Properties

Label 4830.2.a.bt.1.1
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.5677441763\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 4830.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.46410 q^{11} -1.00000 q^{12} -3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -5.46410 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -3.46410 q^{26} -1.00000 q^{27} -1.00000 q^{28} -7.46410 q^{29} +1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +5.46410 q^{33} -2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +4.92820 q^{37} +4.00000 q^{38} +3.46410 q^{39} -1.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} +5.46410 q^{43} -5.46410 q^{44} -1.00000 q^{45} -1.00000 q^{46} +6.92820 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -3.46410 q^{52} +2.00000 q^{53} -1.00000 q^{54} +5.46410 q^{55} -1.00000 q^{56} -4.00000 q^{57} -7.46410 q^{58} -6.92820 q^{59} +1.00000 q^{60} +10.0000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +3.46410 q^{65} +5.46410 q^{66} +5.46410 q^{67} -2.00000 q^{68} +1.00000 q^{69} +1.00000 q^{70} -2.53590 q^{71} +1.00000 q^{72} +2.00000 q^{73} +4.92820 q^{74} -1.00000 q^{75} +4.00000 q^{76} +5.46410 q^{77} +3.46410 q^{78} -2.92820 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} +2.00000 q^{85} +5.46410 q^{86} +7.46410 q^{87} -5.46410 q^{88} +11.4641 q^{89} -1.00000 q^{90} +3.46410 q^{91} -1.00000 q^{92} -4.00000 q^{93} +6.92820 q^{94} -4.00000 q^{95} -1.00000 q^{96} +0.535898 q^{97} +1.00000 q^{98} -5.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} - 2q^{10} - 4q^{11} - 2q^{12} - 2q^{14} + 2q^{15} + 2q^{16} - 4q^{17} + 2q^{18} + 8q^{19} - 2q^{20} + 2q^{21} - 4q^{22} - 2q^{23} - 2q^{24} + 2q^{25} - 2q^{27} - 2q^{28} - 8q^{29} + 2q^{30} + 8q^{31} + 2q^{32} + 4q^{33} - 4q^{34} + 2q^{35} + 2q^{36} - 4q^{37} + 8q^{38} - 2q^{40} + 4q^{41} + 2q^{42} + 4q^{43} - 4q^{44} - 2q^{45} - 2q^{46} - 2q^{48} + 2q^{49} + 2q^{50} + 4q^{51} + 4q^{53} - 2q^{54} + 4q^{55} - 2q^{56} - 8q^{57} - 8q^{58} + 2q^{60} + 20q^{61} + 8q^{62} - 2q^{63} + 2q^{64} + 4q^{66} + 4q^{67} - 4q^{68} + 2q^{69} + 2q^{70} - 12q^{71} + 2q^{72} + 4q^{73} - 4q^{74} - 2q^{75} + 8q^{76} + 4q^{77} + 8q^{79} - 2q^{80} + 2q^{81} + 4q^{82} + 8q^{83} + 2q^{84} + 4q^{85} + 4q^{86} + 8q^{87} - 4q^{88} + 16q^{89} - 2q^{90} - 2q^{92} - 8q^{93} - 8q^{95} - 2q^{96} + 8q^{97} + 2q^{98} - 4q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −5.46410 −1.64749 −0.823744 0.566961i \(-0.808119\pi\)
−0.823744 + 0.566961i \(0.808119\pi\)
\(12\) −1.00000 −0.288675
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) −5.46410 −1.16495
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −3.46410 −0.679366
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −7.46410 −1.38605 −0.693024 0.720914i \(-0.743722\pi\)
−0.693024 + 0.720914i \(0.743722\pi\)
\(30\) 1.00000 0.182574
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.46410 0.951178
\(34\) −2.00000 −0.342997
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 4.92820 0.810192 0.405096 0.914274i \(-0.367238\pi\)
0.405096 + 0.914274i \(0.367238\pi\)
\(38\) 4.00000 0.648886
\(39\) 3.46410 0.554700
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 1.00000 0.154303
\(43\) 5.46410 0.833268 0.416634 0.909074i \(-0.363210\pi\)
0.416634 + 0.909074i \(0.363210\pi\)
\(44\) −5.46410 −0.823744
\(45\) −1.00000 −0.149071
\(46\) −1.00000 −0.147442
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) −3.46410 −0.480384
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 5.46410 0.736779
\(56\) −1.00000 −0.133631
\(57\) −4.00000 −0.529813
\(58\) −7.46410 −0.980085
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 1.00000 0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 4.00000 0.508001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 3.46410 0.429669
\(66\) 5.46410 0.672584
\(67\) 5.46410 0.667546 0.333773 0.942653i \(-0.391678\pi\)
0.333773 + 0.942653i \(0.391678\pi\)
\(68\) −2.00000 −0.242536
\(69\) 1.00000 0.120386
\(70\) 1.00000 0.119523
\(71\) −2.53590 −0.300956 −0.150478 0.988613i \(-0.548081\pi\)
−0.150478 + 0.988613i \(0.548081\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 4.92820 0.572892
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) 5.46410 0.622692
\(78\) 3.46410 0.392232
\(79\) −2.92820 −0.329449 −0.164724 0.986340i \(-0.552673\pi\)
−0.164724 + 0.986340i \(0.552673\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 1.00000 0.109109
\(85\) 2.00000 0.216930
\(86\) 5.46410 0.589209
\(87\) 7.46410 0.800236
\(88\) −5.46410 −0.582475
\(89\) 11.4641 1.21519 0.607596 0.794246i \(-0.292134\pi\)
0.607596 + 0.794246i \(0.292134\pi\)
\(90\) −1.00000 −0.105409
\(91\) 3.46410 0.363137
\(92\) −1.00000 −0.104257
\(93\) −4.00000 −0.414781
\(94\) 6.92820 0.714590
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 0.535898 0.0544122 0.0272061 0.999630i \(-0.491339\pi\)
0.0272061 + 0.999630i \(0.491339\pi\)
\(98\) 1.00000 0.101015
\(99\) −5.46410 −0.549163
\(100\) 1.00000 0.100000
\(101\) 18.3923 1.83010 0.915051 0.403337i \(-0.132150\pi\)
0.915051 + 0.403337i \(0.132150\pi\)
\(102\) 2.00000 0.198030
\(103\) −13.8564 −1.36531 −0.682656 0.730740i \(-0.739175\pi\)
−0.682656 + 0.730740i \(0.739175\pi\)
\(104\) −3.46410 −0.339683
\(105\) −1.00000 −0.0975900
\(106\) 2.00000 0.194257
\(107\) 10.9282 1.05647 0.528235 0.849098i \(-0.322854\pi\)
0.528235 + 0.849098i \(0.322854\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 5.46410 0.520982
\(111\) −4.92820 −0.467764
\(112\) −1.00000 −0.0944911
\(113\) −19.8564 −1.86793 −0.933967 0.357360i \(-0.883677\pi\)
−0.933967 + 0.357360i \(0.883677\pi\)
\(114\) −4.00000 −0.374634
\(115\) 1.00000 0.0932505
\(116\) −7.46410 −0.693024
\(117\) −3.46410 −0.320256
\(118\) −6.92820 −0.637793
\(119\) 2.00000 0.183340
\(120\) 1.00000 0.0912871
\(121\) 18.8564 1.71422
\(122\) 10.0000 0.905357
\(123\) −2.00000 −0.180334
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −8.39230 −0.744697 −0.372348 0.928093i \(-0.621447\pi\)
−0.372348 + 0.928093i \(0.621447\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.46410 −0.481087
\(130\) 3.46410 0.303822
\(131\) −9.85641 −0.861158 −0.430579 0.902553i \(-0.641691\pi\)
−0.430579 + 0.902553i \(0.641691\pi\)
\(132\) 5.46410 0.475589
\(133\) −4.00000 −0.346844
\(134\) 5.46410 0.472026
\(135\) 1.00000 0.0860663
\(136\) −2.00000 −0.171499
\(137\) −11.8564 −1.01296 −0.506481 0.862251i \(-0.669054\pi\)
−0.506481 + 0.862251i \(0.669054\pi\)
\(138\) 1.00000 0.0851257
\(139\) 17.8564 1.51456 0.757280 0.653090i \(-0.226528\pi\)
0.757280 + 0.653090i \(0.226528\pi\)
\(140\) 1.00000 0.0845154
\(141\) −6.92820 −0.583460
\(142\) −2.53590 −0.212808
\(143\) 18.9282 1.58286
\(144\) 1.00000 0.0833333
\(145\) 7.46410 0.619860
\(146\) 2.00000 0.165521
\(147\) −1.00000 −0.0824786
\(148\) 4.92820 0.405096
\(149\) 4.92820 0.403734 0.201867 0.979413i \(-0.435299\pi\)
0.201867 + 0.979413i \(0.435299\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −2.92820 −0.238294 −0.119147 0.992877i \(-0.538016\pi\)
−0.119147 + 0.992877i \(0.538016\pi\)
\(152\) 4.00000 0.324443
\(153\) −2.00000 −0.161690
\(154\) 5.46410 0.440310
\(155\) −4.00000 −0.321288
\(156\) 3.46410 0.277350
\(157\) −3.85641 −0.307775 −0.153887 0.988088i \(-0.549179\pi\)
−0.153887 + 0.988088i \(0.549179\pi\)
\(158\) −2.92820 −0.232955
\(159\) −2.00000 −0.158610
\(160\) −1.00000 −0.0790569
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) −1.07180 −0.0839496 −0.0419748 0.999119i \(-0.513365\pi\)
−0.0419748 + 0.999119i \(0.513365\pi\)
\(164\) 2.00000 0.156174
\(165\) −5.46410 −0.425380
\(166\) 4.00000 0.310460
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 1.00000 0.0771517
\(169\) −1.00000 −0.0769231
\(170\) 2.00000 0.153393
\(171\) 4.00000 0.305888
\(172\) 5.46410 0.416634
\(173\) 4.92820 0.374684 0.187342 0.982295i \(-0.440013\pi\)
0.187342 + 0.982295i \(0.440013\pi\)
\(174\) 7.46410 0.565852
\(175\) −1.00000 −0.0755929
\(176\) −5.46410 −0.411872
\(177\) 6.92820 0.520756
\(178\) 11.4641 0.859271
\(179\) 22.9282 1.71373 0.856867 0.515537i \(-0.172408\pi\)
0.856867 + 0.515537i \(0.172408\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 3.46410 0.256776
\(183\) −10.0000 −0.739221
\(184\) −1.00000 −0.0737210
\(185\) −4.92820 −0.362329
\(186\) −4.00000 −0.293294
\(187\) 10.9282 0.799149
\(188\) 6.92820 0.505291
\(189\) 1.00000 0.0727393
\(190\) −4.00000 −0.290191
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0.535898 0.0384753
\(195\) −3.46410 −0.248069
\(196\) 1.00000 0.0714286
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −5.46410 −0.388317
\(199\) 13.0718 0.926635 0.463318 0.886192i \(-0.346659\pi\)
0.463318 + 0.886192i \(0.346659\pi\)
\(200\) 1.00000 0.0707107
\(201\) −5.46410 −0.385408
\(202\) 18.3923 1.29408
\(203\) 7.46410 0.523877
\(204\) 2.00000 0.140028
\(205\) −2.00000 −0.139686
\(206\) −13.8564 −0.965422
\(207\) −1.00000 −0.0695048
\(208\) −3.46410 −0.240192
\(209\) −21.8564 −1.51184
\(210\) −1.00000 −0.0690066
\(211\) 20.7846 1.43087 0.715436 0.698679i \(-0.246228\pi\)
0.715436 + 0.698679i \(0.246228\pi\)
\(212\) 2.00000 0.137361
\(213\) 2.53590 0.173757
\(214\) 10.9282 0.747037
\(215\) −5.46410 −0.372649
\(216\) −1.00000 −0.0680414
\(217\) −4.00000 −0.271538
\(218\) 10.0000 0.677285
\(219\) −2.00000 −0.135147
\(220\) 5.46410 0.368390
\(221\) 6.92820 0.466041
\(222\) −4.92820 −0.330759
\(223\) 9.46410 0.633763 0.316882 0.948465i \(-0.397364\pi\)
0.316882 + 0.948465i \(0.397364\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) −19.8564 −1.32083
\(227\) 14.9282 0.990820 0.495410 0.868659i \(-0.335018\pi\)
0.495410 + 0.868659i \(0.335018\pi\)
\(228\) −4.00000 −0.264906
\(229\) −11.8564 −0.783493 −0.391747 0.920073i \(-0.628129\pi\)
−0.391747 + 0.920073i \(0.628129\pi\)
\(230\) 1.00000 0.0659380
\(231\) −5.46410 −0.359511
\(232\) −7.46410 −0.490042
\(233\) −19.8564 −1.30084 −0.650418 0.759576i \(-0.725406\pi\)
−0.650418 + 0.759576i \(0.725406\pi\)
\(234\) −3.46410 −0.226455
\(235\) −6.92820 −0.451946
\(236\) −6.92820 −0.450988
\(237\) 2.92820 0.190207
\(238\) 2.00000 0.129641
\(239\) −8.39230 −0.542853 −0.271427 0.962459i \(-0.587495\pi\)
−0.271427 + 0.962459i \(0.587495\pi\)
\(240\) 1.00000 0.0645497
\(241\) 8.92820 0.575116 0.287558 0.957763i \(-0.407157\pi\)
0.287558 + 0.957763i \(0.407157\pi\)
\(242\) 18.8564 1.21214
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) −1.00000 −0.0638877
\(246\) −2.00000 −0.127515
\(247\) −13.8564 −0.881662
\(248\) 4.00000 0.254000
\(249\) −4.00000 −0.253490
\(250\) −1.00000 −0.0632456
\(251\) 18.2487 1.15185 0.575924 0.817503i \(-0.304642\pi\)
0.575924 + 0.817503i \(0.304642\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 5.46410 0.343525
\(254\) −8.39230 −0.526580
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −5.46410 −0.340180
\(259\) −4.92820 −0.306224
\(260\) 3.46410 0.214834
\(261\) −7.46410 −0.462016
\(262\) −9.85641 −0.608931
\(263\) 5.07180 0.312740 0.156370 0.987699i \(-0.450021\pi\)
0.156370 + 0.987699i \(0.450021\pi\)
\(264\) 5.46410 0.336292
\(265\) −2.00000 −0.122859
\(266\) −4.00000 −0.245256
\(267\) −11.4641 −0.701592
\(268\) 5.46410 0.333773
\(269\) 13.3205 0.812166 0.406083 0.913836i \(-0.366894\pi\)
0.406083 + 0.913836i \(0.366894\pi\)
\(270\) 1.00000 0.0608581
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) −2.00000 −0.121268
\(273\) −3.46410 −0.209657
\(274\) −11.8564 −0.716272
\(275\) −5.46410 −0.329498
\(276\) 1.00000 0.0601929
\(277\) −7.46410 −0.448474 −0.224237 0.974535i \(-0.571989\pi\)
−0.224237 + 0.974535i \(0.571989\pi\)
\(278\) 17.8564 1.07096
\(279\) 4.00000 0.239474
\(280\) 1.00000 0.0597614
\(281\) −28.2487 −1.68518 −0.842588 0.538558i \(-0.818969\pi\)
−0.842588 + 0.538558i \(0.818969\pi\)
\(282\) −6.92820 −0.412568
\(283\) 3.60770 0.214455 0.107228 0.994234i \(-0.465803\pi\)
0.107228 + 0.994234i \(0.465803\pi\)
\(284\) −2.53590 −0.150478
\(285\) 4.00000 0.236940
\(286\) 18.9282 1.11925
\(287\) −2.00000 −0.118056
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 7.46410 0.438307
\(291\) −0.535898 −0.0314149
\(292\) 2.00000 0.117041
\(293\) 2.00000 0.116841 0.0584206 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 6.92820 0.403376
\(296\) 4.92820 0.286446
\(297\) 5.46410 0.317059
\(298\) 4.92820 0.285483
\(299\) 3.46410 0.200334
\(300\) −1.00000 −0.0577350
\(301\) −5.46410 −0.314946
\(302\) −2.92820 −0.168499
\(303\) −18.3923 −1.05661
\(304\) 4.00000 0.229416
\(305\) −10.0000 −0.572598
\(306\) −2.00000 −0.114332
\(307\) −1.85641 −0.105951 −0.0529754 0.998596i \(-0.516870\pi\)
−0.0529754 + 0.998596i \(0.516870\pi\)
\(308\) 5.46410 0.311346
\(309\) 13.8564 0.788263
\(310\) −4.00000 −0.227185
\(311\) −23.3205 −1.32238 −0.661192 0.750216i \(-0.729949\pi\)
−0.661192 + 0.750216i \(0.729949\pi\)
\(312\) 3.46410 0.196116
\(313\) 17.3205 0.979013 0.489506 0.872000i \(-0.337177\pi\)
0.489506 + 0.872000i \(0.337177\pi\)
\(314\) −3.85641 −0.217630
\(315\) 1.00000 0.0563436
\(316\) −2.92820 −0.164724
\(317\) 0.928203 0.0521331 0.0260665 0.999660i \(-0.491702\pi\)
0.0260665 + 0.999660i \(0.491702\pi\)
\(318\) −2.00000 −0.112154
\(319\) 40.7846 2.28350
\(320\) −1.00000 −0.0559017
\(321\) −10.9282 −0.609953
\(322\) 1.00000 0.0557278
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) −3.46410 −0.192154
\(326\) −1.07180 −0.0593613
\(327\) −10.0000 −0.553001
\(328\) 2.00000 0.110432
\(329\) −6.92820 −0.381964
\(330\) −5.46410 −0.300789
\(331\) 9.07180 0.498631 0.249316 0.968422i \(-0.419794\pi\)
0.249316 + 0.968422i \(0.419794\pi\)
\(332\) 4.00000 0.219529
\(333\) 4.92820 0.270064
\(334\) 12.0000 0.656611
\(335\) −5.46410 −0.298536
\(336\) 1.00000 0.0545545
\(337\) 13.3205 0.725614 0.362807 0.931864i \(-0.381818\pi\)
0.362807 + 0.931864i \(0.381818\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 19.8564 1.07845
\(340\) 2.00000 0.108465
\(341\) −21.8564 −1.18359
\(342\) 4.00000 0.216295
\(343\) −1.00000 −0.0539949
\(344\) 5.46410 0.294605
\(345\) −1.00000 −0.0538382
\(346\) 4.92820 0.264942
\(347\) 1.85641 0.0996571 0.0498286 0.998758i \(-0.484133\pi\)
0.0498286 + 0.998758i \(0.484133\pi\)
\(348\) 7.46410 0.400118
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 3.46410 0.184900
\(352\) −5.46410 −0.291238
\(353\) 15.8564 0.843951 0.421976 0.906607i \(-0.361337\pi\)
0.421976 + 0.906607i \(0.361337\pi\)
\(354\) 6.92820 0.368230
\(355\) 2.53590 0.134592
\(356\) 11.4641 0.607596
\(357\) −2.00000 −0.105851
\(358\) 22.9282 1.21179
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) −18.8564 −0.989705
\(364\) 3.46410 0.181568
\(365\) −2.00000 −0.104685
\(366\) −10.0000 −0.522708
\(367\) 35.7128 1.86419 0.932097 0.362209i \(-0.117977\pi\)
0.932097 + 0.362209i \(0.117977\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 2.00000 0.104116
\(370\) −4.92820 −0.256205
\(371\) −2.00000 −0.103835
\(372\) −4.00000 −0.207390
\(373\) 4.92820 0.255173 0.127586 0.991827i \(-0.459277\pi\)
0.127586 + 0.991827i \(0.459277\pi\)
\(374\) 10.9282 0.565084
\(375\) 1.00000 0.0516398
\(376\) 6.92820 0.357295
\(377\) 25.8564 1.33167
\(378\) 1.00000 0.0514344
\(379\) 13.0718 0.671453 0.335727 0.941959i \(-0.391018\pi\)
0.335727 + 0.941959i \(0.391018\pi\)
\(380\) −4.00000 −0.205196
\(381\) 8.39230 0.429951
\(382\) 8.00000 0.409316
\(383\) 5.85641 0.299248 0.149624 0.988743i \(-0.452194\pi\)
0.149624 + 0.988743i \(0.452194\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −5.46410 −0.278476
\(386\) −14.0000 −0.712581
\(387\) 5.46410 0.277756
\(388\) 0.535898 0.0272061
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) −3.46410 −0.175412
\(391\) 2.00000 0.101144
\(392\) 1.00000 0.0505076
\(393\) 9.85641 0.497190
\(394\) −2.00000 −0.100759
\(395\) 2.92820 0.147334
\(396\) −5.46410 −0.274581
\(397\) −9.32051 −0.467783 −0.233891 0.972263i \(-0.575146\pi\)
−0.233891 + 0.972263i \(0.575146\pi\)
\(398\) 13.0718 0.655230
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) −11.4641 −0.572490 −0.286245 0.958156i \(-0.592407\pi\)
−0.286245 + 0.958156i \(0.592407\pi\)
\(402\) −5.46410 −0.272525
\(403\) −13.8564 −0.690237
\(404\) 18.3923 0.915051
\(405\) −1.00000 −0.0496904
\(406\) 7.46410 0.370437
\(407\) −26.9282 −1.33478
\(408\) 2.00000 0.0990148
\(409\) −3.85641 −0.190687 −0.0953435 0.995444i \(-0.530395\pi\)
−0.0953435 + 0.995444i \(0.530395\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 11.8564 0.584833
\(412\) −13.8564 −0.682656
\(413\) 6.92820 0.340915
\(414\) −1.00000 −0.0491473
\(415\) −4.00000 −0.196352
\(416\) −3.46410 −0.169842
\(417\) −17.8564 −0.874432
\(418\) −21.8564 −1.06903
\(419\) 25.4641 1.24400 0.622001 0.783016i \(-0.286320\pi\)
0.622001 + 0.783016i \(0.286320\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 23.8564 1.16269 0.581345 0.813657i \(-0.302527\pi\)
0.581345 + 0.813657i \(0.302527\pi\)
\(422\) 20.7846 1.01178
\(423\) 6.92820 0.336861
\(424\) 2.00000 0.0971286
\(425\) −2.00000 −0.0970143
\(426\) 2.53590 0.122865
\(427\) −10.0000 −0.483934
\(428\) 10.9282 0.528235
\(429\) −18.9282 −0.913862
\(430\) −5.46410 −0.263502
\(431\) 5.85641 0.282093 0.141047 0.990003i \(-0.454953\pi\)
0.141047 + 0.990003i \(0.454953\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 28.2487 1.35755 0.678773 0.734348i \(-0.262512\pi\)
0.678773 + 0.734348i \(0.262512\pi\)
\(434\) −4.00000 −0.192006
\(435\) −7.46410 −0.357876
\(436\) 10.0000 0.478913
\(437\) −4.00000 −0.191346
\(438\) −2.00000 −0.0955637
\(439\) −4.78461 −0.228357 −0.114178 0.993460i \(-0.536424\pi\)
−0.114178 + 0.993460i \(0.536424\pi\)
\(440\) 5.46410 0.260491
\(441\) 1.00000 0.0476190
\(442\) 6.92820 0.329541
\(443\) 33.8564 1.60857 0.804283 0.594246i \(-0.202550\pi\)
0.804283 + 0.594246i \(0.202550\pi\)
\(444\) −4.92820 −0.233882
\(445\) −11.4641 −0.543451
\(446\) 9.46410 0.448138
\(447\) −4.92820 −0.233096
\(448\) −1.00000 −0.0472456
\(449\) −24.9282 −1.17643 −0.588217 0.808703i \(-0.700170\pi\)
−0.588217 + 0.808703i \(0.700170\pi\)
\(450\) 1.00000 0.0471405
\(451\) −10.9282 −0.514589
\(452\) −19.8564 −0.933967
\(453\) 2.92820 0.137579
\(454\) 14.9282 0.700615
\(455\) −3.46410 −0.162400
\(456\) −4.00000 −0.187317
\(457\) −11.4641 −0.536268 −0.268134 0.963382i \(-0.586407\pi\)
−0.268134 + 0.963382i \(0.586407\pi\)
\(458\) −11.8564 −0.554013
\(459\) 2.00000 0.0933520
\(460\) 1.00000 0.0466252
\(461\) 24.2487 1.12938 0.564688 0.825305i \(-0.308997\pi\)
0.564688 + 0.825305i \(0.308997\pi\)
\(462\) −5.46410 −0.254213
\(463\) 0.392305 0.0182320 0.00911598 0.999958i \(-0.497098\pi\)
0.00911598 + 0.999958i \(0.497098\pi\)
\(464\) −7.46410 −0.346512
\(465\) 4.00000 0.185496
\(466\) −19.8564 −0.919830
\(467\) −9.85641 −0.456100 −0.228050 0.973649i \(-0.573235\pi\)
−0.228050 + 0.973649i \(0.573235\pi\)
\(468\) −3.46410 −0.160128
\(469\) −5.46410 −0.252309
\(470\) −6.92820 −0.319574
\(471\) 3.85641 0.177694
\(472\) −6.92820 −0.318896
\(473\) −29.8564 −1.37280
\(474\) 2.92820 0.134497
\(475\) 4.00000 0.183533
\(476\) 2.00000 0.0916698
\(477\) 2.00000 0.0915737
\(478\) −8.39230 −0.383855
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 1.00000 0.0456435
\(481\) −17.0718 −0.778407
\(482\) 8.92820 0.406669
\(483\) −1.00000 −0.0455016
\(484\) 18.8564 0.857109
\(485\) −0.535898 −0.0243339
\(486\) −1.00000 −0.0453609
\(487\) 8.39230 0.380292 0.190146 0.981756i \(-0.439104\pi\)
0.190146 + 0.981756i \(0.439104\pi\)
\(488\) 10.0000 0.452679
\(489\) 1.07180 0.0484683
\(490\) −1.00000 −0.0451754
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 14.9282 0.672332
\(494\) −13.8564 −0.623429
\(495\) 5.46410 0.245593
\(496\) 4.00000 0.179605
\(497\) 2.53590 0.113751
\(498\) −4.00000 −0.179244
\(499\) 1.07180 0.0479802 0.0239901 0.999712i \(-0.492363\pi\)
0.0239901 + 0.999712i \(0.492363\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −12.0000 −0.536120
\(502\) 18.2487 0.814480
\(503\) 2.92820 0.130562 0.0652811 0.997867i \(-0.479206\pi\)
0.0652811 + 0.997867i \(0.479206\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −18.3923 −0.818447
\(506\) 5.46410 0.242909
\(507\) 1.00000 0.0444116
\(508\) −8.39230 −0.372348
\(509\) 24.2487 1.07481 0.537403 0.843326i \(-0.319406\pi\)
0.537403 + 0.843326i \(0.319406\pi\)
\(510\) −2.00000 −0.0885615
\(511\) −2.00000 −0.0884748
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −6.00000 −0.264649
\(515\) 13.8564 0.610586
\(516\) −5.46410 −0.240544
\(517\) −37.8564 −1.66492
\(518\) −4.92820 −0.216533
\(519\) −4.92820 −0.216324
\(520\) 3.46410 0.151911
\(521\) −12.5359 −0.549208 −0.274604 0.961557i \(-0.588547\pi\)
−0.274604 + 0.961557i \(0.588547\pi\)
\(522\) −7.46410 −0.326695
\(523\) 1.46410 0.0640207 0.0320103 0.999488i \(-0.489809\pi\)
0.0320103 + 0.999488i \(0.489809\pi\)
\(524\) −9.85641 −0.430579
\(525\) 1.00000 0.0436436
\(526\) 5.07180 0.221141
\(527\) −8.00000 −0.348485
\(528\) 5.46410 0.237795
\(529\) 1.00000 0.0434783
\(530\) −2.00000 −0.0868744
\(531\) −6.92820 −0.300658
\(532\) −4.00000 −0.173422
\(533\) −6.92820 −0.300094
\(534\) −11.4641 −0.496100
\(535\) −10.9282 −0.472467
\(536\) 5.46410 0.236013
\(537\) −22.9282 −0.989425
\(538\) 13.3205 0.574288
\(539\) −5.46410 −0.235356
\(540\) 1.00000 0.0430331
\(541\) 16.9282 0.727800 0.363900 0.931438i \(-0.381445\pi\)
0.363900 + 0.931438i \(0.381445\pi\)
\(542\) −4.00000 −0.171815
\(543\) −2.00000 −0.0858282
\(544\) −2.00000 −0.0857493
\(545\) −10.0000 −0.428353
\(546\) −3.46410 −0.148250
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) −11.8564 −0.506481
\(549\) 10.0000 0.426790
\(550\) −5.46410 −0.232990
\(551\) −29.8564 −1.27193
\(552\) 1.00000 0.0425628
\(553\) 2.92820 0.124520
\(554\) −7.46410 −0.317119
\(555\) 4.92820 0.209191
\(556\) 17.8564 0.757280
\(557\) −33.7128 −1.42846 −0.714229 0.699912i \(-0.753222\pi\)
−0.714229 + 0.699912i \(0.753222\pi\)
\(558\) 4.00000 0.169334
\(559\) −18.9282 −0.800578
\(560\) 1.00000 0.0422577
\(561\) −10.9282 −0.461389
\(562\) −28.2487 −1.19160
\(563\) −41.8564 −1.76404 −0.882019 0.471215i \(-0.843816\pi\)
−0.882019 + 0.471215i \(0.843816\pi\)
\(564\) −6.92820 −0.291730
\(565\) 19.8564 0.835365
\(566\) 3.60770 0.151643
\(567\) −1.00000 −0.0419961
\(568\) −2.53590 −0.106404
\(569\) 34.3923 1.44180 0.720900 0.693039i \(-0.243729\pi\)
0.720900 + 0.693039i \(0.243729\pi\)
\(570\) 4.00000 0.167542
\(571\) −5.07180 −0.212248 −0.106124 0.994353i \(-0.533844\pi\)
−0.106124 + 0.994353i \(0.533844\pi\)
\(572\) 18.9282 0.791428
\(573\) −8.00000 −0.334205
\(574\) −2.00000 −0.0834784
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −11.8564 −0.493589 −0.246794 0.969068i \(-0.579377\pi\)
−0.246794 + 0.969068i \(0.579377\pi\)
\(578\) −13.0000 −0.540729
\(579\) 14.0000 0.581820
\(580\) 7.46410 0.309930
\(581\) −4.00000 −0.165948
\(582\) −0.535898 −0.0222137
\(583\) −10.9282 −0.452600
\(584\) 2.00000 0.0827606
\(585\) 3.46410 0.143223
\(586\) 2.00000 0.0826192
\(587\) 6.92820 0.285958 0.142979 0.989726i \(-0.454332\pi\)
0.142979 + 0.989726i \(0.454332\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 16.0000 0.659269
\(590\) 6.92820 0.285230
\(591\) 2.00000 0.0822690
\(592\) 4.92820 0.202548
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 5.46410 0.224195
\(595\) −2.00000 −0.0819920
\(596\) 4.92820 0.201867
\(597\) −13.0718 −0.534993
\(598\) 3.46410 0.141658
\(599\) 18.5359 0.757356 0.378678 0.925528i \(-0.376379\pi\)
0.378678 + 0.925528i \(0.376379\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −27.8564 −1.13629 −0.568143 0.822930i \(-0.692338\pi\)
−0.568143 + 0.822930i \(0.692338\pi\)
\(602\) −5.46410 −0.222700
\(603\) 5.46410 0.222515
\(604\) −2.92820 −0.119147
\(605\) −18.8564 −0.766622
\(606\) −18.3923 −0.747136
\(607\) −20.3923 −0.827698 −0.413849 0.910346i \(-0.635816\pi\)
−0.413849 + 0.910346i \(0.635816\pi\)
\(608\) 4.00000 0.162221
\(609\) −7.46410 −0.302461
\(610\) −10.0000 −0.404888
\(611\) −24.0000 −0.970936
\(612\) −2.00000 −0.0808452
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −1.85641 −0.0749185
\(615\) 2.00000 0.0806478
\(616\) 5.46410 0.220155
\(617\) −24.9282 −1.00357 −0.501786 0.864992i \(-0.667323\pi\)
−0.501786 + 0.864992i \(0.667323\pi\)
\(618\) 13.8564 0.557386
\(619\) 33.8564 1.36080 0.680402 0.732839i \(-0.261805\pi\)
0.680402 + 0.732839i \(0.261805\pi\)
\(620\) −4.00000 −0.160644
\(621\) 1.00000 0.0401286
\(622\) −23.3205 −0.935067
\(623\) −11.4641 −0.459300
\(624\) 3.46410 0.138675
\(625\) 1.00000 0.0400000
\(626\) 17.3205 0.692267
\(627\) 21.8564 0.872861
\(628\) −3.85641 −0.153887
\(629\) −9.85641 −0.393001
\(630\) 1.00000 0.0398410
\(631\) 29.0718 1.15733 0.578665 0.815565i \(-0.303574\pi\)
0.578665 + 0.815565i \(0.303574\pi\)
\(632\) −2.92820 −0.116478
\(633\) −20.7846 −0.826114
\(634\) 0.928203 0.0368637
\(635\) 8.39230 0.333038
\(636\) −2.00000 −0.0793052
\(637\) −3.46410 −0.137253
\(638\) 40.7846 1.61468
\(639\) −2.53590 −0.100319
\(640\) −1.00000 −0.0395285
\(641\) −16.5359 −0.653129 −0.326564 0.945175i \(-0.605891\pi\)
−0.326564 + 0.945175i \(0.605891\pi\)
\(642\) −10.9282 −0.431302
\(643\) −34.2487 −1.35064 −0.675319 0.737526i \(-0.735994\pi\)
−0.675319 + 0.737526i \(0.735994\pi\)
\(644\) 1.00000 0.0394055
\(645\) 5.46410 0.215149
\(646\) −8.00000 −0.314756
\(647\) −38.9282 −1.53043 −0.765213 0.643777i \(-0.777366\pi\)
−0.765213 + 0.643777i \(0.777366\pi\)
\(648\) 1.00000 0.0392837
\(649\) 37.8564 1.48599
\(650\) −3.46410 −0.135873
\(651\) 4.00000 0.156772
\(652\) −1.07180 −0.0419748
\(653\) −4.92820 −0.192855 −0.0964277 0.995340i \(-0.530742\pi\)
−0.0964277 + 0.995340i \(0.530742\pi\)
\(654\) −10.0000 −0.391031
\(655\) 9.85641 0.385122
\(656\) 2.00000 0.0780869
\(657\) 2.00000 0.0780274
\(658\) −6.92820 −0.270089
\(659\) 30.2487 1.17832 0.589161 0.808015i \(-0.299458\pi\)
0.589161 + 0.808015i \(0.299458\pi\)
\(660\) −5.46410 −0.212690
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 9.07180 0.352585
\(663\) −6.92820 −0.269069
\(664\) 4.00000 0.155230
\(665\) 4.00000 0.155113
\(666\) 4.92820 0.190964
\(667\) 7.46410 0.289011
\(668\) 12.0000 0.464294
\(669\) −9.46410 −0.365903
\(670\) −5.46410 −0.211097
\(671\) −54.6410 −2.10939
\(672\) 1.00000 0.0385758
\(673\) −40.9282 −1.57767 −0.788833 0.614607i \(-0.789314\pi\)
−0.788833 + 0.614607i \(0.789314\pi\)
\(674\) 13.3205 0.513087
\(675\) −1.00000 −0.0384900
\(676\) −1.00000 −0.0384615
\(677\) −17.7128 −0.680759 −0.340379 0.940288i \(-0.610555\pi\)
−0.340379 + 0.940288i \(0.610555\pi\)
\(678\) 19.8564 0.762581
\(679\) −0.535898 −0.0205659
\(680\) 2.00000 0.0766965
\(681\) −14.9282 −0.572050
\(682\) −21.8564 −0.836925
\(683\) −39.7128 −1.51957 −0.759784 0.650175i \(-0.774695\pi\)
−0.759784 + 0.650175i \(0.774695\pi\)
\(684\) 4.00000 0.152944
\(685\) 11.8564 0.453010
\(686\) −1.00000 −0.0381802
\(687\) 11.8564 0.452350
\(688\) 5.46410 0.208317
\(689\) −6.92820 −0.263944
\(690\) −1.00000 −0.0380693
\(691\) −26.6410 −1.01347 −0.506736 0.862101i \(-0.669148\pi\)
−0.506736 + 0.862101i \(0.669148\pi\)
\(692\) 4.92820 0.187342
\(693\) 5.46410 0.207564
\(694\) 1.85641 0.0704682
\(695\) −17.8564 −0.677332
\(696\) 7.46410 0.282926
\(697\) −4.00000 −0.151511
\(698\) −6.00000 −0.227103
\(699\) 19.8564 0.751038
\(700\) −1.00000 −0.0377964
\(701\) −0.143594 −0.00542345 −0.00271173 0.999996i \(-0.500863\pi\)
−0.00271173 + 0.999996i \(0.500863\pi\)
\(702\) 3.46410 0.130744
\(703\) 19.7128 0.743483
\(704\) −5.46410 −0.205936
\(705\) 6.92820 0.260931
\(706\) 15.8564 0.596764
\(707\) −18.3923 −0.691714
\(708\) 6.92820 0.260378
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 2.53590 0.0951706
\(711\) −2.92820 −0.109816
\(712\) 11.4641 0.429635
\(713\) −4.00000 −0.149801
\(714\) −2.00000 −0.0748481
\(715\) −18.9282 −0.707875
\(716\) 22.9282 0.856867
\(717\) 8.39230 0.313416
\(718\) 0 0
\(719\) 31.3205 1.16806 0.584029 0.811733i \(-0.301475\pi\)
0.584029 + 0.811733i \(0.301475\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 13.8564 0.516040
\(722\) −3.00000 −0.111648
\(723\) −8.92820 −0.332043
\(724\) 2.00000 0.0743294
\(725\) −7.46410 −0.277210
\(726\) −18.8564 −0.699827
\(727\) 38.6410 1.43312 0.716558 0.697528i \(-0.245717\pi\)
0.716558 + 0.697528i \(0.245717\pi\)
\(728\) 3.46410 0.128388
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) −10.9282 −0.404194
\(732\) −10.0000 −0.369611
\(733\) 13.7128 0.506494 0.253247 0.967402i \(-0.418501\pi\)
0.253247 + 0.967402i \(0.418501\pi\)
\(734\) 35.7128 1.31818
\(735\) 1.00000 0.0368856
\(736\) −1.00000 −0.0368605
\(737\) −29.8564 −1.09977
\(738\) 2.00000 0.0736210
\(739\) 52.0000 1.91285 0.956425 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) −4.92820 −0.181164
\(741\) 13.8564 0.509028
\(742\) −2.00000 −0.0734223
\(743\) 8.78461 0.322276 0.161138 0.986932i \(-0.448484\pi\)
0.161138 + 0.986932i \(0.448484\pi\)
\(744\) −4.00000 −0.146647
\(745\) −4.92820 −0.180555
\(746\) 4.92820 0.180434
\(747\) 4.00000 0.146352
\(748\) 10.9282 0.399575
\(749\) −10.9282 −0.399308
\(750\) 1.00000 0.0365148
\(751\) −5.85641 −0.213703 −0.106852 0.994275i \(-0.534077\pi\)
−0.106852 + 0.994275i \(0.534077\pi\)
\(752\) 6.92820 0.252646
\(753\) −18.2487 −0.665020
\(754\) 25.8564 0.941635
\(755\) 2.92820 0.106568
\(756\) 1.00000 0.0363696
\(757\) −14.0000 −0.508839 −0.254419 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) 13.0718 0.474789
\(759\) −5.46410 −0.198334
\(760\) −4.00000 −0.145095
\(761\) 16.6410 0.603236 0.301618 0.953429i \(-0.402473\pi\)
0.301618 + 0.953429i \(0.402473\pi\)
\(762\) 8.39230 0.304021
\(763\) −10.0000 −0.362024
\(764\) 8.00000 0.289430
\(765\) 2.00000 0.0723102
\(766\) 5.85641 0.211601
\(767\) 24.0000 0.866590
\(768\) −1.00000 −0.0360844
\(769\) −29.7128 −1.07147 −0.535736 0.844386i \(-0.679966\pi\)
−0.535736 + 0.844386i \(0.679966\pi\)
\(770\) −5.46410 −0.196913
\(771\) 6.00000 0.216085
\(772\) −14.0000 −0.503871
\(773\) 34.0000 1.22290 0.611448 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) 5.46410 0.196403
\(775\) 4.00000 0.143684
\(776\) 0.535898 0.0192376
\(777\) 4.92820 0.176798
\(778\) 18.0000 0.645331
\(779\) 8.00000 0.286630
\(780\) −3.46410 −0.124035
\(781\) 13.8564 0.495821
\(782\) 2.00000 0.0715199
\(783\) 7.46410 0.266745
\(784\) 1.00000 0.0357143
\(785\) 3.85641 0.137641
\(786\) 9.85641 0.351566
\(787\) −0.679492 −0.0242213 −0.0121106 0.999927i \(-0.503855\pi\)
−0.0121106 + 0.999927i \(0.503855\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −5.07180 −0.180561
\(790\) 2.92820 0.104181
\(791\) 19.8564 0.706013
\(792\) −5.46410 −0.194158
\(793\) −34.6410 −1.23014
\(794\) −9.32051 −0.330772
\(795\) 2.00000 0.0709327
\(796\) 13.0718 0.463318
\(797\) −47.5692 −1.68499 −0.842494 0.538706i \(-0.818914\pi\)
−0.842494 + 0.538706i \(0.818914\pi\)
\(798\) 4.00000 0.141598
\(799\) −13.8564 −0.490204
\(800\) 1.00000 0.0353553
\(801\) 11.4641 0.405064
\(802\) −11.4641 −0.404812
\(803\) −10.9282 −0.385648
\(804\) −5.46410 −0.192704
\(805\) −1.00000 −0.0352454
\(806\) −13.8564 −0.488071
\(807\) −13.3205 −0.468904
\(808\) 18.3923 0.647039
\(809\) −11.8564 −0.416849 −0.208425 0.978038i \(-0.566834\pi\)
−0.208425 + 0.978038i \(0.566834\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −42.6410 −1.49733 −0.748664 0.662949i \(-0.769304\pi\)
−0.748664 + 0.662949i \(0.769304\pi\)
\(812\) 7.46410 0.261939
\(813\) 4.00000 0.140286
\(814\) −26.9282 −0.943833
\(815\) 1.07180 0.0375434
\(816\) 2.00000 0.0700140
\(817\) 21.8564 0.764659
\(818\) −3.85641 −0.134836
\(819\) 3.46410 0.121046
\(820\) −2.00000 −0.0698430
\(821\) −38.1051 −1.32988 −0.664939 0.746898i \(-0.731542\pi\)
−0.664939 + 0.746898i \(0.731542\pi\)
\(822\) 11.8564 0.413540
\(823\) −36.1051 −1.25855 −0.629273 0.777185i \(-0.716647\pi\)
−0.629273 + 0.777185i \(0.716647\pi\)
\(824\) −13.8564 −0.482711
\(825\) 5.46410 0.190236
\(826\) 6.92820 0.241063
\(827\) 10.9282 0.380011 0.190005 0.981783i \(-0.439149\pi\)
0.190005 + 0.981783i \(0.439149\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) −4.00000 −0.138842
\(831\) 7.46410 0.258927
\(832\) −3.46410 −0.120096
\(833\) −2.00000 −0.0692959
\(834\) −17.8564 −0.618317
\(835\) −12.0000 −0.415277
\(836\) −21.8564 −0.755920
\(837\) −4.00000 −0.138260
\(838\) 25.4641 0.879643
\(839\) −29.0718 −1.00367 −0.501835 0.864963i \(-0.667342\pi\)
−0.501835 + 0.864963i \(0.667342\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 26.7128 0.921131
\(842\) 23.8564 0.822146
\(843\) 28.2487 0.972937
\(844\) 20.7846 0.715436
\(845\) 1.00000 0.0344010
\(846\) 6.92820 0.238197
\(847\) −18.8564 −0.647914
\(848\) 2.00000 0.0686803
\(849\) −3.60770 −0.123816
\(850\) −2.00000 −0.0685994
\(851\) −4.92820 −0.168937
\(852\) 2.53590 0.0868784
\(853\) −5.60770 −0.192004 −0.0960019 0.995381i \(-0.530605\pi\)
−0.0960019 + 0.995381i \(0.530605\pi\)
\(854\) −10.0000 −0.342193
\(855\) −4.00000 −0.136797
\(856\) 10.9282 0.373518
\(857\) 51.5692 1.76157 0.880785 0.473515i \(-0.157015\pi\)
0.880785 + 0.473515i \(0.157015\pi\)
\(858\) −18.9282 −0.646198
\(859\) 6.92820 0.236387 0.118194 0.992991i \(-0.462290\pi\)
0.118194 + 0.992991i \(0.462290\pi\)
\(860\) −5.46410 −0.186324
\(861\) 2.00000 0.0681598
\(862\) 5.85641 0.199470
\(863\) 40.7846 1.38832 0.694162 0.719819i \(-0.255775\pi\)
0.694162 + 0.719819i \(0.255775\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −4.92820 −0.167564
\(866\) 28.2487 0.959930
\(867\) 13.0000 0.441503
\(868\) −4.00000 −0.135769
\(869\) 16.0000 0.542763
\(870\) −7.46410 −0.253057
\(871\) −18.9282 −0.641358
\(872\) 10.0000 0.338643
\(873\) 0.535898 0.0181374
\(874\) −4.00000 −0.135302
\(875\) 1.00000 0.0338062
\(876\) −2.00000 −0.0675737
\(877\) −23.4641 −0.792326 −0.396163 0.918180i \(-0.629659\pi\)
−0.396163 + 0.918180i \(0.629659\pi\)
\(878\) −4.78461 −0.161473
\(879\) −2.00000 −0.0674583
\(880\) 5.46410 0.184195
\(881\) −10.3923 −0.350126 −0.175063 0.984557i \(-0.556013\pi\)
−0.175063 + 0.984557i \(0.556013\pi\)
\(882\) 1.00000 0.0336718
\(883\) −20.7846 −0.699458 −0.349729 0.936851i \(-0.613726\pi\)
−0.349729 + 0.936851i \(0.613726\pi\)
\(884\) 6.92820 0.233021
\(885\) −6.92820 −0.232889
\(886\) 33.8564 1.13743
\(887\) −25.0718 −0.841829 −0.420914 0.907100i \(-0.638291\pi\)
−0.420914 + 0.907100i \(0.638291\pi\)
\(888\) −4.92820 −0.165380
\(889\) 8.39230 0.281469
\(890\) −11.4641 −0.384278
\(891\) −5.46410 −0.183054
\(892\) 9.46410 0.316882
\(893\) 27.7128 0.927374
\(894\) −4.92820 −0.164824
\(895\) −22.9282 −0.766405
\(896\) −1.00000 −0.0334077
\(897\) −3.46410 −0.115663
\(898\) −24.9282 −0.831865
\(899\) −29.8564 −0.995767
\(900\) 1.00000 0.0333333
\(901\) −4.00000 −0.133259
\(902\) −10.9282 −0.363869
\(903\) 5.46410 0.181834
\(904\) −19.8564 −0.660414
\(905\) −2.00000 −0.0664822
\(906\) 2.92820 0.0972830
\(907\) 23.6077 0.783881 0.391940 0.919991i \(-0.371804\pi\)
0.391940 + 0.919991i \(0.371804\pi\)
\(908\) 14.9282 0.495410
\(909\) 18.3923 0.610034
\(910\) −3.46410 −0.114834
\(911\) −7.21539 −0.239057 −0.119528 0.992831i \(-0.538138\pi\)
−0.119528 + 0.992831i \(0.538138\pi\)
\(912\) −4.00000 −0.132453
\(913\) −21.8564 −0.723341
\(914\) −11.4641 −0.379199
\(915\) 10.0000 0.330590
\(916\) −11.8564 −0.391747
\(917\) 9.85641 0.325487
\(918\) 2.00000 0.0660098
\(919\) 41.5692 1.37124 0.685621 0.727959i \(-0.259531\pi\)
0.685621 + 0.727959i \(0.259531\pi\)
\(920\) 1.00000 0.0329690
\(921\) 1.85641 0.0611707
\(922\) 24.2487 0.798589
\(923\) 8.78461 0.289149
\(924\) −5.46410 −0.179756
\(925\) 4.92820 0.162038
\(926\) 0.392305 0.0128919
\(927\) −13.8564 −0.455104
\(928\) −7.46410 −0.245021
\(929\) 15.8564 0.520232 0.260116 0.965577i \(-0.416239\pi\)
0.260116 + 0.965577i \(0.416239\pi\)
\(930\) 4.00000 0.131165
\(931\) 4.00000 0.131095
\(932\) −19.8564 −0.650418
\(933\) 23.3205 0.763479
\(934\) −9.85641 −0.322511
\(935\) −10.9282 −0.357390
\(936\) −3.46410 −0.113228
\(937\) 52.2487 1.70689 0.853445 0.521182i \(-0.174509\pi\)
0.853445 + 0.521182i \(0.174509\pi\)
\(938\) −5.46410 −0.178409
\(939\) −17.3205 −0.565233
\(940\) −6.92820 −0.225973
\(941\) −41.7128 −1.35980 −0.679899 0.733305i \(-0.737977\pi\)
−0.679899 + 0.733305i \(0.737977\pi\)
\(942\) 3.85641 0.125649
\(943\) −2.00000 −0.0651290
\(944\) −6.92820 −0.225494
\(945\) −1.00000 −0.0325300
\(946\) −29.8564 −0.970716
\(947\) −3.21539 −0.104486 −0.0522431 0.998634i \(-0.516637\pi\)
−0.0522431 + 0.998634i \(0.516637\pi\)
\(948\) 2.92820 0.0951036
\(949\) −6.92820 −0.224899
\(950\) 4.00000 0.129777
\(951\) −0.928203 −0.0300991
\(952\) 2.00000 0.0648204
\(953\) −19.8564 −0.643212 −0.321606 0.946874i \(-0.604223\pi\)
−0.321606 + 0.946874i \(0.604223\pi\)
\(954\) 2.00000 0.0647524
\(955\) −8.00000 −0.258874
\(956\) −8.39230 −0.271427
\(957\) −40.7846 −1.31838
\(958\) 8.00000 0.258468
\(959\) 11.8564 0.382863
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) −17.0718 −0.550417
\(963\) 10.9282 0.352156
\(964\) 8.92820 0.287558
\(965\) 14.0000 0.450676
\(966\) −1.00000 −0.0321745
\(967\) 44.1051 1.41832 0.709162 0.705045i \(-0.249073\pi\)
0.709162 + 0.705045i \(0.249073\pi\)
\(968\) 18.8564 0.606068
\(969\) 8.00000 0.256997
\(970\) −0.535898 −0.0172067
\(971\) 4.39230 0.140956 0.0704779 0.997513i \(-0.477548\pi\)
0.0704779 + 0.997513i \(0.477548\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −17.8564 −0.572450
\(974\) 8.39230 0.268907
\(975\) 3.46410 0.110940
\(976\) 10.0000 0.320092
\(977\) −13.2154 −0.422798 −0.211399 0.977400i \(-0.567802\pi\)
−0.211399 + 0.977400i \(0.567802\pi\)
\(978\) 1.07180 0.0342723
\(979\) −62.6410 −2.00202
\(980\) −1.00000 −0.0319438
\(981\) 10.0000 0.319275
\(982\) −20.0000 −0.638226
\(983\) 48.0000 1.53096 0.765481 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 2.00000 0.0637253
\(986\) 14.9282 0.475411
\(987\) 6.92820 0.220527
\(988\) −13.8564 −0.440831
\(989\) −5.46410 −0.173748
\(990\) 5.46410 0.173661
\(991\) −16.7846 −0.533181 −0.266590 0.963810i \(-0.585897\pi\)
−0.266590 + 0.963810i \(0.585897\pi\)
\(992\) 4.00000 0.127000
\(993\) −9.07180 −0.287885
\(994\) 2.53590 0.0804338
\(995\) −13.0718 −0.414404
\(996\) −4.00000 −0.126745
\(997\) −25.3205 −0.801909 −0.400954 0.916098i \(-0.631321\pi\)
−0.400954 + 0.916098i \(0.631321\pi\)
\(998\) 1.07180 0.0339271
\(999\) −4.92820 −0.155921
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 4830.2.a.bt.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.bt.1.1 2 1.1 even 1 trivial