Properties

Label 8880.2.a.bh.1.1
Level $8880$
Weight $2$
Character 8880.1
Self dual yes
Analytic conductor $70.907$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8880.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.9071569949\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{113}) \)
Defining polynomial: \(x^{2} - x - 28\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.81507\) of defining polynomial
Character \(\chi\) \(=\) 8880.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.81507 q^{13} -1.00000 q^{15} +1.00000 q^{17} -7.81507 q^{19} +3.00000 q^{21} -5.81507 q^{23} +1.00000 q^{25} -1.00000 q^{27} -6.81507 q^{29} -6.81507 q^{31} -1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} +3.81507 q^{39} -4.81507 q^{41} +4.81507 q^{43} +1.00000 q^{45} +8.00000 q^{47} +2.00000 q^{49} -1.00000 q^{51} +10.6301 q^{53} +1.00000 q^{55} +7.81507 q^{57} -2.00000 q^{59} +6.81507 q^{61} -3.00000 q^{63} -3.81507 q^{65} -7.63015 q^{67} +5.81507 q^{69} +11.6301 q^{71} -5.81507 q^{73} -1.00000 q^{75} -3.00000 q^{77} -12.0000 q^{79} +1.00000 q^{81} +3.81507 q^{83} +1.00000 q^{85} +6.81507 q^{87} +11.8151 q^{89} +11.4452 q^{91} +6.81507 q^{93} -7.81507 q^{95} -4.81507 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} - 6q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} - 6q^{7} + 2q^{9} + 2q^{11} + 3q^{13} - 2q^{15} + 2q^{17} - 5q^{19} + 6q^{21} - q^{23} + 2q^{25} - 2q^{27} - 3q^{29} - 3q^{31} - 2q^{33} - 6q^{35} + 2q^{37} - 3q^{39} + q^{41} - q^{43} + 2q^{45} + 16q^{47} + 4q^{49} - 2q^{51} + 2q^{55} + 5q^{57} - 4q^{59} + 3q^{61} - 6q^{63} + 3q^{65} + 6q^{67} + q^{69} + 2q^{71} - q^{73} - 2q^{75} - 6q^{77} - 24q^{79} + 2q^{81} - 3q^{83} + 2q^{85} + 3q^{87} + 13q^{89} - 9q^{91} + 3q^{93} - 5q^{95} + q^{97} + 2q^{99} + O(q^{100}) \)

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −3.81507 −1.05811 −0.529055 0.848587i \(-0.677454\pi\)
−0.529055 + 0.848587i \(0.677454\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) −7.81507 −1.79290 −0.896450 0.443144i \(-0.853863\pi\)
−0.896450 + 0.443144i \(0.853863\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) −5.81507 −1.21253 −0.606263 0.795264i \(-0.707332\pi\)
−0.606263 + 0.795264i \(0.707332\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.81507 −1.26553 −0.632764 0.774345i \(-0.718080\pi\)
−0.632764 + 0.774345i \(0.718080\pi\)
\(30\) 0 0
\(31\) −6.81507 −1.22402 −0.612012 0.790849i \(-0.709639\pi\)
−0.612012 + 0.790849i \(0.709639\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 3.81507 0.610901
\(40\) 0 0
\(41\) −4.81507 −0.751988 −0.375994 0.926622i \(-0.622699\pi\)
−0.375994 + 0.926622i \(0.622699\pi\)
\(42\) 0 0
\(43\) 4.81507 0.734292 0.367146 0.930163i \(-0.380335\pi\)
0.367146 + 0.930163i \(0.380335\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) 10.6301 1.46016 0.730081 0.683360i \(-0.239482\pi\)
0.730081 + 0.683360i \(0.239482\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 7.81507 1.03513
\(58\) 0 0
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) 6.81507 0.872581 0.436290 0.899806i \(-0.356292\pi\)
0.436290 + 0.899806i \(0.356292\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) −3.81507 −0.473202
\(66\) 0 0
\(67\) −7.63015 −0.932171 −0.466085 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(68\) 0 0
\(69\) 5.81507 0.700053
\(70\) 0 0
\(71\) 11.6301 1.38024 0.690122 0.723693i \(-0.257557\pi\)
0.690122 + 0.723693i \(0.257557\pi\)
\(72\) 0 0
\(73\) −5.81507 −0.680603 −0.340301 0.940316i \(-0.610529\pi\)
−0.340301 + 0.940316i \(0.610529\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.81507 0.418759 0.209379 0.977834i \(-0.432856\pi\)
0.209379 + 0.977834i \(0.432856\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 0 0
\(87\) 6.81507 0.730653
\(88\) 0 0
\(89\) 11.8151 1.25240 0.626198 0.779664i \(-0.284610\pi\)
0.626198 + 0.779664i \(0.284610\pi\)
\(90\) 0 0
\(91\) 11.4452 1.19978
\(92\) 0 0
\(93\) 6.81507 0.706690
\(94\) 0 0
\(95\) −7.81507 −0.801810
\(96\) 0 0
\(97\) −4.81507 −0.488897 −0.244448 0.969662i \(-0.578607\pi\)
−0.244448 + 0.969662i \(0.578607\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 11.6301 1.15724 0.578621 0.815596i \(-0.303591\pi\)
0.578621 + 0.815596i \(0.303591\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) 8.18493 0.791267 0.395633 0.918409i \(-0.370525\pi\)
0.395633 + 0.918409i \(0.370525\pi\)
\(108\) 0 0
\(109\) 13.0000 1.24517 0.622587 0.782551i \(-0.286082\pi\)
0.622587 + 0.782551i \(0.286082\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 0 0
\(113\) −6.81507 −0.641108 −0.320554 0.947230i \(-0.603869\pi\)
−0.320554 + 0.947230i \(0.603869\pi\)
\(114\) 0 0
\(115\) −5.81507 −0.542258
\(116\) 0 0
\(117\) −3.81507 −0.352704
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 4.81507 0.434161
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 21.4452 1.90296 0.951478 0.307718i \(-0.0995652\pi\)
0.951478 + 0.307718i \(0.0995652\pi\)
\(128\) 0 0
\(129\) −4.81507 −0.423944
\(130\) 0 0
\(131\) 9.63015 0.841390 0.420695 0.907202i \(-0.361786\pi\)
0.420695 + 0.907202i \(0.361786\pi\)
\(132\) 0 0
\(133\) 23.4452 2.03296
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) −22.8151 −1.93515 −0.967575 0.252585i \(-0.918719\pi\)
−0.967575 + 0.252585i \(0.918719\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) −3.81507 −0.319032
\(144\) 0 0
\(145\) −6.81507 −0.565961
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) −3.81507 −0.310466 −0.155233 0.987878i \(-0.549613\pi\)
−0.155233 + 0.987878i \(0.549613\pi\)
\(152\) 0 0
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) −6.81507 −0.547400
\(156\) 0 0
\(157\) −12.8151 −1.02275 −0.511377 0.859356i \(-0.670864\pi\)
−0.511377 + 0.859356i \(0.670864\pi\)
\(158\) 0 0
\(159\) −10.6301 −0.843025
\(160\) 0 0
\(161\) 17.4452 1.37488
\(162\) 0 0
\(163\) −16.6301 −1.30257 −0.651287 0.758832i \(-0.725770\pi\)
−0.651287 + 0.758832i \(0.725770\pi\)
\(164\) 0 0
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) 21.8151 1.68810 0.844051 0.536264i \(-0.180165\pi\)
0.844051 + 0.536264i \(0.180165\pi\)
\(168\) 0 0
\(169\) 1.55478 0.119599
\(170\) 0 0
\(171\) −7.81507 −0.597634
\(172\) 0 0
\(173\) −1.00000 −0.0760286 −0.0380143 0.999277i \(-0.512103\pi\)
−0.0380143 + 0.999277i \(0.512103\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 0 0
\(177\) 2.00000 0.150329
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −6.81507 −0.503785
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 1.00000 0.0731272
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) −4.63015 −0.335026 −0.167513 0.985870i \(-0.553574\pi\)
−0.167513 + 0.985870i \(0.553574\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 3.81507 0.273203
\(196\) 0 0
\(197\) −4.18493 −0.298164 −0.149082 0.988825i \(-0.547632\pi\)
−0.149082 + 0.988825i \(0.547632\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 7.63015 0.538189
\(202\) 0 0
\(203\) 20.4452 1.43497
\(204\) 0 0
\(205\) −4.81507 −0.336299
\(206\) 0 0
\(207\) −5.81507 −0.404176
\(208\) 0 0
\(209\) −7.81507 −0.540580
\(210\) 0 0
\(211\) −8.44522 −0.581393 −0.290696 0.956815i \(-0.593887\pi\)
−0.290696 + 0.956815i \(0.593887\pi\)
\(212\) 0 0
\(213\) −11.6301 −0.796884
\(214\) 0 0
\(215\) 4.81507 0.328385
\(216\) 0 0
\(217\) 20.4452 1.38791
\(218\) 0 0
\(219\) 5.81507 0.392946
\(220\) 0 0
\(221\) −3.81507 −0.256630
\(222\) 0 0
\(223\) 16.8151 1.12602 0.563010 0.826450i \(-0.309643\pi\)
0.563010 + 0.826450i \(0.309643\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 24.8151 1.64703 0.823517 0.567291i \(-0.192009\pi\)
0.823517 + 0.567291i \(0.192009\pi\)
\(228\) 0 0
\(229\) 0.369854 0.0244407 0.0122203 0.999925i \(-0.496110\pi\)
0.0122203 + 0.999925i \(0.496110\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) −7.63015 −0.499867 −0.249934 0.968263i \(-0.580409\pi\)
−0.249934 + 0.968263i \(0.580409\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 12.0000 0.779484
\(238\) 0 0
\(239\) −6.44522 −0.416907 −0.208453 0.978032i \(-0.566843\pi\)
−0.208453 + 0.978032i \(0.566843\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 29.8151 1.89709
\(248\) 0 0
\(249\) −3.81507 −0.241770
\(250\) 0 0
\(251\) 11.6301 0.734088 0.367044 0.930204i \(-0.380370\pi\)
0.367044 + 0.930204i \(0.380370\pi\)
\(252\) 0 0
\(253\) −5.81507 −0.365591
\(254\) 0 0
\(255\) −1.00000 −0.0626224
\(256\) 0 0
\(257\) −3.81507 −0.237978 −0.118989 0.992896i \(-0.537965\pi\)
−0.118989 + 0.992896i \(0.537965\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) −6.81507 −0.421842
\(262\) 0 0
\(263\) 30.4452 1.87733 0.938666 0.344827i \(-0.112062\pi\)
0.938666 + 0.344827i \(0.112062\pi\)
\(264\) 0 0
\(265\) 10.6301 0.653005
\(266\) 0 0
\(267\) −11.8151 −0.723071
\(268\) 0 0
\(269\) 9.81507 0.598436 0.299218 0.954185i \(-0.403274\pi\)
0.299218 + 0.954185i \(0.403274\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) −11.4452 −0.692696
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 21.4452 1.28852 0.644259 0.764807i \(-0.277166\pi\)
0.644259 + 0.764807i \(0.277166\pi\)
\(278\) 0 0
\(279\) −6.81507 −0.408008
\(280\) 0 0
\(281\) −19.4452 −1.16000 −0.580002 0.814615i \(-0.696948\pi\)
−0.580002 + 0.814615i \(0.696948\pi\)
\(282\) 0 0
\(283\) 21.8151 1.29677 0.648386 0.761312i \(-0.275444\pi\)
0.648386 + 0.761312i \(0.275444\pi\)
\(284\) 0 0
\(285\) 7.81507 0.462925
\(286\) 0 0
\(287\) 14.4452 0.852674
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 4.81507 0.282265
\(292\) 0 0
\(293\) 28.2603 1.65098 0.825492 0.564414i \(-0.190898\pi\)
0.825492 + 0.564414i \(0.190898\pi\)
\(294\) 0 0
\(295\) −2.00000 −0.116445
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 22.1849 1.28299
\(300\) 0 0
\(301\) −14.4452 −0.832609
\(302\) 0 0
\(303\) −11.6301 −0.668134
\(304\) 0 0
\(305\) 6.81507 0.390230
\(306\) 0 0
\(307\) 18.0000 1.02731 0.513657 0.857996i \(-0.328290\pi\)
0.513657 + 0.857996i \(0.328290\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.4452 −0.592294 −0.296147 0.955142i \(-0.595702\pi\)
−0.296147 + 0.955142i \(0.595702\pi\)
\(312\) 0 0
\(313\) 32.8904 1.85908 0.929539 0.368725i \(-0.120205\pi\)
0.929539 + 0.368725i \(0.120205\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) −2.81507 −0.158110 −0.0790551 0.996870i \(-0.525190\pi\)
−0.0790551 + 0.996870i \(0.525190\pi\)
\(318\) 0 0
\(319\) −6.81507 −0.381571
\(320\) 0 0
\(321\) −8.18493 −0.456838
\(322\) 0 0
\(323\) −7.81507 −0.434842
\(324\) 0 0
\(325\) −3.81507 −0.211622
\(326\) 0 0
\(327\) −13.0000 −0.718902
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 23.2603 1.27850 0.639251 0.768998i \(-0.279245\pi\)
0.639251 + 0.768998i \(0.279245\pi\)
\(332\) 0 0
\(333\) 1.00000 0.0547997
\(334\) 0 0
\(335\) −7.63015 −0.416879
\(336\) 0 0
\(337\) −4.18493 −0.227968 −0.113984 0.993483i \(-0.536361\pi\)
−0.113984 + 0.993483i \(0.536361\pi\)
\(338\) 0 0
\(339\) 6.81507 0.370144
\(340\) 0 0
\(341\) −6.81507 −0.369057
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) 5.81507 0.313073
\(346\) 0 0
\(347\) −11.2603 −0.604484 −0.302242 0.953231i \(-0.597735\pi\)
−0.302242 + 0.953231i \(0.597735\pi\)
\(348\) 0 0
\(349\) 7.26029 0.388635 0.194317 0.980939i \(-0.437751\pi\)
0.194317 + 0.980939i \(0.437751\pi\)
\(350\) 0 0
\(351\) 3.81507 0.203634
\(352\) 0 0
\(353\) −13.1849 −0.701763 −0.350881 0.936420i \(-0.614118\pi\)
−0.350881 + 0.936420i \(0.614118\pi\)
\(354\) 0 0
\(355\) 11.6301 0.617264
\(356\) 0 0
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) −6.36985 −0.336188 −0.168094 0.985771i \(-0.553761\pi\)
−0.168094 + 0.985771i \(0.553761\pi\)
\(360\) 0 0
\(361\) 42.0754 2.21449
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) −5.81507 −0.304375
\(366\) 0 0
\(367\) 21.0000 1.09619 0.548096 0.836416i \(-0.315353\pi\)
0.548096 + 0.836416i \(0.315353\pi\)
\(368\) 0 0
\(369\) −4.81507 −0.250663
\(370\) 0 0
\(371\) −31.8904 −1.65567
\(372\) 0 0
\(373\) −17.2603 −0.893704 −0.446852 0.894608i \(-0.647455\pi\)
−0.446852 + 0.894608i \(0.647455\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 26.0000 1.33907
\(378\) 0 0
\(379\) 19.6301 1.00833 0.504166 0.863607i \(-0.331800\pi\)
0.504166 + 0.863607i \(0.331800\pi\)
\(380\) 0 0
\(381\) −21.4452 −1.09867
\(382\) 0 0
\(383\) −33.4452 −1.70897 −0.854485 0.519475i \(-0.826127\pi\)
−0.854485 + 0.519475i \(0.826127\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) 0 0
\(387\) 4.81507 0.244764
\(388\) 0 0
\(389\) −35.7055 −1.81034 −0.905171 0.425048i \(-0.860257\pi\)
−0.905171 + 0.425048i \(0.860257\pi\)
\(390\) 0 0
\(391\) −5.81507 −0.294081
\(392\) 0 0
\(393\) −9.63015 −0.485777
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −5.63015 −0.282569 −0.141284 0.989969i \(-0.545123\pi\)
−0.141284 + 0.989969i \(0.545123\pi\)
\(398\) 0 0
\(399\) −23.4452 −1.17373
\(400\) 0 0
\(401\) 0.554781 0.0277045 0.0138522 0.999904i \(-0.495591\pi\)
0.0138522 + 0.999904i \(0.495591\pi\)
\(402\) 0 0
\(403\) 26.0000 1.29515
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 1.00000 0.0495682
\(408\) 0 0
\(409\) 6.36985 0.314969 0.157485 0.987521i \(-0.449662\pi\)
0.157485 + 0.987521i \(0.449662\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 3.81507 0.187275
\(416\) 0 0
\(417\) 22.8151 1.11726
\(418\) 0 0
\(419\) 25.4452 1.24308 0.621540 0.783382i \(-0.286507\pi\)
0.621540 + 0.783382i \(0.286507\pi\)
\(420\) 0 0
\(421\) −29.2603 −1.42606 −0.713030 0.701134i \(-0.752678\pi\)
−0.713030 + 0.701134i \(0.752678\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) −20.4452 −0.989413
\(428\) 0 0
\(429\) 3.81507 0.184193
\(430\) 0 0
\(431\) −21.0000 −1.01153 −0.505767 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 0 0
\(433\) 1.81507 0.0872268 0.0436134 0.999048i \(-0.486113\pi\)
0.0436134 + 0.999048i \(0.486113\pi\)
\(434\) 0 0
\(435\) 6.81507 0.326758
\(436\) 0 0
\(437\) 45.4452 2.17394
\(438\) 0 0
\(439\) 18.4452 0.880342 0.440171 0.897914i \(-0.354918\pi\)
0.440171 + 0.897914i \(0.354918\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −13.6301 −0.647588 −0.323794 0.946128i \(-0.604958\pi\)
−0.323794 + 0.946128i \(0.604958\pi\)
\(444\) 0 0
\(445\) 11.8151 0.560088
\(446\) 0 0
\(447\) −2.00000 −0.0945968
\(448\) 0 0
\(449\) 30.8904 1.45781 0.728905 0.684615i \(-0.240030\pi\)
0.728905 + 0.684615i \(0.240030\pi\)
\(450\) 0 0
\(451\) −4.81507 −0.226733
\(452\) 0 0
\(453\) 3.81507 0.179248
\(454\) 0 0
\(455\) 11.4452 0.536560
\(456\) 0 0
\(457\) −28.0754 −1.31331 −0.656655 0.754191i \(-0.728029\pi\)
−0.656655 + 0.754191i \(0.728029\pi\)
\(458\) 0 0
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 34.8151 1.62150 0.810750 0.585393i \(-0.199060\pi\)
0.810750 + 0.585393i \(0.199060\pi\)
\(462\) 0 0
\(463\) 17.6301 0.819342 0.409671 0.912233i \(-0.365643\pi\)
0.409671 + 0.912233i \(0.365643\pi\)
\(464\) 0 0
\(465\) 6.81507 0.316041
\(466\) 0 0
\(467\) −28.8151 −1.33340 −0.666701 0.745325i \(-0.732294\pi\)
−0.666701 + 0.745325i \(0.732294\pi\)
\(468\) 0 0
\(469\) 22.8904 1.05698
\(470\) 0 0
\(471\) 12.8151 0.590487
\(472\) 0 0
\(473\) 4.81507 0.221397
\(474\) 0 0
\(475\) −7.81507 −0.358580
\(476\) 0 0
\(477\) 10.6301 0.486721
\(478\) 0 0
\(479\) −9.81507 −0.448462 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(480\) 0 0
\(481\) −3.81507 −0.173952
\(482\) 0 0
\(483\) −17.4452 −0.793785
\(484\) 0 0
\(485\) −4.81507 −0.218641
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 16.6301 0.752041
\(490\) 0 0
\(491\) −2.18493 −0.0986044 −0.0493022 0.998784i \(-0.515700\pi\)
−0.0493022 + 0.998784i \(0.515700\pi\)
\(492\) 0 0
\(493\) −6.81507 −0.306935
\(494\) 0 0
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) −34.8904 −1.56505
\(498\) 0 0
\(499\) 17.4452 0.780955 0.390478 0.920612i \(-0.372310\pi\)
0.390478 + 0.920612i \(0.372310\pi\)
\(500\) 0 0
\(501\) −21.8151 −0.974626
\(502\) 0 0
\(503\) −0.739708 −0.0329820 −0.0164910 0.999864i \(-0.505249\pi\)
−0.0164910 + 0.999864i \(0.505249\pi\)
\(504\) 0 0
\(505\) 11.6301 0.517535
\(506\) 0 0
\(507\) −1.55478 −0.0690503
\(508\) 0 0
\(509\) 17.4452 0.773246 0.386623 0.922238i \(-0.373642\pi\)
0.386623 + 0.922238i \(0.373642\pi\)
\(510\) 0 0
\(511\) 17.4452 0.771731
\(512\) 0 0
\(513\) 7.81507 0.345044
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) 1.00000 0.0438951
\(520\) 0 0
\(521\) 0.815073 0.0357090 0.0178545 0.999841i \(-0.494316\pi\)
0.0178545 + 0.999841i \(0.494316\pi\)
\(522\) 0 0
\(523\) 0.739708 0.0323452 0.0161726 0.999869i \(-0.494852\pi\)
0.0161726 + 0.999869i \(0.494852\pi\)
\(524\) 0 0
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) −6.81507 −0.296869
\(528\) 0 0
\(529\) 10.8151 0.470221
\(530\) 0 0
\(531\) −2.00000 −0.0867926
\(532\) 0 0
\(533\) 18.3699 0.795687
\(534\) 0 0
\(535\) 8.18493 0.353865
\(536\) 0 0
\(537\) 16.0000 0.690451
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 27.8151 1.19586 0.597932 0.801547i \(-0.295989\pi\)
0.597932 + 0.801547i \(0.295989\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) 13.0000 0.556859
\(546\) 0 0
\(547\) −23.8904 −1.02148 −0.510741 0.859735i \(-0.670629\pi\)
−0.510741 + 0.859735i \(0.670629\pi\)
\(548\) 0 0
\(549\) 6.81507 0.290860
\(550\) 0 0
\(551\) 53.2603 2.26896
\(552\) 0 0
\(553\) 36.0000 1.53088
\(554\) 0 0
\(555\) −1.00000 −0.0424476
\(556\) 0 0
\(557\) −21.6301 −0.916499 −0.458249 0.888824i \(-0.651523\pi\)
−0.458249 + 0.888824i \(0.651523\pi\)
\(558\) 0 0
\(559\) −18.3699 −0.776962
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) 0 0
\(563\) 8.44522 0.355924 0.177962 0.984037i \(-0.443050\pi\)
0.177962 + 0.984037i \(0.443050\pi\)
\(564\) 0 0
\(565\) −6.81507 −0.286712
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) −35.8151 −1.50145 −0.750723 0.660617i \(-0.770295\pi\)
−0.750723 + 0.660617i \(0.770295\pi\)
\(570\) 0 0
\(571\) 14.8151 0.619992 0.309996 0.950738i \(-0.399672\pi\)
0.309996 + 0.950738i \(0.399672\pi\)
\(572\) 0 0
\(573\) 4.63015 0.193427
\(574\) 0 0
\(575\) −5.81507 −0.242505
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 0 0
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) −11.4452 −0.474828
\(582\) 0 0
\(583\) 10.6301 0.440256
\(584\) 0 0
\(585\) −3.81507 −0.157734
\(586\) 0 0
\(587\) −20.8151 −0.859130 −0.429565 0.903036i \(-0.641333\pi\)
−0.429565 + 0.903036i \(0.641333\pi\)
\(588\) 0 0
\(589\) 53.2603 2.19455
\(590\) 0 0
\(591\) 4.18493 0.172145
\(592\) 0 0
\(593\) −13.6301 −0.559723 −0.279862 0.960040i \(-0.590289\pi\)
−0.279862 + 0.960040i \(0.590289\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −19.3699 −0.790113 −0.395056 0.918657i \(-0.629275\pi\)
−0.395056 + 0.918657i \(0.629275\pi\)
\(602\) 0 0
\(603\) −7.63015 −0.310724
\(604\) 0 0
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) 37.2603 1.51235 0.756174 0.654370i \(-0.227066\pi\)
0.756174 + 0.654370i \(0.227066\pi\)
\(608\) 0 0
\(609\) −20.4452 −0.828482
\(610\) 0 0
\(611\) −30.5206 −1.23473
\(612\) 0 0
\(613\) −6.07536 −0.245382 −0.122691 0.992445i \(-0.539152\pi\)
−0.122691 + 0.992445i \(0.539152\pi\)
\(614\) 0 0
\(615\) 4.81507 0.194162
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 13.5548 0.544813 0.272406 0.962182i \(-0.412181\pi\)
0.272406 + 0.962182i \(0.412181\pi\)
\(620\) 0 0
\(621\) 5.81507 0.233351
\(622\) 0 0
\(623\) −35.4452 −1.42008
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.81507 0.312104
\(628\) 0 0
\(629\) 1.00000 0.0398726
\(630\) 0 0
\(631\) −33.7055 −1.34180 −0.670898 0.741550i \(-0.734091\pi\)
−0.670898 + 0.741550i \(0.734091\pi\)
\(632\) 0 0
\(633\) 8.44522 0.335667
\(634\) 0 0
\(635\) 21.4452 0.851028
\(636\) 0 0
\(637\) −7.63015 −0.302317
\(638\) 0 0
\(639\) 11.6301 0.460081
\(640\) 0 0
\(641\) −13.5548 −0.535382 −0.267691 0.963505i \(-0.586261\pi\)
−0.267691 + 0.963505i \(0.586261\pi\)
\(642\) 0 0
\(643\) −17.0000 −0.670415 −0.335207 0.942144i \(-0.608806\pi\)
−0.335207 + 0.942144i \(0.608806\pi\)
\(644\) 0 0
\(645\) −4.81507 −0.189593
\(646\) 0 0
\(647\) −25.0754 −0.985814 −0.492907 0.870082i \(-0.664066\pi\)
−0.492907 + 0.870082i \(0.664066\pi\)
\(648\) 0 0
\(649\) −2.00000 −0.0785069
\(650\) 0 0
\(651\) −20.4452 −0.801311
\(652\) 0 0
\(653\) 4.00000 0.156532 0.0782660 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) 0 0
\(655\) 9.63015 0.376281
\(656\) 0 0
\(657\) −5.81507 −0.226868
\(658\) 0 0
\(659\) 31.2603 1.21773 0.608864 0.793275i \(-0.291625\pi\)
0.608864 + 0.793275i \(0.291625\pi\)
\(660\) 0 0
\(661\) 3.73971 0.145458 0.0727289 0.997352i \(-0.476829\pi\)
0.0727289 + 0.997352i \(0.476829\pi\)
\(662\) 0 0
\(663\) 3.81507 0.148165
\(664\) 0 0
\(665\) 23.4452 0.909167
\(666\) 0 0
\(667\) 39.6301 1.53449
\(668\) 0 0
\(669\) −16.8151 −0.650108
\(670\) 0 0
\(671\) 6.81507 0.263093
\(672\) 0 0
\(673\) 6.18493 0.238411 0.119206 0.992870i \(-0.461965\pi\)
0.119206 + 0.992870i \(0.461965\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 47.0754 1.80925 0.904627 0.426205i \(-0.140150\pi\)
0.904627 + 0.426205i \(0.140150\pi\)
\(678\) 0 0
\(679\) 14.4452 0.554357
\(680\) 0 0
\(681\) −24.8151 −0.950916
\(682\) 0 0
\(683\) 47.3357 1.81125 0.905624 0.424081i \(-0.139403\pi\)
0.905624 + 0.424081i \(0.139403\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) −0.369854 −0.0141108
\(688\) 0 0
\(689\) −40.5548 −1.54501
\(690\) 0 0
\(691\) 34.0754 1.29629 0.648144 0.761518i \(-0.275545\pi\)
0.648144 + 0.761518i \(0.275545\pi\)
\(692\) 0 0
\(693\) −3.00000 −0.113961
\(694\) 0 0
\(695\) −22.8151 −0.865425
\(696\) 0 0
\(697\) −4.81507 −0.182384
\(698\) 0 0
\(699\) 7.63015 0.288599
\(700\) 0 0
\(701\) 33.6301 1.27019 0.635097 0.772433i \(-0.280960\pi\)
0.635097 + 0.772433i \(0.280960\pi\)
\(702\) 0 0
\(703\) −7.81507 −0.294751
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) −34.8904 −1.31219
\(708\) 0 0
\(709\) 20.6301 0.774781 0.387391 0.921916i \(-0.373376\pi\)
0.387391 + 0.921916i \(0.373376\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 0 0
\(713\) 39.6301 1.48416
\(714\) 0 0
\(715\) −3.81507 −0.142676
\(716\) 0 0
\(717\) 6.44522 0.240701
\(718\) 0 0
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 26.0000 0.966950
\(724\) 0 0
\(725\) −6.81507 −0.253105
\(726\) 0 0
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.81507 0.178092
\(732\) 0 0
\(733\) 40.4452 1.49388 0.746939 0.664892i \(-0.231523\pi\)
0.746939 + 0.664892i \(0.231523\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) −7.63015 −0.281060
\(738\) 0 0
\(739\) −34.0754 −1.25348 −0.626741 0.779227i \(-0.715612\pi\)
−0.626741 + 0.779227i \(0.715612\pi\)
\(740\) 0 0
\(741\) −29.8151 −1.09528
\(742\) 0 0
\(743\) −4.44522 −0.163079 −0.0815396 0.996670i \(-0.525984\pi\)
−0.0815396 + 0.996670i \(0.525984\pi\)
\(744\) 0 0
\(745\) 2.00000 0.0732743
\(746\) 0 0
\(747\) 3.81507 0.139586
\(748\) 0 0
\(749\) −24.5548 −0.897212
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) −11.6301 −0.423826
\(754\) 0 0
\(755\) −3.81507 −0.138845
\(756\) 0 0
\(757\) 23.8151 0.865574 0.432787 0.901496i \(-0.357530\pi\)
0.432787 + 0.901496i \(0.357530\pi\)
\(758\) 0 0
\(759\) 5.81507 0.211074
\(760\) 0 0
\(761\) −9.18493 −0.332953 −0.166477 0.986045i \(-0.553239\pi\)
−0.166477 + 0.986045i \(0.553239\pi\)
\(762\) 0 0
\(763\) −39.0000 −1.41189
\(764\) 0 0
\(765\) 1.00000 0.0361551
\(766\) 0 0
\(767\) 7.63015 0.275509
\(768\) 0 0
\(769\) 49.2603 1.77637 0.888186 0.459485i \(-0.151966\pi\)
0.888186 + 0.459485i \(0.151966\pi\)
\(770\) 0 0
\(771\) 3.81507 0.137396
\(772\) 0 0
\(773\) −7.36985 −0.265075 −0.132538 0.991178i \(-0.542313\pi\)
−0.132538 + 0.991178i \(0.542313\pi\)
\(774\) 0 0
\(775\) −6.81507 −0.244805
\(776\) 0 0
\(777\) 3.00000 0.107624
\(778\) 0 0
\(779\) 37.6301 1.34824
\(780\) 0 0
\(781\) 11.6301 0.416159
\(782\) 0 0
\(783\) 6.81507 0.243551
\(784\) 0 0
\(785\) −12.8151 −0.457390
\(786\) 0 0
\(787\) 39.6301 1.41266 0.706331 0.707882i \(-0.250349\pi\)
0.706331 + 0.707882i \(0.250349\pi\)
\(788\) 0 0
\(789\) −30.4452 −1.08388
\(790\) 0 0
\(791\) 20.4452 0.726948
\(792\) 0 0
\(793\) −26.0000 −0.923287
\(794\) 0 0
\(795\) −10.6301 −0.377012
\(796\) 0 0
\(797\) −34.8904 −1.23588 −0.617941 0.786224i \(-0.712033\pi\)
−0.617941 + 0.786224i \(0.712033\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 11.8151 0.417465
\(802\) 0 0
\(803\) −5.81507 −0.205209
\(804\) 0 0
\(805\) 17.4452 0.614863
\(806\) 0 0
\(807\) −9.81507 −0.345507
\(808\) 0 0
\(809\) −13.0754 −0.459705 −0.229853 0.973225i \(-0.573824\pi\)
−0.229853 + 0.973225i \(0.573824\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) −16.6301 −0.582529
\(816\) 0 0
\(817\) −37.6301 −1.31651
\(818\) 0 0
\(819\) 11.4452 0.399928
\(820\) 0 0
\(821\) 11.0754 0.386533 0.193266 0.981146i \(-0.438092\pi\)
0.193266 + 0.981146i \(0.438092\pi\)
\(822\) 0 0
\(823\) −41.0754 −1.43180 −0.715899 0.698204i \(-0.753983\pi\)
−0.715899 + 0.698204i \(0.753983\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 28.0754 0.976276 0.488138 0.872766i \(-0.337676\pi\)
0.488138 + 0.872766i \(0.337676\pi\)
\(828\) 0 0
\(829\) −53.5206 −1.85885 −0.929423 0.369015i \(-0.879695\pi\)
−0.929423 + 0.369015i \(0.879695\pi\)
\(830\) 0 0
\(831\) −21.4452 −0.743926
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 21.8151 0.754942
\(836\) 0 0
\(837\) 6.81507 0.235563
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) 17.4452 0.601559
\(842\) 0 0
\(843\) 19.4452 0.669729
\(844\) 0 0
\(845\) 1.55478 0.0534861
\(846\) 0 0
\(847\) 30.0000 1.03081
\(848\) 0 0
\(849\) −21.8151 −0.748691
\(850\) 0 0
\(851\) −5.81507 −0.199338
\(852\) 0 0
\(853\) −11.0754 −0.379213 −0.189607 0.981860i \(-0.560721\pi\)
−0.189607 + 0.981860i \(0.560721\pi\)
\(854\) 0 0
\(855\) −7.81507 −0.267270
\(856\) 0 0
\(857\) 30.2603 1.03367 0.516836 0.856084i \(-0.327110\pi\)
0.516836 + 0.856084i \(0.327110\pi\)
\(858\) 0 0
\(859\) −31.4452 −1.07290 −0.536449 0.843933i \(-0.680234\pi\)
−0.536449 + 0.843933i \(0.680234\pi\)
\(860\) 0 0
\(861\) −14.4452 −0.492292
\(862\) 0 0
\(863\) 1.55478 0.0529254 0.0264627 0.999650i \(-0.491576\pi\)
0.0264627 + 0.999650i \(0.491576\pi\)
\(864\) 0 0
\(865\) −1.00000 −0.0340010
\(866\) 0 0
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) 29.1096 0.986340
\(872\) 0 0
\(873\) −4.81507 −0.162966
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) 49.7055 1.67844 0.839218 0.543795i \(-0.183013\pi\)
0.839218 + 0.543795i \(0.183013\pi\)
\(878\) 0 0
\(879\) −28.2603 −0.953196
\(880\) 0 0
\(881\) 43.1849 1.45494 0.727469 0.686141i \(-0.240697\pi\)
0.727469 + 0.686141i \(0.240697\pi\)
\(882\) 0 0
\(883\) −27.0000 −0.908622 −0.454311 0.890843i \(-0.650115\pi\)
−0.454311 + 0.890843i \(0.650115\pi\)
\(884\) 0 0
\(885\) 2.00000 0.0672293
\(886\) 0 0
\(887\) 14.0754 0.472604 0.236302 0.971680i \(-0.424064\pi\)
0.236302 + 0.971680i \(0.424064\pi\)
\(888\) 0 0
\(889\) −64.3357 −2.15775
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −62.5206 −2.09217
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) 0 0
\(897\) −22.1849 −0.740733
\(898\) 0 0
\(899\) 46.4452 1.54903
\(900\) 0 0
\(901\) 10.6301 0.354142
\(902\) 0 0
\(903\) 14.4452 0.480707
\(904\) 0 0
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) 45.8151 1.52126 0.760632 0.649183i \(-0.224889\pi\)
0.760632 + 0.649183i \(0.224889\pi\)
\(908\) 0 0
\(909\) 11.6301 0.385748
\(910\) 0 0
\(911\) 54.5206 1.80635 0.903174 0.429275i \(-0.141231\pi\)
0.903174 + 0.429275i \(0.141231\pi\)
\(912\) 0 0
\(913\) 3.81507 0.126260
\(914\) 0 0
\(915\) −6.81507 −0.225299
\(916\) 0 0
\(917\) −28.8904 −0.954046
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) −18.0000 −0.593120
\(922\) 0 0
\(923\) −44.3699 −1.46045
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 24.4452 0.802022 0.401011 0.916073i \(-0.368659\pi\)
0.401011 + 0.916073i \(0.368659\pi\)
\(930\) 0 0
\(931\) −15.6301 −0.512257
\(932\) 0 0
\(933\) 10.4452 0.341961
\(934\) 0 0
\(935\) 1.00000 0.0327035
\(936\) 0 0
\(937\) −11.6301 −0.379940 −0.189970 0.981790i \(-0.560839\pi\)
−0.189970 + 0.981790i \(0.560839\pi\)
\(938\) 0 0
\(939\) −32.8904 −1.07334
\(940\) 0 0
\(941\) −1.26029 −0.0410843 −0.0205422 0.999789i \(-0.506539\pi\)
−0.0205422 + 0.999789i \(0.506539\pi\)
\(942\) 0 0
\(943\) 28.0000 0.911805
\(944\) 0 0
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) 25.5548 0.830419 0.415209 0.909726i \(-0.363708\pi\)
0.415209 + 0.909726i \(0.363708\pi\)
\(948\) 0 0
\(949\) 22.1849 0.720153
\(950\) 0 0
\(951\) 2.81507 0.0912850
\(952\) 0 0
\(953\) 14.8904 0.482349 0.241174 0.970482i \(-0.422467\pi\)
0.241174 + 0.970482i \(0.422467\pi\)
\(954\) 0 0
\(955\) −4.63015 −0.149828
\(956\) 0 0
\(957\) 6.81507 0.220300
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 15.4452 0.498233
\(962\) 0 0
\(963\) 8.18493 0.263756
\(964\) 0 0
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −7.63015 −0.245369 −0.122684 0.992446i \(-0.539150\pi\)
−0.122684 + 0.992446i \(0.539150\pi\)
\(968\) 0 0
\(969\) 7.81507 0.251056
\(970\) 0 0
\(971\) 48.0754 1.54281 0.771406 0.636343i \(-0.219554\pi\)
0.771406 + 0.636343i \(0.219554\pi\)
\(972\) 0 0
\(973\) 68.4452 2.19425
\(974\) 0 0
\(975\) 3.81507 0.122180
\(976\) 0 0
\(977\) 21.0000 0.671850 0.335925 0.941889i \(-0.390951\pi\)
0.335925 + 0.941889i \(0.390951\pi\)
\(978\) 0 0
\(979\) 11.8151 0.377611
\(980\) 0 0
\(981\) 13.0000 0.415058
\(982\) 0 0
\(983\) −57.3357 −1.82872 −0.914362 0.404898i \(-0.867307\pi\)
−0.914362 + 0.404898i \(0.867307\pi\)
\(984\) 0 0
\(985\) −4.18493 −0.133343
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) −28.0000 −0.890348
\(990\) 0 0
\(991\) 38.4452 1.22125 0.610626 0.791919i \(-0.290918\pi\)
0.610626 + 0.791919i \(0.290918\pi\)
\(992\) 0 0
\(993\) −23.2603 −0.738143
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) 55.0754 1.74425 0.872127 0.489279i \(-0.162740\pi\)
0.872127 + 0.489279i \(0.162740\pi\)
\(998\) 0 0
\(999\) −1.00000 −0.0316386
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 8880.2.a.bh.1.1 2
4.3 odd 2 1110.2.a.q.1.1 2
12.11 even 2 3330.2.a.bf.1.1 2
20.19 odd 2 5550.2.a.bx.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.q.1.1 2 4.3 odd 2
3330.2.a.bf.1.1 2 12.11 even 2
5550.2.a.bx.1.2 2 20.19 odd 2
8880.2.a.bh.1.1 2 1.1 even 1 trivial