Properties

Label 7098.2.a.bp.1.1
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 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.732051 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.732051 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.732051 q^{10} -1.73205 q^{11} +1.00000 q^{12} -1.00000 q^{14} -0.732051 q^{15} +1.00000 q^{16} +3.73205 q^{17} -1.00000 q^{18} -1.00000 q^{19} -0.732051 q^{20} +1.00000 q^{21} +1.73205 q^{22} -3.46410 q^{23} -1.00000 q^{24} -4.46410 q^{25} +1.00000 q^{27} +1.00000 q^{28} +6.46410 q^{29} +0.732051 q^{30} -2.19615 q^{31} -1.00000 q^{32} -1.73205 q^{33} -3.73205 q^{34} -0.732051 q^{35} +1.00000 q^{36} -6.73205 q^{37} +1.00000 q^{38} +0.732051 q^{40} +3.00000 q^{41} -1.00000 q^{42} +3.26795 q^{43} -1.73205 q^{44} -0.732051 q^{45} +3.46410 q^{46} +2.46410 q^{47} +1.00000 q^{48} +1.00000 q^{49} +4.46410 q^{50} +3.73205 q^{51} +7.00000 q^{53} -1.00000 q^{54} +1.26795 q^{55} -1.00000 q^{56} -1.00000 q^{57} -6.46410 q^{58} +0.928203 q^{59} -0.732051 q^{60} +5.19615 q^{61} +2.19615 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.73205 q^{66} +8.92820 q^{67} +3.73205 q^{68} -3.46410 q^{69} +0.732051 q^{70} +2.19615 q^{71} -1.00000 q^{72} -5.46410 q^{73} +6.73205 q^{74} -4.46410 q^{75} -1.00000 q^{76} -1.73205 q^{77} +2.07180 q^{79} -0.732051 q^{80} +1.00000 q^{81} -3.00000 q^{82} -0.196152 q^{83} +1.00000 q^{84} -2.73205 q^{85} -3.26795 q^{86} +6.46410 q^{87} +1.73205 q^{88} +10.4641 q^{89} +0.732051 q^{90} -3.46410 q^{92} -2.19615 q^{93} -2.46410 q^{94} +0.732051 q^{95} -1.00000 q^{96} -15.6603 q^{97} -1.00000 q^{98} -1.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} - 2q^{10} + 2q^{12} - 2q^{14} + 2q^{15} + 2q^{16} + 4q^{17} - 2q^{18} - 2q^{19} + 2q^{20} + 2q^{21} - 2q^{24} - 2q^{25} + 2q^{27} + 2q^{28} + 6q^{29} - 2q^{30} + 6q^{31} - 2q^{32} - 4q^{34} + 2q^{35} + 2q^{36} - 10q^{37} + 2q^{38} - 2q^{40} + 6q^{41} - 2q^{42} + 10q^{43} + 2q^{45} - 2q^{47} + 2q^{48} + 2q^{49} + 2q^{50} + 4q^{51} + 14q^{53} - 2q^{54} + 6q^{55} - 2q^{56} - 2q^{57} - 6q^{58} - 12q^{59} + 2q^{60} - 6q^{62} + 2q^{63} + 2q^{64} + 4q^{67} + 4q^{68} - 2q^{70} - 6q^{71} - 2q^{72} - 4q^{73} + 10q^{74} - 2q^{75} - 2q^{76} + 18q^{79} + 2q^{80} + 2q^{81} - 6q^{82} + 10q^{83} + 2q^{84} - 2q^{85} - 10q^{86} + 6q^{87} + 14q^{89} - 2q^{90} + 6q^{93} + 2q^{94} - 2q^{95} - 2q^{96} - 14q^{97} - 2q^{98} + O(q^{100}) \)

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
\(5\) −0.732051 −0.327383 −0.163692 0.986512i \(-0.552340\pi\)
−0.163692 + 0.986512i \(0.552340\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0.732051 0.231495
\(11\) −1.73205 −0.522233 −0.261116 0.965307i \(-0.584091\pi\)
−0.261116 + 0.965307i \(0.584091\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −0.732051 −0.189015
\(16\) 1.00000 0.250000
\(17\) 3.73205 0.905155 0.452578 0.891725i \(-0.350505\pi\)
0.452578 + 0.891725i \(0.350505\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −0.732051 −0.163692
\(21\) 1.00000 0.218218
\(22\) 1.73205 0.369274
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.46410 −0.892820
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 6.46410 1.20035 0.600177 0.799867i \(-0.295097\pi\)
0.600177 + 0.799867i \(0.295097\pi\)
\(30\) 0.732051 0.133654
\(31\) −2.19615 −0.394441 −0.197220 0.980359i \(-0.563191\pi\)
−0.197220 + 0.980359i \(0.563191\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.73205 −0.301511
\(34\) −3.73205 −0.640041
\(35\) −0.732051 −0.123739
\(36\) 1.00000 0.166667
\(37\) −6.73205 −1.10674 −0.553371 0.832935i \(-0.686659\pi\)
−0.553371 + 0.832935i \(0.686659\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0.732051 0.115747
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) −1.00000 −0.154303
\(43\) 3.26795 0.498358 0.249179 0.968458i \(-0.419839\pi\)
0.249179 + 0.968458i \(0.419839\pi\)
\(44\) −1.73205 −0.261116
\(45\) −0.732051 −0.109128
\(46\) 3.46410 0.510754
\(47\) 2.46410 0.359426 0.179713 0.983719i \(-0.442483\pi\)
0.179713 + 0.983719i \(0.442483\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 4.46410 0.631319
\(51\) 3.73205 0.522592
\(52\) 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
\(55\) 1.26795 0.170970
\(56\) −1.00000 −0.133631
\(57\) −1.00000 −0.132453
\(58\) −6.46410 −0.848778
\(59\) 0.928203 0.120842 0.0604209 0.998173i \(-0.480756\pi\)
0.0604209 + 0.998173i \(0.480756\pi\)
\(60\) −0.732051 −0.0945074
\(61\) 5.19615 0.665299 0.332650 0.943051i \(-0.392057\pi\)
0.332650 + 0.943051i \(0.392057\pi\)
\(62\) 2.19615 0.278912
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.73205 0.213201
\(67\) 8.92820 1.09075 0.545377 0.838191i \(-0.316387\pi\)
0.545377 + 0.838191i \(0.316387\pi\)
\(68\) 3.73205 0.452578
\(69\) −3.46410 −0.417029
\(70\) 0.732051 0.0874968
\(71\) 2.19615 0.260635 0.130318 0.991472i \(-0.458400\pi\)
0.130318 + 0.991472i \(0.458400\pi\)
\(72\) −1.00000 −0.117851
\(73\) −5.46410 −0.639525 −0.319762 0.947498i \(-0.603603\pi\)
−0.319762 + 0.947498i \(0.603603\pi\)
\(74\) 6.73205 0.782585
\(75\) −4.46410 −0.515470
\(76\) −1.00000 −0.114708
\(77\) −1.73205 −0.197386
\(78\) 0 0
\(79\) 2.07180 0.233095 0.116548 0.993185i \(-0.462817\pi\)
0.116548 + 0.993185i \(0.462817\pi\)
\(80\) −0.732051 −0.0818458
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) −0.196152 −0.0215305 −0.0107653 0.999942i \(-0.503427\pi\)
−0.0107653 + 0.999942i \(0.503427\pi\)
\(84\) 1.00000 0.109109
\(85\) −2.73205 −0.296333
\(86\) −3.26795 −0.352392
\(87\) 6.46410 0.693024
\(88\) 1.73205 0.184637
\(89\) 10.4641 1.10919 0.554596 0.832120i \(-0.312873\pi\)
0.554596 + 0.832120i \(0.312873\pi\)
\(90\) 0.732051 0.0771649
\(91\) 0 0
\(92\) −3.46410 −0.361158
\(93\) −2.19615 −0.227730
\(94\) −2.46410 −0.254153
\(95\) 0.732051 0.0751068
\(96\) −1.00000 −0.102062
\(97\) −15.6603 −1.59006 −0.795029 0.606571i \(-0.792544\pi\)
−0.795029 + 0.606571i \(0.792544\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.73205 −0.174078
\(100\) −4.46410 −0.446410
\(101\) 3.07180 0.305655 0.152828 0.988253i \(-0.451162\pi\)
0.152828 + 0.988253i \(0.451162\pi\)
\(102\) −3.73205 −0.369528
\(103\) 14.7321 1.45159 0.725796 0.687910i \(-0.241472\pi\)
0.725796 + 0.687910i \(0.241472\pi\)
\(104\) 0 0
\(105\) −0.732051 −0.0714408
\(106\) −7.00000 −0.679900
\(107\) 10.8564 1.04953 0.524764 0.851248i \(-0.324153\pi\)
0.524764 + 0.851248i \(0.324153\pi\)
\(108\) 1.00000 0.0962250
\(109\) −4.73205 −0.453248 −0.226624 0.973982i \(-0.572769\pi\)
−0.226624 + 0.973982i \(0.572769\pi\)
\(110\) −1.26795 −0.120894
\(111\) −6.73205 −0.638978
\(112\) 1.00000 0.0944911
\(113\) 15.2679 1.43629 0.718144 0.695895i \(-0.244992\pi\)
0.718144 + 0.695895i \(0.244992\pi\)
\(114\) 1.00000 0.0936586
\(115\) 2.53590 0.236474
\(116\) 6.46410 0.600177
\(117\) 0 0
\(118\) −0.928203 −0.0854480
\(119\) 3.73205 0.342117
\(120\) 0.732051 0.0668268
\(121\) −8.00000 −0.727273
\(122\) −5.19615 −0.470438
\(123\) 3.00000 0.270501
\(124\) −2.19615 −0.197220
\(125\) 6.92820 0.619677
\(126\) −1.00000 −0.0890871
\(127\) −15.4641 −1.37222 −0.686109 0.727499i \(-0.740683\pi\)
−0.686109 + 0.727499i \(0.740683\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 3.26795 0.287727
\(130\) 0 0
\(131\) −13.8564 −1.21064 −0.605320 0.795982i \(-0.706955\pi\)
−0.605320 + 0.795982i \(0.706955\pi\)
\(132\) −1.73205 −0.150756
\(133\) −1.00000 −0.0867110
\(134\) −8.92820 −0.771279
\(135\) −0.732051 −0.0630049
\(136\) −3.73205 −0.320021
\(137\) 14.3923 1.22962 0.614809 0.788676i \(-0.289233\pi\)
0.614809 + 0.788676i \(0.289233\pi\)
\(138\) 3.46410 0.294884
\(139\) 11.5885 0.982920 0.491460 0.870900i \(-0.336463\pi\)
0.491460 + 0.870900i \(0.336463\pi\)
\(140\) −0.732051 −0.0618696
\(141\) 2.46410 0.207515
\(142\) −2.19615 −0.184297
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −4.73205 −0.392975
\(146\) 5.46410 0.452212
\(147\) 1.00000 0.0824786
\(148\) −6.73205 −0.553371
\(149\) −13.8564 −1.13516 −0.567581 0.823318i \(-0.692120\pi\)
−0.567581 + 0.823318i \(0.692120\pi\)
\(150\) 4.46410 0.364492
\(151\) 5.19615 0.422857 0.211428 0.977393i \(-0.432188\pi\)
0.211428 + 0.977393i \(0.432188\pi\)
\(152\) 1.00000 0.0811107
\(153\) 3.73205 0.301718
\(154\) 1.73205 0.139573
\(155\) 1.60770 0.129133
\(156\) 0 0
\(157\) −0.535898 −0.0427693 −0.0213847 0.999771i \(-0.506807\pi\)
−0.0213847 + 0.999771i \(0.506807\pi\)
\(158\) −2.07180 −0.164823
\(159\) 7.00000 0.555136
\(160\) 0.732051 0.0578737
\(161\) −3.46410 −0.273009
\(162\) −1.00000 −0.0785674
\(163\) 2.73205 0.213991 0.106995 0.994260i \(-0.465877\pi\)
0.106995 + 0.994260i \(0.465877\pi\)
\(164\) 3.00000 0.234261
\(165\) 1.26795 0.0987097
\(166\) 0.196152 0.0152244
\(167\) 13.8564 1.07224 0.536120 0.844141i \(-0.319889\pi\)
0.536120 + 0.844141i \(0.319889\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) 2.73205 0.209539
\(171\) −1.00000 −0.0764719
\(172\) 3.26795 0.249179
\(173\) −9.12436 −0.693712 −0.346856 0.937918i \(-0.612751\pi\)
−0.346856 + 0.937918i \(0.612751\pi\)
\(174\) −6.46410 −0.490042
\(175\) −4.46410 −0.337454
\(176\) −1.73205 −0.130558
\(177\) 0.928203 0.0697680
\(178\) −10.4641 −0.784318
\(179\) 11.4641 0.856867 0.428434 0.903573i \(-0.359066\pi\)
0.428434 + 0.903573i \(0.359066\pi\)
\(180\) −0.732051 −0.0545638
\(181\) −2.66025 −0.197735 −0.0988676 0.995101i \(-0.531522\pi\)
−0.0988676 + 0.995101i \(0.531522\pi\)
\(182\) 0 0
\(183\) 5.19615 0.384111
\(184\) 3.46410 0.255377
\(185\) 4.92820 0.362329
\(186\) 2.19615 0.161030
\(187\) −6.46410 −0.472702
\(188\) 2.46410 0.179713
\(189\) 1.00000 0.0727393
\(190\) −0.732051 −0.0531085
\(191\) 23.1244 1.67322 0.836610 0.547799i \(-0.184534\pi\)
0.836610 + 0.547799i \(0.184534\pi\)
\(192\) 1.00000 0.0721688
\(193\) 1.19615 0.0861009 0.0430505 0.999073i \(-0.486292\pi\)
0.0430505 + 0.999073i \(0.486292\pi\)
\(194\) 15.6603 1.12434
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 26.6603 1.89946 0.949732 0.313063i \(-0.101355\pi\)
0.949732 + 0.313063i \(0.101355\pi\)
\(198\) 1.73205 0.123091
\(199\) −16.1962 −1.14811 −0.574057 0.818815i \(-0.694631\pi\)
−0.574057 + 0.818815i \(0.694631\pi\)
\(200\) 4.46410 0.315660
\(201\) 8.92820 0.629747
\(202\) −3.07180 −0.216131
\(203\) 6.46410 0.453691
\(204\) 3.73205 0.261296
\(205\) −2.19615 −0.153386
\(206\) −14.7321 −1.02643
\(207\) −3.46410 −0.240772
\(208\) 0 0
\(209\) 1.73205 0.119808
\(210\) 0.732051 0.0505163
\(211\) 26.0000 1.78991 0.894957 0.446153i \(-0.147206\pi\)
0.894957 + 0.446153i \(0.147206\pi\)
\(212\) 7.00000 0.480762
\(213\) 2.19615 0.150478
\(214\) −10.8564 −0.742129
\(215\) −2.39230 −0.163154
\(216\) −1.00000 −0.0680414
\(217\) −2.19615 −0.149085
\(218\) 4.73205 0.320495
\(219\) −5.46410 −0.369230
\(220\) 1.26795 0.0854851
\(221\) 0 0
\(222\) 6.73205 0.451826
\(223\) 24.3923 1.63343 0.816715 0.577042i \(-0.195793\pi\)
0.816715 + 0.577042i \(0.195793\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −4.46410 −0.297607
\(226\) −15.2679 −1.01561
\(227\) −18.3923 −1.22074 −0.610370 0.792116i \(-0.708979\pi\)
−0.610370 + 0.792116i \(0.708979\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −19.9282 −1.31689 −0.658446 0.752628i \(-0.728786\pi\)
−0.658446 + 0.752628i \(0.728786\pi\)
\(230\) −2.53590 −0.167212
\(231\) −1.73205 −0.113961
\(232\) −6.46410 −0.424389
\(233\) −9.12436 −0.597756 −0.298878 0.954291i \(-0.596612\pi\)
−0.298878 + 0.954291i \(0.596612\pi\)
\(234\) 0 0
\(235\) −1.80385 −0.117670
\(236\) 0.928203 0.0604209
\(237\) 2.07180 0.134578
\(238\) −3.73205 −0.241913
\(239\) −0.732051 −0.0473524 −0.0236762 0.999720i \(-0.507537\pi\)
−0.0236762 + 0.999720i \(0.507537\pi\)
\(240\) −0.732051 −0.0472537
\(241\) −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
\(245\) −0.732051 −0.0467690
\(246\) −3.00000 −0.191273
\(247\) 0 0
\(248\) 2.19615 0.139456
\(249\) −0.196152 −0.0124307
\(250\) −6.92820 −0.438178
\(251\) 13.1244 0.828402 0.414201 0.910185i \(-0.364061\pi\)
0.414201 + 0.910185i \(0.364061\pi\)
\(252\) 1.00000 0.0629941
\(253\) 6.00000 0.377217
\(254\) 15.4641 0.970304
\(255\) −2.73205 −0.171088
\(256\) 1.00000 0.0625000
\(257\) 20.6603 1.28875 0.644376 0.764709i \(-0.277117\pi\)
0.644376 + 0.764709i \(0.277117\pi\)
\(258\) −3.26795 −0.203454
\(259\) −6.73205 −0.418309
\(260\) 0 0
\(261\) 6.46410 0.400118
\(262\) 13.8564 0.856052
\(263\) −15.1244 −0.932608 −0.466304 0.884625i \(-0.654415\pi\)
−0.466304 + 0.884625i \(0.654415\pi\)
\(264\) 1.73205 0.106600
\(265\) −5.12436 −0.314787
\(266\) 1.00000 0.0613139
\(267\) 10.4641 0.640393
\(268\) 8.92820 0.545377
\(269\) −9.85641 −0.600956 −0.300478 0.953789i \(-0.597146\pi\)
−0.300478 + 0.953789i \(0.597146\pi\)
\(270\) 0.732051 0.0445512
\(271\) 28.0526 1.70407 0.852036 0.523484i \(-0.175368\pi\)
0.852036 + 0.523484i \(0.175368\pi\)
\(272\) 3.73205 0.226289
\(273\) 0 0
\(274\) −14.3923 −0.869471
\(275\) 7.73205 0.466260
\(276\) −3.46410 −0.208514
\(277\) 9.32051 0.560015 0.280008 0.959998i \(-0.409663\pi\)
0.280008 + 0.959998i \(0.409663\pi\)
\(278\) −11.5885 −0.695029
\(279\) −2.19615 −0.131480
\(280\) 0.732051 0.0437484
\(281\) −0.339746 −0.0202675 −0.0101338 0.999949i \(-0.503226\pi\)
−0.0101338 + 0.999949i \(0.503226\pi\)
\(282\) −2.46410 −0.146735
\(283\) −9.85641 −0.585903 −0.292951 0.956127i \(-0.594637\pi\)
−0.292951 + 0.956127i \(0.594637\pi\)
\(284\) 2.19615 0.130318
\(285\) 0.732051 0.0433629
\(286\) 0 0
\(287\) 3.00000 0.177084
\(288\) −1.00000 −0.0589256
\(289\) −3.07180 −0.180694
\(290\) 4.73205 0.277876
\(291\) −15.6603 −0.918020
\(292\) −5.46410 −0.319762
\(293\) −17.0718 −0.997345 −0.498673 0.866790i \(-0.666179\pi\)
−0.498673 + 0.866790i \(0.666179\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −0.679492 −0.0395615
\(296\) 6.73205 0.391293
\(297\) −1.73205 −0.100504
\(298\) 13.8564 0.802680
\(299\) 0 0
\(300\) −4.46410 −0.257735
\(301\) 3.26795 0.188361
\(302\) −5.19615 −0.299005
\(303\) 3.07180 0.176470
\(304\) −1.00000 −0.0573539
\(305\) −3.80385 −0.217808
\(306\) −3.73205 −0.213347
\(307\) 8.60770 0.491267 0.245634 0.969363i \(-0.421004\pi\)
0.245634 + 0.969363i \(0.421004\pi\)
\(308\) −1.73205 −0.0986928
\(309\) 14.7321 0.838077
\(310\) −1.60770 −0.0913109
\(311\) 4.26795 0.242013 0.121007 0.992652i \(-0.461388\pi\)
0.121007 + 0.992652i \(0.461388\pi\)
\(312\) 0 0
\(313\) 13.5167 0.764007 0.382003 0.924161i \(-0.375234\pi\)
0.382003 + 0.924161i \(0.375234\pi\)
\(314\) 0.535898 0.0302425
\(315\) −0.732051 −0.0412464
\(316\) 2.07180 0.116548
\(317\) −11.8564 −0.665922 −0.332961 0.942941i \(-0.608048\pi\)
−0.332961 + 0.942941i \(0.608048\pi\)
\(318\) −7.00000 −0.392541
\(319\) −11.1962 −0.626864
\(320\) −0.732051 −0.0409229
\(321\) 10.8564 0.605946
\(322\) 3.46410 0.193047
\(323\) −3.73205 −0.207657
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −2.73205 −0.151314
\(327\) −4.73205 −0.261683
\(328\) −3.00000 −0.165647
\(329\) 2.46410 0.135850
\(330\) −1.26795 −0.0697983
\(331\) −27.8564 −1.53113 −0.765563 0.643361i \(-0.777540\pi\)
−0.765563 + 0.643361i \(0.777540\pi\)
\(332\) −0.196152 −0.0107653
\(333\) −6.73205 −0.368914
\(334\) −13.8564 −0.758189
\(335\) −6.53590 −0.357094
\(336\) 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
\(339\) 15.2679 0.829241
\(340\) −2.73205 −0.148166
\(341\) 3.80385 0.205990
\(342\) 1.00000 0.0540738
\(343\) 1.00000 0.0539949
\(344\) −3.26795 −0.176196
\(345\) 2.53590 0.136528
\(346\) 9.12436 0.490528
\(347\) 5.39230 0.289474 0.144737 0.989470i \(-0.453766\pi\)
0.144737 + 0.989470i \(0.453766\pi\)
\(348\) 6.46410 0.346512
\(349\) 23.7128 1.26932 0.634659 0.772792i \(-0.281141\pi\)
0.634659 + 0.772792i \(0.281141\pi\)
\(350\) 4.46410 0.238616
\(351\) 0 0
\(352\) 1.73205 0.0923186
\(353\) −9.46410 −0.503723 −0.251862 0.967763i \(-0.581043\pi\)
−0.251862 + 0.967763i \(0.581043\pi\)
\(354\) −0.928203 −0.0493334
\(355\) −1.60770 −0.0853276
\(356\) 10.4641 0.554596
\(357\) 3.73205 0.197521
\(358\) −11.4641 −0.605897
\(359\) −3.80385 −0.200759 −0.100380 0.994949i \(-0.532006\pi\)
−0.100380 + 0.994949i \(0.532006\pi\)
\(360\) 0.732051 0.0385825
\(361\) −18.0000 −0.947368
\(362\) 2.66025 0.139820
\(363\) −8.00000 −0.419891
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) −5.19615 −0.271607
\(367\) 25.3205 1.32172 0.660860 0.750509i \(-0.270192\pi\)
0.660860 + 0.750509i \(0.270192\pi\)
\(368\) −3.46410 −0.180579
\(369\) 3.00000 0.156174
\(370\) −4.92820 −0.256205
\(371\) 7.00000 0.363422
\(372\) −2.19615 −0.113865
\(373\) 9.66025 0.500189 0.250094 0.968221i \(-0.419538\pi\)
0.250094 + 0.968221i \(0.419538\pi\)
\(374\) 6.46410 0.334251
\(375\) 6.92820 0.357771
\(376\) −2.46410 −0.127076
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) 28.4449 1.46111 0.730557 0.682851i \(-0.239260\pi\)
0.730557 + 0.682851i \(0.239260\pi\)
\(380\) 0.732051 0.0375534
\(381\) −15.4641 −0.792250
\(382\) −23.1244 −1.18314
\(383\) 5.53590 0.282871 0.141436 0.989947i \(-0.454828\pi\)
0.141436 + 0.989947i \(0.454828\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 1.26795 0.0646207
\(386\) −1.19615 −0.0608826
\(387\) 3.26795 0.166119
\(388\) −15.6603 −0.795029
\(389\) 8.39230 0.425507 0.212753 0.977106i \(-0.431757\pi\)
0.212753 + 0.977106i \(0.431757\pi\)
\(390\) 0 0
\(391\) −12.9282 −0.653807
\(392\) −1.00000 −0.0505076
\(393\) −13.8564 −0.698963
\(394\) −26.6603 −1.34312
\(395\) −1.51666 −0.0763115
\(396\) −1.73205 −0.0870388
\(397\) 8.07180 0.405112 0.202556 0.979271i \(-0.435075\pi\)
0.202556 + 0.979271i \(0.435075\pi\)
\(398\) 16.1962 0.811840
\(399\) −1.00000 −0.0500626
\(400\) −4.46410 −0.223205
\(401\) 38.7846 1.93681 0.968405 0.249381i \(-0.0802271\pi\)
0.968405 + 0.249381i \(0.0802271\pi\)
\(402\) −8.92820 −0.445298
\(403\) 0 0
\(404\) 3.07180 0.152828
\(405\) −0.732051 −0.0363759
\(406\) −6.46410 −0.320808
\(407\) 11.6603 0.577977
\(408\) −3.73205 −0.184764
\(409\) 38.7321 1.91518 0.957588 0.288140i \(-0.0930368\pi\)
0.957588 + 0.288140i \(0.0930368\pi\)
\(410\) 2.19615 0.108460
\(411\) 14.3923 0.709920
\(412\) 14.7321 0.725796
\(413\) 0.928203 0.0456739
\(414\) 3.46410 0.170251
\(415\) 0.143594 0.00704873
\(416\) 0 0
\(417\) 11.5885 0.567489
\(418\) −1.73205 −0.0847174
\(419\) 0.732051 0.0357630 0.0178815 0.999840i \(-0.494308\pi\)
0.0178815 + 0.999840i \(0.494308\pi\)
\(420\) −0.732051 −0.0357204
\(421\) −22.3923 −1.09133 −0.545667 0.838002i \(-0.683724\pi\)
−0.545667 + 0.838002i \(0.683724\pi\)
\(422\) −26.0000 −1.26566
\(423\) 2.46410 0.119809
\(424\) −7.00000 −0.339950
\(425\) −16.6603 −0.808141
\(426\) −2.19615 −0.106404
\(427\) 5.19615 0.251459
\(428\) 10.8564 0.524764
\(429\) 0 0
\(430\) 2.39230 0.115367
\(431\) −14.1962 −0.683805 −0.341902 0.939736i \(-0.611071\pi\)
−0.341902 + 0.939736i \(0.611071\pi\)
\(432\) 1.00000 0.0481125
\(433\) −27.0718 −1.30099 −0.650494 0.759512i \(-0.725438\pi\)
−0.650494 + 0.759512i \(0.725438\pi\)
\(434\) 2.19615 0.105419
\(435\) −4.73205 −0.226884
\(436\) −4.73205 −0.226624
\(437\) 3.46410 0.165710
\(438\) 5.46410 0.261085
\(439\) −25.1769 −1.20163 −0.600814 0.799389i \(-0.705157\pi\)
−0.600814 + 0.799389i \(0.705157\pi\)
\(440\) −1.26795 −0.0604471
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −33.6410 −1.59833 −0.799166 0.601110i \(-0.794725\pi\)
−0.799166 + 0.601110i \(0.794725\pi\)
\(444\) −6.73205 −0.319489
\(445\) −7.66025 −0.363131
\(446\) −24.3923 −1.15501
\(447\) −13.8564 −0.655386
\(448\) 1.00000 0.0472456
\(449\) 20.5885 0.971629 0.485815 0.874062i \(-0.338523\pi\)
0.485815 + 0.874062i \(0.338523\pi\)
\(450\) 4.46410 0.210440
\(451\) −5.19615 −0.244677
\(452\) 15.2679 0.718144
\(453\) 5.19615 0.244137
\(454\) 18.3923 0.863194
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 11.8564 0.554619 0.277310 0.960781i \(-0.410557\pi\)
0.277310 + 0.960781i \(0.410557\pi\)
\(458\) 19.9282 0.931184
\(459\) 3.73205 0.174197
\(460\) 2.53590 0.118237
\(461\) 26.9282 1.25417 0.627086 0.778950i \(-0.284248\pi\)
0.627086 + 0.778950i \(0.284248\pi\)
\(462\) 1.73205 0.0805823
\(463\) −6.26795 −0.291296 −0.145648 0.989336i \(-0.546527\pi\)
−0.145648 + 0.989336i \(0.546527\pi\)
\(464\) 6.46410 0.300088
\(465\) 1.60770 0.0745551
\(466\) 9.12436 0.422678
\(467\) −13.8564 −0.641198 −0.320599 0.947215i \(-0.603884\pi\)
−0.320599 + 0.947215i \(0.603884\pi\)
\(468\) 0 0
\(469\) 8.92820 0.412266
\(470\) 1.80385 0.0832053
\(471\) −0.535898 −0.0246929
\(472\) −0.928203 −0.0427240
\(473\) −5.66025 −0.260259
\(474\) −2.07180 −0.0951608
\(475\) 4.46410 0.204827
\(476\) 3.73205 0.171058
\(477\) 7.00000 0.320508
\(478\) 0.732051 0.0334832
\(479\) 32.4641 1.48332 0.741661 0.670775i \(-0.234038\pi\)
0.741661 + 0.670775i \(0.234038\pi\)
\(480\) 0.732051 0.0334134
\(481\) 0 0
\(482\) 8.00000 0.364390
\(483\) −3.46410 −0.157622
\(484\) −8.00000 −0.363636
\(485\) 11.4641 0.520558
\(486\) −1.00000 −0.0453609
\(487\) 16.6603 0.754948 0.377474 0.926020i \(-0.376793\pi\)
0.377474 + 0.926020i \(0.376793\pi\)
\(488\) −5.19615 −0.235219
\(489\) 2.73205 0.123548
\(490\) 0.732051 0.0330707
\(491\) 35.4641 1.60047 0.800236 0.599685i \(-0.204707\pi\)
0.800236 + 0.599685i \(0.204707\pi\)
\(492\) 3.00000 0.135250
\(493\) 24.1244 1.08651
\(494\) 0 0
\(495\) 1.26795 0.0569901
\(496\) −2.19615 −0.0986102
\(497\) 2.19615 0.0985109
\(498\) 0.196152 0.00878980
\(499\) 11.5167 0.515557 0.257778 0.966204i \(-0.417010\pi\)
0.257778 + 0.966204i \(0.417010\pi\)
\(500\) 6.92820 0.309839
\(501\) 13.8564 0.619059
\(502\) −13.1244 −0.585769
\(503\) 6.92820 0.308913 0.154457 0.988000i \(-0.450637\pi\)
0.154457 + 0.988000i \(0.450637\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −2.24871 −0.100066
\(506\) −6.00000 −0.266733
\(507\) 0 0
\(508\) −15.4641 −0.686109
\(509\) −22.3397 −0.990192 −0.495096 0.868838i \(-0.664867\pi\)
−0.495096 + 0.868838i \(0.664867\pi\)
\(510\) 2.73205 0.120977
\(511\) −5.46410 −0.241718
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −20.6603 −0.911285
\(515\) −10.7846 −0.475227
\(516\) 3.26795 0.143863
\(517\) −4.26795 −0.187704
\(518\) 6.73205 0.295789
\(519\) −9.12436 −0.400515
\(520\) 0 0
\(521\) 7.05256 0.308978 0.154489 0.987994i \(-0.450627\pi\)
0.154489 + 0.987994i \(0.450627\pi\)
\(522\) −6.46410 −0.282926
\(523\) −13.8756 −0.606740 −0.303370 0.952873i \(-0.598112\pi\)
−0.303370 + 0.952873i \(0.598112\pi\)
\(524\) −13.8564 −0.605320
\(525\) −4.46410 −0.194829
\(526\) 15.1244 0.659453
\(527\) −8.19615 −0.357030
\(528\) −1.73205 −0.0753778
\(529\) −11.0000 −0.478261
\(530\) 5.12436 0.222588
\(531\) 0.928203 0.0402806
\(532\) −1.00000 −0.0433555
\(533\) 0 0
\(534\) −10.4641 −0.452826
\(535\) −7.94744 −0.343598
\(536\) −8.92820 −0.385640
\(537\) 11.4641 0.494713
\(538\) 9.85641 0.424940
\(539\) −1.73205 −0.0746047
\(540\) −0.732051 −0.0315025
\(541\) −24.3397 −1.04645 −0.523224 0.852195i \(-0.675271\pi\)
−0.523224 + 0.852195i \(0.675271\pi\)
\(542\) −28.0526 −1.20496
\(543\) −2.66025 −0.114162
\(544\) −3.73205 −0.160010
\(545\) 3.46410 0.148386
\(546\) 0 0
\(547\) −1.26795 −0.0542136 −0.0271068 0.999633i \(-0.508629\pi\)
−0.0271068 + 0.999633i \(0.508629\pi\)
\(548\) 14.3923 0.614809
\(549\) 5.19615 0.221766
\(550\) −7.73205 −0.329696
\(551\) −6.46410 −0.275380
\(552\) 3.46410 0.147442
\(553\) 2.07180 0.0881018
\(554\) −9.32051 −0.395990
\(555\) 4.92820 0.209191
\(556\) 11.5885 0.491460
\(557\) −13.0526 −0.553055 −0.276527 0.961006i \(-0.589184\pi\)
−0.276527 + 0.961006i \(0.589184\pi\)
\(558\) 2.19615 0.0929705
\(559\) 0 0
\(560\) −0.732051 −0.0309348
\(561\) −6.46410 −0.272915
\(562\) 0.339746 0.0143313
\(563\) −36.1962 −1.52549 −0.762743 0.646702i \(-0.776148\pi\)
−0.762743 + 0.646702i \(0.776148\pi\)
\(564\) 2.46410 0.103757
\(565\) −11.1769 −0.470216
\(566\) 9.85641 0.414296
\(567\) 1.00000 0.0419961
\(568\) −2.19615 −0.0921485
\(569\) −38.4449 −1.61169 −0.805846 0.592125i \(-0.798289\pi\)
−0.805846 + 0.592125i \(0.798289\pi\)
\(570\) −0.732051 −0.0306622
\(571\) 16.9808 0.710623 0.355311 0.934748i \(-0.384375\pi\)
0.355311 + 0.934748i \(0.384375\pi\)
\(572\) 0 0
\(573\) 23.1244 0.966034
\(574\) −3.00000 −0.125218
\(575\) 15.4641 0.644898
\(576\) 1.00000 0.0416667
\(577\) 10.3397 0.430449 0.215225 0.976565i \(-0.430952\pi\)
0.215225 + 0.976565i \(0.430952\pi\)
\(578\) 3.07180 0.127770
\(579\) 1.19615 0.0497104
\(580\) −4.73205 −0.196488
\(581\) −0.196152 −0.00813777
\(582\) 15.6603 0.649138
\(583\) −12.1244 −0.502140
\(584\) 5.46410 0.226106
\(585\) 0 0
\(586\) 17.0718 0.705229
\(587\) 12.3397 0.509316 0.254658 0.967031i \(-0.418037\pi\)
0.254658 + 0.967031i \(0.418037\pi\)
\(588\) 1.00000 0.0412393
\(589\) 2.19615 0.0904909
\(590\) 0.679492 0.0279742
\(591\) 26.6603 1.09666
\(592\) −6.73205 −0.276686
\(593\) 0.464102 0.0190584 0.00952918 0.999955i \(-0.496967\pi\)
0.00952918 + 0.999955i \(0.496967\pi\)
\(594\) 1.73205 0.0710669
\(595\) −2.73205 −0.112003
\(596\) −13.8564 −0.567581
\(597\) −16.1962 −0.662864
\(598\) 0 0
\(599\) −28.6410 −1.17024 −0.585120 0.810947i \(-0.698953\pi\)
−0.585120 + 0.810947i \(0.698953\pi\)
\(600\) 4.46410 0.182246
\(601\) −22.7321 −0.927260 −0.463630 0.886029i \(-0.653453\pi\)
−0.463630 + 0.886029i \(0.653453\pi\)
\(602\) −3.26795 −0.133192
\(603\) 8.92820 0.363585
\(604\) 5.19615 0.211428
\(605\) 5.85641 0.238097
\(606\) −3.07180 −0.124783
\(607\) −31.5167 −1.27922 −0.639611 0.768699i \(-0.720905\pi\)
−0.639611 + 0.768699i \(0.720905\pi\)
\(608\) 1.00000 0.0405554
\(609\) 6.46410 0.261939
\(610\) 3.80385 0.154013
\(611\) 0 0
\(612\) 3.73205 0.150859
\(613\) 4.00000 0.161558 0.0807792 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) −8.60770 −0.347378
\(615\) −2.19615 −0.0885574
\(616\) 1.73205 0.0697863
\(617\) −2.19615 −0.0884138 −0.0442069 0.999022i \(-0.514076\pi\)
−0.0442069 + 0.999022i \(0.514076\pi\)
\(618\) −14.7321 −0.592610
\(619\) 22.0718 0.887140 0.443570 0.896240i \(-0.353712\pi\)
0.443570 + 0.896240i \(0.353712\pi\)
\(620\) 1.60770 0.0645666
\(621\) −3.46410 −0.139010
\(622\) −4.26795 −0.171129
\(623\) 10.4641 0.419235
\(624\) 0 0
\(625\) 17.2487 0.689948
\(626\) −13.5167 −0.540234
\(627\) 1.73205 0.0691714
\(628\) −0.535898 −0.0213847
\(629\) −25.1244 −1.00177
\(630\) 0.732051 0.0291656
\(631\) 16.8038 0.668951 0.334475 0.942405i \(-0.391441\pi\)
0.334475 + 0.942405i \(0.391441\pi\)
\(632\) −2.07180 −0.0824117
\(633\) 26.0000 1.03341
\(634\) 11.8564 0.470878
\(635\) 11.3205 0.449241
\(636\) 7.00000 0.277568
\(637\) 0 0
\(638\) 11.1962 0.443260
\(639\) 2.19615 0.0868784
\(640\) 0.732051 0.0289368
\(641\) 31.1769 1.23141 0.615707 0.787975i \(-0.288870\pi\)
0.615707 + 0.787975i \(0.288870\pi\)
\(642\) −10.8564 −0.428468
\(643\) −32.8564 −1.29573 −0.647865 0.761755i \(-0.724338\pi\)
−0.647865 + 0.761755i \(0.724338\pi\)
\(644\) −3.46410 −0.136505
\(645\) −2.39230 −0.0941969
\(646\) 3.73205 0.146836
\(647\) 35.8372 1.40890 0.704452 0.709751i \(-0.251193\pi\)
0.704452 + 0.709751i \(0.251193\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −1.60770 −0.0631076
\(650\) 0 0
\(651\) −2.19615 −0.0860740
\(652\) 2.73205 0.106995
\(653\) 46.7128 1.82801 0.914007 0.405699i \(-0.132972\pi\)
0.914007 + 0.405699i \(0.132972\pi\)
\(654\) 4.73205 0.185038
\(655\) 10.1436 0.396343
\(656\) 3.00000 0.117130
\(657\) −5.46410 −0.213175
\(658\) −2.46410 −0.0960607
\(659\) 18.6077 0.724853 0.362426 0.932012i \(-0.381948\pi\)
0.362426 + 0.932012i \(0.381948\pi\)
\(660\) 1.26795 0.0493549
\(661\) −12.3923 −0.482005 −0.241002 0.970525i \(-0.577476\pi\)
−0.241002 + 0.970525i \(0.577476\pi\)
\(662\) 27.8564 1.08267
\(663\) 0 0
\(664\) 0.196152 0.00761219
\(665\) 0.732051 0.0283877
\(666\) 6.73205 0.260862
\(667\) −22.3923 −0.867034
\(668\) 13.8564 0.536120
\(669\) 24.3923 0.943061
\(670\) 6.53590 0.252504
\(671\) −9.00000 −0.347441
\(672\) −1.00000 −0.0385758
\(673\) −32.7128 −1.26099 −0.630493 0.776195i \(-0.717147\pi\)
−0.630493 + 0.776195i \(0.717147\pi\)
\(674\) −19.0000 −0.731853
\(675\) −4.46410 −0.171823
\(676\) 0 0
\(677\) 36.7321 1.41173 0.705864 0.708348i \(-0.250559\pi\)
0.705864 + 0.708348i \(0.250559\pi\)
\(678\) −15.2679 −0.586362
\(679\) −15.6603 −0.600985
\(680\) 2.73205 0.104769
\(681\) −18.3923 −0.704795
\(682\) −3.80385 −0.145657
\(683\) 29.6077 1.13291 0.566453 0.824094i \(-0.308315\pi\)
0.566453 + 0.824094i \(0.308315\pi\)
\(684\) −1.00000 −0.0382360
\(685\) −10.5359 −0.402556
\(686\) −1.00000 −0.0381802
\(687\) −19.9282 −0.760308
\(688\) 3.26795 0.124589
\(689\) 0 0
\(690\) −2.53590 −0.0965400
\(691\) 43.7128 1.66291 0.831457 0.555589i \(-0.187507\pi\)
0.831457 + 0.555589i \(0.187507\pi\)
\(692\) −9.12436 −0.346856
\(693\) −1.73205 −0.0657952
\(694\) −5.39230 −0.204689
\(695\) −8.48334 −0.321791
\(696\) −6.46410 −0.245021
\(697\) 11.1962 0.424085
\(698\) −23.7128 −0.897543
\(699\) −9.12436 −0.345115
\(700\) −4.46410 −0.168727
\(701\) 1.14359 0.0431929 0.0215965 0.999767i \(-0.493125\pi\)
0.0215965 + 0.999767i \(0.493125\pi\)
\(702\) 0 0
\(703\) 6.73205 0.253904
\(704\) −1.73205 −0.0652791
\(705\) −1.80385 −0.0679368
\(706\) 9.46410 0.356186
\(707\) 3.07180 0.115527
\(708\) 0.928203 0.0348840
\(709\) −32.0526 −1.20376 −0.601880 0.798587i \(-0.705581\pi\)
−0.601880 + 0.798587i \(0.705581\pi\)
\(710\) 1.60770 0.0603357
\(711\) 2.07180 0.0776984
\(712\) −10.4641 −0.392159
\(713\) 7.60770 0.284910
\(714\) −3.73205 −0.139668
\(715\) 0 0
\(716\) 11.4641 0.428434
\(717\) −0.732051 −0.0273389
\(718\) 3.80385 0.141958
\(719\) 26.5167 0.988905 0.494452 0.869205i \(-0.335369\pi\)
0.494452 + 0.869205i \(0.335369\pi\)
\(720\) −0.732051 −0.0272819
\(721\) 14.7321 0.548650
\(722\) 18.0000 0.669891
\(723\) −8.00000 −0.297523
\(724\) −2.66025 −0.0988676
\(725\) −28.8564 −1.07170
\(726\) 8.00000 0.296908
\(727\) −19.3205 −0.716558 −0.358279 0.933615i \(-0.616636\pi\)
−0.358279 + 0.933615i \(0.616636\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) 12.1962 0.451091
\(732\) 5.19615 0.192055
\(733\) 48.1769 1.77945 0.889727 0.456492i \(-0.150894\pi\)
0.889727 + 0.456492i \(0.150894\pi\)
\(734\) −25.3205 −0.934597
\(735\) −0.732051 −0.0270021
\(736\) 3.46410 0.127688
\(737\) −15.4641 −0.569628
\(738\) −3.00000 −0.110432
\(739\) −15.1769 −0.558292 −0.279146 0.960249i \(-0.590051\pi\)
−0.279146 + 0.960249i \(0.590051\pi\)
\(740\) 4.92820 0.181164
\(741\) 0 0
\(742\) −7.00000 −0.256978
\(743\) −14.0526 −0.515538 −0.257769 0.966207i \(-0.582987\pi\)
−0.257769 + 0.966207i \(0.582987\pi\)
\(744\) 2.19615 0.0805149
\(745\) 10.1436 0.371633
\(746\) −9.66025 −0.353687
\(747\) −0.196152 −0.00717684
\(748\) −6.46410 −0.236351
\(749\) 10.8564 0.396684
\(750\) −6.92820 −0.252982
\(751\) −19.1436 −0.698560 −0.349280 0.937018i \(-0.613574\pi\)
−0.349280 + 0.937018i \(0.613574\pi\)
\(752\) 2.46410 0.0898565
\(753\) 13.1244 0.478278
\(754\) 0 0
\(755\) −3.80385 −0.138436
\(756\) 1.00000 0.0363696
\(757\) −0.588457 −0.0213878 −0.0106939 0.999943i \(-0.503404\pi\)
−0.0106939 + 0.999943i \(0.503404\pi\)
\(758\) −28.4449 −1.03316
\(759\) 6.00000 0.217786
\(760\) −0.732051 −0.0265543
\(761\) 15.3205 0.555368 0.277684 0.960672i \(-0.410433\pi\)
0.277684 + 0.960672i \(0.410433\pi\)
\(762\) 15.4641 0.560205
\(763\) −4.73205 −0.171312
\(764\) 23.1244 0.836610
\(765\) −2.73205 −0.0987775
\(766\) −5.53590 −0.200020
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 42.3013 1.52542 0.762711 0.646739i \(-0.223868\pi\)
0.762711 + 0.646739i \(0.223868\pi\)
\(770\) −1.26795 −0.0456937
\(771\) 20.6603 0.744061
\(772\) 1.19615 0.0430505
\(773\) 47.1769 1.69684 0.848418 0.529327i \(-0.177556\pi\)
0.848418 + 0.529327i \(0.177556\pi\)
\(774\) −3.26795 −0.117464
\(775\) 9.80385 0.352165
\(776\) 15.6603 0.562170
\(777\) −6.73205 −0.241511
\(778\) −8.39230 −0.300879
\(779\) −3.00000 −0.107486
\(780\) 0 0
\(781\) −3.80385 −0.136112
\(782\) 12.9282 0.462312
\(783\) 6.46410 0.231008
\(784\) 1.00000 0.0357143
\(785\) 0.392305 0.0140020
\(786\) 13.8564 0.494242
\(787\) 3.67949 0.131160 0.0655799 0.997847i \(-0.479110\pi\)
0.0655799 + 0.997847i \(0.479110\pi\)
\(788\) 26.6603 0.949732
\(789\) −15.1244 −0.538441
\(790\) 1.51666 0.0539604
\(791\) 15.2679 0.542866
\(792\) 1.73205 0.0615457
\(793\) 0 0
\(794\) −8.07180 −0.286457
\(795\) −5.12436 −0.181742
\(796\) −16.1962 −0.574057
\(797\) −2.87564 −0.101861 −0.0509303 0.998702i \(-0.516219\pi\)
−0.0509303 + 0.998702i \(0.516219\pi\)
\(798\) 1.00000 0.0353996
\(799\) 9.19615 0.325336
\(800\) 4.46410 0.157830
\(801\) 10.4641 0.369731
\(802\) −38.7846 −1.36953
\(803\) 9.46410 0.333981
\(804\) 8.92820 0.314873
\(805\) 2.53590 0.0893787
\(806\) 0 0
\(807\) −9.85641 −0.346962
\(808\) −3.07180 −0.108065
\(809\) −6.44486 −0.226589 −0.113295 0.993561i \(-0.536140\pi\)
−0.113295 + 0.993561i \(0.536140\pi\)
\(810\) 0.732051 0.0257216
\(811\) 24.5359 0.861572 0.430786 0.902454i \(-0.358236\pi\)
0.430786 + 0.902454i \(0.358236\pi\)
\(812\) 6.46410 0.226845
\(813\) 28.0526 0.983846
\(814\) −11.6603 −0.408692
\(815\) −2.00000 −0.0700569
\(816\) 3.73205 0.130648
\(817\) −3.26795 −0.114331
\(818\) −38.7321 −1.35423
\(819\) 0 0
\(820\) −2.19615 −0.0766930
\(821\) −51.8372 −1.80913 −0.904565 0.426336i \(-0.859804\pi\)
−0.904565 + 0.426336i \(0.859804\pi\)
\(822\) −14.3923 −0.501989
\(823\) −6.78461 −0.236497 −0.118248 0.992984i \(-0.537728\pi\)
−0.118248 + 0.992984i \(0.537728\pi\)
\(824\) −14.7321 −0.513215
\(825\) 7.73205 0.269195
\(826\) −0.928203 −0.0322963
\(827\) −7.32051 −0.254559 −0.127280 0.991867i \(-0.540625\pi\)
−0.127280 + 0.991867i \(0.540625\pi\)
\(828\) −3.46410 −0.120386
\(829\) −17.3397 −0.602234 −0.301117 0.953587i \(-0.597360\pi\)
−0.301117 + 0.953587i \(0.597360\pi\)
\(830\) −0.143594 −0.00498420
\(831\) 9.32051 0.323325
\(832\) 0 0
\(833\) 3.73205 0.129308
\(834\) −11.5885 −0.401275
\(835\) −10.1436 −0.351034
\(836\) 1.73205 0.0599042
\(837\) −2.19615 −0.0759101
\(838\) −0.732051 −0.0252883
\(839\) −34.1051 −1.17744 −0.588720 0.808337i \(-0.700368\pi\)
−0.588720 + 0.808337i \(0.700368\pi\)
\(840\) 0.732051 0.0252582
\(841\) 12.7846 0.440849
\(842\) 22.3923 0.771690
\(843\) −0.339746 −0.0117015
\(844\) 26.0000 0.894957
\(845\) 0 0
\(846\) −2.46410 −0.0847176
\(847\) −8.00000 −0.274883
\(848\) 7.00000 0.240381
\(849\) −9.85641 −0.338271
\(850\) 16.6603 0.571442
\(851\) 23.3205 0.799417
\(852\) 2.19615 0.0752389
\(853\) −7.92820 −0.271457 −0.135728 0.990746i \(-0.543337\pi\)
−0.135728 + 0.990746i \(0.543337\pi\)
\(854\) −5.19615 −0.177809
\(855\) 0.732051 0.0250356
\(856\) −10.8564 −0.371064
\(857\) −32.5359 −1.11141 −0.555703 0.831381i \(-0.687551\pi\)
−0.555703 + 0.831381i \(0.687551\pi\)
\(858\) 0 0
\(859\) 42.7654 1.45914 0.729568 0.683908i \(-0.239721\pi\)
0.729568 + 0.683908i \(0.239721\pi\)
\(860\) −2.39230 −0.0815769
\(861\) 3.00000 0.102240
\(862\) 14.1962 0.483523
\(863\) −3.07180 −0.104565 −0.0522826 0.998632i \(-0.516650\pi\)
−0.0522826 + 0.998632i \(0.516650\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.67949 0.227110
\(866\) 27.0718 0.919937
\(867\) −3.07180 −0.104324
\(868\) −2.19615 −0.0745423
\(869\) −3.58846 −0.121730
\(870\) 4.73205 0.160432
\(871\) 0 0
\(872\) 4.73205 0.160247
\(873\) −15.6603 −0.530019
\(874\) −3.46410 −0.117175
\(875\) 6.92820 0.234216
\(876\) −5.46410 −0.184615
\(877\) −23.2679 −0.785703 −0.392851 0.919602i \(-0.628511\pi\)
−0.392851 + 0.919602i \(0.628511\pi\)
\(878\) 25.1769 0.849680
\(879\) −17.0718 −0.575817
\(880\) 1.26795 0.0427426
\(881\) 22.9282 0.772471 0.386235 0.922400i \(-0.373775\pi\)
0.386235 + 0.922400i \(0.373775\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −8.05256 −0.270990 −0.135495 0.990778i \(-0.543262\pi\)
−0.135495 + 0.990778i \(0.543262\pi\)
\(884\) 0 0
\(885\) −0.679492 −0.0228409
\(886\) 33.6410 1.13019
\(887\) −15.4449 −0.518588 −0.259294 0.965798i \(-0.583490\pi\)
−0.259294 + 0.965798i \(0.583490\pi\)
\(888\) 6.73205 0.225913
\(889\) −15.4641 −0.518649
\(890\) 7.66025 0.256772
\(891\) −1.73205 −0.0580259
\(892\) 24.3923 0.816715
\(893\) −2.46410 −0.0824580
\(894\) 13.8564 0.463428
\(895\) −8.39230 −0.280524
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −20.5885 −0.687046
\(899\) −14.1962 −0.473468
\(900\) −4.46410 −0.148803
\(901\) 26.1244 0.870328
\(902\) 5.19615 0.173013
\(903\) 3.26795 0.108751
\(904\) −15.2679 −0.507804
\(905\) 1.94744 0.0647351
\(906\) −5.19615 −0.172631
\(907\) −1.60770 −0.0533826 −0.0266913 0.999644i \(-0.508497\pi\)
−0.0266913 + 0.999644i \(0.508497\pi\)
\(908\) −18.3923 −0.610370
\(909\) 3.07180 0.101885
\(910\) 0 0
\(911\) −50.8372 −1.68431 −0.842155 0.539235i \(-0.818713\pi\)
−0.842155 + 0.539235i \(0.818713\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 0.339746 0.0112440
\(914\) −11.8564 −0.392175
\(915\) −3.80385 −0.125751
\(916\) −19.9282 −0.658446
\(917\) −13.8564 −0.457579
\(918\) −3.73205 −0.123176
\(919\) 32.8564 1.08383 0.541916 0.840432i \(-0.317699\pi\)
0.541916 + 0.840432i \(0.317699\pi\)
\(920\) −2.53590 −0.0836061
\(921\) 8.60770 0.283633
\(922\) −26.9282 −0.886833
\(923\) 0 0
\(924\) −1.73205 −0.0569803
\(925\) 30.0526 0.988122
\(926\) 6.26795 0.205978
\(927\) 14.7321 0.483864
\(928\) −6.46410 −0.212195
\(929\) −2.32051 −0.0761334 −0.0380667 0.999275i \(-0.512120\pi\)
−0.0380667 + 0.999275i \(0.512120\pi\)
\(930\) −1.60770 −0.0527184
\(931\) −1.00000 −0.0327737
\(932\) −9.12436 −0.298878
\(933\) 4.26795 0.139726
\(934\) 13.8564 0.453395
\(935\) 4.73205 0.154755
\(936\) 0 0
\(937\) 1.07180 0.0350141 0.0175070 0.999847i \(-0.494427\pi\)
0.0175070 + 0.999847i \(0.494427\pi\)
\(938\) −8.92820 −0.291516
\(939\) 13.5167 0.441100
\(940\) −1.80385 −0.0588350
\(941\) 11.6077 0.378400 0.189200 0.981939i \(-0.439411\pi\)
0.189200 + 0.981939i \(0.439411\pi\)
\(942\) 0.535898 0.0174605
\(943\) −10.3923 −0.338420
\(944\) 0.928203 0.0302104
\(945\) −0.732051 −0.0238136
\(946\) 5.66025 0.184031
\(947\) 32.5167 1.05665 0.528325 0.849042i \(-0.322820\pi\)
0.528325 + 0.849042i \(0.322820\pi\)
\(948\) 2.07180 0.0672888
\(949\) 0 0
\(950\) −4.46410 −0.144835
\(951\) −11.8564 −0.384470
\(952\) −3.73205 −0.120956
\(953\) 34.1962 1.10772 0.553861 0.832609i \(-0.313154\pi\)
0.553861 + 0.832609i \(0.313154\pi\)
\(954\) −7.00000 −0.226633
\(955\) −16.9282 −0.547784
\(956\) −0.732051 −0.0236762
\(957\) −11.1962 −0.361920
\(958\) −32.4641 −1.04887
\(959\) 14.3923 0.464752
\(960\) −0.732051 −0.0236268
\(961\) −26.1769 −0.844417
\(962\) 0 0
\(963\) 10.8564 0.349843
\(964\) −8.00000 −0.257663
\(965\) −0.875644 −0.0281880
\(966\) 3.46410 0.111456
\(967\) −0.287187 −0.00923531 −0.00461766 0.999989i \(-0.501470\pi\)
−0.00461766 + 0.999989i \(0.501470\pi\)
\(968\) 8.00000 0.257130
\(969\) −3.73205 −0.119891
\(970\) −11.4641 −0.368090
\(971\) −54.1051 −1.73632 −0.868158 0.496288i \(-0.834696\pi\)
−0.868158 + 0.496288i \(0.834696\pi\)
\(972\) 1.00000 0.0320750
\(973\) 11.5885 0.371509
\(974\) −16.6603 −0.533829
\(975\) 0 0
\(976\) 5.19615 0.166325
\(977\) −16.3923 −0.524436 −0.262218 0.965009i \(-0.584454\pi\)
−0.262218 + 0.965009i \(0.584454\pi\)
\(978\) −2.73205 −0.0873614
\(979\) −18.1244 −0.579257
\(980\) −0.732051 −0.0233845
\(981\) −4.73205 −0.151083
\(982\) −35.4641 −1.13170
\(983\) −35.8564 −1.14364 −0.571821 0.820379i \(-0.693763\pi\)
−0.571821 + 0.820379i \(0.693763\pi\)
\(984\) −3.00000 −0.0956365
\(985\) −19.5167 −0.621853
\(986\) −24.1244 −0.768276
\(987\) 2.46410 0.0784332
\(988\) 0 0
\(989\) −11.3205 −0.359971
\(990\) −1.26795 −0.0402981
\(991\) −59.2487 −1.88210 −0.941049 0.338271i \(-0.890158\pi\)
−0.941049 + 0.338271i \(0.890158\pi\)
\(992\) 2.19615 0.0697279
\(993\) −27.8564 −0.883996
\(994\) −2.19615 −0.0696577
\(995\) 11.8564 0.375873
\(996\) −0.196152 −0.00621533
\(997\) −21.0526 −0.666741 −0.333371 0.942796i \(-0.608186\pi\)
−0.333371 + 0.942796i \(0.608186\pi\)
\(998\) −11.5167 −0.364554
\(999\) −6.73205 −0.212993
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.1.1 2
13.2 odd 12 546.2.s.a.43.1 4
13.7 odd 12 546.2.s.a.127.1 yes 4
13.12 even 2 7098.2.a.bx.1.2 2
39.2 even 12 1638.2.bj.b.1135.2 4
39.20 even 12 1638.2.bj.b.127.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.s.a.43.1 4 13.2 odd 12
546.2.s.a.127.1 yes 4 13.7 odd 12
1638.2.bj.b.127.2 4 39.20 even 12
1638.2.bj.b.1135.2 4 39.2 even 12
7098.2.a.bp.1.1 2 1.1 even 1 trivial
7098.2.a.bx.1.2 2 13.12 even 2