Properties

Label 8470.2.a.bs.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +1.00000 q^{10} -0.732051 q^{12} +4.00000 q^{13} -1.00000 q^{14} +0.732051 q^{15} +1.00000 q^{16} +7.46410 q^{17} +2.46410 q^{18} +2.73205 q^{19} -1.00000 q^{20} -0.732051 q^{21} -2.19615 q^{23} +0.732051 q^{24} +1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} +9.66025 q^{29} -0.732051 q^{30} +7.46410 q^{31} -1.00000 q^{32} -7.46410 q^{34} -1.00000 q^{35} -2.46410 q^{36} +0.732051 q^{37} -2.73205 q^{38} -2.92820 q^{39} +1.00000 q^{40} -8.19615 q^{41} +0.732051 q^{42} +8.92820 q^{43} +2.46410 q^{45} +2.19615 q^{46} +6.00000 q^{47} -0.732051 q^{48} +1.00000 q^{49} -1.00000 q^{50} -5.46410 q^{51} +4.00000 q^{52} +8.73205 q^{53} -4.00000 q^{54} -1.00000 q^{56} -2.00000 q^{57} -9.66025 q^{58} -12.3923 q^{59} +0.732051 q^{60} -8.00000 q^{61} -7.46410 q^{62} -2.46410 q^{63} +1.00000 q^{64} -4.00000 q^{65} -11.8564 q^{67} +7.46410 q^{68} +1.60770 q^{69} +1.00000 q^{70} +14.9282 q^{71} +2.46410 q^{72} -6.00000 q^{73} -0.732051 q^{74} -0.732051 q^{75} +2.73205 q^{76} +2.92820 q^{78} -9.66025 q^{79} -1.00000 q^{80} +4.46410 q^{81} +8.19615 q^{82} +1.07180 q^{83} -0.732051 q^{84} -7.46410 q^{85} -8.92820 q^{86} -7.07180 q^{87} +4.92820 q^{89} -2.46410 q^{90} +4.00000 q^{91} -2.19615 q^{92} -5.46410 q^{93} -6.00000 q^{94} -2.73205 q^{95} +0.732051 q^{96} -0.732051 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} + 8q^{13} - 2q^{14} - 2q^{15} + 2q^{16} + 8q^{17} - 2q^{18} + 2q^{19} - 2q^{20} + 2q^{21} + 6q^{23} - 2q^{24} + 2q^{25} - 8q^{26} + 8q^{27} + 2q^{28} + 2q^{29} + 2q^{30} + 8q^{31} - 2q^{32} - 8q^{34} - 2q^{35} + 2q^{36} - 2q^{37} - 2q^{38} + 8q^{39} + 2q^{40} - 6q^{41} - 2q^{42} + 4q^{43} - 2q^{45} - 6q^{46} + 12q^{47} + 2q^{48} + 2q^{49} - 2q^{50} - 4q^{51} + 8q^{52} + 14q^{53} - 8q^{54} - 2q^{56} - 4q^{57} - 2q^{58} - 4q^{59} - 2q^{60} - 16q^{61} - 8q^{62} + 2q^{63} + 2q^{64} - 8q^{65} + 4q^{67} + 8q^{68} + 24q^{69} + 2q^{70} + 16q^{71} - 2q^{72} - 12q^{73} + 2q^{74} + 2q^{75} + 2q^{76} - 8q^{78} - 2q^{79} - 2q^{80} + 2q^{81} + 6q^{82} + 16q^{83} + 2q^{84} - 8q^{85} - 4q^{86} - 28q^{87} - 4q^{89} + 2q^{90} + 8q^{91} + 6q^{92} - 4q^{93} - 12q^{94} - 2q^{95} - 2q^{96} + 2q^{97} - 2q^{98} + O(q^{100}) \)

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
\(3\) −0.732051 −0.422650 −0.211325 0.977416i \(-0.567778\pi\)
−0.211325 + 0.977416i \(0.567778\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0.732051 0.298858
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.46410 −0.821367
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −0.732051 −0.211325
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0.732051 0.189015
\(16\) 1.00000 0.250000
\(17\) 7.46410 1.81031 0.905155 0.425081i \(-0.139754\pi\)
0.905155 + 0.425081i \(0.139754\pi\)
\(18\) 2.46410 0.580794
\(19\) 2.73205 0.626775 0.313388 0.949625i \(-0.398536\pi\)
0.313388 + 0.949625i \(0.398536\pi\)
\(20\) −1.00000 −0.223607
\(21\) −0.732051 −0.159747
\(22\) 0 0
\(23\) −2.19615 −0.457929 −0.228965 0.973435i \(-0.573534\pi\)
−0.228965 + 0.973435i \(0.573534\pi\)
\(24\) 0.732051 0.149429
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 4.00000 0.769800
\(28\) 1.00000 0.188982
\(29\) 9.66025 1.79386 0.896932 0.442168i \(-0.145791\pi\)
0.896932 + 0.442168i \(0.145791\pi\)
\(30\) −0.732051 −0.133654
\(31\) 7.46410 1.34059 0.670296 0.742094i \(-0.266167\pi\)
0.670296 + 0.742094i \(0.266167\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −7.46410 −1.28008
\(35\) −1.00000 −0.169031
\(36\) −2.46410 −0.410684
\(37\) 0.732051 0.120348 0.0601742 0.998188i \(-0.480834\pi\)
0.0601742 + 0.998188i \(0.480834\pi\)
\(38\) −2.73205 −0.443197
\(39\) −2.92820 −0.468888
\(40\) 1.00000 0.158114
\(41\) −8.19615 −1.28002 −0.640012 0.768365i \(-0.721071\pi\)
−0.640012 + 0.768365i \(0.721071\pi\)
\(42\) 0.732051 0.112958
\(43\) 8.92820 1.36154 0.680769 0.732498i \(-0.261646\pi\)
0.680769 + 0.732498i \(0.261646\pi\)
\(44\) 0 0
\(45\) 2.46410 0.367327
\(46\) 2.19615 0.323805
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −0.732051 −0.105662
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −5.46410 −0.765127
\(52\) 4.00000 0.554700
\(53\) 8.73205 1.19944 0.599720 0.800210i \(-0.295279\pi\)
0.599720 + 0.800210i \(0.295279\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −2.00000 −0.264906
\(58\) −9.66025 −1.26845
\(59\) −12.3923 −1.61334 −0.806670 0.591002i \(-0.798733\pi\)
−0.806670 + 0.591002i \(0.798733\pi\)
\(60\) 0.732051 0.0945074
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −7.46410 −0.947942
\(63\) −2.46410 −0.310448
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −11.8564 −1.44849 −0.724245 0.689542i \(-0.757812\pi\)
−0.724245 + 0.689542i \(0.757812\pi\)
\(68\) 7.46410 0.905155
\(69\) 1.60770 0.193544
\(70\) 1.00000 0.119523
\(71\) 14.9282 1.77165 0.885826 0.464018i \(-0.153593\pi\)
0.885826 + 0.464018i \(0.153593\pi\)
\(72\) 2.46410 0.290397
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −0.732051 −0.0850992
\(75\) −0.732051 −0.0845299
\(76\) 2.73205 0.313388
\(77\) 0 0
\(78\) 2.92820 0.331554
\(79\) −9.66025 −1.08686 −0.543432 0.839453i \(-0.682875\pi\)
−0.543432 + 0.839453i \(0.682875\pi\)
\(80\) −1.00000 −0.111803
\(81\) 4.46410 0.496011
\(82\) 8.19615 0.905114
\(83\) 1.07180 0.117645 0.0588225 0.998268i \(-0.481265\pi\)
0.0588225 + 0.998268i \(0.481265\pi\)
\(84\) −0.732051 −0.0798733
\(85\) −7.46410 −0.809595
\(86\) −8.92820 −0.962753
\(87\) −7.07180 −0.758176
\(88\) 0 0
\(89\) 4.92820 0.522388 0.261194 0.965286i \(-0.415884\pi\)
0.261194 + 0.965286i \(0.415884\pi\)
\(90\) −2.46410 −0.259739
\(91\) 4.00000 0.419314
\(92\) −2.19615 −0.228965
\(93\) −5.46410 −0.566601
\(94\) −6.00000 −0.618853
\(95\) −2.73205 −0.280302
\(96\) 0.732051 0.0747146
\(97\) −0.732051 −0.0743285 −0.0371642 0.999309i \(-0.511832\pi\)
−0.0371642 + 0.999309i \(0.511832\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 5.46410 0.541027
\(103\) −18.3923 −1.81225 −0.906124 0.423013i \(-0.860973\pi\)
−0.906124 + 0.423013i \(0.860973\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0.732051 0.0714408
\(106\) −8.73205 −0.848132
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 4.00000 0.384900
\(109\) −13.2679 −1.27084 −0.635420 0.772167i \(-0.719173\pi\)
−0.635420 + 0.772167i \(0.719173\pi\)
\(110\) 0 0
\(111\) −0.535898 −0.0508652
\(112\) 1.00000 0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 2.00000 0.187317
\(115\) 2.19615 0.204792
\(116\) 9.66025 0.896932
\(117\) −9.85641 −0.911225
\(118\) 12.3923 1.14080
\(119\) 7.46410 0.684233
\(120\) −0.732051 −0.0668268
\(121\) 0 0
\(122\) 8.00000 0.724286
\(123\) 6.00000 0.541002
\(124\) 7.46410 0.670296
\(125\) −1.00000 −0.0894427
\(126\) 2.46410 0.219520
\(127\) −12.3923 −1.09964 −0.549820 0.835283i \(-0.685304\pi\)
−0.549820 + 0.835283i \(0.685304\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.53590 −0.575454
\(130\) 4.00000 0.350823
\(131\) −0.196152 −0.0171379 −0.00856896 0.999963i \(-0.502728\pi\)
−0.00856896 + 0.999963i \(0.502728\pi\)
\(132\) 0 0
\(133\) 2.73205 0.236899
\(134\) 11.8564 1.02424
\(135\) −4.00000 −0.344265
\(136\) −7.46410 −0.640041
\(137\) 19.8564 1.69645 0.848224 0.529638i \(-0.177672\pi\)
0.848224 + 0.529638i \(0.177672\pi\)
\(138\) −1.60770 −0.136856
\(139\) 13.6603 1.15865 0.579324 0.815097i \(-0.303317\pi\)
0.579324 + 0.815097i \(0.303317\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −4.39230 −0.369899
\(142\) −14.9282 −1.25275
\(143\) 0 0
\(144\) −2.46410 −0.205342
\(145\) −9.66025 −0.802240
\(146\) 6.00000 0.496564
\(147\) −0.732051 −0.0603785
\(148\) 0.732051 0.0601742
\(149\) 12.1962 0.999148 0.499574 0.866271i \(-0.333490\pi\)
0.499574 + 0.866271i \(0.333490\pi\)
\(150\) 0.732051 0.0597717
\(151\) −12.1962 −0.992509 −0.496254 0.868177i \(-0.665292\pi\)
−0.496254 + 0.868177i \(0.665292\pi\)
\(152\) −2.73205 −0.221599
\(153\) −18.3923 −1.48693
\(154\) 0 0
\(155\) −7.46410 −0.599531
\(156\) −2.92820 −0.234444
\(157\) −23.4641 −1.87264 −0.936320 0.351149i \(-0.885791\pi\)
−0.936320 + 0.351149i \(0.885791\pi\)
\(158\) 9.66025 0.768529
\(159\) −6.39230 −0.506943
\(160\) 1.00000 0.0790569
\(161\) −2.19615 −0.173081
\(162\) −4.46410 −0.350733
\(163\) −0.928203 −0.0727025 −0.0363512 0.999339i \(-0.511574\pi\)
−0.0363512 + 0.999339i \(0.511574\pi\)
\(164\) −8.19615 −0.640012
\(165\) 0 0
\(166\) −1.07180 −0.0831876
\(167\) 18.9282 1.46471 0.732354 0.680924i \(-0.238422\pi\)
0.732354 + 0.680924i \(0.238422\pi\)
\(168\) 0.732051 0.0564789
\(169\) 3.00000 0.230769
\(170\) 7.46410 0.572470
\(171\) −6.73205 −0.514813
\(172\) 8.92820 0.680769
\(173\) −8.92820 −0.678799 −0.339399 0.940642i \(-0.610224\pi\)
−0.339399 + 0.940642i \(0.610224\pi\)
\(174\) 7.07180 0.536112
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 9.07180 0.681878
\(178\) −4.92820 −0.369384
\(179\) −14.3923 −1.07573 −0.537866 0.843031i \(-0.680769\pi\)
−0.537866 + 0.843031i \(0.680769\pi\)
\(180\) 2.46410 0.183663
\(181\) 8.92820 0.663628 0.331814 0.943345i \(-0.392339\pi\)
0.331814 + 0.943345i \(0.392339\pi\)
\(182\) −4.00000 −0.296500
\(183\) 5.85641 0.432918
\(184\) 2.19615 0.161903
\(185\) −0.732051 −0.0538214
\(186\) 5.46410 0.400647
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) 4.00000 0.290957
\(190\) 2.73205 0.198204
\(191\) 0.392305 0.0283862 0.0141931 0.999899i \(-0.495482\pi\)
0.0141931 + 0.999899i \(0.495482\pi\)
\(192\) −0.732051 −0.0528312
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 0.732051 0.0525582
\(195\) 2.92820 0.209693
\(196\) 1.00000 0.0714286
\(197\) 3.46410 0.246807 0.123404 0.992357i \(-0.460619\pi\)
0.123404 + 0.992357i \(0.460619\pi\)
\(198\) 0 0
\(199\) 25.8564 1.83291 0.916456 0.400135i \(-0.131037\pi\)
0.916456 + 0.400135i \(0.131037\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.67949 0.612204
\(202\) −8.00000 −0.562878
\(203\) 9.66025 0.678017
\(204\) −5.46410 −0.382564
\(205\) 8.19615 0.572444
\(206\) 18.3923 1.28145
\(207\) 5.41154 0.376128
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) −0.732051 −0.0505163
\(211\) 13.8564 0.953914 0.476957 0.878927i \(-0.341740\pi\)
0.476957 + 0.878927i \(0.341740\pi\)
\(212\) 8.73205 0.599720
\(213\) −10.9282 −0.748788
\(214\) 2.00000 0.136717
\(215\) −8.92820 −0.608898
\(216\) −4.00000 −0.272166
\(217\) 7.46410 0.506696
\(218\) 13.2679 0.898619
\(219\) 4.39230 0.296804
\(220\) 0 0
\(221\) 29.8564 2.00836
\(222\) 0.535898 0.0359671
\(223\) −4.53590 −0.303746 −0.151873 0.988400i \(-0.548531\pi\)
−0.151873 + 0.988400i \(0.548531\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.46410 −0.164273
\(226\) −18.0000 −1.19734
\(227\) 11.3205 0.751369 0.375684 0.926748i \(-0.377408\pi\)
0.375684 + 0.926748i \(0.377408\pi\)
\(228\) −2.00000 −0.132453
\(229\) −11.3205 −0.748080 −0.374040 0.927413i \(-0.622028\pi\)
−0.374040 + 0.927413i \(0.622028\pi\)
\(230\) −2.19615 −0.144810
\(231\) 0 0
\(232\) −9.66025 −0.634227
\(233\) −3.85641 −0.252642 −0.126321 0.991989i \(-0.540317\pi\)
−0.126321 + 0.991989i \(0.540317\pi\)
\(234\) 9.85641 0.644333
\(235\) −6.00000 −0.391397
\(236\) −12.3923 −0.806670
\(237\) 7.07180 0.459363
\(238\) −7.46410 −0.483826
\(239\) −18.0526 −1.16772 −0.583861 0.811853i \(-0.698459\pi\)
−0.583861 + 0.811853i \(0.698459\pi\)
\(240\) 0.732051 0.0472537
\(241\) 28.9808 1.86681 0.933407 0.358818i \(-0.116820\pi\)
0.933407 + 0.358818i \(0.116820\pi\)
\(242\) 0 0
\(243\) −15.2679 −0.979439
\(244\) −8.00000 −0.512148
\(245\) −1.00000 −0.0638877
\(246\) −6.00000 −0.382546
\(247\) 10.9282 0.695345
\(248\) −7.46410 −0.473971
\(249\) −0.784610 −0.0497226
\(250\) 1.00000 0.0632456
\(251\) 16.7846 1.05944 0.529718 0.848174i \(-0.322298\pi\)
0.529718 + 0.848174i \(0.322298\pi\)
\(252\) −2.46410 −0.155224
\(253\) 0 0
\(254\) 12.3923 0.777562
\(255\) 5.46410 0.342175
\(256\) 1.00000 0.0625000
\(257\) −18.1962 −1.13504 −0.567522 0.823358i \(-0.692098\pi\)
−0.567522 + 0.823358i \(0.692098\pi\)
\(258\) 6.53590 0.406907
\(259\) 0.732051 0.0454874
\(260\) −4.00000 −0.248069
\(261\) −23.8038 −1.47342
\(262\) 0.196152 0.0121183
\(263\) 7.60770 0.469111 0.234555 0.972103i \(-0.424637\pi\)
0.234555 + 0.972103i \(0.424637\pi\)
\(264\) 0 0
\(265\) −8.73205 −0.536406
\(266\) −2.73205 −0.167513
\(267\) −3.60770 −0.220787
\(268\) −11.8564 −0.724245
\(269\) 16.3923 0.999456 0.499728 0.866182i \(-0.333433\pi\)
0.499728 + 0.866182i \(0.333433\pi\)
\(270\) 4.00000 0.243432
\(271\) 11.6077 0.705117 0.352559 0.935790i \(-0.385312\pi\)
0.352559 + 0.935790i \(0.385312\pi\)
\(272\) 7.46410 0.452578
\(273\) −2.92820 −0.177223
\(274\) −19.8564 −1.19957
\(275\) 0 0
\(276\) 1.60770 0.0967719
\(277\) 17.3205 1.04069 0.520344 0.853957i \(-0.325804\pi\)
0.520344 + 0.853957i \(0.325804\pi\)
\(278\) −13.6603 −0.819288
\(279\) −18.3923 −1.10112
\(280\) 1.00000 0.0597614
\(281\) 11.3205 0.675325 0.337662 0.941267i \(-0.390364\pi\)
0.337662 + 0.941267i \(0.390364\pi\)
\(282\) 4.39230 0.261558
\(283\) −2.53590 −0.150744 −0.0753718 0.997156i \(-0.524014\pi\)
−0.0753718 + 0.997156i \(0.524014\pi\)
\(284\) 14.9282 0.885826
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) −8.19615 −0.483804
\(288\) 2.46410 0.145199
\(289\) 38.7128 2.27722
\(290\) 9.66025 0.567270
\(291\) 0.535898 0.0314149
\(292\) −6.00000 −0.351123
\(293\) −27.7128 −1.61900 −0.809500 0.587120i \(-0.800262\pi\)
−0.809500 + 0.587120i \(0.800262\pi\)
\(294\) 0.732051 0.0426941
\(295\) 12.3923 0.721508
\(296\) −0.732051 −0.0425496
\(297\) 0 0
\(298\) −12.1962 −0.706504
\(299\) −8.78461 −0.508027
\(300\) −0.732051 −0.0422650
\(301\) 8.92820 0.514613
\(302\) 12.1962 0.701810
\(303\) −5.85641 −0.336442
\(304\) 2.73205 0.156694
\(305\) 8.00000 0.458079
\(306\) 18.3923 1.05142
\(307\) 6.53590 0.373023 0.186512 0.982453i \(-0.440282\pi\)
0.186512 + 0.982453i \(0.440282\pi\)
\(308\) 0 0
\(309\) 13.4641 0.765946
\(310\) 7.46410 0.423932
\(311\) −17.3205 −0.982156 −0.491078 0.871116i \(-0.663397\pi\)
−0.491078 + 0.871116i \(0.663397\pi\)
\(312\) 2.92820 0.165777
\(313\) 0.732051 0.0413780 0.0206890 0.999786i \(-0.493414\pi\)
0.0206890 + 0.999786i \(0.493414\pi\)
\(314\) 23.4641 1.32416
\(315\) 2.46410 0.138836
\(316\) −9.66025 −0.543432
\(317\) 26.5885 1.49336 0.746678 0.665185i \(-0.231647\pi\)
0.746678 + 0.665185i \(0.231647\pi\)
\(318\) 6.39230 0.358463
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 1.46410 0.0817182
\(322\) 2.19615 0.122387
\(323\) 20.3923 1.13466
\(324\) 4.46410 0.248006
\(325\) 4.00000 0.221880
\(326\) 0.928203 0.0514084
\(327\) 9.71281 0.537120
\(328\) 8.19615 0.452557
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 26.3923 1.45065 0.725326 0.688405i \(-0.241689\pi\)
0.725326 + 0.688405i \(0.241689\pi\)
\(332\) 1.07180 0.0588225
\(333\) −1.80385 −0.0988502
\(334\) −18.9282 −1.03571
\(335\) 11.8564 0.647785
\(336\) −0.732051 −0.0399366
\(337\) 0.143594 0.00782204 0.00391102 0.999992i \(-0.498755\pi\)
0.00391102 + 0.999992i \(0.498755\pi\)
\(338\) −3.00000 −0.163178
\(339\) −13.1769 −0.715672
\(340\) −7.46410 −0.404798
\(341\) 0 0
\(342\) 6.73205 0.364028
\(343\) 1.00000 0.0539949
\(344\) −8.92820 −0.481376
\(345\) −1.60770 −0.0865554
\(346\) 8.92820 0.479983
\(347\) −8.92820 −0.479291 −0.239646 0.970860i \(-0.577031\pi\)
−0.239646 + 0.970860i \(0.577031\pi\)
\(348\) −7.07180 −0.379088
\(349\) −1.46410 −0.0783716 −0.0391858 0.999232i \(-0.512476\pi\)
−0.0391858 + 0.999232i \(0.512476\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 16.0000 0.854017
\(352\) 0 0
\(353\) −26.9808 −1.43604 −0.718021 0.696022i \(-0.754952\pi\)
−0.718021 + 0.696022i \(0.754952\pi\)
\(354\) −9.07180 −0.482161
\(355\) −14.9282 −0.792307
\(356\) 4.92820 0.261194
\(357\) −5.46410 −0.289191
\(358\) 14.3923 0.760657
\(359\) −34.4449 −1.81793 −0.908965 0.416872i \(-0.863126\pi\)
−0.908965 + 0.416872i \(0.863126\pi\)
\(360\) −2.46410 −0.129870
\(361\) −11.5359 −0.607153
\(362\) −8.92820 −0.469256
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) 6.00000 0.314054
\(366\) −5.85641 −0.306119
\(367\) 6.39230 0.333676 0.166838 0.985984i \(-0.446644\pi\)
0.166838 + 0.985984i \(0.446644\pi\)
\(368\) −2.19615 −0.114482
\(369\) 20.1962 1.05137
\(370\) 0.732051 0.0380575
\(371\) 8.73205 0.453345
\(372\) −5.46410 −0.283300
\(373\) −1.32051 −0.0683733 −0.0341867 0.999415i \(-0.510884\pi\)
−0.0341867 + 0.999415i \(0.510884\pi\)
\(374\) 0 0
\(375\) 0.732051 0.0378029
\(376\) −6.00000 −0.309426
\(377\) 38.6410 1.99011
\(378\) −4.00000 −0.205738
\(379\) 2.14359 0.110109 0.0550545 0.998483i \(-0.482467\pi\)
0.0550545 + 0.998483i \(0.482467\pi\)
\(380\) −2.73205 −0.140151
\(381\) 9.07180 0.464762
\(382\) −0.392305 −0.0200721
\(383\) −3.46410 −0.177007 −0.0885037 0.996076i \(-0.528208\pi\)
−0.0885037 + 0.996076i \(0.528208\pi\)
\(384\) 0.732051 0.0373573
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) −22.0000 −1.11832
\(388\) −0.732051 −0.0371642
\(389\) 5.60770 0.284321 0.142161 0.989844i \(-0.454595\pi\)
0.142161 + 0.989844i \(0.454595\pi\)
\(390\) −2.92820 −0.148275
\(391\) −16.3923 −0.828994
\(392\) −1.00000 −0.0505076
\(393\) 0.143594 0.00724334
\(394\) −3.46410 −0.174519
\(395\) 9.66025 0.486060
\(396\) 0 0
\(397\) 0.535898 0.0268960 0.0134480 0.999910i \(-0.495719\pi\)
0.0134480 + 0.999910i \(0.495719\pi\)
\(398\) −25.8564 −1.29606
\(399\) −2.00000 −0.100125
\(400\) 1.00000 0.0500000
\(401\) −0.392305 −0.0195908 −0.00979538 0.999952i \(-0.503118\pi\)
−0.00979538 + 0.999952i \(0.503118\pi\)
\(402\) −8.67949 −0.432894
\(403\) 29.8564 1.48725
\(404\) 8.00000 0.398015
\(405\) −4.46410 −0.221823
\(406\) −9.66025 −0.479430
\(407\) 0 0
\(408\) 5.46410 0.270513
\(409\) 19.8038 0.979237 0.489619 0.871937i \(-0.337136\pi\)
0.489619 + 0.871937i \(0.337136\pi\)
\(410\) −8.19615 −0.404779
\(411\) −14.5359 −0.717003
\(412\) −18.3923 −0.906124
\(413\) −12.3923 −0.609785
\(414\) −5.41154 −0.265963
\(415\) −1.07180 −0.0526124
\(416\) −4.00000 −0.196116
\(417\) −10.0000 −0.489702
\(418\) 0 0
\(419\) 10.1436 0.495547 0.247773 0.968818i \(-0.420301\pi\)
0.247773 + 0.968818i \(0.420301\pi\)
\(420\) 0.732051 0.0357204
\(421\) −24.2487 −1.18181 −0.590905 0.806741i \(-0.701229\pi\)
−0.590905 + 0.806741i \(0.701229\pi\)
\(422\) −13.8564 −0.674519
\(423\) −14.7846 −0.718852
\(424\) −8.73205 −0.424066
\(425\) 7.46410 0.362062
\(426\) 10.9282 0.529473
\(427\) −8.00000 −0.387147
\(428\) −2.00000 −0.0966736
\(429\) 0 0
\(430\) 8.92820 0.430556
\(431\) −13.2679 −0.639095 −0.319547 0.947570i \(-0.603531\pi\)
−0.319547 + 0.947570i \(0.603531\pi\)
\(432\) 4.00000 0.192450
\(433\) −32.7321 −1.57300 −0.786501 0.617589i \(-0.788110\pi\)
−0.786501 + 0.617589i \(0.788110\pi\)
\(434\) −7.46410 −0.358288
\(435\) 7.07180 0.339067
\(436\) −13.2679 −0.635420
\(437\) −6.00000 −0.287019
\(438\) −4.39230 −0.209872
\(439\) −22.5359 −1.07558 −0.537790 0.843079i \(-0.680741\pi\)
−0.537790 + 0.843079i \(0.680741\pi\)
\(440\) 0 0
\(441\) −2.46410 −0.117338
\(442\) −29.8564 −1.42012
\(443\) −15.0718 −0.716083 −0.358041 0.933706i \(-0.616555\pi\)
−0.358041 + 0.933706i \(0.616555\pi\)
\(444\) −0.535898 −0.0254326
\(445\) −4.92820 −0.233619
\(446\) 4.53590 0.214781
\(447\) −8.92820 −0.422290
\(448\) 1.00000 0.0472456
\(449\) 38.2487 1.80507 0.902534 0.430618i \(-0.141704\pi\)
0.902534 + 0.430618i \(0.141704\pi\)
\(450\) 2.46410 0.116159
\(451\) 0 0
\(452\) 18.0000 0.846649
\(453\) 8.92820 0.419484
\(454\) −11.3205 −0.531298
\(455\) −4.00000 −0.187523
\(456\) 2.00000 0.0936586
\(457\) 11.8564 0.554619 0.277310 0.960781i \(-0.410557\pi\)
0.277310 + 0.960781i \(0.410557\pi\)
\(458\) 11.3205 0.528973
\(459\) 29.8564 1.39358
\(460\) 2.19615 0.102396
\(461\) −39.3205 −1.83134 −0.915669 0.401932i \(-0.868339\pi\)
−0.915669 + 0.401932i \(0.868339\pi\)
\(462\) 0 0
\(463\) 38.5885 1.79336 0.896679 0.442682i \(-0.145973\pi\)
0.896679 + 0.442682i \(0.145973\pi\)
\(464\) 9.66025 0.448466
\(465\) 5.46410 0.253392
\(466\) 3.85641 0.178645
\(467\) 9.41154 0.435514 0.217757 0.976003i \(-0.430126\pi\)
0.217757 + 0.976003i \(0.430126\pi\)
\(468\) −9.85641 −0.455613
\(469\) −11.8564 −0.547478
\(470\) 6.00000 0.276759
\(471\) 17.1769 0.791470
\(472\) 12.3923 0.570402
\(473\) 0 0
\(474\) −7.07180 −0.324818
\(475\) 2.73205 0.125355
\(476\) 7.46410 0.342117
\(477\) −21.5167 −0.985180
\(478\) 18.0526 0.825705
\(479\) 35.7128 1.63176 0.815880 0.578221i \(-0.196253\pi\)
0.815880 + 0.578221i \(0.196253\pi\)
\(480\) −0.732051 −0.0334134
\(481\) 2.92820 0.133515
\(482\) −28.9808 −1.32004
\(483\) 1.60770 0.0731527
\(484\) 0 0
\(485\) 0.732051 0.0332407
\(486\) 15.2679 0.692568
\(487\) 8.05256 0.364896 0.182448 0.983215i \(-0.441598\pi\)
0.182448 + 0.983215i \(0.441598\pi\)
\(488\) 8.00000 0.362143
\(489\) 0.679492 0.0307277
\(490\) 1.00000 0.0451754
\(491\) −22.9282 −1.03474 −0.517368 0.855763i \(-0.673088\pi\)
−0.517368 + 0.855763i \(0.673088\pi\)
\(492\) 6.00000 0.270501
\(493\) 72.1051 3.24745
\(494\) −10.9282 −0.491683
\(495\) 0 0
\(496\) 7.46410 0.335148
\(497\) 14.9282 0.669621
\(498\) 0.784610 0.0351592
\(499\) −1.60770 −0.0719703 −0.0359852 0.999352i \(-0.511457\pi\)
−0.0359852 + 0.999352i \(0.511457\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −13.8564 −0.619059
\(502\) −16.7846 −0.749134
\(503\) −37.8564 −1.68793 −0.843967 0.536395i \(-0.819786\pi\)
−0.843967 + 0.536395i \(0.819786\pi\)
\(504\) 2.46410 0.109760
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) −2.19615 −0.0975346
\(508\) −12.3923 −0.549820
\(509\) 12.9282 0.573033 0.286516 0.958075i \(-0.407503\pi\)
0.286516 + 0.958075i \(0.407503\pi\)
\(510\) −5.46410 −0.241954
\(511\) −6.00000 −0.265424
\(512\) −1.00000 −0.0441942
\(513\) 10.9282 0.482492
\(514\) 18.1962 0.802598
\(515\) 18.3923 0.810462
\(516\) −6.53590 −0.287727
\(517\) 0 0
\(518\) −0.732051 −0.0321645
\(519\) 6.53590 0.286894
\(520\) 4.00000 0.175412
\(521\) 24.2487 1.06236 0.531178 0.847260i \(-0.321750\pi\)
0.531178 + 0.847260i \(0.321750\pi\)
\(522\) 23.8038 1.04187
\(523\) −21.4641 −0.938560 −0.469280 0.883050i \(-0.655486\pi\)
−0.469280 + 0.883050i \(0.655486\pi\)
\(524\) −0.196152 −0.00856896
\(525\) −0.732051 −0.0319493
\(526\) −7.60770 −0.331711
\(527\) 55.7128 2.42689
\(528\) 0 0
\(529\) −18.1769 −0.790301
\(530\) 8.73205 0.379296
\(531\) 30.5359 1.32515
\(532\) 2.73205 0.118449
\(533\) −32.7846 −1.42006
\(534\) 3.60770 0.156120
\(535\) 2.00000 0.0864675
\(536\) 11.8564 0.512119
\(537\) 10.5359 0.454658
\(538\) −16.3923 −0.706722
\(539\) 0 0
\(540\) −4.00000 −0.172133
\(541\) −9.26795 −0.398460 −0.199230 0.979953i \(-0.563844\pi\)
−0.199230 + 0.979953i \(0.563844\pi\)
\(542\) −11.6077 −0.498593
\(543\) −6.53590 −0.280482
\(544\) −7.46410 −0.320021
\(545\) 13.2679 0.568337
\(546\) 2.92820 0.125316
\(547\) −31.7128 −1.35594 −0.677971 0.735089i \(-0.737141\pi\)
−0.677971 + 0.735089i \(0.737141\pi\)
\(548\) 19.8564 0.848224
\(549\) 19.7128 0.841322
\(550\) 0 0
\(551\) 26.3923 1.12435
\(552\) −1.60770 −0.0684280
\(553\) −9.66025 −0.410796
\(554\) −17.3205 −0.735878
\(555\) 0.535898 0.0227476
\(556\) 13.6603 0.579324
\(557\) 16.9282 0.717271 0.358635 0.933478i \(-0.383242\pi\)
0.358635 + 0.933478i \(0.383242\pi\)
\(558\) 18.3923 0.778608
\(559\) 35.7128 1.51049
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −11.3205 −0.477527
\(563\) −15.7128 −0.662216 −0.331108 0.943593i \(-0.607422\pi\)
−0.331108 + 0.943593i \(0.607422\pi\)
\(564\) −4.39230 −0.184949
\(565\) −18.0000 −0.757266
\(566\) 2.53590 0.106592
\(567\) 4.46410 0.187475
\(568\) −14.9282 −0.626373
\(569\) −6.14359 −0.257553 −0.128776 0.991674i \(-0.541105\pi\)
−0.128776 + 0.991674i \(0.541105\pi\)
\(570\) −2.00000 −0.0837708
\(571\) −15.3205 −0.641143 −0.320572 0.947224i \(-0.603875\pi\)
−0.320572 + 0.947224i \(0.603875\pi\)
\(572\) 0 0
\(573\) −0.287187 −0.0119974
\(574\) 8.19615 0.342101
\(575\) −2.19615 −0.0915859
\(576\) −2.46410 −0.102671
\(577\) 4.73205 0.196998 0.0984989 0.995137i \(-0.468596\pi\)
0.0984989 + 0.995137i \(0.468596\pi\)
\(578\) −38.7128 −1.61024
\(579\) 11.7128 0.486768
\(580\) −9.66025 −0.401120
\(581\) 1.07180 0.0444656
\(582\) −0.535898 −0.0222137
\(583\) 0 0
\(584\) 6.00000 0.248282
\(585\) 9.85641 0.407512
\(586\) 27.7128 1.14481
\(587\) 23.2679 0.960371 0.480186 0.877167i \(-0.340569\pi\)
0.480186 + 0.877167i \(0.340569\pi\)
\(588\) −0.732051 −0.0301893
\(589\) 20.3923 0.840250
\(590\) −12.3923 −0.510183
\(591\) −2.53590 −0.104313
\(592\) 0.732051 0.0300871
\(593\) 40.6410 1.66893 0.834463 0.551064i \(-0.185778\pi\)
0.834463 + 0.551064i \(0.185778\pi\)
\(594\) 0 0
\(595\) −7.46410 −0.305998
\(596\) 12.1962 0.499574
\(597\) −18.9282 −0.774680
\(598\) 8.78461 0.359229
\(599\) −33.0718 −1.35128 −0.675638 0.737233i \(-0.736132\pi\)
−0.675638 + 0.737233i \(0.736132\pi\)
\(600\) 0.732051 0.0298858
\(601\) 27.1244 1.10643 0.553213 0.833040i \(-0.313402\pi\)
0.553213 + 0.833040i \(0.313402\pi\)
\(602\) −8.92820 −0.363886
\(603\) 29.2154 1.18974
\(604\) −12.1962 −0.496254
\(605\) 0 0
\(606\) 5.85641 0.237900
\(607\) 10.0000 0.405887 0.202944 0.979190i \(-0.434949\pi\)
0.202944 + 0.979190i \(0.434949\pi\)
\(608\) −2.73205 −0.110799
\(609\) −7.07180 −0.286564
\(610\) −8.00000 −0.323911
\(611\) 24.0000 0.970936
\(612\) −18.3923 −0.743465
\(613\) −13.7128 −0.553855 −0.276928 0.960891i \(-0.589316\pi\)
−0.276928 + 0.960891i \(0.589316\pi\)
\(614\) −6.53590 −0.263767
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 17.6077 0.708859 0.354430 0.935083i \(-0.384675\pi\)
0.354430 + 0.935083i \(0.384675\pi\)
\(618\) −13.4641 −0.541606
\(619\) 18.9282 0.760789 0.380394 0.924824i \(-0.375788\pi\)
0.380394 + 0.924824i \(0.375788\pi\)
\(620\) −7.46410 −0.299766
\(621\) −8.78461 −0.352514
\(622\) 17.3205 0.694489
\(623\) 4.92820 0.197444
\(624\) −2.92820 −0.117222
\(625\) 1.00000 0.0400000
\(626\) −0.732051 −0.0292586
\(627\) 0 0
\(628\) −23.4641 −0.936320
\(629\) 5.46410 0.217868
\(630\) −2.46410 −0.0981722
\(631\) 1.85641 0.0739024 0.0369512 0.999317i \(-0.488235\pi\)
0.0369512 + 0.999317i \(0.488235\pi\)
\(632\) 9.66025 0.384264
\(633\) −10.1436 −0.403172
\(634\) −26.5885 −1.05596
\(635\) 12.3923 0.491774
\(636\) −6.39230 −0.253471
\(637\) 4.00000 0.158486
\(638\) 0 0
\(639\) −36.7846 −1.45518
\(640\) 1.00000 0.0395285
\(641\) 3.85641 0.152319 0.0761594 0.997096i \(-0.475734\pi\)
0.0761594 + 0.997096i \(0.475734\pi\)
\(642\) −1.46410 −0.0577835
\(643\) −6.19615 −0.244352 −0.122176 0.992508i \(-0.538987\pi\)
−0.122176 + 0.992508i \(0.538987\pi\)
\(644\) −2.19615 −0.0865405
\(645\) 6.53590 0.257351
\(646\) −20.3923 −0.802325
\(647\) −26.3923 −1.03759 −0.518794 0.854899i \(-0.673619\pi\)
−0.518794 + 0.854899i \(0.673619\pi\)
\(648\) −4.46410 −0.175366
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) −5.46410 −0.214155
\(652\) −0.928203 −0.0363512
\(653\) −31.2679 −1.22361 −0.611805 0.791009i \(-0.709556\pi\)
−0.611805 + 0.791009i \(0.709556\pi\)
\(654\) −9.71281 −0.379801
\(655\) 0.196152 0.00766431
\(656\) −8.19615 −0.320006
\(657\) 14.7846 0.576803
\(658\) −6.00000 −0.233904
\(659\) 19.6077 0.763807 0.381904 0.924202i \(-0.375269\pi\)
0.381904 + 0.924202i \(0.375269\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) −26.3923 −1.02577
\(663\) −21.8564 −0.848832
\(664\) −1.07180 −0.0415938
\(665\) −2.73205 −0.105944
\(666\) 1.80385 0.0698977
\(667\) −21.2154 −0.821463
\(668\) 18.9282 0.732354
\(669\) 3.32051 0.128378
\(670\) −11.8564 −0.458053
\(671\) 0 0
\(672\) 0.732051 0.0282395
\(673\) −16.1436 −0.622290 −0.311145 0.950362i \(-0.600712\pi\)
−0.311145 + 0.950362i \(0.600712\pi\)
\(674\) −0.143594 −0.00553102
\(675\) 4.00000 0.153960
\(676\) 3.00000 0.115385
\(677\) 0.928203 0.0356737 0.0178369 0.999841i \(-0.494322\pi\)
0.0178369 + 0.999841i \(0.494322\pi\)
\(678\) 13.1769 0.506056
\(679\) −0.732051 −0.0280935
\(680\) 7.46410 0.286235
\(681\) −8.28719 −0.317566
\(682\) 0 0
\(683\) 22.3923 0.856818 0.428409 0.903585i \(-0.359074\pi\)
0.428409 + 0.903585i \(0.359074\pi\)
\(684\) −6.73205 −0.257406
\(685\) −19.8564 −0.758674
\(686\) −1.00000 −0.0381802
\(687\) 8.28719 0.316176
\(688\) 8.92820 0.340385
\(689\) 34.9282 1.33066
\(690\) 1.60770 0.0612039
\(691\) −1.07180 −0.0407731 −0.0203865 0.999792i \(-0.506490\pi\)
−0.0203865 + 0.999792i \(0.506490\pi\)
\(692\) −8.92820 −0.339399
\(693\) 0 0
\(694\) 8.92820 0.338910
\(695\) −13.6603 −0.518163
\(696\) 7.07180 0.268056
\(697\) −61.1769 −2.31724
\(698\) 1.46410 0.0554171
\(699\) 2.82309 0.106779
\(700\) 1.00000 0.0377964
\(701\) 34.7321 1.31181 0.655906 0.754843i \(-0.272287\pi\)
0.655906 + 0.754843i \(0.272287\pi\)
\(702\) −16.0000 −0.603881
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) 4.39230 0.165424
\(706\) 26.9808 1.01543
\(707\) 8.00000 0.300871
\(708\) 9.07180 0.340939
\(709\) 23.8564 0.895946 0.447973 0.894047i \(-0.352146\pi\)
0.447973 + 0.894047i \(0.352146\pi\)
\(710\) 14.9282 0.560245
\(711\) 23.8038 0.892714
\(712\) −4.92820 −0.184692
\(713\) −16.3923 −0.613897
\(714\) 5.46410 0.204489
\(715\) 0 0
\(716\) −14.3923 −0.537866
\(717\) 13.2154 0.493538
\(718\) 34.4449 1.28547
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 2.46410 0.0918316
\(721\) −18.3923 −0.684965
\(722\) 11.5359 0.429322
\(723\) −21.2154 −0.789009
\(724\) 8.92820 0.331814
\(725\) 9.66025 0.358773
\(726\) 0 0
\(727\) 9.71281 0.360228 0.180114 0.983646i \(-0.442353\pi\)
0.180114 + 0.983646i \(0.442353\pi\)
\(728\) −4.00000 −0.148250
\(729\) −2.21539 −0.0820515
\(730\) −6.00000 −0.222070
\(731\) 66.6410 2.46481
\(732\) 5.85641 0.216459
\(733\) 24.1436 0.891764 0.445882 0.895092i \(-0.352890\pi\)
0.445882 + 0.895092i \(0.352890\pi\)
\(734\) −6.39230 −0.235944
\(735\) 0.732051 0.0270021
\(736\) 2.19615 0.0809513
\(737\) 0 0
\(738\) −20.1962 −0.743431
\(739\) 17.8564 0.656859 0.328429 0.944529i \(-0.393481\pi\)
0.328429 + 0.944529i \(0.393481\pi\)
\(740\) −0.732051 −0.0269107
\(741\) −8.00000 −0.293887
\(742\) −8.73205 −0.320564
\(743\) −24.3923 −0.894867 −0.447433 0.894317i \(-0.647662\pi\)
−0.447433 + 0.894317i \(0.647662\pi\)
\(744\) 5.46410 0.200324
\(745\) −12.1962 −0.446832
\(746\) 1.32051 0.0483472
\(747\) −2.64102 −0.0966297
\(748\) 0 0
\(749\) −2.00000 −0.0730784
\(750\) −0.732051 −0.0267307
\(751\) 49.8564 1.81929 0.909643 0.415391i \(-0.136355\pi\)
0.909643 + 0.415391i \(0.136355\pi\)
\(752\) 6.00000 0.218797
\(753\) −12.2872 −0.447770
\(754\) −38.6410 −1.40722
\(755\) 12.1962 0.443863
\(756\) 4.00000 0.145479
\(757\) −27.6603 −1.00533 −0.502665 0.864482i \(-0.667647\pi\)
−0.502665 + 0.864482i \(0.667647\pi\)
\(758\) −2.14359 −0.0778588
\(759\) 0 0
\(760\) 2.73205 0.0991019
\(761\) −3.41154 −0.123668 −0.0618342 0.998086i \(-0.519695\pi\)
−0.0618342 + 0.998086i \(0.519695\pi\)
\(762\) −9.07180 −0.328637
\(763\) −13.2679 −0.480332
\(764\) 0.392305 0.0141931
\(765\) 18.3923 0.664975
\(766\) 3.46410 0.125163
\(767\) −49.5692 −1.78984
\(768\) −0.732051 −0.0264156
\(769\) 6.73205 0.242764 0.121382 0.992606i \(-0.461267\pi\)
0.121382 + 0.992606i \(0.461267\pi\)
\(770\) 0 0
\(771\) 13.3205 0.479726
\(772\) −16.0000 −0.575853
\(773\) 9.71281 0.349346 0.174673 0.984627i \(-0.444113\pi\)
0.174673 + 0.984627i \(0.444113\pi\)
\(774\) 22.0000 0.790774
\(775\) 7.46410 0.268118
\(776\) 0.732051 0.0262791
\(777\) −0.535898 −0.0192252
\(778\) −5.60770 −0.201046
\(779\) −22.3923 −0.802288
\(780\) 2.92820 0.104846
\(781\) 0 0
\(782\) 16.3923 0.586188
\(783\) 38.6410 1.38092
\(784\) 1.00000 0.0357143
\(785\) 23.4641 0.837470
\(786\) −0.143594 −0.00512181
\(787\) −36.7846 −1.31123 −0.655615 0.755095i \(-0.727590\pi\)
−0.655615 + 0.755095i \(0.727590\pi\)
\(788\) 3.46410 0.123404
\(789\) −5.56922 −0.198269
\(790\) −9.66025 −0.343696
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) −32.0000 −1.13635
\(794\) −0.535898 −0.0190183
\(795\) 6.39230 0.226712
\(796\) 25.8564 0.916456
\(797\) −2.39230 −0.0847398 −0.0423699 0.999102i \(-0.513491\pi\)
−0.0423699 + 0.999102i \(0.513491\pi\)
\(798\) 2.00000 0.0707992
\(799\) 44.7846 1.58437
\(800\) −1.00000 −0.0353553
\(801\) −12.1436 −0.429073
\(802\) 0.392305 0.0138528
\(803\) 0 0
\(804\) 8.67949 0.306102
\(805\) 2.19615 0.0774042
\(806\) −29.8564 −1.05165
\(807\) −12.0000 −0.422420
\(808\) −8.00000 −0.281439
\(809\) 32.1051 1.12876 0.564378 0.825517i \(-0.309116\pi\)
0.564378 + 0.825517i \(0.309116\pi\)
\(810\) 4.46410 0.156853
\(811\) 6.73205 0.236394 0.118197 0.992990i \(-0.462289\pi\)
0.118197 + 0.992990i \(0.462289\pi\)
\(812\) 9.66025 0.339008
\(813\) −8.49742 −0.298018
\(814\) 0 0
\(815\) 0.928203 0.0325135
\(816\) −5.46410 −0.191282
\(817\) 24.3923 0.853379
\(818\) −19.8038 −0.692425
\(819\) −9.85641 −0.344411
\(820\) 8.19615 0.286222
\(821\) 49.3731 1.72313 0.861566 0.507646i \(-0.169484\pi\)
0.861566 + 0.507646i \(0.169484\pi\)
\(822\) 14.5359 0.506998
\(823\) 40.0526 1.39614 0.698072 0.716027i \(-0.254041\pi\)
0.698072 + 0.716027i \(0.254041\pi\)
\(824\) 18.3923 0.640726
\(825\) 0 0
\(826\) 12.3923 0.431183
\(827\) 20.7846 0.722752 0.361376 0.932420i \(-0.382307\pi\)
0.361376 + 0.932420i \(0.382307\pi\)
\(828\) 5.41154 0.188064
\(829\) 24.3923 0.847180 0.423590 0.905854i \(-0.360770\pi\)
0.423590 + 0.905854i \(0.360770\pi\)
\(830\) 1.07180 0.0372026
\(831\) −12.6795 −0.439847
\(832\) 4.00000 0.138675
\(833\) 7.46410 0.258616
\(834\) 10.0000 0.346272
\(835\) −18.9282 −0.655037
\(836\) 0 0
\(837\) 29.8564 1.03199
\(838\) −10.1436 −0.350405
\(839\) 20.2487 0.699063 0.349532 0.936925i \(-0.386341\pi\)
0.349532 + 0.936925i \(0.386341\pi\)
\(840\) −0.732051 −0.0252582
\(841\) 64.3205 2.21795
\(842\) 24.2487 0.835666
\(843\) −8.28719 −0.285426
\(844\) 13.8564 0.476957
\(845\) −3.00000 −0.103203
\(846\) 14.7846 0.508305
\(847\) 0 0
\(848\) 8.73205 0.299860
\(849\) 1.85641 0.0637117
\(850\) −7.46410 −0.256017
\(851\) −1.60770 −0.0551111
\(852\) −10.9282 −0.374394
\(853\) 44.6410 1.52848 0.764240 0.644932i \(-0.223114\pi\)
0.764240 + 0.644932i \(0.223114\pi\)
\(854\) 8.00000 0.273754
\(855\) 6.73205 0.230231
\(856\) 2.00000 0.0683586
\(857\) −11.0718 −0.378205 −0.189103 0.981957i \(-0.560558\pi\)
−0.189103 + 0.981957i \(0.560558\pi\)
\(858\) 0 0
\(859\) 26.9282 0.918778 0.459389 0.888235i \(-0.348068\pi\)
0.459389 + 0.888235i \(0.348068\pi\)
\(860\) −8.92820 −0.304449
\(861\) 6.00000 0.204479
\(862\) 13.2679 0.451908
\(863\) −11.6603 −0.396920 −0.198460 0.980109i \(-0.563594\pi\)
−0.198460 + 0.980109i \(0.563594\pi\)
\(864\) −4.00000 −0.136083
\(865\) 8.92820 0.303568
\(866\) 32.7321 1.11228
\(867\) −28.3397 −0.962468
\(868\) 7.46410 0.253348
\(869\) 0 0
\(870\) −7.07180 −0.239756
\(871\) −47.4256 −1.60696
\(872\) 13.2679 0.449309
\(873\) 1.80385 0.0610510
\(874\) 6.00000 0.202953
\(875\) −1.00000 −0.0338062
\(876\) 4.39230 0.148402
\(877\) −17.6077 −0.594570 −0.297285 0.954789i \(-0.596081\pi\)
−0.297285 + 0.954789i \(0.596081\pi\)
\(878\) 22.5359 0.760550
\(879\) 20.2872 0.684270
\(880\) 0 0
\(881\) 57.7128 1.94439 0.972197 0.234164i \(-0.0752354\pi\)
0.972197 + 0.234164i \(0.0752354\pi\)
\(882\) 2.46410 0.0829706
\(883\) −22.6795 −0.763226 −0.381613 0.924322i \(-0.624631\pi\)
−0.381613 + 0.924322i \(0.624631\pi\)
\(884\) 29.8564 1.00418
\(885\) −9.07180 −0.304945
\(886\) 15.0718 0.506347
\(887\) 22.6410 0.760211 0.380105 0.924943i \(-0.375888\pi\)
0.380105 + 0.924943i \(0.375888\pi\)
\(888\) 0.535898 0.0179836
\(889\) −12.3923 −0.415625
\(890\) 4.92820 0.165194
\(891\) 0 0
\(892\) −4.53590 −0.151873
\(893\) 16.3923 0.548548
\(894\) 8.92820 0.298604
\(895\) 14.3923 0.481082
\(896\) −1.00000 −0.0334077
\(897\) 6.43078 0.214718
\(898\) −38.2487 −1.27638
\(899\) 72.1051 2.40484
\(900\) −2.46410 −0.0821367
\(901\) 65.1769 2.17136
\(902\) 0 0
\(903\) −6.53590 −0.217501
\(904\) −18.0000 −0.598671
\(905\) −8.92820 −0.296784
\(906\) −8.92820 −0.296620
\(907\) 25.3205 0.840754 0.420377 0.907350i \(-0.361898\pi\)
0.420377 + 0.907350i \(0.361898\pi\)
\(908\) 11.3205 0.375684
\(909\) −19.7128 −0.653833
\(910\) 4.00000 0.132599
\(911\) −47.0333 −1.55828 −0.779142 0.626848i \(-0.784345\pi\)
−0.779142 + 0.626848i \(0.784345\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 0 0
\(914\) −11.8564 −0.392175
\(915\) −5.85641 −0.193607
\(916\) −11.3205 −0.374040
\(917\) −0.196152 −0.00647752
\(918\) −29.8564 −0.985408
\(919\) 37.3731 1.23282 0.616412 0.787424i \(-0.288586\pi\)
0.616412 + 0.787424i \(0.288586\pi\)
\(920\) −2.19615 −0.0724050
\(921\) −4.78461 −0.157658
\(922\) 39.3205 1.29495
\(923\) 59.7128 1.96547
\(924\) 0 0
\(925\) 0.732051 0.0240697
\(926\) −38.5885 −1.26810
\(927\) 45.3205 1.48852
\(928\) −9.66025 −0.317113
\(929\) 12.2487 0.401867 0.200934 0.979605i \(-0.435602\pi\)
0.200934 + 0.979605i \(0.435602\pi\)
\(930\) −5.46410 −0.179175
\(931\) 2.73205 0.0895393
\(932\) −3.85641 −0.126321
\(933\) 12.6795 0.415108
\(934\) −9.41154 −0.307955
\(935\) 0 0
\(936\) 9.85641 0.322167
\(937\) −41.3205 −1.34988 −0.674941 0.737872i \(-0.735831\pi\)
−0.674941 + 0.737872i \(0.735831\pi\)
\(938\) 11.8564 0.387125
\(939\) −0.535898 −0.0174884
\(940\) −6.00000 −0.195698
\(941\) 6.92820 0.225853 0.112926 0.993603i \(-0.463978\pi\)
0.112926 + 0.993603i \(0.463978\pi\)
\(942\) −17.1769 −0.559654
\(943\) 18.0000 0.586161
\(944\) −12.3923 −0.403335
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −18.3923 −0.597670 −0.298835 0.954305i \(-0.596598\pi\)
−0.298835 + 0.954305i \(0.596598\pi\)
\(948\) 7.07180 0.229681
\(949\) −24.0000 −0.779073
\(950\) −2.73205 −0.0886394
\(951\) −19.4641 −0.631167
\(952\) −7.46410 −0.241913
\(953\) −46.6410 −1.51085 −0.755425 0.655235i \(-0.772570\pi\)
−0.755425 + 0.655235i \(0.772570\pi\)
\(954\) 21.5167 0.696628
\(955\) −0.392305 −0.0126947
\(956\) −18.0526 −0.583861
\(957\) 0 0
\(958\) −35.7128 −1.15383
\(959\) 19.8564 0.641197
\(960\) 0.732051 0.0236268
\(961\) 24.7128 0.797188
\(962\) −2.92820 −0.0944091
\(963\) 4.92820 0.158809
\(964\) 28.9808 0.933407
\(965\) 16.0000 0.515058
\(966\) −1.60770 −0.0517267
\(967\) 15.6077 0.501910 0.250955 0.967999i \(-0.419255\pi\)
0.250955 + 0.967999i \(0.419255\pi\)
\(968\) 0 0
\(969\) −14.9282 −0.479563
\(970\) −0.732051 −0.0235047
\(971\) 24.3923 0.782786 0.391393 0.920224i \(-0.371993\pi\)
0.391393 + 0.920224i \(0.371993\pi\)
\(972\) −15.2679 −0.489720
\(973\) 13.6603 0.437928
\(974\) −8.05256 −0.258021
\(975\) −2.92820 −0.0937776
\(976\) −8.00000 −0.256074
\(977\) 4.14359 0.132565 0.0662827 0.997801i \(-0.478886\pi\)
0.0662827 + 0.997801i \(0.478886\pi\)
\(978\) −0.679492 −0.0217278
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 32.6936 1.04383
\(982\) 22.9282 0.731668
\(983\) 20.5359 0.654993 0.327497 0.944852i \(-0.393795\pi\)
0.327497 + 0.944852i \(0.393795\pi\)
\(984\) −6.00000 −0.191273
\(985\) −3.46410 −0.110375
\(986\) −72.1051 −2.29629
\(987\) −4.39230 −0.139809
\(988\) 10.9282 0.347672
\(989\) −19.6077 −0.623488
\(990\) 0 0
\(991\) −24.3923 −0.774847 −0.387424 0.921902i \(-0.626635\pi\)
−0.387424 + 0.921902i \(0.626635\pi\)
\(992\) −7.46410 −0.236985
\(993\) −19.3205 −0.613118
\(994\) −14.9282 −0.473494
\(995\) −25.8564 −0.819703
\(996\) −0.784610 −0.0248613
\(997\) −3.85641 −0.122134 −0.0610668 0.998134i \(-0.519450\pi\)
−0.0610668 + 0.998134i \(0.519450\pi\)
\(998\) 1.60770 0.0508907
\(999\) 2.92820 0.0926443
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 8470.2.a.bs.1.1 2
11.10 odd 2 8470.2.a.cd.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.bs.1.1 2 1.1 even 1 trivial
8470.2.a.cd.1.1 yes 2 11.10 odd 2