Properties

Label 8034.2.a.bb.1.7
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.453790\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.453790 q^{5} -1.00000 q^{6} +3.87720 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.453790 q^{5} -1.00000 q^{6} +3.87720 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.453790 q^{10} -1.29295 q^{11} +1.00000 q^{12} -1.00000 q^{13} -3.87720 q^{14} -0.453790 q^{15} +1.00000 q^{16} -0.204049 q^{17} -1.00000 q^{18} -0.475803 q^{19} -0.453790 q^{20} +3.87720 q^{21} +1.29295 q^{22} +1.84848 q^{23} -1.00000 q^{24} -4.79407 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.87720 q^{28} -5.66720 q^{29} +0.453790 q^{30} -7.72539 q^{31} -1.00000 q^{32} -1.29295 q^{33} +0.204049 q^{34} -1.75943 q^{35} +1.00000 q^{36} +9.68872 q^{37} +0.475803 q^{38} -1.00000 q^{39} +0.453790 q^{40} -7.38884 q^{41} -3.87720 q^{42} -4.47310 q^{43} -1.29295 q^{44} -0.453790 q^{45} -1.84848 q^{46} -3.11558 q^{47} +1.00000 q^{48} +8.03267 q^{49} +4.79407 q^{50} -0.204049 q^{51} -1.00000 q^{52} -2.30074 q^{53} -1.00000 q^{54} +0.586726 q^{55} -3.87720 q^{56} -0.475803 q^{57} +5.66720 q^{58} -13.6380 q^{59} -0.453790 q^{60} -11.4248 q^{61} +7.72539 q^{62} +3.87720 q^{63} +1.00000 q^{64} +0.453790 q^{65} +1.29295 q^{66} +0.678171 q^{67} -0.204049 q^{68} +1.84848 q^{69} +1.75943 q^{70} +5.87504 q^{71} -1.00000 q^{72} -3.56157 q^{73} -9.68872 q^{74} -4.79407 q^{75} -0.475803 q^{76} -5.01301 q^{77} +1.00000 q^{78} +0.0379441 q^{79} -0.453790 q^{80} +1.00000 q^{81} +7.38884 q^{82} -7.81001 q^{83} +3.87720 q^{84} +0.0925952 q^{85} +4.47310 q^{86} -5.66720 q^{87} +1.29295 q^{88} +8.08601 q^{89} +0.453790 q^{90} -3.87720 q^{91} +1.84848 q^{92} -7.72539 q^{93} +3.11558 q^{94} +0.215915 q^{95} -1.00000 q^{96} -2.30097 q^{97} -8.03267 q^{98} -1.29295 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −0.453790 −0.202941 −0.101471 0.994839i \(-0.532355\pi\)
−0.101471 + 0.994839i \(0.532355\pi\)
\(6\) −1.00000 −0.408248
\(7\) 3.87720 1.46544 0.732722 0.680528i \(-0.238250\pi\)
0.732722 + 0.680528i \(0.238250\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0.453790 0.143501
\(11\) −1.29295 −0.389838 −0.194919 0.980819i \(-0.562444\pi\)
−0.194919 + 0.980819i \(0.562444\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) −3.87720 −1.03622
\(15\) −0.453790 −0.117168
\(16\) 1.00000 0.250000
\(17\) −0.204049 −0.0494891 −0.0247445 0.999694i \(-0.507877\pi\)
−0.0247445 + 0.999694i \(0.507877\pi\)
\(18\) −1.00000 −0.235702
\(19\) −0.475803 −0.109157 −0.0545784 0.998509i \(-0.517381\pi\)
−0.0545784 + 0.998509i \(0.517381\pi\)
\(20\) −0.453790 −0.101471
\(21\) 3.87720 0.846074
\(22\) 1.29295 0.275657
\(23\) 1.84848 0.385434 0.192717 0.981254i \(-0.438270\pi\)
0.192717 + 0.981254i \(0.438270\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.79407 −0.958815
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 3.87720 0.732722
\(29\) −5.66720 −1.05237 −0.526187 0.850369i \(-0.676379\pi\)
−0.526187 + 0.850369i \(0.676379\pi\)
\(30\) 0.453790 0.0828503
\(31\) −7.72539 −1.38752 −0.693760 0.720206i \(-0.744047\pi\)
−0.693760 + 0.720206i \(0.744047\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.29295 −0.225073
\(34\) 0.204049 0.0349941
\(35\) −1.75943 −0.297399
\(36\) 1.00000 0.166667
\(37\) 9.68872 1.59282 0.796408 0.604760i \(-0.206731\pi\)
0.796408 + 0.604760i \(0.206731\pi\)
\(38\) 0.475803 0.0771855
\(39\) −1.00000 −0.160128
\(40\) 0.453790 0.0717505
\(41\) −7.38884 −1.15394 −0.576972 0.816764i \(-0.695766\pi\)
−0.576972 + 0.816764i \(0.695766\pi\)
\(42\) −3.87720 −0.598265
\(43\) −4.47310 −0.682142 −0.341071 0.940038i \(-0.610790\pi\)
−0.341071 + 0.940038i \(0.610790\pi\)
\(44\) −1.29295 −0.194919
\(45\) −0.453790 −0.0676470
\(46\) −1.84848 −0.272543
\(47\) −3.11558 −0.454455 −0.227227 0.973842i \(-0.572966\pi\)
−0.227227 + 0.973842i \(0.572966\pi\)
\(48\) 1.00000 0.144338
\(49\) 8.03267 1.14752
\(50\) 4.79407 0.677985
\(51\) −0.204049 −0.0285725
\(52\) −1.00000 −0.138675
\(53\) −2.30074 −0.316032 −0.158016 0.987437i \(-0.550510\pi\)
−0.158016 + 0.987437i \(0.550510\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0.586726 0.0791142
\(56\) −3.87720 −0.518112
\(57\) −0.475803 −0.0630217
\(58\) 5.66720 0.744140
\(59\) −13.6380 −1.77552 −0.887761 0.460305i \(-0.847740\pi\)
−0.887761 + 0.460305i \(0.847740\pi\)
\(60\) −0.453790 −0.0585840
\(61\) −11.4248 −1.46280 −0.731400 0.681949i \(-0.761133\pi\)
−0.731400 + 0.681949i \(0.761133\pi\)
\(62\) 7.72539 0.981125
\(63\) 3.87720 0.488481
\(64\) 1.00000 0.125000
\(65\) 0.453790 0.0562857
\(66\) 1.29295 0.159151
\(67\) 0.678171 0.0828518 0.0414259 0.999142i \(-0.486810\pi\)
0.0414259 + 0.999142i \(0.486810\pi\)
\(68\) −0.204049 −0.0247445
\(69\) 1.84848 0.222530
\(70\) 1.75943 0.210293
\(71\) 5.87504 0.697239 0.348620 0.937264i \(-0.386650\pi\)
0.348620 + 0.937264i \(0.386650\pi\)
\(72\) −1.00000 −0.117851
\(73\) −3.56157 −0.416850 −0.208425 0.978038i \(-0.566834\pi\)
−0.208425 + 0.978038i \(0.566834\pi\)
\(74\) −9.68872 −1.12629
\(75\) −4.79407 −0.553572
\(76\) −0.475803 −0.0545784
\(77\) −5.01301 −0.571286
\(78\) 1.00000 0.113228
\(79\) 0.0379441 0.00426905 0.00213452 0.999998i \(-0.499321\pi\)
0.00213452 + 0.999998i \(0.499321\pi\)
\(80\) −0.453790 −0.0507353
\(81\) 1.00000 0.111111
\(82\) 7.38884 0.815961
\(83\) −7.81001 −0.857260 −0.428630 0.903480i \(-0.641004\pi\)
−0.428630 + 0.903480i \(0.641004\pi\)
\(84\) 3.87720 0.423037
\(85\) 0.0925952 0.0100434
\(86\) 4.47310 0.482347
\(87\) −5.66720 −0.607588
\(88\) 1.29295 0.137829
\(89\) 8.08601 0.857115 0.428558 0.903514i \(-0.359022\pi\)
0.428558 + 0.903514i \(0.359022\pi\)
\(90\) 0.453790 0.0478337
\(91\) −3.87720 −0.406441
\(92\) 1.84848 0.192717
\(93\) −7.72539 −0.801085
\(94\) 3.11558 0.321348
\(95\) 0.215915 0.0221524
\(96\) −1.00000 −0.102062
\(97\) −2.30097 −0.233628 −0.116814 0.993154i \(-0.537268\pi\)
−0.116814 + 0.993154i \(0.537268\pi\)
\(98\) −8.03267 −0.811422
\(99\) −1.29295 −0.129946
\(100\) −4.79407 −0.479407
\(101\) −3.28191 −0.326562 −0.163281 0.986580i \(-0.552208\pi\)
−0.163281 + 0.986580i \(0.552208\pi\)
\(102\) 0.204049 0.0202038
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) −1.75943 −0.171703
\(106\) 2.30074 0.223468
\(107\) −13.4166 −1.29703 −0.648515 0.761202i \(-0.724610\pi\)
−0.648515 + 0.761202i \(0.724610\pi\)
\(108\) 1.00000 0.0962250
\(109\) 13.6736 1.30969 0.654845 0.755763i \(-0.272734\pi\)
0.654845 + 0.755763i \(0.272734\pi\)
\(110\) −0.586726 −0.0559422
\(111\) 9.68872 0.919612
\(112\) 3.87720 0.366361
\(113\) −5.72283 −0.538359 −0.269179 0.963090i \(-0.586752\pi\)
−0.269179 + 0.963090i \(0.586752\pi\)
\(114\) 0.475803 0.0445631
\(115\) −0.838820 −0.0782204
\(116\) −5.66720 −0.526187
\(117\) −1.00000 −0.0924500
\(118\) 13.6380 1.25548
\(119\) −0.791137 −0.0725234
\(120\) 0.453790 0.0414252
\(121\) −9.32829 −0.848026
\(122\) 11.4248 1.03436
\(123\) −7.38884 −0.666230
\(124\) −7.72539 −0.693760
\(125\) 4.44445 0.397524
\(126\) −3.87720 −0.345408
\(127\) 5.32461 0.472483 0.236241 0.971694i \(-0.424084\pi\)
0.236241 + 0.971694i \(0.424084\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.47310 −0.393835
\(130\) −0.453790 −0.0398000
\(131\) −5.11582 −0.446971 −0.223486 0.974707i \(-0.571744\pi\)
−0.223486 + 0.974707i \(0.571744\pi\)
\(132\) −1.29295 −0.112537
\(133\) −1.84478 −0.159963
\(134\) −0.678171 −0.0585851
\(135\) −0.453790 −0.0390560
\(136\) 0.204049 0.0174970
\(137\) −17.4572 −1.49147 −0.745736 0.666241i \(-0.767902\pi\)
−0.745736 + 0.666241i \(0.767902\pi\)
\(138\) −1.84848 −0.157353
\(139\) 15.5994 1.32312 0.661560 0.749892i \(-0.269895\pi\)
0.661560 + 0.749892i \(0.269895\pi\)
\(140\) −1.75943 −0.148699
\(141\) −3.11558 −0.262379
\(142\) −5.87504 −0.493023
\(143\) 1.29295 0.108122
\(144\) 1.00000 0.0833333
\(145\) 2.57172 0.213570
\(146\) 3.56157 0.294758
\(147\) 8.03267 0.662523
\(148\) 9.68872 0.796408
\(149\) −6.75794 −0.553632 −0.276816 0.960923i \(-0.589279\pi\)
−0.276816 + 0.960923i \(0.589279\pi\)
\(150\) 4.79407 0.391435
\(151\) 6.66373 0.542287 0.271143 0.962539i \(-0.412598\pi\)
0.271143 + 0.962539i \(0.412598\pi\)
\(152\) 0.475803 0.0385927
\(153\) −0.204049 −0.0164964
\(154\) 5.01301 0.403960
\(155\) 3.50570 0.281585
\(156\) −1.00000 −0.0800641
\(157\) −13.6895 −1.09254 −0.546271 0.837608i \(-0.683953\pi\)
−0.546271 + 0.837608i \(0.683953\pi\)
\(158\) −0.0379441 −0.00301867
\(159\) −2.30074 −0.182461
\(160\) 0.453790 0.0358753
\(161\) 7.16691 0.564832
\(162\) −1.00000 −0.0785674
\(163\) −3.83818 −0.300630 −0.150315 0.988638i \(-0.548029\pi\)
−0.150315 + 0.988638i \(0.548029\pi\)
\(164\) −7.38884 −0.576972
\(165\) 0.586726 0.0456766
\(166\) 7.81001 0.606174
\(167\) −20.6485 −1.59783 −0.798913 0.601446i \(-0.794591\pi\)
−0.798913 + 0.601446i \(0.794591\pi\)
\(168\) −3.87720 −0.299132
\(169\) 1.00000 0.0769231
\(170\) −0.0925952 −0.00710173
\(171\) −0.475803 −0.0363856
\(172\) −4.47310 −0.341071
\(173\) −5.28951 −0.402154 −0.201077 0.979575i \(-0.564444\pi\)
−0.201077 + 0.979575i \(0.564444\pi\)
\(174\) 5.66720 0.429630
\(175\) −18.5876 −1.40509
\(176\) −1.29295 −0.0974595
\(177\) −13.6380 −1.02510
\(178\) −8.08601 −0.606072
\(179\) 15.5884 1.16513 0.582567 0.812783i \(-0.302049\pi\)
0.582567 + 0.812783i \(0.302049\pi\)
\(180\) −0.453790 −0.0338235
\(181\) 25.5435 1.89863 0.949315 0.314327i \(-0.101779\pi\)
0.949315 + 0.314327i \(0.101779\pi\)
\(182\) 3.87720 0.287397
\(183\) −11.4248 −0.844547
\(184\) −1.84848 −0.136272
\(185\) −4.39664 −0.323248
\(186\) 7.72539 0.566453
\(187\) 0.263824 0.0192927
\(188\) −3.11558 −0.227227
\(189\) 3.87720 0.282025
\(190\) −0.215915 −0.0156641
\(191\) −2.83614 −0.205216 −0.102608 0.994722i \(-0.532719\pi\)
−0.102608 + 0.994722i \(0.532719\pi\)
\(192\) 1.00000 0.0721688
\(193\) −21.2934 −1.53274 −0.766368 0.642402i \(-0.777938\pi\)
−0.766368 + 0.642402i \(0.777938\pi\)
\(194\) 2.30097 0.165200
\(195\) 0.453790 0.0324966
\(196\) 8.03267 0.573762
\(197\) 11.3818 0.810919 0.405459 0.914113i \(-0.367112\pi\)
0.405459 + 0.914113i \(0.367112\pi\)
\(198\) 1.29295 0.0918857
\(199\) 16.0284 1.13622 0.568112 0.822951i \(-0.307674\pi\)
0.568112 + 0.822951i \(0.307674\pi\)
\(200\) 4.79407 0.338992
\(201\) 0.678171 0.0478345
\(202\) 3.28191 0.230914
\(203\) −21.9729 −1.54219
\(204\) −0.204049 −0.0142863
\(205\) 3.35298 0.234183
\(206\) −1.00000 −0.0696733
\(207\) 1.84848 0.128478
\(208\) −1.00000 −0.0693375
\(209\) 0.615188 0.0425535
\(210\) 1.75943 0.121412
\(211\) 18.2423 1.25585 0.627927 0.778273i \(-0.283904\pi\)
0.627927 + 0.778273i \(0.283904\pi\)
\(212\) −2.30074 −0.158016
\(213\) 5.87504 0.402551
\(214\) 13.4166 0.917139
\(215\) 2.02985 0.138435
\(216\) −1.00000 −0.0680414
\(217\) −29.9529 −2.03333
\(218\) −13.6736 −0.926090
\(219\) −3.56157 −0.240669
\(220\) 0.586726 0.0395571
\(221\) 0.204049 0.0137258
\(222\) −9.68872 −0.650264
\(223\) −15.8632 −1.06228 −0.531140 0.847284i \(-0.678236\pi\)
−0.531140 + 0.847284i \(0.678236\pi\)
\(224\) −3.87720 −0.259056
\(225\) −4.79407 −0.319605
\(226\) 5.72283 0.380677
\(227\) 5.41706 0.359543 0.179772 0.983708i \(-0.442464\pi\)
0.179772 + 0.983708i \(0.442464\pi\)
\(228\) −0.475803 −0.0315108
\(229\) 17.2160 1.13767 0.568833 0.822453i \(-0.307395\pi\)
0.568833 + 0.822453i \(0.307395\pi\)
\(230\) 0.838820 0.0553102
\(231\) −5.01301 −0.329832
\(232\) 5.66720 0.372070
\(233\) −2.06406 −0.135221 −0.0676105 0.997712i \(-0.521538\pi\)
−0.0676105 + 0.997712i \(0.521538\pi\)
\(234\) 1.00000 0.0653720
\(235\) 1.41382 0.0922275
\(236\) −13.6380 −0.887761
\(237\) 0.0379441 0.00246474
\(238\) 0.791137 0.0512818
\(239\) −8.92571 −0.577356 −0.288678 0.957426i \(-0.593216\pi\)
−0.288678 + 0.957426i \(0.593216\pi\)
\(240\) −0.453790 −0.0292920
\(241\) 9.50540 0.612296 0.306148 0.951984i \(-0.400960\pi\)
0.306148 + 0.951984i \(0.400960\pi\)
\(242\) 9.32829 0.599645
\(243\) 1.00000 0.0641500
\(244\) −11.4248 −0.731400
\(245\) −3.64515 −0.232880
\(246\) 7.38884 0.471095
\(247\) 0.475803 0.0302746
\(248\) 7.72539 0.490563
\(249\) −7.81001 −0.494939
\(250\) −4.44445 −0.281092
\(251\) −0.527595 −0.0333015 −0.0166508 0.999861i \(-0.505300\pi\)
−0.0166508 + 0.999861i \(0.505300\pi\)
\(252\) 3.87720 0.244241
\(253\) −2.38998 −0.150257
\(254\) −5.32461 −0.334096
\(255\) 0.0925952 0.00579854
\(256\) 1.00000 0.0625000
\(257\) −18.6801 −1.16523 −0.582617 0.812747i \(-0.697971\pi\)
−0.582617 + 0.812747i \(0.697971\pi\)
\(258\) 4.47310 0.278483
\(259\) 37.5651 2.33418
\(260\) 0.453790 0.0281429
\(261\) −5.66720 −0.350791
\(262\) 5.11582 0.316056
\(263\) 15.6935 0.967703 0.483852 0.875150i \(-0.339237\pi\)
0.483852 + 0.875150i \(0.339237\pi\)
\(264\) 1.29295 0.0795754
\(265\) 1.04406 0.0641358
\(266\) 1.84478 0.113111
\(267\) 8.08601 0.494856
\(268\) 0.678171 0.0414259
\(269\) 7.28894 0.444415 0.222207 0.974999i \(-0.428674\pi\)
0.222207 + 0.974999i \(0.428674\pi\)
\(270\) 0.453790 0.0276168
\(271\) −11.6259 −0.706220 −0.353110 0.935582i \(-0.614876\pi\)
−0.353110 + 0.935582i \(0.614876\pi\)
\(272\) −0.204049 −0.0123723
\(273\) −3.87720 −0.234659
\(274\) 17.4572 1.05463
\(275\) 6.19848 0.373783
\(276\) 1.84848 0.111265
\(277\) 19.8613 1.19335 0.596674 0.802484i \(-0.296489\pi\)
0.596674 + 0.802484i \(0.296489\pi\)
\(278\) −15.5994 −0.935588
\(279\) −7.72539 −0.462507
\(280\) 1.75943 0.105146
\(281\) −8.32906 −0.496870 −0.248435 0.968649i \(-0.579916\pi\)
−0.248435 + 0.968649i \(0.579916\pi\)
\(282\) 3.11558 0.185530
\(283\) 10.7185 0.637148 0.318574 0.947898i \(-0.396796\pi\)
0.318574 + 0.947898i \(0.396796\pi\)
\(284\) 5.87504 0.348620
\(285\) 0.215915 0.0127897
\(286\) −1.29295 −0.0764535
\(287\) −28.6480 −1.69104
\(288\) −1.00000 −0.0589256
\(289\) −16.9584 −0.997551
\(290\) −2.57172 −0.151017
\(291\) −2.30097 −0.134885
\(292\) −3.56157 −0.208425
\(293\) 4.33001 0.252962 0.126481 0.991969i \(-0.459632\pi\)
0.126481 + 0.991969i \(0.459632\pi\)
\(294\) −8.03267 −0.468475
\(295\) 6.18881 0.360326
\(296\) −9.68872 −0.563145
\(297\) −1.29295 −0.0750244
\(298\) 6.75794 0.391477
\(299\) −1.84848 −0.106900
\(300\) −4.79407 −0.276786
\(301\) −17.3431 −0.999640
\(302\) −6.66373 −0.383455
\(303\) −3.28191 −0.188541
\(304\) −0.475803 −0.0272892
\(305\) 5.18447 0.296862
\(306\) 0.204049 0.0116647
\(307\) 21.3696 1.21963 0.609813 0.792545i \(-0.291245\pi\)
0.609813 + 0.792545i \(0.291245\pi\)
\(308\) −5.01301 −0.285643
\(309\) 1.00000 0.0568880
\(310\) −3.50570 −0.199111
\(311\) 19.9313 1.13020 0.565099 0.825023i \(-0.308838\pi\)
0.565099 + 0.825023i \(0.308838\pi\)
\(312\) 1.00000 0.0566139
\(313\) 18.7275 1.05854 0.529269 0.848454i \(-0.322466\pi\)
0.529269 + 0.848454i \(0.322466\pi\)
\(314\) 13.6895 0.772544
\(315\) −1.75943 −0.0991329
\(316\) 0.0379441 0.00213452
\(317\) −29.4087 −1.65176 −0.825879 0.563848i \(-0.809321\pi\)
−0.825879 + 0.563848i \(0.809321\pi\)
\(318\) 2.30074 0.129019
\(319\) 7.32739 0.410255
\(320\) −0.453790 −0.0253676
\(321\) −13.4166 −0.748841
\(322\) −7.16691 −0.399396
\(323\) 0.0970870 0.00540207
\(324\) 1.00000 0.0555556
\(325\) 4.79407 0.265927
\(326\) 3.83818 0.212577
\(327\) 13.6736 0.756150
\(328\) 7.38884 0.407981
\(329\) −12.0797 −0.665977
\(330\) −0.586726 −0.0322982
\(331\) −15.8151 −0.869279 −0.434639 0.900605i \(-0.643124\pi\)
−0.434639 + 0.900605i \(0.643124\pi\)
\(332\) −7.81001 −0.428630
\(333\) 9.68872 0.530938
\(334\) 20.6485 1.12983
\(335\) −0.307747 −0.0168140
\(336\) 3.87720 0.211519
\(337\) 14.2786 0.777803 0.388902 0.921279i \(-0.372855\pi\)
0.388902 + 0.921279i \(0.372855\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −5.72283 −0.310822
\(340\) 0.0925952 0.00502168
\(341\) 9.98851 0.540908
\(342\) 0.475803 0.0257285
\(343\) 4.00386 0.216188
\(344\) 4.47310 0.241174
\(345\) −0.838820 −0.0451606
\(346\) 5.28951 0.284366
\(347\) 1.58327 0.0849945 0.0424973 0.999097i \(-0.486469\pi\)
0.0424973 + 0.999097i \(0.486469\pi\)
\(348\) −5.66720 −0.303794
\(349\) −22.0059 −1.17795 −0.588976 0.808151i \(-0.700469\pi\)
−0.588976 + 0.808151i \(0.700469\pi\)
\(350\) 18.5876 0.993548
\(351\) −1.00000 −0.0533761
\(352\) 1.29295 0.0689143
\(353\) 14.1320 0.752171 0.376086 0.926585i \(-0.377270\pi\)
0.376086 + 0.926585i \(0.377270\pi\)
\(354\) 13.6380 0.724854
\(355\) −2.66604 −0.141499
\(356\) 8.08601 0.428558
\(357\) −0.791137 −0.0418714
\(358\) −15.5884 −0.823873
\(359\) −7.97890 −0.421110 −0.210555 0.977582i \(-0.567527\pi\)
−0.210555 + 0.977582i \(0.567527\pi\)
\(360\) 0.453790 0.0239168
\(361\) −18.7736 −0.988085
\(362\) −25.5435 −1.34253
\(363\) −9.32829 −0.489608
\(364\) −3.87720 −0.203220
\(365\) 1.61620 0.0845960
\(366\) 11.4248 0.597185
\(367\) 21.7801 1.13691 0.568455 0.822714i \(-0.307541\pi\)
0.568455 + 0.822714i \(0.307541\pi\)
\(368\) 1.84848 0.0963585
\(369\) −7.38884 −0.384648
\(370\) 4.39664 0.228571
\(371\) −8.92044 −0.463126
\(372\) −7.72539 −0.400543
\(373\) −22.1999 −1.14947 −0.574733 0.818341i \(-0.694894\pi\)
−0.574733 + 0.818341i \(0.694894\pi\)
\(374\) −0.263824 −0.0136420
\(375\) 4.44445 0.229511
\(376\) 3.11558 0.160674
\(377\) 5.66720 0.291876
\(378\) −3.87720 −0.199422
\(379\) 14.0417 0.721271 0.360636 0.932707i \(-0.382560\pi\)
0.360636 + 0.932707i \(0.382560\pi\)
\(380\) 0.215915 0.0110762
\(381\) 5.32461 0.272788
\(382\) 2.83614 0.145109
\(383\) 25.5397 1.30502 0.652508 0.757782i \(-0.273717\pi\)
0.652508 + 0.757782i \(0.273717\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 2.27485 0.115937
\(386\) 21.2934 1.08381
\(387\) −4.47310 −0.227381
\(388\) −2.30097 −0.116814
\(389\) 19.7650 1.00212 0.501062 0.865411i \(-0.332943\pi\)
0.501062 + 0.865411i \(0.332943\pi\)
\(390\) −0.453790 −0.0229786
\(391\) −0.377179 −0.0190748
\(392\) −8.03267 −0.405711
\(393\) −5.11582 −0.258059
\(394\) −11.3818 −0.573406
\(395\) −0.0172187 −0.000866365 0
\(396\) −1.29295 −0.0649730
\(397\) 20.1910 1.01336 0.506680 0.862134i \(-0.330873\pi\)
0.506680 + 0.862134i \(0.330873\pi\)
\(398\) −16.0284 −0.803432
\(399\) −1.84478 −0.0923547
\(400\) −4.79407 −0.239704
\(401\) 12.9075 0.644568 0.322284 0.946643i \(-0.395549\pi\)
0.322284 + 0.946643i \(0.395549\pi\)
\(402\) −0.678171 −0.0338241
\(403\) 7.72539 0.384829
\(404\) −3.28191 −0.163281
\(405\) −0.453790 −0.0225490
\(406\) 21.9729 1.09050
\(407\) −12.5270 −0.620940
\(408\) 0.204049 0.0101019
\(409\) 20.8045 1.02872 0.514358 0.857576i \(-0.328030\pi\)
0.514358 + 0.857576i \(0.328030\pi\)
\(410\) −3.35298 −0.165592
\(411\) −17.4572 −0.861102
\(412\) 1.00000 0.0492665
\(413\) −52.8774 −2.60193
\(414\) −1.84848 −0.0908477
\(415\) 3.54410 0.173973
\(416\) 1.00000 0.0490290
\(417\) 15.5994 0.763904
\(418\) −0.615188 −0.0300898
\(419\) −1.39954 −0.0683720 −0.0341860 0.999415i \(-0.510884\pi\)
−0.0341860 + 0.999415i \(0.510884\pi\)
\(420\) −1.75943 −0.0858516
\(421\) −25.4738 −1.24152 −0.620758 0.784002i \(-0.713175\pi\)
−0.620758 + 0.784002i \(0.713175\pi\)
\(422\) −18.2423 −0.888022
\(423\) −3.11558 −0.151485
\(424\) 2.30074 0.111734
\(425\) 0.978224 0.0474509
\(426\) −5.87504 −0.284647
\(427\) −44.2963 −2.14365
\(428\) −13.4166 −0.648515
\(429\) 1.29295 0.0624241
\(430\) −2.02985 −0.0978881
\(431\) −11.2527 −0.542023 −0.271012 0.962576i \(-0.587358\pi\)
−0.271012 + 0.962576i \(0.587358\pi\)
\(432\) 1.00000 0.0481125
\(433\) −0.443161 −0.0212969 −0.0106485 0.999943i \(-0.503390\pi\)
−0.0106485 + 0.999943i \(0.503390\pi\)
\(434\) 29.9529 1.43778
\(435\) 2.57172 0.123305
\(436\) 13.6736 0.654845
\(437\) −0.879511 −0.0420727
\(438\) 3.56157 0.170178
\(439\) −18.9244 −0.903212 −0.451606 0.892218i \(-0.649149\pi\)
−0.451606 + 0.892218i \(0.649149\pi\)
\(440\) −0.586726 −0.0279711
\(441\) 8.03267 0.382508
\(442\) −0.204049 −0.00970560
\(443\) 4.55574 0.216450 0.108225 0.994126i \(-0.465483\pi\)
0.108225 + 0.994126i \(0.465483\pi\)
\(444\) 9.68872 0.459806
\(445\) −3.66935 −0.173944
\(446\) 15.8632 0.751145
\(447\) −6.75794 −0.319640
\(448\) 3.87720 0.183180
\(449\) −5.07426 −0.239469 −0.119735 0.992806i \(-0.538204\pi\)
−0.119735 + 0.992806i \(0.538204\pi\)
\(450\) 4.79407 0.225995
\(451\) 9.55338 0.449851
\(452\) −5.72283 −0.269179
\(453\) 6.66373 0.313089
\(454\) −5.41706 −0.254235
\(455\) 1.75943 0.0824835
\(456\) 0.475803 0.0222815
\(457\) −5.60898 −0.262377 −0.131189 0.991357i \(-0.541879\pi\)
−0.131189 + 0.991357i \(0.541879\pi\)
\(458\) −17.2160 −0.804451
\(459\) −0.204049 −0.00952417
\(460\) −0.838820 −0.0391102
\(461\) 9.52366 0.443561 0.221781 0.975097i \(-0.428813\pi\)
0.221781 + 0.975097i \(0.428813\pi\)
\(462\) 5.01301 0.233226
\(463\) −12.1793 −0.566021 −0.283011 0.959117i \(-0.591333\pi\)
−0.283011 + 0.959117i \(0.591333\pi\)
\(464\) −5.66720 −0.263093
\(465\) 3.50570 0.162573
\(466\) 2.06406 0.0956157
\(467\) −5.89842 −0.272946 −0.136473 0.990644i \(-0.543577\pi\)
−0.136473 + 0.990644i \(0.543577\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 2.62940 0.121415
\(470\) −1.41382 −0.0652147
\(471\) −13.6895 −0.630779
\(472\) 13.6380 0.627742
\(473\) 5.78349 0.265925
\(474\) −0.0379441 −0.00174283
\(475\) 2.28104 0.104661
\(476\) −0.791137 −0.0362617
\(477\) −2.30074 −0.105344
\(478\) 8.92571 0.408252
\(479\) 0.166160 0.00759207 0.00379603 0.999993i \(-0.498792\pi\)
0.00379603 + 0.999993i \(0.498792\pi\)
\(480\) 0.453790 0.0207126
\(481\) −9.68872 −0.441768
\(482\) −9.50540 −0.432959
\(483\) 7.16691 0.326106
\(484\) −9.32829 −0.424013
\(485\) 1.04416 0.0474128
\(486\) −1.00000 −0.0453609
\(487\) −31.8793 −1.44459 −0.722293 0.691587i \(-0.756912\pi\)
−0.722293 + 0.691587i \(0.756912\pi\)
\(488\) 11.4248 0.517178
\(489\) −3.83818 −0.173569
\(490\) 3.64515 0.164671
\(491\) 35.7593 1.61380 0.806898 0.590691i \(-0.201145\pi\)
0.806898 + 0.590691i \(0.201145\pi\)
\(492\) −7.38884 −0.333115
\(493\) 1.15639 0.0520810
\(494\) −0.475803 −0.0214074
\(495\) 0.586726 0.0263714
\(496\) −7.72539 −0.346880
\(497\) 22.7787 1.02176
\(498\) 7.81001 0.349975
\(499\) 23.4178 1.04832 0.524162 0.851619i \(-0.324379\pi\)
0.524162 + 0.851619i \(0.324379\pi\)
\(500\) 4.44445 0.198762
\(501\) −20.6485 −0.922506
\(502\) 0.527595 0.0235477
\(503\) 2.64065 0.117741 0.0588703 0.998266i \(-0.481250\pi\)
0.0588703 + 0.998266i \(0.481250\pi\)
\(504\) −3.87720 −0.172704
\(505\) 1.48930 0.0662729
\(506\) 2.38998 0.106248
\(507\) 1.00000 0.0444116
\(508\) 5.32461 0.236241
\(509\) 23.2141 1.02895 0.514474 0.857506i \(-0.327987\pi\)
0.514474 + 0.857506i \(0.327987\pi\)
\(510\) −0.0925952 −0.00410019
\(511\) −13.8089 −0.610870
\(512\) −1.00000 −0.0441942
\(513\) −0.475803 −0.0210072
\(514\) 18.6801 0.823945
\(515\) −0.453790 −0.0199964
\(516\) −4.47310 −0.196917
\(517\) 4.02828 0.177164
\(518\) −37.5651 −1.65052
\(519\) −5.28951 −0.232184
\(520\) −0.453790 −0.0199000
\(521\) −18.1187 −0.793793 −0.396897 0.917863i \(-0.629913\pi\)
−0.396897 + 0.917863i \(0.629913\pi\)
\(522\) 5.66720 0.248047
\(523\) −31.7767 −1.38950 −0.694749 0.719252i \(-0.744485\pi\)
−0.694749 + 0.719252i \(0.744485\pi\)
\(524\) −5.11582 −0.223486
\(525\) −18.5876 −0.811228
\(526\) −15.6935 −0.684269
\(527\) 1.57635 0.0686671
\(528\) −1.29295 −0.0562683
\(529\) −19.5831 −0.851441
\(530\) −1.04406 −0.0453509
\(531\) −13.6380 −0.591840
\(532\) −1.84478 −0.0799815
\(533\) 7.38884 0.320046
\(534\) −8.08601 −0.349916
\(535\) 6.08831 0.263221
\(536\) −0.678171 −0.0292925
\(537\) 15.5884 0.672690
\(538\) −7.28894 −0.314249
\(539\) −10.3858 −0.447349
\(540\) −0.453790 −0.0195280
\(541\) −15.8913 −0.683219 −0.341610 0.939842i \(-0.610972\pi\)
−0.341610 + 0.939842i \(0.610972\pi\)
\(542\) 11.6259 0.499373
\(543\) 25.5435 1.09617
\(544\) 0.204049 0.00874851
\(545\) −6.20493 −0.265790
\(546\) 3.87720 0.165929
\(547\) −17.3725 −0.742793 −0.371397 0.928474i \(-0.621121\pi\)
−0.371397 + 0.928474i \(0.621121\pi\)
\(548\) −17.4572 −0.745736
\(549\) −11.4248 −0.487600
\(550\) −6.19848 −0.264304
\(551\) 2.69647 0.114874
\(552\) −1.84848 −0.0786764
\(553\) 0.147117 0.00625605
\(554\) −19.8613 −0.843824
\(555\) −4.39664 −0.186627
\(556\) 15.5994 0.661560
\(557\) −29.2274 −1.23840 −0.619202 0.785232i \(-0.712544\pi\)
−0.619202 + 0.785232i \(0.712544\pi\)
\(558\) 7.72539 0.327042
\(559\) 4.47310 0.189192
\(560\) −1.75943 −0.0743497
\(561\) 0.263824 0.0111387
\(562\) 8.32906 0.351340
\(563\) −2.17941 −0.0918511 −0.0459255 0.998945i \(-0.514624\pi\)
−0.0459255 + 0.998945i \(0.514624\pi\)
\(564\) −3.11558 −0.131190
\(565\) 2.59696 0.109255
\(566\) −10.7185 −0.450532
\(567\) 3.87720 0.162827
\(568\) −5.87504 −0.246511
\(569\) −45.1866 −1.89432 −0.947161 0.320760i \(-0.896062\pi\)
−0.947161 + 0.320760i \(0.896062\pi\)
\(570\) −0.215915 −0.00904368
\(571\) −35.1769 −1.47211 −0.736054 0.676923i \(-0.763313\pi\)
−0.736054 + 0.676923i \(0.763313\pi\)
\(572\) 1.29295 0.0540608
\(573\) −2.83614 −0.118481
\(574\) 28.6480 1.19575
\(575\) −8.86174 −0.369560
\(576\) 1.00000 0.0416667
\(577\) −28.6385 −1.19224 −0.596118 0.802897i \(-0.703291\pi\)
−0.596118 + 0.802897i \(0.703291\pi\)
\(578\) 16.9584 0.705375
\(579\) −21.2934 −0.884925
\(580\) 2.57172 0.106785
\(581\) −30.2810 −1.25627
\(582\) 2.30097 0.0953784
\(583\) 2.97474 0.123201
\(584\) 3.56157 0.147379
\(585\) 0.453790 0.0187619
\(586\) −4.33001 −0.178871
\(587\) −16.6147 −0.685761 −0.342881 0.939379i \(-0.611403\pi\)
−0.342881 + 0.939379i \(0.611403\pi\)
\(588\) 8.03267 0.331262
\(589\) 3.67576 0.151457
\(590\) −6.18881 −0.254789
\(591\) 11.3818 0.468184
\(592\) 9.68872 0.398204
\(593\) 44.3734 1.82220 0.911099 0.412189i \(-0.135236\pi\)
0.911099 + 0.412189i \(0.135236\pi\)
\(594\) 1.29295 0.0530502
\(595\) 0.359010 0.0147180
\(596\) −6.75794 −0.276816
\(597\) 16.0284 0.655999
\(598\) 1.84848 0.0755898
\(599\) 11.1662 0.456239 0.228119 0.973633i \(-0.426742\pi\)
0.228119 + 0.973633i \(0.426742\pi\)
\(600\) 4.79407 0.195717
\(601\) −33.4940 −1.36625 −0.683124 0.730302i \(-0.739379\pi\)
−0.683124 + 0.730302i \(0.739379\pi\)
\(602\) 17.3431 0.706853
\(603\) 0.678171 0.0276173
\(604\) 6.66373 0.271143
\(605\) 4.23308 0.172099
\(606\) 3.28191 0.133318
\(607\) 14.4014 0.584533 0.292267 0.956337i \(-0.405591\pi\)
0.292267 + 0.956337i \(0.405591\pi\)
\(608\) 0.475803 0.0192964
\(609\) −21.9729 −0.890386
\(610\) −5.18447 −0.209913
\(611\) 3.11558 0.126043
\(612\) −0.204049 −0.00824818
\(613\) −22.7741 −0.919836 −0.459918 0.887961i \(-0.652121\pi\)
−0.459918 + 0.887961i \(0.652121\pi\)
\(614\) −21.3696 −0.862406
\(615\) 3.35298 0.135205
\(616\) 5.01301 0.201980
\(617\) 36.7666 1.48017 0.740084 0.672514i \(-0.234786\pi\)
0.740084 + 0.672514i \(0.234786\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 22.8495 0.918398 0.459199 0.888333i \(-0.348136\pi\)
0.459199 + 0.888333i \(0.348136\pi\)
\(620\) 3.50570 0.140792
\(621\) 1.84848 0.0741768
\(622\) −19.9313 −0.799171
\(623\) 31.3511 1.25605
\(624\) −1.00000 −0.0400320
\(625\) 21.9535 0.878141
\(626\) −18.7275 −0.748500
\(627\) 0.615188 0.0245683
\(628\) −13.6895 −0.546271
\(629\) −1.97697 −0.0788269
\(630\) 1.75943 0.0700975
\(631\) 6.03358 0.240193 0.120097 0.992762i \(-0.461680\pi\)
0.120097 + 0.992762i \(0.461680\pi\)
\(632\) −0.0379441 −0.00150934
\(633\) 18.2423 0.725067
\(634\) 29.4087 1.16797
\(635\) −2.41626 −0.0958862
\(636\) −2.30074 −0.0912305
\(637\) −8.03267 −0.318266
\(638\) −7.32739 −0.290094
\(639\) 5.87504 0.232413
\(640\) 0.453790 0.0179376
\(641\) −18.5836 −0.734006 −0.367003 0.930220i \(-0.619616\pi\)
−0.367003 + 0.930220i \(0.619616\pi\)
\(642\) 13.4166 0.529511
\(643\) 25.8742 1.02038 0.510190 0.860062i \(-0.329575\pi\)
0.510190 + 0.860062i \(0.329575\pi\)
\(644\) 7.16691 0.282416
\(645\) 2.02985 0.0799253
\(646\) −0.0970870 −0.00381984
\(647\) 7.09152 0.278797 0.139398 0.990236i \(-0.455483\pi\)
0.139398 + 0.990236i \(0.455483\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 17.6333 0.692166
\(650\) −4.79407 −0.188039
\(651\) −29.9529 −1.17395
\(652\) −3.83818 −0.150315
\(653\) 19.5477 0.764962 0.382481 0.923963i \(-0.375070\pi\)
0.382481 + 0.923963i \(0.375070\pi\)
\(654\) −13.6736 −0.534679
\(655\) 2.32151 0.0907088
\(656\) −7.38884 −0.288486
\(657\) −3.56157 −0.138950
\(658\) 12.0797 0.470917
\(659\) −42.0542 −1.63820 −0.819099 0.573653i \(-0.805526\pi\)
−0.819099 + 0.573653i \(0.805526\pi\)
\(660\) 0.586726 0.0228383
\(661\) −19.6058 −0.762576 −0.381288 0.924456i \(-0.624519\pi\)
−0.381288 + 0.924456i \(0.624519\pi\)
\(662\) 15.8151 0.614673
\(663\) 0.204049 0.00792459
\(664\) 7.81001 0.303087
\(665\) 0.837145 0.0324631
\(666\) −9.68872 −0.375430
\(667\) −10.4757 −0.405621
\(668\) −20.6485 −0.798913
\(669\) −15.8632 −0.613307
\(670\) 0.307747 0.0118893
\(671\) 14.7717 0.570255
\(672\) −3.87720 −0.149566
\(673\) −2.51438 −0.0969222 −0.0484611 0.998825i \(-0.515432\pi\)
−0.0484611 + 0.998825i \(0.515432\pi\)
\(674\) −14.2786 −0.549990
\(675\) −4.79407 −0.184524
\(676\) 1.00000 0.0384615
\(677\) 3.82779 0.147114 0.0735569 0.997291i \(-0.476565\pi\)
0.0735569 + 0.997291i \(0.476565\pi\)
\(678\) 5.72283 0.219784
\(679\) −8.92133 −0.342369
\(680\) −0.0925952 −0.00355087
\(681\) 5.41706 0.207582
\(682\) −9.98851 −0.382480
\(683\) 5.50265 0.210553 0.105276 0.994443i \(-0.466427\pi\)
0.105276 + 0.994443i \(0.466427\pi\)
\(684\) −0.475803 −0.0181928
\(685\) 7.92192 0.302681
\(686\) −4.00386 −0.152868
\(687\) 17.2160 0.656831
\(688\) −4.47310 −0.170536
\(689\) 2.30074 0.0876514
\(690\) 0.838820 0.0319333
\(691\) −20.1534 −0.766670 −0.383335 0.923609i \(-0.625224\pi\)
−0.383335 + 0.923609i \(0.625224\pi\)
\(692\) −5.28951 −0.201077
\(693\) −5.01301 −0.190429
\(694\) −1.58327 −0.0601002
\(695\) −7.07884 −0.268516
\(696\) 5.66720 0.214815
\(697\) 1.50768 0.0571076
\(698\) 22.0059 0.832937
\(699\) −2.06406 −0.0780699
\(700\) −18.5876 −0.702544
\(701\) −25.0943 −0.947798 −0.473899 0.880579i \(-0.657154\pi\)
−0.473899 + 0.880579i \(0.657154\pi\)
\(702\) 1.00000 0.0377426
\(703\) −4.60992 −0.173867
\(704\) −1.29295 −0.0487298
\(705\) 1.41382 0.0532476
\(706\) −14.1320 −0.531865
\(707\) −12.7246 −0.478558
\(708\) −13.6380 −0.512549
\(709\) −2.53504 −0.0952053 −0.0476026 0.998866i \(-0.515158\pi\)
−0.0476026 + 0.998866i \(0.515158\pi\)
\(710\) 2.66604 0.100055
\(711\) 0.0379441 0.00142302
\(712\) −8.08601 −0.303036
\(713\) −14.2802 −0.534798
\(714\) 0.791137 0.0296076
\(715\) −0.586726 −0.0219423
\(716\) 15.5884 0.582567
\(717\) −8.92571 −0.333337
\(718\) 7.97890 0.297770
\(719\) 4.04303 0.150780 0.0753899 0.997154i \(-0.475980\pi\)
0.0753899 + 0.997154i \(0.475980\pi\)
\(720\) −0.453790 −0.0169118
\(721\) 3.87720 0.144394
\(722\) 18.7736 0.698681
\(723\) 9.50540 0.353509
\(724\) 25.5435 0.949315
\(725\) 27.1690 1.00903
\(726\) 9.32829 0.346205
\(727\) 2.60500 0.0966140 0.0483070 0.998833i \(-0.484617\pi\)
0.0483070 + 0.998833i \(0.484617\pi\)
\(728\) 3.87720 0.143699
\(729\) 1.00000 0.0370370
\(730\) −1.61620 −0.0598184
\(731\) 0.912731 0.0337586
\(732\) −11.4248 −0.422274
\(733\) −19.9473 −0.736772 −0.368386 0.929673i \(-0.620089\pi\)
−0.368386 + 0.929673i \(0.620089\pi\)
\(734\) −21.7801 −0.803917
\(735\) −3.64515 −0.134453
\(736\) −1.84848 −0.0681358
\(737\) −0.876839 −0.0322988
\(738\) 7.38884 0.271987
\(739\) −51.6136 −1.89864 −0.949318 0.314316i \(-0.898225\pi\)
−0.949318 + 0.314316i \(0.898225\pi\)
\(740\) −4.39664 −0.161624
\(741\) 0.475803 0.0174791
\(742\) 8.92044 0.327480
\(743\) 21.7795 0.799012 0.399506 0.916730i \(-0.369182\pi\)
0.399506 + 0.916730i \(0.369182\pi\)
\(744\) 7.72539 0.283226
\(745\) 3.06669 0.112355
\(746\) 22.1999 0.812795
\(747\) −7.81001 −0.285753
\(748\) 0.263824 0.00964636
\(749\) −52.0188 −1.90072
\(750\) −4.44445 −0.162288
\(751\) 11.1206 0.405795 0.202898 0.979200i \(-0.434964\pi\)
0.202898 + 0.979200i \(0.434964\pi\)
\(752\) −3.11558 −0.113614
\(753\) −0.527595 −0.0192266
\(754\) −5.66720 −0.206387
\(755\) −3.02394 −0.110052
\(756\) 3.87720 0.141012
\(757\) 36.0724 1.31107 0.655537 0.755163i \(-0.272442\pi\)
0.655537 + 0.755163i \(0.272442\pi\)
\(758\) −14.0417 −0.510016
\(759\) −2.38998 −0.0867508
\(760\) −0.215915 −0.00783205
\(761\) −9.66051 −0.350193 −0.175097 0.984551i \(-0.556024\pi\)
−0.175097 + 0.984551i \(0.556024\pi\)
\(762\) −5.32461 −0.192890
\(763\) 53.0151 1.91928
\(764\) −2.83614 −0.102608
\(765\) 0.0925952 0.00334779
\(766\) −25.5397 −0.922785
\(767\) 13.6380 0.492441
\(768\) 1.00000 0.0360844
\(769\) 20.1651 0.727172 0.363586 0.931561i \(-0.381552\pi\)
0.363586 + 0.931561i \(0.381552\pi\)
\(770\) −2.27485 −0.0819801
\(771\) −18.6801 −0.672748
\(772\) −21.2934 −0.766368
\(773\) 32.0968 1.15444 0.577221 0.816588i \(-0.304137\pi\)
0.577221 + 0.816588i \(0.304137\pi\)
\(774\) 4.47310 0.160782
\(775\) 37.0361 1.33038
\(776\) 2.30097 0.0826001
\(777\) 37.5651 1.34764
\(778\) −19.7650 −0.708609
\(779\) 3.51564 0.125961
\(780\) 0.453790 0.0162483
\(781\) −7.59612 −0.271810
\(782\) 0.377179 0.0134879
\(783\) −5.66720 −0.202529
\(784\) 8.03267 0.286881
\(785\) 6.21216 0.221722
\(786\) 5.11582 0.182475
\(787\) −23.4264 −0.835060 −0.417530 0.908663i \(-0.637104\pi\)
−0.417530 + 0.908663i \(0.637104\pi\)
\(788\) 11.3818 0.405459
\(789\) 15.6935 0.558704
\(790\) 0.0172187 0.000612613 0
\(791\) −22.1886 −0.788934
\(792\) 1.29295 0.0459429
\(793\) 11.4248 0.405707
\(794\) −20.1910 −0.716553
\(795\) 1.04406 0.0370288
\(796\) 16.0284 0.568112
\(797\) −37.8892 −1.34210 −0.671052 0.741410i \(-0.734157\pi\)
−0.671052 + 0.741410i \(0.734157\pi\)
\(798\) 1.84478 0.0653046
\(799\) 0.635730 0.0224905
\(800\) 4.79407 0.169496
\(801\) 8.08601 0.285705
\(802\) −12.9075 −0.455778
\(803\) 4.60492 0.162504
\(804\) 0.678171 0.0239172
\(805\) −3.25227 −0.114628
\(806\) −7.72539 −0.272115
\(807\) 7.28894 0.256583
\(808\) 3.28191 0.115457
\(809\) −26.5488 −0.933407 −0.466703 0.884414i \(-0.654558\pi\)
−0.466703 + 0.884414i \(0.654558\pi\)
\(810\) 0.453790 0.0159446
\(811\) 25.8198 0.906656 0.453328 0.891344i \(-0.350237\pi\)
0.453328 + 0.891344i \(0.350237\pi\)
\(812\) −21.9729 −0.771097
\(813\) −11.6259 −0.407737
\(814\) 12.5270 0.439071
\(815\) 1.74173 0.0610101
\(816\) −0.204049 −0.00714313
\(817\) 2.12832 0.0744604
\(818\) −20.8045 −0.727412
\(819\) −3.87720 −0.135480
\(820\) 3.35298 0.117091
\(821\) −24.6861 −0.861552 −0.430776 0.902459i \(-0.641760\pi\)
−0.430776 + 0.902459i \(0.641760\pi\)
\(822\) 17.4572 0.608891
\(823\) −18.5367 −0.646150 −0.323075 0.946373i \(-0.604717\pi\)
−0.323075 + 0.946373i \(0.604717\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 6.19848 0.215803
\(826\) 52.8774 1.83984
\(827\) 14.6031 0.507799 0.253900 0.967231i \(-0.418287\pi\)
0.253900 + 0.967231i \(0.418287\pi\)
\(828\) 1.84848 0.0642390
\(829\) 51.1255 1.77566 0.887832 0.460168i \(-0.152211\pi\)
0.887832 + 0.460168i \(0.152211\pi\)
\(830\) −3.54410 −0.123018
\(831\) 19.8613 0.688979
\(832\) −1.00000 −0.0346688
\(833\) −1.63906 −0.0567899
\(834\) −15.5994 −0.540162
\(835\) 9.37007 0.324265
\(836\) 0.615188 0.0212767
\(837\) −7.72539 −0.267028
\(838\) 1.39954 0.0483463
\(839\) 15.5369 0.536394 0.268197 0.963364i \(-0.413572\pi\)
0.268197 + 0.963364i \(0.413572\pi\)
\(840\) 1.75943 0.0607062
\(841\) 3.11720 0.107490
\(842\) 25.4738 0.877885
\(843\) −8.32906 −0.286868
\(844\) 18.2423 0.627927
\(845\) −0.453790 −0.0156109
\(846\) 3.11558 0.107116
\(847\) −36.1676 −1.24273
\(848\) −2.30074 −0.0790079
\(849\) 10.7185 0.367858
\(850\) −0.978224 −0.0335528
\(851\) 17.9094 0.613925
\(852\) 5.87504 0.201276
\(853\) 41.9137 1.43510 0.717548 0.696509i \(-0.245264\pi\)
0.717548 + 0.696509i \(0.245264\pi\)
\(854\) 44.2963 1.51579
\(855\) 0.215915 0.00738413
\(856\) 13.4166 0.458570
\(857\) 26.2430 0.896443 0.448222 0.893922i \(-0.352058\pi\)
0.448222 + 0.893922i \(0.352058\pi\)
\(858\) −1.29295 −0.0441405
\(859\) 37.8711 1.29215 0.646073 0.763276i \(-0.276410\pi\)
0.646073 + 0.763276i \(0.276410\pi\)
\(860\) 2.02985 0.0692173
\(861\) −28.6480 −0.976322
\(862\) 11.2527 0.383268
\(863\) −47.3270 −1.61103 −0.805515 0.592575i \(-0.798111\pi\)
−0.805515 + 0.592575i \(0.798111\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 2.40033 0.0816135
\(866\) 0.443161 0.0150592
\(867\) −16.9584 −0.575936
\(868\) −29.9529 −1.01667
\(869\) −0.0490597 −0.00166424
\(870\) −2.57172 −0.0871895
\(871\) −0.678171 −0.0229789
\(872\) −13.6736 −0.463045
\(873\) −2.30097 −0.0778761
\(874\) 0.879511 0.0297499
\(875\) 17.2320 0.582549
\(876\) −3.56157 −0.120334
\(877\) −20.0068 −0.675580 −0.337790 0.941222i \(-0.609679\pi\)
−0.337790 + 0.941222i \(0.609679\pi\)
\(878\) 18.9244 0.638667
\(879\) 4.33001 0.146048
\(880\) 0.586726 0.0197785
\(881\) 31.5624 1.06336 0.531682 0.846944i \(-0.321560\pi\)
0.531682 + 0.846944i \(0.321560\pi\)
\(882\) −8.03267 −0.270474
\(883\) −2.71537 −0.0913795 −0.0456898 0.998956i \(-0.514549\pi\)
−0.0456898 + 0.998956i \(0.514549\pi\)
\(884\) 0.204049 0.00686290
\(885\) 6.18881 0.208034
\(886\) −4.55574 −0.153053
\(887\) 43.4841 1.46005 0.730026 0.683419i \(-0.239508\pi\)
0.730026 + 0.683419i \(0.239508\pi\)
\(888\) −9.68872 −0.325132
\(889\) 20.6446 0.692397
\(890\) 3.66935 0.122997
\(891\) −1.29295 −0.0433153
\(892\) −15.8632 −0.531140
\(893\) 1.48240 0.0496068
\(894\) 6.75794 0.226019
\(895\) −7.07387 −0.236453
\(896\) −3.87720 −0.129528
\(897\) −1.84848 −0.0617188
\(898\) 5.07426 0.169330
\(899\) 43.7813 1.46019
\(900\) −4.79407 −0.159802
\(901\) 0.469464 0.0156401
\(902\) −9.55338 −0.318093
\(903\) −17.3431 −0.577143
\(904\) 5.72283 0.190339
\(905\) −11.5914 −0.385310
\(906\) −6.66373 −0.221388
\(907\) 23.9681 0.795847 0.397923 0.917419i \(-0.369731\pi\)
0.397923 + 0.917419i \(0.369731\pi\)
\(908\) 5.41706 0.179772
\(909\) −3.28191 −0.108854
\(910\) −1.75943 −0.0583247
\(911\) −4.28155 −0.141854 −0.0709271 0.997482i \(-0.522596\pi\)
−0.0709271 + 0.997482i \(0.522596\pi\)
\(912\) −0.475803 −0.0157554
\(913\) 10.0979 0.334193
\(914\) 5.60898 0.185529
\(915\) 5.18447 0.171393
\(916\) 17.2160 0.568833
\(917\) −19.8350 −0.655011
\(918\) 0.204049 0.00673461
\(919\) −59.3392 −1.95742 −0.978709 0.205252i \(-0.934198\pi\)
−0.978709 + 0.205252i \(0.934198\pi\)
\(920\) 0.838820 0.0276551
\(921\) 21.3696 0.704151
\(922\) −9.52366 −0.313645
\(923\) −5.87504 −0.193379
\(924\) −5.01301 −0.164916
\(925\) −46.4484 −1.52722
\(926\) 12.1793 0.400237
\(927\) 1.00000 0.0328443
\(928\) 5.66720 0.186035
\(929\) −49.2330 −1.61528 −0.807641 0.589674i \(-0.799256\pi\)
−0.807641 + 0.589674i \(0.799256\pi\)
\(930\) −3.50570 −0.114957
\(931\) −3.82197 −0.125260
\(932\) −2.06406 −0.0676105
\(933\) 19.9313 0.652520
\(934\) 5.89842 0.193002
\(935\) −0.119721 −0.00391529
\(936\) 1.00000 0.0326860
\(937\) 20.1812 0.659292 0.329646 0.944105i \(-0.393071\pi\)
0.329646 + 0.944105i \(0.393071\pi\)
\(938\) −2.62940 −0.0858531
\(939\) 18.7275 0.611148
\(940\) 1.41382 0.0461137
\(941\) −1.28784 −0.0419823 −0.0209911 0.999780i \(-0.506682\pi\)
−0.0209911 + 0.999780i \(0.506682\pi\)
\(942\) 13.6895 0.446028
\(943\) −13.6581 −0.444769
\(944\) −13.6380 −0.443880
\(945\) −1.75943 −0.0572344
\(946\) −5.78349 −0.188037
\(947\) −11.7884 −0.383071 −0.191535 0.981486i \(-0.561347\pi\)
−0.191535 + 0.981486i \(0.561347\pi\)
\(948\) 0.0379441 0.00123237
\(949\) 3.56157 0.115613
\(950\) −2.28104 −0.0740066
\(951\) −29.4087 −0.953643
\(952\) 0.791137 0.0256409
\(953\) 1.24913 0.0404632 0.0202316 0.999795i \(-0.493560\pi\)
0.0202316 + 0.999795i \(0.493560\pi\)
\(954\) 2.30074 0.0744894
\(955\) 1.28701 0.0416467
\(956\) −8.92571 −0.288678
\(957\) 7.32739 0.236861
\(958\) −0.166160 −0.00536840
\(959\) −67.6852 −2.18567
\(960\) −0.453790 −0.0146460
\(961\) 28.6816 0.925213
\(962\) 9.68872 0.312377
\(963\) −13.4166 −0.432344
\(964\) 9.50540 0.306148
\(965\) 9.66275 0.311055
\(966\) −7.16691 −0.230592
\(967\) −52.6500 −1.69311 −0.846556 0.532300i \(-0.821328\pi\)
−0.846556 + 0.532300i \(0.821328\pi\)
\(968\) 9.32829 0.299823
\(969\) 0.0970870 0.00311888
\(970\) −1.04416 −0.0335259
\(971\) −25.1840 −0.808193 −0.404096 0.914716i \(-0.632414\pi\)
−0.404096 + 0.914716i \(0.632414\pi\)
\(972\) 1.00000 0.0320750
\(973\) 60.4818 1.93896
\(974\) 31.8793 1.02148
\(975\) 4.79407 0.153533
\(976\) −11.4248 −0.365700
\(977\) −19.2605 −0.616197 −0.308098 0.951354i \(-0.599693\pi\)
−0.308098 + 0.951354i \(0.599693\pi\)
\(978\) 3.83818 0.122732
\(979\) −10.4548 −0.334136
\(980\) −3.64515 −0.116440
\(981\) 13.6736 0.436563
\(982\) −35.7593 −1.14113
\(983\) −50.5635 −1.61273 −0.806363 0.591421i \(-0.798567\pi\)
−0.806363 + 0.591421i \(0.798567\pi\)
\(984\) 7.38884 0.235548
\(985\) −5.16494 −0.164569
\(986\) −1.15639 −0.0368268
\(987\) −12.0797 −0.384502
\(988\) 0.475803 0.0151373
\(989\) −8.26843 −0.262921
\(990\) −0.586726 −0.0186474
\(991\) 56.3374 1.78962 0.894808 0.446450i \(-0.147312\pi\)
0.894808 + 0.446450i \(0.147312\pi\)
\(992\) 7.72539 0.245281
\(993\) −15.8151 −0.501878
\(994\) −22.7787 −0.722497
\(995\) −7.27353 −0.230587
\(996\) −7.81001 −0.247470
\(997\) 18.3016 0.579616 0.289808 0.957085i \(-0.406409\pi\)
0.289808 + 0.957085i \(0.406409\pi\)
\(998\) −23.4178 −0.741277
\(999\) 9.68872 0.306537
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.bb.1.7 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bb.1.7 14 1.1 even 1 trivial