Properties

Label 2300.2.c.h.1749.2
Level $2300$
Weight $2$
Character 2300.1749
Analytic conductor $18.366$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3655924649\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 460)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1749.2
Root \(-1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 2300.1749
Dual form 2300.2.c.h.1749.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155i q^{3} -2.56155i q^{7} +0.561553 q^{9} +O(q^{10})\) \(q-1.56155i q^{3} -2.56155i q^{7} +0.561553 q^{9} +2.00000 q^{11} -3.56155i q^{13} -2.56155i q^{17} -6.00000 q^{19} -4.00000 q^{21} +1.00000i q^{23} -5.56155i q^{27} -6.12311 q^{29} +7.24621 q^{31} -3.12311i q^{33} +4.56155i q^{37} -5.56155 q^{39} +4.12311 q^{41} -4.68466i q^{47} +0.438447 q^{49} -4.00000 q^{51} -4.56155i q^{53} +9.36932i q^{57} +3.68466 q^{59} -7.12311 q^{61} -1.43845i q^{63} +8.56155i q^{67} +1.56155 q^{69} +10.1231 q^{71} -4.43845i q^{73} -5.12311i q^{77} -4.87689 q^{79} -7.00000 q^{81} -13.9309i q^{83} +9.56155i q^{87} -14.2462 q^{89} -9.12311 q^{91} -11.3153i q^{93} +13.1231i q^{97} +1.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{9} + O(q^{10}) \) \( 4q - 6q^{9} + 8q^{11} - 24q^{19} - 16q^{21} - 8q^{29} - 4q^{31} - 14q^{39} + 10q^{49} - 16q^{51} - 10q^{59} - 12q^{61} - 2q^{69} + 24q^{71} - 36q^{79} - 28q^{81} - 24q^{89} - 20q^{91} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 1.56155i − 0.901563i −0.892634 0.450781i \(-0.851145\pi\)
0.892634 0.450781i \(-0.148855\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.56155i − 0.968176i −0.875019 0.484088i \(-0.839151\pi\)
0.875019 0.484088i \(-0.160849\pi\)
\(8\) 0 0
\(9\) 0.561553 0.187184
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) − 3.56155i − 0.987797i −0.869520 0.493899i \(-0.835571\pi\)
0.869520 0.493899i \(-0.164429\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.56155i − 0.621268i −0.950530 0.310634i \(-0.899459\pi\)
0.950530 0.310634i \(-0.100541\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.56155i − 1.07032i
\(28\) 0 0
\(29\) −6.12311 −1.13703 −0.568516 0.822672i \(-0.692482\pi\)
−0.568516 + 0.822672i \(0.692482\pi\)
\(30\) 0 0
\(31\) 7.24621 1.30146 0.650729 0.759310i \(-0.274463\pi\)
0.650729 + 0.759310i \(0.274463\pi\)
\(32\) 0 0
\(33\) − 3.12311i − 0.543663i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.56155i 0.749915i 0.927042 + 0.374957i \(0.122343\pi\)
−0.927042 + 0.374957i \(0.877657\pi\)
\(38\) 0 0
\(39\) −5.56155 −0.890561
\(40\) 0 0
\(41\) 4.12311 0.643921 0.321960 0.946753i \(-0.395658\pi\)
0.321960 + 0.946753i \(0.395658\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 4.68466i − 0.683328i −0.939822 0.341664i \(-0.889010\pi\)
0.939822 0.341664i \(-0.110990\pi\)
\(48\) 0 0
\(49\) 0.438447 0.0626353
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) − 4.56155i − 0.626577i −0.949658 0.313289i \(-0.898569\pi\)
0.949658 0.313289i \(-0.101431\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.36932i 1.24100i
\(58\) 0 0
\(59\) 3.68466 0.479702 0.239851 0.970810i \(-0.422901\pi\)
0.239851 + 0.970810i \(0.422901\pi\)
\(60\) 0 0
\(61\) −7.12311 −0.912020 −0.456010 0.889975i \(-0.650722\pi\)
−0.456010 + 0.889975i \(0.650722\pi\)
\(62\) 0 0
\(63\) − 1.43845i − 0.181227i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.56155i 1.04596i 0.852345 + 0.522980i \(0.175180\pi\)
−0.852345 + 0.522980i \(0.824820\pi\)
\(68\) 0 0
\(69\) 1.56155 0.187989
\(70\) 0 0
\(71\) 10.1231 1.20139 0.600696 0.799478i \(-0.294890\pi\)
0.600696 + 0.799478i \(0.294890\pi\)
\(72\) 0 0
\(73\) − 4.43845i − 0.519481i −0.965678 0.259740i \(-0.916363\pi\)
0.965678 0.259740i \(-0.0836370\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.12311i − 0.583832i
\(78\) 0 0
\(79\) −4.87689 −0.548693 −0.274347 0.961631i \(-0.588462\pi\)
−0.274347 + 0.961631i \(0.588462\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) − 13.9309i − 1.52911i −0.644558 0.764556i \(-0.722958\pi\)
0.644558 0.764556i \(-0.277042\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.56155i 1.02511i
\(88\) 0 0
\(89\) −14.2462 −1.51010 −0.755048 0.655670i \(-0.772386\pi\)
−0.755048 + 0.655670i \(0.772386\pi\)
\(90\) 0 0
\(91\) −9.12311 −0.956361
\(92\) 0 0
\(93\) − 11.3153i − 1.17335i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.1231i 1.33245i 0.745751 + 0.666225i \(0.232091\pi\)
−0.745751 + 0.666225i \(0.767909\pi\)
\(98\) 0 0
\(99\) 1.12311 0.112876
\(100\) 0 0
\(101\) −17.6847 −1.75969 −0.879845 0.475261i \(-0.842353\pi\)
−0.879845 + 0.475261i \(0.842353\pi\)
\(102\) 0 0
\(103\) − 16.0000i − 1.57653i −0.615338 0.788263i \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 3.43845i − 0.332407i −0.986091 0.166204i \(-0.946849\pi\)
0.986091 0.166204i \(-0.0531509\pi\)
\(108\) 0 0
\(109\) −15.3693 −1.47211 −0.736057 0.676920i \(-0.763314\pi\)
−0.736057 + 0.676920i \(0.763314\pi\)
\(110\) 0 0
\(111\) 7.12311 0.676095
\(112\) 0 0
\(113\) 12.8078i 1.20485i 0.798174 + 0.602427i \(0.205799\pi\)
−0.798174 + 0.602427i \(0.794201\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2.00000i − 0.184900i
\(118\) 0 0
\(119\) −6.56155 −0.601497
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) − 6.43845i − 0.580535i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 7.80776i − 0.692827i −0.938082 0.346414i \(-0.887399\pi\)
0.938082 0.346414i \(-0.112601\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0.192236 0.0167957 0.00839787 0.999965i \(-0.497327\pi\)
0.00839787 + 0.999965i \(0.497327\pi\)
\(132\) 0 0
\(133\) 15.3693i 1.33269i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 6.87689i − 0.587533i −0.955877 0.293766i \(-0.905091\pi\)
0.955877 0.293766i \(-0.0949088\pi\)
\(138\) 0 0
\(139\) 11.2462 0.953891 0.476946 0.878933i \(-0.341744\pi\)
0.476946 + 0.878933i \(0.341744\pi\)
\(140\) 0 0
\(141\) −7.31534 −0.616063
\(142\) 0 0
\(143\) − 7.12311i − 0.595664i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 0.684658i − 0.0564697i
\(148\) 0 0
\(149\) 9.36932 0.767564 0.383782 0.923424i \(-0.374621\pi\)
0.383782 + 0.923424i \(0.374621\pi\)
\(150\) 0 0
\(151\) −14.0540 −1.14370 −0.571848 0.820359i \(-0.693773\pi\)
−0.571848 + 0.820359i \(0.693773\pi\)
\(152\) 0 0
\(153\) − 1.43845i − 0.116292i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.31534i 0.344402i 0.985062 + 0.172201i \(0.0550878\pi\)
−0.985062 + 0.172201i \(0.944912\pi\)
\(158\) 0 0
\(159\) −7.12311 −0.564899
\(160\) 0 0
\(161\) 2.56155 0.201879
\(162\) 0 0
\(163\) 16.9309i 1.32613i 0.748563 + 0.663064i \(0.230744\pi\)
−0.748563 + 0.663064i \(0.769256\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) 0.315342 0.0242570
\(170\) 0 0
\(171\) −3.36932 −0.257658
\(172\) 0 0
\(173\) 5.36932i 0.408222i 0.978948 + 0.204111i \(0.0654303\pi\)
−0.978948 + 0.204111i \(0.934570\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 5.75379i − 0.432481i
\(178\) 0 0
\(179\) 6.93087 0.518038 0.259019 0.965872i \(-0.416601\pi\)
0.259019 + 0.965872i \(0.416601\pi\)
\(180\) 0 0
\(181\) −5.36932 −0.399098 −0.199549 0.979888i \(-0.563948\pi\)
−0.199549 + 0.979888i \(0.563948\pi\)
\(182\) 0 0
\(183\) 11.1231i 0.822244i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 5.12311i − 0.374639i
\(188\) 0 0
\(189\) −14.2462 −1.03626
\(190\) 0 0
\(191\) 17.3693 1.25680 0.628400 0.777891i \(-0.283710\pi\)
0.628400 + 0.777891i \(0.283710\pi\)
\(192\) 0 0
\(193\) 2.43845i 0.175523i 0.996142 + 0.0877616i \(0.0279714\pi\)
−0.996142 + 0.0877616i \(0.972029\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.6847i 1.33123i 0.746297 + 0.665613i \(0.231830\pi\)
−0.746297 + 0.665613i \(0.768170\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 13.3693 0.942999
\(202\) 0 0
\(203\) 15.6847i 1.10085i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.561553i 0.0390306i
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 0.315342 0.0217090 0.0108545 0.999941i \(-0.496545\pi\)
0.0108545 + 0.999941i \(0.496545\pi\)
\(212\) 0 0
\(213\) − 15.8078i − 1.08313i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 18.5616i − 1.26004i
\(218\) 0 0
\(219\) −6.93087 −0.468345
\(220\) 0 0
\(221\) −9.12311 −0.613686
\(222\) 0 0
\(223\) − 17.6155i − 1.17962i −0.807541 0.589812i \(-0.799202\pi\)
0.807541 0.589812i \(-0.200798\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 26.2462i − 1.74202i −0.491263 0.871011i \(-0.663465\pi\)
0.491263 0.871011i \(-0.336535\pi\)
\(228\) 0 0
\(229\) 26.7386 1.76694 0.883469 0.468489i \(-0.155201\pi\)
0.883469 + 0.468489i \(0.155201\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) − 0.684658i − 0.0448535i −0.999748 0.0224267i \(-0.992861\pi\)
0.999748 0.0224267i \(-0.00713925\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 7.61553i 0.494682i
\(238\) 0 0
\(239\) −2.75379 −0.178128 −0.0890639 0.996026i \(-0.528388\pi\)
−0.0890639 + 0.996026i \(0.528388\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) − 5.75379i − 0.369106i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 21.3693i 1.35970i
\(248\) 0 0
\(249\) −21.7538 −1.37859
\(250\) 0 0
\(251\) −7.36932 −0.465147 −0.232574 0.972579i \(-0.574715\pi\)
−0.232574 + 0.972579i \(0.574715\pi\)
\(252\) 0 0
\(253\) 2.00000i 0.125739i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 31.8078i − 1.98411i −0.125790 0.992057i \(-0.540147\pi\)
0.125790 0.992057i \(-0.459853\pi\)
\(258\) 0 0
\(259\) 11.6847 0.726049
\(260\) 0 0
\(261\) −3.43845 −0.212835
\(262\) 0 0
\(263\) 0.0691303i 0.00426276i 0.999998 + 0.00213138i \(0.000678439\pi\)
−0.999998 + 0.00213138i \(0.999322\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 22.2462i 1.36145i
\(268\) 0 0
\(269\) −23.2462 −1.41735 −0.708673 0.705537i \(-0.750706\pi\)
−0.708673 + 0.705537i \(0.750706\pi\)
\(270\) 0 0
\(271\) 21.9309 1.33221 0.666103 0.745860i \(-0.267961\pi\)
0.666103 + 0.745860i \(0.267961\pi\)
\(272\) 0 0
\(273\) 14.2462i 0.862220i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.68466i 0.401642i 0.979628 + 0.200821i \(0.0643610\pi\)
−0.979628 + 0.200821i \(0.935639\pi\)
\(278\) 0 0
\(279\) 4.06913 0.243612
\(280\) 0 0
\(281\) −6.87689 −0.410241 −0.205121 0.978737i \(-0.565759\pi\)
−0.205121 + 0.978737i \(0.565759\pi\)
\(282\) 0 0
\(283\) 14.8078i 0.880230i 0.897941 + 0.440115i \(0.145062\pi\)
−0.897941 + 0.440115i \(0.854938\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 10.5616i − 0.623429i
\(288\) 0 0
\(289\) 10.4384 0.614026
\(290\) 0 0
\(291\) 20.4924 1.20129
\(292\) 0 0
\(293\) − 16.8078i − 0.981920i −0.871182 0.490960i \(-0.836646\pi\)
0.871182 0.490960i \(-0.163354\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 11.1231i − 0.645428i
\(298\) 0 0
\(299\) 3.56155 0.205970
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 27.6155i 1.58647i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 21.1231i 1.20556i 0.797908 + 0.602780i \(0.205940\pi\)
−0.797908 + 0.602780i \(0.794060\pi\)
\(308\) 0 0
\(309\) −24.9848 −1.42134
\(310\) 0 0
\(311\) 8.68466 0.492462 0.246231 0.969211i \(-0.420808\pi\)
0.246231 + 0.969211i \(0.420808\pi\)
\(312\) 0 0
\(313\) − 0.807764i − 0.0456575i −0.999739 0.0228288i \(-0.992733\pi\)
0.999739 0.0228288i \(-0.00726725\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 15.1231i − 0.849398i −0.905335 0.424699i \(-0.860380\pi\)
0.905335 0.424699i \(-0.139620\pi\)
\(318\) 0 0
\(319\) −12.2462 −0.685656
\(320\) 0 0
\(321\) −5.36932 −0.299686
\(322\) 0 0
\(323\) 15.3693i 0.855172i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 24.0000i 1.32720i
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 28.6155 1.57285 0.786426 0.617685i \(-0.211929\pi\)
0.786426 + 0.617685i \(0.211929\pi\)
\(332\) 0 0
\(333\) 2.56155i 0.140372i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 13.6155i − 0.741685i −0.928696 0.370843i \(-0.879069\pi\)
0.928696 0.370843i \(-0.120931\pi\)
\(338\) 0 0
\(339\) 20.0000 1.08625
\(340\) 0 0
\(341\) 14.4924 0.784809
\(342\) 0 0
\(343\) − 19.0540i − 1.02882i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 16.4924i − 0.885360i −0.896680 0.442680i \(-0.854028\pi\)
0.896680 0.442680i \(-0.145972\pi\)
\(348\) 0 0
\(349\) 26.3693 1.41152 0.705759 0.708452i \(-0.250606\pi\)
0.705759 + 0.708452i \(0.250606\pi\)
\(350\) 0 0
\(351\) −19.8078 −1.05726
\(352\) 0 0
\(353\) − 2.05398i − 0.109322i −0.998505 0.0546610i \(-0.982592\pi\)
0.998505 0.0546610i \(-0.0174078\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 10.2462i 0.542287i
\(358\) 0 0
\(359\) −15.6155 −0.824156 −0.412078 0.911149i \(-0.635197\pi\)
−0.412078 + 0.911149i \(0.635197\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 10.9309i 0.573722i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 22.5616i 1.17770i 0.808241 + 0.588852i \(0.200420\pi\)
−0.808241 + 0.588852i \(0.799580\pi\)
\(368\) 0 0
\(369\) 2.31534 0.120532
\(370\) 0 0
\(371\) −11.6847 −0.606637
\(372\) 0 0
\(373\) − 20.2462i − 1.04831i −0.851623 0.524155i \(-0.824381\pi\)
0.851623 0.524155i \(-0.175619\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 21.8078i 1.12316i
\(378\) 0 0
\(379\) 36.4924 1.87449 0.937245 0.348672i \(-0.113367\pi\)
0.937245 + 0.348672i \(0.113367\pi\)
\(380\) 0 0
\(381\) −12.1922 −0.624627
\(382\) 0 0
\(383\) 3.19224i 0.163116i 0.996669 + 0.0815578i \(0.0259895\pi\)
−0.996669 + 0.0815578i \(0.974010\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10.8769 0.551480 0.275740 0.961232i \(-0.411077\pi\)
0.275740 + 0.961232i \(0.411077\pi\)
\(390\) 0 0
\(391\) 2.56155 0.129543
\(392\) 0 0
\(393\) − 0.300187i − 0.0151424i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 34.5464i − 1.73383i −0.498453 0.866917i \(-0.666098\pi\)
0.498453 0.866917i \(-0.333902\pi\)
\(398\) 0 0
\(399\) 24.0000 1.20150
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) − 25.8078i − 1.28558i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.12311i 0.452216i
\(408\) 0 0
\(409\) 35.9848 1.77934 0.889668 0.456608i \(-0.150936\pi\)
0.889668 + 0.456608i \(0.150936\pi\)
\(410\) 0 0
\(411\) −10.7386 −0.529698
\(412\) 0 0
\(413\) − 9.43845i − 0.464436i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 17.5616i − 0.859993i
\(418\) 0 0
\(419\) 18.8769 0.922197 0.461098 0.887349i \(-0.347456\pi\)
0.461098 + 0.887349i \(0.347456\pi\)
\(420\) 0 0
\(421\) 25.1231 1.22443 0.612213 0.790693i \(-0.290280\pi\)
0.612213 + 0.790693i \(0.290280\pi\)
\(422\) 0 0
\(423\) − 2.63068i − 0.127908i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 18.2462i 0.882996i
\(428\) 0 0
\(429\) −11.1231 −0.537029
\(430\) 0 0
\(431\) −10.2462 −0.493543 −0.246771 0.969074i \(-0.579370\pi\)
−0.246771 + 0.969074i \(0.579370\pi\)
\(432\) 0 0
\(433\) 39.9309i 1.91896i 0.281785 + 0.959478i \(0.409074\pi\)
−0.281785 + 0.959478i \(0.590926\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 6.00000i − 0.287019i
\(438\) 0 0
\(439\) −19.8078 −0.945373 −0.472686 0.881231i \(-0.656716\pi\)
−0.472686 + 0.881231i \(0.656716\pi\)
\(440\) 0 0
\(441\) 0.246211 0.0117243
\(442\) 0 0
\(443\) − 11.8078i − 0.561004i −0.959853 0.280502i \(-0.909499\pi\)
0.959853 0.280502i \(-0.0905009\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 14.6307i − 0.692008i
\(448\) 0 0
\(449\) 21.6847 1.02336 0.511681 0.859175i \(-0.329023\pi\)
0.511681 + 0.859175i \(0.329023\pi\)
\(450\) 0 0
\(451\) 8.24621 0.388299
\(452\) 0 0
\(453\) 21.9460i 1.03111i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 9.43845i − 0.441512i −0.975329 0.220756i \(-0.929148\pi\)
0.975329 0.220756i \(-0.0708524\pi\)
\(458\) 0 0
\(459\) −14.2462 −0.664956
\(460\) 0 0
\(461\) −40.9309 −1.90634 −0.953170 0.302434i \(-0.902201\pi\)
−0.953170 + 0.302434i \(0.902201\pi\)
\(462\) 0 0
\(463\) − 31.3693i − 1.45786i −0.684590 0.728928i \(-0.740019\pi\)
0.684590 0.728928i \(-0.259981\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.3153i 1.03263i 0.856398 + 0.516315i \(0.172697\pi\)
−0.856398 + 0.516315i \(0.827303\pi\)
\(468\) 0 0
\(469\) 21.9309 1.01267
\(470\) 0 0
\(471\) 6.73863 0.310500
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 2.56155i − 0.117285i
\(478\) 0 0
\(479\) 43.2311 1.97528 0.987639 0.156748i \(-0.0501009\pi\)
0.987639 + 0.156748i \(0.0501009\pi\)
\(480\) 0 0
\(481\) 16.2462 0.740763
\(482\) 0 0
\(483\) − 4.00000i − 0.182006i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 7.80776i − 0.353804i −0.984229 0.176902i \(-0.943393\pi\)
0.984229 0.176902i \(-0.0566075\pi\)
\(488\) 0 0
\(489\) 26.4384 1.19559
\(490\) 0 0
\(491\) 11.4924 0.518646 0.259323 0.965791i \(-0.416501\pi\)
0.259323 + 0.965791i \(0.416501\pi\)
\(492\) 0 0
\(493\) 15.6847i 0.706401i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 25.9309i − 1.16316i
\(498\) 0 0
\(499\) −22.6155 −1.01241 −0.506205 0.862413i \(-0.668952\pi\)
−0.506205 + 0.862413i \(0.668952\pi\)
\(500\) 0 0
\(501\) 12.4924 0.558120
\(502\) 0 0
\(503\) − 3.93087i − 0.175269i −0.996153 0.0876344i \(-0.972069\pi\)
0.996153 0.0876344i \(-0.0279307\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 0.492423i − 0.0218693i
\(508\) 0 0
\(509\) 27.1771 1.20460 0.602301 0.798269i \(-0.294251\pi\)
0.602301 + 0.798269i \(0.294251\pi\)
\(510\) 0 0
\(511\) −11.3693 −0.502949
\(512\) 0 0
\(513\) 33.3693i 1.47329i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 9.36932i − 0.412062i
\(518\) 0 0
\(519\) 8.38447 0.368037
\(520\) 0 0
\(521\) −13.6155 −0.596507 −0.298254 0.954487i \(-0.596404\pi\)
−0.298254 + 0.954487i \(0.596404\pi\)
\(522\) 0 0
\(523\) − 14.7386i − 0.644475i −0.946659 0.322238i \(-0.895565\pi\)
0.946659 0.322238i \(-0.104435\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 18.5616i − 0.808554i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 2.06913 0.0897926
\(532\) 0 0
\(533\) − 14.6847i − 0.636063i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 10.8229i − 0.467043i
\(538\) 0 0
\(539\) 0.876894 0.0377705
\(540\) 0 0
\(541\) −2.68466 −0.115422 −0.0577112 0.998333i \(-0.518380\pi\)
−0.0577112 + 0.998333i \(0.518380\pi\)
\(542\) 0 0
\(543\) 8.38447i 0.359812i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 21.1771i 0.905467i 0.891646 + 0.452733i \(0.149551\pi\)
−0.891646 + 0.452733i \(0.850449\pi\)
\(548\) 0 0
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) 36.7386 1.56512
\(552\) 0 0
\(553\) 12.4924i 0.531232i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.6847i 0.664580i 0.943177 + 0.332290i \(0.107821\pi\)
−0.943177 + 0.332290i \(0.892179\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) 32.6695i 1.37686i 0.725305 + 0.688428i \(0.241699\pi\)
−0.725305 + 0.688428i \(0.758301\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 17.9309i 0.753026i
\(568\) 0 0
\(569\) −16.7386 −0.701720 −0.350860 0.936428i \(-0.614111\pi\)
−0.350860 + 0.936428i \(0.614111\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) − 27.1231i − 1.13308i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 29.5616i 1.23066i 0.788268 + 0.615332i \(0.210978\pi\)
−0.788268 + 0.615332i \(0.789022\pi\)
\(578\) 0 0
\(579\) 3.80776 0.158245
\(580\) 0 0
\(581\) −35.6847 −1.48045
\(582\) 0 0
\(583\) − 9.12311i − 0.377840i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 7.06913i − 0.291774i −0.989301 0.145887i \(-0.953396\pi\)
0.989301 0.145887i \(-0.0466036\pi\)
\(588\) 0 0
\(589\) −43.4773 −1.79145
\(590\) 0 0
\(591\) 29.1771 1.20018
\(592\) 0 0
\(593\) 15.6155i 0.641253i 0.947206 + 0.320626i \(0.103893\pi\)
−0.947206 + 0.320626i \(0.896107\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 15.6155i − 0.639101i
\(598\) 0 0
\(599\) 32.9848 1.34772 0.673862 0.738857i \(-0.264634\pi\)
0.673862 + 0.738857i \(0.264634\pi\)
\(600\) 0 0
\(601\) 36.3693 1.48354 0.741768 0.670657i \(-0.233988\pi\)
0.741768 + 0.670657i \(0.233988\pi\)
\(602\) 0 0
\(603\) 4.80776i 0.195787i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 8.49242i − 0.344697i −0.985036 0.172348i \(-0.944865\pi\)
0.985036 0.172348i \(-0.0551355\pi\)
\(608\) 0 0
\(609\) 24.4924 0.992483
\(610\) 0 0
\(611\) −16.6847 −0.674989
\(612\) 0 0
\(613\) − 5.61553i − 0.226809i −0.993549 0.113405i \(-0.963824\pi\)
0.993549 0.113405i \(-0.0361756\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 0.561553i − 0.0226073i −0.999936 0.0113036i \(-0.996402\pi\)
0.999936 0.0113036i \(-0.00359813\pi\)
\(618\) 0 0
\(619\) −12.4924 −0.502113 −0.251056 0.967972i \(-0.580778\pi\)
−0.251056 + 0.967972i \(0.580778\pi\)
\(620\) 0 0
\(621\) 5.56155 0.223177
\(622\) 0 0
\(623\) 36.4924i 1.46204i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 18.7386i 0.748349i
\(628\) 0 0
\(629\) 11.6847 0.465898
\(630\) 0 0
\(631\) 30.2462 1.20408 0.602041 0.798465i \(-0.294354\pi\)
0.602041 + 0.798465i \(0.294354\pi\)
\(632\) 0 0
\(633\) − 0.492423i − 0.0195720i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.56155i − 0.0618710i
\(638\) 0 0
\(639\) 5.68466 0.224882
\(640\) 0 0
\(641\) 6.87689 0.271621 0.135810 0.990735i \(-0.456636\pi\)
0.135810 + 0.990735i \(0.456636\pi\)
\(642\) 0 0
\(643\) 20.4233i 0.805416i 0.915328 + 0.402708i \(0.131931\pi\)
−0.915328 + 0.402708i \(0.868069\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 5.31534i − 0.208968i −0.994527 0.104484i \(-0.966681\pi\)
0.994527 0.104484i \(-0.0333190\pi\)
\(648\) 0 0
\(649\) 7.36932 0.289271
\(650\) 0 0
\(651\) −28.9848 −1.13601
\(652\) 0 0
\(653\) 39.6695i 1.55239i 0.630494 + 0.776194i \(0.282852\pi\)
−0.630494 + 0.776194i \(0.717148\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 2.49242i − 0.0972387i
\(658\) 0 0
\(659\) −15.3693 −0.598704 −0.299352 0.954143i \(-0.596770\pi\)
−0.299352 + 0.954143i \(0.596770\pi\)
\(660\) 0 0
\(661\) −6.49242 −0.252526 −0.126263 0.991997i \(-0.540298\pi\)
−0.126263 + 0.991997i \(0.540298\pi\)
\(662\) 0 0
\(663\) 14.2462i 0.553277i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 6.12311i − 0.237088i
\(668\) 0 0
\(669\) −27.5076 −1.06350
\(670\) 0 0
\(671\) −14.2462 −0.549969
\(672\) 0 0
\(673\) 34.5464i 1.33167i 0.746101 + 0.665833i \(0.231924\pi\)
−0.746101 + 0.665833i \(0.768076\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 26.8078i − 1.03031i −0.857098 0.515153i \(-0.827735\pi\)
0.857098 0.515153i \(-0.172265\pi\)
\(678\) 0 0
\(679\) 33.6155 1.29005
\(680\) 0 0
\(681\) −40.9848 −1.57054
\(682\) 0 0
\(683\) 42.0540i 1.60915i 0.593851 + 0.804575i \(0.297607\pi\)
−0.593851 + 0.804575i \(0.702393\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 41.7538i − 1.59301i
\(688\) 0 0
\(689\) −16.2462 −0.618931
\(690\) 0 0
\(691\) 16.4924 0.627401 0.313701 0.949522i \(-0.398431\pi\)
0.313701 + 0.949522i \(0.398431\pi\)
\(692\) 0 0
\(693\) − 2.87689i − 0.109284i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 10.5616i − 0.400047i
\(698\) 0 0
\(699\) −1.06913 −0.0404382
\(700\) 0 0
\(701\) −1.75379 −0.0662397 −0.0331198 0.999451i \(-0.510544\pi\)
−0.0331198 + 0.999451i \(0.510544\pi\)
\(702\) 0 0
\(703\) − 27.3693i − 1.03225i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 45.3002i 1.70369i
\(708\) 0 0
\(709\) −29.7538 −1.11743 −0.558713 0.829361i \(-0.688705\pi\)
−0.558713 + 0.829361i \(0.688705\pi\)
\(710\) 0 0
\(711\) −2.73863 −0.102707
\(712\) 0 0
\(713\) 7.24621i 0.271373i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.30019i 0.160593i
\(718\) 0 0
\(719\) −19.0540 −0.710593 −0.355297 0.934754i \(-0.615620\pi\)
−0.355297 + 0.934754i \(0.615620\pi\)
\(720\) 0 0
\(721\) −40.9848 −1.52636
\(722\) 0 0
\(723\) − 9.36932i − 0.348449i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 21.1922i 0.785977i 0.919543 + 0.392988i \(0.128559\pi\)
−0.919543 + 0.392988i \(0.871441\pi\)
\(728\) 0 0
\(729\) −29.9848 −1.11055
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0.315342i 0.0116474i 0.999983 + 0.00582370i \(0.00185375\pi\)
−0.999983 + 0.00582370i \(0.998146\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17.1231i 0.630738i
\(738\) 0 0
\(739\) −0.615528 −0.0226426 −0.0113213 0.999936i \(-0.503604\pi\)
−0.0113213 + 0.999936i \(0.503604\pi\)
\(740\) 0 0
\(741\) 33.3693 1.22585
\(742\) 0 0
\(743\) 3.50758i 0.128681i 0.997928 + 0.0643403i \(0.0204943\pi\)
−0.997928 + 0.0643403i \(0.979506\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 7.82292i − 0.286226i
\(748\) 0 0
\(749\) −8.80776 −0.321829
\(750\) 0 0
\(751\) −13.1231 −0.478869 −0.239434 0.970913i \(-0.576962\pi\)
−0.239434 + 0.970913i \(0.576962\pi\)
\(752\) 0 0
\(753\) 11.5076i 0.419359i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 12.5616i − 0.456557i −0.973596 0.228279i \(-0.926690\pi\)
0.973596 0.228279i \(-0.0733097\pi\)
\(758\) 0 0
\(759\) 3.12311 0.113362
\(760\) 0 0
\(761\) −43.9848 −1.59445 −0.797225 0.603683i \(-0.793699\pi\)
−0.797225 + 0.603683i \(0.793699\pi\)
\(762\) 0 0
\(763\) 39.3693i 1.42526i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 13.1231i − 0.473848i
\(768\) 0 0
\(769\) 10.6307 0.383352 0.191676 0.981458i \(-0.438608\pi\)
0.191676 + 0.981458i \(0.438608\pi\)
\(770\) 0 0
\(771\) −49.6695 −1.78880
\(772\) 0 0
\(773\) − 41.1231i − 1.47910i −0.673104 0.739548i \(-0.735039\pi\)
0.673104 0.739548i \(-0.264961\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 18.2462i − 0.654579i
\(778\) 0 0
\(779\) −24.7386 −0.886354
\(780\) 0 0
\(781\) 20.2462 0.724466
\(782\) 0 0
\(783\) 34.0540i 1.21699i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 39.6847i 1.41461i 0.706911 + 0.707303i \(0.250088\pi\)
−0.706911 + 0.707303i \(0.749912\pi\)
\(788\) 0 0
\(789\) 0.107951 0.00384314
\(790\) 0 0
\(791\) 32.8078 1.16651
\(792\) 0 0
\(793\) 25.3693i 0.900891i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 52.4233i 1.85693i 0.371422 + 0.928464i \(0.378870\pi\)
−0.371422 + 0.928464i \(0.621130\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) 0 0
\(803\) − 8.87689i − 0.313259i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 36.3002i 1.27783i
\(808\) 0 0
\(809\) 54.1771 1.90476 0.952382 0.304906i \(-0.0986251\pi\)
0.952382 + 0.304906i \(0.0986251\pi\)
\(810\) 0 0
\(811\) 47.2462 1.65904 0.829519 0.558478i \(-0.188614\pi\)
0.829519 + 0.558478i \(0.188614\pi\)
\(812\) 0 0
\(813\) − 34.2462i − 1.20107i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −5.12311 −0.179016
\(820\) 0 0
\(821\) −34.4924 −1.20379 −0.601897 0.798574i \(-0.705588\pi\)
−0.601897 + 0.798574i \(0.705588\pi\)
\(822\) 0 0
\(823\) − 38.0540i − 1.32648i −0.748408 0.663239i \(-0.769181\pi\)
0.748408 0.663239i \(-0.230819\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 19.6847i − 0.684503i −0.939608 0.342251i \(-0.888811\pi\)
0.939608 0.342251i \(-0.111189\pi\)
\(828\) 0 0
\(829\) 29.5464 1.02619 0.513094 0.858332i \(-0.328499\pi\)
0.513094 + 0.858332i \(0.328499\pi\)
\(830\) 0 0
\(831\) 10.4384 0.362106
\(832\) 0 0
\(833\) − 1.12311i − 0.0389133i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 40.3002i − 1.39298i
\(838\) 0 0
\(839\) 42.8769 1.48027 0.740137 0.672456i \(-0.234760\pi\)
0.740137 + 0.672456i \(0.234760\pi\)
\(840\) 0 0
\(841\) 8.49242 0.292842
\(842\) 0 0
\(843\) 10.7386i 0.369858i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 17.9309i 0.616112i
\(848\) 0 0
\(849\) 23.1231 0.793583
\(850\) 0 0
\(851\) −4.56155 −0.156368
\(852\) 0 0
\(853\) − 20.2462i − 0.693217i −0.938010 0.346609i \(-0.887333\pi\)
0.938010 0.346609i \(-0.112667\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 33.3153i 1.13803i 0.822327 + 0.569015i \(0.192675\pi\)
−0.822327 + 0.569015i \(0.807325\pi\)
\(858\) 0 0
\(859\) 14.5076 0.494992 0.247496 0.968889i \(-0.420392\pi\)
0.247496 + 0.968889i \(0.420392\pi\)
\(860\) 0 0
\(861\) −16.4924 −0.562060
\(862\) 0 0
\(863\) − 7.56155i − 0.257398i −0.991684 0.128699i \(-0.958920\pi\)
0.991684 0.128699i \(-0.0410801\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 16.3002i − 0.553583i
\(868\) 0 0
\(869\) −9.75379 −0.330875
\(870\) 0 0
\(871\) 30.4924 1.03320
\(872\) 0 0
\(873\) 7.36932i 0.249414i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 18.9848i 0.641073i 0.947236 + 0.320536i \(0.103863\pi\)
−0.947236 + 0.320536i \(0.896137\pi\)
\(878\) 0 0
\(879\) −26.2462 −0.885263
\(880\) 0 0
\(881\) 41.4773 1.39740 0.698702 0.715413i \(-0.253761\pi\)
0.698702 + 0.715413i \(0.253761\pi\)
\(882\) 0 0
\(883\) 10.7386i 0.361384i 0.983540 + 0.180692i \(0.0578337\pi\)
−0.983540 + 0.180692i \(0.942166\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 42.7926i − 1.43684i −0.695612 0.718418i \(-0.744867\pi\)
0.695612 0.718418i \(-0.255133\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 0 0
\(891\) −14.0000 −0.469018
\(892\) 0 0
\(893\) 28.1080i 0.940597i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 5.56155i − 0.185695i
\(898\) 0 0
\(899\) −44.3693 −1.47980
\(900\) 0 0
\(901\) −11.6847 −0.389272
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 21.6847i − 0.720027i −0.932947 0.360014i \(-0.882772\pi\)
0.932947 0.360014i \(-0.117228\pi\)
\(908\) 0 0
\(909\) −9.93087 −0.329386
\(910\) 0 0
\(911\) −3.12311 −0.103473 −0.0517366 0.998661i \(-0.516476\pi\)
−0.0517366 + 0.998661i \(0.516476\pi\)
\(912\) 0 0
\(913\) − 27.8617i − 0.922089i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 0.492423i − 0.0162612i
\(918\) 0 0
\(919\) −46.7386 −1.54177 −0.770883 0.636977i \(-0.780185\pi\)
−0.770883 + 0.636977i \(0.780185\pi\)
\(920\) 0 0
\(921\) 32.9848 1.08689
\(922\) 0 0
\(923\) − 36.0540i − 1.18673i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 8.98485i − 0.295101i
\(928\) 0 0
\(929\) 56.7235 1.86104 0.930518 0.366245i \(-0.119357\pi\)
0.930518 + 0.366245i \(0.119357\pi\)
\(930\) 0 0
\(931\) −2.63068 −0.0862172
\(932\) 0 0
\(933\) − 13.5616i − 0.443985i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 16.2462i 0.530741i 0.964147 + 0.265370i \(0.0854942\pi\)
−0.964147 + 0.265370i \(0.914506\pi\)
\(938\) 0 0
\(939\) −1.26137 −0.0411631
\(940\) 0 0
\(941\) −37.7538 −1.23074 −0.615369 0.788239i \(-0.710993\pi\)
−0.615369 + 0.788239i \(0.710993\pi\)
\(942\) 0 0
\(943\) 4.12311i 0.134267i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 56.6847i − 1.84200i −0.389558 0.921002i \(-0.627372\pi\)
0.389558 0.921002i \(-0.372628\pi\)
\(948\) 0 0
\(949\) −15.8078 −0.513142
\(950\) 0 0
\(951\) −23.6155 −0.765786
\(952\) 0 0
\(953\) − 38.4924i − 1.24689i −0.781866 0.623446i \(-0.785732\pi\)
0.781866 0.623446i \(-0.214268\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 19.1231i 0.618162i
\(958\) 0 0
\(959\) −17.6155 −0.568835
\(960\) 0 0
\(961\) 21.5076 0.693793
\(962\) 0 0
\(963\) − 1.93087i − 0.0622214i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 56.6847i 1.82286i 0.411460 + 0.911428i \(0.365019\pi\)
−0.411460 + 0.911428i \(0.634981\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) −22.6307 −0.726253 −0.363127 0.931740i \(-0.618291\pi\)
−0.363127 + 0.931740i \(0.618291\pi\)
\(972\) 0 0
\(973\) − 28.8078i − 0.923535i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 59.7926i − 1.91294i −0.291839 0.956468i \(-0.594267\pi\)
0.291839 0.956468i \(-0.405733\pi\)
\(978\) 0 0
\(979\) −28.4924 −0.910622
\(980\) 0 0
\(981\) −8.63068 −0.275557
\(982\) 0 0
\(983\) − 37.0540i − 1.18184i −0.806731 0.590919i \(-0.798765\pi\)
0.806731 0.590919i \(-0.201235\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 18.7386i 0.596457i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 53.3002 1.69314 0.846568 0.532280i \(-0.178665\pi\)
0.846568 + 0.532280i \(0.178665\pi\)
\(992\) 0 0
\(993\) − 44.6847i − 1.41802i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 55.6155i − 1.76136i −0.473710 0.880681i \(-0.657086\pi\)
0.473710 0.880681i \(-0.342914\pi\)
\(998\) 0 0
\(999\) 25.3693 0.802650
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 2300.2.c.h.1749.2 4
5.2 odd 4 460.2.a.e.1.1 2
5.3 odd 4 2300.2.a.i.1.2 2
5.4 even 2 inner 2300.2.c.h.1749.3 4
15.2 even 4 4140.2.a.m.1.2 2
20.3 even 4 9200.2.a.bv.1.1 2
20.7 even 4 1840.2.a.m.1.2 2
40.27 even 4 7360.2.a.bo.1.1 2
40.37 odd 4 7360.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.a.e.1.1 2 5.2 odd 4
1840.2.a.m.1.2 2 20.7 even 4
2300.2.a.i.1.2 2 5.3 odd 4
2300.2.c.h.1749.2 4 1.1 even 1 trivial
2300.2.c.h.1749.3 4 5.4 even 2 inner
4140.2.a.m.1.2 2 15.2 even 4
7360.2.a.bi.1.2 2 40.37 odd 4
7360.2.a.bo.1.1 2 40.27 even 4
9200.2.a.bv.1.1 2 20.3 even 4