Properties

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.00000i q^{12} +2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} -5.00000 q^{19} +8.00000 q^{21} +3.00000i q^{23} -2.00000 q^{24} -2.00000 q^{26} -4.00000i q^{27} -4.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} +1.00000 q^{36} -1.00000i q^{37} -5.00000i q^{38} +4.00000 q^{39} -9.00000 q^{41} +8.00000i q^{42} -7.00000i q^{43} -3.00000 q^{46} +6.00000i q^{47} -2.00000i q^{48} -9.00000 q^{49} -2.00000i q^{52} +9.00000i q^{53} +4.00000 q^{54} +4.00000 q^{56} +10.0000i q^{57} -6.00000i q^{58} -3.00000 q^{59} +2.00000 q^{61} -4.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -2.00000i q^{67} +6.00000 q^{69} -6.00000 q^{71} +1.00000i q^{72} +11.0000i q^{73} +1.00000 q^{74} +5.00000 q^{76} +4.00000i q^{78} +1.00000 q^{79} -11.0000 q^{81} -9.00000i q^{82} -8.00000 q^{84} +7.00000 q^{86} +12.0000i q^{87} +6.00000 q^{89} -8.00000 q^{91} -3.00000i q^{92} +8.00000i q^{93} -6.00000 q^{94} +2.00000 q^{96} +4.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 8 q^{14} + 2 q^{16} - 10 q^{19} + 16 q^{21} - 4 q^{24} - 4 q^{26} - 12 q^{29} - 8 q^{31} + 2 q^{36} + 8 q^{39} - 18 q^{41} - 6 q^{46} - 18 q^{49} + 8 q^{54} + 8 q^{56} - 6 q^{59} + 4 q^{61} - 2 q^{64} + 12 q^{69} - 12 q^{71} + 2 q^{74} + 10 q^{76} + 2 q^{79} - 22 q^{81} - 16 q^{84} + 14 q^{86} + 12 q^{89} - 16 q^{91} - 12 q^{94} + 4 q^{96}+O(q^{100}) \) 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
\(3\) − 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
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) − 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 \(0.910544\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 8.00000 1.74574
\(22\) 0 0
\(23\) 3.00000i 0.625543i 0.949828 + 0.312772i \(0.101257\pi\)
−0.949828 + 0.312772i \(0.898743\pi\)
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) − 4.00000i − 0.769800i
\(28\) − 4.00000i − 0.755929i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 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
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 1.00000i − 0.164399i
\(38\) − 5.00000i − 0.811107i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 8.00000i 1.23443i
\(43\) − 7.00000i − 1.06749i −0.845645 0.533745i \(-0.820784\pi\)
0.845645 0.533745i \(-0.179216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) − 2.00000i − 0.288675i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) 9.00000i 1.23625i 0.786082 + 0.618123i \(0.212106\pi\)
−0.786082 + 0.618123i \(0.787894\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 10.0000i 1.32453i
\(58\) − 6.00000i − 0.787839i
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) − 4.00000i − 0.503953i
\(64\) −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 \(-0.0389851\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 11.0000i 1.28745i 0.765256 + 0.643726i \(0.222612\pi\)
−0.765256 + 0.643726i \(0.777388\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) 0 0
\(78\) 4.00000i 0.452911i
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) − 9.00000i − 0.993884i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −8.00000 −0.872872
\(85\) 0 0
\(86\) 7.00000 0.754829
\(87\) 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
\(91\) −8.00000 −0.838628
\(92\) − 3.00000i − 0.312772i
\(93\) 8.00000i 0.829561i
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 4.00000i 0.406138i 0.979164 + 0.203069i \(0.0650917\pi\)
−0.979164 + 0.203069i \(0.934908\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) 0 0
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) − 13.0000i − 1.28093i −0.767988 0.640464i \(-0.778742\pi\)
0.767988 0.640464i \(-0.221258\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 18.0000i 1.74013i 0.492941 + 0.870063i \(0.335922\pi\)
−0.492941 + 0.870063i \(0.664078\pi\)
\(108\) 4.00000i 0.384900i
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 4.00000i 0.377964i
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) −10.0000 −0.936586
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) − 2.00000i − 0.184900i
\(118\) − 3.00000i − 0.276172i
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 2.00000i 0.181071i
\(123\) 18.0000i 1.62301i
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −14.0000 −1.23263
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) − 20.0000i − 1.73422i
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 0 0
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 6.00000i 0.510754i
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) − 6.00000i − 0.503509i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −11.0000 −0.910366
\(147\) 18.0000i 1.48461i
\(148\) 1.00000i 0.0821995i
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 5.00000i 0.405554i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 13.0000i 1.03751i 0.854922 + 0.518756i \(0.173605\pi\)
−0.854922 + 0.518756i \(0.826395\pi\)
\(158\) 1.00000i 0.0795557i
\(159\) 18.0000 1.42749
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) − 11.0000i − 0.864242i
\(163\) − 7.00000i − 0.548282i −0.961689 0.274141i \(-0.911606\pi\)
0.961689 0.274141i \(-0.0883936\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000i 0.232147i 0.993241 + 0.116073i \(0.0370308\pi\)
−0.993241 + 0.116073i \(0.962969\pi\)
\(168\) − 8.00000i − 0.617213i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 5.00000 0.382360
\(172\) 7.00000i 0.533745i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) −12.0000 −0.909718
\(175\) 0 0
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 6.00000i 0.449719i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) − 8.00000i − 0.592999i
\(183\) − 4.00000i − 0.295689i
\(184\) 3.00000 0.221163
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) − 6.00000i − 0.437595i
\(189\) 16.0000 1.16383
\(190\) 0 0
\(191\) −9.00000 −0.651217 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) 2.00000i 0.144338i
\(193\) − 4.00000i − 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) − 27.0000i − 1.92367i −0.273629 0.961835i \(-0.588224\pi\)
0.273629 0.961835i \(-0.411776\pi\)
\(198\) 0 0
\(199\) 13.0000 0.921546 0.460773 0.887518i \(-0.347572\pi\)
0.460773 + 0.887518i \(0.347572\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 3.00000i 0.211079i
\(203\) − 24.0000i − 1.68447i
\(204\) 0 0
\(205\) 0 0
\(206\) 13.0000 0.905753
\(207\) − 3.00000i − 0.208514i
\(208\) 2.00000i 0.138675i
\(209\) 0 0
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) − 9.00000i − 0.618123i
\(213\) 12.0000i 0.822226i
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) − 16.0000i − 1.08615i
\(218\) 16.0000i 1.08366i
\(219\) 22.0000 1.48662
\(220\) 0 0
\(221\) 0 0
\(222\) − 2.00000i − 0.134231i
\(223\) 2.00000i 0.133930i 0.997755 + 0.0669650i \(0.0213316\pi\)
−0.997755 + 0.0669650i \(0.978668\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) 9.00000i 0.597351i 0.954355 + 0.298675i \(0.0965448\pi\)
−0.954355 + 0.298675i \(0.903455\pi\)
\(228\) − 10.0000i − 0.662266i
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) − 21.0000i − 1.37576i −0.725826 0.687878i \(-0.758542\pi\)
0.725826 0.687878i \(-0.241458\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 3.00000 0.195283
\(237\) − 2.00000i − 0.129914i
\(238\) 0 0
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) 10.0000i 0.641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −18.0000 −1.14764
\(247\) − 10.0000i − 0.636285i
\(248\) 4.00000i 0.254000i
\(249\) 0 0
\(250\) 0 0
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 4.00000i 0.251976i
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) − 14.0000i − 0.871602i
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) − 12.0000i − 0.741362i
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 20.0000 1.22628
\(267\) − 12.0000i − 0.734388i
\(268\) 2.00000i 0.122169i
\(269\) 21.0000 1.28039 0.640196 0.768211i \(-0.278853\pi\)
0.640196 + 0.768211i \(0.278853\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 16.0000i 0.968364i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 22.0000i 1.32185i 0.750451 + 0.660926i \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) − 14.0000i − 0.839664i
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 12.0000i 0.714590i
\(283\) − 19.0000i − 1.12943i −0.825285 0.564716i \(-0.808986\pi\)
0.825285 0.564716i \(-0.191014\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) − 36.0000i − 2.12501i
\(288\) − 1.00000i − 0.0589256i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) − 11.0000i − 0.643726i
\(293\) − 3.00000i − 0.175262i −0.996153 0.0876309i \(-0.972070\pi\)
0.996153 0.0876309i \(-0.0279296\pi\)
\(294\) −18.0000 −1.04978
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) − 15.0000i − 0.868927i
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 28.0000 1.61389
\(302\) − 4.00000i − 0.230174i
\(303\) − 6.00000i − 0.344691i
\(304\) −5.00000 −0.286770
\(305\) 0 0
\(306\) 0 0
\(307\) − 32.0000i − 1.82634i −0.407583 0.913168i \(-0.633628\pi\)
0.407583 0.913168i \(-0.366372\pi\)
\(308\) 0 0
\(309\) −26.0000 −1.47909
\(310\) 0 0
\(311\) 15.0000 0.850572 0.425286 0.905059i \(-0.360174\pi\)
0.425286 + 0.905059i \(0.360174\pi\)
\(312\) − 4.00000i − 0.226455i
\(313\) − 22.0000i − 1.24351i −0.783210 0.621757i \(-0.786419\pi\)
0.783210 0.621757i \(-0.213581\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) − 33.0000i − 1.85346i −0.375722 0.926732i \(-0.622605\pi\)
0.375722 0.926732i \(-0.377395\pi\)
\(318\) 18.0000i 1.00939i
\(319\) 0 0
\(320\) 0 0
\(321\) 36.0000 2.00932
\(322\) − 12.0000i − 0.668734i
\(323\) 0 0
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) 7.00000 0.387694
\(327\) − 32.0000i − 1.76960i
\(328\) 9.00000i 0.496942i
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 1.00000i 0.0547997i
\(334\) −3.00000 −0.164153
\(335\) 0 0
\(336\) 8.00000 0.436436
\(337\) 25.0000i 1.36184i 0.732359 + 0.680918i \(0.238419\pi\)
−0.732359 + 0.680918i \(0.761581\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 36.0000 1.95525
\(340\) 0 0
\(341\) 0 0
\(342\) 5.00000i 0.270369i
\(343\) − 8.00000i − 0.431959i
\(344\) −7.00000 −0.377415
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) − 15.0000i − 0.805242i −0.915367 0.402621i \(-0.868099\pi\)
0.915367 0.402621i \(-0.131901\pi\)
\(348\) − 12.0000i − 0.643268i
\(349\) 19.0000 1.01705 0.508523 0.861048i \(-0.330192\pi\)
0.508523 + 0.861048i \(0.330192\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) 12.0000i 0.638696i 0.947638 + 0.319348i \(0.103464\pi\)
−0.947638 + 0.319348i \(0.896536\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 12.0000i 0.634220i
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 5.00000i 0.262794i
\(363\) 22.0000i 1.15470i
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) − 2.00000i − 0.104399i −0.998637 0.0521996i \(-0.983377\pi\)
0.998637 0.0521996i \(-0.0166232\pi\)
\(368\) 3.00000i 0.156386i
\(369\) 9.00000 0.468521
\(370\) 0 0
\(371\) −36.0000 −1.86903
\(372\) − 8.00000i − 0.414781i
\(373\) 14.0000i 0.724893i 0.932005 + 0.362446i \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) − 12.0000i − 0.618031i
\(378\) 16.0000i 0.822951i
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 32.0000 1.63941
\(382\) − 9.00000i − 0.460480i
\(383\) − 15.0000i − 0.766464i −0.923652 0.383232i \(-0.874811\pi\)
0.923652 0.383232i \(-0.125189\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 7.00000i 0.355830i
\(388\) − 4.00000i − 0.203069i
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000i 0.454569i
\(393\) 24.0000i 1.21064i
\(394\) 27.0000 1.36024
\(395\) 0 0
\(396\) 0 0
\(397\) 25.0000i 1.25471i 0.778732 + 0.627357i \(0.215863\pi\)
−0.778732 + 0.627357i \(0.784137\pi\)
\(398\) 13.0000i 0.651631i
\(399\) −40.0000 −2.00250
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) − 4.00000i − 0.199502i
\(403\) − 8.00000i − 0.398508i
\(404\) −3.00000 −0.149256
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) 0 0
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 13.0000i 0.640464i
\(413\) − 12.0000i − 0.590481i
\(414\) 3.00000 0.147442
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 28.0000i 1.37117i
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 2.00000i 0.0973585i
\(423\) − 6.00000i − 0.291730i
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) 8.00000i 0.387147i
\(428\) − 18.0000i − 0.870063i
\(429\) 0 0
\(430\) 0 0
\(431\) 15.0000 0.722525 0.361262 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) 26.0000i 1.24948i 0.780833 + 0.624740i \(0.214795\pi\)
−0.780833 + 0.624740i \(0.785205\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) − 15.0000i − 0.717547i
\(438\) 22.0000i 1.05120i
\(439\) 7.00000 0.334092 0.167046 0.985949i \(-0.446577\pi\)
0.167046 + 0.985949i \(0.446577\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 24.0000i 1.14027i 0.821549 + 0.570137i \(0.193110\pi\)
−0.821549 + 0.570137i \(0.806890\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) 30.0000i 1.41895i
\(448\) − 4.00000i − 0.188982i
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 18.0000i − 0.846649i
\(453\) 8.00000i 0.375873i
\(454\) −9.00000 −0.422391
\(455\) 0 0
\(456\) 10.0000 0.468293
\(457\) 28.0000i 1.30978i 0.755722 + 0.654892i \(0.227286\pi\)
−0.755722 + 0.654892i \(0.772714\pi\)
\(458\) 13.0000i 0.607450i
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 32.0000i 1.48717i 0.668644 + 0.743583i \(0.266875\pi\)
−0.668644 + 0.743583i \(0.733125\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 21.0000 0.972806
\(467\) − 12.0000i − 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 26.0000 1.19802
\(472\) 3.00000i 0.138086i
\(473\) 0 0
\(474\) 2.00000 0.0918630
\(475\) 0 0
\(476\) 0 0
\(477\) − 9.00000i − 0.412082i
\(478\) − 9.00000i − 0.411650i
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 14.0000i 0.637683i
\(483\) 24.0000i 1.09204i
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 16.0000i 0.725029i 0.931978 + 0.362515i \(0.118082\pi\)
−0.931978 + 0.362515i \(0.881918\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) −14.0000 −0.633102
\(490\) 0 0
\(491\) −18.0000 −0.812329 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) − 18.0000i − 0.811503i
\(493\) 0 0
\(494\) 10.0000 0.449921
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) − 24.0000i − 1.07655i
\(498\) 0 0
\(499\) 43.0000 1.92494 0.962472 0.271380i \(-0.0874801\pi\)
0.962472 + 0.271380i \(0.0874801\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) − 27.0000i − 1.20507i
\(503\) − 36.0000i − 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) − 18.0000i − 0.799408i
\(508\) − 16.0000i − 0.709885i
\(509\) −9.00000 −0.398918 −0.199459 0.979906i \(-0.563918\pi\)
−0.199459 + 0.979906i \(0.563918\pi\)
\(510\) 0 0
\(511\) −44.0000 −1.94645
\(512\) 1.00000i 0.0441942i
\(513\) 20.0000i 0.883022i
\(514\) 0 0
\(515\) 0 0
\(516\) 14.0000 0.616316
\(517\) 0 0
\(518\) 4.00000i 0.175750i
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) −33.0000 −1.44576 −0.722878 0.690976i \(-0.757181\pi\)
−0.722878 + 0.690976i \(0.757181\pi\)
\(522\) 6.00000i 0.262613i
\(523\) − 28.0000i − 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) 0 0
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 3.00000 0.130189
\(532\) 20.0000i 0.867110i
\(533\) − 18.0000i − 0.779667i
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) − 24.0000i − 1.03568i
\(538\) 21.0000i 0.905374i
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) − 16.0000i − 0.687259i
\(543\) − 10.0000i − 0.429141i
\(544\) 0 0
\(545\) 0 0
\(546\) −16.0000 −0.684737
\(547\) 28.0000i 1.19719i 0.801050 + 0.598597i \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) 6.00000i 0.256307i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 30.0000 1.27804
\(552\) − 6.00000i − 0.255377i
\(553\) 4.00000i 0.170097i
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 12.0000i 0.508456i 0.967144 + 0.254228i \(0.0818214\pi\)
−0.967144 + 0.254228i \(0.918179\pi\)
\(558\) 4.00000i 0.169334i
\(559\) 14.0000 0.592137
\(560\) 0 0
\(561\) 0 0
\(562\) 12.0000i 0.506189i
\(563\) 45.0000i 1.89652i 0.317489 + 0.948262i \(0.397160\pi\)
−0.317489 + 0.948262i \(0.602840\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) 19.0000 0.798630
\(567\) − 44.0000i − 1.84783i
\(568\) 6.00000i 0.251754i
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 2.00000 0.0836974 0.0418487 0.999124i \(-0.486675\pi\)
0.0418487 + 0.999124i \(0.486675\pi\)
\(572\) 0 0
\(573\) 18.0000i 0.751961i
\(574\) 36.0000 1.50261
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 40.0000i 1.66522i 0.553858 + 0.832611i \(0.313155\pi\)
−0.553858 + 0.832611i \(0.686845\pi\)
\(578\) 17.0000i 0.707107i
\(579\) −8.00000 −0.332469
\(580\) 0 0
\(581\) 0 0
\(582\) 8.00000i 0.331611i
\(583\) 0 0
\(584\) 11.0000 0.455183
\(585\) 0 0
\(586\) 3.00000 0.123929
\(587\) 3.00000i 0.123823i 0.998082 + 0.0619116i \(0.0197197\pi\)
−0.998082 + 0.0619116i \(0.980280\pi\)
\(588\) − 18.0000i − 0.742307i
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) −54.0000 −2.22126
\(592\) − 1.00000i − 0.0410997i
\(593\) 15.0000i 0.615976i 0.951390 + 0.307988i \(0.0996557\pi\)
−0.951390 + 0.307988i \(0.900344\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) − 26.0000i − 1.06411i
\(598\) − 6.00000i − 0.245358i
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 28.0000i 1.14119i
\(603\) 2.00000i 0.0814463i
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) − 5.00000i − 0.202777i
\(609\) −48.0000 −1.94506
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) − 25.0000i − 1.00974i −0.863195 0.504870i \(-0.831540\pi\)
0.863195 0.504870i \(-0.168460\pi\)
\(614\) 32.0000 1.29141
\(615\) 0 0
\(616\) 0 0
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) − 26.0000i − 1.04587i
\(619\) 46.0000 1.84890 0.924448 0.381308i \(-0.124526\pi\)
0.924448 + 0.381308i \(0.124526\pi\)
\(620\) 0 0
\(621\) 12.0000 0.481543
\(622\) 15.0000i 0.601445i
\(623\) 24.0000i 0.961540i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) − 13.0000i − 0.518756i
\(629\) 0 0
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) − 1.00000i − 0.0397779i
\(633\) − 4.00000i − 0.158986i
\(634\) 33.0000 1.31060
\(635\) 0 0
\(636\) −18.0000 −0.713746
\(637\) − 18.0000i − 0.713186i
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) −21.0000 −0.829450 −0.414725 0.909947i \(-0.636122\pi\)
−0.414725 + 0.909947i \(0.636122\pi\)
\(642\) 36.0000i 1.42081i
\(643\) − 4.00000i − 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) 0 0
\(647\) 27.0000i 1.06148i 0.847535 + 0.530740i \(0.178086\pi\)
−0.847535 + 0.530740i \(0.821914\pi\)
\(648\) 11.0000i 0.432121i
\(649\) 0 0
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 7.00000i 0.274141i
\(653\) − 6.00000i − 0.234798i −0.993085 0.117399i \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) 32.0000 1.25130
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) − 11.0000i − 0.429151i
\(658\) − 24.0000i − 0.935617i
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 8.00000i 0.310929i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) − 18.0000i − 0.696963i
\(668\) − 3.00000i − 0.116073i
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 0 0
\(672\) 8.00000i 0.308607i
\(673\) − 37.0000i − 1.42625i −0.701039 0.713123i \(-0.747280\pi\)
0.701039 0.713123i \(-0.252720\pi\)
\(674\) −25.0000 −0.962964
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 39.0000i 1.49889i 0.662066 + 0.749446i \(0.269680\pi\)
−0.662066 + 0.749446i \(0.730320\pi\)
\(678\) 36.0000i 1.38257i
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) − 48.0000i − 1.83667i −0.395805 0.918334i \(-0.629534\pi\)
0.395805 0.918334i \(-0.370466\pi\)
\(684\) −5.00000 −0.191180
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) − 26.0000i − 0.991962i
\(688\) − 7.00000i − 0.266872i
\(689\) −18.0000 −0.685745
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) 0 0
\(694\) 15.0000 0.569392
\(695\) 0 0
\(696\) 12.0000 0.454859
\(697\) 0 0
\(698\) 19.0000i 0.719161i
\(699\) −42.0000 −1.58859
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 8.00000i 0.301941i
\(703\) 5.00000i 0.188579i
\(704\) 0 0
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 12.0000i 0.451306i
\(708\) − 6.00000i − 0.225494i
\(709\) 40.0000 1.50223 0.751116 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) − 6.00000i − 0.224860i
\(713\) − 12.0000i − 0.449404i
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 18.0000i 0.672222i
\(718\) − 30.0000i − 1.11959i
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 52.0000 1.93658
\(722\) 6.00000i 0.223297i
\(723\) − 28.0000i − 1.04133i
\(724\) −5.00000 −0.185824
\(725\) 0 0
\(726\) −22.0000 −0.816497
\(727\) 16.0000i 0.593407i 0.954970 + 0.296704i \(0.0958873\pi\)
−0.954970 + 0.296704i \(0.904113\pi\)
\(728\) 8.00000i 0.296500i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 4.00000i 0.147844i
\(733\) − 22.0000i − 0.812589i −0.913742 0.406294i \(-0.866821\pi\)
0.913742 0.406294i \(-0.133179\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) 0 0
\(738\) 9.00000i 0.331295i
\(739\) 34.0000 1.25071 0.625355 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 0 0
\(741\) −20.0000 −0.734718
\(742\) − 36.0000i − 1.32160i
\(743\) − 6.00000i − 0.220119i −0.993925 0.110059i \(-0.964896\pi\)
0.993925 0.110059i \(-0.0351041\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) 0 0
\(749\) −72.0000 −2.63082
\(750\) 0 0
\(751\) 50.0000 1.82453 0.912263 0.409605i \(-0.134333\pi\)
0.912263 + 0.409605i \(0.134333\pi\)
\(752\) 6.00000i 0.218797i
\(753\) 54.0000i 1.96787i
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) −16.0000 −0.581914
\(757\) − 20.0000i − 0.726912i −0.931611 0.363456i \(-0.881597\pi\)
0.931611 0.363456i \(-0.118403\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) 0 0
\(760\) 0 0
\(761\) 45.0000 1.63125 0.815624 0.578582i \(-0.196394\pi\)
0.815624 + 0.578582i \(0.196394\pi\)
\(762\) 32.0000i 1.15924i
\(763\) 64.0000i 2.31696i
\(764\) 9.00000 0.325609
\(765\) 0 0
\(766\) 15.0000 0.541972
\(767\) − 6.00000i − 0.216647i
\(768\) − 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
\(771\) 0 0
\(772\) 4.00000i 0.143963i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) −7.00000 −0.251610
\(775\) 0 0
\(776\) 4.00000 0.143592
\(777\) − 8.00000i − 0.286998i
\(778\) 36.0000i 1.29066i
\(779\) 45.0000 1.61229
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 24.0000i 0.857690i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) −24.0000 −0.856052
\(787\) − 20.0000i − 0.712923i −0.934310 0.356462i \(-0.883983\pi\)
0.934310 0.356462i \(-0.116017\pi\)
\(788\) 27.0000i 0.961835i
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −72.0000 −2.56003
\(792\) 0 0
\(793\) 4.00000i 0.142044i
\(794\) −25.0000 −0.887217
\(795\) 0 0
\(796\) −13.0000 −0.460773
\(797\) 12.0000i 0.425062i 0.977154 + 0.212531i \(0.0681706\pi\)
−0.977154 + 0.212531i \(0.931829\pi\)
\(798\) − 40.0000i − 1.41598i
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 18.0000i 0.635602i
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) − 42.0000i − 1.47847i
\(808\) − 3.00000i − 0.105540i
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 24.0000i 0.842235i
\(813\) 32.0000i 1.12229i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 35.0000i 1.22449i
\(818\) − 20.0000i − 0.699284i
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) 33.0000 1.15171 0.575854 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) − 12.0000i − 0.418548i
\(823\) − 34.0000i − 1.18517i −0.805510 0.592583i \(-0.798108\pi\)
0.805510 0.592583i \(-0.201892\pi\)
\(824\) −13.0000 −0.452876
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 3.00000i 0.104257i
\(829\) −56.0000 −1.94496 −0.972480 0.232986i \(-0.925151\pi\)
−0.972480 + 0.232986i \(0.925151\pi\)
\(830\) 0 0
\(831\) 44.0000 1.52634
\(832\) − 2.00000i − 0.0693375i
\(833\) 0 0
\(834\) −28.0000 −0.969561
\(835\) 0 0
\(836\) 0 0
\(837\) 16.0000i 0.553041i
\(838\) − 12.0000i − 0.414533i
\(839\) 6.00000 0.207143 0.103572 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 26.0000i 0.896019i
\(843\) − 24.0000i − 0.826604i
\(844\) −2.00000 −0.0688428
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) − 44.0000i − 1.51186i
\(848\) 9.00000i 0.309061i
\(849\) −38.0000 −1.30416
\(850\) 0 0
\(851\) 3.00000 0.102839
\(852\) − 12.0000i − 0.411113i
\(853\) − 46.0000i − 1.57501i −0.616308 0.787505i \(-0.711372\pi\)
0.616308 0.787505i \(-0.288628\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 18.0000 0.615227
\(857\) 42.0000i 1.43469i 0.696717 + 0.717346i \(0.254643\pi\)
−0.696717 + 0.717346i \(0.745357\pi\)
\(858\) 0 0
\(859\) −5.00000 −0.170598 −0.0852989 0.996355i \(-0.527185\pi\)
−0.0852989 + 0.996355i \(0.527185\pi\)
\(860\) 0 0
\(861\) −72.0000 −2.45375
\(862\) 15.0000i 0.510902i
\(863\) − 6.00000i − 0.204242i −0.994772 0.102121i \(-0.967437\pi\)
0.994772 0.102121i \(-0.0325630\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) − 34.0000i − 1.15470i
\(868\) 16.0000i 0.543075i
\(869\) 0 0
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) − 16.0000i − 0.541828i
\(873\) − 4.00000i − 0.135379i
\(874\) 15.0000 0.507383
\(875\) 0 0
\(876\) −22.0000 −0.743311
\(877\) 25.0000i 0.844190i 0.906552 + 0.422095i \(0.138705\pi\)
−0.906552 + 0.422095i \(0.861295\pi\)
\(878\) 7.00000i 0.236239i
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −39.0000 −1.31394 −0.656972 0.753915i \(-0.728163\pi\)
−0.656972 + 0.753915i \(0.728163\pi\)
\(882\) 9.00000i 0.303046i
\(883\) 32.0000i 1.07689i 0.842662 + 0.538443i \(0.180987\pi\)
−0.842662 + 0.538443i \(0.819013\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 30.0000i 1.00730i 0.863907 + 0.503651i \(0.168010\pi\)
−0.863907 + 0.503651i \(0.831990\pi\)
\(888\) 2.00000i 0.0671156i
\(889\) −64.0000 −2.14649
\(890\) 0 0
\(891\) 0 0
\(892\) − 2.00000i − 0.0669650i
\(893\) − 30.0000i − 1.00391i
\(894\) −30.0000 −1.00335
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 12.0000i 0.400668i
\(898\) − 6.00000i − 0.200223i
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) − 56.0000i − 1.86356i
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 37.0000i 1.22856i 0.789086 + 0.614282i \(0.210554\pi\)
−0.789086 + 0.614282i \(0.789446\pi\)
\(908\) − 9.00000i − 0.298675i
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) −9.00000 −0.298183 −0.149092 0.988823i \(-0.547635\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(912\) 10.0000i 0.331133i
\(913\) 0 0
\(914\) −28.0000 −0.926158
\(915\) 0 0
\(916\) −13.0000 −0.429532
\(917\) − 48.0000i − 1.58510i
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) −64.0000 −2.10887
\(922\) 0 0
\(923\) − 12.0000i − 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) 13.0000i 0.426976i
\(928\) − 6.00000i − 0.196960i
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 45.0000 1.47482
\(932\) 21.0000i 0.687878i
\(933\) − 30.0000i − 0.982156i
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) − 38.0000i − 1.24141i −0.784046 0.620703i \(-0.786847\pi\)
0.784046 0.620703i \(-0.213153\pi\)
\(938\) 8.00000i 0.261209i
\(939\) −44.0000 −1.43589
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 26.0000i 0.847126i
\(943\) − 27.0000i − 0.879241i
\(944\) −3.00000 −0.0976417
\(945\) 0 0
\(946\) 0 0
\(947\) − 33.0000i − 1.07236i −0.844105 0.536178i \(-0.819868\pi\)
0.844105 0.536178i \(-0.180132\pi\)
\(948\) 2.00000i 0.0649570i
\(949\) −22.0000 −0.714150
\(950\) 0 0
\(951\) −66.0000 −2.14020
\(952\) 0 0
\(953\) 9.00000i 0.291539i 0.989319 + 0.145769i \(0.0465657\pi\)
−0.989319 + 0.145769i \(0.953434\pi\)
\(954\) 9.00000 0.291386
\(955\) 0 0
\(956\) 9.00000 0.291081
\(957\) 0 0
\(958\) − 24.0000i − 0.775405i
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 2.00000i 0.0644826i
\(963\) − 18.0000i − 0.580042i
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) −24.0000 −0.772187
\(967\) 31.0000i 0.996893i 0.866921 + 0.498446i \(0.166096\pi\)
−0.866921 + 0.498446i \(0.833904\pi\)
\(968\) 11.0000i 0.353553i
\(969\) 0 0
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) − 10.0000i − 0.320750i
\(973\) − 56.0000i − 1.79528i
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) − 14.0000i − 0.447671i
\(979\) 0 0
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) − 18.0000i − 0.574403i
\(983\) − 54.0000i − 1.72233i −0.508323 0.861166i \(-0.669735\pi\)
0.508323 0.861166i \(-0.330265\pi\)
\(984\) 18.0000 0.573819
\(985\) 0 0
\(986\) 0 0
\(987\) 48.0000i 1.52786i
\(988\) 10.0000i 0.318142i
\(989\) 21.0000 0.667761
\(990\) 0 0
\(991\) −13.0000 −0.412959 −0.206479 0.978451i \(-0.566201\pi\)
−0.206479 + 0.978451i \(0.566201\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) − 16.0000i − 0.507745i
\(994\) 24.0000 0.761234
\(995\) 0 0
\(996\) 0 0
\(997\) 28.0000i 0.886769i 0.896332 + 0.443384i \(0.146222\pi\)
−0.896332 + 0.443384i \(0.853778\pi\)
\(998\) 43.0000i 1.36114i
\(999\) −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.f.149.2 2
5.2 odd 4 1850.2.a.a.1.1 1
5.3 odd 4 1850.2.a.p.1.1 yes 1
5.4 even 2 inner 1850.2.b.f.149.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.a.1.1 1 5.2 odd 4
1850.2.a.p.1.1 yes 1 5.3 odd 4
1850.2.b.f.149.1 2 5.4 even 2 inner
1850.2.b.f.149.2 2 1.1 even 1 trivial