Properties

Label 2001.2.a.n.1.13
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $16$
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] = [2001,2,Mod(1,2001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.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: \(15.9780654445\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + 5778 x^{8} - 5124 x^{7} - 9405 x^{6} + 8288 x^{5} + 6405 x^{4} - 6032 x^{3} + \cdots - 192 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(-2.00482\) of defining polynomial
Character \(\chi\) \(=\) 2001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00482 q^{2} +1.00000 q^{3} +2.01932 q^{4} +3.94433 q^{5} +2.00482 q^{6} -2.43681 q^{7} +0.0387322 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.00482 q^{2} +1.00000 q^{3} +2.01932 q^{4} +3.94433 q^{5} +2.00482 q^{6} -2.43681 q^{7} +0.0387322 q^{8} +1.00000 q^{9} +7.90768 q^{10} +5.06353 q^{11} +2.01932 q^{12} +0.849677 q^{13} -4.88537 q^{14} +3.94433 q^{15} -3.96099 q^{16} -3.54469 q^{17} +2.00482 q^{18} +2.59053 q^{19} +7.96485 q^{20} -2.43681 q^{21} +10.1515 q^{22} -1.00000 q^{23} +0.0387322 q^{24} +10.5577 q^{25} +1.70345 q^{26} +1.00000 q^{27} -4.92069 q^{28} -1.00000 q^{29} +7.90768 q^{30} -0.555799 q^{31} -8.01855 q^{32} +5.06353 q^{33} -7.10648 q^{34} -9.61156 q^{35} +2.01932 q^{36} -2.47840 q^{37} +5.19356 q^{38} +0.849677 q^{39} +0.152773 q^{40} +2.45770 q^{41} -4.88537 q^{42} -7.48299 q^{43} +10.2249 q^{44} +3.94433 q^{45} -2.00482 q^{46} +4.02053 q^{47} -3.96099 q^{48} -1.06198 q^{49} +21.1663 q^{50} -3.54469 q^{51} +1.71577 q^{52} +3.73104 q^{53} +2.00482 q^{54} +19.9722 q^{55} -0.0943829 q^{56} +2.59053 q^{57} -2.00482 q^{58} +9.66773 q^{59} +7.96485 q^{60} -6.98014 q^{61} -1.11428 q^{62} -2.43681 q^{63} -8.15380 q^{64} +3.35140 q^{65} +10.1515 q^{66} -13.0703 q^{67} -7.15786 q^{68} -1.00000 q^{69} -19.2695 q^{70} +6.82458 q^{71} +0.0387322 q^{72} -15.2749 q^{73} -4.96875 q^{74} +10.5577 q^{75} +5.23112 q^{76} -12.3388 q^{77} +1.70345 q^{78} +8.57302 q^{79} -15.6234 q^{80} +1.00000 q^{81} +4.92726 q^{82} +8.09331 q^{83} -4.92069 q^{84} -13.9814 q^{85} -15.0021 q^{86} -1.00000 q^{87} +0.196122 q^{88} +3.09574 q^{89} +7.90768 q^{90} -2.07050 q^{91} -2.01932 q^{92} -0.555799 q^{93} +8.06045 q^{94} +10.2179 q^{95} -8.01855 q^{96} +0.381632 q^{97} -2.12908 q^{98} +5.06353 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9} + 11 q^{10} + 8 q^{11} + 25 q^{12} + 19 q^{13} + 16 q^{14} + 3 q^{15} + 31 q^{16} - 4 q^{17} - q^{18} + 19 q^{19} + 16 q^{20} + 13 q^{21} + 6 q^{22} - 16 q^{23} + 23 q^{25} - 15 q^{26} + 16 q^{27} + 18 q^{28} - 16 q^{29} + 11 q^{30} + 24 q^{31} - 21 q^{32} + 8 q^{33} - 9 q^{34} - 13 q^{35} + 25 q^{36} + 26 q^{37} + 19 q^{39} - 22 q^{40} - 15 q^{41} + 16 q^{42} + 33 q^{43} + 6 q^{44} + 3 q^{45} + q^{46} + 13 q^{47} + 31 q^{48} + 41 q^{49} + 13 q^{50} - 4 q^{51} - 26 q^{52} + 5 q^{53} - q^{54} + 9 q^{55} + 40 q^{56} + 19 q^{57} + q^{58} + 2 q^{59} + 16 q^{60} + 29 q^{61} - 32 q^{62} + 13 q^{63} + 28 q^{64} + 18 q^{65} + 6 q^{66} + 32 q^{67} - 26 q^{68} - 16 q^{69} + 18 q^{70} + 29 q^{71} + 19 q^{73} - 16 q^{74} + 23 q^{75} + 64 q^{76} - 21 q^{77} - 15 q^{78} + 56 q^{79} + 16 q^{81} + 14 q^{82} + 5 q^{83} + 18 q^{84} + 16 q^{85} - 20 q^{86} - 16 q^{87} + q^{88} + 7 q^{89} + 11 q^{90} - 6 q^{91} - 25 q^{92} + 24 q^{93} - 11 q^{94} + 39 q^{95} - 21 q^{96} + 35 q^{97} - 109 q^{98} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00482 1.41762 0.708812 0.705397i \(-0.249231\pi\)
0.708812 + 0.705397i \(0.249231\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.01932 1.00966
\(5\) 3.94433 1.76396 0.881978 0.471290i \(-0.156212\pi\)
0.881978 + 0.471290i \(0.156212\pi\)
\(6\) 2.00482 0.818466
\(7\) −2.43681 −0.921026 −0.460513 0.887653i \(-0.652335\pi\)
−0.460513 + 0.887653i \(0.652335\pi\)
\(8\) 0.0387322 0.0136939
\(9\) 1.00000 0.333333
\(10\) 7.90768 2.50063
\(11\) 5.06353 1.52671 0.763356 0.645977i \(-0.223550\pi\)
0.763356 + 0.645977i \(0.223550\pi\)
\(12\) 2.01932 0.582927
\(13\) 0.849677 0.235658 0.117829 0.993034i \(-0.462407\pi\)
0.117829 + 0.993034i \(0.462407\pi\)
\(14\) −4.88537 −1.30567
\(15\) 3.94433 1.01842
\(16\) −3.96099 −0.990247
\(17\) −3.54469 −0.859713 −0.429857 0.902897i \(-0.641436\pi\)
−0.429857 + 0.902897i \(0.641436\pi\)
\(18\) 2.00482 0.472542
\(19\) 2.59053 0.594309 0.297155 0.954829i \(-0.403962\pi\)
0.297155 + 0.954829i \(0.403962\pi\)
\(20\) 7.96485 1.78100
\(21\) −2.43681 −0.531755
\(22\) 10.1515 2.16431
\(23\) −1.00000 −0.208514
\(24\) 0.0387322 0.00790619
\(25\) 10.5577 2.11154
\(26\) 1.70345 0.334075
\(27\) 1.00000 0.192450
\(28\) −4.92069 −0.929923
\(29\) −1.00000 −0.185695
\(30\) 7.90768 1.44374
\(31\) −0.555799 −0.0998244 −0.0499122 0.998754i \(-0.515894\pi\)
−0.0499122 + 0.998754i \(0.515894\pi\)
\(32\) −8.01855 −1.41749
\(33\) 5.06353 0.881448
\(34\) −7.10648 −1.21875
\(35\) −9.61156 −1.62465
\(36\) 2.01932 0.336553
\(37\) −2.47840 −0.407446 −0.203723 0.979029i \(-0.565304\pi\)
−0.203723 + 0.979029i \(0.565304\pi\)
\(38\) 5.19356 0.842507
\(39\) 0.849677 0.136057
\(40\) 0.152773 0.0241555
\(41\) 2.45770 0.383829 0.191914 0.981412i \(-0.438530\pi\)
0.191914 + 0.981412i \(0.438530\pi\)
\(42\) −4.88537 −0.753828
\(43\) −7.48299 −1.14114 −0.570572 0.821247i \(-0.693279\pi\)
−0.570572 + 0.821247i \(0.693279\pi\)
\(44\) 10.2249 1.54146
\(45\) 3.94433 0.587985
\(46\) −2.00482 −0.295595
\(47\) 4.02053 0.586454 0.293227 0.956043i \(-0.405271\pi\)
0.293227 + 0.956043i \(0.405271\pi\)
\(48\) −3.96099 −0.571719
\(49\) −1.06198 −0.151711
\(50\) 21.1663 2.99337
\(51\) −3.54469 −0.496356
\(52\) 1.71577 0.237934
\(53\) 3.73104 0.512498 0.256249 0.966611i \(-0.417513\pi\)
0.256249 + 0.966611i \(0.417513\pi\)
\(54\) 2.00482 0.272822
\(55\) 19.9722 2.69305
\(56\) −0.0943829 −0.0126125
\(57\) 2.59053 0.343125
\(58\) −2.00482 −0.263246
\(59\) 9.66773 1.25863 0.629316 0.777150i \(-0.283335\pi\)
0.629316 + 0.777150i \(0.283335\pi\)
\(60\) 7.96485 1.02826
\(61\) −6.98014 −0.893716 −0.446858 0.894605i \(-0.647457\pi\)
−0.446858 + 0.894605i \(0.647457\pi\)
\(62\) −1.11428 −0.141514
\(63\) −2.43681 −0.307009
\(64\) −8.15380 −1.01923
\(65\) 3.35140 0.415691
\(66\) 10.1515 1.24956
\(67\) −13.0703 −1.59679 −0.798394 0.602135i \(-0.794317\pi\)
−0.798394 + 0.602135i \(0.794317\pi\)
\(68\) −7.15786 −0.868018
\(69\) −1.00000 −0.120386
\(70\) −19.2695 −2.30314
\(71\) 6.82458 0.809929 0.404964 0.914332i \(-0.367284\pi\)
0.404964 + 0.914332i \(0.367284\pi\)
\(72\) 0.0387322 0.00456464
\(73\) −15.2749 −1.78779 −0.893894 0.448279i \(-0.852037\pi\)
−0.893894 + 0.448279i \(0.852037\pi\)
\(74\) −4.96875 −0.577606
\(75\) 10.5577 1.21910
\(76\) 5.23112 0.600050
\(77\) −12.3388 −1.40614
\(78\) 1.70345 0.192878
\(79\) 8.57302 0.964541 0.482270 0.876022i \(-0.339812\pi\)
0.482270 + 0.876022i \(0.339812\pi\)
\(80\) −15.6234 −1.74675
\(81\) 1.00000 0.111111
\(82\) 4.92726 0.544125
\(83\) 8.09331 0.888356 0.444178 0.895939i \(-0.353496\pi\)
0.444178 + 0.895939i \(0.353496\pi\)
\(84\) −4.92069 −0.536891
\(85\) −13.9814 −1.51650
\(86\) −15.0021 −1.61771
\(87\) −1.00000 −0.107211
\(88\) 0.196122 0.0209067
\(89\) 3.09574 0.328147 0.164074 0.986448i \(-0.447537\pi\)
0.164074 + 0.986448i \(0.447537\pi\)
\(90\) 7.90768 0.833543
\(91\) −2.07050 −0.217047
\(92\) −2.01932 −0.210529
\(93\) −0.555799 −0.0576336
\(94\) 8.06045 0.831372
\(95\) 10.2179 1.04834
\(96\) −8.01855 −0.818390
\(97\) 0.381632 0.0387488 0.0193744 0.999812i \(-0.493833\pi\)
0.0193744 + 0.999812i \(0.493833\pi\)
\(98\) −2.12908 −0.215070
\(99\) 5.06353 0.508904
\(100\) 21.3194 2.13194
\(101\) −7.72348 −0.768514 −0.384257 0.923226i \(-0.625542\pi\)
−0.384257 + 0.923226i \(0.625542\pi\)
\(102\) −7.10648 −0.703646
\(103\) −12.1128 −1.19351 −0.596756 0.802423i \(-0.703544\pi\)
−0.596756 + 0.802423i \(0.703544\pi\)
\(104\) 0.0329099 0.00322708
\(105\) −9.61156 −0.937992
\(106\) 7.48008 0.726529
\(107\) −12.7158 −1.22928 −0.614640 0.788808i \(-0.710699\pi\)
−0.614640 + 0.788808i \(0.710699\pi\)
\(108\) 2.01932 0.194309
\(109\) −4.06457 −0.389315 −0.194657 0.980871i \(-0.562360\pi\)
−0.194657 + 0.980871i \(0.562360\pi\)
\(110\) 40.0408 3.81774
\(111\) −2.47840 −0.235239
\(112\) 9.65216 0.912043
\(113\) 10.3389 0.972601 0.486301 0.873792i \(-0.338346\pi\)
0.486301 + 0.873792i \(0.338346\pi\)
\(114\) 5.19356 0.486422
\(115\) −3.94433 −0.367810
\(116\) −2.01932 −0.187489
\(117\) 0.849677 0.0785527
\(118\) 19.3821 1.78427
\(119\) 8.63772 0.791818
\(120\) 0.152773 0.0139462
\(121\) 14.6394 1.33085
\(122\) −13.9940 −1.26695
\(123\) 2.45770 0.221604
\(124\) −1.12234 −0.100789
\(125\) 21.9214 1.96071
\(126\) −4.88537 −0.435223
\(127\) 4.63744 0.411507 0.205753 0.978604i \(-0.434036\pi\)
0.205753 + 0.978604i \(0.434036\pi\)
\(128\) −0.309843 −0.0273866
\(129\) −7.48299 −0.658840
\(130\) 6.71898 0.589293
\(131\) −11.3178 −0.988838 −0.494419 0.869224i \(-0.664619\pi\)
−0.494419 + 0.869224i \(0.664619\pi\)
\(132\) 10.2249 0.889963
\(133\) −6.31263 −0.547374
\(134\) −26.2036 −2.26365
\(135\) 3.94433 0.339474
\(136\) −0.137294 −0.0117728
\(137\) 8.44177 0.721229 0.360614 0.932715i \(-0.382567\pi\)
0.360614 + 0.932715i \(0.382567\pi\)
\(138\) −2.00482 −0.170662
\(139\) 0.788881 0.0669120 0.0334560 0.999440i \(-0.489349\pi\)
0.0334560 + 0.999440i \(0.489349\pi\)
\(140\) −19.4088 −1.64034
\(141\) 4.02053 0.338590
\(142\) 13.6821 1.14817
\(143\) 4.30237 0.359782
\(144\) −3.96099 −0.330082
\(145\) −3.94433 −0.327558
\(146\) −30.6234 −2.53441
\(147\) −1.06198 −0.0875905
\(148\) −5.00468 −0.411382
\(149\) −7.78451 −0.637732 −0.318866 0.947800i \(-0.603302\pi\)
−0.318866 + 0.947800i \(0.603302\pi\)
\(150\) 21.1663 1.72822
\(151\) −19.2856 −1.56944 −0.784719 0.619851i \(-0.787193\pi\)
−0.784719 + 0.619851i \(0.787193\pi\)
\(152\) 0.100337 0.00813842
\(153\) −3.54469 −0.286571
\(154\) −24.7372 −1.99338
\(155\) −2.19225 −0.176086
\(156\) 1.71577 0.137372
\(157\) −11.9533 −0.953979 −0.476989 0.878909i \(-0.658272\pi\)
−0.476989 + 0.878909i \(0.658272\pi\)
\(158\) 17.1874 1.36736
\(159\) 3.73104 0.295891
\(160\) −31.6278 −2.50039
\(161\) 2.43681 0.192047
\(162\) 2.00482 0.157514
\(163\) 13.9184 1.09018 0.545088 0.838379i \(-0.316496\pi\)
0.545088 + 0.838379i \(0.316496\pi\)
\(164\) 4.96289 0.387536
\(165\) 19.9722 1.55484
\(166\) 16.2257 1.25936
\(167\) 11.6350 0.900343 0.450171 0.892942i \(-0.351363\pi\)
0.450171 + 0.892942i \(0.351363\pi\)
\(168\) −0.0943829 −0.00728180
\(169\) −12.2780 −0.944465
\(170\) −28.0303 −2.14982
\(171\) 2.59053 0.198103
\(172\) −15.1105 −1.15217
\(173\) −15.0093 −1.14114 −0.570569 0.821250i \(-0.693277\pi\)
−0.570569 + 0.821250i \(0.693277\pi\)
\(174\) −2.00482 −0.151985
\(175\) −25.7271 −1.94478
\(176\) −20.0566 −1.51182
\(177\) 9.66773 0.726671
\(178\) 6.20641 0.465190
\(179\) 0.558696 0.0417589 0.0208794 0.999782i \(-0.493353\pi\)
0.0208794 + 0.999782i \(0.493353\pi\)
\(180\) 7.96485 0.593665
\(181\) 13.6021 1.01104 0.505518 0.862816i \(-0.331301\pi\)
0.505518 + 0.862816i \(0.331301\pi\)
\(182\) −4.15098 −0.307691
\(183\) −6.98014 −0.515987
\(184\) −0.0387322 −0.00285538
\(185\) −9.77562 −0.718718
\(186\) −1.11428 −0.0817029
\(187\) −17.9486 −1.31254
\(188\) 8.11873 0.592119
\(189\) −2.43681 −0.177252
\(190\) 20.4851 1.48615
\(191\) 18.5494 1.34219 0.671093 0.741373i \(-0.265825\pi\)
0.671093 + 0.741373i \(0.265825\pi\)
\(192\) −8.15380 −0.588450
\(193\) 2.30044 0.165589 0.0827947 0.996567i \(-0.473615\pi\)
0.0827947 + 0.996567i \(0.473615\pi\)
\(194\) 0.765104 0.0549313
\(195\) 3.35140 0.239999
\(196\) −2.14447 −0.153177
\(197\) −23.7251 −1.69035 −0.845173 0.534492i \(-0.820503\pi\)
−0.845173 + 0.534492i \(0.820503\pi\)
\(198\) 10.1515 0.721435
\(199\) 10.1146 0.717005 0.358503 0.933529i \(-0.383287\pi\)
0.358503 + 0.933529i \(0.383287\pi\)
\(200\) 0.408924 0.0289153
\(201\) −13.0703 −0.921906
\(202\) −15.4842 −1.08947
\(203\) 2.43681 0.171030
\(204\) −7.15786 −0.501150
\(205\) 9.69398 0.677057
\(206\) −24.2841 −1.69195
\(207\) −1.00000 −0.0695048
\(208\) −3.36556 −0.233360
\(209\) 13.1173 0.907340
\(210\) −19.2695 −1.32972
\(211\) −21.6952 −1.49356 −0.746778 0.665073i \(-0.768400\pi\)
−0.746778 + 0.665073i \(0.768400\pi\)
\(212\) 7.53416 0.517448
\(213\) 6.82458 0.467613
\(214\) −25.4929 −1.74266
\(215\) −29.5153 −2.01293
\(216\) 0.0387322 0.00263540
\(217\) 1.35437 0.0919409
\(218\) −8.14874 −0.551902
\(219\) −15.2749 −1.03218
\(220\) 40.3303 2.71907
\(221\) −3.01184 −0.202598
\(222\) −4.96875 −0.333481
\(223\) −9.48165 −0.634938 −0.317469 0.948269i \(-0.602833\pi\)
−0.317469 + 0.948269i \(0.602833\pi\)
\(224\) 19.5396 1.30555
\(225\) 10.5577 0.703847
\(226\) 20.7277 1.37878
\(227\) 19.8447 1.31714 0.658568 0.752521i \(-0.271162\pi\)
0.658568 + 0.752521i \(0.271162\pi\)
\(228\) 5.23112 0.346439
\(229\) 19.1534 1.26569 0.632847 0.774277i \(-0.281886\pi\)
0.632847 + 0.774277i \(0.281886\pi\)
\(230\) −7.90768 −0.521417
\(231\) −12.3388 −0.811837
\(232\) −0.0387322 −0.00254290
\(233\) 0.846581 0.0554613 0.0277307 0.999615i \(-0.491172\pi\)
0.0277307 + 0.999615i \(0.491172\pi\)
\(234\) 1.70345 0.111358
\(235\) 15.8583 1.03448
\(236\) 19.5222 1.27079
\(237\) 8.57302 0.556878
\(238\) 17.3171 1.12250
\(239\) −0.968657 −0.0626572 −0.0313286 0.999509i \(-0.509974\pi\)
−0.0313286 + 0.999509i \(0.509974\pi\)
\(240\) −15.6234 −1.00849
\(241\) −1.03734 −0.0668212 −0.0334106 0.999442i \(-0.510637\pi\)
−0.0334106 + 0.999442i \(0.510637\pi\)
\(242\) 29.3494 1.88665
\(243\) 1.00000 0.0641500
\(244\) −14.0951 −0.902349
\(245\) −4.18879 −0.267612
\(246\) 4.92726 0.314151
\(247\) 2.20112 0.140054
\(248\) −0.0215273 −0.00136699
\(249\) 8.09331 0.512892
\(250\) 43.9486 2.77955
\(251\) −27.7696 −1.75280 −0.876399 0.481585i \(-0.840061\pi\)
−0.876399 + 0.481585i \(0.840061\pi\)
\(252\) −4.92069 −0.309974
\(253\) −5.06353 −0.318342
\(254\) 9.29726 0.583362
\(255\) −13.9814 −0.875550
\(256\) 15.6864 0.980401
\(257\) −15.0573 −0.939251 −0.469625 0.882866i \(-0.655611\pi\)
−0.469625 + 0.882866i \(0.655611\pi\)
\(258\) −15.0021 −0.933988
\(259\) 6.03938 0.375269
\(260\) 6.76756 0.419706
\(261\) −1.00000 −0.0618984
\(262\) −22.6901 −1.40180
\(263\) 16.2360 1.00115 0.500577 0.865692i \(-0.333121\pi\)
0.500577 + 0.865692i \(0.333121\pi\)
\(264\) 0.196122 0.0120705
\(265\) 14.7164 0.904023
\(266\) −12.6557 −0.775971
\(267\) 3.09574 0.189456
\(268\) −26.3931 −1.61221
\(269\) 19.5471 1.19180 0.595902 0.803057i \(-0.296795\pi\)
0.595902 + 0.803057i \(0.296795\pi\)
\(270\) 7.90768 0.481246
\(271\) 16.0743 0.976444 0.488222 0.872719i \(-0.337646\pi\)
0.488222 + 0.872719i \(0.337646\pi\)
\(272\) 14.0405 0.851328
\(273\) −2.07050 −0.125312
\(274\) 16.9243 1.02243
\(275\) 53.4593 3.22372
\(276\) −2.01932 −0.121549
\(277\) 11.5689 0.695105 0.347553 0.937660i \(-0.387013\pi\)
0.347553 + 0.937660i \(0.387013\pi\)
\(278\) 1.58157 0.0948561
\(279\) −0.555799 −0.0332748
\(280\) −0.372277 −0.0222478
\(281\) 1.08577 0.0647716 0.0323858 0.999475i \(-0.489689\pi\)
0.0323858 + 0.999475i \(0.489689\pi\)
\(282\) 8.06045 0.479993
\(283\) 10.9798 0.652680 0.326340 0.945252i \(-0.394185\pi\)
0.326340 + 0.945252i \(0.394185\pi\)
\(284\) 13.7810 0.817752
\(285\) 10.2179 0.605257
\(286\) 8.62549 0.510036
\(287\) −5.98894 −0.353516
\(288\) −8.01855 −0.472497
\(289\) −4.43519 −0.260893
\(290\) −7.90768 −0.464355
\(291\) 0.381632 0.0223716
\(292\) −30.8448 −1.80506
\(293\) 3.35385 0.195934 0.0979670 0.995190i \(-0.468766\pi\)
0.0979670 + 0.995190i \(0.468766\pi\)
\(294\) −2.12908 −0.124170
\(295\) 38.1327 2.22017
\(296\) −0.0959940 −0.00557954
\(297\) 5.06353 0.293816
\(298\) −15.6066 −0.904064
\(299\) −0.849677 −0.0491381
\(300\) 21.3194 1.23088
\(301\) 18.2346 1.05102
\(302\) −38.6642 −2.22488
\(303\) −7.72348 −0.443702
\(304\) −10.2611 −0.588513
\(305\) −27.5320 −1.57648
\(306\) −7.10648 −0.406250
\(307\) 11.0025 0.627945 0.313972 0.949432i \(-0.398340\pi\)
0.313972 + 0.949432i \(0.398340\pi\)
\(308\) −24.9161 −1.41973
\(309\) −12.1128 −0.689074
\(310\) −4.39508 −0.249624
\(311\) 32.0112 1.81519 0.907594 0.419849i \(-0.137917\pi\)
0.907594 + 0.419849i \(0.137917\pi\)
\(312\) 0.0329099 0.00186316
\(313\) −13.3302 −0.753467 −0.376734 0.926322i \(-0.622953\pi\)
−0.376734 + 0.926322i \(0.622953\pi\)
\(314\) −23.9643 −1.35238
\(315\) −9.61156 −0.541550
\(316\) 17.3117 0.973858
\(317\) −2.22126 −0.124758 −0.0623792 0.998053i \(-0.519869\pi\)
−0.0623792 + 0.998053i \(0.519869\pi\)
\(318\) 7.48008 0.419462
\(319\) −5.06353 −0.283503
\(320\) −32.1613 −1.79787
\(321\) −12.7158 −0.709725
\(322\) 4.88537 0.272251
\(323\) −9.18263 −0.510935
\(324\) 2.01932 0.112184
\(325\) 8.97064 0.497602
\(326\) 27.9040 1.54546
\(327\) −4.06457 −0.224771
\(328\) 0.0951923 0.00525611
\(329\) −9.79725 −0.540140
\(330\) 40.0408 2.20417
\(331\) −6.00619 −0.330130 −0.165065 0.986283i \(-0.552783\pi\)
−0.165065 + 0.986283i \(0.552783\pi\)
\(332\) 16.3430 0.896937
\(333\) −2.47840 −0.135815
\(334\) 23.3261 1.27635
\(335\) −51.5534 −2.81666
\(336\) 9.65216 0.526568
\(337\) 2.98109 0.162390 0.0811952 0.996698i \(-0.474126\pi\)
0.0811952 + 0.996698i \(0.474126\pi\)
\(338\) −24.6153 −1.33890
\(339\) 10.3389 0.561532
\(340\) −28.2329 −1.53115
\(341\) −2.81431 −0.152403
\(342\) 5.19356 0.280836
\(343\) 19.6455 1.06076
\(344\) −0.289833 −0.0156267
\(345\) −3.94433 −0.212355
\(346\) −30.0911 −1.61771
\(347\) −32.2077 −1.72900 −0.864499 0.502634i \(-0.832364\pi\)
−0.864499 + 0.502634i \(0.832364\pi\)
\(348\) −2.01932 −0.108247
\(349\) 26.3675 1.41142 0.705711 0.708500i \(-0.250628\pi\)
0.705711 + 0.708500i \(0.250628\pi\)
\(350\) −51.5783 −2.75697
\(351\) 0.849677 0.0453524
\(352\) −40.6022 −2.16410
\(353\) −28.3671 −1.50983 −0.754914 0.655824i \(-0.772321\pi\)
−0.754914 + 0.655824i \(0.772321\pi\)
\(354\) 19.3821 1.03015
\(355\) 26.9184 1.42868
\(356\) 6.25128 0.331317
\(357\) 8.63772 0.457156
\(358\) 1.12009 0.0591984
\(359\) 33.8035 1.78408 0.892041 0.451955i \(-0.149273\pi\)
0.892041 + 0.451955i \(0.149273\pi\)
\(360\) 0.152773 0.00805182
\(361\) −12.2891 −0.646797
\(362\) 27.2698 1.43327
\(363\) 14.6394 0.768368
\(364\) −4.18100 −0.219144
\(365\) −60.2490 −3.15358
\(366\) −13.9940 −0.731476
\(367\) −25.5711 −1.33480 −0.667399 0.744700i \(-0.732593\pi\)
−0.667399 + 0.744700i \(0.732593\pi\)
\(368\) 3.96099 0.206481
\(369\) 2.45770 0.127943
\(370\) −19.5984 −1.01887
\(371\) −9.09182 −0.472024
\(372\) −1.12234 −0.0581904
\(373\) 0.179366 0.00928724 0.00464362 0.999989i \(-0.498522\pi\)
0.00464362 + 0.999989i \(0.498522\pi\)
\(374\) −35.9839 −1.86068
\(375\) 21.9214 1.13202
\(376\) 0.155724 0.00803086
\(377\) −0.849677 −0.0437606
\(378\) −4.88537 −0.251276
\(379\) −9.59910 −0.493073 −0.246536 0.969134i \(-0.579292\pi\)
−0.246536 + 0.969134i \(0.579292\pi\)
\(380\) 20.6332 1.05846
\(381\) 4.63744 0.237583
\(382\) 37.1882 1.90272
\(383\) 26.0551 1.33136 0.665678 0.746239i \(-0.268143\pi\)
0.665678 + 0.746239i \(0.268143\pi\)
\(384\) −0.309843 −0.0158116
\(385\) −48.6684 −2.48037
\(386\) 4.61198 0.234744
\(387\) −7.48299 −0.380382
\(388\) 0.770636 0.0391231
\(389\) −9.42577 −0.477905 −0.238953 0.971031i \(-0.576804\pi\)
−0.238953 + 0.971031i \(0.576804\pi\)
\(390\) 6.71898 0.340229
\(391\) 3.54469 0.179263
\(392\) −0.0411328 −0.00207752
\(393\) −11.3178 −0.570906
\(394\) −47.5647 −2.39628
\(395\) 33.8148 1.70141
\(396\) 10.2249 0.513820
\(397\) −7.25375 −0.364055 −0.182028 0.983293i \(-0.558266\pi\)
−0.182028 + 0.983293i \(0.558266\pi\)
\(398\) 20.2780 1.01644
\(399\) −6.31263 −0.316027
\(400\) −41.8189 −2.09095
\(401\) −4.37364 −0.218409 −0.109205 0.994019i \(-0.534830\pi\)
−0.109205 + 0.994019i \(0.534830\pi\)
\(402\) −26.2036 −1.30692
\(403\) −0.472250 −0.0235244
\(404\) −15.5962 −0.775938
\(405\) 3.94433 0.195995
\(406\) 4.88537 0.242457
\(407\) −12.5495 −0.622054
\(408\) −0.137294 −0.00679705
\(409\) −17.3026 −0.855556 −0.427778 0.903884i \(-0.640704\pi\)
−0.427778 + 0.903884i \(0.640704\pi\)
\(410\) 19.4347 0.959812
\(411\) 8.44177 0.416402
\(412\) −24.4597 −1.20504
\(413\) −23.5584 −1.15923
\(414\) −2.00482 −0.0985317
\(415\) 31.9226 1.56702
\(416\) −6.81318 −0.334044
\(417\) 0.788881 0.0386317
\(418\) 26.2978 1.28627
\(419\) 36.6204 1.78902 0.894512 0.447044i \(-0.147523\pi\)
0.894512 + 0.447044i \(0.147523\pi\)
\(420\) −19.4088 −0.947053
\(421\) −29.6292 −1.44404 −0.722019 0.691873i \(-0.756786\pi\)
−0.722019 + 0.691873i \(0.756786\pi\)
\(422\) −43.4950 −2.11730
\(423\) 4.02053 0.195485
\(424\) 0.144511 0.00701810
\(425\) −37.4238 −1.81532
\(426\) 13.6821 0.662899
\(427\) 17.0092 0.823135
\(428\) −25.6772 −1.24115
\(429\) 4.30237 0.207720
\(430\) −59.1731 −2.85358
\(431\) 12.4234 0.598413 0.299207 0.954188i \(-0.403278\pi\)
0.299207 + 0.954188i \(0.403278\pi\)
\(432\) −3.96099 −0.190573
\(433\) −37.4749 −1.80093 −0.900463 0.434932i \(-0.856772\pi\)
−0.900463 + 0.434932i \(0.856772\pi\)
\(434\) 2.71528 0.130338
\(435\) −3.94433 −0.189116
\(436\) −8.20766 −0.393076
\(437\) −2.59053 −0.123922
\(438\) −30.6234 −1.46324
\(439\) −33.9757 −1.62157 −0.810787 0.585342i \(-0.800960\pi\)
−0.810787 + 0.585342i \(0.800960\pi\)
\(440\) 0.773569 0.0368785
\(441\) −1.06198 −0.0505704
\(442\) −6.03821 −0.287208
\(443\) −14.2272 −0.675954 −0.337977 0.941154i \(-0.609743\pi\)
−0.337977 + 0.941154i \(0.609743\pi\)
\(444\) −5.00468 −0.237512
\(445\) 12.2106 0.578838
\(446\) −19.0090 −0.900104
\(447\) −7.78451 −0.368195
\(448\) 19.8692 0.938733
\(449\) 23.8102 1.12367 0.561835 0.827249i \(-0.310095\pi\)
0.561835 + 0.827249i \(0.310095\pi\)
\(450\) 21.1663 0.997791
\(451\) 12.4447 0.585996
\(452\) 20.8775 0.981996
\(453\) −19.2856 −0.906116
\(454\) 39.7850 1.86720
\(455\) −8.16672 −0.382862
\(456\) 0.100337 0.00469872
\(457\) 9.04233 0.422982 0.211491 0.977380i \(-0.432168\pi\)
0.211491 + 0.977380i \(0.432168\pi\)
\(458\) 38.3992 1.79428
\(459\) −3.54469 −0.165452
\(460\) −7.96485 −0.371363
\(461\) −24.9696 −1.16295 −0.581475 0.813565i \(-0.697524\pi\)
−0.581475 + 0.813565i \(0.697524\pi\)
\(462\) −24.7372 −1.15088
\(463\) 30.2988 1.40810 0.704051 0.710149i \(-0.251372\pi\)
0.704051 + 0.710149i \(0.251372\pi\)
\(464\) 3.96099 0.183884
\(465\) −2.19225 −0.101663
\(466\) 1.69725 0.0786234
\(467\) 35.3513 1.63586 0.817930 0.575317i \(-0.195121\pi\)
0.817930 + 0.575317i \(0.195121\pi\)
\(468\) 1.71577 0.0793115
\(469\) 31.8497 1.47068
\(470\) 31.7930 1.46650
\(471\) −11.9533 −0.550780
\(472\) 0.374453 0.0172356
\(473\) −37.8904 −1.74220
\(474\) 17.1874 0.789444
\(475\) 27.3501 1.25491
\(476\) 17.4423 0.799467
\(477\) 3.73104 0.170833
\(478\) −1.94199 −0.0888244
\(479\) 10.1693 0.464648 0.232324 0.972638i \(-0.425367\pi\)
0.232324 + 0.972638i \(0.425367\pi\)
\(480\) −31.6278 −1.44360
\(481\) −2.10584 −0.0960180
\(482\) −2.07969 −0.0947274
\(483\) 2.43681 0.110878
\(484\) 29.5616 1.34371
\(485\) 1.50528 0.0683512
\(486\) 2.00482 0.0909407
\(487\) −6.68801 −0.303063 −0.151531 0.988452i \(-0.548420\pi\)
−0.151531 + 0.988452i \(0.548420\pi\)
\(488\) −0.270357 −0.0122385
\(489\) 13.9184 0.629414
\(490\) −8.39779 −0.379373
\(491\) −9.85750 −0.444863 −0.222431 0.974948i \(-0.571399\pi\)
−0.222431 + 0.974948i \(0.571399\pi\)
\(492\) 4.96289 0.223744
\(493\) 3.54469 0.159645
\(494\) 4.41285 0.198544
\(495\) 19.9722 0.897685
\(496\) 2.20151 0.0988508
\(497\) −16.6302 −0.745965
\(498\) 16.2257 0.727089
\(499\) 22.8453 1.02270 0.511348 0.859374i \(-0.329146\pi\)
0.511348 + 0.859374i \(0.329146\pi\)
\(500\) 44.2663 1.97965
\(501\) 11.6350 0.519813
\(502\) −55.6731 −2.48481
\(503\) 6.17445 0.275305 0.137652 0.990481i \(-0.456044\pi\)
0.137652 + 0.990481i \(0.456044\pi\)
\(504\) −0.0943829 −0.00420415
\(505\) −30.4639 −1.35563
\(506\) −10.1515 −0.451289
\(507\) −12.2780 −0.545287
\(508\) 9.36448 0.415482
\(509\) 15.9641 0.707595 0.353798 0.935322i \(-0.384890\pi\)
0.353798 + 0.935322i \(0.384890\pi\)
\(510\) −28.0303 −1.24120
\(511\) 37.2219 1.64660
\(512\) 32.0682 1.41723
\(513\) 2.59053 0.114375
\(514\) −30.1873 −1.33151
\(515\) −47.7769 −2.10530
\(516\) −15.1105 −0.665204
\(517\) 20.3581 0.895347
\(518\) 12.1079 0.531990
\(519\) −15.0093 −0.658837
\(520\) 0.129807 0.00569243
\(521\) 31.0979 1.36243 0.681213 0.732086i \(-0.261453\pi\)
0.681213 + 0.732086i \(0.261453\pi\)
\(522\) −2.00482 −0.0877488
\(523\) 29.9735 1.31065 0.655325 0.755347i \(-0.272532\pi\)
0.655325 + 0.755347i \(0.272532\pi\)
\(524\) −22.8542 −0.998390
\(525\) −25.7271 −1.12282
\(526\) 32.5503 1.41926
\(527\) 1.97013 0.0858204
\(528\) −20.0566 −0.872851
\(529\) 1.00000 0.0434783
\(530\) 29.5039 1.28157
\(531\) 9.66773 0.419544
\(532\) −12.7472 −0.552662
\(533\) 2.08825 0.0904523
\(534\) 6.20641 0.268578
\(535\) −50.1552 −2.16840
\(536\) −0.506241 −0.0218663
\(537\) 0.558696 0.0241095
\(538\) 39.1884 1.68953
\(539\) −5.37736 −0.231619
\(540\) 7.96485 0.342753
\(541\) −25.1201 −1.08000 −0.539998 0.841666i \(-0.681575\pi\)
−0.539998 + 0.841666i \(0.681575\pi\)
\(542\) 32.2261 1.38423
\(543\) 13.6021 0.583722
\(544\) 28.4233 1.21864
\(545\) −16.0320 −0.686734
\(546\) −4.15098 −0.177646
\(547\) 29.3028 1.25290 0.626448 0.779463i \(-0.284508\pi\)
0.626448 + 0.779463i \(0.284508\pi\)
\(548\) 17.0466 0.728196
\(549\) −6.98014 −0.297905
\(550\) 107.177 4.57002
\(551\) −2.59053 −0.110360
\(552\) −0.0387322 −0.00164855
\(553\) −20.8908 −0.888367
\(554\) 23.1935 0.985398
\(555\) −9.77562 −0.414952
\(556\) 1.59300 0.0675584
\(557\) 15.0301 0.636846 0.318423 0.947949i \(-0.396847\pi\)
0.318423 + 0.947949i \(0.396847\pi\)
\(558\) −1.11428 −0.0471712
\(559\) −6.35812 −0.268920
\(560\) 38.0713 1.60880
\(561\) −17.9486 −0.757793
\(562\) 2.17678 0.0918218
\(563\) −28.7347 −1.21102 −0.605511 0.795837i \(-0.707031\pi\)
−0.605511 + 0.795837i \(0.707031\pi\)
\(564\) 8.11873 0.341860
\(565\) 40.7800 1.71563
\(566\) 22.0125 0.925256
\(567\) −2.43681 −0.102336
\(568\) 0.264331 0.0110911
\(569\) −13.5827 −0.569418 −0.284709 0.958614i \(-0.591897\pi\)
−0.284709 + 0.958614i \(0.591897\pi\)
\(570\) 20.4851 0.858027
\(571\) 1.44552 0.0604932 0.0302466 0.999542i \(-0.490371\pi\)
0.0302466 + 0.999542i \(0.490371\pi\)
\(572\) 8.68786 0.363258
\(573\) 18.5494 0.774911
\(574\) −12.0068 −0.501153
\(575\) −10.5577 −0.440287
\(576\) −8.15380 −0.339742
\(577\) 4.53867 0.188947 0.0944736 0.995527i \(-0.469883\pi\)
0.0944736 + 0.995527i \(0.469883\pi\)
\(578\) −8.89177 −0.369849
\(579\) 2.30044 0.0956031
\(580\) −7.96485 −0.330723
\(581\) −19.7218 −0.818199
\(582\) 0.765104 0.0317146
\(583\) 18.8922 0.782437
\(584\) −0.591630 −0.0244818
\(585\) 3.35140 0.138564
\(586\) 6.72388 0.277761
\(587\) −47.3036 −1.95243 −0.976214 0.216810i \(-0.930435\pi\)
−0.976214 + 0.216810i \(0.930435\pi\)
\(588\) −2.14447 −0.0884366
\(589\) −1.43982 −0.0593266
\(590\) 76.4493 3.14737
\(591\) −23.7251 −0.975922
\(592\) 9.81691 0.403473
\(593\) −8.55983 −0.351510 −0.175755 0.984434i \(-0.556237\pi\)
−0.175755 + 0.984434i \(0.556237\pi\)
\(594\) 10.1515 0.416521
\(595\) 34.0700 1.39673
\(596\) −15.7194 −0.643892
\(597\) 10.1146 0.413963
\(598\) −1.70345 −0.0696594
\(599\) −4.50642 −0.184127 −0.0920637 0.995753i \(-0.529346\pi\)
−0.0920637 + 0.995753i \(0.529346\pi\)
\(600\) 0.408924 0.0166942
\(601\) 21.1749 0.863742 0.431871 0.901935i \(-0.357854\pi\)
0.431871 + 0.901935i \(0.357854\pi\)
\(602\) 36.5571 1.48996
\(603\) −13.0703 −0.532263
\(604\) −38.9438 −1.58460
\(605\) 57.7425 2.34757
\(606\) −15.4842 −0.629003
\(607\) −31.5614 −1.28104 −0.640519 0.767942i \(-0.721281\pi\)
−0.640519 + 0.767942i \(0.721281\pi\)
\(608\) −20.7723 −0.842429
\(609\) 2.43681 0.0987443
\(610\) −55.1967 −2.23485
\(611\) 3.41615 0.138203
\(612\) −7.15786 −0.289339
\(613\) 17.3406 0.700379 0.350189 0.936679i \(-0.386117\pi\)
0.350189 + 0.936679i \(0.386117\pi\)
\(614\) 22.0580 0.890190
\(615\) 9.69398 0.390899
\(616\) −0.477911 −0.0192556
\(617\) −13.0619 −0.525853 −0.262927 0.964816i \(-0.584688\pi\)
−0.262927 + 0.964816i \(0.584688\pi\)
\(618\) −24.2841 −0.976849
\(619\) −34.4145 −1.38324 −0.691618 0.722263i \(-0.743102\pi\)
−0.691618 + 0.722263i \(0.743102\pi\)
\(620\) −4.42686 −0.177787
\(621\) −1.00000 −0.0401286
\(622\) 64.1768 2.57325
\(623\) −7.54371 −0.302232
\(624\) −3.36556 −0.134730
\(625\) 33.6766 1.34707
\(626\) −26.7247 −1.06813
\(627\) 13.1173 0.523853
\(628\) −24.1376 −0.963194
\(629\) 8.78515 0.350287
\(630\) −19.2695 −0.767714
\(631\) 40.9589 1.63055 0.815275 0.579075i \(-0.196586\pi\)
0.815275 + 0.579075i \(0.196586\pi\)
\(632\) 0.332052 0.0132083
\(633\) −21.6952 −0.862305
\(634\) −4.45324 −0.176861
\(635\) 18.2916 0.725880
\(636\) 7.53416 0.298749
\(637\) −0.902339 −0.0357520
\(638\) −10.1515 −0.401902
\(639\) 6.82458 0.269976
\(640\) −1.22212 −0.0483087
\(641\) −12.7161 −0.502255 −0.251127 0.967954i \(-0.580801\pi\)
−0.251127 + 0.967954i \(0.580801\pi\)
\(642\) −25.4929 −1.00612
\(643\) −1.86413 −0.0735143 −0.0367571 0.999324i \(-0.511703\pi\)
−0.0367571 + 0.999324i \(0.511703\pi\)
\(644\) 4.92069 0.193902
\(645\) −29.5153 −1.16217
\(646\) −18.4096 −0.724315
\(647\) 40.0695 1.57529 0.787647 0.616126i \(-0.211299\pi\)
0.787647 + 0.616126i \(0.211299\pi\)
\(648\) 0.0387322 0.00152155
\(649\) 48.9529 1.92157
\(650\) 17.9846 0.705413
\(651\) 1.35437 0.0530821
\(652\) 28.1058 1.10071
\(653\) −12.7936 −0.500651 −0.250326 0.968162i \(-0.580538\pi\)
−0.250326 + 0.968162i \(0.580538\pi\)
\(654\) −8.14874 −0.318641
\(655\) −44.6410 −1.74427
\(656\) −9.73493 −0.380085
\(657\) −15.2749 −0.595929
\(658\) −19.6418 −0.765715
\(659\) 37.7280 1.46967 0.734836 0.678245i \(-0.237259\pi\)
0.734836 + 0.678245i \(0.237259\pi\)
\(660\) 40.3303 1.56986
\(661\) 44.0649 1.71393 0.856963 0.515378i \(-0.172348\pi\)
0.856963 + 0.515378i \(0.172348\pi\)
\(662\) −12.0414 −0.468001
\(663\) −3.01184 −0.116970
\(664\) 0.313472 0.0121651
\(665\) −24.8991 −0.965544
\(666\) −4.96875 −0.192535
\(667\) 1.00000 0.0387202
\(668\) 23.4948 0.909040
\(669\) −9.48165 −0.366582
\(670\) −103.356 −3.99297
\(671\) −35.3442 −1.36445
\(672\) 19.5396 0.753758
\(673\) −10.9482 −0.422023 −0.211011 0.977484i \(-0.567676\pi\)
−0.211011 + 0.977484i \(0.567676\pi\)
\(674\) 5.97657 0.230209
\(675\) 10.5577 0.406366
\(676\) −24.7933 −0.953589
\(677\) −13.1905 −0.506954 −0.253477 0.967341i \(-0.581574\pi\)
−0.253477 + 0.967341i \(0.581574\pi\)
\(678\) 20.7277 0.796041
\(679\) −0.929962 −0.0356887
\(680\) −0.541531 −0.0207668
\(681\) 19.8447 0.760449
\(682\) −5.64219 −0.216051
\(683\) 36.1584 1.38356 0.691782 0.722107i \(-0.256826\pi\)
0.691782 + 0.722107i \(0.256826\pi\)
\(684\) 5.23112 0.200017
\(685\) 33.2971 1.27222
\(686\) 39.3857 1.50375
\(687\) 19.1534 0.730749
\(688\) 29.6400 1.13001
\(689\) 3.17018 0.120774
\(690\) −7.90768 −0.301040
\(691\) 39.3807 1.49811 0.749056 0.662507i \(-0.230507\pi\)
0.749056 + 0.662507i \(0.230507\pi\)
\(692\) −30.3086 −1.15216
\(693\) −12.3388 −0.468714
\(694\) −64.5707 −2.45107
\(695\) 3.11160 0.118030
\(696\) −0.0387322 −0.00146814
\(697\) −8.71179 −0.329982
\(698\) 52.8623 2.00087
\(699\) 0.846581 0.0320206
\(700\) −51.9512 −1.96357
\(701\) −16.2919 −0.615336 −0.307668 0.951494i \(-0.599549\pi\)
−0.307668 + 0.951494i \(0.599549\pi\)
\(702\) 1.70345 0.0642927
\(703\) −6.42038 −0.242149
\(704\) −41.2871 −1.55606
\(705\) 15.8583 0.597257
\(706\) −56.8710 −2.14037
\(707\) 18.8206 0.707822
\(708\) 19.5222 0.733691
\(709\) −10.2677 −0.385610 −0.192805 0.981237i \(-0.561758\pi\)
−0.192805 + 0.981237i \(0.561758\pi\)
\(710\) 53.9666 2.02533
\(711\) 8.57302 0.321514
\(712\) 0.119905 0.00449362
\(713\) 0.555799 0.0208148
\(714\) 17.3171 0.648076
\(715\) 16.9699 0.634640
\(716\) 1.12819 0.0421623
\(717\) −0.968657 −0.0361752
\(718\) 67.7701 2.52916
\(719\) −35.4359 −1.32154 −0.660768 0.750590i \(-0.729769\pi\)
−0.660768 + 0.750590i \(0.729769\pi\)
\(720\) −15.6234 −0.582251
\(721\) 29.5166 1.09926
\(722\) −24.6376 −0.916915
\(723\) −1.03734 −0.0385793
\(724\) 27.4670 1.02080
\(725\) −10.5577 −0.392103
\(726\) 29.3494 1.08926
\(727\) 45.3897 1.68341 0.841706 0.539937i \(-0.181552\pi\)
0.841706 + 0.539937i \(0.181552\pi\)
\(728\) −0.0801950 −0.00297223
\(729\) 1.00000 0.0370370
\(730\) −120.789 −4.47059
\(731\) 26.5249 0.981057
\(732\) −14.0951 −0.520971
\(733\) 41.6542 1.53853 0.769266 0.638928i \(-0.220622\pi\)
0.769266 + 0.638928i \(0.220622\pi\)
\(734\) −51.2655 −1.89224
\(735\) −4.18879 −0.154506
\(736\) 8.01855 0.295568
\(737\) −66.1818 −2.43784
\(738\) 4.92726 0.181375
\(739\) 23.7060 0.872038 0.436019 0.899937i \(-0.356388\pi\)
0.436019 + 0.899937i \(0.356388\pi\)
\(740\) −19.7401 −0.725660
\(741\) 2.20112 0.0808601
\(742\) −18.2275 −0.669152
\(743\) −35.7954 −1.31321 −0.656603 0.754236i \(-0.728007\pi\)
−0.656603 + 0.754236i \(0.728007\pi\)
\(744\) −0.0215273 −0.000789230 0
\(745\) −30.7046 −1.12493
\(746\) 0.359598 0.0131658
\(747\) 8.09331 0.296119
\(748\) −36.2441 −1.32521
\(749\) 30.9859 1.13220
\(750\) 43.9486 1.60477
\(751\) 20.2739 0.739806 0.369903 0.929070i \(-0.379391\pi\)
0.369903 + 0.929070i \(0.379391\pi\)
\(752\) −15.9253 −0.580735
\(753\) −27.7696 −1.01198
\(754\) −1.70345 −0.0620361
\(755\) −76.0686 −2.76842
\(756\) −4.92069 −0.178964
\(757\) 15.6157 0.567563 0.283781 0.958889i \(-0.408411\pi\)
0.283781 + 0.958889i \(0.408411\pi\)
\(758\) −19.2445 −0.698992
\(759\) −5.06353 −0.183795
\(760\) 0.395762 0.0143558
\(761\) 47.2992 1.71460 0.857298 0.514821i \(-0.172142\pi\)
0.857298 + 0.514821i \(0.172142\pi\)
\(762\) 9.29726 0.336804
\(763\) 9.90456 0.358569
\(764\) 37.4571 1.35515
\(765\) −13.9814 −0.505499
\(766\) 52.2360 1.88736
\(767\) 8.21445 0.296607
\(768\) 15.6864 0.566035
\(769\) −10.2077 −0.368098 −0.184049 0.982917i \(-0.558920\pi\)
−0.184049 + 0.982917i \(0.558920\pi\)
\(770\) −97.5717 −3.51624
\(771\) −15.0573 −0.542277
\(772\) 4.64533 0.167189
\(773\) −25.6587 −0.922880 −0.461440 0.887171i \(-0.652667\pi\)
−0.461440 + 0.887171i \(0.652667\pi\)
\(774\) −15.0021 −0.539238
\(775\) −5.86796 −0.210783
\(776\) 0.0147814 0.000530623 0
\(777\) 6.03938 0.216661
\(778\) −18.8970 −0.677490
\(779\) 6.36676 0.228113
\(780\) 6.76756 0.242317
\(781\) 34.5565 1.23653
\(782\) 7.10648 0.254127
\(783\) −1.00000 −0.0357371
\(784\) 4.20648 0.150232
\(785\) −47.1478 −1.68278
\(786\) −22.6901 −0.809330
\(787\) −0.441263 −0.0157293 −0.00786466 0.999969i \(-0.502503\pi\)
−0.00786466 + 0.999969i \(0.502503\pi\)
\(788\) −47.9087 −1.70668
\(789\) 16.2360 0.578016
\(790\) 67.7927 2.41196
\(791\) −25.1939 −0.895791
\(792\) 0.196122 0.00696889
\(793\) −5.93087 −0.210611
\(794\) −14.5425 −0.516094
\(795\) 14.7164 0.521938
\(796\) 20.4246 0.723931
\(797\) −41.1212 −1.45659 −0.728295 0.685264i \(-0.759687\pi\)
−0.728295 + 0.685264i \(0.759687\pi\)
\(798\) −12.6557 −0.448007
\(799\) −14.2515 −0.504183
\(800\) −84.6575 −2.99309
\(801\) 3.09574 0.109382
\(802\) −8.76838 −0.309622
\(803\) −77.3448 −2.72944
\(804\) −26.3931 −0.930811
\(805\) 9.61156 0.338763
\(806\) −0.946777 −0.0333488
\(807\) 19.5471 0.688089
\(808\) −0.299147 −0.0105240
\(809\) 28.4603 1.00061 0.500306 0.865849i \(-0.333221\pi\)
0.500306 + 0.865849i \(0.333221\pi\)
\(810\) 7.90768 0.277848
\(811\) −39.4573 −1.38553 −0.692767 0.721161i \(-0.743609\pi\)
−0.692767 + 0.721161i \(0.743609\pi\)
\(812\) 4.92069 0.172682
\(813\) 16.0743 0.563750
\(814\) −25.1595 −0.881839
\(815\) 54.8989 1.92302
\(816\) 14.0405 0.491515
\(817\) −19.3849 −0.678193
\(818\) −34.6886 −1.21286
\(819\) −2.07050 −0.0723491
\(820\) 19.5752 0.683597
\(821\) 31.1954 1.08873 0.544364 0.838849i \(-0.316771\pi\)
0.544364 + 0.838849i \(0.316771\pi\)
\(822\) 16.9243 0.590301
\(823\) −19.8791 −0.692940 −0.346470 0.938061i \(-0.612620\pi\)
−0.346470 + 0.938061i \(0.612620\pi\)
\(824\) −0.469157 −0.0163438
\(825\) 53.4593 1.86121
\(826\) −47.2304 −1.64336
\(827\) 33.4931 1.16467 0.582335 0.812949i \(-0.302139\pi\)
0.582335 + 0.812949i \(0.302139\pi\)
\(828\) −2.01932 −0.0701762
\(829\) 36.8317 1.27922 0.639610 0.768700i \(-0.279096\pi\)
0.639610 + 0.768700i \(0.279096\pi\)
\(830\) 63.9993 2.22145
\(831\) 11.5689 0.401319
\(832\) −6.92810 −0.240189
\(833\) 3.76438 0.130428
\(834\) 1.58157 0.0547652
\(835\) 45.8922 1.58817
\(836\) 26.4879 0.916104
\(837\) −0.555799 −0.0192112
\(838\) 73.4175 2.53617
\(839\) 38.3930 1.32547 0.662737 0.748853i \(-0.269395\pi\)
0.662737 + 0.748853i \(0.269395\pi\)
\(840\) −0.372277 −0.0128448
\(841\) 1.00000 0.0344828
\(842\) −59.4013 −2.04711
\(843\) 1.08577 0.0373959
\(844\) −43.8095 −1.50798
\(845\) −48.4286 −1.66600
\(846\) 8.06045 0.277124
\(847\) −35.6733 −1.22575
\(848\) −14.7786 −0.507499
\(849\) 10.9798 0.376825
\(850\) −75.0281 −2.57344
\(851\) 2.47840 0.0849584
\(852\) 13.7810 0.472130
\(853\) 45.8492 1.56985 0.784923 0.619593i \(-0.212702\pi\)
0.784923 + 0.619593i \(0.212702\pi\)
\(854\) 34.1006 1.16690
\(855\) 10.2179 0.349445
\(856\) −0.492510 −0.0168337
\(857\) 25.3677 0.866544 0.433272 0.901263i \(-0.357359\pi\)
0.433272 + 0.901263i \(0.357359\pi\)
\(858\) 8.62549 0.294470
\(859\) 52.6817 1.79748 0.898738 0.438486i \(-0.144485\pi\)
0.898738 + 0.438486i \(0.144485\pi\)
\(860\) −59.6009 −2.03237
\(861\) −5.98894 −0.204103
\(862\) 24.9067 0.848325
\(863\) −24.0843 −0.819839 −0.409920 0.912122i \(-0.634443\pi\)
−0.409920 + 0.912122i \(0.634443\pi\)
\(864\) −8.01855 −0.272797
\(865\) −59.2017 −2.01292
\(866\) −75.1305 −2.55304
\(867\) −4.43519 −0.150627
\(868\) 2.73491 0.0928290
\(869\) 43.4098 1.47258
\(870\) −7.90768 −0.268095
\(871\) −11.1055 −0.376296
\(872\) −0.157430 −0.00533124
\(873\) 0.381632 0.0129163
\(874\) −5.19356 −0.175675
\(875\) −53.4182 −1.80586
\(876\) −30.8448 −1.04215
\(877\) 48.6597 1.64312 0.821560 0.570122i \(-0.193104\pi\)
0.821560 + 0.570122i \(0.193104\pi\)
\(878\) −68.1154 −2.29878
\(879\) 3.35385 0.113122
\(880\) −79.1098 −2.66679
\(881\) 4.21156 0.141891 0.0709455 0.997480i \(-0.477398\pi\)
0.0709455 + 0.997480i \(0.477398\pi\)
\(882\) −2.12908 −0.0716899
\(883\) −31.2259 −1.05083 −0.525417 0.850845i \(-0.676091\pi\)
−0.525417 + 0.850845i \(0.676091\pi\)
\(884\) −6.08187 −0.204555
\(885\) 38.1327 1.28182
\(886\) −28.5230 −0.958249
\(887\) −10.9743 −0.368480 −0.184240 0.982881i \(-0.558982\pi\)
−0.184240 + 0.982881i \(0.558982\pi\)
\(888\) −0.0959940 −0.00322135
\(889\) −11.3006 −0.379008
\(890\) 24.4801 0.820575
\(891\) 5.06353 0.169635
\(892\) −19.1465 −0.641071
\(893\) 10.4153 0.348535
\(894\) −15.6066 −0.521962
\(895\) 2.20368 0.0736609
\(896\) 0.755028 0.0252237
\(897\) −0.849677 −0.0283699
\(898\) 47.7352 1.59294
\(899\) 0.555799 0.0185369
\(900\) 21.3194 0.710646
\(901\) −13.2254 −0.440601
\(902\) 24.9493 0.830722
\(903\) 18.2346 0.606809
\(904\) 0.400448 0.0133187
\(905\) 53.6511 1.78342
\(906\) −38.6642 −1.28453
\(907\) −6.75010 −0.224133 −0.112067 0.993701i \(-0.535747\pi\)
−0.112067 + 0.993701i \(0.535747\pi\)
\(908\) 40.0727 1.32986
\(909\) −7.72348 −0.256171
\(910\) −16.3728 −0.542754
\(911\) 2.35100 0.0778921 0.0389460 0.999241i \(-0.487600\pi\)
0.0389460 + 0.999241i \(0.487600\pi\)
\(912\) −10.2611 −0.339778
\(913\) 40.9807 1.35626
\(914\) 18.1283 0.599630
\(915\) −27.5320 −0.910178
\(916\) 38.6769 1.27792
\(917\) 27.5792 0.910745
\(918\) −7.10648 −0.234549
\(919\) 57.2879 1.88975 0.944876 0.327427i \(-0.106182\pi\)
0.944876 + 0.327427i \(0.106182\pi\)
\(920\) −0.152773 −0.00503676
\(921\) 11.0025 0.362544
\(922\) −50.0596 −1.64863
\(923\) 5.79869 0.190866
\(924\) −24.9161 −0.819679
\(925\) −26.1662 −0.860340
\(926\) 60.7437 1.99616
\(927\) −12.1128 −0.397837
\(928\) 8.01855 0.263222
\(929\) 13.5839 0.445672 0.222836 0.974856i \(-0.428469\pi\)
0.222836 + 0.974856i \(0.428469\pi\)
\(930\) −4.39508 −0.144120
\(931\) −2.75109 −0.0901634
\(932\) 1.70952 0.0559971
\(933\) 32.0112 1.04800
\(934\) 70.8730 2.31904
\(935\) −70.7953 −2.31525
\(936\) 0.0329099 0.00107569
\(937\) −15.0191 −0.490651 −0.245326 0.969441i \(-0.578895\pi\)
−0.245326 + 0.969441i \(0.578895\pi\)
\(938\) 63.8531 2.08488
\(939\) −13.3302 −0.435015
\(940\) 32.0229 1.04447
\(941\) 10.4773 0.341550 0.170775 0.985310i \(-0.445373\pi\)
0.170775 + 0.985310i \(0.445373\pi\)
\(942\) −23.9643 −0.780799
\(943\) −2.45770 −0.0800338
\(944\) −38.2938 −1.24636
\(945\) −9.61156 −0.312664
\(946\) −75.9635 −2.46979
\(947\) 41.9689 1.36381 0.681903 0.731442i \(-0.261152\pi\)
0.681903 + 0.731442i \(0.261152\pi\)
\(948\) 17.3117 0.562257
\(949\) −12.9787 −0.421307
\(950\) 54.8321 1.77899
\(951\) −2.22126 −0.0720293
\(952\) 0.334558 0.0108431
\(953\) −1.88809 −0.0611613 −0.0305807 0.999532i \(-0.509736\pi\)
−0.0305807 + 0.999532i \(0.509736\pi\)
\(954\) 7.48008 0.242176
\(955\) 73.1648 2.36756
\(956\) −1.95603 −0.0632625
\(957\) −5.06353 −0.163681
\(958\) 20.3877 0.658697
\(959\) −20.5709 −0.664270
\(960\) −32.1613 −1.03800
\(961\) −30.6911 −0.990035
\(962\) −4.22184 −0.136118
\(963\) −12.7158 −0.409760
\(964\) −2.09473 −0.0674667
\(965\) 9.07369 0.292092
\(966\) 4.88537 0.157184
\(967\) 30.5244 0.981599 0.490800 0.871272i \(-0.336705\pi\)
0.490800 + 0.871272i \(0.336705\pi\)
\(968\) 0.567016 0.0182246
\(969\) −9.18263 −0.294989
\(970\) 3.01782 0.0968964
\(971\) 10.0716 0.323212 0.161606 0.986855i \(-0.448333\pi\)
0.161606 + 0.986855i \(0.448333\pi\)
\(972\) 2.01932 0.0647697
\(973\) −1.92235 −0.0616277
\(974\) −13.4083 −0.429629
\(975\) 8.97064 0.287291
\(976\) 27.6483 0.884999
\(977\) 23.5545 0.753576 0.376788 0.926300i \(-0.377029\pi\)
0.376788 + 0.926300i \(0.377029\pi\)
\(978\) 27.9040 0.892272
\(979\) 15.6754 0.500987
\(980\) −8.45850 −0.270197
\(981\) −4.06457 −0.129772
\(982\) −19.7626 −0.630649
\(983\) 17.1703 0.547647 0.273823 0.961780i \(-0.411712\pi\)
0.273823 + 0.961780i \(0.411712\pi\)
\(984\) 0.0951923 0.00303462
\(985\) −93.5797 −2.98170
\(986\) 7.10648 0.226316
\(987\) −9.79725 −0.311850
\(988\) 4.44476 0.141407
\(989\) 7.48299 0.237945
\(990\) 40.0408 1.27258
\(991\) −9.28729 −0.295020 −0.147510 0.989061i \(-0.547126\pi\)
−0.147510 + 0.989061i \(0.547126\pi\)
\(992\) 4.45670 0.141500
\(993\) −6.00619 −0.190601
\(994\) −33.3406 −1.05750
\(995\) 39.8953 1.26477
\(996\) 16.3430 0.517847
\(997\) 19.7598 0.625800 0.312900 0.949786i \(-0.398700\pi\)
0.312900 + 0.949786i \(0.398700\pi\)
\(998\) 45.8008 1.44980
\(999\) −2.47840 −0.0784131
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 2001.2.a.n.1.13 16
3.2 odd 2 6003.2.a.r.1.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.n.1.13 16 1.1 even 1 trivial
6003.2.a.r.1.4 16 3.2 odd 2