Properties

Label 8009.2.a.a.1.18
Level $8009$
Weight $2$
Character 8009.1
Self dual yes
Analytic conductor $63.952$
Analytic rank $1$
Dimension $306$
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] = [8009,2,Mod(1,8009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8009, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8009 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8009.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: \(63.9521869788\)
Analytic rank: \(1\)
Dimension: \(306\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) \(=\) 8009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.52963 q^{2} +0.577486 q^{3} +4.39902 q^{4} -1.57582 q^{5} -1.46082 q^{6} -2.02955 q^{7} -6.06864 q^{8} -2.66651 q^{9} +O(q^{10})\) \(q-2.52963 q^{2} +0.577486 q^{3} +4.39902 q^{4} -1.57582 q^{5} -1.46082 q^{6} -2.02955 q^{7} -6.06864 q^{8} -2.66651 q^{9} +3.98624 q^{10} +1.54104 q^{11} +2.54037 q^{12} -2.97088 q^{13} +5.13402 q^{14} -0.910013 q^{15} +6.55335 q^{16} +5.26544 q^{17} +6.74528 q^{18} +0.106614 q^{19} -6.93206 q^{20} -1.17204 q^{21} -3.89825 q^{22} -3.47995 q^{23} -3.50455 q^{24} -2.51679 q^{25} +7.51522 q^{26} -3.27233 q^{27} -8.92805 q^{28} -0.567612 q^{29} +2.30200 q^{30} +9.64704 q^{31} -4.44028 q^{32} +0.889926 q^{33} -13.3196 q^{34} +3.19821 q^{35} -11.7300 q^{36} +8.29375 q^{37} -0.269693 q^{38} -1.71564 q^{39} +9.56307 q^{40} -5.00824 q^{41} +2.96482 q^{42} -2.17640 q^{43} +6.77905 q^{44} +4.20194 q^{45} +8.80297 q^{46} -1.86539 q^{47} +3.78447 q^{48} -2.88091 q^{49} +6.36655 q^{50} +3.04072 q^{51} -13.0690 q^{52} +13.0403 q^{53} +8.27778 q^{54} -2.42839 q^{55} +12.3166 q^{56} +0.0615680 q^{57} +1.43585 q^{58} -14.4740 q^{59} -4.00317 q^{60} +13.3263 q^{61} -24.4034 q^{62} +5.41182 q^{63} -1.87445 q^{64} +4.68157 q^{65} -2.25118 q^{66} -2.56018 q^{67} +23.1628 q^{68} -2.00962 q^{69} -8.09028 q^{70} -2.02974 q^{71} +16.1821 q^{72} -9.31908 q^{73} -20.9801 q^{74} -1.45341 q^{75} +0.468997 q^{76} -3.12761 q^{77} +4.33993 q^{78} -13.9983 q^{79} -10.3269 q^{80} +6.10981 q^{81} +12.6690 q^{82} -8.30188 q^{83} -5.15582 q^{84} -8.29738 q^{85} +5.50550 q^{86} -0.327788 q^{87} -9.35199 q^{88} +12.4061 q^{89} -10.6293 q^{90} +6.02956 q^{91} -15.3084 q^{92} +5.57103 q^{93} +4.71873 q^{94} -0.168004 q^{95} -2.56420 q^{96} +13.7317 q^{97} +7.28765 q^{98} -4.10919 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9} - 61 q^{10} - 43 q^{11} - 50 q^{12} - 89 q^{13} - 40 q^{14} - 61 q^{15} + 151 q^{16} - 52 q^{17} - 57 q^{18} - 185 q^{19} - 66 q^{20} - 63 q^{21} - 55 q^{22} - 62 q^{23} - 131 q^{24} + 209 q^{25} - 57 q^{26} - 88 q^{27} - 182 q^{28} - 67 q^{29} - 68 q^{30} - 240 q^{31} - 64 q^{32} - 52 q^{33} - 128 q^{34} - 99 q^{35} + 106 q^{36} - 49 q^{37} - 45 q^{38} - 190 q^{39} - 158 q^{40} - 72 q^{41} - 36 q^{42} - 141 q^{43} - 80 q^{44} - 100 q^{45} - 91 q^{46} - 105 q^{47} - 85 q^{48} + 116 q^{49} - 51 q^{50} - 145 q^{51} - 237 q^{52} - 48 q^{53} - 156 q^{54} - 420 q^{55} - 116 q^{56} - 35 q^{57} - 43 q^{58} - 139 q^{59} - 73 q^{60} - 233 q^{61} - 58 q^{62} - 252 q^{63} - 3 q^{64} - 45 q^{65} - 127 q^{66} - 108 q^{67} - 85 q^{68} - 164 q^{69} - 56 q^{70} - 131 q^{71} - 117 q^{72} - 118 q^{73} - 47 q^{74} - 112 q^{75} - 389 q^{76} - 36 q^{77} + 9 q^{78} - 382 q^{79} - 119 q^{80} + 102 q^{81} - 131 q^{82} - 59 q^{83} - 144 q^{84} - 140 q^{85} - 38 q^{86} - 301 q^{87} - 131 q^{88} - 98 q^{89} - 138 q^{90} - 176 q^{91} - 97 q^{92} - 60 q^{93} - 342 q^{94} - 154 q^{95} - 243 q^{96} - 109 q^{97} - 21 q^{98} - 173 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.52963 −1.78872 −0.894359 0.447350i \(-0.852368\pi\)
−0.894359 + 0.447350i \(0.852368\pi\)
\(3\) 0.577486 0.333412 0.166706 0.986007i \(-0.446687\pi\)
0.166706 + 0.986007i \(0.446687\pi\)
\(4\) 4.39902 2.19951
\(5\) −1.57582 −0.704728 −0.352364 0.935863i \(-0.614622\pi\)
−0.352364 + 0.935863i \(0.614622\pi\)
\(6\) −1.46082 −0.596379
\(7\) −2.02955 −0.767099 −0.383549 0.923520i \(-0.625298\pi\)
−0.383549 + 0.923520i \(0.625298\pi\)
\(8\) −6.06864 −2.14559
\(9\) −2.66651 −0.888837
\(10\) 3.98624 1.26056
\(11\) 1.54104 0.464640 0.232320 0.972639i \(-0.425368\pi\)
0.232320 + 0.972639i \(0.425368\pi\)
\(12\) 2.54037 0.733342
\(13\) −2.97088 −0.823974 −0.411987 0.911190i \(-0.635165\pi\)
−0.411987 + 0.911190i \(0.635165\pi\)
\(14\) 5.13402 1.37212
\(15\) −0.910013 −0.234964
\(16\) 6.55335 1.63834
\(17\) 5.26544 1.27706 0.638528 0.769598i \(-0.279543\pi\)
0.638528 + 0.769598i \(0.279543\pi\)
\(18\) 6.74528 1.58988
\(19\) 0.106614 0.0244589 0.0122294 0.999925i \(-0.496107\pi\)
0.0122294 + 0.999925i \(0.496107\pi\)
\(20\) −6.93206 −1.55006
\(21\) −1.17204 −0.255760
\(22\) −3.89825 −0.831110
\(23\) −3.47995 −0.725619 −0.362809 0.931863i \(-0.618182\pi\)
−0.362809 + 0.931863i \(0.618182\pi\)
\(24\) −3.50455 −0.715363
\(25\) −2.51679 −0.503359
\(26\) 7.51522 1.47386
\(27\) −3.27233 −0.629760
\(28\) −8.92805 −1.68724
\(29\) −0.567612 −0.105403 −0.0527014 0.998610i \(-0.516783\pi\)
−0.0527014 + 0.998610i \(0.516783\pi\)
\(30\) 2.30200 0.420285
\(31\) 9.64704 1.73266 0.866330 0.499472i \(-0.166473\pi\)
0.866330 + 0.499472i \(0.166473\pi\)
\(32\) −4.44028 −0.784938
\(33\) 0.889926 0.154916
\(34\) −13.3196 −2.28429
\(35\) 3.19821 0.540596
\(36\) −11.7300 −1.95501
\(37\) 8.29375 1.36348 0.681742 0.731592i \(-0.261222\pi\)
0.681742 + 0.731592i \(0.261222\pi\)
\(38\) −0.269693 −0.0437500
\(39\) −1.71564 −0.274722
\(40\) 9.56307 1.51205
\(41\) −5.00824 −0.782156 −0.391078 0.920357i \(-0.627898\pi\)
−0.391078 + 0.920357i \(0.627898\pi\)
\(42\) 2.96482 0.457482
\(43\) −2.17640 −0.331899 −0.165949 0.986134i \(-0.553069\pi\)
−0.165949 + 0.986134i \(0.553069\pi\)
\(44\) 6.77905 1.02198
\(45\) 4.20194 0.626388
\(46\) 8.80297 1.29793
\(47\) −1.86539 −0.272095 −0.136047 0.990702i \(-0.543440\pi\)
−0.136047 + 0.990702i \(0.543440\pi\)
\(48\) 3.78447 0.546241
\(49\) −2.88091 −0.411559
\(50\) 6.36655 0.900367
\(51\) 3.04072 0.425786
\(52\) −13.0690 −1.81234
\(53\) 13.0403 1.79122 0.895612 0.444837i \(-0.146738\pi\)
0.895612 + 0.444837i \(0.146738\pi\)
\(54\) 8.27778 1.12646
\(55\) −2.42839 −0.327445
\(56\) 12.3166 1.64588
\(57\) 0.0615680 0.00815487
\(58\) 1.43585 0.188536
\(59\) −14.4740 −1.88435 −0.942174 0.335123i \(-0.891222\pi\)
−0.942174 + 0.335123i \(0.891222\pi\)
\(60\) −4.00317 −0.516807
\(61\) 13.3263 1.70626 0.853130 0.521698i \(-0.174701\pi\)
0.853130 + 0.521698i \(0.174701\pi\)
\(62\) −24.4034 −3.09924
\(63\) 5.41182 0.681826
\(64\) −1.87445 −0.234306
\(65\) 4.68157 0.580677
\(66\) −2.25118 −0.277102
\(67\) −2.56018 −0.312776 −0.156388 0.987696i \(-0.549985\pi\)
−0.156388 + 0.987696i \(0.549985\pi\)
\(68\) 23.1628 2.80890
\(69\) −2.00962 −0.241930
\(70\) −8.09028 −0.966974
\(71\) −2.02974 −0.240885 −0.120443 0.992720i \(-0.538431\pi\)
−0.120443 + 0.992720i \(0.538431\pi\)
\(72\) 16.1821 1.90708
\(73\) −9.31908 −1.09072 −0.545358 0.838203i \(-0.683606\pi\)
−0.545358 + 0.838203i \(0.683606\pi\)
\(74\) −20.9801 −2.43889
\(75\) −1.45341 −0.167826
\(76\) 0.468997 0.0537976
\(77\) −3.12761 −0.356425
\(78\) 4.33993 0.491401
\(79\) −13.9983 −1.57494 −0.787468 0.616355i \(-0.788609\pi\)
−0.787468 + 0.616355i \(0.788609\pi\)
\(80\) −10.3269 −1.15458
\(81\) 6.10981 0.678868
\(82\) 12.6690 1.39906
\(83\) −8.30188 −0.911250 −0.455625 0.890172i \(-0.650584\pi\)
−0.455625 + 0.890172i \(0.650584\pi\)
\(84\) −5.15582 −0.562546
\(85\) −8.29738 −0.899978
\(86\) 5.50550 0.593673
\(87\) −0.327788 −0.0351425
\(88\) −9.35199 −0.996925
\(89\) 12.4061 1.31504 0.657521 0.753436i \(-0.271605\pi\)
0.657521 + 0.753436i \(0.271605\pi\)
\(90\) −10.6293 −1.12043
\(91\) 6.02956 0.632070
\(92\) −15.3084 −1.59601
\(93\) 5.57103 0.577689
\(94\) 4.71873 0.486700
\(95\) −0.168004 −0.0172369
\(96\) −2.56420 −0.261707
\(97\) 13.7317 1.39424 0.697121 0.716953i \(-0.254464\pi\)
0.697121 + 0.716953i \(0.254464\pi\)
\(98\) 7.28765 0.736163
\(99\) −4.10919 −0.412989
\(100\) −11.0714 −1.10714
\(101\) 9.47232 0.942532 0.471266 0.881991i \(-0.343797\pi\)
0.471266 + 0.881991i \(0.343797\pi\)
\(102\) −7.69189 −0.761610
\(103\) 5.43537 0.535563 0.267782 0.963480i \(-0.413709\pi\)
0.267782 + 0.963480i \(0.413709\pi\)
\(104\) 18.0292 1.76791
\(105\) 1.84692 0.180241
\(106\) −32.9871 −3.20399
\(107\) 10.9215 1.05582 0.527909 0.849301i \(-0.322976\pi\)
0.527909 + 0.849301i \(0.322976\pi\)
\(108\) −14.3950 −1.38516
\(109\) 11.2703 1.07950 0.539751 0.841825i \(-0.318518\pi\)
0.539751 + 0.841825i \(0.318518\pi\)
\(110\) 6.14294 0.585706
\(111\) 4.78952 0.454602
\(112\) −13.3004 −1.25677
\(113\) 12.1763 1.14545 0.572724 0.819748i \(-0.305887\pi\)
0.572724 + 0.819748i \(0.305887\pi\)
\(114\) −0.155744 −0.0145868
\(115\) 5.48377 0.511364
\(116\) −2.49694 −0.231835
\(117\) 7.92188 0.732378
\(118\) 36.6137 3.37057
\(119\) −10.6865 −0.979629
\(120\) 5.52254 0.504137
\(121\) −8.62521 −0.784110
\(122\) −33.7106 −3.05202
\(123\) −2.89219 −0.260780
\(124\) 42.4376 3.81100
\(125\) 11.8451 1.05946
\(126\) −13.6899 −1.21959
\(127\) −2.85204 −0.253077 −0.126539 0.991962i \(-0.540387\pi\)
−0.126539 + 0.991962i \(0.540387\pi\)
\(128\) 13.6222 1.20405
\(129\) −1.25684 −0.110659
\(130\) −11.8426 −1.03867
\(131\) 15.9974 1.39770 0.698851 0.715268i \(-0.253695\pi\)
0.698851 + 0.715268i \(0.253695\pi\)
\(132\) 3.91481 0.340740
\(133\) −0.216378 −0.0187624
\(134\) 6.47631 0.559468
\(135\) 5.15660 0.443809
\(136\) −31.9540 −2.74004
\(137\) 19.2710 1.64643 0.823215 0.567729i \(-0.192178\pi\)
0.823215 + 0.567729i \(0.192178\pi\)
\(138\) 5.08359 0.432744
\(139\) −11.9024 −1.00955 −0.504774 0.863251i \(-0.668424\pi\)
−0.504774 + 0.863251i \(0.668424\pi\)
\(140\) 14.0690 1.18905
\(141\) −1.07723 −0.0907194
\(142\) 5.13448 0.430876
\(143\) −4.57823 −0.382851
\(144\) −17.4746 −1.45622
\(145\) 0.894454 0.0742804
\(146\) 23.5738 1.95098
\(147\) −1.66369 −0.137219
\(148\) 36.4844 2.99900
\(149\) −4.74061 −0.388366 −0.194183 0.980965i \(-0.562205\pi\)
−0.194183 + 0.980965i \(0.562205\pi\)
\(150\) 3.67659 0.300193
\(151\) −4.97433 −0.404806 −0.202403 0.979302i \(-0.564875\pi\)
−0.202403 + 0.979302i \(0.564875\pi\)
\(152\) −0.647000 −0.0524787
\(153\) −14.0404 −1.13510
\(154\) 7.91170 0.637543
\(155\) −15.2020 −1.22105
\(156\) −7.54714 −0.604255
\(157\) −10.3181 −0.823475 −0.411738 0.911302i \(-0.635078\pi\)
−0.411738 + 0.911302i \(0.635078\pi\)
\(158\) 35.4106 2.81712
\(159\) 7.53059 0.597215
\(160\) 6.99708 0.553167
\(161\) 7.06274 0.556622
\(162\) −15.4555 −1.21430
\(163\) 8.34247 0.653432 0.326716 0.945122i \(-0.394058\pi\)
0.326716 + 0.945122i \(0.394058\pi\)
\(164\) −22.0314 −1.72036
\(165\) −1.40236 −0.109174
\(166\) 21.0007 1.62997
\(167\) 1.14915 0.0889239 0.0444620 0.999011i \(-0.485843\pi\)
0.0444620 + 0.999011i \(0.485843\pi\)
\(168\) 7.11267 0.548754
\(169\) −4.17387 −0.321067
\(170\) 20.9893 1.60981
\(171\) −0.284287 −0.0217400
\(172\) −9.57405 −0.730015
\(173\) 6.33437 0.481593 0.240797 0.970576i \(-0.422591\pi\)
0.240797 + 0.970576i \(0.422591\pi\)
\(174\) 0.829182 0.0628601
\(175\) 5.10797 0.386126
\(176\) 10.0990 0.761237
\(177\) −8.35850 −0.628264
\(178\) −31.3828 −2.35224
\(179\) 0.271048 0.0202591 0.0101295 0.999949i \(-0.496776\pi\)
0.0101295 + 0.999949i \(0.496776\pi\)
\(180\) 18.4844 1.37775
\(181\) −26.2785 −1.95327 −0.976633 0.214915i \(-0.931053\pi\)
−0.976633 + 0.214915i \(0.931053\pi\)
\(182\) −15.2525 −1.13059
\(183\) 7.69576 0.568887
\(184\) 21.1185 1.55688
\(185\) −13.0695 −0.960886
\(186\) −14.0926 −1.03332
\(187\) 8.11424 0.593372
\(188\) −8.20587 −0.598475
\(189\) 6.64136 0.483088
\(190\) 0.424988 0.0308319
\(191\) 12.7717 0.924129 0.462064 0.886846i \(-0.347109\pi\)
0.462064 + 0.886846i \(0.347109\pi\)
\(192\) −1.08247 −0.0781204
\(193\) 10.1400 0.729890 0.364945 0.931029i \(-0.381088\pi\)
0.364945 + 0.931029i \(0.381088\pi\)
\(194\) −34.7361 −2.49391
\(195\) 2.70354 0.193605
\(196\) −12.6732 −0.905229
\(197\) −20.6470 −1.47103 −0.735517 0.677506i \(-0.763061\pi\)
−0.735517 + 0.677506i \(0.763061\pi\)
\(198\) 10.3947 0.738721
\(199\) −3.85601 −0.273345 −0.136673 0.990616i \(-0.543641\pi\)
−0.136673 + 0.990616i \(0.543641\pi\)
\(200\) 15.2735 1.08000
\(201\) −1.47847 −0.104283
\(202\) −23.9615 −1.68592
\(203\) 1.15200 0.0808545
\(204\) 13.3762 0.936520
\(205\) 7.89209 0.551207
\(206\) −13.7495 −0.957971
\(207\) 9.27931 0.644957
\(208\) −19.4692 −1.34995
\(209\) 0.164296 0.0113646
\(210\) −4.67202 −0.322400
\(211\) −14.9307 −1.02787 −0.513937 0.857828i \(-0.671814\pi\)
−0.513937 + 0.857828i \(0.671814\pi\)
\(212\) 57.3646 3.93982
\(213\) −1.17214 −0.0803140
\(214\) −27.6272 −1.88856
\(215\) 3.42962 0.233898
\(216\) 19.8586 1.35120
\(217\) −19.5792 −1.32912
\(218\) −28.5097 −1.93092
\(219\) −5.38163 −0.363657
\(220\) −10.6826 −0.720218
\(221\) −15.6430 −1.05226
\(222\) −12.1157 −0.813154
\(223\) −11.3457 −0.759764 −0.379882 0.925035i \(-0.624035\pi\)
−0.379882 + 0.925035i \(0.624035\pi\)
\(224\) 9.01178 0.602125
\(225\) 6.71105 0.447404
\(226\) −30.8015 −2.04888
\(227\) −14.4999 −0.962394 −0.481197 0.876612i \(-0.659798\pi\)
−0.481197 + 0.876612i \(0.659798\pi\)
\(228\) 0.270839 0.0179367
\(229\) −6.86488 −0.453644 −0.226822 0.973936i \(-0.572834\pi\)
−0.226822 + 0.973936i \(0.572834\pi\)
\(230\) −13.8719 −0.914686
\(231\) −1.80615 −0.118836
\(232\) 3.44463 0.226151
\(233\) 13.8280 0.905903 0.452951 0.891535i \(-0.350371\pi\)
0.452951 + 0.891535i \(0.350371\pi\)
\(234\) −20.0394 −1.31002
\(235\) 2.93951 0.191753
\(236\) −63.6713 −4.14465
\(237\) −8.08384 −0.525102
\(238\) 27.0329 1.75228
\(239\) −6.43481 −0.416233 −0.208117 0.978104i \(-0.566733\pi\)
−0.208117 + 0.978104i \(0.566733\pi\)
\(240\) −5.96364 −0.384951
\(241\) 14.7804 0.952087 0.476044 0.879422i \(-0.342070\pi\)
0.476044 + 0.879422i \(0.342070\pi\)
\(242\) 21.8186 1.40255
\(243\) 13.3453 0.856102
\(244\) 58.6228 3.75294
\(245\) 4.53980 0.290037
\(246\) 7.31617 0.466462
\(247\) −0.316737 −0.0201535
\(248\) −58.5444 −3.71757
\(249\) −4.79422 −0.303821
\(250\) −29.9637 −1.89507
\(251\) −4.44745 −0.280720 −0.140360 0.990101i \(-0.544826\pi\)
−0.140360 + 0.990101i \(0.544826\pi\)
\(252\) 23.8067 1.49968
\(253\) −5.36272 −0.337152
\(254\) 7.21459 0.452684
\(255\) −4.79162 −0.300063
\(256\) −30.7103 −1.91939
\(257\) −23.2825 −1.45232 −0.726161 0.687525i \(-0.758697\pi\)
−0.726161 + 0.687525i \(0.758697\pi\)
\(258\) 3.17935 0.197937
\(259\) −16.8326 −1.04593
\(260\) 20.5943 1.27721
\(261\) 1.51354 0.0936860
\(262\) −40.4675 −2.50009
\(263\) 9.82150 0.605619 0.302810 0.953051i \(-0.402075\pi\)
0.302810 + 0.953051i \(0.402075\pi\)
\(264\) −5.40064 −0.332386
\(265\) −20.5492 −1.26233
\(266\) 0.547357 0.0335606
\(267\) 7.16434 0.438450
\(268\) −11.2623 −0.687954
\(269\) 10.9090 0.665131 0.332566 0.943080i \(-0.392086\pi\)
0.332566 + 0.943080i \(0.392086\pi\)
\(270\) −13.0443 −0.793850
\(271\) 9.42960 0.572808 0.286404 0.958109i \(-0.407540\pi\)
0.286404 + 0.958109i \(0.407540\pi\)
\(272\) 34.5063 2.09225
\(273\) 3.48198 0.210739
\(274\) −48.7484 −2.94500
\(275\) −3.87847 −0.233881
\(276\) −8.84036 −0.532127
\(277\) 13.2115 0.793800 0.396900 0.917862i \(-0.370086\pi\)
0.396900 + 0.917862i \(0.370086\pi\)
\(278\) 30.1087 1.80580
\(279\) −25.7239 −1.54005
\(280\) −19.4088 −1.15990
\(281\) −16.3299 −0.974158 −0.487079 0.873358i \(-0.661938\pi\)
−0.487079 + 0.873358i \(0.661938\pi\)
\(282\) 2.72500 0.162271
\(283\) 12.1223 0.720595 0.360298 0.932837i \(-0.382675\pi\)
0.360298 + 0.932837i \(0.382675\pi\)
\(284\) −8.92886 −0.529830
\(285\) −0.0970200 −0.00574697
\(286\) 11.5812 0.684813
\(287\) 10.1645 0.599991
\(288\) 11.8400 0.697681
\(289\) 10.7249 0.630875
\(290\) −2.26264 −0.132867
\(291\) 7.92986 0.464856
\(292\) −40.9948 −2.39904
\(293\) −16.3448 −0.954871 −0.477436 0.878667i \(-0.658434\pi\)
−0.477436 + 0.878667i \(0.658434\pi\)
\(294\) 4.20851 0.245445
\(295\) 22.8083 1.32795
\(296\) −50.3318 −2.92548
\(297\) −5.04278 −0.292612
\(298\) 11.9920 0.694676
\(299\) 10.3385 0.597891
\(300\) −6.39359 −0.369134
\(301\) 4.41713 0.254599
\(302\) 12.5832 0.724083
\(303\) 5.47013 0.314251
\(304\) 0.698678 0.0400719
\(305\) −20.9999 −1.20245
\(306\) 35.5169 2.03036
\(307\) 12.6810 0.723743 0.361871 0.932228i \(-0.382138\pi\)
0.361871 + 0.932228i \(0.382138\pi\)
\(308\) −13.7584 −0.783960
\(309\) 3.13885 0.178563
\(310\) 38.4554 2.18412
\(311\) 20.0795 1.13860 0.569301 0.822129i \(-0.307214\pi\)
0.569301 + 0.822129i \(0.307214\pi\)
\(312\) 10.4116 0.589441
\(313\) −30.7965 −1.74072 −0.870361 0.492414i \(-0.836115\pi\)
−0.870361 + 0.492414i \(0.836115\pi\)
\(314\) 26.1010 1.47296
\(315\) −8.52806 −0.480502
\(316\) −61.5790 −3.46409
\(317\) −14.4214 −0.809986 −0.404993 0.914320i \(-0.632726\pi\)
−0.404993 + 0.914320i \(0.632726\pi\)
\(318\) −19.0496 −1.06825
\(319\) −0.874711 −0.0489744
\(320\) 2.95379 0.165122
\(321\) 6.30699 0.352022
\(322\) −17.8661 −0.995639
\(323\) 0.561369 0.0312354
\(324\) 26.8772 1.49318
\(325\) 7.47709 0.414754
\(326\) −21.1033 −1.16881
\(327\) 6.50845 0.359918
\(328\) 30.3932 1.67818
\(329\) 3.78590 0.208723
\(330\) 3.54746 0.195281
\(331\) 3.65452 0.200870 0.100435 0.994944i \(-0.467976\pi\)
0.100435 + 0.994944i \(0.467976\pi\)
\(332\) −36.5202 −2.00430
\(333\) −22.1154 −1.21192
\(334\) −2.90692 −0.159060
\(335\) 4.03438 0.220422
\(336\) −7.68078 −0.419021
\(337\) −7.97610 −0.434486 −0.217243 0.976118i \(-0.569706\pi\)
−0.217243 + 0.976118i \(0.569706\pi\)
\(338\) 10.5583 0.574298
\(339\) 7.03163 0.381906
\(340\) −36.5004 −1.97951
\(341\) 14.8664 0.805063
\(342\) 0.719140 0.0388866
\(343\) 20.0538 1.08281
\(344\) 13.2078 0.712117
\(345\) 3.16680 0.170495
\(346\) −16.0236 −0.861434
\(347\) 15.7710 0.846630 0.423315 0.905983i \(-0.360866\pi\)
0.423315 + 0.905983i \(0.360866\pi\)
\(348\) −1.44195 −0.0772964
\(349\) −25.1619 −1.34689 −0.673443 0.739239i \(-0.735185\pi\)
−0.673443 + 0.739239i \(0.735185\pi\)
\(350\) −12.9213 −0.690670
\(351\) 9.72170 0.518906
\(352\) −6.84263 −0.364713
\(353\) 18.7315 0.996977 0.498489 0.866896i \(-0.333889\pi\)
0.498489 + 0.866896i \(0.333889\pi\)
\(354\) 21.1439 1.12379
\(355\) 3.19850 0.169759
\(356\) 54.5746 2.89245
\(357\) −6.17130 −0.326620
\(358\) −0.685652 −0.0362378
\(359\) −20.4138 −1.07740 −0.538698 0.842499i \(-0.681084\pi\)
−0.538698 + 0.842499i \(0.681084\pi\)
\(360\) −25.5000 −1.34397
\(361\) −18.9886 −0.999402
\(362\) 66.4749 3.49384
\(363\) −4.98093 −0.261431
\(364\) 26.5242 1.39024
\(365\) 14.6852 0.768658
\(366\) −19.4674 −1.01758
\(367\) −14.3699 −0.750104 −0.375052 0.927004i \(-0.622375\pi\)
−0.375052 + 0.927004i \(0.622375\pi\)
\(368\) −22.8053 −1.18881
\(369\) 13.3545 0.695209
\(370\) 33.0609 1.71875
\(371\) −26.4660 −1.37405
\(372\) 24.5071 1.27063
\(373\) −15.9217 −0.824392 −0.412196 0.911095i \(-0.635238\pi\)
−0.412196 + 0.911095i \(0.635238\pi\)
\(374\) −20.5260 −1.06137
\(375\) 6.84038 0.353236
\(376\) 11.3203 0.583802
\(377\) 1.68631 0.0868492
\(378\) −16.8002 −0.864108
\(379\) −6.99928 −0.359529 −0.179764 0.983710i \(-0.557534\pi\)
−0.179764 + 0.983710i \(0.557534\pi\)
\(380\) −0.739054 −0.0379127
\(381\) −1.64701 −0.0843789
\(382\) −32.3077 −1.65301
\(383\) 10.3949 0.531153 0.265576 0.964090i \(-0.414438\pi\)
0.265576 + 0.964090i \(0.414438\pi\)
\(384\) 7.86664 0.401443
\(385\) 4.92856 0.251182
\(386\) −25.6503 −1.30557
\(387\) 5.80341 0.295004
\(388\) 60.4060 3.06665
\(389\) −22.1215 −1.12160 −0.560802 0.827950i \(-0.689507\pi\)
−0.560802 + 0.827950i \(0.689507\pi\)
\(390\) −6.83895 −0.346304
\(391\) −18.3235 −0.926657
\(392\) 17.4832 0.883036
\(393\) 9.23828 0.466010
\(394\) 52.2291 2.63127
\(395\) 22.0589 1.10990
\(396\) −18.0764 −0.908374
\(397\) 11.7331 0.588866 0.294433 0.955672i \(-0.404869\pi\)
0.294433 + 0.955672i \(0.404869\pi\)
\(398\) 9.75428 0.488938
\(399\) −0.124955 −0.00625560
\(400\) −16.4934 −0.824672
\(401\) −6.42865 −0.321031 −0.160516 0.987033i \(-0.551316\pi\)
−0.160516 + 0.987033i \(0.551316\pi\)
\(402\) 3.73997 0.186533
\(403\) −28.6602 −1.42767
\(404\) 41.6690 2.07311
\(405\) −9.62795 −0.478417
\(406\) −2.91413 −0.144626
\(407\) 12.7810 0.633529
\(408\) −18.4530 −0.913560
\(409\) 1.19610 0.0591433 0.0295716 0.999563i \(-0.490586\pi\)
0.0295716 + 0.999563i \(0.490586\pi\)
\(410\) −19.9641 −0.985955
\(411\) 11.1287 0.548939
\(412\) 23.9103 1.17798
\(413\) 29.3757 1.44548
\(414\) −23.4732 −1.15365
\(415\) 13.0823 0.642183
\(416\) 13.1915 0.646768
\(417\) −6.87347 −0.336595
\(418\) −0.415607 −0.0203280
\(419\) −6.33558 −0.309513 −0.154757 0.987953i \(-0.549459\pi\)
−0.154757 + 0.987953i \(0.549459\pi\)
\(420\) 8.12464 0.396442
\(421\) −28.6739 −1.39748 −0.698741 0.715375i \(-0.746256\pi\)
−0.698741 + 0.715375i \(0.746256\pi\)
\(422\) 37.7692 1.83858
\(423\) 4.97407 0.241848
\(424\) −79.1369 −3.84323
\(425\) −13.2520 −0.642818
\(426\) 2.96509 0.143659
\(427\) −27.0465 −1.30887
\(428\) 48.0438 2.32228
\(429\) −2.64386 −0.127647
\(430\) −8.67567 −0.418378
\(431\) −12.0146 −0.578724 −0.289362 0.957220i \(-0.593443\pi\)
−0.289362 + 0.957220i \(0.593443\pi\)
\(432\) −21.4447 −1.03176
\(433\) 4.45511 0.214099 0.107050 0.994254i \(-0.465860\pi\)
0.107050 + 0.994254i \(0.465860\pi\)
\(434\) 49.5281 2.37742
\(435\) 0.516534 0.0247659
\(436\) 49.5784 2.37438
\(437\) −0.371010 −0.0177478
\(438\) 13.6135 0.650480
\(439\) −25.7040 −1.22679 −0.613393 0.789778i \(-0.710196\pi\)
−0.613393 + 0.789778i \(0.710196\pi\)
\(440\) 14.7370 0.702561
\(441\) 7.68199 0.365809
\(442\) 39.5710 1.88220
\(443\) −33.4065 −1.58719 −0.793597 0.608444i \(-0.791794\pi\)
−0.793597 + 0.608444i \(0.791794\pi\)
\(444\) 21.0692 0.999901
\(445\) −19.5497 −0.926747
\(446\) 28.7004 1.35900
\(447\) −2.73763 −0.129486
\(448\) 3.80430 0.179736
\(449\) 28.2214 1.33185 0.665924 0.746019i \(-0.268037\pi\)
0.665924 + 0.746019i \(0.268037\pi\)
\(450\) −16.9765 −0.800279
\(451\) −7.71789 −0.363421
\(452\) 53.5637 2.51943
\(453\) −2.87261 −0.134967
\(454\) 36.6794 1.72145
\(455\) −9.50150 −0.445437
\(456\) −0.373634 −0.0174970
\(457\) −0.0250571 −0.00117212 −0.000586061 1.00000i \(-0.500187\pi\)
−0.000586061 1.00000i \(0.500187\pi\)
\(458\) 17.3656 0.811441
\(459\) −17.2303 −0.804239
\(460\) 24.1232 1.12475
\(461\) 30.0540 1.39975 0.699877 0.714263i \(-0.253238\pi\)
0.699877 + 0.714263i \(0.253238\pi\)
\(462\) 4.56890 0.212564
\(463\) 16.9027 0.785535 0.392767 0.919638i \(-0.371518\pi\)
0.392767 + 0.919638i \(0.371518\pi\)
\(464\) −3.71976 −0.172686
\(465\) −8.77894 −0.407113
\(466\) −34.9797 −1.62040
\(467\) 10.7325 0.496640 0.248320 0.968678i \(-0.420121\pi\)
0.248320 + 0.968678i \(0.420121\pi\)
\(468\) 34.8485 1.61087
\(469\) 5.19602 0.239930
\(470\) −7.43587 −0.342991
\(471\) −5.95856 −0.274556
\(472\) 87.8372 4.04303
\(473\) −3.35392 −0.154213
\(474\) 20.4491 0.939259
\(475\) −0.268325 −0.0123116
\(476\) −47.0101 −2.15470
\(477\) −34.7721 −1.59211
\(478\) 16.2777 0.744524
\(479\) 20.9272 0.956190 0.478095 0.878308i \(-0.341328\pi\)
0.478095 + 0.878308i \(0.341328\pi\)
\(480\) 4.04071 0.184432
\(481\) −24.6398 −1.12348
\(482\) −37.3889 −1.70302
\(483\) 4.07863 0.185584
\(484\) −37.9425 −1.72466
\(485\) −21.6387 −0.982561
\(486\) −33.7587 −1.53133
\(487\) −22.9441 −1.03969 −0.519847 0.854259i \(-0.674011\pi\)
−0.519847 + 0.854259i \(0.674011\pi\)
\(488\) −80.8726 −3.66093
\(489\) 4.81766 0.217862
\(490\) −11.4840 −0.518795
\(491\) 15.2557 0.688478 0.344239 0.938882i \(-0.388137\pi\)
0.344239 + 0.938882i \(0.388137\pi\)
\(492\) −12.7228 −0.573588
\(493\) −2.98873 −0.134606
\(494\) 0.801227 0.0360489
\(495\) 6.47534 0.291045
\(496\) 63.2205 2.83868
\(497\) 4.11946 0.184783
\(498\) 12.1276 0.543450
\(499\) 1.70219 0.0762005 0.0381003 0.999274i \(-0.487869\pi\)
0.0381003 + 0.999274i \(0.487869\pi\)
\(500\) 52.1069 2.33029
\(501\) 0.663618 0.0296483
\(502\) 11.2504 0.502130
\(503\) 3.31243 0.147694 0.0738470 0.997270i \(-0.476472\pi\)
0.0738470 + 0.997270i \(0.476472\pi\)
\(504\) −32.8424 −1.46292
\(505\) −14.9267 −0.664228
\(506\) 13.5657 0.603069
\(507\) −2.41035 −0.107047
\(508\) −12.5462 −0.556646
\(509\) −34.6962 −1.53788 −0.768942 0.639319i \(-0.779216\pi\)
−0.768942 + 0.639319i \(0.779216\pi\)
\(510\) 12.1210 0.536728
\(511\) 18.9136 0.836687
\(512\) 50.4411 2.22920
\(513\) −0.348875 −0.0154032
\(514\) 58.8961 2.59779
\(515\) −8.56516 −0.377426
\(516\) −5.52888 −0.243395
\(517\) −2.87463 −0.126426
\(518\) 42.5803 1.87087
\(519\) 3.65801 0.160569
\(520\) −28.4107 −1.24589
\(521\) 8.33417 0.365127 0.182563 0.983194i \(-0.441561\pi\)
0.182563 + 0.983194i \(0.441561\pi\)
\(522\) −3.82870 −0.167578
\(523\) −9.78912 −0.428048 −0.214024 0.976828i \(-0.568657\pi\)
−0.214024 + 0.976828i \(0.568657\pi\)
\(524\) 70.3730 3.07426
\(525\) 2.94978 0.128739
\(526\) −24.8447 −1.08328
\(527\) 50.7959 2.21271
\(528\) 5.83200 0.253805
\(529\) −10.8900 −0.473477
\(530\) 51.9818 2.25794
\(531\) 38.5950 1.67488
\(532\) −0.951853 −0.0412681
\(533\) 14.8789 0.644477
\(534\) −18.1231 −0.784264
\(535\) −17.2103 −0.744064
\(536\) 15.5368 0.671088
\(537\) 0.156526 0.00675462
\(538\) −27.5956 −1.18973
\(539\) −4.43959 −0.191227
\(540\) 22.6840 0.976164
\(541\) −33.2973 −1.43156 −0.715782 0.698324i \(-0.753930\pi\)
−0.715782 + 0.698324i \(0.753930\pi\)
\(542\) −23.8534 −1.02459
\(543\) −15.1755 −0.651241
\(544\) −23.3800 −1.00241
\(545\) −17.7600 −0.760755
\(546\) −8.80813 −0.376953
\(547\) −0.00795725 −0.000340227 0 −0.000170114 1.00000i \(-0.500054\pi\)
−0.000170114 1.00000i \(0.500054\pi\)
\(548\) 84.7735 3.62134
\(549\) −35.5348 −1.51659
\(550\) 9.81109 0.418346
\(551\) −0.0605153 −0.00257804
\(552\) 12.1956 0.519081
\(553\) 28.4104 1.20813
\(554\) −33.4201 −1.41988
\(555\) −7.54743 −0.320370
\(556\) −52.3590 −2.22051
\(557\) 39.2934 1.66492 0.832458 0.554088i \(-0.186933\pi\)
0.832458 + 0.554088i \(0.186933\pi\)
\(558\) 65.0720 2.75472
\(559\) 6.46584 0.273476
\(560\) 20.9590 0.885679
\(561\) 4.68585 0.197837
\(562\) 41.3085 1.74249
\(563\) −45.6737 −1.92492 −0.962458 0.271431i \(-0.912503\pi\)
−0.962458 + 0.271431i \(0.912503\pi\)
\(564\) −4.73878 −0.199538
\(565\) −19.1876 −0.807229
\(566\) −30.6649 −1.28894
\(567\) −12.4002 −0.520759
\(568\) 12.3177 0.516841
\(569\) −25.3923 −1.06450 −0.532251 0.846587i \(-0.678654\pi\)
−0.532251 + 0.846587i \(0.678654\pi\)
\(570\) 0.245425 0.0102797
\(571\) −27.0741 −1.13302 −0.566508 0.824056i \(-0.691706\pi\)
−0.566508 + 0.824056i \(0.691706\pi\)
\(572\) −20.1398 −0.842085
\(573\) 7.37548 0.308115
\(574\) −25.7124 −1.07322
\(575\) 8.75830 0.365247
\(576\) 4.99824 0.208260
\(577\) 1.46468 0.0609752 0.0304876 0.999535i \(-0.490294\pi\)
0.0304876 + 0.999535i \(0.490294\pi\)
\(578\) −27.1299 −1.12846
\(579\) 5.85568 0.243354
\(580\) 3.93472 0.163380
\(581\) 16.8491 0.699019
\(582\) −20.0596 −0.831497
\(583\) 20.0956 0.832274
\(584\) 56.5541 2.34022
\(585\) −12.4835 −0.516127
\(586\) 41.3462 1.70800
\(587\) 22.8204 0.941900 0.470950 0.882160i \(-0.343911\pi\)
0.470950 + 0.882160i \(0.343911\pi\)
\(588\) −7.31860 −0.301814
\(589\) 1.02851 0.0423789
\(590\) −57.6966 −2.37533
\(591\) −11.9233 −0.490460
\(592\) 54.3519 2.23385
\(593\) 8.86170 0.363907 0.181953 0.983307i \(-0.441758\pi\)
0.181953 + 0.983307i \(0.441758\pi\)
\(594\) 12.7564 0.523400
\(595\) 16.8400 0.690372
\(596\) −20.8540 −0.854214
\(597\) −2.22679 −0.0911365
\(598\) −26.1526 −1.06946
\(599\) 28.7869 1.17620 0.588100 0.808788i \(-0.299876\pi\)
0.588100 + 0.808788i \(0.299876\pi\)
\(600\) 8.82023 0.360084
\(601\) 10.6460 0.434258 0.217129 0.976143i \(-0.430331\pi\)
0.217129 + 0.976143i \(0.430331\pi\)
\(602\) −11.1737 −0.455406
\(603\) 6.82675 0.278007
\(604\) −21.8822 −0.890374
\(605\) 13.5918 0.552584
\(606\) −13.8374 −0.562106
\(607\) −39.4007 −1.59923 −0.799613 0.600516i \(-0.794962\pi\)
−0.799613 + 0.600516i \(0.794962\pi\)
\(608\) −0.473395 −0.0191987
\(609\) 0.665263 0.0269578
\(610\) 53.1219 2.15084
\(611\) 5.54184 0.224199
\(612\) −61.7638 −2.49665
\(613\) −17.3233 −0.699682 −0.349841 0.936809i \(-0.613764\pi\)
−0.349841 + 0.936809i \(0.613764\pi\)
\(614\) −32.0782 −1.29457
\(615\) 4.55757 0.183779
\(616\) 18.9804 0.764740
\(617\) 42.8238 1.72402 0.862011 0.506890i \(-0.169205\pi\)
0.862011 + 0.506890i \(0.169205\pi\)
\(618\) −7.94012 −0.319399
\(619\) 24.6484 0.990705 0.495352 0.868692i \(-0.335039\pi\)
0.495352 + 0.868692i \(0.335039\pi\)
\(620\) −66.8739 −2.68572
\(621\) 11.3875 0.456966
\(622\) −50.7936 −2.03664
\(623\) −25.1788 −1.00877
\(624\) −11.2432 −0.450088
\(625\) −6.08179 −0.243271
\(626\) 77.9038 3.11366
\(627\) 0.0948785 0.00378908
\(628\) −45.3896 −1.81124
\(629\) 43.6703 1.74125
\(630\) 21.5728 0.859482
\(631\) −11.5495 −0.459780 −0.229890 0.973217i \(-0.573837\pi\)
−0.229890 + 0.973217i \(0.573837\pi\)
\(632\) 84.9508 3.37916
\(633\) −8.62229 −0.342705
\(634\) 36.4807 1.44884
\(635\) 4.49429 0.178351
\(636\) 33.1272 1.31358
\(637\) 8.55885 0.339114
\(638\) 2.21269 0.0876014
\(639\) 5.41231 0.214108
\(640\) −21.4662 −0.848524
\(641\) −3.19212 −0.126081 −0.0630406 0.998011i \(-0.520080\pi\)
−0.0630406 + 0.998011i \(0.520080\pi\)
\(642\) −15.9543 −0.629668
\(643\) −1.52316 −0.0600676 −0.0300338 0.999549i \(-0.509561\pi\)
−0.0300338 + 0.999549i \(0.509561\pi\)
\(644\) 31.0691 1.22430
\(645\) 1.98056 0.0779844
\(646\) −1.42005 −0.0558713
\(647\) −33.4498 −1.31505 −0.657524 0.753433i \(-0.728396\pi\)
−0.657524 + 0.753433i \(0.728396\pi\)
\(648\) −37.0782 −1.45657
\(649\) −22.3049 −0.875544
\(650\) −18.9143 −0.741879
\(651\) −11.3067 −0.443144
\(652\) 36.6987 1.43723
\(653\) 42.7948 1.67469 0.837345 0.546674i \(-0.184106\pi\)
0.837345 + 0.546674i \(0.184106\pi\)
\(654\) −16.4640 −0.643792
\(655\) −25.2090 −0.984999
\(656\) −32.8208 −1.28144
\(657\) 24.8494 0.969468
\(658\) −9.57692 −0.373347
\(659\) −2.23351 −0.0870051 −0.0435026 0.999053i \(-0.513852\pi\)
−0.0435026 + 0.999053i \(0.513852\pi\)
\(660\) −6.16903 −0.240129
\(661\) 34.1335 1.32764 0.663820 0.747892i \(-0.268934\pi\)
0.663820 + 0.747892i \(0.268934\pi\)
\(662\) −9.24457 −0.359300
\(663\) −9.03361 −0.350836
\(664\) 50.3811 1.95517
\(665\) 0.340973 0.0132224
\(666\) 55.9437 2.16777
\(667\) 1.97526 0.0764823
\(668\) 5.05514 0.195589
\(669\) −6.55198 −0.253314
\(670\) −10.2055 −0.394272
\(671\) 20.5363 0.792797
\(672\) 5.20417 0.200755
\(673\) 20.0627 0.773360 0.386680 0.922214i \(-0.373622\pi\)
0.386680 + 0.922214i \(0.373622\pi\)
\(674\) 20.1766 0.777173
\(675\) 8.23577 0.316995
\(676\) −18.3609 −0.706190
\(677\) −18.0048 −0.691979 −0.345990 0.938238i \(-0.612457\pi\)
−0.345990 + 0.938238i \(0.612457\pi\)
\(678\) −17.7874 −0.683122
\(679\) −27.8692 −1.06952
\(680\) 50.3538 1.93098
\(681\) −8.37350 −0.320873
\(682\) −37.6066 −1.44003
\(683\) −36.7868 −1.40761 −0.703805 0.710394i \(-0.748517\pi\)
−0.703805 + 0.710394i \(0.748517\pi\)
\(684\) −1.25058 −0.0478173
\(685\) −30.3676 −1.16029
\(686\) −50.7288 −1.93683
\(687\) −3.96437 −0.151250
\(688\) −14.2627 −0.543762
\(689\) −38.7412 −1.47592
\(690\) −8.01082 −0.304967
\(691\) −23.9990 −0.912966 −0.456483 0.889732i \(-0.650891\pi\)
−0.456483 + 0.889732i \(0.650891\pi\)
\(692\) 27.8650 1.05927
\(693\) 8.33982 0.316803
\(694\) −39.8947 −1.51438
\(695\) 18.7560 0.711457
\(696\) 1.98922 0.0754014
\(697\) −26.3706 −0.998858
\(698\) 63.6503 2.40920
\(699\) 7.98547 0.302038
\(700\) 22.4701 0.849288
\(701\) 23.5312 0.888760 0.444380 0.895838i \(-0.353424\pi\)
0.444380 + 0.895838i \(0.353424\pi\)
\(702\) −24.5923 −0.928176
\(703\) 0.884229 0.0333493
\(704\) −2.88860 −0.108868
\(705\) 1.69753 0.0639325
\(706\) −47.3837 −1.78331
\(707\) −19.2246 −0.723015
\(708\) −36.7692 −1.38187
\(709\) −14.9623 −0.561921 −0.280960 0.959719i \(-0.590653\pi\)
−0.280960 + 0.959719i \(0.590653\pi\)
\(710\) −8.09101 −0.303650
\(711\) 37.3267 1.39986
\(712\) −75.2880 −2.82154
\(713\) −33.5712 −1.25725
\(714\) 15.6111 0.584230
\(715\) 7.21447 0.269806
\(716\) 1.19235 0.0445601
\(717\) −3.71601 −0.138777
\(718\) 51.6392 1.92716
\(719\) −41.9054 −1.56281 −0.781404 0.624026i \(-0.785496\pi\)
−0.781404 + 0.624026i \(0.785496\pi\)
\(720\) 27.5368 1.02624
\(721\) −11.0314 −0.410830
\(722\) 48.0342 1.78765
\(723\) 8.53545 0.317437
\(724\) −115.600 −4.29623
\(725\) 1.42856 0.0530555
\(726\) 12.5999 0.467627
\(727\) 31.8351 1.18070 0.590350 0.807147i \(-0.298990\pi\)
0.590350 + 0.807147i \(0.298990\pi\)
\(728\) −36.5912 −1.35616
\(729\) −10.6227 −0.393433
\(730\) −37.1481 −1.37491
\(731\) −11.4597 −0.423853
\(732\) 33.8538 1.25127
\(733\) 8.20871 0.303195 0.151598 0.988442i \(-0.451558\pi\)
0.151598 + 0.988442i \(0.451558\pi\)
\(734\) 36.3506 1.34172
\(735\) 2.62167 0.0967018
\(736\) 15.4519 0.569566
\(737\) −3.94533 −0.145328
\(738\) −33.7820 −1.24353
\(739\) 24.0820 0.885872 0.442936 0.896553i \(-0.353937\pi\)
0.442936 + 0.896553i \(0.353937\pi\)
\(740\) −57.4928 −2.11348
\(741\) −0.182911 −0.00671940
\(742\) 66.9491 2.45778
\(743\) −48.0518 −1.76285 −0.881425 0.472324i \(-0.843415\pi\)
−0.881425 + 0.472324i \(0.843415\pi\)
\(744\) −33.8085 −1.23948
\(745\) 7.47034 0.273692
\(746\) 40.2759 1.47460
\(747\) 22.1370 0.809952
\(748\) 35.6947 1.30513
\(749\) −22.1657 −0.809917
\(750\) −17.3036 −0.631839
\(751\) −4.96165 −0.181053 −0.0905267 0.995894i \(-0.528855\pi\)
−0.0905267 + 0.995894i \(0.528855\pi\)
\(752\) −12.2245 −0.445783
\(753\) −2.56834 −0.0935954
\(754\) −4.26573 −0.155349
\(755\) 7.83865 0.285278
\(756\) 29.2155 1.06256
\(757\) −37.2218 −1.35285 −0.676424 0.736512i \(-0.736471\pi\)
−0.676424 + 0.736512i \(0.736471\pi\)
\(758\) 17.7056 0.643096
\(759\) −3.09690 −0.112410
\(760\) 1.01956 0.0369832
\(761\) −53.7809 −1.94955 −0.974777 0.223180i \(-0.928356\pi\)
−0.974777 + 0.223180i \(0.928356\pi\)
\(762\) 4.16632 0.150930
\(763\) −22.8737 −0.828085
\(764\) 56.1830 2.03263
\(765\) 22.1251 0.799933
\(766\) −26.2952 −0.950083
\(767\) 43.0004 1.55265
\(768\) −17.7347 −0.639947
\(769\) 15.5669 0.561358 0.280679 0.959802i \(-0.409440\pi\)
0.280679 + 0.959802i \(0.409440\pi\)
\(770\) −12.4674 −0.449295
\(771\) −13.4453 −0.484221
\(772\) 44.6059 1.60540
\(773\) 34.2086 1.23040 0.615199 0.788372i \(-0.289076\pi\)
0.615199 + 0.788372i \(0.289076\pi\)
\(774\) −14.6805 −0.527678
\(775\) −24.2796 −0.872149
\(776\) −83.3327 −2.99147
\(777\) −9.72059 −0.348724
\(778\) 55.9591 2.00623
\(779\) −0.533948 −0.0191307
\(780\) 11.8929 0.425835
\(781\) −3.12790 −0.111925
\(782\) 46.3515 1.65753
\(783\) 1.85741 0.0663785
\(784\) −18.8797 −0.674273
\(785\) 16.2595 0.580326
\(786\) −23.3694 −0.833560
\(787\) −11.0271 −0.393075 −0.196538 0.980496i \(-0.562970\pi\)
−0.196538 + 0.980496i \(0.562970\pi\)
\(788\) −90.8264 −3.23556
\(789\) 5.67177 0.201921
\(790\) −55.8007 −1.98530
\(791\) −24.7124 −0.878672
\(792\) 24.9372 0.886104
\(793\) −39.5909 −1.40591
\(794\) −29.6803 −1.05332
\(795\) −11.8668 −0.420874
\(796\) −16.9627 −0.601226
\(797\) −22.2583 −0.788428 −0.394214 0.919019i \(-0.628983\pi\)
−0.394214 + 0.919019i \(0.628983\pi\)
\(798\) 0.316091 0.0111895
\(799\) −9.82208 −0.347480
\(800\) 11.1753 0.395105
\(801\) −33.0810 −1.16886
\(802\) 16.2621 0.574235
\(803\) −14.3610 −0.506790
\(804\) −6.50381 −0.229372
\(805\) −11.1296 −0.392267
\(806\) 72.4997 2.55369
\(807\) 6.29977 0.221762
\(808\) −57.4841 −2.02228
\(809\) 25.7295 0.904603 0.452301 0.891865i \(-0.350603\pi\)
0.452301 + 0.891865i \(0.350603\pi\)
\(810\) 24.3551 0.855753
\(811\) −10.4208 −0.365924 −0.182962 0.983120i \(-0.558568\pi\)
−0.182962 + 0.983120i \(0.558568\pi\)
\(812\) 5.06767 0.177840
\(813\) 5.44546 0.190981
\(814\) −32.3311 −1.13321
\(815\) −13.1462 −0.460492
\(816\) 19.9269 0.697581
\(817\) −0.232035 −0.00811787
\(818\) −3.02568 −0.105791
\(819\) −16.0779 −0.561807
\(820\) 34.7175 1.21239
\(821\) −18.4271 −0.643112 −0.321556 0.946891i \(-0.604206\pi\)
−0.321556 + 0.946891i \(0.604206\pi\)
\(822\) −28.1515 −0.981897
\(823\) 9.91467 0.345604 0.172802 0.984957i \(-0.444718\pi\)
0.172802 + 0.984957i \(0.444718\pi\)
\(824\) −32.9853 −1.14910
\(825\) −2.23976 −0.0779785
\(826\) −74.3095 −2.58556
\(827\) 32.0818 1.11559 0.557796 0.829978i \(-0.311647\pi\)
0.557796 + 0.829978i \(0.311647\pi\)
\(828\) 40.8199 1.41859
\(829\) −22.4879 −0.781035 −0.390518 0.920595i \(-0.627704\pi\)
−0.390518 + 0.920595i \(0.627704\pi\)
\(830\) −33.0933 −1.14868
\(831\) 7.62943 0.264662
\(832\) 5.56877 0.193062
\(833\) −15.1693 −0.525585
\(834\) 17.3873 0.602074
\(835\) −1.81085 −0.0626672
\(836\) 0.722741 0.0249965
\(837\) −31.5683 −1.09116
\(838\) 16.0267 0.553632
\(839\) −41.7385 −1.44097 −0.720487 0.693468i \(-0.756082\pi\)
−0.720487 + 0.693468i \(0.756082\pi\)
\(840\) −11.2083 −0.386723
\(841\) −28.6778 −0.988890
\(842\) 72.5344 2.49970
\(843\) −9.43026 −0.324796
\(844\) −65.6807 −2.26082
\(845\) 6.57727 0.226265
\(846\) −12.5826 −0.432597
\(847\) 17.5053 0.601490
\(848\) 85.4577 2.93463
\(849\) 7.00045 0.240255
\(850\) 33.5227 1.14982
\(851\) −28.8618 −0.989370
\(852\) −5.15629 −0.176651
\(853\) −9.97034 −0.341378 −0.170689 0.985325i \(-0.554599\pi\)
−0.170689 + 0.985325i \(0.554599\pi\)
\(854\) 68.4175 2.34120
\(855\) 0.447985 0.0153208
\(856\) −66.2784 −2.26535
\(857\) 6.67851 0.228133 0.114067 0.993473i \(-0.463612\pi\)
0.114067 + 0.993473i \(0.463612\pi\)
\(858\) 6.68800 0.228324
\(859\) −55.5705 −1.89604 −0.948020 0.318210i \(-0.896918\pi\)
−0.948020 + 0.318210i \(0.896918\pi\)
\(860\) 15.0870 0.514462
\(861\) 5.86985 0.200044
\(862\) 30.3926 1.03517
\(863\) −50.2541 −1.71067 −0.855335 0.518076i \(-0.826649\pi\)
−0.855335 + 0.518076i \(0.826649\pi\)
\(864\) 14.5300 0.494322
\(865\) −9.98182 −0.339392
\(866\) −11.2698 −0.382963
\(867\) 6.19346 0.210341
\(868\) −86.1293 −2.92342
\(869\) −21.5720 −0.731778
\(870\) −1.30664 −0.0442993
\(871\) 7.60599 0.257719
\(872\) −68.3955 −2.31616
\(873\) −36.6157 −1.23925
\(874\) 0.938518 0.0317459
\(875\) −24.0403 −0.812710
\(876\) −23.6739 −0.799868
\(877\) −50.0313 −1.68944 −0.844718 0.535212i \(-0.820232\pi\)
−0.844718 + 0.535212i \(0.820232\pi\)
\(878\) 65.0216 2.19437
\(879\) −9.43887 −0.318365
\(880\) −15.9141 −0.536465
\(881\) 41.8745 1.41079 0.705394 0.708816i \(-0.250770\pi\)
0.705394 + 0.708816i \(0.250770\pi\)
\(882\) −19.4326 −0.654329
\(883\) 2.77390 0.0933492 0.0466746 0.998910i \(-0.485138\pi\)
0.0466746 + 0.998910i \(0.485138\pi\)
\(884\) −68.8139 −2.31446
\(885\) 13.1715 0.442755
\(886\) 84.5062 2.83904
\(887\) −43.4158 −1.45776 −0.728881 0.684641i \(-0.759959\pi\)
−0.728881 + 0.684641i \(0.759959\pi\)
\(888\) −29.0659 −0.975387
\(889\) 5.78836 0.194135
\(890\) 49.4536 1.65769
\(891\) 9.41544 0.315429
\(892\) −49.9100 −1.67111
\(893\) −0.198876 −0.00665513
\(894\) 6.92519 0.231613
\(895\) −0.427123 −0.0142772
\(896\) −27.6470 −0.923622
\(897\) 5.97034 0.199344
\(898\) −71.3896 −2.38230
\(899\) −5.47578 −0.182627
\(900\) 29.5221 0.984069
\(901\) 68.6629 2.28749
\(902\) 19.5234 0.650058
\(903\) 2.55083 0.0848863
\(904\) −73.8934 −2.45766
\(905\) 41.4102 1.37652
\(906\) 7.26663 0.241418
\(907\) 27.6191 0.917078 0.458539 0.888674i \(-0.348373\pi\)
0.458539 + 0.888674i \(0.348373\pi\)
\(908\) −63.7855 −2.11680
\(909\) −25.2581 −0.837757
\(910\) 24.0353 0.796761
\(911\) −30.8043 −1.02059 −0.510296 0.859999i \(-0.670464\pi\)
−0.510296 + 0.859999i \(0.670464\pi\)
\(912\) 0.403477 0.0133604
\(913\) −12.7935 −0.423403
\(914\) 0.0633852 0.00209659
\(915\) −12.1271 −0.400910
\(916\) −30.1988 −0.997795
\(917\) −32.4676 −1.07218
\(918\) 43.5861 1.43856
\(919\) −24.3447 −0.803057 −0.401528 0.915847i \(-0.631521\pi\)
−0.401528 + 0.915847i \(0.631521\pi\)
\(920\) −33.2790 −1.09718
\(921\) 7.32309 0.241304
\(922\) −76.0254 −2.50376
\(923\) 6.03010 0.198483
\(924\) −7.94531 −0.261381
\(925\) −20.8737 −0.686322
\(926\) −42.7575 −1.40510
\(927\) −14.4935 −0.476028
\(928\) 2.52036 0.0827347
\(929\) −2.77615 −0.0910826 −0.0455413 0.998962i \(-0.514501\pi\)
−0.0455413 + 0.998962i \(0.514501\pi\)
\(930\) 22.2074 0.728211
\(931\) −0.307145 −0.0100663
\(932\) 60.8297 1.99254
\(933\) 11.5956 0.379623
\(934\) −27.1492 −0.888349
\(935\) −12.7866 −0.418166
\(936\) −48.0750 −1.57138
\(937\) −14.9429 −0.488162 −0.244081 0.969755i \(-0.578486\pi\)
−0.244081 + 0.969755i \(0.578486\pi\)
\(938\) −13.1440 −0.429167
\(939\) −17.7846 −0.580377
\(940\) 12.9310 0.421762
\(941\) −21.4863 −0.700434 −0.350217 0.936669i \(-0.613892\pi\)
−0.350217 + 0.936669i \(0.613892\pi\)
\(942\) 15.0730 0.491103
\(943\) 17.4284 0.567548
\(944\) −94.8529 −3.08720
\(945\) −10.4656 −0.340446
\(946\) 8.48417 0.275844
\(947\) −49.1986 −1.59874 −0.799370 0.600839i \(-0.794833\pi\)
−0.799370 + 0.600839i \(0.794833\pi\)
\(948\) −35.5610 −1.15497
\(949\) 27.6859 0.898721
\(950\) 0.678762 0.0220220
\(951\) −8.32814 −0.270059
\(952\) 64.8524 2.10188
\(953\) −29.3187 −0.949725 −0.474863 0.880060i \(-0.657502\pi\)
−0.474863 + 0.880060i \(0.657502\pi\)
\(954\) 87.9605 2.84783
\(955\) −20.1259 −0.651259
\(956\) −28.3069 −0.915510
\(957\) −0.505133 −0.0163286
\(958\) −52.9381 −1.71035
\(959\) −39.1115 −1.26298
\(960\) 1.70577 0.0550536
\(961\) 62.0654 2.00211
\(962\) 62.3294 2.00958
\(963\) −29.1222 −0.938450
\(964\) 65.0192 2.09413
\(965\) −15.9787 −0.514374
\(966\) −10.3174 −0.331957
\(967\) 53.2020 1.71086 0.855430 0.517918i \(-0.173293\pi\)
0.855430 + 0.517918i \(0.173293\pi\)
\(968\) 52.3432 1.68238
\(969\) 0.324182 0.0104142
\(970\) 54.7378 1.75753
\(971\) 42.3611 1.35943 0.679716 0.733476i \(-0.262103\pi\)
0.679716 + 0.733476i \(0.262103\pi\)
\(972\) 58.7063 1.88301
\(973\) 24.1566 0.774424
\(974\) 58.0399 1.85972
\(975\) 4.31791 0.138284
\(976\) 87.3321 2.79543
\(977\) 49.5988 1.58681 0.793403 0.608697i \(-0.208308\pi\)
0.793403 + 0.608697i \(0.208308\pi\)
\(978\) −12.1869 −0.389693
\(979\) 19.1182 0.611021
\(980\) 19.9707 0.637940
\(981\) −30.0524 −0.959501
\(982\) −38.5911 −1.23149
\(983\) 16.9012 0.539064 0.269532 0.962991i \(-0.413131\pi\)
0.269532 + 0.962991i \(0.413131\pi\)
\(984\) 17.5516 0.559526
\(985\) 32.5359 1.03668
\(986\) 7.56037 0.240771
\(987\) 2.18630 0.0695908
\(988\) −1.39333 −0.0443278
\(989\) 7.57377 0.240832
\(990\) −16.3802 −0.520597
\(991\) −0.262728 −0.00834582 −0.00417291 0.999991i \(-0.501328\pi\)
−0.00417291 + 0.999991i \(0.501328\pi\)
\(992\) −42.8355 −1.36003
\(993\) 2.11043 0.0669725
\(994\) −10.4207 −0.330524
\(995\) 6.07638 0.192634
\(996\) −21.0899 −0.668258
\(997\) −57.6486 −1.82575 −0.912875 0.408240i \(-0.866143\pi\)
−0.912875 + 0.408240i \(0.866143\pi\)
\(998\) −4.30591 −0.136301
\(999\) −27.1399 −0.858668
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 8009.2.a.a.1.18 306
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8009.2.a.a.1.18 306 1.1 even 1 trivial