Properties

Label 1850.2.b.i.149.4
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.4
Root \(2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.i.149.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +3.30278i q^{3} -1.00000 q^{4} -3.30278 q^{6} +2.60555i q^{7} -1.00000i q^{8} -7.90833 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +3.30278i q^{3} -1.00000 q^{4} -3.30278 q^{6} +2.60555i q^{7} -1.00000i q^{8} -7.90833 q^{9} -2.30278 q^{11} -3.30278i q^{12} +1.30278i q^{13} -2.60555 q^{14} +1.00000 q^{16} +6.00000i q^{17} -7.90833i q^{18} -2.00000 q^{19} -8.60555 q^{21} -2.30278i q^{22} +3.90833i q^{23} +3.30278 q^{24} -1.30278 q^{26} -16.2111i q^{27} -2.60555i q^{28} +3.90833 q^{29} -0.302776 q^{31} +1.00000i q^{32} -7.60555i q^{33} -6.00000 q^{34} +7.90833 q^{36} -1.00000i q^{37} -2.00000i q^{38} -4.30278 q^{39} +9.90833 q^{41} -8.60555i q^{42} +0.605551i q^{43} +2.30278 q^{44} -3.90833 q^{46} -4.60555i q^{47} +3.30278i q^{48} +0.211103 q^{49} -19.8167 q^{51} -1.30278i q^{52} -6.00000i q^{53} +16.2111 q^{54} +2.60555 q^{56} -6.60555i q^{57} +3.90833i q^{58} -10.6056 q^{59} +7.51388 q^{61} -0.302776i q^{62} -20.6056i q^{63} -1.00000 q^{64} +7.60555 q^{66} +3.51388i q^{67} -6.00000i q^{68} -12.9083 q^{69} +6.00000 q^{71} +7.90833i q^{72} -12.3028i q^{73} +1.00000 q^{74} +2.00000 q^{76} -6.00000i q^{77} -4.30278i q^{78} -9.11943 q^{79} +29.8167 q^{81} +9.90833i q^{82} +2.78890i q^{83} +8.60555 q^{84} -0.605551 q^{86} +12.9083i q^{87} +2.30278i q^{88} +9.21110 q^{89} -3.39445 q^{91} -3.90833i q^{92} -1.00000i q^{93} +4.60555 q^{94} -3.30278 q^{96} +16.4222i q^{97} +0.211103i q^{98} +18.2111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 2 q^{11} + 4 q^{14} + 4 q^{16} - 8 q^{19} - 20 q^{21} + 6 q^{24} + 2 q^{26} - 6 q^{29} + 6 q^{31} - 24 q^{34} + 10 q^{36} - 10 q^{39} + 18 q^{41} + 2 q^{44} + 6 q^{46} - 28 q^{49} - 36 q^{51} + 36 q^{54} - 4 q^{56} - 28 q^{59} - 6 q^{61} - 4 q^{64} + 16 q^{66} - 30 q^{69} + 24 q^{71} + 4 q^{74} + 8 q^{76} + 14 q^{79} + 76 q^{81} + 20 q^{84} + 12 q^{86} + 8 q^{89} - 28 q^{91} + 4 q^{94} - 6 q^{96} + 44 q^{99}+O(q^{100}) \) 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
\(3\) 3.30278i 1.90686i 0.301617 + 0.953429i \(0.402474\pi\)
−0.301617 + 0.953429i \(0.597526\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.30278 −1.34835
\(7\) 2.60555i 0.984806i 0.870367 + 0.492403i \(0.163881\pi\)
−0.870367 + 0.492403i \(0.836119\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −7.90833 −2.63611
\(10\) 0 0
\(11\) −2.30278 −0.694313 −0.347156 0.937807i \(-0.612853\pi\)
−0.347156 + 0.937807i \(0.612853\pi\)
\(12\) − 3.30278i − 0.953429i
\(13\) 1.30278i 0.361325i 0.983545 + 0.180662i \(0.0578242\pi\)
−0.983545 + 0.180662i \(0.942176\pi\)
\(14\) −2.60555 −0.696363
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) − 7.90833i − 1.86401i
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −8.60555 −1.87789
\(22\) − 2.30278i − 0.490953i
\(23\) 3.90833i 0.814942i 0.913218 + 0.407471i \(0.133589\pi\)
−0.913218 + 0.407471i \(0.866411\pi\)
\(24\) 3.30278 0.674176
\(25\) 0 0
\(26\) −1.30278 −0.255495
\(27\) − 16.2111i − 3.11983i
\(28\) − 2.60555i − 0.492403i
\(29\) 3.90833 0.725758 0.362879 0.931836i \(-0.381794\pi\)
0.362879 + 0.931836i \(0.381794\pi\)
\(30\) 0 0
\(31\) −0.302776 −0.0543801 −0.0271901 0.999630i \(-0.508656\pi\)
−0.0271901 + 0.999630i \(0.508656\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 7.60555i − 1.32396i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 7.90833 1.31805
\(37\) − 1.00000i − 0.164399i
\(38\) − 2.00000i − 0.324443i
\(39\) −4.30278 −0.688996
\(40\) 0 0
\(41\) 9.90833 1.54742 0.773710 0.633540i \(-0.218399\pi\)
0.773710 + 0.633540i \(0.218399\pi\)
\(42\) − 8.60555i − 1.32787i
\(43\) 0.605551i 0.0923457i 0.998933 + 0.0461729i \(0.0147025\pi\)
−0.998933 + 0.0461729i \(0.985297\pi\)
\(44\) 2.30278 0.347156
\(45\) 0 0
\(46\) −3.90833 −0.576251
\(47\) − 4.60555i − 0.671789i −0.941900 0.335894i \(-0.890961\pi\)
0.941900 0.335894i \(-0.109039\pi\)
\(48\) 3.30278i 0.476715i
\(49\) 0.211103 0.0301575
\(50\) 0 0
\(51\) −19.8167 −2.77489
\(52\) − 1.30278i − 0.180662i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 16.2111 2.20605
\(55\) 0 0
\(56\) 2.60555 0.348181
\(57\) − 6.60555i − 0.874927i
\(58\) 3.90833i 0.513188i
\(59\) −10.6056 −1.38073 −0.690363 0.723464i \(-0.742549\pi\)
−0.690363 + 0.723464i \(0.742549\pi\)
\(60\) 0 0
\(61\) 7.51388 0.962054 0.481027 0.876706i \(-0.340264\pi\)
0.481027 + 0.876706i \(0.340264\pi\)
\(62\) − 0.302776i − 0.0384525i
\(63\) − 20.6056i − 2.59606i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 7.60555 0.936179
\(67\) 3.51388i 0.429289i 0.976692 + 0.214644i \(0.0688592\pi\)
−0.976692 + 0.214644i \(0.931141\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) −12.9083 −1.55398
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 7.90833i 0.932005i
\(73\) − 12.3028i − 1.43993i −0.694010 0.719965i \(-0.744158\pi\)
0.694010 0.719965i \(-0.255842\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) − 6.00000i − 0.683763i
\(78\) − 4.30278i − 0.487193i
\(79\) −9.11943 −1.02602 −0.513008 0.858384i \(-0.671469\pi\)
−0.513008 + 0.858384i \(0.671469\pi\)
\(80\) 0 0
\(81\) 29.8167 3.31296
\(82\) 9.90833i 1.09419i
\(83\) 2.78890i 0.306121i 0.988217 + 0.153061i \(0.0489130\pi\)
−0.988217 + 0.153061i \(0.951087\pi\)
\(84\) 8.60555 0.938943
\(85\) 0 0
\(86\) −0.605551 −0.0652983
\(87\) 12.9083i 1.38392i
\(88\) 2.30278i 0.245477i
\(89\) 9.21110 0.976375 0.488187 0.872739i \(-0.337658\pi\)
0.488187 + 0.872739i \(0.337658\pi\)
\(90\) 0 0
\(91\) −3.39445 −0.355835
\(92\) − 3.90833i − 0.407471i
\(93\) − 1.00000i − 0.103695i
\(94\) 4.60555 0.475026
\(95\) 0 0
\(96\) −3.30278 −0.337088
\(97\) 16.4222i 1.66742i 0.552201 + 0.833711i \(0.313788\pi\)
−0.552201 + 0.833711i \(0.686212\pi\)
\(98\) 0.211103i 0.0213246i
\(99\) 18.2111 1.83028
\(100\) 0 0
\(101\) −12.4222 −1.23606 −0.618028 0.786156i \(-0.712068\pi\)
−0.618028 + 0.786156i \(0.712068\pi\)
\(102\) − 19.8167i − 1.96214i
\(103\) − 0.302776i − 0.0298334i −0.999889 0.0149167i \(-0.995252\pi\)
0.999889 0.0149167i \(-0.00474830\pi\)
\(104\) 1.30278 0.127748
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) − 0.697224i − 0.0674032i −0.999432 0.0337016i \(-0.989270\pi\)
0.999432 0.0337016i \(-0.0107296\pi\)
\(108\) 16.2111i 1.55991i
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 3.30278 0.313486
\(112\) 2.60555i 0.246201i
\(113\) − 3.21110i − 0.302075i −0.988528 0.151038i \(-0.951739\pi\)
0.988528 0.151038i \(-0.0482614\pi\)
\(114\) 6.60555 0.618667
\(115\) 0 0
\(116\) −3.90833 −0.362879
\(117\) − 10.3028i − 0.952492i
\(118\) − 10.6056i − 0.976320i
\(119\) −15.6333 −1.43310
\(120\) 0 0
\(121\) −5.69722 −0.517929
\(122\) 7.51388i 0.680275i
\(123\) 32.7250i 2.95071i
\(124\) 0.302776 0.0271901
\(125\) 0 0
\(126\) 20.6056 1.83569
\(127\) 19.2111i 1.70471i 0.522964 + 0.852355i \(0.324826\pi\)
−0.522964 + 0.852355i \(0.675174\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 10.6056 0.926611 0.463306 0.886199i \(-0.346663\pi\)
0.463306 + 0.886199i \(0.346663\pi\)
\(132\) 7.60555i 0.661978i
\(133\) − 5.21110i − 0.451860i
\(134\) −3.51388 −0.303553
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 0.908327i − 0.0776036i −0.999247 0.0388018i \(-0.987646\pi\)
0.999247 0.0388018i \(-0.0123541\pi\)
\(138\) − 12.9083i − 1.09883i
\(139\) 1.90833 0.161862 0.0809311 0.996720i \(-0.474211\pi\)
0.0809311 + 0.996720i \(0.474211\pi\)
\(140\) 0 0
\(141\) 15.2111 1.28101
\(142\) 6.00000i 0.503509i
\(143\) − 3.00000i − 0.250873i
\(144\) −7.90833 −0.659027
\(145\) 0 0
\(146\) 12.3028 1.01818
\(147\) 0.697224i 0.0575061i
\(148\) 1.00000i 0.0821995i
\(149\) −19.8167 −1.62344 −0.811722 0.584044i \(-0.801469\pi\)
−0.811722 + 0.584044i \(0.801469\pi\)
\(150\) 0 0
\(151\) −20.6056 −1.67686 −0.838428 0.545012i \(-0.816525\pi\)
−0.838428 + 0.545012i \(0.816525\pi\)
\(152\) 2.00000i 0.162221i
\(153\) − 47.4500i − 3.83610i
\(154\) 6.00000 0.483494
\(155\) 0 0
\(156\) 4.30278 0.344498
\(157\) 7.21110i 0.575509i 0.957704 + 0.287754i \(0.0929087\pi\)
−0.957704 + 0.287754i \(0.907091\pi\)
\(158\) − 9.11943i − 0.725503i
\(159\) 19.8167 1.57156
\(160\) 0 0
\(161\) −10.1833 −0.802560
\(162\) 29.8167i 2.34262i
\(163\) 8.42221i 0.659678i 0.944037 + 0.329839i \(0.106994\pi\)
−0.944037 + 0.329839i \(0.893006\pi\)
\(164\) −9.90833 −0.773710
\(165\) 0 0
\(166\) −2.78890 −0.216460
\(167\) 5.51388i 0.426677i 0.976978 + 0.213338i \(0.0684337\pi\)
−0.976978 + 0.213338i \(0.931566\pi\)
\(168\) 8.60555i 0.663933i
\(169\) 11.3028 0.869444
\(170\) 0 0
\(171\) 15.8167 1.20953
\(172\) − 0.605551i − 0.0461729i
\(173\) − 8.78890i − 0.668207i −0.942536 0.334104i \(-0.891566\pi\)
0.942536 0.334104i \(-0.108434\pi\)
\(174\) −12.9083 −0.978578
\(175\) 0 0
\(176\) −2.30278 −0.173578
\(177\) − 35.0278i − 2.63285i
\(178\) 9.21110i 0.690401i
\(179\) 13.8167 1.03271 0.516353 0.856376i \(-0.327289\pi\)
0.516353 + 0.856376i \(0.327289\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) − 3.39445i − 0.251613i
\(183\) 24.8167i 1.83450i
\(184\) 3.90833 0.288126
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) − 13.8167i − 1.01037i
\(188\) 4.60555i 0.335894i
\(189\) 42.2389 3.07242
\(190\) 0 0
\(191\) −5.51388 −0.398970 −0.199485 0.979901i \(-0.563927\pi\)
−0.199485 + 0.979901i \(0.563927\pi\)
\(192\) − 3.30278i − 0.238357i
\(193\) − 4.00000i − 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) −16.4222 −1.17905
\(195\) 0 0
\(196\) −0.211103 −0.0150788
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 18.2111i 1.29421i
\(199\) −26.4222 −1.87302 −0.936510 0.350640i \(-0.885964\pi\)
−0.936510 + 0.350640i \(0.885964\pi\)
\(200\) 0 0
\(201\) −11.6056 −0.818592
\(202\) − 12.4222i − 0.874023i
\(203\) 10.1833i 0.714731i
\(204\) 19.8167 1.38744
\(205\) 0 0
\(206\) 0.302776 0.0210954
\(207\) − 30.9083i − 2.14828i
\(208\) 1.30278i 0.0903312i
\(209\) 4.60555 0.318573
\(210\) 0 0
\(211\) 10.3028 0.709272 0.354636 0.935004i \(-0.384605\pi\)
0.354636 + 0.935004i \(0.384605\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 19.8167i 1.35781i
\(214\) 0.697224 0.0476613
\(215\) 0 0
\(216\) −16.2111 −1.10303
\(217\) − 0.788897i − 0.0535538i
\(218\) − 2.00000i − 0.135457i
\(219\) 40.6333 2.74574
\(220\) 0 0
\(221\) −7.81665 −0.525805
\(222\) 3.30278i 0.221668i
\(223\) − 5.81665i − 0.389512i −0.980852 0.194756i \(-0.937609\pi\)
0.980852 0.194756i \(-0.0623915\pi\)
\(224\) −2.60555 −0.174091
\(225\) 0 0
\(226\) 3.21110 0.213599
\(227\) − 13.8167i − 0.917044i −0.888683 0.458522i \(-0.848379\pi\)
0.888683 0.458522i \(-0.151621\pi\)
\(228\) 6.60555i 0.437463i
\(229\) −24.6056 −1.62598 −0.812990 0.582277i \(-0.802162\pi\)
−0.812990 + 0.582277i \(0.802162\pi\)
\(230\) 0 0
\(231\) 19.8167 1.30384
\(232\) − 3.90833i − 0.256594i
\(233\) 8.51388i 0.557763i 0.960326 + 0.278881i \(0.0899636\pi\)
−0.960326 + 0.278881i \(0.910036\pi\)
\(234\) 10.3028 0.673514
\(235\) 0 0
\(236\) 10.6056 0.690363
\(237\) − 30.1194i − 1.95647i
\(238\) − 15.6333i − 1.01336i
\(239\) 17.5139 1.13288 0.566439 0.824103i \(-0.308321\pi\)
0.566439 + 0.824103i \(0.308321\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) − 5.69722i − 0.366231i
\(243\) 49.8444i 3.19752i
\(244\) −7.51388 −0.481027
\(245\) 0 0
\(246\) −32.7250 −2.08647
\(247\) − 2.60555i − 0.165787i
\(248\) 0.302776i 0.0192263i
\(249\) −9.21110 −0.583730
\(250\) 0 0
\(251\) −21.2111 −1.33883 −0.669416 0.742887i \(-0.733456\pi\)
−0.669416 + 0.742887i \(0.733456\pi\)
\(252\) 20.6056i 1.29803i
\(253\) − 9.00000i − 0.565825i
\(254\) −19.2111 −1.20541
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 3.21110i − 0.200303i −0.994972 0.100152i \(-0.968067\pi\)
0.994972 0.100152i \(-0.0319328\pi\)
\(258\) − 2.00000i − 0.124515i
\(259\) 2.60555 0.161901
\(260\) 0 0
\(261\) −30.9083 −1.91318
\(262\) 10.6056i 0.655213i
\(263\) 13.8167i 0.851971i 0.904730 + 0.425986i \(0.140073\pi\)
−0.904730 + 0.425986i \(0.859927\pi\)
\(264\) −7.60555 −0.468089
\(265\) 0 0
\(266\) 5.21110 0.319513
\(267\) 30.4222i 1.86181i
\(268\) − 3.51388i − 0.214644i
\(269\) 21.2111 1.29326 0.646632 0.762802i \(-0.276177\pi\)
0.646632 + 0.762802i \(0.276177\pi\)
\(270\) 0 0
\(271\) −22.4222 −1.36205 −0.681026 0.732259i \(-0.738466\pi\)
−0.681026 + 0.732259i \(0.738466\pi\)
\(272\) 6.00000i 0.363803i
\(273\) − 11.2111i − 0.678527i
\(274\) 0.908327 0.0548740
\(275\) 0 0
\(276\) 12.9083 0.776990
\(277\) − 0.119429i − 0.00717582i −0.999994 0.00358791i \(-0.998858\pi\)
0.999994 0.00358791i \(-0.00114207\pi\)
\(278\) 1.90833i 0.114454i
\(279\) 2.39445 0.143352
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 15.2111i 0.905808i
\(283\) 24.6056i 1.46265i 0.682030 + 0.731324i \(0.261097\pi\)
−0.682030 + 0.731324i \(0.738903\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) 25.8167i 1.52391i
\(288\) − 7.90833i − 0.466003i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −54.2389 −3.17954
\(292\) 12.3028i 0.719965i
\(293\) 11.0278i 0.644248i 0.946697 + 0.322124i \(0.104397\pi\)
−0.946697 + 0.322124i \(0.895603\pi\)
\(294\) −0.697224 −0.0406630
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 37.3305i 2.16614i
\(298\) − 19.8167i − 1.14795i
\(299\) −5.09167 −0.294459
\(300\) 0 0
\(301\) −1.57779 −0.0909426
\(302\) − 20.6056i − 1.18572i
\(303\) − 41.0278i − 2.35698i
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 47.4500 2.71253
\(307\) − 17.9083i − 1.02208i −0.859556 0.511041i \(-0.829260\pi\)
0.859556 0.511041i \(-0.170740\pi\)
\(308\) 6.00000i 0.341882i
\(309\) 1.00000 0.0568880
\(310\) 0 0
\(311\) 15.9083 0.902078 0.451039 0.892504i \(-0.351053\pi\)
0.451039 + 0.892504i \(0.351053\pi\)
\(312\) 4.30278i 0.243597i
\(313\) − 9.02776i − 0.510279i −0.966904 0.255139i \(-0.917879\pi\)
0.966904 0.255139i \(-0.0821214\pi\)
\(314\) −7.21110 −0.406946
\(315\) 0 0
\(316\) 9.11943 0.513008
\(317\) − 9.21110i − 0.517347i −0.965965 0.258674i \(-0.916715\pi\)
0.965965 0.258674i \(-0.0832854\pi\)
\(318\) 19.8167i 1.11126i
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) 2.30278 0.128528
\(322\) − 10.1833i − 0.567496i
\(323\) − 12.0000i − 0.667698i
\(324\) −29.8167 −1.65648
\(325\) 0 0
\(326\) −8.42221 −0.466463
\(327\) − 6.60555i − 0.365288i
\(328\) − 9.90833i − 0.547096i
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −13.2111 −0.726148 −0.363074 0.931760i \(-0.618273\pi\)
−0.363074 + 0.931760i \(0.618273\pi\)
\(332\) − 2.78890i − 0.153061i
\(333\) 7.90833i 0.433374i
\(334\) −5.51388 −0.301706
\(335\) 0 0
\(336\) −8.60555 −0.469471
\(337\) − 6.11943i − 0.333347i −0.986012 0.166673i \(-0.946697\pi\)
0.986012 0.166673i \(-0.0533025\pi\)
\(338\) 11.3028i 0.614790i
\(339\) 10.6056 0.576014
\(340\) 0 0
\(341\) 0.697224 0.0377568
\(342\) 15.8167i 0.855267i
\(343\) 18.7889i 1.01451i
\(344\) 0.605551 0.0326491
\(345\) 0 0
\(346\) 8.78890 0.472494
\(347\) − 10.1833i − 0.546671i −0.961919 0.273335i \(-0.911873\pi\)
0.961919 0.273335i \(-0.0881269\pi\)
\(348\) − 12.9083i − 0.691959i
\(349\) −28.2389 −1.51159 −0.755796 0.654807i \(-0.772750\pi\)
−0.755796 + 0.654807i \(0.772750\pi\)
\(350\) 0 0
\(351\) 21.1194 1.12727
\(352\) − 2.30278i − 0.122738i
\(353\) 10.1833i 0.542005i 0.962579 + 0.271002i \(0.0873551\pi\)
−0.962579 + 0.271002i \(0.912645\pi\)
\(354\) 35.0278 1.86170
\(355\) 0 0
\(356\) −9.21110 −0.488187
\(357\) − 51.6333i − 2.73272i
\(358\) 13.8167i 0.730233i
\(359\) −3.21110 −0.169476 −0.0847378 0.996403i \(-0.527005\pi\)
−0.0847378 + 0.996403i \(0.527005\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 20.0000i 1.05118i
\(363\) − 18.8167i − 0.987618i
\(364\) 3.39445 0.177917
\(365\) 0 0
\(366\) −24.8167 −1.29719
\(367\) − 3.81665i − 0.199228i −0.995026 0.0996139i \(-0.968239\pi\)
0.995026 0.0996139i \(-0.0317607\pi\)
\(368\) 3.90833i 0.203736i
\(369\) −78.3583 −4.07917
\(370\) 0 0
\(371\) 15.6333 0.811641
\(372\) 1.00000i 0.0518476i
\(373\) − 17.8167i − 0.922511i −0.887267 0.461256i \(-0.847399\pi\)
0.887267 0.461256i \(-0.152601\pi\)
\(374\) 13.8167 0.714442
\(375\) 0 0
\(376\) −4.60555 −0.237513
\(377\) 5.09167i 0.262235i
\(378\) 42.2389i 2.17253i
\(379\) −24.3305 −1.24978 −0.624888 0.780715i \(-0.714855\pi\)
−0.624888 + 0.780715i \(0.714855\pi\)
\(380\) 0 0
\(381\) −63.4500 −3.25064
\(382\) − 5.51388i − 0.282115i
\(383\) − 36.8444i − 1.88266i −0.337486 0.941331i \(-0.609576\pi\)
0.337486 0.941331i \(-0.390424\pi\)
\(384\) 3.30278 0.168544
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) − 4.78890i − 0.243433i
\(388\) − 16.4222i − 0.833711i
\(389\) 37.1194 1.88203 0.941015 0.338365i \(-0.109874\pi\)
0.941015 + 0.338365i \(0.109874\pi\)
\(390\) 0 0
\(391\) −23.4500 −1.18592
\(392\) − 0.211103i − 0.0106623i
\(393\) 35.0278i 1.76692i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −18.2111 −0.915142
\(397\) − 6.18335i − 0.310333i −0.987888 0.155167i \(-0.950409\pi\)
0.987888 0.155167i \(-0.0495914\pi\)
\(398\) − 26.4222i − 1.32443i
\(399\) 17.2111 0.861633
\(400\) 0 0
\(401\) −7.81665 −0.390345 −0.195173 0.980769i \(-0.562527\pi\)
−0.195173 + 0.980769i \(0.562527\pi\)
\(402\) − 11.6056i − 0.578832i
\(403\) − 0.394449i − 0.0196489i
\(404\) 12.4222 0.618028
\(405\) 0 0
\(406\) −10.1833 −0.505391
\(407\) 2.30278i 0.114144i
\(408\) 19.8167i 0.981071i
\(409\) −31.0278 −1.53422 −0.767112 0.641513i \(-0.778307\pi\)
−0.767112 + 0.641513i \(0.778307\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 0.302776i 0.0149167i
\(413\) − 27.6333i − 1.35975i
\(414\) 30.9083 1.51906
\(415\) 0 0
\(416\) −1.30278 −0.0638738
\(417\) 6.30278i 0.308648i
\(418\) 4.60555i 0.225265i
\(419\) −36.1472 −1.76591 −0.882953 0.469462i \(-0.844448\pi\)
−0.882953 + 0.469462i \(0.844448\pi\)
\(420\) 0 0
\(421\) −3.72498 −0.181544 −0.0907722 0.995872i \(-0.528934\pi\)
−0.0907722 + 0.995872i \(0.528934\pi\)
\(422\) 10.3028i 0.501531i
\(423\) 36.4222i 1.77091i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −19.8167 −0.960120
\(427\) 19.5778i 0.947436i
\(428\) 0.697224i 0.0337016i
\(429\) 9.90833 0.478379
\(430\) 0 0
\(431\) 9.21110 0.443683 0.221842 0.975083i \(-0.428793\pi\)
0.221842 + 0.975083i \(0.428793\pi\)
\(432\) − 16.2111i − 0.779957i
\(433\) 34.9361i 1.67892i 0.543421 + 0.839461i \(0.317129\pi\)
−0.543421 + 0.839461i \(0.682871\pi\)
\(434\) 0.788897 0.0378683
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) − 7.81665i − 0.373921i
\(438\) 40.6333i 1.94153i
\(439\) −30.3305 −1.44760 −0.723799 0.690011i \(-0.757606\pi\)
−0.723799 + 0.690011i \(0.757606\pi\)
\(440\) 0 0
\(441\) −1.66947 −0.0794985
\(442\) − 7.81665i − 0.371800i
\(443\) 32.7250i 1.55481i 0.629000 + 0.777405i \(0.283465\pi\)
−0.629000 + 0.777405i \(0.716535\pi\)
\(444\) −3.30278 −0.156743
\(445\) 0 0
\(446\) 5.81665 0.275427
\(447\) − 65.4500i − 3.09568i
\(448\) − 2.60555i − 0.123101i
\(449\) 15.2111 0.717856 0.358928 0.933365i \(-0.383142\pi\)
0.358928 + 0.933365i \(0.383142\pi\)
\(450\) 0 0
\(451\) −22.8167 −1.07439
\(452\) 3.21110i 0.151038i
\(453\) − 68.0555i − 3.19753i
\(454\) 13.8167 0.648448
\(455\) 0 0
\(456\) −6.60555 −0.309333
\(457\) 2.60555i 0.121883i 0.998141 + 0.0609413i \(0.0194102\pi\)
−0.998141 + 0.0609413i \(0.980590\pi\)
\(458\) − 24.6056i − 1.14974i
\(459\) 97.2666 4.54002
\(460\) 0 0
\(461\) 12.4222 0.578560 0.289280 0.957245i \(-0.406584\pi\)
0.289280 + 0.957245i \(0.406584\pi\)
\(462\) 19.8167i 0.921954i
\(463\) 26.6972i 1.24073i 0.784315 + 0.620363i \(0.213015\pi\)
−0.784315 + 0.620363i \(0.786985\pi\)
\(464\) 3.90833 0.181440
\(465\) 0 0
\(466\) −8.51388 −0.394398
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 10.3028i 0.476246i
\(469\) −9.15559 −0.422766
\(470\) 0 0
\(471\) −23.8167 −1.09741
\(472\) 10.6056i 0.488160i
\(473\) − 1.39445i − 0.0641168i
\(474\) 30.1194 1.38343
\(475\) 0 0
\(476\) 15.6333 0.716551
\(477\) 47.4500i 2.17258i
\(478\) 17.5139i 0.801066i
\(479\) 13.1194 0.599442 0.299721 0.954027i \(-0.403106\pi\)
0.299721 + 0.954027i \(0.403106\pi\)
\(480\) 0 0
\(481\) 1.30278 0.0594015
\(482\) 8.00000i 0.364390i
\(483\) − 33.6333i − 1.53037i
\(484\) 5.69722 0.258965
\(485\) 0 0
\(486\) −49.8444 −2.26099
\(487\) 37.2111i 1.68620i 0.537760 + 0.843098i \(0.319271\pi\)
−0.537760 + 0.843098i \(0.680729\pi\)
\(488\) − 7.51388i − 0.340137i
\(489\) −27.8167 −1.25791
\(490\) 0 0
\(491\) 17.7250 0.799917 0.399959 0.916533i \(-0.369025\pi\)
0.399959 + 0.916533i \(0.369025\pi\)
\(492\) − 32.7250i − 1.47536i
\(493\) 23.4500i 1.05613i
\(494\) 2.60555 0.117229
\(495\) 0 0
\(496\) −0.302776 −0.0135950
\(497\) 15.6333i 0.701250i
\(498\) − 9.21110i − 0.412759i
\(499\) 42.2389 1.89087 0.945436 0.325809i \(-0.105637\pi\)
0.945436 + 0.325809i \(0.105637\pi\)
\(500\) 0 0
\(501\) −18.2111 −0.813612
\(502\) − 21.2111i − 0.946698i
\(503\) − 6.48612i − 0.289202i −0.989490 0.144601i \(-0.953810\pi\)
0.989490 0.144601i \(-0.0461898\pi\)
\(504\) −20.6056 −0.917844
\(505\) 0 0
\(506\) 9.00000 0.400099
\(507\) 37.3305i 1.65791i
\(508\) − 19.2111i − 0.852355i
\(509\) 4.18335 0.185424 0.0927118 0.995693i \(-0.470446\pi\)
0.0927118 + 0.995693i \(0.470446\pi\)
\(510\) 0 0
\(511\) 32.0555 1.41805
\(512\) 1.00000i 0.0441942i
\(513\) 32.4222i 1.43148i
\(514\) 3.21110 0.141636
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) 10.6056i 0.466432i
\(518\) 2.60555i 0.114481i
\(519\) 29.0278 1.27418
\(520\) 0 0
\(521\) 33.6333 1.47350 0.736751 0.676164i \(-0.236359\pi\)
0.736751 + 0.676164i \(0.236359\pi\)
\(522\) − 30.9083i − 1.35282i
\(523\) − 18.2389i − 0.797530i −0.917053 0.398765i \(-0.869439\pi\)
0.917053 0.398765i \(-0.130561\pi\)
\(524\) −10.6056 −0.463306
\(525\) 0 0
\(526\) −13.8167 −0.602435
\(527\) − 1.81665i − 0.0791347i
\(528\) − 7.60555i − 0.330989i
\(529\) 7.72498 0.335869
\(530\) 0 0
\(531\) 83.8722 3.63974
\(532\) 5.21110i 0.225930i
\(533\) 12.9083i 0.559122i
\(534\) −30.4222 −1.31650
\(535\) 0 0
\(536\) 3.51388 0.151776
\(537\) 45.6333i 1.96922i
\(538\) 21.2111i 0.914476i
\(539\) −0.486122 −0.0209387
\(540\) 0 0
\(541\) 25.9361 1.11508 0.557540 0.830150i \(-0.311745\pi\)
0.557540 + 0.830150i \(0.311745\pi\)
\(542\) − 22.4222i − 0.963116i
\(543\) 66.0555i 2.83471i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 11.2111 0.479791
\(547\) 20.6056i 0.881030i 0.897745 + 0.440515i \(0.145204\pi\)
−0.897745 + 0.440515i \(0.854796\pi\)
\(548\) 0.908327i 0.0388018i
\(549\) −59.4222 −2.53608
\(550\) 0 0
\(551\) −7.81665 −0.333001
\(552\) 12.9083i 0.549415i
\(553\) − 23.7611i − 1.01043i
\(554\) 0.119429 0.00507407
\(555\) 0 0
\(556\) −1.90833 −0.0809311
\(557\) − 11.5139i − 0.487859i −0.969793 0.243929i \(-0.921564\pi\)
0.969793 0.243929i \(-0.0784365\pi\)
\(558\) 2.39445i 0.101365i
\(559\) −0.788897 −0.0333668
\(560\) 0 0
\(561\) 45.6333 1.92664
\(562\) − 12.0000i − 0.506189i
\(563\) − 28.0555i − 1.18240i −0.806525 0.591199i \(-0.798655\pi\)
0.806525 0.591199i \(-0.201345\pi\)
\(564\) −15.2111 −0.640503
\(565\) 0 0
\(566\) −24.6056 −1.03425
\(567\) 77.6888i 3.26262i
\(568\) − 6.00000i − 0.251754i
\(569\) −18.4222 −0.772299 −0.386150 0.922436i \(-0.626195\pi\)
−0.386150 + 0.922436i \(0.626195\pi\)
\(570\) 0 0
\(571\) −16.6972 −0.698757 −0.349379 0.936982i \(-0.613607\pi\)
−0.349379 + 0.936982i \(0.613607\pi\)
\(572\) 3.00000i 0.125436i
\(573\) − 18.2111i − 0.760780i
\(574\) −25.8167 −1.07757
\(575\) 0 0
\(576\) 7.90833 0.329514
\(577\) − 22.2389i − 0.925816i −0.886406 0.462908i \(-0.846806\pi\)
0.886406 0.462908i \(-0.153194\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) 13.2111 0.549035
\(580\) 0 0
\(581\) −7.26662 −0.301470
\(582\) − 54.2389i − 2.24827i
\(583\) 13.8167i 0.572227i
\(584\) −12.3028 −0.509092
\(585\) 0 0
\(586\) −11.0278 −0.455552
\(587\) 45.6333i 1.88349i 0.336330 + 0.941744i \(0.390814\pi\)
−0.336330 + 0.941744i \(0.609186\pi\)
\(588\) − 0.697224i − 0.0287530i
\(589\) 0.605551 0.0249513
\(590\) 0 0
\(591\) −19.8167 −0.815148
\(592\) − 1.00000i − 0.0410997i
\(593\) 18.4861i 0.759134i 0.925164 + 0.379567i \(0.123927\pi\)
−0.925164 + 0.379567i \(0.876073\pi\)
\(594\) −37.3305 −1.53169
\(595\) 0 0
\(596\) 19.8167 0.811722
\(597\) − 87.2666i − 3.57158i
\(598\) − 5.09167i − 0.208214i
\(599\) 20.7889 0.849411 0.424706 0.905331i \(-0.360378\pi\)
0.424706 + 0.905331i \(0.360378\pi\)
\(600\) 0 0
\(601\) −24.3028 −0.991331 −0.495665 0.868514i \(-0.665076\pi\)
−0.495665 + 0.868514i \(0.665076\pi\)
\(602\) − 1.57779i − 0.0643061i
\(603\) − 27.7889i − 1.13165i
\(604\) 20.6056 0.838428
\(605\) 0 0
\(606\) 41.0278 1.66664
\(607\) 13.4861i 0.547385i 0.961817 + 0.273692i \(0.0882450\pi\)
−0.961817 + 0.273692i \(0.911755\pi\)
\(608\) − 2.00000i − 0.0811107i
\(609\) −33.6333 −1.36289
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 47.4500i 1.91805i
\(613\) − 29.8167i − 1.20428i −0.798389 0.602142i \(-0.794314\pi\)
0.798389 0.602142i \(-0.205686\pi\)
\(614\) 17.9083 0.722721
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) 42.5694i 1.71378i 0.515500 + 0.856890i \(0.327606\pi\)
−0.515500 + 0.856890i \(0.672394\pi\)
\(618\) 1.00000i 0.0402259i
\(619\) 6.30278 0.253330 0.126665 0.991946i \(-0.459573\pi\)
0.126665 + 0.991946i \(0.459573\pi\)
\(620\) 0 0
\(621\) 63.3583 2.54248
\(622\) 15.9083i 0.637866i
\(623\) 24.0000i 0.961540i
\(624\) −4.30278 −0.172249
\(625\) 0 0
\(626\) 9.02776 0.360822
\(627\) 15.2111i 0.607473i
\(628\) − 7.21110i − 0.287754i
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 14.6972 0.585087 0.292544 0.956252i \(-0.405498\pi\)
0.292544 + 0.956252i \(0.405498\pi\)
\(632\) 9.11943i 0.362751i
\(633\) 34.0278i 1.35248i
\(634\) 9.21110 0.365820
\(635\) 0 0
\(636\) −19.8167 −0.785781
\(637\) 0.275019i 0.0108967i
\(638\) − 9.00000i − 0.356313i
\(639\) −47.4500 −1.87709
\(640\) 0 0
\(641\) −20.5139 −0.810249 −0.405125 0.914261i \(-0.632772\pi\)
−0.405125 + 0.914261i \(0.632772\pi\)
\(642\) 2.30278i 0.0908833i
\(643\) − 8.18335i − 0.322720i −0.986896 0.161360i \(-0.948412\pi\)
0.986896 0.161360i \(-0.0515880\pi\)
\(644\) 10.1833 0.401280
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) − 20.9361i − 0.823082i −0.911391 0.411541i \(-0.864991\pi\)
0.911391 0.411541i \(-0.135009\pi\)
\(648\) − 29.8167i − 1.17131i
\(649\) 24.4222 0.958655
\(650\) 0 0
\(651\) 2.60555 0.102120
\(652\) − 8.42221i − 0.329839i
\(653\) − 3.90833i − 0.152945i −0.997072 0.0764723i \(-0.975634\pi\)
0.997072 0.0764723i \(-0.0243657\pi\)
\(654\) 6.60555 0.258297
\(655\) 0 0
\(656\) 9.90833 0.386855
\(657\) 97.2944i 3.79581i
\(658\) 12.0000i 0.467809i
\(659\) 16.8806 0.657574 0.328787 0.944404i \(-0.393360\pi\)
0.328787 + 0.944404i \(0.393360\pi\)
\(660\) 0 0
\(661\) −30.5139 −1.18685 −0.593426 0.804888i \(-0.702225\pi\)
−0.593426 + 0.804888i \(0.702225\pi\)
\(662\) − 13.2111i − 0.513464i
\(663\) − 25.8167i − 1.00264i
\(664\) 2.78890 0.108230
\(665\) 0 0
\(666\) −7.90833 −0.306441
\(667\) 15.2750i 0.591451i
\(668\) − 5.51388i − 0.213338i
\(669\) 19.2111 0.742744
\(670\) 0 0
\(671\) −17.3028 −0.667966
\(672\) − 8.60555i − 0.331966i
\(673\) 20.6972i 0.797819i 0.916990 + 0.398910i \(0.130611\pi\)
−0.916990 + 0.398910i \(0.869389\pi\)
\(674\) 6.11943 0.235712
\(675\) 0 0
\(676\) −11.3028 −0.434722
\(677\) − 14.2389i − 0.547244i −0.961837 0.273622i \(-0.911778\pi\)
0.961837 0.273622i \(-0.0882217\pi\)
\(678\) 10.6056i 0.407304i
\(679\) −42.7889 −1.64209
\(680\) 0 0
\(681\) 45.6333 1.74867
\(682\) 0.697224i 0.0266981i
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) −15.8167 −0.604765
\(685\) 0 0
\(686\) −18.7889 −0.717363
\(687\) − 81.2666i − 3.10051i
\(688\) 0.605551i 0.0230864i
\(689\) 7.81665 0.297791
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 8.78890i 0.334104i
\(693\) 47.4500i 1.80247i
\(694\) 10.1833 0.386555
\(695\) 0 0
\(696\) 12.9083 0.489289
\(697\) 59.4500i 2.25183i
\(698\) − 28.2389i − 1.06886i
\(699\) −28.1194 −1.06357
\(700\) 0 0
\(701\) 40.1194 1.51529 0.757645 0.652667i \(-0.226350\pi\)
0.757645 + 0.652667i \(0.226350\pi\)
\(702\) 21.1194i 0.797101i
\(703\) 2.00000i 0.0754314i
\(704\) 2.30278 0.0867891
\(705\) 0 0
\(706\) −10.1833 −0.383255
\(707\) − 32.3667i − 1.21727i
\(708\) 35.0278i 1.31642i
\(709\) 41.3305 1.55220 0.776100 0.630609i \(-0.217195\pi\)
0.776100 + 0.630609i \(0.217195\pi\)
\(710\) 0 0
\(711\) 72.1194 2.70469
\(712\) − 9.21110i − 0.345201i
\(713\) − 1.18335i − 0.0443167i
\(714\) 51.6333 1.93233
\(715\) 0 0
\(716\) −13.8167 −0.516353
\(717\) 57.8444i 2.16024i
\(718\) − 3.21110i − 0.119837i
\(719\) 51.6333 1.92560 0.962799 0.270220i \(-0.0870963\pi\)
0.962799 + 0.270220i \(0.0870963\pi\)
\(720\) 0 0
\(721\) 0.788897 0.0293801
\(722\) − 15.0000i − 0.558242i
\(723\) 26.4222i 0.982652i
\(724\) −20.0000 −0.743294
\(725\) 0 0
\(726\) 18.8167 0.698352
\(727\) − 19.0917i − 0.708071i −0.935232 0.354035i \(-0.884809\pi\)
0.935232 0.354035i \(-0.115191\pi\)
\(728\) 3.39445i 0.125807i
\(729\) −75.1749 −2.78426
\(730\) 0 0
\(731\) −3.63331 −0.134383
\(732\) − 24.8167i − 0.917250i
\(733\) − 13.6333i − 0.503558i −0.967785 0.251779i \(-0.918984\pi\)
0.967785 0.251779i \(-0.0810156\pi\)
\(734\) 3.81665 0.140875
\(735\) 0 0
\(736\) −3.90833 −0.144063
\(737\) − 8.09167i − 0.298061i
\(738\) − 78.3583i − 2.88441i
\(739\) 2.66947 0.0981980 0.0490990 0.998794i \(-0.484365\pi\)
0.0490990 + 0.998794i \(0.484365\pi\)
\(740\) 0 0
\(741\) 8.60555 0.316133
\(742\) 15.6333i 0.573917i
\(743\) − 29.4500i − 1.08041i −0.841532 0.540207i \(-0.818346\pi\)
0.841532 0.540207i \(-0.181654\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) 17.8167 0.652314
\(747\) − 22.0555i − 0.806969i
\(748\) 13.8167i 0.505187i
\(749\) 1.81665 0.0663791
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) − 4.60555i − 0.167947i
\(753\) − 70.0555i − 2.55296i
\(754\) −5.09167 −0.185428
\(755\) 0 0
\(756\) −42.2389 −1.53621
\(757\) − 5.69722i − 0.207069i −0.994626 0.103535i \(-0.966985\pi\)
0.994626 0.103535i \(-0.0330152\pi\)
\(758\) − 24.3305i − 0.883725i
\(759\) 29.7250 1.07895
\(760\) 0 0
\(761\) 16.8806 0.611920 0.305960 0.952044i \(-0.401023\pi\)
0.305960 + 0.952044i \(0.401023\pi\)
\(762\) − 63.4500i − 2.29855i
\(763\) − 5.21110i − 0.188655i
\(764\) 5.51388 0.199485
\(765\) 0 0
\(766\) 36.8444 1.33124
\(767\) − 13.8167i − 0.498890i
\(768\) 3.30278i 0.119179i
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 10.6056 0.381950
\(772\) 4.00000i 0.143963i
\(773\) 22.0555i 0.793282i 0.917974 + 0.396641i \(0.129824\pi\)
−0.917974 + 0.396641i \(0.870176\pi\)
\(774\) 4.78890 0.172133
\(775\) 0 0
\(776\) 16.4222 0.589523
\(777\) 8.60555i 0.308722i
\(778\) 37.1194i 1.33080i
\(779\) −19.8167 −0.710005
\(780\) 0 0
\(781\) −13.8167 −0.494399
\(782\) − 23.4500i − 0.838569i
\(783\) − 63.3583i − 2.26424i
\(784\) 0.211103 0.00753938
\(785\) 0 0
\(786\) −35.0278 −1.24940
\(787\) − 10.7889i − 0.384583i −0.981338 0.192291i \(-0.938408\pi\)
0.981338 0.192291i \(-0.0615919\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) −45.6333 −1.62459
\(790\) 0 0
\(791\) 8.36669 0.297485
\(792\) − 18.2111i − 0.647103i
\(793\) 9.78890i 0.347614i
\(794\) 6.18335 0.219439
\(795\) 0 0
\(796\) 26.4222 0.936510
\(797\) − 22.3305i − 0.790988i −0.918469 0.395494i \(-0.870573\pi\)
0.918469 0.395494i \(-0.129427\pi\)
\(798\) 17.2111i 0.609266i
\(799\) 27.6333 0.977596
\(800\) 0 0
\(801\) −72.8444 −2.57383
\(802\) − 7.81665i − 0.276016i
\(803\) 28.3305i 0.999763i
\(804\) 11.6056 0.409296
\(805\) 0 0
\(806\) 0.394449 0.0138939
\(807\) 70.0555i 2.46607i
\(808\) 12.4222i 0.437012i
\(809\) 35.4500 1.24635 0.623177 0.782081i \(-0.285842\pi\)
0.623177 + 0.782081i \(0.285842\pi\)
\(810\) 0 0
\(811\) −7.14719 −0.250972 −0.125486 0.992095i \(-0.540049\pi\)
−0.125486 + 0.992095i \(0.540049\pi\)
\(812\) − 10.1833i − 0.357365i
\(813\) − 74.0555i − 2.59724i
\(814\) −2.30278 −0.0807122
\(815\) 0 0
\(816\) −19.8167 −0.693722
\(817\) − 1.21110i − 0.0423711i
\(818\) − 31.0278i − 1.08486i
\(819\) 26.8444 0.938020
\(820\) 0 0
\(821\) −3.21110 −0.112068 −0.0560341 0.998429i \(-0.517846\pi\)
−0.0560341 + 0.998429i \(0.517846\pi\)
\(822\) 3.00000i 0.104637i
\(823\) 44.8444i 1.56318i 0.623794 + 0.781589i \(0.285590\pi\)
−0.623794 + 0.781589i \(0.714410\pi\)
\(824\) −0.302776 −0.0105477
\(825\) 0 0
\(826\) 27.6333 0.961486
\(827\) − 34.6056i − 1.20335i −0.798740 0.601676i \(-0.794500\pi\)
0.798740 0.601676i \(-0.205500\pi\)
\(828\) 30.9083i 1.07414i
\(829\) 27.7250 0.962928 0.481464 0.876466i \(-0.340105\pi\)
0.481464 + 0.876466i \(0.340105\pi\)
\(830\) 0 0
\(831\) 0.394449 0.0136833
\(832\) − 1.30278i − 0.0451656i
\(833\) 1.26662i 0.0438856i
\(834\) −6.30278 −0.218247
\(835\) 0 0
\(836\) −4.60555 −0.159286
\(837\) 4.90833i 0.169657i
\(838\) − 36.1472i − 1.24868i
\(839\) 12.9722 0.447852 0.223926 0.974606i \(-0.428113\pi\)
0.223926 + 0.974606i \(0.428113\pi\)
\(840\) 0 0
\(841\) −13.7250 −0.473275
\(842\) − 3.72498i − 0.128371i
\(843\) − 39.6333i − 1.36504i
\(844\) −10.3028 −0.354636
\(845\) 0 0
\(846\) −36.4222 −1.25222
\(847\) − 14.8444i − 0.510060i
\(848\) − 6.00000i − 0.206041i
\(849\) −81.2666 −2.78906
\(850\) 0 0
\(851\) 3.90833 0.133976
\(852\) − 19.8167i − 0.678907i
\(853\) 42.5416i 1.45660i 0.685260 + 0.728299i \(0.259689\pi\)
−0.685260 + 0.728299i \(0.740311\pi\)
\(854\) −19.5778 −0.669938
\(855\) 0 0
\(856\) −0.697224 −0.0238306
\(857\) − 42.8444i − 1.46354i −0.681553 0.731769i \(-0.738695\pi\)
0.681553 0.731769i \(-0.261305\pi\)
\(858\) 9.90833i 0.338265i
\(859\) −48.0555 −1.63963 −0.819816 0.572626i \(-0.805925\pi\)
−0.819816 + 0.572626i \(0.805925\pi\)
\(860\) 0 0
\(861\) −85.2666 −2.90588
\(862\) 9.21110i 0.313731i
\(863\) 12.0000i 0.408485i 0.978920 + 0.204242i \(0.0654731\pi\)
−0.978920 + 0.204242i \(0.934527\pi\)
\(864\) 16.2111 0.551513
\(865\) 0 0
\(866\) −34.9361 −1.18718
\(867\) − 62.7527i − 2.13119i
\(868\) 0.788897i 0.0267769i
\(869\) 21.0000 0.712376
\(870\) 0 0
\(871\) −4.57779 −0.155113
\(872\) 2.00000i 0.0677285i
\(873\) − 129.872i − 4.39551i
\(874\) 7.81665 0.264402
\(875\) 0 0
\(876\) −40.6333 −1.37287
\(877\) 7.21110i 0.243502i 0.992561 + 0.121751i \(0.0388509\pi\)
−0.992561 + 0.121751i \(0.961149\pi\)
\(878\) − 30.3305i − 1.02361i
\(879\) −36.4222 −1.22849
\(880\) 0 0
\(881\) −28.5416 −0.961592 −0.480796 0.876832i \(-0.659652\pi\)
−0.480796 + 0.876832i \(0.659652\pi\)
\(882\) − 1.66947i − 0.0562139i
\(883\) 26.4222i 0.889178i 0.895735 + 0.444589i \(0.146650\pi\)
−0.895735 + 0.444589i \(0.853350\pi\)
\(884\) 7.81665 0.262903
\(885\) 0 0
\(886\) −32.7250 −1.09942
\(887\) 0.422205i 0.0141763i 0.999975 + 0.00708813i \(0.00225624\pi\)
−0.999975 + 0.00708813i \(0.997744\pi\)
\(888\) − 3.30278i − 0.110834i
\(889\) −50.0555 −1.67881
\(890\) 0 0
\(891\) −68.6611 −2.30023
\(892\) 5.81665i 0.194756i
\(893\) 9.21110i 0.308238i
\(894\) 65.4500 2.18897
\(895\) 0 0
\(896\) 2.60555 0.0870454
\(897\) − 16.8167i − 0.561492i
\(898\) 15.2111i 0.507601i
\(899\) −1.18335 −0.0394668
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) − 22.8167i − 0.759711i
\(903\) − 5.21110i − 0.173415i
\(904\) −3.21110 −0.106800
\(905\) 0 0
\(906\) 68.0555 2.26099
\(907\) − 26.0000i − 0.863316i −0.902037 0.431658i \(-0.857929\pi\)
0.902037 0.431658i \(-0.142071\pi\)
\(908\) 13.8167i 0.458522i
\(909\) 98.2389 3.25838
\(910\) 0 0
\(911\) −17.5778 −0.582378 −0.291189 0.956665i \(-0.594051\pi\)
−0.291189 + 0.956665i \(0.594051\pi\)
\(912\) − 6.60555i − 0.218732i
\(913\) − 6.42221i − 0.212544i
\(914\) −2.60555 −0.0861840
\(915\) 0 0
\(916\) 24.6056 0.812990
\(917\) 27.6333i 0.912532i
\(918\) 97.2666i 3.21028i
\(919\) 9.57779 0.315942 0.157971 0.987444i \(-0.449505\pi\)
0.157971 + 0.987444i \(0.449505\pi\)
\(920\) 0 0
\(921\) 59.1472 1.94897
\(922\) 12.4222i 0.409104i
\(923\) 7.81665i 0.257288i
\(924\) −19.8167 −0.651920
\(925\) 0 0
\(926\) −26.6972 −0.877325
\(927\) 2.39445i 0.0786440i
\(928\) 3.90833i 0.128297i
\(929\) 18.4861 0.606510 0.303255 0.952909i \(-0.401927\pi\)
0.303255 + 0.952909i \(0.401927\pi\)
\(930\) 0 0
\(931\) −0.422205 −0.0138372
\(932\) − 8.51388i − 0.278881i
\(933\) 52.5416i 1.72014i
\(934\) 0 0
\(935\) 0 0
\(936\) −10.3028 −0.336757
\(937\) 18.0917i 0.591029i 0.955338 + 0.295515i \(0.0954911\pi\)
−0.955338 + 0.295515i \(0.904509\pi\)
\(938\) − 9.15559i − 0.298941i
\(939\) 29.8167 0.973030
\(940\) 0 0
\(941\) 13.8167 0.450410 0.225205 0.974311i \(-0.427695\pi\)
0.225205 + 0.974311i \(0.427695\pi\)
\(942\) − 23.8167i − 0.775989i
\(943\) 38.7250i 1.26106i
\(944\) −10.6056 −0.345181
\(945\) 0 0
\(946\) 1.39445 0.0453374
\(947\) 3.63331i 0.118067i 0.998256 + 0.0590333i \(0.0188018\pi\)
−0.998256 + 0.0590333i \(0.981198\pi\)
\(948\) 30.1194i 0.978234i
\(949\) 16.0278 0.520283
\(950\) 0 0
\(951\) 30.4222 0.986508
\(952\) 15.6333i 0.506678i
\(953\) 49.7527i 1.61165i 0.592154 + 0.805825i \(0.298278\pi\)
−0.592154 + 0.805825i \(0.701722\pi\)
\(954\) −47.4500 −1.53625
\(955\) 0 0
\(956\) −17.5139 −0.566439
\(957\) − 29.7250i − 0.960872i
\(958\) 13.1194i 0.423870i
\(959\) 2.36669 0.0764245
\(960\) 0 0
\(961\) −30.9083 −0.997043
\(962\) 1.30278i 0.0420032i
\(963\) 5.51388i 0.177682i
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) 33.6333 1.08213
\(967\) 6.72498i 0.216261i 0.994137 + 0.108130i \(0.0344864\pi\)
−0.994137 + 0.108130i \(0.965514\pi\)
\(968\) 5.69722i 0.183116i
\(969\) 39.6333 1.27321
\(970\) 0 0
\(971\) −22.5416 −0.723395 −0.361698 0.932295i \(-0.617803\pi\)
−0.361698 + 0.932295i \(0.617803\pi\)
\(972\) − 49.8444i − 1.59876i
\(973\) 4.97224i 0.159403i
\(974\) −37.2111 −1.19232
\(975\) 0 0
\(976\) 7.51388 0.240513
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) − 27.8167i − 0.889479i
\(979\) −21.2111 −0.677910
\(980\) 0 0
\(981\) 15.8167 0.504987
\(982\) 17.7250i 0.565627i
\(983\) − 12.0000i − 0.382741i −0.981518 0.191370i \(-0.938707\pi\)
0.981518 0.191370i \(-0.0612931\pi\)
\(984\) 32.7250 1.04323
\(985\) 0 0
\(986\) −23.4500 −0.746799
\(987\) 39.6333i 1.26154i
\(988\) 2.60555i 0.0828936i
\(989\) −2.36669 −0.0752564
\(990\) 0 0
\(991\) 50.6972 1.61045 0.805225 0.592969i \(-0.202044\pi\)
0.805225 + 0.592969i \(0.202044\pi\)
\(992\) − 0.302776i − 0.00961314i
\(993\) − 43.6333i − 1.38466i
\(994\) −15.6333 −0.495858
\(995\) 0 0
\(996\) 9.21110 0.291865
\(997\) 52.4222i 1.66023i 0.557594 + 0.830114i \(0.311725\pi\)
−0.557594 + 0.830114i \(0.688275\pi\)
\(998\) 42.2389i 1.33705i
\(999\) −16.2111 −0.512897
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.i.149.4 4
5.2 odd 4 74.2.a.a.1.2 2
5.3 odd 4 1850.2.a.u.1.1 2
5.4 even 2 inner 1850.2.b.i.149.1 4
15.2 even 4 666.2.a.j.1.2 2
20.7 even 4 592.2.a.f.1.1 2
35.27 even 4 3626.2.a.a.1.1 2
40.27 even 4 2368.2.a.ba.1.2 2
40.37 odd 4 2368.2.a.s.1.1 2
55.32 even 4 8954.2.a.p.1.2 2
60.47 odd 4 5328.2.a.bf.1.2 2
185.147 odd 4 2738.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 2 5.2 odd 4
592.2.a.f.1.1 2 20.7 even 4
666.2.a.j.1.2 2 15.2 even 4
1850.2.a.u.1.1 2 5.3 odd 4
1850.2.b.i.149.1 4 5.4 even 2 inner
1850.2.b.i.149.4 4 1.1 even 1 trivial
2368.2.a.s.1.1 2 40.37 odd 4
2368.2.a.ba.1.2 2 40.27 even 4
2738.2.a.l.1.2 2 185.147 odd 4
3626.2.a.a.1.1 2 35.27 even 4
5328.2.a.bf.1.2 2 60.47 odd 4
8954.2.a.p.1.2 2 55.32 even 4