Properties

Label 6825.2.a.be.1.2
Level $6825$
Weight $2$
Character 6825.1
Self dual yes
Analytic conductor $54.498$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6825,2,Mod(1,6825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6825.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6825.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.4978993795\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1365)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.90211\) of defining polynomial
Character \(\chi\) \(=\) 6825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +2.80423 q^{11} +0.618034 q^{12} +1.00000 q^{13} -1.61803 q^{14} -4.85410 q^{16} -2.01062 q^{17} -1.61803 q^{18} -2.34458 q^{19} +1.00000 q^{21} -4.53733 q^{22} -0.939503 q^{23} +2.23607 q^{24} -1.61803 q^{26} +1.00000 q^{27} +0.618034 q^{28} -5.69572 q^{29} +0.895549 q^{31} +3.38197 q^{32} +2.80423 q^{33} +3.25325 q^{34} +0.618034 q^{36} -5.52015 q^{37} +3.79360 q^{38} +1.00000 q^{39} +3.07768 q^{41} -1.61803 q^{42} +5.12454 q^{43} +1.73311 q^{44} +1.52015 q^{46} -5.96261 q^{47} -4.85410 q^{48} +1.00000 q^{49} -2.01062 q^{51} +0.618034 q^{52} -1.30313 q^{53} -1.61803 q^{54} +2.23607 q^{56} -2.34458 q^{57} +9.21586 q^{58} -2.47027 q^{59} -4.15838 q^{61} -1.44903 q^{62} +1.00000 q^{63} +4.23607 q^{64} -4.53733 q^{66} -15.4797 q^{67} -1.24263 q^{68} -0.939503 q^{69} -12.3416 q^{71} +2.23607 q^{72} -2.77455 q^{73} +8.93179 q^{74} -1.44903 q^{76} +2.80423 q^{77} -1.61803 q^{78} -6.16495 q^{79} +1.00000 q^{81} -4.97980 q^{82} -15.5861 q^{83} +0.618034 q^{84} -8.29168 q^{86} -5.69572 q^{87} +6.27044 q^{88} -0.821412 q^{89} +1.00000 q^{91} -0.580644 q^{92} +0.895549 q^{93} +9.64771 q^{94} +3.38197 q^{96} -1.98281 q^{97} -1.61803 q^{98} +2.80423 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 2 q^{12} + 4 q^{13} - 2 q^{14} - 6 q^{16} - 2 q^{17} - 2 q^{18} - 2 q^{19} + 4 q^{21} + 2 q^{22} - 8 q^{23} - 2 q^{26} + 4 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.14412 −0.572061 0.820211i \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) −1.61803 −0.660560
\(7\) 1.00000 0.377964
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.80423 0.845506 0.422753 0.906245i \(-0.361064\pi\)
0.422753 + 0.906245i \(0.361064\pi\)
\(12\) 0.618034 0.178411
\(13\) 1.00000 0.277350
\(14\) −1.61803 −0.432438
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) −2.01062 −0.487647 −0.243824 0.969820i \(-0.578402\pi\)
−0.243824 + 0.969820i \(0.578402\pi\)
\(18\) −1.61803 −0.381374
\(19\) −2.34458 −0.537883 −0.268941 0.963157i \(-0.586674\pi\)
−0.268941 + 0.963157i \(0.586674\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −4.53733 −0.967363
\(23\) −0.939503 −0.195900 −0.0979499 0.995191i \(-0.531228\pi\)
−0.0979499 + 0.995191i \(0.531228\pi\)
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) −1.61803 −0.317323
\(27\) 1.00000 0.192450
\(28\) 0.618034 0.116797
\(29\) −5.69572 −1.05767 −0.528834 0.848725i \(-0.677371\pi\)
−0.528834 + 0.848725i \(0.677371\pi\)
\(30\) 0 0
\(31\) 0.895549 0.160845 0.0804226 0.996761i \(-0.474373\pi\)
0.0804226 + 0.996761i \(0.474373\pi\)
\(32\) 3.38197 0.597853
\(33\) 2.80423 0.488153
\(34\) 3.25325 0.557928
\(35\) 0 0
\(36\) 0.618034 0.103006
\(37\) −5.52015 −0.907507 −0.453753 0.891127i \(-0.649915\pi\)
−0.453753 + 0.891127i \(0.649915\pi\)
\(38\) 3.79360 0.615404
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 3.07768 0.480653 0.240327 0.970692i \(-0.422745\pi\)
0.240327 + 0.970692i \(0.422745\pi\)
\(42\) −1.61803 −0.249668
\(43\) 5.12454 0.781485 0.390743 0.920500i \(-0.372218\pi\)
0.390743 + 0.920500i \(0.372218\pi\)
\(44\) 1.73311 0.261276
\(45\) 0 0
\(46\) 1.52015 0.224133
\(47\) −5.96261 −0.869736 −0.434868 0.900494i \(-0.643205\pi\)
−0.434868 + 0.900494i \(0.643205\pi\)
\(48\) −4.85410 −0.700629
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.01062 −0.281543
\(52\) 0.618034 0.0857059
\(53\) −1.30313 −0.178999 −0.0894993 0.995987i \(-0.528527\pi\)
−0.0894993 + 0.995987i \(0.528527\pi\)
\(54\) −1.61803 −0.220187
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) −2.34458 −0.310547
\(58\) 9.21586 1.21010
\(59\) −2.47027 −0.321602 −0.160801 0.986987i \(-0.551408\pi\)
−0.160801 + 0.986987i \(0.551408\pi\)
\(60\) 0 0
\(61\) −4.15838 −0.532427 −0.266213 0.963914i \(-0.585773\pi\)
−0.266213 + 0.963914i \(0.585773\pi\)
\(62\) −1.44903 −0.184027
\(63\) 1.00000 0.125988
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) −4.53733 −0.558507
\(67\) −15.4797 −1.89115 −0.945576 0.325402i \(-0.894500\pi\)
−0.945576 + 0.325402i \(0.894500\pi\)
\(68\) −1.24263 −0.150691
\(69\) −0.939503 −0.113103
\(70\) 0 0
\(71\) −12.3416 −1.46467 −0.732337 0.680943i \(-0.761570\pi\)
−0.732337 + 0.680943i \(0.761570\pi\)
\(72\) 2.23607 0.263523
\(73\) −2.77455 −0.324737 −0.162368 0.986730i \(-0.551913\pi\)
−0.162368 + 0.986730i \(0.551913\pi\)
\(74\) 8.93179 1.03830
\(75\) 0 0
\(76\) −1.44903 −0.166215
\(77\) 2.80423 0.319571
\(78\) −1.61803 −0.183206
\(79\) −6.16495 −0.693611 −0.346805 0.937937i \(-0.612734\pi\)
−0.346805 + 0.937937i \(0.612734\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −4.97980 −0.549927
\(83\) −15.5861 −1.71079 −0.855396 0.517975i \(-0.826686\pi\)
−0.855396 + 0.517975i \(0.826686\pi\)
\(84\) 0.618034 0.0674330
\(85\) 0 0
\(86\) −8.29168 −0.894115
\(87\) −5.69572 −0.610645
\(88\) 6.27044 0.668431
\(89\) −0.821412 −0.0870695 −0.0435348 0.999052i \(-0.513862\pi\)
−0.0435348 + 0.999052i \(0.513862\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) −0.580644 −0.0605364
\(93\) 0.895549 0.0928641
\(94\) 9.64771 0.995085
\(95\) 0 0
\(96\) 3.38197 0.345170
\(97\) −1.98281 −0.201324 −0.100662 0.994921i \(-0.532096\pi\)
−0.100662 + 0.994921i \(0.532096\pi\)
\(98\) −1.61803 −0.163446
\(99\) 2.80423 0.281835
\(100\) 0 0
\(101\) 7.71883 0.768052 0.384026 0.923322i \(-0.374537\pi\)
0.384026 + 0.923322i \(0.374537\pi\)
\(102\) 3.25325 0.322120
\(103\) −19.3309 −1.90473 −0.952367 0.304954i \(-0.901359\pi\)
−0.952367 + 0.304954i \(0.901359\pi\)
\(104\) 2.23607 0.219265
\(105\) 0 0
\(106\) 2.10851 0.204796
\(107\) 7.81736 0.755732 0.377866 0.925860i \(-0.376658\pi\)
0.377866 + 0.925860i \(0.376658\pi\)
\(108\) 0.618034 0.0594703
\(109\) 1.33686 0.128048 0.0640240 0.997948i \(-0.479607\pi\)
0.0640240 + 0.997948i \(0.479607\pi\)
\(110\) 0 0
\(111\) −5.52015 −0.523949
\(112\) −4.85410 −0.458670
\(113\) −6.79766 −0.639470 −0.319735 0.947507i \(-0.603594\pi\)
−0.319735 + 0.947507i \(0.603594\pi\)
\(114\) 3.79360 0.355304
\(115\) 0 0
\(116\) −3.52015 −0.326837
\(117\) 1.00000 0.0924500
\(118\) 3.99698 0.367952
\(119\) −2.01062 −0.184313
\(120\) 0 0
\(121\) −3.13632 −0.285120
\(122\) 6.72841 0.609161
\(123\) 3.07768 0.277505
\(124\) 0.553479 0.0497039
\(125\) 0 0
\(126\) −1.61803 −0.144146
\(127\) −10.4912 −0.930942 −0.465471 0.885063i \(-0.654115\pi\)
−0.465471 + 0.885063i \(0.654115\pi\)
\(128\) −13.6180 −1.20368
\(129\) 5.12454 0.451191
\(130\) 0 0
\(131\) −2.09319 −0.182883 −0.0914413 0.995810i \(-0.529147\pi\)
−0.0914413 + 0.995810i \(0.529147\pi\)
\(132\) 1.73311 0.150848
\(133\) −2.34458 −0.203301
\(134\) 25.0467 2.16371
\(135\) 0 0
\(136\) −4.49589 −0.385519
\(137\) 12.2704 1.04833 0.524167 0.851615i \(-0.324377\pi\)
0.524167 + 0.851615i \(0.324377\pi\)
\(138\) 1.52015 0.129404
\(139\) 13.4538 1.14114 0.570568 0.821250i \(-0.306723\pi\)
0.570568 + 0.821250i \(0.306723\pi\)
\(140\) 0 0
\(141\) −5.96261 −0.502142
\(142\) 19.9691 1.67577
\(143\) 2.80423 0.234501
\(144\) −4.85410 −0.404508
\(145\) 0 0
\(146\) 4.48932 0.371539
\(147\) 1.00000 0.0824786
\(148\) −3.41164 −0.280435
\(149\) 12.6216 1.03400 0.517000 0.855985i \(-0.327049\pi\)
0.517000 + 0.855985i \(0.327049\pi\)
\(150\) 0 0
\(151\) 0.222429 0.0181010 0.00905051 0.999959i \(-0.497119\pi\)
0.00905051 + 0.999959i \(0.497119\pi\)
\(152\) −5.24263 −0.425234
\(153\) −2.01062 −0.162549
\(154\) −4.53733 −0.365629
\(155\) 0 0
\(156\) 0.618034 0.0494823
\(157\) −19.5013 −1.55637 −0.778187 0.628033i \(-0.783860\pi\)
−0.778187 + 0.628033i \(0.783860\pi\)
\(158\) 9.97510 0.793576
\(159\) −1.30313 −0.103345
\(160\) 0 0
\(161\) −0.939503 −0.0740432
\(162\) −1.61803 −0.127125
\(163\) 1.14712 0.0898494 0.0449247 0.998990i \(-0.485695\pi\)
0.0449247 + 0.998990i \(0.485695\pi\)
\(164\) 1.90211 0.148530
\(165\) 0 0
\(166\) 25.2188 1.95736
\(167\) 17.8232 1.37920 0.689599 0.724191i \(-0.257787\pi\)
0.689599 + 0.724191i \(0.257787\pi\)
\(168\) 2.23607 0.172516
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −2.34458 −0.179294
\(172\) 3.16714 0.241492
\(173\) −2.11443 −0.160757 −0.0803786 0.996764i \(-0.525613\pi\)
−0.0803786 + 0.996764i \(0.525613\pi\)
\(174\) 9.21586 0.698653
\(175\) 0 0
\(176\) −13.6120 −1.02604
\(177\) −2.47027 −0.185677
\(178\) 1.32907 0.0996182
\(179\) 18.6737 1.39574 0.697870 0.716225i \(-0.254131\pi\)
0.697870 + 0.716225i \(0.254131\pi\)
\(180\) 0 0
\(181\) 6.07872 0.451828 0.225914 0.974147i \(-0.427463\pi\)
0.225914 + 0.974147i \(0.427463\pi\)
\(182\) −1.61803 −0.119937
\(183\) −4.15838 −0.307397
\(184\) −2.10079 −0.154872
\(185\) 0 0
\(186\) −1.44903 −0.106248
\(187\) −5.63824 −0.412309
\(188\) −3.68510 −0.268763
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 13.9329 1.00815 0.504076 0.863659i \(-0.331833\pi\)
0.504076 + 0.863659i \(0.331833\pi\)
\(192\) 4.23607 0.305712
\(193\) −3.14339 −0.226266 −0.113133 0.993580i \(-0.536089\pi\)
−0.113133 + 0.993580i \(0.536089\pi\)
\(194\) 3.20826 0.230340
\(195\) 0 0
\(196\) 0.618034 0.0441453
\(197\) 5.11141 0.364173 0.182087 0.983283i \(-0.441715\pi\)
0.182087 + 0.983283i \(0.441715\pi\)
\(198\) −4.53733 −0.322454
\(199\) 16.5135 1.17061 0.585304 0.810814i \(-0.300975\pi\)
0.585304 + 0.810814i \(0.300975\pi\)
\(200\) 0 0
\(201\) −15.4797 −1.09186
\(202\) −12.4893 −0.878746
\(203\) −5.69572 −0.399761
\(204\) −1.24263 −0.0870017
\(205\) 0 0
\(206\) 31.2781 2.17925
\(207\) −0.939503 −0.0652999
\(208\) −4.85410 −0.336571
\(209\) −6.57472 −0.454783
\(210\) 0 0
\(211\) −2.47078 −0.170096 −0.0850478 0.996377i \(-0.527104\pi\)
−0.0850478 + 0.996377i \(0.527104\pi\)
\(212\) −0.805379 −0.0553136
\(213\) −12.3416 −0.845630
\(214\) −12.6487 −0.864650
\(215\) 0 0
\(216\) 2.23607 0.152145
\(217\) 0.895549 0.0607938
\(218\) −2.16308 −0.146503
\(219\) −2.77455 −0.187487
\(220\) 0 0
\(221\) −2.01062 −0.135249
\(222\) 8.93179 0.599462
\(223\) −9.10445 −0.609679 −0.304840 0.952404i \(-0.598603\pi\)
−0.304840 + 0.952404i \(0.598603\pi\)
\(224\) 3.38197 0.225967
\(225\) 0 0
\(226\) 10.9988 0.731632
\(227\) 7.00779 0.465123 0.232562 0.972582i \(-0.425289\pi\)
0.232562 + 0.972582i \(0.425289\pi\)
\(228\) −1.44903 −0.0959642
\(229\) −20.2385 −1.33740 −0.668698 0.743534i \(-0.733148\pi\)
−0.668698 + 0.743534i \(0.733148\pi\)
\(230\) 0 0
\(231\) 2.80423 0.184505
\(232\) −12.7360 −0.836160
\(233\) −14.6750 −0.961391 −0.480696 0.876887i \(-0.659616\pi\)
−0.480696 + 0.876887i \(0.659616\pi\)
\(234\) −1.61803 −0.105774
\(235\) 0 0
\(236\) −1.52671 −0.0993805
\(237\) −6.16495 −0.400456
\(238\) 3.25325 0.210877
\(239\) 23.9963 1.55219 0.776097 0.630613i \(-0.217197\pi\)
0.776097 + 0.630613i \(0.217197\pi\)
\(240\) 0 0
\(241\) 25.2300 1.62521 0.812605 0.582815i \(-0.198049\pi\)
0.812605 + 0.582815i \(0.198049\pi\)
\(242\) 5.07467 0.326212
\(243\) 1.00000 0.0641500
\(244\) −2.57002 −0.164529
\(245\) 0 0
\(246\) −4.97980 −0.317500
\(247\) −2.34458 −0.149182
\(248\) 2.00251 0.127159
\(249\) −15.5861 −0.987726
\(250\) 0 0
\(251\) 11.8992 0.751069 0.375534 0.926808i \(-0.377459\pi\)
0.375534 + 0.926808i \(0.377459\pi\)
\(252\) 0.618034 0.0389325
\(253\) −2.63458 −0.165634
\(254\) 16.9751 1.06511
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) 5.49151 0.342551 0.171276 0.985223i \(-0.445211\pi\)
0.171276 + 0.985223i \(0.445211\pi\)
\(258\) −8.29168 −0.516218
\(259\) −5.52015 −0.343005
\(260\) 0 0
\(261\) −5.69572 −0.352556
\(262\) 3.38685 0.209240
\(263\) −2.38614 −0.147135 −0.0735677 0.997290i \(-0.523439\pi\)
−0.0735677 + 0.997290i \(0.523439\pi\)
\(264\) 6.27044 0.385919
\(265\) 0 0
\(266\) 3.79360 0.232601
\(267\) −0.821412 −0.0502696
\(268\) −9.56701 −0.584398
\(269\) −18.6149 −1.13497 −0.567485 0.823383i \(-0.692084\pi\)
−0.567485 + 0.823383i \(0.692084\pi\)
\(270\) 0 0
\(271\) −2.89400 −0.175798 −0.0878990 0.996129i \(-0.528015\pi\)
−0.0878990 + 0.996129i \(0.528015\pi\)
\(272\) 9.75976 0.591773
\(273\) 1.00000 0.0605228
\(274\) −19.8540 −1.19942
\(275\) 0 0
\(276\) −0.580644 −0.0349507
\(277\) −14.3261 −0.860769 −0.430385 0.902646i \(-0.641622\pi\)
−0.430385 + 0.902646i \(0.641622\pi\)
\(278\) −21.7687 −1.30560
\(279\) 0.895549 0.0536151
\(280\) 0 0
\(281\) 5.57543 0.332603 0.166301 0.986075i \(-0.446818\pi\)
0.166301 + 0.986075i \(0.446818\pi\)
\(282\) 9.64771 0.574513
\(283\) 18.6838 1.11064 0.555319 0.831637i \(-0.312596\pi\)
0.555319 + 0.831637i \(0.312596\pi\)
\(284\) −7.62750 −0.452609
\(285\) 0 0
\(286\) −4.53733 −0.268298
\(287\) 3.07768 0.181670
\(288\) 3.38197 0.199284
\(289\) −12.9574 −0.762200
\(290\) 0 0
\(291\) −1.98281 −0.116235
\(292\) −1.71477 −0.100349
\(293\) −31.0585 −1.81446 −0.907229 0.420637i \(-0.861807\pi\)
−0.907229 + 0.420637i \(0.861807\pi\)
\(294\) −1.61803 −0.0943657
\(295\) 0 0
\(296\) −12.3434 −0.717447
\(297\) 2.80423 0.162718
\(298\) −20.4221 −1.18302
\(299\) −0.939503 −0.0543328
\(300\) 0 0
\(301\) 5.12454 0.295374
\(302\) −0.359898 −0.0207098
\(303\) 7.71883 0.443435
\(304\) 11.3808 0.652734
\(305\) 0 0
\(306\) 3.25325 0.185976
\(307\) −1.72892 −0.0986745 −0.0493373 0.998782i \(-0.515711\pi\)
−0.0493373 + 0.998782i \(0.515711\pi\)
\(308\) 1.73311 0.0987529
\(309\) −19.3309 −1.09970
\(310\) 0 0
\(311\) 17.7385 1.00586 0.502930 0.864327i \(-0.332256\pi\)
0.502930 + 0.864327i \(0.332256\pi\)
\(312\) 2.23607 0.126592
\(313\) −24.8522 −1.40473 −0.702365 0.711817i \(-0.747872\pi\)
−0.702365 + 0.711817i \(0.747872\pi\)
\(314\) 31.5538 1.78068
\(315\) 0 0
\(316\) −3.81015 −0.214338
\(317\) 2.60626 0.146382 0.0731911 0.997318i \(-0.476682\pi\)
0.0731911 + 0.997318i \(0.476682\pi\)
\(318\) 2.10851 0.118239
\(319\) −15.9721 −0.894265
\(320\) 0 0
\(321\) 7.81736 0.436322
\(322\) 1.52015 0.0847145
\(323\) 4.71406 0.262297
\(324\) 0.618034 0.0343352
\(325\) 0 0
\(326\) −1.85608 −0.102799
\(327\) 1.33686 0.0739285
\(328\) 6.88191 0.379990
\(329\) −5.96261 −0.328729
\(330\) 0 0
\(331\) −30.7543 −1.69041 −0.845205 0.534443i \(-0.820522\pi\)
−0.845205 + 0.534443i \(0.820522\pi\)
\(332\) −9.63271 −0.528664
\(333\) −5.52015 −0.302502
\(334\) −28.8385 −1.57797
\(335\) 0 0
\(336\) −4.85410 −0.264813
\(337\) 14.3356 0.780912 0.390456 0.920622i \(-0.372317\pi\)
0.390456 + 0.920622i \(0.372317\pi\)
\(338\) −1.61803 −0.0880094
\(339\) −6.79766 −0.369198
\(340\) 0 0
\(341\) 2.51132 0.135996
\(342\) 3.79360 0.205135
\(343\) 1.00000 0.0539949
\(344\) 11.4588 0.617818
\(345\) 0 0
\(346\) 3.42122 0.183926
\(347\) −28.2949 −1.51895 −0.759475 0.650537i \(-0.774544\pi\)
−0.759475 + 0.650537i \(0.774544\pi\)
\(348\) −3.52015 −0.188700
\(349\) −7.82174 −0.418688 −0.209344 0.977842i \(-0.567133\pi\)
−0.209344 + 0.977842i \(0.567133\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 9.48380 0.505488
\(353\) 12.6656 0.674122 0.337061 0.941483i \(-0.390567\pi\)
0.337061 + 0.941483i \(0.390567\pi\)
\(354\) 3.99698 0.212437
\(355\) 0 0
\(356\) −0.507661 −0.0269060
\(357\) −2.01062 −0.106413
\(358\) −30.2147 −1.59690
\(359\) 4.35926 0.230073 0.115036 0.993361i \(-0.463302\pi\)
0.115036 + 0.993361i \(0.463302\pi\)
\(360\) 0 0
\(361\) −13.5030 −0.710682
\(362\) −9.83558 −0.516947
\(363\) −3.13632 −0.164614
\(364\) 0.618034 0.0323938
\(365\) 0 0
\(366\) 6.72841 0.351699
\(367\) −23.7244 −1.23840 −0.619201 0.785233i \(-0.712543\pi\)
−0.619201 + 0.785233i \(0.712543\pi\)
\(368\) 4.56044 0.237729
\(369\) 3.07768 0.160218
\(370\) 0 0
\(371\) −1.30313 −0.0676551
\(372\) 0.553479 0.0286966
\(373\) −12.2595 −0.634773 −0.317387 0.948296i \(-0.602805\pi\)
−0.317387 + 0.948296i \(0.602805\pi\)
\(374\) 9.12286 0.471732
\(375\) 0 0
\(376\) −13.3328 −0.687587
\(377\) −5.69572 −0.293344
\(378\) −1.61803 −0.0832227
\(379\) −9.44164 −0.484984 −0.242492 0.970153i \(-0.577965\pi\)
−0.242492 + 0.970153i \(0.577965\pi\)
\(380\) 0 0
\(381\) −10.4912 −0.537480
\(382\) −22.5440 −1.15345
\(383\) 1.81922 0.0929578 0.0464789 0.998919i \(-0.485200\pi\)
0.0464789 + 0.998919i \(0.485200\pi\)
\(384\) −13.6180 −0.694942
\(385\) 0 0
\(386\) 5.08611 0.258876
\(387\) 5.12454 0.260495
\(388\) −1.22545 −0.0622126
\(389\) −11.2846 −0.572152 −0.286076 0.958207i \(-0.592351\pi\)
−0.286076 + 0.958207i \(0.592351\pi\)
\(390\) 0 0
\(391\) 1.88898 0.0955300
\(392\) 2.23607 0.112938
\(393\) −2.09319 −0.105587
\(394\) −8.27044 −0.416659
\(395\) 0 0
\(396\) 1.73311 0.0870919
\(397\) 21.6658 1.08738 0.543688 0.839287i \(-0.317027\pi\)
0.543688 + 0.839287i \(0.317027\pi\)
\(398\) −26.7194 −1.33932
\(399\) −2.34458 −0.117376
\(400\) 0 0
\(401\) 15.1128 0.754696 0.377348 0.926072i \(-0.376836\pi\)
0.377348 + 0.926072i \(0.376836\pi\)
\(402\) 25.0467 1.24922
\(403\) 0.895549 0.0446105
\(404\) 4.77050 0.237341
\(405\) 0 0
\(406\) 9.21586 0.457376
\(407\) −15.4797 −0.767302
\(408\) −4.49589 −0.222580
\(409\) 0.0919651 0.00454738 0.00227369 0.999997i \(-0.499276\pi\)
0.00227369 + 0.999997i \(0.499276\pi\)
\(410\) 0 0
\(411\) 12.2704 0.605256
\(412\) −11.9472 −0.588595
\(413\) −2.47027 −0.121554
\(414\) 1.52015 0.0747112
\(415\) 0 0
\(416\) 3.38197 0.165815
\(417\) 13.4538 0.658835
\(418\) 10.6381 0.520328
\(419\) 15.8705 0.775323 0.387661 0.921802i \(-0.373283\pi\)
0.387661 + 0.921802i \(0.373283\pi\)
\(420\) 0 0
\(421\) −13.6257 −0.664076 −0.332038 0.943266i \(-0.607736\pi\)
−0.332038 + 0.943266i \(0.607736\pi\)
\(422\) 3.99781 0.194610
\(423\) −5.96261 −0.289912
\(424\) −2.91389 −0.141511
\(425\) 0 0
\(426\) 19.9691 0.967504
\(427\) −4.15838 −0.201238
\(428\) 4.83139 0.233534
\(429\) 2.80423 0.135389
\(430\) 0 0
\(431\) −25.4756 −1.22712 −0.613559 0.789649i \(-0.710263\pi\)
−0.613559 + 0.789649i \(0.710263\pi\)
\(432\) −4.85410 −0.233543
\(433\) −20.3956 −0.980151 −0.490076 0.871680i \(-0.663031\pi\)
−0.490076 + 0.871680i \(0.663031\pi\)
\(434\) −1.44903 −0.0695556
\(435\) 0 0
\(436\) 0.826225 0.0395690
\(437\) 2.20274 0.105371
\(438\) 4.48932 0.214508
\(439\) 22.8539 1.09076 0.545379 0.838190i \(-0.316386\pi\)
0.545379 + 0.838190i \(0.316386\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 3.25325 0.154742
\(443\) 15.9067 0.755750 0.377875 0.925857i \(-0.376655\pi\)
0.377875 + 0.925857i \(0.376655\pi\)
\(444\) −3.41164 −0.161909
\(445\) 0 0
\(446\) 14.7313 0.697548
\(447\) 12.6216 0.596980
\(448\) 4.23607 0.200135
\(449\) −23.3073 −1.09994 −0.549969 0.835185i \(-0.685360\pi\)
−0.549969 + 0.835185i \(0.685360\pi\)
\(450\) 0 0
\(451\) 8.63052 0.406395
\(452\) −4.20119 −0.197607
\(453\) 0.222429 0.0104506
\(454\) −11.3388 −0.532158
\(455\) 0 0
\(456\) −5.24263 −0.245509
\(457\) 18.1531 0.849164 0.424582 0.905389i \(-0.360421\pi\)
0.424582 + 0.905389i \(0.360421\pi\)
\(458\) 32.7465 1.53014
\(459\) −2.01062 −0.0938478
\(460\) 0 0
\(461\) 30.5416 1.42246 0.711232 0.702957i \(-0.248138\pi\)
0.711232 + 0.702957i \(0.248138\pi\)
\(462\) −4.53733 −0.211096
\(463\) −18.8199 −0.874636 −0.437318 0.899307i \(-0.644072\pi\)
−0.437318 + 0.899307i \(0.644072\pi\)
\(464\) 27.6476 1.28351
\(465\) 0 0
\(466\) 23.7447 1.09995
\(467\) −14.3948 −0.666114 −0.333057 0.942907i \(-0.608080\pi\)
−0.333057 + 0.942907i \(0.608080\pi\)
\(468\) 0.618034 0.0285686
\(469\) −15.4797 −0.714788
\(470\) 0 0
\(471\) −19.5013 −0.898572
\(472\) −5.52369 −0.254249
\(473\) 14.3704 0.660751
\(474\) 9.97510 0.458171
\(475\) 0 0
\(476\) −1.24263 −0.0569560
\(477\) −1.30313 −0.0596662
\(478\) −38.8269 −1.77590
\(479\) −25.3568 −1.15858 −0.579292 0.815120i \(-0.696671\pi\)
−0.579292 + 0.815120i \(0.696671\pi\)
\(480\) 0 0
\(481\) −5.52015 −0.251697
\(482\) −40.8231 −1.85944
\(483\) −0.939503 −0.0427488
\(484\) −1.93835 −0.0881068
\(485\) 0 0
\(486\) −1.61803 −0.0733955
\(487\) −6.55671 −0.297113 −0.148556 0.988904i \(-0.547463\pi\)
−0.148556 + 0.988904i \(0.547463\pi\)
\(488\) −9.29843 −0.420920
\(489\) 1.14712 0.0518746
\(490\) 0 0
\(491\) −36.7728 −1.65954 −0.829768 0.558109i \(-0.811527\pi\)
−0.829768 + 0.558109i \(0.811527\pi\)
\(492\) 1.90211 0.0857539
\(493\) 11.4519 0.515769
\(494\) 3.79360 0.170682
\(495\) 0 0
\(496\) −4.34708 −0.195190
\(497\) −12.3416 −0.553595
\(498\) 25.2188 1.13008
\(499\) 33.8073 1.51342 0.756712 0.653748i \(-0.226804\pi\)
0.756712 + 0.653748i \(0.226804\pi\)
\(500\) 0 0
\(501\) 17.8232 0.796280
\(502\) −19.2533 −0.859315
\(503\) 2.57397 0.114768 0.0573838 0.998352i \(-0.481724\pi\)
0.0573838 + 0.998352i \(0.481724\pi\)
\(504\) 2.23607 0.0996024
\(505\) 0 0
\(506\) 4.26284 0.189506
\(507\) 1.00000 0.0444116
\(508\) −6.48391 −0.287677
\(509\) 14.2906 0.633422 0.316711 0.948522i \(-0.397422\pi\)
0.316711 + 0.948522i \(0.397422\pi\)
\(510\) 0 0
\(511\) −2.77455 −0.122739
\(512\) 5.29180 0.233867
\(513\) −2.34458 −0.103516
\(514\) −8.88546 −0.391921
\(515\) 0 0
\(516\) 3.16714 0.139426
\(517\) −16.7205 −0.735367
\(518\) 8.93179 0.392440
\(519\) −2.11443 −0.0928132
\(520\) 0 0
\(521\) 15.9346 0.698108 0.349054 0.937103i \(-0.386503\pi\)
0.349054 + 0.937103i \(0.386503\pi\)
\(522\) 9.21586 0.403367
\(523\) −42.2459 −1.84729 −0.923643 0.383254i \(-0.874803\pi\)
−0.923643 + 0.383254i \(0.874803\pi\)
\(524\) −1.29366 −0.0565138
\(525\) 0 0
\(526\) 3.86085 0.168341
\(527\) −1.80061 −0.0784358
\(528\) −13.6120 −0.592386
\(529\) −22.1173 −0.961623
\(530\) 0 0
\(531\) −2.47027 −0.107201
\(532\) −1.44903 −0.0628233
\(533\) 3.07768 0.133309
\(534\) 1.32907 0.0575146
\(535\) 0 0
\(536\) −34.6138 −1.49509
\(537\) 18.6737 0.805830
\(538\) 30.1195 1.29855
\(539\) 2.80423 0.120787
\(540\) 0 0
\(541\) −32.0729 −1.37892 −0.689460 0.724324i \(-0.742152\pi\)
−0.689460 + 0.724324i \(0.742152\pi\)
\(542\) 4.68259 0.201134
\(543\) 6.07872 0.260863
\(544\) −6.79985 −0.291541
\(545\) 0 0
\(546\) −1.61803 −0.0692455
\(547\) 19.6201 0.838895 0.419448 0.907779i \(-0.362224\pi\)
0.419448 + 0.907779i \(0.362224\pi\)
\(548\) 7.58355 0.323953
\(549\) −4.15838 −0.177476
\(550\) 0 0
\(551\) 13.3540 0.568901
\(552\) −2.10079 −0.0894156
\(553\) −6.16495 −0.262160
\(554\) 23.1800 0.984826
\(555\) 0 0
\(556\) 8.31490 0.352630
\(557\) 0.768499 0.0325624 0.0162812 0.999867i \(-0.494817\pi\)
0.0162812 + 0.999867i \(0.494817\pi\)
\(558\) −1.44903 −0.0613422
\(559\) 5.12454 0.216745
\(560\) 0 0
\(561\) −5.63824 −0.238047
\(562\) −9.02124 −0.380538
\(563\) 35.0266 1.47620 0.738098 0.674694i \(-0.235724\pi\)
0.738098 + 0.674694i \(0.235724\pi\)
\(564\) −3.68510 −0.155171
\(565\) 0 0
\(566\) −30.2311 −1.27071
\(567\) 1.00000 0.0419961
\(568\) −27.5966 −1.15793
\(569\) 26.3124 1.10307 0.551537 0.834150i \(-0.314042\pi\)
0.551537 + 0.834150i \(0.314042\pi\)
\(570\) 0 0
\(571\) −40.3772 −1.68973 −0.844866 0.534978i \(-0.820320\pi\)
−0.844866 + 0.534978i \(0.820320\pi\)
\(572\) 1.73311 0.0724648
\(573\) 13.9329 0.582057
\(574\) −4.97980 −0.207853
\(575\) 0 0
\(576\) 4.23607 0.176503
\(577\) 33.7497 1.40502 0.702509 0.711675i \(-0.252063\pi\)
0.702509 + 0.711675i \(0.252063\pi\)
\(578\) 20.9655 0.872050
\(579\) −3.14339 −0.130635
\(580\) 0 0
\(581\) −15.5861 −0.646619
\(582\) 3.20826 0.132987
\(583\) −3.65427 −0.151344
\(584\) −6.20409 −0.256727
\(585\) 0 0
\(586\) 50.2537 2.07596
\(587\) 35.8502 1.47970 0.739848 0.672774i \(-0.234897\pi\)
0.739848 + 0.672774i \(0.234897\pi\)
\(588\) 0.618034 0.0254873
\(589\) −2.09968 −0.0865159
\(590\) 0 0
\(591\) 5.11141 0.210255
\(592\) 26.7954 1.10128
\(593\) −28.7642 −1.18120 −0.590602 0.806963i \(-0.701110\pi\)
−0.590602 + 0.806963i \(0.701110\pi\)
\(594\) −4.53733 −0.186169
\(595\) 0 0
\(596\) 7.80057 0.319524
\(597\) 16.5135 0.675851
\(598\) 1.52015 0.0621634
\(599\) −16.1612 −0.660327 −0.330163 0.943924i \(-0.607104\pi\)
−0.330163 + 0.943924i \(0.607104\pi\)
\(600\) 0 0
\(601\) −40.3491 −1.64587 −0.822936 0.568134i \(-0.807666\pi\)
−0.822936 + 0.568134i \(0.807666\pi\)
\(602\) −8.29168 −0.337944
\(603\) −15.4797 −0.630384
\(604\) 0.137469 0.00559352
\(605\) 0 0
\(606\) −12.4893 −0.507344
\(607\) −34.9319 −1.41784 −0.708920 0.705289i \(-0.750817\pi\)
−0.708920 + 0.705289i \(0.750817\pi\)
\(608\) −7.92928 −0.321575
\(609\) −5.69572 −0.230802
\(610\) 0 0
\(611\) −5.96261 −0.241221
\(612\) −1.24263 −0.0502304
\(613\) 36.1928 1.46181 0.730907 0.682477i \(-0.239097\pi\)
0.730907 + 0.682477i \(0.239097\pi\)
\(614\) 2.79745 0.112896
\(615\) 0 0
\(616\) 6.27044 0.252643
\(617\) 27.8387 1.12074 0.560371 0.828242i \(-0.310658\pi\)
0.560371 + 0.828242i \(0.310658\pi\)
\(618\) 31.2781 1.25819
\(619\) 14.9807 0.602126 0.301063 0.953604i \(-0.402659\pi\)
0.301063 + 0.953604i \(0.402659\pi\)
\(620\) 0 0
\(621\) −0.939503 −0.0377009
\(622\) −28.7015 −1.15083
\(623\) −0.821412 −0.0329092
\(624\) −4.85410 −0.194320
\(625\) 0 0
\(626\) 40.2117 1.60718
\(627\) −6.57472 −0.262569
\(628\) −12.0525 −0.480946
\(629\) 11.0989 0.442543
\(630\) 0 0
\(631\) 10.7583 0.428281 0.214140 0.976803i \(-0.431305\pi\)
0.214140 + 0.976803i \(0.431305\pi\)
\(632\) −13.7852 −0.548348
\(633\) −2.47078 −0.0982047
\(634\) −4.21702 −0.167479
\(635\) 0 0
\(636\) −0.805379 −0.0319353
\(637\) 1.00000 0.0396214
\(638\) 25.8434 1.02315
\(639\) −12.3416 −0.488225
\(640\) 0 0
\(641\) −12.9112 −0.509961 −0.254981 0.966946i \(-0.582069\pi\)
−0.254981 + 0.966946i \(0.582069\pi\)
\(642\) −12.6487 −0.499206
\(643\) 20.8057 0.820495 0.410248 0.911974i \(-0.365442\pi\)
0.410248 + 0.911974i \(0.365442\pi\)
\(644\) −0.580644 −0.0228806
\(645\) 0 0
\(646\) −7.62750 −0.300100
\(647\) 25.7610 1.01277 0.506384 0.862308i \(-0.330982\pi\)
0.506384 + 0.862308i \(0.330982\pi\)
\(648\) 2.23607 0.0878410
\(649\) −6.92720 −0.271916
\(650\) 0 0
\(651\) 0.895549 0.0350993
\(652\) 0.708959 0.0277650
\(653\) 33.0209 1.29221 0.646103 0.763250i \(-0.276398\pi\)
0.646103 + 0.763250i \(0.276398\pi\)
\(654\) −2.16308 −0.0845833
\(655\) 0 0
\(656\) −14.9394 −0.583285
\(657\) −2.77455 −0.108246
\(658\) 9.64771 0.376107
\(659\) −36.4979 −1.42176 −0.710879 0.703315i \(-0.751702\pi\)
−0.710879 + 0.703315i \(0.751702\pi\)
\(660\) 0 0
\(661\) 35.1393 1.36676 0.683380 0.730063i \(-0.260509\pi\)
0.683380 + 0.730063i \(0.260509\pi\)
\(662\) 49.7615 1.93404
\(663\) −2.01062 −0.0780861
\(664\) −34.8515 −1.35250
\(665\) 0 0
\(666\) 8.93179 0.346100
\(667\) 5.35114 0.207197
\(668\) 11.0153 0.426196
\(669\) −9.10445 −0.351998
\(670\) 0 0
\(671\) −11.6611 −0.450170
\(672\) 3.38197 0.130462
\(673\) 10.0652 0.387985 0.193992 0.981003i \(-0.437856\pi\)
0.193992 + 0.981003i \(0.437856\pi\)
\(674\) −23.1955 −0.893459
\(675\) 0 0
\(676\) 0.618034 0.0237705
\(677\) −34.4229 −1.32298 −0.661489 0.749955i \(-0.730075\pi\)
−0.661489 + 0.749955i \(0.730075\pi\)
\(678\) 10.9988 0.422408
\(679\) −1.98281 −0.0760934
\(680\) 0 0
\(681\) 7.00779 0.268539
\(682\) −4.06340 −0.155596
\(683\) 7.19340 0.275248 0.137624 0.990485i \(-0.456053\pi\)
0.137624 + 0.990485i \(0.456053\pi\)
\(684\) −1.44903 −0.0554050
\(685\) 0 0
\(686\) −1.61803 −0.0617768
\(687\) −20.2385 −0.772146
\(688\) −24.8750 −0.948352
\(689\) −1.30313 −0.0496453
\(690\) 0 0
\(691\) 2.19402 0.0834645 0.0417323 0.999129i \(-0.486712\pi\)
0.0417323 + 0.999129i \(0.486712\pi\)
\(692\) −1.30679 −0.0496767
\(693\) 2.80423 0.106524
\(694\) 45.7821 1.73787
\(695\) 0 0
\(696\) −12.7360 −0.482757
\(697\) −6.18806 −0.234389
\(698\) 12.6558 0.479031
\(699\) −14.6750 −0.555060
\(700\) 0 0
\(701\) 9.08651 0.343193 0.171596 0.985167i \(-0.445108\pi\)
0.171596 + 0.985167i \(0.445108\pi\)
\(702\) −1.61803 −0.0610688
\(703\) 12.9424 0.488132
\(704\) 11.8789 0.447703
\(705\) 0 0
\(706\) −20.4934 −0.771278
\(707\) 7.71883 0.290296
\(708\) −1.52671 −0.0573773
\(709\) −17.0828 −0.641557 −0.320778 0.947154i \(-0.603944\pi\)
−0.320778 + 0.947154i \(0.603944\pi\)
\(710\) 0 0
\(711\) −6.16495 −0.231204
\(712\) −1.83673 −0.0688345
\(713\) −0.841370 −0.0315096
\(714\) 3.25325 0.121750
\(715\) 0 0
\(716\) 11.5410 0.431307
\(717\) 23.9963 0.896160
\(718\) −7.05342 −0.263231
\(719\) −46.0179 −1.71618 −0.858088 0.513502i \(-0.828348\pi\)
−0.858088 + 0.513502i \(0.828348\pi\)
\(720\) 0 0
\(721\) −19.3309 −0.719922
\(722\) 21.8482 0.813108
\(723\) 25.2300 0.938315
\(724\) 3.75686 0.139623
\(725\) 0 0
\(726\) 5.07467 0.188339
\(727\) −41.4822 −1.53849 −0.769245 0.638954i \(-0.779368\pi\)
−0.769245 + 0.638954i \(0.779368\pi\)
\(728\) 2.23607 0.0828742
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −10.3035 −0.381089
\(732\) −2.57002 −0.0949908
\(733\) −8.95740 −0.330849 −0.165425 0.986222i \(-0.552899\pi\)
−0.165425 + 0.986222i \(0.552899\pi\)
\(734\) 38.3868 1.41688
\(735\) 0 0
\(736\) −3.17737 −0.117119
\(737\) −43.4087 −1.59898
\(738\) −4.97980 −0.183309
\(739\) 41.1448 1.51354 0.756769 0.653683i \(-0.226777\pi\)
0.756769 + 0.653683i \(0.226777\pi\)
\(740\) 0 0
\(741\) −2.34458 −0.0861302
\(742\) 2.10851 0.0774058
\(743\) −51.2608 −1.88058 −0.940288 0.340379i \(-0.889445\pi\)
−0.940288 + 0.340379i \(0.889445\pi\)
\(744\) 2.00251 0.0734155
\(745\) 0 0
\(746\) 19.8363 0.726258
\(747\) −15.5861 −0.570264
\(748\) −3.48462 −0.127410
\(749\) 7.81736 0.285640
\(750\) 0 0
\(751\) −9.69851 −0.353904 −0.176952 0.984220i \(-0.556624\pi\)
−0.176952 + 0.984220i \(0.556624\pi\)
\(752\) 28.9431 1.05545
\(753\) 11.8992 0.433630
\(754\) 9.21586 0.335622
\(755\) 0 0
\(756\) 0.618034 0.0224777
\(757\) −42.0784 −1.52936 −0.764682 0.644408i \(-0.777104\pi\)
−0.764682 + 0.644408i \(0.777104\pi\)
\(758\) 15.2769 0.554882
\(759\) −2.63458 −0.0956291
\(760\) 0 0
\(761\) −20.1288 −0.729668 −0.364834 0.931072i \(-0.618874\pi\)
−0.364834 + 0.931072i \(0.618874\pi\)
\(762\) 16.9751 0.614943
\(763\) 1.33686 0.0483976
\(764\) 8.61103 0.311536
\(765\) 0 0
\(766\) −2.94356 −0.106355
\(767\) −2.47027 −0.0891963
\(768\) 13.5623 0.489388
\(769\) 23.5408 0.848902 0.424451 0.905451i \(-0.360467\pi\)
0.424451 + 0.905451i \(0.360467\pi\)
\(770\) 0 0
\(771\) 5.49151 0.197772
\(772\) −1.94272 −0.0699201
\(773\) 9.93937 0.357494 0.178747 0.983895i \(-0.442796\pi\)
0.178747 + 0.983895i \(0.442796\pi\)
\(774\) −8.29168 −0.298038
\(775\) 0 0
\(776\) −4.43371 −0.159161
\(777\) −5.52015 −0.198034
\(778\) 18.2589 0.654613
\(779\) −7.21586 −0.258535
\(780\) 0 0
\(781\) −34.6085 −1.23839
\(782\) −3.05644 −0.109298
\(783\) −5.69572 −0.203548
\(784\) −4.85410 −0.173361
\(785\) 0 0
\(786\) 3.38685 0.120805
\(787\) 37.9659 1.35334 0.676670 0.736287i \(-0.263423\pi\)
0.676670 + 0.736287i \(0.263423\pi\)
\(788\) 3.15903 0.112536
\(789\) −2.38614 −0.0849487
\(790\) 0 0
\(791\) −6.79766 −0.241697
\(792\) 6.27044 0.222810
\(793\) −4.15838 −0.147669
\(794\) −35.0560 −1.24409
\(795\) 0 0
\(796\) 10.2059 0.361738
\(797\) 42.9372 1.52091 0.760456 0.649389i \(-0.224975\pi\)
0.760456 + 0.649389i \(0.224975\pi\)
\(798\) 3.79360 0.134292
\(799\) 11.9886 0.424125
\(800\) 0 0
\(801\) −0.821412 −0.0290232
\(802\) −24.4530 −0.863464
\(803\) −7.78048 −0.274567
\(804\) −9.56701 −0.337402
\(805\) 0 0
\(806\) −1.44903 −0.0510398
\(807\) −18.6149 −0.655276
\(808\) 17.2598 0.607198
\(809\) 47.5910 1.67321 0.836605 0.547807i \(-0.184537\pi\)
0.836605 + 0.547807i \(0.184537\pi\)
\(810\) 0 0
\(811\) −40.0579 −1.40662 −0.703312 0.710882i \(-0.748296\pi\)
−0.703312 + 0.710882i \(0.748296\pi\)
\(812\) −3.52015 −0.123533
\(813\) −2.89400 −0.101497
\(814\) 25.0467 0.877888
\(815\) 0 0
\(816\) 9.75976 0.341660
\(817\) −12.0149 −0.420347
\(818\) −0.148803 −0.00520276
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −2.41652 −0.0843372 −0.0421686 0.999111i \(-0.513427\pi\)
−0.0421686 + 0.999111i \(0.513427\pi\)
\(822\) −19.8540 −0.692487
\(823\) 57.1896 1.99350 0.996752 0.0805350i \(-0.0256629\pi\)
0.996752 + 0.0805350i \(0.0256629\pi\)
\(824\) −43.2253 −1.50582
\(825\) 0 0
\(826\) 3.99698 0.139073
\(827\) −5.71058 −0.198576 −0.0992881 0.995059i \(-0.531657\pi\)
−0.0992881 + 0.995059i \(0.531657\pi\)
\(828\) −0.580644 −0.0201788
\(829\) −32.2179 −1.11898 −0.559488 0.828839i \(-0.689002\pi\)
−0.559488 + 0.828839i \(0.689002\pi\)
\(830\) 0 0
\(831\) −14.3261 −0.496965
\(832\) 4.23607 0.146859
\(833\) −2.01062 −0.0696639
\(834\) −21.7687 −0.753788
\(835\) 0 0
\(836\) −4.06340 −0.140536
\(837\) 0.895549 0.0309547
\(838\) −25.6789 −0.887065
\(839\) 39.2649 1.35557 0.677787 0.735258i \(-0.262939\pi\)
0.677787 + 0.735258i \(0.262939\pi\)
\(840\) 0 0
\(841\) 3.44120 0.118662
\(842\) 22.0469 0.759785
\(843\) 5.57543 0.192028
\(844\) −1.52703 −0.0525624
\(845\) 0 0
\(846\) 9.64771 0.331695
\(847\) −3.13632 −0.107765
\(848\) 6.32553 0.217219
\(849\) 18.6838 0.641227
\(850\) 0 0
\(851\) 5.18619 0.177780
\(852\) −7.62750 −0.261314
\(853\) −13.4267 −0.459722 −0.229861 0.973224i \(-0.573827\pi\)
−0.229861 + 0.973224i \(0.573827\pi\)
\(854\) 6.72841 0.230241
\(855\) 0 0
\(856\) 17.4801 0.597459
\(857\) −48.4140 −1.65379 −0.826895 0.562356i \(-0.809895\pi\)
−0.826895 + 0.562356i \(0.809895\pi\)
\(858\) −4.53733 −0.154902
\(859\) −16.8574 −0.575168 −0.287584 0.957755i \(-0.592852\pi\)
−0.287584 + 0.957755i \(0.592852\pi\)
\(860\) 0 0
\(861\) 3.07768 0.104887
\(862\) 41.2205 1.40397
\(863\) −47.6860 −1.62325 −0.811625 0.584178i \(-0.801417\pi\)
−0.811625 + 0.584178i \(0.801417\pi\)
\(864\) 3.38197 0.115057
\(865\) 0 0
\(866\) 33.0008 1.12141
\(867\) −12.9574 −0.440056
\(868\) 0.553479 0.0187863
\(869\) −17.2879 −0.586452
\(870\) 0 0
\(871\) −15.4797 −0.524511
\(872\) 2.98931 0.101231
\(873\) −1.98281 −0.0671081
\(874\) −3.56410 −0.120558
\(875\) 0 0
\(876\) −1.71477 −0.0579367
\(877\) 5.74990 0.194160 0.0970801 0.995277i \(-0.469050\pi\)
0.0970801 + 0.995277i \(0.469050\pi\)
\(878\) −36.9784 −1.24796
\(879\) −31.0585 −1.04758
\(880\) 0 0
\(881\) −16.9157 −0.569905 −0.284952 0.958542i \(-0.591978\pi\)
−0.284952 + 0.958542i \(0.591978\pi\)
\(882\) −1.61803 −0.0544820
\(883\) −8.09873 −0.272544 −0.136272 0.990671i \(-0.543512\pi\)
−0.136272 + 0.990671i \(0.543512\pi\)
\(884\) −1.24263 −0.0417943
\(885\) 0 0
\(886\) −25.7376 −0.864671
\(887\) −44.5786 −1.49680 −0.748401 0.663247i \(-0.769178\pi\)
−0.748401 + 0.663247i \(0.769178\pi\)
\(888\) −12.3434 −0.414218
\(889\) −10.4912 −0.351863
\(890\) 0 0
\(891\) 2.80423 0.0939451
\(892\) −5.62686 −0.188401
\(893\) 13.9798 0.467816
\(894\) −20.4221 −0.683019
\(895\) 0 0
\(896\) −13.6180 −0.454947
\(897\) −0.939503 −0.0313691
\(898\) 37.7119 1.25846
\(899\) −5.10079 −0.170121
\(900\) 0 0
\(901\) 2.62010 0.0872882
\(902\) −13.9645 −0.464966
\(903\) 5.12454 0.170534
\(904\) −15.2000 −0.505546
\(905\) 0 0
\(906\) −0.359898 −0.0119568
\(907\) −24.1868 −0.803110 −0.401555 0.915835i \(-0.631530\pi\)
−0.401555 + 0.915835i \(0.631530\pi\)
\(908\) 4.33105 0.143731
\(909\) 7.71883 0.256017
\(910\) 0 0
\(911\) 9.27522 0.307302 0.153651 0.988125i \(-0.450897\pi\)
0.153651 + 0.988125i \(0.450897\pi\)
\(912\) 11.3808 0.376856
\(913\) −43.7068 −1.44648
\(914\) −29.3723 −0.971548
\(915\) 0 0
\(916\) −12.5081 −0.413278
\(917\) −2.09319 −0.0691231
\(918\) 3.25325 0.107373
\(919\) −2.91209 −0.0960611 −0.0480305 0.998846i \(-0.515294\pi\)
−0.0480305 + 0.998846i \(0.515294\pi\)
\(920\) 0 0
\(921\) −1.72892 −0.0569698
\(922\) −49.4173 −1.62747
\(923\) −12.3416 −0.406227
\(924\) 1.73311 0.0570150
\(925\) 0 0
\(926\) 30.4513 1.00069
\(927\) −19.3309 −0.634911
\(928\) −19.2627 −0.632330
\(929\) 35.9359 1.17902 0.589510 0.807761i \(-0.299321\pi\)
0.589510 + 0.807761i \(0.299321\pi\)
\(930\) 0 0
\(931\) −2.34458 −0.0768404
\(932\) −9.06965 −0.297086
\(933\) 17.7385 0.580733
\(934\) 23.2914 0.762116
\(935\) 0 0
\(936\) 2.23607 0.0730882
\(937\) 25.0249 0.817527 0.408763 0.912640i \(-0.365960\pi\)
0.408763 + 0.912640i \(0.365960\pi\)
\(938\) 25.0467 0.817805
\(939\) −24.8522 −0.811021
\(940\) 0 0
\(941\) 12.7170 0.414562 0.207281 0.978281i \(-0.433538\pi\)
0.207281 + 0.978281i \(0.433538\pi\)
\(942\) 31.5538 1.02808
\(943\) −2.89149 −0.0941599
\(944\) 11.9909 0.390272
\(945\) 0 0
\(946\) −23.2518 −0.755980
\(947\) −42.4461 −1.37931 −0.689656 0.724137i \(-0.742238\pi\)
−0.689656 + 0.724137i \(0.742238\pi\)
\(948\) −3.81015 −0.123748
\(949\) −2.77455 −0.0900658
\(950\) 0 0
\(951\) 2.60626 0.0845138
\(952\) −4.49589 −0.145713
\(953\) −44.1291 −1.42948 −0.714741 0.699389i \(-0.753456\pi\)
−0.714741 + 0.699389i \(0.753456\pi\)
\(954\) 2.10851 0.0682655
\(955\) 0 0
\(956\) 14.8306 0.479654
\(957\) −15.9721 −0.516304
\(958\) 41.0282 1.32556
\(959\) 12.2704 0.396233
\(960\) 0 0
\(961\) −30.1980 −0.974129
\(962\) 8.93179 0.287972
\(963\) 7.81736 0.251911
\(964\) 15.5930 0.502217
\(965\) 0 0
\(966\) 1.52015 0.0489099
\(967\) −5.88712 −0.189317 −0.0946585 0.995510i \(-0.530176\pi\)
−0.0946585 + 0.995510i \(0.530176\pi\)
\(968\) −7.01302 −0.225407
\(969\) 4.71406 0.151437
\(970\) 0 0
\(971\) 44.7978 1.43763 0.718814 0.695202i \(-0.244685\pi\)
0.718814 + 0.695202i \(0.244685\pi\)
\(972\) 0.618034 0.0198234
\(973\) 13.4538 0.431309
\(974\) 10.6090 0.339934
\(975\) 0 0
\(976\) 20.1852 0.646113
\(977\) 7.32375 0.234308 0.117154 0.993114i \(-0.462623\pi\)
0.117154 + 0.993114i \(0.462623\pi\)
\(978\) −1.85608 −0.0593509
\(979\) −2.30343 −0.0736178
\(980\) 0 0
\(981\) 1.33686 0.0426826
\(982\) 59.4997 1.89871
\(983\) −15.2176 −0.485367 −0.242683 0.970106i \(-0.578028\pi\)
−0.242683 + 0.970106i \(0.578028\pi\)
\(984\) 6.88191 0.219387
\(985\) 0 0
\(986\) −18.5296 −0.590103
\(987\) −5.96261 −0.189792
\(988\) −1.44903 −0.0460997
\(989\) −4.81452 −0.153093
\(990\) 0 0
\(991\) 1.65833 0.0526785 0.0263393 0.999653i \(-0.491615\pi\)
0.0263393 + 0.999653i \(0.491615\pi\)
\(992\) 3.02871 0.0961618
\(993\) −30.7543 −0.975958
\(994\) 19.9691 0.633380
\(995\) 0 0
\(996\) −9.63271 −0.305224
\(997\) −2.49409 −0.0789887 −0.0394943 0.999220i \(-0.512575\pi\)
−0.0394943 + 0.999220i \(0.512575\pi\)
\(998\) −54.7014 −1.73154
\(999\) −5.52015 −0.174650
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 6825.2.a.be.1.2 4
5.2 odd 4 1365.2.f.c.274.2 8
5.3 odd 4 1365.2.f.c.274.8 yes 8
5.4 even 2 6825.2.a.bm.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1365.2.f.c.274.2 8 5.2 odd 4
1365.2.f.c.274.8 yes 8 5.3 odd 4
6825.2.a.be.1.2 4 1.1 even 1 trivial
6825.2.a.bm.1.4 4 5.4 even 2