Properties

Label 7728.2.a.bq.1.3
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.733.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.17819\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.61186 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.61186 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.43366 q^{11} +0.821806 q^{13} -2.61186 q^{15} -5.79005 q^{17} +7.43366 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.82181 q^{25} -1.00000 q^{27} -5.79005 q^{29} +3.07728 q^{31} -3.43366 q^{33} +2.61186 q^{35} -0.922724 q^{37} -0.821806 q^{39} +0.566335 q^{41} +11.6891 q^{43} +2.61186 q^{45} +2.35639 q^{47} +1.00000 q^{49} +5.79005 q^{51} +0.968247 q^{53} +8.96825 q^{55} -7.43366 q^{57} -1.89908 q^{59} +2.40191 q^{61} +1.00000 q^{63} +2.14644 q^{65} +4.10092 q^{67} +1.00000 q^{69} -1.17819 q^{71} +8.14644 q^{73} -1.82181 q^{75} +3.43366 q^{77} +2.56634 q^{79} +1.00000 q^{81} +6.30099 q^{83} -15.1228 q^{85} +5.79005 q^{87} -7.89097 q^{89} +0.821806 q^{91} -3.07728 q^{93} +19.4157 q^{95} -1.79005 q^{97} +3.43366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 3q^{5} + 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 3q^{5} + 3q^{7} + 3q^{9} + 2q^{11} + 5q^{13} + 3q^{15} - 4q^{17} + 14q^{19} - 3q^{21} - 3q^{23} + 8q^{25} - 3q^{27} - 4q^{29} + 6q^{31} - 2q^{33} - 3q^{35} - 6q^{37} - 5q^{39} + 10q^{41} + 21q^{43} - 3q^{45} + 2q^{47} + 3q^{49} + 4q^{51} - 13q^{53} + 11q^{55} - 14q^{57} - 5q^{59} - 17q^{61} + 3q^{63} - 12q^{65} + 13q^{67} + 3q^{69} - q^{71} + 6q^{73} - 8q^{75} + 2q^{77} + 16q^{79} + 3q^{81} - 6q^{83} - 23q^{85} + 4q^{87} - 11q^{89} + 5q^{91} - 6q^{93} - q^{95} + 8q^{97} + 2q^{99} + O(q^{100}) \)

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
\(5\) 2.61186 1.16806 0.584029 0.811733i \(-0.301475\pi\)
0.584029 + 0.811733i \(0.301475\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.43366 1.03529 0.517644 0.855596i \(-0.326809\pi\)
0.517644 + 0.855596i \(0.326809\pi\)
\(12\) 0 0
\(13\) 0.821806 0.227928 0.113964 0.993485i \(-0.463645\pi\)
0.113964 + 0.993485i \(0.463645\pi\)
\(14\) 0 0
\(15\) −2.61186 −0.674379
\(16\) 0 0
\(17\) −5.79005 −1.40429 −0.702147 0.712032i \(-0.747775\pi\)
−0.702147 + 0.712032i \(0.747775\pi\)
\(18\) 0 0
\(19\) 7.43366 1.70540 0.852700 0.522401i \(-0.174964\pi\)
0.852700 + 0.522401i \(0.174964\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.82181 0.364361
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.79005 −1.07519 −0.537593 0.843205i \(-0.680666\pi\)
−0.537593 + 0.843205i \(0.680666\pi\)
\(30\) 0 0
\(31\) 3.07728 0.552695 0.276348 0.961058i \(-0.410876\pi\)
0.276348 + 0.961058i \(0.410876\pi\)
\(32\) 0 0
\(33\) −3.43366 −0.597724
\(34\) 0 0
\(35\) 2.61186 0.441485
\(36\) 0 0
\(37\) −0.922724 −0.151695 −0.0758474 0.997119i \(-0.524166\pi\)
−0.0758474 + 0.997119i \(0.524166\pi\)
\(38\) 0 0
\(39\) −0.821806 −0.131594
\(40\) 0 0
\(41\) 0.566335 0.0884467 0.0442234 0.999022i \(-0.485919\pi\)
0.0442234 + 0.999022i \(0.485919\pi\)
\(42\) 0 0
\(43\) 11.6891 1.78258 0.891288 0.453437i \(-0.149802\pi\)
0.891288 + 0.453437i \(0.149802\pi\)
\(44\) 0 0
\(45\) 2.61186 0.389353
\(46\) 0 0
\(47\) 2.35639 0.343715 0.171857 0.985122i \(-0.445023\pi\)
0.171857 + 0.985122i \(0.445023\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 5.79005 0.810770
\(52\) 0 0
\(53\) 0.968247 0.132999 0.0664995 0.997786i \(-0.478817\pi\)
0.0664995 + 0.997786i \(0.478817\pi\)
\(54\) 0 0
\(55\) 8.96825 1.20928
\(56\) 0 0
\(57\) −7.43366 −0.984613
\(58\) 0 0
\(59\) −1.89908 −0.247239 −0.123620 0.992330i \(-0.539450\pi\)
−0.123620 + 0.992330i \(0.539450\pi\)
\(60\) 0 0
\(61\) 2.40191 0.307533 0.153767 0.988107i \(-0.450860\pi\)
0.153767 + 0.988107i \(0.450860\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 2.14644 0.266233
\(66\) 0 0
\(67\) 4.10092 0.501007 0.250503 0.968116i \(-0.419404\pi\)
0.250503 + 0.968116i \(0.419404\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −1.17819 −0.139826 −0.0699130 0.997553i \(-0.522272\pi\)
−0.0699130 + 0.997553i \(0.522272\pi\)
\(72\) 0 0
\(73\) 8.14644 0.953469 0.476734 0.879047i \(-0.341820\pi\)
0.476734 + 0.879047i \(0.341820\pi\)
\(74\) 0 0
\(75\) −1.82181 −0.210364
\(76\) 0 0
\(77\) 3.43366 0.391302
\(78\) 0 0
\(79\) 2.56634 0.288735 0.144368 0.989524i \(-0.453885\pi\)
0.144368 + 0.989524i \(0.453885\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.30099 0.691624 0.345812 0.938304i \(-0.387604\pi\)
0.345812 + 0.938304i \(0.387604\pi\)
\(84\) 0 0
\(85\) −15.1228 −1.64030
\(86\) 0 0
\(87\) 5.79005 0.620759
\(88\) 0 0
\(89\) −7.89097 −0.836441 −0.418221 0.908345i \(-0.637346\pi\)
−0.418221 + 0.908345i \(0.637346\pi\)
\(90\) 0 0
\(91\) 0.821806 0.0861487
\(92\) 0 0
\(93\) −3.07728 −0.319099
\(94\) 0 0
\(95\) 19.4157 1.99201
\(96\) 0 0
\(97\) −1.79005 −0.181752 −0.0908762 0.995862i \(-0.528967\pi\)
−0.0908762 + 0.995862i \(0.528967\pi\)
\(98\) 0 0
\(99\) 3.43366 0.345096
\(100\) 0 0
\(101\) 9.12280 0.907753 0.453876 0.891065i \(-0.350041\pi\)
0.453876 + 0.891065i \(0.350041\pi\)
\(102\) 0 0
\(103\) −19.1602 −1.88791 −0.943956 0.330072i \(-0.892927\pi\)
−0.943956 + 0.330072i \(0.892927\pi\)
\(104\) 0 0
\(105\) −2.61186 −0.254891
\(106\) 0 0
\(107\) 14.5565 1.40723 0.703613 0.710583i \(-0.251569\pi\)
0.703613 + 0.710583i \(0.251569\pi\)
\(108\) 0 0
\(109\) 17.0593 1.63398 0.816992 0.576649i \(-0.195640\pi\)
0.816992 + 0.576649i \(0.195640\pi\)
\(110\) 0 0
\(111\) 0.922724 0.0875810
\(112\) 0 0
\(113\) 7.47919 0.703583 0.351791 0.936078i \(-0.385573\pi\)
0.351791 + 0.936078i \(0.385573\pi\)
\(114\) 0 0
\(115\) −2.61186 −0.243557
\(116\) 0 0
\(117\) 0.821806 0.0759760
\(118\) 0 0
\(119\) −5.79005 −0.530773
\(120\) 0 0
\(121\) 0.790053 0.0718230
\(122\) 0 0
\(123\) −0.566335 −0.0510647
\(124\) 0 0
\(125\) −8.30099 −0.742463
\(126\) 0 0
\(127\) −14.4930 −1.28604 −0.643021 0.765849i \(-0.722319\pi\)
−0.643021 + 0.765849i \(0.722319\pi\)
\(128\) 0 0
\(129\) −11.6891 −1.02917
\(130\) 0 0
\(131\) −17.7265 −1.54878 −0.774388 0.632711i \(-0.781942\pi\)
−0.774388 + 0.632711i \(0.781942\pi\)
\(132\) 0 0
\(133\) 7.43366 0.644580
\(134\) 0 0
\(135\) −2.61186 −0.224793
\(136\) 0 0
\(137\) 10.3010 0.880073 0.440037 0.897980i \(-0.354965\pi\)
0.440037 + 0.897980i \(0.354965\pi\)
\(138\) 0 0
\(139\) 19.8356 1.68243 0.841216 0.540700i \(-0.181840\pi\)
0.841216 + 0.540700i \(0.181840\pi\)
\(140\) 0 0
\(141\) −2.35639 −0.198444
\(142\) 0 0
\(143\) 2.82181 0.235971
\(144\) 0 0
\(145\) −15.1228 −1.25588
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 0.0910467 0.00745883 0.00372942 0.999993i \(-0.498813\pi\)
0.00372942 + 0.999993i \(0.498813\pi\)
\(150\) 0 0
\(151\) −11.3783 −0.925951 −0.462975 0.886371i \(-0.653218\pi\)
−0.462975 + 0.886371i \(0.653218\pi\)
\(152\) 0 0
\(153\) −5.79005 −0.468098
\(154\) 0 0
\(155\) 8.03741 0.645580
\(156\) 0 0
\(157\) 5.22372 0.416898 0.208449 0.978033i \(-0.433158\pi\)
0.208449 + 0.978033i \(0.433158\pi\)
\(158\) 0 0
\(159\) −0.968247 −0.0767870
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 6.10092 0.477861 0.238930 0.971037i \(-0.423203\pi\)
0.238930 + 0.971037i \(0.423203\pi\)
\(164\) 0 0
\(165\) −8.96825 −0.698177
\(166\) 0 0
\(167\) 8.30099 0.642350 0.321175 0.947020i \(-0.395922\pi\)
0.321175 + 0.947020i \(0.395922\pi\)
\(168\) 0 0
\(169\) −12.3246 −0.948049
\(170\) 0 0
\(171\) 7.43366 0.568467
\(172\) 0 0
\(173\) 8.14644 0.619362 0.309681 0.950840i \(-0.399778\pi\)
0.309681 + 0.950840i \(0.399778\pi\)
\(174\) 0 0
\(175\) 1.82181 0.137716
\(176\) 0 0
\(177\) 1.89908 0.142744
\(178\) 0 0
\(179\) −8.96825 −0.670318 −0.335159 0.942162i \(-0.608790\pi\)
−0.335159 + 0.942162i \(0.608790\pi\)
\(180\) 0 0
\(181\) −23.6793 −1.76007 −0.880033 0.474913i \(-0.842480\pi\)
−0.880033 + 0.474913i \(0.842480\pi\)
\(182\) 0 0
\(183\) −2.40191 −0.177554
\(184\) 0 0
\(185\) −2.41002 −0.177188
\(186\) 0 0
\(187\) −19.8811 −1.45385
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −4.09105 −0.296018 −0.148009 0.988986i \(-0.547286\pi\)
−0.148009 + 0.988986i \(0.547286\pi\)
\(192\) 0 0
\(193\) 12.3564 0.889432 0.444716 0.895672i \(-0.353305\pi\)
0.444716 + 0.895672i \(0.353305\pi\)
\(194\) 0 0
\(195\) −2.14644 −0.153710
\(196\) 0 0
\(197\) 2.24736 0.160118 0.0800588 0.996790i \(-0.474489\pi\)
0.0800588 + 0.996790i \(0.474489\pi\)
\(198\) 0 0
\(199\) 21.6810 1.53693 0.768463 0.639894i \(-0.221022\pi\)
0.768463 + 0.639894i \(0.221022\pi\)
\(200\) 0 0
\(201\) −4.10092 −0.289256
\(202\) 0 0
\(203\) −5.79005 −0.406382
\(204\) 0 0
\(205\) 1.47919 0.103311
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 25.5247 1.76558
\(210\) 0 0
\(211\) −26.0357 −1.79237 −0.896184 0.443682i \(-0.853672\pi\)
−0.896184 + 0.443682i \(0.853672\pi\)
\(212\) 0 0
\(213\) 1.17819 0.0807285
\(214\) 0 0
\(215\) 30.5304 2.08215
\(216\) 0 0
\(217\) 3.07728 0.208899
\(218\) 0 0
\(219\) −8.14644 −0.550485
\(220\) 0 0
\(221\) −4.75830 −0.320078
\(222\) 0 0
\(223\) 27.8356 1.86401 0.932004 0.362448i \(-0.118059\pi\)
0.932004 + 0.362448i \(0.118059\pi\)
\(224\) 0 0
\(225\) 1.82181 0.121454
\(226\) 0 0
\(227\) −1.32464 −0.0879191 −0.0439596 0.999033i \(-0.513997\pi\)
−0.0439596 + 0.999033i \(0.513997\pi\)
\(228\) 0 0
\(229\) 27.2138 1.79834 0.899171 0.437598i \(-0.144171\pi\)
0.899171 + 0.437598i \(0.144171\pi\)
\(230\) 0 0
\(231\) −3.43366 −0.225919
\(232\) 0 0
\(233\) −20.1920 −1.32282 −0.661410 0.750025i \(-0.730042\pi\)
−0.661410 + 0.750025i \(0.730042\pi\)
\(234\) 0 0
\(235\) 6.15455 0.401479
\(236\) 0 0
\(237\) −2.56634 −0.166701
\(238\) 0 0
\(239\) −21.7721 −1.40832 −0.704159 0.710042i \(-0.748676\pi\)
−0.704159 + 0.710042i \(0.748676\pi\)
\(240\) 0 0
\(241\) −12.6574 −0.815334 −0.407667 0.913131i \(-0.633658\pi\)
−0.407667 + 0.913131i \(0.633658\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.61186 0.166866
\(246\) 0 0
\(247\) 6.10903 0.388708
\(248\) 0 0
\(249\) −6.30099 −0.399309
\(250\) 0 0
\(251\) −8.66549 −0.546961 −0.273481 0.961878i \(-0.588175\pi\)
−0.273481 + 0.961878i \(0.588175\pi\)
\(252\) 0 0
\(253\) −3.43366 −0.215873
\(254\) 0 0
\(255\) 15.1228 0.947026
\(256\) 0 0
\(257\) 20.6493 1.28807 0.644033 0.764998i \(-0.277260\pi\)
0.644033 + 0.764998i \(0.277260\pi\)
\(258\) 0 0
\(259\) −0.922724 −0.0573353
\(260\) 0 0
\(261\) −5.79005 −0.358395
\(262\) 0 0
\(263\) 21.1048 1.30138 0.650689 0.759344i \(-0.274480\pi\)
0.650689 + 0.759344i \(0.274480\pi\)
\(264\) 0 0
\(265\) 2.52892 0.155351
\(266\) 0 0
\(267\) 7.89097 0.482920
\(268\) 0 0
\(269\) −18.8493 −1.14926 −0.574632 0.818412i \(-0.694855\pi\)
−0.574632 + 0.818412i \(0.694855\pi\)
\(270\) 0 0
\(271\) −5.01377 −0.304565 −0.152282 0.988337i \(-0.548662\pi\)
−0.152282 + 0.988337i \(0.548662\pi\)
\(272\) 0 0
\(273\) −0.821806 −0.0497380
\(274\) 0 0
\(275\) 6.25547 0.377219
\(276\) 0 0
\(277\) −7.17819 −0.431296 −0.215648 0.976471i \(-0.569186\pi\)
−0.215648 + 0.976471i \(0.569186\pi\)
\(278\) 0 0
\(279\) 3.07728 0.184232
\(280\) 0 0
\(281\) 1.19618 0.0713579 0.0356790 0.999363i \(-0.488641\pi\)
0.0356790 + 0.999363i \(0.488641\pi\)
\(282\) 0 0
\(283\) 6.96825 0.414219 0.207110 0.978318i \(-0.433594\pi\)
0.207110 + 0.978318i \(0.433594\pi\)
\(284\) 0 0
\(285\) −19.4157 −1.15009
\(286\) 0 0
\(287\) 0.566335 0.0334297
\(288\) 0 0
\(289\) 16.5247 0.972042
\(290\) 0 0
\(291\) 1.79005 0.104935
\(292\) 0 0
\(293\) 22.4474 1.31139 0.655697 0.755025i \(-0.272375\pi\)
0.655697 + 0.755025i \(0.272375\pi\)
\(294\) 0 0
\(295\) −4.96013 −0.288790
\(296\) 0 0
\(297\) −3.43366 −0.199241
\(298\) 0 0
\(299\) −0.821806 −0.0475263
\(300\) 0 0
\(301\) 11.6891 0.673751
\(302\) 0 0
\(303\) −9.12280 −0.524091
\(304\) 0 0
\(305\) 6.27345 0.359217
\(306\) 0 0
\(307\) 22.6574 1.29313 0.646563 0.762861i \(-0.276206\pi\)
0.646563 + 0.762861i \(0.276206\pi\)
\(308\) 0 0
\(309\) 19.1602 1.08999
\(310\) 0 0
\(311\) −23.4238 −1.32824 −0.664121 0.747625i \(-0.731194\pi\)
−0.664121 + 0.747625i \(0.731194\pi\)
\(312\) 0 0
\(313\) −17.7622 −1.00398 −0.501989 0.864874i \(-0.667398\pi\)
−0.501989 + 0.864874i \(0.667398\pi\)
\(314\) 0 0
\(315\) 2.61186 0.147162
\(316\) 0 0
\(317\) 8.19196 0.460107 0.230053 0.973178i \(-0.426110\pi\)
0.230053 + 0.973178i \(0.426110\pi\)
\(318\) 0 0
\(319\) −19.8811 −1.11313
\(320\) 0 0
\(321\) −14.5565 −0.812463
\(322\) 0 0
\(323\) −43.0413 −2.39488
\(324\) 0 0
\(325\) 1.49717 0.0830481
\(326\) 0 0
\(327\) −17.0593 −0.943381
\(328\) 0 0
\(329\) 2.35639 0.129912
\(330\) 0 0
\(331\) −1.13267 −0.0622572 −0.0311286 0.999515i \(-0.509910\pi\)
−0.0311286 + 0.999515i \(0.509910\pi\)
\(332\) 0 0
\(333\) −0.922724 −0.0505649
\(334\) 0 0
\(335\) 10.7110 0.585205
\(336\) 0 0
\(337\) −26.6475 −1.45158 −0.725791 0.687915i \(-0.758526\pi\)
−0.725791 + 0.687915i \(0.758526\pi\)
\(338\) 0 0
\(339\) −7.47919 −0.406214
\(340\) 0 0
\(341\) 10.5663 0.572199
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 2.61186 0.140618
\(346\) 0 0
\(347\) 24.9030 1.33686 0.668431 0.743774i \(-0.266966\pi\)
0.668431 + 0.743774i \(0.266966\pi\)
\(348\) 0 0
\(349\) −21.4157 −1.14636 −0.573178 0.819431i \(-0.694289\pi\)
−0.573178 + 0.819431i \(0.694289\pi\)
\(350\) 0 0
\(351\) −0.821806 −0.0438648
\(352\) 0 0
\(353\) 17.2791 0.919674 0.459837 0.888003i \(-0.347908\pi\)
0.459837 + 0.888003i \(0.347908\pi\)
\(354\) 0 0
\(355\) −3.07728 −0.163325
\(356\) 0 0
\(357\) 5.79005 0.306442
\(358\) 0 0
\(359\) 25.8631 1.36500 0.682502 0.730884i \(-0.260892\pi\)
0.682502 + 0.730884i \(0.260892\pi\)
\(360\) 0 0
\(361\) 36.2594 1.90839
\(362\) 0 0
\(363\) −0.790053 −0.0414670
\(364\) 0 0
\(365\) 21.2774 1.11371
\(366\) 0 0
\(367\) 30.4930 1.59172 0.795860 0.605481i \(-0.207019\pi\)
0.795860 + 0.605481i \(0.207019\pi\)
\(368\) 0 0
\(369\) 0.566335 0.0294822
\(370\) 0 0
\(371\) 0.968247 0.0502689
\(372\) 0 0
\(373\) −13.6355 −0.706019 −0.353010 0.935620i \(-0.614842\pi\)
−0.353010 + 0.935620i \(0.614842\pi\)
\(374\) 0 0
\(375\) 8.30099 0.428661
\(376\) 0 0
\(377\) −4.75830 −0.245065
\(378\) 0 0
\(379\) 2.29288 0.117777 0.0588887 0.998265i \(-0.481244\pi\)
0.0588887 + 0.998265i \(0.481244\pi\)
\(380\) 0 0
\(381\) 14.4930 0.742497
\(382\) 0 0
\(383\) 15.2791 0.780726 0.390363 0.920661i \(-0.372349\pi\)
0.390363 + 0.920661i \(0.372349\pi\)
\(384\) 0 0
\(385\) 8.96825 0.457064
\(386\) 0 0
\(387\) 11.6891 0.594192
\(388\) 0 0
\(389\) −13.2318 −0.670880 −0.335440 0.942062i \(-0.608885\pi\)
−0.335440 + 0.942062i \(0.608885\pi\)
\(390\) 0 0
\(391\) 5.79005 0.292816
\(392\) 0 0
\(393\) 17.7265 0.894186
\(394\) 0 0
\(395\) 6.70291 0.337260
\(396\) 0 0
\(397\) −32.1186 −1.61199 −0.805993 0.591925i \(-0.798368\pi\)
−0.805993 + 0.591925i \(0.798368\pi\)
\(398\) 0 0
\(399\) −7.43366 −0.372149
\(400\) 0 0
\(401\) −6.20995 −0.310110 −0.155055 0.987906i \(-0.549555\pi\)
−0.155055 + 0.987906i \(0.549555\pi\)
\(402\) 0 0
\(403\) 2.52892 0.125975
\(404\) 0 0
\(405\) 2.61186 0.129784
\(406\) 0 0
\(407\) −3.16832 −0.157048
\(408\) 0 0
\(409\) −25.2594 −1.24900 −0.624498 0.781027i \(-0.714696\pi\)
−0.624498 + 0.781027i \(0.714696\pi\)
\(410\) 0 0
\(411\) −10.3010 −0.508111
\(412\) 0 0
\(413\) −1.89908 −0.0934477
\(414\) 0 0
\(415\) 16.4573 0.807857
\(416\) 0 0
\(417\) −19.8356 −0.971352
\(418\) 0 0
\(419\) −24.2830 −1.18630 −0.593151 0.805091i \(-0.702116\pi\)
−0.593151 + 0.805091i \(0.702116\pi\)
\(420\) 0 0
\(421\) 0.465418 0.0226831 0.0113415 0.999936i \(-0.496390\pi\)
0.0113415 + 0.999936i \(0.496390\pi\)
\(422\) 0 0
\(423\) 2.35639 0.114572
\(424\) 0 0
\(425\) −10.5484 −0.511670
\(426\) 0 0
\(427\) 2.40191 0.116237
\(428\) 0 0
\(429\) −2.82181 −0.136238
\(430\) 0 0
\(431\) 5.78018 0.278422 0.139211 0.990263i \(-0.455543\pi\)
0.139211 + 0.990263i \(0.455543\pi\)
\(432\) 0 0
\(433\) 9.19618 0.441940 0.220970 0.975281i \(-0.429078\pi\)
0.220970 + 0.975281i \(0.429078\pi\)
\(434\) 0 0
\(435\) 15.1228 0.725083
\(436\) 0 0
\(437\) −7.43366 −0.355600
\(438\) 0 0
\(439\) 15.7901 0.753618 0.376809 0.926291i \(-0.377021\pi\)
0.376809 + 0.926291i \(0.377021\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −32.1821 −1.52902 −0.764509 0.644613i \(-0.777018\pi\)
−0.764509 + 0.644613i \(0.777018\pi\)
\(444\) 0 0
\(445\) −20.6101 −0.977012
\(446\) 0 0
\(447\) −0.0910467 −0.00430636
\(448\) 0 0
\(449\) −5.41568 −0.255582 −0.127791 0.991801i \(-0.540789\pi\)
−0.127791 + 0.991801i \(0.540789\pi\)
\(450\) 0 0
\(451\) 1.94461 0.0915679
\(452\) 0 0
\(453\) 11.3783 0.534598
\(454\) 0 0
\(455\) 2.14644 0.100627
\(456\) 0 0
\(457\) 26.8412 1.25558 0.627790 0.778383i \(-0.283960\pi\)
0.627790 + 0.778383i \(0.283960\pi\)
\(458\) 0 0
\(459\) 5.79005 0.270257
\(460\) 0 0
\(461\) 0.410023 0.0190967 0.00954835 0.999954i \(-0.496961\pi\)
0.00954835 + 0.999954i \(0.496961\pi\)
\(462\) 0 0
\(463\) 36.6849 1.70489 0.852446 0.522815i \(-0.175118\pi\)
0.852446 + 0.522815i \(0.175118\pi\)
\(464\) 0 0
\(465\) −8.03741 −0.372726
\(466\) 0 0
\(467\) −11.7265 −0.542640 −0.271320 0.962489i \(-0.587460\pi\)
−0.271320 + 0.962489i \(0.587460\pi\)
\(468\) 0 0
\(469\) 4.10092 0.189363
\(470\) 0 0
\(471\) −5.22372 −0.240696
\(472\) 0 0
\(473\) 40.1366 1.84548
\(474\) 0 0
\(475\) 13.5427 0.621381
\(476\) 0 0
\(477\) 0.968247 0.0443330
\(478\) 0 0
\(479\) 20.0829 0.917613 0.458806 0.888536i \(-0.348277\pi\)
0.458806 + 0.888536i \(0.348277\pi\)
\(480\) 0 0
\(481\) −0.758300 −0.0345755
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −4.67536 −0.212297
\(486\) 0 0
\(487\) 4.51905 0.204778 0.102389 0.994744i \(-0.467351\pi\)
0.102389 + 0.994744i \(0.467351\pi\)
\(488\) 0 0
\(489\) −6.10092 −0.275893
\(490\) 0 0
\(491\) 8.26358 0.372930 0.186465 0.982462i \(-0.440297\pi\)
0.186465 + 0.982462i \(0.440297\pi\)
\(492\) 0 0
\(493\) 33.5247 1.50988
\(494\) 0 0
\(495\) 8.96825 0.403093
\(496\) 0 0
\(497\) −1.17819 −0.0528492
\(498\) 0 0
\(499\) 32.9682 1.47586 0.737931 0.674876i \(-0.235803\pi\)
0.737931 + 0.674876i \(0.235803\pi\)
\(500\) 0 0
\(501\) −8.30099 −0.370861
\(502\) 0 0
\(503\) 18.7029 0.833921 0.416961 0.908925i \(-0.363095\pi\)
0.416961 + 0.908925i \(0.363095\pi\)
\(504\) 0 0
\(505\) 23.8275 1.06031
\(506\) 0 0
\(507\) 12.3246 0.547356
\(508\) 0 0
\(509\) −18.5109 −0.820483 −0.410242 0.911977i \(-0.634556\pi\)
−0.410242 + 0.911977i \(0.634556\pi\)
\(510\) 0 0
\(511\) 8.14644 0.360377
\(512\) 0 0
\(513\) −7.43366 −0.328204
\(514\) 0 0
\(515\) −50.0438 −2.20519
\(516\) 0 0
\(517\) 8.09105 0.355844
\(518\) 0 0
\(519\) −8.14644 −0.357589
\(520\) 0 0
\(521\) 35.0219 1.53434 0.767168 0.641446i \(-0.221665\pi\)
0.767168 + 0.641446i \(0.221665\pi\)
\(522\) 0 0
\(523\) 40.3204 1.76309 0.881544 0.472101i \(-0.156504\pi\)
0.881544 + 0.472101i \(0.156504\pi\)
\(524\) 0 0
\(525\) −1.82181 −0.0795101
\(526\) 0 0
\(527\) −17.8176 −0.776147
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −1.89908 −0.0824132
\(532\) 0 0
\(533\) 0.465418 0.0201595
\(534\) 0 0
\(535\) 38.0194 1.64372
\(536\) 0 0
\(537\) 8.96825 0.387008
\(538\) 0 0
\(539\) 3.43366 0.147898
\(540\) 0 0
\(541\) −28.2456 −1.21437 −0.607187 0.794559i \(-0.707702\pi\)
−0.607187 + 0.794559i \(0.707702\pi\)
\(542\) 0 0
\(543\) 23.6793 1.01617
\(544\) 0 0
\(545\) 44.5565 1.90859
\(546\) 0 0
\(547\) 25.6810 1.09804 0.549021 0.835809i \(-0.315001\pi\)
0.549021 + 0.835809i \(0.315001\pi\)
\(548\) 0 0
\(549\) 2.40191 0.102511
\(550\) 0 0
\(551\) −43.0413 −1.83362
\(552\) 0 0
\(553\) 2.56634 0.109132
\(554\) 0 0
\(555\) 2.41002 0.102300
\(556\) 0 0
\(557\) 0.201835 0.00855203 0.00427602 0.999991i \(-0.498639\pi\)
0.00427602 + 0.999991i \(0.498639\pi\)
\(558\) 0 0
\(559\) 9.60620 0.406299
\(560\) 0 0
\(561\) 19.8811 0.839381
\(562\) 0 0
\(563\) −47.4513 −1.99984 −0.999918 0.0128333i \(-0.995915\pi\)
−0.999918 + 0.0128333i \(0.995915\pi\)
\(564\) 0 0
\(565\) 19.5346 0.821826
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −14.4199 −0.604513 −0.302257 0.953227i \(-0.597740\pi\)
−0.302257 + 0.953227i \(0.597740\pi\)
\(570\) 0 0
\(571\) 28.4393 1.19015 0.595074 0.803671i \(-0.297123\pi\)
0.595074 + 0.803671i \(0.297123\pi\)
\(572\) 0 0
\(573\) 4.09105 0.170906
\(574\) 0 0
\(575\) −1.82181 −0.0759746
\(576\) 0 0
\(577\) 28.7565 1.19715 0.598575 0.801067i \(-0.295734\pi\)
0.598575 + 0.801067i \(0.295734\pi\)
\(578\) 0 0
\(579\) −12.3564 −0.513514
\(580\) 0 0
\(581\) 6.30099 0.261409
\(582\) 0 0
\(583\) 3.32464 0.137692
\(584\) 0 0
\(585\) 2.14644 0.0887444
\(586\) 0 0
\(587\) −21.5783 −0.890634 −0.445317 0.895373i \(-0.646909\pi\)
−0.445317 + 0.895373i \(0.646909\pi\)
\(588\) 0 0
\(589\) 22.8754 0.942566
\(590\) 0 0
\(591\) −2.24736 −0.0924440
\(592\) 0 0
\(593\) 11.6436 0.478146 0.239073 0.971002i \(-0.423157\pi\)
0.239073 + 0.971002i \(0.423157\pi\)
\(594\) 0 0
\(595\) −15.1228 −0.619974
\(596\) 0 0
\(597\) −21.6810 −0.887345
\(598\) 0 0
\(599\) −12.0455 −0.492167 −0.246083 0.969249i \(-0.579144\pi\)
−0.246083 + 0.969249i \(0.579144\pi\)
\(600\) 0 0
\(601\) 24.8137 1.01217 0.506086 0.862483i \(-0.331092\pi\)
0.506086 + 0.862483i \(0.331092\pi\)
\(602\) 0 0
\(603\) 4.10092 0.167002
\(604\) 0 0
\(605\) 2.06351 0.0838935
\(606\) 0 0
\(607\) −29.0512 −1.17915 −0.589576 0.807713i \(-0.700705\pi\)
−0.589576 + 0.807713i \(0.700705\pi\)
\(608\) 0 0
\(609\) 5.79005 0.234625
\(610\) 0 0
\(611\) 1.93649 0.0783422
\(612\) 0 0
\(613\) 0.300994 0.0121570 0.00607851 0.999982i \(-0.498065\pi\)
0.00607851 + 0.999982i \(0.498065\pi\)
\(614\) 0 0
\(615\) −1.47919 −0.0596466
\(616\) 0 0
\(617\) 20.8218 0.838254 0.419127 0.907928i \(-0.362336\pi\)
0.419127 + 0.907928i \(0.362336\pi\)
\(618\) 0 0
\(619\) −25.4076 −1.02122 −0.510608 0.859813i \(-0.670580\pi\)
−0.510608 + 0.859813i \(0.670580\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −7.89097 −0.316145
\(624\) 0 0
\(625\) −30.7901 −1.23160
\(626\) 0 0
\(627\) −25.5247 −1.01936
\(628\) 0 0
\(629\) 5.34262 0.213024
\(630\) 0 0
\(631\) 12.8119 0.510035 0.255018 0.966936i \(-0.417919\pi\)
0.255018 + 0.966936i \(0.417919\pi\)
\(632\) 0 0
\(633\) 26.0357 1.03482
\(634\) 0 0
\(635\) −37.8536 −1.50217
\(636\) 0 0
\(637\) 0.821806 0.0325611
\(638\) 0 0
\(639\) −1.17819 −0.0466086
\(640\) 0 0
\(641\) −3.06741 −0.121155 −0.0605776 0.998163i \(-0.519294\pi\)
−0.0605776 + 0.998163i \(0.519294\pi\)
\(642\) 0 0
\(643\) 6.15631 0.242781 0.121391 0.992605i \(-0.461265\pi\)
0.121391 + 0.992605i \(0.461265\pi\)
\(644\) 0 0
\(645\) −30.5304 −1.20213
\(646\) 0 0
\(647\) 12.4376 0.488971 0.244486 0.969653i \(-0.421381\pi\)
0.244486 + 0.969653i \(0.421381\pi\)
\(648\) 0 0
\(649\) −6.52081 −0.255964
\(650\) 0 0
\(651\) −3.07728 −0.120608
\(652\) 0 0
\(653\) −10.0536 −0.393429 −0.196715 0.980461i \(-0.563027\pi\)
−0.196715 + 0.980461i \(0.563027\pi\)
\(654\) 0 0
\(655\) −46.2992 −1.80906
\(656\) 0 0
\(657\) 8.14644 0.317823
\(658\) 0 0
\(659\) 19.1246 0.744987 0.372494 0.928035i \(-0.378503\pi\)
0.372494 + 0.928035i \(0.378503\pi\)
\(660\) 0 0
\(661\) 41.7068 1.62221 0.811103 0.584903i \(-0.198867\pi\)
0.811103 + 0.584903i \(0.198867\pi\)
\(662\) 0 0
\(663\) 4.75830 0.184797
\(664\) 0 0
\(665\) 19.4157 0.752908
\(666\) 0 0
\(667\) 5.79005 0.224192
\(668\) 0 0
\(669\) −27.8356 −1.07619
\(670\) 0 0
\(671\) 8.24736 0.318386
\(672\) 0 0
\(673\) −15.5247 −0.598434 −0.299217 0.954185i \(-0.596725\pi\)
−0.299217 + 0.954185i \(0.596725\pi\)
\(674\) 0 0
\(675\) −1.82181 −0.0701213
\(676\) 0 0
\(677\) −4.24736 −0.163239 −0.0816196 0.996664i \(-0.526009\pi\)
−0.0816196 + 0.996664i \(0.526009\pi\)
\(678\) 0 0
\(679\) −1.79005 −0.0686959
\(680\) 0 0
\(681\) 1.32464 0.0507601
\(682\) 0 0
\(683\) −35.8811 −1.37295 −0.686476 0.727152i \(-0.740843\pi\)
−0.686476 + 0.727152i \(0.740843\pi\)
\(684\) 0 0
\(685\) 26.9047 1.02798
\(686\) 0 0
\(687\) −27.2138 −1.03827
\(688\) 0 0
\(689\) 0.795711 0.0303142
\(690\) 0 0
\(691\) −4.96013 −0.188692 −0.0943462 0.995539i \(-0.530076\pi\)
−0.0943462 + 0.995539i \(0.530076\pi\)
\(692\) 0 0
\(693\) 3.43366 0.130434
\(694\) 0 0
\(695\) 51.8077 1.96518
\(696\) 0 0
\(697\) −3.27911 −0.124205
\(698\) 0 0
\(699\) 20.1920 0.763730
\(700\) 0 0
\(701\) −24.2555 −0.916116 −0.458058 0.888922i \(-0.651455\pi\)
−0.458058 + 0.888922i \(0.651455\pi\)
\(702\) 0 0
\(703\) −6.85922 −0.258700
\(704\) 0 0
\(705\) −6.15455 −0.231794
\(706\) 0 0
\(707\) 9.12280 0.343098
\(708\) 0 0
\(709\) 2.21982 0.0833670 0.0416835 0.999131i \(-0.486728\pi\)
0.0416835 + 0.999131i \(0.486728\pi\)
\(710\) 0 0
\(711\) 2.56634 0.0962451
\(712\) 0 0
\(713\) −3.07728 −0.115245
\(714\) 0 0
\(715\) 7.37016 0.275628
\(716\) 0 0
\(717\) 21.7721 0.813093
\(718\) 0 0
\(719\) 1.23183 0.0459395 0.0229697 0.999736i \(-0.492688\pi\)
0.0229697 + 0.999736i \(0.492688\pi\)
\(720\) 0 0
\(721\) −19.1602 −0.713564
\(722\) 0 0
\(723\) 12.6574 0.470733
\(724\) 0 0
\(725\) −10.5484 −0.391756
\(726\) 0 0
\(727\) 7.88110 0.292294 0.146147 0.989263i \(-0.453313\pi\)
0.146147 + 0.989263i \(0.453313\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −67.6807 −2.50326
\(732\) 0 0
\(733\) 6.64927 0.245596 0.122798 0.992432i \(-0.460813\pi\)
0.122798 + 0.992432i \(0.460813\pi\)
\(734\) 0 0
\(735\) −2.61186 −0.0963399
\(736\) 0 0
\(737\) 14.0812 0.518687
\(738\) 0 0
\(739\) 14.4555 0.531756 0.265878 0.964007i \(-0.414338\pi\)
0.265878 + 0.964007i \(0.414338\pi\)
\(740\) 0 0
\(741\) −6.10903 −0.224421
\(742\) 0 0
\(743\) −8.63940 −0.316949 −0.158474 0.987363i \(-0.550658\pi\)
−0.158474 + 0.987363i \(0.550658\pi\)
\(744\) 0 0
\(745\) 0.237801 0.00871236
\(746\) 0 0
\(747\) 6.30099 0.230541
\(748\) 0 0
\(749\) 14.5565 0.531882
\(750\) 0 0
\(751\) −6.99579 −0.255280 −0.127640 0.991821i \(-0.540740\pi\)
−0.127640 + 0.991821i \(0.540740\pi\)
\(752\) 0 0
\(753\) 8.66549 0.315788
\(754\) 0 0
\(755\) −29.7184 −1.08156
\(756\) 0 0
\(757\) 30.5304 1.10965 0.554823 0.831969i \(-0.312786\pi\)
0.554823 + 0.831969i \(0.312786\pi\)
\(758\) 0 0
\(759\) 3.43366 0.124634
\(760\) 0 0
\(761\) −0.867329 −0.0314407 −0.0157203 0.999876i \(-0.505004\pi\)
−0.0157203 + 0.999876i \(0.505004\pi\)
\(762\) 0 0
\(763\) 17.0593 0.617588
\(764\) 0 0
\(765\) −15.1228 −0.546766
\(766\) 0 0
\(767\) −1.56068 −0.0563528
\(768\) 0 0
\(769\) −27.5166 −0.992274 −0.496137 0.868244i \(-0.665249\pi\)
−0.496137 + 0.868244i \(0.665249\pi\)
\(770\) 0 0
\(771\) −20.6493 −0.743665
\(772\) 0 0
\(773\) −3.58822 −0.129059 −0.0645296 0.997916i \(-0.520555\pi\)
−0.0645296 + 0.997916i \(0.520555\pi\)
\(774\) 0 0
\(775\) 5.60620 0.201381
\(776\) 0 0
\(777\) 0.922724 0.0331025
\(778\) 0 0
\(779\) 4.20995 0.150837
\(780\) 0 0
\(781\) −4.04552 −0.144760
\(782\) 0 0
\(783\) 5.79005 0.206920
\(784\) 0 0
\(785\) 13.6436 0.486961
\(786\) 0 0
\(787\) 32.4100 1.15529 0.577646 0.816287i \(-0.303971\pi\)
0.577646 + 0.816287i \(0.303971\pi\)
\(788\) 0 0
\(789\) −21.1048 −0.751351
\(790\) 0 0
\(791\) 7.47919 0.265929
\(792\) 0 0
\(793\) 1.97391 0.0700954
\(794\) 0 0
\(795\) −2.52892 −0.0896917
\(796\) 0 0
\(797\) −20.8038 −0.736909 −0.368455 0.929646i \(-0.620113\pi\)
−0.368455 + 0.929646i \(0.620113\pi\)
\(798\) 0 0
\(799\) −13.6436 −0.482676
\(800\) 0 0
\(801\) −7.89097 −0.278814
\(802\) 0 0
\(803\) 27.9721 0.987116
\(804\) 0 0
\(805\) −2.61186 −0.0920559
\(806\) 0 0
\(807\) 18.8493 0.663528
\(808\) 0 0
\(809\) −17.1147 −0.601720 −0.300860 0.953668i \(-0.597274\pi\)
−0.300860 + 0.953668i \(0.597274\pi\)
\(810\) 0 0
\(811\) 22.4393 0.787951 0.393976 0.919121i \(-0.371099\pi\)
0.393976 + 0.919121i \(0.371099\pi\)
\(812\) 0 0
\(813\) 5.01377 0.175841
\(814\) 0 0
\(815\) 15.9347 0.558169
\(816\) 0 0
\(817\) 86.8931 3.04001
\(818\) 0 0
\(819\) 0.821806 0.0287162
\(820\) 0 0
\(821\) −6.38393 −0.222801 −0.111400 0.993776i \(-0.535534\pi\)
−0.111400 + 0.993776i \(0.535534\pi\)
\(822\) 0 0
\(823\) −45.8631 −1.59869 −0.799344 0.600874i \(-0.794819\pi\)
−0.799344 + 0.600874i \(0.794819\pi\)
\(824\) 0 0
\(825\) −6.25547 −0.217788
\(826\) 0 0
\(827\) 48.7029 1.69357 0.846783 0.531939i \(-0.178536\pi\)
0.846783 + 0.531939i \(0.178536\pi\)
\(828\) 0 0
\(829\) −2.20995 −0.0767546 −0.0383773 0.999263i \(-0.512219\pi\)
−0.0383773 + 0.999263i \(0.512219\pi\)
\(830\) 0 0
\(831\) 7.17819 0.249009
\(832\) 0 0
\(833\) −5.79005 −0.200613
\(834\) 0 0
\(835\) 21.6810 0.750303
\(836\) 0 0
\(837\) −3.07728 −0.106366
\(838\) 0 0
\(839\) −16.4295 −0.567208 −0.283604 0.958942i \(-0.591530\pi\)
−0.283604 + 0.958942i \(0.591530\pi\)
\(840\) 0 0
\(841\) 4.52471 0.156025
\(842\) 0 0
\(843\) −1.19618 −0.0411985
\(844\) 0 0
\(845\) −32.1902 −1.10738
\(846\) 0 0
\(847\) 0.790053 0.0271465
\(848\) 0 0
\(849\) −6.96825 −0.239150
\(850\) 0 0
\(851\) 0.922724 0.0316306
\(852\) 0 0
\(853\) −10.8038 −0.369916 −0.184958 0.982746i \(-0.559215\pi\)
−0.184958 + 0.982746i \(0.559215\pi\)
\(854\) 0 0
\(855\) 19.4157 0.664002
\(856\) 0 0
\(857\) 1.90895 0.0652086 0.0326043 0.999468i \(-0.489620\pi\)
0.0326043 + 0.999468i \(0.489620\pi\)
\(858\) 0 0
\(859\) −38.0275 −1.29748 −0.648741 0.761009i \(-0.724704\pi\)
−0.648741 + 0.761009i \(0.724704\pi\)
\(860\) 0 0
\(861\) −0.566335 −0.0193007
\(862\) 0 0
\(863\) −36.2018 −1.23232 −0.616162 0.787619i \(-0.711314\pi\)
−0.616162 + 0.787619i \(0.711314\pi\)
\(864\) 0 0
\(865\) 21.2774 0.723452
\(866\) 0 0
\(867\) −16.5247 −0.561209
\(868\) 0 0
\(869\) 8.81193 0.298924
\(870\) 0 0
\(871\) 3.37016 0.114193
\(872\) 0 0
\(873\) −1.79005 −0.0605841
\(874\) 0 0
\(875\) −8.30099 −0.280625
\(876\) 0 0
\(877\) 4.86733 0.164358 0.0821790 0.996618i \(-0.473812\pi\)
0.0821790 + 0.996618i \(0.473812\pi\)
\(878\) 0 0
\(879\) −22.4474 −0.757133
\(880\) 0 0
\(881\) 32.9422 1.10985 0.554925 0.831901i \(-0.312747\pi\)
0.554925 + 0.831901i \(0.312747\pi\)
\(882\) 0 0
\(883\) 14.7858 0.497583 0.248792 0.968557i \(-0.419967\pi\)
0.248792 + 0.968557i \(0.419967\pi\)
\(884\) 0 0
\(885\) 4.96013 0.166733
\(886\) 0 0
\(887\) 14.8966 0.500180 0.250090 0.968223i \(-0.419540\pi\)
0.250090 + 0.968223i \(0.419540\pi\)
\(888\) 0 0
\(889\) −14.4930 −0.486078
\(890\) 0 0
\(891\) 3.43366 0.115032
\(892\) 0 0
\(893\) 17.5166 0.586171
\(894\) 0 0
\(895\) −23.4238 −0.782971
\(896\) 0 0
\(897\) 0.821806 0.0274393
\(898\) 0 0
\(899\) −17.8176 −0.594250
\(900\) 0 0
\(901\) −5.60620 −0.186770
\(902\) 0 0
\(903\) −11.6891 −0.388990
\(904\) 0 0
\(905\) −61.8469 −2.05586
\(906\) 0 0
\(907\) −48.5759 −1.61294 −0.806468 0.591278i \(-0.798624\pi\)
−0.806468 + 0.591278i \(0.798624\pi\)
\(908\) 0 0
\(909\) 9.12280 0.302584
\(910\) 0 0
\(911\) −17.6436 −0.584559 −0.292279 0.956333i \(-0.594414\pi\)
−0.292279 + 0.956333i \(0.594414\pi\)
\(912\) 0 0
\(913\) 21.6355 0.716031
\(914\) 0 0
\(915\) −6.27345 −0.207394
\(916\) 0 0
\(917\) −17.7265 −0.585382
\(918\) 0 0
\(919\) 11.6630 0.384728 0.192364 0.981324i \(-0.438385\pi\)
0.192364 + 0.981324i \(0.438385\pi\)
\(920\) 0 0
\(921\) −22.6574 −0.746586
\(922\) 0 0
\(923\) −0.968247 −0.0318702
\(924\) 0 0
\(925\) −1.68102 −0.0552717
\(926\) 0 0
\(927\) −19.1602 −0.629304
\(928\) 0 0
\(929\) 0.904741 0.0296836 0.0148418 0.999890i \(-0.495276\pi\)
0.0148418 + 0.999890i \(0.495276\pi\)
\(930\) 0 0
\(931\) 7.43366 0.243629
\(932\) 0 0
\(933\) 23.4238 0.766861
\(934\) 0 0
\(935\) −51.9266 −1.69818
\(936\) 0 0
\(937\) 51.4887 1.68206 0.841032 0.540985i \(-0.181949\pi\)
0.841032 + 0.540985i \(0.181949\pi\)
\(938\) 0 0
\(939\) 17.7622 0.579647
\(940\) 0 0
\(941\) 47.4693 1.54746 0.773728 0.633518i \(-0.218390\pi\)
0.773728 + 0.633518i \(0.218390\pi\)
\(942\) 0 0
\(943\) −0.566335 −0.0184424
\(944\) 0 0
\(945\) −2.61186 −0.0849638
\(946\) 0 0
\(947\) 47.5801 1.54615 0.773073 0.634317i \(-0.218719\pi\)
0.773073 + 0.634317i \(0.218719\pi\)
\(948\) 0 0
\(949\) 6.69479 0.217322
\(950\) 0 0
\(951\) −8.19196 −0.265643
\(952\) 0 0
\(953\) −61.6807 −1.99803 −0.999017 0.0443267i \(-0.985886\pi\)
−0.999017 + 0.0443267i \(0.985886\pi\)
\(954\) 0 0
\(955\) −10.6852 −0.345766
\(956\) 0 0
\(957\) 19.8811 0.642665
\(958\) 0 0
\(959\) 10.3010 0.332636
\(960\) 0 0
\(961\) −21.5304 −0.694528
\(962\) 0 0
\(963\) 14.5565 0.469076
\(964\) 0 0
\(965\) 32.2731 1.03891
\(966\) 0 0
\(967\) 18.0275 0.579727 0.289863 0.957068i \(-0.406390\pi\)
0.289863 + 0.957068i \(0.406390\pi\)
\(968\) 0 0
\(969\) 43.0413 1.38269
\(970\) 0 0
\(971\) −19.4873 −0.625377 −0.312689 0.949856i \(-0.601230\pi\)
−0.312689 + 0.949856i \(0.601230\pi\)
\(972\) 0 0
\(973\) 19.8356 0.635899
\(974\) 0 0
\(975\) −1.49717 −0.0479478
\(976\) 0 0
\(977\) 21.1147 0.675519 0.337759 0.941232i \(-0.390331\pi\)
0.337759 + 0.941232i \(0.390331\pi\)
\(978\) 0 0
\(979\) −27.0949 −0.865958
\(980\) 0 0
\(981\) 17.0593 0.544661
\(982\) 0 0
\(983\) −35.8811 −1.14443 −0.572215 0.820104i \(-0.693916\pi\)
−0.572215 + 0.820104i \(0.693916\pi\)
\(984\) 0 0
\(985\) 5.86978 0.187027
\(986\) 0 0
\(987\) −2.35639 −0.0750047
\(988\) 0 0
\(989\) −11.6891 −0.371693
\(990\) 0 0
\(991\) −15.2417 −0.484169 −0.242084 0.970255i \(-0.577831\pi\)
−0.242084 + 0.970255i \(0.577831\pi\)
\(992\) 0 0
\(993\) 1.13267 0.0359442
\(994\) 0 0
\(995\) 56.6278 1.79522
\(996\) 0 0
\(997\) 37.7901 1.19682 0.598411 0.801189i \(-0.295799\pi\)
0.598411 + 0.801189i \(0.295799\pi\)
\(998\) 0 0
\(999\) 0.922724 0.0291937
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.bq.1.3 3
4.3 odd 2 3864.2.a.o.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.o.1.3 3 4.3 odd 2
7728.2.a.bq.1.3 3 1.1 even 1 trivial