Properties

Label 7168.2.a.bb.1.2
Level $7168$
Weight $2$
Character 7168.1
Self dual yes
Analytic conductor $57.237$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7168 = 2^{10} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7168.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.2367681689\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.9433055232.1
Defining polynomial: \(x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 9 x^{4} - 76 x^{3} - 4 x^{2} + 48 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1792)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.38887\) of defining polynomial
Character \(\chi\) \(=\) 7168.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.29954 q^{3} -1.65321 q^{5} +1.00000 q^{7} +2.28788 q^{9} +O(q^{10})\) \(q-2.29954 q^{3} -1.65321 q^{5} +1.00000 q^{7} +2.28788 q^{9} -6.22192 q^{11} +0.634100 q^{13} +3.80163 q^{15} -5.02326 q^{17} -2.11921 q^{19} -2.29954 q^{21} +8.89958 q^{23} -2.26688 q^{25} +1.63755 q^{27} +1.13585 q^{29} +8.27524 q^{31} +14.3075 q^{33} -1.65321 q^{35} +2.19447 q^{37} -1.45814 q^{39} -4.93020 q^{41} +5.27913 q^{43} -3.78235 q^{45} +6.68692 q^{47} +1.00000 q^{49} +11.5512 q^{51} -6.41239 q^{53} +10.2862 q^{55} +4.87321 q^{57} +1.50921 q^{59} +7.23152 q^{61} +2.28788 q^{63} -1.04830 q^{65} -10.5717 q^{67} -20.4649 q^{69} -4.07354 q^{71} +13.5063 q^{73} +5.21278 q^{75} -6.22192 q^{77} -9.19701 q^{79} -10.6293 q^{81} +3.70875 q^{83} +8.30452 q^{85} -2.61192 q^{87} +1.60040 q^{89} +0.634100 q^{91} -19.0292 q^{93} +3.50351 q^{95} -13.0167 q^{97} -14.2350 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + 8q^{7} + 12q^{9} + O(q^{10}) \) \( 8q - 8q^{5} + 8q^{7} + 12q^{9} - 12q^{11} - 20q^{13} + 4q^{17} - 4q^{19} + 8q^{23} + 12q^{25} + 12q^{27} - 8q^{29} - 4q^{31} + 8q^{33} - 8q^{35} - 8q^{37} - 16q^{39} - 12q^{41} + 4q^{43} - 52q^{45} + 20q^{47} + 8q^{49} - 32q^{51} - 40q^{53} + 24q^{55} - 4q^{57} - 4q^{59} + 8q^{61} + 12q^{63} + 36q^{65} - 28q^{67} - 4q^{69} - 16q^{71} + 16q^{73} + 28q^{75} - 12q^{77} + 20q^{81} + 8q^{83} - 16q^{85} + 20q^{87} + 16q^{89} - 20q^{91} - 16q^{93} - 40q^{95} - 36q^{97} + 4q^{99} + O(q^{100}) \)

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
\(3\) −2.29954 −1.32764 −0.663820 0.747893i \(-0.731066\pi\)
−0.663820 + 0.747893i \(0.731066\pi\)
\(4\) 0 0
\(5\) −1.65321 −0.739340 −0.369670 0.929163i \(-0.620529\pi\)
−0.369670 + 0.929163i \(0.620529\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 2.28788 0.762626
\(10\) 0 0
\(11\) −6.22192 −1.87598 −0.937989 0.346664i \(-0.887314\pi\)
−0.937989 + 0.346664i \(0.887314\pi\)
\(12\) 0 0
\(13\) 0.634100 0.175868 0.0879339 0.996126i \(-0.471974\pi\)
0.0879339 + 0.996126i \(0.471974\pi\)
\(14\) 0 0
\(15\) 3.80163 0.981576
\(16\) 0 0
\(17\) −5.02326 −1.21832 −0.609159 0.793048i \(-0.708493\pi\)
−0.609159 + 0.793048i \(0.708493\pi\)
\(18\) 0 0
\(19\) −2.11921 −0.486181 −0.243091 0.970004i \(-0.578161\pi\)
−0.243091 + 0.970004i \(0.578161\pi\)
\(20\) 0 0
\(21\) −2.29954 −0.501800
\(22\) 0 0
\(23\) 8.89958 1.85569 0.927845 0.372965i \(-0.121659\pi\)
0.927845 + 0.372965i \(0.121659\pi\)
\(24\) 0 0
\(25\) −2.26688 −0.453377
\(26\) 0 0
\(27\) 1.63755 0.315148
\(28\) 0 0
\(29\) 1.13585 0.210922 0.105461 0.994423i \(-0.466368\pi\)
0.105461 + 0.994423i \(0.466368\pi\)
\(30\) 0 0
\(31\) 8.27524 1.48628 0.743139 0.669137i \(-0.233336\pi\)
0.743139 + 0.669137i \(0.233336\pi\)
\(32\) 0 0
\(33\) 14.3075 2.49062
\(34\) 0 0
\(35\) −1.65321 −0.279444
\(36\) 0 0
\(37\) 2.19447 0.360769 0.180385 0.983596i \(-0.442266\pi\)
0.180385 + 0.983596i \(0.442266\pi\)
\(38\) 0 0
\(39\) −1.45814 −0.233489
\(40\) 0 0
\(41\) −4.93020 −0.769968 −0.384984 0.922923i \(-0.625793\pi\)
−0.384984 + 0.922923i \(0.625793\pi\)
\(42\) 0 0
\(43\) 5.27913 0.805059 0.402530 0.915407i \(-0.368131\pi\)
0.402530 + 0.915407i \(0.368131\pi\)
\(44\) 0 0
\(45\) −3.78235 −0.563839
\(46\) 0 0
\(47\) 6.68692 0.975387 0.487693 0.873015i \(-0.337838\pi\)
0.487693 + 0.873015i \(0.337838\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 11.5512 1.61749
\(52\) 0 0
\(53\) −6.41239 −0.880809 −0.440405 0.897799i \(-0.645165\pi\)
−0.440405 + 0.897799i \(0.645165\pi\)
\(54\) 0 0
\(55\) 10.2862 1.38699
\(56\) 0 0
\(57\) 4.87321 0.645473
\(58\) 0 0
\(59\) 1.50921 0.196482 0.0982410 0.995163i \(-0.468678\pi\)
0.0982410 + 0.995163i \(0.468678\pi\)
\(60\) 0 0
\(61\) 7.23152 0.925901 0.462951 0.886384i \(-0.346791\pi\)
0.462951 + 0.886384i \(0.346791\pi\)
\(62\) 0 0
\(63\) 2.28788 0.288245
\(64\) 0 0
\(65\) −1.04830 −0.130026
\(66\) 0 0
\(67\) −10.5717 −1.29154 −0.645769 0.763533i \(-0.723463\pi\)
−0.645769 + 0.763533i \(0.723463\pi\)
\(68\) 0 0
\(69\) −20.4649 −2.46369
\(70\) 0 0
\(71\) −4.07354 −0.483440 −0.241720 0.970346i \(-0.577712\pi\)
−0.241720 + 0.970346i \(0.577712\pi\)
\(72\) 0 0
\(73\) 13.5063 1.58079 0.790397 0.612595i \(-0.209874\pi\)
0.790397 + 0.612595i \(0.209874\pi\)
\(74\) 0 0
\(75\) 5.21278 0.601921
\(76\) 0 0
\(77\) −6.22192 −0.709053
\(78\) 0 0
\(79\) −9.19701 −1.03474 −0.517372 0.855760i \(-0.673090\pi\)
−0.517372 + 0.855760i \(0.673090\pi\)
\(80\) 0 0
\(81\) −10.6293 −1.18103
\(82\) 0 0
\(83\) 3.70875 0.407088 0.203544 0.979066i \(-0.434754\pi\)
0.203544 + 0.979066i \(0.434754\pi\)
\(84\) 0 0
\(85\) 8.30452 0.900752
\(86\) 0 0
\(87\) −2.61192 −0.280028
\(88\) 0 0
\(89\) 1.60040 0.169642 0.0848209 0.996396i \(-0.472968\pi\)
0.0848209 + 0.996396i \(0.472968\pi\)
\(90\) 0 0
\(91\) 0.634100 0.0664718
\(92\) 0 0
\(93\) −19.0292 −1.97324
\(94\) 0 0
\(95\) 3.50351 0.359453
\(96\) 0 0
\(97\) −13.0167 −1.32164 −0.660822 0.750543i \(-0.729792\pi\)
−0.660822 + 0.750543i \(0.729792\pi\)
\(98\) 0 0
\(99\) −14.2350 −1.43067
\(100\) 0 0
\(101\) −5.32103 −0.529462 −0.264731 0.964322i \(-0.585283\pi\)
−0.264731 + 0.964322i \(0.585283\pi\)
\(102\) 0 0
\(103\) 7.91886 0.780268 0.390134 0.920758i \(-0.372429\pi\)
0.390134 + 0.920758i \(0.372429\pi\)
\(104\) 0 0
\(105\) 3.80163 0.371001
\(106\) 0 0
\(107\) 19.2277 1.85881 0.929407 0.369056i \(-0.120319\pi\)
0.929407 + 0.369056i \(0.120319\pi\)
\(108\) 0 0
\(109\) −9.94712 −0.952761 −0.476380 0.879239i \(-0.658052\pi\)
−0.476380 + 0.879239i \(0.658052\pi\)
\(110\) 0 0
\(111\) −5.04627 −0.478971
\(112\) 0 0
\(113\) −5.62753 −0.529394 −0.264697 0.964332i \(-0.585272\pi\)
−0.264697 + 0.964332i \(0.585272\pi\)
\(114\) 0 0
\(115\) −14.7129 −1.37199
\(116\) 0 0
\(117\) 1.45074 0.134121
\(118\) 0 0
\(119\) −5.02326 −0.460481
\(120\) 0 0
\(121\) 27.7123 2.51930
\(122\) 0 0
\(123\) 11.3372 1.02224
\(124\) 0 0
\(125\) 12.0137 1.07454
\(126\) 0 0
\(127\) 13.5320 1.20077 0.600386 0.799710i \(-0.295013\pi\)
0.600386 + 0.799710i \(0.295013\pi\)
\(128\) 0 0
\(129\) −12.1396 −1.06883
\(130\) 0 0
\(131\) 2.10195 0.183648 0.0918242 0.995775i \(-0.470730\pi\)
0.0918242 + 0.995775i \(0.470730\pi\)
\(132\) 0 0
\(133\) −2.11921 −0.183759
\(134\) 0 0
\(135\) −2.70723 −0.233001
\(136\) 0 0
\(137\) 11.7463 1.00355 0.501776 0.864998i \(-0.332680\pi\)
0.501776 + 0.864998i \(0.332680\pi\)
\(138\) 0 0
\(139\) 11.3252 0.960590 0.480295 0.877107i \(-0.340530\pi\)
0.480295 + 0.877107i \(0.340530\pi\)
\(140\) 0 0
\(141\) −15.3768 −1.29496
\(142\) 0 0
\(143\) −3.94532 −0.329924
\(144\) 0 0
\(145\) −1.87780 −0.155943
\(146\) 0 0
\(147\) −2.29954 −0.189663
\(148\) 0 0
\(149\) −5.14322 −0.421349 −0.210675 0.977556i \(-0.567566\pi\)
−0.210675 + 0.977556i \(0.567566\pi\)
\(150\) 0 0
\(151\) 15.9697 1.29959 0.649797 0.760108i \(-0.274854\pi\)
0.649797 + 0.760108i \(0.274854\pi\)
\(152\) 0 0
\(153\) −11.4926 −0.929121
\(154\) 0 0
\(155\) −13.6807 −1.09886
\(156\) 0 0
\(157\) 19.8670 1.58556 0.792778 0.609511i \(-0.208634\pi\)
0.792778 + 0.609511i \(0.208634\pi\)
\(158\) 0 0
\(159\) 14.7455 1.16940
\(160\) 0 0
\(161\) 8.89958 0.701385
\(162\) 0 0
\(163\) −14.5377 −1.13868 −0.569341 0.822102i \(-0.692801\pi\)
−0.569341 + 0.822102i \(0.692801\pi\)
\(164\) 0 0
\(165\) −23.6534 −1.84142
\(166\) 0 0
\(167\) −19.4131 −1.50223 −0.751115 0.660172i \(-0.770483\pi\)
−0.751115 + 0.660172i \(0.770483\pi\)
\(168\) 0 0
\(169\) −12.5979 −0.969071
\(170\) 0 0
\(171\) −4.84850 −0.370774
\(172\) 0 0
\(173\) −6.45278 −0.490596 −0.245298 0.969448i \(-0.578886\pi\)
−0.245298 + 0.969448i \(0.578886\pi\)
\(174\) 0 0
\(175\) −2.26688 −0.171360
\(176\) 0 0
\(177\) −3.47048 −0.260857
\(178\) 0 0
\(179\) −23.7770 −1.77717 −0.888587 0.458708i \(-0.848312\pi\)
−0.888587 + 0.458708i \(0.848312\pi\)
\(180\) 0 0
\(181\) −8.85094 −0.657885 −0.328943 0.944350i \(-0.606692\pi\)
−0.328943 + 0.944350i \(0.606692\pi\)
\(182\) 0 0
\(183\) −16.6292 −1.22926
\(184\) 0 0
\(185\) −3.62793 −0.266731
\(186\) 0 0
\(187\) 31.2543 2.28554
\(188\) 0 0
\(189\) 1.63755 0.119115
\(190\) 0 0
\(191\) −15.8222 −1.14486 −0.572428 0.819955i \(-0.693999\pi\)
−0.572428 + 0.819955i \(0.693999\pi\)
\(192\) 0 0
\(193\) −7.50630 −0.540315 −0.270158 0.962816i \(-0.587076\pi\)
−0.270158 + 0.962816i \(0.587076\pi\)
\(194\) 0 0
\(195\) 2.41061 0.172628
\(196\) 0 0
\(197\) 24.3697 1.73627 0.868134 0.496330i \(-0.165319\pi\)
0.868134 + 0.496330i \(0.165319\pi\)
\(198\) 0 0
\(199\) 21.8653 1.54999 0.774995 0.631967i \(-0.217752\pi\)
0.774995 + 0.631967i \(0.217752\pi\)
\(200\) 0 0
\(201\) 24.3100 1.71470
\(202\) 0 0
\(203\) 1.13585 0.0797209
\(204\) 0 0
\(205\) 8.15068 0.569268
\(206\) 0 0
\(207\) 20.3611 1.41520
\(208\) 0 0
\(209\) 13.1856 0.912065
\(210\) 0 0
\(211\) 3.93947 0.271205 0.135602 0.990763i \(-0.456703\pi\)
0.135602 + 0.990763i \(0.456703\pi\)
\(212\) 0 0
\(213\) 9.36726 0.641834
\(214\) 0 0
\(215\) −8.72753 −0.595212
\(216\) 0 0
\(217\) 8.27524 0.561760
\(218\) 0 0
\(219\) −31.0583 −2.09872
\(220\) 0 0
\(221\) −3.18525 −0.214263
\(222\) 0 0
\(223\) −4.64913 −0.311329 −0.155665 0.987810i \(-0.549752\pi\)
−0.155665 + 0.987810i \(0.549752\pi\)
\(224\) 0 0
\(225\) −5.18635 −0.345757
\(226\) 0 0
\(227\) 6.24109 0.414236 0.207118 0.978316i \(-0.433592\pi\)
0.207118 + 0.978316i \(0.433592\pi\)
\(228\) 0 0
\(229\) 20.3814 1.34684 0.673422 0.739258i \(-0.264824\pi\)
0.673422 + 0.739258i \(0.264824\pi\)
\(230\) 0 0
\(231\) 14.3075 0.941367
\(232\) 0 0
\(233\) −11.2765 −0.738746 −0.369373 0.929281i \(-0.620428\pi\)
−0.369373 + 0.929281i \(0.620428\pi\)
\(234\) 0 0
\(235\) −11.0549 −0.721142
\(236\) 0 0
\(237\) 21.1489 1.37377
\(238\) 0 0
\(239\) −0.994605 −0.0643356 −0.0321678 0.999482i \(-0.510241\pi\)
−0.0321678 + 0.999482i \(0.510241\pi\)
\(240\) 0 0
\(241\) −27.1158 −1.74669 −0.873343 0.487107i \(-0.838052\pi\)
−0.873343 + 0.487107i \(0.838052\pi\)
\(242\) 0 0
\(243\) 19.5297 1.25283
\(244\) 0 0
\(245\) −1.65321 −0.105620
\(246\) 0 0
\(247\) −1.34379 −0.0855036
\(248\) 0 0
\(249\) −8.52841 −0.540466
\(250\) 0 0
\(251\) 6.32989 0.399539 0.199769 0.979843i \(-0.435981\pi\)
0.199769 + 0.979843i \(0.435981\pi\)
\(252\) 0 0
\(253\) −55.3724 −3.48124
\(254\) 0 0
\(255\) −19.0966 −1.19587
\(256\) 0 0
\(257\) −7.08743 −0.442102 −0.221051 0.975262i \(-0.570949\pi\)
−0.221051 + 0.975262i \(0.570949\pi\)
\(258\) 0 0
\(259\) 2.19447 0.136358
\(260\) 0 0
\(261\) 2.59868 0.160854
\(262\) 0 0
\(263\) 6.95421 0.428815 0.214408 0.976744i \(-0.431218\pi\)
0.214408 + 0.976744i \(0.431218\pi\)
\(264\) 0 0
\(265\) 10.6010 0.651217
\(266\) 0 0
\(267\) −3.68018 −0.225223
\(268\) 0 0
\(269\) −18.7534 −1.14341 −0.571707 0.820458i \(-0.693719\pi\)
−0.571707 + 0.820458i \(0.693719\pi\)
\(270\) 0 0
\(271\) −14.2967 −0.868463 −0.434231 0.900801i \(-0.642980\pi\)
−0.434231 + 0.900801i \(0.642980\pi\)
\(272\) 0 0
\(273\) −1.45814 −0.0882506
\(274\) 0 0
\(275\) 14.1044 0.850525
\(276\) 0 0
\(277\) −2.20969 −0.132767 −0.0663836 0.997794i \(-0.521146\pi\)
−0.0663836 + 0.997794i \(0.521146\pi\)
\(278\) 0 0
\(279\) 18.9327 1.13347
\(280\) 0 0
\(281\) −6.06619 −0.361879 −0.180939 0.983494i \(-0.557914\pi\)
−0.180939 + 0.983494i \(0.557914\pi\)
\(282\) 0 0
\(283\) 28.7356 1.70815 0.854077 0.520146i \(-0.174123\pi\)
0.854077 + 0.520146i \(0.174123\pi\)
\(284\) 0 0
\(285\) −8.05647 −0.477224
\(286\) 0 0
\(287\) −4.93020 −0.291021
\(288\) 0 0
\(289\) 8.23311 0.484301
\(290\) 0 0
\(291\) 29.9324 1.75467
\(292\) 0 0
\(293\) −31.4461 −1.83710 −0.918549 0.395306i \(-0.870639\pi\)
−0.918549 + 0.395306i \(0.870639\pi\)
\(294\) 0 0
\(295\) −2.49504 −0.145267
\(296\) 0 0
\(297\) −10.1887 −0.591210
\(298\) 0 0
\(299\) 5.64323 0.326356
\(300\) 0 0
\(301\) 5.27913 0.304284
\(302\) 0 0
\(303\) 12.2359 0.702934
\(304\) 0 0
\(305\) −11.9553 −0.684556
\(306\) 0 0
\(307\) −25.1243 −1.43392 −0.716960 0.697115i \(-0.754467\pi\)
−0.716960 + 0.697115i \(0.754467\pi\)
\(308\) 0 0
\(309\) −18.2097 −1.03591
\(310\) 0 0
\(311\) 31.2698 1.77315 0.886574 0.462588i \(-0.153079\pi\)
0.886574 + 0.462588i \(0.153079\pi\)
\(312\) 0 0
\(313\) 18.0095 1.01796 0.508980 0.860778i \(-0.330023\pi\)
0.508980 + 0.860778i \(0.330023\pi\)
\(314\) 0 0
\(315\) −3.78235 −0.213111
\(316\) 0 0
\(317\) −17.3125 −0.972366 −0.486183 0.873857i \(-0.661611\pi\)
−0.486183 + 0.873857i \(0.661611\pi\)
\(318\) 0 0
\(319\) −7.06715 −0.395684
\(320\) 0 0
\(321\) −44.2149 −2.46783
\(322\) 0 0
\(323\) 10.6454 0.592323
\(324\) 0 0
\(325\) −1.43743 −0.0797344
\(326\) 0 0
\(327\) 22.8738 1.26492
\(328\) 0 0
\(329\) 6.68692 0.368662
\(330\) 0 0
\(331\) −16.6108 −0.913014 −0.456507 0.889720i \(-0.650900\pi\)
−0.456507 + 0.889720i \(0.650900\pi\)
\(332\) 0 0
\(333\) 5.02068 0.275132
\(334\) 0 0
\(335\) 17.4773 0.954886
\(336\) 0 0
\(337\) −6.23237 −0.339499 −0.169750 0.985487i \(-0.554296\pi\)
−0.169750 + 0.985487i \(0.554296\pi\)
\(338\) 0 0
\(339\) 12.9407 0.702844
\(340\) 0 0
\(341\) −51.4879 −2.78822
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 33.8329 1.82150
\(346\) 0 0
\(347\) −9.36267 −0.502614 −0.251307 0.967907i \(-0.580860\pi\)
−0.251307 + 0.967907i \(0.580860\pi\)
\(348\) 0 0
\(349\) −20.9244 −1.12006 −0.560029 0.828473i \(-0.689210\pi\)
−0.560029 + 0.828473i \(0.689210\pi\)
\(350\) 0 0
\(351\) 1.03837 0.0554243
\(352\) 0 0
\(353\) −17.7098 −0.942599 −0.471300 0.881973i \(-0.656215\pi\)
−0.471300 + 0.881973i \(0.656215\pi\)
\(354\) 0 0
\(355\) 6.73443 0.357427
\(356\) 0 0
\(357\) 11.5512 0.611353
\(358\) 0 0
\(359\) −8.53643 −0.450535 −0.225268 0.974297i \(-0.572326\pi\)
−0.225268 + 0.974297i \(0.572326\pi\)
\(360\) 0 0
\(361\) −14.5089 −0.763628
\(362\) 0 0
\(363\) −63.7254 −3.34472
\(364\) 0 0
\(365\) −22.3288 −1.16874
\(366\) 0 0
\(367\) 11.3226 0.591037 0.295519 0.955337i \(-0.404508\pi\)
0.295519 + 0.955337i \(0.404508\pi\)
\(368\) 0 0
\(369\) −11.2797 −0.587198
\(370\) 0 0
\(371\) −6.41239 −0.332915
\(372\) 0 0
\(373\) −11.1543 −0.577548 −0.288774 0.957397i \(-0.593248\pi\)
−0.288774 + 0.957397i \(0.593248\pi\)
\(374\) 0 0
\(375\) −27.6260 −1.42660
\(376\) 0 0
\(377\) 0.720241 0.0370943
\(378\) 0 0
\(379\) −13.4809 −0.692466 −0.346233 0.938149i \(-0.612539\pi\)
−0.346233 + 0.938149i \(0.612539\pi\)
\(380\) 0 0
\(381\) −31.1174 −1.59419
\(382\) 0 0
\(383\) −18.9261 −0.967077 −0.483539 0.875323i \(-0.660649\pi\)
−0.483539 + 0.875323i \(0.660649\pi\)
\(384\) 0 0
\(385\) 10.2862 0.524231
\(386\) 0 0
\(387\) 12.0780 0.613959
\(388\) 0 0
\(389\) 9.58885 0.486174 0.243087 0.970004i \(-0.421840\pi\)
0.243087 + 0.970004i \(0.421840\pi\)
\(390\) 0 0
\(391\) −44.7049 −2.26082
\(392\) 0 0
\(393\) −4.83352 −0.243819
\(394\) 0 0
\(395\) 15.2046 0.765028
\(396\) 0 0
\(397\) −2.73804 −0.137418 −0.0687090 0.997637i \(-0.521888\pi\)
−0.0687090 + 0.997637i \(0.521888\pi\)
\(398\) 0 0
\(399\) 4.87321 0.243966
\(400\) 0 0
\(401\) 16.6483 0.831378 0.415689 0.909507i \(-0.363540\pi\)
0.415689 + 0.909507i \(0.363540\pi\)
\(402\) 0 0
\(403\) 5.24733 0.261388
\(404\) 0 0
\(405\) 17.5724 0.873181
\(406\) 0 0
\(407\) −13.6538 −0.676795
\(408\) 0 0
\(409\) −16.8706 −0.834199 −0.417100 0.908861i \(-0.636953\pi\)
−0.417100 + 0.908861i \(0.636953\pi\)
\(410\) 0 0
\(411\) −27.0110 −1.33235
\(412\) 0 0
\(413\) 1.50921 0.0742632
\(414\) 0 0
\(415\) −6.13136 −0.300976
\(416\) 0 0
\(417\) −26.0427 −1.27532
\(418\) 0 0
\(419\) 28.4979 1.39221 0.696107 0.717938i \(-0.254914\pi\)
0.696107 + 0.717938i \(0.254914\pi\)
\(420\) 0 0
\(421\) 9.10775 0.443885 0.221942 0.975060i \(-0.428760\pi\)
0.221942 + 0.975060i \(0.428760\pi\)
\(422\) 0 0
\(423\) 15.2988 0.743855
\(424\) 0 0
\(425\) 11.3871 0.552357
\(426\) 0 0
\(427\) 7.23152 0.349958
\(428\) 0 0
\(429\) 9.07242 0.438020
\(430\) 0 0
\(431\) −28.6481 −1.37993 −0.689965 0.723843i \(-0.742374\pi\)
−0.689965 + 0.723843i \(0.742374\pi\)
\(432\) 0 0
\(433\) 4.69574 0.225663 0.112832 0.993614i \(-0.464008\pi\)
0.112832 + 0.993614i \(0.464008\pi\)
\(434\) 0 0
\(435\) 4.31807 0.207036
\(436\) 0 0
\(437\) −18.8601 −0.902202
\(438\) 0 0
\(439\) −22.6652 −1.08175 −0.540874 0.841103i \(-0.681907\pi\)
−0.540874 + 0.841103i \(0.681907\pi\)
\(440\) 0 0
\(441\) 2.28788 0.108947
\(442\) 0 0
\(443\) −25.1638 −1.19557 −0.597784 0.801657i \(-0.703952\pi\)
−0.597784 + 0.801657i \(0.703952\pi\)
\(444\) 0 0
\(445\) −2.64580 −0.125423
\(446\) 0 0
\(447\) 11.8270 0.559400
\(448\) 0 0
\(449\) −11.2262 −0.529799 −0.264900 0.964276i \(-0.585339\pi\)
−0.264900 + 0.964276i \(0.585339\pi\)
\(450\) 0 0
\(451\) 30.6753 1.44444
\(452\) 0 0
\(453\) −36.7229 −1.72539
\(454\) 0 0
\(455\) −1.04830 −0.0491452
\(456\) 0 0
\(457\) 20.8858 0.976998 0.488499 0.872564i \(-0.337545\pi\)
0.488499 + 0.872564i \(0.337545\pi\)
\(458\) 0 0
\(459\) −8.22586 −0.383950
\(460\) 0 0
\(461\) 2.52649 0.117671 0.0588353 0.998268i \(-0.481261\pi\)
0.0588353 + 0.998268i \(0.481261\pi\)
\(462\) 0 0
\(463\) 21.0754 0.979457 0.489728 0.871875i \(-0.337096\pi\)
0.489728 + 0.871875i \(0.337096\pi\)
\(464\) 0 0
\(465\) 31.4594 1.45889
\(466\) 0 0
\(467\) −11.3928 −0.527194 −0.263597 0.964633i \(-0.584909\pi\)
−0.263597 + 0.964633i \(0.584909\pi\)
\(468\) 0 0
\(469\) −10.5717 −0.488156
\(470\) 0 0
\(471\) −45.6848 −2.10505
\(472\) 0 0
\(473\) −32.8463 −1.51027
\(474\) 0 0
\(475\) 4.80401 0.220423
\(476\) 0 0
\(477\) −14.6708 −0.671728
\(478\) 0 0
\(479\) −19.8741 −0.908070 −0.454035 0.890984i \(-0.650016\pi\)
−0.454035 + 0.890984i \(0.650016\pi\)
\(480\) 0 0
\(481\) 1.39152 0.0634477
\(482\) 0 0
\(483\) −20.4649 −0.931186
\(484\) 0 0
\(485\) 21.5194 0.977144
\(486\) 0 0
\(487\) −19.2744 −0.873406 −0.436703 0.899606i \(-0.643854\pi\)
−0.436703 + 0.899606i \(0.643854\pi\)
\(488\) 0 0
\(489\) 33.4300 1.51176
\(490\) 0 0
\(491\) 5.67386 0.256058 0.128029 0.991770i \(-0.459135\pi\)
0.128029 + 0.991770i \(0.459135\pi\)
\(492\) 0 0
\(493\) −5.70565 −0.256970
\(494\) 0 0
\(495\) 23.5335 1.05775
\(496\) 0 0
\(497\) −4.07354 −0.182723
\(498\) 0 0
\(499\) −5.80598 −0.259912 −0.129956 0.991520i \(-0.541484\pi\)
−0.129956 + 0.991520i \(0.541484\pi\)
\(500\) 0 0
\(501\) 44.6411 1.99442
\(502\) 0 0
\(503\) −19.8588 −0.885460 −0.442730 0.896655i \(-0.645990\pi\)
−0.442730 + 0.896655i \(0.645990\pi\)
\(504\) 0 0
\(505\) 8.79680 0.391452
\(506\) 0 0
\(507\) 28.9694 1.28658
\(508\) 0 0
\(509\) 11.4770 0.508709 0.254355 0.967111i \(-0.418137\pi\)
0.254355 + 0.967111i \(0.418137\pi\)
\(510\) 0 0
\(511\) 13.5063 0.597484
\(512\) 0 0
\(513\) −3.47033 −0.153219
\(514\) 0 0
\(515\) −13.0916 −0.576883
\(516\) 0 0
\(517\) −41.6054 −1.82981
\(518\) 0 0
\(519\) 14.8384 0.651335
\(520\) 0 0
\(521\) −11.8477 −0.519059 −0.259530 0.965735i \(-0.583568\pi\)
−0.259530 + 0.965735i \(0.583568\pi\)
\(522\) 0 0
\(523\) 27.6386 1.20855 0.604276 0.796775i \(-0.293462\pi\)
0.604276 + 0.796775i \(0.293462\pi\)
\(524\) 0 0
\(525\) 5.21278 0.227505
\(526\) 0 0
\(527\) −41.5687 −1.81076
\(528\) 0 0
\(529\) 56.2025 2.44359
\(530\) 0 0
\(531\) 3.45288 0.149842
\(532\) 0 0
\(533\) −3.12624 −0.135413
\(534\) 0 0
\(535\) −31.7876 −1.37430
\(536\) 0 0
\(537\) 54.6760 2.35945
\(538\) 0 0
\(539\) −6.22192 −0.267997
\(540\) 0 0
\(541\) 41.1852 1.77069 0.885345 0.464935i \(-0.153922\pi\)
0.885345 + 0.464935i \(0.153922\pi\)
\(542\) 0 0
\(543\) 20.3531 0.873434
\(544\) 0 0
\(545\) 16.4447 0.704414
\(546\) 0 0
\(547\) −17.9638 −0.768075 −0.384038 0.923317i \(-0.625467\pi\)
−0.384038 + 0.923317i \(0.625467\pi\)
\(548\) 0 0
\(549\) 16.5448 0.706116
\(550\) 0 0
\(551\) −2.40710 −0.102546
\(552\) 0 0
\(553\) −9.19701 −0.391097
\(554\) 0 0
\(555\) 8.34257 0.354122
\(556\) 0 0
\(557\) −25.8596 −1.09570 −0.547852 0.836575i \(-0.684554\pi\)
−0.547852 + 0.836575i \(0.684554\pi\)
\(558\) 0 0
\(559\) 3.34750 0.141584
\(560\) 0 0
\(561\) −71.8704 −3.03437
\(562\) 0 0
\(563\) −1.02037 −0.0430034 −0.0215017 0.999769i \(-0.506845\pi\)
−0.0215017 + 0.999769i \(0.506845\pi\)
\(564\) 0 0
\(565\) 9.30352 0.391402
\(566\) 0 0
\(567\) −10.6293 −0.446387
\(568\) 0 0
\(569\) −33.5555 −1.40672 −0.703360 0.710834i \(-0.748318\pi\)
−0.703360 + 0.710834i \(0.748318\pi\)
\(570\) 0 0
\(571\) 5.37591 0.224975 0.112487 0.993653i \(-0.464118\pi\)
0.112487 + 0.993653i \(0.464118\pi\)
\(572\) 0 0
\(573\) 36.3838 1.51996
\(574\) 0 0
\(575\) −20.1743 −0.841327
\(576\) 0 0
\(577\) −2.22515 −0.0926344 −0.0463172 0.998927i \(-0.514748\pi\)
−0.0463172 + 0.998927i \(0.514748\pi\)
\(578\) 0 0
\(579\) 17.2610 0.717343
\(580\) 0 0
\(581\) 3.70875 0.153865
\(582\) 0 0
\(583\) 39.8973 1.65238
\(584\) 0 0
\(585\) −2.39839 −0.0991612
\(586\) 0 0
\(587\) −6.12882 −0.252964 −0.126482 0.991969i \(-0.540369\pi\)
−0.126482 + 0.991969i \(0.540369\pi\)
\(588\) 0 0
\(589\) −17.5370 −0.722600
\(590\) 0 0
\(591\) −56.0390 −2.30514
\(592\) 0 0
\(593\) 16.8460 0.691784 0.345892 0.938274i \(-0.387576\pi\)
0.345892 + 0.938274i \(0.387576\pi\)
\(594\) 0 0
\(595\) 8.30452 0.340452
\(596\) 0 0
\(597\) −50.2801 −2.05783
\(598\) 0 0
\(599\) 43.9075 1.79401 0.897006 0.442018i \(-0.145737\pi\)
0.897006 + 0.442018i \(0.145737\pi\)
\(600\) 0 0
\(601\) 35.5257 1.44912 0.724562 0.689210i \(-0.242042\pi\)
0.724562 + 0.689210i \(0.242042\pi\)
\(602\) 0 0
\(603\) −24.1867 −0.984960
\(604\) 0 0
\(605\) −45.8143 −1.86262
\(606\) 0 0
\(607\) 15.7700 0.640084 0.320042 0.947403i \(-0.396303\pi\)
0.320042 + 0.947403i \(0.396303\pi\)
\(608\) 0 0
\(609\) −2.61192 −0.105841
\(610\) 0 0
\(611\) 4.24018 0.171539
\(612\) 0 0
\(613\) −11.5565 −0.466765 −0.233382 0.972385i \(-0.574979\pi\)
−0.233382 + 0.972385i \(0.574979\pi\)
\(614\) 0 0
\(615\) −18.7428 −0.755783
\(616\) 0 0
\(617\) −23.3938 −0.941800 −0.470900 0.882187i \(-0.656071\pi\)
−0.470900 + 0.882187i \(0.656071\pi\)
\(618\) 0 0
\(619\) 3.87086 0.155583 0.0777915 0.996970i \(-0.475213\pi\)
0.0777915 + 0.996970i \(0.475213\pi\)
\(620\) 0 0
\(621\) 14.5735 0.584816
\(622\) 0 0
\(623\) 1.60040 0.0641186
\(624\) 0 0
\(625\) −8.52682 −0.341073
\(626\) 0 0
\(627\) −30.3207 −1.21089
\(628\) 0 0
\(629\) −11.0234 −0.439532
\(630\) 0 0
\(631\) −34.9200 −1.39014 −0.695071 0.718941i \(-0.744627\pi\)
−0.695071 + 0.718941i \(0.744627\pi\)
\(632\) 0 0
\(633\) −9.05897 −0.360062
\(634\) 0 0
\(635\) −22.3713 −0.887779
\(636\) 0 0
\(637\) 0.634100 0.0251240
\(638\) 0 0
\(639\) −9.31976 −0.368684
\(640\) 0 0
\(641\) −42.9662 −1.69706 −0.848532 0.529144i \(-0.822513\pi\)
−0.848532 + 0.529144i \(0.822513\pi\)
\(642\) 0 0
\(643\) 24.8794 0.981148 0.490574 0.871399i \(-0.336787\pi\)
0.490574 + 0.871399i \(0.336787\pi\)
\(644\) 0 0
\(645\) 20.0693 0.790227
\(646\) 0 0
\(647\) −33.5883 −1.32049 −0.660246 0.751049i \(-0.729548\pi\)
−0.660246 + 0.751049i \(0.729548\pi\)
\(648\) 0 0
\(649\) −9.39016 −0.368596
\(650\) 0 0
\(651\) −19.0292 −0.745815
\(652\) 0 0
\(653\) −0.397667 −0.0155619 −0.00778095 0.999970i \(-0.502477\pi\)
−0.00778095 + 0.999970i \(0.502477\pi\)
\(654\) 0 0
\(655\) −3.47498 −0.135779
\(656\) 0 0
\(657\) 30.9008 1.20555
\(658\) 0 0
\(659\) 1.77349 0.0690855 0.0345428 0.999403i \(-0.489003\pi\)
0.0345428 + 0.999403i \(0.489003\pi\)
\(660\) 0 0
\(661\) 43.4877 1.69148 0.845738 0.533598i \(-0.179160\pi\)
0.845738 + 0.533598i \(0.179160\pi\)
\(662\) 0 0
\(663\) 7.32460 0.284464
\(664\) 0 0
\(665\) 3.50351 0.135860
\(666\) 0 0
\(667\) 10.1086 0.391405
\(668\) 0 0
\(669\) 10.6909 0.413333
\(670\) 0 0
\(671\) −44.9939 −1.73697
\(672\) 0 0
\(673\) −22.2711 −0.858486 −0.429243 0.903189i \(-0.641220\pi\)
−0.429243 + 0.903189i \(0.641220\pi\)
\(674\) 0 0
\(675\) −3.71214 −0.142881
\(676\) 0 0
\(677\) −41.5083 −1.59529 −0.797646 0.603126i \(-0.793922\pi\)
−0.797646 + 0.603126i \(0.793922\pi\)
\(678\) 0 0
\(679\) −13.0167 −0.499535
\(680\) 0 0
\(681\) −14.3516 −0.549956
\(682\) 0 0
\(683\) −2.53936 −0.0971658 −0.0485829 0.998819i \(-0.515471\pi\)
−0.0485829 + 0.998819i \(0.515471\pi\)
\(684\) 0 0
\(685\) −19.4191 −0.741966
\(686\) 0 0
\(687\) −46.8679 −1.78812
\(688\) 0 0
\(689\) −4.06610 −0.154906
\(690\) 0 0
\(691\) −16.5871 −0.631005 −0.315502 0.948925i \(-0.602173\pi\)
−0.315502 + 0.948925i \(0.602173\pi\)
\(692\) 0 0
\(693\) −14.2350 −0.540742
\(694\) 0 0
\(695\) −18.7230 −0.710202
\(696\) 0 0
\(697\) 24.7657 0.938067
\(698\) 0 0
\(699\) 25.9307 0.980788
\(700\) 0 0
\(701\) 33.2411 1.25550 0.627750 0.778415i \(-0.283976\pi\)
0.627750 + 0.778415i \(0.283976\pi\)
\(702\) 0 0
\(703\) −4.65056 −0.175399
\(704\) 0 0
\(705\) 25.4212 0.957417
\(706\) 0 0
\(707\) −5.32103 −0.200118
\(708\) 0 0
\(709\) 2.20075 0.0826509 0.0413255 0.999146i \(-0.486842\pi\)
0.0413255 + 0.999146i \(0.486842\pi\)
\(710\) 0 0
\(711\) −21.0416 −0.789123
\(712\) 0 0
\(713\) 73.6462 2.75807
\(714\) 0 0
\(715\) 6.52246 0.243926
\(716\) 0 0
\(717\) 2.28713 0.0854145
\(718\) 0 0
\(719\) 6.97452 0.260106 0.130053 0.991507i \(-0.458485\pi\)
0.130053 + 0.991507i \(0.458485\pi\)
\(720\) 0 0
\(721\) 7.91886 0.294914
\(722\) 0 0
\(723\) 62.3539 2.31897
\(724\) 0 0
\(725\) −2.57483 −0.0956269
\(726\) 0 0
\(727\) 12.7285 0.472073 0.236037 0.971744i \(-0.424151\pi\)
0.236037 + 0.971744i \(0.424151\pi\)
\(728\) 0 0
\(729\) −13.0216 −0.482280
\(730\) 0 0
\(731\) −26.5184 −0.980819
\(732\) 0 0
\(733\) −19.9102 −0.735400 −0.367700 0.929945i \(-0.619855\pi\)
−0.367700 + 0.929945i \(0.619855\pi\)
\(734\) 0 0
\(735\) 3.80163 0.140225
\(736\) 0 0
\(737\) 65.7762 2.42290
\(738\) 0 0
\(739\) −52.2947 −1.92369 −0.961846 0.273592i \(-0.911788\pi\)
−0.961846 + 0.273592i \(0.911788\pi\)
\(740\) 0 0
\(741\) 3.09011 0.113518
\(742\) 0 0
\(743\) 10.0453 0.368527 0.184263 0.982877i \(-0.441010\pi\)
0.184263 + 0.982877i \(0.441010\pi\)
\(744\) 0 0
\(745\) 8.50285 0.311520
\(746\) 0 0
\(747\) 8.48516 0.310456
\(748\) 0 0
\(749\) 19.2277 0.702566
\(750\) 0 0
\(751\) −42.4776 −1.55003 −0.775014 0.631944i \(-0.782257\pi\)
−0.775014 + 0.631944i \(0.782257\pi\)
\(752\) 0 0
\(753\) −14.5558 −0.530443
\(754\) 0 0
\(755\) −26.4013 −0.960841
\(756\) 0 0
\(757\) 42.1141 1.53066 0.765332 0.643636i \(-0.222575\pi\)
0.765332 + 0.643636i \(0.222575\pi\)
\(758\) 0 0
\(759\) 127.331 4.62183
\(760\) 0 0
\(761\) −6.54784 −0.237359 −0.118679 0.992933i \(-0.537866\pi\)
−0.118679 + 0.992933i \(0.537866\pi\)
\(762\) 0 0
\(763\) −9.94712 −0.360110
\(764\) 0 0
\(765\) 18.9997 0.686936
\(766\) 0 0
\(767\) 0.956988 0.0345549
\(768\) 0 0
\(769\) 31.6203 1.14026 0.570129 0.821555i \(-0.306893\pi\)
0.570129 + 0.821555i \(0.306893\pi\)
\(770\) 0 0
\(771\) 16.2978 0.586952
\(772\) 0 0
\(773\) −15.0246 −0.540397 −0.270198 0.962805i \(-0.587089\pi\)
−0.270198 + 0.962805i \(0.587089\pi\)
\(774\) 0 0
\(775\) −18.7590 −0.673843
\(776\) 0 0
\(777\) −5.04627 −0.181034
\(778\) 0 0
\(779\) 10.4482 0.374344
\(780\) 0 0
\(781\) 25.3452 0.906924
\(782\) 0 0
\(783\) 1.86001 0.0664714
\(784\) 0 0
\(785\) −32.8443 −1.17226
\(786\) 0 0
\(787\) 48.4507 1.72708 0.863540 0.504280i \(-0.168242\pi\)
0.863540 + 0.504280i \(0.168242\pi\)
\(788\) 0 0
\(789\) −15.9915 −0.569312
\(790\) 0 0
\(791\) −5.62753 −0.200092
\(792\) 0 0
\(793\) 4.58551 0.162836
\(794\) 0 0
\(795\) −24.3775 −0.864581
\(796\) 0 0
\(797\) 12.0940 0.428392 0.214196 0.976791i \(-0.431287\pi\)
0.214196 + 0.976791i \(0.431287\pi\)
\(798\) 0 0
\(799\) −33.5901 −1.18833
\(800\) 0 0
\(801\) 3.66151 0.129373
\(802\) 0 0
\(803\) −84.0352 −2.96554
\(804\) 0 0
\(805\) −14.7129 −0.518562
\(806\) 0 0
\(807\) 43.1241 1.51804
\(808\) 0 0
\(809\) 36.6145 1.28730 0.643649 0.765321i \(-0.277420\pi\)
0.643649 + 0.765321i \(0.277420\pi\)
\(810\) 0 0
\(811\) −36.2833 −1.27408 −0.637040 0.770831i \(-0.719841\pi\)
−0.637040 + 0.770831i \(0.719841\pi\)
\(812\) 0 0
\(813\) 32.8758 1.15300
\(814\) 0 0
\(815\) 24.0340 0.841873
\(816\) 0 0
\(817\) −11.1876 −0.391405
\(818\) 0 0
\(819\) 1.45074 0.0506931
\(820\) 0 0
\(821\) −18.0503 −0.629961 −0.314981 0.949098i \(-0.601998\pi\)
−0.314981 + 0.949098i \(0.601998\pi\)
\(822\) 0 0
\(823\) −36.1501 −1.26011 −0.630057 0.776549i \(-0.716968\pi\)
−0.630057 + 0.776549i \(0.716968\pi\)
\(824\) 0 0
\(825\) −32.4335 −1.12919
\(826\) 0 0
\(827\) 38.5667 1.34110 0.670548 0.741867i \(-0.266059\pi\)
0.670548 + 0.741867i \(0.266059\pi\)
\(828\) 0 0
\(829\) 7.21747 0.250673 0.125337 0.992114i \(-0.459999\pi\)
0.125337 + 0.992114i \(0.459999\pi\)
\(830\) 0 0
\(831\) 5.08126 0.176267
\(832\) 0 0
\(833\) −5.02326 −0.174046
\(834\) 0 0
\(835\) 32.0940 1.11066
\(836\) 0 0
\(837\) 13.5512 0.468397
\(838\) 0 0
\(839\) −34.6198 −1.19521 −0.597604 0.801791i \(-0.703880\pi\)
−0.597604 + 0.801791i \(0.703880\pi\)
\(840\) 0 0
\(841\) −27.7099 −0.955512
\(842\) 0 0
\(843\) 13.9494 0.480444
\(844\) 0 0
\(845\) 20.8271 0.716472
\(846\) 0 0
\(847\) 27.7123 0.952205
\(848\) 0 0
\(849\) −66.0786 −2.26781
\(850\) 0 0
\(851\) 19.5299 0.669476
\(852\) 0 0
\(853\) 52.4072 1.79439 0.897194 0.441636i \(-0.145602\pi\)
0.897194 + 0.441636i \(0.145602\pi\)
\(854\) 0 0
\(855\) 8.01561 0.274128
\(856\) 0 0
\(857\) −18.9647 −0.647823 −0.323912 0.946087i \(-0.604998\pi\)
−0.323912 + 0.946087i \(0.604998\pi\)
\(858\) 0 0
\(859\) −25.3906 −0.866315 −0.433158 0.901318i \(-0.642601\pi\)
−0.433158 + 0.901318i \(0.642601\pi\)
\(860\) 0 0
\(861\) 11.3372 0.386370
\(862\) 0 0
\(863\) 11.5403 0.392837 0.196418 0.980520i \(-0.437069\pi\)
0.196418 + 0.980520i \(0.437069\pi\)
\(864\) 0 0
\(865\) 10.6678 0.362717
\(866\) 0 0
\(867\) −18.9323 −0.642976
\(868\) 0 0
\(869\) 57.2231 1.94116
\(870\) 0 0
\(871\) −6.70352 −0.227140
\(872\) 0 0
\(873\) −29.7806 −1.00792
\(874\) 0 0
\(875\) 12.0137 0.406138
\(876\) 0 0
\(877\) −19.8731 −0.671067 −0.335533 0.942028i \(-0.608917\pi\)
−0.335533 + 0.942028i \(0.608917\pi\)
\(878\) 0 0
\(879\) 72.3114 2.43900
\(880\) 0 0
\(881\) 47.1071 1.58708 0.793539 0.608519i \(-0.208236\pi\)
0.793539 + 0.608519i \(0.208236\pi\)
\(882\) 0 0
\(883\) 24.0255 0.808521 0.404261 0.914644i \(-0.367529\pi\)
0.404261 + 0.914644i \(0.367529\pi\)
\(884\) 0 0
\(885\) 5.73744 0.192862
\(886\) 0 0
\(887\) 33.3126 1.11853 0.559263 0.828990i \(-0.311084\pi\)
0.559263 + 0.828990i \(0.311084\pi\)
\(888\) 0 0
\(889\) 13.5320 0.453850
\(890\) 0 0
\(891\) 66.1343 2.21558
\(892\) 0 0
\(893\) −14.1710 −0.474215
\(894\) 0 0
\(895\) 39.3084 1.31394
\(896\) 0 0
\(897\) −12.9768 −0.433283
\(898\) 0 0
\(899\) 9.39941 0.313488
\(900\) 0 0
\(901\) 32.2111 1.07311
\(902\) 0 0
\(903\) −12.1396 −0.403979
\(904\) 0 0
\(905\) 14.6325 0.486401
\(906\) 0 0
\(907\) −8.00378 −0.265761 −0.132881 0.991132i \(-0.542423\pi\)
−0.132881 + 0.991132i \(0.542423\pi\)
\(908\) 0 0
\(909\) −12.1739 −0.403781
\(910\) 0 0
\(911\) 41.8870 1.38778 0.693889 0.720082i \(-0.255896\pi\)
0.693889 + 0.720082i \(0.255896\pi\)
\(912\) 0 0
\(913\) −23.0755 −0.763689
\(914\) 0 0
\(915\) 27.4916 0.908843
\(916\) 0 0
\(917\) 2.10195 0.0694126
\(918\) 0 0
\(919\) −14.4978 −0.478237 −0.239119 0.970990i \(-0.576858\pi\)
−0.239119 + 0.970990i \(0.576858\pi\)
\(920\) 0 0
\(921\) 57.7743 1.90373
\(922\) 0 0
\(923\) −2.58303 −0.0850216
\(924\) 0 0
\(925\) −4.97461 −0.163564
\(926\) 0 0
\(927\) 18.1174 0.595053
\(928\) 0 0
\(929\) 21.5646 0.707513 0.353756 0.935338i \(-0.384904\pi\)
0.353756 + 0.935338i \(0.384904\pi\)
\(930\) 0 0
\(931\) −2.11921 −0.0694544
\(932\) 0 0
\(933\) −71.9061 −2.35410
\(934\) 0 0
\(935\) −51.6700 −1.68979
\(936\) 0 0
\(937\) −3.51782 −0.114922 −0.0574611 0.998348i \(-0.518301\pi\)
−0.0574611 + 0.998348i \(0.518301\pi\)
\(938\) 0 0
\(939\) −41.4136 −1.35148
\(940\) 0 0
\(941\) 1.08682 0.0354294 0.0177147 0.999843i \(-0.494361\pi\)
0.0177147 + 0.999843i \(0.494361\pi\)
\(942\) 0 0
\(943\) −43.8767 −1.42882
\(944\) 0 0
\(945\) −2.70723 −0.0880661
\(946\) 0 0
\(947\) −16.0164 −0.520462 −0.260231 0.965546i \(-0.583799\pi\)
−0.260231 + 0.965546i \(0.583799\pi\)
\(948\) 0 0
\(949\) 8.56436 0.278011
\(950\) 0 0
\(951\) 39.8107 1.29095
\(952\) 0 0
\(953\) −9.82914 −0.318397 −0.159199 0.987247i \(-0.550891\pi\)
−0.159199 + 0.987247i \(0.550891\pi\)
\(954\) 0 0
\(955\) 26.1575 0.846438
\(956\) 0 0
\(957\) 16.2512 0.525326
\(958\) 0 0
\(959\) 11.7463 0.379307
\(960\) 0 0
\(961\) 37.4796 1.20902
\(962\) 0 0
\(963\) 43.9907 1.41758
\(964\) 0 0
\(965\) 12.4095 0.399476
\(966\) 0 0
\(967\) −5.71372 −0.183741 −0.0918705 0.995771i \(-0.529285\pi\)
−0.0918705 + 0.995771i \(0.529285\pi\)
\(968\) 0 0
\(969\) −24.4794 −0.786392
\(970\) 0 0
\(971\) −18.8818 −0.605946 −0.302973 0.952999i \(-0.597979\pi\)
−0.302973 + 0.952999i \(0.597979\pi\)
\(972\) 0 0
\(973\) 11.3252 0.363069
\(974\) 0 0
\(975\) 3.30543 0.105858
\(976\) 0 0
\(977\) −25.3043 −0.809556 −0.404778 0.914415i \(-0.632651\pi\)
−0.404778 + 0.914415i \(0.632651\pi\)
\(978\) 0 0
\(979\) −9.95755 −0.318245
\(980\) 0 0
\(981\) −22.7578 −0.726600
\(982\) 0 0
\(983\) −12.4206 −0.396157 −0.198078 0.980186i \(-0.563470\pi\)
−0.198078 + 0.980186i \(0.563470\pi\)
\(984\) 0 0
\(985\) −40.2883 −1.28369
\(986\) 0 0
\(987\) −15.3768 −0.489450
\(988\) 0 0
\(989\) 46.9820 1.49394
\(990\) 0 0
\(991\) 30.4013 0.965729 0.482865 0.875695i \(-0.339596\pi\)
0.482865 + 0.875695i \(0.339596\pi\)
\(992\) 0 0
\(993\) 38.1973 1.21215
\(994\) 0 0
\(995\) −36.1480 −1.14597
\(996\) 0 0
\(997\) −12.0040 −0.380171 −0.190085 0.981768i \(-0.560877\pi\)
−0.190085 + 0.981768i \(0.560877\pi\)
\(998\) 0 0
\(999\) 3.59357 0.113695
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 7168.2.a.bb.1.2 8
4.3 odd 2 7168.2.a.ba.1.7 8
8.3 odd 2 7168.2.a.be.1.2 8
8.5 even 2 7168.2.a.bf.1.7 8
32.3 odd 8 1792.2.m.g.1345.6 yes 16
32.5 even 8 1792.2.m.h.449.6 yes 16
32.11 odd 8 1792.2.m.g.449.6 yes 16
32.13 even 8 1792.2.m.h.1345.6 yes 16
32.19 odd 8 1792.2.m.f.1345.3 yes 16
32.21 even 8 1792.2.m.e.449.3 16
32.27 odd 8 1792.2.m.f.449.3 yes 16
32.29 even 8 1792.2.m.e.1345.3 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.e.449.3 16 32.21 even 8
1792.2.m.e.1345.3 yes 16 32.29 even 8
1792.2.m.f.449.3 yes 16 32.27 odd 8
1792.2.m.f.1345.3 yes 16 32.19 odd 8
1792.2.m.g.449.6 yes 16 32.11 odd 8
1792.2.m.g.1345.6 yes 16 32.3 odd 8
1792.2.m.h.449.6 yes 16 32.5 even 8
1792.2.m.h.1345.6 yes 16 32.13 even 8
7168.2.a.ba.1.7 8 4.3 odd 2
7168.2.a.bb.1.2 8 1.1 even 1 trivial
7168.2.a.be.1.2 8 8.3 odd 2
7168.2.a.bf.1.7 8 8.5 even 2