Properties

Label 4598.2.a.w.1.1
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.732051 q^{5} +1.00000 q^{6} +0.267949 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.732051 q^{5} +1.00000 q^{6} +0.267949 q^{7} -1.00000 q^{8} -2.00000 q^{9} +0.732051 q^{10} -1.00000 q^{12} -3.46410 q^{13} -0.267949 q^{14} +0.732051 q^{15} +1.00000 q^{16} -7.46410 q^{17} +2.00000 q^{18} +1.00000 q^{19} -0.732051 q^{20} -0.267949 q^{21} -1.00000 q^{23} +1.00000 q^{24} -4.46410 q^{25} +3.46410 q^{26} +5.00000 q^{27} +0.267949 q^{28} +1.73205 q^{29} -0.732051 q^{30} -1.46410 q^{31} -1.00000 q^{32} +7.46410 q^{34} -0.196152 q^{35} -2.00000 q^{36} -6.46410 q^{37} -1.00000 q^{38} +3.46410 q^{39} +0.732051 q^{40} -10.7321 q^{41} +0.267949 q^{42} +4.92820 q^{43} +1.46410 q^{45} +1.00000 q^{46} +12.4641 q^{47} -1.00000 q^{48} -6.92820 q^{49} +4.46410 q^{50} +7.46410 q^{51} -3.46410 q^{52} -9.92820 q^{53} -5.00000 q^{54} -0.267949 q^{56} -1.00000 q^{57} -1.73205 q^{58} -9.39230 q^{59} +0.732051 q^{60} +1.26795 q^{61} +1.46410 q^{62} -0.535898 q^{63} +1.00000 q^{64} +2.53590 q^{65} +14.0000 q^{67} -7.46410 q^{68} +1.00000 q^{69} +0.196152 q^{70} +14.1962 q^{71} +2.00000 q^{72} +2.00000 q^{73} +6.46410 q^{74} +4.46410 q^{75} +1.00000 q^{76} -3.46410 q^{78} -2.19615 q^{79} -0.732051 q^{80} +1.00000 q^{81} +10.7321 q^{82} -3.80385 q^{83} -0.267949 q^{84} +5.46410 q^{85} -4.92820 q^{86} -1.73205 q^{87} +12.1962 q^{89} -1.46410 q^{90} -0.928203 q^{91} -1.00000 q^{92} +1.46410 q^{93} -12.4641 q^{94} -0.732051 q^{95} +1.00000 q^{96} +13.4641 q^{97} +6.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 4 q^{7} - 2 q^{8} - 4 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 4 q^{7} - 2 q^{8} - 4 q^{9} - 2 q^{10} - 2 q^{12} - 4 q^{14} - 2 q^{15} + 2 q^{16} - 8 q^{17} + 4 q^{18} + 2 q^{19} + 2 q^{20} - 4 q^{21} - 2 q^{23} + 2 q^{24} - 2 q^{25} + 10 q^{27} + 4 q^{28} + 2 q^{30} + 4 q^{31} - 2 q^{32} + 8 q^{34} + 10 q^{35} - 4 q^{36} - 6 q^{37} - 2 q^{38} - 2 q^{40} - 18 q^{41} + 4 q^{42} - 4 q^{43} - 4 q^{45} + 2 q^{46} + 18 q^{47} - 2 q^{48} + 2 q^{50} + 8 q^{51} - 6 q^{53} - 10 q^{54} - 4 q^{56} - 2 q^{57} + 2 q^{59} - 2 q^{60} + 6 q^{61} - 4 q^{62} - 8 q^{63} + 2 q^{64} + 12 q^{65} + 28 q^{67} - 8 q^{68} + 2 q^{69} - 10 q^{70} + 18 q^{71} + 4 q^{72} + 4 q^{73} + 6 q^{74} + 2 q^{75} + 2 q^{76} + 6 q^{79} + 2 q^{80} + 2 q^{81} + 18 q^{82} - 18 q^{83} - 4 q^{84} + 4 q^{85} + 4 q^{86} + 14 q^{89} + 4 q^{90} + 12 q^{91} - 2 q^{92} - 4 q^{93} - 18 q^{94} + 2 q^{95} + 2 q^{96} + 20 q^{97} + O(q^{100}) \)

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 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.732051 −0.327383 −0.163692 0.986512i \(-0.552340\pi\)
−0.163692 + 0.986512i \(0.552340\pi\)
\(6\) 1.00000 0.408248
\(7\) 0.267949 0.101275 0.0506376 0.998717i \(-0.483875\pi\)
0.0506376 + 0.998717i \(0.483875\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 0.732051 0.231495
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) −0.267949 −0.0716124
\(15\) 0.732051 0.189015
\(16\) 1.00000 0.250000
\(17\) −7.46410 −1.81031 −0.905155 0.425081i \(-0.860246\pi\)
−0.905155 + 0.425081i \(0.860246\pi\)
\(18\) 2.00000 0.471405
\(19\) 1.00000 0.229416
\(20\) −0.732051 −0.163692
\(21\) −0.267949 −0.0584713
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.46410 −0.892820
\(26\) 3.46410 0.679366
\(27\) 5.00000 0.962250
\(28\) 0.267949 0.0506376
\(29\) 1.73205 0.321634 0.160817 0.986984i \(-0.448587\pi\)
0.160817 + 0.986984i \(0.448587\pi\)
\(30\) −0.732051 −0.133654
\(31\) −1.46410 −0.262960 −0.131480 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.46410 1.28008
\(35\) −0.196152 −0.0331558
\(36\) −2.00000 −0.333333
\(37\) −6.46410 −1.06269 −0.531346 0.847155i \(-0.678314\pi\)
−0.531346 + 0.847155i \(0.678314\pi\)
\(38\) −1.00000 −0.162221
\(39\) 3.46410 0.554700
\(40\) 0.732051 0.115747
\(41\) −10.7321 −1.67606 −0.838032 0.545621i \(-0.816294\pi\)
−0.838032 + 0.545621i \(0.816294\pi\)
\(42\) 0.267949 0.0413455
\(43\) 4.92820 0.751544 0.375772 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(44\) 0 0
\(45\) 1.46410 0.218255
\(46\) 1.00000 0.147442
\(47\) 12.4641 1.81808 0.909038 0.416713i \(-0.136818\pi\)
0.909038 + 0.416713i \(0.136818\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.92820 −0.989743
\(50\) 4.46410 0.631319
\(51\) 7.46410 1.04518
\(52\) −3.46410 −0.480384
\(53\) −9.92820 −1.36374 −0.681872 0.731472i \(-0.738834\pi\)
−0.681872 + 0.731472i \(0.738834\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) −0.267949 −0.0358062
\(57\) −1.00000 −0.132453
\(58\) −1.73205 −0.227429
\(59\) −9.39230 −1.22277 −0.611387 0.791332i \(-0.709388\pi\)
−0.611387 + 0.791332i \(0.709388\pi\)
\(60\) 0.732051 0.0945074
\(61\) 1.26795 0.162344 0.0811721 0.996700i \(-0.474134\pi\)
0.0811721 + 0.996700i \(0.474134\pi\)
\(62\) 1.46410 0.185941
\(63\) −0.535898 −0.0675169
\(64\) 1.00000 0.125000
\(65\) 2.53590 0.314539
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) −7.46410 −0.905155
\(69\) 1.00000 0.120386
\(70\) 0.196152 0.0234447
\(71\) 14.1962 1.68477 0.842387 0.538874i \(-0.181150\pi\)
0.842387 + 0.538874i \(0.181150\pi\)
\(72\) 2.00000 0.235702
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 6.46410 0.751437
\(75\) 4.46410 0.515470
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −3.46410 −0.392232
\(79\) −2.19615 −0.247086 −0.123543 0.992339i \(-0.539426\pi\)
−0.123543 + 0.992339i \(0.539426\pi\)
\(80\) −0.732051 −0.0818458
\(81\) 1.00000 0.111111
\(82\) 10.7321 1.18516
\(83\) −3.80385 −0.417527 −0.208763 0.977966i \(-0.566944\pi\)
−0.208763 + 0.977966i \(0.566944\pi\)
\(84\) −0.267949 −0.0292357
\(85\) 5.46410 0.592665
\(86\) −4.92820 −0.531422
\(87\) −1.73205 −0.185695
\(88\) 0 0
\(89\) 12.1962 1.29279 0.646395 0.763003i \(-0.276276\pi\)
0.646395 + 0.763003i \(0.276276\pi\)
\(90\) −1.46410 −0.154330
\(91\) −0.928203 −0.0973021
\(92\) −1.00000 −0.104257
\(93\) 1.46410 0.151820
\(94\) −12.4641 −1.28557
\(95\) −0.732051 −0.0751068
\(96\) 1.00000 0.102062
\(97\) 13.4641 1.36707 0.683536 0.729917i \(-0.260441\pi\)
0.683536 + 0.729917i \(0.260441\pi\)
\(98\) 6.92820 0.699854
\(99\) 0 0
\(100\) −4.46410 −0.446410
\(101\) −16.5885 −1.65061 −0.825307 0.564685i \(-0.808998\pi\)
−0.825307 + 0.564685i \(0.808998\pi\)
\(102\) −7.46410 −0.739056
\(103\) −2.73205 −0.269197 −0.134598 0.990900i \(-0.542974\pi\)
−0.134598 + 0.990900i \(0.542974\pi\)
\(104\) 3.46410 0.339683
\(105\) 0.196152 0.0191425
\(106\) 9.92820 0.964312
\(107\) −2.80385 −0.271058 −0.135529 0.990773i \(-0.543273\pi\)
−0.135529 + 0.990773i \(0.543273\pi\)
\(108\) 5.00000 0.481125
\(109\) −6.12436 −0.586607 −0.293303 0.956019i \(-0.594755\pi\)
−0.293303 + 0.956019i \(0.594755\pi\)
\(110\) 0 0
\(111\) 6.46410 0.613545
\(112\) 0.267949 0.0253188
\(113\) −4.73205 −0.445154 −0.222577 0.974915i \(-0.571447\pi\)
−0.222577 + 0.974915i \(0.571447\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0.732051 0.0682641
\(116\) 1.73205 0.160817
\(117\) 6.92820 0.640513
\(118\) 9.39230 0.864632
\(119\) −2.00000 −0.183340
\(120\) −0.732051 −0.0668268
\(121\) 0 0
\(122\) −1.26795 −0.114795
\(123\) 10.7321 0.967676
\(124\) −1.46410 −0.131480
\(125\) 6.92820 0.619677
\(126\) 0.535898 0.0477416
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.92820 −0.433904
\(130\) −2.53590 −0.222413
\(131\) −20.9282 −1.82851 −0.914253 0.405144i \(-0.867221\pi\)
−0.914253 + 0.405144i \(0.867221\pi\)
\(132\) 0 0
\(133\) 0.267949 0.0232341
\(134\) −14.0000 −1.20942
\(135\) −3.66025 −0.315025
\(136\) 7.46410 0.640041
\(137\) 9.46410 0.808573 0.404286 0.914632i \(-0.367520\pi\)
0.404286 + 0.914632i \(0.367520\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −6.53590 −0.554368 −0.277184 0.960817i \(-0.589401\pi\)
−0.277184 + 0.960817i \(0.589401\pi\)
\(140\) −0.196152 −0.0165779
\(141\) −12.4641 −1.04967
\(142\) −14.1962 −1.19131
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) −1.26795 −0.105297
\(146\) −2.00000 −0.165521
\(147\) 6.92820 0.571429
\(148\) −6.46410 −0.531346
\(149\) 11.3205 0.927412 0.463706 0.885989i \(-0.346519\pi\)
0.463706 + 0.885989i \(0.346519\pi\)
\(150\) −4.46410 −0.364492
\(151\) −20.7321 −1.68715 −0.843575 0.537011i \(-0.819553\pi\)
−0.843575 + 0.537011i \(0.819553\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 14.9282 1.20687
\(154\) 0 0
\(155\) 1.07180 0.0860888
\(156\) 3.46410 0.277350
\(157\) −5.26795 −0.420428 −0.210214 0.977655i \(-0.567416\pi\)
−0.210214 + 0.977655i \(0.567416\pi\)
\(158\) 2.19615 0.174717
\(159\) 9.92820 0.787358
\(160\) 0.732051 0.0578737
\(161\) −0.267949 −0.0211174
\(162\) −1.00000 −0.0785674
\(163\) 17.3205 1.35665 0.678323 0.734763i \(-0.262707\pi\)
0.678323 + 0.734763i \(0.262707\pi\)
\(164\) −10.7321 −0.838032
\(165\) 0 0
\(166\) 3.80385 0.295236
\(167\) −3.46410 −0.268060 −0.134030 0.990977i \(-0.542792\pi\)
−0.134030 + 0.990977i \(0.542792\pi\)
\(168\) 0.267949 0.0206727
\(169\) −1.00000 −0.0769231
\(170\) −5.46410 −0.419077
\(171\) −2.00000 −0.152944
\(172\) 4.92820 0.375772
\(173\) −4.12436 −0.313569 −0.156784 0.987633i \(-0.550113\pi\)
−0.156784 + 0.987633i \(0.550113\pi\)
\(174\) 1.73205 0.131306
\(175\) −1.19615 −0.0904206
\(176\) 0 0
\(177\) 9.39230 0.705969
\(178\) −12.1962 −0.914140
\(179\) −2.07180 −0.154853 −0.0774267 0.996998i \(-0.524670\pi\)
−0.0774267 + 0.996998i \(0.524670\pi\)
\(180\) 1.46410 0.109128
\(181\) 24.3923 1.81307 0.906533 0.422135i \(-0.138719\pi\)
0.906533 + 0.422135i \(0.138719\pi\)
\(182\) 0.928203 0.0688030
\(183\) −1.26795 −0.0937295
\(184\) 1.00000 0.0737210
\(185\) 4.73205 0.347907
\(186\) −1.46410 −0.107353
\(187\) 0 0
\(188\) 12.4641 0.909038
\(189\) 1.33975 0.0974522
\(190\) 0.732051 0.0531085
\(191\) −17.3923 −1.25846 −0.629232 0.777218i \(-0.716630\pi\)
−0.629232 + 0.777218i \(0.716630\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 12.5359 0.902354 0.451177 0.892435i \(-0.351004\pi\)
0.451177 + 0.892435i \(0.351004\pi\)
\(194\) −13.4641 −0.966666
\(195\) −2.53590 −0.181599
\(196\) −6.92820 −0.494872
\(197\) 5.07180 0.361351 0.180675 0.983543i \(-0.442172\pi\)
0.180675 + 0.983543i \(0.442172\pi\)
\(198\) 0 0
\(199\) 16.9282 1.20001 0.600004 0.799997i \(-0.295166\pi\)
0.600004 + 0.799997i \(0.295166\pi\)
\(200\) 4.46410 0.315660
\(201\) −14.0000 −0.987484
\(202\) 16.5885 1.16716
\(203\) 0.464102 0.0325735
\(204\) 7.46410 0.522592
\(205\) 7.85641 0.548715
\(206\) 2.73205 0.190351
\(207\) 2.00000 0.139010
\(208\) −3.46410 −0.240192
\(209\) 0 0
\(210\) −0.196152 −0.0135358
\(211\) −4.12436 −0.283932 −0.141966 0.989872i \(-0.545342\pi\)
−0.141966 + 0.989872i \(0.545342\pi\)
\(212\) −9.92820 −0.681872
\(213\) −14.1962 −0.972704
\(214\) 2.80385 0.191667
\(215\) −3.60770 −0.246043
\(216\) −5.00000 −0.340207
\(217\) −0.392305 −0.0266314
\(218\) 6.12436 0.414794
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 25.8564 1.73929
\(222\) −6.46410 −0.433842
\(223\) 13.1244 0.878872 0.439436 0.898274i \(-0.355178\pi\)
0.439436 + 0.898274i \(0.355178\pi\)
\(224\) −0.267949 −0.0179031
\(225\) 8.92820 0.595214
\(226\) 4.73205 0.314771
\(227\) 8.39230 0.557017 0.278508 0.960434i \(-0.410160\pi\)
0.278508 + 0.960434i \(0.410160\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −12.3923 −0.818907 −0.409453 0.912331i \(-0.634281\pi\)
−0.409453 + 0.912331i \(0.634281\pi\)
\(230\) −0.732051 −0.0482700
\(231\) 0 0
\(232\) −1.73205 −0.113715
\(233\) −8.26795 −0.541651 −0.270826 0.962628i \(-0.587297\pi\)
−0.270826 + 0.962628i \(0.587297\pi\)
\(234\) −6.92820 −0.452911
\(235\) −9.12436 −0.595207
\(236\) −9.39230 −0.611387
\(237\) 2.19615 0.142655
\(238\) 2.00000 0.129641
\(239\) −14.1244 −0.913629 −0.456814 0.889562i \(-0.651010\pi\)
−0.456814 + 0.889562i \(0.651010\pi\)
\(240\) 0.732051 0.0472537
\(241\) 11.2679 0.725832 0.362916 0.931822i \(-0.381781\pi\)
0.362916 + 0.931822i \(0.381781\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 1.26795 0.0811721
\(245\) 5.07180 0.324025
\(246\) −10.7321 −0.684251
\(247\) −3.46410 −0.220416
\(248\) 1.46410 0.0929705
\(249\) 3.80385 0.241059
\(250\) −6.92820 −0.438178
\(251\) 8.53590 0.538781 0.269391 0.963031i \(-0.413178\pi\)
0.269391 + 0.963031i \(0.413178\pi\)
\(252\) −0.535898 −0.0337584
\(253\) 0 0
\(254\) −4.00000 −0.250982
\(255\) −5.46410 −0.342175
\(256\) 1.00000 0.0625000
\(257\) −3.32051 −0.207128 −0.103564 0.994623i \(-0.533025\pi\)
−0.103564 + 0.994623i \(0.533025\pi\)
\(258\) 4.92820 0.306817
\(259\) −1.73205 −0.107624
\(260\) 2.53590 0.157270
\(261\) −3.46410 −0.214423
\(262\) 20.9282 1.29295
\(263\) 24.7846 1.52828 0.764142 0.645048i \(-0.223163\pi\)
0.764142 + 0.645048i \(0.223163\pi\)
\(264\) 0 0
\(265\) 7.26795 0.446467
\(266\) −0.267949 −0.0164290
\(267\) −12.1962 −0.746392
\(268\) 14.0000 0.855186
\(269\) −29.7846 −1.81600 −0.908000 0.418970i \(-0.862391\pi\)
−0.908000 + 0.418970i \(0.862391\pi\)
\(270\) 3.66025 0.222756
\(271\) −7.32051 −0.444689 −0.222345 0.974968i \(-0.571371\pi\)
−0.222345 + 0.974968i \(0.571371\pi\)
\(272\) −7.46410 −0.452578
\(273\) 0.928203 0.0561774
\(274\) −9.46410 −0.571747
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 15.1244 0.908734 0.454367 0.890814i \(-0.349865\pi\)
0.454367 + 0.890814i \(0.349865\pi\)
\(278\) 6.53590 0.391997
\(279\) 2.92820 0.175307
\(280\) 0.196152 0.0117223
\(281\) 16.0526 0.957615 0.478808 0.877920i \(-0.341069\pi\)
0.478808 + 0.877920i \(0.341069\pi\)
\(282\) 12.4641 0.742226
\(283\) 21.4641 1.27591 0.637954 0.770074i \(-0.279781\pi\)
0.637954 + 0.770074i \(0.279781\pi\)
\(284\) 14.1962 0.842387
\(285\) 0.732051 0.0433629
\(286\) 0 0
\(287\) −2.87564 −0.169744
\(288\) 2.00000 0.117851
\(289\) 38.7128 2.27722
\(290\) 1.26795 0.0744565
\(291\) −13.4641 −0.789280
\(292\) 2.00000 0.117041
\(293\) 12.1244 0.708312 0.354156 0.935186i \(-0.384768\pi\)
0.354156 + 0.935186i \(0.384768\pi\)
\(294\) −6.92820 −0.404061
\(295\) 6.87564 0.400315
\(296\) 6.46410 0.375718
\(297\) 0 0
\(298\) −11.3205 −0.655779
\(299\) 3.46410 0.200334
\(300\) 4.46410 0.257735
\(301\) 1.32051 0.0761128
\(302\) 20.7321 1.19300
\(303\) 16.5885 0.952982
\(304\) 1.00000 0.0573539
\(305\) −0.928203 −0.0531488
\(306\) −14.9282 −0.853389
\(307\) 15.5885 0.889680 0.444840 0.895610i \(-0.353260\pi\)
0.444840 + 0.895610i \(0.353260\pi\)
\(308\) 0 0
\(309\) 2.73205 0.155421
\(310\) −1.07180 −0.0608740
\(311\) −1.60770 −0.0911640 −0.0455820 0.998961i \(-0.514514\pi\)
−0.0455820 + 0.998961i \(0.514514\pi\)
\(312\) −3.46410 −0.196116
\(313\) 13.5359 0.765094 0.382547 0.923936i \(-0.375047\pi\)
0.382547 + 0.923936i \(0.375047\pi\)
\(314\) 5.26795 0.297288
\(315\) 0.392305 0.0221039
\(316\) −2.19615 −0.123543
\(317\) −20.7846 −1.16738 −0.583690 0.811977i \(-0.698392\pi\)
−0.583690 + 0.811977i \(0.698392\pi\)
\(318\) −9.92820 −0.556746
\(319\) 0 0
\(320\) −0.732051 −0.0409229
\(321\) 2.80385 0.156496
\(322\) 0.267949 0.0149322
\(323\) −7.46410 −0.415314
\(324\) 1.00000 0.0555556
\(325\) 15.4641 0.857794
\(326\) −17.3205 −0.959294
\(327\) 6.12436 0.338678
\(328\) 10.7321 0.592578
\(329\) 3.33975 0.184126
\(330\) 0 0
\(331\) 21.7846 1.19739 0.598695 0.800977i \(-0.295686\pi\)
0.598695 + 0.800977i \(0.295686\pi\)
\(332\) −3.80385 −0.208763
\(333\) 12.9282 0.708461
\(334\) 3.46410 0.189547
\(335\) −10.2487 −0.559947
\(336\) −0.267949 −0.0146178
\(337\) −24.1962 −1.31805 −0.659024 0.752122i \(-0.729031\pi\)
−0.659024 + 0.752122i \(0.729031\pi\)
\(338\) 1.00000 0.0543928
\(339\) 4.73205 0.257010
\(340\) 5.46410 0.296333
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) −3.73205 −0.201512
\(344\) −4.92820 −0.265711
\(345\) −0.732051 −0.0394123
\(346\) 4.12436 0.221727
\(347\) 10.0526 0.539650 0.269825 0.962909i \(-0.413034\pi\)
0.269825 + 0.962909i \(0.413034\pi\)
\(348\) −1.73205 −0.0928477
\(349\) −14.7846 −0.791402 −0.395701 0.918379i \(-0.629498\pi\)
−0.395701 + 0.918379i \(0.629498\pi\)
\(350\) 1.19615 0.0639370
\(351\) −17.3205 −0.924500
\(352\) 0 0
\(353\) −6.46410 −0.344049 −0.172025 0.985093i \(-0.555031\pi\)
−0.172025 + 0.985093i \(0.555031\pi\)
\(354\) −9.39230 −0.499195
\(355\) −10.3923 −0.551566
\(356\) 12.1962 0.646395
\(357\) 2.00000 0.105851
\(358\) 2.07180 0.109498
\(359\) 28.3923 1.49849 0.749244 0.662294i \(-0.230417\pi\)
0.749244 + 0.662294i \(0.230417\pi\)
\(360\) −1.46410 −0.0771649
\(361\) 1.00000 0.0526316
\(362\) −24.3923 −1.28203
\(363\) 0 0
\(364\) −0.928203 −0.0486511
\(365\) −1.46410 −0.0766346
\(366\) 1.26795 0.0662768
\(367\) −22.8564 −1.19309 −0.596547 0.802578i \(-0.703461\pi\)
−0.596547 + 0.802578i \(0.703461\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 21.4641 1.11738
\(370\) −4.73205 −0.246008
\(371\) −2.66025 −0.138114
\(372\) 1.46410 0.0759101
\(373\) −16.8038 −0.870070 −0.435035 0.900413i \(-0.643264\pi\)
−0.435035 + 0.900413i \(0.643264\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) −12.4641 −0.642787
\(377\) −6.00000 −0.309016
\(378\) −1.33975 −0.0689091
\(379\) 23.9282 1.22911 0.614555 0.788874i \(-0.289336\pi\)
0.614555 + 0.788874i \(0.289336\pi\)
\(380\) −0.732051 −0.0375534
\(381\) −4.00000 −0.204926
\(382\) 17.3923 0.889868
\(383\) 33.4641 1.70994 0.854968 0.518681i \(-0.173577\pi\)
0.854968 + 0.518681i \(0.173577\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −12.5359 −0.638060
\(387\) −9.85641 −0.501029
\(388\) 13.4641 0.683536
\(389\) −0.588457 −0.0298360 −0.0149180 0.999889i \(-0.504749\pi\)
−0.0149180 + 0.999889i \(0.504749\pi\)
\(390\) 2.53590 0.128410
\(391\) 7.46410 0.377476
\(392\) 6.92820 0.349927
\(393\) 20.9282 1.05569
\(394\) −5.07180 −0.255513
\(395\) 1.60770 0.0808919
\(396\) 0 0
\(397\) 5.60770 0.281442 0.140721 0.990049i \(-0.455058\pi\)
0.140721 + 0.990049i \(0.455058\pi\)
\(398\) −16.9282 −0.848534
\(399\) −0.267949 −0.0134142
\(400\) −4.46410 −0.223205
\(401\) −16.0526 −0.801627 −0.400813 0.916160i \(-0.631272\pi\)
−0.400813 + 0.916160i \(0.631272\pi\)
\(402\) 14.0000 0.698257
\(403\) 5.07180 0.252644
\(404\) −16.5885 −0.825307
\(405\) −0.732051 −0.0363759
\(406\) −0.464102 −0.0230330
\(407\) 0 0
\(408\) −7.46410 −0.369528
\(409\) 13.4641 0.665757 0.332878 0.942970i \(-0.391980\pi\)
0.332878 + 0.942970i \(0.391980\pi\)
\(410\) −7.85641 −0.388000
\(411\) −9.46410 −0.466830
\(412\) −2.73205 −0.134598
\(413\) −2.51666 −0.123837
\(414\) −2.00000 −0.0982946
\(415\) 2.78461 0.136691
\(416\) 3.46410 0.169842
\(417\) 6.53590 0.320064
\(418\) 0 0
\(419\) −32.4449 −1.58504 −0.792518 0.609849i \(-0.791230\pi\)
−0.792518 + 0.609849i \(0.791230\pi\)
\(420\) 0.196152 0.00957126
\(421\) −1.67949 −0.0818534 −0.0409267 0.999162i \(-0.513031\pi\)
−0.0409267 + 0.999162i \(0.513031\pi\)
\(422\) 4.12436 0.200770
\(423\) −24.9282 −1.21205
\(424\) 9.92820 0.482156
\(425\) 33.3205 1.61628
\(426\) 14.1962 0.687806
\(427\) 0.339746 0.0164415
\(428\) −2.80385 −0.135529
\(429\) 0 0
\(430\) 3.60770 0.173979
\(431\) 23.6603 1.13967 0.569837 0.821758i \(-0.307006\pi\)
0.569837 + 0.821758i \(0.307006\pi\)
\(432\) 5.00000 0.240563
\(433\) 18.7846 0.902731 0.451365 0.892339i \(-0.350937\pi\)
0.451365 + 0.892339i \(0.350937\pi\)
\(434\) 0.392305 0.0188312
\(435\) 1.26795 0.0607935
\(436\) −6.12436 −0.293303
\(437\) −1.00000 −0.0478365
\(438\) 2.00000 0.0955637
\(439\) 17.4641 0.833516 0.416758 0.909017i \(-0.363166\pi\)
0.416758 + 0.909017i \(0.363166\pi\)
\(440\) 0 0
\(441\) 13.8564 0.659829
\(442\) −25.8564 −1.22986
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 6.46410 0.306773
\(445\) −8.92820 −0.423237
\(446\) −13.1244 −0.621456
\(447\) −11.3205 −0.535442
\(448\) 0.267949 0.0126594
\(449\) 27.8564 1.31463 0.657313 0.753618i \(-0.271693\pi\)
0.657313 + 0.753618i \(0.271693\pi\)
\(450\) −8.92820 −0.420880
\(451\) 0 0
\(452\) −4.73205 −0.222577
\(453\) 20.7321 0.974077
\(454\) −8.39230 −0.393870
\(455\) 0.679492 0.0318551
\(456\) 1.00000 0.0468293
\(457\) −25.4449 −1.19026 −0.595130 0.803629i \(-0.702900\pi\)
−0.595130 + 0.803629i \(0.702900\pi\)
\(458\) 12.3923 0.579054
\(459\) −37.3205 −1.74197
\(460\) 0.732051 0.0341320
\(461\) −28.1962 −1.31323 −0.656613 0.754228i \(-0.728011\pi\)
−0.656613 + 0.754228i \(0.728011\pi\)
\(462\) 0 0
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) 1.73205 0.0804084
\(465\) −1.07180 −0.0497034
\(466\) 8.26795 0.383005
\(467\) 33.7128 1.56004 0.780021 0.625753i \(-0.215208\pi\)
0.780021 + 0.625753i \(0.215208\pi\)
\(468\) 6.92820 0.320256
\(469\) 3.75129 0.173218
\(470\) 9.12436 0.420875
\(471\) 5.26795 0.242734
\(472\) 9.39230 0.432316
\(473\) 0 0
\(474\) −2.19615 −0.100873
\(475\) −4.46410 −0.204827
\(476\) −2.00000 −0.0916698
\(477\) 19.8564 0.909162
\(478\) 14.1244 0.646033
\(479\) −31.7321 −1.44987 −0.724937 0.688815i \(-0.758131\pi\)
−0.724937 + 0.688815i \(0.758131\pi\)
\(480\) −0.732051 −0.0334134
\(481\) 22.3923 1.02100
\(482\) −11.2679 −0.513241
\(483\) 0.267949 0.0121921
\(484\) 0 0
\(485\) −9.85641 −0.447556
\(486\) 16.0000 0.725775
\(487\) 27.1769 1.23150 0.615752 0.787940i \(-0.288852\pi\)
0.615752 + 0.787940i \(0.288852\pi\)
\(488\) −1.26795 −0.0573974
\(489\) −17.3205 −0.783260
\(490\) −5.07180 −0.229120
\(491\) 3.60770 0.162813 0.0814065 0.996681i \(-0.474059\pi\)
0.0814065 + 0.996681i \(0.474059\pi\)
\(492\) 10.7321 0.483838
\(493\) −12.9282 −0.582257
\(494\) 3.46410 0.155857
\(495\) 0 0
\(496\) −1.46410 −0.0657401
\(497\) 3.80385 0.170626
\(498\) −3.80385 −0.170454
\(499\) 21.2679 0.952084 0.476042 0.879423i \(-0.342071\pi\)
0.476042 + 0.879423i \(0.342071\pi\)
\(500\) 6.92820 0.309839
\(501\) 3.46410 0.154765
\(502\) −8.53590 −0.380976
\(503\) −0.660254 −0.0294393 −0.0147196 0.999892i \(-0.504686\pi\)
−0.0147196 + 0.999892i \(0.504686\pi\)
\(504\) 0.535898 0.0238708
\(505\) 12.1436 0.540383
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 4.00000 0.177471
\(509\) 33.9282 1.50384 0.751921 0.659254i \(-0.229128\pi\)
0.751921 + 0.659254i \(0.229128\pi\)
\(510\) 5.46410 0.241954
\(511\) 0.535898 0.0237067
\(512\) −1.00000 −0.0441942
\(513\) 5.00000 0.220755
\(514\) 3.32051 0.146461
\(515\) 2.00000 0.0881305
\(516\) −4.92820 −0.216952
\(517\) 0 0
\(518\) 1.73205 0.0761019
\(519\) 4.12436 0.181039
\(520\) −2.53590 −0.111207
\(521\) −17.3205 −0.758825 −0.379413 0.925228i \(-0.623874\pi\)
−0.379413 + 0.925228i \(0.623874\pi\)
\(522\) 3.46410 0.151620
\(523\) 9.33975 0.408399 0.204199 0.978929i \(-0.434541\pi\)
0.204199 + 0.978929i \(0.434541\pi\)
\(524\) −20.9282 −0.914253
\(525\) 1.19615 0.0522044
\(526\) −24.7846 −1.08066
\(527\) 10.9282 0.476040
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) −7.26795 −0.315700
\(531\) 18.7846 0.815183
\(532\) 0.267949 0.0116171
\(533\) 37.1769 1.61031
\(534\) 12.1962 0.527779
\(535\) 2.05256 0.0887399
\(536\) −14.0000 −0.604708
\(537\) 2.07180 0.0894046
\(538\) 29.7846 1.28411
\(539\) 0 0
\(540\) −3.66025 −0.157512
\(541\) 27.1769 1.16843 0.584213 0.811600i \(-0.301403\pi\)
0.584213 + 0.811600i \(0.301403\pi\)
\(542\) 7.32051 0.314443
\(543\) −24.3923 −1.04677
\(544\) 7.46410 0.320021
\(545\) 4.48334 0.192045
\(546\) −0.928203 −0.0397234
\(547\) −4.80385 −0.205398 −0.102699 0.994712i \(-0.532748\pi\)
−0.102699 + 0.994712i \(0.532748\pi\)
\(548\) 9.46410 0.404286
\(549\) −2.53590 −0.108230
\(550\) 0 0
\(551\) 1.73205 0.0737878
\(552\) −1.00000 −0.0425628
\(553\) −0.588457 −0.0250237
\(554\) −15.1244 −0.642572
\(555\) −4.73205 −0.200864
\(556\) −6.53590 −0.277184
\(557\) 23.1769 0.982037 0.491019 0.871149i \(-0.336625\pi\)
0.491019 + 0.871149i \(0.336625\pi\)
\(558\) −2.92820 −0.123961
\(559\) −17.0718 −0.722060
\(560\) −0.196152 −0.00828895
\(561\) 0 0
\(562\) −16.0526 −0.677136
\(563\) −24.6603 −1.03931 −0.519653 0.854377i \(-0.673939\pi\)
−0.519653 + 0.854377i \(0.673939\pi\)
\(564\) −12.4641 −0.524833
\(565\) 3.46410 0.145736
\(566\) −21.4641 −0.902203
\(567\) 0.267949 0.0112528
\(568\) −14.1962 −0.595657
\(569\) 39.7128 1.66485 0.832424 0.554139i \(-0.186953\pi\)
0.832424 + 0.554139i \(0.186953\pi\)
\(570\) −0.732051 −0.0306622
\(571\) 22.9808 0.961715 0.480857 0.876799i \(-0.340325\pi\)
0.480857 + 0.876799i \(0.340325\pi\)
\(572\) 0 0
\(573\) 17.3923 0.726574
\(574\) 2.87564 0.120027
\(575\) 4.46410 0.186166
\(576\) −2.00000 −0.0833333
\(577\) −33.1769 −1.38117 −0.690587 0.723250i \(-0.742648\pi\)
−0.690587 + 0.723250i \(0.742648\pi\)
\(578\) −38.7128 −1.61024
\(579\) −12.5359 −0.520974
\(580\) −1.26795 −0.0526487
\(581\) −1.01924 −0.0422851
\(582\) 13.4641 0.558105
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) −5.07180 −0.209693
\(586\) −12.1244 −0.500853
\(587\) −31.3731 −1.29491 −0.647453 0.762106i \(-0.724166\pi\)
−0.647453 + 0.762106i \(0.724166\pi\)
\(588\) 6.92820 0.285714
\(589\) −1.46410 −0.0603273
\(590\) −6.87564 −0.283066
\(591\) −5.07180 −0.208626
\(592\) −6.46410 −0.265673
\(593\) −1.87564 −0.0770235 −0.0385117 0.999258i \(-0.512262\pi\)
−0.0385117 + 0.999258i \(0.512262\pi\)
\(594\) 0 0
\(595\) 1.46410 0.0600223
\(596\) 11.3205 0.463706
\(597\) −16.9282 −0.692825
\(598\) −3.46410 −0.141658
\(599\) −19.6077 −0.801149 −0.400574 0.916264i \(-0.631189\pi\)
−0.400574 + 0.916264i \(0.631189\pi\)
\(600\) −4.46410 −0.182246
\(601\) −18.7846 −0.766240 −0.383120 0.923699i \(-0.625150\pi\)
−0.383120 + 0.923699i \(0.625150\pi\)
\(602\) −1.32051 −0.0538199
\(603\) −28.0000 −1.14025
\(604\) −20.7321 −0.843575
\(605\) 0 0
\(606\) −16.5885 −0.673860
\(607\) 3.26795 0.132642 0.0663210 0.997798i \(-0.478874\pi\)
0.0663210 + 0.997798i \(0.478874\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −0.464102 −0.0188063
\(610\) 0.928203 0.0375819
\(611\) −43.1769 −1.74675
\(612\) 14.9282 0.603437
\(613\) 22.1962 0.896494 0.448247 0.893910i \(-0.352048\pi\)
0.448247 + 0.893910i \(0.352048\pi\)
\(614\) −15.5885 −0.629099
\(615\) −7.85641 −0.316801
\(616\) 0 0
\(617\) −1.92820 −0.0776265 −0.0388133 0.999246i \(-0.512358\pi\)
−0.0388133 + 0.999246i \(0.512358\pi\)
\(618\) −2.73205 −0.109899
\(619\) 34.3923 1.38234 0.691172 0.722691i \(-0.257095\pi\)
0.691172 + 0.722691i \(0.257095\pi\)
\(620\) 1.07180 0.0430444
\(621\) −5.00000 −0.200643
\(622\) 1.60770 0.0644627
\(623\) 3.26795 0.130928
\(624\) 3.46410 0.138675
\(625\) 17.2487 0.689948
\(626\) −13.5359 −0.541003
\(627\) 0 0
\(628\) −5.26795 −0.210214
\(629\) 48.2487 1.92380
\(630\) −0.392305 −0.0156298
\(631\) −21.2487 −0.845898 −0.422949 0.906154i \(-0.639005\pi\)
−0.422949 + 0.906154i \(0.639005\pi\)
\(632\) 2.19615 0.0873583
\(633\) 4.12436 0.163928
\(634\) 20.7846 0.825462
\(635\) −2.92820 −0.116202
\(636\) 9.92820 0.393679
\(637\) 24.0000 0.950915
\(638\) 0 0
\(639\) −28.3923 −1.12318
\(640\) 0.732051 0.0289368
\(641\) −33.8038 −1.33517 −0.667586 0.744533i \(-0.732672\pi\)
−0.667586 + 0.744533i \(0.732672\pi\)
\(642\) −2.80385 −0.110659
\(643\) −17.7128 −0.698525 −0.349263 0.937025i \(-0.613568\pi\)
−0.349263 + 0.937025i \(0.613568\pi\)
\(644\) −0.267949 −0.0105587
\(645\) 3.60770 0.142053
\(646\) 7.46410 0.293671
\(647\) −3.14359 −0.123587 −0.0617937 0.998089i \(-0.519682\pi\)
−0.0617937 + 0.998089i \(0.519682\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −15.4641 −0.606552
\(651\) 0.392305 0.0153756
\(652\) 17.3205 0.678323
\(653\) −2.67949 −0.104857 −0.0524283 0.998625i \(-0.516696\pi\)
−0.0524283 + 0.998625i \(0.516696\pi\)
\(654\) −6.12436 −0.239481
\(655\) 15.3205 0.598622
\(656\) −10.7321 −0.419016
\(657\) −4.00000 −0.156055
\(658\) −3.33975 −0.130197
\(659\) 17.3205 0.674711 0.337356 0.941377i \(-0.390468\pi\)
0.337356 + 0.941377i \(0.390468\pi\)
\(660\) 0 0
\(661\) 19.2487 0.748688 0.374344 0.927290i \(-0.377868\pi\)
0.374344 + 0.927290i \(0.377868\pi\)
\(662\) −21.7846 −0.846683
\(663\) −25.8564 −1.00418
\(664\) 3.80385 0.147618
\(665\) −0.196152 −0.00760646
\(666\) −12.9282 −0.500958
\(667\) −1.73205 −0.0670653
\(668\) −3.46410 −0.134030
\(669\) −13.1244 −0.507417
\(670\) 10.2487 0.395942
\(671\) 0 0
\(672\) 0.267949 0.0103364
\(673\) −14.5885 −0.562344 −0.281172 0.959657i \(-0.590723\pi\)
−0.281172 + 0.959657i \(0.590723\pi\)
\(674\) 24.1962 0.932001
\(675\) −22.3205 −0.859117
\(676\) −1.00000 −0.0384615
\(677\) 36.6603 1.40897 0.704484 0.709720i \(-0.251178\pi\)
0.704484 + 0.709720i \(0.251178\pi\)
\(678\) −4.73205 −0.181733
\(679\) 3.60770 0.138451
\(680\) −5.46410 −0.209539
\(681\) −8.39230 −0.321594
\(682\) 0 0
\(683\) −22.7846 −0.871829 −0.435914 0.899988i \(-0.643575\pi\)
−0.435914 + 0.899988i \(0.643575\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −6.92820 −0.264713
\(686\) 3.73205 0.142490
\(687\) 12.3923 0.472796
\(688\) 4.92820 0.187886
\(689\) 34.3923 1.31024
\(690\) 0.732051 0.0278687
\(691\) 28.0526 1.06717 0.533585 0.845747i \(-0.320845\pi\)
0.533585 + 0.845747i \(0.320845\pi\)
\(692\) −4.12436 −0.156784
\(693\) 0 0
\(694\) −10.0526 −0.381590
\(695\) 4.78461 0.181491
\(696\) 1.73205 0.0656532
\(697\) 80.1051 3.03420
\(698\) 14.7846 0.559606
\(699\) 8.26795 0.312723
\(700\) −1.19615 −0.0452103
\(701\) −9.46410 −0.357454 −0.178727 0.983899i \(-0.557198\pi\)
−0.178727 + 0.983899i \(0.557198\pi\)
\(702\) 17.3205 0.653720
\(703\) −6.46410 −0.243798
\(704\) 0 0
\(705\) 9.12436 0.343643
\(706\) 6.46410 0.243280
\(707\) −4.44486 −0.167166
\(708\) 9.39230 0.352984
\(709\) 9.85641 0.370165 0.185083 0.982723i \(-0.440745\pi\)
0.185083 + 0.982723i \(0.440745\pi\)
\(710\) 10.3923 0.390016
\(711\) 4.39230 0.164724
\(712\) −12.1962 −0.457070
\(713\) 1.46410 0.0548310
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) −2.07180 −0.0774267
\(717\) 14.1244 0.527484
\(718\) −28.3923 −1.05959
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 1.46410 0.0545638
\(721\) −0.732051 −0.0272630
\(722\) −1.00000 −0.0372161
\(723\) −11.2679 −0.419060
\(724\) 24.3923 0.906533
\(725\) −7.73205 −0.287161
\(726\) 0 0
\(727\) −28.3205 −1.05035 −0.525175 0.850994i \(-0.676000\pi\)
−0.525175 + 0.850994i \(0.676000\pi\)
\(728\) 0.928203 0.0344015
\(729\) 13.0000 0.481481
\(730\) 1.46410 0.0541888
\(731\) −36.7846 −1.36053
\(732\) −1.26795 −0.0468648
\(733\) 5.07180 0.187331 0.0936655 0.995604i \(-0.470142\pi\)
0.0936655 + 0.995604i \(0.470142\pi\)
\(734\) 22.8564 0.843645
\(735\) −5.07180 −0.187076
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) −21.4641 −0.790104
\(739\) −31.6603 −1.16464 −0.582321 0.812959i \(-0.697855\pi\)
−0.582321 + 0.812959i \(0.697855\pi\)
\(740\) 4.73205 0.173954
\(741\) 3.46410 0.127257
\(742\) 2.66025 0.0976610
\(743\) −12.5359 −0.459898 −0.229949 0.973203i \(-0.573856\pi\)
−0.229949 + 0.973203i \(0.573856\pi\)
\(744\) −1.46410 −0.0536766
\(745\) −8.28719 −0.303619
\(746\) 16.8038 0.615233
\(747\) 7.60770 0.278351
\(748\) 0 0
\(749\) −0.751289 −0.0274515
\(750\) 6.92820 0.252982
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 12.4641 0.454519
\(753\) −8.53590 −0.311065
\(754\) 6.00000 0.218507
\(755\) 15.1769 0.552344
\(756\) 1.33975 0.0487261
\(757\) −21.3731 −0.776817 −0.388409 0.921487i \(-0.626975\pi\)
−0.388409 + 0.921487i \(0.626975\pi\)
\(758\) −23.9282 −0.869111
\(759\) 0 0
\(760\) 0.732051 0.0265543
\(761\) −48.7654 −1.76774 −0.883872 0.467729i \(-0.845072\pi\)
−0.883872 + 0.467729i \(0.845072\pi\)
\(762\) 4.00000 0.144905
\(763\) −1.64102 −0.0594088
\(764\) −17.3923 −0.629232
\(765\) −10.9282 −0.395110
\(766\) −33.4641 −1.20911
\(767\) 32.5359 1.17480
\(768\) −1.00000 −0.0360844
\(769\) 27.5885 0.994865 0.497433 0.867503i \(-0.334276\pi\)
0.497433 + 0.867503i \(0.334276\pi\)
\(770\) 0 0
\(771\) 3.32051 0.119585
\(772\) 12.5359 0.451177
\(773\) 6.60770 0.237662 0.118831 0.992914i \(-0.462085\pi\)
0.118831 + 0.992914i \(0.462085\pi\)
\(774\) 9.85641 0.354281
\(775\) 6.53590 0.234776
\(776\) −13.4641 −0.483333
\(777\) 1.73205 0.0621370
\(778\) 0.588457 0.0210972
\(779\) −10.7321 −0.384516
\(780\) −2.53590 −0.0907997
\(781\) 0 0
\(782\) −7.46410 −0.266916
\(783\) 8.66025 0.309492
\(784\) −6.92820 −0.247436
\(785\) 3.85641 0.137641
\(786\) −20.9282 −0.746484
\(787\) −13.0526 −0.465273 −0.232637 0.972564i \(-0.574735\pi\)
−0.232637 + 0.972564i \(0.574735\pi\)
\(788\) 5.07180 0.180675
\(789\) −24.7846 −0.882355
\(790\) −1.60770 −0.0571992
\(791\) −1.26795 −0.0450831
\(792\) 0 0
\(793\) −4.39230 −0.155975
\(794\) −5.60770 −0.199010
\(795\) −7.26795 −0.257768
\(796\) 16.9282 0.600004
\(797\) 11.6077 0.411166 0.205583 0.978640i \(-0.434091\pi\)
0.205583 + 0.978640i \(0.434091\pi\)
\(798\) 0.267949 0.00948530
\(799\) −93.0333 −3.29128
\(800\) 4.46410 0.157830
\(801\) −24.3923 −0.861860
\(802\) 16.0526 0.566836
\(803\) 0 0
\(804\) −14.0000 −0.493742
\(805\) 0.196152 0.00691346
\(806\) −5.07180 −0.178646
\(807\) 29.7846 1.04847
\(808\) 16.5885 0.583580
\(809\) −19.1962 −0.674901 −0.337450 0.941343i \(-0.609565\pi\)
−0.337450 + 0.941343i \(0.609565\pi\)
\(810\) 0.732051 0.0257216
\(811\) 46.7654 1.64215 0.821077 0.570817i \(-0.193374\pi\)
0.821077 + 0.570817i \(0.193374\pi\)
\(812\) 0.464102 0.0162868
\(813\) 7.32051 0.256741
\(814\) 0 0
\(815\) −12.6795 −0.444143
\(816\) 7.46410 0.261296
\(817\) 4.92820 0.172416
\(818\) −13.4641 −0.470761
\(819\) 1.85641 0.0648681
\(820\) 7.85641 0.274358
\(821\) −42.0526 −1.46764 −0.733822 0.679342i \(-0.762265\pi\)
−0.733822 + 0.679342i \(0.762265\pi\)
\(822\) 9.46410 0.330098
\(823\) −21.5359 −0.750694 −0.375347 0.926884i \(-0.622476\pi\)
−0.375347 + 0.926884i \(0.622476\pi\)
\(824\) 2.73205 0.0951755
\(825\) 0 0
\(826\) 2.51666 0.0875658
\(827\) 21.3205 0.741387 0.370693 0.928755i \(-0.379120\pi\)
0.370693 + 0.928755i \(0.379120\pi\)
\(828\) 2.00000 0.0695048
\(829\) 18.6410 0.647429 0.323715 0.946155i \(-0.395068\pi\)
0.323715 + 0.946155i \(0.395068\pi\)
\(830\) −2.78461 −0.0966552
\(831\) −15.1244 −0.524658
\(832\) −3.46410 −0.120096
\(833\) 51.7128 1.79174
\(834\) −6.53590 −0.226320
\(835\) 2.53590 0.0877584
\(836\) 0 0
\(837\) −7.32051 −0.253034
\(838\) 32.4449 1.12079
\(839\) 14.9808 0.517193 0.258597 0.965985i \(-0.416740\pi\)
0.258597 + 0.965985i \(0.416740\pi\)
\(840\) −0.196152 −0.00676790
\(841\) −26.0000 −0.896552
\(842\) 1.67949 0.0578791
\(843\) −16.0526 −0.552879
\(844\) −4.12436 −0.141966
\(845\) 0.732051 0.0251833
\(846\) 24.9282 0.857049
\(847\) 0 0
\(848\) −9.92820 −0.340936
\(849\) −21.4641 −0.736646
\(850\) −33.3205 −1.14288
\(851\) 6.46410 0.221587
\(852\) −14.1962 −0.486352
\(853\) −54.1051 −1.85252 −0.926262 0.376880i \(-0.876997\pi\)
−0.926262 + 0.376880i \(0.876997\pi\)
\(854\) −0.339746 −0.0116259
\(855\) 1.46410 0.0500712
\(856\) 2.80385 0.0958335
\(857\) −16.5359 −0.564856 −0.282428 0.959289i \(-0.591140\pi\)
−0.282428 + 0.959289i \(0.591140\pi\)
\(858\) 0 0
\(859\) −19.9474 −0.680598 −0.340299 0.940317i \(-0.610528\pi\)
−0.340299 + 0.940317i \(0.610528\pi\)
\(860\) −3.60770 −0.123021
\(861\) 2.87564 0.0980017
\(862\) −23.6603 −0.805871
\(863\) −42.9282 −1.46129 −0.730647 0.682756i \(-0.760781\pi\)
−0.730647 + 0.682756i \(0.760781\pi\)
\(864\) −5.00000 −0.170103
\(865\) 3.01924 0.102657
\(866\) −18.7846 −0.638327
\(867\) −38.7128 −1.31476
\(868\) −0.392305 −0.0133157
\(869\) 0 0
\(870\) −1.26795 −0.0429875
\(871\) −48.4974 −1.64327
\(872\) 6.12436 0.207397
\(873\) −26.9282 −0.911382
\(874\) 1.00000 0.0338255
\(875\) 1.85641 0.0627580
\(876\) −2.00000 −0.0675737
\(877\) 37.8564 1.27832 0.639160 0.769074i \(-0.279282\pi\)
0.639160 + 0.769074i \(0.279282\pi\)
\(878\) −17.4641 −0.589385
\(879\) −12.1244 −0.408944
\(880\) 0 0
\(881\) −13.5359 −0.456036 −0.228018 0.973657i \(-0.573225\pi\)
−0.228018 + 0.973657i \(0.573225\pi\)
\(882\) −13.8564 −0.466569
\(883\) −24.5885 −0.827467 −0.413734 0.910398i \(-0.635776\pi\)
−0.413734 + 0.910398i \(0.635776\pi\)
\(884\) 25.8564 0.869645
\(885\) −6.87564 −0.231122
\(886\) −18.0000 −0.604722
\(887\) 29.6603 0.995894 0.497947 0.867208i \(-0.334087\pi\)
0.497947 + 0.867208i \(0.334087\pi\)
\(888\) −6.46410 −0.216921
\(889\) 1.07180 0.0359469
\(890\) 8.92820 0.299274
\(891\) 0 0
\(892\) 13.1244 0.439436
\(893\) 12.4641 0.417095
\(894\) 11.3205 0.378614
\(895\) 1.51666 0.0506964
\(896\) −0.267949 −0.00895155
\(897\) −3.46410 −0.115663
\(898\) −27.8564 −0.929580
\(899\) −2.53590 −0.0845769
\(900\) 8.92820 0.297607
\(901\) 74.1051 2.46880
\(902\) 0 0
\(903\) −1.32051 −0.0439438
\(904\) 4.73205 0.157386
\(905\) −17.8564 −0.593567
\(906\) −20.7321 −0.688776
\(907\) 29.2487 0.971188 0.485594 0.874185i \(-0.338603\pi\)
0.485594 + 0.874185i \(0.338603\pi\)
\(908\) 8.39230 0.278508
\(909\) 33.1769 1.10041
\(910\) −0.679492 −0.0225249
\(911\) 4.58846 0.152022 0.0760112 0.997107i \(-0.475782\pi\)
0.0760112 + 0.997107i \(0.475782\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 0 0
\(914\) 25.4449 0.841641
\(915\) 0.928203 0.0306855
\(916\) −12.3923 −0.409453
\(917\) −5.60770 −0.185182
\(918\) 37.3205 1.23176
\(919\) −54.1051 −1.78476 −0.892382 0.451282i \(-0.850967\pi\)
−0.892382 + 0.451282i \(0.850967\pi\)
\(920\) −0.732051 −0.0241350
\(921\) −15.5885 −0.513657
\(922\) 28.1962 0.928591
\(923\) −49.1769 −1.61868
\(924\) 0 0
\(925\) 28.8564 0.948793
\(926\) −26.0000 −0.854413
\(927\) 5.46410 0.179465
\(928\) −1.73205 −0.0568574
\(929\) −58.0333 −1.90401 −0.952006 0.306080i \(-0.900982\pi\)
−0.952006 + 0.306080i \(0.900982\pi\)
\(930\) 1.07180 0.0351456
\(931\) −6.92820 −0.227063
\(932\) −8.26795 −0.270826
\(933\) 1.60770 0.0526336
\(934\) −33.7128 −1.10312
\(935\) 0 0
\(936\) −6.92820 −0.226455
\(937\) 29.8756 0.975995 0.487997 0.872845i \(-0.337728\pi\)
0.487997 + 0.872845i \(0.337728\pi\)
\(938\) −3.75129 −0.122484
\(939\) −13.5359 −0.441727
\(940\) −9.12436 −0.297604
\(941\) 24.2487 0.790485 0.395243 0.918577i \(-0.370660\pi\)
0.395243 + 0.918577i \(0.370660\pi\)
\(942\) −5.26795 −0.171639
\(943\) 10.7321 0.349484
\(944\) −9.39230 −0.305693
\(945\) −0.980762 −0.0319042
\(946\) 0 0
\(947\) 23.7128 0.770563 0.385281 0.922799i \(-0.374104\pi\)
0.385281 + 0.922799i \(0.374104\pi\)
\(948\) 2.19615 0.0713277
\(949\) −6.92820 −0.224899
\(950\) 4.46410 0.144835
\(951\) 20.7846 0.673987
\(952\) 2.00000 0.0648204
\(953\) −14.9808 −0.485274 −0.242637 0.970117i \(-0.578012\pi\)
−0.242637 + 0.970117i \(0.578012\pi\)
\(954\) −19.8564 −0.642875
\(955\) 12.7321 0.411999
\(956\) −14.1244 −0.456814
\(957\) 0 0
\(958\) 31.7321 1.02522
\(959\) 2.53590 0.0818884
\(960\) 0.732051 0.0236268
\(961\) −28.8564 −0.930852
\(962\) −22.3923 −0.721957
\(963\) 5.60770 0.180705
\(964\) 11.2679 0.362916
\(965\) −9.17691 −0.295415
\(966\) −0.267949 −0.00862112
\(967\) 8.51666 0.273877 0.136939 0.990580i \(-0.456274\pi\)
0.136939 + 0.990580i \(0.456274\pi\)
\(968\) 0 0
\(969\) 7.46410 0.239781
\(970\) 9.85641 0.316470
\(971\) 20.4641 0.656724 0.328362 0.944552i \(-0.393503\pi\)
0.328362 + 0.944552i \(0.393503\pi\)
\(972\) −16.0000 −0.513200
\(973\) −1.75129 −0.0561437
\(974\) −27.1769 −0.870805
\(975\) −15.4641 −0.495248
\(976\) 1.26795 0.0405861
\(977\) −20.5359 −0.657002 −0.328501 0.944504i \(-0.606543\pi\)
−0.328501 + 0.944504i \(0.606543\pi\)
\(978\) 17.3205 0.553849
\(979\) 0 0
\(980\) 5.07180 0.162013
\(981\) 12.2487 0.391071
\(982\) −3.60770 −0.115126
\(983\) 10.9808 0.350232 0.175116 0.984548i \(-0.443970\pi\)
0.175116 + 0.984548i \(0.443970\pi\)
\(984\) −10.7321 −0.342125
\(985\) −3.71281 −0.118300
\(986\) 12.9282 0.411718
\(987\) −3.33975 −0.106305
\(988\) −3.46410 −0.110208
\(989\) −4.92820 −0.156708
\(990\) 0 0
\(991\) −16.0526 −0.509926 −0.254963 0.966951i \(-0.582063\pi\)
−0.254963 + 0.966951i \(0.582063\pi\)
\(992\) 1.46410 0.0464853
\(993\) −21.7846 −0.691314
\(994\) −3.80385 −0.120651
\(995\) −12.3923 −0.392862
\(996\) 3.80385 0.120530
\(997\) 10.3397 0.327463 0.163732 0.986505i \(-0.447647\pi\)
0.163732 + 0.986505i \(0.447647\pi\)
\(998\) −21.2679 −0.673225
\(999\) −32.3205 −1.02258
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 4598.2.a.w.1.1 2
11.10 odd 2 4598.2.a.bf.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.w.1.1 2 1.1 even 1 trivial
4598.2.a.bf.1.1 yes 2 11.10 odd 2