Properties

Label 1900.2.c.e.1749.4
Level $1900$
Weight $2$
Character 1900.1749
Analytic conductor $15.172$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1749.4
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1900.1749
Dual form 1900.2.c.e.1749.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.73205i q^{3} -2.00000i q^{7} -4.46410 q^{9} +O(q^{10})\) \(q+2.73205i q^{3} -2.00000i q^{7} -4.46410 q^{9} -3.46410 q^{11} -2.73205i q^{13} +3.46410i q^{17} -1.00000 q^{19} +5.46410 q^{21} -3.46410i q^{23} -4.00000i q^{27} -3.46410 q^{29} -1.46410 q^{31} -9.46410i q^{33} -6.73205i q^{37} +7.46410 q^{39} -6.00000 q^{41} -4.92820i q^{43} -12.9282i q^{47} +3.00000 q^{49} -9.46410 q^{51} -10.7321i q^{53} -2.73205i q^{57} -6.92820 q^{59} +12.3923 q^{61} +8.92820i q^{63} -6.73205i q^{67} +9.46410 q^{69} -2.53590 q^{71} -0.535898i q^{73} +6.92820i q^{77} -2.92820 q^{79} -2.46410 q^{81} +3.46410i q^{83} -9.46410i q^{87} +15.4641 q^{89} -5.46410 q^{91} -4.00000i q^{93} +16.5885i q^{97} +15.4641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} - 4q^{19} + 8q^{21} + 8q^{31} + 16q^{39} - 24q^{41} + 12q^{49} - 24q^{51} + 8q^{61} + 24q^{69} - 24q^{71} + 16q^{79} + 4q^{81} + 48q^{89} - 8q^{91} + 48q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.73205i 1.57735i 0.614810 + 0.788675i \(0.289233\pi\)
−0.614810 + 0.788675i \(0.710767\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) −4.46410 −1.48803
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) − 2.73205i − 0.757735i −0.925451 0.378867i \(-0.876314\pi\)
0.925451 0.378867i \(-0.123686\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410i 0.840168i 0.907485 + 0.420084i \(0.137999\pi\)
−0.907485 + 0.420084i \(0.862001\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 5.46410 1.19236
\(22\) 0 0
\(23\) − 3.46410i − 0.722315i −0.932505 0.361158i \(-0.882382\pi\)
0.932505 0.361158i \(-0.117618\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) −1.46410 −0.262960 −0.131480 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(32\) 0 0
\(33\) − 9.46410i − 1.64749i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.73205i − 1.10674i −0.832935 0.553371i \(-0.813341\pi\)
0.832935 0.553371i \(-0.186659\pi\)
\(38\) 0 0
\(39\) 7.46410 1.19521
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 4.92820i − 0.751544i −0.926712 0.375772i \(-0.877378\pi\)
0.926712 0.375772i \(-0.122622\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 12.9282i − 1.88577i −0.333115 0.942886i \(-0.608100\pi\)
0.333115 0.942886i \(-0.391900\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −9.46410 −1.32524
\(52\) 0 0
\(53\) − 10.7321i − 1.47416i −0.675805 0.737080i \(-0.736204\pi\)
0.675805 0.737080i \(-0.263796\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.73205i − 0.361869i
\(58\) 0 0
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) 12.3923 1.58667 0.793336 0.608784i \(-0.208342\pi\)
0.793336 + 0.608784i \(0.208342\pi\)
\(62\) 0 0
\(63\) 8.92820i 1.12485i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 6.73205i − 0.822451i −0.911534 0.411225i \(-0.865101\pi\)
0.911534 0.411225i \(-0.134899\pi\)
\(68\) 0 0
\(69\) 9.46410 1.13934
\(70\) 0 0
\(71\) −2.53590 −0.300956 −0.150478 0.988613i \(-0.548081\pi\)
−0.150478 + 0.988613i \(0.548081\pi\)
\(72\) 0 0
\(73\) − 0.535898i − 0.0627222i −0.999508 0.0313611i \(-0.990016\pi\)
0.999508 0.0313611i \(-0.00998418\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.92820i 0.789542i
\(78\) 0 0
\(79\) −2.92820 −0.329449 −0.164724 0.986340i \(-0.552673\pi\)
−0.164724 + 0.986340i \(0.552673\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 0 0
\(83\) 3.46410i 0.380235i 0.981761 + 0.190117i \(0.0608868\pi\)
−0.981761 + 0.190117i \(0.939113\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 9.46410i − 1.01466i
\(88\) 0 0
\(89\) 15.4641 1.63919 0.819596 0.572942i \(-0.194198\pi\)
0.819596 + 0.572942i \(0.194198\pi\)
\(90\) 0 0
\(91\) −5.46410 −0.572793
\(92\) 0 0
\(93\) − 4.00000i − 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.5885i 1.68430i 0.539241 + 0.842151i \(0.318711\pi\)
−0.539241 + 0.842151i \(0.681289\pi\)
\(98\) 0 0
\(99\) 15.4641 1.55420
\(100\) 0 0
\(101\) −16.3923 −1.63110 −0.815548 0.578690i \(-0.803564\pi\)
−0.815548 + 0.578690i \(0.803564\pi\)
\(102\) 0 0
\(103\) 13.6603i 1.34598i 0.739649 + 0.672992i \(0.234991\pi\)
−0.739649 + 0.672992i \(0.765009\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.66025i 0.547197i 0.961844 + 0.273599i \(0.0882140\pi\)
−0.961844 + 0.273599i \(0.911786\pi\)
\(108\) 0 0
\(109\) 14.3923 1.37853 0.689266 0.724508i \(-0.257933\pi\)
0.689266 + 0.724508i \(0.257933\pi\)
\(110\) 0 0
\(111\) 18.3923 1.74572
\(112\) 0 0
\(113\) 12.5885i 1.18422i 0.805856 + 0.592111i \(0.201705\pi\)
−0.805856 + 0.592111i \(0.798295\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 12.1962i 1.12753i
\(118\) 0 0
\(119\) 6.92820 0.635107
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) − 16.3923i − 1.47804i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 20.5885i − 1.82693i −0.406917 0.913465i \(-0.633396\pi\)
0.406917 0.913465i \(-0.366604\pi\)
\(128\) 0 0
\(129\) 13.4641 1.18545
\(130\) 0 0
\(131\) −18.9282 −1.65376 −0.826882 0.562375i \(-0.809888\pi\)
−0.826882 + 0.562375i \(0.809888\pi\)
\(132\) 0 0
\(133\) 2.00000i 0.173422i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 17.3205i − 1.47979i −0.672722 0.739895i \(-0.734875\pi\)
0.672722 0.739895i \(-0.265125\pi\)
\(138\) 0 0
\(139\) −11.4641 −0.972372 −0.486186 0.873855i \(-0.661612\pi\)
−0.486186 + 0.873855i \(0.661612\pi\)
\(140\) 0 0
\(141\) 35.3205 2.97452
\(142\) 0 0
\(143\) 9.46410i 0.791428i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 8.19615i 0.676007i
\(148\) 0 0
\(149\) 4.39230 0.359832 0.179916 0.983682i \(-0.442417\pi\)
0.179916 + 0.983682i \(0.442417\pi\)
\(150\) 0 0
\(151\) −8.39230 −0.682956 −0.341478 0.939890i \(-0.610927\pi\)
−0.341478 + 0.939890i \(0.610927\pi\)
\(152\) 0 0
\(153\) − 15.4641i − 1.25020i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 23.4641i − 1.87264i −0.351149 0.936320i \(-0.614209\pi\)
0.351149 0.936320i \(-0.385791\pi\)
\(158\) 0 0
\(159\) 29.3205 2.32527
\(160\) 0 0
\(161\) −6.92820 −0.546019
\(162\) 0 0
\(163\) 7.07180i 0.553906i 0.960883 + 0.276953i \(0.0893246\pi\)
−0.960883 + 0.276953i \(0.910675\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.7321i 0.830471i 0.909714 + 0.415236i \(0.136301\pi\)
−0.909714 + 0.415236i \(0.863699\pi\)
\(168\) 0 0
\(169\) 5.53590 0.425838
\(170\) 0 0
\(171\) 4.46410 0.341378
\(172\) 0 0
\(173\) 3.80385i 0.289201i 0.989490 + 0.144601i \(0.0461897\pi\)
−0.989490 + 0.144601i \(0.953810\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 18.9282i − 1.42273i
\(178\) 0 0
\(179\) −20.7846 −1.55351 −0.776757 0.629800i \(-0.783137\pi\)
−0.776757 + 0.629800i \(0.783137\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 33.8564i 2.50274i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 12.0000i − 0.877527i
\(188\) 0 0
\(189\) −8.00000 −0.581914
\(190\) 0 0
\(191\) 6.92820 0.501307 0.250654 0.968077i \(-0.419354\pi\)
0.250654 + 0.968077i \(0.419354\pi\)
\(192\) 0 0
\(193\) − 7.80385i − 0.561733i −0.959747 0.280867i \(-0.909378\pi\)
0.959747 0.280867i \(-0.0906219\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 0.928203i − 0.0661317i −0.999453 0.0330659i \(-0.989473\pi\)
0.999453 0.0330659i \(-0.0105271\pi\)
\(198\) 0 0
\(199\) −26.9282 −1.90889 −0.954445 0.298387i \(-0.903551\pi\)
−0.954445 + 0.298387i \(0.903551\pi\)
\(200\) 0 0
\(201\) 18.3923 1.29729
\(202\) 0 0
\(203\) 6.92820i 0.486265i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 15.4641i 1.07483i
\(208\) 0 0
\(209\) 3.46410 0.239617
\(210\) 0 0
\(211\) 19.3205 1.33008 0.665039 0.746808i \(-0.268415\pi\)
0.665039 + 0.746808i \(0.268415\pi\)
\(212\) 0 0
\(213\) − 6.92820i − 0.474713i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.92820i 0.198779i
\(218\) 0 0
\(219\) 1.46410 0.0989348
\(220\) 0 0
\(221\) 9.46410 0.636624
\(222\) 0 0
\(223\) − 9.66025i − 0.646898i −0.946246 0.323449i \(-0.895158\pi\)
0.946246 0.323449i \(-0.104842\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 8.19615i − 0.543998i −0.962298 0.271999i \(-0.912315\pi\)
0.962298 0.271999i \(-0.0876847\pi\)
\(228\) 0 0
\(229\) −17.4641 −1.15406 −0.577030 0.816723i \(-0.695788\pi\)
−0.577030 + 0.816723i \(0.695788\pi\)
\(230\) 0 0
\(231\) −18.9282 −1.24538
\(232\) 0 0
\(233\) − 0.928203i − 0.0608086i −0.999538 0.0304043i \(-0.990321\pi\)
0.999538 0.0304043i \(-0.00967948\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 8.00000i − 0.519656i
\(238\) 0 0
\(239\) 13.8564 0.896296 0.448148 0.893959i \(-0.352084\pi\)
0.448148 + 0.893959i \(0.352084\pi\)
\(240\) 0 0
\(241\) −11.8564 −0.763738 −0.381869 0.924216i \(-0.624719\pi\)
−0.381869 + 0.924216i \(0.624719\pi\)
\(242\) 0 0
\(243\) − 18.7321i − 1.20166i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.73205i 0.173836i
\(248\) 0 0
\(249\) −9.46410 −0.599763
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 12.0000i 0.754434i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 15.1244i 0.943431i 0.881751 + 0.471716i \(0.156365\pi\)
−0.881751 + 0.471716i \(0.843635\pi\)
\(258\) 0 0
\(259\) −13.4641 −0.836619
\(260\) 0 0
\(261\) 15.4641 0.957204
\(262\) 0 0
\(263\) 8.53590i 0.526346i 0.964749 + 0.263173i \(0.0847690\pi\)
−0.964749 + 0.263173i \(0.915231\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 42.2487i 2.58558i
\(268\) 0 0
\(269\) −10.3923 −0.633630 −0.316815 0.948487i \(-0.602613\pi\)
−0.316815 + 0.948487i \(0.602613\pi\)
\(270\) 0 0
\(271\) −2.39230 −0.145322 −0.0726611 0.997357i \(-0.523149\pi\)
−0.0726611 + 0.997357i \(0.523149\pi\)
\(272\) 0 0
\(273\) − 14.9282i − 0.903496i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 18.3923i − 1.10509i −0.833484 0.552543i \(-0.813657\pi\)
0.833484 0.552543i \(-0.186343\pi\)
\(278\) 0 0
\(279\) 6.53590 0.391294
\(280\) 0 0
\(281\) 0.928203 0.0553720 0.0276860 0.999617i \(-0.491186\pi\)
0.0276860 + 0.999617i \(0.491186\pi\)
\(282\) 0 0
\(283\) 18.3923i 1.09331i 0.837358 + 0.546655i \(0.184099\pi\)
−0.837358 + 0.546655i \(0.815901\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000i 0.708338i
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) −45.3205 −2.65674
\(292\) 0 0
\(293\) − 5.66025i − 0.330676i −0.986237 0.165338i \(-0.947129\pi\)
0.986237 0.165338i \(-0.0528714\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 13.8564i 0.804030i
\(298\) 0 0
\(299\) −9.46410 −0.547323
\(300\) 0 0
\(301\) −9.85641 −0.568114
\(302\) 0 0
\(303\) − 44.7846i − 2.57281i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7.80385i 0.445389i 0.974888 + 0.222695i \(0.0714853\pi\)
−0.974888 + 0.222695i \(0.928515\pi\)
\(308\) 0 0
\(309\) −37.3205 −2.12309
\(310\) 0 0
\(311\) −1.60770 −0.0911640 −0.0455820 0.998961i \(-0.514514\pi\)
−0.0455820 + 0.998961i \(0.514514\pi\)
\(312\) 0 0
\(313\) 10.7846i 0.609582i 0.952419 + 0.304791i \(0.0985866\pi\)
−0.952419 + 0.304791i \(0.901413\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 3.12436i − 0.175481i −0.996143 0.0877406i \(-0.972035\pi\)
0.996143 0.0877406i \(-0.0279647\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −15.4641 −0.863122
\(322\) 0 0
\(323\) − 3.46410i − 0.192748i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 39.3205i 2.17443i
\(328\) 0 0
\(329\) −25.8564 −1.42551
\(330\) 0 0
\(331\) −22.2487 −1.22290 −0.611450 0.791283i \(-0.709413\pi\)
−0.611450 + 0.791283i \(0.709413\pi\)
\(332\) 0 0
\(333\) 30.0526i 1.64687i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 17.2679i 0.940645i 0.882495 + 0.470323i \(0.155862\pi\)
−0.882495 + 0.470323i \(0.844138\pi\)
\(338\) 0 0
\(339\) −34.3923 −1.86793
\(340\) 0 0
\(341\) 5.07180 0.274653
\(342\) 0 0
\(343\) − 20.0000i − 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 31.1769i − 1.67366i −0.547459 0.836832i \(-0.684405\pi\)
0.547459 0.836832i \(-0.315595\pi\)
\(348\) 0 0
\(349\) −20.9282 −1.12026 −0.560131 0.828404i \(-0.689249\pi\)
−0.560131 + 0.828404i \(0.689249\pi\)
\(350\) 0 0
\(351\) −10.9282 −0.583304
\(352\) 0 0
\(353\) 1.60770i 0.0855690i 0.999084 + 0.0427845i \(0.0136229\pi\)
−0.999084 + 0.0427845i \(0.986377\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 18.9282i 1.00179i
\(358\) 0 0
\(359\) 8.53590 0.450507 0.225254 0.974300i \(-0.427679\pi\)
0.225254 + 0.974300i \(0.427679\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 2.73205i 0.143395i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 3.85641i − 0.201303i −0.994922 0.100651i \(-0.967907\pi\)
0.994922 0.100651i \(-0.0320927\pi\)
\(368\) 0 0
\(369\) 26.7846 1.39435
\(370\) 0 0
\(371\) −21.4641 −1.11436
\(372\) 0 0
\(373\) − 16.5885i − 0.858918i −0.903086 0.429459i \(-0.858704\pi\)
0.903086 0.429459i \(-0.141296\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.46410i 0.487426i
\(378\) 0 0
\(379\) −14.9282 −0.766810 −0.383405 0.923580i \(-0.625249\pi\)
−0.383405 + 0.923580i \(0.625249\pi\)
\(380\) 0 0
\(381\) 56.2487 2.88171
\(382\) 0 0
\(383\) 6.33975i 0.323946i 0.986795 + 0.161973i \(0.0517857\pi\)
−0.986795 + 0.161973i \(0.948214\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 22.0000i 1.11832i
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) − 51.7128i − 2.60857i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 19.4641i 0.976875i 0.872599 + 0.488438i \(0.162433\pi\)
−0.872599 + 0.488438i \(0.837567\pi\)
\(398\) 0 0
\(399\) −5.46410 −0.273547
\(400\) 0 0
\(401\) 38.7846 1.93681 0.968405 0.249381i \(-0.0802271\pi\)
0.968405 + 0.249381i \(0.0802271\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 23.3205i 1.15595i
\(408\) 0 0
\(409\) 14.3923 0.711654 0.355827 0.934552i \(-0.384199\pi\)
0.355827 + 0.934552i \(0.384199\pi\)
\(410\) 0 0
\(411\) 47.3205 2.33415
\(412\) 0 0
\(413\) 13.8564i 0.681829i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 31.3205i − 1.53377i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 0 0
\(423\) 57.7128i 2.80609i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 24.7846i − 1.19941i
\(428\) 0 0
\(429\) −25.8564 −1.24836
\(430\) 0 0
\(431\) −33.4641 −1.61191 −0.805955 0.591977i \(-0.798347\pi\)
−0.805955 + 0.591977i \(0.798347\pi\)
\(432\) 0 0
\(433\) − 22.3397i − 1.07358i −0.843716 0.536790i \(-0.819637\pi\)
0.843716 0.536790i \(-0.180363\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.46410i 0.165710i
\(438\) 0 0
\(439\) 31.7128 1.51357 0.756785 0.653664i \(-0.226769\pi\)
0.756785 + 0.653664i \(0.226769\pi\)
\(440\) 0 0
\(441\) −13.3923 −0.637729
\(442\) 0 0
\(443\) 13.6077i 0.646521i 0.946310 + 0.323261i \(0.104779\pi\)
−0.946310 + 0.323261i \(0.895221\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 12.0000i 0.567581i
\(448\) 0 0
\(449\) 3.46410 0.163481 0.0817405 0.996654i \(-0.473952\pi\)
0.0817405 + 0.996654i \(0.473952\pi\)
\(450\) 0 0
\(451\) 20.7846 0.978709
\(452\) 0 0
\(453\) − 22.9282i − 1.07726i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 34.7846i − 1.62716i −0.581456 0.813578i \(-0.697517\pi\)
0.581456 0.813578i \(-0.302483\pi\)
\(458\) 0 0
\(459\) 13.8564 0.646762
\(460\) 0 0
\(461\) −21.7128 −1.01127 −0.505633 0.862749i \(-0.668741\pi\)
−0.505633 + 0.862749i \(0.668741\pi\)
\(462\) 0 0
\(463\) 4.53590i 0.210801i 0.994430 + 0.105401i \(0.0336125\pi\)
−0.994430 + 0.105401i \(0.966388\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 29.3205i 1.35679i 0.734697 + 0.678396i \(0.237324\pi\)
−0.734697 + 0.678396i \(0.762676\pi\)
\(468\) 0 0
\(469\) −13.4641 −0.621714
\(470\) 0 0
\(471\) 64.1051 2.95381
\(472\) 0 0
\(473\) 17.0718i 0.784962i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 47.9090i 2.19360i
\(478\) 0 0
\(479\) 22.3923 1.02313 0.511565 0.859244i \(-0.329066\pi\)
0.511565 + 0.859244i \(0.329066\pi\)
\(480\) 0 0
\(481\) −18.3923 −0.838617
\(482\) 0 0
\(483\) − 18.9282i − 0.861263i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.1962i 0.552660i 0.961063 + 0.276330i \(0.0891183\pi\)
−0.961063 + 0.276330i \(0.910882\pi\)
\(488\) 0 0
\(489\) −19.3205 −0.873704
\(490\) 0 0
\(491\) 18.9282 0.854218 0.427109 0.904200i \(-0.359532\pi\)
0.427109 + 0.904200i \(0.359532\pi\)
\(492\) 0 0
\(493\) − 12.0000i − 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.07180i 0.227501i
\(498\) 0 0
\(499\) −23.4641 −1.05040 −0.525199 0.850980i \(-0.676009\pi\)
−0.525199 + 0.850980i \(0.676009\pi\)
\(500\) 0 0
\(501\) −29.3205 −1.30994
\(502\) 0 0
\(503\) 15.4641i 0.689510i 0.938693 + 0.344755i \(0.112038\pi\)
−0.938693 + 0.344755i \(0.887962\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 15.1244i 0.671696i
\(508\) 0 0
\(509\) 5.32051 0.235827 0.117914 0.993024i \(-0.462379\pi\)
0.117914 + 0.993024i \(0.462379\pi\)
\(510\) 0 0
\(511\) −1.07180 −0.0474135
\(512\) 0 0
\(513\) 4.00000i 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 44.7846i 1.96962i
\(518\) 0 0
\(519\) −10.3923 −0.456172
\(520\) 0 0
\(521\) −14.7846 −0.647726 −0.323863 0.946104i \(-0.604982\pi\)
−0.323863 + 0.946104i \(0.604982\pi\)
\(522\) 0 0
\(523\) − 37.3731i − 1.63421i −0.576489 0.817105i \(-0.695578\pi\)
0.576489 0.817105i \(-0.304422\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 5.07180i − 0.220931i
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 30.9282 1.34217
\(532\) 0 0
\(533\) 16.3923i 0.710030i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 56.7846i − 2.45044i
\(538\) 0 0
\(539\) −10.3923 −0.447628
\(540\) 0 0
\(541\) −29.1769 −1.25441 −0.627207 0.778853i \(-0.715802\pi\)
−0.627207 + 0.778853i \(0.715802\pi\)
\(542\) 0 0
\(543\) 38.2487i 1.64141i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 6.73205i − 0.287842i −0.989589 0.143921i \(-0.954029\pi\)
0.989589 0.143921i \(-0.0459711\pi\)
\(548\) 0 0
\(549\) −55.3205 −2.36102
\(550\) 0 0
\(551\) 3.46410 0.147576
\(552\) 0 0
\(553\) 5.85641i 0.249040i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 5.32051i − 0.225437i −0.993627 0.112719i \(-0.964044\pi\)
0.993627 0.112719i \(-0.0359559\pi\)
\(558\) 0 0
\(559\) −13.4641 −0.569471
\(560\) 0 0
\(561\) 32.7846 1.38417
\(562\) 0 0
\(563\) 44.1962i 1.86265i 0.364195 + 0.931323i \(0.381344\pi\)
−0.364195 + 0.931323i \(0.618656\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.92820i 0.206965i
\(568\) 0 0
\(569\) 29.3205 1.22918 0.614590 0.788847i \(-0.289322\pi\)
0.614590 + 0.788847i \(0.289322\pi\)
\(570\) 0 0
\(571\) 18.3923 0.769694 0.384847 0.922980i \(-0.374254\pi\)
0.384847 + 0.922980i \(0.374254\pi\)
\(572\) 0 0
\(573\) 18.9282i 0.790737i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 6.78461i 0.282447i 0.989978 + 0.141223i \(0.0451036\pi\)
−0.989978 + 0.141223i \(0.954896\pi\)
\(578\) 0 0
\(579\) 21.3205 0.886050
\(580\) 0 0
\(581\) 6.92820 0.287430
\(582\) 0 0
\(583\) 37.1769i 1.53971i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 43.1769i − 1.78210i −0.453903 0.891051i \(-0.649969\pi\)
0.453903 0.891051i \(-0.350031\pi\)
\(588\) 0 0
\(589\) 1.46410 0.0603273
\(590\) 0 0
\(591\) 2.53590 0.104313
\(592\) 0 0
\(593\) 31.8564i 1.30819i 0.756414 + 0.654093i \(0.226949\pi\)
−0.756414 + 0.654093i \(0.773051\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 73.5692i − 3.01099i
\(598\) 0 0
\(599\) 20.7846 0.849236 0.424618 0.905373i \(-0.360408\pi\)
0.424618 + 0.905373i \(0.360408\pi\)
\(600\) 0 0
\(601\) 24.6410 1.00513 0.502564 0.864540i \(-0.332390\pi\)
0.502564 + 0.864540i \(0.332390\pi\)
\(602\) 0 0
\(603\) 30.0526i 1.22383i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 39.9090i 1.61985i 0.586530 + 0.809927i \(0.300494\pi\)
−0.586530 + 0.809927i \(0.699506\pi\)
\(608\) 0 0
\(609\) −18.9282 −0.767010
\(610\) 0 0
\(611\) −35.3205 −1.42891
\(612\) 0 0
\(613\) 6.39230i 0.258183i 0.991633 + 0.129091i \(0.0412061\pi\)
−0.991633 + 0.129091i \(0.958794\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 1.60770i − 0.0647234i −0.999476 0.0323617i \(-0.989697\pi\)
0.999476 0.0323617i \(-0.0103028\pi\)
\(618\) 0 0
\(619\) 2.39230 0.0961549 0.0480774 0.998844i \(-0.484691\pi\)
0.0480774 + 0.998844i \(0.484691\pi\)
\(620\) 0 0
\(621\) −13.8564 −0.556038
\(622\) 0 0
\(623\) − 30.9282i − 1.23911i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 9.46410i 0.377960i
\(628\) 0 0
\(629\) 23.3205 0.929850
\(630\) 0 0
\(631\) 11.4641 0.456379 0.228189 0.973617i \(-0.426719\pi\)
0.228189 + 0.973617i \(0.426719\pi\)
\(632\) 0 0
\(633\) 52.7846i 2.09800i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 8.19615i − 0.324743i
\(638\) 0 0
\(639\) 11.3205 0.447832
\(640\) 0 0
\(641\) −24.9282 −0.984605 −0.492302 0.870424i \(-0.663845\pi\)
−0.492302 + 0.870424i \(0.663845\pi\)
\(642\) 0 0
\(643\) − 14.3923i − 0.567577i −0.958887 0.283789i \(-0.908409\pi\)
0.958887 0.283789i \(-0.0915914\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 15.4641i − 0.607957i −0.952679 0.303978i \(-0.901685\pi\)
0.952679 0.303978i \(-0.0983150\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 0 0
\(653\) 27.4641i 1.07475i 0.843342 + 0.537377i \(0.180585\pi\)
−0.843342 + 0.537377i \(0.819415\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.39230i 0.0933327i
\(658\) 0 0
\(659\) −8.78461 −0.342200 −0.171100 0.985254i \(-0.554732\pi\)
−0.171100 + 0.985254i \(0.554732\pi\)
\(660\) 0 0
\(661\) 5.21539 0.202855 0.101428 0.994843i \(-0.467659\pi\)
0.101428 + 0.994843i \(0.467659\pi\)
\(662\) 0 0
\(663\) 25.8564i 1.00418i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.0000i 0.464642i
\(668\) 0 0
\(669\) 26.3923 1.02039
\(670\) 0 0
\(671\) −42.9282 −1.65722
\(672\) 0 0
\(673\) − 50.7321i − 1.95558i −0.209594 0.977788i \(-0.567214\pi\)
0.209594 0.977788i \(-0.432786\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 16.9808i − 0.652624i −0.945262 0.326312i \(-0.894194\pi\)
0.945262 0.326312i \(-0.105806\pi\)
\(678\) 0 0
\(679\) 33.1769 1.27321
\(680\) 0 0
\(681\) 22.3923 0.858075
\(682\) 0 0
\(683\) − 17.6603i − 0.675751i −0.941191 0.337875i \(-0.890292\pi\)
0.941191 0.337875i \(-0.109708\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 47.7128i − 1.82036i
\(688\) 0 0
\(689\) −29.3205 −1.11702
\(690\) 0 0
\(691\) 51.1769 1.94686 0.973431 0.228981i \(-0.0735395\pi\)
0.973431 + 0.228981i \(0.0735395\pi\)
\(692\) 0 0
\(693\) − 30.9282i − 1.17487i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 20.7846i − 0.787273i
\(698\) 0 0
\(699\) 2.53590 0.0959165
\(700\) 0 0
\(701\) −26.5359 −1.00225 −0.501124 0.865376i \(-0.667080\pi\)
−0.501124 + 0.865376i \(0.667080\pi\)
\(702\) 0 0
\(703\) 6.73205i 0.253904i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 32.7846i 1.23299i
\(708\) 0 0
\(709\) 30.7846 1.15614 0.578070 0.815987i \(-0.303806\pi\)
0.578070 + 0.815987i \(0.303806\pi\)
\(710\) 0 0
\(711\) 13.0718 0.490231
\(712\) 0 0
\(713\) 5.07180i 0.189940i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 37.8564i 1.41377i
\(718\) 0 0
\(719\) −17.3205 −0.645946 −0.322973 0.946408i \(-0.604682\pi\)
−0.322973 + 0.946408i \(0.604682\pi\)
\(720\) 0 0
\(721\) 27.3205 1.01747
\(722\) 0 0
\(723\) − 32.3923i − 1.20468i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 0.143594i − 0.00532559i −0.999996 0.00266279i \(-0.999152\pi\)
0.999996 0.00266279i \(-0.000847595\pi\)
\(728\) 0 0
\(729\) 43.7846 1.62165
\(730\) 0 0
\(731\) 17.0718 0.631423
\(732\) 0 0
\(733\) 10.7846i 0.398339i 0.979965 + 0.199169i \(0.0638244\pi\)
−0.979965 + 0.199169i \(0.936176\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 23.3205i 0.859022i
\(738\) 0 0
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) −7.46410 −0.274201
\(742\) 0 0
\(743\) 20.8756i 0.765853i 0.923779 + 0.382927i \(0.125084\pi\)
−0.923779 + 0.382927i \(0.874916\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 15.4641i − 0.565802i
\(748\) 0 0
\(749\) 11.3205 0.413642
\(750\) 0 0
\(751\) −25.4641 −0.929198 −0.464599 0.885521i \(-0.653802\pi\)
−0.464599 + 0.885521i \(0.653802\pi\)
\(752\) 0 0
\(753\) − 65.5692i − 2.38948i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000i 0.363456i 0.983349 + 0.181728i \(0.0581691\pi\)
−0.983349 + 0.181728i \(0.941831\pi\)
\(758\) 0 0
\(759\) −32.7846 −1.19001
\(760\) 0 0
\(761\) 19.6077 0.710778 0.355389 0.934718i \(-0.384348\pi\)
0.355389 + 0.934718i \(0.384348\pi\)
\(762\) 0 0
\(763\) − 28.7846i − 1.04207i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.9282i 0.683458i
\(768\) 0 0
\(769\) 30.5359 1.10115 0.550576 0.834785i \(-0.314408\pi\)
0.550576 + 0.834785i \(0.314408\pi\)
\(770\) 0 0
\(771\) −41.3205 −1.48812
\(772\) 0 0
\(773\) 22.0526i 0.793175i 0.917997 + 0.396588i \(0.129806\pi\)
−0.917997 + 0.396588i \(0.870194\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 36.7846i − 1.31964i
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 8.78461 0.314338
\(782\) 0 0
\(783\) 13.8564i 0.495188i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 12.1962i 0.434746i 0.976089 + 0.217373i \(0.0697488\pi\)
−0.976089 + 0.217373i \(0.930251\pi\)
\(788\) 0 0
\(789\) −23.3205 −0.830232
\(790\) 0 0
\(791\) 25.1769 0.895188
\(792\) 0 0
\(793\) − 33.8564i − 1.20228i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.66025i 0.200496i 0.994962 + 0.100248i \(0.0319637\pi\)
−0.994962 + 0.100248i \(0.968036\pi\)
\(798\) 0 0
\(799\) 44.7846 1.58437
\(800\) 0 0
\(801\) −69.0333 −2.43917
\(802\) 0 0
\(803\) 1.85641i 0.0655112i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 28.3923i − 0.999456i
\(808\) 0 0
\(809\) −9.71281 −0.341484 −0.170742 0.985316i \(-0.554617\pi\)
−0.170742 + 0.985316i \(0.554617\pi\)
\(810\) 0 0
\(811\) 15.6077 0.548060 0.274030 0.961721i \(-0.411643\pi\)
0.274030 + 0.961721i \(0.411643\pi\)
\(812\) 0 0
\(813\) − 6.53590i − 0.229224i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4.92820i 0.172416i
\(818\) 0 0
\(819\) 24.3923 0.852336
\(820\) 0 0
\(821\) −16.1436 −0.563415 −0.281708 0.959500i \(-0.590901\pi\)
−0.281708 + 0.959500i \(0.590901\pi\)
\(822\) 0 0
\(823\) − 39.5692i − 1.37930i −0.724145 0.689648i \(-0.757765\pi\)
0.724145 0.689648i \(-0.242235\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 29.6603i 1.03139i 0.856773 + 0.515694i \(0.172466\pi\)
−0.856773 + 0.515694i \(0.827534\pi\)
\(828\) 0 0
\(829\) −8.24871 −0.286490 −0.143245 0.989687i \(-0.545754\pi\)
−0.143245 + 0.989687i \(0.545754\pi\)
\(830\) 0 0
\(831\) 50.2487 1.74311
\(832\) 0 0
\(833\) 10.3923i 0.360072i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 5.85641i 0.202427i
\(838\) 0 0
\(839\) −18.9282 −0.653474 −0.326737 0.945115i \(-0.605949\pi\)
−0.326737 + 0.945115i \(0.605949\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) 2.53590i 0.0873410i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.00000i − 0.0687208i
\(848\) 0 0
\(849\) −50.2487 −1.72453
\(850\) 0 0
\(851\) −23.3205 −0.799417
\(852\) 0 0
\(853\) − 44.6410i − 1.52848i −0.644932 0.764240i \(-0.723114\pi\)
0.644932 0.764240i \(-0.276886\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 3.80385i − 0.129937i −0.997887 0.0649685i \(-0.979305\pi\)
0.997887 0.0649685i \(-0.0206947\pi\)
\(858\) 0 0
\(859\) −47.7128 −1.62794 −0.813970 0.580907i \(-0.802698\pi\)
−0.813970 + 0.580907i \(0.802698\pi\)
\(860\) 0 0
\(861\) −32.7846 −1.11730
\(862\) 0 0
\(863\) − 37.2679i − 1.26862i −0.773081 0.634308i \(-0.781285\pi\)
0.773081 0.634308i \(-0.218715\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.6603i 0.463927i
\(868\) 0 0
\(869\) 10.1436 0.344098
\(870\) 0 0
\(871\) −18.3923 −0.623199
\(872\) 0 0
\(873\) − 74.0526i − 2.50630i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 0.980762i − 0.0331180i −0.999863 0.0165590i \(-0.994729\pi\)
0.999863 0.0165590i \(-0.00527113\pi\)
\(878\) 0 0
\(879\) 15.4641 0.521591
\(880\) 0 0
\(881\) −0.679492 −0.0228927 −0.0114463 0.999934i \(-0.503644\pi\)
−0.0114463 + 0.999934i \(0.503644\pi\)
\(882\) 0 0
\(883\) − 22.0000i − 0.740359i −0.928960 0.370179i \(-0.879296\pi\)
0.928960 0.370179i \(-0.120704\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 23.4115i − 0.786083i −0.919521 0.393041i \(-0.871423\pi\)
0.919521 0.393041i \(-0.128577\pi\)
\(888\) 0 0
\(889\) −41.1769 −1.38103
\(890\) 0 0
\(891\) 8.53590 0.285963
\(892\) 0 0
\(893\) 12.9282i 0.432626i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 25.8564i − 0.863320i
\(898\) 0 0
\(899\) 5.07180 0.169154
\(900\) 0 0
\(901\) 37.1769 1.23854
\(902\) 0 0
\(903\) − 26.9282i − 0.896114i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 7.41154i − 0.246096i −0.992401 0.123048i \(-0.960733\pi\)
0.992401 0.123048i \(-0.0392670\pi\)
\(908\) 0 0
\(909\) 73.1769 2.42713
\(910\) 0 0
\(911\) 14.5359 0.481596 0.240798 0.970575i \(-0.422591\pi\)
0.240798 + 0.970575i \(0.422591\pi\)
\(912\) 0 0
\(913\) − 12.0000i − 0.397142i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 37.8564i 1.25013i
\(918\) 0 0
\(919\) −44.4974 −1.46783 −0.733917 0.679239i \(-0.762310\pi\)
−0.733917 + 0.679239i \(0.762310\pi\)
\(920\) 0 0
\(921\) −21.3205 −0.702535
\(922\) 0 0
\(923\) 6.92820i 0.228045i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 60.9808i − 2.00287i
\(928\) 0 0
\(929\) 47.5692 1.56070 0.780348 0.625346i \(-0.215042\pi\)
0.780348 + 0.625346i \(0.215042\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 0 0
\(933\) − 4.39230i − 0.143798i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 51.1769i − 1.67188i −0.548823 0.835938i \(-0.684924\pi\)
0.548823 0.835938i \(-0.315076\pi\)
\(938\) 0 0
\(939\) −29.4641 −0.961525
\(940\) 0 0
\(941\) 52.6410 1.71605 0.858024 0.513610i \(-0.171692\pi\)
0.858024 + 0.513610i \(0.171692\pi\)
\(942\) 0 0
\(943\) 20.7846i 0.676840i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 6.67949i − 0.217054i −0.994093 0.108527i \(-0.965387\pi\)
0.994093 0.108527i \(-0.0346134\pi\)
\(948\) 0 0
\(949\) −1.46410 −0.0475267
\(950\) 0 0
\(951\) 8.53590 0.276795
\(952\) 0 0
\(953\) 60.5885i 1.96265i 0.192350 + 0.981326i \(0.438389\pi\)
−0.192350 + 0.981326i \(0.561611\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 32.7846i 1.05978i
\(958\) 0 0
\(959\) −34.6410 −1.11862
\(960\) 0 0
\(961\) −28.8564 −0.930852
\(962\) 0 0
\(963\) − 25.2679i − 0.814248i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 45.3205i 1.45741i 0.684828 + 0.728705i \(0.259877\pi\)
−0.684828 + 0.728705i \(0.740123\pi\)
\(968\) 0 0
\(969\) 9.46410 0.304031
\(970\) 0 0
\(971\) −14.5359 −0.466479 −0.233240 0.972419i \(-0.574933\pi\)
−0.233240 + 0.972419i \(0.574933\pi\)
\(972\) 0 0
\(973\) 22.9282i 0.735044i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 32.1962i − 1.03005i −0.857176 0.515023i \(-0.827783\pi\)
0.857176 0.515023i \(-0.172217\pi\)
\(978\) 0 0
\(979\) −53.5692 −1.71208
\(980\) 0 0
\(981\) −64.2487 −2.05130
\(982\) 0 0
\(983\) − 2.44486i − 0.0779790i −0.999240 0.0389895i \(-0.987586\pi\)
0.999240 0.0389895i \(-0.0124139\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 70.6410i − 2.24853i
\(988\) 0 0
\(989\) −17.0718 −0.542852
\(990\) 0 0
\(991\) −8.39230 −0.266590 −0.133295 0.991076i \(-0.542556\pi\)
−0.133295 + 0.991076i \(0.542556\pi\)
\(992\) 0 0
\(993\) − 60.7846i − 1.92894i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 30.3923i − 0.962534i −0.876574 0.481267i \(-0.840177\pi\)
0.876574 0.481267i \(-0.159823\pi\)
\(998\) 0 0
\(999\) −26.9282 −0.851971
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 1900.2.c.e.1749.4 4
5.2 odd 4 380.2.a.d.1.2 2
5.3 odd 4 1900.2.a.d.1.1 2
5.4 even 2 inner 1900.2.c.e.1749.1 4
15.2 even 4 3420.2.a.h.1.2 2
20.3 even 4 7600.2.a.bf.1.2 2
20.7 even 4 1520.2.a.l.1.1 2
40.27 even 4 6080.2.a.bj.1.2 2
40.37 odd 4 6080.2.a.z.1.1 2
95.37 even 4 7220.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.d.1.2 2 5.2 odd 4
1520.2.a.l.1.1 2 20.7 even 4
1900.2.a.d.1.1 2 5.3 odd 4
1900.2.c.e.1749.1 4 5.4 even 2 inner
1900.2.c.e.1749.4 4 1.1 even 1 trivial
3420.2.a.h.1.2 2 15.2 even 4
6080.2.a.z.1.1 2 40.37 odd 4
6080.2.a.bj.1.2 2 40.27 even 4
7220.2.a.h.1.1 2 95.37 even 4
7600.2.a.bf.1.2 2 20.3 even 4