Properties

Label 2366.2.d.b.337.1
Level 2366
Weight 2
Character 2366.337
Analytic conductor 18.893
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2366.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(1.00000i\)
Character \(\chi\) = 2366.337
Dual form 2366.2.d.b.337.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -2.00000 q^{3} -1.00000 q^{4} +2.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -2.00000 q^{3} -1.00000 q^{4} +2.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +2.00000 q^{12} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -1.00000i q^{18} +2.00000i q^{19} +2.00000i q^{21} -2.00000i q^{24} +5.00000 q^{25} +4.00000 q^{27} +1.00000i q^{28} -6.00000 q^{29} -4.00000i q^{31} -1.00000i q^{32} +6.00000i q^{34} -1.00000 q^{36} -2.00000i q^{37} +2.00000 q^{38} +6.00000i q^{41} +2.00000 q^{42} -8.00000 q^{43} +12.0000i q^{47} -2.00000 q^{48} -1.00000 q^{49} -5.00000i q^{50} +12.0000 q^{51} +6.00000 q^{53} -4.00000i q^{54} +1.00000 q^{56} -4.00000i q^{57} +6.00000i q^{58} +6.00000i q^{59} +8.00000 q^{61} -4.00000 q^{62} -1.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} +6.00000 q^{68} +1.00000i q^{72} -2.00000i q^{73} -2.00000 q^{74} -10.0000 q^{75} -2.00000i q^{76} +8.00000 q^{79} -11.0000 q^{81} +6.00000 q^{82} -6.00000i q^{83} -2.00000i q^{84} +8.00000i q^{86} +12.0000 q^{87} +6.00000i q^{89} +8.00000i q^{93} +12.0000 q^{94} +2.00000i q^{96} -10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} - 2q^{4} + 2q^{9} + 4q^{12} - 2q^{14} + 2q^{16} - 12q^{17} + 10q^{25} + 8q^{27} - 12q^{29} - 2q^{36} + 4q^{38} + 4q^{42} - 16q^{43} - 4q^{48} - 2q^{49} + 24q^{51} + 12q^{53} + 2q^{56} + 16q^{61} - 8q^{62} - 2q^{64} + 12q^{68} - 4q^{74} - 20q^{75} + 16q^{79} - 22q^{81} + 12q^{82} + 24q^{87} + 24q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(2199\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 2.00000i 0.816497i
\(7\) − 1.00000i − 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 2.00000 0.577350
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) − 2.00000i − 0.408248i
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 1.00000i 0.188982i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) − 4.00000i − 0.718421i −0.933257 0.359211i \(-0.883046\pi\)
0.933257 0.359211i \(-0.116954\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 6.00000i 1.02899i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) 2.00000 0.308607
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) −2.00000 −0.288675
\(49\) −1.00000 −0.142857
\(50\) − 5.00000i − 0.707107i
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) − 4.00000i − 0.544331i
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) − 4.00000i − 0.529813i
\(58\) 6.00000i 0.787839i
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −4.00000 −0.508001
\(63\) − 1.00000i − 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) −2.00000 −0.232495
\(75\) −10.0000 −1.15470
\(76\) − 2.00000i − 0.229416i
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) − 2.00000i − 0.218218i
\(85\) 0 0
\(86\) 8.00000i 0.862662i
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 2.00000i 0.204124i
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) − 12.0000i − 1.18818i
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) − 6.00000i − 0.582772i
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −4.00000 −0.384900
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 0 0
\(111\) 4.00000i 0.379663i
\(112\) − 1.00000i − 0.0944911i
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 6.00000i 0.550019i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) − 8.00000i − 0.724286i
\(123\) − 12.0000i − 1.08200i
\(124\) 4.00000i 0.359211i
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 16.0000 1.40872
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) − 6.00000i − 0.514496i
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) − 24.0000i − 2.02116i
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 2.00000 0.164957
\(148\) 2.00000i 0.164399i
\(149\) − 18.0000i − 1.47462i −0.675556 0.737309i \(-0.736096\pi\)
0.675556 0.737309i \(-0.263904\pi\)
\(150\) 10.0000i 0.816497i
\(151\) − 8.00000i − 0.651031i −0.945537 0.325515i \(-0.894462\pi\)
0.945537 0.325515i \(-0.105538\pi\)
\(152\) −2.00000 −0.162221
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 11.0000i 0.864242i
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) − 6.00000i − 0.468521i
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) −2.00000 −0.154303
\(169\) 0 0
\(170\) 0 0
\(171\) 2.00000i 0.152944i
\(172\) 8.00000 0.609994
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) − 12.0000i − 0.909718i
\(175\) − 5.00000i − 0.377964i
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) 6.00000 0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) −16.0000 −1.18275
\(184\) 0 0
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) − 12.0000i − 0.875190i
\(189\) − 4.00000i − 0.290957i
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 2.00000 0.144338
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 5.00000i 0.353553i
\(201\) 8.00000i 0.564276i
\(202\) 0 0
\(203\) 6.00000i 0.421117i
\(204\) −12.0000 −0.840168
\(205\) 0 0
\(206\) − 4.00000i − 0.278693i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) − 12.0000i − 0.820303i
\(215\) 0 0
\(216\) 4.00000i 0.272166i
\(217\) −4.00000 −0.271538
\(218\) 2.00000 0.135457
\(219\) 4.00000i 0.270295i
\(220\) 0 0
\(221\) 0 0
\(222\) 4.00000 0.268462
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 5.00000 0.333333
\(226\) − 6.00000i − 0.399114i
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 4.00000i 0.264327i 0.991228 + 0.132164i \(0.0421925\pi\)
−0.991228 + 0.132164i \(0.957808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 6.00000i − 0.393919i
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) − 6.00000i − 0.390567i
\(237\) −16.0000 −1.03931
\(238\) 6.00000 0.388922
\(239\) 24.0000i 1.55243i 0.630468 + 0.776215i \(0.282863\pi\)
−0.630468 + 0.776215i \(0.717137\pi\)
\(240\) 0 0
\(241\) 10.0000i 0.644157i 0.946713 + 0.322078i \(0.104381\pi\)
−0.946713 + 0.322078i \(0.895619\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) 10.0000 0.641500
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) −12.0000 −0.765092
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 12.0000i 0.760469i
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 1.00000i 0.0629941i
\(253\) 0 0
\(254\) − 16.0000i − 1.00393i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) − 16.0000i − 0.996116i
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) − 18.0000i − 1.11204i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) − 2.00000i − 0.122628i
\(267\) − 12.0000i − 0.734388i
\(268\) 4.00000i 0.244339i
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) 16.0000i 0.971931i 0.873978 + 0.485965i \(0.161532\pi\)
−0.873978 + 0.485965i \(0.838468\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) − 14.0000i − 0.839664i
\(279\) − 4.00000i − 0.239474i
\(280\) 0 0
\(281\) 6.00000i 0.357930i 0.983855 + 0.178965i \(0.0572749\pi\)
−0.983855 + 0.178965i \(0.942725\pi\)
\(282\) −24.0000 −1.42918
\(283\) 22.0000 1.30776 0.653882 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) − 1.00000i − 0.0589256i
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 20.0000i 1.17242i
\(292\) 2.00000i 0.117041i
\(293\) − 24.0000i − 1.40209i −0.713115 0.701047i \(-0.752716\pi\)
0.713115 0.701047i \(-0.247284\pi\)
\(294\) − 2.00000i − 0.116642i
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 10.0000 0.577350
\(301\) 8.00000i 0.461112i
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 2.00000i 0.114708i
\(305\) 0 0
\(306\) 6.00000i 0.342997i
\(307\) − 2.00000i − 0.114146i −0.998370 0.0570730i \(-0.981823\pi\)
0.998370 0.0570730i \(-0.0181768\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 4.00000i 0.225733i
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 12.0000i 0.672927i
\(319\) 0 0
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) − 12.0000i − 0.667698i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) − 4.00000i − 0.221201i
\(328\) −6.00000 −0.331295
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 8.00000i 0.439720i 0.975531 + 0.219860i \(0.0705600\pi\)
−0.975531 + 0.219860i \(0.929440\pi\)
\(332\) 6.00000i 0.329293i
\(333\) − 2.00000i − 0.109599i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 2.00000i 0.109109i
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) 1.00000i 0.0539949i
\(344\) − 8.00000i − 0.431331i
\(345\) 0 0
\(346\) − 12.0000i − 0.645124i
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) −12.0000 −0.643268
\(349\) 28.0000i 1.49881i 0.662114 + 0.749403i \(0.269659\pi\)
−0.662114 + 0.749403i \(0.730341\pi\)
\(350\) −5.00000 −0.267261
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) − 6.00000i − 0.317999i
\(357\) − 12.0000i − 0.635107i
\(358\) − 12.0000i − 0.634220i
\(359\) 24.0000i 1.26667i 0.773877 + 0.633336i \(0.218315\pi\)
−0.773877 + 0.633336i \(0.781685\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 20.0000i 1.05118i
\(363\) −22.0000 −1.15470
\(364\) 0 0
\(365\) 0 0
\(366\) 16.0000i 0.836333i
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 6.00000i 0.312348i
\(370\) 0 0
\(371\) − 6.00000i − 0.311504i
\(372\) − 8.00000i − 0.414781i
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) −4.00000 −0.205738
\(379\) − 16.0000i − 0.821865i −0.911666 0.410932i \(-0.865203\pi\)
0.911666 0.410932i \(-0.134797\pi\)
\(380\) 0 0
\(381\) −32.0000 −1.63941
\(382\) − 24.0000i − 1.22795i
\(383\) 36.0000i 1.83951i 0.392488 + 0.919757i \(0.371614\pi\)
−0.392488 + 0.919757i \(0.628386\pi\)
\(384\) − 2.00000i − 0.102062i
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −8.00000 −0.406663
\(388\) 10.0000i 0.507673i
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 1.00000i − 0.0505076i
\(393\) −36.0000 −1.81596
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) − 20.0000i − 1.00377i −0.864934 0.501886i \(-0.832640\pi\)
0.864934 0.501886i \(-0.167360\pi\)
\(398\) 20.0000i 1.00251i
\(399\) −4.00000 −0.200250
\(400\) 5.00000 0.250000
\(401\) 18.0000i 0.898877i 0.893311 + 0.449439i \(0.148376\pi\)
−0.893311 + 0.449439i \(0.851624\pi\)
\(402\) 8.00000 0.399004
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 0 0
\(408\) 12.0000i 0.594089i
\(409\) 14.0000i 0.692255i 0.938187 + 0.346128i \(0.112504\pi\)
−0.938187 + 0.346128i \(0.887496\pi\)
\(410\) 0 0
\(411\) 36.0000i 1.77575i
\(412\) −4.00000 −0.197066
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −28.0000 −1.37117
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) − 10.0000i − 0.487370i −0.969854 0.243685i \(-0.921644\pi\)
0.969854 0.243685i \(-0.0783563\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 12.0000i 0.583460i
\(424\) 6.00000i 0.291386i
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) − 8.00000i − 0.387147i
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000i 1.15604i 0.816023 + 0.578020i \(0.196174\pi\)
−0.816023 + 0.578020i \(0.803826\pi\)
\(432\) 4.00000 0.192450
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 4.00000i 0.192006i
\(435\) 0 0
\(436\) − 2.00000i − 0.0957826i
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) − 4.00000i − 0.189832i
\(445\) 0 0
\(446\) 8.00000 0.378811
\(447\) 36.0000i 1.70274i
\(448\) 1.00000i 0.0472456i
\(449\) − 18.0000i − 0.849473i −0.905317 0.424736i \(-0.860367\pi\)
0.905317 0.424736i \(-0.139633\pi\)
\(450\) − 5.00000i − 0.235702i
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 16.0000i 0.751746i
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 4.00000 0.186908
\(459\) −24.0000 −1.12022
\(460\) 0 0
\(461\) 12.0000i 0.558896i 0.960161 + 0.279448i \(0.0901514\pi\)
−0.960161 + 0.279448i \(0.909849\pi\)
\(462\) 0 0
\(463\) − 32.0000i − 1.48717i −0.668644 0.743583i \(-0.733125\pi\)
0.668644 0.743583i \(-0.266875\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) − 6.00000i − 0.277945i
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 8.00000 0.368621
\(472\) −6.00000 −0.276172
\(473\) 0 0
\(474\) 16.0000i 0.734904i
\(475\) 10.0000i 0.458831i
\(476\) − 6.00000i − 0.275010i
\(477\) 6.00000 0.274721
\(478\) 24.0000 1.09773
\(479\) 36.0000i 1.64488i 0.568850 + 0.822441i \(0.307388\pi\)
−0.568850 + 0.822441i \(0.692612\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) − 10.0000i − 0.453609i
\(487\) − 16.0000i − 0.725029i −0.931978 0.362515i \(-0.881918\pi\)
0.931978 0.362515i \(-0.118082\pi\)
\(488\) 8.00000i 0.362143i
\(489\) − 32.0000i − 1.44709i
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 12.0000i 0.541002i
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) − 4.00000i − 0.179605i
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) − 4.00000i − 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) 0 0
\(501\) − 24.0000i − 1.07224i
\(502\) − 18.0000i − 0.803379i
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) 36.0000i 1.59567i 0.602875 + 0.797836i \(0.294022\pi\)
−0.602875 + 0.797836i \(0.705978\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) − 1.00000i − 0.0441942i
\(513\) 8.00000i 0.353209i
\(514\) 18.0000i 0.793946i
\(515\) 0 0
\(516\) −16.0000 −0.704361
\(517\) 0 0
\(518\) 2.00000i 0.0878750i
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 6.00000i 0.262613i
\(523\) 2.00000 0.0874539 0.0437269 0.999044i \(-0.486077\pi\)
0.0437269 + 0.999044i \(0.486077\pi\)
\(524\) −18.0000 −0.786334
\(525\) 10.0000i 0.436436i
\(526\) 0 0
\(527\) 24.0000i 1.04546i
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 6.00000i 0.260378i
\(532\) −2.00000 −0.0867110
\(533\) 0 0
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −24.0000 −1.03568
\(538\) 12.0000i 0.517357i
\(539\) 0 0
\(540\) 0 0
\(541\) − 38.0000i − 1.63375i −0.576816 0.816874i \(-0.695705\pi\)
0.576816 0.816874i \(-0.304295\pi\)
\(542\) 16.0000 0.687259
\(543\) 40.0000 1.71656
\(544\) 6.00000i 0.257248i
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 18.0000i 0.768922i
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) − 12.0000i − 0.511217i
\(552\) 0 0
\(553\) − 8.00000i − 0.340195i
\(554\) − 10.0000i − 0.424859i
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) − 6.00000i − 0.254228i −0.991888 0.127114i \(-0.959429\pi\)
0.991888 0.127114i \(-0.0405714\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −30.0000 −1.26435 −0.632175 0.774826i \(-0.717837\pi\)
−0.632175 + 0.774826i \(0.717837\pi\)
\(564\) 24.0000i 1.01058i
\(565\) 0 0
\(566\) − 22.0000i − 0.924729i
\(567\) 11.0000i 0.461957i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) −48.0000 −2.00523
\(574\) − 6.00000i − 0.250435i
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) 28.0000i 1.16364i
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 20.0000 0.829027
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −24.0000 −0.991431
\(587\) − 42.0000i − 1.73353i −0.498721 0.866763i \(-0.666197\pi\)
0.498721 0.866763i \(-0.333803\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 36.0000i 1.48084i
\(592\) − 2.00000i − 0.0821995i
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000i 0.737309i
\(597\) 40.0000 1.63709
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) − 10.0000i − 0.408248i
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 8.00000 0.326056
\(603\) − 4.00000i − 0.162893i
\(604\) 8.00000i 0.325515i
\(605\) 0 0
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 2.00000 0.0811107
\(609\) − 12.0000i − 0.486265i
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 8.00000i 0.321807i
\(619\) − 26.0000i − 1.04503i −0.852631 0.522514i \(-0.824994\pi\)
0.852631 0.522514i \(-0.175006\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 24.0000i − 0.962312i
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 10.0000i 0.399680i
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) 12.0000i 0.478471i
\(630\) 0 0
\(631\) 16.0000i 0.636950i 0.947931 + 0.318475i \(0.103171\pi\)
−0.947931 + 0.318475i \(0.896829\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 8.00000 0.317971
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 24.0000i 0.947204i
\(643\) 14.0000i 0.552106i 0.961142 + 0.276053i \(0.0890266\pi\)
−0.961142 + 0.276053i \(0.910973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) − 11.0000i − 0.432121i
\(649\) 0 0
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) − 16.0000i − 0.626608i
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) 6.00000i 0.234261i
\(657\) − 2.00000i − 0.0780274i
\(658\) − 12.0000i − 0.467809i
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) 40.0000i 1.55582i 0.628376 + 0.777910i \(0.283720\pi\)
−0.628376 + 0.777910i \(0.716280\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) − 12.0000i − 0.464294i
\(669\) − 16.0000i − 0.618596i
\(670\) 0 0
\(671\) 0 0
\(672\) 2.00000 0.0771517
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 14.0000i 0.539260i
\(675\) 20.0000 0.769800
\(676\) 0 0
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 12.0000i 0.460857i
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) − 36.0000i − 1.37952i
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) − 2.00000i − 0.0764719i
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) − 8.00000i − 0.305219i
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) − 46.0000i − 1.74992i −0.484193 0.874961i \(-0.660887\pi\)
0.484193 0.874961i \(-0.339113\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 24.0000i 0.911028i
\(695\) 0 0
\(696\) 12.0000i 0.454859i
\(697\) − 36.0000i − 1.36360i
\(698\) 28.0000 1.05982
\(699\) −12.0000 −0.453882
\(700\) 5.00000i 0.188982i
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 12.0000i 0.450988i
\(709\) 46.0000i 1.72757i 0.503864 + 0.863783i \(0.331911\pi\)
−0.503864 + 0.863783i \(0.668089\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) − 48.0000i − 1.79259i
\(718\) 24.0000 0.895672
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) − 4.00000i − 0.148968i
\(722\) − 15.0000i − 0.558242i
\(723\) − 20.0000i − 0.743808i
\(724\) 20.0000 0.743294
\(725\) −30.0000 −1.11417
\(726\) 22.0000i 0.816497i
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 16.0000 0.591377
\(733\) − 40.0000i − 1.47743i −0.674016 0.738717i \(-0.735432\pi\)
0.674016 0.738717i \(-0.264568\pi\)
\(734\) − 8.00000i − 0.295285i
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) 16.0000i 0.588570i 0.955718 + 0.294285i \(0.0950814\pi\)
−0.955718 + 0.294285i \(0.904919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) − 14.0000i − 0.512576i
\(747\) − 6.00000i − 0.219529i
\(748\) 0 0
\(749\) − 12.0000i − 0.438470i
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 12.0000i 0.437595i
\(753\) −36.0000 −1.31191
\(754\) 0 0
\(755\) 0 0
\(756\) 4.00000i 0.145479i
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000i 0.652499i 0.945284 + 0.326250i \(0.105785\pi\)
−0.945284 + 0.326250i \(0.894215\pi\)
\(762\) 32.0000i 1.15924i
\(763\) 2.00000 0.0724049
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) 0 0
\(768\) −2.00000 −0.0721688
\(769\) 14.0000i 0.504853i 0.967616 + 0.252426i \(0.0812286\pi\)
−0.967616 + 0.252426i \(0.918771\pi\)
\(770\) 0 0
\(771\) 36.0000 1.29651
\(772\) 14.0000i 0.503871i
\(773\) 24.0000i 0.863220i 0.902060 + 0.431610i \(0.142054\pi\)
−0.902060 + 0.431610i \(0.857946\pi\)
\(774\) 8.00000i 0.287554i
\(775\) − 20.0000i − 0.718421i
\(776\) 10.0000 0.358979
\(777\) 4.00000 0.143499
\(778\) 18.0000i 0.645331i
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −24.0000 −0.857690
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 36.0000i 1.28408i
\(787\) 22.0000i 0.784215i 0.919919 + 0.392108i \(0.128254\pi\)
−0.919919 + 0.392108i \(0.871746\pi\)
\(788\) 18.0000i 0.641223i
\(789\) 0 0
\(790\) 0 0
\(791\) − 6.00000i − 0.213335i
\(792\) 0 0
\(793\) 0 0
\(794\) −20.0000 −0.709773
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 4.00000i 0.141598i
\(799\) − 72.0000i − 2.54718i
\(800\) − 5.00000i − 0.176777i
\(801\) 6.00000i 0.212000i
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) − 8.00000i − 0.282138i
\(805\) 0 0
\(806\) 0 0
\(807\) 24.0000 0.844840
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 2.00000i 0.0702295i 0.999383 + 0.0351147i \(0.0111797\pi\)
−0.999383 + 0.0351147i \(0.988820\pi\)
\(812\) − 6.00000i − 0.210559i
\(813\) − 32.0000i − 1.12229i
\(814\) 0 0
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) − 16.0000i − 0.559769i
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000i 0.209401i 0.994504 + 0.104701i \(0.0333885\pi\)
−0.994504 + 0.104701i \(0.966612\pi\)
\(822\) 36.0000 1.25564
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 4.00000i 0.139347i
\(825\) 0 0
\(826\) − 6.00000i − 0.208767i
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) 0 0
\(829\) −56.0000 −1.94496 −0.972480 0.232986i \(-0.925151\pi\)
−0.972480 + 0.232986i \(0.925151\pi\)
\(830\) 0 0
\(831\) −20.0000 −0.693792
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 28.0000i 0.969561i
\(835\) 0 0
\(836\) 0 0
\(837\) − 16.0000i − 0.553041i
\(838\) − 6.00000i − 0.207267i
\(839\) − 12.0000i − 0.414286i −0.978311 0.207143i \(-0.933583\pi\)
0.978311 0.207143i \(-0.0664165\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) − 12.0000i − 0.413302i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) − 11.0000i − 0.377964i
\(848\) 6.00000 0.206041
\(849\) −44.0000 −1.51008
\(850\) 30.0000i 1.02899i
\(851\) 0 0
\(852\) 0 0
\(853\) − 44.0000i − 1.50653i −0.657716 0.753266i \(-0.728477\pi\)
0.657716 0.753266i \(-0.271523\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 12.0000i 0.410152i
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 24.0000 0.817443
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) − 4.00000i − 0.136083i
\(865\) 0 0
\(866\) − 34.0000i − 1.15537i
\(867\) −38.0000 −1.29055
\(868\) 4.00000 0.135769
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −2.00000 −0.0677285
\(873\) − 10.0000i − 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) − 4.00000i − 0.135147i
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) 8.00000i 0.269987i
\(879\) 48.0000i 1.61900i
\(880\) 0 0
\(881\) 54.0000 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) 1.00000i 0.0336718i
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000i 0.403148i
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) −4.00000 −0.134231
\(889\) − 16.0000i − 0.536623i
\(890\) 0 0
\(891\) 0 0
\(892\) − 8.00000i − 0.267860i
\(893\) −24.0000 −0.803129
\(894\) 36.0000 1.20402
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 24.0000i 0.800445i
\(900\) −5.00000 −0.166667
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) − 16.0000i − 0.532447i
\(904\) 6.00000i 0.199557i
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) − 18.0000i − 0.597351i
\(909\) 0 0
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) − 4.00000i − 0.132164i
\(917\) − 18.0000i − 0.594412i
\(918\) 24.0000i 0.792118i
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) 4.00000i 0.131804i
\(922\) 12.0000 0.395199
\(923\) 0 0
\(924\) 0 0
\(925\) − 10.0000i − 0.328798i
\(926\) −32.0000 −1.05159
\(927\) 4.00000 0.131377
\(928\) 6.00000i 0.196960i
\(929\) 6.00000i 0.196854i 0.995144 + 0.0984268i \(0.0313810\pi\)
−0.995144 + 0.0984268i \(0.968619\pi\)
\(930\) 0 0
\(931\) − 2.00000i − 0.0655474i
\(932\) −6.00000 −0.196537
\(933\) −48.0000 −1.57145
\(934\) − 6.00000i − 0.196326i
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 4.00000i 0.130605i
\(939\) 20.0000 0.652675
\(940\) 0 0
\(941\) − 24.0000i − 0.782378i −0.920310 0.391189i \(-0.872064\pi\)
0.920310 0.391189i \(-0.127936\pi\)
\(942\) − 8.00000i − 0.260654i
\(943\) 0 0
\(944\) 6.00000i 0.195283i
\(945\) 0 0
\(946\) 0 0
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) 16.0000 0.519656
\(949\) 0 0
\(950\) 10.0000 0.324443
\(951\) − 12.0000i − 0.389127i
\(952\) −6.00000 −0.194461
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) − 6.00000i − 0.194257i
\(955\) 0 0
\(956\) − 24.0000i − 0.776215i
\(957\) 0 0
\(958\) 36.0000 1.16311
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) − 10.0000i − 0.322078i
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 11.0000i 0.353553i
\(969\) 24.0000i 0.770991i
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) −10.0000 −0.320750
\(973\) − 14.0000i − 0.448819i
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) − 6.00000i − 0.191957i −0.995383 0.0959785i \(-0.969402\pi\)
0.995383 0.0959785i \(-0.0305980\pi\)
\(978\) −32.0000 −1.02325
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000i 0.0638551i
\(982\) − 12.0000i − 0.382935i
\(983\) 36.0000i 1.14822i 0.818778 + 0.574111i \(0.194652\pi\)
−0.818778 + 0.574111i \(0.805348\pi\)
\(984\) 12.0000 0.382546
\(985\) 0 0
\(986\) − 36.0000i − 1.14647i
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −4.00000 −0.127000
\(993\) − 16.0000i − 0.507745i
\(994\) 0 0
\(995\) 0 0
\(996\) − 12.0000i − 0.380235i
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) −4.00000 −0.126618
\(999\) − 8.00000i − 0.253109i
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 2366.2.d.b.337.1 2
13.5 odd 4 14.2.a.a.1.1 1
13.8 odd 4 2366.2.a.j.1.1 1
13.12 even 2 inner 2366.2.d.b.337.2 2
39.5 even 4 126.2.a.b.1.1 1
52.31 even 4 112.2.a.c.1.1 1
65.18 even 4 350.2.c.d.99.2 2
65.44 odd 4 350.2.a.f.1.1 1
65.57 even 4 350.2.c.d.99.1 2
91.5 even 12 98.2.c.a.67.1 2
91.18 odd 12 98.2.c.b.79.1 2
91.31 even 12 98.2.c.a.79.1 2
91.44 odd 12 98.2.c.b.67.1 2
91.83 even 4 98.2.a.a.1.1 1
104.5 odd 4 448.2.a.g.1.1 1
104.83 even 4 448.2.a.a.1.1 1
117.5 even 12 1134.2.f.f.379.1 2
117.31 odd 12 1134.2.f.l.379.1 2
117.70 odd 12 1134.2.f.l.757.1 2
117.83 even 12 1134.2.f.f.757.1 2
143.109 even 4 1694.2.a.e.1.1 1
156.83 odd 4 1008.2.a.h.1.1 1
195.44 even 4 3150.2.a.i.1.1 1
195.83 odd 4 3150.2.g.j.2899.1 2
195.122 odd 4 3150.2.g.j.2899.2 2
208.5 odd 4 1792.2.b.c.897.2 2
208.83 even 4 1792.2.b.g.897.2 2
208.109 odd 4 1792.2.b.c.897.1 2
208.187 even 4 1792.2.b.g.897.1 2
221.135 odd 4 4046.2.a.f.1.1 1
247.18 even 4 5054.2.a.c.1.1 1
260.83 odd 4 2800.2.g.h.449.2 2
260.187 odd 4 2800.2.g.h.449.1 2
260.239 even 4 2800.2.a.g.1.1 1
273.5 odd 12 882.2.g.d.361.1 2
273.44 even 12 882.2.g.c.361.1 2
273.83 odd 4 882.2.a.i.1.1 1
273.122 odd 12 882.2.g.d.667.1 2
273.200 even 12 882.2.g.c.667.1 2
299.252 even 4 7406.2.a.a.1.1 1
312.5 even 4 4032.2.a.w.1.1 1
312.83 odd 4 4032.2.a.r.1.1 1
364.31 odd 12 784.2.i.i.177.1 2
364.83 odd 4 784.2.a.b.1.1 1
364.135 even 12 784.2.i.c.753.1 2
364.187 odd 12 784.2.i.i.753.1 2
364.291 even 12 784.2.i.c.177.1 2
455.83 odd 4 2450.2.c.c.99.2 2
455.174 even 4 2450.2.a.t.1.1 1
455.447 odd 4 2450.2.c.c.99.1 2
728.83 odd 4 3136.2.a.z.1.1 1
728.629 even 4 3136.2.a.e.1.1 1
1092.83 even 4 7056.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 1 13.5 odd 4
98.2.a.a.1.1 1 91.83 even 4
98.2.c.a.67.1 2 91.5 even 12
98.2.c.a.79.1 2 91.31 even 12
98.2.c.b.67.1 2 91.44 odd 12
98.2.c.b.79.1 2 91.18 odd 12
112.2.a.c.1.1 1 52.31 even 4
126.2.a.b.1.1 1 39.5 even 4
350.2.a.f.1.1 1 65.44 odd 4
350.2.c.d.99.1 2 65.57 even 4
350.2.c.d.99.2 2 65.18 even 4
448.2.a.a.1.1 1 104.83 even 4
448.2.a.g.1.1 1 104.5 odd 4
784.2.a.b.1.1 1 364.83 odd 4
784.2.i.c.177.1 2 364.291 even 12
784.2.i.c.753.1 2 364.135 even 12
784.2.i.i.177.1 2 364.31 odd 12
784.2.i.i.753.1 2 364.187 odd 12
882.2.a.i.1.1 1 273.83 odd 4
882.2.g.c.361.1 2 273.44 even 12
882.2.g.c.667.1 2 273.200 even 12
882.2.g.d.361.1 2 273.5 odd 12
882.2.g.d.667.1 2 273.122 odd 12
1008.2.a.h.1.1 1 156.83 odd 4
1134.2.f.f.379.1 2 117.5 even 12
1134.2.f.f.757.1 2 117.83 even 12
1134.2.f.l.379.1 2 117.31 odd 12
1134.2.f.l.757.1 2 117.70 odd 12
1694.2.a.e.1.1 1 143.109 even 4
1792.2.b.c.897.1 2 208.109 odd 4
1792.2.b.c.897.2 2 208.5 odd 4
1792.2.b.g.897.1 2 208.187 even 4
1792.2.b.g.897.2 2 208.83 even 4
2366.2.a.j.1.1 1 13.8 odd 4
2366.2.d.b.337.1 2 1.1 even 1 trivial
2366.2.d.b.337.2 2 13.12 even 2 inner
2450.2.a.t.1.1 1 455.174 even 4
2450.2.c.c.99.1 2 455.447 odd 4
2450.2.c.c.99.2 2 455.83 odd 4
2800.2.a.g.1.1 1 260.239 even 4
2800.2.g.h.449.1 2 260.187 odd 4
2800.2.g.h.449.2 2 260.83 odd 4
3136.2.a.e.1.1 1 728.629 even 4
3136.2.a.z.1.1 1 728.83 odd 4
3150.2.a.i.1.1 1 195.44 even 4
3150.2.g.j.2899.1 2 195.83 odd 4
3150.2.g.j.2899.2 2 195.122 odd 4
4032.2.a.r.1.1 1 312.83 odd 4
4032.2.a.w.1.1 1 312.5 even 4
4046.2.a.f.1.1 1 221.135 odd 4
5054.2.a.c.1.1 1 247.18 even 4
7056.2.a.bd.1.1 1 1092.83 even 4
7406.2.a.a.1.1 1 299.252 even 4