Properties

Label 4009.2.a.c.1.65
Level $4009$
Weight $2$
Character 4009.1
Self dual yes
Analytic conductor $32.012$
Analytic rank $1$
Dimension $71$
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] = [4009,2,Mod(1,4009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4009, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4009 = 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4009.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: \(32.0120261703\)
Analytic rank: \(1\)
Dimension: \(71\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.65
Character \(\chi\) \(=\) 4009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.16719 q^{2} +2.87932 q^{3} +2.69672 q^{4} -4.25842 q^{5} +6.24005 q^{6} -1.70968 q^{7} +1.50993 q^{8} +5.29050 q^{9} +O(q^{10})\) \(q+2.16719 q^{2} +2.87932 q^{3} +2.69672 q^{4} -4.25842 q^{5} +6.24005 q^{6} -1.70968 q^{7} +1.50993 q^{8} +5.29050 q^{9} -9.22881 q^{10} -4.00730 q^{11} +7.76474 q^{12} -2.63185 q^{13} -3.70520 q^{14} -12.2614 q^{15} -2.12113 q^{16} +4.14067 q^{17} +11.4655 q^{18} +1.00000 q^{19} -11.4838 q^{20} -4.92272 q^{21} -8.68458 q^{22} -5.04807 q^{23} +4.34759 q^{24} +13.1341 q^{25} -5.70373 q^{26} +6.59508 q^{27} -4.61053 q^{28} -8.96525 q^{29} -26.5727 q^{30} +2.22075 q^{31} -7.61676 q^{32} -11.5383 q^{33} +8.97362 q^{34} +7.28053 q^{35} +14.2670 q^{36} -7.33592 q^{37} +2.16719 q^{38} -7.57795 q^{39} -6.42993 q^{40} -8.62721 q^{41} -10.6685 q^{42} +6.93765 q^{43} -10.8066 q^{44} -22.5291 q^{45} -10.9401 q^{46} -5.46855 q^{47} -6.10741 q^{48} -4.07700 q^{49} +28.4641 q^{50} +11.9223 q^{51} -7.09738 q^{52} +3.24590 q^{53} +14.2928 q^{54} +17.0647 q^{55} -2.58150 q^{56} +2.87932 q^{57} -19.4294 q^{58} +9.54293 q^{59} -33.0655 q^{60} -2.66329 q^{61} +4.81279 q^{62} -9.04505 q^{63} -12.2647 q^{64} +11.2075 q^{65} -25.0057 q^{66} +6.98673 q^{67} +11.1662 q^{68} -14.5350 q^{69} +15.7783 q^{70} +3.65784 q^{71} +7.98830 q^{72} -2.97541 q^{73} -15.8983 q^{74} +37.8173 q^{75} +2.69672 q^{76} +6.85119 q^{77} -16.4229 q^{78} +14.4890 q^{79} +9.03265 q^{80} +3.11786 q^{81} -18.6968 q^{82} +1.38732 q^{83} -13.2752 q^{84} -17.6327 q^{85} +15.0352 q^{86} -25.8139 q^{87} -6.05076 q^{88} -13.4196 q^{89} -48.8250 q^{90} +4.49963 q^{91} -13.6132 q^{92} +6.39426 q^{93} -11.8514 q^{94} -4.25842 q^{95} -21.9311 q^{96} -0.983902 q^{97} -8.83564 q^{98} -21.2006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.16719 1.53244 0.766218 0.642580i \(-0.222136\pi\)
0.766218 + 0.642580i \(0.222136\pi\)
\(3\) 2.87932 1.66238 0.831189 0.555990i \(-0.187661\pi\)
0.831189 + 0.555990i \(0.187661\pi\)
\(4\) 2.69672 1.34836
\(5\) −4.25842 −1.90442 −0.952211 0.305442i \(-0.901196\pi\)
−0.952211 + 0.305442i \(0.901196\pi\)
\(6\) 6.24005 2.54749
\(7\) −1.70968 −0.646198 −0.323099 0.946365i \(-0.604725\pi\)
−0.323099 + 0.946365i \(0.604725\pi\)
\(8\) 1.50993 0.533842
\(9\) 5.29050 1.76350
\(10\) −9.22881 −2.91840
\(11\) −4.00730 −1.20825 −0.604123 0.796891i \(-0.706476\pi\)
−0.604123 + 0.796891i \(0.706476\pi\)
\(12\) 7.76474 2.24149
\(13\) −2.63185 −0.729945 −0.364972 0.931018i \(-0.618922\pi\)
−0.364972 + 0.931018i \(0.618922\pi\)
\(14\) −3.70520 −0.990258
\(15\) −12.2614 −3.16587
\(16\) −2.12113 −0.530282
\(17\) 4.14067 1.00426 0.502130 0.864792i \(-0.332550\pi\)
0.502130 + 0.864792i \(0.332550\pi\)
\(18\) 11.4655 2.70245
\(19\) 1.00000 0.229416
\(20\) −11.4838 −2.56785
\(21\) −4.92272 −1.07423
\(22\) −8.68458 −1.85156
\(23\) −5.04807 −1.05260 −0.526298 0.850300i \(-0.676420\pi\)
−0.526298 + 0.850300i \(0.676420\pi\)
\(24\) 4.34759 0.887448
\(25\) 13.1341 2.62682
\(26\) −5.70373 −1.11859
\(27\) 6.59508 1.26922
\(28\) −4.61053 −0.871309
\(29\) −8.96525 −1.66481 −0.832403 0.554171i \(-0.813035\pi\)
−0.832403 + 0.554171i \(0.813035\pi\)
\(30\) −26.5727 −4.85149
\(31\) 2.22075 0.398859 0.199429 0.979912i \(-0.436091\pi\)
0.199429 + 0.979912i \(0.436091\pi\)
\(32\) −7.61676 −1.34647
\(33\) −11.5383 −2.00856
\(34\) 8.97362 1.53896
\(35\) 7.28053 1.23063
\(36\) 14.2670 2.37783
\(37\) −7.33592 −1.20602 −0.603009 0.797735i \(-0.706032\pi\)
−0.603009 + 0.797735i \(0.706032\pi\)
\(38\) 2.16719 0.351565
\(39\) −7.57795 −1.21344
\(40\) −6.42993 −1.01666
\(41\) −8.62721 −1.34734 −0.673672 0.739031i \(-0.735284\pi\)
−0.673672 + 0.739031i \(0.735284\pi\)
\(42\) −10.6685 −1.64618
\(43\) 6.93765 1.05798 0.528991 0.848628i \(-0.322571\pi\)
0.528991 + 0.848628i \(0.322571\pi\)
\(44\) −10.8066 −1.62915
\(45\) −22.5291 −3.35844
\(46\) −10.9401 −1.61304
\(47\) −5.46855 −0.797671 −0.398835 0.917023i \(-0.630585\pi\)
−0.398835 + 0.917023i \(0.630585\pi\)
\(48\) −6.10741 −0.881529
\(49\) −4.07700 −0.582428
\(50\) 28.4641 4.02544
\(51\) 11.9223 1.66946
\(52\) −7.09738 −0.984230
\(53\) 3.24590 0.445859 0.222929 0.974835i \(-0.428438\pi\)
0.222929 + 0.974835i \(0.428438\pi\)
\(54\) 14.2928 1.94500
\(55\) 17.0647 2.30101
\(56\) −2.58150 −0.344968
\(57\) 2.87932 0.381376
\(58\) −19.4294 −2.55121
\(59\) 9.54293 1.24238 0.621192 0.783659i \(-0.286649\pi\)
0.621192 + 0.783659i \(0.286649\pi\)
\(60\) −33.0655 −4.26873
\(61\) −2.66329 −0.340999 −0.170500 0.985358i \(-0.554538\pi\)
−0.170500 + 0.985358i \(0.554538\pi\)
\(62\) 4.81279 0.611225
\(63\) −9.04505 −1.13957
\(64\) −12.2647 −1.53309
\(65\) 11.2075 1.39012
\(66\) −25.0057 −3.07799
\(67\) 6.98673 0.853565 0.426783 0.904354i \(-0.359647\pi\)
0.426783 + 0.904354i \(0.359647\pi\)
\(68\) 11.1662 1.35411
\(69\) −14.5350 −1.74981
\(70\) 15.7783 1.88587
\(71\) 3.65784 0.434105 0.217053 0.976160i \(-0.430356\pi\)
0.217053 + 0.976160i \(0.430356\pi\)
\(72\) 7.98830 0.941431
\(73\) −2.97541 −0.348246 −0.174123 0.984724i \(-0.555709\pi\)
−0.174123 + 0.984724i \(0.555709\pi\)
\(74\) −15.8983 −1.84814
\(75\) 37.8173 4.36677
\(76\) 2.69672 0.309335
\(77\) 6.85119 0.780766
\(78\) −16.4229 −1.85953
\(79\) 14.4890 1.63014 0.815071 0.579361i \(-0.196698\pi\)
0.815071 + 0.579361i \(0.196698\pi\)
\(80\) 9.03265 1.00988
\(81\) 3.11786 0.346429
\(82\) −18.6968 −2.06472
\(83\) 1.38732 0.152278 0.0761391 0.997097i \(-0.475741\pi\)
0.0761391 + 0.997097i \(0.475741\pi\)
\(84\) −13.2752 −1.44844
\(85\) −17.6327 −1.91253
\(86\) 15.0352 1.62129
\(87\) −25.8139 −2.76754
\(88\) −6.05076 −0.645013
\(89\) −13.4196 −1.42248 −0.711238 0.702952i \(-0.751865\pi\)
−0.711238 + 0.702952i \(0.751865\pi\)
\(90\) −48.8250 −5.14660
\(91\) 4.49963 0.471689
\(92\) −13.6132 −1.41928
\(93\) 6.39426 0.663053
\(94\) −11.8514 −1.22238
\(95\) −4.25842 −0.436904
\(96\) −21.9311 −2.23834
\(97\) −0.983902 −0.0999001 −0.0499501 0.998752i \(-0.515906\pi\)
−0.0499501 + 0.998752i \(0.515906\pi\)
\(98\) −8.83564 −0.892534
\(99\) −21.2006 −2.13074
\(100\) 35.4191 3.54191
\(101\) −9.86814 −0.981917 −0.490958 0.871183i \(-0.663353\pi\)
−0.490958 + 0.871183i \(0.663353\pi\)
\(102\) 25.8380 2.55834
\(103\) 0.571976 0.0563584 0.0281792 0.999603i \(-0.491029\pi\)
0.0281792 + 0.999603i \(0.491029\pi\)
\(104\) −3.97393 −0.389676
\(105\) 20.9630 2.04578
\(106\) 7.03449 0.683250
\(107\) 7.45932 0.721120 0.360560 0.932736i \(-0.382586\pi\)
0.360560 + 0.932736i \(0.382586\pi\)
\(108\) 17.7851 1.71137
\(109\) 7.75355 0.742656 0.371328 0.928502i \(-0.378903\pi\)
0.371328 + 0.928502i \(0.378903\pi\)
\(110\) 36.9826 3.52615
\(111\) −21.1225 −2.00486
\(112\) 3.62645 0.342667
\(113\) −6.98163 −0.656776 −0.328388 0.944543i \(-0.606505\pi\)
−0.328388 + 0.944543i \(0.606505\pi\)
\(114\) 6.24005 0.584434
\(115\) 21.4968 2.00458
\(116\) −24.1768 −2.24476
\(117\) −13.9238 −1.28726
\(118\) 20.6814 1.90387
\(119\) −7.07921 −0.648950
\(120\) −18.5138 −1.69007
\(121\) 5.05842 0.459857
\(122\) −5.77186 −0.522560
\(123\) −24.8405 −2.23979
\(124\) 5.98875 0.537806
\(125\) −34.6384 −3.09815
\(126\) −19.6024 −1.74632
\(127\) 8.38454 0.744007 0.372004 0.928231i \(-0.378671\pi\)
0.372004 + 0.928231i \(0.378671\pi\)
\(128\) −11.3465 −1.00290
\(129\) 19.9757 1.75876
\(130\) 24.2889 2.13027
\(131\) 12.6908 1.10880 0.554401 0.832250i \(-0.312948\pi\)
0.554401 + 0.832250i \(0.312948\pi\)
\(132\) −31.1156 −2.70827
\(133\) −1.70968 −0.148248
\(134\) 15.1416 1.30803
\(135\) −28.0846 −2.41714
\(136\) 6.25214 0.536116
\(137\) −9.74542 −0.832608 −0.416304 0.909226i \(-0.636675\pi\)
−0.416304 + 0.909226i \(0.636675\pi\)
\(138\) −31.5002 −2.68147
\(139\) 11.7380 0.995605 0.497802 0.867291i \(-0.334140\pi\)
0.497802 + 0.867291i \(0.334140\pi\)
\(140\) 19.6336 1.65934
\(141\) −15.7457 −1.32603
\(142\) 7.92724 0.665239
\(143\) 10.5466 0.881953
\(144\) −11.2218 −0.935152
\(145\) 38.1778 3.17049
\(146\) −6.44830 −0.533665
\(147\) −11.7390 −0.968215
\(148\) −19.7829 −1.62615
\(149\) 7.15271 0.585973 0.292986 0.956117i \(-0.405351\pi\)
0.292986 + 0.956117i \(0.405351\pi\)
\(150\) 81.9574 6.69179
\(151\) 0.483583 0.0393535 0.0196767 0.999806i \(-0.493736\pi\)
0.0196767 + 0.999806i \(0.493736\pi\)
\(152\) 1.50993 0.122472
\(153\) 21.9062 1.77101
\(154\) 14.8479 1.19647
\(155\) −9.45688 −0.759595
\(156\) −20.4357 −1.63616
\(157\) 13.3796 1.06781 0.533905 0.845544i \(-0.320724\pi\)
0.533905 + 0.845544i \(0.320724\pi\)
\(158\) 31.4005 2.49809
\(159\) 9.34599 0.741185
\(160\) 32.4353 2.56424
\(161\) 8.63058 0.680185
\(162\) 6.75701 0.530880
\(163\) −8.68627 −0.680361 −0.340181 0.940360i \(-0.610488\pi\)
−0.340181 + 0.940360i \(0.610488\pi\)
\(164\) −23.2652 −1.81671
\(165\) 49.1349 3.82514
\(166\) 3.00659 0.233357
\(167\) 3.05375 0.236306 0.118153 0.992995i \(-0.462303\pi\)
0.118153 + 0.992995i \(0.462303\pi\)
\(168\) −7.43298 −0.573467
\(169\) −6.07335 −0.467180
\(170\) −38.2134 −2.93084
\(171\) 5.29050 0.404574
\(172\) 18.7089 1.42654
\(173\) −20.1828 −1.53447 −0.767234 0.641367i \(-0.778367\pi\)
−0.767234 + 0.641367i \(0.778367\pi\)
\(174\) −55.9436 −4.24107
\(175\) −22.4551 −1.69745
\(176\) 8.49999 0.640711
\(177\) 27.4772 2.06531
\(178\) −29.0829 −2.17985
\(179\) −21.7815 −1.62802 −0.814012 0.580848i \(-0.802721\pi\)
−0.814012 + 0.580848i \(0.802721\pi\)
\(180\) −60.7548 −4.52840
\(181\) 15.1678 1.12742 0.563708 0.825974i \(-0.309375\pi\)
0.563708 + 0.825974i \(0.309375\pi\)
\(182\) 9.75156 0.722833
\(183\) −7.66847 −0.566869
\(184\) −7.62225 −0.561920
\(185\) 31.2394 2.29676
\(186\) 13.8576 1.01609
\(187\) −16.5929 −1.21339
\(188\) −14.7472 −1.07555
\(189\) −11.2755 −0.820169
\(190\) −9.22881 −0.669528
\(191\) 9.92944 0.718469 0.359234 0.933247i \(-0.383038\pi\)
0.359234 + 0.933247i \(0.383038\pi\)
\(192\) −35.3141 −2.54858
\(193\) −4.68227 −0.337037 −0.168519 0.985698i \(-0.553898\pi\)
−0.168519 + 0.985698i \(0.553898\pi\)
\(194\) −2.13231 −0.153091
\(195\) 32.2701 2.31091
\(196\) −10.9945 −0.785324
\(197\) 12.3738 0.881599 0.440799 0.897606i \(-0.354695\pi\)
0.440799 + 0.897606i \(0.354695\pi\)
\(198\) −45.9458 −3.26522
\(199\) −18.3056 −1.29765 −0.648825 0.760938i \(-0.724739\pi\)
−0.648825 + 0.760938i \(0.724739\pi\)
\(200\) 19.8316 1.40231
\(201\) 20.1171 1.41895
\(202\) −21.3862 −1.50472
\(203\) 15.3277 1.07579
\(204\) 32.1512 2.25103
\(205\) 36.7382 2.56591
\(206\) 1.23958 0.0863657
\(207\) −26.7068 −1.85625
\(208\) 5.58250 0.387077
\(209\) −4.00730 −0.277190
\(210\) 45.4308 3.13502
\(211\) 1.00000 0.0688428
\(212\) 8.75329 0.601179
\(213\) 10.5321 0.721647
\(214\) 16.1658 1.10507
\(215\) −29.5434 −2.01484
\(216\) 9.95813 0.677565
\(217\) −3.79677 −0.257742
\(218\) 16.8034 1.13807
\(219\) −8.56718 −0.578916
\(220\) 46.0189 3.10259
\(221\) −10.8976 −0.733054
\(222\) −45.7765 −3.07231
\(223\) −16.3037 −1.09178 −0.545888 0.837858i \(-0.683808\pi\)
−0.545888 + 0.837858i \(0.683808\pi\)
\(224\) 13.0222 0.870084
\(225\) 69.4859 4.63240
\(226\) −15.1305 −1.00647
\(227\) 22.6846 1.50563 0.752815 0.658232i \(-0.228696\pi\)
0.752815 + 0.658232i \(0.228696\pi\)
\(228\) 7.76474 0.514232
\(229\) −18.3444 −1.21223 −0.606116 0.795376i \(-0.707273\pi\)
−0.606116 + 0.795376i \(0.707273\pi\)
\(230\) 46.5877 3.07190
\(231\) 19.7268 1.29793
\(232\) −13.5369 −0.888744
\(233\) 17.4259 1.14161 0.570805 0.821086i \(-0.306631\pi\)
0.570805 + 0.821086i \(0.306631\pi\)
\(234\) −30.1756 −1.97264
\(235\) 23.2874 1.51910
\(236\) 25.7346 1.67518
\(237\) 41.7185 2.70991
\(238\) −15.3420 −0.994475
\(239\) −29.6240 −1.91622 −0.958110 0.286402i \(-0.907541\pi\)
−0.958110 + 0.286402i \(0.907541\pi\)
\(240\) 26.0079 1.67880
\(241\) −21.8992 −1.41065 −0.705327 0.708882i \(-0.749200\pi\)
−0.705327 + 0.708882i \(0.749200\pi\)
\(242\) 10.9626 0.704701
\(243\) −10.8079 −0.693327
\(244\) −7.18215 −0.459790
\(245\) 17.3615 1.10919
\(246\) −53.8342 −3.43234
\(247\) −2.63185 −0.167461
\(248\) 3.35319 0.212928
\(249\) 3.99454 0.253144
\(250\) −75.0681 −4.74772
\(251\) 14.9288 0.942299 0.471150 0.882053i \(-0.343839\pi\)
0.471150 + 0.882053i \(0.343839\pi\)
\(252\) −24.3920 −1.53655
\(253\) 20.2291 1.27179
\(254\) 18.1709 1.14014
\(255\) −50.7702 −3.17935
\(256\) −0.0606190 −0.00378869
\(257\) −12.5660 −0.783843 −0.391922 0.919999i \(-0.628190\pi\)
−0.391922 + 0.919999i \(0.628190\pi\)
\(258\) 43.2912 2.69519
\(259\) 12.5421 0.779326
\(260\) 30.2236 1.87439
\(261\) −47.4306 −2.93588
\(262\) 27.5034 1.69917
\(263\) 16.7367 1.03203 0.516014 0.856580i \(-0.327415\pi\)
0.516014 + 0.856580i \(0.327415\pi\)
\(264\) −17.4221 −1.07225
\(265\) −13.8224 −0.849103
\(266\) −3.70520 −0.227181
\(267\) −38.6394 −2.36469
\(268\) 18.8413 1.15091
\(269\) −8.52905 −0.520026 −0.260013 0.965605i \(-0.583727\pi\)
−0.260013 + 0.965605i \(0.583727\pi\)
\(270\) −60.8647 −3.70411
\(271\) 9.43623 0.573210 0.286605 0.958049i \(-0.407473\pi\)
0.286605 + 0.958049i \(0.407473\pi\)
\(272\) −8.78289 −0.532541
\(273\) 12.9559 0.784125
\(274\) −21.1202 −1.27592
\(275\) −52.6322 −3.17384
\(276\) −39.1969 −2.35938
\(277\) −7.66038 −0.460267 −0.230134 0.973159i \(-0.573916\pi\)
−0.230134 + 0.973159i \(0.573916\pi\)
\(278\) 25.4385 1.52570
\(279\) 11.7489 0.703387
\(280\) 10.9931 0.656964
\(281\) −12.9359 −0.771694 −0.385847 0.922563i \(-0.626091\pi\)
−0.385847 + 0.922563i \(0.626091\pi\)
\(282\) −34.1240 −2.03206
\(283\) −28.4112 −1.68887 −0.844435 0.535657i \(-0.820064\pi\)
−0.844435 + 0.535657i \(0.820064\pi\)
\(284\) 9.86418 0.585331
\(285\) −12.2614 −0.726300
\(286\) 22.8566 1.35154
\(287\) 14.7498 0.870651
\(288\) −40.2965 −2.37449
\(289\) 0.145128 0.00853695
\(290\) 82.7386 4.85858
\(291\) −2.83297 −0.166072
\(292\) −8.02387 −0.469562
\(293\) 5.53292 0.323237 0.161618 0.986853i \(-0.448329\pi\)
0.161618 + 0.986853i \(0.448329\pi\)
\(294\) −25.4406 −1.48373
\(295\) −40.6377 −2.36602
\(296\) −11.0768 −0.643823
\(297\) −26.4284 −1.53353
\(298\) 15.5013 0.897966
\(299\) 13.2858 0.768336
\(300\) 101.983 5.88798
\(301\) −11.8612 −0.683665
\(302\) 1.04802 0.0603067
\(303\) −28.4136 −1.63232
\(304\) −2.12113 −0.121655
\(305\) 11.3414 0.649406
\(306\) 47.4749 2.71396
\(307\) 18.8491 1.07577 0.537887 0.843017i \(-0.319223\pi\)
0.537887 + 0.843017i \(0.319223\pi\)
\(308\) 18.4758 1.05275
\(309\) 1.64690 0.0936890
\(310\) −20.4949 −1.16403
\(311\) 26.6786 1.51280 0.756402 0.654107i \(-0.226955\pi\)
0.756402 + 0.654107i \(0.226955\pi\)
\(312\) −11.4422 −0.647788
\(313\) 13.6310 0.770468 0.385234 0.922819i \(-0.374121\pi\)
0.385234 + 0.922819i \(0.374121\pi\)
\(314\) 28.9962 1.63635
\(315\) 38.5176 2.17022
\(316\) 39.0729 2.19802
\(317\) −29.6563 −1.66567 −0.832833 0.553524i \(-0.813283\pi\)
−0.832833 + 0.553524i \(0.813283\pi\)
\(318\) 20.2546 1.13582
\(319\) 35.9264 2.01149
\(320\) 52.2283 2.91965
\(321\) 21.4778 1.19877
\(322\) 18.7041 1.04234
\(323\) 4.14067 0.230393
\(324\) 8.40801 0.467112
\(325\) −34.5670 −1.91743
\(326\) −18.8248 −1.04261
\(327\) 22.3250 1.23457
\(328\) −13.0265 −0.719269
\(329\) 9.34947 0.515453
\(330\) 106.485 5.86179
\(331\) 8.17521 0.449350 0.224675 0.974434i \(-0.427868\pi\)
0.224675 + 0.974434i \(0.427868\pi\)
\(332\) 3.74122 0.205326
\(333\) −38.8106 −2.12681
\(334\) 6.61806 0.362124
\(335\) −29.7524 −1.62555
\(336\) 10.4417 0.569642
\(337\) 10.9093 0.594265 0.297132 0.954836i \(-0.403970\pi\)
0.297132 + 0.954836i \(0.403970\pi\)
\(338\) −13.1621 −0.715924
\(339\) −20.1024 −1.09181
\(340\) −47.5505 −2.57879
\(341\) −8.89921 −0.481919
\(342\) 11.4655 0.619985
\(343\) 18.9381 1.02256
\(344\) 10.4754 0.564795
\(345\) 61.8961 3.33238
\(346\) −43.7400 −2.35147
\(347\) −9.01824 −0.484125 −0.242062 0.970261i \(-0.577824\pi\)
−0.242062 + 0.970261i \(0.577824\pi\)
\(348\) −69.6128 −3.73164
\(349\) −14.3748 −0.769464 −0.384732 0.923028i \(-0.625706\pi\)
−0.384732 + 0.923028i \(0.625706\pi\)
\(350\) −48.6645 −2.60123
\(351\) −17.3573 −0.926463
\(352\) 30.5226 1.62686
\(353\) −24.7876 −1.31931 −0.659656 0.751568i \(-0.729298\pi\)
−0.659656 + 0.751568i \(0.729298\pi\)
\(354\) 59.5483 3.16496
\(355\) −15.5766 −0.826720
\(356\) −36.1890 −1.91801
\(357\) −20.3833 −1.07880
\(358\) −47.2047 −2.49484
\(359\) 30.2455 1.59630 0.798149 0.602460i \(-0.205813\pi\)
0.798149 + 0.602460i \(0.205813\pi\)
\(360\) −34.0175 −1.79288
\(361\) 1.00000 0.0526316
\(362\) 32.8716 1.72769
\(363\) 14.5648 0.764456
\(364\) 12.1342 0.636007
\(365\) 12.6706 0.663207
\(366\) −16.6190 −0.868691
\(367\) −20.1918 −1.05400 −0.527001 0.849864i \(-0.676684\pi\)
−0.527001 + 0.849864i \(0.676684\pi\)
\(368\) 10.7076 0.558172
\(369\) −45.6422 −2.37604
\(370\) 67.7018 3.51965
\(371\) −5.54945 −0.288113
\(372\) 17.2435 0.894036
\(373\) −10.6876 −0.553384 −0.276692 0.960959i \(-0.589238\pi\)
−0.276692 + 0.960959i \(0.589238\pi\)
\(374\) −35.9600 −1.85945
\(375\) −99.7351 −5.15030
\(376\) −8.25716 −0.425830
\(377\) 23.5952 1.21522
\(378\) −24.4361 −1.25686
\(379\) 14.9681 0.768861 0.384431 0.923154i \(-0.374398\pi\)
0.384431 + 0.923154i \(0.374398\pi\)
\(380\) −11.4838 −0.589105
\(381\) 24.1418 1.23682
\(382\) 21.5190 1.10101
\(383\) −1.34642 −0.0687988 −0.0343994 0.999408i \(-0.510952\pi\)
−0.0343994 + 0.999408i \(0.510952\pi\)
\(384\) −32.6703 −1.66720
\(385\) −29.1752 −1.48691
\(386\) −10.1474 −0.516488
\(387\) 36.7036 1.86575
\(388\) −2.65331 −0.134702
\(389\) −15.1422 −0.767739 −0.383870 0.923387i \(-0.625409\pi\)
−0.383870 + 0.923387i \(0.625409\pi\)
\(390\) 69.9355 3.54132
\(391\) −20.9024 −1.05708
\(392\) −6.15600 −0.310925
\(393\) 36.5409 1.84325
\(394\) 26.8165 1.35099
\(395\) −61.7003 −3.10448
\(396\) −57.1721 −2.87301
\(397\) −24.7214 −1.24073 −0.620365 0.784314i \(-0.713015\pi\)
−0.620365 + 0.784314i \(0.713015\pi\)
\(398\) −39.6717 −1.98856
\(399\) −4.92272 −0.246444
\(400\) −27.8591 −1.39296
\(401\) −1.13303 −0.0565806 −0.0282903 0.999600i \(-0.509006\pi\)
−0.0282903 + 0.999600i \(0.509006\pi\)
\(402\) 43.5975 2.17445
\(403\) −5.84469 −0.291145
\(404\) −26.6116 −1.32398
\(405\) −13.2771 −0.659747
\(406\) 33.2181 1.64859
\(407\) 29.3972 1.45716
\(408\) 18.0019 0.891228
\(409\) −11.7058 −0.578817 −0.289409 0.957206i \(-0.593459\pi\)
−0.289409 + 0.957206i \(0.593459\pi\)
\(410\) 79.6188 3.93209
\(411\) −28.0602 −1.38411
\(412\) 1.54246 0.0759915
\(413\) −16.3153 −0.802826
\(414\) −57.8788 −2.84459
\(415\) −5.90779 −0.290002
\(416\) 20.0462 0.982846
\(417\) 33.7975 1.65507
\(418\) −8.68458 −0.424777
\(419\) 13.4807 0.658577 0.329289 0.944229i \(-0.393191\pi\)
0.329289 + 0.944229i \(0.393191\pi\)
\(420\) 56.5314 2.75845
\(421\) 22.2022 1.08207 0.541036 0.841000i \(-0.318032\pi\)
0.541036 + 0.841000i \(0.318032\pi\)
\(422\) 2.16719 0.105497
\(423\) −28.9314 −1.40669
\(424\) 4.90110 0.238018
\(425\) 54.3840 2.63801
\(426\) 22.8251 1.10588
\(427\) 4.55337 0.220353
\(428\) 20.1157 0.972330
\(429\) 30.3671 1.46614
\(430\) −64.0262 −3.08762
\(431\) −37.5740 −1.80988 −0.904939 0.425542i \(-0.860084\pi\)
−0.904939 + 0.425542i \(0.860084\pi\)
\(432\) −13.9890 −0.673046
\(433\) 21.1055 1.01426 0.507132 0.861868i \(-0.330706\pi\)
0.507132 + 0.861868i \(0.330706\pi\)
\(434\) −8.22833 −0.394973
\(435\) 109.926 5.27055
\(436\) 20.9092 1.00137
\(437\) −5.04807 −0.241482
\(438\) −18.5667 −0.887152
\(439\) 27.9490 1.33393 0.666965 0.745089i \(-0.267593\pi\)
0.666965 + 0.745089i \(0.267593\pi\)
\(440\) 25.7666 1.22838
\(441\) −21.5693 −1.02711
\(442\) −23.6173 −1.12336
\(443\) 2.46109 0.116930 0.0584648 0.998289i \(-0.481379\pi\)
0.0584648 + 0.998289i \(0.481379\pi\)
\(444\) −56.9615 −2.70327
\(445\) 57.1463 2.70899
\(446\) −35.3333 −1.67308
\(447\) 20.5949 0.974107
\(448\) 20.9688 0.990681
\(449\) 14.3210 0.675851 0.337925 0.941173i \(-0.390275\pi\)
0.337925 + 0.941173i \(0.390275\pi\)
\(450\) 150.589 7.09885
\(451\) 34.5718 1.62792
\(452\) −18.8275 −0.885572
\(453\) 1.39239 0.0654203
\(454\) 49.1619 2.30728
\(455\) −19.1613 −0.898295
\(456\) 4.34759 0.203594
\(457\) −0.831442 −0.0388932 −0.0194466 0.999811i \(-0.506190\pi\)
−0.0194466 + 0.999811i \(0.506190\pi\)
\(458\) −39.7558 −1.85767
\(459\) 27.3080 1.27463
\(460\) 57.9709 2.70291
\(461\) 33.1750 1.54511 0.772557 0.634946i \(-0.218978\pi\)
0.772557 + 0.634946i \(0.218978\pi\)
\(462\) 42.7518 1.98899
\(463\) −20.3065 −0.943724 −0.471862 0.881672i \(-0.656418\pi\)
−0.471862 + 0.881672i \(0.656418\pi\)
\(464\) 19.0165 0.882817
\(465\) −27.2294 −1.26273
\(466\) 37.7653 1.74944
\(467\) −10.5297 −0.487257 −0.243629 0.969869i \(-0.578338\pi\)
−0.243629 + 0.969869i \(0.578338\pi\)
\(468\) −37.5487 −1.73569
\(469\) −11.9451 −0.551572
\(470\) 50.4682 2.32793
\(471\) 38.5242 1.77510
\(472\) 14.4092 0.663237
\(473\) −27.8012 −1.27830
\(474\) 90.4121 4.15277
\(475\) 13.1341 0.602634
\(476\) −19.0907 −0.875020
\(477\) 17.1724 0.786271
\(478\) −64.2010 −2.93648
\(479\) −34.6808 −1.58461 −0.792303 0.610128i \(-0.791118\pi\)
−0.792303 + 0.610128i \(0.791118\pi\)
\(480\) 93.3918 4.26273
\(481\) 19.3071 0.880326
\(482\) −47.4599 −2.16174
\(483\) 24.8502 1.13072
\(484\) 13.6412 0.620053
\(485\) 4.18986 0.190252
\(486\) −23.4228 −1.06248
\(487\) −15.9807 −0.724154 −0.362077 0.932148i \(-0.617932\pi\)
−0.362077 + 0.932148i \(0.617932\pi\)
\(488\) −4.02139 −0.182040
\(489\) −25.0106 −1.13102
\(490\) 37.6258 1.69976
\(491\) −21.4119 −0.966306 −0.483153 0.875536i \(-0.660508\pi\)
−0.483153 + 0.875536i \(0.660508\pi\)
\(492\) −66.9880 −3.02005
\(493\) −37.1221 −1.67190
\(494\) −5.70373 −0.256623
\(495\) 90.2809 4.05783
\(496\) −4.71050 −0.211508
\(497\) −6.25373 −0.280518
\(498\) 8.65695 0.387927
\(499\) −14.4330 −0.646112 −0.323056 0.946380i \(-0.604710\pi\)
−0.323056 + 0.946380i \(0.604710\pi\)
\(500\) −93.4102 −4.17743
\(501\) 8.79272 0.392830
\(502\) 32.3537 1.44401
\(503\) 14.5977 0.650881 0.325440 0.945563i \(-0.394487\pi\)
0.325440 + 0.945563i \(0.394487\pi\)
\(504\) −13.6574 −0.608351
\(505\) 42.0226 1.86998
\(506\) 43.8404 1.94894
\(507\) −17.4871 −0.776630
\(508\) 22.6108 1.00319
\(509\) 26.4523 1.17248 0.586240 0.810138i \(-0.300608\pi\)
0.586240 + 0.810138i \(0.300608\pi\)
\(510\) −110.029 −4.87216
\(511\) 5.08701 0.225036
\(512\) 22.5617 0.997094
\(513\) 6.59508 0.291180
\(514\) −27.2329 −1.20119
\(515\) −2.43571 −0.107330
\(516\) 53.8690 2.37145
\(517\) 21.9141 0.963782
\(518\) 27.1811 1.19427
\(519\) −58.1127 −2.55086
\(520\) 16.9226 0.742107
\(521\) −14.3554 −0.628921 −0.314460 0.949271i \(-0.601824\pi\)
−0.314460 + 0.949271i \(0.601824\pi\)
\(522\) −102.791 −4.49905
\(523\) −25.3097 −1.10672 −0.553359 0.832943i \(-0.686654\pi\)
−0.553359 + 0.832943i \(0.686654\pi\)
\(524\) 34.2236 1.49507
\(525\) −64.6555 −2.82180
\(526\) 36.2716 1.58152
\(527\) 9.19539 0.400557
\(528\) 24.4742 1.06510
\(529\) 2.48300 0.107956
\(530\) −29.9558 −1.30120
\(531\) 50.4868 2.19094
\(532\) −4.61053 −0.199892
\(533\) 22.7055 0.983486
\(534\) −83.7390 −3.62374
\(535\) −31.7649 −1.37332
\(536\) 10.5495 0.455669
\(537\) −62.7159 −2.70639
\(538\) −18.4841 −0.796906
\(539\) 16.3377 0.703716
\(540\) −75.7363 −3.25917
\(541\) 37.4827 1.61151 0.805754 0.592250i \(-0.201760\pi\)
0.805754 + 0.592250i \(0.201760\pi\)
\(542\) 20.4501 0.878408
\(543\) 43.6730 1.87419
\(544\) −31.5385 −1.35220
\(545\) −33.0179 −1.41433
\(546\) 28.0779 1.20162
\(547\) −42.8872 −1.83372 −0.916862 0.399203i \(-0.869287\pi\)
−0.916862 + 0.399203i \(0.869287\pi\)
\(548\) −26.2807 −1.12266
\(549\) −14.0901 −0.601352
\(550\) −114.064 −4.86371
\(551\) −8.96525 −0.381933
\(552\) −21.9469 −0.934123
\(553\) −24.7716 −1.05339
\(554\) −16.6015 −0.705331
\(555\) 89.9482 3.81809
\(556\) 31.6542 1.34244
\(557\) −21.4499 −0.908863 −0.454432 0.890782i \(-0.650158\pi\)
−0.454432 + 0.890782i \(0.650158\pi\)
\(558\) 25.4621 1.07790
\(559\) −18.2589 −0.772268
\(560\) −15.4429 −0.652583
\(561\) −47.7763 −2.01712
\(562\) −28.0347 −1.18257
\(563\) −3.36460 −0.141801 −0.0709005 0.997483i \(-0.522587\pi\)
−0.0709005 + 0.997483i \(0.522587\pi\)
\(564\) −42.4619 −1.78797
\(565\) 29.7307 1.25078
\(566\) −61.5726 −2.58809
\(567\) −5.33054 −0.223862
\(568\) 5.52310 0.231744
\(569\) 40.1048 1.68128 0.840640 0.541595i \(-0.182179\pi\)
0.840640 + 0.541595i \(0.182179\pi\)
\(570\) −26.5727 −1.11301
\(571\) −8.77625 −0.367275 −0.183637 0.982994i \(-0.558787\pi\)
−0.183637 + 0.982994i \(0.558787\pi\)
\(572\) 28.4413 1.18919
\(573\) 28.5900 1.19437
\(574\) 31.9656 1.33422
\(575\) −66.3019 −2.76498
\(576\) −64.8865 −2.70361
\(577\) −32.3697 −1.34757 −0.673784 0.738928i \(-0.735332\pi\)
−0.673784 + 0.738928i \(0.735332\pi\)
\(578\) 0.314521 0.0130823
\(579\) −13.4818 −0.560283
\(580\) 102.955 4.27497
\(581\) −2.37187 −0.0984019
\(582\) −6.13959 −0.254494
\(583\) −13.0073 −0.538707
\(584\) −4.49268 −0.185908
\(585\) 59.2934 2.45148
\(586\) 11.9909 0.495340
\(587\) 9.09005 0.375186 0.187593 0.982247i \(-0.439931\pi\)
0.187593 + 0.982247i \(0.439931\pi\)
\(588\) −31.6568 −1.30550
\(589\) 2.22075 0.0915044
\(590\) −88.0698 −3.62578
\(591\) 35.6282 1.46555
\(592\) 15.5604 0.639529
\(593\) −20.4309 −0.838996 −0.419498 0.907756i \(-0.637794\pi\)
−0.419498 + 0.907756i \(0.637794\pi\)
\(594\) −57.2755 −2.35004
\(595\) 30.1462 1.23588
\(596\) 19.2889 0.790103
\(597\) −52.7077 −2.15718
\(598\) 28.7928 1.17743
\(599\) −18.5797 −0.759147 −0.379573 0.925162i \(-0.623929\pi\)
−0.379573 + 0.925162i \(0.623929\pi\)
\(600\) 57.1017 2.33117
\(601\) 17.2954 0.705493 0.352746 0.935719i \(-0.385248\pi\)
0.352746 + 0.935719i \(0.385248\pi\)
\(602\) −25.7054 −1.04767
\(603\) 36.9633 1.50526
\(604\) 1.30409 0.0530627
\(605\) −21.5409 −0.875761
\(606\) −61.5776 −2.50142
\(607\) 24.4841 0.993778 0.496889 0.867814i \(-0.334476\pi\)
0.496889 + 0.867814i \(0.334476\pi\)
\(608\) −7.61676 −0.308901
\(609\) 44.1334 1.78838
\(610\) 24.5790 0.995174
\(611\) 14.3924 0.582256
\(612\) 59.0749 2.38796
\(613\) −29.0335 −1.17265 −0.586326 0.810075i \(-0.699426\pi\)
−0.586326 + 0.810075i \(0.699426\pi\)
\(614\) 40.8496 1.64855
\(615\) 105.781 4.26551
\(616\) 10.3449 0.416806
\(617\) −28.1947 −1.13508 −0.567538 0.823347i \(-0.692104\pi\)
−0.567538 + 0.823347i \(0.692104\pi\)
\(618\) 3.56915 0.143572
\(619\) 45.9958 1.84873 0.924364 0.381512i \(-0.124597\pi\)
0.924364 + 0.381512i \(0.124597\pi\)
\(620\) −25.5026 −1.02421
\(621\) −33.2924 −1.33598
\(622\) 57.8177 2.31828
\(623\) 22.9432 0.919201
\(624\) 16.0738 0.643468
\(625\) 81.8342 3.27337
\(626\) 29.5409 1.18069
\(627\) −11.5383 −0.460795
\(628\) 36.0811 1.43979
\(629\) −30.3756 −1.21115
\(630\) 83.4750 3.32573
\(631\) 0.940049 0.0374228 0.0187114 0.999825i \(-0.494044\pi\)
0.0187114 + 0.999825i \(0.494044\pi\)
\(632\) 21.8775 0.870239
\(633\) 2.87932 0.114443
\(634\) −64.2710 −2.55253
\(635\) −35.7048 −1.41690
\(636\) 25.2036 0.999386
\(637\) 10.7301 0.425140
\(638\) 77.8595 3.08249
\(639\) 19.3518 0.765544
\(640\) 48.3182 1.90994
\(641\) −29.3880 −1.16076 −0.580379 0.814347i \(-0.697095\pi\)
−0.580379 + 0.814347i \(0.697095\pi\)
\(642\) 46.5465 1.83704
\(643\) −11.6452 −0.459242 −0.229621 0.973280i \(-0.573749\pi\)
−0.229621 + 0.973280i \(0.573749\pi\)
\(644\) 23.2743 0.917135
\(645\) −85.0649 −3.34943
\(646\) 8.97362 0.353063
\(647\) 40.9315 1.60918 0.804591 0.593829i \(-0.202384\pi\)
0.804591 + 0.593829i \(0.202384\pi\)
\(648\) 4.70777 0.184939
\(649\) −38.2413 −1.50110
\(650\) −74.9134 −2.93835
\(651\) −10.9321 −0.428464
\(652\) −23.4245 −0.917373
\(653\) 26.5609 1.03941 0.519704 0.854346i \(-0.326042\pi\)
0.519704 + 0.854346i \(0.326042\pi\)
\(654\) 48.3825 1.89191
\(655\) −54.0428 −2.11162
\(656\) 18.2994 0.714472
\(657\) −15.7414 −0.614131
\(658\) 20.2621 0.789899
\(659\) −29.1176 −1.13426 −0.567130 0.823628i \(-0.691946\pi\)
−0.567130 + 0.823628i \(0.691946\pi\)
\(660\) 132.503 5.15768
\(661\) −34.2846 −1.33352 −0.666759 0.745274i \(-0.732319\pi\)
−0.666759 + 0.745274i \(0.732319\pi\)
\(662\) 17.7173 0.688601
\(663\) −31.3778 −1.21861
\(664\) 2.09476 0.0812926
\(665\) 7.28053 0.282327
\(666\) −84.1101 −3.25920
\(667\) 45.2572 1.75237
\(668\) 8.23511 0.318626
\(669\) −46.9436 −1.81494
\(670\) −64.4792 −2.49105
\(671\) 10.6726 0.412011
\(672\) 37.4952 1.44641
\(673\) −42.7027 −1.64607 −0.823033 0.567993i \(-0.807720\pi\)
−0.823033 + 0.567993i \(0.807720\pi\)
\(674\) 23.6424 0.910673
\(675\) 86.6204 3.33402
\(676\) −16.3781 −0.629928
\(677\) −16.4776 −0.633286 −0.316643 0.948545i \(-0.602556\pi\)
−0.316643 + 0.948545i \(0.602556\pi\)
\(678\) −43.5657 −1.67313
\(679\) 1.68216 0.0645553
\(680\) −26.6242 −1.02099
\(681\) 65.3163 2.50292
\(682\) −19.2863 −0.738510
\(683\) 32.2742 1.23494 0.617468 0.786596i \(-0.288158\pi\)
0.617468 + 0.786596i \(0.288158\pi\)
\(684\) 14.2670 0.545513
\(685\) 41.5001 1.58564
\(686\) 41.0425 1.56701
\(687\) −52.8194 −2.01519
\(688\) −14.7156 −0.561029
\(689\) −8.54273 −0.325452
\(690\) 134.141 5.10666
\(691\) −0.935249 −0.0355786 −0.0177893 0.999842i \(-0.505663\pi\)
−0.0177893 + 0.999842i \(0.505663\pi\)
\(692\) −54.4274 −2.06902
\(693\) 36.2462 1.37688
\(694\) −19.5443 −0.741890
\(695\) −49.9853 −1.89605
\(696\) −38.9772 −1.47743
\(697\) −35.7224 −1.35308
\(698\) −31.1529 −1.17915
\(699\) 50.1748 1.89779
\(700\) −60.5552 −2.28877
\(701\) −18.2432 −0.689036 −0.344518 0.938780i \(-0.611958\pi\)
−0.344518 + 0.938780i \(0.611958\pi\)
\(702\) −37.6166 −1.41975
\(703\) −7.33592 −0.276679
\(704\) 49.1484 1.85235
\(705\) 67.0519 2.52532
\(706\) −53.7196 −2.02176
\(707\) 16.8714 0.634513
\(708\) 74.0983 2.78478
\(709\) 41.1922 1.54701 0.773503 0.633793i \(-0.218503\pi\)
0.773503 + 0.633793i \(0.218503\pi\)
\(710\) −33.7575 −1.26690
\(711\) 76.6541 2.87475
\(712\) −20.2627 −0.759378
\(713\) −11.2105 −0.419837
\(714\) −44.1746 −1.65319
\(715\) −44.9119 −1.67961
\(716\) −58.7386 −2.19517
\(717\) −85.2971 −3.18548
\(718\) 65.5479 2.44623
\(719\) 47.8093 1.78299 0.891493 0.453035i \(-0.149659\pi\)
0.891493 + 0.453035i \(0.149659\pi\)
\(720\) 47.7872 1.78092
\(721\) −0.977895 −0.0364187
\(722\) 2.16719 0.0806546
\(723\) −63.0550 −2.34504
\(724\) 40.9034 1.52016
\(725\) −117.751 −4.37315
\(726\) 31.5648 1.17148
\(727\) 0.453591 0.0168228 0.00841138 0.999965i \(-0.497323\pi\)
0.00841138 + 0.999965i \(0.497323\pi\)
\(728\) 6.79414 0.251808
\(729\) −40.4730 −1.49900
\(730\) 27.4595 1.01632
\(731\) 28.7265 1.06249
\(732\) −20.6797 −0.764345
\(733\) 23.3673 0.863090 0.431545 0.902091i \(-0.357969\pi\)
0.431545 + 0.902091i \(0.357969\pi\)
\(734\) −43.7595 −1.61519
\(735\) 49.9895 1.84389
\(736\) 38.4499 1.41728
\(737\) −27.9979 −1.03132
\(738\) −98.9154 −3.64113
\(739\) 29.2472 1.07588 0.537938 0.842985i \(-0.319204\pi\)
0.537938 + 0.842985i \(0.319204\pi\)
\(740\) 84.2440 3.09687
\(741\) −7.57795 −0.278383
\(742\) −12.0267 −0.441515
\(743\) −35.9322 −1.31822 −0.659112 0.752045i \(-0.729068\pi\)
−0.659112 + 0.752045i \(0.729068\pi\)
\(744\) 9.65491 0.353966
\(745\) −30.4592 −1.11594
\(746\) −23.1621 −0.848026
\(747\) 7.33962 0.268542
\(748\) −44.7464 −1.63609
\(749\) −12.7530 −0.465986
\(750\) −216.145 −7.89251
\(751\) −47.5286 −1.73434 −0.867172 0.498009i \(-0.834065\pi\)
−0.867172 + 0.498009i \(0.834065\pi\)
\(752\) 11.5995 0.422990
\(753\) 42.9849 1.56646
\(754\) 51.1354 1.86224
\(755\) −2.05930 −0.0749456
\(756\) −30.4068 −1.10589
\(757\) 44.2924 1.60984 0.804918 0.593387i \(-0.202209\pi\)
0.804918 + 0.593387i \(0.202209\pi\)
\(758\) 32.4388 1.17823
\(759\) 58.2461 2.11420
\(760\) −6.42993 −0.233238
\(761\) −23.3740 −0.847307 −0.423653 0.905824i \(-0.639253\pi\)
−0.423653 + 0.905824i \(0.639253\pi\)
\(762\) 52.3199 1.89535
\(763\) −13.2561 −0.479903
\(764\) 26.7769 0.968756
\(765\) −93.2856 −3.37275
\(766\) −2.91795 −0.105430
\(767\) −25.1156 −0.906871
\(768\) −0.174542 −0.00629823
\(769\) 24.4875 0.883041 0.441521 0.897251i \(-0.354439\pi\)
0.441521 + 0.897251i \(0.354439\pi\)
\(770\) −63.2283 −2.27859
\(771\) −36.1814 −1.30304
\(772\) −12.6268 −0.454448
\(773\) 23.5308 0.846343 0.423171 0.906050i \(-0.360917\pi\)
0.423171 + 0.906050i \(0.360917\pi\)
\(774\) 79.5438 2.85914
\(775\) 29.1676 1.04773
\(776\) −1.48563 −0.0533309
\(777\) 36.1126 1.29553
\(778\) −32.8160 −1.17651
\(779\) −8.62721 −0.309102
\(780\) 87.0235 3.11594
\(781\) −14.6580 −0.524506
\(782\) −45.2995 −1.61991
\(783\) −59.1265 −2.11301
\(784\) 8.64783 0.308851
\(785\) −56.9760 −2.03356
\(786\) 79.1912 2.82466
\(787\) 22.5911 0.805287 0.402644 0.915357i \(-0.368091\pi\)
0.402644 + 0.915357i \(0.368091\pi\)
\(788\) 33.3688 1.18871
\(789\) 48.1903 1.71562
\(790\) −133.716 −4.75741
\(791\) 11.9363 0.424408
\(792\) −32.0115 −1.13748
\(793\) 7.00939 0.248911
\(794\) −53.5759 −1.90134
\(795\) −39.7991 −1.41153
\(796\) −49.3651 −1.74970
\(797\) 11.7605 0.416577 0.208289 0.978067i \(-0.433211\pi\)
0.208289 + 0.978067i \(0.433211\pi\)
\(798\) −10.6685 −0.377660
\(799\) −22.6435 −0.801068
\(800\) −100.039 −3.53693
\(801\) −70.9964 −2.50853
\(802\) −2.45549 −0.0867062
\(803\) 11.9234 0.420767
\(804\) 54.2501 1.91325
\(805\) −36.7526 −1.29536
\(806\) −12.6666 −0.446161
\(807\) −24.5579 −0.864479
\(808\) −14.9002 −0.524189
\(809\) −49.8893 −1.75401 −0.877006 0.480479i \(-0.840463\pi\)
−0.877006 + 0.480479i \(0.840463\pi\)
\(810\) −28.7741 −1.01102
\(811\) −48.6349 −1.70780 −0.853900 0.520437i \(-0.825769\pi\)
−0.853900 + 0.520437i \(0.825769\pi\)
\(812\) 41.3346 1.45056
\(813\) 27.1699 0.952891
\(814\) 63.7094 2.23301
\(815\) 36.9898 1.29569
\(816\) −25.2888 −0.885284
\(817\) 6.93765 0.242718
\(818\) −25.3688 −0.887000
\(819\) 23.8053 0.831823
\(820\) 99.0729 3.45977
\(821\) −25.0945 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(822\) −60.8119 −2.12106
\(823\) −48.1808 −1.67948 −0.839738 0.542992i \(-0.817291\pi\)
−0.839738 + 0.542992i \(0.817291\pi\)
\(824\) 0.863646 0.0300865
\(825\) −151.545 −5.27613
\(826\) −35.3585 −1.23028
\(827\) −46.7300 −1.62496 −0.812481 0.582988i \(-0.801883\pi\)
−0.812481 + 0.582988i \(0.801883\pi\)
\(828\) −72.0208 −2.50290
\(829\) −32.1946 −1.11817 −0.559083 0.829112i \(-0.688847\pi\)
−0.559083 + 0.829112i \(0.688847\pi\)
\(830\) −12.8033 −0.444410
\(831\) −22.0567 −0.765138
\(832\) 32.2790 1.11907
\(833\) −16.8815 −0.584909
\(834\) 73.2457 2.53629
\(835\) −13.0041 −0.450026
\(836\) −10.8066 −0.373753
\(837\) 14.6460 0.506240
\(838\) 29.2154 1.00923
\(839\) −38.0886 −1.31496 −0.657482 0.753470i \(-0.728378\pi\)
−0.657482 + 0.753470i \(0.728378\pi\)
\(840\) 31.6527 1.09212
\(841\) 51.3758 1.77158
\(842\) 48.1165 1.65821
\(843\) −37.2467 −1.28285
\(844\) 2.69672 0.0928251
\(845\) 25.8628 0.889708
\(846\) −62.6998 −2.15566
\(847\) −8.64828 −0.297159
\(848\) −6.88497 −0.236431
\(849\) −81.8050 −2.80754
\(850\) 117.861 4.04258
\(851\) 37.0322 1.26945
\(852\) 28.4021 0.973041
\(853\) 8.25536 0.282658 0.141329 0.989963i \(-0.454862\pi\)
0.141329 + 0.989963i \(0.454862\pi\)
\(854\) 9.86803 0.337677
\(855\) −22.5291 −0.770480
\(856\) 11.2631 0.384964
\(857\) −14.7310 −0.503201 −0.251600 0.967831i \(-0.580957\pi\)
−0.251600 + 0.967831i \(0.580957\pi\)
\(858\) 65.8114 2.24676
\(859\) −36.6395 −1.25012 −0.625061 0.780576i \(-0.714926\pi\)
−0.625061 + 0.780576i \(0.714926\pi\)
\(860\) −79.6704 −2.71674
\(861\) 42.4693 1.44735
\(862\) −81.4302 −2.77352
\(863\) −9.39637 −0.319856 −0.159928 0.987129i \(-0.551126\pi\)
−0.159928 + 0.987129i \(0.551126\pi\)
\(864\) −50.2331 −1.70897
\(865\) 85.9466 2.92227
\(866\) 45.7396 1.55430
\(867\) 0.417871 0.0141916
\(868\) −10.2388 −0.347529
\(869\) −58.0618 −1.96961
\(870\) 238.231 8.07679
\(871\) −18.3881 −0.623056
\(872\) 11.7074 0.396461
\(873\) −5.20533 −0.176174
\(874\) −10.9401 −0.370056
\(875\) 59.2206 2.00202
\(876\) −23.1033 −0.780589
\(877\) −26.1174 −0.881920 −0.440960 0.897527i \(-0.645362\pi\)
−0.440960 + 0.897527i \(0.645362\pi\)
\(878\) 60.5708 2.04416
\(879\) 15.9311 0.537341
\(880\) −36.1965 −1.22018
\(881\) −29.0872 −0.979974 −0.489987 0.871730i \(-0.662998\pi\)
−0.489987 + 0.871730i \(0.662998\pi\)
\(882\) −46.7449 −1.57398
\(883\) 54.4577 1.83265 0.916324 0.400437i \(-0.131142\pi\)
0.916324 + 0.400437i \(0.131142\pi\)
\(884\) −29.3879 −0.988422
\(885\) −117.009 −3.93322
\(886\) 5.33365 0.179187
\(887\) −40.3775 −1.35574 −0.677872 0.735180i \(-0.737098\pi\)
−0.677872 + 0.735180i \(0.737098\pi\)
\(888\) −31.8935 −1.07028
\(889\) −14.3349 −0.480776
\(890\) 123.847 4.15136
\(891\) −12.4942 −0.418571
\(892\) −43.9666 −1.47211
\(893\) −5.46855 −0.182998
\(894\) 44.6332 1.49276
\(895\) 92.7546 3.10044
\(896\) 19.3989 0.648072
\(897\) 38.2540 1.27727
\(898\) 31.0364 1.03570
\(899\) −19.9096 −0.664022
\(900\) 187.384 6.24615
\(901\) 13.4402 0.447758
\(902\) 74.9237 2.49469
\(903\) −34.1521 −1.13651
\(904\) −10.5418 −0.350615
\(905\) −64.5909 −2.14707
\(906\) 3.01758 0.100252
\(907\) −7.48800 −0.248635 −0.124318 0.992242i \(-0.539674\pi\)
−0.124318 + 0.992242i \(0.539674\pi\)
\(908\) 61.1741 2.03013
\(909\) −52.2074 −1.73161
\(910\) −41.5262 −1.37658
\(911\) −30.2244 −1.00138 −0.500690 0.865627i \(-0.666920\pi\)
−0.500690 + 0.865627i \(0.666920\pi\)
\(912\) −6.10741 −0.202237
\(913\) −5.55941 −0.183989
\(914\) −1.80190 −0.0596014
\(915\) 32.6555 1.07956
\(916\) −49.4698 −1.63453
\(917\) −21.6972 −0.716505
\(918\) 59.1817 1.95329
\(919\) 34.0006 1.12158 0.560788 0.827959i \(-0.310498\pi\)
0.560788 + 0.827959i \(0.310498\pi\)
\(920\) 32.4587 1.07013
\(921\) 54.2726 1.78834
\(922\) 71.8966 2.36779
\(923\) −9.62689 −0.316873
\(924\) 53.1977 1.75008
\(925\) −96.3507 −3.16799
\(926\) −44.0081 −1.44620
\(927\) 3.02603 0.0993880
\(928\) 68.2862 2.24160
\(929\) 22.5268 0.739082 0.369541 0.929214i \(-0.379515\pi\)
0.369541 + 0.929214i \(0.379515\pi\)
\(930\) −59.0114 −1.93506
\(931\) −4.07700 −0.133618
\(932\) 46.9929 1.53930
\(933\) 76.8163 2.51485
\(934\) −22.8199 −0.746691
\(935\) 70.6594 2.31081
\(936\) −21.0240 −0.687192
\(937\) 21.1318 0.690345 0.345172 0.938539i \(-0.387820\pi\)
0.345172 + 0.938539i \(0.387820\pi\)
\(938\) −25.8873 −0.845249
\(939\) 39.2479 1.28081
\(940\) 62.7996 2.04830
\(941\) 2.99282 0.0975632 0.0487816 0.998809i \(-0.484466\pi\)
0.0487816 + 0.998809i \(0.484466\pi\)
\(942\) 83.4894 2.72023
\(943\) 43.5507 1.41821
\(944\) −20.2418 −0.658814
\(945\) 48.0156 1.56195
\(946\) −60.2506 −1.95892
\(947\) −17.4832 −0.568129 −0.284065 0.958805i \(-0.591683\pi\)
−0.284065 + 0.958805i \(0.591683\pi\)
\(948\) 112.503 3.65394
\(949\) 7.83086 0.254200
\(950\) 28.4641 0.923498
\(951\) −85.3901 −2.76897
\(952\) −10.6892 −0.346437
\(953\) 2.09617 0.0679017 0.0339509 0.999424i \(-0.489191\pi\)
0.0339509 + 0.999424i \(0.489191\pi\)
\(954\) 37.2159 1.20491
\(955\) −42.2837 −1.36827
\(956\) −79.8878 −2.58376
\(957\) 103.444 3.34386
\(958\) −75.1600 −2.42831
\(959\) 16.6615 0.538029
\(960\) 150.382 4.85357
\(961\) −26.0683 −0.840912
\(962\) 41.8421 1.34904
\(963\) 39.4635 1.27169
\(964\) −59.0562 −1.90207
\(965\) 19.9390 0.641860
\(966\) 53.8552 1.73276
\(967\) 22.9660 0.738535 0.369268 0.929323i \(-0.379609\pi\)
0.369268 + 0.929323i \(0.379609\pi\)
\(968\) 7.63789 0.245491
\(969\) 11.9223 0.383000
\(970\) 9.08024 0.291549
\(971\) 32.5638 1.04502 0.522511 0.852633i \(-0.324996\pi\)
0.522511 + 0.852633i \(0.324996\pi\)
\(972\) −29.1459 −0.934856
\(973\) −20.0682 −0.643358
\(974\) −34.6332 −1.10972
\(975\) −99.5296 −3.18750
\(976\) 5.64918 0.180826
\(977\) −30.0208 −0.960450 −0.480225 0.877145i \(-0.659445\pi\)
−0.480225 + 0.877145i \(0.659445\pi\)
\(978\) −54.2027 −1.73321
\(979\) 53.7763 1.71870
\(980\) 46.8193 1.49559
\(981\) 41.0202 1.30967
\(982\) −46.4037 −1.48080
\(983\) 46.4133 1.48035 0.740177 0.672412i \(-0.234742\pi\)
0.740177 + 0.672412i \(0.234742\pi\)
\(984\) −37.5075 −1.19570
\(985\) −52.6929 −1.67894
\(986\) −80.4508 −2.56208
\(987\) 26.9201 0.856878
\(988\) −7.09738 −0.225798
\(989\) −35.0217 −1.11363
\(990\) 195.656 6.21836
\(991\) 54.3961 1.72795 0.863974 0.503536i \(-0.167968\pi\)
0.863974 + 0.503536i \(0.167968\pi\)
\(992\) −16.9149 −0.537050
\(993\) 23.5391 0.746990
\(994\) −13.5530 −0.429876
\(995\) 77.9528 2.47127
\(996\) 10.7722 0.341330
\(997\) −42.4442 −1.34422 −0.672111 0.740451i \(-0.734612\pi\)
−0.672111 + 0.740451i \(0.734612\pi\)
\(998\) −31.2792 −0.990125
\(999\) −48.3809 −1.53070
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 4009.2.a.c.1.65 71
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4009.2.a.c.1.65 71 1.1 even 1 trivial