Properties

Label 7098.2.a.bp.1.2
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.73205 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.73205 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.73205 q^{10} +1.73205 q^{11} +1.00000 q^{12} -1.00000 q^{14} +2.73205 q^{15} +1.00000 q^{16} +0.267949 q^{17} -1.00000 q^{18} -1.00000 q^{19} +2.73205 q^{20} +1.00000 q^{21} -1.73205 q^{22} +3.46410 q^{23} -1.00000 q^{24} +2.46410 q^{25} +1.00000 q^{27} +1.00000 q^{28} -0.464102 q^{29} -2.73205 q^{30} +8.19615 q^{31} -1.00000 q^{32} +1.73205 q^{33} -0.267949 q^{34} +2.73205 q^{35} +1.00000 q^{36} -3.26795 q^{37} +1.00000 q^{38} -2.73205 q^{40} +3.00000 q^{41} -1.00000 q^{42} +6.73205 q^{43} +1.73205 q^{44} +2.73205 q^{45} -3.46410 q^{46} -4.46410 q^{47} +1.00000 q^{48} +1.00000 q^{49} -2.46410 q^{50} +0.267949 q^{51} +7.00000 q^{53} -1.00000 q^{54} +4.73205 q^{55} -1.00000 q^{56} -1.00000 q^{57} +0.464102 q^{58} -12.9282 q^{59} +2.73205 q^{60} -5.19615 q^{61} -8.19615 q^{62} +1.00000 q^{63} +1.00000 q^{64} -1.73205 q^{66} -4.92820 q^{67} +0.267949 q^{68} +3.46410 q^{69} -2.73205 q^{70} -8.19615 q^{71} -1.00000 q^{72} +1.46410 q^{73} +3.26795 q^{74} +2.46410 q^{75} -1.00000 q^{76} +1.73205 q^{77} +15.9282 q^{79} +2.73205 q^{80} +1.00000 q^{81} -3.00000 q^{82} +10.1962 q^{83} +1.00000 q^{84} +0.732051 q^{85} -6.73205 q^{86} -0.464102 q^{87} -1.73205 q^{88} +3.53590 q^{89} -2.73205 q^{90} +3.46410 q^{92} +8.19615 q^{93} +4.46410 q^{94} -2.73205 q^{95} -1.00000 q^{96} +1.66025 q^{97} -1.00000 q^{98} +1.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} - 2q^{10} + 2q^{12} - 2q^{14} + 2q^{15} + 2q^{16} + 4q^{17} - 2q^{18} - 2q^{19} + 2q^{20} + 2q^{21} - 2q^{24} - 2q^{25} + 2q^{27} + 2q^{28} + 6q^{29} - 2q^{30} + 6q^{31} - 2q^{32} - 4q^{34} + 2q^{35} + 2q^{36} - 10q^{37} + 2q^{38} - 2q^{40} + 6q^{41} - 2q^{42} + 10q^{43} + 2q^{45} - 2q^{47} + 2q^{48} + 2q^{49} + 2q^{50} + 4q^{51} + 14q^{53} - 2q^{54} + 6q^{55} - 2q^{56} - 2q^{57} - 6q^{58} - 12q^{59} + 2q^{60} - 6q^{62} + 2q^{63} + 2q^{64} + 4q^{67} + 4q^{68} - 2q^{70} - 6q^{71} - 2q^{72} - 4q^{73} + 10q^{74} - 2q^{75} - 2q^{76} + 18q^{79} + 2q^{80} + 2q^{81} - 6q^{82} + 10q^{83} + 2q^{84} - 2q^{85} - 10q^{86} + 6q^{87} + 14q^{89} - 2q^{90} + 6q^{93} + 2q^{94} - 2q^{95} - 2q^{96} - 14q^{97} - 2q^{98} + O(q^{100}) \)

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
\(5\) 2.73205 1.22181 0.610905 0.791704i \(-0.290806\pi\)
0.610905 + 0.791704i \(0.290806\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.73205 −0.863950
\(11\) 1.73205 0.522233 0.261116 0.965307i \(-0.415909\pi\)
0.261116 + 0.965307i \(0.415909\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 2.73205 0.705412
\(16\) 1.00000 0.250000
\(17\) 0.267949 0.0649872 0.0324936 0.999472i \(-0.489655\pi\)
0.0324936 + 0.999472i \(0.489655\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 2.73205 0.610905
\(21\) 1.00000 0.218218
\(22\) −1.73205 −0.369274
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) −1.00000 −0.204124
\(25\) 2.46410 0.492820
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −0.464102 −0.0861815 −0.0430908 0.999071i \(-0.513720\pi\)
−0.0430908 + 0.999071i \(0.513720\pi\)
\(30\) −2.73205 −0.498802
\(31\) 8.19615 1.47207 0.736036 0.676942i \(-0.236695\pi\)
0.736036 + 0.676942i \(0.236695\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.73205 0.301511
\(34\) −0.267949 −0.0459529
\(35\) 2.73205 0.461801
\(36\) 1.00000 0.166667
\(37\) −3.26795 −0.537248 −0.268624 0.963245i \(-0.586569\pi\)
−0.268624 + 0.963245i \(0.586569\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −2.73205 −0.431975
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) −1.00000 −0.154303
\(43\) 6.73205 1.02663 0.513314 0.858201i \(-0.328418\pi\)
0.513314 + 0.858201i \(0.328418\pi\)
\(44\) 1.73205 0.261116
\(45\) 2.73205 0.407270
\(46\) −3.46410 −0.510754
\(47\) −4.46410 −0.651156 −0.325578 0.945515i \(-0.605559\pi\)
−0.325578 + 0.945515i \(0.605559\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −2.46410 −0.348477
\(51\) 0.267949 0.0375204
\(52\) 0 0
\(53\) 7.00000 0.961524 0.480762 0.876851i \(-0.340360\pi\)
0.480762 + 0.876851i \(0.340360\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.73205 0.638070
\(56\) −1.00000 −0.133631
\(57\) −1.00000 −0.132453
\(58\) 0.464102 0.0609395
\(59\) −12.9282 −1.68311 −0.841554 0.540172i \(-0.818359\pi\)
−0.841554 + 0.540172i \(0.818359\pi\)
\(60\) 2.73205 0.352706
\(61\) −5.19615 −0.665299 −0.332650 0.943051i \(-0.607943\pi\)
−0.332650 + 0.943051i \(0.607943\pi\)
\(62\) −8.19615 −1.04091
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.73205 −0.213201
\(67\) −4.92820 −0.602076 −0.301038 0.953612i \(-0.597333\pi\)
−0.301038 + 0.953612i \(0.597333\pi\)
\(68\) 0.267949 0.0324936
\(69\) 3.46410 0.417029
\(70\) −2.73205 −0.326543
\(71\) −8.19615 −0.972704 −0.486352 0.873763i \(-0.661673\pi\)
−0.486352 + 0.873763i \(0.661673\pi\)
\(72\) −1.00000 −0.117851
\(73\) 1.46410 0.171360 0.0856801 0.996323i \(-0.472694\pi\)
0.0856801 + 0.996323i \(0.472694\pi\)
\(74\) 3.26795 0.379891
\(75\) 2.46410 0.284530
\(76\) −1.00000 −0.114708
\(77\) 1.73205 0.197386
\(78\) 0 0
\(79\) 15.9282 1.79206 0.896031 0.443991i \(-0.146438\pi\)
0.896031 + 0.443991i \(0.146438\pi\)
\(80\) 2.73205 0.305453
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) 10.1962 1.11917 0.559587 0.828772i \(-0.310960\pi\)
0.559587 + 0.828772i \(0.310960\pi\)
\(84\) 1.00000 0.109109
\(85\) 0.732051 0.0794021
\(86\) −6.73205 −0.725936
\(87\) −0.464102 −0.0497569
\(88\) −1.73205 −0.184637
\(89\) 3.53590 0.374804 0.187402 0.982283i \(-0.439993\pi\)
0.187402 + 0.982283i \(0.439993\pi\)
\(90\) −2.73205 −0.287983
\(91\) 0 0
\(92\) 3.46410 0.361158
\(93\) 8.19615 0.849901
\(94\) 4.46410 0.460437
\(95\) −2.73205 −0.280302
\(96\) −1.00000 −0.102062
\(97\) 1.66025 0.168573 0.0842866 0.996442i \(-0.473139\pi\)
0.0842866 + 0.996442i \(0.473139\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.73205 0.174078
\(100\) 2.46410 0.246410
\(101\) 16.9282 1.68442 0.842210 0.539150i \(-0.181255\pi\)
0.842210 + 0.539150i \(0.181255\pi\)
\(102\) −0.267949 −0.0265309
\(103\) 11.2679 1.11026 0.555132 0.831762i \(-0.312668\pi\)
0.555132 + 0.831762i \(0.312668\pi\)
\(104\) 0 0
\(105\) 2.73205 0.266621
\(106\) −7.00000 −0.679900
\(107\) −16.8564 −1.62957 −0.814785 0.579763i \(-0.803145\pi\)
−0.814785 + 0.579763i \(0.803145\pi\)
\(108\) 1.00000 0.0962250
\(109\) −1.26795 −0.121448 −0.0607238 0.998155i \(-0.519341\pi\)
−0.0607238 + 0.998155i \(0.519341\pi\)
\(110\) −4.73205 −0.451183
\(111\) −3.26795 −0.310180
\(112\) 1.00000 0.0944911
\(113\) 18.7321 1.76216 0.881082 0.472964i \(-0.156816\pi\)
0.881082 + 0.472964i \(0.156816\pi\)
\(114\) 1.00000 0.0936586
\(115\) 9.46410 0.882532
\(116\) −0.464102 −0.0430908
\(117\) 0 0
\(118\) 12.9282 1.19014
\(119\) 0.267949 0.0245629
\(120\) −2.73205 −0.249401
\(121\) −8.00000 −0.727273
\(122\) 5.19615 0.470438
\(123\) 3.00000 0.270501
\(124\) 8.19615 0.736036
\(125\) −6.92820 −0.619677
\(126\) −1.00000 −0.0890871
\(127\) −8.53590 −0.757438 −0.378719 0.925512i \(-0.623635\pi\)
−0.378719 + 0.925512i \(0.623635\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.73205 0.592724
\(130\) 0 0
\(131\) 13.8564 1.21064 0.605320 0.795982i \(-0.293045\pi\)
0.605320 + 0.795982i \(0.293045\pi\)
\(132\) 1.73205 0.150756
\(133\) −1.00000 −0.0867110
\(134\) 4.92820 0.425732
\(135\) 2.73205 0.235137
\(136\) −0.267949 −0.0229765
\(137\) −6.39230 −0.546131 −0.273066 0.961995i \(-0.588038\pi\)
−0.273066 + 0.961995i \(0.588038\pi\)
\(138\) −3.46410 −0.294884
\(139\) −19.5885 −1.66147 −0.830736 0.556667i \(-0.812080\pi\)
−0.830736 + 0.556667i \(0.812080\pi\)
\(140\) 2.73205 0.230900
\(141\) −4.46410 −0.375945
\(142\) 8.19615 0.687806
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −1.26795 −0.105297
\(146\) −1.46410 −0.121170
\(147\) 1.00000 0.0824786
\(148\) −3.26795 −0.268624
\(149\) 13.8564 1.13516 0.567581 0.823318i \(-0.307880\pi\)
0.567581 + 0.823318i \(0.307880\pi\)
\(150\) −2.46410 −0.201193
\(151\) −5.19615 −0.422857 −0.211428 0.977393i \(-0.567812\pi\)
−0.211428 + 0.977393i \(0.567812\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0.267949 0.0216624
\(154\) −1.73205 −0.139573
\(155\) 22.3923 1.79859
\(156\) 0 0
\(157\) −7.46410 −0.595700 −0.297850 0.954613i \(-0.596270\pi\)
−0.297850 + 0.954613i \(0.596270\pi\)
\(158\) −15.9282 −1.26718
\(159\) 7.00000 0.555136
\(160\) −2.73205 −0.215988
\(161\) 3.46410 0.273009
\(162\) −1.00000 −0.0785674
\(163\) −0.732051 −0.0573386 −0.0286693 0.999589i \(-0.509127\pi\)
−0.0286693 + 0.999589i \(0.509127\pi\)
\(164\) 3.00000 0.234261
\(165\) 4.73205 0.368390
\(166\) −10.1962 −0.791375
\(167\) −13.8564 −1.07224 −0.536120 0.844141i \(-0.680111\pi\)
−0.536120 + 0.844141i \(0.680111\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) −0.732051 −0.0561457
\(171\) −1.00000 −0.0764719
\(172\) 6.73205 0.513314
\(173\) 15.1244 1.14988 0.574942 0.818194i \(-0.305025\pi\)
0.574942 + 0.818194i \(0.305025\pi\)
\(174\) 0.464102 0.0351835
\(175\) 2.46410 0.186269
\(176\) 1.73205 0.130558
\(177\) −12.9282 −0.971743
\(178\) −3.53590 −0.265027
\(179\) 4.53590 0.339029 0.169514 0.985528i \(-0.445780\pi\)
0.169514 + 0.985528i \(0.445780\pi\)
\(180\) 2.73205 0.203635
\(181\) 14.6603 1.08969 0.544844 0.838537i \(-0.316589\pi\)
0.544844 + 0.838537i \(0.316589\pi\)
\(182\) 0 0
\(183\) −5.19615 −0.384111
\(184\) −3.46410 −0.255377
\(185\) −8.92820 −0.656415
\(186\) −8.19615 −0.600971
\(187\) 0.464102 0.0339385
\(188\) −4.46410 −0.325578
\(189\) 1.00000 0.0727393
\(190\) 2.73205 0.198204
\(191\) −1.12436 −0.0813555 −0.0406778 0.999172i \(-0.512952\pi\)
−0.0406778 + 0.999172i \(0.512952\pi\)
\(192\) 1.00000 0.0721688
\(193\) −9.19615 −0.661954 −0.330977 0.943639i \(-0.607378\pi\)
−0.330977 + 0.943639i \(0.607378\pi\)
\(194\) −1.66025 −0.119199
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 9.33975 0.665429 0.332715 0.943028i \(-0.392035\pi\)
0.332715 + 0.943028i \(0.392035\pi\)
\(198\) −1.73205 −0.123091
\(199\) −5.80385 −0.411424 −0.205712 0.978613i \(-0.565951\pi\)
−0.205712 + 0.978613i \(0.565951\pi\)
\(200\) −2.46410 −0.174238
\(201\) −4.92820 −0.347609
\(202\) −16.9282 −1.19106
\(203\) −0.464102 −0.0325735
\(204\) 0.267949 0.0187602
\(205\) 8.19615 0.572444
\(206\) −11.2679 −0.785075
\(207\) 3.46410 0.240772
\(208\) 0 0
\(209\) −1.73205 −0.119808
\(210\) −2.73205 −0.188529
\(211\) 26.0000 1.78991 0.894957 0.446153i \(-0.147206\pi\)
0.894957 + 0.446153i \(0.147206\pi\)
\(212\) 7.00000 0.480762
\(213\) −8.19615 −0.561591
\(214\) 16.8564 1.15228
\(215\) 18.3923 1.25434
\(216\) −1.00000 −0.0680414
\(217\) 8.19615 0.556391
\(218\) 1.26795 0.0858764
\(219\) 1.46410 0.0989348
\(220\) 4.73205 0.319035
\(221\) 0 0
\(222\) 3.26795 0.219330
\(223\) 3.60770 0.241589 0.120795 0.992678i \(-0.461456\pi\)
0.120795 + 0.992678i \(0.461456\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 2.46410 0.164273
\(226\) −18.7321 −1.24604
\(227\) 2.39230 0.158783 0.0793914 0.996844i \(-0.474702\pi\)
0.0793914 + 0.996844i \(0.474702\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −6.07180 −0.401236 −0.200618 0.979670i \(-0.564295\pi\)
−0.200618 + 0.979670i \(0.564295\pi\)
\(230\) −9.46410 −0.624044
\(231\) 1.73205 0.113961
\(232\) 0.464102 0.0304698
\(233\) 15.1244 0.990829 0.495415 0.868657i \(-0.335016\pi\)
0.495415 + 0.868657i \(0.335016\pi\)
\(234\) 0 0
\(235\) −12.1962 −0.795589
\(236\) −12.9282 −0.841554
\(237\) 15.9282 1.03465
\(238\) −0.267949 −0.0173686
\(239\) 2.73205 0.176722 0.0883608 0.996089i \(-0.471837\pi\)
0.0883608 + 0.996089i \(0.471837\pi\)
\(240\) 2.73205 0.176353
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 8.00000 0.514259
\(243\) 1.00000 0.0641500
\(244\) −5.19615 −0.332650
\(245\) 2.73205 0.174544
\(246\) −3.00000 −0.191273
\(247\) 0 0
\(248\) −8.19615 −0.520456
\(249\) 10.1962 0.646155
\(250\) 6.92820 0.438178
\(251\) −11.1244 −0.702163 −0.351082 0.936345i \(-0.614186\pi\)
−0.351082 + 0.936345i \(0.614186\pi\)
\(252\) 1.00000 0.0629941
\(253\) 6.00000 0.377217
\(254\) 8.53590 0.535590
\(255\) 0.732051 0.0458428
\(256\) 1.00000 0.0625000
\(257\) 3.33975 0.208328 0.104164 0.994560i \(-0.466783\pi\)
0.104164 + 0.994560i \(0.466783\pi\)
\(258\) −6.73205 −0.419119
\(259\) −3.26795 −0.203060
\(260\) 0 0
\(261\) −0.464102 −0.0287272
\(262\) −13.8564 −0.856052
\(263\) 9.12436 0.562632 0.281316 0.959615i \(-0.409229\pi\)
0.281316 + 0.959615i \(0.409229\pi\)
\(264\) −1.73205 −0.106600
\(265\) 19.1244 1.17480
\(266\) 1.00000 0.0613139
\(267\) 3.53590 0.216393
\(268\) −4.92820 −0.301038
\(269\) 17.8564 1.08872 0.544362 0.838850i \(-0.316772\pi\)
0.544362 + 0.838850i \(0.316772\pi\)
\(270\) −2.73205 −0.166267
\(271\) −10.0526 −0.610649 −0.305325 0.952248i \(-0.598765\pi\)
−0.305325 + 0.952248i \(0.598765\pi\)
\(272\) 0.267949 0.0162468
\(273\) 0 0
\(274\) 6.39230 0.386173
\(275\) 4.26795 0.257367
\(276\) 3.46410 0.208514
\(277\) −25.3205 −1.52136 −0.760681 0.649126i \(-0.775135\pi\)
−0.760681 + 0.649126i \(0.775135\pi\)
\(278\) 19.5885 1.17484
\(279\) 8.19615 0.490691
\(280\) −2.73205 −0.163271
\(281\) −17.6603 −1.05352 −0.526761 0.850013i \(-0.676594\pi\)
−0.526761 + 0.850013i \(0.676594\pi\)
\(282\) 4.46410 0.265833
\(283\) 17.8564 1.06145 0.530727 0.847543i \(-0.321919\pi\)
0.530727 + 0.847543i \(0.321919\pi\)
\(284\) −8.19615 −0.486352
\(285\) −2.73205 −0.161833
\(286\) 0 0
\(287\) 3.00000 0.177084
\(288\) −1.00000 −0.0589256
\(289\) −16.9282 −0.995777
\(290\) 1.26795 0.0744565
\(291\) 1.66025 0.0973258
\(292\) 1.46410 0.0856801
\(293\) −30.9282 −1.80684 −0.903422 0.428752i \(-0.858954\pi\)
−0.903422 + 0.428752i \(0.858954\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −35.3205 −2.05644
\(296\) 3.26795 0.189946
\(297\) 1.73205 0.100504
\(298\) −13.8564 −0.802680
\(299\) 0 0
\(300\) 2.46410 0.142265
\(301\) 6.73205 0.388029
\(302\) 5.19615 0.299005
\(303\) 16.9282 0.972500
\(304\) −1.00000 −0.0573539
\(305\) −14.1962 −0.812869
\(306\) −0.267949 −0.0153176
\(307\) 29.3923 1.67751 0.838754 0.544511i \(-0.183285\pi\)
0.838754 + 0.544511i \(0.183285\pi\)
\(308\) 1.73205 0.0986928
\(309\) 11.2679 0.641011
\(310\) −22.3923 −1.27180
\(311\) 7.73205 0.438444 0.219222 0.975675i \(-0.429648\pi\)
0.219222 + 0.975675i \(0.429648\pi\)
\(312\) 0 0
\(313\) −31.5167 −1.78143 −0.890713 0.454565i \(-0.849795\pi\)
−0.890713 + 0.454565i \(0.849795\pi\)
\(314\) 7.46410 0.421224
\(315\) 2.73205 0.153934
\(316\) 15.9282 0.896031
\(317\) 15.8564 0.890585 0.445292 0.895385i \(-0.353100\pi\)
0.445292 + 0.895385i \(0.353100\pi\)
\(318\) −7.00000 −0.392541
\(319\) −0.803848 −0.0450068
\(320\) 2.73205 0.152726
\(321\) −16.8564 −0.940833
\(322\) −3.46410 −0.193047
\(323\) −0.267949 −0.0149091
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 0.732051 0.0405445
\(327\) −1.26795 −0.0701178
\(328\) −3.00000 −0.165647
\(329\) −4.46410 −0.246114
\(330\) −4.73205 −0.260491
\(331\) −0.143594 −0.00789261 −0.00394631 0.999992i \(-0.501256\pi\)
−0.00394631 + 0.999992i \(0.501256\pi\)
\(332\) 10.1962 0.559587
\(333\) −3.26795 −0.179083
\(334\) 13.8564 0.758189
\(335\) −13.4641 −0.735622
\(336\) 1.00000 0.0545545
\(337\) 19.0000 1.03500 0.517498 0.855684i \(-0.326864\pi\)
0.517498 + 0.855684i \(0.326864\pi\)
\(338\) 0 0
\(339\) 18.7321 1.01739
\(340\) 0.732051 0.0397010
\(341\) 14.1962 0.768765
\(342\) 1.00000 0.0540738
\(343\) 1.00000 0.0539949
\(344\) −6.73205 −0.362968
\(345\) 9.46410 0.509530
\(346\) −15.1244 −0.813090
\(347\) −15.3923 −0.826302 −0.413151 0.910662i \(-0.635572\pi\)
−0.413151 + 0.910662i \(0.635572\pi\)
\(348\) −0.464102 −0.0248785
\(349\) −31.7128 −1.69755 −0.848774 0.528756i \(-0.822659\pi\)
−0.848774 + 0.528756i \(0.822659\pi\)
\(350\) −2.46410 −0.131712
\(351\) 0 0
\(352\) −1.73205 −0.0923186
\(353\) −2.53590 −0.134972 −0.0674861 0.997720i \(-0.521498\pi\)
−0.0674861 + 0.997720i \(0.521498\pi\)
\(354\) 12.9282 0.687126
\(355\) −22.3923 −1.18846
\(356\) 3.53590 0.187402
\(357\) 0.267949 0.0141814
\(358\) −4.53590 −0.239730
\(359\) −14.1962 −0.749244 −0.374622 0.927178i \(-0.622228\pi\)
−0.374622 + 0.927178i \(0.622228\pi\)
\(360\) −2.73205 −0.143992
\(361\) −18.0000 −0.947368
\(362\) −14.6603 −0.770526
\(363\) −8.00000 −0.419891
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 5.19615 0.271607
\(367\) −9.32051 −0.486527 −0.243263 0.969960i \(-0.578218\pi\)
−0.243263 + 0.969960i \(0.578218\pi\)
\(368\) 3.46410 0.180579
\(369\) 3.00000 0.156174
\(370\) 8.92820 0.464155
\(371\) 7.00000 0.363422
\(372\) 8.19615 0.424951
\(373\) −7.66025 −0.396633 −0.198316 0.980138i \(-0.563547\pi\)
−0.198316 + 0.980138i \(0.563547\pi\)
\(374\) −0.464102 −0.0239981
\(375\) −6.92820 −0.357771
\(376\) 4.46410 0.230218
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) −30.4449 −1.56385 −0.781924 0.623374i \(-0.785761\pi\)
−0.781924 + 0.623374i \(0.785761\pi\)
\(380\) −2.73205 −0.140151
\(381\) −8.53590 −0.437307
\(382\) 1.12436 0.0575270
\(383\) 12.4641 0.636886 0.318443 0.947942i \(-0.396840\pi\)
0.318443 + 0.947942i \(0.396840\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 4.73205 0.241168
\(386\) 9.19615 0.468072
\(387\) 6.73205 0.342209
\(388\) 1.66025 0.0842866
\(389\) −12.3923 −0.628315 −0.314157 0.949371i \(-0.601722\pi\)
−0.314157 + 0.949371i \(0.601722\pi\)
\(390\) 0 0
\(391\) 0.928203 0.0469413
\(392\) −1.00000 −0.0505076
\(393\) 13.8564 0.698963
\(394\) −9.33975 −0.470530
\(395\) 43.5167 2.18956
\(396\) 1.73205 0.0870388
\(397\) 21.9282 1.10054 0.550272 0.834985i \(-0.314524\pi\)
0.550272 + 0.834985i \(0.314524\pi\)
\(398\) 5.80385 0.290921
\(399\) −1.00000 −0.0500626
\(400\) 2.46410 0.123205
\(401\) −2.78461 −0.139057 −0.0695284 0.997580i \(-0.522149\pi\)
−0.0695284 + 0.997580i \(0.522149\pi\)
\(402\) 4.92820 0.245796
\(403\) 0 0
\(404\) 16.9282 0.842210
\(405\) 2.73205 0.135757
\(406\) 0.464102 0.0230330
\(407\) −5.66025 −0.280568
\(408\) −0.267949 −0.0132655
\(409\) 35.2679 1.74389 0.871944 0.489606i \(-0.162859\pi\)
0.871944 + 0.489606i \(0.162859\pi\)
\(410\) −8.19615 −0.404779
\(411\) −6.39230 −0.315309
\(412\) 11.2679 0.555132
\(413\) −12.9282 −0.636155
\(414\) −3.46410 −0.170251
\(415\) 27.8564 1.36742
\(416\) 0 0
\(417\) −19.5885 −0.959251
\(418\) 1.73205 0.0847174
\(419\) −2.73205 −0.133469 −0.0667347 0.997771i \(-0.521258\pi\)
−0.0667347 + 0.997771i \(0.521258\pi\)
\(420\) 2.73205 0.133310
\(421\) −1.60770 −0.0783543 −0.0391771 0.999232i \(-0.512474\pi\)
−0.0391771 + 0.999232i \(0.512474\pi\)
\(422\) −26.0000 −1.26566
\(423\) −4.46410 −0.217052
\(424\) −7.00000 −0.339950
\(425\) 0.660254 0.0320270
\(426\) 8.19615 0.397105
\(427\) −5.19615 −0.251459
\(428\) −16.8564 −0.814785
\(429\) 0 0
\(430\) −18.3923 −0.886956
\(431\) −3.80385 −0.183225 −0.0916124 0.995795i \(-0.529202\pi\)
−0.0916124 + 0.995795i \(0.529202\pi\)
\(432\) 1.00000 0.0481125
\(433\) −40.9282 −1.96688 −0.983442 0.181223i \(-0.941994\pi\)
−0.983442 + 0.181223i \(0.941994\pi\)
\(434\) −8.19615 −0.393428
\(435\) −1.26795 −0.0607935
\(436\) −1.26795 −0.0607238
\(437\) −3.46410 −0.165710
\(438\) −1.46410 −0.0699575
\(439\) 37.1769 1.77436 0.887179 0.461426i \(-0.152662\pi\)
0.887179 + 0.461426i \(0.152662\pi\)
\(440\) −4.73205 −0.225592
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 35.6410 1.69336 0.846678 0.532106i \(-0.178599\pi\)
0.846678 + 0.532106i \(0.178599\pi\)
\(444\) −3.26795 −0.155090
\(445\) 9.66025 0.457940
\(446\) −3.60770 −0.170829
\(447\) 13.8564 0.655386
\(448\) 1.00000 0.0472456
\(449\) −10.5885 −0.499700 −0.249850 0.968285i \(-0.580381\pi\)
−0.249850 + 0.968285i \(0.580381\pi\)
\(450\) −2.46410 −0.116159
\(451\) 5.19615 0.244677
\(452\) 18.7321 0.881082
\(453\) −5.19615 −0.244137
\(454\) −2.39230 −0.112276
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) −15.8564 −0.741731 −0.370866 0.928687i \(-0.620939\pi\)
−0.370866 + 0.928687i \(0.620939\pi\)
\(458\) 6.07180 0.283716
\(459\) 0.267949 0.0125068
\(460\) 9.46410 0.441266
\(461\) 13.0718 0.608814 0.304407 0.952542i \(-0.401542\pi\)
0.304407 + 0.952542i \(0.401542\pi\)
\(462\) −1.73205 −0.0805823
\(463\) −9.73205 −0.452287 −0.226143 0.974094i \(-0.572612\pi\)
−0.226143 + 0.974094i \(0.572612\pi\)
\(464\) −0.464102 −0.0215454
\(465\) 22.3923 1.03842
\(466\) −15.1244 −0.700622
\(467\) 13.8564 0.641198 0.320599 0.947215i \(-0.396116\pi\)
0.320599 + 0.947215i \(0.396116\pi\)
\(468\) 0 0
\(469\) −4.92820 −0.227563
\(470\) 12.1962 0.562567
\(471\) −7.46410 −0.343928
\(472\) 12.9282 0.595069
\(473\) 11.6603 0.536139
\(474\) −15.9282 −0.731607
\(475\) −2.46410 −0.113061
\(476\) 0.267949 0.0122814
\(477\) 7.00000 0.320508
\(478\) −2.73205 −0.124961
\(479\) 25.5359 1.16676 0.583382 0.812198i \(-0.301729\pi\)
0.583382 + 0.812198i \(0.301729\pi\)
\(480\) −2.73205 −0.124700
\(481\) 0 0
\(482\) 8.00000 0.364390
\(483\) 3.46410 0.157622
\(484\) −8.00000 −0.363636
\(485\) 4.53590 0.205965
\(486\) −1.00000 −0.0453609
\(487\) −0.660254 −0.0299190 −0.0149595 0.999888i \(-0.504762\pi\)
−0.0149595 + 0.999888i \(0.504762\pi\)
\(488\) 5.19615 0.235219
\(489\) −0.732051 −0.0331045
\(490\) −2.73205 −0.123421
\(491\) 28.5359 1.28781 0.643904 0.765107i \(-0.277314\pi\)
0.643904 + 0.765107i \(0.277314\pi\)
\(492\) 3.00000 0.135250
\(493\) −0.124356 −0.00560070
\(494\) 0 0
\(495\) 4.73205 0.212690
\(496\) 8.19615 0.368018
\(497\) −8.19615 −0.367648
\(498\) −10.1962 −0.456901
\(499\) −33.5167 −1.50041 −0.750206 0.661204i \(-0.770046\pi\)
−0.750206 + 0.661204i \(0.770046\pi\)
\(500\) −6.92820 −0.309839
\(501\) −13.8564 −0.619059
\(502\) 11.1244 0.496504
\(503\) −6.92820 −0.308913 −0.154457 0.988000i \(-0.549363\pi\)
−0.154457 + 0.988000i \(0.549363\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 46.2487 2.05804
\(506\) −6.00000 −0.266733
\(507\) 0 0
\(508\) −8.53590 −0.378719
\(509\) −39.6603 −1.75791 −0.878955 0.476905i \(-0.841759\pi\)
−0.878955 + 0.476905i \(0.841759\pi\)
\(510\) −0.732051 −0.0324158
\(511\) 1.46410 0.0647680
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −3.33975 −0.147310
\(515\) 30.7846 1.35653
\(516\) 6.73205 0.296362
\(517\) −7.73205 −0.340055
\(518\) 3.26795 0.143585
\(519\) 15.1244 0.663886
\(520\) 0 0
\(521\) −31.0526 −1.36044 −0.680219 0.733009i \(-0.738115\pi\)
−0.680219 + 0.733009i \(0.738115\pi\)
\(522\) 0.464102 0.0203132
\(523\) −38.1244 −1.66706 −0.833531 0.552473i \(-0.813684\pi\)
−0.833531 + 0.552473i \(0.813684\pi\)
\(524\) 13.8564 0.605320
\(525\) 2.46410 0.107542
\(526\) −9.12436 −0.397841
\(527\) 2.19615 0.0956659
\(528\) 1.73205 0.0753778
\(529\) −11.0000 −0.478261
\(530\) −19.1244 −0.830709
\(531\) −12.9282 −0.561036
\(532\) −1.00000 −0.0433555
\(533\) 0 0
\(534\) −3.53590 −0.153013
\(535\) −46.0526 −1.99103
\(536\) 4.92820 0.212866
\(537\) 4.53590 0.195738
\(538\) −17.8564 −0.769844
\(539\) 1.73205 0.0746047
\(540\) 2.73205 0.117569
\(541\) −41.6603 −1.79111 −0.895557 0.444947i \(-0.853223\pi\)
−0.895557 + 0.444947i \(0.853223\pi\)
\(542\) 10.0526 0.431794
\(543\) 14.6603 0.629132
\(544\) −0.267949 −0.0114882
\(545\) −3.46410 −0.148386
\(546\) 0 0
\(547\) −4.73205 −0.202328 −0.101164 0.994870i \(-0.532257\pi\)
−0.101164 + 0.994870i \(0.532257\pi\)
\(548\) −6.39230 −0.273066
\(549\) −5.19615 −0.221766
\(550\) −4.26795 −0.181986
\(551\) 0.464102 0.0197714
\(552\) −3.46410 −0.147442
\(553\) 15.9282 0.677336
\(554\) 25.3205 1.07577
\(555\) −8.92820 −0.378981
\(556\) −19.5885 −0.830736
\(557\) 25.0526 1.06151 0.530755 0.847525i \(-0.321908\pi\)
0.530755 + 0.847525i \(0.321908\pi\)
\(558\) −8.19615 −0.346971
\(559\) 0 0
\(560\) 2.73205 0.115450
\(561\) 0.464102 0.0195944
\(562\) 17.6603 0.744953
\(563\) −25.8038 −1.08750 −0.543751 0.839246i \(-0.682997\pi\)
−0.543751 + 0.839246i \(0.682997\pi\)
\(564\) −4.46410 −0.187973
\(565\) 51.1769 2.15303
\(566\) −17.8564 −0.750561
\(567\) 1.00000 0.0419961
\(568\) 8.19615 0.343903
\(569\) 20.4449 0.857093 0.428547 0.903520i \(-0.359026\pi\)
0.428547 + 0.903520i \(0.359026\pi\)
\(570\) 2.73205 0.114433
\(571\) −34.9808 −1.46390 −0.731950 0.681359i \(-0.761389\pi\)
−0.731950 + 0.681359i \(0.761389\pi\)
\(572\) 0 0
\(573\) −1.12436 −0.0469706
\(574\) −3.00000 −0.125218
\(575\) 8.53590 0.355972
\(576\) 1.00000 0.0416667
\(577\) 27.6603 1.15151 0.575756 0.817622i \(-0.304708\pi\)
0.575756 + 0.817622i \(0.304708\pi\)
\(578\) 16.9282 0.704120
\(579\) −9.19615 −0.382179
\(580\) −1.26795 −0.0526487
\(581\) 10.1962 0.423008
\(582\) −1.66025 −0.0688197
\(583\) 12.1244 0.502140
\(584\) −1.46410 −0.0605850
\(585\) 0 0
\(586\) 30.9282 1.27763
\(587\) 29.6603 1.22421 0.612105 0.790777i \(-0.290323\pi\)
0.612105 + 0.790777i \(0.290323\pi\)
\(588\) 1.00000 0.0412393
\(589\) −8.19615 −0.337717
\(590\) 35.3205 1.45412
\(591\) 9.33975 0.384186
\(592\) −3.26795 −0.134312
\(593\) −6.46410 −0.265449 −0.132724 0.991153i \(-0.542373\pi\)
−0.132724 + 0.991153i \(0.542373\pi\)
\(594\) −1.73205 −0.0710669
\(595\) 0.732051 0.0300112
\(596\) 13.8564 0.567581
\(597\) −5.80385 −0.237536
\(598\) 0 0
\(599\) 40.6410 1.66055 0.830273 0.557356i \(-0.188184\pi\)
0.830273 + 0.557356i \(0.188184\pi\)
\(600\) −2.46410 −0.100597
\(601\) −19.2679 −0.785956 −0.392978 0.919548i \(-0.628555\pi\)
−0.392978 + 0.919548i \(0.628555\pi\)
\(602\) −6.73205 −0.274378
\(603\) −4.92820 −0.200692
\(604\) −5.19615 −0.211428
\(605\) −21.8564 −0.888589
\(606\) −16.9282 −0.687661
\(607\) 13.5167 0.548624 0.274312 0.961641i \(-0.411550\pi\)
0.274312 + 0.961641i \(0.411550\pi\)
\(608\) 1.00000 0.0405554
\(609\) −0.464102 −0.0188063
\(610\) 14.1962 0.574785
\(611\) 0 0
\(612\) 0.267949 0.0108312
\(613\) 4.00000 0.161558 0.0807792 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) −29.3923 −1.18618
\(615\) 8.19615 0.330501
\(616\) −1.73205 −0.0697863
\(617\) 8.19615 0.329965 0.164982 0.986297i \(-0.447243\pi\)
0.164982 + 0.986297i \(0.447243\pi\)
\(618\) −11.2679 −0.453263
\(619\) 35.9282 1.44408 0.722038 0.691853i \(-0.243205\pi\)
0.722038 + 0.691853i \(0.243205\pi\)
\(620\) 22.3923 0.899297
\(621\) 3.46410 0.139010
\(622\) −7.73205 −0.310027
\(623\) 3.53590 0.141663
\(624\) 0 0
\(625\) −31.2487 −1.24995
\(626\) 31.5167 1.25966
\(627\) −1.73205 −0.0691714
\(628\) −7.46410 −0.297850
\(629\) −0.875644 −0.0349142
\(630\) −2.73205 −0.108848
\(631\) 27.1962 1.08266 0.541331 0.840810i \(-0.317921\pi\)
0.541331 + 0.840810i \(0.317921\pi\)
\(632\) −15.9282 −0.633590
\(633\) 26.0000 1.03341
\(634\) −15.8564 −0.629738
\(635\) −23.3205 −0.925446
\(636\) 7.00000 0.277568
\(637\) 0 0
\(638\) 0.803848 0.0318246
\(639\) −8.19615 −0.324235
\(640\) −2.73205 −0.107994
\(641\) −31.1769 −1.23141 −0.615707 0.787975i \(-0.711130\pi\)
−0.615707 + 0.787975i \(0.711130\pi\)
\(642\) 16.8564 0.665269
\(643\) −5.14359 −0.202844 −0.101422 0.994844i \(-0.532339\pi\)
−0.101422 + 0.994844i \(0.532339\pi\)
\(644\) 3.46410 0.136505
\(645\) 18.3923 0.724196
\(646\) 0.267949 0.0105423
\(647\) −43.8372 −1.72342 −0.861708 0.507404i \(-0.830605\pi\)
−0.861708 + 0.507404i \(0.830605\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −22.3923 −0.878975
\(650\) 0 0
\(651\) 8.19615 0.321233
\(652\) −0.732051 −0.0286693
\(653\) −8.71281 −0.340959 −0.170479 0.985361i \(-0.554532\pi\)
−0.170479 + 0.985361i \(0.554532\pi\)
\(654\) 1.26795 0.0495807
\(655\) 37.8564 1.47917
\(656\) 3.00000 0.117130
\(657\) 1.46410 0.0571200
\(658\) 4.46410 0.174029
\(659\) 39.3923 1.53451 0.767253 0.641344i \(-0.221623\pi\)
0.767253 + 0.641344i \(0.221623\pi\)
\(660\) 4.73205 0.184195
\(661\) 8.39230 0.326423 0.163211 0.986591i \(-0.447815\pi\)
0.163211 + 0.986591i \(0.447815\pi\)
\(662\) 0.143594 0.00558092
\(663\) 0 0
\(664\) −10.1962 −0.395687
\(665\) −2.73205 −0.105944
\(666\) 3.26795 0.126630
\(667\) −1.60770 −0.0622502
\(668\) −13.8564 −0.536120
\(669\) 3.60770 0.139482
\(670\) 13.4641 0.520164
\(671\) −9.00000 −0.347441
\(672\) −1.00000 −0.0385758
\(673\) 22.7128 0.875515 0.437757 0.899093i \(-0.355773\pi\)
0.437757 + 0.899093i \(0.355773\pi\)
\(674\) −19.0000 −0.731853
\(675\) 2.46410 0.0948433
\(676\) 0 0
\(677\) 33.2679 1.27859 0.639296 0.768961i \(-0.279226\pi\)
0.639296 + 0.768961i \(0.279226\pi\)
\(678\) −18.7321 −0.719400
\(679\) 1.66025 0.0637147
\(680\) −0.732051 −0.0280729
\(681\) 2.39230 0.0916733
\(682\) −14.1962 −0.543599
\(683\) 50.3923 1.92821 0.964104 0.265525i \(-0.0855453\pi\)
0.964104 + 0.265525i \(0.0855453\pi\)
\(684\) −1.00000 −0.0382360
\(685\) −17.4641 −0.667269
\(686\) −1.00000 −0.0381802
\(687\) −6.07180 −0.231653
\(688\) 6.73205 0.256657
\(689\) 0 0
\(690\) −9.46410 −0.360292
\(691\) −11.7128 −0.445576 −0.222788 0.974867i \(-0.571516\pi\)
−0.222788 + 0.974867i \(0.571516\pi\)
\(692\) 15.1244 0.574942
\(693\) 1.73205 0.0657952
\(694\) 15.3923 0.584284
\(695\) −53.5167 −2.03000
\(696\) 0.464102 0.0175917
\(697\) 0.803848 0.0304479
\(698\) 31.7128 1.20035
\(699\) 15.1244 0.572056
\(700\) 2.46410 0.0931343
\(701\) 28.8564 1.08989 0.544946 0.838471i \(-0.316550\pi\)
0.544946 + 0.838471i \(0.316550\pi\)
\(702\) 0 0
\(703\) 3.26795 0.123253
\(704\) 1.73205 0.0652791
\(705\) −12.1962 −0.459334
\(706\) 2.53590 0.0954398
\(707\) 16.9282 0.636651
\(708\) −12.9282 −0.485872
\(709\) 6.05256 0.227309 0.113654 0.993520i \(-0.463744\pi\)
0.113654 + 0.993520i \(0.463744\pi\)
\(710\) 22.3923 0.840368
\(711\) 15.9282 0.597354
\(712\) −3.53590 −0.132513
\(713\) 28.3923 1.06330
\(714\) −0.267949 −0.0100277
\(715\) 0 0
\(716\) 4.53590 0.169514
\(717\) 2.73205 0.102030
\(718\) 14.1962 0.529796
\(719\) −18.5167 −0.690555 −0.345277 0.938501i \(-0.612215\pi\)
−0.345277 + 0.938501i \(0.612215\pi\)
\(720\) 2.73205 0.101818
\(721\) 11.2679 0.419640
\(722\) 18.0000 0.669891
\(723\) −8.00000 −0.297523
\(724\) 14.6603 0.544844
\(725\) −1.14359 −0.0424720
\(726\) 8.00000 0.296908
\(727\) 15.3205 0.568206 0.284103 0.958794i \(-0.408304\pi\)
0.284103 + 0.958794i \(0.408304\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) 1.80385 0.0667177
\(732\) −5.19615 −0.192055
\(733\) −14.1769 −0.523636 −0.261818 0.965117i \(-0.584322\pi\)
−0.261818 + 0.965117i \(0.584322\pi\)
\(734\) 9.32051 0.344026
\(735\) 2.73205 0.100773
\(736\) −3.46410 −0.127688
\(737\) −8.53590 −0.314424
\(738\) −3.00000 −0.110432
\(739\) 47.1769 1.73543 0.867715 0.497061i \(-0.165588\pi\)
0.867715 + 0.497061i \(0.165588\pi\)
\(740\) −8.92820 −0.328207
\(741\) 0 0
\(742\) −7.00000 −0.256978
\(743\) 24.0526 0.882403 0.441201 0.897408i \(-0.354552\pi\)
0.441201 + 0.897408i \(0.354552\pi\)
\(744\) −8.19615 −0.300486
\(745\) 37.8564 1.38695
\(746\) 7.66025 0.280462
\(747\) 10.1962 0.373058
\(748\) 0.464102 0.0169692
\(749\) −16.8564 −0.615920
\(750\) 6.92820 0.252982
\(751\) −46.8564 −1.70981 −0.854907 0.518781i \(-0.826386\pi\)
−0.854907 + 0.518781i \(0.826386\pi\)
\(752\) −4.46410 −0.162789
\(753\) −11.1244 −0.405394
\(754\) 0 0
\(755\) −14.1962 −0.516651
\(756\) 1.00000 0.0363696
\(757\) 30.5885 1.11176 0.555878 0.831264i \(-0.312382\pi\)
0.555878 + 0.831264i \(0.312382\pi\)
\(758\) 30.4449 1.10581
\(759\) 6.00000 0.217786
\(760\) 2.73205 0.0991019
\(761\) −19.3205 −0.700368 −0.350184 0.936681i \(-0.613881\pi\)
−0.350184 + 0.936681i \(0.613881\pi\)
\(762\) 8.53590 0.309223
\(763\) −1.26795 −0.0459028
\(764\) −1.12436 −0.0406778
\(765\) 0.732051 0.0264674
\(766\) −12.4641 −0.450346
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −44.3013 −1.59754 −0.798772 0.601633i \(-0.794517\pi\)
−0.798772 + 0.601633i \(0.794517\pi\)
\(770\) −4.73205 −0.170531
\(771\) 3.33975 0.120278
\(772\) −9.19615 −0.330977
\(773\) −15.1769 −0.545876 −0.272938 0.962032i \(-0.587995\pi\)
−0.272938 + 0.962032i \(0.587995\pi\)
\(774\) −6.73205 −0.241979
\(775\) 20.1962 0.725467
\(776\) −1.66025 −0.0595996
\(777\) −3.26795 −0.117237
\(778\) 12.3923 0.444286
\(779\) −3.00000 −0.107486
\(780\) 0 0
\(781\) −14.1962 −0.507978
\(782\) −0.928203 −0.0331925
\(783\) −0.464102 −0.0165856
\(784\) 1.00000 0.0357143
\(785\) −20.3923 −0.727833
\(786\) −13.8564 −0.494242
\(787\) 38.3205 1.36598 0.682989 0.730428i \(-0.260680\pi\)
0.682989 + 0.730428i \(0.260680\pi\)
\(788\) 9.33975 0.332715
\(789\) 9.12436 0.324836
\(790\) −43.5167 −1.54825
\(791\) 18.7321 0.666035
\(792\) −1.73205 −0.0615457
\(793\) 0 0
\(794\) −21.9282 −0.778203
\(795\) 19.1244 0.678271
\(796\) −5.80385 −0.205712
\(797\) −27.1244 −0.960794 −0.480397 0.877051i \(-0.659507\pi\)
−0.480397 + 0.877051i \(0.659507\pi\)
\(798\) 1.00000 0.0353996
\(799\) −1.19615 −0.0423168
\(800\) −2.46410 −0.0871191
\(801\) 3.53590 0.124935
\(802\) 2.78461 0.0983280
\(803\) 2.53590 0.0894899
\(804\) −4.92820 −0.173804
\(805\) 9.46410 0.333566
\(806\) 0 0
\(807\) 17.8564 0.628575
\(808\) −16.9282 −0.595532
\(809\) 52.4449 1.84386 0.921932 0.387353i \(-0.126610\pi\)
0.921932 + 0.387353i \(0.126610\pi\)
\(810\) −2.73205 −0.0959945
\(811\) 31.4641 1.10485 0.552427 0.833561i \(-0.313702\pi\)
0.552427 + 0.833561i \(0.313702\pi\)
\(812\) −0.464102 −0.0162868
\(813\) −10.0526 −0.352559
\(814\) 5.66025 0.198392
\(815\) −2.00000 −0.0700569
\(816\) 0.267949 0.00938010
\(817\) −6.73205 −0.235525
\(818\) −35.2679 −1.23311
\(819\) 0 0
\(820\) 8.19615 0.286222
\(821\) 27.8372 0.971524 0.485762 0.874091i \(-0.338542\pi\)
0.485762 + 0.874091i \(0.338542\pi\)
\(822\) 6.39230 0.222957
\(823\) 34.7846 1.21252 0.606258 0.795268i \(-0.292670\pi\)
0.606258 + 0.795268i \(0.292670\pi\)
\(824\) −11.2679 −0.392538
\(825\) 4.26795 0.148591
\(826\) 12.9282 0.449830
\(827\) 27.3205 0.950027 0.475014 0.879978i \(-0.342443\pi\)
0.475014 + 0.879978i \(0.342443\pi\)
\(828\) 3.46410 0.120386
\(829\) −34.6603 −1.20380 −0.601900 0.798571i \(-0.705590\pi\)
−0.601900 + 0.798571i \(0.705590\pi\)
\(830\) −27.8564 −0.966910
\(831\) −25.3205 −0.878359
\(832\) 0 0
\(833\) 0.267949 0.00928389
\(834\) 19.5885 0.678293
\(835\) −37.8564 −1.31007
\(836\) −1.73205 −0.0599042
\(837\) 8.19615 0.283300
\(838\) 2.73205 0.0943771
\(839\) 42.1051 1.45363 0.726815 0.686833i \(-0.241000\pi\)
0.726815 + 0.686833i \(0.241000\pi\)
\(840\) −2.73205 −0.0942647
\(841\) −28.7846 −0.992573
\(842\) 1.60770 0.0554048
\(843\) −17.6603 −0.608251
\(844\) 26.0000 0.894957
\(845\) 0 0
\(846\) 4.46410 0.153479
\(847\) −8.00000 −0.274883
\(848\) 7.00000 0.240381
\(849\) 17.8564 0.612830
\(850\) −0.660254 −0.0226465
\(851\) −11.3205 −0.388062
\(852\) −8.19615 −0.280796
\(853\) 5.92820 0.202978 0.101489 0.994837i \(-0.467639\pi\)
0.101489 + 0.994837i \(0.467639\pi\)
\(854\) 5.19615 0.177809
\(855\) −2.73205 −0.0934342
\(856\) 16.8564 0.576140
\(857\) −39.4641 −1.34807 −0.674034 0.738700i \(-0.735440\pi\)
−0.674034 + 0.738700i \(0.735440\pi\)
\(858\) 0 0
\(859\) −50.7654 −1.73209 −0.866046 0.499964i \(-0.833346\pi\)
−0.866046 + 0.499964i \(0.833346\pi\)
\(860\) 18.3923 0.627172
\(861\) 3.00000 0.102240
\(862\) 3.80385 0.129560
\(863\) −16.9282 −0.576243 −0.288121 0.957594i \(-0.593031\pi\)
−0.288121 + 0.957594i \(0.593031\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 41.3205 1.40494
\(866\) 40.9282 1.39080
\(867\) −16.9282 −0.574912
\(868\) 8.19615 0.278196
\(869\) 27.5885 0.935874
\(870\) 1.26795 0.0429875
\(871\) 0 0
\(872\) 1.26795 0.0429382
\(873\) 1.66025 0.0561911
\(874\) 3.46410 0.117175
\(875\) −6.92820 −0.234216
\(876\) 1.46410 0.0494674
\(877\) −26.7321 −0.902677 −0.451339 0.892353i \(-0.649053\pi\)
−0.451339 + 0.892353i \(0.649053\pi\)
\(878\) −37.1769 −1.25466
\(879\) −30.9282 −1.04318
\(880\) 4.73205 0.159517
\(881\) 9.07180 0.305637 0.152818 0.988254i \(-0.451165\pi\)
0.152818 + 0.988254i \(0.451165\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 30.0526 1.01135 0.505675 0.862724i \(-0.331244\pi\)
0.505675 + 0.862724i \(0.331244\pi\)
\(884\) 0 0
\(885\) −35.3205 −1.18729
\(886\) −35.6410 −1.19738
\(887\) 43.4449 1.45874 0.729368 0.684122i \(-0.239814\pi\)
0.729368 + 0.684122i \(0.239814\pi\)
\(888\) 3.26795 0.109665
\(889\) −8.53590 −0.286285
\(890\) −9.66025 −0.323812
\(891\) 1.73205 0.0580259
\(892\) 3.60770 0.120795
\(893\) 4.46410 0.149385
\(894\) −13.8564 −0.463428
\(895\) 12.3923 0.414229
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 10.5885 0.353341
\(899\) −3.80385 −0.126865
\(900\) 2.46410 0.0821367
\(901\) 1.87564 0.0624868
\(902\) −5.19615 −0.173013
\(903\) 6.73205 0.224029
\(904\) −18.7321 −0.623019
\(905\) 40.0526 1.33139
\(906\) 5.19615 0.172631
\(907\) −22.3923 −0.743524 −0.371762 0.928328i \(-0.621246\pi\)
−0.371762 + 0.928328i \(0.621246\pi\)
\(908\) 2.39230 0.0793914
\(909\) 16.9282 0.561473
\(910\) 0 0
\(911\) 28.8372 0.955418 0.477709 0.878518i \(-0.341467\pi\)
0.477709 + 0.878518i \(0.341467\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 17.6603 0.584469
\(914\) 15.8564 0.524483
\(915\) −14.1962 −0.469310
\(916\) −6.07180 −0.200618
\(917\) 13.8564 0.457579
\(918\) −0.267949 −0.00884364
\(919\) 5.14359 0.169671 0.0848357 0.996395i \(-0.472963\pi\)
0.0848357 + 0.996395i \(0.472963\pi\)
\(920\) −9.46410 −0.312022
\(921\) 29.3923 0.968509
\(922\) −13.0718 −0.430497
\(923\) 0 0
\(924\) 1.73205 0.0569803
\(925\) −8.05256 −0.264767
\(926\) 9.73205 0.319815
\(927\) 11.2679 0.370088
\(928\) 0.464102 0.0152349
\(929\) 32.3205 1.06040 0.530201 0.847872i \(-0.322117\pi\)
0.530201 + 0.847872i \(0.322117\pi\)
\(930\) −22.3923 −0.734273
\(931\) −1.00000 −0.0327737
\(932\) 15.1244 0.495415
\(933\) 7.73205 0.253136
\(934\) −13.8564 −0.453395
\(935\) 1.26795 0.0414664
\(936\) 0 0
\(937\) 14.9282 0.487683 0.243842 0.969815i \(-0.421592\pi\)
0.243842 + 0.969815i \(0.421592\pi\)
\(938\) 4.92820 0.160912
\(939\) −31.5167 −1.02851
\(940\) −12.1962 −0.397795
\(941\) 32.3923 1.05596 0.527979 0.849257i \(-0.322950\pi\)
0.527979 + 0.849257i \(0.322950\pi\)
\(942\) 7.46410 0.243194
\(943\) 10.3923 0.338420
\(944\) −12.9282 −0.420777
\(945\) 2.73205 0.0888736
\(946\) −11.6603 −0.379108
\(947\) −12.5167 −0.406737 −0.203368 0.979102i \(-0.565189\pi\)
−0.203368 + 0.979102i \(0.565189\pi\)
\(948\) 15.9282 0.517324
\(949\) 0 0
\(950\) 2.46410 0.0799460
\(951\) 15.8564 0.514179
\(952\) −0.267949 −0.00868428
\(953\) 23.8038 0.771082 0.385541 0.922691i \(-0.374015\pi\)
0.385541 + 0.922691i \(0.374015\pi\)
\(954\) −7.00000 −0.226633
\(955\) −3.07180 −0.0994010
\(956\) 2.73205 0.0883608
\(957\) −0.803848 −0.0259847
\(958\) −25.5359 −0.825027
\(959\) −6.39230 −0.206418
\(960\) 2.73205 0.0881766
\(961\) 36.1769 1.16700
\(962\) 0 0
\(963\) −16.8564 −0.543190
\(964\) −8.00000 −0.257663
\(965\) −25.1244 −0.808782
\(966\) −3.46410 −0.111456
\(967\) −55.7128 −1.79160 −0.895802 0.444454i \(-0.853398\pi\)
−0.895802 + 0.444454i \(0.853398\pi\)
\(968\) 8.00000 0.257130
\(969\) −0.267949 −0.00860777
\(970\) −4.53590 −0.145639
\(971\) 22.1051 0.709387 0.354693 0.934983i \(-0.384585\pi\)
0.354693 + 0.934983i \(0.384585\pi\)
\(972\) 1.00000 0.0320750
\(973\) −19.5885 −0.627977
\(974\) 0.660254 0.0211559
\(975\) 0 0
\(976\) −5.19615 −0.166325
\(977\) 4.39230 0.140522 0.0702611 0.997529i \(-0.477617\pi\)
0.0702611 + 0.997529i \(0.477617\pi\)
\(978\) 0.732051 0.0234084
\(979\) 6.12436 0.195735
\(980\) 2.73205 0.0872722
\(981\) −1.26795 −0.0404825
\(982\) −28.5359 −0.910617
\(983\) −8.14359 −0.259740 −0.129870 0.991531i \(-0.541456\pi\)
−0.129870 + 0.991531i \(0.541456\pi\)
\(984\) −3.00000 −0.0956365
\(985\) 25.5167 0.813028
\(986\) 0.124356 0.00396029
\(987\) −4.46410 −0.142094
\(988\) 0 0
\(989\) 23.3205 0.741549
\(990\) −4.73205 −0.150394
\(991\) −10.7513 −0.341526 −0.170763 0.985312i \(-0.554623\pi\)
−0.170763 + 0.985312i \(0.554623\pi\)
\(992\) −8.19615 −0.260228
\(993\) −0.143594 −0.00455680
\(994\) 8.19615 0.259966
\(995\) −15.8564 −0.502682
\(996\) 10.1962 0.323077
\(997\) 17.0526 0.540060 0.270030 0.962852i \(-0.412966\pi\)
0.270030 + 0.962852i \(0.412966\pi\)
\(998\) 33.5167 1.06095
\(999\) −3.26795 −0.103393
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.bp.1.2 2
13.6 odd 12 546.2.s.a.127.2 yes 4
13.11 odd 12 546.2.s.a.43.2 4
13.12 even 2 7098.2.a.bx.1.1 2
39.11 even 12 1638.2.bj.b.1135.1 4
39.32 even 12 1638.2.bj.b.127.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.s.a.43.2 4 13.11 odd 12
546.2.s.a.127.2 yes 4 13.6 odd 12
1638.2.bj.b.127.1 4 39.32 even 12
1638.2.bj.b.1135.1 4 39.11 even 12
7098.2.a.bp.1.2 2 1.1 even 1 trivial
7098.2.a.bx.1.1 2 13.12 even 2