Properties

Label 8034.2.a.bc.1.10
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $15$
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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} - 17600 x + 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-1.34829\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.34829 q^{5} -1.00000 q^{6} -1.25683 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.34829 q^{5} -1.00000 q^{6} -1.25683 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.34829 q^{10} +6.18039 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.25683 q^{14} -1.34829 q^{15} +1.00000 q^{16} +0.669960 q^{17} +1.00000 q^{18} +5.57232 q^{19} +1.34829 q^{20} +1.25683 q^{21} +6.18039 q^{22} -6.51348 q^{23} -1.00000 q^{24} -3.18210 q^{25} -1.00000 q^{26} -1.00000 q^{27} -1.25683 q^{28} +7.00067 q^{29} -1.34829 q^{30} +9.94843 q^{31} +1.00000 q^{32} -6.18039 q^{33} +0.669960 q^{34} -1.69458 q^{35} +1.00000 q^{36} -6.41537 q^{37} +5.57232 q^{38} +1.00000 q^{39} +1.34829 q^{40} +3.07063 q^{41} +1.25683 q^{42} +5.34321 q^{43} +6.18039 q^{44} +1.34829 q^{45} -6.51348 q^{46} +6.23707 q^{47} -1.00000 q^{48} -5.42038 q^{49} -3.18210 q^{50} -0.669960 q^{51} -1.00000 q^{52} -8.03980 q^{53} -1.00000 q^{54} +8.33299 q^{55} -1.25683 q^{56} -5.57232 q^{57} +7.00067 q^{58} -3.12465 q^{59} -1.34829 q^{60} +1.33683 q^{61} +9.94843 q^{62} -1.25683 q^{63} +1.00000 q^{64} -1.34829 q^{65} -6.18039 q^{66} +8.66119 q^{67} +0.669960 q^{68} +6.51348 q^{69} -1.69458 q^{70} +6.62891 q^{71} +1.00000 q^{72} -2.20900 q^{73} -6.41537 q^{74} +3.18210 q^{75} +5.57232 q^{76} -7.76770 q^{77} +1.00000 q^{78} -12.7247 q^{79} +1.34829 q^{80} +1.00000 q^{81} +3.07063 q^{82} -0.863190 q^{83} +1.25683 q^{84} +0.903303 q^{85} +5.34321 q^{86} -7.00067 q^{87} +6.18039 q^{88} +4.65035 q^{89} +1.34829 q^{90} +1.25683 q^{91} -6.51348 q^{92} -9.94843 q^{93} +6.23707 q^{94} +7.51313 q^{95} -1.00000 q^{96} -11.6073 q^{97} -5.42038 q^{98} +6.18039 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

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
\(5\) 1.34829 0.602975 0.301488 0.953470i \(-0.402517\pi\)
0.301488 + 0.953470i \(0.402517\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.25683 −0.475037 −0.237518 0.971383i \(-0.576334\pi\)
−0.237518 + 0.971383i \(0.576334\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.34829 0.426368
\(11\) 6.18039 1.86346 0.931729 0.363154i \(-0.118300\pi\)
0.931729 + 0.363154i \(0.118300\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −1.25683 −0.335902
\(15\) −1.34829 −0.348128
\(16\) 1.00000 0.250000
\(17\) 0.669960 0.162489 0.0812445 0.996694i \(-0.474111\pi\)
0.0812445 + 0.996694i \(0.474111\pi\)
\(18\) 1.00000 0.235702
\(19\) 5.57232 1.27838 0.639189 0.769050i \(-0.279270\pi\)
0.639189 + 0.769050i \(0.279270\pi\)
\(20\) 1.34829 0.301488
\(21\) 1.25683 0.274263
\(22\) 6.18039 1.31766
\(23\) −6.51348 −1.35815 −0.679077 0.734067i \(-0.737620\pi\)
−0.679077 + 0.734067i \(0.737620\pi\)
\(24\) −1.00000 −0.204124
\(25\) −3.18210 −0.636421
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −1.25683 −0.237518
\(29\) 7.00067 1.29999 0.649996 0.759938i \(-0.274771\pi\)
0.649996 + 0.759938i \(0.274771\pi\)
\(30\) −1.34829 −0.246164
\(31\) 9.94843 1.78679 0.893396 0.449270i \(-0.148316\pi\)
0.893396 + 0.449270i \(0.148316\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.18039 −1.07587
\(34\) 0.669960 0.114897
\(35\) −1.69458 −0.286436
\(36\) 1.00000 0.166667
\(37\) −6.41537 −1.05468 −0.527341 0.849654i \(-0.676811\pi\)
−0.527341 + 0.849654i \(0.676811\pi\)
\(38\) 5.57232 0.903949
\(39\) 1.00000 0.160128
\(40\) 1.34829 0.213184
\(41\) 3.07063 0.479552 0.239776 0.970828i \(-0.422926\pi\)
0.239776 + 0.970828i \(0.422926\pi\)
\(42\) 1.25683 0.193933
\(43\) 5.34321 0.814832 0.407416 0.913243i \(-0.366430\pi\)
0.407416 + 0.913243i \(0.366430\pi\)
\(44\) 6.18039 0.931729
\(45\) 1.34829 0.200992
\(46\) −6.51348 −0.960360
\(47\) 6.23707 0.909770 0.454885 0.890550i \(-0.349680\pi\)
0.454885 + 0.890550i \(0.349680\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.42038 −0.774340
\(50\) −3.18210 −0.450017
\(51\) −0.669960 −0.0938131
\(52\) −1.00000 −0.138675
\(53\) −8.03980 −1.10435 −0.552176 0.833728i \(-0.686202\pi\)
−0.552176 + 0.833728i \(0.686202\pi\)
\(54\) −1.00000 −0.136083
\(55\) 8.33299 1.12362
\(56\) −1.25683 −0.167951
\(57\) −5.57232 −0.738072
\(58\) 7.00067 0.919233
\(59\) −3.12465 −0.406795 −0.203398 0.979096i \(-0.565198\pi\)
−0.203398 + 0.979096i \(0.565198\pi\)
\(60\) −1.34829 −0.174064
\(61\) 1.33683 0.171163 0.0855815 0.996331i \(-0.472725\pi\)
0.0855815 + 0.996331i \(0.472725\pi\)
\(62\) 9.94843 1.26345
\(63\) −1.25683 −0.158346
\(64\) 1.00000 0.125000
\(65\) −1.34829 −0.167235
\(66\) −6.18039 −0.760754
\(67\) 8.66119 1.05813 0.529067 0.848580i \(-0.322542\pi\)
0.529067 + 0.848580i \(0.322542\pi\)
\(68\) 0.669960 0.0812445
\(69\) 6.51348 0.784131
\(70\) −1.69458 −0.202541
\(71\) 6.62891 0.786707 0.393353 0.919387i \(-0.371315\pi\)
0.393353 + 0.919387i \(0.371315\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.20900 −0.258544 −0.129272 0.991609i \(-0.541264\pi\)
−0.129272 + 0.991609i \(0.541264\pi\)
\(74\) −6.41537 −0.745772
\(75\) 3.18210 0.367438
\(76\) 5.57232 0.639189
\(77\) −7.76770 −0.885211
\(78\) 1.00000 0.113228
\(79\) −12.7247 −1.43164 −0.715822 0.698283i \(-0.753948\pi\)
−0.715822 + 0.698283i \(0.753948\pi\)
\(80\) 1.34829 0.150744
\(81\) 1.00000 0.111111
\(82\) 3.07063 0.339094
\(83\) −0.863190 −0.0947474 −0.0473737 0.998877i \(-0.515085\pi\)
−0.0473737 + 0.998877i \(0.515085\pi\)
\(84\) 1.25683 0.137131
\(85\) 0.903303 0.0979769
\(86\) 5.34321 0.576173
\(87\) −7.00067 −0.750551
\(88\) 6.18039 0.658832
\(89\) 4.65035 0.492936 0.246468 0.969151i \(-0.420730\pi\)
0.246468 + 0.969151i \(0.420730\pi\)
\(90\) 1.34829 0.142123
\(91\) 1.25683 0.131752
\(92\) −6.51348 −0.679077
\(93\) −9.94843 −1.03160
\(94\) 6.23707 0.643305
\(95\) 7.51313 0.770830
\(96\) −1.00000 −0.102062
\(97\) −11.6073 −1.17855 −0.589273 0.807934i \(-0.700586\pi\)
−0.589273 + 0.807934i \(0.700586\pi\)
\(98\) −5.42038 −0.547541
\(99\) 6.18039 0.621153
\(100\) −3.18210 −0.318210
\(101\) 7.53357 0.749618 0.374809 0.927102i \(-0.377708\pi\)
0.374809 + 0.927102i \(0.377708\pi\)
\(102\) −0.669960 −0.0663359
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 1.69458 0.165374
\(106\) −8.03980 −0.780895
\(107\) 10.7444 1.03870 0.519352 0.854560i \(-0.326173\pi\)
0.519352 + 0.854560i \(0.326173\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 0.0532230 0.00509784 0.00254892 0.999997i \(-0.499189\pi\)
0.00254892 + 0.999997i \(0.499189\pi\)
\(110\) 8.33299 0.794519
\(111\) 6.41537 0.608920
\(112\) −1.25683 −0.118759
\(113\) 13.8205 1.30012 0.650060 0.759883i \(-0.274744\pi\)
0.650060 + 0.759883i \(0.274744\pi\)
\(114\) −5.57232 −0.521895
\(115\) −8.78209 −0.818934
\(116\) 7.00067 0.649996
\(117\) −1.00000 −0.0924500
\(118\) −3.12465 −0.287648
\(119\) −0.842025 −0.0771883
\(120\) −1.34829 −0.123082
\(121\) 27.1972 2.47248
\(122\) 1.33683 0.121031
\(123\) −3.07063 −0.276869
\(124\) 9.94843 0.893396
\(125\) −11.0319 −0.986721
\(126\) −1.25683 −0.111967
\(127\) −3.53406 −0.313597 −0.156799 0.987631i \(-0.550117\pi\)
−0.156799 + 0.987631i \(0.550117\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.34321 −0.470443
\(130\) −1.34829 −0.118253
\(131\) 1.91302 0.167142 0.0835708 0.996502i \(-0.473368\pi\)
0.0835708 + 0.996502i \(0.473368\pi\)
\(132\) −6.18039 −0.537934
\(133\) −7.00345 −0.607276
\(134\) 8.66119 0.748213
\(135\) −1.34829 −0.116043
\(136\) 0.669960 0.0574486
\(137\) −5.22718 −0.446588 −0.223294 0.974751i \(-0.571681\pi\)
−0.223294 + 0.974751i \(0.571681\pi\)
\(138\) 6.51348 0.554464
\(139\) −9.43083 −0.799913 −0.399956 0.916534i \(-0.630975\pi\)
−0.399956 + 0.916534i \(0.630975\pi\)
\(140\) −1.69458 −0.143218
\(141\) −6.23707 −0.525256
\(142\) 6.62891 0.556286
\(143\) −6.18039 −0.516830
\(144\) 1.00000 0.0833333
\(145\) 9.43896 0.783863
\(146\) −2.20900 −0.182818
\(147\) 5.42038 0.447065
\(148\) −6.41537 −0.527341
\(149\) −7.36592 −0.603439 −0.301720 0.953397i \(-0.597561\pi\)
−0.301720 + 0.953397i \(0.597561\pi\)
\(150\) 3.18210 0.259818
\(151\) 8.10419 0.659509 0.329755 0.944067i \(-0.393034\pi\)
0.329755 + 0.944067i \(0.393034\pi\)
\(152\) 5.57232 0.451975
\(153\) 0.669960 0.0541630
\(154\) −7.76770 −0.625939
\(155\) 13.4134 1.07739
\(156\) 1.00000 0.0800641
\(157\) −7.16921 −0.572165 −0.286083 0.958205i \(-0.592353\pi\)
−0.286083 + 0.958205i \(0.592353\pi\)
\(158\) −12.7247 −1.01232
\(159\) 8.03980 0.637598
\(160\) 1.34829 0.106592
\(161\) 8.18633 0.645173
\(162\) 1.00000 0.0785674
\(163\) −14.4487 −1.13171 −0.565855 0.824505i \(-0.691454\pi\)
−0.565855 + 0.824505i \(0.691454\pi\)
\(164\) 3.07063 0.239776
\(165\) −8.33299 −0.648722
\(166\) −0.863190 −0.0669965
\(167\) −19.0982 −1.47786 −0.738932 0.673780i \(-0.764670\pi\)
−0.738932 + 0.673780i \(0.764670\pi\)
\(168\) 1.25683 0.0969665
\(169\) 1.00000 0.0769231
\(170\) 0.903303 0.0692801
\(171\) 5.57232 0.426126
\(172\) 5.34321 0.407416
\(173\) 22.9671 1.74616 0.873079 0.487579i \(-0.162120\pi\)
0.873079 + 0.487579i \(0.162120\pi\)
\(174\) −7.00067 −0.530719
\(175\) 3.99936 0.302323
\(176\) 6.18039 0.465865
\(177\) 3.12465 0.234863
\(178\) 4.65035 0.348559
\(179\) 2.88656 0.215752 0.107876 0.994164i \(-0.465595\pi\)
0.107876 + 0.994164i \(0.465595\pi\)
\(180\) 1.34829 0.100496
\(181\) 16.8058 1.24917 0.624584 0.780958i \(-0.285269\pi\)
0.624584 + 0.780958i \(0.285269\pi\)
\(182\) 1.25683 0.0931624
\(183\) −1.33683 −0.0988210
\(184\) −6.51348 −0.480180
\(185\) −8.64981 −0.635947
\(186\) −9.94843 −0.729455
\(187\) 4.14061 0.302792
\(188\) 6.23707 0.454885
\(189\) 1.25683 0.0914209
\(190\) 7.51313 0.545059
\(191\) −4.98127 −0.360432 −0.180216 0.983627i \(-0.557680\pi\)
−0.180216 + 0.983627i \(0.557680\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 7.47451 0.538027 0.269014 0.963136i \(-0.413302\pi\)
0.269014 + 0.963136i \(0.413302\pi\)
\(194\) −11.6073 −0.833358
\(195\) 1.34829 0.0965534
\(196\) −5.42038 −0.387170
\(197\) 16.2736 1.15945 0.579725 0.814812i \(-0.303160\pi\)
0.579725 + 0.814812i \(0.303160\pi\)
\(198\) 6.18039 0.439221
\(199\) −10.9796 −0.778320 −0.389160 0.921170i \(-0.627235\pi\)
−0.389160 + 0.921170i \(0.627235\pi\)
\(200\) −3.18210 −0.225009
\(201\) −8.66119 −0.610913
\(202\) 7.53357 0.530060
\(203\) −8.79865 −0.617544
\(204\) −0.669960 −0.0469066
\(205\) 4.14011 0.289158
\(206\) −1.00000 −0.0696733
\(207\) −6.51348 −0.452718
\(208\) −1.00000 −0.0693375
\(209\) 34.4391 2.38220
\(210\) 1.69458 0.116937
\(211\) −20.4841 −1.41018 −0.705091 0.709117i \(-0.749094\pi\)
−0.705091 + 0.709117i \(0.749094\pi\)
\(212\) −8.03980 −0.552176
\(213\) −6.62891 −0.454205
\(214\) 10.7444 0.734475
\(215\) 7.20422 0.491324
\(216\) −1.00000 −0.0680414
\(217\) −12.5035 −0.848792
\(218\) 0.0532230 0.00360472
\(219\) 2.20900 0.149270
\(220\) 8.33299 0.561810
\(221\) −0.669960 −0.0450664
\(222\) 6.41537 0.430572
\(223\) 3.21166 0.215068 0.107534 0.994201i \(-0.465704\pi\)
0.107534 + 0.994201i \(0.465704\pi\)
\(224\) −1.25683 −0.0839754
\(225\) −3.18210 −0.212140
\(226\) 13.8205 0.919324
\(227\) −28.6424 −1.90106 −0.950530 0.310633i \(-0.899459\pi\)
−0.950530 + 0.310633i \(0.899459\pi\)
\(228\) −5.57232 −0.369036
\(229\) 10.9174 0.721444 0.360722 0.932673i \(-0.382530\pi\)
0.360722 + 0.932673i \(0.382530\pi\)
\(230\) −8.78209 −0.579074
\(231\) 7.76770 0.511077
\(232\) 7.00067 0.459617
\(233\) 22.8998 1.50022 0.750108 0.661316i \(-0.230002\pi\)
0.750108 + 0.661316i \(0.230002\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 8.40941 0.548569
\(236\) −3.12465 −0.203398
\(237\) 12.7247 0.826560
\(238\) −0.842025 −0.0545804
\(239\) 2.97784 0.192621 0.0963103 0.995351i \(-0.469296\pi\)
0.0963103 + 0.995351i \(0.469296\pi\)
\(240\) −1.34829 −0.0870320
\(241\) 13.4226 0.864626 0.432313 0.901724i \(-0.357698\pi\)
0.432313 + 0.901724i \(0.357698\pi\)
\(242\) 27.1972 1.74831
\(243\) −1.00000 −0.0641500
\(244\) 1.33683 0.0855815
\(245\) −7.30827 −0.466908
\(246\) −3.07063 −0.195776
\(247\) −5.57232 −0.354558
\(248\) 9.94843 0.631726
\(249\) 0.863190 0.0547024
\(250\) −11.0319 −0.697717
\(251\) −11.9131 −0.751947 −0.375973 0.926631i \(-0.622692\pi\)
−0.375973 + 0.926631i \(0.622692\pi\)
\(252\) −1.25683 −0.0791728
\(253\) −40.2559 −2.53086
\(254\) −3.53406 −0.221747
\(255\) −0.903303 −0.0565670
\(256\) 1.00000 0.0625000
\(257\) 22.7340 1.41811 0.709054 0.705154i \(-0.249122\pi\)
0.709054 + 0.705154i \(0.249122\pi\)
\(258\) −5.34321 −0.332654
\(259\) 8.06303 0.501012
\(260\) −1.34829 −0.0836177
\(261\) 7.00067 0.433331
\(262\) 1.91302 0.118187
\(263\) 15.9585 0.984046 0.492023 0.870582i \(-0.336258\pi\)
0.492023 + 0.870582i \(0.336258\pi\)
\(264\) −6.18039 −0.380377
\(265\) −10.8400 −0.665897
\(266\) −7.00345 −0.429409
\(267\) −4.65035 −0.284597
\(268\) 8.66119 0.529067
\(269\) 16.0982 0.981526 0.490763 0.871293i \(-0.336718\pi\)
0.490763 + 0.871293i \(0.336718\pi\)
\(270\) −1.34829 −0.0820546
\(271\) 2.40939 0.146360 0.0731801 0.997319i \(-0.476685\pi\)
0.0731801 + 0.997319i \(0.476685\pi\)
\(272\) 0.669960 0.0406223
\(273\) −1.25683 −0.0760668
\(274\) −5.22718 −0.315785
\(275\) −19.6666 −1.18594
\(276\) 6.51348 0.392065
\(277\) 1.30583 0.0784598 0.0392299 0.999230i \(-0.487510\pi\)
0.0392299 + 0.999230i \(0.487510\pi\)
\(278\) −9.43083 −0.565624
\(279\) 9.94843 0.595597
\(280\) −1.69458 −0.101270
\(281\) 12.3979 0.739596 0.369798 0.929112i \(-0.379427\pi\)
0.369798 + 0.929112i \(0.379427\pi\)
\(282\) −6.23707 −0.371412
\(283\) 23.0980 1.37303 0.686516 0.727115i \(-0.259139\pi\)
0.686516 + 0.727115i \(0.259139\pi\)
\(284\) 6.62891 0.393353
\(285\) −7.51313 −0.445039
\(286\) −6.18039 −0.365454
\(287\) −3.85926 −0.227805
\(288\) 1.00000 0.0589256
\(289\) −16.5512 −0.973597
\(290\) 9.43896 0.554275
\(291\) 11.6073 0.680434
\(292\) −2.20900 −0.129272
\(293\) 4.86440 0.284181 0.142091 0.989854i \(-0.454618\pi\)
0.142091 + 0.989854i \(0.454618\pi\)
\(294\) 5.42038 0.316123
\(295\) −4.21295 −0.245288
\(296\) −6.41537 −0.372886
\(297\) −6.18039 −0.358623
\(298\) −7.36592 −0.426696
\(299\) 6.51348 0.376684
\(300\) 3.18210 0.183719
\(301\) −6.71550 −0.387075
\(302\) 8.10419 0.466344
\(303\) −7.53357 −0.432792
\(304\) 5.57232 0.319594
\(305\) 1.80243 0.103207
\(306\) 0.669960 0.0382990
\(307\) −28.0036 −1.59825 −0.799124 0.601167i \(-0.794703\pi\)
−0.799124 + 0.601167i \(0.794703\pi\)
\(308\) −7.76770 −0.442606
\(309\) 1.00000 0.0568880
\(310\) 13.4134 0.761831
\(311\) 25.6298 1.45333 0.726666 0.686991i \(-0.241069\pi\)
0.726666 + 0.686991i \(0.241069\pi\)
\(312\) 1.00000 0.0566139
\(313\) 0.830172 0.0469241 0.0234621 0.999725i \(-0.492531\pi\)
0.0234621 + 0.999725i \(0.492531\pi\)
\(314\) −7.16921 −0.404582
\(315\) −1.69458 −0.0954785
\(316\) −12.7247 −0.715822
\(317\) 15.6607 0.879593 0.439796 0.898098i \(-0.355051\pi\)
0.439796 + 0.898098i \(0.355051\pi\)
\(318\) 8.03980 0.450850
\(319\) 43.2669 2.42248
\(320\) 1.34829 0.0753719
\(321\) −10.7444 −0.599696
\(322\) 8.18633 0.456206
\(323\) 3.73323 0.207722
\(324\) 1.00000 0.0555556
\(325\) 3.18210 0.176511
\(326\) −14.4487 −0.800240
\(327\) −0.0532230 −0.00294324
\(328\) 3.07063 0.169547
\(329\) −7.83893 −0.432174
\(330\) −8.33299 −0.458716
\(331\) 33.8850 1.86249 0.931244 0.364396i \(-0.118725\pi\)
0.931244 + 0.364396i \(0.118725\pi\)
\(332\) −0.863190 −0.0473737
\(333\) −6.41537 −0.351560
\(334\) −19.0982 −1.04501
\(335\) 11.6778 0.638028
\(336\) 1.25683 0.0685657
\(337\) 4.89288 0.266532 0.133266 0.991080i \(-0.457454\pi\)
0.133266 + 0.991080i \(0.457454\pi\)
\(338\) 1.00000 0.0543928
\(339\) −13.8205 −0.750625
\(340\) 0.903303 0.0489885
\(341\) 61.4852 3.32961
\(342\) 5.57232 0.301316
\(343\) 15.6103 0.842877
\(344\) 5.34321 0.288086
\(345\) 8.78209 0.472812
\(346\) 22.9671 1.23472
\(347\) −15.6931 −0.842449 −0.421225 0.906956i \(-0.638400\pi\)
−0.421225 + 0.906956i \(0.638400\pi\)
\(348\) −7.00067 −0.375275
\(349\) −17.0990 −0.915290 −0.457645 0.889135i \(-0.651307\pi\)
−0.457645 + 0.889135i \(0.651307\pi\)
\(350\) 3.99936 0.213775
\(351\) 1.00000 0.0533761
\(352\) 6.18039 0.329416
\(353\) 21.2245 1.12967 0.564833 0.825205i \(-0.308940\pi\)
0.564833 + 0.825205i \(0.308940\pi\)
\(354\) 3.12465 0.166074
\(355\) 8.93772 0.474365
\(356\) 4.65035 0.246468
\(357\) 0.842025 0.0445647
\(358\) 2.88656 0.152560
\(359\) −14.9473 −0.788888 −0.394444 0.918920i \(-0.629063\pi\)
−0.394444 + 0.918920i \(0.629063\pi\)
\(360\) 1.34829 0.0710613
\(361\) 12.0507 0.634249
\(362\) 16.8058 0.883295
\(363\) −27.1972 −1.42749
\(364\) 1.25683 0.0658758
\(365\) −2.97838 −0.155896
\(366\) −1.33683 −0.0698770
\(367\) −12.1126 −0.632272 −0.316136 0.948714i \(-0.602386\pi\)
−0.316136 + 0.948714i \(0.602386\pi\)
\(368\) −6.51348 −0.339539
\(369\) 3.07063 0.159851
\(370\) −8.64981 −0.449682
\(371\) 10.1047 0.524608
\(372\) −9.94843 −0.515802
\(373\) −6.77801 −0.350952 −0.175476 0.984484i \(-0.556146\pi\)
−0.175476 + 0.984484i \(0.556146\pi\)
\(374\) 4.14061 0.214106
\(375\) 11.0319 0.569684
\(376\) 6.23707 0.321652
\(377\) −7.00067 −0.360553
\(378\) 1.25683 0.0646443
\(379\) −4.76965 −0.245001 −0.122500 0.992468i \(-0.539091\pi\)
−0.122500 + 0.992468i \(0.539091\pi\)
\(380\) 7.51313 0.385415
\(381\) 3.53406 0.181055
\(382\) −4.98127 −0.254864
\(383\) −12.0847 −0.617499 −0.308749 0.951143i \(-0.599910\pi\)
−0.308749 + 0.951143i \(0.599910\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −10.4731 −0.533761
\(386\) 7.47451 0.380443
\(387\) 5.34321 0.271611
\(388\) −11.6073 −0.589273
\(389\) 26.8990 1.36383 0.681916 0.731431i \(-0.261147\pi\)
0.681916 + 0.731431i \(0.261147\pi\)
\(390\) 1.34829 0.0682735
\(391\) −4.36377 −0.220685
\(392\) −5.42038 −0.273771
\(393\) −1.91302 −0.0964993
\(394\) 16.2736 0.819854
\(395\) −17.1567 −0.863246
\(396\) 6.18039 0.310576
\(397\) 9.11191 0.457314 0.228657 0.973507i \(-0.426567\pi\)
0.228657 + 0.973507i \(0.426567\pi\)
\(398\) −10.9796 −0.550356
\(399\) 7.00345 0.350611
\(400\) −3.18210 −0.159105
\(401\) 5.41938 0.270631 0.135315 0.990803i \(-0.456795\pi\)
0.135315 + 0.990803i \(0.456795\pi\)
\(402\) −8.66119 −0.431981
\(403\) −9.94843 −0.495567
\(404\) 7.53357 0.374809
\(405\) 1.34829 0.0669973
\(406\) −8.79865 −0.436670
\(407\) −39.6495 −1.96535
\(408\) −0.669960 −0.0331679
\(409\) 23.0164 1.13809 0.569043 0.822308i \(-0.307314\pi\)
0.569043 + 0.822308i \(0.307314\pi\)
\(410\) 4.14011 0.204466
\(411\) 5.22718 0.257838
\(412\) −1.00000 −0.0492665
\(413\) 3.92716 0.193243
\(414\) −6.51348 −0.320120
\(415\) −1.16383 −0.0571303
\(416\) −1.00000 −0.0490290
\(417\) 9.43083 0.461830
\(418\) 34.4391 1.68447
\(419\) 12.9522 0.632758 0.316379 0.948633i \(-0.397533\pi\)
0.316379 + 0.948633i \(0.397533\pi\)
\(420\) 1.69458 0.0826868
\(421\) −30.0337 −1.46375 −0.731877 0.681437i \(-0.761355\pi\)
−0.731877 + 0.681437i \(0.761355\pi\)
\(422\) −20.4841 −0.997149
\(423\) 6.23707 0.303257
\(424\) −8.03980 −0.390447
\(425\) −2.13188 −0.103411
\(426\) −6.62891 −0.321172
\(427\) −1.68016 −0.0813087
\(428\) 10.7444 0.519352
\(429\) 6.18039 0.298392
\(430\) 7.20422 0.347418
\(431\) −27.0278 −1.30188 −0.650942 0.759127i \(-0.725626\pi\)
−0.650942 + 0.759127i \(0.725626\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 38.4811 1.84928 0.924641 0.380841i \(-0.124365\pi\)
0.924641 + 0.380841i \(0.124365\pi\)
\(434\) −12.5035 −0.600186
\(435\) −9.43896 −0.452564
\(436\) 0.0532230 0.00254892
\(437\) −36.2952 −1.73623
\(438\) 2.20900 0.105550
\(439\) −28.9181 −1.38018 −0.690092 0.723722i \(-0.742430\pi\)
−0.690092 + 0.723722i \(0.742430\pi\)
\(440\) 8.33299 0.397260
\(441\) −5.42038 −0.258113
\(442\) −0.669960 −0.0318667
\(443\) 5.38288 0.255748 0.127874 0.991790i \(-0.459185\pi\)
0.127874 + 0.991790i \(0.459185\pi\)
\(444\) 6.41537 0.304460
\(445\) 6.27004 0.297229
\(446\) 3.21166 0.152076
\(447\) 7.36592 0.348396
\(448\) −1.25683 −0.0593796
\(449\) −14.8713 −0.701821 −0.350910 0.936409i \(-0.614128\pi\)
−0.350910 + 0.936409i \(0.614128\pi\)
\(450\) −3.18210 −0.150006
\(451\) 18.9777 0.893625
\(452\) 13.8205 0.650060
\(453\) −8.10419 −0.380768
\(454\) −28.6424 −1.34425
\(455\) 1.69458 0.0794429
\(456\) −5.57232 −0.260948
\(457\) 4.72695 0.221118 0.110559 0.993870i \(-0.464736\pi\)
0.110559 + 0.993870i \(0.464736\pi\)
\(458\) 10.9174 0.510138
\(459\) −0.669960 −0.0312710
\(460\) −8.78209 −0.409467
\(461\) −9.95374 −0.463592 −0.231796 0.972764i \(-0.574460\pi\)
−0.231796 + 0.972764i \(0.574460\pi\)
\(462\) 7.76770 0.361386
\(463\) 29.5029 1.37112 0.685559 0.728018i \(-0.259558\pi\)
0.685559 + 0.728018i \(0.259558\pi\)
\(464\) 7.00067 0.324998
\(465\) −13.4134 −0.622032
\(466\) 22.8998 1.06081
\(467\) −34.5248 −1.59762 −0.798810 0.601584i \(-0.794537\pi\)
−0.798810 + 0.601584i \(0.794537\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −10.8856 −0.502652
\(470\) 8.40941 0.387897
\(471\) 7.16921 0.330340
\(472\) −3.12465 −0.143824
\(473\) 33.0231 1.51840
\(474\) 12.7247 0.584466
\(475\) −17.7317 −0.813586
\(476\) −0.842025 −0.0385941
\(477\) −8.03980 −0.368117
\(478\) 2.97784 0.136203
\(479\) −14.2751 −0.652248 −0.326124 0.945327i \(-0.605743\pi\)
−0.326124 + 0.945327i \(0.605743\pi\)
\(480\) −1.34829 −0.0615409
\(481\) 6.41537 0.292516
\(482\) 13.4226 0.611383
\(483\) −8.18633 −0.372491
\(484\) 27.1972 1.23624
\(485\) −15.6501 −0.710635
\(486\) −1.00000 −0.0453609
\(487\) 11.3377 0.513762 0.256881 0.966443i \(-0.417305\pi\)
0.256881 + 0.966443i \(0.417305\pi\)
\(488\) 1.33683 0.0605153
\(489\) 14.4487 0.653393
\(490\) −7.30827 −0.330154
\(491\) −26.6950 −1.20473 −0.602363 0.798222i \(-0.705774\pi\)
−0.602363 + 0.798222i \(0.705774\pi\)
\(492\) −3.07063 −0.138435
\(493\) 4.69017 0.211234
\(494\) −5.57232 −0.250710
\(495\) 8.33299 0.374540
\(496\) 9.94843 0.446698
\(497\) −8.33141 −0.373715
\(498\) 0.863190 0.0386804
\(499\) 7.46541 0.334197 0.167099 0.985940i \(-0.446560\pi\)
0.167099 + 0.985940i \(0.446560\pi\)
\(500\) −11.0319 −0.493361
\(501\) 19.0982 0.853245
\(502\) −11.9131 −0.531707
\(503\) 17.5149 0.780950 0.390475 0.920614i \(-0.372311\pi\)
0.390475 + 0.920614i \(0.372311\pi\)
\(504\) −1.25683 −0.0559836
\(505\) 10.1575 0.452001
\(506\) −40.2559 −1.78959
\(507\) −1.00000 −0.0444116
\(508\) −3.53406 −0.156799
\(509\) −5.39952 −0.239330 −0.119665 0.992814i \(-0.538182\pi\)
−0.119665 + 0.992814i \(0.538182\pi\)
\(510\) −0.903303 −0.0399989
\(511\) 2.77634 0.122818
\(512\) 1.00000 0.0441942
\(513\) −5.57232 −0.246024
\(514\) 22.7340 1.00275
\(515\) −1.34829 −0.0594129
\(516\) −5.34321 −0.235222
\(517\) 38.5475 1.69532
\(518\) 8.06303 0.354269
\(519\) −22.9671 −1.00814
\(520\) −1.34829 −0.0591266
\(521\) −1.82746 −0.0800624 −0.0400312 0.999198i \(-0.512746\pi\)
−0.0400312 + 0.999198i \(0.512746\pi\)
\(522\) 7.00067 0.306411
\(523\) −4.67744 −0.204530 −0.102265 0.994757i \(-0.532609\pi\)
−0.102265 + 0.994757i \(0.532609\pi\)
\(524\) 1.91302 0.0835708
\(525\) −3.99936 −0.174546
\(526\) 15.9585 0.695825
\(527\) 6.66505 0.290334
\(528\) −6.18039 −0.268967
\(529\) 19.4254 0.844583
\(530\) −10.8400 −0.470860
\(531\) −3.12465 −0.135598
\(532\) −7.00345 −0.303638
\(533\) −3.07063 −0.133004
\(534\) −4.65035 −0.201240
\(535\) 14.4867 0.626313
\(536\) 8.66119 0.374107
\(537\) −2.88656 −0.124564
\(538\) 16.0982 0.694044
\(539\) −33.5001 −1.44295
\(540\) −1.34829 −0.0580213
\(541\) 7.49168 0.322092 0.161046 0.986947i \(-0.448513\pi\)
0.161046 + 0.986947i \(0.448513\pi\)
\(542\) 2.40939 0.103492
\(543\) −16.8058 −0.721207
\(544\) 0.669960 0.0287243
\(545\) 0.0717602 0.00307387
\(546\) −1.25683 −0.0537873
\(547\) 33.3140 1.42440 0.712202 0.701975i \(-0.247698\pi\)
0.712202 + 0.701975i \(0.247698\pi\)
\(548\) −5.22718 −0.223294
\(549\) 1.33683 0.0570543
\(550\) −19.6666 −0.838588
\(551\) 39.0100 1.66188
\(552\) 6.51348 0.277232
\(553\) 15.9928 0.680083
\(554\) 1.30583 0.0554794
\(555\) 8.64981 0.367164
\(556\) −9.43083 −0.399956
\(557\) −20.5170 −0.869331 −0.434666 0.900592i \(-0.643133\pi\)
−0.434666 + 0.900592i \(0.643133\pi\)
\(558\) 9.94843 0.421151
\(559\) −5.34321 −0.225994
\(560\) −1.69458 −0.0716089
\(561\) −4.14061 −0.174817
\(562\) 12.3979 0.522973
\(563\) −10.1882 −0.429383 −0.214691 0.976682i \(-0.568875\pi\)
−0.214691 + 0.976682i \(0.568875\pi\)
\(564\) −6.23707 −0.262628
\(565\) 18.6341 0.783941
\(566\) 23.0980 0.970880
\(567\) −1.25683 −0.0527819
\(568\) 6.62891 0.278143
\(569\) −31.7117 −1.32943 −0.664713 0.747099i \(-0.731446\pi\)
−0.664713 + 0.747099i \(0.731446\pi\)
\(570\) −7.51313 −0.314690
\(571\) 42.4864 1.77800 0.889000 0.457906i \(-0.151401\pi\)
0.889000 + 0.457906i \(0.151401\pi\)
\(572\) −6.18039 −0.258415
\(573\) 4.98127 0.208095
\(574\) −3.85926 −0.161082
\(575\) 20.7266 0.864357
\(576\) 1.00000 0.0416667
\(577\) −20.5607 −0.855951 −0.427976 0.903790i \(-0.640773\pi\)
−0.427976 + 0.903790i \(0.640773\pi\)
\(578\) −16.5512 −0.688437
\(579\) −7.47451 −0.310630
\(580\) 9.43896 0.391932
\(581\) 1.08488 0.0450085
\(582\) 11.6073 0.481140
\(583\) −49.6891 −2.05791
\(584\) −2.20900 −0.0914091
\(585\) −1.34829 −0.0557451
\(586\) 4.86440 0.200946
\(587\) −13.9190 −0.574498 −0.287249 0.957856i \(-0.592741\pi\)
−0.287249 + 0.957856i \(0.592741\pi\)
\(588\) 5.42038 0.223533
\(589\) 55.4358 2.28419
\(590\) −4.21295 −0.173445
\(591\) −16.2736 −0.669408
\(592\) −6.41537 −0.263670
\(593\) −29.1791 −1.19824 −0.599121 0.800658i \(-0.704483\pi\)
−0.599121 + 0.800658i \(0.704483\pi\)
\(594\) −6.18039 −0.253585
\(595\) −1.13530 −0.0465426
\(596\) −7.36592 −0.301720
\(597\) 10.9796 0.449363
\(598\) 6.51348 0.266356
\(599\) 1.96570 0.0803162 0.0401581 0.999193i \(-0.487214\pi\)
0.0401581 + 0.999193i \(0.487214\pi\)
\(600\) 3.18210 0.129909
\(601\) 12.3062 0.501981 0.250991 0.967990i \(-0.419244\pi\)
0.250991 + 0.967990i \(0.419244\pi\)
\(602\) −6.71550 −0.273703
\(603\) 8.66119 0.352711
\(604\) 8.10419 0.329755
\(605\) 36.6699 1.49084
\(606\) −7.53357 −0.306030
\(607\) 14.1524 0.574426 0.287213 0.957867i \(-0.407271\pi\)
0.287213 + 0.957867i \(0.407271\pi\)
\(608\) 5.57232 0.225987
\(609\) 8.79865 0.356539
\(610\) 1.80243 0.0729784
\(611\) −6.23707 −0.252325
\(612\) 0.669960 0.0270815
\(613\) −20.7064 −0.836322 −0.418161 0.908373i \(-0.637325\pi\)
−0.418161 + 0.908373i \(0.637325\pi\)
\(614\) −28.0036 −1.13013
\(615\) −4.14011 −0.166945
\(616\) −7.76770 −0.312969
\(617\) −9.87894 −0.397711 −0.198856 0.980029i \(-0.563722\pi\)
−0.198856 + 0.980029i \(0.563722\pi\)
\(618\) 1.00000 0.0402259
\(619\) 39.0760 1.57060 0.785298 0.619118i \(-0.212510\pi\)
0.785298 + 0.619118i \(0.212510\pi\)
\(620\) 13.4134 0.538696
\(621\) 6.51348 0.261377
\(622\) 25.6298 1.02766
\(623\) −5.84470 −0.234163
\(624\) 1.00000 0.0400320
\(625\) 1.03629 0.0414517
\(626\) 0.830172 0.0331804
\(627\) −34.4391 −1.37537
\(628\) −7.16921 −0.286083
\(629\) −4.29804 −0.171374
\(630\) −1.69458 −0.0675135
\(631\) 42.4635 1.69044 0.845222 0.534415i \(-0.179468\pi\)
0.845222 + 0.534415i \(0.179468\pi\)
\(632\) −12.7247 −0.506162
\(633\) 20.4841 0.814169
\(634\) 15.6607 0.621966
\(635\) −4.76496 −0.189091
\(636\) 8.03980 0.318799
\(637\) 5.42038 0.214763
\(638\) 43.2669 1.71295
\(639\) 6.62891 0.262236
\(640\) 1.34829 0.0532960
\(641\) −41.5522 −1.64121 −0.820606 0.571494i \(-0.806364\pi\)
−0.820606 + 0.571494i \(0.806364\pi\)
\(642\) −10.7444 −0.424049
\(643\) −24.4953 −0.966000 −0.483000 0.875620i \(-0.660453\pi\)
−0.483000 + 0.875620i \(0.660453\pi\)
\(644\) 8.18633 0.322587
\(645\) −7.20422 −0.283666
\(646\) 3.73323 0.146882
\(647\) 23.2463 0.913907 0.456954 0.889491i \(-0.348941\pi\)
0.456954 + 0.889491i \(0.348941\pi\)
\(648\) 1.00000 0.0392837
\(649\) −19.3116 −0.758046
\(650\) 3.18210 0.124812
\(651\) 12.5035 0.490050
\(652\) −14.4487 −0.565855
\(653\) 5.51615 0.215864 0.107932 0.994158i \(-0.465577\pi\)
0.107932 + 0.994158i \(0.465577\pi\)
\(654\) −0.0532230 −0.00208118
\(655\) 2.57932 0.100782
\(656\) 3.07063 0.119888
\(657\) −2.20900 −0.0861813
\(658\) −7.83893 −0.305593
\(659\) 31.6062 1.23120 0.615600 0.788058i \(-0.288914\pi\)
0.615600 + 0.788058i \(0.288914\pi\)
\(660\) −8.33299 −0.324361
\(661\) −40.1139 −1.56025 −0.780124 0.625624i \(-0.784844\pi\)
−0.780124 + 0.625624i \(0.784844\pi\)
\(662\) 33.8850 1.31698
\(663\) 0.669960 0.0260191
\(664\) −0.863190 −0.0334983
\(665\) −9.44272 −0.366173
\(666\) −6.41537 −0.248591
\(667\) −45.5987 −1.76559
\(668\) −19.0982 −0.738932
\(669\) −3.21166 −0.124170
\(670\) 11.6778 0.451154
\(671\) 8.26211 0.318955
\(672\) 1.25683 0.0484832
\(673\) −21.7284 −0.837569 −0.418785 0.908086i \(-0.637544\pi\)
−0.418785 + 0.908086i \(0.637544\pi\)
\(674\) 4.89288 0.188467
\(675\) 3.18210 0.122479
\(676\) 1.00000 0.0384615
\(677\) −40.5339 −1.55785 −0.778923 0.627120i \(-0.784234\pi\)
−0.778923 + 0.627120i \(0.784234\pi\)
\(678\) −13.8205 −0.530772
\(679\) 14.5884 0.559853
\(680\) 0.903303 0.0346401
\(681\) 28.6424 1.09758
\(682\) 61.4852 2.35439
\(683\) 33.7817 1.29262 0.646311 0.763074i \(-0.276311\pi\)
0.646311 + 0.763074i \(0.276311\pi\)
\(684\) 5.57232 0.213063
\(685\) −7.04778 −0.269282
\(686\) 15.6103 0.596004
\(687\) −10.9174 −0.416526
\(688\) 5.34321 0.203708
\(689\) 8.03980 0.306292
\(690\) 8.78209 0.334328
\(691\) −14.7146 −0.559771 −0.279885 0.960033i \(-0.590296\pi\)
−0.279885 + 0.960033i \(0.590296\pi\)
\(692\) 22.9671 0.873079
\(693\) −7.76770 −0.295070
\(694\) −15.6931 −0.595702
\(695\) −12.7155 −0.482328
\(696\) −7.00067 −0.265360
\(697\) 2.05720 0.0779219
\(698\) −17.0990 −0.647208
\(699\) −22.8998 −0.866150
\(700\) 3.99936 0.151162
\(701\) −47.0866 −1.77844 −0.889218 0.457483i \(-0.848751\pi\)
−0.889218 + 0.457483i \(0.848751\pi\)
\(702\) 1.00000 0.0377426
\(703\) −35.7485 −1.34828
\(704\) 6.18039 0.232932
\(705\) −8.40941 −0.316717
\(706\) 21.2245 0.798795
\(707\) −9.46841 −0.356096
\(708\) 3.12465 0.117432
\(709\) −6.68250 −0.250966 −0.125483 0.992096i \(-0.540048\pi\)
−0.125483 + 0.992096i \(0.540048\pi\)
\(710\) 8.93772 0.335427
\(711\) −12.7247 −0.477215
\(712\) 4.65035 0.174279
\(713\) −64.7989 −2.42674
\(714\) 0.842025 0.0315120
\(715\) −8.33299 −0.311636
\(716\) 2.88656 0.107876
\(717\) −2.97784 −0.111210
\(718\) −14.9473 −0.557828
\(719\) 3.95188 0.147380 0.0736902 0.997281i \(-0.476522\pi\)
0.0736902 + 0.997281i \(0.476522\pi\)
\(720\) 1.34829 0.0502480
\(721\) 1.25683 0.0468068
\(722\) 12.0507 0.448482
\(723\) −13.4226 −0.499192
\(724\) 16.8058 0.624584
\(725\) −22.2769 −0.827342
\(726\) −27.1972 −1.00938
\(727\) 29.7334 1.10275 0.551375 0.834257i \(-0.314103\pi\)
0.551375 + 0.834257i \(0.314103\pi\)
\(728\) 1.25683 0.0465812
\(729\) 1.00000 0.0370370
\(730\) −2.97838 −0.110235
\(731\) 3.57973 0.132401
\(732\) −1.33683 −0.0494105
\(733\) −7.34755 −0.271388 −0.135694 0.990751i \(-0.543326\pi\)
−0.135694 + 0.990751i \(0.543326\pi\)
\(734\) −12.1126 −0.447084
\(735\) 7.30827 0.269569
\(736\) −6.51348 −0.240090
\(737\) 53.5296 1.97179
\(738\) 3.07063 0.113031
\(739\) −8.11466 −0.298503 −0.149251 0.988799i \(-0.547686\pi\)
−0.149251 + 0.988799i \(0.547686\pi\)
\(740\) −8.64981 −0.317973
\(741\) 5.57232 0.204704
\(742\) 10.1047 0.370954
\(743\) −26.4538 −0.970497 −0.485248 0.874376i \(-0.661271\pi\)
−0.485248 + 0.874376i \(0.661271\pi\)
\(744\) −9.94843 −0.364727
\(745\) −9.93142 −0.363859
\(746\) −6.77801 −0.248161
\(747\) −0.863190 −0.0315825
\(748\) 4.14061 0.151396
\(749\) −13.5039 −0.493423
\(750\) 11.0319 0.402827
\(751\) 4.89738 0.178708 0.0893539 0.996000i \(-0.471520\pi\)
0.0893539 + 0.996000i \(0.471520\pi\)
\(752\) 6.23707 0.227443
\(753\) 11.9131 0.434137
\(754\) −7.00067 −0.254949
\(755\) 10.9268 0.397668
\(756\) 1.25683 0.0457104
\(757\) −31.6179 −1.14917 −0.574586 0.818444i \(-0.694837\pi\)
−0.574586 + 0.818444i \(0.694837\pi\)
\(758\) −4.76965 −0.173242
\(759\) 40.2559 1.46119
\(760\) 7.51313 0.272530
\(761\) 35.5981 1.29043 0.645214 0.764002i \(-0.276768\pi\)
0.645214 + 0.764002i \(0.276768\pi\)
\(762\) 3.53406 0.128026
\(763\) −0.0668922 −0.00242166
\(764\) −4.98127 −0.180216
\(765\) 0.903303 0.0326590
\(766\) −12.0847 −0.436638
\(767\) 3.12465 0.112825
\(768\) −1.00000 −0.0360844
\(769\) 39.1829 1.41297 0.706486 0.707727i \(-0.250279\pi\)
0.706486 + 0.707727i \(0.250279\pi\)
\(770\) −10.4731 −0.377426
\(771\) −22.7340 −0.818745
\(772\) 7.47451 0.269014
\(773\) −22.3613 −0.804280 −0.402140 0.915578i \(-0.631733\pi\)
−0.402140 + 0.915578i \(0.631733\pi\)
\(774\) 5.34321 0.192058
\(775\) −31.6569 −1.13715
\(776\) −11.6073 −0.416679
\(777\) −8.06303 −0.289260
\(778\) 26.8990 0.964374
\(779\) 17.1105 0.613048
\(780\) 1.34829 0.0482767
\(781\) 40.9693 1.46600
\(782\) −4.36377 −0.156048
\(783\) −7.00067 −0.250184
\(784\) −5.42038 −0.193585
\(785\) −9.66620 −0.345002
\(786\) −1.91302 −0.0682353
\(787\) −54.0364 −1.92619 −0.963094 0.269164i \(-0.913253\pi\)
−0.963094 + 0.269164i \(0.913253\pi\)
\(788\) 16.2736 0.579725
\(789\) −15.9585 −0.568139
\(790\) −17.1567 −0.610407
\(791\) −17.3700 −0.617605
\(792\) 6.18039 0.219611
\(793\) −1.33683 −0.0474721
\(794\) 9.11191 0.323370
\(795\) 10.8400 0.384456
\(796\) −10.9796 −0.389160
\(797\) −5.35792 −0.189787 −0.0948937 0.995487i \(-0.530251\pi\)
−0.0948937 + 0.995487i \(0.530251\pi\)
\(798\) 7.00345 0.247920
\(799\) 4.17858 0.147828
\(800\) −3.18210 −0.112504
\(801\) 4.65035 0.164312
\(802\) 5.41938 0.191365
\(803\) −13.6525 −0.481786
\(804\) −8.66119 −0.305457
\(805\) 11.0376 0.389024
\(806\) −9.94843 −0.350419
\(807\) −16.0982 −0.566684
\(808\) 7.53357 0.265030
\(809\) 47.7356 1.67829 0.839147 0.543904i \(-0.183055\pi\)
0.839147 + 0.543904i \(0.183055\pi\)
\(810\) 1.34829 0.0473742
\(811\) −14.6324 −0.513815 −0.256907 0.966436i \(-0.582704\pi\)
−0.256907 + 0.966436i \(0.582704\pi\)
\(812\) −8.79865 −0.308772
\(813\) −2.40939 −0.0845012
\(814\) −39.6495 −1.38972
\(815\) −19.4811 −0.682394
\(816\) −0.669960 −0.0234533
\(817\) 29.7741 1.04166
\(818\) 23.0164 0.804749
\(819\) 1.25683 0.0439172
\(820\) 4.14011 0.144579
\(821\) −14.8031 −0.516631 −0.258316 0.966061i \(-0.583167\pi\)
−0.258316 + 0.966061i \(0.583167\pi\)
\(822\) 5.22718 0.182319
\(823\) −30.8013 −1.07367 −0.536833 0.843688i \(-0.680380\pi\)
−0.536833 + 0.843688i \(0.680380\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 19.6666 0.684705
\(826\) 3.92716 0.136643
\(827\) −52.7224 −1.83334 −0.916668 0.399649i \(-0.869132\pi\)
−0.916668 + 0.399649i \(0.869132\pi\)
\(828\) −6.51348 −0.226359
\(829\) 9.03526 0.313807 0.156904 0.987614i \(-0.449849\pi\)
0.156904 + 0.987614i \(0.449849\pi\)
\(830\) −1.16383 −0.0403972
\(831\) −1.30583 −0.0452988
\(832\) −1.00000 −0.0346688
\(833\) −3.63144 −0.125822
\(834\) 9.43083 0.326563
\(835\) −25.7500 −0.891116
\(836\) 34.4391 1.19110
\(837\) −9.94843 −0.343868
\(838\) 12.9522 0.447427
\(839\) −42.8789 −1.48034 −0.740172 0.672418i \(-0.765256\pi\)
−0.740172 + 0.672418i \(0.765256\pi\)
\(840\) 1.69458 0.0584684
\(841\) 20.0094 0.689979
\(842\) −30.0337 −1.03503
\(843\) −12.3979 −0.427006
\(844\) −20.4841 −0.705091
\(845\) 1.34829 0.0463827
\(846\) 6.23707 0.214435
\(847\) −34.1823 −1.17452
\(848\) −8.03980 −0.276088
\(849\) −23.0980 −0.792720
\(850\) −2.13188 −0.0731229
\(851\) 41.7864 1.43242
\(852\) −6.62891 −0.227103
\(853\) −40.6980 −1.39347 −0.696737 0.717327i \(-0.745365\pi\)
−0.696737 + 0.717327i \(0.745365\pi\)
\(854\) −1.68016 −0.0574940
\(855\) 7.51313 0.256943
\(856\) 10.7444 0.367237
\(857\) −8.67792 −0.296432 −0.148216 0.988955i \(-0.547353\pi\)
−0.148216 + 0.988955i \(0.547353\pi\)
\(858\) 6.18039 0.210995
\(859\) 16.7189 0.570441 0.285221 0.958462i \(-0.407933\pi\)
0.285221 + 0.958462i \(0.407933\pi\)
\(860\) 7.20422 0.245662
\(861\) 3.85926 0.131523
\(862\) −27.0278 −0.920571
\(863\) −41.7091 −1.41979 −0.709896 0.704306i \(-0.751258\pi\)
−0.709896 + 0.704306i \(0.751258\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 30.9664 1.05289
\(866\) 38.4811 1.30764
\(867\) 16.5512 0.562107
\(868\) −12.5035 −0.424396
\(869\) −78.6438 −2.66781
\(870\) −9.43896 −0.320011
\(871\) −8.66119 −0.293473
\(872\) 0.0532230 0.00180236
\(873\) −11.6073 −0.392849
\(874\) −36.2952 −1.22770
\(875\) 13.8652 0.468729
\(876\) 2.20900 0.0746352
\(877\) 31.6801 1.06976 0.534880 0.844928i \(-0.320357\pi\)
0.534880 + 0.844928i \(0.320357\pi\)
\(878\) −28.9181 −0.975937
\(879\) −4.86440 −0.164072
\(880\) 8.33299 0.280905
\(881\) 39.7532 1.33932 0.669660 0.742668i \(-0.266440\pi\)
0.669660 + 0.742668i \(0.266440\pi\)
\(882\) −5.42038 −0.182514
\(883\) 18.7507 0.631011 0.315506 0.948924i \(-0.397826\pi\)
0.315506 + 0.948924i \(0.397826\pi\)
\(884\) −0.669960 −0.0225332
\(885\) 4.21295 0.141617
\(886\) 5.38288 0.180841
\(887\) −39.1134 −1.31330 −0.656649 0.754196i \(-0.728027\pi\)
−0.656649 + 0.754196i \(0.728027\pi\)
\(888\) 6.41537 0.215286
\(889\) 4.44171 0.148970
\(890\) 6.27004 0.210172
\(891\) 6.18039 0.207051
\(892\) 3.21166 0.107534
\(893\) 34.7549 1.16303
\(894\) 7.36592 0.246353
\(895\) 3.89194 0.130093
\(896\) −1.25683 −0.0419877
\(897\) −6.51348 −0.217479
\(898\) −14.8713 −0.496262
\(899\) 69.6457 2.32281
\(900\) −3.18210 −0.106070
\(901\) −5.38634 −0.179445
\(902\) 18.9777 0.631888
\(903\) 6.71550 0.223478
\(904\) 13.8205 0.459662
\(905\) 22.6592 0.753217
\(906\) −8.10419 −0.269244
\(907\) −0.204614 −0.00679409 −0.00339704 0.999994i \(-0.501081\pi\)
−0.00339704 + 0.999994i \(0.501081\pi\)
\(908\) −28.6424 −0.950530
\(909\) 7.53357 0.249873
\(910\) 1.69458 0.0561746
\(911\) −34.0664 −1.12867 −0.564334 0.825546i \(-0.690867\pi\)
−0.564334 + 0.825546i \(0.690867\pi\)
\(912\) −5.57232 −0.184518
\(913\) −5.33485 −0.176558
\(914\) 4.72695 0.156354
\(915\) −1.80243 −0.0595866
\(916\) 10.9174 0.360722
\(917\) −2.40434 −0.0793984
\(918\) −0.669960 −0.0221120
\(919\) −18.2922 −0.603405 −0.301703 0.953402i \(-0.597555\pi\)
−0.301703 + 0.953402i \(0.597555\pi\)
\(920\) −8.78209 −0.289537
\(921\) 28.0036 0.922749
\(922\) −9.95374 −0.327809
\(923\) −6.62891 −0.218193
\(924\) 7.76770 0.255539
\(925\) 20.4144 0.671221
\(926\) 29.5029 0.969526
\(927\) −1.00000 −0.0328443
\(928\) 7.00067 0.229808
\(929\) 1.22656 0.0402420 0.0201210 0.999798i \(-0.493595\pi\)
0.0201210 + 0.999798i \(0.493595\pi\)
\(930\) −13.4134 −0.439843
\(931\) −30.2041 −0.989899
\(932\) 22.8998 0.750108
\(933\) −25.6298 −0.839082
\(934\) −34.5248 −1.12969
\(935\) 5.58276 0.182576
\(936\) −1.00000 −0.0326860
\(937\) 2.86560 0.0936152 0.0468076 0.998904i \(-0.485095\pi\)
0.0468076 + 0.998904i \(0.485095\pi\)
\(938\) −10.8856 −0.355429
\(939\) −0.830172 −0.0270916
\(940\) 8.40941 0.274285
\(941\) 31.6790 1.03271 0.516353 0.856376i \(-0.327289\pi\)
0.516353 + 0.856376i \(0.327289\pi\)
\(942\) 7.16921 0.233585
\(943\) −20.0005 −0.651305
\(944\) −3.12465 −0.101699
\(945\) 1.69458 0.0551246
\(946\) 33.0231 1.07367
\(947\) 36.8970 1.19899 0.599495 0.800378i \(-0.295368\pi\)
0.599495 + 0.800378i \(0.295368\pi\)
\(948\) 12.7247 0.413280
\(949\) 2.20900 0.0717072
\(950\) −17.7317 −0.575292
\(951\) −15.6607 −0.507833
\(952\) −0.842025 −0.0272902
\(953\) −11.3224 −0.366768 −0.183384 0.983041i \(-0.558705\pi\)
−0.183384 + 0.983041i \(0.558705\pi\)
\(954\) −8.03980 −0.260298
\(955\) −6.71621 −0.217331
\(956\) 2.97784 0.0963103
\(957\) −43.2669 −1.39862
\(958\) −14.2751 −0.461209
\(959\) 6.56967 0.212146
\(960\) −1.34829 −0.0435160
\(961\) 67.9713 2.19262
\(962\) 6.41537 0.206840
\(963\) 10.7444 0.346235
\(964\) 13.4226 0.432313
\(965\) 10.0778 0.324417
\(966\) −8.18633 −0.263391
\(967\) 14.0860 0.452975 0.226487 0.974014i \(-0.427276\pi\)
0.226487 + 0.974014i \(0.427276\pi\)
\(968\) 27.1972 0.874153
\(969\) −3.73323 −0.119929
\(970\) −15.6501 −0.502495
\(971\) −9.68061 −0.310666 −0.155333 0.987862i \(-0.549645\pi\)
−0.155333 + 0.987862i \(0.549645\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 11.8529 0.379988
\(974\) 11.3377 0.363284
\(975\) −3.18210 −0.101909
\(976\) 1.33683 0.0427908
\(977\) −14.3093 −0.457794 −0.228897 0.973451i \(-0.573512\pi\)
−0.228897 + 0.973451i \(0.573512\pi\)
\(978\) 14.4487 0.462019
\(979\) 28.7410 0.918567
\(980\) −7.30827 −0.233454
\(981\) 0.0532230 0.00169928
\(982\) −26.6950 −0.851870
\(983\) 27.6668 0.882433 0.441216 0.897401i \(-0.354547\pi\)
0.441216 + 0.897401i \(0.354547\pi\)
\(984\) −3.07063 −0.0978881
\(985\) 21.9417 0.699120
\(986\) 4.69017 0.149365
\(987\) 7.83893 0.249516
\(988\) −5.57232 −0.177279
\(989\) −34.8029 −1.10667
\(990\) 8.33299 0.264840
\(991\) −7.77320 −0.246924 −0.123462 0.992349i \(-0.539400\pi\)
−0.123462 + 0.992349i \(0.539400\pi\)
\(992\) 9.94843 0.315863
\(993\) −33.8850 −1.07531
\(994\) −8.33141 −0.264256
\(995\) −14.8037 −0.469308
\(996\) 0.863190 0.0273512
\(997\) 3.58907 0.113667 0.0568336 0.998384i \(-0.481900\pi\)
0.0568336 + 0.998384i \(0.481900\pi\)
\(998\) 7.46541 0.236313
\(999\) 6.41537 0.202973
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.bc.1.10 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.10 15 1.1 even 1 trivial