Properties

Label 4600.2.e.v.4049.9
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 18 x^{8} + 103 x^{6} + 239 x^{4} + 197 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.9
Root \(-1.45894i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.v.4049.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.21042i q^{3} +2.22487i q^{7} -1.88594 q^{9} +O(q^{10})\) \(q+2.21042i q^{3} +2.22487i q^{7} -1.88594 q^{9} -1.57300 q^{11} +3.96189i q^{13} -0.294907i q^{17} +7.76572 q^{19} -4.91788 q^{21} -1.00000i q^{23} +2.46253i q^{27} +9.29233 q^{29} +9.18913 q^{31} -3.47698i q^{33} +10.5425i q^{37} -8.75744 q^{39} -2.34251 q^{41} +6.67460i q^{43} +1.38007i q^{47} +2.04998 q^{49} +0.651868 q^{51} -11.0395i q^{53} +17.1655i q^{57} +5.09378 q^{59} -8.91788 q^{61} -4.19597i q^{63} +1.12002i q^{67} +2.21042 q^{69} +7.60168 q^{71} +12.8549i q^{73} -3.49971i q^{77} -11.0211 q^{79} -11.1010 q^{81} -13.5257i q^{83} +20.5399i q^{87} -14.3475 q^{89} -8.81468 q^{91} +20.3118i q^{93} +0.199218i q^{97} +2.96658 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 8q^{9} + O(q^{10}) \) \( 10q - 8q^{9} - 8q^{11} - 8q^{19} - 12q^{21} + 22q^{29} + 8q^{31} - 62q^{39} - 16q^{41} + 4q^{49} - 10q^{51} - 46q^{59} - 52q^{61} - 6q^{69} - 4q^{71} - 86q^{79} - 6q^{81} - 30q^{89} - 38q^{91} - 74q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.21042i 1.27618i 0.769960 + 0.638092i \(0.220276\pi\)
−0.769960 + 0.638092i \(0.779724\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.22487i 0.840920i 0.907311 + 0.420460i \(0.138131\pi\)
−0.907311 + 0.420460i \(0.861869\pi\)
\(8\) 0 0
\(9\) −1.88594 −0.628648
\(10\) 0 0
\(11\) −1.57300 −0.474276 −0.237138 0.971476i \(-0.576209\pi\)
−0.237138 + 0.971476i \(0.576209\pi\)
\(12\) 0 0
\(13\) 3.96189i 1.09883i 0.835549 + 0.549416i \(0.185150\pi\)
−0.835549 + 0.549416i \(0.814850\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.294907i − 0.0715256i −0.999360 0.0357628i \(-0.988614\pi\)
0.999360 0.0357628i \(-0.0113861\pi\)
\(18\) 0 0
\(19\) 7.76572 1.78158 0.890789 0.454418i \(-0.150153\pi\)
0.890789 + 0.454418i \(0.150153\pi\)
\(20\) 0 0
\(21\) −4.91788 −1.07317
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.46253i 0.473914i
\(28\) 0 0
\(29\) 9.29233 1.72554 0.862771 0.505595i \(-0.168727\pi\)
0.862771 + 0.505595i \(0.168727\pi\)
\(30\) 0 0
\(31\) 9.18913 1.65042 0.825208 0.564829i \(-0.191058\pi\)
0.825208 + 0.564829i \(0.191058\pi\)
\(32\) 0 0
\(33\) − 3.47698i − 0.605264i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.5425i 1.73318i 0.499024 + 0.866588i \(0.333692\pi\)
−0.499024 + 0.866588i \(0.666308\pi\)
\(38\) 0 0
\(39\) −8.75744 −1.40231
\(40\) 0 0
\(41\) −2.34251 −0.365839 −0.182920 0.983128i \(-0.558555\pi\)
−0.182920 + 0.983128i \(0.558555\pi\)
\(42\) 0 0
\(43\) 6.67460i 1.01787i 0.860806 + 0.508933i \(0.169960\pi\)
−0.860806 + 0.508933i \(0.830040\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.38007i 0.201304i 0.994922 + 0.100652i \(0.0320928\pi\)
−0.994922 + 0.100652i \(0.967907\pi\)
\(48\) 0 0
\(49\) 2.04998 0.292854
\(50\) 0 0
\(51\) 0.651868 0.0912798
\(52\) 0 0
\(53\) − 11.0395i − 1.51640i −0.652023 0.758199i \(-0.726080\pi\)
0.652023 0.758199i \(-0.273920\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 17.1655i 2.27362i
\(58\) 0 0
\(59\) 5.09378 0.663154 0.331577 0.943428i \(-0.392419\pi\)
0.331577 + 0.943428i \(0.392419\pi\)
\(60\) 0 0
\(61\) −8.91788 −1.14182 −0.570909 0.821014i \(-0.693409\pi\)
−0.570909 + 0.821014i \(0.693409\pi\)
\(62\) 0 0
\(63\) − 4.19597i − 0.528642i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.12002i 0.136832i 0.997657 + 0.0684160i \(0.0217945\pi\)
−0.997657 + 0.0684160i \(0.978205\pi\)
\(68\) 0 0
\(69\) 2.21042 0.266103
\(70\) 0 0
\(71\) 7.60168 0.902154 0.451077 0.892485i \(-0.351040\pi\)
0.451077 + 0.892485i \(0.351040\pi\)
\(72\) 0 0
\(73\) 12.8549i 1.50455i 0.658848 + 0.752276i \(0.271044\pi\)
−0.658848 + 0.752276i \(0.728956\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.49971i − 0.398828i
\(78\) 0 0
\(79\) −11.0211 −1.23997 −0.619984 0.784614i \(-0.712861\pi\)
−0.619984 + 0.784614i \(0.712861\pi\)
\(80\) 0 0
\(81\) −11.1010 −1.23345
\(82\) 0 0
\(83\) − 13.5257i − 1.48464i −0.670048 0.742318i \(-0.733726\pi\)
0.670048 0.742318i \(-0.266274\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 20.5399i 2.20211i
\(88\) 0 0
\(89\) −14.3475 −1.52084 −0.760418 0.649434i \(-0.775006\pi\)
−0.760418 + 0.649434i \(0.775006\pi\)
\(90\) 0 0
\(91\) −8.81468 −0.924030
\(92\) 0 0
\(93\) 20.3118i 2.10624i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.199218i 0.0202275i 0.999949 + 0.0101137i \(0.00321936\pi\)
−0.999949 + 0.0101137i \(0.996781\pi\)
\(98\) 0 0
\(99\) 2.96658 0.298153
\(100\) 0 0
\(101\) −6.07595 −0.604580 −0.302290 0.953216i \(-0.597751\pi\)
−0.302290 + 0.953216i \(0.597751\pi\)
\(102\) 0 0
\(103\) − 11.9423i − 1.17671i −0.808604 0.588353i \(-0.799777\pi\)
0.808604 0.588353i \(-0.200223\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.99058i − 0.772479i −0.922399 0.386239i \(-0.873774\pi\)
0.922399 0.386239i \(-0.126226\pi\)
\(108\) 0 0
\(109\) 4.36170 0.417775 0.208888 0.977940i \(-0.433016\pi\)
0.208888 + 0.977940i \(0.433016\pi\)
\(110\) 0 0
\(111\) −23.3033 −2.21185
\(112\) 0 0
\(113\) 12.9156i 1.21500i 0.794319 + 0.607501i \(0.207828\pi\)
−0.794319 + 0.607501i \(0.792172\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 7.47191i − 0.690778i
\(118\) 0 0
\(119\) 0.656129 0.0601473
\(120\) 0 0
\(121\) −8.52568 −0.775062
\(122\) 0 0
\(123\) − 5.17793i − 0.466878i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.3759i − 1.45313i −0.687098 0.726565i \(-0.741116\pi\)
0.687098 0.726565i \(-0.258884\pi\)
\(128\) 0 0
\(129\) −14.7536 −1.29899
\(130\) 0 0
\(131\) 9.62242 0.840715 0.420357 0.907359i \(-0.361905\pi\)
0.420357 + 0.907359i \(0.361905\pi\)
\(132\) 0 0
\(133\) 17.2777i 1.49816i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.78721i 0.579871i 0.957046 + 0.289935i \(0.0936338\pi\)
−0.957046 + 0.289935i \(0.906366\pi\)
\(138\) 0 0
\(139\) −6.55788 −0.556232 −0.278116 0.960547i \(-0.589710\pi\)
−0.278116 + 0.960547i \(0.589710\pi\)
\(140\) 0 0
\(141\) −3.05053 −0.256901
\(142\) 0 0
\(143\) − 6.23205i − 0.521150i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.53130i 0.373735i
\(148\) 0 0
\(149\) 13.8647 1.13584 0.567918 0.823085i \(-0.307749\pi\)
0.567918 + 0.823085i \(0.307749\pi\)
\(150\) 0 0
\(151\) 2.42205 0.197104 0.0985520 0.995132i \(-0.468579\pi\)
0.0985520 + 0.995132i \(0.468579\pi\)
\(152\) 0 0
\(153\) 0.556179i 0.0449644i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 20.2638i 1.61723i 0.588341 + 0.808613i \(0.299781\pi\)
−0.588341 + 0.808613i \(0.700219\pi\)
\(158\) 0 0
\(159\) 24.4020 1.93520
\(160\) 0 0
\(161\) 2.22487 0.175344
\(162\) 0 0
\(163\) − 6.91982i − 0.542002i −0.962579 0.271001i \(-0.912645\pi\)
0.962579 0.271001i \(-0.0873547\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.8296i 1.45708i 0.685005 + 0.728539i \(0.259800\pi\)
−0.685005 + 0.728539i \(0.740200\pi\)
\(168\) 0 0
\(169\) −2.69661 −0.207431
\(170\) 0 0
\(171\) −14.6457 −1.11998
\(172\) 0 0
\(173\) − 23.2017i − 1.76399i −0.471255 0.881997i \(-0.656199\pi\)
0.471255 0.881997i \(-0.343801\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 11.2594i 0.846307i
\(178\) 0 0
\(179\) 10.7011 0.799837 0.399918 0.916551i \(-0.369038\pi\)
0.399918 + 0.916551i \(0.369038\pi\)
\(180\) 0 0
\(181\) −7.27484 −0.540735 −0.270367 0.962757i \(-0.587145\pi\)
−0.270367 + 0.962757i \(0.587145\pi\)
\(182\) 0 0
\(183\) − 19.7122i − 1.45717i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.463888i 0.0339229i
\(188\) 0 0
\(189\) −5.47880 −0.398524
\(190\) 0 0
\(191\) −3.74624 −0.271068 −0.135534 0.990773i \(-0.543275\pi\)
−0.135534 + 0.990773i \(0.543275\pi\)
\(192\) 0 0
\(193\) − 11.8470i − 0.852763i −0.904543 0.426381i \(-0.859788\pi\)
0.904543 0.426381i \(-0.140212\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 10.0064i − 0.712924i −0.934310 0.356462i \(-0.883983\pi\)
0.934310 0.356462i \(-0.116017\pi\)
\(198\) 0 0
\(199\) −6.19597 −0.439221 −0.219610 0.975588i \(-0.570479\pi\)
−0.219610 + 0.975588i \(0.570479\pi\)
\(200\) 0 0
\(201\) −2.47571 −0.174623
\(202\) 0 0
\(203\) 20.6742i 1.45104i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.88594i 0.131082i
\(208\) 0 0
\(209\) −12.2154 −0.844960
\(210\) 0 0
\(211\) −0.825746 −0.0568467 −0.0284234 0.999596i \(-0.509049\pi\)
−0.0284234 + 0.999596i \(0.509049\pi\)
\(212\) 0 0
\(213\) 16.8029i 1.15132i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 20.4446i 1.38787i
\(218\) 0 0
\(219\) −28.4147 −1.92009
\(220\) 0 0
\(221\) 1.16839 0.0785946
\(222\) 0 0
\(223\) − 18.2911i − 1.22486i −0.790525 0.612430i \(-0.790192\pi\)
0.790525 0.612430i \(-0.209808\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.5556i 1.43069i 0.698769 + 0.715347i \(0.253732\pi\)
−0.698769 + 0.715347i \(0.746268\pi\)
\(228\) 0 0
\(229\) −9.65960 −0.638324 −0.319162 0.947700i \(-0.603401\pi\)
−0.319162 + 0.947700i \(0.603401\pi\)
\(230\) 0 0
\(231\) 7.73581 0.508979
\(232\) 0 0
\(233\) − 0.879774i − 0.0576359i −0.999585 0.0288179i \(-0.990826\pi\)
0.999585 0.0288179i \(-0.00917431\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 24.3612i − 1.58243i
\(238\) 0 0
\(239\) −22.3240 −1.44402 −0.722010 0.691883i \(-0.756782\pi\)
−0.722010 + 0.691883i \(0.756782\pi\)
\(240\) 0 0
\(241\) −7.43761 −0.479098 −0.239549 0.970884i \(-0.577000\pi\)
−0.239549 + 0.970884i \(0.577000\pi\)
\(242\) 0 0
\(243\) − 17.1504i − 1.10020i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 30.7669i 1.95765i
\(248\) 0 0
\(249\) 29.8974 1.89467
\(250\) 0 0
\(251\) 21.4221 1.35215 0.676074 0.736833i \(-0.263680\pi\)
0.676074 + 0.736833i \(0.263680\pi\)
\(252\) 0 0
\(253\) 1.57300i 0.0988935i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 10.9788i − 0.684838i −0.939547 0.342419i \(-0.888754\pi\)
0.939547 0.342419i \(-0.111246\pi\)
\(258\) 0 0
\(259\) −23.4556 −1.45746
\(260\) 0 0
\(261\) −17.5248 −1.08476
\(262\) 0 0
\(263\) − 11.3268i − 0.698440i −0.937041 0.349220i \(-0.886447\pi\)
0.937041 0.349220i \(-0.113553\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 31.7141i − 1.94087i
\(268\) 0 0
\(269\) −31.1010 −1.89626 −0.948130 0.317883i \(-0.897028\pi\)
−0.948130 + 0.317883i \(0.897028\pi\)
\(270\) 0 0
\(271\) 29.1151 1.76862 0.884308 0.466905i \(-0.154631\pi\)
0.884308 + 0.466905i \(0.154631\pi\)
\(272\) 0 0
\(273\) − 19.4841i − 1.17923i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.6829i 1.36288i 0.731873 + 0.681441i \(0.238646\pi\)
−0.731873 + 0.681441i \(0.761354\pi\)
\(278\) 0 0
\(279\) −17.3302 −1.03753
\(280\) 0 0
\(281\) 11.2531 0.671305 0.335653 0.941986i \(-0.391043\pi\)
0.335653 + 0.941986i \(0.391043\pi\)
\(282\) 0 0
\(283\) 0.419014i 0.0249078i 0.999922 + 0.0124539i \(0.00396431\pi\)
−0.999922 + 0.0124539i \(0.996036\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 5.21177i − 0.307641i
\(288\) 0 0
\(289\) 16.9130 0.994884
\(290\) 0 0
\(291\) −0.440354 −0.0258140
\(292\) 0 0
\(293\) 1.06844i 0.0624190i 0.999513 + 0.0312095i \(0.00993591\pi\)
−0.999513 + 0.0312095i \(0.990064\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 3.87355i − 0.224766i
\(298\) 0 0
\(299\) 3.96189 0.229122
\(300\) 0 0
\(301\) −14.8501 −0.855944
\(302\) 0 0
\(303\) − 13.4304i − 0.771555i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 22.3991i − 1.27838i −0.769048 0.639191i \(-0.779269\pi\)
0.769048 0.639191i \(-0.220731\pi\)
\(308\) 0 0
\(309\) 26.3974 1.50169
\(310\) 0 0
\(311\) −19.9717 −1.13249 −0.566245 0.824237i \(-0.691604\pi\)
−0.566245 + 0.824237i \(0.691604\pi\)
\(312\) 0 0
\(313\) 14.1975i 0.802488i 0.915971 + 0.401244i \(0.131422\pi\)
−0.915971 + 0.401244i \(0.868578\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.48205i 0.420234i 0.977676 + 0.210117i \(0.0673844\pi\)
−0.977676 + 0.210117i \(0.932616\pi\)
\(318\) 0 0
\(319\) −14.6168 −0.818384
\(320\) 0 0
\(321\) 17.6625 0.985826
\(322\) 0 0
\(323\) − 2.29017i − 0.127428i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 9.64118i 0.533158i
\(328\) 0 0
\(329\) −3.07047 −0.169280
\(330\) 0 0
\(331\) 6.56494 0.360842 0.180421 0.983590i \(-0.442254\pi\)
0.180421 + 0.983590i \(0.442254\pi\)
\(332\) 0 0
\(333\) − 19.8826i − 1.08956i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 17.2484i − 0.939583i −0.882777 0.469791i \(-0.844329\pi\)
0.882777 0.469791i \(-0.155671\pi\)
\(338\) 0 0
\(339\) −28.5490 −1.55057
\(340\) 0 0
\(341\) −14.4545 −0.782753
\(342\) 0 0
\(343\) 20.1350i 1.08719i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 4.18016i − 0.224403i −0.993685 0.112201i \(-0.964210\pi\)
0.993685 0.112201i \(-0.0357902\pi\)
\(348\) 0 0
\(349\) −11.6150 −0.621736 −0.310868 0.950453i \(-0.600620\pi\)
−0.310868 + 0.950453i \(0.600620\pi\)
\(350\) 0 0
\(351\) −9.75628 −0.520752
\(352\) 0 0
\(353\) 15.8241i 0.842233i 0.907006 + 0.421117i \(0.138362\pi\)
−0.907006 + 0.421117i \(0.861638\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.45032i 0.0767590i
\(358\) 0 0
\(359\) 6.25850 0.330311 0.165156 0.986268i \(-0.447187\pi\)
0.165156 + 0.986268i \(0.447187\pi\)
\(360\) 0 0
\(361\) 41.3064 2.17402
\(362\) 0 0
\(363\) − 18.8453i − 0.989122i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5.01500i 0.261781i 0.991397 + 0.130890i \(0.0417836\pi\)
−0.991397 + 0.130890i \(0.958216\pi\)
\(368\) 0 0
\(369\) 4.41785 0.229984
\(370\) 0 0
\(371\) 24.5615 1.27517
\(372\) 0 0
\(373\) − 20.1804i − 1.04490i −0.852670 0.522450i \(-0.825018\pi\)
0.852670 0.522450i \(-0.174982\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 36.8152i 1.89608i
\(378\) 0 0
\(379\) −10.9860 −0.564314 −0.282157 0.959368i \(-0.591050\pi\)
−0.282157 + 0.959368i \(0.591050\pi\)
\(380\) 0 0
\(381\) 36.1977 1.85446
\(382\) 0 0
\(383\) − 23.9968i − 1.22618i −0.790014 0.613088i \(-0.789927\pi\)
0.790014 0.613088i \(-0.210073\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 12.5879i − 0.639879i
\(388\) 0 0
\(389\) −19.3913 −0.983180 −0.491590 0.870827i \(-0.663584\pi\)
−0.491590 + 0.870827i \(0.663584\pi\)
\(390\) 0 0
\(391\) −0.294907 −0.0149141
\(392\) 0 0
\(393\) 21.2696i 1.07291i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 10.0550i 0.504644i 0.967643 + 0.252322i \(0.0811942\pi\)
−0.967643 + 0.252322i \(0.918806\pi\)
\(398\) 0 0
\(399\) −38.1909 −1.91193
\(400\) 0 0
\(401\) −16.6178 −0.829854 −0.414927 0.909855i \(-0.636193\pi\)
−0.414927 + 0.909855i \(0.636193\pi\)
\(402\) 0 0
\(403\) 36.4064i 1.81353i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 16.5833i − 0.822005i
\(408\) 0 0
\(409\) −7.92642 −0.391936 −0.195968 0.980610i \(-0.562785\pi\)
−0.195968 + 0.980610i \(0.562785\pi\)
\(410\) 0 0
\(411\) −15.0026 −0.740022
\(412\) 0 0
\(413\) 11.3330i 0.557659i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 14.4956i − 0.709855i
\(418\) 0 0
\(419\) −40.4944 −1.97828 −0.989141 0.146967i \(-0.953049\pi\)
−0.989141 + 0.146967i \(0.953049\pi\)
\(420\) 0 0
\(421\) 7.67870 0.374237 0.187118 0.982337i \(-0.440085\pi\)
0.187118 + 0.982337i \(0.440085\pi\)
\(422\) 0 0
\(423\) − 2.60273i − 0.126549i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 19.8411i − 0.960177i
\(428\) 0 0
\(429\) 13.7754 0.665084
\(430\) 0 0
\(431\) −36.1674 −1.74212 −0.871061 0.491175i \(-0.836567\pi\)
−0.871061 + 0.491175i \(0.836567\pi\)
\(432\) 0 0
\(433\) − 9.42856i − 0.453108i −0.973999 0.226554i \(-0.927254\pi\)
0.973999 0.226554i \(-0.0727459\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 7.76572i − 0.371485i
\(438\) 0 0
\(439\) −24.4443 −1.16666 −0.583332 0.812234i \(-0.698251\pi\)
−0.583332 + 0.812234i \(0.698251\pi\)
\(440\) 0 0
\(441\) −3.86614 −0.184102
\(442\) 0 0
\(443\) − 35.5786i − 1.69039i −0.534458 0.845195i \(-0.679484\pi\)
0.534458 0.845195i \(-0.320516\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 30.6467i 1.44954i
\(448\) 0 0
\(449\) 0.660651 0.0311780 0.0155890 0.999878i \(-0.495038\pi\)
0.0155890 + 0.999878i \(0.495038\pi\)
\(450\) 0 0
\(451\) 3.68476 0.173509
\(452\) 0 0
\(453\) 5.35375i 0.251541i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.62187i 0.403314i 0.979456 + 0.201657i \(0.0646326\pi\)
−0.979456 + 0.201657i \(0.935367\pi\)
\(458\) 0 0
\(459\) 0.726219 0.0338970
\(460\) 0 0
\(461\) −5.54580 −0.258294 −0.129147 0.991625i \(-0.541224\pi\)
−0.129147 + 0.991625i \(0.541224\pi\)
\(462\) 0 0
\(463\) 6.15748i 0.286162i 0.989711 + 0.143081i \(0.0457010\pi\)
−0.989711 + 0.143081i \(0.954299\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 2.93879i − 0.135991i −0.997686 0.0679954i \(-0.978340\pi\)
0.997686 0.0679954i \(-0.0216603\pi\)
\(468\) 0 0
\(469\) −2.49189 −0.115065
\(470\) 0 0
\(471\) −44.7914 −2.06388
\(472\) 0 0
\(473\) − 10.4991i − 0.482750i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 20.8200i 0.953280i
\(478\) 0 0
\(479\) −13.4939 −0.616550 −0.308275 0.951297i \(-0.599752\pi\)
−0.308275 + 0.951297i \(0.599752\pi\)
\(480\) 0 0
\(481\) −41.7683 −1.90447
\(482\) 0 0
\(483\) 4.91788i 0.223771i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.28554i 0.148882i 0.997225 + 0.0744410i \(0.0237173\pi\)
−0.997225 + 0.0744410i \(0.976283\pi\)
\(488\) 0 0
\(489\) 15.2957 0.691695
\(490\) 0 0
\(491\) 24.2119 1.09267 0.546335 0.837567i \(-0.316023\pi\)
0.546335 + 0.837567i \(0.316023\pi\)
\(492\) 0 0
\(493\) − 2.74038i − 0.123420i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.9127i 0.758639i
\(498\) 0 0
\(499\) 37.5956 1.68301 0.841506 0.540248i \(-0.181669\pi\)
0.841506 + 0.540248i \(0.181669\pi\)
\(500\) 0 0
\(501\) −41.6212 −1.85950
\(502\) 0 0
\(503\) 11.0448i 0.492464i 0.969211 + 0.246232i \(0.0791925\pi\)
−0.969211 + 0.246232i \(0.920807\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 5.96062i − 0.264720i
\(508\) 0 0
\(509\) 31.4409 1.39359 0.696796 0.717269i \(-0.254608\pi\)
0.696796 + 0.717269i \(0.254608\pi\)
\(510\) 0 0
\(511\) −28.6004 −1.26521
\(512\) 0 0
\(513\) 19.1233i 0.844315i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.17084i − 0.0954735i
\(518\) 0 0
\(519\) 51.2855 2.25118
\(520\) 0 0
\(521\) 35.3591 1.54911 0.774555 0.632506i \(-0.217974\pi\)
0.774555 + 0.632506i \(0.217974\pi\)
\(522\) 0 0
\(523\) 41.8119i 1.82830i 0.405371 + 0.914152i \(0.367142\pi\)
−0.405371 + 0.914152i \(0.632858\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 2.70994i − 0.118047i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −9.60658 −0.416890
\(532\) 0 0
\(533\) − 9.28079i − 0.401996i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 23.6539i 1.02074i
\(538\) 0 0
\(539\) −3.22460 −0.138894
\(540\) 0 0
\(541\) 1.43188 0.0615612 0.0307806 0.999526i \(-0.490201\pi\)
0.0307806 + 0.999526i \(0.490201\pi\)
\(542\) 0 0
\(543\) − 16.0804i − 0.690077i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 44.8700i − 1.91850i −0.282555 0.959251i \(-0.591182\pi\)
0.282555 0.959251i \(-0.408818\pi\)
\(548\) 0 0
\(549\) 16.8186 0.717801
\(550\) 0 0
\(551\) 72.1616 3.07419
\(552\) 0 0
\(553\) − 24.5204i − 1.04271i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 22.3106i 0.945329i 0.881242 + 0.472665i \(0.156708\pi\)
−0.881242 + 0.472665i \(0.843292\pi\)
\(558\) 0 0
\(559\) −26.4440 −1.11846
\(560\) 0 0
\(561\) −1.02539 −0.0432919
\(562\) 0 0
\(563\) − 1.41928i − 0.0598153i −0.999553 0.0299077i \(-0.990479\pi\)
0.999553 0.0299077i \(-0.00952132\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 24.6983i − 1.03723i
\(568\) 0 0
\(569\) 21.4950 0.901116 0.450558 0.892747i \(-0.351225\pi\)
0.450558 + 0.892747i \(0.351225\pi\)
\(570\) 0 0
\(571\) 41.0272 1.71694 0.858468 0.512868i \(-0.171417\pi\)
0.858468 + 0.512868i \(0.171417\pi\)
\(572\) 0 0
\(573\) − 8.28075i − 0.345933i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 37.1477i − 1.54648i −0.634114 0.773239i \(-0.718635\pi\)
0.634114 0.773239i \(-0.281365\pi\)
\(578\) 0 0
\(579\) 26.1867 1.08828
\(580\) 0 0
\(581\) 30.0928 1.24846
\(582\) 0 0
\(583\) 17.3652i 0.719192i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 12.1838i − 0.502881i −0.967873 0.251440i \(-0.919096\pi\)
0.967873 0.251440i \(-0.0809043\pi\)
\(588\) 0 0
\(589\) 71.3602 2.94034
\(590\) 0 0
\(591\) 22.1183 0.909823
\(592\) 0 0
\(593\) 40.1421i 1.64844i 0.566270 + 0.824220i \(0.308386\pi\)
−0.566270 + 0.824220i \(0.691614\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 13.6957i − 0.560527i
\(598\) 0 0
\(599\) 16.4377 0.671624 0.335812 0.941929i \(-0.390989\pi\)
0.335812 + 0.941929i \(0.390989\pi\)
\(600\) 0 0
\(601\) −4.98809 −0.203469 −0.101734 0.994812i \(-0.532439\pi\)
−0.101734 + 0.994812i \(0.532439\pi\)
\(602\) 0 0
\(603\) − 2.11229i − 0.0860191i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.09352i 0.0443846i 0.999754 + 0.0221923i \(0.00706461\pi\)
−0.999754 + 0.0221923i \(0.992935\pi\)
\(608\) 0 0
\(609\) −45.6986 −1.85180
\(610\) 0 0
\(611\) −5.46768 −0.221199
\(612\) 0 0
\(613\) − 16.4617i − 0.664880i −0.943124 0.332440i \(-0.892128\pi\)
0.943124 0.332440i \(-0.107872\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.9090i 0.962540i 0.876572 + 0.481270i \(0.159824\pi\)
−0.876572 + 0.481270i \(0.840176\pi\)
\(618\) 0 0
\(619\) 26.7498 1.07517 0.537583 0.843211i \(-0.319337\pi\)
0.537583 + 0.843211i \(0.319337\pi\)
\(620\) 0 0
\(621\) 2.46253 0.0988179
\(622\) 0 0
\(623\) − 31.9213i − 1.27890i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 27.0012i − 1.07833i
\(628\) 0 0
\(629\) 3.10906 0.123966
\(630\) 0 0
\(631\) 0.317634 0.0126448 0.00632241 0.999980i \(-0.497988\pi\)
0.00632241 + 0.999980i \(0.497988\pi\)
\(632\) 0 0
\(633\) − 1.82524i − 0.0725469i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8.12179i 0.321797i
\(638\) 0 0
\(639\) −14.3363 −0.567137
\(640\) 0 0
\(641\) 19.9895 0.789539 0.394769 0.918780i \(-0.370825\pi\)
0.394769 + 0.918780i \(0.370825\pi\)
\(642\) 0 0
\(643\) 20.0837i 0.792023i 0.918246 + 0.396011i \(0.129606\pi\)
−0.918246 + 0.396011i \(0.870394\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22.0801i 0.868060i 0.900898 + 0.434030i \(0.142909\pi\)
−0.900898 + 0.434030i \(0.857091\pi\)
\(648\) 0 0
\(649\) −8.01250 −0.314518
\(650\) 0 0
\(651\) −45.1910 −1.77118
\(652\) 0 0
\(653\) − 33.2385i − 1.30072i −0.759625 0.650362i \(-0.774617\pi\)
0.759625 0.650362i \(-0.225383\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 24.2436i − 0.945833i
\(658\) 0 0
\(659\) −17.3023 −0.674002 −0.337001 0.941504i \(-0.609413\pi\)
−0.337001 + 0.941504i \(0.609413\pi\)
\(660\) 0 0
\(661\) −28.9521 −1.12611 −0.563053 0.826421i \(-0.690373\pi\)
−0.563053 + 0.826421i \(0.690373\pi\)
\(662\) 0 0
\(663\) 2.58263i 0.100301i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 9.29233i − 0.359800i
\(668\) 0 0
\(669\) 40.4309 1.56315
\(670\) 0 0
\(671\) 14.0278 0.541537
\(672\) 0 0
\(673\) 10.7860i 0.415771i 0.978153 + 0.207885i \(0.0666581\pi\)
−0.978153 + 0.207885i \(0.933342\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 16.1146i 0.619336i 0.950845 + 0.309668i \(0.100218\pi\)
−0.950845 + 0.309668i \(0.899782\pi\)
\(678\) 0 0
\(679\) −0.443232 −0.0170097
\(680\) 0 0
\(681\) −47.6468 −1.82583
\(682\) 0 0
\(683\) 12.1106i 0.463399i 0.972787 + 0.231699i \(0.0744286\pi\)
−0.972787 + 0.231699i \(0.925571\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 21.3517i − 0.814620i
\(688\) 0 0
\(689\) 43.7375 1.66627
\(690\) 0 0
\(691\) −30.1463 −1.14682 −0.573410 0.819269i \(-0.694380\pi\)
−0.573410 + 0.819269i \(0.694380\pi\)
\(692\) 0 0
\(693\) 6.60025i 0.250723i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.690824i 0.0261668i
\(698\) 0 0
\(699\) 1.94467 0.0735540
\(700\) 0 0
\(701\) 14.8464 0.560741 0.280371 0.959892i \(-0.409543\pi\)
0.280371 + 0.959892i \(0.409543\pi\)
\(702\) 0 0
\(703\) 81.8701i 3.08779i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 13.5182i − 0.508403i
\(708\) 0 0
\(709\) −37.0163 −1.39018 −0.695088 0.718924i \(-0.744635\pi\)
−0.695088 + 0.718924i \(0.744635\pi\)
\(710\) 0 0
\(711\) 20.7851 0.779503
\(712\) 0 0
\(713\) − 9.18913i − 0.344136i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 49.3454i − 1.84284i
\(718\) 0 0
\(719\) −7.63459 −0.284722 −0.142361 0.989815i \(-0.545469\pi\)
−0.142361 + 0.989815i \(0.545469\pi\)
\(720\) 0 0
\(721\) 26.5699 0.989515
\(722\) 0 0
\(723\) − 16.4402i − 0.611418i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 29.1278i 1.08029i 0.841571 + 0.540146i \(0.181631\pi\)
−0.841571 + 0.540146i \(0.818369\pi\)
\(728\) 0 0
\(729\) 4.60629 0.170603
\(730\) 0 0
\(731\) 1.96839 0.0728034
\(732\) 0 0
\(733\) − 7.30466i − 0.269804i −0.990859 0.134902i \(-0.956928\pi\)
0.990859 0.134902i \(-0.0430719\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1.76178i − 0.0648962i
\(738\) 0 0
\(739\) 10.5109 0.386649 0.193324 0.981135i \(-0.438073\pi\)
0.193324 + 0.981135i \(0.438073\pi\)
\(740\) 0 0
\(741\) −68.0078 −2.49833
\(742\) 0 0
\(743\) 11.4448i 0.419869i 0.977715 + 0.209934i \(0.0673250\pi\)
−0.977715 + 0.209934i \(0.932675\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 25.5087i 0.933313i
\(748\) 0 0
\(749\) 17.7780 0.649593
\(750\) 0 0
\(751\) −7.62069 −0.278083 −0.139041 0.990287i \(-0.544402\pi\)
−0.139041 + 0.990287i \(0.544402\pi\)
\(752\) 0 0
\(753\) 47.3517i 1.72559i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 41.3725i − 1.50371i −0.659329 0.751855i \(-0.729159\pi\)
0.659329 0.751855i \(-0.270841\pi\)
\(758\) 0 0
\(759\) −3.47698 −0.126206
\(760\) 0 0
\(761\) 12.5358 0.454421 0.227211 0.973846i \(-0.427039\pi\)
0.227211 + 0.973846i \(0.427039\pi\)
\(762\) 0 0
\(763\) 9.70420i 0.351315i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.1810i 0.728694i
\(768\) 0 0
\(769\) 25.0182 0.902180 0.451090 0.892479i \(-0.351035\pi\)
0.451090 + 0.892479i \(0.351035\pi\)
\(770\) 0 0
\(771\) 24.2677 0.873980
\(772\) 0 0
\(773\) − 37.1948i − 1.33781i −0.743350 0.668903i \(-0.766764\pi\)
0.743350 0.668903i \(-0.233236\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 51.8467i − 1.85999i
\(778\) 0 0
\(779\) −18.1913 −0.651771
\(780\) 0 0
\(781\) −11.9574 −0.427870
\(782\) 0 0
\(783\) 22.8826i 0.817759i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 7.64645i 0.272567i 0.990670 + 0.136283i \(0.0435157\pi\)
−0.990670 + 0.136283i \(0.956484\pi\)
\(788\) 0 0
\(789\) 25.0369 0.891338
\(790\) 0 0
\(791\) −28.7356 −1.02172
\(792\) 0 0
\(793\) − 35.3317i − 1.25467i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.38544i 0.0844966i 0.999107 + 0.0422483i \(0.0134521\pi\)
−0.999107 + 0.0422483i \(0.986548\pi\)
\(798\) 0 0
\(799\) 0.406992 0.0143984
\(800\) 0 0
\(801\) 27.0586 0.956070
\(802\) 0 0
\(803\) − 20.2207i − 0.713573i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 68.7461i − 2.41998i
\(808\) 0 0
\(809\) −15.4937 −0.544730 −0.272365 0.962194i \(-0.587806\pi\)
−0.272365 + 0.962194i \(0.587806\pi\)
\(810\) 0 0
\(811\) 55.3850 1.94483 0.972415 0.233259i \(-0.0749389\pi\)
0.972415 + 0.233259i \(0.0749389\pi\)
\(812\) 0 0
\(813\) 64.3565i 2.25708i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 51.8330i 1.81341i
\(818\) 0 0
\(819\) 16.6240 0.580889
\(820\) 0 0
\(821\) −38.7241 −1.35148 −0.675740 0.737140i \(-0.736176\pi\)
−0.675740 + 0.737140i \(0.736176\pi\)
\(822\) 0 0
\(823\) − 0.237837i − 0.00829047i −0.999991 0.00414524i \(-0.998681\pi\)
0.999991 0.00414524i \(-0.00131947\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17.5821i 0.611391i 0.952129 + 0.305695i \(0.0988889\pi\)
−0.952129 + 0.305695i \(0.901111\pi\)
\(828\) 0 0
\(829\) −20.5607 −0.714103 −0.357051 0.934085i \(-0.616218\pi\)
−0.357051 + 0.934085i \(0.616218\pi\)
\(830\) 0 0
\(831\) −50.1386 −1.73929
\(832\) 0 0
\(833\) − 0.604553i − 0.0209465i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 22.6285i 0.782156i
\(838\) 0 0
\(839\) 15.1584 0.523325 0.261663 0.965159i \(-0.415729\pi\)
0.261663 + 0.965159i \(0.415729\pi\)
\(840\) 0 0
\(841\) 57.3474 1.97750
\(842\) 0 0
\(843\) 24.8741i 0.856709i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 18.9685i − 0.651765i
\(848\) 0 0
\(849\) −0.926197 −0.0317870
\(850\) 0 0
\(851\) 10.5425 0.361392
\(852\) 0 0
\(853\) 18.7788i 0.642973i 0.946914 + 0.321486i \(0.104182\pi\)
−0.946914 + 0.321486i \(0.895818\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 29.4670i 1.00657i 0.864120 + 0.503286i \(0.167876\pi\)
−0.864120 + 0.503286i \(0.832124\pi\)
\(858\) 0 0
\(859\) 50.7562 1.73178 0.865889 0.500236i \(-0.166753\pi\)
0.865889 + 0.500236i \(0.166753\pi\)
\(860\) 0 0
\(861\) 11.5202 0.392607
\(862\) 0 0
\(863\) 19.5166i 0.664351i 0.943218 + 0.332176i \(0.107783\pi\)
−0.943218 + 0.332176i \(0.892217\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 37.3848i 1.26966i
\(868\) 0 0
\(869\) 17.3361 0.588088
\(870\) 0 0
\(871\) −4.43739 −0.150355
\(872\) 0 0
\(873\) − 0.375713i − 0.0127160i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 6.68909i 0.225874i 0.993602 + 0.112937i \(0.0360259\pi\)
−0.993602 + 0.112937i \(0.963974\pi\)
\(878\) 0 0
\(879\) −2.36170 −0.0796582
\(880\) 0 0
\(881\) 47.1211 1.58755 0.793775 0.608212i \(-0.208113\pi\)
0.793775 + 0.608212i \(0.208113\pi\)
\(882\) 0 0
\(883\) − 40.5934i − 1.36608i −0.730383 0.683038i \(-0.760658\pi\)
0.730383 0.683038i \(-0.239342\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.89101i 0.332108i 0.986117 + 0.166054i \(0.0531025\pi\)
−0.986117 + 0.166054i \(0.946897\pi\)
\(888\) 0 0
\(889\) 36.4343 1.22197
\(890\) 0 0
\(891\) 17.4619 0.584996
\(892\) 0 0
\(893\) 10.7172i 0.358638i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 8.75744i 0.292402i
\(898\) 0 0
\(899\) 85.3884 2.84786
\(900\) 0 0
\(901\) −3.25564 −0.108461
\(902\) 0 0
\(903\) − 32.8249i − 1.09234i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 42.3250i − 1.40538i −0.711497 0.702689i \(-0.751983\pi\)
0.711497 0.702689i \(-0.248017\pi\)
\(908\) 0 0
\(909\) 11.4589 0.380068
\(910\) 0 0
\(911\) 5.48129 0.181603 0.0908016 0.995869i \(-0.471057\pi\)
0.0908016 + 0.995869i \(0.471057\pi\)
\(912\) 0 0
\(913\) 21.2759i 0.704128i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 21.4086i 0.706974i
\(918\) 0 0
\(919\) −16.6386 −0.548856 −0.274428 0.961608i \(-0.588488\pi\)
−0.274428 + 0.961608i \(0.588488\pi\)
\(920\) 0 0
\(921\) 49.5113 1.63145
\(922\) 0 0
\(923\) 30.1171i 0.991315i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 22.5224i 0.739733i
\(928\) 0 0
\(929\) 30.2326 0.991900 0.495950 0.868351i \(-0.334820\pi\)
0.495950 + 0.868351i \(0.334820\pi\)
\(930\) 0 0
\(931\) 15.9195 0.521741
\(932\) 0 0
\(933\) − 44.1457i − 1.44527i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 27.8718i 0.910532i 0.890356 + 0.455266i \(0.150456\pi\)
−0.890356 + 0.455266i \(0.849544\pi\)
\(938\) 0 0
\(939\) −31.3823 −1.02412
\(940\) 0 0
\(941\) 26.8768 0.876159 0.438079 0.898936i \(-0.355659\pi\)
0.438079 + 0.898936i \(0.355659\pi\)
\(942\) 0 0
\(943\) 2.34251i 0.0762827i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 23.1773i − 0.753161i −0.926384 0.376581i \(-0.877100\pi\)
0.926384 0.376581i \(-0.122900\pi\)
\(948\) 0 0
\(949\) −50.9297 −1.65325
\(950\) 0 0
\(951\) −16.5384 −0.536296
\(952\) 0 0
\(953\) 4.21870i 0.136657i 0.997663 + 0.0683285i \(0.0217666\pi\)
−0.997663 + 0.0683285i \(0.978233\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 32.3092i − 1.04441i
\(958\) 0 0
\(959\) −15.1006 −0.487625
\(960\) 0 0
\(961\) 53.4401 1.72387
\(962\) 0 0
\(963\) 15.0698i 0.485617i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 17.8055i 0.572588i 0.958142 + 0.286294i \(0.0924233\pi\)
−0.958142 + 0.286294i \(0.907577\pi\)
\(968\) 0 0
\(969\) 5.06223 0.162622
\(970\) 0 0
\(971\) 17.9741 0.576815 0.288408 0.957508i \(-0.406874\pi\)
0.288408 + 0.957508i \(0.406874\pi\)
\(972\) 0 0
\(973\) − 14.5904i − 0.467747i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 58.6275i 1.87566i 0.347097 + 0.937829i \(0.387167\pi\)
−0.347097 + 0.937829i \(0.612833\pi\)
\(978\) 0 0
\(979\) 22.5686 0.721297
\(980\) 0 0
\(981\) −8.22592 −0.262633
\(982\) 0 0
\(983\) − 7.84103i − 0.250090i −0.992151 0.125045i \(-0.960092\pi\)
0.992151 0.125045i \(-0.0399075\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 6.78701i − 0.216033i
\(988\) 0 0
\(989\) 6.67460 0.212240
\(990\) 0 0
\(991\) −33.1872 −1.05423 −0.527114 0.849795i \(-0.676726\pi\)
−0.527114 + 0.849795i \(0.676726\pi\)
\(992\) 0 0
\(993\) 14.5112i 0.460500i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 60.8220i − 1.92625i −0.269049 0.963127i \(-0.586709\pi\)
0.269049 0.963127i \(-0.413291\pi\)
\(998\) 0 0
\(999\) −25.9612 −0.821377
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 4600.2.e.v.4049.9 10
5.2 odd 4 4600.2.a.bc.1.5 5
5.3 odd 4 4600.2.a.bg.1.1 yes 5
5.4 even 2 inner 4600.2.e.v.4049.2 10
20.3 even 4 9200.2.a.cs.1.5 5
20.7 even 4 9200.2.a.cw.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.5 5 5.2 odd 4
4600.2.a.bg.1.1 yes 5 5.3 odd 4
4600.2.e.v.4049.2 10 5.4 even 2 inner
4600.2.e.v.4049.9 10 1.1 even 1 trivial
9200.2.a.cs.1.5 5 20.3 even 4
9200.2.a.cw.1.1 5 20.7 even 4