Properties

Label 8034.2.a.ba.1.9
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} - 4126 x^{5} - 22759 x^{4} + 2997 x^{3} + 10504 x^{2} - 506 x - 1492\)
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 1.9
Root \(-0.643572\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.643572 q^{5} +1.00000 q^{6} +1.17705 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.643572 q^{5} +1.00000 q^{6} +1.17705 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.643572 q^{10} -1.62256 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.17705 q^{14} -0.643572 q^{15} +1.00000 q^{16} +1.49860 q^{17} -1.00000 q^{18} +6.90649 q^{19} +0.643572 q^{20} -1.17705 q^{21} +1.62256 q^{22} -6.05927 q^{23} +1.00000 q^{24} -4.58582 q^{25} +1.00000 q^{26} -1.00000 q^{27} +1.17705 q^{28} +6.15424 q^{29} +0.643572 q^{30} -2.72186 q^{31} -1.00000 q^{32} +1.62256 q^{33} -1.49860 q^{34} +0.757514 q^{35} +1.00000 q^{36} +9.41176 q^{37} -6.90649 q^{38} +1.00000 q^{39} -0.643572 q^{40} -7.56422 q^{41} +1.17705 q^{42} +8.05306 q^{43} -1.62256 q^{44} +0.643572 q^{45} +6.05927 q^{46} -6.76522 q^{47} -1.00000 q^{48} -5.61456 q^{49} +4.58582 q^{50} -1.49860 q^{51} -1.00000 q^{52} -5.26991 q^{53} +1.00000 q^{54} -1.04423 q^{55} -1.17705 q^{56} -6.90649 q^{57} -6.15424 q^{58} +8.48262 q^{59} -0.643572 q^{60} +12.9422 q^{61} +2.72186 q^{62} +1.17705 q^{63} +1.00000 q^{64} -0.643572 q^{65} -1.62256 q^{66} -3.48887 q^{67} +1.49860 q^{68} +6.05927 q^{69} -0.757514 q^{70} +9.90543 q^{71} -1.00000 q^{72} +2.07783 q^{73} -9.41176 q^{74} +4.58582 q^{75} +6.90649 q^{76} -1.90982 q^{77} -1.00000 q^{78} -0.329478 q^{79} +0.643572 q^{80} +1.00000 q^{81} +7.56422 q^{82} -1.69775 q^{83} -1.17705 q^{84} +0.964456 q^{85} -8.05306 q^{86} -6.15424 q^{87} +1.62256 q^{88} -1.76809 q^{89} -0.643572 q^{90} -1.17705 q^{91} -6.05927 q^{92} +2.72186 q^{93} +6.76522 q^{94} +4.44482 q^{95} +1.00000 q^{96} -15.6706 q^{97} +5.61456 q^{98} -1.62256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + q^{10} - q^{11} - 14q^{12} - 14q^{13} - 5q^{14} + q^{15} + 14q^{16} - 12q^{17} - 14q^{18} + 10q^{19} - q^{20} - 5q^{21} + q^{22} - q^{23} + 14q^{24} + 19q^{25} + 14q^{26} - 14q^{27} + 5q^{28} + 6q^{29} - q^{30} + 20q^{31} - 14q^{32} + q^{33} + 12q^{34} - 16q^{35} + 14q^{36} - 3q^{37} - 10q^{38} + 14q^{39} + q^{40} + q^{41} + 5q^{42} + 6q^{43} - q^{44} - q^{45} + q^{46} - 13q^{47} - 14q^{48} + 9q^{49} - 19q^{50} + 12q^{51} - 14q^{52} - 27q^{53} + 14q^{54} + 10q^{55} - 5q^{56} - 10q^{57} - 6q^{58} - 6q^{59} + q^{60} - 4q^{61} - 20q^{62} + 5q^{63} + 14q^{64} + q^{65} - q^{66} + 13q^{67} - 12q^{68} + q^{69} + 16q^{70} + 18q^{71} - 14q^{72} + 11q^{73} + 3q^{74} - 19q^{75} + 10q^{76} - 15q^{77} - 14q^{78} + 33q^{79} - q^{80} + 14q^{81} - q^{82} - 25q^{83} - 5q^{84} + 25q^{85} - 6q^{86} - 6q^{87} + q^{88} + 3q^{89} + q^{90} - 5q^{91} - q^{92} - 20q^{93} + 13q^{94} + 30q^{95} + 14q^{96} + 11q^{97} - 9q^{98} - q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0.643572 0.287814 0.143907 0.989591i \(-0.454033\pi\)
0.143907 + 0.989591i \(0.454033\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.17705 0.444882 0.222441 0.974946i \(-0.428598\pi\)
0.222441 + 0.974946i \(0.428598\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −0.643572 −0.203515
\(11\) −1.62256 −0.489219 −0.244610 0.969622i \(-0.578660\pi\)
−0.244610 + 0.969622i \(0.578660\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −1.17705 −0.314579
\(15\) −0.643572 −0.166169
\(16\) 1.00000 0.250000
\(17\) 1.49860 0.363464 0.181732 0.983348i \(-0.441830\pi\)
0.181732 + 0.983348i \(0.441830\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.90649 1.58446 0.792228 0.610225i \(-0.208921\pi\)
0.792228 + 0.610225i \(0.208921\pi\)
\(20\) 0.643572 0.143907
\(21\) −1.17705 −0.256853
\(22\) 1.62256 0.345930
\(23\) −6.05927 −1.26345 −0.631723 0.775194i \(-0.717652\pi\)
−0.631723 + 0.775194i \(0.717652\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.58582 −0.917163
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 1.17705 0.222441
\(29\) 6.15424 1.14281 0.571407 0.820667i \(-0.306398\pi\)
0.571407 + 0.820667i \(0.306398\pi\)
\(30\) 0.643572 0.117500
\(31\) −2.72186 −0.488861 −0.244430 0.969667i \(-0.578601\pi\)
−0.244430 + 0.969667i \(0.578601\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.62256 0.282451
\(34\) −1.49860 −0.257008
\(35\) 0.757514 0.128043
\(36\) 1.00000 0.166667
\(37\) 9.41176 1.54728 0.773642 0.633624i \(-0.218433\pi\)
0.773642 + 0.633624i \(0.218433\pi\)
\(38\) −6.90649 −1.12038
\(39\) 1.00000 0.160128
\(40\) −0.643572 −0.101758
\(41\) −7.56422 −1.18133 −0.590667 0.806916i \(-0.701135\pi\)
−0.590667 + 0.806916i \(0.701135\pi\)
\(42\) 1.17705 0.181622
\(43\) 8.05306 1.22808 0.614040 0.789275i \(-0.289543\pi\)
0.614040 + 0.789275i \(0.289543\pi\)
\(44\) −1.62256 −0.244610
\(45\) 0.643572 0.0959380
\(46\) 6.05927 0.893391
\(47\) −6.76522 −0.986808 −0.493404 0.869800i \(-0.664248\pi\)
−0.493404 + 0.869800i \(0.664248\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.61456 −0.802080
\(50\) 4.58582 0.648532
\(51\) −1.49860 −0.209846
\(52\) −1.00000 −0.138675
\(53\) −5.26991 −0.723878 −0.361939 0.932202i \(-0.617885\pi\)
−0.361939 + 0.932202i \(0.617885\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.04423 −0.140804
\(56\) −1.17705 −0.157289
\(57\) −6.90649 −0.914787
\(58\) −6.15424 −0.808091
\(59\) 8.48262 1.10434 0.552172 0.833730i \(-0.313799\pi\)
0.552172 + 0.833730i \(0.313799\pi\)
\(60\) −0.643572 −0.0830847
\(61\) 12.9422 1.65708 0.828540 0.559929i \(-0.189172\pi\)
0.828540 + 0.559929i \(0.189172\pi\)
\(62\) 2.72186 0.345677
\(63\) 1.17705 0.148294
\(64\) 1.00000 0.125000
\(65\) −0.643572 −0.0798252
\(66\) −1.62256 −0.199723
\(67\) −3.48887 −0.426233 −0.213117 0.977027i \(-0.568361\pi\)
−0.213117 + 0.977027i \(0.568361\pi\)
\(68\) 1.49860 0.181732
\(69\) 6.05927 0.729451
\(70\) −0.757514 −0.0905402
\(71\) 9.90543 1.17556 0.587779 0.809022i \(-0.300002\pi\)
0.587779 + 0.809022i \(0.300002\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.07783 0.243192 0.121596 0.992580i \(-0.461199\pi\)
0.121596 + 0.992580i \(0.461199\pi\)
\(74\) −9.41176 −1.09409
\(75\) 4.58582 0.529524
\(76\) 6.90649 0.792228
\(77\) −1.90982 −0.217645
\(78\) −1.00000 −0.113228
\(79\) −0.329478 −0.0370692 −0.0185346 0.999828i \(-0.505900\pi\)
−0.0185346 + 0.999828i \(0.505900\pi\)
\(80\) 0.643572 0.0719535
\(81\) 1.00000 0.111111
\(82\) 7.56422 0.835329
\(83\) −1.69775 −0.186353 −0.0931764 0.995650i \(-0.529702\pi\)
−0.0931764 + 0.995650i \(0.529702\pi\)
\(84\) −1.17705 −0.128426
\(85\) 0.964456 0.104610
\(86\) −8.05306 −0.868384
\(87\) −6.15424 −0.659804
\(88\) 1.62256 0.172965
\(89\) −1.76809 −0.187417 −0.0937083 0.995600i \(-0.529872\pi\)
−0.0937083 + 0.995600i \(0.529872\pi\)
\(90\) −0.643572 −0.0678384
\(91\) −1.17705 −0.123388
\(92\) −6.05927 −0.631723
\(93\) 2.72186 0.282244
\(94\) 6.76522 0.697779
\(95\) 4.44482 0.456029
\(96\) 1.00000 0.102062
\(97\) −15.6706 −1.59111 −0.795556 0.605881i \(-0.792821\pi\)
−0.795556 + 0.605881i \(0.792821\pi\)
\(98\) 5.61456 0.567156
\(99\) −1.62256 −0.163073
\(100\) −4.58582 −0.458582
\(101\) −5.93932 −0.590985 −0.295492 0.955345i \(-0.595484\pi\)
−0.295492 + 0.955345i \(0.595484\pi\)
\(102\) 1.49860 0.148383
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) −0.757514 −0.0739258
\(106\) 5.26991 0.511859
\(107\) 9.16659 0.886168 0.443084 0.896480i \(-0.353884\pi\)
0.443084 + 0.896480i \(0.353884\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −0.979696 −0.0938379 −0.0469189 0.998899i \(-0.514940\pi\)
−0.0469189 + 0.998899i \(0.514940\pi\)
\(110\) 1.04423 0.0995636
\(111\) −9.41176 −0.893324
\(112\) 1.17705 0.111220
\(113\) 13.0922 1.23161 0.615807 0.787897i \(-0.288830\pi\)
0.615807 + 0.787897i \(0.288830\pi\)
\(114\) 6.90649 0.646852
\(115\) −3.89957 −0.363637
\(116\) 6.15424 0.571407
\(117\) −1.00000 −0.0924500
\(118\) −8.48262 −0.780889
\(119\) 1.76392 0.161698
\(120\) 0.643572 0.0587498
\(121\) −8.36731 −0.760665
\(122\) −12.9422 −1.17173
\(123\) 7.56422 0.682043
\(124\) −2.72186 −0.244430
\(125\) −6.16916 −0.551786
\(126\) −1.17705 −0.104860
\(127\) 4.42014 0.392224 0.196112 0.980582i \(-0.437168\pi\)
0.196112 + 0.980582i \(0.437168\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.05306 −0.709032
\(130\) 0.643572 0.0564450
\(131\) 3.53496 0.308850 0.154425 0.988004i \(-0.450647\pi\)
0.154425 + 0.988004i \(0.450647\pi\)
\(132\) 1.62256 0.141225
\(133\) 8.12926 0.704896
\(134\) 3.48887 0.301392
\(135\) −0.643572 −0.0553898
\(136\) −1.49860 −0.128504
\(137\) 14.5896 1.24647 0.623235 0.782034i \(-0.285818\pi\)
0.623235 + 0.782034i \(0.285818\pi\)
\(138\) −6.05927 −0.515799
\(139\) −0.829863 −0.0703881 −0.0351940 0.999380i \(-0.511205\pi\)
−0.0351940 + 0.999380i \(0.511205\pi\)
\(140\) 0.757514 0.0640216
\(141\) 6.76522 0.569734
\(142\) −9.90543 −0.831245
\(143\) 1.62256 0.135685
\(144\) 1.00000 0.0833333
\(145\) 3.96069 0.328918
\(146\) −2.07783 −0.171962
\(147\) 5.61456 0.463081
\(148\) 9.41176 0.773642
\(149\) 7.83483 0.641854 0.320927 0.947104i \(-0.396006\pi\)
0.320927 + 0.947104i \(0.396006\pi\)
\(150\) −4.58582 −0.374430
\(151\) 4.95195 0.402984 0.201492 0.979490i \(-0.435421\pi\)
0.201492 + 0.979490i \(0.435421\pi\)
\(152\) −6.90649 −0.560190
\(153\) 1.49860 0.121155
\(154\) 1.90982 0.153898
\(155\) −1.75171 −0.140701
\(156\) 1.00000 0.0800641
\(157\) 4.49425 0.358680 0.179340 0.983787i \(-0.442604\pi\)
0.179340 + 0.983787i \(0.442604\pi\)
\(158\) 0.329478 0.0262119
\(159\) 5.26991 0.417931
\(160\) −0.643572 −0.0508788
\(161\) −7.13205 −0.562084
\(162\) −1.00000 −0.0785674
\(163\) 13.0819 1.02466 0.512328 0.858790i \(-0.328783\pi\)
0.512328 + 0.858790i \(0.328783\pi\)
\(164\) −7.56422 −0.590667
\(165\) 1.04423 0.0812933
\(166\) 1.69775 0.131771
\(167\) −0.842027 −0.0651580 −0.0325790 0.999469i \(-0.510372\pi\)
−0.0325790 + 0.999469i \(0.510372\pi\)
\(168\) 1.17705 0.0908111
\(169\) 1.00000 0.0769231
\(170\) −0.964456 −0.0739704
\(171\) 6.90649 0.528152
\(172\) 8.05306 0.614040
\(173\) −12.9042 −0.981084 −0.490542 0.871417i \(-0.663201\pi\)
−0.490542 + 0.871417i \(0.663201\pi\)
\(174\) 6.15424 0.466552
\(175\) −5.39772 −0.408029
\(176\) −1.62256 −0.122305
\(177\) −8.48262 −0.637593
\(178\) 1.76809 0.132524
\(179\) 2.76289 0.206508 0.103254 0.994655i \(-0.467075\pi\)
0.103254 + 0.994655i \(0.467075\pi\)
\(180\) 0.643572 0.0479690
\(181\) 8.64089 0.642273 0.321136 0.947033i \(-0.395935\pi\)
0.321136 + 0.947033i \(0.395935\pi\)
\(182\) 1.17705 0.0872485
\(183\) −12.9422 −0.956716
\(184\) 6.05927 0.446695
\(185\) 6.05714 0.445330
\(186\) −2.72186 −0.199577
\(187\) −2.43156 −0.177813
\(188\) −6.76522 −0.493404
\(189\) −1.17705 −0.0856175
\(190\) −4.44482 −0.322461
\(191\) −1.08471 −0.0784870 −0.0392435 0.999230i \(-0.512495\pi\)
−0.0392435 + 0.999230i \(0.512495\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 18.1254 1.30469 0.652347 0.757921i \(-0.273785\pi\)
0.652347 + 0.757921i \(0.273785\pi\)
\(194\) 15.6706 1.12509
\(195\) 0.643572 0.0460871
\(196\) −5.61456 −0.401040
\(197\) −19.3889 −1.38140 −0.690702 0.723139i \(-0.742698\pi\)
−0.690702 + 0.723139i \(0.742698\pi\)
\(198\) 1.62256 0.115310
\(199\) 13.7592 0.975367 0.487684 0.873020i \(-0.337842\pi\)
0.487684 + 0.873020i \(0.337842\pi\)
\(200\) 4.58582 0.324266
\(201\) 3.48887 0.246086
\(202\) 5.93932 0.417889
\(203\) 7.24383 0.508417
\(204\) −1.49860 −0.104923
\(205\) −4.86812 −0.340004
\(206\) −1.00000 −0.0696733
\(207\) −6.05927 −0.421148
\(208\) −1.00000 −0.0693375
\(209\) −11.2062 −0.775147
\(210\) 0.757514 0.0522734
\(211\) 28.7921 1.98213 0.991063 0.133392i \(-0.0425870\pi\)
0.991063 + 0.133392i \(0.0425870\pi\)
\(212\) −5.26991 −0.361939
\(213\) −9.90543 −0.678709
\(214\) −9.16659 −0.626615
\(215\) 5.18272 0.353459
\(216\) 1.00000 0.0680414
\(217\) −3.20376 −0.217485
\(218\) 0.979696 0.0663534
\(219\) −2.07783 −0.140407
\(220\) −1.04423 −0.0704021
\(221\) −1.49860 −0.100807
\(222\) 9.41176 0.631676
\(223\) −12.7364 −0.852892 −0.426446 0.904513i \(-0.640235\pi\)
−0.426446 + 0.904513i \(0.640235\pi\)
\(224\) −1.17705 −0.0786447
\(225\) −4.58582 −0.305721
\(226\) −13.0922 −0.870883
\(227\) −22.9067 −1.52037 −0.760185 0.649706i \(-0.774892\pi\)
−0.760185 + 0.649706i \(0.774892\pi\)
\(228\) −6.90649 −0.457393
\(229\) 10.6201 0.701794 0.350897 0.936414i \(-0.385877\pi\)
0.350897 + 0.936414i \(0.385877\pi\)
\(230\) 3.89957 0.257130
\(231\) 1.90982 0.125657
\(232\) −6.15424 −0.404046
\(233\) 6.61868 0.433604 0.216802 0.976216i \(-0.430437\pi\)
0.216802 + 0.976216i \(0.430437\pi\)
\(234\) 1.00000 0.0653720
\(235\) −4.35390 −0.284017
\(236\) 8.48262 0.552172
\(237\) 0.329478 0.0214019
\(238\) −1.76392 −0.114338
\(239\) −16.6591 −1.07759 −0.538794 0.842438i \(-0.681120\pi\)
−0.538794 + 0.842438i \(0.681120\pi\)
\(240\) −0.643572 −0.0415424
\(241\) −13.5837 −0.875000 −0.437500 0.899218i \(-0.644136\pi\)
−0.437500 + 0.899218i \(0.644136\pi\)
\(242\) 8.36731 0.537871
\(243\) −1.00000 −0.0641500
\(244\) 12.9422 0.828540
\(245\) −3.61337 −0.230850
\(246\) −7.56422 −0.482277
\(247\) −6.90649 −0.439449
\(248\) 2.72186 0.172838
\(249\) 1.69775 0.107591
\(250\) 6.16916 0.390172
\(251\) 23.5935 1.48921 0.744603 0.667508i \(-0.232639\pi\)
0.744603 + 0.667508i \(0.232639\pi\)
\(252\) 1.17705 0.0741470
\(253\) 9.83151 0.618102
\(254\) −4.42014 −0.277344
\(255\) −0.964456 −0.0603966
\(256\) 1.00000 0.0625000
\(257\) 0.0762573 0.00475680 0.00237840 0.999997i \(-0.499243\pi\)
0.00237840 + 0.999997i \(0.499243\pi\)
\(258\) 8.05306 0.501362
\(259\) 11.0781 0.688358
\(260\) −0.643572 −0.0399126
\(261\) 6.15424 0.380938
\(262\) −3.53496 −0.218390
\(263\) −26.4812 −1.63290 −0.816451 0.577414i \(-0.804062\pi\)
−0.816451 + 0.577414i \(0.804062\pi\)
\(264\) −1.62256 −0.0998615
\(265\) −3.39157 −0.208342
\(266\) −8.12926 −0.498437
\(267\) 1.76809 0.108205
\(268\) −3.48887 −0.213117
\(269\) −23.4459 −1.42952 −0.714761 0.699368i \(-0.753465\pi\)
−0.714761 + 0.699368i \(0.753465\pi\)
\(270\) 0.643572 0.0391665
\(271\) 19.3142 1.17325 0.586626 0.809858i \(-0.300456\pi\)
0.586626 + 0.809858i \(0.300456\pi\)
\(272\) 1.49860 0.0908659
\(273\) 1.17705 0.0712381
\(274\) −14.5896 −0.881388
\(275\) 7.44075 0.448694
\(276\) 6.05927 0.364725
\(277\) 17.3607 1.04311 0.521553 0.853219i \(-0.325353\pi\)
0.521553 + 0.853219i \(0.325353\pi\)
\(278\) 0.829863 0.0497719
\(279\) −2.72186 −0.162954
\(280\) −0.757514 −0.0452701
\(281\) −16.4966 −0.984106 −0.492053 0.870565i \(-0.663753\pi\)
−0.492053 + 0.870565i \(0.663753\pi\)
\(282\) −6.76522 −0.402863
\(283\) −17.2117 −1.02313 −0.511564 0.859245i \(-0.670934\pi\)
−0.511564 + 0.859245i \(0.670934\pi\)
\(284\) 9.90543 0.587779
\(285\) −4.44482 −0.263288
\(286\) −1.62256 −0.0959438
\(287\) −8.90344 −0.525554
\(288\) −1.00000 −0.0589256
\(289\) −14.7542 −0.867894
\(290\) −3.96069 −0.232580
\(291\) 15.6706 0.918628
\(292\) 2.07783 0.121596
\(293\) 4.91838 0.287335 0.143667 0.989626i \(-0.454110\pi\)
0.143667 + 0.989626i \(0.454110\pi\)
\(294\) −5.61456 −0.327448
\(295\) 5.45918 0.317845
\(296\) −9.41176 −0.547047
\(297\) 1.62256 0.0941503
\(298\) −7.83483 −0.453860
\(299\) 6.05927 0.350417
\(300\) 4.58582 0.264762
\(301\) 9.47883 0.546350
\(302\) −4.95195 −0.284952
\(303\) 5.93932 0.341205
\(304\) 6.90649 0.396114
\(305\) 8.32924 0.476931
\(306\) −1.49860 −0.0856692
\(307\) −3.35221 −0.191321 −0.0956603 0.995414i \(-0.530496\pi\)
−0.0956603 + 0.995414i \(0.530496\pi\)
\(308\) −1.90982 −0.108822
\(309\) −1.00000 −0.0568880
\(310\) 1.75171 0.0994906
\(311\) 28.2832 1.60379 0.801897 0.597462i \(-0.203824\pi\)
0.801897 + 0.597462i \(0.203824\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 22.0633 1.24709 0.623547 0.781786i \(-0.285691\pi\)
0.623547 + 0.781786i \(0.285691\pi\)
\(314\) −4.49425 −0.253625
\(315\) 0.757514 0.0426811
\(316\) −0.329478 −0.0185346
\(317\) 6.69370 0.375956 0.187978 0.982173i \(-0.439807\pi\)
0.187978 + 0.982173i \(0.439807\pi\)
\(318\) −5.26991 −0.295522
\(319\) −9.98560 −0.559086
\(320\) 0.643572 0.0359767
\(321\) −9.16659 −0.511629
\(322\) 7.13205 0.397453
\(323\) 10.3501 0.575892
\(324\) 1.00000 0.0555556
\(325\) 4.58582 0.254375
\(326\) −13.0819 −0.724542
\(327\) 0.979696 0.0541773
\(328\) 7.56422 0.417664
\(329\) −7.96298 −0.439013
\(330\) −1.04423 −0.0574830
\(331\) 15.8048 0.868710 0.434355 0.900742i \(-0.356976\pi\)
0.434355 + 0.900742i \(0.356976\pi\)
\(332\) −1.69775 −0.0931764
\(333\) 9.41176 0.515761
\(334\) 0.842027 0.0460737
\(335\) −2.24534 −0.122676
\(336\) −1.17705 −0.0642132
\(337\) −28.1213 −1.53187 −0.765934 0.642919i \(-0.777723\pi\)
−0.765934 + 0.642919i \(0.777723\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −13.0922 −0.711073
\(340\) 0.964456 0.0523050
\(341\) 4.41637 0.239160
\(342\) −6.90649 −0.373460
\(343\) −14.8479 −0.801713
\(344\) −8.05306 −0.434192
\(345\) 3.89957 0.209946
\(346\) 12.9042 0.693731
\(347\) 10.8696 0.583509 0.291755 0.956493i \(-0.405761\pi\)
0.291755 + 0.956493i \(0.405761\pi\)
\(348\) −6.15424 −0.329902
\(349\) −21.7245 −1.16289 −0.581443 0.813587i \(-0.697512\pi\)
−0.581443 + 0.813587i \(0.697512\pi\)
\(350\) 5.39772 0.288520
\(351\) 1.00000 0.0533761
\(352\) 1.62256 0.0864826
\(353\) 17.5463 0.933894 0.466947 0.884285i \(-0.345354\pi\)
0.466947 + 0.884285i \(0.345354\pi\)
\(354\) 8.48262 0.450846
\(355\) 6.37485 0.338342
\(356\) −1.76809 −0.0937083
\(357\) −1.76392 −0.0933566
\(358\) −2.76289 −0.146023
\(359\) 25.6817 1.35543 0.677713 0.735326i \(-0.262971\pi\)
0.677713 + 0.735326i \(0.262971\pi\)
\(360\) −0.643572 −0.0339192
\(361\) 28.6996 1.51050
\(362\) −8.64089 −0.454155
\(363\) 8.36731 0.439170
\(364\) −1.17705 −0.0616940
\(365\) 1.33723 0.0699939
\(366\) 12.9422 0.676500
\(367\) −31.1996 −1.62861 −0.814304 0.580439i \(-0.802881\pi\)
−0.814304 + 0.580439i \(0.802881\pi\)
\(368\) −6.05927 −0.315861
\(369\) −7.56422 −0.393778
\(370\) −6.05714 −0.314896
\(371\) −6.20293 −0.322040
\(372\) 2.72186 0.141122
\(373\) 14.9563 0.774407 0.387204 0.921994i \(-0.373441\pi\)
0.387204 + 0.921994i \(0.373441\pi\)
\(374\) 2.43156 0.125733
\(375\) 6.16916 0.318574
\(376\) 6.76522 0.348889
\(377\) −6.15424 −0.316959
\(378\) 1.17705 0.0605407
\(379\) 15.4787 0.795085 0.397543 0.917584i \(-0.369863\pi\)
0.397543 + 0.917584i \(0.369863\pi\)
\(380\) 4.44482 0.228014
\(381\) −4.42014 −0.226450
\(382\) 1.08471 0.0554987
\(383\) 2.23651 0.114280 0.0571402 0.998366i \(-0.481802\pi\)
0.0571402 + 0.998366i \(0.481802\pi\)
\(384\) 1.00000 0.0510310
\(385\) −1.22911 −0.0626412
\(386\) −18.1254 −0.922557
\(387\) 8.05306 0.409360
\(388\) −15.6706 −0.795556
\(389\) −12.8220 −0.650102 −0.325051 0.945697i \(-0.605381\pi\)
−0.325051 + 0.945697i \(0.605381\pi\)
\(390\) −0.643572 −0.0325885
\(391\) −9.08042 −0.459216
\(392\) 5.61456 0.283578
\(393\) −3.53496 −0.178315
\(394\) 19.3889 0.976801
\(395\) −0.212043 −0.0106690
\(396\) −1.62256 −0.0815365
\(397\) 20.6229 1.03503 0.517516 0.855674i \(-0.326857\pi\)
0.517516 + 0.855674i \(0.326857\pi\)
\(398\) −13.7592 −0.689689
\(399\) −8.12926 −0.406972
\(400\) −4.58582 −0.229291
\(401\) −3.88998 −0.194256 −0.0971282 0.995272i \(-0.530966\pi\)
−0.0971282 + 0.995272i \(0.530966\pi\)
\(402\) −3.48887 −0.174009
\(403\) 2.72186 0.135586
\(404\) −5.93932 −0.295492
\(405\) 0.643572 0.0319793
\(406\) −7.24383 −0.359505
\(407\) −15.2711 −0.756961
\(408\) 1.49860 0.0741917
\(409\) 12.0257 0.594631 0.297315 0.954779i \(-0.403909\pi\)
0.297315 + 0.954779i \(0.403909\pi\)
\(410\) 4.86812 0.240419
\(411\) −14.5896 −0.719650
\(412\) 1.00000 0.0492665
\(413\) 9.98444 0.491302
\(414\) 6.05927 0.297797
\(415\) −1.09263 −0.0536349
\(416\) 1.00000 0.0490290
\(417\) 0.829863 0.0406386
\(418\) 11.2062 0.548112
\(419\) −9.30539 −0.454598 −0.227299 0.973825i \(-0.572989\pi\)
−0.227299 + 0.973825i \(0.572989\pi\)
\(420\) −0.757514 −0.0369629
\(421\) 37.1227 1.80925 0.904624 0.426210i \(-0.140152\pi\)
0.904624 + 0.426210i \(0.140152\pi\)
\(422\) −28.7921 −1.40158
\(423\) −6.76522 −0.328936
\(424\) 5.26991 0.255930
\(425\) −6.87230 −0.333355
\(426\) 9.90543 0.479919
\(427\) 15.2336 0.737205
\(428\) 9.16659 0.443084
\(429\) −1.62256 −0.0783378
\(430\) −5.18272 −0.249933
\(431\) −4.29157 −0.206718 −0.103359 0.994644i \(-0.532959\pi\)
−0.103359 + 0.994644i \(0.532959\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 9.74101 0.468123 0.234062 0.972222i \(-0.424798\pi\)
0.234062 + 0.972222i \(0.424798\pi\)
\(434\) 3.20376 0.153785
\(435\) −3.96069 −0.189901
\(436\) −0.979696 −0.0469189
\(437\) −41.8483 −2.00187
\(438\) 2.07783 0.0992826
\(439\) 13.6511 0.651532 0.325766 0.945450i \(-0.394378\pi\)
0.325766 + 0.945450i \(0.394378\pi\)
\(440\) 1.04423 0.0497818
\(441\) −5.61456 −0.267360
\(442\) 1.49860 0.0712811
\(443\) −22.2391 −1.05661 −0.528306 0.849054i \(-0.677173\pi\)
−0.528306 + 0.849054i \(0.677173\pi\)
\(444\) −9.41176 −0.446662
\(445\) −1.13789 −0.0539411
\(446\) 12.7364 0.603086
\(447\) −7.83483 −0.370575
\(448\) 1.17705 0.0556102
\(449\) −19.2668 −0.909259 −0.454629 0.890681i \(-0.650228\pi\)
−0.454629 + 0.890681i \(0.650228\pi\)
\(450\) 4.58582 0.216177
\(451\) 12.2734 0.577931
\(452\) 13.0922 0.615807
\(453\) −4.95195 −0.232663
\(454\) 22.9067 1.07506
\(455\) −0.757514 −0.0355128
\(456\) 6.90649 0.323426
\(457\) −40.3189 −1.88604 −0.943019 0.332739i \(-0.892027\pi\)
−0.943019 + 0.332739i \(0.892027\pi\)
\(458\) −10.6201 −0.496243
\(459\) −1.49860 −0.0699486
\(460\) −3.89957 −0.181819
\(461\) −9.53646 −0.444157 −0.222079 0.975029i \(-0.571284\pi\)
−0.222079 + 0.975029i \(0.571284\pi\)
\(462\) −1.90982 −0.0888531
\(463\) 21.8752 1.01663 0.508313 0.861172i \(-0.330269\pi\)
0.508313 + 0.861172i \(0.330269\pi\)
\(464\) 6.15424 0.285703
\(465\) 1.75171 0.0812337
\(466\) −6.61868 −0.306605
\(467\) 41.0475 1.89945 0.949726 0.313083i \(-0.101362\pi\)
0.949726 + 0.313083i \(0.101362\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −4.10656 −0.189623
\(470\) 4.35390 0.200831
\(471\) −4.49425 −0.207084
\(472\) −8.48262 −0.390444
\(473\) −13.0665 −0.600800
\(474\) −0.329478 −0.0151334
\(475\) −31.6719 −1.45321
\(476\) 1.76392 0.0808492
\(477\) −5.26991 −0.241293
\(478\) 16.6591 0.761970
\(479\) 13.1109 0.599051 0.299526 0.954088i \(-0.403172\pi\)
0.299526 + 0.954088i \(0.403172\pi\)
\(480\) 0.643572 0.0293749
\(481\) −9.41176 −0.429139
\(482\) 13.5837 0.618719
\(483\) 7.13205 0.324519
\(484\) −8.36731 −0.380332
\(485\) −10.0852 −0.457944
\(486\) 1.00000 0.0453609
\(487\) −9.34915 −0.423650 −0.211825 0.977308i \(-0.567941\pi\)
−0.211825 + 0.977308i \(0.567941\pi\)
\(488\) −12.9422 −0.585867
\(489\) −13.0819 −0.591586
\(490\) 3.61337 0.163236
\(491\) −3.89868 −0.175945 −0.0879726 0.996123i \(-0.528039\pi\)
−0.0879726 + 0.996123i \(0.528039\pi\)
\(492\) 7.56422 0.341022
\(493\) 9.22274 0.415371
\(494\) 6.90649 0.310738
\(495\) −1.04423 −0.0469347
\(496\) −2.72186 −0.122215
\(497\) 11.6591 0.522984
\(498\) −1.69775 −0.0760782
\(499\) −25.0610 −1.12188 −0.560942 0.827855i \(-0.689561\pi\)
−0.560942 + 0.827855i \(0.689561\pi\)
\(500\) −6.16916 −0.275893
\(501\) 0.842027 0.0376190
\(502\) −23.5935 −1.05303
\(503\) 18.4213 0.821364 0.410682 0.911779i \(-0.365291\pi\)
0.410682 + 0.911779i \(0.365291\pi\)
\(504\) −1.17705 −0.0524298
\(505\) −3.82238 −0.170094
\(506\) −9.83151 −0.437064
\(507\) −1.00000 −0.0444116
\(508\) 4.42014 0.196112
\(509\) 42.1686 1.86909 0.934546 0.355842i \(-0.115806\pi\)
0.934546 + 0.355842i \(0.115806\pi\)
\(510\) 0.964456 0.0427068
\(511\) 2.44570 0.108192
\(512\) −1.00000 −0.0441942
\(513\) −6.90649 −0.304929
\(514\) −0.0762573 −0.00336356
\(515\) 0.643572 0.0283592
\(516\) −8.05306 −0.354516
\(517\) 10.9769 0.482766
\(518\) −11.0781 −0.486743
\(519\) 12.9042 0.566429
\(520\) 0.643572 0.0282225
\(521\) −4.06054 −0.177895 −0.0889477 0.996036i \(-0.528350\pi\)
−0.0889477 + 0.996036i \(0.528350\pi\)
\(522\) −6.15424 −0.269364
\(523\) −38.7979 −1.69651 −0.848256 0.529587i \(-0.822347\pi\)
−0.848256 + 0.529587i \(0.822347\pi\)
\(524\) 3.53496 0.154425
\(525\) 5.39772 0.235576
\(526\) 26.4812 1.15464
\(527\) −4.07898 −0.177683
\(528\) 1.62256 0.0706127
\(529\) 13.7148 0.596294
\(530\) 3.39157 0.147320
\(531\) 8.48262 0.368114
\(532\) 8.12926 0.352448
\(533\) 7.56422 0.327643
\(534\) −1.76809 −0.0765125
\(535\) 5.89936 0.255051
\(536\) 3.48887 0.150696
\(537\) −2.76289 −0.119227
\(538\) 23.4459 1.01083
\(539\) 9.10994 0.392393
\(540\) −0.643572 −0.0276949
\(541\) 18.9115 0.813067 0.406534 0.913636i \(-0.366737\pi\)
0.406534 + 0.913636i \(0.366737\pi\)
\(542\) −19.3142 −0.829614
\(543\) −8.64089 −0.370816
\(544\) −1.49860 −0.0642519
\(545\) −0.630505 −0.0270079
\(546\) −1.17705 −0.0503729
\(547\) −2.99158 −0.127911 −0.0639554 0.997953i \(-0.520372\pi\)
−0.0639554 + 0.997953i \(0.520372\pi\)
\(548\) 14.5896 0.623235
\(549\) 12.9422 0.552360
\(550\) −7.44075 −0.317274
\(551\) 42.5042 1.81074
\(552\) −6.05927 −0.257900
\(553\) −0.387811 −0.0164914
\(554\) −17.3607 −0.737587
\(555\) −6.05714 −0.257111
\(556\) −0.829863 −0.0351940
\(557\) 20.2906 0.859739 0.429870 0.902891i \(-0.358560\pi\)
0.429870 + 0.902891i \(0.358560\pi\)
\(558\) 2.72186 0.115226
\(559\) −8.05306 −0.340608
\(560\) 0.757514 0.0320108
\(561\) 2.43156 0.102661
\(562\) 16.4966 0.695868
\(563\) 17.4691 0.736237 0.368118 0.929779i \(-0.380002\pi\)
0.368118 + 0.929779i \(0.380002\pi\)
\(564\) 6.76522 0.284867
\(565\) 8.42579 0.354476
\(566\) 17.2117 0.723460
\(567\) 1.17705 0.0494313
\(568\) −9.90543 −0.415622
\(569\) −2.10370 −0.0881918 −0.0440959 0.999027i \(-0.514041\pi\)
−0.0440959 + 0.999027i \(0.514041\pi\)
\(570\) 4.44482 0.186173
\(571\) 32.1980 1.34744 0.673721 0.738985i \(-0.264695\pi\)
0.673721 + 0.738985i \(0.264695\pi\)
\(572\) 1.62256 0.0678425
\(573\) 1.08471 0.0453145
\(574\) 8.90344 0.371623
\(575\) 27.7867 1.15879
\(576\) 1.00000 0.0416667
\(577\) 37.6482 1.56731 0.783657 0.621194i \(-0.213352\pi\)
0.783657 + 0.621194i \(0.213352\pi\)
\(578\) 14.7542 0.613694
\(579\) −18.1254 −0.753265
\(580\) 3.96069 0.164459
\(581\) −1.99834 −0.0829050
\(582\) −15.6706 −0.649568
\(583\) 8.55073 0.354135
\(584\) −2.07783 −0.0859812
\(585\) −0.643572 −0.0266084
\(586\) −4.91838 −0.203176
\(587\) 36.8823 1.52230 0.761148 0.648578i \(-0.224636\pi\)
0.761148 + 0.648578i \(0.224636\pi\)
\(588\) 5.61456 0.231541
\(589\) −18.7985 −0.774579
\(590\) −5.45918 −0.224751
\(591\) 19.3889 0.797554
\(592\) 9.41176 0.386821
\(593\) 38.8654 1.59601 0.798004 0.602652i \(-0.205889\pi\)
0.798004 + 0.602652i \(0.205889\pi\)
\(594\) −1.62256 −0.0665743
\(595\) 1.13521 0.0465390
\(596\) 7.83483 0.320927
\(597\) −13.7592 −0.563129
\(598\) −6.05927 −0.247782
\(599\) 2.56135 0.104654 0.0523270 0.998630i \(-0.483336\pi\)
0.0523270 + 0.998630i \(0.483336\pi\)
\(600\) −4.58582 −0.187215
\(601\) 38.8543 1.58490 0.792451 0.609936i \(-0.208805\pi\)
0.792451 + 0.609936i \(0.208805\pi\)
\(602\) −9.47883 −0.386328
\(603\) −3.48887 −0.142078
\(604\) 4.95195 0.201492
\(605\) −5.38496 −0.218930
\(606\) −5.93932 −0.241269
\(607\) −0.129186 −0.00524350 −0.00262175 0.999997i \(-0.500835\pi\)
−0.00262175 + 0.999997i \(0.500835\pi\)
\(608\) −6.90649 −0.280095
\(609\) −7.24383 −0.293535
\(610\) −8.32924 −0.337241
\(611\) 6.76522 0.273691
\(612\) 1.49860 0.0605773
\(613\) 1.47528 0.0595858 0.0297929 0.999556i \(-0.490515\pi\)
0.0297929 + 0.999556i \(0.490515\pi\)
\(614\) 3.35221 0.135284
\(615\) 4.86812 0.196302
\(616\) 1.90982 0.0769490
\(617\) −10.5489 −0.424684 −0.212342 0.977195i \(-0.568109\pi\)
−0.212342 + 0.977195i \(0.568109\pi\)
\(618\) 1.00000 0.0402259
\(619\) 25.3604 1.01932 0.509660 0.860376i \(-0.329771\pi\)
0.509660 + 0.860376i \(0.329771\pi\)
\(620\) −1.75171 −0.0703505
\(621\) 6.05927 0.243150
\(622\) −28.2832 −1.13405
\(623\) −2.08112 −0.0833783
\(624\) 1.00000 0.0400320
\(625\) 18.9588 0.758351
\(626\) −22.0633 −0.881829
\(627\) 11.2062 0.447531
\(628\) 4.49425 0.179340
\(629\) 14.1044 0.562381
\(630\) −0.757514 −0.0301801
\(631\) 28.5992 1.13852 0.569258 0.822159i \(-0.307231\pi\)
0.569258 + 0.822159i \(0.307231\pi\)
\(632\) 0.329478 0.0131059
\(633\) −28.7921 −1.14438
\(634\) −6.69370 −0.265841
\(635\) 2.84467 0.112887
\(636\) 5.26991 0.208966
\(637\) 5.61456 0.222457
\(638\) 9.98560 0.395334
\(639\) 9.90543 0.391853
\(640\) −0.643572 −0.0254394
\(641\) −29.4248 −1.16221 −0.581106 0.813828i \(-0.697380\pi\)
−0.581106 + 0.813828i \(0.697380\pi\)
\(642\) 9.16659 0.361776
\(643\) 21.1208 0.832923 0.416461 0.909153i \(-0.363270\pi\)
0.416461 + 0.909153i \(0.363270\pi\)
\(644\) −7.13205 −0.281042
\(645\) −5.18272 −0.204069
\(646\) −10.3501 −0.407217
\(647\) −13.6705 −0.537443 −0.268722 0.963218i \(-0.586601\pi\)
−0.268722 + 0.963218i \(0.586601\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −13.7635 −0.540266
\(650\) −4.58582 −0.179870
\(651\) 3.20376 0.125565
\(652\) 13.0819 0.512328
\(653\) 21.2825 0.832847 0.416424 0.909171i \(-0.363283\pi\)
0.416424 + 0.909171i \(0.363283\pi\)
\(654\) −0.979696 −0.0383092
\(655\) 2.27500 0.0888915
\(656\) −7.56422 −0.295333
\(657\) 2.07783 0.0810639
\(658\) 7.96298 0.310429
\(659\) 6.56859 0.255876 0.127938 0.991782i \(-0.459164\pi\)
0.127938 + 0.991782i \(0.459164\pi\)
\(660\) 1.04423 0.0406466
\(661\) −20.1598 −0.784124 −0.392062 0.919939i \(-0.628238\pi\)
−0.392062 + 0.919939i \(0.628238\pi\)
\(662\) −15.8048 −0.614270
\(663\) 1.49860 0.0582008
\(664\) 1.69775 0.0658857
\(665\) 5.23176 0.202879
\(666\) −9.41176 −0.364698
\(667\) −37.2902 −1.44388
\(668\) −0.842027 −0.0325790
\(669\) 12.7364 0.492417
\(670\) 2.24534 0.0867449
\(671\) −20.9995 −0.810676
\(672\) 1.17705 0.0454056
\(673\) 33.0805 1.27516 0.637579 0.770385i \(-0.279936\pi\)
0.637579 + 0.770385i \(0.279936\pi\)
\(674\) 28.1213 1.08319
\(675\) 4.58582 0.176508
\(676\) 1.00000 0.0384615
\(677\) −3.63576 −0.139734 −0.0698668 0.997556i \(-0.522257\pi\)
−0.0698668 + 0.997556i \(0.522257\pi\)
\(678\) 13.0922 0.502804
\(679\) −18.4451 −0.707856
\(680\) −0.964456 −0.0369852
\(681\) 22.9067 0.877786
\(682\) −4.41637 −0.169112
\(683\) 23.9793 0.917542 0.458771 0.888555i \(-0.348290\pi\)
0.458771 + 0.888555i \(0.348290\pi\)
\(684\) 6.90649 0.264076
\(685\) 9.38943 0.358752
\(686\) 14.8479 0.566897
\(687\) −10.6201 −0.405181
\(688\) 8.05306 0.307020
\(689\) 5.26991 0.200768
\(690\) −3.89957 −0.148454
\(691\) 10.4362 0.397010 0.198505 0.980100i \(-0.436391\pi\)
0.198505 + 0.980100i \(0.436391\pi\)
\(692\) −12.9042 −0.490542
\(693\) −1.90982 −0.0725482
\(694\) −10.8696 −0.412603
\(695\) −0.534076 −0.0202587
\(696\) 6.15424 0.233276
\(697\) −11.3357 −0.429372
\(698\) 21.7245 0.822285
\(699\) −6.61868 −0.250342
\(700\) −5.39772 −0.204015
\(701\) −10.1995 −0.385230 −0.192615 0.981274i \(-0.561697\pi\)
−0.192615 + 0.981274i \(0.561697\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 65.0022 2.45160
\(704\) −1.62256 −0.0611524
\(705\) 4.35390 0.163977
\(706\) −17.5463 −0.660363
\(707\) −6.99086 −0.262918
\(708\) −8.48262 −0.318796
\(709\) 34.4092 1.29226 0.646132 0.763226i \(-0.276386\pi\)
0.646132 + 0.763226i \(0.276386\pi\)
\(710\) −6.37485 −0.239244
\(711\) −0.329478 −0.0123564
\(712\) 1.76809 0.0662618
\(713\) 16.4925 0.617649
\(714\) 1.76392 0.0660131
\(715\) 1.04423 0.0390520
\(716\) 2.76289 0.103254
\(717\) 16.6591 0.622146
\(718\) −25.6817 −0.958431
\(719\) 41.7873 1.55840 0.779202 0.626773i \(-0.215625\pi\)
0.779202 + 0.626773i \(0.215625\pi\)
\(720\) 0.643572 0.0239845
\(721\) 1.17705 0.0438355
\(722\) −28.6996 −1.06809
\(723\) 13.5837 0.505182
\(724\) 8.64089 0.321136
\(725\) −28.2222 −1.04815
\(726\) −8.36731 −0.310540
\(727\) 11.9237 0.442224 0.221112 0.975248i \(-0.429031\pi\)
0.221112 + 0.975248i \(0.429031\pi\)
\(728\) 1.17705 0.0436243
\(729\) 1.00000 0.0370370
\(730\) −1.33723 −0.0494932
\(731\) 12.0683 0.446362
\(732\) −12.9422 −0.478358
\(733\) 46.1861 1.70592 0.852962 0.521974i \(-0.174804\pi\)
0.852962 + 0.521974i \(0.174804\pi\)
\(734\) 31.1996 1.15160
\(735\) 3.61337 0.133281
\(736\) 6.05927 0.223348
\(737\) 5.66089 0.208521
\(738\) 7.56422 0.278443
\(739\) 10.1331 0.372752 0.186376 0.982479i \(-0.440326\pi\)
0.186376 + 0.982479i \(0.440326\pi\)
\(740\) 6.05714 0.222665
\(741\) 6.90649 0.253716
\(742\) 6.20293 0.227717
\(743\) 20.5959 0.755589 0.377795 0.925889i \(-0.376683\pi\)
0.377795 + 0.925889i \(0.376683\pi\)
\(744\) −2.72186 −0.0997883
\(745\) 5.04228 0.184735
\(746\) −14.9563 −0.547589
\(747\) −1.69775 −0.0621176
\(748\) −2.43156 −0.0889067
\(749\) 10.7895 0.394240
\(750\) −6.16916 −0.225266
\(751\) 33.3868 1.21830 0.609150 0.793055i \(-0.291511\pi\)
0.609150 + 0.793055i \(0.291511\pi\)
\(752\) −6.76522 −0.246702
\(753\) −23.5935 −0.859793
\(754\) 6.15424 0.224124
\(755\) 3.18693 0.115984
\(756\) −1.17705 −0.0428088
\(757\) 19.0021 0.690643 0.345321 0.938484i \(-0.387770\pi\)
0.345321 + 0.938484i \(0.387770\pi\)
\(758\) −15.4787 −0.562210
\(759\) −9.83151 −0.356861
\(760\) −4.44482 −0.161231
\(761\) −12.9138 −0.468126 −0.234063 0.972221i \(-0.575202\pi\)
−0.234063 + 0.972221i \(0.575202\pi\)
\(762\) 4.42014 0.160125
\(763\) −1.15315 −0.0417468
\(764\) −1.08471 −0.0392435
\(765\) 0.964456 0.0348700
\(766\) −2.23651 −0.0808084
\(767\) −8.48262 −0.306290
\(768\) −1.00000 −0.0360844
\(769\) 23.2573 0.838681 0.419341 0.907829i \(-0.362261\pi\)
0.419341 + 0.907829i \(0.362261\pi\)
\(770\) 1.22911 0.0442940
\(771\) −0.0762573 −0.00274634
\(772\) 18.1254 0.652347
\(773\) −36.7853 −1.32307 −0.661537 0.749912i \(-0.730096\pi\)
−0.661537 + 0.749912i \(0.730096\pi\)
\(774\) −8.05306 −0.289461
\(775\) 12.4820 0.448365
\(776\) 15.6706 0.562543
\(777\) −11.0781 −0.397424
\(778\) 12.8220 0.459691
\(779\) −52.2422 −1.87177
\(780\) 0.643572 0.0230436
\(781\) −16.0721 −0.575105
\(782\) 9.08042 0.324715
\(783\) −6.15424 −0.219935
\(784\) −5.61456 −0.200520
\(785\) 2.89237 0.103233
\(786\) 3.53496 0.126088
\(787\) −5.48590 −0.195551 −0.0977756 0.995208i \(-0.531173\pi\)
−0.0977756 + 0.995208i \(0.531173\pi\)
\(788\) −19.3889 −0.690702
\(789\) 26.4812 0.942757
\(790\) 0.212043 0.00754415
\(791\) 15.4102 0.547923
\(792\) 1.62256 0.0576550
\(793\) −12.9422 −0.459592
\(794\) −20.6229 −0.731878
\(795\) 3.39157 0.120286
\(796\) 13.7592 0.487684
\(797\) −47.6326 −1.68723 −0.843617 0.536946i \(-0.819578\pi\)
−0.843617 + 0.536946i \(0.819578\pi\)
\(798\) 8.12926 0.287773
\(799\) −10.1383 −0.358669
\(800\) 4.58582 0.162133
\(801\) −1.76809 −0.0624722
\(802\) 3.88998 0.137360
\(803\) −3.37140 −0.118974
\(804\) 3.48887 0.123043
\(805\) −4.58998 −0.161776
\(806\) −2.72186 −0.0958735
\(807\) 23.4459 0.825335
\(808\) 5.93932 0.208945
\(809\) −38.1522 −1.34136 −0.670679 0.741747i \(-0.733997\pi\)
−0.670679 + 0.741747i \(0.733997\pi\)
\(810\) −0.643572 −0.0226128
\(811\) −16.4021 −0.575957 −0.287978 0.957637i \(-0.592983\pi\)
−0.287978 + 0.957637i \(0.592983\pi\)
\(812\) 7.24383 0.254208
\(813\) −19.3142 −0.677377
\(814\) 15.2711 0.535252
\(815\) 8.41916 0.294910
\(816\) −1.49860 −0.0524614
\(817\) 55.6184 1.94584
\(818\) −12.0257 −0.420468
\(819\) −1.17705 −0.0411293
\(820\) −4.86812 −0.170002
\(821\) 28.5588 0.996710 0.498355 0.866973i \(-0.333938\pi\)
0.498355 + 0.866973i \(0.333938\pi\)
\(822\) 14.5896 0.508870
\(823\) 27.8798 0.971829 0.485915 0.874006i \(-0.338487\pi\)
0.485915 + 0.874006i \(0.338487\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −7.44075 −0.259053
\(826\) −9.98444 −0.347403
\(827\) 29.5089 1.02613 0.513063 0.858351i \(-0.328511\pi\)
0.513063 + 0.858351i \(0.328511\pi\)
\(828\) −6.05927 −0.210574
\(829\) 11.4832 0.398829 0.199415 0.979915i \(-0.436096\pi\)
0.199415 + 0.979915i \(0.436096\pi\)
\(830\) 1.09263 0.0379256
\(831\) −17.3607 −0.602237
\(832\) −1.00000 −0.0346688
\(833\) −8.41397 −0.291527
\(834\) −0.829863 −0.0287358
\(835\) −0.541905 −0.0187534
\(836\) −11.2062 −0.387573
\(837\) 2.72186 0.0940813
\(838\) 9.30539 0.321449
\(839\) 34.7745 1.20055 0.600274 0.799795i \(-0.295058\pi\)
0.600274 + 0.799795i \(0.295058\pi\)
\(840\) 0.757514 0.0261367
\(841\) 8.87466 0.306023
\(842\) −37.1227 −1.27933
\(843\) 16.4966 0.568174
\(844\) 28.7921 0.991063
\(845\) 0.643572 0.0221395
\(846\) 6.76522 0.232593
\(847\) −9.84871 −0.338406
\(848\) −5.26991 −0.180970
\(849\) 17.2117 0.590703
\(850\) 6.87230 0.235718
\(851\) −57.0284 −1.95491
\(852\) −9.90543 −0.339354
\(853\) −33.8518 −1.15906 −0.579532 0.814950i \(-0.696765\pi\)
−0.579532 + 0.814950i \(0.696765\pi\)
\(854\) −15.2336 −0.521283
\(855\) 4.44482 0.152010
\(856\) −9.16659 −0.313308
\(857\) 0.260357 0.00889363 0.00444682 0.999990i \(-0.498585\pi\)
0.00444682 + 0.999990i \(0.498585\pi\)
\(858\) 1.62256 0.0553932
\(859\) 19.4697 0.664297 0.332149 0.943227i \(-0.392226\pi\)
0.332149 + 0.943227i \(0.392226\pi\)
\(860\) 5.18272 0.176729
\(861\) 8.90344 0.303429
\(862\) 4.29157 0.146171
\(863\) 37.6400 1.28128 0.640641 0.767841i \(-0.278669\pi\)
0.640641 + 0.767841i \(0.278669\pi\)
\(864\) 1.00000 0.0340207
\(865\) −8.30475 −0.282370
\(866\) −9.74101 −0.331013
\(867\) 14.7542 0.501079
\(868\) −3.20376 −0.108743
\(869\) 0.534597 0.0181350
\(870\) 3.96069 0.134280
\(871\) 3.48887 0.118216
\(872\) 0.979696 0.0331767
\(873\) −15.6706 −0.530370
\(874\) 41.8483 1.41554
\(875\) −7.26139 −0.245480
\(876\) −2.07783 −0.0702034
\(877\) −10.7653 −0.363518 −0.181759 0.983343i \(-0.558179\pi\)
−0.181759 + 0.983343i \(0.558179\pi\)
\(878\) −13.6511 −0.460703
\(879\) −4.91838 −0.165893
\(880\) −1.04423 −0.0352010
\(881\) 0.252673 0.00851276 0.00425638 0.999991i \(-0.498645\pi\)
0.00425638 + 0.999991i \(0.498645\pi\)
\(882\) 5.61456 0.189052
\(883\) 23.2546 0.782579 0.391289 0.920268i \(-0.372029\pi\)
0.391289 + 0.920268i \(0.372029\pi\)
\(884\) −1.49860 −0.0504033
\(885\) −5.45918 −0.183508
\(886\) 22.2391 0.747138
\(887\) 4.32823 0.145328 0.0726639 0.997356i \(-0.476850\pi\)
0.0726639 + 0.997356i \(0.476850\pi\)
\(888\) 9.41176 0.315838
\(889\) 5.20271 0.174493
\(890\) 1.13789 0.0381421
\(891\) −1.62256 −0.0543577
\(892\) −12.7364 −0.426446
\(893\) −46.7239 −1.56356
\(894\) 7.83483 0.262036
\(895\) 1.77812 0.0594359
\(896\) −1.17705 −0.0393224
\(897\) −6.05927 −0.202313
\(898\) 19.2668 0.642943
\(899\) −16.7510 −0.558677
\(900\) −4.58582 −0.152861
\(901\) −7.89749 −0.263103
\(902\) −12.2734 −0.408659
\(903\) −9.47883 −0.315436
\(904\) −13.0922 −0.435441
\(905\) 5.56103 0.184855
\(906\) 4.95195 0.164517
\(907\) 21.2902 0.706928 0.353464 0.935448i \(-0.385004\pi\)
0.353464 + 0.935448i \(0.385004\pi\)
\(908\) −22.9067 −0.760185
\(909\) −5.93932 −0.196995
\(910\) 0.757514 0.0251113
\(911\) 0.346872 0.0114924 0.00574619 0.999983i \(-0.498171\pi\)
0.00574619 + 0.999983i \(0.498171\pi\)
\(912\) −6.90649 −0.228697
\(913\) 2.75470 0.0911674
\(914\) 40.3189 1.33363
\(915\) −8.32924 −0.275356
\(916\) 10.6201 0.350897
\(917\) 4.16081 0.137402
\(918\) 1.49860 0.0494611
\(919\) −16.6945 −0.550700 −0.275350 0.961344i \(-0.588794\pi\)
−0.275350 + 0.961344i \(0.588794\pi\)
\(920\) 3.89957 0.128565
\(921\) 3.35221 0.110459
\(922\) 9.53646 0.314066
\(923\) −9.90543 −0.326041
\(924\) 1.90982 0.0628286
\(925\) −43.1606 −1.41911
\(926\) −21.8752 −0.718863
\(927\) 1.00000 0.0328443
\(928\) −6.15424 −0.202023
\(929\) −37.0289 −1.21488 −0.607439 0.794366i \(-0.707803\pi\)
−0.607439 + 0.794366i \(0.707803\pi\)
\(930\) −1.75171 −0.0574409
\(931\) −38.7769 −1.27086
\(932\) 6.61868 0.216802
\(933\) −28.2832 −0.925951
\(934\) −41.0475 −1.34311
\(935\) −1.56488 −0.0511772
\(936\) 1.00000 0.0326860
\(937\) 29.0955 0.950507 0.475254 0.879849i \(-0.342356\pi\)
0.475254 + 0.879849i \(0.342356\pi\)
\(938\) 4.10656 0.134084
\(939\) −22.0633 −0.720010
\(940\) −4.35390 −0.142009
\(941\) 3.95886 0.129055 0.0645276 0.997916i \(-0.479446\pi\)
0.0645276 + 0.997916i \(0.479446\pi\)
\(942\) 4.49425 0.146431
\(943\) 45.8337 1.49255
\(944\) 8.48262 0.276086
\(945\) −0.757514 −0.0246419
\(946\) 13.0665 0.424830
\(947\) −9.69321 −0.314987 −0.157493 0.987520i \(-0.550341\pi\)
−0.157493 + 0.987520i \(0.550341\pi\)
\(948\) 0.329478 0.0107010
\(949\) −2.07783 −0.0674492
\(950\) 31.6719 1.02757
\(951\) −6.69370 −0.217058
\(952\) −1.76392 −0.0571690
\(953\) −37.8939 −1.22750 −0.613752 0.789499i \(-0.710341\pi\)
−0.613752 + 0.789499i \(0.710341\pi\)
\(954\) 5.26991 0.170620
\(955\) −0.698090 −0.0225897
\(956\) −16.6591 −0.538794
\(957\) 9.98560 0.322789
\(958\) −13.1109 −0.423593
\(959\) 17.1726 0.554532
\(960\) −0.643572 −0.0207712
\(961\) −23.5915 −0.761015
\(962\) 9.41176 0.303447
\(963\) 9.16659 0.295389
\(964\) −13.5837 −0.437500
\(965\) 11.6650 0.375509
\(966\) −7.13205 −0.229470
\(967\) −20.9496 −0.673695 −0.336847 0.941559i \(-0.609361\pi\)
−0.336847 + 0.941559i \(0.609361\pi\)
\(968\) 8.36731 0.268936
\(969\) −10.3501 −0.332492
\(970\) 10.0852 0.323815
\(971\) −31.2211 −1.00193 −0.500966 0.865467i \(-0.667022\pi\)
−0.500966 + 0.865467i \(0.667022\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −0.976788 −0.0313144
\(974\) 9.34915 0.299566
\(975\) −4.58582 −0.146864
\(976\) 12.9422 0.414270
\(977\) 31.4890 1.00742 0.503711 0.863872i \(-0.331967\pi\)
0.503711 + 0.863872i \(0.331967\pi\)
\(978\) 13.0819 0.418314
\(979\) 2.86882 0.0916878
\(980\) −3.61337 −0.115425
\(981\) −0.979696 −0.0312793
\(982\) 3.89868 0.124412
\(983\) −45.3072 −1.44507 −0.722537 0.691332i \(-0.757024\pi\)
−0.722537 + 0.691332i \(0.757024\pi\)
\(984\) −7.56422 −0.241139
\(985\) −12.4782 −0.397588
\(986\) −9.22274 −0.293712
\(987\) 7.96298 0.253464
\(988\) −6.90649 −0.219725
\(989\) −48.7957 −1.55161
\(990\) 1.04423 0.0331879
\(991\) −2.35314 −0.0747498 −0.0373749 0.999301i \(-0.511900\pi\)
−0.0373749 + 0.999301i \(0.511900\pi\)
\(992\) 2.72186 0.0864192
\(993\) −15.8048 −0.501550
\(994\) −11.6591 −0.369806
\(995\) 8.85506 0.280724
\(996\) 1.69775 0.0537954
\(997\) 45.3234 1.43541 0.717703 0.696349i \(-0.245193\pi\)
0.717703 + 0.696349i \(0.245193\pi\)
\(998\) 25.0610 0.793292
\(999\) −9.41176 −0.297775
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 8034.2.a.ba.1.9 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.ba.1.9 14 1.1 even 1 trivial