Properties

Label 8034.2.a.bc.1.12
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 1.12
Root \(-1.97610\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.97610 q^{5} -1.00000 q^{6} +4.65524 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.97610 q^{5} -1.00000 q^{6} +4.65524 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.97610 q^{10} +2.35768 q^{11} -1.00000 q^{12} -1.00000 q^{13} +4.65524 q^{14} -1.97610 q^{15} +1.00000 q^{16} -1.40007 q^{17} +1.00000 q^{18} -4.22562 q^{19} +1.97610 q^{20} -4.65524 q^{21} +2.35768 q^{22} -1.68831 q^{23} -1.00000 q^{24} -1.09504 q^{25} -1.00000 q^{26} -1.00000 q^{27} +4.65524 q^{28} +0.920725 q^{29} -1.97610 q^{30} -4.23306 q^{31} +1.00000 q^{32} -2.35768 q^{33} -1.40007 q^{34} +9.19921 q^{35} +1.00000 q^{36} +1.22354 q^{37} -4.22562 q^{38} +1.00000 q^{39} +1.97610 q^{40} -8.28255 q^{41} -4.65524 q^{42} +9.40003 q^{43} +2.35768 q^{44} +1.97610 q^{45} -1.68831 q^{46} +13.2833 q^{47} -1.00000 q^{48} +14.6713 q^{49} -1.09504 q^{50} +1.40007 q^{51} -1.00000 q^{52} +11.7500 q^{53} -1.00000 q^{54} +4.65900 q^{55} +4.65524 q^{56} +4.22562 q^{57} +0.920725 q^{58} +4.69835 q^{59} -1.97610 q^{60} -1.74167 q^{61} -4.23306 q^{62} +4.65524 q^{63} +1.00000 q^{64} -1.97610 q^{65} -2.35768 q^{66} +10.0174 q^{67} -1.40007 q^{68} +1.68831 q^{69} +9.19921 q^{70} +6.96979 q^{71} +1.00000 q^{72} +11.5943 q^{73} +1.22354 q^{74} +1.09504 q^{75} -4.22562 q^{76} +10.9756 q^{77} +1.00000 q^{78} -4.59328 q^{79} +1.97610 q^{80} +1.00000 q^{81} -8.28255 q^{82} -2.82344 q^{83} -4.65524 q^{84} -2.76667 q^{85} +9.40003 q^{86} -0.920725 q^{87} +2.35768 q^{88} +3.99760 q^{89} +1.97610 q^{90} -4.65524 q^{91} -1.68831 q^{92} +4.23306 q^{93} +13.2833 q^{94} -8.35023 q^{95} -1.00000 q^{96} -11.8358 q^{97} +14.6713 q^{98} +2.35768 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

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
\(5\) 1.97610 0.883738 0.441869 0.897080i \(-0.354316\pi\)
0.441869 + 0.897080i \(0.354316\pi\)
\(6\) −1.00000 −0.408248
\(7\) 4.65524 1.75952 0.879758 0.475422i \(-0.157705\pi\)
0.879758 + 0.475422i \(0.157705\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.97610 0.624897
\(11\) 2.35768 0.710867 0.355433 0.934702i \(-0.384333\pi\)
0.355433 + 0.934702i \(0.384333\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 4.65524 1.24417
\(15\) −1.97610 −0.510226
\(16\) 1.00000 0.250000
\(17\) −1.40007 −0.339566 −0.169783 0.985481i \(-0.554307\pi\)
−0.169783 + 0.985481i \(0.554307\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.22562 −0.969423 −0.484712 0.874674i \(-0.661075\pi\)
−0.484712 + 0.874674i \(0.661075\pi\)
\(20\) 1.97610 0.441869
\(21\) −4.65524 −1.01586
\(22\) 2.35768 0.502659
\(23\) −1.68831 −0.352036 −0.176018 0.984387i \(-0.556322\pi\)
−0.176018 + 0.984387i \(0.556322\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.09504 −0.219008
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 4.65524 0.879758
\(29\) 0.920725 0.170974 0.0854872 0.996339i \(-0.472755\pi\)
0.0854872 + 0.996339i \(0.472755\pi\)
\(30\) −1.97610 −0.360784
\(31\) −4.23306 −0.760280 −0.380140 0.924929i \(-0.624124\pi\)
−0.380140 + 0.924929i \(0.624124\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.35768 −0.410419
\(34\) −1.40007 −0.240110
\(35\) 9.19921 1.55495
\(36\) 1.00000 0.166667
\(37\) 1.22354 0.201149 0.100575 0.994930i \(-0.467932\pi\)
0.100575 + 0.994930i \(0.467932\pi\)
\(38\) −4.22562 −0.685486
\(39\) 1.00000 0.160128
\(40\) 1.97610 0.312448
\(41\) −8.28255 −1.29352 −0.646759 0.762695i \(-0.723876\pi\)
−0.646759 + 0.762695i \(0.723876\pi\)
\(42\) −4.65524 −0.718319
\(43\) 9.40003 1.43349 0.716746 0.697335i \(-0.245631\pi\)
0.716746 + 0.697335i \(0.245631\pi\)
\(44\) 2.35768 0.355433
\(45\) 1.97610 0.294579
\(46\) −1.68831 −0.248927
\(47\) 13.2833 1.93757 0.968783 0.247909i \(-0.0797434\pi\)
0.968783 + 0.247909i \(0.0797434\pi\)
\(48\) −1.00000 −0.144338
\(49\) 14.6713 2.09590
\(50\) −1.09504 −0.154862
\(51\) 1.40007 0.196049
\(52\) −1.00000 −0.138675
\(53\) 11.7500 1.61399 0.806996 0.590556i \(-0.201092\pi\)
0.806996 + 0.590556i \(0.201092\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.65900 0.628220
\(56\) 4.65524 0.622083
\(57\) 4.22562 0.559697
\(58\) 0.920725 0.120897
\(59\) 4.69835 0.611673 0.305837 0.952084i \(-0.401064\pi\)
0.305837 + 0.952084i \(0.401064\pi\)
\(60\) −1.97610 −0.255113
\(61\) −1.74167 −0.222998 −0.111499 0.993765i \(-0.535565\pi\)
−0.111499 + 0.993765i \(0.535565\pi\)
\(62\) −4.23306 −0.537599
\(63\) 4.65524 0.586505
\(64\) 1.00000 0.125000
\(65\) −1.97610 −0.245105
\(66\) −2.35768 −0.290210
\(67\) 10.0174 1.22383 0.611913 0.790925i \(-0.290400\pi\)
0.611913 + 0.790925i \(0.290400\pi\)
\(68\) −1.40007 −0.169783
\(69\) 1.68831 0.203248
\(70\) 9.19921 1.09952
\(71\) 6.96979 0.827162 0.413581 0.910467i \(-0.364278\pi\)
0.413581 + 0.910467i \(0.364278\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.5943 1.35701 0.678505 0.734595i \(-0.262628\pi\)
0.678505 + 0.734595i \(0.262628\pi\)
\(74\) 1.22354 0.142234
\(75\) 1.09504 0.126444
\(76\) −4.22562 −0.484712
\(77\) 10.9756 1.25078
\(78\) 1.00000 0.113228
\(79\) −4.59328 −0.516785 −0.258392 0.966040i \(-0.583193\pi\)
−0.258392 + 0.966040i \(0.583193\pi\)
\(80\) 1.97610 0.220934
\(81\) 1.00000 0.111111
\(82\) −8.28255 −0.914655
\(83\) −2.82344 −0.309913 −0.154956 0.987921i \(-0.549524\pi\)
−0.154956 + 0.987921i \(0.549524\pi\)
\(84\) −4.65524 −0.507928
\(85\) −2.76667 −0.300087
\(86\) 9.40003 1.01363
\(87\) −0.920725 −0.0987121
\(88\) 2.35768 0.251329
\(89\) 3.99760 0.423745 0.211873 0.977297i \(-0.432044\pi\)
0.211873 + 0.977297i \(0.432044\pi\)
\(90\) 1.97610 0.208299
\(91\) −4.65524 −0.488002
\(92\) −1.68831 −0.176018
\(93\) 4.23306 0.438948
\(94\) 13.2833 1.37007
\(95\) −8.35023 −0.856716
\(96\) −1.00000 −0.102062
\(97\) −11.8358 −1.20174 −0.600872 0.799345i \(-0.705180\pi\)
−0.600872 + 0.799345i \(0.705180\pi\)
\(98\) 14.6713 1.48202
\(99\) 2.35768 0.236956
\(100\) −1.09504 −0.109504
\(101\) −7.65739 −0.761939 −0.380969 0.924588i \(-0.624410\pi\)
−0.380969 + 0.924588i \(0.624410\pi\)
\(102\) 1.40007 0.138627
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −9.19921 −0.897751
\(106\) 11.7500 1.14127
\(107\) −2.19049 −0.211763 −0.105882 0.994379i \(-0.533766\pi\)
−0.105882 + 0.994379i \(0.533766\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 15.4443 1.47930 0.739649 0.672993i \(-0.234991\pi\)
0.739649 + 0.672993i \(0.234991\pi\)
\(110\) 4.65900 0.444218
\(111\) −1.22354 −0.116134
\(112\) 4.65524 0.439879
\(113\) 16.2250 1.52632 0.763159 0.646211i \(-0.223647\pi\)
0.763159 + 0.646211i \(0.223647\pi\)
\(114\) 4.22562 0.395765
\(115\) −3.33626 −0.311108
\(116\) 0.920725 0.0854872
\(117\) −1.00000 −0.0924500
\(118\) 4.69835 0.432518
\(119\) −6.51765 −0.597472
\(120\) −1.97610 −0.180392
\(121\) −5.44135 −0.494668
\(122\) −1.74167 −0.157684
\(123\) 8.28255 0.746813
\(124\) −4.23306 −0.380140
\(125\) −12.0444 −1.07728
\(126\) 4.65524 0.414722
\(127\) 12.1773 1.08056 0.540279 0.841486i \(-0.318319\pi\)
0.540279 + 0.841486i \(0.318319\pi\)
\(128\) 1.00000 0.0883883
\(129\) −9.40003 −0.827627
\(130\) −1.97610 −0.173315
\(131\) −11.5772 −1.01151 −0.505754 0.862678i \(-0.668786\pi\)
−0.505754 + 0.862678i \(0.668786\pi\)
\(132\) −2.35768 −0.205210
\(133\) −19.6713 −1.70572
\(134\) 10.0174 0.865375
\(135\) −1.97610 −0.170075
\(136\) −1.40007 −0.120055
\(137\) 5.67371 0.484738 0.242369 0.970184i \(-0.422076\pi\)
0.242369 + 0.970184i \(0.422076\pi\)
\(138\) 1.68831 0.143718
\(139\) −19.7870 −1.67831 −0.839154 0.543894i \(-0.816949\pi\)
−0.839154 + 0.543894i \(0.816949\pi\)
\(140\) 9.19921 0.777475
\(141\) −13.2833 −1.11865
\(142\) 6.96979 0.584892
\(143\) −2.35768 −0.197159
\(144\) 1.00000 0.0833333
\(145\) 1.81944 0.151096
\(146\) 11.5943 0.959552
\(147\) −14.6713 −1.21007
\(148\) 1.22354 0.100575
\(149\) −19.8323 −1.62473 −0.812364 0.583151i \(-0.801820\pi\)
−0.812364 + 0.583151i \(0.801820\pi\)
\(150\) 1.09504 0.0894096
\(151\) −16.6789 −1.35731 −0.678656 0.734456i \(-0.737437\pi\)
−0.678656 + 0.734456i \(0.737437\pi\)
\(152\) −4.22562 −0.342743
\(153\) −1.40007 −0.113189
\(154\) 10.9756 0.884436
\(155\) −8.36493 −0.671888
\(156\) 1.00000 0.0800641
\(157\) 7.72650 0.616641 0.308321 0.951282i \(-0.400233\pi\)
0.308321 + 0.951282i \(0.400233\pi\)
\(158\) −4.59328 −0.365422
\(159\) −11.7500 −0.931839
\(160\) 1.97610 0.156224
\(161\) −7.85948 −0.619414
\(162\) 1.00000 0.0785674
\(163\) 19.9507 1.56266 0.781330 0.624118i \(-0.214542\pi\)
0.781330 + 0.624118i \(0.214542\pi\)
\(164\) −8.28255 −0.646759
\(165\) −4.65900 −0.362703
\(166\) −2.82344 −0.219141
\(167\) 15.6969 1.21466 0.607330 0.794449i \(-0.292240\pi\)
0.607330 + 0.794449i \(0.292240\pi\)
\(168\) −4.65524 −0.359160
\(169\) 1.00000 0.0769231
\(170\) −2.76667 −0.212194
\(171\) −4.22562 −0.323141
\(172\) 9.40003 0.716746
\(173\) −8.94426 −0.680019 −0.340010 0.940422i \(-0.610430\pi\)
−0.340010 + 0.940422i \(0.610430\pi\)
\(174\) −0.920725 −0.0698000
\(175\) −5.09767 −0.385348
\(176\) 2.35768 0.177717
\(177\) −4.69835 −0.353150
\(178\) 3.99760 0.299633
\(179\) 22.5180 1.68307 0.841535 0.540202i \(-0.181652\pi\)
0.841535 + 0.540202i \(0.181652\pi\)
\(180\) 1.97610 0.147290
\(181\) −4.81147 −0.357634 −0.178817 0.983882i \(-0.557227\pi\)
−0.178817 + 0.983882i \(0.557227\pi\)
\(182\) −4.65524 −0.345069
\(183\) 1.74167 0.128748
\(184\) −1.68831 −0.124464
\(185\) 2.41784 0.177763
\(186\) 4.23306 0.310383
\(187\) −3.30091 −0.241386
\(188\) 13.2833 0.968783
\(189\) −4.65524 −0.338619
\(190\) −8.35023 −0.605790
\(191\) 7.01017 0.507238 0.253619 0.967304i \(-0.418379\pi\)
0.253619 + 0.967304i \(0.418379\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 12.1251 0.872781 0.436391 0.899757i \(-0.356257\pi\)
0.436391 + 0.899757i \(0.356257\pi\)
\(194\) −11.8358 −0.849761
\(195\) 1.97610 0.141511
\(196\) 14.6713 1.04795
\(197\) −17.9383 −1.27805 −0.639026 0.769185i \(-0.720662\pi\)
−0.639026 + 0.769185i \(0.720662\pi\)
\(198\) 2.35768 0.167553
\(199\) −11.7186 −0.830710 −0.415355 0.909659i \(-0.636343\pi\)
−0.415355 + 0.909659i \(0.636343\pi\)
\(200\) −1.09504 −0.0774310
\(201\) −10.0174 −0.706576
\(202\) −7.65739 −0.538772
\(203\) 4.28620 0.300832
\(204\) 1.40007 0.0980243
\(205\) −16.3671 −1.14313
\(206\) −1.00000 −0.0696733
\(207\) −1.68831 −0.117345
\(208\) −1.00000 −0.0693375
\(209\) −9.96265 −0.689131
\(210\) −9.19921 −0.634806
\(211\) 15.2946 1.05293 0.526463 0.850198i \(-0.323518\pi\)
0.526463 + 0.850198i \(0.323518\pi\)
\(212\) 11.7500 0.806996
\(213\) −6.96979 −0.477562
\(214\) −2.19049 −0.149739
\(215\) 18.5754 1.26683
\(216\) −1.00000 −0.0680414
\(217\) −19.7059 −1.33772
\(218\) 15.4443 1.04602
\(219\) −11.5943 −0.783471
\(220\) 4.65900 0.314110
\(221\) 1.40007 0.0941787
\(222\) −1.22354 −0.0821189
\(223\) 14.1192 0.945489 0.472744 0.881200i \(-0.343263\pi\)
0.472744 + 0.881200i \(0.343263\pi\)
\(224\) 4.65524 0.311041
\(225\) −1.09504 −0.0730026
\(226\) 16.2250 1.07927
\(227\) −1.43072 −0.0949605 −0.0474803 0.998872i \(-0.515119\pi\)
−0.0474803 + 0.998872i \(0.515119\pi\)
\(228\) 4.22562 0.279848
\(229\) 25.0949 1.65832 0.829158 0.559015i \(-0.188820\pi\)
0.829158 + 0.559015i \(0.188820\pi\)
\(230\) −3.33626 −0.219986
\(231\) −10.9756 −0.722139
\(232\) 0.920725 0.0604486
\(233\) −0.642463 −0.0420892 −0.0210446 0.999779i \(-0.506699\pi\)
−0.0210446 + 0.999779i \(0.506699\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 26.2491 1.71230
\(236\) 4.69835 0.305837
\(237\) 4.59328 0.298366
\(238\) −6.51765 −0.422477
\(239\) −8.20641 −0.530829 −0.265414 0.964134i \(-0.585509\pi\)
−0.265414 + 0.964134i \(0.585509\pi\)
\(240\) −1.97610 −0.127557
\(241\) 6.48254 0.417577 0.208789 0.977961i \(-0.433048\pi\)
0.208789 + 0.977961i \(0.433048\pi\)
\(242\) −5.44135 −0.349783
\(243\) −1.00000 −0.0641500
\(244\) −1.74167 −0.111499
\(245\) 28.9919 1.85222
\(246\) 8.28255 0.528076
\(247\) 4.22562 0.268870
\(248\) −4.23306 −0.268799
\(249\) 2.82344 0.178928
\(250\) −12.0444 −0.761754
\(251\) −29.4254 −1.85732 −0.928658 0.370936i \(-0.879037\pi\)
−0.928658 + 0.370936i \(0.879037\pi\)
\(252\) 4.65524 0.293253
\(253\) −3.98049 −0.250251
\(254\) 12.1773 0.764070
\(255\) 2.76667 0.173256
\(256\) 1.00000 0.0625000
\(257\) 2.86093 0.178460 0.0892300 0.996011i \(-0.471559\pi\)
0.0892300 + 0.996011i \(0.471559\pi\)
\(258\) −9.40003 −0.585221
\(259\) 5.69589 0.353925
\(260\) −1.97610 −0.122552
\(261\) 0.920725 0.0569915
\(262\) −11.5772 −0.715244
\(263\) −3.22757 −0.199021 −0.0995103 0.995037i \(-0.531728\pi\)
−0.0995103 + 0.995037i \(0.531728\pi\)
\(264\) −2.35768 −0.145105
\(265\) 23.2192 1.42635
\(266\) −19.6713 −1.20612
\(267\) −3.99760 −0.244649
\(268\) 10.0174 0.611913
\(269\) −2.39878 −0.146256 −0.0731281 0.997323i \(-0.523298\pi\)
−0.0731281 + 0.997323i \(0.523298\pi\)
\(270\) −1.97610 −0.120261
\(271\) 0.749348 0.0455197 0.0227598 0.999741i \(-0.492755\pi\)
0.0227598 + 0.999741i \(0.492755\pi\)
\(272\) −1.40007 −0.0848916
\(273\) 4.65524 0.281748
\(274\) 5.67371 0.342761
\(275\) −2.58175 −0.155685
\(276\) 1.68831 0.101624
\(277\) 21.0829 1.26675 0.633374 0.773846i \(-0.281669\pi\)
0.633374 + 0.773846i \(0.281669\pi\)
\(278\) −19.7870 −1.18674
\(279\) −4.23306 −0.253427
\(280\) 9.19921 0.549758
\(281\) −22.8003 −1.36015 −0.680077 0.733141i \(-0.738054\pi\)
−0.680077 + 0.733141i \(0.738054\pi\)
\(282\) −13.2833 −0.791008
\(283\) −31.8124 −1.89105 −0.945525 0.325550i \(-0.894451\pi\)
−0.945525 + 0.325550i \(0.894451\pi\)
\(284\) 6.96979 0.413581
\(285\) 8.35023 0.494625
\(286\) −2.35768 −0.139412
\(287\) −38.5573 −2.27596
\(288\) 1.00000 0.0589256
\(289\) −15.0398 −0.884695
\(290\) 1.81944 0.106841
\(291\) 11.8358 0.693827
\(292\) 11.5943 0.678505
\(293\) 23.9481 1.39906 0.699530 0.714603i \(-0.253393\pi\)
0.699530 + 0.714603i \(0.253393\pi\)
\(294\) −14.6713 −0.855646
\(295\) 9.28440 0.540559
\(296\) 1.22354 0.0711170
\(297\) −2.35768 −0.136806
\(298\) −19.8323 −1.14886
\(299\) 1.68831 0.0976373
\(300\) 1.09504 0.0632221
\(301\) 43.7594 2.52225
\(302\) −16.6789 −0.959765
\(303\) 7.65739 0.439906
\(304\) −4.22562 −0.242356
\(305\) −3.44172 −0.197072
\(306\) −1.40007 −0.0800365
\(307\) −19.9043 −1.13600 −0.567999 0.823029i \(-0.692282\pi\)
−0.567999 + 0.823029i \(0.692282\pi\)
\(308\) 10.9756 0.625391
\(309\) 1.00000 0.0568880
\(310\) −8.36493 −0.475096
\(311\) −3.07394 −0.174307 −0.0871536 0.996195i \(-0.527777\pi\)
−0.0871536 + 0.996195i \(0.527777\pi\)
\(312\) 1.00000 0.0566139
\(313\) 17.0139 0.961681 0.480841 0.876808i \(-0.340332\pi\)
0.480841 + 0.876808i \(0.340332\pi\)
\(314\) 7.72650 0.436031
\(315\) 9.19921 0.518317
\(316\) −4.59328 −0.258392
\(317\) −15.5410 −0.872867 −0.436434 0.899736i \(-0.643759\pi\)
−0.436434 + 0.899736i \(0.643759\pi\)
\(318\) −11.7500 −0.658910
\(319\) 2.17077 0.121540
\(320\) 1.97610 0.110467
\(321\) 2.19049 0.122261
\(322\) −7.85948 −0.437991
\(323\) 5.91615 0.329183
\(324\) 1.00000 0.0555556
\(325\) 1.09504 0.0607419
\(326\) 19.9507 1.10497
\(327\) −15.4443 −0.854073
\(328\) −8.28255 −0.457327
\(329\) 61.8369 3.40918
\(330\) −4.65900 −0.256470
\(331\) −15.1150 −0.830794 −0.415397 0.909640i \(-0.636357\pi\)
−0.415397 + 0.909640i \(0.636357\pi\)
\(332\) −2.82344 −0.154956
\(333\) 1.22354 0.0670498
\(334\) 15.6969 0.858895
\(335\) 19.7954 1.08154
\(336\) −4.65524 −0.253964
\(337\) −3.85788 −0.210152 −0.105076 0.994464i \(-0.533509\pi\)
−0.105076 + 0.994464i \(0.533509\pi\)
\(338\) 1.00000 0.0543928
\(339\) −16.2250 −0.881220
\(340\) −2.76667 −0.150044
\(341\) −9.98019 −0.540458
\(342\) −4.22562 −0.228495
\(343\) 35.7116 1.92825
\(344\) 9.40003 0.506816
\(345\) 3.33626 0.179618
\(346\) −8.94426 −0.480846
\(347\) −0.410196 −0.0220205 −0.0110102 0.999939i \(-0.503505\pi\)
−0.0110102 + 0.999939i \(0.503505\pi\)
\(348\) −0.920725 −0.0493560
\(349\) 24.9299 1.33447 0.667233 0.744849i \(-0.267478\pi\)
0.667233 + 0.744849i \(0.267478\pi\)
\(350\) −5.09767 −0.272482
\(351\) 1.00000 0.0533761
\(352\) 2.35768 0.125665
\(353\) −1.62571 −0.0865279 −0.0432639 0.999064i \(-0.513776\pi\)
−0.0432639 + 0.999064i \(0.513776\pi\)
\(354\) −4.69835 −0.249715
\(355\) 13.7730 0.730994
\(356\) 3.99760 0.211873
\(357\) 6.51765 0.344951
\(358\) 22.5180 1.19011
\(359\) −3.25356 −0.171716 −0.0858582 0.996307i \(-0.527363\pi\)
−0.0858582 + 0.996307i \(0.527363\pi\)
\(360\) 1.97610 0.104149
\(361\) −1.14415 −0.0602184
\(362\) −4.81147 −0.252885
\(363\) 5.44135 0.285597
\(364\) −4.65524 −0.244001
\(365\) 22.9115 1.19924
\(366\) 1.74167 0.0910387
\(367\) −15.5365 −0.810998 −0.405499 0.914095i \(-0.632902\pi\)
−0.405499 + 0.914095i \(0.632902\pi\)
\(368\) −1.68831 −0.0880091
\(369\) −8.28255 −0.431172
\(370\) 2.41784 0.125698
\(371\) 54.6993 2.83985
\(372\) 4.23306 0.219474
\(373\) −12.8180 −0.663692 −0.331846 0.943334i \(-0.607671\pi\)
−0.331846 + 0.943334i \(0.607671\pi\)
\(374\) −3.30091 −0.170686
\(375\) 12.0444 0.621970
\(376\) 13.2833 0.685033
\(377\) −0.920725 −0.0474198
\(378\) −4.65524 −0.239440
\(379\) −26.0379 −1.33748 −0.668739 0.743497i \(-0.733166\pi\)
−0.668739 + 0.743497i \(0.733166\pi\)
\(380\) −8.35023 −0.428358
\(381\) −12.1773 −0.623860
\(382\) 7.01017 0.358672
\(383\) −37.9053 −1.93687 −0.968436 0.249262i \(-0.919812\pi\)
−0.968436 + 0.249262i \(0.919812\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 21.6888 1.10536
\(386\) 12.1251 0.617150
\(387\) 9.40003 0.477831
\(388\) −11.8358 −0.600872
\(389\) −25.9319 −1.31480 −0.657399 0.753542i \(-0.728344\pi\)
−0.657399 + 0.753542i \(0.728344\pi\)
\(390\) 1.97610 0.100064
\(391\) 2.36374 0.119540
\(392\) 14.6713 0.741011
\(393\) 11.5772 0.583995
\(394\) −17.9383 −0.903719
\(395\) −9.07677 −0.456702
\(396\) 2.35768 0.118478
\(397\) 13.6737 0.686264 0.343132 0.939287i \(-0.388512\pi\)
0.343132 + 0.939287i \(0.388512\pi\)
\(398\) −11.7186 −0.587401
\(399\) 19.6713 0.984795
\(400\) −1.09504 −0.0547520
\(401\) −1.62541 −0.0811692 −0.0405846 0.999176i \(-0.512922\pi\)
−0.0405846 + 0.999176i \(0.512922\pi\)
\(402\) −10.0174 −0.499624
\(403\) 4.23306 0.210864
\(404\) −7.65739 −0.380969
\(405\) 1.97610 0.0981931
\(406\) 4.28620 0.212720
\(407\) 2.88472 0.142990
\(408\) 1.40007 0.0693137
\(409\) 12.6629 0.626142 0.313071 0.949730i \(-0.398642\pi\)
0.313071 + 0.949730i \(0.398642\pi\)
\(410\) −16.3671 −0.808315
\(411\) −5.67371 −0.279863
\(412\) −1.00000 −0.0492665
\(413\) 21.8720 1.07625
\(414\) −1.68831 −0.0829758
\(415\) −5.57939 −0.273882
\(416\) −1.00000 −0.0490290
\(417\) 19.7870 0.968971
\(418\) −9.96265 −0.487289
\(419\) −26.5504 −1.29707 −0.648535 0.761185i \(-0.724618\pi\)
−0.648535 + 0.761185i \(0.724618\pi\)
\(420\) −9.19921 −0.448875
\(421\) −31.5279 −1.53658 −0.768288 0.640104i \(-0.778891\pi\)
−0.768288 + 0.640104i \(0.778891\pi\)
\(422\) 15.2946 0.744531
\(423\) 13.2833 0.645856
\(424\) 11.7500 0.570633
\(425\) 1.53313 0.0743677
\(426\) −6.96979 −0.337688
\(427\) −8.10791 −0.392369
\(428\) −2.19049 −0.105882
\(429\) 2.35768 0.113830
\(430\) 18.5754 0.895784
\(431\) −0.0732573 −0.00352868 −0.00176434 0.999998i \(-0.500562\pi\)
−0.00176434 + 0.999998i \(0.500562\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −28.0122 −1.34618 −0.673091 0.739560i \(-0.735034\pi\)
−0.673091 + 0.739560i \(0.735034\pi\)
\(434\) −19.7059 −0.945914
\(435\) −1.81944 −0.0872356
\(436\) 15.4443 0.739649
\(437\) 7.13414 0.341272
\(438\) −11.5943 −0.553997
\(439\) 35.9419 1.71542 0.857708 0.514137i \(-0.171888\pi\)
0.857708 + 0.514137i \(0.171888\pi\)
\(440\) 4.65900 0.222109
\(441\) 14.6713 0.698632
\(442\) 1.40007 0.0665944
\(443\) 7.72845 0.367190 0.183595 0.983002i \(-0.441227\pi\)
0.183595 + 0.983002i \(0.441227\pi\)
\(444\) −1.22354 −0.0580668
\(445\) 7.89965 0.374480
\(446\) 14.1192 0.668562
\(447\) 19.8323 0.938037
\(448\) 4.65524 0.219939
\(449\) −40.5691 −1.91458 −0.957288 0.289135i \(-0.906632\pi\)
−0.957288 + 0.289135i \(0.906632\pi\)
\(450\) −1.09504 −0.0516207
\(451\) −19.5276 −0.919519
\(452\) 16.2250 0.763159
\(453\) 16.6789 0.783645
\(454\) −1.43072 −0.0671472
\(455\) −9.19921 −0.431266
\(456\) 4.22562 0.197883
\(457\) 13.7808 0.644638 0.322319 0.946631i \(-0.395538\pi\)
0.322319 + 0.946631i \(0.395538\pi\)
\(458\) 25.0949 1.17261
\(459\) 1.40007 0.0653495
\(460\) −3.33626 −0.155554
\(461\) −15.1040 −0.703465 −0.351732 0.936101i \(-0.614407\pi\)
−0.351732 + 0.936101i \(0.614407\pi\)
\(462\) −10.9756 −0.510629
\(463\) −19.0640 −0.885980 −0.442990 0.896527i \(-0.646082\pi\)
−0.442990 + 0.896527i \(0.646082\pi\)
\(464\) 0.920725 0.0427436
\(465\) 8.36493 0.387914
\(466\) −0.642463 −0.0297615
\(467\) −30.1782 −1.39648 −0.698240 0.715864i \(-0.746033\pi\)
−0.698240 + 0.715864i \(0.746033\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 46.6336 2.15334
\(470\) 26.2491 1.21078
\(471\) −7.72650 −0.356018
\(472\) 4.69835 0.216259
\(473\) 22.1623 1.01902
\(474\) 4.59328 0.210976
\(475\) 4.62722 0.212311
\(476\) −6.51765 −0.298736
\(477\) 11.7500 0.537998
\(478\) −8.20641 −0.375353
\(479\) 16.5702 0.757111 0.378556 0.925578i \(-0.376421\pi\)
0.378556 + 0.925578i \(0.376421\pi\)
\(480\) −1.97610 −0.0901961
\(481\) −1.22354 −0.0557888
\(482\) 6.48254 0.295272
\(483\) 7.85948 0.357619
\(484\) −5.44135 −0.247334
\(485\) −23.3887 −1.06203
\(486\) −1.00000 −0.0453609
\(487\) 2.30340 0.104377 0.0521884 0.998637i \(-0.483380\pi\)
0.0521884 + 0.998637i \(0.483380\pi\)
\(488\) −1.74167 −0.0788418
\(489\) −19.9507 −0.902202
\(490\) 28.9919 1.30972
\(491\) 33.2881 1.50227 0.751134 0.660149i \(-0.229507\pi\)
0.751134 + 0.660149i \(0.229507\pi\)
\(492\) 8.28255 0.373406
\(493\) −1.28908 −0.0580571
\(494\) 4.22562 0.190120
\(495\) 4.65900 0.209407
\(496\) −4.23306 −0.190070
\(497\) 32.4461 1.45541
\(498\) 2.82344 0.126521
\(499\) 16.7316 0.749009 0.374504 0.927225i \(-0.377813\pi\)
0.374504 + 0.927225i \(0.377813\pi\)
\(500\) −12.0444 −0.538642
\(501\) −15.6969 −0.701285
\(502\) −29.4254 −1.31332
\(503\) 5.43676 0.242413 0.121206 0.992627i \(-0.461324\pi\)
0.121206 + 0.992627i \(0.461324\pi\)
\(504\) 4.65524 0.207361
\(505\) −15.1317 −0.673354
\(506\) −3.98049 −0.176954
\(507\) −1.00000 −0.0444116
\(508\) 12.1773 0.540279
\(509\) −34.1782 −1.51492 −0.757461 0.652880i \(-0.773561\pi\)
−0.757461 + 0.652880i \(0.773561\pi\)
\(510\) 2.76667 0.122510
\(511\) 53.9743 2.38768
\(512\) 1.00000 0.0441942
\(513\) 4.22562 0.186566
\(514\) 2.86093 0.126190
\(515\) −1.97610 −0.0870773
\(516\) −9.40003 −0.413813
\(517\) 31.3177 1.37735
\(518\) 5.69589 0.250263
\(519\) 8.94426 0.392609
\(520\) −1.97610 −0.0866576
\(521\) 25.9293 1.13599 0.567993 0.823034i \(-0.307720\pi\)
0.567993 + 0.823034i \(0.307720\pi\)
\(522\) 0.920725 0.0402990
\(523\) 6.35658 0.277954 0.138977 0.990296i \(-0.455619\pi\)
0.138977 + 0.990296i \(0.455619\pi\)
\(524\) −11.5772 −0.505754
\(525\) 5.09767 0.222481
\(526\) −3.22757 −0.140729
\(527\) 5.92657 0.258165
\(528\) −2.35768 −0.102605
\(529\) −20.1496 −0.876070
\(530\) 23.2192 1.00858
\(531\) 4.69835 0.203891
\(532\) −19.6713 −0.852858
\(533\) 8.28255 0.358757
\(534\) −3.99760 −0.172993
\(535\) −4.32863 −0.187143
\(536\) 10.0174 0.432688
\(537\) −22.5180 −0.971721
\(538\) −2.39878 −0.103419
\(539\) 34.5901 1.48990
\(540\) −1.97610 −0.0850377
\(541\) −17.6478 −0.758737 −0.379369 0.925246i \(-0.623859\pi\)
−0.379369 + 0.925246i \(0.623859\pi\)
\(542\) 0.749348 0.0321873
\(543\) 4.81147 0.206480
\(544\) −1.40007 −0.0600274
\(545\) 30.5195 1.30731
\(546\) 4.65524 0.199226
\(547\) −14.6376 −0.625859 −0.312929 0.949776i \(-0.601310\pi\)
−0.312929 + 0.949776i \(0.601310\pi\)
\(548\) 5.67371 0.242369
\(549\) −1.74167 −0.0743328
\(550\) −2.58175 −0.110086
\(551\) −3.89063 −0.165747
\(552\) 1.68831 0.0718591
\(553\) −21.3828 −0.909291
\(554\) 21.0829 0.895727
\(555\) −2.41784 −0.102632
\(556\) −19.7870 −0.839154
\(557\) −30.4552 −1.29043 −0.645213 0.764002i \(-0.723232\pi\)
−0.645213 + 0.764002i \(0.723232\pi\)
\(558\) −4.23306 −0.179200
\(559\) −9.40003 −0.397579
\(560\) 9.19921 0.388738
\(561\) 3.30091 0.139364
\(562\) −22.8003 −0.961774
\(563\) −6.82548 −0.287660 −0.143830 0.989602i \(-0.545942\pi\)
−0.143830 + 0.989602i \(0.545942\pi\)
\(564\) −13.2833 −0.559327
\(565\) 32.0622 1.34886
\(566\) −31.8124 −1.33717
\(567\) 4.65524 0.195502
\(568\) 6.96979 0.292446
\(569\) 2.03472 0.0853001 0.0426501 0.999090i \(-0.486420\pi\)
0.0426501 + 0.999090i \(0.486420\pi\)
\(570\) 8.35023 0.349753
\(571\) 9.03720 0.378195 0.189098 0.981958i \(-0.439444\pi\)
0.189098 + 0.981958i \(0.439444\pi\)
\(572\) −2.35768 −0.0985795
\(573\) −7.01017 −0.292854
\(574\) −38.5573 −1.60935
\(575\) 1.84876 0.0770987
\(576\) 1.00000 0.0416667
\(577\) 13.3567 0.556046 0.278023 0.960574i \(-0.410321\pi\)
0.278023 + 0.960574i \(0.410321\pi\)
\(578\) −15.0398 −0.625574
\(579\) −12.1251 −0.503901
\(580\) 1.81944 0.0755482
\(581\) −13.1438 −0.545296
\(582\) 11.8358 0.490610
\(583\) 27.7028 1.14733
\(584\) 11.5943 0.479776
\(585\) −1.97610 −0.0817016
\(586\) 23.9481 0.989285
\(587\) −41.3212 −1.70551 −0.852755 0.522311i \(-0.825070\pi\)
−0.852755 + 0.522311i \(0.825070\pi\)
\(588\) −14.6713 −0.605033
\(589\) 17.8873 0.737033
\(590\) 9.28440 0.382233
\(591\) 17.9383 0.737883
\(592\) 1.22354 0.0502873
\(593\) 9.26916 0.380639 0.190319 0.981722i \(-0.439048\pi\)
0.190319 + 0.981722i \(0.439048\pi\)
\(594\) −2.35768 −0.0967367
\(595\) −12.8795 −0.528009
\(596\) −19.8323 −0.812364
\(597\) 11.7186 0.479611
\(598\) 1.68831 0.0690400
\(599\) 2.06596 0.0844128 0.0422064 0.999109i \(-0.486561\pi\)
0.0422064 + 0.999109i \(0.486561\pi\)
\(600\) 1.09504 0.0447048
\(601\) 41.7134 1.70152 0.850762 0.525552i \(-0.176141\pi\)
0.850762 + 0.525552i \(0.176141\pi\)
\(602\) 43.7594 1.78350
\(603\) 10.0174 0.407942
\(604\) −16.6789 −0.678656
\(605\) −10.7526 −0.437157
\(606\) 7.65739 0.311060
\(607\) 4.01768 0.163073 0.0815364 0.996670i \(-0.474017\pi\)
0.0815364 + 0.996670i \(0.474017\pi\)
\(608\) −4.22562 −0.171371
\(609\) −4.28620 −0.173685
\(610\) −3.44172 −0.139351
\(611\) −13.2833 −0.537384
\(612\) −1.40007 −0.0565944
\(613\) −33.7009 −1.36117 −0.680583 0.732671i \(-0.738274\pi\)
−0.680583 + 0.732671i \(0.738274\pi\)
\(614\) −19.9043 −0.803272
\(615\) 16.3671 0.659986
\(616\) 10.9756 0.442218
\(617\) 20.1016 0.809259 0.404630 0.914481i \(-0.367400\pi\)
0.404630 + 0.914481i \(0.367400\pi\)
\(618\) 1.00000 0.0402259
\(619\) 9.78828 0.393424 0.196712 0.980461i \(-0.436974\pi\)
0.196712 + 0.980461i \(0.436974\pi\)
\(620\) −8.36493 −0.335944
\(621\) 1.68831 0.0677494
\(622\) −3.07394 −0.123254
\(623\) 18.6098 0.745586
\(624\) 1.00000 0.0400320
\(625\) −18.3257 −0.733028
\(626\) 17.0139 0.680011
\(627\) 9.96265 0.397870
\(628\) 7.72650 0.308321
\(629\) −1.71304 −0.0683035
\(630\) 9.19921 0.366505
\(631\) −26.1972 −1.04289 −0.521447 0.853284i \(-0.674608\pi\)
−0.521447 + 0.853284i \(0.674608\pi\)
\(632\) −4.59328 −0.182711
\(633\) −15.2946 −0.607907
\(634\) −15.5410 −0.617210
\(635\) 24.0635 0.954929
\(636\) −11.7500 −0.465920
\(637\) −14.6713 −0.581297
\(638\) 2.17077 0.0859418
\(639\) 6.96979 0.275721
\(640\) 1.97610 0.0781121
\(641\) −17.6597 −0.697516 −0.348758 0.937213i \(-0.613397\pi\)
−0.348758 + 0.937213i \(0.613397\pi\)
\(642\) 2.19049 0.0864519
\(643\) −41.3631 −1.63120 −0.815601 0.578615i \(-0.803593\pi\)
−0.815601 + 0.578615i \(0.803593\pi\)
\(644\) −7.85948 −0.309707
\(645\) −18.5754 −0.731405
\(646\) 5.91615 0.232768
\(647\) −48.1061 −1.89124 −0.945622 0.325267i \(-0.894546\pi\)
−0.945622 + 0.325267i \(0.894546\pi\)
\(648\) 1.00000 0.0392837
\(649\) 11.0772 0.434818
\(650\) 1.09504 0.0429510
\(651\) 19.7059 0.772335
\(652\) 19.9507 0.781330
\(653\) −20.4563 −0.800517 −0.400259 0.916402i \(-0.631080\pi\)
−0.400259 + 0.916402i \(0.631080\pi\)
\(654\) −15.4443 −0.603921
\(655\) −22.8778 −0.893908
\(656\) −8.28255 −0.323379
\(657\) 11.5943 0.452337
\(658\) 61.8369 2.41065
\(659\) −12.7352 −0.496092 −0.248046 0.968748i \(-0.579788\pi\)
−0.248046 + 0.968748i \(0.579788\pi\)
\(660\) −4.65900 −0.181351
\(661\) −0.422485 −0.0164328 −0.00821639 0.999966i \(-0.502615\pi\)
−0.00821639 + 0.999966i \(0.502615\pi\)
\(662\) −15.1150 −0.587460
\(663\) −1.40007 −0.0543741
\(664\) −2.82344 −0.109571
\(665\) −38.8723 −1.50740
\(666\) 1.22354 0.0474113
\(667\) −1.55447 −0.0601892
\(668\) 15.6969 0.607330
\(669\) −14.1192 −0.545878
\(670\) 19.7954 0.764764
\(671\) −4.10631 −0.158522
\(672\) −4.65524 −0.179580
\(673\) 13.2581 0.511064 0.255532 0.966801i \(-0.417749\pi\)
0.255532 + 0.966801i \(0.417749\pi\)
\(674\) −3.85788 −0.148600
\(675\) 1.09504 0.0421481
\(676\) 1.00000 0.0384615
\(677\) 44.2611 1.70109 0.850545 0.525902i \(-0.176272\pi\)
0.850545 + 0.525902i \(0.176272\pi\)
\(678\) −16.2250 −0.623117
\(679\) −55.0985 −2.11449
\(680\) −2.76667 −0.106097
\(681\) 1.43072 0.0548255
\(682\) −9.98019 −0.382161
\(683\) 11.9060 0.455571 0.227786 0.973711i \(-0.426851\pi\)
0.227786 + 0.973711i \(0.426851\pi\)
\(684\) −4.22562 −0.161571
\(685\) 11.2118 0.428381
\(686\) 35.7116 1.36348
\(687\) −25.0949 −0.957429
\(688\) 9.40003 0.358373
\(689\) −11.7500 −0.447641
\(690\) 3.33626 0.127009
\(691\) 17.3712 0.660831 0.330415 0.943836i \(-0.392811\pi\)
0.330415 + 0.943836i \(0.392811\pi\)
\(692\) −8.94426 −0.340010
\(693\) 10.9756 0.416927
\(694\) −0.410196 −0.0155708
\(695\) −39.1009 −1.48318
\(696\) −0.920725 −0.0349000
\(697\) 11.5961 0.439235
\(698\) 24.9299 0.943610
\(699\) 0.642463 0.0243002
\(700\) −5.09767 −0.192674
\(701\) 49.4216 1.86663 0.933315 0.359059i \(-0.116902\pi\)
0.933315 + 0.359059i \(0.116902\pi\)
\(702\) 1.00000 0.0377426
\(703\) −5.17023 −0.194999
\(704\) 2.35768 0.0888584
\(705\) −26.2491 −0.988597
\(706\) −1.62571 −0.0611844
\(707\) −35.6470 −1.34064
\(708\) −4.69835 −0.176575
\(709\) 2.66753 0.100181 0.0500907 0.998745i \(-0.484049\pi\)
0.0500907 + 0.998745i \(0.484049\pi\)
\(710\) 13.7730 0.516891
\(711\) −4.59328 −0.172262
\(712\) 3.99760 0.149817
\(713\) 7.14670 0.267646
\(714\) 6.51765 0.243917
\(715\) −4.65900 −0.174237
\(716\) 22.5180 0.841535
\(717\) 8.20641 0.306474
\(718\) −3.25356 −0.121422
\(719\) −32.1454 −1.19882 −0.599410 0.800442i \(-0.704598\pi\)
−0.599410 + 0.800442i \(0.704598\pi\)
\(720\) 1.97610 0.0736448
\(721\) −4.65524 −0.173370
\(722\) −1.14415 −0.0425808
\(723\) −6.48254 −0.241088
\(724\) −4.81147 −0.178817
\(725\) −1.00823 −0.0374447
\(726\) 5.44135 0.201947
\(727\) −12.3347 −0.457469 −0.228735 0.973489i \(-0.573459\pi\)
−0.228735 + 0.973489i \(0.573459\pi\)
\(728\) −4.65524 −0.172535
\(729\) 1.00000 0.0370370
\(730\) 22.9115 0.847992
\(731\) −13.1607 −0.486765
\(732\) 1.74167 0.0643741
\(733\) 9.97138 0.368301 0.184151 0.982898i \(-0.441047\pi\)
0.184151 + 0.982898i \(0.441047\pi\)
\(734\) −15.5365 −0.573462
\(735\) −28.9919 −1.06938
\(736\) −1.68831 −0.0622318
\(737\) 23.6179 0.869977
\(738\) −8.28255 −0.304885
\(739\) 21.2278 0.780877 0.390439 0.920629i \(-0.372323\pi\)
0.390439 + 0.920629i \(0.372323\pi\)
\(740\) 2.41784 0.0888816
\(741\) −4.22562 −0.155232
\(742\) 54.6993 2.00807
\(743\) 16.2868 0.597505 0.298752 0.954331i \(-0.403430\pi\)
0.298752 + 0.954331i \(0.403430\pi\)
\(744\) 4.23306 0.155191
\(745\) −39.1906 −1.43583
\(746\) −12.8180 −0.469301
\(747\) −2.82344 −0.103304
\(748\) −3.30091 −0.120693
\(749\) −10.1973 −0.372600
\(750\) 12.0444 0.439799
\(751\) 28.2516 1.03092 0.515458 0.856915i \(-0.327622\pi\)
0.515458 + 0.856915i \(0.327622\pi\)
\(752\) 13.2833 0.484392
\(753\) 29.4254 1.07232
\(754\) −0.920725 −0.0335308
\(755\) −32.9592 −1.19951
\(756\) −4.65524 −0.169309
\(757\) 42.2881 1.53699 0.768494 0.639857i \(-0.221006\pi\)
0.768494 + 0.639857i \(0.221006\pi\)
\(758\) −26.0379 −0.945740
\(759\) 3.98049 0.144482
\(760\) −8.35023 −0.302895
\(761\) −31.0413 −1.12525 −0.562623 0.826714i \(-0.690208\pi\)
−0.562623 + 0.826714i \(0.690208\pi\)
\(762\) −12.1773 −0.441136
\(763\) 71.8971 2.60285
\(764\) 7.01017 0.253619
\(765\) −2.76667 −0.100029
\(766\) −37.9053 −1.36958
\(767\) −4.69835 −0.169648
\(768\) −1.00000 −0.0360844
\(769\) −52.1811 −1.88170 −0.940850 0.338824i \(-0.889971\pi\)
−0.940850 + 0.338824i \(0.889971\pi\)
\(770\) 21.6888 0.781609
\(771\) −2.86093 −0.103034
\(772\) 12.1251 0.436391
\(773\) 29.9114 1.07584 0.537918 0.842997i \(-0.319211\pi\)
0.537918 + 0.842997i \(0.319211\pi\)
\(774\) 9.40003 0.337877
\(775\) 4.63536 0.166507
\(776\) −11.8358 −0.424881
\(777\) −5.69589 −0.204339
\(778\) −25.9319 −0.929703
\(779\) 34.9989 1.25397
\(780\) 1.97610 0.0707556
\(781\) 16.4325 0.588002
\(782\) 2.36374 0.0845273
\(783\) −0.920725 −0.0329040
\(784\) 14.6713 0.523974
\(785\) 15.2683 0.544949
\(786\) 11.5772 0.412946
\(787\) 21.5388 0.767776 0.383888 0.923380i \(-0.374585\pi\)
0.383888 + 0.923380i \(0.374585\pi\)
\(788\) −17.9383 −0.639026
\(789\) 3.22757 0.114905
\(790\) −9.07677 −0.322937
\(791\) 75.5312 2.68558
\(792\) 2.35768 0.0837765
\(793\) 1.74167 0.0618486
\(794\) 13.6737 0.485262
\(795\) −23.2192 −0.823501
\(796\) −11.7186 −0.415355
\(797\) 39.2712 1.39106 0.695528 0.718499i \(-0.255170\pi\)
0.695528 + 0.718499i \(0.255170\pi\)
\(798\) 19.6713 0.696355
\(799\) −18.5975 −0.657932
\(800\) −1.09504 −0.0387155
\(801\) 3.99760 0.141248
\(802\) −1.62541 −0.0573953
\(803\) 27.3356 0.964654
\(804\) −10.0174 −0.353288
\(805\) −15.5311 −0.547399
\(806\) 4.23306 0.149103
\(807\) 2.39878 0.0844410
\(808\) −7.65739 −0.269386
\(809\) 44.9800 1.58141 0.790706 0.612196i \(-0.209714\pi\)
0.790706 + 0.612196i \(0.209714\pi\)
\(810\) 1.97610 0.0694330
\(811\) 16.1384 0.566696 0.283348 0.959017i \(-0.408555\pi\)
0.283348 + 0.959017i \(0.408555\pi\)
\(812\) 4.28620 0.150416
\(813\) −0.749348 −0.0262808
\(814\) 2.88472 0.101109
\(815\) 39.4245 1.38098
\(816\) 1.40007 0.0490122
\(817\) −39.7210 −1.38966
\(818\) 12.6629 0.442749
\(819\) −4.65524 −0.162667
\(820\) −16.3671 −0.571565
\(821\) 1.55416 0.0542404 0.0271202 0.999632i \(-0.491366\pi\)
0.0271202 + 0.999632i \(0.491366\pi\)
\(822\) −5.67371 −0.197893
\(823\) −51.6454 −1.80025 −0.900123 0.435636i \(-0.856523\pi\)
−0.900123 + 0.435636i \(0.856523\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 2.58175 0.0898850
\(826\) 21.8720 0.761023
\(827\) 19.0442 0.662233 0.331117 0.943590i \(-0.392575\pi\)
0.331117 + 0.943590i \(0.392575\pi\)
\(828\) −1.68831 −0.0586727
\(829\) −26.9045 −0.934431 −0.467215 0.884144i \(-0.654743\pi\)
−0.467215 + 0.884144i \(0.654743\pi\)
\(830\) −5.57939 −0.193664
\(831\) −21.0829 −0.731358
\(832\) −1.00000 −0.0346688
\(833\) −20.5408 −0.711695
\(834\) 19.7870 0.685166
\(835\) 31.0185 1.07344
\(836\) −9.96265 −0.344565
\(837\) 4.23306 0.146316
\(838\) −26.5504 −0.917167
\(839\) −41.9742 −1.44911 −0.724555 0.689217i \(-0.757955\pi\)
−0.724555 + 0.689217i \(0.757955\pi\)
\(840\) −9.19921 −0.317403
\(841\) −28.1523 −0.970768
\(842\) −31.5279 −1.08652
\(843\) 22.8003 0.785285
\(844\) 15.2946 0.526463
\(845\) 1.97610 0.0679798
\(846\) 13.2833 0.456689
\(847\) −25.3308 −0.870377
\(848\) 11.7500 0.403498
\(849\) 31.8124 1.09180
\(850\) 1.53313 0.0525859
\(851\) −2.06572 −0.0708119
\(852\) −6.96979 −0.238781
\(853\) −2.02827 −0.0694467 −0.0347234 0.999397i \(-0.511055\pi\)
−0.0347234 + 0.999397i \(0.511055\pi\)
\(854\) −8.10791 −0.277447
\(855\) −8.35023 −0.285572
\(856\) −2.19049 −0.0748695
\(857\) −10.4055 −0.355446 −0.177723 0.984081i \(-0.556873\pi\)
−0.177723 + 0.984081i \(0.556873\pi\)
\(858\) 2.35768 0.0804898
\(859\) 29.9199 1.02085 0.510427 0.859921i \(-0.329487\pi\)
0.510427 + 0.859921i \(0.329487\pi\)
\(860\) 18.5754 0.633415
\(861\) 38.5573 1.31403
\(862\) −0.0732573 −0.00249515
\(863\) 1.86961 0.0636422 0.0318211 0.999494i \(-0.489869\pi\)
0.0318211 + 0.999494i \(0.489869\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −17.6747 −0.600959
\(866\) −28.0122 −0.951894
\(867\) 15.0398 0.510779
\(868\) −19.7059 −0.668862
\(869\) −10.8295 −0.367365
\(870\) −1.81944 −0.0616849
\(871\) −10.0174 −0.339428
\(872\) 15.4443 0.523011
\(873\) −11.8358 −0.400581
\(874\) 7.13414 0.241316
\(875\) −56.0695 −1.89550
\(876\) −11.5943 −0.391735
\(877\) −9.92590 −0.335174 −0.167587 0.985857i \(-0.553597\pi\)
−0.167587 + 0.985857i \(0.553597\pi\)
\(878\) 35.9419 1.21298
\(879\) −23.9481 −0.807748
\(880\) 4.65900 0.157055
\(881\) −24.5217 −0.826158 −0.413079 0.910695i \(-0.635547\pi\)
−0.413079 + 0.910695i \(0.635547\pi\)
\(882\) 14.6713 0.494007
\(883\) 27.9675 0.941180 0.470590 0.882352i \(-0.344041\pi\)
0.470590 + 0.882352i \(0.344041\pi\)
\(884\) 1.40007 0.0470894
\(885\) −9.28440 −0.312092
\(886\) 7.72845 0.259642
\(887\) 34.0164 1.14216 0.571080 0.820895i \(-0.306525\pi\)
0.571080 + 0.820895i \(0.306525\pi\)
\(888\) −1.22354 −0.0410594
\(889\) 56.6881 1.90126
\(890\) 7.89965 0.264797
\(891\) 2.35768 0.0789852
\(892\) 14.1192 0.472744
\(893\) −56.1301 −1.87832
\(894\) 19.8323 0.663292
\(895\) 44.4977 1.48739
\(896\) 4.65524 0.155521
\(897\) −1.68831 −0.0563709
\(898\) −40.5691 −1.35381
\(899\) −3.89748 −0.129988
\(900\) −1.09504 −0.0365013
\(901\) −16.4509 −0.548057
\(902\) −19.5276 −0.650198
\(903\) −43.7594 −1.45622
\(904\) 16.2250 0.539635
\(905\) −9.50794 −0.316054
\(906\) 16.6789 0.554121
\(907\) −15.4638 −0.513466 −0.256733 0.966482i \(-0.582646\pi\)
−0.256733 + 0.966482i \(0.582646\pi\)
\(908\) −1.43072 −0.0474803
\(909\) −7.65739 −0.253980
\(910\) −9.19921 −0.304951
\(911\) −5.47641 −0.181442 −0.0907208 0.995876i \(-0.528917\pi\)
−0.0907208 + 0.995876i \(0.528917\pi\)
\(912\) 4.22562 0.139924
\(913\) −6.65676 −0.220307
\(914\) 13.7808 0.455828
\(915\) 3.44172 0.113780
\(916\) 25.0949 0.829158
\(917\) −53.8948 −1.77976
\(918\) 1.40007 0.0462091
\(919\) −24.7430 −0.816195 −0.408098 0.912938i \(-0.633808\pi\)
−0.408098 + 0.912938i \(0.633808\pi\)
\(920\) −3.33626 −0.109993
\(921\) 19.9043 0.655869
\(922\) −15.1040 −0.497425
\(923\) −6.96979 −0.229414
\(924\) −10.9756 −0.361070
\(925\) −1.33983 −0.0440533
\(926\) −19.0640 −0.626482
\(927\) −1.00000 −0.0328443
\(928\) 0.920725 0.0302243
\(929\) −51.9988 −1.70602 −0.853012 0.521891i \(-0.825227\pi\)
−0.853012 + 0.521891i \(0.825227\pi\)
\(930\) 8.36493 0.274297
\(931\) −61.9952 −2.03181
\(932\) −0.642463 −0.0210446
\(933\) 3.07394 0.100636
\(934\) −30.1782 −0.987460
\(935\) −6.52292 −0.213322
\(936\) −1.00000 −0.0326860
\(937\) −47.7692 −1.56055 −0.780276 0.625435i \(-0.784921\pi\)
−0.780276 + 0.625435i \(0.784921\pi\)
\(938\) 46.6336 1.52264
\(939\) −17.0139 −0.555227
\(940\) 26.2491 0.856150
\(941\) 49.4890 1.61330 0.806648 0.591032i \(-0.201279\pi\)
0.806648 + 0.591032i \(0.201279\pi\)
\(942\) −7.72650 −0.251743
\(943\) 13.9835 0.455365
\(944\) 4.69835 0.152918
\(945\) −9.19921 −0.299250
\(946\) 22.1623 0.720557
\(947\) 9.12725 0.296596 0.148298 0.988943i \(-0.452621\pi\)
0.148298 + 0.988943i \(0.452621\pi\)
\(948\) 4.59328 0.149183
\(949\) −11.5943 −0.376367
\(950\) 4.62722 0.150127
\(951\) 15.5410 0.503950
\(952\) −6.51765 −0.211238
\(953\) 29.1733 0.945016 0.472508 0.881326i \(-0.343349\pi\)
0.472508 + 0.881326i \(0.343349\pi\)
\(954\) 11.7500 0.380422
\(955\) 13.8528 0.448265
\(956\) −8.20641 −0.265414
\(957\) −2.17077 −0.0701712
\(958\) 16.5702 0.535359
\(959\) 26.4125 0.852903
\(960\) −1.97610 −0.0637783
\(961\) −13.0812 −0.421975
\(962\) −1.22354 −0.0394486
\(963\) −2.19049 −0.0705877
\(964\) 6.48254 0.208789
\(965\) 23.9603 0.771310
\(966\) 7.85948 0.252875
\(967\) −30.1231 −0.968693 −0.484346 0.874876i \(-0.660943\pi\)
−0.484346 + 0.874876i \(0.660943\pi\)
\(968\) −5.44135 −0.174892
\(969\) −5.91615 −0.190054
\(970\) −23.3887 −0.750966
\(971\) −21.9567 −0.704622 −0.352311 0.935883i \(-0.614604\pi\)
−0.352311 + 0.935883i \(0.614604\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −92.1130 −2.95301
\(974\) 2.30340 0.0738056
\(975\) −1.09504 −0.0350693
\(976\) −1.74167 −0.0557496
\(977\) 36.3325 1.16238 0.581190 0.813768i \(-0.302587\pi\)
0.581190 + 0.813768i \(0.302587\pi\)
\(978\) −19.9507 −0.637953
\(979\) 9.42507 0.301226
\(980\) 28.9919 0.926111
\(981\) 15.4443 0.493100
\(982\) 33.2881 1.06226
\(983\) −60.1263 −1.91773 −0.958866 0.283858i \(-0.908385\pi\)
−0.958866 + 0.283858i \(0.908385\pi\)
\(984\) 8.28255 0.264038
\(985\) −35.4478 −1.12946
\(986\) −1.28908 −0.0410526
\(987\) −61.8369 −1.96829
\(988\) 4.22562 0.134435
\(989\) −15.8701 −0.504641
\(990\) 4.65900 0.148073
\(991\) 43.3331 1.37652 0.688261 0.725463i \(-0.258374\pi\)
0.688261 + 0.725463i \(0.258374\pi\)
\(992\) −4.23306 −0.134400
\(993\) 15.1150 0.479659
\(994\) 32.4461 1.02913
\(995\) −23.1571 −0.734130
\(996\) 2.82344 0.0894641
\(997\) 4.68320 0.148318 0.0741592 0.997246i \(-0.476373\pi\)
0.0741592 + 0.997246i \(0.476373\pi\)
\(998\) 16.7316 0.529629
\(999\) −1.22354 −0.0387112
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.1.12 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.12 15 1.1 even 1 trivial