Properties

Label 1850.2.a.ba.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.21432 q^{3} +1.00000 q^{4} +1.21432 q^{6} +3.59210 q^{7} -1.00000 q^{8} -1.52543 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.21432 q^{3} +1.00000 q^{4} +1.21432 q^{6} +3.59210 q^{7} -1.00000 q^{8} -1.52543 q^{9} -4.73975 q^{11} -1.21432 q^{12} +1.78568 q^{13} -3.59210 q^{14} +1.00000 q^{16} +1.83654 q^{17} +1.52543 q^{18} +3.28100 q^{19} -4.36196 q^{21} +4.73975 q^{22} -1.80642 q^{23} +1.21432 q^{24} -1.78568 q^{26} +5.49532 q^{27} +3.59210 q^{28} -0.755569 q^{29} -2.06668 q^{31} -1.00000 q^{32} +5.75557 q^{33} -1.83654 q^{34} -1.52543 q^{36} +1.00000 q^{37} -3.28100 q^{38} -2.16839 q^{39} -2.57136 q^{41} +4.36196 q^{42} +9.19850 q^{43} -4.73975 q^{44} +1.80642 q^{46} +1.24443 q^{47} -1.21432 q^{48} +5.90321 q^{49} -2.23014 q^{51} +1.78568 q^{52} +6.56199 q^{53} -5.49532 q^{54} -3.59210 q^{56} -3.98418 q^{57} +0.755569 q^{58} +9.19850 q^{59} -9.39853 q^{61} +2.06668 q^{62} -5.47949 q^{63} +1.00000 q^{64} -5.75557 q^{66} -2.39853 q^{67} +1.83654 q^{68} +2.19358 q^{69} -9.39853 q^{71} +1.52543 q^{72} +8.09679 q^{73} -1.00000 q^{74} +3.28100 q^{76} -17.0257 q^{77} +2.16839 q^{78} +11.8064 q^{79} -2.09679 q^{81} +2.57136 q^{82} -15.4035 q^{83} -4.36196 q^{84} -9.19850 q^{86} +0.917502 q^{87} +4.73975 q^{88} -0.933323 q^{89} +6.41435 q^{91} -1.80642 q^{92} +2.50961 q^{93} -1.24443 q^{94} +1.21432 q^{96} -2.42864 q^{97} -5.90321 q^{98} +7.23014 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 2 q^{9} - q^{11} + 3 q^{12} + 12 q^{13} - 4 q^{14} + 3 q^{16} - q^{17} - 2 q^{18} + 3 q^{19} + q^{22} + 8 q^{23} - 3 q^{24} - 12 q^{26} + 3 q^{27} + 4 q^{28} - 2 q^{29} - 6 q^{31} - 3 q^{32} + 17 q^{33} + q^{34} + 2 q^{36} + 3 q^{37} - 3 q^{38} + 20 q^{39} - 21 q^{41} + 8 q^{43} - q^{44} - 8 q^{46} + 4 q^{47} + 3 q^{48} + 11 q^{49} - 13 q^{51} + 12 q^{52} + 6 q^{53} - 3 q^{54} - 4 q^{56} + q^{57} + 2 q^{58} + 8 q^{59} - 8 q^{61} + 6 q^{62} + 10 q^{63} + 3 q^{64} - 17 q^{66} + 13 q^{67} - q^{68} + 20 q^{69} - 8 q^{71} - 2 q^{72} + 31 q^{73} - 3 q^{74} + 3 q^{76} + 2 q^{77} - 20 q^{78} + 22 q^{79} - 13 q^{81} + 21 q^{82} + 7 q^{83} - 8 q^{86} - 10 q^{87} + q^{88} - 3 q^{89} + 12 q^{91} + 8 q^{92} - 12 q^{93} - 4 q^{94} - 3 q^{96} + 6 q^{97} - 11 q^{98} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.21432 −0.701088 −0.350544 0.936546i \(-0.614003\pi\)
−0.350544 + 0.936546i \(0.614003\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.21432 0.495744
\(7\) 3.59210 1.35769 0.678844 0.734283i \(-0.262481\pi\)
0.678844 + 0.734283i \(0.262481\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.52543 −0.508476
\(10\) 0 0
\(11\) −4.73975 −1.42909 −0.714544 0.699591i \(-0.753366\pi\)
−0.714544 + 0.699591i \(0.753366\pi\)
\(12\) −1.21432 −0.350544
\(13\) 1.78568 0.495259 0.247629 0.968855i \(-0.420348\pi\)
0.247629 + 0.968855i \(0.420348\pi\)
\(14\) −3.59210 −0.960030
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.83654 0.445425 0.222713 0.974884i \(-0.428509\pi\)
0.222713 + 0.974884i \(0.428509\pi\)
\(18\) 1.52543 0.359547
\(19\) 3.28100 0.752712 0.376356 0.926475i \(-0.377177\pi\)
0.376356 + 0.926475i \(0.377177\pi\)
\(20\) 0 0
\(21\) −4.36196 −0.951858
\(22\) 4.73975 1.01052
\(23\) −1.80642 −0.376665 −0.188333 0.982105i \(-0.560308\pi\)
−0.188333 + 0.982105i \(0.560308\pi\)
\(24\) 1.21432 0.247872
\(25\) 0 0
\(26\) −1.78568 −0.350201
\(27\) 5.49532 1.05757
\(28\) 3.59210 0.678844
\(29\) −0.755569 −0.140306 −0.0701528 0.997536i \(-0.522349\pi\)
−0.0701528 + 0.997536i \(0.522349\pi\)
\(30\) 0 0
\(31\) −2.06668 −0.371186 −0.185593 0.982627i \(-0.559421\pi\)
−0.185593 + 0.982627i \(0.559421\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.75557 1.00192
\(34\) −1.83654 −0.314963
\(35\) 0 0
\(36\) −1.52543 −0.254238
\(37\) 1.00000 0.164399
\(38\) −3.28100 −0.532248
\(39\) −2.16839 −0.347220
\(40\) 0 0
\(41\) −2.57136 −0.401579 −0.200790 0.979634i \(-0.564351\pi\)
−0.200790 + 0.979634i \(0.564351\pi\)
\(42\) 4.36196 0.673065
\(43\) 9.19850 1.40276 0.701379 0.712789i \(-0.252568\pi\)
0.701379 + 0.712789i \(0.252568\pi\)
\(44\) −4.73975 −0.714544
\(45\) 0 0
\(46\) 1.80642 0.266343
\(47\) 1.24443 0.181519 0.0907595 0.995873i \(-0.471071\pi\)
0.0907595 + 0.995873i \(0.471071\pi\)
\(48\) −1.21432 −0.175272
\(49\) 5.90321 0.843316
\(50\) 0 0
\(51\) −2.23014 −0.312282
\(52\) 1.78568 0.247629
\(53\) 6.56199 0.901359 0.450680 0.892686i \(-0.351182\pi\)
0.450680 + 0.892686i \(0.351182\pi\)
\(54\) −5.49532 −0.747818
\(55\) 0 0
\(56\) −3.59210 −0.480015
\(57\) −3.98418 −0.527717
\(58\) 0.755569 0.0992110
\(59\) 9.19850 1.19754 0.598771 0.800920i \(-0.295656\pi\)
0.598771 + 0.800920i \(0.295656\pi\)
\(60\) 0 0
\(61\) −9.39853 −1.20336 −0.601679 0.798738i \(-0.705501\pi\)
−0.601679 + 0.798738i \(0.705501\pi\)
\(62\) 2.06668 0.262468
\(63\) −5.47949 −0.690351
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.75557 −0.708462
\(67\) −2.39853 −0.293027 −0.146513 0.989209i \(-0.546805\pi\)
−0.146513 + 0.989209i \(0.546805\pi\)
\(68\) 1.83654 0.222713
\(69\) 2.19358 0.264076
\(70\) 0 0
\(71\) −9.39853 −1.11540 −0.557700 0.830043i \(-0.688316\pi\)
−0.557700 + 0.830043i \(0.688316\pi\)
\(72\) 1.52543 0.179773
\(73\) 8.09679 0.947657 0.473829 0.880617i \(-0.342872\pi\)
0.473829 + 0.880617i \(0.342872\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 3.28100 0.376356
\(77\) −17.0257 −1.94025
\(78\) 2.16839 0.245521
\(79\) 11.8064 1.32833 0.664163 0.747588i \(-0.268788\pi\)
0.664163 + 0.747588i \(0.268788\pi\)
\(80\) 0 0
\(81\) −2.09679 −0.232976
\(82\) 2.57136 0.283959
\(83\) −15.4035 −1.69075 −0.845374 0.534175i \(-0.820622\pi\)
−0.845374 + 0.534175i \(0.820622\pi\)
\(84\) −4.36196 −0.475929
\(85\) 0 0
\(86\) −9.19850 −0.991900
\(87\) 0.917502 0.0983665
\(88\) 4.73975 0.505259
\(89\) −0.933323 −0.0989321 −0.0494660 0.998776i \(-0.515752\pi\)
−0.0494660 + 0.998776i \(0.515752\pi\)
\(90\) 0 0
\(91\) 6.41435 0.672407
\(92\) −1.80642 −0.188333
\(93\) 2.50961 0.260234
\(94\) −1.24443 −0.128353
\(95\) 0 0
\(96\) 1.21432 0.123936
\(97\) −2.42864 −0.246591 −0.123295 0.992370i \(-0.539346\pi\)
−0.123295 + 0.992370i \(0.539346\pi\)
\(98\) −5.90321 −0.596314
\(99\) 7.23014 0.726657
\(100\) 0 0
\(101\) 8.04149 0.800158 0.400079 0.916481i \(-0.368983\pi\)
0.400079 + 0.916481i \(0.368983\pi\)
\(102\) 2.23014 0.220817
\(103\) 0.495316 0.0488049 0.0244025 0.999702i \(-0.492232\pi\)
0.0244025 + 0.999702i \(0.492232\pi\)
\(104\) −1.78568 −0.175100
\(105\) 0 0
\(106\) −6.56199 −0.637357
\(107\) 13.4494 1.30020 0.650100 0.759848i \(-0.274727\pi\)
0.650100 + 0.759848i \(0.274727\pi\)
\(108\) 5.49532 0.528787
\(109\) 10.2351 0.980341 0.490171 0.871626i \(-0.336934\pi\)
0.490171 + 0.871626i \(0.336934\pi\)
\(110\) 0 0
\(111\) −1.21432 −0.115258
\(112\) 3.59210 0.339422
\(113\) −7.49532 −0.705100 −0.352550 0.935793i \(-0.614685\pi\)
−0.352550 + 0.935793i \(0.614685\pi\)
\(114\) 3.98418 0.373153
\(115\) 0 0
\(116\) −0.755569 −0.0701528
\(117\) −2.72393 −0.251827
\(118\) −9.19850 −0.846790
\(119\) 6.59703 0.604748
\(120\) 0 0
\(121\) 11.4652 1.04229
\(122\) 9.39853 0.850903
\(123\) 3.12245 0.281542
\(124\) −2.06668 −0.185593
\(125\) 0 0
\(126\) 5.47949 0.488152
\(127\) 3.45875 0.306915 0.153457 0.988155i \(-0.450959\pi\)
0.153457 + 0.988155i \(0.450959\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −11.1699 −0.983456
\(130\) 0 0
\(131\) 16.0874 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(132\) 5.75557 0.500958
\(133\) 11.7857 1.02195
\(134\) 2.39853 0.207201
\(135\) 0 0
\(136\) −1.83654 −0.157482
\(137\) 15.9906 1.36617 0.683086 0.730338i \(-0.260637\pi\)
0.683086 + 0.730338i \(0.260637\pi\)
\(138\) −2.19358 −0.186730
\(139\) 11.0350 0.935979 0.467990 0.883734i \(-0.344978\pi\)
0.467990 + 0.883734i \(0.344978\pi\)
\(140\) 0 0
\(141\) −1.51114 −0.127261
\(142\) 9.39853 0.788707
\(143\) −8.46367 −0.707768
\(144\) −1.52543 −0.127119
\(145\) 0 0
\(146\) −8.09679 −0.670095
\(147\) −7.16839 −0.591239
\(148\) 1.00000 0.0821995
\(149\) 13.9748 1.14486 0.572431 0.819953i \(-0.306001\pi\)
0.572431 + 0.819953i \(0.306001\pi\)
\(150\) 0 0
\(151\) 8.42864 0.685913 0.342956 0.939351i \(-0.388572\pi\)
0.342956 + 0.939351i \(0.388572\pi\)
\(152\) −3.28100 −0.266124
\(153\) −2.80150 −0.226488
\(154\) 17.0257 1.37197
\(155\) 0 0
\(156\) −2.16839 −0.173610
\(157\) 9.17775 0.732465 0.366232 0.930523i \(-0.380648\pi\)
0.366232 + 0.930523i \(0.380648\pi\)
\(158\) −11.8064 −0.939269
\(159\) −7.96836 −0.631932
\(160\) 0 0
\(161\) −6.48886 −0.511394
\(162\) 2.09679 0.164739
\(163\) 13.9447 1.09223 0.546117 0.837709i \(-0.316106\pi\)
0.546117 + 0.837709i \(0.316106\pi\)
\(164\) −2.57136 −0.200790
\(165\) 0 0
\(166\) 15.4035 1.19554
\(167\) 17.6795 1.36808 0.684041 0.729443i \(-0.260221\pi\)
0.684041 + 0.729443i \(0.260221\pi\)
\(168\) 4.36196 0.336533
\(169\) −9.81135 −0.754719
\(170\) 0 0
\(171\) −5.00492 −0.382736
\(172\) 9.19850 0.701379
\(173\) 15.2859 1.16217 0.581083 0.813844i \(-0.302629\pi\)
0.581083 + 0.813844i \(0.302629\pi\)
\(174\) −0.917502 −0.0695556
\(175\) 0 0
\(176\) −4.73975 −0.357272
\(177\) −11.1699 −0.839582
\(178\) 0.933323 0.0699556
\(179\) 18.2859 1.36675 0.683377 0.730066i \(-0.260511\pi\)
0.683377 + 0.730066i \(0.260511\pi\)
\(180\) 0 0
\(181\) −8.23506 −0.612107 −0.306054 0.952014i \(-0.599009\pi\)
−0.306054 + 0.952014i \(0.599009\pi\)
\(182\) −6.41435 −0.475463
\(183\) 11.4128 0.843660
\(184\) 1.80642 0.133171
\(185\) 0 0
\(186\) −2.50961 −0.184013
\(187\) −8.70471 −0.636552
\(188\) 1.24443 0.0907595
\(189\) 19.7397 1.43586
\(190\) 0 0
\(191\) 17.3461 1.25512 0.627561 0.778567i \(-0.284053\pi\)
0.627561 + 0.778567i \(0.284053\pi\)
\(192\) −1.21432 −0.0876360
\(193\) −12.0257 −0.865626 −0.432813 0.901484i \(-0.642479\pi\)
−0.432813 + 0.901484i \(0.642479\pi\)
\(194\) 2.42864 0.174366
\(195\) 0 0
\(196\) 5.90321 0.421658
\(197\) 13.9146 0.991373 0.495687 0.868501i \(-0.334916\pi\)
0.495687 + 0.868501i \(0.334916\pi\)
\(198\) −7.23014 −0.513824
\(199\) 24.5718 1.74185 0.870926 0.491415i \(-0.163520\pi\)
0.870926 + 0.491415i \(0.163520\pi\)
\(200\) 0 0
\(201\) 2.91258 0.205438
\(202\) −8.04149 −0.565797
\(203\) −2.71408 −0.190491
\(204\) −2.23014 −0.156141
\(205\) 0 0
\(206\) −0.495316 −0.0345103
\(207\) 2.75557 0.191525
\(208\) 1.78568 0.123815
\(209\) −15.5511 −1.07569
\(210\) 0 0
\(211\) −14.3383 −0.987090 −0.493545 0.869720i \(-0.664299\pi\)
−0.493545 + 0.869720i \(0.664299\pi\)
\(212\) 6.56199 0.450680
\(213\) 11.4128 0.781993
\(214\) −13.4494 −0.919381
\(215\) 0 0
\(216\) −5.49532 −0.373909
\(217\) −7.42372 −0.503955
\(218\) −10.2351 −0.693206
\(219\) −9.83209 −0.664391
\(220\) 0 0
\(221\) 3.27946 0.220601
\(222\) 1.21432 0.0814998
\(223\) −9.09234 −0.608868 −0.304434 0.952533i \(-0.598467\pi\)
−0.304434 + 0.952533i \(0.598467\pi\)
\(224\) −3.59210 −0.240008
\(225\) 0 0
\(226\) 7.49532 0.498581
\(227\) −21.3319 −1.41584 −0.707922 0.706290i \(-0.750367\pi\)
−0.707922 + 0.706290i \(0.750367\pi\)
\(228\) −3.98418 −0.263859
\(229\) −24.0098 −1.58662 −0.793308 0.608821i \(-0.791643\pi\)
−0.793308 + 0.608821i \(0.791643\pi\)
\(230\) 0 0
\(231\) 20.6746 1.36029
\(232\) 0.755569 0.0496055
\(233\) 8.28100 0.542506 0.271253 0.962508i \(-0.412562\pi\)
0.271253 + 0.962508i \(0.412562\pi\)
\(234\) 2.72393 0.178069
\(235\) 0 0
\(236\) 9.19850 0.598771
\(237\) −14.3368 −0.931274
\(238\) −6.59703 −0.427622
\(239\) −25.5812 −1.65471 −0.827355 0.561679i \(-0.810156\pi\)
−0.827355 + 0.561679i \(0.810156\pi\)
\(240\) 0 0
\(241\) 2.72546 0.175562 0.0877811 0.996140i \(-0.472022\pi\)
0.0877811 + 0.996140i \(0.472022\pi\)
\(242\) −11.4652 −0.737011
\(243\) −13.9398 −0.894237
\(244\) −9.39853 −0.601679
\(245\) 0 0
\(246\) −3.12245 −0.199080
\(247\) 5.85881 0.372787
\(248\) 2.06668 0.131234
\(249\) 18.7047 1.18536
\(250\) 0 0
\(251\) −10.2859 −0.649241 −0.324621 0.945844i \(-0.605237\pi\)
−0.324621 + 0.945844i \(0.605237\pi\)
\(252\) −5.47949 −0.345176
\(253\) 8.56199 0.538288
\(254\) −3.45875 −0.217021
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 5.14272 0.320794 0.160397 0.987053i \(-0.448723\pi\)
0.160397 + 0.987053i \(0.448723\pi\)
\(258\) 11.1699 0.695409
\(259\) 3.59210 0.223202
\(260\) 0 0
\(261\) 1.15257 0.0713420
\(262\) −16.0874 −0.993884
\(263\) 7.09234 0.437333 0.218666 0.975800i \(-0.429829\pi\)
0.218666 + 0.975800i \(0.429829\pi\)
\(264\) −5.75557 −0.354231
\(265\) 0 0
\(266\) −11.7857 −0.722626
\(267\) 1.13335 0.0693601
\(268\) −2.39853 −0.146513
\(269\) −26.0830 −1.59031 −0.795154 0.606408i \(-0.792610\pi\)
−0.795154 + 0.606408i \(0.792610\pi\)
\(270\) 0 0
\(271\) 1.84590 0.112131 0.0560653 0.998427i \(-0.482144\pi\)
0.0560653 + 0.998427i \(0.482144\pi\)
\(272\) 1.83654 0.111356
\(273\) −7.78907 −0.471416
\(274\) −15.9906 −0.966029
\(275\) 0 0
\(276\) 2.19358 0.132038
\(277\) −22.9906 −1.38137 −0.690687 0.723154i \(-0.742692\pi\)
−0.690687 + 0.723154i \(0.742692\pi\)
\(278\) −11.0350 −0.661837
\(279\) 3.15257 0.188739
\(280\) 0 0
\(281\) 8.60793 0.513506 0.256753 0.966477i \(-0.417347\pi\)
0.256753 + 0.966477i \(0.417347\pi\)
\(282\) 1.51114 0.0899869
\(283\) 30.7605 1.82852 0.914261 0.405126i \(-0.132772\pi\)
0.914261 + 0.405126i \(0.132772\pi\)
\(284\) −9.39853 −0.557700
\(285\) 0 0
\(286\) 8.46367 0.500467
\(287\) −9.23659 −0.545219
\(288\) 1.52543 0.0898867
\(289\) −13.6271 −0.801596
\(290\) 0 0
\(291\) 2.94914 0.172882
\(292\) 8.09679 0.473829
\(293\) −14.2286 −0.831244 −0.415622 0.909537i \(-0.636436\pi\)
−0.415622 + 0.909537i \(0.636436\pi\)
\(294\) 7.16839 0.418069
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −26.0464 −1.51137
\(298\) −13.9748 −0.809539
\(299\) −3.22570 −0.186547
\(300\) 0 0
\(301\) 33.0420 1.90451
\(302\) −8.42864 −0.485014
\(303\) −9.76494 −0.560981
\(304\) 3.28100 0.188178
\(305\) 0 0
\(306\) 2.80150 0.160151
\(307\) 29.2514 1.66946 0.834732 0.550657i \(-0.185623\pi\)
0.834732 + 0.550657i \(0.185623\pi\)
\(308\) −17.0257 −0.970127
\(309\) −0.601472 −0.0342165
\(310\) 0 0
\(311\) −14.5970 −0.827721 −0.413861 0.910340i \(-0.635820\pi\)
−0.413861 + 0.910340i \(0.635820\pi\)
\(312\) 2.16839 0.122761
\(313\) −0.516060 −0.0291694 −0.0145847 0.999894i \(-0.504643\pi\)
−0.0145847 + 0.999894i \(0.504643\pi\)
\(314\) −9.17775 −0.517931
\(315\) 0 0
\(316\) 11.8064 0.664163
\(317\) 28.8736 1.62170 0.810851 0.585253i \(-0.199005\pi\)
0.810851 + 0.585253i \(0.199005\pi\)
\(318\) 7.96836 0.446843
\(319\) 3.58120 0.200509
\(320\) 0 0
\(321\) −16.3319 −0.911555
\(322\) 6.48886 0.361610
\(323\) 6.02567 0.335277
\(324\) −2.09679 −0.116488
\(325\) 0 0
\(326\) −13.9447 −0.772325
\(327\) −12.4286 −0.687305
\(328\) 2.57136 0.141980
\(329\) 4.47013 0.246446
\(330\) 0 0
\(331\) 6.55707 0.360409 0.180205 0.983629i \(-0.442324\pi\)
0.180205 + 0.983629i \(0.442324\pi\)
\(332\) −15.4035 −0.845374
\(333\) −1.52543 −0.0835929
\(334\) −17.6795 −0.967381
\(335\) 0 0
\(336\) −4.36196 −0.237965
\(337\) −29.0098 −1.58027 −0.790133 0.612935i \(-0.789989\pi\)
−0.790133 + 0.612935i \(0.789989\pi\)
\(338\) 9.81135 0.533667
\(339\) 9.10171 0.494337
\(340\) 0 0
\(341\) 9.79552 0.530457
\(342\) 5.00492 0.270635
\(343\) −3.93978 −0.212728
\(344\) −9.19850 −0.495950
\(345\) 0 0
\(346\) −15.2859 −0.821776
\(347\) 24.8479 1.33391 0.666953 0.745100i \(-0.267598\pi\)
0.666953 + 0.745100i \(0.267598\pi\)
\(348\) 0.917502 0.0491833
\(349\) 24.0765 1.28879 0.644393 0.764694i \(-0.277110\pi\)
0.644393 + 0.764694i \(0.277110\pi\)
\(350\) 0 0
\(351\) 9.81288 0.523773
\(352\) 4.73975 0.252629
\(353\) −32.5433 −1.73210 −0.866051 0.499955i \(-0.833350\pi\)
−0.866051 + 0.499955i \(0.833350\pi\)
\(354\) 11.1699 0.593674
\(355\) 0 0
\(356\) −0.933323 −0.0494660
\(357\) −8.01090 −0.423982
\(358\) −18.2859 −0.966441
\(359\) −23.0923 −1.21877 −0.609384 0.792876i \(-0.708583\pi\)
−0.609384 + 0.792876i \(0.708583\pi\)
\(360\) 0 0
\(361\) −8.23506 −0.433424
\(362\) 8.23506 0.432825
\(363\) −13.9224 −0.730738
\(364\) 6.41435 0.336203
\(365\) 0 0
\(366\) −11.4128 −0.596558
\(367\) 15.4717 0.807614 0.403807 0.914844i \(-0.367687\pi\)
0.403807 + 0.914844i \(0.367687\pi\)
\(368\) −1.80642 −0.0941664
\(369\) 3.92242 0.204193
\(370\) 0 0
\(371\) 23.5714 1.22376
\(372\) 2.50961 0.130117
\(373\) 17.6064 0.911625 0.455812 0.890076i \(-0.349349\pi\)
0.455812 + 0.890076i \(0.349349\pi\)
\(374\) 8.70471 0.450110
\(375\) 0 0
\(376\) −1.24443 −0.0641766
\(377\) −1.34920 −0.0694875
\(378\) −19.7397 −1.01530
\(379\) −9.68445 −0.497457 −0.248728 0.968573i \(-0.580013\pi\)
−0.248728 + 0.968573i \(0.580013\pi\)
\(380\) 0 0
\(381\) −4.20003 −0.215174
\(382\) −17.3461 −0.887506
\(383\) −1.50468 −0.0768858 −0.0384429 0.999261i \(-0.512240\pi\)
−0.0384429 + 0.999261i \(0.512240\pi\)
\(384\) 1.21432 0.0619680
\(385\) 0 0
\(386\) 12.0257 0.612090
\(387\) −14.0316 −0.713268
\(388\) −2.42864 −0.123295
\(389\) 5.59210 0.283531 0.141765 0.989900i \(-0.454722\pi\)
0.141765 + 0.989900i \(0.454722\pi\)
\(390\) 0 0
\(391\) −3.31756 −0.167776
\(392\) −5.90321 −0.298157
\(393\) −19.5353 −0.985424
\(394\) −13.9146 −0.701007
\(395\) 0 0
\(396\) 7.23014 0.363328
\(397\) 27.5274 1.38156 0.690781 0.723064i \(-0.257267\pi\)
0.690781 + 0.723064i \(0.257267\pi\)
\(398\) −24.5718 −1.23167
\(399\) −14.3116 −0.716475
\(400\) 0 0
\(401\) −4.83161 −0.241279 −0.120640 0.992696i \(-0.538495\pi\)
−0.120640 + 0.992696i \(0.538495\pi\)
\(402\) −2.91258 −0.145266
\(403\) −3.69042 −0.183833
\(404\) 8.04149 0.400079
\(405\) 0 0
\(406\) 2.71408 0.134698
\(407\) −4.73975 −0.234941
\(408\) 2.23014 0.110408
\(409\) −33.7447 −1.66857 −0.834283 0.551336i \(-0.814118\pi\)
−0.834283 + 0.551336i \(0.814118\pi\)
\(410\) 0 0
\(411\) −19.4177 −0.957806
\(412\) 0.495316 0.0244025
\(413\) 33.0420 1.62589
\(414\) −2.75557 −0.135429
\(415\) 0 0
\(416\) −1.78568 −0.0875502
\(417\) −13.4001 −0.656204
\(418\) 15.5511 0.760629
\(419\) 15.0479 0.735140 0.367570 0.929996i \(-0.380190\pi\)
0.367570 + 0.929996i \(0.380190\pi\)
\(420\) 0 0
\(421\) −21.2859 −1.03741 −0.518706 0.854953i \(-0.673586\pi\)
−0.518706 + 0.854953i \(0.673586\pi\)
\(422\) 14.3383 0.697978
\(423\) −1.89829 −0.0922980
\(424\) −6.56199 −0.318679
\(425\) 0 0
\(426\) −11.4128 −0.552953
\(427\) −33.7605 −1.63378
\(428\) 13.4494 0.650100
\(429\) 10.2776 0.496207
\(430\) 0 0
\(431\) 35.8479 1.72673 0.863367 0.504577i \(-0.168352\pi\)
0.863367 + 0.504577i \(0.168352\pi\)
\(432\) 5.49532 0.264394
\(433\) −19.1847 −0.921957 −0.460979 0.887411i \(-0.652501\pi\)
−0.460979 + 0.887411i \(0.652501\pi\)
\(434\) 7.42372 0.356350
\(435\) 0 0
\(436\) 10.2351 0.490171
\(437\) −5.92687 −0.283521
\(438\) 9.83209 0.469795
\(439\) −11.1427 −0.531813 −0.265907 0.963999i \(-0.585671\pi\)
−0.265907 + 0.963999i \(0.585671\pi\)
\(440\) 0 0
\(441\) −9.00492 −0.428806
\(442\) −3.27946 −0.155988
\(443\) 2.13627 0.101497 0.0507486 0.998711i \(-0.483839\pi\)
0.0507486 + 0.998711i \(0.483839\pi\)
\(444\) −1.21432 −0.0576291
\(445\) 0 0
\(446\) 9.09234 0.430535
\(447\) −16.9699 −0.802648
\(448\) 3.59210 0.169711
\(449\) −0.711167 −0.0335621 −0.0167810 0.999859i \(-0.505342\pi\)
−0.0167810 + 0.999859i \(0.505342\pi\)
\(450\) 0 0
\(451\) 12.1876 0.573892
\(452\) −7.49532 −0.352550
\(453\) −10.2351 −0.480885
\(454\) 21.3319 1.00115
\(455\) 0 0
\(456\) 3.98418 0.186576
\(457\) −29.8321 −1.39549 −0.697743 0.716348i \(-0.745812\pi\)
−0.697743 + 0.716348i \(0.745812\pi\)
\(458\) 24.0098 1.12191
\(459\) 10.0923 0.471070
\(460\) 0 0
\(461\) 4.10171 0.191036 0.0955178 0.995428i \(-0.469549\pi\)
0.0955178 + 0.995428i \(0.469549\pi\)
\(462\) −20.6746 −0.961870
\(463\) 13.2924 0.617749 0.308874 0.951103i \(-0.400048\pi\)
0.308874 + 0.951103i \(0.400048\pi\)
\(464\) −0.755569 −0.0350764
\(465\) 0 0
\(466\) −8.28100 −0.383610
\(467\) −34.7052 −1.60596 −0.802982 0.596003i \(-0.796755\pi\)
−0.802982 + 0.596003i \(0.796755\pi\)
\(468\) −2.72393 −0.125914
\(469\) −8.61576 −0.397839
\(470\) 0 0
\(471\) −11.1447 −0.513522
\(472\) −9.19850 −0.423395
\(473\) −43.5986 −2.00466
\(474\) 14.3368 0.658510
\(475\) 0 0
\(476\) 6.59703 0.302374
\(477\) −10.0098 −0.458319
\(478\) 25.5812 1.17006
\(479\) 2.89231 0.132153 0.0660766 0.997815i \(-0.478952\pi\)
0.0660766 + 0.997815i \(0.478952\pi\)
\(480\) 0 0
\(481\) 1.78568 0.0814200
\(482\) −2.72546 −0.124141
\(483\) 7.87955 0.358532
\(484\) 11.4652 0.521146
\(485\) 0 0
\(486\) 13.9398 0.632321
\(487\) 39.5560 1.79245 0.896227 0.443596i \(-0.146297\pi\)
0.896227 + 0.443596i \(0.146297\pi\)
\(488\) 9.39853 0.425451
\(489\) −16.9333 −0.765751
\(490\) 0 0
\(491\) 26.5018 1.19601 0.598004 0.801493i \(-0.295961\pi\)
0.598004 + 0.801493i \(0.295961\pi\)
\(492\) 3.12245 0.140771
\(493\) −1.38763 −0.0624957
\(494\) −5.85881 −0.263600
\(495\) 0 0
\(496\) −2.06668 −0.0927965
\(497\) −33.7605 −1.51436
\(498\) −18.7047 −0.838178
\(499\) −41.3319 −1.85027 −0.925134 0.379641i \(-0.876048\pi\)
−0.925134 + 0.379641i \(0.876048\pi\)
\(500\) 0 0
\(501\) −21.4686 −0.959146
\(502\) 10.2859 0.459083
\(503\) −2.87601 −0.128235 −0.0641176 0.997942i \(-0.520423\pi\)
−0.0641176 + 0.997942i \(0.520423\pi\)
\(504\) 5.47949 0.244076
\(505\) 0 0
\(506\) −8.56199 −0.380627
\(507\) 11.9141 0.529124
\(508\) 3.45875 0.153457
\(509\) −11.3713 −0.504025 −0.252013 0.967724i \(-0.581092\pi\)
−0.252013 + 0.967724i \(0.581092\pi\)
\(510\) 0 0
\(511\) 29.0845 1.28662
\(512\) −1.00000 −0.0441942
\(513\) 18.0301 0.796049
\(514\) −5.14272 −0.226836
\(515\) 0 0
\(516\) −11.1699 −0.491728
\(517\) −5.89829 −0.259406
\(518\) −3.59210 −0.157828
\(519\) −18.5620 −0.814781
\(520\) 0 0
\(521\) −19.2908 −0.845147 −0.422574 0.906329i \(-0.638873\pi\)
−0.422574 + 0.906329i \(0.638873\pi\)
\(522\) −1.15257 −0.0504464
\(523\) 12.9813 0.567631 0.283816 0.958879i \(-0.408400\pi\)
0.283816 + 0.958879i \(0.408400\pi\)
\(524\) 16.0874 0.702782
\(525\) 0 0
\(526\) −7.09234 −0.309241
\(527\) −3.79552 −0.165336
\(528\) 5.75557 0.250479
\(529\) −19.7368 −0.858123
\(530\) 0 0
\(531\) −14.0316 −0.608921
\(532\) 11.7857 0.510974
\(533\) −4.59163 −0.198885
\(534\) −1.13335 −0.0490450
\(535\) 0 0
\(536\) 2.39853 0.103601
\(537\) −22.2050 −0.958214
\(538\) 26.0830 1.12452
\(539\) −27.9797 −1.20517
\(540\) 0 0
\(541\) −16.3160 −0.701481 −0.350740 0.936473i \(-0.614070\pi\)
−0.350740 + 0.936473i \(0.614070\pi\)
\(542\) −1.84590 −0.0792883
\(543\) 10.0000 0.429141
\(544\) −1.83654 −0.0787408
\(545\) 0 0
\(546\) 7.78907 0.333341
\(547\) 0.677517 0.0289685 0.0144843 0.999895i \(-0.495389\pi\)
0.0144843 + 0.999895i \(0.495389\pi\)
\(548\) 15.9906 0.683086
\(549\) 14.3368 0.611879
\(550\) 0 0
\(551\) −2.47902 −0.105610
\(552\) −2.19358 −0.0933648
\(553\) 42.4099 1.80345
\(554\) 22.9906 0.976778
\(555\) 0 0
\(556\) 11.0350 0.467990
\(557\) −3.11108 −0.131821 −0.0659103 0.997826i \(-0.520995\pi\)
−0.0659103 + 0.997826i \(0.520995\pi\)
\(558\) −3.15257 −0.133459
\(559\) 16.4256 0.694728
\(560\) 0 0
\(561\) 10.5703 0.446279
\(562\) −8.60793 −0.363103
\(563\) −22.1891 −0.935160 −0.467580 0.883951i \(-0.654874\pi\)
−0.467580 + 0.883951i \(0.654874\pi\)
\(564\) −1.51114 −0.0636304
\(565\) 0 0
\(566\) −30.7605 −1.29296
\(567\) −7.53188 −0.316309
\(568\) 9.39853 0.394353
\(569\) 17.5783 0.736920 0.368460 0.929644i \(-0.379885\pi\)
0.368460 + 0.929644i \(0.379885\pi\)
\(570\) 0 0
\(571\) 30.5303 1.27766 0.638828 0.769350i \(-0.279420\pi\)
0.638828 + 0.769350i \(0.279420\pi\)
\(572\) −8.46367 −0.353884
\(573\) −21.0638 −0.879951
\(574\) 9.23659 0.385528
\(575\) 0 0
\(576\) −1.52543 −0.0635595
\(577\) 14.2335 0.592550 0.296275 0.955103i \(-0.404256\pi\)
0.296275 + 0.955103i \(0.404256\pi\)
\(578\) 13.6271 0.566814
\(579\) 14.6030 0.606880
\(580\) 0 0
\(581\) −55.3308 −2.29551
\(582\) −2.94914 −0.122246
\(583\) −31.1022 −1.28812
\(584\) −8.09679 −0.335047
\(585\) 0 0
\(586\) 14.2286 0.587778
\(587\) −10.5254 −0.434431 −0.217215 0.976124i \(-0.569697\pi\)
−0.217215 + 0.976124i \(0.569697\pi\)
\(588\) −7.16839 −0.295619
\(589\) −6.78076 −0.279396
\(590\) 0 0
\(591\) −16.8968 −0.695040
\(592\) 1.00000 0.0410997
\(593\) −22.8163 −0.936952 −0.468476 0.883476i \(-0.655197\pi\)
−0.468476 + 0.883476i \(0.655197\pi\)
\(594\) 26.0464 1.06870
\(595\) 0 0
\(596\) 13.9748 0.572431
\(597\) −29.8381 −1.22119
\(598\) 3.22570 0.131908
\(599\) −7.23659 −0.295679 −0.147840 0.989011i \(-0.547232\pi\)
−0.147840 + 0.989011i \(0.547232\pi\)
\(600\) 0 0
\(601\) −17.2623 −0.704142 −0.352071 0.935973i \(-0.614522\pi\)
−0.352071 + 0.935973i \(0.614522\pi\)
\(602\) −33.0420 −1.34669
\(603\) 3.65878 0.148997
\(604\) 8.42864 0.342956
\(605\) 0 0
\(606\) 9.76494 0.396673
\(607\) 30.8825 1.25348 0.626740 0.779228i \(-0.284389\pi\)
0.626740 + 0.779228i \(0.284389\pi\)
\(608\) −3.28100 −0.133062
\(609\) 3.29576 0.133551
\(610\) 0 0
\(611\) 2.22216 0.0898988
\(612\) −2.80150 −0.113244
\(613\) 16.3180 0.659079 0.329540 0.944142i \(-0.393106\pi\)
0.329540 + 0.944142i \(0.393106\pi\)
\(614\) −29.2514 −1.18049
\(615\) 0 0
\(616\) 17.0257 0.685984
\(617\) −3.16992 −0.127616 −0.0638080 0.997962i \(-0.520325\pi\)
−0.0638080 + 0.997962i \(0.520325\pi\)
\(618\) 0.601472 0.0241948
\(619\) 30.5018 1.22597 0.612985 0.790095i \(-0.289969\pi\)
0.612985 + 0.790095i \(0.289969\pi\)
\(620\) 0 0
\(621\) −9.92687 −0.398352
\(622\) 14.5970 0.585287
\(623\) −3.35260 −0.134319
\(624\) −2.16839 −0.0868049
\(625\) 0 0
\(626\) 0.516060 0.0206259
\(627\) 18.8840 0.754154
\(628\) 9.17775 0.366232
\(629\) 1.83654 0.0732275
\(630\) 0 0
\(631\) −13.3274 −0.530556 −0.265278 0.964172i \(-0.585464\pi\)
−0.265278 + 0.964172i \(0.585464\pi\)
\(632\) −11.8064 −0.469634
\(633\) 17.4113 0.692037
\(634\) −28.8736 −1.14672
\(635\) 0 0
\(636\) −7.96836 −0.315966
\(637\) 10.5412 0.417659
\(638\) −3.58120 −0.141781
\(639\) 14.3368 0.567154
\(640\) 0 0
\(641\) −26.9403 −1.06408 −0.532038 0.846720i \(-0.678574\pi\)
−0.532038 + 0.846720i \(0.678574\pi\)
\(642\) 16.3319 0.644567
\(643\) −28.0370 −1.10567 −0.552836 0.833290i \(-0.686455\pi\)
−0.552836 + 0.833290i \(0.686455\pi\)
\(644\) −6.48886 −0.255697
\(645\) 0 0
\(646\) −6.02567 −0.237077
\(647\) −3.81288 −0.149900 −0.0749498 0.997187i \(-0.523880\pi\)
−0.0749498 + 0.997187i \(0.523880\pi\)
\(648\) 2.09679 0.0823696
\(649\) −43.5986 −1.71139
\(650\) 0 0
\(651\) 9.01477 0.353317
\(652\) 13.9447 0.546117
\(653\) 1.53188 0.0599471 0.0299736 0.999551i \(-0.490458\pi\)
0.0299736 + 0.999551i \(0.490458\pi\)
\(654\) 12.4286 0.485998
\(655\) 0 0
\(656\) −2.57136 −0.100395
\(657\) −12.3511 −0.481861
\(658\) −4.47013 −0.174264
\(659\) 30.9101 1.20409 0.602044 0.798463i \(-0.294353\pi\)
0.602044 + 0.798463i \(0.294353\pi\)
\(660\) 0 0
\(661\) −11.8163 −0.459600 −0.229800 0.973238i \(-0.573807\pi\)
−0.229800 + 0.973238i \(0.573807\pi\)
\(662\) −6.55707 −0.254848
\(663\) −3.98232 −0.154660
\(664\) 15.4035 0.597770
\(665\) 0 0
\(666\) 1.52543 0.0591091
\(667\) 1.36488 0.0528483
\(668\) 17.6795 0.684041
\(669\) 11.0410 0.426870
\(670\) 0 0
\(671\) 44.5466 1.71970
\(672\) 4.36196 0.168266
\(673\) 10.6780 0.411606 0.205803 0.978593i \(-0.434019\pi\)
0.205803 + 0.978593i \(0.434019\pi\)
\(674\) 29.0098 1.11742
\(675\) 0 0
\(676\) −9.81135 −0.377359
\(677\) 0.828699 0.0318495 0.0159247 0.999873i \(-0.494931\pi\)
0.0159247 + 0.999873i \(0.494931\pi\)
\(678\) −9.10171 −0.349549
\(679\) −8.72393 −0.334794
\(680\) 0 0
\(681\) 25.9037 0.992631
\(682\) −9.79552 −0.375090
\(683\) −12.3921 −0.474170 −0.237085 0.971489i \(-0.576192\pi\)
−0.237085 + 0.971489i \(0.576192\pi\)
\(684\) −5.00492 −0.191368
\(685\) 0 0
\(686\) 3.93978 0.150421
\(687\) 29.1556 1.11236
\(688\) 9.19850 0.350689
\(689\) 11.7176 0.446406
\(690\) 0 0
\(691\) 22.9146 0.871712 0.435856 0.900016i \(-0.356446\pi\)
0.435856 + 0.900016i \(0.356446\pi\)
\(692\) 15.2859 0.581083
\(693\) 25.9714 0.986573
\(694\) −24.8479 −0.943214
\(695\) 0 0
\(696\) −0.917502 −0.0347778
\(697\) −4.72239 −0.178873
\(698\) −24.0765 −0.911310
\(699\) −10.0558 −0.380344
\(700\) 0 0
\(701\) −42.7161 −1.61336 −0.806682 0.590985i \(-0.798739\pi\)
−0.806682 + 0.590985i \(0.798739\pi\)
\(702\) −9.81288 −0.370363
\(703\) 3.28100 0.123745
\(704\) −4.73975 −0.178636
\(705\) 0 0
\(706\) 32.5433 1.22478
\(707\) 28.8859 1.08636
\(708\) −11.1699 −0.419791
\(709\) −47.0005 −1.76514 −0.882570 0.470181i \(-0.844189\pi\)
−0.882570 + 0.470181i \(0.844189\pi\)
\(710\) 0 0
\(711\) −18.0098 −0.675422
\(712\) 0.933323 0.0349778
\(713\) 3.73329 0.139813
\(714\) 8.01090 0.299800
\(715\) 0 0
\(716\) 18.2859 0.683377
\(717\) 31.0638 1.16010
\(718\) 23.0923 0.861799
\(719\) 5.82717 0.217317 0.108658 0.994079i \(-0.465345\pi\)
0.108658 + 0.994079i \(0.465345\pi\)
\(720\) 0 0
\(721\) 1.77923 0.0662619
\(722\) 8.23506 0.306477
\(723\) −3.30958 −0.123084
\(724\) −8.23506 −0.306054
\(725\) 0 0
\(726\) 13.9224 0.516710
\(727\) −15.4479 −0.572929 −0.286465 0.958091i \(-0.592480\pi\)
−0.286465 + 0.958091i \(0.592480\pi\)
\(728\) −6.41435 −0.237732
\(729\) 23.2177 0.859915
\(730\) 0 0
\(731\) 16.8934 0.624824
\(732\) 11.4128 0.421830
\(733\) 15.4479 0.570579 0.285290 0.958441i \(-0.407910\pi\)
0.285290 + 0.958441i \(0.407910\pi\)
\(734\) −15.4717 −0.571069
\(735\) 0 0
\(736\) 1.80642 0.0665857
\(737\) 11.3684 0.418761
\(738\) −3.92242 −0.144386
\(739\) 52.2034 1.92033 0.960167 0.279427i \(-0.0901445\pi\)
0.960167 + 0.279427i \(0.0901445\pi\)
\(740\) 0 0
\(741\) −7.11447 −0.261357
\(742\) −23.5714 −0.865332
\(743\) −16.7556 −0.614702 −0.307351 0.951596i \(-0.599443\pi\)
−0.307351 + 0.951596i \(0.599443\pi\)
\(744\) −2.50961 −0.0920066
\(745\) 0 0
\(746\) −17.6064 −0.644616
\(747\) 23.4968 0.859705
\(748\) −8.70471 −0.318276
\(749\) 48.3116 1.76527
\(750\) 0 0
\(751\) 9.67307 0.352975 0.176488 0.984303i \(-0.443526\pi\)
0.176488 + 0.984303i \(0.443526\pi\)
\(752\) 1.24443 0.0453797
\(753\) 12.4904 0.455175
\(754\) 1.34920 0.0491351
\(755\) 0 0
\(756\) 19.7397 0.717928
\(757\) −5.98126 −0.217393 −0.108696 0.994075i \(-0.534668\pi\)
−0.108696 + 0.994075i \(0.534668\pi\)
\(758\) 9.68445 0.351755
\(759\) −10.3970 −0.377387
\(760\) 0 0
\(761\) −27.6365 −1.00182 −0.500911 0.865499i \(-0.667002\pi\)
−0.500911 + 0.865499i \(0.667002\pi\)
\(762\) 4.20003 0.152151
\(763\) 36.7654 1.33100
\(764\) 17.3461 0.627561
\(765\) 0 0
\(766\) 1.50468 0.0543664
\(767\) 16.4256 0.593093
\(768\) −1.21432 −0.0438180
\(769\) −9.64740 −0.347894 −0.173947 0.984755i \(-0.555652\pi\)
−0.173947 + 0.984755i \(0.555652\pi\)
\(770\) 0 0
\(771\) −6.24491 −0.224905
\(772\) −12.0257 −0.432813
\(773\) 4.56845 0.164316 0.0821578 0.996619i \(-0.473819\pi\)
0.0821578 + 0.996619i \(0.473819\pi\)
\(774\) 14.0316 0.504357
\(775\) 0 0
\(776\) 2.42864 0.0871831
\(777\) −4.36196 −0.156485
\(778\) −5.59210 −0.200487
\(779\) −8.43662 −0.302273
\(780\) 0 0
\(781\) 44.5466 1.59400
\(782\) 3.31756 0.118636
\(783\) −4.15209 −0.148384
\(784\) 5.90321 0.210829
\(785\) 0 0
\(786\) 19.5353 0.696800
\(787\) 25.3274 0.902825 0.451412 0.892316i \(-0.350920\pi\)
0.451412 + 0.892316i \(0.350920\pi\)
\(788\) 13.9146 0.495687
\(789\) −8.61237 −0.306609
\(790\) 0 0
\(791\) −26.9240 −0.957306
\(792\) −7.23014 −0.256912
\(793\) −16.7828 −0.595973
\(794\) −27.5274 −0.976912
\(795\) 0 0
\(796\) 24.5718 0.870926
\(797\) 8.98862 0.318393 0.159197 0.987247i \(-0.449110\pi\)
0.159197 + 0.987247i \(0.449110\pi\)
\(798\) 14.3116 0.506625
\(799\) 2.28544 0.0808531
\(800\) 0 0
\(801\) 1.42372 0.0503046
\(802\) 4.83161 0.170610
\(803\) −38.3767 −1.35429
\(804\) 2.91258 0.102719
\(805\) 0 0
\(806\) 3.69042 0.129990
\(807\) 31.6731 1.11494
\(808\) −8.04149 −0.282899
\(809\) 36.8256 1.29472 0.647360 0.762184i \(-0.275873\pi\)
0.647360 + 0.762184i \(0.275873\pi\)
\(810\) 0 0
\(811\) 16.4286 0.576888 0.288444 0.957497i \(-0.406862\pi\)
0.288444 + 0.957497i \(0.406862\pi\)
\(812\) −2.71408 −0.0952456
\(813\) −2.24152 −0.0786134
\(814\) 4.73975 0.166128
\(815\) 0 0
\(816\) −2.23014 −0.0780706
\(817\) 30.1802 1.05587
\(818\) 33.7447 1.17985
\(819\) −9.78463 −0.341902
\(820\) 0 0
\(821\) 42.5881 1.48634 0.743168 0.669105i \(-0.233322\pi\)
0.743168 + 0.669105i \(0.233322\pi\)
\(822\) 19.4177 0.677271
\(823\) −3.79552 −0.132304 −0.0661518 0.997810i \(-0.521072\pi\)
−0.0661518 + 0.997810i \(0.521072\pi\)
\(824\) −0.495316 −0.0172552
\(825\) 0 0
\(826\) −33.0420 −1.14968
\(827\) −9.61777 −0.334443 −0.167221 0.985919i \(-0.553479\pi\)
−0.167221 + 0.985919i \(0.553479\pi\)
\(828\) 2.75557 0.0957626
\(829\) −30.2657 −1.05117 −0.525585 0.850741i \(-0.676153\pi\)
−0.525585 + 0.850741i \(0.676153\pi\)
\(830\) 0 0
\(831\) 27.9180 0.968464
\(832\) 1.78568 0.0619073
\(833\) 10.8415 0.375634
\(834\) 13.4001 0.464006
\(835\) 0 0
\(836\) −15.5511 −0.537846
\(837\) −11.3570 −0.392557
\(838\) −15.0479 −0.519822
\(839\) 8.05239 0.277999 0.139000 0.990292i \(-0.455611\pi\)
0.139000 + 0.990292i \(0.455611\pi\)
\(840\) 0 0
\(841\) −28.4291 −0.980314
\(842\) 21.2859 0.733561
\(843\) −10.4528 −0.360013
\(844\) −14.3383 −0.493545
\(845\) 0 0
\(846\) 1.89829 0.0652645
\(847\) 41.1842 1.41511
\(848\) 6.56199 0.225340
\(849\) −37.3531 −1.28195
\(850\) 0 0
\(851\) −1.80642 −0.0619234
\(852\) 11.4128 0.390997
\(853\) −25.0845 −0.858877 −0.429439 0.903096i \(-0.641289\pi\)
−0.429439 + 0.903096i \(0.641289\pi\)
\(854\) 33.7605 1.15526
\(855\) 0 0
\(856\) −13.4494 −0.459690
\(857\) 20.5506 0.701996 0.350998 0.936376i \(-0.385842\pi\)
0.350998 + 0.936376i \(0.385842\pi\)
\(858\) −10.2776 −0.350872
\(859\) −49.2810 −1.68145 −0.840723 0.541466i \(-0.817870\pi\)
−0.840723 + 0.541466i \(0.817870\pi\)
\(860\) 0 0
\(861\) 11.2162 0.382246
\(862\) −35.8479 −1.22098
\(863\) −37.6849 −1.28281 −0.641405 0.767203i \(-0.721648\pi\)
−0.641405 + 0.767203i \(0.721648\pi\)
\(864\) −5.49532 −0.186954
\(865\) 0 0
\(866\) 19.1847 0.651922
\(867\) 16.5477 0.561989
\(868\) −7.42372 −0.251977
\(869\) −55.9595 −1.89829
\(870\) 0 0
\(871\) −4.28300 −0.145124
\(872\) −10.2351 −0.346603
\(873\) 3.70471 0.125386
\(874\) 5.92687 0.200479
\(875\) 0 0
\(876\) −9.83209 −0.332196
\(877\) 50.9022 1.71884 0.859422 0.511267i \(-0.170824\pi\)
0.859422 + 0.511267i \(0.170824\pi\)
\(878\) 11.1427 0.376049
\(879\) 17.2781 0.582775
\(880\) 0 0
\(881\) 5.39207 0.181664 0.0908318 0.995866i \(-0.471047\pi\)
0.0908318 + 0.995866i \(0.471047\pi\)
\(882\) 9.00492 0.303212
\(883\) −30.6356 −1.03097 −0.515485 0.856899i \(-0.672388\pi\)
−0.515485 + 0.856899i \(0.672388\pi\)
\(884\) 3.27946 0.110300
\(885\) 0 0
\(886\) −2.13627 −0.0717693
\(887\) 37.3087 1.25270 0.626351 0.779541i \(-0.284548\pi\)
0.626351 + 0.779541i \(0.284548\pi\)
\(888\) 1.21432 0.0407499
\(889\) 12.4242 0.416694
\(890\) 0 0
\(891\) 9.93825 0.332944
\(892\) −9.09234 −0.304434
\(893\) 4.08297 0.136632
\(894\) 16.9699 0.567558
\(895\) 0 0
\(896\) −3.59210 −0.120004
\(897\) 3.91703 0.130786
\(898\) 0.711167 0.0237320
\(899\) 1.56152 0.0520795
\(900\) 0 0
\(901\) 12.0513 0.401488
\(902\) −12.1876 −0.405803
\(903\) −40.1235 −1.33523
\(904\) 7.49532 0.249291
\(905\) 0 0
\(906\) 10.2351 0.340037
\(907\) −18.0973 −0.600910 −0.300455 0.953796i \(-0.597138\pi\)
−0.300455 + 0.953796i \(0.597138\pi\)
\(908\) −21.3319 −0.707922
\(909\) −12.2667 −0.406861
\(910\) 0 0
\(911\) −44.9753 −1.49010 −0.745049 0.667010i \(-0.767574\pi\)
−0.745049 + 0.667010i \(0.767574\pi\)
\(912\) −3.98418 −0.131929
\(913\) 73.0085 2.41623
\(914\) 29.8321 0.986758
\(915\) 0 0
\(916\) −24.0098 −0.793308
\(917\) 57.7877 1.90832
\(918\) −10.0923 −0.333097
\(919\) 5.87955 0.193949 0.0969743 0.995287i \(-0.469084\pi\)
0.0969743 + 0.995287i \(0.469084\pi\)
\(920\) 0 0
\(921\) −35.5205 −1.17044
\(922\) −4.10171 −0.135083
\(923\) −16.7828 −0.552411
\(924\) 20.6746 0.680144
\(925\) 0 0
\(926\) −13.2924 −0.436814
\(927\) −0.755569 −0.0248161
\(928\) 0.755569 0.0248028
\(929\) 39.0910 1.28253 0.641266 0.767318i \(-0.278409\pi\)
0.641266 + 0.767318i \(0.278409\pi\)
\(930\) 0 0
\(931\) 19.3684 0.634774
\(932\) 8.28100 0.271253
\(933\) 17.7255 0.580305
\(934\) 34.7052 1.13559
\(935\) 0 0
\(936\) 2.72393 0.0890343
\(937\) 4.36349 0.142549 0.0712746 0.997457i \(-0.477293\pi\)
0.0712746 + 0.997457i \(0.477293\pi\)
\(938\) 8.61576 0.281315
\(939\) 0.626661 0.0204503
\(940\) 0 0
\(941\) −12.8671 −0.419456 −0.209728 0.977760i \(-0.567258\pi\)
−0.209728 + 0.977760i \(0.567258\pi\)
\(942\) 11.1447 0.363115
\(943\) 4.64497 0.151261
\(944\) 9.19850 0.299386
\(945\) 0 0
\(946\) 43.5986 1.41751
\(947\) −17.2444 −0.560369 −0.280184 0.959946i \(-0.590396\pi\)
−0.280184 + 0.959946i \(0.590396\pi\)
\(948\) −14.3368 −0.465637
\(949\) 14.4583 0.469335
\(950\) 0 0
\(951\) −35.0618 −1.13696
\(952\) −6.59703 −0.213811
\(953\) −24.6874 −0.799702 −0.399851 0.916580i \(-0.630938\pi\)
−0.399851 + 0.916580i \(0.630938\pi\)
\(954\) 10.0098 0.324081
\(955\) 0 0
\(956\) −25.5812 −0.827355
\(957\) −4.34873 −0.140574
\(958\) −2.89231 −0.0934464
\(959\) 57.4400 1.85483
\(960\) 0 0
\(961\) −26.7288 −0.862221
\(962\) −1.78568 −0.0575726
\(963\) −20.5161 −0.661121
\(964\) 2.72546 0.0877811
\(965\) 0 0
\(966\) −7.87955 −0.253520
\(967\) −36.1748 −1.16330 −0.581652 0.813438i \(-0.697594\pi\)
−0.581652 + 0.813438i \(0.697594\pi\)
\(968\) −11.4652 −0.368506
\(969\) −7.31708 −0.235059
\(970\) 0 0
\(971\) −33.8419 −1.08604 −0.543020 0.839720i \(-0.682719\pi\)
−0.543020 + 0.839720i \(0.682719\pi\)
\(972\) −13.9398 −0.447119
\(973\) 39.6390 1.27077
\(974\) −39.5560 −1.26746
\(975\) 0 0
\(976\) −9.39853 −0.300840
\(977\) 38.7768 1.24058 0.620290 0.784373i \(-0.287015\pi\)
0.620290 + 0.784373i \(0.287015\pi\)
\(978\) 16.9333 0.541468
\(979\) 4.42372 0.141383
\(980\) 0 0
\(981\) −15.6128 −0.498480
\(982\) −26.5018 −0.845705
\(983\) −59.3975 −1.89449 −0.947243 0.320518i \(-0.896143\pi\)
−0.947243 + 0.320518i \(0.896143\pi\)
\(984\) −3.12245 −0.0995402
\(985\) 0 0
\(986\) 1.38763 0.0441911
\(987\) −5.42816 −0.172780
\(988\) 5.85881 0.186394
\(989\) −16.6164 −0.528370
\(990\) 0 0
\(991\) 36.3245 1.15389 0.576943 0.816785i \(-0.304246\pi\)
0.576943 + 0.816785i \(0.304246\pi\)
\(992\) 2.06668 0.0656170
\(993\) −7.96238 −0.252678
\(994\) 33.7605 1.07082
\(995\) 0 0
\(996\) 18.7047 0.592682
\(997\) 32.7732 1.03794 0.518970 0.854793i \(-0.326316\pi\)
0.518970 + 0.854793i \(0.326316\pi\)
\(998\) 41.3319 1.30834
\(999\) 5.49532 0.173864
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 1850.2.a.ba.1.1 3
5.2 odd 4 1850.2.b.n.149.3 6
5.3 odd 4 1850.2.b.n.149.4 6
5.4 even 2 1850.2.a.bb.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.ba.1.1 3 1.1 even 1 trivial
1850.2.a.bb.1.3 yes 3 5.4 even 2
1850.2.b.n.149.3 6 5.2 odd 4
1850.2.b.n.149.4 6 5.3 odd 4