Properties

Label 1850.2.a.y.1.2
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1524.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 1\)
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 \(0.140435\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.140435 q^{3} +1.00000 q^{4} +0.140435 q^{6} -1.14044 q^{7} -1.00000 q^{8} -2.98028 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.140435 q^{3} +1.00000 q^{4} +0.140435 q^{6} -1.14044 q^{7} -1.00000 q^{8} -2.98028 q^{9} +3.12071 q^{11} -0.140435 q^{12} -2.85956 q^{13} +1.14044 q^{14} +1.00000 q^{16} +2.14044 q^{17} +2.98028 q^{18} +4.26115 q^{19} +0.160157 q^{21} -3.12071 q^{22} -1.71913 q^{23} +0.140435 q^{24} +2.85956 q^{26} +0.839843 q^{27} -1.14044 q^{28} -4.28087 q^{29} +6.40158 q^{31} -1.00000 q^{32} -0.438259 q^{33} -2.14044 q^{34} -2.98028 q^{36} -1.00000 q^{37} -4.26115 q^{38} +0.401584 q^{39} -3.24143 q^{41} -0.160157 q^{42} -10.6994 q^{43} +3.12071 q^{44} +1.71913 q^{46} -0.140435 q^{48} -5.69941 q^{49} -0.300593 q^{51} -2.85956 q^{52} -7.96056 q^{53} -0.839843 q^{54} +1.14044 q^{56} -0.598416 q^{57} +4.28087 q^{58} +2.69941 q^{59} +4.57869 q^{61} -6.40158 q^{62} +3.39881 q^{63} +1.00000 q^{64} +0.438259 q^{66} -10.3819 q^{67} +2.14044 q^{68} +0.241427 q^{69} -7.10099 q^{71} +2.98028 q^{72} +0.261149 q^{73} +1.00000 q^{74} +4.26115 q^{76} -3.55897 q^{77} -0.401584 q^{78} -8.52230 q^{79} +8.82289 q^{81} +3.24143 q^{82} +1.16016 q^{83} +0.160157 q^{84} +10.6994 q^{86} +0.601186 q^{87} -3.12071 q^{88} -13.6430 q^{89} +3.26115 q^{91} -1.71913 q^{92} -0.899009 q^{93} +0.140435 q^{96} -14.2414 q^{97} +5.69941 q^{98} -9.30059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 6 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 6 q^{9} - 5 q^{11} - q^{12} - 8 q^{13} + 4 q^{14} + 3 q^{16} + 7 q^{17} - 6 q^{18} - q^{19} + 16 q^{21} + 5 q^{22} - 4 q^{23} + q^{24} + 8 q^{26} - 13 q^{27} - 4 q^{28} - 14 q^{29} + 6 q^{31} - 3 q^{32} + q^{33} - 7 q^{34} + 6 q^{36} - 3 q^{37} + q^{38} - 12 q^{39} + 19 q^{41} - 16 q^{42} - 16 q^{43} - 5 q^{44} + 4 q^{46} - q^{48} - q^{49} - 17 q^{51} - 8 q^{52} + 6 q^{53} + 13 q^{54} + 4 q^{56} - 15 q^{57} + 14 q^{58} - 8 q^{59} + 12 q^{61} - 6 q^{62} - 22 q^{63} + 3 q^{64} - q^{66} - 3 q^{67} + 7 q^{68} - 28 q^{69} + 8 q^{71} - 6 q^{72} - 13 q^{73} + 3 q^{74} - q^{76} + 6 q^{77} + 12 q^{78} + 2 q^{79} + 15 q^{81} - 19 q^{82} + 19 q^{83} + 16 q^{84} + 16 q^{86} + 34 q^{87} + 5 q^{88} + q^{89} - 4 q^{91} - 4 q^{92} - 32 q^{93} + q^{96} - 14 q^{97} + q^{98} - 44 q^{99} + O(q^{100}) \)

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
\(3\) −0.140435 −0.0810804 −0.0405402 0.999178i \(-0.512908\pi\)
−0.0405402 + 0.999178i \(0.512908\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0.140435 0.0573325
\(7\) −1.14044 −0.431044 −0.215522 0.976499i \(-0.569145\pi\)
−0.215522 + 0.976499i \(0.569145\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.98028 −0.993426
\(10\) 0 0
\(11\) 3.12071 0.940930 0.470465 0.882419i \(-0.344086\pi\)
0.470465 + 0.882419i \(0.344086\pi\)
\(12\) −0.140435 −0.0405402
\(13\) −2.85956 −0.793101 −0.396550 0.918013i \(-0.629793\pi\)
−0.396550 + 0.918013i \(0.629793\pi\)
\(14\) 1.14044 0.304794
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.14044 0.519132 0.259566 0.965725i \(-0.416421\pi\)
0.259566 + 0.965725i \(0.416421\pi\)
\(18\) 2.98028 0.702458
\(19\) 4.26115 0.977575 0.488787 0.872403i \(-0.337439\pi\)
0.488787 + 0.872403i \(0.337439\pi\)
\(20\) 0 0
\(21\) 0.160157 0.0349492
\(22\) −3.12071 −0.665338
\(23\) −1.71913 −0.358463 −0.179232 0.983807i \(-0.557361\pi\)
−0.179232 + 0.983807i \(0.557361\pi\)
\(24\) 0.140435 0.0286662
\(25\) 0 0
\(26\) 2.85956 0.560807
\(27\) 0.839843 0.161628
\(28\) −1.14044 −0.215522
\(29\) −4.28087 −0.794938 −0.397469 0.917616i \(-0.630111\pi\)
−0.397469 + 0.917616i \(0.630111\pi\)
\(30\) 0 0
\(31\) 6.40158 1.14976 0.574879 0.818238i \(-0.305049\pi\)
0.574879 + 0.818238i \(0.305049\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.438259 −0.0762910
\(34\) −2.14044 −0.367082
\(35\) 0 0
\(36\) −2.98028 −0.496713
\(37\) −1.00000 −0.164399
\(38\) −4.26115 −0.691250
\(39\) 0.401584 0.0643049
\(40\) 0 0
\(41\) −3.24143 −0.506226 −0.253113 0.967437i \(-0.581454\pi\)
−0.253113 + 0.967437i \(0.581454\pi\)
\(42\) −0.160157 −0.0247128
\(43\) −10.6994 −1.63164 −0.815822 0.578303i \(-0.803715\pi\)
−0.815822 + 0.578303i \(0.803715\pi\)
\(44\) 3.12071 0.470465
\(45\) 0 0
\(46\) 1.71913 0.253472
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −0.140435 −0.0202701
\(49\) −5.69941 −0.814201
\(50\) 0 0
\(51\) −0.300593 −0.0420914
\(52\) −2.85956 −0.396550
\(53\) −7.96056 −1.09347 −0.546733 0.837307i \(-0.684129\pi\)
−0.546733 + 0.837307i \(0.684129\pi\)
\(54\) −0.839843 −0.114288
\(55\) 0 0
\(56\) 1.14044 0.152397
\(57\) −0.598416 −0.0792621
\(58\) 4.28087 0.562106
\(59\) 2.69941 0.351433 0.175716 0.984441i \(-0.443776\pi\)
0.175716 + 0.984441i \(0.443776\pi\)
\(60\) 0 0
\(61\) 4.57869 0.586242 0.293121 0.956075i \(-0.405306\pi\)
0.293121 + 0.956075i \(0.405306\pi\)
\(62\) −6.40158 −0.813002
\(63\) 3.39881 0.428210
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.438259 0.0539459
\(67\) −10.3819 −1.26835 −0.634173 0.773191i \(-0.718659\pi\)
−0.634173 + 0.773191i \(0.718659\pi\)
\(68\) 2.14044 0.259566
\(69\) 0.241427 0.0290643
\(70\) 0 0
\(71\) −7.10099 −0.842733 −0.421366 0.906891i \(-0.638449\pi\)
−0.421366 + 0.906891i \(0.638449\pi\)
\(72\) 2.98028 0.351229
\(73\) 0.261149 0.0305651 0.0152826 0.999883i \(-0.495135\pi\)
0.0152826 + 0.999883i \(0.495135\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 4.26115 0.488787
\(77\) −3.55897 −0.405582
\(78\) −0.401584 −0.0454704
\(79\) −8.52230 −0.958833 −0.479417 0.877587i \(-0.659152\pi\)
−0.479417 + 0.877587i \(0.659152\pi\)
\(80\) 0 0
\(81\) 8.82289 0.980321
\(82\) 3.24143 0.357956
\(83\) 1.16016 0.127344 0.0636719 0.997971i \(-0.479719\pi\)
0.0636719 + 0.997971i \(0.479719\pi\)
\(84\) 0.160157 0.0174746
\(85\) 0 0
\(86\) 10.6994 1.15375
\(87\) 0.601186 0.0644539
\(88\) −3.12071 −0.332669
\(89\) −13.6430 −1.44616 −0.723078 0.690766i \(-0.757273\pi\)
−0.723078 + 0.690766i \(0.757273\pi\)
\(90\) 0 0
\(91\) 3.26115 0.341861
\(92\) −1.71913 −0.179232
\(93\) −0.899009 −0.0932229
\(94\) 0 0
\(95\) 0 0
\(96\) 0.140435 0.0143331
\(97\) −14.2414 −1.44600 −0.722999 0.690849i \(-0.757237\pi\)
−0.722999 + 0.690849i \(0.757237\pi\)
\(98\) 5.69941 0.575727
\(99\) −9.30059 −0.934745
\(100\) 0 0
\(101\) 13.9606 1.38913 0.694564 0.719431i \(-0.255597\pi\)
0.694564 + 0.719431i \(0.255597\pi\)
\(102\) 0.300593 0.0297631
\(103\) 10.3621 1.02101 0.510506 0.859874i \(-0.329458\pi\)
0.510506 + 0.859874i \(0.329458\pi\)
\(104\) 2.85956 0.280403
\(105\) 0 0
\(106\) 7.96056 0.773198
\(107\) −4.70218 −0.454577 −0.227288 0.973828i \(-0.572986\pi\)
−0.227288 + 0.973828i \(0.572986\pi\)
\(108\) 0.839843 0.0808139
\(109\) −5.96056 −0.570918 −0.285459 0.958391i \(-0.592146\pi\)
−0.285459 + 0.958391i \(0.592146\pi\)
\(110\) 0 0
\(111\) 0.140435 0.0133295
\(112\) −1.14044 −0.107761
\(113\) 4.83984 0.455294 0.227647 0.973744i \(-0.426897\pi\)
0.227647 + 0.973744i \(0.426897\pi\)
\(114\) 0.598416 0.0560468
\(115\) 0 0
\(116\) −4.28087 −0.397469
\(117\) 8.52230 0.787887
\(118\) −2.69941 −0.248501
\(119\) −2.44103 −0.223769
\(120\) 0 0
\(121\) −1.26115 −0.114650
\(122\) −4.57869 −0.414535
\(123\) 0.455211 0.0410450
\(124\) 6.40158 0.574879
\(125\) 0 0
\(126\) −3.39881 −0.302790
\(127\) −0.899009 −0.0797741 −0.0398871 0.999204i \(-0.512700\pi\)
−0.0398871 + 0.999204i \(0.512700\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.50258 0.132294
\(130\) 0 0
\(131\) −0.980278 −0.0856473 −0.0428236 0.999083i \(-0.513635\pi\)
−0.0428236 + 0.999083i \(0.513635\pi\)
\(132\) −0.438259 −0.0381455
\(133\) −4.85956 −0.421378
\(134\) 10.3819 0.896856
\(135\) 0 0
\(136\) −2.14044 −0.183541
\(137\) 6.67969 0.570684 0.285342 0.958426i \(-0.407893\pi\)
0.285342 + 0.958426i \(0.407893\pi\)
\(138\) −0.241427 −0.0205516
\(139\) −9.68245 −0.821255 −0.410628 0.911803i \(-0.634690\pi\)
−0.410628 + 0.911803i \(0.634690\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.10099 0.595902
\(143\) −8.92388 −0.746252
\(144\) −2.98028 −0.248356
\(145\) 0 0
\(146\) −0.261149 −0.0216128
\(147\) 0.800398 0.0660157
\(148\) −1.00000 −0.0821995
\(149\) −1.59842 −0.130947 −0.0654737 0.997854i \(-0.520856\pi\)
−0.0654737 + 0.997854i \(0.520856\pi\)
\(150\) 0 0
\(151\) −16.4829 −1.34136 −0.670678 0.741749i \(-0.733997\pi\)
−0.670678 + 0.741749i \(0.733997\pi\)
\(152\) −4.26115 −0.345625
\(153\) −6.37909 −0.515719
\(154\) 3.55897 0.286790
\(155\) 0 0
\(156\) 0.401584 0.0321525
\(157\) 10.1207 0.807721 0.403860 0.914821i \(-0.367668\pi\)
0.403860 + 0.914821i \(0.367668\pi\)
\(158\) 8.52230 0.677998
\(159\) 1.11794 0.0886587
\(160\) 0 0
\(161\) 1.96056 0.154513
\(162\) −8.82289 −0.693192
\(163\) −11.7389 −0.919458 −0.459729 0.888059i \(-0.652053\pi\)
−0.459729 + 0.888059i \(0.652053\pi\)
\(164\) −3.24143 −0.253113
\(165\) 0 0
\(166\) −1.16016 −0.0900457
\(167\) −10.6430 −0.823581 −0.411790 0.911279i \(-0.635096\pi\)
−0.411790 + 0.911279i \(0.635096\pi\)
\(168\) −0.160157 −0.0123564
\(169\) −4.82289 −0.370992
\(170\) 0 0
\(171\) −12.6994 −0.971148
\(172\) −10.6994 −0.815822
\(173\) 11.3988 0.866636 0.433318 0.901241i \(-0.357343\pi\)
0.433318 + 0.901241i \(0.357343\pi\)
\(174\) −0.601186 −0.0455758
\(175\) 0 0
\(176\) 3.12071 0.235233
\(177\) −0.379092 −0.0284943
\(178\) 13.6430 1.02259
\(179\) −18.9211 −1.41423 −0.707115 0.707098i \(-0.750004\pi\)
−0.707115 + 0.707098i \(0.750004\pi\)
\(180\) 0 0
\(181\) −18.8032 −1.39763 −0.698814 0.715303i \(-0.746289\pi\)
−0.698814 + 0.715303i \(0.746289\pi\)
\(182\) −3.26115 −0.241732
\(183\) −0.643011 −0.0475327
\(184\) 1.71913 0.126736
\(185\) 0 0
\(186\) 0.899009 0.0659185
\(187\) 6.67969 0.488467
\(188\) 0 0
\(189\) −0.957786 −0.0696687
\(190\) 0 0
\(191\) 15.0446 1.08859 0.544294 0.838894i \(-0.316797\pi\)
0.544294 + 0.838894i \(0.316797\pi\)
\(192\) −0.140435 −0.0101350
\(193\) −25.3227 −1.82277 −0.911384 0.411558i \(-0.864985\pi\)
−0.911384 + 0.411558i \(0.864985\pi\)
\(194\) 14.2414 1.02247
\(195\) 0 0
\(196\) −5.69941 −0.407101
\(197\) 2.68245 0.191117 0.0955585 0.995424i \(-0.469536\pi\)
0.0955585 + 0.995424i \(0.469536\pi\)
\(198\) 9.30059 0.660964
\(199\) 2.24143 0.158891 0.0794453 0.996839i \(-0.474685\pi\)
0.0794453 + 0.996839i \(0.474685\pi\)
\(200\) 0 0
\(201\) 1.45798 0.102838
\(202\) −13.9606 −0.982261
\(203\) 4.88206 0.342653
\(204\) −0.300593 −0.0210457
\(205\) 0 0
\(206\) −10.3621 −0.721964
\(207\) 5.12348 0.356107
\(208\) −2.85956 −0.198275
\(209\) 13.2978 0.919830
\(210\) 0 0
\(211\) −1.61814 −0.111397 −0.0556986 0.998448i \(-0.517739\pi\)
−0.0556986 + 0.998448i \(0.517739\pi\)
\(212\) −7.96056 −0.546733
\(213\) 0.997230 0.0683291
\(214\) 4.70218 0.321434
\(215\) 0 0
\(216\) −0.839843 −0.0571440
\(217\) −7.30059 −0.495597
\(218\) 5.96056 0.403700
\(219\) −0.0366745 −0.00247823
\(220\) 0 0
\(221\) −6.12071 −0.411724
\(222\) −0.140435 −0.00942540
\(223\) 3.39881 0.227601 0.113801 0.993504i \(-0.463697\pi\)
0.113801 + 0.993504i \(0.463697\pi\)
\(224\) 1.14044 0.0761985
\(225\) 0 0
\(226\) −4.83984 −0.321942
\(227\) −2.45798 −0.163142 −0.0815710 0.996668i \(-0.525994\pi\)
−0.0815710 + 0.996668i \(0.525994\pi\)
\(228\) −0.598416 −0.0396311
\(229\) 12.7637 0.843451 0.421725 0.906724i \(-0.361425\pi\)
0.421725 + 0.906724i \(0.361425\pi\)
\(230\) 0 0
\(231\) 0.499806 0.0328848
\(232\) 4.28087 0.281053
\(233\) 0.103761 0.00679760 0.00339880 0.999994i \(-0.498918\pi\)
0.00339880 + 0.999994i \(0.498918\pi\)
\(234\) −8.52230 −0.557120
\(235\) 0 0
\(236\) 2.69941 0.175716
\(237\) 1.19683 0.0777426
\(238\) 2.44103 0.158228
\(239\) −3.67969 −0.238019 −0.119010 0.992893i \(-0.537972\pi\)
−0.119010 + 0.992893i \(0.537972\pi\)
\(240\) 0 0
\(241\) −6.14044 −0.395540 −0.197770 0.980248i \(-0.563370\pi\)
−0.197770 + 0.980248i \(0.563370\pi\)
\(242\) 1.26115 0.0810697
\(243\) −3.75857 −0.241113
\(244\) 4.57869 0.293121
\(245\) 0 0
\(246\) −0.455211 −0.0290232
\(247\) −12.1850 −0.775315
\(248\) −6.40158 −0.406501
\(249\) −0.162927 −0.0103251
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 3.39881 0.214105
\(253\) −5.36491 −0.337289
\(254\) 0.899009 0.0564088
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.5223 1.03063 0.515316 0.857000i \(-0.327674\pi\)
0.515316 + 0.857000i \(0.327674\pi\)
\(258\) −1.50258 −0.0935462
\(259\) 1.14044 0.0708632
\(260\) 0 0
\(261\) 12.7582 0.789712
\(262\) 0.980278 0.0605618
\(263\) 22.5223 1.38878 0.694392 0.719597i \(-0.255673\pi\)
0.694392 + 0.719597i \(0.255673\pi\)
\(264\) 0.438259 0.0269729
\(265\) 0 0
\(266\) 4.85956 0.297959
\(267\) 1.91596 0.117255
\(268\) −10.3819 −0.634173
\(269\) 27.2860 1.66366 0.831829 0.555032i \(-0.187294\pi\)
0.831829 + 0.555032i \(0.187294\pi\)
\(270\) 0 0
\(271\) −26.1456 −1.58823 −0.794116 0.607767i \(-0.792066\pi\)
−0.794116 + 0.607767i \(0.792066\pi\)
\(272\) 2.14044 0.129783
\(273\) −0.457981 −0.0277182
\(274\) −6.67969 −0.403535
\(275\) 0 0
\(276\) 0.241427 0.0145322
\(277\) 20.5223 1.23307 0.616533 0.787329i \(-0.288537\pi\)
0.616533 + 0.787329i \(0.288537\pi\)
\(278\) 9.68245 0.580715
\(279\) −19.0785 −1.14220
\(280\) 0 0
\(281\) −14.4580 −0.862491 −0.431245 0.902235i \(-0.641926\pi\)
−0.431245 + 0.902235i \(0.641926\pi\)
\(282\) 0 0
\(283\) −6.85679 −0.407594 −0.203797 0.979013i \(-0.565328\pi\)
−0.203797 + 0.979013i \(0.565328\pi\)
\(284\) −7.10099 −0.421366
\(285\) 0 0
\(286\) 8.92388 0.527680
\(287\) 3.69664 0.218206
\(288\) 2.98028 0.175615
\(289\) −12.4185 −0.730502
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0.261149 0.0152826
\(293\) 19.2048 1.12195 0.560977 0.827832i \(-0.310426\pi\)
0.560977 + 0.827832i \(0.310426\pi\)
\(294\) −0.800398 −0.0466802
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 2.62091 0.152080
\(298\) 1.59842 0.0925938
\(299\) 4.91596 0.284297
\(300\) 0 0
\(301\) 12.2020 0.703311
\(302\) 16.4829 0.948482
\(303\) −1.96056 −0.112631
\(304\) 4.26115 0.244394
\(305\) 0 0
\(306\) 6.37909 0.364668
\(307\) −16.2781 −0.929040 −0.464520 0.885563i \(-0.653773\pi\)
−0.464520 + 0.885563i \(0.653773\pi\)
\(308\) −3.55897 −0.202791
\(309\) −1.45521 −0.0827841
\(310\) 0 0
\(311\) −25.4801 −1.44484 −0.722421 0.691453i \(-0.756971\pi\)
−0.722421 + 0.691453i \(0.756971\pi\)
\(312\) −0.401584 −0.0227352
\(313\) −19.0197 −1.07506 −0.537529 0.843245i \(-0.680642\pi\)
−0.537529 + 0.843245i \(0.680642\pi\)
\(314\) −10.1207 −0.571145
\(315\) 0 0
\(316\) −8.52230 −0.479417
\(317\) −15.1653 −0.851769 −0.425884 0.904778i \(-0.640037\pi\)
−0.425884 + 0.904778i \(0.640037\pi\)
\(318\) −1.11794 −0.0626912
\(319\) −13.3594 −0.747981
\(320\) 0 0
\(321\) 0.660352 0.0368573
\(322\) −1.96056 −0.109258
\(323\) 9.12071 0.507490
\(324\) 8.82289 0.490161
\(325\) 0 0
\(326\) 11.7389 0.650155
\(327\) 0.837073 0.0462902
\(328\) 3.24143 0.178978
\(329\) 0 0
\(330\) 0 0
\(331\) −23.4185 −1.28720 −0.643600 0.765362i \(-0.722560\pi\)
−0.643600 + 0.765362i \(0.722560\pi\)
\(332\) 1.16016 0.0636719
\(333\) 2.98028 0.163318
\(334\) 10.6430 0.582360
\(335\) 0 0
\(336\) 0.160157 0.00873731
\(337\) 28.9211 1.57543 0.787717 0.616038i \(-0.211263\pi\)
0.787717 + 0.616038i \(0.211263\pi\)
\(338\) 4.82289 0.262331
\(339\) −0.679685 −0.0369154
\(340\) 0 0
\(341\) 19.9775 1.08184
\(342\) 12.6994 0.686705
\(343\) 14.4829 0.782001
\(344\) 10.6994 0.576873
\(345\) 0 0
\(346\) −11.3988 −0.612804
\(347\) 32.8817 1.76518 0.882590 0.470143i \(-0.155798\pi\)
0.882590 + 0.470143i \(0.155798\pi\)
\(348\) 0.601186 0.0322269
\(349\) 15.2048 0.813892 0.406946 0.913452i \(-0.366594\pi\)
0.406946 + 0.913452i \(0.366594\pi\)
\(350\) 0 0
\(351\) −2.40158 −0.128187
\(352\) −3.12071 −0.166335
\(353\) 18.4829 0.983743 0.491872 0.870668i \(-0.336313\pi\)
0.491872 + 0.870668i \(0.336313\pi\)
\(354\) 0.379092 0.0201485
\(355\) 0 0
\(356\) −13.6430 −0.723078
\(357\) 0.342807 0.0181433
\(358\) 18.9211 1.00001
\(359\) 22.5223 1.18868 0.594341 0.804213i \(-0.297413\pi\)
0.594341 + 0.804213i \(0.297413\pi\)
\(360\) 0 0
\(361\) −0.842612 −0.0443480
\(362\) 18.8032 0.988273
\(363\) 0.177110 0.00929586
\(364\) 3.26115 0.170931
\(365\) 0 0
\(366\) 0.643011 0.0336107
\(367\) −15.3424 −0.800868 −0.400434 0.916326i \(-0.631141\pi\)
−0.400434 + 0.916326i \(0.631141\pi\)
\(368\) −1.71913 −0.0896158
\(369\) 9.66035 0.502898
\(370\) 0 0
\(371\) 9.07850 0.471332
\(372\) −0.899009 −0.0466114
\(373\) −12.6430 −0.654630 −0.327315 0.944915i \(-0.606144\pi\)
−0.327315 + 0.944915i \(0.606144\pi\)
\(374\) −6.67969 −0.345398
\(375\) 0 0
\(376\) 0 0
\(377\) 12.2414 0.630466
\(378\) 0.957786 0.0492632
\(379\) 15.7807 0.810599 0.405299 0.914184i \(-0.367167\pi\)
0.405299 + 0.914184i \(0.367167\pi\)
\(380\) 0 0
\(381\) 0.126253 0.00646812
\(382\) −15.0446 −0.769748
\(383\) 27.8004 1.42053 0.710267 0.703932i \(-0.248574\pi\)
0.710267 + 0.703932i \(0.248574\pi\)
\(384\) 0.140435 0.00716656
\(385\) 0 0
\(386\) 25.3227 1.28889
\(387\) 31.8872 1.62092
\(388\) −14.2414 −0.722999
\(389\) 9.66273 0.489920 0.244960 0.969533i \(-0.421225\pi\)
0.244960 + 0.969533i \(0.421225\pi\)
\(390\) 0 0
\(391\) −3.67969 −0.186090
\(392\) 5.69941 0.287864
\(393\) 0.137666 0.00694432
\(394\) −2.68245 −0.135140
\(395\) 0 0
\(396\) −9.30059 −0.467372
\(397\) 23.4801 1.17843 0.589216 0.807976i \(-0.299437\pi\)
0.589216 + 0.807976i \(0.299437\pi\)
\(398\) −2.24143 −0.112353
\(399\) 0.682455 0.0341655
\(400\) 0 0
\(401\) 20.2781 1.01264 0.506320 0.862346i \(-0.331005\pi\)
0.506320 + 0.862346i \(0.331005\pi\)
\(402\) −1.45798 −0.0727175
\(403\) −18.3057 −0.911874
\(404\) 13.9606 0.694564
\(405\) 0 0
\(406\) −4.88206 −0.242292
\(407\) −3.12071 −0.154688
\(408\) 0.300593 0.0148816
\(409\) 4.17434 0.206408 0.103204 0.994660i \(-0.467091\pi\)
0.103204 + 0.994660i \(0.467091\pi\)
\(410\) 0 0
\(411\) −0.938064 −0.0462713
\(412\) 10.3621 0.510506
\(413\) −3.07850 −0.151483
\(414\) −5.12348 −0.251805
\(415\) 0 0
\(416\) 2.85956 0.140202
\(417\) 1.35976 0.0665877
\(418\) −13.2978 −0.650418
\(419\) −12.3175 −0.601751 −0.300876 0.953663i \(-0.597279\pi\)
−0.300876 + 0.953663i \(0.597279\pi\)
\(420\) 0 0
\(421\) 23.8872 1.16419 0.582096 0.813120i \(-0.302233\pi\)
0.582096 + 0.813120i \(0.302233\pi\)
\(422\) 1.61814 0.0787697
\(423\) 0 0
\(424\) 7.96056 0.386599
\(425\) 0 0
\(426\) −0.997230 −0.0483160
\(427\) −5.22170 −0.252696
\(428\) −4.70218 −0.227288
\(429\) 1.25323 0.0605064
\(430\) 0 0
\(431\) −2.80317 −0.135024 −0.0675119 0.997718i \(-0.521506\pi\)
−0.0675119 + 0.997718i \(0.521506\pi\)
\(432\) 0.839843 0.0404069
\(433\) −5.00000 −0.240285 −0.120142 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) 7.30059 0.350440
\(435\) 0 0
\(436\) −5.96056 −0.285459
\(437\) −7.32547 −0.350425
\(438\) 0.0366745 0.00175238
\(439\) 30.7976 1.46989 0.734945 0.678126i \(-0.237208\pi\)
0.734945 + 0.678126i \(0.237208\pi\)
\(440\) 0 0
\(441\) 16.9858 0.808848
\(442\) 6.12071 0.291133
\(443\) 16.5984 0.788615 0.394307 0.918979i \(-0.370985\pi\)
0.394307 + 0.918979i \(0.370985\pi\)
\(444\) 0.140435 0.00666477
\(445\) 0 0
\(446\) −3.39881 −0.160939
\(447\) 0.224474 0.0106173
\(448\) −1.14044 −0.0538805
\(449\) 9.75580 0.460405 0.230202 0.973143i \(-0.426061\pi\)
0.230202 + 0.973143i \(0.426061\pi\)
\(450\) 0 0
\(451\) −10.1156 −0.476323
\(452\) 4.83984 0.227647
\(453\) 2.31478 0.108758
\(454\) 2.45798 0.115359
\(455\) 0 0
\(456\) 0.598416 0.0280234
\(457\) 7.99723 0.374095 0.187047 0.982351i \(-0.440108\pi\)
0.187047 + 0.982351i \(0.440108\pi\)
\(458\) −12.7637 −0.596410
\(459\) 1.79763 0.0839061
\(460\) 0 0
\(461\) −20.6848 −0.963389 −0.481694 0.876339i \(-0.659978\pi\)
−0.481694 + 0.876339i \(0.659978\pi\)
\(462\) −0.499806 −0.0232531
\(463\) 7.52507 0.349720 0.174860 0.984593i \(-0.444053\pi\)
0.174860 + 0.984593i \(0.444053\pi\)
\(464\) −4.28087 −0.198734
\(465\) 0 0
\(466\) −0.103761 −0.00480663
\(467\) −6.84261 −0.316638 −0.158319 0.987388i \(-0.550607\pi\)
−0.158319 + 0.987388i \(0.550607\pi\)
\(468\) 8.52230 0.393943
\(469\) 11.8398 0.546713
\(470\) 0 0
\(471\) −1.42131 −0.0654903
\(472\) −2.69941 −0.124250
\(473\) −33.3898 −1.53526
\(474\) −1.19683 −0.0549723
\(475\) 0 0
\(476\) −2.44103 −0.111884
\(477\) 23.7247 1.08628
\(478\) 3.67969 0.168305
\(479\) −16.4016 −0.749408 −0.374704 0.927145i \(-0.622256\pi\)
−0.374704 + 0.927145i \(0.622256\pi\)
\(480\) 0 0
\(481\) 2.85956 0.130385
\(482\) 6.14044 0.279689
\(483\) −0.275331 −0.0125280
\(484\) −1.26115 −0.0573249
\(485\) 0 0
\(486\) 3.75857 0.170492
\(487\) −13.7270 −0.622032 −0.311016 0.950405i \(-0.600669\pi\)
−0.311016 + 0.950405i \(0.600669\pi\)
\(488\) −4.57869 −0.207268
\(489\) 1.64855 0.0745500
\(490\) 0 0
\(491\) 3.40435 0.153636 0.0768182 0.997045i \(-0.475524\pi\)
0.0768182 + 0.997045i \(0.475524\pi\)
\(492\) 0.455211 0.0205225
\(493\) −9.16293 −0.412677
\(494\) 12.1850 0.548230
\(495\) 0 0
\(496\) 6.40158 0.287440
\(497\) 8.09822 0.363255
\(498\) 0.162927 0.00730094
\(499\) −29.4631 −1.31895 −0.659475 0.751726i \(-0.729222\pi\)
−0.659475 + 0.751726i \(0.729222\pi\)
\(500\) 0 0
\(501\) 1.49466 0.0667763
\(502\) 9.00000 0.401690
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −3.39881 −0.151395
\(505\) 0 0
\(506\) 5.36491 0.238499
\(507\) 0.677304 0.0300801
\(508\) −0.899009 −0.0398871
\(509\) 12.9633 0.574589 0.287295 0.957842i \(-0.407244\pi\)
0.287295 + 0.957842i \(0.407244\pi\)
\(510\) 0 0
\(511\) −0.297823 −0.0131749
\(512\) −1.00000 −0.0441942
\(513\) 3.57869 0.158003
\(514\) −16.5223 −0.728767
\(515\) 0 0
\(516\) 1.50258 0.0661472
\(517\) 0 0
\(518\) −1.14044 −0.0501079
\(519\) −1.60080 −0.0702672
\(520\) 0 0
\(521\) −2.01972 −0.0884856 −0.0442428 0.999021i \(-0.514088\pi\)
−0.0442428 + 0.999021i \(0.514088\pi\)
\(522\) −12.7582 −0.558411
\(523\) −15.5562 −0.680225 −0.340113 0.940385i \(-0.610465\pi\)
−0.340113 + 0.940385i \(0.610465\pi\)
\(524\) −0.980278 −0.0428236
\(525\) 0 0
\(526\) −22.5223 −0.982019
\(527\) 13.7022 0.596876
\(528\) −0.438259 −0.0190728
\(529\) −20.0446 −0.871504
\(530\) 0 0
\(531\) −8.04498 −0.349123
\(532\) −4.85956 −0.210689
\(533\) 9.26907 0.401488
\(534\) −1.91596 −0.0829118
\(535\) 0 0
\(536\) 10.3819 0.448428
\(537\) 2.65719 0.114666
\(538\) −27.2860 −1.17638
\(539\) −17.7862 −0.766107
\(540\) 0 0
\(541\) −0.544789 −0.0234223 −0.0117112 0.999931i \(-0.503728\pi\)
−0.0117112 + 0.999931i \(0.503728\pi\)
\(542\) 26.1456 1.12305
\(543\) 2.64063 0.113320
\(544\) −2.14044 −0.0917704
\(545\) 0 0
\(546\) 0.457981 0.0195998
\(547\) 35.6261 1.52326 0.761630 0.648012i \(-0.224399\pi\)
0.761630 + 0.648012i \(0.224399\pi\)
\(548\) 6.67969 0.285342
\(549\) −13.6458 −0.582388
\(550\) 0 0
\(551\) −18.2414 −0.777111
\(552\) −0.241427 −0.0102758
\(553\) 9.71913 0.413299
\(554\) −20.5223 −0.871909
\(555\) 0 0
\(556\) −9.68245 −0.410628
\(557\) −31.9606 −1.35421 −0.677106 0.735885i \(-0.736766\pi\)
−0.677106 + 0.735885i \(0.736766\pi\)
\(558\) 19.0785 0.807657
\(559\) 30.5956 1.29406
\(560\) 0 0
\(561\) −0.938064 −0.0396051
\(562\) 14.4580 0.609873
\(563\) −14.6655 −0.618077 −0.309039 0.951049i \(-0.600007\pi\)
−0.309039 + 0.951049i \(0.600007\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6.85679 0.288213
\(567\) −10.0619 −0.422562
\(568\) 7.10099 0.297951
\(569\) 9.99723 0.419106 0.209553 0.977797i \(-0.432799\pi\)
0.209553 + 0.977797i \(0.432799\pi\)
\(570\) 0 0
\(571\) 27.6008 1.15506 0.577529 0.816370i \(-0.304017\pi\)
0.577529 + 0.816370i \(0.304017\pi\)
\(572\) −8.92388 −0.373126
\(573\) −2.11279 −0.0882632
\(574\) −3.69664 −0.154295
\(575\) 0 0
\(576\) −2.98028 −0.124178
\(577\) 3.78622 0.157622 0.0788111 0.996890i \(-0.474888\pi\)
0.0788111 + 0.996890i \(0.474888\pi\)
\(578\) 12.4185 0.516543
\(579\) 3.55620 0.147791
\(580\) 0 0
\(581\) −1.32308 −0.0548908
\(582\) −2.00000 −0.0829027
\(583\) −24.8426 −1.02888
\(584\) −0.261149 −0.0108064
\(585\) 0 0
\(586\) −19.2048 −0.793341
\(587\) −19.1771 −0.791524 −0.395762 0.918353i \(-0.629519\pi\)
−0.395762 + 0.918353i \(0.629519\pi\)
\(588\) 0.800398 0.0330079
\(589\) 27.2781 1.12397
\(590\) 0 0
\(591\) −0.376712 −0.0154958
\(592\) −1.00000 −0.0410997
\(593\) 12.4383 0.510778 0.255389 0.966838i \(-0.417796\pi\)
0.255389 + 0.966838i \(0.417796\pi\)
\(594\) −2.62091 −0.107537
\(595\) 0 0
\(596\) −1.59842 −0.0654737
\(597\) −0.314776 −0.0128829
\(598\) −4.91596 −0.201029
\(599\) −35.3819 −1.44566 −0.722832 0.691024i \(-0.757160\pi\)
−0.722832 + 0.691024i \(0.757160\pi\)
\(600\) 0 0
\(601\) 22.2951 0.909434 0.454717 0.890636i \(-0.349740\pi\)
0.454717 + 0.890636i \(0.349740\pi\)
\(602\) −12.2020 −0.497316
\(603\) 30.9408 1.26001
\(604\) −16.4829 −0.670678
\(605\) 0 0
\(606\) 1.96056 0.0796421
\(607\) 9.72705 0.394809 0.197404 0.980322i \(-0.436749\pi\)
0.197404 + 0.980322i \(0.436749\pi\)
\(608\) −4.26115 −0.172812
\(609\) −0.685613 −0.0277825
\(610\) 0 0
\(611\) 0 0
\(612\) −6.37909 −0.257860
\(613\) −10.3203 −0.416834 −0.208417 0.978040i \(-0.566831\pi\)
−0.208417 + 0.978040i \(0.566831\pi\)
\(614\) 16.2781 0.656931
\(615\) 0 0
\(616\) 3.55897 0.143395
\(617\) 3.78345 0.152316 0.0761579 0.997096i \(-0.475735\pi\)
0.0761579 + 0.997096i \(0.475735\pi\)
\(618\) 1.45521 0.0585372
\(619\) 27.1179 1.08996 0.544981 0.838448i \(-0.316537\pi\)
0.544981 + 0.838448i \(0.316537\pi\)
\(620\) 0 0
\(621\) −1.44380 −0.0579376
\(622\) 25.4801 1.02166
\(623\) 15.5590 0.623357
\(624\) 0.401584 0.0160762
\(625\) 0 0
\(626\) 19.0197 0.760181
\(627\) −1.86748 −0.0745802
\(628\) 10.1207 0.403860
\(629\) −2.14044 −0.0853447
\(630\) 0 0
\(631\) −11.1179 −0.442598 −0.221299 0.975206i \(-0.571030\pi\)
−0.221299 + 0.975206i \(0.571030\pi\)
\(632\) 8.52230 0.338999
\(633\) 0.227244 0.00903213
\(634\) 15.1653 0.602291
\(635\) 0 0
\(636\) 1.11794 0.0443293
\(637\) 16.2978 0.645743
\(638\) 13.3594 0.528903
\(639\) 21.1629 0.837192
\(640\) 0 0
\(641\) 27.0446 1.06820 0.534099 0.845422i \(-0.320651\pi\)
0.534099 + 0.845422i \(0.320651\pi\)
\(642\) −0.660352 −0.0260620
\(643\) −29.7834 −1.17454 −0.587272 0.809389i \(-0.699798\pi\)
−0.587272 + 0.809389i \(0.699798\pi\)
\(644\) 1.96056 0.0772567
\(645\) 0 0
\(646\) −9.12071 −0.358850
\(647\) −19.3570 −0.761002 −0.380501 0.924781i \(-0.624248\pi\)
−0.380501 + 0.924781i \(0.624248\pi\)
\(648\) −8.82289 −0.346596
\(649\) 8.42408 0.330674
\(650\) 0 0
\(651\) 1.02526 0.0401832
\(652\) −11.7389 −0.459729
\(653\) 10.5392 0.412433 0.206216 0.978506i \(-0.433885\pi\)
0.206216 + 0.978506i \(0.433885\pi\)
\(654\) −0.837073 −0.0327321
\(655\) 0 0
\(656\) −3.24143 −0.126556
\(657\) −0.778296 −0.0303642
\(658\) 0 0
\(659\) 9.22447 0.359334 0.179667 0.983727i \(-0.442498\pi\)
0.179667 + 0.983727i \(0.442498\pi\)
\(660\) 0 0
\(661\) 27.3594 1.06416 0.532078 0.846695i \(-0.321411\pi\)
0.532078 + 0.846695i \(0.321411\pi\)
\(662\) 23.4185 0.910187
\(663\) 0.859565 0.0333827
\(664\) −1.16016 −0.0450228
\(665\) 0 0
\(666\) −2.98028 −0.115483
\(667\) 7.35937 0.284956
\(668\) −10.6430 −0.411790
\(669\) −0.477314 −0.0184540
\(670\) 0 0
\(671\) 14.2888 0.551613
\(672\) −0.160157 −0.00617821
\(673\) 13.1484 0.506832 0.253416 0.967357i \(-0.418446\pi\)
0.253416 + 0.967357i \(0.418446\pi\)
\(674\) −28.9211 −1.11400
\(675\) 0 0
\(676\) −4.82289 −0.185496
\(677\) −29.8817 −1.14845 −0.574223 0.818699i \(-0.694696\pi\)
−0.574223 + 0.818699i \(0.694696\pi\)
\(678\) 0.679685 0.0261031
\(679\) 16.2414 0.623289
\(680\) 0 0
\(681\) 0.345187 0.0132276
\(682\) −19.9775 −0.764978
\(683\) 26.9854 1.03257 0.516284 0.856417i \(-0.327315\pi\)
0.516284 + 0.856417i \(0.327315\pi\)
\(684\) −12.6994 −0.485574
\(685\) 0 0
\(686\) −14.4829 −0.552958
\(687\) −1.79248 −0.0683873
\(688\) −10.6994 −0.407911
\(689\) 22.7637 0.867229
\(690\) 0 0
\(691\) −30.7270 −1.16891 −0.584456 0.811425i \(-0.698692\pi\)
−0.584456 + 0.811425i \(0.698692\pi\)
\(692\) 11.3988 0.433318
\(693\) 10.6067 0.402916
\(694\) −32.8817 −1.24817
\(695\) 0 0
\(696\) −0.601186 −0.0227879
\(697\) −6.93806 −0.262798
\(698\) −15.2048 −0.575508
\(699\) −0.0145717 −0.000551152 0
\(700\) 0 0
\(701\) −27.7858 −1.04946 −0.524728 0.851270i \(-0.675833\pi\)
−0.524728 + 0.851270i \(0.675833\pi\)
\(702\) 2.40158 0.0906419
\(703\) −4.26115 −0.160712
\(704\) 3.12071 0.117616
\(705\) 0 0
\(706\) −18.4829 −0.695611
\(707\) −15.9211 −0.598775
\(708\) −0.379092 −0.0142472
\(709\) 9.71913 0.365010 0.182505 0.983205i \(-0.441579\pi\)
0.182505 + 0.983205i \(0.441579\pi\)
\(710\) 0 0
\(711\) 25.3988 0.952530
\(712\) 13.6430 0.511293
\(713\) −11.0052 −0.412146
\(714\) −0.342807 −0.0128292
\(715\) 0 0
\(716\) −18.9211 −0.707115
\(717\) 0.516758 0.0192987
\(718\) −22.5223 −0.840525
\(719\) 5.41577 0.201974 0.100987 0.994888i \(-0.467800\pi\)
0.100987 + 0.994888i \(0.467800\pi\)
\(720\) 0 0
\(721\) −11.8174 −0.440101
\(722\) 0.842612 0.0313588
\(723\) 0.862334 0.0320706
\(724\) −18.8032 −0.698814
\(725\) 0 0
\(726\) −0.177110 −0.00657316
\(727\) −13.7586 −0.510277 −0.255139 0.966904i \(-0.582121\pi\)
−0.255139 + 0.966904i \(0.582121\pi\)
\(728\) −3.26115 −0.120866
\(729\) −25.9408 −0.960772
\(730\) 0 0
\(731\) −22.9014 −0.847038
\(732\) −0.643011 −0.0237664
\(733\) −52.4095 −1.93579 −0.967895 0.251356i \(-0.919123\pi\)
−0.967895 + 0.251356i \(0.919123\pi\)
\(734\) 15.3424 0.566299
\(735\) 0 0
\(736\) 1.71913 0.0633679
\(737\) −32.3988 −1.19343
\(738\) −9.66035 −0.355602
\(739\) 50.0892 1.84256 0.921280 0.388899i \(-0.127145\pi\)
0.921280 + 0.388899i \(0.127145\pi\)
\(740\) 0 0
\(741\) 1.71121 0.0628628
\(742\) −9.07850 −0.333282
\(743\) 40.7976 1.49672 0.748360 0.663293i \(-0.230842\pi\)
0.748360 + 0.663293i \(0.230842\pi\)
\(744\) 0.899009 0.0329593
\(745\) 0 0
\(746\) 12.6430 0.462894
\(747\) −3.45759 −0.126507
\(748\) 6.67969 0.244233
\(749\) 5.36253 0.195943
\(750\) 0 0
\(751\) 2.84261 0.103728 0.0518642 0.998654i \(-0.483484\pi\)
0.0518642 + 0.998654i \(0.483484\pi\)
\(752\) 0 0
\(753\) 1.26392 0.0460597
\(754\) −12.2414 −0.445806
\(755\) 0 0
\(756\) −0.957786 −0.0348343
\(757\) −11.4383 −0.415731 −0.207865 0.978157i \(-0.566652\pi\)
−0.207865 + 0.978157i \(0.566652\pi\)
\(758\) −15.7807 −0.573180
\(759\) 0.753423 0.0273475
\(760\) 0 0
\(761\) 43.2663 1.56840 0.784201 0.620507i \(-0.213073\pi\)
0.784201 + 0.620507i \(0.213073\pi\)
\(762\) −0.126253 −0.00457365
\(763\) 6.79763 0.246091
\(764\) 15.0446 0.544294
\(765\) 0 0
\(766\) −27.8004 −1.00447
\(767\) −7.71913 −0.278722
\(768\) −0.140435 −0.00506752
\(769\) 26.5590 0.957741 0.478871 0.877886i \(-0.341046\pi\)
0.478871 + 0.877886i \(0.341046\pi\)
\(770\) 0 0
\(771\) −2.32031 −0.0835641
\(772\) −25.3227 −0.911384
\(773\) −12.3621 −0.444635 −0.222318 0.974974i \(-0.571362\pi\)
−0.222318 + 0.974974i \(0.571362\pi\)
\(774\) −31.8872 −1.14616
\(775\) 0 0
\(776\) 14.2414 0.511237
\(777\) −0.160157 −0.00574562
\(778\) −9.66273 −0.346426
\(779\) −13.8122 −0.494873
\(780\) 0 0
\(781\) −22.1602 −0.792953
\(782\) 3.67969 0.131585
\(783\) −3.59526 −0.128484
\(784\) −5.69941 −0.203550
\(785\) 0 0
\(786\) −0.137666 −0.00491037
\(787\) −6.03944 −0.215283 −0.107641 0.994190i \(-0.534330\pi\)
−0.107641 + 0.994190i \(0.534330\pi\)
\(788\) 2.68245 0.0955585
\(789\) −3.16293 −0.112603
\(790\) 0 0
\(791\) −5.51953 −0.196252
\(792\) 9.30059 0.330482
\(793\) −13.0931 −0.464949
\(794\) −23.4801 −0.833277
\(795\) 0 0
\(796\) 2.24143 0.0794453
\(797\) −35.6233 −1.26184 −0.630921 0.775847i \(-0.717323\pi\)
−0.630921 + 0.775847i \(0.717323\pi\)
\(798\) −0.682455 −0.0241586
\(799\) 0 0
\(800\) 0 0
\(801\) 40.6600 1.43665
\(802\) −20.2781 −0.716045
\(803\) 0.814970 0.0287597
\(804\) 1.45798 0.0514190
\(805\) 0 0
\(806\) 18.3057 0.644792
\(807\) −3.83192 −0.134890
\(808\) −13.9606 −0.491131
\(809\) −23.6402 −0.831147 −0.415573 0.909560i \(-0.636419\pi\)
−0.415573 + 0.909560i \(0.636419\pi\)
\(810\) 0 0
\(811\) −3.27572 −0.115026 −0.0575130 0.998345i \(-0.518317\pi\)
−0.0575130 + 0.998345i \(0.518317\pi\)
\(812\) 4.88206 0.171327
\(813\) 3.67176 0.128774
\(814\) 3.12071 0.109381
\(815\) 0 0
\(816\) −0.300593 −0.0105229
\(817\) −45.5918 −1.59505
\(818\) −4.17434 −0.145952
\(819\) −9.71913 −0.339614
\(820\) 0 0
\(821\) −30.3621 −1.05965 −0.529823 0.848108i \(-0.677742\pi\)
−0.529823 + 0.848108i \(0.677742\pi\)
\(822\) 0.938064 0.0327187
\(823\) 14.2978 0.498391 0.249195 0.968453i \(-0.419834\pi\)
0.249195 + 0.968453i \(0.419834\pi\)
\(824\) −10.3621 −0.360982
\(825\) 0 0
\(826\) 3.07850 0.107115
\(827\) −7.90139 −0.274758 −0.137379 0.990519i \(-0.543868\pi\)
−0.137379 + 0.990519i \(0.543868\pi\)
\(828\) 5.12348 0.178053
\(829\) −43.3030 −1.50397 −0.751987 0.659178i \(-0.770905\pi\)
−0.751987 + 0.659178i \(0.770905\pi\)
\(830\) 0 0
\(831\) −2.88206 −0.0999774
\(832\) −2.85956 −0.0991376
\(833\) −12.1992 −0.422678
\(834\) −1.35976 −0.0470846
\(835\) 0 0
\(836\) 13.2978 0.459915
\(837\) 5.37632 0.185833
\(838\) 12.3175 0.425503
\(839\) 11.3819 0.392946 0.196473 0.980509i \(-0.437051\pi\)
0.196473 + 0.980509i \(0.437051\pi\)
\(840\) 0 0
\(841\) −10.6741 −0.368074
\(842\) −23.8872 −0.823208
\(843\) 2.03041 0.0699311
\(844\) −1.61814 −0.0556986
\(845\) 0 0
\(846\) 0 0
\(847\) 1.43826 0.0494191
\(848\) −7.96056 −0.273367
\(849\) 0.962937 0.0330479
\(850\) 0 0
\(851\) 1.71913 0.0589310
\(852\) 0.997230 0.0341645
\(853\) −10.7862 −0.369313 −0.184656 0.982803i \(-0.559117\pi\)
−0.184656 + 0.982803i \(0.559117\pi\)
\(854\) 5.22170 0.178683
\(855\) 0 0
\(856\) 4.70218 0.160717
\(857\) 2.38186 0.0813629 0.0406814 0.999172i \(-0.487047\pi\)
0.0406814 + 0.999172i \(0.487047\pi\)
\(858\) −1.25323 −0.0427845
\(859\) −48.1428 −1.64261 −0.821306 0.570488i \(-0.806754\pi\)
−0.821306 + 0.570488i \(0.806754\pi\)
\(860\) 0 0
\(861\) −0.519139 −0.0176922
\(862\) 2.80317 0.0954763
\(863\) −29.8308 −1.01545 −0.507726 0.861518i \(-0.669514\pi\)
−0.507726 + 0.861518i \(0.669514\pi\)
\(864\) −0.839843 −0.0285720
\(865\) 0 0
\(866\) 5.00000 0.169907
\(867\) 1.74400 0.0592294
\(868\) −7.30059 −0.247798
\(869\) −26.5956 −0.902196
\(870\) 0 0
\(871\) 29.6876 1.00593
\(872\) 5.96056 0.201850
\(873\) 42.4434 1.43649
\(874\) 7.32547 0.247788
\(875\) 0 0
\(876\) −0.0366745 −0.00123912
\(877\) 48.9578 1.65319 0.826593 0.562799i \(-0.190276\pi\)
0.826593 + 0.562799i \(0.190276\pi\)
\(878\) −30.7976 −1.03937
\(879\) −2.69703 −0.0909684
\(880\) 0 0
\(881\) 3.50811 0.118191 0.0590957 0.998252i \(-0.481178\pi\)
0.0590957 + 0.998252i \(0.481178\pi\)
\(882\) −16.9858 −0.571942
\(883\) 48.8817 1.64500 0.822500 0.568766i \(-0.192579\pi\)
0.822500 + 0.568766i \(0.192579\pi\)
\(884\) −6.12071 −0.205862
\(885\) 0 0
\(886\) −16.5984 −0.557635
\(887\) −46.3701 −1.55695 −0.778477 0.627673i \(-0.784008\pi\)
−0.778477 + 0.627673i \(0.784008\pi\)
\(888\) −0.140435 −0.00471270
\(889\) 1.02526 0.0343862
\(890\) 0 0
\(891\) 27.5337 0.922414
\(892\) 3.39881 0.113801
\(893\) 0 0
\(894\) −0.224474 −0.00750754
\(895\) 0 0
\(896\) 1.14044 0.0380993
\(897\) −0.690375 −0.0230509
\(898\) −9.75580 −0.325555
\(899\) −27.4044 −0.913986
\(900\) 0 0
\(901\) −17.0391 −0.567653
\(902\) 10.1156 0.336811
\(903\) −1.71359 −0.0570247
\(904\) −4.83984 −0.160971
\(905\) 0 0
\(906\) −2.31478 −0.0769033
\(907\) −45.4631 −1.50958 −0.754789 0.655967i \(-0.772261\pi\)
−0.754789 + 0.655967i \(0.772261\pi\)
\(908\) −2.45798 −0.0815710
\(909\) −41.6063 −1.38000
\(910\) 0 0
\(911\) 17.7610 0.588447 0.294223 0.955737i \(-0.404939\pi\)
0.294223 + 0.955737i \(0.404939\pi\)
\(912\) −0.598416 −0.0198155
\(913\) 3.62052 0.119822
\(914\) −7.99723 −0.264525
\(915\) 0 0
\(916\) 12.7637 0.421725
\(917\) 1.11794 0.0369178
\(918\) −1.79763 −0.0593306
\(919\) 26.4829 0.873589 0.436794 0.899561i \(-0.356114\pi\)
0.436794 + 0.899561i \(0.356114\pi\)
\(920\) 0 0
\(921\) 2.28602 0.0753270
\(922\) 20.6848 0.681219
\(923\) 20.3057 0.668372
\(924\) 0.499806 0.0164424
\(925\) 0 0
\(926\) −7.52507 −0.247289
\(927\) −30.8821 −1.01430
\(928\) 4.28087 0.140526
\(929\) 59.3842 1.94833 0.974167 0.225829i \(-0.0725091\pi\)
0.974167 + 0.225829i \(0.0725091\pi\)
\(930\) 0 0
\(931\) −24.2860 −0.795942
\(932\) 0.103761 0.00339880
\(933\) 3.57830 0.117148
\(934\) 6.84261 0.223897
\(935\) 0 0
\(936\) −8.52230 −0.278560
\(937\) −15.1771 −0.495815 −0.247907 0.968784i \(-0.579743\pi\)
−0.247907 + 0.968784i \(0.579743\pi\)
\(938\) −11.8398 −0.386585
\(939\) 2.67104 0.0871662
\(940\) 0 0
\(941\) 12.8426 0.418657 0.209329 0.977845i \(-0.432872\pi\)
0.209329 + 0.977845i \(0.432872\pi\)
\(942\) 1.42131 0.0463087
\(943\) 5.57243 0.181463
\(944\) 2.69941 0.0878582
\(945\) 0 0
\(946\) 33.3898 1.08560
\(947\) 9.04459 0.293910 0.146955 0.989143i \(-0.453053\pi\)
0.146955 + 0.989143i \(0.453053\pi\)
\(948\) 1.19683 0.0388713
\(949\) −0.746771 −0.0242412
\(950\) 0 0
\(951\) 2.12975 0.0690617
\(952\) 2.44103 0.0791142
\(953\) 13.7783 0.446323 0.223161 0.974782i \(-0.428362\pi\)
0.223161 + 0.974782i \(0.428362\pi\)
\(954\) −23.7247 −0.768115
\(955\) 0 0
\(956\) −3.67969 −0.119010
\(957\) 1.87613 0.0606466
\(958\) 16.4016 0.529911
\(959\) −7.61775 −0.245990
\(960\) 0 0
\(961\) 9.98028 0.321944
\(962\) −2.85956 −0.0921961
\(963\) 14.0138 0.451588
\(964\) −6.14044 −0.197770
\(965\) 0 0
\(966\) 0.275331 0.00885864
\(967\) −60.1964 −1.93579 −0.967894 0.251360i \(-0.919122\pi\)
−0.967894 + 0.251360i \(0.919122\pi\)
\(968\) 1.26115 0.0405349
\(969\) −1.28087 −0.0411475
\(970\) 0 0
\(971\) −28.4123 −0.911793 −0.455897 0.890033i \(-0.650681\pi\)
−0.455897 + 0.890033i \(0.650681\pi\)
\(972\) −3.75857 −0.120556
\(973\) 11.0422 0.353997
\(974\) 13.7270 0.439843
\(975\) 0 0
\(976\) 4.57869 0.146560
\(977\) 55.7468 1.78350 0.891749 0.452531i \(-0.149479\pi\)
0.891749 + 0.452531i \(0.149479\pi\)
\(978\) −1.64855 −0.0527148
\(979\) −42.5759 −1.36073
\(980\) 0 0
\(981\) 17.7641 0.567164
\(982\) −3.40435 −0.108637
\(983\) −37.1179 −1.18388 −0.591939 0.805983i \(-0.701637\pi\)
−0.591939 + 0.805983i \(0.701637\pi\)
\(984\) −0.455211 −0.0145116
\(985\) 0 0
\(986\) 9.16293 0.291807
\(987\) 0 0
\(988\) −12.1850 −0.387657
\(989\) 18.3937 0.584884
\(990\) 0 0
\(991\) −38.0418 −1.20844 −0.604219 0.796818i \(-0.706515\pi\)
−0.604219 + 0.796818i \(0.706515\pi\)
\(992\) −6.40158 −0.203250
\(993\) 3.28879 0.104367
\(994\) −8.09822 −0.256860
\(995\) 0 0
\(996\) −0.162927 −0.00516254
\(997\) −11.6288 −0.368289 −0.184144 0.982899i \(-0.558951\pi\)
−0.184144 + 0.982899i \(0.558951\pi\)
\(998\) 29.4631 0.932639
\(999\) −0.839843 −0.0265714
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.y.1.2 3
5.2 odd 4 1850.2.b.p.149.2 6
5.3 odd 4 1850.2.b.p.149.5 6
5.4 even 2 1850.2.a.bc.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.y.1.2 3 1.1 even 1 trivial
1850.2.a.bc.1.2 yes 3 5.4 even 2
1850.2.b.p.149.2 6 5.2 odd 4
1850.2.b.p.149.5 6 5.3 odd 4