Properties

Label 370.2.a.f.1.1
Level $370$
Weight $2$
Character 370.1
Self dual yes
Analytic conductor $2.954$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 370.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -4.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -4.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.37228 q^{11} +2.00000 q^{12} +6.74456 q^{13} -4.37228 q^{14} +2.00000 q^{15} +1.00000 q^{16} +0.372281 q^{17} +1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} -8.74456 q^{21} +2.37228 q^{22} -4.74456 q^{23} +2.00000 q^{24} +1.00000 q^{25} +6.74456 q^{26} -4.00000 q^{27} -4.37228 q^{28} -9.11684 q^{29} +2.00000 q^{30} -8.37228 q^{31} +1.00000 q^{32} +4.74456 q^{33} +0.372281 q^{34} -4.37228 q^{35} +1.00000 q^{36} +1.00000 q^{37} -2.00000 q^{38} +13.4891 q^{39} +1.00000 q^{40} -0.372281 q^{41} -8.74456 q^{42} +1.62772 q^{43} +2.37228 q^{44} +1.00000 q^{45} -4.74456 q^{46} +2.74456 q^{47} +2.00000 q^{48} +12.1168 q^{49} +1.00000 q^{50} +0.744563 q^{51} +6.74456 q^{52} -4.37228 q^{53} -4.00000 q^{54} +2.37228 q^{55} -4.37228 q^{56} -4.00000 q^{57} -9.11684 q^{58} +1.25544 q^{59} +2.00000 q^{60} +0.372281 q^{61} -8.37228 q^{62} -4.37228 q^{63} +1.00000 q^{64} +6.74456 q^{65} +4.74456 q^{66} -6.74456 q^{67} +0.372281 q^{68} -9.48913 q^{69} -4.37228 q^{70} +4.74456 q^{71} +1.00000 q^{72} -2.74456 q^{73} +1.00000 q^{74} +2.00000 q^{75} -2.00000 q^{76} -10.3723 q^{77} +13.4891 q^{78} +6.74456 q^{79} +1.00000 q^{80} -11.0000 q^{81} -0.372281 q^{82} +10.7446 q^{83} -8.74456 q^{84} +0.372281 q^{85} +1.62772 q^{86} -18.2337 q^{87} +2.37228 q^{88} +10.0000 q^{89} +1.00000 q^{90} -29.4891 q^{91} -4.74456 q^{92} -16.7446 q^{93} +2.74456 q^{94} -2.00000 q^{95} +2.00000 q^{96} +17.1168 q^{97} +12.1168 q^{98} +2.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} - q^{11} + 4 q^{12} + 2 q^{13} - 3 q^{14} + 4 q^{15} + 2 q^{16} - 5 q^{17} + 2 q^{18} - 4 q^{19} + 2 q^{20} - 6 q^{21} - q^{22} + 2 q^{23} + 4 q^{24} + 2 q^{25} + 2 q^{26} - 8 q^{27} - 3 q^{28} - q^{29} + 4 q^{30} - 11 q^{31} + 2 q^{32} - 2 q^{33} - 5 q^{34} - 3 q^{35} + 2 q^{36} + 2 q^{37} - 4 q^{38} + 4 q^{39} + 2 q^{40} + 5 q^{41} - 6 q^{42} + 9 q^{43} - q^{44} + 2 q^{45} + 2 q^{46} - 6 q^{47} + 4 q^{48} + 7 q^{49} + 2 q^{50} - 10 q^{51} + 2 q^{52} - 3 q^{53} - 8 q^{54} - q^{55} - 3 q^{56} - 8 q^{57} - q^{58} + 14 q^{59} + 4 q^{60} - 5 q^{61} - 11 q^{62} - 3 q^{63} + 2 q^{64} + 2 q^{65} - 2 q^{66} - 2 q^{67} - 5 q^{68} + 4 q^{69} - 3 q^{70} - 2 q^{71} + 2 q^{72} + 6 q^{73} + 2 q^{74} + 4 q^{75} - 4 q^{76} - 15 q^{77} + 4 q^{78} + 2 q^{79} + 2 q^{80} - 22 q^{81} + 5 q^{82} + 10 q^{83} - 6 q^{84} - 5 q^{85} + 9 q^{86} - 2 q^{87} - q^{88} + 20 q^{89} + 2 q^{90} - 36 q^{91} + 2 q^{92} - 22 q^{93} - 6 q^{94} - 4 q^{95} + 4 q^{96} + 17 q^{97} + 7 q^{98} - q^{99}+O(q^{100}) \) 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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.00000 0.816497
\(7\) −4.37228 −1.65257 −0.826284 0.563254i \(-0.809549\pi\)
−0.826284 + 0.563254i \(0.809549\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 2.37228 0.715270 0.357635 0.933862i \(-0.383583\pi\)
0.357635 + 0.933862i \(0.383583\pi\)
\(12\) 2.00000 0.577350
\(13\) 6.74456 1.87061 0.935303 0.353849i \(-0.115127\pi\)
0.935303 + 0.353849i \(0.115127\pi\)
\(14\) −4.37228 −1.16854
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 0.372281 0.0902915 0.0451457 0.998980i \(-0.485625\pi\)
0.0451457 + 0.998980i \(0.485625\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 1.00000 0.223607
\(21\) −8.74456 −1.90822
\(22\) 2.37228 0.505772
\(23\) −4.74456 −0.989310 −0.494655 0.869090i \(-0.664706\pi\)
−0.494655 + 0.869090i \(0.664706\pi\)
\(24\) 2.00000 0.408248
\(25\) 1.00000 0.200000
\(26\) 6.74456 1.32272
\(27\) −4.00000 −0.769800
\(28\) −4.37228 −0.826284
\(29\) −9.11684 −1.69296 −0.846478 0.532424i \(-0.821281\pi\)
−0.846478 + 0.532424i \(0.821281\pi\)
\(30\) 2.00000 0.365148
\(31\) −8.37228 −1.50371 −0.751853 0.659331i \(-0.770840\pi\)
−0.751853 + 0.659331i \(0.770840\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.74456 0.825922
\(34\) 0.372281 0.0638457
\(35\) −4.37228 −0.739050
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) −2.00000 −0.324443
\(39\) 13.4891 2.15999
\(40\) 1.00000 0.158114
\(41\) −0.372281 −0.0581406 −0.0290703 0.999577i \(-0.509255\pi\)
−0.0290703 + 0.999577i \(0.509255\pi\)
\(42\) −8.74456 −1.34932
\(43\) 1.62772 0.248225 0.124112 0.992268i \(-0.460392\pi\)
0.124112 + 0.992268i \(0.460392\pi\)
\(44\) 2.37228 0.357635
\(45\) 1.00000 0.149071
\(46\) −4.74456 −0.699548
\(47\) 2.74456 0.400336 0.200168 0.979762i \(-0.435851\pi\)
0.200168 + 0.979762i \(0.435851\pi\)
\(48\) 2.00000 0.288675
\(49\) 12.1168 1.73098
\(50\) 1.00000 0.141421
\(51\) 0.744563 0.104260
\(52\) 6.74456 0.935303
\(53\) −4.37228 −0.600579 −0.300290 0.953848i \(-0.597083\pi\)
−0.300290 + 0.953848i \(0.597083\pi\)
\(54\) −4.00000 −0.544331
\(55\) 2.37228 0.319878
\(56\) −4.37228 −0.584271
\(57\) −4.00000 −0.529813
\(58\) −9.11684 −1.19710
\(59\) 1.25544 0.163444 0.0817220 0.996655i \(-0.473958\pi\)
0.0817220 + 0.996655i \(0.473958\pi\)
\(60\) 2.00000 0.258199
\(61\) 0.372281 0.0476657 0.0238329 0.999716i \(-0.492413\pi\)
0.0238329 + 0.999716i \(0.492413\pi\)
\(62\) −8.37228 −1.06328
\(63\) −4.37228 −0.550856
\(64\) 1.00000 0.125000
\(65\) 6.74456 0.836560
\(66\) 4.74456 0.584015
\(67\) −6.74456 −0.823979 −0.411990 0.911188i \(-0.635166\pi\)
−0.411990 + 0.911188i \(0.635166\pi\)
\(68\) 0.372281 0.0451457
\(69\) −9.48913 −1.14236
\(70\) −4.37228 −0.522588
\(71\) 4.74456 0.563076 0.281538 0.959550i \(-0.409155\pi\)
0.281538 + 0.959550i \(0.409155\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.74456 −0.321227 −0.160613 0.987017i \(-0.551347\pi\)
−0.160613 + 0.987017i \(0.551347\pi\)
\(74\) 1.00000 0.116248
\(75\) 2.00000 0.230940
\(76\) −2.00000 −0.229416
\(77\) −10.3723 −1.18203
\(78\) 13.4891 1.52734
\(79\) 6.74456 0.758823 0.379411 0.925228i \(-0.376127\pi\)
0.379411 + 0.925228i \(0.376127\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) −0.372281 −0.0411116
\(83\) 10.7446 1.17937 0.589684 0.807634i \(-0.299252\pi\)
0.589684 + 0.807634i \(0.299252\pi\)
\(84\) −8.74456 −0.954110
\(85\) 0.372281 0.0403796
\(86\) 1.62772 0.175521
\(87\) −18.2337 −1.95486
\(88\) 2.37228 0.252886
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 1.00000 0.105409
\(91\) −29.4891 −3.09130
\(92\) −4.74456 −0.494655
\(93\) −16.7446 −1.73633
\(94\) 2.74456 0.283080
\(95\) −2.00000 −0.205196
\(96\) 2.00000 0.204124
\(97\) 17.1168 1.73795 0.868976 0.494854i \(-0.164778\pi\)
0.868976 + 0.494854i \(0.164778\pi\)
\(98\) 12.1168 1.22399
\(99\) 2.37228 0.238423
\(100\) 1.00000 0.100000
\(101\) 11.4891 1.14321 0.571605 0.820529i \(-0.306321\pi\)
0.571605 + 0.820529i \(0.306321\pi\)
\(102\) 0.744563 0.0737227
\(103\) 13.4891 1.32912 0.664562 0.747234i \(-0.268618\pi\)
0.664562 + 0.747234i \(0.268618\pi\)
\(104\) 6.74456 0.661359
\(105\) −8.74456 −0.853382
\(106\) −4.37228 −0.424674
\(107\) 19.4891 1.88408 0.942042 0.335494i \(-0.108903\pi\)
0.942042 + 0.335494i \(0.108903\pi\)
\(108\) −4.00000 −0.384900
\(109\) −17.1168 −1.63950 −0.819748 0.572724i \(-0.805887\pi\)
−0.819748 + 0.572724i \(0.805887\pi\)
\(110\) 2.37228 0.226188
\(111\) 2.00000 0.189832
\(112\) −4.37228 −0.413142
\(113\) 11.6277 1.09384 0.546922 0.837184i \(-0.315799\pi\)
0.546922 + 0.837184i \(0.315799\pi\)
\(114\) −4.00000 −0.374634
\(115\) −4.74456 −0.442433
\(116\) −9.11684 −0.846478
\(117\) 6.74456 0.623535
\(118\) 1.25544 0.115572
\(119\) −1.62772 −0.149213
\(120\) 2.00000 0.182574
\(121\) −5.37228 −0.488389
\(122\) 0.372281 0.0337048
\(123\) −0.744563 −0.0671350
\(124\) −8.37228 −0.751853
\(125\) 1.00000 0.0894427
\(126\) −4.37228 −0.389514
\(127\) −5.25544 −0.466345 −0.233172 0.972435i \(-0.574911\pi\)
−0.233172 + 0.972435i \(0.574911\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.25544 0.286625
\(130\) 6.74456 0.591537
\(131\) −14.7446 −1.28824 −0.644119 0.764925i \(-0.722776\pi\)
−0.644119 + 0.764925i \(0.722776\pi\)
\(132\) 4.74456 0.412961
\(133\) 8.74456 0.758250
\(134\) −6.74456 −0.582641
\(135\) −4.00000 −0.344265
\(136\) 0.372281 0.0319229
\(137\) 5.25544 0.449002 0.224501 0.974474i \(-0.427925\pi\)
0.224501 + 0.974474i \(0.427925\pi\)
\(138\) −9.48913 −0.807768
\(139\) −19.1168 −1.62147 −0.810735 0.585414i \(-0.800932\pi\)
−0.810735 + 0.585414i \(0.800932\pi\)
\(140\) −4.37228 −0.369525
\(141\) 5.48913 0.462268
\(142\) 4.74456 0.398155
\(143\) 16.0000 1.33799
\(144\) 1.00000 0.0833333
\(145\) −9.11684 −0.757113
\(146\) −2.74456 −0.227142
\(147\) 24.2337 1.99876
\(148\) 1.00000 0.0821995
\(149\) 11.4891 0.941226 0.470613 0.882340i \(-0.344033\pi\)
0.470613 + 0.882340i \(0.344033\pi\)
\(150\) 2.00000 0.163299
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0.372281 0.0300972
\(154\) −10.3723 −0.835822
\(155\) −8.37228 −0.672478
\(156\) 13.4891 1.07999
\(157\) 11.6277 0.927993 0.463996 0.885837i \(-0.346415\pi\)
0.463996 + 0.885837i \(0.346415\pi\)
\(158\) 6.74456 0.536569
\(159\) −8.74456 −0.693489
\(160\) 1.00000 0.0790569
\(161\) 20.7446 1.63490
\(162\) −11.0000 −0.864242
\(163\) 13.6277 1.06741 0.533703 0.845672i \(-0.320800\pi\)
0.533703 + 0.845672i \(0.320800\pi\)
\(164\) −0.372281 −0.0290703
\(165\) 4.74456 0.369364
\(166\) 10.7446 0.833940
\(167\) −21.4891 −1.66288 −0.831439 0.555616i \(-0.812483\pi\)
−0.831439 + 0.555616i \(0.812483\pi\)
\(168\) −8.74456 −0.674658
\(169\) 32.4891 2.49916
\(170\) 0.372281 0.0285527
\(171\) −2.00000 −0.152944
\(172\) 1.62772 0.124112
\(173\) −5.86141 −0.445634 −0.222817 0.974860i \(-0.571525\pi\)
−0.222817 + 0.974860i \(0.571525\pi\)
\(174\) −18.2337 −1.38229
\(175\) −4.37228 −0.330513
\(176\) 2.37228 0.178817
\(177\) 2.51087 0.188729
\(178\) 10.0000 0.749532
\(179\) 8.23369 0.615415 0.307707 0.951481i \(-0.400438\pi\)
0.307707 + 0.951481i \(0.400438\pi\)
\(180\) 1.00000 0.0745356
\(181\) −15.4891 −1.15130 −0.575649 0.817697i \(-0.695250\pi\)
−0.575649 + 0.817697i \(0.695250\pi\)
\(182\) −29.4891 −2.18588
\(183\) 0.744563 0.0550397
\(184\) −4.74456 −0.349774
\(185\) 1.00000 0.0735215
\(186\) −16.7446 −1.22777
\(187\) 0.883156 0.0645828
\(188\) 2.74456 0.200168
\(189\) 17.4891 1.27215
\(190\) −2.00000 −0.145095
\(191\) 24.3723 1.76352 0.881758 0.471702i \(-0.156360\pi\)
0.881758 + 0.471702i \(0.156360\pi\)
\(192\) 2.00000 0.144338
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 17.1168 1.22892
\(195\) 13.4891 0.965976
\(196\) 12.1168 0.865489
\(197\) 4.51087 0.321387 0.160693 0.987004i \(-0.448627\pi\)
0.160693 + 0.987004i \(0.448627\pi\)
\(198\) 2.37228 0.168591
\(199\) −22.7446 −1.61232 −0.806160 0.591698i \(-0.798458\pi\)
−0.806160 + 0.591698i \(0.798458\pi\)
\(200\) 1.00000 0.0707107
\(201\) −13.4891 −0.951450
\(202\) 11.4891 0.808372
\(203\) 39.8614 2.79772
\(204\) 0.744563 0.0521298
\(205\) −0.372281 −0.0260013
\(206\) 13.4891 0.939832
\(207\) −4.74456 −0.329770
\(208\) 6.74456 0.467651
\(209\) −4.74456 −0.328188
\(210\) −8.74456 −0.603432
\(211\) 3.11684 0.214572 0.107286 0.994228i \(-0.465784\pi\)
0.107286 + 0.994228i \(0.465784\pi\)
\(212\) −4.37228 −0.300290
\(213\) 9.48913 0.650184
\(214\) 19.4891 1.33225
\(215\) 1.62772 0.111009
\(216\) −4.00000 −0.272166
\(217\) 36.6060 2.48498
\(218\) −17.1168 −1.15930
\(219\) −5.48913 −0.370921
\(220\) 2.37228 0.159939
\(221\) 2.51087 0.168900
\(222\) 2.00000 0.134231
\(223\) 4.37228 0.292790 0.146395 0.989226i \(-0.453233\pi\)
0.146395 + 0.989226i \(0.453233\pi\)
\(224\) −4.37228 −0.292135
\(225\) 1.00000 0.0666667
\(226\) 11.6277 0.773464
\(227\) −5.62772 −0.373525 −0.186762 0.982405i \(-0.559799\pi\)
−0.186762 + 0.982405i \(0.559799\pi\)
\(228\) −4.00000 −0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −4.74456 −0.312847
\(231\) −20.7446 −1.36489
\(232\) −9.11684 −0.598550
\(233\) −28.2337 −1.84965 −0.924825 0.380392i \(-0.875789\pi\)
−0.924825 + 0.380392i \(0.875789\pi\)
\(234\) 6.74456 0.440906
\(235\) 2.74456 0.179036
\(236\) 1.25544 0.0817220
\(237\) 13.4891 0.876213
\(238\) −1.62772 −0.105509
\(239\) −22.6060 −1.46226 −0.731129 0.682239i \(-0.761006\pi\)
−0.731129 + 0.682239i \(0.761006\pi\)
\(240\) 2.00000 0.129099
\(241\) 24.2337 1.56103 0.780515 0.625138i \(-0.214957\pi\)
0.780515 + 0.625138i \(0.214957\pi\)
\(242\) −5.37228 −0.345343
\(243\) −10.0000 −0.641500
\(244\) 0.372281 0.0238329
\(245\) 12.1168 0.774117
\(246\) −0.744563 −0.0474716
\(247\) −13.4891 −0.858292
\(248\) −8.37228 −0.531640
\(249\) 21.4891 1.36182
\(250\) 1.00000 0.0632456
\(251\) −11.4891 −0.725187 −0.362594 0.931947i \(-0.618109\pi\)
−0.362594 + 0.931947i \(0.618109\pi\)
\(252\) −4.37228 −0.275428
\(253\) −11.2554 −0.707623
\(254\) −5.25544 −0.329755
\(255\) 0.744563 0.0466263
\(256\) 1.00000 0.0625000
\(257\) 24.9783 1.55810 0.779050 0.626962i \(-0.215702\pi\)
0.779050 + 0.626962i \(0.215702\pi\)
\(258\) 3.25544 0.202675
\(259\) −4.37228 −0.271680
\(260\) 6.74456 0.418280
\(261\) −9.11684 −0.564318
\(262\) −14.7446 −0.910922
\(263\) −17.1168 −1.05547 −0.527735 0.849409i \(-0.676959\pi\)
−0.527735 + 0.849409i \(0.676959\pi\)
\(264\) 4.74456 0.292008
\(265\) −4.37228 −0.268587
\(266\) 8.74456 0.536164
\(267\) 20.0000 1.22398
\(268\) −6.74456 −0.411990
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) −4.00000 −0.243432
\(271\) 4.74456 0.288212 0.144106 0.989562i \(-0.453969\pi\)
0.144106 + 0.989562i \(0.453969\pi\)
\(272\) 0.372281 0.0225729
\(273\) −58.9783 −3.56953
\(274\) 5.25544 0.317493
\(275\) 2.37228 0.143054
\(276\) −9.48913 −0.571178
\(277\) −10.7446 −0.645578 −0.322789 0.946471i \(-0.604620\pi\)
−0.322789 + 0.946471i \(0.604620\pi\)
\(278\) −19.1168 −1.14655
\(279\) −8.37228 −0.501235
\(280\) −4.37228 −0.261294
\(281\) 13.2554 0.790753 0.395377 0.918519i \(-0.370614\pi\)
0.395377 + 0.918519i \(0.370614\pi\)
\(282\) 5.48913 0.326873
\(283\) −17.4891 −1.03962 −0.519810 0.854282i \(-0.673997\pi\)
−0.519810 + 0.854282i \(0.673997\pi\)
\(284\) 4.74456 0.281538
\(285\) −4.00000 −0.236940
\(286\) 16.0000 0.946100
\(287\) 1.62772 0.0960812
\(288\) 1.00000 0.0589256
\(289\) −16.8614 −0.991847
\(290\) −9.11684 −0.535360
\(291\) 34.2337 2.00681
\(292\) −2.74456 −0.160613
\(293\) 9.86141 0.576110 0.288055 0.957614i \(-0.406991\pi\)
0.288055 + 0.957614i \(0.406991\pi\)
\(294\) 24.2337 1.41334
\(295\) 1.25544 0.0730944
\(296\) 1.00000 0.0581238
\(297\) −9.48913 −0.550615
\(298\) 11.4891 0.665547
\(299\) −32.0000 −1.85061
\(300\) 2.00000 0.115470
\(301\) −7.11684 −0.410208
\(302\) −20.0000 −1.15087
\(303\) 22.9783 1.32007
\(304\) −2.00000 −0.114708
\(305\) 0.372281 0.0213168
\(306\) 0.372281 0.0212819
\(307\) −0.510875 −0.0291572 −0.0145786 0.999894i \(-0.504641\pi\)
−0.0145786 + 0.999894i \(0.504641\pi\)
\(308\) −10.3723 −0.591016
\(309\) 26.9783 1.53474
\(310\) −8.37228 −0.475514
\(311\) 4.37228 0.247929 0.123965 0.992287i \(-0.460439\pi\)
0.123965 + 0.992287i \(0.460439\pi\)
\(312\) 13.4891 0.763671
\(313\) 19.4891 1.10159 0.550795 0.834640i \(-0.314325\pi\)
0.550795 + 0.834640i \(0.314325\pi\)
\(314\) 11.6277 0.656190
\(315\) −4.37228 −0.246350
\(316\) 6.74456 0.379411
\(317\) 19.6277 1.10240 0.551201 0.834372i \(-0.314170\pi\)
0.551201 + 0.834372i \(0.314170\pi\)
\(318\) −8.74456 −0.490371
\(319\) −21.6277 −1.21092
\(320\) 1.00000 0.0559017
\(321\) 38.9783 2.17555
\(322\) 20.7446 1.15605
\(323\) −0.744563 −0.0414286
\(324\) −11.0000 −0.611111
\(325\) 6.74456 0.374121
\(326\) 13.6277 0.754770
\(327\) −34.2337 −1.89313
\(328\) −0.372281 −0.0205558
\(329\) −12.0000 −0.661581
\(330\) 4.74456 0.261180
\(331\) 0.510875 0.0280802 0.0140401 0.999901i \(-0.495531\pi\)
0.0140401 + 0.999901i \(0.495531\pi\)
\(332\) 10.7446 0.589684
\(333\) 1.00000 0.0547997
\(334\) −21.4891 −1.17583
\(335\) −6.74456 −0.368495
\(336\) −8.74456 −0.477055
\(337\) −18.7446 −1.02108 −0.510541 0.859854i \(-0.670555\pi\)
−0.510541 + 0.859854i \(0.670555\pi\)
\(338\) 32.4891 1.76718
\(339\) 23.2554 1.26306
\(340\) 0.372281 0.0201898
\(341\) −19.8614 −1.07556
\(342\) −2.00000 −0.108148
\(343\) −22.3723 −1.20799
\(344\) 1.62772 0.0877607
\(345\) −9.48913 −0.510877
\(346\) −5.86141 −0.315111
\(347\) 22.9783 1.23354 0.616769 0.787145i \(-0.288441\pi\)
0.616769 + 0.787145i \(0.288441\pi\)
\(348\) −18.2337 −0.977428
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) −4.37228 −0.233708
\(351\) −26.9783 −1.43999
\(352\) 2.37228 0.126443
\(353\) 5.86141 0.311971 0.155986 0.987759i \(-0.450145\pi\)
0.155986 + 0.987759i \(0.450145\pi\)
\(354\) 2.51087 0.133451
\(355\) 4.74456 0.251815
\(356\) 10.0000 0.529999
\(357\) −3.25544 −0.172296
\(358\) 8.23369 0.435164
\(359\) 14.9783 0.790522 0.395261 0.918569i \(-0.370654\pi\)
0.395261 + 0.918569i \(0.370654\pi\)
\(360\) 1.00000 0.0527046
\(361\) −15.0000 −0.789474
\(362\) −15.4891 −0.814090
\(363\) −10.7446 −0.563943
\(364\) −29.4891 −1.54565
\(365\) −2.74456 −0.143657
\(366\) 0.744563 0.0389189
\(367\) 21.1168 1.10229 0.551145 0.834409i \(-0.314191\pi\)
0.551145 + 0.834409i \(0.314191\pi\)
\(368\) −4.74456 −0.247327
\(369\) −0.372281 −0.0193802
\(370\) 1.00000 0.0519875
\(371\) 19.1168 0.992497
\(372\) −16.7446 −0.868165
\(373\) −8.51087 −0.440676 −0.220338 0.975424i \(-0.570716\pi\)
−0.220338 + 0.975424i \(0.570716\pi\)
\(374\) 0.883156 0.0456669
\(375\) 2.00000 0.103280
\(376\) 2.74456 0.141540
\(377\) −61.4891 −3.16685
\(378\) 17.4891 0.899544
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) −2.00000 −0.102598
\(381\) −10.5109 −0.538488
\(382\) 24.3723 1.24699
\(383\) 9.48913 0.484872 0.242436 0.970167i \(-0.422054\pi\)
0.242436 + 0.970167i \(0.422054\pi\)
\(384\) 2.00000 0.102062
\(385\) −10.3723 −0.528620
\(386\) 2.00000 0.101797
\(387\) 1.62772 0.0827416
\(388\) 17.1168 0.868976
\(389\) −13.8614 −0.702801 −0.351401 0.936225i \(-0.614294\pi\)
−0.351401 + 0.936225i \(0.614294\pi\)
\(390\) 13.4891 0.683048
\(391\) −1.76631 −0.0893262
\(392\) 12.1168 0.611993
\(393\) −29.4891 −1.48753
\(394\) 4.51087 0.227255
\(395\) 6.74456 0.339356
\(396\) 2.37228 0.119212
\(397\) −20.9783 −1.05287 −0.526434 0.850216i \(-0.676471\pi\)
−0.526434 + 0.850216i \(0.676471\pi\)
\(398\) −22.7446 −1.14008
\(399\) 17.4891 0.875551
\(400\) 1.00000 0.0500000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) −13.4891 −0.672776
\(403\) −56.4674 −2.81284
\(404\) 11.4891 0.571605
\(405\) −11.0000 −0.546594
\(406\) 39.8614 1.97829
\(407\) 2.37228 0.117590
\(408\) 0.744563 0.0368613
\(409\) −28.2337 −1.39607 −0.698033 0.716066i \(-0.745941\pi\)
−0.698033 + 0.716066i \(0.745941\pi\)
\(410\) −0.372281 −0.0183857
\(411\) 10.5109 0.518463
\(412\) 13.4891 0.664562
\(413\) −5.48913 −0.270102
\(414\) −4.74456 −0.233183
\(415\) 10.7446 0.527430
\(416\) 6.74456 0.330679
\(417\) −38.2337 −1.87231
\(418\) −4.74456 −0.232064
\(419\) −9.48913 −0.463574 −0.231787 0.972767i \(-0.574457\pi\)
−0.231787 + 0.972767i \(0.574457\pi\)
\(420\) −8.74456 −0.426691
\(421\) 15.4891 0.754894 0.377447 0.926031i \(-0.376802\pi\)
0.377447 + 0.926031i \(0.376802\pi\)
\(422\) 3.11684 0.151726
\(423\) 2.74456 0.133445
\(424\) −4.37228 −0.212337
\(425\) 0.372281 0.0180583
\(426\) 9.48913 0.459750
\(427\) −1.62772 −0.0787708
\(428\) 19.4891 0.942042
\(429\) 32.0000 1.54497
\(430\) 1.62772 0.0784956
\(431\) 4.37228 0.210605 0.105303 0.994440i \(-0.466419\pi\)
0.105303 + 0.994440i \(0.466419\pi\)
\(432\) −4.00000 −0.192450
\(433\) −32.9783 −1.58483 −0.792417 0.609980i \(-0.791177\pi\)
−0.792417 + 0.609980i \(0.791177\pi\)
\(434\) 36.6060 1.75714
\(435\) −18.2337 −0.874238
\(436\) −17.1168 −0.819748
\(437\) 9.48913 0.453926
\(438\) −5.48913 −0.262281
\(439\) −7.62772 −0.364051 −0.182026 0.983294i \(-0.558265\pi\)
−0.182026 + 0.983294i \(0.558265\pi\)
\(440\) 2.37228 0.113094
\(441\) 12.1168 0.576993
\(442\) 2.51087 0.119430
\(443\) 28.9783 1.37680 0.688399 0.725332i \(-0.258314\pi\)
0.688399 + 0.725332i \(0.258314\pi\)
\(444\) 2.00000 0.0949158
\(445\) 10.0000 0.474045
\(446\) 4.37228 0.207034
\(447\) 22.9783 1.08683
\(448\) −4.37228 −0.206571
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 1.00000 0.0471405
\(451\) −0.883156 −0.0415862
\(452\) 11.6277 0.546922
\(453\) −40.0000 −1.87936
\(454\) −5.62772 −0.264122
\(455\) −29.4891 −1.38247
\(456\) −4.00000 −0.187317
\(457\) −9.86141 −0.461297 −0.230649 0.973037i \(-0.574085\pi\)
−0.230649 + 0.973037i \(0.574085\pi\)
\(458\) −10.0000 −0.467269
\(459\) −1.48913 −0.0695064
\(460\) −4.74456 −0.221216
\(461\) 24.0951 1.12222 0.561110 0.827741i \(-0.310374\pi\)
0.561110 + 0.827741i \(0.310374\pi\)
\(462\) −20.7446 −0.965124
\(463\) 17.4891 0.812789 0.406394 0.913698i \(-0.366786\pi\)
0.406394 + 0.913698i \(0.366786\pi\)
\(464\) −9.11684 −0.423239
\(465\) −16.7446 −0.776510
\(466\) −28.2337 −1.30790
\(467\) −21.3505 −0.987985 −0.493992 0.869466i \(-0.664463\pi\)
−0.493992 + 0.869466i \(0.664463\pi\)
\(468\) 6.74456 0.311768
\(469\) 29.4891 1.36168
\(470\) 2.74456 0.126597
\(471\) 23.2554 1.07155
\(472\) 1.25544 0.0577862
\(473\) 3.86141 0.177548
\(474\) 13.4891 0.619576
\(475\) −2.00000 −0.0917663
\(476\) −1.62772 −0.0746064
\(477\) −4.37228 −0.200193
\(478\) −22.6060 −1.03397
\(479\) 14.7446 0.673696 0.336848 0.941559i \(-0.390639\pi\)
0.336848 + 0.941559i \(0.390639\pi\)
\(480\) 2.00000 0.0912871
\(481\) 6.74456 0.307526
\(482\) 24.2337 1.10381
\(483\) 41.4891 1.88782
\(484\) −5.37228 −0.244195
\(485\) 17.1168 0.777236
\(486\) −10.0000 −0.453609
\(487\) 19.7228 0.893726 0.446863 0.894602i \(-0.352541\pi\)
0.446863 + 0.894602i \(0.352541\pi\)
\(488\) 0.372281 0.0168524
\(489\) 27.2554 1.23253
\(490\) 12.1168 0.547383
\(491\) −30.9783 −1.39803 −0.699014 0.715108i \(-0.746378\pi\)
−0.699014 + 0.715108i \(0.746378\pi\)
\(492\) −0.744563 −0.0335675
\(493\) −3.39403 −0.152859
\(494\) −13.4891 −0.606904
\(495\) 2.37228 0.106626
\(496\) −8.37228 −0.375927
\(497\) −20.7446 −0.930521
\(498\) 21.4891 0.962951
\(499\) 20.9783 0.939115 0.469558 0.882902i \(-0.344413\pi\)
0.469558 + 0.882902i \(0.344413\pi\)
\(500\) 1.00000 0.0447214
\(501\) −42.9783 −1.92013
\(502\) −11.4891 −0.512785
\(503\) 6.51087 0.290306 0.145153 0.989409i \(-0.453633\pi\)
0.145153 + 0.989409i \(0.453633\pi\)
\(504\) −4.37228 −0.194757
\(505\) 11.4891 0.511259
\(506\) −11.2554 −0.500365
\(507\) 64.9783 2.88579
\(508\) −5.25544 −0.233172
\(509\) −22.7446 −1.00814 −0.504068 0.863664i \(-0.668164\pi\)
−0.504068 + 0.863664i \(0.668164\pi\)
\(510\) 0.744563 0.0329698
\(511\) 12.0000 0.530849
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) 24.9783 1.10174
\(515\) 13.4891 0.594402
\(516\) 3.25544 0.143313
\(517\) 6.51087 0.286348
\(518\) −4.37228 −0.192107
\(519\) −11.7228 −0.514574
\(520\) 6.74456 0.295769
\(521\) −36.0951 −1.58135 −0.790677 0.612233i \(-0.790271\pi\)
−0.790677 + 0.612233i \(0.790271\pi\)
\(522\) −9.11684 −0.399033
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) −14.7446 −0.644119
\(525\) −8.74456 −0.381644
\(526\) −17.1168 −0.746330
\(527\) −3.11684 −0.135772
\(528\) 4.74456 0.206481
\(529\) −0.489125 −0.0212663
\(530\) −4.37228 −0.189920
\(531\) 1.25544 0.0544813
\(532\) 8.74456 0.379125
\(533\) −2.51087 −0.108758
\(534\) 20.0000 0.865485
\(535\) 19.4891 0.842588
\(536\) −6.74456 −0.291321
\(537\) 16.4674 0.710620
\(538\) 10.0000 0.431131
\(539\) 28.7446 1.23812
\(540\) −4.00000 −0.172133
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 4.74456 0.203796
\(543\) −30.9783 −1.32940
\(544\) 0.372281 0.0159614
\(545\) −17.1168 −0.733205
\(546\) −58.9783 −2.52404
\(547\) 33.6277 1.43782 0.718909 0.695104i \(-0.244642\pi\)
0.718909 + 0.695104i \(0.244642\pi\)
\(548\) 5.25544 0.224501
\(549\) 0.372281 0.0158886
\(550\) 2.37228 0.101154
\(551\) 18.2337 0.776781
\(552\) −9.48913 −0.403884
\(553\) −29.4891 −1.25401
\(554\) −10.7446 −0.456493
\(555\) 2.00000 0.0848953
\(556\) −19.1168 −0.810735
\(557\) −2.74456 −0.116291 −0.0581454 0.998308i \(-0.518519\pi\)
−0.0581454 + 0.998308i \(0.518519\pi\)
\(558\) −8.37228 −0.354427
\(559\) 10.9783 0.464331
\(560\) −4.37228 −0.184763
\(561\) 1.76631 0.0745738
\(562\) 13.2554 0.559147
\(563\) −30.0951 −1.26836 −0.634179 0.773187i \(-0.718662\pi\)
−0.634179 + 0.773187i \(0.718662\pi\)
\(564\) 5.48913 0.231134
\(565\) 11.6277 0.489182
\(566\) −17.4891 −0.735123
\(567\) 48.0951 2.01980
\(568\) 4.74456 0.199077
\(569\) 12.9783 0.544077 0.272038 0.962286i \(-0.412302\pi\)
0.272038 + 0.962286i \(0.412302\pi\)
\(570\) −4.00000 −0.167542
\(571\) 12.6060 0.527543 0.263772 0.964585i \(-0.415033\pi\)
0.263772 + 0.964585i \(0.415033\pi\)
\(572\) 16.0000 0.668994
\(573\) 48.7446 2.03633
\(574\) 1.62772 0.0679397
\(575\) −4.74456 −0.197862
\(576\) 1.00000 0.0416667
\(577\) 15.4891 0.644821 0.322410 0.946600i \(-0.395507\pi\)
0.322410 + 0.946600i \(0.395507\pi\)
\(578\) −16.8614 −0.701342
\(579\) 4.00000 0.166234
\(580\) −9.11684 −0.378556
\(581\) −46.9783 −1.94899
\(582\) 34.2337 1.41903
\(583\) −10.3723 −0.429576
\(584\) −2.74456 −0.113571
\(585\) 6.74456 0.278853
\(586\) 9.86141 0.407371
\(587\) −26.8397 −1.10779 −0.553896 0.832586i \(-0.686859\pi\)
−0.553896 + 0.832586i \(0.686859\pi\)
\(588\) 24.2337 0.999380
\(589\) 16.7446 0.689948
\(590\) 1.25544 0.0516855
\(591\) 9.02175 0.371105
\(592\) 1.00000 0.0410997
\(593\) 24.2337 0.995158 0.497579 0.867419i \(-0.334222\pi\)
0.497579 + 0.867419i \(0.334222\pi\)
\(594\) −9.48913 −0.389344
\(595\) −1.62772 −0.0667300
\(596\) 11.4891 0.470613
\(597\) −45.4891 −1.86175
\(598\) −32.0000 −1.30858
\(599\) −5.48913 −0.224280 −0.112140 0.993692i \(-0.535770\pi\)
−0.112140 + 0.993692i \(0.535770\pi\)
\(600\) 2.00000 0.0816497
\(601\) 0.372281 0.0151857 0.00759284 0.999971i \(-0.497583\pi\)
0.00759284 + 0.999971i \(0.497583\pi\)
\(602\) −7.11684 −0.290061
\(603\) −6.74456 −0.274660
\(604\) −20.0000 −0.813788
\(605\) −5.37228 −0.218414
\(606\) 22.9783 0.933428
\(607\) −39.7228 −1.61230 −0.806150 0.591712i \(-0.798452\pi\)
−0.806150 + 0.591712i \(0.798452\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 79.7228 3.23053
\(610\) 0.372281 0.0150732
\(611\) 18.5109 0.748870
\(612\) 0.372281 0.0150486
\(613\) −20.0951 −0.811633 −0.405817 0.913955i \(-0.633013\pi\)
−0.405817 + 0.913955i \(0.633013\pi\)
\(614\) −0.510875 −0.0206172
\(615\) −0.744563 −0.0300237
\(616\) −10.3723 −0.417911
\(617\) 25.7228 1.03556 0.517781 0.855513i \(-0.326758\pi\)
0.517781 + 0.855513i \(0.326758\pi\)
\(618\) 26.9783 1.08522
\(619\) 12.8832 0.517818 0.258909 0.965902i \(-0.416637\pi\)
0.258909 + 0.965902i \(0.416637\pi\)
\(620\) −8.37228 −0.336239
\(621\) 18.9783 0.761571
\(622\) 4.37228 0.175313
\(623\) −43.7228 −1.75172
\(624\) 13.4891 0.539997
\(625\) 1.00000 0.0400000
\(626\) 19.4891 0.778942
\(627\) −9.48913 −0.378959
\(628\) 11.6277 0.463996
\(629\) 0.372281 0.0148438
\(630\) −4.37228 −0.174196
\(631\) 44.0951 1.75540 0.877699 0.479212i \(-0.159078\pi\)
0.877699 + 0.479212i \(0.159078\pi\)
\(632\) 6.74456 0.268284
\(633\) 6.23369 0.247767
\(634\) 19.6277 0.779516
\(635\) −5.25544 −0.208556
\(636\) −8.74456 −0.346744
\(637\) 81.7228 3.23798
\(638\) −21.6277 −0.856250
\(639\) 4.74456 0.187692
\(640\) 1.00000 0.0395285
\(641\) −9.39403 −0.371042 −0.185521 0.982640i \(-0.559397\pi\)
−0.185521 + 0.982640i \(0.559397\pi\)
\(642\) 38.9783 1.53835
\(643\) −2.37228 −0.0935536 −0.0467768 0.998905i \(-0.514895\pi\)
−0.0467768 + 0.998905i \(0.514895\pi\)
\(644\) 20.7446 0.817450
\(645\) 3.25544 0.128183
\(646\) −0.744563 −0.0292944
\(647\) 9.48913 0.373056 0.186528 0.982450i \(-0.440276\pi\)
0.186528 + 0.982450i \(0.440276\pi\)
\(648\) −11.0000 −0.432121
\(649\) 2.97825 0.116907
\(650\) 6.74456 0.264544
\(651\) 73.2119 2.86940
\(652\) 13.6277 0.533703
\(653\) 23.4891 0.919201 0.459600 0.888126i \(-0.347993\pi\)
0.459600 + 0.888126i \(0.347993\pi\)
\(654\) −34.2337 −1.33864
\(655\) −14.7446 −0.576118
\(656\) −0.372281 −0.0145351
\(657\) −2.74456 −0.107076
\(658\) −12.0000 −0.467809
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 4.74456 0.184682
\(661\) −47.3505 −1.84172 −0.920861 0.389891i \(-0.872513\pi\)
−0.920861 + 0.389891i \(0.872513\pi\)
\(662\) 0.510875 0.0198557
\(663\) 5.02175 0.195029
\(664\) 10.7446 0.416970
\(665\) 8.74456 0.339100
\(666\) 1.00000 0.0387492
\(667\) 43.2554 1.67486
\(668\) −21.4891 −0.831439
\(669\) 8.74456 0.338084
\(670\) −6.74456 −0.260565
\(671\) 0.883156 0.0340939
\(672\) −8.74456 −0.337329
\(673\) −31.4891 −1.21382 −0.606908 0.794772i \(-0.707590\pi\)
−0.606908 + 0.794772i \(0.707590\pi\)
\(674\) −18.7446 −0.722014
\(675\) −4.00000 −0.153960
\(676\) 32.4891 1.24958
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 23.2554 0.893120
\(679\) −74.8397 −2.87208
\(680\) 0.372281 0.0142763
\(681\) −11.2554 −0.431309
\(682\) −19.8614 −0.760533
\(683\) 14.3723 0.549940 0.274970 0.961453i \(-0.411332\pi\)
0.274970 + 0.961453i \(0.411332\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 5.25544 0.200800
\(686\) −22.3723 −0.854178
\(687\) −20.0000 −0.763048
\(688\) 1.62772 0.0620562
\(689\) −29.4891 −1.12345
\(690\) −9.48913 −0.361245
\(691\) −29.3505 −1.11655 −0.558273 0.829657i \(-0.688536\pi\)
−0.558273 + 0.829657i \(0.688536\pi\)
\(692\) −5.86141 −0.222817
\(693\) −10.3723 −0.394010
\(694\) 22.9783 0.872242
\(695\) −19.1168 −0.725143
\(696\) −18.2337 −0.691146
\(697\) −0.138593 −0.00524960
\(698\) −22.0000 −0.832712
\(699\) −56.4674 −2.13579
\(700\) −4.37228 −0.165257
\(701\) 42.4674 1.60397 0.801985 0.597344i \(-0.203777\pi\)
0.801985 + 0.597344i \(0.203777\pi\)
\(702\) −26.9783 −1.01823
\(703\) −2.00000 −0.0754314
\(704\) 2.37228 0.0894087
\(705\) 5.48913 0.206732
\(706\) 5.86141 0.220597
\(707\) −50.2337 −1.88923
\(708\) 2.51087 0.0943645
\(709\) 22.8832 0.859395 0.429697 0.902973i \(-0.358620\pi\)
0.429697 + 0.902973i \(0.358620\pi\)
\(710\) 4.74456 0.178060
\(711\) 6.74456 0.252941
\(712\) 10.0000 0.374766
\(713\) 39.7228 1.48763
\(714\) −3.25544 −0.121832
\(715\) 16.0000 0.598366
\(716\) 8.23369 0.307707
\(717\) −45.2119 −1.68847
\(718\) 14.9783 0.558983
\(719\) −23.2554 −0.867281 −0.433641 0.901086i \(-0.642771\pi\)
−0.433641 + 0.901086i \(0.642771\pi\)
\(720\) 1.00000 0.0372678
\(721\) −58.9783 −2.19646
\(722\) −15.0000 −0.558242
\(723\) 48.4674 1.80252
\(724\) −15.4891 −0.575649
\(725\) −9.11684 −0.338591
\(726\) −10.7446 −0.398768
\(727\) −48.0000 −1.78022 −0.890111 0.455744i \(-0.849373\pi\)
−0.890111 + 0.455744i \(0.849373\pi\)
\(728\) −29.4891 −1.09294
\(729\) 13.0000 0.481481
\(730\) −2.74456 −0.101581
\(731\) 0.605969 0.0224126
\(732\) 0.744563 0.0275198
\(733\) −23.6277 −0.872710 −0.436355 0.899775i \(-0.643731\pi\)
−0.436355 + 0.899775i \(0.643731\pi\)
\(734\) 21.1168 0.779437
\(735\) 24.2337 0.893873
\(736\) −4.74456 −0.174887
\(737\) −16.0000 −0.589368
\(738\) −0.372281 −0.0137039
\(739\) −8.13859 −0.299383 −0.149691 0.988733i \(-0.547828\pi\)
−0.149691 + 0.988733i \(0.547828\pi\)
\(740\) 1.00000 0.0367607
\(741\) −26.9783 −0.991071
\(742\) 19.1168 0.701801
\(743\) 0.372281 0.0136577 0.00682884 0.999977i \(-0.497826\pi\)
0.00682884 + 0.999977i \(0.497826\pi\)
\(744\) −16.7446 −0.613885
\(745\) 11.4891 0.420929
\(746\) −8.51087 −0.311605
\(747\) 10.7446 0.393123
\(748\) 0.883156 0.0322914
\(749\) −85.2119 −3.11358
\(750\) 2.00000 0.0730297
\(751\) −0.744563 −0.0271695 −0.0135847 0.999908i \(-0.504324\pi\)
−0.0135847 + 0.999908i \(0.504324\pi\)
\(752\) 2.74456 0.100084
\(753\) −22.9783 −0.837374
\(754\) −61.4891 −2.23930
\(755\) −20.0000 −0.727875
\(756\) 17.4891 0.636073
\(757\) −47.9565 −1.74301 −0.871504 0.490388i \(-0.836855\pi\)
−0.871504 + 0.490388i \(0.836855\pi\)
\(758\) −8.00000 −0.290573
\(759\) −22.5109 −0.817093
\(760\) −2.00000 −0.0725476
\(761\) −0.372281 −0.0134952 −0.00674759 0.999977i \(-0.502148\pi\)
−0.00674759 + 0.999977i \(0.502148\pi\)
\(762\) −10.5109 −0.380769
\(763\) 74.8397 2.70938
\(764\) 24.3723 0.881758
\(765\) 0.372281 0.0134599
\(766\) 9.48913 0.342856
\(767\) 8.46738 0.305739
\(768\) 2.00000 0.0721688
\(769\) 11.7663 0.424304 0.212152 0.977237i \(-0.431953\pi\)
0.212152 + 0.977237i \(0.431953\pi\)
\(770\) −10.3723 −0.373791
\(771\) 49.9565 1.79914
\(772\) 2.00000 0.0719816
\(773\) 27.3505 0.983730 0.491865 0.870671i \(-0.336315\pi\)
0.491865 + 0.870671i \(0.336315\pi\)
\(774\) 1.62772 0.0585071
\(775\) −8.37228 −0.300741
\(776\) 17.1168 0.614459
\(777\) −8.74456 −0.313709
\(778\) −13.8614 −0.496956
\(779\) 0.744563 0.0266767
\(780\) 13.4891 0.482988
\(781\) 11.2554 0.402751
\(782\) −1.76631 −0.0631632
\(783\) 36.4674 1.30324
\(784\) 12.1168 0.432744
\(785\) 11.6277 0.415011
\(786\) −29.4891 −1.05184
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 4.51087 0.160693
\(789\) −34.2337 −1.21875
\(790\) 6.74456 0.239961
\(791\) −50.8397 −1.80765
\(792\) 2.37228 0.0842953
\(793\) 2.51087 0.0891638
\(794\) −20.9783 −0.744490
\(795\) −8.74456 −0.310138
\(796\) −22.7446 −0.806160
\(797\) −4.51087 −0.159783 −0.0798917 0.996804i \(-0.525457\pi\)
−0.0798917 + 0.996804i \(0.525457\pi\)
\(798\) 17.4891 0.619108
\(799\) 1.02175 0.0361469
\(800\) 1.00000 0.0353553
\(801\) 10.0000 0.353333
\(802\) 10.0000 0.353112
\(803\) −6.51087 −0.229764
\(804\) −13.4891 −0.475725
\(805\) 20.7446 0.731150
\(806\) −56.4674 −1.98898
\(807\) 20.0000 0.704033
\(808\) 11.4891 0.404186
\(809\) −28.5109 −1.00239 −0.501194 0.865335i \(-0.667106\pi\)
−0.501194 + 0.865335i \(0.667106\pi\)
\(810\) −11.0000 −0.386501
\(811\) 26.5109 0.930923 0.465461 0.885068i \(-0.345888\pi\)
0.465461 + 0.885068i \(0.345888\pi\)
\(812\) 39.8614 1.39886
\(813\) 9.48913 0.332798
\(814\) 2.37228 0.0831484
\(815\) 13.6277 0.477358
\(816\) 0.744563 0.0260649
\(817\) −3.25544 −0.113893
\(818\) −28.2337 −0.987168
\(819\) −29.4891 −1.03043
\(820\) −0.372281 −0.0130006
\(821\) −21.2554 −0.741820 −0.370910 0.928669i \(-0.620954\pi\)
−0.370910 + 0.928669i \(0.620954\pi\)
\(822\) 10.5109 0.366609
\(823\) 17.7228 0.617778 0.308889 0.951098i \(-0.400043\pi\)
0.308889 + 0.951098i \(0.400043\pi\)
\(824\) 13.4891 0.469916
\(825\) 4.74456 0.165184
\(826\) −5.48913 −0.190991
\(827\) −2.64947 −0.0921310 −0.0460655 0.998938i \(-0.514668\pi\)
−0.0460655 + 0.998938i \(0.514668\pi\)
\(828\) −4.74456 −0.164885
\(829\) 11.3505 0.394220 0.197110 0.980381i \(-0.436844\pi\)
0.197110 + 0.980381i \(0.436844\pi\)
\(830\) 10.7446 0.372949
\(831\) −21.4891 −0.745449
\(832\) 6.74456 0.233826
\(833\) 4.51087 0.156293
\(834\) −38.2337 −1.32392
\(835\) −21.4891 −0.743662
\(836\) −4.74456 −0.164094
\(837\) 33.4891 1.15755
\(838\) −9.48913 −0.327796
\(839\) −6.51087 −0.224780 −0.112390 0.993664i \(-0.535851\pi\)
−0.112390 + 0.993664i \(0.535851\pi\)
\(840\) −8.74456 −0.301716
\(841\) 54.1168 1.86610
\(842\) 15.4891 0.533791
\(843\) 26.5109 0.913083
\(844\) 3.11684 0.107286
\(845\) 32.4891 1.11766
\(846\) 2.74456 0.0943600
\(847\) 23.4891 0.807096
\(848\) −4.37228 −0.150145
\(849\) −34.9783 −1.20045
\(850\) 0.372281 0.0127691
\(851\) −4.74456 −0.162642
\(852\) 9.48913 0.325092
\(853\) 12.5109 0.428364 0.214182 0.976794i \(-0.431291\pi\)
0.214182 + 0.976794i \(0.431291\pi\)
\(854\) −1.62772 −0.0556994
\(855\) −2.00000 −0.0683986
\(856\) 19.4891 0.666125
\(857\) −57.8614 −1.97651 −0.988254 0.152820i \(-0.951164\pi\)
−0.988254 + 0.152820i \(0.951164\pi\)
\(858\) 32.0000 1.09246
\(859\) 37.7228 1.28709 0.643543 0.765410i \(-0.277464\pi\)
0.643543 + 0.765410i \(0.277464\pi\)
\(860\) 1.62772 0.0555047
\(861\) 3.25544 0.110945
\(862\) 4.37228 0.148920
\(863\) −20.8397 −0.709390 −0.354695 0.934982i \(-0.615415\pi\)
−0.354695 + 0.934982i \(0.615415\pi\)
\(864\) −4.00000 −0.136083
\(865\) −5.86141 −0.199294
\(866\) −32.9783 −1.12065
\(867\) −33.7228 −1.14529
\(868\) 36.6060 1.24249
\(869\) 16.0000 0.542763
\(870\) −18.2337 −0.618180
\(871\) −45.4891 −1.54134
\(872\) −17.1168 −0.579649
\(873\) 17.1168 0.579317
\(874\) 9.48913 0.320974
\(875\) −4.37228 −0.147810
\(876\) −5.48913 −0.185460
\(877\) 43.6277 1.47320 0.736602 0.676327i \(-0.236429\pi\)
0.736602 + 0.676327i \(0.236429\pi\)
\(878\) −7.62772 −0.257423
\(879\) 19.7228 0.665234
\(880\) 2.37228 0.0799696
\(881\) −4.37228 −0.147306 −0.0736530 0.997284i \(-0.523466\pi\)
−0.0736530 + 0.997284i \(0.523466\pi\)
\(882\) 12.1168 0.407995
\(883\) −5.62772 −0.189388 −0.0946939 0.995506i \(-0.530187\pi\)
−0.0946939 + 0.995506i \(0.530187\pi\)
\(884\) 2.51087 0.0844499
\(885\) 2.51087 0.0844021
\(886\) 28.9783 0.973543
\(887\) −19.6277 −0.659034 −0.329517 0.944150i \(-0.606886\pi\)
−0.329517 + 0.944150i \(0.606886\pi\)
\(888\) 2.00000 0.0671156
\(889\) 22.9783 0.770666
\(890\) 10.0000 0.335201
\(891\) −26.0951 −0.874219
\(892\) 4.37228 0.146395
\(893\) −5.48913 −0.183687
\(894\) 22.9783 0.768508
\(895\) 8.23369 0.275222
\(896\) −4.37228 −0.146068
\(897\) −64.0000 −2.13690
\(898\) 18.0000 0.600668
\(899\) 76.3288 2.54571
\(900\) 1.00000 0.0333333
\(901\) −1.62772 −0.0542272
\(902\) −0.883156 −0.0294059
\(903\) −14.2337 −0.473667
\(904\) 11.6277 0.386732
\(905\) −15.4891 −0.514876
\(906\) −40.0000 −1.32891
\(907\) −28.4674 −0.945244 −0.472622 0.881265i \(-0.656692\pi\)
−0.472622 + 0.881265i \(0.656692\pi\)
\(908\) −5.62772 −0.186762
\(909\) 11.4891 0.381070
\(910\) −29.4891 −0.977555
\(911\) 52.2337 1.73058 0.865290 0.501272i \(-0.167134\pi\)
0.865290 + 0.501272i \(0.167134\pi\)
\(912\) −4.00000 −0.132453
\(913\) 25.4891 0.843567
\(914\) −9.86141 −0.326186
\(915\) 0.744563 0.0246145
\(916\) −10.0000 −0.330409
\(917\) 64.4674 2.12890
\(918\) −1.48913 −0.0491485
\(919\) −49.7228 −1.64020 −0.820102 0.572217i \(-0.806083\pi\)
−0.820102 + 0.572217i \(0.806083\pi\)
\(920\) −4.74456 −0.156424
\(921\) −1.02175 −0.0336678
\(922\) 24.0951 0.793530
\(923\) 32.0000 1.05329
\(924\) −20.7446 −0.682446
\(925\) 1.00000 0.0328798
\(926\) 17.4891 0.574728
\(927\) 13.4891 0.443041
\(928\) −9.11684 −0.299275
\(929\) −32.0951 −1.05301 −0.526503 0.850173i \(-0.676497\pi\)
−0.526503 + 0.850173i \(0.676497\pi\)
\(930\) −16.7446 −0.549076
\(931\) −24.2337 −0.794227
\(932\) −28.2337 −0.924825
\(933\) 8.74456 0.286284
\(934\) −21.3505 −0.698611
\(935\) 0.883156 0.0288823
\(936\) 6.74456 0.220453
\(937\) −8.97825 −0.293307 −0.146653 0.989188i \(-0.546850\pi\)
−0.146653 + 0.989188i \(0.546850\pi\)
\(938\) 29.4891 0.962854
\(939\) 38.9783 1.27201
\(940\) 2.74456 0.0895178
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 23.2554 0.757703
\(943\) 1.76631 0.0575190
\(944\) 1.25544 0.0408610
\(945\) 17.4891 0.568921
\(946\) 3.86141 0.125545
\(947\) −26.0951 −0.847977 −0.423988 0.905668i \(-0.639370\pi\)
−0.423988 + 0.905668i \(0.639370\pi\)
\(948\) 13.4891 0.438106
\(949\) −18.5109 −0.600888
\(950\) −2.00000 −0.0648886
\(951\) 39.2554 1.27294
\(952\) −1.62772 −0.0527547
\(953\) 11.7663 0.381148 0.190574 0.981673i \(-0.438965\pi\)
0.190574 + 0.981673i \(0.438965\pi\)
\(954\) −4.37228 −0.141558
\(955\) 24.3723 0.788669
\(956\) −22.6060 −0.731129
\(957\) −43.2554 −1.39825
\(958\) 14.7446 0.476375
\(959\) −22.9783 −0.742006
\(960\) 2.00000 0.0645497
\(961\) 39.0951 1.26113
\(962\) 6.74456 0.217453
\(963\) 19.4891 0.628028
\(964\) 24.2337 0.780515
\(965\) 2.00000 0.0643823
\(966\) 41.4891 1.33489
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) −5.37228 −0.172672
\(969\) −1.48913 −0.0478376
\(970\) 17.1168 0.549589
\(971\) 20.6060 0.661277 0.330639 0.943757i \(-0.392736\pi\)
0.330639 + 0.943757i \(0.392736\pi\)
\(972\) −10.0000 −0.320750
\(973\) 83.5842 2.67959
\(974\) 19.7228 0.631960
\(975\) 13.4891 0.431998
\(976\) 0.372281 0.0119164
\(977\) −29.1168 −0.931530 −0.465765 0.884908i \(-0.654221\pi\)
−0.465765 + 0.884908i \(0.654221\pi\)
\(978\) 27.2554 0.871533
\(979\) 23.7228 0.758184
\(980\) 12.1168 0.387058
\(981\) −17.1168 −0.546499
\(982\) −30.9783 −0.988556
\(983\) 2.13859 0.0682105 0.0341053 0.999418i \(-0.489142\pi\)
0.0341053 + 0.999418i \(0.489142\pi\)
\(984\) −0.744563 −0.0237358
\(985\) 4.51087 0.143728
\(986\) −3.39403 −0.108088
\(987\) −24.0000 −0.763928
\(988\) −13.4891 −0.429146
\(989\) −7.72281 −0.245571
\(990\) 2.37228 0.0753960
\(991\) −14.6060 −0.463974 −0.231987 0.972719i \(-0.574523\pi\)
−0.231987 + 0.972719i \(0.574523\pi\)
\(992\) −8.37228 −0.265820
\(993\) 1.02175 0.0324242
\(994\) −20.7446 −0.657978
\(995\) −22.7446 −0.721051
\(996\) 21.4891 0.680909
\(997\) −8.23369 −0.260764 −0.130382 0.991464i \(-0.541620\pi\)
−0.130382 + 0.991464i \(0.541620\pi\)
\(998\) 20.9783 0.664055
\(999\) −4.00000 −0.126554
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 370.2.a.f.1.1 2
3.2 odd 2 3330.2.a.bb.1.1 2
4.3 odd 2 2960.2.a.o.1.2 2
5.2 odd 4 1850.2.b.m.149.3 4
5.3 odd 4 1850.2.b.m.149.2 4
5.4 even 2 1850.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.f.1.1 2 1.1 even 1 trivial
1850.2.a.q.1.2 2 5.4 even 2
1850.2.b.m.149.2 4 5.3 odd 4
1850.2.b.m.149.3 4 5.2 odd 4
2960.2.a.o.1.2 2 4.3 odd 2
3330.2.a.bb.1.1 2 3.2 odd 2