Properties

Label 1536.2.d.d.769.4
Level $1536$
Weight $2$
Character 1536.769
Analytic conductor $12.265$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2650217505\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.4
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1536.769
Dual form 1536.2.d.d.769.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.41421i q^{5} +1.41421 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.41421i q^{5} +1.41421 q^{7} -1.00000 q^{9} -2.00000i q^{11} -1.41421 q^{15} +2.00000 q^{17} +4.00000i q^{19} +1.41421i q^{21} +2.82843 q^{23} +3.00000 q^{25} -1.00000i q^{27} +9.89949i q^{29} +7.07107 q^{31} +2.00000 q^{33} +2.00000i q^{35} +8.48528i q^{37} -6.00000 q^{41} -8.00000i q^{43} -1.41421i q^{45} -2.82843 q^{47} -5.00000 q^{49} +2.00000i q^{51} +1.41421i q^{53} +2.82843 q^{55} -4.00000 q^{57} -12.0000i q^{59} +14.1421i q^{61} -1.41421 q^{63} +8.00000i q^{67} +2.82843i q^{69} -14.1421 q^{71} +8.00000 q^{73} +3.00000i q^{75} -2.82843i q^{77} +4.24264 q^{79} +1.00000 q^{81} +6.00000i q^{83} +2.82843i q^{85} -9.89949 q^{87} -2.00000 q^{89} +7.07107i q^{93} -5.65685 q^{95} -14.0000 q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{9} + 8 q^{17} + 12 q^{25} + 8 q^{33} - 24 q^{41} - 20 q^{49} - 16 q^{57} + 32 q^{73} + 4 q^{81} - 8 q^{89} - 56 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(517\) \(1025\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.41421i 0.632456i 0.948683 + 0.316228i \(0.102416\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 0 0
\(7\) 1.41421 0.534522 0.267261 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 1.41421i 0.308607i
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 9.89949i 1.83829i 0.393919 + 0.919145i \(0.371119\pi\)
−0.393919 + 0.919145i \(0.628881\pi\)
\(30\) 0 0
\(31\) 7.07107 1.27000 0.635001 0.772512i \(-0.281000\pi\)
0.635001 + 0.772512i \(0.281000\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 2.00000i 0.338062i
\(36\) 0 0
\(37\) 8.48528i 1.39497i 0.716599 + 0.697486i \(0.245698\pi\)
−0.716599 + 0.697486i \(0.754302\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) − 1.41421i − 0.210819i
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 2.00000i 0.280056i
\(52\) 0 0
\(53\) 1.41421i 0.194257i 0.995272 + 0.0971286i \(0.0309658\pi\)
−0.995272 + 0.0971286i \(0.969034\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) − 12.0000i − 1.56227i −0.624364 0.781133i \(-0.714642\pi\)
0.624364 0.781133i \(-0.285358\pi\)
\(60\) 0 0
\(61\) 14.1421i 1.81071i 0.424650 + 0.905357i \(0.360397\pi\)
−0.424650 + 0.905357i \(0.639603\pi\)
\(62\) 0 0
\(63\) −1.41421 −0.178174
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 2.82843i 0.340503i
\(70\) 0 0
\(71\) −14.1421 −1.67836 −0.839181 0.543852i \(-0.816965\pi\)
−0.839181 + 0.543852i \(0.816965\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 3.00000i 0.346410i
\(76\) 0 0
\(77\) − 2.82843i − 0.322329i
\(78\) 0 0
\(79\) 4.24264 0.477334 0.238667 0.971101i \(-0.423290\pi\)
0.238667 + 0.971101i \(0.423290\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 2.82843i 0.306786i
\(86\) 0 0
\(87\) −9.89949 −1.06134
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7.07107i 0.733236i
\(94\) 0 0
\(95\) −5.65685 −0.580381
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 2.00000i 0.201008i
\(100\) 0 0
\(101\) 9.89949i 0.985037i 0.870302 + 0.492518i \(0.163924\pi\)
−0.870302 + 0.492518i \(0.836076\pi\)
\(102\) 0 0
\(103\) 12.7279 1.25412 0.627060 0.778971i \(-0.284258\pi\)
0.627060 + 0.778971i \(0.284258\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 0 0
\(109\) − 11.3137i − 1.08366i −0.840489 0.541828i \(-0.817732\pi\)
0.840489 0.541828i \(-0.182268\pi\)
\(110\) 0 0
\(111\) −8.48528 −0.805387
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 4.00000i 0.373002i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.82843 0.259281
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) − 6.00000i − 0.541002i
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) 4.24264 0.376473 0.188237 0.982124i \(-0.439723\pi\)
0.188237 + 0.982124i \(0.439723\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 12.0000i 1.04844i 0.851581 + 0.524222i \(0.175644\pi\)
−0.851581 + 0.524222i \(0.824356\pi\)
\(132\) 0 0
\(133\) 5.65685i 0.490511i
\(134\) 0 0
\(135\) 1.41421 0.121716
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) − 2.82843i − 0.238197i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −14.0000 −1.16264
\(146\) 0 0
\(147\) − 5.00000i − 0.412393i
\(148\) 0 0
\(149\) − 4.24264i − 0.347571i −0.984784 0.173785i \(-0.944400\pi\)
0.984784 0.173785i \(-0.0555999\pi\)
\(150\) 0 0
\(151\) 4.24264 0.345261 0.172631 0.984987i \(-0.444773\pi\)
0.172631 + 0.984987i \(0.444773\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 10.0000i 0.803219i
\(156\) 0 0
\(157\) 14.1421i 1.12867i 0.825547 + 0.564333i \(0.190866\pi\)
−0.825547 + 0.564333i \(0.809134\pi\)
\(158\) 0 0
\(159\) −1.41421 −0.112154
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 0 0
\(165\) 2.82843i 0.220193i
\(166\) 0 0
\(167\) −11.3137 −0.875481 −0.437741 0.899101i \(-0.644221\pi\)
−0.437741 + 0.899101i \(0.644221\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) − 4.00000i − 0.305888i
\(172\) 0 0
\(173\) − 15.5563i − 1.18273i −0.806405 0.591364i \(-0.798590\pi\)
0.806405 0.591364i \(-0.201410\pi\)
\(174\) 0 0
\(175\) 4.24264 0.320713
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) − 4.00000i − 0.298974i −0.988764 0.149487i \(-0.952238\pi\)
0.988764 0.149487i \(-0.0477622\pi\)
\(180\) 0 0
\(181\) − 11.3137i − 0.840941i −0.907306 0.420471i \(-0.861865\pi\)
0.907306 0.420471i \(-0.138135\pi\)
\(182\) 0 0
\(183\) −14.1421 −1.04542
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) − 4.00000i − 0.292509i
\(188\) 0 0
\(189\) − 1.41421i − 0.102869i
\(190\) 0 0
\(191\) −22.6274 −1.63726 −0.818631 0.574320i \(-0.805267\pi\)
−0.818631 + 0.574320i \(0.805267\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.07107i 0.503793i 0.967754 + 0.251896i \(0.0810542\pi\)
−0.967754 + 0.251896i \(0.918946\pi\)
\(198\) 0 0
\(199\) 24.0416 1.70427 0.852133 0.523325i \(-0.175309\pi\)
0.852133 + 0.523325i \(0.175309\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) 14.0000i 0.982607i
\(204\) 0 0
\(205\) − 8.48528i − 0.592638i
\(206\) 0 0
\(207\) −2.82843 −0.196589
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) − 14.1421i − 0.969003i
\(214\) 0 0
\(215\) 11.3137 0.771589
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 0 0
\(219\) 8.00000i 0.540590i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 12.7279 0.852325 0.426162 0.904647i \(-0.359865\pi\)
0.426162 + 0.904647i \(0.359865\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) − 22.0000i − 1.46019i −0.683345 0.730096i \(-0.739475\pi\)
0.683345 0.730096i \(-0.260525\pi\)
\(228\) 0 0
\(229\) − 16.9706i − 1.12145i −0.828003 0.560723i \(-0.810523\pi\)
0.828003 0.560723i \(-0.189477\pi\)
\(230\) 0 0
\(231\) 2.82843 0.186097
\(232\) 0 0
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) − 4.00000i − 0.260931i
\(236\) 0 0
\(237\) 4.24264i 0.275589i
\(238\) 0 0
\(239\) 22.6274 1.46365 0.731823 0.681495i \(-0.238670\pi\)
0.731823 + 0.681495i \(0.238670\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) − 7.07107i − 0.451754i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 6.00000i 0.378717i 0.981908 + 0.189358i \(0.0606408\pi\)
−0.981908 + 0.189358i \(0.939359\pi\)
\(252\) 0 0
\(253\) − 5.65685i − 0.355643i
\(254\) 0 0
\(255\) −2.82843 −0.177123
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 12.0000i 0.745644i
\(260\) 0 0
\(261\) − 9.89949i − 0.612763i
\(262\) 0 0
\(263\) −28.2843 −1.74408 −0.872041 0.489432i \(-0.837204\pi\)
−0.872041 + 0.489432i \(0.837204\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) − 2.00000i − 0.122398i
\(268\) 0 0
\(269\) − 21.2132i − 1.29339i −0.762748 0.646696i \(-0.776150\pi\)
0.762748 0.646696i \(-0.223850\pi\)
\(270\) 0 0
\(271\) 24.0416 1.46043 0.730213 0.683220i \(-0.239421\pi\)
0.730213 + 0.683220i \(0.239421\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 6.00000i − 0.361814i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −7.07107 −0.423334
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 20.0000i 1.18888i 0.804141 + 0.594438i \(0.202626\pi\)
−0.804141 + 0.594438i \(0.797374\pi\)
\(284\) 0 0
\(285\) − 5.65685i − 0.335083i
\(286\) 0 0
\(287\) −8.48528 −0.500870
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 14.0000i − 0.820695i
\(292\) 0 0
\(293\) − 9.89949i − 0.578335i −0.957279 0.289167i \(-0.906622\pi\)
0.957279 0.289167i \(-0.0933784\pi\)
\(294\) 0 0
\(295\) 16.9706 0.988064
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 11.3137i − 0.652111i
\(302\) 0 0
\(303\) −9.89949 −0.568711
\(304\) 0 0
\(305\) −20.0000 −1.14520
\(306\) 0 0
\(307\) − 32.0000i − 1.82634i −0.407583 0.913168i \(-0.633628\pi\)
0.407583 0.913168i \(-0.366372\pi\)
\(308\) 0 0
\(309\) 12.7279i 0.724066i
\(310\) 0 0
\(311\) 22.6274 1.28308 0.641542 0.767088i \(-0.278295\pi\)
0.641542 + 0.767088i \(0.278295\pi\)
\(312\) 0 0
\(313\) 4.00000 0.226093 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(314\) 0 0
\(315\) − 2.00000i − 0.112687i
\(316\) 0 0
\(317\) − 9.89949i − 0.556011i −0.960579 0.278006i \(-0.910327\pi\)
0.960579 0.278006i \(-0.0896734\pi\)
\(318\) 0 0
\(319\) 19.7990 1.10853
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.3137 0.625650
\(328\) 0 0
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 32.0000i 1.75888i 0.476011 + 0.879440i \(0.342082\pi\)
−0.476011 + 0.879440i \(0.657918\pi\)
\(332\) 0 0
\(333\) − 8.48528i − 0.464991i
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) 6.00000i 0.325875i
\(340\) 0 0
\(341\) − 14.1421i − 0.765840i
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) 26.0000i 1.39575i 0.716218 + 0.697877i \(0.245872\pi\)
−0.716218 + 0.697877i \(0.754128\pi\)
\(348\) 0 0
\(349\) − 25.4558i − 1.36262i −0.731995 0.681310i \(-0.761411\pi\)
0.731995 0.681310i \(-0.238589\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) − 20.0000i − 1.06149i
\(356\) 0 0
\(357\) 2.82843i 0.149696i
\(358\) 0 0
\(359\) −19.7990 −1.04495 −0.522475 0.852654i \(-0.674991\pi\)
−0.522475 + 0.852654i \(0.674991\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 11.3137i 0.592187i
\(366\) 0 0
\(367\) −4.24264 −0.221464 −0.110732 0.993850i \(-0.535320\pi\)
−0.110732 + 0.993850i \(0.535320\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 2.00000i 0.103835i
\(372\) 0 0
\(373\) − 19.7990i − 1.02515i −0.858642 0.512576i \(-0.828691\pi\)
0.858642 0.512576i \(-0.171309\pi\)
\(374\) 0 0
\(375\) −11.3137 −0.584237
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 20.0000i − 1.02733i −0.857991 0.513665i \(-0.828287\pi\)
0.857991 0.513665i \(-0.171713\pi\)
\(380\) 0 0
\(381\) 4.24264i 0.217357i
\(382\) 0 0
\(383\) 5.65685 0.289052 0.144526 0.989501i \(-0.453834\pi\)
0.144526 + 0.989501i \(0.453834\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 8.00000i 0.406663i
\(388\) 0 0
\(389\) − 24.0416i − 1.21896i −0.792802 0.609480i \(-0.791378\pi\)
0.792802 0.609480i \(-0.208622\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 6.00000i 0.301893i
\(396\) 0 0
\(397\) 2.82843i 0.141955i 0.997478 + 0.0709773i \(0.0226118\pi\)
−0.997478 + 0.0709773i \(0.977388\pi\)
\(398\) 0 0
\(399\) −5.65685 −0.283197
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.41421i 0.0702728i
\(406\) 0 0
\(407\) 16.9706 0.841200
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 10.0000i 0.493264i
\(412\) 0 0
\(413\) − 16.9706i − 0.835067i
\(414\) 0 0
\(415\) −8.48528 −0.416526
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 30.0000i − 1.46560i −0.680446 0.732798i \(-0.738214\pi\)
0.680446 0.732798i \(-0.261786\pi\)
\(420\) 0 0
\(421\) − 5.65685i − 0.275698i −0.990453 0.137849i \(-0.955981\pi\)
0.990453 0.137849i \(-0.0440189\pi\)
\(422\) 0 0
\(423\) 2.82843 0.137523
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 20.0000i 0.967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.82843 −0.136241 −0.0681203 0.997677i \(-0.521700\pi\)
−0.0681203 + 0.997677i \(0.521700\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) − 14.0000i − 0.671249i
\(436\) 0 0
\(437\) 11.3137i 0.541208i
\(438\) 0 0
\(439\) −4.24264 −0.202490 −0.101245 0.994862i \(-0.532283\pi\)
−0.101245 + 0.994862i \(0.532283\pi\)
\(440\) 0 0
\(441\) 5.00000 0.238095
\(442\) 0 0
\(443\) − 30.0000i − 1.42534i −0.701498 0.712672i \(-0.747485\pi\)
0.701498 0.712672i \(-0.252515\pi\)
\(444\) 0 0
\(445\) − 2.82843i − 0.134080i
\(446\) 0 0
\(447\) 4.24264 0.200670
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) 12.0000i 0.565058i
\(452\) 0 0
\(453\) 4.24264i 0.199337i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) 0 0
\(459\) − 2.00000i − 0.0933520i
\(460\) 0 0
\(461\) 12.7279i 0.592798i 0.955064 + 0.296399i \(0.0957859\pi\)
−0.955064 + 0.296399i \(0.904214\pi\)
\(462\) 0 0
\(463\) −35.3553 −1.64310 −0.821551 0.570135i \(-0.806891\pi\)
−0.821551 + 0.570135i \(0.806891\pi\)
\(464\) 0 0
\(465\) −10.0000 −0.463739
\(466\) 0 0
\(467\) − 22.0000i − 1.01804i −0.860755 0.509019i \(-0.830008\pi\)
0.860755 0.509019i \(-0.169992\pi\)
\(468\) 0 0
\(469\) 11.3137i 0.522419i
\(470\) 0 0
\(471\) −14.1421 −0.651635
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 12.0000i 0.550598i
\(476\) 0 0
\(477\) − 1.41421i − 0.0647524i
\(478\) 0 0
\(479\) −25.4558 −1.16311 −0.581554 0.813508i \(-0.697555\pi\)
−0.581554 + 0.813508i \(0.697555\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 4.00000i 0.182006i
\(484\) 0 0
\(485\) − 19.7990i − 0.899026i
\(486\) 0 0
\(487\) −4.24264 −0.192252 −0.0961262 0.995369i \(-0.530645\pi\)
−0.0961262 + 0.995369i \(0.530645\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) − 28.0000i − 1.26362i −0.775122 0.631811i \(-0.782312\pi\)
0.775122 0.631811i \(-0.217688\pi\)
\(492\) 0 0
\(493\) 19.7990i 0.891702i
\(494\) 0 0
\(495\) −2.82843 −0.127128
\(496\) 0 0
\(497\) −20.0000 −0.897123
\(498\) 0 0
\(499\) 20.0000i 0.895323i 0.894203 + 0.447661i \(0.147743\pi\)
−0.894203 + 0.447661i \(0.852257\pi\)
\(500\) 0 0
\(501\) − 11.3137i − 0.505459i
\(502\) 0 0
\(503\) 2.82843 0.126113 0.0630567 0.998010i \(-0.479915\pi\)
0.0630567 + 0.998010i \(0.479915\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 0 0
\(507\) 13.0000i 0.577350i
\(508\) 0 0
\(509\) − 24.0416i − 1.06563i −0.846233 0.532813i \(-0.821135\pi\)
0.846233 0.532813i \(-0.178865\pi\)
\(510\) 0 0
\(511\) 11.3137 0.500489
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) 18.0000i 0.793175i
\(516\) 0 0
\(517\) 5.65685i 0.248788i
\(518\) 0 0
\(519\) 15.5563 0.682848
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 0 0
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 0 0
\(525\) 4.24264i 0.185164i
\(526\) 0 0
\(527\) 14.1421 0.616041
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 12.0000i 0.520756i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 5.65685 0.244567
\(536\) 0 0
\(537\) 4.00000 0.172613
\(538\) 0 0
\(539\) 10.0000i 0.430730i
\(540\) 0 0
\(541\) − 28.2843i − 1.21604i −0.793923 0.608018i \(-0.791965\pi\)
0.793923 0.608018i \(-0.208035\pi\)
\(542\) 0 0
\(543\) 11.3137 0.485518
\(544\) 0 0
\(545\) 16.0000 0.685365
\(546\) 0 0
\(547\) − 16.0000i − 0.684111i −0.939680 0.342055i \(-0.888877\pi\)
0.939680 0.342055i \(-0.111123\pi\)
\(548\) 0 0
\(549\) − 14.1421i − 0.603572i
\(550\) 0 0
\(551\) −39.5980 −1.68693
\(552\) 0 0
\(553\) 6.00000 0.255146
\(554\) 0 0
\(555\) − 12.0000i − 0.509372i
\(556\) 0 0
\(557\) 38.1838i 1.61790i 0.587879 + 0.808949i \(0.299963\pi\)
−0.587879 + 0.808949i \(0.700037\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) 0 0
\(563\) − 6.00000i − 0.252870i −0.991975 0.126435i \(-0.959647\pi\)
0.991975 0.126435i \(-0.0403535\pi\)
\(564\) 0 0
\(565\) 8.48528i 0.356978i
\(566\) 0 0
\(567\) 1.41421 0.0593914
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) − 28.0000i − 1.17176i −0.810397 0.585882i \(-0.800748\pi\)
0.810397 0.585882i \(-0.199252\pi\)
\(572\) 0 0
\(573\) − 22.6274i − 0.945274i
\(574\) 0 0
\(575\) 8.48528 0.353861
\(576\) 0 0
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) 0 0
\(579\) − 4.00000i − 0.166234i
\(580\) 0 0
\(581\) 8.48528i 0.352029i
\(582\) 0 0
\(583\) 2.82843 0.117141
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.0000i 1.15568i 0.816149 + 0.577842i \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) 0 0
\(589\) 28.2843i 1.16543i
\(590\) 0 0
\(591\) −7.07107 −0.290865
\(592\) 0 0
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) 4.00000i 0.163984i
\(596\) 0 0
\(597\) 24.0416i 0.983958i
\(598\) 0 0
\(599\) 36.7696 1.50236 0.751182 0.660096i \(-0.229484\pi\)
0.751182 + 0.660096i \(0.229484\pi\)
\(600\) 0 0
\(601\) −44.0000 −1.79480 −0.897399 0.441221i \(-0.854546\pi\)
−0.897399 + 0.441221i \(0.854546\pi\)
\(602\) 0 0
\(603\) − 8.00000i − 0.325785i
\(604\) 0 0
\(605\) 9.89949i 0.402472i
\(606\) 0 0
\(607\) 4.24264 0.172203 0.0861017 0.996286i \(-0.472559\pi\)
0.0861017 + 0.996286i \(0.472559\pi\)
\(608\) 0 0
\(609\) −14.0000 −0.567309
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.48528i 0.342717i 0.985209 + 0.171359i \(0.0548157\pi\)
−0.985209 + 0.171359i \(0.945184\pi\)
\(614\) 0 0
\(615\) 8.48528 0.342160
\(616\) 0 0
\(617\) −46.0000 −1.85189 −0.925945 0.377658i \(-0.876729\pi\)
−0.925945 + 0.377658i \(0.876729\pi\)
\(618\) 0 0
\(619\) − 36.0000i − 1.44696i −0.690344 0.723481i \(-0.742541\pi\)
0.690344 0.723481i \(-0.257459\pi\)
\(620\) 0 0
\(621\) − 2.82843i − 0.113501i
\(622\) 0 0
\(623\) −2.82843 −0.113319
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 8.00000i 0.319489i
\(628\) 0 0
\(629\) 16.9706i 0.676661i
\(630\) 0 0
\(631\) −15.5563 −0.619288 −0.309644 0.950852i \(-0.600210\pi\)
−0.309644 + 0.950852i \(0.600210\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.00000i 0.238103i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 14.1421 0.559454
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 40.0000i 1.57745i 0.614749 + 0.788723i \(0.289257\pi\)
−0.614749 + 0.788723i \(0.710743\pi\)
\(644\) 0 0
\(645\) 11.3137i 0.445477i
\(646\) 0 0
\(647\) −14.1421 −0.555985 −0.277992 0.960583i \(-0.589669\pi\)
−0.277992 + 0.960583i \(0.589669\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 10.0000i 0.391931i
\(652\) 0 0
\(653\) 29.6985i 1.16219i 0.813835 + 0.581096i \(0.197376\pi\)
−0.813835 + 0.581096i \(0.802624\pi\)
\(654\) 0 0
\(655\) −16.9706 −0.663095
\(656\) 0 0
\(657\) −8.00000 −0.312110
\(658\) 0 0
\(659\) 44.0000i 1.71400i 0.515319 + 0.856998i \(0.327673\pi\)
−0.515319 + 0.856998i \(0.672327\pi\)
\(660\) 0 0
\(661\) 19.7990i 0.770091i 0.922897 + 0.385046i \(0.125814\pi\)
−0.922897 + 0.385046i \(0.874186\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 28.0000i 1.08416i
\(668\) 0 0
\(669\) 12.7279i 0.492090i
\(670\) 0 0
\(671\) 28.2843 1.09190
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 0 0
\(675\) − 3.00000i − 0.115470i
\(676\) 0 0
\(677\) 24.0416i 0.923995i 0.886881 + 0.461997i \(0.152867\pi\)
−0.886881 + 0.461997i \(0.847133\pi\)
\(678\) 0 0
\(679\) −19.7990 −0.759815
\(680\) 0 0
\(681\) 22.0000 0.843042
\(682\) 0 0
\(683\) − 26.0000i − 0.994862i −0.867503 0.497431i \(-0.834277\pi\)
0.867503 0.497431i \(-0.165723\pi\)
\(684\) 0 0
\(685\) 14.1421i 0.540343i
\(686\) 0 0
\(687\) 16.9706 0.647467
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 8.00000i − 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) 0 0
\(693\) 2.82843i 0.107443i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 0 0
\(699\) − 22.0000i − 0.832116i
\(700\) 0 0
\(701\) − 4.24264i − 0.160242i −0.996785 0.0801212i \(-0.974469\pi\)
0.996785 0.0801212i \(-0.0255307\pi\)
\(702\) 0 0
\(703\) −33.9411 −1.28011
\(704\) 0 0
\(705\) 4.00000 0.150649
\(706\) 0 0
\(707\) 14.0000i 0.526524i
\(708\) 0 0
\(709\) − 33.9411i − 1.27469i −0.770580 0.637343i \(-0.780034\pi\)
0.770580 0.637343i \(-0.219966\pi\)
\(710\) 0 0
\(711\) −4.24264 −0.159111
\(712\) 0 0
\(713\) 20.0000 0.749006
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 22.6274i 0.845036i
\(718\) 0 0
\(719\) 25.4558 0.949343 0.474671 0.880163i \(-0.342567\pi\)
0.474671 + 0.880163i \(0.342567\pi\)
\(720\) 0 0
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 29.6985i 1.10297i
\(726\) 0 0
\(727\) −24.0416 −0.891655 −0.445827 0.895119i \(-0.647090\pi\)
−0.445827 + 0.895119i \(0.647090\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 16.0000i − 0.591781i
\(732\) 0 0
\(733\) − 11.3137i − 0.417881i −0.977928 0.208941i \(-0.932998\pi\)
0.977928 0.208941i \(-0.0670016\pi\)
\(734\) 0 0
\(735\) 7.07107 0.260820
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 12.0000i 0.441427i 0.975339 + 0.220714i \(0.0708386\pi\)
−0.975339 + 0.220714i \(0.929161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.2843 1.03765 0.518825 0.854881i \(-0.326370\pi\)
0.518825 + 0.854881i \(0.326370\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) − 6.00000i − 0.219529i
\(748\) 0 0
\(749\) − 5.65685i − 0.206697i
\(750\) 0 0
\(751\) −12.7279 −0.464448 −0.232224 0.972662i \(-0.574600\pi\)
−0.232224 + 0.972662i \(0.574600\pi\)
\(752\) 0 0
\(753\) −6.00000 −0.218652
\(754\) 0 0
\(755\) 6.00000i 0.218362i
\(756\) 0 0
\(757\) 50.9117i 1.85042i 0.379459 + 0.925208i \(0.376110\pi\)
−0.379459 + 0.925208i \(0.623890\pi\)
\(758\) 0 0
\(759\) 5.65685 0.205331
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) − 16.0000i − 0.579239i
\(764\) 0 0
\(765\) − 2.82843i − 0.102262i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 20.0000 0.721218 0.360609 0.932717i \(-0.382569\pi\)
0.360609 + 0.932717i \(0.382569\pi\)
\(770\) 0 0
\(771\) 18.0000i 0.648254i
\(772\) 0 0
\(773\) 15.5563i 0.559523i 0.960070 + 0.279761i \(0.0902554\pi\)
−0.960070 + 0.279761i \(0.909745\pi\)
\(774\) 0 0
\(775\) 21.2132 0.762001
\(776\) 0 0
\(777\) −12.0000 −0.430498
\(778\) 0 0
\(779\) − 24.0000i − 0.859889i
\(780\) 0 0
\(781\) 28.2843i 1.01209i
\(782\) 0 0
\(783\) 9.89949 0.353779
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 0 0
\(789\) − 28.2843i − 1.00695i
\(790\) 0 0
\(791\) 8.48528 0.301702
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) − 2.00000i − 0.0709327i
\(796\) 0 0
\(797\) − 1.41421i − 0.0500940i −0.999686 0.0250470i \(-0.992026\pi\)
0.999686 0.0250470i \(-0.00797354\pi\)
\(798\) 0 0
\(799\) −5.65685 −0.200125
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 0 0
\(803\) − 16.0000i − 0.564628i
\(804\) 0 0
\(805\) 5.65685i 0.199378i
\(806\) 0 0
\(807\) 21.2132 0.746740
\(808\) 0 0
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) − 28.0000i − 0.983213i −0.870817 0.491606i \(-0.836410\pi\)
0.870817 0.491606i \(-0.163590\pi\)
\(812\) 0 0
\(813\) 24.0416i 0.843177i
\(814\) 0 0
\(815\) 22.6274 0.792604
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9.89949i 0.345495i 0.984966 + 0.172747i \(0.0552644\pi\)
−0.984966 + 0.172747i \(0.944736\pi\)
\(822\) 0 0
\(823\) −21.2132 −0.739446 −0.369723 0.929142i \(-0.620547\pi\)
−0.369723 + 0.929142i \(0.620547\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) − 28.0000i − 0.973655i −0.873498 0.486828i \(-0.838154\pi\)
0.873498 0.486828i \(-0.161846\pi\)
\(828\) 0 0
\(829\) 11.3137i 0.392941i 0.980510 + 0.196471i \(0.0629480\pi\)
−0.980510 + 0.196471i \(0.937052\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −10.0000 −0.346479
\(834\) 0 0
\(835\) − 16.0000i − 0.553703i
\(836\) 0 0
\(837\) − 7.07107i − 0.244412i
\(838\) 0 0
\(839\) 14.1421 0.488241 0.244120 0.969745i \(-0.421501\pi\)
0.244120 + 0.969745i \(0.421501\pi\)
\(840\) 0 0
\(841\) −69.0000 −2.37931
\(842\) 0 0
\(843\) 2.00000i 0.0688837i
\(844\) 0 0
\(845\) 18.3848i 0.632456i
\(846\) 0 0
\(847\) 9.89949 0.340151
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 24.0000i 0.822709i
\(852\) 0 0
\(853\) − 14.1421i − 0.484218i −0.970249 0.242109i \(-0.922161\pi\)
0.970249 0.242109i \(-0.0778391\pi\)
\(854\) 0 0
\(855\) 5.65685 0.193460
\(856\) 0 0
\(857\) 26.0000 0.888143 0.444072 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) 0 0
\(859\) − 8.00000i − 0.272956i −0.990643 0.136478i \(-0.956422\pi\)
0.990643 0.136478i \(-0.0435784\pi\)
\(860\) 0 0
\(861\) − 8.48528i − 0.289178i
\(862\) 0 0
\(863\) −22.6274 −0.770246 −0.385123 0.922865i \(-0.625841\pi\)
−0.385123 + 0.922865i \(0.625841\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 0 0
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) − 8.48528i − 0.287843i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 16.0000i 0.540899i
\(876\) 0 0
\(877\) 42.4264i 1.43264i 0.697773 + 0.716319i \(0.254174\pi\)
−0.697773 + 0.716319i \(0.745826\pi\)
\(878\) 0 0
\(879\) 9.89949 0.333902
\(880\) 0 0
\(881\) −22.0000 −0.741199 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) 0 0
\(883\) 48.0000i 1.61533i 0.589643 + 0.807664i \(0.299269\pi\)
−0.589643 + 0.807664i \(0.700731\pi\)
\(884\) 0 0
\(885\) 16.9706i 0.570459i
\(886\) 0 0
\(887\) 50.9117 1.70945 0.854724 0.519083i \(-0.173727\pi\)
0.854724 + 0.519083i \(0.173727\pi\)
\(888\) 0 0
\(889\) 6.00000 0.201234
\(890\) 0 0
\(891\) − 2.00000i − 0.0670025i
\(892\) 0 0
\(893\) − 11.3137i − 0.378599i
\(894\) 0 0
\(895\) 5.65685 0.189088
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 70.0000i 2.33463i
\(900\) 0 0
\(901\) 2.82843i 0.0942286i
\(902\) 0 0
\(903\) 11.3137 0.376497
\(904\) 0 0
\(905\) 16.0000 0.531858
\(906\) 0 0
\(907\) − 8.00000i − 0.265636i −0.991140 0.132818i \(-0.957597\pi\)
0.991140 0.132818i \(-0.0424025\pi\)
\(908\) 0 0
\(909\) − 9.89949i − 0.328346i
\(910\) 0 0
\(911\) −5.65685 −0.187420 −0.0937100 0.995600i \(-0.529873\pi\)
−0.0937100 + 0.995600i \(0.529873\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) − 20.0000i − 0.661180i
\(916\) 0 0
\(917\) 16.9706i 0.560417i
\(918\) 0 0
\(919\) −21.2132 −0.699759 −0.349880 0.936795i \(-0.613777\pi\)
−0.349880 + 0.936795i \(0.613777\pi\)
\(920\) 0 0
\(921\) 32.0000 1.05444
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 25.4558i 0.836983i
\(926\) 0 0
\(927\) −12.7279 −0.418040
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) − 20.0000i − 0.655474i
\(932\) 0 0
\(933\) 22.6274i 0.740788i
\(934\) 0 0
\(935\) 5.65685 0.184999
\(936\) 0 0
\(937\) 18.0000 0.588034 0.294017 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) 0 0
\(939\) 4.00000i 0.130535i
\(940\) 0 0
\(941\) 1.41421i 0.0461020i 0.999734 + 0.0230510i \(0.00733802\pi\)
−0.999734 + 0.0230510i \(0.992662\pi\)
\(942\) 0 0
\(943\) −16.9706 −0.552638
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) − 28.0000i − 0.909878i −0.890523 0.454939i \(-0.849661\pi\)
0.890523 0.454939i \(-0.150339\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 9.89949 0.321013
\(952\) 0 0
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) − 32.0000i − 1.03550i
\(956\) 0 0
\(957\) 19.7990i 0.640010i
\(958\) 0 0
\(959\) 14.1421 0.456673
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 4.00000i 0.128898i
\(964\) 0 0
\(965\) − 5.65685i − 0.182101i
\(966\) 0 0
\(967\) −52.3259 −1.68269 −0.841344 0.540500i \(-0.818235\pi\)
−0.841344 + 0.540500i \(0.818235\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) 22.0000i 0.706014i 0.935621 + 0.353007i \(0.114841\pi\)
−0.935621 + 0.353007i \(0.885159\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 50.0000 1.59964 0.799821 0.600239i \(-0.204928\pi\)
0.799821 + 0.600239i \(0.204928\pi\)
\(978\) 0 0
\(979\) 4.00000i 0.127841i
\(980\) 0 0
\(981\) 11.3137i 0.361219i
\(982\) 0 0
\(983\) −50.9117 −1.62383 −0.811915 0.583775i \(-0.801575\pi\)
−0.811915 + 0.583775i \(0.801575\pi\)
\(984\) 0 0
\(985\) −10.0000 −0.318626
\(986\) 0 0
\(987\) − 4.00000i − 0.127321i
\(988\) 0 0
\(989\) − 22.6274i − 0.719510i
\(990\) 0 0
\(991\) −46.6690 −1.48249 −0.741246 0.671234i \(-0.765765\pi\)
−0.741246 + 0.671234i \(0.765765\pi\)
\(992\) 0 0
\(993\) −32.0000 −1.01549
\(994\) 0 0
\(995\) 34.0000i 1.07787i
\(996\) 0 0
\(997\) 31.1127i 0.985349i 0.870214 + 0.492675i \(0.163981\pi\)
−0.870214 + 0.492675i \(0.836019\pi\)
\(998\) 0 0
\(999\) 8.48528 0.268462
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 1536.2.d.d.769.4 4
3.2 odd 2 4608.2.d.g.2305.2 4
4.3 odd 2 inner 1536.2.d.d.769.2 4
8.3 odd 2 inner 1536.2.d.d.769.3 4
8.5 even 2 inner 1536.2.d.d.769.1 4
12.11 even 2 4608.2.d.g.2305.1 4
16.3 odd 4 1536.2.a.j.1.2 yes 2
16.5 even 4 1536.2.a.j.1.1 yes 2
16.11 odd 4 1536.2.a.c.1.1 2
16.13 even 4 1536.2.a.c.1.2 yes 2
24.5 odd 2 4608.2.d.g.2305.4 4
24.11 even 2 4608.2.d.g.2305.3 4
48.5 odd 4 4608.2.a.h.1.2 2
48.11 even 4 4608.2.a.j.1.2 2
48.29 odd 4 4608.2.a.j.1.1 2
48.35 even 4 4608.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.c.1.1 2 16.11 odd 4
1536.2.a.c.1.2 yes 2 16.13 even 4
1536.2.a.j.1.1 yes 2 16.5 even 4
1536.2.a.j.1.2 yes 2 16.3 odd 4
1536.2.d.d.769.1 4 8.5 even 2 inner
1536.2.d.d.769.2 4 4.3 odd 2 inner
1536.2.d.d.769.3 4 8.3 odd 2 inner
1536.2.d.d.769.4 4 1.1 even 1 trivial
4608.2.a.h.1.1 2 48.35 even 4
4608.2.a.h.1.2 2 48.5 odd 4
4608.2.a.j.1.1 2 48.29 odd 4
4608.2.a.j.1.2 2 48.11 even 4
4608.2.d.g.2305.1 4 12.11 even 2
4608.2.d.g.2305.2 4 3.2 odd 2
4608.2.d.g.2305.3 4 24.11 even 2
4608.2.d.g.2305.4 4 24.5 odd 2