Properties

Label 1850.2.a.ba.1.2
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 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.53919 q^{3} +1.00000 q^{4} -1.53919 q^{6} -2.87936 q^{7} -1.00000 q^{8} -0.630898 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.53919 q^{3} +1.00000 q^{4} -1.53919 q^{6} -2.87936 q^{7} -1.00000 q^{8} -0.630898 q^{9} -1.09171 q^{11} +1.53919 q^{12} +4.53919 q^{13} +2.87936 q^{14} +1.00000 q^{16} +2.80098 q^{17} +0.630898 q^{18} -5.04945 q^{19} -4.43188 q^{21} +1.09171 q^{22} +7.41855 q^{23} -1.53919 q^{24} -4.53919 q^{26} -5.58864 q^{27} -2.87936 q^{28} +6.68035 q^{29} +3.51026 q^{31} -1.00000 q^{32} -1.68035 q^{33} -2.80098 q^{34} -0.630898 q^{36} +1.00000 q^{37} +5.04945 q^{38} +6.98667 q^{39} -8.07838 q^{41} +4.43188 q^{42} +10.2329 q^{43} -1.09171 q^{44} -7.41855 q^{46} +8.68035 q^{47} +1.53919 q^{48} +1.29072 q^{49} +4.31124 q^{51} +4.53919 q^{52} -10.0989 q^{53} +5.58864 q^{54} +2.87936 q^{56} -7.77205 q^{57} -6.68035 q^{58} +10.2329 q^{59} +6.29791 q^{61} -3.51026 q^{62} +1.81658 q^{63} +1.00000 q^{64} +1.68035 q^{66} +13.2979 q^{67} +2.80098 q^{68} +11.4186 q^{69} +6.29791 q^{71} +0.630898 q^{72} +12.7093 q^{73} -1.00000 q^{74} -5.04945 q^{76} +3.14342 q^{77} -6.98667 q^{78} +2.58145 q^{79} -6.70928 q^{81} +8.07838 q^{82} +8.48360 q^{83} -4.43188 q^{84} -10.2329 q^{86} +10.2823 q^{87} +1.09171 q^{88} -6.51026 q^{89} -13.0700 q^{91} +7.41855 q^{92} +5.40295 q^{93} -8.68035 q^{94} -1.53919 q^{96} +3.07838 q^{97} -1.29072 q^{98} +0.688756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 2 q^{9}+O(q^{10}) \) 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
\(3\) 1.53919 0.888651 0.444326 0.895865i \(-0.353443\pi\)
0.444326 + 0.895865i \(0.353443\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.53919 −0.628371
\(7\) −2.87936 −1.08830 −0.544148 0.838989i \(-0.683147\pi\)
−0.544148 + 0.838989i \(0.683147\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.630898 −0.210299
\(10\) 0 0
\(11\) −1.09171 −0.329163 −0.164581 0.986364i \(-0.552627\pi\)
−0.164581 + 0.986364i \(0.552627\pi\)
\(12\) 1.53919 0.444326
\(13\) 4.53919 1.25894 0.629472 0.777023i \(-0.283271\pi\)
0.629472 + 0.777023i \(0.283271\pi\)
\(14\) 2.87936 0.769542
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.80098 0.679338 0.339669 0.940545i \(-0.389685\pi\)
0.339669 + 0.940545i \(0.389685\pi\)
\(18\) 0.630898 0.148704
\(19\) −5.04945 −1.15842 −0.579211 0.815177i \(-0.696639\pi\)
−0.579211 + 0.815177i \(0.696639\pi\)
\(20\) 0 0
\(21\) −4.43188 −0.967116
\(22\) 1.09171 0.232753
\(23\) 7.41855 1.54687 0.773437 0.633873i \(-0.218536\pi\)
0.773437 + 0.633873i \(0.218536\pi\)
\(24\) −1.53919 −0.314186
\(25\) 0 0
\(26\) −4.53919 −0.890208
\(27\) −5.58864 −1.07553
\(28\) −2.87936 −0.544148
\(29\) 6.68035 1.24051 0.620255 0.784401i \(-0.287029\pi\)
0.620255 + 0.784401i \(0.287029\pi\)
\(30\) 0 0
\(31\) 3.51026 0.630461 0.315231 0.949015i \(-0.397918\pi\)
0.315231 + 0.949015i \(0.397918\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.68035 −0.292511
\(34\) −2.80098 −0.480365
\(35\) 0 0
\(36\) −0.630898 −0.105150
\(37\) 1.00000 0.164399
\(38\) 5.04945 0.819129
\(39\) 6.98667 1.11876
\(40\) 0 0
\(41\) −8.07838 −1.26163 −0.630815 0.775933i \(-0.717279\pi\)
−0.630815 + 0.775933i \(0.717279\pi\)
\(42\) 4.43188 0.683854
\(43\) 10.2329 1.56050 0.780249 0.625469i \(-0.215092\pi\)
0.780249 + 0.625469i \(0.215092\pi\)
\(44\) −1.09171 −0.164581
\(45\) 0 0
\(46\) −7.41855 −1.09381
\(47\) 8.68035 1.26616 0.633079 0.774087i \(-0.281791\pi\)
0.633079 + 0.774087i \(0.281791\pi\)
\(48\) 1.53919 0.222163
\(49\) 1.29072 0.184389
\(50\) 0 0
\(51\) 4.31124 0.603695
\(52\) 4.53919 0.629472
\(53\) −10.0989 −1.38719 −0.693595 0.720365i \(-0.743974\pi\)
−0.693595 + 0.720365i \(0.743974\pi\)
\(54\) 5.58864 0.760517
\(55\) 0 0
\(56\) 2.87936 0.384771
\(57\) −7.77205 −1.02943
\(58\) −6.68035 −0.877172
\(59\) 10.2329 1.33221 0.666103 0.745860i \(-0.267961\pi\)
0.666103 + 0.745860i \(0.267961\pi\)
\(60\) 0 0
\(61\) 6.29791 0.806365 0.403183 0.915120i \(-0.367904\pi\)
0.403183 + 0.915120i \(0.367904\pi\)
\(62\) −3.51026 −0.445803
\(63\) 1.81658 0.228868
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.68035 0.206836
\(67\) 13.2979 1.62460 0.812299 0.583241i \(-0.198216\pi\)
0.812299 + 0.583241i \(0.198216\pi\)
\(68\) 2.80098 0.339669
\(69\) 11.4186 1.37463
\(70\) 0 0
\(71\) 6.29791 0.747425 0.373712 0.927545i \(-0.378085\pi\)
0.373712 + 0.927545i \(0.378085\pi\)
\(72\) 0.630898 0.0743520
\(73\) 12.7093 1.48751 0.743754 0.668453i \(-0.233043\pi\)
0.743754 + 0.668453i \(0.233043\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −5.04945 −0.579211
\(77\) 3.14342 0.358226
\(78\) −6.98667 −0.791084
\(79\) 2.58145 0.290436 0.145218 0.989400i \(-0.453612\pi\)
0.145218 + 0.989400i \(0.453612\pi\)
\(80\) 0 0
\(81\) −6.70928 −0.745475
\(82\) 8.07838 0.892108
\(83\) 8.48360 0.931196 0.465598 0.884996i \(-0.345839\pi\)
0.465598 + 0.884996i \(0.345839\pi\)
\(84\) −4.43188 −0.483558
\(85\) 0 0
\(86\) −10.2329 −1.10344
\(87\) 10.2823 1.10238
\(88\) 1.09171 0.116377
\(89\) −6.51026 −0.690086 −0.345043 0.938587i \(-0.612136\pi\)
−0.345043 + 0.938587i \(0.612136\pi\)
\(90\) 0 0
\(91\) −13.0700 −1.37010
\(92\) 7.41855 0.773437
\(93\) 5.40295 0.560260
\(94\) −8.68035 −0.895309
\(95\) 0 0
\(96\) −1.53919 −0.157093
\(97\) 3.07838 0.312562 0.156281 0.987713i \(-0.450049\pi\)
0.156281 + 0.987713i \(0.450049\pi\)
\(98\) −1.29072 −0.130383
\(99\) 0.688756 0.0692226
\(100\) 0 0
\(101\) −15.9155 −1.58365 −0.791825 0.610748i \(-0.790869\pi\)
−0.791825 + 0.610748i \(0.790869\pi\)
\(102\) −4.31124 −0.426877
\(103\) −10.5886 −1.04333 −0.521665 0.853151i \(-0.674689\pi\)
−0.521665 + 0.853151i \(0.674689\pi\)
\(104\) −4.53919 −0.445104
\(105\) 0 0
\(106\) 10.0989 0.980892
\(107\) −4.03612 −0.390186 −0.195093 0.980785i \(-0.562501\pi\)
−0.195093 + 0.980785i \(0.562501\pi\)
\(108\) −5.58864 −0.537767
\(109\) −4.49693 −0.430728 −0.215364 0.976534i \(-0.569094\pi\)
−0.215364 + 0.976534i \(0.569094\pi\)
\(110\) 0 0
\(111\) 1.53919 0.146093
\(112\) −2.87936 −0.272074
\(113\) 3.58864 0.337591 0.168795 0.985651i \(-0.446012\pi\)
0.168795 + 0.985651i \(0.446012\pi\)
\(114\) 7.77205 0.727920
\(115\) 0 0
\(116\) 6.68035 0.620255
\(117\) −2.86376 −0.264755
\(118\) −10.2329 −0.942012
\(119\) −8.06505 −0.739322
\(120\) 0 0
\(121\) −9.80817 −0.891652
\(122\) −6.29791 −0.570186
\(123\) −12.4341 −1.12115
\(124\) 3.51026 0.315231
\(125\) 0 0
\(126\) −1.81658 −0.161834
\(127\) 8.14116 0.722411 0.361205 0.932486i \(-0.382365\pi\)
0.361205 + 0.932486i \(0.382365\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 15.7503 1.38674
\(130\) 0 0
\(131\) −1.46800 −0.128260 −0.0641298 0.997942i \(-0.520427\pi\)
−0.0641298 + 0.997942i \(0.520427\pi\)
\(132\) −1.68035 −0.146255
\(133\) 14.5392 1.26071
\(134\) −13.2979 −1.14876
\(135\) 0 0
\(136\) −2.80098 −0.240182
\(137\) −6.17727 −0.527760 −0.263880 0.964555i \(-0.585002\pi\)
−0.263880 + 0.964555i \(0.585002\pi\)
\(138\) −11.4186 −0.972012
\(139\) 13.0338 1.10552 0.552758 0.833342i \(-0.313575\pi\)
0.552758 + 0.833342i \(0.313575\pi\)
\(140\) 0 0
\(141\) 13.3607 1.12517
\(142\) −6.29791 −0.528509
\(143\) −4.95547 −0.414397
\(144\) −0.630898 −0.0525748
\(145\) 0 0
\(146\) −12.7093 −1.05183
\(147\) 1.98667 0.163858
\(148\) 1.00000 0.0821995
\(149\) −4.40522 −0.360890 −0.180445 0.983585i \(-0.557754\pi\)
−0.180445 + 0.983585i \(0.557754\pi\)
\(150\) 0 0
\(151\) 2.92162 0.237758 0.118879 0.992909i \(-0.462070\pi\)
0.118879 + 0.992909i \(0.462070\pi\)
\(152\) 5.04945 0.409564
\(153\) −1.76713 −0.142864
\(154\) −3.14342 −0.253304
\(155\) 0 0
\(156\) 6.98667 0.559381
\(157\) 22.1906 1.77100 0.885502 0.464636i \(-0.153815\pi\)
0.885502 + 0.464636i \(0.153815\pi\)
\(158\) −2.58145 −0.205369
\(159\) −15.5441 −1.23273
\(160\) 0 0
\(161\) −21.3607 −1.68346
\(162\) 6.70928 0.527130
\(163\) −14.6248 −1.14550 −0.572750 0.819730i \(-0.694123\pi\)
−0.572750 + 0.819730i \(0.694123\pi\)
\(164\) −8.07838 −0.630815
\(165\) 0 0
\(166\) −8.48360 −0.658455
\(167\) −6.34736 −0.491174 −0.245587 0.969375i \(-0.578981\pi\)
−0.245587 + 0.969375i \(0.578981\pi\)
\(168\) 4.43188 0.341927
\(169\) 7.60424 0.584941
\(170\) 0 0
\(171\) 3.18568 0.243615
\(172\) 10.2329 0.780249
\(173\) −1.23513 −0.0939054 −0.0469527 0.998897i \(-0.514951\pi\)
−0.0469527 + 0.998897i \(0.514951\pi\)
\(174\) −10.2823 −0.779500
\(175\) 0 0
\(176\) −1.09171 −0.0822906
\(177\) 15.7503 1.18387
\(178\) 6.51026 0.487965
\(179\) 1.76487 0.131912 0.0659562 0.997823i \(-0.478990\pi\)
0.0659562 + 0.997823i \(0.478990\pi\)
\(180\) 0 0
\(181\) 6.49693 0.482913 0.241456 0.970412i \(-0.422375\pi\)
0.241456 + 0.970412i \(0.422375\pi\)
\(182\) 13.0700 0.968810
\(183\) 9.69368 0.716577
\(184\) −7.41855 −0.546903
\(185\) 0 0
\(186\) −5.40295 −0.396164
\(187\) −3.05786 −0.223613
\(188\) 8.68035 0.633079
\(189\) 16.0917 1.17050
\(190\) 0 0
\(191\) 21.2039 1.53426 0.767131 0.641490i \(-0.221683\pi\)
0.767131 + 0.641490i \(0.221683\pi\)
\(192\) 1.53919 0.111081
\(193\) 8.14342 0.586177 0.293088 0.956085i \(-0.405317\pi\)
0.293088 + 0.956085i \(0.405317\pi\)
\(194\) −3.07838 −0.221015
\(195\) 0 0
\(196\) 1.29072 0.0921946
\(197\) −24.8443 −1.77008 −0.885041 0.465513i \(-0.845870\pi\)
−0.885041 + 0.465513i \(0.845870\pi\)
\(198\) −0.688756 −0.0489478
\(199\) −8.47027 −0.600441 −0.300221 0.953870i \(-0.597060\pi\)
−0.300221 + 0.953870i \(0.597060\pi\)
\(200\) 0 0
\(201\) 20.4680 1.44370
\(202\) 15.9155 1.11981
\(203\) −19.2351 −1.35004
\(204\) 4.31124 0.301847
\(205\) 0 0
\(206\) 10.5886 0.737745
\(207\) −4.68035 −0.325307
\(208\) 4.53919 0.314736
\(209\) 5.51253 0.381309
\(210\) 0 0
\(211\) 21.7370 1.49644 0.748218 0.663453i \(-0.230910\pi\)
0.748218 + 0.663453i \(0.230910\pi\)
\(212\) −10.0989 −0.693595
\(213\) 9.69368 0.664200
\(214\) 4.03612 0.275903
\(215\) 0 0
\(216\) 5.58864 0.380259
\(217\) −10.1073 −0.686129
\(218\) 4.49693 0.304570
\(219\) 19.5620 1.32188
\(220\) 0 0
\(221\) 12.7142 0.855249
\(222\) −1.53919 −0.103304
\(223\) 16.6537 1.11521 0.557607 0.830105i \(-0.311720\pi\)
0.557607 + 0.830105i \(0.311720\pi\)
\(224\) 2.87936 0.192385
\(225\) 0 0
\(226\) −3.58864 −0.238713
\(227\) −11.2123 −0.744190 −0.372095 0.928195i \(-0.621360\pi\)
−0.372095 + 0.928195i \(0.621360\pi\)
\(228\) −7.77205 −0.514717
\(229\) −7.62863 −0.504114 −0.252057 0.967712i \(-0.581107\pi\)
−0.252057 + 0.967712i \(0.581107\pi\)
\(230\) 0 0
\(231\) 4.83832 0.318338
\(232\) −6.68035 −0.438586
\(233\) −0.0494483 −0.00323947 −0.00161973 0.999999i \(-0.500516\pi\)
−0.00161973 + 0.999999i \(0.500516\pi\)
\(234\) 2.86376 0.187210
\(235\) 0 0
\(236\) 10.2329 0.666103
\(237\) 3.97334 0.258096
\(238\) 8.06505 0.522779
\(239\) −14.7070 −0.951317 −0.475659 0.879630i \(-0.657790\pi\)
−0.475659 + 0.879630i \(0.657790\pi\)
\(240\) 0 0
\(241\) −14.8999 −0.959786 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(242\) 9.80817 0.630493
\(243\) 6.43907 0.413067
\(244\) 6.29791 0.403183
\(245\) 0 0
\(246\) 12.4341 0.792772
\(247\) −22.9204 −1.45839
\(248\) −3.51026 −0.222902
\(249\) 13.0579 0.827508
\(250\) 0 0
\(251\) 6.23513 0.393558 0.196779 0.980448i \(-0.436952\pi\)
0.196779 + 0.980448i \(0.436952\pi\)
\(252\) 1.81658 0.114434
\(253\) −8.09890 −0.509173
\(254\) −8.14116 −0.510822
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.1568 1.00783 0.503915 0.863753i \(-0.331892\pi\)
0.503915 + 0.863753i \(0.331892\pi\)
\(258\) −15.7503 −0.980572
\(259\) −2.87936 −0.178915
\(260\) 0 0
\(261\) −4.21461 −0.260878
\(262\) 1.46800 0.0906933
\(263\) −18.6537 −1.15024 −0.575118 0.818071i \(-0.695044\pi\)
−0.575118 + 0.818071i \(0.695044\pi\)
\(264\) 1.68035 0.103418
\(265\) 0 0
\(266\) −14.5392 −0.891455
\(267\) −10.0205 −0.613246
\(268\) 13.2979 0.812299
\(269\) 21.8310 1.33106 0.665529 0.746372i \(-0.268206\pi\)
0.665529 + 0.746372i \(0.268206\pi\)
\(270\) 0 0
\(271\) 24.9783 1.51732 0.758661 0.651486i \(-0.225854\pi\)
0.758661 + 0.651486i \(0.225854\pi\)
\(272\) 2.80098 0.169835
\(273\) −20.1171 −1.21755
\(274\) 6.17727 0.373183
\(275\) 0 0
\(276\) 11.4186 0.687316
\(277\) −0.822726 −0.0494328 −0.0247164 0.999695i \(-0.507868\pi\)
−0.0247164 + 0.999695i \(0.507868\pi\)
\(278\) −13.0338 −0.781718
\(279\) −2.21461 −0.132585
\(280\) 0 0
\(281\) −1.65142 −0.0985153 −0.0492576 0.998786i \(-0.515686\pi\)
−0.0492576 + 0.998786i \(0.515686\pi\)
\(282\) −13.3607 −0.795618
\(283\) 15.1340 0.899621 0.449811 0.893124i \(-0.351492\pi\)
0.449811 + 0.893124i \(0.351492\pi\)
\(284\) 6.29791 0.373712
\(285\) 0 0
\(286\) 4.95547 0.293023
\(287\) 23.2606 1.37303
\(288\) 0.630898 0.0371760
\(289\) −9.15449 −0.538499
\(290\) 0 0
\(291\) 4.73820 0.277758
\(292\) 12.7093 0.743754
\(293\) −25.4524 −1.48695 −0.743473 0.668766i \(-0.766823\pi\)
−0.743473 + 0.668766i \(0.766823\pi\)
\(294\) −1.98667 −0.115865
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 6.10116 0.354025
\(298\) 4.40522 0.255188
\(299\) 33.6742 1.94743
\(300\) 0 0
\(301\) −29.4641 −1.69828
\(302\) −2.92162 −0.168120
\(303\) −24.4969 −1.40731
\(304\) −5.04945 −0.289606
\(305\) 0 0
\(306\) 1.76713 0.101020
\(307\) −27.8176 −1.58764 −0.793818 0.608155i \(-0.791910\pi\)
−0.793818 + 0.608155i \(0.791910\pi\)
\(308\) 3.14342 0.179113
\(309\) −16.2979 −0.927156
\(310\) 0 0
\(311\) 0.0650468 0.00368846 0.00184423 0.999998i \(-0.499413\pi\)
0.00184423 + 0.999998i \(0.499413\pi\)
\(312\) −6.98667 −0.395542
\(313\) 22.5464 1.27440 0.637198 0.770700i \(-0.280093\pi\)
0.637198 + 0.770700i \(0.280093\pi\)
\(314\) −22.1906 −1.25229
\(315\) 0 0
\(316\) 2.58145 0.145218
\(317\) −24.4775 −1.37479 −0.687395 0.726283i \(-0.741246\pi\)
−0.687395 + 0.726283i \(0.741246\pi\)
\(318\) 15.5441 0.871670
\(319\) −7.29299 −0.408329
\(320\) 0 0
\(321\) −6.21235 −0.346739
\(322\) 21.3607 1.19038
\(323\) −14.1434 −0.786961
\(324\) −6.70928 −0.372738
\(325\) 0 0
\(326\) 14.6248 0.809990
\(327\) −6.92162 −0.382767
\(328\) 8.07838 0.446054
\(329\) −24.9939 −1.37796
\(330\) 0 0
\(331\) −1.91321 −0.105160 −0.0525798 0.998617i \(-0.516744\pi\)
−0.0525798 + 0.998617i \(0.516744\pi\)
\(332\) 8.48360 0.465598
\(333\) −0.630898 −0.0345730
\(334\) 6.34736 0.347312
\(335\) 0 0
\(336\) −4.43188 −0.241779
\(337\) −12.6286 −0.687925 −0.343963 0.938983i \(-0.611769\pi\)
−0.343963 + 0.938983i \(0.611769\pi\)
\(338\) −7.60424 −0.413616
\(339\) 5.52359 0.300000
\(340\) 0 0
\(341\) −3.83218 −0.207524
\(342\) −3.18568 −0.172262
\(343\) 16.4391 0.887626
\(344\) −10.2329 −0.551719
\(345\) 0 0
\(346\) 1.23513 0.0664012
\(347\) −8.33403 −0.447394 −0.223697 0.974659i \(-0.571813\pi\)
−0.223697 + 0.974659i \(0.571813\pi\)
\(348\) 10.2823 0.551190
\(349\) 2.11837 0.113394 0.0566969 0.998391i \(-0.481943\pi\)
0.0566969 + 0.998391i \(0.481943\pi\)
\(350\) 0 0
\(351\) −25.3679 −1.35404
\(352\) 1.09171 0.0581883
\(353\) 28.4534 1.51442 0.757212 0.653169i \(-0.226561\pi\)
0.757212 + 0.653169i \(0.226561\pi\)
\(354\) −15.7503 −0.837120
\(355\) 0 0
\(356\) −6.51026 −0.345043
\(357\) −12.4136 −0.656999
\(358\) −1.76487 −0.0932761
\(359\) 2.65368 0.140056 0.0700280 0.997545i \(-0.477691\pi\)
0.0700280 + 0.997545i \(0.477691\pi\)
\(360\) 0 0
\(361\) 6.49693 0.341944
\(362\) −6.49693 −0.341471
\(363\) −15.0966 −0.792368
\(364\) −13.0700 −0.685052
\(365\) 0 0
\(366\) −9.69368 −0.506697
\(367\) −31.7575 −1.65773 −0.828864 0.559450i \(-0.811012\pi\)
−0.828864 + 0.559450i \(0.811012\pi\)
\(368\) 7.41855 0.386719
\(369\) 5.09663 0.265320
\(370\) 0 0
\(371\) 29.0784 1.50967
\(372\) 5.40295 0.280130
\(373\) 25.1122 1.30026 0.650131 0.759822i \(-0.274714\pi\)
0.650131 + 0.759822i \(0.274714\pi\)
\(374\) 3.05786 0.158118
\(375\) 0 0
\(376\) −8.68035 −0.447655
\(377\) 30.3234 1.56173
\(378\) −16.0917 −0.827668
\(379\) 22.5330 1.15744 0.578722 0.815525i \(-0.303551\pi\)
0.578722 + 0.815525i \(0.303551\pi\)
\(380\) 0 0
\(381\) 12.5308 0.641971
\(382\) −21.2039 −1.08489
\(383\) −12.5886 −0.643249 −0.321625 0.946867i \(-0.604229\pi\)
−0.321625 + 0.946867i \(0.604229\pi\)
\(384\) −1.53919 −0.0785464
\(385\) 0 0
\(386\) −8.14342 −0.414489
\(387\) −6.45589 −0.328171
\(388\) 3.07838 0.156281
\(389\) −0.879362 −0.0445854 −0.0222927 0.999751i \(-0.507097\pi\)
−0.0222927 + 0.999751i \(0.507097\pi\)
\(390\) 0 0
\(391\) 20.7792 1.05085
\(392\) −1.29072 −0.0651914
\(393\) −2.25953 −0.113978
\(394\) 24.8443 1.25164
\(395\) 0 0
\(396\) 0.688756 0.0346113
\(397\) −29.6814 −1.48967 −0.744833 0.667251i \(-0.767471\pi\)
−0.744833 + 0.667251i \(0.767471\pi\)
\(398\) 8.47027 0.424576
\(399\) 22.3786 1.12033
\(400\) 0 0
\(401\) −13.9867 −0.698461 −0.349230 0.937037i \(-0.613557\pi\)
−0.349230 + 0.937037i \(0.613557\pi\)
\(402\) −20.4680 −1.02085
\(403\) 15.9337 0.793716
\(404\) −15.9155 −0.791825
\(405\) 0 0
\(406\) 19.2351 0.954624
\(407\) −1.09171 −0.0541140
\(408\) −4.31124 −0.213438
\(409\) −21.9060 −1.08318 −0.541592 0.840642i \(-0.682178\pi\)
−0.541592 + 0.840642i \(0.682178\pi\)
\(410\) 0 0
\(411\) −9.50799 −0.468995
\(412\) −10.5886 −0.521665
\(413\) −29.4641 −1.44983
\(414\) 4.68035 0.230026
\(415\) 0 0
\(416\) −4.53919 −0.222552
\(417\) 20.0616 0.982419
\(418\) −5.51253 −0.269627
\(419\) −34.8648 −1.70326 −0.851629 0.524146i \(-0.824385\pi\)
−0.851629 + 0.524146i \(0.824385\pi\)
\(420\) 0 0
\(421\) −4.76487 −0.232225 −0.116113 0.993236i \(-0.537043\pi\)
−0.116113 + 0.993236i \(0.537043\pi\)
\(422\) −21.7370 −1.05814
\(423\) −5.47641 −0.266272
\(424\) 10.0989 0.490446
\(425\) 0 0
\(426\) −9.69368 −0.469660
\(427\) −18.1340 −0.877564
\(428\) −4.03612 −0.195093
\(429\) −7.62741 −0.368255
\(430\) 0 0
\(431\) 2.66597 0.128415 0.0642076 0.997937i \(-0.479548\pi\)
0.0642076 + 0.997937i \(0.479548\pi\)
\(432\) −5.58864 −0.268883
\(433\) 32.3074 1.55259 0.776297 0.630368i \(-0.217096\pi\)
0.776297 + 0.630368i \(0.217096\pi\)
\(434\) 10.1073 0.485166
\(435\) 0 0
\(436\) −4.49693 −0.215364
\(437\) −37.4596 −1.79194
\(438\) −19.5620 −0.934707
\(439\) −22.1568 −1.05748 −0.528742 0.848783i \(-0.677336\pi\)
−0.528742 + 0.848783i \(0.677336\pi\)
\(440\) 0 0
\(441\) −0.814315 −0.0387769
\(442\) −12.7142 −0.604753
\(443\) 39.1061 1.85799 0.928993 0.370097i \(-0.120676\pi\)
0.928993 + 0.370097i \(0.120676\pi\)
\(444\) 1.53919 0.0730467
\(445\) 0 0
\(446\) −16.6537 −0.788575
\(447\) −6.78047 −0.320705
\(448\) −2.87936 −0.136037
\(449\) 30.8915 1.45786 0.728929 0.684589i \(-0.240018\pi\)
0.728929 + 0.684589i \(0.240018\pi\)
\(450\) 0 0
\(451\) 8.81924 0.415282
\(452\) 3.58864 0.168795
\(453\) 4.49693 0.211284
\(454\) 11.2123 0.526222
\(455\) 0 0
\(456\) 7.77205 0.363960
\(457\) −0.438025 −0.0204899 −0.0102450 0.999948i \(-0.503261\pi\)
−0.0102450 + 0.999948i \(0.503261\pi\)
\(458\) 7.62863 0.356462
\(459\) −15.6537 −0.730651
\(460\) 0 0
\(461\) 0.523590 0.0243860 0.0121930 0.999926i \(-0.496119\pi\)
0.0121930 + 0.999926i \(0.496119\pi\)
\(462\) −4.83832 −0.225099
\(463\) −29.1845 −1.35632 −0.678158 0.734916i \(-0.737222\pi\)
−0.678158 + 0.734916i \(0.737222\pi\)
\(464\) 6.68035 0.310127
\(465\) 0 0
\(466\) 0.0494483 0.00229065
\(467\) 9.49079 0.439181 0.219591 0.975592i \(-0.429528\pi\)
0.219591 + 0.975592i \(0.429528\pi\)
\(468\) −2.86376 −0.132378
\(469\) −38.2895 −1.76804
\(470\) 0 0
\(471\) 34.1555 1.57380
\(472\) −10.2329 −0.471006
\(473\) −11.1713 −0.513657
\(474\) −3.97334 −0.182501
\(475\) 0 0
\(476\) −8.06505 −0.369661
\(477\) 6.37137 0.291725
\(478\) 14.7070 0.672683
\(479\) −6.12291 −0.279763 −0.139881 0.990168i \(-0.544672\pi\)
−0.139881 + 0.990168i \(0.544672\pi\)
\(480\) 0 0
\(481\) 4.53919 0.206969
\(482\) 14.8999 0.678671
\(483\) −32.8781 −1.49601
\(484\) −9.80817 −0.445826
\(485\) 0 0
\(486\) −6.43907 −0.292082
\(487\) 10.3018 0.466819 0.233409 0.972379i \(-0.425012\pi\)
0.233409 + 0.972379i \(0.425012\pi\)
\(488\) −6.29791 −0.285093
\(489\) −22.5103 −1.01795
\(490\) 0 0
\(491\) −10.5380 −0.475572 −0.237786 0.971318i \(-0.576422\pi\)
−0.237786 + 0.971318i \(0.576422\pi\)
\(492\) −12.4341 −0.560575
\(493\) 18.7115 0.842726
\(494\) 22.9204 1.03124
\(495\) 0 0
\(496\) 3.51026 0.157615
\(497\) −18.1340 −0.813420
\(498\) −13.0579 −0.585137
\(499\) −31.2123 −1.39726 −0.698628 0.715485i \(-0.746206\pi\)
−0.698628 + 0.715485i \(0.746206\pi\)
\(500\) 0 0
\(501\) −9.76979 −0.436482
\(502\) −6.23513 −0.278288
\(503\) −36.1978 −1.61398 −0.806990 0.590565i \(-0.798905\pi\)
−0.806990 + 0.590565i \(0.798905\pi\)
\(504\) −1.81658 −0.0809170
\(505\) 0 0
\(506\) 8.09890 0.360040
\(507\) 11.7044 0.519809
\(508\) 8.14116 0.361205
\(509\) −33.6092 −1.48970 −0.744850 0.667232i \(-0.767479\pi\)
−0.744850 + 0.667232i \(0.767479\pi\)
\(510\) 0 0
\(511\) −36.5946 −1.61885
\(512\) −1.00000 −0.0441942
\(513\) 28.2195 1.24592
\(514\) −16.1568 −0.712644
\(515\) 0 0
\(516\) 15.7503 0.693369
\(517\) −9.47641 −0.416772
\(518\) 2.87936 0.126512
\(519\) −1.90110 −0.0834492
\(520\) 0 0
\(521\) 5.42082 0.237490 0.118745 0.992925i \(-0.462113\pi\)
0.118745 + 0.992925i \(0.462113\pi\)
\(522\) 4.21461 0.184469
\(523\) −31.3545 −1.37104 −0.685519 0.728054i \(-0.740425\pi\)
−0.685519 + 0.728054i \(0.740425\pi\)
\(524\) −1.46800 −0.0641298
\(525\) 0 0
\(526\) 18.6537 0.813339
\(527\) 9.83218 0.428297
\(528\) −1.68035 −0.0731277
\(529\) 32.0349 1.39282
\(530\) 0 0
\(531\) −6.45589 −0.280162
\(532\) 14.5392 0.630354
\(533\) −36.6693 −1.58832
\(534\) 10.0205 0.433630
\(535\) 0 0
\(536\) −13.2979 −0.574382
\(537\) 2.71646 0.117224
\(538\) −21.8310 −0.941199
\(539\) −1.40910 −0.0606940
\(540\) 0 0
\(541\) −9.98440 −0.429263 −0.214631 0.976695i \(-0.568855\pi\)
−0.214631 + 0.976695i \(0.568855\pi\)
\(542\) −24.9783 −1.07291
\(543\) 10.0000 0.429141
\(544\) −2.80098 −0.120091
\(545\) 0 0
\(546\) 20.1171 0.860934
\(547\) 32.9649 1.40948 0.704739 0.709466i \(-0.251064\pi\)
0.704739 + 0.709466i \(0.251064\pi\)
\(548\) −6.17727 −0.263880
\(549\) −3.97334 −0.169578
\(550\) 0 0
\(551\) −33.7321 −1.43703
\(552\) −11.4186 −0.486006
\(553\) −7.43293 −0.316080
\(554\) 0.822726 0.0349543
\(555\) 0 0
\(556\) 13.0338 0.552758
\(557\) −21.7009 −0.919495 −0.459748 0.888050i \(-0.652060\pi\)
−0.459748 + 0.888050i \(0.652060\pi\)
\(558\) 2.21461 0.0937521
\(559\) 46.4489 1.96458
\(560\) 0 0
\(561\) −4.70662 −0.198714
\(562\) 1.65142 0.0696608
\(563\) −1.05559 −0.0444879 −0.0222439 0.999753i \(-0.507081\pi\)
−0.0222439 + 0.999753i \(0.507081\pi\)
\(564\) 13.3607 0.562587
\(565\) 0 0
\(566\) −15.1340 −0.636128
\(567\) 19.3184 0.811298
\(568\) −6.29791 −0.264255
\(569\) −41.4196 −1.73640 −0.868200 0.496215i \(-0.834723\pi\)
−0.868200 + 0.496215i \(0.834723\pi\)
\(570\) 0 0
\(571\) 21.4452 0.897454 0.448727 0.893669i \(-0.351878\pi\)
0.448727 + 0.893669i \(0.351878\pi\)
\(572\) −4.95547 −0.207199
\(573\) 32.6369 1.36342
\(574\) −23.2606 −0.970878
\(575\) 0 0
\(576\) −0.630898 −0.0262874
\(577\) 17.2667 0.718823 0.359411 0.933179i \(-0.382977\pi\)
0.359411 + 0.933179i \(0.382977\pi\)
\(578\) 9.15449 0.380777
\(579\) 12.5343 0.520906
\(580\) 0 0
\(581\) −24.4273 −1.01342
\(582\) −4.73820 −0.196405
\(583\) 11.0251 0.456611
\(584\) −12.7093 −0.525914
\(585\) 0 0
\(586\) 25.4524 1.05143
\(587\) −9.63090 −0.397510 −0.198755 0.980049i \(-0.563690\pi\)
−0.198755 + 0.980049i \(0.563690\pi\)
\(588\) 1.98667 0.0819288
\(589\) −17.7249 −0.730341
\(590\) 0 0
\(591\) −38.2401 −1.57299
\(592\) 1.00000 0.0410997
\(593\) 2.78992 0.114568 0.0572842 0.998358i \(-0.481756\pi\)
0.0572842 + 0.998358i \(0.481756\pi\)
\(594\) −6.10116 −0.250334
\(595\) 0 0
\(596\) −4.40522 −0.180445
\(597\) −13.0373 −0.533583
\(598\) −33.6742 −1.37704
\(599\) 25.2606 1.03212 0.516060 0.856553i \(-0.327398\pi\)
0.516060 + 0.856553i \(0.327398\pi\)
\(600\) 0 0
\(601\) 35.4040 1.44416 0.722080 0.691810i \(-0.243186\pi\)
0.722080 + 0.691810i \(0.243186\pi\)
\(602\) 29.4641 1.20087
\(603\) −8.38962 −0.341652
\(604\) 2.92162 0.118879
\(605\) 0 0
\(606\) 24.4969 0.995120
\(607\) 38.2485 1.55246 0.776229 0.630452i \(-0.217130\pi\)
0.776229 + 0.630452i \(0.217130\pi\)
\(608\) 5.04945 0.204782
\(609\) −29.6065 −1.19972
\(610\) 0 0
\(611\) 39.4017 1.59402
\(612\) −1.76713 −0.0714322
\(613\) −46.3279 −1.87117 −0.935583 0.353107i \(-0.885125\pi\)
−0.935583 + 0.353107i \(0.885125\pi\)
\(614\) 27.8176 1.12263
\(615\) 0 0
\(616\) −3.14342 −0.126652
\(617\) 23.7503 0.956152 0.478076 0.878319i \(-0.341334\pi\)
0.478076 + 0.878319i \(0.341334\pi\)
\(618\) 16.2979 0.655598
\(619\) −6.53797 −0.262783 −0.131392 0.991331i \(-0.541945\pi\)
−0.131392 + 0.991331i \(0.541945\pi\)
\(620\) 0 0
\(621\) −41.4596 −1.66372
\(622\) −0.0650468 −0.00260814
\(623\) 18.7454 0.751018
\(624\) 6.98667 0.279691
\(625\) 0 0
\(626\) −22.5464 −0.901134
\(627\) 8.48482 0.338851
\(628\) 22.1906 0.885502
\(629\) 2.80098 0.111683
\(630\) 0 0
\(631\) 27.1506 1.08085 0.540424 0.841393i \(-0.318264\pi\)
0.540424 + 0.841393i \(0.318264\pi\)
\(632\) −2.58145 −0.102685
\(633\) 33.4573 1.32981
\(634\) 24.4775 0.972124
\(635\) 0 0
\(636\) −15.5441 −0.616364
\(637\) 5.85884 0.232136
\(638\) 7.29299 0.288732
\(639\) −3.97334 −0.157183
\(640\) 0 0
\(641\) 31.9877 1.26344 0.631719 0.775197i \(-0.282350\pi\)
0.631719 + 0.775197i \(0.282350\pi\)
\(642\) 6.21235 0.245182
\(643\) 26.2784 1.03632 0.518160 0.855284i \(-0.326617\pi\)
0.518160 + 0.855284i \(0.326617\pi\)
\(644\) −21.3607 −0.841729
\(645\) 0 0
\(646\) 14.1434 0.556466
\(647\) 31.3679 1.23320 0.616599 0.787277i \(-0.288510\pi\)
0.616599 + 0.787277i \(0.288510\pi\)
\(648\) 6.70928 0.263565
\(649\) −11.1713 −0.438512
\(650\) 0 0
\(651\) −15.5571 −0.609729
\(652\) −14.6248 −0.572750
\(653\) −25.3184 −0.990787 −0.495393 0.868669i \(-0.664976\pi\)
−0.495393 + 0.868669i \(0.664976\pi\)
\(654\) 6.92162 0.270657
\(655\) 0 0
\(656\) −8.07838 −0.315408
\(657\) −8.01825 −0.312822
\(658\) 24.9939 0.974362
\(659\) −38.2072 −1.48834 −0.744172 0.667989i \(-0.767156\pi\)
−0.744172 + 0.667989i \(0.767156\pi\)
\(660\) 0 0
\(661\) 13.7899 0.536366 0.268183 0.963368i \(-0.413577\pi\)
0.268183 + 0.963368i \(0.413577\pi\)
\(662\) 1.91321 0.0743591
\(663\) 19.5695 0.760018
\(664\) −8.48360 −0.329227
\(665\) 0 0
\(666\) 0.630898 0.0244468
\(667\) 49.5585 1.91891
\(668\) −6.34736 −0.245587
\(669\) 25.6332 0.991035
\(670\) 0 0
\(671\) −6.87549 −0.265425
\(672\) 4.43188 0.170964
\(673\) 4.41628 0.170235 0.0851176 0.996371i \(-0.472873\pi\)
0.0851176 + 0.996371i \(0.472873\pi\)
\(674\) 12.6286 0.486437
\(675\) 0 0
\(676\) 7.60424 0.292471
\(677\) −38.1399 −1.46584 −0.732918 0.680317i \(-0.761842\pi\)
−0.732918 + 0.680317i \(0.761842\pi\)
\(678\) −5.52359 −0.212132
\(679\) −8.86376 −0.340160
\(680\) 0 0
\(681\) −17.2579 −0.661325
\(682\) 3.83218 0.146742
\(683\) −22.6514 −0.866732 −0.433366 0.901218i \(-0.642674\pi\)
−0.433366 + 0.901218i \(0.642674\pi\)
\(684\) 3.18568 0.121808
\(685\) 0 0
\(686\) −16.4391 −0.627647
\(687\) −11.7419 −0.447982
\(688\) 10.2329 0.390124
\(689\) −45.8408 −1.74640
\(690\) 0 0
\(691\) −15.8443 −0.602745 −0.301373 0.953506i \(-0.597445\pi\)
−0.301373 + 0.953506i \(0.597445\pi\)
\(692\) −1.23513 −0.0469527
\(693\) −1.98318 −0.0753347
\(694\) 8.33403 0.316355
\(695\) 0 0
\(696\) −10.2823 −0.389750
\(697\) −22.6274 −0.857074
\(698\) −2.11837 −0.0801815
\(699\) −0.0761103 −0.00287876
\(700\) 0 0
\(701\) −2.92284 −0.110394 −0.0551972 0.998475i \(-0.517579\pi\)
−0.0551972 + 0.998475i \(0.517579\pi\)
\(702\) 25.3679 0.957449
\(703\) −5.04945 −0.190444
\(704\) −1.09171 −0.0411453
\(705\) 0 0
\(706\) −28.4534 −1.07086
\(707\) 45.8264 1.72348
\(708\) 15.7503 0.591933
\(709\) −8.45136 −0.317397 −0.158699 0.987327i \(-0.550730\pi\)
−0.158699 + 0.987327i \(0.550730\pi\)
\(710\) 0 0
\(711\) −1.62863 −0.0610784
\(712\) 6.51026 0.243982
\(713\) 26.0410 0.975245
\(714\) 12.4136 0.464568
\(715\) 0 0
\(716\) 1.76487 0.0659562
\(717\) −22.6369 −0.845389
\(718\) −2.65368 −0.0990346
\(719\) −15.3763 −0.573439 −0.286719 0.958015i \(-0.592565\pi\)
−0.286719 + 0.958015i \(0.592565\pi\)
\(720\) 0 0
\(721\) 30.4885 1.13545
\(722\) −6.49693 −0.241791
\(723\) −22.9337 −0.852915
\(724\) 6.49693 0.241456
\(725\) 0 0
\(726\) 15.0966 0.560288
\(727\) −15.7275 −0.583302 −0.291651 0.956525i \(-0.594205\pi\)
−0.291651 + 0.956525i \(0.594205\pi\)
\(728\) 13.0700 0.484405
\(729\) 30.0388 1.11255
\(730\) 0 0
\(731\) 28.6621 1.06011
\(732\) 9.69368 0.358289
\(733\) 15.7275 0.580909 0.290455 0.956889i \(-0.406193\pi\)
0.290455 + 0.956889i \(0.406193\pi\)
\(734\) 31.7575 1.17219
\(735\) 0 0
\(736\) −7.41855 −0.273451
\(737\) −14.5174 −0.534757
\(738\) −5.09663 −0.187610
\(739\) 45.0472 1.65709 0.828544 0.559924i \(-0.189170\pi\)
0.828544 + 0.559924i \(0.189170\pi\)
\(740\) 0 0
\(741\) −35.2788 −1.29600
\(742\) −29.0784 −1.06750
\(743\) −9.31965 −0.341905 −0.170952 0.985279i \(-0.554684\pi\)
−0.170952 + 0.985279i \(0.554684\pi\)
\(744\) −5.40295 −0.198082
\(745\) 0 0
\(746\) −25.1122 −0.919424
\(747\) −5.35228 −0.195830
\(748\) −3.05786 −0.111806
\(749\) 11.6214 0.424638
\(750\) 0 0
\(751\) 11.6020 0.423362 0.211681 0.977339i \(-0.432106\pi\)
0.211681 + 0.977339i \(0.432106\pi\)
\(752\) 8.68035 0.316540
\(753\) 9.59705 0.349736
\(754\) −30.3234 −1.10431
\(755\) 0 0
\(756\) 16.0917 0.585250
\(757\) 38.3545 1.39402 0.697010 0.717062i \(-0.254513\pi\)
0.697010 + 0.717062i \(0.254513\pi\)
\(758\) −22.5330 −0.818437
\(759\) −12.4657 −0.452477
\(760\) 0 0
\(761\) −45.3318 −1.64328 −0.821638 0.570010i \(-0.806939\pi\)
−0.821638 + 0.570010i \(0.806939\pi\)
\(762\) −12.5308 −0.453942
\(763\) 12.9483 0.468759
\(764\) 21.2039 0.767131
\(765\) 0 0
\(766\) 12.5886 0.454846
\(767\) 46.4489 1.67717
\(768\) 1.53919 0.0555407
\(769\) −31.7454 −1.14477 −0.572384 0.819986i \(-0.693981\pi\)
−0.572384 + 0.819986i \(0.693981\pi\)
\(770\) 0 0
\(771\) 24.8683 0.895610
\(772\) 8.14342 0.293088
\(773\) −38.0482 −1.36850 −0.684250 0.729248i \(-0.739870\pi\)
−0.684250 + 0.729248i \(0.739870\pi\)
\(774\) 6.45589 0.232052
\(775\) 0 0
\(776\) −3.07838 −0.110507
\(777\) −4.43188 −0.158993
\(778\) 0.879362 0.0315266
\(779\) 40.7914 1.46150
\(780\) 0 0
\(781\) −6.87549 −0.246024
\(782\) −20.7792 −0.743064
\(783\) −37.3340 −1.33421
\(784\) 1.29072 0.0460973
\(785\) 0 0
\(786\) 2.25953 0.0805947
\(787\) −15.1506 −0.540061 −0.270031 0.962852i \(-0.587034\pi\)
−0.270031 + 0.962852i \(0.587034\pi\)
\(788\) −24.8443 −0.885041
\(789\) −28.7115 −1.02216
\(790\) 0 0
\(791\) −10.3330 −0.367399
\(792\) −0.688756 −0.0244739
\(793\) 28.5874 1.01517
\(794\) 29.6814 1.05335
\(795\) 0 0
\(796\) −8.47027 −0.300221
\(797\) 43.1350 1.52792 0.763960 0.645263i \(-0.223252\pi\)
0.763960 + 0.645263i \(0.223252\pi\)
\(798\) −22.3786 −0.792192
\(799\) 24.3135 0.860150
\(800\) 0 0
\(801\) 4.10731 0.145125
\(802\) 13.9867 0.493886
\(803\) −13.8748 −0.489632
\(804\) 20.4680 0.721851
\(805\) 0 0
\(806\) −15.9337 −0.561242
\(807\) 33.6020 1.18285
\(808\) 15.9155 0.559905
\(809\) 33.3874 1.17384 0.586918 0.809646i \(-0.300341\pi\)
0.586918 + 0.809646i \(0.300341\pi\)
\(810\) 0 0
\(811\) 10.9216 0.383510 0.191755 0.981443i \(-0.438582\pi\)
0.191755 + 0.981443i \(0.438582\pi\)
\(812\) −19.2351 −0.675021
\(813\) 38.4463 1.34837
\(814\) 1.09171 0.0382644
\(815\) 0 0
\(816\) 4.31124 0.150924
\(817\) −51.6703 −1.80772
\(818\) 21.9060 0.765926
\(819\) 8.24581 0.288132
\(820\) 0 0
\(821\) −32.7910 −1.14441 −0.572206 0.820110i \(-0.693912\pi\)
−0.572206 + 0.820110i \(0.693912\pi\)
\(822\) 9.50799 0.331629
\(823\) 9.83218 0.342728 0.171364 0.985208i \(-0.445183\pi\)
0.171364 + 0.985208i \(0.445183\pi\)
\(824\) 10.5886 0.368873
\(825\) 0 0
\(826\) 29.4641 1.02519
\(827\) 17.0228 0.591940 0.295970 0.955197i \(-0.404357\pi\)
0.295970 + 0.955197i \(0.404357\pi\)
\(828\) −4.68035 −0.162653
\(829\) 12.8260 0.445467 0.222733 0.974879i \(-0.428502\pi\)
0.222733 + 0.974879i \(0.428502\pi\)
\(830\) 0 0
\(831\) −1.26633 −0.0439285
\(832\) 4.53919 0.157368
\(833\) 3.61530 0.125263
\(834\) −20.0616 −0.694675
\(835\) 0 0
\(836\) 5.51253 0.190655
\(837\) −19.6176 −0.678082
\(838\) 34.8648 1.20438
\(839\) −11.5018 −0.397088 −0.198544 0.980092i \(-0.563621\pi\)
−0.198544 + 0.980092i \(0.563621\pi\)
\(840\) 0 0
\(841\) 15.6270 0.538863
\(842\) 4.76487 0.164208
\(843\) −2.54184 −0.0875457
\(844\) 21.7370 0.748218
\(845\) 0 0
\(846\) 5.47641 0.188283
\(847\) 28.2413 0.970382
\(848\) −10.0989 −0.346798
\(849\) 23.2940 0.799449
\(850\) 0 0
\(851\) 7.41855 0.254305
\(852\) 9.69368 0.332100
\(853\) 40.5946 1.38993 0.694966 0.719042i \(-0.255419\pi\)
0.694966 + 0.719042i \(0.255419\pi\)
\(854\) 18.1340 0.620532
\(855\) 0 0
\(856\) 4.03612 0.137952
\(857\) 38.0361 1.29929 0.649645 0.760238i \(-0.274918\pi\)
0.649645 + 0.760238i \(0.274918\pi\)
\(858\) 7.62741 0.260395
\(859\) −40.9506 −1.39721 −0.698607 0.715505i \(-0.746197\pi\)
−0.698607 + 0.715505i \(0.746197\pi\)
\(860\) 0 0
\(861\) 35.8024 1.22014
\(862\) −2.66597 −0.0908033
\(863\) 33.0817 1.12611 0.563057 0.826418i \(-0.309625\pi\)
0.563057 + 0.826418i \(0.309625\pi\)
\(864\) 5.58864 0.190129
\(865\) 0 0
\(866\) −32.3074 −1.09785
\(867\) −14.0905 −0.478538
\(868\) −10.1073 −0.343064
\(869\) −2.81819 −0.0956006
\(870\) 0 0
\(871\) 60.3617 2.04528
\(872\) 4.49693 0.152285
\(873\) −1.94214 −0.0657315
\(874\) 37.4596 1.26709
\(875\) 0 0
\(876\) 19.5620 0.660938
\(877\) 25.5057 0.861267 0.430634 0.902527i \(-0.358290\pi\)
0.430634 + 0.902527i \(0.358290\pi\)
\(878\) 22.1568 0.747754
\(879\) −39.1761 −1.32138
\(880\) 0 0
\(881\) 15.6514 0.527310 0.263655 0.964617i \(-0.415072\pi\)
0.263655 + 0.964617i \(0.415072\pi\)
\(882\) 0.814315 0.0274194
\(883\) 56.1071 1.88816 0.944078 0.329723i \(-0.106955\pi\)
0.944078 + 0.329723i \(0.106955\pi\)
\(884\) 12.7142 0.427625
\(885\) 0 0
\(886\) −39.1061 −1.31379
\(887\) −47.5052 −1.59507 −0.797534 0.603275i \(-0.793862\pi\)
−0.797534 + 0.603275i \(0.793862\pi\)
\(888\) −1.53919 −0.0516518
\(889\) −23.4413 −0.786197
\(890\) 0 0
\(891\) 7.32457 0.245382
\(892\) 16.6537 0.557607
\(893\) −43.8310 −1.46675
\(894\) 6.78047 0.226773
\(895\) 0 0
\(896\) 2.87936 0.0961927
\(897\) 51.8310 1.73059
\(898\) −30.8915 −1.03086
\(899\) 23.4497 0.782093
\(900\) 0 0
\(901\) −28.2868 −0.942372
\(902\) −8.81924 −0.293648
\(903\) −45.3509 −1.50918
\(904\) −3.58864 −0.119356
\(905\) 0 0
\(906\) −4.49693 −0.149400
\(907\) 15.8394 0.525938 0.262969 0.964804i \(-0.415298\pi\)
0.262969 + 0.964804i \(0.415298\pi\)
\(908\) −11.2123 −0.372095
\(909\) 10.0410 0.333040
\(910\) 0 0
\(911\) 11.9539 0.396049 0.198025 0.980197i \(-0.436547\pi\)
0.198025 + 0.980197i \(0.436547\pi\)
\(912\) −7.77205 −0.257358
\(913\) −9.26162 −0.306515
\(914\) 0.438025 0.0144886
\(915\) 0 0
\(916\) −7.62863 −0.252057
\(917\) 4.22690 0.139585
\(918\) 15.6537 0.516649
\(919\) −34.8781 −1.15052 −0.575262 0.817969i \(-0.695100\pi\)
−0.575262 + 0.817969i \(0.695100\pi\)
\(920\) 0 0
\(921\) −42.8166 −1.41085
\(922\) −0.523590 −0.0172435
\(923\) 28.5874 0.940966
\(924\) 4.83832 0.159169
\(925\) 0 0
\(926\) 29.1845 0.959061
\(927\) 6.68035 0.219411
\(928\) −6.68035 −0.219293
\(929\) −52.5439 −1.72391 −0.861955 0.506984i \(-0.830760\pi\)
−0.861955 + 0.506984i \(0.830760\pi\)
\(930\) 0 0
\(931\) −6.51745 −0.213601
\(932\) −0.0494483 −0.00161973
\(933\) 0.100119 0.00327776
\(934\) −9.49079 −0.310548
\(935\) 0 0
\(936\) 2.86376 0.0936050
\(937\) −13.3318 −0.435530 −0.217765 0.976001i \(-0.569877\pi\)
−0.217765 + 0.976001i \(0.569877\pi\)
\(938\) 38.2895 1.25020
\(939\) 34.7031 1.13249
\(940\) 0 0
\(941\) 14.5281 0.473603 0.236802 0.971558i \(-0.423901\pi\)
0.236802 + 0.971558i \(0.423901\pi\)
\(942\) −34.1555 −1.11285
\(943\) −59.9299 −1.95158
\(944\) 10.2329 0.333051
\(945\) 0 0
\(946\) 11.1713 0.363211
\(947\) −24.6803 −0.802003 −0.401002 0.916077i \(-0.631338\pi\)
−0.401002 + 0.916077i \(0.631338\pi\)
\(948\) 3.97334 0.129048
\(949\) 57.6898 1.87269
\(950\) 0 0
\(951\) −37.6754 −1.22171
\(952\) 8.06505 0.261390
\(953\) −40.5936 −1.31495 −0.657477 0.753474i \(-0.728376\pi\)
−0.657477 + 0.753474i \(0.728376\pi\)
\(954\) −6.37137 −0.206281
\(955\) 0 0
\(956\) −14.7070 −0.475659
\(957\) −11.2253 −0.362862
\(958\) 6.12291 0.197822
\(959\) 17.7866 0.574360
\(960\) 0 0
\(961\) −18.6781 −0.602519
\(962\) −4.53919 −0.146349
\(963\) 2.54638 0.0820558
\(964\) −14.8999 −0.479893
\(965\) 0 0
\(966\) 32.8781 1.05784
\(967\) −1.06400 −0.0342160 −0.0171080 0.999854i \(-0.505446\pi\)
−0.0171080 + 0.999854i \(0.505446\pi\)
\(968\) 9.80817 0.315247
\(969\) −21.7694 −0.699334
\(970\) 0 0
\(971\) 11.9333 0.382959 0.191480 0.981497i \(-0.438671\pi\)
0.191480 + 0.981497i \(0.438671\pi\)
\(972\) 6.43907 0.206533
\(973\) −37.5292 −1.20313
\(974\) −10.3018 −0.330091
\(975\) 0 0
\(976\) 6.29791 0.201591
\(977\) −19.1867 −0.613838 −0.306919 0.951736i \(-0.599298\pi\)
−0.306919 + 0.951736i \(0.599298\pi\)
\(978\) 22.5103 0.719799
\(979\) 7.10731 0.227151
\(980\) 0 0
\(981\) 2.83710 0.0905817
\(982\) 10.5380 0.336280
\(983\) −22.9171 −0.730942 −0.365471 0.930823i \(-0.619092\pi\)
−0.365471 + 0.930823i \(0.619092\pi\)
\(984\) 12.4341 0.396386
\(985\) 0 0
\(986\) −18.7115 −0.595897
\(987\) −38.4703 −1.22452
\(988\) −22.9204 −0.729195
\(989\) 75.9130 2.41389
\(990\) 0 0
\(991\) −52.2772 −1.66064 −0.830320 0.557287i \(-0.811842\pi\)
−0.830320 + 0.557287i \(0.811842\pi\)
\(992\) −3.51026 −0.111451
\(993\) −2.94479 −0.0934502
\(994\) 18.1340 0.575175
\(995\) 0 0
\(996\) 13.0579 0.413754
\(997\) 48.8892 1.54834 0.774168 0.632980i \(-0.218168\pi\)
0.774168 + 0.632980i \(0.218168\pi\)
\(998\) 31.2123 0.988010
\(999\) −5.58864 −0.176817
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.1.2 3
5.2 odd 4 1850.2.b.n.149.2 6
5.3 odd 4 1850.2.b.n.149.5 6
5.4 even 2 1850.2.a.bb.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.ba.1.2 3 1.1 even 1 trivial
1850.2.a.bb.1.2 yes 3 5.4 even 2
1850.2.b.n.149.2 6 5.2 odd 4
1850.2.b.n.149.5 6 5.3 odd 4