Properties

Label 4650.2.a.bp.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.1304369399\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4650.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -6.00000 q^{26} +1.00000 q^{27} +2.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -6.00000 q^{39} -6.00000 q^{41} +4.00000 q^{43} -4.00000 q^{44} +4.00000 q^{46} +1.00000 q^{48} -7.00000 q^{49} -2.00000 q^{51} -6.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -4.00000 q^{57} +2.00000 q^{58} -4.00000 q^{59} -6.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -4.00000 q^{66} -16.0000 q^{67} -2.00000 q^{68} +4.00000 q^{69} -12.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} -4.00000 q^{76} -6.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +4.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} -18.0000 q^{89} +4.00000 q^{92} -1.00000 q^{93} +1.00000 q^{96} +14.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 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
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) −6.00000 −0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 2.00000 0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) −16.0000 −1.95471 −0.977356 0.211604i \(-0.932131\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) −2.00000 −0.242536
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) −4.00000 −0.426401
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −7.00000 −0.707107
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −2.00000 −0.198030
\(103\) −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
\(106\) −2.00000 −0.194257
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) −6.00000 −0.554700
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −6.00000 −0.543214
\(123\) −6.00000 −0.541002
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 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
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −16.0000 −1.38219
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 4.00000 0.340503
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 24.0000 2.00698
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) −7.00000 −0.577350
\(148\) 2.00000 0.164399
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −4.00000 −0.324443
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −16.0000 −1.27289
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 24.0000 1.87983 0.939913 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −4.00000 −0.300658
\(178\) −18.0000 −1.34916
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 1.00000 0.0721688
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −4.00000 −0.284268
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 4.00000 0.278019
\(208\) −6.00000 −0.416025
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −2.00000 −0.137361
\(213\) −12.0000 −0.822226
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 6.00000 0.406371
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 2.00000 0.134231
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) −4.00000 −0.264906
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 24.0000 1.52708
\(248\) −1.00000 −0.0635001
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) −4.00000 −0.247121
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) −16.0000 −0.977356
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 8.00000 0.479808
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 6.00000 0.351123
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) −4.00000 −0.232104
\(298\) 10.0000 0.579284
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 2.00000 0.114897
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) −6.00000 −0.339683
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −2.00000 −0.112154
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 24.0000 1.32924
\(327\) 6.00000 0.331801
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) 12.0000 0.658586
\(333\) 2.00000 0.109599
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 23.0000 1.25104
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 2.00000 0.107211
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) −4.00000 −0.213201
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −6.00000 −0.315353
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 4.00000 0.208514
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) −1.00000 −0.0518476
\(373\) 38.0000 1.96757 0.983783 0.179364i \(-0.0574041\pi\)
0.983783 + 0.179364i \(0.0574041\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 4.00000 0.204658
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) 4.00000 0.203331
\(388\) 14.0000 0.710742
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) −7.00000 −0.353553
\(393\) −4.00000 −0.201773
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) 14.0000 0.699127 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) −16.0000 −0.798007
\(403\) 6.00000 0.298881
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) −2.00000 −0.0990148
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) 8.00000 0.391762
\(418\) 16.0000 0.782586
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) 24.0000 1.15873
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000 0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) −16.0000 −0.765384
\(438\) 6.00000 0.286691
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 12.0000 0.570782
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 8.00000 0.378811
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 10.0000 0.470360
\(453\) 16.0000 0.751746
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 10.0000 0.467269
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) −6.00000 −0.277350
\(469\) 0 0
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) −4.00000 −0.184115
\(473\) −16.0000 −0.735681
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 40.0000 1.81257 0.906287 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) −6.00000 −0.271607
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) −6.00000 −0.270501
\(493\) −4.00000 −0.180151
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 20.0000 0.892644
\(503\) −32.0000 −1.42681 −0.713405 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) 23.0000 1.02147
\(508\) 8.00000 0.354943
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −22.0000 −0.970378
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 2.00000 0.0875376
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 2.00000 0.0871214
\(528\) −4.00000 −0.174078
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) −18.0000 −0.778936
\(535\) 0 0
\(536\) −16.0000 −0.691095
\(537\) −4.00000 −0.172613
\(538\) 18.0000 0.776035
\(539\) 28.0000 1.20605
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −8.00000 −0.343629
\(543\) −6.00000 −0.257485
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) −18.0000 −0.768922
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 4.00000 0.170251
\(553\) 0 0
\(554\) −6.00000 −0.254916
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) −1.00000 −0.0423334
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 10.0000 0.421825
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 24.0000 1.00349
\(573\) 4.00000 0.167102
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) −13.0000 −0.540729
\(579\) 22.0000 0.914289
\(580\) 0 0
\(581\) 0 0
\(582\) 14.0000 0.580319
\(583\) 8.00000 0.331326
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) −7.00000 −0.288675
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 2.00000 0.0821995
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 16.0000 0.654836
\(598\) −24.0000 −0.981433
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) −16.0000 −0.651570
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) −46.0000 −1.85189 −0.925945 0.377658i \(-0.876729\pi\)
−0.925945 + 0.377658i \(0.876729\pi\)
\(618\) −8.00000 −0.321807
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −28.0000 −1.12270
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) 16.0000 0.638978
\(628\) −18.0000 −0.718278
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −16.0000 −0.636446
\(633\) −4.00000 −0.158986
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) 42.0000 1.66410
\(638\) −8.00000 −0.316723
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) −4.00000 −0.157867
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 24.0000 0.939913
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −34.0000 −1.32245 −0.661223 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) 24.0000 0.932786
\(663\) 12.0000 0.466041
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 8.00000 0.309761
\(668\) −12.0000 −0.464294
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 10.0000 0.384048
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 4.00000 0.153168
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 4.00000 0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) 2.00000 0.0758098
\(697\) 12.0000 0.454532
\(698\) −2.00000 −0.0757011
\(699\) −22.0000 −0.832116
\(700\) 0 0
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) −6.00000 −0.226455
\(703\) −8.00000 −0.301726
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) −18.0000 −0.674579
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −20.0000 −0.746393
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −14.0000 −0.520666
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) −6.00000 −0.221766
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 64.0000 2.35747
\(738\) −6.00000 −0.220863
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 24.0000 0.881662
\(742\) 0 0
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) 38.0000 1.39128
\(747\) 12.0000 0.439057
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 20.0000 0.728841
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) −20.0000 −0.726433
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 8.00000 0.289809
\(763\) 0 0
\(764\) 4.00000 0.144715
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 22.0000 0.791797
\(773\) −50.0000 −1.79838 −0.899188 0.437564i \(-0.855842\pi\)
−0.899188 + 0.437564i \(0.855842\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) −8.00000 −0.286079
\(783\) 2.00000 0.0714742
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) 36.0000 1.27840
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) 14.0000 0.494357
\(803\) −24.0000 −0.846942
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) 6.00000 0.211341
\(807\) 18.0000 0.633630
\(808\) 2.00000 0.0703598
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) −16.0000 −0.559769
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) 0 0
\(821\) −22.0000 −0.767805 −0.383903 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) −18.0000 −0.627822
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) −52.0000 −1.80822 −0.904109 0.427303i \(-0.859464\pi\)
−0.904109 + 0.427303i \(0.859464\pi\)
\(828\) 4.00000 0.139010
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) −6.00000 −0.208013
\(833\) 14.0000 0.485071
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) −1.00000 −0.0345651
\(838\) −12.0000 −0.414533
\(839\) 28.0000 0.966667 0.483334 0.875436i \(-0.339426\pi\)
0.483334 + 0.875436i \(0.339426\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 14.0000 0.482472
\(843\) 10.0000 0.344418
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) −12.0000 −0.411113
\(853\) −58.0000 −1.98588 −0.992941 0.118609i \(-0.962157\pi\)
−0.992941 + 0.118609i \(0.962157\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) −54.0000 −1.84460 −0.922302 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 24.0000 0.819346
\(859\) 48.0000 1.63774 0.818869 0.573980i \(-0.194601\pi\)
0.818869 + 0.573980i \(0.194601\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) 0 0
\(871\) 96.0000 3.25284
\(872\) 6.00000 0.203186
\(873\) 14.0000 0.473828
\(874\) −16.0000 −0.541208
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) −32.0000 −1.07995
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) −7.00000 −0.235702
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) −18.0000 −0.600668
\(899\) −2.00000 −0.0667037
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 56.0000 1.85945 0.929725 0.368255i \(-0.120045\pi\)
0.929725 + 0.368255i \(0.120045\pi\)
\(908\) −20.0000 −0.663723
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) −4.00000 −0.132453
\(913\) −48.0000 −1.58857
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) −14.0000 −0.461065
\(923\) 72.0000 2.36991
\(924\) 0 0
\(925\) 0 0
\(926\) 24.0000 0.788689
\(927\) −8.00000 −0.262754
\(928\) 2.00000 0.0656532
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) −22.0000 −0.720634
\(933\) −28.0000 −0.916679
\(934\) 4.00000 0.130884
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −18.0000 −0.586472
\(943\) −24.0000 −0.781548
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 44.0000 1.42981 0.714904 0.699223i \(-0.246470\pi\)
0.714904 + 0.699223i \(0.246470\pi\)
\(948\) −16.0000 −0.519656
\(949\) −36.0000 −1.16861
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 0 0
\(957\) −8.00000 −0.258603
\(958\) 36.0000 1.16311
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −12.0000 −0.386896
\(963\) −4.00000 −0.128898
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 5.00000 0.160706
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) −22.0000 −0.703842 −0.351921 0.936030i \(-0.614471\pi\)
−0.351921 + 0.936030i \(0.614471\pi\)
\(978\) 24.0000 0.767435
\(979\) 72.0000 2.30113
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 20.0000 0.638226
\(983\) 4.00000 0.127580 0.0637901 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 24.0000 0.763542
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 24.0000 0.761617
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 16.0000 0.506471
\(999\) 2.00000 0.0632772
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 4650.2.a.bp.1.1 1
5.2 odd 4 4650.2.d.o.3349.2 2
5.3 odd 4 4650.2.d.o.3349.1 2
5.4 even 2 930.2.a.b.1.1 1
15.14 odd 2 2790.2.a.ba.1.1 1
20.19 odd 2 7440.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.b.1.1 1 5.4 even 2
2790.2.a.ba.1.1 1 15.14 odd 2
4650.2.a.bp.1.1 1 1.1 even 1 trivial
4650.2.d.o.3349.1 2 5.3 odd 4
4650.2.d.o.3349.2 2 5.2 odd 4
7440.2.a.q.1.1 1 20.19 odd 2