Properties

Label 1680.2.t.h.1009.1
Level $1680$
Weight $2$
Character 1680.1009
Analytic conductor $13.415$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.t (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.1
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1680.1009
Dual form 1680.2.t.h.1009.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} -2.23607 q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -2.23607 q^{5} -1.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} -4.47214i q^{13} +2.23607i q^{15} -2.47214i q^{17} +2.00000 q^{19} -1.00000 q^{21} +4.00000i q^{23} +5.00000 q^{25} +1.00000i q^{27} -0.472136 q^{29} -8.47214 q^{31} -2.00000i q^{33} +2.23607i q^{35} -6.47214i q^{37} -4.47214 q^{39} -12.4721 q^{41} +6.47214i q^{43} +2.23607 q^{45} +2.47214i q^{47} -1.00000 q^{49} -2.47214 q^{51} -2.00000i q^{53} -4.47214 q^{55} -2.00000i q^{57} -12.4721 q^{61} +1.00000i q^{63} +10.0000i q^{65} -10.4721i q^{67} +4.00000 q^{69} -3.52786 q^{71} -16.4721i q^{73} -5.00000i q^{75} -2.00000i q^{77} -8.94427 q^{79} +1.00000 q^{81} +12.9443i q^{83} +5.52786i q^{85} +0.472136i q^{87} -9.41641 q^{89} -4.47214 q^{91} +8.47214i q^{93} -4.47214 q^{95} +12.4721i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 8q^{11} + 8q^{19} - 4q^{21} + 20q^{25} + 16q^{29} - 16q^{31} - 32q^{41} - 4q^{49} + 8q^{51} - 32q^{61} + 16q^{69} - 32q^{71} + 4q^{81} + 16q^{89} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(1\)

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
\(4\) 0 0
\(5\) −2.23607 −1.00000
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) − 4.47214i − 1.24035i −0.784465 0.620174i \(-0.787062\pi\)
0.784465 0.620174i \(-0.212938\pi\)
\(14\) 0 0
\(15\) 2.23607i 0.577350i
\(16\) 0 0
\(17\) − 2.47214i − 0.599581i −0.954005 0.299791i \(-0.903083\pi\)
0.954005 0.299791i \(-0.0969168\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −0.472136 −0.0876734 −0.0438367 0.999039i \(-0.513958\pi\)
−0.0438367 + 0.999039i \(0.513958\pi\)
\(30\) 0 0
\(31\) −8.47214 −1.52164 −0.760820 0.648963i \(-0.775203\pi\)
−0.760820 + 0.648963i \(0.775203\pi\)
\(32\) 0 0
\(33\) − 2.00000i − 0.348155i
\(34\) 0 0
\(35\) 2.23607i 0.377964i
\(36\) 0 0
\(37\) − 6.47214i − 1.06401i −0.846740 0.532006i \(-0.821438\pi\)
0.846740 0.532006i \(-0.178562\pi\)
\(38\) 0 0
\(39\) −4.47214 −0.716115
\(40\) 0 0
\(41\) −12.4721 −1.94782 −0.973910 0.226934i \(-0.927130\pi\)
−0.973910 + 0.226934i \(0.927130\pi\)
\(42\) 0 0
\(43\) 6.47214i 0.986991i 0.869748 + 0.493496i \(0.164281\pi\)
−0.869748 + 0.493496i \(0.835719\pi\)
\(44\) 0 0
\(45\) 2.23607 0.333333
\(46\) 0 0
\(47\) 2.47214i 0.360598i 0.983612 + 0.180299i \(0.0577065\pi\)
−0.983612 + 0.180299i \(0.942293\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −2.47214 −0.346168
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) −4.47214 −0.603023
\(56\) 0 0
\(57\) − 2.00000i − 0.264906i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −12.4721 −1.59689 −0.798447 0.602066i \(-0.794345\pi\)
−0.798447 + 0.602066i \(0.794345\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 10.0000i 1.24035i
\(66\) 0 0
\(67\) − 10.4721i − 1.27938i −0.768635 0.639688i \(-0.779064\pi\)
0.768635 0.639688i \(-0.220936\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −3.52786 −0.418680 −0.209340 0.977843i \(-0.567132\pi\)
−0.209340 + 0.977843i \(0.567132\pi\)
\(72\) 0 0
\(73\) − 16.4721i − 1.92792i −0.266051 0.963959i \(-0.585719\pi\)
0.266051 0.963959i \(-0.414281\pi\)
\(74\) 0 0
\(75\) − 5.00000i − 0.577350i
\(76\) 0 0
\(77\) − 2.00000i − 0.227921i
\(78\) 0 0
\(79\) −8.94427 −1.00631 −0.503155 0.864196i \(-0.667827\pi\)
−0.503155 + 0.864196i \(0.667827\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.9443i 1.42082i 0.703789 + 0.710409i \(0.251490\pi\)
−0.703789 + 0.710409i \(0.748510\pi\)
\(84\) 0 0
\(85\) 5.52786i 0.599581i
\(86\) 0 0
\(87\) 0.472136i 0.0506183i
\(88\) 0 0
\(89\) −9.41641 −0.998137 −0.499069 0.866562i \(-0.666324\pi\)
−0.499069 + 0.866562i \(0.666324\pi\)
\(90\) 0 0
\(91\) −4.47214 −0.468807
\(92\) 0 0
\(93\) 8.47214i 0.878520i
\(94\) 0 0
\(95\) −4.47214 −0.458831
\(96\) 0 0
\(97\) 12.4721i 1.26635i 0.774007 + 0.633177i \(0.218249\pi\)
−0.774007 + 0.633177i \(0.781751\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −4.47214 −0.444994 −0.222497 0.974933i \(-0.571421\pi\)
−0.222497 + 0.974933i \(0.571421\pi\)
\(102\) 0 0
\(103\) − 4.94427i − 0.487174i −0.969879 0.243587i \(-0.921676\pi\)
0.969879 0.243587i \(-0.0783241\pi\)
\(104\) 0 0
\(105\) 2.23607 0.218218
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 2.94427 0.282010 0.141005 0.990009i \(-0.454967\pi\)
0.141005 + 0.990009i \(0.454967\pi\)
\(110\) 0 0
\(111\) −6.47214 −0.614308
\(112\) 0 0
\(113\) 2.94427i 0.276974i 0.990364 + 0.138487i \(0.0442239\pi\)
−0.990364 + 0.138487i \(0.955776\pi\)
\(114\) 0 0
\(115\) − 8.94427i − 0.834058i
\(116\) 0 0
\(117\) 4.47214i 0.413449i
\(118\) 0 0
\(119\) −2.47214 −0.226620
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 12.4721i 1.12457i
\(124\) 0 0
\(125\) −11.1803 −1.00000
\(126\) 0 0
\(127\) − 4.00000i − 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) 0 0
\(129\) 6.47214 0.569840
\(130\) 0 0
\(131\) 21.8885 1.91241 0.956205 0.292696i \(-0.0945525\pi\)
0.956205 + 0.292696i \(0.0945525\pi\)
\(132\) 0 0
\(133\) − 2.00000i − 0.173422i
\(134\) 0 0
\(135\) − 2.23607i − 0.192450i
\(136\) 0 0
\(137\) 15.8885i 1.35745i 0.734393 + 0.678725i \(0.237467\pi\)
−0.734393 + 0.678725i \(0.762533\pi\)
\(138\) 0 0
\(139\) 14.9443 1.26756 0.633778 0.773515i \(-0.281503\pi\)
0.633778 + 0.773515i \(0.281503\pi\)
\(140\) 0 0
\(141\) 2.47214 0.208191
\(142\) 0 0
\(143\) − 8.94427i − 0.747958i
\(144\) 0 0
\(145\) 1.05573 0.0876734
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) 3.52786 0.289014 0.144507 0.989504i \(-0.453840\pi\)
0.144507 + 0.989504i \(0.453840\pi\)
\(150\) 0 0
\(151\) −17.8885 −1.45575 −0.727875 0.685710i \(-0.759492\pi\)
−0.727875 + 0.685710i \(0.759492\pi\)
\(152\) 0 0
\(153\) 2.47214i 0.199860i
\(154\) 0 0
\(155\) 18.9443 1.52164
\(156\) 0 0
\(157\) − 17.4164i − 1.38998i −0.719019 0.694990i \(-0.755409\pi\)
0.719019 0.694990i \(-0.244591\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) − 3.41641i − 0.267594i −0.991009 0.133797i \(-0.957283\pi\)
0.991009 0.133797i \(-0.0427170\pi\)
\(164\) 0 0
\(165\) 4.47214i 0.348155i
\(166\) 0 0
\(167\) − 1.52786i − 0.118230i −0.998251 0.0591148i \(-0.981172\pi\)
0.998251 0.0591148i \(-0.0188278\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) − 12.0000i − 0.912343i −0.889892 0.456172i \(-0.849220\pi\)
0.889892 0.456172i \(-0.150780\pi\)
\(174\) 0 0
\(175\) − 5.00000i − 0.377964i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.05573 −0.377883 −0.188941 0.981988i \(-0.560506\pi\)
−0.188941 + 0.981988i \(0.560506\pi\)
\(180\) 0 0
\(181\) 7.52786 0.559542 0.279771 0.960067i \(-0.409742\pi\)
0.279771 + 0.960067i \(0.409742\pi\)
\(182\) 0 0
\(183\) 12.4721i 0.921967i
\(184\) 0 0
\(185\) 14.4721i 1.06401i
\(186\) 0 0
\(187\) − 4.94427i − 0.361561i
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −9.41641 −0.681347 −0.340674 0.940182i \(-0.610655\pi\)
−0.340674 + 0.940182i \(0.610655\pi\)
\(192\) 0 0
\(193\) 4.94427i 0.355896i 0.984040 + 0.177948i \(0.0569460\pi\)
−0.984040 + 0.177948i \(0.943054\pi\)
\(194\) 0 0
\(195\) 10.0000 0.716115
\(196\) 0 0
\(197\) 15.8885i 1.13201i 0.824401 + 0.566006i \(0.191512\pi\)
−0.824401 + 0.566006i \(0.808488\pi\)
\(198\) 0 0
\(199\) 11.5279 0.817189 0.408594 0.912716i \(-0.366019\pi\)
0.408594 + 0.912716i \(0.366019\pi\)
\(200\) 0 0
\(201\) −10.4721 −0.738648
\(202\) 0 0
\(203\) 0.472136i 0.0331374i
\(204\) 0 0
\(205\) 27.8885 1.94782
\(206\) 0 0
\(207\) − 4.00000i − 0.278019i
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 9.88854 0.680755 0.340378 0.940289i \(-0.389445\pi\)
0.340378 + 0.940289i \(0.389445\pi\)
\(212\) 0 0
\(213\) 3.52786i 0.241725i
\(214\) 0 0
\(215\) − 14.4721i − 0.986991i
\(216\) 0 0
\(217\) 8.47214i 0.575126i
\(218\) 0 0
\(219\) −16.4721 −1.11308
\(220\) 0 0
\(221\) −11.0557 −0.743689
\(222\) 0 0
\(223\) 3.05573i 0.204627i 0.994752 + 0.102313i \(0.0326244\pi\)
−0.994752 + 0.102313i \(0.967376\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) − 25.8885i − 1.71828i −0.511738 0.859142i \(-0.670998\pi\)
0.511738 0.859142i \(-0.329002\pi\)
\(228\) 0 0
\(229\) −18.3607 −1.21331 −0.606654 0.794966i \(-0.707489\pi\)
−0.606654 + 0.794966i \(0.707489\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) − 14.9443i − 0.979032i −0.871994 0.489516i \(-0.837174\pi\)
0.871994 0.489516i \(-0.162826\pi\)
\(234\) 0 0
\(235\) − 5.52786i − 0.360598i
\(236\) 0 0
\(237\) 8.94427i 0.580993i
\(238\) 0 0
\(239\) 24.4721 1.58297 0.791485 0.611188i \(-0.209308\pi\)
0.791485 + 0.611188i \(0.209308\pi\)
\(240\) 0 0
\(241\) 23.8885 1.53880 0.769398 0.638769i \(-0.220556\pi\)
0.769398 + 0.638769i \(0.220556\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 2.23607 0.142857
\(246\) 0 0
\(247\) − 8.94427i − 0.569110i
\(248\) 0 0
\(249\) 12.9443 0.820310
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 8.00000i 0.502956i
\(254\) 0 0
\(255\) 5.52786 0.346168
\(256\) 0 0
\(257\) − 3.41641i − 0.213110i −0.994307 0.106555i \(-0.966018\pi\)
0.994307 0.106555i \(-0.0339820\pi\)
\(258\) 0 0
\(259\) −6.47214 −0.402159
\(260\) 0 0
\(261\) 0.472136 0.0292245
\(262\) 0 0
\(263\) 4.00000i 0.246651i 0.992366 + 0.123325i \(0.0393559\pi\)
−0.992366 + 0.123325i \(0.960644\pi\)
\(264\) 0 0
\(265\) 4.47214i 0.274721i
\(266\) 0 0
\(267\) 9.41641i 0.576275i
\(268\) 0 0
\(269\) 14.3607 0.875586 0.437793 0.899076i \(-0.355760\pi\)
0.437793 + 0.899076i \(0.355760\pi\)
\(270\) 0 0
\(271\) −15.5279 −0.943251 −0.471625 0.881799i \(-0.656332\pi\)
−0.471625 + 0.881799i \(0.656332\pi\)
\(272\) 0 0
\(273\) 4.47214i 0.270666i
\(274\) 0 0
\(275\) 10.0000 0.603023
\(276\) 0 0
\(277\) − 3.41641i − 0.205272i −0.994719 0.102636i \(-0.967272\pi\)
0.994719 0.102636i \(-0.0327277\pi\)
\(278\) 0 0
\(279\) 8.47214 0.507214
\(280\) 0 0
\(281\) −14.9443 −0.891501 −0.445750 0.895157i \(-0.647063\pi\)
−0.445750 + 0.895157i \(0.647063\pi\)
\(282\) 0 0
\(283\) − 0.944272i − 0.0561311i −0.999606 0.0280656i \(-0.991065\pi\)
0.999606 0.0280656i \(-0.00893472\pi\)
\(284\) 0 0
\(285\) 4.47214i 0.264906i
\(286\) 0 0
\(287\) 12.4721i 0.736207i
\(288\) 0 0
\(289\) 10.8885 0.640503
\(290\) 0 0
\(291\) 12.4721 0.731130
\(292\) 0 0
\(293\) − 7.05573i − 0.412200i −0.978531 0.206100i \(-0.933923\pi\)
0.978531 0.206100i \(-0.0660772\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.00000i 0.116052i
\(298\) 0 0
\(299\) 17.8885 1.03452
\(300\) 0 0
\(301\) 6.47214 0.373048
\(302\) 0 0
\(303\) 4.47214i 0.256917i
\(304\) 0 0
\(305\) 27.8885 1.59689
\(306\) 0 0
\(307\) − 20.0000i − 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 0 0
\(309\) −4.94427 −0.281270
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) − 1.41641i − 0.0800601i −0.999198 0.0400301i \(-0.987255\pi\)
0.999198 0.0400301i \(-0.0127454\pi\)
\(314\) 0 0
\(315\) − 2.23607i − 0.125988i
\(316\) 0 0
\(317\) − 10.9443i − 0.614692i −0.951598 0.307346i \(-0.900559\pi\)
0.951598 0.307346i \(-0.0994408\pi\)
\(318\) 0 0
\(319\) −0.944272 −0.0528691
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 4.94427i − 0.275107i
\(324\) 0 0
\(325\) − 22.3607i − 1.24035i
\(326\) 0 0
\(327\) − 2.94427i − 0.162819i
\(328\) 0 0
\(329\) 2.47214 0.136293
\(330\) 0 0
\(331\) 1.88854 0.103804 0.0519019 0.998652i \(-0.483472\pi\)
0.0519019 + 0.998652i \(0.483472\pi\)
\(332\) 0 0
\(333\) 6.47214i 0.354671i
\(334\) 0 0
\(335\) 23.4164i 1.27938i
\(336\) 0 0
\(337\) − 28.9443i − 1.57669i −0.615230 0.788347i \(-0.710937\pi\)
0.615230 0.788347i \(-0.289063\pi\)
\(338\) 0 0
\(339\) 2.94427 0.159911
\(340\) 0 0
\(341\) −16.9443 −0.917584
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −8.94427 −0.481543
\(346\) 0 0
\(347\) 22.8328i 1.22573i 0.790188 + 0.612865i \(0.209983\pi\)
−0.790188 + 0.612865i \(0.790017\pi\)
\(348\) 0 0
\(349\) 1.41641 0.0758186 0.0379093 0.999281i \(-0.487930\pi\)
0.0379093 + 0.999281i \(0.487930\pi\)
\(350\) 0 0
\(351\) 4.47214 0.238705
\(352\) 0 0
\(353\) − 26.4721i − 1.40897i −0.709719 0.704485i \(-0.751178\pi\)
0.709719 0.704485i \(-0.248822\pi\)
\(354\) 0 0
\(355\) 7.88854 0.418680
\(356\) 0 0
\(357\) 2.47214i 0.130839i
\(358\) 0 0
\(359\) −13.4164 −0.708091 −0.354045 0.935228i \(-0.615194\pi\)
−0.354045 + 0.935228i \(0.615194\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 36.8328i 1.92792i
\(366\) 0 0
\(367\) − 9.88854i − 0.516178i −0.966121 0.258089i \(-0.916907\pi\)
0.966121 0.258089i \(-0.0830927\pi\)
\(368\) 0 0
\(369\) 12.4721 0.649273
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) − 28.3607i − 1.46846i −0.678901 0.734230i \(-0.737543\pi\)
0.678901 0.734230i \(-0.262457\pi\)
\(374\) 0 0
\(375\) 11.1803i 0.577350i
\(376\) 0 0
\(377\) 2.11146i 0.108746i
\(378\) 0 0
\(379\) 29.8885 1.53527 0.767636 0.640886i \(-0.221433\pi\)
0.767636 + 0.640886i \(0.221433\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 24.3607i 1.24477i 0.782710 + 0.622386i \(0.213837\pi\)
−0.782710 + 0.622386i \(0.786163\pi\)
\(384\) 0 0
\(385\) 4.47214i 0.227921i
\(386\) 0 0
\(387\) − 6.47214i − 0.328997i
\(388\) 0 0
\(389\) 29.4164 1.49147 0.745736 0.666242i \(-0.232098\pi\)
0.745736 + 0.666242i \(0.232098\pi\)
\(390\) 0 0
\(391\) 9.88854 0.500085
\(392\) 0 0
\(393\) − 21.8885i − 1.10413i
\(394\) 0 0
\(395\) 20.0000 1.00631
\(396\) 0 0
\(397\) − 16.4721i − 0.826713i −0.910569 0.413356i \(-0.864356\pi\)
0.910569 0.413356i \(-0.135644\pi\)
\(398\) 0 0
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) 19.8885 0.993186 0.496593 0.867983i \(-0.334584\pi\)
0.496593 + 0.867983i \(0.334584\pi\)
\(402\) 0 0
\(403\) 37.8885i 1.88736i
\(404\) 0 0
\(405\) −2.23607 −0.111111
\(406\) 0 0
\(407\) − 12.9443i − 0.641624i
\(408\) 0 0
\(409\) −27.8885 −1.37900 −0.689500 0.724286i \(-0.742170\pi\)
−0.689500 + 0.724286i \(0.742170\pi\)
\(410\) 0 0
\(411\) 15.8885 0.783724
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 28.9443i − 1.42082i
\(416\) 0 0
\(417\) − 14.9443i − 0.731824i
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) 6.94427 0.338443 0.169222 0.985578i \(-0.445875\pi\)
0.169222 + 0.985578i \(0.445875\pi\)
\(422\) 0 0
\(423\) − 2.47214i − 0.120199i
\(424\) 0 0
\(425\) − 12.3607i − 0.599581i
\(426\) 0 0
\(427\) 12.4721i 0.603569i
\(428\) 0 0
\(429\) −8.94427 −0.431834
\(430\) 0 0
\(431\) −5.41641 −0.260899 −0.130450 0.991455i \(-0.541642\pi\)
−0.130450 + 0.991455i \(0.541642\pi\)
\(432\) 0 0
\(433\) 5.41641i 0.260296i 0.991495 + 0.130148i \(0.0415452\pi\)
−0.991495 + 0.130148i \(0.958455\pi\)
\(434\) 0 0
\(435\) − 1.05573i − 0.0506183i
\(436\) 0 0
\(437\) 8.00000i 0.382692i
\(438\) 0 0
\(439\) 40.4721 1.93163 0.965815 0.259233i \(-0.0834697\pi\)
0.965815 + 0.259233i \(0.0834697\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) − 19.0557i − 0.905365i −0.891672 0.452682i \(-0.850467\pi\)
0.891672 0.452682i \(-0.149533\pi\)
\(444\) 0 0
\(445\) 21.0557 0.998137
\(446\) 0 0
\(447\) − 3.52786i − 0.166862i
\(448\) 0 0
\(449\) 26.9443 1.27158 0.635789 0.771863i \(-0.280675\pi\)
0.635789 + 0.771863i \(0.280675\pi\)
\(450\) 0 0
\(451\) −24.9443 −1.17458
\(452\) 0 0
\(453\) 17.8885i 0.840477i
\(454\) 0 0
\(455\) 10.0000 0.468807
\(456\) 0 0
\(457\) 8.94427i 0.418395i 0.977873 + 0.209198i \(0.0670852\pi\)
−0.977873 + 0.209198i \(0.932915\pi\)
\(458\) 0 0
\(459\) 2.47214 0.115389
\(460\) 0 0
\(461\) −3.52786 −0.164309 −0.0821545 0.996620i \(-0.526180\pi\)
−0.0821545 + 0.996620i \(0.526180\pi\)
\(462\) 0 0
\(463\) 18.8328i 0.875235i 0.899161 + 0.437618i \(0.144178\pi\)
−0.899161 + 0.437618i \(0.855822\pi\)
\(464\) 0 0
\(465\) − 18.9443i − 0.878520i
\(466\) 0 0
\(467\) 36.9443i 1.70958i 0.518976 + 0.854789i \(0.326313\pi\)
−0.518976 + 0.854789i \(0.673687\pi\)
\(468\) 0 0
\(469\) −10.4721 −0.483558
\(470\) 0 0
\(471\) −17.4164 −0.802506
\(472\) 0 0
\(473\) 12.9443i 0.595178i
\(474\) 0 0
\(475\) 10.0000 0.458831
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −28.9443 −1.31975
\(482\) 0 0
\(483\) − 4.00000i − 0.182006i
\(484\) 0 0
\(485\) − 27.8885i − 1.26635i
\(486\) 0 0
\(487\) 21.8885i 0.991865i 0.868361 + 0.495932i \(0.165174\pi\)
−0.868361 + 0.495932i \(0.834826\pi\)
\(488\) 0 0
\(489\) −3.41641 −0.154495
\(490\) 0 0
\(491\) −18.9443 −0.854943 −0.427472 0.904029i \(-0.640596\pi\)
−0.427472 + 0.904029i \(0.640596\pi\)
\(492\) 0 0
\(493\) 1.16718i 0.0525673i
\(494\) 0 0
\(495\) 4.47214 0.201008
\(496\) 0 0
\(497\) 3.52786i 0.158246i
\(498\) 0 0
\(499\) 29.8885 1.33799 0.668997 0.743265i \(-0.266724\pi\)
0.668997 + 0.743265i \(0.266724\pi\)
\(500\) 0 0
\(501\) −1.52786 −0.0682599
\(502\) 0 0
\(503\) − 12.5836i − 0.561075i −0.959843 0.280537i \(-0.909487\pi\)
0.959843 0.280537i \(-0.0905126\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 7.00000i 0.310881i
\(508\) 0 0
\(509\) 37.4164 1.65845 0.829227 0.558913i \(-0.188781\pi\)
0.829227 + 0.558913i \(0.188781\pi\)
\(510\) 0 0
\(511\) −16.4721 −0.728684
\(512\) 0 0
\(513\) 2.00000i 0.0883022i
\(514\) 0 0
\(515\) 11.0557i 0.487174i
\(516\) 0 0
\(517\) 4.94427i 0.217449i
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 17.4164 0.763027 0.381513 0.924363i \(-0.375403\pi\)
0.381513 + 0.924363i \(0.375403\pi\)
\(522\) 0 0
\(523\) − 24.9443i − 1.09074i −0.838196 0.545368i \(-0.816390\pi\)
0.838196 0.545368i \(-0.183610\pi\)
\(524\) 0 0
\(525\) −5.00000 −0.218218
\(526\) 0 0
\(527\) 20.9443i 0.912347i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 55.7771i 2.41597i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 5.05573i 0.218171i
\(538\) 0 0
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −5.05573 −0.217363 −0.108681 0.994077i \(-0.534663\pi\)
−0.108681 + 0.994077i \(0.534663\pi\)
\(542\) 0 0
\(543\) − 7.52786i − 0.323052i
\(544\) 0 0
\(545\) −6.58359 −0.282010
\(546\) 0 0
\(547\) 25.3050i 1.08196i 0.841035 + 0.540981i \(0.181947\pi\)
−0.841035 + 0.540981i \(0.818053\pi\)
\(548\) 0 0
\(549\) 12.4721 0.532298
\(550\) 0 0
\(551\) −0.944272 −0.0402273
\(552\) 0 0
\(553\) 8.94427i 0.380349i
\(554\) 0 0
\(555\) 14.4721 0.614308
\(556\) 0 0
\(557\) 22.9443i 0.972180i 0.873909 + 0.486090i \(0.161577\pi\)
−0.873909 + 0.486090i \(0.838423\pi\)
\(558\) 0 0
\(559\) 28.9443 1.22421
\(560\) 0 0
\(561\) −4.94427 −0.208747
\(562\) 0 0
\(563\) − 18.8328i − 0.793709i −0.917882 0.396854i \(-0.870102\pi\)
0.917882 0.396854i \(-0.129898\pi\)
\(564\) 0 0
\(565\) − 6.58359i − 0.276974i
\(566\) 0 0
\(567\) − 1.00000i − 0.0419961i
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 45.8885 1.92038 0.960188 0.279355i \(-0.0901206\pi\)
0.960188 + 0.279355i \(0.0901206\pi\)
\(572\) 0 0
\(573\) 9.41641i 0.393376i
\(574\) 0 0
\(575\) 20.0000i 0.834058i
\(576\) 0 0
\(577\) 15.5279i 0.646433i 0.946325 + 0.323217i \(0.104764\pi\)
−0.946325 + 0.323217i \(0.895236\pi\)
\(578\) 0 0
\(579\) 4.94427 0.205477
\(580\) 0 0
\(581\) 12.9443 0.537019
\(582\) 0 0
\(583\) − 4.00000i − 0.165663i
\(584\) 0 0
\(585\) − 10.0000i − 0.413449i
\(586\) 0 0
\(587\) 32.9443i 1.35976i 0.733325 + 0.679878i \(0.237967\pi\)
−0.733325 + 0.679878i \(0.762033\pi\)
\(588\) 0 0
\(589\) −16.9443 −0.698177
\(590\) 0 0
\(591\) 15.8885 0.653567
\(592\) 0 0
\(593\) − 41.3050i − 1.69619i −0.529843 0.848096i \(-0.677749\pi\)
0.529843 0.848096i \(-0.322251\pi\)
\(594\) 0 0
\(595\) 5.52786 0.226620
\(596\) 0 0
\(597\) − 11.5279i − 0.471804i
\(598\) 0 0
\(599\) −26.3607 −1.07707 −0.538534 0.842604i \(-0.681022\pi\)
−0.538534 + 0.842604i \(0.681022\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 10.4721i 0.426458i
\(604\) 0 0
\(605\) 15.6525 0.636364
\(606\) 0 0
\(607\) − 12.9443i − 0.525392i −0.964879 0.262696i \(-0.915388\pi\)
0.964879 0.262696i \(-0.0846116\pi\)
\(608\) 0 0
\(609\) 0.472136 0.0191319
\(610\) 0 0
\(611\) 11.0557 0.447267
\(612\) 0 0
\(613\) − 19.4164i − 0.784221i −0.919918 0.392111i \(-0.871745\pi\)
0.919918 0.392111i \(-0.128255\pi\)
\(614\) 0 0
\(615\) − 27.8885i − 1.12457i
\(616\) 0 0
\(617\) − 12.1115i − 0.487589i −0.969827 0.243794i \(-0.921608\pi\)
0.969827 0.243794i \(-0.0783922\pi\)
\(618\) 0 0
\(619\) −23.8885 −0.960162 −0.480081 0.877224i \(-0.659393\pi\)
−0.480081 + 0.877224i \(0.659393\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 9.41641i 0.377260i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) − 4.00000i − 0.159745i
\(628\) 0 0
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −40.9443 −1.62997 −0.814983 0.579485i \(-0.803254\pi\)
−0.814983 + 0.579485i \(0.803254\pi\)
\(632\) 0 0
\(633\) − 9.88854i − 0.393034i
\(634\) 0 0
\(635\) 8.94427i 0.354943i
\(636\) 0 0
\(637\) 4.47214i 0.177192i
\(638\) 0 0
\(639\) 3.52786 0.139560
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) − 24.9443i − 0.983706i −0.870678 0.491853i \(-0.836320\pi\)
0.870678 0.491853i \(-0.163680\pi\)
\(644\) 0 0
\(645\) −14.4721 −0.569840
\(646\) 0 0
\(647\) − 42.2492i − 1.66099i −0.557027 0.830494i \(-0.688058\pi\)
0.557027 0.830494i \(-0.311942\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 8.47214 0.332049
\(652\) 0 0
\(653\) − 18.9443i − 0.741347i −0.928763 0.370673i \(-0.879127\pi\)
0.928763 0.370673i \(-0.120873\pi\)
\(654\) 0 0
\(655\) −48.9443 −1.91241
\(656\) 0 0
\(657\) 16.4721i 0.642639i
\(658\) 0 0
\(659\) −47.8885 −1.86547 −0.932736 0.360559i \(-0.882586\pi\)
−0.932736 + 0.360559i \(0.882586\pi\)
\(660\) 0 0
\(661\) −32.2492 −1.25435 −0.627175 0.778879i \(-0.715789\pi\)
−0.627175 + 0.778879i \(0.715789\pi\)
\(662\) 0 0
\(663\) 11.0557i 0.429369i
\(664\) 0 0
\(665\) 4.47214i 0.173422i
\(666\) 0 0
\(667\) − 1.88854i − 0.0731247i
\(668\) 0 0
\(669\) 3.05573 0.118141
\(670\) 0 0
\(671\) −24.9443 −0.962963
\(672\) 0 0
\(673\) 13.8885i 0.535364i 0.963507 + 0.267682i \(0.0862577\pi\)
−0.963507 + 0.267682i \(0.913742\pi\)
\(674\) 0 0
\(675\) 5.00000i 0.192450i
\(676\) 0 0
\(677\) 37.8885i 1.45618i 0.685484 + 0.728088i \(0.259591\pi\)
−0.685484 + 0.728088i \(0.740409\pi\)
\(678\) 0 0
\(679\) 12.4721 0.478637
\(680\) 0 0
\(681\) −25.8885 −0.992051
\(682\) 0 0
\(683\) − 49.8885i − 1.90893i −0.298320 0.954466i \(-0.596426\pi\)
0.298320 0.954466i \(-0.403574\pi\)
\(684\) 0 0
\(685\) − 35.5279i − 1.35745i
\(686\) 0 0
\(687\) 18.3607i 0.700504i
\(688\) 0 0
\(689\) −8.94427 −0.340750
\(690\) 0 0
\(691\) −4.83282 −0.183849 −0.0919245 0.995766i \(-0.529302\pi\)
−0.0919245 + 0.995766i \(0.529302\pi\)
\(692\) 0 0
\(693\) 2.00000i 0.0759737i
\(694\) 0 0
\(695\) −33.4164 −1.26756
\(696\) 0 0
\(697\) 30.8328i 1.16788i
\(698\) 0 0
\(699\) −14.9443 −0.565244
\(700\) 0 0
\(701\) 8.47214 0.319988 0.159994 0.987118i \(-0.448852\pi\)
0.159994 + 0.987118i \(0.448852\pi\)
\(702\) 0 0
\(703\) − 12.9443i − 0.488202i
\(704\) 0 0
\(705\) −5.52786 −0.208191
\(706\) 0 0
\(707\) 4.47214i 0.168192i
\(708\) 0 0
\(709\) −0.111456 −0.00418582 −0.00209291 0.999998i \(-0.500666\pi\)
−0.00209291 + 0.999998i \(0.500666\pi\)
\(710\) 0 0
\(711\) 8.94427 0.335436
\(712\) 0 0
\(713\) − 33.8885i − 1.26914i
\(714\) 0 0
\(715\) 20.0000i 0.747958i
\(716\) 0 0
\(717\) − 24.4721i − 0.913929i
\(718\) 0 0
\(719\) −8.94427 −0.333565 −0.166783 0.985994i \(-0.553338\pi\)
−0.166783 + 0.985994i \(0.553338\pi\)
\(720\) 0 0
\(721\) −4.94427 −0.184134
\(722\) 0 0
\(723\) − 23.8885i − 0.888425i
\(724\) 0 0
\(725\) −2.36068 −0.0876734
\(726\) 0 0
\(727\) 36.9443i 1.37019i 0.728455 + 0.685094i \(0.240239\pi\)
−0.728455 + 0.685094i \(0.759761\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) 50.3607i 1.86011i 0.367415 + 0.930057i \(0.380243\pi\)
−0.367415 + 0.930057i \(0.619757\pi\)
\(734\) 0 0
\(735\) − 2.23607i − 0.0824786i
\(736\) 0 0
\(737\) − 20.9443i − 0.771492i
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −8.94427 −0.328576
\(742\) 0 0
\(743\) − 50.8328i − 1.86488i −0.361331 0.932438i \(-0.617678\pi\)
0.361331 0.932438i \(-0.382322\pi\)
\(744\) 0 0
\(745\) −7.88854 −0.289014
\(746\) 0 0
\(747\) − 12.9443i − 0.473606i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 26.8328 0.979143 0.489572 0.871963i \(-0.337153\pi\)
0.489572 + 0.871963i \(0.337153\pi\)
\(752\) 0 0
\(753\) 24.0000i 0.874609i
\(754\) 0 0
\(755\) 40.0000 1.45575
\(756\) 0 0
\(757\) − 21.5279i − 0.782444i −0.920296 0.391222i \(-0.872053\pi\)
0.920296 0.391222i \(-0.127947\pi\)
\(758\) 0 0
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) −15.5279 −0.562885 −0.281442 0.959578i \(-0.590813\pi\)
−0.281442 + 0.959578i \(0.590813\pi\)
\(762\) 0 0
\(763\) − 2.94427i − 0.106590i
\(764\) 0 0
\(765\) − 5.52786i − 0.199860i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −18.9443 −0.683148 −0.341574 0.939855i \(-0.610960\pi\)
−0.341574 + 0.939855i \(0.610960\pi\)
\(770\) 0 0
\(771\) −3.41641 −0.123039
\(772\) 0 0
\(773\) − 0.944272i − 0.0339631i −0.999856 0.0169815i \(-0.994594\pi\)
0.999856 0.0169815i \(-0.00540566\pi\)
\(774\) 0 0
\(775\) −42.3607 −1.52164
\(776\) 0 0
\(777\) 6.47214i 0.232187i
\(778\) 0 0
\(779\) −24.9443 −0.893721
\(780\) 0 0
\(781\) −7.05573 −0.252474
\(782\) 0 0
\(783\) − 0.472136i − 0.0168728i
\(784\) 0 0
\(785\) 38.9443i 1.38998i
\(786\) 0 0
\(787\) − 36.0000i − 1.28326i −0.767014 0.641631i \(-0.778258\pi\)
0.767014 0.641631i \(-0.221742\pi\)
\(788\) 0 0
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) 2.94427 0.104686
\(792\) 0 0
\(793\) 55.7771i 1.98070i
\(794\) 0 0
\(795\) 4.47214 0.158610
\(796\) 0 0
\(797\) 28.0000i 0.991811i 0.868377 + 0.495905i \(0.165164\pi\)
−0.868377 + 0.495905i \(0.834836\pi\)
\(798\) 0 0
\(799\) 6.11146 0.216208
\(800\) 0 0
\(801\) 9.41641 0.332712
\(802\) 0 0
\(803\) − 32.9443i − 1.16258i
\(804\) 0 0
\(805\) −8.94427 −0.315244
\(806\) 0 0
\(807\) − 14.3607i − 0.505520i
\(808\) 0 0
\(809\) −28.8328 −1.01371 −0.506854 0.862032i \(-0.669192\pi\)
−0.506854 + 0.862032i \(0.669192\pi\)
\(810\) 0 0
\(811\) −30.9443 −1.08660 −0.543300 0.839539i \(-0.682825\pi\)
−0.543300 + 0.839539i \(0.682825\pi\)
\(812\) 0 0
\(813\) 15.5279i 0.544586i
\(814\) 0 0
\(815\) 7.63932i 0.267594i
\(816\) 0 0
\(817\) 12.9443i 0.452863i
\(818\) 0 0
\(819\) 4.47214 0.156269
\(820\) 0 0
\(821\) −43.3050 −1.51135 −0.755677 0.654945i \(-0.772692\pi\)
−0.755677 + 0.654945i \(0.772692\pi\)
\(822\) 0 0
\(823\) 32.9443i 1.14837i 0.818727 + 0.574183i \(0.194680\pi\)
−0.818727 + 0.574183i \(0.805320\pi\)
\(824\) 0 0
\(825\) − 10.0000i − 0.348155i
\(826\) 0 0
\(827\) − 56.7214i − 1.97239i −0.165573 0.986197i \(-0.552947\pi\)
0.165573 0.986197i \(-0.447053\pi\)
\(828\) 0 0
\(829\) −12.4721 −0.433175 −0.216588 0.976263i \(-0.569493\pi\)
−0.216588 + 0.976263i \(0.569493\pi\)
\(830\) 0 0
\(831\) −3.41641 −0.118514
\(832\) 0 0
\(833\) 2.47214i 0.0856544i
\(834\) 0 0
\(835\) 3.41641i 0.118230i
\(836\) 0 0
\(837\) − 8.47214i − 0.292840i
\(838\) 0 0
\(839\) 15.0557 0.519781 0.259891 0.965638i \(-0.416313\pi\)
0.259891 + 0.965638i \(0.416313\pi\)
\(840\) 0 0
\(841\) −28.7771 −0.992313
\(842\) 0 0
\(843\) 14.9443i 0.514708i
\(844\) 0 0
\(845\) 15.6525 0.538462
\(846\) 0 0
\(847\) 7.00000i 0.240523i
\(848\) 0 0
\(849\) −0.944272 −0.0324073
\(850\) 0 0
\(851\) 25.8885 0.887448
\(852\) 0 0
\(853\) 40.4721i 1.38574i 0.721063 + 0.692870i \(0.243654\pi\)
−0.721063 + 0.692870i \(0.756346\pi\)
\(854\) 0 0
\(855\) 4.47214 0.152944
\(856\) 0 0
\(857\) − 43.4164i − 1.48308i −0.670911 0.741538i \(-0.734097\pi\)
0.670911 0.741538i \(-0.265903\pi\)
\(858\) 0 0
\(859\) −38.9443 −1.32876 −0.664381 0.747394i \(-0.731305\pi\)
−0.664381 + 0.747394i \(0.731305\pi\)
\(860\) 0 0
\(861\) 12.4721 0.425049
\(862\) 0 0
\(863\) − 34.8328i − 1.18572i −0.805305 0.592861i \(-0.797998\pi\)
0.805305 0.592861i \(-0.202002\pi\)
\(864\) 0 0
\(865\) 26.8328i 0.912343i
\(866\) 0 0
\(867\) − 10.8885i − 0.369794i
\(868\) 0 0
\(869\) −17.8885 −0.606827
\(870\) 0 0
\(871\) −46.8328 −1.58687
\(872\) 0 0
\(873\) − 12.4721i − 0.422118i
\(874\) 0 0
\(875\) 11.1803i 0.377964i
\(876\) 0 0
\(877\) − 13.3050i − 0.449276i −0.974442 0.224638i \(-0.927880\pi\)
0.974442 0.224638i \(-0.0721200\pi\)
\(878\) 0 0
\(879\) −7.05573 −0.237984
\(880\) 0 0
\(881\) 36.4721 1.22878 0.614389 0.789003i \(-0.289403\pi\)
0.614389 + 0.789003i \(0.289403\pi\)
\(882\) 0 0
\(883\) 28.3607i 0.954413i 0.878791 + 0.477206i \(0.158351\pi\)
−0.878791 + 0.477206i \(0.841649\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 31.4164i 1.05486i 0.849599 + 0.527430i \(0.176844\pi\)
−0.849599 + 0.527430i \(0.823156\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) 4.94427i 0.165454i
\(894\) 0 0
\(895\) 11.3050 0.377883
\(896\) 0 0
\(897\) − 17.8885i − 0.597281i
\(898\) 0 0
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) −4.94427 −0.164718
\(902\) 0 0
\(903\) − 6.47214i − 0.215379i
\(904\) 0 0
\(905\) −16.8328 −0.559542
\(906\) 0 0
\(907\) − 28.3607i − 0.941701i −0.882213 0.470850i \(-0.843947\pi\)
0.882213 0.470850i \(-0.156053\pi\)
\(908\) 0 0
\(909\) 4.47214 0.148331
\(910\) 0 0
\(911\) −1.41641 −0.0469277 −0.0234638 0.999725i \(-0.507469\pi\)
−0.0234638 + 0.999725i \(0.507469\pi\)
\(912\) 0 0
\(913\) 25.8885i 0.856786i
\(914\) 0 0
\(915\) − 27.8885i − 0.921967i
\(916\) 0 0
\(917\) − 21.8885i − 0.722823i
\(918\) 0 0
\(919\) 52.7214 1.73912 0.869559 0.493830i \(-0.164403\pi\)
0.869559 + 0.493830i \(0.164403\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) 15.7771i 0.519309i
\(924\) 0 0
\(925\) − 32.3607i − 1.06401i
\(926\) 0 0
\(927\) 4.94427i 0.162391i
\(928\) 0 0
\(929\) 43.3050 1.42079 0.710395 0.703804i \(-0.248516\pi\)
0.710395 + 0.703804i \(0.248516\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) 16.0000i 0.523816i
\(934\) 0 0
\(935\) 11.0557i 0.361561i
\(936\) 0 0
\(937\) 47.3050i 1.54539i 0.634780 + 0.772693i \(0.281091\pi\)
−0.634780 + 0.772693i \(0.718909\pi\)
\(938\) 0 0
\(939\) −1.41641 −0.0462227
\(940\) 0 0
\(941\) 44.4721 1.44975 0.724875 0.688880i \(-0.241897\pi\)
0.724875 + 0.688880i \(0.241897\pi\)
\(942\) 0 0
\(943\) − 49.8885i − 1.62459i
\(944\) 0 0
\(945\) −2.23607 −0.0727393
\(946\) 0 0
\(947\) 16.0000i 0.519930i 0.965618 + 0.259965i \(0.0837111\pi\)
−0.965618 + 0.259965i \(0.916289\pi\)
\(948\) 0 0
\(949\) −73.6656 −2.39129
\(950\) 0 0
\(951\) −10.9443 −0.354892
\(952\) 0 0
\(953\) − 18.0000i − 0.583077i −0.956559 0.291539i \(-0.905833\pi\)
0.956559 0.291539i \(-0.0941672\pi\)
\(954\) 0 0
\(955\) 21.0557 0.681347
\(956\) 0 0
\(957\) 0.944272i 0.0305240i
\(958\) 0 0
\(959\) 15.8885 0.513068
\(960\) 0 0
\(961\) 40.7771 1.31539
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 11.0557i − 0.355896i
\(966\) 0 0
\(967\) 9.88854i 0.317994i 0.987279 + 0.158997i \(0.0508260\pi\)
−0.987279 + 0.158997i \(0.949174\pi\)
\(968\) 0 0
\(969\) −4.94427 −0.158833
\(970\) 0 0
\(971\) −1.88854 −0.0606063 −0.0303031 0.999541i \(-0.509647\pi\)
−0.0303031 + 0.999541i \(0.509647\pi\)
\(972\) 0 0
\(973\) − 14.9443i − 0.479091i
\(974\) 0 0
\(975\) −22.3607 −0.716115
\(976\) 0 0
\(977\) 0.832816i 0.0266441i 0.999911 + 0.0133221i \(0.00424067\pi\)
−0.999911 + 0.0133221i \(0.995759\pi\)
\(978\) 0 0
\(979\) −18.8328 −0.601899
\(980\) 0 0
\(981\) −2.94427 −0.0940034
\(982\) 0 0
\(983\) − 34.4721i − 1.09949i −0.835332 0.549745i \(-0.814725\pi\)
0.835332 0.549745i \(-0.185275\pi\)
\(984\) 0 0
\(985\) − 35.5279i − 1.13201i
\(986\) 0 0
\(987\) − 2.47214i − 0.0786890i
\(988\) 0 0
\(989\) −25.8885 −0.823208
\(990\) 0 0
\(991\) 39.0557 1.24065 0.620323 0.784346i \(-0.287002\pi\)
0.620323 + 0.784346i \(0.287002\pi\)
\(992\) 0 0
\(993\) − 1.88854i − 0.0599311i
\(994\) 0 0
\(995\) −25.7771 −0.817189
\(996\) 0 0
\(997\) − 50.3607i − 1.59494i −0.603359 0.797469i \(-0.706172\pi\)
0.603359 0.797469i \(-0.293828\pi\)
\(998\) 0 0
\(999\) 6.47214 0.204769
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 1680.2.t.h.1009.1 4
3.2 odd 2 5040.2.t.u.1009.3 4
4.3 odd 2 840.2.t.c.169.3 yes 4
5.2 odd 4 8400.2.a.cz.1.1 2
5.3 odd 4 8400.2.a.db.1.2 2
5.4 even 2 inner 1680.2.t.h.1009.3 4
12.11 even 2 2520.2.t.f.1009.4 4
15.14 odd 2 5040.2.t.u.1009.4 4
20.3 even 4 4200.2.a.bj.1.2 2
20.7 even 4 4200.2.a.bk.1.1 2
20.19 odd 2 840.2.t.c.169.1 4
60.59 even 2 2520.2.t.f.1009.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.c.169.1 4 20.19 odd 2
840.2.t.c.169.3 yes 4 4.3 odd 2
1680.2.t.h.1009.1 4 1.1 even 1 trivial
1680.2.t.h.1009.3 4 5.4 even 2 inner
2520.2.t.f.1009.3 4 60.59 even 2
2520.2.t.f.1009.4 4 12.11 even 2
4200.2.a.bj.1.2 2 20.3 even 4
4200.2.a.bk.1.1 2 20.7 even 4
5040.2.t.u.1009.3 4 3.2 odd 2
5040.2.t.u.1009.4 4 15.14 odd 2
8400.2.a.cz.1.1 2 5.2 odd 4
8400.2.a.db.1.2 2 5.3 odd 4