Properties

Label 4235.2.a.bc.1.4
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $5$
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] = [4235,2,Mod(1,4235)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4235.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.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: \(33.8166452560\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.288385.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 7x^{3} - 2x^{2} + 10x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.58597\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.58597 q^{2} -3.01851 q^{3} +0.515286 q^{4} +1.00000 q^{5} -4.78725 q^{6} -1.00000 q^{7} -2.35471 q^{8} +6.11138 q^{9} +O(q^{10})\) \(q+1.58597 q^{2} -3.01851 q^{3} +0.515286 q^{4} +1.00000 q^{5} -4.78725 q^{6} -1.00000 q^{7} -2.35471 q^{8} +6.11138 q^{9} +1.58597 q^{10} -1.55539 q^{12} -3.17071 q^{13} -1.58597 q^{14} -3.01851 q^{15} -4.76505 q^{16} +6.29047 q^{17} +9.69244 q^{18} -1.15342 q^{19} +0.515286 q^{20} +3.01851 q^{21} +5.96633 q^{23} +7.10769 q^{24} +1.00000 q^{25} -5.02863 q^{26} -9.39172 q^{27} -0.515286 q^{28} -8.32104 q^{29} -4.78725 q^{30} +8.69244 q^{31} -2.84780 q^{32} +9.97646 q^{34} -1.00000 q^{35} +3.14911 q^{36} +3.37140 q^{37} -1.82929 q^{38} +9.57080 q^{39} -2.35471 q^{40} -6.78009 q^{41} +4.78725 q^{42} -5.11794 q^{43} +6.11138 q^{45} +9.46240 q^{46} +10.3868 q^{47} +14.3833 q^{48} +1.00000 q^{49} +1.58597 q^{50} -18.9878 q^{51} -1.63382 q^{52} -8.18922 q^{53} -14.8949 q^{54} +2.35471 q^{56} +3.48162 q^{57} -13.1969 q^{58} +7.10422 q^{59} -1.55539 q^{60} -7.27196 q^{61} +13.7859 q^{62} -6.11138 q^{63} +5.01360 q^{64} -3.17071 q^{65} -8.71238 q^{67} +3.24139 q^{68} -18.0094 q^{69} -1.58597 q^{70} -12.5319 q^{71} -14.3905 q^{72} -4.48594 q^{73} +5.34692 q^{74} -3.01851 q^{75} -0.594343 q^{76} +15.1790 q^{78} +3.30007 q^{79} -4.76505 q^{80} +10.0148 q^{81} -10.7530 q^{82} +7.14492 q^{83} +1.55539 q^{84} +6.29047 q^{85} -8.11688 q^{86} +25.1171 q^{87} +0.654265 q^{89} +9.69244 q^{90} +3.17071 q^{91} +3.07437 q^{92} -26.2382 q^{93} +16.4731 q^{94} -1.15342 q^{95} +8.59609 q^{96} +8.73507 q^{97} +1.58597 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{3} + 4 q^{4} + 5 q^{5} - 9 q^{6} - 5 q^{7} - 6 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{3} + 4 q^{4} + 5 q^{5} - 9 q^{6} - 5 q^{7} - 6 q^{8} + 7 q^{9} + 3 q^{12} - 6 q^{13} + 2 q^{15} - 6 q^{16} - 2 q^{17} + 3 q^{18} - 7 q^{19} + 4 q^{20} - 2 q^{21} + 5 q^{23} - 8 q^{24} + 5 q^{25} + 9 q^{26} - 7 q^{27} - 4 q^{28} - 11 q^{29} - 9 q^{30} - 2 q^{31} - 7 q^{32} + 8 q^{34} - 5 q^{35} + q^{36} + 2 q^{37} - 19 q^{38} - 2 q^{39} - 6 q^{40} - 13 q^{41} + 9 q^{42} - 10 q^{43} + 7 q^{45} - 2 q^{46} - 2 q^{47} + 32 q^{48} + 5 q^{49} - 30 q^{51} + 8 q^{52} - 14 q^{53} - 16 q^{54} + 6 q^{56} - 6 q^{57} + 4 q^{58} + 6 q^{59} + 3 q^{60} - 20 q^{61} + 15 q^{62} - 7 q^{63} - 6 q^{64} - 6 q^{65} - 9 q^{67} - 3 q^{68} - 30 q^{69} - 10 q^{71} - 38 q^{72} - 15 q^{73} + 19 q^{74} + 2 q^{75} + 5 q^{76} + 21 q^{78} - 23 q^{79} - 6 q^{80} + 13 q^{81} - 16 q^{82} - 8 q^{83} - 3 q^{84} - 2 q^{85} - 29 q^{86} + 22 q^{87} + 7 q^{89} + 3 q^{90} + 6 q^{91} + 26 q^{92} - 45 q^{93} + 14 q^{94} - 7 q^{95} + 18 q^{96} + 21 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.58597 1.12145 0.560723 0.828003i \(-0.310523\pi\)
0.560723 + 0.828003i \(0.310523\pi\)
\(3\) −3.01851 −1.74274 −0.871368 0.490631i \(-0.836766\pi\)
−0.871368 + 0.490631i \(0.836766\pi\)
\(4\) 0.515286 0.257643
\(5\) 1.00000 0.447214
\(6\) −4.78725 −1.95438
\(7\) −1.00000 −0.377964
\(8\) −2.35471 −0.832514
\(9\) 6.11138 2.03713
\(10\) 1.58597 0.501526
\(11\) 0 0
\(12\) −1.55539 −0.449003
\(13\) −3.17071 −0.879396 −0.439698 0.898146i \(-0.644915\pi\)
−0.439698 + 0.898146i \(0.644915\pi\)
\(14\) −1.58597 −0.423867
\(15\) −3.01851 −0.779375
\(16\) −4.76505 −1.19126
\(17\) 6.29047 1.52566 0.762831 0.646598i \(-0.223809\pi\)
0.762831 + 0.646598i \(0.223809\pi\)
\(18\) 9.69244 2.28453
\(19\) −1.15342 −0.264614 −0.132307 0.991209i \(-0.542238\pi\)
−0.132307 + 0.991209i \(0.542238\pi\)
\(20\) 0.515286 0.115221
\(21\) 3.01851 0.658692
\(22\) 0 0
\(23\) 5.96633 1.24407 0.622033 0.782991i \(-0.286307\pi\)
0.622033 + 0.782991i \(0.286307\pi\)
\(24\) 7.10769 1.45085
\(25\) 1.00000 0.200000
\(26\) −5.02863 −0.986196
\(27\) −9.39172 −1.80744
\(28\) −0.515286 −0.0973799
\(29\) −8.32104 −1.54518 −0.772589 0.634907i \(-0.781039\pi\)
−0.772589 + 0.634907i \(0.781039\pi\)
\(30\) −4.78725 −0.874028
\(31\) 8.69244 1.56121 0.780604 0.625026i \(-0.214912\pi\)
0.780604 + 0.625026i \(0.214912\pi\)
\(32\) −2.84780 −0.503424
\(33\) 0 0
\(34\) 9.97646 1.71095
\(35\) −1.00000 −0.169031
\(36\) 3.14911 0.524851
\(37\) 3.37140 0.554254 0.277127 0.960833i \(-0.410618\pi\)
0.277127 + 0.960833i \(0.410618\pi\)
\(38\) −1.82929 −0.296750
\(39\) 9.57080 1.53256
\(40\) −2.35471 −0.372312
\(41\) −6.78009 −1.05887 −0.529436 0.848350i \(-0.677596\pi\)
−0.529436 + 0.848350i \(0.677596\pi\)
\(42\) 4.78725 0.738688
\(43\) −5.11794 −0.780479 −0.390240 0.920713i \(-0.627608\pi\)
−0.390240 + 0.920713i \(0.627608\pi\)
\(44\) 0 0
\(45\) 6.11138 0.911031
\(46\) 9.46240 1.39515
\(47\) 10.3868 1.51507 0.757536 0.652794i \(-0.226403\pi\)
0.757536 + 0.652794i \(0.226403\pi\)
\(48\) 14.3833 2.07606
\(49\) 1.00000 0.142857
\(50\) 1.58597 0.224289
\(51\) −18.9878 −2.65883
\(52\) −1.63382 −0.226570
\(53\) −8.18922 −1.12488 −0.562438 0.826840i \(-0.690136\pi\)
−0.562438 + 0.826840i \(0.690136\pi\)
\(54\) −14.8949 −2.02694
\(55\) 0 0
\(56\) 2.35471 0.314661
\(57\) 3.48162 0.461152
\(58\) −13.1969 −1.73283
\(59\) 7.10422 0.924891 0.462446 0.886648i \(-0.346972\pi\)
0.462446 + 0.886648i \(0.346972\pi\)
\(60\) −1.55539 −0.200800
\(61\) −7.27196 −0.931079 −0.465540 0.885027i \(-0.654140\pi\)
−0.465540 + 0.885027i \(0.654140\pi\)
\(62\) 13.7859 1.75081
\(63\) −6.11138 −0.769961
\(64\) 5.01360 0.626700
\(65\) −3.17071 −0.393278
\(66\) 0 0
\(67\) −8.71238 −1.06439 −0.532193 0.846623i \(-0.678632\pi\)
−0.532193 + 0.846623i \(0.678632\pi\)
\(68\) 3.24139 0.393076
\(69\) −18.0094 −2.16808
\(70\) −1.58597 −0.189559
\(71\) −12.5319 −1.48726 −0.743629 0.668593i \(-0.766897\pi\)
−0.743629 + 0.668593i \(0.766897\pi\)
\(72\) −14.3905 −1.69594
\(73\) −4.48594 −0.525039 −0.262520 0.964927i \(-0.584553\pi\)
−0.262520 + 0.964927i \(0.584553\pi\)
\(74\) 5.34692 0.621567
\(75\) −3.01851 −0.348547
\(76\) −0.594343 −0.0681759
\(77\) 0 0
\(78\) 15.1790 1.71868
\(79\) 3.30007 0.371286 0.185643 0.982617i \(-0.440563\pi\)
0.185643 + 0.982617i \(0.440563\pi\)
\(80\) −4.76505 −0.532749
\(81\) 10.0148 1.11276
\(82\) −10.7530 −1.18747
\(83\) 7.14492 0.784257 0.392129 0.919910i \(-0.371739\pi\)
0.392129 + 0.919910i \(0.371739\pi\)
\(84\) 1.55539 0.169707
\(85\) 6.29047 0.682297
\(86\) −8.11688 −0.875266
\(87\) 25.1171 2.69284
\(88\) 0 0
\(89\) 0.654265 0.0693520 0.0346760 0.999399i \(-0.488960\pi\)
0.0346760 + 0.999399i \(0.488960\pi\)
\(90\) 9.69244 1.02167
\(91\) 3.17071 0.332381
\(92\) 3.07437 0.320525
\(93\) −26.2382 −2.72077
\(94\) 16.4731 1.69907
\(95\) −1.15342 −0.118339
\(96\) 8.59609 0.877335
\(97\) 8.73507 0.886912 0.443456 0.896296i \(-0.353752\pi\)
0.443456 + 0.896296i \(0.353752\pi\)
\(98\) 1.58597 0.160207
\(99\) 0 0
\(100\) 0.515286 0.0515286
\(101\) −4.93648 −0.491198 −0.245599 0.969372i \(-0.578985\pi\)
−0.245599 + 0.969372i \(0.578985\pi\)
\(102\) −30.1140 −2.98173
\(103\) −11.5887 −1.14187 −0.570933 0.820996i \(-0.693419\pi\)
−0.570933 + 0.820996i \(0.693419\pi\)
\(104\) 7.46608 0.732110
\(105\) 3.01851 0.294576
\(106\) −12.9878 −1.26149
\(107\) 15.4407 1.49271 0.746356 0.665547i \(-0.231802\pi\)
0.746356 + 0.665547i \(0.231802\pi\)
\(108\) −4.83942 −0.465673
\(109\) −18.6914 −1.79031 −0.895156 0.445753i \(-0.852936\pi\)
−0.895156 + 0.445753i \(0.852936\pi\)
\(110\) 0 0
\(111\) −10.1766 −0.965919
\(112\) 4.76505 0.450255
\(113\) 2.74889 0.258594 0.129297 0.991606i \(-0.458728\pi\)
0.129297 + 0.991606i \(0.458728\pi\)
\(114\) 5.52173 0.517157
\(115\) 5.96633 0.556363
\(116\) −4.28771 −0.398104
\(117\) −19.3774 −1.79144
\(118\) 11.2671 1.03722
\(119\) −6.29047 −0.576646
\(120\) 7.10769 0.648841
\(121\) 0 0
\(122\) −11.5331 −1.04416
\(123\) 20.4657 1.84533
\(124\) 4.47909 0.402234
\(125\) 1.00000 0.0894427
\(126\) −9.69244 −0.863471
\(127\) 0.801691 0.0711386 0.0355693 0.999367i \(-0.488676\pi\)
0.0355693 + 0.999367i \(0.488676\pi\)
\(128\) 13.6470 1.20623
\(129\) 15.4485 1.36017
\(130\) −5.02863 −0.441040
\(131\) −12.4209 −1.08522 −0.542611 0.839984i \(-0.682564\pi\)
−0.542611 + 0.839984i \(0.682564\pi\)
\(132\) 0 0
\(133\) 1.15342 0.100015
\(134\) −13.8175 −1.19365
\(135\) −9.39172 −0.808310
\(136\) −14.8122 −1.27014
\(137\) −19.1100 −1.63268 −0.816339 0.577573i \(-0.804000\pi\)
−0.816339 + 0.577573i \(0.804000\pi\)
\(138\) −28.5623 −2.43138
\(139\) 2.87837 0.244140 0.122070 0.992521i \(-0.461047\pi\)
0.122070 + 0.992521i \(0.461047\pi\)
\(140\) −0.515286 −0.0435496
\(141\) −31.3526 −2.64037
\(142\) −19.8751 −1.66788
\(143\) 0 0
\(144\) −29.1210 −2.42675
\(145\) −8.32104 −0.691025
\(146\) −7.11454 −0.588803
\(147\) −3.01851 −0.248962
\(148\) 1.73723 0.142800
\(149\) −15.4427 −1.26511 −0.632556 0.774514i \(-0.717994\pi\)
−0.632556 + 0.774514i \(0.717994\pi\)
\(150\) −4.78725 −0.390877
\(151\) 12.8668 1.04709 0.523544 0.851999i \(-0.324610\pi\)
0.523544 + 0.851999i \(0.324610\pi\)
\(152\) 2.71597 0.220295
\(153\) 38.4434 3.10797
\(154\) 0 0
\(155\) 8.69244 0.698193
\(156\) 4.93170 0.394852
\(157\) −2.64189 −0.210846 −0.105423 0.994427i \(-0.533620\pi\)
−0.105423 + 0.994427i \(0.533620\pi\)
\(158\) 5.23379 0.416378
\(159\) 24.7192 1.96036
\(160\) −2.84780 −0.225138
\(161\) −5.96633 −0.470213
\(162\) 15.8832 1.24790
\(163\) −13.5443 −1.06087 −0.530435 0.847725i \(-0.677972\pi\)
−0.530435 + 0.847725i \(0.677972\pi\)
\(164\) −3.49368 −0.272811
\(165\) 0 0
\(166\) 11.3316 0.879503
\(167\) −20.4083 −1.57924 −0.789620 0.613596i \(-0.789722\pi\)
−0.789620 + 0.613596i \(0.789722\pi\)
\(168\) −7.10769 −0.548370
\(169\) −2.94661 −0.226662
\(170\) 9.97646 0.765160
\(171\) −7.04901 −0.539052
\(172\) −2.63720 −0.201085
\(173\) −12.5397 −0.953378 −0.476689 0.879072i \(-0.658163\pi\)
−0.476689 + 0.879072i \(0.658163\pi\)
\(174\) 39.8349 3.01987
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −21.4441 −1.61184
\(178\) 1.03764 0.0777745
\(179\) −11.5313 −0.861893 −0.430947 0.902377i \(-0.641820\pi\)
−0.430947 + 0.902377i \(0.641820\pi\)
\(180\) 3.14911 0.234721
\(181\) −2.25692 −0.167756 −0.0838779 0.996476i \(-0.526731\pi\)
−0.0838779 + 0.996476i \(0.526731\pi\)
\(182\) 5.02863 0.372747
\(183\) 21.9505 1.62262
\(184\) −14.0490 −1.03570
\(185\) 3.37140 0.247870
\(186\) −41.6128 −3.05120
\(187\) 0 0
\(188\) 5.35218 0.390348
\(189\) 9.39172 0.683147
\(190\) −1.82929 −0.132711
\(191\) −9.31441 −0.673967 −0.336984 0.941511i \(-0.609407\pi\)
−0.336984 + 0.941511i \(0.609407\pi\)
\(192\) −15.1336 −1.09217
\(193\) −3.67986 −0.264882 −0.132441 0.991191i \(-0.542282\pi\)
−0.132441 + 0.991191i \(0.542282\pi\)
\(194\) 13.8535 0.994625
\(195\) 9.57080 0.685380
\(196\) 0.515286 0.0368061
\(197\) −8.44369 −0.601588 −0.300794 0.953689i \(-0.597252\pi\)
−0.300794 + 0.953689i \(0.597252\pi\)
\(198\) 0 0
\(199\) 21.8381 1.54806 0.774029 0.633150i \(-0.218238\pi\)
0.774029 + 0.633150i \(0.218238\pi\)
\(200\) −2.35471 −0.166503
\(201\) 26.2984 1.85494
\(202\) −7.82908 −0.550852
\(203\) 8.32104 0.584022
\(204\) −9.78415 −0.685028
\(205\) −6.78009 −0.473542
\(206\) −18.3792 −1.28054
\(207\) 36.4625 2.53432
\(208\) 15.1086 1.04759
\(209\) 0 0
\(210\) 4.78725 0.330351
\(211\) −11.1893 −0.770302 −0.385151 0.922854i \(-0.625851\pi\)
−0.385151 + 0.922854i \(0.625851\pi\)
\(212\) −4.21979 −0.289816
\(213\) 37.8275 2.59190
\(214\) 24.4885 1.67400
\(215\) −5.11794 −0.349041
\(216\) 22.1147 1.50472
\(217\) −8.69244 −0.590081
\(218\) −29.6439 −2.00774
\(219\) 13.5408 0.915004
\(220\) 0 0
\(221\) −19.9452 −1.34166
\(222\) −16.1397 −1.08323
\(223\) −9.08965 −0.608688 −0.304344 0.952562i \(-0.598437\pi\)
−0.304344 + 0.952562i \(0.598437\pi\)
\(224\) 2.84780 0.190276
\(225\) 6.11138 0.407425
\(226\) 4.35964 0.289999
\(227\) −0.378617 −0.0251297 −0.0125649 0.999921i \(-0.504000\pi\)
−0.0125649 + 0.999921i \(0.504000\pi\)
\(228\) 1.79403 0.118812
\(229\) −3.92667 −0.259481 −0.129741 0.991548i \(-0.541415\pi\)
−0.129741 + 0.991548i \(0.541415\pi\)
\(230\) 9.46240 0.623932
\(231\) 0 0
\(232\) 19.5936 1.28638
\(233\) 20.1205 1.31814 0.659070 0.752082i \(-0.270950\pi\)
0.659070 + 0.752082i \(0.270950\pi\)
\(234\) −30.7319 −2.00901
\(235\) 10.3868 0.677561
\(236\) 3.66071 0.238292
\(237\) −9.96127 −0.647054
\(238\) −9.97646 −0.646678
\(239\) −23.1335 −1.49638 −0.748190 0.663484i \(-0.769077\pi\)
−0.748190 + 0.663484i \(0.769077\pi\)
\(240\) 14.3833 0.928441
\(241\) −21.8666 −1.40855 −0.704276 0.709926i \(-0.748728\pi\)
−0.704276 + 0.709926i \(0.748728\pi\)
\(242\) 0 0
\(243\) −2.05464 −0.131805
\(244\) −3.74714 −0.239886
\(245\) 1.00000 0.0638877
\(246\) 32.4580 2.06944
\(247\) 3.65717 0.232700
\(248\) −20.4681 −1.29973
\(249\) −21.5670 −1.36675
\(250\) 1.58597 0.100305
\(251\) −23.5940 −1.48924 −0.744620 0.667489i \(-0.767369\pi\)
−0.744620 + 0.667489i \(0.767369\pi\)
\(252\) −3.14911 −0.198375
\(253\) 0 0
\(254\) 1.27145 0.0797781
\(255\) −18.9878 −1.18906
\(256\) 11.6165 0.726028
\(257\) 3.74757 0.233767 0.116884 0.993146i \(-0.462710\pi\)
0.116884 + 0.993146i \(0.462710\pi\)
\(258\) 24.5009 1.52536
\(259\) −3.37140 −0.209488
\(260\) −1.63382 −0.101325
\(261\) −50.8530 −3.14772
\(262\) −19.6992 −1.21702
\(263\) 20.4872 1.26330 0.631648 0.775256i \(-0.282379\pi\)
0.631648 + 0.775256i \(0.282379\pi\)
\(264\) 0 0
\(265\) −8.18922 −0.503059
\(266\) 1.82929 0.112161
\(267\) −1.97490 −0.120862
\(268\) −4.48937 −0.274232
\(269\) −29.4456 −1.79533 −0.897665 0.440678i \(-0.854738\pi\)
−0.897665 + 0.440678i \(0.854738\pi\)
\(270\) −14.8949 −0.906477
\(271\) 18.6959 1.13569 0.567847 0.823134i \(-0.307777\pi\)
0.567847 + 0.823134i \(0.307777\pi\)
\(272\) −29.9744 −1.81746
\(273\) −9.57080 −0.579251
\(274\) −30.3078 −1.83096
\(275\) 0 0
\(276\) −9.28000 −0.558590
\(277\) −19.1250 −1.14911 −0.574554 0.818467i \(-0.694824\pi\)
−0.574554 + 0.818467i \(0.694824\pi\)
\(278\) 4.56499 0.273790
\(279\) 53.1228 3.18038
\(280\) 2.35471 0.140721
\(281\) 3.09043 0.184360 0.0921798 0.995742i \(-0.470617\pi\)
0.0921798 + 0.995742i \(0.470617\pi\)
\(282\) −49.7242 −2.96103
\(283\) 14.4606 0.859592 0.429796 0.902926i \(-0.358585\pi\)
0.429796 + 0.902926i \(0.358585\pi\)
\(284\) −6.45749 −0.383181
\(285\) 3.48162 0.206233
\(286\) 0 0
\(287\) 6.78009 0.400216
\(288\) −17.4040 −1.02554
\(289\) 22.5700 1.32764
\(290\) −13.1969 −0.774947
\(291\) −26.3669 −1.54565
\(292\) −2.31154 −0.135273
\(293\) 15.8628 0.926713 0.463357 0.886172i \(-0.346645\pi\)
0.463357 + 0.886172i \(0.346645\pi\)
\(294\) −4.78725 −0.279198
\(295\) 7.10422 0.413624
\(296\) −7.93865 −0.461424
\(297\) 0 0
\(298\) −24.4915 −1.41876
\(299\) −18.9175 −1.09403
\(300\) −1.55539 −0.0898007
\(301\) 5.11794 0.294993
\(302\) 20.4063 1.17425
\(303\) 14.9008 0.856028
\(304\) 5.49613 0.315225
\(305\) −7.27196 −0.416391
\(306\) 60.9699 3.48542
\(307\) 31.9573 1.82390 0.911950 0.410301i \(-0.134576\pi\)
0.911950 + 0.410301i \(0.134576\pi\)
\(308\) 0 0
\(309\) 34.9805 1.98997
\(310\) 13.7859 0.782987
\(311\) −34.1773 −1.93802 −0.969009 0.247024i \(-0.920547\pi\)
−0.969009 + 0.247024i \(0.920547\pi\)
\(312\) −22.5364 −1.27587
\(313\) 7.10835 0.401788 0.200894 0.979613i \(-0.435615\pi\)
0.200894 + 0.979613i \(0.435615\pi\)
\(314\) −4.18995 −0.236452
\(315\) −6.11138 −0.344337
\(316\) 1.70048 0.0956593
\(317\) −7.93860 −0.445876 −0.222938 0.974833i \(-0.571565\pi\)
−0.222938 + 0.974833i \(0.571565\pi\)
\(318\) 39.2038 2.19844
\(319\) 0 0
\(320\) 5.01360 0.280269
\(321\) −46.6079 −2.60140
\(322\) −9.46240 −0.527319
\(323\) −7.25558 −0.403711
\(324\) 5.16049 0.286694
\(325\) −3.17071 −0.175879
\(326\) −21.4808 −1.18971
\(327\) 56.4201 3.12004
\(328\) 15.9651 0.881526
\(329\) −10.3868 −0.572643
\(330\) 0 0
\(331\) −26.9984 −1.48397 −0.741984 0.670417i \(-0.766115\pi\)
−0.741984 + 0.670417i \(0.766115\pi\)
\(332\) 3.68168 0.202058
\(333\) 20.6039 1.12909
\(334\) −32.3668 −1.77103
\(335\) −8.71238 −0.476008
\(336\) −14.3833 −0.784675
\(337\) −5.99694 −0.326674 −0.163337 0.986570i \(-0.552226\pi\)
−0.163337 + 0.986570i \(0.552226\pi\)
\(338\) −4.67321 −0.254189
\(339\) −8.29754 −0.450660
\(340\) 3.24139 0.175789
\(341\) 0 0
\(342\) −11.1795 −0.604518
\(343\) −1.00000 −0.0539949
\(344\) 12.0512 0.649760
\(345\) −18.0094 −0.969594
\(346\) −19.8876 −1.06916
\(347\) 1.59687 0.0857246 0.0428623 0.999081i \(-0.486352\pi\)
0.0428623 + 0.999081i \(0.486352\pi\)
\(348\) 12.9425 0.693790
\(349\) 1.62486 0.0869768 0.0434884 0.999054i \(-0.486153\pi\)
0.0434884 + 0.999054i \(0.486153\pi\)
\(350\) −1.58597 −0.0847734
\(351\) 29.7784 1.58945
\(352\) 0 0
\(353\) 10.6170 0.565088 0.282544 0.959254i \(-0.408822\pi\)
0.282544 + 0.959254i \(0.408822\pi\)
\(354\) −34.0097 −1.80759
\(355\) −12.5319 −0.665122
\(356\) 0.337134 0.0178680
\(357\) 18.9878 1.00494
\(358\) −18.2883 −0.966568
\(359\) −17.8274 −0.940895 −0.470447 0.882428i \(-0.655907\pi\)
−0.470447 + 0.882428i \(0.655907\pi\)
\(360\) −14.3905 −0.758446
\(361\) −17.6696 −0.929980
\(362\) −3.57940 −0.188129
\(363\) 0 0
\(364\) 1.63382 0.0856355
\(365\) −4.48594 −0.234805
\(366\) 34.8127 1.81969
\(367\) 24.9036 1.29996 0.649978 0.759953i \(-0.274778\pi\)
0.649978 + 0.759953i \(0.274778\pi\)
\(368\) −28.4299 −1.48201
\(369\) −41.4357 −2.15706
\(370\) 5.34692 0.277973
\(371\) 8.18922 0.425163
\(372\) −13.5202 −0.700988
\(373\) −2.62079 −0.135700 −0.0678498 0.997696i \(-0.521614\pi\)
−0.0678498 + 0.997696i \(0.521614\pi\)
\(374\) 0 0
\(375\) −3.01851 −0.155875
\(376\) −24.4579 −1.26132
\(377\) 26.3836 1.35882
\(378\) 14.8949 0.766113
\(379\) −1.16480 −0.0598316 −0.0299158 0.999552i \(-0.509524\pi\)
−0.0299158 + 0.999552i \(0.509524\pi\)
\(380\) −0.594343 −0.0304892
\(381\) −2.41991 −0.123976
\(382\) −14.7723 −0.755818
\(383\) −12.6727 −0.647545 −0.323773 0.946135i \(-0.604951\pi\)
−0.323773 + 0.946135i \(0.604951\pi\)
\(384\) −41.1935 −2.10215
\(385\) 0 0
\(386\) −5.83614 −0.297052
\(387\) −31.2777 −1.58993
\(388\) 4.50106 0.228507
\(389\) 26.0879 1.32271 0.661355 0.750073i \(-0.269982\pi\)
0.661355 + 0.750073i \(0.269982\pi\)
\(390\) 15.1790 0.768617
\(391\) 37.5310 1.89803
\(392\) −2.35471 −0.118931
\(393\) 37.4927 1.89126
\(394\) −13.3914 −0.674649
\(395\) 3.30007 0.166044
\(396\) 0 0
\(397\) 24.8879 1.24909 0.624543 0.780990i \(-0.285285\pi\)
0.624543 + 0.780990i \(0.285285\pi\)
\(398\) 34.6344 1.73607
\(399\) −3.48162 −0.174299
\(400\) −4.76505 −0.238253
\(401\) 29.2391 1.46013 0.730066 0.683376i \(-0.239489\pi\)
0.730066 + 0.683376i \(0.239489\pi\)
\(402\) 41.7083 2.08022
\(403\) −27.5612 −1.37292
\(404\) −2.54370 −0.126554
\(405\) 10.0148 0.497640
\(406\) 13.1969 0.654950
\(407\) 0 0
\(408\) 44.7107 2.21351
\(409\) −2.22494 −0.110016 −0.0550082 0.998486i \(-0.517518\pi\)
−0.0550082 + 0.998486i \(0.517518\pi\)
\(410\) −10.7530 −0.531052
\(411\) 57.6837 2.84533
\(412\) −5.97148 −0.294194
\(413\) −7.10422 −0.349576
\(414\) 57.8283 2.84211
\(415\) 7.14492 0.350730
\(416\) 9.02954 0.442709
\(417\) −8.68837 −0.425472
\(418\) 0 0
\(419\) 26.8546 1.31193 0.655966 0.754791i \(-0.272262\pi\)
0.655966 + 0.754791i \(0.272262\pi\)
\(420\) 1.55539 0.0758954
\(421\) 34.3522 1.67422 0.837112 0.547031i \(-0.184242\pi\)
0.837112 + 0.547031i \(0.184242\pi\)
\(422\) −17.7458 −0.863852
\(423\) 63.4777 3.08639
\(424\) 19.2832 0.936474
\(425\) 6.29047 0.305132
\(426\) 59.9931 2.90667
\(427\) 7.27196 0.351915
\(428\) 7.95639 0.384587
\(429\) 0 0
\(430\) −8.11688 −0.391431
\(431\) 26.6494 1.28366 0.641829 0.766848i \(-0.278176\pi\)
0.641829 + 0.766848i \(0.278176\pi\)
\(432\) 44.7520 2.15313
\(433\) −0.137853 −0.00662478 −0.00331239 0.999995i \(-0.501054\pi\)
−0.00331239 + 0.999995i \(0.501054\pi\)
\(434\) −13.7859 −0.661744
\(435\) 25.1171 1.20427
\(436\) −9.63142 −0.461261
\(437\) −6.88171 −0.329197
\(438\) 21.4753 1.02613
\(439\) −3.30273 −0.157631 −0.0788153 0.996889i \(-0.525114\pi\)
−0.0788153 + 0.996889i \(0.525114\pi\)
\(440\) 0 0
\(441\) 6.11138 0.291018
\(442\) −31.6325 −1.50460
\(443\) −4.46330 −0.212058 −0.106029 0.994363i \(-0.533814\pi\)
−0.106029 + 0.994363i \(0.533814\pi\)
\(444\) −5.24385 −0.248862
\(445\) 0.654265 0.0310151
\(446\) −14.4159 −0.682611
\(447\) 46.6138 2.20476
\(448\) −5.01360 −0.236870
\(449\) −18.8425 −0.889230 −0.444615 0.895722i \(-0.646660\pi\)
−0.444615 + 0.895722i \(0.646660\pi\)
\(450\) 9.69244 0.456906
\(451\) 0 0
\(452\) 1.41646 0.0666248
\(453\) −38.8386 −1.82480
\(454\) −0.600474 −0.0281817
\(455\) 3.17071 0.148645
\(456\) −8.19819 −0.383915
\(457\) 4.41629 0.206585 0.103293 0.994651i \(-0.467062\pi\)
0.103293 + 0.994651i \(0.467062\pi\)
\(458\) −6.22756 −0.290995
\(459\) −59.0783 −2.75754
\(460\) 3.07437 0.143343
\(461\) 15.0090 0.699040 0.349520 0.936929i \(-0.386345\pi\)
0.349520 + 0.936929i \(0.386345\pi\)
\(462\) 0 0
\(463\) 5.25808 0.244364 0.122182 0.992508i \(-0.461011\pi\)
0.122182 + 0.992508i \(0.461011\pi\)
\(464\) 39.6502 1.84071
\(465\) −26.2382 −1.21677
\(466\) 31.9105 1.47822
\(467\) 9.86973 0.456717 0.228358 0.973577i \(-0.426664\pi\)
0.228358 + 0.973577i \(0.426664\pi\)
\(468\) −9.98490 −0.461552
\(469\) 8.71238 0.402300
\(470\) 16.4731 0.759848
\(471\) 7.97456 0.367448
\(472\) −16.7283 −0.769985
\(473\) 0 0
\(474\) −15.7982 −0.725637
\(475\) −1.15342 −0.0529227
\(476\) −3.24139 −0.148569
\(477\) −50.0474 −2.29151
\(478\) −36.6889 −1.67811
\(479\) −18.1109 −0.827508 −0.413754 0.910389i \(-0.635783\pi\)
−0.413754 + 0.910389i \(0.635783\pi\)
\(480\) 8.59609 0.392356
\(481\) −10.6897 −0.487409
\(482\) −34.6797 −1.57962
\(483\) 18.0094 0.819457
\(484\) 0 0
\(485\) 8.73507 0.396639
\(486\) −3.25859 −0.147812
\(487\) 20.8870 0.946479 0.473240 0.880934i \(-0.343084\pi\)
0.473240 + 0.880934i \(0.343084\pi\)
\(488\) 17.1233 0.775136
\(489\) 40.8835 1.84882
\(490\) 1.58597 0.0716466
\(491\) 28.0947 1.26790 0.633948 0.773376i \(-0.281433\pi\)
0.633948 + 0.773376i \(0.281433\pi\)
\(492\) 10.5457 0.475437
\(493\) −52.3432 −2.35742
\(494\) 5.80015 0.260961
\(495\) 0 0
\(496\) −41.4199 −1.85981
\(497\) 12.5319 0.562130
\(498\) −34.2045 −1.53274
\(499\) 1.87174 0.0837906 0.0418953 0.999122i \(-0.486660\pi\)
0.0418953 + 0.999122i \(0.486660\pi\)
\(500\) 0.515286 0.0230443
\(501\) 61.6025 2.75220
\(502\) −37.4193 −1.67010
\(503\) 10.7887 0.481044 0.240522 0.970644i \(-0.422681\pi\)
0.240522 + 0.970644i \(0.422681\pi\)
\(504\) 14.3905 0.641004
\(505\) −4.93648 −0.219670
\(506\) 0 0
\(507\) 8.89435 0.395012
\(508\) 0.413100 0.0183284
\(509\) 40.7394 1.80574 0.902870 0.429913i \(-0.141456\pi\)
0.902870 + 0.429913i \(0.141456\pi\)
\(510\) −30.1140 −1.33347
\(511\) 4.48594 0.198446
\(512\) −8.87068 −0.392032
\(513\) 10.8326 0.478273
\(514\) 5.94352 0.262157
\(515\) −11.5887 −0.510658
\(516\) 7.96042 0.350438
\(517\) 0 0
\(518\) −5.34692 −0.234930
\(519\) 37.8513 1.66149
\(520\) 7.46608 0.327409
\(521\) −23.2302 −1.01773 −0.508867 0.860845i \(-0.669935\pi\)
−0.508867 + 0.860845i \(0.669935\pi\)
\(522\) −80.6511 −3.53000
\(523\) 6.91423 0.302338 0.151169 0.988508i \(-0.451696\pi\)
0.151169 + 0.988508i \(0.451696\pi\)
\(524\) −6.40034 −0.279600
\(525\) 3.01851 0.131738
\(526\) 32.4920 1.41672
\(527\) 54.6795 2.38188
\(528\) 0 0
\(529\) 12.5971 0.547701
\(530\) −12.9878 −0.564154
\(531\) 43.4166 1.88412
\(532\) 0.594343 0.0257681
\(533\) 21.4977 0.931168
\(534\) −3.13213 −0.135540
\(535\) 15.4407 0.667561
\(536\) 20.5151 0.886117
\(537\) 34.8074 1.50205
\(538\) −46.6997 −2.01337
\(539\) 0 0
\(540\) −4.83942 −0.208255
\(541\) −38.9010 −1.67249 −0.836243 0.548359i \(-0.815253\pi\)
−0.836243 + 0.548359i \(0.815253\pi\)
\(542\) 29.6510 1.27362
\(543\) 6.81254 0.292354
\(544\) −17.9140 −0.768055
\(545\) −18.6914 −0.800652
\(546\) −15.1790 −0.649600
\(547\) −13.4121 −0.573458 −0.286729 0.958012i \(-0.592568\pi\)
−0.286729 + 0.958012i \(0.592568\pi\)
\(548\) −9.84712 −0.420648
\(549\) −44.4417 −1.89673
\(550\) 0 0
\(551\) 9.59769 0.408875
\(552\) 42.4069 1.80496
\(553\) −3.30007 −0.140333
\(554\) −30.3315 −1.28866
\(555\) −10.1766 −0.431972
\(556\) 1.48318 0.0629010
\(557\) −17.5394 −0.743167 −0.371583 0.928400i \(-0.621185\pi\)
−0.371583 + 0.928400i \(0.621185\pi\)
\(558\) 84.2509 3.56662
\(559\) 16.2275 0.686350
\(560\) 4.76505 0.201360
\(561\) 0 0
\(562\) 4.90131 0.206749
\(563\) −32.5660 −1.37249 −0.686246 0.727369i \(-0.740743\pi\)
−0.686246 + 0.727369i \(0.740743\pi\)
\(564\) −16.1556 −0.680272
\(565\) 2.74889 0.115647
\(566\) 22.9340 0.963987
\(567\) −10.0148 −0.420583
\(568\) 29.5088 1.23816
\(569\) −23.3009 −0.976825 −0.488413 0.872613i \(-0.662424\pi\)
−0.488413 + 0.872613i \(0.662424\pi\)
\(570\) 5.52173 0.231280
\(571\) −40.6108 −1.69951 −0.849754 0.527179i \(-0.823250\pi\)
−0.849754 + 0.527179i \(0.823250\pi\)
\(572\) 0 0
\(573\) 28.1156 1.17455
\(574\) 10.7530 0.448821
\(575\) 5.96633 0.248813
\(576\) 30.6400 1.27667
\(577\) −13.8348 −0.575949 −0.287974 0.957638i \(-0.592982\pi\)
−0.287974 + 0.957638i \(0.592982\pi\)
\(578\) 35.7952 1.48888
\(579\) 11.1077 0.461620
\(580\) −4.28771 −0.178038
\(581\) −7.14492 −0.296421
\(582\) −41.8169 −1.73337
\(583\) 0 0
\(584\) 10.5631 0.437102
\(585\) −19.3774 −0.801157
\(586\) 25.1578 1.03926
\(587\) 4.36970 0.180357 0.0901784 0.995926i \(-0.471256\pi\)
0.0901784 + 0.995926i \(0.471256\pi\)
\(588\) −1.55539 −0.0641433
\(589\) −10.0261 −0.413117
\(590\) 11.2671 0.463857
\(591\) 25.4873 1.04841
\(592\) −16.0649 −0.660263
\(593\) −31.5379 −1.29511 −0.647553 0.762020i \(-0.724208\pi\)
−0.647553 + 0.762020i \(0.724208\pi\)
\(594\) 0 0
\(595\) −6.29047 −0.257884
\(596\) −7.95739 −0.325947
\(597\) −65.9183 −2.69786
\(598\) −30.0025 −1.22689
\(599\) −8.49529 −0.347108 −0.173554 0.984824i \(-0.555525\pi\)
−0.173554 + 0.984824i \(0.555525\pi\)
\(600\) 7.10769 0.290170
\(601\) −33.3257 −1.35938 −0.679692 0.733497i \(-0.737887\pi\)
−0.679692 + 0.733497i \(0.737887\pi\)
\(602\) 8.11688 0.330819
\(603\) −53.2447 −2.16829
\(604\) 6.63010 0.269775
\(605\) 0 0
\(606\) 23.6321 0.959990
\(607\) 5.85092 0.237481 0.118741 0.992925i \(-0.462114\pi\)
0.118741 + 0.992925i \(0.462114\pi\)
\(608\) 3.28472 0.133213
\(609\) −25.1171 −1.01780
\(610\) −11.5331 −0.466961
\(611\) −32.9335 −1.33235
\(612\) 19.8094 0.800746
\(613\) −15.9665 −0.644879 −0.322440 0.946590i \(-0.604503\pi\)
−0.322440 + 0.946590i \(0.604503\pi\)
\(614\) 50.6832 2.04541
\(615\) 20.4657 0.825258
\(616\) 0 0
\(617\) −31.3079 −1.26041 −0.630205 0.776429i \(-0.717029\pi\)
−0.630205 + 0.776429i \(0.717029\pi\)
\(618\) 55.4779 2.23165
\(619\) 19.9543 0.802032 0.401016 0.916071i \(-0.368657\pi\)
0.401016 + 0.916071i \(0.368657\pi\)
\(620\) 4.47909 0.179885
\(621\) −56.0341 −2.24857
\(622\) −54.2041 −2.17339
\(623\) −0.654265 −0.0262126
\(624\) −45.6054 −1.82568
\(625\) 1.00000 0.0400000
\(626\) 11.2736 0.450583
\(627\) 0 0
\(628\) −1.36133 −0.0543229
\(629\) 21.2077 0.845605
\(630\) −9.69244 −0.386156
\(631\) −44.6493 −1.77746 −0.888730 0.458431i \(-0.848412\pi\)
−0.888730 + 0.458431i \(0.848412\pi\)
\(632\) −7.77068 −0.309101
\(633\) 33.7749 1.34243
\(634\) −12.5903 −0.500026
\(635\) 0.801691 0.0318141
\(636\) 12.7375 0.505073
\(637\) −3.17071 −0.125628
\(638\) 0 0
\(639\) −76.5869 −3.02973
\(640\) 13.6470 0.539444
\(641\) 30.5351 1.20607 0.603033 0.797716i \(-0.293959\pi\)
0.603033 + 0.797716i \(0.293959\pi\)
\(642\) −73.9186 −2.91733
\(643\) 40.8926 1.61265 0.806324 0.591474i \(-0.201454\pi\)
0.806324 + 0.591474i \(0.201454\pi\)
\(644\) −3.07437 −0.121147
\(645\) 15.4485 0.608286
\(646\) −11.5071 −0.452741
\(647\) −16.6431 −0.654307 −0.327154 0.944971i \(-0.606089\pi\)
−0.327154 + 0.944971i \(0.606089\pi\)
\(648\) −23.5819 −0.926386
\(649\) 0 0
\(650\) −5.02863 −0.197239
\(651\) 26.2382 1.02835
\(652\) −6.97918 −0.273326
\(653\) −4.44966 −0.174129 −0.0870644 0.996203i \(-0.527749\pi\)
−0.0870644 + 0.996203i \(0.527749\pi\)
\(654\) 89.4804 3.49896
\(655\) −12.4209 −0.485326
\(656\) 32.3075 1.26140
\(657\) −27.4153 −1.06957
\(658\) −16.4731 −0.642189
\(659\) 35.0740 1.36629 0.683145 0.730283i \(-0.260612\pi\)
0.683145 + 0.730283i \(0.260612\pi\)
\(660\) 0 0
\(661\) 30.9685 1.20453 0.602267 0.798295i \(-0.294264\pi\)
0.602267 + 0.798295i \(0.294264\pi\)
\(662\) −42.8186 −1.66419
\(663\) 60.2048 2.33816
\(664\) −16.8242 −0.652905
\(665\) 1.15342 0.0447279
\(666\) 32.6771 1.26621
\(667\) −49.6461 −1.92230
\(668\) −10.5161 −0.406880
\(669\) 27.4372 1.06078
\(670\) −13.8175 −0.533818
\(671\) 0 0
\(672\) −8.59609 −0.331602
\(673\) 9.33892 0.359989 0.179995 0.983668i \(-0.442392\pi\)
0.179995 + 0.983668i \(0.442392\pi\)
\(674\) −9.51093 −0.366347
\(675\) −9.39172 −0.361487
\(676\) −1.51834 −0.0583978
\(677\) −20.4271 −0.785076 −0.392538 0.919736i \(-0.628403\pi\)
−0.392538 + 0.919736i \(0.628403\pi\)
\(678\) −13.1596 −0.505392
\(679\) −8.73507 −0.335221
\(680\) −14.8122 −0.568022
\(681\) 1.14286 0.0437945
\(682\) 0 0
\(683\) 27.7278 1.06098 0.530488 0.847693i \(-0.322009\pi\)
0.530488 + 0.847693i \(0.322009\pi\)
\(684\) −3.63226 −0.138883
\(685\) −19.1100 −0.730156
\(686\) −1.58597 −0.0605524
\(687\) 11.8527 0.452208
\(688\) 24.3873 0.929756
\(689\) 25.9656 0.989211
\(690\) −28.5623 −1.08735
\(691\) 36.1009 1.37334 0.686672 0.726967i \(-0.259071\pi\)
0.686672 + 0.726967i \(0.259071\pi\)
\(692\) −6.46155 −0.245631
\(693\) 0 0
\(694\) 2.53259 0.0961356
\(695\) 2.87837 0.109183
\(696\) −59.1434 −2.24182
\(697\) −42.6499 −1.61548
\(698\) 2.57697 0.0975398
\(699\) −60.7340 −2.29717
\(700\) −0.515286 −0.0194760
\(701\) 42.8713 1.61923 0.809614 0.586963i \(-0.199677\pi\)
0.809614 + 0.586963i \(0.199677\pi\)
\(702\) 47.2275 1.78249
\(703\) −3.88865 −0.146663
\(704\) 0 0
\(705\) −31.3526 −1.18081
\(706\) 16.8383 0.633716
\(707\) 4.93648 0.185655
\(708\) −11.0499 −0.415279
\(709\) −30.1475 −1.13221 −0.566106 0.824333i \(-0.691551\pi\)
−0.566106 + 0.824333i \(0.691551\pi\)
\(710\) −19.8751 −0.745899
\(711\) 20.1680 0.756357
\(712\) −1.54060 −0.0577365
\(713\) 51.8620 1.94225
\(714\) 30.1140 1.12699
\(715\) 0 0
\(716\) −5.94194 −0.222061
\(717\) 69.8286 2.60780
\(718\) −28.2736 −1.05516
\(719\) −45.5833 −1.69997 −0.849986 0.526805i \(-0.823390\pi\)
−0.849986 + 0.526805i \(0.823390\pi\)
\(720\) −29.1210 −1.08528
\(721\) 11.5887 0.431585
\(722\) −28.0234 −1.04292
\(723\) 66.0045 2.45473
\(724\) −1.16296 −0.0432211
\(725\) −8.32104 −0.309036
\(726\) 0 0
\(727\) 16.4698 0.610831 0.305416 0.952219i \(-0.401205\pi\)
0.305416 + 0.952219i \(0.401205\pi\)
\(728\) −7.46608 −0.276711
\(729\) −23.8425 −0.883056
\(730\) −7.11454 −0.263321
\(731\) −32.1943 −1.19075
\(732\) 11.3108 0.418058
\(733\) −19.9387 −0.736453 −0.368227 0.929736i \(-0.620035\pi\)
−0.368227 + 0.929736i \(0.620035\pi\)
\(734\) 39.4962 1.45783
\(735\) −3.01851 −0.111339
\(736\) −16.9909 −0.626293
\(737\) 0 0
\(738\) −65.7156 −2.41902
\(739\) −7.42054 −0.272969 −0.136485 0.990642i \(-0.543580\pi\)
−0.136485 + 0.990642i \(0.543580\pi\)
\(740\) 1.73723 0.0638620
\(741\) −11.0392 −0.405535
\(742\) 12.9878 0.476798
\(743\) −19.5947 −0.718859 −0.359429 0.933172i \(-0.617029\pi\)
−0.359429 + 0.933172i \(0.617029\pi\)
\(744\) 61.7831 2.26508
\(745\) −15.4427 −0.565776
\(746\) −4.15649 −0.152180
\(747\) 43.6653 1.59763
\(748\) 0 0
\(749\) −15.4407 −0.564192
\(750\) −4.78725 −0.174806
\(751\) −26.0399 −0.950209 −0.475105 0.879929i \(-0.657590\pi\)
−0.475105 + 0.879929i \(0.657590\pi\)
\(752\) −49.4937 −1.80485
\(753\) 71.2186 2.59535
\(754\) 41.8435 1.52385
\(755\) 12.8668 0.468272
\(756\) 4.83942 0.176008
\(757\) 50.1682 1.82339 0.911697 0.410862i \(-0.134772\pi\)
0.911697 + 0.410862i \(0.134772\pi\)
\(758\) −1.84733 −0.0670979
\(759\) 0 0
\(760\) 2.71597 0.0985187
\(761\) 49.1453 1.78152 0.890759 0.454477i \(-0.150174\pi\)
0.890759 + 0.454477i \(0.150174\pi\)
\(762\) −3.83789 −0.139032
\(763\) 18.6914 0.676674
\(764\) −4.79958 −0.173643
\(765\) 38.4434 1.38992
\(766\) −20.0985 −0.726187
\(767\) −22.5254 −0.813346
\(768\) −35.0643 −1.26527
\(769\) −12.7595 −0.460120 −0.230060 0.973176i \(-0.573892\pi\)
−0.230060 + 0.973176i \(0.573892\pi\)
\(770\) 0 0
\(771\) −11.3121 −0.407394
\(772\) −1.89618 −0.0682451
\(773\) −15.4282 −0.554915 −0.277457 0.960738i \(-0.589492\pi\)
−0.277457 + 0.960738i \(0.589492\pi\)
\(774\) −49.6053 −1.78303
\(775\) 8.69244 0.312241
\(776\) −20.5685 −0.738367
\(777\) 10.1766 0.365083
\(778\) 41.3745 1.48335
\(779\) 7.82032 0.280192
\(780\) 4.93170 0.176583
\(781\) 0 0
\(782\) 59.5229 2.12853
\(783\) 78.1488 2.79281
\(784\) −4.76505 −0.170180
\(785\) −2.64189 −0.0942931
\(786\) 59.4621 2.12094
\(787\) −19.1854 −0.683885 −0.341942 0.939721i \(-0.611085\pi\)
−0.341942 + 0.939721i \(0.611085\pi\)
\(788\) −4.35092 −0.154995
\(789\) −61.8408 −2.20159
\(790\) 5.23379 0.186210
\(791\) −2.74889 −0.0977392
\(792\) 0 0
\(793\) 23.0573 0.818788
\(794\) 39.4713 1.40078
\(795\) 24.7192 0.876700
\(796\) 11.2528 0.398846
\(797\) −35.4716 −1.25647 −0.628235 0.778024i \(-0.716222\pi\)
−0.628235 + 0.778024i \(0.716222\pi\)
\(798\) −5.52173 −0.195467
\(799\) 65.3379 2.31149
\(800\) −2.84780 −0.100685
\(801\) 3.99846 0.141279
\(802\) 46.3722 1.63746
\(803\) 0 0
\(804\) 13.5512 0.477913
\(805\) −5.96633 −0.210286
\(806\) −43.7111 −1.53966
\(807\) 88.8817 3.12879
\(808\) 11.6239 0.408929
\(809\) 22.3975 0.787453 0.393726 0.919228i \(-0.371186\pi\)
0.393726 + 0.919228i \(0.371186\pi\)
\(810\) 15.8832 0.558077
\(811\) −5.90988 −0.207524 −0.103762 0.994602i \(-0.533088\pi\)
−0.103762 + 0.994602i \(0.533088\pi\)
\(812\) 4.28771 0.150469
\(813\) −56.4336 −1.97921
\(814\) 0 0
\(815\) −13.5443 −0.474436
\(816\) 90.4779 3.16736
\(817\) 5.90316 0.206525
\(818\) −3.52868 −0.123378
\(819\) 19.3774 0.677101
\(820\) −3.49368 −0.122005
\(821\) 47.8343 1.66943 0.834715 0.550682i \(-0.185632\pi\)
0.834715 + 0.550682i \(0.185632\pi\)
\(822\) 91.4843 3.19088
\(823\) 9.18303 0.320100 0.160050 0.987109i \(-0.448834\pi\)
0.160050 + 0.987109i \(0.448834\pi\)
\(824\) 27.2879 0.950620
\(825\) 0 0
\(826\) −11.2671 −0.392031
\(827\) −34.8116 −1.21052 −0.605259 0.796028i \(-0.706931\pi\)
−0.605259 + 0.796028i \(0.706931\pi\)
\(828\) 18.7886 0.652950
\(829\) 21.3407 0.741194 0.370597 0.928794i \(-0.379153\pi\)
0.370597 + 0.928794i \(0.379153\pi\)
\(830\) 11.3316 0.393326
\(831\) 57.7288 2.00259
\(832\) −15.8967 −0.551117
\(833\) 6.29047 0.217952
\(834\) −13.7795 −0.477144
\(835\) −20.4083 −0.706258
\(836\) 0 0
\(837\) −81.6369 −2.82178
\(838\) 42.5904 1.47126
\(839\) −24.9743 −0.862209 −0.431105 0.902302i \(-0.641876\pi\)
−0.431105 + 0.902302i \(0.641876\pi\)
\(840\) −7.10769 −0.245239
\(841\) 40.2397 1.38757
\(842\) 54.4814 1.87755
\(843\) −9.32848 −0.321290
\(844\) −5.76568 −0.198463
\(845\) −2.94661 −0.101366
\(846\) 100.673 3.46123
\(847\) 0 0
\(848\) 39.0220 1.34002
\(849\) −43.6494 −1.49804
\(850\) 9.97646 0.342190
\(851\) 20.1149 0.689529
\(852\) 19.4920 0.667784
\(853\) 0.765445 0.0262083 0.0131042 0.999914i \(-0.495829\pi\)
0.0131042 + 0.999914i \(0.495829\pi\)
\(854\) 11.5331 0.394654
\(855\) −7.04901 −0.241071
\(856\) −36.3584 −1.24270
\(857\) 0.632460 0.0216044 0.0108022 0.999942i \(-0.496561\pi\)
0.0108022 + 0.999942i \(0.496561\pi\)
\(858\) 0 0
\(859\) 7.43917 0.253821 0.126911 0.991914i \(-0.459494\pi\)
0.126911 + 0.991914i \(0.459494\pi\)
\(860\) −2.63720 −0.0899279
\(861\) −20.4657 −0.697471
\(862\) 42.2651 1.43955
\(863\) −58.4772 −1.99059 −0.995294 0.0968976i \(-0.969108\pi\)
−0.995294 + 0.0968976i \(0.969108\pi\)
\(864\) 26.7457 0.909907
\(865\) −12.5397 −0.426364
\(866\) −0.218630 −0.00742934
\(867\) −68.1276 −2.31373
\(868\) −4.47909 −0.152030
\(869\) 0 0
\(870\) 39.8349 1.35053
\(871\) 27.6244 0.936018
\(872\) 44.0128 1.49046
\(873\) 53.3833 1.80675
\(874\) −10.9142 −0.369177
\(875\) −1.00000 −0.0338062
\(876\) 6.97740 0.235744
\(877\) 29.1727 0.985093 0.492546 0.870286i \(-0.336066\pi\)
0.492546 + 0.870286i \(0.336066\pi\)
\(878\) −5.23801 −0.176774
\(879\) −47.8819 −1.61502
\(880\) 0 0
\(881\) 0.474047 0.0159711 0.00798553 0.999968i \(-0.497458\pi\)
0.00798553 + 0.999968i \(0.497458\pi\)
\(882\) 9.69244 0.326361
\(883\) 6.21786 0.209248 0.104624 0.994512i \(-0.466636\pi\)
0.104624 + 0.994512i \(0.466636\pi\)
\(884\) −10.2775 −0.345670
\(885\) −21.4441 −0.720837
\(886\) −7.07864 −0.237811
\(887\) 5.89043 0.197781 0.0988906 0.995098i \(-0.468471\pi\)
0.0988906 + 0.995098i \(0.468471\pi\)
\(888\) 23.9629 0.804141
\(889\) −0.801691 −0.0268879
\(890\) 1.03764 0.0347818
\(891\) 0 0
\(892\) −4.68377 −0.156824
\(893\) −11.9804 −0.400909
\(894\) 73.9279 2.47252
\(895\) −11.5313 −0.385450
\(896\) −13.6470 −0.455914
\(897\) 57.1026 1.90660
\(898\) −29.8835 −0.997225
\(899\) −72.3301 −2.41234
\(900\) 3.14911 0.104970
\(901\) −51.5140 −1.71618
\(902\) 0 0
\(903\) −15.4485 −0.514095
\(904\) −6.47282 −0.215283
\(905\) −2.25692 −0.0750226
\(906\) −61.5967 −2.04641
\(907\) 6.97336 0.231547 0.115773 0.993276i \(-0.463065\pi\)
0.115773 + 0.993276i \(0.463065\pi\)
\(908\) −0.195096 −0.00647450
\(909\) −30.1687 −1.00063
\(910\) 5.02863 0.166698
\(911\) −25.4630 −0.843626 −0.421813 0.906683i \(-0.638606\pi\)
−0.421813 + 0.906683i \(0.638606\pi\)
\(912\) −16.5901 −0.549353
\(913\) 0 0
\(914\) 7.00408 0.231674
\(915\) 21.9505 0.725660
\(916\) −2.02336 −0.0668536
\(917\) 12.4209 0.410176
\(918\) −93.6961 −3.09243
\(919\) −27.0003 −0.890658 −0.445329 0.895367i \(-0.646913\pi\)
−0.445329 + 0.895367i \(0.646913\pi\)
\(920\) −14.0490 −0.463180
\(921\) −96.4633 −3.17858
\(922\) 23.8038 0.783936
\(923\) 39.7349 1.30789
\(924\) 0 0
\(925\) 3.37140 0.110851
\(926\) 8.33914 0.274041
\(927\) −70.8228 −2.32613
\(928\) 23.6966 0.777880
\(929\) 0.622063 0.0204092 0.0102046 0.999948i \(-0.496752\pi\)
0.0102046 + 0.999948i \(0.496752\pi\)
\(930\) −41.6128 −1.36454
\(931\) −1.15342 −0.0378020
\(932\) 10.3678 0.339609
\(933\) 103.164 3.37745
\(934\) 15.6531 0.512183
\(935\) 0 0
\(936\) 45.6281 1.49140
\(937\) −29.3023 −0.957263 −0.478631 0.878016i \(-0.658867\pi\)
−0.478631 + 0.878016i \(0.658867\pi\)
\(938\) 13.8175 0.451159
\(939\) −21.4566 −0.700209
\(940\) 5.35218 0.174569
\(941\) −8.77735 −0.286133 −0.143067 0.989713i \(-0.545696\pi\)
−0.143067 + 0.989713i \(0.545696\pi\)
\(942\) 12.6474 0.412074
\(943\) −40.4523 −1.31731
\(944\) −33.8520 −1.10179
\(945\) 9.39172 0.305513
\(946\) 0 0
\(947\) −42.4838 −1.38054 −0.690269 0.723553i \(-0.742508\pi\)
−0.690269 + 0.723553i \(0.742508\pi\)
\(948\) −5.13290 −0.166709
\(949\) 14.2236 0.461717
\(950\) −1.82929 −0.0593500
\(951\) 23.9627 0.777044
\(952\) 14.8122 0.480066
\(953\) −29.1406 −0.943958 −0.471979 0.881610i \(-0.656460\pi\)
−0.471979 + 0.881610i \(0.656460\pi\)
\(954\) −79.3734 −2.56981
\(955\) −9.31441 −0.301407
\(956\) −11.9204 −0.385532
\(957\) 0 0
\(958\) −28.7232 −0.928006
\(959\) 19.1100 0.617094
\(960\) −15.1336 −0.488434
\(961\) 44.5584 1.43737
\(962\) −16.9535 −0.546604
\(963\) 94.3642 3.04084
\(964\) −11.2676 −0.362904
\(965\) −3.67986 −0.118459
\(966\) 28.5623 0.918977
\(967\) 4.05314 0.130340 0.0651701 0.997874i \(-0.479241\pi\)
0.0651701 + 0.997874i \(0.479241\pi\)
\(968\) 0 0
\(969\) 21.9010 0.703562
\(970\) 13.8535 0.444810
\(971\) −5.05440 −0.162203 −0.0811016 0.996706i \(-0.525844\pi\)
−0.0811016 + 0.996706i \(0.525844\pi\)
\(972\) −1.05873 −0.0339587
\(973\) −2.87837 −0.0922763
\(974\) 33.1260 1.06143
\(975\) 9.57080 0.306511
\(976\) 34.6513 1.10916
\(977\) −10.3457 −0.330987 −0.165494 0.986211i \(-0.552922\pi\)
−0.165494 + 0.986211i \(0.552922\pi\)
\(978\) 64.8399 2.07335
\(979\) 0 0
\(980\) 0.515286 0.0164602
\(981\) −114.230 −3.64709
\(982\) 44.5572 1.42188
\(983\) 35.5145 1.13274 0.566369 0.824152i \(-0.308348\pi\)
0.566369 + 0.824152i \(0.308348\pi\)
\(984\) −48.1908 −1.53627
\(985\) −8.44369 −0.269038
\(986\) −83.0145 −2.64372
\(987\) 31.3526 0.997966
\(988\) 1.88449 0.0599536
\(989\) −30.5354 −0.970968
\(990\) 0 0
\(991\) 52.9460 1.68188 0.840942 0.541126i \(-0.182002\pi\)
0.840942 + 0.541126i \(0.182002\pi\)
\(992\) −24.7543 −0.785950
\(993\) 81.4949 2.58616
\(994\) 19.8751 0.630399
\(995\) 21.8381 0.692313
\(996\) −11.1132 −0.352134
\(997\) 23.2468 0.736233 0.368117 0.929780i \(-0.380003\pi\)
0.368117 + 0.929780i \(0.380003\pi\)
\(998\) 2.96852 0.0939667
\(999\) −31.6632 −1.00178
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 4235.2.a.bc.1.4 5
11.10 odd 2 4235.2.a.bd.1.2 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4235.2.a.bc.1.4 5 1.1 even 1 trivial
4235.2.a.bd.1.2 yes 5 11.10 odd 2