Properties

Label 882.3.b.c.197.1
Level $882$
Weight $3$
Character 882.197
Analytic conductor $24.033$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 882.197
Dual form 882.3.b.c.197.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} +2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} +2.82843i q^{8} -2.00000 q^{10} -7.07107i q^{11} +15.0000 q^{13} +4.00000 q^{16} -11.3137i q^{17} -13.0000 q^{19} +2.82843i q^{20} -10.0000 q^{22} +22.6274i q^{23} +23.0000 q^{25} -21.2132i q^{26} -22.6274i q^{29} +3.00000 q^{31} -5.65685i q^{32} -16.0000 q^{34} +17.0000 q^{37} +18.3848i q^{38} +4.00000 q^{40} -80.6102i q^{41} -85.0000 q^{43} +14.1421i q^{44} +32.0000 q^{46} -72.1249i q^{47} -32.5269i q^{50} -30.0000 q^{52} +33.9411i q^{53} -10.0000 q^{55} -32.0000 q^{58} -90.5097i q^{59} -72.0000 q^{61} -4.24264i q^{62} -8.00000 q^{64} -21.2132i q^{65} +43.0000 q^{67} +22.6274i q^{68} +52.3259i q^{71} -95.0000 q^{73} -24.0416i q^{74} +26.0000 q^{76} +69.0000 q^{79} -5.65685i q^{80} -114.000 q^{82} +60.8112i q^{83} -16.0000 q^{85} +120.208i q^{86} +20.0000 q^{88} -135.765i q^{89} -45.2548i q^{92} -102.000 q^{94} +18.3848i q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} - 4 q^{10} + 30 q^{13} + 8 q^{16} - 26 q^{19} - 20 q^{22} + 46 q^{25} + 6 q^{31} - 32 q^{34} + 34 q^{37} + 8 q^{40} - 170 q^{43} + 64 q^{46} - 60 q^{52} - 20 q^{55} - 64 q^{58} - 144 q^{61} - 16 q^{64} + 86 q^{67} - 190 q^{73} + 52 q^{76} + 138 q^{79} - 228 q^{82} - 32 q^{85} + 40 q^{88} - 204 q^{94} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 0.707107i
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) − 1.41421i − 0.282843i −0.989949 0.141421i \(-0.954833\pi\)
0.989949 0.141421i \(-0.0451672\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.82843i 0.353553i
\(9\) 0 0
\(10\) −2.00000 −0.200000
\(11\) − 7.07107i − 0.642824i −0.946939 0.321412i \(-0.895843\pi\)
0.946939 0.321412i \(-0.104157\pi\)
\(12\) 0 0
\(13\) 15.0000 1.15385 0.576923 0.816798i \(-0.304253\pi\)
0.576923 + 0.816798i \(0.304253\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) − 11.3137i − 0.665512i −0.943013 0.332756i \(-0.892021\pi\)
0.943013 0.332756i \(-0.107979\pi\)
\(18\) 0 0
\(19\) −13.0000 −0.684211 −0.342105 0.939662i \(-0.611140\pi\)
−0.342105 + 0.939662i \(0.611140\pi\)
\(20\) 2.82843i 0.141421i
\(21\) 0 0
\(22\) −10.0000 −0.454545
\(23\) 22.6274i 0.983801i 0.870651 + 0.491900i \(0.163698\pi\)
−0.870651 + 0.491900i \(0.836302\pi\)
\(24\) 0 0
\(25\) 23.0000 0.920000
\(26\) − 21.2132i − 0.815892i
\(27\) 0 0
\(28\) 0 0
\(29\) − 22.6274i − 0.780256i −0.920761 0.390128i \(-0.872431\pi\)
0.920761 0.390128i \(-0.127569\pi\)
\(30\) 0 0
\(31\) 3.00000 0.0967742 0.0483871 0.998829i \(-0.484592\pi\)
0.0483871 + 0.998829i \(0.484592\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) 0 0
\(34\) −16.0000 −0.470588
\(35\) 0 0
\(36\) 0 0
\(37\) 17.0000 0.459459 0.229730 0.973254i \(-0.426216\pi\)
0.229730 + 0.973254i \(0.426216\pi\)
\(38\) 18.3848i 0.483810i
\(39\) 0 0
\(40\) 4.00000 0.100000
\(41\) − 80.6102i − 1.96610i −0.183333 0.983051i \(-0.558689\pi\)
0.183333 0.983051i \(-0.441311\pi\)
\(42\) 0 0
\(43\) −85.0000 −1.97674 −0.988372 0.152055i \(-0.951411\pi\)
−0.988372 + 0.152055i \(0.951411\pi\)
\(44\) 14.1421i 0.321412i
\(45\) 0 0
\(46\) 32.0000 0.695652
\(47\) − 72.1249i − 1.53457i −0.641305 0.767286i \(-0.721607\pi\)
0.641305 0.767286i \(-0.278393\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 32.5269i − 0.650538i
\(51\) 0 0
\(52\) −30.0000 −0.576923
\(53\) 33.9411i 0.640399i 0.947350 + 0.320199i \(0.103750\pi\)
−0.947350 + 0.320199i \(0.896250\pi\)
\(54\) 0 0
\(55\) −10.0000 −0.181818
\(56\) 0 0
\(57\) 0 0
\(58\) −32.0000 −0.551724
\(59\) − 90.5097i − 1.53406i −0.641610 0.767031i \(-0.721733\pi\)
0.641610 0.767031i \(-0.278267\pi\)
\(60\) 0 0
\(61\) −72.0000 −1.18033 −0.590164 0.807283i \(-0.700937\pi\)
−0.590164 + 0.807283i \(0.700937\pi\)
\(62\) − 4.24264i − 0.0684297i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) − 21.2132i − 0.326357i
\(66\) 0 0
\(67\) 43.0000 0.641791 0.320896 0.947115i \(-0.396016\pi\)
0.320896 + 0.947115i \(0.396016\pi\)
\(68\) 22.6274i 0.332756i
\(69\) 0 0
\(70\) 0 0
\(71\) 52.3259i 0.736985i 0.929631 + 0.368492i \(0.120126\pi\)
−0.929631 + 0.368492i \(0.879874\pi\)
\(72\) 0 0
\(73\) −95.0000 −1.30137 −0.650685 0.759348i \(-0.725518\pi\)
−0.650685 + 0.759348i \(0.725518\pi\)
\(74\) − 24.0416i − 0.324887i
\(75\) 0 0
\(76\) 26.0000 0.342105
\(77\) 0 0
\(78\) 0 0
\(79\) 69.0000 0.873418 0.436709 0.899603i \(-0.356144\pi\)
0.436709 + 0.899603i \(0.356144\pi\)
\(80\) − 5.65685i − 0.0707107i
\(81\) 0 0
\(82\) −114.000 −1.39024
\(83\) 60.8112i 0.732665i 0.930484 + 0.366332i \(0.119387\pi\)
−0.930484 + 0.366332i \(0.880613\pi\)
\(84\) 0 0
\(85\) −16.0000 −0.188235
\(86\) 120.208i 1.39777i
\(87\) 0 0
\(88\) 20.0000 0.227273
\(89\) − 135.765i − 1.52544i −0.646727 0.762722i \(-0.723863\pi\)
0.646727 0.762722i \(-0.276137\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 45.2548i − 0.491900i
\(93\) 0 0
\(94\) −102.000 −1.08511
\(95\) 18.3848i 0.193524i
\(96\) 0 0
\(97\) 16.0000 0.164948 0.0824742 0.996593i \(-0.473718\pi\)
0.0824742 + 0.996593i \(0.473718\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −46.0000 −0.460000
\(101\) − 69.2965i − 0.686104i −0.939316 0.343052i \(-0.888539\pi\)
0.939316 0.343052i \(-0.111461\pi\)
\(102\) 0 0
\(103\) −61.0000 −0.592233 −0.296117 0.955152i \(-0.595692\pi\)
−0.296117 + 0.955152i \(0.595692\pi\)
\(104\) 42.4264i 0.407946i
\(105\) 0 0
\(106\) 48.0000 0.452830
\(107\) − 169.706i − 1.58603i −0.609200 0.793017i \(-0.708509\pi\)
0.609200 0.793017i \(-0.291491\pi\)
\(108\) 0 0
\(109\) −65.0000 −0.596330 −0.298165 0.954514i \(-0.596375\pi\)
−0.298165 + 0.954514i \(0.596375\pi\)
\(110\) 14.1421i 0.128565i
\(111\) 0 0
\(112\) 0 0
\(113\) − 137.179i − 1.21397i −0.794713 0.606985i \(-0.792379\pi\)
0.794713 0.606985i \(-0.207621\pi\)
\(114\) 0 0
\(115\) 32.0000 0.278261
\(116\) 45.2548i 0.390128i
\(117\) 0 0
\(118\) −128.000 −1.08475
\(119\) 0 0
\(120\) 0 0
\(121\) 71.0000 0.586777
\(122\) 101.823i 0.834618i
\(123\) 0 0
\(124\) −6.00000 −0.0483871
\(125\) − 67.8823i − 0.543058i
\(126\) 0 0
\(127\) −171.000 −1.34646 −0.673228 0.739435i \(-0.735093\pi\)
−0.673228 + 0.739435i \(0.735093\pi\)
\(128\) 11.3137i 0.0883883i
\(129\) 0 0
\(130\) −30.0000 −0.230769
\(131\) 117.380i 0.896028i 0.894026 + 0.448014i \(0.147869\pi\)
−0.894026 + 0.448014i \(0.852131\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 60.8112i − 0.453815i
\(135\) 0 0
\(136\) 32.0000 0.235294
\(137\) 214.960i 1.56905i 0.620094 + 0.784527i \(0.287094\pi\)
−0.620094 + 0.784527i \(0.712906\pi\)
\(138\) 0 0
\(139\) −83.0000 −0.597122 −0.298561 0.954391i \(-0.596507\pi\)
−0.298561 + 0.954391i \(0.596507\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 74.0000 0.521127
\(143\) − 106.066i − 0.741720i
\(144\) 0 0
\(145\) −32.0000 −0.220690
\(146\) 134.350i 0.920207i
\(147\) 0 0
\(148\) −34.0000 −0.229730
\(149\) 101.823i 0.683378i 0.939813 + 0.341689i \(0.110999\pi\)
−0.939813 + 0.341689i \(0.889001\pi\)
\(150\) 0 0
\(151\) 40.0000 0.264901 0.132450 0.991190i \(-0.457715\pi\)
0.132450 + 0.991190i \(0.457715\pi\)
\(152\) − 36.7696i − 0.241905i
\(153\) 0 0
\(154\) 0 0
\(155\) − 4.24264i − 0.0273719i
\(156\) 0 0
\(157\) 296.000 1.88535 0.942675 0.333712i \(-0.108301\pi\)
0.942675 + 0.333712i \(0.108301\pi\)
\(158\) − 97.5807i − 0.617600i
\(159\) 0 0
\(160\) −8.00000 −0.0500000
\(161\) 0 0
\(162\) 0 0
\(163\) 128.000 0.785276 0.392638 0.919693i \(-0.371563\pi\)
0.392638 + 0.919693i \(0.371563\pi\)
\(164\) 161.220i 0.983051i
\(165\) 0 0
\(166\) 86.0000 0.518072
\(167\) − 60.8112i − 0.364139i −0.983286 0.182069i \(-0.941720\pi\)
0.983286 0.182069i \(-0.0582796\pi\)
\(168\) 0 0
\(169\) 56.0000 0.331361
\(170\) 22.6274i 0.133102i
\(171\) 0 0
\(172\) 170.000 0.988372
\(173\) 113.137i 0.653972i 0.945029 + 0.326986i \(0.106033\pi\)
−0.945029 + 0.326986i \(0.893967\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 28.2843i − 0.160706i
\(177\) 0 0
\(178\) −192.000 −1.07865
\(179\) − 29.6985i − 0.165913i −0.996553 0.0829567i \(-0.973564\pi\)
0.996553 0.0829567i \(-0.0264363\pi\)
\(180\) 0 0
\(181\) 81.0000 0.447514 0.223757 0.974645i \(-0.428168\pi\)
0.223757 + 0.974645i \(0.428168\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −64.0000 −0.347826
\(185\) − 24.0416i − 0.129955i
\(186\) 0 0
\(187\) −80.0000 −0.427807
\(188\) 144.250i 0.767286i
\(189\) 0 0
\(190\) 26.0000 0.136842
\(191\) − 63.6396i − 0.333192i −0.986025 0.166596i \(-0.946722\pi\)
0.986025 0.166596i \(-0.0532775\pi\)
\(192\) 0 0
\(193\) −223.000 −1.15544 −0.577720 0.816235i \(-0.696058\pi\)
−0.577720 + 0.816235i \(0.696058\pi\)
\(194\) − 22.6274i − 0.116636i
\(195\) 0 0
\(196\) 0 0
\(197\) 158.392i 0.804020i 0.915635 + 0.402010i \(0.131688\pi\)
−0.915635 + 0.402010i \(0.868312\pi\)
\(198\) 0 0
\(199\) −136.000 −0.683417 −0.341709 0.939806i \(-0.611006\pi\)
−0.341709 + 0.939806i \(0.611006\pi\)
\(200\) 65.0538i 0.325269i
\(201\) 0 0
\(202\) −98.0000 −0.485149
\(203\) 0 0
\(204\) 0 0
\(205\) −114.000 −0.556098
\(206\) 86.2670i 0.418772i
\(207\) 0 0
\(208\) 60.0000 0.288462
\(209\) 91.9239i 0.439827i
\(210\) 0 0
\(211\) 272.000 1.28910 0.644550 0.764562i \(-0.277045\pi\)
0.644550 + 0.764562i \(0.277045\pi\)
\(212\) − 67.8823i − 0.320199i
\(213\) 0 0
\(214\) −240.000 −1.12150
\(215\) 120.208i 0.559108i
\(216\) 0 0
\(217\) 0 0
\(218\) 91.9239i 0.421669i
\(219\) 0 0
\(220\) 20.0000 0.0909091
\(221\) − 169.706i − 0.767899i
\(222\) 0 0
\(223\) 248.000 1.11211 0.556054 0.831146i \(-0.312315\pi\)
0.556054 + 0.831146i \(0.312315\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −194.000 −0.858407
\(227\) 165.463i 0.728912i 0.931221 + 0.364456i \(0.118745\pi\)
−0.931221 + 0.364456i \(0.881255\pi\)
\(228\) 0 0
\(229\) 433.000 1.89083 0.945415 0.325869i \(-0.105657\pi\)
0.945415 + 0.325869i \(0.105657\pi\)
\(230\) − 45.2548i − 0.196760i
\(231\) 0 0
\(232\) 64.0000 0.275862
\(233\) − 227.688i − 0.977203i −0.872507 0.488602i \(-0.837507\pi\)
0.872507 0.488602i \(-0.162493\pi\)
\(234\) 0 0
\(235\) −102.000 −0.434043
\(236\) 181.019i 0.767031i
\(237\) 0 0
\(238\) 0 0
\(239\) 343.654i 1.43788i 0.695071 + 0.718941i \(0.255373\pi\)
−0.695071 + 0.718941i \(0.744627\pi\)
\(240\) 0 0
\(241\) 158.000 0.655602 0.327801 0.944747i \(-0.393693\pi\)
0.327801 + 0.944747i \(0.393693\pi\)
\(242\) − 100.409i − 0.414914i
\(243\) 0 0
\(244\) 144.000 0.590164
\(245\) 0 0
\(246\) 0 0
\(247\) −195.000 −0.789474
\(248\) 8.48528i 0.0342148i
\(249\) 0 0
\(250\) −96.0000 −0.384000
\(251\) − 237.588i − 0.946565i −0.880911 0.473283i \(-0.843069\pi\)
0.880911 0.473283i \(-0.156931\pi\)
\(252\) 0 0
\(253\) 160.000 0.632411
\(254\) 241.831i 0.952089i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) − 318.198i − 1.23812i −0.785342 0.619062i \(-0.787513\pi\)
0.785342 0.619062i \(-0.212487\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 42.4264i 0.163178i
\(261\) 0 0
\(262\) 166.000 0.633588
\(263\) − 169.706i − 0.645269i −0.946524 0.322634i \(-0.895432\pi\)
0.946524 0.322634i \(-0.104568\pi\)
\(264\) 0 0
\(265\) 48.0000 0.181132
\(266\) 0 0
\(267\) 0 0
\(268\) −86.0000 −0.320896
\(269\) − 485.075i − 1.80325i −0.432515 0.901627i \(-0.642374\pi\)
0.432515 0.901627i \(-0.357626\pi\)
\(270\) 0 0
\(271\) 216.000 0.797048 0.398524 0.917158i \(-0.369523\pi\)
0.398524 + 0.917158i \(0.369523\pi\)
\(272\) − 45.2548i − 0.166378i
\(273\) 0 0
\(274\) 304.000 1.10949
\(275\) − 162.635i − 0.591398i
\(276\) 0 0
\(277\) 305.000 1.10108 0.550542 0.834808i \(-0.314421\pi\)
0.550542 + 0.834808i \(0.314421\pi\)
\(278\) 117.380i 0.422229i
\(279\) 0 0
\(280\) 0 0
\(281\) 22.6274i 0.0805246i 0.999189 + 0.0402623i \(0.0128194\pi\)
−0.999189 + 0.0402623i \(0.987181\pi\)
\(282\) 0 0
\(283\) 157.000 0.554770 0.277385 0.960759i \(-0.410532\pi\)
0.277385 + 0.960759i \(0.410532\pi\)
\(284\) − 104.652i − 0.368492i
\(285\) 0 0
\(286\) −150.000 −0.524476
\(287\) 0 0
\(288\) 0 0
\(289\) 161.000 0.557093
\(290\) 45.2548i 0.156051i
\(291\) 0 0
\(292\) 190.000 0.650685
\(293\) 101.823i 0.347520i 0.984788 + 0.173760i \(0.0555917\pi\)
−0.984788 + 0.173760i \(0.944408\pi\)
\(294\) 0 0
\(295\) −128.000 −0.433898
\(296\) 48.0833i 0.162443i
\(297\) 0 0
\(298\) 144.000 0.483221
\(299\) 339.411i 1.13515i
\(300\) 0 0
\(301\) 0 0
\(302\) − 56.5685i − 0.187313i
\(303\) 0 0
\(304\) −52.0000 −0.171053
\(305\) 101.823i 0.333847i
\(306\) 0 0
\(307\) −11.0000 −0.0358306 −0.0179153 0.999840i \(-0.505703\pi\)
−0.0179153 + 0.999840i \(0.505703\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −6.00000 −0.0193548
\(311\) 524.673i 1.68705i 0.537088 + 0.843526i \(0.319524\pi\)
−0.537088 + 0.843526i \(0.680476\pi\)
\(312\) 0 0
\(313\) 553.000 1.76677 0.883387 0.468645i \(-0.155258\pi\)
0.883387 + 0.468645i \(0.155258\pi\)
\(314\) − 418.607i − 1.33314i
\(315\) 0 0
\(316\) −138.000 −0.436709
\(317\) 135.765i 0.428279i 0.976803 + 0.214140i \(0.0686947\pi\)
−0.976803 + 0.214140i \(0.931305\pi\)
\(318\) 0 0
\(319\) −160.000 −0.501567
\(320\) 11.3137i 0.0353553i
\(321\) 0 0
\(322\) 0 0
\(323\) 147.078i 0.455350i
\(324\) 0 0
\(325\) 345.000 1.06154
\(326\) − 181.019i − 0.555274i
\(327\) 0 0
\(328\) 228.000 0.695122
\(329\) 0 0
\(330\) 0 0
\(331\) −61.0000 −0.184290 −0.0921450 0.995746i \(-0.529372\pi\)
−0.0921450 + 0.995746i \(0.529372\pi\)
\(332\) − 121.622i − 0.366332i
\(333\) 0 0
\(334\) −86.0000 −0.257485
\(335\) − 60.8112i − 0.181526i
\(336\) 0 0
\(337\) 135.000 0.400593 0.200297 0.979735i \(-0.435809\pi\)
0.200297 + 0.979735i \(0.435809\pi\)
\(338\) − 79.1960i − 0.234308i
\(339\) 0 0
\(340\) 32.0000 0.0941176
\(341\) − 21.2132i − 0.0622088i
\(342\) 0 0
\(343\) 0 0
\(344\) − 240.416i − 0.698885i
\(345\) 0 0
\(346\) 160.000 0.462428
\(347\) 101.823i 0.293439i 0.989178 + 0.146720i \(0.0468715\pi\)
−0.989178 + 0.146720i \(0.953129\pi\)
\(348\) 0 0
\(349\) −152.000 −0.435530 −0.217765 0.976001i \(-0.569877\pi\)
−0.217765 + 0.976001i \(0.569877\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −40.0000 −0.113636
\(353\) 386.080i 1.09371i 0.837227 + 0.546856i \(0.184175\pi\)
−0.837227 + 0.546856i \(0.815825\pi\)
\(354\) 0 0
\(355\) 74.0000 0.208451
\(356\) 271.529i 0.762722i
\(357\) 0 0
\(358\) −42.0000 −0.117318
\(359\) 45.2548i 0.126058i 0.998012 + 0.0630290i \(0.0200761\pi\)
−0.998012 + 0.0630290i \(0.979924\pi\)
\(360\) 0 0
\(361\) −192.000 −0.531856
\(362\) − 114.551i − 0.316440i
\(363\) 0 0
\(364\) 0 0
\(365\) 134.350i 0.368083i
\(366\) 0 0
\(367\) 101.000 0.275204 0.137602 0.990488i \(-0.456060\pi\)
0.137602 + 0.990488i \(0.456060\pi\)
\(368\) 90.5097i 0.245950i
\(369\) 0 0
\(370\) −34.0000 −0.0918919
\(371\) 0 0
\(372\) 0 0
\(373\) −311.000 −0.833780 −0.416890 0.908957i \(-0.636880\pi\)
−0.416890 + 0.908957i \(0.636880\pi\)
\(374\) 113.137i 0.302506i
\(375\) 0 0
\(376\) 204.000 0.542553
\(377\) − 339.411i − 0.900295i
\(378\) 0 0
\(379\) 91.0000 0.240106 0.120053 0.992768i \(-0.461694\pi\)
0.120053 + 0.992768i \(0.461694\pi\)
\(380\) − 36.7696i − 0.0967620i
\(381\) 0 0
\(382\) −90.0000 −0.235602
\(383\) 328.098i 0.856652i 0.903624 + 0.428326i \(0.140896\pi\)
−0.903624 + 0.428326i \(0.859104\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 315.370i 0.817020i
\(387\) 0 0
\(388\) −32.0000 −0.0824742
\(389\) − 688.722i − 1.77049i −0.465122 0.885247i \(-0.653989\pi\)
0.465122 0.885247i \(-0.346011\pi\)
\(390\) 0 0
\(391\) 256.000 0.654731
\(392\) 0 0
\(393\) 0 0
\(394\) 224.000 0.568528
\(395\) − 97.5807i − 0.247040i
\(396\) 0 0
\(397\) −279.000 −0.702771 −0.351385 0.936231i \(-0.614289\pi\)
−0.351385 + 0.936231i \(0.614289\pi\)
\(398\) 192.333i 0.483249i
\(399\) 0 0
\(400\) 92.0000 0.230000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 45.0000 0.111663
\(404\) 138.593i 0.343052i
\(405\) 0 0
\(406\) 0 0
\(407\) − 120.208i − 0.295352i
\(408\) 0 0
\(409\) −223.000 −0.545232 −0.272616 0.962123i \(-0.587889\pi\)
−0.272616 + 0.962123i \(0.587889\pi\)
\(410\) 161.220i 0.393220i
\(411\) 0 0
\(412\) 122.000 0.296117
\(413\) 0 0
\(414\) 0 0
\(415\) 86.0000 0.207229
\(416\) − 84.8528i − 0.203973i
\(417\) 0 0
\(418\) 130.000 0.311005
\(419\) 581.242i 1.38721i 0.720355 + 0.693606i \(0.243979\pi\)
−0.720355 + 0.693606i \(0.756021\pi\)
\(420\) 0 0
\(421\) 153.000 0.363420 0.181710 0.983352i \(-0.441837\pi\)
0.181710 + 0.983352i \(0.441837\pi\)
\(422\) − 384.666i − 0.911531i
\(423\) 0 0
\(424\) −96.0000 −0.226415
\(425\) − 260.215i − 0.612271i
\(426\) 0 0
\(427\) 0 0
\(428\) 339.411i 0.793017i
\(429\) 0 0
\(430\) 170.000 0.395349
\(431\) − 140.007i − 0.324843i −0.986722 0.162421i \(-0.948070\pi\)
0.986722 0.162421i \(-0.0519304\pi\)
\(432\) 0 0
\(433\) 137.000 0.316397 0.158199 0.987407i \(-0.449431\pi\)
0.158199 + 0.987407i \(0.449431\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 130.000 0.298165
\(437\) − 294.156i − 0.673127i
\(438\) 0 0
\(439\) −600.000 −1.36674 −0.683371 0.730071i \(-0.739487\pi\)
−0.683371 + 0.730071i \(0.739487\pi\)
\(440\) − 28.2843i − 0.0642824i
\(441\) 0 0
\(442\) −240.000 −0.542986
\(443\) 633.568i 1.43018i 0.699035 + 0.715088i \(0.253613\pi\)
−0.699035 + 0.715088i \(0.746387\pi\)
\(444\) 0 0
\(445\) −192.000 −0.431461
\(446\) − 350.725i − 0.786379i
\(447\) 0 0
\(448\) 0 0
\(449\) − 383.252i − 0.853568i −0.904354 0.426784i \(-0.859647\pi\)
0.904354 0.426784i \(-0.140353\pi\)
\(450\) 0 0
\(451\) −570.000 −1.26386
\(452\) 274.357i 0.606985i
\(453\) 0 0
\(454\) 234.000 0.515419
\(455\) 0 0
\(456\) 0 0
\(457\) −239.000 −0.522976 −0.261488 0.965207i \(-0.584213\pi\)
−0.261488 + 0.965207i \(0.584213\pi\)
\(458\) − 612.354i − 1.33702i
\(459\) 0 0
\(460\) −64.0000 −0.139130
\(461\) 452.548i 0.981667i 0.871253 + 0.490833i \(0.163308\pi\)
−0.871253 + 0.490833i \(0.836692\pi\)
\(462\) 0 0
\(463\) 211.000 0.455724 0.227862 0.973693i \(-0.426827\pi\)
0.227862 + 0.973693i \(0.426827\pi\)
\(464\) − 90.5097i − 0.195064i
\(465\) 0 0
\(466\) −322.000 −0.690987
\(467\) 312.541i 0.669253i 0.942351 + 0.334627i \(0.108610\pi\)
−0.942351 + 0.334627i \(0.891390\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 144.250i 0.306914i
\(471\) 0 0
\(472\) 256.000 0.542373
\(473\) 601.041i 1.27070i
\(474\) 0 0
\(475\) −299.000 −0.629474
\(476\) 0 0
\(477\) 0 0
\(478\) 486.000 1.01674
\(479\) 701.450i 1.46440i 0.681087 + 0.732202i \(0.261508\pi\)
−0.681087 + 0.732202i \(0.738492\pi\)
\(480\) 0 0
\(481\) 255.000 0.530146
\(482\) − 223.446i − 0.463580i
\(483\) 0 0
\(484\) −142.000 −0.293388
\(485\) − 22.6274i − 0.0466545i
\(486\) 0 0
\(487\) 419.000 0.860370 0.430185 0.902741i \(-0.358448\pi\)
0.430185 + 0.902741i \(0.358448\pi\)
\(488\) − 203.647i − 0.417309i
\(489\) 0 0
\(490\) 0 0
\(491\) − 169.706i − 0.345633i −0.984954 0.172816i \(-0.944713\pi\)
0.984954 0.172816i \(-0.0552867\pi\)
\(492\) 0 0
\(493\) −256.000 −0.519270
\(494\) 275.772i 0.558242i
\(495\) 0 0
\(496\) 12.0000 0.0241935
\(497\) 0 0
\(498\) 0 0
\(499\) 187.000 0.374749 0.187375 0.982289i \(-0.440002\pi\)
0.187375 + 0.982289i \(0.440002\pi\)
\(500\) 135.765i 0.271529i
\(501\) 0 0
\(502\) −336.000 −0.669323
\(503\) − 173.948i − 0.345822i −0.984937 0.172911i \(-0.944683\pi\)
0.984937 0.172911i \(-0.0553172\pi\)
\(504\) 0 0
\(505\) −98.0000 −0.194059
\(506\) − 226.274i − 0.447182i
\(507\) 0 0
\(508\) 342.000 0.673228
\(509\) − 258.801i − 0.508450i −0.967145 0.254225i \(-0.918180\pi\)
0.967145 0.254225i \(-0.0818204\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 22.6274i − 0.0441942i
\(513\) 0 0
\(514\) −450.000 −0.875486
\(515\) 86.2670i 0.167509i
\(516\) 0 0
\(517\) −510.000 −0.986460
\(518\) 0 0
\(519\) 0 0
\(520\) 60.0000 0.115385
\(521\) − 407.294i − 0.781753i −0.920443 0.390877i \(-0.872172\pi\)
0.920443 0.390877i \(-0.127828\pi\)
\(522\) 0 0
\(523\) −811.000 −1.55067 −0.775335 0.631551i \(-0.782419\pi\)
−0.775335 + 0.631551i \(0.782419\pi\)
\(524\) − 234.759i − 0.448014i
\(525\) 0 0
\(526\) −240.000 −0.456274
\(527\) − 33.9411i − 0.0644044i
\(528\) 0 0
\(529\) 17.0000 0.0321361
\(530\) − 67.8823i − 0.128080i
\(531\) 0 0
\(532\) 0 0
\(533\) − 1209.15i − 2.26858i
\(534\) 0 0
\(535\) −240.000 −0.448598
\(536\) 121.622i 0.226907i
\(537\) 0 0
\(538\) −686.000 −1.27509
\(539\) 0 0
\(540\) 0 0
\(541\) −345.000 −0.637708 −0.318854 0.947804i \(-0.603298\pi\)
−0.318854 + 0.947804i \(0.603298\pi\)
\(542\) − 305.470i − 0.563598i
\(543\) 0 0
\(544\) −64.0000 −0.117647
\(545\) 91.9239i 0.168668i
\(546\) 0 0
\(547\) −864.000 −1.57952 −0.789762 0.613413i \(-0.789796\pi\)
−0.789762 + 0.613413i \(0.789796\pi\)
\(548\) − 429.921i − 0.784527i
\(549\) 0 0
\(550\) −230.000 −0.418182
\(551\) 294.156i 0.533859i
\(552\) 0 0
\(553\) 0 0
\(554\) − 431.335i − 0.778583i
\(555\) 0 0
\(556\) 166.000 0.298561
\(557\) 948.937i 1.70366i 0.523820 + 0.851829i \(0.324506\pi\)
−0.523820 + 0.851829i \(0.675494\pi\)
\(558\) 0 0
\(559\) −1275.00 −2.28086
\(560\) 0 0
\(561\) 0 0
\(562\) 32.0000 0.0569395
\(563\) 739.634i 1.31374i 0.754005 + 0.656868i \(0.228119\pi\)
−0.754005 + 0.656868i \(0.771881\pi\)
\(564\) 0 0
\(565\) −194.000 −0.343363
\(566\) − 222.032i − 0.392282i
\(567\) 0 0
\(568\) −148.000 −0.260563
\(569\) 767.918i 1.34959i 0.738004 + 0.674796i \(0.235768\pi\)
−0.738004 + 0.674796i \(0.764232\pi\)
\(570\) 0 0
\(571\) 477.000 0.835377 0.417688 0.908590i \(-0.362840\pi\)
0.417688 + 0.908590i \(0.362840\pi\)
\(572\) 212.132i 0.370860i
\(573\) 0 0
\(574\) 0 0
\(575\) 520.431i 0.905097i
\(576\) 0 0
\(577\) −263.000 −0.455806 −0.227903 0.973684i \(-0.573187\pi\)
−0.227903 + 0.973684i \(0.573187\pi\)
\(578\) − 227.688i − 0.393925i
\(579\) 0 0
\(580\) 64.0000 0.110345
\(581\) 0 0
\(582\) 0 0
\(583\) 240.000 0.411664
\(584\) − 268.701i − 0.460104i
\(585\) 0 0
\(586\) 144.000 0.245734
\(587\) 531.744i 0.905868i 0.891544 + 0.452934i \(0.149623\pi\)
−0.891544 + 0.452934i \(0.850377\pi\)
\(588\) 0 0
\(589\) −39.0000 −0.0662139
\(590\) 181.019i 0.306812i
\(591\) 0 0
\(592\) 68.0000 0.114865
\(593\) − 801.859i − 1.35221i −0.736806 0.676104i \(-0.763667\pi\)
0.736806 0.676104i \(-0.236333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 203.647i − 0.341689i
\(597\) 0 0
\(598\) 480.000 0.802676
\(599\) − 780.646i − 1.30325i −0.758542 0.651624i \(-0.774088\pi\)
0.758542 0.651624i \(-0.225912\pi\)
\(600\) 0 0
\(601\) 383.000 0.637271 0.318636 0.947877i \(-0.396775\pi\)
0.318636 + 0.947877i \(0.396775\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −80.0000 −0.132450
\(605\) − 100.409i − 0.165966i
\(606\) 0 0
\(607\) −765.000 −1.26030 −0.630148 0.776475i \(-0.717006\pi\)
−0.630148 + 0.776475i \(0.717006\pi\)
\(608\) 73.5391i 0.120952i
\(609\) 0 0
\(610\) 144.000 0.236066
\(611\) − 1081.87i − 1.77066i
\(612\) 0 0
\(613\) 408.000 0.665579 0.332790 0.943001i \(-0.392010\pi\)
0.332790 + 0.943001i \(0.392010\pi\)
\(614\) 15.5563i 0.0253361i
\(615\) 0 0
\(616\) 0 0
\(617\) − 914.996i − 1.48298i −0.670966 0.741488i \(-0.734120\pi\)
0.670966 0.741488i \(-0.265880\pi\)
\(618\) 0 0
\(619\) −947.000 −1.52989 −0.764943 0.644097i \(-0.777233\pi\)
−0.764943 + 0.644097i \(0.777233\pi\)
\(620\) 8.48528i 0.0136859i
\(621\) 0 0
\(622\) 742.000 1.19293
\(623\) 0 0
\(624\) 0 0
\(625\) 479.000 0.766400
\(626\) − 782.060i − 1.24930i
\(627\) 0 0
\(628\) −592.000 −0.942675
\(629\) − 192.333i − 0.305776i
\(630\) 0 0
\(631\) −760.000 −1.20444 −0.602219 0.798331i \(-0.705716\pi\)
−0.602219 + 0.798331i \(0.705716\pi\)
\(632\) 195.161i 0.308800i
\(633\) 0 0
\(634\) 192.000 0.302839
\(635\) 241.831i 0.380835i
\(636\) 0 0
\(637\) 0 0
\(638\) 226.274i 0.354662i
\(639\) 0 0
\(640\) 16.0000 0.0250000
\(641\) 463.862i 0.723654i 0.932245 + 0.361827i \(0.117847\pi\)
−0.932245 + 0.361827i \(0.882153\pi\)
\(642\) 0 0
\(643\) −19.0000 −0.0295490 −0.0147745 0.999891i \(-0.504703\pi\)
−0.0147745 + 0.999891i \(0.504703\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 208.000 0.321981
\(647\) − 1158.24i − 1.79017i −0.445894 0.895086i \(-0.647114\pi\)
0.445894 0.895086i \(-0.352886\pi\)
\(648\) 0 0
\(649\) −640.000 −0.986133
\(650\) − 487.904i − 0.750621i
\(651\) 0 0
\(652\) −256.000 −0.392638
\(653\) 1030.96i 1.57881i 0.613874 + 0.789404i \(0.289610\pi\)
−0.613874 + 0.789404i \(0.710390\pi\)
\(654\) 0 0
\(655\) 166.000 0.253435
\(656\) − 322.441i − 0.491525i
\(657\) 0 0
\(658\) 0 0
\(659\) 972.979i 1.47645i 0.674556 + 0.738224i \(0.264335\pi\)
−0.674556 + 0.738224i \(0.735665\pi\)
\(660\) 0 0
\(661\) −401.000 −0.606657 −0.303328 0.952886i \(-0.598098\pi\)
−0.303328 + 0.952886i \(0.598098\pi\)
\(662\) 86.2670i 0.130313i
\(663\) 0 0
\(664\) −172.000 −0.259036
\(665\) 0 0
\(666\) 0 0
\(667\) 512.000 0.767616
\(668\) 121.622i 0.182069i
\(669\) 0 0
\(670\) −86.0000 −0.128358
\(671\) 509.117i 0.758743i
\(672\) 0 0
\(673\) 665.000 0.988113 0.494056 0.869430i \(-0.335514\pi\)
0.494056 + 0.869430i \(0.335514\pi\)
\(674\) − 190.919i − 0.283262i
\(675\) 0 0
\(676\) −112.000 −0.165680
\(677\) − 916.410i − 1.35363i −0.736151 0.676817i \(-0.763359\pi\)
0.736151 0.676817i \(-0.236641\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) − 45.2548i − 0.0665512i
\(681\) 0 0
\(682\) −30.0000 −0.0439883
\(683\) 135.765i 0.198777i 0.995049 + 0.0993884i \(0.0316886\pi\)
−0.995049 + 0.0993884i \(0.968311\pi\)
\(684\) 0 0
\(685\) 304.000 0.443796
\(686\) 0 0
\(687\) 0 0
\(688\) −340.000 −0.494186
\(689\) 509.117i 0.738921i
\(690\) 0 0
\(691\) 221.000 0.319826 0.159913 0.987131i \(-0.448879\pi\)
0.159913 + 0.987131i \(0.448879\pi\)
\(692\) − 226.274i − 0.326986i
\(693\) 0 0
\(694\) 144.000 0.207493
\(695\) 117.380i 0.168892i
\(696\) 0 0
\(697\) −912.000 −1.30846
\(698\) 214.960i 0.307966i
\(699\) 0 0
\(700\) 0 0
\(701\) − 192.333i − 0.274370i −0.990545 0.137185i \(-0.956195\pi\)
0.990545 0.137185i \(-0.0438054\pi\)
\(702\) 0 0
\(703\) −221.000 −0.314367
\(704\) 56.5685i 0.0803530i
\(705\) 0 0
\(706\) 546.000 0.773371
\(707\) 0 0
\(708\) 0 0
\(709\) −328.000 −0.462623 −0.231312 0.972880i \(-0.574302\pi\)
−0.231312 + 0.972880i \(0.574302\pi\)
\(710\) − 104.652i − 0.147397i
\(711\) 0 0
\(712\) 384.000 0.539326
\(713\) 67.8823i 0.0952065i
\(714\) 0 0
\(715\) −150.000 −0.209790
\(716\) 59.3970i 0.0829567i
\(717\) 0 0
\(718\) 64.0000 0.0891365
\(719\) − 1013.99i − 1.41028i −0.709068 0.705140i \(-0.750884\pi\)
0.709068 0.705140i \(-0.249116\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 271.529i 0.376079i
\(723\) 0 0
\(724\) −162.000 −0.223757
\(725\) − 520.431i − 0.717835i
\(726\) 0 0
\(727\) 1069.00 1.47043 0.735213 0.677836i \(-0.237082\pi\)
0.735213 + 0.677836i \(0.237082\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 190.000 0.260274
\(731\) 961.665i 1.31555i
\(732\) 0 0
\(733\) 495.000 0.675307 0.337653 0.941270i \(-0.390367\pi\)
0.337653 + 0.941270i \(0.390367\pi\)
\(734\) − 142.836i − 0.194599i
\(735\) 0 0
\(736\) 128.000 0.173913
\(737\) − 304.056i − 0.412559i
\(738\) 0 0
\(739\) 453.000 0.612991 0.306495 0.951872i \(-0.400844\pi\)
0.306495 + 0.951872i \(0.400844\pi\)
\(740\) 48.0833i 0.0649774i
\(741\) 0 0
\(742\) 0 0
\(743\) − 538.815i − 0.725189i −0.931947 0.362594i \(-0.881891\pi\)
0.931947 0.362594i \(-0.118109\pi\)
\(744\) 0 0
\(745\) 144.000 0.193289
\(746\) 439.820i 0.589572i
\(747\) 0 0
\(748\) 160.000 0.213904
\(749\) 0 0
\(750\) 0 0
\(751\) −339.000 −0.451398 −0.225699 0.974197i \(-0.572467\pi\)
−0.225699 + 0.974197i \(0.572467\pi\)
\(752\) − 288.500i − 0.383643i
\(753\) 0 0
\(754\) −480.000 −0.636605
\(755\) − 56.5685i − 0.0749252i
\(756\) 0 0
\(757\) 198.000 0.261559 0.130779 0.991411i \(-0.458252\pi\)
0.130779 + 0.991411i \(0.458252\pi\)
\(758\) − 128.693i − 0.169780i
\(759\) 0 0
\(760\) −52.0000 −0.0684211
\(761\) 362.039i 0.475741i 0.971297 + 0.237870i \(0.0764493\pi\)
−0.971297 + 0.237870i \(0.923551\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 127.279i 0.166596i
\(765\) 0 0
\(766\) 464.000 0.605744
\(767\) − 1357.65i − 1.77007i
\(768\) 0 0
\(769\) −929.000 −1.20806 −0.604031 0.796961i \(-0.706440\pi\)
−0.604031 + 0.796961i \(0.706440\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 446.000 0.577720
\(773\) − 171.120i − 0.221371i −0.993855 0.110686i \(-0.964695\pi\)
0.993855 0.110686i \(-0.0353046\pi\)
\(774\) 0 0
\(775\) 69.0000 0.0890323
\(776\) 45.2548i 0.0583181i
\(777\) 0 0
\(778\) −974.000 −1.25193
\(779\) 1047.93i 1.34523i
\(780\) 0 0
\(781\) 370.000 0.473752
\(782\) − 362.039i − 0.462965i
\(783\) 0 0
\(784\) 0 0
\(785\) − 418.607i − 0.533258i
\(786\) 0 0
\(787\) 90.0000 0.114358 0.0571792 0.998364i \(-0.481789\pi\)
0.0571792 + 0.998364i \(0.481789\pi\)
\(788\) − 316.784i − 0.402010i
\(789\) 0 0
\(790\) −138.000 −0.174684
\(791\) 0 0
\(792\) 0 0
\(793\) −1080.00 −1.36192
\(794\) 394.566i 0.496934i
\(795\) 0 0
\(796\) 272.000 0.341709
\(797\) − 1154.00i − 1.44793i −0.689838 0.723964i \(-0.742318\pi\)
0.689838 0.723964i \(-0.257682\pi\)
\(798\) 0 0
\(799\) −816.000 −1.02128
\(800\) − 130.108i − 0.162635i
\(801\) 0 0
\(802\) 0 0
\(803\) 671.751i 0.836552i
\(804\) 0 0
\(805\) 0 0
\(806\) − 63.6396i − 0.0789573i
\(807\) 0 0
\(808\) 196.000 0.242574
\(809\) − 171.120i − 0.211520i −0.994392 0.105760i \(-0.966272\pi\)
0.994392 0.105760i \(-0.0337276\pi\)
\(810\) 0 0
\(811\) −554.000 −0.683107 −0.341554 0.939862i \(-0.610953\pi\)
−0.341554 + 0.939862i \(0.610953\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −170.000 −0.208845
\(815\) − 181.019i − 0.222110i
\(816\) 0 0
\(817\) 1105.00 1.35251
\(818\) 315.370i 0.385537i
\(819\) 0 0
\(820\) 228.000 0.278049
\(821\) 1062.07i 1.29364i 0.762645 + 0.646818i \(0.223900\pi\)
−0.762645 + 0.646818i \(0.776100\pi\)
\(822\) 0 0
\(823\) 856.000 1.04010 0.520049 0.854137i \(-0.325914\pi\)
0.520049 + 0.854137i \(0.325914\pi\)
\(824\) − 172.534i − 0.209386i
\(825\) 0 0
\(826\) 0 0
\(827\) 1022.48i 1.23637i 0.786033 + 0.618184i \(0.212131\pi\)
−0.786033 + 0.618184i \(0.787869\pi\)
\(828\) 0 0
\(829\) 1039.00 1.25332 0.626659 0.779294i \(-0.284422\pi\)
0.626659 + 0.779294i \(0.284422\pi\)
\(830\) − 121.622i − 0.146533i
\(831\) 0 0
\(832\) −120.000 −0.144231
\(833\) 0 0
\(834\) 0 0
\(835\) −86.0000 −0.102994
\(836\) − 183.848i − 0.219914i
\(837\) 0 0
\(838\) 822.000 0.980907
\(839\) − 656.195i − 0.782116i −0.920366 0.391058i \(-0.872109\pi\)
0.920366 0.391058i \(-0.127891\pi\)
\(840\) 0 0
\(841\) 329.000 0.391201
\(842\) − 216.375i − 0.256977i
\(843\) 0 0
\(844\) −544.000 −0.644550
\(845\) − 79.1960i − 0.0937230i
\(846\) 0 0
\(847\) 0 0
\(848\) 135.765i 0.160100i
\(849\) 0 0
\(850\) −368.000 −0.432941
\(851\) 384.666i 0.452017i
\(852\) 0 0
\(853\) 1463.00 1.71512 0.857562 0.514381i \(-0.171978\pi\)
0.857562 + 0.514381i \(0.171978\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 480.000 0.560748
\(857\) − 67.8823i − 0.0792092i −0.999215 0.0396046i \(-0.987390\pi\)
0.999215 0.0396046i \(-0.0126098\pi\)
\(858\) 0 0
\(859\) −1462.00 −1.70198 −0.850990 0.525183i \(-0.823997\pi\)
−0.850990 + 0.525183i \(0.823997\pi\)
\(860\) − 240.416i − 0.279554i
\(861\) 0 0
\(862\) −198.000 −0.229698
\(863\) − 807.516i − 0.935708i −0.883806 0.467854i \(-0.845027\pi\)
0.883806 0.467854i \(-0.154973\pi\)
\(864\) 0 0
\(865\) 160.000 0.184971
\(866\) − 193.747i − 0.223727i
\(867\) 0 0
\(868\) 0 0
\(869\) − 487.904i − 0.561454i
\(870\) 0 0
\(871\) 645.000 0.740528
\(872\) − 183.848i − 0.210835i
\(873\) 0 0
\(874\) −416.000 −0.475973
\(875\) 0 0
\(876\) 0 0
\(877\) 1480.00 1.68757 0.843786 0.536680i \(-0.180322\pi\)
0.843786 + 0.536680i \(0.180322\pi\)
\(878\) 848.528i 0.966433i
\(879\) 0 0
\(880\) −40.0000 −0.0454545
\(881\) − 712.764i − 0.809039i −0.914529 0.404520i \(-0.867439\pi\)
0.914529 0.404520i \(-0.132561\pi\)
\(882\) 0 0
\(883\) −115.000 −0.130238 −0.0651189 0.997878i \(-0.520743\pi\)
−0.0651189 + 0.997878i \(0.520743\pi\)
\(884\) 339.411i 0.383949i
\(885\) 0 0
\(886\) 896.000 1.01129
\(887\) 1192.18i 1.34406i 0.740524 + 0.672030i \(0.234578\pi\)
−0.740524 + 0.672030i \(0.765422\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 271.529i 0.305089i
\(891\) 0 0
\(892\) −496.000 −0.556054
\(893\) 937.624i 1.04997i
\(894\) 0 0
\(895\) −42.0000 −0.0469274
\(896\) 0 0
\(897\) 0 0
\(898\) −542.000 −0.603563
\(899\) − 67.8823i − 0.0755086i
\(900\) 0 0
\(901\) 384.000 0.426193
\(902\) 806.102i 0.893683i
\(903\) 0 0
\(904\) 388.000 0.429204
\(905\) − 114.551i − 0.126576i
\(906\) 0 0
\(907\) −501.000 −0.552370 −0.276185 0.961104i \(-0.589070\pi\)
−0.276185 + 0.961104i \(0.589070\pi\)
\(908\) − 330.926i − 0.364456i
\(909\) 0 0
\(910\) 0 0
\(911\) 203.647i 0.223542i 0.993734 + 0.111771i \(0.0356523\pi\)
−0.993734 + 0.111771i \(0.964348\pi\)
\(912\) 0 0
\(913\) 430.000 0.470975
\(914\) 337.997i 0.369800i
\(915\) 0 0
\(916\) −866.000 −0.945415
\(917\) 0 0
\(918\) 0 0
\(919\) −653.000 −0.710555 −0.355277 0.934761i \(-0.615614\pi\)
−0.355277 + 0.934761i \(0.615614\pi\)
\(920\) 90.5097i 0.0983801i
\(921\) 0 0
\(922\) 640.000 0.694143
\(923\) 784.889i 0.850367i
\(924\) 0 0
\(925\) 391.000 0.422703
\(926\) − 298.399i − 0.322245i
\(927\) 0 0
\(928\) −128.000 −0.137931
\(929\) 12.7279i 0.0137007i 0.999977 + 0.00685033i \(0.00218055\pi\)
−0.999977 + 0.00685033i \(0.997819\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 455.377i 0.488602i
\(933\) 0 0
\(934\) 442.000 0.473233
\(935\) 113.137i 0.121002i
\(936\) 0 0
\(937\) 761.000 0.812166 0.406083 0.913836i \(-0.366894\pi\)
0.406083 + 0.913836i \(0.366894\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 204.000 0.217021
\(941\) 282.843i 0.300577i 0.988642 + 0.150288i \(0.0480202\pi\)
−0.988642 + 0.150288i \(0.951980\pi\)
\(942\) 0 0
\(943\) 1824.00 1.93425
\(944\) − 362.039i − 0.383516i
\(945\) 0 0
\(946\) 850.000 0.898520
\(947\) − 1079.04i − 1.13944i −0.821841 0.569718i \(-0.807053\pi\)
0.821841 0.569718i \(-0.192947\pi\)
\(948\) 0 0
\(949\) −1425.00 −1.50158
\(950\) 422.850i 0.445105i
\(951\) 0 0
\(952\) 0 0
\(953\) 1074.80i 1.12781i 0.825840 + 0.563905i \(0.190701\pi\)
−0.825840 + 0.563905i \(0.809299\pi\)
\(954\) 0 0
\(955\) −90.0000 −0.0942408
\(956\) − 687.308i − 0.718941i
\(957\) 0 0
\(958\) 992.000 1.03549
\(959\) 0 0
\(960\) 0 0
\(961\) −952.000 −0.990635
\(962\) − 360.624i − 0.374869i
\(963\) 0 0
\(964\) −316.000 −0.327801
\(965\) 315.370i 0.326808i
\(966\) 0 0
\(967\) 1163.00 1.20269 0.601344 0.798990i \(-0.294632\pi\)
0.601344 + 0.798990i \(0.294632\pi\)
\(968\) 200.818i 0.207457i
\(969\) 0 0
\(970\) −32.0000 −0.0329897
\(971\) 1561.29i 1.60792i 0.594682 + 0.803961i \(0.297278\pi\)
−0.594682 + 0.803961i \(0.702722\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 592.555i − 0.608373i
\(975\) 0 0
\(976\) −288.000 −0.295082
\(977\) − 1132.79i − 1.15945i −0.814811 0.579726i \(-0.803160\pi\)
0.814811 0.579726i \(-0.196840\pi\)
\(978\) 0 0
\(979\) −960.000 −0.980592
\(980\) 0 0
\(981\) 0 0
\(982\) −240.000 −0.244399
\(983\) 305.470i 0.310753i 0.987855 + 0.155376i \(0.0496591\pi\)
−0.987855 + 0.155376i \(0.950341\pi\)
\(984\) 0 0
\(985\) 224.000 0.227411
\(986\) 362.039i 0.367179i
\(987\) 0 0
\(988\) 390.000 0.394737
\(989\) − 1923.33i − 1.94472i
\(990\) 0 0
\(991\) −973.000 −0.981837 −0.490918 0.871206i \(-0.663339\pi\)
−0.490918 + 0.871206i \(0.663339\pi\)
\(992\) − 16.9706i − 0.0171074i
\(993\) 0 0
\(994\) 0 0
\(995\) 192.333i 0.193300i
\(996\) 0 0
\(997\) −79.0000 −0.0792377 −0.0396189 0.999215i \(-0.512614\pi\)
−0.0396189 + 0.999215i \(0.512614\pi\)
\(998\) − 264.458i − 0.264988i
\(999\) 0 0
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 882.3.b.c.197.1 2
3.2 odd 2 inner 882.3.b.c.197.2 2
7.2 even 3 126.3.s.b.53.1 4
7.3 odd 6 882.3.s.c.863.2 4
7.4 even 3 126.3.s.b.107.2 yes 4
7.5 odd 6 882.3.s.c.557.1 4
7.6 odd 2 882.3.b.d.197.1 2
21.2 odd 6 126.3.s.b.53.2 yes 4
21.5 even 6 882.3.s.c.557.2 4
21.11 odd 6 126.3.s.b.107.1 yes 4
21.17 even 6 882.3.s.c.863.1 4
21.20 even 2 882.3.b.d.197.2 2
28.11 odd 6 1008.3.dc.a.737.2 4
28.23 odd 6 1008.3.dc.a.305.1 4
84.11 even 6 1008.3.dc.a.737.1 4
84.23 even 6 1008.3.dc.a.305.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.s.b.53.1 4 7.2 even 3
126.3.s.b.53.2 yes 4 21.2 odd 6
126.3.s.b.107.1 yes 4 21.11 odd 6
126.3.s.b.107.2 yes 4 7.4 even 3
882.3.b.c.197.1 2 1.1 even 1 trivial
882.3.b.c.197.2 2 3.2 odd 2 inner
882.3.b.d.197.1 2 7.6 odd 2
882.3.b.d.197.2 2 21.20 even 2
882.3.s.c.557.1 4 7.5 odd 6
882.3.s.c.557.2 4 21.5 even 6
882.3.s.c.863.1 4 21.17 even 6
882.3.s.c.863.2 4 7.3 odd 6
1008.3.dc.a.305.1 4 28.23 odd 6
1008.3.dc.a.305.2 4 84.23 even 6
1008.3.dc.a.737.1 4 84.11 even 6
1008.3.dc.a.737.2 4 28.11 odd 6