Properties

Label 91.2.a.d.1.2
Level $91$
Weight $2$
Character 91.1
Self dual yes
Analytic conductor $0.727$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.470683 q^{2} +2.24914 q^{3} -1.77846 q^{4} +0.529317 q^{5} +1.05863 q^{6} -1.00000 q^{7} -1.77846 q^{8} +2.05863 q^{9} +O(q^{10})\) \(q+0.470683 q^{2} +2.24914 q^{3} -1.77846 q^{4} +0.529317 q^{5} +1.05863 q^{6} -1.00000 q^{7} -1.77846 q^{8} +2.05863 q^{9} +0.249141 q^{10} -2.24914 q^{11} -4.00000 q^{12} +1.00000 q^{13} -0.470683 q^{14} +1.19051 q^{15} +2.71982 q^{16} -1.30777 q^{17} +0.968964 q^{18} -1.47068 q^{19} -0.941367 q^{20} -2.24914 q^{21} -1.05863 q^{22} +5.83709 q^{23} -4.00000 q^{24} -4.71982 q^{25} +0.470683 q^{26} -2.11727 q^{27} +1.77846 q^{28} +5.22154 q^{29} +0.560352 q^{30} -7.02760 q^{31} +4.83709 q^{32} -5.05863 q^{33} -0.615547 q^{34} -0.529317 q^{35} -3.66119 q^{36} -2.36641 q^{37} -0.692226 q^{38} +2.24914 q^{39} -0.941367 q^{40} +6.49828 q^{41} -1.05863 q^{42} +11.3940 q^{43} +4.00000 q^{44} +1.08967 q^{45} +2.74742 q^{46} +8.58451 q^{47} +6.11727 q^{48} +1.00000 q^{49} -2.22154 q^{50} -2.94137 q^{51} -1.77846 q^{52} +11.2767 q^{53} -0.996562 q^{54} -1.19051 q^{55} +1.77846 q^{56} -3.30777 q^{57} +2.45769 q^{58} -12.1725 q^{59} -2.11727 q^{60} -2.00000 q^{61} -3.30777 q^{62} -2.05863 q^{63} -3.16291 q^{64} +0.529317 q^{65} -2.38101 q^{66} -15.9379 q^{67} +2.32582 q^{68} +13.1284 q^{69} -0.249141 q^{70} +1.19051 q^{71} -3.66119 q^{72} +7.64315 q^{73} -1.11383 q^{74} -10.6155 q^{75} +2.61555 q^{76} +2.24914 q^{77} +1.05863 q^{78} -1.33881 q^{79} +1.43965 q^{80} -10.9379 q^{81} +3.05863 q^{82} -16.3500 q^{83} +4.00000 q^{84} -0.692226 q^{85} +5.36297 q^{86} +11.7440 q^{87} +4.00000 q^{88} +6.91033 q^{89} +0.512889 q^{90} -1.00000 q^{91} -10.3810 q^{92} -15.8061 q^{93} +4.04059 q^{94} -0.778457 q^{95} +10.8793 q^{96} -3.47068 q^{97} +0.470683 q^{98} -4.63016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} - 2q^{3} + 3q^{4} + 2q^{5} + 4q^{6} - 3q^{7} + 3q^{8} + 7q^{9} + O(q^{10}) \) \( 3q + q^{2} - 2q^{3} + 3q^{4} + 2q^{5} + 4q^{6} - 3q^{7} + 3q^{8} + 7q^{9} - 8q^{10} + 2q^{11} - 12q^{12} + 3q^{13} - q^{14} - 6q^{15} - q^{16} + 4q^{17} - 15q^{18} - 4q^{19} - 2q^{20} + 2q^{21} - 4q^{22} + 10q^{23} - 12q^{24} - 5q^{25} + q^{26} - 8q^{27} - 3q^{28} + 24q^{29} + 20q^{30} - 4q^{31} + 7q^{32} - 16q^{33} + 14q^{34} - 2q^{35} - q^{36} - 10q^{38} - 2q^{39} - 2q^{40} + 2q^{41} - 4q^{42} + 10q^{43} + 12q^{44} + 22q^{45} - 18q^{46} - 8q^{47} + 20q^{48} + 3q^{49} - 15q^{50} - 8q^{51} + 3q^{52} + 8q^{53} + 32q^{54} + 6q^{55} - 3q^{56} - 2q^{57} + 12q^{58} - 4q^{59} - 8q^{60} - 6q^{61} - 2q^{62} - 7q^{63} - 17q^{64} + 2q^{65} + 12q^{66} - 12q^{67} + 22q^{68} - 6q^{69} + 8q^{70} - 6q^{71} - q^{72} - 10q^{73} + 30q^{74} - 16q^{75} - 8q^{76} - 2q^{77} + 4q^{78} - 14q^{79} - 14q^{80} + 3q^{81} + 10q^{82} - 12q^{83} + 12q^{84} - 10q^{85} - 26q^{86} - 26q^{87} + 12q^{88} + 2q^{89} - 28q^{90} - 3q^{91} - 12q^{92} - 22q^{93} - 10q^{94} + 6q^{95} - 4q^{96} - 10q^{97} + q^{98} + 14q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.470683 0.332823 0.166412 0.986056i \(-0.446782\pi\)
0.166412 + 0.986056i \(0.446782\pi\)
\(3\) 2.24914 1.29854 0.649271 0.760557i \(-0.275074\pi\)
0.649271 + 0.760557i \(0.275074\pi\)
\(4\) −1.77846 −0.889229
\(5\) 0.529317 0.236718 0.118359 0.992971i \(-0.462237\pi\)
0.118359 + 0.992971i \(0.462237\pi\)
\(6\) 1.05863 0.432185
\(7\) −1.00000 −0.377964
\(8\) −1.77846 −0.628780
\(9\) 2.05863 0.686211
\(10\) 0.249141 0.0787852
\(11\) −2.24914 −0.678141 −0.339071 0.940761i \(-0.610113\pi\)
−0.339071 + 0.940761i \(0.610113\pi\)
\(12\) −4.00000 −1.15470
\(13\) 1.00000 0.277350
\(14\) −0.470683 −0.125795
\(15\) 1.19051 0.307388
\(16\) 2.71982 0.679956
\(17\) −1.30777 −0.317182 −0.158591 0.987344i \(-0.550695\pi\)
−0.158591 + 0.987344i \(0.550695\pi\)
\(18\) 0.968964 0.228387
\(19\) −1.47068 −0.337398 −0.168699 0.985668i \(-0.553957\pi\)
−0.168699 + 0.985668i \(0.553957\pi\)
\(20\) −0.941367 −0.210496
\(21\) −2.24914 −0.490803
\(22\) −1.05863 −0.225701
\(23\) 5.83709 1.21712 0.608559 0.793509i \(-0.291748\pi\)
0.608559 + 0.793509i \(0.291748\pi\)
\(24\) −4.00000 −0.816497
\(25\) −4.71982 −0.943965
\(26\) 0.470683 0.0923086
\(27\) −2.11727 −0.407468
\(28\) 1.77846 0.336097
\(29\) 5.22154 0.969616 0.484808 0.874621i \(-0.338889\pi\)
0.484808 + 0.874621i \(0.338889\pi\)
\(30\) 0.560352 0.102306
\(31\) −7.02760 −1.26219 −0.631097 0.775704i \(-0.717395\pi\)
−0.631097 + 0.775704i \(0.717395\pi\)
\(32\) 4.83709 0.855085
\(33\) −5.05863 −0.880595
\(34\) −0.615547 −0.105566
\(35\) −0.529317 −0.0894708
\(36\) −3.66119 −0.610198
\(37\) −2.36641 −0.389035 −0.194517 0.980899i \(-0.562314\pi\)
−0.194517 + 0.980899i \(0.562314\pi\)
\(38\) −0.692226 −0.112294
\(39\) 2.24914 0.360151
\(40\) −0.941367 −0.148843
\(41\) 6.49828 1.01486 0.507431 0.861693i \(-0.330595\pi\)
0.507431 + 0.861693i \(0.330595\pi\)
\(42\) −1.05863 −0.163351
\(43\) 11.3940 1.73757 0.868785 0.495190i \(-0.164902\pi\)
0.868785 + 0.495190i \(0.164902\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.08967 0.162438
\(46\) 2.74742 0.405085
\(47\) 8.58451 1.25218 0.626090 0.779751i \(-0.284654\pi\)
0.626090 + 0.779751i \(0.284654\pi\)
\(48\) 6.11727 0.882951
\(49\) 1.00000 0.142857
\(50\) −2.22154 −0.314174
\(51\) −2.94137 −0.411874
\(52\) −1.77846 −0.246628
\(53\) 11.2767 1.54898 0.774490 0.632587i \(-0.218007\pi\)
0.774490 + 0.632587i \(0.218007\pi\)
\(54\) −0.996562 −0.135615
\(55\) −1.19051 −0.160528
\(56\) 1.77846 0.237656
\(57\) −3.30777 −0.438125
\(58\) 2.45769 0.322711
\(59\) −12.1725 −1.58472 −0.792360 0.610054i \(-0.791148\pi\)
−0.792360 + 0.610054i \(0.791148\pi\)
\(60\) −2.11727 −0.273338
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −3.30777 −0.420088
\(63\) −2.05863 −0.259363
\(64\) −3.16291 −0.395364
\(65\) 0.529317 0.0656536
\(66\) −2.38101 −0.293083
\(67\) −15.9379 −1.94713 −0.973564 0.228415i \(-0.926646\pi\)
−0.973564 + 0.228415i \(0.926646\pi\)
\(68\) 2.32582 0.282047
\(69\) 13.1284 1.58048
\(70\) −0.249141 −0.0297780
\(71\) 1.19051 0.141287 0.0706436 0.997502i \(-0.477495\pi\)
0.0706436 + 0.997502i \(0.477495\pi\)
\(72\) −3.66119 −0.431475
\(73\) 7.64315 0.894562 0.447281 0.894393i \(-0.352392\pi\)
0.447281 + 0.894393i \(0.352392\pi\)
\(74\) −1.11383 −0.129480
\(75\) −10.6155 −1.22578
\(76\) 2.61555 0.300024
\(77\) 2.24914 0.256313
\(78\) 1.05863 0.119867
\(79\) −1.33881 −0.150628 −0.0753139 0.997160i \(-0.523996\pi\)
−0.0753139 + 0.997160i \(0.523996\pi\)
\(80\) 1.43965 0.160958
\(81\) −10.9379 −1.21533
\(82\) 3.05863 0.337770
\(83\) −16.3500 −1.79464 −0.897322 0.441377i \(-0.854490\pi\)
−0.897322 + 0.441377i \(0.854490\pi\)
\(84\) 4.00000 0.436436
\(85\) −0.692226 −0.0750825
\(86\) 5.36297 0.578304
\(87\) 11.7440 1.25909
\(88\) 4.00000 0.426401
\(89\) 6.91033 0.732494 0.366247 0.930518i \(-0.380643\pi\)
0.366247 + 0.930518i \(0.380643\pi\)
\(90\) 0.512889 0.0540632
\(91\) −1.00000 −0.104828
\(92\) −10.3810 −1.08230
\(93\) −15.8061 −1.63901
\(94\) 4.04059 0.416755
\(95\) −0.778457 −0.0798680
\(96\) 10.8793 1.11036
\(97\) −3.47068 −0.352395 −0.176197 0.984355i \(-0.556380\pi\)
−0.176197 + 0.984355i \(0.556380\pi\)
\(98\) 0.470683 0.0475462
\(99\) −4.63016 −0.465348
\(100\) 8.39400 0.839400
\(101\) 7.75086 0.771239 0.385620 0.922658i \(-0.373988\pi\)
0.385620 + 0.922658i \(0.373988\pi\)
\(102\) −1.38445 −0.137081
\(103\) −16.9966 −1.67472 −0.837361 0.546651i \(-0.815902\pi\)
−0.837361 + 0.546651i \(0.815902\pi\)
\(104\) −1.77846 −0.174392
\(105\) −1.19051 −0.116182
\(106\) 5.30777 0.515537
\(107\) −5.55691 −0.537207 −0.268604 0.963251i \(-0.586562\pi\)
−0.268604 + 0.963251i \(0.586562\pi\)
\(108\) 3.76547 0.362332
\(109\) 7.92332 0.758917 0.379458 0.925209i \(-0.376110\pi\)
0.379458 + 0.925209i \(0.376110\pi\)
\(110\) −0.560352 −0.0534275
\(111\) −5.32238 −0.505178
\(112\) −2.71982 −0.256999
\(113\) −9.89229 −0.930588 −0.465294 0.885156i \(-0.654051\pi\)
−0.465294 + 0.885156i \(0.654051\pi\)
\(114\) −1.55691 −0.145818
\(115\) 3.08967 0.288113
\(116\) −9.28629 −0.862210
\(117\) 2.05863 0.190321
\(118\) −5.72938 −0.527432
\(119\) 1.30777 0.119883
\(120\) −2.11727 −0.193279
\(121\) −5.94137 −0.540124
\(122\) −0.941367 −0.0852273
\(123\) 14.6155 1.31784
\(124\) 12.4983 1.12238
\(125\) −5.14486 −0.460171
\(126\) −0.968964 −0.0863222
\(127\) 0.824101 0.0731271 0.0365635 0.999331i \(-0.488359\pi\)
0.0365635 + 0.999331i \(0.488359\pi\)
\(128\) −11.1629 −0.986671
\(129\) 25.6267 2.25631
\(130\) 0.249141 0.0218511
\(131\) 10.6155 0.927485 0.463742 0.885970i \(-0.346506\pi\)
0.463742 + 0.885970i \(0.346506\pi\)
\(132\) 8.99656 0.783050
\(133\) 1.47068 0.127524
\(134\) −7.50172 −0.648050
\(135\) −1.12070 −0.0964549
\(136\) 2.32582 0.199437
\(137\) 11.3630 0.970804 0.485402 0.874291i \(-0.338673\pi\)
0.485402 + 0.874291i \(0.338673\pi\)
\(138\) 6.17934 0.526020
\(139\) −13.9233 −1.18096 −0.590480 0.807052i \(-0.701062\pi\)
−0.590480 + 0.807052i \(0.701062\pi\)
\(140\) 0.941367 0.0795600
\(141\) 19.3078 1.62601
\(142\) 0.560352 0.0470237
\(143\) −2.24914 −0.188083
\(144\) 5.59912 0.466593
\(145\) 2.76385 0.229525
\(146\) 3.59750 0.297731
\(147\) 2.24914 0.185506
\(148\) 4.20855 0.345941
\(149\) 9.30777 0.762523 0.381261 0.924467i \(-0.375490\pi\)
0.381261 + 0.924467i \(0.375490\pi\)
\(150\) −4.99656 −0.407968
\(151\) 7.07324 0.575612 0.287806 0.957689i \(-0.407074\pi\)
0.287806 + 0.957689i \(0.407074\pi\)
\(152\) 2.61555 0.212149
\(153\) −2.69223 −0.217654
\(154\) 1.05863 0.0853071
\(155\) −3.71982 −0.298783
\(156\) −4.00000 −0.320256
\(157\) −6.04059 −0.482091 −0.241046 0.970514i \(-0.577490\pi\)
−0.241046 + 0.970514i \(0.577490\pi\)
\(158\) −0.630155 −0.0501325
\(159\) 25.3630 2.01141
\(160\) 2.56035 0.202414
\(161\) −5.83709 −0.460027
\(162\) −5.14830 −0.404489
\(163\) 6.38101 0.499800 0.249900 0.968272i \(-0.419602\pi\)
0.249900 + 0.968272i \(0.419602\pi\)
\(164\) −11.5569 −0.902443
\(165\) −2.67762 −0.208452
\(166\) −7.69566 −0.597299
\(167\) −16.5845 −1.28335 −0.641674 0.766977i \(-0.721760\pi\)
−0.641674 + 0.766977i \(0.721760\pi\)
\(168\) 4.00000 0.308607
\(169\) 1.00000 0.0769231
\(170\) −0.325819 −0.0249892
\(171\) −3.02760 −0.231526
\(172\) −20.2637 −1.54510
\(173\) −23.3009 −1.77153 −0.885767 0.464130i \(-0.846367\pi\)
−0.885767 + 0.464130i \(0.846367\pi\)
\(174\) 5.52770 0.419054
\(175\) 4.71982 0.356785
\(176\) −6.11727 −0.461106
\(177\) −27.3776 −2.05782
\(178\) 3.25258 0.243791
\(179\) 21.0422 1.57277 0.786384 0.617738i \(-0.211951\pi\)
0.786384 + 0.617738i \(0.211951\pi\)
\(180\) −1.93793 −0.144445
\(181\) 16.7474 1.24483 0.622413 0.782689i \(-0.286152\pi\)
0.622413 + 0.782689i \(0.286152\pi\)
\(182\) −0.470683 −0.0348894
\(183\) −4.49828 −0.332523
\(184\) −10.3810 −0.765299
\(185\) −1.25258 −0.0920914
\(186\) −7.43965 −0.545501
\(187\) 2.94137 0.215094
\(188\) −15.2672 −1.11347
\(189\) 2.11727 0.154008
\(190\) −0.366407 −0.0265819
\(191\) −7.43965 −0.538314 −0.269157 0.963096i \(-0.586745\pi\)
−0.269157 + 0.963096i \(0.586745\pi\)
\(192\) −7.11383 −0.513396
\(193\) −1.50172 −0.108096 −0.0540480 0.998538i \(-0.517212\pi\)
−0.0540480 + 0.998538i \(0.517212\pi\)
\(194\) −1.63359 −0.117285
\(195\) 1.19051 0.0852540
\(196\) −1.77846 −0.127033
\(197\) 23.9931 1.70944 0.854720 0.519090i \(-0.173729\pi\)
0.854720 + 0.519090i \(0.173729\pi\)
\(198\) −2.17934 −0.154879
\(199\) −2.01461 −0.142812 −0.0714059 0.997447i \(-0.522749\pi\)
−0.0714059 + 0.997447i \(0.522749\pi\)
\(200\) 8.39400 0.593546
\(201\) −35.8466 −2.52843
\(202\) 3.64820 0.256687
\(203\) −5.22154 −0.366480
\(204\) 5.23109 0.366250
\(205\) 3.43965 0.240235
\(206\) −8.00000 −0.557386
\(207\) 12.0164 0.835199
\(208\) 2.71982 0.188586
\(209\) 3.30777 0.228803
\(210\) −0.560352 −0.0386680
\(211\) 10.1008 0.695370 0.347685 0.937611i \(-0.386968\pi\)
0.347685 + 0.937611i \(0.386968\pi\)
\(212\) −20.0552 −1.37740
\(213\) 2.67762 0.183467
\(214\) −2.61555 −0.178795
\(215\) 6.03104 0.411313
\(216\) 3.76547 0.256208
\(217\) 7.02760 0.477064
\(218\) 3.72938 0.252585
\(219\) 17.1905 1.16163
\(220\) 2.11727 0.142746
\(221\) −1.30777 −0.0879704
\(222\) −2.50516 −0.168135
\(223\) 10.1414 0.679120 0.339560 0.940584i \(-0.389722\pi\)
0.339560 + 0.940584i \(0.389722\pi\)
\(224\) −4.83709 −0.323192
\(225\) −9.71639 −0.647759
\(226\) −4.65613 −0.309721
\(227\) −5.38445 −0.357379 −0.178689 0.983906i \(-0.557186\pi\)
−0.178689 + 0.983906i \(0.557186\pi\)
\(228\) 5.88273 0.389594
\(229\) 3.32238 0.219549 0.109775 0.993957i \(-0.464987\pi\)
0.109775 + 0.993957i \(0.464987\pi\)
\(230\) 1.45426 0.0958908
\(231\) 5.05863 0.332834
\(232\) −9.28629 −0.609675
\(233\) 13.7198 0.898816 0.449408 0.893327i \(-0.351635\pi\)
0.449408 + 0.893327i \(0.351635\pi\)
\(234\) 0.968964 0.0633432
\(235\) 4.54392 0.296413
\(236\) 21.6482 1.40918
\(237\) −3.01117 −0.195597
\(238\) 0.615547 0.0399000
\(239\) −3.50172 −0.226507 −0.113254 0.993566i \(-0.536127\pi\)
−0.113254 + 0.993566i \(0.536127\pi\)
\(240\) 3.23797 0.209010
\(241\) 1.58795 0.102289 0.0511444 0.998691i \(-0.483713\pi\)
0.0511444 + 0.998691i \(0.483713\pi\)
\(242\) −2.79650 −0.179766
\(243\) −18.2491 −1.17068
\(244\) 3.55691 0.227708
\(245\) 0.529317 0.0338168
\(246\) 6.87930 0.438608
\(247\) −1.47068 −0.0935773
\(248\) 12.4983 0.793642
\(249\) −36.7734 −2.33042
\(250\) −2.42160 −0.153156
\(251\) −4.92676 −0.310974 −0.155487 0.987838i \(-0.549695\pi\)
−0.155487 + 0.987838i \(0.549695\pi\)
\(252\) 3.66119 0.230633
\(253\) −13.1284 −0.825378
\(254\) 0.387890 0.0243384
\(255\) −1.55691 −0.0974978
\(256\) 1.07162 0.0669764
\(257\) −8.01461 −0.499938 −0.249969 0.968254i \(-0.580420\pi\)
−0.249969 + 0.968254i \(0.580420\pi\)
\(258\) 12.0621 0.750952
\(259\) 2.36641 0.147041
\(260\) −0.941367 −0.0583811
\(261\) 10.7492 0.665361
\(262\) 4.99656 0.308689
\(263\) 1.60256 0.0988179 0.0494090 0.998779i \(-0.484266\pi\)
0.0494090 + 0.998779i \(0.484266\pi\)
\(264\) 8.99656 0.553700
\(265\) 5.96896 0.366671
\(266\) 0.692226 0.0424431
\(267\) 15.5423 0.951174
\(268\) 28.3449 1.73144
\(269\) −11.8207 −0.720719 −0.360359 0.932814i \(-0.617346\pi\)
−0.360359 + 0.932814i \(0.617346\pi\)
\(270\) −0.527497 −0.0321024
\(271\) 21.8827 1.32928 0.664641 0.747163i \(-0.268585\pi\)
0.664641 + 0.747163i \(0.268585\pi\)
\(272\) −3.55691 −0.215670
\(273\) −2.24914 −0.136124
\(274\) 5.34836 0.323106
\(275\) 10.6155 0.640142
\(276\) −23.3484 −1.40541
\(277\) −10.2181 −0.613946 −0.306973 0.951718i \(-0.599316\pi\)
−0.306973 + 0.951718i \(0.599316\pi\)
\(278\) −6.55348 −0.393051
\(279\) −14.4672 −0.866131
\(280\) 0.941367 0.0562574
\(281\) 1.54231 0.0920063 0.0460031 0.998941i \(-0.485352\pi\)
0.0460031 + 0.998941i \(0.485352\pi\)
\(282\) 9.08785 0.541174
\(283\) 15.8466 0.941985 0.470993 0.882137i \(-0.343896\pi\)
0.470993 + 0.882137i \(0.343896\pi\)
\(284\) −2.11727 −0.125637
\(285\) −1.75086 −0.103712
\(286\) −1.05863 −0.0625983
\(287\) −6.49828 −0.383581
\(288\) 9.95779 0.586769
\(289\) −15.2897 −0.899396
\(290\) 1.30090 0.0763914
\(291\) −7.80605 −0.457599
\(292\) −13.5930 −0.795470
\(293\) 11.0828 0.647464 0.323732 0.946149i \(-0.395062\pi\)
0.323732 + 0.946149i \(0.395062\pi\)
\(294\) 1.05863 0.0617407
\(295\) −6.44309 −0.375131
\(296\) 4.20855 0.244617
\(297\) 4.76203 0.276321
\(298\) 4.38101 0.253785
\(299\) 5.83709 0.337568
\(300\) 18.8793 1.09000
\(301\) −11.3940 −0.656740
\(302\) 3.32926 0.191577
\(303\) 17.4328 1.00149
\(304\) −4.00000 −0.229416
\(305\) −1.05863 −0.0606172
\(306\) −1.26719 −0.0724402
\(307\) 20.4121 1.16498 0.582489 0.812839i \(-0.302079\pi\)
0.582489 + 0.812839i \(0.302079\pi\)
\(308\) −4.00000 −0.227921
\(309\) −38.2277 −2.17470
\(310\) −1.75086 −0.0994421
\(311\) −1.92332 −0.109062 −0.0545308 0.998512i \(-0.517366\pi\)
−0.0545308 + 0.998512i \(0.517366\pi\)
\(312\) −4.00000 −0.226455
\(313\) −14.3664 −0.812037 −0.406019 0.913865i \(-0.633083\pi\)
−0.406019 + 0.913865i \(0.633083\pi\)
\(314\) −2.84320 −0.160451
\(315\) −1.08967 −0.0613959
\(316\) 2.38101 0.133943
\(317\) 15.5569 0.873763 0.436882 0.899519i \(-0.356083\pi\)
0.436882 + 0.899519i \(0.356083\pi\)
\(318\) 11.9379 0.669446
\(319\) −11.7440 −0.657537
\(320\) −1.67418 −0.0935895
\(321\) −12.4983 −0.697586
\(322\) −2.74742 −0.153108
\(323\) 1.92332 0.107016
\(324\) 19.4526 1.08070
\(325\) −4.71982 −0.261809
\(326\) 3.00344 0.166345
\(327\) 17.8207 0.985485
\(328\) −11.5569 −0.638124
\(329\) −8.58451 −0.473279
\(330\) −1.26031 −0.0693778
\(331\) 31.5500 1.73415 0.867073 0.498180i \(-0.165998\pi\)
0.867073 + 0.498180i \(0.165998\pi\)
\(332\) 29.0777 1.59585
\(333\) −4.87156 −0.266960
\(334\) −7.80605 −0.427128
\(335\) −8.43621 −0.460919
\(336\) −6.11727 −0.333724
\(337\) −8.42666 −0.459029 −0.229515 0.973305i \(-0.573714\pi\)
−0.229515 + 0.973305i \(0.573714\pi\)
\(338\) 0.470683 0.0256018
\(339\) −22.2491 −1.20841
\(340\) 1.23109 0.0667655
\(341\) 15.8061 0.855946
\(342\) −1.42504 −0.0770573
\(343\) −1.00000 −0.0539949
\(344\) −20.2637 −1.09255
\(345\) 6.94910 0.374127
\(346\) −10.9673 −0.589608
\(347\) 19.3484 1.03867 0.519337 0.854569i \(-0.326179\pi\)
0.519337 + 0.854569i \(0.326179\pi\)
\(348\) −20.8862 −1.11962
\(349\) 27.2553 1.45894 0.729470 0.684013i \(-0.239767\pi\)
0.729470 + 0.684013i \(0.239767\pi\)
\(350\) 2.22154 0.118746
\(351\) −2.11727 −0.113011
\(352\) −10.8793 −0.579868
\(353\) −25.6742 −1.36650 −0.683249 0.730185i \(-0.739434\pi\)
−0.683249 + 0.730185i \(0.739434\pi\)
\(354\) −12.8862 −0.684892
\(355\) 0.630155 0.0334452
\(356\) −12.2897 −0.651354
\(357\) 2.94137 0.155674
\(358\) 9.90422 0.523454
\(359\) −23.4182 −1.23596 −0.617982 0.786192i \(-0.712049\pi\)
−0.617982 + 0.786192i \(0.712049\pi\)
\(360\) −1.93793 −0.102138
\(361\) −16.8371 −0.886163
\(362\) 7.88273 0.414307
\(363\) −13.3630 −0.701374
\(364\) 1.77846 0.0932165
\(365\) 4.04564 0.211759
\(366\) −2.11727 −0.110671
\(367\) −14.6854 −0.766569 −0.383285 0.923630i \(-0.625207\pi\)
−0.383285 + 0.923630i \(0.625207\pi\)
\(368\) 15.8759 0.827586
\(369\) 13.3776 0.696409
\(370\) −0.589568 −0.0306502
\(371\) −11.2767 −0.585459
\(372\) 28.1104 1.45746
\(373\) −23.6673 −1.22545 −0.612723 0.790298i \(-0.709926\pi\)
−0.612723 + 0.790298i \(0.709926\pi\)
\(374\) 1.38445 0.0715883
\(375\) −11.5715 −0.597551
\(376\) −15.2672 −0.787345
\(377\) 5.22154 0.268923
\(378\) 0.996562 0.0512576
\(379\) −32.7405 −1.68177 −0.840884 0.541215i \(-0.817965\pi\)
−0.840884 + 0.541215i \(0.817965\pi\)
\(380\) 1.38445 0.0710209
\(381\) 1.85352 0.0949586
\(382\) −3.50172 −0.179164
\(383\) −22.6155 −1.15560 −0.577800 0.816178i \(-0.696089\pi\)
−0.577800 + 0.816178i \(0.696089\pi\)
\(384\) −25.1070 −1.28123
\(385\) 1.19051 0.0606739
\(386\) −0.706834 −0.0359769
\(387\) 23.4561 1.19234
\(388\) 6.17246 0.313359
\(389\) 38.0483 1.92913 0.964563 0.263852i \(-0.0849930\pi\)
0.964563 + 0.263852i \(0.0849930\pi\)
\(390\) 0.560352 0.0283745
\(391\) −7.63359 −0.386047
\(392\) −1.77846 −0.0898256
\(393\) 23.8759 1.20438
\(394\) 11.2932 0.568941
\(395\) −0.708654 −0.0356562
\(396\) 8.23453 0.413801
\(397\) −11.7052 −0.587468 −0.293734 0.955887i \(-0.594898\pi\)
−0.293734 + 0.955887i \(0.594898\pi\)
\(398\) −0.948243 −0.0475311
\(399\) 3.30777 0.165596
\(400\) −12.8371 −0.641855
\(401\) 3.55691 0.177624 0.0888119 0.996048i \(-0.471693\pi\)
0.0888119 + 0.996048i \(0.471693\pi\)
\(402\) −16.8724 −0.841520
\(403\) −7.02760 −0.350070
\(404\) −13.7846 −0.685808
\(405\) −5.78963 −0.287689
\(406\) −2.45769 −0.121973
\(407\) 5.32238 0.263821
\(408\) 5.23109 0.258978
\(409\) 5.26213 0.260196 0.130098 0.991501i \(-0.458471\pi\)
0.130098 + 0.991501i \(0.458471\pi\)
\(410\) 1.61899 0.0799560
\(411\) 25.5569 1.26063
\(412\) 30.2277 1.48921
\(413\) 12.1725 0.598968
\(414\) 5.65593 0.277974
\(415\) −8.65432 −0.424824
\(416\) 4.83709 0.237158
\(417\) −31.3155 −1.53353
\(418\) 1.55691 0.0761512
\(419\) 26.0337 1.27183 0.635915 0.771759i \(-0.280623\pi\)
0.635915 + 0.771759i \(0.280623\pi\)
\(420\) 2.11727 0.103312
\(421\) 22.2423 1.08402 0.542011 0.840372i \(-0.317663\pi\)
0.542011 + 0.840372i \(0.317663\pi\)
\(422\) 4.75430 0.231436
\(423\) 17.6724 0.859260
\(424\) −20.0552 −0.973966
\(425\) 6.17246 0.299408
\(426\) 1.26031 0.0610622
\(427\) 2.00000 0.0967868
\(428\) 9.88273 0.477700
\(429\) −5.05863 −0.244233
\(430\) 2.83871 0.136895
\(431\) 27.6742 1.33302 0.666509 0.745497i \(-0.267788\pi\)
0.666509 + 0.745497i \(0.267788\pi\)
\(432\) −5.75859 −0.277060
\(433\) −12.7880 −0.614552 −0.307276 0.951620i \(-0.599418\pi\)
−0.307276 + 0.951620i \(0.599418\pi\)
\(434\) 3.30777 0.158778
\(435\) 6.21629 0.298048
\(436\) −14.0913 −0.674850
\(437\) −8.58451 −0.410653
\(438\) 8.09129 0.386617
\(439\) −18.1656 −0.866996 −0.433498 0.901155i \(-0.642721\pi\)
−0.433498 + 0.901155i \(0.642721\pi\)
\(440\) 2.11727 0.100937
\(441\) 2.05863 0.0980302
\(442\) −0.615547 −0.0292786
\(443\) 0.107714 0.00511767 0.00255883 0.999997i \(-0.499185\pi\)
0.00255883 + 0.999997i \(0.499185\pi\)
\(444\) 9.46563 0.449219
\(445\) 3.65775 0.173394
\(446\) 4.77340 0.226027
\(447\) 20.9345 0.990167
\(448\) 3.16291 0.149433
\(449\) −22.1725 −1.04638 −0.523192 0.852215i \(-0.675259\pi\)
−0.523192 + 0.852215i \(0.675259\pi\)
\(450\) −4.57334 −0.215589
\(451\) −14.6155 −0.688219
\(452\) 17.5930 0.827505
\(453\) 15.9087 0.747457
\(454\) −2.53437 −0.118944
\(455\) −0.529317 −0.0248147
\(456\) 5.88273 0.275484
\(457\) 4.35953 0.203930 0.101965 0.994788i \(-0.467487\pi\)
0.101965 + 0.994788i \(0.467487\pi\)
\(458\) 1.56379 0.0730711
\(459\) 2.76891 0.129241
\(460\) −5.49484 −0.256198
\(461\) 32.3810 1.50813 0.754067 0.656797i \(-0.228089\pi\)
0.754067 + 0.656797i \(0.228089\pi\)
\(462\) 2.38101 0.110775
\(463\) 8.36641 0.388820 0.194410 0.980920i \(-0.437721\pi\)
0.194410 + 0.980920i \(0.437721\pi\)
\(464\) 14.2017 0.659296
\(465\) −8.36641 −0.387983
\(466\) 6.45769 0.299147
\(467\) −19.5423 −0.904310 −0.452155 0.891939i \(-0.649345\pi\)
−0.452155 + 0.891939i \(0.649345\pi\)
\(468\) −3.66119 −0.169239
\(469\) 15.9379 0.735945
\(470\) 2.13875 0.0986532
\(471\) −13.5861 −0.626016
\(472\) 21.6482 0.996439
\(473\) −25.6267 −1.17832
\(474\) −1.41731 −0.0650991
\(475\) 6.94137 0.318492
\(476\) −2.32582 −0.106604
\(477\) 23.2147 1.06293
\(478\) −1.64820 −0.0753870
\(479\) −28.5224 −1.30322 −0.651612 0.758553i \(-0.725907\pi\)
−0.651612 + 0.758553i \(0.725907\pi\)
\(480\) 5.75859 0.262843
\(481\) −2.36641 −0.107899
\(482\) 0.747422 0.0340441
\(483\) −13.1284 −0.597365
\(484\) 10.5665 0.480294
\(485\) −1.83709 −0.0834180
\(486\) −8.58957 −0.389631
\(487\) 24.8241 1.12489 0.562444 0.826836i \(-0.309861\pi\)
0.562444 + 0.826836i \(0.309861\pi\)
\(488\) 3.55691 0.161014
\(489\) 14.3518 0.649011
\(490\) 0.249141 0.0112550
\(491\) 29.1690 1.31638 0.658190 0.752852i \(-0.271322\pi\)
0.658190 + 0.752852i \(0.271322\pi\)
\(492\) −25.9931 −1.17186
\(493\) −6.82860 −0.307545
\(494\) −0.692226 −0.0311447
\(495\) −2.45082 −0.110156
\(496\) −19.1138 −0.858236
\(497\) −1.19051 −0.0534016
\(498\) −17.3086 −0.775618
\(499\) −33.3009 −1.49075 −0.745376 0.666644i \(-0.767730\pi\)
−0.745376 + 0.666644i \(0.767730\pi\)
\(500\) 9.14992 0.409197
\(501\) −37.3009 −1.66648
\(502\) −2.31894 −0.103500
\(503\) −12.3258 −0.549581 −0.274791 0.961504i \(-0.588609\pi\)
−0.274791 + 0.961504i \(0.588609\pi\)
\(504\) 3.66119 0.163082
\(505\) 4.10266 0.182566
\(506\) −6.17934 −0.274705
\(507\) 2.24914 0.0998878
\(508\) −1.46563 −0.0650267
\(509\) 23.7052 1.05072 0.525358 0.850882i \(-0.323932\pi\)
0.525358 + 0.850882i \(0.323932\pi\)
\(510\) −0.732814 −0.0324495
\(511\) −7.64315 −0.338113
\(512\) 22.8302 1.00896
\(513\) 3.11383 0.137479
\(514\) −3.77234 −0.166391
\(515\) −8.99656 −0.396436
\(516\) −45.5760 −2.00637
\(517\) −19.3078 −0.849155
\(518\) 1.11383 0.0489388
\(519\) −52.4070 −2.30041
\(520\) −0.941367 −0.0412817
\(521\) 43.9018 1.92337 0.961687 0.274149i \(-0.0883962\pi\)
0.961687 + 0.274149i \(0.0883962\pi\)
\(522\) 5.05949 0.221448
\(523\) 37.4328 1.63682 0.818410 0.574634i \(-0.194856\pi\)
0.818410 + 0.574634i \(0.194856\pi\)
\(524\) −18.8793 −0.824746
\(525\) 10.6155 0.463300
\(526\) 0.754297 0.0328889
\(527\) 9.19051 0.400345
\(528\) −13.7586 −0.598766
\(529\) 11.0716 0.481375
\(530\) 2.80949 0.122037
\(531\) −25.0586 −1.08745
\(532\) −2.61555 −0.113398
\(533\) 6.49828 0.281472
\(534\) 7.31551 0.316573
\(535\) −2.94137 −0.127166
\(536\) 28.3449 1.22431
\(537\) 47.3269 2.04231
\(538\) −5.56379 −0.239872
\(539\) −2.24914 −0.0968773
\(540\) 1.99312 0.0857704
\(541\) −34.9751 −1.50370 −0.751848 0.659336i \(-0.770837\pi\)
−0.751848 + 0.659336i \(0.770837\pi\)
\(542\) 10.2998 0.442416
\(543\) 37.6673 1.61646
\(544\) −6.32582 −0.271217
\(545\) 4.19395 0.179649
\(546\) −1.05863 −0.0453053
\(547\) 6.50783 0.278255 0.139127 0.990274i \(-0.455570\pi\)
0.139127 + 0.990274i \(0.455570\pi\)
\(548\) −20.2086 −0.863267
\(549\) −4.11727 −0.175721
\(550\) 4.99656 0.213054
\(551\) −7.67924 −0.327146
\(552\) −23.3484 −0.993772
\(553\) 1.33881 0.0569320
\(554\) −4.80949 −0.204336
\(555\) −2.81722 −0.119585
\(556\) 24.7620 1.05014
\(557\) −43.4328 −1.84031 −0.920153 0.391559i \(-0.871936\pi\)
−0.920153 + 0.391559i \(0.871936\pi\)
\(558\) −6.80949 −0.288269
\(559\) 11.3940 0.481915
\(560\) −1.43965 −0.0608362
\(561\) 6.61555 0.279309
\(562\) 0.725938 0.0306218
\(563\) −33.8827 −1.42799 −0.713993 0.700152i \(-0.753115\pi\)
−0.713993 + 0.700152i \(0.753115\pi\)
\(564\) −34.3380 −1.44589
\(565\) −5.23615 −0.220287
\(566\) 7.45875 0.313515
\(567\) 10.9379 0.459350
\(568\) −2.11727 −0.0888385
\(569\) 19.2147 0.805521 0.402760 0.915305i \(-0.368051\pi\)
0.402760 + 0.915305i \(0.368051\pi\)
\(570\) −0.824101 −0.0345178
\(571\) 20.8268 0.871573 0.435787 0.900050i \(-0.356470\pi\)
0.435787 + 0.900050i \(0.356470\pi\)
\(572\) 4.00000 0.167248
\(573\) −16.7328 −0.699023
\(574\) −3.05863 −0.127665
\(575\) −27.5500 −1.14892
\(576\) −6.51127 −0.271303
\(577\) 28.6448 1.19250 0.596249 0.802800i \(-0.296657\pi\)
0.596249 + 0.802800i \(0.296657\pi\)
\(578\) −7.19662 −0.299340
\(579\) −3.37758 −0.140367
\(580\) −4.91539 −0.204100
\(581\) 16.3500 0.678311
\(582\) −3.67418 −0.152300
\(583\) −25.3630 −1.05043
\(584\) −13.5930 −0.562483
\(585\) 1.08967 0.0450523
\(586\) 5.21649 0.215491
\(587\) −4.32076 −0.178337 −0.0891685 0.996017i \(-0.528421\pi\)
−0.0891685 + 0.996017i \(0.528421\pi\)
\(588\) −4.00000 −0.164957
\(589\) 10.3354 0.425862
\(590\) −3.03265 −0.124852
\(591\) 53.9639 2.21978
\(592\) −6.43621 −0.264527
\(593\) −15.9690 −0.655767 −0.327883 0.944718i \(-0.606335\pi\)
−0.327883 + 0.944718i \(0.606335\pi\)
\(594\) 2.24141 0.0919661
\(595\) 0.692226 0.0283785
\(596\) −16.5535 −0.678057
\(597\) −4.53114 −0.185447
\(598\) 2.74742 0.112350
\(599\) −16.8697 −0.689279 −0.344640 0.938735i \(-0.611999\pi\)
−0.344640 + 0.938735i \(0.611999\pi\)
\(600\) 18.8793 0.770744
\(601\) −15.3415 −0.625792 −0.312896 0.949787i \(-0.601299\pi\)
−0.312896 + 0.949787i \(0.601299\pi\)
\(602\) −5.36297 −0.218578
\(603\) −32.8103 −1.33614
\(604\) −12.5795 −0.511851
\(605\) −3.14486 −0.127857
\(606\) 8.20532 0.333318
\(607\) 35.8353 1.45451 0.727254 0.686368i \(-0.240796\pi\)
0.727254 + 0.686368i \(0.240796\pi\)
\(608\) −7.11383 −0.288504
\(609\) −11.7440 −0.475890
\(610\) −0.498281 −0.0201748
\(611\) 8.58451 0.347292
\(612\) 4.78801 0.193544
\(613\) 19.6673 0.794355 0.397177 0.917742i \(-0.369990\pi\)
0.397177 + 0.917742i \(0.369990\pi\)
\(614\) 9.60761 0.387732
\(615\) 7.73625 0.311956
\(616\) −4.00000 −0.161165
\(617\) −41.4588 −1.66907 −0.834533 0.550958i \(-0.814263\pi\)
−0.834533 + 0.550958i \(0.814263\pi\)
\(618\) −17.9931 −0.723790
\(619\) 10.8793 0.437276 0.218638 0.975806i \(-0.429839\pi\)
0.218638 + 0.975806i \(0.429839\pi\)
\(620\) 6.61555 0.265687
\(621\) −12.3587 −0.495937
\(622\) −0.905275 −0.0362982
\(623\) −6.91033 −0.276857
\(624\) 6.11727 0.244887
\(625\) 20.8759 0.835034
\(626\) −6.76203 −0.270265
\(627\) 7.43965 0.297111
\(628\) 10.7429 0.428689
\(629\) 3.09472 0.123395
\(630\) −0.512889 −0.0204340
\(631\) −31.4396 −1.25159 −0.625796 0.779987i \(-0.715226\pi\)
−0.625796 + 0.779987i \(0.715226\pi\)
\(632\) 2.38101 0.0947117
\(633\) 22.7182 0.902968
\(634\) 7.32238 0.290809
\(635\) 0.436210 0.0173105
\(636\) −45.1070 −1.78861
\(637\) 1.00000 0.0396214
\(638\) −5.52770 −0.218844
\(639\) 2.45082 0.0969529
\(640\) −5.90871 −0.233562
\(641\) 3.04221 0.120160 0.0600799 0.998194i \(-0.480864\pi\)
0.0600799 + 0.998194i \(0.480864\pi\)
\(642\) −5.88273 −0.232173
\(643\) 8.02922 0.316641 0.158321 0.987388i \(-0.449392\pi\)
0.158321 + 0.987388i \(0.449392\pi\)
\(644\) 10.3810 0.409069
\(645\) 13.5646 0.534107
\(646\) 0.905275 0.0356176
\(647\) 7.07324 0.278078 0.139039 0.990287i \(-0.455599\pi\)
0.139039 + 0.990287i \(0.455599\pi\)
\(648\) 19.4526 0.764172
\(649\) 27.3776 1.07466
\(650\) −2.22154 −0.0871361
\(651\) 15.8061 0.619488
\(652\) −11.3484 −0.444436
\(653\) −15.7586 −0.616681 −0.308341 0.951276i \(-0.599774\pi\)
−0.308341 + 0.951276i \(0.599774\pi\)
\(654\) 8.38789 0.327992
\(655\) 5.61899 0.219552
\(656\) 17.6742 0.690061
\(657\) 15.7344 0.613859
\(658\) −4.04059 −0.157518
\(659\) −12.2181 −0.475950 −0.237975 0.971271i \(-0.576484\pi\)
−0.237975 + 0.971271i \(0.576484\pi\)
\(660\) 4.76203 0.185362
\(661\) 3.73443 0.145253 0.0726263 0.997359i \(-0.476862\pi\)
0.0726263 + 0.997359i \(0.476862\pi\)
\(662\) 14.8501 0.577165
\(663\) −2.94137 −0.114233
\(664\) 29.0777 1.12844
\(665\) 0.778457 0.0301873
\(666\) −2.29296 −0.0888506
\(667\) 30.4786 1.18014
\(668\) 29.4948 1.14119
\(669\) 22.8095 0.881866
\(670\) −3.97078 −0.153405
\(671\) 4.49828 0.173654
\(672\) −10.8793 −0.419678
\(673\) −5.65775 −0.218090 −0.109045 0.994037i \(-0.534779\pi\)
−0.109045 + 0.994037i \(0.534779\pi\)
\(674\) −3.96629 −0.152776
\(675\) 9.99312 0.384636
\(676\) −1.77846 −0.0684022
\(677\) 9.39906 0.361235 0.180618 0.983553i \(-0.442190\pi\)
0.180618 + 0.983553i \(0.442190\pi\)
\(678\) −10.4723 −0.402186
\(679\) 3.47068 0.133193
\(680\) 1.23109 0.0472103
\(681\) −12.1104 −0.464071
\(682\) 7.43965 0.284879
\(683\) −20.7328 −0.793319 −0.396660 0.917966i \(-0.629831\pi\)
−0.396660 + 0.917966i \(0.629831\pi\)
\(684\) 5.38445 0.205880
\(685\) 6.01461 0.229806
\(686\) −0.470683 −0.0179708
\(687\) 7.47250 0.285094
\(688\) 30.9897 1.18147
\(689\) 11.2767 0.429610
\(690\) 3.27083 0.124518
\(691\) 16.0862 0.611949 0.305975 0.952040i \(-0.401018\pi\)
0.305975 + 0.952040i \(0.401018\pi\)
\(692\) 41.4396 1.57530
\(693\) 4.63016 0.175885
\(694\) 9.10695 0.345695
\(695\) −7.36984 −0.279554
\(696\) −20.8862 −0.791688
\(697\) −8.49828 −0.321895
\(698\) 12.8286 0.485570
\(699\) 30.8578 1.16715
\(700\) −8.39400 −0.317264
\(701\) −6.98013 −0.263636 −0.131818 0.991274i \(-0.542081\pi\)
−0.131818 + 0.991274i \(0.542081\pi\)
\(702\) −0.996562 −0.0376128
\(703\) 3.48024 0.131260
\(704\) 7.11383 0.268112
\(705\) 10.2199 0.384905
\(706\) −12.0844 −0.454803
\(707\) −7.75086 −0.291501
\(708\) 48.6898 1.82988
\(709\) 8.39239 0.315183 0.157591 0.987504i \(-0.449627\pi\)
0.157591 + 0.987504i \(0.449627\pi\)
\(710\) 0.296604 0.0111313
\(711\) −2.75612 −0.103362
\(712\) −12.2897 −0.460577
\(713\) −41.0207 −1.53624
\(714\) 1.38445 0.0518118
\(715\) −1.19051 −0.0445225
\(716\) −37.4227 −1.39855
\(717\) −7.87586 −0.294129
\(718\) −11.0225 −0.411358
\(719\) −5.16129 −0.192484 −0.0962418 0.995358i \(-0.530682\pi\)
−0.0962418 + 0.995358i \(0.530682\pi\)
\(720\) 2.96371 0.110451
\(721\) 16.9966 0.632985
\(722\) −7.92494 −0.294936
\(723\) 3.57152 0.132826
\(724\) −29.7846 −1.10693
\(725\) −24.6448 −0.915284
\(726\) −6.28973 −0.233434
\(727\) −40.4362 −1.49970 −0.749848 0.661610i \(-0.769873\pi\)
−0.749848 + 0.661610i \(0.769873\pi\)
\(728\) 1.77846 0.0659140
\(729\) −8.23109 −0.304855
\(730\) 1.90422 0.0704782
\(731\) −14.9008 −0.551125
\(732\) 8.00000 0.295689
\(733\) −39.1311 −1.44534 −0.722670 0.691193i \(-0.757085\pi\)
−0.722670 + 0.691193i \(0.757085\pi\)
\(734\) −6.91215 −0.255132
\(735\) 1.19051 0.0439125
\(736\) 28.2345 1.04074
\(737\) 35.8466 1.32043
\(738\) 6.29660 0.231781
\(739\) −7.13531 −0.262477 −0.131238 0.991351i \(-0.541895\pi\)
−0.131238 + 0.991351i \(0.541895\pi\)
\(740\) 2.22766 0.0818903
\(741\) −3.30777 −0.121514
\(742\) −5.30777 −0.194855
\(743\) 13.8827 0.509308 0.254654 0.967032i \(-0.418038\pi\)
0.254654 + 0.967032i \(0.418038\pi\)
\(744\) 28.1104 1.03058
\(745\) 4.92676 0.180502
\(746\) −11.1398 −0.407857
\(747\) −33.6586 −1.23150
\(748\) −5.23109 −0.191268
\(749\) 5.55691 0.203045
\(750\) −5.44652 −0.198879
\(751\) −37.3251 −1.36201 −0.681005 0.732278i \(-0.738457\pi\)
−0.681005 + 0.732278i \(0.738457\pi\)
\(752\) 23.3484 0.851427
\(753\) −11.0810 −0.403813
\(754\) 2.45769 0.0895039
\(755\) 3.74398 0.136258
\(756\) −3.76547 −0.136949
\(757\) −7.10428 −0.258209 −0.129105 0.991631i \(-0.541210\pi\)
−0.129105 + 0.991631i \(0.541210\pi\)
\(758\) −15.4104 −0.559732
\(759\) −29.5277 −1.07179
\(760\) 1.38445 0.0502194
\(761\) −25.9621 −0.941125 −0.470562 0.882367i \(-0.655949\pi\)
−0.470562 + 0.882367i \(0.655949\pi\)
\(762\) 0.872420 0.0316044
\(763\) −7.92332 −0.286843
\(764\) 13.2311 0.478684
\(765\) −1.42504 −0.0515224
\(766\) −10.6448 −0.384611
\(767\) −12.1725 −0.439522
\(768\) 2.41023 0.0869717
\(769\) −21.4638 −0.774005 −0.387002 0.922079i \(-0.626489\pi\)
−0.387002 + 0.922079i \(0.626489\pi\)
\(770\) 0.560352 0.0201937
\(771\) −18.0260 −0.649190
\(772\) 2.67074 0.0961221
\(773\) −40.4914 −1.45637 −0.728187 0.685378i \(-0.759637\pi\)
−0.728187 + 0.685378i \(0.759637\pi\)
\(774\) 11.0404 0.396838
\(775\) 33.1690 1.19147
\(776\) 6.17246 0.221578
\(777\) 5.32238 0.190939
\(778\) 17.9087 0.642058
\(779\) −9.55691 −0.342412
\(780\) −2.11727 −0.0758103
\(781\) −2.67762 −0.0958127
\(782\) −3.59301 −0.128486
\(783\) −11.0554 −0.395088
\(784\) 2.71982 0.0971366
\(785\) −3.19738 −0.114119
\(786\) 11.2380 0.400845
\(787\) −5.02072 −0.178969 −0.0894847 0.995988i \(-0.528522\pi\)
−0.0894847 + 0.995988i \(0.528522\pi\)
\(788\) −42.6707 −1.52008
\(789\) 3.60438 0.128319
\(790\) −0.333552 −0.0118672
\(791\) 9.89229 0.351729
\(792\) 8.23453 0.292601
\(793\) −2.00000 −0.0710221
\(794\) −5.50945 −0.195523
\(795\) 13.4250 0.476137
\(796\) 3.58289 0.126992
\(797\) 19.7002 0.697815 0.348908 0.937157i \(-0.386553\pi\)
0.348908 + 0.937157i \(0.386553\pi\)
\(798\) 1.55691 0.0551142
\(799\) −11.2266 −0.397169
\(800\) −22.8302 −0.807170
\(801\) 14.2258 0.502645
\(802\) 1.67418 0.0591174
\(803\) −17.1905 −0.606640
\(804\) 63.7517 2.24835
\(805\) −3.08967 −0.108897
\(806\) −3.30777 −0.116511
\(807\) −26.5863 −0.935883
\(808\) −13.7846 −0.484940
\(809\) −2.57678 −0.0905947 −0.0452974 0.998974i \(-0.514424\pi\)
−0.0452974 + 0.998974i \(0.514424\pi\)
\(810\) −2.72508 −0.0957496
\(811\) 41.2311 1.44782 0.723910 0.689895i \(-0.242343\pi\)
0.723910 + 0.689895i \(0.242343\pi\)
\(812\) 9.28629 0.325885
\(813\) 49.2173 1.72613
\(814\) 2.50516 0.0878057
\(815\) 3.37758 0.118311
\(816\) −8.00000 −0.280056
\(817\) −16.7570 −0.586252
\(818\) 2.47680 0.0865992
\(819\) −2.05863 −0.0719345
\(820\) −6.11727 −0.213624
\(821\) 5.26719 0.183826 0.0919130 0.995767i \(-0.470702\pi\)
0.0919130 + 0.995767i \(0.470702\pi\)
\(822\) 12.0292 0.419567
\(823\) 36.0191 1.25555 0.627774 0.778396i \(-0.283966\pi\)
0.627774 + 0.778396i \(0.283966\pi\)
\(824\) 30.2277 1.05303
\(825\) 23.8759 0.831251
\(826\) 5.72938 0.199350
\(827\) −16.3157 −0.567353 −0.283676 0.958920i \(-0.591554\pi\)
−0.283676 + 0.958920i \(0.591554\pi\)
\(828\) −21.3707 −0.742683
\(829\) −30.3956 −1.05568 −0.527842 0.849343i \(-0.676999\pi\)
−0.527842 + 0.849343i \(0.676999\pi\)
\(830\) −4.07344 −0.141391
\(831\) −22.9820 −0.797235
\(832\) −3.16291 −0.109654
\(833\) −1.30777 −0.0453117
\(834\) −14.7397 −0.510394
\(835\) −8.77846 −0.303791
\(836\) −5.88273 −0.203459
\(837\) 14.8793 0.514304
\(838\) 12.2536 0.423295
\(839\) 29.8398 1.03018 0.515092 0.857135i \(-0.327758\pi\)
0.515092 + 0.857135i \(0.327758\pi\)
\(840\) 2.11727 0.0730526
\(841\) −1.73549 −0.0598445
\(842\) 10.4691 0.360788
\(843\) 3.46886 0.119474
\(844\) −17.9639 −0.618343
\(845\) 0.529317 0.0182090
\(846\) 8.31809 0.285982
\(847\) 5.94137 0.204148
\(848\) 30.6707 1.05324
\(849\) 35.6413 1.22321
\(850\) 2.90528 0.0996501
\(851\) −13.8129 −0.473501
\(852\) −4.76203 −0.163144
\(853\) −0.203497 −0.00696761 −0.00348380 0.999994i \(-0.501109\pi\)
−0.00348380 + 0.999994i \(0.501109\pi\)
\(854\) 0.941367 0.0322129
\(855\) −1.60256 −0.0548063
\(856\) 9.88273 0.337785
\(857\) −12.6155 −0.430939 −0.215469 0.976511i \(-0.569128\pi\)
−0.215469 + 0.976511i \(0.569128\pi\)
\(858\) −2.38101 −0.0812865
\(859\) −27.9671 −0.954227 −0.477113 0.878842i \(-0.658317\pi\)
−0.477113 + 0.878842i \(0.658317\pi\)
\(860\) −10.7259 −0.365751
\(861\) −14.6155 −0.498097
\(862\) 13.0258 0.443660
\(863\) −2.76891 −0.0942546 −0.0471273 0.998889i \(-0.515007\pi\)
−0.0471273 + 0.998889i \(0.515007\pi\)
\(864\) −10.2414 −0.348420
\(865\) −12.3336 −0.419353
\(866\) −6.01910 −0.204537
\(867\) −34.3887 −1.16790
\(868\) −12.4983 −0.424219
\(869\) 3.01117 0.102147
\(870\) 2.92590 0.0991974
\(871\) −15.9379 −0.540036
\(872\) −14.0913 −0.477191
\(873\) −7.14486 −0.241817
\(874\) −4.04059 −0.136675
\(875\) 5.14486 0.173928
\(876\) −30.5726 −1.03295
\(877\) 6.71133 0.226626 0.113313 0.993559i \(-0.463854\pi\)
0.113313 + 0.993559i \(0.463854\pi\)
\(878\) −8.55024 −0.288557
\(879\) 24.9268 0.840759
\(880\) −3.23797 −0.109152
\(881\) −18.3741 −0.619040 −0.309520 0.950893i \(-0.600168\pi\)
−0.309520 + 0.950893i \(0.600168\pi\)
\(882\) 0.968964 0.0326267
\(883\) 9.93105 0.334207 0.167103 0.985939i \(-0.446559\pi\)
0.167103 + 0.985939i \(0.446559\pi\)
\(884\) 2.32582 0.0782258
\(885\) −14.4914 −0.487123
\(886\) 0.0506994 0.00170328
\(887\) −51.3776 −1.72509 −0.862545 0.505980i \(-0.831131\pi\)
−0.862545 + 0.505980i \(0.831131\pi\)
\(888\) 9.46563 0.317646
\(889\) −0.824101 −0.0276394
\(890\) 1.72164 0.0577096
\(891\) 24.6009 0.824162
\(892\) −18.0361 −0.603893
\(893\) −12.6251 −0.422483
\(894\) 9.85352 0.329551
\(895\) 11.1380 0.372302
\(896\) 11.1629 0.372927
\(897\) 13.1284 0.438346
\(898\) −10.4362 −0.348261
\(899\) −36.6949 −1.22384
\(900\) 17.2802 0.576006
\(901\) −14.7474 −0.491308
\(902\) −6.87930 −0.229055
\(903\) −25.6267 −0.852804
\(904\) 17.5930 0.585135
\(905\) 8.86469 0.294672
\(906\) 7.48797 0.248771
\(907\) 4.34225 0.144182 0.0720910 0.997398i \(-0.477033\pi\)
0.0720910 + 0.997398i \(0.477033\pi\)
\(908\) 9.57602 0.317791
\(909\) 15.9562 0.529233
\(910\) −0.249141 −0.00825893
\(911\) 31.4853 1.04315 0.521577 0.853204i \(-0.325344\pi\)
0.521577 + 0.853204i \(0.325344\pi\)
\(912\) −8.99656 −0.297906
\(913\) 36.7734 1.21702
\(914\) 2.05196 0.0678728
\(915\) −2.38101 −0.0787139
\(916\) −5.90871 −0.195229
\(917\) −10.6155 −0.350556
\(918\) 1.30328 0.0430146
\(919\) 43.4588 1.43357 0.716786 0.697293i \(-0.245612\pi\)
0.716786 + 0.697293i \(0.245612\pi\)
\(920\) −5.49484 −0.181160
\(921\) 45.9096 1.51277
\(922\) 15.2412 0.501942
\(923\) 1.19051 0.0391860
\(924\) −8.99656 −0.295965
\(925\) 11.1690 0.367235
\(926\) 3.93793 0.129408
\(927\) −34.9897 −1.14921
\(928\) 25.2571 0.829104
\(929\) 4.40517 0.144529 0.0722645 0.997385i \(-0.476977\pi\)
0.0722645 + 0.997385i \(0.476977\pi\)
\(930\) −3.93793 −0.129130
\(931\) −1.47068 −0.0481997
\(932\) −24.4001 −0.799252
\(933\) −4.32582 −0.141621
\(934\) −9.19824 −0.300976
\(935\) 1.55691 0.0509165
\(936\) −3.66119 −0.119670
\(937\) −34.0990 −1.11397 −0.556983 0.830524i \(-0.688041\pi\)
−0.556983 + 0.830524i \(0.688041\pi\)
\(938\) 7.50172 0.244940
\(939\) −32.3121 −1.05446
\(940\) −8.08117 −0.263579
\(941\) 44.4672 1.44959 0.724795 0.688964i \(-0.241934\pi\)
0.724795 + 0.688964i \(0.241934\pi\)
\(942\) −6.39477 −0.208353
\(943\) 37.9311 1.23521
\(944\) −33.1070 −1.07754
\(945\) 1.12070 0.0364565
\(946\) −12.0621 −0.392172
\(947\) 57.9311 1.88251 0.941253 0.337702i \(-0.109650\pi\)
0.941253 + 0.337702i \(0.109650\pi\)
\(948\) 5.35524 0.173930
\(949\) 7.64315 0.248107
\(950\) 3.26719 0.106002
\(951\) 34.9897 1.13462
\(952\) −2.32582 −0.0753802
\(953\) 35.3060 1.14367 0.571836 0.820368i \(-0.306231\pi\)
0.571836 + 0.820368i \(0.306231\pi\)
\(954\) 10.9268 0.353767
\(955\) −3.93793 −0.127428
\(956\) 6.22766 0.201417
\(957\) −26.4139 −0.853839
\(958\) −13.4250 −0.433743
\(959\) −11.3630 −0.366929
\(960\) −3.76547 −0.121530
\(961\) 18.3871 0.593133
\(962\) −1.11383 −0.0359113
\(963\) −11.4396 −0.368638
\(964\) −2.82410 −0.0909582
\(965\) −0.794885 −0.0255882
\(966\) −6.17934 −0.198817
\(967\) 23.7148 0.762616 0.381308 0.924448i \(-0.375474\pi\)
0.381308 + 0.924448i \(0.375474\pi\)
\(968\) 10.5665 0.339619
\(969\) 4.32582 0.138965
\(970\) −0.864688 −0.0277635
\(971\) −23.9379 −0.768205 −0.384102 0.923291i \(-0.625489\pi\)
−0.384102 + 0.923291i \(0.625489\pi\)
\(972\) 32.4553 1.04100
\(973\) 13.9233 0.446361
\(974\) 11.6843 0.374389
\(975\) −10.6155 −0.339970
\(976\) −5.43965 −0.174119
\(977\) −16.1871 −0.517870 −0.258935 0.965895i \(-0.583372\pi\)
−0.258935 + 0.965895i \(0.583372\pi\)
\(978\) 6.75515 0.216006
\(979\) −15.5423 −0.496734
\(980\) −0.941367 −0.0300709
\(981\) 16.3112 0.520777
\(982\) 13.7294 0.438122
\(983\) 45.7243 1.45838 0.729190 0.684312i \(-0.239897\pi\)
0.729190 + 0.684312i \(0.239897\pi\)
\(984\) −25.9931 −0.828631
\(985\) 12.7000 0.404654
\(986\) −3.21411 −0.102358
\(987\) −19.3078 −0.614573
\(988\) 2.61555 0.0832116
\(989\) 66.5078 2.11483
\(990\) −1.15356 −0.0366625
\(991\) −5.40356 −0.171650 −0.0858248 0.996310i \(-0.527353\pi\)
−0.0858248 + 0.996310i \(0.527353\pi\)
\(992\) −33.9931 −1.07928
\(993\) 70.9605 2.25186
\(994\) −0.560352 −0.0177733
\(995\) −1.06637 −0.0338061
\(996\) 65.3999 2.07228
\(997\) 6.04832 0.191552 0.0957761 0.995403i \(-0.469467\pi\)
0.0957761 + 0.995403i \(0.469467\pi\)
\(998\) −15.6742 −0.496158
\(999\) 5.01031 0.158519
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 91.2.a.d.1.2 3
3.2 odd 2 819.2.a.i.1.2 3
4.3 odd 2 1456.2.a.t.1.1 3
5.4 even 2 2275.2.a.m.1.2 3
7.2 even 3 637.2.e.j.508.2 6
7.3 odd 6 637.2.e.i.79.2 6
7.4 even 3 637.2.e.j.79.2 6
7.5 odd 6 637.2.e.i.508.2 6
7.6 odd 2 637.2.a.j.1.2 3
8.3 odd 2 5824.2.a.bs.1.3 3
8.5 even 2 5824.2.a.by.1.1 3
13.5 odd 4 1183.2.c.f.337.3 6
13.8 odd 4 1183.2.c.f.337.4 6
13.12 even 2 1183.2.a.i.1.2 3
21.20 even 2 5733.2.a.x.1.2 3
91.90 odd 2 8281.2.a.bg.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.2 3 1.1 even 1 trivial
637.2.a.j.1.2 3 7.6 odd 2
637.2.e.i.79.2 6 7.3 odd 6
637.2.e.i.508.2 6 7.5 odd 6
637.2.e.j.79.2 6 7.4 even 3
637.2.e.j.508.2 6 7.2 even 3
819.2.a.i.1.2 3 3.2 odd 2
1183.2.a.i.1.2 3 13.12 even 2
1183.2.c.f.337.3 6 13.5 odd 4
1183.2.c.f.337.4 6 13.8 odd 4
1456.2.a.t.1.1 3 4.3 odd 2
2275.2.a.m.1.2 3 5.4 even 2
5733.2.a.x.1.2 3 21.20 even 2
5824.2.a.bs.1.3 3 8.3 odd 2
5824.2.a.by.1.1 3 8.5 even 2
8281.2.a.bg.1.2 3 91.90 odd 2