0183287\pi\)
−0.998343 + 0.0575496i \(0.981671\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.52213i 0.383347i 0.981459 + 0.191673i \(0.0613914\pi\)
−0.981459 + 0.191673i \(0.938609\pi\)
\(618\) 0 0
\(619\) 23.9762 0.963683 0.481841 0.876258i \(-0.339968\pi\)
0.481841 + 0.876258i \(0.339968\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 56.6313i 2.26889i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 33.1604i − 1.32430i
\(628\) 0 0
\(629\) 1.74053 0.0693993
\(630\) 0 0
\(631\) −17.8789 −0.711747 −0.355873 0.934534i \(-0.615817\pi\)
−0.355873 + 0.934534i \(0.615817\pi\)
\(632\) 0 0
\(633\) 5.95388i 0.236646i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 19.7816i 0.783776i
\(638\) 0 0
\(639\) 12.3618 0.489025
\(640\) 0 0
\(641\) 2.93692 0.116001 0.0580007 0.998317i \(-0.481527\pi\)
0.0580007 + 0.998317i \(0.481527\pi\)
\(642\) 0 0
\(643\) − 0.895840i − 0.0353285i −0.999844 0.0176642i \(-0.994377\pi\)
0.999844 0.0176642i \(-0.00562299\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21.4659i 0.843913i 0.906616 + 0.421957i \(0.138657\pi\)
−0.906616 + 0.421957i \(0.861343\pi\)
\(648\) 0 0
\(649\) −4.31566 −0.169405
\(650\) 0 0
\(651\) −28.8958 −1.13252
\(652\) 0 0
\(653\) − 4.84469i − 0.189587i −0.995497 0.0947936i \(-0.969781\pi\)
0.995497 0.0947936i \(-0.0302191\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 16.5341i − 0.645055i
\(658\) 0 0
\(659\) 18.0973 0.704970 0.352485 0.935818i \(-0.385337\pi\)
0.352485 + 0.935818i \(0.385337\pi\)
\(660\) 0 0
\(661\) 0.747413 0.0290710 0.0145355 0.999894i \(-0.495373\pi\)
0.0145355 + 0.999894i \(0.495373\pi\)
\(662\) 0 0
\(663\) − 2.53406i − 0.0984146i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.890804i 0.0344921i
\(668\) 0 0
\(669\) 11.9078 0.460381
\(670\) 0 0
\(671\) 41.7917 1.61335
\(672\) 0 0
\(673\) − 19.1892i − 0.739691i −0.929093 0.369845i \(-0.879411\pi\)
0.929093 0.369845i \(-0.120589\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.17731i 0.160547i 0.996773 + 0.0802735i \(0.0255794\pi\)
−0.996773 + 0.0802735i \(0.974421\pi\)
\(678\) 0 0
\(679\) −64.4469 −2.47324
\(680\) 0 0
\(681\) 7.37874 0.282754
\(682\) 0 0
\(683\) − 26.0050i − 0.995055i −0.867448 0.497528i \(-0.834241\pi\)
0.867448 0.497528i \(-0.165759\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 23.1604i − 0.883622i
\(688\) 0 0
\(689\) 35.1654 1.33969
\(690\) 0 0
\(691\) 26.8497 1.02141 0.510706 0.859756i \(-0.329384\pi\)
0.510706 + 0.859756i \(0.329384\pi\)
\(692\) 0 0
\(693\) − 18.0973i − 0.687459i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 11.0119i − 0.417106i
\(698\) 0 0
\(699\) 22.9420 0.867745
\(700\) 0 0
\(701\) 11.9027 0.449560 0.224780 0.974410i \(-0.427834\pi\)
0.224780 + 0.974410i \(0.427834\pi\)
\(702\) 0 0
\(703\) − 13.3737i − 0.504399i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 9.98277i − 0.375441i
\(708\) 0 0
\(709\) 39.3737 1.47871 0.739355 0.673315i \(-0.235130\pi\)
0.739355 + 0.673315i \(0.235130\pi\)
\(710\) 0 0
\(711\) 13.4710 0.505201
\(712\) 0 0
\(713\) 7.73549i 0.289696i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 0.361783i − 0.0135110i
\(718\) 0 0
\(719\) −23.6382 −0.881557 −0.440778 0.897616i \(-0.645298\pi\)
−0.440778 + 0.897616i \(0.645298\pi\)
\(720\) 0 0
\(721\) −23.2287 −0.865083
\(722\) 0 0
\(723\) 28.5341i 1.06119i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 12.5513i − 0.465502i −0.972536 0.232751i \(-0.925227\pi\)
0.972536 0.232751i \(-0.0747727\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −3.09196 −0.114360
\(732\) 0 0
\(733\) 39.1142i 1.44472i 0.691519 + 0.722359i \(0.256942\pi\)
−0.691519 + 0.722359i \(0.743058\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 37.4760i − 1.38045i
\(738\) 0 0
\(739\) −35.7456 −1.31492 −0.657461 0.753489i \(-0.728370\pi\)
−0.657461 + 0.753489i \(0.728370\pi\)
\(740\) 0 0
\(741\) −19.4710 −0.715284
\(742\) 0 0
\(743\) 20.4368i 0.749753i 0.927075 + 0.374876i \(0.122315\pi\)
−0.927075 + 0.374876i \(0.877685\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4.58018i 0.167580i
\(748\) 0 0
\(749\) −55.2799 −2.01988
\(750\) 0 0
\(751\) −0.534057 −0.0194880 −0.00974401 0.999953i \(-0.503102\pi\)
−0.00974401 + 0.999953i \(0.503102\pi\)
\(752\) 0 0
\(753\) − 30.8497i − 1.12423i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 29.5171i 1.07282i 0.843958 + 0.536409i \(0.180219\pi\)
−0.843958 + 0.536409i \(0.819781\pi\)
\(758\) 0 0
\(759\) −4.84469 −0.175851
\(760\) 0 0
\(761\) 38.1434 1.38270 0.691348 0.722522i \(-0.257017\pi\)
0.691348 + 0.722522i \(0.257017\pi\)
\(762\) 0 0
\(763\) 46.8208i 1.69503i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.53406i 0.0914995i
\(768\) 0 0
\(769\) −20.8786 −0.752902 −0.376451 0.926437i \(-0.622856\pi\)
−0.376451 + 0.926437i \(0.622856\pi\)
\(770\) 0 0
\(771\) −12.7524 −0.459268
\(772\) 0 0
\(773\) 8.31063i 0.298913i 0.988768 + 0.149456i \(0.0477523\pi\)
−0.988768 + 0.149456i \(0.952248\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 7.29870i − 0.261840i
\(778\) 0 0
\(779\) −84.6125 −3.03156
\(780\) 0 0
\(781\) −59.8890 −2.14300
\(782\) 0 0
\(783\) − 0.890804i − 0.0318347i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.95388i 0.0696484i 0.999393 + 0.0348242i \(0.0110871\pi\)
−0.999393 + 0.0348242i \(0.988913\pi\)
\(788\) 0 0
\(789\) −6.36178 −0.226485
\(790\) 0 0
\(791\) −46.1773 −1.64188
\(792\) 0 0
\(793\) − 24.5391i − 0.871409i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 16.1773i 0.573030i 0.958076 + 0.286515i \(0.0924968\pi\)
−0.958076 + 0.286515i \(0.907503\pi\)
\(798\) 0 0
\(799\) 5.90273 0.208823
\(800\) 0 0
\(801\) −15.1604 −0.535665
\(802\) 0 0
\(803\) 80.1023i 2.82675i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 10.5802i − 0.372440i
\(808\) 0 0
\(809\) 26.8670 0.944592 0.472296 0.881440i \(-0.343425\pi\)
0.472296 + 0.881440i \(0.343425\pi\)
\(810\) 0 0
\(811\) 2.13835 0.0750878 0.0375439 0.999295i \(-0.488047\pi\)
0.0375439 + 0.999295i \(0.488047\pi\)
\(812\) 0 0
\(813\) − 4.26451i − 0.149563i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 23.7578i 0.831179i
\(818\) 0 0
\(819\) −10.6263 −0.371313
\(820\) 0 0
\(821\) 15.1604 0.529100 0.264550 0.964372i \(-0.414777\pi\)
0.264550 + 0.964372i \(0.414777\pi\)
\(822\) 0 0
\(823\) − 23.1264i − 0.806137i −0.915170 0.403068i \(-0.867944\pi\)
0.915170 0.403068i \(-0.132056\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 47.4299i − 1.64930i −0.565644 0.824650i \(-0.691372\pi\)
0.565644 0.824650i \(-0.308628\pi\)
\(828\) 0 0
\(829\) 30.1723 1.04793 0.523963 0.851741i \(-0.324453\pi\)
0.523963 + 0.851741i \(0.324453\pi\)
\(830\) 0 0
\(831\) −29.3787 −1.01914
\(832\) 0 0
\(833\) 6.19454i 0.214628i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 7.73549i − 0.267378i
\(838\) 0 0
\(839\) −18.8497 −0.650765 −0.325382 0.945583i \(-0.605493\pi\)
−0.325382 + 0.945583i \(0.605493\pi\)
\(840\) 0 0
\(841\) −28.2065 −0.972637
\(842\) 0 0
\(843\) − 18.7524i − 0.645869i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 46.5852i 1.60069i
\(848\) 0 0
\(849\) 12.8958 0.442584
\(850\) 0 0
\(851\) −1.95388 −0.0669782
\(852\) 0 0
\(853\) 29.7578i 1.01889i 0.860504 + 0.509443i \(0.170149\pi\)
−0.860504 + 0.509443i \(0.829851\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15.2865i 0.522177i 0.965315 + 0.261089i \(0.0840815\pi\)
−0.965315 + 0.261089i \(0.915919\pi\)
\(858\) 0 0
\(859\) −33.5171 −1.14359 −0.571794 0.820397i \(-0.693752\pi\)
−0.571794 + 0.820397i \(0.693752\pi\)
\(860\) 0 0
\(861\) −46.1773 −1.57372
\(862\) 0 0
\(863\) − 31.0443i − 1.05676i −0.849008 0.528380i \(-0.822800\pi\)
0.849008 0.528380i \(-0.177200\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 16.2065i 0.550401i
\(868\) 0 0
\(869\) −65.2627 −2.21388
\(870\) 0 0
\(871\) −22.0050 −0.745612
\(872\) 0 0
\(873\) − 17.2526i − 0.583912i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 10.6313i − 0.358994i −0.983758 0.179497i \(-0.942553\pi\)
0.983758 0.179497i \(-0.0574471\pi\)
\(878\) 0 0
\(879\) −8.79857 −0.296768
\(880\) 0 0
\(881\) 20.5341 0.691810 0.345905 0.938270i \(-0.387572\pi\)
0.345905 + 0.938270i \(0.387572\pi\)
\(882\) 0 0
\(883\) 36.0390i 1.21281i 0.795157 + 0.606404i \(0.207388\pi\)
−0.795157 + 0.606404i \(0.792612\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 24.9759i 0.838608i 0.907846 + 0.419304i \(0.137726\pi\)
−0.907846 + 0.419304i \(0.862274\pi\)
\(888\) 0 0
\(889\) 44.0289 1.47668
\(890\) 0 0
\(891\) 4.84469 0.162303
\(892\) 0 0
\(893\) − 45.3549i − 1.51774i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.84469i 0.0949813i
\(898\) 0 0
\(899\) 6.89080 0.229821
\(900\) 0 0
\(901\) 11.0119 0.366860
\(902\) 0 0
\(903\) 12.9658i 0.431475i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 41.6194i − 1.38195i −0.722879 0.690975i \(-0.757182\pi\)
0.722879 0.690975i \(-0.242818\pi\)
\(908\) 0 0
\(909\) 2.67241 0.0886383
\(910\) 0 0
\(911\) 3.81553 0.126414 0.0632070 0.998000i \(-0.479867\pi\)
0.0632070 + 0.998000i \(0.479867\pi\)
\(912\) 0 0
\(913\) − 22.1895i − 0.734366i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 65.2627i − 2.15516i
\(918\) 0 0
\(919\) 39.7917 1.31261 0.656303 0.754497i \(-0.272119\pi\)
0.656303 + 0.754497i \(0.272119\pi\)
\(920\) 0 0
\(921\) 10.0050 0.329677
\(922\) 0 0
\(923\) 35.1654i 1.15748i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 6.21839i − 0.204239i
\(928\) 0 0
\(929\) −12.6141 −0.413855 −0.206928 0.978356i \(-0.566346\pi\)
−0.206928 + 0.978356i \(0.566346\pi\)
\(930\) 0 0
\(931\) 47.5971 1.55993
\(932\) 0 0
\(933\) − 18.5290i − 0.606613i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.1262i 0.722830i 0.932405 + 0.361415i \(0.117706\pi\)
−0.932405 + 0.361415i \(0.882294\pi\)
\(938\) 0 0
\(939\) −2.39067 −0.0780164
\(940\) 0 0
\(941\) −11.0392 −0.359869 −0.179934 0.983679i \(-0.557589\pi\)
−0.179934 + 0.983679i \(0.557589\pi\)
\(942\) 0 0
\(943\) 12.3618i 0.402555i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 9.68937i − 0.314862i −0.987530 0.157431i \(-0.949679\pi\)
0.987530 0.157431i \(-0.0503212\pi\)
\(948\) 0 0
\(949\) 47.0342 1.52679
\(950\) 0 0
\(951\) −16.3157 −0.529072
\(952\) 0 0
\(953\) − 14.1845i − 0.459480i −0.973252 0.229740i \(-0.926212\pi\)
0.973252 0.229740i \(-0.0737876\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.31566i 0.139506i
\(958\) 0 0
\(959\) 56.6313 1.82872
\(960\) 0 0
\(961\) 28.8378 0.930252
\(962\) 0 0
\(963\) − 14.7986i − 0.476877i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 12.8447i − 0.413057i −0.978441 0.206529i \(-0.933783\pi\)
0.978441 0.206529i \(-0.0662167\pi\)
\(968\) 0 0
\(969\) −6.09727 −0.195873
\(970\) 0 0
\(971\) −24.1945 −0.776440 −0.388220 0.921567i \(-0.626910\pi\)
−0.388220 + 0.921567i \(0.626910\pi\)
\(972\) 0 0
\(973\) 20.6091i 0.660696i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 18.4879i − 0.591482i −0.955268 0.295741i \(-0.904434\pi\)
0.955268 0.295741i \(-0.0955665\pi\)
\(978\) 0 0
\(979\) 73.4471 2.34738
\(980\) 0 0
\(981\) −12.5341 −0.400182
\(982\) 0 0
\(983\) − 28.8325i − 0.919614i −0.888019 0.459807i \(-0.847919\pi\)
0.888019 0.459807i \(-0.152081\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 24.7524i − 0.787879i
\(988\) 0 0
\(989\) 3.47098 0.110371
\(990\) 0 0
\(991\) 40.8958 1.29910 0.649550 0.760319i \(-0.274957\pi\)
0.649550 + 0.760319i \(0.274957\pi\)
\(992\) 0 0
\(993\) − 24.8958i − 0.790046i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 4.65518i 0.147431i 0.997279 + 0.0737155i \(0.0234857\pi\)
−0.997279 + 0.0737155i \(0.976514\pi\)
\(998\) 0 0
\(999\) 1.95388 0.0618181
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 6900.2.f.r.6349.6 6
5.2 odd 4 1380.2.a.j.1.1 3
5.3 odd 4 6900.2.a.x.1.3 3
5.4 even 2 inner 6900.2.f.r.6349.1 6
15.2 even 4 4140.2.a.s.1.1 3
20.7 even 4 5520.2.a.bv.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.j.1.1 3 5.2 odd 4
4140.2.a.s.1.1 3 15.2 even 4
5520.2.a.bv.1.3 3 20.7 even 4
6900.2.a.x.1.3 3 5.3 odd 4
6900.2.f.r.6349.1 6 5.4 even 2 inner
6900.2.f.r.6349.6 6 1.1 even 1 trivial