Properties

Label 2925.2.c.e.2224.1
Level $2925$
Weight $2$
Character 2925.2224
Analytic conductor $23.356$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.3562425912\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2224.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2925.2224
Dual form 2925.2.c.e.2224.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{4} -4.00000i q^{7} -3.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{4} -4.00000i q^{7} -3.00000i q^{8} -4.00000 q^{11} -1.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +4.00000i q^{22} -1.00000 q^{26} -4.00000i q^{28} -10.0000 q^{29} +4.00000 q^{31} -5.00000i q^{32} -2.00000 q^{34} -2.00000i q^{37} -6.00000 q^{41} +12.0000i q^{43} -4.00000 q^{44} -9.00000 q^{49} -1.00000i q^{52} +6.00000i q^{53} -12.0000 q^{56} +10.0000i q^{58} +12.0000 q^{59} -2.00000 q^{61} -4.00000i q^{62} -7.00000 q^{64} -8.00000i q^{67} -2.00000i q^{68} -2.00000i q^{73} -2.00000 q^{74} +16.0000i q^{77} -8.00000 q^{79} +6.00000i q^{82} +4.00000i q^{83} +12.0000 q^{86} +12.0000i q^{88} -2.00000 q^{89} -4.00000 q^{91} +10.0000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 8 q^{11} - 8 q^{14} - 2 q^{16} - 2 q^{26} - 20 q^{29} + 8 q^{31} - 4 q^{34} - 12 q^{41} - 8 q^{44} - 18 q^{49} - 24 q^{56} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 4 q^{74} - 16 q^{79} + 24 q^{86} - 4 q^{89} - 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(2251\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i −0.935414 0.353553i \(-0.884973\pi\)
0.935414 0.353553i \(-0.115027\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) − 4.00000i − 0.755929i
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 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
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) − 1.00000i − 0.138675i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −12.0000 −1.60357
\(57\) 0 0
\(58\) 10.0000i 1.31306i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) 16.0000i 1.82337i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.00000i 0.662589i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) 12.0000i 1.27920i
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.00000i 0.377964i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) 0 0
\(118\) − 12.0000i − 1.10469i
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 2.00000i 0.181071i
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) − 3.00000i − 0.265165i
\(129\) 0 0
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000i 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) − 2.00000i − 0.164399i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) 0 0
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 8.00000i − 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 12.0000i 0.914991i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 2.00000i 0.149906i
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 4.00000i 0.296500i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) − 18.0000i − 1.29567i −0.761781 0.647834i \(-0.775675\pi\)
0.761781 0.647834i \(-0.224325\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 18.0000i − 1.26648i
\(203\) 40.0000i 2.80745i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.00000i 0.0693375i
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) − 16.0000i − 1.08615i
\(218\) − 2.00000i − 0.135457i
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) − 4.00000i − 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) −20.0000 −1.33631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 30.0000i 1.96960i
\(233\) − 14.0000i − 0.917170i −0.888650 0.458585i \(-0.848356\pi\)
0.888650 0.458585i \(-0.151644\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 8.00000i 0.518563i
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) − 12.0000i − 0.762001i
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) − 26.0000i − 1.62184i −0.585160 0.810918i \(-0.698968\pi\)
0.585160 0.810918i \(-0.301032\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) 4.00000i 0.247121i
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) − 8.00000i − 0.488678i
\(269\) 22.0000 1.34136 0.670682 0.741745i \(-0.266002\pi\)
0.670682 + 0.741745i \(0.266002\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 2.00000i 0.121268i
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) 12.0000i 0.719712i
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) − 12.0000i − 0.713326i −0.934233 0.356663i \(-0.883914\pi\)
0.934233 0.356663i \(-0.116086\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 24.0000i 1.41668i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) − 2.00000i − 0.117041i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 6.00000i 0.347571i
\(299\) 0 0
\(300\) 0 0
\(301\) 48.0000 2.76667
\(302\) − 4.00000i − 0.230174i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 16.0000i − 0.913168i −0.889680 0.456584i \(-0.849073\pi\)
0.889680 0.456584i \(-0.150927\pi\)
\(308\) 16.0000i 0.911685i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) − 26.0000i − 1.46031i −0.683284 0.730153i \(-0.739449\pi\)
0.683284 0.730153i \(-0.260551\pi\)
\(318\) 0 0
\(319\) 40.0000 2.23957
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) 18.0000i 0.993884i
\(329\) 0 0
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 4.00000i 0.219529i
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) 18.0000i 0.980522i 0.871576 + 0.490261i \(0.163099\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 1.00000i 0.0543928i
\(339\) 0 0
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 36.0000 1.94099
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 20.0000i 1.06600i
\(353\) − 2.00000i − 0.106449i −0.998583 0.0532246i \(-0.983050\pi\)
0.998583 0.0532246i \(-0.0169499\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) − 4.00000i − 0.211407i
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 10.0000i 0.525588i
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) 16.0000i 0.835193i 0.908633 + 0.417597i \(0.137127\pi\)
−0.908633 + 0.417597i \(0.862873\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) 26.0000i 1.34623i 0.739538 + 0.673114i \(0.235044\pi\)
−0.739538 + 0.673114i \(0.764956\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) 10.0000i 0.515026i
\(378\) 0 0
\(379\) 24.0000 1.23280 0.616399 0.787434i \(-0.288591\pi\)
0.616399 + 0.787434i \(0.288591\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.00000i 0.409316i
\(383\) − 16.0000i − 0.817562i −0.912633 0.408781i \(-0.865954\pi\)
0.912633 0.408781i \(-0.134046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) 0 0
\(388\) 10.0000i 0.507673i
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 27.0000i 1.36371i
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) 38.0000i 1.90717i 0.301131 + 0.953583i \(0.402636\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 0 0
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) − 4.00000i − 0.199254i
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 40.0000 1.98517
\(407\) 8.00000i 0.396545i
\(408\) 0 0
\(409\) −34.0000 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 48.0000i − 2.36193i
\(414\) 0 0
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 20.0000i 0.973585i
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000i 0.387147i
\(428\) − 12.0000i − 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) − 34.0000i − 1.63394i −0.576683 0.816968i \(-0.695653\pi\)
0.576683 0.816968i \(-0.304347\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.00000i 0.0951303i
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) 28.0000i 1.32288i
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) − 6.00000i − 0.282216i
\(453\) 0 0
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) 2.00000i 0.0935561i 0.998905 + 0.0467780i \(0.0148953\pi\)
−0.998905 + 0.0467780i \(0.985105\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) 0 0
\(460\) 0 0
\(461\) 38.0000 1.76984 0.884918 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) 0 0
\(463\) − 4.00000i − 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) 10.0000 0.464238
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) − 12.0000i − 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) 0 0
\(472\) − 36.0000i − 1.65703i
\(473\) − 48.0000i − 2.20704i
\(474\) 0 0
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) 0 0
\(478\) 24.0000i 1.09773i
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) − 10.0000i − 0.455488i
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) 12.0000i 0.543772i 0.962329 + 0.271886i \(0.0876473\pi\)
−0.962329 + 0.271886i \(0.912353\pi\)
\(488\) 6.00000i 0.271607i
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 20.0000i 0.900755i
\(494\) 0 0
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 12.0000i − 0.535586i
\(503\) − 8.00000i − 0.356702i −0.983967 0.178351i \(-0.942924\pi\)
0.983967 0.178351i \(-0.0570763\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) − 16.0000i − 0.709885i
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 11.0000i 0.486136i
\(513\) 0 0
\(514\) −26.0000 −1.14681
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 8.00000i 0.351500i
\(519\) 0 0
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) − 44.0000i − 1.92399i −0.273075 0.961993i \(-0.588041\pi\)
0.273075 0.961993i \(-0.411959\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) − 8.00000i − 0.348485i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.00000i 0.259889i
\(534\) 0 0
\(535\) 0 0
\(536\) −24.0000 −1.03664
\(537\) 0 0
\(538\) − 22.0000i − 0.948487i
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 12.0000i 0.515444i
\(543\) 0 0
\(544\) −10.0000 −0.428746
\(545\) 0 0
\(546\) 0 0
\(547\) 4.00000i 0.171028i 0.996337 + 0.0855138i \(0.0272532\pi\)
−0.996337 + 0.0855138i \(0.972747\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) − 10.0000i − 0.421825i
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 4.00000i 0.167248i
\(573\) 0 0
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 0 0
\(577\) − 46.0000i − 1.91501i −0.288425 0.957503i \(-0.593132\pi\)
0.288425 0.957503i \(-0.406868\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) 0 0
\(580\) 0 0
\(581\) 16.0000 0.663792
\(582\) 0 0
\(583\) − 24.0000i − 0.993978i
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) − 28.0000i − 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000i 0.0821995i
\(593\) − 26.0000i − 1.06769i −0.845582 0.533846i \(-0.820746\pi\)
0.845582 0.533846i \(-0.179254\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) − 48.0000i − 1.95633i
\(603\) 0 0
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) 0 0
\(607\) − 16.0000i − 0.649420i −0.945814 0.324710i \(-0.894733\pi\)
0.945814 0.324710i \(-0.105267\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 48.0000 1.93398
\(617\) − 22.0000i − 0.885687i −0.896599 0.442843i \(-0.853970\pi\)
0.896599 0.442843i \(-0.146030\pi\)
\(618\) 0 0
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.00000i 0.320513i
\(624\) 0 0
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) − 18.0000i − 0.718278i
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 24.0000i 0.954669i
\(633\) 0 0
\(634\) −26.0000 −1.03259
\(635\) 0 0
\(636\) 0 0
\(637\) 9.00000i 0.356593i
\(638\) − 40.0000i − 1.58362i
\(639\) 0 0
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) 40.0000i 1.57745i 0.614749 + 0.788723i \(0.289257\pi\)
−0.614749 + 0.788723i \(0.710743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.00000i 0.314512i 0.987558 + 0.157256i \(0.0502649\pi\)
−0.987558 + 0.157256i \(0.949735\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) − 8.00000i − 0.313304i
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 16.0000i 0.621858i
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 8.00000i 0.309529i
\(669\) 0 0
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 14.0000i 0.539660i 0.962908 + 0.269830i \(0.0869676\pi\)
−0.962908 + 0.269830i \(0.913032\pi\)
\(674\) 18.0000 0.693334
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) − 22.0000i − 0.845529i −0.906240 0.422764i \(-0.861060\pi\)
0.906240 0.422764i \(-0.138940\pi\)
\(678\) 0 0
\(679\) 40.0000 1.53506
\(680\) 0 0
\(681\) 0 0
\(682\) 16.0000i 0.612672i
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) − 12.0000i − 0.457496i
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −24.0000 −0.913003 −0.456502 0.889723i \(-0.650898\pi\)
−0.456502 + 0.889723i \(0.650898\pi\)
\(692\) 6.00000i 0.228086i
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 12.0000i 0.454532i
\(698\) − 26.0000i − 0.984115i
\(699\) 0 0
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) − 72.0000i − 2.70784i
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.00000i 0.224860i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) − 24.0000i − 0.895672i
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 19.0000i 0.707107i
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 0 0
\(727\) 8.00000i 0.296704i 0.988935 + 0.148352i \(0.0473968\pi\)
−0.988935 + 0.148352i \(0.952603\pi\)
\(728\) 12.0000i 0.444750i
\(729\) 0 0
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) − 30.0000i − 1.10808i −0.832492 0.554038i \(-0.813086\pi\)
0.832492 0.554038i \(-0.186914\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 0 0
\(737\) 32.0000i 1.17874i
\(738\) 0 0
\(739\) −32.0000 −1.17714 −0.588570 0.808447i \(-0.700309\pi\)
−0.588570 + 0.808447i \(0.700309\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 24.0000i − 0.881068i
\(743\) 48.0000i 1.76095i 0.474093 + 0.880475i \(0.342776\pi\)
−0.474093 + 0.880475i \(0.657224\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 26.0000 0.951928
\(747\) 0 0
\(748\) 8.00000i 0.292509i
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 10.0000 0.364179
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000i 0.799604i 0.916602 + 0.399802i \(0.130921\pi\)
−0.916602 + 0.399802i \(0.869079\pi\)
\(758\) − 24.0000i − 0.871719i
\(759\) 0 0
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) − 8.00000i − 0.289619i
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) − 12.0000i − 0.433295i
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 18.0000i − 0.647834i
\(773\) 10.0000i 0.359675i 0.983696 + 0.179838i \(0.0575572\pi\)
−0.983696 + 0.179838i \(0.942443\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 30.0000 1.07694
\(777\) 0 0
\(778\) − 22.0000i − 0.788738i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) − 32.0000i − 1.14068i −0.821410 0.570338i \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) − 18.0000i − 0.641223i
\(789\) 0 0
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) 2.00000i 0.0710221i
\(794\) 38.0000 1.34857
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) − 46.0000i − 1.62940i −0.579880 0.814702i \(-0.696901\pi\)
0.579880 0.814702i \(-0.303099\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 22.0000i 0.776847i
\(803\) 8.00000i 0.282314i
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) − 54.0000i − 1.89971i
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 40.0000i 1.40372i
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 34.0000i 1.18878i
\(819\) 0 0
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) − 4.00000i − 0.139094i −0.997579 0.0695468i \(-0.977845\pi\)
0.997579 0.0695468i \(-0.0221553\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 7.00000i 0.242681i
\(833\) 18.0000i 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) − 4.00000i − 0.138178i
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 10.0000i 0.344623i
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) − 20.0000i − 0.687208i
\(848\) − 6.00000i − 0.206041i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 30.0000i − 1.02718i −0.858036 0.513590i \(-0.828315\pi\)
0.858036 0.513590i \(-0.171685\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) 46.0000i 1.57133i 0.618652 + 0.785665i \(0.287679\pi\)
−0.618652 + 0.785665i \(0.712321\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.0000i 0.544646i 0.962206 + 0.272323i \(0.0877920\pi\)
−0.962206 + 0.272323i \(0.912208\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) − 16.0000i − 0.543075i
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) − 6.00000i − 0.203186i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 10.0000i − 0.337676i −0.985644 0.168838i \(-0.945999\pi\)
0.985644 0.168838i \(-0.0540015\pi\)
\(878\) 32.0000i 1.07995i
\(879\) 0 0
\(880\) 0 0
\(881\) −58.0000 −1.95407 −0.977035 0.213080i \(-0.931651\pi\)
−0.977035 + 0.213080i \(0.931651\pi\)
\(882\) 0 0
\(883\) 44.0000i 1.48072i 0.672212 + 0.740359i \(0.265344\pi\)
−0.672212 + 0.740359i \(0.734656\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 48.0000i 1.61168i 0.592132 + 0.805841i \(0.298286\pi\)
−0.592132 + 0.805841i \(0.701714\pi\)
\(888\) 0 0
\(889\) −64.0000 −2.14649
\(890\) 0 0
\(891\) 0 0
\(892\) − 4.00000i − 0.133930i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −12.0000 −0.400892
\(897\) 0 0
\(898\) − 22.0000i − 0.734150i
\(899\) −40.0000 −1.33407
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) − 24.0000i − 0.799113i
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 0 0
\(907\) 12.0000i 0.398453i 0.979953 + 0.199227i \(0.0638430\pi\)
−0.979953 + 0.199227i \(0.936157\pi\)
\(908\) 20.0000i 0.663723i
\(909\) 0 0
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) − 16.0000i − 0.529523i
\(914\) 2.00000 0.0661541
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 16.0000i 0.528367i
\(918\) 0 0
\(919\) 48.0000 1.58337 0.791687 0.610927i \(-0.209203\pi\)
0.791687 + 0.610927i \(0.209203\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 38.0000i − 1.25146i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) 50.0000i 1.64133i
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 14.0000i − 0.458585i
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000i 0.849383i 0.905338 + 0.424691i \(0.139617\pi\)
−0.905338 + 0.424691i \(0.860383\pi\)
\(938\) 32.0000i 1.04484i
\(939\) 0 0
\(940\) 0 0
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) 60.0000i 1.94974i 0.222779 + 0.974869i \(0.428487\pi\)
−0.222779 + 0.974869i \(0.571513\pi\)
\(948\) 0 0
\(949\) −2.00000 −0.0649227
\(950\) 0 0
\(951\) 0 0
\(952\) 24.0000i 0.777844i
\(953\) − 30.0000i − 0.971795i −0.874016 0.485898i \(-0.838493\pi\)
0.874016 0.485898i \(-0.161507\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 24.0000i 0.775405i
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 2.00000i 0.0644826i
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) − 52.0000i − 1.67221i −0.548572 0.836104i \(-0.684828\pi\)
0.548572 0.836104i \(-0.315172\pi\)
\(968\) − 15.0000i − 0.482118i
\(969\) 0 0
\(970\) 0 0
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) 0 0
\(973\) 48.0000i 1.53881i
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 0 0
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) 0 0
\(982\) − 12.0000i − 0.382935i
\(983\) 16.0000i 0.510321i 0.966899 + 0.255160i \(0.0821283\pi\)
−0.966899 + 0.255160i \(0.917872\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 20.0000 0.636930
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) − 20.0000i − 0.635001i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 26.0000i − 0.823428i −0.911313 0.411714i \(-0.864930\pi\)
0.911313 0.411714i \(-0.135070\pi\)
\(998\) − 24.0000i − 0.759707i
\(999\) 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 2925.2.c.e.2224.1 2
3.2 odd 2 975.2.c.f.274.2 2
5.2 odd 4 2925.2.a.p.1.1 1
5.3 odd 4 117.2.a.a.1.1 1
5.4 even 2 inner 2925.2.c.e.2224.2 2
15.2 even 4 975.2.a.f.1.1 1
15.8 even 4 39.2.a.a.1.1 1
15.14 odd 2 975.2.c.f.274.1 2
20.3 even 4 1872.2.a.h.1.1 1
35.13 even 4 5733.2.a.e.1.1 1
40.3 even 4 7488.2.a.by.1.1 1
40.13 odd 4 7488.2.a.bl.1.1 1
45.13 odd 12 1053.2.e.d.703.1 2
45.23 even 12 1053.2.e.b.703.1 2
45.38 even 12 1053.2.e.b.352.1 2
45.43 odd 12 1053.2.e.d.352.1 2
60.23 odd 4 624.2.a.i.1.1 1
65.8 even 4 1521.2.b.b.1351.1 2
65.18 even 4 1521.2.b.b.1351.2 2
65.38 odd 4 1521.2.a.e.1.1 1
105.83 odd 4 1911.2.a.f.1.1 1
120.53 even 4 2496.2.a.q.1.1 1
120.83 odd 4 2496.2.a.e.1.1 1
165.98 odd 4 4719.2.a.c.1.1 1
195.8 odd 4 507.2.b.a.337.2 2
195.23 even 12 507.2.e.b.22.1 2
195.38 even 4 507.2.a.a.1.1 1
195.68 even 12 507.2.e.a.22.1 2
195.83 odd 4 507.2.b.a.337.1 2
195.98 odd 12 507.2.j.e.361.2 4
195.113 even 12 507.2.e.a.484.1 2
195.128 odd 12 507.2.j.e.316.1 4
195.158 odd 12 507.2.j.e.316.2 4
195.173 even 12 507.2.e.b.484.1 2
195.188 odd 12 507.2.j.e.361.1 4
780.623 odd 4 8112.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.a.1.1 1 15.8 even 4
117.2.a.a.1.1 1 5.3 odd 4
507.2.a.a.1.1 1 195.38 even 4
507.2.b.a.337.1 2 195.83 odd 4
507.2.b.a.337.2 2 195.8 odd 4
507.2.e.a.22.1 2 195.68 even 12
507.2.e.a.484.1 2 195.113 even 12
507.2.e.b.22.1 2 195.23 even 12
507.2.e.b.484.1 2 195.173 even 12
507.2.j.e.316.1 4 195.128 odd 12
507.2.j.e.316.2 4 195.158 odd 12
507.2.j.e.361.1 4 195.188 odd 12
507.2.j.e.361.2 4 195.98 odd 12
624.2.a.i.1.1 1 60.23 odd 4
975.2.a.f.1.1 1 15.2 even 4
975.2.c.f.274.1 2 15.14 odd 2
975.2.c.f.274.2 2 3.2 odd 2
1053.2.e.b.352.1 2 45.38 even 12
1053.2.e.b.703.1 2 45.23 even 12
1053.2.e.d.352.1 2 45.43 odd 12
1053.2.e.d.703.1 2 45.13 odd 12
1521.2.a.e.1.1 1 65.38 odd 4
1521.2.b.b.1351.1 2 65.8 even 4
1521.2.b.b.1351.2 2 65.18 even 4
1872.2.a.h.1.1 1 20.3 even 4
1911.2.a.f.1.1 1 105.83 odd 4
2496.2.a.e.1.1 1 120.83 odd 4
2496.2.a.q.1.1 1 120.53 even 4
2925.2.a.p.1.1 1 5.2 odd 4
2925.2.c.e.2224.1 2 1.1 even 1 trivial
2925.2.c.e.2224.2 2 5.4 even 2 inner
4719.2.a.c.1.1 1 165.98 odd 4
5733.2.a.e.1.1 1 35.13 even 4
7488.2.a.bl.1.1 1 40.13 odd 4
7488.2.a.by.1.1 1 40.3 even 4
8112.2.a.s.1.1 1 780.623 odd 4