Properties

Label 8034.2.a.bb.1.9
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 1.9
Root \(-0.480706\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.480706 q^{5} -1.00000 q^{6} +0.765483 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.480706 q^{5} -1.00000 q^{6} +0.765483 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.480706 q^{10} -1.12221 q^{11} +1.00000 q^{12} -1.00000 q^{13} -0.765483 q^{14} +0.480706 q^{15} +1.00000 q^{16} +0.131422 q^{17} -1.00000 q^{18} +7.23270 q^{19} +0.480706 q^{20} +0.765483 q^{21} +1.12221 q^{22} -6.47882 q^{23} -1.00000 q^{24} -4.76892 q^{25} +1.00000 q^{26} +1.00000 q^{27} +0.765483 q^{28} -5.92565 q^{29} -0.480706 q^{30} +0.823617 q^{31} -1.00000 q^{32} -1.12221 q^{33} -0.131422 q^{34} +0.367973 q^{35} +1.00000 q^{36} -5.74456 q^{37} -7.23270 q^{38} -1.00000 q^{39} -0.480706 q^{40} +9.71768 q^{41} -0.765483 q^{42} -1.79479 q^{43} -1.12221 q^{44} +0.480706 q^{45} +6.47882 q^{46} -1.21402 q^{47} +1.00000 q^{48} -6.41404 q^{49} +4.76892 q^{50} +0.131422 q^{51} -1.00000 q^{52} -6.16740 q^{53} -1.00000 q^{54} -0.539454 q^{55} -0.765483 q^{56} +7.23270 q^{57} +5.92565 q^{58} -7.84426 q^{59} +0.480706 q^{60} +8.36385 q^{61} -0.823617 q^{62} +0.765483 q^{63} +1.00000 q^{64} -0.480706 q^{65} +1.12221 q^{66} -12.4809 q^{67} +0.131422 q^{68} -6.47882 q^{69} -0.367973 q^{70} -16.6763 q^{71} -1.00000 q^{72} -2.44935 q^{73} +5.74456 q^{74} -4.76892 q^{75} +7.23270 q^{76} -0.859034 q^{77} +1.00000 q^{78} +7.55695 q^{79} +0.480706 q^{80} +1.00000 q^{81} -9.71768 q^{82} +14.6910 q^{83} +0.765483 q^{84} +0.0631755 q^{85} +1.79479 q^{86} -5.92565 q^{87} +1.12221 q^{88} -16.8623 q^{89} -0.480706 q^{90} -0.765483 q^{91} -6.47882 q^{92} +0.823617 q^{93} +1.21402 q^{94} +3.47680 q^{95} -1.00000 q^{96} +13.3568 q^{97} +6.41404 q^{98} -1.12221 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) \)

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
\(5\) 0.480706 0.214978 0.107489 0.994206i \(-0.465719\pi\)
0.107489 + 0.994206i \(0.465719\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0.765483 0.289326 0.144663 0.989481i \(-0.453790\pi\)
0.144663 + 0.989481i \(0.453790\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −0.480706 −0.152013
\(11\) −1.12221 −0.338359 −0.169180 0.985585i \(-0.554112\pi\)
−0.169180 + 0.985585i \(0.554112\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) −0.765483 −0.204584
\(15\) 0.480706 0.124118
\(16\) 1.00000 0.250000
\(17\) 0.131422 0.0318746 0.0159373 0.999873i \(-0.494927\pi\)
0.0159373 + 0.999873i \(0.494927\pi\)
\(18\) −1.00000 −0.235702
\(19\) 7.23270 1.65929 0.829647 0.558288i \(-0.188542\pi\)
0.829647 + 0.558288i \(0.188542\pi\)
\(20\) 0.480706 0.107489
\(21\) 0.765483 0.167042
\(22\) 1.12221 0.239256
\(23\) −6.47882 −1.35093 −0.675464 0.737393i \(-0.736057\pi\)
−0.675464 + 0.737393i \(0.736057\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.76892 −0.953784
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 0.765483 0.144663
\(29\) −5.92565 −1.10036 −0.550182 0.835044i \(-0.685442\pi\)
−0.550182 + 0.835044i \(0.685442\pi\)
\(30\) −0.480706 −0.0877645
\(31\) 0.823617 0.147926 0.0739630 0.997261i \(-0.476435\pi\)
0.0739630 + 0.997261i \(0.476435\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.12221 −0.195352
\(34\) −0.131422 −0.0225387
\(35\) 0.367973 0.0621987
\(36\) 1.00000 0.166667
\(37\) −5.74456 −0.944400 −0.472200 0.881491i \(-0.656540\pi\)
−0.472200 + 0.881491i \(0.656540\pi\)
\(38\) −7.23270 −1.17330
\(39\) −1.00000 −0.160128
\(40\) −0.480706 −0.0760063
\(41\) 9.71768 1.51765 0.758823 0.651297i \(-0.225775\pi\)
0.758823 + 0.651297i \(0.225775\pi\)
\(42\) −0.765483 −0.118117
\(43\) −1.79479 −0.273703 −0.136851 0.990592i \(-0.543698\pi\)
−0.136851 + 0.990592i \(0.543698\pi\)
\(44\) −1.12221 −0.169180
\(45\) 0.480706 0.0716594
\(46\) 6.47882 0.955250
\(47\) −1.21402 −0.177083 −0.0885414 0.996073i \(-0.528221\pi\)
−0.0885414 + 0.996073i \(0.528221\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.41404 −0.916291
\(50\) 4.76892 0.674427
\(51\) 0.131422 0.0184028
\(52\) −1.00000 −0.138675
\(53\) −6.16740 −0.847158 −0.423579 0.905859i \(-0.639226\pi\)
−0.423579 + 0.905859i \(0.639226\pi\)
\(54\) −1.00000 −0.136083
\(55\) −0.539454 −0.0727399
\(56\) −0.765483 −0.102292
\(57\) 7.23270 0.957994
\(58\) 5.92565 0.778076
\(59\) −7.84426 −1.02124 −0.510618 0.859808i \(-0.670583\pi\)
−0.510618 + 0.859808i \(0.670583\pi\)
\(60\) 0.480706 0.0620589
\(61\) 8.36385 1.07088 0.535441 0.844573i \(-0.320145\pi\)
0.535441 + 0.844573i \(0.320145\pi\)
\(62\) −0.823617 −0.104599
\(63\) 0.765483 0.0964418
\(64\) 1.00000 0.125000
\(65\) −0.480706 −0.0596243
\(66\) 1.12221 0.138135
\(67\) −12.4809 −1.52478 −0.762392 0.647115i \(-0.775975\pi\)
−0.762392 + 0.647115i \(0.775975\pi\)
\(68\) 0.131422 0.0159373
\(69\) −6.47882 −0.779959
\(70\) −0.367973 −0.0439811
\(71\) −16.6763 −1.97911 −0.989557 0.144140i \(-0.953958\pi\)
−0.989557 + 0.144140i \(0.953958\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.44935 −0.286675 −0.143337 0.989674i \(-0.545783\pi\)
−0.143337 + 0.989674i \(0.545783\pi\)
\(74\) 5.74456 0.667792
\(75\) −4.76892 −0.550668
\(76\) 7.23270 0.829647
\(77\) −0.859034 −0.0978960
\(78\) 1.00000 0.113228
\(79\) 7.55695 0.850223 0.425111 0.905141i \(-0.360235\pi\)
0.425111 + 0.905141i \(0.360235\pi\)
\(80\) 0.480706 0.0537446
\(81\) 1.00000 0.111111
\(82\) −9.71768 −1.07314
\(83\) 14.6910 1.61255 0.806273 0.591544i \(-0.201481\pi\)
0.806273 + 0.591544i \(0.201481\pi\)
\(84\) 0.765483 0.0835211
\(85\) 0.0631755 0.00685234
\(86\) 1.79479 0.193537
\(87\) −5.92565 −0.635296
\(88\) 1.12221 0.119628
\(89\) −16.8623 −1.78740 −0.893702 0.448661i \(-0.851901\pi\)
−0.893702 + 0.448661i \(0.851901\pi\)
\(90\) −0.480706 −0.0506709
\(91\) −0.765483 −0.0802445
\(92\) −6.47882 −0.675464
\(93\) 0.823617 0.0854051
\(94\) 1.21402 0.125216
\(95\) 3.47680 0.356712
\(96\) −1.00000 −0.102062
\(97\) 13.3568 1.35618 0.678089 0.734980i \(-0.262809\pi\)
0.678089 + 0.734980i \(0.262809\pi\)
\(98\) 6.41404 0.647915
\(99\) −1.12221 −0.112786
\(100\) −4.76892 −0.476892
\(101\) −17.0194 −1.69350 −0.846748 0.531994i \(-0.821443\pi\)
−0.846748 + 0.531994i \(0.821443\pi\)
\(102\) −0.131422 −0.0130127
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 0.367973 0.0359104
\(106\) 6.16740 0.599031
\(107\) −4.57541 −0.442321 −0.221161 0.975237i \(-0.570985\pi\)
−0.221161 + 0.975237i \(0.570985\pi\)
\(108\) 1.00000 0.0962250
\(109\) 4.28510 0.410438 0.205219 0.978716i \(-0.434209\pi\)
0.205219 + 0.978716i \(0.434209\pi\)
\(110\) 0.539454 0.0514349
\(111\) −5.74456 −0.545249
\(112\) 0.765483 0.0723314
\(113\) 18.0957 1.70230 0.851151 0.524921i \(-0.175905\pi\)
0.851151 + 0.524921i \(0.175905\pi\)
\(114\) −7.23270 −0.677404
\(115\) −3.11441 −0.290420
\(116\) −5.92565 −0.550182
\(117\) −1.00000 −0.0924500
\(118\) 7.84426 0.722123
\(119\) 0.100602 0.00922213
\(120\) −0.480706 −0.0438823
\(121\) −9.74064 −0.885513
\(122\) −8.36385 −0.757228
\(123\) 9.71768 0.876214
\(124\) 0.823617 0.0739630
\(125\) −4.69598 −0.420021
\(126\) −0.765483 −0.0681947
\(127\) 1.93648 0.171834 0.0859172 0.996302i \(-0.472618\pi\)
0.0859172 + 0.996302i \(0.472618\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.79479 −0.158022
\(130\) 0.480706 0.0421607
\(131\) 17.0455 1.48928 0.744638 0.667468i \(-0.232622\pi\)
0.744638 + 0.667468i \(0.232622\pi\)
\(132\) −1.12221 −0.0976760
\(133\) 5.53651 0.480076
\(134\) 12.4809 1.07819
\(135\) 0.480706 0.0413726
\(136\) −0.131422 −0.0112694
\(137\) −13.6581 −1.16689 −0.583447 0.812151i \(-0.698296\pi\)
−0.583447 + 0.812151i \(0.698296\pi\)
\(138\) 6.47882 0.551514
\(139\) −16.1582 −1.37052 −0.685259 0.728300i \(-0.740311\pi\)
−0.685259 + 0.728300i \(0.740311\pi\)
\(140\) 0.367973 0.0310994
\(141\) −1.21402 −0.102239
\(142\) 16.6763 1.39945
\(143\) 1.12221 0.0938440
\(144\) 1.00000 0.0833333
\(145\) −2.84849 −0.236555
\(146\) 2.44935 0.202710
\(147\) −6.41404 −0.529021
\(148\) −5.74456 −0.472200
\(149\) −6.45756 −0.529024 −0.264512 0.964382i \(-0.585211\pi\)
−0.264512 + 0.964382i \(0.585211\pi\)
\(150\) 4.76892 0.389381
\(151\) 6.56350 0.534130 0.267065 0.963679i \(-0.413946\pi\)
0.267065 + 0.963679i \(0.413946\pi\)
\(152\) −7.23270 −0.586649
\(153\) 0.131422 0.0106249
\(154\) 0.859034 0.0692230
\(155\) 0.395918 0.0318009
\(156\) −1.00000 −0.0800641
\(157\) −4.46960 −0.356713 −0.178357 0.983966i \(-0.557078\pi\)
−0.178357 + 0.983966i \(0.557078\pi\)
\(158\) −7.55695 −0.601198
\(159\) −6.16740 −0.489107
\(160\) −0.480706 −0.0380032
\(161\) −4.95943 −0.390858
\(162\) −1.00000 −0.0785674
\(163\) 11.9349 0.934817 0.467408 0.884041i \(-0.345188\pi\)
0.467408 + 0.884041i \(0.345188\pi\)
\(164\) 9.71768 0.758823
\(165\) −0.539454 −0.0419964
\(166\) −14.6910 −1.14024
\(167\) 11.7525 0.909437 0.454719 0.890635i \(-0.349740\pi\)
0.454719 + 0.890635i \(0.349740\pi\)
\(168\) −0.765483 −0.0590583
\(169\) 1.00000 0.0769231
\(170\) −0.0631755 −0.00484534
\(171\) 7.23270 0.553098
\(172\) −1.79479 −0.136851
\(173\) −6.62095 −0.503381 −0.251691 0.967808i \(-0.580987\pi\)
−0.251691 + 0.967808i \(0.580987\pi\)
\(174\) 5.92565 0.449222
\(175\) −3.65053 −0.275954
\(176\) −1.12221 −0.0845899
\(177\) −7.84426 −0.589611
\(178\) 16.8623 1.26389
\(179\) −2.90096 −0.216828 −0.108414 0.994106i \(-0.534577\pi\)
−0.108414 + 0.994106i \(0.534577\pi\)
\(180\) 0.480706 0.0358297
\(181\) 7.61769 0.566218 0.283109 0.959088i \(-0.408634\pi\)
0.283109 + 0.959088i \(0.408634\pi\)
\(182\) 0.765483 0.0567414
\(183\) 8.36385 0.618274
\(184\) 6.47882 0.477625
\(185\) −2.76145 −0.203025
\(186\) −0.823617 −0.0603905
\(187\) −0.147484 −0.0107851
\(188\) −1.21402 −0.0885414
\(189\) 0.765483 0.0556807
\(190\) −3.47680 −0.252234
\(191\) −21.2603 −1.53834 −0.769169 0.639045i \(-0.779330\pi\)
−0.769169 + 0.639045i \(0.779330\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.66290 0.191680 0.0958400 0.995397i \(-0.469446\pi\)
0.0958400 + 0.995397i \(0.469446\pi\)
\(194\) −13.3568 −0.958962
\(195\) −0.480706 −0.0344241
\(196\) −6.41404 −0.458145
\(197\) 1.87363 0.133491 0.0667453 0.997770i \(-0.478738\pi\)
0.0667453 + 0.997770i \(0.478738\pi\)
\(198\) 1.12221 0.0797521
\(199\) 3.88713 0.275551 0.137776 0.990463i \(-0.456005\pi\)
0.137776 + 0.990463i \(0.456005\pi\)
\(200\) 4.76892 0.337214
\(201\) −12.4809 −0.880335
\(202\) 17.0194 1.19748
\(203\) −4.53598 −0.318364
\(204\) 0.131422 0.00920140
\(205\) 4.67135 0.326261
\(206\) −1.00000 −0.0696733
\(207\) −6.47882 −0.450309
\(208\) −1.00000 −0.0693375
\(209\) −8.11661 −0.561438
\(210\) −0.367973 −0.0253925
\(211\) −12.2619 −0.844143 −0.422072 0.906562i \(-0.638697\pi\)
−0.422072 + 0.906562i \(0.638697\pi\)
\(212\) −6.16740 −0.423579
\(213\) −16.6763 −1.14264
\(214\) 4.57541 0.312768
\(215\) −0.862766 −0.0588402
\(216\) −1.00000 −0.0680414
\(217\) 0.630465 0.0427987
\(218\) −4.28510 −0.290224
\(219\) −2.44935 −0.165512
\(220\) −0.539454 −0.0363700
\(221\) −0.131422 −0.00884042
\(222\) 5.74456 0.385550
\(223\) −18.6606 −1.24960 −0.624802 0.780783i \(-0.714820\pi\)
−0.624802 + 0.780783i \(0.714820\pi\)
\(224\) −0.765483 −0.0511460
\(225\) −4.76892 −0.317928
\(226\) −18.0957 −1.20371
\(227\) −4.40861 −0.292610 −0.146305 0.989240i \(-0.546738\pi\)
−0.146305 + 0.989240i \(0.546738\pi\)
\(228\) 7.23270 0.478997
\(229\) −5.24559 −0.346639 −0.173319 0.984866i \(-0.555449\pi\)
−0.173319 + 0.984866i \(0.555449\pi\)
\(230\) 3.11441 0.205358
\(231\) −0.859034 −0.0565203
\(232\) 5.92565 0.389038
\(233\) 7.38317 0.483687 0.241844 0.970315i \(-0.422248\pi\)
0.241844 + 0.970315i \(0.422248\pi\)
\(234\) 1.00000 0.0653720
\(235\) −0.583586 −0.0380689
\(236\) −7.84426 −0.510618
\(237\) 7.55695 0.490876
\(238\) −0.100602 −0.00652103
\(239\) 11.7084 0.757351 0.378676 0.925529i \(-0.376380\pi\)
0.378676 + 0.925529i \(0.376380\pi\)
\(240\) 0.480706 0.0310294
\(241\) −24.2815 −1.56411 −0.782054 0.623211i \(-0.785828\pi\)
−0.782054 + 0.623211i \(0.785828\pi\)
\(242\) 9.74064 0.626152
\(243\) 1.00000 0.0641500
\(244\) 8.36385 0.535441
\(245\) −3.08327 −0.196983
\(246\) −9.71768 −0.619577
\(247\) −7.23270 −0.460205
\(248\) −0.823617 −0.0522997
\(249\) 14.6910 0.931003
\(250\) 4.69598 0.297000
\(251\) 20.9442 1.32198 0.660992 0.750393i \(-0.270136\pi\)
0.660992 + 0.750393i \(0.270136\pi\)
\(252\) 0.765483 0.0482209
\(253\) 7.27061 0.457099
\(254\) −1.93648 −0.121505
\(255\) 0.0631755 0.00395620
\(256\) 1.00000 0.0625000
\(257\) −18.5827 −1.15916 −0.579578 0.814917i \(-0.696783\pi\)
−0.579578 + 0.814917i \(0.696783\pi\)
\(258\) 1.79479 0.111739
\(259\) −4.39737 −0.273239
\(260\) −0.480706 −0.0298121
\(261\) −5.92565 −0.366788
\(262\) −17.0455 −1.05308
\(263\) 0.246017 0.0151700 0.00758502 0.999971i \(-0.497586\pi\)
0.00758502 + 0.999971i \(0.497586\pi\)
\(264\) 1.12221 0.0690673
\(265\) −2.96471 −0.182121
\(266\) −5.53651 −0.339465
\(267\) −16.8623 −1.03196
\(268\) −12.4809 −0.762392
\(269\) −21.0144 −1.28127 −0.640635 0.767845i \(-0.721329\pi\)
−0.640635 + 0.767845i \(0.721329\pi\)
\(270\) −0.480706 −0.0292548
\(271\) −30.8832 −1.87602 −0.938011 0.346605i \(-0.887334\pi\)
−0.938011 + 0.346605i \(0.887334\pi\)
\(272\) 0.131422 0.00796865
\(273\) −0.765483 −0.0463292
\(274\) 13.6581 0.825119
\(275\) 5.35174 0.322722
\(276\) −6.47882 −0.389979
\(277\) 12.7814 0.767960 0.383980 0.923341i \(-0.374553\pi\)
0.383980 + 0.923341i \(0.374553\pi\)
\(278\) 16.1582 0.969102
\(279\) 0.823617 0.0493086
\(280\) −0.367973 −0.0219906
\(281\) 11.4614 0.683732 0.341866 0.939749i \(-0.388941\pi\)
0.341866 + 0.939749i \(0.388941\pi\)
\(282\) 1.21402 0.0722937
\(283\) 23.9695 1.42484 0.712418 0.701755i \(-0.247600\pi\)
0.712418 + 0.701755i \(0.247600\pi\)
\(284\) −16.6763 −0.989557
\(285\) 3.47680 0.205948
\(286\) −1.12221 −0.0663578
\(287\) 7.43872 0.439094
\(288\) −1.00000 −0.0589256
\(289\) −16.9827 −0.998984
\(290\) 2.84849 0.167269
\(291\) 13.3568 0.782989
\(292\) −2.44935 −0.143337
\(293\) −27.6256 −1.61390 −0.806951 0.590618i \(-0.798884\pi\)
−0.806951 + 0.590618i \(0.798884\pi\)
\(294\) 6.41404 0.374074
\(295\) −3.77078 −0.219543
\(296\) 5.74456 0.333896
\(297\) −1.12221 −0.0651173
\(298\) 6.45756 0.374077
\(299\) 6.47882 0.374680
\(300\) −4.76892 −0.275334
\(301\) −1.37388 −0.0791892
\(302\) −6.56350 −0.377687
\(303\) −17.0194 −0.977740
\(304\) 7.23270 0.414824
\(305\) 4.02056 0.230216
\(306\) −0.131422 −0.00751291
\(307\) 31.0759 1.77360 0.886799 0.462155i \(-0.152924\pi\)
0.886799 + 0.462155i \(0.152924\pi\)
\(308\) −0.859034 −0.0489480
\(309\) 1.00000 0.0568880
\(310\) −0.395918 −0.0224866
\(311\) −12.4535 −0.706171 −0.353085 0.935591i \(-0.614867\pi\)
−0.353085 + 0.935591i \(0.614867\pi\)
\(312\) 1.00000 0.0566139
\(313\) 8.68625 0.490976 0.245488 0.969400i \(-0.421052\pi\)
0.245488 + 0.969400i \(0.421052\pi\)
\(314\) 4.46960 0.252234
\(315\) 0.367973 0.0207329
\(316\) 7.55695 0.425111
\(317\) 6.18684 0.347488 0.173744 0.984791i \(-0.444414\pi\)
0.173744 + 0.984791i \(0.444414\pi\)
\(318\) 6.16740 0.345851
\(319\) 6.64983 0.372319
\(320\) 0.480706 0.0268723
\(321\) −4.57541 −0.255374
\(322\) 4.95943 0.276378
\(323\) 0.950537 0.0528893
\(324\) 1.00000 0.0555556
\(325\) 4.76892 0.264532
\(326\) −11.9349 −0.661015
\(327\) 4.28510 0.236967
\(328\) −9.71768 −0.536569
\(329\) −0.929311 −0.0512346
\(330\) 0.539454 0.0296960
\(331\) −29.0460 −1.59651 −0.798256 0.602318i \(-0.794244\pi\)
−0.798256 + 0.602318i \(0.794244\pi\)
\(332\) 14.6910 0.806273
\(333\) −5.74456 −0.314800
\(334\) −11.7525 −0.643069
\(335\) −5.99964 −0.327796
\(336\) 0.765483 0.0417605
\(337\) −0.492714 −0.0268398 −0.0134199 0.999910i \(-0.504272\pi\)
−0.0134199 + 0.999910i \(0.504272\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 18.0957 0.982824
\(340\) 0.0631755 0.00342617
\(341\) −0.924272 −0.0500521
\(342\) −7.23270 −0.391099
\(343\) −10.2682 −0.554432
\(344\) 1.79479 0.0967685
\(345\) −3.11441 −0.167674
\(346\) 6.62095 0.355944
\(347\) 20.9494 1.12462 0.562311 0.826926i \(-0.309912\pi\)
0.562311 + 0.826926i \(0.309912\pi\)
\(348\) −5.92565 −0.317648
\(349\) −29.6109 −1.58503 −0.792517 0.609850i \(-0.791230\pi\)
−0.792517 + 0.609850i \(0.791230\pi\)
\(350\) 3.65053 0.195129
\(351\) −1.00000 −0.0533761
\(352\) 1.12221 0.0598141
\(353\) 12.6044 0.670865 0.335433 0.942064i \(-0.391117\pi\)
0.335433 + 0.942064i \(0.391117\pi\)
\(354\) 7.84426 0.416918
\(355\) −8.01641 −0.425467
\(356\) −16.8623 −0.893702
\(357\) 0.100602 0.00532440
\(358\) 2.90096 0.153320
\(359\) 27.7883 1.46661 0.733306 0.679899i \(-0.237976\pi\)
0.733306 + 0.679899i \(0.237976\pi\)
\(360\) −0.480706 −0.0253354
\(361\) 33.3119 1.75326
\(362\) −7.61769 −0.400377
\(363\) −9.74064 −0.511251
\(364\) −0.765483 −0.0401222
\(365\) −1.17742 −0.0616288
\(366\) −8.36385 −0.437186
\(367\) 2.77067 0.144628 0.0723140 0.997382i \(-0.476962\pi\)
0.0723140 + 0.997382i \(0.476962\pi\)
\(368\) −6.47882 −0.337732
\(369\) 9.71768 0.505882
\(370\) 2.76145 0.143561
\(371\) −4.72104 −0.245104
\(372\) 0.823617 0.0427025
\(373\) 25.4710 1.31884 0.659419 0.751776i \(-0.270802\pi\)
0.659419 + 0.751776i \(0.270802\pi\)
\(374\) 0.147484 0.00762619
\(375\) −4.69598 −0.242499
\(376\) 1.21402 0.0626082
\(377\) 5.92565 0.305186
\(378\) −0.765483 −0.0393722
\(379\) −7.25237 −0.372529 −0.186265 0.982500i \(-0.559638\pi\)
−0.186265 + 0.982500i \(0.559638\pi\)
\(380\) 3.47680 0.178356
\(381\) 1.93648 0.0992087
\(382\) 21.2603 1.08777
\(383\) −20.9833 −1.07219 −0.536097 0.844156i \(-0.680102\pi\)
−0.536097 + 0.844156i \(0.680102\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −0.412943 −0.0210455
\(386\) −2.66290 −0.135538
\(387\) −1.79479 −0.0912343
\(388\) 13.3568 0.678089
\(389\) 1.84882 0.0937390 0.0468695 0.998901i \(-0.485076\pi\)
0.0468695 + 0.998901i \(0.485076\pi\)
\(390\) 0.480706 0.0243415
\(391\) −0.851462 −0.0430603
\(392\) 6.41404 0.323958
\(393\) 17.0455 0.859834
\(394\) −1.87363 −0.0943921
\(395\) 3.63267 0.182779
\(396\) −1.12221 −0.0563932
\(397\) 38.9381 1.95425 0.977123 0.212675i \(-0.0682177\pi\)
0.977123 + 0.212675i \(0.0682177\pi\)
\(398\) −3.88713 −0.194844
\(399\) 5.53651 0.277172
\(400\) −4.76892 −0.238446
\(401\) −18.6730 −0.932487 −0.466243 0.884657i \(-0.654393\pi\)
−0.466243 + 0.884657i \(0.654393\pi\)
\(402\) 12.4809 0.622491
\(403\) −0.823617 −0.0410273
\(404\) −17.0194 −0.846748
\(405\) 0.480706 0.0238865
\(406\) 4.53598 0.225117
\(407\) 6.44661 0.319547
\(408\) −0.131422 −0.00650637
\(409\) 26.4718 1.30895 0.654474 0.756084i \(-0.272890\pi\)
0.654474 + 0.756084i \(0.272890\pi\)
\(410\) −4.67135 −0.230701
\(411\) −13.6581 −0.673707
\(412\) 1.00000 0.0492665
\(413\) −6.00465 −0.295470
\(414\) 6.47882 0.318417
\(415\) 7.06205 0.346662
\(416\) 1.00000 0.0490290
\(417\) −16.1582 −0.791269
\(418\) 8.11661 0.396997
\(419\) 14.5263 0.709654 0.354827 0.934932i \(-0.384540\pi\)
0.354827 + 0.934932i \(0.384540\pi\)
\(420\) 0.367973 0.0179552
\(421\) −13.3600 −0.651129 −0.325564 0.945520i \(-0.605554\pi\)
−0.325564 + 0.945520i \(0.605554\pi\)
\(422\) 12.2619 0.596899
\(423\) −1.21402 −0.0590276
\(424\) 6.16740 0.299516
\(425\) −0.626743 −0.0304015
\(426\) 16.6763 0.807970
\(427\) 6.40239 0.309833
\(428\) −4.57541 −0.221161
\(429\) 1.12221 0.0541809
\(430\) 0.862766 0.0416063
\(431\) −40.1593 −1.93441 −0.967203 0.254006i \(-0.918252\pi\)
−0.967203 + 0.254006i \(0.918252\pi\)
\(432\) 1.00000 0.0481125
\(433\) 15.9722 0.767577 0.383789 0.923421i \(-0.374619\pi\)
0.383789 + 0.923421i \(0.374619\pi\)
\(434\) −0.630465 −0.0302633
\(435\) −2.84849 −0.136575
\(436\) 4.28510 0.205219
\(437\) −46.8593 −2.24159
\(438\) 2.44935 0.117034
\(439\) −9.11229 −0.434906 −0.217453 0.976071i \(-0.569775\pi\)
−0.217453 + 0.976071i \(0.569775\pi\)
\(440\) 0.539454 0.0257175
\(441\) −6.41404 −0.305430
\(442\) 0.131422 0.00625112
\(443\) 7.39207 0.351208 0.175604 0.984461i \(-0.443812\pi\)
0.175604 + 0.984461i \(0.443812\pi\)
\(444\) −5.74456 −0.272625
\(445\) −8.10583 −0.384253
\(446\) 18.6606 0.883604
\(447\) −6.45756 −0.305432
\(448\) 0.765483 0.0361657
\(449\) 29.3660 1.38587 0.692933 0.721002i \(-0.256318\pi\)
0.692933 + 0.721002i \(0.256318\pi\)
\(450\) 4.76892 0.224809
\(451\) −10.9053 −0.513510
\(452\) 18.0957 0.851151
\(453\) 6.56350 0.308380
\(454\) 4.40861 0.206906
\(455\) −0.367973 −0.0172508
\(456\) −7.23270 −0.338702
\(457\) 24.8708 1.16341 0.581703 0.813401i \(-0.302387\pi\)
0.581703 + 0.813401i \(0.302387\pi\)
\(458\) 5.24559 0.245110
\(459\) 0.131422 0.00613427
\(460\) −3.11441 −0.145210
\(461\) −26.9015 −1.25293 −0.626464 0.779451i \(-0.715498\pi\)
−0.626464 + 0.779451i \(0.715498\pi\)
\(462\) 0.859034 0.0399659
\(463\) −19.5006 −0.906268 −0.453134 0.891442i \(-0.649694\pi\)
−0.453134 + 0.891442i \(0.649694\pi\)
\(464\) −5.92565 −0.275091
\(465\) 0.395918 0.0183602
\(466\) −7.38317 −0.342019
\(467\) −15.4476 −0.714828 −0.357414 0.933946i \(-0.616342\pi\)
−0.357414 + 0.933946i \(0.616342\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −9.55392 −0.441159
\(470\) 0.583586 0.0269188
\(471\) −4.46960 −0.205948
\(472\) 7.84426 0.361061
\(473\) 2.01413 0.0926099
\(474\) −7.55695 −0.347102
\(475\) −34.4922 −1.58261
\(476\) 0.100602 0.00461107
\(477\) −6.16740 −0.282386
\(478\) −11.7084 −0.535528
\(479\) −27.7327 −1.26714 −0.633569 0.773686i \(-0.718411\pi\)
−0.633569 + 0.773686i \(0.718411\pi\)
\(480\) −0.480706 −0.0219411
\(481\) 5.74456 0.261929
\(482\) 24.2815 1.10599
\(483\) −4.95943 −0.225662
\(484\) −9.74064 −0.442756
\(485\) 6.42069 0.291549
\(486\) −1.00000 −0.0453609
\(487\) 33.1502 1.50218 0.751090 0.660200i \(-0.229529\pi\)
0.751090 + 0.660200i \(0.229529\pi\)
\(488\) −8.36385 −0.378614
\(489\) 11.9349 0.539717
\(490\) 3.08327 0.139288
\(491\) 6.41368 0.289445 0.144723 0.989472i \(-0.453771\pi\)
0.144723 + 0.989472i \(0.453771\pi\)
\(492\) 9.71768 0.438107
\(493\) −0.778762 −0.0350737
\(494\) 7.23270 0.325414
\(495\) −0.539454 −0.0242466
\(496\) 0.823617 0.0369815
\(497\) −12.7654 −0.572608
\(498\) −14.6910 −0.658319
\(499\) −24.8825 −1.11389 −0.556947 0.830548i \(-0.688027\pi\)
−0.556947 + 0.830548i \(0.688027\pi\)
\(500\) −4.69598 −0.210011
\(501\) 11.7525 0.525064
\(502\) −20.9442 −0.934783
\(503\) 5.87473 0.261941 0.130971 0.991386i \(-0.458191\pi\)
0.130971 + 0.991386i \(0.458191\pi\)
\(504\) −0.765483 −0.0340973
\(505\) −8.18134 −0.364065
\(506\) −7.27061 −0.323218
\(507\) 1.00000 0.0444116
\(508\) 1.93648 0.0859172
\(509\) −11.1462 −0.494046 −0.247023 0.969010i \(-0.579452\pi\)
−0.247023 + 0.969010i \(0.579452\pi\)
\(510\) −0.0631755 −0.00279746
\(511\) −1.87494 −0.0829423
\(512\) −1.00000 −0.0441942
\(513\) 7.23270 0.319331
\(514\) 18.5827 0.819647
\(515\) 0.480706 0.0211824
\(516\) −1.79479 −0.0790112
\(517\) 1.36238 0.0599176
\(518\) 4.39737 0.193209
\(519\) −6.62095 −0.290627
\(520\) 0.480706 0.0210804
\(521\) 31.9050 1.39778 0.698891 0.715228i \(-0.253677\pi\)
0.698891 + 0.715228i \(0.253677\pi\)
\(522\) 5.92565 0.259359
\(523\) −4.89722 −0.214140 −0.107070 0.994251i \(-0.534147\pi\)
−0.107070 + 0.994251i \(0.534147\pi\)
\(524\) 17.0455 0.744638
\(525\) −3.65053 −0.159322
\(526\) −0.246017 −0.0107268
\(527\) 0.108242 0.00471508
\(528\) −1.12221 −0.0488380
\(529\) 18.9751 0.825006
\(530\) 2.96471 0.128779
\(531\) −7.84426 −0.340412
\(532\) 5.53651 0.240038
\(533\) −9.71768 −0.420919
\(534\) 16.8623 0.729704
\(535\) −2.19943 −0.0950895
\(536\) 12.4809 0.539093
\(537\) −2.90096 −0.125185
\(538\) 21.0144 0.905995
\(539\) 7.19790 0.310036
\(540\) 0.480706 0.0206863
\(541\) 36.2873 1.56011 0.780057 0.625708i \(-0.215190\pi\)
0.780057 + 0.625708i \(0.215190\pi\)
\(542\) 30.8832 1.32655
\(543\) 7.61769 0.326906
\(544\) −0.131422 −0.00563468
\(545\) 2.05987 0.0882353
\(546\) 0.765483 0.0327597
\(547\) 27.5912 1.17971 0.589856 0.807508i \(-0.299184\pi\)
0.589856 + 0.807508i \(0.299184\pi\)
\(548\) −13.6581 −0.583447
\(549\) 8.36385 0.356961
\(550\) −5.35174 −0.228199
\(551\) −42.8584 −1.82583
\(552\) 6.47882 0.275757
\(553\) 5.78472 0.245991
\(554\) −12.7814 −0.543030
\(555\) −2.76145 −0.117217
\(556\) −16.1582 −0.685259
\(557\) −33.3217 −1.41189 −0.705943 0.708269i \(-0.749476\pi\)
−0.705943 + 0.708269i \(0.749476\pi\)
\(558\) −0.823617 −0.0348665
\(559\) 1.79479 0.0759115
\(560\) 0.367973 0.0155497
\(561\) −0.147484 −0.00622676
\(562\) −11.4614 −0.483471
\(563\) −30.0026 −1.26446 −0.632230 0.774781i \(-0.717860\pi\)
−0.632230 + 0.774781i \(0.717860\pi\)
\(564\) −1.21402 −0.0511194
\(565\) 8.69872 0.365958
\(566\) −23.9695 −1.00751
\(567\) 0.765483 0.0321473
\(568\) 16.6763 0.699723
\(569\) 15.9275 0.667713 0.333857 0.942624i \(-0.391650\pi\)
0.333857 + 0.942624i \(0.391650\pi\)
\(570\) −3.47680 −0.145627
\(571\) −22.7744 −0.953079 −0.476539 0.879153i \(-0.658109\pi\)
−0.476539 + 0.879153i \(0.658109\pi\)
\(572\) 1.12221 0.0469220
\(573\) −21.2603 −0.888160
\(574\) −7.43872 −0.310486
\(575\) 30.8970 1.28849
\(576\) 1.00000 0.0416667
\(577\) −29.6358 −1.23376 −0.616878 0.787059i \(-0.711603\pi\)
−0.616878 + 0.787059i \(0.711603\pi\)
\(578\) 16.9827 0.706388
\(579\) 2.66290 0.110667
\(580\) −2.84849 −0.118277
\(581\) 11.2457 0.466551
\(582\) −13.3568 −0.553657
\(583\) 6.92113 0.286644
\(584\) 2.44935 0.101355
\(585\) −0.480706 −0.0198748
\(586\) 27.6256 1.14120
\(587\) −18.3847 −0.758816 −0.379408 0.925229i \(-0.623872\pi\)
−0.379408 + 0.925229i \(0.623872\pi\)
\(588\) −6.41404 −0.264510
\(589\) 5.95697 0.245453
\(590\) 3.77078 0.155241
\(591\) 1.87363 0.0770709
\(592\) −5.74456 −0.236100
\(593\) −42.2315 −1.73424 −0.867121 0.498098i \(-0.834032\pi\)
−0.867121 + 0.498098i \(0.834032\pi\)
\(594\) 1.12221 0.0460449
\(595\) 0.0483598 0.00198256
\(596\) −6.45756 −0.264512
\(597\) 3.88713 0.159090
\(598\) −6.47882 −0.264939
\(599\) −30.6689 −1.25310 −0.626549 0.779382i \(-0.715533\pi\)
−0.626549 + 0.779382i \(0.715533\pi\)
\(600\) 4.76892 0.194690
\(601\) 4.48766 0.183055 0.0915276 0.995803i \(-0.470825\pi\)
0.0915276 + 0.995803i \(0.470825\pi\)
\(602\) 1.37388 0.0559952
\(603\) −12.4809 −0.508261
\(604\) 6.56350 0.267065
\(605\) −4.68239 −0.190366
\(606\) 17.0194 0.691367
\(607\) 31.0647 1.26088 0.630439 0.776239i \(-0.282875\pi\)
0.630439 + 0.776239i \(0.282875\pi\)
\(608\) −7.23270 −0.293325
\(609\) −4.53598 −0.183807
\(610\) −4.02056 −0.162788
\(611\) 1.21402 0.0491139
\(612\) 0.131422 0.00531243
\(613\) 26.9085 1.08682 0.543412 0.839466i \(-0.317132\pi\)
0.543412 + 0.839466i \(0.317132\pi\)
\(614\) −31.0759 −1.25412
\(615\) 4.67135 0.188367
\(616\) 0.859034 0.0346115
\(617\) −39.4434 −1.58793 −0.793966 0.607962i \(-0.791987\pi\)
−0.793966 + 0.607962i \(0.791987\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −11.9302 −0.479515 −0.239758 0.970833i \(-0.577068\pi\)
−0.239758 + 0.970833i \(0.577068\pi\)
\(620\) 0.395918 0.0159004
\(621\) −6.47882 −0.259986
\(622\) 12.4535 0.499338
\(623\) −12.9078 −0.517141
\(624\) −1.00000 −0.0400320
\(625\) 21.5872 0.863489
\(626\) −8.68625 −0.347172
\(627\) −8.11661 −0.324146
\(628\) −4.46960 −0.178357
\(629\) −0.754963 −0.0301023
\(630\) −0.367973 −0.0146604
\(631\) 22.2362 0.885209 0.442604 0.896717i \(-0.354055\pi\)
0.442604 + 0.896717i \(0.354055\pi\)
\(632\) −7.55695 −0.300599
\(633\) −12.2619 −0.487366
\(634\) −6.18684 −0.245711
\(635\) 0.930876 0.0369407
\(636\) −6.16740 −0.244553
\(637\) 6.41404 0.254133
\(638\) −6.64983 −0.263269
\(639\) −16.6763 −0.659705
\(640\) −0.480706 −0.0190016
\(641\) −41.1262 −1.62439 −0.812194 0.583387i \(-0.801727\pi\)
−0.812194 + 0.583387i \(0.801727\pi\)
\(642\) 4.57541 0.180577
\(643\) 9.83517 0.387861 0.193931 0.981015i \(-0.437876\pi\)
0.193931 + 0.981015i \(0.437876\pi\)
\(644\) −4.95943 −0.195429
\(645\) −0.862766 −0.0339714
\(646\) −0.950537 −0.0373984
\(647\) 33.4506 1.31508 0.657540 0.753419i \(-0.271597\pi\)
0.657540 + 0.753419i \(0.271597\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 8.80292 0.345545
\(650\) −4.76892 −0.187052
\(651\) 0.630465 0.0247099
\(652\) 11.9349 0.467408
\(653\) 23.3997 0.915701 0.457850 0.889029i \(-0.348620\pi\)
0.457850 + 0.889029i \(0.348620\pi\)
\(654\) −4.28510 −0.167561
\(655\) 8.19390 0.320162
\(656\) 9.71768 0.379412
\(657\) −2.44935 −0.0955582
\(658\) 0.929311 0.0362283
\(659\) −19.9669 −0.777798 −0.388899 0.921280i \(-0.627145\pi\)
−0.388899 + 0.921280i \(0.627145\pi\)
\(660\) −0.539454 −0.0209982
\(661\) −12.5468 −0.488015 −0.244007 0.969773i \(-0.578462\pi\)
−0.244007 + 0.969773i \(0.578462\pi\)
\(662\) 29.0460 1.12890
\(663\) −0.131422 −0.00510402
\(664\) −14.6910 −0.570121
\(665\) 2.66143 0.103206
\(666\) 5.74456 0.222597
\(667\) 38.3912 1.48651
\(668\) 11.7525 0.454719
\(669\) −18.6606 −0.721459
\(670\) 5.99964 0.231786
\(671\) −9.38601 −0.362343
\(672\) −0.765483 −0.0295292
\(673\) 14.3712 0.553968 0.276984 0.960874i \(-0.410665\pi\)
0.276984 + 0.960874i \(0.410665\pi\)
\(674\) 0.492714 0.0189786
\(675\) −4.76892 −0.183556
\(676\) 1.00000 0.0384615
\(677\) −31.6177 −1.21517 −0.607584 0.794255i \(-0.707861\pi\)
−0.607584 + 0.794255i \(0.707861\pi\)
\(678\) −18.0957 −0.694962
\(679\) 10.2244 0.392377
\(680\) −0.0631755 −0.00242267
\(681\) −4.40861 −0.168938
\(682\) 0.924272 0.0353922
\(683\) −42.9732 −1.64432 −0.822162 0.569254i \(-0.807232\pi\)
−0.822162 + 0.569254i \(0.807232\pi\)
\(684\) 7.23270 0.276549
\(685\) −6.56555 −0.250857
\(686\) 10.2682 0.392043
\(687\) −5.24559 −0.200132
\(688\) −1.79479 −0.0684257
\(689\) 6.16740 0.234959
\(690\) 3.11441 0.118564
\(691\) −37.8566 −1.44013 −0.720066 0.693905i \(-0.755889\pi\)
−0.720066 + 0.693905i \(0.755889\pi\)
\(692\) −6.62095 −0.251691
\(693\) −0.859034 −0.0326320
\(694\) −20.9494 −0.795227
\(695\) −7.76733 −0.294631
\(696\) 5.92565 0.224611
\(697\) 1.27712 0.0483743
\(698\) 29.6109 1.12079
\(699\) 7.38317 0.279257
\(700\) −3.65053 −0.137977
\(701\) 45.8771 1.73275 0.866377 0.499391i \(-0.166443\pi\)
0.866377 + 0.499391i \(0.166443\pi\)
\(702\) 1.00000 0.0377426
\(703\) −41.5487 −1.56704
\(704\) −1.12221 −0.0422949
\(705\) −0.583586 −0.0219791
\(706\) −12.6044 −0.474374
\(707\) −13.0281 −0.489972
\(708\) −7.84426 −0.294805
\(709\) 4.87459 0.183069 0.0915345 0.995802i \(-0.470823\pi\)
0.0915345 + 0.995802i \(0.470823\pi\)
\(710\) 8.01641 0.300850
\(711\) 7.55695 0.283408
\(712\) 16.8623 0.631943
\(713\) −5.33607 −0.199837
\(714\) −0.100602 −0.00376492
\(715\) 0.539454 0.0201744
\(716\) −2.90096 −0.108414
\(717\) 11.7084 0.437257
\(718\) −27.7883 −1.03705
\(719\) −29.5494 −1.10201 −0.551003 0.834503i \(-0.685755\pi\)
−0.551003 + 0.834503i \(0.685755\pi\)
\(720\) 0.480706 0.0179149
\(721\) 0.765483 0.0285081
\(722\) −33.3119 −1.23974
\(723\) −24.2815 −0.903038
\(724\) 7.61769 0.283109
\(725\) 28.2589 1.04951
\(726\) 9.74064 0.361509
\(727\) 29.9698 1.11152 0.555759 0.831344i \(-0.312428\pi\)
0.555759 + 0.831344i \(0.312428\pi\)
\(728\) 0.765483 0.0283707
\(729\) 1.00000 0.0370370
\(730\) 1.17742 0.0435782
\(731\) −0.235875 −0.00872416
\(732\) 8.36385 0.309137
\(733\) −16.5241 −0.610331 −0.305165 0.952299i \(-0.598712\pi\)
−0.305165 + 0.952299i \(0.598712\pi\)
\(734\) −2.77067 −0.102267
\(735\) −3.08327 −0.113728
\(736\) 6.47882 0.238813
\(737\) 14.0062 0.515925
\(738\) −9.71768 −0.357713
\(739\) −34.7278 −1.27748 −0.638742 0.769421i \(-0.720545\pi\)
−0.638742 + 0.769421i \(0.720545\pi\)
\(740\) −2.76145 −0.101513
\(741\) −7.23270 −0.265700
\(742\) 4.72104 0.173315
\(743\) −1.58390 −0.0581077 −0.0290538 0.999578i \(-0.509249\pi\)
−0.0290538 + 0.999578i \(0.509249\pi\)
\(744\) −0.823617 −0.0301953
\(745\) −3.10419 −0.113729
\(746\) −25.4710 −0.932559
\(747\) 14.6910 0.537515
\(748\) −0.147484 −0.00539253
\(749\) −3.50240 −0.127975
\(750\) 4.69598 0.171473
\(751\) 11.0699 0.403945 0.201973 0.979391i \(-0.435265\pi\)
0.201973 + 0.979391i \(0.435265\pi\)
\(752\) −1.21402 −0.0442707
\(753\) 20.9442 0.763247
\(754\) −5.92565 −0.215799
\(755\) 3.15511 0.114826
\(756\) 0.765483 0.0278404
\(757\) −51.6099 −1.87579 −0.937897 0.346915i \(-0.887229\pi\)
−0.937897 + 0.346915i \(0.887229\pi\)
\(758\) 7.25237 0.263418
\(759\) 7.27061 0.263906
\(760\) −3.47680 −0.126117
\(761\) −19.1241 −0.693246 −0.346623 0.938004i \(-0.612672\pi\)
−0.346623 + 0.938004i \(0.612672\pi\)
\(762\) −1.93648 −0.0701511
\(763\) 3.28017 0.118750
\(764\) −21.2603 −0.769169
\(765\) 0.0631755 0.00228411
\(766\) 20.9833 0.758156
\(767\) 7.84426 0.283240
\(768\) 1.00000 0.0360844
\(769\) 1.19596 0.0431276 0.0215638 0.999767i \(-0.493135\pi\)
0.0215638 + 0.999767i \(0.493135\pi\)
\(770\) 0.412943 0.0148814
\(771\) −18.5827 −0.669239
\(772\) 2.66290 0.0958400
\(773\) 18.2539 0.656547 0.328274 0.944583i \(-0.393533\pi\)
0.328274 + 0.944583i \(0.393533\pi\)
\(774\) 1.79479 0.0645124
\(775\) −3.92776 −0.141089
\(776\) −13.3568 −0.479481
\(777\) −4.39737 −0.157755
\(778\) −1.84882 −0.0662835
\(779\) 70.2850 2.51822
\(780\) −0.480706 −0.0172120
\(781\) 18.7144 0.669652
\(782\) 0.851462 0.0304482
\(783\) −5.92565 −0.211765
\(784\) −6.41404 −0.229073
\(785\) −2.14857 −0.0766856
\(786\) −17.0455 −0.607995
\(787\) −7.19066 −0.256319 −0.128160 0.991754i \(-0.540907\pi\)
−0.128160 + 0.991754i \(0.540907\pi\)
\(788\) 1.87363 0.0667453
\(789\) 0.246017 0.00875842
\(790\) −3.63267 −0.129245
\(791\) 13.8520 0.492519
\(792\) 1.12221 0.0398760
\(793\) −8.36385 −0.297009
\(794\) −38.9381 −1.38186
\(795\) −2.96471 −0.105147
\(796\) 3.88713 0.137776
\(797\) −48.8498 −1.73035 −0.865175 0.501470i \(-0.832793\pi\)
−0.865175 + 0.501470i \(0.832793\pi\)
\(798\) −5.53651 −0.195990
\(799\) −0.159549 −0.00564444
\(800\) 4.76892 0.168607
\(801\) −16.8623 −0.595801
\(802\) 18.6730 0.659368
\(803\) 2.74869 0.0969991
\(804\) −12.4809 −0.440167
\(805\) −2.38403 −0.0840260
\(806\) 0.823617 0.0290107
\(807\) −21.0144 −0.739742
\(808\) 17.0194 0.598741
\(809\) 44.5321 1.56566 0.782832 0.622233i \(-0.213774\pi\)
0.782832 + 0.622233i \(0.213774\pi\)
\(810\) −0.480706 −0.0168903
\(811\) 46.2106 1.62267 0.811337 0.584579i \(-0.198740\pi\)
0.811337 + 0.584579i \(0.198740\pi\)
\(812\) −4.53598 −0.159182
\(813\) −30.8832 −1.08312
\(814\) −6.44661 −0.225954
\(815\) 5.73720 0.200965
\(816\) 0.131422 0.00460070
\(817\) −12.9812 −0.454153
\(818\) −26.4718 −0.925566
\(819\) −0.765483 −0.0267482
\(820\) 4.67135 0.163131
\(821\) 36.7399 1.28223 0.641115 0.767445i \(-0.278472\pi\)
0.641115 + 0.767445i \(0.278472\pi\)
\(822\) 13.6581 0.476383
\(823\) −22.6385 −0.789128 −0.394564 0.918869i \(-0.629104\pi\)
−0.394564 + 0.918869i \(0.629104\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 5.35174 0.186324
\(826\) 6.00465 0.208928
\(827\) −9.97795 −0.346967 −0.173484 0.984837i \(-0.555502\pi\)
−0.173484 + 0.984837i \(0.555502\pi\)
\(828\) −6.47882 −0.225155
\(829\) 10.5754 0.367298 0.183649 0.982992i \(-0.441209\pi\)
0.183649 + 0.982992i \(0.441209\pi\)
\(830\) −7.06205 −0.245127
\(831\) 12.7814 0.443382
\(832\) −1.00000 −0.0346688
\(833\) −0.842947 −0.0292064
\(834\) 16.1582 0.559511
\(835\) 5.64951 0.195509
\(836\) −8.11661 −0.280719
\(837\) 0.823617 0.0284684
\(838\) −14.5263 −0.501801
\(839\) 24.4530 0.844212 0.422106 0.906547i \(-0.361291\pi\)
0.422106 + 0.906547i \(0.361291\pi\)
\(840\) −0.367973 −0.0126963
\(841\) 6.11329 0.210803
\(842\) 13.3600 0.460417
\(843\) 11.4614 0.394753
\(844\) −12.2619 −0.422072
\(845\) 0.480706 0.0165368
\(846\) 1.21402 0.0417388
\(847\) −7.45630 −0.256201
\(848\) −6.16740 −0.211789
\(849\) 23.9695 0.822630
\(850\) 0.626743 0.0214971
\(851\) 37.2180 1.27582
\(852\) −16.6763 −0.571321
\(853\) 22.0285 0.754243 0.377121 0.926164i \(-0.376914\pi\)
0.377121 + 0.926164i \(0.376914\pi\)
\(854\) −6.40239 −0.219085
\(855\) 3.47680 0.118904
\(856\) 4.57541 0.156384
\(857\) 12.5241 0.427816 0.213908 0.976854i \(-0.431381\pi\)
0.213908 + 0.976854i \(0.431381\pi\)
\(858\) −1.12221 −0.0383117
\(859\) −39.6779 −1.35379 −0.676897 0.736078i \(-0.736676\pi\)
−0.676897 + 0.736078i \(0.736676\pi\)
\(860\) −0.862766 −0.0294201
\(861\) 7.43872 0.253511
\(862\) 40.1593 1.36783
\(863\) −28.9247 −0.984609 −0.492305 0.870423i \(-0.663845\pi\)
−0.492305 + 0.870423i \(0.663845\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −3.18273 −0.108216
\(866\) −15.9722 −0.542759
\(867\) −16.9827 −0.576764
\(868\) 0.630465 0.0213994
\(869\) −8.48049 −0.287681
\(870\) 2.84849 0.0965730
\(871\) 12.4809 0.422899
\(872\) −4.28510 −0.145112
\(873\) 13.3568 0.452059
\(874\) 46.8593 1.58504
\(875\) −3.59470 −0.121523
\(876\) −2.44935 −0.0827558
\(877\) −0.160892 −0.00543295 −0.00271648 0.999996i \(-0.500865\pi\)
−0.00271648 + 0.999996i \(0.500865\pi\)
\(878\) 9.11229 0.307525
\(879\) −27.6256 −0.931787
\(880\) −0.539454 −0.0181850
\(881\) 1.77982 0.0599636 0.0299818 0.999550i \(-0.490455\pi\)
0.0299818 + 0.999550i \(0.490455\pi\)
\(882\) 6.41404 0.215972
\(883\) 56.4860 1.90090 0.950452 0.310871i \(-0.100621\pi\)
0.950452 + 0.310871i \(0.100621\pi\)
\(884\) −0.131422 −0.00442021
\(885\) −3.77078 −0.126753
\(886\) −7.39207 −0.248342
\(887\) −37.2982 −1.25235 −0.626175 0.779682i \(-0.715380\pi\)
−0.626175 + 0.779682i \(0.715380\pi\)
\(888\) 5.74456 0.192775
\(889\) 1.48234 0.0497161
\(890\) 8.10583 0.271708
\(891\) −1.12221 −0.0375955
\(892\) −18.6606 −0.624802
\(893\) −8.78062 −0.293832
\(894\) 6.45756 0.215973
\(895\) −1.39451 −0.0466132
\(896\) −0.765483 −0.0255730
\(897\) 6.47882 0.216322
\(898\) −29.3660 −0.979955
\(899\) −4.88046 −0.162773
\(900\) −4.76892 −0.158964
\(901\) −0.810534 −0.0270028
\(902\) 10.9053 0.363106
\(903\) −1.37388 −0.0457199
\(904\) −18.0957 −0.601854
\(905\) 3.66187 0.121725
\(906\) −6.56350 −0.218058
\(907\) 27.4272 0.910706 0.455353 0.890311i \(-0.349513\pi\)
0.455353 + 0.890311i \(0.349513\pi\)
\(908\) −4.40861 −0.146305
\(909\) −17.0194 −0.564499
\(910\) 0.367973 0.0121982
\(911\) −14.0108 −0.464200 −0.232100 0.972692i \(-0.574560\pi\)
−0.232100 + 0.972692i \(0.574560\pi\)
\(912\) 7.23270 0.239498
\(913\) −16.4864 −0.545620
\(914\) −24.8708 −0.822652
\(915\) 4.02056 0.132915
\(916\) −5.24559 −0.173319
\(917\) 13.0481 0.430886
\(918\) −0.131422 −0.00433758
\(919\) −12.8254 −0.423071 −0.211535 0.977370i \(-0.567846\pi\)
−0.211535 + 0.977370i \(0.567846\pi\)
\(920\) 3.11441 0.102679
\(921\) 31.0759 1.02399
\(922\) 26.9015 0.885953
\(923\) 16.6763 0.548908
\(924\) −0.859034 −0.0282602
\(925\) 27.3954 0.900754
\(926\) 19.5006 0.640828
\(927\) 1.00000 0.0328443
\(928\) 5.92565 0.194519
\(929\) 58.8254 1.93000 0.964999 0.262254i \(-0.0844659\pi\)
0.964999 + 0.262254i \(0.0844659\pi\)
\(930\) −0.395918 −0.0129826
\(931\) −46.3908 −1.52040
\(932\) 7.38317 0.241844
\(933\) −12.4535 −0.407708
\(934\) 15.4476 0.505460
\(935\) −0.0708963 −0.00231856
\(936\) 1.00000 0.0326860
\(937\) −35.1385 −1.14792 −0.573962 0.818882i \(-0.694594\pi\)
−0.573962 + 0.818882i \(0.694594\pi\)
\(938\) 9.55392 0.311947
\(939\) 8.68625 0.283465
\(940\) −0.583586 −0.0190345
\(941\) −33.1669 −1.08121 −0.540606 0.841276i \(-0.681805\pi\)
−0.540606 + 0.841276i \(0.681805\pi\)
\(942\) 4.46960 0.145628
\(943\) −62.9591 −2.05023
\(944\) −7.84426 −0.255309
\(945\) 0.367973 0.0119701
\(946\) −2.01413 −0.0654851
\(947\) −12.6520 −0.411134 −0.205567 0.978643i \(-0.565904\pi\)
−0.205567 + 0.978643i \(0.565904\pi\)
\(948\) 7.55695 0.245438
\(949\) 2.44935 0.0795092
\(950\) 34.4922 1.11907
\(951\) 6.18684 0.200622
\(952\) −0.100602 −0.00326052
\(953\) 43.0310 1.39391 0.696956 0.717114i \(-0.254537\pi\)
0.696956 + 0.717114i \(0.254537\pi\)
\(954\) 6.16740 0.199677
\(955\) −10.2199 −0.330709
\(956\) 11.7084 0.378676
\(957\) 6.64983 0.214958
\(958\) 27.7327 0.896003
\(959\) −10.4551 −0.337612
\(960\) 0.480706 0.0155147
\(961\) −30.3217 −0.978118
\(962\) −5.74456 −0.185212
\(963\) −4.57541 −0.147440
\(964\) −24.2815 −0.782054
\(965\) 1.28007 0.0412070
\(966\) 4.95943 0.159567
\(967\) 44.2881 1.42421 0.712105 0.702073i \(-0.247742\pi\)
0.712105 + 0.702073i \(0.247742\pi\)
\(968\) 9.74064 0.313076
\(969\) 0.950537 0.0305357
\(970\) −6.42069 −0.206156
\(971\) 35.4241 1.13681 0.568406 0.822748i \(-0.307560\pi\)
0.568406 + 0.822748i \(0.307560\pi\)
\(972\) 1.00000 0.0320750
\(973\) −12.3688 −0.396526
\(974\) −33.1502 −1.06220
\(975\) 4.76892 0.152728
\(976\) 8.36385 0.267720
\(977\) 24.0530 0.769524 0.384762 0.923016i \(-0.374283\pi\)
0.384762 + 0.923016i \(0.374283\pi\)
\(978\) −11.9349 −0.381637
\(979\) 18.9231 0.604785
\(980\) −3.08327 −0.0984913
\(981\) 4.28510 0.136813
\(982\) −6.41368 −0.204669
\(983\) 25.4509 0.811758 0.405879 0.913927i \(-0.366965\pi\)
0.405879 + 0.913927i \(0.366965\pi\)
\(984\) −9.71768 −0.309788
\(985\) 0.900666 0.0286976
\(986\) 0.778762 0.0248008
\(987\) −0.929311 −0.0295803
\(988\) −7.23270 −0.230103
\(989\) 11.6281 0.369753
\(990\) 0.539454 0.0171450
\(991\) 18.3877 0.584104 0.292052 0.956402i \(-0.405662\pi\)
0.292052 + 0.956402i \(0.405662\pi\)
\(992\) −0.823617 −0.0261499
\(993\) −29.0460 −0.921747
\(994\) 12.7654 0.404895
\(995\) 1.86857 0.0592376
\(996\) 14.6910 0.465502
\(997\) −15.3558 −0.486322 −0.243161 0.969986i \(-0.578184\pi\)
−0.243161 + 0.969986i \(0.578184\pi\)
\(998\) 24.8825 0.787642
\(999\) −5.74456 −0.181750
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.1.9 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bb.1.9 14 1.1 even 1 trivial