Properties

Label 7728.2.a.bz.1.3
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.39605.1
Defining polynomial: \(x^{4} - 2 x^{3} - 8 x^{2} + 9 x - 2\)
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 \(0.681685\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.318315 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.318315 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.71255 q^{11} -2.66749 q^{13} +0.318315 q^{15} +2.21699 q^{17} +2.34918 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.89868 q^{25} -1.00000 q^{27} +5.41527 q^{29} -8.21699 q^{31} +3.71255 q^{33} -0.318315 q^{35} +7.28308 q^{37} +2.66749 q^{39} +4.77864 q^{41} +8.59703 q^{43} -0.318315 q^{45} +3.63226 q^{47} +1.00000 q^{49} -2.21699 q^{51} -0.615591 q^{53} +1.18176 q^{55} -2.34918 q^{57} -9.81839 q^{59} +10.9364 q^{61} +1.00000 q^{63} +0.849103 q^{65} -0.747779 q^{67} -1.00000 q^{69} -13.5309 q^{71} -3.84473 q^{73} +4.89868 q^{75} -3.71255 q^{77} -15.1797 q^{79} +1.00000 q^{81} +1.11799 q^{83} -0.705701 q^{85} -5.41527 q^{87} +6.32814 q^{89} -2.66749 q^{91} +8.21699 q^{93} -0.747779 q^{95} -16.1136 q^{97} -3.71255 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 2q^{5} + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 2q^{5} + 4q^{7} + 4q^{9} + 2q^{13} + 2q^{15} - 10q^{17} - 4q^{19} - 4q^{21} + 4q^{23} - 4q^{27} + 11q^{29} - 14q^{31} - 2q^{35} + 13q^{37} - 2q^{39} + 7q^{41} - 12q^{43} - 2q^{45} - 15q^{47} + 4q^{49} + 10q^{51} + q^{53} - 31q^{55} + 4q^{57} - 5q^{59} + 3q^{61} + 4q^{63} + 25q^{65} - 5q^{67} - 4q^{69} - 5q^{71} - 6q^{73} - 26q^{79} + 4q^{81} - 2q^{83} + 5q^{85} - 11q^{87} + 7q^{89} + 2q^{91} + 14q^{93} - 5q^{95} - 27q^{97} + O(q^{100}) \)

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
\(5\) −0.318315 −0.142355 −0.0711774 0.997464i \(-0.522676\pi\)
−0.0711774 + 0.997464i \(0.522676\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.71255 −1.11938 −0.559688 0.828704i \(-0.689079\pi\)
−0.559688 + 0.828704i \(0.689079\pi\)
\(12\) 0 0
\(13\) −2.66749 −0.739829 −0.369915 0.929066i \(-0.620613\pi\)
−0.369915 + 0.929066i \(0.620613\pi\)
\(14\) 0 0
\(15\) 0.318315 0.0821886
\(16\) 0 0
\(17\) 2.21699 0.537699 0.268850 0.963182i \(-0.413357\pi\)
0.268850 + 0.963182i \(0.413357\pi\)
\(18\) 0 0
\(19\) 2.34918 0.538938 0.269469 0.963009i \(-0.413152\pi\)
0.269469 + 0.963009i \(0.413152\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.89868 −0.979735
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.41527 1.00559 0.502795 0.864406i \(-0.332305\pi\)
0.502795 + 0.864406i \(0.332305\pi\)
\(30\) 0 0
\(31\) −8.21699 −1.47582 −0.737908 0.674902i \(-0.764186\pi\)
−0.737908 + 0.674902i \(0.764186\pi\)
\(32\) 0 0
\(33\) 3.71255 0.646272
\(34\) 0 0
\(35\) −0.318315 −0.0538051
\(36\) 0 0
\(37\) 7.28308 1.19733 0.598666 0.800999i \(-0.295698\pi\)
0.598666 + 0.800999i \(0.295698\pi\)
\(38\) 0 0
\(39\) 2.66749 0.427141
\(40\) 0 0
\(41\) 4.77864 0.746298 0.373149 0.927771i \(-0.378278\pi\)
0.373149 + 0.927771i \(0.378278\pi\)
\(42\) 0 0
\(43\) 8.59703 1.31103 0.655517 0.755180i \(-0.272451\pi\)
0.655517 + 0.755180i \(0.272451\pi\)
\(44\) 0 0
\(45\) −0.318315 −0.0474516
\(46\) 0 0
\(47\) 3.63226 0.529820 0.264910 0.964273i \(-0.414658\pi\)
0.264910 + 0.964273i \(0.414658\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.21699 −0.310441
\(52\) 0 0
\(53\) −0.615591 −0.0845580 −0.0422790 0.999106i \(-0.513462\pi\)
−0.0422790 + 0.999106i \(0.513462\pi\)
\(54\) 0 0
\(55\) 1.18176 0.159348
\(56\) 0 0
\(57\) −2.34918 −0.311156
\(58\) 0 0
\(59\) −9.81839 −1.27825 −0.639123 0.769105i \(-0.720702\pi\)
−0.639123 + 0.769105i \(0.720702\pi\)
\(60\) 0 0
\(61\) 10.9364 1.40026 0.700130 0.714015i \(-0.253125\pi\)
0.700130 + 0.714015i \(0.253125\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0.849103 0.105318
\(66\) 0 0
\(67\) −0.747779 −0.0913557 −0.0456778 0.998956i \(-0.514545\pi\)
−0.0456778 + 0.998956i \(0.514545\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −13.5309 −1.60583 −0.802913 0.596096i \(-0.796718\pi\)
−0.802913 + 0.596096i \(0.796718\pi\)
\(72\) 0 0
\(73\) −3.84473 −0.449992 −0.224996 0.974360i \(-0.572237\pi\)
−0.224996 + 0.974360i \(0.572237\pi\)
\(74\) 0 0
\(75\) 4.89868 0.565650
\(76\) 0 0
\(77\) −3.71255 −0.423084
\(78\) 0 0
\(79\) −15.1797 −1.70785 −0.853926 0.520395i \(-0.825785\pi\)
−0.853926 + 0.520395i \(0.825785\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.11799 0.122716 0.0613579 0.998116i \(-0.480457\pi\)
0.0613579 + 0.998116i \(0.480457\pi\)
\(84\) 0 0
\(85\) −0.705701 −0.0765441
\(86\) 0 0
\(87\) −5.41527 −0.580578
\(88\) 0 0
\(89\) 6.32814 0.670781 0.335391 0.942079i \(-0.391132\pi\)
0.335391 + 0.942079i \(0.391132\pi\)
\(90\) 0 0
\(91\) −2.66749 −0.279629
\(92\) 0 0
\(93\) 8.21699 0.852062
\(94\) 0 0
\(95\) −0.747779 −0.0767205
\(96\) 0 0
\(97\) −16.1136 −1.63609 −0.818045 0.575154i \(-0.804942\pi\)
−0.818045 + 0.575154i \(0.804942\pi\)
\(98\) 0 0
\(99\) −3.71255 −0.373125
\(100\) 0 0
\(101\) −13.2435 −1.31778 −0.658888 0.752241i \(-0.728973\pi\)
−0.658888 + 0.752241i \(0.728973\pi\)
\(102\) 0 0
\(103\) 3.12782 0.308193 0.154097 0.988056i \(-0.450753\pi\)
0.154097 + 0.988056i \(0.450753\pi\)
\(104\) 0 0
\(105\) 0.318315 0.0310644
\(106\) 0 0
\(107\) 15.6631 1.51421 0.757106 0.653292i \(-0.226613\pi\)
0.757106 + 0.653292i \(0.226613\pi\)
\(108\) 0 0
\(109\) 8.75230 0.838318 0.419159 0.907913i \(-0.362325\pi\)
0.419159 + 0.907913i \(0.362325\pi\)
\(110\) 0 0
\(111\) −7.28308 −0.691280
\(112\) 0 0
\(113\) −15.8051 −1.48682 −0.743411 0.668835i \(-0.766793\pi\)
−0.743411 + 0.668835i \(0.766793\pi\)
\(114\) 0 0
\(115\) −0.318315 −0.0296830
\(116\) 0 0
\(117\) −2.66749 −0.246610
\(118\) 0 0
\(119\) 2.21699 0.203231
\(120\) 0 0
\(121\) 2.78301 0.253001
\(122\) 0 0
\(123\) −4.77864 −0.430876
\(124\) 0 0
\(125\) 3.15090 0.281825
\(126\) 0 0
\(127\) −5.92706 −0.525942 −0.262971 0.964804i \(-0.584702\pi\)
−0.262971 + 0.964804i \(0.584702\pi\)
\(128\) 0 0
\(129\) −8.59703 −0.756926
\(130\) 0 0
\(131\) −8.78316 −0.767388 −0.383694 0.923460i \(-0.625348\pi\)
−0.383694 + 0.923460i \(0.625348\pi\)
\(132\) 0 0
\(133\) 2.34918 0.203700
\(134\) 0 0
\(135\) 0.318315 0.0273962
\(136\) 0 0
\(137\) 18.1333 1.54923 0.774615 0.632433i \(-0.217944\pi\)
0.774615 + 0.632433i \(0.217944\pi\)
\(138\) 0 0
\(139\) 0.884483 0.0750209 0.0375104 0.999296i \(-0.488057\pi\)
0.0375104 + 0.999296i \(0.488057\pi\)
\(140\) 0 0
\(141\) −3.63226 −0.305892
\(142\) 0 0
\(143\) 9.90319 0.828147
\(144\) 0 0
\(145\) −1.72376 −0.143151
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 19.7690 1.61954 0.809768 0.586750i \(-0.199593\pi\)
0.809768 + 0.586750i \(0.199593\pi\)
\(150\) 0 0
\(151\) −20.7930 −1.69211 −0.846054 0.533096i \(-0.821028\pi\)
−0.846054 + 0.533096i \(0.821028\pi\)
\(152\) 0 0
\(153\) 2.21699 0.179233
\(154\) 0 0
\(155\) 2.61559 0.210089
\(156\) 0 0
\(157\) 11.3350 0.904630 0.452315 0.891858i \(-0.350598\pi\)
0.452315 + 0.891858i \(0.350598\pi\)
\(158\) 0 0
\(159\) 0.615591 0.0488196
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −23.1534 −1.81351 −0.906756 0.421655i \(-0.861449\pi\)
−0.906756 + 0.421655i \(0.861449\pi\)
\(164\) 0 0
\(165\) −1.18176 −0.0919999
\(166\) 0 0
\(167\) 2.34918 0.181785 0.0908924 0.995861i \(-0.471028\pi\)
0.0908924 + 0.995861i \(0.471028\pi\)
\(168\) 0 0
\(169\) −5.88448 −0.452653
\(170\) 0 0
\(171\) 2.34918 0.179646
\(172\) 0 0
\(173\) −9.74589 −0.740966 −0.370483 0.928839i \(-0.620808\pi\)
−0.370483 + 0.928839i \(0.620808\pi\)
\(174\) 0 0
\(175\) −4.89868 −0.370305
\(176\) 0 0
\(177\) 9.81839 0.737995
\(178\) 0 0
\(179\) 26.0168 1.94459 0.972294 0.233761i \(-0.0751032\pi\)
0.972294 + 0.233761i \(0.0751032\pi\)
\(180\) 0 0
\(181\) 18.8449 1.40073 0.700365 0.713785i \(-0.253021\pi\)
0.700365 + 0.713785i \(0.253021\pi\)
\(182\) 0 0
\(183\) −10.9364 −0.808441
\(184\) 0 0
\(185\) −2.31832 −0.170446
\(186\) 0 0
\(187\) −8.23068 −0.601887
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −10.1939 −0.737606 −0.368803 0.929508i \(-0.620232\pi\)
−0.368803 + 0.929508i \(0.620232\pi\)
\(192\) 0 0
\(193\) 11.7361 0.844780 0.422390 0.906414i \(-0.361191\pi\)
0.422390 + 0.906414i \(0.361191\pi\)
\(194\) 0 0
\(195\) −0.849103 −0.0608055
\(196\) 0 0
\(197\) 15.4560 1.10119 0.550596 0.834772i \(-0.314401\pi\)
0.550596 + 0.834772i \(0.314401\pi\)
\(198\) 0 0
\(199\) −16.8424 −1.19393 −0.596963 0.802269i \(-0.703626\pi\)
−0.596963 + 0.802269i \(0.703626\pi\)
\(200\) 0 0
\(201\) 0.747779 0.0527442
\(202\) 0 0
\(203\) 5.41527 0.380078
\(204\) 0 0
\(205\) −1.52111 −0.106239
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −8.72143 −0.603274
\(210\) 0 0
\(211\) −17.6842 −1.21743 −0.608714 0.793390i \(-0.708314\pi\)
−0.608714 + 0.793390i \(0.708314\pi\)
\(212\) 0 0
\(213\) 13.5309 0.927125
\(214\) 0 0
\(215\) −2.73656 −0.186632
\(216\) 0 0
\(217\) −8.21699 −0.557806
\(218\) 0 0
\(219\) 3.84473 0.259803
\(220\) 0 0
\(221\) −5.91381 −0.397806
\(222\) 0 0
\(223\) 5.90850 0.395662 0.197831 0.980236i \(-0.436610\pi\)
0.197831 + 0.980236i \(0.436610\pi\)
\(224\) 0 0
\(225\) −4.89868 −0.326578
\(226\) 0 0
\(227\) −19.0500 −1.26439 −0.632197 0.774808i \(-0.717847\pi\)
−0.632197 + 0.774808i \(0.717847\pi\)
\(228\) 0 0
\(229\) −22.1819 −1.46582 −0.732911 0.680325i \(-0.761839\pi\)
−0.732911 + 0.680325i \(0.761839\pi\)
\(230\) 0 0
\(231\) 3.71255 0.244268
\(232\) 0 0
\(233\) −14.8712 −0.974247 −0.487123 0.873333i \(-0.661954\pi\)
−0.487123 + 0.873333i \(0.661954\pi\)
\(234\) 0 0
\(235\) −1.15620 −0.0754224
\(236\) 0 0
\(237\) 15.1797 0.986029
\(238\) 0 0
\(239\) 14.8008 0.957382 0.478691 0.877983i \(-0.341111\pi\)
0.478691 + 0.877983i \(0.341111\pi\)
\(240\) 0 0
\(241\) −27.6887 −1.78358 −0.891792 0.452445i \(-0.850552\pi\)
−0.891792 + 0.452445i \(0.850552\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.318315 −0.0203364
\(246\) 0 0
\(247\) −6.26641 −0.398722
\(248\) 0 0
\(249\) −1.11799 −0.0708500
\(250\) 0 0
\(251\) 25.1895 1.58995 0.794975 0.606642i \(-0.207484\pi\)
0.794975 + 0.606642i \(0.207484\pi\)
\(252\) 0 0
\(253\) −3.71255 −0.233406
\(254\) 0 0
\(255\) 0.705701 0.0441927
\(256\) 0 0
\(257\) 4.89227 0.305171 0.152586 0.988290i \(-0.451240\pi\)
0.152586 + 0.988290i \(0.451240\pi\)
\(258\) 0 0
\(259\) 7.28308 0.452549
\(260\) 0 0
\(261\) 5.41527 0.335197
\(262\) 0 0
\(263\) −23.1755 −1.42906 −0.714531 0.699603i \(-0.753360\pi\)
−0.714531 + 0.699603i \(0.753360\pi\)
\(264\) 0 0
\(265\) 0.195952 0.0120372
\(266\) 0 0
\(267\) −6.32814 −0.387276
\(268\) 0 0
\(269\) −24.3811 −1.48654 −0.743272 0.668990i \(-0.766727\pi\)
−0.743272 + 0.668990i \(0.766727\pi\)
\(270\) 0 0
\(271\) −5.42853 −0.329759 −0.164880 0.986314i \(-0.552724\pi\)
−0.164880 + 0.986314i \(0.552724\pi\)
\(272\) 0 0
\(273\) 2.66749 0.161444
\(274\) 0 0
\(275\) 18.1866 1.09669
\(276\) 0 0
\(277\) −15.5730 −0.935692 −0.467846 0.883810i \(-0.654970\pi\)
−0.467846 + 0.883810i \(0.654970\pi\)
\(278\) 0 0
\(279\) −8.21699 −0.491938
\(280\) 0 0
\(281\) 3.05736 0.182387 0.0911933 0.995833i \(-0.470932\pi\)
0.0911933 + 0.995833i \(0.470932\pi\)
\(282\) 0 0
\(283\) −4.24334 −0.252240 −0.126120 0.992015i \(-0.540252\pi\)
−0.126120 + 0.992015i \(0.540252\pi\)
\(284\) 0 0
\(285\) 0.747779 0.0442946
\(286\) 0 0
\(287\) 4.77864 0.282074
\(288\) 0 0
\(289\) −12.0850 −0.710880
\(290\) 0 0
\(291\) 16.1136 0.944598
\(292\) 0 0
\(293\) −18.6984 −1.09237 −0.546185 0.837665i \(-0.683920\pi\)
−0.546185 + 0.837665i \(0.683920\pi\)
\(294\) 0 0
\(295\) 3.12534 0.181964
\(296\) 0 0
\(297\) 3.71255 0.215424
\(298\) 0 0
\(299\) −2.66749 −0.154265
\(300\) 0 0
\(301\) 8.59703 0.495525
\(302\) 0 0
\(303\) 13.2435 0.760818
\(304\) 0 0
\(305\) −3.48122 −0.199334
\(306\) 0 0
\(307\) −25.3448 −1.44650 −0.723252 0.690584i \(-0.757354\pi\)
−0.723252 + 0.690584i \(0.757354\pi\)
\(308\) 0 0
\(309\) −3.12782 −0.177935
\(310\) 0 0
\(311\) −14.5221 −0.823470 −0.411735 0.911304i \(-0.635077\pi\)
−0.411735 + 0.911304i \(0.635077\pi\)
\(312\) 0 0
\(313\) 4.53778 0.256491 0.128245 0.991742i \(-0.459066\pi\)
0.128245 + 0.991742i \(0.459066\pi\)
\(314\) 0 0
\(315\) −0.318315 −0.0179350
\(316\) 0 0
\(317\) 3.61996 0.203317 0.101659 0.994819i \(-0.467585\pi\)
0.101659 + 0.994819i \(0.467585\pi\)
\(318\) 0 0
\(319\) −20.1045 −1.12563
\(320\) 0 0
\(321\) −15.6631 −0.874230
\(322\) 0 0
\(323\) 5.20810 0.289787
\(324\) 0 0
\(325\) 13.0672 0.724837
\(326\) 0 0
\(327\) −8.75230 −0.484003
\(328\) 0 0
\(329\) 3.63226 0.200253
\(330\) 0 0
\(331\) −12.5749 −0.691179 −0.345590 0.938386i \(-0.612321\pi\)
−0.345590 + 0.938386i \(0.612321\pi\)
\(332\) 0 0
\(333\) 7.28308 0.399111
\(334\) 0 0
\(335\) 0.238029 0.0130049
\(336\) 0 0
\(337\) 1.07265 0.0584310 0.0292155 0.999573i \(-0.490699\pi\)
0.0292155 + 0.999573i \(0.490699\pi\)
\(338\) 0 0
\(339\) 15.8051 0.858417
\(340\) 0 0
\(341\) 30.5060 1.65199
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0.318315 0.0171375
\(346\) 0 0
\(347\) 11.3732 0.610545 0.305273 0.952265i \(-0.401252\pi\)
0.305273 + 0.952265i \(0.401252\pi\)
\(348\) 0 0
\(349\) 32.3580 1.73209 0.866043 0.499969i \(-0.166655\pi\)
0.866043 + 0.499969i \(0.166655\pi\)
\(350\) 0 0
\(351\) 2.66749 0.142380
\(352\) 0 0
\(353\) −12.3447 −0.657040 −0.328520 0.944497i \(-0.606550\pi\)
−0.328520 + 0.944497i \(0.606550\pi\)
\(354\) 0 0
\(355\) 4.30710 0.228597
\(356\) 0 0
\(357\) −2.21699 −0.117336
\(358\) 0 0
\(359\) −28.7239 −1.51599 −0.757995 0.652260i \(-0.773821\pi\)
−0.757995 + 0.652260i \(0.773821\pi\)
\(360\) 0 0
\(361\) −13.4814 −0.709546
\(362\) 0 0
\(363\) −2.78301 −0.146070
\(364\) 0 0
\(365\) 1.22384 0.0640585
\(366\) 0 0
\(367\) 12.4365 0.649178 0.324589 0.945855i \(-0.394774\pi\)
0.324589 + 0.945855i \(0.394774\pi\)
\(368\) 0 0
\(369\) 4.77864 0.248766
\(370\) 0 0
\(371\) −0.615591 −0.0319599
\(372\) 0 0
\(373\) −5.97692 −0.309473 −0.154737 0.987956i \(-0.549453\pi\)
−0.154737 + 0.987956i \(0.549453\pi\)
\(374\) 0 0
\(375\) −3.15090 −0.162712
\(376\) 0 0
\(377\) −14.4452 −0.743965
\(378\) 0 0
\(379\) 19.1369 0.982994 0.491497 0.870879i \(-0.336450\pi\)
0.491497 + 0.870879i \(0.336450\pi\)
\(380\) 0 0
\(381\) 5.92706 0.303652
\(382\) 0 0
\(383\) 15.4482 0.789365 0.394682 0.918818i \(-0.370855\pi\)
0.394682 + 0.918818i \(0.370855\pi\)
\(384\) 0 0
\(385\) 1.18176 0.0602280
\(386\) 0 0
\(387\) 8.59703 0.437012
\(388\) 0 0
\(389\) −23.9065 −1.21211 −0.606053 0.795424i \(-0.707248\pi\)
−0.606053 + 0.795424i \(0.707248\pi\)
\(390\) 0 0
\(391\) 2.21699 0.112118
\(392\) 0 0
\(393\) 8.78316 0.443052
\(394\) 0 0
\(395\) 4.83193 0.243121
\(396\) 0 0
\(397\) −34.8551 −1.74933 −0.874665 0.484728i \(-0.838919\pi\)
−0.874665 + 0.484728i \(0.838919\pi\)
\(398\) 0 0
\(399\) −2.34918 −0.117606
\(400\) 0 0
\(401\) −0.481365 −0.0240382 −0.0120191 0.999928i \(-0.503826\pi\)
−0.0120191 + 0.999928i \(0.503826\pi\)
\(402\) 0 0
\(403\) 21.9188 1.09185
\(404\) 0 0
\(405\) −0.318315 −0.0158172
\(406\) 0 0
\(407\) −27.0388 −1.34026
\(408\) 0 0
\(409\) 30.1621 1.49142 0.745710 0.666271i \(-0.232111\pi\)
0.745710 + 0.666271i \(0.232111\pi\)
\(410\) 0 0
\(411\) −18.1333 −0.894448
\(412\) 0 0
\(413\) −9.81839 −0.483131
\(414\) 0 0
\(415\) −0.355874 −0.0174692
\(416\) 0 0
\(417\) −0.884483 −0.0433133
\(418\) 0 0
\(419\) −24.7390 −1.20858 −0.604291 0.796764i \(-0.706543\pi\)
−0.604291 + 0.796764i \(0.706543\pi\)
\(420\) 0 0
\(421\) −16.4452 −0.801490 −0.400745 0.916190i \(-0.631249\pi\)
−0.400745 + 0.916190i \(0.631249\pi\)
\(422\) 0 0
\(423\) 3.63226 0.176607
\(424\) 0 0
\(425\) −10.8603 −0.526803
\(426\) 0 0
\(427\) 10.9364 0.529249
\(428\) 0 0
\(429\) −9.90319 −0.478131
\(430\) 0 0
\(431\) 40.6061 1.95593 0.977963 0.208780i \(-0.0669492\pi\)
0.977963 + 0.208780i \(0.0669492\pi\)
\(432\) 0 0
\(433\) 5.36789 0.257964 0.128982 0.991647i \(-0.458829\pi\)
0.128982 + 0.991647i \(0.458829\pi\)
\(434\) 0 0
\(435\) 1.72376 0.0826481
\(436\) 0 0
\(437\) 2.34918 0.112376
\(438\) 0 0
\(439\) 23.7655 1.13427 0.567134 0.823626i \(-0.308052\pi\)
0.567134 + 0.823626i \(0.308052\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −16.6655 −0.791800 −0.395900 0.918294i \(-0.629567\pi\)
−0.395900 + 0.918294i \(0.629567\pi\)
\(444\) 0 0
\(445\) −2.01434 −0.0954890
\(446\) 0 0
\(447\) −19.7690 −0.935040
\(448\) 0 0
\(449\) 14.6910 0.693312 0.346656 0.937992i \(-0.387317\pi\)
0.346656 + 0.937992i \(0.387317\pi\)
\(450\) 0 0
\(451\) −17.7409 −0.835388
\(452\) 0 0
\(453\) 20.7930 0.976940
\(454\) 0 0
\(455\) 0.849103 0.0398066
\(456\) 0 0
\(457\) 18.2346 0.852979 0.426489 0.904493i \(-0.359750\pi\)
0.426489 + 0.904493i \(0.359750\pi\)
\(458\) 0 0
\(459\) −2.21699 −0.103480
\(460\) 0 0
\(461\) −21.1200 −0.983658 −0.491829 0.870692i \(-0.663671\pi\)
−0.491829 + 0.870692i \(0.663671\pi\)
\(462\) 0 0
\(463\) −8.75026 −0.406659 −0.203329 0.979110i \(-0.565176\pi\)
−0.203329 + 0.979110i \(0.565176\pi\)
\(464\) 0 0
\(465\) −2.61559 −0.121295
\(466\) 0 0
\(467\) 33.2743 1.53975 0.769877 0.638193i \(-0.220318\pi\)
0.769877 + 0.638193i \(0.220318\pi\)
\(468\) 0 0
\(469\) −0.747779 −0.0345292
\(470\) 0 0
\(471\) −11.3350 −0.522289
\(472\) 0 0
\(473\) −31.9169 −1.46754
\(474\) 0 0
\(475\) −11.5079 −0.528017
\(476\) 0 0
\(477\) −0.615591 −0.0281860
\(478\) 0 0
\(479\) 13.9155 0.635815 0.317908 0.948122i \(-0.397020\pi\)
0.317908 + 0.948122i \(0.397020\pi\)
\(480\) 0 0
\(481\) −19.4276 −0.885821
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 5.12921 0.232905
\(486\) 0 0
\(487\) 27.9306 1.26566 0.632829 0.774292i \(-0.281894\pi\)
0.632829 + 0.774292i \(0.281894\pi\)
\(488\) 0 0
\(489\) 23.1534 1.04703
\(490\) 0 0
\(491\) −24.3047 −1.09686 −0.548428 0.836198i \(-0.684774\pi\)
−0.548428 + 0.836198i \(0.684774\pi\)
\(492\) 0 0
\(493\) 12.0056 0.540705
\(494\) 0 0
\(495\) 1.18176 0.0531161
\(496\) 0 0
\(497\) −13.5309 −0.606945
\(498\) 0 0
\(499\) −13.7763 −0.616712 −0.308356 0.951271i \(-0.599779\pi\)
−0.308356 + 0.951271i \(0.599779\pi\)
\(500\) 0 0
\(501\) −2.34918 −0.104954
\(502\) 0 0
\(503\) 19.2572 0.858635 0.429318 0.903154i \(-0.358754\pi\)
0.429318 + 0.903154i \(0.358754\pi\)
\(504\) 0 0
\(505\) 4.21560 0.187592
\(506\) 0 0
\(507\) 5.88448 0.261339
\(508\) 0 0
\(509\) −5.18503 −0.229822 −0.114911 0.993376i \(-0.536658\pi\)
−0.114911 + 0.993376i \(0.536658\pi\)
\(510\) 0 0
\(511\) −3.84473 −0.170081
\(512\) 0 0
\(513\) −2.34918 −0.103719
\(514\) 0 0
\(515\) −0.995632 −0.0438728
\(516\) 0 0
\(517\) −13.4849 −0.593067
\(518\) 0 0
\(519\) 9.74589 0.427797
\(520\) 0 0
\(521\) −33.1520 −1.45241 −0.726207 0.687476i \(-0.758719\pi\)
−0.726207 + 0.687476i \(0.758719\pi\)
\(522\) 0 0
\(523\) −11.1323 −0.486783 −0.243392 0.969928i \(-0.578260\pi\)
−0.243392 + 0.969928i \(0.578260\pi\)
\(524\) 0 0
\(525\) 4.89868 0.213796
\(526\) 0 0
\(527\) −18.2170 −0.793545
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −9.81839 −0.426082
\(532\) 0 0
\(533\) −12.7470 −0.552133
\(534\) 0 0
\(535\) −4.98581 −0.215555
\(536\) 0 0
\(537\) −26.0168 −1.12271
\(538\) 0 0
\(539\) −3.71255 −0.159911
\(540\) 0 0
\(541\) −10.6192 −0.456553 −0.228277 0.973596i \(-0.573309\pi\)
−0.228277 + 0.973596i \(0.573309\pi\)
\(542\) 0 0
\(543\) −18.8449 −0.808711
\(544\) 0 0
\(545\) −2.78599 −0.119339
\(546\) 0 0
\(547\) −36.1819 −1.54703 −0.773513 0.633780i \(-0.781502\pi\)
−0.773513 + 0.633780i \(0.781502\pi\)
\(548\) 0 0
\(549\) 10.9364 0.466754
\(550\) 0 0
\(551\) 12.7214 0.541951
\(552\) 0 0
\(553\) −15.1797 −0.645507
\(554\) 0 0
\(555\) 2.31832 0.0984070
\(556\) 0 0
\(557\) 0.524091 0.0222065 0.0111032 0.999938i \(-0.496466\pi\)
0.0111032 + 0.999938i \(0.496466\pi\)
\(558\) 0 0
\(559\) −22.9325 −0.969942
\(560\) 0 0
\(561\) 8.23068 0.347500
\(562\) 0 0
\(563\) −4.30427 −0.181403 −0.0907017 0.995878i \(-0.528911\pi\)
−0.0907017 + 0.995878i \(0.528911\pi\)
\(564\) 0 0
\(565\) 5.03101 0.211656
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 36.2318 1.51891 0.759457 0.650557i \(-0.225465\pi\)
0.759457 + 0.650557i \(0.225465\pi\)
\(570\) 0 0
\(571\) −30.5480 −1.27840 −0.639198 0.769042i \(-0.720734\pi\)
−0.639198 + 0.769042i \(0.720734\pi\)
\(572\) 0 0
\(573\) 10.1939 0.425857
\(574\) 0 0
\(575\) −4.89868 −0.204289
\(576\) 0 0
\(577\) −30.3878 −1.26506 −0.632531 0.774535i \(-0.717984\pi\)
−0.632531 + 0.774535i \(0.717984\pi\)
\(578\) 0 0
\(579\) −11.7361 −0.487734
\(580\) 0 0
\(581\) 1.11799 0.0463822
\(582\) 0 0
\(583\) 2.28541 0.0946521
\(584\) 0 0
\(585\) 0.849103 0.0351061
\(586\) 0 0
\(587\) −39.4654 −1.62891 −0.814456 0.580225i \(-0.802965\pi\)
−0.814456 + 0.580225i \(0.802965\pi\)
\(588\) 0 0
\(589\) −19.3032 −0.795373
\(590\) 0 0
\(591\) −15.4560 −0.635773
\(592\) 0 0
\(593\) −22.7930 −0.935996 −0.467998 0.883730i \(-0.655024\pi\)
−0.467998 + 0.883730i \(0.655024\pi\)
\(594\) 0 0
\(595\) −0.705701 −0.0289309
\(596\) 0 0
\(597\) 16.8424 0.689314
\(598\) 0 0
\(599\) 35.8287 1.46392 0.731960 0.681348i \(-0.238606\pi\)
0.731960 + 0.681348i \(0.238606\pi\)
\(600\) 0 0
\(601\) −39.3669 −1.60581 −0.802905 0.596106i \(-0.796714\pi\)
−0.802905 + 0.596106i \(0.796714\pi\)
\(602\) 0 0
\(603\) −0.747779 −0.0304519
\(604\) 0 0
\(605\) −0.885874 −0.0360159
\(606\) 0 0
\(607\) −30.4076 −1.23421 −0.617104 0.786882i \(-0.711694\pi\)
−0.617104 + 0.786882i \(0.711694\pi\)
\(608\) 0 0
\(609\) −5.41527 −0.219438
\(610\) 0 0
\(611\) −9.68903 −0.391976
\(612\) 0 0
\(613\) −38.5252 −1.55602 −0.778009 0.628254i \(-0.783770\pi\)
−0.778009 + 0.628254i \(0.783770\pi\)
\(614\) 0 0
\(615\) 1.52111 0.0613372
\(616\) 0 0
\(617\) −11.9032 −0.479205 −0.239602 0.970871i \(-0.577017\pi\)
−0.239602 + 0.970871i \(0.577017\pi\)
\(618\) 0 0
\(619\) −10.8522 −0.436188 −0.218094 0.975928i \(-0.569984\pi\)
−0.218094 + 0.975928i \(0.569984\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 6.32814 0.253532
\(624\) 0 0
\(625\) 23.4904 0.939616
\(626\) 0 0
\(627\) 8.72143 0.348301
\(628\) 0 0
\(629\) 16.1465 0.643804
\(630\) 0 0
\(631\) 21.3250 0.848936 0.424468 0.905443i \(-0.360461\pi\)
0.424468 + 0.905443i \(0.360461\pi\)
\(632\) 0 0
\(633\) 17.6842 0.702882
\(634\) 0 0
\(635\) 1.88667 0.0748703
\(636\) 0 0
\(637\) −2.66749 −0.105690
\(638\) 0 0
\(639\) −13.5309 −0.535276
\(640\) 0 0
\(641\) −3.33251 −0.131626 −0.0658131 0.997832i \(-0.520964\pi\)
−0.0658131 + 0.997832i \(0.520964\pi\)
\(642\) 0 0
\(643\) −45.0776 −1.77769 −0.888844 0.458211i \(-0.848491\pi\)
−0.888844 + 0.458211i \(0.848491\pi\)
\(644\) 0 0
\(645\) 2.73656 0.107752
\(646\) 0 0
\(647\) −15.7616 −0.619653 −0.309827 0.950793i \(-0.600271\pi\)
−0.309827 + 0.950793i \(0.600271\pi\)
\(648\) 0 0
\(649\) 36.4512 1.43084
\(650\) 0 0
\(651\) 8.21699 0.322049
\(652\) 0 0
\(653\) 40.9836 1.60381 0.801907 0.597449i \(-0.203819\pi\)
0.801907 + 0.597449i \(0.203819\pi\)
\(654\) 0 0
\(655\) 2.79581 0.109241
\(656\) 0 0
\(657\) −3.84473 −0.149997
\(658\) 0 0
\(659\) 15.1376 0.589679 0.294839 0.955547i \(-0.404734\pi\)
0.294839 + 0.955547i \(0.404734\pi\)
\(660\) 0 0
\(661\) −46.2700 −1.79969 −0.899847 0.436206i \(-0.856322\pi\)
−0.899847 + 0.436206i \(0.856322\pi\)
\(662\) 0 0
\(663\) 5.91381 0.229673
\(664\) 0 0
\(665\) −0.747779 −0.0289976
\(666\) 0 0
\(667\) 5.41527 0.209680
\(668\) 0 0
\(669\) −5.90850 −0.228436
\(670\) 0 0
\(671\) −40.6018 −1.56742
\(672\) 0 0
\(673\) −10.1177 −0.390009 −0.195004 0.980802i \(-0.562472\pi\)
−0.195004 + 0.980802i \(0.562472\pi\)
\(674\) 0 0
\(675\) 4.89868 0.188550
\(676\) 0 0
\(677\) −16.0373 −0.616362 −0.308181 0.951328i \(-0.599720\pi\)
−0.308181 + 0.951328i \(0.599720\pi\)
\(678\) 0 0
\(679\) −16.1136 −0.618384
\(680\) 0 0
\(681\) 19.0500 0.729998
\(682\) 0 0
\(683\) 39.6137 1.51578 0.757888 0.652385i \(-0.226231\pi\)
0.757888 + 0.652385i \(0.226231\pi\)
\(684\) 0 0
\(685\) −5.77209 −0.220540
\(686\) 0 0
\(687\) 22.1819 0.846293
\(688\) 0 0
\(689\) 1.64209 0.0625585
\(690\) 0 0
\(691\) −6.69210 −0.254579 −0.127290 0.991866i \(-0.540628\pi\)
−0.127290 + 0.991866i \(0.540628\pi\)
\(692\) 0 0
\(693\) −3.71255 −0.141028
\(694\) 0 0
\(695\) −0.281544 −0.0106796
\(696\) 0 0
\(697\) 10.5942 0.401284
\(698\) 0 0
\(699\) 14.8712 0.562482
\(700\) 0 0
\(701\) −42.1778 −1.59303 −0.796517 0.604616i \(-0.793327\pi\)
−0.796517 + 0.604616i \(0.793327\pi\)
\(702\) 0 0
\(703\) 17.1093 0.645288
\(704\) 0 0
\(705\) 1.15620 0.0435451
\(706\) 0 0
\(707\) −13.2435 −0.498073
\(708\) 0 0
\(709\) 9.18707 0.345028 0.172514 0.985007i \(-0.444811\pi\)
0.172514 + 0.985007i \(0.444811\pi\)
\(710\) 0 0
\(711\) −15.1797 −0.569284
\(712\) 0 0
\(713\) −8.21699 −0.307729
\(714\) 0 0
\(715\) −3.15234 −0.117891
\(716\) 0 0
\(717\) −14.8008 −0.552745
\(718\) 0 0
\(719\) −15.1883 −0.566429 −0.283214 0.959057i \(-0.591401\pi\)
−0.283214 + 0.959057i \(0.591401\pi\)
\(720\) 0 0
\(721\) 3.12782 0.116486
\(722\) 0 0
\(723\) 27.6887 1.02975
\(724\) 0 0
\(725\) −26.5277 −0.985212
\(726\) 0 0
\(727\) −17.7830 −0.659535 −0.329768 0.944062i \(-0.606970\pi\)
−0.329768 + 0.944062i \(0.606970\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 19.0595 0.704942
\(732\) 0 0
\(733\) −13.6564 −0.504412 −0.252206 0.967674i \(-0.581156\pi\)
−0.252206 + 0.967674i \(0.581156\pi\)
\(734\) 0 0
\(735\) 0.318315 0.0117412
\(736\) 0 0
\(737\) 2.77616 0.102261
\(738\) 0 0
\(739\) 22.7127 0.835500 0.417750 0.908562i \(-0.362819\pi\)
0.417750 + 0.908562i \(0.362819\pi\)
\(740\) 0 0
\(741\) 6.26641 0.230202
\(742\) 0 0
\(743\) 6.81824 0.250137 0.125068 0.992148i \(-0.460085\pi\)
0.125068 + 0.992148i \(0.460085\pi\)
\(744\) 0 0
\(745\) −6.29276 −0.230549
\(746\) 0 0
\(747\) 1.11799 0.0409053
\(748\) 0 0
\(749\) 15.6631 0.572318
\(750\) 0 0
\(751\) 7.78505 0.284080 0.142040 0.989861i \(-0.454634\pi\)
0.142040 + 0.989861i \(0.454634\pi\)
\(752\) 0 0
\(753\) −25.1895 −0.917958
\(754\) 0 0
\(755\) 6.61872 0.240880
\(756\) 0 0
\(757\) 41.1096 1.49415 0.747076 0.664738i \(-0.231457\pi\)
0.747076 + 0.664738i \(0.231457\pi\)
\(758\) 0 0
\(759\) 3.71255 0.134757
\(760\) 0 0
\(761\) −29.8485 −1.08201 −0.541003 0.841021i \(-0.681955\pi\)
−0.541003 + 0.841021i \(0.681955\pi\)
\(762\) 0 0
\(763\) 8.75230 0.316854
\(764\) 0 0
\(765\) −0.705701 −0.0255147
\(766\) 0 0
\(767\) 26.1905 0.945683
\(768\) 0 0
\(769\) −33.6148 −1.21218 −0.606090 0.795396i \(-0.707263\pi\)
−0.606090 + 0.795396i \(0.707263\pi\)
\(770\) 0 0
\(771\) −4.89227 −0.176191
\(772\) 0 0
\(773\) 14.3212 0.515099 0.257549 0.966265i \(-0.417085\pi\)
0.257549 + 0.966265i \(0.417085\pi\)
\(774\) 0 0
\(775\) 40.2524 1.44591
\(776\) 0 0
\(777\) −7.28308 −0.261279
\(778\) 0 0
\(779\) 11.2259 0.402209
\(780\) 0 0
\(781\) 50.2342 1.79752
\(782\) 0 0
\(783\) −5.41527 −0.193526
\(784\) 0 0
\(785\) −3.60810 −0.128778
\(786\) 0 0
\(787\) 6.71066 0.239209 0.119605 0.992822i \(-0.461837\pi\)
0.119605 + 0.992822i \(0.461837\pi\)
\(788\) 0 0
\(789\) 23.1755 0.825070
\(790\) 0 0
\(791\) −15.8051 −0.561966
\(792\) 0 0
\(793\) −29.1727 −1.03595
\(794\) 0 0
\(795\) −0.195952 −0.00694970
\(796\) 0 0
\(797\) −18.4248 −0.652640 −0.326320 0.945259i \(-0.605809\pi\)
−0.326320 + 0.945259i \(0.605809\pi\)
\(798\) 0 0
\(799\) 8.05269 0.284884
\(800\) 0 0
\(801\) 6.32814 0.223594
\(802\) 0 0
\(803\) 14.2738 0.503710
\(804\) 0 0
\(805\) −0.318315 −0.0112191
\(806\) 0 0
\(807\) 24.3811 0.858256
\(808\) 0 0
\(809\) 18.6917 0.657164 0.328582 0.944475i \(-0.393429\pi\)
0.328582 + 0.944475i \(0.393429\pi\)
\(810\) 0 0
\(811\) −24.7543 −0.869242 −0.434621 0.900613i \(-0.643118\pi\)
−0.434621 + 0.900613i \(0.643118\pi\)
\(812\) 0 0
\(813\) 5.42853 0.190387
\(814\) 0 0
\(815\) 7.37007 0.258162
\(816\) 0 0
\(817\) 20.1960 0.706567
\(818\) 0 0
\(819\) −2.66749 −0.0932097
\(820\) 0 0
\(821\) 7.24866 0.252980 0.126490 0.991968i \(-0.459629\pi\)
0.126490 + 0.991968i \(0.459629\pi\)
\(822\) 0 0
\(823\) −11.1352 −0.388147 −0.194074 0.980987i \(-0.562170\pi\)
−0.194074 + 0.980987i \(0.562170\pi\)
\(824\) 0 0
\(825\) −18.1866 −0.633175
\(826\) 0 0
\(827\) 23.7059 0.824333 0.412167 0.911108i \(-0.364772\pi\)
0.412167 + 0.911108i \(0.364772\pi\)
\(828\) 0 0
\(829\) −8.28250 −0.287663 −0.143832 0.989602i \(-0.545942\pi\)
−0.143832 + 0.989602i \(0.545942\pi\)
\(830\) 0 0
\(831\) 15.5730 0.540222
\(832\) 0 0
\(833\) 2.21699 0.0768142
\(834\) 0 0
\(835\) −0.747779 −0.0258779
\(836\) 0 0
\(837\) 8.21699 0.284021
\(838\) 0 0
\(839\) 45.7672 1.58006 0.790030 0.613068i \(-0.210065\pi\)
0.790030 + 0.613068i \(0.210065\pi\)
\(840\) 0 0
\(841\) 0.325161 0.0112125
\(842\) 0 0
\(843\) −3.05736 −0.105301
\(844\) 0 0
\(845\) 1.87312 0.0644373
\(846\) 0 0
\(847\) 2.78301 0.0956253
\(848\) 0 0
\(849\) 4.24334 0.145631
\(850\) 0 0
\(851\) 7.28308 0.249661
\(852\) 0 0
\(853\) −54.8222 −1.87708 −0.938539 0.345173i \(-0.887820\pi\)
−0.938539 + 0.345173i \(0.887820\pi\)
\(854\) 0 0
\(855\) −0.747779 −0.0255735
\(856\) 0 0
\(857\) −45.7841 −1.56395 −0.781977 0.623307i \(-0.785789\pi\)
−0.781977 + 0.623307i \(0.785789\pi\)
\(858\) 0 0
\(859\) 28.8351 0.983840 0.491920 0.870641i \(-0.336295\pi\)
0.491920 + 0.870641i \(0.336295\pi\)
\(860\) 0 0
\(861\) −4.77864 −0.162856
\(862\) 0 0
\(863\) 21.1744 0.720785 0.360393 0.932801i \(-0.382643\pi\)
0.360393 + 0.932801i \(0.382643\pi\)
\(864\) 0 0
\(865\) 3.10226 0.105480
\(866\) 0 0
\(867\) 12.0850 0.410427
\(868\) 0 0
\(869\) 56.3554 1.91173
\(870\) 0 0
\(871\) 1.99469 0.0675876
\(872\) 0 0
\(873\) −16.1136 −0.545364
\(874\) 0 0
\(875\) 3.15090 0.106520
\(876\) 0 0
\(877\) 7.55728 0.255191 0.127596 0.991826i \(-0.459274\pi\)
0.127596 + 0.991826i \(0.459274\pi\)
\(878\) 0 0
\(879\) 18.6984 0.630680
\(880\) 0 0
\(881\) 11.0422 0.372022 0.186011 0.982548i \(-0.440444\pi\)
0.186011 + 0.982548i \(0.440444\pi\)
\(882\) 0 0
\(883\) 49.5381 1.66709 0.833544 0.552453i \(-0.186308\pi\)
0.833544 + 0.552453i \(0.186308\pi\)
\(884\) 0 0
\(885\) −3.12534 −0.105057
\(886\) 0 0
\(887\) −36.4475 −1.22379 −0.611894 0.790940i \(-0.709592\pi\)
−0.611894 + 0.790940i \(0.709592\pi\)
\(888\) 0 0
\(889\) −5.92706 −0.198787
\(890\) 0 0
\(891\) −3.71255 −0.124375
\(892\) 0 0
\(893\) 8.53283 0.285540
\(894\) 0 0
\(895\) −8.28154 −0.276821
\(896\) 0 0
\(897\) 2.66749 0.0890650
\(898\) 0 0
\(899\) −44.4972 −1.48407
\(900\) 0 0
\(901\) −1.36476 −0.0454668
\(902\) 0 0
\(903\) −8.59703 −0.286091
\(904\) 0 0
\(905\) −5.99861 −0.199401
\(906\) 0 0
\(907\) −10.5496 −0.350295 −0.175148 0.984542i \(-0.556040\pi\)
−0.175148 + 0.984542i \(0.556040\pi\)
\(908\) 0 0
\(909\) −13.2435 −0.439259
\(910\) 0 0
\(911\) 26.6343 0.882433 0.441217 0.897401i \(-0.354547\pi\)
0.441217 + 0.897401i \(0.354547\pi\)
\(912\) 0 0
\(913\) −4.15061 −0.137365
\(914\) 0 0
\(915\) 3.48122 0.115085
\(916\) 0 0
\(917\) −8.78316 −0.290045
\(918\) 0 0
\(919\) −13.8918 −0.458247 −0.229124 0.973397i \(-0.573586\pi\)
−0.229124 + 0.973397i \(0.573586\pi\)
\(920\) 0 0
\(921\) 25.3448 0.835140
\(922\) 0 0
\(923\) 36.0937 1.18804
\(924\) 0 0
\(925\) −35.6775 −1.17307
\(926\) 0 0
\(927\) 3.12782 0.102731
\(928\) 0 0
\(929\) −9.69538 −0.318095 −0.159048 0.987271i \(-0.550842\pi\)
−0.159048 + 0.987271i \(0.550842\pi\)
\(930\) 0 0
\(931\) 2.34918 0.0769912
\(932\) 0 0
\(933\) 14.5221 0.475431
\(934\) 0 0
\(935\) 2.61995 0.0856815
\(936\) 0 0
\(937\) −24.3715 −0.796181 −0.398091 0.917346i \(-0.630327\pi\)
−0.398091 + 0.917346i \(0.630327\pi\)
\(938\) 0 0
\(939\) −4.53778 −0.148085
\(940\) 0 0
\(941\) −14.1171 −0.460203 −0.230101 0.973167i \(-0.573906\pi\)
−0.230101 + 0.973167i \(0.573906\pi\)
\(942\) 0 0
\(943\) 4.77864 0.155614
\(944\) 0 0
\(945\) 0.318315 0.0103548
\(946\) 0 0
\(947\) −14.6097 −0.474751 −0.237375 0.971418i \(-0.576287\pi\)
−0.237375 + 0.971418i \(0.576287\pi\)
\(948\) 0 0
\(949\) 10.2558 0.332917
\(950\) 0 0
\(951\) −3.61996 −0.117385
\(952\) 0 0
\(953\) 17.5328 0.567944 0.283972 0.958833i \(-0.408348\pi\)
0.283972 + 0.958833i \(0.408348\pi\)
\(954\) 0 0
\(955\) 3.24488 0.105002
\(956\) 0 0
\(957\) 20.1045 0.649885
\(958\) 0 0
\(959\) 18.1333 0.585554
\(960\) 0 0
\(961\) 36.5189 1.17803
\(962\) 0 0
\(963\) 15.6631 0.504737
\(964\) 0 0
\(965\) −3.73577 −0.120259
\(966\) 0 0
\(967\) −46.8221 −1.50570 −0.752849 0.658194i \(-0.771321\pi\)
−0.752849 + 0.658194i \(0.771321\pi\)
\(968\) 0 0
\(969\) −5.20810 −0.167308
\(970\) 0 0
\(971\) 55.9608 1.79587 0.897934 0.440129i \(-0.145067\pi\)
0.897934 + 0.440129i \(0.145067\pi\)
\(972\) 0 0
\(973\) 0.884483 0.0283552
\(974\) 0 0
\(975\) −13.0672 −0.418485
\(976\) 0 0
\(977\) 12.7251 0.407113 0.203557 0.979063i \(-0.434750\pi\)
0.203557 + 0.979063i \(0.434750\pi\)
\(978\) 0 0
\(979\) −23.4935 −0.750856
\(980\) 0 0
\(981\) 8.75230 0.279439
\(982\) 0 0
\(983\) 28.5147 0.909478 0.454739 0.890625i \(-0.349733\pi\)
0.454739 + 0.890625i \(0.349733\pi\)
\(984\) 0 0
\(985\) −4.91986 −0.156760
\(986\) 0 0
\(987\) −3.63226 −0.115616
\(988\) 0 0
\(989\) 8.59703 0.273370
\(990\) 0 0
\(991\) −0.993154 −0.0315486 −0.0157743 0.999876i \(-0.505021\pi\)
−0.0157743 + 0.999876i \(0.505021\pi\)
\(992\) 0 0
\(993\) 12.5749 0.399053
\(994\) 0 0
\(995\) 5.36119 0.169961
\(996\) 0 0
\(997\) 39.8410 1.26178 0.630888 0.775874i \(-0.282691\pi\)
0.630888 + 0.775874i \(0.282691\pi\)
\(998\) 0 0
\(999\) −7.28308 −0.230427
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.bz.1.3 4
4.3 odd 2 3864.2.a.t.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.t.1.3 4 4.3 odd 2
7728.2.a.bz.1.3 4 1.1 even 1 trivial