Properties

Label 1850.2.b.a.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: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.a.149.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} +1.00000i q^{8} -6.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} +1.00000i q^{8} -6.00000 q^{9} -1.00000 q^{11} +3.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} +7.00000i q^{17} +6.00000i q^{18} -5.00000 q^{19} +1.00000i q^{22} +6.00000i q^{23} +3.00000 q^{24} -2.00000 q^{26} +9.00000i q^{27} -4.00000 q^{31} -1.00000i q^{32} +3.00000i q^{33} +7.00000 q^{34} +6.00000 q^{36} -1.00000i q^{37} +5.00000i q^{38} -6.00000 q^{39} -3.00000 q^{41} -4.00000i q^{43} +1.00000 q^{44} +6.00000 q^{46} +4.00000i q^{47} -3.00000i q^{48} +7.00000 q^{49} +21.0000 q^{51} +2.00000i q^{52} +2.00000i q^{53} +9.00000 q^{54} +15.0000i q^{57} -4.00000 q^{59} -8.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +3.00000 q^{66} +13.0000i q^{67} -7.00000i q^{68} +18.0000 q^{69} -6.00000 q^{71} -6.00000i q^{72} -7.00000i q^{73} -1.00000 q^{74} +5.00000 q^{76} +6.00000i q^{78} -14.0000 q^{79} +9.00000 q^{81} +3.00000i q^{82} +3.00000i q^{83} -4.00000 q^{86} -1.00000i q^{88} +7.00000 q^{89} -6.00000i q^{92} +12.0000i q^{93} +4.00000 q^{94} -3.00000 q^{96} +18.0000i q^{97} -7.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9} - 2 q^{11} + 2 q^{16} - 10 q^{19} + 6 q^{24} - 4 q^{26} - 8 q^{31} + 14 q^{34} + 12 q^{36} - 12 q^{39} - 6 q^{41} + 2 q^{44} + 12 q^{46} + 14 q^{49} + 42 q^{51} + 18 q^{54} - 8 q^{59} - 16 q^{61} - 2 q^{64} + 6 q^{66} + 36 q^{69} - 12 q^{71} - 2 q^{74} + 10 q^{76} - 28 q^{79} + 18 q^{81} - 8 q^{86} + 14 q^{89} + 8 q^{94} - 6 q^{96} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) − 3.00000i − 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 3.00000i 0.866025i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000i 1.69775i 0.528594 + 0.848875i \(0.322719\pi\)
−0.528594 + 0.848875i \(0.677281\pi\)
\(18\) 6.00000i 1.41421i
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000i 0.213201i
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 9.00000i 1.73205i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 3.00000i 0.522233i
\(34\) 7.00000 1.20049
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) − 1.00000i − 0.164399i
\(38\) 5.00000i 0.811107i
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) − 3.00000i − 0.433013i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 21.0000 2.94059
\(52\) 2.00000i 0.277350i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) 0 0
\(57\) 15.0000i 1.98680i
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 13.0000i 1.58820i 0.607785 + 0.794101i \(0.292058\pi\)
−0.607785 + 0.794101i \(0.707942\pi\)
\(68\) − 7.00000i − 0.848875i
\(69\) 18.0000 2.16695
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) − 6.00000i − 0.707107i
\(73\) − 7.00000i − 0.819288i −0.912245 0.409644i \(-0.865653\pi\)
0.912245 0.409644i \(-0.134347\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) 0 0
\(78\) 6.00000i 0.679366i
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 3.00000i 0.331295i
\(83\) 3.00000i 0.329293i 0.986353 + 0.164646i \(0.0526483\pi\)
−0.986353 + 0.164646i \(0.947352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) − 1.00000i − 0.106600i
\(89\) 7.00000 0.741999 0.370999 0.928633i \(-0.379015\pi\)
0.370999 + 0.928633i \(0.379015\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 6.00000i − 0.625543i
\(93\) 12.0000i 1.24434i
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) 18.0000i 1.82762i 0.406138 + 0.913812i \(0.366875\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) − 21.0000i − 2.07931i
\(103\) − 18.0000i − 1.77359i −0.462160 0.886796i \(-0.652926\pi\)
0.462160 0.886796i \(-0.347074\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) − 5.00000i − 0.483368i −0.970355 0.241684i \(-0.922300\pi\)
0.970355 0.241684i \(-0.0776998\pi\)
\(108\) − 9.00000i − 0.866025i
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) 0 0
\(113\) − 17.0000i − 1.59923i −0.600516 0.799613i \(-0.705038\pi\)
0.600516 0.799613i \(-0.294962\pi\)
\(114\) 15.0000 1.40488
\(115\) 0 0
\(116\) 0 0
\(117\) 12.0000i 1.10940i
\(118\) 4.00000i 0.368230i
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 8.00000i 0.724286i
\(123\) 9.00000i 0.811503i
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) − 4.00000i − 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) − 3.00000i − 0.261116i
\(133\) 0 0
\(134\) 13.0000 1.12303
\(135\) 0 0
\(136\) −7.00000 −0.600245
\(137\) − 15.0000i − 1.28154i −0.767734 0.640768i \(-0.778616\pi\)
0.767734 0.640768i \(-0.221384\pi\)
\(138\) − 18.0000i − 1.53226i
\(139\) 1.00000 0.0848189 0.0424094 0.999100i \(-0.486497\pi\)
0.0424094 + 0.999100i \(0.486497\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 6.00000i 0.503509i
\(143\) 2.00000i 0.167248i
\(144\) −6.00000 −0.500000
\(145\) 0 0
\(146\) −7.00000 −0.579324
\(147\) − 21.0000i − 1.73205i
\(148\) 1.00000i 0.0821995i
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) −18.0000 −1.46482 −0.732410 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) − 5.00000i − 0.405554i
\(153\) − 42.0000i − 3.39550i
\(154\) 0 0
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) − 12.0000i − 0.957704i −0.877896 0.478852i \(-0.841053\pi\)
0.877896 0.478852i \(-0.158947\pi\)
\(158\) 14.0000i 1.11378i
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) − 9.00000i − 0.707107i
\(163\) 1.00000i 0.0783260i 0.999233 + 0.0391630i \(0.0124692\pi\)
−0.999233 + 0.0391630i \(0.987531\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 3.00000 0.232845
\(167\) 2.00000i 0.154765i 0.997001 + 0.0773823i \(0.0246562\pi\)
−0.997001 + 0.0773823i \(0.975344\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 30.0000 2.29416
\(172\) 4.00000i 0.304997i
\(173\) 8.00000i 0.608229i 0.952636 + 0.304114i \(0.0983605\pi\)
−0.952636 + 0.304114i \(0.901639\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 12.0000i 0.901975i
\(178\) − 7.00000i − 0.524672i
\(179\) −13.0000 −0.971666 −0.485833 0.874052i \(-0.661484\pi\)
−0.485833 + 0.874052i \(0.661484\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 24.0000i 1.77413i
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 12.0000 0.879883
\(187\) − 7.00000i − 0.511891i
\(188\) − 4.00000i − 0.291730i
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 3.00000i 0.216506i
\(193\) 25.0000i 1.79954i 0.436365 + 0.899770i \(0.356266\pi\)
−0.436365 + 0.899770i \(0.643734\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 8.00000i 0.569976i 0.958531 + 0.284988i \(0.0919897\pi\)
−0.958531 + 0.284988i \(0.908010\pi\)
\(198\) − 6.00000i − 0.426401i
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 39.0000 2.75085
\(202\) − 6.00000i − 0.422159i
\(203\) 0 0
\(204\) −21.0000 −1.47029
\(205\) 0 0
\(206\) −18.0000 −1.25412
\(207\) − 36.0000i − 2.50217i
\(208\) − 2.00000i − 0.138675i
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) − 2.00000i − 0.137361i
\(213\) 18.0000i 1.23334i
\(214\) −5.00000 −0.341793
\(215\) 0 0
\(216\) −9.00000 −0.612372
\(217\) 0 0
\(218\) − 6.00000i − 0.406371i
\(219\) −21.0000 −1.41905
\(220\) 0 0
\(221\) 14.0000 0.941742
\(222\) 3.00000i 0.201347i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −17.0000 −1.13082
\(227\) − 28.0000i − 1.85843i −0.369546 0.929213i \(-0.620487\pi\)
0.369546 0.929213i \(-0.379513\pi\)
\(228\) − 15.0000i − 0.993399i
\(229\) −24.0000 −1.58596 −0.792982 0.609245i \(-0.791473\pi\)
−0.792982 + 0.609245i \(0.791473\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 14.0000i − 0.917170i −0.888650 0.458585i \(-0.848356\pi\)
0.888650 0.458585i \(-0.151644\pi\)
\(234\) 12.0000 0.784465
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 42.0000i 2.72819i
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −25.0000 −1.61039 −0.805196 0.593009i \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) 10.0000i 0.642824i
\(243\) 0 0
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 9.00000 0.573819
\(247\) 10.0000i 0.636285i
\(248\) − 4.00000i − 0.254000i
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) −1.00000 −0.0631194 −0.0315597 0.999502i \(-0.510047\pi\)
−0.0315597 + 0.999502i \(0.510047\pi\)
\(252\) 0 0
\(253\) − 6.00000i − 0.377217i
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000i 0.124757i 0.998053 + 0.0623783i \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) 12.0000i 0.747087i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000i 0.741362i
\(263\) 14.0000i 0.863277i 0.902047 + 0.431638i \(0.142064\pi\)
−0.902047 + 0.431638i \(0.857936\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) 0 0
\(267\) − 21.0000i − 1.28518i
\(268\) − 13.0000i − 0.794101i
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 7.00000i 0.424437i
\(273\) 0 0
\(274\) −15.0000 −0.906183
\(275\) 0 0
\(276\) −18.0000 −1.08347
\(277\) 20.0000i 1.20168i 0.799368 + 0.600842i \(0.205168\pi\)
−0.799368 + 0.600842i \(0.794832\pi\)
\(278\) − 1.00000i − 0.0599760i
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) − 12.0000i − 0.714590i
\(283\) − 7.00000i − 0.416107i −0.978117 0.208053i \(-0.933287\pi\)
0.978117 0.208053i \(-0.0667128\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 6.00000i 0.353553i
\(289\) −32.0000 −1.88235
\(290\) 0 0
\(291\) 54.0000 3.16554
\(292\) 7.00000i 0.409644i
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) −21.0000 −1.22474
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) − 9.00000i − 0.522233i
\(298\) − 22.0000i − 1.27443i
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 0 0
\(302\) 18.0000i 1.03578i
\(303\) − 18.0000i − 1.03407i
\(304\) −5.00000 −0.286770
\(305\) 0 0
\(306\) −42.0000 −2.40098
\(307\) 7.00000i 0.399511i 0.979846 + 0.199756i \(0.0640148\pi\)
−0.979846 + 0.199756i \(0.935985\pi\)
\(308\) 0 0
\(309\) −54.0000 −3.07195
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) − 6.00000i − 0.339683i
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) −12.0000 −0.677199
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 0 0
\(320\) 0 0
\(321\) −15.0000 −0.837218
\(322\) 0 0
\(323\) − 35.0000i − 1.94745i
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) 1.00000 0.0553849
\(327\) − 18.0000i − 0.995402i
\(328\) − 3.00000i − 0.165647i
\(329\) 0 0
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) − 3.00000i − 0.164646i
\(333\) 6.00000i 0.328798i
\(334\) 2.00000 0.109435
\(335\) 0 0
\(336\) 0 0
\(337\) 31.0000i 1.68868i 0.535810 + 0.844339i \(0.320006\pi\)
−0.535810 + 0.844339i \(0.679994\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) −51.0000 −2.76994
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) − 30.0000i − 1.62221i
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 8.00000 0.430083
\(347\) − 23.0000i − 1.23470i −0.786687 0.617352i \(-0.788205\pi\)
0.786687 0.617352i \(-0.211795\pi\)
\(348\) 0 0
\(349\) 12.0000 0.642345 0.321173 0.947021i \(-0.395923\pi\)
0.321173 + 0.947021i \(0.395923\pi\)
\(350\) 0 0
\(351\) 18.0000 0.960769
\(352\) 1.00000i 0.0533002i
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −7.00000 −0.370999
\(357\) 0 0
\(358\) 13.0000i 0.687071i
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 2.00000i 0.105118i
\(363\) 30.0000i 1.57459i
\(364\) 0 0
\(365\) 0 0
\(366\) 24.0000 1.25450
\(367\) 26.0000i 1.35719i 0.734513 + 0.678594i \(0.237411\pi\)
−0.734513 + 0.678594i \(0.762589\pi\)
\(368\) 6.00000i 0.312772i
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) 0 0
\(372\) − 12.0000i − 0.622171i
\(373\) 32.0000i 1.65690i 0.560065 + 0.828449i \(0.310776\pi\)
−0.560065 + 0.828449i \(0.689224\pi\)
\(374\) −7.00000 −0.361961
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) 0 0
\(378\) 0 0
\(379\) 19.0000 0.975964 0.487982 0.872854i \(-0.337733\pi\)
0.487982 + 0.872854i \(0.337733\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) − 6.00000i − 0.306987i
\(383\) 36.0000i 1.83951i 0.392488 + 0.919757i \(0.371614\pi\)
−0.392488 + 0.919757i \(0.628386\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 25.0000 1.27247
\(387\) 24.0000i 1.21999i
\(388\) − 18.0000i − 0.913812i
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) −42.0000 −2.12403
\(392\) 7.00000i 0.353553i
\(393\) 36.0000i 1.81596i
\(394\) 8.00000 0.403034
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) − 2.00000i − 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 16.0000i 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) 23.0000 1.14857 0.574283 0.818657i \(-0.305281\pi\)
0.574283 + 0.818657i \(0.305281\pi\)
\(402\) − 39.0000i − 1.94514i
\(403\) 8.00000i 0.398508i
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 1.00000i 0.0495682i
\(408\) 21.0000i 1.03965i
\(409\) −11.0000 −0.543915 −0.271957 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) 0 0
\(411\) −45.0000 −2.21969
\(412\) 18.0000i 0.886796i
\(413\) 0 0
\(414\) −36.0000 −1.76930
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) − 3.00000i − 0.146911i
\(418\) − 5.00000i − 0.244558i
\(419\) 17.0000 0.830504 0.415252 0.909706i \(-0.363693\pi\)
0.415252 + 0.909706i \(0.363693\pi\)
\(420\) 0 0
\(421\) 24.0000 1.16969 0.584844 0.811146i \(-0.301156\pi\)
0.584844 + 0.811146i \(0.301156\pi\)
\(422\) − 13.0000i − 0.632830i
\(423\) − 24.0000i − 1.16692i
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) 18.0000 0.872103
\(427\) 0 0
\(428\) 5.00000i 0.241684i
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) −38.0000 −1.83040 −0.915198 0.403005i \(-0.867966\pi\)
−0.915198 + 0.403005i \(0.867966\pi\)
\(432\) 9.00000i 0.433013i
\(433\) 11.0000i 0.528626i 0.964437 + 0.264313i \(0.0851452\pi\)
−0.964437 + 0.264313i \(0.914855\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) − 30.0000i − 1.43509i
\(438\) 21.0000i 1.00342i
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −42.0000 −2.00000
\(442\) − 14.0000i − 0.665912i
\(443\) 15.0000i 0.712672i 0.934358 + 0.356336i \(0.115974\pi\)
−0.934358 + 0.356336i \(0.884026\pi\)
\(444\) 3.00000 0.142374
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) − 66.0000i − 3.12169i
\(448\) 0 0
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 3.00000 0.141264
\(452\) 17.0000i 0.799613i
\(453\) 54.0000i 2.53714i
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) −15.0000 −0.702439
\(457\) − 21.0000i − 0.982339i −0.871064 0.491169i \(-0.836570\pi\)
0.871064 0.491169i \(-0.163430\pi\)
\(458\) 24.0000i 1.12145i
\(459\) −63.0000 −2.94059
\(460\) 0 0
\(461\) 4.00000 0.186299 0.0931493 0.995652i \(-0.470307\pi\)
0.0931493 + 0.995652i \(0.470307\pi\)
\(462\) 0 0
\(463\) − 26.0000i − 1.20832i −0.796862 0.604161i \(-0.793508\pi\)
0.796862 0.604161i \(-0.206492\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) 4.00000i 0.185098i 0.995708 + 0.0925490i \(0.0295015\pi\)
−0.995708 + 0.0925490i \(0.970499\pi\)
\(468\) − 12.0000i − 0.554700i
\(469\) 0 0
\(470\) 0 0
\(471\) −36.0000 −1.65879
\(472\) − 4.00000i − 0.184115i
\(473\) 4.00000i 0.183920i
\(474\) 42.0000 1.92912
\(475\) 0 0
\(476\) 0 0
\(477\) − 12.0000i − 0.549442i
\(478\) 12.0000i 0.548867i
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 25.0000i 1.13872i
\(483\) 0 0
\(484\) 10.0000 0.454545
\(485\) 0 0
\(486\) 0 0
\(487\) − 16.0000i − 0.725029i −0.931978 0.362515i \(-0.881918\pi\)
0.931978 0.362515i \(-0.118082\pi\)
\(488\) − 8.00000i − 0.362143i
\(489\) 3.00000 0.135665
\(490\) 0 0
\(491\) 44.0000 1.98569 0.992846 0.119401i \(-0.0380974\pi\)
0.992846 + 0.119401i \(0.0380974\pi\)
\(492\) − 9.00000i − 0.405751i
\(493\) 0 0
\(494\) 10.0000 0.449921
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) − 9.00000i − 0.403300i
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) 1.00000i 0.0446322i
\(503\) − 36.0000i − 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) − 27.0000i − 1.19911i
\(508\) 4.00000i 0.177471i
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) − 45.0000i − 1.98680i
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) − 4.00000i − 0.175920i
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) −35.0000 −1.53338 −0.766689 0.642019i \(-0.778097\pi\)
−0.766689 + 0.642019i \(0.778097\pi\)
\(522\) 0 0
\(523\) − 45.0000i − 1.96771i −0.178960 0.983856i \(-0.557273\pi\)
0.178960 0.983856i \(-0.442727\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 14.0000 0.610429
\(527\) − 28.0000i − 1.21970i
\(528\) 3.00000i 0.130558i
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) 6.00000i 0.259889i
\(534\) −21.0000 −0.908759
\(535\) 0 0
\(536\) −13.0000 −0.561514
\(537\) 39.0000i 1.68297i
\(538\) − 4.00000i − 0.172452i
\(539\) −7.00000 −0.301511
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) − 16.0000i − 0.687259i
\(543\) 6.00000i 0.257485i
\(544\) 7.00000 0.300123
\(545\) 0 0
\(546\) 0 0
\(547\) − 11.0000i − 0.470326i −0.971956 0.235163i \(-0.924438\pi\)
0.971956 0.235163i \(-0.0755624\pi\)
\(548\) 15.0000i 0.640768i
\(549\) 48.0000 2.04859
\(550\) 0 0
\(551\) 0 0
\(552\) 18.0000i 0.766131i
\(553\) 0 0
\(554\) 20.0000 0.849719
\(555\) 0 0
\(556\) −1.00000 −0.0424094
\(557\) − 40.0000i − 1.69485i −0.530912 0.847427i \(-0.678150\pi\)
0.530912 0.847427i \(-0.321850\pi\)
\(558\) − 24.0000i − 1.01600i
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −21.0000 −0.886621
\(562\) 22.0000i 0.928014i
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) −7.00000 −0.294232
\(567\) 0 0
\(568\) − 6.00000i − 0.251754i
\(569\) −39.0000 −1.63497 −0.817483 0.575953i \(-0.804631\pi\)
−0.817483 + 0.575953i \(0.804631\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) − 2.00000i − 0.0836242i
\(573\) − 18.0000i − 0.751961i
\(574\) 0 0
\(575\) 0 0
\(576\) 6.00000 0.250000
\(577\) − 11.0000i − 0.457936i −0.973434 0.228968i \(-0.926465\pi\)
0.973434 0.228968i \(-0.0735351\pi\)
\(578\) 32.0000i 1.33102i
\(579\) 75.0000 3.11689
\(580\) 0 0
\(581\) 0 0
\(582\) − 54.0000i − 2.23837i
\(583\) − 2.00000i − 0.0828315i
\(584\) 7.00000 0.289662
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 3.00000i 0.123823i 0.998082 + 0.0619116i \(0.0197197\pi\)
−0.998082 + 0.0619116i \(0.980280\pi\)
\(588\) 21.0000i 0.866025i
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) − 1.00000i − 0.0410997i
\(593\) 41.0000i 1.68367i 0.539736 + 0.841834i \(0.318524\pi\)
−0.539736 + 0.841834i \(0.681476\pi\)
\(594\) −9.00000 −0.369274
\(595\) 0 0
\(596\) −22.0000 −0.901155
\(597\) 48.0000i 1.96451i
\(598\) − 12.0000i − 0.490716i
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −17.0000 −0.693444 −0.346722 0.937968i \(-0.612705\pi\)
−0.346722 + 0.937968i \(0.612705\pi\)
\(602\) 0 0
\(603\) − 78.0000i − 3.17641i
\(604\) 18.0000 0.732410
\(605\) 0 0
\(606\) −18.0000 −0.731200
\(607\) 18.0000i 0.730597i 0.930890 + 0.365299i \(0.119033\pi\)
−0.930890 + 0.365299i \(0.880967\pi\)
\(608\) 5.00000i 0.202777i
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 42.0000i 1.69775i
\(613\) − 42.0000i − 1.69636i −0.529705 0.848182i \(-0.677697\pi\)
0.529705 0.848182i \(-0.322303\pi\)
\(614\) 7.00000 0.282497
\(615\) 0 0
\(616\) 0 0
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 54.0000i 2.17220i
\(619\) −40.0000 −1.60774 −0.803868 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) 0 0
\(621\) −54.0000 −2.16695
\(622\) 24.0000i 0.962312i
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) − 15.0000i − 0.599042i
\(628\) 12.0000i 0.478852i
\(629\) 7.00000 0.279108
\(630\) 0 0
\(631\) 30.0000 1.19428 0.597141 0.802137i \(-0.296303\pi\)
0.597141 + 0.802137i \(0.296303\pi\)
\(632\) − 14.0000i − 0.556890i
\(633\) − 39.0000i − 1.55011i
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) − 14.0000i − 0.554700i
\(638\) 0 0
\(639\) 36.0000 1.42414
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 15.0000i 0.592003i
\(643\) − 28.0000i − 1.10421i −0.833774 0.552106i \(-0.813824\pi\)
0.833774 0.552106i \(-0.186176\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −35.0000 −1.37706
\(647\) 38.0000i 1.49393i 0.664861 + 0.746967i \(0.268491\pi\)
−0.664861 + 0.746967i \(0.731509\pi\)
\(648\) 9.00000i 0.353553i
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) − 1.00000i − 0.0391630i
\(653\) − 40.0000i − 1.56532i −0.622449 0.782660i \(-0.713862\pi\)
0.622449 0.782660i \(-0.286138\pi\)
\(654\) −18.0000 −0.703856
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) 42.0000i 1.63858i
\(658\) 0 0
\(659\) −5.00000 −0.194772 −0.0973862 0.995247i \(-0.531048\pi\)
−0.0973862 + 0.995247i \(0.531048\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 19.0000i 0.738456i
\(663\) − 42.0000i − 1.63114i
\(664\) −3.00000 −0.116423
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 0 0
\(668\) − 2.00000i − 0.0773823i
\(669\) 48.0000 1.85579
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) − 46.0000i − 1.77317i −0.462566 0.886585i \(-0.653071\pi\)
0.462566 0.886585i \(-0.346929\pi\)
\(674\) 31.0000 1.19408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 8.00000i 0.307465i 0.988113 + 0.153732i \(0.0491294\pi\)
−0.988113 + 0.153732i \(0.950871\pi\)
\(678\) 51.0000i 1.95864i
\(679\) 0 0
\(680\) 0 0
\(681\) −84.0000 −3.21889
\(682\) − 4.00000i − 0.153168i
\(683\) − 9.00000i − 0.344375i −0.985064 0.172188i \(-0.944916\pi\)
0.985064 0.172188i \(-0.0550836\pi\)
\(684\) −30.0000 −1.14708
\(685\) 0 0
\(686\) 0 0
\(687\) 72.0000i 2.74697i
\(688\) − 4.00000i − 0.152499i
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) −5.00000 −0.190209 −0.0951045 0.995467i \(-0.530319\pi\)
−0.0951045 + 0.995467i \(0.530319\pi\)
\(692\) − 8.00000i − 0.304114i
\(693\) 0 0
\(694\) −23.0000 −0.873068
\(695\) 0 0
\(696\) 0 0
\(697\) − 21.0000i − 0.795432i
\(698\) − 12.0000i − 0.454207i
\(699\) −42.0000 −1.58859
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) − 18.0000i − 0.679366i
\(703\) 5.00000i 0.188579i
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) − 12.0000i − 0.450988i
\(709\) −24.0000 −0.901339 −0.450669 0.892691i \(-0.648815\pi\)
−0.450669 + 0.892691i \(0.648815\pi\)
\(710\) 0 0
\(711\) 84.0000 3.15025
\(712\) 7.00000i 0.262336i
\(713\) − 24.0000i − 0.898807i
\(714\) 0 0
\(715\) 0 0
\(716\) 13.0000 0.485833
\(717\) 36.0000i 1.34444i
\(718\) − 6.00000i − 0.223918i
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 6.00000i − 0.223297i
\(723\) 75.0000i 2.78928i
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 30.0000 1.11340
\(727\) − 20.0000i − 0.741759i −0.928681 0.370879i \(-0.879056\pi\)
0.928681 0.370879i \(-0.120944\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 28.0000 1.03562
\(732\) − 24.0000i − 0.887066i
\(733\) 32.0000i 1.18195i 0.806691 + 0.590973i \(0.201256\pi\)
−0.806691 + 0.590973i \(0.798744\pi\)
\(734\) 26.0000 0.959678
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) − 13.0000i − 0.478861i
\(738\) − 18.0000i − 0.662589i
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 30.0000 1.10208
\(742\) 0 0
\(743\) − 54.0000i − 1.98107i −0.137268 0.990534i \(-0.543832\pi\)
0.137268 0.990534i \(-0.456168\pi\)
\(744\) −12.0000 −0.439941
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) − 18.0000i − 0.658586i
\(748\) 7.00000i 0.255945i
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 4.00000i 0.145865i
\(753\) 3.00000i 0.109326i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 32.0000i − 1.16306i −0.813525 0.581530i \(-0.802454\pi\)
0.813525 0.581530i \(-0.197546\pi\)
\(758\) − 19.0000i − 0.690111i
\(759\) −18.0000 −0.653359
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 12.0000i 0.434714i
\(763\) 0 0
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) 8.00000i 0.288863i
\(768\) − 3.00000i − 0.108253i
\(769\) 21.0000 0.757279 0.378640 0.925544i \(-0.376392\pi\)
0.378640 + 0.925544i \(0.376392\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) − 25.0000i − 0.899770i
\(773\) 22.0000i 0.791285i 0.918405 + 0.395643i \(0.129478\pi\)
−0.918405 + 0.395643i \(0.870522\pi\)
\(774\) 24.0000 0.862662
\(775\) 0 0
\(776\) −18.0000 −0.646162
\(777\) 0 0
\(778\) − 8.00000i − 0.286814i
\(779\) 15.0000 0.537431
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 42.0000i 1.50192i
\(783\) 0 0
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) 36.0000 1.28408
\(787\) 24.0000i 0.855508i 0.903895 + 0.427754i \(0.140695\pi\)
−0.903895 + 0.427754i \(0.859305\pi\)
\(788\) − 8.00000i − 0.284988i
\(789\) 42.0000 1.49524
\(790\) 0 0
\(791\) 0 0
\(792\) 6.00000i 0.213201i
\(793\) 16.0000i 0.568177i
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) − 12.0000i − 0.425062i −0.977154 0.212531i \(-0.931829\pi\)
0.977154 0.212531i \(-0.0681706\pi\)
\(798\) 0 0
\(799\) −28.0000 −0.990569
\(800\) 0 0
\(801\) −42.0000 −1.48400
\(802\) − 23.0000i − 0.812158i
\(803\) 7.00000i 0.247025i
\(804\) −39.0000 −1.37542
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) − 12.0000i − 0.422420i
\(808\) 6.00000i 0.211079i
\(809\) 22.0000 0.773479 0.386739 0.922189i \(-0.373601\pi\)
0.386739 + 0.922189i \(0.373601\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) − 48.0000i − 1.68343i
\(814\) 1.00000 0.0350500
\(815\) 0 0
\(816\) 21.0000 0.735147
\(817\) 20.0000i 0.699711i
\(818\) 11.0000i 0.384606i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 45.0000i 1.56956i
\(823\) 34.0000i 1.18517i 0.805510 + 0.592583i \(0.201892\pi\)
−0.805510 + 0.592583i \(0.798108\pi\)
\(824\) 18.0000 0.627060
\(825\) 0 0
\(826\) 0 0
\(827\) − 33.0000i − 1.14752i −0.819023 0.573761i \(-0.805484\pi\)
0.819023 0.573761i \(-0.194516\pi\)
\(828\) 36.0000i 1.25109i
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 60.0000 2.08138
\(832\) 2.00000i 0.0693375i
\(833\) 49.0000i 1.69775i
\(834\) −3.00000 −0.103882
\(835\) 0 0
\(836\) −5.00000 −0.172929
\(837\) − 36.0000i − 1.24434i
\(838\) − 17.0000i − 0.587255i
\(839\) 6.00000 0.207143 0.103572 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) − 24.0000i − 0.827095i
\(843\) 66.0000i 2.27316i
\(844\) −13.0000 −0.447478
\(845\) 0 0
\(846\) −24.0000 −0.825137
\(847\) 0 0
\(848\) 2.00000i 0.0686803i
\(849\) −21.0000 −0.720718
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) − 18.0000i − 0.616670i
\(853\) 20.0000i 0.684787i 0.939557 + 0.342393i \(0.111238\pi\)
−0.939557 + 0.342393i \(0.888762\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 5.00000 0.170896
\(857\) − 9.00000i − 0.307434i −0.988115 0.153717i \(-0.950876\pi\)
0.988115 0.153717i \(-0.0491244\pi\)
\(858\) − 6.00000i − 0.204837i
\(859\) −41.0000 −1.39890 −0.699451 0.714681i \(-0.746572\pi\)
−0.699451 + 0.714681i \(0.746572\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 38.0000i 1.29429i
\(863\) 6.00000i 0.204242i 0.994772 + 0.102121i \(0.0325630\pi\)
−0.994772 + 0.102121i \(0.967437\pi\)
\(864\) 9.00000 0.306186
\(865\) 0 0
\(866\) 11.0000 0.373795
\(867\) 96.0000i 3.26033i
\(868\) 0 0
\(869\) 14.0000 0.474917
\(870\) 0 0
\(871\) 26.0000 0.880976
\(872\) 6.00000i 0.203186i
\(873\) − 108.000i − 3.65525i
\(874\) −30.0000 −1.01477
\(875\) 0 0
\(876\) 21.0000 0.709524
\(877\) − 42.0000i − 1.41824i −0.705088 0.709120i \(-0.749093\pi\)
0.705088 0.709120i \(-0.250907\pi\)
\(878\) − 32.0000i − 1.07995i
\(879\) 72.0000 2.42850
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 42.0000i 1.41421i
\(883\) 7.00000i 0.235569i 0.993039 + 0.117784i \(0.0375792\pi\)
−0.993039 + 0.117784i \(0.962421\pi\)
\(884\) −14.0000 −0.470871
\(885\) 0 0
\(886\) 15.0000 0.503935
\(887\) − 52.0000i − 1.74599i −0.487730 0.872995i \(-0.662175\pi\)
0.487730 0.872995i \(-0.337825\pi\)
\(888\) − 3.00000i − 0.100673i
\(889\) 0 0
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) − 16.0000i − 0.535720i
\(893\) − 20.0000i − 0.669274i
\(894\) −66.0000 −2.20737
\(895\) 0 0
\(896\) 0 0
\(897\) − 36.0000i − 1.20201i
\(898\) 15.0000i 0.500556i
\(899\) 0 0
\(900\) 0 0
\(901\) −14.0000 −0.466408
\(902\) − 3.00000i − 0.0998891i
\(903\) 0 0
\(904\) 17.0000 0.565412
\(905\) 0 0
\(906\) 54.0000 1.79403
\(907\) 28.0000i 0.929725i 0.885383 + 0.464862i \(0.153896\pi\)
−0.885383 + 0.464862i \(0.846104\pi\)
\(908\) 28.0000i 0.929213i
\(909\) −36.0000 −1.19404
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 15.0000i 0.496700i
\(913\) − 3.00000i − 0.0992855i
\(914\) −21.0000 −0.694618
\(915\) 0 0
\(916\) 24.0000 0.792982
\(917\) 0 0
\(918\) 63.0000i 2.07931i
\(919\) −14.0000 −0.461817 −0.230909 0.972975i \(-0.574170\pi\)
−0.230909 + 0.972975i \(0.574170\pi\)
\(920\) 0 0
\(921\) 21.0000 0.691974
\(922\) − 4.00000i − 0.131733i
\(923\) 12.0000i 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) 108.000i 3.54719i
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −35.0000 −1.14708
\(932\) 14.0000i 0.458585i
\(933\) 72.0000i 2.35717i
\(934\) 4.00000 0.130884
\(935\) 0 0
\(936\) −12.0000 −0.392232
\(937\) 7.00000i 0.228680i 0.993442 + 0.114340i \(0.0364753\pi\)
−0.993442 + 0.114340i \(0.963525\pi\)
\(938\) 0 0
\(939\) 30.0000 0.979013
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 36.0000i 1.17294i
\(943\) − 18.0000i − 0.586161i
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 56.0000i 1.81976i 0.414876 + 0.909878i \(0.363825\pi\)
−0.414876 + 0.909878i \(0.636175\pi\)
\(948\) − 42.0000i − 1.36410i
\(949\) −14.0000 −0.454459
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) − 9.00000i − 0.291539i −0.989319 0.145769i \(-0.953434\pi\)
0.989319 0.145769i \(-0.0465657\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 4.00000i 0.129234i
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 2.00000i 0.0644826i
\(963\) 30.0000i 0.966736i
\(964\) 25.0000 0.805196
\(965\) 0 0
\(966\) 0 0
\(967\) − 32.0000i − 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) − 10.0000i − 0.321412i
\(969\) −105.000 −3.37309
\(970\) 0 0
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) 43.0000i 1.37569i 0.725857 + 0.687846i \(0.241444\pi\)
−0.725857 + 0.687846i \(0.758556\pi\)
\(978\) − 3.00000i − 0.0959294i
\(979\) −7.00000 −0.223721
\(980\) 0 0
\(981\) −36.0000 −1.14939
\(982\) − 44.0000i − 1.40410i
\(983\) − 42.0000i − 1.33959i −0.742545 0.669796i \(-0.766382\pi\)
0.742545 0.669796i \(-0.233618\pi\)
\(984\) −9.00000 −0.286910
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) − 10.0000i − 0.318142i
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 4.00000i 0.127000i
\(993\) 57.0000i 1.80884i
\(994\) 0 0
\(995\) 0 0
\(996\) −9.00000 −0.285176
\(997\) 10.0000i 0.316703i 0.987383 + 0.158352i \(0.0506179\pi\)
−0.987383 + 0.158352i \(0.949382\pi\)
\(998\) 16.0000i 0.506471i
\(999\) 9.00000 0.284747
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.a.149.1 2
5.2 odd 4 1850.2.a.h.1.1 yes 1
5.3 odd 4 1850.2.a.g.1.1 1
5.4 even 2 inner 1850.2.b.a.149.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.g.1.1 1 5.3 odd 4
1850.2.a.h.1.1 yes 1 5.2 odd 4
1850.2.b.a.149.1 2 1.1 even 1 trivial
1850.2.b.a.149.2 2 5.4 even 2 inner