Properties

Label 9408.2.a.do.1.2
Level $9408$
Weight $2$
Character 9408.1
Self dual yes
Analytic conductor $75.123$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{9} -1.46410 q^{11} +2.00000 q^{13} -3.46410 q^{15} -0.535898 q^{17} -6.92820 q^{19} +1.46410 q^{23} +7.00000 q^{25} -1.00000 q^{27} +4.92820 q^{29} -10.9282 q^{31} +1.46410 q^{33} +2.00000 q^{37} -2.00000 q^{39} -11.4641 q^{41} -8.00000 q^{43} +3.46410 q^{45} +10.9282 q^{47} +0.535898 q^{51} +2.00000 q^{53} -5.07180 q^{55} +6.92820 q^{57} +1.07180 q^{59} -8.92820 q^{61} +6.92820 q^{65} -2.92820 q^{67} -1.46410 q^{69} -9.46410 q^{71} -12.9282 q^{73} -7.00000 q^{75} +10.9282 q^{79} +1.00000 q^{81} +4.00000 q^{83} -1.85641 q^{85} -4.92820 q^{87} -3.46410 q^{89} +10.9282 q^{93} -24.0000 q^{95} +8.92820 q^{97} -1.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} + 4q^{11} + 4q^{13} - 8q^{17} - 4q^{23} + 14q^{25} - 2q^{27} - 4q^{29} - 8q^{31} - 4q^{33} + 4q^{37} - 4q^{39} - 16q^{41} - 16q^{43} + 8q^{47} + 8q^{51} + 4q^{53} - 24q^{55} + 16q^{59} - 4q^{61} + 8q^{67} + 4q^{69} - 12q^{71} - 12q^{73} - 14q^{75} + 8q^{79} + 2q^{81} + 8q^{83} + 24q^{85} + 4q^{87} + 8q^{93} - 48q^{95} + 4q^{97} + 4q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.46410 −0.441443 −0.220722 0.975337i \(-0.570841\pi\)
−0.220722 + 0.975337i \(0.570841\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −3.46410 −0.894427
\(16\) 0 0
\(17\) −0.535898 −0.129974 −0.0649872 0.997886i \(-0.520701\pi\)
−0.0649872 + 0.997886i \(0.520701\pi\)
\(18\) 0 0
\(19\) −6.92820 −1.58944 −0.794719 0.606977i \(-0.792382\pi\)
−0.794719 + 0.606977i \(0.792382\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.46410 0.305286 0.152643 0.988281i \(-0.451221\pi\)
0.152643 + 0.988281i \(0.451221\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.92820 0.915144 0.457572 0.889172i \(-0.348719\pi\)
0.457572 + 0.889172i \(0.348719\pi\)
\(30\) 0 0
\(31\) −10.9282 −1.96276 −0.981382 0.192068i \(-0.938481\pi\)
−0.981382 + 0.192068i \(0.938481\pi\)
\(32\) 0 0
\(33\) 1.46410 0.254867
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −11.4641 −1.79039 −0.895196 0.445673i \(-0.852964\pi\)
−0.895196 + 0.445673i \(0.852964\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 3.46410 0.516398
\(46\) 0 0
\(47\) 10.9282 1.59404 0.797021 0.603951i \(-0.206408\pi\)
0.797021 + 0.603951i \(0.206408\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0.535898 0.0750408
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −5.07180 −0.683881
\(56\) 0 0
\(57\) 6.92820 0.917663
\(58\) 0 0
\(59\) 1.07180 0.139536 0.0697680 0.997563i \(-0.477774\pi\)
0.0697680 + 0.997563i \(0.477774\pi\)
\(60\) 0 0
\(61\) −8.92820 −1.14314 −0.571570 0.820554i \(-0.693665\pi\)
−0.571570 + 0.820554i \(0.693665\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.92820 0.859338
\(66\) 0 0
\(67\) −2.92820 −0.357737 −0.178868 0.983873i \(-0.557244\pi\)
−0.178868 + 0.983873i \(0.557244\pi\)
\(68\) 0 0
\(69\) −1.46410 −0.176257
\(70\) 0 0
\(71\) −9.46410 −1.12318 −0.561591 0.827415i \(-0.689811\pi\)
−0.561591 + 0.827415i \(0.689811\pi\)
\(72\) 0 0
\(73\) −12.9282 −1.51313 −0.756566 0.653917i \(-0.773124\pi\)
−0.756566 + 0.653917i \(0.773124\pi\)
\(74\) 0 0
\(75\) −7.00000 −0.808290
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.9282 1.22952 0.614759 0.788715i \(-0.289253\pi\)
0.614759 + 0.788715i \(0.289253\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −1.85641 −0.201356
\(86\) 0 0
\(87\) −4.92820 −0.528359
\(88\) 0 0
\(89\) −3.46410 −0.367194 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 10.9282 1.13320
\(94\) 0 0
\(95\) −24.0000 −2.46235
\(96\) 0 0
\(97\) 8.92820 0.906522 0.453261 0.891378i \(-0.350261\pi\)
0.453261 + 0.891378i \(0.350261\pi\)
\(98\) 0 0
\(99\) −1.46410 −0.147148
\(100\) 0 0
\(101\) −15.4641 −1.53874 −0.769368 0.638806i \(-0.779429\pi\)
−0.769368 + 0.638806i \(0.779429\pi\)
\(102\) 0 0
\(103\) −10.9282 −1.07679 −0.538394 0.842693i \(-0.680969\pi\)
−0.538394 + 0.842693i \(0.680969\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.3923 1.19801 0.599005 0.800746i \(-0.295563\pi\)
0.599005 + 0.800746i \(0.295563\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −19.8564 −1.86793 −0.933967 0.357360i \(-0.883677\pi\)
−0.933967 + 0.357360i \(0.883677\pi\)
\(114\) 0 0
\(115\) 5.07180 0.472947
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.85641 −0.805128
\(122\) 0 0
\(123\) 11.4641 1.03368
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) −2.92820 −0.259836 −0.129918 0.991525i \(-0.541471\pi\)
−0.129918 + 0.991525i \(0.541471\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.46410 −0.298142
\(136\) 0 0
\(137\) −8.92820 −0.762788 −0.381394 0.924413i \(-0.624556\pi\)
−0.381394 + 0.924413i \(0.624556\pi\)
\(138\) 0 0
\(139\) 17.8564 1.51456 0.757280 0.653090i \(-0.226528\pi\)
0.757280 + 0.653090i \(0.226528\pi\)
\(140\) 0 0
\(141\) −10.9282 −0.920321
\(142\) 0 0
\(143\) −2.92820 −0.244869
\(144\) 0 0
\(145\) 17.0718 1.41774
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −21.8564 −1.77865 −0.889325 0.457277i \(-0.848825\pi\)
−0.889325 + 0.457277i \(0.848825\pi\)
\(152\) 0 0
\(153\) −0.535898 −0.0433248
\(154\) 0 0
\(155\) −37.8564 −3.04070
\(156\) 0 0
\(157\) −0.928203 −0.0740787 −0.0370393 0.999314i \(-0.511793\pi\)
−0.0370393 + 0.999314i \(0.511793\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.9282 0.855963 0.427981 0.903788i \(-0.359225\pi\)
0.427981 + 0.903788i \(0.359225\pi\)
\(164\) 0 0
\(165\) 5.07180 0.394839
\(166\) 0 0
\(167\) −10.9282 −0.845650 −0.422825 0.906211i \(-0.638961\pi\)
−0.422825 + 0.906211i \(0.638961\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −6.92820 −0.529813
\(172\) 0 0
\(173\) 22.3923 1.70246 0.851228 0.524797i \(-0.175859\pi\)
0.851228 + 0.524797i \(0.175859\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.07180 −0.0805612
\(178\) 0 0
\(179\) 9.46410 0.707380 0.353690 0.935363i \(-0.384927\pi\)
0.353690 + 0.935363i \(0.384927\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 8.92820 0.659992
\(184\) 0 0
\(185\) 6.92820 0.509372
\(186\) 0 0
\(187\) 0.784610 0.0573763
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.46410 −0.684798 −0.342399 0.939555i \(-0.611240\pi\)
−0.342399 + 0.939555i \(0.611240\pi\)
\(192\) 0 0
\(193\) 15.8564 1.14137 0.570685 0.821169i \(-0.306678\pi\)
0.570685 + 0.821169i \(0.306678\pi\)
\(194\) 0 0
\(195\) −6.92820 −0.496139
\(196\) 0 0
\(197\) 7.85641 0.559746 0.279873 0.960037i \(-0.409708\pi\)
0.279873 + 0.960037i \(0.409708\pi\)
\(198\) 0 0
\(199\) 5.85641 0.415150 0.207575 0.978219i \(-0.433443\pi\)
0.207575 + 0.978219i \(0.433443\pi\)
\(200\) 0 0
\(201\) 2.92820 0.206540
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −39.7128 −2.77366
\(206\) 0 0
\(207\) 1.46410 0.101762
\(208\) 0 0
\(209\) 10.1436 0.701647
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 9.46410 0.648470
\(214\) 0 0
\(215\) −27.7128 −1.89000
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 12.9282 0.873607
\(220\) 0 0
\(221\) −1.07180 −0.0720969
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) 22.9282 1.52180 0.760899 0.648870i \(-0.224758\pi\)
0.760899 + 0.648870i \(0.224758\pi\)
\(228\) 0 0
\(229\) −27.8564 −1.84080 −0.920402 0.390974i \(-0.872138\pi\)
−0.920402 + 0.390974i \(0.872138\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.9282 0.846955 0.423477 0.905907i \(-0.360809\pi\)
0.423477 + 0.905907i \(0.360809\pi\)
\(234\) 0 0
\(235\) 37.8564 2.46948
\(236\) 0 0
\(237\) −10.9282 −0.709863
\(238\) 0 0
\(239\) −14.5359 −0.940249 −0.470125 0.882600i \(-0.655791\pi\)
−0.470125 + 0.882600i \(0.655791\pi\)
\(240\) 0 0
\(241\) 3.07180 0.197872 0.0989359 0.995094i \(-0.468456\pi\)
0.0989359 + 0.995094i \(0.468456\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −13.8564 −0.881662
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 9.07180 0.572607 0.286303 0.958139i \(-0.407573\pi\)
0.286303 + 0.958139i \(0.407573\pi\)
\(252\) 0 0
\(253\) −2.14359 −0.134767
\(254\) 0 0
\(255\) 1.85641 0.116253
\(256\) 0 0
\(257\) −11.4641 −0.715111 −0.357556 0.933892i \(-0.616390\pi\)
−0.357556 + 0.933892i \(0.616390\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 4.92820 0.305048
\(262\) 0 0
\(263\) −12.3923 −0.764142 −0.382071 0.924133i \(-0.624789\pi\)
−0.382071 + 0.924133i \(0.624789\pi\)
\(264\) 0 0
\(265\) 6.92820 0.425596
\(266\) 0 0
\(267\) 3.46410 0.212000
\(268\) 0 0
\(269\) 3.46410 0.211210 0.105605 0.994408i \(-0.466322\pi\)
0.105605 + 0.994408i \(0.466322\pi\)
\(270\) 0 0
\(271\) 2.92820 0.177876 0.0889378 0.996037i \(-0.471653\pi\)
0.0889378 + 0.996037i \(0.471653\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10.2487 −0.618021
\(276\) 0 0
\(277\) −0.143594 −0.00862770 −0.00431385 0.999991i \(-0.501373\pi\)
−0.00431385 + 0.999991i \(0.501373\pi\)
\(278\) 0 0
\(279\) −10.9282 −0.654254
\(280\) 0 0
\(281\) 12.9282 0.771232 0.385616 0.922659i \(-0.373989\pi\)
0.385616 + 0.922659i \(0.373989\pi\)
\(282\) 0 0
\(283\) 14.9282 0.887390 0.443695 0.896178i \(-0.353667\pi\)
0.443695 + 0.896178i \(0.353667\pi\)
\(284\) 0 0
\(285\) 24.0000 1.42164
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.7128 −0.983107
\(290\) 0 0
\(291\) −8.92820 −0.523381
\(292\) 0 0
\(293\) 16.5359 0.966037 0.483019 0.875610i \(-0.339540\pi\)
0.483019 + 0.875610i \(0.339540\pi\)
\(294\) 0 0
\(295\) 3.71281 0.216168
\(296\) 0 0
\(297\) 1.46410 0.0849558
\(298\) 0 0
\(299\) 2.92820 0.169342
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 15.4641 0.888389
\(304\) 0 0
\(305\) −30.9282 −1.77094
\(306\) 0 0
\(307\) −22.9282 −1.30858 −0.654291 0.756243i \(-0.727033\pi\)
−0.654291 + 0.756243i \(0.727033\pi\)
\(308\) 0 0
\(309\) 10.9282 0.621684
\(310\) 0 0
\(311\) 18.9282 1.07332 0.536660 0.843799i \(-0.319686\pi\)
0.536660 + 0.843799i \(0.319686\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.143594 −0.00806502 −0.00403251 0.999992i \(-0.501284\pi\)
−0.00403251 + 0.999992i \(0.501284\pi\)
\(318\) 0 0
\(319\) −7.21539 −0.403984
\(320\) 0 0
\(321\) −12.3923 −0.691671
\(322\) 0 0
\(323\) 3.71281 0.206586
\(324\) 0 0
\(325\) 14.0000 0.776580
\(326\) 0 0
\(327\) −2.00000 −0.110600
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −10.1436 −0.554204
\(336\) 0 0
\(337\) 7.85641 0.427966 0.213983 0.976837i \(-0.431356\pi\)
0.213983 + 0.976837i \(0.431356\pi\)
\(338\) 0 0
\(339\) 19.8564 1.07845
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −5.07180 −0.273056
\(346\) 0 0
\(347\) 23.3205 1.25191 0.625955 0.779859i \(-0.284709\pi\)
0.625955 + 0.779859i \(0.284709\pi\)
\(348\) 0 0
\(349\) 10.7846 0.577287 0.288643 0.957437i \(-0.406796\pi\)
0.288643 + 0.957437i \(0.406796\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −8.53590 −0.454320 −0.227160 0.973857i \(-0.572944\pi\)
−0.227160 + 0.973857i \(0.572944\pi\)
\(354\) 0 0
\(355\) −32.7846 −1.74003
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7.32051 −0.386362 −0.193181 0.981163i \(-0.561880\pi\)
−0.193181 + 0.981163i \(0.561880\pi\)
\(360\) 0 0
\(361\) 29.0000 1.52632
\(362\) 0 0
\(363\) 8.85641 0.464841
\(364\) 0 0
\(365\) −44.7846 −2.34413
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) −11.4641 −0.596797
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −30.0000 −1.55334 −0.776671 0.629907i \(-0.783093\pi\)
−0.776671 + 0.629907i \(0.783093\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) 9.85641 0.507631
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 2.92820 0.150016
\(382\) 0 0
\(383\) 35.7128 1.82484 0.912420 0.409256i \(-0.134212\pi\)
0.912420 + 0.409256i \(0.134212\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) −14.7846 −0.749609 −0.374805 0.927104i \(-0.622290\pi\)
−0.374805 + 0.927104i \(0.622290\pi\)
\(390\) 0 0
\(391\) −0.784610 −0.0396794
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 37.8564 1.90476
\(396\) 0 0
\(397\) 10.7846 0.541264 0.270632 0.962683i \(-0.412767\pi\)
0.270632 + 0.962683i \(0.412767\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.92820 0.246103 0.123051 0.992400i \(-0.460732\pi\)
0.123051 + 0.992400i \(0.460732\pi\)
\(402\) 0 0
\(403\) −21.8564 −1.08875
\(404\) 0 0
\(405\) 3.46410 0.172133
\(406\) 0 0
\(407\) −2.92820 −0.145146
\(408\) 0 0
\(409\) −4.92820 −0.243684 −0.121842 0.992550i \(-0.538880\pi\)
−0.121842 + 0.992550i \(0.538880\pi\)
\(410\) 0 0
\(411\) 8.92820 0.440396
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 13.8564 0.680184
\(416\) 0 0
\(417\) −17.8564 −0.874432
\(418\) 0 0
\(419\) −17.0718 −0.834012 −0.417006 0.908904i \(-0.636921\pi\)
−0.417006 + 0.908904i \(0.636921\pi\)
\(420\) 0 0
\(421\) 4.14359 0.201946 0.100973 0.994889i \(-0.467804\pi\)
0.100973 + 0.994889i \(0.467804\pi\)
\(422\) 0 0
\(423\) 10.9282 0.531347
\(424\) 0 0
\(425\) −3.75129 −0.181964
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 2.92820 0.141375
\(430\) 0 0
\(431\) 10.2487 0.493663 0.246832 0.969058i \(-0.420611\pi\)
0.246832 + 0.969058i \(0.420611\pi\)
\(432\) 0 0
\(433\) 27.8564 1.33869 0.669347 0.742950i \(-0.266574\pi\)
0.669347 + 0.742950i \(0.266574\pi\)
\(434\) 0 0
\(435\) −17.0718 −0.818530
\(436\) 0 0
\(437\) −10.1436 −0.485234
\(438\) 0 0
\(439\) 13.8564 0.661330 0.330665 0.943748i \(-0.392727\pi\)
0.330665 + 0.943748i \(0.392727\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −28.3923 −1.34896 −0.674480 0.738294i \(-0.735632\pi\)
−0.674480 + 0.738294i \(0.735632\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) 0 0
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) 7.85641 0.370767 0.185383 0.982666i \(-0.440647\pi\)
0.185383 + 0.982666i \(0.440647\pi\)
\(450\) 0 0
\(451\) 16.7846 0.790356
\(452\) 0 0
\(453\) 21.8564 1.02690
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 37.7128 1.76413 0.882065 0.471127i \(-0.156153\pi\)
0.882065 + 0.471127i \(0.156153\pi\)
\(458\) 0 0
\(459\) 0.535898 0.0250136
\(460\) 0 0
\(461\) −27.1769 −1.26576 −0.632878 0.774252i \(-0.718126\pi\)
−0.632878 + 0.774252i \(0.718126\pi\)
\(462\) 0 0
\(463\) −32.7846 −1.52363 −0.761815 0.647795i \(-0.775691\pi\)
−0.761815 + 0.647795i \(0.775691\pi\)
\(464\) 0 0
\(465\) 37.8564 1.75555
\(466\) 0 0
\(467\) −20.7846 −0.961797 −0.480899 0.876776i \(-0.659689\pi\)
−0.480899 + 0.876776i \(0.659689\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.928203 0.0427693
\(472\) 0 0
\(473\) 11.7128 0.538556
\(474\) 0 0
\(475\) −48.4974 −2.22521
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) −5.07180 −0.231736 −0.115868 0.993265i \(-0.536965\pi\)
−0.115868 + 0.993265i \(0.536965\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 30.9282 1.40438
\(486\) 0 0
\(487\) 27.7128 1.25579 0.627894 0.778299i \(-0.283917\pi\)
0.627894 + 0.778299i \(0.283917\pi\)
\(488\) 0 0
\(489\) −10.9282 −0.494190
\(490\) 0 0
\(491\) −6.53590 −0.294961 −0.147480 0.989065i \(-0.547116\pi\)
−0.147480 + 0.989065i \(0.547116\pi\)
\(492\) 0 0
\(493\) −2.64102 −0.118945
\(494\) 0 0
\(495\) −5.07180 −0.227960
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −21.8564 −0.978427 −0.489214 0.872164i \(-0.662716\pi\)
−0.489214 + 0.872164i \(0.662716\pi\)
\(500\) 0 0
\(501\) 10.9282 0.488236
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −53.5692 −2.38380
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) −40.2487 −1.78399 −0.891996 0.452043i \(-0.850696\pi\)
−0.891996 + 0.452043i \(0.850696\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 6.92820 0.305888
\(514\) 0 0
\(515\) −37.8564 −1.66815
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) −22.3923 −0.982913
\(520\) 0 0
\(521\) −30.3923 −1.33151 −0.665756 0.746170i \(-0.731891\pi\)
−0.665756 + 0.746170i \(0.731891\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.85641 0.255109
\(528\) 0 0
\(529\) −20.8564 −0.906800
\(530\) 0 0
\(531\) 1.07180 0.0465120
\(532\) 0 0
\(533\) −22.9282 −0.993131
\(534\) 0 0
\(535\) 42.9282 1.85595
\(536\) 0 0
\(537\) −9.46410 −0.408406
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.85641 0.337773 0.168887 0.985635i \(-0.445983\pi\)
0.168887 + 0.985635i \(0.445983\pi\)
\(542\) 0 0
\(543\) 6.00000 0.257485
\(544\) 0 0
\(545\) 6.92820 0.296772
\(546\) 0 0
\(547\) 10.9282 0.467256 0.233628 0.972326i \(-0.424940\pi\)
0.233628 + 0.972326i \(0.424940\pi\)
\(548\) 0 0
\(549\) −8.92820 −0.381046
\(550\) 0 0
\(551\) −34.1436 −1.45457
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −6.92820 −0.294086
\(556\) 0 0
\(557\) 4.14359 0.175570 0.0877848 0.996139i \(-0.472021\pi\)
0.0877848 + 0.996139i \(0.472021\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) −0.784610 −0.0331262
\(562\) 0 0
\(563\) −33.0718 −1.39381 −0.696905 0.717163i \(-0.745440\pi\)
−0.696905 + 0.717163i \(0.745440\pi\)
\(564\) 0 0
\(565\) −68.7846 −2.89379
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.9282 −1.04504 −0.522522 0.852626i \(-0.675009\pi\)
−0.522522 + 0.852626i \(0.675009\pi\)
\(570\) 0 0
\(571\) 16.7846 0.702414 0.351207 0.936298i \(-0.385771\pi\)
0.351207 + 0.936298i \(0.385771\pi\)
\(572\) 0 0
\(573\) 9.46410 0.395369
\(574\) 0 0
\(575\) 10.2487 0.427401
\(576\) 0 0
\(577\) −23.8564 −0.993155 −0.496578 0.867992i \(-0.665410\pi\)
−0.496578 + 0.867992i \(0.665410\pi\)
\(578\) 0 0
\(579\) −15.8564 −0.658970
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2.92820 −0.121274
\(584\) 0 0
\(585\) 6.92820 0.286446
\(586\) 0 0
\(587\) 20.7846 0.857873 0.428936 0.903335i \(-0.358888\pi\)
0.428936 + 0.903335i \(0.358888\pi\)
\(588\) 0 0
\(589\) 75.7128 3.11969
\(590\) 0 0
\(591\) −7.85641 −0.323169
\(592\) 0 0
\(593\) −8.53590 −0.350527 −0.175264 0.984522i \(-0.556078\pi\)
−0.175264 + 0.984522i \(0.556078\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.85641 −0.239687
\(598\) 0 0
\(599\) 28.3923 1.16008 0.580039 0.814589i \(-0.303037\pi\)
0.580039 + 0.814589i \(0.303037\pi\)
\(600\) 0 0
\(601\) −43.5692 −1.77723 −0.888613 0.458658i \(-0.848330\pi\)
−0.888613 + 0.458658i \(0.848330\pi\)
\(602\) 0 0
\(603\) −2.92820 −0.119246
\(604\) 0 0
\(605\) −30.6795 −1.24730
\(606\) 0 0
\(607\) 13.8564 0.562414 0.281207 0.959647i \(-0.409265\pi\)
0.281207 + 0.959647i \(0.409265\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 21.8564 0.884216
\(612\) 0 0
\(613\) −27.8564 −1.12511 −0.562555 0.826760i \(-0.690181\pi\)
−0.562555 + 0.826760i \(0.690181\pi\)
\(614\) 0 0
\(615\) 39.7128 1.60138
\(616\) 0 0
\(617\) 18.7846 0.756240 0.378120 0.925757i \(-0.376571\pi\)
0.378120 + 0.925757i \(0.376571\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −1.46410 −0.0587524
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) −10.1436 −0.405096
\(628\) 0 0
\(629\) −1.07180 −0.0427353
\(630\) 0 0
\(631\) −26.9282 −1.07199 −0.535997 0.844220i \(-0.680064\pi\)
−0.535997 + 0.844220i \(0.680064\pi\)
\(632\) 0 0
\(633\) 16.0000 0.635943
\(634\) 0 0
\(635\) −10.1436 −0.402536
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −9.46410 −0.374394
\(640\) 0 0
\(641\) −6.78461 −0.267976 −0.133988 0.990983i \(-0.542778\pi\)
−0.133988 + 0.990983i \(0.542778\pi\)
\(642\) 0 0
\(643\) 12.7846 0.504176 0.252088 0.967704i \(-0.418883\pi\)
0.252088 + 0.967704i \(0.418883\pi\)
\(644\) 0 0
\(645\) 27.7128 1.09119
\(646\) 0 0
\(647\) −5.07180 −0.199393 −0.0996965 0.995018i \(-0.531787\pi\)
−0.0996965 + 0.995018i \(0.531787\pi\)
\(648\) 0 0
\(649\) −1.56922 −0.0615972
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20.9282 0.818984 0.409492 0.912314i \(-0.365706\pi\)
0.409492 + 0.912314i \(0.365706\pi\)
\(654\) 0 0
\(655\) −41.5692 −1.62424
\(656\) 0 0
\(657\) −12.9282 −0.504377
\(658\) 0 0
\(659\) 14.5359 0.566238 0.283119 0.959085i \(-0.408631\pi\)
0.283119 + 0.959085i \(0.408631\pi\)
\(660\) 0 0
\(661\) 26.7846 1.04180 0.520900 0.853618i \(-0.325596\pi\)
0.520900 + 0.853618i \(0.325596\pi\)
\(662\) 0 0
\(663\) 1.07180 0.0416251
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.21539 0.279381
\(668\) 0 0
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) 13.0718 0.504631
\(672\) 0 0
\(673\) 31.8564 1.22797 0.613987 0.789316i \(-0.289565\pi\)
0.613987 + 0.789316i \(0.289565\pi\)
\(674\) 0 0
\(675\) −7.00000 −0.269430
\(676\) 0 0
\(677\) −1.60770 −0.0617887 −0.0308944 0.999523i \(-0.509836\pi\)
−0.0308944 + 0.999523i \(0.509836\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −22.9282 −0.878611
\(682\) 0 0
\(683\) −18.2487 −0.698268 −0.349134 0.937073i \(-0.613524\pi\)
−0.349134 + 0.937073i \(0.613524\pi\)
\(684\) 0 0
\(685\) −30.9282 −1.18171
\(686\) 0 0
\(687\) 27.8564 1.06279
\(688\) 0 0
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 41.8564 1.59229 0.796146 0.605104i \(-0.206869\pi\)
0.796146 + 0.605104i \(0.206869\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 61.8564 2.34635
\(696\) 0 0
\(697\) 6.14359 0.232705
\(698\) 0 0
\(699\) −12.9282 −0.488990
\(700\) 0 0
\(701\) −28.6410 −1.08176 −0.540878 0.841101i \(-0.681908\pi\)
−0.540878 + 0.841101i \(0.681908\pi\)
\(702\) 0 0
\(703\) −13.8564 −0.522604
\(704\) 0 0
\(705\) −37.8564 −1.42575
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −17.7128 −0.665219 −0.332609 0.943065i \(-0.607929\pi\)
−0.332609 + 0.943065i \(0.607929\pi\)
\(710\) 0 0
\(711\) 10.9282 0.409840
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) −10.1436 −0.379349
\(716\) 0 0
\(717\) 14.5359 0.542853
\(718\) 0 0
\(719\) −13.8564 −0.516757 −0.258378 0.966044i \(-0.583188\pi\)
−0.258378 + 0.966044i \(0.583188\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −3.07180 −0.114241
\(724\) 0 0
\(725\) 34.4974 1.28120
\(726\) 0 0
\(727\) 21.0718 0.781510 0.390755 0.920495i \(-0.372214\pi\)
0.390755 + 0.920495i \(0.372214\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.28719 0.158567
\(732\) 0 0
\(733\) 5.71281 0.211008 0.105504 0.994419i \(-0.466354\pi\)
0.105504 + 0.994419i \(0.466354\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.28719 0.157921
\(738\) 0 0
\(739\) −37.0718 −1.36371 −0.681854 0.731488i \(-0.738826\pi\)
−0.681854 + 0.731488i \(0.738826\pi\)
\(740\) 0 0
\(741\) 13.8564 0.509028
\(742\) 0 0
\(743\) −11.6077 −0.425845 −0.212923 0.977069i \(-0.568298\pi\)
−0.212923 + 0.977069i \(0.568298\pi\)
\(744\) 0 0
\(745\) 62.3538 2.28447
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −27.7128 −1.01125 −0.505627 0.862752i \(-0.668739\pi\)
−0.505627 + 0.862752i \(0.668739\pi\)
\(752\) 0 0
\(753\) −9.07180 −0.330595
\(754\) 0 0
\(755\) −75.7128 −2.75547
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) 2.14359 0.0778075
\(760\) 0 0
\(761\) −27.4641 −0.995573 −0.497786 0.867300i \(-0.665854\pi\)
−0.497786 + 0.867300i \(0.665854\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.85641 −0.0671185
\(766\) 0 0
\(767\) 2.14359 0.0774007
\(768\) 0 0
\(769\) −4.14359 −0.149422 −0.0747109 0.997205i \(-0.523803\pi\)
−0.0747109 + 0.997205i \(0.523803\pi\)
\(770\) 0 0
\(771\) 11.4641 0.412870
\(772\) 0 0
\(773\) −5.32051 −0.191365 −0.0956827 0.995412i \(-0.530503\pi\)
−0.0956827 + 0.995412i \(0.530503\pi\)
\(774\) 0 0
\(775\) −76.4974 −2.74787
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 79.4256 2.84572
\(780\) 0 0
\(781\) 13.8564 0.495821
\(782\) 0 0
\(783\) −4.92820 −0.176120
\(784\) 0 0
\(785\) −3.21539 −0.114762
\(786\) 0 0
\(787\) −23.7128 −0.845270 −0.422635 0.906300i \(-0.638895\pi\)
−0.422635 + 0.906300i \(0.638895\pi\)
\(788\) 0 0
\(789\) 12.3923 0.441178
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −17.8564 −0.634100
\(794\) 0 0
\(795\) −6.92820 −0.245718
\(796\) 0 0
\(797\) 6.39230 0.226427 0.113214 0.993571i \(-0.463886\pi\)
0.113214 + 0.993571i \(0.463886\pi\)
\(798\) 0 0
\(799\) −5.85641 −0.207185
\(800\) 0 0
\(801\) −3.46410 −0.122398
\(802\) 0 0
\(803\) 18.9282 0.667962
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −3.46410 −0.121942
\(808\) 0 0
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) 37.5692 1.31923 0.659617 0.751602i \(-0.270719\pi\)
0.659617 + 0.751602i \(0.270719\pi\)
\(812\) 0 0
\(813\) −2.92820 −0.102697
\(814\) 0 0
\(815\) 37.8564 1.32605
\(816\) 0 0
\(817\) 55.4256 1.93910
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.1436 0.423814 0.211907 0.977290i \(-0.432033\pi\)
0.211907 + 0.977290i \(0.432033\pi\)
\(822\) 0 0
\(823\) 21.0718 0.734517 0.367258 0.930119i \(-0.380296\pi\)
0.367258 + 0.930119i \(0.380296\pi\)
\(824\) 0 0
\(825\) 10.2487 0.356814
\(826\) 0 0
\(827\) −37.1769 −1.29277 −0.646384 0.763012i \(-0.723720\pi\)
−0.646384 + 0.763012i \(0.723720\pi\)
\(828\) 0 0
\(829\) 21.7128 0.754117 0.377059 0.926189i \(-0.376936\pi\)
0.377059 + 0.926189i \(0.376936\pi\)
\(830\) 0 0
\(831\) 0.143594 0.00498120
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −37.8564 −1.31007
\(836\) 0 0
\(837\) 10.9282 0.377734
\(838\) 0 0
\(839\) −10.9282 −0.377283 −0.188642 0.982046i \(-0.560408\pi\)
−0.188642 + 0.982046i \(0.560408\pi\)
\(840\) 0 0
\(841\) −4.71281 −0.162511
\(842\) 0 0
\(843\) −12.9282 −0.445271
\(844\) 0 0
\(845\) −31.1769 −1.07252
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −14.9282 −0.512335
\(850\) 0 0
\(851\) 2.92820 0.100378
\(852\) 0 0
\(853\) 23.0718 0.789963 0.394982 0.918689i \(-0.370751\pi\)
0.394982 + 0.918689i \(0.370751\pi\)
\(854\) 0 0
\(855\) −24.0000 −0.820783
\(856\) 0 0
\(857\) −55.1769 −1.88481 −0.942404 0.334477i \(-0.891440\pi\)
−0.942404 + 0.334477i \(0.891440\pi\)
\(858\) 0 0
\(859\) 38.9282 1.32821 0.664107 0.747638i \(-0.268812\pi\)
0.664107 + 0.747638i \(0.268812\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9.46410 0.322162 0.161081 0.986941i \(-0.448502\pi\)
0.161081 + 0.986941i \(0.448502\pi\)
\(864\) 0 0
\(865\) 77.5692 2.63743
\(866\) 0 0
\(867\) 16.7128 0.567597
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −5.85641 −0.198437
\(872\) 0 0
\(873\) 8.92820 0.302174
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −8.14359 −0.274990 −0.137495 0.990502i \(-0.543905\pi\)
−0.137495 + 0.990502i \(0.543905\pi\)
\(878\) 0 0
\(879\) −16.5359 −0.557742
\(880\) 0 0
\(881\) 13.3205 0.448779 0.224390 0.974500i \(-0.427961\pi\)
0.224390 + 0.974500i \(0.427961\pi\)
\(882\) 0 0
\(883\) −33.5692 −1.12969 −0.564847 0.825196i \(-0.691065\pi\)
−0.564847 + 0.825196i \(0.691065\pi\)
\(884\) 0 0
\(885\) −3.71281 −0.124805
\(886\) 0 0
\(887\) −26.9282 −0.904161 −0.452080 0.891977i \(-0.649318\pi\)
−0.452080 + 0.891977i \(0.649318\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.46410 −0.0490492
\(892\) 0 0
\(893\) −75.7128 −2.53363
\(894\) 0 0
\(895\) 32.7846 1.09587
\(896\) 0 0
\(897\) −2.92820 −0.0977699
\(898\) 0 0
\(899\) −53.8564 −1.79621
\(900\) 0 0
\(901\) −1.07180 −0.0357067
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −20.7846 −0.690904
\(906\) 0 0
\(907\) −43.7128 −1.45146 −0.725730 0.687980i \(-0.758498\pi\)
−0.725730 + 0.687980i \(0.758498\pi\)
\(908\) 0 0
\(909\) −15.4641 −0.512912
\(910\) 0 0
\(911\) −29.1769 −0.966674 −0.483337 0.875434i \(-0.660575\pi\)
−0.483337 + 0.875434i \(0.660575\pi\)
\(912\) 0 0
\(913\) −5.85641 −0.193819
\(914\) 0 0
\(915\) 30.9282 1.02245
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 22.9282 0.755510
\(922\) 0 0
\(923\) −18.9282 −0.623029
\(924\) 0 0
\(925\) 14.0000 0.460317
\(926\) 0 0
\(927\) −10.9282 −0.358929
\(928\) 0 0
\(929\) −13.6077 −0.446454 −0.223227 0.974766i \(-0.571659\pi\)
−0.223227 + 0.974766i \(0.571659\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −18.9282 −0.619682
\(934\) 0 0
\(935\) 2.71797 0.0888870
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) −36.5359 −1.19104 −0.595518 0.803342i \(-0.703053\pi\)
−0.595518 + 0.803342i \(0.703053\pi\)
\(942\) 0 0
\(943\) −16.7846 −0.546582
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.2487 0.852969 0.426484 0.904495i \(-0.359752\pi\)
0.426484 + 0.904495i \(0.359752\pi\)
\(948\) 0 0
\(949\) −25.8564 −0.839334
\(950\) 0 0
\(951\) 0.143594 0.00465634
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 0 0
\(955\) −32.7846 −1.06089
\(956\) 0 0
\(957\) 7.21539 0.233240
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 88.4256 2.85244
\(962\) 0 0
\(963\) 12.3923 0.399336
\(964\) 0 0
\(965\) 54.9282 1.76820
\(966\) 0 0
\(967\) −34.9282 −1.12322 −0.561608 0.827404i \(-0.689817\pi\)
−0.561608 + 0.827404i \(0.689817\pi\)
\(968\) 0 0
\(969\) −3.71281 −0.119273
\(970\) 0 0
\(971\) −1.85641 −0.0595749 −0.0297875 0.999556i \(-0.509483\pi\)
−0.0297875 + 0.999556i \(0.509483\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −14.0000 −0.448359
\(976\) 0 0
\(977\) 38.4974 1.23164 0.615821 0.787886i \(-0.288825\pi\)
0.615821 + 0.787886i \(0.288825\pi\)
\(978\) 0 0
\(979\) 5.07180 0.162095
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) −3.71281 −0.118420 −0.0592102 0.998246i \(-0.518858\pi\)
−0.0592102 + 0.998246i \(0.518858\pi\)
\(984\) 0 0
\(985\) 27.2154 0.867154
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −11.7128 −0.372446
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) −24.0000 −0.761617
\(994\) 0 0
\(995\) 20.2872 0.643147
\(996\) 0 0
\(997\) 31.0718 0.984054 0.492027 0.870580i \(-0.336256\pi\)
0.492027 + 0.870580i \(0.336256\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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 9408.2.a.do.1.2 2
4.3 odd 2 9408.2.a.dx.1.2 2
7.6 odd 2 1344.2.a.v.1.1 2
8.3 odd 2 4704.2.a.bm.1.1 2
8.5 even 2 4704.2.a.bn.1.1 2
21.20 even 2 4032.2.a.bs.1.2 2
28.27 even 2 1344.2.a.u.1.1 2
56.13 odd 2 672.2.a.i.1.2 2
56.27 even 2 672.2.a.j.1.2 yes 2
84.83 odd 2 4032.2.a.br.1.2 2
112.13 odd 4 5376.2.c.bh.2689.4 4
112.27 even 4 5376.2.c.bn.2689.3 4
112.69 odd 4 5376.2.c.bh.2689.1 4
112.83 even 4 5376.2.c.bn.2689.2 4
168.83 odd 2 2016.2.a.s.1.1 2
168.125 even 2 2016.2.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.a.i.1.2 2 56.13 odd 2
672.2.a.j.1.2 yes 2 56.27 even 2
1344.2.a.u.1.1 2 28.27 even 2
1344.2.a.v.1.1 2 7.6 odd 2
2016.2.a.s.1.1 2 168.83 odd 2
2016.2.a.t.1.1 2 168.125 even 2
4032.2.a.br.1.2 2 84.83 odd 2
4032.2.a.bs.1.2 2 21.20 even 2
4704.2.a.bm.1.1 2 8.3 odd 2
4704.2.a.bn.1.1 2 8.5 even 2
5376.2.c.bh.2689.1 4 112.69 odd 4
5376.2.c.bh.2689.4 4 112.13 odd 4
5376.2.c.bn.2689.2 4 112.83 even 4
5376.2.c.bn.2689.3 4 112.27 even 4
9408.2.a.do.1.2 2 1.1 even 1 trivial
9408.2.a.dx.1.2 2 4.3 odd 2