Properties

Label 8470.2.a.x.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -3.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} -2.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{28} -4.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -1.00000 q^{35} -3.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} -1.00000 q^{40} -10.0000 q^{41} -4.00000 q^{43} +3.00000 q^{45} -2.00000 q^{46} +1.00000 q^{49} +1.00000 q^{50} -4.00000 q^{52} -10.0000 q^{53} +1.00000 q^{56} -4.00000 q^{58} +6.00000 q^{59} -2.00000 q^{61} +2.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +14.0000 q^{67} +4.00000 q^{68} -1.00000 q^{70} -12.0000 q^{71} -3.00000 q^{72} +6.00000 q^{74} +4.00000 q^{76} +8.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} -10.0000 q^{82} -16.0000 q^{83} -4.00000 q^{85} -4.00000 q^{86} -14.0000 q^{89} +3.00000 q^{90} -4.00000 q^{91} -2.00000 q^{92} -4.00000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −3.00000 −1.00000
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −3.00000 −0.707107
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −1.00000 −0.169031
\(36\) −3.00000 −0.500000
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) −2.00000 −0.294884
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 2.00000 0.254000
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −3.00000 −0.353553
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 9.00000 1.00000
\(82\) −10.0000 −1.10432
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 3.00000 0.316228
\(91\) −4.00000 −0.419314
\(92\) −2.00000 −0.208514
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) −4.00000 −0.371391
\(117\) 12.0000 1.10940
\(118\) 6.00000 0.552345
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 0 0
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) −1.00000 −0.0894427
\(126\) −3.00000 −0.267261
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) −3.00000 −0.250000
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 4.00000 0.324443
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −2.00000 −0.157622
\(162\) 9.00000 0.707107
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −16.0000 −1.24184
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −4.00000 −0.306786
\(171\) −12.0000 −0.917663
\(172\) −4.00000 −0.304997
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 3.00000 0.223607
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) −2.00000 −0.147442
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) −8.00000 −0.541828
\(219\) 0 0
\(220\) 0 0
\(221\) −16.0000 −1.07628
\(222\) 0 0
\(223\) 20.0000 1.33930 0.669650 0.742677i \(-0.266444\pi\)
0.669650 + 0.742677i \(0.266444\pi\)
\(224\) 1.00000 0.0668153
\(225\) −3.00000 −0.200000
\(226\) −2.00000 −0.133038
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 12.0000 0.784465
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) −3.00000 −0.188982
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 4.00000 0.248069
\(261\) 12.0000 0.742781
\(262\) 4.00000 0.247121
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) 14.0000 0.855186
\(269\) 22.0000 1.34136 0.670682 0.741745i \(-0.266002\pi\)
0.670682 + 0.741745i \(0.266002\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 12.0000 0.719712
\(279\) −6.00000 −0.359211
\(280\) −1.00000 −0.0597614
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) −10.0000 −0.590281
\(288\) −3.00000 −0.176777
\(289\) −1.00000 −0.0588235
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) 0 0
\(293\) −8.00000 −0.467365 −0.233682 0.972313i \(-0.575078\pi\)
−0.233682 + 0.972313i \(0.575078\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 4.00000 0.231714
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 2.00000 0.114520
\(306\) −12.0000 −0.685994
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.00000 −0.113592
\(311\) −22.0000 −1.24751 −0.623753 0.781622i \(-0.714393\pi\)
−0.623753 + 0.781622i \(0.714393\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −10.0000 −0.564333
\(315\) 3.00000 0.169031
\(316\) 8.00000 0.450035
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −2.00000 −0.111456
\(323\) 16.0000 0.890264
\(324\) 9.00000 0.500000
\(325\) −4.00000 −0.221880
\(326\) −2.00000 −0.110770
\(327\) 0 0
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 0 0
\(331\) −24.0000 −1.31916 −0.659580 0.751635i \(-0.729266\pi\)
−0.659580 + 0.751635i \(0.729266\pi\)
\(332\) −16.0000 −0.878114
\(333\) −18.0000 −0.986394
\(334\) 12.0000 0.656611
\(335\) −14.0000 −0.764902
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) 0 0
\(342\) −12.0000 −0.648886
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −24.0000 −1.29025
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 3.00000 0.158114
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) −2.00000 −0.104257
\(369\) 30.0000 1.56174
\(370\) −6.00000 −0.311925
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.0000 0.824042
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 12.0000 0.609994
\(388\) −14.0000 −0.710742
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 6.00000 0.298511
\(405\) −9.00000 −0.447214
\(406\) −4.00000 −0.198517
\(407\) 0 0
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 10.0000 0.493865
\(411\) 0 0
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 6.00000 0.294884
\(415\) 16.0000 0.785409
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −16.0000 −0.761042
\(443\) 34.0000 1.61539 0.807694 0.589601i \(-0.200715\pi\)
0.807694 + 0.589601i \(0.200715\pi\)
\(444\) 0 0
\(445\) 14.0000 0.663664
\(446\) 20.0000 0.947027
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) −3.00000 −0.141421
\(451\) 0 0
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −26.0000 −1.21490
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) 22.0000 1.02243 0.511213 0.859454i \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 12.0000 0.554700
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 4.00000 0.183340
\(477\) 30.0000 1.37361
\(478\) 20.0000 0.914779
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 0 0
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) 10.0000 0.453143 0.226572 0.973995i \(-0.427248\pi\)
0.226572 + 0.973995i \(0.427248\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) −16.0000 −0.720604
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −2.00000 −0.0892644
\(503\) −20.0000 −0.891756 −0.445878 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) −3.00000 −0.133631
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 6.00000 0.263625
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 12.0000 0.525226
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 10.0000 0.434372
\(531\) −18.0000 −0.781133
\(532\) 4.00000 0.173422
\(533\) 40.0000 1.73259
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 14.0000 0.604708
\(537\) 0 0
\(538\) 22.0000 0.948487
\(539\) 0 0
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) 8.00000 0.342682
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) −18.0000 −0.768922
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) −16.0000 −0.681623
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −42.0000 −1.77960 −0.889799 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) −6.00000 −0.254000
\(559\) 16.0000 0.676728
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) −4.00000 −0.168133
\(567\) 9.00000 0.377964
\(568\) −12.0000 −0.503509
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −10.0000 −0.417392
\(575\) −2.00000 −0.0834058
\(576\) −3.00000 −0.125000
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −12.0000 −0.496139
\(586\) −8.00000 −0.330477
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) −6.00000 −0.247016
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) −16.0000 −0.657041 −0.328521 0.944497i \(-0.606550\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) −4.00000 −0.163028
\(603\) −42.0000 −1.71037
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) 0 0
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) 0 0
\(612\) −12.0000 −0.485071
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 0 0
\(622\) −22.0000 −0.882120
\(623\) −14.0000 −0.560898
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 24.0000 0.956943
\(630\) 3.00000 0.119523
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) 36.0000 1.42414
\(640\) −1.00000 −0.0395285
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) 16.0000 0.629512
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 9.00000 0.353553
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) −2.00000 −0.0783260
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) −6.00000 −0.233373 −0.116686 0.993169i \(-0.537227\pi\)
−0.116686 + 0.993169i \(0.537227\pi\)
\(662\) −24.0000 −0.932786
\(663\) 0 0
\(664\) −16.0000 −0.620920
\(665\) −4.00000 −0.155113
\(666\) −18.0000 −0.697486
\(667\) 8.00000 0.309761
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) −14.0000 −0.540867
\(671\) 0 0
\(672\) 0 0
\(673\) 6.00000 0.231283 0.115642 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 20.0000 0.768662 0.384331 0.923195i \(-0.374432\pi\)
0.384331 + 0.923195i \(0.374432\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) −4.00000 −0.153393
\(681\) 0 0
\(682\) 0 0
\(683\) 22.0000 0.841807 0.420903 0.907106i \(-0.361713\pi\)
0.420903 + 0.907106i \(0.361713\pi\)
\(684\) −12.0000 −0.458831
\(685\) 18.0000 0.687745
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 40.0000 1.52388
\(690\) 0 0
\(691\) 18.0000 0.684752 0.342376 0.939563i \(-0.388768\pi\)
0.342376 + 0.939563i \(0.388768\pi\)
\(692\) −24.0000 −0.912343
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −40.0000 −1.51511
\(698\) −18.0000 −0.681310
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 12.0000 0.450352
\(711\) −24.0000 −0.900070
\(712\) −14.0000 −0.524672
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) −10.0000 −0.372937 −0.186469 0.982461i \(-0.559704\pi\)
−0.186469 + 0.982461i \(0.559704\pi\)
\(720\) 3.00000 0.111803
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) −4.00000 −0.148250
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) −2.00000 −0.0737210
\(737\) 0 0
\(738\) 30.0000 1.10432
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) −6.00000 −0.220564
\(741\) 0 0
\(742\) −10.0000 −0.367112
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 0 0
\(745\) −4.00000 −0.146549
\(746\) 22.0000 0.805477
\(747\) 48.0000 1.75623
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 16.0000 0.582686
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) −26.0000 −0.942499 −0.471250 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) 12.0000 0.434145
\(765\) 12.0000 0.433861
\(766\) −28.0000 −1.01168
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.0000 0.359908
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 12.0000 0.431331
\(775\) 2.00000 0.0718421
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) 22.0000 0.788738
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) 0 0
\(782\) −8.00000 −0.286079
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 42.0000 1.48400
\(802\) 2.00000 0.0706225
\(803\) 0 0
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 36.0000 1.26569 0.632846 0.774277i \(-0.281886\pi\)
0.632846 + 0.774277i \(0.281886\pi\)
\(810\) −9.00000 −0.316228
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) −4.00000 −0.140372
\(813\) 0 0
\(814\) 0 0
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) −10.0000 −0.349642
\(819\) 12.0000 0.419314
\(820\) 10.0000 0.349215
\(821\) 48.0000 1.67521 0.837606 0.546275i \(-0.183955\pi\)
0.837606 + 0.546275i \(0.183955\pi\)
\(822\) 0 0
\(823\) −26.0000 −0.906303 −0.453152 0.891434i \(-0.649700\pi\)
−0.453152 + 0.891434i \(0.649700\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 6.00000 0.208514
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 16.0000 0.555368
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) 0 0
\(837\) 0 0
\(838\) −30.0000 −1.03633
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 6.00000 0.206774
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) 4.00000 0.137199
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) −32.0000 −1.09566 −0.547830 0.836590i \(-0.684546\pi\)
−0.547830 + 0.836590i \(0.684546\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 12.0000 0.410391
\(856\) −4.00000 −0.136717
\(857\) 16.0000 0.546550 0.273275 0.961936i \(-0.411893\pi\)
0.273275 + 0.961936i \(0.411893\pi\)
\(858\) 0 0
\(859\) 6.00000 0.204717 0.102359 0.994748i \(-0.467361\pi\)
0.102359 + 0.994748i \(0.467361\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 32.0000 1.08992
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) 24.0000 0.816024
\(866\) 14.0000 0.475739
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) −56.0000 −1.89749
\(872\) −8.00000 −0.270914
\(873\) 42.0000 1.42148
\(874\) −8.00000 −0.270604
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 26.0000 0.877958 0.438979 0.898497i \(-0.355340\pi\)
0.438979 + 0.898497i \(0.355340\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) 0 0
\(881\) −58.0000 −1.95407 −0.977035 0.213080i \(-0.931651\pi\)
−0.977035 + 0.213080i \(0.931651\pi\)
\(882\) −3.00000 −0.101015
\(883\) −46.0000 −1.54802 −0.774012 0.633171i \(-0.781753\pi\)
−0.774012 + 0.633171i \(0.781753\pi\)
\(884\) −16.0000 −0.538138
\(885\) 0 0
\(886\) 34.0000 1.14225
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 14.0000 0.469281
\(891\) 0 0
\(892\) 20.0000 0.669650
\(893\) 0 0
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) −8.00000 −0.266815
\(900\) −3.00000 −0.100000
\(901\) −40.0000 −1.33259
\(902\) 0 0
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) 54.0000 1.79304 0.896520 0.443003i \(-0.146087\pi\)
0.896520 + 0.443003i \(0.146087\pi\)
\(908\) −12.0000 −0.398234
\(909\) −18.0000 −0.597022
\(910\) 4.00000 0.132599
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 2.00000 0.0659380
\(921\) 0 0
\(922\) 26.0000 0.856264
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 22.0000 0.722965
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) 4.00000 0.130884
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) −28.0000 −0.914720 −0.457360 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) 14.0000 0.457116
\(939\) 0 0
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 20.0000 0.651290
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) −38.0000 −1.23483 −0.617417 0.786636i \(-0.711821\pi\)
−0.617417 + 0.786636i \(0.711821\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 4.00000 0.129641
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 30.0000 0.971286
\(955\) −12.0000 −0.388311
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) 0 0
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −24.0000 −0.773791
\(963\) 12.0000 0.386695
\(964\) −10.0000 −0.322078
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 14.0000 0.449513
\(971\) −22.0000 −0.706014 −0.353007 0.935621i \(-0.614841\pi\)
−0.353007 + 0.935621i \(0.614841\pi\)
\(972\) 0 0
\(973\) 12.0000 0.384702
\(974\) 10.0000 0.320421
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 24.0000 0.766261
\(982\) 28.0000 0.893516
\(983\) −48.0000 −1.53096 −0.765481 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) −16.0000 −0.509544
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 10.0000 0.317021
\(996\) 0 0
\(997\) 48.0000 1.52018 0.760088 0.649821i \(-0.225156\pi\)
0.760088 + 0.649821i \(0.225156\pi\)
\(998\) −4.00000 −0.126618
\(999\) 0 0
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 8470.2.a.x.1.1 yes 1
11.10 odd 2 8470.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.i.1.1 1 11.10 odd 2
8470.2.a.x.1.1 yes 1 1.1 even 1 trivial