Properties

Label 1850.2.a.u.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) 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)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.30278 q^{3} +1.00000 q^{4} -3.30278 q^{6} +2.60555 q^{7} +1.00000 q^{8} +7.90833 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.30278 q^{3} +1.00000 q^{4} -3.30278 q^{6} +2.60555 q^{7} +1.00000 q^{8} +7.90833 q^{9} -2.30278 q^{11} -3.30278 q^{12} -1.30278 q^{13} +2.60555 q^{14} +1.00000 q^{16} +6.00000 q^{17} +7.90833 q^{18} +2.00000 q^{19} -8.60555 q^{21} -2.30278 q^{22} -3.90833 q^{23} -3.30278 q^{24} -1.30278 q^{26} -16.2111 q^{27} +2.60555 q^{28} -3.90833 q^{29} -0.302776 q^{31} +1.00000 q^{32} +7.60555 q^{33} +6.00000 q^{34} +7.90833 q^{36} -1.00000 q^{37} +2.00000 q^{38} +4.30278 q^{39} +9.90833 q^{41} -8.60555 q^{42} -0.605551 q^{43} -2.30278 q^{44} -3.90833 q^{46} -4.60555 q^{47} -3.30278 q^{48} -0.211103 q^{49} -19.8167 q^{51} -1.30278 q^{52} +6.00000 q^{53} -16.2111 q^{54} +2.60555 q^{56} -6.60555 q^{57} -3.90833 q^{58} +10.6056 q^{59} +7.51388 q^{61} -0.302776 q^{62} +20.6056 q^{63} +1.00000 q^{64} +7.60555 q^{66} +3.51388 q^{67} +6.00000 q^{68} +12.9083 q^{69} +6.00000 q^{71} +7.90833 q^{72} +12.3028 q^{73} -1.00000 q^{74} +2.00000 q^{76} -6.00000 q^{77} +4.30278 q^{78} +9.11943 q^{79} +29.8167 q^{81} +9.90833 q^{82} -2.78890 q^{83} -8.60555 q^{84} -0.605551 q^{86} +12.9083 q^{87} -2.30278 q^{88} -9.21110 q^{89} -3.39445 q^{91} -3.90833 q^{92} +1.00000 q^{93} -4.60555 q^{94} -3.30278 q^{96} +16.4222 q^{97} -0.211103 q^{98} -18.2111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} - q^{11} - 3 q^{12} + q^{13} - 2 q^{14} + 2 q^{16} + 12 q^{17} + 5 q^{18} + 4 q^{19} - 10 q^{21} - q^{22} + 3 q^{23} - 3 q^{24} + q^{26} - 18 q^{27} - 2 q^{28} + 3 q^{29} + 3 q^{31} + 2 q^{32} + 8 q^{33} + 12 q^{34} + 5 q^{36} - 2 q^{37} + 4 q^{38} + 5 q^{39} + 9 q^{41} - 10 q^{42} + 6 q^{43} - q^{44} + 3 q^{46} - 2 q^{47} - 3 q^{48} + 14 q^{49} - 18 q^{51} + q^{52} + 12 q^{53} - 18 q^{54} - 2 q^{56} - 6 q^{57} + 3 q^{58} + 14 q^{59} - 3 q^{61} + 3 q^{62} + 34 q^{63} + 2 q^{64} + 8 q^{66} - 11 q^{67} + 12 q^{68} + 15 q^{69} + 12 q^{71} + 5 q^{72} + 21 q^{73} - 2 q^{74} + 4 q^{76} - 12 q^{77} + 5 q^{78} - 7 q^{79} + 38 q^{81} + 9 q^{82} - 20 q^{83} - 10 q^{84} + 6 q^{86} + 15 q^{87} - q^{88} - 4 q^{89} - 14 q^{91} + 3 q^{92} + 2 q^{93} - 2 q^{94} - 3 q^{96} + 4 q^{97} + 14 q^{98} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.30278 −1.90686 −0.953429 0.301617i \(-0.902474\pi\)
−0.953429 + 0.301617i \(0.902474\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −3.30278 −1.34835
\(7\) 2.60555 0.984806 0.492403 0.870367i \(-0.336119\pi\)
0.492403 + 0.870367i \(0.336119\pi\)
\(8\) 1.00000 0.353553
\(9\) 7.90833 2.63611
\(10\) 0 0
\(11\) −2.30278 −0.694313 −0.347156 0.937807i \(-0.612853\pi\)
−0.347156 + 0.937807i \(0.612853\pi\)
\(12\) −3.30278 −0.953429
\(13\) −1.30278 −0.361325 −0.180662 0.983545i \(-0.557824\pi\)
−0.180662 + 0.983545i \(0.557824\pi\)
\(14\) 2.60555 0.696363
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 7.90833 1.86401
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −8.60555 −1.87789
\(22\) −2.30278 −0.490953
\(23\) −3.90833 −0.814942 −0.407471 0.913218i \(-0.633589\pi\)
−0.407471 + 0.913218i \(0.633589\pi\)
\(24\) −3.30278 −0.674176
\(25\) 0 0
\(26\) −1.30278 −0.255495
\(27\) −16.2111 −3.11983
\(28\) 2.60555 0.492403
\(29\) −3.90833 −0.725758 −0.362879 0.931836i \(-0.618206\pi\)
−0.362879 + 0.931836i \(0.618206\pi\)
\(30\) 0 0
\(31\) −0.302776 −0.0543801 −0.0271901 0.999630i \(-0.508656\pi\)
−0.0271901 + 0.999630i \(0.508656\pi\)
\(32\) 1.00000 0.176777
\(33\) 7.60555 1.32396
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 7.90833 1.31805
\(37\) −1.00000 −0.164399
\(38\) 2.00000 0.324443
\(39\) 4.30278 0.688996
\(40\) 0 0
\(41\) 9.90833 1.54742 0.773710 0.633540i \(-0.218399\pi\)
0.773710 + 0.633540i \(0.218399\pi\)
\(42\) −8.60555 −1.32787
\(43\) −0.605551 −0.0923457 −0.0461729 0.998933i \(-0.514703\pi\)
−0.0461729 + 0.998933i \(0.514703\pi\)
\(44\) −2.30278 −0.347156
\(45\) 0 0
\(46\) −3.90833 −0.576251
\(47\) −4.60555 −0.671789 −0.335894 0.941900i \(-0.609039\pi\)
−0.335894 + 0.941900i \(0.609039\pi\)
\(48\) −3.30278 −0.476715
\(49\) −0.211103 −0.0301575
\(50\) 0 0
\(51\) −19.8167 −2.77489
\(52\) −1.30278 −0.180662
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −16.2111 −2.20605
\(55\) 0 0
\(56\) 2.60555 0.348181
\(57\) −6.60555 −0.874927
\(58\) −3.90833 −0.513188
\(59\) 10.6056 1.38073 0.690363 0.723464i \(-0.257451\pi\)
0.690363 + 0.723464i \(0.257451\pi\)
\(60\) 0 0
\(61\) 7.51388 0.962054 0.481027 0.876706i \(-0.340264\pi\)
0.481027 + 0.876706i \(0.340264\pi\)
\(62\) −0.302776 −0.0384525
\(63\) 20.6056 2.59606
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 7.60555 0.936179
\(67\) 3.51388 0.429289 0.214644 0.976692i \(-0.431141\pi\)
0.214644 + 0.976692i \(0.431141\pi\)
\(68\) 6.00000 0.727607
\(69\) 12.9083 1.55398
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 7.90833 0.932005
\(73\) 12.3028 1.43993 0.719965 0.694010i \(-0.244158\pi\)
0.719965 + 0.694010i \(0.244158\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) −6.00000 −0.683763
\(78\) 4.30278 0.487193
\(79\) 9.11943 1.02602 0.513008 0.858384i \(-0.328531\pi\)
0.513008 + 0.858384i \(0.328531\pi\)
\(80\) 0 0
\(81\) 29.8167 3.31296
\(82\) 9.90833 1.09419
\(83\) −2.78890 −0.306121 −0.153061 0.988217i \(-0.548913\pi\)
−0.153061 + 0.988217i \(0.548913\pi\)
\(84\) −8.60555 −0.938943
\(85\) 0 0
\(86\) −0.605551 −0.0652983
\(87\) 12.9083 1.38392
\(88\) −2.30278 −0.245477
\(89\) −9.21110 −0.976375 −0.488187 0.872739i \(-0.662342\pi\)
−0.488187 + 0.872739i \(0.662342\pi\)
\(90\) 0 0
\(91\) −3.39445 −0.355835
\(92\) −3.90833 −0.407471
\(93\) 1.00000 0.103695
\(94\) −4.60555 −0.475026
\(95\) 0 0
\(96\) −3.30278 −0.337088
\(97\) 16.4222 1.66742 0.833711 0.552201i \(-0.186212\pi\)
0.833711 + 0.552201i \(0.186212\pi\)
\(98\) −0.211103 −0.0213246
\(99\) −18.2111 −1.83028
\(100\) 0 0
\(101\) −12.4222 −1.23606 −0.618028 0.786156i \(-0.712068\pi\)
−0.618028 + 0.786156i \(0.712068\pi\)
\(102\) −19.8167 −1.96214
\(103\) 0.302776 0.0298334 0.0149167 0.999889i \(-0.495252\pi\)
0.0149167 + 0.999889i \(0.495252\pi\)
\(104\) −1.30278 −0.127748
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −0.697224 −0.0674032 −0.0337016 0.999432i \(-0.510730\pi\)
−0.0337016 + 0.999432i \(0.510730\pi\)
\(108\) −16.2111 −1.55991
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 3.30278 0.313486
\(112\) 2.60555 0.246201
\(113\) 3.21110 0.302075 0.151038 0.988528i \(-0.451739\pi\)
0.151038 + 0.988528i \(0.451739\pi\)
\(114\) −6.60555 −0.618667
\(115\) 0 0
\(116\) −3.90833 −0.362879
\(117\) −10.3028 −0.952492
\(118\) 10.6056 0.976320
\(119\) 15.6333 1.43310
\(120\) 0 0
\(121\) −5.69722 −0.517929
\(122\) 7.51388 0.680275
\(123\) −32.7250 −2.95071
\(124\) −0.302776 −0.0271901
\(125\) 0 0
\(126\) 20.6056 1.83569
\(127\) 19.2111 1.70471 0.852355 0.522964i \(-0.175174\pi\)
0.852355 + 0.522964i \(0.175174\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 10.6056 0.926611 0.463306 0.886199i \(-0.346663\pi\)
0.463306 + 0.886199i \(0.346663\pi\)
\(132\) 7.60555 0.661978
\(133\) 5.21110 0.451860
\(134\) 3.51388 0.303553
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −0.908327 −0.0776036 −0.0388018 0.999247i \(-0.512354\pi\)
−0.0388018 + 0.999247i \(0.512354\pi\)
\(138\) 12.9083 1.09883
\(139\) −1.90833 −0.161862 −0.0809311 0.996720i \(-0.525789\pi\)
−0.0809311 + 0.996720i \(0.525789\pi\)
\(140\) 0 0
\(141\) 15.2111 1.28101
\(142\) 6.00000 0.503509
\(143\) 3.00000 0.250873
\(144\) 7.90833 0.659027
\(145\) 0 0
\(146\) 12.3028 1.01818
\(147\) 0.697224 0.0575061
\(148\) −1.00000 −0.0821995
\(149\) 19.8167 1.62344 0.811722 0.584044i \(-0.198531\pi\)
0.811722 + 0.584044i \(0.198531\pi\)
\(150\) 0 0
\(151\) −20.6056 −1.67686 −0.838428 0.545012i \(-0.816525\pi\)
−0.838428 + 0.545012i \(0.816525\pi\)
\(152\) 2.00000 0.162221
\(153\) 47.4500 3.83610
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) 4.30278 0.344498
\(157\) 7.21110 0.575509 0.287754 0.957704i \(-0.407091\pi\)
0.287754 + 0.957704i \(0.407091\pi\)
\(158\) 9.11943 0.725503
\(159\) −19.8167 −1.57156
\(160\) 0 0
\(161\) −10.1833 −0.802560
\(162\) 29.8167 2.34262
\(163\) −8.42221 −0.659678 −0.329839 0.944037i \(-0.606994\pi\)
−0.329839 + 0.944037i \(0.606994\pi\)
\(164\) 9.90833 0.773710
\(165\) 0 0
\(166\) −2.78890 −0.216460
\(167\) 5.51388 0.426677 0.213338 0.976978i \(-0.431566\pi\)
0.213338 + 0.976978i \(0.431566\pi\)
\(168\) −8.60555 −0.663933
\(169\) −11.3028 −0.869444
\(170\) 0 0
\(171\) 15.8167 1.20953
\(172\) −0.605551 −0.0461729
\(173\) 8.78890 0.668207 0.334104 0.942536i \(-0.391566\pi\)
0.334104 + 0.942536i \(0.391566\pi\)
\(174\) 12.9083 0.978578
\(175\) 0 0
\(176\) −2.30278 −0.173578
\(177\) −35.0278 −2.63285
\(178\) −9.21110 −0.690401
\(179\) −13.8167 −1.03271 −0.516353 0.856376i \(-0.672711\pi\)
−0.516353 + 0.856376i \(0.672711\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) −3.39445 −0.251613
\(183\) −24.8167 −1.83450
\(184\) −3.90833 −0.288126
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) −13.8167 −1.01037
\(188\) −4.60555 −0.335894
\(189\) −42.2389 −3.07242
\(190\) 0 0
\(191\) −5.51388 −0.398970 −0.199485 0.979901i \(-0.563927\pi\)
−0.199485 + 0.979901i \(0.563927\pi\)
\(192\) −3.30278 −0.238357
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 16.4222 1.17905
\(195\) 0 0
\(196\) −0.211103 −0.0150788
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −18.2111 −1.29421
\(199\) 26.4222 1.87302 0.936510 0.350640i \(-0.114036\pi\)
0.936510 + 0.350640i \(0.114036\pi\)
\(200\) 0 0
\(201\) −11.6056 −0.818592
\(202\) −12.4222 −0.874023
\(203\) −10.1833 −0.714731
\(204\) −19.8167 −1.38744
\(205\) 0 0
\(206\) 0.302776 0.0210954
\(207\) −30.9083 −2.14828
\(208\) −1.30278 −0.0903312
\(209\) −4.60555 −0.318573
\(210\) 0 0
\(211\) 10.3028 0.709272 0.354636 0.935004i \(-0.384605\pi\)
0.354636 + 0.935004i \(0.384605\pi\)
\(212\) 6.00000 0.412082
\(213\) −19.8167 −1.35781
\(214\) −0.697224 −0.0476613
\(215\) 0 0
\(216\) −16.2111 −1.10303
\(217\) −0.788897 −0.0535538
\(218\) 2.00000 0.135457
\(219\) −40.6333 −2.74574
\(220\) 0 0
\(221\) −7.81665 −0.525805
\(222\) 3.30278 0.221668
\(223\) 5.81665 0.389512 0.194756 0.980852i \(-0.437609\pi\)
0.194756 + 0.980852i \(0.437609\pi\)
\(224\) 2.60555 0.174091
\(225\) 0 0
\(226\) 3.21110 0.213599
\(227\) −13.8167 −0.917044 −0.458522 0.888683i \(-0.651621\pi\)
−0.458522 + 0.888683i \(0.651621\pi\)
\(228\) −6.60555 −0.437463
\(229\) 24.6056 1.62598 0.812990 0.582277i \(-0.197838\pi\)
0.812990 + 0.582277i \(0.197838\pi\)
\(230\) 0 0
\(231\) 19.8167 1.30384
\(232\) −3.90833 −0.256594
\(233\) −8.51388 −0.557763 −0.278881 0.960326i \(-0.589964\pi\)
−0.278881 + 0.960326i \(0.589964\pi\)
\(234\) −10.3028 −0.673514
\(235\) 0 0
\(236\) 10.6056 0.690363
\(237\) −30.1194 −1.95647
\(238\) 15.6333 1.01336
\(239\) −17.5139 −1.13288 −0.566439 0.824103i \(-0.691679\pi\)
−0.566439 + 0.824103i \(0.691679\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −5.69722 −0.366231
\(243\) −49.8444 −3.19752
\(244\) 7.51388 0.481027
\(245\) 0 0
\(246\) −32.7250 −2.08647
\(247\) −2.60555 −0.165787
\(248\) −0.302776 −0.0192263
\(249\) 9.21110 0.583730
\(250\) 0 0
\(251\) −21.2111 −1.33883 −0.669416 0.742887i \(-0.733456\pi\)
−0.669416 + 0.742887i \(0.733456\pi\)
\(252\) 20.6056 1.29803
\(253\) 9.00000 0.565825
\(254\) 19.2111 1.20541
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −3.21110 −0.200303 −0.100152 0.994972i \(-0.531933\pi\)
−0.100152 + 0.994972i \(0.531933\pi\)
\(258\) 2.00000 0.124515
\(259\) −2.60555 −0.161901
\(260\) 0 0
\(261\) −30.9083 −1.91318
\(262\) 10.6056 0.655213
\(263\) −13.8167 −0.851971 −0.425986 0.904730i \(-0.640073\pi\)
−0.425986 + 0.904730i \(0.640073\pi\)
\(264\) 7.60555 0.468089
\(265\) 0 0
\(266\) 5.21110 0.319513
\(267\) 30.4222 1.86181
\(268\) 3.51388 0.214644
\(269\) −21.2111 −1.29326 −0.646632 0.762802i \(-0.723823\pi\)
−0.646632 + 0.762802i \(0.723823\pi\)
\(270\) 0 0
\(271\) −22.4222 −1.36205 −0.681026 0.732259i \(-0.738466\pi\)
−0.681026 + 0.732259i \(0.738466\pi\)
\(272\) 6.00000 0.363803
\(273\) 11.2111 0.678527
\(274\) −0.908327 −0.0548740
\(275\) 0 0
\(276\) 12.9083 0.776990
\(277\) −0.119429 −0.00717582 −0.00358791 0.999994i \(-0.501142\pi\)
−0.00358791 + 0.999994i \(0.501142\pi\)
\(278\) −1.90833 −0.114454
\(279\) −2.39445 −0.143352
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 15.2111 0.905808
\(283\) −24.6056 −1.46265 −0.731324 0.682030i \(-0.761097\pi\)
−0.731324 + 0.682030i \(0.761097\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) 25.8167 1.52391
\(288\) 7.90833 0.466003
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −54.2389 −3.17954
\(292\) 12.3028 0.719965
\(293\) −11.0278 −0.644248 −0.322124 0.946697i \(-0.604397\pi\)
−0.322124 + 0.946697i \(0.604397\pi\)
\(294\) 0.697224 0.0406630
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 37.3305 2.16614
\(298\) 19.8167 1.14795
\(299\) 5.09167 0.294459
\(300\) 0 0
\(301\) −1.57779 −0.0909426
\(302\) −20.6056 −1.18572
\(303\) 41.0278 2.35698
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 47.4500 2.71253
\(307\) −17.9083 −1.02208 −0.511041 0.859556i \(-0.670740\pi\)
−0.511041 + 0.859556i \(0.670740\pi\)
\(308\) −6.00000 −0.341882
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) 15.9083 0.902078 0.451039 0.892504i \(-0.351053\pi\)
0.451039 + 0.892504i \(0.351053\pi\)
\(312\) 4.30278 0.243597
\(313\) 9.02776 0.510279 0.255139 0.966904i \(-0.417879\pi\)
0.255139 + 0.966904i \(0.417879\pi\)
\(314\) 7.21110 0.406946
\(315\) 0 0
\(316\) 9.11943 0.513008
\(317\) −9.21110 −0.517347 −0.258674 0.965965i \(-0.583285\pi\)
−0.258674 + 0.965965i \(0.583285\pi\)
\(318\) −19.8167 −1.11126
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) 2.30278 0.128528
\(322\) −10.1833 −0.567496
\(323\) 12.0000 0.667698
\(324\) 29.8167 1.65648
\(325\) 0 0
\(326\) −8.42221 −0.466463
\(327\) −6.60555 −0.365288
\(328\) 9.90833 0.547096
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −13.2111 −0.726148 −0.363074 0.931760i \(-0.618273\pi\)
−0.363074 + 0.931760i \(0.618273\pi\)
\(332\) −2.78890 −0.153061
\(333\) −7.90833 −0.433374
\(334\) 5.51388 0.301706
\(335\) 0 0
\(336\) −8.60555 −0.469471
\(337\) −6.11943 −0.333347 −0.166673 0.986012i \(-0.553303\pi\)
−0.166673 + 0.986012i \(0.553303\pi\)
\(338\) −11.3028 −0.614790
\(339\) −10.6056 −0.576014
\(340\) 0 0
\(341\) 0.697224 0.0377568
\(342\) 15.8167 0.855267
\(343\) −18.7889 −1.01451
\(344\) −0.605551 −0.0326491
\(345\) 0 0
\(346\) 8.78890 0.472494
\(347\) −10.1833 −0.546671 −0.273335 0.961919i \(-0.588127\pi\)
−0.273335 + 0.961919i \(0.588127\pi\)
\(348\) 12.9083 0.691959
\(349\) 28.2389 1.51159 0.755796 0.654807i \(-0.227250\pi\)
0.755796 + 0.654807i \(0.227250\pi\)
\(350\) 0 0
\(351\) 21.1194 1.12727
\(352\) −2.30278 −0.122738
\(353\) −10.1833 −0.542005 −0.271002 0.962579i \(-0.587355\pi\)
−0.271002 + 0.962579i \(0.587355\pi\)
\(354\) −35.0278 −1.86170
\(355\) 0 0
\(356\) −9.21110 −0.488187
\(357\) −51.6333 −2.73272
\(358\) −13.8167 −0.730233
\(359\) 3.21110 0.169476 0.0847378 0.996403i \(-0.472995\pi\)
0.0847378 + 0.996403i \(0.472995\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 20.0000 1.05118
\(363\) 18.8167 0.987618
\(364\) −3.39445 −0.177917
\(365\) 0 0
\(366\) −24.8167 −1.29719
\(367\) −3.81665 −0.199228 −0.0996139 0.995026i \(-0.531761\pi\)
−0.0996139 + 0.995026i \(0.531761\pi\)
\(368\) −3.90833 −0.203736
\(369\) 78.3583 4.07917
\(370\) 0 0
\(371\) 15.6333 0.811641
\(372\) 1.00000 0.0518476
\(373\) 17.8167 0.922511 0.461256 0.887267i \(-0.347399\pi\)
0.461256 + 0.887267i \(0.347399\pi\)
\(374\) −13.8167 −0.714442
\(375\) 0 0
\(376\) −4.60555 −0.237513
\(377\) 5.09167 0.262235
\(378\) −42.2389 −2.17253
\(379\) 24.3305 1.24978 0.624888 0.780715i \(-0.285145\pi\)
0.624888 + 0.780715i \(0.285145\pi\)
\(380\) 0 0
\(381\) −63.4500 −3.25064
\(382\) −5.51388 −0.282115
\(383\) 36.8444 1.88266 0.941331 0.337486i \(-0.109576\pi\)
0.941331 + 0.337486i \(0.109576\pi\)
\(384\) −3.30278 −0.168544
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) −4.78890 −0.243433
\(388\) 16.4222 0.833711
\(389\) −37.1194 −1.88203 −0.941015 0.338365i \(-0.890126\pi\)
−0.941015 + 0.338365i \(0.890126\pi\)
\(390\) 0 0
\(391\) −23.4500 −1.18592
\(392\) −0.211103 −0.0106623
\(393\) −35.0278 −1.76692
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −18.2111 −0.915142
\(397\) −6.18335 −0.310333 −0.155167 0.987888i \(-0.549591\pi\)
−0.155167 + 0.987888i \(0.549591\pi\)
\(398\) 26.4222 1.32443
\(399\) −17.2111 −0.861633
\(400\) 0 0
\(401\) −7.81665 −0.390345 −0.195173 0.980769i \(-0.562527\pi\)
−0.195173 + 0.980769i \(0.562527\pi\)
\(402\) −11.6056 −0.578832
\(403\) 0.394449 0.0196489
\(404\) −12.4222 −0.618028
\(405\) 0 0
\(406\) −10.1833 −0.505391
\(407\) 2.30278 0.114144
\(408\) −19.8167 −0.981071
\(409\) 31.0278 1.53422 0.767112 0.641513i \(-0.221693\pi\)
0.767112 + 0.641513i \(0.221693\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 0.302776 0.0149167
\(413\) 27.6333 1.35975
\(414\) −30.9083 −1.51906
\(415\) 0 0
\(416\) −1.30278 −0.0638738
\(417\) 6.30278 0.308648
\(418\) −4.60555 −0.225265
\(419\) 36.1472 1.76591 0.882953 0.469462i \(-0.155552\pi\)
0.882953 + 0.469462i \(0.155552\pi\)
\(420\) 0 0
\(421\) −3.72498 −0.181544 −0.0907722 0.995872i \(-0.528934\pi\)
−0.0907722 + 0.995872i \(0.528934\pi\)
\(422\) 10.3028 0.501531
\(423\) −36.4222 −1.77091
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −19.8167 −0.960120
\(427\) 19.5778 0.947436
\(428\) −0.697224 −0.0337016
\(429\) −9.90833 −0.478379
\(430\) 0 0
\(431\) 9.21110 0.443683 0.221842 0.975083i \(-0.428793\pi\)
0.221842 + 0.975083i \(0.428793\pi\)
\(432\) −16.2111 −0.779957
\(433\) −34.9361 −1.67892 −0.839461 0.543421i \(-0.817129\pi\)
−0.839461 + 0.543421i \(0.817129\pi\)
\(434\) −0.788897 −0.0378683
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −7.81665 −0.373921
\(438\) −40.6333 −1.94153
\(439\) 30.3305 1.44760 0.723799 0.690011i \(-0.242394\pi\)
0.723799 + 0.690011i \(0.242394\pi\)
\(440\) 0 0
\(441\) −1.66947 −0.0794985
\(442\) −7.81665 −0.371800
\(443\) −32.7250 −1.55481 −0.777405 0.629000i \(-0.783465\pi\)
−0.777405 + 0.629000i \(0.783465\pi\)
\(444\) 3.30278 0.156743
\(445\) 0 0
\(446\) 5.81665 0.275427
\(447\) −65.4500 −3.09568
\(448\) 2.60555 0.123101
\(449\) −15.2111 −0.717856 −0.358928 0.933365i \(-0.616858\pi\)
−0.358928 + 0.933365i \(0.616858\pi\)
\(450\) 0 0
\(451\) −22.8167 −1.07439
\(452\) 3.21110 0.151038
\(453\) 68.0555 3.19753
\(454\) −13.8167 −0.648448
\(455\) 0 0
\(456\) −6.60555 −0.309333
\(457\) 2.60555 0.121883 0.0609413 0.998141i \(-0.480590\pi\)
0.0609413 + 0.998141i \(0.480590\pi\)
\(458\) 24.6056 1.14974
\(459\) −97.2666 −4.54002
\(460\) 0 0
\(461\) 12.4222 0.578560 0.289280 0.957245i \(-0.406584\pi\)
0.289280 + 0.957245i \(0.406584\pi\)
\(462\) 19.8167 0.921954
\(463\) −26.6972 −1.24073 −0.620363 0.784315i \(-0.713015\pi\)
−0.620363 + 0.784315i \(0.713015\pi\)
\(464\) −3.90833 −0.181440
\(465\) 0 0
\(466\) −8.51388 −0.394398
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) −10.3028 −0.476246
\(469\) 9.15559 0.422766
\(470\) 0 0
\(471\) −23.8167 −1.09741
\(472\) 10.6056 0.488160
\(473\) 1.39445 0.0641168
\(474\) −30.1194 −1.38343
\(475\) 0 0
\(476\) 15.6333 0.716551
\(477\) 47.4500 2.17258
\(478\) −17.5139 −0.801066
\(479\) −13.1194 −0.599442 −0.299721 0.954027i \(-0.596894\pi\)
−0.299721 + 0.954027i \(0.596894\pi\)
\(480\) 0 0
\(481\) 1.30278 0.0594015
\(482\) 8.00000 0.364390
\(483\) 33.6333 1.53037
\(484\) −5.69722 −0.258965
\(485\) 0 0
\(486\) −49.8444 −2.26099
\(487\) 37.2111 1.68620 0.843098 0.537760i \(-0.180729\pi\)
0.843098 + 0.537760i \(0.180729\pi\)
\(488\) 7.51388 0.340137
\(489\) 27.8167 1.25791
\(490\) 0 0
\(491\) 17.7250 0.799917 0.399959 0.916533i \(-0.369025\pi\)
0.399959 + 0.916533i \(0.369025\pi\)
\(492\) −32.7250 −1.47536
\(493\) −23.4500 −1.05613
\(494\) −2.60555 −0.117229
\(495\) 0 0
\(496\) −0.302776 −0.0135950
\(497\) 15.6333 0.701250
\(498\) 9.21110 0.412759
\(499\) −42.2389 −1.89087 −0.945436 0.325809i \(-0.894363\pi\)
−0.945436 + 0.325809i \(0.894363\pi\)
\(500\) 0 0
\(501\) −18.2111 −0.813612
\(502\) −21.2111 −0.946698
\(503\) 6.48612 0.289202 0.144601 0.989490i \(-0.453810\pi\)
0.144601 + 0.989490i \(0.453810\pi\)
\(504\) 20.6056 0.917844
\(505\) 0 0
\(506\) 9.00000 0.400099
\(507\) 37.3305 1.65791
\(508\) 19.2111 0.852355
\(509\) −4.18335 −0.185424 −0.0927118 0.995693i \(-0.529554\pi\)
−0.0927118 + 0.995693i \(0.529554\pi\)
\(510\) 0 0
\(511\) 32.0555 1.41805
\(512\) 1.00000 0.0441942
\(513\) −32.4222 −1.43148
\(514\) −3.21110 −0.141636
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) 10.6056 0.466432
\(518\) −2.60555 −0.114481
\(519\) −29.0278 −1.27418
\(520\) 0 0
\(521\) 33.6333 1.47350 0.736751 0.676164i \(-0.236359\pi\)
0.736751 + 0.676164i \(0.236359\pi\)
\(522\) −30.9083 −1.35282
\(523\) 18.2389 0.797530 0.398765 0.917053i \(-0.369439\pi\)
0.398765 + 0.917053i \(0.369439\pi\)
\(524\) 10.6056 0.463306
\(525\) 0 0
\(526\) −13.8167 −0.602435
\(527\) −1.81665 −0.0791347
\(528\) 7.60555 0.330989
\(529\) −7.72498 −0.335869
\(530\) 0 0
\(531\) 83.8722 3.63974
\(532\) 5.21110 0.225930
\(533\) −12.9083 −0.559122
\(534\) 30.4222 1.31650
\(535\) 0 0
\(536\) 3.51388 0.151776
\(537\) 45.6333 1.96922
\(538\) −21.2111 −0.914476
\(539\) 0.486122 0.0209387
\(540\) 0 0
\(541\) 25.9361 1.11508 0.557540 0.830150i \(-0.311745\pi\)
0.557540 + 0.830150i \(0.311745\pi\)
\(542\) −22.4222 −0.963116
\(543\) −66.0555 −2.83471
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 11.2111 0.479791
\(547\) 20.6056 0.881030 0.440515 0.897745i \(-0.354796\pi\)
0.440515 + 0.897745i \(0.354796\pi\)
\(548\) −0.908327 −0.0388018
\(549\) 59.4222 2.53608
\(550\) 0 0
\(551\) −7.81665 −0.333001
\(552\) 12.9083 0.549415
\(553\) 23.7611 1.01043
\(554\) −0.119429 −0.00507407
\(555\) 0 0
\(556\) −1.90833 −0.0809311
\(557\) −11.5139 −0.487859 −0.243929 0.969793i \(-0.578436\pi\)
−0.243929 + 0.969793i \(0.578436\pi\)
\(558\) −2.39445 −0.101365
\(559\) 0.788897 0.0333668
\(560\) 0 0
\(561\) 45.6333 1.92664
\(562\) −12.0000 −0.506189
\(563\) 28.0555 1.18240 0.591199 0.806525i \(-0.298655\pi\)
0.591199 + 0.806525i \(0.298655\pi\)
\(564\) 15.2111 0.640503
\(565\) 0 0
\(566\) −24.6056 −1.03425
\(567\) 77.6888 3.26262
\(568\) 6.00000 0.251754
\(569\) 18.4222 0.772299 0.386150 0.922436i \(-0.373805\pi\)
0.386150 + 0.922436i \(0.373805\pi\)
\(570\) 0 0
\(571\) −16.6972 −0.698757 −0.349379 0.936982i \(-0.613607\pi\)
−0.349379 + 0.936982i \(0.613607\pi\)
\(572\) 3.00000 0.125436
\(573\) 18.2111 0.760780
\(574\) 25.8167 1.07757
\(575\) 0 0
\(576\) 7.90833 0.329514
\(577\) −22.2389 −0.925816 −0.462908 0.886406i \(-0.653194\pi\)
−0.462908 + 0.886406i \(0.653194\pi\)
\(578\) 19.0000 0.790296
\(579\) −13.2111 −0.549035
\(580\) 0 0
\(581\) −7.26662 −0.301470
\(582\) −54.2389 −2.24827
\(583\) −13.8167 −0.572227
\(584\) 12.3028 0.509092
\(585\) 0 0
\(586\) −11.0278 −0.455552
\(587\) 45.6333 1.88349 0.941744 0.336330i \(-0.109186\pi\)
0.941744 + 0.336330i \(0.109186\pi\)
\(588\) 0.697224 0.0287530
\(589\) −0.605551 −0.0249513
\(590\) 0 0
\(591\) −19.8167 −0.815148
\(592\) −1.00000 −0.0410997
\(593\) −18.4861 −0.759134 −0.379567 0.925164i \(-0.623927\pi\)
−0.379567 + 0.925164i \(0.623927\pi\)
\(594\) 37.3305 1.53169
\(595\) 0 0
\(596\) 19.8167 0.811722
\(597\) −87.2666 −3.57158
\(598\) 5.09167 0.208214
\(599\) −20.7889 −0.849411 −0.424706 0.905331i \(-0.639622\pi\)
−0.424706 + 0.905331i \(0.639622\pi\)
\(600\) 0 0
\(601\) −24.3028 −0.991331 −0.495665 0.868514i \(-0.665076\pi\)
−0.495665 + 0.868514i \(0.665076\pi\)
\(602\) −1.57779 −0.0643061
\(603\) 27.7889 1.13165
\(604\) −20.6056 −0.838428
\(605\) 0 0
\(606\) 41.0278 1.66664
\(607\) 13.4861 0.547385 0.273692 0.961817i \(-0.411755\pi\)
0.273692 + 0.961817i \(0.411755\pi\)
\(608\) 2.00000 0.0811107
\(609\) 33.6333 1.36289
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 47.4500 1.91805
\(613\) 29.8167 1.20428 0.602142 0.798389i \(-0.294314\pi\)
0.602142 + 0.798389i \(0.294314\pi\)
\(614\) −17.9083 −0.722721
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) 42.5694 1.71378 0.856890 0.515500i \(-0.172394\pi\)
0.856890 + 0.515500i \(0.172394\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −6.30278 −0.253330 −0.126665 0.991946i \(-0.540427\pi\)
−0.126665 + 0.991946i \(0.540427\pi\)
\(620\) 0 0
\(621\) 63.3583 2.54248
\(622\) 15.9083 0.637866
\(623\) −24.0000 −0.961540
\(624\) 4.30278 0.172249
\(625\) 0 0
\(626\) 9.02776 0.360822
\(627\) 15.2111 0.607473
\(628\) 7.21110 0.287754
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 14.6972 0.585087 0.292544 0.956252i \(-0.405498\pi\)
0.292544 + 0.956252i \(0.405498\pi\)
\(632\) 9.11943 0.362751
\(633\) −34.0278 −1.35248
\(634\) −9.21110 −0.365820
\(635\) 0 0
\(636\) −19.8167 −0.785781
\(637\) 0.275019 0.0108967
\(638\) 9.00000 0.356313
\(639\) 47.4500 1.87709
\(640\) 0 0
\(641\) −20.5139 −0.810249 −0.405125 0.914261i \(-0.632772\pi\)
−0.405125 + 0.914261i \(0.632772\pi\)
\(642\) 2.30278 0.0908833
\(643\) 8.18335 0.322720 0.161360 0.986896i \(-0.448412\pi\)
0.161360 + 0.986896i \(0.448412\pi\)
\(644\) −10.1833 −0.401280
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −20.9361 −0.823082 −0.411541 0.911391i \(-0.635009\pi\)
−0.411541 + 0.911391i \(0.635009\pi\)
\(648\) 29.8167 1.17131
\(649\) −24.4222 −0.958655
\(650\) 0 0
\(651\) 2.60555 0.102120
\(652\) −8.42221 −0.329839
\(653\) 3.90833 0.152945 0.0764723 0.997072i \(-0.475634\pi\)
0.0764723 + 0.997072i \(0.475634\pi\)
\(654\) −6.60555 −0.258297
\(655\) 0 0
\(656\) 9.90833 0.386855
\(657\) 97.2944 3.79581
\(658\) −12.0000 −0.467809
\(659\) −16.8806 −0.657574 −0.328787 0.944404i \(-0.606640\pi\)
−0.328787 + 0.944404i \(0.606640\pi\)
\(660\) 0 0
\(661\) −30.5139 −1.18685 −0.593426 0.804888i \(-0.702225\pi\)
−0.593426 + 0.804888i \(0.702225\pi\)
\(662\) −13.2111 −0.513464
\(663\) 25.8167 1.00264
\(664\) −2.78890 −0.108230
\(665\) 0 0
\(666\) −7.90833 −0.306441
\(667\) 15.2750 0.591451
\(668\) 5.51388 0.213338
\(669\) −19.2111 −0.742744
\(670\) 0 0
\(671\) −17.3028 −0.667966
\(672\) −8.60555 −0.331966
\(673\) −20.6972 −0.797819 −0.398910 0.916990i \(-0.630611\pi\)
−0.398910 + 0.916990i \(0.630611\pi\)
\(674\) −6.11943 −0.235712
\(675\) 0 0
\(676\) −11.3028 −0.434722
\(677\) −14.2389 −0.547244 −0.273622 0.961837i \(-0.588222\pi\)
−0.273622 + 0.961837i \(0.588222\pi\)
\(678\) −10.6056 −0.407304
\(679\) 42.7889 1.64209
\(680\) 0 0
\(681\) 45.6333 1.74867
\(682\) 0.697224 0.0266981
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 15.8167 0.604765
\(685\) 0 0
\(686\) −18.7889 −0.717363
\(687\) −81.2666 −3.10051
\(688\) −0.605551 −0.0230864
\(689\) −7.81665 −0.297791
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 8.78890 0.334104
\(693\) −47.4500 −1.80247
\(694\) −10.1833 −0.386555
\(695\) 0 0
\(696\) 12.9083 0.489289
\(697\) 59.4500 2.25183
\(698\) 28.2389 1.06886
\(699\) 28.1194 1.06357
\(700\) 0 0
\(701\) 40.1194 1.51529 0.757645 0.652667i \(-0.226350\pi\)
0.757645 + 0.652667i \(0.226350\pi\)
\(702\) 21.1194 0.797101
\(703\) −2.00000 −0.0754314
\(704\) −2.30278 −0.0867891
\(705\) 0 0
\(706\) −10.1833 −0.383255
\(707\) −32.3667 −1.21727
\(708\) −35.0278 −1.31642
\(709\) −41.3305 −1.55220 −0.776100 0.630609i \(-0.782805\pi\)
−0.776100 + 0.630609i \(0.782805\pi\)
\(710\) 0 0
\(711\) 72.1194 2.70469
\(712\) −9.21110 −0.345201
\(713\) 1.18335 0.0443167
\(714\) −51.6333 −1.93233
\(715\) 0 0
\(716\) −13.8167 −0.516353
\(717\) 57.8444 2.16024
\(718\) 3.21110 0.119837
\(719\) −51.6333 −1.92560 −0.962799 0.270220i \(-0.912904\pi\)
−0.962799 + 0.270220i \(0.912904\pi\)
\(720\) 0 0
\(721\) 0.788897 0.0293801
\(722\) −15.0000 −0.558242
\(723\) −26.4222 −0.982652
\(724\) 20.0000 0.743294
\(725\) 0 0
\(726\) 18.8167 0.698352
\(727\) −19.0917 −0.708071 −0.354035 0.935232i \(-0.615191\pi\)
−0.354035 + 0.935232i \(0.615191\pi\)
\(728\) −3.39445 −0.125807
\(729\) 75.1749 2.78426
\(730\) 0 0
\(731\) −3.63331 −0.134383
\(732\) −24.8167 −0.917250
\(733\) 13.6333 0.503558 0.251779 0.967785i \(-0.418984\pi\)
0.251779 + 0.967785i \(0.418984\pi\)
\(734\) −3.81665 −0.140875
\(735\) 0 0
\(736\) −3.90833 −0.144063
\(737\) −8.09167 −0.298061
\(738\) 78.3583 2.88441
\(739\) −2.66947 −0.0981980 −0.0490990 0.998794i \(-0.515635\pi\)
−0.0490990 + 0.998794i \(0.515635\pi\)
\(740\) 0 0
\(741\) 8.60555 0.316133
\(742\) 15.6333 0.573917
\(743\) 29.4500 1.08041 0.540207 0.841532i \(-0.318346\pi\)
0.540207 + 0.841532i \(0.318346\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) 17.8167 0.652314
\(747\) −22.0555 −0.806969
\(748\) −13.8167 −0.505187
\(749\) −1.81665 −0.0663791
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) −4.60555 −0.167947
\(753\) 70.0555 2.55296
\(754\) 5.09167 0.185428
\(755\) 0 0
\(756\) −42.2389 −1.53621
\(757\) −5.69722 −0.207069 −0.103535 0.994626i \(-0.533015\pi\)
−0.103535 + 0.994626i \(0.533015\pi\)
\(758\) 24.3305 0.883725
\(759\) −29.7250 −1.07895
\(760\) 0 0
\(761\) 16.8806 0.611920 0.305960 0.952044i \(-0.401023\pi\)
0.305960 + 0.952044i \(0.401023\pi\)
\(762\) −63.4500 −2.29855
\(763\) 5.21110 0.188655
\(764\) −5.51388 −0.199485
\(765\) 0 0
\(766\) 36.8444 1.33124
\(767\) −13.8167 −0.498890
\(768\) −3.30278 −0.119179
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 10.6056 0.381950
\(772\) 4.00000 0.143963
\(773\) −22.0555 −0.793282 −0.396641 0.917974i \(-0.629824\pi\)
−0.396641 + 0.917974i \(0.629824\pi\)
\(774\) −4.78890 −0.172133
\(775\) 0 0
\(776\) 16.4222 0.589523
\(777\) 8.60555 0.308722
\(778\) −37.1194 −1.33080
\(779\) 19.8167 0.710005
\(780\) 0 0
\(781\) −13.8167 −0.494399
\(782\) −23.4500 −0.838569
\(783\) 63.3583 2.26424
\(784\) −0.211103 −0.00753938
\(785\) 0 0
\(786\) −35.0278 −1.24940
\(787\) −10.7889 −0.384583 −0.192291 0.981338i \(-0.561592\pi\)
−0.192291 + 0.981338i \(0.561592\pi\)
\(788\) 6.00000 0.213741
\(789\) 45.6333 1.62459
\(790\) 0 0
\(791\) 8.36669 0.297485
\(792\) −18.2111 −0.647103
\(793\) −9.78890 −0.347614
\(794\) −6.18335 −0.219439
\(795\) 0 0
\(796\) 26.4222 0.936510
\(797\) −22.3305 −0.790988 −0.395494 0.918469i \(-0.629427\pi\)
−0.395494 + 0.918469i \(0.629427\pi\)
\(798\) −17.2111 −0.609266
\(799\) −27.6333 −0.977596
\(800\) 0 0
\(801\) −72.8444 −2.57383
\(802\) −7.81665 −0.276016
\(803\) −28.3305 −0.999763
\(804\) −11.6056 −0.409296
\(805\) 0 0
\(806\) 0.394449 0.0138939
\(807\) 70.0555 2.46607
\(808\) −12.4222 −0.437012
\(809\) −35.4500 −1.24635 −0.623177 0.782081i \(-0.714158\pi\)
−0.623177 + 0.782081i \(0.714158\pi\)
\(810\) 0 0
\(811\) −7.14719 −0.250972 −0.125486 0.992095i \(-0.540049\pi\)
−0.125486 + 0.992095i \(0.540049\pi\)
\(812\) −10.1833 −0.357365
\(813\) 74.0555 2.59724
\(814\) 2.30278 0.0807122
\(815\) 0 0
\(816\) −19.8167 −0.693722
\(817\) −1.21110 −0.0423711
\(818\) 31.0278 1.08486
\(819\) −26.8444 −0.938020
\(820\) 0 0
\(821\) −3.21110 −0.112068 −0.0560341 0.998429i \(-0.517846\pi\)
−0.0560341 + 0.998429i \(0.517846\pi\)
\(822\) 3.00000 0.104637
\(823\) −44.8444 −1.56318 −0.781589 0.623794i \(-0.785590\pi\)
−0.781589 + 0.623794i \(0.785590\pi\)
\(824\) 0.302776 0.0105477
\(825\) 0 0
\(826\) 27.6333 0.961486
\(827\) −34.6056 −1.20335 −0.601676 0.798740i \(-0.705500\pi\)
−0.601676 + 0.798740i \(0.705500\pi\)
\(828\) −30.9083 −1.07414
\(829\) −27.7250 −0.962928 −0.481464 0.876466i \(-0.659895\pi\)
−0.481464 + 0.876466i \(0.659895\pi\)
\(830\) 0 0
\(831\) 0.394449 0.0136833
\(832\) −1.30278 −0.0451656
\(833\) −1.26662 −0.0438856
\(834\) 6.30278 0.218247
\(835\) 0 0
\(836\) −4.60555 −0.159286
\(837\) 4.90833 0.169657
\(838\) 36.1472 1.24868
\(839\) −12.9722 −0.447852 −0.223926 0.974606i \(-0.571887\pi\)
−0.223926 + 0.974606i \(0.571887\pi\)
\(840\) 0 0
\(841\) −13.7250 −0.473275
\(842\) −3.72498 −0.128371
\(843\) 39.6333 1.36504
\(844\) 10.3028 0.354636
\(845\) 0 0
\(846\) −36.4222 −1.25222
\(847\) −14.8444 −0.510060
\(848\) 6.00000 0.206041
\(849\) 81.2666 2.78906
\(850\) 0 0
\(851\) 3.90833 0.133976
\(852\) −19.8167 −0.678907
\(853\) −42.5416 −1.45660 −0.728299 0.685260i \(-0.759689\pi\)
−0.728299 + 0.685260i \(0.759689\pi\)
\(854\) 19.5778 0.669938
\(855\) 0 0
\(856\) −0.697224 −0.0238306
\(857\) −42.8444 −1.46354 −0.731769 0.681553i \(-0.761305\pi\)
−0.731769 + 0.681553i \(0.761305\pi\)
\(858\) −9.90833 −0.338265
\(859\) 48.0555 1.63963 0.819816 0.572626i \(-0.194075\pi\)
0.819816 + 0.572626i \(0.194075\pi\)
\(860\) 0 0
\(861\) −85.2666 −2.90588
\(862\) 9.21110 0.313731
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) −16.2111 −0.551513
\(865\) 0 0
\(866\) −34.9361 −1.18718
\(867\) −62.7527 −2.13119
\(868\) −0.788897 −0.0267769
\(869\) −21.0000 −0.712376
\(870\) 0 0
\(871\) −4.57779 −0.155113
\(872\) 2.00000 0.0677285
\(873\) 129.872 4.39551
\(874\) −7.81665 −0.264402
\(875\) 0 0
\(876\) −40.6333 −1.37287
\(877\) 7.21110 0.243502 0.121751 0.992561i \(-0.461149\pi\)
0.121751 + 0.992561i \(0.461149\pi\)
\(878\) 30.3305 1.02361
\(879\) 36.4222 1.22849
\(880\) 0 0
\(881\) −28.5416 −0.961592 −0.480796 0.876832i \(-0.659652\pi\)
−0.480796 + 0.876832i \(0.659652\pi\)
\(882\) −1.66947 −0.0562139
\(883\) −26.4222 −0.889178 −0.444589 0.895735i \(-0.646650\pi\)
−0.444589 + 0.895735i \(0.646650\pi\)
\(884\) −7.81665 −0.262903
\(885\) 0 0
\(886\) −32.7250 −1.09942
\(887\) 0.422205 0.0141763 0.00708813 0.999975i \(-0.497744\pi\)
0.00708813 + 0.999975i \(0.497744\pi\)
\(888\) 3.30278 0.110834
\(889\) 50.0555 1.67881
\(890\) 0 0
\(891\) −68.6611 −2.30023
\(892\) 5.81665 0.194756
\(893\) −9.21110 −0.308238
\(894\) −65.4500 −2.18897
\(895\) 0 0
\(896\) 2.60555 0.0870454
\(897\) −16.8167 −0.561492
\(898\) −15.2111 −0.507601
\(899\) 1.18335 0.0394668
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) −22.8167 −0.759711
\(903\) 5.21110 0.173415
\(904\) 3.21110 0.106800
\(905\) 0 0
\(906\) 68.0555 2.26099
\(907\) −26.0000 −0.863316 −0.431658 0.902037i \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) −13.8167 −0.458522
\(909\) −98.2389 −3.25838
\(910\) 0 0
\(911\) −17.5778 −0.582378 −0.291189 0.956665i \(-0.594051\pi\)
−0.291189 + 0.956665i \(0.594051\pi\)
\(912\) −6.60555 −0.218732
\(913\) 6.42221 0.212544
\(914\) 2.60555 0.0861840
\(915\) 0 0
\(916\) 24.6056 0.812990
\(917\) 27.6333 0.912532
\(918\) −97.2666 −3.21028
\(919\) −9.57779 −0.315942 −0.157971 0.987444i \(-0.550495\pi\)
−0.157971 + 0.987444i \(0.550495\pi\)
\(920\) 0 0
\(921\) 59.1472 1.94897
\(922\) 12.4222 0.409104
\(923\) −7.81665 −0.257288
\(924\) 19.8167 0.651920
\(925\) 0 0
\(926\) −26.6972 −0.877325
\(927\) 2.39445 0.0786440
\(928\) −3.90833 −0.128297
\(929\) −18.4861 −0.606510 −0.303255 0.952909i \(-0.598073\pi\)
−0.303255 + 0.952909i \(0.598073\pi\)
\(930\) 0 0
\(931\) −0.422205 −0.0138372
\(932\) −8.51388 −0.278881
\(933\) −52.5416 −1.72014
\(934\) 0 0
\(935\) 0 0
\(936\) −10.3028 −0.336757
\(937\) 18.0917 0.591029 0.295515 0.955338i \(-0.404509\pi\)
0.295515 + 0.955338i \(0.404509\pi\)
\(938\) 9.15559 0.298941
\(939\) −29.8167 −0.973030
\(940\) 0 0
\(941\) 13.8167 0.450410 0.225205 0.974311i \(-0.427695\pi\)
0.225205 + 0.974311i \(0.427695\pi\)
\(942\) −23.8167 −0.775989
\(943\) −38.7250 −1.26106
\(944\) 10.6056 0.345181
\(945\) 0 0
\(946\) 1.39445 0.0453374
\(947\) 3.63331 0.118067 0.0590333 0.998256i \(-0.481198\pi\)
0.0590333 + 0.998256i \(0.481198\pi\)
\(948\) −30.1194 −0.978234
\(949\) −16.0278 −0.520283
\(950\) 0 0
\(951\) 30.4222 0.986508
\(952\) 15.6333 0.506678
\(953\) −49.7527 −1.61165 −0.805825 0.592154i \(-0.798278\pi\)
−0.805825 + 0.592154i \(0.798278\pi\)
\(954\) 47.4500 1.53625
\(955\) 0 0
\(956\) −17.5139 −0.566439
\(957\) −29.7250 −0.960872
\(958\) −13.1194 −0.423870
\(959\) −2.36669 −0.0764245
\(960\) 0 0
\(961\) −30.9083 −0.997043
\(962\) 1.30278 0.0420032
\(963\) −5.51388 −0.177682
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) 33.6333 1.08213
\(967\) 6.72498 0.216261 0.108130 0.994137i \(-0.465514\pi\)
0.108130 + 0.994137i \(0.465514\pi\)
\(968\) −5.69722 −0.183116
\(969\) −39.6333 −1.27321
\(970\) 0 0
\(971\) −22.5416 −0.723395 −0.361698 0.932295i \(-0.617803\pi\)
−0.361698 + 0.932295i \(0.617803\pi\)
\(972\) −49.8444 −1.59876
\(973\) −4.97224 −0.159403
\(974\) 37.2111 1.19232
\(975\) 0 0
\(976\) 7.51388 0.240513
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 27.8167 0.889479
\(979\) 21.2111 0.677910
\(980\) 0 0
\(981\) 15.8167 0.504987
\(982\) 17.7250 0.565627
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) −32.7250 −1.04323
\(985\) 0 0
\(986\) −23.4500 −0.746799
\(987\) 39.6333 1.26154
\(988\) −2.60555 −0.0828936
\(989\) 2.36669 0.0752564
\(990\) 0 0
\(991\) 50.6972 1.61045 0.805225 0.592969i \(-0.202044\pi\)
0.805225 + 0.592969i \(0.202044\pi\)
\(992\) −0.302776 −0.00961314
\(993\) 43.6333 1.38466
\(994\) 15.6333 0.495858
\(995\) 0 0
\(996\) 9.21110 0.291865
\(997\) 52.4222 1.66023 0.830114 0.557594i \(-0.188275\pi\)
0.830114 + 0.557594i \(0.188275\pi\)
\(998\) −42.2389 −1.33705
\(999\) 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.a.u.1.1 2
5.2 odd 4 1850.2.b.i.149.4 4
5.3 odd 4 1850.2.b.i.149.1 4
5.4 even 2 74.2.a.a.1.2 2
15.14 odd 2 666.2.a.j.1.2 2
20.19 odd 2 592.2.a.f.1.1 2
35.34 odd 2 3626.2.a.a.1.1 2
40.19 odd 2 2368.2.a.ba.1.2 2
40.29 even 2 2368.2.a.s.1.1 2
55.54 odd 2 8954.2.a.p.1.2 2
60.59 even 2 5328.2.a.bf.1.2 2
185.184 even 2 2738.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 2 5.4 even 2
592.2.a.f.1.1 2 20.19 odd 2
666.2.a.j.1.2 2 15.14 odd 2
1850.2.a.u.1.1 2 1.1 even 1 trivial
1850.2.b.i.149.1 4 5.3 odd 4
1850.2.b.i.149.4 4 5.2 odd 4
2368.2.a.s.1.1 2 40.29 even 2
2368.2.a.ba.1.2 2 40.19 odd 2
2738.2.a.l.1.2 2 185.184 even 2
3626.2.a.a.1.1 2 35.34 odd 2
5328.2.a.bf.1.2 2 60.59 even 2
8954.2.a.p.1.2 2 55.54 odd 2