Properties

Label 4650.2.a.by.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.1304369399\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 4650.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.627719 q^{11} -1.00000 q^{12} +2.00000 q^{13} +3.37228 q^{14} +1.00000 q^{16} +4.74456 q^{17} -1.00000 q^{18} -0.627719 q^{19} +3.37228 q^{21} -0.627719 q^{22} -3.37228 q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -3.37228 q^{28} -8.74456 q^{29} -1.00000 q^{31} -1.00000 q^{32} -0.627719 q^{33} -4.74456 q^{34} +1.00000 q^{36} +0.744563 q^{37} +0.627719 q^{38} -2.00000 q^{39} +0.744563 q^{41} -3.37228 q^{42} -0.627719 q^{43} +0.627719 q^{44} +3.37228 q^{46} +6.74456 q^{47} -1.00000 q^{48} +4.37228 q^{49} -4.74456 q^{51} +2.00000 q^{52} -1.37228 q^{53} +1.00000 q^{54} +3.37228 q^{56} +0.627719 q^{57} +8.74456 q^{58} +2.74456 q^{59} +11.4891 q^{61} +1.00000 q^{62} -3.37228 q^{63} +1.00000 q^{64} +0.627719 q^{66} -10.7446 q^{67} +4.74456 q^{68} +3.37228 q^{69} +3.37228 q^{71} -1.00000 q^{72} +8.11684 q^{73} -0.744563 q^{74} -0.627719 q^{76} -2.11684 q^{77} +2.00000 q^{78} -4.62772 q^{79} +1.00000 q^{81} -0.744563 q^{82} +12.0000 q^{83} +3.37228 q^{84} +0.627719 q^{86} +8.74456 q^{87} -0.627719 q^{88} -1.37228 q^{89} -6.74456 q^{91} -3.37228 q^{92} +1.00000 q^{93} -6.74456 q^{94} +1.00000 q^{96} -2.00000 q^{97} -4.37228 q^{98} +0.627719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + 7 q^{11} - 2 q^{12} + 4 q^{13} + q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 7 q^{19} + q^{21} - 7 q^{22} - q^{23} + 2 q^{24} - 4 q^{26} - 2 q^{27} - q^{28} - 6 q^{29} - 2 q^{31} - 2 q^{32} - 7 q^{33} + 2 q^{34} + 2 q^{36} - 10 q^{37} + 7 q^{38} - 4 q^{39} - 10 q^{41} - q^{42} - 7 q^{43} + 7 q^{44} + q^{46} + 2 q^{47} - 2 q^{48} + 3 q^{49} + 2 q^{51} + 4 q^{52} + 3 q^{53} + 2 q^{54} + q^{56} + 7 q^{57} + 6 q^{58} - 6 q^{59} + 2 q^{62} - q^{63} + 2 q^{64} + 7 q^{66} - 10 q^{67} - 2 q^{68} + q^{69} + q^{71} - 2 q^{72} - q^{73} + 10 q^{74} - 7 q^{76} + 13 q^{77} + 4 q^{78} - 15 q^{79} + 2 q^{81} + 10 q^{82} + 24 q^{83} + q^{84} + 7 q^{86} + 6 q^{87} - 7 q^{88} + 3 q^{89} - 2 q^{91} - q^{92} + 2 q^{93} - 2 q^{94} + 2 q^{96} - 4 q^{97} - 3 q^{98} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −3.37228 −1.27460 −0.637301 0.770615i \(-0.719949\pi\)
−0.637301 + 0.770615i \(0.719949\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.627719 0.189264 0.0946322 0.995512i \(-0.469833\pi\)
0.0946322 + 0.995512i \(0.469833\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 3.37228 0.901280
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.74456 1.15073 0.575363 0.817898i \(-0.304861\pi\)
0.575363 + 0.817898i \(0.304861\pi\)
\(18\) −1.00000 −0.235702
\(19\) −0.627719 −0.144009 −0.0720043 0.997404i \(-0.522940\pi\)
−0.0720043 + 0.997404i \(0.522940\pi\)
\(20\) 0 0
\(21\) 3.37228 0.735892
\(22\) −0.627719 −0.133830
\(23\) −3.37228 −0.703169 −0.351585 0.936156i \(-0.614357\pi\)
−0.351585 + 0.936156i \(0.614357\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −3.37228 −0.637301
\(29\) −8.74456 −1.62382 −0.811912 0.583779i \(-0.801573\pi\)
−0.811912 + 0.583779i \(0.801573\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) −1.00000 −0.176777
\(33\) −0.627719 −0.109272
\(34\) −4.74456 −0.813686
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0.744563 0.122405 0.0612027 0.998125i \(-0.480506\pi\)
0.0612027 + 0.998125i \(0.480506\pi\)
\(38\) 0.627719 0.101829
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 0.744563 0.116281 0.0581406 0.998308i \(-0.481483\pi\)
0.0581406 + 0.998308i \(0.481483\pi\)
\(42\) −3.37228 −0.520354
\(43\) −0.627719 −0.0957262 −0.0478631 0.998854i \(-0.515241\pi\)
−0.0478631 + 0.998854i \(0.515241\pi\)
\(44\) 0.627719 0.0946322
\(45\) 0 0
\(46\) 3.37228 0.497216
\(47\) 6.74456 0.983796 0.491898 0.870653i \(-0.336303\pi\)
0.491898 + 0.870653i \(0.336303\pi\)
\(48\) −1.00000 −0.144338
\(49\) 4.37228 0.624612
\(50\) 0 0
\(51\) −4.74456 −0.664372
\(52\) 2.00000 0.277350
\(53\) −1.37228 −0.188497 −0.0942487 0.995549i \(-0.530045\pi\)
−0.0942487 + 0.995549i \(0.530045\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 3.37228 0.450640
\(57\) 0.627719 0.0831434
\(58\) 8.74456 1.14822
\(59\) 2.74456 0.357312 0.178656 0.983912i \(-0.442825\pi\)
0.178656 + 0.983912i \(0.442825\pi\)
\(60\) 0 0
\(61\) 11.4891 1.47103 0.735516 0.677507i \(-0.236940\pi\)
0.735516 + 0.677507i \(0.236940\pi\)
\(62\) 1.00000 0.127000
\(63\) −3.37228 −0.424868
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.627719 0.0772668
\(67\) −10.7446 −1.31266 −0.656329 0.754475i \(-0.727892\pi\)
−0.656329 + 0.754475i \(0.727892\pi\)
\(68\) 4.74456 0.575363
\(69\) 3.37228 0.405975
\(70\) 0 0
\(71\) 3.37228 0.400216 0.200108 0.979774i \(-0.435871\pi\)
0.200108 + 0.979774i \(0.435871\pi\)
\(72\) −1.00000 −0.117851
\(73\) 8.11684 0.950005 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(74\) −0.744563 −0.0865536
\(75\) 0 0
\(76\) −0.627719 −0.0720043
\(77\) −2.11684 −0.241237
\(78\) 2.00000 0.226455
\(79\) −4.62772 −0.520659 −0.260330 0.965520i \(-0.583831\pi\)
−0.260330 + 0.965520i \(0.583831\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −0.744563 −0.0822232
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 3.37228 0.367946
\(85\) 0 0
\(86\) 0.627719 0.0676886
\(87\) 8.74456 0.937516
\(88\) −0.627719 −0.0669150
\(89\) −1.37228 −0.145462 −0.0727308 0.997352i \(-0.523171\pi\)
−0.0727308 + 0.997352i \(0.523171\pi\)
\(90\) 0 0
\(91\) −6.74456 −0.707022
\(92\) −3.37228 −0.351585
\(93\) 1.00000 0.103695
\(94\) −6.74456 −0.695649
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −4.37228 −0.441667
\(99\) 0.627719 0.0630881
\(100\) 0 0
\(101\) 8.11684 0.807656 0.403828 0.914835i \(-0.367679\pi\)
0.403828 + 0.914835i \(0.367679\pi\)
\(102\) 4.74456 0.469782
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 1.37228 0.133288
\(107\) −0.627719 −0.0606839 −0.0303419 0.999540i \(-0.509660\pi\)
−0.0303419 + 0.999540i \(0.509660\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.74456 0.454447 0.227223 0.973843i \(-0.427035\pi\)
0.227223 + 0.973843i \(0.427035\pi\)
\(110\) 0 0
\(111\) −0.744563 −0.0706708
\(112\) −3.37228 −0.318651
\(113\) 2.62772 0.247195 0.123597 0.992332i \(-0.460557\pi\)
0.123597 + 0.992332i \(0.460557\pi\)
\(114\) −0.627719 −0.0587912
\(115\) 0 0
\(116\) −8.74456 −0.811912
\(117\) 2.00000 0.184900
\(118\) −2.74456 −0.252657
\(119\) −16.0000 −1.46672
\(120\) 0 0
\(121\) −10.6060 −0.964179
\(122\) −11.4891 −1.04018
\(123\) −0.744563 −0.0671350
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 3.37228 0.300427
\(127\) −13.4891 −1.19697 −0.598483 0.801135i \(-0.704230\pi\)
−0.598483 + 0.801135i \(0.704230\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.627719 0.0552675
\(130\) 0 0
\(131\) −16.2337 −1.41834 −0.709172 0.705036i \(-0.750931\pi\)
−0.709172 + 0.705036i \(0.750931\pi\)
\(132\) −0.627719 −0.0546359
\(133\) 2.11684 0.183554
\(134\) 10.7446 0.928189
\(135\) 0 0
\(136\) −4.74456 −0.406843
\(137\) 3.48913 0.298096 0.149048 0.988830i \(-0.452379\pi\)
0.149048 + 0.988830i \(0.452379\pi\)
\(138\) −3.37228 −0.287068
\(139\) −10.7446 −0.911342 −0.455671 0.890148i \(-0.650601\pi\)
−0.455671 + 0.890148i \(0.650601\pi\)
\(140\) 0 0
\(141\) −6.74456 −0.567995
\(142\) −3.37228 −0.282996
\(143\) 1.25544 0.104985
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −8.11684 −0.671755
\(147\) −4.37228 −0.360620
\(148\) 0.744563 0.0612027
\(149\) −12.1168 −0.992651 −0.496325 0.868137i \(-0.665318\pi\)
−0.496325 + 0.868137i \(0.665318\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0.627719 0.0509147
\(153\) 4.74456 0.383575
\(154\) 2.11684 0.170580
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −9.37228 −0.747989 −0.373995 0.927431i \(-0.622012\pi\)
−0.373995 + 0.927431i \(0.622012\pi\)
\(158\) 4.62772 0.368162
\(159\) 1.37228 0.108829
\(160\) 0 0
\(161\) 11.3723 0.896261
\(162\) −1.00000 −0.0785674
\(163\) −24.2337 −1.89813 −0.949064 0.315082i \(-0.897968\pi\)
−0.949064 + 0.315082i \(0.897968\pi\)
\(164\) 0.744563 0.0581406
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 12.6277 0.977162 0.488581 0.872518i \(-0.337515\pi\)
0.488581 + 0.872518i \(0.337515\pi\)
\(168\) −3.37228 −0.260177
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −0.627719 −0.0480028
\(172\) −0.627719 −0.0478631
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −8.74456 −0.662924
\(175\) 0 0
\(176\) 0.627719 0.0473161
\(177\) −2.74456 −0.206294
\(178\) 1.37228 0.102857
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 14.8614 1.10464 0.552320 0.833632i \(-0.313743\pi\)
0.552320 + 0.833632i \(0.313743\pi\)
\(182\) 6.74456 0.499940
\(183\) −11.4891 −0.849301
\(184\) 3.37228 0.248608
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) 2.97825 0.217791
\(188\) 6.74456 0.491898
\(189\) 3.37228 0.245297
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −15.4891 −1.11493 −0.557466 0.830200i \(-0.688226\pi\)
−0.557466 + 0.830200i \(0.688226\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 4.37228 0.312306
\(197\) −19.4891 −1.38854 −0.694271 0.719713i \(-0.744273\pi\)
−0.694271 + 0.719713i \(0.744273\pi\)
\(198\) −0.627719 −0.0446100
\(199\) −12.6277 −0.895155 −0.447578 0.894245i \(-0.647713\pi\)
−0.447578 + 0.894245i \(0.647713\pi\)
\(200\) 0 0
\(201\) 10.7446 0.757863
\(202\) −8.11684 −0.571099
\(203\) 29.4891 2.06973
\(204\) −4.74456 −0.332186
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) −3.37228 −0.234390
\(208\) 2.00000 0.138675
\(209\) −0.394031 −0.0272557
\(210\) 0 0
\(211\) −0.627719 −0.0432139 −0.0216070 0.999767i \(-0.506878\pi\)
−0.0216070 + 0.999767i \(0.506878\pi\)
\(212\) −1.37228 −0.0942487
\(213\) −3.37228 −0.231065
\(214\) 0.627719 0.0429100
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 3.37228 0.228925
\(218\) −4.74456 −0.321342
\(219\) −8.11684 −0.548485
\(220\) 0 0
\(221\) 9.48913 0.638308
\(222\) 0.744563 0.0499718
\(223\) −9.25544 −0.619790 −0.309895 0.950771i \(-0.600294\pi\)
−0.309895 + 0.950771i \(0.600294\pi\)
\(224\) 3.37228 0.225320
\(225\) 0 0
\(226\) −2.62772 −0.174793
\(227\) −6.11684 −0.405989 −0.202995 0.979180i \(-0.565067\pi\)
−0.202995 + 0.979180i \(0.565067\pi\)
\(228\) 0.627719 0.0415717
\(229\) 3.88316 0.256606 0.128303 0.991735i \(-0.459047\pi\)
0.128303 + 0.991735i \(0.459047\pi\)
\(230\) 0 0
\(231\) 2.11684 0.139278
\(232\) 8.74456 0.574109
\(233\) 14.8614 0.973603 0.486802 0.873513i \(-0.338163\pi\)
0.486802 + 0.873513i \(0.338163\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 2.74456 0.178656
\(237\) 4.62772 0.300603
\(238\) 16.0000 1.03713
\(239\) −29.4891 −1.90749 −0.953746 0.300612i \(-0.902809\pi\)
−0.953746 + 0.300612i \(0.902809\pi\)
\(240\) 0 0
\(241\) 4.51087 0.290571 0.145285 0.989390i \(-0.453590\pi\)
0.145285 + 0.989390i \(0.453590\pi\)
\(242\) 10.6060 0.681778
\(243\) −1.00000 −0.0641500
\(244\) 11.4891 0.735516
\(245\) 0 0
\(246\) 0.744563 0.0474716
\(247\) −1.25544 −0.0798816
\(248\) 1.00000 0.0635001
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) −3.37228 −0.212434
\(253\) −2.11684 −0.133085
\(254\) 13.4891 0.846383
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −13.3723 −0.834140 −0.417070 0.908874i \(-0.636943\pi\)
−0.417070 + 0.908874i \(0.636943\pi\)
\(258\) −0.627719 −0.0390801
\(259\) −2.51087 −0.156018
\(260\) 0 0
\(261\) −8.74456 −0.541275
\(262\) 16.2337 1.00292
\(263\) 18.9783 1.17025 0.585125 0.810943i \(-0.301046\pi\)
0.585125 + 0.810943i \(0.301046\pi\)
\(264\) 0.627719 0.0386334
\(265\) 0 0
\(266\) −2.11684 −0.129792
\(267\) 1.37228 0.0839823
\(268\) −10.7446 −0.656329
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −13.8832 −0.843342 −0.421671 0.906749i \(-0.638556\pi\)
−0.421671 + 0.906749i \(0.638556\pi\)
\(272\) 4.74456 0.287681
\(273\) 6.74456 0.408200
\(274\) −3.48913 −0.210786
\(275\) 0 0
\(276\) 3.37228 0.202987
\(277\) −15.2554 −0.916610 −0.458305 0.888795i \(-0.651543\pi\)
−0.458305 + 0.888795i \(0.651543\pi\)
\(278\) 10.7446 0.644416
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 16.7446 0.998897 0.499448 0.866344i \(-0.333536\pi\)
0.499448 + 0.866344i \(0.333536\pi\)
\(282\) 6.74456 0.401633
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 3.37228 0.200108
\(285\) 0 0
\(286\) −1.25544 −0.0742356
\(287\) −2.51087 −0.148212
\(288\) −1.00000 −0.0589256
\(289\) 5.51087 0.324169
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 8.11684 0.475002
\(293\) 7.48913 0.437519 0.218760 0.975779i \(-0.429799\pi\)
0.218760 + 0.975779i \(0.429799\pi\)
\(294\) 4.37228 0.254997
\(295\) 0 0
\(296\) −0.744563 −0.0432768
\(297\) −0.627719 −0.0364239
\(298\) 12.1168 0.701910
\(299\) −6.74456 −0.390048
\(300\) 0 0
\(301\) 2.11684 0.122013
\(302\) 8.00000 0.460348
\(303\) −8.11684 −0.466301
\(304\) −0.627719 −0.0360021
\(305\) 0 0
\(306\) −4.74456 −0.271229
\(307\) −30.9783 −1.76802 −0.884011 0.467466i \(-0.845167\pi\)
−0.884011 + 0.467466i \(0.845167\pi\)
\(308\) −2.11684 −0.120618
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 2.00000 0.113228
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 9.37228 0.528908
\(315\) 0 0
\(316\) −4.62772 −0.260330
\(317\) 24.7446 1.38979 0.694897 0.719110i \(-0.255450\pi\)
0.694897 + 0.719110i \(0.255450\pi\)
\(318\) −1.37228 −0.0769537
\(319\) −5.48913 −0.307332
\(320\) 0 0
\(321\) 0.627719 0.0350358
\(322\) −11.3723 −0.633752
\(323\) −2.97825 −0.165714
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 24.2337 1.34218
\(327\) −4.74456 −0.262375
\(328\) −0.744563 −0.0411116
\(329\) −22.7446 −1.25395
\(330\) 0 0
\(331\) −1.48913 −0.0818497 −0.0409249 0.999162i \(-0.513030\pi\)
−0.0409249 + 0.999162i \(0.513030\pi\)
\(332\) 12.0000 0.658586
\(333\) 0.744563 0.0408018
\(334\) −12.6277 −0.690958
\(335\) 0 0
\(336\) 3.37228 0.183973
\(337\) −28.9783 −1.57855 −0.789273 0.614043i \(-0.789542\pi\)
−0.789273 + 0.614043i \(0.789542\pi\)
\(338\) 9.00000 0.489535
\(339\) −2.62772 −0.142718
\(340\) 0 0
\(341\) −0.627719 −0.0339929
\(342\) 0.627719 0.0339431
\(343\) 8.86141 0.478471
\(344\) 0.627719 0.0338443
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −36.4674 −1.95767 −0.978836 0.204648i \(-0.934395\pi\)
−0.978836 + 0.204648i \(0.934395\pi\)
\(348\) 8.74456 0.468758
\(349\) 7.25544 0.388375 0.194187 0.980964i \(-0.437793\pi\)
0.194187 + 0.980964i \(0.437793\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −0.627719 −0.0334575
\(353\) −15.4891 −0.824403 −0.412201 0.911093i \(-0.635240\pi\)
−0.412201 + 0.911093i \(0.635240\pi\)
\(354\) 2.74456 0.145872
\(355\) 0 0
\(356\) −1.37228 −0.0727308
\(357\) 16.0000 0.846810
\(358\) 12.0000 0.634220
\(359\) 3.37228 0.177982 0.0889911 0.996032i \(-0.471636\pi\)
0.0889911 + 0.996032i \(0.471636\pi\)
\(360\) 0 0
\(361\) −18.6060 −0.979262
\(362\) −14.8614 −0.781098
\(363\) 10.6060 0.556669
\(364\) −6.74456 −0.353511
\(365\) 0 0
\(366\) 11.4891 0.600546
\(367\) 20.2337 1.05619 0.528095 0.849185i \(-0.322907\pi\)
0.528095 + 0.849185i \(0.322907\pi\)
\(368\) −3.37228 −0.175792
\(369\) 0.744563 0.0387604
\(370\) 0 0
\(371\) 4.62772 0.240259
\(372\) 1.00000 0.0518476
\(373\) −14.8614 −0.769494 −0.384747 0.923022i \(-0.625711\pi\)
−0.384747 + 0.923022i \(0.625711\pi\)
\(374\) −2.97825 −0.154002
\(375\) 0 0
\(376\) −6.74456 −0.347824
\(377\) −17.4891 −0.900736
\(378\) −3.37228 −0.173451
\(379\) −28.8614 −1.48251 −0.741255 0.671223i \(-0.765769\pi\)
−0.741255 + 0.671223i \(0.765769\pi\)
\(380\) 0 0
\(381\) 13.4891 0.691069
\(382\) −16.0000 −0.818631
\(383\) −13.4891 −0.689262 −0.344631 0.938738i \(-0.611996\pi\)
−0.344631 + 0.938738i \(0.611996\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 15.4891 0.788376
\(387\) −0.627719 −0.0319087
\(388\) −2.00000 −0.101535
\(389\) 16.9783 0.860831 0.430416 0.902631i \(-0.358367\pi\)
0.430416 + 0.902631i \(0.358367\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) −4.37228 −0.220834
\(393\) 16.2337 0.818881
\(394\) 19.4891 0.981848
\(395\) 0 0
\(396\) 0.627719 0.0315441
\(397\) 1.60597 0.0806013 0.0403006 0.999188i \(-0.487168\pi\)
0.0403006 + 0.999188i \(0.487168\pi\)
\(398\) 12.6277 0.632970
\(399\) −2.11684 −0.105975
\(400\) 0 0
\(401\) 22.6277 1.12997 0.564987 0.825100i \(-0.308881\pi\)
0.564987 + 0.825100i \(0.308881\pi\)
\(402\) −10.7446 −0.535890
\(403\) −2.00000 −0.0996271
\(404\) 8.11684 0.403828
\(405\) 0 0
\(406\) −29.4891 −1.46352
\(407\) 0.467376 0.0231670
\(408\) 4.74456 0.234891
\(409\) −10.2337 −0.506023 −0.253012 0.967463i \(-0.581421\pi\)
−0.253012 + 0.967463i \(0.581421\pi\)
\(410\) 0 0
\(411\) −3.48913 −0.172106
\(412\) 8.00000 0.394132
\(413\) −9.25544 −0.455430
\(414\) 3.37228 0.165739
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 10.7446 0.526163
\(418\) 0.394031 0.0192727
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 19.4891 0.949842 0.474921 0.880028i \(-0.342477\pi\)
0.474921 + 0.880028i \(0.342477\pi\)
\(422\) 0.627719 0.0305569
\(423\) 6.74456 0.327932
\(424\) 1.37228 0.0666439
\(425\) 0 0
\(426\) 3.37228 0.163388
\(427\) −38.7446 −1.87498
\(428\) −0.627719 −0.0303419
\(429\) −1.25544 −0.0606131
\(430\) 0 0
\(431\) −26.9783 −1.29950 −0.649748 0.760149i \(-0.725126\pi\)
−0.649748 + 0.760149i \(0.725126\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −36.1168 −1.73566 −0.867832 0.496857i \(-0.834487\pi\)
−0.867832 + 0.496857i \(0.834487\pi\)
\(434\) −3.37228 −0.161875
\(435\) 0 0
\(436\) 4.74456 0.227223
\(437\) 2.11684 0.101262
\(438\) 8.11684 0.387838
\(439\) 14.7446 0.703720 0.351860 0.936053i \(-0.385549\pi\)
0.351860 + 0.936053i \(0.385549\pi\)
\(440\) 0 0
\(441\) 4.37228 0.208204
\(442\) −9.48913 −0.451352
\(443\) −34.3505 −1.63204 −0.816022 0.578022i \(-0.803825\pi\)
−0.816022 + 0.578022i \(0.803825\pi\)
\(444\) −0.744563 −0.0353354
\(445\) 0 0
\(446\) 9.25544 0.438258
\(447\) 12.1168 0.573107
\(448\) −3.37228 −0.159325
\(449\) −8.97825 −0.423710 −0.211855 0.977301i \(-0.567950\pi\)
−0.211855 + 0.977301i \(0.567950\pi\)
\(450\) 0 0
\(451\) 0.467376 0.0220079
\(452\) 2.62772 0.123597
\(453\) 8.00000 0.375873
\(454\) 6.11684 0.287078
\(455\) 0 0
\(456\) −0.627719 −0.0293956
\(457\) −34.4674 −1.61232 −0.806158 0.591700i \(-0.798457\pi\)
−0.806158 + 0.591700i \(0.798457\pi\)
\(458\) −3.88316 −0.181448
\(459\) −4.74456 −0.221457
\(460\) 0 0
\(461\) −19.7228 −0.918583 −0.459291 0.888286i \(-0.651897\pi\)
−0.459291 + 0.888286i \(0.651897\pi\)
\(462\) −2.11684 −0.0984845
\(463\) −2.51087 −0.116690 −0.0583451 0.998296i \(-0.518582\pi\)
−0.0583451 + 0.998296i \(0.518582\pi\)
\(464\) −8.74456 −0.405956
\(465\) 0 0
\(466\) −14.8614 −0.688441
\(467\) −6.51087 −0.301287 −0.150644 0.988588i \(-0.548135\pi\)
−0.150644 + 0.988588i \(0.548135\pi\)
\(468\) 2.00000 0.0924500
\(469\) 36.2337 1.67312
\(470\) 0 0
\(471\) 9.37228 0.431852
\(472\) −2.74456 −0.126329
\(473\) −0.394031 −0.0181176
\(474\) −4.62772 −0.212558
\(475\) 0 0
\(476\) −16.0000 −0.733359
\(477\) −1.37228 −0.0628324
\(478\) 29.4891 1.34880
\(479\) 8.86141 0.404888 0.202444 0.979294i \(-0.435112\pi\)
0.202444 + 0.979294i \(0.435112\pi\)
\(480\) 0 0
\(481\) 1.48913 0.0678983
\(482\) −4.51087 −0.205465
\(483\) −11.3723 −0.517457
\(484\) −10.6060 −0.482090
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 37.4891 1.69879 0.849397 0.527754i \(-0.176966\pi\)
0.849397 + 0.527754i \(0.176966\pi\)
\(488\) −11.4891 −0.520088
\(489\) 24.2337 1.09589
\(490\) 0 0
\(491\) 27.6060 1.24584 0.622920 0.782286i \(-0.285946\pi\)
0.622920 + 0.782286i \(0.285946\pi\)
\(492\) −0.744563 −0.0335675
\(493\) −41.4891 −1.86858
\(494\) 1.25544 0.0564848
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) −11.3723 −0.510117
\(498\) 12.0000 0.537733
\(499\) 33.4891 1.49918 0.749590 0.661903i \(-0.230251\pi\)
0.749590 + 0.661903i \(0.230251\pi\)
\(500\) 0 0
\(501\) −12.6277 −0.564165
\(502\) 4.00000 0.178529
\(503\) −5.48913 −0.244748 −0.122374 0.992484i \(-0.539051\pi\)
−0.122374 + 0.992484i \(0.539051\pi\)
\(504\) 3.37228 0.150213
\(505\) 0 0
\(506\) 2.11684 0.0941052
\(507\) 9.00000 0.399704
\(508\) −13.4891 −0.598483
\(509\) −27.2554 −1.20808 −0.604038 0.796956i \(-0.706442\pi\)
−0.604038 + 0.796956i \(0.706442\pi\)
\(510\) 0 0
\(511\) −27.3723 −1.21088
\(512\) −1.00000 −0.0441942
\(513\) 0.627719 0.0277145
\(514\) 13.3723 0.589826
\(515\) 0 0
\(516\) 0.627719 0.0276338
\(517\) 4.23369 0.186197
\(518\) 2.51087 0.110322
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 36.9783 1.62005 0.810023 0.586398i \(-0.199454\pi\)
0.810023 + 0.586398i \(0.199454\pi\)
\(522\) 8.74456 0.382739
\(523\) 12.8614 0.562390 0.281195 0.959651i \(-0.409269\pi\)
0.281195 + 0.959651i \(0.409269\pi\)
\(524\) −16.2337 −0.709172
\(525\) 0 0
\(526\) −18.9783 −0.827491
\(527\) −4.74456 −0.206676
\(528\) −0.627719 −0.0273179
\(529\) −11.6277 −0.505553
\(530\) 0 0
\(531\) 2.74456 0.119104
\(532\) 2.11684 0.0917768
\(533\) 1.48913 0.0645012
\(534\) −1.37228 −0.0593844
\(535\) 0 0
\(536\) 10.7446 0.464094
\(537\) 12.0000 0.517838
\(538\) 2.00000 0.0862261
\(539\) 2.74456 0.118217
\(540\) 0 0
\(541\) −15.4891 −0.665930 −0.332965 0.942939i \(-0.608049\pi\)
−0.332965 + 0.942939i \(0.608049\pi\)
\(542\) 13.8832 0.596333
\(543\) −14.8614 −0.637764
\(544\) −4.74456 −0.203421
\(545\) 0 0
\(546\) −6.74456 −0.288641
\(547\) 2.74456 0.117349 0.0586745 0.998277i \(-0.481313\pi\)
0.0586745 + 0.998277i \(0.481313\pi\)
\(548\) 3.48913 0.149048
\(549\) 11.4891 0.490344
\(550\) 0 0
\(551\) 5.48913 0.233845
\(552\) −3.37228 −0.143534
\(553\) 15.6060 0.663633
\(554\) 15.2554 0.648141
\(555\) 0 0
\(556\) −10.7446 −0.455671
\(557\) −11.8832 −0.503505 −0.251753 0.967792i \(-0.581007\pi\)
−0.251753 + 0.967792i \(0.581007\pi\)
\(558\) 1.00000 0.0423334
\(559\) −1.25544 −0.0530993
\(560\) 0 0
\(561\) −2.97825 −0.125742
\(562\) −16.7446 −0.706327
\(563\) 6.97825 0.294098 0.147049 0.989129i \(-0.453022\pi\)
0.147049 + 0.989129i \(0.453022\pi\)
\(564\) −6.74456 −0.283997
\(565\) 0 0
\(566\) 12.0000 0.504398
\(567\) −3.37228 −0.141623
\(568\) −3.37228 −0.141498
\(569\) −1.37228 −0.0575290 −0.0287645 0.999586i \(-0.509157\pi\)
−0.0287645 + 0.999586i \(0.509157\pi\)
\(570\) 0 0
\(571\) −26.7446 −1.11923 −0.559613 0.828754i \(-0.689050\pi\)
−0.559613 + 0.828754i \(0.689050\pi\)
\(572\) 1.25544 0.0524925
\(573\) −16.0000 −0.668410
\(574\) 2.51087 0.104802
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.23369 0.0929896 0.0464948 0.998919i \(-0.485195\pi\)
0.0464948 + 0.998919i \(0.485195\pi\)
\(578\) −5.51087 −0.229222
\(579\) 15.4891 0.643706
\(580\) 0 0
\(581\) −40.4674 −1.67887
\(582\) −2.00000 −0.0829027
\(583\) −0.861407 −0.0356758
\(584\) −8.11684 −0.335877
\(585\) 0 0
\(586\) −7.48913 −0.309373
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −4.37228 −0.180310
\(589\) 0.627719 0.0258647
\(590\) 0 0
\(591\) 19.4891 0.801675
\(592\) 0.744563 0.0306013
\(593\) −44.9783 −1.84704 −0.923518 0.383556i \(-0.874699\pi\)
−0.923518 + 0.383556i \(0.874699\pi\)
\(594\) 0.627719 0.0257556
\(595\) 0 0
\(596\) −12.1168 −0.496325
\(597\) 12.6277 0.516818
\(598\) 6.74456 0.275806
\(599\) 10.1168 0.413363 0.206682 0.978408i \(-0.433734\pi\)
0.206682 + 0.978408i \(0.433734\pi\)
\(600\) 0 0
\(601\) 12.5109 0.510329 0.255165 0.966898i \(-0.417870\pi\)
0.255165 + 0.966898i \(0.417870\pi\)
\(602\) −2.11684 −0.0862761
\(603\) −10.7446 −0.437552
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 8.11684 0.329724
\(607\) 4.62772 0.187833 0.0939167 0.995580i \(-0.470061\pi\)
0.0939167 + 0.995580i \(0.470061\pi\)
\(608\) 0.627719 0.0254574
\(609\) −29.4891 −1.19496
\(610\) 0 0
\(611\) 13.4891 0.545712
\(612\) 4.74456 0.191788
\(613\) −35.4891 −1.43339 −0.716696 0.697386i \(-0.754347\pi\)
−0.716696 + 0.697386i \(0.754347\pi\)
\(614\) 30.9783 1.25018
\(615\) 0 0
\(616\) 2.11684 0.0852901
\(617\) −7.88316 −0.317364 −0.158682 0.987330i \(-0.550724\pi\)
−0.158682 + 0.987330i \(0.550724\pi\)
\(618\) 8.00000 0.321807
\(619\) 16.2337 0.652487 0.326244 0.945286i \(-0.394217\pi\)
0.326244 + 0.945286i \(0.394217\pi\)
\(620\) 0 0
\(621\) 3.37228 0.135325
\(622\) −8.00000 −0.320771
\(623\) 4.62772 0.185406
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 0.394031 0.0157361
\(628\) −9.37228 −0.373995
\(629\) 3.53262 0.140855
\(630\) 0 0
\(631\) −46.3505 −1.84519 −0.922593 0.385775i \(-0.873934\pi\)
−0.922593 + 0.385775i \(0.873934\pi\)
\(632\) 4.62772 0.184081
\(633\) 0.627719 0.0249496
\(634\) −24.7446 −0.982732
\(635\) 0 0
\(636\) 1.37228 0.0544145
\(637\) 8.74456 0.346472
\(638\) 5.48913 0.217317
\(639\) 3.37228 0.133405
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −0.627719 −0.0247741
\(643\) 25.0951 0.989654 0.494827 0.868992i \(-0.335231\pi\)
0.494827 + 0.868992i \(0.335231\pi\)
\(644\) 11.3723 0.448131
\(645\) 0 0
\(646\) 2.97825 0.117178
\(647\) −50.5842 −1.98867 −0.994335 0.106287i \(-0.966104\pi\)
−0.994335 + 0.106287i \(0.966104\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 1.72281 0.0676263
\(650\) 0 0
\(651\) −3.37228 −0.132170
\(652\) −24.2337 −0.949064
\(653\) −40.9783 −1.60360 −0.801801 0.597591i \(-0.796125\pi\)
−0.801801 + 0.597591i \(0.796125\pi\)
\(654\) 4.74456 0.185527
\(655\) 0 0
\(656\) 0.744563 0.0290703
\(657\) 8.11684 0.316668
\(658\) 22.7446 0.886675
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −4.97825 −0.193632 −0.0968158 0.995302i \(-0.530866\pi\)
−0.0968158 + 0.995302i \(0.530866\pi\)
\(662\) 1.48913 0.0578765
\(663\) −9.48913 −0.368527
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −0.744563 −0.0288512
\(667\) 29.4891 1.14182
\(668\) 12.6277 0.488581
\(669\) 9.25544 0.357836
\(670\) 0 0
\(671\) 7.21194 0.278414
\(672\) −3.37228 −0.130089
\(673\) 0.510875 0.0196928 0.00984639 0.999952i \(-0.496866\pi\)
0.00984639 + 0.999952i \(0.496866\pi\)
\(674\) 28.9783 1.11620
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 41.6060 1.59905 0.799524 0.600635i \(-0.205085\pi\)
0.799524 + 0.600635i \(0.205085\pi\)
\(678\) 2.62772 0.100917
\(679\) 6.74456 0.258833
\(680\) 0 0
\(681\) 6.11684 0.234398
\(682\) 0.627719 0.0240366
\(683\) 12.8614 0.492128 0.246064 0.969254i \(-0.420863\pi\)
0.246064 + 0.969254i \(0.420863\pi\)
\(684\) −0.627719 −0.0240014
\(685\) 0 0
\(686\) −8.86141 −0.338330
\(687\) −3.88316 −0.148152
\(688\) −0.627719 −0.0239316
\(689\) −2.74456 −0.104560
\(690\) 0 0
\(691\) −19.1386 −0.728066 −0.364033 0.931386i \(-0.618601\pi\)
−0.364033 + 0.931386i \(0.618601\pi\)
\(692\) 18.0000 0.684257
\(693\) −2.11684 −0.0804123
\(694\) 36.4674 1.38428
\(695\) 0 0
\(696\) −8.74456 −0.331462
\(697\) 3.53262 0.133808
\(698\) −7.25544 −0.274622
\(699\) −14.8614 −0.562110
\(700\) 0 0
\(701\) 45.6060 1.72251 0.861257 0.508170i \(-0.169678\pi\)
0.861257 + 0.508170i \(0.169678\pi\)
\(702\) 2.00000 0.0754851
\(703\) −0.467376 −0.0176274
\(704\) 0.627719 0.0236580
\(705\) 0 0
\(706\) 15.4891 0.582941
\(707\) −27.3723 −1.02944
\(708\) −2.74456 −0.103147
\(709\) 40.1168 1.50662 0.753310 0.657666i \(-0.228456\pi\)
0.753310 + 0.657666i \(0.228456\pi\)
\(710\) 0 0
\(711\) −4.62772 −0.173553
\(712\) 1.37228 0.0514284
\(713\) 3.37228 0.126293
\(714\) −16.0000 −0.598785
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 29.4891 1.10129
\(718\) −3.37228 −0.125852
\(719\) 38.7446 1.44493 0.722464 0.691408i \(-0.243009\pi\)
0.722464 + 0.691408i \(0.243009\pi\)
\(720\) 0 0
\(721\) −26.9783 −1.00472
\(722\) 18.6060 0.692442
\(723\) −4.51087 −0.167761
\(724\) 14.8614 0.552320
\(725\) 0 0
\(726\) −10.6060 −0.393624
\(727\) −19.3723 −0.718478 −0.359239 0.933246i \(-0.616964\pi\)
−0.359239 + 0.933246i \(0.616964\pi\)
\(728\) 6.74456 0.249970
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −2.97825 −0.110155
\(732\) −11.4891 −0.424650
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −20.2337 −0.746839
\(735\) 0 0
\(736\) 3.37228 0.124304
\(737\) −6.74456 −0.248439
\(738\) −0.744563 −0.0274077
\(739\) −22.9783 −0.845269 −0.422634 0.906300i \(-0.638895\pi\)
−0.422634 + 0.906300i \(0.638895\pi\)
\(740\) 0 0
\(741\) 1.25544 0.0461196
\(742\) −4.62772 −0.169889
\(743\) 35.3723 1.29768 0.648842 0.760924i \(-0.275254\pi\)
0.648842 + 0.760924i \(0.275254\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) 14.8614 0.544115
\(747\) 12.0000 0.439057
\(748\) 2.97825 0.108896
\(749\) 2.11684 0.0773478
\(750\) 0 0
\(751\) −38.7446 −1.41381 −0.706905 0.707309i \(-0.749909\pi\)
−0.706905 + 0.707309i \(0.749909\pi\)
\(752\) 6.74456 0.245949
\(753\) 4.00000 0.145768
\(754\) 17.4891 0.636916
\(755\) 0 0
\(756\) 3.37228 0.122649
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 28.8614 1.04829
\(759\) 2.11684 0.0768366
\(760\) 0 0
\(761\) −46.8614 −1.69872 −0.849362 0.527810i \(-0.823013\pi\)
−0.849362 + 0.527810i \(0.823013\pi\)
\(762\) −13.4891 −0.488659
\(763\) −16.0000 −0.579239
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 13.4891 0.487382
\(767\) 5.48913 0.198201
\(768\) −1.00000 −0.0360844
\(769\) −9.37228 −0.337973 −0.168987 0.985618i \(-0.554049\pi\)
−0.168987 + 0.985618i \(0.554049\pi\)
\(770\) 0 0
\(771\) 13.3723 0.481591
\(772\) −15.4891 −0.557466
\(773\) −21.6060 −0.777113 −0.388556 0.921425i \(-0.627026\pi\)
−0.388556 + 0.921425i \(0.627026\pi\)
\(774\) 0.627719 0.0225629
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 2.51087 0.0900771
\(778\) −16.9783 −0.608700
\(779\) −0.467376 −0.0167455
\(780\) 0 0
\(781\) 2.11684 0.0757466
\(782\) 16.0000 0.572159
\(783\) 8.74456 0.312505
\(784\) 4.37228 0.156153
\(785\) 0 0
\(786\) −16.2337 −0.579036
\(787\) 9.88316 0.352296 0.176148 0.984364i \(-0.443636\pi\)
0.176148 + 0.984364i \(0.443636\pi\)
\(788\) −19.4891 −0.694271
\(789\) −18.9783 −0.675644
\(790\) 0 0
\(791\) −8.86141 −0.315075
\(792\) −0.627719 −0.0223050
\(793\) 22.9783 0.815982
\(794\) −1.60597 −0.0569937
\(795\) 0 0
\(796\) −12.6277 −0.447578
\(797\) 20.5109 0.726532 0.363266 0.931685i \(-0.381662\pi\)
0.363266 + 0.931685i \(0.381662\pi\)
\(798\) 2.11684 0.0749355
\(799\) 32.0000 1.13208
\(800\) 0 0
\(801\) −1.37228 −0.0484872
\(802\) −22.6277 −0.799013
\(803\) 5.09509 0.179802
\(804\) 10.7446 0.378932
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) 2.00000 0.0704033
\(808\) −8.11684 −0.285550
\(809\) 9.60597 0.337728 0.168864 0.985639i \(-0.445990\pi\)
0.168864 + 0.985639i \(0.445990\pi\)
\(810\) 0 0
\(811\) −40.6277 −1.42663 −0.713316 0.700842i \(-0.752808\pi\)
−0.713316 + 0.700842i \(0.752808\pi\)
\(812\) 29.4891 1.03487
\(813\) 13.8832 0.486904
\(814\) −0.467376 −0.0163815
\(815\) 0 0
\(816\) −4.74456 −0.166093
\(817\) 0.394031 0.0137854
\(818\) 10.2337 0.357813
\(819\) −6.74456 −0.235674
\(820\) 0 0
\(821\) 15.2554 0.532418 0.266209 0.963915i \(-0.414229\pi\)
0.266209 + 0.963915i \(0.414229\pi\)
\(822\) 3.48913 0.121697
\(823\) 17.2554 0.601487 0.300743 0.953705i \(-0.402765\pi\)
0.300743 + 0.953705i \(0.402765\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 9.25544 0.322038
\(827\) 17.4891 0.608156 0.304078 0.952647i \(-0.401652\pi\)
0.304078 + 0.952647i \(0.401652\pi\)
\(828\) −3.37228 −0.117195
\(829\) −31.0951 −1.07998 −0.539989 0.841672i \(-0.681571\pi\)
−0.539989 + 0.841672i \(0.681571\pi\)
\(830\) 0 0
\(831\) 15.2554 0.529205
\(832\) 2.00000 0.0693375
\(833\) 20.7446 0.718756
\(834\) −10.7446 −0.372054
\(835\) 0 0
\(836\) −0.394031 −0.0136278
\(837\) 1.00000 0.0345651
\(838\) 12.0000 0.414533
\(839\) 16.8614 0.582120 0.291060 0.956705i \(-0.405992\pi\)
0.291060 + 0.956705i \(0.405992\pi\)
\(840\) 0 0
\(841\) 47.4674 1.63681
\(842\) −19.4891 −0.671640
\(843\) −16.7446 −0.576713
\(844\) −0.627719 −0.0216070
\(845\) 0 0
\(846\) −6.74456 −0.231883
\(847\) 35.7663 1.22895
\(848\) −1.37228 −0.0471243
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) −2.51087 −0.0860717
\(852\) −3.37228 −0.115532
\(853\) 39.0951 1.33859 0.669295 0.742997i \(-0.266596\pi\)
0.669295 + 0.742997i \(0.266596\pi\)
\(854\) 38.7446 1.32581
\(855\) 0 0
\(856\) 0.627719 0.0214550
\(857\) −7.48913 −0.255824 −0.127912 0.991786i \(-0.540827\pi\)
−0.127912 + 0.991786i \(0.540827\pi\)
\(858\) 1.25544 0.0428599
\(859\) −17.4891 −0.596721 −0.298361 0.954453i \(-0.596440\pi\)
−0.298361 + 0.954453i \(0.596440\pi\)
\(860\) 0 0
\(861\) 2.51087 0.0855704
\(862\) 26.9783 0.918883
\(863\) 6.35053 0.216175 0.108087 0.994141i \(-0.465527\pi\)
0.108087 + 0.994141i \(0.465527\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 36.1168 1.22730
\(867\) −5.51087 −0.187159
\(868\) 3.37228 0.114463
\(869\) −2.90491 −0.0985422
\(870\) 0 0
\(871\) −21.4891 −0.728131
\(872\) −4.74456 −0.160671
\(873\) −2.00000 −0.0676897
\(874\) −2.11684 −0.0716033
\(875\) 0 0
\(876\) −8.11684 −0.274243
\(877\) 44.9783 1.51881 0.759404 0.650620i \(-0.225491\pi\)
0.759404 + 0.650620i \(0.225491\pi\)
\(878\) −14.7446 −0.497605
\(879\) −7.48913 −0.252602
\(880\) 0 0
\(881\) −3.02175 −0.101805 −0.0509027 0.998704i \(-0.516210\pi\)
−0.0509027 + 0.998704i \(0.516210\pi\)
\(882\) −4.37228 −0.147222
\(883\) −44.8614 −1.50971 −0.754853 0.655894i \(-0.772292\pi\)
−0.754853 + 0.655894i \(0.772292\pi\)
\(884\) 9.48913 0.319154
\(885\) 0 0
\(886\) 34.3505 1.15403
\(887\) 30.7446 1.03230 0.516151 0.856498i \(-0.327364\pi\)
0.516151 + 0.856498i \(0.327364\pi\)
\(888\) 0.744563 0.0249859
\(889\) 45.4891 1.52566
\(890\) 0 0
\(891\) 0.627719 0.0210294
\(892\) −9.25544 −0.309895
\(893\) −4.23369 −0.141675
\(894\) −12.1168 −0.405248
\(895\) 0 0
\(896\) 3.37228 0.112660
\(897\) 6.74456 0.225194
\(898\) 8.97825 0.299608
\(899\) 8.74456 0.291647
\(900\) 0 0
\(901\) −6.51087 −0.216909
\(902\) −0.467376 −0.0155619
\(903\) −2.11684 −0.0704442
\(904\) −2.62772 −0.0873966
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 17.4891 0.580717 0.290358 0.956918i \(-0.406225\pi\)
0.290358 + 0.956918i \(0.406225\pi\)
\(908\) −6.11684 −0.202995
\(909\) 8.11684 0.269219
\(910\) 0 0
\(911\) −2.51087 −0.0831890 −0.0415945 0.999135i \(-0.513244\pi\)
−0.0415945 + 0.999135i \(0.513244\pi\)
\(912\) 0.627719 0.0207858
\(913\) 7.53262 0.249293
\(914\) 34.4674 1.14008
\(915\) 0 0
\(916\) 3.88316 0.128303
\(917\) 54.7446 1.80782
\(918\) 4.74456 0.156594
\(919\) −12.2337 −0.403552 −0.201776 0.979432i \(-0.564671\pi\)
−0.201776 + 0.979432i \(0.564671\pi\)
\(920\) 0 0
\(921\) 30.9783 1.02077
\(922\) 19.7228 0.649536
\(923\) 6.74456 0.222000
\(924\) 2.11684 0.0696391
\(925\) 0 0
\(926\) 2.51087 0.0825125
\(927\) 8.00000 0.262754
\(928\) 8.74456 0.287054
\(929\) −16.1168 −0.528776 −0.264388 0.964416i \(-0.585170\pi\)
−0.264388 + 0.964416i \(0.585170\pi\)
\(930\) 0 0
\(931\) −2.74456 −0.0899494
\(932\) 14.8614 0.486802
\(933\) −8.00000 −0.261908
\(934\) 6.51087 0.213042
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 31.2554 1.02107 0.510535 0.859857i \(-0.329447\pi\)
0.510535 + 0.859857i \(0.329447\pi\)
\(938\) −36.2337 −1.18307
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) −19.7228 −0.642945 −0.321473 0.946919i \(-0.604178\pi\)
−0.321473 + 0.946919i \(0.604178\pi\)
\(942\) −9.37228 −0.305365
\(943\) −2.51087 −0.0817653
\(944\) 2.74456 0.0893279
\(945\) 0 0
\(946\) 0.394031 0.0128110
\(947\) 2.74456 0.0891863 0.0445932 0.999005i \(-0.485801\pi\)
0.0445932 + 0.999005i \(0.485801\pi\)
\(948\) 4.62772 0.150301
\(949\) 16.2337 0.526968
\(950\) 0 0
\(951\) −24.7446 −0.802397
\(952\) 16.0000 0.518563
\(953\) −11.7228 −0.379739 −0.189870 0.981809i \(-0.560807\pi\)
−0.189870 + 0.981809i \(0.560807\pi\)
\(954\) 1.37228 0.0444292
\(955\) 0 0
\(956\) −29.4891 −0.953746
\(957\) 5.48913 0.177438
\(958\) −8.86141 −0.286299
\(959\) −11.7663 −0.379954
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −1.48913 −0.0480113
\(963\) −0.627719 −0.0202280
\(964\) 4.51087 0.145285
\(965\) 0 0
\(966\) 11.3723 0.365897
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 10.6060 0.340889
\(969\) 2.97825 0.0956752
\(970\) 0 0
\(971\) −48.7011 −1.56289 −0.781446 0.623973i \(-0.785517\pi\)
−0.781446 + 0.623973i \(0.785517\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 36.2337 1.16160
\(974\) −37.4891 −1.20123
\(975\) 0 0
\(976\) 11.4891 0.367758
\(977\) 46.0000 1.47167 0.735835 0.677161i \(-0.236790\pi\)
0.735835 + 0.677161i \(0.236790\pi\)
\(978\) −24.2337 −0.774908
\(979\) −0.861407 −0.0275307
\(980\) 0 0
\(981\) 4.74456 0.151482
\(982\) −27.6060 −0.880942
\(983\) 2.97825 0.0949914 0.0474957 0.998871i \(-0.484876\pi\)
0.0474957 + 0.998871i \(0.484876\pi\)
\(984\) 0.744563 0.0237358
\(985\) 0 0
\(986\) 41.4891 1.32128
\(987\) 22.7446 0.723967
\(988\) −1.25544 −0.0399408
\(989\) 2.11684 0.0673117
\(990\) 0 0
\(991\) −47.6060 −1.51225 −0.756127 0.654425i \(-0.772911\pi\)
−0.756127 + 0.654425i \(0.772911\pi\)
\(992\) 1.00000 0.0317500
\(993\) 1.48913 0.0472560
\(994\) 11.3723 0.360707
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) −33.4891 −1.06008
\(999\) −0.744563 −0.0235569
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 4650.2.a.by.1.1 2
5.2 odd 4 4650.2.d.bh.3349.1 4
5.3 odd 4 4650.2.d.bh.3349.4 4
5.4 even 2 930.2.a.r.1.2 2
15.14 odd 2 2790.2.a.bd.1.2 2
20.19 odd 2 7440.2.a.bg.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.r.1.2 2 5.4 even 2
2790.2.a.bd.1.2 2 15.14 odd 2
4650.2.a.by.1.1 2 1.1 even 1 trivial
4650.2.d.bh.3349.1 4 5.2 odd 4
4650.2.d.bh.3349.4 4 5.3 odd 4
7440.2.a.bg.1.1 2 20.19 odd 2