Properties

Label 1840.2.a.k.1.2
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155 q^{3} +1.00000 q^{5} +3.12311 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q+1.56155 q^{3} +1.00000 q^{5} +3.12311 q^{7} -0.561553 q^{9} +4.00000 q^{11} +3.56155 q^{13} +1.56155 q^{15} +5.12311 q^{17} -4.00000 q^{19} +4.87689 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.56155 q^{27} -4.43845 q^{29} -5.56155 q^{31} +6.24621 q^{33} +3.12311 q^{35} +1.12311 q^{37} +5.56155 q^{39} -3.56155 q^{41} +0.876894 q^{43} -0.561553 q^{45} -8.68466 q^{47} +2.75379 q^{49} +8.00000 q^{51} +12.2462 q^{53} +4.00000 q^{55} -6.24621 q^{57} -10.2462 q^{59} +2.87689 q^{61} -1.75379 q^{63} +3.56155 q^{65} +10.2462 q^{67} -1.56155 q^{69} +8.68466 q^{71} +12.4384 q^{73} +1.56155 q^{75} +12.4924 q^{77} -6.24621 q^{79} -7.00000 q^{81} -12.0000 q^{83} +5.12311 q^{85} -6.93087 q^{87} +10.0000 q^{89} +11.1231 q^{91} -8.68466 q^{93} -4.00000 q^{95} +0.246211 q^{97} -2.24621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} + O(q^{10}) \) \( 2 q - q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} + 8 q^{11} + 3 q^{13} - q^{15} + 2 q^{17} - 8 q^{19} + 18 q^{21} - 2 q^{23} + 2 q^{25} - 7 q^{27} - 13 q^{29} - 7 q^{31} - 4 q^{33} - 2 q^{35} - 6 q^{37} + 7 q^{39} - 3 q^{41} + 10 q^{43} + 3 q^{45} - 5 q^{47} + 22 q^{49} + 16 q^{51} + 8 q^{53} + 8 q^{55} + 4 q^{57} - 4 q^{59} + 14 q^{61} - 20 q^{63} + 3 q^{65} + 4 q^{67} + q^{69} + 5 q^{71} + 29 q^{73} - q^{75} - 8 q^{77} + 4 q^{79} - 14 q^{81} - 24 q^{83} + 2 q^{85} + 15 q^{87} + 20 q^{89} + 14 q^{91} - 5 q^{93} - 8 q^{95} - 16 q^{97} + 12 q^{99} + O(q^{100}) \)

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
\(3\) 1.56155 0.901563 0.450781 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.12311 1.18042 0.590211 0.807249i \(-0.299044\pi\)
0.590211 + 0.807249i \(0.299044\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 3.56155 0.987797 0.493899 0.869520i \(-0.335571\pi\)
0.493899 + 0.869520i \(0.335571\pi\)
\(14\) 0 0
\(15\) 1.56155 0.403191
\(16\) 0 0
\(17\) 5.12311 1.24254 0.621268 0.783598i \(-0.286618\pi\)
0.621268 + 0.783598i \(0.286618\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 4.87689 1.06423
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) −4.43845 −0.824199 −0.412099 0.911139i \(-0.635204\pi\)
−0.412099 + 0.911139i \(0.635204\pi\)
\(30\) 0 0
\(31\) −5.56155 −0.998884 −0.499442 0.866347i \(-0.666462\pi\)
−0.499442 + 0.866347i \(0.666462\pi\)
\(32\) 0 0
\(33\) 6.24621 1.08733
\(34\) 0 0
\(35\) 3.12311 0.527901
\(36\) 0 0
\(37\) 1.12311 0.184637 0.0923187 0.995730i \(-0.470572\pi\)
0.0923187 + 0.995730i \(0.470572\pi\)
\(38\) 0 0
\(39\) 5.56155 0.890561
\(40\) 0 0
\(41\) −3.56155 −0.556221 −0.278111 0.960549i \(-0.589708\pi\)
−0.278111 + 0.960549i \(0.589708\pi\)
\(42\) 0 0
\(43\) 0.876894 0.133725 0.0668626 0.997762i \(-0.478701\pi\)
0.0668626 + 0.997762i \(0.478701\pi\)
\(44\) 0 0
\(45\) −0.561553 −0.0837114
\(46\) 0 0
\(47\) −8.68466 −1.26679 −0.633394 0.773830i \(-0.718339\pi\)
−0.633394 + 0.773830i \(0.718339\pi\)
\(48\) 0 0
\(49\) 2.75379 0.393398
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) 12.2462 1.68215 0.841073 0.540921i \(-0.181924\pi\)
0.841073 + 0.540921i \(0.181924\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −6.24621 −0.827331
\(58\) 0 0
\(59\) −10.2462 −1.33394 −0.666972 0.745083i \(-0.732410\pi\)
−0.666972 + 0.745083i \(0.732410\pi\)
\(60\) 0 0
\(61\) 2.87689 0.368349 0.184174 0.982894i \(-0.441039\pi\)
0.184174 + 0.982894i \(0.441039\pi\)
\(62\) 0 0
\(63\) −1.75379 −0.220957
\(64\) 0 0
\(65\) 3.56155 0.441756
\(66\) 0 0
\(67\) 10.2462 1.25177 0.625887 0.779914i \(-0.284737\pi\)
0.625887 + 0.779914i \(0.284737\pi\)
\(68\) 0 0
\(69\) −1.56155 −0.187989
\(70\) 0 0
\(71\) 8.68466 1.03068 0.515340 0.856986i \(-0.327666\pi\)
0.515340 + 0.856986i \(0.327666\pi\)
\(72\) 0 0
\(73\) 12.4384 1.45581 0.727905 0.685678i \(-0.240494\pi\)
0.727905 + 0.685678i \(0.240494\pi\)
\(74\) 0 0
\(75\) 1.56155 0.180313
\(76\) 0 0
\(77\) 12.4924 1.42364
\(78\) 0 0
\(79\) −6.24621 −0.702754 −0.351377 0.936234i \(-0.614286\pi\)
−0.351377 + 0.936234i \(0.614286\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 5.12311 0.555679
\(86\) 0 0
\(87\) −6.93087 −0.743067
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 11.1231 1.16602
\(92\) 0 0
\(93\) −8.68466 −0.900557
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 0.246211 0.0249990 0.0124995 0.999922i \(-0.496021\pi\)
0.0124995 + 0.999922i \(0.496021\pi\)
\(98\) 0 0
\(99\) −2.24621 −0.225753
\(100\) 0 0
\(101\) −16.2462 −1.61656 −0.808279 0.588799i \(-0.799601\pi\)
−0.808279 + 0.588799i \(0.799601\pi\)
\(102\) 0 0
\(103\) −14.2462 −1.40372 −0.701860 0.712314i \(-0.747647\pi\)
−0.701860 + 0.712314i \(0.747647\pi\)
\(104\) 0 0
\(105\) 4.87689 0.475936
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −8.24621 −0.789844 −0.394922 0.918715i \(-0.629228\pi\)
−0.394922 + 0.918715i \(0.629228\pi\)
\(110\) 0 0
\(111\) 1.75379 0.166462
\(112\) 0 0
\(113\) 3.75379 0.353127 0.176563 0.984289i \(-0.443502\pi\)
0.176563 + 0.984289i \(0.443502\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −5.56155 −0.501468
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −3.80776 −0.337884 −0.168942 0.985626i \(-0.554035\pi\)
−0.168942 + 0.985626i \(0.554035\pi\)
\(128\) 0 0
\(129\) 1.36932 0.120562
\(130\) 0 0
\(131\) 18.9309 1.65400 0.826999 0.562204i \(-0.190046\pi\)
0.826999 + 0.562204i \(0.190046\pi\)
\(132\) 0 0
\(133\) −12.4924 −1.08323
\(134\) 0 0
\(135\) −5.56155 −0.478662
\(136\) 0 0
\(137\) 6.87689 0.587533 0.293766 0.955877i \(-0.405091\pi\)
0.293766 + 0.955877i \(0.405091\pi\)
\(138\) 0 0
\(139\) −12.6847 −1.07590 −0.537949 0.842977i \(-0.680801\pi\)
−0.537949 + 0.842977i \(0.680801\pi\)
\(140\) 0 0
\(141\) −13.5616 −1.14209
\(142\) 0 0
\(143\) 14.2462 1.19133
\(144\) 0 0
\(145\) −4.43845 −0.368593
\(146\) 0 0
\(147\) 4.30019 0.354673
\(148\) 0 0
\(149\) 12.2462 1.00325 0.501624 0.865086i \(-0.332736\pi\)
0.501624 + 0.865086i \(0.332736\pi\)
\(150\) 0 0
\(151\) −3.80776 −0.309871 −0.154936 0.987925i \(-0.549517\pi\)
−0.154936 + 0.987925i \(0.549517\pi\)
\(152\) 0 0
\(153\) −2.87689 −0.232583
\(154\) 0 0
\(155\) −5.56155 −0.446715
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 19.1231 1.51656
\(160\) 0 0
\(161\) −3.12311 −0.246135
\(162\) 0 0
\(163\) 12.6847 0.993539 0.496770 0.867882i \(-0.334519\pi\)
0.496770 + 0.867882i \(0.334519\pi\)
\(164\) 0 0
\(165\) 6.24621 0.486267
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −0.315342 −0.0242570
\(170\) 0 0
\(171\) 2.24621 0.171772
\(172\) 0 0
\(173\) −22.4924 −1.71007 −0.855034 0.518573i \(-0.826464\pi\)
−0.855034 + 0.518573i \(0.826464\pi\)
\(174\) 0 0
\(175\) 3.12311 0.236085
\(176\) 0 0
\(177\) −16.0000 −1.20263
\(178\) 0 0
\(179\) −4.68466 −0.350148 −0.175074 0.984555i \(-0.556016\pi\)
−0.175074 + 0.984555i \(0.556016\pi\)
\(180\) 0 0
\(181\) −5.12311 −0.380797 −0.190399 0.981707i \(-0.560978\pi\)
−0.190399 + 0.981707i \(0.560978\pi\)
\(182\) 0 0
\(183\) 4.49242 0.332089
\(184\) 0 0
\(185\) 1.12311 0.0825724
\(186\) 0 0
\(187\) 20.4924 1.49855
\(188\) 0 0
\(189\) −17.3693 −1.26343
\(190\) 0 0
\(191\) 11.1231 0.804840 0.402420 0.915455i \(-0.368169\pi\)
0.402420 + 0.915455i \(0.368169\pi\)
\(192\) 0 0
\(193\) 20.4384 1.47119 0.735596 0.677421i \(-0.236902\pi\)
0.735596 + 0.677421i \(0.236902\pi\)
\(194\) 0 0
\(195\) 5.56155 0.398271
\(196\) 0 0
\(197\) −15.5616 −1.10871 −0.554357 0.832279i \(-0.687036\pi\)
−0.554357 + 0.832279i \(0.687036\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 0 0
\(203\) −13.8617 −0.972903
\(204\) 0 0
\(205\) −3.56155 −0.248750
\(206\) 0 0
\(207\) 0.561553 0.0390306
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −2.24621 −0.154636 −0.0773178 0.997006i \(-0.524636\pi\)
−0.0773178 + 0.997006i \(0.524636\pi\)
\(212\) 0 0
\(213\) 13.5616 0.929222
\(214\) 0 0
\(215\) 0.876894 0.0598037
\(216\) 0 0
\(217\) −17.3693 −1.17911
\(218\) 0 0
\(219\) 19.4233 1.31250
\(220\) 0 0
\(221\) 18.2462 1.22737
\(222\) 0 0
\(223\) −12.4924 −0.836554 −0.418277 0.908319i \(-0.637366\pi\)
−0.418277 + 0.908319i \(0.637366\pi\)
\(224\) 0 0
\(225\) −0.561553 −0.0374369
\(226\) 0 0
\(227\) −0.876894 −0.0582015 −0.0291008 0.999576i \(-0.509264\pi\)
−0.0291008 + 0.999576i \(0.509264\pi\)
\(228\) 0 0
\(229\) −16.2462 −1.07358 −0.536790 0.843716i \(-0.680363\pi\)
−0.536790 + 0.843716i \(0.680363\pi\)
\(230\) 0 0
\(231\) 19.5076 1.28350
\(232\) 0 0
\(233\) −8.43845 −0.552821 −0.276411 0.961040i \(-0.589145\pi\)
−0.276411 + 0.961040i \(0.589145\pi\)
\(234\) 0 0
\(235\) −8.68466 −0.566525
\(236\) 0 0
\(237\) −9.75379 −0.633577
\(238\) 0 0
\(239\) −11.8078 −0.763781 −0.381890 0.924208i \(-0.624727\pi\)
−0.381890 + 0.924208i \(0.624727\pi\)
\(240\) 0 0
\(241\) 14.4924 0.933539 0.466769 0.884379i \(-0.345418\pi\)
0.466769 + 0.884379i \(0.345418\pi\)
\(242\) 0 0
\(243\) 5.75379 0.369106
\(244\) 0 0
\(245\) 2.75379 0.175933
\(246\) 0 0
\(247\) −14.2462 −0.906465
\(248\) 0 0
\(249\) −18.7386 −1.18751
\(250\) 0 0
\(251\) 10.2462 0.646735 0.323368 0.946273i \(-0.395185\pi\)
0.323368 + 0.946273i \(0.395185\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) 8.00000 0.500979
\(256\) 0 0
\(257\) 28.0540 1.74996 0.874979 0.484160i \(-0.160875\pi\)
0.874979 + 0.484160i \(0.160875\pi\)
\(258\) 0 0
\(259\) 3.50758 0.217950
\(260\) 0 0
\(261\) 2.49242 0.154277
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) 12.2462 0.752279
\(266\) 0 0
\(267\) 15.6155 0.955655
\(268\) 0 0
\(269\) 30.3002 1.84743 0.923717 0.383074i \(-0.125135\pi\)
0.923717 + 0.383074i \(0.125135\pi\)
\(270\) 0 0
\(271\) −14.2462 −0.865396 −0.432698 0.901539i \(-0.642438\pi\)
−0.432698 + 0.901539i \(0.642438\pi\)
\(272\) 0 0
\(273\) 17.3693 1.05124
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −15.5616 −0.935003 −0.467502 0.883992i \(-0.654846\pi\)
−0.467502 + 0.883992i \(0.654846\pi\)
\(278\) 0 0
\(279\) 3.12311 0.186975
\(280\) 0 0
\(281\) −2.49242 −0.148685 −0.0743427 0.997233i \(-0.523686\pi\)
−0.0743427 + 0.997233i \(0.523686\pi\)
\(282\) 0 0
\(283\) −31.1231 −1.85008 −0.925038 0.379874i \(-0.875967\pi\)
−0.925038 + 0.379874i \(0.875967\pi\)
\(284\) 0 0
\(285\) −6.24621 −0.369994
\(286\) 0 0
\(287\) −11.1231 −0.656576
\(288\) 0 0
\(289\) 9.24621 0.543895
\(290\) 0 0
\(291\) 0.384472 0.0225381
\(292\) 0 0
\(293\) −21.1231 −1.23403 −0.617013 0.786953i \(-0.711657\pi\)
−0.617013 + 0.786953i \(0.711657\pi\)
\(294\) 0 0
\(295\) −10.2462 −0.596557
\(296\) 0 0
\(297\) −22.2462 −1.29086
\(298\) 0 0
\(299\) −3.56155 −0.205970
\(300\) 0 0
\(301\) 2.73863 0.157852
\(302\) 0 0
\(303\) −25.3693 −1.45743
\(304\) 0 0
\(305\) 2.87689 0.164730
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) −22.2462 −1.26554
\(310\) 0 0
\(311\) 14.9309 0.846652 0.423326 0.905977i \(-0.360863\pi\)
0.423326 + 0.905977i \(0.360863\pi\)
\(312\) 0 0
\(313\) 6.87689 0.388705 0.194353 0.980932i \(-0.437739\pi\)
0.194353 + 0.980932i \(0.437739\pi\)
\(314\) 0 0
\(315\) −1.75379 −0.0988148
\(316\) 0 0
\(317\) −28.7386 −1.61412 −0.807061 0.590468i \(-0.798943\pi\)
−0.807061 + 0.590468i \(0.798943\pi\)
\(318\) 0 0
\(319\) −17.7538 −0.994021
\(320\) 0 0
\(321\) 18.7386 1.04589
\(322\) 0 0
\(323\) −20.4924 −1.14023
\(324\) 0 0
\(325\) 3.56155 0.197559
\(326\) 0 0
\(327\) −12.8769 −0.712094
\(328\) 0 0
\(329\) −27.1231 −1.49535
\(330\) 0 0
\(331\) −6.43845 −0.353889 −0.176945 0.984221i \(-0.556621\pi\)
−0.176945 + 0.984221i \(0.556621\pi\)
\(332\) 0 0
\(333\) −0.630683 −0.0345612
\(334\) 0 0
\(335\) 10.2462 0.559810
\(336\) 0 0
\(337\) 32.2462 1.75656 0.878282 0.478144i \(-0.158690\pi\)
0.878282 + 0.478144i \(0.158690\pi\)
\(338\) 0 0
\(339\) 5.86174 0.318366
\(340\) 0 0
\(341\) −22.2462 −1.20470
\(342\) 0 0
\(343\) −13.2614 −0.716046
\(344\) 0 0
\(345\) −1.56155 −0.0840712
\(346\) 0 0
\(347\) −16.4924 −0.885360 −0.442680 0.896680i \(-0.645972\pi\)
−0.442680 + 0.896680i \(0.645972\pi\)
\(348\) 0 0
\(349\) 17.8078 0.953228 0.476614 0.879113i \(-0.341864\pi\)
0.476614 + 0.879113i \(0.341864\pi\)
\(350\) 0 0
\(351\) −19.8078 −1.05726
\(352\) 0 0
\(353\) 14.1922 0.755377 0.377688 0.925933i \(-0.376719\pi\)
0.377688 + 0.925933i \(0.376719\pi\)
\(354\) 0 0
\(355\) 8.68466 0.460934
\(356\) 0 0
\(357\) 24.9848 1.32234
\(358\) 0 0
\(359\) 30.2462 1.59633 0.798167 0.602436i \(-0.205803\pi\)
0.798167 + 0.602436i \(0.205803\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 7.80776 0.409801
\(364\) 0 0
\(365\) 12.4384 0.651058
\(366\) 0 0
\(367\) 12.4924 0.652099 0.326050 0.945353i \(-0.394282\pi\)
0.326050 + 0.945353i \(0.394282\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 38.2462 1.98564
\(372\) 0 0
\(373\) 24.7386 1.28092 0.640459 0.767992i \(-0.278744\pi\)
0.640459 + 0.767992i \(0.278744\pi\)
\(374\) 0 0
\(375\) 1.56155 0.0806382
\(376\) 0 0
\(377\) −15.8078 −0.814141
\(378\) 0 0
\(379\) −2.24621 −0.115380 −0.0576901 0.998335i \(-0.518374\pi\)
−0.0576901 + 0.998335i \(0.518374\pi\)
\(380\) 0 0
\(381\) −5.94602 −0.304624
\(382\) 0 0
\(383\) 7.61553 0.389135 0.194568 0.980889i \(-0.437670\pi\)
0.194568 + 0.980889i \(0.437670\pi\)
\(384\) 0 0
\(385\) 12.4924 0.636673
\(386\) 0 0
\(387\) −0.492423 −0.0250312
\(388\) 0 0
\(389\) −33.6155 −1.70437 −0.852187 0.523237i \(-0.824724\pi\)
−0.852187 + 0.523237i \(0.824724\pi\)
\(390\) 0 0
\(391\) −5.12311 −0.259087
\(392\) 0 0
\(393\) 29.5616 1.49118
\(394\) 0 0
\(395\) −6.24621 −0.314281
\(396\) 0 0
\(397\) −24.9309 −1.25124 −0.625622 0.780126i \(-0.715155\pi\)
−0.625622 + 0.780126i \(0.715155\pi\)
\(398\) 0 0
\(399\) −19.5076 −0.976600
\(400\) 0 0
\(401\) 20.7386 1.03564 0.517819 0.855490i \(-0.326744\pi\)
0.517819 + 0.855490i \(0.326744\pi\)
\(402\) 0 0
\(403\) −19.8078 −0.986695
\(404\) 0 0
\(405\) −7.00000 −0.347833
\(406\) 0 0
\(407\) 4.49242 0.222681
\(408\) 0 0
\(409\) −14.6847 −0.726110 −0.363055 0.931768i \(-0.618266\pi\)
−0.363055 + 0.931768i \(0.618266\pi\)
\(410\) 0 0
\(411\) 10.7386 0.529698
\(412\) 0 0
\(413\) −32.0000 −1.57462
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) −19.8078 −0.969990
\(418\) 0 0
\(419\) 0.876894 0.0428391 0.0214195 0.999771i \(-0.493181\pi\)
0.0214195 + 0.999771i \(0.493181\pi\)
\(420\) 0 0
\(421\) −6.49242 −0.316421 −0.158211 0.987405i \(-0.550573\pi\)
−0.158211 + 0.987405i \(0.550573\pi\)
\(422\) 0 0
\(423\) 4.87689 0.237123
\(424\) 0 0
\(425\) 5.12311 0.248507
\(426\) 0 0
\(427\) 8.98485 0.434807
\(428\) 0 0
\(429\) 22.2462 1.07406
\(430\) 0 0
\(431\) −7.61553 −0.366827 −0.183414 0.983036i \(-0.558715\pi\)
−0.183414 + 0.983036i \(0.558715\pi\)
\(432\) 0 0
\(433\) −24.7386 −1.18886 −0.594431 0.804146i \(-0.702623\pi\)
−0.594431 + 0.804146i \(0.702623\pi\)
\(434\) 0 0
\(435\) −6.93087 −0.332310
\(436\) 0 0
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 32.3002 1.54160 0.770802 0.637075i \(-0.219856\pi\)
0.770802 + 0.637075i \(0.219856\pi\)
\(440\) 0 0
\(441\) −1.54640 −0.0736380
\(442\) 0 0
\(443\) 3.31534 0.157517 0.0787583 0.996894i \(-0.474904\pi\)
0.0787583 + 0.996894i \(0.474904\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) 19.1231 0.904492
\(448\) 0 0
\(449\) −32.7386 −1.54503 −0.772516 0.634996i \(-0.781002\pi\)
−0.772516 + 0.634996i \(0.781002\pi\)
\(450\) 0 0
\(451\) −14.2462 −0.670828
\(452\) 0 0
\(453\) −5.94602 −0.279369
\(454\) 0 0
\(455\) 11.1231 0.521459
\(456\) 0 0
\(457\) −9.12311 −0.426761 −0.213380 0.976969i \(-0.568447\pi\)
−0.213380 + 0.976969i \(0.568447\pi\)
\(458\) 0 0
\(459\) −28.4924 −1.32991
\(460\) 0 0
\(461\) 27.5616 1.28367 0.641835 0.766843i \(-0.278174\pi\)
0.641835 + 0.766843i \(0.278174\pi\)
\(462\) 0 0
\(463\) 6.24621 0.290286 0.145143 0.989411i \(-0.453636\pi\)
0.145143 + 0.989411i \(0.453636\pi\)
\(464\) 0 0
\(465\) −8.68466 −0.402741
\(466\) 0 0
\(467\) −10.6307 −0.491929 −0.245965 0.969279i \(-0.579105\pi\)
−0.245965 + 0.969279i \(0.579105\pi\)
\(468\) 0 0
\(469\) 32.0000 1.47762
\(470\) 0 0
\(471\) −3.12311 −0.143905
\(472\) 0 0
\(473\) 3.50758 0.161279
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −6.87689 −0.314871
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) −4.87689 −0.221906
\(484\) 0 0
\(485\) 0.246211 0.0111799
\(486\) 0 0
\(487\) −34.0540 −1.54313 −0.771566 0.636149i \(-0.780526\pi\)
−0.771566 + 0.636149i \(0.780526\pi\)
\(488\) 0 0
\(489\) 19.8078 0.895738
\(490\) 0 0
\(491\) −6.43845 −0.290563 −0.145282 0.989390i \(-0.546409\pi\)
−0.145282 + 0.989390i \(0.546409\pi\)
\(492\) 0 0
\(493\) −22.7386 −1.02410
\(494\) 0 0
\(495\) −2.24621 −0.100960
\(496\) 0 0
\(497\) 27.1231 1.21664
\(498\) 0 0
\(499\) −38.0540 −1.70353 −0.851765 0.523924i \(-0.824468\pi\)
−0.851765 + 0.523924i \(0.824468\pi\)
\(500\) 0 0
\(501\) 12.4924 0.558120
\(502\) 0 0
\(503\) 15.6155 0.696262 0.348131 0.937446i \(-0.386816\pi\)
0.348131 + 0.937446i \(0.386816\pi\)
\(504\) 0 0
\(505\) −16.2462 −0.722947
\(506\) 0 0
\(507\) −0.492423 −0.0218693
\(508\) 0 0
\(509\) −4.43845 −0.196731 −0.0983654 0.995150i \(-0.531361\pi\)
−0.0983654 + 0.995150i \(0.531361\pi\)
\(510\) 0 0
\(511\) 38.8466 1.71847
\(512\) 0 0
\(513\) 22.2462 0.982194
\(514\) 0 0
\(515\) −14.2462 −0.627763
\(516\) 0 0
\(517\) −34.7386 −1.52780
\(518\) 0 0
\(519\) −35.1231 −1.54173
\(520\) 0 0
\(521\) 39.8617 1.74637 0.873187 0.487385i \(-0.162049\pi\)
0.873187 + 0.487385i \(0.162049\pi\)
\(522\) 0 0
\(523\) −33.8617 −1.48067 −0.740335 0.672238i \(-0.765333\pi\)
−0.740335 + 0.672238i \(0.765333\pi\)
\(524\) 0 0
\(525\) 4.87689 0.212845
\(526\) 0 0
\(527\) −28.4924 −1.24115
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 5.75379 0.249693
\(532\) 0 0
\(533\) −12.6847 −0.549434
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) −7.31534 −0.315680
\(538\) 0 0
\(539\) 11.0152 0.474456
\(540\) 0 0
\(541\) 21.3153 0.916418 0.458209 0.888844i \(-0.348491\pi\)
0.458209 + 0.888844i \(0.348491\pi\)
\(542\) 0 0
\(543\) −8.00000 −0.343313
\(544\) 0 0
\(545\) −8.24621 −0.353229
\(546\) 0 0
\(547\) −38.0540 −1.62707 −0.813535 0.581516i \(-0.802460\pi\)
−0.813535 + 0.581516i \(0.802460\pi\)
\(548\) 0 0
\(549\) −1.61553 −0.0689491
\(550\) 0 0
\(551\) 17.7538 0.756337
\(552\) 0 0
\(553\) −19.5076 −0.829547
\(554\) 0 0
\(555\) 1.75379 0.0744442
\(556\) 0 0
\(557\) −24.2462 −1.02734 −0.513672 0.857986i \(-0.671715\pi\)
−0.513672 + 0.857986i \(0.671715\pi\)
\(558\) 0 0
\(559\) 3.12311 0.132093
\(560\) 0 0
\(561\) 32.0000 1.35104
\(562\) 0 0
\(563\) −43.2311 −1.82197 −0.910986 0.412438i \(-0.864678\pi\)
−0.910986 + 0.412438i \(0.864678\pi\)
\(564\) 0 0
\(565\) 3.75379 0.157923
\(566\) 0 0
\(567\) −21.8617 −0.918107
\(568\) 0 0
\(569\) 28.7386 1.20479 0.602393 0.798200i \(-0.294214\pi\)
0.602393 + 0.798200i \(0.294214\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 17.3693 0.725614
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 29.4233 1.22491 0.612454 0.790506i \(-0.290183\pi\)
0.612454 + 0.790506i \(0.290183\pi\)
\(578\) 0 0
\(579\) 31.9157 1.32637
\(580\) 0 0
\(581\) −37.4773 −1.55482
\(582\) 0 0
\(583\) 48.9848 2.02874
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) −9.17708 −0.378779 −0.189389 0.981902i \(-0.560651\pi\)
−0.189389 + 0.981902i \(0.560651\pi\)
\(588\) 0 0
\(589\) 22.2462 0.916639
\(590\) 0 0
\(591\) −24.3002 −0.999576
\(592\) 0 0
\(593\) −38.9848 −1.60092 −0.800458 0.599389i \(-0.795410\pi\)
−0.800458 + 0.599389i \(0.795410\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) 0 0
\(597\) −37.4773 −1.53384
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −27.1771 −1.10858 −0.554288 0.832325i \(-0.687009\pi\)
−0.554288 + 0.832325i \(0.687009\pi\)
\(602\) 0 0
\(603\) −5.75379 −0.234312
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) −21.6458 −0.877134
\(610\) 0 0
\(611\) −30.9309 −1.25133
\(612\) 0 0
\(613\) −5.12311 −0.206920 −0.103460 0.994634i \(-0.532991\pi\)
−0.103460 + 0.994634i \(0.532991\pi\)
\(614\) 0 0
\(615\) −5.56155 −0.224263
\(616\) 0 0
\(617\) −5.61553 −0.226073 −0.113036 0.993591i \(-0.536058\pi\)
−0.113036 + 0.993591i \(0.536058\pi\)
\(618\) 0 0
\(619\) −36.9848 −1.48655 −0.743273 0.668988i \(-0.766728\pi\)
−0.743273 + 0.668988i \(0.766728\pi\)
\(620\) 0 0
\(621\) 5.56155 0.223177
\(622\) 0 0
\(623\) 31.2311 1.25125
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −24.9848 −0.997799
\(628\) 0 0
\(629\) 5.75379 0.229419
\(630\) 0 0
\(631\) 6.63068 0.263963 0.131982 0.991252i \(-0.457866\pi\)
0.131982 + 0.991252i \(0.457866\pi\)
\(632\) 0 0
\(633\) −3.50758 −0.139414
\(634\) 0 0
\(635\) −3.80776 −0.151107
\(636\) 0 0
\(637\) 9.80776 0.388598
\(638\) 0 0
\(639\) −4.87689 −0.192927
\(640\) 0 0
\(641\) 40.2462 1.58963 0.794815 0.606852i \(-0.207568\pi\)
0.794815 + 0.606852i \(0.207568\pi\)
\(642\) 0 0
\(643\) 46.3542 1.82803 0.914015 0.405681i \(-0.132966\pi\)
0.914015 + 0.405681i \(0.132966\pi\)
\(644\) 0 0
\(645\) 1.36932 0.0539168
\(646\) 0 0
\(647\) 5.56155 0.218647 0.109324 0.994006i \(-0.465132\pi\)
0.109324 + 0.994006i \(0.465132\pi\)
\(648\) 0 0
\(649\) −40.9848 −1.60880
\(650\) 0 0
\(651\) −27.1231 −1.06304
\(652\) 0 0
\(653\) 50.7926 1.98767 0.993834 0.110876i \(-0.0353658\pi\)
0.993834 + 0.110876i \(0.0353658\pi\)
\(654\) 0 0
\(655\) 18.9309 0.739690
\(656\) 0 0
\(657\) −6.98485 −0.272505
\(658\) 0 0
\(659\) 2.24621 0.0875000 0.0437500 0.999043i \(-0.486070\pi\)
0.0437500 + 0.999043i \(0.486070\pi\)
\(660\) 0 0
\(661\) −22.4924 −0.874854 −0.437427 0.899254i \(-0.644110\pi\)
−0.437427 + 0.899254i \(0.644110\pi\)
\(662\) 0 0
\(663\) 28.4924 1.10655
\(664\) 0 0
\(665\) −12.4924 −0.484435
\(666\) 0 0
\(667\) 4.43845 0.171857
\(668\) 0 0
\(669\) −19.5076 −0.754207
\(670\) 0 0
\(671\) 11.5076 0.444245
\(672\) 0 0
\(673\) −0.438447 −0.0169009 −0.00845045 0.999964i \(-0.502690\pi\)
−0.00845045 + 0.999964i \(0.502690\pi\)
\(674\) 0 0
\(675\) −5.56155 −0.214064
\(676\) 0 0
\(677\) −28.7386 −1.10452 −0.552258 0.833673i \(-0.686234\pi\)
−0.552258 + 0.833673i \(0.686234\pi\)
\(678\) 0 0
\(679\) 0.768944 0.0295094
\(680\) 0 0
\(681\) −1.36932 −0.0524723
\(682\) 0 0
\(683\) 14.4384 0.552472 0.276236 0.961090i \(-0.410913\pi\)
0.276236 + 0.961090i \(0.410913\pi\)
\(684\) 0 0
\(685\) 6.87689 0.262753
\(686\) 0 0
\(687\) −25.3693 −0.967900
\(688\) 0 0
\(689\) 43.6155 1.66162
\(690\) 0 0
\(691\) 10.2462 0.389784 0.194892 0.980825i \(-0.437564\pi\)
0.194892 + 0.980825i \(0.437564\pi\)
\(692\) 0 0
\(693\) −7.01515 −0.266484
\(694\) 0 0
\(695\) −12.6847 −0.481157
\(696\) 0 0
\(697\) −18.2462 −0.691125
\(698\) 0 0
\(699\) −13.1771 −0.498403
\(700\) 0 0
\(701\) −26.9848 −1.01920 −0.509602 0.860410i \(-0.670207\pi\)
−0.509602 + 0.860410i \(0.670207\pi\)
\(702\) 0 0
\(703\) −4.49242 −0.169435
\(704\) 0 0
\(705\) −13.5616 −0.510758
\(706\) 0 0
\(707\) −50.7386 −1.90822
\(708\) 0 0
\(709\) 8.73863 0.328186 0.164093 0.986445i \(-0.447530\pi\)
0.164093 + 0.986445i \(0.447530\pi\)
\(710\) 0 0
\(711\) 3.50758 0.131544
\(712\) 0 0
\(713\) 5.56155 0.208282
\(714\) 0 0
\(715\) 14.2462 0.532778
\(716\) 0 0
\(717\) −18.4384 −0.688596
\(718\) 0 0
\(719\) 10.7386 0.400483 0.200242 0.979747i \(-0.435827\pi\)
0.200242 + 0.979747i \(0.435827\pi\)
\(720\) 0 0
\(721\) −44.4924 −1.65698
\(722\) 0 0
\(723\) 22.6307 0.841644
\(724\) 0 0
\(725\) −4.43845 −0.164840
\(726\) 0 0
\(727\) −28.8769 −1.07098 −0.535492 0.844540i \(-0.679874\pi\)
−0.535492 + 0.844540i \(0.679874\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) 4.49242 0.166158
\(732\) 0 0
\(733\) 2.87689 0.106261 0.0531303 0.998588i \(-0.483080\pi\)
0.0531303 + 0.998588i \(0.483080\pi\)
\(734\) 0 0
\(735\) 4.30019 0.158615
\(736\) 0 0
\(737\) 40.9848 1.50970
\(738\) 0 0
\(739\) 42.5464 1.56509 0.782547 0.622591i \(-0.213920\pi\)
0.782547 + 0.622591i \(0.213920\pi\)
\(740\) 0 0
\(741\) −22.2462 −0.817235
\(742\) 0 0
\(743\) 30.2462 1.10963 0.554813 0.831975i \(-0.312790\pi\)
0.554813 + 0.831975i \(0.312790\pi\)
\(744\) 0 0
\(745\) 12.2462 0.448666
\(746\) 0 0
\(747\) 6.73863 0.246554
\(748\) 0 0
\(749\) 37.4773 1.36939
\(750\) 0 0
\(751\) −8.38447 −0.305954 −0.152977 0.988230i \(-0.548886\pi\)
−0.152977 + 0.988230i \(0.548886\pi\)
\(752\) 0 0
\(753\) 16.0000 0.583072
\(754\) 0 0
\(755\) −3.80776 −0.138579
\(756\) 0 0
\(757\) 33.1231 1.20388 0.601940 0.798541i \(-0.294395\pi\)
0.601940 + 0.798541i \(0.294395\pi\)
\(758\) 0 0
\(759\) −6.24621 −0.226723
\(760\) 0 0
\(761\) 42.3002 1.53338 0.766690 0.642017i \(-0.221902\pi\)
0.766690 + 0.642017i \(0.221902\pi\)
\(762\) 0 0
\(763\) −25.7538 −0.932350
\(764\) 0 0
\(765\) −2.87689 −0.104014
\(766\) 0 0
\(767\) −36.4924 −1.31767
\(768\) 0 0
\(769\) 5.50758 0.198608 0.0993042 0.995057i \(-0.468338\pi\)
0.0993042 + 0.995057i \(0.468338\pi\)
\(770\) 0 0
\(771\) 43.8078 1.57770
\(772\) 0 0
\(773\) 12.2462 0.440466 0.220233 0.975447i \(-0.429318\pi\)
0.220233 + 0.975447i \(0.429318\pi\)
\(774\) 0 0
\(775\) −5.56155 −0.199777
\(776\) 0 0
\(777\) 5.47727 0.196496
\(778\) 0 0
\(779\) 14.2462 0.510423
\(780\) 0 0
\(781\) 34.7386 1.24305
\(782\) 0 0
\(783\) 24.6847 0.882158
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 29.7538 1.06061 0.530304 0.847808i \(-0.322078\pi\)
0.530304 + 0.847808i \(0.322078\pi\)
\(788\) 0 0
\(789\) 12.4924 0.444742
\(790\) 0 0
\(791\) 11.7235 0.416839
\(792\) 0 0
\(793\) 10.2462 0.363854
\(794\) 0 0
\(795\) 19.1231 0.678227
\(796\) 0 0
\(797\) 18.8769 0.668654 0.334327 0.942457i \(-0.391491\pi\)
0.334327 + 0.942457i \(0.391491\pi\)
\(798\) 0 0
\(799\) −44.4924 −1.57403
\(800\) 0 0
\(801\) −5.61553 −0.198415
\(802\) 0 0
\(803\) 49.7538 1.75577
\(804\) 0 0
\(805\) −3.12311 −0.110075
\(806\) 0 0
\(807\) 47.3153 1.66558
\(808\) 0 0
\(809\) 22.4924 0.790791 0.395396 0.918511i \(-0.370607\pi\)
0.395396 + 0.918511i \(0.370607\pi\)
\(810\) 0 0
\(811\) 20.6847 0.726337 0.363168 0.931724i \(-0.381695\pi\)
0.363168 + 0.931724i \(0.381695\pi\)
\(812\) 0 0
\(813\) −22.2462 −0.780209
\(814\) 0 0
\(815\) 12.6847 0.444324
\(816\) 0 0
\(817\) −3.50758 −0.122715
\(818\) 0 0
\(819\) −6.24621 −0.218260
\(820\) 0 0
\(821\) −48.2462 −1.68380 −0.841902 0.539630i \(-0.818564\pi\)
−0.841902 + 0.539630i \(0.818564\pi\)
\(822\) 0 0
\(823\) 21.5616 0.751588 0.375794 0.926703i \(-0.377370\pi\)
0.375794 + 0.926703i \(0.377370\pi\)
\(824\) 0 0
\(825\) 6.24621 0.217465
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 17.5076 0.608063 0.304032 0.952662i \(-0.401667\pi\)
0.304032 + 0.952662i \(0.401667\pi\)
\(830\) 0 0
\(831\) −24.3002 −0.842964
\(832\) 0 0
\(833\) 14.1080 0.488812
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 30.9309 1.06913
\(838\) 0 0
\(839\) 31.6155 1.09149 0.545745 0.837952i \(-0.316247\pi\)
0.545745 + 0.837952i \(0.316247\pi\)
\(840\) 0 0
\(841\) −9.30019 −0.320696
\(842\) 0 0
\(843\) −3.89205 −0.134049
\(844\) 0 0
\(845\) −0.315342 −0.0108481
\(846\) 0 0
\(847\) 15.6155 0.536556
\(848\) 0 0
\(849\) −48.6004 −1.66796
\(850\) 0 0
\(851\) −1.12311 −0.0384996
\(852\) 0 0
\(853\) −42.9848 −1.47177 −0.735887 0.677105i \(-0.763234\pi\)
−0.735887 + 0.677105i \(0.763234\pi\)
\(854\) 0 0
\(855\) 2.24621 0.0768188
\(856\) 0 0
\(857\) −11.1771 −0.381802 −0.190901 0.981609i \(-0.561141\pi\)
−0.190901 + 0.981609i \(0.561141\pi\)
\(858\) 0 0
\(859\) 23.4233 0.799192 0.399596 0.916691i \(-0.369150\pi\)
0.399596 + 0.916691i \(0.369150\pi\)
\(860\) 0 0
\(861\) −17.3693 −0.591945
\(862\) 0 0
\(863\) 43.4233 1.47815 0.739073 0.673625i \(-0.235264\pi\)
0.739073 + 0.673625i \(0.235264\pi\)
\(864\) 0 0
\(865\) −22.4924 −0.764765
\(866\) 0 0
\(867\) 14.4384 0.490355
\(868\) 0 0
\(869\) −24.9848 −0.847553
\(870\) 0 0
\(871\) 36.4924 1.23650
\(872\) 0 0
\(873\) −0.138261 −0.00467941
\(874\) 0 0
\(875\) 3.12311 0.105580
\(876\) 0 0
\(877\) 8.73863 0.295083 0.147541 0.989056i \(-0.452864\pi\)
0.147541 + 0.989056i \(0.452864\pi\)
\(878\) 0 0
\(879\) −32.9848 −1.11255
\(880\) 0 0
\(881\) −50.1080 −1.68818 −0.844090 0.536202i \(-0.819859\pi\)
−0.844090 + 0.536202i \(0.819859\pi\)
\(882\) 0 0
\(883\) 7.50758 0.252650 0.126325 0.991989i \(-0.459682\pi\)
0.126325 + 0.991989i \(0.459682\pi\)
\(884\) 0 0
\(885\) −16.0000 −0.537834
\(886\) 0 0
\(887\) 37.5616 1.26119 0.630597 0.776111i \(-0.282810\pi\)
0.630597 + 0.776111i \(0.282810\pi\)
\(888\) 0 0
\(889\) −11.8920 −0.398847
\(890\) 0 0
\(891\) −28.0000 −0.938035
\(892\) 0 0
\(893\) 34.7386 1.16248
\(894\) 0 0
\(895\) −4.68466 −0.156591
\(896\) 0 0
\(897\) −5.56155 −0.185695
\(898\) 0 0
\(899\) 24.6847 0.823279
\(900\) 0 0
\(901\) 62.7386 2.09013
\(902\) 0 0
\(903\) 4.27652 0.142314
\(904\) 0 0
\(905\) −5.12311 −0.170298
\(906\) 0 0
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 0 0
\(909\) 9.12311 0.302594
\(910\) 0 0
\(911\) −12.4924 −0.413892 −0.206946 0.978352i \(-0.566353\pi\)
−0.206946 + 0.978352i \(0.566353\pi\)
\(912\) 0 0
\(913\) −48.0000 −1.58857
\(914\) 0 0
\(915\) 4.49242 0.148515
\(916\) 0 0
\(917\) 59.1231 1.95242
\(918\) 0 0
\(919\) −21.8617 −0.721152 −0.360576 0.932730i \(-0.617420\pi\)
−0.360576 + 0.932730i \(0.617420\pi\)
\(920\) 0 0
\(921\) −18.7386 −0.617459
\(922\) 0 0
\(923\) 30.9309 1.01810
\(924\) 0 0
\(925\) 1.12311 0.0369275
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) −24.0540 −0.789185 −0.394593 0.918856i \(-0.629114\pi\)
−0.394593 + 0.918856i \(0.629114\pi\)
\(930\) 0 0
\(931\) −11.0152 −0.361007
\(932\) 0 0
\(933\) 23.3153 0.763310
\(934\) 0 0
\(935\) 20.4924 0.670174
\(936\) 0 0
\(937\) 16.6307 0.543301 0.271650 0.962396i \(-0.412431\pi\)
0.271650 + 0.962396i \(0.412431\pi\)
\(938\) 0 0
\(939\) 10.7386 0.350442
\(940\) 0 0
\(941\) 46.6004 1.51913 0.759564 0.650432i \(-0.225412\pi\)
0.759564 + 0.650432i \(0.225412\pi\)
\(942\) 0 0
\(943\) 3.56155 0.115980
\(944\) 0 0
\(945\) −17.3693 −0.565024
\(946\) 0 0
\(947\) 55.8078 1.81351 0.906754 0.421659i \(-0.138552\pi\)
0.906754 + 0.421659i \(0.138552\pi\)
\(948\) 0 0
\(949\) 44.3002 1.43804
\(950\) 0 0
\(951\) −44.8769 −1.45523
\(952\) 0 0
\(953\) 46.4924 1.50604 0.753019 0.657999i \(-0.228597\pi\)
0.753019 + 0.657999i \(0.228597\pi\)
\(954\) 0 0
\(955\) 11.1231 0.359935
\(956\) 0 0
\(957\) −27.7235 −0.896173
\(958\) 0 0
\(959\) 21.4773 0.693537
\(960\) 0 0
\(961\) −0.0691303 −0.00223001
\(962\) 0 0
\(963\) −6.73863 −0.217149
\(964\) 0 0
\(965\) 20.4384 0.657937
\(966\) 0 0
\(967\) 45.1771 1.45280 0.726398 0.687274i \(-0.241193\pi\)
0.726398 + 0.687274i \(0.241193\pi\)
\(968\) 0 0
\(969\) −32.0000 −1.02799
\(970\) 0 0
\(971\) 21.3693 0.685774 0.342887 0.939377i \(-0.388595\pi\)
0.342887 + 0.939377i \(0.388595\pi\)
\(972\) 0 0
\(973\) −39.6155 −1.27002
\(974\) 0 0
\(975\) 5.56155 0.178112
\(976\) 0 0
\(977\) −26.1080 −0.835267 −0.417634 0.908615i \(-0.637140\pi\)
−0.417634 + 0.908615i \(0.637140\pi\)
\(978\) 0 0
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) 4.63068 0.147846
\(982\) 0 0
\(983\) −28.8769 −0.921030 −0.460515 0.887652i \(-0.652335\pi\)
−0.460515 + 0.887652i \(0.652335\pi\)
\(984\) 0 0
\(985\) −15.5616 −0.495832
\(986\) 0 0
\(987\) −42.3542 −1.34815
\(988\) 0 0
\(989\) −0.876894 −0.0278836
\(990\) 0 0
\(991\) −20.4924 −0.650963 −0.325482 0.945548i \(-0.605526\pi\)
−0.325482 + 0.945548i \(0.605526\pi\)
\(992\) 0 0
\(993\) −10.0540 −0.319053
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) 0.738634 0.0233928 0.0116964 0.999932i \(-0.496277\pi\)
0.0116964 + 0.999932i \(0.496277\pi\)
\(998\) 0 0
\(999\) −6.24621 −0.197621
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 1840.2.a.k.1.2 2
4.3 odd 2 920.2.a.f.1.1 2
5.4 even 2 9200.2.a.bx.1.1 2
8.3 odd 2 7360.2.a.bj.1.2 2
8.5 even 2 7360.2.a.bm.1.1 2
12.11 even 2 8280.2.a.bb.1.1 2
20.3 even 4 4600.2.e.m.4049.2 4
20.7 even 4 4600.2.e.m.4049.3 4
20.19 odd 2 4600.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.f.1.1 2 4.3 odd 2
1840.2.a.k.1.2 2 1.1 even 1 trivial
4600.2.a.r.1.2 2 20.19 odd 2
4600.2.e.m.4049.2 4 20.3 even 4
4600.2.e.m.4049.3 4 20.7 even 4
7360.2.a.bj.1.2 2 8.3 odd 2
7360.2.a.bm.1.1 2 8.5 even 2
8280.2.a.bb.1.1 2 12.11 even 2
9200.2.a.bx.1.1 2 5.4 even 2