Properties

Label 7728.2.a.ch.1.5
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 3 x^{5} - 11 x^{4} + 23 x^{3} + 9 x^{2} - 23 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.23871\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.34309 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.34309 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.36801 q^{11} +1.50992 q^{13} +3.34309 q^{15} +5.56451 q^{17} +4.36801 q^{19} +1.00000 q^{21} +1.00000 q^{23} +6.17627 q^{25} +1.00000 q^{27} -5.73398 q^{29} -10.9233 q^{31} +4.36801 q^{33} +3.34309 q^{35} -0.336292 q^{37} +1.50992 q^{39} +7.82598 q^{41} +9.58018 q^{43} +3.34309 q^{45} -9.39049 q^{47} +1.00000 q^{49} +5.56451 q^{51} +1.31138 q^{53} +14.6027 q^{55} +4.36801 q^{57} +2.02287 q^{59} +4.87588 q^{61} +1.00000 q^{63} +5.04779 q^{65} -12.1987 q^{67} +1.00000 q^{69} -12.4154 q^{71} +9.23086 q^{73} +6.17627 q^{75} +4.36801 q^{77} -0.878321 q^{79} +1.00000 q^{81} -12.9622 q^{83} +18.6027 q^{85} -5.73398 q^{87} -13.1016 q^{89} +1.50992 q^{91} -10.9233 q^{93} +14.6027 q^{95} +8.31098 q^{97} +4.36801 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} + 2q^{5} + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} + 2q^{5} + 6q^{7} + 6q^{9} + 3q^{11} + 2q^{15} + 6q^{17} + 3q^{19} + 6q^{21} + 6q^{23} + 10q^{25} + 6q^{27} + 5q^{29} + 4q^{31} + 3q^{33} + 2q^{35} + q^{37} + 12q^{41} + 6q^{43} + 2q^{45} + 6q^{47} + 6q^{49} + 6q^{51} + 10q^{53} - 3q^{55} + 3q^{57} + 14q^{59} + 4q^{61} + 6q^{63} + 27q^{65} - 7q^{67} + 6q^{69} - 7q^{71} + 10q^{73} + 10q^{75} + 3q^{77} - 14q^{79} + 6q^{81} - 14q^{83} + 21q^{85} + 5q^{87} + 25q^{89} + 4q^{93} - 3q^{95} + 11q^{97} + 3q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 3.34309 1.49508 0.747538 0.664219i \(-0.231236\pi\)
0.747538 + 0.664219i \(0.231236\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.36801 1.31700 0.658502 0.752579i \(-0.271190\pi\)
0.658502 + 0.752579i \(0.271190\pi\)
\(12\) 0 0
\(13\) 1.50992 0.418775 0.209388 0.977833i \(-0.432853\pi\)
0.209388 + 0.977833i \(0.432853\pi\)
\(14\) 0 0
\(15\) 3.34309 0.863183
\(16\) 0 0
\(17\) 5.56451 1.34959 0.674796 0.738005i \(-0.264232\pi\)
0.674796 + 0.738005i \(0.264232\pi\)
\(18\) 0 0
\(19\) 4.36801 1.00209 0.501045 0.865421i \(-0.332949\pi\)
0.501045 + 0.865421i \(0.332949\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 6.17627 1.23525
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −5.73398 −1.06477 −0.532386 0.846502i \(-0.678705\pi\)
−0.532386 + 0.846502i \(0.678705\pi\)
\(30\) 0 0
\(31\) −10.9233 −1.96188 −0.980939 0.194315i \(-0.937752\pi\)
−0.980939 + 0.194315i \(0.937752\pi\)
\(32\) 0 0
\(33\) 4.36801 0.760373
\(34\) 0 0
\(35\) 3.34309 0.565086
\(36\) 0 0
\(37\) −0.336292 −0.0552861 −0.0276430 0.999618i \(-0.508800\pi\)
−0.0276430 + 0.999618i \(0.508800\pi\)
\(38\) 0 0
\(39\) 1.50992 0.241780
\(40\) 0 0
\(41\) 7.82598 1.22221 0.611106 0.791548i \(-0.290725\pi\)
0.611106 + 0.791548i \(0.290725\pi\)
\(42\) 0 0
\(43\) 9.58018 1.46096 0.730482 0.682932i \(-0.239295\pi\)
0.730482 + 0.682932i \(0.239295\pi\)
\(44\) 0 0
\(45\) 3.34309 0.498359
\(46\) 0 0
\(47\) −9.39049 −1.36974 −0.684872 0.728664i \(-0.740142\pi\)
−0.684872 + 0.728664i \(0.740142\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 5.56451 0.779187
\(52\) 0 0
\(53\) 1.31138 0.180131 0.0900656 0.995936i \(-0.471292\pi\)
0.0900656 + 0.995936i \(0.471292\pi\)
\(54\) 0 0
\(55\) 14.6027 1.96902
\(56\) 0 0
\(57\) 4.36801 0.578557
\(58\) 0 0
\(59\) 2.02287 0.263356 0.131678 0.991293i \(-0.457964\pi\)
0.131678 + 0.991293i \(0.457964\pi\)
\(60\) 0 0
\(61\) 4.87588 0.624293 0.312146 0.950034i \(-0.398952\pi\)
0.312146 + 0.950034i \(0.398952\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 5.04779 0.626101
\(66\) 0 0
\(67\) −12.1987 −1.49031 −0.745157 0.666889i \(-0.767626\pi\)
−0.745157 + 0.666889i \(0.767626\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −12.4154 −1.47344 −0.736719 0.676199i \(-0.763626\pi\)
−0.736719 + 0.676199i \(0.763626\pi\)
\(72\) 0 0
\(73\) 9.23086 1.08039 0.540195 0.841540i \(-0.318350\pi\)
0.540195 + 0.841540i \(0.318350\pi\)
\(74\) 0 0
\(75\) 6.17627 0.713174
\(76\) 0 0
\(77\) 4.36801 0.497781
\(78\) 0 0
\(79\) −0.878321 −0.0988188 −0.0494094 0.998779i \(-0.515734\pi\)
−0.0494094 + 0.998779i \(0.515734\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.9622 −1.42279 −0.711393 0.702795i \(-0.751935\pi\)
−0.711393 + 0.702795i \(0.751935\pi\)
\(84\) 0 0
\(85\) 18.6027 2.01774
\(86\) 0 0
\(87\) −5.73398 −0.614747
\(88\) 0 0
\(89\) −13.1016 −1.38877 −0.694383 0.719606i \(-0.744323\pi\)
−0.694383 + 0.719606i \(0.744323\pi\)
\(90\) 0 0
\(91\) 1.50992 0.158282
\(92\) 0 0
\(93\) −10.9233 −1.13269
\(94\) 0 0
\(95\) 14.6027 1.49820
\(96\) 0 0
\(97\) 8.31098 0.843852 0.421926 0.906630i \(-0.361354\pi\)
0.421926 + 0.906630i \(0.361354\pi\)
\(98\) 0 0
\(99\) 4.36801 0.439001
\(100\) 0 0
\(101\) −2.70906 −0.269561 −0.134781 0.990875i \(-0.543033\pi\)
−0.134781 + 0.990875i \(0.543033\pi\)
\(102\) 0 0
\(103\) −8.41032 −0.828693 −0.414347 0.910119i \(-0.635990\pi\)
−0.414347 + 0.910119i \(0.635990\pi\)
\(104\) 0 0
\(105\) 3.34309 0.326252
\(106\) 0 0
\(107\) 12.6977 1.22753 0.613765 0.789489i \(-0.289654\pi\)
0.613765 + 0.789489i \(0.289654\pi\)
\(108\) 0 0
\(109\) −8.45850 −0.810178 −0.405089 0.914277i \(-0.632759\pi\)
−0.405089 + 0.914277i \(0.632759\pi\)
\(110\) 0 0
\(111\) −0.336292 −0.0319194
\(112\) 0 0
\(113\) −16.8356 −1.58376 −0.791878 0.610679i \(-0.790897\pi\)
−0.791878 + 0.610679i \(0.790897\pi\)
\(114\) 0 0
\(115\) 3.34309 0.311745
\(116\) 0 0
\(117\) 1.50992 0.139592
\(118\) 0 0
\(119\) 5.56451 0.510097
\(120\) 0 0
\(121\) 8.07951 0.734501
\(122\) 0 0
\(123\) 7.82598 0.705645
\(124\) 0 0
\(125\) 3.93238 0.351723
\(126\) 0 0
\(127\) −10.3837 −0.921403 −0.460702 0.887555i \(-0.652402\pi\)
−0.460702 + 0.887555i \(0.652402\pi\)
\(128\) 0 0
\(129\) 9.58018 0.843488
\(130\) 0 0
\(131\) −9.76569 −0.853233 −0.426616 0.904433i \(-0.640295\pi\)
−0.426616 + 0.904433i \(0.640295\pi\)
\(132\) 0 0
\(133\) 4.36801 0.378754
\(134\) 0 0
\(135\) 3.34309 0.287728
\(136\) 0 0
\(137\) −6.51217 −0.556372 −0.278186 0.960527i \(-0.589733\pi\)
−0.278186 + 0.960527i \(0.589733\pi\)
\(138\) 0 0
\(139\) −9.69717 −0.822504 −0.411252 0.911522i \(-0.634908\pi\)
−0.411252 + 0.911522i \(0.634908\pi\)
\(140\) 0 0
\(141\) −9.39049 −0.790822
\(142\) 0 0
\(143\) 6.59533 0.551529
\(144\) 0 0
\(145\) −19.1692 −1.59192
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 10.1623 0.832526 0.416263 0.909244i \(-0.363340\pi\)
0.416263 + 0.909244i \(0.363340\pi\)
\(150\) 0 0
\(151\) −4.98811 −0.405927 −0.202964 0.979186i \(-0.565057\pi\)
−0.202964 + 0.979186i \(0.565057\pi\)
\(152\) 0 0
\(153\) 5.56451 0.449864
\(154\) 0 0
\(155\) −36.5175 −2.93316
\(156\) 0 0
\(157\) −19.6843 −1.57098 −0.785488 0.618877i \(-0.787588\pi\)
−0.785488 + 0.618877i \(0.787588\pi\)
\(158\) 0 0
\(159\) 1.31138 0.103999
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −6.31606 −0.494712 −0.247356 0.968925i \(-0.579562\pi\)
−0.247356 + 0.968925i \(0.579562\pi\)
\(164\) 0 0
\(165\) 14.6027 1.13682
\(166\) 0 0
\(167\) −13.0153 −1.00715 −0.503576 0.863951i \(-0.667983\pi\)
−0.503576 + 0.863951i \(0.667983\pi\)
\(168\) 0 0
\(169\) −10.7202 −0.824627
\(170\) 0 0
\(171\) 4.36801 0.334030
\(172\) 0 0
\(173\) 7.61434 0.578908 0.289454 0.957192i \(-0.406526\pi\)
0.289454 + 0.957192i \(0.406526\pi\)
\(174\) 0 0
\(175\) 6.17627 0.466882
\(176\) 0 0
\(177\) 2.02287 0.152048
\(178\) 0 0
\(179\) 19.9196 1.48886 0.744431 0.667699i \(-0.232721\pi\)
0.744431 + 0.667699i \(0.232721\pi\)
\(180\) 0 0
\(181\) 6.22617 0.462788 0.231394 0.972860i \(-0.425671\pi\)
0.231394 + 0.972860i \(0.425671\pi\)
\(182\) 0 0
\(183\) 4.87588 0.360436
\(184\) 0 0
\(185\) −1.12426 −0.0826569
\(186\) 0 0
\(187\) 24.3058 1.77742
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −1.46517 −0.106016 −0.0530079 0.998594i \(-0.516881\pi\)
−0.0530079 + 0.998594i \(0.516881\pi\)
\(192\) 0 0
\(193\) 4.65856 0.335330 0.167665 0.985844i \(-0.446377\pi\)
0.167665 + 0.985844i \(0.446377\pi\)
\(194\) 0 0
\(195\) 5.04779 0.361480
\(196\) 0 0
\(197\) 21.3115 1.51838 0.759191 0.650868i \(-0.225595\pi\)
0.759191 + 0.650868i \(0.225595\pi\)
\(198\) 0 0
\(199\) 15.2715 1.08257 0.541285 0.840839i \(-0.317938\pi\)
0.541285 + 0.840839i \(0.317938\pi\)
\(200\) 0 0
\(201\) −12.1987 −0.860433
\(202\) 0 0
\(203\) −5.73398 −0.402446
\(204\) 0 0
\(205\) 26.1630 1.82730
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 19.0795 1.31976
\(210\) 0 0
\(211\) 12.4779 0.859017 0.429508 0.903063i \(-0.358687\pi\)
0.429508 + 0.903063i \(0.358687\pi\)
\(212\) 0 0
\(213\) −12.4154 −0.850689
\(214\) 0 0
\(215\) 32.0274 2.18425
\(216\) 0 0
\(217\) −10.9233 −0.741520
\(218\) 0 0
\(219\) 9.23086 0.623764
\(220\) 0 0
\(221\) 8.40194 0.565175
\(222\) 0 0
\(223\) −5.93495 −0.397434 −0.198717 0.980057i \(-0.563677\pi\)
−0.198717 + 0.980057i \(0.563677\pi\)
\(224\) 0 0
\(225\) 6.17627 0.411751
\(226\) 0 0
\(227\) −15.0544 −0.999196 −0.499598 0.866257i \(-0.666519\pi\)
−0.499598 + 0.866257i \(0.666519\pi\)
\(228\) 0 0
\(229\) 9.40616 0.621577 0.310788 0.950479i \(-0.399407\pi\)
0.310788 + 0.950479i \(0.399407\pi\)
\(230\) 0 0
\(231\) 4.36801 0.287394
\(232\) 0 0
\(233\) 4.83730 0.316902 0.158451 0.987367i \(-0.449350\pi\)
0.158451 + 0.987367i \(0.449350\pi\)
\(234\) 0 0
\(235\) −31.3933 −2.04787
\(236\) 0 0
\(237\) −0.878321 −0.0570531
\(238\) 0 0
\(239\) −11.7812 −0.762065 −0.381032 0.924562i \(-0.624431\pi\)
−0.381032 + 0.924562i \(0.624431\pi\)
\(240\) 0 0
\(241\) −0.910038 −0.0586207 −0.0293104 0.999570i \(-0.509331\pi\)
−0.0293104 + 0.999570i \(0.509331\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.34309 0.213582
\(246\) 0 0
\(247\) 6.59533 0.419651
\(248\) 0 0
\(249\) −12.9622 −0.821445
\(250\) 0 0
\(251\) −23.5808 −1.48840 −0.744202 0.667955i \(-0.767170\pi\)
−0.744202 + 0.667955i \(0.767170\pi\)
\(252\) 0 0
\(253\) 4.36801 0.274614
\(254\) 0 0
\(255\) 18.6027 1.16494
\(256\) 0 0
\(257\) −1.17119 −0.0730565 −0.0365283 0.999333i \(-0.511630\pi\)
−0.0365283 + 0.999333i \(0.511630\pi\)
\(258\) 0 0
\(259\) −0.336292 −0.0208962
\(260\) 0 0
\(261\) −5.73398 −0.354924
\(262\) 0 0
\(263\) −29.2939 −1.80634 −0.903171 0.429282i \(-0.858767\pi\)
−0.903171 + 0.429282i \(0.858767\pi\)
\(264\) 0 0
\(265\) 4.38405 0.269310
\(266\) 0 0
\(267\) −13.1016 −0.801804
\(268\) 0 0
\(269\) 17.4092 1.06146 0.530728 0.847542i \(-0.321919\pi\)
0.530728 + 0.847542i \(0.321919\pi\)
\(270\) 0 0
\(271\) 13.3068 0.808328 0.404164 0.914687i \(-0.367563\pi\)
0.404164 + 0.914687i \(0.367563\pi\)
\(272\) 0 0
\(273\) 1.50992 0.0913843
\(274\) 0 0
\(275\) 26.9780 1.62683
\(276\) 0 0
\(277\) 13.5866 0.816339 0.408170 0.912906i \(-0.366167\pi\)
0.408170 + 0.912906i \(0.366167\pi\)
\(278\) 0 0
\(279\) −10.9233 −0.653959
\(280\) 0 0
\(281\) 1.09451 0.0652931 0.0326466 0.999467i \(-0.489606\pi\)
0.0326466 + 0.999467i \(0.489606\pi\)
\(282\) 0 0
\(283\) 32.0003 1.90222 0.951112 0.308848i \(-0.0999433\pi\)
0.951112 + 0.308848i \(0.0999433\pi\)
\(284\) 0 0
\(285\) 14.6027 0.864987
\(286\) 0 0
\(287\) 7.82598 0.461953
\(288\) 0 0
\(289\) 13.9637 0.821396
\(290\) 0 0
\(291\) 8.31098 0.487198
\(292\) 0 0
\(293\) 15.6909 0.916671 0.458335 0.888779i \(-0.348446\pi\)
0.458335 + 0.888779i \(0.348446\pi\)
\(294\) 0 0
\(295\) 6.76265 0.393737
\(296\) 0 0
\(297\) 4.36801 0.253458
\(298\) 0 0
\(299\) 1.50992 0.0873207
\(300\) 0 0
\(301\) 9.58018 0.552193
\(302\) 0 0
\(303\) −2.70906 −0.155631
\(304\) 0 0
\(305\) 16.3005 0.933365
\(306\) 0 0
\(307\) −29.4453 −1.68053 −0.840266 0.542175i \(-0.817601\pi\)
−0.840266 + 0.542175i \(0.817601\pi\)
\(308\) 0 0
\(309\) −8.41032 −0.478446
\(310\) 0 0
\(311\) −7.85680 −0.445518 −0.222759 0.974874i \(-0.571506\pi\)
−0.222759 + 0.974874i \(0.571506\pi\)
\(312\) 0 0
\(313\) −27.3253 −1.54452 −0.772259 0.635308i \(-0.780873\pi\)
−0.772259 + 0.635308i \(0.780873\pi\)
\(314\) 0 0
\(315\) 3.34309 0.188362
\(316\) 0 0
\(317\) −14.0716 −0.790337 −0.395169 0.918609i \(-0.629314\pi\)
−0.395169 + 0.918609i \(0.629314\pi\)
\(318\) 0 0
\(319\) −25.0461 −1.40231
\(320\) 0 0
\(321\) 12.6977 0.708715
\(322\) 0 0
\(323\) 24.3058 1.35241
\(324\) 0 0
\(325\) 9.32565 0.517294
\(326\) 0 0
\(327\) −8.45850 −0.467756
\(328\) 0 0
\(329\) −9.39049 −0.517714
\(330\) 0 0
\(331\) −8.18758 −0.450030 −0.225015 0.974355i \(-0.572243\pi\)
−0.225015 + 0.974355i \(0.572243\pi\)
\(332\) 0 0
\(333\) −0.336292 −0.0184287
\(334\) 0 0
\(335\) −40.7815 −2.22813
\(336\) 0 0
\(337\) −17.3409 −0.944617 −0.472308 0.881433i \(-0.656579\pi\)
−0.472308 + 0.881433i \(0.656579\pi\)
\(338\) 0 0
\(339\) −16.8356 −0.914382
\(340\) 0 0
\(341\) −47.7130 −2.58380
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 3.34309 0.179986
\(346\) 0 0
\(347\) −1.77852 −0.0954758 −0.0477379 0.998860i \(-0.515201\pi\)
−0.0477379 + 0.998860i \(0.515201\pi\)
\(348\) 0 0
\(349\) 14.9499 0.800250 0.400125 0.916461i \(-0.368967\pi\)
0.400125 + 0.916461i \(0.368967\pi\)
\(350\) 0 0
\(351\) 1.50992 0.0805934
\(352\) 0 0
\(353\) 21.1188 1.12404 0.562020 0.827123i \(-0.310024\pi\)
0.562020 + 0.827123i \(0.310024\pi\)
\(354\) 0 0
\(355\) −41.5058 −2.20290
\(356\) 0 0
\(357\) 5.56451 0.294505
\(358\) 0 0
\(359\) 24.2586 1.28032 0.640159 0.768243i \(-0.278869\pi\)
0.640159 + 0.768243i \(0.278869\pi\)
\(360\) 0 0
\(361\) 0.0795069 0.00418458
\(362\) 0 0
\(363\) 8.07951 0.424064
\(364\) 0 0
\(365\) 30.8596 1.61527
\(366\) 0 0
\(367\) 15.2920 0.798238 0.399119 0.916899i \(-0.369316\pi\)
0.399119 + 0.916899i \(0.369316\pi\)
\(368\) 0 0
\(369\) 7.82598 0.407404
\(370\) 0 0
\(371\) 1.31138 0.0680832
\(372\) 0 0
\(373\) −10.2219 −0.529272 −0.264636 0.964348i \(-0.585252\pi\)
−0.264636 + 0.964348i \(0.585252\pi\)
\(374\) 0 0
\(375\) 3.93238 0.203067
\(376\) 0 0
\(377\) −8.65782 −0.445900
\(378\) 0 0
\(379\) 9.82297 0.504572 0.252286 0.967653i \(-0.418818\pi\)
0.252286 + 0.967653i \(0.418818\pi\)
\(380\) 0 0
\(381\) −10.3837 −0.531972
\(382\) 0 0
\(383\) −28.3009 −1.44611 −0.723055 0.690790i \(-0.757263\pi\)
−0.723055 + 0.690790i \(0.757263\pi\)
\(384\) 0 0
\(385\) 14.6027 0.744221
\(386\) 0 0
\(387\) 9.58018 0.486988
\(388\) 0 0
\(389\) −7.23769 −0.366966 −0.183483 0.983023i \(-0.558737\pi\)
−0.183483 + 0.983023i \(0.558737\pi\)
\(390\) 0 0
\(391\) 5.56451 0.281409
\(392\) 0 0
\(393\) −9.76569 −0.492614
\(394\) 0 0
\(395\) −2.93631 −0.147742
\(396\) 0 0
\(397\) −12.3472 −0.619691 −0.309845 0.950787i \(-0.600277\pi\)
−0.309845 + 0.950787i \(0.600277\pi\)
\(398\) 0 0
\(399\) 4.36801 0.218674
\(400\) 0 0
\(401\) −2.14429 −0.107081 −0.0535405 0.998566i \(-0.517051\pi\)
−0.0535405 + 0.998566i \(0.517051\pi\)
\(402\) 0 0
\(403\) −16.4932 −0.821586
\(404\) 0 0
\(405\) 3.34309 0.166120
\(406\) 0 0
\(407\) −1.46893 −0.0728120
\(408\) 0 0
\(409\) 11.9027 0.588548 0.294274 0.955721i \(-0.404922\pi\)
0.294274 + 0.955721i \(0.404922\pi\)
\(410\) 0 0
\(411\) −6.51217 −0.321221
\(412\) 0 0
\(413\) 2.02287 0.0995391
\(414\) 0 0
\(415\) −43.3338 −2.12717
\(416\) 0 0
\(417\) −9.69717 −0.474873
\(418\) 0 0
\(419\) −10.1436 −0.495548 −0.247774 0.968818i \(-0.579699\pi\)
−0.247774 + 0.968818i \(0.579699\pi\)
\(420\) 0 0
\(421\) 14.6197 0.712521 0.356260 0.934387i \(-0.384052\pi\)
0.356260 + 0.934387i \(0.384052\pi\)
\(422\) 0 0
\(423\) −9.39049 −0.456581
\(424\) 0 0
\(425\) 34.3679 1.66709
\(426\) 0 0
\(427\) 4.87588 0.235960
\(428\) 0 0
\(429\) 6.59533 0.318425
\(430\) 0 0
\(431\) −13.9585 −0.672360 −0.336180 0.941798i \(-0.609135\pi\)
−0.336180 + 0.941798i \(0.609135\pi\)
\(432\) 0 0
\(433\) −0.975653 −0.0468869 −0.0234435 0.999725i \(-0.507463\pi\)
−0.0234435 + 0.999725i \(0.507463\pi\)
\(434\) 0 0
\(435\) −19.1692 −0.919093
\(436\) 0 0
\(437\) 4.36801 0.208950
\(438\) 0 0
\(439\) −10.5178 −0.501988 −0.250994 0.967989i \(-0.580757\pi\)
−0.250994 + 0.967989i \(0.580757\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 21.8350 1.03741 0.518707 0.854952i \(-0.326414\pi\)
0.518707 + 0.854952i \(0.326414\pi\)
\(444\) 0 0
\(445\) −43.7998 −2.07631
\(446\) 0 0
\(447\) 10.1623 0.480659
\(448\) 0 0
\(449\) 3.43821 0.162259 0.0811295 0.996704i \(-0.474147\pi\)
0.0811295 + 0.996704i \(0.474147\pi\)
\(450\) 0 0
\(451\) 34.1840 1.60966
\(452\) 0 0
\(453\) −4.98811 −0.234362
\(454\) 0 0
\(455\) 5.04779 0.236644
\(456\) 0 0
\(457\) 0.249057 0.0116504 0.00582519 0.999983i \(-0.498146\pi\)
0.00582519 + 0.999983i \(0.498146\pi\)
\(458\) 0 0
\(459\) 5.56451 0.259729
\(460\) 0 0
\(461\) 28.3729 1.32146 0.660729 0.750625i \(-0.270247\pi\)
0.660729 + 0.750625i \(0.270247\pi\)
\(462\) 0 0
\(463\) −9.97074 −0.463379 −0.231690 0.972790i \(-0.574425\pi\)
−0.231690 + 0.972790i \(0.574425\pi\)
\(464\) 0 0
\(465\) −36.5175 −1.69346
\(466\) 0 0
\(467\) 20.0525 0.927918 0.463959 0.885857i \(-0.346428\pi\)
0.463959 + 0.885857i \(0.346428\pi\)
\(468\) 0 0
\(469\) −12.1987 −0.563286
\(470\) 0 0
\(471\) −19.6843 −0.907004
\(472\) 0 0
\(473\) 41.8463 1.92410
\(474\) 0 0
\(475\) 26.9780 1.23784
\(476\) 0 0
\(477\) 1.31138 0.0600438
\(478\) 0 0
\(479\) 5.17151 0.236292 0.118146 0.992996i \(-0.462305\pi\)
0.118146 + 0.992996i \(0.462305\pi\)
\(480\) 0 0
\(481\) −0.507773 −0.0231525
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 27.7844 1.26162
\(486\) 0 0
\(487\) 33.3508 1.51127 0.755634 0.654994i \(-0.227329\pi\)
0.755634 + 0.654994i \(0.227329\pi\)
\(488\) 0 0
\(489\) −6.31606 −0.285622
\(490\) 0 0
\(491\) 39.5448 1.78463 0.892317 0.451410i \(-0.149079\pi\)
0.892317 + 0.451410i \(0.149079\pi\)
\(492\) 0 0
\(493\) −31.9067 −1.43701
\(494\) 0 0
\(495\) 14.6027 0.656341
\(496\) 0 0
\(497\) −12.4154 −0.556907
\(498\) 0 0
\(499\) 7.48316 0.334992 0.167496 0.985873i \(-0.446432\pi\)
0.167496 + 0.985873i \(0.446432\pi\)
\(500\) 0 0
\(501\) −13.0153 −0.581480
\(502\) 0 0
\(503\) −35.0864 −1.56442 −0.782212 0.623013i \(-0.785908\pi\)
−0.782212 + 0.623013i \(0.785908\pi\)
\(504\) 0 0
\(505\) −9.05663 −0.403015
\(506\) 0 0
\(507\) −10.7202 −0.476099
\(508\) 0 0
\(509\) −9.36959 −0.415300 −0.207650 0.978203i \(-0.566581\pi\)
−0.207650 + 0.978203i \(0.566581\pi\)
\(510\) 0 0
\(511\) 9.23086 0.408349
\(512\) 0 0
\(513\) 4.36801 0.192852
\(514\) 0 0
\(515\) −28.1165 −1.23896
\(516\) 0 0
\(517\) −41.0177 −1.80396
\(518\) 0 0
\(519\) 7.61434 0.334232
\(520\) 0 0
\(521\) −18.1405 −0.794751 −0.397376 0.917656i \(-0.630079\pi\)
−0.397376 + 0.917656i \(0.630079\pi\)
\(522\) 0 0
\(523\) 24.6685 1.07868 0.539340 0.842088i \(-0.318674\pi\)
0.539340 + 0.842088i \(0.318674\pi\)
\(524\) 0 0
\(525\) 6.17627 0.269555
\(526\) 0 0
\(527\) −60.7826 −2.64773
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 2.02287 0.0877852
\(532\) 0 0
\(533\) 11.8166 0.511833
\(534\) 0 0
\(535\) 42.4495 1.83525
\(536\) 0 0
\(537\) 19.9196 0.859595
\(538\) 0 0
\(539\) 4.36801 0.188143
\(540\) 0 0
\(541\) 22.9043 0.984734 0.492367 0.870388i \(-0.336132\pi\)
0.492367 + 0.870388i \(0.336132\pi\)
\(542\) 0 0
\(543\) 6.22617 0.267191
\(544\) 0 0
\(545\) −28.2776 −1.21128
\(546\) 0 0
\(547\) 1.40263 0.0599721 0.0299861 0.999550i \(-0.490454\pi\)
0.0299861 + 0.999550i \(0.490454\pi\)
\(548\) 0 0
\(549\) 4.87588 0.208098
\(550\) 0 0
\(551\) −25.0461 −1.06700
\(552\) 0 0
\(553\) −0.878321 −0.0373500
\(554\) 0 0
\(555\) −1.12426 −0.0477220
\(556\) 0 0
\(557\) 32.8328 1.39117 0.695584 0.718444i \(-0.255146\pi\)
0.695584 + 0.718444i \(0.255146\pi\)
\(558\) 0 0
\(559\) 14.4653 0.611816
\(560\) 0 0
\(561\) 24.3058 1.02619
\(562\) 0 0
\(563\) 32.0271 1.34978 0.674891 0.737918i \(-0.264191\pi\)
0.674891 + 0.737918i \(0.264191\pi\)
\(564\) 0 0
\(565\) −56.2829 −2.36784
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 26.8483 1.12554 0.562769 0.826614i \(-0.309736\pi\)
0.562769 + 0.826614i \(0.309736\pi\)
\(570\) 0 0
\(571\) −10.4695 −0.438136 −0.219068 0.975710i \(-0.570302\pi\)
−0.219068 + 0.975710i \(0.570302\pi\)
\(572\) 0 0
\(573\) −1.46517 −0.0612082
\(574\) 0 0
\(575\) 6.17627 0.257568
\(576\) 0 0
\(577\) −25.9984 −1.08233 −0.541164 0.840917i \(-0.682016\pi\)
−0.541164 + 0.840917i \(0.682016\pi\)
\(578\) 0 0
\(579\) 4.65856 0.193603
\(580\) 0 0
\(581\) −12.9622 −0.537762
\(582\) 0 0
\(583\) 5.72810 0.237234
\(584\) 0 0
\(585\) 5.04779 0.208700
\(586\) 0 0
\(587\) 23.1003 0.953451 0.476725 0.879052i \(-0.341824\pi\)
0.476725 + 0.879052i \(0.341824\pi\)
\(588\) 0 0
\(589\) −47.7130 −1.96598
\(590\) 0 0
\(591\) 21.3115 0.876639
\(592\) 0 0
\(593\) 26.5087 1.08858 0.544291 0.838897i \(-0.316799\pi\)
0.544291 + 0.838897i \(0.316799\pi\)
\(594\) 0 0
\(595\) 18.6027 0.762635
\(596\) 0 0
\(597\) 15.2715 0.625022
\(598\) 0 0
\(599\) 42.2215 1.72512 0.862562 0.505952i \(-0.168859\pi\)
0.862562 + 0.505952i \(0.168859\pi\)
\(600\) 0 0
\(601\) −28.4707 −1.16134 −0.580672 0.814137i \(-0.697210\pi\)
−0.580672 + 0.814137i \(0.697210\pi\)
\(602\) 0 0
\(603\) −12.1987 −0.496771
\(604\) 0 0
\(605\) 27.0105 1.09813
\(606\) 0 0
\(607\) 37.2462 1.51178 0.755889 0.654700i \(-0.227205\pi\)
0.755889 + 0.654700i \(0.227205\pi\)
\(608\) 0 0
\(609\) −5.73398 −0.232352
\(610\) 0 0
\(611\) −14.1788 −0.573615
\(612\) 0 0
\(613\) 1.95309 0.0788845 0.0394422 0.999222i \(-0.487442\pi\)
0.0394422 + 0.999222i \(0.487442\pi\)
\(614\) 0 0
\(615\) 26.1630 1.05499
\(616\) 0 0
\(617\) 39.6809 1.59749 0.798746 0.601669i \(-0.205497\pi\)
0.798746 + 0.601669i \(0.205497\pi\)
\(618\) 0 0
\(619\) −43.3165 −1.74104 −0.870518 0.492136i \(-0.836216\pi\)
−0.870518 + 0.492136i \(0.836216\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −13.1016 −0.524904
\(624\) 0 0
\(625\) −17.7350 −0.709402
\(626\) 0 0
\(627\) 19.0795 0.761962
\(628\) 0 0
\(629\) −1.87130 −0.0746136
\(630\) 0 0
\(631\) 49.8769 1.98557 0.992783 0.119922i \(-0.0382644\pi\)
0.992783 + 0.119922i \(0.0382644\pi\)
\(632\) 0 0
\(633\) 12.4779 0.495954
\(634\) 0 0
\(635\) −34.7136 −1.37757
\(636\) 0 0
\(637\) 1.50992 0.0598251
\(638\) 0 0
\(639\) −12.4154 −0.491146
\(640\) 0 0
\(641\) 20.7445 0.819360 0.409680 0.912229i \(-0.365640\pi\)
0.409680 + 0.912229i \(0.365640\pi\)
\(642\) 0 0
\(643\) −11.9002 −0.469299 −0.234649 0.972080i \(-0.575394\pi\)
−0.234649 + 0.972080i \(0.575394\pi\)
\(644\) 0 0
\(645\) 32.0274 1.26108
\(646\) 0 0
\(647\) 20.5754 0.808902 0.404451 0.914560i \(-0.367463\pi\)
0.404451 + 0.914560i \(0.367463\pi\)
\(648\) 0 0
\(649\) 8.83593 0.346841
\(650\) 0 0
\(651\) −10.9233 −0.428117
\(652\) 0 0
\(653\) 13.8467 0.541862 0.270931 0.962599i \(-0.412668\pi\)
0.270931 + 0.962599i \(0.412668\pi\)
\(654\) 0 0
\(655\) −32.6476 −1.27565
\(656\) 0 0
\(657\) 9.23086 0.360130
\(658\) 0 0
\(659\) −8.77396 −0.341785 −0.170892 0.985290i \(-0.554665\pi\)
−0.170892 + 0.985290i \(0.554665\pi\)
\(660\) 0 0
\(661\) −45.3931 −1.76559 −0.882794 0.469760i \(-0.844340\pi\)
−0.882794 + 0.469760i \(0.844340\pi\)
\(662\) 0 0
\(663\) 8.40194 0.326304
\(664\) 0 0
\(665\) 14.6027 0.566267
\(666\) 0 0
\(667\) −5.73398 −0.222020
\(668\) 0 0
\(669\) −5.93495 −0.229459
\(670\) 0 0
\(671\) 21.2979 0.822196
\(672\) 0 0
\(673\) −21.7908 −0.839974 −0.419987 0.907530i \(-0.637965\pi\)
−0.419987 + 0.907530i \(0.637965\pi\)
\(674\) 0 0
\(675\) 6.17627 0.237725
\(676\) 0 0
\(677\) −44.3602 −1.70490 −0.852451 0.522807i \(-0.824885\pi\)
−0.852451 + 0.522807i \(0.824885\pi\)
\(678\) 0 0
\(679\) 8.31098 0.318946
\(680\) 0 0
\(681\) −15.0544 −0.576886
\(682\) 0 0
\(683\) 10.1053 0.386668 0.193334 0.981133i \(-0.438070\pi\)
0.193334 + 0.981133i \(0.438070\pi\)
\(684\) 0 0
\(685\) −21.7708 −0.831819
\(686\) 0 0
\(687\) 9.40616 0.358868
\(688\) 0 0
\(689\) 1.98007 0.0754345
\(690\) 0 0
\(691\) 26.7112 1.01614 0.508070 0.861316i \(-0.330359\pi\)
0.508070 + 0.861316i \(0.330359\pi\)
\(692\) 0 0
\(693\) 4.36801 0.165927
\(694\) 0 0
\(695\) −32.4186 −1.22971
\(696\) 0 0
\(697\) 43.5477 1.64949
\(698\) 0 0
\(699\) 4.83730 0.182963
\(700\) 0 0
\(701\) 5.61896 0.212225 0.106113 0.994354i \(-0.466160\pi\)
0.106113 + 0.994354i \(0.466160\pi\)
\(702\) 0 0
\(703\) −1.46893 −0.0554016
\(704\) 0 0
\(705\) −31.3933 −1.18234
\(706\) 0 0
\(707\) −2.70906 −0.101885
\(708\) 0 0
\(709\) −20.2873 −0.761904 −0.380952 0.924595i \(-0.624404\pi\)
−0.380952 + 0.924595i \(0.624404\pi\)
\(710\) 0 0
\(711\) −0.878321 −0.0329396
\(712\) 0 0
\(713\) −10.9233 −0.409080
\(714\) 0 0
\(715\) 22.0488 0.824578
\(716\) 0 0
\(717\) −11.7812 −0.439978
\(718\) 0 0
\(719\) 45.4985 1.69681 0.848405 0.529348i \(-0.177563\pi\)
0.848405 + 0.529348i \(0.177563\pi\)
\(720\) 0 0
\(721\) −8.41032 −0.313217
\(722\) 0 0
\(723\) −0.910038 −0.0338447
\(724\) 0 0
\(725\) −35.4146 −1.31526
\(726\) 0 0
\(727\) 38.7942 1.43880 0.719400 0.694597i \(-0.244417\pi\)
0.719400 + 0.694597i \(0.244417\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 53.3090 1.97170
\(732\) 0 0
\(733\) −33.5370 −1.23872 −0.619359 0.785108i \(-0.712607\pi\)
−0.619359 + 0.785108i \(0.712607\pi\)
\(734\) 0 0
\(735\) 3.34309 0.123312
\(736\) 0 0
\(737\) −53.2842 −1.96275
\(738\) 0 0
\(739\) −20.3756 −0.749530 −0.374765 0.927120i \(-0.622277\pi\)
−0.374765 + 0.927120i \(0.622277\pi\)
\(740\) 0 0
\(741\) 6.59533 0.242285
\(742\) 0 0
\(743\) 37.6960 1.38293 0.691466 0.722409i \(-0.256965\pi\)
0.691466 + 0.722409i \(0.256965\pi\)
\(744\) 0 0
\(745\) 33.9734 1.24469
\(746\) 0 0
\(747\) −12.9622 −0.474262
\(748\) 0 0
\(749\) 12.6977 0.463963
\(750\) 0 0
\(751\) −11.0991 −0.405013 −0.202507 0.979281i \(-0.564909\pi\)
−0.202507 + 0.979281i \(0.564909\pi\)
\(752\) 0 0
\(753\) −23.5808 −0.859330
\(754\) 0 0
\(755\) −16.6757 −0.606892
\(756\) 0 0
\(757\) −16.7983 −0.610545 −0.305272 0.952265i \(-0.598747\pi\)
−0.305272 + 0.952265i \(0.598747\pi\)
\(758\) 0 0
\(759\) 4.36801 0.158549
\(760\) 0 0
\(761\) 28.6568 1.03881 0.519405 0.854528i \(-0.326154\pi\)
0.519405 + 0.854528i \(0.326154\pi\)
\(762\) 0 0
\(763\) −8.45850 −0.306218
\(764\) 0 0
\(765\) 18.6027 0.672581
\(766\) 0 0
\(767\) 3.05437 0.110287
\(768\) 0 0
\(769\) 46.3103 1.66999 0.834996 0.550256i \(-0.185470\pi\)
0.834996 + 0.550256i \(0.185470\pi\)
\(770\) 0 0
\(771\) −1.17119 −0.0421792
\(772\) 0 0
\(773\) −13.7491 −0.494521 −0.247261 0.968949i \(-0.579530\pi\)
−0.247261 + 0.968949i \(0.579530\pi\)
\(774\) 0 0
\(775\) −67.4651 −2.42342
\(776\) 0 0
\(777\) −0.336292 −0.0120644
\(778\) 0 0
\(779\) 34.1840 1.22477
\(780\) 0 0
\(781\) −54.2306 −1.94052
\(782\) 0 0
\(783\) −5.73398 −0.204916
\(784\) 0 0
\(785\) −65.8064 −2.34873
\(786\) 0 0
\(787\) 35.9759 1.28240 0.641200 0.767373i \(-0.278437\pi\)
0.641200 + 0.767373i \(0.278437\pi\)
\(788\) 0 0
\(789\) −29.2939 −1.04289
\(790\) 0 0
\(791\) −16.8356 −0.598604
\(792\) 0 0
\(793\) 7.36217 0.261438
\(794\) 0 0
\(795\) 4.38405 0.155486
\(796\) 0 0
\(797\) 47.8653 1.69547 0.847737 0.530417i \(-0.177964\pi\)
0.847737 + 0.530417i \(0.177964\pi\)
\(798\) 0 0
\(799\) −52.2534 −1.84859
\(800\) 0 0
\(801\) −13.1016 −0.462922
\(802\) 0 0
\(803\) 40.3205 1.42288
\(804\) 0 0
\(805\) 3.34309 0.117829
\(806\) 0 0
\(807\) 17.4092 0.612832
\(808\) 0 0
\(809\) −35.4802 −1.24742 −0.623709 0.781657i \(-0.714375\pi\)
−0.623709 + 0.781657i \(0.714375\pi\)
\(810\) 0 0
\(811\) 19.0909 0.670372 0.335186 0.942152i \(-0.391201\pi\)
0.335186 + 0.942152i \(0.391201\pi\)
\(812\) 0 0
\(813\) 13.3068 0.466688
\(814\) 0 0
\(815\) −21.1152 −0.739633
\(816\) 0 0
\(817\) 41.8463 1.46402
\(818\) 0 0
\(819\) 1.50992 0.0527607
\(820\) 0 0
\(821\) 33.5983 1.17259 0.586295 0.810098i \(-0.300586\pi\)
0.586295 + 0.810098i \(0.300586\pi\)
\(822\) 0 0
\(823\) −19.4558 −0.678187 −0.339093 0.940753i \(-0.610120\pi\)
−0.339093 + 0.940753i \(0.610120\pi\)
\(824\) 0 0
\(825\) 26.9780 0.939254
\(826\) 0 0
\(827\) −12.7785 −0.444353 −0.222177 0.975006i \(-0.571316\pi\)
−0.222177 + 0.975006i \(0.571316\pi\)
\(828\) 0 0
\(829\) −23.4640 −0.814940 −0.407470 0.913219i \(-0.633589\pi\)
−0.407470 + 0.913219i \(0.633589\pi\)
\(830\) 0 0
\(831\) 13.5866 0.471314
\(832\) 0 0
\(833\) 5.56451 0.192799
\(834\) 0 0
\(835\) −43.5113 −1.50577
\(836\) 0 0
\(837\) −10.9233 −0.377564
\(838\) 0 0
\(839\) 49.5145 1.70943 0.854716 0.519096i \(-0.173731\pi\)
0.854716 + 0.519096i \(0.173731\pi\)
\(840\) 0 0
\(841\) 3.87847 0.133740
\(842\) 0 0
\(843\) 1.09451 0.0376970
\(844\) 0 0
\(845\) −35.8385 −1.23288
\(846\) 0 0
\(847\) 8.07951 0.277615
\(848\) 0 0
\(849\) 32.0003 1.09825
\(850\) 0 0
\(851\) −0.336292 −0.0115279
\(852\) 0 0
\(853\) 16.8514 0.576982 0.288491 0.957483i \(-0.406846\pi\)
0.288491 + 0.957483i \(0.406846\pi\)
\(854\) 0 0
\(855\) 14.6027 0.499400
\(856\) 0 0
\(857\) 5.93126 0.202608 0.101304 0.994856i \(-0.467699\pi\)
0.101304 + 0.994856i \(0.467699\pi\)
\(858\) 0 0
\(859\) −52.5264 −1.79218 −0.896088 0.443876i \(-0.853603\pi\)
−0.896088 + 0.443876i \(0.853603\pi\)
\(860\) 0 0
\(861\) 7.82598 0.266709
\(862\) 0 0
\(863\) −26.3038 −0.895391 −0.447695 0.894186i \(-0.647755\pi\)
−0.447695 + 0.894186i \(0.647755\pi\)
\(864\) 0 0
\(865\) 25.4554 0.865511
\(866\) 0 0
\(867\) 13.9637 0.474233
\(868\) 0 0
\(869\) −3.83651 −0.130145
\(870\) 0 0
\(871\) −18.4191 −0.624107
\(872\) 0 0
\(873\) 8.31098 0.281284
\(874\) 0 0
\(875\) 3.93238 0.132939
\(876\) 0 0
\(877\) −41.9167 −1.41543 −0.707713 0.706500i \(-0.750273\pi\)
−0.707713 + 0.706500i \(0.750273\pi\)
\(878\) 0 0
\(879\) 15.6909 0.529240
\(880\) 0 0
\(881\) −12.8183 −0.431861 −0.215931 0.976409i \(-0.569278\pi\)
−0.215931 + 0.976409i \(0.569278\pi\)
\(882\) 0 0
\(883\) −54.4558 −1.83258 −0.916291 0.400513i \(-0.868832\pi\)
−0.916291 + 0.400513i \(0.868832\pi\)
\(884\) 0 0
\(885\) 6.76265 0.227324
\(886\) 0 0
\(887\) −47.1571 −1.58338 −0.791691 0.610922i \(-0.790799\pi\)
−0.791691 + 0.610922i \(0.790799\pi\)
\(888\) 0 0
\(889\) −10.3837 −0.348258
\(890\) 0 0
\(891\) 4.36801 0.146334
\(892\) 0 0
\(893\) −41.0177 −1.37261
\(894\) 0 0
\(895\) 66.5931 2.22596
\(896\) 0 0
\(897\) 1.50992 0.0504146
\(898\) 0 0
\(899\) 62.6338 2.08895
\(900\) 0 0
\(901\) 7.29716 0.243104
\(902\) 0 0
\(903\) 9.58018 0.318809
\(904\) 0 0
\(905\) 20.8147 0.691903
\(906\) 0 0
\(907\) 56.9065 1.88955 0.944775 0.327720i \(-0.106280\pi\)
0.944775 + 0.327720i \(0.106280\pi\)
\(908\) 0 0
\(909\) −2.70906 −0.0898538
\(910\) 0 0
\(911\) −26.9942 −0.894358 −0.447179 0.894445i \(-0.647571\pi\)
−0.447179 + 0.894445i \(0.647571\pi\)
\(912\) 0 0
\(913\) −56.6190 −1.87381
\(914\) 0 0
\(915\) 16.3005 0.538879
\(916\) 0 0
\(917\) −9.76569 −0.322492
\(918\) 0 0
\(919\) 39.7594 1.31154 0.655770 0.754960i \(-0.272344\pi\)
0.655770 + 0.754960i \(0.272344\pi\)
\(920\) 0 0
\(921\) −29.4453 −0.970255
\(922\) 0 0
\(923\) −18.7462 −0.617039
\(924\) 0 0
\(925\) −2.07703 −0.0682924
\(926\) 0 0
\(927\) −8.41032 −0.276231
\(928\) 0 0
\(929\) −27.7938 −0.911884 −0.455942 0.890010i \(-0.650697\pi\)
−0.455942 + 0.890010i \(0.650697\pi\)
\(930\) 0 0
\(931\) 4.36801 0.143156
\(932\) 0 0
\(933\) −7.85680 −0.257220
\(934\) 0 0
\(935\) 81.2566 2.65737
\(936\) 0 0
\(937\) −58.2369 −1.90252 −0.951259 0.308393i \(-0.900209\pi\)
−0.951259 + 0.308393i \(0.900209\pi\)
\(938\) 0 0
\(939\) −27.3253 −0.891728
\(940\) 0 0
\(941\) 33.8951 1.10495 0.552475 0.833530i \(-0.313684\pi\)
0.552475 + 0.833530i \(0.313684\pi\)
\(942\) 0 0
\(943\) 7.82598 0.254849
\(944\) 0 0
\(945\) 3.34309 0.108751
\(946\) 0 0
\(947\) 49.0637 1.59436 0.797178 0.603744i \(-0.206325\pi\)
0.797178 + 0.603744i \(0.206325\pi\)
\(948\) 0 0
\(949\) 13.9378 0.452441
\(950\) 0 0
\(951\) −14.0716 −0.456301
\(952\) 0 0
\(953\) 18.7705 0.608035 0.304017 0.952667i \(-0.401672\pi\)
0.304017 + 0.952667i \(0.401672\pi\)
\(954\) 0 0
\(955\) −4.89819 −0.158502
\(956\) 0 0
\(957\) −25.0461 −0.809624
\(958\) 0 0
\(959\) −6.51217 −0.210289
\(960\) 0 0
\(961\) 88.3180 2.84897
\(962\) 0 0
\(963\) 12.6977 0.409177
\(964\) 0 0
\(965\) 15.5740 0.501344
\(966\) 0 0
\(967\) 15.2423 0.490161 0.245080 0.969503i \(-0.421186\pi\)
0.245080 + 0.969503i \(0.421186\pi\)
\(968\) 0 0
\(969\) 24.3058 0.780815
\(970\) 0 0
\(971\) −10.7150 −0.343861 −0.171931 0.985109i \(-0.555000\pi\)
−0.171931 + 0.985109i \(0.555000\pi\)
\(972\) 0 0
\(973\) −9.69717 −0.310877
\(974\) 0 0
\(975\) 9.32565 0.298660
\(976\) 0 0
\(977\) −10.8224 −0.346240 −0.173120 0.984901i \(-0.555385\pi\)
−0.173120 + 0.984901i \(0.555385\pi\)
\(978\) 0 0
\(979\) −57.2279 −1.82901
\(980\) 0 0
\(981\) −8.45850 −0.270059
\(982\) 0 0
\(983\) 49.0256 1.56368 0.781838 0.623482i \(-0.214283\pi\)
0.781838 + 0.623482i \(0.214283\pi\)
\(984\) 0 0
\(985\) 71.2464 2.27010
\(986\) 0 0
\(987\) −9.39049 −0.298903
\(988\) 0 0
\(989\) 9.58018 0.304632
\(990\) 0 0
\(991\) 21.1406 0.671553 0.335777 0.941942i \(-0.391001\pi\)
0.335777 + 0.941942i \(0.391001\pi\)
\(992\) 0 0
\(993\) −8.18758 −0.259825
\(994\) 0 0
\(995\) 51.0541 1.61852
\(996\) 0 0
\(997\) 0.223489 0.00707797 0.00353899 0.999994i \(-0.498874\pi\)
0.00353899 + 0.999994i \(0.498874\pi\)
\(998\) 0 0
\(999\) −0.336292 −0.0106398
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 7728.2.a.ch.1.5 6
4.3 odd 2 3864.2.a.w.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.w.1.5 6 4.3 odd 2
7728.2.a.ch.1.5 6 1.1 even 1 trivial