Properties

Label 6930.2.a.r.1.1
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -6.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{35} -12.0000 q^{37} +6.00000 q^{38} -1.00000 q^{40} -2.00000 q^{41} +6.00000 q^{43} -1.00000 q^{44} +4.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} -6.00000 q^{53} +1.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} +4.00000 q^{59} +8.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +10.0000 q^{67} +1.00000 q^{70} +10.0000 q^{71} -10.0000 q^{73} -12.0000 q^{74} +6.00000 q^{76} +1.00000 q^{77} +8.00000 q^{79} -1.00000 q^{80} -2.00000 q^{82} +6.00000 q^{86} -1.00000 q^{88} -6.00000 q^{89} +6.00000 q^{91} +4.00000 q^{92} +6.00000 q^{94} -6.00000 q^{95} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −12.0000 −1.97279 −0.986394 0.164399i \(-0.947432\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −12.0000 −1.39497
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 1.00000 0.113961
\(78\) 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
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 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
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 6.00000 0.526235
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 10.0000 0.863868
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 10.0000 0.839181
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −12.0000 −0.986394
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 0 0
\(167\) −10.0000 −0.773823 −0.386912 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 6.00000 0.457496
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 6.00000 0.444750
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) 2.00000 0.144715 0.0723575 0.997379i \(-0.476948\pi\)
0.0723575 + 0.997379i \(0.476948\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) −6.00000 −0.406371
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) 0 0
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 0 0
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) 18.0000 1.16432 0.582162 0.813073i \(-0.302207\pi\)
0.582162 + 0.813073i \(0.302207\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 8.00000 0.512148
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −36.0000 −2.29063
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) 20.0000 1.23560
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −6.00000 −0.367884
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 2.00000 0.119952
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −28.0000 −1.67034 −0.835170 0.549992i \(-0.814631\pi\)
−0.835170 + 0.549992i \(0.814631\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) −12.0000 −0.697486
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −4.00000 −0.222911
\(323\) 0 0
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) 10.0000 0.553849
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −10.0000 −0.547176
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 23.0000 1.25104
\(339\) 0 0
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) −10.0000 −0.530745
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) 2.00000 0.105556 0.0527780 0.998606i \(-0.483192\pi\)
0.0527780 + 0.998606i \(0.483192\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −12.0000 −0.630706
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 12.0000 0.623850
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −6.00000 −0.307794
\(381\) 0 0
\(382\) 2.00000 0.102329
\(383\) −34.0000 −1.73732 −0.868659 0.495410i \(-0.835018\pi\)
−0.868659 + 0.495410i \(0.835018\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 10.0000 0.503793
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −24.0000 −1.19553
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 2.00000 0.0987730
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) −2.00000 −0.0963366 −0.0481683 0.998839i \(-0.515338\pi\)
−0.0481683 + 0.998839i \(0.515338\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) −24.0000 −1.13643
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 28.0000 1.32140 0.660701 0.750649i \(-0.270259\pi\)
0.660701 + 0.750649i \(0.270259\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 10.0000 0.470360
\(453\) 0 0
\(454\) −28.0000 −1.31411
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 8.00000 0.373815
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −10.0000 −0.461757
\(470\) −6.00000 −0.276759
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) 0 0
\(478\) 18.0000 0.823301
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) 72.0000 3.28292
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 8.00000 0.362143
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −36.0000 −1.61972
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −10.0000 −0.448561
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 20.0000 0.892644
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 8.00000 0.352865
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −6.00000 −0.263880
\(518\) 12.0000 0.527250
\(519\) 0 0
\(520\) 6.00000 0.263117
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) −6.00000 −0.260133
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 10.0000 0.431934
\(537\) 0 0
\(538\) 14.0000 0.603583
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 20.0000 0.859074
\(543\) 0 0
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −28.0000 −1.18111
\(563\) 28.0000 1.18006 0.590030 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 0 0
\(565\) −10.0000 −0.420703
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) 10.0000 0.419591
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 6.00000 0.250873
\(573\) 0 0
\(574\) 2.00000 0.0834784
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 46.0000 1.91501 0.957503 0.288425i \(-0.0931316\pi\)
0.957503 + 0.288425i \(0.0931316\pi\)
\(578\) −17.0000 −0.707107
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) 0 0
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) −4.00000 −0.164677
\(591\) 0 0
\(592\) −12.0000 −0.493197
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) 10.0000 0.408589 0.204294 0.978909i \(-0.434510\pi\)
0.204294 + 0.978909i \(0.434510\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −6.00000 −0.244542
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) −8.00000 −0.323911
\(611\) −36.0000 −1.45640
\(612\) 0 0
\(613\) −24.0000 −0.969351 −0.484675 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) 0 0
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 14.0000 0.556011
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −4.00000 −0.157991 −0.0789953 0.996875i \(-0.525171\pi\)
−0.0789953 + 0.996875i \(0.525171\pi\)
\(642\) 0 0
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) 0 0
\(655\) −20.0000 −0.781465
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) −6.00000 −0.233904
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) −32.0000 −1.24372
\(663\) 0 0
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) −10.0000 −0.386912
\(669\) 0 0
\(670\) −10.0000 −0.386334
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 6.00000 0.228748
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −6.00000 −0.228251 −0.114125 0.993466i \(-0.536407\pi\)
−0.114125 + 0.993466i \(0.536407\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) −2.00000 −0.0758643
\(696\) 0 0
\(697\) 0 0
\(698\) −20.0000 −0.757011
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 0 0
\(703\) −72.0000 −2.71553
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 18.0000 0.676004 0.338002 0.941145i \(-0.390249\pi\)
0.338002 + 0.941145i \(0.390249\pi\)
\(710\) −10.0000 −0.375293
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) 24.0000 0.896922
\(717\) 0 0
\(718\) 2.00000 0.0746393
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) −12.0000 −0.445976
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 6.00000 0.222375
\(729\) 0 0
\(730\) 10.0000 0.370117
\(731\) 0 0
\(732\) 0 0
\(733\) 10.0000 0.369358 0.184679 0.982799i \(-0.440875\pi\)
0.184679 + 0.982799i \(0.440875\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −10.0000 −0.368355
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 32.0000 1.17397 0.586983 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 32.0000 1.17160
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −44.0000 −1.59921 −0.799604 0.600528i \(-0.794957\pi\)
−0.799604 + 0.600528i \(0.794957\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) −6.00000 −0.217643
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 6.00000 0.217215
\(764\) 2.00000 0.0723575
\(765\) 0 0
\(766\) −34.0000 −1.22847
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −10.0000 −0.357828
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 6.00000 0.214149
\(786\) 0 0
\(787\) −24.0000 −0.855508 −0.427754 0.903895i \(-0.640695\pi\)
−0.427754 + 0.903895i \(0.640695\pi\)
\(788\) 10.0000 0.356235
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) −48.0000 −1.70453
\(794\) −10.0000 −0.354887
\(795\) 0 0
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 0 0
\(803\) 10.0000 0.352892
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) −24.0000 −0.845364
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) 10.0000 0.351147 0.175574 0.984466i \(-0.443822\pi\)
0.175574 + 0.984466i \(0.443822\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 12.0000 0.420600
\(815\) −10.0000 −0.350285
\(816\) 0 0
\(817\) 36.0000 1.25948
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) −50.0000 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 8.00000 0.277851 0.138926 0.990303i \(-0.455635\pi\)
0.138926 + 0.990303i \(0.455635\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) 0 0
\(834\) 0 0
\(835\) 10.0000 0.346064
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) −4.00000 −0.138178
\(839\) 52.0000 1.79524 0.897620 0.440771i \(-0.145295\pi\)
0.897620 + 0.440771i \(0.145295\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −18.0000 −0.620321
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 0 0
\(851\) −48.0000 −1.64542
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 12.0000 0.409912 0.204956 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(858\) 0 0
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) −2.00000 −0.0681203
\(863\) −40.0000 −1.36162 −0.680808 0.732462i \(-0.738371\pi\)
−0.680808 + 0.732462i \(0.738371\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 26.0000 0.883516
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −60.0000 −2.03302
\(872\) −6.00000 −0.203186
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 28.0000 0.945493 0.472746 0.881199i \(-0.343263\pi\)
0.472746 + 0.881199i \(0.343263\pi\)
\(878\) −20.0000 −0.674967
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) −10.0000 −0.336909 −0.168454 0.985709i \(-0.553878\pi\)
−0.168454 + 0.985709i \(0.553878\pi\)
\(882\) 0 0
\(883\) 10.0000 0.336527 0.168263 0.985742i \(-0.446184\pi\)
0.168263 + 0.985742i \(0.446184\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) −26.0000 −0.872995 −0.436497 0.899706i \(-0.643781\pi\)
−0.436497 + 0.899706i \(0.643781\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −24.0000 −0.803579
\(893\) 36.0000 1.20469
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 28.0000 0.934372
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 0 0
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 12.0000 0.398893
\(906\) 0 0
\(907\) 18.0000 0.597680 0.298840 0.954303i \(-0.403400\pi\)
0.298840 + 0.954303i \(0.403400\pi\)
\(908\) −28.0000 −0.929213
\(909\) 0 0
\(910\) −6.00000 −0.198898
\(911\) −22.0000 −0.728893 −0.364446 0.931224i \(-0.618742\pi\)
−0.364446 + 0.931224i \(0.618742\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) −20.0000 −0.660458
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) 14.0000 0.461065
\(923\) −60.0000 −1.97492
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) 4.00000 0.131448
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −54.0000 −1.76410 −0.882052 0.471153i \(-0.843838\pi\)
−0.882052 + 0.471153i \(0.843838\pi\)
\(938\) −10.0000 −0.326512
\(939\) 0 0
\(940\) −6.00000 −0.195698
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 60.0000 1.94768
\(950\) 6.00000 0.194666
\(951\) 0 0
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) −2.00000 −0.0647185
\(956\) 18.0000 0.582162
\(957\) 0 0
\(958\) 28.0000 0.904639
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 72.0000 2.32137
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) 0 0
\(973\) −2.00000 −0.0641171
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 0 0
\(983\) 22.0000 0.701691 0.350846 0.936433i \(-0.385894\pi\)
0.350846 + 0.936433i \(0.385894\pi\)
\(984\) 0 0
\(985\) −10.0000 −0.318626
\(986\) 0 0
\(987\) 0 0
\(988\) −36.0000 −1.14531
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) −10.0000 −0.317181
\(995\) 0 0
\(996\) 0 0
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) 32.0000 1.01294
\(999\) 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 6930.2.a.r.1.1 yes 1
3.2 odd 2 6930.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6930.2.a.k.1.1 1 3.2 odd 2
6930.2.a.r.1.1 yes 1 1.1 even 1 trivial