Properties

Label 8034.2.a.bc.1.15
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 1.15
Root \(-4.10857\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.10857 q^{5} -1.00000 q^{6} +1.08136 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.10857 q^{5} -1.00000 q^{6} +1.08136 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.10857 q^{10} +1.94311 q^{11} -1.00000 q^{12} -1.00000 q^{13} +1.08136 q^{14} -4.10857 q^{15} +1.00000 q^{16} +3.29001 q^{17} +1.00000 q^{18} +5.56641 q^{19} +4.10857 q^{20} -1.08136 q^{21} +1.94311 q^{22} +4.23927 q^{23} -1.00000 q^{24} +11.8804 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.08136 q^{28} +5.73866 q^{29} -4.10857 q^{30} -9.76349 q^{31} +1.00000 q^{32} -1.94311 q^{33} +3.29001 q^{34} +4.44286 q^{35} +1.00000 q^{36} -0.721621 q^{37} +5.56641 q^{38} +1.00000 q^{39} +4.10857 q^{40} +2.73623 q^{41} -1.08136 q^{42} -1.00708 q^{43} +1.94311 q^{44} +4.10857 q^{45} +4.23927 q^{46} -7.53102 q^{47} -1.00000 q^{48} -5.83065 q^{49} +11.8804 q^{50} -3.29001 q^{51} -1.00000 q^{52} +6.31473 q^{53} -1.00000 q^{54} +7.98340 q^{55} +1.08136 q^{56} -5.56641 q^{57} +5.73866 q^{58} +5.34007 q^{59} -4.10857 q^{60} -9.12177 q^{61} -9.76349 q^{62} +1.08136 q^{63} +1.00000 q^{64} -4.10857 q^{65} -1.94311 q^{66} -3.91535 q^{67} +3.29001 q^{68} -4.23927 q^{69} +4.44286 q^{70} -3.70178 q^{71} +1.00000 q^{72} -12.7190 q^{73} -0.721621 q^{74} -11.8804 q^{75} +5.56641 q^{76} +2.10121 q^{77} +1.00000 q^{78} +7.86754 q^{79} +4.10857 q^{80} +1.00000 q^{81} +2.73623 q^{82} -11.0081 q^{83} -1.08136 q^{84} +13.5172 q^{85} -1.00708 q^{86} -5.73866 q^{87} +1.94311 q^{88} +3.37812 q^{89} +4.10857 q^{90} -1.08136 q^{91} +4.23927 q^{92} +9.76349 q^{93} -7.53102 q^{94} +22.8700 q^{95} -1.00000 q^{96} -3.56041 q^{97} -5.83065 q^{98} +1.94311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

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
\(5\) 4.10857 1.83741 0.918704 0.394946i \(-0.129237\pi\)
0.918704 + 0.394946i \(0.129237\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.08136 0.408717 0.204359 0.978896i \(-0.434489\pi\)
0.204359 + 0.978896i \(0.434489\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.10857 1.29924
\(11\) 1.94311 0.585870 0.292935 0.956132i \(-0.405368\pi\)
0.292935 + 0.956132i \(0.405368\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 1.08136 0.289007
\(15\) −4.10857 −1.06083
\(16\) 1.00000 0.250000
\(17\) 3.29001 0.797944 0.398972 0.916963i \(-0.369367\pi\)
0.398972 + 0.916963i \(0.369367\pi\)
\(18\) 1.00000 0.235702
\(19\) 5.56641 1.27702 0.638511 0.769613i \(-0.279551\pi\)
0.638511 + 0.769613i \(0.279551\pi\)
\(20\) 4.10857 0.918704
\(21\) −1.08136 −0.235973
\(22\) 1.94311 0.414272
\(23\) 4.23927 0.883949 0.441974 0.897028i \(-0.354278\pi\)
0.441974 + 0.897028i \(0.354278\pi\)
\(24\) −1.00000 −0.204124
\(25\) 11.8804 2.37607
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 1.08136 0.204359
\(29\) 5.73866 1.06564 0.532821 0.846228i \(-0.321132\pi\)
0.532821 + 0.846228i \(0.321132\pi\)
\(30\) −4.10857 −0.750119
\(31\) −9.76349 −1.75358 −0.876788 0.480878i \(-0.840318\pi\)
−0.876788 + 0.480878i \(0.840318\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.94311 −0.338252
\(34\) 3.29001 0.564231
\(35\) 4.44286 0.750980
\(36\) 1.00000 0.166667
\(37\) −0.721621 −0.118634 −0.0593168 0.998239i \(-0.518892\pi\)
−0.0593168 + 0.998239i \(0.518892\pi\)
\(38\) 5.56641 0.902990
\(39\) 1.00000 0.160128
\(40\) 4.10857 0.649622
\(41\) 2.73623 0.427327 0.213663 0.976907i \(-0.431460\pi\)
0.213663 + 0.976907i \(0.431460\pi\)
\(42\) −1.08136 −0.166858
\(43\) −1.00708 −0.153579 −0.0767895 0.997047i \(-0.524467\pi\)
−0.0767895 + 0.997047i \(0.524467\pi\)
\(44\) 1.94311 0.292935
\(45\) 4.10857 0.612470
\(46\) 4.23927 0.625046
\(47\) −7.53102 −1.09851 −0.549256 0.835654i \(-0.685089\pi\)
−0.549256 + 0.835654i \(0.685089\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.83065 −0.832950
\(50\) 11.8804 1.68014
\(51\) −3.29001 −0.460693
\(52\) −1.00000 −0.138675
\(53\) 6.31473 0.867395 0.433697 0.901059i \(-0.357209\pi\)
0.433697 + 0.901059i \(0.357209\pi\)
\(54\) −1.00000 −0.136083
\(55\) 7.98340 1.07648
\(56\) 1.08136 0.144503
\(57\) −5.56641 −0.737289
\(58\) 5.73866 0.753523
\(59\) 5.34007 0.695218 0.347609 0.937640i \(-0.386994\pi\)
0.347609 + 0.937640i \(0.386994\pi\)
\(60\) −4.10857 −0.530414
\(61\) −9.12177 −1.16792 −0.583962 0.811781i \(-0.698498\pi\)
−0.583962 + 0.811781i \(0.698498\pi\)
\(62\) −9.76349 −1.23996
\(63\) 1.08136 0.136239
\(64\) 1.00000 0.125000
\(65\) −4.10857 −0.509605
\(66\) −1.94311 −0.239180
\(67\) −3.91535 −0.478336 −0.239168 0.970978i \(-0.576875\pi\)
−0.239168 + 0.970978i \(0.576875\pi\)
\(68\) 3.29001 0.398972
\(69\) −4.23927 −0.510348
\(70\) 4.44286 0.531023
\(71\) −3.70178 −0.439320 −0.219660 0.975576i \(-0.570495\pi\)
−0.219660 + 0.975576i \(0.570495\pi\)
\(72\) 1.00000 0.117851
\(73\) −12.7190 −1.48864 −0.744320 0.667823i \(-0.767226\pi\)
−0.744320 + 0.667823i \(0.767226\pi\)
\(74\) −0.721621 −0.0838867
\(75\) −11.8804 −1.37182
\(76\) 5.56641 0.638511
\(77\) 2.10121 0.239455
\(78\) 1.00000 0.113228
\(79\) 7.86754 0.885168 0.442584 0.896727i \(-0.354062\pi\)
0.442584 + 0.896727i \(0.354062\pi\)
\(80\) 4.10857 0.459352
\(81\) 1.00000 0.111111
\(82\) 2.73623 0.302166
\(83\) −11.0081 −1.20830 −0.604148 0.796872i \(-0.706487\pi\)
−0.604148 + 0.796872i \(0.706487\pi\)
\(84\) −1.08136 −0.117986
\(85\) 13.5172 1.46615
\(86\) −1.00708 −0.108597
\(87\) −5.73866 −0.615249
\(88\) 1.94311 0.207136
\(89\) 3.37812 0.358080 0.179040 0.983842i \(-0.442701\pi\)
0.179040 + 0.983842i \(0.442701\pi\)
\(90\) 4.10857 0.433081
\(91\) −1.08136 −0.113358
\(92\) 4.23927 0.441974
\(93\) 9.76349 1.01243
\(94\) −7.53102 −0.776766
\(95\) 22.8700 2.34641
\(96\) −1.00000 −0.102062
\(97\) −3.56041 −0.361505 −0.180753 0.983529i \(-0.557853\pi\)
−0.180753 + 0.983529i \(0.557853\pi\)
\(98\) −5.83065 −0.588985
\(99\) 1.94311 0.195290
\(100\) 11.8804 1.18804
\(101\) −6.29949 −0.626822 −0.313411 0.949618i \(-0.601472\pi\)
−0.313411 + 0.949618i \(0.601472\pi\)
\(102\) −3.29001 −0.325759
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −4.44286 −0.433579
\(106\) 6.31473 0.613341
\(107\) 0.737374 0.0712846 0.0356423 0.999365i \(-0.488652\pi\)
0.0356423 + 0.999365i \(0.488652\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 7.22763 0.692282 0.346141 0.938183i \(-0.387492\pi\)
0.346141 + 0.938183i \(0.387492\pi\)
\(110\) 7.98340 0.761188
\(111\) 0.721621 0.0684932
\(112\) 1.08136 0.102179
\(113\) −0.412595 −0.0388137 −0.0194068 0.999812i \(-0.506178\pi\)
−0.0194068 + 0.999812i \(0.506178\pi\)
\(114\) −5.56641 −0.521342
\(115\) 17.4173 1.62418
\(116\) 5.73866 0.532821
\(117\) −1.00000 −0.0924500
\(118\) 5.34007 0.491593
\(119\) 3.55769 0.326133
\(120\) −4.10857 −0.375059
\(121\) −7.22432 −0.656757
\(122\) −9.12177 −0.825847
\(123\) −2.73623 −0.246717
\(124\) −9.76349 −0.876788
\(125\) 28.2684 2.52840
\(126\) 1.08136 0.0963356
\(127\) 8.14381 0.722646 0.361323 0.932441i \(-0.382325\pi\)
0.361323 + 0.932441i \(0.382325\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.00708 0.0886689
\(130\) −4.10857 −0.360345
\(131\) −1.82445 −0.159403 −0.0797014 0.996819i \(-0.525397\pi\)
−0.0797014 + 0.996819i \(0.525397\pi\)
\(132\) −1.94311 −0.169126
\(133\) 6.01931 0.521941
\(134\) −3.91535 −0.338235
\(135\) −4.10857 −0.353609
\(136\) 3.29001 0.282116
\(137\) 16.2132 1.38518 0.692592 0.721330i \(-0.256469\pi\)
0.692592 + 0.721330i \(0.256469\pi\)
\(138\) −4.23927 −0.360871
\(139\) −15.7184 −1.33321 −0.666607 0.745410i \(-0.732254\pi\)
−0.666607 + 0.745410i \(0.732254\pi\)
\(140\) 4.44286 0.375490
\(141\) 7.53102 0.634227
\(142\) −3.70178 −0.310646
\(143\) −1.94311 −0.162491
\(144\) 1.00000 0.0833333
\(145\) 23.5777 1.95802
\(146\) −12.7190 −1.05263
\(147\) 5.83065 0.480904
\(148\) −0.721621 −0.0593168
\(149\) 3.39227 0.277906 0.138953 0.990299i \(-0.455626\pi\)
0.138953 + 0.990299i \(0.455626\pi\)
\(150\) −11.8804 −0.970027
\(151\) −1.86945 −0.152134 −0.0760670 0.997103i \(-0.524236\pi\)
−0.0760670 + 0.997103i \(0.524236\pi\)
\(152\) 5.56641 0.451495
\(153\) 3.29001 0.265981
\(154\) 2.10121 0.169320
\(155\) −40.1140 −3.22203
\(156\) 1.00000 0.0800641
\(157\) 2.95582 0.235900 0.117950 0.993020i \(-0.462368\pi\)
0.117950 + 0.993020i \(0.462368\pi\)
\(158\) 7.86754 0.625908
\(159\) −6.31473 −0.500791
\(160\) 4.10857 0.324811
\(161\) 4.58419 0.361285
\(162\) 1.00000 0.0785674
\(163\) −6.68930 −0.523946 −0.261973 0.965075i \(-0.584373\pi\)
−0.261973 + 0.965075i \(0.584373\pi\)
\(164\) 2.73623 0.213663
\(165\) −7.98340 −0.621507
\(166\) −11.0081 −0.854395
\(167\) −5.25050 −0.406296 −0.203148 0.979148i \(-0.565117\pi\)
−0.203148 + 0.979148i \(0.565117\pi\)
\(168\) −1.08136 −0.0834290
\(169\) 1.00000 0.0769231
\(170\) 13.5172 1.03672
\(171\) 5.56641 0.425674
\(172\) −1.00708 −0.0767895
\(173\) −8.82486 −0.670941 −0.335471 0.942051i \(-0.608895\pi\)
−0.335471 + 0.942051i \(0.608895\pi\)
\(174\) −5.73866 −0.435047
\(175\) 12.8470 0.971141
\(176\) 1.94311 0.146467
\(177\) −5.34007 −0.401384
\(178\) 3.37812 0.253201
\(179\) 1.87031 0.139793 0.0698966 0.997554i \(-0.477733\pi\)
0.0698966 + 0.997554i \(0.477733\pi\)
\(180\) 4.10857 0.306235
\(181\) −21.6013 −1.60561 −0.802806 0.596241i \(-0.796660\pi\)
−0.802806 + 0.596241i \(0.796660\pi\)
\(182\) −1.08136 −0.0801560
\(183\) 9.12177 0.674301
\(184\) 4.23927 0.312523
\(185\) −2.96483 −0.217979
\(186\) 9.76349 0.715894
\(187\) 6.39284 0.467491
\(188\) −7.53102 −0.549256
\(189\) −1.08136 −0.0786577
\(190\) 22.8700 1.65916
\(191\) −17.5991 −1.27342 −0.636712 0.771102i \(-0.719706\pi\)
−0.636712 + 0.771102i \(0.719706\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −7.66750 −0.551919 −0.275959 0.961169i \(-0.588996\pi\)
−0.275959 + 0.961169i \(0.588996\pi\)
\(194\) −3.56041 −0.255623
\(195\) 4.10857 0.294221
\(196\) −5.83065 −0.416475
\(197\) 12.4756 0.888849 0.444424 0.895816i \(-0.353408\pi\)
0.444424 + 0.895816i \(0.353408\pi\)
\(198\) 1.94311 0.138091
\(199\) 3.77199 0.267389 0.133695 0.991023i \(-0.457316\pi\)
0.133695 + 0.991023i \(0.457316\pi\)
\(200\) 11.8804 0.840068
\(201\) 3.91535 0.276167
\(202\) −6.29949 −0.443230
\(203\) 6.20558 0.435546
\(204\) −3.29001 −0.230346
\(205\) 11.2420 0.785174
\(206\) −1.00000 −0.0696733
\(207\) 4.23927 0.294650
\(208\) −1.00000 −0.0693375
\(209\) 10.8161 0.748168
\(210\) −4.44286 −0.306586
\(211\) −11.0391 −0.759966 −0.379983 0.924993i \(-0.624070\pi\)
−0.379983 + 0.924993i \(0.624070\pi\)
\(212\) 6.31473 0.433697
\(213\) 3.70178 0.253642
\(214\) 0.737374 0.0504058
\(215\) −4.13768 −0.282187
\(216\) −1.00000 −0.0680414
\(217\) −10.5579 −0.716716
\(218\) 7.22763 0.489517
\(219\) 12.7190 0.859467
\(220\) 7.98340 0.538241
\(221\) −3.29001 −0.221310
\(222\) 0.721621 0.0484320
\(223\) −19.7509 −1.32262 −0.661308 0.750115i \(-0.729998\pi\)
−0.661308 + 0.750115i \(0.729998\pi\)
\(224\) 1.08136 0.0722517
\(225\) 11.8804 0.792023
\(226\) −0.412595 −0.0274454
\(227\) −15.9220 −1.05678 −0.528389 0.849003i \(-0.677204\pi\)
−0.528389 + 0.849003i \(0.677204\pi\)
\(228\) −5.56641 −0.368644
\(229\) 14.3924 0.951074 0.475537 0.879696i \(-0.342254\pi\)
0.475537 + 0.879696i \(0.342254\pi\)
\(230\) 17.4173 1.14847
\(231\) −2.10121 −0.138249
\(232\) 5.73866 0.376762
\(233\) 21.0680 1.38021 0.690105 0.723709i \(-0.257564\pi\)
0.690105 + 0.723709i \(0.257564\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −30.9417 −2.01842
\(236\) 5.34007 0.347609
\(237\) −7.86754 −0.511052
\(238\) 3.55769 0.230611
\(239\) 24.6679 1.59563 0.797816 0.602900i \(-0.205988\pi\)
0.797816 + 0.602900i \(0.205988\pi\)
\(240\) −4.10857 −0.265207
\(241\) 12.8439 0.827350 0.413675 0.910425i \(-0.364245\pi\)
0.413675 + 0.910425i \(0.364245\pi\)
\(242\) −7.22432 −0.464397
\(243\) −1.00000 −0.0641500
\(244\) −9.12177 −0.583962
\(245\) −23.9556 −1.53047
\(246\) −2.73623 −0.174455
\(247\) −5.56641 −0.354182
\(248\) −9.76349 −0.619982
\(249\) 11.0081 0.697610
\(250\) 28.2684 1.78785
\(251\) −24.4841 −1.54543 −0.772713 0.634756i \(-0.781101\pi\)
−0.772713 + 0.634756i \(0.781101\pi\)
\(252\) 1.08136 0.0681195
\(253\) 8.23737 0.517879
\(254\) 8.14381 0.510988
\(255\) −13.5172 −0.846481
\(256\) 1.00000 0.0625000
\(257\) 3.27567 0.204330 0.102165 0.994767i \(-0.467423\pi\)
0.102165 + 0.994767i \(0.467423\pi\)
\(258\) 1.00708 0.0626984
\(259\) −0.780335 −0.0484876
\(260\) −4.10857 −0.254803
\(261\) 5.73866 0.355214
\(262\) −1.82445 −0.112715
\(263\) 8.28156 0.510663 0.255331 0.966854i \(-0.417815\pi\)
0.255331 + 0.966854i \(0.417815\pi\)
\(264\) −1.94311 −0.119590
\(265\) 25.9445 1.59376
\(266\) 6.01931 0.369068
\(267\) −3.37812 −0.206738
\(268\) −3.91535 −0.239168
\(269\) 20.5589 1.25350 0.626749 0.779221i \(-0.284385\pi\)
0.626749 + 0.779221i \(0.284385\pi\)
\(270\) −4.10857 −0.250040
\(271\) −11.0715 −0.672544 −0.336272 0.941765i \(-0.609166\pi\)
−0.336272 + 0.941765i \(0.609166\pi\)
\(272\) 3.29001 0.199486
\(273\) 1.08136 0.0654471
\(274\) 16.2132 0.979473
\(275\) 23.0848 1.39207
\(276\) −4.23927 −0.255174
\(277\) −28.3155 −1.70131 −0.850656 0.525723i \(-0.823795\pi\)
−0.850656 + 0.525723i \(0.823795\pi\)
\(278\) −15.7184 −0.942724
\(279\) −9.76349 −0.584525
\(280\) 4.44286 0.265512
\(281\) 24.4791 1.46030 0.730152 0.683285i \(-0.239449\pi\)
0.730152 + 0.683285i \(0.239449\pi\)
\(282\) 7.53102 0.448466
\(283\) 1.81030 0.107611 0.0538055 0.998551i \(-0.482865\pi\)
0.0538055 + 0.998551i \(0.482865\pi\)
\(284\) −3.70178 −0.219660
\(285\) −22.8700 −1.35470
\(286\) −1.94311 −0.114898
\(287\) 2.95886 0.174656
\(288\) 1.00000 0.0589256
\(289\) −6.17586 −0.363286
\(290\) 23.5777 1.38453
\(291\) 3.56041 0.208715
\(292\) −12.7190 −0.744320
\(293\) −19.4393 −1.13566 −0.567829 0.823146i \(-0.692217\pi\)
−0.567829 + 0.823146i \(0.692217\pi\)
\(294\) 5.83065 0.340051
\(295\) 21.9401 1.27740
\(296\) −0.721621 −0.0419433
\(297\) −1.94311 −0.112751
\(298\) 3.39227 0.196509
\(299\) −4.23927 −0.245163
\(300\) −11.8804 −0.685912
\(301\) −1.08903 −0.0627704
\(302\) −1.86945 −0.107575
\(303\) 6.29949 0.361896
\(304\) 5.56641 0.319255
\(305\) −37.4774 −2.14595
\(306\) 3.29001 0.188077
\(307\) −22.9394 −1.30922 −0.654611 0.755966i \(-0.727168\pi\)
−0.654611 + 0.755966i \(0.727168\pi\)
\(308\) 2.10121 0.119727
\(309\) 1.00000 0.0568880
\(310\) −40.1140 −2.27832
\(311\) 30.3781 1.72258 0.861291 0.508112i \(-0.169656\pi\)
0.861291 + 0.508112i \(0.169656\pi\)
\(312\) 1.00000 0.0566139
\(313\) 9.29840 0.525577 0.262788 0.964853i \(-0.415358\pi\)
0.262788 + 0.964853i \(0.415358\pi\)
\(314\) 2.95582 0.166807
\(315\) 4.44286 0.250327
\(316\) 7.86754 0.442584
\(317\) 27.1474 1.52475 0.762374 0.647137i \(-0.224034\pi\)
0.762374 + 0.647137i \(0.224034\pi\)
\(318\) −6.31473 −0.354112
\(319\) 11.1508 0.624328
\(320\) 4.10857 0.229676
\(321\) −0.737374 −0.0411562
\(322\) 4.58419 0.255467
\(323\) 18.3135 1.01899
\(324\) 1.00000 0.0555556
\(325\) −11.8804 −0.659003
\(326\) −6.68930 −0.370486
\(327\) −7.22763 −0.399689
\(328\) 2.73623 0.151083
\(329\) −8.14378 −0.448981
\(330\) −7.98340 −0.439472
\(331\) −18.1767 −0.999081 −0.499541 0.866290i \(-0.666498\pi\)
−0.499541 + 0.866290i \(0.666498\pi\)
\(332\) −11.0081 −0.604148
\(333\) −0.721621 −0.0395446
\(334\) −5.25050 −0.287295
\(335\) −16.0865 −0.878899
\(336\) −1.08136 −0.0589932
\(337\) −5.52986 −0.301230 −0.150615 0.988592i \(-0.548125\pi\)
−0.150615 + 0.988592i \(0.548125\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0.412595 0.0224091
\(340\) 13.5172 0.733074
\(341\) −18.9715 −1.02737
\(342\) 5.56641 0.300997
\(343\) −13.8746 −0.749158
\(344\) −1.00708 −0.0542984
\(345\) −17.4173 −0.937718
\(346\) −8.82486 −0.474427
\(347\) 28.9834 1.55591 0.777955 0.628320i \(-0.216257\pi\)
0.777955 + 0.628320i \(0.216257\pi\)
\(348\) −5.73866 −0.307625
\(349\) 3.47838 0.186193 0.0930967 0.995657i \(-0.470323\pi\)
0.0930967 + 0.995657i \(0.470323\pi\)
\(350\) 12.8470 0.686700
\(351\) 1.00000 0.0533761
\(352\) 1.94311 0.103568
\(353\) 8.39501 0.446822 0.223411 0.974724i \(-0.428281\pi\)
0.223411 + 0.974724i \(0.428281\pi\)
\(354\) −5.34007 −0.283822
\(355\) −15.2090 −0.807211
\(356\) 3.37812 0.179040
\(357\) −3.55769 −0.188293
\(358\) 1.87031 0.0988487
\(359\) −20.5902 −1.08671 −0.543355 0.839503i \(-0.682846\pi\)
−0.543355 + 0.839503i \(0.682846\pi\)
\(360\) 4.10857 0.216541
\(361\) 11.9849 0.630783
\(362\) −21.6013 −1.13534
\(363\) 7.22432 0.379179
\(364\) −1.08136 −0.0566789
\(365\) −52.2567 −2.73524
\(366\) 9.12177 0.476803
\(367\) −20.1506 −1.05185 −0.525927 0.850530i \(-0.676281\pi\)
−0.525927 + 0.850530i \(0.676281\pi\)
\(368\) 4.23927 0.220987
\(369\) 2.73623 0.142442
\(370\) −2.96483 −0.154134
\(371\) 6.82852 0.354519
\(372\) 9.76349 0.506214
\(373\) 7.45519 0.386015 0.193008 0.981197i \(-0.438176\pi\)
0.193008 + 0.981197i \(0.438176\pi\)
\(374\) 6.39284 0.330566
\(375\) −28.2684 −1.45977
\(376\) −7.53102 −0.388383
\(377\) −5.73866 −0.295556
\(378\) −1.08136 −0.0556194
\(379\) 6.30499 0.323866 0.161933 0.986802i \(-0.448227\pi\)
0.161933 + 0.986802i \(0.448227\pi\)
\(380\) 22.8700 1.17320
\(381\) −8.14381 −0.417220
\(382\) −17.5991 −0.900447
\(383\) −3.64218 −0.186107 −0.0930534 0.995661i \(-0.529663\pi\)
−0.0930534 + 0.995661i \(0.529663\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 8.63297 0.439977
\(386\) −7.66750 −0.390265
\(387\) −1.00708 −0.0511930
\(388\) −3.56041 −0.180753
\(389\) −24.6613 −1.25038 −0.625189 0.780473i \(-0.714978\pi\)
−0.625189 + 0.780473i \(0.714978\pi\)
\(390\) 4.10857 0.208046
\(391\) 13.9472 0.705341
\(392\) −5.83065 −0.294492
\(393\) 1.82445 0.0920313
\(394\) 12.4756 0.628511
\(395\) 32.3244 1.62642
\(396\) 1.94311 0.0976449
\(397\) −18.1777 −0.912314 −0.456157 0.889899i \(-0.650775\pi\)
−0.456157 + 0.889899i \(0.650775\pi\)
\(398\) 3.77199 0.189073
\(399\) −6.01931 −0.301342
\(400\) 11.8804 0.594018
\(401\) −19.3316 −0.965372 −0.482686 0.875793i \(-0.660339\pi\)
−0.482686 + 0.875793i \(0.660339\pi\)
\(402\) 3.91535 0.195280
\(403\) 9.76349 0.486354
\(404\) −6.29949 −0.313411
\(405\) 4.10857 0.204157
\(406\) 6.20558 0.307978
\(407\) −1.40219 −0.0695039
\(408\) −3.29001 −0.162880
\(409\) 30.9524 1.53050 0.765250 0.643734i \(-0.222616\pi\)
0.765250 + 0.643734i \(0.222616\pi\)
\(410\) 11.2420 0.555202
\(411\) −16.2132 −0.799736
\(412\) −1.00000 −0.0492665
\(413\) 5.77456 0.284148
\(414\) 4.23927 0.208349
\(415\) −45.2276 −2.22013
\(416\) −1.00000 −0.0490290
\(417\) 15.7184 0.769731
\(418\) 10.8161 0.529035
\(419\) 19.3432 0.944977 0.472489 0.881337i \(-0.343356\pi\)
0.472489 + 0.881337i \(0.343356\pi\)
\(420\) −4.44286 −0.216789
\(421\) 38.1559 1.85960 0.929801 0.368062i \(-0.119979\pi\)
0.929801 + 0.368062i \(0.119979\pi\)
\(422\) −11.0391 −0.537377
\(423\) −7.53102 −0.366171
\(424\) 6.31473 0.306670
\(425\) 39.0864 1.89597
\(426\) 3.70178 0.179352
\(427\) −9.86396 −0.477350
\(428\) 0.737374 0.0356423
\(429\) 1.94311 0.0938142
\(430\) −4.13768 −0.199537
\(431\) −9.67949 −0.466245 −0.233122 0.972447i \(-0.574894\pi\)
−0.233122 + 0.972447i \(0.574894\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −19.9384 −0.958180 −0.479090 0.877766i \(-0.659033\pi\)
−0.479090 + 0.877766i \(0.659033\pi\)
\(434\) −10.5579 −0.506795
\(435\) −23.5777 −1.13046
\(436\) 7.22763 0.346141
\(437\) 23.5975 1.12882
\(438\) 12.7190 0.607735
\(439\) 28.0392 1.33824 0.669119 0.743155i \(-0.266672\pi\)
0.669119 + 0.743155i \(0.266672\pi\)
\(440\) 7.98340 0.380594
\(441\) −5.83065 −0.277650
\(442\) −3.29001 −0.156490
\(443\) −22.6671 −1.07695 −0.538474 0.842642i \(-0.680999\pi\)
−0.538474 + 0.842642i \(0.680999\pi\)
\(444\) 0.721621 0.0342466
\(445\) 13.8793 0.657940
\(446\) −19.7509 −0.935230
\(447\) −3.39227 −0.160449
\(448\) 1.08136 0.0510896
\(449\) 19.7650 0.932770 0.466385 0.884582i \(-0.345556\pi\)
0.466385 + 0.884582i \(0.345556\pi\)
\(450\) 11.8804 0.560045
\(451\) 5.31679 0.250358
\(452\) −0.412595 −0.0194068
\(453\) 1.86945 0.0878346
\(454\) −15.9220 −0.747254
\(455\) −4.44286 −0.208284
\(456\) −5.56641 −0.260671
\(457\) −29.8725 −1.39738 −0.698688 0.715427i \(-0.746232\pi\)
−0.698688 + 0.715427i \(0.746232\pi\)
\(458\) 14.3924 0.672511
\(459\) −3.29001 −0.153564
\(460\) 17.4173 0.812088
\(461\) 8.02794 0.373899 0.186949 0.982370i \(-0.440140\pi\)
0.186949 + 0.982370i \(0.440140\pi\)
\(462\) −2.10121 −0.0977571
\(463\) 16.1898 0.752404 0.376202 0.926538i \(-0.377230\pi\)
0.376202 + 0.926538i \(0.377230\pi\)
\(464\) 5.73866 0.266411
\(465\) 40.1140 1.86024
\(466\) 21.0680 0.975956
\(467\) 15.1940 0.703095 0.351548 0.936170i \(-0.385656\pi\)
0.351548 + 0.936170i \(0.385656\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −4.23392 −0.195504
\(470\) −30.9417 −1.42724
\(471\) −2.95582 −0.136197
\(472\) 5.34007 0.245797
\(473\) −1.95688 −0.0899773
\(474\) −7.86754 −0.361368
\(475\) 66.1309 3.03429
\(476\) 3.55769 0.163067
\(477\) 6.31473 0.289132
\(478\) 24.6679 1.12828
\(479\) −8.06166 −0.368347 −0.184173 0.982894i \(-0.558961\pi\)
−0.184173 + 0.982894i \(0.558961\pi\)
\(480\) −4.10857 −0.187530
\(481\) 0.721621 0.0329031
\(482\) 12.8439 0.585025
\(483\) −4.58419 −0.208588
\(484\) −7.22432 −0.328378
\(485\) −14.6282 −0.664233
\(486\) −1.00000 −0.0453609
\(487\) 33.4341 1.51504 0.757521 0.652810i \(-0.226410\pi\)
0.757521 + 0.652810i \(0.226410\pi\)
\(488\) −9.12177 −0.412923
\(489\) 6.68930 0.302501
\(490\) −23.9556 −1.08221
\(491\) −8.29680 −0.374429 −0.187215 0.982319i \(-0.559946\pi\)
−0.187215 + 0.982319i \(0.559946\pi\)
\(492\) −2.73623 −0.123359
\(493\) 18.8802 0.850323
\(494\) −5.56641 −0.250444
\(495\) 7.98340 0.358827
\(496\) −9.76349 −0.438394
\(497\) −4.00297 −0.179558
\(498\) 11.0081 0.493285
\(499\) 26.0825 1.16761 0.583807 0.811892i \(-0.301562\pi\)
0.583807 + 0.811892i \(0.301562\pi\)
\(500\) 28.2684 1.26420
\(501\) 5.25050 0.234575
\(502\) −24.4841 −1.09278
\(503\) −34.7353 −1.54877 −0.774385 0.632715i \(-0.781941\pi\)
−0.774385 + 0.632715i \(0.781941\pi\)
\(504\) 1.08136 0.0481678
\(505\) −25.8819 −1.15173
\(506\) 8.23737 0.366196
\(507\) −1.00000 −0.0444116
\(508\) 8.14381 0.361323
\(509\) 24.7532 1.09717 0.548584 0.836096i \(-0.315167\pi\)
0.548584 + 0.836096i \(0.315167\pi\)
\(510\) −13.5172 −0.598553
\(511\) −13.7538 −0.608433
\(512\) 1.00000 0.0441942
\(513\) −5.56641 −0.245763
\(514\) 3.27567 0.144483
\(515\) −4.10857 −0.181045
\(516\) 1.00708 0.0443344
\(517\) −14.6336 −0.643585
\(518\) −0.780335 −0.0342859
\(519\) 8.82486 0.387368
\(520\) −4.10857 −0.180173
\(521\) 35.0415 1.53519 0.767597 0.640933i \(-0.221452\pi\)
0.767597 + 0.640933i \(0.221452\pi\)
\(522\) 5.73866 0.251174
\(523\) 11.4032 0.498627 0.249314 0.968423i \(-0.419795\pi\)
0.249314 + 0.968423i \(0.419795\pi\)
\(524\) −1.82445 −0.0797014
\(525\) −12.8470 −0.560688
\(526\) 8.28156 0.361093
\(527\) −32.1219 −1.39925
\(528\) −1.94311 −0.0845630
\(529\) −5.02859 −0.218634
\(530\) 25.9445 1.12696
\(531\) 5.34007 0.231739
\(532\) 6.01931 0.260970
\(533\) −2.73623 −0.118519
\(534\) −3.37812 −0.146186
\(535\) 3.02955 0.130979
\(536\) −3.91535 −0.169117
\(537\) −1.87031 −0.0807096
\(538\) 20.5589 0.886357
\(539\) −11.3296 −0.488000
\(540\) −4.10857 −0.176805
\(541\) −28.2165 −1.21312 −0.606562 0.795036i \(-0.707452\pi\)
−0.606562 + 0.795036i \(0.707452\pi\)
\(542\) −11.0715 −0.475561
\(543\) 21.6013 0.927000
\(544\) 3.29001 0.141058
\(545\) 29.6952 1.27200
\(546\) 1.08136 0.0462781
\(547\) 37.0522 1.58424 0.792118 0.610368i \(-0.208978\pi\)
0.792118 + 0.610368i \(0.208978\pi\)
\(548\) 16.2132 0.692592
\(549\) −9.12177 −0.389308
\(550\) 23.0848 0.984340
\(551\) 31.9437 1.36085
\(552\) −4.23927 −0.180435
\(553\) 8.50768 0.361783
\(554\) −28.3155 −1.20301
\(555\) 2.96483 0.125850
\(556\) −15.7184 −0.666607
\(557\) 1.47666 0.0625682 0.0312841 0.999511i \(-0.490040\pi\)
0.0312841 + 0.999511i \(0.490040\pi\)
\(558\) −9.76349 −0.413322
\(559\) 1.00708 0.0425951
\(560\) 4.44286 0.187745
\(561\) −6.39284 −0.269906
\(562\) 24.4791 1.03259
\(563\) 34.5977 1.45812 0.729059 0.684451i \(-0.239958\pi\)
0.729059 + 0.684451i \(0.239958\pi\)
\(564\) 7.53102 0.317113
\(565\) −1.69518 −0.0713166
\(566\) 1.81030 0.0760924
\(567\) 1.08136 0.0454130
\(568\) −3.70178 −0.155323
\(569\) −5.04434 −0.211470 −0.105735 0.994394i \(-0.533719\pi\)
−0.105735 + 0.994394i \(0.533719\pi\)
\(570\) −22.8700 −0.957918
\(571\) −8.80528 −0.368489 −0.184245 0.982880i \(-0.558984\pi\)
−0.184245 + 0.982880i \(0.558984\pi\)
\(572\) −1.94311 −0.0812455
\(573\) 17.5991 0.735212
\(574\) 2.95886 0.123500
\(575\) 50.3640 2.10032
\(576\) 1.00000 0.0416667
\(577\) 29.0192 1.20809 0.604044 0.796951i \(-0.293555\pi\)
0.604044 + 0.796951i \(0.293555\pi\)
\(578\) −6.17586 −0.256882
\(579\) 7.66750 0.318650
\(580\) 23.5777 0.979010
\(581\) −11.9038 −0.493852
\(582\) 3.56041 0.147584
\(583\) 12.2702 0.508180
\(584\) −12.7190 −0.526314
\(585\) −4.10857 −0.169868
\(586\) −19.4393 −0.803032
\(587\) 12.3121 0.508174 0.254087 0.967181i \(-0.418225\pi\)
0.254087 + 0.967181i \(0.418225\pi\)
\(588\) 5.83065 0.240452
\(589\) −54.3476 −2.23935
\(590\) 21.9401 0.903258
\(591\) −12.4756 −0.513177
\(592\) −0.721621 −0.0296584
\(593\) 11.5334 0.473621 0.236811 0.971556i \(-0.423898\pi\)
0.236811 + 0.971556i \(0.423898\pi\)
\(594\) −1.94311 −0.0797268
\(595\) 14.6170 0.599240
\(596\) 3.39227 0.138953
\(597\) −3.77199 −0.154377
\(598\) −4.23927 −0.173357
\(599\) −16.5041 −0.674341 −0.337170 0.941444i \(-0.609470\pi\)
−0.337170 + 0.941444i \(0.609470\pi\)
\(600\) −11.8804 −0.485013
\(601\) −29.2661 −1.19379 −0.596894 0.802320i \(-0.703599\pi\)
−0.596894 + 0.802320i \(0.703599\pi\)
\(602\) −1.08903 −0.0443854
\(603\) −3.91535 −0.159445
\(604\) −1.86945 −0.0760670
\(605\) −29.6816 −1.20673
\(606\) 6.29949 0.255899
\(607\) −26.7272 −1.08482 −0.542412 0.840112i \(-0.682489\pi\)
−0.542412 + 0.840112i \(0.682489\pi\)
\(608\) 5.56641 0.225748
\(609\) −6.20558 −0.251463
\(610\) −37.4774 −1.51742
\(611\) 7.53102 0.304673
\(612\) 3.29001 0.132991
\(613\) 8.94000 0.361083 0.180542 0.983567i \(-0.442215\pi\)
0.180542 + 0.983567i \(0.442215\pi\)
\(614\) −22.9394 −0.925760
\(615\) −11.2420 −0.453320
\(616\) 2.10121 0.0846601
\(617\) −16.2807 −0.655435 −0.327718 0.944776i \(-0.606279\pi\)
−0.327718 + 0.944776i \(0.606279\pi\)
\(618\) 1.00000 0.0402259
\(619\) −11.9378 −0.479819 −0.239910 0.970795i \(-0.577118\pi\)
−0.239910 + 0.970795i \(0.577118\pi\)
\(620\) −40.1140 −1.61102
\(621\) −4.23927 −0.170116
\(622\) 30.3781 1.21805
\(623\) 3.65298 0.146354
\(624\) 1.00000 0.0400320
\(625\) 56.7410 2.26964
\(626\) 9.29840 0.371639
\(627\) −10.8161 −0.431955
\(628\) 2.95582 0.117950
\(629\) −2.37414 −0.0946630
\(630\) 4.44286 0.177008
\(631\) 34.0937 1.35725 0.678624 0.734486i \(-0.262577\pi\)
0.678624 + 0.734486i \(0.262577\pi\)
\(632\) 7.86754 0.312954
\(633\) 11.0391 0.438767
\(634\) 27.1474 1.07816
\(635\) 33.4594 1.32780
\(636\) −6.31473 −0.250395
\(637\) 5.83065 0.231019
\(638\) 11.1508 0.441466
\(639\) −3.70178 −0.146440
\(640\) 4.10857 0.162406
\(641\) 48.1232 1.90075 0.950377 0.311102i \(-0.100698\pi\)
0.950377 + 0.311102i \(0.100698\pi\)
\(642\) −0.737374 −0.0291018
\(643\) 2.07504 0.0818315 0.0409157 0.999163i \(-0.486972\pi\)
0.0409157 + 0.999163i \(0.486972\pi\)
\(644\) 4.58419 0.180643
\(645\) 4.13768 0.162921
\(646\) 18.3135 0.720535
\(647\) −17.5737 −0.690894 −0.345447 0.938438i \(-0.612273\pi\)
−0.345447 + 0.938438i \(0.612273\pi\)
\(648\) 1.00000 0.0392837
\(649\) 10.3763 0.407307
\(650\) −11.8804 −0.465986
\(651\) 10.5579 0.413796
\(652\) −6.68930 −0.261973
\(653\) 41.4486 1.62201 0.811006 0.585039i \(-0.198921\pi\)
0.811006 + 0.585039i \(0.198921\pi\)
\(654\) −7.22763 −0.282623
\(655\) −7.49588 −0.292888
\(656\) 2.73623 0.106832
\(657\) −12.7190 −0.496214
\(658\) −8.14378 −0.317478
\(659\) −15.7497 −0.613520 −0.306760 0.951787i \(-0.599245\pi\)
−0.306760 + 0.951787i \(0.599245\pi\)
\(660\) −7.98340 −0.310754
\(661\) 17.7941 0.692109 0.346054 0.938214i \(-0.387521\pi\)
0.346054 + 0.938214i \(0.387521\pi\)
\(662\) −18.1767 −0.706457
\(663\) 3.29001 0.127773
\(664\) −11.0081 −0.427197
\(665\) 24.7308 0.959018
\(666\) −0.721621 −0.0279622
\(667\) 24.3277 0.941974
\(668\) −5.25050 −0.203148
\(669\) 19.7509 0.763612
\(670\) −16.0865 −0.621475
\(671\) −17.7246 −0.684251
\(672\) −1.08136 −0.0417145
\(673\) −4.75654 −0.183351 −0.0916757 0.995789i \(-0.529222\pi\)
−0.0916757 + 0.995789i \(0.529222\pi\)
\(674\) −5.52986 −0.213002
\(675\) −11.8804 −0.457275
\(676\) 1.00000 0.0384615
\(677\) 7.37777 0.283551 0.141775 0.989899i \(-0.454719\pi\)
0.141775 + 0.989899i \(0.454719\pi\)
\(678\) 0.412595 0.0158456
\(679\) −3.85010 −0.147753
\(680\) 13.5172 0.518362
\(681\) 15.9220 0.610131
\(682\) −18.9715 −0.726458
\(683\) −13.5732 −0.519365 −0.259683 0.965694i \(-0.583618\pi\)
−0.259683 + 0.965694i \(0.583618\pi\)
\(684\) 5.56641 0.212837
\(685\) 66.6129 2.54515
\(686\) −13.8746 −0.529735
\(687\) −14.3924 −0.549103
\(688\) −1.00708 −0.0383947
\(689\) −6.31473 −0.240572
\(690\) −17.4173 −0.663067
\(691\) −9.36218 −0.356154 −0.178077 0.984017i \(-0.556988\pi\)
−0.178077 + 0.984017i \(0.556988\pi\)
\(692\) −8.82486 −0.335471
\(693\) 2.10121 0.0798183
\(694\) 28.9834 1.10019
\(695\) −64.5800 −2.44966
\(696\) −5.73866 −0.217523
\(697\) 9.00220 0.340983
\(698\) 3.47838 0.131659
\(699\) −21.0680 −0.796865
\(700\) 12.8470 0.485570
\(701\) −19.9820 −0.754711 −0.377356 0.926068i \(-0.623167\pi\)
−0.377356 + 0.926068i \(0.623167\pi\)
\(702\) 1.00000 0.0377426
\(703\) −4.01683 −0.151498
\(704\) 1.94311 0.0732337
\(705\) 30.9417 1.16533
\(706\) 8.39501 0.315951
\(707\) −6.81204 −0.256193
\(708\) −5.34007 −0.200692
\(709\) −17.8979 −0.672171 −0.336086 0.941831i \(-0.609103\pi\)
−0.336086 + 0.941831i \(0.609103\pi\)
\(710\) −15.2090 −0.570784
\(711\) 7.86754 0.295056
\(712\) 3.37812 0.126601
\(713\) −41.3901 −1.55007
\(714\) −3.55769 −0.133143
\(715\) −7.98340 −0.298562
\(716\) 1.87031 0.0698966
\(717\) −24.6679 −0.921239
\(718\) −20.5902 −0.768420
\(719\) 4.97052 0.185369 0.0926845 0.995696i \(-0.470455\pi\)
0.0926845 + 0.995696i \(0.470455\pi\)
\(720\) 4.10857 0.153117
\(721\) −1.08136 −0.0402721
\(722\) 11.9849 0.446031
\(723\) −12.8439 −0.477671
\(724\) −21.6013 −0.802806
\(725\) 68.1773 2.53204
\(726\) 7.22432 0.268120
\(727\) −19.5779 −0.726103 −0.363052 0.931769i \(-0.618265\pi\)
−0.363052 + 0.931769i \(0.618265\pi\)
\(728\) −1.08136 −0.0400780
\(729\) 1.00000 0.0370370
\(730\) −52.2567 −1.93411
\(731\) −3.31331 −0.122547
\(732\) 9.12177 0.337150
\(733\) 43.6858 1.61357 0.806786 0.590843i \(-0.201205\pi\)
0.806786 + 0.590843i \(0.201205\pi\)
\(734\) −20.1506 −0.743773
\(735\) 23.9556 0.883617
\(736\) 4.23927 0.156262
\(737\) −7.60795 −0.280243
\(738\) 2.73623 0.100722
\(739\) 18.0459 0.663829 0.331914 0.943309i \(-0.392305\pi\)
0.331914 + 0.943309i \(0.392305\pi\)
\(740\) −2.96483 −0.108989
\(741\) 5.56641 0.204487
\(742\) 6.82852 0.250683
\(743\) 1.11335 0.0408449 0.0204224 0.999791i \(-0.493499\pi\)
0.0204224 + 0.999791i \(0.493499\pi\)
\(744\) 9.76349 0.357947
\(745\) 13.9374 0.510626
\(746\) 7.45519 0.272954
\(747\) −11.0081 −0.402765
\(748\) 6.39284 0.233745
\(749\) 0.797369 0.0291352
\(750\) −28.2684 −1.03222
\(751\) 14.9853 0.546820 0.273410 0.961898i \(-0.411848\pi\)
0.273410 + 0.961898i \(0.411848\pi\)
\(752\) −7.53102 −0.274628
\(753\) 24.4841 0.892252
\(754\) −5.73866 −0.208990
\(755\) −7.68078 −0.279532
\(756\) −1.08136 −0.0393288
\(757\) 33.5533 1.21952 0.609758 0.792587i \(-0.291266\pi\)
0.609758 + 0.792587i \(0.291266\pi\)
\(758\) 6.30499 0.229008
\(759\) −8.23737 −0.298997
\(760\) 22.8700 0.829581
\(761\) 8.51222 0.308568 0.154284 0.988027i \(-0.450693\pi\)
0.154284 + 0.988027i \(0.450693\pi\)
\(762\) −8.14381 −0.295019
\(763\) 7.81570 0.282947
\(764\) −17.5991 −0.636712
\(765\) 13.5172 0.488716
\(766\) −3.64218 −0.131597
\(767\) −5.34007 −0.192819
\(768\) −1.00000 −0.0360844
\(769\) −33.8906 −1.22213 −0.611063 0.791582i \(-0.709258\pi\)
−0.611063 + 0.791582i \(0.709258\pi\)
\(770\) 8.63297 0.311110
\(771\) −3.27567 −0.117970
\(772\) −7.66750 −0.275959
\(773\) −54.3237 −1.95389 −0.976943 0.213499i \(-0.931514\pi\)
−0.976943 + 0.213499i \(0.931514\pi\)
\(774\) −1.00708 −0.0361989
\(775\) −115.994 −4.16662
\(776\) −3.56041 −0.127811
\(777\) 0.780335 0.0279943
\(778\) −24.6613 −0.884151
\(779\) 15.2309 0.545705
\(780\) 4.10857 0.147110
\(781\) −7.19296 −0.257384
\(782\) 13.9472 0.498752
\(783\) −5.73866 −0.205083
\(784\) −5.83065 −0.208238
\(785\) 12.1442 0.433445
\(786\) 1.82445 0.0650760
\(787\) 46.3832 1.65338 0.826691 0.562656i \(-0.190220\pi\)
0.826691 + 0.562656i \(0.190220\pi\)
\(788\) 12.4756 0.444424
\(789\) −8.28156 −0.294831
\(790\) 32.3244 1.15005
\(791\) −0.446166 −0.0158638
\(792\) 1.94311 0.0690454
\(793\) 9.12177 0.323924
\(794\) −18.1777 −0.645104
\(795\) −25.9445 −0.920157
\(796\) 3.77199 0.133695
\(797\) −34.1125 −1.20833 −0.604163 0.796861i \(-0.706492\pi\)
−0.604163 + 0.796861i \(0.706492\pi\)
\(798\) −6.01931 −0.213081
\(799\) −24.7771 −0.876551
\(800\) 11.8804 0.420034
\(801\) 3.37812 0.119360
\(802\) −19.3316 −0.682621
\(803\) −24.7143 −0.872149
\(804\) 3.91535 0.138084
\(805\) 18.8345 0.663828
\(806\) 9.76349 0.343904
\(807\) −20.5589 −0.723708
\(808\) −6.29949 −0.221615
\(809\) 22.7610 0.800235 0.400118 0.916464i \(-0.368969\pi\)
0.400118 + 0.916464i \(0.368969\pi\)
\(810\) 4.10857 0.144360
\(811\) 10.7541 0.377626 0.188813 0.982013i \(-0.439536\pi\)
0.188813 + 0.982013i \(0.439536\pi\)
\(812\) 6.20558 0.217773
\(813\) 11.0715 0.388294
\(814\) −1.40219 −0.0491467
\(815\) −27.4835 −0.962704
\(816\) −3.29001 −0.115173
\(817\) −5.60584 −0.196124
\(818\) 30.9524 1.08223
\(819\) −1.08136 −0.0377859
\(820\) 11.2420 0.392587
\(821\) −2.79374 −0.0975023 −0.0487511 0.998811i \(-0.515524\pi\)
−0.0487511 + 0.998811i \(0.515524\pi\)
\(822\) −16.2132 −0.565499
\(823\) −27.1977 −0.948053 −0.474026 0.880511i \(-0.657200\pi\)
−0.474026 + 0.880511i \(0.657200\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −23.0848 −0.803710
\(826\) 5.77456 0.200923
\(827\) 8.39731 0.292003 0.146002 0.989284i \(-0.453360\pi\)
0.146002 + 0.989284i \(0.453360\pi\)
\(828\) 4.23927 0.147325
\(829\) −39.2512 −1.36325 −0.681626 0.731701i \(-0.738727\pi\)
−0.681626 + 0.731701i \(0.738727\pi\)
\(830\) −45.2276 −1.56987
\(831\) 28.3155 0.982253
\(832\) −1.00000 −0.0346688
\(833\) −19.1829 −0.664647
\(834\) 15.7184 0.544282
\(835\) −21.5720 −0.746531
\(836\) 10.8161 0.374084
\(837\) 9.76349 0.337476
\(838\) 19.3432 0.668200
\(839\) −28.4064 −0.980697 −0.490348 0.871526i \(-0.663130\pi\)
−0.490348 + 0.871526i \(0.663130\pi\)
\(840\) −4.44286 −0.153293
\(841\) 3.93223 0.135594
\(842\) 38.1559 1.31494
\(843\) −24.4791 −0.843106
\(844\) −11.0391 −0.379983
\(845\) 4.10857 0.141339
\(846\) −7.53102 −0.258922
\(847\) −7.81212 −0.268428
\(848\) 6.31473 0.216849
\(849\) −1.81030 −0.0621292
\(850\) 39.0864 1.34065
\(851\) −3.05914 −0.104866
\(852\) 3.70178 0.126821
\(853\) −9.60722 −0.328945 −0.164472 0.986382i \(-0.552592\pi\)
−0.164472 + 0.986382i \(0.552592\pi\)
\(854\) −9.86396 −0.337538
\(855\) 22.8700 0.782137
\(856\) 0.737374 0.0252029
\(857\) 27.9151 0.953562 0.476781 0.879022i \(-0.341803\pi\)
0.476781 + 0.879022i \(0.341803\pi\)
\(858\) 1.94311 0.0663367
\(859\) −27.7473 −0.946727 −0.473363 0.880867i \(-0.656960\pi\)
−0.473363 + 0.880867i \(0.656960\pi\)
\(860\) −4.13768 −0.141094
\(861\) −2.95886 −0.100838
\(862\) −9.67949 −0.329685
\(863\) −32.8331 −1.11765 −0.558825 0.829285i \(-0.688748\pi\)
−0.558825 + 0.829285i \(0.688748\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −36.2575 −1.23279
\(866\) −19.9384 −0.677536
\(867\) 6.17586 0.209743
\(868\) −10.5579 −0.358358
\(869\) 15.2875 0.518593
\(870\) −23.5777 −0.799359
\(871\) 3.91535 0.132667
\(872\) 7.22763 0.244759
\(873\) −3.56041 −0.120502
\(874\) 23.5975 0.798197
\(875\) 30.5684 1.03340
\(876\) 12.7190 0.429734
\(877\) −9.31727 −0.314622 −0.157311 0.987549i \(-0.550283\pi\)
−0.157311 + 0.987549i \(0.550283\pi\)
\(878\) 28.0392 0.946277
\(879\) 19.4393 0.655673
\(880\) 7.98340 0.269120
\(881\) 8.29518 0.279472 0.139736 0.990189i \(-0.455375\pi\)
0.139736 + 0.990189i \(0.455375\pi\)
\(882\) −5.83065 −0.196328
\(883\) −16.6831 −0.561432 −0.280716 0.959791i \(-0.590572\pi\)
−0.280716 + 0.959791i \(0.590572\pi\)
\(884\) −3.29001 −0.110655
\(885\) −21.9401 −0.737507
\(886\) −22.6671 −0.761517
\(887\) −37.9842 −1.27539 −0.637693 0.770291i \(-0.720111\pi\)
−0.637693 + 0.770291i \(0.720111\pi\)
\(888\) 0.721621 0.0242160
\(889\) 8.80643 0.295358
\(890\) 13.8793 0.465234
\(891\) 1.94311 0.0650966
\(892\) −19.7509 −0.661308
\(893\) −41.9207 −1.40282
\(894\) −3.39227 −0.113454
\(895\) 7.68428 0.256857
\(896\) 1.08136 0.0361258
\(897\) 4.23927 0.141545
\(898\) 19.7650 0.659568
\(899\) −56.0294 −1.86868
\(900\) 11.8804 0.396012
\(901\) 20.7755 0.692132
\(902\) 5.31679 0.177030
\(903\) 1.08903 0.0362405
\(904\) −0.412595 −0.0137227
\(905\) −88.7504 −2.95016
\(906\) 1.86945 0.0621084
\(907\) 11.2104 0.372235 0.186117 0.982527i \(-0.440409\pi\)
0.186117 + 0.982527i \(0.440409\pi\)
\(908\) −15.9220 −0.528389
\(909\) −6.29949 −0.208941
\(910\) −4.44286 −0.147279
\(911\) −10.5819 −0.350595 −0.175297 0.984516i \(-0.556089\pi\)
−0.175297 + 0.984516i \(0.556089\pi\)
\(912\) −5.56641 −0.184322
\(913\) −21.3900 −0.707904
\(914\) −29.8725 −0.988094
\(915\) 37.4774 1.23897
\(916\) 14.3924 0.475537
\(917\) −1.97289 −0.0651507
\(918\) −3.29001 −0.108586
\(919\) 0.249990 0.00824640 0.00412320 0.999991i \(-0.498688\pi\)
0.00412320 + 0.999991i \(0.498688\pi\)
\(920\) 17.4173 0.574233
\(921\) 22.9394 0.755880
\(922\) 8.02794 0.264386
\(923\) 3.70178 0.121846
\(924\) −2.10121 −0.0691247
\(925\) −8.57311 −0.281882
\(926\) 16.1898 0.532030
\(927\) −1.00000 −0.0328443
\(928\) 5.73866 0.188381
\(929\) −8.91055 −0.292346 −0.146173 0.989259i \(-0.546696\pi\)
−0.146173 + 0.989259i \(0.546696\pi\)
\(930\) 40.1140 1.31539
\(931\) −32.4558 −1.06370
\(932\) 21.0680 0.690105
\(933\) −30.3781 −0.994533
\(934\) 15.1940 0.497163
\(935\) 26.2654 0.858972
\(936\) −1.00000 −0.0326860
\(937\) 50.1970 1.63986 0.819932 0.572462i \(-0.194011\pi\)
0.819932 + 0.572462i \(0.194011\pi\)
\(938\) −4.23392 −0.138242
\(939\) −9.29840 −0.303442
\(940\) −30.9417 −1.00921
\(941\) −21.3817 −0.697025 −0.348512 0.937304i \(-0.613313\pi\)
−0.348512 + 0.937304i \(0.613313\pi\)
\(942\) −2.95582 −0.0963059
\(943\) 11.5996 0.377735
\(944\) 5.34007 0.173805
\(945\) −4.44286 −0.144526
\(946\) −1.95688 −0.0636235
\(947\) −13.4273 −0.436328 −0.218164 0.975912i \(-0.570007\pi\)
−0.218164 + 0.975912i \(0.570007\pi\)
\(948\) −7.86754 −0.255526
\(949\) 12.7190 0.412875
\(950\) 66.1309 2.14557
\(951\) −27.1474 −0.880313
\(952\) 3.55769 0.115306
\(953\) 25.9491 0.840574 0.420287 0.907391i \(-0.361929\pi\)
0.420287 + 0.907391i \(0.361929\pi\)
\(954\) 6.31473 0.204447
\(955\) −72.3070 −2.33980
\(956\) 24.6679 0.797816
\(957\) −11.1508 −0.360456
\(958\) −8.06166 −0.260460
\(959\) 17.5323 0.566148
\(960\) −4.10857 −0.132604
\(961\) 64.3258 2.07503
\(962\) 0.721621 0.0232660
\(963\) 0.737374 0.0237615
\(964\) 12.8439 0.413675
\(965\) −31.5025 −1.01410
\(966\) −4.58419 −0.147494
\(967\) 51.5075 1.65637 0.828184 0.560456i \(-0.189374\pi\)
0.828184 + 0.560456i \(0.189374\pi\)
\(968\) −7.22432 −0.232199
\(969\) −18.3135 −0.588315
\(970\) −14.6282 −0.469683
\(971\) 5.76955 0.185154 0.0925769 0.995706i \(-0.470490\pi\)
0.0925769 + 0.995706i \(0.470490\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −16.9973 −0.544907
\(974\) 33.4341 1.07130
\(975\) 11.8804 0.380476
\(976\) −9.12177 −0.291981
\(977\) −30.2599 −0.968099 −0.484050 0.875040i \(-0.660835\pi\)
−0.484050 + 0.875040i \(0.660835\pi\)
\(978\) 6.68930 0.213900
\(979\) 6.56407 0.209788
\(980\) −23.9556 −0.765235
\(981\) 7.22763 0.230761
\(982\) −8.29680 −0.264762
\(983\) −6.72633 −0.214537 −0.107268 0.994230i \(-0.534210\pi\)
−0.107268 + 0.994230i \(0.534210\pi\)
\(984\) −2.73623 −0.0872277
\(985\) 51.2568 1.63318
\(986\) 18.8802 0.601269
\(987\) 8.14378 0.259219
\(988\) −5.56641 −0.177091
\(989\) −4.26930 −0.135756
\(990\) 7.98340 0.253729
\(991\) −15.4282 −0.490094 −0.245047 0.969511i \(-0.578803\pi\)
−0.245047 + 0.969511i \(0.578803\pi\)
\(992\) −9.76349 −0.309991
\(993\) 18.1767 0.576820
\(994\) −4.00297 −0.126967
\(995\) 15.4975 0.491303
\(996\) 11.0081 0.348805
\(997\) −22.4185 −0.710002 −0.355001 0.934866i \(-0.615520\pi\)
−0.355001 + 0.934866i \(0.615520\pi\)
\(998\) 26.0825 0.825628
\(999\) 0.721621 0.0228311
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.1.15 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.15 15 1.1 even 1 trivial