Properties

Label 770.2.a.f.1.1
Level $770$
Weight $2$
Character 770.1
Self dual yes
Analytic conductor $6.148$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.14848095564\)
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 \(\chi\) \(=\) 770.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} -1.00000 q^{22} -2.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +2.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} +8.00000 q^{39} -1.00000 q^{40} -12.0000 q^{41} -2.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +6.00000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.00000 q^{52} -6.00000 q^{53} +4.00000 q^{54} +1.00000 q^{55} +1.00000 q^{56} +8.00000 q^{57} -6.00000 q^{58} -6.00000 q^{59} +2.00000 q^{60} -4.00000 q^{61} -10.0000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +2.00000 q^{66} -4.00000 q^{67} -1.00000 q^{70} +12.0000 q^{71} +1.00000 q^{72} -4.00000 q^{73} +2.00000 q^{74} -2.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} +8.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -12.0000 q^{82} +12.0000 q^{83} -2.00000 q^{84} -4.00000 q^{86} +12.0000 q^{87} -1.00000 q^{88} +18.0000 q^{89} -1.00000 q^{90} -4.00000 q^{91} +20.0000 q^{93} +6.00000 q^{94} +4.00000 q^{95} -2.00000 q^{96} -10.0000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) −2.00000 −0.577350
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.00000 −0.436436
\(22\) −1.00000 −0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −2.00000 −0.408248
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 4.00000 0.769800
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −4.00000 −0.648886
\(39\) 8.00000 1.28103
\(40\) −1.00000 −0.158114
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) −2.00000 −0.308607
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 4.00000 0.544331
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 8.00000 1.05963
\(58\) −6.00000 −0.787839
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 2.00000 0.258199
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) −10.0000 −1.27000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 2.00000 0.246183
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 2.00000 0.232495
\(75\) −2.00000 −0.230940
\(76\) −4.00000 −0.458831
\(77\) −1.00000 −0.113961
\(78\) 8.00000 0.905822
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) −12.0000 −1.32518
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 12.0000 1.28654
\(88\) −1.00000 −0.106600
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) −1.00000 −0.105409
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 20.0000 2.07390
\(94\) 6.00000 0.618853
\(95\) 4.00000 0.410391
\(96\) −2.00000 −0.204124
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −4.00000 −0.392232
\(105\) 2.00000 0.195180
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 4.00000 0.384900
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 1.00000 0.0953463
\(111\) −4.00000 −0.379663
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −4.00000 −0.369800
\(118\) −6.00000 −0.552345
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) −4.00000 −0.362143
\(123\) 24.0000 2.16401
\(124\) −10.0000 −0.898027
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 4.00000 0.350823
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 2.00000 0.174078
\(133\) −4.00000 −0.346844
\(134\) −4.00000 −0.345547
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −12.0000 −1.01058
\(142\) 12.0000 1.00702
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) −4.00000 −0.331042
\(147\) −2.00000 −0.164957
\(148\) 2.00000 0.164399
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) −2.00000 −0.163299
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 10.0000 0.803219
\(156\) 8.00000 0.640513
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 8.00000 0.636446
\(159\) 12.0000 0.951662
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −11.0000 −0.864242
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −12.0000 −0.937043
\(165\) −2.00000 −0.155700
\(166\) 12.0000 0.931381
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) −2.00000 −0.154303
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 12.0000 0.909718
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 12.0000 0.901975
\(178\) 18.0000 1.34916
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −4.00000 −0.296500
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 20.0000 1.46647
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) 4.00000 0.290957
\(190\) 4.00000 0.290191
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −2.00000 −0.144338
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −10.0000 −0.717958
\(195\) −8.00000 −0.572892
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 14.0000 0.975426
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) 4.00000 0.276686
\(210\) 2.00000 0.138013
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) −24.0000 −1.64445
\(214\) 12.0000 0.820303
\(215\) 4.00000 0.272798
\(216\) 4.00000 0.272166
\(217\) −10.0000 −0.678844
\(218\) 2.00000 0.135457
\(219\) 8.00000 0.540590
\(220\) 1.00000 0.0674200
\(221\) 0 0
\(222\) −4.00000 −0.268462
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 8.00000 0.529813
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) −6.00000 −0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −4.00000 −0.261488
\(235\) −6.00000 −0.391397
\(236\) −6.00000 −0.390567
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 2.00000 0.129099
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 1.00000 0.0642824
\(243\) 10.0000 0.641500
\(244\) −4.00000 −0.256074
\(245\) −1.00000 −0.0638877
\(246\) 24.0000 1.53018
\(247\) 16.0000 1.01806
\(248\) −10.0000 −0.635001
\(249\) −24.0000 −1.52094
\(250\) −1.00000 −0.0632456
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 8.00000 0.498058
\(259\) 2.00000 0.124274
\(260\) 4.00000 0.248069
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 2.00000 0.123091
\(265\) 6.00000 0.368577
\(266\) −4.00000 −0.245256
\(267\) −36.0000 −2.20316
\(268\) −4.00000 −0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −4.00000 −0.243432
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 6.00000 0.362473
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −4.00000 −0.239904
\(279\) −10.0000 −0.598684
\(280\) −1.00000 −0.0597614
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −12.0000 −0.714590
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 12.0000 0.712069
\(285\) −8.00000 −0.473879
\(286\) 4.00000 0.236525
\(287\) −12.0000 −0.708338
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 6.00000 0.352332
\(291\) 20.0000 1.17242
\(292\) −4.00000 −0.234082
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) −2.00000 −0.116642
\(295\) 6.00000 0.349334
\(296\) 2.00000 0.116248
\(297\) −4.00000 −0.232104
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) −2.00000 −0.115470
\(301\) −4.00000 −0.230556
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) 32.0000 1.82634 0.913168 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −28.0000 −1.59286
\(310\) 10.0000 0.567962
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 8.00000 0.452911
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −22.0000 −1.24153
\(315\) −1.00000 −0.0563436
\(316\) 8.00000 0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 12.0000 0.672927
\(319\) 6.00000 0.335936
\(320\) −1.00000 −0.0559017
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) −4.00000 −0.221880
\(326\) −4.00000 −0.221540
\(327\) −4.00000 −0.221201
\(328\) −12.0000 −0.662589
\(329\) 6.00000 0.330791
\(330\) −2.00000 −0.110096
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 12.0000 0.658586
\(333\) 2.00000 0.109599
\(334\) 24.0000 1.31322
\(335\) 4.00000 0.218543
\(336\) −2.00000 −0.109109
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 3.00000 0.163178
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 10.0000 0.541530
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 12.0000 0.643268
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) 1.00000 0.0534522
\(351\) −16.0000 −0.854017
\(352\) −1.00000 −0.0533002
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 12.0000 0.637793
\(355\) −12.0000 −0.636894
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) −2.00000 −0.104973
\(364\) −4.00000 −0.209657
\(365\) 4.00000 0.209370
\(366\) 8.00000 0.418167
\(367\) −34.0000 −1.77479 −0.887393 0.461014i \(-0.847486\pi\)
−0.887393 + 0.461014i \(0.847486\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) −2.00000 −0.103975
\(371\) −6.00000 −0.311504
\(372\) 20.0000 1.03695
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 2.00000 0.103280
\(376\) 6.00000 0.309426
\(377\) 24.0000 1.23606
\(378\) 4.00000 0.205738
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 4.00000 0.205196
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) −2.00000 −0.102062
\(385\) 1.00000 0.0509647
\(386\) 14.0000 0.712581
\(387\) −4.00000 −0.203331
\(388\) −10.0000 −0.507673
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) −8.00000 −0.405096
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) −8.00000 −0.402524
\(396\) −1.00000 −0.0502519
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 2.00000 0.100251
\(399\) 8.00000 0.400501
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 8.00000 0.399004
\(403\) 40.0000 1.99254
\(404\) 0 0
\(405\) 11.0000 0.546594
\(406\) −6.00000 −0.297775
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 12.0000 0.592638
\(411\) −12.0000 −0.591916
\(412\) 14.0000 0.689730
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) −4.00000 −0.196116
\(417\) 8.00000 0.391762
\(418\) 4.00000 0.195646
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 2.00000 0.0975900
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −4.00000 −0.194717
\(423\) 6.00000 0.291730
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −24.0000 −1.16280
\(427\) −4.00000 −0.193574
\(428\) 12.0000 0.580042
\(429\) −8.00000 −0.386244
\(430\) 4.00000 0.192897
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 4.00000 0.192450
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −10.0000 −0.480015
\(435\) −12.0000 −0.575356
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 8.00000 0.382255
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 1.00000 0.0476731
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) −4.00000 −0.189832
\(445\) −18.0000 −0.853282
\(446\) −10.0000 −0.473514
\(447\) 36.0000 1.70274
\(448\) 1.00000 0.0472456
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 1.00000 0.0471405
\(451\) 12.0000 0.565058
\(452\) −6.00000 −0.282216
\(453\) 32.0000 1.50349
\(454\) −24.0000 −1.12638
\(455\) 4.00000 0.187523
\(456\) 8.00000 0.374634
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 0 0
\(461\) −36.0000 −1.67669 −0.838344 0.545142i \(-0.816476\pi\)
−0.838344 + 0.545142i \(0.816476\pi\)
\(462\) 2.00000 0.0930484
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) −6.00000 −0.278543
\(465\) −20.0000 −0.927478
\(466\) −6.00000 −0.277945
\(467\) 42.0000 1.94353 0.971764 0.235954i \(-0.0758216\pi\)
0.971764 + 0.235954i \(0.0758216\pi\)
\(468\) −4.00000 −0.184900
\(469\) −4.00000 −0.184703
\(470\) −6.00000 −0.276759
\(471\) 44.0000 2.02741
\(472\) −6.00000 −0.276172
\(473\) 4.00000 0.183920
\(474\) −16.0000 −0.734904
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −24.0000 −1.09773
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 2.00000 0.0912871
\(481\) −8.00000 −0.364769
\(482\) −4.00000 −0.182195
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 10.0000 0.454077
\(486\) 10.0000 0.453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −4.00000 −0.181071
\(489\) 8.00000 0.361773
\(490\) −1.00000 −0.0451754
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 24.0000 1.08200
\(493\) 0 0
\(494\) 16.0000 0.719874
\(495\) 1.00000 0.0449467
\(496\) −10.0000 −0.449013
\(497\) 12.0000 0.538274
\(498\) −24.0000 −1.07547
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −48.0000 −2.14448
\(502\) −18.0000 −0.803379
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) −6.00000 −0.266469
\(508\) 8.00000 0.354943
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 1.00000 0.0441942
\(513\) −16.0000 −0.706417
\(514\) 18.0000 0.793946
\(515\) −14.0000 −0.616914
\(516\) 8.00000 0.352180
\(517\) −6.00000 −0.263880
\(518\) 2.00000 0.0878750
\(519\) −24.0000 −1.05348
\(520\) 4.00000 0.175412
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) −6.00000 −0.262613
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) −2.00000 −0.0872872
\(526\) 0 0
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) −23.0000 −1.00000
\(530\) 6.00000 0.260623
\(531\) −6.00000 −0.260378
\(532\) −4.00000 −0.173422
\(533\) 48.0000 2.07911
\(534\) −36.0000 −1.55787
\(535\) −12.0000 −0.518805
\(536\) −4.00000 −0.172774
\(537\) −24.0000 −1.03568
\(538\) 6.00000 0.258678
\(539\) −1.00000 −0.0430730
\(540\) −4.00000 −0.172133
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 20.0000 0.859074
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) −2.00000 −0.0856706
\(546\) 8.00000 0.342368
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 6.00000 0.256307
\(549\) −4.00000 −0.170716
\(550\) −1.00000 −0.0426401
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −22.0000 −0.934690
\(555\) 4.00000 0.169791
\(556\) −4.00000 −0.169638
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) −10.0000 −0.423334
\(559\) 16.0000 0.676728
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) −12.0000 −0.505291
\(565\) 6.00000 0.252422
\(566\) −16.0000 −0.672530
\(567\) −11.0000 −0.461957
\(568\) 12.0000 0.503509
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) −8.00000 −0.335083
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −17.0000 −0.707107
\(579\) −28.0000 −1.16364
\(580\) 6.00000 0.249136
\(581\) 12.0000 0.497844
\(582\) 20.0000 0.829027
\(583\) 6.00000 0.248495
\(584\) −4.00000 −0.165521
\(585\) 4.00000 0.165380
\(586\) 24.0000 0.991431
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 40.0000 1.64817
\(590\) 6.00000 0.247016
\(591\) 12.0000 0.493614
\(592\) 2.00000 0.0821995
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) −2.00000 −0.0816497
\(601\) −4.00000 −0.163163 −0.0815817 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) −4.00000 −0.163028
\(603\) −4.00000 −0.162893
\(604\) −16.0000 −0.651031
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) −4.00000 −0.162221
\(609\) 12.0000 0.486265
\(610\) 4.00000 0.161955
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 32.0000 1.29141
\(615\) −24.0000 −0.967773
\(616\) −1.00000 −0.0402911
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) −28.0000 −1.12633
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 10.0000 0.401610
\(621\) 0 0
\(622\) 18.0000 0.721734
\(623\) 18.0000 0.721155
\(624\) 8.00000 0.320256
\(625\) 1.00000 0.0400000
\(626\) 2.00000 0.0799361
\(627\) −8.00000 −0.319489
\(628\) −22.0000 −0.877896
\(629\) 0 0
\(630\) −1.00000 −0.0398410
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 8.00000 0.318223
\(633\) 8.00000 0.317971
\(634\) 6.00000 0.238290
\(635\) −8.00000 −0.317470
\(636\) 12.0000 0.475831
\(637\) −4.00000 −0.158486
\(638\) 6.00000 0.237542
\(639\) 12.0000 0.474713
\(640\) −1.00000 −0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) −24.0000 −0.947204
\(643\) 2.00000 0.0788723 0.0394362 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) −11.0000 −0.432121
\(649\) 6.00000 0.235521
\(650\) −4.00000 −0.156893
\(651\) 20.0000 0.783862
\(652\) −4.00000 −0.156652
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) −12.0000 −0.468521
\(657\) −4.00000 −0.156055
\(658\) 6.00000 0.233904
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 4.00000 0.155113
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) 24.0000 0.928588
\(669\) 20.0000 0.773245
\(670\) 4.00000 0.154533
\(671\) 4.00000 0.154418
\(672\) −2.00000 −0.0771517
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −22.0000 −0.847408
\(675\) 4.00000 0.153960
\(676\) 3.00000 0.115385
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 12.0000 0.460857
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 48.0000 1.83936
\(682\) 10.0000 0.382920
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −4.00000 −0.152944
\(685\) −6.00000 −0.229248
\(686\) 1.00000 0.0381802
\(687\) 20.0000 0.763048
\(688\) −4.00000 −0.152499
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) 12.0000 0.456172
\(693\) −1.00000 −0.0379869
\(694\) 12.0000 0.455514
\(695\) 4.00000 0.151729
\(696\) 12.0000 0.454859
\(697\) 0 0
\(698\) 20.0000 0.757011
\(699\) 12.0000 0.453882
\(700\) 1.00000 0.0377964
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) −16.0000 −0.603881
\(703\) −8.00000 −0.301726
\(704\) −1.00000 −0.0376889
\(705\) 12.0000 0.451946
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) −12.0000 −0.450352
\(711\) 8.00000 0.300023
\(712\) 18.0000 0.674579
\(713\) 0 0
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 12.0000 0.448461
\(717\) 48.0000 1.79259
\(718\) −24.0000 −0.895672
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 14.0000 0.521387
\(722\) −3.00000 −0.111648
\(723\) 8.00000 0.297523
\(724\) −10.0000 −0.371647
\(725\) −6.00000 −0.222834
\(726\) −2.00000 −0.0742270
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) −4.00000 −0.148250
\(729\) 13.0000 0.481481
\(730\) 4.00000 0.148047
\(731\) 0 0
\(732\) 8.00000 0.295689
\(733\) −40.0000 −1.47743 −0.738717 0.674016i \(-0.764568\pi\)
−0.738717 + 0.674016i \(0.764568\pi\)
\(734\) −34.0000 −1.25496
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) −12.0000 −0.441726
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −32.0000 −1.17555
\(742\) −6.00000 −0.220267
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 20.0000 0.733236
\(745\) 18.0000 0.659469
\(746\) −22.0000 −0.805477
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 2.00000 0.0730297
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 6.00000 0.218797
\(753\) 36.0000 1.31191
\(754\) 24.0000 0.874028
\(755\) 16.0000 0.582300
\(756\) 4.00000 0.145479
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) 24.0000 0.869999 0.435000 0.900431i \(-0.356748\pi\)
0.435000 + 0.900431i \(0.356748\pi\)
\(762\) −16.0000 −0.579619
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) 24.0000 0.866590
\(768\) −2.00000 −0.0721688
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 1.00000 0.0360375
\(771\) −36.0000 −1.29651
\(772\) 14.0000 0.503871
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) −4.00000 −0.143777
\(775\) −10.0000 −0.359211
\(776\) −10.0000 −0.358979
\(777\) −4.00000 −0.143499
\(778\) 30.0000 1.07555
\(779\) 48.0000 1.71978
\(780\) −8.00000 −0.286446
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) −24.0000 −0.857690
\(784\) 1.00000 0.0357143
\(785\) 22.0000 0.785214
\(786\) 0 0
\(787\) 44.0000 1.56843 0.784215 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −6.00000 −0.213335
\(792\) −1.00000 −0.0355335
\(793\) 16.0000 0.568177
\(794\) 2.00000 0.0709773
\(795\) −12.0000 −0.425596
\(796\) 2.00000 0.0708881
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 8.00000 0.283197
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 18.0000 0.635999
\(802\) −6.00000 −0.211867
\(803\) 4.00000 0.141157
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 40.0000 1.40894
\(807\) −12.0000 −0.422420
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 11.0000 0.386501
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) −6.00000 −0.210559
\(813\) −40.0000 −1.40286
\(814\) −2.00000 −0.0701000
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 32.0000 1.11885
\(819\) −4.00000 −0.139771
\(820\) 12.0000 0.419058
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −12.0000 −0.418548
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 14.0000 0.487713
\(825\) 2.00000 0.0696311
\(826\) −6.00000 −0.208767
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) −12.0000 −0.416526
\(831\) 44.0000 1.52634
\(832\) −4.00000 −0.138675
\(833\) 0 0
\(834\) 8.00000 0.277017
\(835\) −24.0000 −0.830554
\(836\) 4.00000 0.138343
\(837\) −40.0000 −1.38260
\(838\) −6.00000 −0.207267
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 2.00000 0.0690066
\(841\) 7.00000 0.241379
\(842\) 26.0000 0.896019
\(843\) 36.0000 1.23991
\(844\) −4.00000 −0.137686
\(845\) −3.00000 −0.103203
\(846\) 6.00000 0.206284
\(847\) 1.00000 0.0343604
\(848\) −6.00000 −0.206041
\(849\) 32.0000 1.09824
\(850\) 0 0
\(851\) 0 0
\(852\) −24.0000 −0.822226
\(853\) −4.00000 −0.136957 −0.0684787 0.997653i \(-0.521815\pi\)
−0.0684787 + 0.997653i \(0.521815\pi\)
\(854\) −4.00000 −0.136877
\(855\) 4.00000 0.136797
\(856\) 12.0000 0.410152
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) −8.00000 −0.273115
\(859\) −10.0000 −0.341196 −0.170598 0.985341i \(-0.554570\pi\)
−0.170598 + 0.985341i \(0.554570\pi\)
\(860\) 4.00000 0.136399
\(861\) 24.0000 0.817918
\(862\) 24.0000 0.817443
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 4.00000 0.136083
\(865\) −12.0000 −0.408012
\(866\) 14.0000 0.475739
\(867\) 34.0000 1.15470
\(868\) −10.0000 −0.339422
\(869\) −8.00000 −0.271381
\(870\) −12.0000 −0.406838
\(871\) 16.0000 0.542139
\(872\) 2.00000 0.0677285
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 8.00000 0.270295
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) −28.0000 −0.944954
\(879\) −48.0000 −1.61900
\(880\) 1.00000 0.0337100
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 1.00000 0.0336718
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) −24.0000 −0.806296
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) −4.00000 −0.134231
\(889\) 8.00000 0.268311
\(890\) −18.0000 −0.603361
\(891\) 11.0000 0.368514
\(892\) −10.0000 −0.334825
\(893\) −24.0000 −0.803129
\(894\) 36.0000 1.20402
\(895\) −12.0000 −0.401116
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) 60.0000 2.00111
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 12.0000 0.399556
\(903\) 8.00000 0.266223
\(904\) −6.00000 −0.199557
\(905\) 10.0000 0.332411
\(906\) 32.0000 1.06313
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −24.0000 −0.796468
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 8.00000 0.264906
\(913\) −12.0000 −0.397142
\(914\) −10.0000 −0.330771
\(915\) −8.00000 −0.264472
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) −64.0000 −2.10887
\(922\) −36.0000 −1.18560
\(923\) −48.0000 −1.57994
\(924\) 2.00000 0.0657952
\(925\) 2.00000 0.0657596
\(926\) −40.0000 −1.31448
\(927\) 14.0000 0.459820
\(928\) −6.00000 −0.196960
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) −20.0000 −0.655826
\(931\) −4.00000 −0.131095
\(932\) −6.00000 −0.196537
\(933\) −36.0000 −1.17859
\(934\) 42.0000 1.37428
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) −52.0000 −1.69877 −0.849383 0.527777i \(-0.823026\pi\)
−0.849383 + 0.527777i \(0.823026\pi\)
\(938\) −4.00000 −0.130605
\(939\) −4.00000 −0.130535
\(940\) −6.00000 −0.195698
\(941\) −36.0000 −1.17357 −0.586783 0.809744i \(-0.699606\pi\)
−0.586783 + 0.809744i \(0.699606\pi\)
\(942\) 44.0000 1.43360
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) −4.00000 −0.130120
\(946\) 4.00000 0.130051
\(947\) 48.0000 1.55979 0.779895 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) −16.0000 −0.519656
\(949\) 16.0000 0.519382
\(950\) −4.00000 −0.129777
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) −12.0000 −0.387905
\(958\) 12.0000 0.387702
\(959\) 6.00000 0.193750
\(960\) 2.00000 0.0645497
\(961\) 69.0000 2.22581
\(962\) −8.00000 −0.257930
\(963\) 12.0000 0.386695
\(964\) −4.00000 −0.128831
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) 30.0000 0.962746 0.481373 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) 10.0000 0.320750
\(973\) −4.00000 −0.128234
\(974\) 32.0000 1.02535
\(975\) 8.00000 0.256205
\(976\) −4.00000 −0.128037
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 8.00000 0.255812
\(979\) −18.0000 −0.575282
\(980\) −1.00000 −0.0319438
\(981\) 2.00000 0.0638551
\(982\) −12.0000 −0.382935
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 24.0000 0.765092
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) −12.0000 −0.381964
\(988\) 16.0000 0.509028
\(989\) 0 0
\(990\) 1.00000 0.0317821
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −10.0000 −0.317500
\(993\) −40.0000 −1.26936
\(994\) 12.0000 0.380617
\(995\) −2.00000 −0.0634043
\(996\) −24.0000 −0.760469
\(997\) −52.0000 −1.64686 −0.823428 0.567420i \(-0.807941\pi\)
−0.823428 + 0.567420i \(0.807941\pi\)
\(998\) −40.0000 −1.26618
\(999\) 8.00000 0.253109
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 770.2.a.f.1.1 1
3.2 odd 2 6930.2.a.o.1.1 1
4.3 odd 2 6160.2.a.j.1.1 1
5.2 odd 4 3850.2.c.b.1849.2 2
5.3 odd 4 3850.2.c.b.1849.1 2
5.4 even 2 3850.2.a.k.1.1 1
7.6 odd 2 5390.2.a.bj.1.1 1
11.10 odd 2 8470.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.f.1.1 1 1.1 even 1 trivial
3850.2.a.k.1.1 1 5.4 even 2
3850.2.c.b.1849.1 2 5.3 odd 4
3850.2.c.b.1849.2 2 5.2 odd 4
5390.2.a.bj.1.1 1 7.6 odd 2
6160.2.a.j.1.1 1 4.3 odd 2
6930.2.a.o.1.1 1 3.2 odd 2
8470.2.a.c.1.1 1 11.10 odd 2