Properties

Label 2738.2.a.l.1.2
Level $2738$
Weight $2$
Character 2738.1
Self dual yes
Analytic conductor $21.863$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.8630400734\)
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 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 2738.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.30278 q^{3} +1.00000 q^{4} +2.30278 q^{5} +3.30278 q^{6} -2.60555 q^{7} +1.00000 q^{8} +7.90833 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.30278 q^{3} +1.00000 q^{4} +2.30278 q^{5} +3.30278 q^{6} -2.60555 q^{7} +1.00000 q^{8} +7.90833 q^{9} +2.30278 q^{10} -2.30278 q^{11} +3.30278 q^{12} -1.30278 q^{13} -2.60555 q^{14} +7.60555 q^{15} +1.00000 q^{16} +6.00000 q^{17} +7.90833 q^{18} -2.00000 q^{19} +2.30278 q^{20} -8.60555 q^{21} -2.30278 q^{22} -3.90833 q^{23} +3.30278 q^{24} +0.302776 q^{25} -1.30278 q^{26} +16.2111 q^{27} -2.60555 q^{28} +3.90833 q^{29} +7.60555 q^{30} +0.302776 q^{31} +1.00000 q^{32} -7.60555 q^{33} +6.00000 q^{34} -6.00000 q^{35} +7.90833 q^{36} -2.00000 q^{38} -4.30278 q^{39} +2.30278 q^{40} +9.90833 q^{41} -8.60555 q^{42} -0.605551 q^{43} -2.30278 q^{44} +18.2111 q^{45} -3.90833 q^{46} +4.60555 q^{47} +3.30278 q^{48} -0.211103 q^{49} +0.302776 q^{50} +19.8167 q^{51} -1.30278 q^{52} -6.00000 q^{53} +16.2111 q^{54} -5.30278 q^{55} -2.60555 q^{56} -6.60555 q^{57} +3.90833 q^{58} -10.6056 q^{59} +7.60555 q^{60} -7.51388 q^{61} +0.302776 q^{62} -20.6056 q^{63} +1.00000 q^{64} -3.00000 q^{65} -7.60555 q^{66} -3.51388 q^{67} +6.00000 q^{68} -12.9083 q^{69} -6.00000 q^{70} +6.00000 q^{71} +7.90833 q^{72} -12.3028 q^{73} +1.00000 q^{75} -2.00000 q^{76} +6.00000 q^{77} -4.30278 q^{78} -9.11943 q^{79} +2.30278 q^{80} +29.8167 q^{81} +9.90833 q^{82} +2.78890 q^{83} -8.60555 q^{84} +13.8167 q^{85} -0.605551 q^{86} +12.9083 q^{87} -2.30278 q^{88} +9.21110 q^{89} +18.2111 q^{90} +3.39445 q^{91} -3.90833 q^{92} +1.00000 q^{93} +4.60555 q^{94} -4.60555 q^{95} +3.30278 q^{96} +16.4222 q^{97} -0.211103 q^{98} -18.2111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + 5 q^{9} + q^{10} - q^{11} + 3 q^{12} + q^{13} + 2 q^{14} + 8 q^{15} + 2 q^{16} + 12 q^{17} + 5 q^{18} - 4 q^{19} + q^{20} - 10 q^{21} - q^{22} + 3 q^{23} + 3 q^{24} - 3 q^{25} + q^{26} + 18 q^{27} + 2 q^{28} - 3 q^{29} + 8 q^{30} - 3 q^{31} + 2 q^{32} - 8 q^{33} + 12 q^{34} - 12 q^{35} + 5 q^{36} - 4 q^{38} - 5 q^{39} + q^{40} + 9 q^{41} - 10 q^{42} + 6 q^{43} - q^{44} + 22 q^{45} + 3 q^{46} + 2 q^{47} + 3 q^{48} + 14 q^{49} - 3 q^{50} + 18 q^{51} + q^{52} - 12 q^{53} + 18 q^{54} - 7 q^{55} + 2 q^{56} - 6 q^{57} - 3 q^{58} - 14 q^{59} + 8 q^{60} + 3 q^{61} - 3 q^{62} - 34 q^{63} + 2 q^{64} - 6 q^{65} - 8 q^{66} + 11 q^{67} + 12 q^{68} - 15 q^{69} - 12 q^{70} + 12 q^{71} + 5 q^{72} - 21 q^{73} + 2 q^{75} - 4 q^{76} + 12 q^{77} - 5 q^{78} + 7 q^{79} + q^{80} + 38 q^{81} + 9 q^{82} + 20 q^{83} - 10 q^{84} + 6 q^{85} + 6 q^{86} + 15 q^{87} - q^{88} + 4 q^{89} + 22 q^{90} + 14 q^{91} + 3 q^{92} + 2 q^{93} + 2 q^{94} - 2 q^{95} + 3 q^{96} + 4 q^{97} + 14 q^{98} - 22 q^{99}+O(q^{100}) \) 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 \(-0.0975264\pi\)
0.953429 + 0.301617i \(0.0975264\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.30278 1.02983 0.514916 0.857240i \(-0.327823\pi\)
0.514916 + 0.857240i \(0.327823\pi\)
\(6\) 3.30278 1.34835
\(7\) −2.60555 −0.984806 −0.492403 0.870367i \(-0.663881\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) 1.00000 0.353553
\(9\) 7.90833 2.63611
\(10\) 2.30278 0.728202
\(11\) −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
\(15\) 7.60555 1.96374
\(16\) 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 \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.30278 0.514916
\(21\) −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
\(25\) 0.302776 0.0605551
\(26\) −1.30278 −0.255495
\(27\) 16.2111 3.11983
\(28\) −2.60555 −0.492403
\(29\) 3.90833 0.725758 0.362879 0.931836i \(-0.381794\pi\)
0.362879 + 0.931836i \(0.381794\pi\)
\(30\) 7.60555 1.38858
\(31\) 0.302776 0.0543801 0.0271901 0.999630i \(-0.491344\pi\)
0.0271901 + 0.999630i \(0.491344\pi\)
\(32\) 1.00000 0.176777
\(33\) −7.60555 −1.32396
\(34\) 6.00000 1.02899
\(35\) −6.00000 −1.01419
\(36\) 7.90833 1.31805
\(37\) 0 0
\(38\) −2.00000 −0.324443
\(39\) −4.30278 −0.688996
\(40\) 2.30278 0.364101
\(41\) 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
\(45\) 18.2111 2.71475
\(46\) −3.90833 −0.576251
\(47\) 4.60555 0.671789 0.335894 0.941900i \(-0.390961\pi\)
0.335894 + 0.941900i \(0.390961\pi\)
\(48\) 3.30278 0.476715
\(49\) −0.211103 −0.0301575
\(50\) 0.302776 0.0428189
\(51\) 19.8167 2.77489
\(52\) −1.30278 −0.180662
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 16.2111 2.20605
\(55\) −5.30278 −0.715026
\(56\) −2.60555 −0.348181
\(57\) −6.60555 −0.874927
\(58\) 3.90833 0.513188
\(59\) −10.6056 −1.38073 −0.690363 0.723464i \(-0.742549\pi\)
−0.690363 + 0.723464i \(0.742549\pi\)
\(60\) 7.60555 0.981872
\(61\) −7.51388 −0.962054 −0.481027 0.876706i \(-0.659736\pi\)
−0.481027 + 0.876706i \(0.659736\pi\)
\(62\) 0.302776 0.0384525
\(63\) −20.6056 −2.59606
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) −7.60555 −0.936179
\(67\) −3.51388 −0.429289 −0.214644 0.976692i \(-0.568859\pi\)
−0.214644 + 0.976692i \(0.568859\pi\)
\(68\) 6.00000 0.727607
\(69\) −12.9083 −1.55398
\(70\) −6.00000 −0.717137
\(71\) 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 \(-0.755842\pi\)
−0.719965 + 0.694010i \(0.755842\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) −2.00000 −0.229416
\(77\) 6.00000 0.683763
\(78\) −4.30278 −0.487193
\(79\) −9.11943 −1.02602 −0.513008 0.858384i \(-0.671469\pi\)
−0.513008 + 0.858384i \(0.671469\pi\)
\(80\) 2.30278 0.257458
\(81\) 29.8167 3.31296
\(82\) 9.90833 1.09419
\(83\) 2.78890 0.306121 0.153061 0.988217i \(-0.451087\pi\)
0.153061 + 0.988217i \(0.451087\pi\)
\(84\) −8.60555 −0.938943
\(85\) 13.8167 1.49863
\(86\) −0.605551 −0.0652983
\(87\) 12.9083 1.38392
\(88\) −2.30278 −0.245477
\(89\) 9.21110 0.976375 0.488187 0.872739i \(-0.337658\pi\)
0.488187 + 0.872739i \(0.337658\pi\)
\(90\) 18.2111 1.91962
\(91\) 3.39445 0.355835
\(92\) −3.90833 −0.407471
\(93\) 1.00000 0.103695
\(94\) 4.60555 0.475026
\(95\) −4.60555 −0.472520
\(96\) 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
\(100\) 0.302776 0.0302776
\(101\) −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
\(105\) −19.8167 −1.93391
\(106\) −6.00000 −0.582772
\(107\) 0.697224 0.0674032 0.0337016 0.999432i \(-0.489270\pi\)
0.0337016 + 0.999432i \(0.489270\pi\)
\(108\) 16.2111 1.55991
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −5.30278 −0.505600
\(111\) 0 0
\(112\) −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
\(115\) −9.00000 −0.839254
\(116\) 3.90833 0.362879
\(117\) −10.3028 −0.952492
\(118\) −10.6056 −0.976320
\(119\) −15.6333 −1.43310
\(120\) 7.60555 0.694289
\(121\) −5.69722 −0.517929
\(122\) −7.51388 −0.680275
\(123\) 32.7250 2.95071
\(124\) 0.302776 0.0271901
\(125\) −10.8167 −0.967471
\(126\) −20.6056 −1.83569
\(127\) −19.2111 −1.70471 −0.852355 0.522964i \(-0.824826\pi\)
−0.852355 + 0.522964i \(0.824826\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.00000 −0.176090
\(130\) −3.00000 −0.263117
\(131\) −10.6056 −0.926611 −0.463306 0.886199i \(-0.653337\pi\)
−0.463306 + 0.886199i \(0.653337\pi\)
\(132\) −7.60555 −0.661978
\(133\) 5.21110 0.451860
\(134\) −3.51388 −0.303553
\(135\) 37.3305 3.21290
\(136\) 6.00000 0.514496
\(137\) 0.908327 0.0776036 0.0388018 0.999247i \(-0.487646\pi\)
0.0388018 + 0.999247i \(0.487646\pi\)
\(138\) −12.9083 −1.09883
\(139\) −1.90833 −0.161862 −0.0809311 0.996720i \(-0.525789\pi\)
−0.0809311 + 0.996720i \(0.525789\pi\)
\(140\) −6.00000 −0.507093
\(141\) 15.2111 1.28101
\(142\) 6.00000 0.503509
\(143\) 3.00000 0.250873
\(144\) 7.90833 0.659027
\(145\) 9.00000 0.747409
\(146\) −12.3028 −1.01818
\(147\) −0.697224 −0.0575061
\(148\) 0 0
\(149\) 19.8167 1.62344 0.811722 0.584044i \(-0.198531\pi\)
0.811722 + 0.584044i \(0.198531\pi\)
\(150\) 1.00000 0.0816497
\(151\) −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
\(155\) 0.697224 0.0560024
\(156\) −4.30278 −0.344498
\(157\) −7.21110 −0.575509 −0.287754 0.957704i \(-0.592909\pi\)
−0.287754 + 0.957704i \(0.592909\pi\)
\(158\) −9.11943 −0.725503
\(159\) −19.8167 −1.57156
\(160\) 2.30278 0.182050
\(161\) 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
\(165\) −17.5139 −1.36345
\(166\) 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
\(170\) 13.8167 1.05969
\(171\) −15.8167 −1.20953
\(172\) −0.605551 −0.0461729
\(173\) −8.78890 −0.668207 −0.334104 0.942536i \(-0.608434\pi\)
−0.334104 + 0.942536i \(0.608434\pi\)
\(174\) 12.9083 0.978578
\(175\) −0.788897 −0.0596350
\(176\) −2.30278 −0.173578
\(177\) −35.0278 −2.63285
\(178\) 9.21110 0.690401
\(179\) 13.8167 1.03271 0.516353 0.856376i \(-0.327289\pi\)
0.516353 + 0.856376i \(0.327289\pi\)
\(180\) 18.2111 1.35738
\(181\) 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
\(190\) −4.60555 −0.334122
\(191\) 5.51388 0.398970 0.199485 0.979901i \(-0.436073\pi\)
0.199485 + 0.979901i \(0.436073\pi\)
\(192\) 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
\(195\) −9.90833 −0.709550
\(196\) −0.211103 −0.0150788
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −18.2111 −1.29421
\(199\) −26.4222 −1.87302 −0.936510 0.350640i \(-0.885964\pi\)
−0.936510 + 0.350640i \(0.885964\pi\)
\(200\) 0.302776 0.0214095
\(201\) −11.6056 −0.818592
\(202\) −12.4222 −0.874023
\(203\) −10.1833 −0.714731
\(204\) 19.8167 1.38744
\(205\) 22.8167 1.59358
\(206\) 0.302776 0.0210954
\(207\) −30.9083 −2.14828
\(208\) −1.30278 −0.0903312
\(209\) 4.60555 0.318573
\(210\) −19.8167 −1.36748
\(211\) 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
\(215\) −1.39445 −0.0951006
\(216\) 16.2111 1.10303
\(217\) −0.788897 −0.0535538
\(218\) −2.00000 −0.135457
\(219\) −40.6333 −2.74574
\(220\) −5.30278 −0.357513
\(221\) −7.81665 −0.525805
\(222\) 0 0
\(223\) −5.81665 −0.389512 −0.194756 0.980852i \(-0.562391\pi\)
−0.194756 + 0.980852i \(0.562391\pi\)
\(224\) −2.60555 −0.174091
\(225\) 2.39445 0.159630
\(226\) 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 \(0.197838\pi\)
\(230\) −9.00000 −0.593442
\(231\) 19.8167 1.30384
\(232\) 3.90833 0.256594
\(233\) 8.51388 0.557763 0.278881 0.960326i \(-0.410036\pi\)
0.278881 + 0.960326i \(0.410036\pi\)
\(234\) −10.3028 −0.673514
\(235\) 10.6056 0.691830
\(236\) −10.6056 −0.690363
\(237\) −30.1194 −1.95647
\(238\) −15.6333 −1.01336
\(239\) 17.5139 1.13288 0.566439 0.824103i \(-0.308321\pi\)
0.566439 + 0.824103i \(0.308321\pi\)
\(240\) 7.60555 0.490936
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) −5.69722 −0.366231
\(243\) 49.8444 3.19752
\(244\) −7.51388 −0.481027
\(245\) −0.486122 −0.0310572
\(246\) 32.7250 2.08647
\(247\) 2.60555 0.165787
\(248\) 0.302776 0.0192263
\(249\) 9.21110 0.583730
\(250\) −10.8167 −0.684105
\(251\) 21.2111 1.33883 0.669416 0.742887i \(-0.266544\pi\)
0.669416 + 0.742887i \(0.266544\pi\)
\(252\) −20.6056 −1.29803
\(253\) 9.00000 0.565825
\(254\) −19.2111 −1.20541
\(255\) 45.6333 2.85767
\(256\) 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
\(259\) 0 0
\(260\) −3.00000 −0.186052
\(261\) 30.9083 1.91318
\(262\) −10.6056 −0.655213
\(263\) 13.8167 0.851971 0.425986 0.904730i \(-0.359927\pi\)
0.425986 + 0.904730i \(0.359927\pi\)
\(264\) −7.60555 −0.468089
\(265\) −13.8167 −0.848750
\(266\) 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 \(0.723823\pi\)
\(270\) 37.3305 2.27186
\(271\) −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
\(275\) −0.697224 −0.0420442
\(276\) −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
\(280\) −6.00000 −0.358569
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 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
\(285\) −15.2111 −0.901028
\(286\) 3.00000 0.177394
\(287\) −25.8167 −1.52391
\(288\) 7.90833 0.466003
\(289\) 19.0000 1.11765
\(290\) 9.00000 0.528498
\(291\) 54.2389 3.17954
\(292\) −12.3028 −0.719965
\(293\) 11.0278 0.644248 0.322124 0.946697i \(-0.395603\pi\)
0.322124 + 0.946697i \(0.395603\pi\)
\(294\) −0.697224 −0.0406630
\(295\) −24.4222 −1.42192
\(296\) 0 0
\(297\) −37.3305 −2.16614
\(298\) 19.8167 1.14795
\(299\) 5.09167 0.294459
\(300\) 1.00000 0.0577350
\(301\) 1.57779 0.0909426
\(302\) −20.6056 −1.18572
\(303\) −41.0278 −2.35698
\(304\) −2.00000 −0.114708
\(305\) −17.3028 −0.990754
\(306\) 47.4500 2.71253
\(307\) 17.9083 1.02208 0.511041 0.859556i \(-0.329260\pi\)
0.511041 + 0.859556i \(0.329260\pi\)
\(308\) 6.00000 0.341882
\(309\) 1.00000 0.0568880
\(310\) 0.697224 0.0395997
\(311\) −15.9083 −0.902078 −0.451039 0.892504i \(-0.648947\pi\)
−0.451039 + 0.892504i \(0.648947\pi\)
\(312\) −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
\(315\) −47.4500 −2.67350
\(316\) −9.11943 −0.513008
\(317\) 9.21110 0.517347 0.258674 0.965965i \(-0.416715\pi\)
0.258674 + 0.965965i \(0.416715\pi\)
\(318\) −19.8167 −1.11126
\(319\) −9.00000 −0.503903
\(320\) 2.30278 0.128729
\(321\) 2.30278 0.128528
\(322\) 10.1833 0.567496
\(323\) −12.0000 −0.667698
\(324\) 29.8167 1.65648
\(325\) −0.394449 −0.0218801
\(326\) −8.42221 −0.466463
\(327\) −6.60555 −0.365288
\(328\) 9.90833 0.547096
\(329\) −12.0000 −0.661581
\(330\) −17.5139 −0.964107
\(331\) 13.2111 0.726148 0.363074 0.931760i \(-0.381727\pi\)
0.363074 + 0.931760i \(0.381727\pi\)
\(332\) 2.78890 0.153061
\(333\) 0 0
\(334\) 5.51388 0.301706
\(335\) −8.09167 −0.442095
\(336\) −8.60555 −0.469471
\(337\) 6.11943 0.333347 0.166673 0.986012i \(-0.446697\pi\)
0.166673 + 0.986012i \(0.446697\pi\)
\(338\) −11.3028 −0.614790
\(339\) 10.6056 0.576014
\(340\) 13.8167 0.749313
\(341\) −0.697224 −0.0377568
\(342\) −15.8167 −0.855267
\(343\) 18.7889 1.01451
\(344\) −0.605551 −0.0326491
\(345\) −29.7250 −1.60034
\(346\) −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 \(0.227250\pi\)
\(350\) −0.788897 −0.0421683
\(351\) −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
\(355\) 13.8167 0.733312
\(356\) 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 \(0.472995\pi\)
\(360\) 18.2111 0.959809
\(361\) −15.0000 −0.789474
\(362\) 20.0000 1.05118
\(363\) −18.8167 −0.987618
\(364\) 3.39445 0.177917
\(365\) −28.3305 −1.48289
\(366\) −24.8167 −1.29719
\(367\) 3.81665 0.199228 0.0996139 0.995026i \(-0.468239\pi\)
0.0996139 + 0.995026i \(0.468239\pi\)
\(368\) −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 \(-0.652601\pi\)
−0.461256 + 0.887267i \(0.652601\pi\)
\(374\) −13.8167 −0.714442
\(375\) −35.7250 −1.84483
\(376\) 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 \(0.285145\pi\)
\(380\) −4.60555 −0.236260
\(381\) −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
\(385\) 13.8167 0.704162
\(386\) 4.00000 0.203595
\(387\) −4.78890 −0.243433
\(388\) 16.4222 0.833711
\(389\) 37.1194 1.88203 0.941015 0.338365i \(-0.109874\pi\)
0.941015 + 0.338365i \(0.109874\pi\)
\(390\) −9.90833 −0.501728
\(391\) −23.4500 −1.18592
\(392\) −0.211103 −0.0106623
\(393\) −35.0278 −1.76692
\(394\) −6.00000 −0.302276
\(395\) −21.0000 −1.05662
\(396\) −18.2111 −0.915142
\(397\) 6.18335 0.310333 0.155167 0.987888i \(-0.450409\pi\)
0.155167 + 0.987888i \(0.450409\pi\)
\(398\) −26.4222 −1.32443
\(399\) 17.2111 0.861633
\(400\) 0.302776 0.0151388
\(401\) 7.81665 0.390345 0.195173 0.980769i \(-0.437473\pi\)
0.195173 + 0.980769i \(0.437473\pi\)
\(402\) −11.6056 −0.578832
\(403\) −0.394449 −0.0196489
\(404\) −12.4222 −0.618028
\(405\) 68.6611 3.41180
\(406\) −10.1833 −0.505391
\(407\) 0 0
\(408\) 19.8167 0.981071
\(409\) −31.0278 −1.53422 −0.767112 0.641513i \(-0.778307\pi\)
−0.767112 + 0.641513i \(0.778307\pi\)
\(410\) 22.8167 1.12683
\(411\) 3.00000 0.147979
\(412\) 0.302776 0.0149167
\(413\) 27.6333 1.35975
\(414\) −30.9083 −1.51906
\(415\) 6.42221 0.315254
\(416\) −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 \(0.155552\pi\)
\(420\) −19.8167 −0.966954
\(421\) 3.72498 0.181544 0.0907722 0.995872i \(-0.471066\pi\)
0.0907722 + 0.995872i \(0.471066\pi\)
\(422\) 10.3028 0.501531
\(423\) 36.4222 1.77091
\(424\) −6.00000 −0.291386
\(425\) 1.81665 0.0881207
\(426\) 19.8167 0.960120
\(427\) 19.5778 0.947436
\(428\) 0.697224 0.0337016
\(429\) 9.90833 0.478379
\(430\) −1.39445 −0.0672463
\(431\) −9.21110 −0.443683 −0.221842 0.975083i \(-0.571207\pi\)
−0.221842 + 0.975083i \(0.571207\pi\)
\(432\) 16.2111 0.779957
\(433\) 34.9361 1.67892 0.839461 0.543421i \(-0.182871\pi\)
0.839461 + 0.543421i \(0.182871\pi\)
\(434\) −0.788897 −0.0378683
\(435\) 29.7250 1.42520
\(436\) −2.00000 −0.0957826
\(437\) 7.81665 0.373921
\(438\) −40.6333 −1.94153
\(439\) −30.3305 −1.44760 −0.723799 0.690011i \(-0.757606\pi\)
−0.723799 + 0.690011i \(0.757606\pi\)
\(440\) −5.30278 −0.252800
\(441\) −1.66947 −0.0794985
\(442\) −7.81665 −0.371800
\(443\) 32.7250 1.55481 0.777405 0.629000i \(-0.216535\pi\)
0.777405 + 0.629000i \(0.216535\pi\)
\(444\) 0 0
\(445\) 21.2111 1.00550
\(446\) −5.81665 −0.275427
\(447\) 65.4500 3.09568
\(448\) −2.60555 −0.123101
\(449\) 15.2111 0.717856 0.358928 0.933365i \(-0.383142\pi\)
0.358928 + 0.933365i \(0.383142\pi\)
\(450\) 2.39445 0.112875
\(451\) −22.8167 −1.07439
\(452\) 3.21110 0.151038
\(453\) −68.0555 −3.19753
\(454\) −13.8167 −0.648448
\(455\) 7.81665 0.366450
\(456\) −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
\(460\) −9.00000 −0.419627
\(461\) −12.4222 −0.578560 −0.289280 0.957245i \(-0.593416\pi\)
−0.289280 + 0.957245i \(0.593416\pi\)
\(462\) 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
\(465\) 2.30278 0.106789
\(466\) 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
\(470\) 10.6056 0.489198
\(471\) −23.8167 −1.09741
\(472\) −10.6056 −0.488160
\(473\) 1.39445 0.0641168
\(474\) −30.1194 −1.38343
\(475\) −0.605551 −0.0277846
\(476\) −15.6333 −0.716551
\(477\) −47.4500 −2.17258
\(478\) 17.5139 0.801066
\(479\) 13.1194 0.599442 0.299721 0.954027i \(-0.403106\pi\)
0.299721 + 0.954027i \(0.403106\pi\)
\(480\) 7.60555 0.347144
\(481\) 0 0
\(482\) −8.00000 −0.364390
\(483\) 33.6333 1.53037
\(484\) −5.69722 −0.258965
\(485\) 37.8167 1.71717
\(486\) 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
\(490\) −0.486122 −0.0219607
\(491\) 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
\(495\) −41.9361 −1.88489
\(496\) 0.302776 0.0135950
\(497\) −15.6333 −0.701250
\(498\) 9.21110 0.412759
\(499\) 42.2389 1.89087 0.945436 0.325809i \(-0.105637\pi\)
0.945436 + 0.325809i \(0.105637\pi\)
\(500\) −10.8167 −0.483735
\(501\) 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
\(505\) −28.6056 −1.27293
\(506\) 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 \(0.529554\pi\)
\(510\) 45.6333 2.02068
\(511\) 32.0555 1.41805
\(512\) 1.00000 0.0441942
\(513\) −32.4222 −1.43148
\(514\) −3.21110 −0.141636
\(515\) 0.697224 0.0307234
\(516\) −2.00000 −0.0880451
\(517\) −10.6056 −0.466432
\(518\) 0 0
\(519\) −29.0278 −1.27418
\(520\) −3.00000 −0.131559
\(521\) 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
\(525\) −2.60555 −0.113716
\(526\) 13.8167 0.602435
\(527\) 1.81665 0.0791347
\(528\) −7.60555 −0.330989
\(529\) −7.72498 −0.335869
\(530\) −13.8167 −0.600157
\(531\) −83.8722 −3.63974
\(532\) 5.21110 0.225930
\(533\) −12.9083 −0.559122
\(534\) 30.4222 1.31650
\(535\) 1.60555 0.0694140
\(536\) −3.51388 −0.151776
\(537\) 45.6333 1.96922
\(538\) −21.2111 −0.914476
\(539\) 0.486122 0.0209387
\(540\) 37.3305 1.60645
\(541\) −25.9361 −1.11508 −0.557540 0.830150i \(-0.688255\pi\)
−0.557540 + 0.830150i \(0.688255\pi\)
\(542\) −22.4222 −0.963116
\(543\) 66.0555 2.83471
\(544\) 6.00000 0.257248
\(545\) −4.60555 −0.197280
\(546\) 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
\(550\) −0.697224 −0.0297297
\(551\) −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
\(560\) −6.00000 −0.253546
\(561\) −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
\(565\) 7.39445 0.311087
\(566\) −24.6056 −1.03425
\(567\) −77.6888 −3.26262
\(568\) 6.00000 0.251754
\(569\) −18.4222 −0.772299 −0.386150 0.922436i \(-0.626195\pi\)
−0.386150 + 0.922436i \(0.626195\pi\)
\(570\) −15.2111 −0.637123
\(571\) −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
\(575\) −1.18335 −0.0493489
\(576\) 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
\(580\) 9.00000 0.373705
\(581\) −7.26662 −0.301470
\(582\) 54.2389 2.24827
\(583\) 13.8167 0.572227
\(584\) −12.3028 −0.509092
\(585\) −23.7250 −0.980907
\(586\) 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
\(590\) −24.4222 −1.00545
\(591\) −19.8167 −0.815148
\(592\) 0 0
\(593\) 18.4861 0.759134 0.379567 0.925164i \(-0.376073\pi\)
0.379567 + 0.925164i \(0.376073\pi\)
\(594\) −37.3305 −1.53169
\(595\) −36.0000 −1.47586
\(596\) 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 \(0.639622\pi\)
\(600\) 1.00000 0.0408248
\(601\) −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
\(605\) −13.1194 −0.533381
\(606\) −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
\(610\) −17.3028 −0.700569
\(611\) −6.00000 −0.242734
\(612\) 47.4500 1.91805
\(613\) −29.8167 −1.20428 −0.602142 0.798389i \(-0.705686\pi\)
−0.602142 + 0.798389i \(0.705686\pi\)
\(614\) 17.9083 0.722721
\(615\) 75.3583 3.03874
\(616\) 6.00000 0.241747
\(617\) −42.5694 −1.71378 −0.856890 0.515500i \(-0.827606\pi\)
−0.856890 + 0.515500i \(0.827606\pi\)
\(618\) 1.00000 0.0402259
\(619\) −6.30278 −0.253330 −0.126665 0.991946i \(-0.540427\pi\)
−0.126665 + 0.991946i \(0.540427\pi\)
\(620\) 0.697224 0.0280012
\(621\) −63.3583 −2.54248
\(622\) −15.9083 −0.637866
\(623\) −24.0000 −0.961540
\(624\) −4.30278 −0.172249
\(625\) −26.4222 −1.05689
\(626\) 9.02776 0.360822
\(627\) 15.2111 0.607473
\(628\) −7.21110 −0.287754
\(629\) 0 0
\(630\) −47.4500 −1.89045
\(631\) −14.6972 −0.585087 −0.292544 0.956252i \(-0.594502\pi\)
−0.292544 + 0.956252i \(0.594502\pi\)
\(632\) −9.11943 −0.362751
\(633\) 34.0278 1.35248
\(634\) 9.21110 0.365820
\(635\) −44.2389 −1.75557
\(636\) −19.8167 −0.785781
\(637\) 0.275019 0.0108967
\(638\) −9.00000 −0.356313
\(639\) 47.4500 1.87709
\(640\) 2.30278 0.0910252
\(641\) −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
\(645\) −4.60555 −0.181343
\(646\) −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
\(650\) −0.394449 −0.0154716
\(651\) −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
\(655\) −24.4222 −0.954255
\(656\) 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 \(0.606640\pi\)
\(660\) −17.5139 −0.681727
\(661\) 30.5139 1.18685 0.593426 0.804888i \(-0.297775\pi\)
0.593426 + 0.804888i \(0.297775\pi\)
\(662\) 13.2111 0.513464
\(663\) −25.8167 −1.00264
\(664\) 2.78890 0.108230
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) −15.2750 −0.591451
\(668\) 5.51388 0.213338
\(669\) −19.2111 −0.742744
\(670\) −8.09167 −0.312609
\(671\) 17.3028 0.667966
\(672\) −8.60555 −0.331966
\(673\) 20.6972 0.797819 0.398910 0.916990i \(-0.369389\pi\)
0.398910 + 0.916990i \(0.369389\pi\)
\(674\) 6.11943 0.235712
\(675\) 4.90833 0.188922
\(676\) −11.3028 −0.434722
\(677\) 14.2389 0.547244 0.273622 0.961837i \(-0.411778\pi\)
0.273622 + 0.961837i \(0.411778\pi\)
\(678\) 10.6056 0.407304
\(679\) −42.7889 −1.64209
\(680\) 13.8167 0.529844
\(681\) −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
\(685\) 2.09167 0.0799187
\(686\) 18.7889 0.717363
\(687\) 81.2666 3.10051
\(688\) −0.605551 −0.0230864
\(689\) 7.81665 0.297791
\(690\) −29.7250 −1.13161
\(691\) 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
\(695\) −4.39445 −0.166691
\(696\) 12.9083 0.489289
\(697\) 59.4500 2.25183
\(698\) 28.2389 1.06886
\(699\) 28.1194 1.06357
\(700\) −0.788897 −0.0298175
\(701\) −40.1194 −1.51529 −0.757645 0.652667i \(-0.773650\pi\)
−0.757645 + 0.652667i \(0.773650\pi\)
\(702\) −21.1194 −0.797101
\(703\) 0 0
\(704\) −2.30278 −0.0867891
\(705\) 35.0278 1.31922
\(706\) −10.1833 −0.383255
\(707\) 32.3667 1.21727
\(708\) −35.0278 −1.31642
\(709\) 41.3305 1.55220 0.776100 0.630609i \(-0.217195\pi\)
0.776100 + 0.630609i \(0.217195\pi\)
\(710\) 13.8167 0.518530
\(711\) −72.1194 −2.70469
\(712\) 9.21110 0.345201
\(713\) −1.18335 −0.0443167
\(714\) −51.6333 −1.93233
\(715\) 6.90833 0.258357
\(716\) 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 \(0.912904\pi\)
\(720\) 18.2111 0.678688
\(721\) −0.788897 −0.0293801
\(722\) −15.0000 −0.558242
\(723\) −26.4222 −0.982652
\(724\) 20.0000 0.743294
\(725\) 1.18335 0.0439484
\(726\) −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
\(730\) −28.3305 −1.04856
\(731\) −3.63331 −0.134383
\(732\) −24.8167 −0.917250
\(733\) −13.6333 −0.503558 −0.251779 0.967785i \(-0.581016\pi\)
−0.251779 + 0.967785i \(0.581016\pi\)
\(734\) 3.81665 0.140875
\(735\) −1.60555 −0.0592217
\(736\) −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 \(-0.681654\pi\)
−0.540207 + 0.841532i \(0.681654\pi\)
\(744\) 1.00000 0.0366618
\(745\) 45.6333 1.67188
\(746\) −17.8167 −0.652314
\(747\) 22.0555 0.806969
\(748\) −13.8167 −0.505187
\(749\) −1.81665 −0.0663791
\(750\) −35.7250 −1.30449
\(751\) 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
\(755\) −47.4500 −1.72688
\(756\) −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
\(760\) −4.60555 −0.167061
\(761\) 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
\(765\) 109.267 3.95054
\(766\) 36.8444 1.33124
\(767\) 13.8167 0.498890
\(768\) 3.30278 0.119179
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 13.8167 0.497918
\(771\) −10.6056 −0.381950
\(772\) 4.00000 0.143963
\(773\) 22.0555 0.793282 0.396641 0.917974i \(-0.370176\pi\)
0.396641 + 0.917974i \(0.370176\pi\)
\(774\) −4.78890 −0.172133
\(775\) 0.0916731 0.00329299
\(776\) 16.4222 0.589523
\(777\) 0 0
\(778\) 37.1194 1.33080
\(779\) −19.8167 −0.710005
\(780\) −9.90833 −0.354775
\(781\) −13.8167 −0.494399
\(782\) −23.4500 −0.838569
\(783\) 63.3583 2.26424
\(784\) −0.211103 −0.00753938
\(785\) −16.6056 −0.592678
\(786\) −35.0278 −1.24940
\(787\) 10.7889 0.384583 0.192291 0.981338i \(-0.438408\pi\)
0.192291 + 0.981338i \(0.438408\pi\)
\(788\) −6.00000 −0.213741
\(789\) 45.6333 1.62459
\(790\) −21.0000 −0.747146
\(791\) −8.36669 −0.297485
\(792\) −18.2111 −0.647103
\(793\) 9.78890 0.347614
\(794\) 6.18335 0.219439
\(795\) −45.6333 −1.61845
\(796\) −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
\(800\) 0.302776 0.0107047
\(801\) 72.8444 2.57383
\(802\) 7.81665 0.276016
\(803\) 28.3305 0.999763
\(804\) −11.6056 −0.409296
\(805\) 23.4500 0.826503
\(806\) −0.394449 −0.0138939
\(807\) −70.0555 −2.46607
\(808\) −12.4222 −0.437012
\(809\) 35.4500 1.24635 0.623177 0.782081i \(-0.285842\pi\)
0.623177 + 0.782081i \(0.285842\pi\)
\(810\) 68.6611 2.41250
\(811\) −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
\(814\) 0 0
\(815\) −19.3944 −0.679358
\(816\) 19.8167 0.693722
\(817\) 1.21110 0.0423711
\(818\) −31.0278 −1.08486
\(819\) 26.8444 0.938020
\(820\) 22.8167 0.796792
\(821\) −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 \(-0.214410\pi\)
0.781589 + 0.623794i \(0.214410\pi\)
\(824\) 0.302776 0.0105477
\(825\) −2.30278 −0.0801724
\(826\) 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 \(-0.340105\pi\)
0.481464 + 0.876466i \(0.340105\pi\)
\(830\) 6.42221 0.222918
\(831\) −0.394449 −0.0136833
\(832\) −1.30278 −0.0451656
\(833\) −1.26662 −0.0438856
\(834\) −6.30278 −0.218247
\(835\) 12.6972 0.439406
\(836\) 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 \(0.571887\pi\)
\(840\) −19.8167 −0.683740
\(841\) −13.7250 −0.473275
\(842\) 3.72498 0.128371
\(843\) 39.6333 1.36504
\(844\) 10.3028 0.354636
\(845\) −26.0278 −0.895382
\(846\) 36.4222 1.25222
\(847\) 14.8444 0.510060
\(848\) −6.00000 −0.206041
\(849\) −81.2666 −2.78906
\(850\) 1.81665 0.0623107
\(851\) 0 0
\(852\) 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
\(855\) −36.4222 −1.24561
\(856\) 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 \(-0.805925\pi\)
−0.819816 + 0.572626i \(0.805925\pi\)
\(860\) −1.39445 −0.0475503
\(861\) −85.2666 −2.90588
\(862\) −9.21110 −0.313731
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 16.2111 0.551513
\(865\) −20.2389 −0.688142
\(866\) 34.9361 1.18718
\(867\) 62.7527 2.13119
\(868\) −0.788897 −0.0267769
\(869\) 21.0000 0.712376
\(870\) 29.7250 1.00777
\(871\) 4.57779 0.155113
\(872\) −2.00000 −0.0677285
\(873\) 129.872 4.39551
\(874\) 7.81665 0.264402
\(875\) 28.1833 0.952771
\(876\) −40.6333 −1.37287
\(877\) −7.21110 −0.243502 −0.121751 0.992561i \(-0.538851\pi\)
−0.121751 + 0.992561i \(0.538851\pi\)
\(878\) −30.3305 −1.02361
\(879\) 36.4222 1.22849
\(880\) −5.30278 −0.178757
\(881\) −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
\(885\) −80.6611 −2.71139
\(886\) 32.7250 1.09942
\(887\) −0.422205 −0.0141763 −0.00708813 0.999975i \(-0.502256\pi\)
−0.00708813 + 0.999975i \(0.502256\pi\)
\(888\) 0 0
\(889\) 50.0555 1.67881
\(890\) 21.2111 0.710998
\(891\) −68.6611 −2.30023
\(892\) −5.81665 −0.194756
\(893\) −9.21110 −0.308238
\(894\) 65.4500 2.18897
\(895\) 31.8167 1.06351
\(896\) −2.60555 −0.0870454
\(897\) 16.8167 0.561492
\(898\) 15.2111 0.507601
\(899\) 1.18335 0.0394668
\(900\) 2.39445 0.0798150
\(901\) −36.0000 −1.19933
\(902\) −22.8167 −0.759711
\(903\) 5.21110 0.173415
\(904\) 3.21110 0.106800
\(905\) 46.0555 1.53094
\(906\) −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
\(910\) 7.81665 0.259120
\(911\) 17.5778 0.582378 0.291189 0.956665i \(-0.405949\pi\)
0.291189 + 0.956665i \(0.405949\pi\)
\(912\) −6.60555 −0.218732
\(913\) −6.42221 −0.212544
\(914\) 2.60555 0.0861840
\(915\) −57.1472 −1.88923
\(916\) 24.6056 0.812990
\(917\) 27.6333 0.912532
\(918\) 97.2666 3.21028
\(919\) 9.57779 0.315942 0.157971 0.987444i \(-0.449505\pi\)
0.157971 + 0.987444i \(0.449505\pi\)
\(920\) −9.00000 −0.296721
\(921\) 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 \(0.598073\pi\)
\(930\) 2.30278 0.0755110
\(931\) 0.422205 0.0138372
\(932\) 8.51388 0.278881
\(933\) −52.5416 −1.72014
\(934\) 0 0
\(935\) −31.8167 −1.04052
\(936\) −10.3028 −0.336757
\(937\) −18.0917 −0.591029 −0.295515 0.955338i \(-0.595491\pi\)
−0.295515 + 0.955338i \(0.595491\pi\)
\(938\) 9.15559 0.298941
\(939\) 29.8167 0.973030
\(940\) 10.6056 0.345915
\(941\) 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
\(945\) −97.2666 −3.16408
\(946\) 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
\(950\) −0.605551 −0.0196467
\(951\) 30.4222 0.986508
\(952\) −15.6333 −0.506678
\(953\) 49.7527 1.61165 0.805825 0.592154i \(-0.201722\pi\)
0.805825 + 0.592154i \(0.201722\pi\)
\(954\) −47.4500 −1.53625
\(955\) 12.6972 0.410873
\(956\) 17.5139 0.566439
\(957\) −29.7250 −0.960872
\(958\) 13.1194 0.423870
\(959\) −2.36669 −0.0764245
\(960\) 7.60555 0.245468
\(961\) −30.9083 −0.997043
\(962\) 0 0
\(963\) 5.51388 0.177682
\(964\) −8.00000 −0.257663
\(965\) 9.21110 0.296516
\(966\) 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
\(970\) 37.8167 1.21422
\(971\) −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
\(975\) −1.30278 −0.0417222
\(976\) −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
\(980\) −0.486122 −0.0155286
\(981\) −15.8167 −0.504987
\(982\) 17.7250 0.565627
\(983\) −12.0000 −0.382741 −0.191370 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 32.7250 1.04323
\(985\) −13.8167 −0.440235
\(986\) 23.4500 0.746799
\(987\) −39.6333 −1.26154
\(988\) 2.60555 0.0828936
\(989\) 2.36669 0.0752564
\(990\) −41.9361 −1.33282
\(991\) −50.6972 −1.61045 −0.805225 0.592969i \(-0.797956\pi\)
−0.805225 + 0.592969i \(0.797956\pi\)
\(992\) 0.302776 0.00961314
\(993\) 43.6333 1.38466
\(994\) −15.6333 −0.495858
\(995\) −60.8444 −1.92890
\(996\) 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
\(999\) 0 0
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 2738.2.a.l.1.2 2
37.36 even 2 74.2.a.a.1.2 2
111.110 odd 2 666.2.a.j.1.2 2
148.147 odd 2 592.2.a.f.1.1 2
185.73 odd 4 1850.2.b.i.149.4 4
185.147 odd 4 1850.2.b.i.149.1 4
185.184 even 2 1850.2.a.u.1.1 2
259.258 odd 2 3626.2.a.a.1.1 2
296.147 odd 2 2368.2.a.ba.1.2 2
296.221 even 2 2368.2.a.s.1.1 2
407.406 odd 2 8954.2.a.p.1.2 2
444.443 even 2 5328.2.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 2 37.36 even 2
592.2.a.f.1.1 2 148.147 odd 2
666.2.a.j.1.2 2 111.110 odd 2
1850.2.a.u.1.1 2 185.184 even 2
1850.2.b.i.149.1 4 185.147 odd 4
1850.2.b.i.149.4 4 185.73 odd 4
2368.2.a.s.1.1 2 296.221 even 2
2368.2.a.ba.1.2 2 296.147 odd 2
2738.2.a.l.1.2 2 1.1 even 1 trivial
3626.2.a.a.1.1 2 259.258 odd 2
5328.2.a.bf.1.2 2 444.443 even 2
8954.2.a.p.1.2 2 407.406 odd 2