Properties

Label 7098.2.a.ct.1.5
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.48406561.1
Defining polynomial: \(x^{6} - 3 x^{5} - 17 x^{4} + 39 x^{3} + 111 x^{2} - 131 x - 281\)
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 1.5
Root \(3.10863\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.10863 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.10863 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.10863 q^{10} +1.55496 q^{11} +1.00000 q^{12} +1.00000 q^{14} +3.10863 q^{15} +1.00000 q^{16} -1.24698 q^{17} +1.00000 q^{18} -4.73805 q^{19} +3.10863 q^{20} +1.00000 q^{21} +1.55496 q^{22} +5.10817 q^{23} +1.00000 q^{24} +4.66359 q^{25} +1.00000 q^{27} +1.00000 q^{28} -1.05926 q^{29} +3.10863 q^{30} -2.46553 q^{31} +1.00000 q^{32} +1.55496 q^{33} -1.24698 q^{34} +3.10863 q^{35} +1.00000 q^{36} +7.92429 q^{37} -4.73805 q^{38} +3.10863 q^{40} +2.71654 q^{41} +1.00000 q^{42} -2.63128 q^{43} +1.55496 q^{44} +3.10863 q^{45} +5.10817 q^{46} +5.06412 q^{47} +1.00000 q^{48} +1.00000 q^{49} +4.66359 q^{50} -1.24698 q^{51} -7.09392 q^{53} +1.00000 q^{54} +4.83379 q^{55} +1.00000 q^{56} -4.73805 q^{57} -1.05926 q^{58} -3.41870 q^{59} +3.10863 q^{60} +13.2814 q^{61} -2.46553 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.55496 q^{66} -8.48684 q^{67} -1.24698 q^{68} +5.10817 q^{69} +3.10863 q^{70} -5.31350 q^{71} +1.00000 q^{72} +9.45056 q^{73} +7.92429 q^{74} +4.66359 q^{75} -4.73805 q^{76} +1.55496 q^{77} +11.1942 q^{79} +3.10863 q^{80} +1.00000 q^{81} +2.71654 q^{82} -4.40061 q^{83} +1.00000 q^{84} -3.87640 q^{85} -2.63128 q^{86} -1.05926 q^{87} +1.55496 q^{88} +5.76399 q^{89} +3.10863 q^{90} +5.10817 q^{92} -2.46553 q^{93} +5.06412 q^{94} -14.7289 q^{95} +1.00000 q^{96} +10.8007 q^{97} +1.00000 q^{98} +1.55496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 6q^{3} + 6q^{4} + 3q^{5} + 6q^{6} + 6q^{7} + 6q^{8} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 6q^{3} + 6q^{4} + 3q^{5} + 6q^{6} + 6q^{7} + 6q^{8} + 6q^{9} + 3q^{10} + 10q^{11} + 6q^{12} + 6q^{14} + 3q^{15} + 6q^{16} + 2q^{17} + 6q^{18} + 2q^{19} + 3q^{20} + 6q^{21} + 10q^{22} + 4q^{23} + 6q^{24} + 13q^{25} + 6q^{27} + 6q^{28} + 2q^{29} + 3q^{30} + 9q^{31} + 6q^{32} + 10q^{33} + 2q^{34} + 3q^{35} + 6q^{36} + 7q^{37} + 2q^{38} + 3q^{40} + 11q^{41} + 6q^{42} - 5q^{43} + 10q^{44} + 3q^{45} + 4q^{46} + 5q^{47} + 6q^{48} + 6q^{49} + 13q^{50} + 2q^{51} - 6q^{53} + 6q^{54} + 5q^{55} + 6q^{56} + 2q^{57} + 2q^{58} + 28q^{59} + 3q^{60} + 23q^{61} + 9q^{62} + 6q^{63} + 6q^{64} + 10q^{66} - 10q^{67} + 2q^{68} + 4q^{69} + 3q^{70} + 21q^{71} + 6q^{72} - 7q^{73} + 7q^{74} + 13q^{75} + 2q^{76} + 10q^{77} - 14q^{79} + 3q^{80} + 6q^{81} + 11q^{82} + 17q^{83} + 6q^{84} + q^{85} - 5q^{86} + 2q^{87} + 10q^{88} + 17q^{89} + 3q^{90} + 4q^{92} + 9q^{93} + 5q^{94} - 22q^{95} + 6q^{96} + 6q^{98} + 10q^{99} + O(q^{100}) \)

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
\(5\) 3.10863 1.39022 0.695111 0.718902i \(-0.255355\pi\)
0.695111 + 0.718902i \(0.255355\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.10863 0.983036
\(11\) 1.55496 0.468838 0.234419 0.972136i \(-0.424681\pi\)
0.234419 + 0.972136i \(0.424681\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 3.10863 0.802645
\(16\) 1.00000 0.250000
\(17\) −1.24698 −0.302437 −0.151218 0.988500i \(-0.548320\pi\)
−0.151218 + 0.988500i \(0.548320\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.73805 −1.08698 −0.543492 0.839414i \(-0.682898\pi\)
−0.543492 + 0.839414i \(0.682898\pi\)
\(20\) 3.10863 0.695111
\(21\) 1.00000 0.218218
\(22\) 1.55496 0.331518
\(23\) 5.10817 1.06513 0.532564 0.846390i \(-0.321229\pi\)
0.532564 + 0.846390i \(0.321229\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.66359 0.932718
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −1.05926 −0.196699 −0.0983494 0.995152i \(-0.531356\pi\)
−0.0983494 + 0.995152i \(0.531356\pi\)
\(30\) 3.10863 0.567556
\(31\) −2.46553 −0.442822 −0.221411 0.975181i \(-0.571066\pi\)
−0.221411 + 0.975181i \(0.571066\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.55496 0.270683
\(34\) −1.24698 −0.213855
\(35\) 3.10863 0.525455
\(36\) 1.00000 0.166667
\(37\) 7.92429 1.30274 0.651372 0.758758i \(-0.274194\pi\)
0.651372 + 0.758758i \(0.274194\pi\)
\(38\) −4.73805 −0.768614
\(39\) 0 0
\(40\) 3.10863 0.491518
\(41\) 2.71654 0.424252 0.212126 0.977242i \(-0.431961\pi\)
0.212126 + 0.977242i \(0.431961\pi\)
\(42\) 1.00000 0.154303
\(43\) −2.63128 −0.401266 −0.200633 0.979666i \(-0.564300\pi\)
−0.200633 + 0.979666i \(0.564300\pi\)
\(44\) 1.55496 0.234419
\(45\) 3.10863 0.463407
\(46\) 5.10817 0.753159
\(47\) 5.06412 0.738678 0.369339 0.929295i \(-0.379584\pi\)
0.369339 + 0.929295i \(0.379584\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 4.66359 0.659531
\(51\) −1.24698 −0.174612
\(52\) 0 0
\(53\) −7.09392 −0.974425 −0.487212 0.873284i \(-0.661986\pi\)
−0.487212 + 0.873284i \(0.661986\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.83379 0.651788
\(56\) 1.00000 0.133631
\(57\) −4.73805 −0.627570
\(58\) −1.05926 −0.139087
\(59\) −3.41870 −0.445077 −0.222539 0.974924i \(-0.571434\pi\)
−0.222539 + 0.974924i \(0.571434\pi\)
\(60\) 3.10863 0.401323
\(61\) 13.2814 1.70051 0.850254 0.526373i \(-0.176448\pi\)
0.850254 + 0.526373i \(0.176448\pi\)
\(62\) −2.46553 −0.313122
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.55496 0.191402
\(67\) −8.48684 −1.03683 −0.518416 0.855128i \(-0.673478\pi\)
−0.518416 + 0.855128i \(0.673478\pi\)
\(68\) −1.24698 −0.151218
\(69\) 5.10817 0.614952
\(70\) 3.10863 0.371553
\(71\) −5.31350 −0.630596 −0.315298 0.948993i \(-0.602104\pi\)
−0.315298 + 0.948993i \(0.602104\pi\)
\(72\) 1.00000 0.117851
\(73\) 9.45056 1.10610 0.553052 0.833147i \(-0.313463\pi\)
0.553052 + 0.833147i \(0.313463\pi\)
\(74\) 7.92429 0.921180
\(75\) 4.66359 0.538505
\(76\) −4.73805 −0.543492
\(77\) 1.55496 0.177204
\(78\) 0 0
\(79\) 11.1942 1.25945 0.629725 0.776818i \(-0.283167\pi\)
0.629725 + 0.776818i \(0.283167\pi\)
\(80\) 3.10863 0.347556
\(81\) 1.00000 0.111111
\(82\) 2.71654 0.299991
\(83\) −4.40061 −0.483030 −0.241515 0.970397i \(-0.577644\pi\)
−0.241515 + 0.970397i \(0.577644\pi\)
\(84\) 1.00000 0.109109
\(85\) −3.87640 −0.420455
\(86\) −2.63128 −0.283738
\(87\) −1.05926 −0.113564
\(88\) 1.55496 0.165759
\(89\) 5.76399 0.610982 0.305491 0.952195i \(-0.401179\pi\)
0.305491 + 0.952195i \(0.401179\pi\)
\(90\) 3.10863 0.327679
\(91\) 0 0
\(92\) 5.10817 0.532564
\(93\) −2.46553 −0.255663
\(94\) 5.06412 0.522325
\(95\) −14.7289 −1.51115
\(96\) 1.00000 0.102062
\(97\) 10.8007 1.09664 0.548322 0.836267i \(-0.315267\pi\)
0.548322 + 0.836267i \(0.315267\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.55496 0.156279
\(100\) 4.66359 0.466359
\(101\) 8.82958 0.878576 0.439288 0.898346i \(-0.355231\pi\)
0.439288 + 0.898346i \(0.355231\pi\)
\(102\) −1.24698 −0.123469
\(103\) 1.89793 0.187009 0.0935044 0.995619i \(-0.470193\pi\)
0.0935044 + 0.995619i \(0.470193\pi\)
\(104\) 0 0
\(105\) 3.10863 0.303371
\(106\) −7.09392 −0.689022
\(107\) −3.58727 −0.346795 −0.173397 0.984852i \(-0.555474\pi\)
−0.173397 + 0.984852i \(0.555474\pi\)
\(108\) 1.00000 0.0962250
\(109\) −20.3167 −1.94599 −0.972995 0.230826i \(-0.925857\pi\)
−0.972995 + 0.230826i \(0.925857\pi\)
\(110\) 4.83379 0.460884
\(111\) 7.92429 0.752140
\(112\) 1.00000 0.0944911
\(113\) −6.85060 −0.644450 −0.322225 0.946663i \(-0.604431\pi\)
−0.322225 + 0.946663i \(0.604431\pi\)
\(114\) −4.73805 −0.443759
\(115\) 15.8794 1.48076
\(116\) −1.05926 −0.0983494
\(117\) 0 0
\(118\) −3.41870 −0.314717
\(119\) −1.24698 −0.114310
\(120\) 3.10863 0.283778
\(121\) −8.58211 −0.780191
\(122\) 13.2814 1.20244
\(123\) 2.71654 0.244942
\(124\) −2.46553 −0.221411
\(125\) −1.04578 −0.0935370
\(126\) 1.00000 0.0890871
\(127\) 8.46378 0.751039 0.375520 0.926814i \(-0.377464\pi\)
0.375520 + 0.926814i \(0.377464\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.63128 −0.231671
\(130\) 0 0
\(131\) −5.07860 −0.443719 −0.221860 0.975079i \(-0.571213\pi\)
−0.221860 + 0.975079i \(0.571213\pi\)
\(132\) 1.55496 0.135342
\(133\) −4.73805 −0.410841
\(134\) −8.48684 −0.733151
\(135\) 3.10863 0.267548
\(136\) −1.24698 −0.106928
\(137\) 20.8764 1.78359 0.891795 0.452440i \(-0.149446\pi\)
0.891795 + 0.452440i \(0.149446\pi\)
\(138\) 5.10817 0.434837
\(139\) 17.7255 1.50346 0.751729 0.659472i \(-0.229220\pi\)
0.751729 + 0.659472i \(0.229220\pi\)
\(140\) 3.10863 0.262727
\(141\) 5.06412 0.426476
\(142\) −5.31350 −0.445899
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −3.29284 −0.273455
\(146\) 9.45056 0.782134
\(147\) 1.00000 0.0824786
\(148\) 7.92429 0.651372
\(149\) 0.829877 0.0679862 0.0339931 0.999422i \(-0.489178\pi\)
0.0339931 + 0.999422i \(0.489178\pi\)
\(150\) 4.66359 0.380780
\(151\) −3.39170 −0.276012 −0.138006 0.990431i \(-0.544069\pi\)
−0.138006 + 0.990431i \(0.544069\pi\)
\(152\) −4.73805 −0.384307
\(153\) −1.24698 −0.100812
\(154\) 1.55496 0.125302
\(155\) −7.66442 −0.615621
\(156\) 0 0
\(157\) −23.7381 −1.89450 −0.947251 0.320492i \(-0.896152\pi\)
−0.947251 + 0.320492i \(0.896152\pi\)
\(158\) 11.1942 0.890566
\(159\) −7.09392 −0.562584
\(160\) 3.10863 0.245759
\(161\) 5.10817 0.402580
\(162\) 1.00000 0.0785674
\(163\) −6.35482 −0.497748 −0.248874 0.968536i \(-0.580060\pi\)
−0.248874 + 0.968536i \(0.580060\pi\)
\(164\) 2.71654 0.212126
\(165\) 4.83379 0.376310
\(166\) −4.40061 −0.341554
\(167\) −7.50114 −0.580456 −0.290228 0.956958i \(-0.593731\pi\)
−0.290228 + 0.956958i \(0.593731\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) −3.87640 −0.297306
\(171\) −4.73805 −0.362328
\(172\) −2.63128 −0.200633
\(173\) −16.1881 −1.23076 −0.615378 0.788232i \(-0.710997\pi\)
−0.615378 + 0.788232i \(0.710997\pi\)
\(174\) −1.05926 −0.0803020
\(175\) 4.66359 0.352534
\(176\) 1.55496 0.117209
\(177\) −3.41870 −0.256966
\(178\) 5.76399 0.432030
\(179\) 1.95421 0.146064 0.0730322 0.997330i \(-0.476732\pi\)
0.0730322 + 0.997330i \(0.476732\pi\)
\(180\) 3.10863 0.231704
\(181\) −1.81969 −0.135256 −0.0676281 0.997711i \(-0.521543\pi\)
−0.0676281 + 0.997711i \(0.521543\pi\)
\(182\) 0 0
\(183\) 13.2814 0.981788
\(184\) 5.10817 0.376580
\(185\) 24.6337 1.81110
\(186\) −2.46553 −0.180781
\(187\) −1.93900 −0.141794
\(188\) 5.06412 0.369339
\(189\) 1.00000 0.0727393
\(190\) −14.7289 −1.06854
\(191\) −23.3210 −1.68745 −0.843723 0.536779i \(-0.819641\pi\)
−0.843723 + 0.536779i \(0.819641\pi\)
\(192\) 1.00000 0.0721688
\(193\) 7.99277 0.575332 0.287666 0.957731i \(-0.407121\pi\)
0.287666 + 0.957731i \(0.407121\pi\)
\(194\) 10.8007 0.775444
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −2.24858 −0.160205 −0.0801024 0.996787i \(-0.525525\pi\)
−0.0801024 + 0.996787i \(0.525525\pi\)
\(198\) 1.55496 0.110506
\(199\) −25.4768 −1.80600 −0.903001 0.429638i \(-0.858641\pi\)
−0.903001 + 0.429638i \(0.858641\pi\)
\(200\) 4.66359 0.329766
\(201\) −8.48684 −0.598615
\(202\) 8.82958 0.621247
\(203\) −1.05926 −0.0743452
\(204\) −1.24698 −0.0873060
\(205\) 8.44471 0.589804
\(206\) 1.89793 0.132235
\(207\) 5.10817 0.355043
\(208\) 0 0
\(209\) −7.36747 −0.509619
\(210\) 3.10863 0.214516
\(211\) 2.01914 0.139003 0.0695016 0.997582i \(-0.477859\pi\)
0.0695016 + 0.997582i \(0.477859\pi\)
\(212\) −7.09392 −0.487212
\(213\) −5.31350 −0.364075
\(214\) −3.58727 −0.245221
\(215\) −8.17967 −0.557849
\(216\) 1.00000 0.0680414
\(217\) −2.46553 −0.167371
\(218\) −20.3167 −1.37602
\(219\) 9.45056 0.638610
\(220\) 4.83379 0.325894
\(221\) 0 0
\(222\) 7.92429 0.531843
\(223\) −13.1491 −0.880532 −0.440266 0.897867i \(-0.645116\pi\)
−0.440266 + 0.897867i \(0.645116\pi\)
\(224\) 1.00000 0.0668153
\(225\) 4.66359 0.310906
\(226\) −6.85060 −0.455695
\(227\) 1.90722 0.126587 0.0632935 0.997995i \(-0.479840\pi\)
0.0632935 + 0.997995i \(0.479840\pi\)
\(228\) −4.73805 −0.313785
\(229\) −1.32344 −0.0874554 −0.0437277 0.999043i \(-0.513923\pi\)
−0.0437277 + 0.999043i \(0.513923\pi\)
\(230\) 15.8794 1.04706
\(231\) 1.55496 0.102309
\(232\) −1.05926 −0.0695435
\(233\) −23.7561 −1.55631 −0.778157 0.628069i \(-0.783845\pi\)
−0.778157 + 0.628069i \(0.783845\pi\)
\(234\) 0 0
\(235\) 15.7425 1.02693
\(236\) −3.41870 −0.222539
\(237\) 11.1942 0.727144
\(238\) −1.24698 −0.0808297
\(239\) 27.3308 1.76789 0.883943 0.467595i \(-0.154879\pi\)
0.883943 + 0.467595i \(0.154879\pi\)
\(240\) 3.10863 0.200661
\(241\) 2.68328 0.172846 0.0864228 0.996259i \(-0.472456\pi\)
0.0864228 + 0.996259i \(0.472456\pi\)
\(242\) −8.58211 −0.551679
\(243\) 1.00000 0.0641500
\(244\) 13.2814 0.850254
\(245\) 3.10863 0.198603
\(246\) 2.71654 0.173200
\(247\) 0 0
\(248\) −2.46553 −0.156561
\(249\) −4.40061 −0.278877
\(250\) −1.04578 −0.0661407
\(251\) −30.8962 −1.95015 −0.975077 0.221867i \(-0.928785\pi\)
−0.975077 + 0.221867i \(0.928785\pi\)
\(252\) 1.00000 0.0629941
\(253\) 7.94300 0.499372
\(254\) 8.46378 0.531065
\(255\) −3.87640 −0.242750
\(256\) 1.00000 0.0625000
\(257\) 27.6822 1.72677 0.863385 0.504546i \(-0.168340\pi\)
0.863385 + 0.504546i \(0.168340\pi\)
\(258\) −2.63128 −0.163816
\(259\) 7.92429 0.492391
\(260\) 0 0
\(261\) −1.05926 −0.0655663
\(262\) −5.07860 −0.313757
\(263\) 21.8601 1.34795 0.673975 0.738754i \(-0.264586\pi\)
0.673975 + 0.738754i \(0.264586\pi\)
\(264\) 1.55496 0.0957011
\(265\) −22.0524 −1.35467
\(266\) −4.73805 −0.290509
\(267\) 5.76399 0.352751
\(268\) −8.48684 −0.518416
\(269\) −17.0766 −1.04118 −0.520590 0.853807i \(-0.674288\pi\)
−0.520590 + 0.853807i \(0.674288\pi\)
\(270\) 3.10863 0.189185
\(271\) 27.8718 1.69309 0.846547 0.532315i \(-0.178678\pi\)
0.846547 + 0.532315i \(0.178678\pi\)
\(272\) −1.24698 −0.0756092
\(273\) 0 0
\(274\) 20.8764 1.26119
\(275\) 7.25169 0.437293
\(276\) 5.10817 0.307476
\(277\) −31.9253 −1.91820 −0.959102 0.283061i \(-0.908650\pi\)
−0.959102 + 0.283061i \(0.908650\pi\)
\(278\) 17.7255 1.06311
\(279\) −2.46553 −0.147607
\(280\) 3.10863 0.185776
\(281\) −11.3692 −0.678229 −0.339114 0.940745i \(-0.610127\pi\)
−0.339114 + 0.940745i \(0.610127\pi\)
\(282\) 5.06412 0.301564
\(283\) −16.6264 −0.988334 −0.494167 0.869367i \(-0.664527\pi\)
−0.494167 + 0.869367i \(0.664527\pi\)
\(284\) −5.31350 −0.315298
\(285\) −14.7289 −0.872462
\(286\) 0 0
\(287\) 2.71654 0.160352
\(288\) 1.00000 0.0589256
\(289\) −15.4450 −0.908532
\(290\) −3.29284 −0.193362
\(291\) 10.8007 0.633147
\(292\) 9.45056 0.553052
\(293\) −5.41358 −0.316265 −0.158132 0.987418i \(-0.550547\pi\)
−0.158132 + 0.987418i \(0.550547\pi\)
\(294\) 1.00000 0.0583212
\(295\) −10.6275 −0.618757
\(296\) 7.92429 0.460590
\(297\) 1.55496 0.0902278
\(298\) 0.829877 0.0480735
\(299\) 0 0
\(300\) 4.66359 0.269252
\(301\) −2.63128 −0.151664
\(302\) −3.39170 −0.195170
\(303\) 8.82958 0.507246
\(304\) −4.73805 −0.271746
\(305\) 41.2869 2.36408
\(306\) −1.24698 −0.0712851
\(307\) 6.53538 0.372994 0.186497 0.982456i \(-0.440287\pi\)
0.186497 + 0.982456i \(0.440287\pi\)
\(308\) 1.55496 0.0886020
\(309\) 1.89793 0.107970
\(310\) −7.66442 −0.435310
\(311\) −27.7921 −1.57595 −0.787973 0.615710i \(-0.788869\pi\)
−0.787973 + 0.615710i \(0.788869\pi\)
\(312\) 0 0
\(313\) −4.35209 −0.245995 −0.122997 0.992407i \(-0.539251\pi\)
−0.122997 + 0.992407i \(0.539251\pi\)
\(314\) −23.7381 −1.33962
\(315\) 3.10863 0.175152
\(316\) 11.1942 0.629725
\(317\) 12.5835 0.706761 0.353381 0.935480i \(-0.385032\pi\)
0.353381 + 0.935480i \(0.385032\pi\)
\(318\) −7.09392 −0.397807
\(319\) −1.64710 −0.0922198
\(320\) 3.10863 0.173778
\(321\) −3.58727 −0.200222
\(322\) 5.10817 0.284667
\(323\) 5.90825 0.328744
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −6.35482 −0.351961
\(327\) −20.3167 −1.12352
\(328\) 2.71654 0.149996
\(329\) 5.06412 0.279194
\(330\) 4.83379 0.266091
\(331\) 19.8866 1.09307 0.546533 0.837437i \(-0.315947\pi\)
0.546533 + 0.837437i \(0.315947\pi\)
\(332\) −4.40061 −0.241515
\(333\) 7.92429 0.434248
\(334\) −7.50114 −0.410444
\(335\) −26.3825 −1.44143
\(336\) 1.00000 0.0545545
\(337\) 0.500733 0.0272767 0.0136383 0.999907i \(-0.495659\pi\)
0.0136383 + 0.999907i \(0.495659\pi\)
\(338\) 0 0
\(339\) −6.85060 −0.372074
\(340\) −3.87640 −0.210227
\(341\) −3.83379 −0.207611
\(342\) −4.73805 −0.256205
\(343\) 1.00000 0.0539949
\(344\) −2.63128 −0.141869
\(345\) 15.8794 0.854920
\(346\) −16.1881 −0.870276
\(347\) −24.7464 −1.32846 −0.664229 0.747530i \(-0.731240\pi\)
−0.664229 + 0.747530i \(0.731240\pi\)
\(348\) −1.05926 −0.0567821
\(349\) −4.30578 −0.230483 −0.115241 0.993338i \(-0.536764\pi\)
−0.115241 + 0.993338i \(0.536764\pi\)
\(350\) 4.66359 0.249279
\(351\) 0 0
\(352\) 1.55496 0.0828795
\(353\) −4.18501 −0.222746 −0.111373 0.993779i \(-0.535525\pi\)
−0.111373 + 0.993779i \(0.535525\pi\)
\(354\) −3.41870 −0.181702
\(355\) −16.5177 −0.876668
\(356\) 5.76399 0.305491
\(357\) −1.24698 −0.0659972
\(358\) 1.95421 0.103283
\(359\) −6.33135 −0.334156 −0.167078 0.985944i \(-0.553433\pi\)
−0.167078 + 0.985944i \(0.553433\pi\)
\(360\) 3.10863 0.163839
\(361\) 3.44914 0.181533
\(362\) −1.81969 −0.0956406
\(363\) −8.58211 −0.450444
\(364\) 0 0
\(365\) 29.3783 1.53773
\(366\) 13.2814 0.694229
\(367\) 19.8935 1.03843 0.519216 0.854643i \(-0.326224\pi\)
0.519216 + 0.854643i \(0.326224\pi\)
\(368\) 5.10817 0.266282
\(369\) 2.71654 0.141417
\(370\) 24.6337 1.28064
\(371\) −7.09392 −0.368298
\(372\) −2.46553 −0.127832
\(373\) −35.9969 −1.86385 −0.931923 0.362655i \(-0.881870\pi\)
−0.931923 + 0.362655i \(0.881870\pi\)
\(374\) −1.93900 −0.100263
\(375\) −1.04578 −0.0540036
\(376\) 5.06412 0.261162
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) 29.5252 1.51661 0.758304 0.651901i \(-0.226028\pi\)
0.758304 + 0.651901i \(0.226028\pi\)
\(380\) −14.7289 −0.755574
\(381\) 8.46378 0.433613
\(382\) −23.3210 −1.19320
\(383\) −0.770083 −0.0393494 −0.0196747 0.999806i \(-0.506263\pi\)
−0.0196747 + 0.999806i \(0.506263\pi\)
\(384\) 1.00000 0.0510310
\(385\) 4.83379 0.246353
\(386\) 7.99277 0.406821
\(387\) −2.63128 −0.133755
\(388\) 10.8007 0.548322
\(389\) 18.6621 0.946208 0.473104 0.881006i \(-0.343133\pi\)
0.473104 + 0.881006i \(0.343133\pi\)
\(390\) 0 0
\(391\) −6.36979 −0.322134
\(392\) 1.00000 0.0505076
\(393\) −5.07860 −0.256181
\(394\) −2.24858 −0.113282
\(395\) 34.7988 1.75092
\(396\) 1.55496 0.0781396
\(397\) 17.7403 0.890360 0.445180 0.895441i \(-0.353140\pi\)
0.445180 + 0.895441i \(0.353140\pi\)
\(398\) −25.4768 −1.27704
\(399\) −4.73805 −0.237199
\(400\) 4.66359 0.233179
\(401\) 18.1461 0.906172 0.453086 0.891467i \(-0.350323\pi\)
0.453086 + 0.891467i \(0.350323\pi\)
\(402\) −8.48684 −0.423285
\(403\) 0 0
\(404\) 8.82958 0.439288
\(405\) 3.10863 0.154469
\(406\) −1.05926 −0.0525700
\(407\) 12.3219 0.610776
\(408\) −1.24698 −0.0617347
\(409\) 0.936649 0.0463143 0.0231572 0.999732i \(-0.492628\pi\)
0.0231572 + 0.999732i \(0.492628\pi\)
\(410\) 8.44471 0.417054
\(411\) 20.8764 1.02976
\(412\) 1.89793 0.0935044
\(413\) −3.41870 −0.168223
\(414\) 5.10817 0.251053
\(415\) −13.6799 −0.671519
\(416\) 0 0
\(417\) 17.7255 0.868022
\(418\) −7.36747 −0.360355
\(419\) −10.0234 −0.489675 −0.244838 0.969564i \(-0.578735\pi\)
−0.244838 + 0.969564i \(0.578735\pi\)
\(420\) 3.10863 0.151686
\(421\) −19.7171 −0.960951 −0.480475 0.877008i \(-0.659536\pi\)
−0.480475 + 0.877008i \(0.659536\pi\)
\(422\) 2.01914 0.0982902
\(423\) 5.06412 0.246226
\(424\) −7.09392 −0.344511
\(425\) −5.81540 −0.282088
\(426\) −5.31350 −0.257440
\(427\) 13.2814 0.642731
\(428\) −3.58727 −0.173397
\(429\) 0 0
\(430\) −8.17967 −0.394459
\(431\) −31.5252 −1.51852 −0.759259 0.650789i \(-0.774438\pi\)
−0.759259 + 0.650789i \(0.774438\pi\)
\(432\) 1.00000 0.0481125
\(433\) 19.6284 0.943281 0.471641 0.881791i \(-0.343662\pi\)
0.471641 + 0.881791i \(0.343662\pi\)
\(434\) −2.46553 −0.118349
\(435\) −3.29284 −0.157879
\(436\) −20.3167 −0.972995
\(437\) −24.2028 −1.15778
\(438\) 9.45056 0.451565
\(439\) 34.8525 1.66342 0.831708 0.555213i \(-0.187363\pi\)
0.831708 + 0.555213i \(0.187363\pi\)
\(440\) 4.83379 0.230442
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −9.65305 −0.458630 −0.229315 0.973352i \(-0.573649\pi\)
−0.229315 + 0.973352i \(0.573649\pi\)
\(444\) 7.92429 0.376070
\(445\) 17.9181 0.849401
\(446\) −13.1491 −0.622630
\(447\) 0.829877 0.0392518
\(448\) 1.00000 0.0472456
\(449\) 15.2830 0.721249 0.360625 0.932711i \(-0.382564\pi\)
0.360625 + 0.932711i \(0.382564\pi\)
\(450\) 4.66359 0.219844
\(451\) 4.22410 0.198905
\(452\) −6.85060 −0.322225
\(453\) −3.39170 −0.159356
\(454\) 1.90722 0.0895105
\(455\) 0 0
\(456\) −4.73805 −0.221880
\(457\) 16.0318 0.749937 0.374968 0.927038i \(-0.377654\pi\)
0.374968 + 0.927038i \(0.377654\pi\)
\(458\) −1.32344 −0.0618403
\(459\) −1.24698 −0.0582040
\(460\) 15.8794 0.740382
\(461\) 28.6119 1.33259 0.666294 0.745689i \(-0.267879\pi\)
0.666294 + 0.745689i \(0.267879\pi\)
\(462\) 1.55496 0.0723432
\(463\) −35.8620 −1.66665 −0.833325 0.552783i \(-0.813566\pi\)
−0.833325 + 0.552783i \(0.813566\pi\)
\(464\) −1.05926 −0.0491747
\(465\) −7.66442 −0.355429
\(466\) −23.7561 −1.10048
\(467\) −14.9263 −0.690708 −0.345354 0.938473i \(-0.612241\pi\)
−0.345354 + 0.938473i \(0.612241\pi\)
\(468\) 0 0
\(469\) −8.48684 −0.391886
\(470\) 15.7425 0.726147
\(471\) −23.7381 −1.09379
\(472\) −3.41870 −0.157359
\(473\) −4.09153 −0.188129
\(474\) 11.1942 0.514169
\(475\) −22.0963 −1.01385
\(476\) −1.24698 −0.0571552
\(477\) −7.09392 −0.324808
\(478\) 27.3308 1.25008
\(479\) −14.4348 −0.659541 −0.329771 0.944061i \(-0.606971\pi\)
−0.329771 + 0.944061i \(0.606971\pi\)
\(480\) 3.10863 0.141889
\(481\) 0 0
\(482\) 2.68328 0.122220
\(483\) 5.10817 0.232430
\(484\) −8.58211 −0.390096
\(485\) 33.5754 1.52458
\(486\) 1.00000 0.0453609
\(487\) −24.8740 −1.12715 −0.563575 0.826065i \(-0.690575\pi\)
−0.563575 + 0.826065i \(0.690575\pi\)
\(488\) 13.2814 0.601220
\(489\) −6.35482 −0.287375
\(490\) 3.10863 0.140434
\(491\) −10.6142 −0.479014 −0.239507 0.970895i \(-0.576986\pi\)
−0.239507 + 0.970895i \(0.576986\pi\)
\(492\) 2.71654 0.122471
\(493\) 1.32087 0.0594890
\(494\) 0 0
\(495\) 4.83379 0.217263
\(496\) −2.46553 −0.110705
\(497\) −5.31350 −0.238343
\(498\) −4.40061 −0.197196
\(499\) 0.950418 0.0425465 0.0212733 0.999774i \(-0.493228\pi\)
0.0212733 + 0.999774i \(0.493228\pi\)
\(500\) −1.04578 −0.0467685
\(501\) −7.50114 −0.335126
\(502\) −30.8962 −1.37897
\(503\) 34.9853 1.55992 0.779958 0.625832i \(-0.215240\pi\)
0.779958 + 0.625832i \(0.215240\pi\)
\(504\) 1.00000 0.0445435
\(505\) 27.4479 1.22142
\(506\) 7.94300 0.353109
\(507\) 0 0
\(508\) 8.46378 0.375520
\(509\) 26.0425 1.15431 0.577157 0.816634i \(-0.304162\pi\)
0.577157 + 0.816634i \(0.304162\pi\)
\(510\) −3.87640 −0.171650
\(511\) 9.45056 0.418068
\(512\) 1.00000 0.0441942
\(513\) −4.73805 −0.209190
\(514\) 27.6822 1.22101
\(515\) 5.89997 0.259984
\(516\) −2.63128 −0.115836
\(517\) 7.87450 0.346320
\(518\) 7.92429 0.348173
\(519\) −16.1881 −0.710578
\(520\) 0 0
\(521\) 39.0890 1.71252 0.856261 0.516544i \(-0.172782\pi\)
0.856261 + 0.516544i \(0.172782\pi\)
\(522\) −1.05926 −0.0463624
\(523\) −6.41000 −0.280290 −0.140145 0.990131i \(-0.544757\pi\)
−0.140145 + 0.990131i \(0.544757\pi\)
\(524\) −5.07860 −0.221860
\(525\) 4.66359 0.203536
\(526\) 21.8601 0.953144
\(527\) 3.07446 0.133926
\(528\) 1.55496 0.0676709
\(529\) 3.09343 0.134497
\(530\) −22.0524 −0.957894
\(531\) −3.41870 −0.148359
\(532\) −4.73805 −0.205421
\(533\) 0 0
\(534\) 5.76399 0.249432
\(535\) −11.1515 −0.482122
\(536\) −8.48684 −0.366576
\(537\) 1.95421 0.0843303
\(538\) −17.0766 −0.736225
\(539\) 1.55496 0.0669768
\(540\) 3.10863 0.133774
\(541\) 20.7891 0.893792 0.446896 0.894586i \(-0.352529\pi\)
0.446896 + 0.894586i \(0.352529\pi\)
\(542\) 27.8718 1.19720
\(543\) −1.81969 −0.0780902
\(544\) −1.24698 −0.0534638
\(545\) −63.1572 −2.70536
\(546\) 0 0
\(547\) 12.9402 0.553284 0.276642 0.960973i \(-0.410778\pi\)
0.276642 + 0.960973i \(0.410778\pi\)
\(548\) 20.8764 0.891795
\(549\) 13.2814 0.566836
\(550\) 7.25169 0.309213
\(551\) 5.01881 0.213808
\(552\) 5.10817 0.217418
\(553\) 11.1942 0.476028
\(554\) −31.9253 −1.35637
\(555\) 24.6337 1.04564
\(556\) 17.7255 0.751729
\(557\) −24.0823 −1.02040 −0.510200 0.860056i \(-0.670429\pi\)
−0.510200 + 0.860056i \(0.670429\pi\)
\(558\) −2.46553 −0.104374
\(559\) 0 0
\(560\) 3.10863 0.131364
\(561\) −1.93900 −0.0818647
\(562\) −11.3692 −0.479580
\(563\) −27.0146 −1.13853 −0.569266 0.822154i \(-0.692772\pi\)
−0.569266 + 0.822154i \(0.692772\pi\)
\(564\) 5.06412 0.213238
\(565\) −21.2960 −0.895929
\(566\) −16.6264 −0.698858
\(567\) 1.00000 0.0419961
\(568\) −5.31350 −0.222949
\(569\) −17.1952 −0.720859 −0.360429 0.932787i \(-0.617370\pi\)
−0.360429 + 0.932787i \(0.617370\pi\)
\(570\) −14.7289 −0.616924
\(571\) −21.5316 −0.901071 −0.450536 0.892758i \(-0.648767\pi\)
−0.450536 + 0.892758i \(0.648767\pi\)
\(572\) 0 0
\(573\) −23.3210 −0.974247
\(574\) 2.71654 0.113386
\(575\) 23.8224 0.993464
\(576\) 1.00000 0.0416667
\(577\) −30.2104 −1.25767 −0.628837 0.777537i \(-0.716469\pi\)
−0.628837 + 0.777537i \(0.716469\pi\)
\(578\) −15.4450 −0.642429
\(579\) 7.99277 0.332168
\(580\) −3.29284 −0.136728
\(581\) −4.40061 −0.182568
\(582\) 10.8007 0.447703
\(583\) −11.0307 −0.456847
\(584\) 9.45056 0.391067
\(585\) 0 0
\(586\) −5.41358 −0.223633
\(587\) −39.7362 −1.64009 −0.820045 0.572299i \(-0.806052\pi\)
−0.820045 + 0.572299i \(0.806052\pi\)
\(588\) 1.00000 0.0412393
\(589\) 11.6818 0.481340
\(590\) −10.6275 −0.437527
\(591\) −2.24858 −0.0924943
\(592\) 7.92429 0.325686
\(593\) 37.1150 1.52413 0.762066 0.647499i \(-0.224185\pi\)
0.762066 + 0.647499i \(0.224185\pi\)
\(594\) 1.55496 0.0638007
\(595\) −3.87640 −0.158917
\(596\) 0.829877 0.0339931
\(597\) −25.4768 −1.04270
\(598\) 0 0
\(599\) 10.5957 0.432928 0.216464 0.976291i \(-0.430548\pi\)
0.216464 + 0.976291i \(0.430548\pi\)
\(600\) 4.66359 0.190390
\(601\) 0.577212 0.0235450 0.0117725 0.999931i \(-0.496253\pi\)
0.0117725 + 0.999931i \(0.496253\pi\)
\(602\) −2.63128 −0.107243
\(603\) −8.48684 −0.345611
\(604\) −3.39170 −0.138006
\(605\) −26.6786 −1.08464
\(606\) 8.82958 0.358677
\(607\) 3.75798 0.152532 0.0762659 0.997088i \(-0.475700\pi\)
0.0762659 + 0.997088i \(0.475700\pi\)
\(608\) −4.73805 −0.192153
\(609\) −1.05926 −0.0429232
\(610\) 41.2869 1.67166
\(611\) 0 0
\(612\) −1.24698 −0.0504062
\(613\) 8.92496 0.360476 0.180238 0.983623i \(-0.442313\pi\)
0.180238 + 0.983623i \(0.442313\pi\)
\(614\) 6.53538 0.263746
\(615\) 8.44471 0.340524
\(616\) 1.55496 0.0626510
\(617\) −8.90864 −0.358648 −0.179324 0.983790i \(-0.557391\pi\)
−0.179324 + 0.983790i \(0.557391\pi\)
\(618\) 1.89793 0.0763460
\(619\) −8.63105 −0.346911 −0.173456 0.984842i \(-0.555493\pi\)
−0.173456 + 0.984842i \(0.555493\pi\)
\(620\) −7.66442 −0.307810
\(621\) 5.10817 0.204984
\(622\) −27.7921 −1.11436
\(623\) 5.76399 0.230930
\(624\) 0 0
\(625\) −26.5689 −1.06276
\(626\) −4.35209 −0.173945
\(627\) −7.36747 −0.294229
\(628\) −23.7381 −0.947251
\(629\) −9.88142 −0.393998
\(630\) 3.10863 0.123851
\(631\) −22.0935 −0.879528 −0.439764 0.898113i \(-0.644938\pi\)
−0.439764 + 0.898113i \(0.644938\pi\)
\(632\) 11.1942 0.445283
\(633\) 2.01914 0.0802536
\(634\) 12.5835 0.499756
\(635\) 26.3108 1.04411
\(636\) −7.09392 −0.281292
\(637\) 0 0
\(638\) −1.64710 −0.0652092
\(639\) −5.31350 −0.210199
\(640\) 3.10863 0.122879
\(641\) 10.0871 0.398416 0.199208 0.979957i \(-0.436163\pi\)
0.199208 + 0.979957i \(0.436163\pi\)
\(642\) −3.58727 −0.141578
\(643\) 8.84091 0.348652 0.174326 0.984688i \(-0.444225\pi\)
0.174326 + 0.984688i \(0.444225\pi\)
\(644\) 5.10817 0.201290
\(645\) −8.17967 −0.322074
\(646\) 5.90825 0.232457
\(647\) 1.73856 0.0683497 0.0341748 0.999416i \(-0.489120\pi\)
0.0341748 + 0.999416i \(0.489120\pi\)
\(648\) 1.00000 0.0392837
\(649\) −5.31594 −0.208669
\(650\) 0 0
\(651\) −2.46553 −0.0966316
\(652\) −6.35482 −0.248874
\(653\) −45.8052 −1.79250 −0.896248 0.443554i \(-0.853717\pi\)
−0.896248 + 0.443554i \(0.853717\pi\)
\(654\) −20.3167 −0.794447
\(655\) −15.7875 −0.616868
\(656\) 2.71654 0.106063
\(657\) 9.45056 0.368701
\(658\) 5.06412 0.197420
\(659\) 11.7599 0.458099 0.229050 0.973415i \(-0.426438\pi\)
0.229050 + 0.973415i \(0.426438\pi\)
\(660\) 4.83379 0.188155
\(661\) −13.5257 −0.526089 −0.263044 0.964784i \(-0.584727\pi\)
−0.263044 + 0.964784i \(0.584727\pi\)
\(662\) 19.8866 0.772915
\(663\) 0 0
\(664\) −4.40061 −0.170777
\(665\) −14.7289 −0.571161
\(666\) 7.92429 0.307060
\(667\) −5.41086 −0.209509
\(668\) −7.50114 −0.290228
\(669\) −13.1491 −0.508375
\(670\) −26.3825 −1.01924
\(671\) 20.6520 0.797262
\(672\) 1.00000 0.0385758
\(673\) 12.0762 0.465505 0.232752 0.972536i \(-0.425227\pi\)
0.232752 + 0.972536i \(0.425227\pi\)
\(674\) 0.500733 0.0192875
\(675\) 4.66359 0.179502
\(676\) 0 0
\(677\) 4.22192 0.162262 0.0811308 0.996703i \(-0.474147\pi\)
0.0811308 + 0.996703i \(0.474147\pi\)
\(678\) −6.85060 −0.263096
\(679\) 10.8007 0.414492
\(680\) −3.87640 −0.148653
\(681\) 1.90722 0.0730850
\(682\) −3.83379 −0.146803
\(683\) 46.8330 1.79201 0.896007 0.444039i \(-0.146455\pi\)
0.896007 + 0.444039i \(0.146455\pi\)
\(684\) −4.73805 −0.181164
\(685\) 64.8970 2.47959
\(686\) 1.00000 0.0381802
\(687\) −1.32344 −0.0504924
\(688\) −2.63128 −0.100316
\(689\) 0 0
\(690\) 15.8794 0.604519
\(691\) −19.2883 −0.733762 −0.366881 0.930268i \(-0.619574\pi\)
−0.366881 + 0.930268i \(0.619574\pi\)
\(692\) −16.1881 −0.615378
\(693\) 1.55496 0.0590680
\(694\) −24.7464 −0.939361
\(695\) 55.1021 2.09014
\(696\) −1.05926 −0.0401510
\(697\) −3.38747 −0.128309
\(698\) −4.30578 −0.162976
\(699\) −23.7561 −0.898539
\(700\) 4.66359 0.176267
\(701\) −16.9306 −0.639459 −0.319729 0.947509i \(-0.603592\pi\)
−0.319729 + 0.947509i \(0.603592\pi\)
\(702\) 0 0
\(703\) −37.5457 −1.41606
\(704\) 1.55496 0.0586047
\(705\) 15.7425 0.592897
\(706\) −4.18501 −0.157505
\(707\) 8.82958 0.332070
\(708\) −3.41870 −0.128483
\(709\) −7.93895 −0.298153 −0.149077 0.988826i \(-0.547630\pi\)
−0.149077 + 0.988826i \(0.547630\pi\)
\(710\) −16.5177 −0.619898
\(711\) 11.1942 0.419817
\(712\) 5.76399 0.216015
\(713\) −12.5943 −0.471662
\(714\) −1.24698 −0.0466670
\(715\) 0 0
\(716\) 1.95421 0.0730322
\(717\) 27.3308 1.02069
\(718\) −6.33135 −0.236284
\(719\) 31.5702 1.17737 0.588684 0.808363i \(-0.299646\pi\)
0.588684 + 0.808363i \(0.299646\pi\)
\(720\) 3.10863 0.115852
\(721\) 1.89793 0.0706827
\(722\) 3.44914 0.128364
\(723\) 2.68328 0.0997924
\(724\) −1.81969 −0.0676281
\(725\) −4.93993 −0.183465
\(726\) −8.58211 −0.318512
\(727\) 10.8589 0.402733 0.201367 0.979516i \(-0.435462\pi\)
0.201367 + 0.979516i \(0.435462\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 29.3783 1.08734
\(731\) 3.28115 0.121358
\(732\) 13.2814 0.490894
\(733\) 2.22682 0.0822495 0.0411247 0.999154i \(-0.486906\pi\)
0.0411247 + 0.999154i \(0.486906\pi\)
\(734\) 19.8935 0.734282
\(735\) 3.10863 0.114664
\(736\) 5.10817 0.188290
\(737\) −13.1967 −0.486106
\(738\) 2.71654 0.0999971
\(739\) 39.5399 1.45450 0.727249 0.686374i \(-0.240799\pi\)
0.727249 + 0.686374i \(0.240799\pi\)
\(740\) 24.6337 0.905552
\(741\) 0 0
\(742\) −7.09392 −0.260426
\(743\) −35.0085 −1.28434 −0.642169 0.766563i \(-0.721965\pi\)
−0.642169 + 0.766563i \(0.721965\pi\)
\(744\) −2.46553 −0.0903906
\(745\) 2.57978 0.0945159
\(746\) −35.9969 −1.31794
\(747\) −4.40061 −0.161010
\(748\) −1.93900 −0.0708969
\(749\) −3.58727 −0.131076
\(750\) −1.04578 −0.0381863
\(751\) 3.08816 0.112689 0.0563443 0.998411i \(-0.482056\pi\)
0.0563443 + 0.998411i \(0.482056\pi\)
\(752\) 5.06412 0.184670
\(753\) −30.8962 −1.12592
\(754\) 0 0
\(755\) −10.5435 −0.383719
\(756\) 1.00000 0.0363696
\(757\) −31.1069 −1.13060 −0.565300 0.824885i \(-0.691240\pi\)
−0.565300 + 0.824885i \(0.691240\pi\)
\(758\) 29.5252 1.07240
\(759\) 7.94300 0.288312
\(760\) −14.7289 −0.534272
\(761\) 34.3881 1.24657 0.623284 0.781996i \(-0.285798\pi\)
0.623284 + 0.781996i \(0.285798\pi\)
\(762\) 8.46378 0.306611
\(763\) −20.3167 −0.735515
\(764\) −23.3210 −0.843723
\(765\) −3.87640 −0.140152
\(766\) −0.770083 −0.0278242
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 28.7352 1.03622 0.518109 0.855314i \(-0.326636\pi\)
0.518109 + 0.855314i \(0.326636\pi\)
\(770\) 4.83379 0.174198
\(771\) 27.6822 0.996951
\(772\) 7.99277 0.287666
\(773\) −43.0766 −1.54936 −0.774679 0.632354i \(-0.782089\pi\)
−0.774679 + 0.632354i \(0.782089\pi\)
\(774\) −2.63128 −0.0945793
\(775\) −11.4982 −0.413028
\(776\) 10.8007 0.387722
\(777\) 7.92429 0.284282
\(778\) 18.6621 0.669070
\(779\) −12.8711 −0.461155
\(780\) 0 0
\(781\) −8.26226 −0.295647
\(782\) −6.36979 −0.227783
\(783\) −1.05926 −0.0378547
\(784\) 1.00000 0.0357143
\(785\) −73.7929 −2.63378
\(786\) −5.07860 −0.181148
\(787\) −12.5432 −0.447117 −0.223558 0.974691i \(-0.571767\pi\)
−0.223558 + 0.974691i \(0.571767\pi\)
\(788\) −2.24858 −0.0801024
\(789\) 21.8601 0.778239
\(790\) 34.7988 1.23808
\(791\) −6.85060 −0.243579
\(792\) 1.55496 0.0552530
\(793\) 0 0
\(794\) 17.7403 0.629580
\(795\) −22.0524 −0.782117
\(796\) −25.4768 −0.903001
\(797\) 35.9473 1.27332 0.636659 0.771146i \(-0.280316\pi\)
0.636659 + 0.771146i \(0.280316\pi\)
\(798\) −4.73805 −0.167725
\(799\) −6.31486 −0.223404
\(800\) 4.66359 0.164883
\(801\) 5.76399 0.203661
\(802\) 18.1461 0.640761
\(803\) 14.6952 0.518583
\(804\) −8.48684 −0.299308
\(805\) 15.8794 0.559676
\(806\) 0 0
\(807\) −17.0766 −0.601125
\(808\) 8.82958 0.310623
\(809\) 38.3283 1.34755 0.673776 0.738936i \(-0.264671\pi\)
0.673776 + 0.738936i \(0.264671\pi\)
\(810\) 3.10863 0.109226
\(811\) −55.4449 −1.94693 −0.973466 0.228831i \(-0.926510\pi\)
−0.973466 + 0.228831i \(0.926510\pi\)
\(812\) −1.05926 −0.0371726
\(813\) 27.8718 0.977508
\(814\) 12.3219 0.431884
\(815\) −19.7548 −0.691980
\(816\) −1.24698 −0.0436530
\(817\) 12.4671 0.436170
\(818\) 0.936649 0.0327492
\(819\) 0 0
\(820\) 8.44471 0.294902
\(821\) 30.7208 1.07216 0.536082 0.844166i \(-0.319904\pi\)
0.536082 + 0.844166i \(0.319904\pi\)
\(822\) 20.8764 0.728148
\(823\) −29.4586 −1.02686 −0.513431 0.858131i \(-0.671626\pi\)
−0.513431 + 0.858131i \(0.671626\pi\)
\(824\) 1.89793 0.0661176
\(825\) 7.25169 0.252471
\(826\) −3.41870 −0.118952
\(827\) 39.5955 1.37687 0.688435 0.725298i \(-0.258298\pi\)
0.688435 + 0.725298i \(0.258298\pi\)
\(828\) 5.10817 0.177521
\(829\) 52.6258 1.82777 0.913884 0.405975i \(-0.133068\pi\)
0.913884 + 0.405975i \(0.133068\pi\)
\(830\) −13.6799 −0.474836
\(831\) −31.9253 −1.10748
\(832\) 0 0
\(833\) −1.24698 −0.0432053
\(834\) 17.7255 0.613784
\(835\) −23.3183 −0.806962
\(836\) −7.36747 −0.254809
\(837\) −2.46553 −0.0852211
\(838\) −10.0234 −0.346253
\(839\) −31.0141 −1.07073 −0.535363 0.844622i \(-0.679825\pi\)
−0.535363 + 0.844622i \(0.679825\pi\)
\(840\) 3.10863 0.107258
\(841\) −27.8780 −0.961310
\(842\) −19.7171 −0.679495
\(843\) −11.3692 −0.391576
\(844\) 2.01914 0.0695016
\(845\) 0 0
\(846\) 5.06412 0.174108
\(847\) −8.58211 −0.294885
\(848\) −7.09392 −0.243606
\(849\) −16.6264 −0.570615
\(850\) −5.81540 −0.199467
\(851\) 40.4786 1.38759
\(852\) −5.31350 −0.182037
\(853\) 55.8464 1.91214 0.956072 0.293133i \(-0.0946979\pi\)
0.956072 + 0.293133i \(0.0946979\pi\)
\(854\) 13.2814 0.454480
\(855\) −14.7289 −0.503716
\(856\) −3.58727 −0.122610
\(857\) 26.7950 0.915299 0.457650 0.889133i \(-0.348691\pi\)
0.457650 + 0.889133i \(0.348691\pi\)
\(858\) 0 0
\(859\) 13.5325 0.461724 0.230862 0.972986i \(-0.425845\pi\)
0.230862 + 0.972986i \(0.425845\pi\)
\(860\) −8.17967 −0.278924
\(861\) 2.71654 0.0925793
\(862\) −31.5252 −1.07375
\(863\) 34.8946 1.18783 0.593914 0.804529i \(-0.297582\pi\)
0.593914 + 0.804529i \(0.297582\pi\)
\(864\) 1.00000 0.0340207
\(865\) −50.3228 −1.71103
\(866\) 19.6284 0.667000
\(867\) −15.4450 −0.524541
\(868\) −2.46553 −0.0836855
\(869\) 17.4066 0.590478
\(870\) −3.29284 −0.111638
\(871\) 0 0
\(872\) −20.3167 −0.688011
\(873\) 10.8007 0.365548
\(874\) −24.2028 −0.818672
\(875\) −1.04578 −0.0353537
\(876\) 9.45056 0.319305
\(877\) 51.7834 1.74860 0.874300 0.485386i \(-0.161321\pi\)
0.874300 + 0.485386i \(0.161321\pi\)
\(878\) 34.8525 1.17621
\(879\) −5.41358 −0.182596
\(880\) 4.83379 0.162947
\(881\) −2.60278 −0.0876900 −0.0438450 0.999038i \(-0.513961\pi\)
−0.0438450 + 0.999038i \(0.513961\pi\)
\(882\) 1.00000 0.0336718
\(883\) −38.5127 −1.29605 −0.648027 0.761617i \(-0.724406\pi\)
−0.648027 + 0.761617i \(0.724406\pi\)
\(884\) 0 0
\(885\) −10.6275 −0.357239
\(886\) −9.65305 −0.324300
\(887\) 15.6703 0.526156 0.263078 0.964775i \(-0.415262\pi\)
0.263078 + 0.964775i \(0.415262\pi\)
\(888\) 7.92429 0.265922
\(889\) 8.46378 0.283866
\(890\) 17.9181 0.600617
\(891\) 1.55496 0.0520931
\(892\) −13.1491 −0.440266
\(893\) −23.9941 −0.802931
\(894\) 0.829877 0.0277552
\(895\) 6.07491 0.203062
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 15.2830 0.510000
\(899\) 2.61162 0.0871025
\(900\) 4.66359 0.155453
\(901\) 8.84597 0.294702
\(902\) 4.22410 0.140647
\(903\) −2.63128 −0.0875634
\(904\) −6.85060 −0.227848
\(905\) −5.65673 −0.188036
\(906\) −3.39170 −0.112682
\(907\) −30.6745 −1.01853 −0.509265 0.860610i \(-0.670083\pi\)
−0.509265 + 0.860610i \(0.670083\pi\)
\(908\) 1.90722 0.0632935
\(909\) 8.82958 0.292859
\(910\) 0 0
\(911\) 32.8588 1.08866 0.544331 0.838870i \(-0.316784\pi\)
0.544331 + 0.838870i \(0.316784\pi\)
\(912\) −4.73805 −0.156893
\(913\) −6.84277 −0.226463
\(914\) 16.0318 0.530285
\(915\) 41.2869 1.36490
\(916\) −1.32344 −0.0437277
\(917\) −5.07860 −0.167710
\(918\) −1.24698 −0.0411565
\(919\) −10.6383 −0.350925 −0.175462 0.984486i \(-0.556142\pi\)
−0.175462 + 0.984486i \(0.556142\pi\)
\(920\) 15.8794 0.523529
\(921\) 6.53538 0.215348
\(922\) 28.6119 0.942282
\(923\) 0 0
\(924\) 1.55496 0.0511544
\(925\) 36.9556 1.21509
\(926\) −35.8620 −1.17850
\(927\) 1.89793 0.0623363
\(928\) −1.05926 −0.0347718
\(929\) 3.54100 0.116176 0.0580882 0.998311i \(-0.481500\pi\)
0.0580882 + 0.998311i \(0.481500\pi\)
\(930\) −7.66442 −0.251326
\(931\) −4.73805 −0.155283
\(932\) −23.7561 −0.778157
\(933\) −27.7921 −0.909873
\(934\) −14.9263 −0.488404
\(935\) −6.02764 −0.197125
\(936\) 0 0
\(937\) −15.5489 −0.507959 −0.253980 0.967210i \(-0.581740\pi\)
−0.253980 + 0.967210i \(0.581740\pi\)
\(938\) −8.48684 −0.277105
\(939\) −4.35209 −0.142025
\(940\) 15.7425 0.513464
\(941\) 34.3011 1.11818 0.559092 0.829106i \(-0.311150\pi\)
0.559092 + 0.829106i \(0.311150\pi\)
\(942\) −23.7381 −0.773427
\(943\) 13.8765 0.451882
\(944\) −3.41870 −0.111269
\(945\) 3.10863 0.101124
\(946\) −4.09153 −0.133027
\(947\) −45.7712 −1.48737 −0.743683 0.668533i \(-0.766923\pi\)
−0.743683 + 0.668533i \(0.766923\pi\)
\(948\) 11.1942 0.363572
\(949\) 0 0
\(950\) −22.0963 −0.716900
\(951\) 12.5835 0.408049
\(952\) −1.24698 −0.0404148
\(953\) 38.1938 1.23722 0.618609 0.785699i \(-0.287697\pi\)
0.618609 + 0.785699i \(0.287697\pi\)
\(954\) −7.09392 −0.229674
\(955\) −72.4963 −2.34592
\(956\) 27.3308 0.883943
\(957\) −1.64710 −0.0532431
\(958\) −14.4348 −0.466366
\(959\) 20.8764 0.674134
\(960\) 3.10863 0.100331
\(961\) −24.9212 −0.803909
\(962\) 0 0
\(963\) −3.58727 −0.115598
\(964\) 2.68328 0.0864228
\(965\) 24.8466 0.799839
\(966\) 5.10817 0.164353
\(967\) −1.46535 −0.0471223 −0.0235612 0.999722i \(-0.507500\pi\)
−0.0235612 + 0.999722i \(0.507500\pi\)
\(968\) −8.58211 −0.275839
\(969\) 5.90825 0.189800
\(970\) 33.5754 1.07804
\(971\) −15.1475 −0.486106 −0.243053 0.970013i \(-0.578149\pi\)
−0.243053 + 0.970013i \(0.578149\pi\)
\(972\) 1.00000 0.0320750
\(973\) 17.7255 0.568254
\(974\) −24.8740 −0.797016
\(975\) 0 0
\(976\) 13.2814 0.425127
\(977\) 30.6974 0.982098 0.491049 0.871132i \(-0.336614\pi\)
0.491049 + 0.871132i \(0.336614\pi\)
\(978\) −6.35482 −0.203205
\(979\) 8.96277 0.286451
\(980\) 3.10863 0.0993016
\(981\) −20.3167 −0.648663
\(982\) −10.6142 −0.338714
\(983\) 13.4997 0.430574 0.215287 0.976551i \(-0.430931\pi\)
0.215287 + 0.976551i \(0.430931\pi\)
\(984\) 2.71654 0.0866000
\(985\) −6.99001 −0.222720
\(986\) 1.32087 0.0420651
\(987\) 5.06412 0.161193
\(988\) 0 0
\(989\) −13.4410 −0.427400
\(990\) 4.83379 0.153628
\(991\) −39.7612 −1.26306 −0.631529 0.775352i \(-0.717572\pi\)
−0.631529 + 0.775352i \(0.717572\pi\)
\(992\) −2.46553 −0.0782806
\(993\) 19.8866 0.631082
\(994\) −5.31350 −0.168534
\(995\) −79.1980 −2.51075
\(996\) −4.40061 −0.139439
\(997\) 5.54069 0.175475 0.0877377 0.996144i \(-0.472036\pi\)
0.0877377 + 0.996144i \(0.472036\pi\)
\(998\) 0.950418 0.0300849
\(999\) 7.92429 0.250713
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 7098.2.a.ct.1.5 yes 6
13.12 even 2 7098.2.a.cr.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.cr.1.2 6 13.12 even 2
7098.2.a.ct.1.5 yes 6 1.1 even 1 trivial