Properties

Label 9408.2.a.ed.1.2
Level 9408
Weight 2
Character 9408.1
Self dual yes
Analytic conductor 75.123
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1176)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.41421 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.41421 q^{5} +1.00000 q^{9} +0.828427 q^{11} -4.24264 q^{13} +3.41421 q^{15} -7.41421 q^{17} -6.82843 q^{19} +4.82843 q^{23} +6.65685 q^{25} +1.00000 q^{27} -2.82843 q^{29} -2.82843 q^{31} +0.828427 q^{33} +1.65685 q^{37} -4.24264 q^{39} -10.2426 q^{41} -11.3137 q^{43} +3.41421 q^{45} -4.48528 q^{47} -7.41421 q^{51} +2.00000 q^{53} +2.82843 q^{55} -6.82843 q^{57} -8.48528 q^{59} +11.0711 q^{61} -14.4853 q^{65} +11.3137 q^{67} +4.82843 q^{69} +10.4853 q^{71} -7.75736 q^{73} +6.65685 q^{75} -13.6569 q^{79} +1.00000 q^{81} +4.00000 q^{83} -25.3137 q^{85} -2.82843 q^{87} -5.75736 q^{89} -2.82843 q^{93} -23.3137 q^{95} +0.242641 q^{97} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 4q^{5} + 2q^{9} - 4q^{11} + 4q^{15} - 12q^{17} - 8q^{19} + 4q^{23} + 2q^{25} + 2q^{27} - 4q^{33} - 8q^{37} - 12q^{41} + 4q^{45} + 8q^{47} - 12q^{51} + 4q^{53} - 8q^{57} + 8q^{61} - 12q^{65} + 4q^{69} + 4q^{71} - 24q^{73} + 2q^{75} - 16q^{79} + 2q^{81} + 8q^{83} - 28q^{85} - 20q^{89} - 24q^{95} - 8q^{97} - 4q^{99} + O(q^{100}) \)

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
\(5\) 3.41421 1.52688 0.763441 0.645877i \(-0.223508\pi\)
0.763441 + 0.645877i \(0.223508\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.828427 0.249780 0.124890 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) 3.41421 0.881546
\(16\) 0 0
\(17\) −7.41421 −1.79821 −0.899105 0.437732i \(-0.855782\pi\)
−0.899105 + 0.437732i \(0.855782\pi\)
\(18\) 0 0
\(19\) −6.82843 −1.56655 −0.783274 0.621676i \(-0.786452\pi\)
−0.783274 + 0.621676i \(0.786452\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.82843 1.00680 0.503398 0.864054i \(-0.332083\pi\)
0.503398 + 0.864054i \(0.332083\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.82843 −0.525226 −0.262613 0.964901i \(-0.584584\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) 0 0
\(33\) 0.828427 0.144211
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.65685 0.272385 0.136193 0.990682i \(-0.456513\pi\)
0.136193 + 0.990682i \(0.456513\pi\)
\(38\) 0 0
\(39\) −4.24264 −0.679366
\(40\) 0 0
\(41\) −10.2426 −1.59963 −0.799816 0.600245i \(-0.795070\pi\)
−0.799816 + 0.600245i \(0.795070\pi\)
\(42\) 0 0
\(43\) −11.3137 −1.72532 −0.862662 0.505781i \(-0.831205\pi\)
−0.862662 + 0.505781i \(0.831205\pi\)
\(44\) 0 0
\(45\) 3.41421 0.508961
\(46\) 0 0
\(47\) −4.48528 −0.654246 −0.327123 0.944982i \(-0.606079\pi\)
−0.327123 + 0.944982i \(0.606079\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −7.41421 −1.03820
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) 0 0
\(57\) −6.82843 −0.904447
\(58\) 0 0
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 0 0
\(61\) 11.0711 1.41750 0.708752 0.705457i \(-0.249258\pi\)
0.708752 + 0.705457i \(0.249258\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −14.4853 −1.79668
\(66\) 0 0
\(67\) 11.3137 1.38219 0.691095 0.722764i \(-0.257129\pi\)
0.691095 + 0.722764i \(0.257129\pi\)
\(68\) 0 0
\(69\) 4.82843 0.581274
\(70\) 0 0
\(71\) 10.4853 1.24437 0.622187 0.782869i \(-0.286244\pi\)
0.622187 + 0.782869i \(0.286244\pi\)
\(72\) 0 0
\(73\) −7.75736 −0.907930 −0.453965 0.891019i \(-0.649991\pi\)
−0.453965 + 0.891019i \(0.649991\pi\)
\(74\) 0 0
\(75\) 6.65685 0.768667
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.6569 −1.53652 −0.768258 0.640140i \(-0.778876\pi\)
−0.768258 + 0.640140i \(0.778876\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −25.3137 −2.74566
\(86\) 0 0
\(87\) −2.82843 −0.303239
\(88\) 0 0
\(89\) −5.75736 −0.610279 −0.305139 0.952308i \(-0.598703\pi\)
−0.305139 + 0.952308i \(0.598703\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.82843 −0.293294
\(94\) 0 0
\(95\) −23.3137 −2.39194
\(96\) 0 0
\(97\) 0.242641 0.0246364 0.0123182 0.999924i \(-0.496079\pi\)
0.0123182 + 0.999924i \(0.496079\pi\)
\(98\) 0 0
\(99\) 0.828427 0.0832601
\(100\) 0 0
\(101\) −10.7279 −1.06747 −0.533734 0.845652i \(-0.679212\pi\)
−0.533734 + 0.845652i \(0.679212\pi\)
\(102\) 0 0
\(103\) 14.1421 1.39347 0.696733 0.717331i \(-0.254636\pi\)
0.696733 + 0.717331i \(0.254636\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.8284 1.24017 0.620085 0.784534i \(-0.287098\pi\)
0.620085 + 0.784534i \(0.287098\pi\)
\(108\) 0 0
\(109\) 3.31371 0.317396 0.158698 0.987327i \(-0.449270\pi\)
0.158698 + 0.987327i \(0.449270\pi\)
\(110\) 0 0
\(111\) 1.65685 0.157262
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 16.4853 1.53726
\(116\) 0 0
\(117\) −4.24264 −0.392232
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 0 0
\(123\) −10.2426 −0.923548
\(124\) 0 0
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) −11.3137 −0.996116
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.41421 0.293849
\(136\) 0 0
\(137\) −9.17157 −0.783580 −0.391790 0.920055i \(-0.628144\pi\)
−0.391790 + 0.920055i \(0.628144\pi\)
\(138\) 0 0
\(139\) 20.9706 1.77870 0.889350 0.457227i \(-0.151157\pi\)
0.889350 + 0.457227i \(0.151157\pi\)
\(140\) 0 0
\(141\) −4.48528 −0.377729
\(142\) 0 0
\(143\) −3.51472 −0.293916
\(144\) 0 0
\(145\) −9.65685 −0.801958
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.31371 −0.107623 −0.0538116 0.998551i \(-0.517137\pi\)
−0.0538116 + 0.998551i \(0.517137\pi\)
\(150\) 0 0
\(151\) 9.65685 0.785864 0.392932 0.919568i \(-0.371461\pi\)
0.392932 + 0.919568i \(0.371461\pi\)
\(152\) 0 0
\(153\) −7.41421 −0.599404
\(154\) 0 0
\(155\) −9.65685 −0.775657
\(156\) 0 0
\(157\) 0.242641 0.0193648 0.00968242 0.999953i \(-0.496918\pi\)
0.00968242 + 0.999953i \(0.496918\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 5.65685 0.443079 0.221540 0.975151i \(-0.428892\pi\)
0.221540 + 0.975151i \(0.428892\pi\)
\(164\) 0 0
\(165\) 2.82843 0.220193
\(166\) 0 0
\(167\) −14.8284 −1.14746 −0.573729 0.819045i \(-0.694504\pi\)
−0.573729 + 0.819045i \(0.694504\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) −6.82843 −0.522183
\(172\) 0 0
\(173\) −16.5858 −1.26099 −0.630497 0.776192i \(-0.717149\pi\)
−0.630497 + 0.776192i \(0.717149\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −8.48528 −0.637793
\(178\) 0 0
\(179\) −6.48528 −0.484733 −0.242366 0.970185i \(-0.577924\pi\)
−0.242366 + 0.970185i \(0.577924\pi\)
\(180\) 0 0
\(181\) −7.07107 −0.525588 −0.262794 0.964852i \(-0.584644\pi\)
−0.262794 + 0.964852i \(0.584644\pi\)
\(182\) 0 0
\(183\) 11.0711 0.818397
\(184\) 0 0
\(185\) 5.65685 0.415900
\(186\) 0 0
\(187\) −6.14214 −0.449157
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.1421 −0.878574 −0.439287 0.898347i \(-0.644769\pi\)
−0.439287 + 0.898347i \(0.644769\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) −14.4853 −1.03731
\(196\) 0 0
\(197\) 2.68629 0.191390 0.0956952 0.995411i \(-0.469493\pi\)
0.0956952 + 0.995411i \(0.469493\pi\)
\(198\) 0 0
\(199\) −5.65685 −0.401004 −0.200502 0.979693i \(-0.564257\pi\)
−0.200502 + 0.979693i \(0.564257\pi\)
\(200\) 0 0
\(201\) 11.3137 0.798007
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −34.9706 −2.44245
\(206\) 0 0
\(207\) 4.82843 0.335599
\(208\) 0 0
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) 1.65685 0.114063 0.0570313 0.998372i \(-0.481837\pi\)
0.0570313 + 0.998372i \(0.481837\pi\)
\(212\) 0 0
\(213\) 10.4853 0.718440
\(214\) 0 0
\(215\) −38.6274 −2.63437
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −7.75736 −0.524194
\(220\) 0 0
\(221\) 31.4558 2.11595
\(222\) 0 0
\(223\) 13.6569 0.914531 0.457265 0.889330i \(-0.348829\pi\)
0.457265 + 0.889330i \(0.348829\pi\)
\(224\) 0 0
\(225\) 6.65685 0.443790
\(226\) 0 0
\(227\) 5.17157 0.343249 0.171625 0.985162i \(-0.445098\pi\)
0.171625 + 0.985162i \(0.445098\pi\)
\(228\) 0 0
\(229\) −25.4142 −1.67942 −0.839709 0.543036i \(-0.817275\pi\)
−0.839709 + 0.543036i \(0.817275\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.8284 −0.971443 −0.485721 0.874114i \(-0.661443\pi\)
−0.485721 + 0.874114i \(0.661443\pi\)
\(234\) 0 0
\(235\) −15.3137 −0.998956
\(236\) 0 0
\(237\) −13.6569 −0.887108
\(238\) 0 0
\(239\) 12.8284 0.829802 0.414901 0.909867i \(-0.363816\pi\)
0.414901 + 0.909867i \(0.363816\pi\)
\(240\) 0 0
\(241\) −13.8995 −0.895345 −0.447673 0.894198i \(-0.647747\pi\)
−0.447673 + 0.894198i \(0.647747\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 28.9706 1.84335
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 6.14214 0.387688 0.193844 0.981032i \(-0.437904\pi\)
0.193844 + 0.981032i \(0.437904\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) −25.3137 −1.58521
\(256\) 0 0
\(257\) 3.41421 0.212973 0.106486 0.994314i \(-0.466040\pi\)
0.106486 + 0.994314i \(0.466040\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.82843 −0.175075
\(262\) 0 0
\(263\) 20.1421 1.24202 0.621009 0.783804i \(-0.286723\pi\)
0.621009 + 0.783804i \(0.286723\pi\)
\(264\) 0 0
\(265\) 6.82843 0.419467
\(266\) 0 0
\(267\) −5.75736 −0.352345
\(268\) 0 0
\(269\) −0.100505 −0.00612790 −0.00306395 0.999995i \(-0.500975\pi\)
−0.00306395 + 0.999995i \(0.500975\pi\)
\(270\) 0 0
\(271\) 6.14214 0.373108 0.186554 0.982445i \(-0.440268\pi\)
0.186554 + 0.982445i \(0.440268\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.51472 0.332550
\(276\) 0 0
\(277\) 28.6274 1.72005 0.860027 0.510248i \(-0.170446\pi\)
0.860027 + 0.510248i \(0.170446\pi\)
\(278\) 0 0
\(279\) −2.82843 −0.169334
\(280\) 0 0
\(281\) 1.17157 0.0698902 0.0349451 0.999389i \(-0.488874\pi\)
0.0349451 + 0.999389i \(0.488874\pi\)
\(282\) 0 0
\(283\) 26.1421 1.55399 0.776994 0.629508i \(-0.216743\pi\)
0.776994 + 0.629508i \(0.216743\pi\)
\(284\) 0 0
\(285\) −23.3137 −1.38098
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 37.9706 2.23356
\(290\) 0 0
\(291\) 0.242641 0.0142238
\(292\) 0 0
\(293\) −1.75736 −0.102666 −0.0513330 0.998682i \(-0.516347\pi\)
−0.0513330 + 0.998682i \(0.516347\pi\)
\(294\) 0 0
\(295\) −28.9706 −1.68673
\(296\) 0 0
\(297\) 0.828427 0.0480702
\(298\) 0 0
\(299\) −20.4853 −1.18469
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −10.7279 −0.616303
\(304\) 0 0
\(305\) 37.7990 2.16436
\(306\) 0 0
\(307\) 11.5147 0.657180 0.328590 0.944473i \(-0.393427\pi\)
0.328590 + 0.944473i \(0.393427\pi\)
\(308\) 0 0
\(309\) 14.1421 0.804518
\(310\) 0 0
\(311\) 23.7990 1.34952 0.674758 0.738039i \(-0.264248\pi\)
0.674758 + 0.738039i \(0.264248\pi\)
\(312\) 0 0
\(313\) −28.7279 −1.62380 −0.811899 0.583798i \(-0.801566\pi\)
−0.811899 + 0.583798i \(0.801566\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) −2.34315 −0.131191
\(320\) 0 0
\(321\) 12.8284 0.716013
\(322\) 0 0
\(323\) 50.6274 2.81698
\(324\) 0 0
\(325\) −28.2426 −1.56662
\(326\) 0 0
\(327\) 3.31371 0.183248
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7.31371 0.401998 0.200999 0.979591i \(-0.435581\pi\)
0.200999 + 0.979591i \(0.435581\pi\)
\(332\) 0 0
\(333\) 1.65685 0.0907951
\(334\) 0 0
\(335\) 38.6274 2.11044
\(336\) 0 0
\(337\) 10.3431 0.563427 0.281714 0.959499i \(-0.409097\pi\)
0.281714 + 0.959499i \(0.409097\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) −2.34315 −0.126888
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 16.4853 0.887538
\(346\) 0 0
\(347\) −2.48528 −0.133417 −0.0667084 0.997773i \(-0.521250\pi\)
−0.0667084 + 0.997773i \(0.521250\pi\)
\(348\) 0 0
\(349\) 10.5858 0.566644 0.283322 0.959025i \(-0.408563\pi\)
0.283322 + 0.959025i \(0.408563\pi\)
\(350\) 0 0
\(351\) −4.24264 −0.226455
\(352\) 0 0
\(353\) 27.6985 1.47424 0.737121 0.675761i \(-0.236185\pi\)
0.737121 + 0.675761i \(0.236185\pi\)
\(354\) 0 0
\(355\) 35.7990 1.90001
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −28.8284 −1.52151 −0.760753 0.649041i \(-0.775170\pi\)
−0.760753 + 0.649041i \(0.775170\pi\)
\(360\) 0 0
\(361\) 27.6274 1.45407
\(362\) 0 0
\(363\) −10.3137 −0.541329
\(364\) 0 0
\(365\) −26.4853 −1.38630
\(366\) 0 0
\(367\) 4.68629 0.244622 0.122311 0.992492i \(-0.460969\pi\)
0.122311 + 0.992492i \(0.460969\pi\)
\(368\) 0 0
\(369\) −10.2426 −0.533211
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −5.31371 −0.275133 −0.137567 0.990493i \(-0.543928\pi\)
−0.137567 + 0.990493i \(0.543928\pi\)
\(374\) 0 0
\(375\) 5.65685 0.292119
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −23.3137 −1.19754 −0.598772 0.800919i \(-0.704345\pi\)
−0.598772 + 0.800919i \(0.704345\pi\)
\(380\) 0 0
\(381\) −20.0000 −1.02463
\(382\) 0 0
\(383\) −8.97056 −0.458374 −0.229187 0.973382i \(-0.573607\pi\)
−0.229187 + 0.973382i \(0.573607\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −11.3137 −0.575108
\(388\) 0 0
\(389\) 35.7990 1.81508 0.907540 0.419965i \(-0.137958\pi\)
0.907540 + 0.419965i \(0.137958\pi\)
\(390\) 0 0
\(391\) −35.7990 −1.81043
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) −46.6274 −2.34608
\(396\) 0 0
\(397\) −16.2426 −0.815195 −0.407597 0.913162i \(-0.633633\pi\)
−0.407597 + 0.913162i \(0.633633\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.17157 0.0585056 0.0292528 0.999572i \(-0.490687\pi\)
0.0292528 + 0.999572i \(0.490687\pi\)
\(402\) 0 0
\(403\) 12.0000 0.597763
\(404\) 0 0
\(405\) 3.41421 0.169654
\(406\) 0 0
\(407\) 1.37258 0.0680364
\(408\) 0 0
\(409\) 8.24264 0.407572 0.203786 0.979015i \(-0.434675\pi\)
0.203786 + 0.979015i \(0.434675\pi\)
\(410\) 0 0
\(411\) −9.17157 −0.452400
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 13.6569 0.670389
\(416\) 0 0
\(417\) 20.9706 1.02693
\(418\) 0 0
\(419\) −7.51472 −0.367118 −0.183559 0.983009i \(-0.558762\pi\)
−0.183559 + 0.983009i \(0.558762\pi\)
\(420\) 0 0
\(421\) 25.3137 1.23371 0.616857 0.787075i \(-0.288406\pi\)
0.616857 + 0.787075i \(0.288406\pi\)
\(422\) 0 0
\(423\) −4.48528 −0.218082
\(424\) 0 0
\(425\) −49.3553 −2.39409
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −3.51472 −0.169692
\(430\) 0 0
\(431\) 8.14214 0.392193 0.196096 0.980585i \(-0.437173\pi\)
0.196096 + 0.980585i \(0.437173\pi\)
\(432\) 0 0
\(433\) −0.928932 −0.0446416 −0.0223208 0.999751i \(-0.507106\pi\)
−0.0223208 + 0.999751i \(0.507106\pi\)
\(434\) 0 0
\(435\) −9.65685 −0.463011
\(436\) 0 0
\(437\) −32.9706 −1.57720
\(438\) 0 0
\(439\) 12.6863 0.605484 0.302742 0.953073i \(-0.402098\pi\)
0.302742 + 0.953073i \(0.402098\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −11.1716 −0.530777 −0.265389 0.964141i \(-0.585500\pi\)
−0.265389 + 0.964141i \(0.585500\pi\)
\(444\) 0 0
\(445\) −19.6569 −0.931824
\(446\) 0 0
\(447\) −1.31371 −0.0621363
\(448\) 0 0
\(449\) −16.6274 −0.784696 −0.392348 0.919817i \(-0.628337\pi\)
−0.392348 + 0.919817i \(0.628337\pi\)
\(450\) 0 0
\(451\) −8.48528 −0.399556
\(452\) 0 0
\(453\) 9.65685 0.453719
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.6274 −1.15202 −0.576011 0.817442i \(-0.695391\pi\)
−0.576011 + 0.817442i \(0.695391\pi\)
\(458\) 0 0
\(459\) −7.41421 −0.346066
\(460\) 0 0
\(461\) 16.3848 0.763115 0.381558 0.924345i \(-0.375388\pi\)
0.381558 + 0.924345i \(0.375388\pi\)
\(462\) 0 0
\(463\) −1.65685 −0.0770005 −0.0385003 0.999259i \(-0.512258\pi\)
−0.0385003 + 0.999259i \(0.512258\pi\)
\(464\) 0 0
\(465\) −9.65685 −0.447826
\(466\) 0 0
\(467\) 18.8284 0.871276 0.435638 0.900122i \(-0.356523\pi\)
0.435638 + 0.900122i \(0.356523\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.242641 0.0111803
\(472\) 0 0
\(473\) −9.37258 −0.430952
\(474\) 0 0
\(475\) −45.4558 −2.08566
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) 22.8284 1.04306 0.521529 0.853234i \(-0.325362\pi\)
0.521529 + 0.853234i \(0.325362\pi\)
\(480\) 0 0
\(481\) −7.02944 −0.320515
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.828427 0.0376169
\(486\) 0 0
\(487\) −4.97056 −0.225238 −0.112619 0.993638i \(-0.535924\pi\)
−0.112619 + 0.993638i \(0.535924\pi\)
\(488\) 0 0
\(489\) 5.65685 0.255812
\(490\) 0 0
\(491\) 19.1716 0.865201 0.432600 0.901586i \(-0.357596\pi\)
0.432600 + 0.901586i \(0.357596\pi\)
\(492\) 0 0
\(493\) 20.9706 0.944467
\(494\) 0 0
\(495\) 2.82843 0.127128
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −24.9706 −1.11784 −0.558918 0.829223i \(-0.688783\pi\)
−0.558918 + 0.829223i \(0.688783\pi\)
\(500\) 0 0
\(501\) −14.8284 −0.662485
\(502\) 0 0
\(503\) −19.3137 −0.861156 −0.430578 0.902553i \(-0.641690\pi\)
−0.430578 + 0.902553i \(0.641690\pi\)
\(504\) 0 0
\(505\) −36.6274 −1.62990
\(506\) 0 0
\(507\) 5.00000 0.222058
\(508\) 0 0
\(509\) −36.3848 −1.61273 −0.806363 0.591420i \(-0.798567\pi\)
−0.806363 + 0.591420i \(0.798567\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −6.82843 −0.301482
\(514\) 0 0
\(515\) 48.2843 2.12766
\(516\) 0 0
\(517\) −3.71573 −0.163418
\(518\) 0 0
\(519\) −16.5858 −0.728035
\(520\) 0 0
\(521\) −14.2426 −0.623981 −0.311991 0.950085i \(-0.600996\pi\)
−0.311991 + 0.950085i \(0.600996\pi\)
\(522\) 0 0
\(523\) 4.97056 0.217348 0.108674 0.994077i \(-0.465340\pi\)
0.108674 + 0.994077i \(0.465340\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.9706 0.913492
\(528\) 0 0
\(529\) 0.313708 0.0136395
\(530\) 0 0
\(531\) −8.48528 −0.368230
\(532\) 0 0
\(533\) 43.4558 1.88228
\(534\) 0 0
\(535\) 43.7990 1.89360
\(536\) 0 0
\(537\) −6.48528 −0.279861
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 0 0
\(543\) −7.07107 −0.303449
\(544\) 0 0
\(545\) 11.3137 0.484626
\(546\) 0 0
\(547\) −3.02944 −0.129529 −0.0647647 0.997901i \(-0.520630\pi\)
−0.0647647 + 0.997901i \(0.520630\pi\)
\(548\) 0 0
\(549\) 11.0711 0.472502
\(550\) 0 0
\(551\) 19.3137 0.822792
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 5.65685 0.240120
\(556\) 0 0
\(557\) −32.6274 −1.38247 −0.691234 0.722631i \(-0.742933\pi\)
−0.691234 + 0.722631i \(0.742933\pi\)
\(558\) 0 0
\(559\) 48.0000 2.03018
\(560\) 0 0
\(561\) −6.14214 −0.259321
\(562\) 0 0
\(563\) −1.85786 −0.0782996 −0.0391498 0.999233i \(-0.512465\pi\)
−0.0391498 + 0.999233i \(0.512465\pi\)
\(564\) 0 0
\(565\) −34.1421 −1.43637
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.85786 −0.245574 −0.122787 0.992433i \(-0.539183\pi\)
−0.122787 + 0.992433i \(0.539183\pi\)
\(570\) 0 0
\(571\) −41.6569 −1.74329 −0.871643 0.490142i \(-0.836945\pi\)
−0.871643 + 0.490142i \(0.836945\pi\)
\(572\) 0 0
\(573\) −12.1421 −0.507245
\(574\) 0 0
\(575\) 32.1421 1.34042
\(576\) 0 0
\(577\) −13.2132 −0.550073 −0.275036 0.961434i \(-0.588690\pi\)
−0.275036 + 0.961434i \(0.588690\pi\)
\(578\) 0 0
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.65685 0.0686199
\(584\) 0 0
\(585\) −14.4853 −0.598893
\(586\) 0 0
\(587\) 35.7990 1.47758 0.738791 0.673934i \(-0.235397\pi\)
0.738791 + 0.673934i \(0.235397\pi\)
\(588\) 0 0
\(589\) 19.3137 0.795807
\(590\) 0 0
\(591\) 2.68629 0.110499
\(592\) 0 0
\(593\) −47.4142 −1.94707 −0.973534 0.228541i \(-0.926604\pi\)
−0.973534 + 0.228541i \(0.926604\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.65685 −0.231520
\(598\) 0 0
\(599\) −23.4558 −0.958380 −0.479190 0.877711i \(-0.659069\pi\)
−0.479190 + 0.877711i \(0.659069\pi\)
\(600\) 0 0
\(601\) 12.2426 0.499388 0.249694 0.968325i \(-0.419670\pi\)
0.249694 + 0.968325i \(0.419670\pi\)
\(602\) 0 0
\(603\) 11.3137 0.460730
\(604\) 0 0
\(605\) −35.2132 −1.43162
\(606\) 0 0
\(607\) 24.9706 1.01352 0.506762 0.862086i \(-0.330842\pi\)
0.506762 + 0.862086i \(0.330842\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 19.0294 0.769849
\(612\) 0 0
\(613\) 10.3431 0.417756 0.208878 0.977942i \(-0.433019\pi\)
0.208878 + 0.977942i \(0.433019\pi\)
\(614\) 0 0
\(615\) −34.9706 −1.41015
\(616\) 0 0
\(617\) −4.48528 −0.180571 −0.0902853 0.995916i \(-0.528778\pi\)
−0.0902853 + 0.995916i \(0.528778\pi\)
\(618\) 0 0
\(619\) −33.6569 −1.35278 −0.676392 0.736542i \(-0.736457\pi\)
−0.676392 + 0.736542i \(0.736457\pi\)
\(620\) 0 0
\(621\) 4.82843 0.193758
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) −5.65685 −0.225913
\(628\) 0 0
\(629\) −12.2843 −0.489806
\(630\) 0 0
\(631\) 3.02944 0.120600 0.0603000 0.998180i \(-0.480794\pi\)
0.0603000 + 0.998180i \(0.480794\pi\)
\(632\) 0 0
\(633\) 1.65685 0.0658540
\(634\) 0 0
\(635\) −68.2843 −2.70978
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 10.4853 0.414791
\(640\) 0 0
\(641\) 35.1127 1.38687 0.693434 0.720520i \(-0.256097\pi\)
0.693434 + 0.720520i \(0.256097\pi\)
\(642\) 0 0
\(643\) −31.7990 −1.25403 −0.627015 0.779007i \(-0.715723\pi\)
−0.627015 + 0.779007i \(0.715723\pi\)
\(644\) 0 0
\(645\) −38.6274 −1.52095
\(646\) 0 0
\(647\) −0.201010 −0.00790252 −0.00395126 0.999992i \(-0.501258\pi\)
−0.00395126 + 0.999992i \(0.501258\pi\)
\(648\) 0 0
\(649\) −7.02944 −0.275930
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −46.1421 −1.80568 −0.902841 0.429975i \(-0.858522\pi\)
−0.902841 + 0.429975i \(0.858522\pi\)
\(654\) 0 0
\(655\) 13.6569 0.533617
\(656\) 0 0
\(657\) −7.75736 −0.302643
\(658\) 0 0
\(659\) 21.5147 0.838094 0.419047 0.907964i \(-0.362364\pi\)
0.419047 + 0.907964i \(0.362364\pi\)
\(660\) 0 0
\(661\) 4.92893 0.191713 0.0958566 0.995395i \(-0.469441\pi\)
0.0958566 + 0.995395i \(0.469441\pi\)
\(662\) 0 0
\(663\) 31.4558 1.22164
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −13.6569 −0.528796
\(668\) 0 0
\(669\) 13.6569 0.528004
\(670\) 0 0
\(671\) 9.17157 0.354065
\(672\) 0 0
\(673\) −23.3137 −0.898677 −0.449339 0.893361i \(-0.648340\pi\)
−0.449339 + 0.893361i \(0.648340\pi\)
\(674\) 0 0
\(675\) 6.65685 0.256222
\(676\) 0 0
\(677\) 27.6985 1.06454 0.532270 0.846575i \(-0.321339\pi\)
0.532270 + 0.846575i \(0.321339\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 5.17157 0.198175
\(682\) 0 0
\(683\) −1.79899 −0.0688364 −0.0344182 0.999408i \(-0.510958\pi\)
−0.0344182 + 0.999408i \(0.510958\pi\)
\(684\) 0 0
\(685\) −31.3137 −1.19644
\(686\) 0 0
\(687\) −25.4142 −0.969613
\(688\) 0 0
\(689\) −8.48528 −0.323263
\(690\) 0 0
\(691\) −8.68629 −0.330442 −0.165221 0.986257i \(-0.552834\pi\)
−0.165221 + 0.986257i \(0.552834\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 71.5980 2.71587
\(696\) 0 0
\(697\) 75.9411 2.87648
\(698\) 0 0
\(699\) −14.8284 −0.560863
\(700\) 0 0
\(701\) 6.14214 0.231985 0.115993 0.993250i \(-0.462995\pi\)
0.115993 + 0.993250i \(0.462995\pi\)
\(702\) 0 0
\(703\) −11.3137 −0.426705
\(704\) 0 0
\(705\) −15.3137 −0.576748
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.62742 −0.248898 −0.124449 0.992226i \(-0.539716\pi\)
−0.124449 + 0.992226i \(0.539716\pi\)
\(710\) 0 0
\(711\) −13.6569 −0.512172
\(712\) 0 0
\(713\) −13.6569 −0.511453
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 0 0
\(717\) 12.8284 0.479086
\(718\) 0 0
\(719\) −6.62742 −0.247161 −0.123580 0.992335i \(-0.539438\pi\)
−0.123580 + 0.992335i \(0.539438\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −13.8995 −0.516928
\(724\) 0 0
\(725\) −18.8284 −0.699270
\(726\) 0 0
\(727\) −9.85786 −0.365608 −0.182804 0.983149i \(-0.558517\pi\)
−0.182804 + 0.983149i \(0.558517\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 83.8823 3.10250
\(732\) 0 0
\(733\) −8.04163 −0.297024 −0.148512 0.988911i \(-0.547448\pi\)
−0.148512 + 0.988911i \(0.547448\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.37258 0.345244
\(738\) 0 0
\(739\) 32.9706 1.21284 0.606421 0.795144i \(-0.292605\pi\)
0.606421 + 0.795144i \(0.292605\pi\)
\(740\) 0 0
\(741\) 28.9706 1.06426
\(742\) 0 0
\(743\) 4.82843 0.177138 0.0885689 0.996070i \(-0.471771\pi\)
0.0885689 + 0.996070i \(0.471771\pi\)
\(744\) 0 0
\(745\) −4.48528 −0.164328
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −20.2843 −0.740184 −0.370092 0.928995i \(-0.620674\pi\)
−0.370092 + 0.928995i \(0.620674\pi\)
\(752\) 0 0
\(753\) 6.14214 0.223832
\(754\) 0 0
\(755\) 32.9706 1.19992
\(756\) 0 0
\(757\) −41.9411 −1.52438 −0.762188 0.647356i \(-0.775875\pi\)
−0.762188 + 0.647356i \(0.775875\pi\)
\(758\) 0 0
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 31.8995 1.15636 0.578178 0.815911i \(-0.303764\pi\)
0.578178 + 0.815911i \(0.303764\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −25.3137 −0.915219
\(766\) 0 0
\(767\) 36.0000 1.29988
\(768\) 0 0
\(769\) −34.8701 −1.25745 −0.628723 0.777629i \(-0.716422\pi\)
−0.628723 + 0.777629i \(0.716422\pi\)
\(770\) 0 0
\(771\) 3.41421 0.122960
\(772\) 0 0
\(773\) 33.3553 1.19971 0.599854 0.800109i \(-0.295225\pi\)
0.599854 + 0.800109i \(0.295225\pi\)
\(774\) 0 0
\(775\) −18.8284 −0.676337
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 69.9411 2.50590
\(780\) 0 0
\(781\) 8.68629 0.310820
\(782\) 0 0
\(783\) −2.82843 −0.101080
\(784\) 0 0
\(785\) 0.828427 0.0295678
\(786\) 0 0
\(787\) −15.3137 −0.545875 −0.272937 0.962032i \(-0.587995\pi\)
−0.272937 + 0.962032i \(0.587995\pi\)
\(788\) 0 0
\(789\) 20.1421 0.717079
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −46.9706 −1.66797
\(794\) 0 0
\(795\) 6.82843 0.242179
\(796\) 0 0
\(797\) −39.4142 −1.39612 −0.698062 0.716038i \(-0.745954\pi\)
−0.698062 + 0.716038i \(0.745954\pi\)
\(798\) 0 0
\(799\) 33.2548 1.17647
\(800\) 0 0
\(801\) −5.75736 −0.203426
\(802\) 0 0
\(803\) −6.42641 −0.226783
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −0.100505 −0.00353795
\(808\) 0 0
\(809\) −4.62742 −0.162691 −0.0813457 0.996686i \(-0.525922\pi\)
−0.0813457 + 0.996686i \(0.525922\pi\)
\(810\) 0 0
\(811\) −25.6569 −0.900934 −0.450467 0.892793i \(-0.648743\pi\)
−0.450467 + 0.892793i \(0.648743\pi\)
\(812\) 0 0
\(813\) 6.14214 0.215414
\(814\) 0 0
\(815\) 19.3137 0.676530
\(816\) 0 0
\(817\) 77.2548 2.70280
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.6863 0.372954 0.186477 0.982459i \(-0.440293\pi\)
0.186477 + 0.982459i \(0.440293\pi\)
\(822\) 0 0
\(823\) 24.9706 0.870419 0.435210 0.900329i \(-0.356674\pi\)
0.435210 + 0.900329i \(0.356674\pi\)
\(824\) 0 0
\(825\) 5.51472 0.191998
\(826\) 0 0
\(827\) 40.1421 1.39588 0.697939 0.716157i \(-0.254100\pi\)
0.697939 + 0.716157i \(0.254100\pi\)
\(828\) 0 0
\(829\) 34.3848 1.19423 0.597116 0.802155i \(-0.296313\pi\)
0.597116 + 0.802155i \(0.296313\pi\)
\(830\) 0 0
\(831\) 28.6274 0.993074
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −50.6274 −1.75203
\(836\) 0 0
\(837\) −2.82843 −0.0977647
\(838\) 0 0
\(839\) 23.7990 0.821632 0.410816 0.911718i \(-0.365244\pi\)
0.410816 + 0.911718i \(0.365244\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) 1.17157 0.0403511
\(844\) 0 0
\(845\) 17.0711 0.587263
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 26.1421 0.897196
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 4.92893 0.168763 0.0843817 0.996434i \(-0.473108\pi\)
0.0843817 + 0.996434i \(0.473108\pi\)
\(854\) 0 0
\(855\) −23.3137 −0.797312
\(856\) 0 0
\(857\) 11.2132 0.383036 0.191518 0.981489i \(-0.438659\pi\)
0.191518 + 0.981489i \(0.438659\pi\)
\(858\) 0 0
\(859\) −2.54416 −0.0868055 −0.0434027 0.999058i \(-0.513820\pi\)
−0.0434027 + 0.999058i \(0.513820\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −54.0833 −1.84102 −0.920508 0.390724i \(-0.872225\pi\)
−0.920508 + 0.390724i \(0.872225\pi\)
\(864\) 0 0
\(865\) −56.6274 −1.92539
\(866\) 0 0
\(867\) 37.9706 1.28955
\(868\) 0 0
\(869\) −11.3137 −0.383791
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 0 0
\(873\) 0.242641 0.00821214
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 28.2843 0.955092 0.477546 0.878607i \(-0.341526\pi\)
0.477546 + 0.878607i \(0.341526\pi\)
\(878\) 0 0
\(879\) −1.75736 −0.0592743
\(880\) 0 0
\(881\) −30.0416 −1.01213 −0.506064 0.862496i \(-0.668900\pi\)
−0.506064 + 0.862496i \(0.668900\pi\)
\(882\) 0 0
\(883\) −10.3431 −0.348075 −0.174037 0.984739i \(-0.555681\pi\)
−0.174037 + 0.984739i \(0.555681\pi\)
\(884\) 0 0
\(885\) −28.9706 −0.973835
\(886\) 0 0
\(887\) −39.7990 −1.33632 −0.668160 0.744018i \(-0.732918\pi\)
−0.668160 + 0.744018i \(0.732918\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.828427 0.0277534
\(892\) 0 0
\(893\) 30.6274 1.02491
\(894\) 0 0
\(895\) −22.1421 −0.740130
\(896\) 0 0
\(897\) −20.4853 −0.683984
\(898\) 0 0
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −14.8284 −0.494007
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −24.1421 −0.802512
\(906\) 0 0
\(907\) 53.6569 1.78165 0.890823 0.454350i \(-0.150128\pi\)
0.890823 + 0.454350i \(0.150128\pi\)
\(908\) 0 0
\(909\) −10.7279 −0.355823
\(910\) 0 0
\(911\) −6.48528 −0.214867 −0.107433 0.994212i \(-0.534263\pi\)
−0.107433 + 0.994212i \(0.534263\pi\)
\(912\) 0 0
\(913\) 3.31371 0.109668
\(914\) 0 0
\(915\) 37.7990 1.24960
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −31.3137 −1.03294 −0.516472 0.856304i \(-0.672755\pi\)
−0.516472 + 0.856304i \(0.672755\pi\)
\(920\) 0 0
\(921\) 11.5147 0.379423
\(922\) 0 0
\(923\) −44.4853 −1.46425
\(924\) 0 0
\(925\) 11.0294 0.362646
\(926\) 0 0
\(927\) 14.1421 0.464489
\(928\) 0 0
\(929\) 15.8995 0.521646 0.260823 0.965387i \(-0.416006\pi\)
0.260823 + 0.965387i \(0.416006\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 23.7990 0.779144
\(934\) 0 0
\(935\) −20.9706 −0.685811
\(936\) 0 0
\(937\) −51.3553 −1.67771 −0.838853 0.544358i \(-0.816773\pi\)
−0.838853 + 0.544358i \(0.816773\pi\)
\(938\) 0 0
\(939\) −28.7279 −0.937500
\(940\) 0 0
\(941\) −14.7279 −0.480117 −0.240058 0.970758i \(-0.577167\pi\)
−0.240058 + 0.970758i \(0.577167\pi\)
\(942\) 0 0
\(943\) −49.4558 −1.61050
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −58.7696 −1.90975 −0.954877 0.297002i \(-0.904013\pi\)
−0.954877 + 0.297002i \(0.904013\pi\)
\(948\) 0 0
\(949\) 32.9117 1.06836
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) 0 0
\(955\) −41.4558 −1.34148
\(956\) 0 0
\(957\) −2.34315 −0.0757431
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) 12.8284 0.413390
\(964\) 0 0
\(965\) 6.82843 0.219815
\(966\) 0 0
\(967\) 31.3137 1.00698 0.503490 0.864001i \(-0.332049\pi\)
0.503490 + 0.864001i \(0.332049\pi\)
\(968\) 0 0
\(969\) 50.6274 1.62639
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −28.2426 −0.904488
\(976\) 0 0
\(977\) −7.79899 −0.249512 −0.124756 0.992187i \(-0.539815\pi\)
−0.124756 + 0.992187i \(0.539815\pi\)
\(978\) 0 0
\(979\) −4.76955 −0.152436
\(980\) 0 0
\(981\) 3.31371 0.105799
\(982\) 0 0
\(983\) 1.37258 0.0437786 0.0218893 0.999760i \(-0.493032\pi\)
0.0218893 + 0.999760i \(0.493032\pi\)
\(984\) 0 0
\(985\) 9.17157 0.292231
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −54.6274 −1.73705
\(990\) 0 0
\(991\) 18.6274 0.591719 0.295860 0.955231i \(-0.404394\pi\)
0.295860 + 0.955231i \(0.404394\pi\)
\(992\) 0 0
\(993\) 7.31371 0.232094
\(994\) 0 0
\(995\) −19.3137 −0.612286
\(996\) 0 0
\(997\) −54.3848 −1.72238 −0.861192 0.508281i \(-0.830281\pi\)
−0.861192 + 0.508281i \(0.830281\pi\)
\(998\) 0 0
\(999\) 1.65685 0.0524205
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 9408.2.a.ed.1.2 2
4.3 odd 2 9408.2.a.dr.1.2 2
7.6 odd 2 9408.2.a.dh.1.1 2
8.3 odd 2 1176.2.a.m.1.1 yes 2
8.5 even 2 2352.2.a.z.1.1 2
24.5 odd 2 7056.2.a.cw.1.2 2
24.11 even 2 3528.2.a.bm.1.2 2
28.27 even 2 9408.2.a.dv.1.1 2
56.3 even 6 1176.2.q.n.961.1 4
56.5 odd 6 2352.2.q.ba.1537.1 4
56.11 odd 6 1176.2.q.m.961.2 4
56.13 odd 2 2352.2.a.bg.1.2 2
56.19 even 6 1176.2.q.n.361.1 4
56.27 even 2 1176.2.a.l.1.2 2
56.37 even 6 2352.2.q.bg.1537.2 4
56.45 odd 6 2352.2.q.ba.961.1 4
56.51 odd 6 1176.2.q.m.361.2 4
56.53 even 6 2352.2.q.bg.961.2 4
168.11 even 6 3528.2.s.bc.3313.1 4
168.59 odd 6 3528.2.s.bl.3313.2 4
168.83 odd 2 3528.2.a.bc.1.1 2
168.107 even 6 3528.2.s.bc.361.1 4
168.125 even 2 7056.2.a.ce.1.1 2
168.131 odd 6 3528.2.s.bl.361.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.2.a.l.1.2 2 56.27 even 2
1176.2.a.m.1.1 yes 2 8.3 odd 2
1176.2.q.m.361.2 4 56.51 odd 6
1176.2.q.m.961.2 4 56.11 odd 6
1176.2.q.n.361.1 4 56.19 even 6
1176.2.q.n.961.1 4 56.3 even 6
2352.2.a.z.1.1 2 8.5 even 2
2352.2.a.bg.1.2 2 56.13 odd 2
2352.2.q.ba.961.1 4 56.45 odd 6
2352.2.q.ba.1537.1 4 56.5 odd 6
2352.2.q.bg.961.2 4 56.53 even 6
2352.2.q.bg.1537.2 4 56.37 even 6
3528.2.a.bc.1.1 2 168.83 odd 2
3528.2.a.bm.1.2 2 24.11 even 2
3528.2.s.bc.361.1 4 168.107 even 6
3528.2.s.bc.3313.1 4 168.11 even 6
3528.2.s.bl.361.2 4 168.131 odd 6
3528.2.s.bl.3313.2 4 168.59 odd 6
7056.2.a.ce.1.1 2 168.125 even 2
7056.2.a.cw.1.2 2 24.5 odd 2
9408.2.a.dh.1.1 2 7.6 odd 2
9408.2.a.dr.1.2 2 4.3 odd 2
9408.2.a.dv.1.1 2 28.27 even 2
9408.2.a.ed.1.2 2 1.1 even 1 trivial