Properties

Label 370.2.a.d.1.1
Level $370$
Weight $2$
Character 370.1
Self dual yes
Analytic conductor $2.954$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.95446487479\)
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 \(\chi\) \(=\) 370.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{12} +2.00000 q^{13} +2.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} -2.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} -4.00000 q^{43} +1.00000 q^{45} -6.00000 q^{47} -2.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -12.0000 q^{51} +2.00000 q^{52} +6.00000 q^{53} +4.00000 q^{54} +2.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} -6.00000 q^{59} -2.00000 q^{60} -10.0000 q^{61} -10.0000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{67} +6.00000 q^{68} +2.00000 q^{70} +1.00000 q^{72} +2.00000 q^{73} +1.00000 q^{74} -2.00000 q^{75} +2.00000 q^{76} -4.00000 q^{78} -10.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -6.00000 q^{82} -6.00000 q^{83} -4.00000 q^{84} +6.00000 q^{85} -4.00000 q^{86} -12.0000 q^{87} -6.00000 q^{89} +1.00000 q^{90} +4.00000 q^{91} +20.0000 q^{93} -6.00000 q^{94} +2.00000 q^{95} -2.00000 q^{96} +2.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
\(3\) −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
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −2.00000 −0.577350
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 2.00000 0.534522
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 1.00000 0.223607
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −2.00000 −0.408248
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 4.00000 0.769800
\(28\) 2.00000 0.377964
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −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
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) 2.00000 0.324443
\(39\) −4.00000 −0.640513
\(40\) 1.00000 0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −4.00000 −0.617213
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −2.00000 −0.288675
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) −12.0000 −1.68034
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) −4.00000 −0.529813
\(58\) 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
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −10.0000 −1.27000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 1.00000 0.116248
\(75\) −2.00000 −0.230940
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) −6.00000 −0.662589
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −4.00000 −0.436436
\(85\) 6.00000 0.650791
\(86\) −4.00000 −0.431331
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 1.00000 0.105409
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 20.0000 2.07390
\(94\) −6.00000 −0.618853
\(95\) 2.00000 0.205196
\(96\) −2.00000 −0.204124
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) −12.0000 −1.18818
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 2.00000 0.196116
\(105\) −4.00000 −0.390360
\(106\) 6.00000 0.582772
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 4.00000 0.384900
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 2.00000 0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) −6.00000 −0.552345
\(119\) 12.0000 1.10004
\(120\) −2.00000 −0.182574
\(121\) −11.0000 −1.00000
\(122\) −10.0000 −0.905357
\(123\) 12.0000 1.08200
\(124\) −10.0000 −0.898027
\(125\) 1.00000 0.0894427
\(126\) 2.00000 0.178174
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 2.00000 0.175412
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 2.00000 0.172774
\(135\) 4.00000 0.344265
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 2.00000 0.169031
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 2.00000 0.165521
\(147\) 6.00000 0.494872
\(148\) 1.00000 0.0821995
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −2.00000 −0.163299
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 2.00000 0.162221
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −10.0000 −0.803219
\(156\) −4.00000 −0.320256
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −10.0000 −0.795557
\(159\) −12.0000 −0.951662
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −11.0000 −0.864242
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 2.00000 0.152944
\(172\) −4.00000 −0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −12.0000 −0.909718
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) −6.00000 −0.449719
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 1.00000 0.0745356
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 4.00000 0.296500
\(183\) 20.0000 1.47844
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 20.0000 1.46647
\(187\) 0 0
\(188\) −6.00000 −0.437595
\(189\) 8.00000 0.581914
\(190\) 2.00000 0.145095
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) −2.00000 −0.144338
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 2.00000 0.143592
\(195\) −4.00000 −0.286446
\(196\) −3.00000 −0.214286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −22.0000 −1.55954 −0.779769 0.626067i \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) 18.0000 1.26648
\(203\) 12.0000 0.842235
\(204\) −12.0000 −0.840168
\(205\) −6.00000 −0.419058
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) −4.00000 −0.276026
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) −4.00000 −0.272798
\(216\) 4.00000 0.272166
\(217\) −20.0000 −1.35769
\(218\) 14.0000 0.948200
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) −2.00000 −0.134231
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 2.00000 0.133631
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −4.00000 −0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 2.00000 0.130744
\(235\) −6.00000 −0.391397
\(236\) −6.00000 −0.390567
\(237\) 20.0000 1.29914
\(238\) 12.0000 0.777844
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) −2.00000 −0.129099
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −11.0000 −0.707107
\(243\) 10.0000 0.641500
\(244\) −10.0000 −0.640184
\(245\) −3.00000 −0.191663
\(246\) 12.0000 0.765092
\(247\) 4.00000 0.254514
\(248\) −10.0000 −0.635001
\(249\) 12.0000 0.760469
\(250\) 1.00000 0.0632456
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) 2.00000 0.125491
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 8.00000 0.498058
\(259\) 2.00000 0.124274
\(260\) 2.00000 0.124035
\(261\) 6.00000 0.371391
\(262\) −6.00000 −0.370681
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 4.00000 0.245256
\(267\) 12.0000 0.734388
\(268\) 2.00000 0.122169
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 4.00000 0.243432
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 6.00000 0.363803
\(273\) −8.00000 −0.484182
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −4.00000 −0.239904
\(279\) −10.0000 −0.598684
\(280\) 2.00000 0.119523
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 12.0000 0.714590
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) −4.00000 −0.234484
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 6.00000 0.349927
\(295\) −6.00000 −0.349334
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) −2.00000 −0.115470
\(301\) −8.00000 −0.461112
\(302\) 20.0000 1.15087
\(303\) −36.0000 −2.06815
\(304\) 2.00000 0.114708
\(305\) −10.0000 −0.572598
\(306\) 6.00000 0.342997
\(307\) 26.0000 1.48390 0.741949 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) −10.0000 −0.567962
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −4.00000 −0.226455
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −10.0000 −0.564333
\(315\) 2.00000 0.112687
\(316\) −10.0000 −0.562544
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −12.0000 −0.672927
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) −11.0000 −0.611111
\(325\) 2.00000 0.110940
\(326\) −16.0000 −0.886158
\(327\) −28.0000 −1.54840
\(328\) −6.00000 −0.331295
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) −6.00000 −0.329293
\(333\) 1.00000 0.0547997
\(334\) 12.0000 0.656611
\(335\) 2.00000 0.109272
\(336\) −4.00000 −0.218218
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −9.00000 −0.489535
\(339\) −12.0000 −0.651751
\(340\) 6.00000 0.325396
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) −20.0000 −1.07990
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −12.0000 −0.643268
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 2.00000 0.106904
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −24.0000 −1.27021
\(358\) 6.00000 0.317110
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 1.00000 0.0527046
\(361\) −15.0000 −0.789474
\(362\) −22.0000 −1.15629
\(363\) 22.0000 1.15470
\(364\) 4.00000 0.209657
\(365\) 2.00000 0.104685
\(366\) 20.0000 1.04542
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 1.00000 0.0519875
\(371\) 12.0000 0.623009
\(372\) 20.0000 1.03695
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) −6.00000 −0.309426
\(377\) 12.0000 0.618031
\(378\) 8.00000 0.411476
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 2.00000 0.102598
\(381\) −4.00000 −0.204926
\(382\) −6.00000 −0.306987
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) −4.00000 −0.203331
\(388\) 2.00000 0.101535
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −4.00000 −0.202548
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) 12.0000 0.605320
\(394\) −18.0000 −0.906827
\(395\) −10.0000 −0.503155
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −22.0000 −1.10276
\(399\) −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 \(0.547867\pi\)
\(402\) −4.00000 −0.199502
\(403\) −20.0000 −0.996271
\(404\) 18.0000 0.895533
\(405\) −11.0000 −0.546594
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) −12.0000 −0.594089
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) −6.00000 −0.296319
\(411\) 12.0000 0.591916
\(412\) −4.00000 −0.197066
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 2.00000 0.0980581
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) −4.00000 −0.195180
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 20.0000 0.973585
\(423\) −6.00000 −0.291730
\(424\) 6.00000 0.291386
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) −20.0000 −0.967868
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 4.00000 0.192450
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −20.0000 −0.960031
\(435\) −12.0000 −0.575356
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) −4.00000 −0.191127
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 12.0000 0.570782
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −6.00000 −0.284427
\(446\) 14.0000 0.662919
\(447\) −36.0000 −1.70274
\(448\) 2.00000 0.0944911
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) −40.0000 −1.87936
\(454\) 24.0000 1.12638
\(455\) 4.00000 0.187523
\(456\) −4.00000 −0.187317
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −10.0000 −0.467269
\(459\) 24.0000 1.12022
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −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
\(466\) 18.0000 0.833834
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 2.00000 0.0924500
\(469\) 4.00000 0.184703
\(470\) −6.00000 −0.276759
\(471\) 20.0000 0.921551
\(472\) −6.00000 −0.276172
\(473\) 0 0
\(474\) 20.0000 0.918630
\(475\) 2.00000 0.0917663
\(476\) 12.0000 0.550019
\(477\) 6.00000 0.274721
\(478\) 6.00000 0.274434
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 2.00000 0.0911922
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 2.00000 0.0908153
\(486\) 10.0000 0.453609
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) −10.0000 −0.452679
\(489\) 32.0000 1.44709
\(490\) −3.00000 −0.135526
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 12.0000 0.541002
\(493\) 36.0000 1.62136
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 1.00000 0.0447214
\(501\) −24.0000 −1.07224
\(502\) 18.0000 0.803379
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 2.00000 0.0890871
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 18.0000 0.799408
\(508\) 2.00000 0.0887357
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) −12.0000 −0.531369
\(511\) 4.00000 0.176950
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) 6.00000 0.264649
\(515\) −4.00000 −0.176261
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) 36.0000 1.58022
\(520\) 2.00000 0.0877058
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 6.00000 0.262613
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −6.00000 −0.262111
\(525\) −4.00000 −0.174574
\(526\) −6.00000 −0.261612
\(527\) −60.0000 −2.61364
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 6.00000 0.260623
\(531\) −6.00000 −0.260378
\(532\) 4.00000 0.173422
\(533\) −12.0000 −0.519778
\(534\) 12.0000 0.519291
\(535\) 6.00000 0.259403
\(536\) 2.00000 0.0863868
\(537\) −12.0000 −0.517838
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) 4.00000 0.172133
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) −16.0000 −0.687259
\(543\) 44.0000 1.88822
\(544\) 6.00000 0.257248
\(545\) 14.0000 0.599694
\(546\) −8.00000 −0.342368
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) −6.00000 −0.256307
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) −20.0000 −0.850487
\(554\) 26.0000 1.10463
\(555\) −2.00000 −0.0848953
\(556\) −4.00000 −0.169638
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) −10.0000 −0.423334
\(559\) −8.00000 −0.338364
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 12.0000 0.505291
\(565\) 6.00000 0.252422
\(566\) −16.0000 −0.672530
\(567\) −22.0000 −0.923913
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −4.00000 −0.167542
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) −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 \(0.209571\pi\)
\(578\) 19.0000 0.790296
\(579\) −4.00000 −0.166234
\(580\) 6.00000 0.249136
\(581\) −12.0000 −0.497844
\(582\) −4.00000 −0.165805
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) 2.00000 0.0826898
\(586\) 6.00000 0.247858
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 6.00000 0.247436
\(589\) −20.0000 −0.824086
\(590\) −6.00000 −0.247016
\(591\) 36.0000 1.48084
\(592\) 1.00000 0.0410997
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) 18.0000 0.737309
\(597\) 44.0000 1.80080
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) −2.00000 −0.0816497
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) −8.00000 −0.326056
\(603\) 2.00000 0.0814463
\(604\) 20.0000 0.813788
\(605\) −11.0000 −0.447214
\(606\) −36.0000 −1.46240
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 2.00000 0.0811107
\(609\) −24.0000 −0.972529
\(610\) −10.0000 −0.404888
\(611\) −12.0000 −0.485468
\(612\) 6.00000 0.242536
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 26.0000 1.04927
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 8.00000 0.321807
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −10.0000 −0.401610
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) −12.0000 −0.480770
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 6.00000 0.239236
\(630\) 2.00000 0.0796819
\(631\) 14.0000 0.557331 0.278666 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) −10.0000 −0.397779
\(633\) −40.0000 −1.58986
\(634\) −18.0000 −0.714871
\(635\) 2.00000 0.0793676
\(636\) −12.0000 −0.475831
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) −12.0000 −0.473602
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 12.0000 0.472134
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 40.0000 1.56772
\(652\) −16.0000 −0.626608
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −28.0000 −1.09489
\(655\) −6.00000 −0.234439
\(656\) −6.00000 −0.234261
\(657\) 2.00000 0.0780274
\(658\) −12.0000 −0.467809
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −10.0000 −0.388661
\(663\) −24.0000 −0.932083
\(664\) −6.00000 −0.232845
\(665\) 4.00000 0.155113
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) −28.0000 −1.08254
\(670\) 2.00000 0.0772667
\(671\) 0 0
\(672\) −4.00000 −0.154303
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 2.00000 0.0770371
\(675\) 4.00000 0.153960
\(676\) −9.00000 −0.346154
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) −12.0000 −0.460857
\(679\) 4.00000 0.153506
\(680\) 6.00000 0.230089
\(681\) −48.0000 −1.83936
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 2.00000 0.0764719
\(685\) −6.00000 −0.229248
\(686\) −20.0000 −0.763604
\(687\) 20.0000 0.763048
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −4.00000 −0.151729
\(696\) −12.0000 −0.454859
\(697\) −36.0000 −1.36360
\(698\) 26.0000 0.984115
\(699\) −36.0000 −1.36165
\(700\) 2.00000 0.0755929
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 8.00000 0.301941
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) −30.0000 −1.12906
\(707\) 36.0000 1.35392
\(708\) 12.0000 0.450988
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) −24.0000 −0.898177
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) −12.0000 −0.448148
\(718\) 36.0000 1.34351
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 1.00000 0.0372678
\(721\) −8.00000 −0.297936
\(722\) −15.0000 −0.558242
\(723\) 44.0000 1.63638
\(724\) −22.0000 −0.817624
\(725\) 6.00000 0.222834
\(726\) 22.0000 0.816497
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 4.00000 0.148250
\(729\) 13.0000 0.481481
\(730\) 2.00000 0.0740233
\(731\) −24.0000 −0.887672
\(732\) 20.0000 0.739221
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 2.00000 0.0738213
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 1.00000 0.0367607
\(741\) −8.00000 −0.293887
\(742\) 12.0000 0.440534
\(743\) 18.0000 0.660356 0.330178 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 20.0000 0.733236
\(745\) 18.0000 0.659469
\(746\) −34.0000 −1.24483
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) −2.00000 −0.0730297
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −6.00000 −0.218797
\(753\) −36.0000 −1.31191
\(754\) 12.0000 0.437014
\(755\) 20.0000 0.727875
\(756\) 8.00000 0.290957
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −4.00000 −0.144905
\(763\) 28.0000 1.01367
\(764\) −6.00000 −0.217072
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) −2.00000 −0.0721688
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 2.00000 0.0719816
\(773\) −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
\(776\) 2.00000 0.0717958
\(777\) −4.00000 −0.143499
\(778\) 6.00000 0.215110
\(779\) −12.0000 −0.429945
\(780\) −4.00000 −0.143223
\(781\) 0 0
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) −3.00000 −0.107143
\(785\) −10.0000 −0.356915
\(786\) 12.0000 0.428026
\(787\) −10.0000 −0.356462 −0.178231 0.983989i \(-0.557037\pi\)
−0.178231 + 0.983989i \(0.557037\pi\)
\(788\) −18.0000 −0.641223
\(789\) 12.0000 0.427211
\(790\) −10.0000 −0.355784
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 14.0000 0.496841
\(795\) −12.0000 −0.425596
\(796\) −22.0000 −0.779769
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) −8.00000 −0.283197
\(799\) −36.0000 −1.27359
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) −6.00000 −0.211867
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −20.0000 −0.704470
\(807\) 12.0000 0.422420
\(808\) 18.0000 0.633238
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) −11.0000 −0.386501
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 12.0000 0.421117
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) −16.0000 −0.560456
\(816\) −12.0000 −0.420084
\(817\) −8.00000 −0.279885
\(818\) 2.00000 0.0699284
\(819\) 4.00000 0.139771
\(820\) −6.00000 −0.209529
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 12.0000 0.418548
\(823\) −34.0000 −1.18517 −0.592583 0.805510i \(-0.701892\pi\)
−0.592583 + 0.805510i \(0.701892\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 24.0000 0.834562 0.417281 0.908778i \(-0.362983\pi\)
0.417281 + 0.908778i \(0.362983\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −6.00000 −0.208263
\(831\) −52.0000 −1.80386
\(832\) 2.00000 0.0693375
\(833\) −18.0000 −0.623663
\(834\) 8.00000 0.277017
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) −40.0000 −1.38260
\(838\) 24.0000 0.829066
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −4.00000 −0.138013
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) −36.0000 −1.23991
\(844\) 20.0000 0.688428
\(845\) −9.00000 −0.309609
\(846\) −6.00000 −0.206284
\(847\) −22.0000 −0.755929
\(848\) 6.00000 0.206041
\(849\) 32.0000 1.09824
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) −20.0000 −0.684386
\(855\) 2.00000 0.0683986
\(856\) 6.00000 0.205076
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) −4.00000 −0.136399
\(861\) 24.0000 0.817918
\(862\) 30.0000 1.02180
\(863\) 54.0000 1.83818 0.919091 0.394046i \(-0.128925\pi\)
0.919091 + 0.394046i \(0.128925\pi\)
\(864\) 4.00000 0.136083
\(865\) −18.0000 −0.612018
\(866\) 2.00000 0.0679628
\(867\) −38.0000 −1.29055
\(868\) −20.0000 −0.678844
\(869\) 0 0
\(870\) −12.0000 −0.406838
\(871\) 4.00000 0.135535
\(872\) 14.0000 0.474100
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) −4.00000 −0.135147
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) −22.0000 −0.742464
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) −3.00000 −0.101015
\(883\) 56.0000 1.88455 0.942275 0.334840i \(-0.108682\pi\)
0.942275 + 0.334840i \(0.108682\pi\)
\(884\) 12.0000 0.403604
\(885\) 12.0000 0.403376
\(886\) 6.00000 0.201574
\(887\) −42.0000 −1.41022 −0.705111 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 4.00000 0.134156
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) 14.0000 0.468755
\(893\) −12.0000 −0.401565
\(894\) −36.0000 −1.20402
\(895\) 6.00000 0.200558
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −60.0000 −2.00111
\(900\) 1.00000 0.0333333
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) 6.00000 0.199557
\(905\) −22.0000 −0.731305
\(906\) −40.0000 −1.32891
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 24.0000 0.796468
\(909\) 18.0000 0.597022
\(910\) 4.00000 0.132599
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) −4.00000 −0.132453
\(913\) 0 0
\(914\) 26.0000 0.860004
\(915\) 20.0000 0.661180
\(916\) −10.0000 −0.330409
\(917\) −12.0000 −0.396275
\(918\) 24.0000 0.792118
\(919\) −46.0000 −1.51740 −0.758700 0.651440i \(-0.774165\pi\)
−0.758700 + 0.651440i \(0.774165\pi\)
\(920\) 0 0
\(921\) −52.0000 −1.71346
\(922\) 6.00000 0.197599
\(923\) 0 0
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) −40.0000 −1.31448
\(927\) −4.00000 −0.131377
\(928\) 6.00000 0.196960
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 20.0000 0.655826
\(931\) −6.00000 −0.196642
\(932\) 18.0000 0.589610
\(933\) 36.0000 1.17859
\(934\) 0 0
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 4.00000 0.130605
\(939\) 44.0000 1.43589
\(940\) −6.00000 −0.195698
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 20.0000 0.651635
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 8.00000 0.260240
\(946\) 0 0
\(947\) 48.0000 1.55979 0.779895 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) 20.0000 0.649570
\(949\) 4.00000 0.129845
\(950\) 2.00000 0.0648886
\(951\) 36.0000 1.16738
\(952\) 12.0000 0.388922
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 6.00000 0.194257
\(955\) −6.00000 −0.194155
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) −18.0000 −0.581554
\(959\) −12.0000 −0.387500
\(960\) −2.00000 −0.0645497
\(961\) 69.0000 2.22581
\(962\) 2.00000 0.0644826
\(963\) 6.00000 0.193347
\(964\) −22.0000 −0.708572
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) −11.0000 −0.353553
\(969\) −24.0000 −0.770991
\(970\) 2.00000 0.0642161
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 10.0000 0.320750
\(973\) −8.00000 −0.256468
\(974\) 20.0000 0.640841
\(975\) −4.00000 −0.128103
\(976\) −10.0000 −0.320092
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 32.0000 1.02325
\(979\) 0 0
\(980\) −3.00000 −0.0958315
\(981\) 14.0000 0.446986
\(982\) 36.0000 1.14881
\(983\) 18.0000 0.574111 0.287055 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) 12.0000 0.382546
\(985\) −18.0000 −0.573528
\(986\) 36.0000 1.14647
\(987\) 24.0000 0.763928
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) −34.0000 −1.08005 −0.540023 0.841650i \(-0.681584\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(992\) −10.0000 −0.317500
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) −22.0000 −0.697447
\(996\) 12.0000 0.380235
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 14.0000 0.443162
\(999\) 4.00000 0.126554
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 370.2.a.d.1.1 1
3.2 odd 2 3330.2.a.d.1.1 1
4.3 odd 2 2960.2.a.m.1.1 1
5.2 odd 4 1850.2.b.b.149.2 2
5.3 odd 4 1850.2.b.b.149.1 2
5.4 even 2 1850.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.d.1.1 1 1.1 even 1 trivial
1850.2.a.f.1.1 1 5.4 even 2
1850.2.b.b.149.1 2 5.3 odd 4
1850.2.b.b.149.2 2 5.2 odd 4
2960.2.a.m.1.1 1 4.3 odd 2
3330.2.a.d.1.1 1 3.2 odd 2