Properties

Label 8045.2.a.c.1.12
Level $8045$
Weight $2$
Character 8045.1
Self dual yes
Analytic conductor $64.240$
Analytic rank $1$
Dimension $127$
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] = [8045,2,Mod(1,8045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8045, 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("8045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8045 = 5 \cdot 1609 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8045.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: \(64.2396484261\)
Analytic rank: \(1\)
Dimension: \(127\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) \(=\) 8045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.49863 q^{2} +2.41476 q^{3} +4.24316 q^{4} +1.00000 q^{5} -6.03361 q^{6} +1.44699 q^{7} -5.60483 q^{8} +2.83108 q^{9} +O(q^{10})\) \(q-2.49863 q^{2} +2.41476 q^{3} +4.24316 q^{4} +1.00000 q^{5} -6.03361 q^{6} +1.44699 q^{7} -5.60483 q^{8} +2.83108 q^{9} -2.49863 q^{10} -2.48494 q^{11} +10.2462 q^{12} -5.50824 q^{13} -3.61550 q^{14} +2.41476 q^{15} +5.51809 q^{16} +1.30802 q^{17} -7.07384 q^{18} +7.78323 q^{19} +4.24316 q^{20} +3.49414 q^{21} +6.20895 q^{22} -4.73454 q^{23} -13.5343 q^{24} +1.00000 q^{25} +13.7631 q^{26} -0.407891 q^{27} +6.13981 q^{28} +3.55204 q^{29} -6.03361 q^{30} +0.736339 q^{31} -2.57800 q^{32} -6.00055 q^{33} -3.26825 q^{34} +1.44699 q^{35} +12.0127 q^{36} -8.95720 q^{37} -19.4474 q^{38} -13.3011 q^{39} -5.60483 q^{40} +1.00361 q^{41} -8.73057 q^{42} -10.8969 q^{43} -10.5440 q^{44} +2.83108 q^{45} +11.8299 q^{46} +3.12224 q^{47} +13.3249 q^{48} -4.90622 q^{49} -2.49863 q^{50} +3.15855 q^{51} -23.3723 q^{52} -11.6301 q^{53} +1.01917 q^{54} -2.48494 q^{55} -8.11014 q^{56} +18.7947 q^{57} -8.87523 q^{58} +0.923057 q^{59} +10.2462 q^{60} -9.46226 q^{61} -1.83984 q^{62} +4.09655 q^{63} -4.59469 q^{64} -5.50824 q^{65} +14.9932 q^{66} -3.31802 q^{67} +5.55012 q^{68} -11.4328 q^{69} -3.61550 q^{70} +10.2493 q^{71} -15.8677 q^{72} -7.50674 q^{73} +22.3807 q^{74} +2.41476 q^{75} +33.0255 q^{76} -3.59569 q^{77} +33.2345 q^{78} +4.07064 q^{79} +5.51809 q^{80} -9.47821 q^{81} -2.50765 q^{82} +8.23799 q^{83} +14.8262 q^{84} +1.30802 q^{85} +27.2273 q^{86} +8.57733 q^{87} +13.9277 q^{88} -4.44209 q^{89} -7.07384 q^{90} -7.97037 q^{91} -20.0894 q^{92} +1.77808 q^{93} -7.80132 q^{94} +7.78323 q^{95} -6.22527 q^{96} +4.92639 q^{97} +12.2588 q^{98} -7.03508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9} - 20 q^{10} - 32 q^{11} - 65 q^{12} - 49 q^{13} - 4 q^{14} - 31 q^{15} + 98 q^{16} - 53 q^{17} - 60 q^{18} - 126 q^{19} + 116 q^{20} - 24 q^{21} - 46 q^{22} - 121 q^{23} - 51 q^{24} + 127 q^{25} - 9 q^{26} - 112 q^{27} - 123 q^{28} - 6 q^{29} - 17 q^{30} - 54 q^{31} - 119 q^{32} - 57 q^{33} - 53 q^{34} - 63 q^{35} + 133 q^{36} - 61 q^{37} - 27 q^{38} - 33 q^{39} - 57 q^{40} - 8 q^{41} - 46 q^{42} - 208 q^{43} - 54 q^{44} + 122 q^{45} - 34 q^{46} - 116 q^{47} - 106 q^{48} + 110 q^{49} - 20 q^{50} - 54 q^{51} - 142 q^{52} - 60 q^{53} - 62 q^{54} - 32 q^{55} + 33 q^{56} - 56 q^{57} - 87 q^{58} - 53 q^{59} - 65 q^{60} - 76 q^{61} - 84 q^{62} - 215 q^{63} + 67 q^{64} - 49 q^{65} + 9 q^{66} - 145 q^{67} - 133 q^{68} + q^{69} - 4 q^{70} - 4 q^{71} - 167 q^{72} - 155 q^{73} - 14 q^{74} - 31 q^{75} - 199 q^{76} - 97 q^{77} - 24 q^{78} - 73 q^{79} + 98 q^{80} + 127 q^{81} - 69 q^{82} - 225 q^{83} - 59 q^{84} - 53 q^{85} + 30 q^{86} - 179 q^{87} - 119 q^{88} - 25 q^{89} - 60 q^{90} - 160 q^{91} - 188 q^{92} - 44 q^{93} - 32 q^{94} - 126 q^{95} - 43 q^{96} - 72 q^{97} - 111 q^{98} - 141 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.49863 −1.76680 −0.883400 0.468620i \(-0.844751\pi\)
−0.883400 + 0.468620i \(0.844751\pi\)
\(3\) 2.41476 1.39416 0.697082 0.716991i \(-0.254481\pi\)
0.697082 + 0.716991i \(0.254481\pi\)
\(4\) 4.24316 2.12158
\(5\) 1.00000 0.447214
\(6\) −6.03361 −2.46321
\(7\) 1.44699 0.546911 0.273456 0.961885i \(-0.411833\pi\)
0.273456 + 0.961885i \(0.411833\pi\)
\(8\) −5.60483 −1.98161
\(9\) 2.83108 0.943695
\(10\) −2.49863 −0.790137
\(11\) −2.48494 −0.749238 −0.374619 0.927179i \(-0.622226\pi\)
−0.374619 + 0.927179i \(0.622226\pi\)
\(12\) 10.2462 2.95783
\(13\) −5.50824 −1.52771 −0.763855 0.645388i \(-0.776696\pi\)
−0.763855 + 0.645388i \(0.776696\pi\)
\(14\) −3.61550 −0.966282
\(15\) 2.41476 0.623489
\(16\) 5.51809 1.37952
\(17\) 1.30802 0.317241 0.158620 0.987340i \(-0.449295\pi\)
0.158620 + 0.987340i \(0.449295\pi\)
\(18\) −7.07384 −1.66732
\(19\) 7.78323 1.78560 0.892798 0.450457i \(-0.148739\pi\)
0.892798 + 0.450457i \(0.148739\pi\)
\(20\) 4.24316 0.948799
\(21\) 3.49414 0.762484
\(22\) 6.20895 1.32375
\(23\) −4.73454 −0.987220 −0.493610 0.869683i \(-0.664323\pi\)
−0.493610 + 0.869683i \(0.664323\pi\)
\(24\) −13.5343 −2.76269
\(25\) 1.00000 0.200000
\(26\) 13.7631 2.69916
\(27\) −0.407891 −0.0784987
\(28\) 6.13981 1.16032
\(29\) 3.55204 0.659597 0.329798 0.944051i \(-0.393019\pi\)
0.329798 + 0.944051i \(0.393019\pi\)
\(30\) −6.03361 −1.10158
\(31\) 0.736339 0.132250 0.0661252 0.997811i \(-0.478936\pi\)
0.0661252 + 0.997811i \(0.478936\pi\)
\(32\) −2.57800 −0.455731
\(33\) −6.00055 −1.04456
\(34\) −3.26825 −0.560501
\(35\) 1.44699 0.244586
\(36\) 12.0127 2.00212
\(37\) −8.95720 −1.47255 −0.736277 0.676680i \(-0.763418\pi\)
−0.736277 + 0.676680i \(0.763418\pi\)
\(38\) −19.4474 −3.15479
\(39\) −13.3011 −2.12988
\(40\) −5.60483 −0.886201
\(41\) 1.00361 0.156738 0.0783689 0.996924i \(-0.475029\pi\)
0.0783689 + 0.996924i \(0.475029\pi\)
\(42\) −8.73057 −1.34716
\(43\) −10.8969 −1.66176 −0.830880 0.556452i \(-0.812162\pi\)
−0.830880 + 0.556452i \(0.812162\pi\)
\(44\) −10.5440 −1.58957
\(45\) 2.83108 0.422033
\(46\) 11.8299 1.74422
\(47\) 3.12224 0.455425 0.227713 0.973728i \(-0.426875\pi\)
0.227713 + 0.973728i \(0.426875\pi\)
\(48\) 13.3249 1.92328
\(49\) −4.90622 −0.700888
\(50\) −2.49863 −0.353360
\(51\) 3.15855 0.442286
\(52\) −23.3723 −3.24116
\(53\) −11.6301 −1.59752 −0.798761 0.601649i \(-0.794511\pi\)
−0.798761 + 0.601649i \(0.794511\pi\)
\(54\) 1.01917 0.138691
\(55\) −2.48494 −0.335069
\(56\) −8.11014 −1.08376
\(57\) 18.7947 2.48942
\(58\) −8.87523 −1.16537
\(59\) 0.923057 0.120172 0.0600859 0.998193i \(-0.480863\pi\)
0.0600859 + 0.998193i \(0.480863\pi\)
\(60\) 10.2462 1.32278
\(61\) −9.46226 −1.21152 −0.605759 0.795648i \(-0.707131\pi\)
−0.605759 + 0.795648i \(0.707131\pi\)
\(62\) −1.83984 −0.233660
\(63\) 4.09655 0.516117
\(64\) −4.59469 −0.574336
\(65\) −5.50824 −0.683213
\(66\) 14.9932 1.84553
\(67\) −3.31802 −0.405361 −0.202680 0.979245i \(-0.564965\pi\)
−0.202680 + 0.979245i \(0.564965\pi\)
\(68\) 5.55012 0.673051
\(69\) −11.4328 −1.37635
\(70\) −3.61550 −0.432134
\(71\) 10.2493 1.21637 0.608183 0.793797i \(-0.291899\pi\)
0.608183 + 0.793797i \(0.291899\pi\)
\(72\) −15.8677 −1.87003
\(73\) −7.50674 −0.878597 −0.439298 0.898341i \(-0.644773\pi\)
−0.439298 + 0.898341i \(0.644773\pi\)
\(74\) 22.3807 2.60171
\(75\) 2.41476 0.278833
\(76\) 33.0255 3.78829
\(77\) −3.59569 −0.409767
\(78\) 33.2345 3.76307
\(79\) 4.07064 0.457983 0.228991 0.973428i \(-0.426457\pi\)
0.228991 + 0.973428i \(0.426457\pi\)
\(80\) 5.51809 0.616941
\(81\) −9.47821 −1.05313
\(82\) −2.50765 −0.276924
\(83\) 8.23799 0.904236 0.452118 0.891958i \(-0.350669\pi\)
0.452118 + 0.891958i \(0.350669\pi\)
\(84\) 14.8262 1.61767
\(85\) 1.30802 0.141874
\(86\) 27.2273 2.93600
\(87\) 8.57733 0.919586
\(88\) 13.9277 1.48470
\(89\) −4.44209 −0.470861 −0.235431 0.971891i \(-0.575650\pi\)
−0.235431 + 0.971891i \(0.575650\pi\)
\(90\) −7.07384 −0.745648
\(91\) −7.97037 −0.835522
\(92\) −20.0894 −2.09447
\(93\) 1.77808 0.184379
\(94\) −7.80132 −0.804645
\(95\) 7.78323 0.798543
\(96\) −6.22527 −0.635364
\(97\) 4.92639 0.500199 0.250100 0.968220i \(-0.419537\pi\)
0.250100 + 0.968220i \(0.419537\pi\)
\(98\) 12.2588 1.23833
\(99\) −7.03508 −0.707052
\(100\) 4.24316 0.424316
\(101\) −9.94164 −0.989230 −0.494615 0.869112i \(-0.664691\pi\)
−0.494615 + 0.869112i \(0.664691\pi\)
\(102\) −7.89206 −0.781430
\(103\) 5.84798 0.576219 0.288110 0.957597i \(-0.406973\pi\)
0.288110 + 0.957597i \(0.406973\pi\)
\(104\) 30.8727 3.02732
\(105\) 3.49414 0.340993
\(106\) 29.0594 2.82250
\(107\) 8.68309 0.839426 0.419713 0.907657i \(-0.362131\pi\)
0.419713 + 0.907657i \(0.362131\pi\)
\(108\) −1.73075 −0.166541
\(109\) −19.6761 −1.88463 −0.942313 0.334733i \(-0.891354\pi\)
−0.942313 + 0.334733i \(0.891354\pi\)
\(110\) 6.20895 0.592000
\(111\) −21.6295 −2.05298
\(112\) 7.98462 0.754476
\(113\) −11.2022 −1.05381 −0.526905 0.849924i \(-0.676648\pi\)
−0.526905 + 0.849924i \(0.676648\pi\)
\(114\) −46.9610 −4.39830
\(115\) −4.73454 −0.441498
\(116\) 15.0719 1.39939
\(117\) −15.5943 −1.44169
\(118\) −2.30638 −0.212320
\(119\) 1.89269 0.173502
\(120\) −13.5343 −1.23551
\(121\) −4.82507 −0.438642
\(122\) 23.6427 2.14051
\(123\) 2.42348 0.218518
\(124\) 3.12440 0.280580
\(125\) 1.00000 0.0894427
\(126\) −10.2358 −0.911875
\(127\) 13.8649 1.23031 0.615155 0.788406i \(-0.289094\pi\)
0.615155 + 0.788406i \(0.289094\pi\)
\(128\) 16.6364 1.47047
\(129\) −26.3134 −2.31677
\(130\) 13.7631 1.20710
\(131\) 12.5570 1.09711 0.548555 0.836114i \(-0.315178\pi\)
0.548555 + 0.836114i \(0.315178\pi\)
\(132\) −25.4613 −2.21612
\(133\) 11.2623 0.976562
\(134\) 8.29052 0.716191
\(135\) −0.407891 −0.0351057
\(136\) −7.33121 −0.628646
\(137\) −1.25640 −0.107342 −0.0536709 0.998559i \(-0.517092\pi\)
−0.0536709 + 0.998559i \(0.517092\pi\)
\(138\) 28.5664 2.43173
\(139\) −14.3477 −1.21696 −0.608480 0.793569i \(-0.708220\pi\)
−0.608480 + 0.793569i \(0.708220\pi\)
\(140\) 6.13981 0.518909
\(141\) 7.53947 0.634938
\(142\) −25.6092 −2.14908
\(143\) 13.6877 1.14462
\(144\) 15.6222 1.30185
\(145\) 3.55204 0.294981
\(146\) 18.7566 1.55230
\(147\) −11.8474 −0.977154
\(148\) −38.0068 −3.12414
\(149\) 1.37793 0.112884 0.0564422 0.998406i \(-0.482024\pi\)
0.0564422 + 0.998406i \(0.482024\pi\)
\(150\) −6.03361 −0.492642
\(151\) −6.34924 −0.516693 −0.258347 0.966052i \(-0.583178\pi\)
−0.258347 + 0.966052i \(0.583178\pi\)
\(152\) −43.6237 −3.53835
\(153\) 3.70311 0.299378
\(154\) 8.98430 0.723975
\(155\) 0.736339 0.0591441
\(156\) −56.4387 −4.51871
\(157\) 1.26479 0.100942 0.0504708 0.998726i \(-0.483928\pi\)
0.0504708 + 0.998726i \(0.483928\pi\)
\(158\) −10.1710 −0.809163
\(159\) −28.0840 −2.22721
\(160\) −2.57800 −0.203809
\(161\) −6.85084 −0.539922
\(162\) 23.6826 1.86068
\(163\) 8.32875 0.652358 0.326179 0.945308i \(-0.394239\pi\)
0.326179 + 0.945308i \(0.394239\pi\)
\(164\) 4.25848 0.332532
\(165\) −6.00055 −0.467142
\(166\) −20.5837 −1.59760
\(167\) −14.1115 −1.09198 −0.545991 0.837791i \(-0.683847\pi\)
−0.545991 + 0.837791i \(0.683847\pi\)
\(168\) −19.5841 −1.51094
\(169\) 17.3407 1.33390
\(170\) −3.26825 −0.250663
\(171\) 22.0350 1.68506
\(172\) −46.2372 −3.52556
\(173\) −5.18347 −0.394092 −0.197046 0.980394i \(-0.563135\pi\)
−0.197046 + 0.980394i \(0.563135\pi\)
\(174\) −21.4316 −1.62472
\(175\) 1.44699 0.109382
\(176\) −13.7121 −1.03359
\(177\) 2.22897 0.167539
\(178\) 11.0992 0.831917
\(179\) −12.9026 −0.964383 −0.482191 0.876066i \(-0.660159\pi\)
−0.482191 + 0.876066i \(0.660159\pi\)
\(180\) 12.0127 0.895377
\(181\) 10.1403 0.753721 0.376860 0.926270i \(-0.377004\pi\)
0.376860 + 0.926270i \(0.377004\pi\)
\(182\) 19.9150 1.47620
\(183\) −22.8491 −1.68906
\(184\) 26.5363 1.95628
\(185\) −8.95720 −0.658547
\(186\) −4.44278 −0.325760
\(187\) −3.25034 −0.237689
\(188\) 13.2482 0.966221
\(189\) −0.590214 −0.0429318
\(190\) −19.4474 −1.41087
\(191\) 24.9694 1.80672 0.903361 0.428882i \(-0.141092\pi\)
0.903361 + 0.428882i \(0.141092\pi\)
\(192\) −11.0951 −0.800719
\(193\) −16.3647 −1.17795 −0.588977 0.808150i \(-0.700469\pi\)
−0.588977 + 0.808150i \(0.700469\pi\)
\(194\) −12.3092 −0.883752
\(195\) −13.3011 −0.952511
\(196\) −20.8179 −1.48699
\(197\) −2.82979 −0.201614 −0.100807 0.994906i \(-0.532142\pi\)
−0.100807 + 0.994906i \(0.532142\pi\)
\(198\) 17.5781 1.24922
\(199\) 13.0368 0.924156 0.462078 0.886839i \(-0.347104\pi\)
0.462078 + 0.886839i \(0.347104\pi\)
\(200\) −5.60483 −0.396321
\(201\) −8.01224 −0.565140
\(202\) 24.8405 1.74777
\(203\) 5.13976 0.360741
\(204\) 13.4022 0.938344
\(205\) 1.00361 0.0700952
\(206\) −14.6120 −1.01806
\(207\) −13.4039 −0.931635
\(208\) −30.3949 −2.10751
\(209\) −19.3409 −1.33784
\(210\) −8.73057 −0.602467
\(211\) −10.0705 −0.693284 −0.346642 0.937997i \(-0.612678\pi\)
−0.346642 + 0.937997i \(0.612678\pi\)
\(212\) −49.3485 −3.38927
\(213\) 24.7496 1.69582
\(214\) −21.6958 −1.48310
\(215\) −10.8969 −0.743162
\(216\) 2.28616 0.155553
\(217\) 1.06547 0.0723292
\(218\) 49.1633 3.32976
\(219\) −18.1270 −1.22491
\(220\) −10.5440 −0.710877
\(221\) −7.20487 −0.484652
\(222\) 54.0442 3.62721
\(223\) −4.07100 −0.272614 −0.136307 0.990667i \(-0.543523\pi\)
−0.136307 + 0.990667i \(0.543523\pi\)
\(224\) −3.73035 −0.249244
\(225\) 2.83108 0.188739
\(226\) 27.9900 1.86187
\(227\) −12.6440 −0.839214 −0.419607 0.907706i \(-0.637832\pi\)
−0.419607 + 0.907706i \(0.637832\pi\)
\(228\) 79.7488 5.28149
\(229\) 27.5577 1.82106 0.910531 0.413440i \(-0.135673\pi\)
0.910531 + 0.413440i \(0.135673\pi\)
\(230\) 11.8299 0.780039
\(231\) −8.68273 −0.571282
\(232\) −19.9086 −1.30706
\(233\) 14.5411 0.952620 0.476310 0.879277i \(-0.341974\pi\)
0.476310 + 0.879277i \(0.341974\pi\)
\(234\) 38.9644 2.54718
\(235\) 3.12224 0.203672
\(236\) 3.91668 0.254954
\(237\) 9.82963 0.638503
\(238\) −4.72913 −0.306544
\(239\) 24.7594 1.60155 0.800776 0.598964i \(-0.204421\pi\)
0.800776 + 0.598964i \(0.204421\pi\)
\(240\) 13.3249 0.860117
\(241\) −26.4374 −1.70298 −0.851490 0.524370i \(-0.824301\pi\)
−0.851490 + 0.524370i \(0.824301\pi\)
\(242\) 12.0561 0.774993
\(243\) −21.6640 −1.38974
\(244\) −40.1499 −2.57033
\(245\) −4.90622 −0.313447
\(246\) −6.05539 −0.386078
\(247\) −42.8719 −2.72787
\(248\) −4.12705 −0.262068
\(249\) 19.8928 1.26065
\(250\) −2.49863 −0.158027
\(251\) −7.73280 −0.488090 −0.244045 0.969764i \(-0.578474\pi\)
−0.244045 + 0.969764i \(0.578474\pi\)
\(252\) 17.3823 1.09498
\(253\) 11.7651 0.739663
\(254\) −34.6432 −2.17371
\(255\) 3.15855 0.197796
\(256\) −32.3790 −2.02369
\(257\) 11.1312 0.694343 0.347172 0.937802i \(-0.387142\pi\)
0.347172 + 0.937802i \(0.387142\pi\)
\(258\) 65.7475 4.09326
\(259\) −12.9610 −0.805357
\(260\) −23.3723 −1.44949
\(261\) 10.0561 0.622458
\(262\) −31.3753 −1.93837
\(263\) 20.4844 1.26312 0.631562 0.775325i \(-0.282414\pi\)
0.631562 + 0.775325i \(0.282414\pi\)
\(264\) 33.6320 2.06991
\(265\) −11.6301 −0.714433
\(266\) −28.1403 −1.72539
\(267\) −10.7266 −0.656458
\(268\) −14.0789 −0.860006
\(269\) 28.3194 1.72667 0.863333 0.504634i \(-0.168373\pi\)
0.863333 + 0.504634i \(0.168373\pi\)
\(270\) 1.01917 0.0620247
\(271\) −19.6254 −1.19216 −0.596078 0.802926i \(-0.703275\pi\)
−0.596078 + 0.802926i \(0.703275\pi\)
\(272\) 7.21775 0.437640
\(273\) −19.2466 −1.16485
\(274\) 3.13929 0.189652
\(275\) −2.48494 −0.149848
\(276\) −48.5112 −2.92003
\(277\) −10.5941 −0.636536 −0.318268 0.948001i \(-0.603101\pi\)
−0.318268 + 0.948001i \(0.603101\pi\)
\(278\) 35.8497 2.15012
\(279\) 2.08464 0.124804
\(280\) −8.11014 −0.484673
\(281\) −13.8012 −0.823312 −0.411656 0.911339i \(-0.635049\pi\)
−0.411656 + 0.911339i \(0.635049\pi\)
\(282\) −18.8384 −1.12181
\(283\) −31.6037 −1.87865 −0.939324 0.343032i \(-0.888546\pi\)
−0.939324 + 0.343032i \(0.888546\pi\)
\(284\) 43.4894 2.58062
\(285\) 18.7947 1.11330
\(286\) −34.2004 −2.02231
\(287\) 1.45222 0.0857216
\(288\) −7.29855 −0.430071
\(289\) −15.2891 −0.899358
\(290\) −8.87523 −0.521171
\(291\) 11.8961 0.697360
\(292\) −31.8523 −1.86401
\(293\) −21.3344 −1.24637 −0.623184 0.782075i \(-0.714161\pi\)
−0.623184 + 0.782075i \(0.714161\pi\)
\(294\) 29.6022 1.72643
\(295\) 0.923057 0.0537425
\(296\) 50.2036 2.91802
\(297\) 1.01359 0.0588142
\(298\) −3.44294 −0.199444
\(299\) 26.0790 1.50819
\(300\) 10.2462 0.591566
\(301\) −15.7677 −0.908835
\(302\) 15.8644 0.912894
\(303\) −24.0067 −1.37915
\(304\) 42.9486 2.46327
\(305\) −9.46226 −0.541807
\(306\) −9.25270 −0.528941
\(307\) −0.936988 −0.0534767 −0.0267384 0.999642i \(-0.508512\pi\)
−0.0267384 + 0.999642i \(0.508512\pi\)
\(308\) −15.2571 −0.869352
\(309\) 14.1215 0.803344
\(310\) −1.83984 −0.104496
\(311\) −9.18545 −0.520859 −0.260430 0.965493i \(-0.583864\pi\)
−0.260430 + 0.965493i \(0.583864\pi\)
\(312\) 74.5504 4.22058
\(313\) −22.4962 −1.27156 −0.635782 0.771869i \(-0.719322\pi\)
−0.635782 + 0.771869i \(0.719322\pi\)
\(314\) −3.16026 −0.178344
\(315\) 4.09655 0.230815
\(316\) 17.2724 0.971647
\(317\) −29.0285 −1.63041 −0.815203 0.579176i \(-0.803375\pi\)
−0.815203 + 0.579176i \(0.803375\pi\)
\(318\) 70.1716 3.93503
\(319\) −8.82660 −0.494195
\(320\) −4.59469 −0.256851
\(321\) 20.9676 1.17030
\(322\) 17.1177 0.953933
\(323\) 10.1806 0.566464
\(324\) −40.2176 −2.23431
\(325\) −5.50824 −0.305542
\(326\) −20.8105 −1.15259
\(327\) −47.5131 −2.62748
\(328\) −5.62507 −0.310592
\(329\) 4.51785 0.249077
\(330\) 14.9932 0.825346
\(331\) −1.60625 −0.0882876 −0.0441438 0.999025i \(-0.514056\pi\)
−0.0441438 + 0.999025i \(0.514056\pi\)
\(332\) 34.9551 1.91841
\(333\) −25.3586 −1.38964
\(334\) 35.2595 1.92931
\(335\) −3.31802 −0.181283
\(336\) 19.2810 1.05186
\(337\) 31.4333 1.71228 0.856141 0.516743i \(-0.172856\pi\)
0.856141 + 0.516743i \(0.172856\pi\)
\(338\) −43.3280 −2.35673
\(339\) −27.0505 −1.46918
\(340\) 5.55012 0.300998
\(341\) −1.82976 −0.0990870
\(342\) −55.0573 −2.97716
\(343\) −17.2282 −0.930235
\(344\) 61.0752 3.29295
\(345\) −11.4328 −0.615521
\(346\) 12.9516 0.696281
\(347\) −26.1957 −1.40626 −0.703129 0.711062i \(-0.748214\pi\)
−0.703129 + 0.711062i \(0.748214\pi\)
\(348\) 36.3950 1.95098
\(349\) −5.04512 −0.270059 −0.135030 0.990842i \(-0.543113\pi\)
−0.135030 + 0.990842i \(0.543113\pi\)
\(350\) −3.61550 −0.193256
\(351\) 2.24676 0.119923
\(352\) 6.40619 0.341451
\(353\) 26.2670 1.39805 0.699026 0.715096i \(-0.253617\pi\)
0.699026 + 0.715096i \(0.253617\pi\)
\(354\) −5.56936 −0.296008
\(355\) 10.2493 0.543976
\(356\) −18.8485 −0.998969
\(357\) 4.57039 0.241891
\(358\) 32.2388 1.70387
\(359\) −13.7718 −0.726847 −0.363423 0.931624i \(-0.618392\pi\)
−0.363423 + 0.931624i \(0.618392\pi\)
\(360\) −15.8677 −0.836304
\(361\) 41.5787 2.18835
\(362\) −25.3368 −1.33167
\(363\) −11.6514 −0.611540
\(364\) −33.8196 −1.77263
\(365\) −7.50674 −0.392920
\(366\) 57.0915 2.98422
\(367\) 17.6511 0.921379 0.460689 0.887561i \(-0.347602\pi\)
0.460689 + 0.887561i \(0.347602\pi\)
\(368\) −26.1256 −1.36189
\(369\) 2.84131 0.147913
\(370\) 22.3807 1.16352
\(371\) −16.8287 −0.873702
\(372\) 7.54469 0.391174
\(373\) −17.4992 −0.906073 −0.453036 0.891492i \(-0.649659\pi\)
−0.453036 + 0.891492i \(0.649659\pi\)
\(374\) 8.12141 0.419948
\(375\) 2.41476 0.124698
\(376\) −17.4996 −0.902474
\(377\) −19.5655 −1.00767
\(378\) 1.47473 0.0758518
\(379\) 34.2792 1.76081 0.880403 0.474226i \(-0.157272\pi\)
0.880403 + 0.474226i \(0.157272\pi\)
\(380\) 33.0255 1.69417
\(381\) 33.4804 1.71525
\(382\) −62.3893 −3.19211
\(383\) 35.7192 1.82517 0.912584 0.408890i \(-0.134084\pi\)
0.912584 + 0.408890i \(0.134084\pi\)
\(384\) 40.1731 2.05007
\(385\) −3.59569 −0.183253
\(386\) 40.8892 2.08121
\(387\) −30.8500 −1.56819
\(388\) 20.9035 1.06121
\(389\) −6.08946 −0.308748 −0.154374 0.988013i \(-0.549336\pi\)
−0.154374 + 0.988013i \(0.549336\pi\)
\(390\) 33.2345 1.68290
\(391\) −6.19286 −0.313186
\(392\) 27.4985 1.38889
\(393\) 30.3222 1.52955
\(394\) 7.07060 0.356212
\(395\) 4.07064 0.204816
\(396\) −29.8510 −1.50007
\(397\) −23.2515 −1.16696 −0.583481 0.812127i \(-0.698310\pi\)
−0.583481 + 0.812127i \(0.698310\pi\)
\(398\) −32.5742 −1.63280
\(399\) 27.1957 1.36149
\(400\) 5.51809 0.275904
\(401\) −14.2343 −0.710828 −0.355414 0.934709i \(-0.615660\pi\)
−0.355414 + 0.934709i \(0.615660\pi\)
\(402\) 20.0196 0.998489
\(403\) −4.05593 −0.202040
\(404\) −42.1840 −2.09873
\(405\) −9.47821 −0.470976
\(406\) −12.8424 −0.637356
\(407\) 22.2581 1.10329
\(408\) −17.7031 −0.876436
\(409\) 30.9854 1.53213 0.766064 0.642765i \(-0.222213\pi\)
0.766064 + 0.642765i \(0.222213\pi\)
\(410\) −2.50765 −0.123844
\(411\) −3.03392 −0.149652
\(412\) 24.8139 1.22249
\(413\) 1.33566 0.0657233
\(414\) 33.4914 1.64601
\(415\) 8.23799 0.404387
\(416\) 14.2003 0.696225
\(417\) −34.6464 −1.69664
\(418\) 48.3257 2.36369
\(419\) −18.7604 −0.916505 −0.458253 0.888822i \(-0.651525\pi\)
−0.458253 + 0.888822i \(0.651525\pi\)
\(420\) 14.8262 0.723444
\(421\) 7.06630 0.344391 0.172195 0.985063i \(-0.444914\pi\)
0.172195 + 0.985063i \(0.444914\pi\)
\(422\) 25.1626 1.22489
\(423\) 8.83932 0.429782
\(424\) 65.1849 3.16566
\(425\) 1.30802 0.0634481
\(426\) −61.8402 −2.99617
\(427\) −13.6918 −0.662593
\(428\) 36.8437 1.78091
\(429\) 33.0524 1.59579
\(430\) 27.2273 1.31302
\(431\) 14.6287 0.704641 0.352321 0.935879i \(-0.385393\pi\)
0.352321 + 0.935879i \(0.385393\pi\)
\(432\) −2.25078 −0.108291
\(433\) −10.1692 −0.488702 −0.244351 0.969687i \(-0.578575\pi\)
−0.244351 + 0.969687i \(0.578575\pi\)
\(434\) −2.66223 −0.127791
\(435\) 8.57733 0.411251
\(436\) −83.4887 −3.99838
\(437\) −36.8501 −1.76278
\(438\) 45.2927 2.16417
\(439\) 13.4253 0.640755 0.320378 0.947290i \(-0.396190\pi\)
0.320378 + 0.947290i \(0.396190\pi\)
\(440\) 13.9277 0.663976
\(441\) −13.8899 −0.661425
\(442\) 18.0023 0.856283
\(443\) −3.13989 −0.149181 −0.0745903 0.997214i \(-0.523765\pi\)
−0.0745903 + 0.997214i \(0.523765\pi\)
\(444\) −91.7775 −4.35557
\(445\) −4.44209 −0.210575
\(446\) 10.1719 0.481654
\(447\) 3.32737 0.157379
\(448\) −6.64847 −0.314111
\(449\) −9.03784 −0.426522 −0.213261 0.976995i \(-0.568408\pi\)
−0.213261 + 0.976995i \(0.568408\pi\)
\(450\) −7.07384 −0.333464
\(451\) −2.49391 −0.117434
\(452\) −47.5325 −2.23574
\(453\) −15.3319 −0.720356
\(454\) 31.5928 1.48272
\(455\) −7.97037 −0.373657
\(456\) −105.341 −4.93304
\(457\) −7.66439 −0.358525 −0.179263 0.983801i \(-0.557371\pi\)
−0.179263 + 0.983801i \(0.557371\pi\)
\(458\) −68.8565 −3.21745
\(459\) −0.533528 −0.0249030
\(460\) −20.0894 −0.936674
\(461\) 9.53198 0.443948 0.221974 0.975053i \(-0.428750\pi\)
0.221974 + 0.975053i \(0.428750\pi\)
\(462\) 21.6950 1.00934
\(463\) 16.3712 0.760835 0.380417 0.924815i \(-0.375780\pi\)
0.380417 + 0.924815i \(0.375780\pi\)
\(464\) 19.6004 0.909928
\(465\) 1.77808 0.0824567
\(466\) −36.3329 −1.68309
\(467\) −22.4664 −1.03962 −0.519811 0.854281i \(-0.673998\pi\)
−0.519811 + 0.854281i \(0.673998\pi\)
\(468\) −66.1691 −3.05867
\(469\) −4.80115 −0.221696
\(470\) −7.80132 −0.359848
\(471\) 3.05418 0.140729
\(472\) −5.17358 −0.238133
\(473\) 27.0781 1.24505
\(474\) −24.5606 −1.12811
\(475\) 7.78323 0.357119
\(476\) 8.03098 0.368099
\(477\) −32.9259 −1.50757
\(478\) −61.8646 −2.82962
\(479\) −23.7555 −1.08542 −0.542709 0.839921i \(-0.682601\pi\)
−0.542709 + 0.839921i \(0.682601\pi\)
\(480\) −6.22527 −0.284143
\(481\) 49.3384 2.24964
\(482\) 66.0573 3.00883
\(483\) −16.5432 −0.752740
\(484\) −20.4735 −0.930615
\(485\) 4.92639 0.223696
\(486\) 54.1303 2.45540
\(487\) 28.8998 1.30957 0.654786 0.755814i \(-0.272759\pi\)
0.654786 + 0.755814i \(0.272759\pi\)
\(488\) 53.0344 2.40075
\(489\) 20.1120 0.909494
\(490\) 12.2588 0.553798
\(491\) 13.7481 0.620445 0.310222 0.950664i \(-0.399597\pi\)
0.310222 + 0.950664i \(0.399597\pi\)
\(492\) 10.2832 0.463604
\(493\) 4.64612 0.209251
\(494\) 107.121 4.81961
\(495\) −7.03508 −0.316203
\(496\) 4.06318 0.182442
\(497\) 14.8306 0.665244
\(498\) −49.7048 −2.22732
\(499\) 11.2546 0.503825 0.251913 0.967750i \(-0.418940\pi\)
0.251913 + 0.967750i \(0.418940\pi\)
\(500\) 4.24316 0.189760
\(501\) −34.0760 −1.52240
\(502\) 19.3214 0.862357
\(503\) −16.7179 −0.745415 −0.372707 0.927949i \(-0.621570\pi\)
−0.372707 + 0.927949i \(0.621570\pi\)
\(504\) −22.9605 −1.02274
\(505\) −9.94164 −0.442397
\(506\) −29.3966 −1.30684
\(507\) 41.8737 1.85968
\(508\) 58.8309 2.61020
\(509\) −35.5775 −1.57694 −0.788472 0.615071i \(-0.789127\pi\)
−0.788472 + 0.615071i \(0.789127\pi\)
\(510\) −7.89206 −0.349466
\(511\) −10.8622 −0.480514
\(512\) 47.6302 2.10498
\(513\) −3.17471 −0.140167
\(514\) −27.8127 −1.22677
\(515\) 5.84798 0.257693
\(516\) −111.652 −4.91521
\(517\) −7.75858 −0.341222
\(518\) 32.3847 1.42290
\(519\) −12.5169 −0.549429
\(520\) 30.8727 1.35386
\(521\) 38.7174 1.69624 0.848120 0.529804i \(-0.177735\pi\)
0.848120 + 0.529804i \(0.177735\pi\)
\(522\) −25.1265 −1.09976
\(523\) −3.10085 −0.135591 −0.0677954 0.997699i \(-0.521597\pi\)
−0.0677954 + 0.997699i \(0.521597\pi\)
\(524\) 53.2814 2.32761
\(525\) 3.49414 0.152497
\(526\) −51.1830 −2.23169
\(527\) 0.963143 0.0419552
\(528\) −33.1115 −1.44099
\(529\) −0.584110 −0.0253961
\(530\) 29.0594 1.26226
\(531\) 2.61325 0.113406
\(532\) 47.7876 2.07186
\(533\) −5.52813 −0.239450
\(534\) 26.8018 1.15983
\(535\) 8.68309 0.375403
\(536\) 18.5970 0.803266
\(537\) −31.1566 −1.34451
\(538\) −70.7598 −3.05067
\(539\) 12.1917 0.525132
\(540\) −1.73075 −0.0744795
\(541\) 12.7434 0.547880 0.273940 0.961747i \(-0.411673\pi\)
0.273940 + 0.961747i \(0.411673\pi\)
\(542\) 49.0366 2.10630
\(543\) 24.4864 1.05081
\(544\) −3.37207 −0.144576
\(545\) −19.6761 −0.842830
\(546\) 48.0901 2.05806
\(547\) −36.0931 −1.54323 −0.771614 0.636091i \(-0.780550\pi\)
−0.771614 + 0.636091i \(0.780550\pi\)
\(548\) −5.33112 −0.227734
\(549\) −26.7885 −1.14330
\(550\) 6.20895 0.264751
\(551\) 27.6463 1.17777
\(552\) 64.0789 2.72738
\(553\) 5.89017 0.250476
\(554\) 26.4707 1.12463
\(555\) −21.6295 −0.918122
\(556\) −60.8798 −2.58188
\(557\) 30.3294 1.28510 0.642549 0.766245i \(-0.277877\pi\)
0.642549 + 0.766245i \(0.277877\pi\)
\(558\) −5.20874 −0.220504
\(559\) 60.0227 2.53869
\(560\) 7.98462 0.337412
\(561\) −7.84882 −0.331377
\(562\) 34.4842 1.45463
\(563\) 11.5831 0.488170 0.244085 0.969754i \(-0.421512\pi\)
0.244085 + 0.969754i \(0.421512\pi\)
\(564\) 31.9912 1.34707
\(565\) −11.2022 −0.471278
\(566\) 78.9661 3.31919
\(567\) −13.7149 −0.575971
\(568\) −57.4455 −2.41036
\(569\) 7.89884 0.331137 0.165568 0.986198i \(-0.447054\pi\)
0.165568 + 0.986198i \(0.447054\pi\)
\(570\) −46.9610 −1.96698
\(571\) 12.0596 0.504677 0.252339 0.967639i \(-0.418800\pi\)
0.252339 + 0.967639i \(0.418800\pi\)
\(572\) 58.0789 2.42840
\(573\) 60.2952 2.51887
\(574\) −3.62855 −0.151453
\(575\) −4.73454 −0.197444
\(576\) −13.0080 −0.541998
\(577\) −2.82176 −0.117471 −0.0587356 0.998274i \(-0.518707\pi\)
−0.0587356 + 0.998274i \(0.518707\pi\)
\(578\) 38.2018 1.58899
\(579\) −39.5168 −1.64226
\(580\) 15.0719 0.625825
\(581\) 11.9203 0.494537
\(582\) −29.7239 −1.23210
\(583\) 28.9002 1.19692
\(584\) 42.0740 1.74103
\(585\) −15.5943 −0.644745
\(586\) 53.3068 2.20208
\(587\) −22.9085 −0.945533 −0.472767 0.881188i \(-0.656745\pi\)
−0.472767 + 0.881188i \(0.656745\pi\)
\(588\) −50.2702 −2.07311
\(589\) 5.73110 0.236146
\(590\) −2.30638 −0.0949522
\(591\) −6.83327 −0.281083
\(592\) −49.4266 −2.03142
\(593\) −36.3653 −1.49334 −0.746672 0.665192i \(-0.768350\pi\)
−0.746672 + 0.665192i \(0.768350\pi\)
\(594\) −2.53258 −0.103913
\(595\) 1.89269 0.0775926
\(596\) 5.84677 0.239493
\(597\) 31.4809 1.28843
\(598\) −65.1618 −2.66466
\(599\) 13.6674 0.558436 0.279218 0.960228i \(-0.409925\pi\)
0.279218 + 0.960228i \(0.409925\pi\)
\(600\) −13.5343 −0.552537
\(601\) −47.3929 −1.93320 −0.966599 0.256293i \(-0.917499\pi\)
−0.966599 + 0.256293i \(0.917499\pi\)
\(602\) 39.3977 1.60573
\(603\) −9.39360 −0.382537
\(604\) −26.9408 −1.09621
\(605\) −4.82507 −0.196167
\(606\) 59.9839 2.43668
\(607\) 26.1737 1.06236 0.531180 0.847259i \(-0.321749\pi\)
0.531180 + 0.847259i \(0.321749\pi\)
\(608\) −20.0652 −0.813752
\(609\) 12.4113 0.502932
\(610\) 23.6427 0.957265
\(611\) −17.1980 −0.695758
\(612\) 15.7129 0.635155
\(613\) −13.1757 −0.532163 −0.266082 0.963951i \(-0.585729\pi\)
−0.266082 + 0.963951i \(0.585729\pi\)
\(614\) 2.34119 0.0944827
\(615\) 2.42348 0.0977243
\(616\) 20.1532 0.811996
\(617\) −31.8421 −1.28192 −0.640958 0.767576i \(-0.721463\pi\)
−0.640958 + 0.767576i \(0.721463\pi\)
\(618\) −35.2844 −1.41935
\(619\) −24.1610 −0.971113 −0.485556 0.874205i \(-0.661383\pi\)
−0.485556 + 0.874205i \(0.661383\pi\)
\(620\) 3.12440 0.125479
\(621\) 1.93118 0.0774955
\(622\) 22.9511 0.920254
\(623\) −6.42767 −0.257519
\(624\) −73.3966 −2.93822
\(625\) 1.00000 0.0400000
\(626\) 56.2098 2.24660
\(627\) −46.7037 −1.86516
\(628\) 5.36673 0.214156
\(629\) −11.7162 −0.467154
\(630\) −10.2358 −0.407803
\(631\) 22.8875 0.911139 0.455569 0.890200i \(-0.349436\pi\)
0.455569 + 0.890200i \(0.349436\pi\)
\(632\) −22.8152 −0.907541
\(633\) −24.3180 −0.966553
\(634\) 72.5316 2.88060
\(635\) 13.8649 0.550211
\(636\) −119.165 −4.72520
\(637\) 27.0246 1.07075
\(638\) 22.0544 0.873143
\(639\) 29.0166 1.14788
\(640\) 16.6364 0.657613
\(641\) −26.3132 −1.03931 −0.519654 0.854377i \(-0.673939\pi\)
−0.519654 + 0.854377i \(0.673939\pi\)
\(642\) −52.3903 −2.06768
\(643\) 31.5128 1.24274 0.621372 0.783516i \(-0.286576\pi\)
0.621372 + 0.783516i \(0.286576\pi\)
\(644\) −29.0692 −1.14549
\(645\) −26.3134 −1.03609
\(646\) −25.4376 −1.00083
\(647\) 19.8520 0.780464 0.390232 0.920717i \(-0.372395\pi\)
0.390232 + 0.920717i \(0.372395\pi\)
\(648\) 53.1238 2.08690
\(649\) −2.29374 −0.0900373
\(650\) 13.7631 0.539832
\(651\) 2.57287 0.100839
\(652\) 35.3402 1.38403
\(653\) 24.7533 0.968673 0.484337 0.874882i \(-0.339061\pi\)
0.484337 + 0.874882i \(0.339061\pi\)
\(654\) 118.718 4.64223
\(655\) 12.5570 0.490643
\(656\) 5.53801 0.216223
\(657\) −21.2522 −0.829127
\(658\) −11.2884 −0.440069
\(659\) 40.6886 1.58500 0.792502 0.609869i \(-0.208778\pi\)
0.792502 + 0.609869i \(0.208778\pi\)
\(660\) −25.4613 −0.991079
\(661\) −24.9617 −0.970896 −0.485448 0.874265i \(-0.661344\pi\)
−0.485448 + 0.874265i \(0.661344\pi\)
\(662\) 4.01343 0.155986
\(663\) −17.3981 −0.675684
\(664\) −46.1725 −1.79184
\(665\) 11.2623 0.436732
\(666\) 63.3618 2.45522
\(667\) −16.8173 −0.651167
\(668\) −59.8774 −2.31673
\(669\) −9.83050 −0.380069
\(670\) 8.29052 0.320291
\(671\) 23.5132 0.907716
\(672\) −9.00791 −0.347488
\(673\) 14.8449 0.572230 0.286115 0.958195i \(-0.407636\pi\)
0.286115 + 0.958195i \(0.407636\pi\)
\(674\) −78.5403 −3.02526
\(675\) −0.407891 −0.0156997
\(676\) 73.5793 2.82997
\(677\) 12.6868 0.487593 0.243797 0.969826i \(-0.421607\pi\)
0.243797 + 0.969826i \(0.421607\pi\)
\(678\) 67.5894 2.59575
\(679\) 7.12845 0.273565
\(680\) −7.33121 −0.281139
\(681\) −30.5324 −1.17000
\(682\) 4.57189 0.175067
\(683\) −0.00966065 −0.000369654 0 −0.000184827 1.00000i \(-0.500059\pi\)
−0.000184827 1.00000i \(0.500059\pi\)
\(684\) 93.4980 3.57499
\(685\) −1.25640 −0.0480047
\(686\) 43.0469 1.64354
\(687\) 66.5453 2.53886
\(688\) −60.1300 −2.29243
\(689\) 64.0615 2.44055
\(690\) 28.5664 1.08750
\(691\) 22.1972 0.844422 0.422211 0.906498i \(-0.361254\pi\)
0.422211 + 0.906498i \(0.361254\pi\)
\(692\) −21.9943 −0.836097
\(693\) −10.1797 −0.386695
\(694\) 65.4533 2.48458
\(695\) −14.3477 −0.544241
\(696\) −48.0745 −1.82226
\(697\) 1.31274 0.0497236
\(698\) 12.6059 0.477140
\(699\) 35.1134 1.32811
\(700\) 6.13981 0.232063
\(701\) 12.6821 0.478995 0.239497 0.970897i \(-0.423017\pi\)
0.239497 + 0.970897i \(0.423017\pi\)
\(702\) −5.61383 −0.211880
\(703\) −69.7160 −2.62939
\(704\) 11.4175 0.430315
\(705\) 7.53947 0.283953
\(706\) −65.6316 −2.47008
\(707\) −14.3855 −0.541021
\(708\) 9.45786 0.355448
\(709\) 39.8334 1.49597 0.747987 0.663713i \(-0.231020\pi\)
0.747987 + 0.663713i \(0.231020\pi\)
\(710\) −25.6092 −0.961096
\(711\) 11.5243 0.432196
\(712\) 24.8972 0.933061
\(713\) −3.48623 −0.130560
\(714\) −11.4197 −0.427373
\(715\) 13.6877 0.511889
\(716\) −54.7476 −2.04602
\(717\) 59.7881 2.23283
\(718\) 34.4106 1.28419
\(719\) 48.8639 1.82232 0.911158 0.412057i \(-0.135190\pi\)
0.911158 + 0.412057i \(0.135190\pi\)
\(720\) 15.6222 0.582204
\(721\) 8.46198 0.315141
\(722\) −103.890 −3.86638
\(723\) −63.8400 −2.37424
\(724\) 43.0268 1.59908
\(725\) 3.55204 0.131919
\(726\) 29.1125 1.08047
\(727\) 5.38627 0.199766 0.0998828 0.994999i \(-0.468153\pi\)
0.0998828 + 0.994999i \(0.468153\pi\)
\(728\) 44.6726 1.65568
\(729\) −23.8787 −0.884398
\(730\) 18.7566 0.694212
\(731\) −14.2533 −0.527178
\(732\) −96.9525 −3.58347
\(733\) 24.8205 0.916768 0.458384 0.888754i \(-0.348428\pi\)
0.458384 + 0.888754i \(0.348428\pi\)
\(734\) −44.1035 −1.62789
\(735\) −11.8474 −0.436996
\(736\) 12.2057 0.449907
\(737\) 8.24509 0.303712
\(738\) −7.09938 −0.261332
\(739\) −33.4440 −1.23026 −0.615130 0.788426i \(-0.710896\pi\)
−0.615130 + 0.788426i \(0.710896\pi\)
\(740\) −38.0068 −1.39716
\(741\) −103.526 −3.80311
\(742\) 42.0487 1.54366
\(743\) −16.4660 −0.604080 −0.302040 0.953295i \(-0.597668\pi\)
−0.302040 + 0.953295i \(0.597668\pi\)
\(744\) −9.96586 −0.365366
\(745\) 1.37793 0.0504834
\(746\) 43.7240 1.60085
\(747\) 23.3224 0.853323
\(748\) −13.7917 −0.504276
\(749\) 12.5643 0.459091
\(750\) −6.03361 −0.220316
\(751\) −37.2968 −1.36098 −0.680490 0.732757i \(-0.738233\pi\)
−0.680490 + 0.732757i \(0.738233\pi\)
\(752\) 17.2288 0.628269
\(753\) −18.6729 −0.680477
\(754\) 48.8869 1.78036
\(755\) −6.34924 −0.231072
\(756\) −2.50437 −0.0910832
\(757\) 3.47328 0.126239 0.0631193 0.998006i \(-0.479895\pi\)
0.0631193 + 0.998006i \(0.479895\pi\)
\(758\) −85.6512 −3.11099
\(759\) 28.4098 1.03121
\(760\) −43.6237 −1.58240
\(761\) −24.2763 −0.880014 −0.440007 0.897994i \(-0.645024\pi\)
−0.440007 + 0.897994i \(0.645024\pi\)
\(762\) −83.6552 −3.03051
\(763\) −28.4711 −1.03072
\(764\) 105.949 3.83310
\(765\) 3.70311 0.133886
\(766\) −89.2492 −3.22470
\(767\) −5.08442 −0.183588
\(768\) −78.1876 −2.82135
\(769\) −48.4764 −1.74810 −0.874051 0.485834i \(-0.838516\pi\)
−0.874051 + 0.485834i \(0.838516\pi\)
\(770\) 8.98430 0.323772
\(771\) 26.8792 0.968029
\(772\) −69.4378 −2.49912
\(773\) −18.1920 −0.654322 −0.327161 0.944969i \(-0.606092\pi\)
−0.327161 + 0.944969i \(0.606092\pi\)
\(774\) 77.0828 2.77068
\(775\) 0.736339 0.0264501
\(776\) −27.6116 −0.991199
\(777\) −31.2977 −1.12280
\(778\) 15.2153 0.545495
\(779\) 7.81134 0.279870
\(780\) −56.4387 −2.02083
\(781\) −25.4689 −0.911348
\(782\) 15.4737 0.553338
\(783\) −1.44884 −0.0517774
\(784\) −27.0729 −0.966891
\(785\) 1.26479 0.0451425
\(786\) −75.7640 −2.70241
\(787\) −33.0744 −1.17897 −0.589487 0.807778i \(-0.700670\pi\)
−0.589487 + 0.807778i \(0.700670\pi\)
\(788\) −12.0072 −0.427740
\(789\) 49.4651 1.76100
\(790\) −10.1710 −0.361869
\(791\) −16.2094 −0.576340
\(792\) 39.4304 1.40110
\(793\) 52.1204 1.85085
\(794\) 58.0970 2.06179
\(795\) −28.0840 −0.996037
\(796\) 55.3173 1.96067
\(797\) 37.4388 1.32615 0.663075 0.748553i \(-0.269251\pi\)
0.663075 + 0.748553i \(0.269251\pi\)
\(798\) −67.9521 −2.40548
\(799\) 4.08394 0.144479
\(800\) −2.57800 −0.0911462
\(801\) −12.5759 −0.444349
\(802\) 35.5663 1.25589
\(803\) 18.6538 0.658278
\(804\) −33.9972 −1.19899
\(805\) −6.85084 −0.241460
\(806\) 10.1343 0.356964
\(807\) 68.3848 2.40726
\(808\) 55.7212 1.96027
\(809\) −14.7661 −0.519147 −0.259573 0.965723i \(-0.583582\pi\)
−0.259573 + 0.965723i \(0.583582\pi\)
\(810\) 23.6826 0.832121
\(811\) 44.8113 1.57354 0.786769 0.617247i \(-0.211752\pi\)
0.786769 + 0.617247i \(0.211752\pi\)
\(812\) 21.8088 0.765340
\(813\) −47.3906 −1.66206
\(814\) −55.6148 −1.94930
\(815\) 8.32875 0.291743
\(816\) 17.4292 0.610143
\(817\) −84.8130 −2.96723
\(818\) −77.4210 −2.70696
\(819\) −22.5648 −0.788478
\(820\) 4.25848 0.148713
\(821\) −36.6367 −1.27863 −0.639315 0.768945i \(-0.720782\pi\)
−0.639315 + 0.768945i \(0.720782\pi\)
\(822\) 7.58065 0.264405
\(823\) −18.8314 −0.656420 −0.328210 0.944605i \(-0.606445\pi\)
−0.328210 + 0.944605i \(0.606445\pi\)
\(824\) −32.7770 −1.14184
\(825\) −6.00055 −0.208912
\(826\) −3.33731 −0.116120
\(827\) −25.7635 −0.895884 −0.447942 0.894063i \(-0.647843\pi\)
−0.447942 + 0.894063i \(0.647843\pi\)
\(828\) −56.8748 −1.97654
\(829\) −33.6683 −1.16935 −0.584675 0.811268i \(-0.698778\pi\)
−0.584675 + 0.811268i \(0.698778\pi\)
\(830\) −20.5837 −0.714470
\(831\) −25.5822 −0.887436
\(832\) 25.3087 0.877420
\(833\) −6.41742 −0.222350
\(834\) 86.5686 2.99763
\(835\) −14.1115 −0.488349
\(836\) −82.0665 −2.83833
\(837\) −0.300346 −0.0103815
\(838\) 46.8753 1.61928
\(839\) 31.8624 1.10001 0.550005 0.835161i \(-0.314626\pi\)
0.550005 + 0.835161i \(0.314626\pi\)
\(840\) −19.5841 −0.675714
\(841\) −16.3830 −0.564932
\(842\) −17.6561 −0.608469
\(843\) −33.3267 −1.14783
\(844\) −42.7309 −1.47086
\(845\) 17.3407 0.596538
\(846\) −22.0862 −0.759339
\(847\) −6.98183 −0.239898
\(848\) −64.1761 −2.20381
\(849\) −76.3156 −2.61914
\(850\) −3.26825 −0.112100
\(851\) 42.4083 1.45374
\(852\) 105.017 3.59781
\(853\) −1.60687 −0.0550182 −0.0275091 0.999622i \(-0.508758\pi\)
−0.0275091 + 0.999622i \(0.508758\pi\)
\(854\) 34.2108 1.17067
\(855\) 22.0350 0.753581
\(856\) −48.6672 −1.66341
\(857\) 19.6947 0.672758 0.336379 0.941727i \(-0.390798\pi\)
0.336379 + 0.941727i \(0.390798\pi\)
\(858\) −82.5859 −2.81944
\(859\) −11.7540 −0.401040 −0.200520 0.979690i \(-0.564263\pi\)
−0.200520 + 0.979690i \(0.564263\pi\)
\(860\) −46.2372 −1.57668
\(861\) 3.50676 0.119510
\(862\) −36.5518 −1.24496
\(863\) −32.2841 −1.09896 −0.549482 0.835505i \(-0.685175\pi\)
−0.549482 + 0.835505i \(0.685175\pi\)
\(864\) 1.05154 0.0357743
\(865\) −5.18347 −0.176243
\(866\) 25.4091 0.863438
\(867\) −36.9195 −1.25385
\(868\) 4.52098 0.153452
\(869\) −10.1153 −0.343138
\(870\) −21.4316 −0.726599
\(871\) 18.2765 0.619274
\(872\) 110.281 3.73459
\(873\) 13.9470 0.472036
\(874\) 92.0747 3.11447
\(875\) 1.44699 0.0489172
\(876\) −76.9157 −2.59874
\(877\) −40.5928 −1.37072 −0.685361 0.728203i \(-0.740356\pi\)
−0.685361 + 0.728203i \(0.740356\pi\)
\(878\) −33.5449 −1.13209
\(879\) −51.5175 −1.73764
\(880\) −13.7121 −0.462235
\(881\) 15.6056 0.525765 0.262882 0.964828i \(-0.415327\pi\)
0.262882 + 0.964828i \(0.415327\pi\)
\(882\) 34.7058 1.16860
\(883\) −14.2212 −0.478581 −0.239291 0.970948i \(-0.576915\pi\)
−0.239291 + 0.970948i \(0.576915\pi\)
\(884\) −30.5714 −1.02823
\(885\) 2.22897 0.0749259
\(886\) 7.84542 0.263572
\(887\) 55.0102 1.84706 0.923530 0.383526i \(-0.125290\pi\)
0.923530 + 0.383526i \(0.125290\pi\)
\(888\) 121.230 4.06821
\(889\) 20.0624 0.672870
\(890\) 11.0992 0.372045
\(891\) 23.5528 0.789049
\(892\) −17.2739 −0.578373
\(893\) 24.3011 0.813206
\(894\) −8.31388 −0.278058
\(895\) −12.9026 −0.431285
\(896\) 24.0728 0.804215
\(897\) 62.9746 2.10266
\(898\) 22.5822 0.753579
\(899\) 2.61550 0.0872319
\(900\) 12.0127 0.400425
\(901\) −15.2124 −0.506799
\(902\) 6.23137 0.207482
\(903\) −38.0753 −1.26707
\(904\) 62.7862 2.08824
\(905\) 10.1403 0.337074
\(906\) 38.3088 1.27272
\(907\) 11.4747 0.381012 0.190506 0.981686i \(-0.438987\pi\)
0.190506 + 0.981686i \(0.438987\pi\)
\(908\) −53.6507 −1.78046
\(909\) −28.1456 −0.933531
\(910\) 19.9150 0.660176
\(911\) −34.1664 −1.13198 −0.565991 0.824411i \(-0.691506\pi\)
−0.565991 + 0.824411i \(0.691506\pi\)
\(912\) 103.711 3.43420
\(913\) −20.4709 −0.677488
\(914\) 19.1505 0.633442
\(915\) −22.8491 −0.755369
\(916\) 116.932 3.86353
\(917\) 18.1699 0.600022
\(918\) 1.33309 0.0439985
\(919\) 17.8279 0.588088 0.294044 0.955792i \(-0.404999\pi\)
0.294044 + 0.955792i \(0.404999\pi\)
\(920\) 26.5363 0.874876
\(921\) −2.26261 −0.0745554
\(922\) −23.8169 −0.784368
\(923\) −56.4555 −1.85826
\(924\) −36.8422 −1.21202
\(925\) −8.95720 −0.294511
\(926\) −40.9056 −1.34424
\(927\) 16.5561 0.543775
\(928\) −9.15716 −0.300599
\(929\) −0.900419 −0.0295418 −0.0147709 0.999891i \(-0.504702\pi\)
−0.0147709 + 0.999891i \(0.504702\pi\)
\(930\) −4.44278 −0.145684
\(931\) −38.1862 −1.25150
\(932\) 61.7003 2.02106
\(933\) −22.1807 −0.726163
\(934\) 56.1353 1.83680
\(935\) −3.25034 −0.106298
\(936\) 87.4033 2.85687
\(937\) −20.2468 −0.661434 −0.330717 0.943730i \(-0.607291\pi\)
−0.330717 + 0.943730i \(0.607291\pi\)
\(938\) 11.9963 0.391693
\(939\) −54.3231 −1.77277
\(940\) 13.2482 0.432107
\(941\) 2.58717 0.0843394 0.0421697 0.999110i \(-0.486573\pi\)
0.0421697 + 0.999110i \(0.486573\pi\)
\(942\) −7.63127 −0.248640
\(943\) −4.75164 −0.154735
\(944\) 5.09351 0.165780
\(945\) −0.590214 −0.0191997
\(946\) −67.6583 −2.19976
\(947\) −13.8701 −0.450719 −0.225359 0.974276i \(-0.572356\pi\)
−0.225359 + 0.974276i \(0.572356\pi\)
\(948\) 41.7087 1.35464
\(949\) 41.3489 1.34224
\(950\) −19.4474 −0.630958
\(951\) −70.0971 −2.27305
\(952\) −10.6082 −0.343814
\(953\) −33.2057 −1.07564 −0.537820 0.843060i \(-0.680752\pi\)
−0.537820 + 0.843060i \(0.680752\pi\)
\(954\) 82.2696 2.66358
\(955\) 24.9694 0.807990
\(956\) 105.058 3.39782
\(957\) −21.3142 −0.688989
\(958\) 59.3563 1.91772
\(959\) −1.81801 −0.0587064
\(960\) −11.0951 −0.358093
\(961\) −30.4578 −0.982510
\(962\) −123.279 −3.97466
\(963\) 24.5826 0.792162
\(964\) −112.178 −3.61301
\(965\) −16.3647 −0.526797
\(966\) 41.3353 1.32994
\(967\) −1.84359 −0.0592858 −0.0296429 0.999561i \(-0.509437\pi\)
−0.0296429 + 0.999561i \(0.509437\pi\)
\(968\) 27.0437 0.869217
\(969\) 24.5837 0.789744
\(970\) −12.3092 −0.395226
\(971\) 34.3365 1.10191 0.550956 0.834534i \(-0.314263\pi\)
0.550956 + 0.834534i \(0.314263\pi\)
\(972\) −91.9237 −2.94845
\(973\) −20.7610 −0.665569
\(974\) −72.2098 −2.31375
\(975\) −13.3011 −0.425976
\(976\) −52.2136 −1.67132
\(977\) −1.78151 −0.0569954 −0.0284977 0.999594i \(-0.509072\pi\)
−0.0284977 + 0.999594i \(0.509072\pi\)
\(978\) −50.2524 −1.60689
\(979\) 11.0383 0.352787
\(980\) −20.8179 −0.665002
\(981\) −55.7046 −1.77851
\(982\) −34.3515 −1.09620
\(983\) 62.4345 1.99135 0.995675 0.0929015i \(-0.0296142\pi\)
0.995675 + 0.0929015i \(0.0296142\pi\)
\(984\) −13.5832 −0.433017
\(985\) −2.82979 −0.0901645
\(986\) −11.6090 −0.369704
\(987\) 10.9095 0.347254
\(988\) −181.912 −5.78740
\(989\) 51.5918 1.64052
\(990\) 17.5781 0.558668
\(991\) −36.3717 −1.15538 −0.577692 0.816255i \(-0.696047\pi\)
−0.577692 + 0.816255i \(0.696047\pi\)
\(992\) −1.89828 −0.0602706
\(993\) −3.87872 −0.123087
\(994\) −37.0563 −1.17535
\(995\) 13.0368 0.413295
\(996\) 84.4083 2.67458
\(997\) 39.5482 1.25250 0.626251 0.779621i \(-0.284588\pi\)
0.626251 + 0.779621i \(0.284588\pi\)
\(998\) −28.1211 −0.890158
\(999\) 3.65356 0.115594
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 8045.2.a.c.1.12 127
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8045.2.a.c.1.12 127 1.1 even 1 trivial