Properties

Label 1850.2.b.p.149.2
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.37161216.1
Defining polynomial: \( x^{6} + 15x^{4} + 51x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.2
Root \(0.140435i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.p.149.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +0.140435i q^{3} -1.00000 q^{4} +0.140435 q^{6} -1.14044i q^{7} +1.00000i q^{8} +2.98028 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +0.140435i q^{3} -1.00000 q^{4} +0.140435 q^{6} -1.14044i q^{7} +1.00000i q^{8} +2.98028 q^{9} +3.12071 q^{11} -0.140435i q^{12} +2.85956i q^{13} -1.14044 q^{14} +1.00000 q^{16} +2.14044i q^{17} -2.98028i q^{18} -4.26115 q^{19} +0.160157 q^{21} -3.12071i q^{22} +1.71913i q^{23} -0.140435 q^{24} +2.85956 q^{26} +0.839843i q^{27} +1.14044i q^{28} +4.28087 q^{29} +6.40158 q^{31} -1.00000i q^{32} +0.438259i q^{33} +2.14044 q^{34} -2.98028 q^{36} -1.00000i q^{37} +4.26115i q^{38} -0.401584 q^{39} -3.24143 q^{41} -0.160157i q^{42} +10.6994i q^{43} -3.12071 q^{44} +1.71913 q^{46} +0.140435i q^{48} +5.69941 q^{49} -0.300593 q^{51} -2.85956i q^{52} +7.96056i q^{53} +0.839843 q^{54} +1.14044 q^{56} -0.598416i q^{57} -4.28087i q^{58} -2.69941 q^{59} +4.57869 q^{61} -6.40158i q^{62} -3.39881i q^{63} -1.00000 q^{64} +0.438259 q^{66} -10.3819i q^{67} -2.14044i q^{68} -0.241427 q^{69} -7.10099 q^{71} +2.98028i q^{72} -0.261149i q^{73} -1.00000 q^{74} +4.26115 q^{76} -3.55897i q^{77} +0.401584i q^{78} +8.52230 q^{79} +8.82289 q^{81} +3.24143i q^{82} -1.16016i q^{83} -0.160157 q^{84} +10.6994 q^{86} +0.601186i q^{87} +3.12071i q^{88} +13.6430 q^{89} +3.26115 q^{91} -1.71913i q^{92} +0.899009i q^{93} +0.140435 q^{96} -14.2414i q^{97} -5.69941i q^{98} +9.30059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 2 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 2 q^{6} - 12 q^{9} - 10 q^{11} - 8 q^{14} + 6 q^{16} + 2 q^{19} + 32 q^{21} - 2 q^{24} + 16 q^{26} + 28 q^{29} + 12 q^{31} + 14 q^{34} + 12 q^{36} + 24 q^{39} + 38 q^{41} + 10 q^{44} + 8 q^{46} + 2 q^{49} - 34 q^{51} - 26 q^{54} + 8 q^{56} + 16 q^{59} + 24 q^{61} - 6 q^{64} - 2 q^{66} + 56 q^{69} + 16 q^{71} - 6 q^{74} - 2 q^{76} - 4 q^{79} + 30 q^{81} - 32 q^{84} + 32 q^{86} - 2 q^{89} - 8 q^{91} + 2 q^{96} + 88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0.140435i 0.0810804i 0.999178 + 0.0405402i \(0.0129079\pi\)
−0.999178 + 0.0405402i \(0.987092\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0.140435 0.0573325
\(7\) − 1.14044i − 0.431044i −0.976499 0.215522i \(-0.930855\pi\)
0.976499 0.215522i \(-0.0691453\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.98028 0.993426
\(10\) 0 0
\(11\) 3.12071 0.940930 0.470465 0.882419i \(-0.344086\pi\)
0.470465 + 0.882419i \(0.344086\pi\)
\(12\) − 0.140435i − 0.0405402i
\(13\) 2.85956i 0.793101i 0.918013 + 0.396550i \(0.129793\pi\)
−0.918013 + 0.396550i \(0.870207\pi\)
\(14\) −1.14044 −0.304794
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.14044i 0.519132i 0.965725 + 0.259566i \(0.0835795\pi\)
−0.965725 + 0.259566i \(0.916421\pi\)
\(18\) − 2.98028i − 0.702458i
\(19\) −4.26115 −0.977575 −0.488787 0.872403i \(-0.662561\pi\)
−0.488787 + 0.872403i \(0.662561\pi\)
\(20\) 0 0
\(21\) 0.160157 0.0349492
\(22\) − 3.12071i − 0.665338i
\(23\) 1.71913i 0.358463i 0.983807 + 0.179232i \(0.0573611\pi\)
−0.983807 + 0.179232i \(0.942639\pi\)
\(24\) −0.140435 −0.0286662
\(25\) 0 0
\(26\) 2.85956 0.560807
\(27\) 0.839843i 0.161628i
\(28\) 1.14044i 0.215522i
\(29\) 4.28087 0.794938 0.397469 0.917616i \(-0.369889\pi\)
0.397469 + 0.917616i \(0.369889\pi\)
\(30\) 0 0
\(31\) 6.40158 1.14976 0.574879 0.818238i \(-0.305049\pi\)
0.574879 + 0.818238i \(0.305049\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0.438259i 0.0762910i
\(34\) 2.14044 0.367082
\(35\) 0 0
\(36\) −2.98028 −0.496713
\(37\) − 1.00000i − 0.164399i
\(38\) 4.26115i 0.691250i
\(39\) −0.401584 −0.0643049
\(40\) 0 0
\(41\) −3.24143 −0.506226 −0.253113 0.967437i \(-0.581454\pi\)
−0.253113 + 0.967437i \(0.581454\pi\)
\(42\) − 0.160157i − 0.0247128i
\(43\) 10.6994i 1.63164i 0.578303 + 0.815822i \(0.303715\pi\)
−0.578303 + 0.815822i \(0.696285\pi\)
\(44\) −3.12071 −0.470465
\(45\) 0 0
\(46\) 1.71913 0.253472
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0.140435i 0.0202701i
\(49\) 5.69941 0.814201
\(50\) 0 0
\(51\) −0.300593 −0.0420914
\(52\) − 2.85956i − 0.396550i
\(53\) 7.96056i 1.09347i 0.837307 + 0.546733i \(0.184129\pi\)
−0.837307 + 0.546733i \(0.815871\pi\)
\(54\) 0.839843 0.114288
\(55\) 0 0
\(56\) 1.14044 0.152397
\(57\) − 0.598416i − 0.0792621i
\(58\) − 4.28087i − 0.562106i
\(59\) −2.69941 −0.351433 −0.175716 0.984441i \(-0.556224\pi\)
−0.175716 + 0.984441i \(0.556224\pi\)
\(60\) 0 0
\(61\) 4.57869 0.586242 0.293121 0.956075i \(-0.405306\pi\)
0.293121 + 0.956075i \(0.405306\pi\)
\(62\) − 6.40158i − 0.813002i
\(63\) − 3.39881i − 0.428210i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0.438259 0.0539459
\(67\) − 10.3819i − 1.26835i −0.773191 0.634173i \(-0.781341\pi\)
0.773191 0.634173i \(-0.218659\pi\)
\(68\) − 2.14044i − 0.259566i
\(69\) −0.241427 −0.0290643
\(70\) 0 0
\(71\) −7.10099 −0.842733 −0.421366 0.906891i \(-0.638449\pi\)
−0.421366 + 0.906891i \(0.638449\pi\)
\(72\) 2.98028i 0.351229i
\(73\) − 0.261149i − 0.0305651i −0.999883 0.0152826i \(-0.995135\pi\)
0.999883 0.0152826i \(-0.00486478\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 4.26115 0.488787
\(77\) − 3.55897i − 0.405582i
\(78\) 0.401584i 0.0454704i
\(79\) 8.52230 0.958833 0.479417 0.877587i \(-0.340848\pi\)
0.479417 + 0.877587i \(0.340848\pi\)
\(80\) 0 0
\(81\) 8.82289 0.980321
\(82\) 3.24143i 0.357956i
\(83\) − 1.16016i − 0.127344i −0.997971 0.0636719i \(-0.979719\pi\)
0.997971 0.0636719i \(-0.0202811\pi\)
\(84\) −0.160157 −0.0174746
\(85\) 0 0
\(86\) 10.6994 1.15375
\(87\) 0.601186i 0.0644539i
\(88\) 3.12071i 0.332669i
\(89\) 13.6430 1.44616 0.723078 0.690766i \(-0.242727\pi\)
0.723078 + 0.690766i \(0.242727\pi\)
\(90\) 0 0
\(91\) 3.26115 0.341861
\(92\) − 1.71913i − 0.179232i
\(93\) 0.899009i 0.0932229i
\(94\) 0 0
\(95\) 0 0
\(96\) 0.140435 0.0143331
\(97\) − 14.2414i − 1.44600i −0.690849 0.722999i \(-0.742763\pi\)
0.690849 0.722999i \(-0.257237\pi\)
\(98\) − 5.69941i − 0.575727i
\(99\) 9.30059 0.934745
\(100\) 0 0
\(101\) 13.9606 1.38913 0.694564 0.719431i \(-0.255597\pi\)
0.694564 + 0.719431i \(0.255597\pi\)
\(102\) 0.300593i 0.0297631i
\(103\) − 10.3621i − 1.02101i −0.859874 0.510506i \(-0.829458\pi\)
0.859874 0.510506i \(-0.170542\pi\)
\(104\) −2.85956 −0.280403
\(105\) 0 0
\(106\) 7.96056 0.773198
\(107\) − 4.70218i − 0.454577i −0.973828 0.227288i \(-0.927014\pi\)
0.973828 0.227288i \(-0.0729860\pi\)
\(108\) − 0.839843i − 0.0808139i
\(109\) 5.96056 0.570918 0.285459 0.958391i \(-0.407854\pi\)
0.285459 + 0.958391i \(0.407854\pi\)
\(110\) 0 0
\(111\) 0.140435 0.0133295
\(112\) − 1.14044i − 0.107761i
\(113\) − 4.83984i − 0.455294i −0.973744 0.227647i \(-0.926897\pi\)
0.973744 0.227647i \(-0.0731032\pi\)
\(114\) −0.598416 −0.0560468
\(115\) 0 0
\(116\) −4.28087 −0.397469
\(117\) 8.52230i 0.787887i
\(118\) 2.69941i 0.248501i
\(119\) 2.44103 0.223769
\(120\) 0 0
\(121\) −1.26115 −0.114650
\(122\) − 4.57869i − 0.414535i
\(123\) − 0.455211i − 0.0410450i
\(124\) −6.40158 −0.574879
\(125\) 0 0
\(126\) −3.39881 −0.302790
\(127\) − 0.899009i − 0.0797741i −0.999204 0.0398871i \(-0.987300\pi\)
0.999204 0.0398871i \(-0.0126998\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −1.50258 −0.132294
\(130\) 0 0
\(131\) −0.980278 −0.0856473 −0.0428236 0.999083i \(-0.513635\pi\)
−0.0428236 + 0.999083i \(0.513635\pi\)
\(132\) − 0.438259i − 0.0381455i
\(133\) 4.85956i 0.421378i
\(134\) −10.3819 −0.896856
\(135\) 0 0
\(136\) −2.14044 −0.183541
\(137\) 6.67969i 0.570684i 0.958426 + 0.285342i \(0.0921072\pi\)
−0.958426 + 0.285342i \(0.907893\pi\)
\(138\) 0.241427i 0.0205516i
\(139\) 9.68245 0.821255 0.410628 0.911803i \(-0.365310\pi\)
0.410628 + 0.911803i \(0.365310\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.10099i 0.595902i
\(143\) 8.92388i 0.746252i
\(144\) 2.98028 0.248356
\(145\) 0 0
\(146\) −0.261149 −0.0216128
\(147\) 0.800398i 0.0660157i
\(148\) 1.00000i 0.0821995i
\(149\) 1.59842 0.130947 0.0654737 0.997854i \(-0.479144\pi\)
0.0654737 + 0.997854i \(0.479144\pi\)
\(150\) 0 0
\(151\) −16.4829 −1.34136 −0.670678 0.741749i \(-0.733997\pi\)
−0.670678 + 0.741749i \(0.733997\pi\)
\(152\) − 4.26115i − 0.345625i
\(153\) 6.37909i 0.515719i
\(154\) −3.55897 −0.286790
\(155\) 0 0
\(156\) 0.401584 0.0321525
\(157\) 10.1207i 0.807721i 0.914821 + 0.403860i \(0.132332\pi\)
−0.914821 + 0.403860i \(0.867668\pi\)
\(158\) − 8.52230i − 0.677998i
\(159\) −1.11794 −0.0886587
\(160\) 0 0
\(161\) 1.96056 0.154513
\(162\) − 8.82289i − 0.693192i
\(163\) 11.7389i 0.919458i 0.888059 + 0.459729i \(0.152053\pi\)
−0.888059 + 0.459729i \(0.847947\pi\)
\(164\) 3.24143 0.253113
\(165\) 0 0
\(166\) −1.16016 −0.0900457
\(167\) − 10.6430i − 0.823581i −0.911279 0.411790i \(-0.864904\pi\)
0.911279 0.411790i \(-0.135096\pi\)
\(168\) 0.160157i 0.0123564i
\(169\) 4.82289 0.370992
\(170\) 0 0
\(171\) −12.6994 −0.971148
\(172\) − 10.6994i − 0.815822i
\(173\) − 11.3988i − 0.866636i −0.901241 0.433318i \(-0.857343\pi\)
0.901241 0.433318i \(-0.142657\pi\)
\(174\) 0.601186 0.0455758
\(175\) 0 0
\(176\) 3.12071 0.235233
\(177\) − 0.379092i − 0.0284943i
\(178\) − 13.6430i − 1.02259i
\(179\) 18.9211 1.41423 0.707115 0.707098i \(-0.249996\pi\)
0.707115 + 0.707098i \(0.249996\pi\)
\(180\) 0 0
\(181\) −18.8032 −1.39763 −0.698814 0.715303i \(-0.746289\pi\)
−0.698814 + 0.715303i \(0.746289\pi\)
\(182\) − 3.26115i − 0.241732i
\(183\) 0.643011i 0.0475327i
\(184\) −1.71913 −0.126736
\(185\) 0 0
\(186\) 0.899009 0.0659185
\(187\) 6.67969i 0.488467i
\(188\) 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 \(0.316797\pi\)
\(192\) − 0.140435i − 0.0101350i
\(193\) 25.3227i 1.82277i 0.411558 + 0.911384i \(0.364985\pi\)
−0.411558 + 0.911384i \(0.635015\pi\)
\(194\) −14.2414 −1.02247
\(195\) 0 0
\(196\) −5.69941 −0.407101
\(197\) 2.68245i 0.191117i 0.995424 + 0.0955585i \(0.0304637\pi\)
−0.995424 + 0.0955585i \(0.969536\pi\)
\(198\) − 9.30059i − 0.660964i
\(199\) −2.24143 −0.158891 −0.0794453 0.996839i \(-0.525315\pi\)
−0.0794453 + 0.996839i \(0.525315\pi\)
\(200\) 0 0
\(201\) 1.45798 0.102838
\(202\) − 13.9606i − 0.982261i
\(203\) − 4.88206i − 0.342653i
\(204\) 0.300593 0.0210457
\(205\) 0 0
\(206\) −10.3621 −0.721964
\(207\) 5.12348i 0.356107i
\(208\) 2.85956i 0.198275i
\(209\) −13.2978 −0.919830
\(210\) 0 0
\(211\) −1.61814 −0.111397 −0.0556986 0.998448i \(-0.517739\pi\)
−0.0556986 + 0.998448i \(0.517739\pi\)
\(212\) − 7.96056i − 0.546733i
\(213\) − 0.997230i − 0.0683291i
\(214\) −4.70218 −0.321434
\(215\) 0 0
\(216\) −0.839843 −0.0571440
\(217\) − 7.30059i − 0.495597i
\(218\) − 5.96056i − 0.403700i
\(219\) 0.0366745 0.00247823
\(220\) 0 0
\(221\) −6.12071 −0.411724
\(222\) − 0.140435i − 0.00942540i
\(223\) − 3.39881i − 0.227601i −0.993504 0.113801i \(-0.963697\pi\)
0.993504 0.113801i \(-0.0363025\pi\)
\(224\) −1.14044 −0.0761985
\(225\) 0 0
\(226\) −4.83984 −0.321942
\(227\) − 2.45798i − 0.163142i −0.996668 0.0815710i \(-0.974006\pi\)
0.996668 0.0815710i \(-0.0259937\pi\)
\(228\) 0.598416i 0.0396311i
\(229\) −12.7637 −0.843451 −0.421725 0.906724i \(-0.638575\pi\)
−0.421725 + 0.906724i \(0.638575\pi\)
\(230\) 0 0
\(231\) 0.499806 0.0328848
\(232\) 4.28087i 0.281053i
\(233\) − 0.103761i − 0.00679760i −0.999994 0.00339880i \(-0.998918\pi\)
0.999994 0.00339880i \(-0.00108187\pi\)
\(234\) 8.52230 0.557120
\(235\) 0 0
\(236\) 2.69941 0.175716
\(237\) 1.19683i 0.0777426i
\(238\) − 2.44103i − 0.158228i
\(239\) 3.67969 0.238019 0.119010 0.992893i \(-0.462028\pi\)
0.119010 + 0.992893i \(0.462028\pi\)
\(240\) 0 0
\(241\) −6.14044 −0.395540 −0.197770 0.980248i \(-0.563370\pi\)
−0.197770 + 0.980248i \(0.563370\pi\)
\(242\) 1.26115i 0.0810697i
\(243\) 3.75857i 0.241113i
\(244\) −4.57869 −0.293121
\(245\) 0 0
\(246\) −0.455211 −0.0290232
\(247\) − 12.1850i − 0.775315i
\(248\) 6.40158i 0.406501i
\(249\) 0.162927 0.0103251
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 3.39881i 0.214105i
\(253\) 5.36491i 0.337289i
\(254\) −0.899009 −0.0564088
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.5223i 1.03063i 0.857000 + 0.515316i \(0.172326\pi\)
−0.857000 + 0.515316i \(0.827674\pi\)
\(258\) 1.50258i 0.0935462i
\(259\) −1.14044 −0.0708632
\(260\) 0 0
\(261\) 12.7582 0.789712
\(262\) 0.980278i 0.0605618i
\(263\) − 22.5223i − 1.38878i −0.719597 0.694392i \(-0.755673\pi\)
0.719597 0.694392i \(-0.244327\pi\)
\(264\) −0.438259 −0.0269729
\(265\) 0 0
\(266\) 4.85956 0.297959
\(267\) 1.91596i 0.117255i
\(268\) 10.3819i 0.634173i
\(269\) −27.2860 −1.66366 −0.831829 0.555032i \(-0.812706\pi\)
−0.831829 + 0.555032i \(0.812706\pi\)
\(270\) 0 0
\(271\) −26.1456 −1.58823 −0.794116 0.607767i \(-0.792066\pi\)
−0.794116 + 0.607767i \(0.792066\pi\)
\(272\) 2.14044i 0.129783i
\(273\) 0.457981i 0.0277182i
\(274\) 6.67969 0.403535
\(275\) 0 0
\(276\) 0.241427 0.0145322
\(277\) 20.5223i 1.23307i 0.787329 + 0.616533i \(0.211463\pi\)
−0.787329 + 0.616533i \(0.788537\pi\)
\(278\) − 9.68245i − 0.580715i
\(279\) 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
\(283\) 6.85679i 0.407594i 0.979013 + 0.203797i \(0.0653283\pi\)
−0.979013 + 0.203797i \(0.934672\pi\)
\(284\) 7.10099 0.421366
\(285\) 0 0
\(286\) 8.92388 0.527680
\(287\) 3.69664i 0.218206i
\(288\) − 2.98028i − 0.175615i
\(289\) 12.4185 0.730502
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0.261149i 0.0152826i
\(293\) − 19.2048i − 1.12195i −0.827832 0.560977i \(-0.810426\pi\)
0.827832 0.560977i \(-0.189574\pi\)
\(294\) 0.800398 0.0466802
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 2.62091i 0.152080i
\(298\) − 1.59842i − 0.0925938i
\(299\) −4.91596 −0.284297
\(300\) 0 0
\(301\) 12.2020 0.703311
\(302\) 16.4829i 0.948482i
\(303\) 1.96056i 0.112631i
\(304\) −4.26115 −0.244394
\(305\) 0 0
\(306\) 6.37909 0.364668
\(307\) − 16.2781i − 0.929040i −0.885563 0.464520i \(-0.846227\pi\)
0.885563 0.464520i \(-0.153773\pi\)
\(308\) 3.55897i 0.202791i
\(309\) 1.45521 0.0827841
\(310\) 0 0
\(311\) −25.4801 −1.44484 −0.722421 0.691453i \(-0.756971\pi\)
−0.722421 + 0.691453i \(0.756971\pi\)
\(312\) − 0.401584i − 0.0227352i
\(313\) 19.0197i 1.07506i 0.843245 + 0.537529i \(0.180642\pi\)
−0.843245 + 0.537529i \(0.819358\pi\)
\(314\) 10.1207 0.571145
\(315\) 0 0
\(316\) −8.52230 −0.479417
\(317\) − 15.1653i − 0.851769i −0.904778 0.425884i \(-0.859963\pi\)
0.904778 0.425884i \(-0.140037\pi\)
\(318\) 1.11794i 0.0626912i
\(319\) 13.3594 0.747981
\(320\) 0 0
\(321\) 0.660352 0.0368573
\(322\) − 1.96056i − 0.109258i
\(323\) − 9.12071i − 0.507490i
\(324\) −8.82289 −0.490161
\(325\) 0 0
\(326\) 11.7389 0.650155
\(327\) 0.837073i 0.0462902i
\(328\) − 3.24143i − 0.178978i
\(329\) 0 0
\(330\) 0 0
\(331\) −23.4185 −1.28720 −0.643600 0.765362i \(-0.722560\pi\)
−0.643600 + 0.765362i \(0.722560\pi\)
\(332\) 1.16016i 0.0636719i
\(333\) − 2.98028i − 0.163318i
\(334\) −10.6430 −0.582360
\(335\) 0 0
\(336\) 0.160157 0.00873731
\(337\) 28.9211i 1.57543i 0.616038 + 0.787717i \(0.288737\pi\)
−0.616038 + 0.787717i \(0.711263\pi\)
\(338\) − 4.82289i − 0.262331i
\(339\) 0.679685 0.0369154
\(340\) 0 0
\(341\) 19.9775 1.08184
\(342\) 12.6994i 0.686705i
\(343\) − 14.4829i − 0.782001i
\(344\) −10.6994 −0.576873
\(345\) 0 0
\(346\) −11.3988 −0.612804
\(347\) 32.8817i 1.76518i 0.470143 + 0.882590i \(0.344202\pi\)
−0.470143 + 0.882590i \(0.655798\pi\)
\(348\) − 0.601186i − 0.0322269i
\(349\) −15.2048 −0.813892 −0.406946 0.913452i \(-0.633406\pi\)
−0.406946 + 0.913452i \(0.633406\pi\)
\(350\) 0 0
\(351\) −2.40158 −0.128187
\(352\) − 3.12071i − 0.166335i
\(353\) − 18.4829i − 0.983743i −0.870668 0.491872i \(-0.836313\pi\)
0.870668 0.491872i \(-0.163687\pi\)
\(354\) −0.379092 −0.0201485
\(355\) 0 0
\(356\) −13.6430 −0.723078
\(357\) 0.342807i 0.0181433i
\(358\) − 18.9211i − 1.00001i
\(359\) −22.5223 −1.18868 −0.594341 0.804213i \(-0.702587\pi\)
−0.594341 + 0.804213i \(0.702587\pi\)
\(360\) 0 0
\(361\) −0.842612 −0.0443480
\(362\) 18.8032i 0.988273i
\(363\) − 0.177110i − 0.00929586i
\(364\) −3.26115 −0.170931
\(365\) 0 0
\(366\) 0.643011 0.0336107
\(367\) − 15.3424i − 0.800868i −0.916326 0.400434i \(-0.868859\pi\)
0.916326 0.400434i \(-0.131141\pi\)
\(368\) 1.71913i 0.0896158i
\(369\) −9.66035 −0.502898
\(370\) 0 0
\(371\) 9.07850 0.471332
\(372\) − 0.899009i − 0.0466114i
\(373\) 12.6430i 0.654630i 0.944915 + 0.327315i \(0.106144\pi\)
−0.944915 + 0.327315i \(0.893856\pi\)
\(374\) 6.67969 0.345398
\(375\) 0 0
\(376\) 0 0
\(377\) 12.2414i 0.630466i
\(378\) − 0.957786i − 0.0492632i
\(379\) −15.7807 −0.810599 −0.405299 0.914184i \(-0.632833\pi\)
−0.405299 + 0.914184i \(0.632833\pi\)
\(380\) 0 0
\(381\) 0.126253 0.00646812
\(382\) − 15.0446i − 0.769748i
\(383\) − 27.8004i − 1.42053i −0.703932 0.710267i \(-0.748574\pi\)
0.703932 0.710267i \(-0.251426\pi\)
\(384\) −0.140435 −0.00716656
\(385\) 0 0
\(386\) 25.3227 1.28889
\(387\) 31.8872i 1.62092i
\(388\) 14.2414i 0.722999i
\(389\) −9.66273 −0.489920 −0.244960 0.969533i \(-0.578775\pi\)
−0.244960 + 0.969533i \(0.578775\pi\)
\(390\) 0 0
\(391\) −3.67969 −0.186090
\(392\) 5.69941i 0.287864i
\(393\) − 0.137666i − 0.00694432i
\(394\) 2.68245 0.135140
\(395\) 0 0
\(396\) −9.30059 −0.467372
\(397\) 23.4801i 1.17843i 0.807976 + 0.589216i \(0.200563\pi\)
−0.807976 + 0.589216i \(0.799437\pi\)
\(398\) 2.24143i 0.112353i
\(399\) −0.682455 −0.0341655
\(400\) 0 0
\(401\) 20.2781 1.01264 0.506320 0.862346i \(-0.331005\pi\)
0.506320 + 0.862346i \(0.331005\pi\)
\(402\) − 1.45798i − 0.0727175i
\(403\) 18.3057i 0.911874i
\(404\) −13.9606 −0.694564
\(405\) 0 0
\(406\) −4.88206 −0.242292
\(407\) − 3.12071i − 0.154688i
\(408\) − 0.300593i − 0.0148816i
\(409\) −4.17434 −0.206408 −0.103204 0.994660i \(-0.532909\pi\)
−0.103204 + 0.994660i \(0.532909\pi\)
\(410\) 0 0
\(411\) −0.938064 −0.0462713
\(412\) 10.3621i 0.510506i
\(413\) 3.07850i 0.151483i
\(414\) 5.12348 0.251805
\(415\) 0 0
\(416\) 2.85956 0.140202
\(417\) 1.35976i 0.0665877i
\(418\) 13.2978i 0.650418i
\(419\) 12.3175 0.601751 0.300876 0.953663i \(-0.402721\pi\)
0.300876 + 0.953663i \(0.402721\pi\)
\(420\) 0 0
\(421\) 23.8872 1.16419 0.582096 0.813120i \(-0.302233\pi\)
0.582096 + 0.813120i \(0.302233\pi\)
\(422\) 1.61814i 0.0787697i
\(423\) 0 0
\(424\) −7.96056 −0.386599
\(425\) 0 0
\(426\) −0.997230 −0.0483160
\(427\) − 5.22170i − 0.252696i
\(428\) 4.70218i 0.227288i
\(429\) −1.25323 −0.0605064
\(430\) 0 0
\(431\) −2.80317 −0.135024 −0.0675119 0.997718i \(-0.521506\pi\)
−0.0675119 + 0.997718i \(0.521506\pi\)
\(432\) 0.839843i 0.0404069i
\(433\) 5.00000i 0.240285i 0.992757 + 0.120142i \(0.0383351\pi\)
−0.992757 + 0.120142i \(0.961665\pi\)
\(434\) −7.30059 −0.350440
\(435\) 0 0
\(436\) −5.96056 −0.285459
\(437\) − 7.32547i − 0.350425i
\(438\) − 0.0366745i − 0.00175238i
\(439\) −30.7976 −1.46989 −0.734945 0.678126i \(-0.762792\pi\)
−0.734945 + 0.678126i \(0.762792\pi\)
\(440\) 0 0
\(441\) 16.9858 0.808848
\(442\) 6.12071i 0.291133i
\(443\) − 16.5984i − 0.788615i −0.918979 0.394307i \(-0.870985\pi\)
0.918979 0.394307i \(-0.129015\pi\)
\(444\) −0.140435 −0.00666477
\(445\) 0 0
\(446\) −3.39881 −0.160939
\(447\) 0.224474i 0.0106173i
\(448\) 1.14044i 0.0538805i
\(449\) −9.75580 −0.460405 −0.230202 0.973143i \(-0.573939\pi\)
−0.230202 + 0.973143i \(0.573939\pi\)
\(450\) 0 0
\(451\) −10.1156 −0.476323
\(452\) 4.83984i 0.227647i
\(453\) − 2.31478i − 0.108758i
\(454\) −2.45798 −0.115359
\(455\) 0 0
\(456\) 0.598416 0.0280234
\(457\) 7.99723i 0.374095i 0.982351 + 0.187047i \(0.0598918\pi\)
−0.982351 + 0.187047i \(0.940108\pi\)
\(458\) 12.7637i 0.596410i
\(459\) −1.79763 −0.0839061
\(460\) 0 0
\(461\) −20.6848 −0.963389 −0.481694 0.876339i \(-0.659978\pi\)
−0.481694 + 0.876339i \(0.659978\pi\)
\(462\) − 0.499806i − 0.0232531i
\(463\) − 7.52507i − 0.349720i −0.984593 0.174860i \(-0.944053\pi\)
0.984593 0.174860i \(-0.0559472\pi\)
\(464\) 4.28087 0.198734
\(465\) 0 0
\(466\) −0.103761 −0.00480663
\(467\) − 6.84261i − 0.316638i −0.987388 0.158319i \(-0.949393\pi\)
0.987388 0.158319i \(-0.0506075\pi\)
\(468\) − 8.52230i − 0.393943i
\(469\) −11.8398 −0.546713
\(470\) 0 0
\(471\) −1.42131 −0.0654903
\(472\) − 2.69941i − 0.124250i
\(473\) 33.3898i 1.53526i
\(474\) 1.19683 0.0549723
\(475\) 0 0
\(476\) −2.44103 −0.111884
\(477\) 23.7247i 1.08628i
\(478\) − 3.67969i − 0.168305i
\(479\) 16.4016 0.749408 0.374704 0.927145i \(-0.377744\pi\)
0.374704 + 0.927145i \(0.377744\pi\)
\(480\) 0 0
\(481\) 2.85956 0.130385
\(482\) 6.14044i 0.279689i
\(483\) 0.275331i 0.0125280i
\(484\) 1.26115 0.0573249
\(485\) 0 0
\(486\) 3.75857 0.170492
\(487\) − 13.7270i − 0.622032i −0.950405 0.311016i \(-0.899331\pi\)
0.950405 0.311016i \(-0.100669\pi\)
\(488\) 4.57869i 0.207268i
\(489\) −1.64855 −0.0745500
\(490\) 0 0
\(491\) 3.40435 0.153636 0.0768182 0.997045i \(-0.475524\pi\)
0.0768182 + 0.997045i \(0.475524\pi\)
\(492\) 0.455211i 0.0205225i
\(493\) 9.16293i 0.412677i
\(494\) −12.1850 −0.548230
\(495\) 0 0
\(496\) 6.40158 0.287440
\(497\) 8.09822i 0.363255i
\(498\) − 0.162927i − 0.00730094i
\(499\) 29.4631 1.31895 0.659475 0.751726i \(-0.270778\pi\)
0.659475 + 0.751726i \(0.270778\pi\)
\(500\) 0 0
\(501\) 1.49466 0.0667763
\(502\) 9.00000i 0.401690i
\(503\) − 24.0000i − 1.07011i −0.844818 0.535054i \(-0.820291\pi\)
0.844818 0.535054i \(-0.179709\pi\)
\(504\) 3.39881 0.151395
\(505\) 0 0
\(506\) 5.36491 0.238499
\(507\) 0.677304i 0.0300801i
\(508\) 0.899009i 0.0398871i
\(509\) −12.9633 −0.574589 −0.287295 0.957842i \(-0.592756\pi\)
−0.287295 + 0.957842i \(0.592756\pi\)
\(510\) 0 0
\(511\) −0.297823 −0.0131749
\(512\) − 1.00000i − 0.0441942i
\(513\) − 3.57869i − 0.158003i
\(514\) 16.5223 0.728767
\(515\) 0 0
\(516\) 1.50258 0.0661472
\(517\) 0 0
\(518\) 1.14044i 0.0501079i
\(519\) 1.60080 0.0702672
\(520\) 0 0
\(521\) −2.01972 −0.0884856 −0.0442428 0.999021i \(-0.514088\pi\)
−0.0442428 + 0.999021i \(0.514088\pi\)
\(522\) − 12.7582i − 0.558411i
\(523\) 15.5562i 0.680225i 0.940385 + 0.340113i \(0.110465\pi\)
−0.940385 + 0.340113i \(0.889535\pi\)
\(524\) 0.980278 0.0428236
\(525\) 0 0
\(526\) −22.5223 −0.982019
\(527\) 13.7022i 0.596876i
\(528\) 0.438259i 0.0190728i
\(529\) 20.0446 0.871504
\(530\) 0 0
\(531\) −8.04498 −0.349123
\(532\) − 4.85956i − 0.210689i
\(533\) − 9.26907i − 0.401488i
\(534\) 1.91596 0.0829118
\(535\) 0 0
\(536\) 10.3819 0.448428
\(537\) 2.65719i 0.114666i
\(538\) 27.2860i 1.17638i
\(539\) 17.7862 0.766107
\(540\) 0 0
\(541\) −0.544789 −0.0234223 −0.0117112 0.999931i \(-0.503728\pi\)
−0.0117112 + 0.999931i \(0.503728\pi\)
\(542\) 26.1456i 1.12305i
\(543\) − 2.64063i − 0.113320i
\(544\) 2.14044 0.0917704
\(545\) 0 0
\(546\) 0.457981 0.0195998
\(547\) 35.6261i 1.52326i 0.648012 + 0.761630i \(0.275601\pi\)
−0.648012 + 0.761630i \(0.724399\pi\)
\(548\) − 6.67969i − 0.285342i
\(549\) 13.6458 0.582388
\(550\) 0 0
\(551\) −18.2414 −0.777111
\(552\) − 0.241427i − 0.0102758i
\(553\) − 9.71913i − 0.413299i
\(554\) 20.5223 0.871909
\(555\) 0 0
\(556\) −9.68245 −0.410628
\(557\) − 31.9606i − 1.35421i −0.735885 0.677106i \(-0.763234\pi\)
0.735885 0.677106i \(-0.236766\pi\)
\(558\) − 19.0785i − 0.807657i
\(559\) −30.5956 −1.29406
\(560\) 0 0
\(561\) −0.938064 −0.0396051
\(562\) 14.4580i 0.609873i
\(563\) 14.6655i 0.618077i 0.951049 + 0.309039i \(0.100007\pi\)
−0.951049 + 0.309039i \(0.899993\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6.85679 0.288213
\(567\) − 10.0619i − 0.422562i
\(568\) − 7.10099i − 0.297951i
\(569\) −9.99723 −0.419106 −0.209553 0.977797i \(-0.567201\pi\)
−0.209553 + 0.977797i \(0.567201\pi\)
\(570\) 0 0
\(571\) 27.6008 1.15506 0.577529 0.816370i \(-0.304017\pi\)
0.577529 + 0.816370i \(0.304017\pi\)
\(572\) − 8.92388i − 0.373126i
\(573\) 2.11279i 0.0882632i
\(574\) 3.69664 0.154295
\(575\) 0 0
\(576\) −2.98028 −0.124178
\(577\) 3.78622i 0.157622i 0.996890 + 0.0788111i \(0.0251124\pi\)
−0.996890 + 0.0788111i \(0.974888\pi\)
\(578\) − 12.4185i − 0.516543i
\(579\) −3.55620 −0.147791
\(580\) 0 0
\(581\) −1.32308 −0.0548908
\(582\) − 2.00000i − 0.0829027i
\(583\) 24.8426i 1.02888i
\(584\) 0.261149 0.0108064
\(585\) 0 0
\(586\) −19.2048 −0.793341
\(587\) − 19.1771i − 0.791524i −0.918353 0.395762i \(-0.870481\pi\)
0.918353 0.395762i \(-0.129519\pi\)
\(588\) − 0.800398i − 0.0330079i
\(589\) −27.2781 −1.12397
\(590\) 0 0
\(591\) −0.376712 −0.0154958
\(592\) − 1.00000i − 0.0410997i
\(593\) − 12.4383i − 0.510778i −0.966838 0.255389i \(-0.917796\pi\)
0.966838 0.255389i \(-0.0822035\pi\)
\(594\) 2.62091 0.107537
\(595\) 0 0
\(596\) −1.59842 −0.0654737
\(597\) − 0.314776i − 0.0128829i
\(598\) 4.91596i 0.201029i
\(599\) 35.3819 1.44566 0.722832 0.691024i \(-0.242840\pi\)
0.722832 + 0.691024i \(0.242840\pi\)
\(600\) 0 0
\(601\) 22.2951 0.909434 0.454717 0.890636i \(-0.349740\pi\)
0.454717 + 0.890636i \(0.349740\pi\)
\(602\) − 12.2020i − 0.497316i
\(603\) − 30.9408i − 1.26001i
\(604\) 16.4829 0.670678
\(605\) 0 0
\(606\) 1.96056 0.0796421
\(607\) 9.72705i 0.394809i 0.980322 + 0.197404i \(0.0632512\pi\)
−0.980322 + 0.197404i \(0.936749\pi\)
\(608\) 4.26115i 0.172812i
\(609\) 0.685613 0.0277825
\(610\) 0 0
\(611\) 0 0
\(612\) − 6.37909i − 0.257860i
\(613\) 10.3203i 0.416834i 0.978040 + 0.208417i \(0.0668310\pi\)
−0.978040 + 0.208417i \(0.933169\pi\)
\(614\) −16.2781 −0.656931
\(615\) 0 0
\(616\) 3.55897 0.143395
\(617\) 3.78345i 0.152316i 0.997096 + 0.0761579i \(0.0242653\pi\)
−0.997096 + 0.0761579i \(0.975735\pi\)
\(618\) − 1.45521i − 0.0585372i
\(619\) −27.1179 −1.08996 −0.544981 0.838448i \(-0.683463\pi\)
−0.544981 + 0.838448i \(0.683463\pi\)
\(620\) 0 0
\(621\) −1.44380 −0.0579376
\(622\) 25.4801i 1.02166i
\(623\) − 15.5590i − 0.623357i
\(624\) −0.401584 −0.0160762
\(625\) 0 0
\(626\) 19.0197 0.760181
\(627\) − 1.86748i − 0.0745802i
\(628\) − 10.1207i − 0.403860i
\(629\) 2.14044 0.0853447
\(630\) 0 0
\(631\) −11.1179 −0.442598 −0.221299 0.975206i \(-0.571030\pi\)
−0.221299 + 0.975206i \(0.571030\pi\)
\(632\) 8.52230i 0.338999i
\(633\) − 0.227244i − 0.00903213i
\(634\) −15.1653 −0.602291
\(635\) 0 0
\(636\) 1.11794 0.0443293
\(637\) 16.2978i 0.645743i
\(638\) − 13.3594i − 0.528903i
\(639\) −21.1629 −0.837192
\(640\) 0 0
\(641\) 27.0446 1.06820 0.534099 0.845422i \(-0.320651\pi\)
0.534099 + 0.845422i \(0.320651\pi\)
\(642\) − 0.660352i − 0.0260620i
\(643\) 29.7834i 1.17454i 0.809389 + 0.587272i \(0.199798\pi\)
−0.809389 + 0.587272i \(0.800202\pi\)
\(644\) −1.96056 −0.0772567
\(645\) 0 0
\(646\) −9.12071 −0.358850
\(647\) − 19.3570i − 0.761002i −0.924781 0.380501i \(-0.875752\pi\)
0.924781 0.380501i \(-0.124248\pi\)
\(648\) 8.82289i 0.346596i
\(649\) −8.42408 −0.330674
\(650\) 0 0
\(651\) 1.02526 0.0401832
\(652\) − 11.7389i − 0.459729i
\(653\) − 10.5392i − 0.412433i −0.978506 0.206216i \(-0.933885\pi\)
0.978506 0.206216i \(-0.0661151\pi\)
\(654\) 0.837073 0.0327321
\(655\) 0 0
\(656\) −3.24143 −0.126556
\(657\) − 0.778296i − 0.0303642i
\(658\) 0 0
\(659\) −9.22447 −0.359334 −0.179667 0.983727i \(-0.557502\pi\)
−0.179667 + 0.983727i \(0.557502\pi\)
\(660\) 0 0
\(661\) 27.3594 1.06416 0.532078 0.846695i \(-0.321411\pi\)
0.532078 + 0.846695i \(0.321411\pi\)
\(662\) 23.4185i 0.910187i
\(663\) − 0.859565i − 0.0333827i
\(664\) 1.16016 0.0450228
\(665\) 0 0
\(666\) −2.98028 −0.115483
\(667\) 7.35937i 0.284956i
\(668\) 10.6430i 0.411790i
\(669\) 0.477314 0.0184540
\(670\) 0 0
\(671\) 14.2888 0.551613
\(672\) − 0.160157i − 0.00617821i
\(673\) − 13.1484i − 0.506832i −0.967357 0.253416i \(-0.918446\pi\)
0.967357 0.253416i \(-0.0815541\pi\)
\(674\) 28.9211 1.11400
\(675\) 0 0
\(676\) −4.82289 −0.185496
\(677\) − 29.8817i − 1.14845i −0.818699 0.574223i \(-0.805304\pi\)
0.818699 0.574223i \(-0.194696\pi\)
\(678\) − 0.679685i − 0.0261031i
\(679\) −16.2414 −0.623289
\(680\) 0 0
\(681\) 0.345187 0.0132276
\(682\) − 19.9775i − 0.764978i
\(683\) − 26.9854i − 1.03257i −0.856417 0.516284i \(-0.827315\pi\)
0.856417 0.516284i \(-0.172685\pi\)
\(684\) 12.6994 0.485574
\(685\) 0 0
\(686\) −14.4829 −0.552958
\(687\) − 1.79248i − 0.0683873i
\(688\) 10.6994i 0.407911i
\(689\) −22.7637 −0.867229
\(690\) 0 0
\(691\) −30.7270 −1.16891 −0.584456 0.811425i \(-0.698692\pi\)
−0.584456 + 0.811425i \(0.698692\pi\)
\(692\) 11.3988i 0.433318i
\(693\) − 10.6067i − 0.402916i
\(694\) 32.8817 1.24817
\(695\) 0 0
\(696\) −0.601186 −0.0227879
\(697\) − 6.93806i − 0.262798i
\(698\) 15.2048i 0.575508i
\(699\) 0.0145717 0.000551152 0
\(700\) 0 0
\(701\) −27.7858 −1.04946 −0.524728 0.851270i \(-0.675833\pi\)
−0.524728 + 0.851270i \(0.675833\pi\)
\(702\) 2.40158i 0.0906419i
\(703\) 4.26115i 0.160712i
\(704\) −3.12071 −0.117616
\(705\) 0 0
\(706\) −18.4829 −0.695611
\(707\) − 15.9211i − 0.598775i
\(708\) 0.379092i 0.0142472i
\(709\) −9.71913 −0.365010 −0.182505 0.983205i \(-0.558421\pi\)
−0.182505 + 0.983205i \(0.558421\pi\)
\(710\) 0 0
\(711\) 25.3988 0.952530
\(712\) 13.6430i 0.511293i
\(713\) 11.0052i 0.412146i
\(714\) 0.342807 0.0128292
\(715\) 0 0
\(716\) −18.9211 −0.707115
\(717\) 0.516758i 0.0192987i
\(718\) 22.5223i 0.840525i
\(719\) −5.41577 −0.201974 −0.100987 0.994888i \(-0.532200\pi\)
−0.100987 + 0.994888i \(0.532200\pi\)
\(720\) 0 0
\(721\) −11.8174 −0.440101
\(722\) 0.842612i 0.0313588i
\(723\) − 0.862334i − 0.0320706i
\(724\) 18.8032 0.698814
\(725\) 0 0
\(726\) −0.177110 −0.00657316
\(727\) − 13.7586i − 0.510277i −0.966904 0.255139i \(-0.917879\pi\)
0.966904 0.255139i \(-0.0821211\pi\)
\(728\) 3.26115i 0.120866i
\(729\) 25.9408 0.960772
\(730\) 0 0
\(731\) −22.9014 −0.847038
\(732\) − 0.643011i − 0.0237664i
\(733\) 52.4095i 1.93579i 0.251356 + 0.967895i \(0.419123\pi\)
−0.251356 + 0.967895i \(0.580877\pi\)
\(734\) −15.3424 −0.566299
\(735\) 0 0
\(736\) 1.71913 0.0633679
\(737\) − 32.3988i − 1.19343i
\(738\) 9.66035i 0.355602i
\(739\) −50.0892 −1.84256 −0.921280 0.388899i \(-0.872855\pi\)
−0.921280 + 0.388899i \(0.872855\pi\)
\(740\) 0 0
\(741\) 1.71121 0.0628628
\(742\) − 9.07850i − 0.333282i
\(743\) − 40.7976i − 1.49672i −0.663293 0.748360i \(-0.730842\pi\)
0.663293 0.748360i \(-0.269158\pi\)
\(744\) −0.899009 −0.0329593
\(745\) 0 0
\(746\) 12.6430 0.462894
\(747\) − 3.45759i − 0.126507i
\(748\) − 6.67969i − 0.244233i
\(749\) −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
\(753\) − 1.26392i − 0.0460597i
\(754\) 12.2414 0.445806
\(755\) 0 0
\(756\) −0.957786 −0.0348343
\(757\) − 11.4383i − 0.415731i −0.978157 0.207865i \(-0.933348\pi\)
0.978157 0.207865i \(-0.0666516\pi\)
\(758\) 15.7807i 0.573180i
\(759\) −0.753423 −0.0273475
\(760\) 0 0
\(761\) 43.2663 1.56840 0.784201 0.620507i \(-0.213073\pi\)
0.784201 + 0.620507i \(0.213073\pi\)
\(762\) − 0.126253i − 0.00457365i
\(763\) − 6.79763i − 0.246091i
\(764\) −15.0446 −0.544294
\(765\) 0 0
\(766\) −27.8004 −1.00447
\(767\) − 7.71913i − 0.278722i
\(768\) 0.140435i 0.00506752i
\(769\) −26.5590 −0.957741 −0.478871 0.877886i \(-0.658954\pi\)
−0.478871 + 0.877886i \(0.658954\pi\)
\(770\) 0 0
\(771\) −2.32031 −0.0835641
\(772\) − 25.3227i − 0.911384i
\(773\) 12.3621i 0.444635i 0.974974 + 0.222318i \(0.0713622\pi\)
−0.974974 + 0.222318i \(0.928638\pi\)
\(774\) 31.8872 1.14616
\(775\) 0 0
\(776\) 14.2414 0.511237
\(777\) − 0.160157i − 0.00574562i
\(778\) 9.66273i 0.346426i
\(779\) 13.8122 0.494873
\(780\) 0 0
\(781\) −22.1602 −0.792953
\(782\) 3.67969i 0.131585i
\(783\) 3.59526i 0.128484i
\(784\) 5.69941 0.203550
\(785\) 0 0
\(786\) −0.137666 −0.00491037
\(787\) − 6.03944i − 0.215283i −0.994190 0.107641i \(-0.965670\pi\)
0.994190 0.107641i \(-0.0343299\pi\)
\(788\) − 2.68245i − 0.0955585i
\(789\) 3.16293 0.112603
\(790\) 0 0
\(791\) −5.51953 −0.196252
\(792\) 9.30059i 0.330482i
\(793\) 13.0931i 0.464949i
\(794\) 23.4801 0.833277
\(795\) 0 0
\(796\) 2.24143 0.0794453
\(797\) − 35.6233i − 1.26184i −0.775847 0.630921i \(-0.782677\pi\)
0.775847 0.630921i \(-0.217323\pi\)
\(798\) 0.682455i 0.0241586i
\(799\) 0 0
\(800\) 0 0
\(801\) 40.6600 1.43665
\(802\) − 20.2781i − 0.716045i
\(803\) − 0.814970i − 0.0287597i
\(804\) −1.45798 −0.0514190
\(805\) 0 0
\(806\) 18.3057 0.644792
\(807\) − 3.83192i − 0.134890i
\(808\) 13.9606i 0.491131i
\(809\) 23.6402 0.831147 0.415573 0.909560i \(-0.363581\pi\)
0.415573 + 0.909560i \(0.363581\pi\)
\(810\) 0 0
\(811\) −3.27572 −0.115026 −0.0575130 0.998345i \(-0.518317\pi\)
−0.0575130 + 0.998345i \(0.518317\pi\)
\(812\) 4.88206i 0.171327i
\(813\) − 3.67176i − 0.128774i
\(814\) −3.12071 −0.109381
\(815\) 0 0
\(816\) −0.300593 −0.0105229
\(817\) − 45.5918i − 1.59505i
\(818\) 4.17434i 0.145952i
\(819\) 9.71913 0.339614
\(820\) 0 0
\(821\) −30.3621 −1.05965 −0.529823 0.848108i \(-0.677742\pi\)
−0.529823 + 0.848108i \(0.677742\pi\)
\(822\) 0.938064i 0.0327187i
\(823\) − 14.2978i − 0.498391i −0.968453 0.249195i \(-0.919834\pi\)
0.968453 0.249195i \(-0.0801661\pi\)
\(824\) 10.3621 0.360982
\(825\) 0 0
\(826\) 3.07850 0.107115
\(827\) − 7.90139i − 0.274758i −0.990519 0.137379i \(-0.956132\pi\)
0.990519 0.137379i \(-0.0438679\pi\)
\(828\) − 5.12348i − 0.178053i
\(829\) 43.3030 1.50397 0.751987 0.659178i \(-0.229095\pi\)
0.751987 + 0.659178i \(0.229095\pi\)
\(830\) 0 0
\(831\) −2.88206 −0.0999774
\(832\) − 2.85956i − 0.0991376i
\(833\) 12.1992i 0.422678i
\(834\) 1.35976 0.0470846
\(835\) 0 0
\(836\) 13.2978 0.459915
\(837\) 5.37632i 0.185833i
\(838\) − 12.3175i − 0.425503i
\(839\) −11.3819 −0.392946 −0.196473 0.980509i \(-0.562949\pi\)
−0.196473 + 0.980509i \(0.562949\pi\)
\(840\) 0 0
\(841\) −10.6741 −0.368074
\(842\) − 23.8872i − 0.823208i
\(843\) − 2.03041i − 0.0699311i
\(844\) 1.61814 0.0556986
\(845\) 0 0
\(846\) 0 0
\(847\) 1.43826i 0.0494191i
\(848\) 7.96056i 0.273367i
\(849\) −0.962937 −0.0330479
\(850\) 0 0
\(851\) 1.71913 0.0589310
\(852\) 0.997230i 0.0341645i
\(853\) 10.7862i 0.369313i 0.982803 + 0.184656i \(0.0591173\pi\)
−0.982803 + 0.184656i \(0.940883\pi\)
\(854\) −5.22170 −0.178683
\(855\) 0 0
\(856\) 4.70218 0.160717
\(857\) 2.38186i 0.0813629i 0.999172 + 0.0406814i \(0.0129529\pi\)
−0.999172 + 0.0406814i \(0.987047\pi\)
\(858\) 1.25323i 0.0427845i
\(859\) 48.1428 1.64261 0.821306 0.570488i \(-0.193246\pi\)
0.821306 + 0.570488i \(0.193246\pi\)
\(860\) 0 0
\(861\) −0.519139 −0.0176922
\(862\) 2.80317i 0.0954763i
\(863\) 29.8308i 1.01545i 0.861518 + 0.507726i \(0.169514\pi\)
−0.861518 + 0.507726i \(0.830486\pi\)
\(864\) 0.839843 0.0285720
\(865\) 0 0
\(866\) 5.00000 0.169907
\(867\) 1.74400i 0.0592294i
\(868\) 7.30059i 0.247798i
\(869\) 26.5956 0.902196
\(870\) 0 0
\(871\) 29.6876 1.00593
\(872\) 5.96056i 0.201850i
\(873\) − 42.4434i − 1.43649i
\(874\) −7.32547 −0.247788
\(875\) 0 0
\(876\) −0.0366745 −0.00123912
\(877\) 48.9578i 1.65319i 0.562799 + 0.826593i \(0.309724\pi\)
−0.562799 + 0.826593i \(0.690276\pi\)
\(878\) 30.7976i 1.03937i
\(879\) 2.69703 0.0909684
\(880\) 0 0
\(881\) 3.50811 0.118191 0.0590957 0.998252i \(-0.481178\pi\)
0.0590957 + 0.998252i \(0.481178\pi\)
\(882\) − 16.9858i − 0.571942i
\(883\) − 48.8817i − 1.64500i −0.568766 0.822500i \(-0.692579\pi\)
0.568766 0.822500i \(-0.307421\pi\)
\(884\) 6.12071 0.205862
\(885\) 0 0
\(886\) −16.5984 −0.557635
\(887\) − 46.3701i − 1.55695i −0.627673 0.778477i \(-0.715992\pi\)
0.627673 0.778477i \(-0.284008\pi\)
\(888\) 0.140435i 0.00471270i
\(889\) −1.02526 −0.0343862
\(890\) 0 0
\(891\) 27.5337 0.922414
\(892\) 3.39881i 0.113801i
\(893\) 0 0
\(894\) 0.224474 0.00750754
\(895\) 0 0
\(896\) 1.14044 0.0380993
\(897\) − 0.690375i − 0.0230509i
\(898\) 9.75580i 0.325555i
\(899\) 27.4044 0.913986
\(900\) 0 0
\(901\) −17.0391 −0.567653
\(902\) 10.1156i 0.336811i
\(903\) 1.71359i 0.0570247i
\(904\) 4.83984 0.160971
\(905\) 0 0
\(906\) −2.31478 −0.0769033
\(907\) − 45.4631i − 1.50958i −0.655967 0.754789i \(-0.727739\pi\)
0.655967 0.754789i \(-0.272261\pi\)
\(908\) 2.45798i 0.0815710i
\(909\) 41.6063 1.38000
\(910\) 0 0
\(911\) 17.7610 0.588447 0.294223 0.955737i \(-0.404939\pi\)
0.294223 + 0.955737i \(0.404939\pi\)
\(912\) − 0.598416i − 0.0198155i
\(913\) − 3.62052i − 0.119822i
\(914\) 7.99723 0.264525
\(915\) 0 0
\(916\) 12.7637 0.421725
\(917\) 1.11794i 0.0369178i
\(918\) 1.79763i 0.0593306i
\(919\) −26.4829 −0.873589 −0.436794 0.899561i \(-0.643886\pi\)
−0.436794 + 0.899561i \(0.643886\pi\)
\(920\) 0 0
\(921\) 2.28602 0.0753270
\(922\) 20.6848i 0.681219i
\(923\) − 20.3057i − 0.668372i
\(924\) −0.499806 −0.0164424
\(925\) 0 0
\(926\) −7.52507 −0.247289
\(927\) − 30.8821i − 1.01430i
\(928\) − 4.28087i − 0.140526i
\(929\) −59.3842 −1.94833 −0.974167 0.225829i \(-0.927491\pi\)
−0.974167 + 0.225829i \(0.927491\pi\)
\(930\) 0 0
\(931\) −24.2860 −0.795942
\(932\) 0.103761i 0.00339880i
\(933\) − 3.57830i − 0.117148i
\(934\) −6.84261 −0.223897
\(935\) 0 0
\(936\) −8.52230 −0.278560
\(937\) − 15.1771i − 0.495815i −0.968784 0.247907i \(-0.920257\pi\)
0.968784 0.247907i \(-0.0797428\pi\)
\(938\) 11.8398i 0.386585i
\(939\) −2.67104 −0.0871662
\(940\) 0 0
\(941\) 12.8426 0.418657 0.209329 0.977845i \(-0.432872\pi\)
0.209329 + 0.977845i \(0.432872\pi\)
\(942\) 1.42131i 0.0463087i
\(943\) − 5.57243i − 0.181463i
\(944\) −2.69941 −0.0878582
\(945\) 0 0
\(946\) 33.3898 1.08560
\(947\) 9.04459i 0.293910i 0.989143 + 0.146955i \(0.0469472\pi\)
−0.989143 + 0.146955i \(0.953053\pi\)
\(948\) − 1.19683i − 0.0388713i
\(949\) 0.746771 0.0242412
\(950\) 0 0
\(951\) 2.12975 0.0690617
\(952\) 2.44103i 0.0791142i
\(953\) − 13.7783i − 0.446323i −0.974782 0.223161i \(-0.928362\pi\)
0.974782 0.223161i \(-0.0716377\pi\)
\(954\) 23.7247 0.768115
\(955\) 0 0
\(956\) −3.67969 −0.119010
\(957\) 1.87613i 0.0606466i
\(958\) − 16.4016i − 0.529911i
\(959\) 7.61775 0.245990
\(960\) 0 0
\(961\) 9.98028 0.321944
\(962\) − 2.85956i − 0.0921961i
\(963\) − 14.0138i − 0.451588i
\(964\) 6.14044 0.197770
\(965\) 0 0
\(966\) 0.275331 0.00885864
\(967\) − 60.1964i − 1.93579i −0.251360 0.967894i \(-0.580878\pi\)
0.251360 0.967894i \(-0.419122\pi\)
\(968\) − 1.26115i − 0.0405349i
\(969\) 1.28087 0.0411475
\(970\) 0 0
\(971\) −28.4123 −0.911793 −0.455897 0.890033i \(-0.650681\pi\)
−0.455897 + 0.890033i \(0.650681\pi\)
\(972\) − 3.75857i − 0.120556i
\(973\) − 11.0422i − 0.353997i
\(974\) −13.7270 −0.439843
\(975\) 0 0
\(976\) 4.57869 0.146560
\(977\) 55.7468i 1.78350i 0.452531 + 0.891749i \(0.350521\pi\)
−0.452531 + 0.891749i \(0.649479\pi\)
\(978\) 1.64855i 0.0527148i
\(979\) 42.5759 1.36073
\(980\) 0 0
\(981\) 17.7641 0.567164
\(982\) − 3.40435i − 0.108637i
\(983\) 37.1179i 1.18388i 0.805983 + 0.591939i \(0.201637\pi\)
−0.805983 + 0.591939i \(0.798363\pi\)
\(984\) 0.455211 0.0145116
\(985\) 0 0
\(986\) 9.16293 0.291807
\(987\) 0 0
\(988\) 12.1850i 0.387657i
\(989\) −18.3937 −0.584884
\(990\) 0 0
\(991\) −38.0418 −1.20844 −0.604219 0.796818i \(-0.706515\pi\)
−0.604219 + 0.796818i \(0.706515\pi\)
\(992\) − 6.40158i − 0.203250i
\(993\) − 3.28879i − 0.104367i
\(994\) 8.09822 0.256860
\(995\) 0 0
\(996\) −0.162927 −0.00516254
\(997\) − 11.6288i − 0.368289i −0.982899 0.184144i \(-0.941049\pi\)
0.982899 0.184144i \(-0.0589514\pi\)
\(998\) − 29.4631i − 0.932639i
\(999\) 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.b.p.149.2 6
5.2 odd 4 1850.2.a.bc.1.2 yes 3
5.3 odd 4 1850.2.a.y.1.2 3
5.4 even 2 inner 1850.2.b.p.149.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.y.1.2 3 5.3 odd 4
1850.2.a.bc.1.2 yes 3 5.2 odd 4
1850.2.b.p.149.2 6 1.1 even 1 trivial
1850.2.b.p.149.5 6 5.4 even 2 inner