Properties

Label 8034.2.a.ba.1.6
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} - 4126 x^{5} - 22759 x^{4} + 2997 x^{3} + 10504 x^{2} - 506 x - 1492\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.891518\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.891518 q^{5} +1.00000 q^{6} +4.07845 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.891518 q^{5} +1.00000 q^{6} +4.07845 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.891518 q^{10} -4.41011 q^{11} -1.00000 q^{12} -1.00000 q^{13} -4.07845 q^{14} +0.891518 q^{15} +1.00000 q^{16} -3.83133 q^{17} -1.00000 q^{18} +1.31800 q^{19} -0.891518 q^{20} -4.07845 q^{21} +4.41011 q^{22} -6.87898 q^{23} +1.00000 q^{24} -4.20520 q^{25} +1.00000 q^{26} -1.00000 q^{27} +4.07845 q^{28} -7.06622 q^{29} -0.891518 q^{30} -4.56821 q^{31} -1.00000 q^{32} +4.41011 q^{33} +3.83133 q^{34} -3.63601 q^{35} +1.00000 q^{36} -8.36395 q^{37} -1.31800 q^{38} +1.00000 q^{39} +0.891518 q^{40} -0.436547 q^{41} +4.07845 q^{42} -3.87678 q^{43} -4.41011 q^{44} -0.891518 q^{45} +6.87898 q^{46} +1.61571 q^{47} -1.00000 q^{48} +9.63379 q^{49} +4.20520 q^{50} +3.83133 q^{51} -1.00000 q^{52} +4.54487 q^{53} +1.00000 q^{54} +3.93169 q^{55} -4.07845 q^{56} -1.31800 q^{57} +7.06622 q^{58} +3.89094 q^{59} +0.891518 q^{60} +10.3612 q^{61} +4.56821 q^{62} +4.07845 q^{63} +1.00000 q^{64} +0.891518 q^{65} -4.41011 q^{66} +3.87397 q^{67} -3.83133 q^{68} +6.87898 q^{69} +3.63601 q^{70} -8.92491 q^{71} -1.00000 q^{72} +9.37758 q^{73} +8.36395 q^{74} +4.20520 q^{75} +1.31800 q^{76} -17.9864 q^{77} -1.00000 q^{78} +8.17661 q^{79} -0.891518 q^{80} +1.00000 q^{81} +0.436547 q^{82} +4.84486 q^{83} -4.07845 q^{84} +3.41570 q^{85} +3.87678 q^{86} +7.06622 q^{87} +4.41011 q^{88} +10.7038 q^{89} +0.891518 q^{90} -4.07845 q^{91} -6.87898 q^{92} +4.56821 q^{93} -1.61571 q^{94} -1.17502 q^{95} +1.00000 q^{96} -1.63498 q^{97} -9.63379 q^{98} -4.41011 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + q^{10} - q^{11} - 14q^{12} - 14q^{13} - 5q^{14} + q^{15} + 14q^{16} - 12q^{17} - 14q^{18} + 10q^{19} - q^{20} - 5q^{21} + q^{22} - q^{23} + 14q^{24} + 19q^{25} + 14q^{26} - 14q^{27} + 5q^{28} + 6q^{29} - q^{30} + 20q^{31} - 14q^{32} + q^{33} + 12q^{34} - 16q^{35} + 14q^{36} - 3q^{37} - 10q^{38} + 14q^{39} + q^{40} + q^{41} + 5q^{42} + 6q^{43} - q^{44} - q^{45} + q^{46} - 13q^{47} - 14q^{48} + 9q^{49} - 19q^{50} + 12q^{51} - 14q^{52} - 27q^{53} + 14q^{54} + 10q^{55} - 5q^{56} - 10q^{57} - 6q^{58} - 6q^{59} + q^{60} - 4q^{61} - 20q^{62} + 5q^{63} + 14q^{64} + q^{65} - q^{66} + 13q^{67} - 12q^{68} + q^{69} + 16q^{70} + 18q^{71} - 14q^{72} + 11q^{73} + 3q^{74} - 19q^{75} + 10q^{76} - 15q^{77} - 14q^{78} + 33q^{79} - q^{80} + 14q^{81} - q^{82} - 25q^{83} - 5q^{84} + 25q^{85} - 6q^{86} - 6q^{87} + q^{88} + 3q^{89} + q^{90} - 5q^{91} - q^{92} - 20q^{93} + 13q^{94} + 30q^{95} + 14q^{96} + 11q^{97} - 9q^{98} - q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −0.891518 −0.398699 −0.199349 0.979928i \(-0.563883\pi\)
−0.199349 + 0.979928i \(0.563883\pi\)
\(6\) 1.00000 0.408248
\(7\) 4.07845 1.54151 0.770755 0.637131i \(-0.219879\pi\)
0.770755 + 0.637131i \(0.219879\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0.891518 0.281923
\(11\) −4.41011 −1.32970 −0.664849 0.746978i \(-0.731504\pi\)
−0.664849 + 0.746978i \(0.731504\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −4.07845 −1.09001
\(15\) 0.891518 0.230189
\(16\) 1.00000 0.250000
\(17\) −3.83133 −0.929235 −0.464617 0.885512i \(-0.653808\pi\)
−0.464617 + 0.885512i \(0.653808\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.31800 0.302371 0.151185 0.988505i \(-0.451691\pi\)
0.151185 + 0.988505i \(0.451691\pi\)
\(20\) −0.891518 −0.199349
\(21\) −4.07845 −0.889992
\(22\) 4.41011 0.940238
\(23\) −6.87898 −1.43437 −0.717183 0.696885i \(-0.754569\pi\)
−0.717183 + 0.696885i \(0.754569\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.20520 −0.841039
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 4.07845 0.770755
\(29\) −7.06622 −1.31216 −0.656082 0.754690i \(-0.727787\pi\)
−0.656082 + 0.754690i \(0.727787\pi\)
\(30\) −0.891518 −0.162768
\(31\) −4.56821 −0.820475 −0.410238 0.911979i \(-0.634554\pi\)
−0.410238 + 0.911979i \(0.634554\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.41011 0.767701
\(34\) 3.83133 0.657068
\(35\) −3.63601 −0.614599
\(36\) 1.00000 0.166667
\(37\) −8.36395 −1.37502 −0.687512 0.726173i \(-0.741297\pi\)
−0.687512 + 0.726173i \(0.741297\pi\)
\(38\) −1.31800 −0.213808
\(39\) 1.00000 0.160128
\(40\) 0.891518 0.140961
\(41\) −0.436547 −0.0681771 −0.0340886 0.999419i \(-0.510853\pi\)
−0.0340886 + 0.999419i \(0.510853\pi\)
\(42\) 4.07845 0.629319
\(43\) −3.87678 −0.591204 −0.295602 0.955311i \(-0.595520\pi\)
−0.295602 + 0.955311i \(0.595520\pi\)
\(44\) −4.41011 −0.664849
\(45\) −0.891518 −0.132900
\(46\) 6.87898 1.01425
\(47\) 1.61571 0.235675 0.117838 0.993033i \(-0.462404\pi\)
0.117838 + 0.993033i \(0.462404\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.63379 1.37626
\(50\) 4.20520 0.594704
\(51\) 3.83133 0.536494
\(52\) −1.00000 −0.138675
\(53\) 4.54487 0.624286 0.312143 0.950035i \(-0.398953\pi\)
0.312143 + 0.950035i \(0.398953\pi\)
\(54\) 1.00000 0.136083
\(55\) 3.93169 0.530149
\(56\) −4.07845 −0.545006
\(57\) −1.31800 −0.174574
\(58\) 7.06622 0.927840
\(59\) 3.89094 0.506557 0.253279 0.967393i \(-0.418491\pi\)
0.253279 + 0.967393i \(0.418491\pi\)
\(60\) 0.891518 0.115094
\(61\) 10.3612 1.32662 0.663308 0.748346i \(-0.269152\pi\)
0.663308 + 0.748346i \(0.269152\pi\)
\(62\) 4.56821 0.580164
\(63\) 4.07845 0.513837
\(64\) 1.00000 0.125000
\(65\) 0.891518 0.110579
\(66\) −4.41011 −0.542847
\(67\) 3.87397 0.473281 0.236641 0.971597i \(-0.423954\pi\)
0.236641 + 0.971597i \(0.423954\pi\)
\(68\) −3.83133 −0.464617
\(69\) 6.87898 0.828132
\(70\) 3.63601 0.434587
\(71\) −8.92491 −1.05919 −0.529596 0.848250i \(-0.677657\pi\)
−0.529596 + 0.848250i \(0.677657\pi\)
\(72\) −1.00000 −0.117851
\(73\) 9.37758 1.09756 0.548781 0.835966i \(-0.315092\pi\)
0.548781 + 0.835966i \(0.315092\pi\)
\(74\) 8.36395 0.972289
\(75\) 4.20520 0.485574
\(76\) 1.31800 0.151185
\(77\) −17.9864 −2.04974
\(78\) −1.00000 −0.113228
\(79\) 8.17661 0.919941 0.459970 0.887934i \(-0.347860\pi\)
0.459970 + 0.887934i \(0.347860\pi\)
\(80\) −0.891518 −0.0996747
\(81\) 1.00000 0.111111
\(82\) 0.436547 0.0482085
\(83\) 4.84486 0.531793 0.265896 0.964002i \(-0.414332\pi\)
0.265896 + 0.964002i \(0.414332\pi\)
\(84\) −4.07845 −0.444996
\(85\) 3.41570 0.370485
\(86\) 3.87678 0.418044
\(87\) 7.06622 0.757578
\(88\) 4.41011 0.470119
\(89\) 10.7038 1.13460 0.567301 0.823511i \(-0.307988\pi\)
0.567301 + 0.823511i \(0.307988\pi\)
\(90\) 0.891518 0.0939742
\(91\) −4.07845 −0.427538
\(92\) −6.87898 −0.717183
\(93\) 4.56821 0.473702
\(94\) −1.61571 −0.166647
\(95\) −1.17502 −0.120555
\(96\) 1.00000 0.102062
\(97\) −1.63498 −0.166007 −0.0830037 0.996549i \(-0.526451\pi\)
−0.0830037 + 0.996549i \(0.526451\pi\)
\(98\) −9.63379 −0.973160
\(99\) −4.41011 −0.443232
\(100\) −4.20520 −0.420520
\(101\) 7.15302 0.711752 0.355876 0.934533i \(-0.384182\pi\)
0.355876 + 0.934533i \(0.384182\pi\)
\(102\) −3.83133 −0.379358
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 3.63601 0.354839
\(106\) −4.54487 −0.441437
\(107\) −6.28356 −0.607455 −0.303728 0.952759i \(-0.598231\pi\)
−0.303728 + 0.952759i \(0.598231\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −3.77713 −0.361784 −0.180892 0.983503i \(-0.557898\pi\)
−0.180892 + 0.983503i \(0.557898\pi\)
\(110\) −3.93169 −0.374872
\(111\) 8.36395 0.793871
\(112\) 4.07845 0.385378
\(113\) 10.7015 1.00671 0.503355 0.864079i \(-0.332099\pi\)
0.503355 + 0.864079i \(0.332099\pi\)
\(114\) 1.31800 0.123442
\(115\) 6.13273 0.571880
\(116\) −7.06622 −0.656082
\(117\) −1.00000 −0.0924500
\(118\) −3.89094 −0.358190
\(119\) −15.6259 −1.43243
\(120\) −0.891518 −0.0813841
\(121\) 8.44905 0.768095
\(122\) −10.3612 −0.938060
\(123\) 0.436547 0.0393621
\(124\) −4.56821 −0.410238
\(125\) 8.20660 0.734020
\(126\) −4.07845 −0.363338
\(127\) 14.7723 1.31083 0.655417 0.755268i \(-0.272493\pi\)
0.655417 + 0.755268i \(0.272493\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 3.87678 0.341332
\(130\) −0.891518 −0.0781913
\(131\) −0.121373 −0.0106044 −0.00530219 0.999986i \(-0.501688\pi\)
−0.00530219 + 0.999986i \(0.501688\pi\)
\(132\) 4.41011 0.383851
\(133\) 5.37541 0.466108
\(134\) −3.87397 −0.334660
\(135\) 0.891518 0.0767296
\(136\) 3.83133 0.328534
\(137\) −18.9964 −1.62297 −0.811485 0.584373i \(-0.801340\pi\)
−0.811485 + 0.584373i \(0.801340\pi\)
\(138\) −6.87898 −0.585577
\(139\) 15.1465 1.28471 0.642354 0.766408i \(-0.277958\pi\)
0.642354 + 0.766408i \(0.277958\pi\)
\(140\) −3.63601 −0.307299
\(141\) −1.61571 −0.136067
\(142\) 8.92491 0.748962
\(143\) 4.41011 0.368792
\(144\) 1.00000 0.0833333
\(145\) 6.29966 0.523158
\(146\) −9.37758 −0.776094
\(147\) −9.63379 −0.794582
\(148\) −8.36395 −0.687512
\(149\) −4.78470 −0.391978 −0.195989 0.980606i \(-0.562792\pi\)
−0.195989 + 0.980606i \(0.562792\pi\)
\(150\) −4.20520 −0.343353
\(151\) −21.4491 −1.74550 −0.872751 0.488167i \(-0.837666\pi\)
−0.872751 + 0.488167i \(0.837666\pi\)
\(152\) −1.31800 −0.106904
\(153\) −3.83133 −0.309745
\(154\) 17.9864 1.44939
\(155\) 4.07264 0.327123
\(156\) 1.00000 0.0800641
\(157\) 10.9038 0.870219 0.435110 0.900377i \(-0.356710\pi\)
0.435110 + 0.900377i \(0.356710\pi\)
\(158\) −8.17661 −0.650496
\(159\) −4.54487 −0.360432
\(160\) 0.891518 0.0704807
\(161\) −28.0556 −2.21109
\(162\) −1.00000 −0.0785674
\(163\) −5.98549 −0.468820 −0.234410 0.972138i \(-0.575316\pi\)
−0.234410 + 0.972138i \(0.575316\pi\)
\(164\) −0.436547 −0.0340886
\(165\) −3.93169 −0.306082
\(166\) −4.84486 −0.376034
\(167\) 18.9623 1.46735 0.733674 0.679501i \(-0.237804\pi\)
0.733674 + 0.679501i \(0.237804\pi\)
\(168\) 4.07845 0.314660
\(169\) 1.00000 0.0769231
\(170\) −3.41570 −0.261972
\(171\) 1.31800 0.100790
\(172\) −3.87678 −0.295602
\(173\) 19.1488 1.45586 0.727928 0.685654i \(-0.240483\pi\)
0.727928 + 0.685654i \(0.240483\pi\)
\(174\) −7.06622 −0.535689
\(175\) −17.1507 −1.29647
\(176\) −4.41011 −0.332424
\(177\) −3.89094 −0.292461
\(178\) −10.7038 −0.802284
\(179\) 14.7047 1.09908 0.549540 0.835467i \(-0.314803\pi\)
0.549540 + 0.835467i \(0.314803\pi\)
\(180\) −0.891518 −0.0664498
\(181\) 5.74905 0.427323 0.213662 0.976908i \(-0.431461\pi\)
0.213662 + 0.976908i \(0.431461\pi\)
\(182\) 4.07845 0.302315
\(183\) −10.3612 −0.765923
\(184\) 6.87898 0.507125
\(185\) 7.45661 0.548221
\(186\) −4.56821 −0.334958
\(187\) 16.8966 1.23560
\(188\) 1.61571 0.117838
\(189\) −4.07845 −0.296664
\(190\) 1.17502 0.0852451
\(191\) 22.7543 1.64644 0.823222 0.567719i \(-0.192174\pi\)
0.823222 + 0.567719i \(0.192174\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 12.9628 0.933084 0.466542 0.884499i \(-0.345500\pi\)
0.466542 + 0.884499i \(0.345500\pi\)
\(194\) 1.63498 0.117385
\(195\) −0.891518 −0.0638429
\(196\) 9.63379 0.688128
\(197\) 13.9660 0.995033 0.497517 0.867454i \(-0.334245\pi\)
0.497517 + 0.867454i \(0.334245\pi\)
\(198\) 4.41011 0.313413
\(199\) 18.9472 1.34313 0.671567 0.740944i \(-0.265622\pi\)
0.671567 + 0.740944i \(0.265622\pi\)
\(200\) 4.20520 0.297352
\(201\) −3.87397 −0.273249
\(202\) −7.15302 −0.503285
\(203\) −28.8193 −2.02271
\(204\) 3.83133 0.268247
\(205\) 0.389189 0.0271822
\(206\) −1.00000 −0.0696733
\(207\) −6.87898 −0.478122
\(208\) −1.00000 −0.0693375
\(209\) −5.81253 −0.402061
\(210\) −3.63601 −0.250909
\(211\) −25.0073 −1.72157 −0.860785 0.508968i \(-0.830027\pi\)
−0.860785 + 0.508968i \(0.830027\pi\)
\(212\) 4.54487 0.312143
\(213\) 8.92491 0.611525
\(214\) 6.28356 0.429536
\(215\) 3.45622 0.235712
\(216\) 1.00000 0.0680414
\(217\) −18.6312 −1.26477
\(218\) 3.77713 0.255820
\(219\) −9.37758 −0.633678
\(220\) 3.93169 0.265074
\(221\) 3.83133 0.257723
\(222\) −8.36395 −0.561351
\(223\) 12.6477 0.846950 0.423475 0.905908i \(-0.360810\pi\)
0.423475 + 0.905908i \(0.360810\pi\)
\(224\) −4.07845 −0.272503
\(225\) −4.20520 −0.280346
\(226\) −10.7015 −0.711852
\(227\) 6.84365 0.454229 0.227115 0.973868i \(-0.427071\pi\)
0.227115 + 0.973868i \(0.427071\pi\)
\(228\) −1.31800 −0.0872869
\(229\) −1.06504 −0.0703799 −0.0351900 0.999381i \(-0.511204\pi\)
−0.0351900 + 0.999381i \(0.511204\pi\)
\(230\) −6.13273 −0.404380
\(231\) 17.9864 1.18342
\(232\) 7.06622 0.463920
\(233\) −4.95796 −0.324807 −0.162403 0.986724i \(-0.551925\pi\)
−0.162403 + 0.986724i \(0.551925\pi\)
\(234\) 1.00000 0.0653720
\(235\) −1.44043 −0.0939634
\(236\) 3.89094 0.253279
\(237\) −8.17661 −0.531128
\(238\) 15.6259 1.01288
\(239\) −17.9214 −1.15924 −0.579621 0.814886i \(-0.696799\pi\)
−0.579621 + 0.814886i \(0.696799\pi\)
\(240\) 0.891518 0.0575472
\(241\) 22.2265 1.43174 0.715868 0.698235i \(-0.246031\pi\)
0.715868 + 0.698235i \(0.246031\pi\)
\(242\) −8.44905 −0.543125
\(243\) −1.00000 −0.0641500
\(244\) 10.3612 0.663308
\(245\) −8.58870 −0.548712
\(246\) −0.436547 −0.0278332
\(247\) −1.31800 −0.0838625
\(248\) 4.56821 0.290082
\(249\) −4.84486 −0.307031
\(250\) −8.20660 −0.519031
\(251\) −19.5551 −1.23431 −0.617153 0.786843i \(-0.711714\pi\)
−0.617153 + 0.786843i \(0.711714\pi\)
\(252\) 4.07845 0.256918
\(253\) 30.3370 1.90727
\(254\) −14.7723 −0.926899
\(255\) −3.41570 −0.213900
\(256\) 1.00000 0.0625000
\(257\) −6.39342 −0.398810 −0.199405 0.979917i \(-0.563901\pi\)
−0.199405 + 0.979917i \(0.563901\pi\)
\(258\) −3.87678 −0.241358
\(259\) −34.1120 −2.11961
\(260\) 0.891518 0.0552896
\(261\) −7.06622 −0.437388
\(262\) 0.121373 0.00749843
\(263\) 1.21355 0.0748306 0.0374153 0.999300i \(-0.488088\pi\)
0.0374153 + 0.999300i \(0.488088\pi\)
\(264\) −4.41011 −0.271423
\(265\) −4.05184 −0.248902
\(266\) −5.37541 −0.329588
\(267\) −10.7038 −0.655062
\(268\) 3.87397 0.236641
\(269\) 15.0912 0.920127 0.460064 0.887886i \(-0.347827\pi\)
0.460064 + 0.887886i \(0.347827\pi\)
\(270\) −0.891518 −0.0542561
\(271\) 26.4715 1.60803 0.804015 0.594610i \(-0.202693\pi\)
0.804015 + 0.594610i \(0.202693\pi\)
\(272\) −3.83133 −0.232309
\(273\) 4.07845 0.246839
\(274\) 18.9964 1.14761
\(275\) 18.5454 1.11833
\(276\) 6.87898 0.414066
\(277\) −22.7248 −1.36540 −0.682699 0.730699i \(-0.739194\pi\)
−0.682699 + 0.730699i \(0.739194\pi\)
\(278\) −15.1465 −0.908426
\(279\) −4.56821 −0.273492
\(280\) 3.63601 0.217293
\(281\) 9.21995 0.550016 0.275008 0.961442i \(-0.411319\pi\)
0.275008 + 0.961442i \(0.411319\pi\)
\(282\) 1.61571 0.0962139
\(283\) −21.5212 −1.27930 −0.639650 0.768666i \(-0.720921\pi\)
−0.639650 + 0.768666i \(0.720921\pi\)
\(284\) −8.92491 −0.529596
\(285\) 1.17502 0.0696024
\(286\) −4.41011 −0.260775
\(287\) −1.78044 −0.105096
\(288\) −1.00000 −0.0589256
\(289\) −2.32089 −0.136523
\(290\) −6.29966 −0.369929
\(291\) 1.63498 0.0958444
\(292\) 9.37758 0.548781
\(293\) −24.1063 −1.40831 −0.704154 0.710047i \(-0.748673\pi\)
−0.704154 + 0.710047i \(0.748673\pi\)
\(294\) 9.63379 0.561854
\(295\) −3.46884 −0.201964
\(296\) 8.36395 0.486144
\(297\) 4.41011 0.255900
\(298\) 4.78470 0.277170
\(299\) 6.87898 0.397822
\(300\) 4.20520 0.242787
\(301\) −15.8113 −0.911347
\(302\) 21.4491 1.23426
\(303\) −7.15302 −0.410930
\(304\) 1.31800 0.0755927
\(305\) −9.23720 −0.528921
\(306\) 3.83133 0.219023
\(307\) −28.1373 −1.60588 −0.802941 0.596059i \(-0.796733\pi\)
−0.802941 + 0.596059i \(0.796733\pi\)
\(308\) −17.9864 −1.02487
\(309\) −1.00000 −0.0568880
\(310\) −4.07264 −0.231311
\(311\) 22.9358 1.30057 0.650285 0.759691i \(-0.274650\pi\)
0.650285 + 0.759691i \(0.274650\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −21.1390 −1.19485 −0.597424 0.801925i \(-0.703809\pi\)
−0.597424 + 0.801925i \(0.703809\pi\)
\(314\) −10.9038 −0.615338
\(315\) −3.63601 −0.204866
\(316\) 8.17661 0.459970
\(317\) −12.2082 −0.685680 −0.342840 0.939394i \(-0.611389\pi\)
−0.342840 + 0.939394i \(0.611389\pi\)
\(318\) 4.54487 0.254864
\(319\) 31.1628 1.74478
\(320\) −0.891518 −0.0498374
\(321\) 6.28356 0.350714
\(322\) 28.0556 1.56348
\(323\) −5.04971 −0.280973
\(324\) 1.00000 0.0555556
\(325\) 4.20520 0.233262
\(326\) 5.98549 0.331506
\(327\) 3.77713 0.208876
\(328\) 0.436547 0.0241043
\(329\) 6.58959 0.363296
\(330\) 3.93169 0.216432
\(331\) −1.90976 −0.104970 −0.0524849 0.998622i \(-0.516714\pi\)
−0.0524849 + 0.998622i \(0.516714\pi\)
\(332\) 4.84486 0.265896
\(333\) −8.36395 −0.458341
\(334\) −18.9623 −1.03757
\(335\) −3.45372 −0.188697
\(336\) −4.07845 −0.222498
\(337\) 3.62376 0.197399 0.0986994 0.995117i \(-0.468532\pi\)
0.0986994 + 0.995117i \(0.468532\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −10.7015 −0.581225
\(340\) 3.41570 0.185242
\(341\) 20.1463 1.09098
\(342\) −1.31800 −0.0712694
\(343\) 10.7418 0.580002
\(344\) 3.87678 0.209022
\(345\) −6.13273 −0.330175
\(346\) −19.1488 −1.02945
\(347\) −2.20372 −0.118302 −0.0591509 0.998249i \(-0.518839\pi\)
−0.0591509 + 0.998249i \(0.518839\pi\)
\(348\) 7.06622 0.378789
\(349\) −4.15341 −0.222327 −0.111164 0.993802i \(-0.535458\pi\)
−0.111164 + 0.993802i \(0.535458\pi\)
\(350\) 17.1507 0.916743
\(351\) 1.00000 0.0533761
\(352\) 4.41011 0.235060
\(353\) 15.1988 0.808949 0.404474 0.914549i \(-0.367454\pi\)
0.404474 + 0.914549i \(0.367454\pi\)
\(354\) 3.89094 0.206801
\(355\) 7.95672 0.422299
\(356\) 10.7038 0.567301
\(357\) 15.6259 0.827011
\(358\) −14.7047 −0.777167
\(359\) −8.81818 −0.465406 −0.232703 0.972548i \(-0.574757\pi\)
−0.232703 + 0.972548i \(0.574757\pi\)
\(360\) 0.891518 0.0469871
\(361\) −17.2629 −0.908572
\(362\) −5.74905 −0.302163
\(363\) −8.44905 −0.443460
\(364\) −4.07845 −0.213769
\(365\) −8.36028 −0.437597
\(366\) 10.3612 0.541589
\(367\) −22.3664 −1.16752 −0.583758 0.811928i \(-0.698418\pi\)
−0.583758 + 0.811928i \(0.698418\pi\)
\(368\) −6.87898 −0.358591
\(369\) −0.436547 −0.0227257
\(370\) −7.45661 −0.387651
\(371\) 18.5361 0.962344
\(372\) 4.56821 0.236851
\(373\) −24.7009 −1.27896 −0.639481 0.768807i \(-0.720851\pi\)
−0.639481 + 0.768807i \(0.720851\pi\)
\(374\) −16.8966 −0.873702
\(375\) −8.20660 −0.423787
\(376\) −1.61571 −0.0833237
\(377\) 7.06622 0.363929
\(378\) 4.07845 0.209773
\(379\) −24.3122 −1.24884 −0.624418 0.781091i \(-0.714664\pi\)
−0.624418 + 0.781091i \(0.714664\pi\)
\(380\) −1.17502 −0.0602774
\(381\) −14.7723 −0.756810
\(382\) −22.7543 −1.16421
\(383\) 5.30660 0.271155 0.135577 0.990767i \(-0.456711\pi\)
0.135577 + 0.990767i \(0.456711\pi\)
\(384\) 1.00000 0.0510310
\(385\) 16.0352 0.817230
\(386\) −12.9628 −0.659790
\(387\) −3.87678 −0.197068
\(388\) −1.63498 −0.0830037
\(389\) −28.2582 −1.43275 −0.716374 0.697716i \(-0.754200\pi\)
−0.716374 + 0.697716i \(0.754200\pi\)
\(390\) 0.891518 0.0451438
\(391\) 26.3557 1.33286
\(392\) −9.63379 −0.486580
\(393\) 0.121373 0.00612244
\(394\) −13.9660 −0.703595
\(395\) −7.28959 −0.366779
\(396\) −4.41011 −0.221616
\(397\) 5.52509 0.277296 0.138648 0.990342i \(-0.455724\pi\)
0.138648 + 0.990342i \(0.455724\pi\)
\(398\) −18.9472 −0.949739
\(399\) −5.37541 −0.269107
\(400\) −4.20520 −0.210260
\(401\) 20.3794 1.01770 0.508850 0.860855i \(-0.330071\pi\)
0.508850 + 0.860855i \(0.330071\pi\)
\(402\) 3.87397 0.193216
\(403\) 4.56821 0.227559
\(404\) 7.15302 0.355876
\(405\) −0.891518 −0.0442999
\(406\) 28.8193 1.43028
\(407\) 36.8859 1.82837
\(408\) −3.83133 −0.189679
\(409\) 12.8602 0.635896 0.317948 0.948108i \(-0.397006\pi\)
0.317948 + 0.948108i \(0.397006\pi\)
\(410\) −0.389189 −0.0192207
\(411\) 18.9964 0.937022
\(412\) 1.00000 0.0492665
\(413\) 15.8690 0.780864
\(414\) 6.87898 0.338083
\(415\) −4.31928 −0.212025
\(416\) 1.00000 0.0490290
\(417\) −15.1465 −0.741727
\(418\) 5.81253 0.284300
\(419\) 17.2438 0.842413 0.421207 0.906965i \(-0.361607\pi\)
0.421207 + 0.906965i \(0.361607\pi\)
\(420\) 3.63601 0.177419
\(421\) −8.60431 −0.419348 −0.209674 0.977771i \(-0.567240\pi\)
−0.209674 + 0.977771i \(0.567240\pi\)
\(422\) 25.0073 1.21733
\(423\) 1.61571 0.0785584
\(424\) −4.54487 −0.220719
\(425\) 16.1115 0.781523
\(426\) −8.92491 −0.432413
\(427\) 42.2577 2.04499
\(428\) −6.28356 −0.303728
\(429\) −4.41011 −0.212922
\(430\) −3.45622 −0.166674
\(431\) 29.0333 1.39848 0.699241 0.714886i \(-0.253521\pi\)
0.699241 + 0.714886i \(0.253521\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 13.7831 0.662373 0.331186 0.943565i \(-0.392551\pi\)
0.331186 + 0.943565i \(0.392551\pi\)
\(434\) 18.6312 0.894328
\(435\) −6.29966 −0.302046
\(436\) −3.77713 −0.180892
\(437\) −9.06651 −0.433710
\(438\) 9.37758 0.448078
\(439\) −14.3736 −0.686013 −0.343007 0.939333i \(-0.611445\pi\)
−0.343007 + 0.939333i \(0.611445\pi\)
\(440\) −3.93169 −0.187436
\(441\) 9.63379 0.458752
\(442\) −3.83133 −0.182238
\(443\) −4.89352 −0.232498 −0.116249 0.993220i \(-0.537087\pi\)
−0.116249 + 0.993220i \(0.537087\pi\)
\(444\) 8.36395 0.396935
\(445\) −9.54264 −0.452364
\(446\) −12.6477 −0.598884
\(447\) 4.78470 0.226309
\(448\) 4.07845 0.192689
\(449\) 15.7086 0.741337 0.370668 0.928765i \(-0.379129\pi\)
0.370668 + 0.928765i \(0.379129\pi\)
\(450\) 4.20520 0.198235
\(451\) 1.92522 0.0906550
\(452\) 10.7015 0.503355
\(453\) 21.4491 1.00777
\(454\) −6.84365 −0.321188
\(455\) 3.63601 0.170459
\(456\) 1.31800 0.0617211
\(457\) 39.5699 1.85100 0.925500 0.378747i \(-0.123645\pi\)
0.925500 + 0.378747i \(0.123645\pi\)
\(458\) 1.06504 0.0497661
\(459\) 3.83133 0.178831
\(460\) 6.13273 0.285940
\(461\) −26.8050 −1.24843 −0.624216 0.781252i \(-0.714582\pi\)
−0.624216 + 0.781252i \(0.714582\pi\)
\(462\) −17.9864 −0.836804
\(463\) 21.2693 0.988470 0.494235 0.869328i \(-0.335448\pi\)
0.494235 + 0.869328i \(0.335448\pi\)
\(464\) −7.06622 −0.328041
\(465\) −4.07264 −0.188864
\(466\) 4.95796 0.229673
\(467\) −0.247842 −0.0114688 −0.00573438 0.999984i \(-0.501825\pi\)
−0.00573438 + 0.999984i \(0.501825\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 15.7998 0.729568
\(470\) 1.44043 0.0664422
\(471\) −10.9038 −0.502421
\(472\) −3.89094 −0.179095
\(473\) 17.0970 0.786122
\(474\) 8.17661 0.375564
\(475\) −5.54246 −0.254306
\(476\) −15.6259 −0.716213
\(477\) 4.54487 0.208095
\(478\) 17.9214 0.819708
\(479\) −6.39802 −0.292333 −0.146167 0.989260i \(-0.546694\pi\)
−0.146167 + 0.989260i \(0.546694\pi\)
\(480\) −0.891518 −0.0406920
\(481\) 8.36395 0.381363
\(482\) −22.2265 −1.01239
\(483\) 28.0556 1.27657
\(484\) 8.44905 0.384048
\(485\) 1.45762 0.0661869
\(486\) 1.00000 0.0453609
\(487\) −13.1572 −0.596208 −0.298104 0.954533i \(-0.596354\pi\)
−0.298104 + 0.954533i \(0.596354\pi\)
\(488\) −10.3612 −0.469030
\(489\) 5.98549 0.270673
\(490\) 8.58870 0.387998
\(491\) 17.1405 0.773538 0.386769 0.922177i \(-0.373591\pi\)
0.386769 + 0.922177i \(0.373591\pi\)
\(492\) 0.436547 0.0196810
\(493\) 27.0730 1.21931
\(494\) 1.31800 0.0592998
\(495\) 3.93169 0.176716
\(496\) −4.56821 −0.205119
\(497\) −36.3999 −1.63276
\(498\) 4.84486 0.217103
\(499\) −6.32801 −0.283281 −0.141640 0.989918i \(-0.545238\pi\)
−0.141640 + 0.989918i \(0.545238\pi\)
\(500\) 8.20660 0.367010
\(501\) −18.9623 −0.847174
\(502\) 19.5551 0.872786
\(503\) −9.22271 −0.411220 −0.205610 0.978634i \(-0.565918\pi\)
−0.205610 + 0.978634i \(0.565918\pi\)
\(504\) −4.07845 −0.181669
\(505\) −6.37705 −0.283775
\(506\) −30.3370 −1.34865
\(507\) −1.00000 −0.0444116
\(508\) 14.7723 0.655417
\(509\) −4.34576 −0.192622 −0.0963111 0.995351i \(-0.530704\pi\)
−0.0963111 + 0.995351i \(0.530704\pi\)
\(510\) 3.41570 0.151250
\(511\) 38.2460 1.69190
\(512\) −1.00000 −0.0441942
\(513\) −1.31800 −0.0581913
\(514\) 6.39342 0.282002
\(515\) −0.891518 −0.0392850
\(516\) 3.87678 0.170666
\(517\) −7.12544 −0.313377
\(518\) 34.1120 1.49879
\(519\) −19.1488 −0.840539
\(520\) −0.891518 −0.0390956
\(521\) −19.0186 −0.833220 −0.416610 0.909085i \(-0.636782\pi\)
−0.416610 + 0.909085i \(0.636782\pi\)
\(522\) 7.06622 0.309280
\(523\) 29.5410 1.29174 0.645868 0.763449i \(-0.276496\pi\)
0.645868 + 0.763449i \(0.276496\pi\)
\(524\) −0.121373 −0.00530219
\(525\) 17.1507 0.748518
\(526\) −1.21355 −0.0529132
\(527\) 17.5023 0.762414
\(528\) 4.41011 0.191925
\(529\) 24.3203 1.05741
\(530\) 4.05184 0.176001
\(531\) 3.89094 0.168852
\(532\) 5.37541 0.233054
\(533\) 0.436547 0.0189089
\(534\) 10.7038 0.463199
\(535\) 5.60191 0.242192
\(536\) −3.87397 −0.167330
\(537\) −14.7047 −0.634554
\(538\) −15.0912 −0.650628
\(539\) −42.4860 −1.83000
\(540\) 0.891518 0.0383648
\(541\) 21.7968 0.937117 0.468559 0.883432i \(-0.344773\pi\)
0.468559 + 0.883432i \(0.344773\pi\)
\(542\) −26.4715 −1.13705
\(543\) −5.74905 −0.246715
\(544\) 3.83133 0.164267
\(545\) 3.36738 0.144243
\(546\) −4.07845 −0.174542
\(547\) −4.43820 −0.189764 −0.0948819 0.995489i \(-0.530247\pi\)
−0.0948819 + 0.995489i \(0.530247\pi\)
\(548\) −18.9964 −0.811485
\(549\) 10.3612 0.442206
\(550\) −18.5454 −0.790777
\(551\) −9.31330 −0.396760
\(552\) −6.87898 −0.292789
\(553\) 33.3479 1.41810
\(554\) 22.7248 0.965483
\(555\) −7.45661 −0.316515
\(556\) 15.1465 0.642354
\(557\) −0.959232 −0.0406439 −0.0203220 0.999793i \(-0.506469\pi\)
−0.0203220 + 0.999793i \(0.506469\pi\)
\(558\) 4.56821 0.193388
\(559\) 3.87678 0.163970
\(560\) −3.63601 −0.153650
\(561\) −16.8966 −0.713375
\(562\) −9.21995 −0.388920
\(563\) −7.04973 −0.297111 −0.148555 0.988904i \(-0.547462\pi\)
−0.148555 + 0.988904i \(0.547462\pi\)
\(564\) −1.61571 −0.0680335
\(565\) −9.54056 −0.401375
\(566\) 21.5212 0.904602
\(567\) 4.07845 0.171279
\(568\) 8.92491 0.374481
\(569\) 26.5564 1.11330 0.556651 0.830746i \(-0.312086\pi\)
0.556651 + 0.830746i \(0.312086\pi\)
\(570\) −1.17502 −0.0492163
\(571\) −2.11440 −0.0884849 −0.0442424 0.999021i \(-0.514087\pi\)
−0.0442424 + 0.999021i \(0.514087\pi\)
\(572\) 4.41011 0.184396
\(573\) −22.7543 −0.950575
\(574\) 1.78044 0.0743140
\(575\) 28.9274 1.20636
\(576\) 1.00000 0.0416667
\(577\) 5.24850 0.218498 0.109249 0.994014i \(-0.465155\pi\)
0.109249 + 0.994014i \(0.465155\pi\)
\(578\) 2.32089 0.0965362
\(579\) −12.9628 −0.538716
\(580\) 6.29966 0.261579
\(581\) 19.7595 0.819764
\(582\) −1.63498 −0.0677722
\(583\) −20.0434 −0.830112
\(584\) −9.37758 −0.388047
\(585\) 0.891518 0.0368597
\(586\) 24.1063 0.995824
\(587\) 14.6107 0.603049 0.301524 0.953458i \(-0.402505\pi\)
0.301524 + 0.953458i \(0.402505\pi\)
\(588\) −9.63379 −0.397291
\(589\) −6.02092 −0.248088
\(590\) 3.46884 0.142810
\(591\) −13.9660 −0.574483
\(592\) −8.36395 −0.343756
\(593\) 1.63231 0.0670308 0.0335154 0.999438i \(-0.489330\pi\)
0.0335154 + 0.999438i \(0.489330\pi\)
\(594\) −4.41011 −0.180949
\(595\) 13.9308 0.571106
\(596\) −4.78470 −0.195989
\(597\) −18.9472 −0.775459
\(598\) −6.87898 −0.281302
\(599\) 1.16804 0.0477249 0.0238625 0.999715i \(-0.492404\pi\)
0.0238625 + 0.999715i \(0.492404\pi\)
\(600\) −4.20520 −0.171676
\(601\) −12.9978 −0.530192 −0.265096 0.964222i \(-0.585404\pi\)
−0.265096 + 0.964222i \(0.585404\pi\)
\(602\) 15.8113 0.644420
\(603\) 3.87397 0.157760
\(604\) −21.4491 −0.872751
\(605\) −7.53248 −0.306239
\(606\) 7.15302 0.290572
\(607\) 24.9606 1.01312 0.506560 0.862205i \(-0.330917\pi\)
0.506560 + 0.862205i \(0.330917\pi\)
\(608\) −1.31800 −0.0534521
\(609\) 28.8193 1.16781
\(610\) 9.23720 0.374003
\(611\) −1.61571 −0.0653645
\(612\) −3.83133 −0.154872
\(613\) 10.9244 0.441232 0.220616 0.975361i \(-0.429193\pi\)
0.220616 + 0.975361i \(0.429193\pi\)
\(614\) 28.1373 1.13553
\(615\) −0.389189 −0.0156936
\(616\) 17.9864 0.724694
\(617\) 9.26005 0.372796 0.186398 0.982474i \(-0.440319\pi\)
0.186398 + 0.982474i \(0.440319\pi\)
\(618\) 1.00000 0.0402259
\(619\) −3.74311 −0.150448 −0.0752242 0.997167i \(-0.523967\pi\)
−0.0752242 + 0.997167i \(0.523967\pi\)
\(620\) 4.07264 0.163561
\(621\) 6.87898 0.276044
\(622\) −22.9358 −0.919641
\(623\) 43.6550 1.74900
\(624\) 1.00000 0.0400320
\(625\) 13.7097 0.548386
\(626\) 21.1390 0.844885
\(627\) 5.81253 0.232130
\(628\) 10.9038 0.435110
\(629\) 32.0451 1.27772
\(630\) 3.63601 0.144862
\(631\) 20.5453 0.817894 0.408947 0.912558i \(-0.365896\pi\)
0.408947 + 0.912558i \(0.365896\pi\)
\(632\) −8.17661 −0.325248
\(633\) 25.0073 0.993949
\(634\) 12.2082 0.484849
\(635\) −13.1698 −0.522628
\(636\) −4.54487 −0.180216
\(637\) −9.63379 −0.381705
\(638\) −31.1628 −1.23375
\(639\) −8.92491 −0.353064
\(640\) 0.891518 0.0352403
\(641\) −4.29432 −0.169616 −0.0848078 0.996397i \(-0.527028\pi\)
−0.0848078 + 0.996397i \(0.527028\pi\)
\(642\) −6.28356 −0.247992
\(643\) 31.3180 1.23506 0.617530 0.786547i \(-0.288133\pi\)
0.617530 + 0.786547i \(0.288133\pi\)
\(644\) −28.0556 −1.10555
\(645\) −3.45622 −0.136089
\(646\) 5.04971 0.198678
\(647\) 19.3978 0.762606 0.381303 0.924450i \(-0.375475\pi\)
0.381303 + 0.924450i \(0.375475\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −17.1595 −0.673568
\(650\) −4.20520 −0.164941
\(651\) 18.6312 0.730216
\(652\) −5.98549 −0.234410
\(653\) 12.7751 0.499927 0.249964 0.968255i \(-0.419581\pi\)
0.249964 + 0.968255i \(0.419581\pi\)
\(654\) −3.77713 −0.147698
\(655\) 0.108206 0.00422796
\(656\) −0.436547 −0.0170443
\(657\) 9.37758 0.365854
\(658\) −6.58959 −0.256889
\(659\) 5.95057 0.231801 0.115901 0.993261i \(-0.463025\pi\)
0.115901 + 0.993261i \(0.463025\pi\)
\(660\) −3.93169 −0.153041
\(661\) −21.1806 −0.823831 −0.411915 0.911222i \(-0.635140\pi\)
−0.411915 + 0.911222i \(0.635140\pi\)
\(662\) 1.90976 0.0742248
\(663\) −3.83133 −0.148797
\(664\) −4.84486 −0.188017
\(665\) −4.79228 −0.185837
\(666\) 8.36395 0.324096
\(667\) 48.6084 1.88212
\(668\) 18.9623 0.733674
\(669\) −12.6477 −0.488987
\(670\) 3.45372 0.133429
\(671\) −45.6940 −1.76400
\(672\) 4.07845 0.157330
\(673\) 23.9055 0.921489 0.460744 0.887533i \(-0.347583\pi\)
0.460744 + 0.887533i \(0.347583\pi\)
\(674\) −3.62376 −0.139582
\(675\) 4.20520 0.161858
\(676\) 1.00000 0.0384615
\(677\) −4.68892 −0.180210 −0.0901049 0.995932i \(-0.528720\pi\)
−0.0901049 + 0.995932i \(0.528720\pi\)
\(678\) 10.7015 0.410988
\(679\) −6.66820 −0.255902
\(680\) −3.41570 −0.130986
\(681\) −6.84365 −0.262249
\(682\) −20.1463 −0.771442
\(683\) 29.1205 1.11426 0.557132 0.830424i \(-0.311902\pi\)
0.557132 + 0.830424i \(0.311902\pi\)
\(684\) 1.31800 0.0503951
\(685\) 16.9356 0.647076
\(686\) −10.7418 −0.410123
\(687\) 1.06504 0.0406339
\(688\) −3.87678 −0.147801
\(689\) −4.54487 −0.173146
\(690\) 6.13273 0.233469
\(691\) 36.7897 1.39955 0.699773 0.714365i \(-0.253285\pi\)
0.699773 + 0.714365i \(0.253285\pi\)
\(692\) 19.1488 0.727928
\(693\) −17.9864 −0.683248
\(694\) 2.20372 0.0836520
\(695\) −13.5034 −0.512212
\(696\) −7.06622 −0.267844
\(697\) 1.67256 0.0633526
\(698\) 4.15341 0.157209
\(699\) 4.95796 0.187527
\(700\) −17.1507 −0.648235
\(701\) −23.1946 −0.876049 −0.438025 0.898963i \(-0.644322\pi\)
−0.438025 + 0.898963i \(0.644322\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −11.0237 −0.415767
\(704\) −4.41011 −0.166212
\(705\) 1.44043 0.0542498
\(706\) −15.1988 −0.572013
\(707\) 29.1733 1.09717
\(708\) −3.89094 −0.146231
\(709\) −10.0809 −0.378596 −0.189298 0.981920i \(-0.560621\pi\)
−0.189298 + 0.981920i \(0.560621\pi\)
\(710\) −7.95672 −0.298610
\(711\) 8.17661 0.306647
\(712\) −10.7038 −0.401142
\(713\) 31.4246 1.17686
\(714\) −15.6259 −0.584785
\(715\) −3.93169 −0.147037
\(716\) 14.7047 0.549540
\(717\) 17.9214 0.669289
\(718\) 8.81818 0.329092
\(719\) −15.2349 −0.568168 −0.284084 0.958799i \(-0.591689\pi\)
−0.284084 + 0.958799i \(0.591689\pi\)
\(720\) −0.891518 −0.0332249
\(721\) 4.07845 0.151890
\(722\) 17.2629 0.642457
\(723\) −22.2265 −0.826614
\(724\) 5.74905 0.213662
\(725\) 29.7148 1.10358
\(726\) 8.44905 0.313574
\(727\) −20.5445 −0.761953 −0.380976 0.924585i \(-0.624412\pi\)
−0.380976 + 0.924585i \(0.624412\pi\)
\(728\) 4.07845 0.151158
\(729\) 1.00000 0.0370370
\(730\) 8.36028 0.309428
\(731\) 14.8532 0.549367
\(732\) −10.3612 −0.382961
\(733\) −11.0053 −0.406489 −0.203244 0.979128i \(-0.565149\pi\)
−0.203244 + 0.979128i \(0.565149\pi\)
\(734\) 22.3664 0.825558
\(735\) 8.58870 0.316799
\(736\) 6.87898 0.253562
\(737\) −17.0846 −0.629321
\(738\) 0.436547 0.0160695
\(739\) 10.7796 0.396532 0.198266 0.980148i \(-0.436469\pi\)
0.198266 + 0.980148i \(0.436469\pi\)
\(740\) 7.45661 0.274110
\(741\) 1.31800 0.0484180
\(742\) −18.5361 −0.680480
\(743\) 21.4755 0.787860 0.393930 0.919140i \(-0.371115\pi\)
0.393930 + 0.919140i \(0.371115\pi\)
\(744\) −4.56821 −0.167479
\(745\) 4.26565 0.156281
\(746\) 24.7009 0.904363
\(747\) 4.84486 0.177264
\(748\) 16.8966 0.617801
\(749\) −25.6272 −0.936399
\(750\) 8.20660 0.299663
\(751\) −44.2278 −1.61389 −0.806947 0.590623i \(-0.798882\pi\)
−0.806947 + 0.590623i \(0.798882\pi\)
\(752\) 1.61571 0.0589188
\(753\) 19.5551 0.712626
\(754\) −7.06622 −0.257337
\(755\) 19.1222 0.695929
\(756\) −4.07845 −0.148332
\(757\) 23.2092 0.843551 0.421776 0.906700i \(-0.361407\pi\)
0.421776 + 0.906700i \(0.361407\pi\)
\(758\) 24.3122 0.883060
\(759\) −30.3370 −1.10116
\(760\) 1.17502 0.0426226
\(761\) 28.0527 1.01691 0.508455 0.861088i \(-0.330217\pi\)
0.508455 + 0.861088i \(0.330217\pi\)
\(762\) 14.7723 0.535145
\(763\) −15.4049 −0.557694
\(764\) 22.7543 0.823222
\(765\) 3.41570 0.123495
\(766\) −5.30660 −0.191735
\(767\) −3.89094 −0.140494
\(768\) −1.00000 −0.0360844
\(769\) −43.5951 −1.57208 −0.786040 0.618176i \(-0.787872\pi\)
−0.786040 + 0.618176i \(0.787872\pi\)
\(770\) −16.0352 −0.577869
\(771\) 6.39342 0.230253
\(772\) 12.9628 0.466542
\(773\) −4.86642 −0.175033 −0.0875165 0.996163i \(-0.527893\pi\)
−0.0875165 + 0.996163i \(0.527893\pi\)
\(774\) 3.87678 0.139348
\(775\) 19.2102 0.690052
\(776\) 1.63498 0.0586924
\(777\) 34.1120 1.22376
\(778\) 28.2582 1.01311
\(779\) −0.575370 −0.0206148
\(780\) −0.891518 −0.0319215
\(781\) 39.3598 1.40841
\(782\) −26.3557 −0.942476
\(783\) 7.06622 0.252526
\(784\) 9.63379 0.344064
\(785\) −9.72095 −0.346956
\(786\) −0.121373 −0.00432922
\(787\) 46.0753 1.64241 0.821203 0.570636i \(-0.193303\pi\)
0.821203 + 0.570636i \(0.193303\pi\)
\(788\) 13.9660 0.497517
\(789\) −1.21355 −0.0432035
\(790\) 7.28959 0.259352
\(791\) 43.6455 1.55186
\(792\) 4.41011 0.156706
\(793\) −10.3612 −0.367937
\(794\) −5.52509 −0.196078
\(795\) 4.05184 0.143704
\(796\) 18.9472 0.671567
\(797\) −41.4859 −1.46951 −0.734753 0.678335i \(-0.762702\pi\)
−0.734753 + 0.678335i \(0.762702\pi\)
\(798\) 5.37541 0.190288
\(799\) −6.19031 −0.218997
\(800\) 4.20520 0.148676
\(801\) 10.7038 0.378200
\(802\) −20.3794 −0.719622
\(803\) −41.3561 −1.45943
\(804\) −3.87397 −0.136625
\(805\) 25.0121 0.881559
\(806\) −4.56821 −0.160908
\(807\) −15.0912 −0.531236
\(808\) −7.15302 −0.251642
\(809\) −40.9959 −1.44134 −0.720670 0.693279i \(-0.756166\pi\)
−0.720670 + 0.693279i \(0.756166\pi\)
\(810\) 0.891518 0.0313247
\(811\) 3.82293 0.134241 0.0671207 0.997745i \(-0.478619\pi\)
0.0671207 + 0.997745i \(0.478619\pi\)
\(812\) −28.8193 −1.01136
\(813\) −26.4715 −0.928396
\(814\) −36.8859 −1.29285
\(815\) 5.33617 0.186918
\(816\) 3.83133 0.134123
\(817\) −5.10961 −0.178763
\(818\) −12.8602 −0.449646
\(819\) −4.07845 −0.142513
\(820\) 0.389189 0.0135911
\(821\) −13.3049 −0.464344 −0.232172 0.972675i \(-0.574583\pi\)
−0.232172 + 0.972675i \(0.574583\pi\)
\(822\) −18.9964 −0.662575
\(823\) 42.6312 1.48603 0.743016 0.669274i \(-0.233395\pi\)
0.743016 + 0.669274i \(0.233395\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −18.5454 −0.645667
\(826\) −15.8690 −0.552154
\(827\) −36.4742 −1.26833 −0.634166 0.773197i \(-0.718656\pi\)
−0.634166 + 0.773197i \(0.718656\pi\)
\(828\) −6.87898 −0.239061
\(829\) 28.4917 0.989556 0.494778 0.869019i \(-0.335249\pi\)
0.494778 + 0.869019i \(0.335249\pi\)
\(830\) 4.31928 0.149924
\(831\) 22.7248 0.788313
\(832\) −1.00000 −0.0346688
\(833\) −36.9103 −1.27886
\(834\) 15.1465 0.524480
\(835\) −16.9053 −0.585030
\(836\) −5.81253 −0.201031
\(837\) 4.56821 0.157901
\(838\) −17.2438 −0.595676
\(839\) −28.5577 −0.985921 −0.492961 0.870052i \(-0.664085\pi\)
−0.492961 + 0.870052i \(0.664085\pi\)
\(840\) −3.63601 −0.125454
\(841\) 20.9314 0.721774
\(842\) 8.60431 0.296524
\(843\) −9.21995 −0.317552
\(844\) −25.0073 −0.860785
\(845\) −0.891518 −0.0306691
\(846\) −1.61571 −0.0555491
\(847\) 34.4591 1.18403
\(848\) 4.54487 0.156072
\(849\) 21.5212 0.738604
\(850\) −16.1115 −0.552620
\(851\) 57.5354 1.97229
\(852\) 8.92491 0.305762
\(853\) 2.56693 0.0878902 0.0439451 0.999034i \(-0.486007\pi\)
0.0439451 + 0.999034i \(0.486007\pi\)
\(854\) −42.2577 −1.44603
\(855\) −1.17502 −0.0401849
\(856\) 6.28356 0.214768
\(857\) −21.7233 −0.742055 −0.371027 0.928622i \(-0.620994\pi\)
−0.371027 + 0.928622i \(0.620994\pi\)
\(858\) 4.41011 0.150559
\(859\) −5.66069 −0.193140 −0.0965701 0.995326i \(-0.530787\pi\)
−0.0965701 + 0.995326i \(0.530787\pi\)
\(860\) 3.45622 0.117856
\(861\) 1.78044 0.0606771
\(862\) −29.0333 −0.988877
\(863\) 48.7380 1.65906 0.829531 0.558461i \(-0.188608\pi\)
0.829531 + 0.558461i \(0.188608\pi\)
\(864\) 1.00000 0.0340207
\(865\) −17.0715 −0.580448
\(866\) −13.7831 −0.468368
\(867\) 2.32089 0.0788215
\(868\) −18.6312 −0.632386
\(869\) −36.0597 −1.22324
\(870\) 6.29966 0.213578
\(871\) −3.87397 −0.131265
\(872\) 3.77713 0.127910
\(873\) −1.63498 −0.0553358
\(874\) 9.06651 0.306679
\(875\) 33.4702 1.13150
\(876\) −9.37758 −0.316839
\(877\) −34.9677 −1.18077 −0.590387 0.807120i \(-0.701025\pi\)
−0.590387 + 0.807120i \(0.701025\pi\)
\(878\) 14.3736 0.485085
\(879\) 24.1063 0.813087
\(880\) 3.93169 0.132537
\(881\) −41.9907 −1.41470 −0.707351 0.706863i \(-0.750110\pi\)
−0.707351 + 0.706863i \(0.750110\pi\)
\(882\) −9.63379 −0.324387
\(883\) −55.5234 −1.86851 −0.934256 0.356604i \(-0.883935\pi\)
−0.934256 + 0.356604i \(0.883935\pi\)
\(884\) 3.83133 0.128862
\(885\) 3.46884 0.116604
\(886\) 4.89352 0.164401
\(887\) 17.4782 0.586862 0.293431 0.955980i \(-0.405203\pi\)
0.293431 + 0.955980i \(0.405203\pi\)
\(888\) −8.36395 −0.280676
\(889\) 60.2483 2.02066
\(890\) 9.54264 0.319870
\(891\) −4.41011 −0.147744
\(892\) 12.6477 0.423475
\(893\) 2.12951 0.0712612
\(894\) −4.78470 −0.160024
\(895\) −13.1095 −0.438202
\(896\) −4.07845 −0.136252
\(897\) −6.87898 −0.229682
\(898\) −15.7086 −0.524204
\(899\) 32.2800 1.07660
\(900\) −4.20520 −0.140173
\(901\) −17.4129 −0.580109
\(902\) −1.92522 −0.0641027
\(903\) 15.8113 0.526166
\(904\) −10.7015 −0.355926
\(905\) −5.12538 −0.170373
\(906\) −21.4491 −0.712598
\(907\) −5.52710 −0.183524 −0.0917622 0.995781i \(-0.529250\pi\)
−0.0917622 + 0.995781i \(0.529250\pi\)
\(908\) 6.84365 0.227115
\(909\) 7.15302 0.237251
\(910\) −3.63601 −0.120533
\(911\) 33.9205 1.12384 0.561918 0.827193i \(-0.310064\pi\)
0.561918 + 0.827193i \(0.310064\pi\)
\(912\) −1.31800 −0.0436434
\(913\) −21.3664 −0.707123
\(914\) −39.5699 −1.30885
\(915\) 9.23720 0.305373
\(916\) −1.06504 −0.0351900
\(917\) −0.495013 −0.0163468
\(918\) −3.83133 −0.126453
\(919\) −10.8130 −0.356689 −0.178344 0.983968i \(-0.557074\pi\)
−0.178344 + 0.983968i \(0.557074\pi\)
\(920\) −6.13273 −0.202190
\(921\) 28.1373 0.927156
\(922\) 26.8050 0.882775
\(923\) 8.92491 0.293767
\(924\) 17.9864 0.591710
\(925\) 35.1720 1.15645
\(926\) −21.2693 −0.698954
\(927\) 1.00000 0.0328443
\(928\) 7.06622 0.231960
\(929\) 8.27943 0.271639 0.135820 0.990734i \(-0.456633\pi\)
0.135820 + 0.990734i \(0.456633\pi\)
\(930\) 4.07264 0.133547
\(931\) 12.6974 0.416139
\(932\) −4.95796 −0.162403
\(933\) −22.9358 −0.750884
\(934\) 0.247842 0.00810964
\(935\) −15.0636 −0.492633
\(936\) 1.00000 0.0326860
\(937\) 23.5881 0.770591 0.385295 0.922793i \(-0.374100\pi\)
0.385295 + 0.922793i \(0.374100\pi\)
\(938\) −15.7998 −0.515883
\(939\) 21.1390 0.689846
\(940\) −1.44043 −0.0469817
\(941\) 21.6294 0.705097 0.352548 0.935794i \(-0.385315\pi\)
0.352548 + 0.935794i \(0.385315\pi\)
\(942\) 10.9038 0.355266
\(943\) 3.00299 0.0977910
\(944\) 3.89094 0.126639
\(945\) 3.63601 0.118280
\(946\) −17.0970 −0.555872
\(947\) 40.0713 1.30214 0.651071 0.759016i \(-0.274320\pi\)
0.651071 + 0.759016i \(0.274320\pi\)
\(948\) −8.17661 −0.265564
\(949\) −9.37758 −0.304409
\(950\) 5.54246 0.179821
\(951\) 12.2082 0.395878
\(952\) 15.6259 0.506439
\(953\) −38.4990 −1.24710 −0.623552 0.781782i \(-0.714311\pi\)
−0.623552 + 0.781782i \(0.714311\pi\)
\(954\) −4.54487 −0.147146
\(955\) −20.2859 −0.656436
\(956\) −17.9214 −0.579621
\(957\) −31.1628 −1.00735
\(958\) 6.39802 0.206711
\(959\) −77.4759 −2.50183
\(960\) 0.891518 0.0287736
\(961\) −10.1314 −0.326820
\(962\) −8.36395 −0.269664
\(963\) −6.28356 −0.202485
\(964\) 22.2265 0.715868
\(965\) −11.5566 −0.372020
\(966\) −28.0556 −0.902674
\(967\) −44.5554 −1.43280 −0.716402 0.697687i \(-0.754212\pi\)
−0.716402 + 0.697687i \(0.754212\pi\)
\(968\) −8.44905 −0.271563
\(969\) 5.04971 0.162220
\(970\) −1.45762 −0.0468012
\(971\) 49.9507 1.60299 0.801497 0.597999i \(-0.204037\pi\)
0.801497 + 0.597999i \(0.204037\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 61.7743 1.98039
\(974\) 13.1572 0.421583
\(975\) −4.20520 −0.134674
\(976\) 10.3612 0.331654
\(977\) −8.48621 −0.271498 −0.135749 0.990743i \(-0.543344\pi\)
−0.135749 + 0.990743i \(0.543344\pi\)
\(978\) −5.98549 −0.191395
\(979\) −47.2049 −1.50868
\(980\) −8.58870 −0.274356
\(981\) −3.77713 −0.120595
\(982\) −17.1405 −0.546974
\(983\) −3.73281 −0.119058 −0.0595291 0.998227i \(-0.518960\pi\)
−0.0595291 + 0.998227i \(0.518960\pi\)
\(984\) −0.436547 −0.0139166
\(985\) −12.4509 −0.396719
\(986\) −27.0730 −0.862181
\(987\) −6.58959 −0.209749
\(988\) −1.31800 −0.0419313
\(989\) 26.6683 0.848002
\(990\) −3.93169 −0.124957
\(991\) 43.0792 1.36846 0.684228 0.729268i \(-0.260139\pi\)
0.684228 + 0.729268i \(0.260139\pi\)
\(992\) 4.56821 0.145041
\(993\) 1.90976 0.0606043
\(994\) 36.3999 1.15453
\(995\) −16.8918 −0.535506
\(996\) −4.84486 −0.153515
\(997\) −11.8723 −0.376000 −0.188000 0.982169i \(-0.560200\pi\)
−0.188000 + 0.982169i \(0.560200\pi\)
\(998\) 6.32801 0.200310
\(999\) 8.36395 0.264624
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8034.2.a.ba.1.6 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.ba.1.6 14 1.1 even 1 trivial