Properties

Label 507.2.b.f.337.6
Level $507$
Weight $2$
Character 507.337
Analytic conductor $4.048$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.6
Root \(0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.2.b.f.337.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.69202i q^{2} -1.00000 q^{3} -5.24698 q^{4} -1.04892i q^{5} -2.69202i q^{6} -0.554958i q^{7} -8.74094i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.69202i q^{2} -1.00000 q^{3} -5.24698 q^{4} -1.04892i q^{5} -2.69202i q^{6} -0.554958i q^{7} -8.74094i q^{8} +1.00000 q^{9} +2.82371 q^{10} -2.91185i q^{11} +5.24698 q^{12} +1.49396 q^{14} +1.04892i q^{15} +13.0368 q^{16} +4.85086 q^{17} +2.69202i q^{18} -0.753020i q^{19} +5.50365i q^{20} +0.554958i q^{21} +7.83877 q^{22} -5.76271 q^{23} +8.74094i q^{24} +3.89977 q^{25} -1.00000 q^{27} +2.91185i q^{28} -1.91185 q^{29} -2.82371 q^{30} -9.51573i q^{31} +17.6136i q^{32} +2.91185i q^{33} +13.0586i q^{34} -0.582105 q^{35} -5.24698 q^{36} -5.75302i q^{37} +2.02715 q^{38} -9.16852 q^{40} +4.91185i q^{41} -1.49396 q^{42} +11.0978 q^{43} +15.2784i q^{44} -1.04892i q^{45} -15.5133i q^{46} -0.753020i q^{47} -13.0368 q^{48} +6.69202 q^{49} +10.4983i q^{50} -4.85086 q^{51} -7.58211 q^{53} -2.69202i q^{54} -3.05429 q^{55} -4.85086 q^{56} +0.753020i q^{57} -5.14675i q^{58} -4.09783i q^{59} -5.50365i q^{60} -3.42327 q^{61} +25.6165 q^{62} -0.554958i q^{63} -21.3424 q^{64} -7.83877 q^{66} -1.87263i q^{67} -25.4523 q^{68} +5.76271 q^{69} -1.56704i q^{70} -10.5036i q^{71} -8.74094i q^{72} -10.4765i q^{73} +15.4873 q^{74} -3.89977 q^{75} +3.95108i q^{76} -1.61596 q^{77} +1.33513 q^{79} -13.6746i q^{80} +1.00000 q^{81} -13.2228 q^{82} +2.64310i q^{83} -2.91185i q^{84} -5.08815i q^{85} +29.8756i q^{86} +1.91185 q^{87} -25.4523 q^{88} -9.92692i q^{89} +2.82371 q^{90} +30.2368 q^{92} +9.51573i q^{93} +2.02715 q^{94} -0.789856 q^{95} -17.6136i q^{96} +17.0737i q^{97} +18.0151i q^{98} -2.91185i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 22q^{4} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 22q^{4} + 6q^{9} + 2q^{10} + 22q^{12} - 10q^{14} + 22q^{16} + 2q^{17} - 18q^{22} - 22q^{25} - 6q^{27} - 4q^{29} - 2q^{30} + 8q^{35} - 22q^{36} + 6q^{40} + 10q^{42} + 30q^{43} - 22q^{48} + 30q^{49} - 2q^{51} - 34q^{53} + 6q^{55} - 2q^{56} - 26q^{61} + 4q^{62} + 18q^{66} - 26q^{68} + 30q^{74} + 22q^{75} - 30q^{77} + 6q^{79} + 6q^{81} + 6q^{82} + 4q^{87} - 26q^{88} + 2q^{90} + 14q^{92} + 42q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.69202i 1.90355i 0.306802 + 0.951773i \(0.400741\pi\)
−0.306802 + 0.951773i \(0.599259\pi\)
\(3\) −1.00000 −0.577350
\(4\) −5.24698 −2.62349
\(5\) − 1.04892i − 0.469090i −0.972105 0.234545i \(-0.924640\pi\)
0.972105 0.234545i \(-0.0753600\pi\)
\(6\) − 2.69202i − 1.09901i
\(7\) − 0.554958i − 0.209754i −0.994485 0.104877i \(-0.966555\pi\)
0.994485 0.104877i \(-0.0334450\pi\)
\(8\) − 8.74094i − 3.09039i
\(9\) 1.00000 0.333333
\(10\) 2.82371 0.892935
\(11\) − 2.91185i − 0.877957i −0.898498 0.438979i \(-0.855340\pi\)
0.898498 0.438979i \(-0.144660\pi\)
\(12\) 5.24698 1.51467
\(13\) 0 0
\(14\) 1.49396 0.399277
\(15\) 1.04892i 0.270829i
\(16\) 13.0368 3.25921
\(17\) 4.85086 1.17651 0.588253 0.808677i \(-0.299816\pi\)
0.588253 + 0.808677i \(0.299816\pi\)
\(18\) 2.69202i 0.634516i
\(19\) − 0.753020i − 0.172755i −0.996262 0.0863774i \(-0.972471\pi\)
0.996262 0.0863774i \(-0.0275291\pi\)
\(20\) 5.50365i 1.23065i
\(21\) 0.554958i 0.121102i
\(22\) 7.83877 1.67123
\(23\) −5.76271 −1.20161 −0.600804 0.799396i \(-0.705153\pi\)
−0.600804 + 0.799396i \(0.705153\pi\)
\(24\) 8.74094i 1.78424i
\(25\) 3.89977 0.779954
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.91185i 0.550289i
\(29\) −1.91185 −0.355022 −0.177511 0.984119i \(-0.556805\pi\)
−0.177511 + 0.984119i \(0.556805\pi\)
\(30\) −2.82371 −0.515536
\(31\) − 9.51573i − 1.70908i −0.519389 0.854538i \(-0.673841\pi\)
0.519389 0.854538i \(-0.326159\pi\)
\(32\) 17.6136i 3.11367i
\(33\) 2.91185i 0.506889i
\(34\) 13.0586i 2.23953i
\(35\) −0.582105 −0.0983937
\(36\) −5.24698 −0.874497
\(37\) − 5.75302i − 0.945791i −0.881119 0.472895i \(-0.843209\pi\)
0.881119 0.472895i \(-0.156791\pi\)
\(38\) 2.02715 0.328847
\(39\) 0 0
\(40\) −9.16852 −1.44967
\(41\) 4.91185i 0.767103i 0.923520 + 0.383551i \(0.125299\pi\)
−0.923520 + 0.383551i \(0.874701\pi\)
\(42\) −1.49396 −0.230523
\(43\) 11.0978 1.69240 0.846202 0.532862i \(-0.178884\pi\)
0.846202 + 0.532862i \(0.178884\pi\)
\(44\) 15.2784i 2.30331i
\(45\) − 1.04892i − 0.156363i
\(46\) − 15.5133i − 2.28732i
\(47\) − 0.753020i − 0.109839i −0.998491 0.0549197i \(-0.982510\pi\)
0.998491 0.0549197i \(-0.0174903\pi\)
\(48\) −13.0368 −1.88171
\(49\) 6.69202 0.956003
\(50\) 10.4983i 1.48468i
\(51\) −4.85086 −0.679256
\(52\) 0 0
\(53\) −7.58211 −1.04148 −0.520741 0.853715i \(-0.674344\pi\)
−0.520741 + 0.853715i \(0.674344\pi\)
\(54\) − 2.69202i − 0.366338i
\(55\) −3.05429 −0.411841
\(56\) −4.85086 −0.648223
\(57\) 0.753020i 0.0997400i
\(58\) − 5.14675i − 0.675802i
\(59\) − 4.09783i − 0.533493i −0.963767 0.266746i \(-0.914051\pi\)
0.963767 0.266746i \(-0.0859486\pi\)
\(60\) − 5.50365i − 0.710518i
\(61\) −3.42327 −0.438305 −0.219153 0.975691i \(-0.570329\pi\)
−0.219153 + 0.975691i \(0.570329\pi\)
\(62\) 25.6165 3.25330
\(63\) − 0.554958i − 0.0699182i
\(64\) −21.3424 −2.66780
\(65\) 0 0
\(66\) −7.83877 −0.964886
\(67\) − 1.87263i − 0.228778i −0.993436 0.114389i \(-0.963509\pi\)
0.993436 0.114389i \(-0.0364910\pi\)
\(68\) −25.4523 −3.08655
\(69\) 5.76271 0.693749
\(70\) − 1.56704i − 0.187297i
\(71\) − 10.5036i − 1.24655i −0.782001 0.623277i \(-0.785801\pi\)
0.782001 0.623277i \(-0.214199\pi\)
\(72\) − 8.74094i − 1.03013i
\(73\) − 10.4765i − 1.22618i −0.790012 0.613091i \(-0.789926\pi\)
0.790012 0.613091i \(-0.210074\pi\)
\(74\) 15.4873 1.80036
\(75\) −3.89977 −0.450307
\(76\) 3.95108i 0.453220i
\(77\) −1.61596 −0.184155
\(78\) 0 0
\(79\) 1.33513 0.150213 0.0751067 0.997176i \(-0.476070\pi\)
0.0751067 + 0.997176i \(0.476070\pi\)
\(80\) − 13.6746i − 1.52886i
\(81\) 1.00000 0.111111
\(82\) −13.2228 −1.46022
\(83\) 2.64310i 0.290118i 0.989423 + 0.145059i \(0.0463373\pi\)
−0.989423 + 0.145059i \(0.953663\pi\)
\(84\) − 2.91185i − 0.317709i
\(85\) − 5.08815i − 0.551887i
\(86\) 29.8756i 3.22157i
\(87\) 1.91185 0.204972
\(88\) −25.4523 −2.71323
\(89\) − 9.92692i − 1.05225i −0.850407 0.526126i \(-0.823644\pi\)
0.850407 0.526126i \(-0.176356\pi\)
\(90\) 2.82371 0.297645
\(91\) 0 0
\(92\) 30.2368 3.15241
\(93\) 9.51573i 0.986735i
\(94\) 2.02715 0.209084
\(95\) −0.789856 −0.0810375
\(96\) − 17.6136i − 1.79768i
\(97\) 17.0737i 1.73357i 0.498683 + 0.866784i \(0.333817\pi\)
−0.498683 + 0.866784i \(0.666183\pi\)
\(98\) 18.0151i 1.81980i
\(99\) − 2.91185i − 0.292652i
\(100\) −20.4620 −2.04620
\(101\) −7.32304 −0.728670 −0.364335 0.931268i \(-0.618704\pi\)
−0.364335 + 0.931268i \(0.618704\pi\)
\(102\) − 13.0586i − 1.29299i
\(103\) 4.21983 0.415792 0.207896 0.978151i \(-0.433338\pi\)
0.207896 + 0.978151i \(0.433338\pi\)
\(104\) 0 0
\(105\) 0.582105 0.0568077
\(106\) − 20.4112i − 1.98251i
\(107\) 6.39373 0.618105 0.309053 0.951045i \(-0.399988\pi\)
0.309053 + 0.951045i \(0.399988\pi\)
\(108\) 5.24698 0.504891
\(109\) − 3.46011i − 0.331418i −0.986175 0.165709i \(-0.947009\pi\)
0.986175 0.165709i \(-0.0529913\pi\)
\(110\) − 8.22223i − 0.783958i
\(111\) 5.75302i 0.546053i
\(112\) − 7.23490i − 0.683634i
\(113\) 9.35690 0.880223 0.440111 0.897943i \(-0.354939\pi\)
0.440111 + 0.897943i \(0.354939\pi\)
\(114\) −2.02715 −0.189860
\(115\) 6.04461i 0.563662i
\(116\) 10.0315 0.931398
\(117\) 0 0
\(118\) 11.0315 1.01553
\(119\) − 2.69202i − 0.246777i
\(120\) 9.16852 0.836968
\(121\) 2.52111 0.229191
\(122\) − 9.21552i − 0.834334i
\(123\) − 4.91185i − 0.442887i
\(124\) 49.9288i 4.48374i
\(125\) − 9.33513i − 0.834959i
\(126\) 1.49396 0.133092
\(127\) 4.48188 0.397702 0.198851 0.980030i \(-0.436279\pi\)
0.198851 + 0.980030i \(0.436279\pi\)
\(128\) − 22.2271i − 1.96462i
\(129\) −11.0978 −0.977110
\(130\) 0 0
\(131\) −9.21744 −0.805331 −0.402666 0.915347i \(-0.631916\pi\)
−0.402666 + 0.915347i \(0.631916\pi\)
\(132\) − 15.2784i − 1.32982i
\(133\) −0.417895 −0.0362361
\(134\) 5.04115 0.435489
\(135\) 1.04892i 0.0902764i
\(136\) − 42.4010i − 3.63586i
\(137\) 7.46980i 0.638188i 0.947723 + 0.319094i \(0.103379\pi\)
−0.947723 + 0.319094i \(0.896621\pi\)
\(138\) 15.5133i 1.32058i
\(139\) −17.9976 −1.52654 −0.763269 0.646081i \(-0.776407\pi\)
−0.763269 + 0.646081i \(0.776407\pi\)
\(140\) 3.05429 0.258135
\(141\) 0.753020i 0.0634158i
\(142\) 28.2760 2.37287
\(143\) 0 0
\(144\) 13.0368 1.08640
\(145\) 2.00538i 0.166537i
\(146\) 28.2030 2.33409
\(147\) −6.69202 −0.551949
\(148\) 30.1860i 2.48127i
\(149\) 15.3351i 1.25630i 0.778091 + 0.628151i \(0.216188\pi\)
−0.778091 + 0.628151i \(0.783812\pi\)
\(150\) − 10.4983i − 0.857180i
\(151\) − 2.53079i − 0.205953i −0.994684 0.102977i \(-0.967163\pi\)
0.994684 0.102977i \(-0.0328367\pi\)
\(152\) −6.58211 −0.533879
\(153\) 4.85086 0.392168
\(154\) − 4.35019i − 0.350548i
\(155\) −9.98121 −0.801710
\(156\) 0 0
\(157\) 17.2392 1.37584 0.687919 0.725787i \(-0.258524\pi\)
0.687919 + 0.725787i \(0.258524\pi\)
\(158\) 3.59419i 0.285938i
\(159\) 7.58211 0.601300
\(160\) 18.4752 1.46059
\(161\) 3.19806i 0.252043i
\(162\) 2.69202i 0.211505i
\(163\) − 15.7071i − 1.23027i −0.788420 0.615137i \(-0.789101\pi\)
0.788420 0.615137i \(-0.210899\pi\)
\(164\) − 25.7724i − 2.01249i
\(165\) 3.05429 0.237776
\(166\) −7.11529 −0.552254
\(167\) − 5.39612i − 0.417565i −0.977962 0.208782i \(-0.933050\pi\)
0.977962 0.208782i \(-0.0669500\pi\)
\(168\) 4.85086 0.374252
\(169\) 0 0
\(170\) 13.6974 1.05054
\(171\) − 0.753020i − 0.0575849i
\(172\) −58.2301 −4.44000
\(173\) −23.9420 −1.82028 −0.910138 0.414306i \(-0.864024\pi\)
−0.910138 + 0.414306i \(0.864024\pi\)
\(174\) 5.14675i 0.390174i
\(175\) − 2.16421i − 0.163599i
\(176\) − 37.9614i − 2.86145i
\(177\) 4.09783i 0.308012i
\(178\) 26.7235 2.00301
\(179\) −18.4088 −1.37594 −0.687969 0.725740i \(-0.741498\pi\)
−0.687969 + 0.725740i \(0.741498\pi\)
\(180\) 5.50365i 0.410218i
\(181\) 3.63342 0.270070 0.135035 0.990841i \(-0.456885\pi\)
0.135035 + 0.990841i \(0.456885\pi\)
\(182\) 0 0
\(183\) 3.42327 0.253056
\(184\) 50.3715i 3.71344i
\(185\) −6.03444 −0.443661
\(186\) −25.6165 −1.87830
\(187\) − 14.1250i − 1.03292i
\(188\) 3.95108i 0.288162i
\(189\) 0.554958i 0.0403673i
\(190\) − 2.12631i − 0.154259i
\(191\) −21.1782 −1.53240 −0.766201 0.642601i \(-0.777855\pi\)
−0.766201 + 0.642601i \(0.777855\pi\)
\(192\) 21.3424 1.54026
\(193\) 17.6112i 1.26768i 0.773464 + 0.633840i \(0.218522\pi\)
−0.773464 + 0.633840i \(0.781478\pi\)
\(194\) −45.9627 −3.29993
\(195\) 0 0
\(196\) −35.1129 −2.50806
\(197\) − 4.66248i − 0.332188i −0.986110 0.166094i \(-0.946884\pi\)
0.986110 0.166094i \(-0.0531155\pi\)
\(198\) 7.83877 0.557077
\(199\) −15.0368 −1.06593 −0.532967 0.846136i \(-0.678923\pi\)
−0.532967 + 0.846136i \(0.678923\pi\)
\(200\) − 34.0877i − 2.41036i
\(201\) 1.87263i 0.132085i
\(202\) − 19.7138i − 1.38706i
\(203\) 1.06100i 0.0744675i
\(204\) 25.4523 1.78202
\(205\) 5.15213 0.359840
\(206\) 11.3599i 0.791480i
\(207\) −5.76271 −0.400536
\(208\) 0 0
\(209\) −2.19269 −0.151671
\(210\) 1.56704i 0.108136i
\(211\) −0.460107 −0.0316751 −0.0158375 0.999875i \(-0.505041\pi\)
−0.0158375 + 0.999875i \(0.505041\pi\)
\(212\) 39.7832 2.73232
\(213\) 10.5036i 0.719698i
\(214\) 17.2121i 1.17659i
\(215\) − 11.6407i − 0.793890i
\(216\) 8.74094i 0.594746i
\(217\) −5.28083 −0.358486
\(218\) 9.31468 0.630870
\(219\) 10.4765i 0.707936i
\(220\) 16.0258 1.08046
\(221\) 0 0
\(222\) −15.4873 −1.03944
\(223\) − 16.3502i − 1.09489i −0.836842 0.547445i \(-0.815601\pi\)
0.836842 0.547445i \(-0.184399\pi\)
\(224\) 9.77479 0.653106
\(225\) 3.89977 0.259985
\(226\) 25.1890i 1.67555i
\(227\) 6.56033i 0.435425i 0.976013 + 0.217712i \(0.0698595\pi\)
−0.976013 + 0.217712i \(0.930141\pi\)
\(228\) − 3.95108i − 0.261667i
\(229\) 3.95539i 0.261380i 0.991423 + 0.130690i \(0.0417192\pi\)
−0.991423 + 0.130690i \(0.958281\pi\)
\(230\) −16.2722 −1.07296
\(231\) 1.61596 0.106322
\(232\) 16.7114i 1.09716i
\(233\) −8.35690 −0.547478 −0.273739 0.961804i \(-0.588261\pi\)
−0.273739 + 0.961804i \(0.588261\pi\)
\(234\) 0 0
\(235\) −0.789856 −0.0515245
\(236\) 21.5013i 1.39961i
\(237\) −1.33513 −0.0867257
\(238\) 7.24698 0.469752
\(239\) 20.1008i 1.30021i 0.759843 + 0.650107i \(0.225276\pi\)
−0.759843 + 0.650107i \(0.774724\pi\)
\(240\) 13.6746i 0.882689i
\(241\) 19.0127i 1.22471i 0.790581 + 0.612357i \(0.209778\pi\)
−0.790581 + 0.612357i \(0.790222\pi\)
\(242\) 6.78687i 0.436277i
\(243\) −1.00000 −0.0641500
\(244\) 17.9618 1.14989
\(245\) − 7.01938i − 0.448452i
\(246\) 13.2228 0.843056
\(247\) 0 0
\(248\) −83.1764 −5.28171
\(249\) − 2.64310i − 0.167500i
\(250\) 25.1304 1.58938
\(251\) 0.763774 0.0482090 0.0241045 0.999709i \(-0.492327\pi\)
0.0241045 + 0.999709i \(0.492327\pi\)
\(252\) 2.91185i 0.183430i
\(253\) 16.7802i 1.05496i
\(254\) 12.0653i 0.757045i
\(255\) 5.08815i 0.318632i
\(256\) 17.1511 1.07194
\(257\) 13.0911 0.816602 0.408301 0.912847i \(-0.366121\pi\)
0.408301 + 0.912847i \(0.366121\pi\)
\(258\) − 29.8756i − 1.85997i
\(259\) −3.19269 −0.198384
\(260\) 0 0
\(261\) −1.91185 −0.118341
\(262\) − 24.8135i − 1.53299i
\(263\) 18.3773 1.13320 0.566598 0.823995i \(-0.308259\pi\)
0.566598 + 0.823995i \(0.308259\pi\)
\(264\) 25.4523 1.56648
\(265\) 7.95300i 0.488549i
\(266\) − 1.12498i − 0.0689771i
\(267\) 9.92692i 0.607518i
\(268\) 9.82563i 0.600196i
\(269\) 23.6625 1.44273 0.721363 0.692557i \(-0.243516\pi\)
0.721363 + 0.692557i \(0.243516\pi\)
\(270\) −2.82371 −0.171845
\(271\) 19.7530i 1.19991i 0.800034 + 0.599955i \(0.204815\pi\)
−0.800034 + 0.599955i \(0.795185\pi\)
\(272\) 63.2398 3.83448
\(273\) 0 0
\(274\) −20.1089 −1.21482
\(275\) − 11.3556i − 0.684767i
\(276\) −30.2368 −1.82004
\(277\) −1.77777 −0.106816 −0.0534081 0.998573i \(-0.517008\pi\)
−0.0534081 + 0.998573i \(0.517008\pi\)
\(278\) − 48.4499i − 2.90583i
\(279\) − 9.51573i − 0.569692i
\(280\) 5.08815i 0.304075i
\(281\) − 1.62133i − 0.0967207i −0.998830 0.0483603i \(-0.984600\pi\)
0.998830 0.0483603i \(-0.0153996\pi\)
\(282\) −2.02715 −0.120715
\(283\) −4.98361 −0.296245 −0.148122 0.988969i \(-0.547323\pi\)
−0.148122 + 0.988969i \(0.547323\pi\)
\(284\) 55.1124i 3.27032i
\(285\) 0.789856 0.0467870
\(286\) 0 0
\(287\) 2.72587 0.160903
\(288\) 17.6136i 1.03789i
\(289\) 6.53079 0.384164
\(290\) −5.39852 −0.317012
\(291\) − 17.0737i − 1.00088i
\(292\) 54.9700i 3.21688i
\(293\) 0.0717525i 0.00419183i 0.999998 + 0.00209591i \(0.000667151\pi\)
−0.999998 + 0.00209591i \(0.999333\pi\)
\(294\) − 18.0151i − 1.05066i
\(295\) −4.29829 −0.250256
\(296\) −50.2868 −2.92286
\(297\) 2.91185i 0.168963i
\(298\) −41.2825 −2.39143
\(299\) 0 0
\(300\) 20.4620 1.18138
\(301\) − 6.15883i − 0.354989i
\(302\) 6.81295 0.392041
\(303\) 7.32304 0.420698
\(304\) − 9.81700i − 0.563044i
\(305\) 3.59073i 0.205605i
\(306\) 13.0586i 0.746511i
\(307\) 5.19806i 0.296669i 0.988937 + 0.148335i \(0.0473912\pi\)
−0.988937 + 0.148335i \(0.952609\pi\)
\(308\) 8.47889 0.483130
\(309\) −4.21983 −0.240058
\(310\) − 26.8696i − 1.52609i
\(311\) 22.5429 1.27829 0.639145 0.769087i \(-0.279289\pi\)
0.639145 + 0.769087i \(0.279289\pi\)
\(312\) 0 0
\(313\) 22.6612 1.28088 0.640442 0.768006i \(-0.278751\pi\)
0.640442 + 0.768006i \(0.278751\pi\)
\(314\) 46.4083i 2.61897i
\(315\) −0.582105 −0.0327979
\(316\) −7.00538 −0.394083
\(317\) 26.3424i 1.47954i 0.672861 + 0.739769i \(0.265065\pi\)
−0.672861 + 0.739769i \(0.734935\pi\)
\(318\) 20.4112i 1.14460i
\(319\) 5.56704i 0.311694i
\(320\) 22.3864i 1.25144i
\(321\) −6.39373 −0.356863
\(322\) −8.60925 −0.479775
\(323\) − 3.65279i − 0.203247i
\(324\) −5.24698 −0.291499
\(325\) 0 0
\(326\) 42.2838 2.34188
\(327\) 3.46011i 0.191344i
\(328\) 42.9342 2.37065
\(329\) −0.417895 −0.0230393
\(330\) 8.22223i 0.452619i
\(331\) − 11.2295i − 0.617230i −0.951187 0.308615i \(-0.900134\pi\)
0.951187 0.308615i \(-0.0998655\pi\)
\(332\) − 13.8683i − 0.761123i
\(333\) − 5.75302i − 0.315264i
\(334\) 14.5265 0.794854
\(335\) −1.96423 −0.107317
\(336\) 7.23490i 0.394696i
\(337\) −2.30798 −0.125724 −0.0628618 0.998022i \(-0.520023\pi\)
−0.0628618 + 0.998022i \(0.520023\pi\)
\(338\) 0 0
\(339\) −9.35690 −0.508197
\(340\) 26.6974i 1.44787i
\(341\) −27.7084 −1.50049
\(342\) 2.02715 0.109616
\(343\) − 7.59850i − 0.410280i
\(344\) − 97.0055i − 5.23019i
\(345\) − 6.04461i − 0.325431i
\(346\) − 64.4523i − 3.46498i
\(347\) −9.13706 −0.490503 −0.245252 0.969459i \(-0.578871\pi\)
−0.245252 + 0.969459i \(0.578871\pi\)
\(348\) −10.0315 −0.537743
\(349\) − 23.9758i − 1.28340i −0.766957 0.641699i \(-0.778230\pi\)
0.766957 0.641699i \(-0.221770\pi\)
\(350\) 5.82610 0.311418
\(351\) 0 0
\(352\) 51.2881 2.73367
\(353\) − 27.1239i − 1.44366i −0.692070 0.721830i \(-0.743301\pi\)
0.692070 0.721830i \(-0.256699\pi\)
\(354\) −11.0315 −0.586315
\(355\) −11.0175 −0.584746
\(356\) 52.0863i 2.76057i
\(357\) 2.69202i 0.142477i
\(358\) − 49.5569i − 2.61916i
\(359\) 26.0790i 1.37640i 0.725521 + 0.688200i \(0.241599\pi\)
−0.725521 + 0.688200i \(0.758401\pi\)
\(360\) −9.16852 −0.483224
\(361\) 18.4330 0.970156
\(362\) 9.78123i 0.514090i
\(363\) −2.52111 −0.132324
\(364\) 0 0
\(365\) −10.9890 −0.575190
\(366\) 9.21552i 0.481703i
\(367\) 9.57434 0.499776 0.249888 0.968275i \(-0.419606\pi\)
0.249888 + 0.968275i \(0.419606\pi\)
\(368\) −75.1275 −3.91629
\(369\) 4.91185i 0.255701i
\(370\) − 16.2448i − 0.844530i
\(371\) 4.20775i 0.218456i
\(372\) − 49.9288i − 2.58869i
\(373\) 28.1497 1.45754 0.728769 0.684760i \(-0.240093\pi\)
0.728769 + 0.684760i \(0.240093\pi\)
\(374\) 38.0248 1.96621
\(375\) 9.33513i 0.482064i
\(376\) −6.58211 −0.339446
\(377\) 0 0
\(378\) −1.49396 −0.0768410
\(379\) 16.0465i 0.824255i 0.911126 + 0.412127i \(0.135214\pi\)
−0.911126 + 0.412127i \(0.864786\pi\)
\(380\) 4.14436 0.212601
\(381\) −4.48188 −0.229614
\(382\) − 57.0122i − 2.91700i
\(383\) − 24.6165i − 1.25785i −0.777467 0.628923i \(-0.783496\pi\)
0.777467 0.628923i \(-0.216504\pi\)
\(384\) 22.2271i 1.13427i
\(385\) 1.69501i 0.0863855i
\(386\) −47.4097 −2.41309
\(387\) 11.0978 0.564135
\(388\) − 89.5852i − 4.54800i
\(389\) 17.2198 0.873080 0.436540 0.899685i \(-0.356204\pi\)
0.436540 + 0.899685i \(0.356204\pi\)
\(390\) 0 0
\(391\) −27.9541 −1.41370
\(392\) − 58.4946i − 2.95442i
\(393\) 9.21744 0.464958
\(394\) 12.5515 0.632335
\(395\) − 1.40044i − 0.0704636i
\(396\) 15.2784i 0.767770i
\(397\) 2.03923i 0.102346i 0.998690 + 0.0511730i \(0.0162960\pi\)
−0.998690 + 0.0511730i \(0.983704\pi\)
\(398\) − 40.4795i − 2.02905i
\(399\) 0.417895 0.0209209
\(400\) 50.8407 2.54203
\(401\) 1.46144i 0.0729806i 0.999334 + 0.0364903i \(0.0116178\pi\)
−0.999334 + 0.0364903i \(0.988382\pi\)
\(402\) −5.04115 −0.251430
\(403\) 0 0
\(404\) 38.4239 1.91166
\(405\) − 1.04892i − 0.0521211i
\(406\) −2.85623 −0.141752
\(407\) −16.7520 −0.830364
\(408\) 42.4010i 2.09916i
\(409\) − 29.9390i − 1.48039i −0.672393 0.740194i \(-0.734734\pi\)
0.672393 0.740194i \(-0.265266\pi\)
\(410\) 13.8696i 0.684973i
\(411\) − 7.46980i − 0.368458i
\(412\) −22.1414 −1.09083
\(413\) −2.27413 −0.111902
\(414\) − 15.5133i − 0.762439i
\(415\) 2.77240 0.136092
\(416\) 0 0
\(417\) 17.9976 0.881347
\(418\) − 5.90276i − 0.288713i
\(419\) 6.64742 0.324748 0.162374 0.986729i \(-0.448085\pi\)
0.162374 + 0.986729i \(0.448085\pi\)
\(420\) −3.05429 −0.149034
\(421\) 13.5646i 0.661100i 0.943788 + 0.330550i \(0.107234\pi\)
−0.943788 + 0.330550i \(0.892766\pi\)
\(422\) − 1.23862i − 0.0602950i
\(423\) − 0.753020i − 0.0366131i
\(424\) 66.2747i 3.21858i
\(425\) 18.9172 0.917620
\(426\) −28.2760 −1.36998
\(427\) 1.89977i 0.0919364i
\(428\) −33.5478 −1.62159
\(429\) 0 0
\(430\) 31.3370 1.51121
\(431\) 35.9463i 1.73147i 0.500501 + 0.865736i \(0.333149\pi\)
−0.500501 + 0.865736i \(0.666851\pi\)
\(432\) −13.0368 −0.627235
\(433\) 32.4741 1.56061 0.780303 0.625402i \(-0.215065\pi\)
0.780303 + 0.625402i \(0.215065\pi\)
\(434\) − 14.2161i − 0.682395i
\(435\) − 2.00538i − 0.0961505i
\(436\) 18.1551i 0.869472i
\(437\) 4.33944i 0.207583i
\(438\) −28.2030 −1.34759
\(439\) −12.8321 −0.612441 −0.306221 0.951961i \(-0.599065\pi\)
−0.306221 + 0.951961i \(0.599065\pi\)
\(440\) 26.6974i 1.27275i
\(441\) 6.69202 0.318668
\(442\) 0 0
\(443\) 11.9608 0.568273 0.284137 0.958784i \(-0.408293\pi\)
0.284137 + 0.958784i \(0.408293\pi\)
\(444\) − 30.1860i − 1.43256i
\(445\) −10.4125 −0.493601
\(446\) 44.0151 2.08417
\(447\) − 15.3351i − 0.725327i
\(448\) 11.8442i 0.559584i
\(449\) 12.4789i 0.588915i 0.955665 + 0.294458i \(0.0951390\pi\)
−0.955665 + 0.294458i \(0.904861\pi\)
\(450\) 10.4983i 0.494893i
\(451\) 14.3026 0.673483
\(452\) −49.0954 −2.30926
\(453\) 2.53079i 0.118907i
\(454\) −17.6606 −0.828851
\(455\) 0 0
\(456\) 6.58211 0.308235
\(457\) 32.4523i 1.51806i 0.651058 + 0.759028i \(0.274326\pi\)
−0.651058 + 0.759028i \(0.725674\pi\)
\(458\) −10.6480 −0.497549
\(459\) −4.85086 −0.226419
\(460\) − 31.7159i − 1.47876i
\(461\) 24.4034i 1.13658i 0.822828 + 0.568290i \(0.192395\pi\)
−0.822828 + 0.568290i \(0.807605\pi\)
\(462\) 4.35019i 0.202389i
\(463\) 33.1836i 1.54217i 0.636731 + 0.771086i \(0.280286\pi\)
−0.636731 + 0.771086i \(0.719714\pi\)
\(464\) −24.9245 −1.15709
\(465\) 9.98121 0.462868
\(466\) − 22.4969i − 1.04215i
\(467\) 38.5206 1.78252 0.891261 0.453490i \(-0.149821\pi\)
0.891261 + 0.453490i \(0.149821\pi\)
\(468\) 0 0
\(469\) −1.03923 −0.0479871
\(470\) − 2.12631i − 0.0980794i
\(471\) −17.2392 −0.794341
\(472\) −35.8189 −1.64870
\(473\) − 32.3153i − 1.48586i
\(474\) − 3.59419i − 0.165086i
\(475\) − 2.93661i − 0.134741i
\(476\) 14.1250i 0.647417i
\(477\) −7.58211 −0.347161
\(478\) −54.1118 −2.47502
\(479\) 8.34481i 0.381284i 0.981660 + 0.190642i \(0.0610570\pi\)
−0.981660 + 0.190642i \(0.938943\pi\)
\(480\) −18.4752 −0.843272
\(481\) 0 0
\(482\) −51.1825 −2.33130
\(483\) − 3.19806i − 0.145517i
\(484\) −13.2282 −0.601282
\(485\) 17.9089 0.813200
\(486\) − 2.69202i − 0.122113i
\(487\) − 14.8586i − 0.673309i −0.941628 0.336654i \(-0.890705\pi\)
0.941628 0.336654i \(-0.109295\pi\)
\(488\) 29.9226i 1.35453i
\(489\) 15.7071i 0.710299i
\(490\) 18.8963 0.853648
\(491\) −4.99894 −0.225599 −0.112799 0.993618i \(-0.535982\pi\)
−0.112799 + 0.993618i \(0.535982\pi\)
\(492\) 25.7724i 1.16191i
\(493\) −9.27413 −0.417686
\(494\) 0 0
\(495\) −3.05429 −0.137280
\(496\) − 124.055i − 5.57023i
\(497\) −5.82908 −0.261470
\(498\) 7.11529 0.318844
\(499\) − 0.385371i − 0.0172516i −0.999963 0.00862579i \(-0.997254\pi\)
0.999963 0.00862579i \(-0.00274571\pi\)
\(500\) 48.9812i 2.19051i
\(501\) 5.39612i 0.241081i
\(502\) 2.05610i 0.0917681i
\(503\) 22.6179 1.00848 0.504241 0.863563i \(-0.331772\pi\)
0.504241 + 0.863563i \(0.331772\pi\)
\(504\) −4.85086 −0.216074
\(505\) 7.68127i 0.341812i
\(506\) −45.1726 −2.00817
\(507\) 0 0
\(508\) −23.5163 −1.04337
\(509\) − 11.6039i − 0.514333i −0.966367 0.257166i \(-0.917211\pi\)
0.966367 0.257166i \(-0.0827888\pi\)
\(510\) −13.6974 −0.606531
\(511\) −5.81402 −0.257197
\(512\) 1.71678i 0.0758715i
\(513\) 0.753020i 0.0332467i
\(514\) 35.2416i 1.55444i
\(515\) − 4.42626i − 0.195044i
\(516\) 58.2301 2.56344
\(517\) −2.19269 −0.0964342
\(518\) − 8.59478i − 0.377633i
\(519\) 23.9420 1.05094
\(520\) 0 0
\(521\) −1.62671 −0.0712675 −0.0356337 0.999365i \(-0.511345\pi\)
−0.0356337 + 0.999365i \(0.511345\pi\)
\(522\) − 5.14675i − 0.225267i
\(523\) 10.0718 0.440407 0.220203 0.975454i \(-0.429328\pi\)
0.220203 + 0.975454i \(0.429328\pi\)
\(524\) 48.3637 2.11278
\(525\) 2.16421i 0.0944539i
\(526\) 49.4722i 2.15709i
\(527\) − 46.1594i − 2.01074i
\(528\) 37.9614i 1.65206i
\(529\) 10.2088 0.443862
\(530\) −21.4097 −0.929976
\(531\) − 4.09783i − 0.177831i
\(532\) 2.19269 0.0950650
\(533\) 0 0
\(534\) −26.7235 −1.15644
\(535\) − 6.70650i − 0.289947i
\(536\) −16.3685 −0.707012
\(537\) 18.4088 0.794398
\(538\) 63.6999i 2.74630i
\(539\) − 19.4862i − 0.839330i
\(540\) − 5.50365i − 0.236839i
\(541\) 20.4674i 0.879962i 0.898007 + 0.439981i \(0.145015\pi\)
−0.898007 + 0.439981i \(0.854985\pi\)
\(542\) −53.1756 −2.28409
\(543\) −3.63342 −0.155925
\(544\) 85.4408i 3.66325i
\(545\) −3.62937 −0.155465
\(546\) 0 0
\(547\) −27.5478 −1.17786 −0.588929 0.808185i \(-0.700450\pi\)
−0.588929 + 0.808185i \(0.700450\pi\)
\(548\) − 39.1939i − 1.67428i
\(549\) −3.42327 −0.146102
\(550\) 30.5694 1.30348
\(551\) 1.43967i 0.0613318i
\(552\) − 50.3715i − 2.14395i
\(553\) − 0.740939i − 0.0315079i
\(554\) − 4.78581i − 0.203329i
\(555\) 6.03444 0.256148
\(556\) 94.4331 4.00485
\(557\) − 37.9855i − 1.60950i −0.593615 0.804749i \(-0.702300\pi\)
0.593615 0.804749i \(-0.297700\pi\)
\(558\) 25.6165 1.08443
\(559\) 0 0
\(560\) −7.58881 −0.320686
\(561\) 14.1250i 0.596357i
\(562\) 4.36467 0.184112
\(563\) 29.7724 1.25476 0.627378 0.778714i \(-0.284128\pi\)
0.627378 + 0.778714i \(0.284128\pi\)
\(564\) − 3.95108i − 0.166371i
\(565\) − 9.81461i − 0.412904i
\(566\) − 13.4160i − 0.563916i
\(567\) − 0.554958i − 0.0233061i
\(568\) −91.8117 −3.85234
\(569\) 21.9541 0.920362 0.460181 0.887825i \(-0.347784\pi\)
0.460181 + 0.887825i \(0.347784\pi\)
\(570\) 2.12631i 0.0890613i
\(571\) −2.46575 −0.103188 −0.0515942 0.998668i \(-0.516430\pi\)
−0.0515942 + 0.998668i \(0.516430\pi\)
\(572\) 0 0
\(573\) 21.1782 0.884732
\(574\) 7.33811i 0.306287i
\(575\) −22.4733 −0.937199
\(576\) −21.3424 −0.889268
\(577\) 17.4547i 0.726650i 0.931662 + 0.363325i \(0.118359\pi\)
−0.931662 + 0.363325i \(0.881641\pi\)
\(578\) 17.5810i 0.731275i
\(579\) − 17.6112i − 0.731895i
\(580\) − 10.5222i − 0.436909i
\(581\) 1.46681 0.0608536
\(582\) 45.9627 1.90521
\(583\) 22.0780i 0.914377i
\(584\) −91.5745 −3.78938
\(585\) 0 0
\(586\) −0.193159 −0.00797934
\(587\) 6.26337i 0.258517i 0.991611 + 0.129259i \(0.0412597\pi\)
−0.991611 + 0.129259i \(0.958740\pi\)
\(588\) 35.1129 1.44803
\(589\) −7.16554 −0.295251
\(590\) − 11.5711i − 0.476374i
\(591\) 4.66248i 0.191789i
\(592\) − 75.0012i − 3.08253i
\(593\) − 22.8745i − 0.939345i −0.882841 0.469672i \(-0.844372\pi\)
0.882841 0.469672i \(-0.155628\pi\)
\(594\) −7.83877 −0.321629
\(595\) −2.82371 −0.115761
\(596\) − 80.4631i − 3.29590i
\(597\) 15.0368 0.615417
\(598\) 0 0
\(599\) −1.05621 −0.0431557 −0.0215778 0.999767i \(-0.506869\pi\)
−0.0215778 + 0.999767i \(0.506869\pi\)
\(600\) 34.0877i 1.39162i
\(601\) −33.3236 −1.35930 −0.679650 0.733537i \(-0.737868\pi\)
−0.679650 + 0.733537i \(0.737868\pi\)
\(602\) 16.5797 0.675739
\(603\) − 1.87263i − 0.0762592i
\(604\) 13.2790i 0.540316i
\(605\) − 2.64443i − 0.107511i
\(606\) 19.7138i 0.800818i
\(607\) 16.2403 0.659172 0.329586 0.944125i \(-0.393091\pi\)
0.329586 + 0.944125i \(0.393091\pi\)
\(608\) 13.2634 0.537901
\(609\) − 1.06100i − 0.0429938i
\(610\) −9.66632 −0.391378
\(611\) 0 0
\(612\) −25.4523 −1.02885
\(613\) 11.0479i 0.446219i 0.974793 + 0.223109i \(0.0716207\pi\)
−0.974793 + 0.223109i \(0.928379\pi\)
\(614\) −13.9933 −0.564723
\(615\) −5.15213 −0.207754
\(616\) 14.1250i 0.569112i
\(617\) − 4.65950i − 0.187584i −0.995592 0.0937922i \(-0.970101\pi\)
0.995592 0.0937922i \(-0.0298989\pi\)
\(618\) − 11.3599i − 0.456961i
\(619\) − 31.9259i − 1.28321i −0.767036 0.641604i \(-0.778269\pi\)
0.767036 0.641604i \(-0.221731\pi\)
\(620\) 52.3712 2.10328
\(621\) 5.76271 0.231250
\(622\) 60.6859i 2.43328i
\(623\) −5.50902 −0.220714
\(624\) 0 0
\(625\) 9.70709 0.388283
\(626\) 61.0043i 2.43822i
\(627\) 2.19269 0.0875674
\(628\) −90.4538 −3.60950
\(629\) − 27.9071i − 1.11273i
\(630\) − 1.56704i − 0.0624324i
\(631\) 39.4413i 1.57013i 0.619411 + 0.785067i \(0.287372\pi\)
−0.619411 + 0.785067i \(0.712628\pi\)
\(632\) − 11.6703i − 0.464218i
\(633\) 0.460107 0.0182876
\(634\) −70.9144 −2.81637
\(635\) − 4.70112i − 0.186558i
\(636\) −39.7832 −1.57750
\(637\) 0 0
\(638\) −14.9866 −0.593325
\(639\) − 10.5036i − 0.415518i
\(640\) −23.3144 −0.921583
\(641\) −45.4510 −1.79521 −0.897603 0.440804i \(-0.854693\pi\)
−0.897603 + 0.440804i \(0.854693\pi\)
\(642\) − 17.2121i − 0.679306i
\(643\) − 29.7469i − 1.17310i −0.809912 0.586552i \(-0.800485\pi\)
0.809912 0.586552i \(-0.199515\pi\)
\(644\) − 16.7802i − 0.661231i
\(645\) 11.6407i 0.458353i
\(646\) 9.83340 0.386890
\(647\) 16.9312 0.665635 0.332818 0.942991i \(-0.392001\pi\)
0.332818 + 0.942991i \(0.392001\pi\)
\(648\) − 8.74094i − 0.343377i
\(649\) −11.9323 −0.468384
\(650\) 0 0
\(651\) 5.28083 0.206972
\(652\) 82.4148i 3.22761i
\(653\) −33.1976 −1.29912 −0.649561 0.760309i \(-0.725047\pi\)
−0.649561 + 0.760309i \(0.725047\pi\)
\(654\) −9.31468 −0.364233
\(655\) 9.66833i 0.377773i
\(656\) 64.0350i 2.50015i
\(657\) − 10.4765i − 0.408727i
\(658\) − 1.12498i − 0.0438564i
\(659\) −42.6571 −1.66168 −0.830842 0.556508i \(-0.812141\pi\)
−0.830842 + 0.556508i \(0.812141\pi\)
\(660\) −16.0258 −0.623804
\(661\) − 38.6902i − 1.50488i −0.658664 0.752438i \(-0.728878\pi\)
0.658664 0.752438i \(-0.271122\pi\)
\(662\) 30.2301 1.17493
\(663\) 0 0
\(664\) 23.1032 0.896578
\(665\) 0.438337i 0.0169980i
\(666\) 15.4873 0.600119
\(667\) 11.0175 0.426598
\(668\) 28.3134i 1.09548i
\(669\) 16.3502i 0.632135i
\(670\) − 5.28775i − 0.204283i
\(671\) 9.96807i 0.384813i
\(672\) −9.77479 −0.377071
\(673\) 2.59419 0.0999986 0.0499993 0.998749i \(-0.484078\pi\)
0.0499993 + 0.998749i \(0.484078\pi\)
\(674\) − 6.21313i − 0.239321i
\(675\) −3.89977 −0.150102
\(676\) 0 0
\(677\) 1.75302 0.0673740 0.0336870 0.999432i \(-0.489275\pi\)
0.0336870 + 0.999432i \(0.489275\pi\)
\(678\) − 25.1890i − 0.967376i
\(679\) 9.47517 0.363624
\(680\) −44.4752 −1.70555
\(681\) − 6.56033i − 0.251393i
\(682\) − 74.5916i − 2.85626i
\(683\) − 16.3351i − 0.625046i −0.949910 0.312523i \(-0.898826\pi\)
0.949910 0.312523i \(-0.101174\pi\)
\(684\) 3.95108i 0.151073i
\(685\) 7.83520 0.299368
\(686\) 20.4553 0.780988
\(687\) − 3.95539i − 0.150908i
\(688\) 144.681 5.51590
\(689\) 0 0
\(690\) 16.2722 0.619472
\(691\) 15.9105i 0.605265i 0.953107 + 0.302632i \(0.0978655\pi\)
−0.953107 + 0.302632i \(0.902135\pi\)
\(692\) 125.623 4.77547
\(693\) −1.61596 −0.0613851
\(694\) − 24.5972i − 0.933696i
\(695\) 18.8780i 0.716083i
\(696\) − 16.7114i − 0.633444i
\(697\) 23.8267i 0.902500i
\(698\) 64.5435 2.44301
\(699\) 8.35690 0.316087
\(700\) 11.3556i 0.429200i
\(701\) 20.8635 0.788005 0.394002 0.919109i \(-0.371090\pi\)
0.394002 + 0.919109i \(0.371090\pi\)
\(702\) 0 0
\(703\) −4.33214 −0.163390
\(704\) 62.1460i 2.34222i
\(705\) 0.789856 0.0297477
\(706\) 73.0182 2.74807
\(707\) 4.06398i 0.152842i
\(708\) − 21.5013i − 0.808067i
\(709\) − 15.6485i − 0.587691i −0.955853 0.293846i \(-0.905065\pi\)
0.955853 0.293846i \(-0.0949351\pi\)
\(710\) − 29.6592i − 1.11309i
\(711\) 1.33513 0.0500711
\(712\) −86.7706 −3.25187
\(713\) 54.8364i 2.05364i
\(714\) −7.24698 −0.271211
\(715\) 0 0
\(716\) 96.5906 3.60976
\(717\) − 20.1008i − 0.750679i
\(718\) −70.2054 −2.62004
\(719\) 27.1594 1.01288 0.506438 0.862276i \(-0.330962\pi\)
0.506438 + 0.862276i \(0.330962\pi\)
\(720\) − 13.6746i − 0.509621i
\(721\) − 2.34183i − 0.0872143i
\(722\) 49.6219i 1.84674i
\(723\) − 19.0127i − 0.707089i
\(724\) −19.0645 −0.708525
\(725\) −7.45580 −0.276901
\(726\) − 6.78687i − 0.251884i
\(727\) −31.7784 −1.17859 −0.589297 0.807916i \(-0.700595\pi\)
−0.589297 + 0.807916i \(0.700595\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 29.5826i − 1.09490i
\(731\) 53.8340 1.99112
\(732\) −17.9618 −0.663889
\(733\) 46.8907i 1.73195i 0.500090 + 0.865973i \(0.333300\pi\)
−0.500090 + 0.865973i \(0.666700\pi\)
\(734\) 25.7743i 0.951347i
\(735\) 7.01938i 0.258914i
\(736\) − 101.502i − 3.74141i
\(737\) −5.45281 −0.200857
\(738\) −13.2228 −0.486739
\(739\) − 17.0479i − 0.627115i −0.949569 0.313558i \(-0.898479\pi\)
0.949569 0.313558i \(-0.101521\pi\)
\(740\) 31.6626 1.16394
\(741\) 0 0
\(742\) −11.3274 −0.415840
\(743\) − 11.6324i − 0.426750i −0.976970 0.213375i \(-0.931554\pi\)
0.976970 0.213375i \(-0.0684455\pi\)
\(744\) 83.1764 3.04940
\(745\) 16.0853 0.589319
\(746\) 75.7797i 2.77449i
\(747\) 2.64310i 0.0967061i
\(748\) 74.1135i 2.70986i
\(749\) − 3.54825i − 0.129650i
\(750\) −25.1304 −0.917631
\(751\) 13.0295 0.475455 0.237727 0.971332i \(-0.423598\pi\)
0.237727 + 0.971332i \(0.423598\pi\)
\(752\) − 9.81700i − 0.357989i
\(753\) −0.763774 −0.0278335
\(754\) 0 0
\(755\) −2.65459 −0.0966106
\(756\) − 2.91185i − 0.105903i
\(757\) 22.7899 0.828311 0.414156 0.910206i \(-0.364077\pi\)
0.414156 + 0.910206i \(0.364077\pi\)
\(758\) −43.1976 −1.56901
\(759\) − 16.7802i − 0.609081i
\(760\) 6.90408i 0.250437i
\(761\) 38.3424i 1.38991i 0.719052 + 0.694956i \(0.244576\pi\)
−0.719052 + 0.694956i \(0.755424\pi\)
\(762\) − 12.0653i − 0.437080i
\(763\) −1.92021 −0.0695164
\(764\) 111.122 4.02024
\(765\) − 5.08815i − 0.183962i
\(766\) 66.2683 2.39437
\(767\) 0 0
\(768\) −17.1511 −0.618886
\(769\) − 3.63879i − 0.131218i −0.997845 0.0656091i \(-0.979101\pi\)
0.997845 0.0656091i \(-0.0208990\pi\)
\(770\) −4.56299 −0.164439
\(771\) −13.0911 −0.471466
\(772\) − 92.4055i − 3.32575i
\(773\) − 39.3424i − 1.41505i −0.706689 0.707524i \(-0.749812\pi\)
0.706689 0.707524i \(-0.250188\pi\)
\(774\) 29.8756i 1.07386i
\(775\) − 37.1092i − 1.33300i
\(776\) 149.240 5.35740
\(777\) 3.19269 0.114537
\(778\) 46.3562i 1.66195i
\(779\) 3.69873 0.132521
\(780\) 0 0
\(781\) −30.5851 −1.09442
\(782\) − 75.2529i − 2.69104i
\(783\) 1.91185 0.0683241
\(784\) 87.2428 3.11581
\(785\) − 18.0825i − 0.645392i
\(786\) 24.8135i 0.885070i
\(787\) 25.4252i 0.906310i 0.891432 + 0.453155i \(0.149702\pi\)
−0.891432 + 0.453155i \(0.850298\pi\)
\(788\) 24.4639i 0.871492i
\(789\) −18.3773 −0.654251
\(790\) 3.77000 0.134131
\(791\) − 5.19269i − 0.184631i
\(792\) −25.4523 −0.904409
\(793\) 0 0
\(794\) −5.48965 −0.194820
\(795\) − 7.95300i − 0.282064i
\(796\) 78.8980 2.79646
\(797\) −20.7138 −0.733720 −0.366860 0.930276i \(-0.619567\pi\)
−0.366860 + 0.930276i \(0.619567\pi\)
\(798\) 1.12498i 0.0398239i
\(799\) − 3.65279i − 0.129227i
\(800\) 68.6889i 2.42852i
\(801\) − 9.92692i − 0.350750i
\(802\) −3.93422 −0.138922
\(803\) −30.5060 −1.07653
\(804\) − 9.82563i − 0.346523i
\(805\) 3.35450 0.118231
\(806\) 0 0
\(807\) −23.6625 −0.832959
\(808\) 64.0103i 2.25187i
\(809\) 11.0978 0.390179 0.195090 0.980785i \(-0.437500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(810\) 2.82371 0.0992150
\(811\) − 4.84223i − 0.170034i −0.996380 0.0850169i \(-0.972906\pi\)
0.996380 0.0850169i \(-0.0270944\pi\)
\(812\) − 5.56704i − 0.195365i
\(813\) − 19.7530i − 0.692769i
\(814\) − 45.0966i − 1.58064i
\(815\) −16.4754 −0.577109
\(816\) −63.2398 −2.21384
\(817\) − 8.35690i − 0.292371i
\(818\) 80.5964 2.81799
\(819\) 0 0
\(820\) −27.0331 −0.944037
\(821\) − 43.7381i − 1.52647i −0.646121 0.763235i \(-0.723610\pi\)
0.646121 0.763235i \(-0.276390\pi\)
\(822\) 20.1089 0.701377
\(823\) −12.0998 −0.421771 −0.210885 0.977511i \(-0.567635\pi\)
−0.210885 + 0.977511i \(0.567635\pi\)
\(824\) − 36.8853i − 1.28496i
\(825\) 11.3556i 0.395350i
\(826\) − 6.12200i − 0.213012i
\(827\) 35.3212i 1.22824i 0.789213 + 0.614120i \(0.210489\pi\)
−0.789213 + 0.614120i \(0.789511\pi\)
\(828\) 30.2368 1.05080
\(829\) −53.4946 −1.85794 −0.928971 0.370152i \(-0.879306\pi\)
−0.928971 + 0.370152i \(0.879306\pi\)
\(830\) 7.46335i 0.259057i
\(831\) 1.77777 0.0616703
\(832\) 0 0
\(833\) 32.4620 1.12474
\(834\) 48.4499i 1.67768i
\(835\) −5.66009 −0.195875
\(836\) 11.5050 0.397908
\(837\) 9.51573i 0.328912i
\(838\) 17.8950i 0.618172i
\(839\) − 23.1521i − 0.799300i −0.916668 0.399650i \(-0.869132\pi\)
0.916668 0.399650i \(-0.130868\pi\)
\(840\) − 5.08815i − 0.175558i
\(841\) −25.3448 −0.873959
\(842\) −36.5163 −1.25844
\(843\) 1.62133i 0.0558417i
\(844\) 2.41417 0.0830993
\(845\) 0 0
\(846\) 2.02715 0.0696948
\(847\) − 1.39911i − 0.0480739i
\(848\) −98.8467 −3.39441
\(849\) 4.98361 0.171037
\(850\) 50.9256i 1.74673i
\(851\) 33.1530i 1.13647i
\(852\) − 55.1124i − 1.88812i
\(853\) − 26.7265i − 0.915097i −0.889185 0.457548i \(-0.848728\pi\)
0.889185 0.457548i \(-0.151272\pi\)
\(854\) −5.11423 −0.175005
\(855\) −0.789856 −0.0270125
\(856\) − 55.8872i − 1.91019i
\(857\) −42.6064 −1.45541 −0.727703 0.685892i \(-0.759412\pi\)
−0.727703 + 0.685892i \(0.759412\pi\)
\(858\) 0 0
\(859\) 33.6079 1.14669 0.573344 0.819315i \(-0.305646\pi\)
0.573344 + 0.819315i \(0.305646\pi\)
\(860\) 61.0786i 2.08276i
\(861\) −2.72587 −0.0928975
\(862\) −96.7682 −3.29594
\(863\) 18.7047i 0.636715i 0.947971 + 0.318358i \(0.103131\pi\)
−0.947971 + 0.318358i \(0.896869\pi\)
\(864\) − 17.6136i − 0.599226i
\(865\) 25.1132i 0.853873i
\(866\) 87.4210i 2.97069i
\(867\) −6.53079 −0.221797
\(868\) 27.7084 0.940485
\(869\) − 3.88769i − 0.131881i
\(870\) 5.39852 0.183027
\(871\) 0 0
\(872\) −30.2446 −1.02421
\(873\) 17.0737i 0.577856i
\(874\) −11.6819 −0.395145
\(875\) −5.18060 −0.175136
\(876\) − 54.9700i − 1.85726i
\(877\) 36.8237i 1.24345i 0.783236 + 0.621724i \(0.213567\pi\)
−0.783236 + 0.621724i \(0.786433\pi\)
\(878\) − 34.5442i − 1.16581i
\(879\) − 0.0717525i − 0.00242015i
\(880\) −39.8183 −1.34228
\(881\) 41.1250 1.38554 0.692768 0.721161i \(-0.256391\pi\)
0.692768 + 0.721161i \(0.256391\pi\)
\(882\) 18.0151i 0.606599i
\(883\) −30.7482 −1.03476 −0.517380 0.855756i \(-0.673093\pi\)
−0.517380 + 0.855756i \(0.673093\pi\)
\(884\) 0 0
\(885\) 4.29829 0.144485
\(886\) 32.1987i 1.08173i
\(887\) 7.58940 0.254827 0.127414 0.991850i \(-0.459332\pi\)
0.127414 + 0.991850i \(0.459332\pi\)
\(888\) 50.2868 1.68751
\(889\) − 2.48725i − 0.0834198i
\(890\) − 28.0307i − 0.939592i
\(891\) − 2.91185i − 0.0975508i
\(892\) 85.7891i 2.87243i
\(893\) −0.567040 −0.0189753
\(894\) 41.2825 1.38069
\(895\) 19.3093i 0.645439i
\(896\) −12.3351 −0.412088
\(897\) 0 0
\(898\) −33.5934 −1.12103
\(899\) 18.1927i 0.606760i
\(900\) −20.4620 −0.682068
\(901\) −36.7797 −1.22531
\(902\) 38.5029i 1.28201i
\(903\) 6.15883i 0.204953i
\(904\) − 81.7881i − 2.72023i
\(905\) − 3.81115i − 0.126687i
\(906\) −6.81295 −0.226345
\(907\) −19.3333 −0.641952 −0.320976 0.947087i \(-0.604011\pi\)
−0.320976 + 0.947087i \(0.604011\pi\)
\(908\) − 34.4219i − 1.14233i
\(909\) −7.32304 −0.242890
\(910\) 0 0
\(911\) 22.7149 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(912\) 9.81700i 0.325073i
\(913\) 7.69633 0.254711
\(914\) −87.3624 −2.88969
\(915\) − 3.59073i − 0.118706i
\(916\) − 20.7539i − 0.685727i
\(917\) 5.11529i 0.168922i
\(918\) − 13.0586i − 0.430998i
\(919\) 14.2911 0.471420 0.235710 0.971823i \(-0.424258\pi\)
0.235710 + 0.971823i \(0.424258\pi\)
\(920\) 52.8355 1.74194
\(921\) − 5.19806i − 0.171282i
\(922\) −65.6945 −2.16353
\(923\) 0 0
\(924\) −8.47889 −0.278935
\(925\) − 22.4355i − 0.737674i
\(926\) −89.3309 −2.93560
\(927\) 4.21983 0.138597
\(928\) − 33.6746i − 1.10542i
\(929\) 32.7211i 1.07354i 0.843727 + 0.536772i \(0.180356\pi\)
−0.843727 + 0.536772i \(0.819644\pi\)
\(930\) 26.8696i 0.881090i
\(931\) − 5.03923i − 0.165154i
\(932\) 43.8485 1.43630
\(933\) −22.5429 −0.738021
\(934\) 103.698i 3.39311i
\(935\) −14.8159 −0.484533
\(936\) 0 0
\(937\) −4.01400 −0.131132 −0.0655658 0.997848i \(-0.520885\pi\)
−0.0655658 + 0.997848i \(0.520885\pi\)
\(938\) − 2.79763i − 0.0913457i
\(939\) −22.6612 −0.739519
\(940\) 4.14436 0.135174
\(941\) − 28.2669i − 0.921476i −0.887536 0.460738i \(-0.847585\pi\)
0.887536 0.460738i \(-0.152415\pi\)
\(942\) − 46.4083i − 1.51206i
\(943\) − 28.3056i − 0.921757i
\(944\) − 53.4228i − 1.73876i
\(945\) 0.582105 0.0189359
\(946\) 86.9934 2.82840
\(947\) 37.8455i 1.22981i 0.788600 + 0.614906i \(0.210806\pi\)
−0.788600 + 0.614906i \(0.789194\pi\)
\(948\) 7.00538 0.227524
\(949\) 0 0
\(950\) 7.90541 0.256485
\(951\) − 26.3424i − 0.854212i
\(952\) −23.5308 −0.762637
\(953\) 40.8256 1.32247 0.661236 0.750178i \(-0.270032\pi\)
0.661236 + 0.750178i \(0.270032\pi\)
\(954\) − 20.4112i − 0.660837i
\(955\) 22.2142i 0.718834i
\(956\) − 105.469i − 3.41110i
\(957\) − 5.56704i − 0.179957i
\(958\) −22.4644 −0.725792
\(959\) 4.14542 0.133863
\(960\) − 22.3864i − 0.722519i
\(961\) −59.5491 −1.92094
\(962\) 0 0
\(963\) 6.39373 0.206035
\(964\) − 99.7591i − 3.21302i
\(965\) 18.4727 0.594656
\(966\) 8.60925 0.276998
\(967\) 10.2798i 0.330575i 0.986245 + 0.165288i \(0.0528552\pi\)
−0.986245 + 0.165288i \(0.947145\pi\)
\(968\) − 22.0368i − 0.708291i
\(969\) 3.65279i 0.117345i
\(970\) 48.2111i 1.54796i
\(971\) −19.4805 −0.625161 −0.312580 0.949891i \(-0.601193\pi\)
−0.312580 + 0.949891i \(0.601193\pi\)
\(972\) 5.24698 0.168297
\(973\) 9.98792i 0.320198i
\(974\) 39.9997 1.28167
\(975\) 0 0
\(976\) −44.6286 −1.42853
\(977\) 47.0538i 1.50539i 0.658372 + 0.752693i \(0.271245\pi\)
−0.658372 + 0.752693i \(0.728755\pi\)
\(978\) −42.2838 −1.35209
\(979\) −28.9057 −0.923831
\(980\) 36.8305i 1.17651i
\(981\) − 3.46011i − 0.110473i
\(982\) − 13.4572i − 0.429438i
\(983\) − 34.4295i − 1.09813i −0.835779 0.549065i \(-0.814984\pi\)
0.835779 0.549065i \(-0.185016\pi\)
\(984\) −42.9342 −1.36869
\(985\) −4.89056 −0.155826
\(986\) − 24.9661i − 0.795084i
\(987\) 0.417895 0.0133017
\(988\) 0 0
\(989\) −63.9536 −2.03361
\(990\) − 8.22223i − 0.261319i
\(991\) 7.30798 0.232146 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(992\) 167.606 5.32149
\(993\) 11.2295i 0.356358i
\(994\) − 15.6920i − 0.497721i
\(995\) 15.7724i 0.500019i
\(996\) 13.8683i 0.439434i
\(997\) −35.6915 −1.13036 −0.565181 0.824967i \(-0.691194\pi\)
−0.565181 + 0.824967i \(0.691194\pi\)
\(998\) 1.03743 0.0328392
\(999\) 5.75302i 0.182018i
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 507.2.b.f.337.6 6
3.2 odd 2 1521.2.b.k.1351.1 6
13.2 odd 12 507.2.e.i.22.1 6
13.3 even 3 507.2.j.i.316.1 12
13.4 even 6 507.2.j.i.361.1 12
13.5 odd 4 507.2.a.l.1.3 yes 3
13.6 odd 12 507.2.e.i.484.1 6
13.7 odd 12 507.2.e.l.484.3 6
13.8 odd 4 507.2.a.i.1.1 3
13.9 even 3 507.2.j.i.361.6 12
13.10 even 6 507.2.j.i.316.6 12
13.11 odd 12 507.2.e.l.22.3 6
13.12 even 2 inner 507.2.b.f.337.1 6
39.5 even 4 1521.2.a.n.1.1 3
39.8 even 4 1521.2.a.s.1.3 3
39.38 odd 2 1521.2.b.k.1351.6 6
52.31 even 4 8112.2.a.cp.1.1 3
52.47 even 4 8112.2.a.cg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.1 3 13.8 odd 4
507.2.a.l.1.3 yes 3 13.5 odd 4
507.2.b.f.337.1 6 13.12 even 2 inner
507.2.b.f.337.6 6 1.1 even 1 trivial
507.2.e.i.22.1 6 13.2 odd 12
507.2.e.i.484.1 6 13.6 odd 12
507.2.e.l.22.3 6 13.11 odd 12
507.2.e.l.484.3 6 13.7 odd 12
507.2.j.i.316.1 12 13.3 even 3
507.2.j.i.316.6 12 13.10 even 6
507.2.j.i.361.1 12 13.4 even 6
507.2.j.i.361.6 12 13.9 even 3
1521.2.a.n.1.1 3 39.5 even 4
1521.2.a.s.1.3 3 39.8 even 4
1521.2.b.k.1351.1 6 3.2 odd 2
1521.2.b.k.1351.6 6 39.38 odd 2
8112.2.a.cg.1.3 3 52.47 even 4
8112.2.a.cp.1.1 3 52.31 even 4