Properties

Label 1850.2.b.o.149.2
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.3182656.1
Defining polynomial: \( x^{6} - 2x^{3} + 25x^{2} - 10x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.2
Root \(0.203364 - 0.203364i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.o.149.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +0.406728i q^{3} -1.00000 q^{4} +0.406728 q^{6} -2.91729i q^{7} +1.00000i q^{8} +2.83457 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +0.406728i q^{3} -1.00000 q^{4} +0.406728 q^{6} -2.91729i q^{7} +1.00000i q^{8} +2.83457 q^{9} +6.51056 q^{11} -0.406728i q^{12} -0.813457i q^{13} -2.91729 q^{14} +1.00000 q^{16} +2.51056i q^{17} -2.83457i q^{18} -0.406728 q^{19} +1.18654 q^{21} -6.51056i q^{22} +5.02112i q^{23} -0.406728 q^{24} -0.813457 q^{26} +2.37309i q^{27} +2.91729i q^{28} +5.32401 q^{29} -8.75186 q^{31} -1.00000i q^{32} +2.64803i q^{33} +2.51056 q^{34} -2.83457 q^{36} +1.00000i q^{37} +0.406728i q^{38} +0.330856 q^{39} +6.34513 q^{41} -1.18654i q^{42} -7.32401i q^{43} -6.51056 q^{44} +5.02112 q^{46} +5.42784i q^{47} +0.406728i q^{48} -1.51056 q^{49} -1.02112 q^{51} +0.813457i q^{52} -2.34513i q^{53} +2.37309 q^{54} +2.91729 q^{56} -0.165428i q^{57} -5.32401i q^{58} +1.42784 q^{59} -1.32401 q^{61} +8.75186i q^{62} -8.26926i q^{63} -1.00000 q^{64} +2.64803 q^{66} +5.42784i q^{67} -2.51056i q^{68} -2.04223 q^{69} +14.6480 q^{71} +2.83457i q^{72} -11.0211i q^{73} +1.00000 q^{74} +0.406728 q^{76} -18.9932i q^{77} -0.330856i q^{78} +1.75870 q^{79} +7.53851 q^{81} -6.34513i q^{82} -7.05476i q^{83} -1.18654 q^{84} -7.32401 q^{86} +2.16543i q^{87} +6.51056i q^{88} -6.00000 q^{89} -2.37309 q^{91} -5.02112i q^{92} -3.55963i q^{93} +5.42784 q^{94} +0.406728 q^{96} -2.34513i q^{97} +1.51056i q^{98} +18.4546 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} - 22 q^{9} + 22 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{21} + 10 q^{29} + 6 q^{31} - 2 q^{34} + 22 q^{36} + 80 q^{39} - 18 q^{41} - 22 q^{44} - 4 q^{46} + 8 q^{49} + 28 q^{51} + 24 q^{54} - 2 q^{56} - 28 q^{59} + 14 q^{61} - 6 q^{64} - 28 q^{66} + 56 q^{69} + 44 q^{71} + 6 q^{74} + 52 q^{79} + 94 q^{81} - 12 q^{84} - 22 q^{86} - 36 q^{89} - 24 q^{91} - 4 q^{94} - 38 q^{99}+O(q^{100}) \) 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
\(3\) 0.406728i 0.234825i 0.993083 + 0.117412i \(0.0374599\pi\)
−0.993083 + 0.117412i \(0.962540\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0.406728 0.166046
\(7\) − 2.91729i − 1.10263i −0.834297 0.551315i \(-0.814126\pi\)
0.834297 0.551315i \(-0.185874\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.83457 0.944857
\(10\) 0 0
\(11\) 6.51056 1.96301 0.981503 0.191444i \(-0.0613171\pi\)
0.981503 + 0.191444i \(0.0613171\pi\)
\(12\) − 0.406728i − 0.117412i
\(13\) − 0.813457i − 0.225612i −0.993617 0.112806i \(-0.964016\pi\)
0.993617 0.112806i \(-0.0359839\pi\)
\(14\) −2.91729 −0.779677
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.51056i 0.608900i 0.952528 + 0.304450i \(0.0984726\pi\)
−0.952528 + 0.304450i \(0.901527\pi\)
\(18\) − 2.83457i − 0.668115i
\(19\) −0.406728 −0.0933099 −0.0466550 0.998911i \(-0.514856\pi\)
−0.0466550 + 0.998911i \(0.514856\pi\)
\(20\) 0 0
\(21\) 1.18654 0.258925
\(22\) − 6.51056i − 1.38806i
\(23\) 5.02112i 1.04697i 0.852033 + 0.523487i \(0.175369\pi\)
−0.852033 + 0.523487i \(0.824631\pi\)
\(24\) −0.406728 −0.0830231
\(25\) 0 0
\(26\) −0.813457 −0.159532
\(27\) 2.37309i 0.456701i
\(28\) 2.91729i 0.551315i
\(29\) 5.32401 0.988645 0.494322 0.869279i \(-0.335416\pi\)
0.494322 + 0.869279i \(0.335416\pi\)
\(30\) 0 0
\(31\) −8.75186 −1.57188 −0.785940 0.618303i \(-0.787821\pi\)
−0.785940 + 0.618303i \(0.787821\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 2.64803i 0.460963i
\(34\) 2.51056 0.430557
\(35\) 0 0
\(36\) −2.83457 −0.472429
\(37\) 1.00000i 0.164399i
\(38\) 0.406728i 0.0659801i
\(39\) 0.330856 0.0529794
\(40\) 0 0
\(41\) 6.34513 0.990943 0.495471 0.868624i \(-0.334995\pi\)
0.495471 + 0.868624i \(0.334995\pi\)
\(42\) − 1.18654i − 0.183088i
\(43\) − 7.32401i − 1.11690i −0.829538 0.558451i \(-0.811396\pi\)
0.829538 0.558451i \(-0.188604\pi\)
\(44\) −6.51056 −0.981503
\(45\) 0 0
\(46\) 5.02112 0.740323
\(47\) 5.42784i 0.791732i 0.918308 + 0.395866i \(0.129556\pi\)
−0.918308 + 0.395866i \(0.870444\pi\)
\(48\) 0.406728i 0.0587062i
\(49\) −1.51056 −0.215794
\(50\) 0 0
\(51\) −1.02112 −0.142985
\(52\) 0.813457i 0.112806i
\(53\) − 2.34513i − 0.322128i −0.986944 0.161064i \(-0.948507\pi\)
0.986944 0.161064i \(-0.0514926\pi\)
\(54\) 2.37309 0.322936
\(55\) 0 0
\(56\) 2.91729 0.389839
\(57\) − 0.165428i − 0.0219115i
\(58\) − 5.32401i − 0.699077i
\(59\) 1.42784 0.185889 0.0929447 0.995671i \(-0.470372\pi\)
0.0929447 + 0.995671i \(0.470372\pi\)
\(60\) 0 0
\(61\) −1.32401 −0.169523 −0.0847613 0.996401i \(-0.527013\pi\)
−0.0847613 + 0.996401i \(0.527013\pi\)
\(62\) 8.75186i 1.11149i
\(63\) − 8.26926i − 1.04183i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.64803 0.325950
\(67\) 5.42784i 0.663117i 0.943435 + 0.331558i \(0.107574\pi\)
−0.943435 + 0.331558i \(0.892426\pi\)
\(68\) − 2.51056i − 0.304450i
\(69\) −2.04223 −0.245856
\(70\) 0 0
\(71\) 14.6480 1.73840 0.869201 0.494460i \(-0.164634\pi\)
0.869201 + 0.494460i \(0.164634\pi\)
\(72\) 2.83457i 0.334058i
\(73\) − 11.0211i − 1.28992i −0.764215 0.644962i \(-0.776873\pi\)
0.764215 0.644962i \(-0.223127\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 0.406728 0.0466550
\(77\) − 18.9932i − 2.16447i
\(78\) − 0.330856i − 0.0374621i
\(79\) 1.75870 0.197869 0.0989346 0.995094i \(-0.468457\pi\)
0.0989346 + 0.995094i \(0.468457\pi\)
\(80\) 0 0
\(81\) 7.53851 0.837613
\(82\) − 6.34513i − 0.700702i
\(83\) − 7.05476i − 0.774360i −0.922004 0.387180i \(-0.873449\pi\)
0.922004 0.387180i \(-0.126551\pi\)
\(84\) −1.18654 −0.129462
\(85\) 0 0
\(86\) −7.32401 −0.789769
\(87\) 2.16543i 0.232158i
\(88\) 6.51056i 0.694028i
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −2.37309 −0.248767
\(92\) − 5.02112i − 0.523487i
\(93\) − 3.55963i − 0.369116i
\(94\) 5.42784 0.559839
\(95\) 0 0
\(96\) 0.406728 0.0415115
\(97\) − 2.34513i − 0.238112i −0.992888 0.119056i \(-0.962013\pi\)
0.992888 0.119056i \(-0.0379868\pi\)
\(98\) 1.51056i 0.152589i
\(99\) 18.4546 1.85476
\(100\) 0 0
\(101\) −8.20766 −0.816693 −0.408346 0.912827i \(-0.633894\pi\)
−0.408346 + 0.912827i \(0.633894\pi\)
\(102\) 1.02112i 0.101105i
\(103\) − 8.81346i − 0.868416i −0.900813 0.434208i \(-0.857028\pi\)
0.900813 0.434208i \(-0.142972\pi\)
\(104\) 0.813457 0.0797660
\(105\) 0 0
\(106\) −2.34513 −0.227779
\(107\) − 15.2624i − 1.47547i −0.675089 0.737737i \(-0.735895\pi\)
0.675089 0.737737i \(-0.264105\pi\)
\(108\) − 2.37309i − 0.228350i
\(109\) −4.30290 −0.412143 −0.206072 0.978537i \(-0.566068\pi\)
−0.206072 + 0.978537i \(0.566068\pi\)
\(110\) 0 0
\(111\) −0.406728 −0.0386050
\(112\) − 2.91729i − 0.275658i
\(113\) − 9.15859i − 0.861567i −0.902455 0.430784i \(-0.858237\pi\)
0.902455 0.430784i \(-0.141763\pi\)
\(114\) −0.165428 −0.0154938
\(115\) 0 0
\(116\) −5.32401 −0.494322
\(117\) − 2.30580i − 0.213171i
\(118\) − 1.42784i − 0.131444i
\(119\) 7.32401 0.671391
\(120\) 0 0
\(121\) 31.3874 2.85340
\(122\) 1.32401i 0.119871i
\(123\) 2.58074i 0.232698i
\(124\) 8.75186 0.785940
\(125\) 0 0
\(126\) −8.26926 −0.736684
\(127\) 8.61439i 0.764403i 0.924079 + 0.382202i \(0.124834\pi\)
−0.924079 + 0.382202i \(0.875166\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 2.97888 0.262276
\(130\) 0 0
\(131\) 13.4278 1.17320 0.586598 0.809878i \(-0.300467\pi\)
0.586598 + 0.809878i \(0.300467\pi\)
\(132\) − 2.64803i − 0.230481i
\(133\) 1.18654i 0.102886i
\(134\) 5.42784 0.468894
\(135\) 0 0
\(136\) −2.51056 −0.215279
\(137\) 14.6903i 1.25507i 0.778587 + 0.627537i \(0.215937\pi\)
−0.778587 + 0.627537i \(0.784063\pi\)
\(138\) 2.04223i 0.173846i
\(139\) −17.3662 −1.47299 −0.736493 0.676445i \(-0.763519\pi\)
−0.736493 + 0.676445i \(0.763519\pi\)
\(140\) 0 0
\(141\) −2.20766 −0.185918
\(142\) − 14.6480i − 1.22924i
\(143\) − 5.29606i − 0.442879i
\(144\) 2.83457 0.236214
\(145\) 0 0
\(146\) −11.0211 −0.912114
\(147\) − 0.614387i − 0.0506738i
\(148\) − 1.00000i − 0.0821995i
\(149\) 0.207658 0.0170120 0.00850601 0.999964i \(-0.497292\pi\)
0.00850601 + 0.999964i \(0.497292\pi\)
\(150\) 0 0
\(151\) 0.813457 0.0661982 0.0330991 0.999452i \(-0.489462\pi\)
0.0330991 + 0.999452i \(0.489462\pi\)
\(152\) − 0.406728i − 0.0329900i
\(153\) 7.11636i 0.575323i
\(154\) −18.9932 −1.53051
\(155\) 0 0
\(156\) −0.330856 −0.0264897
\(157\) − 5.32401i − 0.424903i −0.977172 0.212451i \(-0.931855\pi\)
0.977172 0.212451i \(-0.0681447\pi\)
\(158\) − 1.75870i − 0.139915i
\(159\) 0.953831 0.0756437
\(160\) 0 0
\(161\) 14.6480 1.15443
\(162\) − 7.53851i − 0.592282i
\(163\) 15.2008i 1.19062i 0.803496 + 0.595310i \(0.202971\pi\)
−0.803496 + 0.595310i \(0.797029\pi\)
\(164\) −6.34513 −0.495471
\(165\) 0 0
\(166\) −7.05476 −0.547555
\(167\) 12.4826i 0.965933i 0.875639 + 0.482966i \(0.160441\pi\)
−0.875639 + 0.482966i \(0.839559\pi\)
\(168\) 1.18654i 0.0915438i
\(169\) 12.3383 0.949099
\(170\) 0 0
\(171\) −1.15290 −0.0881645
\(172\) 7.32401i 0.558451i
\(173\) 23.0354i 1.75135i 0.482903 + 0.875674i \(0.339583\pi\)
−0.482903 + 0.875674i \(0.660417\pi\)
\(174\) 2.16543 0.164161
\(175\) 0 0
\(176\) 6.51056 0.490752
\(177\) 0.580745i 0.0436514i
\(178\) 6.00000i 0.449719i
\(179\) −24.2835 −1.81504 −0.907518 0.420013i \(-0.862026\pi\)
−0.907518 + 0.420013i \(0.862026\pi\)
\(180\) 0 0
\(181\) 2.16543 0.160955 0.0804775 0.996756i \(-0.474355\pi\)
0.0804775 + 0.996756i \(0.474355\pi\)
\(182\) 2.37309i 0.175905i
\(183\) − 0.538514i − 0.0398081i
\(184\) −5.02112 −0.370162
\(185\) 0 0
\(186\) −3.55963 −0.261005
\(187\) 16.3451i 1.19527i
\(188\) − 5.42784i − 0.395866i
\(189\) 6.92297 0.503572
\(190\) 0 0
\(191\) −19.3326 −1.39886 −0.699429 0.714702i \(-0.746562\pi\)
−0.699429 + 0.714702i \(0.746562\pi\)
\(192\) − 0.406728i − 0.0293531i
\(193\) − 20.0422i − 1.44267i −0.692586 0.721336i \(-0.743529\pi\)
0.692586 0.721336i \(-0.256471\pi\)
\(194\) −2.34513 −0.168370
\(195\) 0 0
\(196\) 1.51056 0.107897
\(197\) − 15.6269i − 1.11337i −0.830723 0.556686i \(-0.812073\pi\)
0.830723 0.556686i \(-0.187927\pi\)
\(198\) − 18.4546i − 1.31151i
\(199\) −20.2835 −1.43786 −0.718931 0.695082i \(-0.755368\pi\)
−0.718931 + 0.695082i \(0.755368\pi\)
\(200\) 0 0
\(201\) −2.20766 −0.155716
\(202\) 8.20766i 0.577489i
\(203\) − 15.5317i − 1.09011i
\(204\) 1.02112 0.0714924
\(205\) 0 0
\(206\) −8.81346 −0.614063
\(207\) 14.2327i 0.989242i
\(208\) − 0.813457i − 0.0564031i
\(209\) −2.64803 −0.183168
\(210\) 0 0
\(211\) 18.7182 1.28862 0.644308 0.764766i \(-0.277146\pi\)
0.644308 + 0.764766i \(0.277146\pi\)
\(212\) 2.34513i 0.161064i
\(213\) 5.95777i 0.408220i
\(214\) −15.2624 −1.04332
\(215\) 0 0
\(216\) −2.37309 −0.161468
\(217\) 25.5317i 1.73320i
\(218\) 4.30290i 0.291429i
\(219\) 4.48260 0.302906
\(220\) 0 0
\(221\) 2.04223 0.137375
\(222\) 0.406728i 0.0272978i
\(223\) − 15.7307i − 1.05341i −0.850049 0.526704i \(-0.823428\pi\)
0.850049 0.526704i \(-0.176572\pi\)
\(224\) −2.91729 −0.194919
\(225\) 0 0
\(226\) −9.15859 −0.609220
\(227\) − 12.8836i − 0.855117i −0.903988 0.427559i \(-0.859374\pi\)
0.903988 0.427559i \(-0.140626\pi\)
\(228\) 0.165428i 0.0109557i
\(229\) −5.25383 −0.347183 −0.173591 0.984818i \(-0.555537\pi\)
−0.173591 + 0.984818i \(0.555537\pi\)
\(230\) 0 0
\(231\) 7.72506 0.508271
\(232\) 5.32401i 0.349539i
\(233\) 28.3172i 1.85512i 0.373675 + 0.927560i \(0.378098\pi\)
−0.373675 + 0.927560i \(0.621902\pi\)
\(234\) −2.30580 −0.150735
\(235\) 0 0
\(236\) −1.42784 −0.0929447
\(237\) 0.715313i 0.0464646i
\(238\) − 7.32401i − 0.474745i
\(239\) −14.3788 −0.930085 −0.465043 0.885288i \(-0.653961\pi\)
−0.465043 + 0.885288i \(0.653961\pi\)
\(240\) 0 0
\(241\) 18.2749 1.17719 0.588596 0.808427i \(-0.299681\pi\)
0.588596 + 0.808427i \(0.299681\pi\)
\(242\) − 31.3874i − 2.01766i
\(243\) 10.1854i 0.653393i
\(244\) 1.32401 0.0847613
\(245\) 0 0
\(246\) 2.58074 0.164542
\(247\) 0.330856i 0.0210519i
\(248\) − 8.75186i − 0.555744i
\(249\) 2.86937 0.181839
\(250\) 0 0
\(251\) −1.22019 −0.0770174 −0.0385087 0.999258i \(-0.512261\pi\)
−0.0385087 + 0.999258i \(0.512261\pi\)
\(252\) 8.26926i 0.520914i
\(253\) 32.6903i 2.05522i
\(254\) 8.61439 0.540515
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 27.1306i − 1.69236i −0.532895 0.846181i \(-0.678896\pi\)
0.532895 0.846181i \(-0.321104\pi\)
\(258\) − 2.97888i − 0.185457i
\(259\) 2.91729 0.181271
\(260\) 0 0
\(261\) 15.0913 0.934128
\(262\) − 13.4278i − 0.829575i
\(263\) − 20.2133i − 1.24641i −0.782059 0.623204i \(-0.785831\pi\)
0.782059 0.623204i \(-0.214169\pi\)
\(264\) −2.64803 −0.162975
\(265\) 0 0
\(266\) 1.18654 0.0727516
\(267\) − 2.44037i − 0.149348i
\(268\) − 5.42784i − 0.331558i
\(269\) 0.207658 0.0126611 0.00633057 0.999980i \(-0.497985\pi\)
0.00633057 + 0.999980i \(0.497985\pi\)
\(270\) 0 0
\(271\) −30.6480 −1.86174 −0.930868 0.365357i \(-0.880947\pi\)
−0.930868 + 0.365357i \(0.880947\pi\)
\(272\) 2.51056i 0.152225i
\(273\) − 0.965202i − 0.0584167i
\(274\) 14.6903 0.887471
\(275\) 0 0
\(276\) 2.04223 0.122928
\(277\) 16.8135i 1.01022i 0.863054 + 0.505111i \(0.168549\pi\)
−0.863054 + 0.505111i \(0.831451\pi\)
\(278\) 17.3662i 1.04156i
\(279\) −24.8078 −1.48520
\(280\) 0 0
\(281\) −14.2749 −0.851572 −0.425786 0.904824i \(-0.640002\pi\)
−0.425786 + 0.904824i \(0.640002\pi\)
\(282\) 2.20766i 0.131464i
\(283\) − 13.2288i − 0.786369i −0.919460 0.393184i \(-0.871373\pi\)
0.919460 0.393184i \(-0.128627\pi\)
\(284\) −14.6480 −0.869201
\(285\) 0 0
\(286\) −5.29606 −0.313162
\(287\) − 18.5106i − 1.09264i
\(288\) − 2.83457i − 0.167029i
\(289\) 10.6971 0.629241
\(290\) 0 0
\(291\) 0.953831 0.0559146
\(292\) 11.0211i 0.644962i
\(293\) 2.34513i 0.137004i 0.997651 + 0.0685020i \(0.0218219\pi\)
−0.997651 + 0.0685020i \(0.978178\pi\)
\(294\) −0.614387 −0.0358318
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 15.4501i 0.896507i
\(298\) − 0.207658i − 0.0120293i
\(299\) 4.08446 0.236210
\(300\) 0 0
\(301\) −21.3662 −1.23153
\(302\) − 0.813457i − 0.0468092i
\(303\) − 3.33829i − 0.191780i
\(304\) −0.406728 −0.0233275
\(305\) 0 0
\(306\) 7.11636 0.406815
\(307\) 23.2624i 1.32766i 0.747885 + 0.663828i \(0.231069\pi\)
−0.747885 + 0.663828i \(0.768931\pi\)
\(308\) 18.9932i 1.08224i
\(309\) 3.58468 0.203926
\(310\) 0 0
\(311\) −31.3999 −1.78052 −0.890262 0.455449i \(-0.849479\pi\)
−0.890262 + 0.455449i \(0.849479\pi\)
\(312\) 0.330856i 0.0187310i
\(313\) − 8.04223i − 0.454574i −0.973828 0.227287i \(-0.927014\pi\)
0.973828 0.227287i \(-0.0729855\pi\)
\(314\) −5.32401 −0.300452
\(315\) 0 0
\(316\) −1.75870 −0.0989346
\(317\) 12.3029i 0.691000i 0.938419 + 0.345500i \(0.112291\pi\)
−0.938419 + 0.345500i \(0.887709\pi\)
\(318\) − 0.953831i − 0.0534882i
\(319\) 34.6623 1.94072
\(320\) 0 0
\(321\) 6.20766 0.346478
\(322\) − 14.6480i − 0.816303i
\(323\) − 1.02112i − 0.0568164i
\(324\) −7.53851 −0.418806
\(325\) 0 0
\(326\) 15.2008 0.841895
\(327\) − 1.75011i − 0.0967814i
\(328\) 6.34513i 0.350351i
\(329\) 15.8346 0.872988
\(330\) 0 0
\(331\) −6.77981 −0.372652 −0.186326 0.982488i \(-0.559658\pi\)
−0.186326 + 0.982488i \(0.559658\pi\)
\(332\) 7.05476i 0.387180i
\(333\) 2.83457i 0.155334i
\(334\) 12.4826 0.683018
\(335\) 0 0
\(336\) 1.18654 0.0647312
\(337\) − 10.6903i − 0.582336i −0.956672 0.291168i \(-0.905956\pi\)
0.956672 0.291168i \(-0.0940438\pi\)
\(338\) − 12.3383i − 0.671114i
\(339\) 3.72506 0.202317
\(340\) 0 0
\(341\) −56.9795 −3.08561
\(342\) 1.15290i 0.0623417i
\(343\) − 16.0143i − 0.864689i
\(344\) 7.32401 0.394884
\(345\) 0 0
\(346\) 23.0354 1.23839
\(347\) − 5.62691i − 0.302069i −0.988529 0.151034i \(-0.951740\pi\)
0.988529 0.151034i \(-0.0482604\pi\)
\(348\) − 2.16543i − 0.116079i
\(349\) −15.1306 −0.809924 −0.404962 0.914334i \(-0.632715\pi\)
−0.404962 + 0.914334i \(0.632715\pi\)
\(350\) 0 0
\(351\) 1.93040 0.103037
\(352\) − 6.51056i − 0.347014i
\(353\) 35.7816i 1.90446i 0.305381 + 0.952230i \(0.401216\pi\)
−0.305381 + 0.952230i \(0.598784\pi\)
\(354\) 0.580745 0.0308662
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 2.97888i 0.157659i
\(358\) 24.2835i 1.28342i
\(359\) −4.48260 −0.236583 −0.118291 0.992979i \(-0.537742\pi\)
−0.118291 + 0.992979i \(0.537742\pi\)
\(360\) 0 0
\(361\) −18.8346 −0.991293
\(362\) − 2.16543i − 0.113812i
\(363\) 12.7661i 0.670048i
\(364\) 2.37309 0.124384
\(365\) 0 0
\(366\) −0.538514 −0.0281486
\(367\) 21.5653i 1.12570i 0.826559 + 0.562850i \(0.190295\pi\)
−0.826559 + 0.562850i \(0.809705\pi\)
\(368\) 5.02112i 0.261744i
\(369\) 17.9857 0.936300
\(370\) 0 0
\(371\) −6.84141 −0.355188
\(372\) 3.55963i 0.184558i
\(373\) 16.9230i 0.876238i 0.898917 + 0.438119i \(0.144355\pi\)
−0.898917 + 0.438119i \(0.855645\pi\)
\(374\) 16.3451 0.845187
\(375\) 0 0
\(376\) −5.42784 −0.279920
\(377\) − 4.33086i − 0.223050i
\(378\) − 6.92297i − 0.356079i
\(379\) −31.2710 −1.60628 −0.803142 0.595788i \(-0.796840\pi\)
−0.803142 + 0.595788i \(0.796840\pi\)
\(380\) 0 0
\(381\) −3.50372 −0.179501
\(382\) 19.3326i 0.989142i
\(383\) 1.62691i 0.0831314i 0.999136 + 0.0415657i \(0.0132346\pi\)
−0.999136 + 0.0415657i \(0.986765\pi\)
\(384\) −0.406728 −0.0207558
\(385\) 0 0
\(386\) −20.0422 −1.02012
\(387\) − 20.7604i − 1.05531i
\(388\) 2.34513i 0.119056i
\(389\) 31.6412 1.60427 0.802136 0.597141i \(-0.203697\pi\)
0.802136 + 0.597141i \(0.203697\pi\)
\(390\) 0 0
\(391\) −12.6058 −0.637503
\(392\) − 1.51056i − 0.0762947i
\(393\) 5.46149i 0.275496i
\(394\) −15.6269 −0.787273
\(395\) 0 0
\(396\) −18.4546 −0.927381
\(397\) 39.2961i 1.97221i 0.166116 + 0.986106i \(0.446878\pi\)
−0.166116 + 0.986106i \(0.553122\pi\)
\(398\) 20.2835i 1.01672i
\(399\) −0.482601 −0.0241603
\(400\) 0 0
\(401\) −9.66914 −0.482854 −0.241427 0.970419i \(-0.577615\pi\)
−0.241427 + 0.970419i \(0.577615\pi\)
\(402\) 2.20766i 0.110108i
\(403\) 7.11926i 0.354636i
\(404\) 8.20766 0.408346
\(405\) 0 0
\(406\) −15.5317 −0.770824
\(407\) 6.51056i 0.322716i
\(408\) − 1.02112i − 0.0505527i
\(409\) 26.2749 1.29921 0.649606 0.760271i \(-0.274934\pi\)
0.649606 + 0.760271i \(0.274934\pi\)
\(410\) 0 0
\(411\) −5.97495 −0.294722
\(412\) 8.81346i 0.434208i
\(413\) − 4.16543i − 0.204967i
\(414\) 14.2327 0.699500
\(415\) 0 0
\(416\) −0.813457 −0.0398830
\(417\) − 7.06335i − 0.345894i
\(418\) 2.64803i 0.129519i
\(419\) 30.1940 1.47507 0.737536 0.675308i \(-0.235989\pi\)
0.737536 + 0.675308i \(0.235989\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) − 18.7182i − 0.911188i
\(423\) 15.3856i 0.748074i
\(424\) 2.34513 0.113890
\(425\) 0 0
\(426\) 5.95777 0.288655
\(427\) 3.86253i 0.186921i
\(428\) 15.2624i 0.737737i
\(429\) 2.15406 0.103999
\(430\) 0 0
\(431\) 25.5653 1.23144 0.615719 0.787966i \(-0.288866\pi\)
0.615719 + 0.787966i \(0.288866\pi\)
\(432\) 2.37309i 0.114175i
\(433\) 32.0422i 1.53985i 0.638134 + 0.769926i \(0.279707\pi\)
−0.638134 + 0.769926i \(0.720293\pi\)
\(434\) 25.5317 1.22556
\(435\) 0 0
\(436\) 4.30290 0.206072
\(437\) − 2.04223i − 0.0976931i
\(438\) − 4.48260i − 0.214187i
\(439\) −3.93840 −0.187970 −0.0939848 0.995574i \(-0.529961\pi\)
−0.0939848 + 0.995574i \(0.529961\pi\)
\(440\) 0 0
\(441\) −4.28178 −0.203894
\(442\) − 2.04223i − 0.0971390i
\(443\) − 0.0758724i − 0.00360481i −0.999998 0.00180240i \(-0.999426\pi\)
0.999998 0.00180240i \(-0.000573723\pi\)
\(444\) 0.406728 0.0193025
\(445\) 0 0
\(446\) −15.7307 −0.744872
\(447\) 0.0844605i 0.00399485i
\(448\) 2.91729i 0.137829i
\(449\) −19.2961 −0.910637 −0.455319 0.890329i \(-0.650475\pi\)
−0.455319 + 0.890329i \(0.650475\pi\)
\(450\) 0 0
\(451\) 41.3103 1.94523
\(452\) 9.15859i 0.430784i
\(453\) 0.330856i 0.0155450i
\(454\) −12.8836 −0.604659
\(455\) 0 0
\(456\) 0.165428 0.00774688
\(457\) − 3.00684i − 0.140654i −0.997524 0.0703271i \(-0.977596\pi\)
0.997524 0.0703271i \(-0.0224043\pi\)
\(458\) 5.25383i 0.245495i
\(459\) −5.95777 −0.278085
\(460\) 0 0
\(461\) −0.633755 −0.0295169 −0.0147585 0.999891i \(-0.504698\pi\)
−0.0147585 + 0.999891i \(0.504698\pi\)
\(462\) − 7.72506i − 0.359402i
\(463\) 18.0422i 0.838494i 0.907872 + 0.419247i \(0.137706\pi\)
−0.907872 + 0.419247i \(0.862294\pi\)
\(464\) 5.32401 0.247161
\(465\) 0 0
\(466\) 28.3172 1.31177
\(467\) 23.4758i 1.08633i 0.839626 + 0.543164i \(0.182774\pi\)
−0.839626 + 0.543164i \(0.817226\pi\)
\(468\) 2.30580i 0.106586i
\(469\) 15.8346 0.731173
\(470\) 0 0
\(471\) 2.16543 0.0997777
\(472\) 1.42784i 0.0657218i
\(473\) − 47.6834i − 2.19249i
\(474\) 0.715313 0.0328554
\(475\) 0 0
\(476\) −7.32401 −0.335696
\(477\) − 6.64744i − 0.304365i
\(478\) 14.3788i 0.657670i
\(479\) 8.19907 0.374625 0.187313 0.982300i \(-0.440022\pi\)
0.187313 + 0.982300i \(0.440022\pi\)
\(480\) 0 0
\(481\) 0.813457 0.0370904
\(482\) − 18.2749i − 0.832401i
\(483\) 5.95777i 0.271088i
\(484\) −31.3874 −1.42670
\(485\) 0 0
\(486\) 10.1854 0.462019
\(487\) − 0.953831i − 0.0432222i −0.999766 0.0216111i \(-0.993120\pi\)
0.999766 0.0216111i \(-0.00687956\pi\)
\(488\) − 1.32401i − 0.0599353i
\(489\) −6.18260 −0.279587
\(490\) 0 0
\(491\) 19.3383 0.872725 0.436362 0.899771i \(-0.356267\pi\)
0.436362 + 0.899771i \(0.356267\pi\)
\(492\) − 2.58074i − 0.116349i
\(493\) 13.3662i 0.601985i
\(494\) 0.330856 0.0148859
\(495\) 0 0
\(496\) −8.75186 −0.392970
\(497\) − 42.7325i − 1.91681i
\(498\) − 2.86937i − 0.128580i
\(499\) 5.63550 0.252280 0.126140 0.992012i \(-0.459741\pi\)
0.126140 + 0.992012i \(0.459741\pi\)
\(500\) 0 0
\(501\) −5.07703 −0.226825
\(502\) 1.22019i 0.0544595i
\(503\) − 33.6269i − 1.49935i −0.661806 0.749675i \(-0.730210\pi\)
0.661806 0.749675i \(-0.269790\pi\)
\(504\) 8.26926 0.368342
\(505\) 0 0
\(506\) 32.6903 1.45326
\(507\) 5.01833i 0.222872i
\(508\) − 8.61439i − 0.382202i
\(509\) 18.6903 0.828431 0.414216 0.910179i \(-0.364056\pi\)
0.414216 + 0.910179i \(0.364056\pi\)
\(510\) 0 0
\(511\) −32.1517 −1.42231
\(512\) − 1.00000i − 0.0441942i
\(513\) − 0.965202i − 0.0426147i
\(514\) −27.1306 −1.19668
\(515\) 0 0
\(516\) −2.97888 −0.131138
\(517\) 35.3383i 1.55418i
\(518\) − 2.91729i − 0.128178i
\(519\) −9.36915 −0.411260
\(520\) 0 0
\(521\) 25.1586 1.10222 0.551109 0.834433i \(-0.314205\pi\)
0.551109 + 0.834433i \(0.314205\pi\)
\(522\) − 15.0913i − 0.660528i
\(523\) − 7.25383i − 0.317188i −0.987344 0.158594i \(-0.949304\pi\)
0.987344 0.158594i \(-0.0506960\pi\)
\(524\) −13.4278 −0.586598
\(525\) 0 0
\(526\) −20.2133 −0.881344
\(527\) − 21.9720i − 0.957117i
\(528\) 2.64803i 0.115241i
\(529\) −2.21160 −0.0961564
\(530\) 0 0
\(531\) 4.04733 0.175639
\(532\) − 1.18654i − 0.0514432i
\(533\) − 5.16149i − 0.223569i
\(534\) −2.44037 −0.105605
\(535\) 0 0
\(536\) −5.42784 −0.234447
\(537\) − 9.87680i − 0.426215i
\(538\) − 0.207658i − 0.00895278i
\(539\) −9.83457 −0.423605
\(540\) 0 0
\(541\) 16.0422 0.689709 0.344855 0.938656i \(-0.387928\pi\)
0.344855 + 0.938656i \(0.387928\pi\)
\(542\) 30.6480i 1.31645i
\(543\) 0.880741i 0.0377962i
\(544\) 2.51056 0.107639
\(545\) 0 0
\(546\) −0.965202 −0.0413068
\(547\) − 16.0143i − 0.684721i −0.939569 0.342360i \(-0.888774\pi\)
0.939569 0.342360i \(-0.111226\pi\)
\(548\) − 14.6903i − 0.627537i
\(549\) −3.75301 −0.160175
\(550\) 0 0
\(551\) −2.16543 −0.0922503
\(552\) − 2.04223i − 0.0869231i
\(553\) − 5.13063i − 0.218177i
\(554\) 16.8135 0.714335
\(555\) 0 0
\(556\) 17.3662 0.736493
\(557\) 10.8557i 0.459970i 0.973194 + 0.229985i \(0.0738678\pi\)
−0.973194 + 0.229985i \(0.926132\pi\)
\(558\) 24.8078i 1.05020i
\(559\) −5.95777 −0.251987
\(560\) 0 0
\(561\) −6.64803 −0.280680
\(562\) 14.2749i 0.602152i
\(563\) 31.0776i 1.30977i 0.755731 + 0.654883i \(0.227282\pi\)
−0.755731 + 0.654883i \(0.772718\pi\)
\(564\) 2.20766 0.0929592
\(565\) 0 0
\(566\) −13.2288 −0.556047
\(567\) − 21.9920i − 0.923577i
\(568\) 14.6480i 0.614618i
\(569\) −30.0845 −1.26121 −0.630603 0.776105i \(-0.717192\pi\)
−0.630603 + 0.776105i \(0.717192\pi\)
\(570\) 0 0
\(571\) 5.55673 0.232542 0.116271 0.993218i \(-0.462906\pi\)
0.116271 + 0.993218i \(0.462906\pi\)
\(572\) 5.29606i 0.221439i
\(573\) − 7.86312i − 0.328487i
\(574\) −18.5106 −0.772616
\(575\) 0 0
\(576\) −2.83457 −0.118107
\(577\) 22.2499i 0.926275i 0.886286 + 0.463137i \(0.153276\pi\)
−0.886286 + 0.463137i \(0.846724\pi\)
\(578\) − 10.6971i − 0.444941i
\(579\) 8.15174 0.338775
\(580\) 0 0
\(581\) −20.5807 −0.853833
\(582\) − 0.953831i − 0.0395376i
\(583\) − 15.2681i − 0.632340i
\(584\) 11.0211 0.456057
\(585\) 0 0
\(586\) 2.34513 0.0968764
\(587\) − 45.3103i − 1.87016i −0.354440 0.935079i \(-0.615328\pi\)
0.354440 0.935079i \(-0.384672\pi\)
\(588\) 0.614387i 0.0253369i
\(589\) 3.55963 0.146672
\(590\) 0 0
\(591\) 6.35591 0.261447
\(592\) 1.00000i 0.0410997i
\(593\) − 0.232712i − 0.00955635i −0.999989 0.00477817i \(-0.998479\pi\)
0.999989 0.00477817i \(-0.00152095\pi\)
\(594\) 15.4501 0.633926
\(595\) 0 0
\(596\) −0.207658 −0.00850601
\(597\) − 8.24989i − 0.337645i
\(598\) − 4.08446i − 0.167026i
\(599\) −43.4056 −1.77350 −0.886752 0.462246i \(-0.847044\pi\)
−0.886752 + 0.462246i \(0.847044\pi\)
\(600\) 0 0
\(601\) 35.5317 1.44937 0.724684 0.689082i \(-0.241986\pi\)
0.724684 + 0.689082i \(0.241986\pi\)
\(602\) 21.3662i 0.870823i
\(603\) 15.3856i 0.626551i
\(604\) −0.813457 −0.0330991
\(605\) 0 0
\(606\) −3.33829 −0.135609
\(607\) − 26.7325i − 1.08504i −0.840043 0.542519i \(-0.817471\pi\)
0.840043 0.542519i \(-0.182529\pi\)
\(608\) 0.406728i 0.0164950i
\(609\) 6.31717 0.255985
\(610\) 0 0
\(611\) 4.41532 0.178625
\(612\) − 7.11636i − 0.287662i
\(613\) 48.0565i 1.94098i 0.241134 + 0.970492i \(0.422481\pi\)
−0.241134 + 0.970492i \(0.577519\pi\)
\(614\) 23.2624 0.938795
\(615\) 0 0
\(616\) 18.9932 0.765256
\(617\) − 12.9789i − 0.522510i −0.965270 0.261255i \(-0.915864\pi\)
0.965270 0.261255i \(-0.0841364\pi\)
\(618\) − 3.58468i − 0.144197i
\(619\) −25.3662 −1.01956 −0.509778 0.860306i \(-0.670272\pi\)
−0.509778 + 0.860306i \(0.670272\pi\)
\(620\) 0 0
\(621\) −11.9155 −0.478154
\(622\) 31.3999i 1.25902i
\(623\) 17.5037i 0.701272i
\(624\) 0.330856 0.0132448
\(625\) 0 0
\(626\) −8.04223 −0.321432
\(627\) − 1.07703i − 0.0430124i
\(628\) 5.32401i 0.212451i
\(629\) −2.51056 −0.100102
\(630\) 0 0
\(631\) −31.6075 −1.25828 −0.629138 0.777293i \(-0.716592\pi\)
−0.629138 + 0.777293i \(0.716592\pi\)
\(632\) 1.75870i 0.0699573i
\(633\) 7.61323i 0.302599i
\(634\) 12.3029 0.488611
\(635\) 0 0
\(636\) −0.953831 −0.0378219
\(637\) 1.22877i 0.0486858i
\(638\) − 34.6623i − 1.37229i
\(639\) 41.5209 1.64254
\(640\) 0 0
\(641\) −23.8625 −0.942513 −0.471257 0.881996i \(-0.656199\pi\)
−0.471257 + 0.881996i \(0.656199\pi\)
\(642\) − 6.20766i − 0.244997i
\(643\) 32.8277i 1.29460i 0.762236 + 0.647300i \(0.224102\pi\)
−0.762236 + 0.647300i \(0.775898\pi\)
\(644\) −14.6480 −0.577213
\(645\) 0 0
\(646\) −1.02112 −0.0401752
\(647\) 13.9578i 0.548737i 0.961625 + 0.274368i \(0.0884687\pi\)
−0.961625 + 0.274368i \(0.911531\pi\)
\(648\) 7.53851i 0.296141i
\(649\) 9.29606 0.364902
\(650\) 0 0
\(651\) −10.3845 −0.406999
\(652\) − 15.2008i − 0.595310i
\(653\) 28.5921i 1.11890i 0.828865 + 0.559448i \(0.188987\pi\)
−0.828865 + 0.559448i \(0.811013\pi\)
\(654\) −1.75011 −0.0684348
\(655\) 0 0
\(656\) 6.34513 0.247736
\(657\) − 31.2401i − 1.21879i
\(658\) − 15.8346i − 0.617296i
\(659\) 5.62691 0.219193 0.109597 0.993976i \(-0.465044\pi\)
0.109597 + 0.993976i \(0.465044\pi\)
\(660\) 0 0
\(661\) 13.3240 0.518244 0.259122 0.965845i \(-0.416567\pi\)
0.259122 + 0.965845i \(0.416567\pi\)
\(662\) 6.77981i 0.263505i
\(663\) 0.830633i 0.0322591i
\(664\) 7.05476 0.273778
\(665\) 0 0
\(666\) 2.83457 0.109837
\(667\) 26.7325i 1.03509i
\(668\) − 12.4826i − 0.482966i
\(669\) 6.39814 0.247366
\(670\) 0 0
\(671\) −8.62007 −0.332774
\(672\) − 1.18654i − 0.0457719i
\(673\) 29.6691i 1.14366i 0.820372 + 0.571831i \(0.193767\pi\)
−0.820372 + 0.571831i \(0.806233\pi\)
\(674\) −10.6903 −0.411773
\(675\) 0 0
\(676\) −12.3383 −0.474550
\(677\) 1.25383i 0.0481885i 0.999710 + 0.0240943i \(0.00767018\pi\)
−0.999710 + 0.0240943i \(0.992330\pi\)
\(678\) − 3.72506i − 0.143060i
\(679\) −6.84141 −0.262549
\(680\) 0 0
\(681\) 5.24014 0.200803
\(682\) 56.9795i 2.18186i
\(683\) − 4.76045i − 0.182153i −0.995844 0.0910767i \(-0.970969\pi\)
0.995844 0.0910767i \(-0.0290308\pi\)
\(684\) 1.15290 0.0440823
\(685\) 0 0
\(686\) −16.0143 −0.611428
\(687\) − 2.13688i − 0.0815271i
\(688\) − 7.32401i − 0.279225i
\(689\) −1.90766 −0.0726761
\(690\) 0 0
\(691\) −20.2219 −0.769279 −0.384639 0.923067i \(-0.625674\pi\)
−0.384639 + 0.923067i \(0.625674\pi\)
\(692\) − 23.0354i − 0.875674i
\(693\) − 53.8375i − 2.04512i
\(694\) −5.62691 −0.213595
\(695\) 0 0
\(696\) −2.16543 −0.0820803
\(697\) 15.9298i 0.603385i
\(698\) 15.1306i 0.572703i
\(699\) −11.5174 −0.435628
\(700\) 0 0
\(701\) −22.0845 −0.834119 −0.417059 0.908879i \(-0.636939\pi\)
−0.417059 + 0.908879i \(0.636939\pi\)
\(702\) − 1.93040i − 0.0728584i
\(703\) − 0.406728i − 0.0153401i
\(704\) −6.51056 −0.245376
\(705\) 0 0
\(706\) 35.7816 1.34666
\(707\) 23.9441i 0.900510i
\(708\) − 0.580745i − 0.0218257i
\(709\) −5.59896 −0.210273 −0.105137 0.994458i \(-0.533528\pi\)
−0.105137 + 0.994458i \(0.533528\pi\)
\(710\) 0 0
\(711\) 4.98516 0.186958
\(712\) − 6.00000i − 0.224860i
\(713\) − 43.9441i − 1.64572i
\(714\) 2.97888 0.111482
\(715\) 0 0
\(716\) 24.2835 0.907518
\(717\) − 5.84826i − 0.218407i
\(718\) 4.48260i 0.167289i
\(719\) −40.2921 −1.50264 −0.751321 0.659937i \(-0.770583\pi\)
−0.751321 + 0.659937i \(0.770583\pi\)
\(720\) 0 0
\(721\) −25.7114 −0.957542
\(722\) 18.8346i 0.700950i
\(723\) 7.43294i 0.276434i
\(724\) −2.16543 −0.0804775
\(725\) 0 0
\(726\) 12.7661 0.473796
\(727\) − 11.2538i − 0.417381i −0.977982 0.208691i \(-0.933080\pi\)
0.977982 0.208691i \(-0.0669202\pi\)
\(728\) − 2.37309i − 0.0879524i
\(729\) 18.4729 0.684180
\(730\) 0 0
\(731\) 18.3874 0.680081
\(732\) 0.538514i 0.0199041i
\(733\) − 42.2892i − 1.56199i −0.624539 0.780994i \(-0.714713\pi\)
0.624539 0.780994i \(-0.285287\pi\)
\(734\) 21.5653 0.795990
\(735\) 0 0
\(736\) 5.02112 0.185081
\(737\) 35.3383i 1.30170i
\(738\) − 17.9857i − 0.662064i
\(739\) 29.3103 1.07820 0.539099 0.842242i \(-0.318765\pi\)
0.539099 + 0.842242i \(0.318765\pi\)
\(740\) 0 0
\(741\) −0.134569 −0.00494350
\(742\) 6.84141i 0.251156i
\(743\) − 4.39245i − 0.161144i −0.996749 0.0805718i \(-0.974325\pi\)
0.996749 0.0805718i \(-0.0256746\pi\)
\(744\) 3.55963 0.130502
\(745\) 0 0
\(746\) 16.9230 0.619594
\(747\) − 19.9972i − 0.731660i
\(748\) − 16.3451i − 0.597637i
\(749\) −44.5248 −1.62690
\(750\) 0 0
\(751\) 2.58074 0.0941727 0.0470864 0.998891i \(-0.485006\pi\)
0.0470864 + 0.998891i \(0.485006\pi\)
\(752\) 5.42784i 0.197933i
\(753\) − 0.496284i − 0.0180856i
\(754\) −4.33086 −0.157720
\(755\) 0 0
\(756\) −6.92297 −0.251786
\(757\) − 27.3805i − 0.995162i −0.867418 0.497581i \(-0.834222\pi\)
0.867418 0.497581i \(-0.165778\pi\)
\(758\) 31.2710i 1.13581i
\(759\) −13.2961 −0.482616
\(760\) 0 0
\(761\) −42.9509 −1.55697 −0.778485 0.627663i \(-0.784011\pi\)
−0.778485 + 0.627663i \(0.784011\pi\)
\(762\) 3.50372i 0.126926i
\(763\) 12.5528i 0.454441i
\(764\) 19.3326 0.699429
\(765\) 0 0
\(766\) 1.62691 0.0587828
\(767\) − 1.16149i − 0.0419389i
\(768\) 0.406728i 0.0146765i
\(769\) 13.0633 0.471076 0.235538 0.971865i \(-0.424315\pi\)
0.235538 + 0.971865i \(0.424315\pi\)
\(770\) 0 0
\(771\) 11.0348 0.397409
\(772\) 20.0422i 0.721336i
\(773\) − 15.3662i − 0.552685i −0.961059 0.276343i \(-0.910878\pi\)
0.961059 0.276343i \(-0.0891225\pi\)
\(774\) −20.7604 −0.746219
\(775\) 0 0
\(776\) 2.34513 0.0841852
\(777\) 1.18654i 0.0425670i
\(778\) − 31.6412i − 1.13439i
\(779\) −2.58074 −0.0924648
\(780\) 0 0
\(781\) 95.3668 3.41249
\(782\) 12.6058i 0.450782i
\(783\) 12.6343i 0.451515i
\(784\) −1.51056 −0.0539485
\(785\) 0 0
\(786\) 5.46149 0.194805
\(787\) − 10.9316i − 0.389668i −0.980836 0.194834i \(-0.937583\pi\)
0.980836 0.194834i \(-0.0624168\pi\)
\(788\) 15.6269i 0.556686i
\(789\) 8.22134 0.292688
\(790\) 0 0
\(791\) −26.7182 −0.949990
\(792\) 18.4546i 0.655757i
\(793\) 1.07703i 0.0382464i
\(794\) 39.2961 1.39456
\(795\) 0 0
\(796\) 20.2835 0.718931
\(797\) 24.2921i 0.860471i 0.902717 + 0.430235i \(0.141569\pi\)
−0.902717 + 0.430235i \(0.858431\pi\)
\(798\) 0.482601i 0.0170839i
\(799\) −13.6269 −0.482086
\(800\) 0 0
\(801\) −17.0074 −0.600928
\(802\) 9.66914i 0.341429i
\(803\) − 71.7536i − 2.53213i
\(804\) 2.20766 0.0778581
\(805\) 0 0
\(806\) 7.11926 0.250765
\(807\) 0.0844605i 0.00297315i
\(808\) − 8.20766i − 0.288744i
\(809\) 40.0422 1.40781 0.703905 0.710294i \(-0.251438\pi\)
0.703905 + 0.710294i \(0.251438\pi\)
\(810\) 0 0
\(811\) −40.6343 −1.42686 −0.713432 0.700724i \(-0.752860\pi\)
−0.713432 + 0.700724i \(0.752860\pi\)
\(812\) 15.5317i 0.545055i
\(813\) − 12.4654i − 0.437182i
\(814\) 6.51056 0.228195
\(815\) 0 0
\(816\) −1.02112 −0.0357462
\(817\) 2.97888i 0.104218i
\(818\) − 26.2749i − 0.918682i
\(819\) −6.72668 −0.235049
\(820\) 0 0
\(821\) −41.4787 −1.44762 −0.723808 0.690002i \(-0.757610\pi\)
−0.723808 + 0.690002i \(0.757610\pi\)
\(822\) 5.97495i 0.208400i
\(823\) − 5.27610i − 0.183913i −0.995763 0.0919566i \(-0.970688\pi\)
0.995763 0.0919566i \(-0.0293121\pi\)
\(824\) 8.81346 0.307031
\(825\) 0 0
\(826\) −4.16543 −0.144934
\(827\) − 1.76438i − 0.0613537i −0.999529 0.0306768i \(-0.990234\pi\)
0.999529 0.0306768i \(-0.00976627\pi\)
\(828\) − 14.2327i − 0.494621i
\(829\) −16.3029 −0.566223 −0.283112 0.959087i \(-0.591367\pi\)
−0.283112 + 0.959087i \(0.591367\pi\)
\(830\) 0 0
\(831\) −6.83851 −0.237225
\(832\) 0.813457i 0.0282015i
\(833\) − 3.79234i − 0.131397i
\(834\) −7.06335 −0.244584
\(835\) 0 0
\(836\) 2.64803 0.0915840
\(837\) − 20.7689i − 0.717879i
\(838\) − 30.1940i − 1.04303i
\(839\) −14.3731 −0.496214 −0.248107 0.968733i \(-0.579808\pi\)
−0.248107 + 0.968733i \(0.579808\pi\)
\(840\) 0 0
\(841\) −0.654870 −0.0225817
\(842\) − 22.0000i − 0.758170i
\(843\) − 5.80602i − 0.199970i
\(844\) −18.7182 −0.644308
\(845\) 0 0
\(846\) 15.3856 0.528968
\(847\) − 91.5659i − 3.14624i
\(848\) − 2.34513i − 0.0805321i
\(849\) 5.38052 0.184659
\(850\) 0 0
\(851\) −5.02112 −0.172122
\(852\) − 5.95777i − 0.204110i
\(853\) 43.2961i 1.48243i 0.671268 + 0.741214i \(0.265750\pi\)
−0.671268 + 0.741214i \(0.734250\pi\)
\(854\) 3.86253 0.132173
\(855\) 0 0
\(856\) 15.2624 0.521659
\(857\) 49.0776i 1.67646i 0.545317 + 0.838230i \(0.316409\pi\)
−0.545317 + 0.838230i \(0.683591\pi\)
\(858\) − 2.15406i − 0.0735383i
\(859\) −15.5374 −0.530128 −0.265064 0.964231i \(-0.585393\pi\)
−0.265064 + 0.964231i \(0.585393\pi\)
\(860\) 0 0
\(861\) 7.52877 0.256580
\(862\) − 25.5653i − 0.870758i
\(863\) − 36.0901i − 1.22852i −0.789103 0.614261i \(-0.789454\pi\)
0.789103 0.614261i \(-0.210546\pi\)
\(864\) 2.37309 0.0807340
\(865\) 0 0
\(866\) 32.0422 1.08884
\(867\) 4.35081i 0.147761i
\(868\) − 25.5317i − 0.866601i
\(869\) 11.4501 0.388419
\(870\) 0 0
\(871\) 4.41532 0.149607
\(872\) − 4.30290i − 0.145715i
\(873\) − 6.64744i − 0.224982i
\(874\) −2.04223 −0.0690795
\(875\) 0 0
\(876\) −4.48260 −0.151453
\(877\) − 29.3240i − 0.990202i −0.868836 0.495101i \(-0.835131\pi\)
0.868836 0.495101i \(-0.164869\pi\)
\(878\) 3.93840i 0.132915i
\(879\) −0.953831 −0.0321719
\(880\) 0 0
\(881\) 23.8066 0.802065 0.401033 0.916064i \(-0.368651\pi\)
0.401033 + 0.916064i \(0.368651\pi\)
\(882\) 4.28178i 0.144175i
\(883\) − 3.94699i − 0.132827i −0.997792 0.0664134i \(-0.978844\pi\)
0.997792 0.0664134i \(-0.0211556\pi\)
\(884\) −2.04223 −0.0686876
\(885\) 0 0
\(886\) −0.0758724 −0.00254898
\(887\) − 1.96346i − 0.0659264i −0.999457 0.0329632i \(-0.989506\pi\)
0.999457 0.0329632i \(-0.0104944\pi\)
\(888\) − 0.406728i − 0.0136489i
\(889\) 25.1306 0.842854
\(890\) 0 0
\(891\) 49.0799 1.64424
\(892\) 15.7307i 0.526704i
\(893\) − 2.20766i − 0.0738765i
\(894\) 0.0844605 0.00282478
\(895\) 0 0
\(896\) 2.91729 0.0974597
\(897\) 1.66127i 0.0554681i
\(898\) 19.2961i 0.643918i
\(899\) −46.5950 −1.55403
\(900\) 0 0
\(901\) 5.88758 0.196144
\(902\) − 41.3103i − 1.37548i
\(903\) − 8.69026i − 0.289194i
\(904\) 9.15859 0.304610
\(905\) 0 0
\(906\) 0.330856 0.0109920
\(907\) − 5.89049i − 0.195590i −0.995207 0.0977952i \(-0.968821\pi\)
0.995207 0.0977952i \(-0.0311790\pi\)
\(908\) 12.8836i 0.427559i
\(909\) −23.2652 −0.771658
\(910\) 0 0
\(911\) −27.9527 −0.926113 −0.463057 0.886329i \(-0.653247\pi\)
−0.463057 + 0.886329i \(0.653247\pi\)
\(912\) − 0.165428i − 0.00547787i
\(913\) − 45.9304i − 1.52007i
\(914\) −3.00684 −0.0994575
\(915\) 0 0
\(916\) 5.25383 0.173591
\(917\) − 39.1729i − 1.29360i
\(918\) 5.95777i 0.196636i
\(919\) −17.7587 −0.585805 −0.292903 0.956142i \(-0.594621\pi\)
−0.292903 + 0.956142i \(0.594621\pi\)
\(920\) 0 0
\(921\) −9.46149 −0.311767
\(922\) 0.633755i 0.0208716i
\(923\) − 11.9155i − 0.392205i
\(924\) −7.72506 −0.254136
\(925\) 0 0
\(926\) 18.0422 0.592904
\(927\) − 24.9824i − 0.820529i
\(928\) − 5.32401i − 0.174769i
\(929\) −34.2892 −1.12499 −0.562496 0.826800i \(-0.690159\pi\)
−0.562496 + 0.826800i \(0.690159\pi\)
\(930\) 0 0
\(931\) 0.614387 0.0201357
\(932\) − 28.3172i − 0.927560i
\(933\) − 12.7712i − 0.418111i
\(934\) 23.4758 0.768150
\(935\) 0 0
\(936\) 2.30580 0.0753675
\(937\) 14.0845i 0.460119i 0.973177 + 0.230060i \(0.0738921\pi\)
−0.973177 + 0.230060i \(0.926108\pi\)
\(938\) − 15.8346i − 0.517017i
\(939\) 3.27100 0.106745
\(940\) 0 0
\(941\) 14.2499 0.464533 0.232267 0.972652i \(-0.425386\pi\)
0.232267 + 0.972652i \(0.425386\pi\)
\(942\) − 2.16543i − 0.0705535i
\(943\) 31.8596i 1.03749i
\(944\) 1.42784 0.0464723
\(945\) 0 0
\(946\) −47.6834 −1.55032
\(947\) 29.8203i 0.969029i 0.874783 + 0.484515i \(0.161004\pi\)
−0.874783 + 0.484515i \(0.838996\pi\)
\(948\) − 0.715313i − 0.0232323i
\(949\) −8.96520 −0.291023
\(950\) 0 0
\(951\) −5.00394 −0.162264
\(952\) 7.32401i 0.237373i
\(953\) 8.64803i 0.280137i 0.990142 + 0.140069i \(0.0447323\pi\)
−0.990142 + 0.140069i \(0.955268\pi\)
\(954\) −6.64744 −0.215219
\(955\) 0 0
\(956\) 14.3788 0.465043
\(957\) 14.0981i 0.455728i
\(958\) − 8.19907i − 0.264900i
\(959\) 42.8557 1.38388
\(960\) 0 0
\(961\) 45.5950 1.47081
\(962\) − 0.813457i − 0.0262269i
\(963\) − 43.2624i − 1.39411i
\(964\) −18.2749 −0.588596
\(965\) 0 0
\(966\) 5.95777 0.191688
\(967\) 45.0325i 1.44815i 0.689723 + 0.724074i \(0.257732\pi\)
−0.689723 + 0.724074i \(0.742268\pi\)
\(968\) 31.3874i 1.00883i
\(969\) 0.415317 0.0133419
\(970\) 0 0
\(971\) 26.3029 0.844100 0.422050 0.906573i \(-0.361311\pi\)
0.422050 + 0.906573i \(0.361311\pi\)
\(972\) − 10.1854i − 0.326696i
\(973\) 50.6623i 1.62416i
\(974\) −0.953831 −0.0305627
\(975\) 0 0
\(976\) −1.32401 −0.0423807
\(977\) − 25.6549i − 0.820772i −0.911912 0.410386i \(-0.865394\pi\)
0.911912 0.410386i \(-0.134606\pi\)
\(978\) 6.18260i 0.197698i
\(979\) −39.0633 −1.24847
\(980\) 0 0
\(981\) −12.1969 −0.389416
\(982\) − 19.3383i − 0.617110i
\(983\) − 55.3611i − 1.76575i −0.469611 0.882873i \(-0.655606\pi\)
0.469611 0.882873i \(-0.344394\pi\)
\(984\) −2.58074 −0.0822711
\(985\) 0 0
\(986\) 13.3662 0.425668
\(987\) 6.44037i 0.204999i
\(988\) − 0.330856i − 0.0105259i
\(989\) 36.7747 1.16937
\(990\) 0 0
\(991\) −3.24814 −0.103181 −0.0515903 0.998668i \(-0.516429\pi\)
−0.0515903 + 0.998668i \(0.516429\pi\)
\(992\) 8.75186i 0.277872i
\(993\) − 2.75754i − 0.0875080i
\(994\) −42.7325 −1.35539
\(995\) 0 0
\(996\) −2.86937 −0.0909195
\(997\) 12.6480i 0.400567i 0.979738 + 0.200284i \(0.0641863\pi\)
−0.979738 + 0.200284i \(0.935814\pi\)
\(998\) − 5.63550i − 0.178389i
\(999\) −2.37309 −0.0750811
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.o.149.2 6
5.2 odd 4 370.2.a.g.1.2 3
5.3 odd 4 1850.2.a.z.1.2 3
5.4 even 2 inner 1850.2.b.o.149.5 6
15.2 even 4 3330.2.a.bg.1.3 3
20.7 even 4 2960.2.a.u.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.2 3 5.2 odd 4
1850.2.a.z.1.2 3 5.3 odd 4
1850.2.b.o.149.2 6 1.1 even 1 trivial
1850.2.b.o.149.5 6 5.4 even 2 inner
2960.2.a.u.1.2 3 20.7 even 4
3330.2.a.bg.1.3 3 15.2 even 4