Properties

Label 1850.2.a.be.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.1791440.1
Defining polynomial: \(x^{5} - 9 x^{3} + 13 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.72987\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.72987 q^{3} +1.00000 q^{4} -2.72987 q^{6} -4.14336 q^{7} +1.00000 q^{8} +4.45216 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.72987 q^{3} +1.00000 q^{4} -2.72987 q^{6} -4.14336 q^{7} +1.00000 q^{8} +4.45216 q^{9} -4.76853 q^{11} -2.72987 q^{12} -3.91744 q^{13} -4.14336 q^{14} +1.00000 q^{16} +3.31637 q^{17} +4.45216 q^{18} -1.85109 q^{19} +11.3108 q^{21} -4.76853 q^{22} -1.54229 q^{23} -2.72987 q^{24} -3.91744 q^{26} -3.96421 q^{27} -4.14336 q^{28} +8.87323 q^{29} +9.75286 q^{31} +1.00000 q^{32} +13.0174 q^{33} +3.31637 q^{34} +4.45216 q^{36} +1.00000 q^{37} -1.85109 q^{38} +10.6941 q^{39} -5.06977 q^{41} +11.3108 q^{42} -9.99446 q^{43} -4.76853 q^{44} -1.54229 q^{46} +4.82700 q^{47} -2.72987 q^{48} +10.1675 q^{49} -9.05324 q^{51} -3.91744 q^{52} -5.13611 q^{53} -3.96421 q^{54} -4.14336 q^{56} +5.05324 q^{57} +8.87323 q^{58} -1.05324 q^{59} -4.14422 q^{61} +9.75286 q^{62} -18.4469 q^{63} +1.00000 q^{64} +13.0174 q^{66} +1.00811 q^{67} +3.31637 q^{68} +4.21025 q^{69} +6.45248 q^{71} +4.45216 q^{72} +10.7714 q^{73} +1.00000 q^{74} -1.85109 q^{76} +19.7578 q^{77} +10.6941 q^{78} -1.19856 q^{79} -2.53473 q^{81} -5.06977 q^{82} +10.6932 q^{83} +11.3108 q^{84} -9.99446 q^{86} -24.2227 q^{87} -4.76853 q^{88} +7.29569 q^{89} +16.2314 q^{91} -1.54229 q^{92} -26.6240 q^{93} +4.82700 q^{94} -2.72987 q^{96} +14.8988 q^{97} +10.1675 q^{98} -21.2303 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{2} + 5q^{4} - q^{7} + 5q^{8} + 3q^{9} + O(q^{10}) \) \( 5q + 5q^{2} + 5q^{4} - q^{7} + 5q^{8} + 3q^{9} + 3q^{11} - 6q^{13} - q^{14} + 5q^{16} + 9q^{17} + 3q^{18} + 4q^{19} + 16q^{21} + 3q^{22} + 6q^{23} - 6q^{26} - q^{28} + 11q^{29} + 23q^{31} + 5q^{32} + 20q^{33} + 9q^{34} + 3q^{36} + 5q^{37} + 4q^{38} + 20q^{39} - 7q^{41} + 16q^{42} - 17q^{43} + 3q^{44} + 6q^{46} + 12q^{47} + 30q^{49} - 20q^{51} - 6q^{52} - 7q^{53} - q^{56} + 11q^{58} + 20q^{59} - 9q^{61} + 23q^{62} - 33q^{63} + 5q^{64} + 20q^{66} + 12q^{67} + 9q^{68} + 16q^{69} + 6q^{71} + 3q^{72} - 6q^{73} + 5q^{74} + 4q^{76} - q^{77} + 20q^{78} + 20q^{79} - 7q^{81} - 7q^{82} + 12q^{83} + 16q^{84} - 17q^{86} - 34q^{87} + 3q^{88} + 12q^{89} + 16q^{91} + 6q^{92} - 4q^{93} + 12q^{94} + 3q^{97} + 30q^{98} - 11q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.72987 −1.57609 −0.788044 0.615619i \(-0.788906\pi\)
−0.788044 + 0.615619i \(0.788906\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.72987 −1.11446
\(7\) −4.14336 −1.56604 −0.783022 0.621994i \(-0.786323\pi\)
−0.783022 + 0.621994i \(0.786323\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.45216 1.48405
\(10\) 0 0
\(11\) −4.76853 −1.43777 −0.718883 0.695131i \(-0.755346\pi\)
−0.718883 + 0.695131i \(0.755346\pi\)
\(12\) −2.72987 −0.788044
\(13\) −3.91744 −1.08650 −0.543251 0.839570i \(-0.682807\pi\)
−0.543251 + 0.839570i \(0.682807\pi\)
\(14\) −4.14336 −1.10736
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.31637 0.804337 0.402169 0.915566i \(-0.368257\pi\)
0.402169 + 0.915566i \(0.368257\pi\)
\(18\) 4.45216 1.04939
\(19\) −1.85109 −0.424670 −0.212335 0.977197i \(-0.568107\pi\)
−0.212335 + 0.977197i \(0.568107\pi\)
\(20\) 0 0
\(21\) 11.3108 2.46822
\(22\) −4.76853 −1.01665
\(23\) −1.54229 −0.321590 −0.160795 0.986988i \(-0.551406\pi\)
−0.160795 + 0.986988i \(0.551406\pi\)
\(24\) −2.72987 −0.557231
\(25\) 0 0
\(26\) −3.91744 −0.768273
\(27\) −3.96421 −0.762913
\(28\) −4.14336 −0.783022
\(29\) 8.87323 1.64772 0.823859 0.566795i \(-0.191817\pi\)
0.823859 + 0.566795i \(0.191817\pi\)
\(30\) 0 0
\(31\) 9.75286 1.75166 0.875832 0.482615i \(-0.160313\pi\)
0.875832 + 0.482615i \(0.160313\pi\)
\(32\) 1.00000 0.176777
\(33\) 13.0174 2.26605
\(34\) 3.31637 0.568752
\(35\) 0 0
\(36\) 4.45216 0.742027
\(37\) 1.00000 0.164399
\(38\) −1.85109 −0.300287
\(39\) 10.6941 1.71242
\(40\) 0 0
\(41\) −5.06977 −0.791764 −0.395882 0.918301i \(-0.629561\pi\)
−0.395882 + 0.918301i \(0.629561\pi\)
\(42\) 11.3108 1.74530
\(43\) −9.99446 −1.52414 −0.762070 0.647494i \(-0.775817\pi\)
−0.762070 + 0.647494i \(0.775817\pi\)
\(44\) −4.76853 −0.718883
\(45\) 0 0
\(46\) −1.54229 −0.227399
\(47\) 4.82700 0.704090 0.352045 0.935983i \(-0.385486\pi\)
0.352045 + 0.935983i \(0.385486\pi\)
\(48\) −2.72987 −0.394022
\(49\) 10.1675 1.45249
\(50\) 0 0
\(51\) −9.05324 −1.26771
\(52\) −3.91744 −0.543251
\(53\) −5.13611 −0.705499 −0.352750 0.935718i \(-0.614753\pi\)
−0.352750 + 0.935718i \(0.614753\pi\)
\(54\) −3.96421 −0.539461
\(55\) 0 0
\(56\) −4.14336 −0.553680
\(57\) 5.05324 0.669317
\(58\) 8.87323 1.16511
\(59\) −1.05324 −0.137120 −0.0685598 0.997647i \(-0.521840\pi\)
−0.0685598 + 0.997647i \(0.521840\pi\)
\(60\) 0 0
\(61\) −4.14422 −0.530613 −0.265306 0.964164i \(-0.585473\pi\)
−0.265306 + 0.964164i \(0.585473\pi\)
\(62\) 9.75286 1.23861
\(63\) −18.4469 −2.32410
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 13.0174 1.60234
\(67\) 1.00811 0.123160 0.0615800 0.998102i \(-0.480386\pi\)
0.0615800 + 0.998102i \(0.480386\pi\)
\(68\) 3.31637 0.402169
\(69\) 4.21025 0.506855
\(70\) 0 0
\(71\) 6.45248 0.765768 0.382884 0.923796i \(-0.374931\pi\)
0.382884 + 0.923796i \(0.374931\pi\)
\(72\) 4.45216 0.524693
\(73\) 10.7714 1.26070 0.630349 0.776312i \(-0.282912\pi\)
0.630349 + 0.776312i \(0.282912\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −1.85109 −0.212335
\(77\) 19.7578 2.25161
\(78\) 10.6941 1.21087
\(79\) −1.19856 −0.134849 −0.0674243 0.997724i \(-0.521478\pi\)
−0.0674243 + 0.997724i \(0.521478\pi\)
\(80\) 0 0
\(81\) −2.53473 −0.281636
\(82\) −5.06977 −0.559862
\(83\) 10.6932 1.17373 0.586867 0.809683i \(-0.300361\pi\)
0.586867 + 0.809683i \(0.300361\pi\)
\(84\) 11.3108 1.23411
\(85\) 0 0
\(86\) −9.99446 −1.07773
\(87\) −24.2227 −2.59695
\(88\) −4.76853 −0.508327
\(89\) 7.29569 0.773342 0.386671 0.922218i \(-0.373625\pi\)
0.386671 + 0.922218i \(0.373625\pi\)
\(90\) 0 0
\(91\) 16.2314 1.70151
\(92\) −1.54229 −0.160795
\(93\) −26.6240 −2.76078
\(94\) 4.82700 0.497867
\(95\) 0 0
\(96\) −2.72987 −0.278616
\(97\) 14.8988 1.51274 0.756371 0.654143i \(-0.226970\pi\)
0.756371 + 0.654143i \(0.226970\pi\)
\(98\) 10.1675 1.02707
\(99\) −21.2303 −2.13372
\(100\) 0 0
\(101\) 18.5903 1.84980 0.924902 0.380206i \(-0.124147\pi\)
0.924902 + 0.380206i \(0.124147\pi\)
\(102\) −9.05324 −0.896404
\(103\) −13.3033 −1.31081 −0.655404 0.755278i \(-0.727502\pi\)
−0.655404 + 0.755278i \(0.727502\pi\)
\(104\) −3.91744 −0.384136
\(105\) 0 0
\(106\) −5.13611 −0.498863
\(107\) 12.4763 1.20613 0.603066 0.797691i \(-0.293946\pi\)
0.603066 + 0.797691i \(0.293946\pi\)
\(108\) −3.96421 −0.381457
\(109\) −4.35447 −0.417083 −0.208541 0.978014i \(-0.566872\pi\)
−0.208541 + 0.978014i \(0.566872\pi\)
\(110\) 0 0
\(111\) −2.72987 −0.259107
\(112\) −4.14336 −0.391511
\(113\) 9.54261 0.897693 0.448846 0.893609i \(-0.351835\pi\)
0.448846 + 0.893609i \(0.351835\pi\)
\(114\) 5.05324 0.473279
\(115\) 0 0
\(116\) 8.87323 0.823859
\(117\) −17.4411 −1.61243
\(118\) −1.05324 −0.0969582
\(119\) −13.7409 −1.25963
\(120\) 0 0
\(121\) 11.7389 1.06717
\(122\) −4.14422 −0.375200
\(123\) 13.8398 1.24789
\(124\) 9.75286 0.875832
\(125\) 0 0
\(126\) −18.4469 −1.64338
\(127\) 8.33492 0.739605 0.369802 0.929110i \(-0.379425\pi\)
0.369802 + 0.929110i \(0.379425\pi\)
\(128\) 1.00000 0.0883883
\(129\) 27.2835 2.40218
\(130\) 0 0
\(131\) 3.68597 0.322045 0.161022 0.986951i \(-0.448521\pi\)
0.161022 + 0.986951i \(0.448521\pi\)
\(132\) 13.0174 1.13302
\(133\) 7.66975 0.665052
\(134\) 1.00811 0.0870873
\(135\) 0 0
\(136\) 3.31637 0.284376
\(137\) −12.3027 −1.05109 −0.525546 0.850765i \(-0.676139\pi\)
−0.525546 + 0.850765i \(0.676139\pi\)
\(138\) 4.21025 0.358400
\(139\) −0.842442 −0.0714550 −0.0357275 0.999362i \(-0.511375\pi\)
−0.0357275 + 0.999362i \(0.511375\pi\)
\(140\) 0 0
\(141\) −13.1770 −1.10971
\(142\) 6.45248 0.541480
\(143\) 18.6804 1.56214
\(144\) 4.45216 0.371014
\(145\) 0 0
\(146\) 10.7714 0.891448
\(147\) −27.7558 −2.28926
\(148\) 1.00000 0.0821995
\(149\) −16.7723 −1.37404 −0.687019 0.726640i \(-0.741081\pi\)
−0.687019 + 0.726640i \(0.741081\pi\)
\(150\) 0 0
\(151\) −7.59755 −0.618280 −0.309140 0.951017i \(-0.600041\pi\)
−0.309140 + 0.951017i \(0.600041\pi\)
\(152\) −1.85109 −0.150143
\(153\) 14.7650 1.19368
\(154\) 19.7578 1.59213
\(155\) 0 0
\(156\) 10.6941 0.856211
\(157\) 4.45631 0.355652 0.177826 0.984062i \(-0.443094\pi\)
0.177826 + 0.984062i \(0.443094\pi\)
\(158\) −1.19856 −0.0953523
\(159\) 14.0209 1.11193
\(160\) 0 0
\(161\) 6.39028 0.503624
\(162\) −2.53473 −0.199147
\(163\) 19.4599 1.52422 0.762110 0.647447i \(-0.224163\pi\)
0.762110 + 0.647447i \(0.224163\pi\)
\(164\) −5.06977 −0.395882
\(165\) 0 0
\(166\) 10.6932 0.829955
\(167\) 7.13813 0.552365 0.276183 0.961105i \(-0.410931\pi\)
0.276183 + 0.961105i \(0.410931\pi\)
\(168\) 11.3108 0.872649
\(169\) 2.34632 0.180486
\(170\) 0 0
\(171\) −8.24137 −0.630233
\(172\) −9.99446 −0.762070
\(173\) −11.6199 −0.883448 −0.441724 0.897151i \(-0.645633\pi\)
−0.441724 + 0.897151i \(0.645633\pi\)
\(174\) −24.2227 −1.83632
\(175\) 0 0
\(176\) −4.76853 −0.359442
\(177\) 2.87519 0.216113
\(178\) 7.29569 0.546835
\(179\) −17.2224 −1.28726 −0.643631 0.765336i \(-0.722573\pi\)
−0.643631 + 0.765336i \(0.722573\pi\)
\(180\) 0 0
\(181\) −23.5098 −1.74747 −0.873733 0.486405i \(-0.838308\pi\)
−0.873733 + 0.486405i \(0.838308\pi\)
\(182\) 16.2314 1.20315
\(183\) 11.3132 0.836293
\(184\) −1.54229 −0.113699
\(185\) 0 0
\(186\) −26.6240 −1.95217
\(187\) −15.8142 −1.15645
\(188\) 4.82700 0.352045
\(189\) 16.4252 1.19476
\(190\) 0 0
\(191\) 1.16660 0.0844125 0.0422063 0.999109i \(-0.486561\pi\)
0.0422063 + 0.999109i \(0.486561\pi\)
\(192\) −2.72987 −0.197011
\(193\) −9.36727 −0.674271 −0.337135 0.941456i \(-0.609458\pi\)
−0.337135 + 0.941456i \(0.609458\pi\)
\(194\) 14.8988 1.06967
\(195\) 0 0
\(196\) 10.1675 0.726247
\(197\) 3.21836 0.229299 0.114649 0.993406i \(-0.463426\pi\)
0.114649 + 0.993406i \(0.463426\pi\)
\(198\) −21.2303 −1.50877
\(199\) −2.99104 −0.212029 −0.106014 0.994365i \(-0.533809\pi\)
−0.106014 + 0.994365i \(0.533809\pi\)
\(200\) 0 0
\(201\) −2.75200 −0.194111
\(202\) 18.5903 1.30801
\(203\) −36.7650 −2.58040
\(204\) −9.05324 −0.633853
\(205\) 0 0
\(206\) −13.3033 −0.926882
\(207\) −6.86654 −0.477257
\(208\) −3.91744 −0.271625
\(209\) 8.82700 0.610576
\(210\) 0 0
\(211\) 4.75340 0.327237 0.163619 0.986524i \(-0.447683\pi\)
0.163619 + 0.986524i \(0.447683\pi\)
\(212\) −5.13611 −0.352750
\(213\) −17.6144 −1.20692
\(214\) 12.4763 0.852864
\(215\) 0 0
\(216\) −3.96421 −0.269731
\(217\) −40.4096 −2.74318
\(218\) −4.35447 −0.294922
\(219\) −29.4045 −1.98697
\(220\) 0 0
\(221\) −12.9917 −0.873914
\(222\) −2.72987 −0.183217
\(223\) −6.95635 −0.465832 −0.232916 0.972497i \(-0.574827\pi\)
−0.232916 + 0.972497i \(0.574827\pi\)
\(224\) −4.14336 −0.276840
\(225\) 0 0
\(226\) 9.54261 0.634765
\(227\) −22.2980 −1.47997 −0.739986 0.672622i \(-0.765168\pi\)
−0.739986 + 0.672622i \(0.765168\pi\)
\(228\) 5.05324 0.334659
\(229\) 12.9648 0.856739 0.428370 0.903604i \(-0.359088\pi\)
0.428370 + 0.903604i \(0.359088\pi\)
\(230\) 0 0
\(231\) −53.9360 −3.54873
\(232\) 8.87323 0.582556
\(233\) −10.4327 −0.683468 −0.341734 0.939797i \(-0.611014\pi\)
−0.341734 + 0.939797i \(0.611014\pi\)
\(234\) −17.4411 −1.14016
\(235\) 0 0
\(236\) −1.05324 −0.0685598
\(237\) 3.27191 0.212533
\(238\) −13.7409 −0.890691
\(239\) 1.58711 0.102661 0.0513307 0.998682i \(-0.483654\pi\)
0.0513307 + 0.998682i \(0.483654\pi\)
\(240\) 0 0
\(241\) 16.2576 1.04724 0.523622 0.851951i \(-0.324581\pi\)
0.523622 + 0.851951i \(0.324581\pi\)
\(242\) 11.7389 0.754604
\(243\) 18.8121 1.20680
\(244\) −4.14422 −0.265306
\(245\) 0 0
\(246\) 13.8398 0.882392
\(247\) 7.25154 0.461405
\(248\) 9.75286 0.619307
\(249\) −29.1911 −1.84991
\(250\) 0 0
\(251\) 14.9486 0.943547 0.471774 0.881720i \(-0.343614\pi\)
0.471774 + 0.881720i \(0.343614\pi\)
\(252\) −18.4469 −1.16205
\(253\) 7.35447 0.462372
\(254\) 8.33492 0.522979
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.9968 0.935474 0.467737 0.883868i \(-0.345069\pi\)
0.467737 + 0.883868i \(0.345069\pi\)
\(258\) 27.2835 1.69860
\(259\) −4.14336 −0.257456
\(260\) 0 0
\(261\) 39.5051 2.44530
\(262\) 3.68597 0.227720
\(263\) 22.9610 1.41583 0.707917 0.706295i \(-0.249635\pi\)
0.707917 + 0.706295i \(0.249635\pi\)
\(264\) 13.0174 0.801169
\(265\) 0 0
\(266\) 7.66975 0.470263
\(267\) −19.9163 −1.21885
\(268\) 1.00811 0.0615800
\(269\) 10.0896 0.615175 0.307588 0.951520i \(-0.400478\pi\)
0.307588 + 0.951520i \(0.400478\pi\)
\(270\) 0 0
\(271\) 3.30678 0.200872 0.100436 0.994943i \(-0.467976\pi\)
0.100436 + 0.994943i \(0.467976\pi\)
\(272\) 3.31637 0.201084
\(273\) −44.3095 −2.68173
\(274\) −12.3027 −0.743234
\(275\) 0 0
\(276\) 4.21025 0.253427
\(277\) −15.9046 −0.955617 −0.477809 0.878464i \(-0.658569\pi\)
−0.477809 + 0.878464i \(0.658569\pi\)
\(278\) −0.842442 −0.0505263
\(279\) 43.4213 2.59957
\(280\) 0 0
\(281\) 11.3609 0.677732 0.338866 0.940835i \(-0.389957\pi\)
0.338866 + 0.940835i \(0.389957\pi\)
\(282\) −13.1770 −0.784682
\(283\) 20.1530 1.19797 0.598984 0.800761i \(-0.295571\pi\)
0.598984 + 0.800761i \(0.295571\pi\)
\(284\) 6.45248 0.382884
\(285\) 0 0
\(286\) 18.6804 1.10460
\(287\) 21.0059 1.23994
\(288\) 4.45216 0.262346
\(289\) −6.00171 −0.353042
\(290\) 0 0
\(291\) −40.6717 −2.38422
\(292\) 10.7714 0.630349
\(293\) −21.0517 −1.22986 −0.614928 0.788583i \(-0.710815\pi\)
−0.614928 + 0.788583i \(0.710815\pi\)
\(294\) −27.7558 −1.61875
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 18.9035 1.09689
\(298\) −16.7723 −0.971591
\(299\) 6.04184 0.349408
\(300\) 0 0
\(301\) 41.4107 2.38687
\(302\) −7.59755 −0.437190
\(303\) −50.7490 −2.91545
\(304\) −1.85109 −0.106167
\(305\) 0 0
\(306\) 14.7650 0.844059
\(307\) 17.0946 0.975638 0.487819 0.872945i \(-0.337793\pi\)
0.487819 + 0.872945i \(0.337793\pi\)
\(308\) 19.7578 1.12580
\(309\) 36.3161 2.06595
\(310\) 0 0
\(311\) 20.0503 1.13695 0.568473 0.822702i \(-0.307534\pi\)
0.568473 + 0.822702i \(0.307534\pi\)
\(312\) 10.6941 0.605433
\(313\) 13.3400 0.754019 0.377010 0.926209i \(-0.376952\pi\)
0.377010 + 0.926209i \(0.376952\pi\)
\(314\) 4.45631 0.251484
\(315\) 0 0
\(316\) −1.19856 −0.0674243
\(317\) 12.8528 0.721885 0.360943 0.932588i \(-0.382455\pi\)
0.360943 + 0.932588i \(0.382455\pi\)
\(318\) 14.0209 0.786252
\(319\) −42.3123 −2.36903
\(320\) 0 0
\(321\) −34.0587 −1.90097
\(322\) 6.39028 0.356116
\(323\) −6.13890 −0.341578
\(324\) −2.53473 −0.140818
\(325\) 0 0
\(326\) 19.4599 1.07779
\(327\) 11.8871 0.657359
\(328\) −5.06977 −0.279931
\(329\) −20.0000 −1.10264
\(330\) 0 0
\(331\) −23.5835 −1.29627 −0.648134 0.761526i \(-0.724451\pi\)
−0.648134 + 0.761526i \(0.724451\pi\)
\(332\) 10.6932 0.586867
\(333\) 4.45216 0.243977
\(334\) 7.13813 0.390581
\(335\) 0 0
\(336\) 11.3108 0.617056
\(337\) 26.6074 1.44940 0.724698 0.689067i \(-0.241979\pi\)
0.724698 + 0.689067i \(0.241979\pi\)
\(338\) 2.34632 0.127623
\(339\) −26.0500 −1.41484
\(340\) 0 0
\(341\) −46.5068 −2.51848
\(342\) −8.24137 −0.445642
\(343\) −13.1239 −0.708626
\(344\) −9.99446 −0.538865
\(345\) 0 0
\(346\) −11.6199 −0.624692
\(347\) −17.2626 −0.926707 −0.463353 0.886174i \(-0.653354\pi\)
−0.463353 + 0.886174i \(0.653354\pi\)
\(348\) −24.2227 −1.29847
\(349\) 11.1619 0.597484 0.298742 0.954334i \(-0.403433\pi\)
0.298742 + 0.954334i \(0.403433\pi\)
\(350\) 0 0
\(351\) 15.5296 0.828906
\(352\) −4.76853 −0.254164
\(353\) −12.6563 −0.673628 −0.336814 0.941571i \(-0.609349\pi\)
−0.336814 + 0.941571i \(0.609349\pi\)
\(354\) 2.87519 0.152815
\(355\) 0 0
\(356\) 7.29569 0.386671
\(357\) 37.5108 1.98528
\(358\) −17.2224 −0.910232
\(359\) −23.2778 −1.22855 −0.614277 0.789091i \(-0.710552\pi\)
−0.614277 + 0.789091i \(0.710552\pi\)
\(360\) 0 0
\(361\) −15.5735 −0.819655
\(362\) −23.5098 −1.23565
\(363\) −32.0456 −1.68196
\(364\) 16.2314 0.850755
\(365\) 0 0
\(366\) 11.3132 0.591348
\(367\) −9.09417 −0.474712 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(368\) −1.54229 −0.0803976
\(369\) −22.5714 −1.17502
\(370\) 0 0
\(371\) 21.2808 1.10484
\(372\) −26.6240 −1.38039
\(373\) 5.75976 0.298229 0.149114 0.988820i \(-0.452358\pi\)
0.149114 + 0.988820i \(0.452358\pi\)
\(374\) −15.8142 −0.817733
\(375\) 0 0
\(376\) 4.82700 0.248933
\(377\) −34.7603 −1.79025
\(378\) 16.4252 0.844820
\(379\) 18.0902 0.929229 0.464614 0.885513i \(-0.346193\pi\)
0.464614 + 0.885513i \(0.346193\pi\)
\(380\) 0 0
\(381\) −22.7532 −1.16568
\(382\) 1.16660 0.0596887
\(383\) 3.54752 0.181270 0.0906350 0.995884i \(-0.471110\pi\)
0.0906350 + 0.995884i \(0.471110\pi\)
\(384\) −2.72987 −0.139308
\(385\) 0 0
\(386\) −9.36727 −0.476781
\(387\) −44.4970 −2.26191
\(388\) 14.8988 0.756371
\(389\) 25.6053 1.29824 0.649119 0.760687i \(-0.275138\pi\)
0.649119 + 0.760687i \(0.275138\pi\)
\(390\) 0 0
\(391\) −5.11481 −0.258667
\(392\) 10.1675 0.513534
\(393\) −10.0622 −0.507571
\(394\) 3.21836 0.162139
\(395\) 0 0
\(396\) −21.2303 −1.06686
\(397\) 31.4597 1.57892 0.789459 0.613803i \(-0.210361\pi\)
0.789459 + 0.613803i \(0.210361\pi\)
\(398\) −2.99104 −0.149927
\(399\) −20.9374 −1.04818
\(400\) 0 0
\(401\) −22.9940 −1.14826 −0.574132 0.818763i \(-0.694660\pi\)
−0.574132 + 0.818763i \(0.694660\pi\)
\(402\) −2.75200 −0.137257
\(403\) −38.2062 −1.90319
\(404\) 18.5903 0.924902
\(405\) 0 0
\(406\) −36.7650 −1.82462
\(407\) −4.76853 −0.236367
\(408\) −9.05324 −0.448202
\(409\) −33.3828 −1.65067 −0.825337 0.564640i \(-0.809015\pi\)
−0.825337 + 0.564640i \(0.809015\pi\)
\(410\) 0 0
\(411\) 33.5848 1.65661
\(412\) −13.3033 −0.655404
\(413\) 4.36394 0.214735
\(414\) −6.86654 −0.337472
\(415\) 0 0
\(416\) −3.91744 −0.192068
\(417\) 2.29975 0.112619
\(418\) 8.82700 0.431743
\(419\) 8.32565 0.406734 0.203367 0.979103i \(-0.434811\pi\)
0.203367 + 0.979103i \(0.434811\pi\)
\(420\) 0 0
\(421\) −8.41521 −0.410132 −0.205066 0.978748i \(-0.565741\pi\)
−0.205066 + 0.978748i \(0.565741\pi\)
\(422\) 4.75340 0.231392
\(423\) 21.4906 1.04491
\(424\) −5.13611 −0.249432
\(425\) 0 0
\(426\) −17.6144 −0.853420
\(427\) 17.1710 0.830963
\(428\) 12.4763 0.603066
\(429\) −50.9950 −2.46206
\(430\) 0 0
\(431\) 3.04769 0.146802 0.0734011 0.997303i \(-0.476615\pi\)
0.0734011 + 0.997303i \(0.476615\pi\)
\(432\) −3.96421 −0.190728
\(433\) 22.9439 1.10262 0.551308 0.834302i \(-0.314129\pi\)
0.551308 + 0.834302i \(0.314129\pi\)
\(434\) −40.4096 −1.93972
\(435\) 0 0
\(436\) −4.35447 −0.208541
\(437\) 2.85493 0.136570
\(438\) −29.4045 −1.40500
\(439\) 10.3033 0.491751 0.245875 0.969301i \(-0.420925\pi\)
0.245875 + 0.969301i \(0.420925\pi\)
\(440\) 0 0
\(441\) 45.2672 2.15558
\(442\) −12.9917 −0.617950
\(443\) −4.84236 −0.230067 −0.115034 0.993362i \(-0.536698\pi\)
−0.115034 + 0.993362i \(0.536698\pi\)
\(444\) −2.72987 −0.129554
\(445\) 0 0
\(446\) −6.95635 −0.329393
\(447\) 45.7860 2.16560
\(448\) −4.14336 −0.195756
\(449\) 25.3149 1.19468 0.597341 0.801987i \(-0.296224\pi\)
0.597341 + 0.801987i \(0.296224\pi\)
\(450\) 0 0
\(451\) 24.1753 1.13837
\(452\) 9.54261 0.448846
\(453\) 20.7403 0.974464
\(454\) −22.2980 −1.04650
\(455\) 0 0
\(456\) 5.05324 0.236639
\(457\) 11.9945 0.561077 0.280539 0.959843i \(-0.409487\pi\)
0.280539 + 0.959843i \(0.409487\pi\)
\(458\) 12.9648 0.605806
\(459\) −13.1468 −0.613639
\(460\) 0 0
\(461\) −23.6570 −1.10182 −0.550909 0.834565i \(-0.685719\pi\)
−0.550909 + 0.834565i \(0.685719\pi\)
\(462\) −53.9360 −2.50933
\(463\) −17.3667 −0.807099 −0.403550 0.914958i \(-0.632224\pi\)
−0.403550 + 0.914958i \(0.632224\pi\)
\(464\) 8.87323 0.411929
\(465\) 0 0
\(466\) −10.4327 −0.483285
\(467\) −20.4476 −0.946200 −0.473100 0.881009i \(-0.656865\pi\)
−0.473100 + 0.881009i \(0.656865\pi\)
\(468\) −17.4411 −0.806214
\(469\) −4.17696 −0.192874
\(470\) 0 0
\(471\) −12.1651 −0.560539
\(472\) −1.05324 −0.0484791
\(473\) 47.6589 2.19136
\(474\) 3.27191 0.150284
\(475\) 0 0
\(476\) −13.7409 −0.629814
\(477\) −22.8668 −1.04700
\(478\) 1.58711 0.0725925
\(479\) −34.1325 −1.55955 −0.779777 0.626057i \(-0.784668\pi\)
−0.779777 + 0.626057i \(0.784668\pi\)
\(480\) 0 0
\(481\) −3.91744 −0.178620
\(482\) 16.2576 0.740513
\(483\) −17.4446 −0.793757
\(484\) 11.7389 0.533586
\(485\) 0 0
\(486\) 18.8121 0.853334
\(487\) 6.16917 0.279552 0.139776 0.990183i \(-0.455362\pi\)
0.139776 + 0.990183i \(0.455362\pi\)
\(488\) −4.14422 −0.187600
\(489\) −53.1230 −2.40231
\(490\) 0 0
\(491\) −11.8290 −0.533836 −0.266918 0.963719i \(-0.586005\pi\)
−0.266918 + 0.963719i \(0.586005\pi\)
\(492\) 13.8398 0.623945
\(493\) 29.4269 1.32532
\(494\) 7.25154 0.326262
\(495\) 0 0
\(496\) 9.75286 0.437916
\(497\) −26.7350 −1.19923
\(498\) −29.1911 −1.30808
\(499\) 31.4004 1.40568 0.702839 0.711349i \(-0.251916\pi\)
0.702839 + 0.711349i \(0.251916\pi\)
\(500\) 0 0
\(501\) −19.4861 −0.870577
\(502\) 14.9486 0.667189
\(503\) 39.7743 1.77345 0.886724 0.462299i \(-0.152975\pi\)
0.886724 + 0.462299i \(0.152975\pi\)
\(504\) −18.4469 −0.821692
\(505\) 0 0
\(506\) 7.35447 0.326946
\(507\) −6.40514 −0.284462
\(508\) 8.33492 0.369802
\(509\) −6.82812 −0.302651 −0.151326 0.988484i \(-0.548354\pi\)
−0.151326 + 0.988484i \(0.548354\pi\)
\(510\) 0 0
\(511\) −44.6299 −1.97431
\(512\) 1.00000 0.0441942
\(513\) 7.33813 0.323986
\(514\) 14.9968 0.661480
\(515\) 0 0
\(516\) 27.2835 1.20109
\(517\) −23.0177 −1.01232
\(518\) −4.14336 −0.182049
\(519\) 31.7209 1.39239
\(520\) 0 0
\(521\) −30.6638 −1.34341 −0.671703 0.740821i \(-0.734437\pi\)
−0.671703 + 0.740821i \(0.734437\pi\)
\(522\) 39.5051 1.72909
\(523\) 12.1618 0.531800 0.265900 0.964001i \(-0.414331\pi\)
0.265900 + 0.964001i \(0.414331\pi\)
\(524\) 3.68597 0.161022
\(525\) 0 0
\(526\) 22.9610 1.00115
\(527\) 32.3441 1.40893
\(528\) 13.0174 0.566512
\(529\) −20.6213 −0.896580
\(530\) 0 0
\(531\) −4.68918 −0.203493
\(532\) 7.66975 0.332526
\(533\) 19.8605 0.860253
\(534\) −19.9163 −0.861861
\(535\) 0 0
\(536\) 1.00811 0.0435437
\(537\) 47.0148 2.02884
\(538\) 10.0896 0.434995
\(539\) −48.4839 −2.08835
\(540\) 0 0
\(541\) 10.5469 0.453446 0.226723 0.973959i \(-0.427199\pi\)
0.226723 + 0.973959i \(0.427199\pi\)
\(542\) 3.30678 0.142038
\(543\) 64.1785 2.75416
\(544\) 3.31637 0.142188
\(545\) 0 0
\(546\) −44.3095 −1.89627
\(547\) 4.09517 0.175097 0.0875484 0.996160i \(-0.472097\pi\)
0.0875484 + 0.996160i \(0.472097\pi\)
\(548\) −12.3027 −0.525546
\(549\) −18.4507 −0.787459
\(550\) 0 0
\(551\) −16.4252 −0.699736
\(552\) 4.21025 0.179200
\(553\) 4.96607 0.211179
\(554\) −15.9046 −0.675723
\(555\) 0 0
\(556\) −0.842442 −0.0357275
\(557\) −2.50711 −0.106230 −0.0531149 0.998588i \(-0.516915\pi\)
−0.0531149 + 0.998588i \(0.516915\pi\)
\(558\) 43.4213 1.83817
\(559\) 39.1527 1.65598
\(560\) 0 0
\(561\) 43.1706 1.82267
\(562\) 11.3609 0.479229
\(563\) 9.87532 0.416195 0.208097 0.978108i \(-0.433273\pi\)
0.208097 + 0.978108i \(0.433273\pi\)
\(564\) −13.1770 −0.554854
\(565\) 0 0
\(566\) 20.1530 0.847092
\(567\) 10.5023 0.441055
\(568\) 6.45248 0.270740
\(569\) 12.5198 0.524858 0.262429 0.964951i \(-0.415477\pi\)
0.262429 + 0.964951i \(0.415477\pi\)
\(570\) 0 0
\(571\) 20.3156 0.850179 0.425090 0.905151i \(-0.360243\pi\)
0.425090 + 0.905151i \(0.360243\pi\)
\(572\) 18.6804 0.781068
\(573\) −3.18467 −0.133042
\(574\) 21.0059 0.876769
\(575\) 0 0
\(576\) 4.45216 0.185507
\(577\) 20.0756 0.835759 0.417880 0.908502i \(-0.362773\pi\)
0.417880 + 0.908502i \(0.362773\pi\)
\(578\) −6.00171 −0.249638
\(579\) 25.5714 1.06271
\(580\) 0 0
\(581\) −44.3059 −1.83812
\(582\) −40.6717 −1.68590
\(583\) 24.4917 1.01434
\(584\) 10.7714 0.445724
\(585\) 0 0
\(586\) −21.0517 −0.869639
\(587\) 3.71178 0.153201 0.0766007 0.997062i \(-0.475593\pi\)
0.0766007 + 0.997062i \(0.475593\pi\)
\(588\) −27.7558 −1.14463
\(589\) −18.0534 −0.743879
\(590\) 0 0
\(591\) −8.78569 −0.361395
\(592\) 1.00000 0.0410997
\(593\) 36.2899 1.49025 0.745124 0.666926i \(-0.232390\pi\)
0.745124 + 0.666926i \(0.232390\pi\)
\(594\) 18.9035 0.775619
\(595\) 0 0
\(596\) −16.7723 −0.687019
\(597\) 8.16512 0.334176
\(598\) 6.04184 0.247069
\(599\) 23.7528 0.970515 0.485257 0.874371i \(-0.338726\pi\)
0.485257 + 0.874371i \(0.338726\pi\)
\(600\) 0 0
\(601\) 15.9718 0.651502 0.325751 0.945456i \(-0.394383\pi\)
0.325751 + 0.945456i \(0.394383\pi\)
\(602\) 41.4107 1.68777
\(603\) 4.48827 0.182776
\(604\) −7.59755 −0.309140
\(605\) 0 0
\(606\) −50.7490 −2.06154
\(607\) 38.0139 1.54294 0.771469 0.636267i \(-0.219522\pi\)
0.771469 + 0.636267i \(0.219522\pi\)
\(608\) −1.85109 −0.0750717
\(609\) 100.364 4.06694
\(610\) 0 0
\(611\) −18.9095 −0.764995
\(612\) 14.7650 0.596840
\(613\) −8.40204 −0.339355 −0.169678 0.985500i \(-0.554273\pi\)
−0.169678 + 0.985500i \(0.554273\pi\)
\(614\) 17.0946 0.689880
\(615\) 0 0
\(616\) 19.7578 0.796063
\(617\) −6.72707 −0.270822 −0.135411 0.990790i \(-0.543235\pi\)
−0.135411 + 0.990790i \(0.543235\pi\)
\(618\) 36.3161 1.46085
\(619\) 24.3689 0.979470 0.489735 0.871871i \(-0.337094\pi\)
0.489735 + 0.871871i \(0.337094\pi\)
\(620\) 0 0
\(621\) 6.11398 0.245345
\(622\) 20.0503 0.803943
\(623\) −30.2287 −1.21109
\(624\) 10.6941 0.428106
\(625\) 0 0
\(626\) 13.3400 0.533172
\(627\) −24.0965 −0.962322
\(628\) 4.45631 0.177826
\(629\) 3.31637 0.132232
\(630\) 0 0
\(631\) 34.7737 1.38432 0.692160 0.721744i \(-0.256659\pi\)
0.692160 + 0.721744i \(0.256659\pi\)
\(632\) −1.19856 −0.0476762
\(633\) −12.9761 −0.515755
\(634\) 12.8528 0.510450
\(635\) 0 0
\(636\) 14.0209 0.555964
\(637\) −39.8304 −1.57814
\(638\) −42.3123 −1.67516
\(639\) 28.7275 1.13644
\(640\) 0 0
\(641\) −27.8544 −1.10018 −0.550091 0.835105i \(-0.685407\pi\)
−0.550091 + 0.835105i \(0.685407\pi\)
\(642\) −34.0587 −1.34419
\(643\) −35.4013 −1.39609 −0.698045 0.716054i \(-0.745947\pi\)
−0.698045 + 0.716054i \(0.745947\pi\)
\(644\) 6.39028 0.251812
\(645\) 0 0
\(646\) −6.13890 −0.241532
\(647\) 16.1343 0.634305 0.317152 0.948375i \(-0.397273\pi\)
0.317152 + 0.948375i \(0.397273\pi\)
\(648\) −2.53473 −0.0995734
\(649\) 5.02239 0.197146
\(650\) 0 0
\(651\) 110.313 4.32350
\(652\) 19.4599 0.762110
\(653\) 11.9146 0.466256 0.233128 0.972446i \(-0.425104\pi\)
0.233128 + 0.972446i \(0.425104\pi\)
\(654\) 11.8871 0.464823
\(655\) 0 0
\(656\) −5.06977 −0.197941
\(657\) 47.9561 1.87095
\(658\) −20.0000 −0.779681
\(659\) −30.5124 −1.18859 −0.594297 0.804246i \(-0.702569\pi\)
−0.594297 + 0.804246i \(0.702569\pi\)
\(660\) 0 0
\(661\) 42.0973 1.63740 0.818698 0.574224i \(-0.194696\pi\)
0.818698 + 0.574224i \(0.194696\pi\)
\(662\) −23.5835 −0.916600
\(663\) 35.4655 1.37737
\(664\) 10.6932 0.414977
\(665\) 0 0
\(666\) 4.45216 0.172518
\(667\) −13.6851 −0.529890
\(668\) 7.13813 0.276183
\(669\) 18.9899 0.734192
\(670\) 0 0
\(671\) 19.7618 0.762897
\(672\) 11.3108 0.436324
\(673\) −41.1540 −1.58637 −0.793184 0.608982i \(-0.791578\pi\)
−0.793184 + 0.608982i \(0.791578\pi\)
\(674\) 26.6074 1.02488
\(675\) 0 0
\(676\) 2.34632 0.0902431
\(677\) −10.3914 −0.399373 −0.199686 0.979860i \(-0.563992\pi\)
−0.199686 + 0.979860i \(0.563992\pi\)
\(678\) −26.0500 −1.00045
\(679\) −61.7311 −2.36902
\(680\) 0 0
\(681\) 60.8706 2.33257
\(682\) −46.5068 −1.78084
\(683\) 5.05345 0.193365 0.0966824 0.995315i \(-0.469177\pi\)
0.0966824 + 0.995315i \(0.469177\pi\)
\(684\) −8.24137 −0.315117
\(685\) 0 0
\(686\) −13.1239 −0.501074
\(687\) −35.3922 −1.35030
\(688\) −9.99446 −0.381035
\(689\) 20.1204 0.766526
\(690\) 0 0
\(691\) 13.7857 0.524432 0.262216 0.965009i \(-0.415547\pi\)
0.262216 + 0.965009i \(0.415547\pi\)
\(692\) −11.6199 −0.441724
\(693\) 87.9648 3.34151
\(694\) −17.2626 −0.655280
\(695\) 0 0
\(696\) −24.2227 −0.918160
\(697\) −16.8132 −0.636846
\(698\) 11.1619 0.422485
\(699\) 28.4798 1.07721
\(700\) 0 0
\(701\) −2.20512 −0.0832862 −0.0416431 0.999133i \(-0.513259\pi\)
−0.0416431 + 0.999133i \(0.513259\pi\)
\(702\) 15.5296 0.586125
\(703\) −1.85109 −0.0698153
\(704\) −4.76853 −0.179721
\(705\) 0 0
\(706\) −12.6563 −0.476327
\(707\) −77.0264 −2.89687
\(708\) 2.87519 0.108056
\(709\) −12.7194 −0.477687 −0.238843 0.971058i \(-0.576768\pi\)
−0.238843 + 0.971058i \(0.576768\pi\)
\(710\) 0 0
\(711\) −5.33619 −0.200123
\(712\) 7.29569 0.273418
\(713\) −15.0418 −0.563318
\(714\) 37.5108 1.40381
\(715\) 0 0
\(716\) −17.2224 −0.643631
\(717\) −4.33258 −0.161803
\(718\) −23.2778 −0.868718
\(719\) 34.9654 1.30399 0.651996 0.758223i \(-0.273932\pi\)
0.651996 + 0.758223i \(0.273932\pi\)
\(720\) 0 0
\(721\) 55.1202 2.05278
\(722\) −15.5735 −0.579584
\(723\) −44.3810 −1.65055
\(724\) −23.5098 −0.873733
\(725\) 0 0
\(726\) −32.0456 −1.18932
\(727\) −9.90635 −0.367406 −0.183703 0.982982i \(-0.558809\pi\)
−0.183703 + 0.982982i \(0.558809\pi\)
\(728\) 16.2314 0.601575
\(729\) −43.7503 −1.62038
\(730\) 0 0
\(731\) −33.1453 −1.22592
\(732\) 11.3132 0.418146
\(733\) 37.3826 1.38076 0.690379 0.723448i \(-0.257444\pi\)
0.690379 + 0.723448i \(0.257444\pi\)
\(734\) −9.09417 −0.335672
\(735\) 0 0
\(736\) −1.54229 −0.0568497
\(737\) −4.80720 −0.177075
\(738\) −22.5714 −0.830866
\(739\) 14.4487 0.531506 0.265753 0.964041i \(-0.414380\pi\)
0.265753 + 0.964041i \(0.414380\pi\)
\(740\) 0 0
\(741\) −19.7957 −0.727215
\(742\) 21.2808 0.781242
\(743\) 45.1826 1.65759 0.828794 0.559553i \(-0.189027\pi\)
0.828794 + 0.559553i \(0.189027\pi\)
\(744\) −26.6240 −0.976083
\(745\) 0 0
\(746\) 5.75976 0.210880
\(747\) 47.6080 1.74188
\(748\) −15.8142 −0.578224
\(749\) −51.6939 −1.88886
\(750\) 0 0
\(751\) 49.3127 1.79945 0.899723 0.436461i \(-0.143768\pi\)
0.899723 + 0.436461i \(0.143768\pi\)
\(752\) 4.82700 0.176022
\(753\) −40.8077 −1.48711
\(754\) −34.7603 −1.26590
\(755\) 0 0
\(756\) 16.4252 0.597378
\(757\) 21.5911 0.784741 0.392370 0.919807i \(-0.371655\pi\)
0.392370 + 0.919807i \(0.371655\pi\)
\(758\) 18.0902 0.657064
\(759\) −20.0767 −0.728738
\(760\) 0 0
\(761\) 18.4253 0.667917 0.333959 0.942588i \(-0.391615\pi\)
0.333959 + 0.942588i \(0.391615\pi\)
\(762\) −22.7532 −0.824262
\(763\) 18.0422 0.653170
\(764\) 1.16660 0.0422063
\(765\) 0 0
\(766\) 3.54752 0.128177
\(767\) 4.12598 0.148981
\(768\) −2.72987 −0.0985055
\(769\) 6.50829 0.234695 0.117348 0.993091i \(-0.462561\pi\)
0.117348 + 0.993091i \(0.462561\pi\)
\(770\) 0 0
\(771\) −40.9392 −1.47439
\(772\) −9.36727 −0.337135
\(773\) −40.5853 −1.45975 −0.729875 0.683581i \(-0.760422\pi\)
−0.729875 + 0.683581i \(0.760422\pi\)
\(774\) −44.4970 −1.59941
\(775\) 0 0
\(776\) 14.8988 0.534835
\(777\) 11.3108 0.405774
\(778\) 25.6053 0.917993
\(779\) 9.38461 0.336239
\(780\) 0 0
\(781\) −30.7688 −1.10100
\(782\) −5.11481 −0.182905
\(783\) −35.1754 −1.25707
\(784\) 10.1675 0.363124
\(785\) 0 0
\(786\) −10.0622 −0.358907
\(787\) −37.5389 −1.33812 −0.669059 0.743210i \(-0.733302\pi\)
−0.669059 + 0.743210i \(0.733302\pi\)
\(788\) 3.21836 0.114649
\(789\) −62.6804 −2.23148
\(790\) 0 0
\(791\) −39.5385 −1.40583
\(792\) −21.2303 −0.754385
\(793\) 16.2347 0.576512
\(794\) 31.4597 1.11646
\(795\) 0 0
\(796\) −2.99104 −0.106014
\(797\) 28.8746 1.02279 0.511396 0.859345i \(-0.329129\pi\)
0.511396 + 0.859345i \(0.329129\pi\)
\(798\) −20.9374 −0.741176
\(799\) 16.0081 0.566326
\(800\) 0 0
\(801\) 32.4816 1.14768
\(802\) −22.9940 −0.811945
\(803\) −51.3638 −1.81259
\(804\) −2.75200 −0.0970556
\(805\) 0 0
\(806\) −38.2062 −1.34576
\(807\) −27.5433 −0.969571
\(808\) 18.5903 0.654004
\(809\) −8.98633 −0.315943 −0.157971 0.987444i \(-0.550495\pi\)
−0.157971 + 0.987444i \(0.550495\pi\)
\(810\) 0 0
\(811\) −24.4949 −0.860134 −0.430067 0.902797i \(-0.641510\pi\)
−0.430067 + 0.902797i \(0.641510\pi\)
\(812\) −36.7650 −1.29020
\(813\) −9.02706 −0.316593
\(814\) −4.76853 −0.167137
\(815\) 0 0
\(816\) −9.05324 −0.316927
\(817\) 18.5007 0.647257
\(818\) −33.3828 −1.16720
\(819\) 72.2647 2.52513
\(820\) 0 0
\(821\) 27.1346 0.947005 0.473502 0.880793i \(-0.342990\pi\)
0.473502 + 0.880793i \(0.342990\pi\)
\(822\) 33.5848 1.17140
\(823\) −10.7460 −0.374584 −0.187292 0.982304i \(-0.559971\pi\)
−0.187292 + 0.982304i \(0.559971\pi\)
\(824\) −13.3033 −0.463441
\(825\) 0 0
\(826\) 4.36394 0.151841
\(827\) 19.1035 0.664293 0.332147 0.943228i \(-0.392227\pi\)
0.332147 + 0.943228i \(0.392227\pi\)
\(828\) −6.86654 −0.238629
\(829\) 51.1250 1.77565 0.887823 0.460185i \(-0.152217\pi\)
0.887823 + 0.460185i \(0.152217\pi\)
\(830\) 0 0
\(831\) 43.4175 1.50614
\(832\) −3.91744 −0.135813
\(833\) 33.7190 1.16830
\(834\) 2.29975 0.0796340
\(835\) 0 0
\(836\) 8.82700 0.305288
\(837\) −38.6624 −1.33637
\(838\) 8.32565 0.287605
\(839\) −42.1947 −1.45672 −0.728361 0.685193i \(-0.759718\pi\)
−0.728361 + 0.685193i \(0.759718\pi\)
\(840\) 0 0
\(841\) 49.7342 1.71497
\(842\) −8.41521 −0.290007
\(843\) −31.0136 −1.06816
\(844\) 4.75340 0.163619
\(845\) 0 0
\(846\) 21.4906 0.738861
\(847\) −48.6385 −1.67124
\(848\) −5.13611 −0.176375
\(849\) −55.0148 −1.88810
\(850\) 0 0
\(851\) −1.54229 −0.0528691
\(852\) −17.6144 −0.603459
\(853\) 42.5489 1.45685 0.728423 0.685128i \(-0.240254\pi\)
0.728423 + 0.685128i \(0.240254\pi\)
\(854\) 17.1710 0.587580
\(855\) 0 0
\(856\) 12.4763 0.426432
\(857\) 29.7588 1.01654 0.508271 0.861197i \(-0.330285\pi\)
0.508271 + 0.861197i \(0.330285\pi\)
\(858\) −50.9950 −1.74094
\(859\) 8.40182 0.286666 0.143333 0.989674i \(-0.454218\pi\)
0.143333 + 0.989674i \(0.454218\pi\)
\(860\) 0 0
\(861\) −57.3432 −1.95425
\(862\) 3.04769 0.103805
\(863\) −36.3794 −1.23837 −0.619185 0.785245i \(-0.712537\pi\)
−0.619185 + 0.785245i \(0.712537\pi\)
\(864\) −3.96421 −0.134865
\(865\) 0 0
\(866\) 22.9439 0.779667
\(867\) 16.3839 0.556425
\(868\) −40.4096 −1.37159
\(869\) 5.71537 0.193881
\(870\) 0 0
\(871\) −3.94920 −0.133814
\(872\) −4.35447 −0.147461
\(873\) 66.3318 2.24499
\(874\) 2.85493 0.0965694
\(875\) 0 0
\(876\) −29.4045 −0.993486
\(877\) 41.3380 1.39588 0.697942 0.716154i \(-0.254099\pi\)
0.697942 + 0.716154i \(0.254099\pi\)
\(878\) 10.3033 0.347720
\(879\) 57.4684 1.93836
\(880\) 0 0
\(881\) 21.1354 0.712071 0.356035 0.934472i \(-0.384128\pi\)
0.356035 + 0.934472i \(0.384128\pi\)
\(882\) 45.2672 1.52423
\(883\) −10.7365 −0.361312 −0.180656 0.983546i \(-0.557822\pi\)
−0.180656 + 0.983546i \(0.557822\pi\)
\(884\) −12.9917 −0.436957
\(885\) 0 0
\(886\) −4.84236 −0.162682
\(887\) −11.1798 −0.375379 −0.187690 0.982228i \(-0.560100\pi\)
−0.187690 + 0.982228i \(0.560100\pi\)
\(888\) −2.72987 −0.0916083
\(889\) −34.5346 −1.15825
\(890\) 0 0
\(891\) 12.0869 0.404927
\(892\) −6.95635 −0.232916
\(893\) −8.93522 −0.299006
\(894\) 45.7860 1.53131
\(895\) 0 0
\(896\) −4.14336 −0.138420
\(897\) −16.4934 −0.550698
\(898\) 25.3149 0.844768
\(899\) 86.5393 2.88625
\(900\) 0 0
\(901\) −17.0332 −0.567459
\(902\) 24.1753 0.804951
\(903\) −113.046 −3.76192
\(904\) 9.54261 0.317382
\(905\) 0 0
\(906\) 20.7403 0.689050
\(907\) −20.5343 −0.681831 −0.340915 0.940094i \(-0.610737\pi\)
−0.340915 + 0.940094i \(0.610737\pi\)
\(908\) −22.2980 −0.739986
\(909\) 82.7671 2.74521
\(910\) 0 0
\(911\) 17.5197 0.580454 0.290227 0.956958i \(-0.406269\pi\)
0.290227 + 0.956958i \(0.406269\pi\)
\(912\) 5.05324 0.167329
\(913\) −50.9910 −1.68755
\(914\) 11.9945 0.396741
\(915\) 0 0
\(916\) 12.9648 0.428370
\(917\) −15.2723 −0.504336
\(918\) −13.1468 −0.433909
\(919\) 43.8760 1.44734 0.723668 0.690149i \(-0.242455\pi\)
0.723668 + 0.690149i \(0.242455\pi\)
\(920\) 0 0
\(921\) −46.6658 −1.53769
\(922\) −23.6570 −0.779103
\(923\) −25.2772 −0.832009
\(924\) −53.9360 −1.77436
\(925\) 0 0
\(926\) −17.3667 −0.570705
\(927\) −59.2283 −1.94531
\(928\) 8.87323 0.291278
\(929\) −21.9800 −0.721142 −0.360571 0.932732i \(-0.617418\pi\)
−0.360571 + 0.932732i \(0.617418\pi\)
\(930\) 0 0
\(931\) −18.8209 −0.616831
\(932\) −10.4327 −0.341734
\(933\) −54.7346 −1.79193
\(934\) −20.4476 −0.669065
\(935\) 0 0
\(936\) −17.4411 −0.570079
\(937\) −9.62735 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(938\) −4.17696 −0.136383
\(939\) −36.4163 −1.18840
\(940\) 0 0
\(941\) −28.3548 −0.924340 −0.462170 0.886791i \(-0.652929\pi\)
−0.462170 + 0.886791i \(0.652929\pi\)
\(942\) −12.1651 −0.396361
\(943\) 7.81906 0.254624
\(944\) −1.05324 −0.0342799
\(945\) 0 0
\(946\) 47.6589 1.54952
\(947\) −12.5179 −0.406778 −0.203389 0.979098i \(-0.565196\pi\)
−0.203389 + 0.979098i \(0.565196\pi\)
\(948\) 3.27191 0.106267
\(949\) −42.1963 −1.36975
\(950\) 0 0
\(951\) −35.0864 −1.13776
\(952\) −13.7409 −0.445346
\(953\) −3.39730 −0.110049 −0.0550247 0.998485i \(-0.517524\pi\)
−0.0550247 + 0.998485i \(0.517524\pi\)
\(954\) −22.8668 −0.740340
\(955\) 0 0
\(956\) 1.58711 0.0513307
\(957\) 115.507 3.73380
\(958\) −34.1325 −1.10277
\(959\) 50.9746 1.64606
\(960\) 0 0
\(961\) 64.1182 2.06833
\(962\) −3.91744 −0.126303
\(963\) 55.5466 1.78997
\(964\) 16.2576 0.523622
\(965\) 0 0
\(966\) −17.4446 −0.561271
\(967\) 2.54955 0.0819879 0.0409940 0.999159i \(-0.486948\pi\)
0.0409940 + 0.999159i \(0.486948\pi\)
\(968\) 11.7389 0.377302
\(969\) 16.7584 0.538357
\(970\) 0 0
\(971\) −35.2223 −1.13034 −0.565168 0.824976i \(-0.691189\pi\)
−0.565168 + 0.824976i \(0.691189\pi\)
\(972\) 18.8121 0.603398
\(973\) 3.49054 0.111902
\(974\) 6.16917 0.197673
\(975\) 0 0
\(976\) −4.14422 −0.132653
\(977\) −5.79786 −0.185490 −0.0927450 0.995690i \(-0.529564\pi\)
−0.0927450 + 0.995690i \(0.529564\pi\)
\(978\) −53.1230 −1.69869
\(979\) −34.7897 −1.11188
\(980\) 0 0
\(981\) −19.3868 −0.618973
\(982\) −11.8290 −0.377479
\(983\) −20.6825 −0.659671 −0.329835 0.944038i \(-0.606993\pi\)
−0.329835 + 0.944038i \(0.606993\pi\)
\(984\) 13.8398 0.441196
\(985\) 0 0
\(986\) 29.4269 0.937143
\(987\) 54.5973 1.73785
\(988\) 7.25154 0.230702
\(989\) 15.4144 0.490149
\(990\) 0 0
\(991\) 49.7158 1.57927 0.789637 0.613574i \(-0.210269\pi\)
0.789637 + 0.613574i \(0.210269\pi\)
\(992\) 9.75286 0.309654
\(993\) 64.3799 2.04303
\(994\) −26.7350 −0.847981
\(995\) 0 0
\(996\) −29.1911 −0.924954
\(997\) −53.1550 −1.68344 −0.841718 0.539918i \(-0.818455\pi\)
−0.841718 + 0.539918i \(0.818455\pi\)
\(998\) 31.4004 0.993964
\(999\) −3.96421 −0.125422
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.be.1.1 5
5.2 odd 4 370.2.b.d.149.10 yes 10
5.3 odd 4 370.2.b.d.149.1 10
5.4 even 2 1850.2.a.bd.1.5 5
15.2 even 4 3330.2.d.p.1999.5 10
15.8 even 4 3330.2.d.p.1999.10 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.1 10 5.3 odd 4
370.2.b.d.149.10 yes 10 5.2 odd 4
1850.2.a.bd.1.5 5 5.4 even 2
1850.2.a.be.1.1 5 1.1 even 1 trivial
3330.2.d.p.1999.5 10 15.2 even 4
3330.2.d.p.1999.10 10 15.8 even 4