Properties

Label 1850.2.b.b.149.1
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.b.149.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.00000i q^{12} +2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -2.00000 q^{19} -4.00000 q^{21} +2.00000 q^{24} +2.00000 q^{26} -4.00000i q^{27} +2.00000i q^{28} -6.00000 q^{29} -10.0000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} -1.00000i q^{37} +2.00000i q^{38} +4.00000 q^{39} -6.00000 q^{41} +4.00000i q^{42} -4.00000i q^{43} +6.00000i q^{47} -2.00000i q^{48} +3.00000 q^{49} -12.0000 q^{51} -2.00000i q^{52} +6.00000i q^{53} -4.00000 q^{54} +2.00000 q^{56} +4.00000i q^{57} +6.00000i q^{58} +6.00000 q^{59} -10.0000 q^{61} +10.0000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -2.00000i q^{67} +6.00000i q^{68} -1.00000i q^{72} +2.00000i q^{73} -1.00000 q^{74} +2.00000 q^{76} -4.00000i q^{78} +10.0000 q^{79} -11.0000 q^{81} +6.00000i q^{82} -6.00000i q^{83} +4.00000 q^{84} -4.00000 q^{86} +12.0000i q^{87} +6.00000 q^{89} +4.00000 q^{91} +20.0000i q^{93} +6.00000 q^{94} -2.00000 q^{96} -2.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{6} - 2q^{9} - 4q^{14} + 2q^{16} - 4q^{19} - 8q^{21} + 4q^{24} + 4q^{26} - 12q^{29} - 20q^{31} - 12q^{34} + 2q^{36} + 8q^{39} - 12q^{41} + 6q^{49} - 24q^{51} - 8q^{54} + 4q^{56} + 12q^{59} - 20q^{61} - 2q^{64} - 2q^{74} + 4q^{76} + 20q^{79} - 22q^{81} + 8q^{84} - 8q^{86} + 12q^{89} + 8q^{91} + 12q^{94} - 4q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 2.00000i 0.577350i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) − 4.00000i − 0.769800i
\(28\) 2.00000i 0.377964i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 1.00000i − 0.164399i
\(38\) 2.00000i 0.324443i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000i 0.617213i
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) − 2.00000i − 0.288675i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) − 2.00000i − 0.277350i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 4.00000i 0.529813i
\(58\) 6.00000i 0.787839i
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 10.0000i 1.27000i
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.00000i − 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) − 4.00000i − 0.452911i
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 6.00000i 0.662589i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 12.0000i 1.28654i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 20.0000i 2.07390i
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 12.0000i 1.18818i
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 4.00000i 0.384900i
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) − 2.00000i − 0.188982i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) − 2.00000i − 0.184900i
\(118\) − 6.00000i − 0.552345i
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 10.0000i 0.905357i
\(123\) 12.0000i 1.08200i
\(124\) 10.0000 0.898027
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) − 2.00000i − 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) − 6.00000i − 0.494872i
\(148\) 1.00000i 0.0821995i
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) − 2.00000i − 0.162221i
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) − 10.0000i − 0.795557i
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 11.0000i 0.864242i
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) − 4.00000i − 0.308607i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 4.00000i 0.304997i
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 12.0000 0.909718
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) − 6.00000i − 0.449719i
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) − 4.00000i − 0.296500i
\(183\) 20.0000i 1.47844i
\(184\) 0 0
\(185\) 0 0
\(186\) 20.0000 1.46647
\(187\) 0 0
\(188\) − 6.00000i − 0.437595i
\(189\) −8.00000 −0.581914
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 2.00000i 0.144338i
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 0 0
\(199\) 22.0000 1.55954 0.779769 0.626067i \(-0.215336\pi\)
0.779769 + 0.626067i \(0.215336\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) − 18.0000i − 1.26648i
\(203\) 12.0000i 0.842235i
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 2.00000i 0.138675i
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) 0 0
\(214\) −6.00000 −0.410152
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) 20.0000i 1.35769i
\(218\) 14.0000i 0.948200i
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 2.00000i 0.134231i
\(223\) 14.0000i 0.937509i 0.883328 + 0.468755i \(0.155297\pi\)
−0.883328 + 0.468755i \(0.844703\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) − 24.0000i − 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) − 4.00000i − 0.264906i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 6.00000i − 0.393919i
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) − 20.0000i − 1.29914i
\(238\) 12.0000i 0.777844i
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 11.0000i 0.707107i
\(243\) 10.0000i 0.641500i
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) − 4.00000i − 0.254514i
\(248\) − 10.0000i − 0.635001i
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 0 0
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 8.00000i 0.498058i
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 6.00000i 0.370681i
\(263\) − 6.00000i − 0.369976i −0.982741 0.184988i \(-0.940775\pi\)
0.982741 0.184988i \(-0.0592246\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) − 12.0000i − 0.734388i
\(268\) 2.00000i 0.122169i
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) − 8.00000i − 0.484182i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) − 26.0000i − 1.56219i −0.624413 0.781094i \(-0.714662\pi\)
0.624413 0.781094i \(-0.285338\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) − 12.0000i − 0.714590i
\(283\) − 16.0000i − 0.951101i −0.879688 0.475551i \(-0.842249\pi\)
0.879688 0.475551i \(-0.157751\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000i 0.708338i
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) − 2.00000i − 0.117041i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) −6.00000 −0.349927
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 18.0000i 1.04271i
\(299\) 0 0
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) − 20.0000i − 1.15087i
\(303\) − 36.0000i − 2.06815i
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 26.0000i − 1.48390i −0.670456 0.741949i \(-0.733902\pi\)
0.670456 0.741949i \(-0.266098\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 4.00000i 0.226455i
\(313\) − 22.0000i − 1.24351i −0.783210 0.621757i \(-0.786419\pi\)
0.783210 0.621757i \(-0.213581\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) − 12.0000i − 0.672927i
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 12.0000i 0.667698i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 28.0000i 1.54840i
\(328\) − 6.00000i − 0.331295i
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 6.00000i 0.329293i
\(333\) 1.00000i 0.0547997i
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) − 2.00000i − 0.108947i −0.998515 0.0544735i \(-0.982652\pi\)
0.998515 0.0544735i \(-0.0173480\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) − 2.00000i − 0.108148i
\(343\) − 20.0000i − 1.07990i
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) − 12.0000i − 0.643268i
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 24.0000i 1.27021i
\(358\) 6.00000i 0.317110i
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 22.0000i 1.15629i
\(363\) 22.0000i 1.15470i
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 20.0000 1.04542
\(367\) − 2.00000i − 0.104399i −0.998637 0.0521996i \(-0.983377\pi\)
0.998637 0.0521996i \(-0.0166232\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) − 20.0000i − 1.03695i
\(373\) − 34.0000i − 1.76045i −0.474554 0.880227i \(-0.657390\pi\)
0.474554 0.880227i \(-0.342610\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) − 12.0000i − 0.618031i
\(378\) 8.00000i 0.411476i
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 6.00000i 0.306987i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 4.00000i 0.203331i
\(388\) 2.00000i 0.101535i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 3.00000i 0.151523i
\(393\) 12.0000i 0.605320i
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) − 14.0000i − 0.702640i −0.936255 0.351320i \(-0.885733\pi\)
0.936255 0.351320i \(-0.114267\pi\)
\(398\) − 22.0000i − 1.10276i
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 4.00000i 0.199502i
\(403\) − 20.0000i − 0.996271i
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) − 12.0000i − 0.594089i
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 4.00000i 0.197066i
\(413\) − 12.0000i − 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) − 8.00000i − 0.391762i
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) − 20.0000i − 0.973585i
\(423\) − 6.00000i − 0.291730i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 20.0000i 0.967868i
\(428\) 6.00000i 0.290021i
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) 2.00000i 0.0961139i 0.998845 + 0.0480569i \(0.0153029\pi\)
−0.998845 + 0.0480569i \(0.984697\pi\)
\(434\) 20.0000 0.960031
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) − 4.00000i − 0.191127i
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) − 12.0000i − 0.570782i
\(443\) 6.00000i 0.285069i 0.989790 + 0.142534i \(0.0455251\pi\)
−0.989790 + 0.142534i \(0.954475\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 14.0000 0.662919
\(447\) 36.0000i 1.70274i
\(448\) 2.00000i 0.0944911i
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 6.00000i − 0.282216i
\(453\) − 40.0000i − 1.87936i
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) − 26.0000i − 1.21623i −0.793849 0.608114i \(-0.791926\pi\)
0.793849 0.608114i \(-0.208074\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) −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
\(463\) − 40.0000i − 1.85896i −0.368875 0.929479i \(-0.620257\pi\)
0.368875 0.929479i \(-0.379743\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 20.0000 0.921551
\(472\) 6.00000i 0.276172i
\(473\) 0 0
\(474\) −20.0000 −0.918630
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) − 6.00000i − 0.274721i
\(478\) 6.00000i 0.274434i
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 22.0000i 1.00207i
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 10.0000 0.453609
\(487\) − 20.0000i − 0.906287i −0.891438 0.453143i \(-0.850303\pi\)
0.891438 0.453143i \(-0.149697\pi\)
\(488\) − 10.0000i − 0.452679i
\(489\) −32.0000 −1.44709
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) − 12.0000i − 0.541002i
\(493\) 36.0000i 1.62136i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 0 0
\(498\) 12.0000i 0.537733i
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) − 18.0000i − 0.803379i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) − 18.0000i − 0.799408i
\(508\) 2.00000i 0.0887357i
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) − 1.00000i − 0.0441942i
\(513\) 8.00000i 0.353209i
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 2.00000i 0.0878750i
\(519\) −36.0000 −1.58022
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) − 6.00000i − 0.262613i
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) 60.0000i 2.61364i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) − 4.00000i − 0.173422i
\(533\) − 12.0000i − 0.519778i
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) 12.0000i 0.517838i
\(538\) − 6.00000i − 0.258678i
\(539\) 0 0
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 16.0000i 0.687259i
\(543\) 44.0000i 1.88822i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) − 44.0000i − 1.88130i −0.339372 0.940652i \(-0.610215\pi\)
0.339372 0.940652i \(-0.389785\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) − 20.0000i − 0.850487i
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) − 10.0000i − 0.423334i
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) − 18.0000i − 0.759284i
\(563\) 36.0000i 1.51722i 0.651546 + 0.758610i \(0.274121\pi\)
−0.651546 + 0.758610i \(0.725879\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) 22.0000i 0.923913i
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 12.0000i 0.501307i
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 38.0000i − 1.58196i −0.611842 0.790980i \(-0.709571\pi\)
0.611842 0.790980i \(-0.290429\pi\)
\(578\) 19.0000i 0.790296i
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 4.00000i 0.165805i
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) 6.00000i 0.247436i
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) 36.0000 1.48084
\(592\) − 1.00000i − 0.0410997i
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) − 44.0000i − 1.80080i
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 2.00000i 0.0814463i
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) −36.0000 −1.46240
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) 2.00000i 0.0811107i
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) − 6.00000i − 0.242536i
\(613\) − 34.0000i − 1.37325i −0.727013 0.686624i \(-0.759092\pi\)
0.727013 0.686624i \(-0.240908\pi\)
\(614\) −26.0000 −1.04927
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) 8.00000i 0.321807i
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18.0000i 0.721734i
\(623\) − 12.0000i − 0.480770i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) − 10.0000i − 0.399043i
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 10.0000i 0.397779i
\(633\) − 40.0000i − 1.58986i
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −12.0000 −0.475831
\(637\) 6.00000i 0.237729i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 32.0000i 1.26196i 0.775800 + 0.630978i \(0.217346\pi\)
−0.775800 + 0.630978i \(0.782654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) − 48.0000i − 1.88707i −0.331266 0.943537i \(-0.607476\pi\)
0.331266 0.943537i \(-0.392524\pi\)
\(648\) − 11.0000i − 0.432121i
\(649\) 0 0
\(650\) 0 0
\(651\) 40.0000 1.56772
\(652\) 16.0000i 0.626608i
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 28.0000 1.09489
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) − 2.00000i − 0.0780274i
\(658\) − 12.0000i − 0.467809i
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 10.0000i 0.388661i
\(663\) − 24.0000i − 0.932083i
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) 12.0000i 0.464294i
\(669\) 28.0000 1.08254
\(670\) 0 0
\(671\) 0 0
\(672\) 4.00000i 0.154303i
\(673\) 26.0000i 1.00223i 0.865382 + 0.501113i \(0.167076\pi\)
−0.865382 + 0.501113i \(0.832924\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) − 12.0000i − 0.460857i
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) −48.0000 −1.83936
\(682\) 0 0
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) − 20.0000i − 0.763048i
\(688\) − 4.00000i − 0.152499i
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 18.0000i 0.684257i
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −12.0000 −0.454859
\(697\) 36.0000i 1.36360i
\(698\) 26.0000i 0.984115i
\(699\) 36.0000 1.36165
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) − 8.00000i − 0.301941i
\(703\) 2.00000i 0.0754314i
\(704\) 0 0
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) − 36.0000i − 1.35392i
\(708\) 12.0000i 0.450988i
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 6.00000i 0.224860i
\(713\) 0 0
\(714\) 24.0000 0.898177
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) 12.0000i 0.448148i
\(718\) 36.0000i 1.34351i
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 15.0000i 0.558242i
\(723\) 44.0000i 1.63638i
\(724\) 22.0000 0.817624
\(725\) 0 0
\(726\) 22.0000 0.816497
\(727\) − 32.0000i − 1.18681i −0.804902 0.593407i \(-0.797782\pi\)
0.804902 0.593407i \(-0.202218\pi\)
\(728\) 4.00000i 0.148250i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) − 20.0000i − 0.739221i
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) − 6.00000i − 0.220863i
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) − 12.0000i − 0.440534i
\(743\) 18.0000i 0.660356i 0.943919 + 0.330178i \(0.107109\pi\)
−0.943919 + 0.330178i \(0.892891\pi\)
\(744\) −20.0000 −0.733236
\(745\) 0 0
\(746\) −34.0000 −1.24483
\(747\) 6.00000i 0.219529i
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 6.00000i 0.218797i
\(753\) − 36.0000i − 1.31191i
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 8.00000 0.290957
\(757\) 10.0000i 0.363456i 0.983349 + 0.181728i \(0.0581691\pi\)
−0.983349 + 0.181728i \(0.941831\pi\)
\(758\) − 16.0000i − 0.581146i
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 4.00000i 0.144905i
\(763\) 28.0000i 1.01367i
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000i 0.433295i
\(768\) − 2.00000i − 0.0721688i
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) − 2.00000i − 0.0719816i
\(773\) − 18.0000i − 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 4.00000i 0.143499i
\(778\) 6.00000i 0.215110i
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 24.0000i 0.857690i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 10.0000i 0.356462i 0.983989 + 0.178231i \(0.0570374\pi\)
−0.983989 + 0.178231i \(0.942963\pi\)
\(788\) − 18.0000i − 0.641223i
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) − 20.0000i − 0.710221i
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −22.0000 −0.779769
\(797\) 30.0000i 1.06265i 0.847167 + 0.531327i \(0.178307\pi\)
−0.847167 + 0.531327i \(0.821693\pi\)
\(798\) − 8.00000i − 0.283197i
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 6.00000i 0.211867i
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −20.0000 −0.704470
\(807\) − 12.0000i − 0.422420i
\(808\) 18.0000i 0.633238i
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) − 12.0000i − 0.421117i
\(813\) 32.0000i 1.12229i
\(814\) 0 0
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) 8.00000i 0.279885i
\(818\) 2.00000i 0.0699284i
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) − 12.0000i − 0.418548i
\(823\) − 34.0000i − 1.18517i −0.805510 0.592583i \(-0.798108\pi\)
0.805510 0.592583i \(-0.201892\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) − 24.0000i − 0.834562i −0.908778 0.417281i \(-0.862983\pi\)
0.908778 0.417281i \(-0.137017\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) −52.0000 −1.80386
\(832\) − 2.00000i − 0.0693375i
\(833\) − 18.0000i − 0.623663i
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 0 0
\(837\) 40.0000i 1.38260i
\(838\) 24.0000i 0.829066i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 10.0000i 0.344623i
\(843\) − 36.0000i − 1.23991i
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) 22.0000i 0.755929i
\(848\) 6.00000i 0.206041i
\(849\) −32.0000 −1.09824
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 10.0000i − 0.342393i −0.985237 0.171197i \(-0.945237\pi\)
0.985237 0.171197i \(-0.0547634\pi\)
\(854\) 20.0000 0.684386
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) − 42.0000i − 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) 0 0
\(859\) −50.0000 −1.70598 −0.852989 0.521929i \(-0.825213\pi\)
−0.852989 + 0.521929i \(0.825213\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) − 30.0000i − 1.02180i
\(863\) 54.0000i 1.83818i 0.394046 + 0.919091i \(0.371075\pi\)
−0.394046 + 0.919091i \(0.628925\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 38.0000i 1.29055i
\(868\) − 20.0000i − 0.678844i
\(869\) 0 0
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) − 14.0000i − 0.474100i
\(873\) 2.00000i 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 10.0000i 0.337676i 0.985644 + 0.168838i \(0.0540015\pi\)
−0.985644 + 0.168838i \(0.945999\pi\)
\(878\) − 22.0000i − 0.742464i
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 56.0000i 1.88455i 0.334840 + 0.942275i \(0.391318\pi\)
−0.334840 + 0.942275i \(0.608682\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) 42.0000i 1.41022i 0.709097 + 0.705111i \(0.249103\pi\)
−0.709097 + 0.705111i \(0.750897\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) 0 0
\(892\) − 14.0000i − 0.468755i
\(893\) − 12.0000i − 0.401565i
\(894\) 36.0000 1.20402
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) − 30.0000i − 1.00111i
\(899\) 60.0000 2.00111
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 16.0000i 0.532447i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −40.0000 −1.32891
\(907\) − 8.00000i − 0.265636i −0.991140 0.132818i \(-0.957597\pi\)
0.991140 0.132818i \(-0.0424025\pi\)
\(908\) 24.0000i 0.796468i
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 4.00000i 0.132453i
\(913\) 0 0
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 12.0000i 0.396275i
\(918\) 24.0000i 0.792118i
\(919\) 46.0000 1.51740 0.758700 0.651440i \(-0.225835\pi\)
0.758700 + 0.651440i \(0.225835\pi\)
\(920\) 0 0
\(921\) −52.0000 −1.71346
\(922\) − 6.00000i − 0.197599i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −40.0000 −1.31448
\(927\) 4.00000i 0.131377i
\(928\) 6.00000i 0.196960i
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) − 18.0000i − 0.589610i
\(933\) 36.0000i 1.17859i
\(934\) 0 0
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) − 26.0000i − 0.849383i −0.905338 0.424691i \(-0.860383\pi\)
0.905338 0.424691i \(-0.139617\pi\)
\(938\) 4.00000i 0.130605i
\(939\) −44.0000 −1.43589
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) − 20.0000i − 0.651635i
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) − 48.0000i − 1.55979i −0.625910 0.779895i \(-0.715272\pi\)
0.625910 0.779895i \(-0.284728\pi\)
\(948\) 20.0000i 0.649570i
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 36.0000 1.16738
\(952\) − 12.0000i − 0.388922i
\(953\) 42.0000i 1.36051i 0.732974 + 0.680257i \(0.238132\pi\)
−0.732974 + 0.680257i \(0.761868\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) − 18.0000i − 0.581554i
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) − 2.00000i − 0.0644826i
\(963\) 6.00000i 0.193347i
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) 4.00000i 0.128631i 0.997930 + 0.0643157i \(0.0204865\pi\)
−0.997930 + 0.0643157i \(0.979514\pi\)
\(968\) − 11.0000i − 0.353553i
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) − 10.0000i − 0.320750i
\(973\) − 8.00000i − 0.256468i
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) 32.0000i 1.02325i
\(979\) 0 0
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) − 36.0000i − 1.14881i
\(983\) 18.0000i 0.574111i 0.957914 + 0.287055i \(0.0926764\pi\)
−0.957914 + 0.287055i \(0.907324\pi\)
\(984\) −12.0000 −0.382546
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) − 24.0000i − 0.763928i
\(988\) 4.00000i 0.127257i
\(989\) 0 0
\(990\) 0 0
\(991\) −34.0000 −1.08005 −0.540023 0.841650i \(-0.681584\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(992\) 10.0000i 0.317500i
\(993\) 20.0000i 0.634681i
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 10.0000i 0.316703i 0.987383 + 0.158352i \(0.0506179\pi\)
−0.987383 + 0.158352i \(0.949382\pi\)
\(998\) 14.0000i 0.443162i
\(999\) −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 1850.2.b.b.149.1 2
5.2 odd 4 370.2.a.d.1.1 1
5.3 odd 4 1850.2.a.f.1.1 1
5.4 even 2 inner 1850.2.b.b.149.2 2
15.2 even 4 3330.2.a.d.1.1 1
20.7 even 4 2960.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.d.1.1 1 5.2 odd 4
1850.2.a.f.1.1 1 5.3 odd 4
1850.2.b.b.149.1 2 1.1 even 1 trivial
1850.2.b.b.149.2 2 5.4 even 2 inner
2960.2.a.m.1.1 1 20.7 even 4
3330.2.a.d.1.1 1 15.2 even 4