Properties

Label 8880.2.a.bh.1.2
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 1.2
Root \(5.81507\) of defining polynomial
Character \(\chi\) \(=\) 8880.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +6.81507 q^{13} -1.00000 q^{15} +1.00000 q^{17} +2.81507 q^{19} +3.00000 q^{21} +4.81507 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.81507 q^{29} +3.81507 q^{31} -1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} -6.81507 q^{39} +5.81507 q^{41} -5.81507 q^{43} +1.00000 q^{45} +8.00000 q^{47} +2.00000 q^{49} -1.00000 q^{51} -10.6301 q^{53} +1.00000 q^{55} -2.81507 q^{57} -2.00000 q^{59} -3.81507 q^{61} -3.00000 q^{63} +6.81507 q^{65} +13.6301 q^{67} -4.81507 q^{69} -9.63015 q^{71} +4.81507 q^{73} -1.00000 q^{75} -3.00000 q^{77} -12.0000 q^{79} +1.00000 q^{81} -6.81507 q^{83} +1.00000 q^{85} -3.81507 q^{87} +1.18493 q^{89} -20.4452 q^{91} -3.81507 q^{93} +2.81507 q^{95} +5.81507 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} - 6q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} - 6q^{7} + 2q^{9} + 2q^{11} + 3q^{13} - 2q^{15} + 2q^{17} - 5q^{19} + 6q^{21} - q^{23} + 2q^{25} - 2q^{27} - 3q^{29} - 3q^{31} - 2q^{33} - 6q^{35} + 2q^{37} - 3q^{39} + q^{41} - q^{43} + 2q^{45} + 16q^{47} + 4q^{49} - 2q^{51} + 2q^{55} + 5q^{57} - 4q^{59} + 3q^{61} - 6q^{63} + 3q^{65} + 6q^{67} + q^{69} + 2q^{71} - q^{73} - 2q^{75} - 6q^{77} - 24q^{79} + 2q^{81} - 3q^{83} + 2q^{85} + 3q^{87} + 13q^{89} - 9q^{91} + 3q^{93} - 5q^{95} + q^{97} + 2q^{99} + O(q^{100}) \)

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
\(13\) 6.81507 1.89016 0.945081 0.326837i \(-0.105983\pi\)
0.945081 + 0.326837i \(0.105983\pi\)
\(14\) 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
\(19\) 2.81507 0.645822 0.322911 0.946429i \(-0.395339\pi\)
0.322911 + 0.946429i \(0.395339\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 4.81507 1.00401 0.502006 0.864864i \(-0.332596\pi\)
0.502006 + 0.864864i \(0.332596\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.81507 0.708441 0.354221 0.935162i \(-0.384746\pi\)
0.354221 + 0.935162i \(0.384746\pi\)
\(30\) 0 0
\(31\) 3.81507 0.685207 0.342604 0.939480i \(-0.388691\pi\)
0.342604 + 0.939480i \(0.388691\pi\)
\(32\) 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
\(39\) −6.81507 −1.09129
\(40\) 0 0
\(41\) 5.81507 0.908162 0.454081 0.890960i \(-0.349968\pi\)
0.454081 + 0.890960i \(0.349968\pi\)
\(42\) 0 0
\(43\) −5.81507 −0.886790 −0.443395 0.896326i \(-0.646226\pi\)
−0.443395 + 0.896326i \(0.646226\pi\)
\(44\) 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 \(-0.760518\pi\)
−0.730081 + 0.683360i \(0.760518\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −2.81507 −0.372866
\(58\) 0 0
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) −3.81507 −0.488470 −0.244235 0.969716i \(-0.578537\pi\)
−0.244235 + 0.969716i \(0.578537\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) 6.81507 0.845306
\(66\) 0 0
\(67\) 13.6301 1.66519 0.832594 0.553884i \(-0.186855\pi\)
0.832594 + 0.553884i \(0.186855\pi\)
\(68\) 0 0
\(69\) −4.81507 −0.579667
\(70\) 0 0
\(71\) −9.63015 −1.14289 −0.571444 0.820641i \(-0.693617\pi\)
−0.571444 + 0.820641i \(0.693617\pi\)
\(72\) 0 0
\(73\) 4.81507 0.563562 0.281781 0.959479i \(-0.409075\pi\)
0.281781 + 0.959479i \(0.409075\pi\)
\(74\) 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
\(83\) −6.81507 −0.748051 −0.374026 0.927418i \(-0.622023\pi\)
−0.374026 + 0.927418i \(0.622023\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 0 0
\(87\) −3.81507 −0.409019
\(88\) 0 0
\(89\) 1.18493 0.125602 0.0628010 0.998026i \(-0.479997\pi\)
0.0628010 + 0.998026i \(0.479997\pi\)
\(90\) 0 0
\(91\) −20.4452 −2.14324
\(92\) 0 0
\(93\) −3.81507 −0.395605
\(94\) 0 0
\(95\) 2.81507 0.288820
\(96\) 0 0
\(97\) 5.81507 0.590431 0.295216 0.955431i \(-0.404609\pi\)
0.295216 + 0.955431i \(0.404609\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −9.63015 −0.958235 −0.479118 0.877751i \(-0.659043\pi\)
−0.479118 + 0.877751i \(0.659043\pi\)
\(102\) 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
\(107\) 18.8151 1.81892 0.909461 0.415790i \(-0.136495\pi\)
0.909461 + 0.415790i \(0.136495\pi\)
\(108\) 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
\(113\) 3.81507 0.358892 0.179446 0.983768i \(-0.442570\pi\)
0.179446 + 0.983768i \(0.442570\pi\)
\(114\) 0 0
\(115\) 4.81507 0.449008
\(116\) 0 0
\(117\) 6.81507 0.630054
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) −5.81507 −0.524327
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −10.4452 −0.926863 −0.463432 0.886133i \(-0.653382\pi\)
−0.463432 + 0.886133i \(0.653382\pi\)
\(128\) 0 0
\(129\) 5.81507 0.511989
\(130\) 0 0
\(131\) −11.6301 −1.01613 −0.508065 0.861319i \(-0.669639\pi\)
−0.508065 + 0.861319i \(0.669639\pi\)
\(132\) 0 0
\(133\) −8.44522 −0.732293
\(134\) 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
\(139\) −12.1849 −1.03351 −0.516756 0.856133i \(-0.672861\pi\)
−0.516756 + 0.856133i \(0.672861\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 6.81507 0.569905
\(144\) 0 0
\(145\) 3.81507 0.316825
\(146\) 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
\(151\) 6.81507 0.554603 0.277301 0.960783i \(-0.410560\pi\)
0.277301 + 0.960783i \(0.410560\pi\)
\(152\) 0 0
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) 3.81507 0.306434
\(156\) 0 0
\(157\) −2.18493 −0.174376 −0.0871881 0.996192i \(-0.527788\pi\)
−0.0871881 + 0.996192i \(0.527788\pi\)
\(158\) 0 0
\(159\) 10.6301 0.843025
\(160\) 0 0
\(161\) −14.4452 −1.13844
\(162\) 0 0
\(163\) 4.63015 0.362661 0.181331 0.983422i \(-0.441960\pi\)
0.181331 + 0.983422i \(0.441960\pi\)
\(164\) 0 0
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) 11.1849 0.865516 0.432758 0.901510i \(-0.357541\pi\)
0.432758 + 0.901510i \(0.357541\pi\)
\(168\) 0 0
\(169\) 33.4452 2.57271
\(170\) 0 0
\(171\) 2.81507 0.215274
\(172\) 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
\(183\) 3.81507 0.282018
\(184\) 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
\(191\) 16.6301 1.20332 0.601658 0.798754i \(-0.294507\pi\)
0.601658 + 0.798754i \(0.294507\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) −6.81507 −0.488038
\(196\) 0 0
\(197\) −14.8151 −1.05553 −0.527765 0.849390i \(-0.676970\pi\)
−0.527765 + 0.849390i \(0.676970\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −13.6301 −0.961396
\(202\) 0 0
\(203\) −11.4452 −0.803297
\(204\) 0 0
\(205\) 5.81507 0.406142
\(206\) 0 0
\(207\) 4.81507 0.334671
\(208\) 0 0
\(209\) 2.81507 0.194723
\(210\) 0 0
\(211\) 23.4452 1.61404 0.807018 0.590527i \(-0.201080\pi\)
0.807018 + 0.590527i \(0.201080\pi\)
\(212\) 0 0
\(213\) 9.63015 0.659847
\(214\) 0 0
\(215\) −5.81507 −0.396585
\(216\) 0 0
\(217\) −11.4452 −0.776952
\(218\) 0 0
\(219\) −4.81507 −0.325372
\(220\) 0 0
\(221\) 6.81507 0.458431
\(222\) 0 0
\(223\) 6.18493 0.414173 0.207087 0.978323i \(-0.433602\pi\)
0.207087 + 0.978323i \(0.433602\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 14.1849 0.941487 0.470743 0.882270i \(-0.343986\pi\)
0.470743 + 0.882270i \(0.343986\pi\)
\(228\) 0 0
\(229\) 21.6301 1.42936 0.714680 0.699451i \(-0.246572\pi\)
0.714680 + 0.699451i \(0.246572\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) 13.6301 0.892941 0.446470 0.894798i \(-0.352681\pi\)
0.446470 + 0.894798i \(0.352681\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 12.0000 0.779484
\(238\) 0 0
\(239\) 25.4452 1.64591 0.822957 0.568103i \(-0.192323\pi\)
0.822957 + 0.568103i \(0.192323\pi\)
\(240\) 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
\(247\) 19.1849 1.22071
\(248\) 0 0
\(249\) 6.81507 0.431888
\(250\) 0 0
\(251\) −9.63015 −0.607849 −0.303925 0.952696i \(-0.598297\pi\)
−0.303925 + 0.952696i \(0.598297\pi\)
\(252\) 0 0
\(253\) 4.81507 0.302721
\(254\) 0 0
\(255\) −1.00000 −0.0626224
\(256\) 0 0
\(257\) 6.81507 0.425113 0.212556 0.977149i \(-0.431821\pi\)
0.212556 + 0.977149i \(0.431821\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 3.81507 0.236147
\(262\) 0 0
\(263\) −1.44522 −0.0891160 −0.0445580 0.999007i \(-0.514188\pi\)
−0.0445580 + 0.999007i \(0.514188\pi\)
\(264\) 0 0
\(265\) −10.6301 −0.653005
\(266\) 0 0
\(267\) −1.18493 −0.0725164
\(268\) 0 0
\(269\) −0.815073 −0.0496959 −0.0248479 0.999691i \(-0.507910\pi\)
−0.0248479 + 0.999691i \(0.507910\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 20.4452 1.23740
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −10.4452 −0.627592 −0.313796 0.949490i \(-0.601601\pi\)
−0.313796 + 0.949490i \(0.601601\pi\)
\(278\) 0 0
\(279\) 3.81507 0.228402
\(280\) 0 0
\(281\) 12.4452 0.742420 0.371210 0.928549i \(-0.378943\pi\)
0.371210 + 0.928549i \(0.378943\pi\)
\(282\) 0 0
\(283\) 11.1849 0.664875 0.332437 0.943125i \(-0.392129\pi\)
0.332437 + 0.943125i \(0.392129\pi\)
\(284\) 0 0
\(285\) −2.81507 −0.166751
\(286\) 0 0
\(287\) −17.4452 −1.02976
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −5.81507 −0.340886
\(292\) 0 0
\(293\) −14.2603 −0.833095 −0.416548 0.909114i \(-0.636760\pi\)
−0.416548 + 0.909114i \(0.636760\pi\)
\(294\) 0 0
\(295\) −2.00000 −0.116445
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 32.8151 1.89774
\(300\) 0 0
\(301\) 17.4452 1.00553
\(302\) 0 0
\(303\) 9.63015 0.553237
\(304\) 0 0
\(305\) −3.81507 −0.218450
\(306\) 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
\(311\) 21.4452 1.21605 0.608023 0.793919i \(-0.291963\pi\)
0.608023 + 0.793919i \(0.291963\pi\)
\(312\) 0 0
\(313\) −30.8904 −1.74603 −0.873015 0.487693i \(-0.837839\pi\)
−0.873015 + 0.487693i \(0.837839\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) 7.81507 0.438938 0.219469 0.975619i \(-0.429567\pi\)
0.219469 + 0.975619i \(0.429567\pi\)
\(318\) 0 0
\(319\) 3.81507 0.213603
\(320\) 0 0
\(321\) −18.8151 −1.05015
\(322\) 0 0
\(323\) 2.81507 0.156635
\(324\) 0 0
\(325\) 6.81507 0.378032
\(326\) 0 0
\(327\) −13.0000 −0.718902
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) −19.2603 −1.05864 −0.529321 0.848422i \(-0.677553\pi\)
−0.529321 + 0.848422i \(0.677553\pi\)
\(332\) 0 0
\(333\) 1.00000 0.0547997
\(334\) 0 0
\(335\) 13.6301 0.744694
\(336\) 0 0
\(337\) −14.8151 −0.807028 −0.403514 0.914973i \(-0.632211\pi\)
−0.403514 + 0.914973i \(0.632211\pi\)
\(338\) 0 0
\(339\) −3.81507 −0.207206
\(340\) 0 0
\(341\) 3.81507 0.206598
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −4.81507 −0.259235
\(346\) 0 0
\(347\) 31.2603 1.67814 0.839070 0.544023i \(-0.183100\pi\)
0.839070 + 0.544023i \(0.183100\pi\)
\(348\) 0 0
\(349\) −35.2603 −1.88744 −0.943720 0.330745i \(-0.892700\pi\)
−0.943720 + 0.330745i \(0.892700\pi\)
\(350\) 0 0
\(351\) −6.81507 −0.363762
\(352\) 0 0
\(353\) −23.8151 −1.26755 −0.633774 0.773518i \(-0.718495\pi\)
−0.633774 + 0.773518i \(0.718495\pi\)
\(354\) 0 0
\(355\) −9.63015 −0.511115
\(356\) 0 0
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) −27.6301 −1.45826 −0.729132 0.684373i \(-0.760076\pi\)
−0.729132 + 0.684373i \(0.760076\pi\)
\(360\) 0 0
\(361\) −11.0754 −0.582914
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 4.81507 0.252032
\(366\) 0 0
\(367\) 21.0000 1.09619 0.548096 0.836416i \(-0.315353\pi\)
0.548096 + 0.836416i \(0.315353\pi\)
\(368\) 0 0
\(369\) 5.81507 0.302721
\(370\) 0 0
\(371\) 31.8904 1.65567
\(372\) 0 0
\(373\) 25.2603 1.30793 0.653964 0.756526i \(-0.273105\pi\)
0.653964 + 0.756526i \(0.273105\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 26.0000 1.33907
\(378\) 0 0
\(379\) −1.63015 −0.0837350 −0.0418675 0.999123i \(-0.513331\pi\)
−0.0418675 + 0.999123i \(0.513331\pi\)
\(380\) 0 0
\(381\) 10.4452 0.535125
\(382\) 0 0
\(383\) −1.55478 −0.0794456 −0.0397228 0.999211i \(-0.512647\pi\)
−0.0397228 + 0.999211i \(0.512647\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) 0 0
\(387\) −5.81507 −0.295597
\(388\) 0 0
\(389\) 38.7055 1.96245 0.981224 0.192873i \(-0.0617807\pi\)
0.981224 + 0.192873i \(0.0617807\pi\)
\(390\) 0 0
\(391\) 4.81507 0.243509
\(392\) 0 0
\(393\) 11.6301 0.586663
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 15.6301 0.784455 0.392227 0.919868i \(-0.371705\pi\)
0.392227 + 0.919868i \(0.371705\pi\)
\(398\) 0 0
\(399\) 8.44522 0.422790
\(400\) 0 0
\(401\) 32.4452 1.62024 0.810118 0.586266i \(-0.199403\pi\)
0.810118 + 0.586266i \(0.199403\pi\)
\(402\) 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
\(409\) 27.6301 1.36622 0.683111 0.730314i \(-0.260626\pi\)
0.683111 + 0.730314i \(0.260626\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) −6.81507 −0.334539
\(416\) 0 0
\(417\) 12.1849 0.596698
\(418\) 0 0
\(419\) −6.44522 −0.314870 −0.157435 0.987529i \(-0.550322\pi\)
−0.157435 + 0.987529i \(0.550322\pi\)
\(420\) 0 0
\(421\) 13.2603 0.646267 0.323134 0.946353i \(-0.395264\pi\)
0.323134 + 0.946353i \(0.395264\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) 11.4452 0.553873
\(428\) 0 0
\(429\) −6.81507 −0.329035
\(430\) 0 0
\(431\) −21.0000 −1.01153 −0.505767 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 0 0
\(433\) −8.81507 −0.423625 −0.211813 0.977310i \(-0.567937\pi\)
−0.211813 + 0.977310i \(0.567937\pi\)
\(434\) 0 0
\(435\) −3.81507 −0.182919
\(436\) 0 0
\(437\) 13.5548 0.648413
\(438\) 0 0
\(439\) −13.4452 −0.641705 −0.320853 0.947129i \(-0.603969\pi\)
−0.320853 + 0.947129i \(0.603969\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 7.63015 0.362519 0.181260 0.983435i \(-0.441983\pi\)
0.181260 + 0.983435i \(0.441983\pi\)
\(444\) 0 0
\(445\) 1.18493 0.0561709
\(446\) 0 0
\(447\) −2.00000 −0.0945968
\(448\) 0 0
\(449\) −32.8904 −1.55220 −0.776098 0.630612i \(-0.782804\pi\)
−0.776098 + 0.630612i \(0.782804\pi\)
\(450\) 0 0
\(451\) 5.81507 0.273821
\(452\) 0 0
\(453\) −6.81507 −0.320200
\(454\) 0 0
\(455\) −20.4452 −0.958487
\(456\) 0 0
\(457\) 25.0754 1.17298 0.586488 0.809958i \(-0.300510\pi\)
0.586488 + 0.809958i \(0.300510\pi\)
\(458\) 0 0
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 24.1849 1.12640 0.563202 0.826319i \(-0.309569\pi\)
0.563202 + 0.826319i \(0.309569\pi\)
\(462\) 0 0
\(463\) −3.63015 −0.168707 −0.0843536 0.996436i \(-0.526883\pi\)
−0.0843536 + 0.996436i \(0.526883\pi\)
\(464\) 0 0
\(465\) −3.81507 −0.176920
\(466\) 0 0
\(467\) −18.1849 −0.841498 −0.420749 0.907177i \(-0.638233\pi\)
−0.420749 + 0.907177i \(0.638233\pi\)
\(468\) 0 0
\(469\) −40.8904 −1.88814
\(470\) 0 0
\(471\) 2.18493 0.100676
\(472\) 0 0
\(473\) −5.81507 −0.267377
\(474\) 0 0
\(475\) 2.81507 0.129164
\(476\) 0 0
\(477\) −10.6301 −0.486721
\(478\) 0 0
\(479\) 0.815073 0.0372416 0.0186208 0.999827i \(-0.494072\pi\)
0.0186208 + 0.999827i \(0.494072\pi\)
\(480\) 0 0
\(481\) 6.81507 0.310741
\(482\) 0 0
\(483\) 14.4452 0.657280
\(484\) 0 0
\(485\) 5.81507 0.264049
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) −4.63015 −0.209382
\(490\) 0 0
\(491\) −12.8151 −0.578336 −0.289168 0.957278i \(-0.593379\pi\)
−0.289168 + 0.957278i \(0.593379\pi\)
\(492\) 0 0
\(493\) 3.81507 0.171822
\(494\) 0 0
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) 28.8904 1.29591
\(498\) 0 0
\(499\) −14.4452 −0.646657 −0.323328 0.946287i \(-0.604802\pi\)
−0.323328 + 0.946287i \(0.604802\pi\)
\(500\) 0 0
\(501\) −11.1849 −0.499706
\(502\) 0 0
\(503\) −43.2603 −1.92888 −0.964441 0.264300i \(-0.914859\pi\)
−0.964441 + 0.264300i \(0.914859\pi\)
\(504\) 0 0
\(505\) −9.63015 −0.428536
\(506\) 0 0
\(507\) −33.4452 −1.48535
\(508\) 0 0
\(509\) −14.4452 −0.640273 −0.320137 0.947371i \(-0.603729\pi\)
−0.320137 + 0.947371i \(0.603729\pi\)
\(510\) 0 0
\(511\) −14.4452 −0.639019
\(512\) 0 0
\(513\) −2.81507 −0.124289
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) 1.00000 0.0438951
\(520\) 0 0
\(521\) −9.81507 −0.430006 −0.215003 0.976613i \(-0.568976\pi\)
−0.215003 + 0.976613i \(0.568976\pi\)
\(522\) 0 0
\(523\) 43.2603 1.89164 0.945820 0.324691i \(-0.105260\pi\)
0.945820 + 0.324691i \(0.105260\pi\)
\(524\) 0 0
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) 3.81507 0.166187
\(528\) 0 0
\(529\) 0.184927 0.00804031
\(530\) 0 0
\(531\) −2.00000 −0.0867926
\(532\) 0 0
\(533\) 39.6301 1.71657
\(534\) 0 0
\(535\) 18.8151 0.813447
\(536\) 0 0
\(537\) 16.0000 0.690451
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 17.1849 0.738838 0.369419 0.929263i \(-0.379557\pi\)
0.369419 + 0.929263i \(0.379557\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) 13.0000 0.556859
\(546\) 0 0
\(547\) 39.8904 1.70559 0.852796 0.522244i \(-0.174905\pi\)
0.852796 + 0.522244i \(0.174905\pi\)
\(548\) 0 0
\(549\) −3.81507 −0.162823
\(550\) 0 0
\(551\) 10.7397 0.457527
\(552\) 0 0
\(553\) 36.0000 1.53088
\(554\) 0 0
\(555\) −1.00000 −0.0424476
\(556\) 0 0
\(557\) −0.369854 −0.0156712 −0.00783561 0.999969i \(-0.502494\pi\)
−0.00783561 + 0.999969i \(0.502494\pi\)
\(558\) 0 0
\(559\) −39.6301 −1.67618
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) 0 0
\(563\) −23.4452 −0.988098 −0.494049 0.869434i \(-0.664484\pi\)
−0.494049 + 0.869434i \(0.664484\pi\)
\(564\) 0 0
\(565\) 3.81507 0.160501
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) −25.1849 −1.05581 −0.527904 0.849304i \(-0.677022\pi\)
−0.527904 + 0.849304i \(0.677022\pi\)
\(570\) 0 0
\(571\) 4.18493 0.175134 0.0875669 0.996159i \(-0.472091\pi\)
0.0875669 + 0.996159i \(0.472091\pi\)
\(572\) 0 0
\(573\) −16.6301 −0.694734
\(574\) 0 0
\(575\) 4.81507 0.200802
\(576\) 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
\(581\) 20.4452 0.848211
\(582\) 0 0
\(583\) −10.6301 −0.440256
\(584\) 0 0
\(585\) 6.81507 0.281769
\(586\) 0 0
\(587\) −10.1849 −0.420377 −0.210188 0.977661i \(-0.567408\pi\)
−0.210188 + 0.977661i \(0.567408\pi\)
\(588\) 0 0
\(589\) 10.7397 0.442522
\(590\) 0 0
\(591\) 14.8151 0.609411
\(592\) 0 0
\(593\) 7.63015 0.313333 0.156666 0.987652i \(-0.449925\pi\)
0.156666 + 0.987652i \(0.449925\pi\)
\(594\) 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
\(601\) −40.6301 −1.65734 −0.828669 0.559739i \(-0.810901\pi\)
−0.828669 + 0.559739i \(0.810901\pi\)
\(602\) 0 0
\(603\) 13.6301 0.555062
\(604\) 0 0
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) −5.26029 −0.213509 −0.106754 0.994285i \(-0.534046\pi\)
−0.106754 + 0.994285i \(0.534046\pi\)
\(608\) 0 0
\(609\) 11.4452 0.463784
\(610\) 0 0
\(611\) 54.5206 2.20567
\(612\) 0 0
\(613\) 47.0754 1.90136 0.950678 0.310179i \(-0.100389\pi\)
0.950678 + 0.310179i \(0.100389\pi\)
\(614\) 0 0
\(615\) −5.81507 −0.234486
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 45.4452 1.82660 0.913299 0.407290i \(-0.133526\pi\)
0.913299 + 0.407290i \(0.133526\pi\)
\(620\) 0 0
\(621\) −4.81507 −0.193222
\(622\) 0 0
\(623\) −3.55478 −0.142419
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −2.81507 −0.112423
\(628\) 0 0
\(629\) 1.00000 0.0398726
\(630\) 0 0
\(631\) 40.7055 1.62046 0.810230 0.586112i \(-0.199342\pi\)
0.810230 + 0.586112i \(0.199342\pi\)
\(632\) 0 0
\(633\) −23.4452 −0.931864
\(634\) 0 0
\(635\) −10.4452 −0.414506
\(636\) 0 0
\(637\) 13.6301 0.540046
\(638\) 0 0
\(639\) −9.63015 −0.380963
\(640\) 0 0
\(641\) −45.4452 −1.79498 −0.897489 0.441037i \(-0.854611\pi\)
−0.897489 + 0.441037i \(0.854611\pi\)
\(642\) 0 0
\(643\) −17.0000 −0.670415 −0.335207 0.942144i \(-0.608806\pi\)
−0.335207 + 0.942144i \(0.608806\pi\)
\(644\) 0 0
\(645\) 5.81507 0.228968
\(646\) 0 0
\(647\) 28.0754 1.10376 0.551878 0.833925i \(-0.313911\pi\)
0.551878 + 0.833925i \(0.313911\pi\)
\(648\) 0 0
\(649\) −2.00000 −0.0785069
\(650\) 0 0
\(651\) 11.4452 0.448573
\(652\) 0 0
\(653\) 4.00000 0.156532 0.0782660 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) 0 0
\(655\) −11.6301 −0.454427
\(656\) 0 0
\(657\) 4.81507 0.187854
\(658\) 0 0
\(659\) −11.2603 −0.438639 −0.219319 0.975653i \(-0.570384\pi\)
−0.219319 + 0.975653i \(0.570384\pi\)
\(660\) 0 0
\(661\) 46.2603 1.79932 0.899658 0.436595i \(-0.143816\pi\)
0.899658 + 0.436595i \(0.143816\pi\)
\(662\) 0 0
\(663\) −6.81507 −0.264675
\(664\) 0 0
\(665\) −8.44522 −0.327492
\(666\) 0 0
\(667\) 18.3699 0.711284
\(668\) 0 0
\(669\) −6.18493 −0.239123
\(670\) 0 0
\(671\) −3.81507 −0.147279
\(672\) 0 0
\(673\) 16.8151 0.648173 0.324087 0.946027i \(-0.394943\pi\)
0.324087 + 0.946027i \(0.394943\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −6.07536 −0.233495 −0.116748 0.993162i \(-0.537247\pi\)
−0.116748 + 0.993162i \(0.537247\pi\)
\(678\) 0 0
\(679\) −17.4452 −0.669486
\(680\) 0 0
\(681\) −14.1849 −0.543568
\(682\) 0 0
\(683\) −48.3357 −1.84951 −0.924756 0.380560i \(-0.875731\pi\)
−0.924756 + 0.380560i \(0.875731\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) −21.6301 −0.825242
\(688\) 0 0
\(689\) −72.4452 −2.75994
\(690\) 0 0
\(691\) −19.0754 −0.725661 −0.362831 0.931855i \(-0.618190\pi\)
−0.362831 + 0.931855i \(0.618190\pi\)
\(692\) 0 0
\(693\) −3.00000 −0.113961
\(694\) 0 0
\(695\) −12.1849 −0.462201
\(696\) 0 0
\(697\) 5.81507 0.220262
\(698\) 0 0
\(699\) −13.6301 −0.515539
\(700\) 0 0
\(701\) 12.3699 0.467203 0.233601 0.972332i \(-0.424949\pi\)
0.233601 + 0.972332i \(0.424949\pi\)
\(702\) 0 0
\(703\) 2.81507 0.106172
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) 28.8904 1.08654
\(708\) 0 0
\(709\) −0.630146 −0.0236656 −0.0118328 0.999930i \(-0.503767\pi\)
−0.0118328 + 0.999930i \(0.503767\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 0 0
\(713\) 18.3699 0.687956
\(714\) 0 0
\(715\) 6.81507 0.254869
\(716\) 0 0
\(717\) −25.4452 −0.950269
\(718\) 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
\(725\) 3.81507 0.141688
\(726\) 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
\(731\) −5.81507 −0.215078
\(732\) 0 0
\(733\) 8.55478 0.315978 0.157989 0.987441i \(-0.449499\pi\)
0.157989 + 0.987441i \(0.449499\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) 13.6301 0.502073
\(738\) 0 0
\(739\) 19.0754 0.701699 0.350849 0.936432i \(-0.385893\pi\)
0.350849 + 0.936432i \(0.385893\pi\)
\(740\) 0 0
\(741\) −19.1849 −0.704776
\(742\) 0 0
\(743\) 27.4452 1.00687 0.503434 0.864034i \(-0.332070\pi\)
0.503434 + 0.864034i \(0.332070\pi\)
\(744\) 0 0
\(745\) 2.00000 0.0732743
\(746\) 0 0
\(747\) −6.81507 −0.249350
\(748\) 0 0
\(749\) −56.4452 −2.06246
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 9.63015 0.350942
\(754\) 0 0
\(755\) 6.81507 0.248026
\(756\) 0 0
\(757\) 13.1849 0.479214 0.239607 0.970870i \(-0.422981\pi\)
0.239607 + 0.970870i \(0.422981\pi\)
\(758\) 0 0
\(759\) −4.81507 −0.174776
\(760\) 0 0
\(761\) −19.8151 −0.718296 −0.359148 0.933281i \(-0.616933\pi\)
−0.359148 + 0.933281i \(0.616933\pi\)
\(762\) 0 0
\(763\) −39.0000 −1.41189
\(764\) 0 0
\(765\) 1.00000 0.0361551
\(766\) 0 0
\(767\) −13.6301 −0.492156
\(768\) 0 0
\(769\) 6.73971 0.243040 0.121520 0.992589i \(-0.461223\pi\)
0.121520 + 0.992589i \(0.461223\pi\)
\(770\) 0 0
\(771\) −6.81507 −0.245439
\(772\) 0 0
\(773\) −28.6301 −1.02975 −0.514877 0.857264i \(-0.672163\pi\)
−0.514877 + 0.857264i \(0.672163\pi\)
\(774\) 0 0
\(775\) 3.81507 0.137041
\(776\) 0 0
\(777\) 3.00000 0.107624
\(778\) 0 0
\(779\) 16.3699 0.586511
\(780\) 0 0
\(781\) −9.63015 −0.344594
\(782\) 0 0
\(783\) −3.81507 −0.136340
\(784\) 0 0
\(785\) −2.18493 −0.0779834
\(786\) 0 0
\(787\) 18.3699 0.654815 0.327407 0.944883i \(-0.393825\pi\)
0.327407 + 0.944883i \(0.393825\pi\)
\(788\) 0 0
\(789\) 1.44522 0.0514511
\(790\) 0 0
\(791\) −11.4452 −0.406945
\(792\) 0 0
\(793\) −26.0000 −0.923287
\(794\) 0 0
\(795\) 10.6301 0.377012
\(796\) 0 0
\(797\) 28.8904 1.02335 0.511676 0.859179i \(-0.329025\pi\)
0.511676 + 0.859179i \(0.329025\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 1.18493 0.0418673
\(802\) 0 0
\(803\) 4.81507 0.169920
\(804\) 0 0
\(805\) −14.4452 −0.509127
\(806\) 0 0
\(807\) 0.815073 0.0286919
\(808\) 0 0
\(809\) 40.0754 1.40897 0.704487 0.709717i \(-0.251177\pi\)
0.704487 + 0.709717i \(0.251177\pi\)
\(810\) 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
\(815\) 4.63015 0.162187
\(816\) 0 0
\(817\) −16.3699 −0.572709
\(818\) 0 0
\(819\) −20.4452 −0.714414
\(820\) 0 0
\(821\) −42.0754 −1.46844 −0.734220 0.678911i \(-0.762452\pi\)
−0.734220 + 0.678911i \(0.762452\pi\)
\(822\) 0 0
\(823\) 12.0754 0.420921 0.210460 0.977602i \(-0.432504\pi\)
0.210460 + 0.977602i \(0.432504\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) −25.0754 −0.871956 −0.435978 0.899957i \(-0.643597\pi\)
−0.435978 + 0.899957i \(0.643597\pi\)
\(828\) 0 0
\(829\) 31.5206 1.09476 0.547378 0.836886i \(-0.315626\pi\)
0.547378 + 0.836886i \(0.315626\pi\)
\(830\) 0 0
\(831\) 10.4452 0.362341
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 11.1849 0.387070
\(836\) 0 0
\(837\) −3.81507 −0.131868
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) −14.4452 −0.498111
\(842\) 0 0
\(843\) −12.4452 −0.428636
\(844\) 0 0
\(845\) 33.4452 1.15055
\(846\) 0 0
\(847\) 30.0000 1.03081
\(848\) 0 0
\(849\) −11.1849 −0.383866
\(850\) 0 0
\(851\) 4.81507 0.165059
\(852\) 0 0
\(853\) 42.0754 1.44063 0.720317 0.693646i \(-0.243997\pi\)
0.720317 + 0.693646i \(0.243997\pi\)
\(854\) 0 0
\(855\) 2.81507 0.0962735
\(856\) 0 0
\(857\) −12.2603 −0.418804 −0.209402 0.977830i \(-0.567152\pi\)
−0.209402 + 0.977830i \(0.567152\pi\)
\(858\) 0 0
\(859\) 0.445219 0.0151907 0.00759533 0.999971i \(-0.497582\pi\)
0.00759533 + 0.999971i \(0.497582\pi\)
\(860\) 0 0
\(861\) 17.4452 0.594531
\(862\) 0 0
\(863\) 33.4452 1.13849 0.569244 0.822168i \(-0.307236\pi\)
0.569244 + 0.822168i \(0.307236\pi\)
\(864\) 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
\(871\) 92.8904 3.14747
\(872\) 0 0
\(873\) 5.81507 0.196810
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −24.7055 −0.834246 −0.417123 0.908850i \(-0.636962\pi\)
−0.417123 + 0.908850i \(0.636962\pi\)
\(878\) 0 0
\(879\) 14.2603 0.480988
\(880\) 0 0
\(881\) 53.8151 1.81308 0.906538 0.422124i \(-0.138715\pi\)
0.906538 + 0.422124i \(0.138715\pi\)
\(882\) 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
\(887\) −39.0754 −1.31202 −0.656011 0.754751i \(-0.727758\pi\)
−0.656011 + 0.754751i \(0.727758\pi\)
\(888\) 0 0
\(889\) 31.3357 1.05096
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 22.5206 0.753623
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) 0 0
\(897\) −32.8151 −1.09566
\(898\) 0 0
\(899\) 14.5548 0.485429
\(900\) 0 0
\(901\) −10.6301 −0.354142
\(902\) 0 0
\(903\) −17.4452 −0.580541
\(904\) 0 0
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) 35.1849 1.16830 0.584148 0.811647i \(-0.301429\pi\)
0.584148 + 0.811647i \(0.301429\pi\)
\(908\) 0 0
\(909\) −9.63015 −0.319412
\(910\) 0 0
\(911\) −30.5206 −1.01119 −0.505596 0.862770i \(-0.668727\pi\)
−0.505596 + 0.862770i \(0.668727\pi\)
\(912\) 0 0
\(913\) −6.81507 −0.225546
\(914\) 0 0
\(915\) 3.81507 0.126122
\(916\) 0 0
\(917\) 34.8904 1.15218
\(918\) 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
\(923\) −65.6301 −2.16024
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −7.44522 −0.244270 −0.122135 0.992514i \(-0.538974\pi\)
−0.122135 + 0.992514i \(0.538974\pi\)
\(930\) 0 0
\(931\) 5.63015 0.184521
\(932\) 0 0
\(933\) −21.4452 −0.702085
\(934\) 0 0
\(935\) 1.00000 0.0327035
\(936\) 0 0
\(937\) 9.63015 0.314603 0.157302 0.987551i \(-0.449721\pi\)
0.157302 + 0.987551i \(0.449721\pi\)
\(938\) 0 0
\(939\) 30.8904 1.00807
\(940\) 0 0
\(941\) 41.2603 1.34505 0.672524 0.740076i \(-0.265210\pi\)
0.672524 + 0.740076i \(0.265210\pi\)
\(942\) 0 0
\(943\) 28.0000 0.911805
\(944\) 0 0
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) 57.4452 1.86672 0.933359 0.358943i \(-0.116863\pi\)
0.933359 + 0.358943i \(0.116863\pi\)
\(948\) 0 0
\(949\) 32.8151 1.06522
\(950\) 0 0
\(951\) −7.81507 −0.253421
\(952\) 0 0
\(953\) −48.8904 −1.58372 −0.791858 0.610705i \(-0.790886\pi\)
−0.791858 + 0.610705i \(0.790886\pi\)
\(954\) 0 0
\(955\) 16.6301 0.538139
\(956\) 0 0
\(957\) −3.81507 −0.123324
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −16.4452 −0.530491
\(962\) 0 0
\(963\) 18.8151 0.606307
\(964\) 0 0
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) 13.6301 0.438316 0.219158 0.975689i \(-0.429669\pi\)
0.219158 + 0.975689i \(0.429669\pi\)
\(968\) 0 0
\(969\) −2.81507 −0.0904332
\(970\) 0 0
\(971\) −5.07536 −0.162876 −0.0814381 0.996678i \(-0.525951\pi\)
−0.0814381 + 0.996678i \(0.525951\pi\)
\(972\) 0 0
\(973\) 36.5548 1.17189
\(974\) 0 0
\(975\) −6.81507 −0.218257
\(976\) 0 0
\(977\) 21.0000 0.671850 0.335925 0.941889i \(-0.390951\pi\)
0.335925 + 0.941889i \(0.390951\pi\)
\(978\) 0 0
\(979\) 1.18493 0.0378704
\(980\) 0 0
\(981\) 13.0000 0.415058
\(982\) 0 0
\(983\) 38.3357 1.22272 0.611359 0.791354i \(-0.290623\pi\)
0.611359 + 0.791354i \(0.290623\pi\)
\(984\) 0 0
\(985\) −14.8151 −0.472047
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) −28.0000 −0.890348
\(990\) 0 0
\(991\) 6.55478 0.208219 0.104110 0.994566i \(-0.466801\pi\)
0.104110 + 0.994566i \(0.466801\pi\)
\(992\) 0 0
\(993\) 19.2603 0.611207
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) 1.92464 0.0609538 0.0304769 0.999535i \(-0.490297\pi\)
0.0304769 + 0.999535i \(0.490297\pi\)
\(998\) 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.1.2 2
4.3 odd 2 1110.2.a.q.1.2 2
12.11 even 2 3330.2.a.bf.1.2 2
20.19 odd 2 5550.2.a.bx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.q.1.2 2 4.3 odd 2
3330.2.a.bf.1.2 2 12.11 even 2
5550.2.a.bx.1.1 2 20.19 odd 2
8880.2.a.bh.1.2 2 1.1 even 1 trivial