Properties

Label 8085.2.a.cp.1.8
Level $8085$
Weight $2$
Character 8085.1
Self dual yes
Analytic conductor $64.559$
Analytic rank $0$
Dimension $14$
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] = [8085,2,Mod(1,8085)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8085, 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("8085.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8085.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: \(64.5590500342\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} - 16 x^{12} + 76 x^{11} + 78 x^{10} - 532 x^{9} - 56 x^{8} + 1684 x^{7} - 471 x^{6} - 2352 x^{5} + 950 x^{4} + 1184 x^{3} - 340 x^{2} - 152 x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.537407\) of defining polynomial
Character \(\chi\) \(=\) 8085.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.537407 q^{2} +1.00000 q^{3} -1.71119 q^{4} +1.00000 q^{5} +0.537407 q^{6} -1.99442 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.537407 q^{2} +1.00000 q^{3} -1.71119 q^{4} +1.00000 q^{5} +0.537407 q^{6} -1.99442 q^{8} +1.00000 q^{9} +0.537407 q^{10} +1.00000 q^{11} -1.71119 q^{12} +1.59143 q^{13} +1.00000 q^{15} +2.35057 q^{16} +0.957367 q^{17} +0.537407 q^{18} +0.301599 q^{19} -1.71119 q^{20} +0.537407 q^{22} +5.18215 q^{23} -1.99442 q^{24} +1.00000 q^{25} +0.855247 q^{26} +1.00000 q^{27} +5.89540 q^{29} +0.537407 q^{30} +2.28121 q^{31} +5.25206 q^{32} +1.00000 q^{33} +0.514496 q^{34} -1.71119 q^{36} -9.90813 q^{37} +0.162081 q^{38} +1.59143 q^{39} -1.99442 q^{40} +11.2799 q^{41} -0.309330 q^{43} -1.71119 q^{44} +1.00000 q^{45} +2.78492 q^{46} -6.44995 q^{47} +2.35057 q^{48} +0.537407 q^{50} +0.957367 q^{51} -2.72325 q^{52} -12.4653 q^{53} +0.537407 q^{54} +1.00000 q^{55} +0.301599 q^{57} +3.16823 q^{58} +10.5444 q^{59} -1.71119 q^{60} -10.9417 q^{61} +1.22594 q^{62} -1.87864 q^{64} +1.59143 q^{65} +0.537407 q^{66} -4.92273 q^{67} -1.63824 q^{68} +5.18215 q^{69} -16.6715 q^{71} -1.99442 q^{72} -12.5376 q^{73} -5.32470 q^{74} +1.00000 q^{75} -0.516093 q^{76} +0.855247 q^{78} +8.17883 q^{79} +2.35057 q^{80} +1.00000 q^{81} +6.06191 q^{82} -4.70769 q^{83} +0.957367 q^{85} -0.166236 q^{86} +5.89540 q^{87} -1.99442 q^{88} +15.4510 q^{89} +0.537407 q^{90} -8.86766 q^{92} +2.28121 q^{93} -3.46625 q^{94} +0.301599 q^{95} +5.25206 q^{96} +18.2513 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 4 q^{2} + 14 q^{3} + 20 q^{4} + 14 q^{5} + 4 q^{6} + 12 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 4 q^{2} + 14 q^{3} + 20 q^{4} + 14 q^{5} + 4 q^{6} + 12 q^{8} + 14 q^{9} + 4 q^{10} + 14 q^{11} + 20 q^{12} + 8 q^{13} + 14 q^{15} + 32 q^{16} - 6 q^{17} + 4 q^{18} + 10 q^{19} + 20 q^{20} + 4 q^{22} + 22 q^{23} + 12 q^{24} + 14 q^{25} + 12 q^{26} + 14 q^{27} + 10 q^{29} + 4 q^{30} + 20 q^{31} + 28 q^{32} + 14 q^{33} + 16 q^{34} + 20 q^{36} + 20 q^{37} - 16 q^{38} + 8 q^{39} + 12 q^{40} + 22 q^{43} + 20 q^{44} + 14 q^{45} + 4 q^{46} - 8 q^{47} + 32 q^{48} + 4 q^{50} - 6 q^{51} + 4 q^{52} + 34 q^{53} + 4 q^{54} + 14 q^{55} + 10 q^{57} + 20 q^{58} + 22 q^{59} + 20 q^{60} + 6 q^{61} - 28 q^{62} + 32 q^{64} + 8 q^{65} + 4 q^{66} + 32 q^{67} - 56 q^{68} + 22 q^{69} + 4 q^{71} + 12 q^{72} + 8 q^{73} + 36 q^{74} + 14 q^{75} + 24 q^{76} + 12 q^{78} + 12 q^{79} + 32 q^{80} + 14 q^{81} - 28 q^{82} - 26 q^{83} - 6 q^{85} + 40 q^{86} + 10 q^{87} + 12 q^{88} + 10 q^{89} + 4 q^{90} + 8 q^{92} + 20 q^{93} + 28 q^{94} + 10 q^{95} + 28 q^{96} + 14 q^{97} + 14 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.537407 0.380004 0.190002 0.981784i \(-0.439150\pi\)
0.190002 + 0.981784i \(0.439150\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.71119 −0.855597
\(5\) 1.00000 0.447214
\(6\) 0.537407 0.219396
\(7\) 0 0
\(8\) −1.99442 −0.705135
\(9\) 1.00000 0.333333
\(10\) 0.537407 0.169943
\(11\) 1.00000 0.301511
\(12\) −1.71119 −0.493979
\(13\) 1.59143 0.441384 0.220692 0.975344i \(-0.429169\pi\)
0.220692 + 0.975344i \(0.429169\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 2.35057 0.587642
\(17\) 0.957367 0.232196 0.116098 0.993238i \(-0.462961\pi\)
0.116098 + 0.993238i \(0.462961\pi\)
\(18\) 0.537407 0.126668
\(19\) 0.301599 0.0691915 0.0345957 0.999401i \(-0.488986\pi\)
0.0345957 + 0.999401i \(0.488986\pi\)
\(20\) −1.71119 −0.382634
\(21\) 0 0
\(22\) 0.537407 0.114576
\(23\) 5.18215 1.08055 0.540276 0.841488i \(-0.318320\pi\)
0.540276 + 0.841488i \(0.318320\pi\)
\(24\) −1.99442 −0.407110
\(25\) 1.00000 0.200000
\(26\) 0.855247 0.167728
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.89540 1.09475 0.547374 0.836888i \(-0.315627\pi\)
0.547374 + 0.836888i \(0.315627\pi\)
\(30\) 0.537407 0.0981167
\(31\) 2.28121 0.409717 0.204858 0.978792i \(-0.434327\pi\)
0.204858 + 0.978792i \(0.434327\pi\)
\(32\) 5.25206 0.928442
\(33\) 1.00000 0.174078
\(34\) 0.514496 0.0882354
\(35\) 0 0
\(36\) −1.71119 −0.285199
\(37\) −9.90813 −1.62889 −0.814444 0.580243i \(-0.802958\pi\)
−0.814444 + 0.580243i \(0.802958\pi\)
\(38\) 0.162081 0.0262931
\(39\) 1.59143 0.254833
\(40\) −1.99442 −0.315346
\(41\) 11.2799 1.76163 0.880813 0.473465i \(-0.156997\pi\)
0.880813 + 0.473465i \(0.156997\pi\)
\(42\) 0 0
\(43\) −0.309330 −0.0471724 −0.0235862 0.999722i \(-0.507508\pi\)
−0.0235862 + 0.999722i \(0.507508\pi\)
\(44\) −1.71119 −0.257972
\(45\) 1.00000 0.149071
\(46\) 2.78492 0.410615
\(47\) −6.44995 −0.940822 −0.470411 0.882447i \(-0.655894\pi\)
−0.470411 + 0.882447i \(0.655894\pi\)
\(48\) 2.35057 0.339275
\(49\) 0 0
\(50\) 0.537407 0.0760009
\(51\) 0.957367 0.134058
\(52\) −2.72325 −0.377646
\(53\) −12.4653 −1.71224 −0.856119 0.516780i \(-0.827131\pi\)
−0.856119 + 0.516780i \(0.827131\pi\)
\(54\) 0.537407 0.0731319
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 0.301599 0.0399477
\(58\) 3.16823 0.416009
\(59\) 10.5444 1.37277 0.686385 0.727239i \(-0.259197\pi\)
0.686385 + 0.727239i \(0.259197\pi\)
\(60\) −1.71119 −0.220914
\(61\) −10.9417 −1.40094 −0.700471 0.713681i \(-0.747026\pi\)
−0.700471 + 0.713681i \(0.747026\pi\)
\(62\) 1.22594 0.155694
\(63\) 0 0
\(64\) −1.87864 −0.234831
\(65\) 1.59143 0.197393
\(66\) 0.537407 0.0661503
\(67\) −4.92273 −0.601407 −0.300703 0.953718i \(-0.597221\pi\)
−0.300703 + 0.953718i \(0.597221\pi\)
\(68\) −1.63824 −0.198666
\(69\) 5.18215 0.623857
\(70\) 0 0
\(71\) −16.6715 −1.97855 −0.989274 0.146070i \(-0.953337\pi\)
−0.989274 + 0.146070i \(0.953337\pi\)
\(72\) −1.99442 −0.235045
\(73\) −12.5376 −1.46742 −0.733710 0.679463i \(-0.762213\pi\)
−0.733710 + 0.679463i \(0.762213\pi\)
\(74\) −5.32470 −0.618984
\(75\) 1.00000 0.115470
\(76\) −0.516093 −0.0592000
\(77\) 0 0
\(78\) 0.855247 0.0968376
\(79\) 8.17883 0.920190 0.460095 0.887870i \(-0.347815\pi\)
0.460095 + 0.887870i \(0.347815\pi\)
\(80\) 2.35057 0.262802
\(81\) 1.00000 0.111111
\(82\) 6.06191 0.669425
\(83\) −4.70769 −0.516736 −0.258368 0.966047i \(-0.583185\pi\)
−0.258368 + 0.966047i \(0.583185\pi\)
\(84\) 0 0
\(85\) 0.957367 0.103841
\(86\) −0.166236 −0.0179257
\(87\) 5.89540 0.632053
\(88\) −1.99442 −0.212606
\(89\) 15.4510 1.63780 0.818902 0.573934i \(-0.194583\pi\)
0.818902 + 0.573934i \(0.194583\pi\)
\(90\) 0.537407 0.0566477
\(91\) 0 0
\(92\) −8.86766 −0.924517
\(93\) 2.28121 0.236550
\(94\) −3.46625 −0.357516
\(95\) 0.301599 0.0309434
\(96\) 5.25206 0.536036
\(97\) 18.2513 1.85314 0.926571 0.376119i \(-0.122742\pi\)
0.926571 + 0.376119i \(0.122742\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −1.71119 −0.171119
\(101\) −10.8839 −1.08299 −0.541493 0.840705i \(-0.682141\pi\)
−0.541493 + 0.840705i \(0.682141\pi\)
\(102\) 0.514496 0.0509427
\(103\) 2.06192 0.203167 0.101584 0.994827i \(-0.467609\pi\)
0.101584 + 0.994827i \(0.467609\pi\)
\(104\) −3.17399 −0.311235
\(105\) 0 0
\(106\) −6.69893 −0.650658
\(107\) 7.26476 0.702311 0.351155 0.936317i \(-0.385789\pi\)
0.351155 + 0.936317i \(0.385789\pi\)
\(108\) −1.71119 −0.164660
\(109\) 17.0326 1.63142 0.815711 0.578459i \(-0.196346\pi\)
0.815711 + 0.578459i \(0.196346\pi\)
\(110\) 0.537407 0.0512398
\(111\) −9.90813 −0.940439
\(112\) 0 0
\(113\) 16.2281 1.52661 0.763304 0.646039i \(-0.223576\pi\)
0.763304 + 0.646039i \(0.223576\pi\)
\(114\) 0.162081 0.0151803
\(115\) 5.18215 0.483238
\(116\) −10.0882 −0.936663
\(117\) 1.59143 0.147128
\(118\) 5.66666 0.521658
\(119\) 0 0
\(120\) −1.99442 −0.182065
\(121\) 1.00000 0.0909091
\(122\) −5.88015 −0.532364
\(123\) 11.2799 1.01708
\(124\) −3.90358 −0.350552
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 7.49535 0.665105 0.332553 0.943085i \(-0.392090\pi\)
0.332553 + 0.943085i \(0.392090\pi\)
\(128\) −11.5137 −1.01768
\(129\) −0.309330 −0.0272350
\(130\) 0.855247 0.0750101
\(131\) 16.3884 1.43186 0.715932 0.698170i \(-0.246002\pi\)
0.715932 + 0.698170i \(0.246002\pi\)
\(132\) −1.71119 −0.148940
\(133\) 0 0
\(134\) −2.64551 −0.228537
\(135\) 1.00000 0.0860663
\(136\) −1.90940 −0.163729
\(137\) 5.84770 0.499603 0.249801 0.968297i \(-0.419635\pi\)
0.249801 + 0.968297i \(0.419635\pi\)
\(138\) 2.78492 0.237068
\(139\) 22.3715 1.89752 0.948762 0.315992i \(-0.102337\pi\)
0.948762 + 0.315992i \(0.102337\pi\)
\(140\) 0 0
\(141\) −6.44995 −0.543184
\(142\) −8.95941 −0.751857
\(143\) 1.59143 0.133082
\(144\) 2.35057 0.195881
\(145\) 5.89540 0.489586
\(146\) −6.73782 −0.557626
\(147\) 0 0
\(148\) 16.9547 1.39367
\(149\) 22.0198 1.80393 0.901965 0.431809i \(-0.142125\pi\)
0.901965 + 0.431809i \(0.142125\pi\)
\(150\) 0.537407 0.0438791
\(151\) −0.293489 −0.0238838 −0.0119419 0.999929i \(-0.503801\pi\)
−0.0119419 + 0.999929i \(0.503801\pi\)
\(152\) −0.601515 −0.0487893
\(153\) 0.957367 0.0773986
\(154\) 0 0
\(155\) 2.28121 0.183231
\(156\) −2.72325 −0.218034
\(157\) −4.13745 −0.330204 −0.165102 0.986276i \(-0.552795\pi\)
−0.165102 + 0.986276i \(0.552795\pi\)
\(158\) 4.39536 0.349676
\(159\) −12.4653 −0.988561
\(160\) 5.25206 0.415212
\(161\) 0 0
\(162\) 0.537407 0.0422227
\(163\) −6.19238 −0.485024 −0.242512 0.970148i \(-0.577971\pi\)
−0.242512 + 0.970148i \(0.577971\pi\)
\(164\) −19.3021 −1.50724
\(165\) 1.00000 0.0778499
\(166\) −2.52995 −0.196362
\(167\) 4.74324 0.367043 0.183522 0.983016i \(-0.441250\pi\)
0.183522 + 0.983016i \(0.441250\pi\)
\(168\) 0 0
\(169\) −10.4673 −0.805180
\(170\) 0.514496 0.0394601
\(171\) 0.301599 0.0230638
\(172\) 0.529324 0.0403606
\(173\) −23.5426 −1.78991 −0.894956 0.446154i \(-0.852793\pi\)
−0.894956 + 0.446154i \(0.852793\pi\)
\(174\) 3.16823 0.240183
\(175\) 0 0
\(176\) 2.35057 0.177181
\(177\) 10.5444 0.792569
\(178\) 8.30348 0.622372
\(179\) 3.34159 0.249762 0.124881 0.992172i \(-0.460145\pi\)
0.124881 + 0.992172i \(0.460145\pi\)
\(180\) −1.71119 −0.127545
\(181\) 13.2594 0.985560 0.492780 0.870154i \(-0.335981\pi\)
0.492780 + 0.870154i \(0.335981\pi\)
\(182\) 0 0
\(183\) −10.9417 −0.808834
\(184\) −10.3354 −0.761935
\(185\) −9.90813 −0.728461
\(186\) 1.22594 0.0898901
\(187\) 0.957367 0.0700096
\(188\) 11.0371 0.804964
\(189\) 0 0
\(190\) 0.162081 0.0117586
\(191\) 21.6487 1.56645 0.783223 0.621741i \(-0.213574\pi\)
0.783223 + 0.621741i \(0.213574\pi\)
\(192\) −1.87864 −0.135579
\(193\) 19.5443 1.40683 0.703416 0.710779i \(-0.251657\pi\)
0.703416 + 0.710779i \(0.251657\pi\)
\(194\) 9.80840 0.704202
\(195\) 1.59143 0.113965
\(196\) 0 0
\(197\) 14.0102 0.998189 0.499094 0.866548i \(-0.333666\pi\)
0.499094 + 0.866548i \(0.333666\pi\)
\(198\) 0.537407 0.0381919
\(199\) −9.00502 −0.638349 −0.319174 0.947696i \(-0.603406\pi\)
−0.319174 + 0.947696i \(0.603406\pi\)
\(200\) −1.99442 −0.141027
\(201\) −4.92273 −0.347222
\(202\) −5.84907 −0.411539
\(203\) 0 0
\(204\) −1.63824 −0.114700
\(205\) 11.2799 0.787823
\(206\) 1.10809 0.0772045
\(207\) 5.18215 0.360184
\(208\) 3.74077 0.259376
\(209\) 0.301599 0.0208620
\(210\) 0 0
\(211\) 8.40675 0.578745 0.289372 0.957217i \(-0.406553\pi\)
0.289372 + 0.957217i \(0.406553\pi\)
\(212\) 21.3305 1.46498
\(213\) −16.6715 −1.14232
\(214\) 3.90414 0.266881
\(215\) −0.309330 −0.0210961
\(216\) −1.99442 −0.135703
\(217\) 0 0
\(218\) 9.15342 0.619948
\(219\) −12.5376 −0.847215
\(220\) −1.71119 −0.115369
\(221\) 1.52358 0.102487
\(222\) −5.32470 −0.357371
\(223\) −3.96213 −0.265324 −0.132662 0.991161i \(-0.542352\pi\)
−0.132662 + 0.991161i \(0.542352\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 8.72109 0.580118
\(227\) 4.72377 0.313527 0.156764 0.987636i \(-0.449894\pi\)
0.156764 + 0.987636i \(0.449894\pi\)
\(228\) −0.516093 −0.0341791
\(229\) −3.44504 −0.227654 −0.113827 0.993501i \(-0.536311\pi\)
−0.113827 + 0.993501i \(0.536311\pi\)
\(230\) 2.78492 0.183632
\(231\) 0 0
\(232\) −11.7579 −0.771945
\(233\) 17.5457 1.14946 0.574728 0.818345i \(-0.305108\pi\)
0.574728 + 0.818345i \(0.305108\pi\)
\(234\) 0.855247 0.0559092
\(235\) −6.44995 −0.420748
\(236\) −18.0436 −1.17454
\(237\) 8.17883 0.531272
\(238\) 0 0
\(239\) −22.3788 −1.44757 −0.723783 0.690027i \(-0.757598\pi\)
−0.723783 + 0.690027i \(0.757598\pi\)
\(240\) 2.35057 0.151729
\(241\) −12.6591 −0.815442 −0.407721 0.913107i \(-0.633676\pi\)
−0.407721 + 0.913107i \(0.633676\pi\)
\(242\) 0.537407 0.0345459
\(243\) 1.00000 0.0641500
\(244\) 18.7234 1.19864
\(245\) 0 0
\(246\) 6.06191 0.386493
\(247\) 0.479973 0.0305400
\(248\) −4.54969 −0.288906
\(249\) −4.70769 −0.298338
\(250\) 0.537407 0.0339886
\(251\) −9.48355 −0.598597 −0.299298 0.954160i \(-0.596753\pi\)
−0.299298 + 0.954160i \(0.596753\pi\)
\(252\) 0 0
\(253\) 5.18215 0.325799
\(254\) 4.02806 0.252743
\(255\) 0.957367 0.0599527
\(256\) −2.43027 −0.151892
\(257\) 9.41243 0.587131 0.293566 0.955939i \(-0.405158\pi\)
0.293566 + 0.955939i \(0.405158\pi\)
\(258\) −0.166236 −0.0103494
\(259\) 0 0
\(260\) −2.72325 −0.168889
\(261\) 5.89540 0.364916
\(262\) 8.80727 0.544115
\(263\) −9.97123 −0.614852 −0.307426 0.951572i \(-0.599468\pi\)
−0.307426 + 0.951572i \(0.599468\pi\)
\(264\) −1.99442 −0.122748
\(265\) −12.4653 −0.765736
\(266\) 0 0
\(267\) 15.4510 0.945586
\(268\) 8.42373 0.514562
\(269\) 13.9922 0.853122 0.426561 0.904459i \(-0.359725\pi\)
0.426561 + 0.904459i \(0.359725\pi\)
\(270\) 0.537407 0.0327056
\(271\) 17.8855 1.08646 0.543232 0.839583i \(-0.317200\pi\)
0.543232 + 0.839583i \(0.317200\pi\)
\(272\) 2.25036 0.136448
\(273\) 0 0
\(274\) 3.14260 0.189851
\(275\) 1.00000 0.0603023
\(276\) −8.86766 −0.533770
\(277\) −4.08065 −0.245183 −0.122591 0.992457i \(-0.539120\pi\)
−0.122591 + 0.992457i \(0.539120\pi\)
\(278\) 12.0226 0.721067
\(279\) 2.28121 0.136572
\(280\) 0 0
\(281\) 1.48034 0.0883094 0.0441547 0.999025i \(-0.485941\pi\)
0.0441547 + 0.999025i \(0.485941\pi\)
\(282\) −3.46625 −0.206412
\(283\) −22.3349 −1.32767 −0.663836 0.747878i \(-0.731073\pi\)
−0.663836 + 0.747878i \(0.731073\pi\)
\(284\) 28.5282 1.69284
\(285\) 0.301599 0.0178652
\(286\) 0.855247 0.0505718
\(287\) 0 0
\(288\) 5.25206 0.309481
\(289\) −16.0834 −0.946085
\(290\) 3.16823 0.186045
\(291\) 18.2513 1.06991
\(292\) 21.4543 1.25552
\(293\) −3.68648 −0.215367 −0.107683 0.994185i \(-0.534343\pi\)
−0.107683 + 0.994185i \(0.534343\pi\)
\(294\) 0 0
\(295\) 10.5444 0.613921
\(296\) 19.7610 1.14859
\(297\) 1.00000 0.0580259
\(298\) 11.8336 0.685501
\(299\) 8.24703 0.476938
\(300\) −1.71119 −0.0987958
\(301\) 0 0
\(302\) −0.157723 −0.00907596
\(303\) −10.8839 −0.625262
\(304\) 0.708928 0.0406598
\(305\) −10.9417 −0.626520
\(306\) 0.514496 0.0294118
\(307\) −2.54070 −0.145005 −0.0725027 0.997368i \(-0.523099\pi\)
−0.0725027 + 0.997368i \(0.523099\pi\)
\(308\) 0 0
\(309\) 2.06192 0.117299
\(310\) 1.22594 0.0696285
\(311\) −5.06535 −0.287229 −0.143615 0.989634i \(-0.545873\pi\)
−0.143615 + 0.989634i \(0.545873\pi\)
\(312\) −3.17399 −0.179692
\(313\) 0.628632 0.0355324 0.0177662 0.999842i \(-0.494345\pi\)
0.0177662 + 0.999842i \(0.494345\pi\)
\(314\) −2.22349 −0.125479
\(315\) 0 0
\(316\) −13.9956 −0.787312
\(317\) 0.181949 0.0102193 0.00510963 0.999987i \(-0.498374\pi\)
0.00510963 + 0.999987i \(0.498374\pi\)
\(318\) −6.69893 −0.375657
\(319\) 5.89540 0.330079
\(320\) −1.87864 −0.105019
\(321\) 7.26476 0.405479
\(322\) 0 0
\(323\) 0.288741 0.0160660
\(324\) −1.71119 −0.0950663
\(325\) 1.59143 0.0882767
\(326\) −3.32783 −0.184311
\(327\) 17.0326 0.941902
\(328\) −22.4969 −1.24218
\(329\) 0 0
\(330\) 0.537407 0.0295833
\(331\) 26.8960 1.47834 0.739170 0.673519i \(-0.235218\pi\)
0.739170 + 0.673519i \(0.235218\pi\)
\(332\) 8.05577 0.442118
\(333\) −9.90813 −0.542962
\(334\) 2.54905 0.139478
\(335\) −4.92273 −0.268957
\(336\) 0 0
\(337\) −20.1187 −1.09594 −0.547969 0.836499i \(-0.684599\pi\)
−0.547969 + 0.836499i \(0.684599\pi\)
\(338\) −5.62523 −0.305972
\(339\) 16.2281 0.881388
\(340\) −1.63824 −0.0888461
\(341\) 2.28121 0.123534
\(342\) 0.162081 0.00876435
\(343\) 0 0
\(344\) 0.616935 0.0332629
\(345\) 5.18215 0.278997
\(346\) −12.6520 −0.680175
\(347\) −2.20167 −0.118192 −0.0590958 0.998252i \(-0.518822\pi\)
−0.0590958 + 0.998252i \(0.518822\pi\)
\(348\) −10.0882 −0.540783
\(349\) −24.3330 −1.30252 −0.651258 0.758857i \(-0.725758\pi\)
−0.651258 + 0.758857i \(0.725758\pi\)
\(350\) 0 0
\(351\) 1.59143 0.0849443
\(352\) 5.25206 0.279936
\(353\) 10.0855 0.536795 0.268397 0.963308i \(-0.413506\pi\)
0.268397 + 0.963308i \(0.413506\pi\)
\(354\) 5.66666 0.301180
\(355\) −16.6715 −0.884834
\(356\) −26.4397 −1.40130
\(357\) 0 0
\(358\) 1.79580 0.0949108
\(359\) 10.2313 0.539988 0.269994 0.962862i \(-0.412978\pi\)
0.269994 + 0.962862i \(0.412978\pi\)
\(360\) −1.99442 −0.105115
\(361\) −18.9090 −0.995213
\(362\) 7.12567 0.374517
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −12.5376 −0.656250
\(366\) −5.88015 −0.307360
\(367\) −25.6154 −1.33711 −0.668555 0.743662i \(-0.733087\pi\)
−0.668555 + 0.743662i \(0.733087\pi\)
\(368\) 12.1810 0.634978
\(369\) 11.2799 0.587209
\(370\) −5.32470 −0.276818
\(371\) 0 0
\(372\) −3.90358 −0.202391
\(373\) 25.7975 1.33574 0.667871 0.744277i \(-0.267206\pi\)
0.667871 + 0.744277i \(0.267206\pi\)
\(374\) 0.514496 0.0266040
\(375\) 1.00000 0.0516398
\(376\) 12.8639 0.663406
\(377\) 9.38212 0.483204
\(378\) 0 0
\(379\) 18.8896 0.970293 0.485146 0.874433i \(-0.338766\pi\)
0.485146 + 0.874433i \(0.338766\pi\)
\(380\) −0.516093 −0.0264750
\(381\) 7.49535 0.383999
\(382\) 11.6342 0.595257
\(383\) −28.7713 −1.47014 −0.735072 0.677990i \(-0.762851\pi\)
−0.735072 + 0.677990i \(0.762851\pi\)
\(384\) −11.5137 −0.587557
\(385\) 0 0
\(386\) 10.5033 0.534602
\(387\) −0.309330 −0.0157241
\(388\) −31.2316 −1.58554
\(389\) 15.1536 0.768318 0.384159 0.923267i \(-0.374491\pi\)
0.384159 + 0.923267i \(0.374491\pi\)
\(390\) 0.855247 0.0433071
\(391\) 4.96122 0.250900
\(392\) 0 0
\(393\) 16.3884 0.826687
\(394\) 7.52921 0.379316
\(395\) 8.17883 0.411522
\(396\) −1.71119 −0.0859907
\(397\) 2.35650 0.118269 0.0591346 0.998250i \(-0.481166\pi\)
0.0591346 + 0.998250i \(0.481166\pi\)
\(398\) −4.83936 −0.242575
\(399\) 0 0
\(400\) 2.35057 0.117528
\(401\) 15.4658 0.772325 0.386163 0.922431i \(-0.373800\pi\)
0.386163 + 0.922431i \(0.373800\pi\)
\(402\) −2.64551 −0.131946
\(403\) 3.63038 0.180842
\(404\) 18.6244 0.926599
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −9.90813 −0.491128
\(408\) −1.90940 −0.0945292
\(409\) −5.06959 −0.250675 −0.125338 0.992114i \(-0.540001\pi\)
−0.125338 + 0.992114i \(0.540001\pi\)
\(410\) 6.06191 0.299376
\(411\) 5.84770 0.288446
\(412\) −3.52835 −0.173829
\(413\) 0 0
\(414\) 2.78492 0.136872
\(415\) −4.70769 −0.231091
\(416\) 8.35829 0.409799
\(417\) 22.3715 1.09554
\(418\) 0.162081 0.00792765
\(419\) 39.1469 1.91245 0.956226 0.292629i \(-0.0945302\pi\)
0.956226 + 0.292629i \(0.0945302\pi\)
\(420\) 0 0
\(421\) 15.0025 0.731177 0.365589 0.930777i \(-0.380868\pi\)
0.365589 + 0.930777i \(0.380868\pi\)
\(422\) 4.51785 0.219926
\(423\) −6.44995 −0.313607
\(424\) 24.8610 1.20736
\(425\) 0.957367 0.0464391
\(426\) −8.95941 −0.434085
\(427\) 0 0
\(428\) −12.4314 −0.600895
\(429\) 1.59143 0.0768350
\(430\) −0.166236 −0.00801663
\(431\) −17.0025 −0.818983 −0.409492 0.912314i \(-0.634294\pi\)
−0.409492 + 0.912314i \(0.634294\pi\)
\(432\) 2.35057 0.113092
\(433\) 20.2420 0.972768 0.486384 0.873745i \(-0.338316\pi\)
0.486384 + 0.873745i \(0.338316\pi\)
\(434\) 0 0
\(435\) 5.89540 0.282663
\(436\) −29.1460 −1.39584
\(437\) 1.56293 0.0747650
\(438\) −6.73782 −0.321946
\(439\) −9.22630 −0.440347 −0.220174 0.975461i \(-0.570662\pi\)
−0.220174 + 0.975461i \(0.570662\pi\)
\(440\) −1.99442 −0.0950804
\(441\) 0 0
\(442\) 0.818786 0.0389457
\(443\) 34.2426 1.62692 0.813458 0.581624i \(-0.197582\pi\)
0.813458 + 0.581624i \(0.197582\pi\)
\(444\) 16.9547 0.804636
\(445\) 15.4510 0.732448
\(446\) −2.12928 −0.100824
\(447\) 22.0198 1.04150
\(448\) 0 0
\(449\) −3.40604 −0.160741 −0.0803704 0.996765i \(-0.525610\pi\)
−0.0803704 + 0.996765i \(0.525610\pi\)
\(450\) 0.537407 0.0253336
\(451\) 11.2799 0.531150
\(452\) −27.7694 −1.30616
\(453\) −0.293489 −0.0137893
\(454\) 2.53859 0.119142
\(455\) 0 0
\(456\) −0.601515 −0.0281685
\(457\) −19.8711 −0.929529 −0.464764 0.885434i \(-0.653861\pi\)
−0.464764 + 0.885434i \(0.653861\pi\)
\(458\) −1.85139 −0.0865097
\(459\) 0.957367 0.0446861
\(460\) −8.86766 −0.413457
\(461\) −16.1882 −0.753961 −0.376981 0.926221i \(-0.623038\pi\)
−0.376981 + 0.926221i \(0.623038\pi\)
\(462\) 0 0
\(463\) 37.5479 1.74500 0.872499 0.488616i \(-0.162498\pi\)
0.872499 + 0.488616i \(0.162498\pi\)
\(464\) 13.8575 0.643321
\(465\) 2.28121 0.105788
\(466\) 9.42918 0.436798
\(467\) 16.6119 0.768709 0.384355 0.923186i \(-0.374424\pi\)
0.384355 + 0.923186i \(0.374424\pi\)
\(468\) −2.72325 −0.125882
\(469\) 0 0
\(470\) −3.46625 −0.159886
\(471\) −4.13745 −0.190644
\(472\) −21.0301 −0.967987
\(473\) −0.309330 −0.0142230
\(474\) 4.39536 0.201886
\(475\) 0.301599 0.0138383
\(476\) 0 0
\(477\) −12.4653 −0.570746
\(478\) −12.0265 −0.550082
\(479\) −27.8995 −1.27476 −0.637381 0.770549i \(-0.719982\pi\)
−0.637381 + 0.770549i \(0.719982\pi\)
\(480\) 5.25206 0.239723
\(481\) −15.7681 −0.718964
\(482\) −6.80308 −0.309872
\(483\) 0 0
\(484\) −1.71119 −0.0777815
\(485\) 18.2513 0.828751
\(486\) 0.537407 0.0243773
\(487\) 0.319844 0.0144935 0.00724675 0.999974i \(-0.497693\pi\)
0.00724675 + 0.999974i \(0.497693\pi\)
\(488\) 21.8224 0.987852
\(489\) −6.19238 −0.280029
\(490\) 0 0
\(491\) 24.6414 1.11205 0.556025 0.831166i \(-0.312326\pi\)
0.556025 + 0.831166i \(0.312326\pi\)
\(492\) −19.3021 −0.870206
\(493\) 5.64406 0.254196
\(494\) 0.257941 0.0116053
\(495\) 1.00000 0.0449467
\(496\) 5.36213 0.240767
\(497\) 0 0
\(498\) −2.52995 −0.113370
\(499\) 18.9623 0.848870 0.424435 0.905458i \(-0.360473\pi\)
0.424435 + 0.905458i \(0.360473\pi\)
\(500\) −1.71119 −0.0765269
\(501\) 4.74324 0.211912
\(502\) −5.09653 −0.227469
\(503\) −20.5865 −0.917906 −0.458953 0.888461i \(-0.651775\pi\)
−0.458953 + 0.888461i \(0.651775\pi\)
\(504\) 0 0
\(505\) −10.8839 −0.484326
\(506\) 2.78492 0.123805
\(507\) −10.4673 −0.464871
\(508\) −12.8260 −0.569062
\(509\) −24.6176 −1.09115 −0.545577 0.838061i \(-0.683689\pi\)
−0.545577 + 0.838061i \(0.683689\pi\)
\(510\) 0.514496 0.0227823
\(511\) 0 0
\(512\) 21.7214 0.959959
\(513\) 0.301599 0.0133159
\(514\) 5.05831 0.223112
\(515\) 2.06192 0.0908592
\(516\) 0.529324 0.0233022
\(517\) −6.44995 −0.283668
\(518\) 0 0
\(519\) −23.5426 −1.03341
\(520\) −3.17399 −0.139189
\(521\) −24.1063 −1.05612 −0.528058 0.849209i \(-0.677079\pi\)
−0.528058 + 0.849209i \(0.677079\pi\)
\(522\) 3.16823 0.138670
\(523\) −37.5949 −1.64391 −0.821955 0.569553i \(-0.807116\pi\)
−0.821955 + 0.569553i \(0.807116\pi\)
\(524\) −28.0438 −1.22510
\(525\) 0 0
\(526\) −5.35861 −0.233647
\(527\) 2.18395 0.0951345
\(528\) 2.35057 0.102295
\(529\) 3.85465 0.167594
\(530\) −6.69893 −0.290983
\(531\) 10.5444 0.457590
\(532\) 0 0
\(533\) 17.9512 0.777553
\(534\) 8.30348 0.359327
\(535\) 7.26476 0.314083
\(536\) 9.81799 0.424073
\(537\) 3.34159 0.144200
\(538\) 7.51953 0.324190
\(539\) 0 0
\(540\) −1.71119 −0.0736380
\(541\) −14.3326 −0.616206 −0.308103 0.951353i \(-0.599694\pi\)
−0.308103 + 0.951353i \(0.599694\pi\)
\(542\) 9.61177 0.412861
\(543\) 13.2594 0.569013
\(544\) 5.02815 0.215580
\(545\) 17.0326 0.729595
\(546\) 0 0
\(547\) 37.3357 1.59636 0.798179 0.602421i \(-0.205797\pi\)
0.798179 + 0.602421i \(0.205797\pi\)
\(548\) −10.0065 −0.427458
\(549\) −10.9417 −0.466980
\(550\) 0.537407 0.0229151
\(551\) 1.77804 0.0757472
\(552\) −10.3354 −0.439903
\(553\) 0 0
\(554\) −2.19297 −0.0931705
\(555\) −9.90813 −0.420577
\(556\) −38.2819 −1.62351
\(557\) 1.72433 0.0730622 0.0365311 0.999333i \(-0.488369\pi\)
0.0365311 + 0.999333i \(0.488369\pi\)
\(558\) 1.22594 0.0518980
\(559\) −0.492278 −0.0208211
\(560\) 0 0
\(561\) 0.957367 0.0404201
\(562\) 0.795543 0.0335580
\(563\) −27.1865 −1.14577 −0.572887 0.819634i \(-0.694177\pi\)
−0.572887 + 0.819634i \(0.694177\pi\)
\(564\) 11.0371 0.464746
\(565\) 16.2281 0.682720
\(566\) −12.0029 −0.504521
\(567\) 0 0
\(568\) 33.2501 1.39514
\(569\) 10.1029 0.423534 0.211767 0.977320i \(-0.432078\pi\)
0.211767 + 0.977320i \(0.432078\pi\)
\(570\) 0.162081 0.00678884
\(571\) −37.7805 −1.58106 −0.790532 0.612421i \(-0.790196\pi\)
−0.790532 + 0.612421i \(0.790196\pi\)
\(572\) −2.72325 −0.113865
\(573\) 21.6487 0.904388
\(574\) 0 0
\(575\) 5.18215 0.216110
\(576\) −1.87864 −0.0782768
\(577\) 35.9847 1.49806 0.749031 0.662535i \(-0.230519\pi\)
0.749031 + 0.662535i \(0.230519\pi\)
\(578\) −8.64336 −0.359516
\(579\) 19.5443 0.812234
\(580\) −10.0882 −0.418888
\(581\) 0 0
\(582\) 9.80840 0.406571
\(583\) −12.4653 −0.516259
\(584\) 25.0054 1.03473
\(585\) 1.59143 0.0657976
\(586\) −1.98114 −0.0818403
\(587\) −29.1862 −1.20464 −0.602322 0.798253i \(-0.705758\pi\)
−0.602322 + 0.798253i \(0.705758\pi\)
\(588\) 0 0
\(589\) 0.688009 0.0283489
\(590\) 5.66666 0.233293
\(591\) 14.0102 0.576305
\(592\) −23.2898 −0.957203
\(593\) −20.2522 −0.831660 −0.415830 0.909442i \(-0.636509\pi\)
−0.415830 + 0.909442i \(0.636509\pi\)
\(594\) 0.537407 0.0220501
\(595\) 0 0
\(596\) −37.6801 −1.54344
\(597\) −9.00502 −0.368551
\(598\) 4.43202 0.181239
\(599\) 0.363495 0.0148520 0.00742600 0.999972i \(-0.497636\pi\)
0.00742600 + 0.999972i \(0.497636\pi\)
\(600\) −1.99442 −0.0814220
\(601\) −31.8648 −1.29979 −0.649896 0.760023i \(-0.725187\pi\)
−0.649896 + 0.760023i \(0.725187\pi\)
\(602\) 0 0
\(603\) −4.92273 −0.200469
\(604\) 0.502217 0.0204349
\(605\) 1.00000 0.0406558
\(606\) −5.84907 −0.237602
\(607\) 39.4818 1.60252 0.801258 0.598318i \(-0.204164\pi\)
0.801258 + 0.598318i \(0.204164\pi\)
\(608\) 1.58401 0.0642402
\(609\) 0 0
\(610\) −5.88015 −0.238080
\(611\) −10.2647 −0.415263
\(612\) −1.63824 −0.0662220
\(613\) −38.1007 −1.53887 −0.769436 0.638724i \(-0.779463\pi\)
−0.769436 + 0.638724i \(0.779463\pi\)
\(614\) −1.36539 −0.0551027
\(615\) 11.2799 0.454850
\(616\) 0 0
\(617\) 22.7915 0.917552 0.458776 0.888552i \(-0.348288\pi\)
0.458776 + 0.888552i \(0.348288\pi\)
\(618\) 1.10809 0.0445740
\(619\) 40.4851 1.62723 0.813617 0.581401i \(-0.197495\pi\)
0.813617 + 0.581401i \(0.197495\pi\)
\(620\) −3.90358 −0.156772
\(621\) 5.18215 0.207952
\(622\) −2.72215 −0.109148
\(623\) 0 0
\(624\) 3.74077 0.149751
\(625\) 1.00000 0.0400000
\(626\) 0.337832 0.0135025
\(627\) 0.301599 0.0120447
\(628\) 7.07997 0.282522
\(629\) −9.48573 −0.378221
\(630\) 0 0
\(631\) −35.2789 −1.40443 −0.702216 0.711964i \(-0.747806\pi\)
−0.702216 + 0.711964i \(0.747806\pi\)
\(632\) −16.3120 −0.648858
\(633\) 8.40675 0.334138
\(634\) 0.0977805 0.00388336
\(635\) 7.49535 0.297444
\(636\) 21.3305 0.845809
\(637\) 0 0
\(638\) 3.16823 0.125431
\(639\) −16.6715 −0.659516
\(640\) −11.5137 −0.455119
\(641\) 8.37635 0.330846 0.165423 0.986223i \(-0.447101\pi\)
0.165423 + 0.986223i \(0.447101\pi\)
\(642\) 3.90414 0.154084
\(643\) −6.75016 −0.266200 −0.133100 0.991103i \(-0.542493\pi\)
−0.133100 + 0.991103i \(0.542493\pi\)
\(644\) 0 0
\(645\) −0.309330 −0.0121799
\(646\) 0.155171 0.00610514
\(647\) −18.3045 −0.719625 −0.359812 0.933025i \(-0.617159\pi\)
−0.359812 + 0.933025i \(0.617159\pi\)
\(648\) −1.99442 −0.0783483
\(649\) 10.5444 0.413905
\(650\) 0.855247 0.0335455
\(651\) 0 0
\(652\) 10.5964 0.414985
\(653\) −34.9119 −1.36621 −0.683104 0.730321i \(-0.739370\pi\)
−0.683104 + 0.730321i \(0.739370\pi\)
\(654\) 9.15342 0.357927
\(655\) 16.3884 0.640349
\(656\) 26.5142 1.03521
\(657\) −12.5376 −0.489140
\(658\) 0 0
\(659\) −22.5509 −0.878456 −0.439228 0.898376i \(-0.644748\pi\)
−0.439228 + 0.898376i \(0.644748\pi\)
\(660\) −1.71119 −0.0666081
\(661\) −11.6028 −0.451297 −0.225649 0.974209i \(-0.572450\pi\)
−0.225649 + 0.974209i \(0.572450\pi\)
\(662\) 14.4541 0.561776
\(663\) 1.52358 0.0591711
\(664\) 9.38912 0.364369
\(665\) 0 0
\(666\) −5.32470 −0.206328
\(667\) 30.5508 1.18293
\(668\) −8.11660 −0.314041
\(669\) −3.96213 −0.153185
\(670\) −2.64551 −0.102205
\(671\) −10.9417 −0.422400
\(672\) 0 0
\(673\) −9.60158 −0.370114 −0.185057 0.982728i \(-0.559247\pi\)
−0.185057 + 0.982728i \(0.559247\pi\)
\(674\) −10.8120 −0.416461
\(675\) 1.00000 0.0384900
\(676\) 17.9117 0.688910
\(677\) 3.45175 0.132662 0.0663308 0.997798i \(-0.478871\pi\)
0.0663308 + 0.997798i \(0.478871\pi\)
\(678\) 8.72109 0.334931
\(679\) 0 0
\(680\) −1.90940 −0.0732220
\(681\) 4.72377 0.181015
\(682\) 1.22594 0.0469436
\(683\) −4.50424 −0.172350 −0.0861750 0.996280i \(-0.527464\pi\)
−0.0861750 + 0.996280i \(0.527464\pi\)
\(684\) −0.516093 −0.0197333
\(685\) 5.84770 0.223429
\(686\) 0 0
\(687\) −3.44504 −0.131436
\(688\) −0.727102 −0.0277205
\(689\) −19.8376 −0.755753
\(690\) 2.78492 0.106020
\(691\) 16.0030 0.608781 0.304391 0.952547i \(-0.401547\pi\)
0.304391 + 0.952547i \(0.401547\pi\)
\(692\) 40.2860 1.53144
\(693\) 0 0
\(694\) −1.18319 −0.0449133
\(695\) 22.3715 0.848598
\(696\) −11.7579 −0.445683
\(697\) 10.7990 0.409042
\(698\) −13.0767 −0.494962
\(699\) 17.5457 0.663639
\(700\) 0 0
\(701\) −27.0747 −1.02260 −0.511299 0.859403i \(-0.670835\pi\)
−0.511299 + 0.859403i \(0.670835\pi\)
\(702\) 0.855247 0.0322792
\(703\) −2.98828 −0.112705
\(704\) −1.87864 −0.0708041
\(705\) −6.44995 −0.242919
\(706\) 5.42000 0.203984
\(707\) 0 0
\(708\) −18.0436 −0.678119
\(709\) 4.02201 0.151050 0.0755248 0.997144i \(-0.475937\pi\)
0.0755248 + 0.997144i \(0.475937\pi\)
\(710\) −8.95941 −0.336241
\(711\) 8.17883 0.306730
\(712\) −30.8158 −1.15487
\(713\) 11.8215 0.442720
\(714\) 0 0
\(715\) 1.59143 0.0595162
\(716\) −5.71811 −0.213696
\(717\) −22.3788 −0.835753
\(718\) 5.49838 0.205198
\(719\) −29.1357 −1.08658 −0.543289 0.839546i \(-0.682821\pi\)
−0.543289 + 0.839546i \(0.682821\pi\)
\(720\) 2.35057 0.0876006
\(721\) 0 0
\(722\) −10.1619 −0.378185
\(723\) −12.6591 −0.470796
\(724\) −22.6893 −0.843242
\(725\) 5.89540 0.218950
\(726\) 0.537407 0.0199451
\(727\) 13.5927 0.504127 0.252064 0.967711i \(-0.418891\pi\)
0.252064 + 0.967711i \(0.418891\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.73782 −0.249378
\(731\) −0.296143 −0.0109532
\(732\) 18.7234 0.692035
\(733\) 42.9606 1.58679 0.793393 0.608710i \(-0.208313\pi\)
0.793393 + 0.608710i \(0.208313\pi\)
\(734\) −13.7659 −0.508108
\(735\) 0 0
\(736\) 27.2169 1.00323
\(737\) −4.92273 −0.181331
\(738\) 6.06191 0.223142
\(739\) 34.1334 1.25562 0.627809 0.778367i \(-0.283952\pi\)
0.627809 + 0.778367i \(0.283952\pi\)
\(740\) 16.9547 0.623268
\(741\) 0.479973 0.0176323
\(742\) 0 0
\(743\) 19.1862 0.703874 0.351937 0.936024i \(-0.385523\pi\)
0.351937 + 0.936024i \(0.385523\pi\)
\(744\) −4.54969 −0.166800
\(745\) 22.0198 0.806742
\(746\) 13.8638 0.507588
\(747\) −4.70769 −0.172245
\(748\) −1.63824 −0.0599000
\(749\) 0 0
\(750\) 0.537407 0.0196233
\(751\) 26.8910 0.981267 0.490633 0.871366i \(-0.336765\pi\)
0.490633 + 0.871366i \(0.336765\pi\)
\(752\) −15.1611 −0.552867
\(753\) −9.48355 −0.345600
\(754\) 5.04202 0.183620
\(755\) −0.293489 −0.0106812
\(756\) 0 0
\(757\) 36.8100 1.33788 0.668941 0.743316i \(-0.266748\pi\)
0.668941 + 0.743316i \(0.266748\pi\)
\(758\) 10.1514 0.368715
\(759\) 5.18215 0.188100
\(760\) −0.601515 −0.0218192
\(761\) −7.34622 −0.266300 −0.133150 0.991096i \(-0.542509\pi\)
−0.133150 + 0.991096i \(0.542509\pi\)
\(762\) 4.02806 0.145921
\(763\) 0 0
\(764\) −37.0451 −1.34025
\(765\) 0.957367 0.0346137
\(766\) −15.4619 −0.558661
\(767\) 16.7808 0.605918
\(768\) −2.43027 −0.0876947
\(769\) 33.4817 1.20738 0.603690 0.797219i \(-0.293697\pi\)
0.603690 + 0.797219i \(0.293697\pi\)
\(770\) 0 0
\(771\) 9.41243 0.338980
\(772\) −33.4441 −1.20368
\(773\) −17.1652 −0.617389 −0.308694 0.951161i \(-0.599892\pi\)
−0.308694 + 0.951161i \(0.599892\pi\)
\(774\) −0.166236 −0.00597524
\(775\) 2.28121 0.0819433
\(776\) −36.4009 −1.30672
\(777\) 0 0
\(778\) 8.14366 0.291964
\(779\) 3.40200 0.121889
\(780\) −2.72325 −0.0975079
\(781\) −16.6715 −0.596555
\(782\) 2.66620 0.0953430
\(783\) 5.89540 0.210684
\(784\) 0 0
\(785\) −4.13745 −0.147672
\(786\) 8.80727 0.314145
\(787\) −11.5253 −0.410834 −0.205417 0.978675i \(-0.565855\pi\)
−0.205417 + 0.978675i \(0.565855\pi\)
\(788\) −23.9742 −0.854047
\(789\) −9.97123 −0.354985
\(790\) 4.39536 0.156380
\(791\) 0 0
\(792\) −1.99442 −0.0708687
\(793\) −17.4130 −0.618352
\(794\) 1.26640 0.0449428
\(795\) −12.4653 −0.442098
\(796\) 15.4093 0.546169
\(797\) 27.8120 0.985152 0.492576 0.870269i \(-0.336055\pi\)
0.492576 + 0.870269i \(0.336055\pi\)
\(798\) 0 0
\(799\) −6.17497 −0.218455
\(800\) 5.25206 0.185688
\(801\) 15.4510 0.545934
\(802\) 8.31144 0.293487
\(803\) −12.5376 −0.442444
\(804\) 8.42373 0.297082
\(805\) 0 0
\(806\) 1.95099 0.0687208
\(807\) 13.9922 0.492550
\(808\) 21.7070 0.763651
\(809\) −9.79928 −0.344524 −0.172262 0.985051i \(-0.555108\pi\)
−0.172262 + 0.985051i \(0.555108\pi\)
\(810\) 0.537407 0.0188826
\(811\) −28.0466 −0.984849 −0.492424 0.870355i \(-0.663889\pi\)
−0.492424 + 0.870355i \(0.663889\pi\)
\(812\) 0 0
\(813\) 17.8855 0.627270
\(814\) −5.32470 −0.186631
\(815\) −6.19238 −0.216909
\(816\) 2.25036 0.0787783
\(817\) −0.0932935 −0.00326393
\(818\) −2.72444 −0.0952577
\(819\) 0 0
\(820\) −19.3021 −0.674059
\(821\) 2.89729 0.101116 0.0505581 0.998721i \(-0.483900\pi\)
0.0505581 + 0.998721i \(0.483900\pi\)
\(822\) 3.14260 0.109611
\(823\) −5.71777 −0.199309 −0.0996544 0.995022i \(-0.531774\pi\)
−0.0996544 + 0.995022i \(0.531774\pi\)
\(824\) −4.11235 −0.143260
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) 14.6027 0.507786 0.253893 0.967232i \(-0.418289\pi\)
0.253893 + 0.967232i \(0.418289\pi\)
\(828\) −8.86766 −0.308172
\(829\) −18.5862 −0.645524 −0.322762 0.946480i \(-0.604611\pi\)
−0.322762 + 0.946480i \(0.604611\pi\)
\(830\) −2.52995 −0.0878157
\(831\) −4.08065 −0.141556
\(832\) −2.98973 −0.103650
\(833\) 0 0
\(834\) 12.0226 0.416308
\(835\) 4.74324 0.164147
\(836\) −0.516093 −0.0178495
\(837\) 2.28121 0.0788500
\(838\) 21.0378 0.726740
\(839\) 7.03801 0.242979 0.121490 0.992593i \(-0.461233\pi\)
0.121490 + 0.992593i \(0.461233\pi\)
\(840\) 0 0
\(841\) 5.75575 0.198474
\(842\) 8.06246 0.277851
\(843\) 1.48034 0.0509855
\(844\) −14.3856 −0.495172
\(845\) −10.4673 −0.360088
\(846\) −3.46625 −0.119172
\(847\) 0 0
\(848\) −29.3005 −1.00618
\(849\) −22.3349 −0.766532
\(850\) 0.514496 0.0176471
\(851\) −51.3454 −1.76010
\(852\) 28.5282 0.977361
\(853\) 9.35268 0.320230 0.160115 0.987098i \(-0.448814\pi\)
0.160115 + 0.987098i \(0.448814\pi\)
\(854\) 0 0
\(855\) 0.301599 0.0103145
\(856\) −14.4890 −0.495224
\(857\) −50.5947 −1.72828 −0.864141 0.503250i \(-0.832137\pi\)
−0.864141 + 0.503250i \(0.832137\pi\)
\(858\) 0.855247 0.0291976
\(859\) −10.8366 −0.369740 −0.184870 0.982763i \(-0.559186\pi\)
−0.184870 + 0.982763i \(0.559186\pi\)
\(860\) 0.529324 0.0180498
\(861\) 0 0
\(862\) −9.13728 −0.311217
\(863\) 34.7467 1.18279 0.591396 0.806381i \(-0.298577\pi\)
0.591396 + 0.806381i \(0.298577\pi\)
\(864\) 5.25206 0.178679
\(865\) −23.5426 −0.800473
\(866\) 10.8782 0.369656
\(867\) −16.0834 −0.546223
\(868\) 0 0
\(869\) 8.17883 0.277448
\(870\) 3.16823 0.107413
\(871\) −7.83418 −0.265451
\(872\) −33.9701 −1.15037
\(873\) 18.2513 0.617714
\(874\) 0.839929 0.0284110
\(875\) 0 0
\(876\) 21.4543 0.724875
\(877\) −40.3760 −1.36340 −0.681701 0.731631i \(-0.738759\pi\)
−0.681701 + 0.731631i \(0.738759\pi\)
\(878\) −4.95828 −0.167334
\(879\) −3.68648 −0.124342
\(880\) 2.35057 0.0792377
\(881\) −45.4623 −1.53166 −0.765832 0.643040i \(-0.777673\pi\)
−0.765832 + 0.643040i \(0.777673\pi\)
\(882\) 0 0
\(883\) 32.6461 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(884\) −2.60715 −0.0876879
\(885\) 10.5444 0.354447
\(886\) 18.4022 0.618235
\(887\) 37.1818 1.24844 0.624222 0.781247i \(-0.285416\pi\)
0.624222 + 0.781247i \(0.285416\pi\)
\(888\) 19.7610 0.663136
\(889\) 0 0
\(890\) 8.30348 0.278333
\(891\) 1.00000 0.0335013
\(892\) 6.77997 0.227010
\(893\) −1.94530 −0.0650968
\(894\) 11.8336 0.395774
\(895\) 3.34159 0.111697
\(896\) 0 0
\(897\) 8.24703 0.275360
\(898\) −1.83043 −0.0610822
\(899\) 13.4486 0.448537
\(900\) −1.71119 −0.0570398
\(901\) −11.9338 −0.397574
\(902\) 6.06191 0.201839
\(903\) 0 0
\(904\) −32.3656 −1.07647
\(905\) 13.2594 0.440756
\(906\) −0.157723 −0.00524001
\(907\) −19.8449 −0.658938 −0.329469 0.944166i \(-0.606870\pi\)
−0.329469 + 0.944166i \(0.606870\pi\)
\(908\) −8.08328 −0.268253
\(909\) −10.8839 −0.360995
\(910\) 0 0
\(911\) 11.6268 0.385214 0.192607 0.981276i \(-0.438306\pi\)
0.192607 + 0.981276i \(0.438306\pi\)
\(912\) 0.708928 0.0234750
\(913\) −4.70769 −0.155802
\(914\) −10.6789 −0.353225
\(915\) −10.9417 −0.361721
\(916\) 5.89512 0.194780
\(917\) 0 0
\(918\) 0.514496 0.0169809
\(919\) 10.0408 0.331214 0.165607 0.986192i \(-0.447042\pi\)
0.165607 + 0.986192i \(0.447042\pi\)
\(920\) −10.3354 −0.340748
\(921\) −2.54070 −0.0837189
\(922\) −8.69967 −0.286509
\(923\) −26.5316 −0.873299
\(924\) 0 0
\(925\) −9.90813 −0.325777
\(926\) 20.1785 0.663107
\(927\) 2.06192 0.0677225
\(928\) 30.9630 1.01641
\(929\) −26.9895 −0.885496 −0.442748 0.896646i \(-0.645996\pi\)
−0.442748 + 0.896646i \(0.645996\pi\)
\(930\) 1.22594 0.0402001
\(931\) 0 0
\(932\) −30.0241 −0.983471
\(933\) −5.06535 −0.165832
\(934\) 8.92738 0.292113
\(935\) 0.957367 0.0313093
\(936\) −3.17399 −0.103745
\(937\) 34.3873 1.12338 0.561691 0.827347i \(-0.310151\pi\)
0.561691 + 0.827347i \(0.310151\pi\)
\(938\) 0 0
\(939\) 0.628632 0.0205146
\(940\) 11.0371 0.359991
\(941\) 30.4791 0.993589 0.496794 0.867868i \(-0.334510\pi\)
0.496794 + 0.867868i \(0.334510\pi\)
\(942\) −2.22349 −0.0724454
\(943\) 58.4541 1.90353
\(944\) 24.7854 0.806697
\(945\) 0 0
\(946\) −0.166236 −0.00540481
\(947\) 60.0385 1.95099 0.975495 0.220023i \(-0.0706133\pi\)
0.975495 + 0.220023i \(0.0706133\pi\)
\(948\) −13.9956 −0.454555
\(949\) −19.9528 −0.647695
\(950\) 0.162081 0.00525861
\(951\) 0.181949 0.00590009
\(952\) 0 0
\(953\) −11.7675 −0.381188 −0.190594 0.981669i \(-0.561041\pi\)
−0.190594 + 0.981669i \(0.561041\pi\)
\(954\) −6.69893 −0.216886
\(955\) 21.6487 0.700536
\(956\) 38.2945 1.23853
\(957\) 5.89540 0.190571
\(958\) −14.9934 −0.484415
\(959\) 0 0
\(960\) −1.87864 −0.0606330
\(961\) −25.7961 −0.832132
\(962\) −8.47390 −0.273210
\(963\) 7.26476 0.234104
\(964\) 21.6621 0.697690
\(965\) 19.5443 0.629154
\(966\) 0 0
\(967\) −15.6712 −0.503953 −0.251976 0.967733i \(-0.581081\pi\)
−0.251976 + 0.967733i \(0.581081\pi\)
\(968\) −1.99442 −0.0641032
\(969\) 0.288741 0.00927569
\(970\) 9.80840 0.314929
\(971\) −39.0248 −1.25237 −0.626183 0.779676i \(-0.715384\pi\)
−0.626183 + 0.779676i \(0.715384\pi\)
\(972\) −1.71119 −0.0548866
\(973\) 0 0
\(974\) 0.171886 0.00550759
\(975\) 1.59143 0.0509666
\(976\) −25.7192 −0.823252
\(977\) −52.9257 −1.69324 −0.846621 0.532196i \(-0.821367\pi\)
−0.846621 + 0.532196i \(0.821367\pi\)
\(978\) −3.32783 −0.106412
\(979\) 15.4510 0.493816
\(980\) 0 0
\(981\) 17.0326 0.543808
\(982\) 13.2425 0.422584
\(983\) 7.76273 0.247592 0.123796 0.992308i \(-0.460493\pi\)
0.123796 + 0.992308i \(0.460493\pi\)
\(984\) −22.4969 −0.717175
\(985\) 14.0102 0.446404
\(986\) 3.03316 0.0965955
\(987\) 0 0
\(988\) −0.821327 −0.0261299
\(989\) −1.60299 −0.0509723
\(990\) 0.537407 0.0170799
\(991\) 13.6086 0.432290 0.216145 0.976361i \(-0.430652\pi\)
0.216145 + 0.976361i \(0.430652\pi\)
\(992\) 11.9810 0.380398
\(993\) 26.8960 0.853520
\(994\) 0 0
\(995\) −9.00502 −0.285478
\(996\) 8.05577 0.255257
\(997\) 46.0890 1.45965 0.729827 0.683632i \(-0.239601\pi\)
0.729827 + 0.683632i \(0.239601\pi\)
\(998\) 10.1905 0.322574
\(999\) −9.90813 −0.313480
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 8085.2.a.cp.1.8 yes 14
7.6 odd 2 8085.2.a.co.1.8 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8085.2.a.co.1.8 14 7.6 odd 2
8085.2.a.cp.1.8 yes 14 1.1 even 1 trivial