Properties

Label 1014.2.b.e.337.3
Level $1014$
Weight $2$
Character 1014.337
Analytic conductor $8.097$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.3
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.337
Dual form 1014.2.b.e.337.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +0.267949i q^{5} +1.00000i q^{6} -0.732051i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +0.267949i q^{5} +1.00000i q^{6} -0.732051i q^{7} -1.00000i q^{8} +1.00000 q^{9} -0.267949 q^{10} -4.73205i q^{11} -1.00000 q^{12} +0.732051 q^{14} +0.267949i q^{15} +1.00000 q^{16} +2.26795 q^{17} +1.00000i q^{18} +1.26795i q^{19} -0.267949i q^{20} -0.732051i q^{21} +4.73205 q^{22} +6.19615 q^{23} -1.00000i q^{24} +4.92820 q^{25} +1.00000 q^{27} +0.732051i q^{28} +2.46410 q^{29} -0.267949 q^{30} -5.46410i q^{31} +1.00000i q^{32} -4.73205i q^{33} +2.26795i q^{34} +0.196152 q^{35} -1.00000 q^{36} -10.4641i q^{37} -1.26795 q^{38} +0.267949 q^{40} +11.3923i q^{41} +0.732051 q^{42} -7.66025 q^{43} +4.73205i q^{44} +0.267949i q^{45} +6.19615i q^{46} +8.19615i q^{47} +1.00000 q^{48} +6.46410 q^{49} +4.92820i q^{50} +2.26795 q^{51} +0.464102 q^{53} +1.00000i q^{54} +1.26795 q^{55} -0.732051 q^{56} +1.26795i q^{57} +2.46410i q^{58} -8.00000i q^{59} -0.267949i q^{60} +1.19615 q^{61} +5.46410 q^{62} -0.732051i q^{63} -1.00000 q^{64} +4.73205 q^{66} -11.1244i q^{67} -2.26795 q^{68} +6.19615 q^{69} +0.196152i q^{70} +1.26795i q^{71} -1.00000i q^{72} +9.73205i q^{73} +10.4641 q^{74} +4.92820 q^{75} -1.26795i q^{76} -3.46410 q^{77} -9.46410 q^{79} +0.267949i q^{80} +1.00000 q^{81} -11.3923 q^{82} +10.1962i q^{83} +0.732051i q^{84} +0.607695i q^{85} -7.66025i q^{86} +2.46410 q^{87} -4.73205 q^{88} +2.53590i q^{89} -0.267949 q^{90} -6.19615 q^{92} -5.46410i q^{93} -8.19615 q^{94} -0.339746 q^{95} +1.00000i q^{96} -6.00000i q^{97} +6.46410i q^{98} -4.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 8 q^{10} - 4 q^{12} - 4 q^{14} + 4 q^{16} + 16 q^{17} + 12 q^{22} + 4 q^{23} - 8 q^{25} + 4 q^{27} - 4 q^{29} - 8 q^{30} - 20 q^{35} - 4 q^{36} - 12 q^{38} + 8 q^{40} - 4 q^{42} + 4 q^{43} + 4 q^{48} + 12 q^{49} + 16 q^{51} - 12 q^{53} + 12 q^{55} + 4 q^{56} - 16 q^{61} + 8 q^{62} - 4 q^{64} + 12 q^{66} - 16 q^{68} + 4 q^{69} + 28 q^{74} - 8 q^{75} - 24 q^{79} + 4 q^{81} - 4 q^{82} - 4 q^{87} - 12 q^{88} - 8 q^{90} - 4 q^{92} - 12 q^{94} - 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0.267949i 0.119831i 0.998203 + 0.0599153i \(0.0190830\pi\)
−0.998203 + 0.0599153i \(0.980917\pi\)
\(6\) 1.00000i 0.408248i
\(7\) − 0.732051i − 0.276689i −0.990384 0.138345i \(-0.955822\pi\)
0.990384 0.138345i \(-0.0441781\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) −0.267949 −0.0847330
\(11\) − 4.73205i − 1.42677i −0.700774 0.713384i \(-0.747162\pi\)
0.700774 0.713384i \(-0.252838\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 0.732051 0.195649
\(15\) 0.267949i 0.0691842i
\(16\) 1.00000 0.250000
\(17\) 2.26795 0.550058 0.275029 0.961436i \(-0.411312\pi\)
0.275029 + 0.961436i \(0.411312\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 1.26795i 0.290887i 0.989367 + 0.145444i \(0.0464610\pi\)
−0.989367 + 0.145444i \(0.953539\pi\)
\(20\) − 0.267949i − 0.0599153i
\(21\) − 0.732051i − 0.159747i
\(22\) 4.73205 1.00888
\(23\) 6.19615 1.29199 0.645994 0.763343i \(-0.276443\pi\)
0.645994 + 0.763343i \(0.276443\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) 4.92820 0.985641
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0.732051i 0.138345i
\(29\) 2.46410 0.457572 0.228786 0.973477i \(-0.426524\pi\)
0.228786 + 0.973477i \(0.426524\pi\)
\(30\) −0.267949 −0.0489206
\(31\) − 5.46410i − 0.981382i −0.871334 0.490691i \(-0.836744\pi\)
0.871334 0.490691i \(-0.163256\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 4.73205i − 0.823744i
\(34\) 2.26795i 0.388950i
\(35\) 0.196152 0.0331558
\(36\) −1.00000 −0.166667
\(37\) − 10.4641i − 1.72029i −0.510052 0.860144i \(-0.670374\pi\)
0.510052 0.860144i \(-0.329626\pi\)
\(38\) −1.26795 −0.205689
\(39\) 0 0
\(40\) 0.267949 0.0423665
\(41\) 11.3923i 1.77918i 0.456761 + 0.889590i \(0.349010\pi\)
−0.456761 + 0.889590i \(0.650990\pi\)
\(42\) 0.732051 0.112958
\(43\) −7.66025 −1.16818 −0.584089 0.811690i \(-0.698548\pi\)
−0.584089 + 0.811690i \(0.698548\pi\)
\(44\) 4.73205i 0.713384i
\(45\) 0.267949i 0.0399435i
\(46\) 6.19615i 0.913573i
\(47\) 8.19615i 1.19553i 0.801671 + 0.597766i \(0.203945\pi\)
−0.801671 + 0.597766i \(0.796055\pi\)
\(48\) 1.00000 0.144338
\(49\) 6.46410 0.923443
\(50\) 4.92820i 0.696953i
\(51\) 2.26795 0.317576
\(52\) 0 0
\(53\) 0.464102 0.0637493 0.0318746 0.999492i \(-0.489852\pi\)
0.0318746 + 0.999492i \(0.489852\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 1.26795 0.170970
\(56\) −0.732051 −0.0978244
\(57\) 1.26795i 0.167944i
\(58\) 2.46410i 0.323552i
\(59\) − 8.00000i − 1.04151i −0.853706 0.520756i \(-0.825650\pi\)
0.853706 0.520756i \(-0.174350\pi\)
\(60\) − 0.267949i − 0.0345921i
\(61\) 1.19615 0.153152 0.0765758 0.997064i \(-0.475601\pi\)
0.0765758 + 0.997064i \(0.475601\pi\)
\(62\) 5.46410 0.693942
\(63\) − 0.732051i − 0.0922297i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.73205 0.582475
\(67\) − 11.1244i − 1.35906i −0.733649 0.679528i \(-0.762185\pi\)
0.733649 0.679528i \(-0.237815\pi\)
\(68\) −2.26795 −0.275029
\(69\) 6.19615 0.745929
\(70\) 0.196152i 0.0234447i
\(71\) 1.26795i 0.150478i 0.997166 + 0.0752389i \(0.0239720\pi\)
−0.997166 + 0.0752389i \(0.976028\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 9.73205i 1.13905i 0.821974 + 0.569525i \(0.192873\pi\)
−0.821974 + 0.569525i \(0.807127\pi\)
\(74\) 10.4641 1.21643
\(75\) 4.92820 0.569060
\(76\) − 1.26795i − 0.145444i
\(77\) −3.46410 −0.394771
\(78\) 0 0
\(79\) −9.46410 −1.06479 −0.532397 0.846495i \(-0.678709\pi\)
−0.532397 + 0.846495i \(0.678709\pi\)
\(80\) 0.267949i 0.0299576i
\(81\) 1.00000 0.111111
\(82\) −11.3923 −1.25807
\(83\) 10.1962i 1.11917i 0.828772 + 0.559587i \(0.189040\pi\)
−0.828772 + 0.559587i \(0.810960\pi\)
\(84\) 0.732051i 0.0798733i
\(85\) 0.607695i 0.0659138i
\(86\) − 7.66025i − 0.826026i
\(87\) 2.46410 0.264179
\(88\) −4.73205 −0.504438
\(89\) 2.53590i 0.268805i 0.990927 + 0.134402i \(0.0429115\pi\)
−0.990927 + 0.134402i \(0.957089\pi\)
\(90\) −0.267949 −0.0282443
\(91\) 0 0
\(92\) −6.19615 −0.645994
\(93\) − 5.46410i − 0.566601i
\(94\) −8.19615 −0.845369
\(95\) −0.339746 −0.0348572
\(96\) 1.00000i 0.102062i
\(97\) − 6.00000i − 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) 6.46410i 0.652973i
\(99\) − 4.73205i − 0.475589i
\(100\) −4.92820 −0.492820
\(101\) 11.9282 1.18690 0.593450 0.804871i \(-0.297765\pi\)
0.593450 + 0.804871i \(0.297765\pi\)
\(102\) 2.26795i 0.224560i
\(103\) 18.7321 1.84572 0.922862 0.385131i \(-0.125844\pi\)
0.922862 + 0.385131i \(0.125844\pi\)
\(104\) 0 0
\(105\) 0.196152 0.0191425
\(106\) 0.464102i 0.0450775i
\(107\) −0.196152 −0.0189628 −0.00948139 0.999955i \(-0.503018\pi\)
−0.00948139 + 0.999955i \(0.503018\pi\)
\(108\) −1.00000 −0.0962250
\(109\) − 5.46410i − 0.523366i −0.965154 0.261683i \(-0.915723\pi\)
0.965154 0.261683i \(-0.0842775\pi\)
\(110\) 1.26795i 0.120894i
\(111\) − 10.4641i − 0.993209i
\(112\) − 0.732051i − 0.0691723i
\(113\) −18.6603 −1.75541 −0.877705 0.479202i \(-0.840926\pi\)
−0.877705 + 0.479202i \(0.840926\pi\)
\(114\) −1.26795 −0.118754
\(115\) 1.66025i 0.154819i
\(116\) −2.46410 −0.228786
\(117\) 0 0
\(118\) 8.00000 0.736460
\(119\) − 1.66025i − 0.152195i
\(120\) 0.267949 0.0244603
\(121\) −11.3923 −1.03566
\(122\) 1.19615i 0.108295i
\(123\) 11.3923i 1.02721i
\(124\) 5.46410i 0.490691i
\(125\) 2.66025i 0.237940i
\(126\) 0.732051 0.0652163
\(127\) −17.8564 −1.58450 −0.792250 0.610197i \(-0.791090\pi\)
−0.792250 + 0.610197i \(0.791090\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −7.66025 −0.674448
\(130\) 0 0
\(131\) 13.4641 1.17636 0.588182 0.808729i \(-0.299844\pi\)
0.588182 + 0.808729i \(0.299844\pi\)
\(132\) 4.73205i 0.411872i
\(133\) 0.928203 0.0804854
\(134\) 11.1244 0.960998
\(135\) 0.267949i 0.0230614i
\(136\) − 2.26795i − 0.194475i
\(137\) 1.92820i 0.164738i 0.996602 + 0.0823688i \(0.0262485\pi\)
−0.996602 + 0.0823688i \(0.973751\pi\)
\(138\) 6.19615i 0.527452i
\(139\) −9.85641 −0.836009 −0.418005 0.908445i \(-0.637270\pi\)
−0.418005 + 0.908445i \(0.637270\pi\)
\(140\) −0.196152 −0.0165779
\(141\) 8.19615i 0.690241i
\(142\) −1.26795 −0.106404
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0.660254i 0.0548311i
\(146\) −9.73205 −0.805430
\(147\) 6.46410 0.533150
\(148\) 10.4641i 0.860144i
\(149\) 2.80385i 0.229700i 0.993383 + 0.114850i \(0.0366388\pi\)
−0.993383 + 0.114850i \(0.963361\pi\)
\(150\) 4.92820i 0.402386i
\(151\) − 3.26795i − 0.265942i −0.991120 0.132971i \(-0.957548\pi\)
0.991120 0.132971i \(-0.0424517\pi\)
\(152\) 1.26795 0.102844
\(153\) 2.26795 0.183353
\(154\) − 3.46410i − 0.279145i
\(155\) 1.46410 0.117599
\(156\) 0 0
\(157\) −23.5885 −1.88256 −0.941282 0.337622i \(-0.890378\pi\)
−0.941282 + 0.337622i \(0.890378\pi\)
\(158\) − 9.46410i − 0.752923i
\(159\) 0.464102 0.0368057
\(160\) −0.267949 −0.0211832
\(161\) − 4.53590i − 0.357479i
\(162\) 1.00000i 0.0785674i
\(163\) 6.53590i 0.511931i 0.966686 + 0.255966i \(0.0823934\pi\)
−0.966686 + 0.255966i \(0.917607\pi\)
\(164\) − 11.3923i − 0.889590i
\(165\) 1.26795 0.0987097
\(166\) −10.1962 −0.791375
\(167\) 2.53590i 0.196234i 0.995175 + 0.0981169i \(0.0312819\pi\)
−0.995175 + 0.0981169i \(0.968718\pi\)
\(168\) −0.732051 −0.0564789
\(169\) 0 0
\(170\) −0.607695 −0.0466081
\(171\) 1.26795i 0.0969625i
\(172\) 7.66025 0.584089
\(173\) 16.3923 1.24628 0.623142 0.782109i \(-0.285856\pi\)
0.623142 + 0.782109i \(0.285856\pi\)
\(174\) 2.46410i 0.186803i
\(175\) − 3.60770i − 0.272716i
\(176\) − 4.73205i − 0.356692i
\(177\) − 8.00000i − 0.601317i
\(178\) −2.53590 −0.190074
\(179\) −22.0526 −1.64829 −0.824143 0.566382i \(-0.808343\pi\)
−0.824143 + 0.566382i \(0.808343\pi\)
\(180\) − 0.267949i − 0.0199718i
\(181\) −8.80385 −0.654385 −0.327192 0.944958i \(-0.606103\pi\)
−0.327192 + 0.944958i \(0.606103\pi\)
\(182\) 0 0
\(183\) 1.19615 0.0884221
\(184\) − 6.19615i − 0.456786i
\(185\) 2.80385 0.206143
\(186\) 5.46410 0.400647
\(187\) − 10.7321i − 0.784805i
\(188\) − 8.19615i − 0.597766i
\(189\) − 0.732051i − 0.0532489i
\(190\) − 0.339746i − 0.0246478i
\(191\) 6.92820 0.501307 0.250654 0.968077i \(-0.419354\pi\)
0.250654 + 0.968077i \(0.419354\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 8.26795i 0.595140i 0.954700 + 0.297570i \(0.0961762\pi\)
−0.954700 + 0.297570i \(0.903824\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −6.46410 −0.461722
\(197\) 9.85641i 0.702240i 0.936330 + 0.351120i \(0.114199\pi\)
−0.936330 + 0.351120i \(0.885801\pi\)
\(198\) 4.73205 0.336292
\(199\) 3.80385 0.269648 0.134824 0.990870i \(-0.456953\pi\)
0.134824 + 0.990870i \(0.456953\pi\)
\(200\) − 4.92820i − 0.348477i
\(201\) − 11.1244i − 0.784652i
\(202\) 11.9282i 0.839265i
\(203\) − 1.80385i − 0.126605i
\(204\) −2.26795 −0.158788
\(205\) −3.05256 −0.213200
\(206\) 18.7321i 1.30512i
\(207\) 6.19615 0.430662
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0.196152i 0.0135358i
\(211\) −4.39230 −0.302379 −0.151189 0.988505i \(-0.548310\pi\)
−0.151189 + 0.988505i \(0.548310\pi\)
\(212\) −0.464102 −0.0318746
\(213\) 1.26795i 0.0868784i
\(214\) − 0.196152i − 0.0134087i
\(215\) − 2.05256i − 0.139983i
\(216\) − 1.00000i − 0.0680414i
\(217\) −4.00000 −0.271538
\(218\) 5.46410 0.370076
\(219\) 9.73205i 0.657631i
\(220\) −1.26795 −0.0854851
\(221\) 0 0
\(222\) 10.4641 0.702305
\(223\) − 13.0718i − 0.875352i −0.899133 0.437676i \(-0.855802\pi\)
0.899133 0.437676i \(-0.144198\pi\)
\(224\) 0.732051 0.0489122
\(225\) 4.92820 0.328547
\(226\) − 18.6603i − 1.24126i
\(227\) − 1.80385i − 0.119726i −0.998207 0.0598628i \(-0.980934\pi\)
0.998207 0.0598628i \(-0.0190663\pi\)
\(228\) − 1.26795i − 0.0839720i
\(229\) − 15.8564i − 1.04782i −0.851773 0.523910i \(-0.824473\pi\)
0.851773 0.523910i \(-0.175527\pi\)
\(230\) −1.66025 −0.109474
\(231\) −3.46410 −0.227921
\(232\) − 2.46410i − 0.161776i
\(233\) −19.8564 −1.30084 −0.650418 0.759576i \(-0.725406\pi\)
−0.650418 + 0.759576i \(0.725406\pi\)
\(234\) 0 0
\(235\) −2.19615 −0.143261
\(236\) 8.00000i 0.520756i
\(237\) −9.46410 −0.614759
\(238\) 1.66025 0.107618
\(239\) − 9.66025i − 0.624870i −0.949939 0.312435i \(-0.898855\pi\)
0.949939 0.312435i \(-0.101145\pi\)
\(240\) 0.267949i 0.0172960i
\(241\) 17.5885i 1.13297i 0.824071 + 0.566486i \(0.191698\pi\)
−0.824071 + 0.566486i \(0.808302\pi\)
\(242\) − 11.3923i − 0.732325i
\(243\) 1.00000 0.0641500
\(244\) −1.19615 −0.0765758
\(245\) 1.73205i 0.110657i
\(246\) −11.3923 −0.726347
\(247\) 0 0
\(248\) −5.46410 −0.346971
\(249\) 10.1962i 0.646155i
\(250\) −2.66025 −0.168249
\(251\) −6.53590 −0.412542 −0.206271 0.978495i \(-0.566133\pi\)
−0.206271 + 0.978495i \(0.566133\pi\)
\(252\) 0.732051i 0.0461149i
\(253\) − 29.3205i − 1.84336i
\(254\) − 17.8564i − 1.12041i
\(255\) 0.607695i 0.0380553i
\(256\) 1.00000 0.0625000
\(257\) −26.6603 −1.66302 −0.831510 0.555509i \(-0.812523\pi\)
−0.831510 + 0.555509i \(0.812523\pi\)
\(258\) − 7.66025i − 0.476907i
\(259\) −7.66025 −0.475985
\(260\) 0 0
\(261\) 2.46410 0.152524
\(262\) 13.4641i 0.831815i
\(263\) 28.0526 1.72979 0.864897 0.501949i \(-0.167383\pi\)
0.864897 + 0.501949i \(0.167383\pi\)
\(264\) −4.73205 −0.291238
\(265\) 0.124356i 0.00763911i
\(266\) 0.928203i 0.0569118i
\(267\) 2.53590i 0.155194i
\(268\) 11.1244i 0.679528i
\(269\) −1.46410 −0.0892679 −0.0446339 0.999003i \(-0.514212\pi\)
−0.0446339 + 0.999003i \(0.514212\pi\)
\(270\) −0.267949 −0.0163069
\(271\) − 5.85641i − 0.355751i −0.984053 0.177876i \(-0.943078\pi\)
0.984053 0.177876i \(-0.0569225\pi\)
\(272\) 2.26795 0.137515
\(273\) 0 0
\(274\) −1.92820 −0.116487
\(275\) − 23.3205i − 1.40628i
\(276\) −6.19615 −0.372965
\(277\) 2.26795 0.136268 0.0681339 0.997676i \(-0.478295\pi\)
0.0681339 + 0.997676i \(0.478295\pi\)
\(278\) − 9.85641i − 0.591148i
\(279\) − 5.46410i − 0.327127i
\(280\) − 0.196152i − 0.0117223i
\(281\) 22.3205i 1.33153i 0.746162 + 0.665765i \(0.231895\pi\)
−0.746162 + 0.665765i \(0.768105\pi\)
\(282\) −8.19615 −0.488074
\(283\) −8.33975 −0.495746 −0.247873 0.968792i \(-0.579732\pi\)
−0.247873 + 0.968792i \(0.579732\pi\)
\(284\) − 1.26795i − 0.0752389i
\(285\) −0.339746 −0.0201248
\(286\) 0 0
\(287\) 8.33975 0.492280
\(288\) 1.00000i 0.0589256i
\(289\) −11.8564 −0.697436
\(290\) −0.660254 −0.0387715
\(291\) − 6.00000i − 0.351726i
\(292\) − 9.73205i − 0.569525i
\(293\) 14.5167i 0.848072i 0.905645 + 0.424036i \(0.139387\pi\)
−0.905645 + 0.424036i \(0.860613\pi\)
\(294\) 6.46410i 0.376994i
\(295\) 2.14359 0.124805
\(296\) −10.4641 −0.608214
\(297\) − 4.73205i − 0.274581i
\(298\) −2.80385 −0.162423
\(299\) 0 0
\(300\) −4.92820 −0.284530
\(301\) 5.60770i 0.323222i
\(302\) 3.26795 0.188049
\(303\) 11.9282 0.685257
\(304\) 1.26795i 0.0727219i
\(305\) 0.320508i 0.0183522i
\(306\) 2.26795i 0.129650i
\(307\) − 8.58846i − 0.490169i −0.969502 0.245085i \(-0.921184\pi\)
0.969502 0.245085i \(-0.0788157\pi\)
\(308\) 3.46410 0.197386
\(309\) 18.7321 1.06563
\(310\) 1.46410i 0.0831554i
\(311\) −15.6603 −0.888012 −0.444006 0.896024i \(-0.646443\pi\)
−0.444006 + 0.896024i \(0.646443\pi\)
\(312\) 0 0
\(313\) 13.4641 0.761036 0.380518 0.924774i \(-0.375746\pi\)
0.380518 + 0.924774i \(0.375746\pi\)
\(314\) − 23.5885i − 1.33117i
\(315\) 0.196152 0.0110519
\(316\) 9.46410 0.532397
\(317\) 3.33975i 0.187579i 0.995592 + 0.0937894i \(0.0298980\pi\)
−0.995592 + 0.0937894i \(0.970102\pi\)
\(318\) 0.464102i 0.0260255i
\(319\) − 11.6603i − 0.652849i
\(320\) − 0.267949i − 0.0149788i
\(321\) −0.196152 −0.0109482
\(322\) 4.53590 0.252776
\(323\) 2.87564i 0.160005i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −6.53590 −0.361990
\(327\) − 5.46410i − 0.302166i
\(328\) 11.3923 0.629035
\(329\) 6.00000 0.330791
\(330\) 1.26795i 0.0697983i
\(331\) 20.0000i 1.09930i 0.835395 + 0.549650i \(0.185239\pi\)
−0.835395 + 0.549650i \(0.814761\pi\)
\(332\) − 10.1962i − 0.559587i
\(333\) − 10.4641i − 0.573429i
\(334\) −2.53590 −0.138758
\(335\) 2.98076 0.162856
\(336\) − 0.732051i − 0.0399366i
\(337\) −6.85641 −0.373492 −0.186746 0.982408i \(-0.559794\pi\)
−0.186746 + 0.982408i \(0.559794\pi\)
\(338\) 0 0
\(339\) −18.6603 −1.01349
\(340\) − 0.607695i − 0.0329569i
\(341\) −25.8564 −1.40020
\(342\) −1.26795 −0.0685628
\(343\) − 9.85641i − 0.532196i
\(344\) 7.66025i 0.413013i
\(345\) 1.66025i 0.0893851i
\(346\) 16.3923i 0.881256i
\(347\) 8.87564 0.476470 0.238235 0.971208i \(-0.423431\pi\)
0.238235 + 0.971208i \(0.423431\pi\)
\(348\) −2.46410 −0.132090
\(349\) 19.3205i 1.03420i 0.855924 + 0.517102i \(0.172989\pi\)
−0.855924 + 0.517102i \(0.827011\pi\)
\(350\) 3.60770 0.192839
\(351\) 0 0
\(352\) 4.73205 0.252219
\(353\) 19.7846i 1.05303i 0.850166 + 0.526514i \(0.176501\pi\)
−0.850166 + 0.526514i \(0.823499\pi\)
\(354\) 8.00000 0.425195
\(355\) −0.339746 −0.0180318
\(356\) − 2.53590i − 0.134402i
\(357\) − 1.66025i − 0.0878700i
\(358\) − 22.0526i − 1.16551i
\(359\) − 23.1244i − 1.22046i −0.792226 0.610228i \(-0.791078\pi\)
0.792226 0.610228i \(-0.208922\pi\)
\(360\) 0.267949 0.0141222
\(361\) 17.3923 0.915384
\(362\) − 8.80385i − 0.462720i
\(363\) −11.3923 −0.597941
\(364\) 0 0
\(365\) −2.60770 −0.136493
\(366\) 1.19615i 0.0625239i
\(367\) −14.7321 −0.769007 −0.384503 0.923124i \(-0.625627\pi\)
−0.384503 + 0.923124i \(0.625627\pi\)
\(368\) 6.19615 0.322997
\(369\) 11.3923i 0.593060i
\(370\) 2.80385i 0.145765i
\(371\) − 0.339746i − 0.0176387i
\(372\) 5.46410i 0.283300i
\(373\) −10.2679 −0.531654 −0.265827 0.964021i \(-0.585645\pi\)
−0.265827 + 0.964021i \(0.585645\pi\)
\(374\) 10.7321 0.554941
\(375\) 2.66025i 0.137375i
\(376\) 8.19615 0.422684
\(377\) 0 0
\(378\) 0.732051 0.0376526
\(379\) 1.46410i 0.0752058i 0.999293 + 0.0376029i \(0.0119722\pi\)
−0.999293 + 0.0376029i \(0.988028\pi\)
\(380\) 0.339746 0.0174286
\(381\) −17.8564 −0.914811
\(382\) 6.92820i 0.354478i
\(383\) 5.46410i 0.279203i 0.990208 + 0.139601i \(0.0445821\pi\)
−0.990208 + 0.139601i \(0.955418\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) − 0.928203i − 0.0473056i
\(386\) −8.26795 −0.420828
\(387\) −7.66025 −0.389393
\(388\) 6.00000i 0.304604i
\(389\) 29.7846 1.51014 0.755070 0.655644i \(-0.227603\pi\)
0.755070 + 0.655644i \(0.227603\pi\)
\(390\) 0 0
\(391\) 14.0526 0.710668
\(392\) − 6.46410i − 0.326486i
\(393\) 13.4641 0.679174
\(394\) −9.85641 −0.496559
\(395\) − 2.53590i − 0.127595i
\(396\) 4.73205i 0.237795i
\(397\) 0.392305i 0.0196892i 0.999952 + 0.00984461i \(0.00313369\pi\)
−0.999952 + 0.00984461i \(0.996866\pi\)
\(398\) 3.80385i 0.190670i
\(399\) 0.928203 0.0464683
\(400\) 4.92820 0.246410
\(401\) − 21.9282i − 1.09504i −0.836792 0.547521i \(-0.815572\pi\)
0.836792 0.547521i \(-0.184428\pi\)
\(402\) 11.1244 0.554832
\(403\) 0 0
\(404\) −11.9282 −0.593450
\(405\) 0.267949i 0.0133145i
\(406\) 1.80385 0.0895235
\(407\) −49.5167 −2.45445
\(408\) − 2.26795i − 0.112280i
\(409\) 14.2679i 0.705505i 0.935717 + 0.352752i \(0.114754\pi\)
−0.935717 + 0.352752i \(0.885246\pi\)
\(410\) − 3.05256i − 0.150755i
\(411\) 1.92820i 0.0951113i
\(412\) −18.7321 −0.922862
\(413\) −5.85641 −0.288175
\(414\) 6.19615i 0.304524i
\(415\) −2.73205 −0.134111
\(416\) 0 0
\(417\) −9.85641 −0.482670
\(418\) 6.00000i 0.293470i
\(419\) −10.5359 −0.514712 −0.257356 0.966317i \(-0.582851\pi\)
−0.257356 + 0.966317i \(0.582851\pi\)
\(420\) −0.196152 −0.00957126
\(421\) 32.7128i 1.59432i 0.603765 + 0.797162i \(0.293667\pi\)
−0.603765 + 0.797162i \(0.706333\pi\)
\(422\) − 4.39230i − 0.213814i
\(423\) 8.19615i 0.398511i
\(424\) − 0.464102i − 0.0225388i
\(425\) 11.1769 0.542160
\(426\) −1.26795 −0.0614323
\(427\) − 0.875644i − 0.0423754i
\(428\) 0.196152 0.00948139
\(429\) 0 0
\(430\) 2.05256 0.0989832
\(431\) − 11.1244i − 0.535841i −0.963441 0.267921i \(-0.913663\pi\)
0.963441 0.267921i \(-0.0863365\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.8564 0.713953 0.356977 0.934113i \(-0.383808\pi\)
0.356977 + 0.934113i \(0.383808\pi\)
\(434\) − 4.00000i − 0.192006i
\(435\) 0.660254i 0.0316568i
\(436\) 5.46410i 0.261683i
\(437\) 7.85641i 0.375823i
\(438\) −9.73205 −0.465015
\(439\) 17.6603 0.842878 0.421439 0.906857i \(-0.361525\pi\)
0.421439 + 0.906857i \(0.361525\pi\)
\(440\) − 1.26795i − 0.0604471i
\(441\) 6.46410 0.307814
\(442\) 0 0
\(443\) 36.3923 1.72905 0.864525 0.502589i \(-0.167619\pi\)
0.864525 + 0.502589i \(0.167619\pi\)
\(444\) 10.4641i 0.496604i
\(445\) −0.679492 −0.0322110
\(446\) 13.0718 0.618968
\(447\) 2.80385i 0.132617i
\(448\) 0.732051i 0.0345861i
\(449\) 23.3205i 1.10056i 0.834979 + 0.550281i \(0.185480\pi\)
−0.834979 + 0.550281i \(0.814520\pi\)
\(450\) 4.92820i 0.232318i
\(451\) 53.9090 2.53847
\(452\) 18.6603 0.877705
\(453\) − 3.26795i − 0.153542i
\(454\) 1.80385 0.0846588
\(455\) 0 0
\(456\) 1.26795 0.0593772
\(457\) 18.6603i 0.872890i 0.899731 + 0.436445i \(0.143763\pi\)
−0.899731 + 0.436445i \(0.856237\pi\)
\(458\) 15.8564 0.740921
\(459\) 2.26795 0.105859
\(460\) − 1.66025i − 0.0774097i
\(461\) 25.7321i 1.19846i 0.800577 + 0.599231i \(0.204527\pi\)
−0.800577 + 0.599231i \(0.795473\pi\)
\(462\) − 3.46410i − 0.161165i
\(463\) 28.0526i 1.30371i 0.758342 + 0.651856i \(0.226010\pi\)
−0.758342 + 0.651856i \(0.773990\pi\)
\(464\) 2.46410 0.114393
\(465\) 1.46410 0.0678961
\(466\) − 19.8564i − 0.919830i
\(467\) 12.5885 0.582524 0.291262 0.956643i \(-0.405925\pi\)
0.291262 + 0.956643i \(0.405925\pi\)
\(468\) 0 0
\(469\) −8.14359 −0.376036
\(470\) − 2.19615i − 0.101301i
\(471\) −23.5885 −1.08690
\(472\) −8.00000 −0.368230
\(473\) 36.2487i 1.66672i
\(474\) − 9.46410i − 0.434701i
\(475\) 6.24871i 0.286711i
\(476\) 1.66025i 0.0760976i
\(477\) 0.464102 0.0212498
\(478\) 9.66025 0.441850
\(479\) − 26.5359i − 1.21246i −0.795291 0.606228i \(-0.792682\pi\)
0.795291 0.606228i \(-0.207318\pi\)
\(480\) −0.267949 −0.0122302
\(481\) 0 0
\(482\) −17.5885 −0.801132
\(483\) − 4.53590i − 0.206391i
\(484\) 11.3923 0.517832
\(485\) 1.60770 0.0730017
\(486\) 1.00000i 0.0453609i
\(487\) 21.1244i 0.957236i 0.878023 + 0.478618i \(0.158862\pi\)
−0.878023 + 0.478618i \(0.841138\pi\)
\(488\) − 1.19615i − 0.0541473i
\(489\) 6.53590i 0.295564i
\(490\) −1.73205 −0.0782461
\(491\) −5.26795 −0.237739 −0.118870 0.992910i \(-0.537927\pi\)
−0.118870 + 0.992910i \(0.537927\pi\)
\(492\) − 11.3923i − 0.513605i
\(493\) 5.58846 0.251691
\(494\) 0 0
\(495\) 1.26795 0.0569901
\(496\) − 5.46410i − 0.245345i
\(497\) 0.928203 0.0416356
\(498\) −10.1962 −0.456901
\(499\) 32.0000i 1.43252i 0.697835 + 0.716258i \(0.254147\pi\)
−0.697835 + 0.716258i \(0.745853\pi\)
\(500\) − 2.66025i − 0.118970i
\(501\) 2.53590i 0.113296i
\(502\) − 6.53590i − 0.291711i
\(503\) −10.9808 −0.489608 −0.244804 0.969573i \(-0.578724\pi\)
−0.244804 + 0.969573i \(0.578724\pi\)
\(504\) −0.732051 −0.0326081
\(505\) 3.19615i 0.142227i
\(506\) 29.3205 1.30346
\(507\) 0 0
\(508\) 17.8564 0.792250
\(509\) 10.2679i 0.455119i 0.973764 + 0.227559i \(0.0730746\pi\)
−0.973764 + 0.227559i \(0.926925\pi\)
\(510\) −0.607695 −0.0269092
\(511\) 7.12436 0.315163
\(512\) 1.00000i 0.0441942i
\(513\) 1.26795i 0.0559813i
\(514\) − 26.6603i − 1.17593i
\(515\) 5.01924i 0.221174i
\(516\) 7.66025 0.337224
\(517\) 38.7846 1.70575
\(518\) − 7.66025i − 0.336572i
\(519\) 16.3923 0.719542
\(520\) 0 0
\(521\) −17.4449 −0.764273 −0.382137 0.924106i \(-0.624812\pi\)
−0.382137 + 0.924106i \(0.624812\pi\)
\(522\) 2.46410i 0.107851i
\(523\) 36.4449 1.59362 0.796811 0.604228i \(-0.206518\pi\)
0.796811 + 0.604228i \(0.206518\pi\)
\(524\) −13.4641 −0.588182
\(525\) − 3.60770i − 0.157453i
\(526\) 28.0526i 1.22315i
\(527\) − 12.3923i − 0.539817i
\(528\) − 4.73205i − 0.205936i
\(529\) 15.3923 0.669231
\(530\) −0.124356 −0.00540166
\(531\) − 8.00000i − 0.347170i
\(532\) −0.928203 −0.0402427
\(533\) 0 0
\(534\) −2.53590 −0.109739
\(535\) − 0.0525589i − 0.00227232i
\(536\) −11.1244 −0.480499
\(537\) −22.0526 −0.951638
\(538\) − 1.46410i − 0.0631219i
\(539\) − 30.5885i − 1.31754i
\(540\) − 0.267949i − 0.0115307i
\(541\) − 40.3205i − 1.73351i −0.498731 0.866757i \(-0.666200\pi\)
0.498731 0.866757i \(-0.333800\pi\)
\(542\) 5.85641 0.251554
\(543\) −8.80385 −0.377809
\(544\) 2.26795i 0.0972375i
\(545\) 1.46410 0.0627152
\(546\) 0 0
\(547\) 6.19615 0.264928 0.132464 0.991188i \(-0.457711\pi\)
0.132464 + 0.991188i \(0.457711\pi\)
\(548\) − 1.92820i − 0.0823688i
\(549\) 1.19615 0.0510505
\(550\) 23.3205 0.994390
\(551\) 3.12436i 0.133102i
\(552\) − 6.19615i − 0.263726i
\(553\) 6.92820i 0.294617i
\(554\) 2.26795i 0.0963559i
\(555\) 2.80385 0.119017
\(556\) 9.85641 0.418005
\(557\) 30.3731i 1.28695i 0.765468 + 0.643474i \(0.222508\pi\)
−0.765468 + 0.643474i \(0.777492\pi\)
\(558\) 5.46410 0.231314
\(559\) 0 0
\(560\) 0.196152 0.00828895
\(561\) − 10.7321i − 0.453108i
\(562\) −22.3205 −0.941534
\(563\) −21.0718 −0.888070 −0.444035 0.896009i \(-0.646453\pi\)
−0.444035 + 0.896009i \(0.646453\pi\)
\(564\) − 8.19615i − 0.345120i
\(565\) − 5.00000i − 0.210352i
\(566\) − 8.33975i − 0.350546i
\(567\) − 0.732051i − 0.0307432i
\(568\) 1.26795 0.0532020
\(569\) 38.6410 1.61992 0.809958 0.586488i \(-0.199490\pi\)
0.809958 + 0.586488i \(0.199490\pi\)
\(570\) − 0.339746i − 0.0142304i
\(571\) 24.0526 1.00657 0.503284 0.864121i \(-0.332125\pi\)
0.503284 + 0.864121i \(0.332125\pi\)
\(572\) 0 0
\(573\) 6.92820 0.289430
\(574\) 8.33975i 0.348094i
\(575\) 30.5359 1.27343
\(576\) −1.00000 −0.0416667
\(577\) 0.267949i 0.0111549i 0.999984 + 0.00557744i \(0.00177536\pi\)
−0.999984 + 0.00557744i \(0.998225\pi\)
\(578\) − 11.8564i − 0.493161i
\(579\) 8.26795i 0.343604i
\(580\) − 0.660254i − 0.0274156i
\(581\) 7.46410 0.309663
\(582\) 6.00000 0.248708
\(583\) − 2.19615i − 0.0909553i
\(584\) 9.73205 0.402715
\(585\) 0 0
\(586\) −14.5167 −0.599678
\(587\) 16.0000i 0.660391i 0.943913 + 0.330195i \(0.107115\pi\)
−0.943913 + 0.330195i \(0.892885\pi\)
\(588\) −6.46410 −0.266575
\(589\) 6.92820 0.285472
\(590\) 2.14359i 0.0882503i
\(591\) 9.85641i 0.405438i
\(592\) − 10.4641i − 0.430072i
\(593\) 36.8564i 1.51351i 0.653698 + 0.756756i \(0.273217\pi\)
−0.653698 + 0.756756i \(0.726783\pi\)
\(594\) 4.73205 0.194158
\(595\) 0.444864 0.0182376
\(596\) − 2.80385i − 0.114850i
\(597\) 3.80385 0.155681
\(598\) 0 0
\(599\) −9.46410 −0.386693 −0.193346 0.981131i \(-0.561934\pi\)
−0.193346 + 0.981131i \(0.561934\pi\)
\(600\) − 4.92820i − 0.201193i
\(601\) −5.92820 −0.241816 −0.120908 0.992664i \(-0.538581\pi\)
−0.120908 + 0.992664i \(0.538581\pi\)
\(602\) −5.60770 −0.228553
\(603\) − 11.1244i − 0.453019i
\(604\) 3.26795i 0.132971i
\(605\) − 3.05256i − 0.124104i
\(606\) 11.9282i 0.484550i
\(607\) 0.784610 0.0318463 0.0159232 0.999873i \(-0.494931\pi\)
0.0159232 + 0.999873i \(0.494931\pi\)
\(608\) −1.26795 −0.0514221
\(609\) − 1.80385i − 0.0730956i
\(610\) −0.320508 −0.0129770
\(611\) 0 0
\(612\) −2.26795 −0.0916764
\(613\) 11.3923i 0.460131i 0.973175 + 0.230065i \(0.0738940\pi\)
−0.973175 + 0.230065i \(0.926106\pi\)
\(614\) 8.58846 0.346602
\(615\) −3.05256 −0.123091
\(616\) 3.46410i 0.139573i
\(617\) 35.2487i 1.41906i 0.704675 + 0.709530i \(0.251093\pi\)
−0.704675 + 0.709530i \(0.748907\pi\)
\(618\) 18.7321i 0.753514i
\(619\) − 10.5359i − 0.423474i −0.977327 0.211737i \(-0.932088\pi\)
0.977327 0.211737i \(-0.0679119\pi\)
\(620\) −1.46410 −0.0587997
\(621\) 6.19615 0.248643
\(622\) − 15.6603i − 0.627919i
\(623\) 1.85641 0.0743754
\(624\) 0 0
\(625\) 23.9282 0.957128
\(626\) 13.4641i 0.538134i
\(627\) 6.00000 0.239617
\(628\) 23.5885 0.941282
\(629\) − 23.7321i − 0.946259i
\(630\) 0.196152i 0.00781490i
\(631\) − 47.7128i − 1.89942i −0.313135 0.949709i \(-0.601379\pi\)
0.313135 0.949709i \(-0.398621\pi\)
\(632\) 9.46410i 0.376462i
\(633\) −4.39230 −0.174578
\(634\) −3.33975 −0.132638
\(635\) − 4.78461i − 0.189871i
\(636\) −0.464102 −0.0184028
\(637\) 0 0
\(638\) 11.6603 0.461634
\(639\) 1.26795i 0.0501593i
\(640\) 0.267949 0.0105916
\(641\) −25.9808 −1.02618 −0.513089 0.858335i \(-0.671499\pi\)
−0.513089 + 0.858335i \(0.671499\pi\)
\(642\) − 0.196152i − 0.00774152i
\(643\) 13.8564i 0.546443i 0.961951 + 0.273222i \(0.0880892\pi\)
−0.961951 + 0.273222i \(0.911911\pi\)
\(644\) 4.53590i 0.178739i
\(645\) − 2.05256i − 0.0808194i
\(646\) −2.87564 −0.113141
\(647\) −26.2487 −1.03194 −0.515972 0.856606i \(-0.672569\pi\)
−0.515972 + 0.856606i \(0.672569\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −37.8564 −1.48599
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) − 6.53590i − 0.255966i
\(653\) −10.5359 −0.412302 −0.206151 0.978520i \(-0.566094\pi\)
−0.206151 + 0.978520i \(0.566094\pi\)
\(654\) 5.46410 0.213663
\(655\) 3.60770i 0.140964i
\(656\) 11.3923i 0.444795i
\(657\) 9.73205i 0.379683i
\(658\) 6.00000i 0.233904i
\(659\) 38.2487 1.48996 0.744979 0.667088i \(-0.232459\pi\)
0.744979 + 0.667088i \(0.232459\pi\)
\(660\) −1.26795 −0.0493549
\(661\) 9.39230i 0.365318i 0.983176 + 0.182659i \(0.0584705\pi\)
−0.983176 + 0.182659i \(0.941530\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 10.1962 0.395687
\(665\) 0.248711i 0.00964461i
\(666\) 10.4641 0.405476
\(667\) 15.2679 0.591177
\(668\) − 2.53590i − 0.0981169i
\(669\) − 13.0718i − 0.505385i
\(670\) 2.98076i 0.115157i
\(671\) − 5.66025i − 0.218512i
\(672\) 0.732051 0.0282395
\(673\) −14.0718 −0.542428 −0.271214 0.962519i \(-0.587425\pi\)
−0.271214 + 0.962519i \(0.587425\pi\)
\(674\) − 6.85641i − 0.264099i
\(675\) 4.92820 0.189687
\(676\) 0 0
\(677\) −38.5359 −1.48105 −0.740527 0.672026i \(-0.765424\pi\)
−0.740527 + 0.672026i \(0.765424\pi\)
\(678\) − 18.6603i − 0.716643i
\(679\) −4.39230 −0.168561
\(680\) 0.607695 0.0233040
\(681\) − 1.80385i − 0.0691236i
\(682\) − 25.8564i − 0.990093i
\(683\) 37.8564i 1.44854i 0.689519 + 0.724268i \(0.257822\pi\)
−0.689519 + 0.724268i \(0.742178\pi\)
\(684\) − 1.26795i − 0.0484812i
\(685\) −0.516660 −0.0197406
\(686\) 9.85641 0.376319
\(687\) − 15.8564i − 0.604960i
\(688\) −7.66025 −0.292044
\(689\) 0 0
\(690\) −1.66025 −0.0632048
\(691\) 26.3397i 1.00201i 0.865444 + 0.501006i \(0.167036\pi\)
−0.865444 + 0.501006i \(0.832964\pi\)
\(692\) −16.3923 −0.623142
\(693\) −3.46410 −0.131590
\(694\) 8.87564i 0.336915i
\(695\) − 2.64102i − 0.100179i
\(696\) − 2.46410i − 0.0934015i
\(697\) 25.8372i 0.978653i
\(698\) −19.3205 −0.731292
\(699\) −19.8564 −0.751038
\(700\) 3.60770i 0.136358i
\(701\) −31.3205 −1.18296 −0.591480 0.806320i \(-0.701456\pi\)
−0.591480 + 0.806320i \(0.701456\pi\)
\(702\) 0 0
\(703\) 13.2679 0.500410
\(704\) 4.73205i 0.178346i
\(705\) −2.19615 −0.0827119
\(706\) −19.7846 −0.744604
\(707\) − 8.73205i − 0.328403i
\(708\) 8.00000i 0.300658i
\(709\) − 40.8564i − 1.53439i −0.641411 0.767197i \(-0.721651\pi\)
0.641411 0.767197i \(-0.278349\pi\)
\(710\) − 0.339746i − 0.0127504i
\(711\) −9.46410 −0.354932
\(712\) 2.53590 0.0950368
\(713\) − 33.8564i − 1.26793i
\(714\) 1.66025 0.0621334
\(715\) 0 0
\(716\) 22.0526 0.824143
\(717\) − 9.66025i − 0.360769i
\(718\) 23.1244 0.862993
\(719\) 22.5359 0.840447 0.420224 0.907421i \(-0.361952\pi\)
0.420224 + 0.907421i \(0.361952\pi\)
\(720\) 0.267949i 0.00998588i
\(721\) − 13.7128i − 0.510692i
\(722\) 17.3923i 0.647275i
\(723\) 17.5885i 0.654122i
\(724\) 8.80385 0.327192
\(725\) 12.1436 0.451002
\(726\) − 11.3923i − 0.422808i
\(727\) −20.9808 −0.778133 −0.389067 0.921210i \(-0.627202\pi\)
−0.389067 + 0.921210i \(0.627202\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 2.60770i − 0.0965151i
\(731\) −17.3731 −0.642566
\(732\) −1.19615 −0.0442111
\(733\) 19.0000i 0.701781i 0.936416 + 0.350891i \(0.114121\pi\)
−0.936416 + 0.350891i \(0.885879\pi\)
\(734\) − 14.7321i − 0.543770i
\(735\) 1.73205i 0.0638877i
\(736\) 6.19615i 0.228393i
\(737\) −52.6410 −1.93906
\(738\) −11.3923 −0.419357
\(739\) 10.9282i 0.402000i 0.979591 + 0.201000i \(0.0644192\pi\)
−0.979591 + 0.201000i \(0.935581\pi\)
\(740\) −2.80385 −0.103071
\(741\) 0 0
\(742\) 0.339746 0.0124725
\(743\) − 27.6077i − 1.01283i −0.862290 0.506414i \(-0.830971\pi\)
0.862290 0.506414i \(-0.169029\pi\)
\(744\) −5.46410 −0.200324
\(745\) −0.751289 −0.0275251
\(746\) − 10.2679i − 0.375936i
\(747\) 10.1962i 0.373058i
\(748\) 10.7321i 0.392403i
\(749\) 0.143594i 0.00524679i
\(750\) −2.66025 −0.0971387
\(751\) −15.9090 −0.580526 −0.290263 0.956947i \(-0.593743\pi\)
−0.290263 + 0.956947i \(0.593743\pi\)
\(752\) 8.19615i 0.298883i
\(753\) −6.53590 −0.238181
\(754\) 0 0
\(755\) 0.875644 0.0318680
\(756\) 0.732051i 0.0266244i
\(757\) 7.07180 0.257029 0.128514 0.991708i \(-0.458979\pi\)
0.128514 + 0.991708i \(0.458979\pi\)
\(758\) −1.46410 −0.0531786
\(759\) − 29.3205i − 1.06427i
\(760\) 0.339746i 0.0123239i
\(761\) 23.3205i 0.845368i 0.906277 + 0.422684i \(0.138912\pi\)
−0.906277 + 0.422684i \(0.861088\pi\)
\(762\) − 17.8564i − 0.646869i
\(763\) −4.00000 −0.144810
\(764\) −6.92820 −0.250654
\(765\) 0.607695i 0.0219713i
\(766\) −5.46410 −0.197426
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) − 16.1436i − 0.582153i −0.956700 0.291076i \(-0.905987\pi\)
0.956700 0.291076i \(-0.0940134\pi\)
\(770\) 0.928203 0.0334501
\(771\) −26.6603 −0.960146
\(772\) − 8.26795i − 0.297570i
\(773\) − 35.0718i − 1.26144i −0.776009 0.630722i \(-0.782759\pi\)
0.776009 0.630722i \(-0.217241\pi\)
\(774\) − 7.66025i − 0.275342i
\(775\) − 26.9282i − 0.967290i
\(776\) −6.00000 −0.215387
\(777\) −7.66025 −0.274810
\(778\) 29.7846i 1.06783i
\(779\) −14.4449 −0.517541
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 14.0526i 0.502518i
\(783\) 2.46410 0.0880598
\(784\) 6.46410 0.230861
\(785\) − 6.32051i − 0.225589i
\(786\) 13.4641i 0.480249i
\(787\) − 39.3205i − 1.40162i −0.713346 0.700812i \(-0.752821\pi\)
0.713346 0.700812i \(-0.247179\pi\)
\(788\) − 9.85641i − 0.351120i
\(789\) 28.0526 0.998698
\(790\) 2.53590 0.0902232
\(791\) 13.6603i 0.485703i
\(792\) −4.73205 −0.168146
\(793\) 0 0
\(794\) −0.392305 −0.0139224
\(795\) 0.124356i 0.00441044i
\(796\) −3.80385 −0.134824
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) 0.928203i 0.0328580i
\(799\) 18.5885i 0.657612i
\(800\) 4.92820i 0.174238i
\(801\) 2.53590i 0.0896016i
\(802\) 21.9282 0.774312
\(803\) 46.0526 1.62516
\(804\) 11.1244i 0.392326i
\(805\) 1.21539 0.0428369
\(806\) 0 0
\(807\) −1.46410 −0.0515388
\(808\) − 11.9282i − 0.419633i
\(809\) −22.4115 −0.787948 −0.393974 0.919122i \(-0.628900\pi\)
−0.393974 + 0.919122i \(0.628900\pi\)
\(810\) −0.267949 −0.00941477
\(811\) − 45.1769i − 1.58638i −0.608977 0.793188i \(-0.708420\pi\)
0.608977 0.793188i \(-0.291580\pi\)
\(812\) 1.80385i 0.0633026i
\(813\) − 5.85641i − 0.205393i
\(814\) − 49.5167i − 1.73556i
\(815\) −1.75129 −0.0613450
\(816\) 2.26795 0.0793941
\(817\) − 9.71281i − 0.339808i
\(818\) −14.2679 −0.498867
\(819\) 0 0
\(820\) 3.05256 0.106600
\(821\) 12.9282i 0.451197i 0.974220 + 0.225599i \(0.0724338\pi\)
−0.974220 + 0.225599i \(0.927566\pi\)
\(822\) −1.92820 −0.0672538
\(823\) 41.5692 1.44901 0.724506 0.689269i \(-0.242068\pi\)
0.724506 + 0.689269i \(0.242068\pi\)
\(824\) − 18.7321i − 0.652562i
\(825\) − 23.3205i − 0.811916i
\(826\) − 5.85641i − 0.203770i
\(827\) − 33.4641i − 1.16366i −0.813310 0.581830i \(-0.802337\pi\)
0.813310 0.581830i \(-0.197663\pi\)
\(828\) −6.19615 −0.215331
\(829\) −12.1244 −0.421096 −0.210548 0.977583i \(-0.567525\pi\)
−0.210548 + 0.977583i \(0.567525\pi\)
\(830\) − 2.73205i − 0.0948309i
\(831\) 2.26795 0.0786743
\(832\) 0 0
\(833\) 14.6603 0.507948
\(834\) − 9.85641i − 0.341299i
\(835\) −0.679492 −0.0235148
\(836\) −6.00000 −0.207514
\(837\) − 5.46410i − 0.188867i
\(838\) − 10.5359i − 0.363957i
\(839\) − 14.1436i − 0.488291i −0.969739 0.244146i \(-0.921493\pi\)
0.969739 0.244146i \(-0.0785075\pi\)
\(840\) − 0.196152i − 0.00676790i
\(841\) −22.9282 −0.790628
\(842\) −32.7128 −1.12736
\(843\) 22.3205i 0.768759i
\(844\) 4.39230 0.151189
\(845\) 0 0
\(846\) −8.19615 −0.281790
\(847\) 8.33975i 0.286557i
\(848\) 0.464102 0.0159373
\(849\) −8.33975 −0.286219
\(850\) 11.1769i 0.383365i
\(851\) − 64.8372i − 2.22259i
\(852\) − 1.26795i − 0.0434392i
\(853\) − 8.17691i − 0.279972i −0.990153 0.139986i \(-0.955294\pi\)
0.990153 0.139986i \(-0.0447058\pi\)
\(854\) 0.875644 0.0299639
\(855\) −0.339746 −0.0116191
\(856\) 0.196152i 0.00670435i
\(857\) 19.4449 0.664224 0.332112 0.943240i \(-0.392239\pi\)
0.332112 + 0.943240i \(0.392239\pi\)
\(858\) 0 0
\(859\) −22.8756 −0.780507 −0.390253 0.920707i \(-0.627613\pi\)
−0.390253 + 0.920707i \(0.627613\pi\)
\(860\) 2.05256i 0.0699917i
\(861\) 8.33975 0.284218
\(862\) 11.1244 0.378897
\(863\) 7.12436i 0.242516i 0.992621 + 0.121258i \(0.0386928\pi\)
−0.992621 + 0.121258i \(0.961307\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 4.39230i 0.149343i
\(866\) 14.8564i 0.504841i
\(867\) −11.8564 −0.402665
\(868\) 4.00000 0.135769
\(869\) 44.7846i 1.51921i
\(870\) −0.660254 −0.0223847
\(871\) 0 0
\(872\) −5.46410 −0.185038
\(873\) − 6.00000i − 0.203069i
\(874\) −7.85641 −0.265747
\(875\) 1.94744 0.0658355
\(876\) − 9.73205i − 0.328816i
\(877\) 10.0718i 0.340100i 0.985435 + 0.170050i \(0.0543930\pi\)
−0.985435 + 0.170050i \(0.945607\pi\)
\(878\) 17.6603i 0.596005i
\(879\) 14.5167i 0.489635i
\(880\) 1.26795 0.0427426
\(881\) −51.8372 −1.74644 −0.873219 0.487327i \(-0.837972\pi\)
−0.873219 + 0.487327i \(0.837972\pi\)
\(882\) 6.46410i 0.217658i
\(883\) −29.0718 −0.978344 −0.489172 0.872187i \(-0.662701\pi\)
−0.489172 + 0.872187i \(0.662701\pi\)
\(884\) 0 0
\(885\) 2.14359 0.0720561
\(886\) 36.3923i 1.22262i
\(887\) 10.1436 0.340589 0.170294 0.985393i \(-0.445528\pi\)
0.170294 + 0.985393i \(0.445528\pi\)
\(888\) −10.4641 −0.351152
\(889\) 13.0718i 0.438414i
\(890\) − 0.679492i − 0.0227766i
\(891\) − 4.73205i − 0.158530i
\(892\) 13.0718i 0.437676i
\(893\) −10.3923 −0.347765
\(894\) −2.80385 −0.0937747
\(895\) − 5.90897i − 0.197515i
\(896\) −0.732051 −0.0244561
\(897\) 0 0
\(898\) −23.3205 −0.778215
\(899\) − 13.4641i − 0.449053i
\(900\) −4.92820 −0.164273
\(901\) 1.05256 0.0350658
\(902\) 53.9090i 1.79497i
\(903\) 5.60770i 0.186612i
\(904\) 18.6603i 0.620631i
\(905\) − 2.35898i − 0.0784153i
\(906\) 3.26795 0.108570
\(907\) −15.6077 −0.518245 −0.259123 0.965844i \(-0.583433\pi\)
−0.259123 + 0.965844i \(0.583433\pi\)
\(908\) 1.80385i 0.0598628i
\(909\) 11.9282 0.395634
\(910\) 0 0
\(911\) −9.46410 −0.313560 −0.156780 0.987634i \(-0.550111\pi\)
−0.156780 + 0.987634i \(0.550111\pi\)
\(912\) 1.26795i 0.0419860i
\(913\) 48.2487 1.59680
\(914\) −18.6603 −0.617226
\(915\) 0.320508i 0.0105957i
\(916\) 15.8564i 0.523910i
\(917\) − 9.85641i − 0.325487i
\(918\) 2.26795i 0.0748535i
\(919\) −57.9615 −1.91197 −0.955987 0.293409i \(-0.905210\pi\)
−0.955987 + 0.293409i \(0.905210\pi\)
\(920\) 1.66025 0.0547370
\(921\) − 8.58846i − 0.282999i
\(922\) −25.7321 −0.847440
\(923\) 0 0
\(924\) 3.46410 0.113961
\(925\) − 51.5692i − 1.69559i
\(926\) −28.0526 −0.921864
\(927\) 18.7321 0.615241
\(928\) 2.46410i 0.0808881i
\(929\) 9.24871i 0.303440i 0.988423 + 0.151720i \(0.0484813\pi\)
−0.988423 + 0.151720i \(0.951519\pi\)
\(930\) 1.46410i 0.0480098i
\(931\) 8.19615i 0.268618i
\(932\) 19.8564 0.650418
\(933\) −15.6603 −0.512694
\(934\) 12.5885i 0.411907i
\(935\) 2.87564 0.0940436
\(936\) 0 0
\(937\) 43.2487 1.41287 0.706437 0.707776i \(-0.250301\pi\)
0.706437 + 0.707776i \(0.250301\pi\)
\(938\) − 8.14359i − 0.265898i
\(939\) 13.4641 0.439384
\(940\) 2.19615 0.0716306
\(941\) − 56.6410i − 1.84644i −0.384267 0.923222i \(-0.625546\pi\)
0.384267 0.923222i \(-0.374454\pi\)
\(942\) − 23.5885i − 0.768553i
\(943\) 70.5885i 2.29868i
\(944\) − 8.00000i − 0.260378i
\(945\) 0.196152 0.00638084
\(946\) −36.2487 −1.17855
\(947\) − 34.9282i − 1.13501i −0.823369 0.567507i \(-0.807908\pi\)
0.823369 0.567507i \(-0.192092\pi\)
\(948\) 9.46410 0.307380
\(949\) 0 0
\(950\) −6.24871 −0.202735
\(951\) 3.33975i 0.108299i
\(952\) −1.66025 −0.0538091
\(953\) 41.5692 1.34656 0.673280 0.739388i \(-0.264885\pi\)
0.673280 + 0.739388i \(0.264885\pi\)
\(954\) 0.464102i 0.0150258i
\(955\) 1.85641i 0.0600719i
\(956\) 9.66025i 0.312435i
\(957\) − 11.6603i − 0.376922i
\(958\) 26.5359 0.857336
\(959\) 1.41154 0.0455811
\(960\) − 0.267949i − 0.00864802i
\(961\) 1.14359 0.0368901
\(962\) 0 0
\(963\) −0.196152 −0.00632092
\(964\) − 17.5885i − 0.566486i
\(965\) −2.21539 −0.0713159
\(966\) 4.53590 0.145940
\(967\) − 18.8756i − 0.607000i −0.952831 0.303500i \(-0.901845\pi\)
0.952831 0.303500i \(-0.0981552\pi\)
\(968\) 11.3923i 0.366163i
\(969\) 2.87564i 0.0923790i
\(970\) 1.60770i 0.0516200i
\(971\) −18.2487 −0.585629 −0.292815 0.956169i \(-0.594592\pi\)
−0.292815 + 0.956169i \(0.594592\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 7.21539i 0.231315i
\(974\) −21.1244 −0.676868
\(975\) 0 0
\(976\) 1.19615 0.0382879
\(977\) 32.0718i 1.02607i 0.858368 + 0.513034i \(0.171478\pi\)
−0.858368 + 0.513034i \(0.828522\pi\)
\(978\) −6.53590 −0.208995
\(979\) 12.0000 0.383522
\(980\) − 1.73205i − 0.0553283i
\(981\) − 5.46410i − 0.174455i
\(982\) − 5.26795i − 0.168107i
\(983\) 20.7846i 0.662926i 0.943468 + 0.331463i \(0.107542\pi\)
−0.943468 + 0.331463i \(0.892458\pi\)
\(984\) 11.3923 0.363173
\(985\) −2.64102 −0.0841498
\(986\) 5.58846i 0.177973i
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) −47.4641 −1.50927
\(990\) 1.26795i 0.0402981i
\(991\) −8.58846 −0.272821 −0.136411 0.990652i \(-0.543557\pi\)
−0.136411 + 0.990652i \(0.543557\pi\)
\(992\) 5.46410 0.173485
\(993\) 20.0000i 0.634681i
\(994\) 0.928203i 0.0294408i
\(995\) 1.01924i 0.0323120i
\(996\) − 10.1962i − 0.323077i
\(997\) 38.6603 1.22438 0.612191 0.790710i \(-0.290288\pi\)
0.612191 + 0.790710i \(0.290288\pi\)
\(998\) −32.0000 −1.01294
\(999\) − 10.4641i − 0.331070i
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 1014.2.b.e.337.3 4
3.2 odd 2 3042.2.b.i.1351.2 4
13.2 odd 12 1014.2.e.g.529.1 4
13.3 even 3 1014.2.i.a.823.1 4
13.4 even 6 1014.2.i.a.361.1 4
13.5 odd 4 1014.2.a.k.1.1 2
13.6 odd 12 1014.2.e.g.991.1 4
13.7 odd 12 1014.2.e.i.991.2 4
13.8 odd 4 1014.2.a.i.1.2 2
13.9 even 3 78.2.i.a.49.2 yes 4
13.10 even 6 78.2.i.a.43.2 4
13.11 odd 12 1014.2.e.i.529.2 4
13.12 even 2 inner 1014.2.b.e.337.2 4
39.5 even 4 3042.2.a.p.1.2 2
39.8 even 4 3042.2.a.y.1.1 2
39.23 odd 6 234.2.l.c.199.1 4
39.35 odd 6 234.2.l.c.127.1 4
39.38 odd 2 3042.2.b.i.1351.3 4
52.23 odd 6 624.2.bv.e.433.1 4
52.31 even 4 8112.2.a.bp.1.1 2
52.35 odd 6 624.2.bv.e.49.2 4
52.47 even 4 8112.2.a.bj.1.2 2
65.9 even 6 1950.2.bc.d.751.1 4
65.22 odd 12 1950.2.y.g.49.1 4
65.23 odd 12 1950.2.y.g.199.1 4
65.48 odd 12 1950.2.y.b.49.2 4
65.49 even 6 1950.2.bc.d.901.1 4
65.62 odd 12 1950.2.y.b.199.2 4
156.23 even 6 1872.2.by.h.433.2 4
156.35 even 6 1872.2.by.h.1297.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.2 4 13.10 even 6
78.2.i.a.49.2 yes 4 13.9 even 3
234.2.l.c.127.1 4 39.35 odd 6
234.2.l.c.199.1 4 39.23 odd 6
624.2.bv.e.49.2 4 52.35 odd 6
624.2.bv.e.433.1 4 52.23 odd 6
1014.2.a.i.1.2 2 13.8 odd 4
1014.2.a.k.1.1 2 13.5 odd 4
1014.2.b.e.337.2 4 13.12 even 2 inner
1014.2.b.e.337.3 4 1.1 even 1 trivial
1014.2.e.g.529.1 4 13.2 odd 12
1014.2.e.g.991.1 4 13.6 odd 12
1014.2.e.i.529.2 4 13.11 odd 12
1014.2.e.i.991.2 4 13.7 odd 12
1014.2.i.a.361.1 4 13.4 even 6
1014.2.i.a.823.1 4 13.3 even 3
1872.2.by.h.433.2 4 156.23 even 6
1872.2.by.h.1297.1 4 156.35 even 6
1950.2.y.b.49.2 4 65.48 odd 12
1950.2.y.b.199.2 4 65.62 odd 12
1950.2.y.g.49.1 4 65.22 odd 12
1950.2.y.g.199.1 4 65.23 odd 12
1950.2.bc.d.751.1 4 65.9 even 6
1950.2.bc.d.901.1 4 65.49 even 6
3042.2.a.p.1.2 2 39.5 even 4
3042.2.a.y.1.1 2 39.8 even 4
3042.2.b.i.1351.2 4 3.2 odd 2
3042.2.b.i.1351.3 4 39.38 odd 2
8112.2.a.bj.1.2 2 52.47 even 4
8112.2.a.bp.1.1 2 52.31 even 4