Properties

Label 3072.2.a.n.1.4
Level $3072$
Weight $2$
Character 3072.1
Self dual yes
Analytic conductor $24.530$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 48)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.68554\) of defining polynomial
Character \(\chi\) \(=\) 3072.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.79793 q^{5} -2.15894 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.79793 q^{5} -2.15894 q^{7} +1.00000 q^{9} -2.54266 q^{11} -1.95687 q^{13} -3.79793 q^{15} +0.224777 q^{17} +0.224777 q^{19} +2.15894 q^{21} +2.82843 q^{23} +9.42429 q^{25} -1.00000 q^{27} -2.62636 q^{29} -1.84106 q^{31} +2.54266 q^{33} -8.19951 q^{35} +5.18944 q^{37} +1.95687 q^{39} +5.88163 q^{41} +10.9670 q^{43} +3.79793 q^{45} -2.82843 q^{47} -2.33897 q^{49} -0.224777 q^{51} +10.6264 q^{53} -9.65685 q^{55} -0.224777 q^{57} +5.65685 q^{59} +8.46742 q^{61} -2.15894 q^{63} -7.43208 q^{65} +14.7422 q^{67} -2.82843 q^{69} +4.31788 q^{71} +5.97474 q^{73} -9.42429 q^{75} +5.48946 q^{77} +15.0075 q^{79} +1.00000 q^{81} -14.3059 q^{83} +0.853690 q^{85} +2.62636 q^{87} -1.42847 q^{89} +4.22478 q^{91} +1.84106 q^{93} +0.853690 q^{95} -16.3990 q^{97} -2.54266 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{5} - 4q^{7} + 4q^{9} + 8q^{13} - 4q^{15} + 4q^{21} + 4q^{25} - 4q^{27} + 12q^{29} - 12q^{31} + 16q^{37} - 8q^{39} + 4q^{45} + 4q^{49} + 20q^{53} - 16q^{55} + 16q^{61} - 4q^{63} - 8q^{65} + 16q^{67} + 8q^{71} - 8q^{73} - 4q^{75} + 24q^{77} - 12q^{79} + 4q^{81} + 24q^{85} - 12q^{87} - 8q^{89} + 16q^{91} + 12q^{93} + 24q^{95} + O(q^{100}) \)

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
\(5\) 3.79793 1.69849 0.849244 0.528001i \(-0.177058\pi\)
0.849244 + 0.528001i \(0.177058\pi\)
\(6\) 0 0
\(7\) −2.15894 −0.816003 −0.408002 0.912981i \(-0.633774\pi\)
−0.408002 + 0.912981i \(0.633774\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.54266 −0.766641 −0.383321 0.923615i \(-0.625220\pi\)
−0.383321 + 0.923615i \(0.625220\pi\)
\(12\) 0 0
\(13\) −1.95687 −0.542739 −0.271370 0.962475i \(-0.587477\pi\)
−0.271370 + 0.962475i \(0.587477\pi\)
\(14\) 0 0
\(15\) −3.79793 −0.980622
\(16\) 0 0
\(17\) 0.224777 0.0545165 0.0272583 0.999628i \(-0.491322\pi\)
0.0272583 + 0.999628i \(0.491322\pi\)
\(18\) 0 0
\(19\) 0.224777 0.0515675 0.0257837 0.999668i \(-0.491792\pi\)
0.0257837 + 0.999668i \(0.491792\pi\)
\(20\) 0 0
\(21\) 2.15894 0.471120
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) 9.42429 1.88486
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.62636 −0.487703 −0.243851 0.969813i \(-0.578411\pi\)
−0.243851 + 0.969813i \(0.578411\pi\)
\(30\) 0 0
\(31\) −1.84106 −0.330664 −0.165332 0.986238i \(-0.552870\pi\)
−0.165332 + 0.986238i \(0.552870\pi\)
\(32\) 0 0
\(33\) 2.54266 0.442620
\(34\) 0 0
\(35\) −8.19951 −1.38597
\(36\) 0 0
\(37\) 5.18944 0.853138 0.426569 0.904455i \(-0.359722\pi\)
0.426569 + 0.904455i \(0.359722\pi\)
\(38\) 0 0
\(39\) 1.95687 0.313351
\(40\) 0 0
\(41\) 5.88163 0.918557 0.459278 0.888292i \(-0.348108\pi\)
0.459278 + 0.888292i \(0.348108\pi\)
\(42\) 0 0
\(43\) 10.9670 1.67244 0.836222 0.548391i \(-0.184759\pi\)
0.836222 + 0.548391i \(0.184759\pi\)
\(44\) 0 0
\(45\) 3.79793 0.566162
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −2.33897 −0.334139
\(50\) 0 0
\(51\) −0.224777 −0.0314751
\(52\) 0 0
\(53\) 10.6264 1.45964 0.729821 0.683638i \(-0.239603\pi\)
0.729821 + 0.683638i \(0.239603\pi\)
\(54\) 0 0
\(55\) −9.65685 −1.30213
\(56\) 0 0
\(57\) −0.224777 −0.0297725
\(58\) 0 0
\(59\) 5.65685 0.736460 0.368230 0.929735i \(-0.379964\pi\)
0.368230 + 0.929735i \(0.379964\pi\)
\(60\) 0 0
\(61\) 8.46742 1.08414 0.542071 0.840333i \(-0.317640\pi\)
0.542071 + 0.840333i \(0.317640\pi\)
\(62\) 0 0
\(63\) −2.15894 −0.272001
\(64\) 0 0
\(65\) −7.43208 −0.921836
\(66\) 0 0
\(67\) 14.7422 1.80104 0.900522 0.434811i \(-0.143185\pi\)
0.900522 + 0.434811i \(0.143185\pi\)
\(68\) 0 0
\(69\) −2.82843 −0.340503
\(70\) 0 0
\(71\) 4.31788 0.512438 0.256219 0.966619i \(-0.417523\pi\)
0.256219 + 0.966619i \(0.417523\pi\)
\(72\) 0 0
\(73\) 5.97474 0.699290 0.349645 0.936882i \(-0.386302\pi\)
0.349645 + 0.936882i \(0.386302\pi\)
\(74\) 0 0
\(75\) −9.42429 −1.08822
\(76\) 0 0
\(77\) 5.48946 0.625582
\(78\) 0 0
\(79\) 15.0075 1.68848 0.844239 0.535966i \(-0.180053\pi\)
0.844239 + 0.535966i \(0.180053\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.3059 −1.57028 −0.785140 0.619319i \(-0.787409\pi\)
−0.785140 + 0.619319i \(0.787409\pi\)
\(84\) 0 0
\(85\) 0.853690 0.0925956
\(86\) 0 0
\(87\) 2.62636 0.281575
\(88\) 0 0
\(89\) −1.42847 −0.151417 −0.0757086 0.997130i \(-0.524122\pi\)
−0.0757086 + 0.997130i \(0.524122\pi\)
\(90\) 0 0
\(91\) 4.22478 0.442877
\(92\) 0 0
\(93\) 1.84106 0.190909
\(94\) 0 0
\(95\) 0.853690 0.0875867
\(96\) 0 0
\(97\) −16.3990 −1.66507 −0.832535 0.553973i \(-0.813111\pi\)
−0.832535 + 0.553973i \(0.813111\pi\)
\(98\) 0 0
\(99\) −2.54266 −0.255547
\(100\) 0 0
\(101\) −0.115816 −0.0115241 −0.00576206 0.999983i \(-0.501834\pi\)
−0.00576206 + 0.999983i \(0.501834\pi\)
\(102\) 0 0
\(103\) −13.3507 −1.31548 −0.657740 0.753245i \(-0.728488\pi\)
−0.657740 + 0.753245i \(0.728488\pi\)
\(104\) 0 0
\(105\) 8.19951 0.800191
\(106\) 0 0
\(107\) 10.2926 0.995025 0.497513 0.867457i \(-0.334247\pi\)
0.497513 + 0.867457i \(0.334247\pi\)
\(108\) 0 0
\(109\) 9.95687 0.953696 0.476848 0.878986i \(-0.341779\pi\)
0.476848 + 0.878986i \(0.341779\pi\)
\(110\) 0 0
\(111\) −5.18944 −0.492559
\(112\) 0 0
\(113\) −18.8486 −1.77313 −0.886563 0.462608i \(-0.846914\pi\)
−0.886563 + 0.462608i \(0.846914\pi\)
\(114\) 0 0
\(115\) 10.7422 1.00171
\(116\) 0 0
\(117\) −1.95687 −0.180913
\(118\) 0 0
\(119\) −0.485281 −0.0444857
\(120\) 0 0
\(121\) −4.53488 −0.412261
\(122\) 0 0
\(123\) −5.88163 −0.530329
\(124\) 0 0
\(125\) 16.8032 1.50292
\(126\) 0 0
\(127\) 3.81580 0.338597 0.169299 0.985565i \(-0.445850\pi\)
0.169299 + 0.985565i \(0.445850\pi\)
\(128\) 0 0
\(129\) −10.9670 −0.965586
\(130\) 0 0
\(131\) 1.08532 0.0948250 0.0474125 0.998875i \(-0.484902\pi\)
0.0474125 + 0.998875i \(0.484902\pi\)
\(132\) 0 0
\(133\) −0.485281 −0.0420792
\(134\) 0 0
\(135\) −3.79793 −0.326874
\(136\) 0 0
\(137\) −5.31010 −0.453672 −0.226836 0.973933i \(-0.572838\pi\)
−0.226836 + 0.973933i \(0.572838\pi\)
\(138\) 0 0
\(139\) −12.3990 −1.05167 −0.525836 0.850586i \(-0.676247\pi\)
−0.525836 + 0.850586i \(0.676247\pi\)
\(140\) 0 0
\(141\) 2.82843 0.238197
\(142\) 0 0
\(143\) 4.97567 0.416086
\(144\) 0 0
\(145\) −9.97474 −0.828357
\(146\) 0 0
\(147\) 2.33897 0.192915
\(148\) 0 0
\(149\) −1.45479 −0.119181 −0.0595904 0.998223i \(-0.518979\pi\)
−0.0595904 + 0.998223i \(0.518979\pi\)
\(150\) 0 0
\(151\) −2.03696 −0.165766 −0.0828829 0.996559i \(-0.526413\pi\)
−0.0828829 + 0.996559i \(0.526413\pi\)
\(152\) 0 0
\(153\) 0.224777 0.0181722
\(154\) 0 0
\(155\) −6.99222 −0.561628
\(156\) 0 0
\(157\) 8.61790 0.687784 0.343892 0.939009i \(-0.388255\pi\)
0.343892 + 0.939009i \(0.388255\pi\)
\(158\) 0 0
\(159\) −10.6264 −0.842725
\(160\) 0 0
\(161\) −6.10641 −0.481252
\(162\) 0 0
\(163\) −4.86054 −0.380707 −0.190354 0.981716i \(-0.560963\pi\)
−0.190354 + 0.981716i \(0.560963\pi\)
\(164\) 0 0
\(165\) 9.65685 0.751785
\(166\) 0 0
\(167\) 21.7023 1.67937 0.839686 0.543072i \(-0.182739\pi\)
0.839686 + 0.543072i \(0.182739\pi\)
\(168\) 0 0
\(169\) −9.17064 −0.705434
\(170\) 0 0
\(171\) 0.224777 0.0171892
\(172\) 0 0
\(173\) 12.3695 0.940433 0.470217 0.882551i \(-0.344176\pi\)
0.470217 + 0.882551i \(0.344176\pi\)
\(174\) 0 0
\(175\) −20.3465 −1.53805
\(176\) 0 0
\(177\) −5.65685 −0.425195
\(178\) 0 0
\(179\) 11.6413 0.870111 0.435055 0.900404i \(-0.356729\pi\)
0.435055 + 0.900404i \(0.356729\pi\)
\(180\) 0 0
\(181\) 9.50732 0.706673 0.353337 0.935496i \(-0.385047\pi\)
0.353337 + 0.935496i \(0.385047\pi\)
\(182\) 0 0
\(183\) −8.46742 −0.625930
\(184\) 0 0
\(185\) 19.7091 1.44904
\(186\) 0 0
\(187\) −0.571533 −0.0417946
\(188\) 0 0
\(189\) 2.15894 0.157040
\(190\) 0 0
\(191\) 20.8032 1.50526 0.752632 0.658441i \(-0.228784\pi\)
0.752632 + 0.658441i \(0.228784\pi\)
\(192\) 0 0
\(193\) 14.1454 1.01821 0.509103 0.860705i \(-0.329977\pi\)
0.509103 + 0.860705i \(0.329977\pi\)
\(194\) 0 0
\(195\) 7.43208 0.532222
\(196\) 0 0
\(197\) 3.43463 0.244707 0.122354 0.992487i \(-0.460956\pi\)
0.122354 + 0.992487i \(0.460956\pi\)
\(198\) 0 0
\(199\) −0.306182 −0.0217047 −0.0108523 0.999941i \(-0.503454\pi\)
−0.0108523 + 0.999941i \(0.503454\pi\)
\(200\) 0 0
\(201\) −14.7422 −1.03983
\(202\) 0 0
\(203\) 5.67016 0.397967
\(204\) 0 0
\(205\) 22.3380 1.56016
\(206\) 0 0
\(207\) 2.82843 0.196589
\(208\) 0 0
\(209\) −0.571533 −0.0395337
\(210\) 0 0
\(211\) 10.2284 0.704151 0.352076 0.935972i \(-0.385476\pi\)
0.352076 + 0.935972i \(0.385476\pi\)
\(212\) 0 0
\(213\) −4.31788 −0.295856
\(214\) 0 0
\(215\) 41.6517 2.84063
\(216\) 0 0
\(217\) 3.97474 0.269823
\(218\) 0 0
\(219\) −5.97474 −0.403735
\(220\) 0 0
\(221\) −0.439861 −0.0295883
\(222\) 0 0
\(223\) 1.71908 0.115118 0.0575591 0.998342i \(-0.481668\pi\)
0.0575591 + 0.998342i \(0.481668\pi\)
\(224\) 0 0
\(225\) 9.42429 0.628286
\(226\) 0 0
\(227\) −14.3059 −0.949518 −0.474759 0.880116i \(-0.657465\pi\)
−0.474759 + 0.880116i \(0.657465\pi\)
\(228\) 0 0
\(229\) 16.9981 1.12327 0.561634 0.827386i \(-0.310173\pi\)
0.561634 + 0.827386i \(0.310173\pi\)
\(230\) 0 0
\(231\) −5.48946 −0.361180
\(232\) 0 0
\(233\) 13.3779 0.876418 0.438209 0.898873i \(-0.355613\pi\)
0.438209 + 0.898873i \(0.355613\pi\)
\(234\) 0 0
\(235\) −10.7422 −0.700742
\(236\) 0 0
\(237\) −15.0075 −0.974844
\(238\) 0 0
\(239\) 13.3675 0.864670 0.432335 0.901713i \(-0.357690\pi\)
0.432335 + 0.901713i \(0.357690\pi\)
\(240\) 0 0
\(241\) −0.211474 −0.0136222 −0.00681112 0.999977i \(-0.502168\pi\)
−0.00681112 + 0.999977i \(0.502168\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −8.88325 −0.567530
\(246\) 0 0
\(247\) −0.439861 −0.0279877
\(248\) 0 0
\(249\) 14.3059 0.906601
\(250\) 0 0
\(251\) 14.7555 0.931358 0.465679 0.884954i \(-0.345810\pi\)
0.465679 + 0.884954i \(0.345810\pi\)
\(252\) 0 0
\(253\) −7.19173 −0.452140
\(254\) 0 0
\(255\) −0.853690 −0.0534601
\(256\) 0 0
\(257\) −0.742176 −0.0462957 −0.0231478 0.999732i \(-0.507369\pi\)
−0.0231478 + 0.999732i \(0.507369\pi\)
\(258\) 0 0
\(259\) −11.2037 −0.696163
\(260\) 0 0
\(261\) −2.62636 −0.162568
\(262\) 0 0
\(263\) −5.48435 −0.338180 −0.169090 0.985601i \(-0.554083\pi\)
−0.169090 + 0.985601i \(0.554083\pi\)
\(264\) 0 0
\(265\) 40.3582 2.47918
\(266\) 0 0
\(267\) 1.42847 0.0874208
\(268\) 0 0
\(269\) −20.4694 −1.24804 −0.624021 0.781407i \(-0.714502\pi\)
−0.624021 + 0.781407i \(0.714502\pi\)
\(270\) 0 0
\(271\) −14.0370 −0.852685 −0.426342 0.904562i \(-0.640198\pi\)
−0.426342 + 0.904562i \(0.640198\pi\)
\(272\) 0 0
\(273\) −4.22478 −0.255695
\(274\) 0 0
\(275\) −23.9628 −1.44501
\(276\) 0 0
\(277\) 13.4211 0.806394 0.403197 0.915113i \(-0.367899\pi\)
0.403197 + 0.915113i \(0.367899\pi\)
\(278\) 0 0
\(279\) −1.84106 −0.110221
\(280\) 0 0
\(281\) 3.89359 0.232272 0.116136 0.993233i \(-0.462949\pi\)
0.116136 + 0.993233i \(0.462949\pi\)
\(282\) 0 0
\(283\) −17.6569 −1.04959 −0.524796 0.851228i \(-0.675858\pi\)
−0.524796 + 0.851228i \(0.675858\pi\)
\(284\) 0 0
\(285\) −0.853690 −0.0505682
\(286\) 0 0
\(287\) −12.6981 −0.749545
\(288\) 0 0
\(289\) −16.9495 −0.997028
\(290\) 0 0
\(291\) 16.3990 0.961328
\(292\) 0 0
\(293\) 15.7759 0.921639 0.460819 0.887494i \(-0.347556\pi\)
0.460819 + 0.887494i \(0.347556\pi\)
\(294\) 0 0
\(295\) 21.4844 1.25087
\(296\) 0 0
\(297\) 2.54266 0.147540
\(298\) 0 0
\(299\) −5.53488 −0.320090
\(300\) 0 0
\(301\) −23.6770 −1.36472
\(302\) 0 0
\(303\) 0.115816 0.00665345
\(304\) 0 0
\(305\) 32.1587 1.84140
\(306\) 0 0
\(307\) −7.64129 −0.436111 −0.218056 0.975936i \(-0.569971\pi\)
−0.218056 + 0.975936i \(0.569971\pi\)
\(308\) 0 0
\(309\) 13.3507 0.759493
\(310\) 0 0
\(311\) −24.1623 −1.37012 −0.685059 0.728488i \(-0.740224\pi\)
−0.685059 + 0.728488i \(0.740224\pi\)
\(312\) 0 0
\(313\) −16.6105 −0.938881 −0.469441 0.882964i \(-0.655544\pi\)
−0.469441 + 0.882964i \(0.655544\pi\)
\(314\) 0 0
\(315\) −8.19951 −0.461990
\(316\) 0 0
\(317\) 2.56213 0.143903 0.0719517 0.997408i \(-0.477077\pi\)
0.0719517 + 0.997408i \(0.477077\pi\)
\(318\) 0 0
\(319\) 6.67794 0.373893
\(320\) 0 0
\(321\) −10.2926 −0.574478
\(322\) 0 0
\(323\) 0.0505249 0.00281128
\(324\) 0 0
\(325\) −18.4422 −1.02299
\(326\) 0 0
\(327\) −9.95687 −0.550616
\(328\) 0 0
\(329\) 6.10641 0.336657
\(330\) 0 0
\(331\) −19.1275 −1.05134 −0.525671 0.850688i \(-0.676186\pi\)
−0.525671 + 0.850688i \(0.676186\pi\)
\(332\) 0 0
\(333\) 5.18944 0.284379
\(334\) 0 0
\(335\) 55.9898 3.05905
\(336\) 0 0
\(337\) 1.12615 0.0613454 0.0306727 0.999529i \(-0.490235\pi\)
0.0306727 + 0.999529i \(0.490235\pi\)
\(338\) 0 0
\(339\) 18.8486 1.02371
\(340\) 0 0
\(341\) 4.68119 0.253500
\(342\) 0 0
\(343\) 20.1623 1.08866
\(344\) 0 0
\(345\) −10.7422 −0.578339
\(346\) 0 0
\(347\) 29.4068 1.57864 0.789320 0.613982i \(-0.210433\pi\)
0.789320 + 0.613982i \(0.210433\pi\)
\(348\) 0 0
\(349\) −27.2738 −1.45993 −0.729967 0.683482i \(-0.760465\pi\)
−0.729967 + 0.683482i \(0.760465\pi\)
\(350\) 0 0
\(351\) 1.95687 0.104450
\(352\) 0 0
\(353\) 25.5908 1.36206 0.681029 0.732256i \(-0.261533\pi\)
0.681029 + 0.732256i \(0.261533\pi\)
\(354\) 0 0
\(355\) 16.3990 0.870370
\(356\) 0 0
\(357\) 0.485281 0.0256838
\(358\) 0 0
\(359\) −3.77296 −0.199129 −0.0995645 0.995031i \(-0.531745\pi\)
−0.0995645 + 0.995031i \(0.531745\pi\)
\(360\) 0 0
\(361\) −18.9495 −0.997341
\(362\) 0 0
\(363\) 4.53488 0.238019
\(364\) 0 0
\(365\) 22.6917 1.18774
\(366\) 0 0
\(367\) −27.4474 −1.43274 −0.716371 0.697720i \(-0.754198\pi\)
−0.716371 + 0.697720i \(0.754198\pi\)
\(368\) 0 0
\(369\) 5.88163 0.306186
\(370\) 0 0
\(371\) −22.9417 −1.19107
\(372\) 0 0
\(373\) 17.8518 0.924332 0.462166 0.886794i \(-0.347072\pi\)
0.462166 + 0.886794i \(0.347072\pi\)
\(374\) 0 0
\(375\) −16.8032 −0.867712
\(376\) 0 0
\(377\) 5.13946 0.264695
\(378\) 0 0
\(379\) −16.5018 −0.847642 −0.423821 0.905746i \(-0.639311\pi\)
−0.423821 + 0.905746i \(0.639311\pi\)
\(380\) 0 0
\(381\) −3.81580 −0.195489
\(382\) 0 0
\(383\) 17.1885 0.878291 0.439145 0.898416i \(-0.355281\pi\)
0.439145 + 0.898416i \(0.355281\pi\)
\(384\) 0 0
\(385\) 20.8486 1.06254
\(386\) 0 0
\(387\) 10.9670 0.557482
\(388\) 0 0
\(389\) 2.66209 0.134973 0.0674866 0.997720i \(-0.478502\pi\)
0.0674866 + 0.997720i \(0.478502\pi\)
\(390\) 0 0
\(391\) 0.635767 0.0321521
\(392\) 0 0
\(393\) −1.08532 −0.0547472
\(394\) 0 0
\(395\) 56.9976 2.86786
\(396\) 0 0
\(397\) −11.8959 −0.597037 −0.298519 0.954404i \(-0.596493\pi\)
−0.298519 + 0.954404i \(0.596493\pi\)
\(398\) 0 0
\(399\) 0.485281 0.0242945
\(400\) 0 0
\(401\) −1.12389 −0.0561242 −0.0280621 0.999606i \(-0.508934\pi\)
−0.0280621 + 0.999606i \(0.508934\pi\)
\(402\) 0 0
\(403\) 3.60272 0.179464
\(404\) 0 0
\(405\) 3.79793 0.188721
\(406\) 0 0
\(407\) −13.1950 −0.654051
\(408\) 0 0
\(409\) 13.7211 0.678464 0.339232 0.940703i \(-0.389833\pi\)
0.339232 + 0.940703i \(0.389833\pi\)
\(410\) 0 0
\(411\) 5.31010 0.261928
\(412\) 0 0
\(413\) −12.2128 −0.600953
\(414\) 0 0
\(415\) −54.3329 −2.66710
\(416\) 0 0
\(417\) 12.3990 0.607183
\(418\) 0 0
\(419\) 13.1629 0.643048 0.321524 0.946901i \(-0.395805\pi\)
0.321524 + 0.946901i \(0.395805\pi\)
\(420\) 0 0
\(421\) −11.9413 −0.581984 −0.290992 0.956726i \(-0.593985\pi\)
−0.290992 + 0.956726i \(0.593985\pi\)
\(422\) 0 0
\(423\) −2.82843 −0.137523
\(424\) 0 0
\(425\) 2.11837 0.102756
\(426\) 0 0
\(427\) −18.2807 −0.884663
\(428\) 0 0
\(429\) −4.97567 −0.240227
\(430\) 0 0
\(431\) −30.6054 −1.47421 −0.737105 0.675778i \(-0.763808\pi\)
−0.737105 + 0.675778i \(0.763808\pi\)
\(432\) 0 0
\(433\) 15.3137 0.735930 0.367965 0.929840i \(-0.380055\pi\)
0.367965 + 0.929840i \(0.380055\pi\)
\(434\) 0 0
\(435\) 9.97474 0.478252
\(436\) 0 0
\(437\) 0.635767 0.0304128
\(438\) 0 0
\(439\) −33.3676 −1.59255 −0.796274 0.604936i \(-0.793199\pi\)
−0.796274 + 0.604936i \(0.793199\pi\)
\(440\) 0 0
\(441\) −2.33897 −0.111380
\(442\) 0 0
\(443\) 3.23617 0.153755 0.0768776 0.997041i \(-0.475505\pi\)
0.0768776 + 0.997041i \(0.475505\pi\)
\(444\) 0 0
\(445\) −5.42522 −0.257180
\(446\) 0 0
\(447\) 1.45479 0.0688091
\(448\) 0 0
\(449\) −27.4165 −1.29387 −0.646933 0.762547i \(-0.723948\pi\)
−0.646933 + 0.762547i \(0.723948\pi\)
\(450\) 0 0
\(451\) −14.9550 −0.704203
\(452\) 0 0
\(453\) 2.03696 0.0957049
\(454\) 0 0
\(455\) 16.0454 0.752221
\(456\) 0 0
\(457\) −10.9147 −0.510567 −0.255284 0.966866i \(-0.582169\pi\)
−0.255284 + 0.966866i \(0.582169\pi\)
\(458\) 0 0
\(459\) −0.224777 −0.0104917
\(460\) 0 0
\(461\) −25.2181 −1.17452 −0.587261 0.809398i \(-0.699794\pi\)
−0.587261 + 0.809398i \(0.699794\pi\)
\(462\) 0 0
\(463\) 22.4937 1.04537 0.522686 0.852525i \(-0.324930\pi\)
0.522686 + 0.852525i \(0.324930\pi\)
\(464\) 0 0
\(465\) 6.99222 0.324256
\(466\) 0 0
\(467\) 34.2482 1.58482 0.792408 0.609991i \(-0.208827\pi\)
0.792408 + 0.609991i \(0.208827\pi\)
\(468\) 0 0
\(469\) −31.8275 −1.46966
\(470\) 0 0
\(471\) −8.61790 −0.397092
\(472\) 0 0
\(473\) −27.8852 −1.28216
\(474\) 0 0
\(475\) 2.11837 0.0971974
\(476\) 0 0
\(477\) 10.6264 0.486548
\(478\) 0 0
\(479\) 36.2362 1.65568 0.827838 0.560968i \(-0.189571\pi\)
0.827838 + 0.560968i \(0.189571\pi\)
\(480\) 0 0
\(481\) −10.1551 −0.463032
\(482\) 0 0
\(483\) 6.10641 0.277851
\(484\) 0 0
\(485\) −62.2824 −2.82810
\(486\) 0 0
\(487\) −16.8200 −0.762186 −0.381093 0.924537i \(-0.624452\pi\)
−0.381093 + 0.924537i \(0.624452\pi\)
\(488\) 0 0
\(489\) 4.86054 0.219801
\(490\) 0 0
\(491\) −8.63577 −0.389727 −0.194863 0.980830i \(-0.562426\pi\)
−0.194863 + 0.980830i \(0.562426\pi\)
\(492\) 0 0
\(493\) −0.590346 −0.0265879
\(494\) 0 0
\(495\) −9.65685 −0.434043
\(496\) 0 0
\(497\) −9.32206 −0.418151
\(498\) 0 0
\(499\) −27.8275 −1.24573 −0.622865 0.782329i \(-0.714031\pi\)
−0.622865 + 0.782329i \(0.714031\pi\)
\(500\) 0 0
\(501\) −21.7023 −0.969586
\(502\) 0 0
\(503\) 25.7308 1.14728 0.573639 0.819108i \(-0.305531\pi\)
0.573639 + 0.819108i \(0.305531\pi\)
\(504\) 0 0
\(505\) −0.439861 −0.0195736
\(506\) 0 0
\(507\) 9.17064 0.407283
\(508\) 0 0
\(509\) 2.45386 0.108765 0.0543826 0.998520i \(-0.482681\pi\)
0.0543826 + 0.998520i \(0.482681\pi\)
\(510\) 0 0
\(511\) −12.8991 −0.570623
\(512\) 0 0
\(513\) −0.224777 −0.00992417
\(514\) 0 0
\(515\) −50.7050 −2.23433
\(516\) 0 0
\(517\) 7.19173 0.316292
\(518\) 0 0
\(519\) −12.3695 −0.542959
\(520\) 0 0
\(521\) −33.5944 −1.47180 −0.735898 0.677092i \(-0.763240\pi\)
−0.735898 + 0.677092i \(0.763240\pi\)
\(522\) 0 0
\(523\) 30.8522 1.34907 0.674536 0.738242i \(-0.264344\pi\)
0.674536 + 0.738242i \(0.264344\pi\)
\(524\) 0 0
\(525\) 20.3465 0.887994
\(526\) 0 0
\(527\) −0.413828 −0.0180266
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 5.65685 0.245487
\(532\) 0 0
\(533\) −11.5096 −0.498537
\(534\) 0 0
\(535\) 39.0907 1.69004
\(536\) 0 0
\(537\) −11.6413 −0.502359
\(538\) 0 0
\(539\) 5.94721 0.256164
\(540\) 0 0
\(541\) 38.4825 1.65449 0.827245 0.561841i \(-0.189907\pi\)
0.827245 + 0.561841i \(0.189907\pi\)
\(542\) 0 0
\(543\) −9.50732 −0.407998
\(544\) 0 0
\(545\) 37.8155 1.61984
\(546\) 0 0
\(547\) 9.61829 0.411248 0.205624 0.978631i \(-0.434078\pi\)
0.205624 + 0.978631i \(0.434078\pi\)
\(548\) 0 0
\(549\) 8.46742 0.361381
\(550\) 0 0
\(551\) −0.590346 −0.0251496
\(552\) 0 0
\(553\) −32.4004 −1.37780
\(554\) 0 0
\(555\) −19.7091 −0.836606
\(556\) 0 0
\(557\) 6.07174 0.257268 0.128634 0.991692i \(-0.458941\pi\)
0.128634 + 0.991692i \(0.458941\pi\)
\(558\) 0 0
\(559\) −21.4609 −0.907701
\(560\) 0 0
\(561\) 0.571533 0.0241301
\(562\) 0 0
\(563\) −14.2554 −0.600793 −0.300397 0.953814i \(-0.597119\pi\)
−0.300397 + 0.953814i \(0.597119\pi\)
\(564\) 0 0
\(565\) −71.5857 −3.01163
\(566\) 0 0
\(567\) −2.15894 −0.0906670
\(568\) 0 0
\(569\) −32.5018 −1.36255 −0.681274 0.732029i \(-0.738574\pi\)
−0.681274 + 0.732029i \(0.738574\pi\)
\(570\) 0 0
\(571\) −12.9706 −0.542801 −0.271401 0.962466i \(-0.587487\pi\)
−0.271401 + 0.962466i \(0.587487\pi\)
\(572\) 0 0
\(573\) −20.8032 −0.869065
\(574\) 0 0
\(575\) 26.6559 1.11163
\(576\) 0 0
\(577\) 11.7536 0.489308 0.244654 0.969611i \(-0.421326\pi\)
0.244654 + 0.969611i \(0.421326\pi\)
\(578\) 0 0
\(579\) −14.1454 −0.587862
\(580\) 0 0
\(581\) 30.8857 1.28135
\(582\) 0 0
\(583\) −27.0192 −1.11902
\(584\) 0 0
\(585\) −7.43208 −0.307279
\(586\) 0 0
\(587\) −9.13585 −0.377077 −0.188538 0.982066i \(-0.560375\pi\)
−0.188538 + 0.982066i \(0.560375\pi\)
\(588\) 0 0
\(589\) −0.413828 −0.0170515
\(590\) 0 0
\(591\) −3.43463 −0.141282
\(592\) 0 0
\(593\) −5.49270 −0.225558 −0.112779 0.993620i \(-0.535975\pi\)
−0.112779 + 0.993620i \(0.535975\pi\)
\(594\) 0 0
\(595\) −1.84307 −0.0755583
\(596\) 0 0
\(597\) 0.306182 0.0125312
\(598\) 0 0
\(599\) 36.4348 1.48868 0.744342 0.667799i \(-0.232763\pi\)
0.744342 + 0.667799i \(0.232763\pi\)
\(600\) 0 0
\(601\) −9.97474 −0.406878 −0.203439 0.979088i \(-0.565212\pi\)
−0.203439 + 0.979088i \(0.565212\pi\)
\(602\) 0 0
\(603\) 14.7422 0.600348
\(604\) 0 0
\(605\) −17.2232 −0.700221
\(606\) 0 0
\(607\) −4.51900 −0.183421 −0.0917103 0.995786i \(-0.529233\pi\)
−0.0917103 + 0.995786i \(0.529233\pi\)
\(608\) 0 0
\(609\) −5.67016 −0.229766
\(610\) 0 0
\(611\) 5.53488 0.223917
\(612\) 0 0
\(613\) 11.9316 0.481913 0.240957 0.970536i \(-0.422539\pi\)
0.240957 + 0.970536i \(0.422539\pi\)
\(614\) 0 0
\(615\) −22.3380 −0.900757
\(616\) 0 0
\(617\) −32.1201 −1.29311 −0.646554 0.762869i \(-0.723790\pi\)
−0.646554 + 0.762869i \(0.723790\pi\)
\(618\) 0 0
\(619\) −21.2715 −0.854975 −0.427488 0.904021i \(-0.640601\pi\)
−0.427488 + 0.904021i \(0.640601\pi\)
\(620\) 0 0
\(621\) −2.82843 −0.113501
\(622\) 0 0
\(623\) 3.08398 0.123557
\(624\) 0 0
\(625\) 16.6958 0.667833
\(626\) 0 0
\(627\) 0.571533 0.0228248
\(628\) 0 0
\(629\) 1.16647 0.0465101
\(630\) 0 0
\(631\) −36.4685 −1.45179 −0.725894 0.687807i \(-0.758574\pi\)
−0.725894 + 0.687807i \(0.758574\pi\)
\(632\) 0 0
\(633\) −10.2284 −0.406542
\(634\) 0 0
\(635\) 14.4921 0.575103
\(636\) 0 0
\(637\) 4.57707 0.181350
\(638\) 0 0
\(639\) 4.31788 0.170813
\(640\) 0 0
\(641\) 14.0036 0.553109 0.276555 0.960998i \(-0.410807\pi\)
0.276555 + 0.960998i \(0.410807\pi\)
\(642\) 0 0
\(643\) −23.4807 −0.925990 −0.462995 0.886361i \(-0.653225\pi\)
−0.462995 + 0.886361i \(0.653225\pi\)
\(644\) 0 0
\(645\) −41.6517 −1.64004
\(646\) 0 0
\(647\) 12.1908 0.479270 0.239635 0.970863i \(-0.422972\pi\)
0.239635 + 0.970863i \(0.422972\pi\)
\(648\) 0 0
\(649\) −14.3835 −0.564600
\(650\) 0 0
\(651\) −3.97474 −0.155782
\(652\) 0 0
\(653\) −1.39055 −0.0544165 −0.0272083 0.999630i \(-0.508662\pi\)
−0.0272083 + 0.999630i \(0.508662\pi\)
\(654\) 0 0
\(655\) 4.12198 0.161059
\(656\) 0 0
\(657\) 5.97474 0.233097
\(658\) 0 0
\(659\) −25.5349 −0.994698 −0.497349 0.867551i \(-0.665693\pi\)
−0.497349 + 0.867551i \(0.665693\pi\)
\(660\) 0 0
\(661\) −6.44726 −0.250769 −0.125385 0.992108i \(-0.540017\pi\)
−0.125385 + 0.992108i \(0.540017\pi\)
\(662\) 0 0
\(663\) 0.439861 0.0170828
\(664\) 0 0
\(665\) −1.84307 −0.0714710
\(666\) 0 0
\(667\) −7.42847 −0.287631
\(668\) 0 0
\(669\) −1.71908 −0.0664635
\(670\) 0 0
\(671\) −21.5298 −0.831148
\(672\) 0 0
\(673\) −10.8569 −0.418504 −0.209252 0.977862i \(-0.567103\pi\)
−0.209252 + 0.977862i \(0.567103\pi\)
\(674\) 0 0
\(675\) −9.42429 −0.362741
\(676\) 0 0
\(677\) −33.5262 −1.28852 −0.644259 0.764807i \(-0.722834\pi\)
−0.644259 + 0.764807i \(0.722834\pi\)
\(678\) 0 0
\(679\) 35.4045 1.35870
\(680\) 0 0
\(681\) 14.3059 0.548204
\(682\) 0 0
\(683\) 25.2206 0.965040 0.482520 0.875885i \(-0.339722\pi\)
0.482520 + 0.875885i \(0.339722\pi\)
\(684\) 0 0
\(685\) −20.1674 −0.770557
\(686\) 0 0
\(687\) −16.9981 −0.648519
\(688\) 0 0
\(689\) −20.7945 −0.792205
\(690\) 0 0
\(691\) 15.3523 0.584028 0.292014 0.956414i \(-0.405675\pi\)
0.292014 + 0.956414i \(0.405675\pi\)
\(692\) 0 0
\(693\) 5.48946 0.208527
\(694\) 0 0
\(695\) −47.0907 −1.78625
\(696\) 0 0
\(697\) 1.32206 0.0500765
\(698\) 0 0
\(699\) −13.3779 −0.506000
\(700\) 0 0
\(701\) −8.61079 −0.325225 −0.162613 0.986690i \(-0.551992\pi\)
−0.162613 + 0.986690i \(0.551992\pi\)
\(702\) 0 0
\(703\) 1.16647 0.0439942
\(704\) 0 0
\(705\) 10.7422 0.404574
\(706\) 0 0
\(707\) 0.250040 0.00940372
\(708\) 0 0
\(709\) −32.3624 −1.21539 −0.607697 0.794169i \(-0.707906\pi\)
−0.607697 + 0.794169i \(0.707906\pi\)
\(710\) 0 0
\(711\) 15.0075 0.562826
\(712\) 0 0
\(713\) −5.20730 −0.195015
\(714\) 0 0
\(715\) 18.8972 0.706717
\(716\) 0 0
\(717\) −13.3675 −0.499218
\(718\) 0 0
\(719\) −1.46744 −0.0547262 −0.0273631 0.999626i \(-0.508711\pi\)
−0.0273631 + 0.999626i \(0.508711\pi\)
\(720\) 0 0
\(721\) 28.8233 1.07344
\(722\) 0 0
\(723\) 0.211474 0.00786481
\(724\) 0 0
\(725\) −24.7516 −0.919251
\(726\) 0 0
\(727\) 15.3928 0.570889 0.285445 0.958395i \(-0.407859\pi\)
0.285445 + 0.958395i \(0.407859\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.46512 0.0911759
\(732\) 0 0
\(733\) −17.5624 −0.648683 −0.324342 0.945940i \(-0.605143\pi\)
−0.324342 + 0.945940i \(0.605143\pi\)
\(734\) 0 0
\(735\) 8.88325 0.327664
\(736\) 0 0
\(737\) −37.4844 −1.38075
\(738\) 0 0
\(739\) 20.7266 0.762441 0.381220 0.924484i \(-0.375504\pi\)
0.381220 + 0.924484i \(0.375504\pi\)
\(740\) 0 0
\(741\) 0.439861 0.0161587
\(742\) 0 0
\(743\) 31.7821 1.16597 0.582986 0.812482i \(-0.301884\pi\)
0.582986 + 0.812482i \(0.301884\pi\)
\(744\) 0 0
\(745\) −5.52518 −0.202427
\(746\) 0 0
\(747\) −14.3059 −0.523426
\(748\) 0 0
\(749\) −22.2212 −0.811944
\(750\) 0 0
\(751\) 29.7594 1.08594 0.542968 0.839753i \(-0.317301\pi\)
0.542968 + 0.839753i \(0.317301\pi\)
\(752\) 0 0
\(753\) −14.7555 −0.537720
\(754\) 0 0
\(755\) −7.73625 −0.281551
\(756\) 0 0
\(757\) 22.1119 0.803672 0.401836 0.915712i \(-0.368372\pi\)
0.401836 + 0.915712i \(0.368372\pi\)
\(758\) 0 0
\(759\) 7.19173 0.261043
\(760\) 0 0
\(761\) −4.55957 −0.165284 −0.0826422 0.996579i \(-0.526336\pi\)
−0.0826422 + 0.996579i \(0.526336\pi\)
\(762\) 0 0
\(763\) −21.4963 −0.778219
\(764\) 0 0
\(765\) 0.853690 0.0308652
\(766\) 0 0
\(767\) −11.0698 −0.399706
\(768\) 0 0
\(769\) 36.5794 1.31909 0.659543 0.751667i \(-0.270750\pi\)
0.659543 + 0.751667i \(0.270750\pi\)
\(770\) 0 0
\(771\) 0.742176 0.0267288
\(772\) 0 0
\(773\) 26.4611 0.951739 0.475869 0.879516i \(-0.342133\pi\)
0.475869 + 0.879516i \(0.342133\pi\)
\(774\) 0 0
\(775\) −17.3507 −0.623255
\(776\) 0 0
\(777\) 11.2037 0.401930
\(778\) 0 0
\(779\) 1.32206 0.0473676
\(780\) 0 0
\(781\) −10.9789 −0.392856
\(782\) 0 0
\(783\) 2.62636 0.0938584
\(784\) 0 0
\(785\) 32.7302 1.16819
\(786\) 0 0
\(787\) 18.9164 0.674298 0.337149 0.941451i \(-0.390537\pi\)
0.337149 + 0.941451i \(0.390537\pi\)
\(788\) 0 0
\(789\) 5.48435 0.195248
\(790\) 0 0
\(791\) 40.6930 1.44688
\(792\) 0 0
\(793\) −16.5697 −0.588406
\(794\) 0 0
\(795\) −40.3582 −1.43136
\(796\) 0 0
\(797\) −47.8065 −1.69339 −0.846697 0.532075i \(-0.821412\pi\)
−0.846697 + 0.532075i \(0.821412\pi\)
\(798\) 0 0
\(799\) −0.635767 −0.0224918
\(800\) 0 0
\(801\) −1.42847 −0.0504724
\(802\) 0 0
\(803\) −15.1917 −0.536105
\(804\) 0 0
\(805\) −23.1917 −0.817401
\(806\) 0 0
\(807\) 20.4694 0.720558
\(808\) 0 0
\(809\) 29.9862 1.05426 0.527129 0.849785i \(-0.323268\pi\)
0.527129 + 0.849785i \(0.323268\pi\)
\(810\) 0 0
\(811\) 11.3899 0.399954 0.199977 0.979801i \(-0.435913\pi\)
0.199977 + 0.979801i \(0.435913\pi\)
\(812\) 0 0
\(813\) 14.0370 0.492298
\(814\) 0 0
\(815\) −18.4600 −0.646626
\(816\) 0 0
\(817\) 2.46512 0.0862438
\(818\) 0 0
\(819\) 4.22478 0.147626
\(820\) 0 0
\(821\) 19.6929 0.687286 0.343643 0.939100i \(-0.388339\pi\)
0.343643 + 0.939100i \(0.388339\pi\)
\(822\) 0 0
\(823\) −22.4666 −0.783137 −0.391568 0.920149i \(-0.628067\pi\)
−0.391568 + 0.920149i \(0.628067\pi\)
\(824\) 0 0
\(825\) 23.9628 0.834277
\(826\) 0 0
\(827\) 10.7927 0.375299 0.187649 0.982236i \(-0.439913\pi\)
0.187649 + 0.982236i \(0.439913\pi\)
\(828\) 0 0
\(829\) −45.9421 −1.59563 −0.797817 0.602900i \(-0.794012\pi\)
−0.797817 + 0.602900i \(0.794012\pi\)
\(830\) 0 0
\(831\) −13.4211 −0.465572
\(832\) 0 0
\(833\) −0.525748 −0.0182161
\(834\) 0 0
\(835\) 82.4238 2.85239
\(836\) 0 0
\(837\) 1.84106 0.0636363
\(838\) 0 0
\(839\) −22.9142 −0.791085 −0.395542 0.918448i \(-0.629443\pi\)
−0.395542 + 0.918448i \(0.629443\pi\)
\(840\) 0 0
\(841\) −22.1022 −0.762146
\(842\) 0 0
\(843\) −3.89359 −0.134102
\(844\) 0 0
\(845\) −34.8295 −1.19817
\(846\) 0 0
\(847\) 9.79053 0.336407
\(848\) 0 0
\(849\) 17.6569 0.605982
\(850\) 0 0
\(851\) 14.6779 0.503153
\(852\) 0 0
\(853\) 55.0728 1.88566 0.942829 0.333278i \(-0.108155\pi\)
0.942829 + 0.333278i \(0.108155\pi\)
\(854\) 0 0
\(855\) 0.853690 0.0291956
\(856\) 0 0
\(857\) −1.79079 −0.0611723 −0.0305861 0.999532i \(-0.509737\pi\)
−0.0305861 + 0.999532i \(0.509737\pi\)
\(858\) 0 0
\(859\) 25.5468 0.871647 0.435823 0.900032i \(-0.356457\pi\)
0.435823 + 0.900032i \(0.356457\pi\)
\(860\) 0 0
\(861\) 12.6981 0.432750
\(862\) 0 0
\(863\) −42.1150 −1.43361 −0.716806 0.697273i \(-0.754397\pi\)
−0.716806 + 0.697273i \(0.754397\pi\)
\(864\) 0 0
\(865\) 46.9784 1.59731
\(866\) 0 0
\(867\) 16.9495 0.575634
\(868\) 0 0
\(869\) −38.1590 −1.29446
\(870\) 0 0
\(871\) −28.8486 −0.977497
\(872\) 0 0
\(873\) −16.3990 −0.555023
\(874\) 0 0
\(875\) −36.2771 −1.22639
\(876\) 0 0
\(877\) −1.01044 −0.0341203 −0.0170601 0.999854i \(-0.505431\pi\)
−0.0170601 + 0.999854i \(0.505431\pi\)
\(878\) 0 0
\(879\) −15.7759 −0.532108
\(880\) 0 0
\(881\) 44.3972 1.49578 0.747889 0.663823i \(-0.231067\pi\)
0.747889 + 0.663823i \(0.231067\pi\)
\(882\) 0 0
\(883\) 1.82389 0.0613787 0.0306894 0.999529i \(-0.490230\pi\)
0.0306894 + 0.999529i \(0.490230\pi\)
\(884\) 0 0
\(885\) −21.4844 −0.722189
\(886\) 0 0
\(887\) −4.38532 −0.147245 −0.0736223 0.997286i \(-0.523456\pi\)
−0.0736223 + 0.997286i \(0.523456\pi\)
\(888\) 0 0
\(889\) −8.23808 −0.276296
\(890\) 0 0
\(891\) −2.54266 −0.0851823
\(892\) 0 0
\(893\) −0.635767 −0.0212751
\(894\) 0 0
\(895\) 44.2128 1.47787
\(896\) 0 0
\(897\) 5.53488 0.184804
\(898\) 0 0
\(899\) 4.83528 0.161266
\(900\) 0 0
\(901\) 2.38857 0.0795747
\(902\) 0 0
\(903\) 23.6770 0.787922
\(904\) 0 0
\(905\) 36.1082 1.20028
\(906\) 0 0
\(907\) 4.06248 0.134893 0.0674463 0.997723i \(-0.478515\pi\)
0.0674463 + 0.997723i \(0.478515\pi\)
\(908\) 0 0
\(909\) −0.115816 −0.00384137
\(910\) 0 0
\(911\) 42.3784 1.40406 0.702029 0.712149i \(-0.252278\pi\)
0.702029 + 0.712149i \(0.252278\pi\)
\(912\) 0 0
\(913\) 36.3751 1.20384
\(914\) 0 0
\(915\) −32.1587 −1.06313
\(916\) 0 0
\(917\) −2.34315 −0.0773775
\(918\) 0 0
\(919\) −44.8603 −1.47980 −0.739902 0.672715i \(-0.765128\pi\)
−0.739902 + 0.672715i \(0.765128\pi\)
\(920\) 0 0
\(921\) 7.64129 0.251789
\(922\) 0 0
\(923\) −8.44955 −0.278120
\(924\) 0 0
\(925\) 48.9068 1.60804
\(926\) 0 0
\(927\) −13.3507 −0.438494
\(928\) 0 0
\(929\) −2.96695 −0.0973426 −0.0486713 0.998815i \(-0.515499\pi\)
−0.0486713 + 0.998815i \(0.515499\pi\)
\(930\) 0 0
\(931\) −0.525748 −0.0172307
\(932\) 0 0
\(933\) 24.1623 0.791038
\(934\) 0 0
\(935\) −2.17064 −0.0709876
\(936\) 0 0
\(937\) 54.7669 1.78916 0.894579 0.446910i \(-0.147476\pi\)
0.894579 + 0.446910i \(0.147476\pi\)
\(938\) 0 0
\(939\) 16.6105 0.542063
\(940\) 0 0
\(941\) 8.89558 0.289988 0.144994 0.989433i \(-0.453684\pi\)
0.144994 + 0.989433i \(0.453684\pi\)
\(942\) 0 0
\(943\) 16.6358 0.541735
\(944\) 0 0
\(945\) 8.19951 0.266730
\(946\) 0 0
\(947\) −15.8541 −0.515189 −0.257595 0.966253i \(-0.582930\pi\)
−0.257595 + 0.966253i \(0.582930\pi\)
\(948\) 0 0
\(949\) −11.6918 −0.379532
\(950\) 0 0
\(951\) −2.56213 −0.0830826
\(952\) 0 0
\(953\) −30.2807 −0.980887 −0.490443 0.871473i \(-0.663165\pi\)
−0.490443 + 0.871473i \(0.663165\pi\)
\(954\) 0 0
\(955\) 79.0090 2.55667
\(956\) 0 0
\(957\) −6.67794 −0.215867
\(958\) 0 0
\(959\) 11.4642 0.370198
\(960\) 0 0
\(961\) −27.6105 −0.890661
\(962\) 0 0
\(963\) 10.2926 0.331675
\(964\) 0 0
\(965\) 53.7232 1.72941
\(966\) 0 0
\(967\) 10.5273 0.338537 0.169268 0.985570i \(-0.445860\pi\)
0.169268 + 0.985570i \(0.445860\pi\)
\(968\) 0 0
\(969\) −0.0505249 −0.00162309
\(970\) 0 0
\(971\) −38.7050 −1.24210 −0.621051 0.783771i \(-0.713294\pi\)
−0.621051 + 0.783771i \(0.713294\pi\)
\(972\) 0 0
\(973\) 26.7688 0.858168
\(974\) 0 0
\(975\) 18.4422 0.590622
\(976\) 0 0
\(977\) 16.9009 0.540706 0.270353 0.962761i \(-0.412860\pi\)
0.270353 + 0.962761i \(0.412860\pi\)
\(978\) 0 0
\(979\) 3.63211 0.116083
\(980\) 0 0
\(981\) 9.95687 0.317899
\(982\) 0 0
\(983\) 10.0798 0.321496 0.160748 0.986995i \(-0.448609\pi\)
0.160748 + 0.986995i \(0.448609\pi\)
\(984\) 0 0
\(985\) 13.0445 0.415632
\(986\) 0 0
\(987\) −6.10641 −0.194369
\(988\) 0 0
\(989\) 31.0192 0.986354
\(990\) 0 0
\(991\) 42.3446 1.34512 0.672561 0.740042i \(-0.265194\pi\)
0.672561 + 0.740042i \(0.265194\pi\)
\(992\) 0 0
\(993\) 19.1275 0.606993
\(994\) 0 0
\(995\) −1.16286 −0.0368651
\(996\) 0 0
\(997\) 47.6132 1.50793 0.753963 0.656917i \(-0.228140\pi\)
0.753963 + 0.656917i \(0.228140\pi\)
\(998\) 0 0
\(999\) −5.18944 −0.164186
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 3072.2.a.n.1.4 4
3.2 odd 2 9216.2.a.x.1.1 4
4.3 odd 2 3072.2.a.t.1.4 4
8.3 odd 2 3072.2.a.i.1.1 4
8.5 even 2 3072.2.a.o.1.1 4
12.11 even 2 9216.2.a.y.1.1 4
16.3 odd 4 3072.2.d.f.1537.5 8
16.5 even 4 3072.2.d.i.1537.8 8
16.11 odd 4 3072.2.d.f.1537.4 8
16.13 even 4 3072.2.d.i.1537.1 8
24.5 odd 2 9216.2.a.bn.1.4 4
24.11 even 2 9216.2.a.bo.1.4 4
32.3 odd 8 48.2.j.a.13.3 8
32.5 even 8 384.2.j.a.97.4 8
32.11 odd 8 48.2.j.a.37.3 yes 8
32.13 even 8 384.2.j.a.289.4 8
32.19 odd 8 384.2.j.b.289.2 8
32.21 even 8 192.2.j.a.49.1 8
32.27 odd 8 384.2.j.b.97.2 8
32.29 even 8 192.2.j.a.145.1 8
96.5 odd 8 1152.2.k.f.865.1 8
96.11 even 8 144.2.k.b.37.2 8
96.29 odd 8 576.2.k.b.145.4 8
96.35 even 8 144.2.k.b.109.2 8
96.53 odd 8 576.2.k.b.433.4 8
96.59 even 8 1152.2.k.c.865.1 8
96.77 odd 8 1152.2.k.f.289.1 8
96.83 even 8 1152.2.k.c.289.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.3 8 32.3 odd 8
48.2.j.a.37.3 yes 8 32.11 odd 8
144.2.k.b.37.2 8 96.11 even 8
144.2.k.b.109.2 8 96.35 even 8
192.2.j.a.49.1 8 32.21 even 8
192.2.j.a.145.1 8 32.29 even 8
384.2.j.a.97.4 8 32.5 even 8
384.2.j.a.289.4 8 32.13 even 8
384.2.j.b.97.2 8 32.27 odd 8
384.2.j.b.289.2 8 32.19 odd 8
576.2.k.b.145.4 8 96.29 odd 8
576.2.k.b.433.4 8 96.53 odd 8
1152.2.k.c.289.1 8 96.83 even 8
1152.2.k.c.865.1 8 96.59 even 8
1152.2.k.f.289.1 8 96.77 odd 8
1152.2.k.f.865.1 8 96.5 odd 8
3072.2.a.i.1.1 4 8.3 odd 2
3072.2.a.n.1.4 4 1.1 even 1 trivial
3072.2.a.o.1.1 4 8.5 even 2
3072.2.a.t.1.4 4 4.3 odd 2
3072.2.d.f.1537.4 8 16.11 odd 4
3072.2.d.f.1537.5 8 16.3 odd 4
3072.2.d.i.1537.1 8 16.13 even 4
3072.2.d.i.1537.8 8 16.5 even 4
9216.2.a.x.1.1 4 3.2 odd 2
9216.2.a.y.1.1 4 12.11 even 2
9216.2.a.bn.1.4 4 24.5 odd 2
9216.2.a.bo.1.4 4 24.11 even 2