Properties

Label 6001.2.a.a.1.19
Level $6001$
Weight $2$
Character 6001.1
Self dual yes
Analytic conductor $47.918$
Analytic rank $1$
Dimension $113$
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] = [6001,2,Mod(1,6001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6001, 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("6001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6001 = 17 \cdot 353 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6001.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: \(47.9182262530\)
Analytic rank: \(1\)
Dimension: \(113\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Character \(\chi\) \(=\) 6001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.14256 q^{2} -2.11238 q^{3} +2.59055 q^{4} +1.25903 q^{5} +4.52589 q^{6} +1.44603 q^{7} -1.26528 q^{8} +1.46215 q^{9} +O(q^{10})\) \(q-2.14256 q^{2} -2.11238 q^{3} +2.59055 q^{4} +1.25903 q^{5} +4.52589 q^{6} +1.44603 q^{7} -1.26528 q^{8} +1.46215 q^{9} -2.69753 q^{10} +4.85189 q^{11} -5.47222 q^{12} -5.39622 q^{13} -3.09820 q^{14} -2.65954 q^{15} -2.47016 q^{16} -1.00000 q^{17} -3.13274 q^{18} -0.860517 q^{19} +3.26157 q^{20} -3.05457 q^{21} -10.3955 q^{22} +2.81301 q^{23} +2.67275 q^{24} -3.41485 q^{25} +11.5617 q^{26} +3.24853 q^{27} +3.74601 q^{28} +4.40407 q^{29} +5.69822 q^{30} +4.93268 q^{31} +7.82302 q^{32} -10.2490 q^{33} +2.14256 q^{34} +1.82059 q^{35} +3.78777 q^{36} +4.46799 q^{37} +1.84371 q^{38} +11.3989 q^{39} -1.59302 q^{40} -1.43011 q^{41} +6.54458 q^{42} -12.0014 q^{43} +12.5691 q^{44} +1.84088 q^{45} -6.02702 q^{46} +9.34116 q^{47} +5.21791 q^{48} -4.90900 q^{49} +7.31651 q^{50} +2.11238 q^{51} -13.9792 q^{52} +4.19720 q^{53} -6.96015 q^{54} +6.10866 q^{55} -1.82964 q^{56} +1.81774 q^{57} -9.43596 q^{58} -11.5846 q^{59} -6.88967 q^{60} -11.6645 q^{61} -10.5685 q^{62} +2.11431 q^{63} -11.8209 q^{64} -6.79398 q^{65} +21.9591 q^{66} +10.9973 q^{67} -2.59055 q^{68} -5.94214 q^{69} -3.90072 q^{70} -10.7868 q^{71} -1.85003 q^{72} -5.27905 q^{73} -9.57291 q^{74} +7.21347 q^{75} -2.22921 q^{76} +7.01598 q^{77} -24.4227 q^{78} -11.9667 q^{79} -3.10999 q^{80} -11.2486 q^{81} +3.06409 q^{82} -0.0276943 q^{83} -7.91300 q^{84} -1.25903 q^{85} +25.7137 q^{86} -9.30306 q^{87} -6.13901 q^{88} -15.4679 q^{89} -3.94420 q^{90} -7.80309 q^{91} +7.28722 q^{92} -10.4197 q^{93} -20.0140 q^{94} -1.08341 q^{95} -16.5252 q^{96} +1.90368 q^{97} +10.5178 q^{98} +7.09419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 113 q - 11 q^{2} - 11 q^{3} + 103 q^{4} - 19 q^{5} - 13 q^{6} + 11 q^{7} - 36 q^{8} + 94 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 113 q - 11 q^{2} - 11 q^{3} + 103 q^{4} - 19 q^{5} - 13 q^{6} + 11 q^{7} - 36 q^{8} + 94 q^{9} - 5 q^{10} - 40 q^{11} - 19 q^{12} - 18 q^{13} - 48 q^{14} - 63 q^{15} + 79 q^{16} - 113 q^{17} - 32 q^{18} - 46 q^{19} - 56 q^{20} - 46 q^{21} + 14 q^{22} - 35 q^{23} - 42 q^{24} + 88 q^{25} - 89 q^{26} - 41 q^{27} + 20 q^{28} - 51 q^{29} - 18 q^{30} - 57 q^{31} - 93 q^{32} - 40 q^{33} + 11 q^{34} - 69 q^{35} + 18 q^{36} + 16 q^{37} - 74 q^{38} - 51 q^{39} + 2 q^{40} - 87 q^{41} - 23 q^{42} - 32 q^{43} - 110 q^{44} - 17 q^{45} - 17 q^{46} - 161 q^{47} - 36 q^{48} + 56 q^{49} - 69 q^{50} + 11 q^{51} - 49 q^{52} - 48 q^{53} - 38 q^{54} - 79 q^{55} - 171 q^{56} + 20 q^{57} + 13 q^{58} - 174 q^{59} - 146 q^{60} - 34 q^{61} - 34 q^{62} - 14 q^{63} + 62 q^{64} - 22 q^{65} - 60 q^{66} - 50 q^{67} - 103 q^{68} - 59 q^{69} - 58 q^{70} - 189 q^{71} - 123 q^{72} - 4 q^{73} - 24 q^{74} - 106 q^{75} - 92 q^{76} - 78 q^{77} - 42 q^{78} + 8 q^{79} - 150 q^{80} + 13 q^{81} + 6 q^{82} - 109 q^{83} - 114 q^{84} + 19 q^{85} - 116 q^{86} - 106 q^{87} + 54 q^{88} - 170 q^{89} - q^{90} - 43 q^{91} - 94 q^{92} - 69 q^{93} - 35 q^{94} - 78 q^{95} - 44 q^{96} - 3 q^{97} - 68 q^{98} - 119 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.14256 −1.51502 −0.757508 0.652826i \(-0.773583\pi\)
−0.757508 + 0.652826i \(0.773583\pi\)
\(3\) −2.11238 −1.21958 −0.609792 0.792562i \(-0.708747\pi\)
−0.609792 + 0.792562i \(0.708747\pi\)
\(4\) 2.59055 1.29527
\(5\) 1.25903 0.563054 0.281527 0.959553i \(-0.409159\pi\)
0.281527 + 0.959553i \(0.409159\pi\)
\(6\) 4.52589 1.84769
\(7\) 1.44603 0.546548 0.273274 0.961936i \(-0.411893\pi\)
0.273274 + 0.961936i \(0.411893\pi\)
\(8\) −1.26528 −0.447345
\(9\) 1.46215 0.487383
\(10\) −2.69753 −0.853035
\(11\) 4.85189 1.46290 0.731450 0.681895i \(-0.238844\pi\)
0.731450 + 0.681895i \(0.238844\pi\)
\(12\) −5.47222 −1.57969
\(13\) −5.39622 −1.49664 −0.748320 0.663337i \(-0.769139\pi\)
−0.748320 + 0.663337i \(0.769139\pi\)
\(14\) −3.09820 −0.828029
\(15\) −2.65954 −0.686691
\(16\) −2.47016 −0.617540
\(17\) −1.00000 −0.242536
\(18\) −3.13274 −0.738393
\(19\) −0.860517 −0.197416 −0.0987081 0.995116i \(-0.531471\pi\)
−0.0987081 + 0.995116i \(0.531471\pi\)
\(20\) 3.26157 0.729309
\(21\) −3.05457 −0.666561
\(22\) −10.3955 −2.21632
\(23\) 2.81301 0.586552 0.293276 0.956028i \(-0.405254\pi\)
0.293276 + 0.956028i \(0.405254\pi\)
\(24\) 2.67275 0.545574
\(25\) −3.41485 −0.682971
\(26\) 11.5617 2.26744
\(27\) 3.24853 0.625179
\(28\) 3.74601 0.707929
\(29\) 4.40407 0.817815 0.408907 0.912576i \(-0.365910\pi\)
0.408907 + 0.912576i \(0.365910\pi\)
\(30\) 5.69822 1.04035
\(31\) 4.93268 0.885935 0.442968 0.896538i \(-0.353926\pi\)
0.442968 + 0.896538i \(0.353926\pi\)
\(32\) 7.82302 1.38293
\(33\) −10.2490 −1.78413
\(34\) 2.14256 0.367445
\(35\) 1.82059 0.307736
\(36\) 3.78777 0.631294
\(37\) 4.46799 0.734533 0.367266 0.930116i \(-0.380294\pi\)
0.367266 + 0.930116i \(0.380294\pi\)
\(38\) 1.84371 0.299089
\(39\) 11.3989 1.82528
\(40\) −1.59302 −0.251879
\(41\) −1.43011 −0.223346 −0.111673 0.993745i \(-0.535621\pi\)
−0.111673 + 0.993745i \(0.535621\pi\)
\(42\) 6.54458 1.00985
\(43\) −12.0014 −1.83020 −0.915098 0.403231i \(-0.867887\pi\)
−0.915098 + 0.403231i \(0.867887\pi\)
\(44\) 12.5691 1.89486
\(45\) 1.84088 0.274423
\(46\) −6.02702 −0.888636
\(47\) 9.34116 1.36255 0.681274 0.732029i \(-0.261426\pi\)
0.681274 + 0.732029i \(0.261426\pi\)
\(48\) 5.21791 0.753141
\(49\) −4.90900 −0.701285
\(50\) 7.31651 1.03471
\(51\) 2.11238 0.295792
\(52\) −13.9792 −1.93856
\(53\) 4.19720 0.576530 0.288265 0.957551i \(-0.406922\pi\)
0.288265 + 0.957551i \(0.406922\pi\)
\(54\) −6.96015 −0.947156
\(55\) 6.10866 0.823691
\(56\) −1.82964 −0.244495
\(57\) 1.81774 0.240765
\(58\) −9.43596 −1.23900
\(59\) −11.5846 −1.50819 −0.754094 0.656766i \(-0.771924\pi\)
−0.754094 + 0.656766i \(0.771924\pi\)
\(60\) −6.88967 −0.889452
\(61\) −11.6645 −1.49349 −0.746746 0.665110i \(-0.768385\pi\)
−0.746746 + 0.665110i \(0.768385\pi\)
\(62\) −10.5685 −1.34221
\(63\) 2.11431 0.266378
\(64\) −11.8209 −1.47762
\(65\) −6.79398 −0.842689
\(66\) 21.9591 2.70298
\(67\) 10.9973 1.34353 0.671767 0.740763i \(-0.265536\pi\)
0.671767 + 0.740763i \(0.265536\pi\)
\(68\) −2.59055 −0.314150
\(69\) −5.94214 −0.715349
\(70\) −3.90072 −0.466225
\(71\) −10.7868 −1.28016 −0.640081 0.768307i \(-0.721099\pi\)
−0.640081 + 0.768307i \(0.721099\pi\)
\(72\) −1.85003 −0.218028
\(73\) −5.27905 −0.617866 −0.308933 0.951084i \(-0.599972\pi\)
−0.308933 + 0.951084i \(0.599972\pi\)
\(74\) −9.57291 −1.11283
\(75\) 7.21347 0.832939
\(76\) −2.22921 −0.255708
\(77\) 7.01598 0.799546
\(78\) −24.4227 −2.76533
\(79\) −11.9667 −1.34636 −0.673181 0.739478i \(-0.735073\pi\)
−0.673181 + 0.739478i \(0.735073\pi\)
\(80\) −3.10999 −0.347708
\(81\) −11.2486 −1.24984
\(82\) 3.06409 0.338372
\(83\) −0.0276943 −0.00303985 −0.00151992 0.999999i \(-0.500484\pi\)
−0.00151992 + 0.999999i \(0.500484\pi\)
\(84\) −7.91300 −0.863379
\(85\) −1.25903 −0.136561
\(86\) 25.7137 2.77278
\(87\) −9.30306 −0.997393
\(88\) −6.13901 −0.654421
\(89\) −15.4679 −1.63959 −0.819796 0.572656i \(-0.805913\pi\)
−0.819796 + 0.572656i \(0.805913\pi\)
\(90\) −3.94420 −0.415755
\(91\) −7.80309 −0.817986
\(92\) 7.28722 0.759746
\(93\) −10.4197 −1.08047
\(94\) −20.0140 −2.06428
\(95\) −1.08341 −0.111156
\(96\) −16.5252 −1.68659
\(97\) 1.90368 0.193289 0.0966447 0.995319i \(-0.469189\pi\)
0.0966447 + 0.995319i \(0.469189\pi\)
\(98\) 10.5178 1.06246
\(99\) 7.09419 0.712993
\(100\) −8.84634 −0.884634
\(101\) 14.8528 1.47791 0.738957 0.673753i \(-0.235319\pi\)
0.738957 + 0.673753i \(0.235319\pi\)
\(102\) −4.52589 −0.448130
\(103\) 8.13194 0.801264 0.400632 0.916239i \(-0.368791\pi\)
0.400632 + 0.916239i \(0.368791\pi\)
\(104\) 6.82773 0.669514
\(105\) −3.84578 −0.375310
\(106\) −8.99274 −0.873452
\(107\) 11.7474 1.13566 0.567831 0.823145i \(-0.307783\pi\)
0.567831 + 0.823145i \(0.307783\pi\)
\(108\) 8.41546 0.809778
\(109\) −7.55948 −0.724067 −0.362033 0.932165i \(-0.617917\pi\)
−0.362033 + 0.932165i \(0.617917\pi\)
\(110\) −13.0881 −1.24791
\(111\) −9.43809 −0.895823
\(112\) −3.57192 −0.337515
\(113\) −8.61277 −0.810221 −0.405110 0.914268i \(-0.632767\pi\)
−0.405110 + 0.914268i \(0.632767\pi\)
\(114\) −3.89461 −0.364763
\(115\) 3.54165 0.330260
\(116\) 11.4089 1.05929
\(117\) −7.89007 −0.729437
\(118\) 24.8207 2.28493
\(119\) −1.44603 −0.132557
\(120\) 3.36507 0.307187
\(121\) 12.5409 1.14008
\(122\) 24.9919 2.26266
\(123\) 3.02094 0.272389
\(124\) 12.7783 1.14753
\(125\) −10.5945 −0.947603
\(126\) −4.53003 −0.403567
\(127\) 20.3232 1.80339 0.901697 0.432369i \(-0.142322\pi\)
0.901697 + 0.432369i \(0.142322\pi\)
\(128\) 9.68099 0.855687
\(129\) 25.3515 2.23208
\(130\) 14.5565 1.27669
\(131\) 6.50414 0.568269 0.284135 0.958784i \(-0.408294\pi\)
0.284135 + 0.958784i \(0.408294\pi\)
\(132\) −26.5506 −2.31094
\(133\) −1.24433 −0.107897
\(134\) −23.5623 −2.03547
\(135\) 4.08998 0.352009
\(136\) 1.26528 0.108497
\(137\) −2.78341 −0.237803 −0.118902 0.992906i \(-0.537937\pi\)
−0.118902 + 0.992906i \(0.537937\pi\)
\(138\) 12.7314 1.08377
\(139\) −11.4915 −0.974697 −0.487349 0.873208i \(-0.662036\pi\)
−0.487349 + 0.873208i \(0.662036\pi\)
\(140\) 4.71633 0.398602
\(141\) −19.7321 −1.66174
\(142\) 23.1114 1.93947
\(143\) −26.1819 −2.18944
\(144\) −3.61174 −0.300978
\(145\) 5.54484 0.460474
\(146\) 11.3107 0.936076
\(147\) 10.3697 0.855275
\(148\) 11.5745 0.951421
\(149\) −18.3441 −1.50281 −0.751405 0.659841i \(-0.770624\pi\)
−0.751405 + 0.659841i \(0.770624\pi\)
\(150\) −15.4553 −1.26192
\(151\) 2.12540 0.172963 0.0864814 0.996253i \(-0.472438\pi\)
0.0864814 + 0.996253i \(0.472438\pi\)
\(152\) 1.08880 0.0883130
\(153\) −1.46215 −0.118208
\(154\) −15.0321 −1.21132
\(155\) 6.21037 0.498829
\(156\) 29.5293 2.36423
\(157\) 4.01364 0.320323 0.160162 0.987091i \(-0.448798\pi\)
0.160162 + 0.987091i \(0.448798\pi\)
\(158\) 25.6394 2.03976
\(159\) −8.86608 −0.703126
\(160\) 9.84938 0.778662
\(161\) 4.06769 0.320579
\(162\) 24.1007 1.89353
\(163\) 12.1998 0.955561 0.477781 0.878479i \(-0.341441\pi\)
0.477781 + 0.878479i \(0.341441\pi\)
\(164\) −3.70477 −0.289294
\(165\) −12.9038 −1.00456
\(166\) 0.0593367 0.00460542
\(167\) −15.5593 −1.20401 −0.602006 0.798492i \(-0.705632\pi\)
−0.602006 + 0.798492i \(0.705632\pi\)
\(168\) 3.86489 0.298182
\(169\) 16.1191 1.23993
\(170\) 2.69753 0.206891
\(171\) −1.25820 −0.0962173
\(172\) −31.0902 −2.37060
\(173\) 4.00343 0.304375 0.152187 0.988352i \(-0.451368\pi\)
0.152187 + 0.988352i \(0.451368\pi\)
\(174\) 19.9323 1.51107
\(175\) −4.93798 −0.373276
\(176\) −11.9849 −0.903399
\(177\) 24.4711 1.83936
\(178\) 33.1408 2.48401
\(179\) −20.7365 −1.54992 −0.774960 0.632010i \(-0.782230\pi\)
−0.774960 + 0.632010i \(0.782230\pi\)
\(180\) 4.76890 0.355453
\(181\) −12.6564 −0.940739 −0.470370 0.882469i \(-0.655879\pi\)
−0.470370 + 0.882469i \(0.655879\pi\)
\(182\) 16.7186 1.23926
\(183\) 24.6399 1.82144
\(184\) −3.55924 −0.262391
\(185\) 5.62531 0.413581
\(186\) 22.3248 1.63693
\(187\) −4.85189 −0.354805
\(188\) 24.1987 1.76487
\(189\) 4.69747 0.341690
\(190\) 2.32127 0.168403
\(191\) 22.8301 1.65193 0.825963 0.563725i \(-0.190632\pi\)
0.825963 + 0.563725i \(0.190632\pi\)
\(192\) 24.9703 1.80208
\(193\) 17.6885 1.27324 0.636622 0.771176i \(-0.280331\pi\)
0.636622 + 0.771176i \(0.280331\pi\)
\(194\) −4.07874 −0.292837
\(195\) 14.3515 1.02773
\(196\) −12.7170 −0.908356
\(197\) 8.87651 0.632425 0.316213 0.948688i \(-0.397589\pi\)
0.316213 + 0.948688i \(0.397589\pi\)
\(198\) −15.1997 −1.08020
\(199\) 21.3147 1.51096 0.755478 0.655174i \(-0.227405\pi\)
0.755478 + 0.655174i \(0.227405\pi\)
\(200\) 4.32075 0.305523
\(201\) −23.2305 −1.63855
\(202\) −31.8231 −2.23906
\(203\) 6.36842 0.446975
\(204\) 5.47222 0.383132
\(205\) −1.80055 −0.125756
\(206\) −17.4231 −1.21393
\(207\) 4.11303 0.285876
\(208\) 13.3295 0.924235
\(209\) −4.17514 −0.288800
\(210\) 8.23980 0.568600
\(211\) −10.0900 −0.694623 −0.347312 0.937750i \(-0.612905\pi\)
−0.347312 + 0.937750i \(0.612905\pi\)
\(212\) 10.8730 0.746764
\(213\) 22.7859 1.56126
\(214\) −25.1694 −1.72055
\(215\) −15.1101 −1.03050
\(216\) −4.11030 −0.279670
\(217\) 7.13280 0.484206
\(218\) 16.1966 1.09697
\(219\) 11.1514 0.753538
\(220\) 15.8248 1.06691
\(221\) 5.39622 0.362989
\(222\) 20.2216 1.35719
\(223\) −10.7813 −0.721968 −0.360984 0.932572i \(-0.617559\pi\)
−0.360984 + 0.932572i \(0.617559\pi\)
\(224\) 11.3123 0.755836
\(225\) −4.99302 −0.332868
\(226\) 18.4533 1.22750
\(227\) 5.18809 0.344346 0.172173 0.985067i \(-0.444921\pi\)
0.172173 + 0.985067i \(0.444921\pi\)
\(228\) 4.70894 0.311857
\(229\) −19.6840 −1.30076 −0.650378 0.759611i \(-0.725389\pi\)
−0.650378 + 0.759611i \(0.725389\pi\)
\(230\) −7.58818 −0.500350
\(231\) −14.8204 −0.975112
\(232\) −5.57239 −0.365845
\(233\) 2.73430 0.179130 0.0895650 0.995981i \(-0.471452\pi\)
0.0895650 + 0.995981i \(0.471452\pi\)
\(234\) 16.9049 1.10511
\(235\) 11.7608 0.767187
\(236\) −30.0105 −1.95352
\(237\) 25.2783 1.64200
\(238\) 3.09820 0.200827
\(239\) 6.00842 0.388653 0.194326 0.980937i \(-0.437748\pi\)
0.194326 + 0.980937i \(0.437748\pi\)
\(240\) 6.56949 0.424059
\(241\) −4.29472 −0.276647 −0.138324 0.990387i \(-0.544171\pi\)
−0.138324 + 0.990387i \(0.544171\pi\)
\(242\) −26.8695 −1.72724
\(243\) 14.0157 0.899106
\(244\) −30.2175 −1.93448
\(245\) −6.18055 −0.394861
\(246\) −6.47253 −0.412673
\(247\) 4.64353 0.295461
\(248\) −6.24122 −0.396318
\(249\) 0.0585010 0.00370735
\(250\) 22.6994 1.43563
\(251\) −1.17799 −0.0743539 −0.0371770 0.999309i \(-0.511837\pi\)
−0.0371770 + 0.999309i \(0.511837\pi\)
\(252\) 5.47722 0.345033
\(253\) 13.6484 0.858067
\(254\) −43.5436 −2.73217
\(255\) 2.65954 0.166547
\(256\) 2.89981 0.181238
\(257\) −20.6085 −1.28553 −0.642763 0.766065i \(-0.722212\pi\)
−0.642763 + 0.766065i \(0.722212\pi\)
\(258\) −54.3170 −3.38163
\(259\) 6.46084 0.401457
\(260\) −17.6001 −1.09151
\(261\) 6.43940 0.398589
\(262\) −13.9355 −0.860937
\(263\) −21.3263 −1.31504 −0.657518 0.753439i \(-0.728394\pi\)
−0.657518 + 0.753439i \(0.728394\pi\)
\(264\) 12.9679 0.798120
\(265\) 5.28438 0.324617
\(266\) 2.66606 0.163466
\(267\) 32.6740 1.99962
\(268\) 28.4890 1.74024
\(269\) −22.5415 −1.37438 −0.687189 0.726478i \(-0.741156\pi\)
−0.687189 + 0.726478i \(0.741156\pi\)
\(270\) −8.76301 −0.533300
\(271\) 13.2512 0.804953 0.402477 0.915430i \(-0.368149\pi\)
0.402477 + 0.915430i \(0.368149\pi\)
\(272\) 2.47016 0.149775
\(273\) 16.4831 0.997602
\(274\) 5.96362 0.360276
\(275\) −16.5685 −0.999118
\(276\) −15.3934 −0.926573
\(277\) 6.16242 0.370264 0.185132 0.982714i \(-0.440729\pi\)
0.185132 + 0.982714i \(0.440729\pi\)
\(278\) 24.6212 1.47668
\(279\) 7.21231 0.431790
\(280\) −2.30356 −0.137664
\(281\) 10.1465 0.605288 0.302644 0.953104i \(-0.402131\pi\)
0.302644 + 0.953104i \(0.402131\pi\)
\(282\) 42.2771 2.51756
\(283\) −16.1457 −0.959762 −0.479881 0.877333i \(-0.659320\pi\)
−0.479881 + 0.877333i \(0.659320\pi\)
\(284\) −27.9438 −1.65816
\(285\) 2.28858 0.135564
\(286\) 56.0961 3.31703
\(287\) −2.06798 −0.122069
\(288\) 11.4384 0.674015
\(289\) 1.00000 0.0588235
\(290\) −11.8801 −0.697625
\(291\) −4.02130 −0.235733
\(292\) −13.6756 −0.800305
\(293\) 29.1061 1.70040 0.850198 0.526462i \(-0.176482\pi\)
0.850198 + 0.526462i \(0.176482\pi\)
\(294\) −22.2176 −1.29576
\(295\) −14.5853 −0.849191
\(296\) −5.65326 −0.328589
\(297\) 15.7615 0.914575
\(298\) 39.3034 2.27678
\(299\) −15.1796 −0.877858
\(300\) 18.6868 1.07888
\(301\) −17.3544 −1.00029
\(302\) −4.55379 −0.262041
\(303\) −31.3749 −1.80244
\(304\) 2.12561 0.121912
\(305\) −14.6860 −0.840916
\(306\) 3.13274 0.179087
\(307\) −16.4458 −0.938609 −0.469304 0.883036i \(-0.655495\pi\)
−0.469304 + 0.883036i \(0.655495\pi\)
\(308\) 18.1752 1.03563
\(309\) −17.1777 −0.977207
\(310\) −13.3061 −0.755734
\(311\) 24.9437 1.41443 0.707213 0.707000i \(-0.249952\pi\)
0.707213 + 0.707000i \(0.249952\pi\)
\(312\) −14.4228 −0.816528
\(313\) −28.2448 −1.59649 −0.798246 0.602331i \(-0.794239\pi\)
−0.798246 + 0.602331i \(0.794239\pi\)
\(314\) −8.59945 −0.485295
\(315\) 2.66197 0.149985
\(316\) −31.0004 −1.74391
\(317\) −27.4623 −1.54244 −0.771219 0.636570i \(-0.780352\pi\)
−0.771219 + 0.636570i \(0.780352\pi\)
\(318\) 18.9961 1.06525
\(319\) 21.3681 1.19638
\(320\) −14.8829 −0.831978
\(321\) −24.8149 −1.38503
\(322\) −8.71526 −0.485682
\(323\) 0.860517 0.0478804
\(324\) −29.1399 −1.61889
\(325\) 18.4273 1.02216
\(326\) −26.1387 −1.44769
\(327\) 15.9685 0.883060
\(328\) 1.80949 0.0999125
\(329\) 13.5076 0.744698
\(330\) 27.6471 1.52192
\(331\) −4.50743 −0.247751 −0.123875 0.992298i \(-0.539532\pi\)
−0.123875 + 0.992298i \(0.539532\pi\)
\(332\) −0.0717435 −0.00393744
\(333\) 6.53286 0.357999
\(334\) 33.3366 1.82410
\(335\) 13.8459 0.756481
\(336\) 7.54526 0.411628
\(337\) 2.12518 0.115766 0.0578831 0.998323i \(-0.481565\pi\)
0.0578831 + 0.998323i \(0.481565\pi\)
\(338\) −34.5362 −1.87852
\(339\) 18.1934 0.988132
\(340\) −3.26157 −0.176883
\(341\) 23.9328 1.29603
\(342\) 2.69577 0.145771
\(343\) −17.2208 −0.929834
\(344\) 15.1851 0.818728
\(345\) −7.48131 −0.402780
\(346\) −8.57757 −0.461133
\(347\) −11.2330 −0.603018 −0.301509 0.953463i \(-0.597490\pi\)
−0.301509 + 0.953463i \(0.597490\pi\)
\(348\) −24.1000 −1.29190
\(349\) 26.0958 1.39688 0.698439 0.715670i \(-0.253878\pi\)
0.698439 + 0.715670i \(0.253878\pi\)
\(350\) 10.5799 0.565520
\(351\) −17.5297 −0.935669
\(352\) 37.9564 2.02308
\(353\) −1.00000 −0.0532246
\(354\) −52.4307 −2.78666
\(355\) −13.5809 −0.720800
\(356\) −40.0703 −2.12372
\(357\) 3.05457 0.161665
\(358\) 44.4292 2.34815
\(359\) 32.2363 1.70137 0.850684 0.525677i \(-0.176188\pi\)
0.850684 + 0.525677i \(0.176188\pi\)
\(360\) −2.32924 −0.122762
\(361\) −18.2595 −0.961027
\(362\) 27.1170 1.42524
\(363\) −26.4911 −1.39042
\(364\) −20.2143 −1.05952
\(365\) −6.64646 −0.347891
\(366\) −52.7924 −2.75951
\(367\) −6.89914 −0.360132 −0.180066 0.983654i \(-0.557631\pi\)
−0.180066 + 0.983654i \(0.557631\pi\)
\(368\) −6.94857 −0.362219
\(369\) −2.09103 −0.108855
\(370\) −12.0525 −0.626582
\(371\) 6.06928 0.315101
\(372\) −26.9927 −1.39951
\(373\) 19.6813 1.01906 0.509529 0.860454i \(-0.329820\pi\)
0.509529 + 0.860454i \(0.329820\pi\)
\(374\) 10.3955 0.537536
\(375\) 22.3797 1.15568
\(376\) −11.8192 −0.609528
\(377\) −23.7653 −1.22398
\(378\) −10.0646 −0.517667
\(379\) −28.6601 −1.47217 −0.736085 0.676889i \(-0.763328\pi\)
−0.736085 + 0.676889i \(0.763328\pi\)
\(380\) −2.80663 −0.143977
\(381\) −42.9304 −2.19939
\(382\) −48.9147 −2.50269
\(383\) −19.9905 −1.02147 −0.510734 0.859739i \(-0.670626\pi\)
−0.510734 + 0.859739i \(0.670626\pi\)
\(384\) −20.4499 −1.04358
\(385\) 8.83331 0.450187
\(386\) −37.8985 −1.92898
\(387\) −17.5478 −0.892006
\(388\) 4.93157 0.250363
\(389\) 25.8923 1.31279 0.656396 0.754417i \(-0.272080\pi\)
0.656396 + 0.754417i \(0.272080\pi\)
\(390\) −30.7488 −1.55703
\(391\) −2.81301 −0.142260
\(392\) 6.21126 0.313716
\(393\) −13.7392 −0.693051
\(394\) −19.0184 −0.958135
\(395\) −15.0664 −0.758074
\(396\) 18.3778 0.923521
\(397\) 32.8009 1.64623 0.823115 0.567874i \(-0.192234\pi\)
0.823115 + 0.567874i \(0.192234\pi\)
\(398\) −45.6679 −2.28912
\(399\) 2.62851 0.131590
\(400\) 8.43523 0.421761
\(401\) 21.5362 1.07546 0.537732 0.843116i \(-0.319281\pi\)
0.537732 + 0.843116i \(0.319281\pi\)
\(402\) 49.7726 2.48243
\(403\) −26.6178 −1.32593
\(404\) 38.4770 1.91430
\(405\) −14.1622 −0.703727
\(406\) −13.6447 −0.677175
\(407\) 21.6782 1.07455
\(408\) −2.67275 −0.132321
\(409\) 6.95199 0.343754 0.171877 0.985118i \(-0.445017\pi\)
0.171877 + 0.985118i \(0.445017\pi\)
\(410\) 3.85777 0.190522
\(411\) 5.87963 0.290021
\(412\) 21.0662 1.03786
\(413\) −16.7517 −0.824297
\(414\) −8.81240 −0.433106
\(415\) −0.0348679 −0.00171160
\(416\) −42.2147 −2.06975
\(417\) 24.2744 1.18872
\(418\) 8.94546 0.437537
\(419\) 6.48010 0.316574 0.158287 0.987393i \(-0.449403\pi\)
0.158287 + 0.987393i \(0.449403\pi\)
\(420\) −9.96267 −0.486129
\(421\) 14.8094 0.721768 0.360884 0.932611i \(-0.382475\pi\)
0.360884 + 0.932611i \(0.382475\pi\)
\(422\) 21.6184 1.05237
\(423\) 13.6582 0.664082
\(424\) −5.31064 −0.257907
\(425\) 3.41485 0.165645
\(426\) −48.8201 −2.36534
\(427\) −16.8673 −0.816265
\(428\) 30.4321 1.47099
\(429\) 55.3060 2.67020
\(430\) 32.3742 1.56122
\(431\) 13.2015 0.635895 0.317948 0.948108i \(-0.397006\pi\)
0.317948 + 0.948108i \(0.397006\pi\)
\(432\) −8.02437 −0.386073
\(433\) −3.37603 −0.162241 −0.0811207 0.996704i \(-0.525850\pi\)
−0.0811207 + 0.996704i \(0.525850\pi\)
\(434\) −15.2824 −0.733580
\(435\) −11.7128 −0.561586
\(436\) −19.5832 −0.937865
\(437\) −2.42064 −0.115795
\(438\) −23.8924 −1.14162
\(439\) −1.52581 −0.0728227 −0.0364114 0.999337i \(-0.511593\pi\)
−0.0364114 + 0.999337i \(0.511593\pi\)
\(440\) −7.72917 −0.368474
\(441\) −7.17768 −0.341794
\(442\) −11.5617 −0.549934
\(443\) 10.0461 0.477306 0.238653 0.971105i \(-0.423294\pi\)
0.238653 + 0.971105i \(0.423294\pi\)
\(444\) −24.4498 −1.16034
\(445\) −19.4745 −0.923178
\(446\) 23.0995 1.09379
\(447\) 38.7498 1.83280
\(448\) −17.0934 −0.807589
\(449\) −26.6886 −1.25951 −0.629756 0.776793i \(-0.716845\pi\)
−0.629756 + 0.776793i \(0.716845\pi\)
\(450\) 10.6978 0.504301
\(451\) −6.93874 −0.326733
\(452\) −22.3118 −1.04946
\(453\) −4.48966 −0.210943
\(454\) −11.1158 −0.521689
\(455\) −9.82430 −0.460570
\(456\) −2.29995 −0.107705
\(457\) −6.03441 −0.282278 −0.141139 0.989990i \(-0.545076\pi\)
−0.141139 + 0.989990i \(0.545076\pi\)
\(458\) 42.1741 1.97067
\(459\) −3.24853 −0.151628
\(460\) 9.17481 0.427778
\(461\) 15.0232 0.699699 0.349849 0.936806i \(-0.386233\pi\)
0.349849 + 0.936806i \(0.386233\pi\)
\(462\) 31.7536 1.47731
\(463\) 4.78332 0.222300 0.111150 0.993804i \(-0.464547\pi\)
0.111150 + 0.993804i \(0.464547\pi\)
\(464\) −10.8787 −0.505033
\(465\) −13.1187 −0.608363
\(466\) −5.85840 −0.271385
\(467\) −17.1328 −0.792810 −0.396405 0.918076i \(-0.629742\pi\)
−0.396405 + 0.918076i \(0.629742\pi\)
\(468\) −20.4396 −0.944821
\(469\) 15.9024 0.734306
\(470\) −25.1981 −1.16230
\(471\) −8.47834 −0.390661
\(472\) 14.6578 0.674680
\(473\) −58.2295 −2.67739
\(474\) −54.1601 −2.48766
\(475\) 2.93854 0.134829
\(476\) −3.74601 −0.171698
\(477\) 6.13693 0.280991
\(478\) −12.8734 −0.588815
\(479\) 22.9247 1.04746 0.523729 0.851885i \(-0.324541\pi\)
0.523729 + 0.851885i \(0.324541\pi\)
\(480\) −20.8056 −0.949643
\(481\) −24.1102 −1.09933
\(482\) 9.20167 0.419125
\(483\) −8.59251 −0.390973
\(484\) 32.4877 1.47671
\(485\) 2.39678 0.108832
\(486\) −30.0294 −1.36216
\(487\) −9.04444 −0.409843 −0.204921 0.978778i \(-0.565694\pi\)
−0.204921 + 0.978778i \(0.565694\pi\)
\(488\) 14.7589 0.668105
\(489\) −25.7706 −1.16539
\(490\) 13.2422 0.598221
\(491\) −8.42855 −0.380375 −0.190188 0.981748i \(-0.560910\pi\)
−0.190188 + 0.981748i \(0.560910\pi\)
\(492\) 7.82588 0.352818
\(493\) −4.40407 −0.198349
\(494\) −9.94904 −0.447628
\(495\) 8.93177 0.401453
\(496\) −12.1845 −0.547100
\(497\) −15.5981 −0.699671
\(498\) −0.125342 −0.00561669
\(499\) 7.46498 0.334178 0.167089 0.985942i \(-0.446563\pi\)
0.167089 + 0.985942i \(0.446563\pi\)
\(500\) −27.4456 −1.22740
\(501\) 32.8671 1.46839
\(502\) 2.52391 0.112647
\(503\) −9.87757 −0.440419 −0.220210 0.975453i \(-0.570674\pi\)
−0.220210 + 0.975453i \(0.570674\pi\)
\(504\) −2.67520 −0.119163
\(505\) 18.7001 0.832145
\(506\) −29.2425 −1.29999
\(507\) −34.0497 −1.51220
\(508\) 52.6483 2.33589
\(509\) −25.0389 −1.10983 −0.554914 0.831908i \(-0.687249\pi\)
−0.554914 + 0.831908i \(0.687249\pi\)
\(510\) −5.69822 −0.252321
\(511\) −7.63366 −0.337693
\(512\) −25.5750 −1.13027
\(513\) −2.79541 −0.123420
\(514\) 44.1550 1.94759
\(515\) 10.2383 0.451154
\(516\) 65.6743 2.89115
\(517\) 45.3223 1.99327
\(518\) −13.8427 −0.608214
\(519\) −8.45676 −0.371210
\(520\) 8.59629 0.376972
\(521\) −16.4977 −0.722777 −0.361389 0.932415i \(-0.617697\pi\)
−0.361389 + 0.932415i \(0.617697\pi\)
\(522\) −13.7968 −0.603869
\(523\) −9.83004 −0.429837 −0.214919 0.976632i \(-0.568949\pi\)
−0.214919 + 0.976632i \(0.568949\pi\)
\(524\) 16.8493 0.736064
\(525\) 10.4309 0.455241
\(526\) 45.6928 1.99230
\(527\) −4.93268 −0.214871
\(528\) 25.3168 1.10177
\(529\) −15.0870 −0.655957
\(530\) −11.3221 −0.491800
\(531\) −16.9384 −0.735065
\(532\) −3.22351 −0.139757
\(533\) 7.71718 0.334268
\(534\) −70.0059 −3.02945
\(535\) 14.7903 0.639438
\(536\) −13.9147 −0.601022
\(537\) 43.8034 1.89026
\(538\) 48.2964 2.08221
\(539\) −23.8179 −1.02591
\(540\) 10.5953 0.455949
\(541\) 31.7712 1.36595 0.682976 0.730441i \(-0.260685\pi\)
0.682976 + 0.730441i \(0.260685\pi\)
\(542\) −28.3914 −1.21952
\(543\) 26.7350 1.14731
\(544\) −7.82302 −0.335409
\(545\) −9.51759 −0.407689
\(546\) −35.3160 −1.51138
\(547\) 6.28505 0.268729 0.134365 0.990932i \(-0.457101\pi\)
0.134365 + 0.990932i \(0.457101\pi\)
\(548\) −7.21057 −0.308020
\(549\) −17.0553 −0.727902
\(550\) 35.4989 1.51368
\(551\) −3.78978 −0.161450
\(552\) 7.51847 0.320008
\(553\) −17.3043 −0.735852
\(554\) −13.2033 −0.560956
\(555\) −11.8828 −0.504397
\(556\) −29.7693 −1.26250
\(557\) −1.62184 −0.0687196 −0.0343598 0.999410i \(-0.510939\pi\)
−0.0343598 + 0.999410i \(0.510939\pi\)
\(558\) −15.4528 −0.654168
\(559\) 64.7621 2.73915
\(560\) −4.49715 −0.190039
\(561\) 10.2490 0.432715
\(562\) −21.7394 −0.917021
\(563\) 24.9610 1.05198 0.525991 0.850490i \(-0.323694\pi\)
0.525991 + 0.850490i \(0.323694\pi\)
\(564\) −51.1169 −2.15241
\(565\) −10.8437 −0.456198
\(566\) 34.5931 1.45406
\(567\) −16.2658 −0.683098
\(568\) 13.6484 0.572674
\(569\) −32.9797 −1.38258 −0.691291 0.722576i \(-0.742958\pi\)
−0.691291 + 0.722576i \(0.742958\pi\)
\(570\) −4.90341 −0.205381
\(571\) −43.3587 −1.81451 −0.907253 0.420585i \(-0.861825\pi\)
−0.907253 + 0.420585i \(0.861825\pi\)
\(572\) −67.8253 −2.83592
\(573\) −48.2258 −2.01466
\(574\) 4.43077 0.184937
\(575\) −9.60600 −0.400598
\(576\) −17.2840 −0.720165
\(577\) 34.0482 1.41745 0.708723 0.705487i \(-0.249272\pi\)
0.708723 + 0.705487i \(0.249272\pi\)
\(578\) −2.14256 −0.0891186
\(579\) −37.3648 −1.55283
\(580\) 14.3642 0.596439
\(581\) −0.0400469 −0.00166142
\(582\) 8.61585 0.357139
\(583\) 20.3644 0.843406
\(584\) 6.67948 0.276399
\(585\) −9.93380 −0.410712
\(586\) −62.3615 −2.57613
\(587\) −37.5840 −1.55126 −0.775628 0.631190i \(-0.782567\pi\)
−0.775628 + 0.631190i \(0.782567\pi\)
\(588\) 26.8631 1.10782
\(589\) −4.24465 −0.174898
\(590\) 31.2499 1.28654
\(591\) −18.7506 −0.771295
\(592\) −11.0366 −0.453603
\(593\) 6.91019 0.283768 0.141884 0.989883i \(-0.454684\pi\)
0.141884 + 0.989883i \(0.454684\pi\)
\(594\) −33.7699 −1.38560
\(595\) −1.82059 −0.0746369
\(596\) −47.5214 −1.94655
\(597\) −45.0247 −1.84274
\(598\) 32.5231 1.32997
\(599\) −28.2377 −1.15376 −0.576880 0.816829i \(-0.695730\pi\)
−0.576880 + 0.816829i \(0.695730\pi\)
\(600\) −9.12707 −0.372611
\(601\) 34.3753 1.40220 0.701100 0.713063i \(-0.252693\pi\)
0.701100 + 0.713063i \(0.252693\pi\)
\(602\) 37.1828 1.51546
\(603\) 16.0797 0.654815
\(604\) 5.50596 0.224034
\(605\) 15.7893 0.641925
\(606\) 67.2224 2.73072
\(607\) −28.4971 −1.15666 −0.578332 0.815802i \(-0.696296\pi\)
−0.578332 + 0.815802i \(0.696296\pi\)
\(608\) −6.73184 −0.273012
\(609\) −13.4525 −0.545123
\(610\) 31.4655 1.27400
\(611\) −50.4069 −2.03924
\(612\) −3.78777 −0.153111
\(613\) −40.8237 −1.64885 −0.824426 0.565969i \(-0.808502\pi\)
−0.824426 + 0.565969i \(0.808502\pi\)
\(614\) 35.2359 1.42201
\(615\) 3.80344 0.153369
\(616\) −8.87719 −0.357672
\(617\) −4.03369 −0.162390 −0.0811951 0.996698i \(-0.525874\pi\)
−0.0811951 + 0.996698i \(0.525874\pi\)
\(618\) 36.8043 1.48049
\(619\) 4.27327 0.171757 0.0858786 0.996306i \(-0.472630\pi\)
0.0858786 + 0.996306i \(0.472630\pi\)
\(620\) 16.0883 0.646120
\(621\) 9.13812 0.366700
\(622\) −53.4432 −2.14288
\(623\) −22.3670 −0.896116
\(624\) −28.1570 −1.12718
\(625\) 3.73549 0.149419
\(626\) 60.5161 2.41871
\(627\) 8.81947 0.352216
\(628\) 10.3975 0.414907
\(629\) −4.46799 −0.178150
\(630\) −5.70343 −0.227230
\(631\) −11.1461 −0.443719 −0.221860 0.975079i \(-0.571213\pi\)
−0.221860 + 0.975079i \(0.571213\pi\)
\(632\) 15.1413 0.602288
\(633\) 21.3139 0.847151
\(634\) 58.8396 2.33682
\(635\) 25.5875 1.01541
\(636\) −22.9680 −0.910741
\(637\) 26.4900 1.04957
\(638\) −45.7823 −1.81254
\(639\) −15.7720 −0.623930
\(640\) 12.1886 0.481797
\(641\) −40.5431 −1.60135 −0.800677 0.599096i \(-0.795527\pi\)
−0.800677 + 0.599096i \(0.795527\pi\)
\(642\) 53.1674 2.09835
\(643\) −9.29223 −0.366450 −0.183225 0.983071i \(-0.558654\pi\)
−0.183225 + 0.983071i \(0.558654\pi\)
\(644\) 10.5375 0.415238
\(645\) 31.9182 1.25678
\(646\) −1.84371 −0.0725396
\(647\) −29.6720 −1.16653 −0.583263 0.812283i \(-0.698224\pi\)
−0.583263 + 0.812283i \(0.698224\pi\)
\(648\) 14.2326 0.559109
\(649\) −56.2073 −2.20633
\(650\) −39.4815 −1.54859
\(651\) −15.0672 −0.590530
\(652\) 31.6041 1.23771
\(653\) 27.3242 1.06928 0.534638 0.845081i \(-0.320448\pi\)
0.534638 + 0.845081i \(0.320448\pi\)
\(654\) −34.2134 −1.33785
\(655\) 8.18888 0.319966
\(656\) 3.53260 0.137925
\(657\) −7.71875 −0.301137
\(658\) −28.9408 −1.12823
\(659\) −0.301844 −0.0117582 −0.00587909 0.999983i \(-0.501871\pi\)
−0.00587909 + 0.999983i \(0.501871\pi\)
\(660\) −33.4279 −1.30118
\(661\) −42.6379 −1.65842 −0.829210 0.558937i \(-0.811209\pi\)
−0.829210 + 0.558937i \(0.811209\pi\)
\(662\) 9.65743 0.375347
\(663\) −11.3989 −0.442695
\(664\) 0.0350411 0.00135986
\(665\) −1.56665 −0.0607520
\(666\) −13.9970 −0.542374
\(667\) 12.3887 0.479691
\(668\) −40.3070 −1.55952
\(669\) 22.7742 0.880500
\(670\) −29.6656 −1.14608
\(671\) −56.5951 −2.18483
\(672\) −23.8959 −0.921805
\(673\) 3.89457 0.150125 0.0750623 0.997179i \(-0.476084\pi\)
0.0750623 + 0.997179i \(0.476084\pi\)
\(674\) −4.55333 −0.175388
\(675\) −11.0932 −0.426979
\(676\) 41.7574 1.60605
\(677\) −23.3401 −0.897034 −0.448517 0.893774i \(-0.648048\pi\)
−0.448517 + 0.893774i \(0.648048\pi\)
\(678\) −38.9805 −1.49704
\(679\) 2.75278 0.105642
\(680\) 1.59302 0.0610896
\(681\) −10.9592 −0.419958
\(682\) −51.2774 −1.96351
\(683\) −38.7929 −1.48437 −0.742185 0.670195i \(-0.766211\pi\)
−0.742185 + 0.670195i \(0.766211\pi\)
\(684\) −3.25944 −0.124628
\(685\) −3.50439 −0.133896
\(686\) 36.8965 1.40871
\(687\) 41.5801 1.58638
\(688\) 29.6454 1.13022
\(689\) −22.6490 −0.862858
\(690\) 16.0291 0.610218
\(691\) 3.05375 0.116170 0.0580850 0.998312i \(-0.481501\pi\)
0.0580850 + 0.998312i \(0.481501\pi\)
\(692\) 10.3711 0.394249
\(693\) 10.2584 0.389685
\(694\) 24.0673 0.913583
\(695\) −14.4681 −0.548807
\(696\) 11.7710 0.446178
\(697\) 1.43011 0.0541693
\(698\) −55.9118 −2.11629
\(699\) −5.77588 −0.218464
\(700\) −12.7921 −0.483495
\(701\) 4.41130 0.166612 0.0833062 0.996524i \(-0.473452\pi\)
0.0833062 + 0.996524i \(0.473452\pi\)
\(702\) 37.5585 1.41755
\(703\) −3.84478 −0.145009
\(704\) −57.3539 −2.16161
\(705\) −24.8432 −0.935649
\(706\) 2.14256 0.0806362
\(707\) 21.4777 0.807751
\(708\) 63.3936 2.38248
\(709\) −12.7835 −0.480096 −0.240048 0.970761i \(-0.577163\pi\)
−0.240048 + 0.970761i \(0.577163\pi\)
\(710\) 29.0979 1.09202
\(711\) −17.4971 −0.656194
\(712\) 19.5712 0.733462
\(713\) 13.8756 0.519647
\(714\) −6.54458 −0.244925
\(715\) −32.9636 −1.23277
\(716\) −53.7190 −2.00757
\(717\) −12.6921 −0.473994
\(718\) −69.0681 −2.57760
\(719\) −4.46156 −0.166388 −0.0831940 0.996533i \(-0.526512\pi\)
−0.0831940 + 0.996533i \(0.526512\pi\)
\(720\) −4.54727 −0.169467
\(721\) 11.7590 0.437929
\(722\) 39.1220 1.45597
\(723\) 9.07207 0.337394
\(724\) −32.7869 −1.21852
\(725\) −15.0392 −0.558544
\(726\) 56.7586 2.10651
\(727\) −8.58580 −0.318430 −0.159215 0.987244i \(-0.550896\pi\)
−0.159215 + 0.987244i \(0.550896\pi\)
\(728\) 9.87311 0.365922
\(729\) 4.13928 0.153307
\(730\) 14.2404 0.527061
\(731\) 12.0014 0.443888
\(732\) 63.8309 2.35926
\(733\) 13.2110 0.487958 0.243979 0.969780i \(-0.421547\pi\)
0.243979 + 0.969780i \(0.421547\pi\)
\(734\) 14.7818 0.545606
\(735\) 13.0557 0.481566
\(736\) 22.0062 0.811159
\(737\) 53.3577 1.96546
\(738\) 4.48016 0.164917
\(739\) 45.7908 1.68444 0.842220 0.539133i \(-0.181248\pi\)
0.842220 + 0.539133i \(0.181248\pi\)
\(740\) 14.5726 0.535701
\(741\) −9.80891 −0.360339
\(742\) −13.0038 −0.477383
\(743\) −43.5356 −1.59717 −0.798583 0.601884i \(-0.794417\pi\)
−0.798583 + 0.601884i \(0.794417\pi\)
\(744\) 13.1838 0.483343
\(745\) −23.0958 −0.846163
\(746\) −42.1682 −1.54389
\(747\) −0.0404932 −0.00148157
\(748\) −12.5691 −0.459570
\(749\) 16.9871 0.620694
\(750\) −47.9497 −1.75087
\(751\) −45.7345 −1.66888 −0.834438 0.551102i \(-0.814208\pi\)
−0.834438 + 0.551102i \(0.814208\pi\)
\(752\) −23.0741 −0.841427
\(753\) 2.48836 0.0906808
\(754\) 50.9185 1.85434
\(755\) 2.67594 0.0973873
\(756\) 12.1690 0.442583
\(757\) 35.1277 1.27674 0.638368 0.769731i \(-0.279610\pi\)
0.638368 + 0.769731i \(0.279610\pi\)
\(758\) 61.4059 2.23036
\(759\) −28.8306 −1.04648
\(760\) 1.37082 0.0497250
\(761\) −22.1947 −0.804559 −0.402279 0.915517i \(-0.631782\pi\)
−0.402279 + 0.915517i \(0.631782\pi\)
\(762\) 91.9807 3.33211
\(763\) −10.9312 −0.395737
\(764\) 59.1424 2.13970
\(765\) −1.84088 −0.0665573
\(766\) 42.8308 1.54754
\(767\) 62.5131 2.25722
\(768\) −6.12550 −0.221035
\(769\) −19.4933 −0.702948 −0.351474 0.936198i \(-0.614319\pi\)
−0.351474 + 0.936198i \(0.614319\pi\)
\(770\) −18.9259 −0.682041
\(771\) 43.5331 1.56781
\(772\) 45.8228 1.64920
\(773\) 30.2999 1.08981 0.544905 0.838498i \(-0.316566\pi\)
0.544905 + 0.838498i \(0.316566\pi\)
\(774\) 37.5972 1.35140
\(775\) −16.8444 −0.605068
\(776\) −2.40869 −0.0864670
\(777\) −13.6478 −0.489611
\(778\) −55.4757 −1.98890
\(779\) 1.23063 0.0440920
\(780\) 37.1781 1.33119
\(781\) −52.3366 −1.87275
\(782\) 6.02702 0.215526
\(783\) 14.3067 0.511281
\(784\) 12.1260 0.433071
\(785\) 5.05328 0.180359
\(786\) 29.4370 1.04998
\(787\) −10.7649 −0.383727 −0.191864 0.981422i \(-0.561453\pi\)
−0.191864 + 0.981422i \(0.561453\pi\)
\(788\) 22.9950 0.819164
\(789\) 45.0493 1.60380
\(790\) 32.2807 1.14849
\(791\) −12.4543 −0.442825
\(792\) −8.97614 −0.318953
\(793\) 62.9444 2.23522
\(794\) −70.2778 −2.49407
\(795\) −11.1626 −0.395898
\(796\) 55.2166 1.95710
\(797\) −5.71963 −0.202600 −0.101300 0.994856i \(-0.532300\pi\)
−0.101300 + 0.994856i \(0.532300\pi\)
\(798\) −5.63172 −0.199361
\(799\) −9.34116 −0.330466
\(800\) −26.7145 −0.944498
\(801\) −22.6163 −0.799109
\(802\) −46.1424 −1.62935
\(803\) −25.6134 −0.903876
\(804\) −60.1796 −2.12237
\(805\) 5.12133 0.180503
\(806\) 57.0301 2.00880
\(807\) 47.6162 1.67617
\(808\) −18.7930 −0.661137
\(809\) 53.2172 1.87102 0.935508 0.353305i \(-0.114942\pi\)
0.935508 + 0.353305i \(0.114942\pi\)
\(810\) 30.3434 1.06616
\(811\) −17.7778 −0.624264 −0.312132 0.950039i \(-0.601043\pi\)
−0.312132 + 0.950039i \(0.601043\pi\)
\(812\) 16.4977 0.578955
\(813\) −27.9916 −0.981707
\(814\) −46.4467 −1.62796
\(815\) 15.3599 0.538032
\(816\) −5.21791 −0.182663
\(817\) 10.3274 0.361310
\(818\) −14.8950 −0.520792
\(819\) −11.4093 −0.398673
\(820\) −4.66440 −0.162888
\(821\) 16.8662 0.588635 0.294317 0.955708i \(-0.404908\pi\)
0.294317 + 0.955708i \(0.404908\pi\)
\(822\) −12.5974 −0.439386
\(823\) 6.16622 0.214941 0.107470 0.994208i \(-0.465725\pi\)
0.107470 + 0.994208i \(0.465725\pi\)
\(824\) −10.2892 −0.358441
\(825\) 34.9990 1.21851
\(826\) 35.8915 1.24882
\(827\) 0.932231 0.0324168 0.0162084 0.999869i \(-0.494840\pi\)
0.0162084 + 0.999869i \(0.494840\pi\)
\(828\) 10.6550 0.370287
\(829\) 8.45248 0.293567 0.146783 0.989169i \(-0.453108\pi\)
0.146783 + 0.989169i \(0.453108\pi\)
\(830\) 0.0747064 0.00259310
\(831\) −13.0174 −0.451568
\(832\) 63.7883 2.21146
\(833\) 4.90900 0.170087
\(834\) −52.0093 −1.80094
\(835\) −19.5895 −0.677923
\(836\) −10.8159 −0.374075
\(837\) 16.0239 0.553868
\(838\) −13.8840 −0.479614
\(839\) −7.05848 −0.243686 −0.121843 0.992549i \(-0.538880\pi\)
−0.121843 + 0.992549i \(0.538880\pi\)
\(840\) 4.86599 0.167893
\(841\) −9.60419 −0.331179
\(842\) −31.7301 −1.09349
\(843\) −21.4332 −0.738199
\(844\) −26.1386 −0.899728
\(845\) 20.2944 0.698149
\(846\) −29.2634 −1.00610
\(847\) 18.1345 0.623107
\(848\) −10.3677 −0.356030
\(849\) 34.1059 1.17051
\(850\) −7.31651 −0.250954
\(851\) 12.5685 0.430842
\(852\) 59.0280 2.02227
\(853\) −10.4495 −0.357783 −0.178891 0.983869i \(-0.557251\pi\)
−0.178891 + 0.983869i \(0.557251\pi\)
\(854\) 36.1391 1.23665
\(855\) −1.58411 −0.0541755
\(856\) −14.8637 −0.508032
\(857\) −12.6675 −0.432714 −0.216357 0.976314i \(-0.569418\pi\)
−0.216357 + 0.976314i \(0.569418\pi\)
\(858\) −118.496 −4.04540
\(859\) −20.0566 −0.684324 −0.342162 0.939641i \(-0.611159\pi\)
−0.342162 + 0.939641i \(0.611159\pi\)
\(860\) −39.1434 −1.33478
\(861\) 4.36837 0.148873
\(862\) −28.2850 −0.963391
\(863\) 21.0799 0.717569 0.358785 0.933420i \(-0.383191\pi\)
0.358785 + 0.933420i \(0.383191\pi\)
\(864\) 25.4133 0.864577
\(865\) 5.04042 0.171379
\(866\) 7.23332 0.245798
\(867\) −2.11238 −0.0717402
\(868\) 18.4779 0.627180
\(869\) −58.0613 −1.96959
\(870\) 25.0953 0.850812
\(871\) −59.3438 −2.01079
\(872\) 9.56487 0.323907
\(873\) 2.78346 0.0942060
\(874\) 5.18636 0.175431
\(875\) −15.3200 −0.517911
\(876\) 28.8881 0.976039
\(877\) 37.7023 1.27311 0.636557 0.771229i \(-0.280358\pi\)
0.636557 + 0.771229i \(0.280358\pi\)
\(878\) 3.26912 0.110328
\(879\) −61.4831 −2.07378
\(880\) −15.0894 −0.508662
\(881\) 44.9407 1.51409 0.757046 0.653362i \(-0.226642\pi\)
0.757046 + 0.653362i \(0.226642\pi\)
\(882\) 15.3786 0.517824
\(883\) −38.5616 −1.29770 −0.648851 0.760915i \(-0.724750\pi\)
−0.648851 + 0.760915i \(0.724750\pi\)
\(884\) 13.9792 0.470170
\(885\) 30.8098 1.03566
\(886\) −21.5244 −0.723126
\(887\) −24.5934 −0.825765 −0.412882 0.910784i \(-0.635478\pi\)
−0.412882 + 0.910784i \(0.635478\pi\)
\(888\) 11.9418 0.400742
\(889\) 29.3880 0.985642
\(890\) 41.7251 1.39863
\(891\) −54.5768 −1.82839
\(892\) −27.9294 −0.935146
\(893\) −8.03822 −0.268989
\(894\) −83.0236 −2.77673
\(895\) −26.1078 −0.872689
\(896\) 13.9990 0.467674
\(897\) 32.0650 1.07062
\(898\) 57.1818 1.90818
\(899\) 21.7238 0.724531
\(900\) −12.9347 −0.431155
\(901\) −4.19720 −0.139829
\(902\) 14.8666 0.495005
\(903\) 36.6591 1.21994
\(904\) 10.8976 0.362448
\(905\) −15.9347 −0.529687
\(906\) 9.61934 0.319581
\(907\) −29.1769 −0.968804 −0.484402 0.874845i \(-0.660963\pi\)
−0.484402 + 0.874845i \(0.660963\pi\)
\(908\) 13.4400 0.446022
\(909\) 21.7171 0.720310
\(910\) 21.0491 0.697771
\(911\) −39.5502 −1.31036 −0.655178 0.755474i \(-0.727406\pi\)
−0.655178 + 0.755474i \(0.727406\pi\)
\(912\) −4.49010 −0.148682
\(913\) −0.134370 −0.00444700
\(914\) 12.9291 0.427655
\(915\) 31.0223 1.02557
\(916\) −50.9924 −1.68483
\(917\) 9.40518 0.310586
\(918\) 6.96015 0.229719
\(919\) −5.00651 −0.165149 −0.0825747 0.996585i \(-0.526314\pi\)
−0.0825747 + 0.996585i \(0.526314\pi\)
\(920\) −4.48118 −0.147740
\(921\) 34.7397 1.14471
\(922\) −32.1880 −1.06005
\(923\) 58.2081 1.91594
\(924\) −38.3930 −1.26304
\(925\) −15.2575 −0.501664
\(926\) −10.2485 −0.336788
\(927\) 11.8901 0.390522
\(928\) 34.4531 1.13098
\(929\) −52.8100 −1.73264 −0.866320 0.499490i \(-0.833521\pi\)
−0.866320 + 0.499490i \(0.833521\pi\)
\(930\) 28.1075 0.921680
\(931\) 4.22427 0.138445
\(932\) 7.08334 0.232022
\(933\) −52.6905 −1.72501
\(934\) 36.7079 1.20112
\(935\) −6.10866 −0.199775
\(936\) 9.98316 0.326310
\(937\) −25.5732 −0.835439 −0.417720 0.908576i \(-0.637171\pi\)
−0.417720 + 0.908576i \(0.637171\pi\)
\(938\) −34.0718 −1.11248
\(939\) 59.6638 1.94705
\(940\) 30.4668 0.993718
\(941\) 15.2943 0.498581 0.249290 0.968429i \(-0.419803\pi\)
0.249290 + 0.968429i \(0.419803\pi\)
\(942\) 18.1653 0.591858
\(943\) −4.02291 −0.131004
\(944\) 28.6158 0.931366
\(945\) 5.91423 0.192390
\(946\) 124.760 4.05630
\(947\) 24.4424 0.794273 0.397136 0.917760i \(-0.370004\pi\)
0.397136 + 0.917760i \(0.370004\pi\)
\(948\) 65.4846 2.12684
\(949\) 28.4869 0.924723
\(950\) −6.29599 −0.204269
\(951\) 58.0108 1.88113
\(952\) 1.82964 0.0592988
\(953\) −50.6478 −1.64064 −0.820321 0.571903i \(-0.806205\pi\)
−0.820321 + 0.571903i \(0.806205\pi\)
\(954\) −13.1487 −0.425706
\(955\) 28.7437 0.930123
\(956\) 15.5651 0.503411
\(957\) −45.1375 −1.45909
\(958\) −49.1175 −1.58691
\(959\) −4.02490 −0.129971
\(960\) 31.4383 1.01467
\(961\) −6.66869 −0.215119
\(962\) 51.6575 1.66550
\(963\) 17.1764 0.553502
\(964\) −11.1257 −0.358334
\(965\) 22.2702 0.716904
\(966\) 18.4099 0.592330
\(967\) −26.7278 −0.859507 −0.429754 0.902946i \(-0.641400\pi\)
−0.429754 + 0.902946i \(0.641400\pi\)
\(968\) −15.8677 −0.510008
\(969\) −1.81774 −0.0583942
\(970\) −5.13524 −0.164883
\(971\) −20.2491 −0.649824 −0.324912 0.945744i \(-0.605335\pi\)
−0.324912 + 0.945744i \(0.605335\pi\)
\(972\) 36.3083 1.16459
\(973\) −16.6171 −0.532719
\(974\) 19.3782 0.620918
\(975\) −38.9254 −1.24661
\(976\) 28.8133 0.922290
\(977\) −13.0510 −0.417538 −0.208769 0.977965i \(-0.566946\pi\)
−0.208769 + 0.977965i \(0.566946\pi\)
\(978\) 55.2149 1.76558
\(979\) −75.0485 −2.39856
\(980\) −16.0110 −0.511453
\(981\) −11.0531 −0.352898
\(982\) 18.0586 0.576275
\(983\) −27.7124 −0.883888 −0.441944 0.897043i \(-0.645711\pi\)
−0.441944 + 0.897043i \(0.645711\pi\)
\(984\) −3.82233 −0.121852
\(985\) 11.1758 0.356089
\(986\) 9.43596 0.300502
\(987\) −28.5332 −0.908221
\(988\) 12.0293 0.382703
\(989\) −33.7600 −1.07351
\(990\) −19.1368 −0.608208
\(991\) 57.4191 1.82398 0.911988 0.410216i \(-0.134547\pi\)
0.911988 + 0.410216i \(0.134547\pi\)
\(992\) 38.5884 1.22518
\(993\) 9.52141 0.302153
\(994\) 33.4198 1.06001
\(995\) 26.8357 0.850749
\(996\) 0.151550 0.00480203
\(997\) −11.8212 −0.374380 −0.187190 0.982324i \(-0.559938\pi\)
−0.187190 + 0.982324i \(0.559938\pi\)
\(998\) −15.9941 −0.506286
\(999\) 14.5144 0.459214
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 6001.2.a.a.1.19 113
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6001.2.a.a.1.19 113 1.1 even 1 trivial