Properties

Label 1400.2.g.f.449.1
Level $1400$
Weight $2$
Character 1400.449
Analytic conductor $11.179$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1400.449
Dual form 1400.2.g.f.449.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} -1.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -1.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +6.00000i q^{13} +7.00000i q^{17} -1.00000 q^{19} -1.00000 q^{21} +8.00000i q^{23} -5.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} +1.00000i q^{33} -8.00000i q^{37} +6.00000 q^{39} -5.00000 q^{41} -6.00000i q^{47} -1.00000 q^{49} +7.00000 q^{51} +4.00000i q^{53} +1.00000i q^{57} +4.00000 q^{59} +6.00000 q^{61} -2.00000i q^{63} +5.00000i q^{67} +8.00000 q^{69} +14.0000 q^{71} +15.0000i q^{73} +1.00000i q^{77} -14.0000 q^{79} +1.00000 q^{81} +1.00000i q^{83} -6.00000i q^{87} +3.00000 q^{89} +6.00000 q^{91} -4.00000i q^{93} +6.00000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{9} - 2q^{11} - 2q^{19} - 2q^{21} + 12q^{29} + 8q^{31} + 12q^{39} - 10q^{41} - 2q^{49} + 14q^{51} + 8q^{59} + 12q^{61} + 16q^{69} + 28q^{71} - 28q^{79} + 2q^{81} + 6q^{89} + 12q^{91} - 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.00000i 1.69775i 0.528594 + 0.848875i \(0.322719\pi\)
−0.528594 + 0.848875i \(0.677281\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 8.00000i 1.66812i 0.551677 + 0.834058i \(0.313988\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 1.00000i 0.174078i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 7.00000 0.980196
\(52\) 0 0
\(53\) 4.00000i 0.549442i 0.961524 + 0.274721i \(0.0885855\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) − 2.00000i − 0.251976i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.00000i 0.610847i 0.952217 + 0.305424i \(0.0987981\pi\)
−0.952217 + 0.305424i \(0.901202\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) 15.0000i 1.75562i 0.479012 + 0.877809i \(0.340995\pi\)
−0.479012 + 0.877809i \(0.659005\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000i 0.113961i
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.00000i 0.109764i 0.998493 + 0.0548821i \(0.0174783\pi\)
−0.998493 + 0.0548821i \(0.982522\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 6.00000i − 0.643268i
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) − 4.00000i − 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.00000i − 0.870063i −0.900415 0.435031i \(-0.856737\pi\)
0.900415 0.435031i \(-0.143263\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 1.00000i 0.0940721i 0.998893 + 0.0470360i \(0.0149776\pi\)
−0.998893 + 0.0470360i \(0.985022\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 12.0000i 1.10940i
\(118\) 0 0
\(119\) 7.00000 0.641689
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 5.00000i 0.450835i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 22.0000i − 1.95218i −0.217357 0.976092i \(-0.569744\pi\)
0.217357 0.976092i \(-0.430256\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) 1.00000i 0.0867110i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 3.00000i − 0.256307i −0.991754 0.128154i \(-0.959095\pi\)
0.991754 0.128154i \(-0.0409051\pi\)
\(138\) 0 0
\(139\) −1.00000 −0.0848189 −0.0424094 0.999100i \(-0.513503\pi\)
−0.0424094 + 0.999100i \(0.513503\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) − 6.00000i − 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 14.0000i 1.13183i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.00000i − 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) 0 0
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) − 25.0000i − 1.95815i −0.203497 0.979076i \(-0.565231\pi\)
0.203497 0.979076i \(-0.434769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.0000i 1.54765i 0.633402 + 0.773823i \(0.281658\pi\)
−0.633402 + 0.773823i \(0.718342\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) − 10.0000i − 0.760286i −0.924928 0.380143i \(-0.875875\pi\)
0.924928 0.380143i \(-0.124125\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 4.00000i − 0.300658i
\(178\) 0 0
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) − 6.00000i − 0.443533i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 7.00000i − 0.511891i
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) 11.0000i 0.791797i 0.918294 + 0.395899i \(0.129567\pi\)
−0.918294 + 0.395899i \(0.870433\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 5.00000 0.352673
\(202\) 0 0
\(203\) − 6.00000i − 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 16.0000i 1.11208i
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −19.0000 −1.30801 −0.654007 0.756489i \(-0.726913\pi\)
−0.654007 + 0.756489i \(0.726913\pi\)
\(212\) 0 0
\(213\) − 14.0000i − 0.959264i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 4.00000i − 0.271538i
\(218\) 0 0
\(219\) 15.0000 1.01361
\(220\) 0 0
\(221\) −42.0000 −2.82523
\(222\) 0 0
\(223\) 10.0000i 0.669650i 0.942280 + 0.334825i \(0.108677\pi\)
−0.942280 + 0.334825i \(0.891323\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 28.0000i 1.85843i 0.369546 + 0.929213i \(0.379513\pi\)
−0.369546 + 0.929213i \(0.620487\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) 10.0000i 0.655122i 0.944830 + 0.327561i \(0.106227\pi\)
−0.944830 + 0.327561i \(0.893773\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 14.0000i 0.909398i
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −13.0000 −0.837404 −0.418702 0.908124i \(-0.637515\pi\)
−0.418702 + 0.908124i \(0.637515\pi\)
\(242\) 0 0
\(243\) − 16.0000i − 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 6.00000i − 0.381771i
\(248\) 0 0
\(249\) 1.00000 0.0633724
\(250\) 0 0
\(251\) −25.0000 −1.57799 −0.788993 0.614402i \(-0.789397\pi\)
−0.788993 + 0.614402i \(0.789397\pi\)
\(252\) 0 0
\(253\) − 8.00000i − 0.502956i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) − 14.0000i − 0.863277i −0.902047 0.431638i \(-0.857936\pi\)
0.902047 0.431638i \(-0.142064\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 3.00000i − 0.183597i
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 0 0
\(273\) − 6.00000i − 0.363137i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 1.00000i 0.0594438i 0.999558 + 0.0297219i \(0.00946217\pi\)
−0.999558 + 0.0297219i \(0.990538\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.00000i 0.295141i
\(288\) 0 0
\(289\) −32.0000 −1.88235
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.00000i 0.290129i
\(298\) 0 0
\(299\) −48.0000 −2.77591
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 15.0000i − 0.856095i −0.903756 0.428048i \(-0.859202\pi\)
0.903756 0.428048i \(-0.140798\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 0 0
\(313\) − 22.0000i − 1.24351i −0.783210 0.621757i \(-0.786419\pi\)
0.783210 0.621757i \(-0.213581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 28.0000i − 1.57264i −0.617822 0.786318i \(-0.711985\pi\)
0.617822 0.786318i \(-0.288015\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) −9.00000 −0.502331
\(322\) 0 0
\(323\) − 7.00000i − 0.389490i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 10.0000i − 0.553001i
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 3.00000 0.164895 0.0824475 0.996595i \(-0.473726\pi\)
0.0824475 + 0.996595i \(0.473726\pi\)
\(332\) 0 0
\(333\) − 16.0000i − 0.876795i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 19.0000i 1.03500i 0.855684 + 0.517498i \(0.173136\pi\)
−0.855684 + 0.517498i \(0.826864\pi\)
\(338\) 0 0
\(339\) 1.00000 0.0543125
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 23.0000i 1.23470i 0.786687 + 0.617352i \(0.211795\pi\)
−0.786687 + 0.617352i \(0.788205\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) 0 0
\(351\) 30.0000 1.60128
\(352\) 0 0
\(353\) − 14.0000i − 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 7.00000i − 0.370479i
\(358\) 0 0
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 10.0000i 0.524864i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 8.00000i − 0.417597i −0.977959 0.208798i \(-0.933045\pi\)
0.977959 0.208798i \(-0.0669552\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) − 20.0000i − 1.03556i −0.855514 0.517780i \(-0.826758\pi\)
0.855514 0.517780i \(-0.173242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 36.0000i 1.85409i
\(378\) 0 0
\(379\) 3.00000 0.154100 0.0770498 0.997027i \(-0.475450\pi\)
0.0770498 + 0.997027i \(0.475450\pi\)
\(380\) 0 0
\(381\) −22.0000 −1.12709
\(382\) 0 0
\(383\) 6.00000i 0.306586i 0.988181 + 0.153293i \(0.0489878\pi\)
−0.988181 + 0.153293i \(0.951012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.0000 0.811232 0.405616 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) 0 0
\(391\) −56.0000 −2.83204
\(392\) 0 0
\(393\) − 16.0000i − 0.807093i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 30.0000i − 1.50566i −0.658217 0.752828i \(-0.728689\pi\)
0.658217 0.752828i \(-0.271311\pi\)
\(398\) 0 0
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 13.0000 0.649189 0.324595 0.945853i \(-0.394772\pi\)
0.324595 + 0.945853i \(0.394772\pi\)
\(402\) 0 0
\(403\) 24.0000i 1.19553i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8.00000i 0.396545i
\(408\) 0 0
\(409\) −27.0000 −1.33506 −0.667532 0.744581i \(-0.732649\pi\)
−0.667532 + 0.744581i \(0.732649\pi\)
\(410\) 0 0
\(411\) −3.00000 −0.147979
\(412\) 0 0
\(413\) − 4.00000i − 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.00000i 0.0489702i
\(418\) 0 0
\(419\) 35.0000 1.70986 0.854931 0.518742i \(-0.173599\pi\)
0.854931 + 0.518742i \(0.173599\pi\)
\(420\) 0 0
\(421\) 36.0000 1.75453 0.877266 0.480004i \(-0.159365\pi\)
0.877266 + 0.480004i \(0.159365\pi\)
\(422\) 0 0
\(423\) − 12.0000i − 0.583460i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 6.00000i − 0.290360i
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) 0 0
\(433\) 9.00000i 0.432512i 0.976337 + 0.216256i \(0.0693846\pi\)
−0.976337 + 0.216256i \(0.930615\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 8.00000i − 0.382692i
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) − 25.0000i − 1.18779i −0.804544 0.593893i \(-0.797590\pi\)
0.804544 0.593893i \(-0.202410\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4.00000i 0.189194i
\(448\) 0 0
\(449\) 23.0000 1.08544 0.542719 0.839915i \(-0.317395\pi\)
0.542719 + 0.839915i \(0.317395\pi\)
\(450\) 0 0
\(451\) 5.00000 0.235441
\(452\) 0 0
\(453\) − 8.00000i − 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 7.00000i − 0.327446i −0.986506 0.163723i \(-0.947650\pi\)
0.986506 0.163723i \(-0.0523504\pi\)
\(458\) 0 0
\(459\) 35.0000 1.63366
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) − 24.0000i − 1.11537i −0.830051 0.557687i \(-0.811689\pi\)
0.830051 0.557687i \(-0.188311\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 28.0000i − 1.29569i −0.761774 0.647843i \(-0.775671\pi\)
0.761774 0.647843i \(-0.224329\pi\)
\(468\) 0 0
\(469\) 5.00000 0.230879
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 8.00000i 0.366295i
\(478\) 0 0
\(479\) −2.00000 −0.0913823 −0.0456912 0.998956i \(-0.514549\pi\)
−0.0456912 + 0.998956i \(0.514549\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) 0 0
\(483\) − 8.00000i − 0.364013i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 22.0000i 0.996915i 0.866914 + 0.498458i \(0.166100\pi\)
−0.866914 + 0.498458i \(0.833900\pi\)
\(488\) 0 0
\(489\) −25.0000 −1.13054
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) 42.0000i 1.89158i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 14.0000i − 0.627986i
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 20.0000 0.893534
\(502\) 0 0
\(503\) 14.0000i 0.624229i 0.950044 + 0.312115i \(0.101037\pi\)
−0.950044 + 0.312115i \(0.898963\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 23.0000i 1.02147i
\(508\) 0 0
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 15.0000 0.663561
\(512\) 0 0
\(513\) 5.00000i 0.220755i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 6.00000i 0.263880i
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) 27.0000 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) 0 0
\(523\) 23.0000i 1.00572i 0.864368 + 0.502860i \(0.167719\pi\)
−0.864368 + 0.502860i \(0.832281\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 28.0000i 1.21970i
\(528\) 0 0
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) − 30.0000i − 1.29944i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.0000i 0.647298i
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) 22.0000i 0.944110i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 41.0000i − 1.75303i −0.481371 0.876517i \(-0.659861\pi\)
0.481371 0.876517i \(-0.340139\pi\)
\(548\) 0 0
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 14.0000i 0.595341i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 36.0000i − 1.52537i −0.646771 0.762684i \(-0.723881\pi\)
0.646771 0.762684i \(-0.276119\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −7.00000 −0.295540
\(562\) 0 0
\(563\) 36.0000i 1.51722i 0.651546 + 0.758610i \(0.274121\pi\)
−0.651546 + 0.758610i \(0.725879\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1.00000i − 0.0419961i
\(568\) 0 0
\(569\) 11.0000 0.461144 0.230572 0.973055i \(-0.425940\pi\)
0.230572 + 0.973055i \(0.425940\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) 0 0
\(573\) − 18.0000i − 0.751961i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 19.0000i − 0.790980i −0.918470 0.395490i \(-0.870575\pi\)
0.918470 0.395490i \(-0.129425\pi\)
\(578\) 0 0
\(579\) 11.0000 0.457144
\(580\) 0 0
\(581\) 1.00000 0.0414870
\(582\) 0 0
\(583\) − 4.00000i − 0.165663i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.0000i 0.619116i 0.950881 + 0.309558i \(0.100181\pi\)
−0.950881 + 0.309558i \(0.899819\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) − 9.00000i − 0.369586i −0.982777 0.184793i \(-0.940839\pi\)
0.982777 0.184793i \(-0.0591614\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 10.0000i − 0.409273i
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) 0 0
\(603\) 10.0000i 0.407231i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 36.0000i − 1.46119i −0.682808 0.730597i \(-0.739242\pi\)
0.682808 0.730597i \(-0.260758\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 36.0000 1.45640
\(612\) 0 0
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 40.0000 1.60514
\(622\) 0 0
\(623\) − 3.00000i − 0.120192i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 1.00000i − 0.0399362i
\(628\) 0 0
\(629\) 56.0000 2.23287
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 19.0000i 0.755182i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 6.00000i − 0.237729i
\(638\) 0 0
\(639\) 28.0000 1.10766
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 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
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 0 0
\(653\) − 36.0000i − 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 30.0000i 1.17041i
\(658\) 0 0
\(659\) −29.0000 −1.12968 −0.564840 0.825201i \(-0.691062\pi\)
−0.564840 + 0.825201i \(0.691062\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 0 0
\(663\) 42.0000i 1.63114i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 48.0000i 1.85857i
\(668\) 0 0
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) −6.00000 −0.231627
\(672\) 0 0
\(673\) 30.0000i 1.15642i 0.815890 + 0.578208i \(0.196248\pi\)
−0.815890 + 0.578208i \(0.803752\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 48.0000i 1.84479i 0.386248 + 0.922395i \(0.373771\pi\)
−0.386248 + 0.922395i \(0.626229\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) 0 0
\(683\) 9.00000i 0.344375i 0.985064 + 0.172188i \(0.0550836\pi\)
−0.985064 + 0.172188i \(0.944916\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 22.0000i − 0.839352i
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −43.0000 −1.63580 −0.817899 0.575362i \(-0.804861\pi\)
−0.817899 + 0.575362i \(0.804861\pi\)
\(692\) 0 0
\(693\) 2.00000i 0.0759737i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 35.0000i − 1.32572i
\(698\) 0 0
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 24.0000 0.906467 0.453234 0.891392i \(-0.350270\pi\)
0.453234 + 0.891392i \(0.350270\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 12.0000 0.450669 0.225335 0.974281i \(-0.427652\pi\)
0.225335 + 0.974281i \(0.427652\pi\)
\(710\) 0 0
\(711\) −28.0000 −1.05008
\(712\) 0 0
\(713\) 32.0000i 1.19841i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 12.0000i 0.448148i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) 13.0000i 0.483475i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 14.0000i 0.519231i 0.965712 + 0.259616i \(0.0835959\pi\)
−0.965712 + 0.259616i \(0.916404\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) − 16.0000i − 0.590973i −0.955347 0.295487i \(-0.904518\pi\)
0.955347 0.295487i \(-0.0954818\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 5.00000i − 0.184177i
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) 0 0
\(743\) 48.0000i 1.76095i 0.474093 + 0.880475i \(0.342776\pi\)
−0.474093 + 0.880475i \(0.657224\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.00000i 0.0731762i
\(748\) 0 0
\(749\) −9.00000 −0.328853
\(750\) 0 0
\(751\) 42.0000 1.53260 0.766301 0.642482i \(-0.222095\pi\)
0.766301 + 0.642482i \(0.222095\pi\)
\(752\) 0 0
\(753\) 25.0000i 0.911051i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 42.0000i − 1.52652i −0.646094 0.763258i \(-0.723599\pi\)
0.646094 0.763258i \(-0.276401\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) 0 0
\(763\) − 10.0000i − 0.362024i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000i 0.866590i
\(768\) 0 0
\(769\) 53.0000 1.91123 0.955614 0.294620i \(-0.0951931\pi\)
0.955614 + 0.294620i \(0.0951931\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) − 12.0000i − 0.431610i −0.976436 0.215805i \(-0.930762\pi\)
0.976436 0.215805i \(-0.0692376\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 8.00000i 0.286998i
\(778\) 0 0
\(779\) 5.00000 0.179144
\(780\) 0 0
\(781\) −14.0000 −0.500959
\(782\) 0 0
\(783\) − 30.0000i − 1.07211i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 36.0000i − 1.28326i −0.767014 0.641631i \(-0.778258\pi\)
0.767014 0.641631i \(-0.221742\pi\)
\(788\) 0 0
\(789\) −14.0000 −0.498413
\(790\) 0 0
\(791\) 1.00000 0.0355559
\(792\) 0 0
\(793\) 36.0000i 1.27840i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 22.0000i − 0.779280i −0.920967 0.389640i \(-0.872599\pi\)
0.920967 0.389640i \(-0.127401\pi\)
\(798\) 0 0
\(799\) 42.0000 1.48585
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) − 15.0000i − 0.529339i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 6.00000i − 0.211210i
\(808\) 0 0
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 14.0000i 0.491001i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) − 10.0000i − 0.348578i −0.984695 0.174289i \(-0.944237\pi\)
0.984695 0.174289i \(-0.0557627\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13.0000i 0.452054i 0.974121 + 0.226027i \(0.0725738\pi\)
−0.974121 + 0.226027i \(0.927426\pi\)
\(828\) 0 0
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) − 7.00000i − 0.242536i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 20.0000i − 0.691301i
\(838\) 0 0
\(839\) −34.0000 −1.17381 −0.586905 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) − 30.0000i − 1.03325i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000i 0.343604i
\(848\) 0 0
\(849\) 1.00000 0.0343199
\(850\) 0 0
\(851\) 64.0000 2.19389
\(852\) 0 0
\(853\) − 14.0000i − 0.479351i −0.970853 0.239675i \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 45.0000i − 1.53717i −0.639747 0.768585i \(-0.720961\pi\)
0.639747 0.768585i \(-0.279039\pi\)
\(858\) 0 0
\(859\) 23.0000 0.784750 0.392375 0.919805i \(-0.371654\pi\)
0.392375 + 0.919805i \(0.371654\pi\)
\(860\) 0 0
\(861\) 5.00000 0.170400
\(862\) 0 0
\(863\) 4.00000i 0.136162i 0.997680 + 0.0680808i \(0.0216876\pi\)
−0.997680 + 0.0680808i \(0.978312\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 32.0000i 1.08678i
\(868\) 0 0
\(869\) 14.0000 0.474917
\(870\) 0 0
\(871\) −30.0000 −1.01651
\(872\) 0 0
\(873\) 12.0000i 0.406138i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 48.0000i − 1.62084i −0.585846 0.810422i \(-0.699238\pi\)
0.585846 0.810422i \(-0.300762\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 0 0
\(883\) − 11.0000i − 0.370179i −0.982722 0.185090i \(-0.940742\pi\)
0.982722 0.185090i \(-0.0592576\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 18.0000i − 0.604381i −0.953248 0.302190i \(-0.902282\pi\)
0.953248 0.302190i \(-0.0977178\pi\)
\(888\) 0 0
\(889\) −22.0000 −0.737856
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 6.00000i 0.200782i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 48.0000i 1.60267i
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −28.0000 −0.932815
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) − 1.00000i − 0.0330952i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 16.0000i − 0.528367i
\(918\) 0 0
\(919\) 42.0000 1.38545 0.692726 0.721201i \(-0.256409\pi\)
0.692726 + 0.721201i \(0.256409\pi\)
\(920\) 0 0
\(921\) −15.0000 −0.494267
\(922\) 0 0
\(923\) 84.0000i 2.76489i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 8.00000i 0.262754i
\(928\) 0 0
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) 10.0000i 0.327385i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.00000i 0.0326686i 0.999867 + 0.0163343i \(0.00519960\pi\)
−0.999867 + 0.0163343i \(0.994800\pi\)
\(938\) 0 0
\(939\) −22.0000 −0.717943
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 0 0
\(943\) − 40.0000i − 1.30258i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20.0000i 0.649913i 0.945729 + 0.324956i \(0.105350\pi\)
−0.945729 + 0.324956i \(0.894650\pi\)
\(948\) 0 0
\(949\) −90.0000 −2.92152
\(950\) 0 0
\(951\) −28.0000 −0.907962
\(952\) 0 0
\(953\) − 25.0000i − 0.809829i −0.914354 0.404915i \(-0.867301\pi\)
0.914354 0.404915i \(-0.132699\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 6.00000i 0.193952i
\(958\) 0 0
\(959\) −3.00000 −0.0968751
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 18.0000i − 0.580042i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 2.00000i − 0.0643157i −0.999483 0.0321578i \(-0.989762\pi\)
0.999483 0.0321578i \(-0.0102379\pi\)
\(968\) 0 0
\(969\) −7.00000 −0.224872
\(970\) 0 0
\(971\) −23.0000 −0.738105 −0.369053 0.929409i \(-0.620318\pi\)
−0.369053 + 0.929409i \(0.620318\pi\)
\(972\) 0 0
\(973\) 1.00000i 0.0320585i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 45.0000i 1.43968i 0.694141 + 0.719839i \(0.255784\pi\)
−0.694141 + 0.719839i \(0.744216\pi\)
\(978\) 0 0
\(979\) −3.00000 −0.0958804
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) 0 0
\(983\) 36.0000i 1.14822i 0.818778 + 0.574111i \(0.194652\pi\)
−0.818778 + 0.574111i \(0.805348\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.00000i 0.190982i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 0 0
\(993\) − 3.00000i − 0.0952021i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 2.00000i − 0.0633406i −0.999498 0.0316703i \(-0.989917\pi\)
0.999498 0.0316703i \(-0.0100827\pi\)
\(998\) 0 0
\(999\) −40.0000 −1.26554
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 1400.2.g.f.449.1 2
4.3 odd 2 2800.2.g.k.449.2 2
5.2 odd 4 1400.2.a.e.1.1 1
5.3 odd 4 1400.2.a.i.1.1 yes 1
5.4 even 2 inner 1400.2.g.f.449.2 2
20.3 even 4 2800.2.a.j.1.1 1
20.7 even 4 2800.2.a.v.1.1 1
20.19 odd 2 2800.2.g.k.449.1 2
35.13 even 4 9800.2.a.q.1.1 1
35.27 even 4 9800.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.a.e.1.1 1 5.2 odd 4
1400.2.a.i.1.1 yes 1 5.3 odd 4
1400.2.g.f.449.1 2 1.1 even 1 trivial
1400.2.g.f.449.2 2 5.4 even 2 inner
2800.2.a.j.1.1 1 20.3 even 4
2800.2.a.v.1.1 1 20.7 even 4
2800.2.g.k.449.1 2 20.19 odd 2
2800.2.g.k.449.2 2 4.3 odd 2
9800.2.a.q.1.1 1 35.13 even 4
9800.2.a.ba.1.1 1 35.27 even 4