Properties

Label 8034.2.a.ba.1.4
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 1.4
Root \(2.71343\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.71343 q^{5} +1.00000 q^{6} -2.34175 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.71343 q^{5} +1.00000 q^{6} -2.34175 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.71343 q^{10} +0.805528 q^{11} -1.00000 q^{12} -1.00000 q^{13} +2.34175 q^{14} +2.71343 q^{15} +1.00000 q^{16} -7.05126 q^{17} -1.00000 q^{18} +3.05101 q^{19} -2.71343 q^{20} +2.34175 q^{21} -0.805528 q^{22} -3.54988 q^{23} +1.00000 q^{24} +2.36272 q^{25} +1.00000 q^{26} -1.00000 q^{27} -2.34175 q^{28} -1.98812 q^{29} -2.71343 q^{30} -7.06425 q^{31} -1.00000 q^{32} -0.805528 q^{33} +7.05126 q^{34} +6.35418 q^{35} +1.00000 q^{36} +1.93939 q^{37} -3.05101 q^{38} +1.00000 q^{39} +2.71343 q^{40} -1.08094 q^{41} -2.34175 q^{42} -3.41372 q^{43} +0.805528 q^{44} -2.71343 q^{45} +3.54988 q^{46} -12.0993 q^{47} -1.00000 q^{48} -1.51621 q^{49} -2.36272 q^{50} +7.05126 q^{51} -1.00000 q^{52} +6.23971 q^{53} +1.00000 q^{54} -2.18575 q^{55} +2.34175 q^{56} -3.05101 q^{57} +1.98812 q^{58} +1.79179 q^{59} +2.71343 q^{60} -2.31314 q^{61} +7.06425 q^{62} -2.34175 q^{63} +1.00000 q^{64} +2.71343 q^{65} +0.805528 q^{66} +4.26635 q^{67} -7.05126 q^{68} +3.54988 q^{69} -6.35418 q^{70} -0.749357 q^{71} -1.00000 q^{72} -4.06343 q^{73} -1.93939 q^{74} -2.36272 q^{75} +3.05101 q^{76} -1.88635 q^{77} -1.00000 q^{78} +4.75496 q^{79} -2.71343 q^{80} +1.00000 q^{81} +1.08094 q^{82} -12.8746 q^{83} +2.34175 q^{84} +19.1331 q^{85} +3.41372 q^{86} +1.98812 q^{87} -0.805528 q^{88} -7.49391 q^{89} +2.71343 q^{90} +2.34175 q^{91} -3.54988 q^{92} +7.06425 q^{93} +12.0993 q^{94} -8.27870 q^{95} +1.00000 q^{96} -9.03214 q^{97} +1.51621 q^{98} +0.805528 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + q^{10} - q^{11} - 14q^{12} - 14q^{13} - 5q^{14} + q^{15} + 14q^{16} - 12q^{17} - 14q^{18} + 10q^{19} - q^{20} - 5q^{21} + q^{22} - q^{23} + 14q^{24} + 19q^{25} + 14q^{26} - 14q^{27} + 5q^{28} + 6q^{29} - q^{30} + 20q^{31} - 14q^{32} + q^{33} + 12q^{34} - 16q^{35} + 14q^{36} - 3q^{37} - 10q^{38} + 14q^{39} + q^{40} + q^{41} + 5q^{42} + 6q^{43} - q^{44} - q^{45} + q^{46} - 13q^{47} - 14q^{48} + 9q^{49} - 19q^{50} + 12q^{51} - 14q^{52} - 27q^{53} + 14q^{54} + 10q^{55} - 5q^{56} - 10q^{57} - 6q^{58} - 6q^{59} + q^{60} - 4q^{61} - 20q^{62} + 5q^{63} + 14q^{64} + q^{65} - q^{66} + 13q^{67} - 12q^{68} + q^{69} + 16q^{70} + 18q^{71} - 14q^{72} + 11q^{73} + 3q^{74} - 19q^{75} + 10q^{76} - 15q^{77} - 14q^{78} + 33q^{79} - q^{80} + 14q^{81} - q^{82} - 25q^{83} - 5q^{84} + 25q^{85} - 6q^{86} - 6q^{87} + q^{88} + 3q^{89} + q^{90} - 5q^{91} - q^{92} - 20q^{93} + 13q^{94} + 30q^{95} + 14q^{96} + 11q^{97} - 9q^{98} - q^{99} + O(q^{100}) \)

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
\(5\) −2.71343 −1.21348 −0.606742 0.794899i \(-0.707524\pi\)
−0.606742 + 0.794899i \(0.707524\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.34175 −0.885098 −0.442549 0.896744i \(-0.645926\pi\)
−0.442549 + 0.896744i \(0.645926\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.71343 0.858063
\(11\) 0.805528 0.242876 0.121438 0.992599i \(-0.461249\pi\)
0.121438 + 0.992599i \(0.461249\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 2.34175 0.625859
\(15\) 2.71343 0.700606
\(16\) 1.00000 0.250000
\(17\) −7.05126 −1.71018 −0.855091 0.518478i \(-0.826499\pi\)
−0.855091 + 0.518478i \(0.826499\pi\)
\(18\) −1.00000 −0.235702
\(19\) 3.05101 0.699949 0.349974 0.936759i \(-0.386190\pi\)
0.349974 + 0.936759i \(0.386190\pi\)
\(20\) −2.71343 −0.606742
\(21\) 2.34175 0.511012
\(22\) −0.805528 −0.171739
\(23\) −3.54988 −0.740202 −0.370101 0.928992i \(-0.620677\pi\)
−0.370101 + 0.928992i \(0.620677\pi\)
\(24\) 1.00000 0.204124
\(25\) 2.36272 0.472544
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) −2.34175 −0.442549
\(29\) −1.98812 −0.369185 −0.184592 0.982815i \(-0.559097\pi\)
−0.184592 + 0.982815i \(0.559097\pi\)
\(30\) −2.71343 −0.495403
\(31\) −7.06425 −1.26878 −0.634389 0.773014i \(-0.718748\pi\)
−0.634389 + 0.773014i \(0.718748\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.805528 −0.140225
\(34\) 7.05126 1.20928
\(35\) 6.35418 1.07405
\(36\) 1.00000 0.166667
\(37\) 1.93939 0.318834 0.159417 0.987211i \(-0.449039\pi\)
0.159417 + 0.987211i \(0.449039\pi\)
\(38\) −3.05101 −0.494939
\(39\) 1.00000 0.160128
\(40\) 2.71343 0.429032
\(41\) −1.08094 −0.168814 −0.0844069 0.996431i \(-0.526900\pi\)
−0.0844069 + 0.996431i \(0.526900\pi\)
\(42\) −2.34175 −0.361340
\(43\) −3.41372 −0.520587 −0.260294 0.965530i \(-0.583819\pi\)
−0.260294 + 0.965530i \(0.583819\pi\)
\(44\) 0.805528 0.121438
\(45\) −2.71343 −0.404495
\(46\) 3.54988 0.523402
\(47\) −12.0993 −1.76486 −0.882432 0.470440i \(-0.844095\pi\)
−0.882432 + 0.470440i \(0.844095\pi\)
\(48\) −1.00000 −0.144338
\(49\) −1.51621 −0.216601
\(50\) −2.36272 −0.334139
\(51\) 7.05126 0.987374
\(52\) −1.00000 −0.138675
\(53\) 6.23971 0.857089 0.428545 0.903521i \(-0.359026\pi\)
0.428545 + 0.903521i \(0.359026\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.18575 −0.294726
\(56\) 2.34175 0.312930
\(57\) −3.05101 −0.404116
\(58\) 1.98812 0.261053
\(59\) 1.79179 0.233271 0.116635 0.993175i \(-0.462789\pi\)
0.116635 + 0.993175i \(0.462789\pi\)
\(60\) 2.71343 0.350303
\(61\) −2.31314 −0.296167 −0.148084 0.988975i \(-0.547310\pi\)
−0.148084 + 0.988975i \(0.547310\pi\)
\(62\) 7.06425 0.897161
\(63\) −2.34175 −0.295033
\(64\) 1.00000 0.125000
\(65\) 2.71343 0.336560
\(66\) 0.805528 0.0991537
\(67\) 4.26635 0.521218 0.260609 0.965444i \(-0.416077\pi\)
0.260609 + 0.965444i \(0.416077\pi\)
\(68\) −7.05126 −0.855091
\(69\) 3.54988 0.427356
\(70\) −6.35418 −0.759470
\(71\) −0.749357 −0.0889323 −0.0444661 0.999011i \(-0.514159\pi\)
−0.0444661 + 0.999011i \(0.514159\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.06343 −0.475589 −0.237794 0.971316i \(-0.576424\pi\)
−0.237794 + 0.971316i \(0.576424\pi\)
\(74\) −1.93939 −0.225450
\(75\) −2.36272 −0.272824
\(76\) 3.05101 0.349974
\(77\) −1.88635 −0.214969
\(78\) −1.00000 −0.113228
\(79\) 4.75496 0.534974 0.267487 0.963561i \(-0.413807\pi\)
0.267487 + 0.963561i \(0.413807\pi\)
\(80\) −2.71343 −0.303371
\(81\) 1.00000 0.111111
\(82\) 1.08094 0.119369
\(83\) −12.8746 −1.41317 −0.706584 0.707629i \(-0.749765\pi\)
−0.706584 + 0.707629i \(0.749765\pi\)
\(84\) 2.34175 0.255506
\(85\) 19.1331 2.07528
\(86\) 3.41372 0.368111
\(87\) 1.98812 0.213149
\(88\) −0.805528 −0.0858696
\(89\) −7.49391 −0.794353 −0.397176 0.917742i \(-0.630010\pi\)
−0.397176 + 0.917742i \(0.630010\pi\)
\(90\) 2.71343 0.286021
\(91\) 2.34175 0.245482
\(92\) −3.54988 −0.370101
\(93\) 7.06425 0.732529
\(94\) 12.0993 1.24795
\(95\) −8.27870 −0.849377
\(96\) 1.00000 0.102062
\(97\) −9.03214 −0.917075 −0.458537 0.888675i \(-0.651627\pi\)
−0.458537 + 0.888675i \(0.651627\pi\)
\(98\) 1.51621 0.153160
\(99\) 0.805528 0.0809587
\(100\) 2.36272 0.236272
\(101\) −8.39612 −0.835445 −0.417723 0.908575i \(-0.637172\pi\)
−0.417723 + 0.908575i \(0.637172\pi\)
\(102\) −7.05126 −0.698179
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) −6.35418 −0.620105
\(106\) −6.23971 −0.606054
\(107\) −6.34500 −0.613394 −0.306697 0.951807i \(-0.599224\pi\)
−0.306697 + 0.951807i \(0.599224\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −3.27488 −0.313676 −0.156838 0.987624i \(-0.550130\pi\)
−0.156838 + 0.987624i \(0.550130\pi\)
\(110\) 2.18575 0.208403
\(111\) −1.93939 −0.184079
\(112\) −2.34175 −0.221275
\(113\) −11.0842 −1.04271 −0.521355 0.853340i \(-0.674573\pi\)
−0.521355 + 0.853340i \(0.674573\pi\)
\(114\) 3.05101 0.285753
\(115\) 9.63237 0.898223
\(116\) −1.98812 −0.184592
\(117\) −1.00000 −0.0924500
\(118\) −1.79179 −0.164947
\(119\) 16.5123 1.51368
\(120\) −2.71343 −0.247701
\(121\) −10.3511 −0.941011
\(122\) 2.31314 0.209422
\(123\) 1.08094 0.0974647
\(124\) −7.06425 −0.634389
\(125\) 7.15608 0.640059
\(126\) 2.34175 0.208620
\(127\) −20.2440 −1.79636 −0.898181 0.439627i \(-0.855111\pi\)
−0.898181 + 0.439627i \(0.855111\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 3.41372 0.300561
\(130\) −2.71343 −0.237984
\(131\) −18.2985 −1.59875 −0.799374 0.600833i \(-0.794835\pi\)
−0.799374 + 0.600833i \(0.794835\pi\)
\(132\) −0.805528 −0.0701123
\(133\) −7.14469 −0.619524
\(134\) −4.26635 −0.368557
\(135\) 2.71343 0.233535
\(136\) 7.05126 0.604641
\(137\) −9.99842 −0.854222 −0.427111 0.904199i \(-0.640469\pi\)
−0.427111 + 0.904199i \(0.640469\pi\)
\(138\) −3.54988 −0.302186
\(139\) 8.86959 0.752309 0.376154 0.926557i \(-0.377246\pi\)
0.376154 + 0.926557i \(0.377246\pi\)
\(140\) 6.35418 0.537026
\(141\) 12.0993 1.01894
\(142\) 0.749357 0.0628846
\(143\) −0.805528 −0.0673617
\(144\) 1.00000 0.0833333
\(145\) 5.39464 0.448000
\(146\) 4.06343 0.336292
\(147\) 1.51621 0.125055
\(148\) 1.93939 0.159417
\(149\) 1.38681 0.113612 0.0568058 0.998385i \(-0.481908\pi\)
0.0568058 + 0.998385i \(0.481908\pi\)
\(150\) 2.36272 0.192915
\(151\) 14.3074 1.16432 0.582159 0.813075i \(-0.302208\pi\)
0.582159 + 0.813075i \(0.302208\pi\)
\(152\) −3.05101 −0.247469
\(153\) −7.05126 −0.570061
\(154\) 1.88635 0.152006
\(155\) 19.1684 1.53964
\(156\) 1.00000 0.0800641
\(157\) −8.73671 −0.697265 −0.348633 0.937259i \(-0.613354\pi\)
−0.348633 + 0.937259i \(0.613354\pi\)
\(158\) −4.75496 −0.378284
\(159\) −6.23971 −0.494841
\(160\) 2.71343 0.214516
\(161\) 8.31294 0.655151
\(162\) −1.00000 −0.0785674
\(163\) 23.6512 1.85251 0.926253 0.376902i \(-0.123011\pi\)
0.926253 + 0.376902i \(0.123011\pi\)
\(164\) −1.08094 −0.0844069
\(165\) 2.18575 0.170160
\(166\) 12.8746 0.999260
\(167\) 0.498550 0.0385790 0.0192895 0.999814i \(-0.493860\pi\)
0.0192895 + 0.999814i \(0.493860\pi\)
\(168\) −2.34175 −0.180670
\(169\) 1.00000 0.0769231
\(170\) −19.1331 −1.46744
\(171\) 3.05101 0.233316
\(172\) −3.41372 −0.260294
\(173\) 1.59065 0.120935 0.0604676 0.998170i \(-0.480741\pi\)
0.0604676 + 0.998170i \(0.480741\pi\)
\(174\) −1.98812 −0.150719
\(175\) −5.53290 −0.418248
\(176\) 0.805528 0.0607190
\(177\) −1.79179 −0.134679
\(178\) 7.49391 0.561692
\(179\) −19.1240 −1.42939 −0.714696 0.699436i \(-0.753435\pi\)
−0.714696 + 0.699436i \(0.753435\pi\)
\(180\) −2.71343 −0.202247
\(181\) 13.3384 0.991436 0.495718 0.868483i \(-0.334905\pi\)
0.495718 + 0.868483i \(0.334905\pi\)
\(182\) −2.34175 −0.173582
\(183\) 2.31314 0.170992
\(184\) 3.54988 0.261701
\(185\) −5.26241 −0.386900
\(186\) −7.06425 −0.517976
\(187\) −5.67999 −0.415362
\(188\) −12.0993 −0.882432
\(189\) 2.34175 0.170337
\(190\) 8.27870 0.600600
\(191\) −20.3164 −1.47004 −0.735022 0.678043i \(-0.762828\pi\)
−0.735022 + 0.678043i \(0.762828\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −19.0532 −1.37148 −0.685740 0.727847i \(-0.740521\pi\)
−0.685740 + 0.727847i \(0.740521\pi\)
\(194\) 9.03214 0.648470
\(195\) −2.71343 −0.194313
\(196\) −1.51621 −0.108300
\(197\) 0.799715 0.0569774 0.0284887 0.999594i \(-0.490931\pi\)
0.0284887 + 0.999594i \(0.490931\pi\)
\(198\) −0.805528 −0.0572464
\(199\) −11.3388 −0.803786 −0.401893 0.915687i \(-0.631648\pi\)
−0.401893 + 0.915687i \(0.631648\pi\)
\(200\) −2.36272 −0.167070
\(201\) −4.26635 −0.300925
\(202\) 8.39612 0.590749
\(203\) 4.65568 0.326765
\(204\) 7.05126 0.493687
\(205\) 2.93305 0.204853
\(206\) −1.00000 −0.0696733
\(207\) −3.54988 −0.246734
\(208\) −1.00000 −0.0693375
\(209\) 2.45767 0.170001
\(210\) 6.35418 0.438480
\(211\) −18.5295 −1.27562 −0.637811 0.770193i \(-0.720160\pi\)
−0.637811 + 0.770193i \(0.720160\pi\)
\(212\) 6.23971 0.428545
\(213\) 0.749357 0.0513451
\(214\) 6.34500 0.433735
\(215\) 9.26290 0.631724
\(216\) 1.00000 0.0680414
\(217\) 16.5427 1.12299
\(218\) 3.27488 0.221803
\(219\) 4.06343 0.274581
\(220\) −2.18575 −0.147363
\(221\) 7.05126 0.474319
\(222\) 1.93939 0.130163
\(223\) −21.4736 −1.43798 −0.718988 0.695022i \(-0.755395\pi\)
−0.718988 + 0.695022i \(0.755395\pi\)
\(224\) 2.34175 0.156465
\(225\) 2.36272 0.157515
\(226\) 11.0842 0.737307
\(227\) −5.45603 −0.362129 −0.181065 0.983471i \(-0.557954\pi\)
−0.181065 + 0.983471i \(0.557954\pi\)
\(228\) −3.05101 −0.202058
\(229\) 2.36284 0.156141 0.0780703 0.996948i \(-0.475124\pi\)
0.0780703 + 0.996948i \(0.475124\pi\)
\(230\) −9.63237 −0.635140
\(231\) 1.88635 0.124112
\(232\) 1.98812 0.130527
\(233\) 4.80781 0.314970 0.157485 0.987521i \(-0.449661\pi\)
0.157485 + 0.987521i \(0.449661\pi\)
\(234\) 1.00000 0.0653720
\(235\) 32.8306 2.14163
\(236\) 1.79179 0.116635
\(237\) −4.75496 −0.308868
\(238\) −16.5123 −1.07033
\(239\) −25.2866 −1.63565 −0.817826 0.575465i \(-0.804821\pi\)
−0.817826 + 0.575465i \(0.804821\pi\)
\(240\) 2.71343 0.175151
\(241\) 3.99306 0.257215 0.128608 0.991696i \(-0.458949\pi\)
0.128608 + 0.991696i \(0.458949\pi\)
\(242\) 10.3511 0.665395
\(243\) −1.00000 −0.0641500
\(244\) −2.31314 −0.148084
\(245\) 4.11413 0.262842
\(246\) −1.08094 −0.0689180
\(247\) −3.05101 −0.194131
\(248\) 7.06425 0.448580
\(249\) 12.8746 0.815893
\(250\) −7.15608 −0.452590
\(251\) −2.60067 −0.164153 −0.0820764 0.996626i \(-0.526155\pi\)
−0.0820764 + 0.996626i \(0.526155\pi\)
\(252\) −2.34175 −0.147516
\(253\) −2.85953 −0.179777
\(254\) 20.2440 1.27022
\(255\) −19.1331 −1.19816
\(256\) 1.00000 0.0625000
\(257\) 1.93178 0.120501 0.0602506 0.998183i \(-0.480810\pi\)
0.0602506 + 0.998183i \(0.480810\pi\)
\(258\) −3.41372 −0.212529
\(259\) −4.54157 −0.282200
\(260\) 2.71343 0.168280
\(261\) −1.98812 −0.123062
\(262\) 18.2985 1.13049
\(263\) 19.7974 1.22076 0.610380 0.792109i \(-0.291017\pi\)
0.610380 + 0.792109i \(0.291017\pi\)
\(264\) 0.805528 0.0495769
\(265\) −16.9310 −1.04006
\(266\) 7.14469 0.438069
\(267\) 7.49391 0.458620
\(268\) 4.26635 0.260609
\(269\) 20.6511 1.25912 0.629560 0.776952i \(-0.283235\pi\)
0.629560 + 0.776952i \(0.283235\pi\)
\(270\) −2.71343 −0.165134
\(271\) 3.84564 0.233606 0.116803 0.993155i \(-0.462735\pi\)
0.116803 + 0.993155i \(0.462735\pi\)
\(272\) −7.05126 −0.427546
\(273\) −2.34175 −0.141729
\(274\) 9.99842 0.604026
\(275\) 1.90324 0.114770
\(276\) 3.54988 0.213678
\(277\) −0.0114465 −0.000687752 0 −0.000343876 1.00000i \(-0.500109\pi\)
−0.000343876 1.00000i \(0.500109\pi\)
\(278\) −8.86959 −0.531963
\(279\) −7.06425 −0.422926
\(280\) −6.35418 −0.379735
\(281\) 33.0434 1.97120 0.985601 0.169089i \(-0.0540824\pi\)
0.985601 + 0.169089i \(0.0540824\pi\)
\(282\) −12.0993 −0.720503
\(283\) 26.6102 1.58181 0.790906 0.611938i \(-0.209610\pi\)
0.790906 + 0.611938i \(0.209610\pi\)
\(284\) −0.749357 −0.0444661
\(285\) 8.27870 0.490388
\(286\) 0.805528 0.0476319
\(287\) 2.53128 0.149417
\(288\) −1.00000 −0.0589256
\(289\) 32.7203 1.92472
\(290\) −5.39464 −0.316784
\(291\) 9.03214 0.529473
\(292\) −4.06343 −0.237794
\(293\) −1.54531 −0.0902778 −0.0451389 0.998981i \(-0.514373\pi\)
−0.0451389 + 0.998981i \(0.514373\pi\)
\(294\) −1.51621 −0.0884270
\(295\) −4.86189 −0.283070
\(296\) −1.93939 −0.112725
\(297\) −0.805528 −0.0467415
\(298\) −1.38681 −0.0803356
\(299\) 3.54988 0.205295
\(300\) −2.36272 −0.136412
\(301\) 7.99407 0.460771
\(302\) −14.3074 −0.823297
\(303\) 8.39612 0.482345
\(304\) 3.05101 0.174987
\(305\) 6.27655 0.359394
\(306\) 7.05126 0.403094
\(307\) −12.1150 −0.691442 −0.345721 0.938337i \(-0.612366\pi\)
−0.345721 + 0.938337i \(0.612366\pi\)
\(308\) −1.88635 −0.107485
\(309\) −1.00000 −0.0568880
\(310\) −19.1684 −1.08869
\(311\) 23.1713 1.31393 0.656963 0.753923i \(-0.271841\pi\)
0.656963 + 0.753923i \(0.271841\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −24.8428 −1.40420 −0.702098 0.712080i \(-0.747753\pi\)
−0.702098 + 0.712080i \(0.747753\pi\)
\(314\) 8.73671 0.493041
\(315\) 6.35418 0.358018
\(316\) 4.75496 0.267487
\(317\) 3.82838 0.215023 0.107512 0.994204i \(-0.465712\pi\)
0.107512 + 0.994204i \(0.465712\pi\)
\(318\) 6.23971 0.349905
\(319\) −1.60149 −0.0896661
\(320\) −2.71343 −0.151686
\(321\) 6.34500 0.354143
\(322\) −8.31294 −0.463262
\(323\) −21.5134 −1.19704
\(324\) 1.00000 0.0555556
\(325\) −2.36272 −0.131060
\(326\) −23.6512 −1.30992
\(327\) 3.27488 0.181101
\(328\) 1.08094 0.0596847
\(329\) 28.3335 1.56208
\(330\) −2.18575 −0.120321
\(331\) −14.1012 −0.775071 −0.387536 0.921855i \(-0.626674\pi\)
−0.387536 + 0.921855i \(0.626674\pi\)
\(332\) −12.8746 −0.706584
\(333\) 1.93939 0.106278
\(334\) −0.498550 −0.0272795
\(335\) −11.5765 −0.632489
\(336\) 2.34175 0.127753
\(337\) 4.39388 0.239350 0.119675 0.992813i \(-0.461815\pi\)
0.119675 + 0.992813i \(0.461815\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 11.0842 0.602009
\(340\) 19.1331 1.03764
\(341\) −5.69046 −0.308155
\(342\) −3.05101 −0.164980
\(343\) 19.9428 1.07681
\(344\) 3.41372 0.184055
\(345\) −9.63237 −0.518589
\(346\) −1.59065 −0.0855141
\(347\) −22.7230 −1.21984 −0.609918 0.792465i \(-0.708797\pi\)
−0.609918 + 0.792465i \(0.708797\pi\)
\(348\) 1.98812 0.106575
\(349\) 10.5744 0.566033 0.283017 0.959115i \(-0.408665\pi\)
0.283017 + 0.959115i \(0.408665\pi\)
\(350\) 5.53290 0.295746
\(351\) 1.00000 0.0533761
\(352\) −0.805528 −0.0429348
\(353\) −22.2749 −1.18557 −0.592787 0.805359i \(-0.701972\pi\)
−0.592787 + 0.805359i \(0.701972\pi\)
\(354\) 1.79179 0.0952324
\(355\) 2.03333 0.107918
\(356\) −7.49391 −0.397176
\(357\) −16.5123 −0.873923
\(358\) 19.1240 1.01073
\(359\) 37.2802 1.96757 0.983786 0.179344i \(-0.0573974\pi\)
0.983786 + 0.179344i \(0.0573974\pi\)
\(360\) 2.71343 0.143011
\(361\) −9.69136 −0.510072
\(362\) −13.3384 −0.701051
\(363\) 10.3511 0.543293
\(364\) 2.34175 0.122741
\(365\) 11.0259 0.577120
\(366\) −2.31314 −0.120910
\(367\) 31.1367 1.62532 0.812662 0.582736i \(-0.198018\pi\)
0.812662 + 0.582736i \(0.198018\pi\)
\(368\) −3.54988 −0.185050
\(369\) −1.08094 −0.0562713
\(370\) 5.26241 0.273580
\(371\) −14.6118 −0.758608
\(372\) 7.06425 0.366264
\(373\) 19.0317 0.985426 0.492713 0.870192i \(-0.336005\pi\)
0.492713 + 0.870192i \(0.336005\pi\)
\(374\) 5.67999 0.293705
\(375\) −7.15608 −0.369538
\(376\) 12.0993 0.623974
\(377\) 1.98812 0.102393
\(378\) −2.34175 −0.120447
\(379\) 16.7080 0.858235 0.429117 0.903249i \(-0.358825\pi\)
0.429117 + 0.903249i \(0.358825\pi\)
\(380\) −8.27870 −0.424688
\(381\) 20.2440 1.03713
\(382\) 20.3164 1.03948
\(383\) 20.9445 1.07022 0.535108 0.844784i \(-0.320271\pi\)
0.535108 + 0.844784i \(0.320271\pi\)
\(384\) 1.00000 0.0510310
\(385\) 5.11848 0.260862
\(386\) 19.0532 0.969783
\(387\) −3.41372 −0.173529
\(388\) −9.03214 −0.458537
\(389\) −12.5690 −0.637273 −0.318636 0.947877i \(-0.603225\pi\)
−0.318636 + 0.947877i \(0.603225\pi\)
\(390\) 2.71343 0.137400
\(391\) 25.0311 1.26588
\(392\) 1.51621 0.0765800
\(393\) 18.2985 0.923038
\(394\) −0.799715 −0.0402891
\(395\) −12.9023 −0.649183
\(396\) 0.805528 0.0404793
\(397\) −13.8447 −0.694843 −0.347422 0.937709i \(-0.612943\pi\)
−0.347422 + 0.937709i \(0.612943\pi\)
\(398\) 11.3388 0.568363
\(399\) 7.14469 0.357682
\(400\) 2.36272 0.118136
\(401\) 2.71854 0.135758 0.0678788 0.997694i \(-0.478377\pi\)
0.0678788 + 0.997694i \(0.478377\pi\)
\(402\) 4.26635 0.212786
\(403\) 7.06425 0.351895
\(404\) −8.39612 −0.417723
\(405\) −2.71343 −0.134832
\(406\) −4.65568 −0.231058
\(407\) 1.56224 0.0774371
\(408\) −7.05126 −0.349090
\(409\) 27.6900 1.36918 0.684591 0.728927i \(-0.259981\pi\)
0.684591 + 0.728927i \(0.259981\pi\)
\(410\) −2.93305 −0.144853
\(411\) 9.99842 0.493186
\(412\) 1.00000 0.0492665
\(413\) −4.19592 −0.206468
\(414\) 3.54988 0.174467
\(415\) 34.9343 1.71486
\(416\) 1.00000 0.0490290
\(417\) −8.86959 −0.434346
\(418\) −2.45767 −0.120209
\(419\) −3.53590 −0.172740 −0.0863700 0.996263i \(-0.527527\pi\)
−0.0863700 + 0.996263i \(0.527527\pi\)
\(420\) −6.35418 −0.310052
\(421\) 1.25978 0.0613979 0.0306990 0.999529i \(-0.490227\pi\)
0.0306990 + 0.999529i \(0.490227\pi\)
\(422\) 18.5295 0.902001
\(423\) −12.0993 −0.588288
\(424\) −6.23971 −0.303027
\(425\) −16.6602 −0.808137
\(426\) −0.749357 −0.0363065
\(427\) 5.41680 0.262137
\(428\) −6.34500 −0.306697
\(429\) 0.805528 0.0388913
\(430\) −9.26290 −0.446697
\(431\) 1.84789 0.0890098 0.0445049 0.999009i \(-0.485829\pi\)
0.0445049 + 0.999009i \(0.485829\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −13.8258 −0.664427 −0.332213 0.943204i \(-0.607795\pi\)
−0.332213 + 0.943204i \(0.607795\pi\)
\(434\) −16.5427 −0.794076
\(435\) −5.39464 −0.258653
\(436\) −3.27488 −0.156838
\(437\) −10.8307 −0.518103
\(438\) −4.06343 −0.194158
\(439\) 17.6012 0.840059 0.420030 0.907510i \(-0.362020\pi\)
0.420030 + 0.907510i \(0.362020\pi\)
\(440\) 2.18575 0.104201
\(441\) −1.51621 −0.0722003
\(442\) −7.05126 −0.335394
\(443\) −15.0423 −0.714681 −0.357341 0.933974i \(-0.616316\pi\)
−0.357341 + 0.933974i \(0.616316\pi\)
\(444\) −1.93939 −0.0920395
\(445\) 20.3342 0.963935
\(446\) 21.4736 1.01680
\(447\) −1.38681 −0.0655937
\(448\) −2.34175 −0.110637
\(449\) 11.1989 0.528509 0.264255 0.964453i \(-0.414874\pi\)
0.264255 + 0.964453i \(0.414874\pi\)
\(450\) −2.36272 −0.111380
\(451\) −0.870725 −0.0410008
\(452\) −11.0842 −0.521355
\(453\) −14.3074 −0.672219
\(454\) 5.45603 0.256064
\(455\) −6.35418 −0.297889
\(456\) 3.05101 0.142876
\(457\) 8.67449 0.405775 0.202888 0.979202i \(-0.434967\pi\)
0.202888 + 0.979202i \(0.434967\pi\)
\(458\) −2.36284 −0.110408
\(459\) 7.05126 0.329125
\(460\) 9.63237 0.449111
\(461\) 10.0534 0.468233 0.234116 0.972209i \(-0.424780\pi\)
0.234116 + 0.972209i \(0.424780\pi\)
\(462\) −1.88635 −0.0877608
\(463\) 33.8559 1.57342 0.786708 0.617325i \(-0.211784\pi\)
0.786708 + 0.617325i \(0.211784\pi\)
\(464\) −1.98812 −0.0922962
\(465\) −19.1684 −0.888912
\(466\) −4.80781 −0.222717
\(467\) −18.4086 −0.851848 −0.425924 0.904759i \(-0.640051\pi\)
−0.425924 + 0.904759i \(0.640051\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −9.99073 −0.461329
\(470\) −32.8306 −1.51436
\(471\) 8.73671 0.402566
\(472\) −1.79179 −0.0824737
\(473\) −2.74985 −0.126438
\(474\) 4.75496 0.218402
\(475\) 7.20868 0.330757
\(476\) 16.5123 0.756840
\(477\) 6.23971 0.285696
\(478\) 25.2866 1.15658
\(479\) −33.3664 −1.52455 −0.762275 0.647254i \(-0.775917\pi\)
−0.762275 + 0.647254i \(0.775917\pi\)
\(480\) −2.71343 −0.123851
\(481\) −1.93939 −0.0884287
\(482\) −3.99306 −0.181879
\(483\) −8.31294 −0.378252
\(484\) −10.3511 −0.470506
\(485\) 24.5081 1.11286
\(486\) 1.00000 0.0453609
\(487\) 16.6033 0.752367 0.376184 0.926545i \(-0.377236\pi\)
0.376184 + 0.926545i \(0.377236\pi\)
\(488\) 2.31314 0.104711
\(489\) −23.6512 −1.06955
\(490\) −4.11413 −0.185857
\(491\) 21.0116 0.948242 0.474121 0.880460i \(-0.342766\pi\)
0.474121 + 0.880460i \(0.342766\pi\)
\(492\) 1.08094 0.0487324
\(493\) 14.0188 0.631374
\(494\) 3.05101 0.137271
\(495\) −2.18575 −0.0982421
\(496\) −7.06425 −0.317194
\(497\) 1.75481 0.0787138
\(498\) −12.8746 −0.576923
\(499\) 9.64276 0.431669 0.215834 0.976430i \(-0.430753\pi\)
0.215834 + 0.976430i \(0.430753\pi\)
\(500\) 7.15608 0.320030
\(501\) −0.498550 −0.0222736
\(502\) 2.60067 0.116074
\(503\) −10.0013 −0.445938 −0.222969 0.974826i \(-0.571575\pi\)
−0.222969 + 0.974826i \(0.571575\pi\)
\(504\) 2.34175 0.104310
\(505\) 22.7823 1.01380
\(506\) 2.85953 0.127122
\(507\) −1.00000 −0.0444116
\(508\) −20.2440 −0.898181
\(509\) 11.4095 0.505718 0.252859 0.967503i \(-0.418629\pi\)
0.252859 + 0.967503i \(0.418629\pi\)
\(510\) 19.1331 0.847229
\(511\) 9.51554 0.420943
\(512\) −1.00000 −0.0441942
\(513\) −3.05101 −0.134705
\(514\) −1.93178 −0.0852072
\(515\) −2.71343 −0.119568
\(516\) 3.41372 0.150281
\(517\) −9.74633 −0.428643
\(518\) 4.54157 0.199545
\(519\) −1.59065 −0.0698220
\(520\) −2.71343 −0.118992
\(521\) −26.1445 −1.14541 −0.572707 0.819760i \(-0.694107\pi\)
−0.572707 + 0.819760i \(0.694107\pi\)
\(522\) 1.98812 0.0870177
\(523\) −33.5042 −1.46503 −0.732517 0.680748i \(-0.761655\pi\)
−0.732517 + 0.680748i \(0.761655\pi\)
\(524\) −18.2985 −0.799374
\(525\) 5.53290 0.241476
\(526\) −19.7974 −0.863208
\(527\) 49.8119 2.16984
\(528\) −0.805528 −0.0350561
\(529\) −10.3983 −0.452102
\(530\) 16.9310 0.735437
\(531\) 1.79179 0.0777569
\(532\) −7.14469 −0.309762
\(533\) 1.08094 0.0468205
\(534\) −7.49391 −0.324293
\(535\) 17.2167 0.744344
\(536\) −4.26635 −0.184278
\(537\) 19.1240 0.825260
\(538\) −20.6511 −0.890333
\(539\) −1.22135 −0.0526072
\(540\) 2.71343 0.116768
\(541\) −35.4070 −1.52226 −0.761132 0.648597i \(-0.775356\pi\)
−0.761132 + 0.648597i \(0.775356\pi\)
\(542\) −3.84564 −0.165184
\(543\) −13.3384 −0.572406
\(544\) 7.05126 0.302320
\(545\) 8.88616 0.380641
\(546\) 2.34175 0.100218
\(547\) 42.6935 1.82544 0.912722 0.408582i \(-0.133977\pi\)
0.912722 + 0.408582i \(0.133977\pi\)
\(548\) −9.99842 −0.427111
\(549\) −2.31314 −0.0987224
\(550\) −1.90324 −0.0811544
\(551\) −6.06577 −0.258411
\(552\) −3.54988 −0.151093
\(553\) −11.1349 −0.473505
\(554\) 0.0114465 0.000486314 0
\(555\) 5.26241 0.223377
\(556\) 8.86959 0.376154
\(557\) 33.9092 1.43678 0.718390 0.695640i \(-0.244879\pi\)
0.718390 + 0.695640i \(0.244879\pi\)
\(558\) 7.06425 0.299054
\(559\) 3.41372 0.144385
\(560\) 6.35418 0.268513
\(561\) 5.67999 0.239809
\(562\) −33.0434 −1.39385
\(563\) 3.43746 0.144872 0.0724359 0.997373i \(-0.476923\pi\)
0.0724359 + 0.997373i \(0.476923\pi\)
\(564\) 12.0993 0.509472
\(565\) 30.0761 1.26531
\(566\) −26.6102 −1.11851
\(567\) −2.34175 −0.0983443
\(568\) 0.749357 0.0314423
\(569\) −29.7936 −1.24901 −0.624507 0.781019i \(-0.714700\pi\)
−0.624507 + 0.781019i \(0.714700\pi\)
\(570\) −8.27870 −0.346757
\(571\) 2.28764 0.0957349 0.0478675 0.998854i \(-0.484757\pi\)
0.0478675 + 0.998854i \(0.484757\pi\)
\(572\) −0.805528 −0.0336808
\(573\) 20.3164 0.848731
\(574\) −2.53128 −0.105654
\(575\) −8.38738 −0.349778
\(576\) 1.00000 0.0416667
\(577\) −11.9061 −0.495657 −0.247828 0.968804i \(-0.579717\pi\)
−0.247828 + 0.968804i \(0.579717\pi\)
\(578\) −32.7203 −1.36099
\(579\) 19.0532 0.791824
\(580\) 5.39464 0.224000
\(581\) 30.1490 1.25079
\(582\) −9.03214 −0.374394
\(583\) 5.02626 0.208166
\(584\) 4.06343 0.168146
\(585\) 2.71343 0.112187
\(586\) 1.54531 0.0638361
\(587\) −29.3687 −1.21218 −0.606088 0.795397i \(-0.707262\pi\)
−0.606088 + 0.795397i \(0.707262\pi\)
\(588\) 1.51621 0.0625273
\(589\) −21.5531 −0.888079
\(590\) 4.86189 0.200161
\(591\) −0.799715 −0.0328959
\(592\) 1.93939 0.0797085
\(593\) −24.4812 −1.00532 −0.502661 0.864484i \(-0.667646\pi\)
−0.502661 + 0.864484i \(0.667646\pi\)
\(594\) 0.805528 0.0330512
\(595\) −44.8050 −1.83683
\(596\) 1.38681 0.0568058
\(597\) 11.3388 0.464066
\(598\) −3.54988 −0.145165
\(599\) −21.3771 −0.873445 −0.436722 0.899596i \(-0.643861\pi\)
−0.436722 + 0.899596i \(0.643861\pi\)
\(600\) 2.36272 0.0964577
\(601\) −10.3517 −0.422253 −0.211127 0.977459i \(-0.567713\pi\)
−0.211127 + 0.977459i \(0.567713\pi\)
\(602\) −7.99407 −0.325814
\(603\) 4.26635 0.173739
\(604\) 14.3074 0.582159
\(605\) 28.0871 1.14190
\(606\) −8.39612 −0.341069
\(607\) −8.26118 −0.335311 −0.167655 0.985846i \(-0.553620\pi\)
−0.167655 + 0.985846i \(0.553620\pi\)
\(608\) −3.05101 −0.123735
\(609\) −4.65568 −0.188658
\(610\) −6.27655 −0.254130
\(611\) 12.0993 0.489485
\(612\) −7.05126 −0.285030
\(613\) 23.6435 0.954951 0.477476 0.878645i \(-0.341552\pi\)
0.477476 + 0.878645i \(0.341552\pi\)
\(614\) 12.1150 0.488923
\(615\) −2.93305 −0.118272
\(616\) 1.88635 0.0760031
\(617\) −6.83016 −0.274972 −0.137486 0.990504i \(-0.543902\pi\)
−0.137486 + 0.990504i \(0.543902\pi\)
\(618\) 1.00000 0.0402259
\(619\) 9.60961 0.386243 0.193121 0.981175i \(-0.438139\pi\)
0.193121 + 0.981175i \(0.438139\pi\)
\(620\) 19.1684 0.769821
\(621\) 3.54988 0.142452
\(622\) −23.1713 −0.929086
\(623\) 17.5489 0.703080
\(624\) 1.00000 0.0400320
\(625\) −31.2312 −1.24925
\(626\) 24.8428 0.992917
\(627\) −2.45767 −0.0981500
\(628\) −8.73671 −0.348633
\(629\) −13.6752 −0.545264
\(630\) −6.35418 −0.253157
\(631\) 25.4180 1.01188 0.505938 0.862570i \(-0.331146\pi\)
0.505938 + 0.862570i \(0.331146\pi\)
\(632\) −4.75496 −0.189142
\(633\) 18.5295 0.736481
\(634\) −3.82838 −0.152044
\(635\) 54.9306 2.17986
\(636\) −6.23971 −0.247420
\(637\) 1.51621 0.0600743
\(638\) 1.60149 0.0634035
\(639\) −0.749357 −0.0296441
\(640\) 2.71343 0.107258
\(641\) −34.7435 −1.37229 −0.686143 0.727467i \(-0.740698\pi\)
−0.686143 + 0.727467i \(0.740698\pi\)
\(642\) −6.34500 −0.250417
\(643\) −7.13330 −0.281310 −0.140655 0.990059i \(-0.544921\pi\)
−0.140655 + 0.990059i \(0.544921\pi\)
\(644\) 8.31294 0.327576
\(645\) −9.26290 −0.364726
\(646\) 21.5134 0.846435
\(647\) −1.15514 −0.0454131 −0.0227066 0.999742i \(-0.507228\pi\)
−0.0227066 + 0.999742i \(0.507228\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 1.44334 0.0566559
\(650\) 2.36272 0.0926736
\(651\) −16.5427 −0.648360
\(652\) 23.6512 0.926253
\(653\) 6.30977 0.246920 0.123460 0.992350i \(-0.460601\pi\)
0.123460 + 0.992350i \(0.460601\pi\)
\(654\) −3.27488 −0.128058
\(655\) 49.6518 1.94006
\(656\) −1.08094 −0.0422035
\(657\) −4.06343 −0.158530
\(658\) −28.3335 −1.10456
\(659\) −0.690152 −0.0268845 −0.0134423 0.999910i \(-0.504279\pi\)
−0.0134423 + 0.999910i \(0.504279\pi\)
\(660\) 2.18575 0.0850801
\(661\) 23.7004 0.921839 0.460919 0.887442i \(-0.347520\pi\)
0.460919 + 0.887442i \(0.347520\pi\)
\(662\) 14.1012 0.548058
\(663\) −7.05126 −0.273848
\(664\) 12.8746 0.499630
\(665\) 19.3867 0.751782
\(666\) −1.93939 −0.0751499
\(667\) 7.05760 0.273271
\(668\) 0.498550 0.0192895
\(669\) 21.4736 0.830216
\(670\) 11.5765 0.447238
\(671\) −1.86330 −0.0719319
\(672\) −2.34175 −0.0903350
\(673\) 0.0367792 0.00141773 0.000708867 1.00000i \(-0.499774\pi\)
0.000708867 1.00000i \(0.499774\pi\)
\(674\) −4.39388 −0.169246
\(675\) −2.36272 −0.0909412
\(676\) 1.00000 0.0384615
\(677\) −33.0504 −1.27023 −0.635115 0.772417i \(-0.719047\pi\)
−0.635115 + 0.772417i \(0.719047\pi\)
\(678\) −11.0842 −0.425684
\(679\) 21.1510 0.811701
\(680\) −19.1331 −0.733722
\(681\) 5.45603 0.209076
\(682\) 5.69046 0.217899
\(683\) 35.3472 1.35252 0.676261 0.736662i \(-0.263599\pi\)
0.676261 + 0.736662i \(0.263599\pi\)
\(684\) 3.05101 0.116658
\(685\) 27.1300 1.03659
\(686\) −19.9428 −0.761421
\(687\) −2.36284 −0.0901478
\(688\) −3.41372 −0.130147
\(689\) −6.23971 −0.237714
\(690\) 9.63237 0.366698
\(691\) 26.1666 0.995423 0.497711 0.867343i \(-0.334174\pi\)
0.497711 + 0.867343i \(0.334174\pi\)
\(692\) 1.59065 0.0604676
\(693\) −1.88635 −0.0716564
\(694\) 22.7230 0.862554
\(695\) −24.0670 −0.912915
\(696\) −1.98812 −0.0753596
\(697\) 7.62196 0.288702
\(698\) −10.5744 −0.400246
\(699\) −4.80781 −0.181848
\(700\) −5.53290 −0.209124
\(701\) 1.48336 0.0560259 0.0280130 0.999608i \(-0.491082\pi\)
0.0280130 + 0.999608i \(0.491082\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 5.91710 0.223168
\(704\) 0.805528 0.0303595
\(705\) −32.8306 −1.23647
\(706\) 22.2749 0.838328
\(707\) 19.6616 0.739451
\(708\) −1.79179 −0.0673395
\(709\) 34.3436 1.28980 0.644901 0.764266i \(-0.276899\pi\)
0.644901 + 0.764266i \(0.276899\pi\)
\(710\) −2.03333 −0.0763095
\(711\) 4.75496 0.178325
\(712\) 7.49391 0.280846
\(713\) 25.0773 0.939151
\(714\) 16.5123 0.617957
\(715\) 2.18575 0.0817423
\(716\) −19.1240 −0.714696
\(717\) 25.2866 0.944344
\(718\) −37.2802 −1.39128
\(719\) −21.8973 −0.816632 −0.408316 0.912841i \(-0.633884\pi\)
−0.408316 + 0.912841i \(0.633884\pi\)
\(720\) −2.71343 −0.101124
\(721\) −2.34175 −0.0872113
\(722\) 9.69136 0.360675
\(723\) −3.99306 −0.148503
\(724\) 13.3384 0.495718
\(725\) −4.69738 −0.174456
\(726\) −10.3511 −0.384166
\(727\) −11.0035 −0.408096 −0.204048 0.978961i \(-0.565410\pi\)
−0.204048 + 0.978961i \(0.565410\pi\)
\(728\) −2.34175 −0.0867910
\(729\) 1.00000 0.0370370
\(730\) −11.0259 −0.408085
\(731\) 24.0710 0.890299
\(732\) 2.31314 0.0854961
\(733\) −18.8770 −0.697238 −0.348619 0.937265i \(-0.613349\pi\)
−0.348619 + 0.937265i \(0.613349\pi\)
\(734\) −31.1367 −1.14928
\(735\) −4.11413 −0.151752
\(736\) 3.54988 0.130850
\(737\) 3.43667 0.126591
\(738\) 1.08094 0.0397898
\(739\) 34.2502 1.25991 0.629956 0.776631i \(-0.283073\pi\)
0.629956 + 0.776631i \(0.283073\pi\)
\(740\) −5.26241 −0.193450
\(741\) 3.05101 0.112082
\(742\) 14.6118 0.536417
\(743\) −9.59805 −0.352118 −0.176059 0.984380i \(-0.556335\pi\)
−0.176059 + 0.984380i \(0.556335\pi\)
\(744\) −7.06425 −0.258988
\(745\) −3.76301 −0.137866
\(746\) −19.0317 −0.696802
\(747\) −12.8746 −0.471056
\(748\) −5.67999 −0.207681
\(749\) 14.8584 0.542914
\(750\) 7.15608 0.261303
\(751\) 31.5449 1.15109 0.575545 0.817770i \(-0.304790\pi\)
0.575545 + 0.817770i \(0.304790\pi\)
\(752\) −12.0993 −0.441216
\(753\) 2.60067 0.0947737
\(754\) −1.98812 −0.0724031
\(755\) −38.8221 −1.41288
\(756\) 2.34175 0.0851686
\(757\) 4.66137 0.169420 0.0847101 0.996406i \(-0.473004\pi\)
0.0847101 + 0.996406i \(0.473004\pi\)
\(758\) −16.7080 −0.606864
\(759\) 2.85953 0.103794
\(760\) 8.27870 0.300300
\(761\) −21.2740 −0.771181 −0.385590 0.922670i \(-0.626002\pi\)
−0.385590 + 0.922670i \(0.626002\pi\)
\(762\) −20.2440 −0.733361
\(763\) 7.66894 0.277634
\(764\) −20.3164 −0.735022
\(765\) 19.1331 0.691760
\(766\) −20.9445 −0.756757
\(767\) −1.79179 −0.0646977
\(768\) −1.00000 −0.0360844
\(769\) 39.3228 1.41802 0.709008 0.705201i \(-0.249143\pi\)
0.709008 + 0.705201i \(0.249143\pi\)
\(770\) −5.11848 −0.184457
\(771\) −1.93178 −0.0695714
\(772\) −19.0532 −0.685740
\(773\) −9.02272 −0.324525 −0.162262 0.986748i \(-0.551879\pi\)
−0.162262 + 0.986748i \(0.551879\pi\)
\(774\) 3.41372 0.122704
\(775\) −16.6909 −0.599553
\(776\) 9.03214 0.324235
\(777\) 4.54157 0.162928
\(778\) 12.5690 0.450620
\(779\) −3.29794 −0.118161
\(780\) −2.71343 −0.0971565
\(781\) −0.603628 −0.0215995
\(782\) −25.0311 −0.895112
\(783\) 1.98812 0.0710497
\(784\) −1.51621 −0.0541502
\(785\) 23.7065 0.846120
\(786\) −18.2985 −0.652686
\(787\) −23.0968 −0.823312 −0.411656 0.911339i \(-0.635050\pi\)
−0.411656 + 0.911339i \(0.635050\pi\)
\(788\) 0.799715 0.0284887
\(789\) −19.7974 −0.704806
\(790\) 12.9023 0.459042
\(791\) 25.9563 0.922900
\(792\) −0.805528 −0.0286232
\(793\) 2.31314 0.0821420
\(794\) 13.8447 0.491328
\(795\) 16.9310 0.600482
\(796\) −11.3388 −0.401893
\(797\) −34.9077 −1.23650 −0.618248 0.785983i \(-0.712157\pi\)
−0.618248 + 0.785983i \(0.712157\pi\)
\(798\) −7.14469 −0.252919
\(799\) 85.3153 3.01824
\(800\) −2.36272 −0.0835348
\(801\) −7.49391 −0.264784
\(802\) −2.71854 −0.0959951
\(803\) −3.27321 −0.115509
\(804\) −4.26635 −0.150463
\(805\) −22.5566 −0.795016
\(806\) −7.06425 −0.248828
\(807\) −20.6511 −0.726953
\(808\) 8.39612 0.295374
\(809\) 33.8401 1.18975 0.594877 0.803817i \(-0.297201\pi\)
0.594877 + 0.803817i \(0.297201\pi\)
\(810\) 2.71343 0.0953403
\(811\) −46.2855 −1.62530 −0.812651 0.582751i \(-0.801976\pi\)
−0.812651 + 0.582751i \(0.801976\pi\)
\(812\) 4.65568 0.163382
\(813\) −3.84564 −0.134872
\(814\) −1.56224 −0.0547563
\(815\) −64.1760 −2.24799
\(816\) 7.05126 0.246844
\(817\) −10.4153 −0.364384
\(818\) −27.6900 −0.968158
\(819\) 2.34175 0.0818274
\(820\) 2.93305 0.102426
\(821\) −20.1519 −0.703305 −0.351653 0.936131i \(-0.614380\pi\)
−0.351653 + 0.936131i \(0.614380\pi\)
\(822\) −9.99842 −0.348735
\(823\) −20.7515 −0.723353 −0.361676 0.932304i \(-0.617795\pi\)
−0.361676 + 0.932304i \(0.617795\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −1.90324 −0.0662623
\(826\) 4.19592 0.145995
\(827\) −3.60766 −0.125451 −0.0627253 0.998031i \(-0.519979\pi\)
−0.0627253 + 0.998031i \(0.519979\pi\)
\(828\) −3.54988 −0.123367
\(829\) −13.0412 −0.452941 −0.226471 0.974018i \(-0.572719\pi\)
−0.226471 + 0.974018i \(0.572719\pi\)
\(830\) −34.9343 −1.21259
\(831\) 0.0114465 0.000397074 0
\(832\) −1.00000 −0.0346688
\(833\) 10.6912 0.370427
\(834\) 8.86959 0.307129
\(835\) −1.35278 −0.0468150
\(836\) 2.45767 0.0850004
\(837\) 7.06425 0.244176
\(838\) 3.53590 0.122146
\(839\) 16.3968 0.566081 0.283041 0.959108i \(-0.408657\pi\)
0.283041 + 0.959108i \(0.408657\pi\)
\(840\) 6.35418 0.219240
\(841\) −25.0474 −0.863702
\(842\) −1.25978 −0.0434149
\(843\) −33.0434 −1.13807
\(844\) −18.5295 −0.637811
\(845\) −2.71343 −0.0933450
\(846\) 12.0993 0.415982
\(847\) 24.2397 0.832888
\(848\) 6.23971 0.214272
\(849\) −26.6102 −0.913259
\(850\) 16.6602 0.571439
\(851\) −6.88461 −0.236001
\(852\) 0.749357 0.0256725
\(853\) −41.3915 −1.41722 −0.708610 0.705601i \(-0.750677\pi\)
−0.708610 + 0.705601i \(0.750677\pi\)
\(854\) −5.41680 −0.185359
\(855\) −8.27870 −0.283126
\(856\) 6.34500 0.216867
\(857\) 19.7619 0.675055 0.337527 0.941316i \(-0.390409\pi\)
0.337527 + 0.941316i \(0.390409\pi\)
\(858\) −0.805528 −0.0275003
\(859\) 38.9296 1.32826 0.664130 0.747617i \(-0.268802\pi\)
0.664130 + 0.747617i \(0.268802\pi\)
\(860\) 9.26290 0.315862
\(861\) −2.53128 −0.0862659
\(862\) −1.84789 −0.0629395
\(863\) 53.6130 1.82501 0.912504 0.409067i \(-0.134146\pi\)
0.912504 + 0.409067i \(0.134146\pi\)
\(864\) 1.00000 0.0340207
\(865\) −4.31614 −0.146753
\(866\) 13.8258 0.469821
\(867\) −32.7203 −1.11124
\(868\) 16.5427 0.561496
\(869\) 3.83025 0.129932
\(870\) 5.39464 0.182895
\(871\) −4.26635 −0.144560
\(872\) 3.27488 0.110901
\(873\) −9.03214 −0.305692
\(874\) 10.8307 0.366354
\(875\) −16.7578 −0.566515
\(876\) 4.06343 0.137291
\(877\) −40.5319 −1.36867 −0.684333 0.729170i \(-0.739906\pi\)
−0.684333 + 0.729170i \(0.739906\pi\)
\(878\) −17.6012 −0.594011
\(879\) 1.54531 0.0521219
\(880\) −2.18575 −0.0736815
\(881\) −25.9077 −0.872854 −0.436427 0.899740i \(-0.643756\pi\)
−0.436427 + 0.899740i \(0.643756\pi\)
\(882\) 1.51621 0.0510533
\(883\) 10.4702 0.352351 0.176175 0.984359i \(-0.443627\pi\)
0.176175 + 0.984359i \(0.443627\pi\)
\(884\) 7.05126 0.237160
\(885\) 4.86189 0.163431
\(886\) 15.0423 0.505356
\(887\) 9.90172 0.332467 0.166234 0.986086i \(-0.446839\pi\)
0.166234 + 0.986086i \(0.446839\pi\)
\(888\) 1.93939 0.0650817
\(889\) 47.4063 1.58996
\(890\) −20.3342 −0.681605
\(891\) 0.805528 0.0269862
\(892\) −21.4736 −0.718988
\(893\) −36.9150 −1.23531
\(894\) 1.38681 0.0463818
\(895\) 51.8916 1.73454
\(896\) 2.34175 0.0782324
\(897\) −3.54988 −0.118527
\(898\) −11.1989 −0.373713
\(899\) 14.0446 0.468413
\(900\) 2.36272 0.0787574
\(901\) −43.9978 −1.46578
\(902\) 0.870725 0.0289920
\(903\) −7.99407 −0.266026
\(904\) 11.0842 0.368653
\(905\) −36.1929 −1.20309
\(906\) 14.3074 0.475331
\(907\) 8.30387 0.275726 0.137863 0.990451i \(-0.455977\pi\)
0.137863 + 0.990451i \(0.455977\pi\)
\(908\) −5.45603 −0.181065
\(909\) −8.39612 −0.278482
\(910\) 6.35418 0.210639
\(911\) 3.19275 0.105780 0.0528902 0.998600i \(-0.483157\pi\)
0.0528902 + 0.998600i \(0.483157\pi\)
\(912\) −3.05101 −0.101029
\(913\) −10.3708 −0.343224
\(914\) −8.67449 −0.286927
\(915\) −6.27655 −0.207496
\(916\) 2.36284 0.0780703
\(917\) 42.8505 1.41505
\(918\) −7.05126 −0.232726
\(919\) −2.87135 −0.0947171 −0.0473586 0.998878i \(-0.515080\pi\)
−0.0473586 + 0.998878i \(0.515080\pi\)
\(920\) −9.63237 −0.317570
\(921\) 12.1150 0.399204
\(922\) −10.0534 −0.331090
\(923\) 0.749357 0.0246654
\(924\) 1.88635 0.0620562
\(925\) 4.58224 0.150663
\(926\) −33.8559 −1.11257
\(927\) 1.00000 0.0328443
\(928\) 1.98812 0.0652633
\(929\) 27.2997 0.895673 0.447836 0.894116i \(-0.352195\pi\)
0.447836 + 0.894116i \(0.352195\pi\)
\(930\) 19.1684 0.628556
\(931\) −4.62596 −0.151610
\(932\) 4.80781 0.157485
\(933\) −23.1713 −0.758595
\(934\) 18.4086 0.602347
\(935\) 15.4123 0.504036
\(936\) 1.00000 0.0326860
\(937\) −17.2594 −0.563839 −0.281920 0.959438i \(-0.590971\pi\)
−0.281920 + 0.959438i \(0.590971\pi\)
\(938\) 9.99073 0.326209
\(939\) 24.8428 0.810713
\(940\) 32.8306 1.07082
\(941\) 11.7910 0.384376 0.192188 0.981358i \(-0.438442\pi\)
0.192188 + 0.981358i \(0.438442\pi\)
\(942\) −8.73671 −0.284657
\(943\) 3.83720 0.124956
\(944\) 1.79179 0.0583177
\(945\) −6.35418 −0.206702
\(946\) 2.74985 0.0894052
\(947\) −1.42209 −0.0462116 −0.0231058 0.999733i \(-0.507355\pi\)
−0.0231058 + 0.999733i \(0.507355\pi\)
\(948\) −4.75496 −0.154434
\(949\) 4.06343 0.131905
\(950\) −7.20868 −0.233880
\(951\) −3.82838 −0.124144
\(952\) −16.5123 −0.535167
\(953\) −5.98592 −0.193903 −0.0969515 0.995289i \(-0.530909\pi\)
−0.0969515 + 0.995289i \(0.530909\pi\)
\(954\) −6.23971 −0.202018
\(955\) 55.1273 1.78388
\(956\) −25.2866 −0.817826
\(957\) 1.60149 0.0517688
\(958\) 33.3664 1.07802
\(959\) 23.4138 0.756071
\(960\) 2.71343 0.0875757
\(961\) 18.9037 0.609795
\(962\) 1.93939 0.0625285
\(963\) −6.34500 −0.204465
\(964\) 3.99306 0.128608
\(965\) 51.6996 1.66427
\(966\) 8.31294 0.267464
\(967\) 42.6878 1.37275 0.686374 0.727249i \(-0.259201\pi\)
0.686374 + 0.727249i \(0.259201\pi\)
\(968\) 10.3511 0.332698
\(969\) 21.5134 0.691111
\(970\) −24.5081 −0.786908
\(971\) 45.7866 1.46936 0.734682 0.678412i \(-0.237332\pi\)
0.734682 + 0.678412i \(0.237332\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −20.7704 −0.665867
\(974\) −16.6033 −0.532004
\(975\) 2.36272 0.0756676
\(976\) −2.31314 −0.0740418
\(977\) 34.8691 1.11556 0.557781 0.829988i \(-0.311653\pi\)
0.557781 + 0.829988i \(0.311653\pi\)
\(978\) 23.6512 0.756283
\(979\) −6.03656 −0.192929
\(980\) 4.11413 0.131421
\(981\) −3.27488 −0.104559
\(982\) −21.0116 −0.670508
\(983\) −10.3386 −0.329749 −0.164874 0.986315i \(-0.552722\pi\)
−0.164874 + 0.986315i \(0.552722\pi\)
\(984\) −1.08094 −0.0344590
\(985\) −2.16997 −0.0691411
\(986\) −14.0188 −0.446449
\(987\) −28.3335 −0.901866
\(988\) −3.05101 −0.0970654
\(989\) 12.1183 0.385339
\(990\) 2.18575 0.0694676
\(991\) 20.6801 0.656926 0.328463 0.944517i \(-0.393469\pi\)
0.328463 + 0.944517i \(0.393469\pi\)
\(992\) 7.06425 0.224290
\(993\) 14.1012 0.447488
\(994\) −1.75481 −0.0556591
\(995\) 30.7671 0.975382
\(996\) 12.8746 0.407946
\(997\) 10.6927 0.338642 0.169321 0.985561i \(-0.445843\pi\)
0.169321 + 0.985561i \(0.445843\pi\)
\(998\) −9.64276 −0.305236
\(999\) −1.93939 −0.0613596
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.1.4 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.ba.1.4 14 1.1 even 1 trivial