Properties

Label 2001.2.a.o.1.13
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + 14484 x^{12} - 24566 x^{11} - 36791 x^{10} + 59410 x^{9} + 52109 x^{8} + \cdots - 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(0.762638\) 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)
 
\(f(q)\) \(=\) \(q+0.762638 q^{2} +1.00000 q^{3} -1.41838 q^{4} -4.10409 q^{5} +0.762638 q^{6} -2.37592 q^{7} -2.60699 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.762638 q^{2} +1.00000 q^{3} -1.41838 q^{4} -4.10409 q^{5} +0.762638 q^{6} -2.37592 q^{7} -2.60699 q^{8} +1.00000 q^{9} -3.12993 q^{10} -4.76863 q^{11} -1.41838 q^{12} +5.19051 q^{13} -1.81197 q^{14} -4.10409 q^{15} +0.848580 q^{16} -2.80912 q^{17} +0.762638 q^{18} -0.644865 q^{19} +5.82117 q^{20} -2.37592 q^{21} -3.63674 q^{22} +1.00000 q^{23} -2.60699 q^{24} +11.8435 q^{25} +3.95848 q^{26} +1.00000 q^{27} +3.36997 q^{28} +1.00000 q^{29} -3.12993 q^{30} +0.195443 q^{31} +5.86114 q^{32} -4.76863 q^{33} -2.14234 q^{34} +9.75098 q^{35} -1.41838 q^{36} +4.46849 q^{37} -0.491798 q^{38} +5.19051 q^{39} +10.6993 q^{40} +2.20248 q^{41} -1.81197 q^{42} +10.7899 q^{43} +6.76374 q^{44} -4.10409 q^{45} +0.762638 q^{46} -10.3878 q^{47} +0.848580 q^{48} -1.35501 q^{49} +9.03232 q^{50} -2.80912 q^{51} -7.36213 q^{52} -3.71349 q^{53} +0.762638 q^{54} +19.5709 q^{55} +6.19400 q^{56} -0.644865 q^{57} +0.762638 q^{58} -6.79452 q^{59} +5.82117 q^{60} +12.5542 q^{61} +0.149052 q^{62} -2.37592 q^{63} +2.77276 q^{64} -21.3023 q^{65} -3.63674 q^{66} -2.80619 q^{67} +3.98441 q^{68} +1.00000 q^{69} +7.43647 q^{70} +14.2651 q^{71} -2.60699 q^{72} -7.86909 q^{73} +3.40784 q^{74} +11.8435 q^{75} +0.914665 q^{76} +11.3299 q^{77} +3.95848 q^{78} -9.62495 q^{79} -3.48264 q^{80} +1.00000 q^{81} +1.67969 q^{82} +14.6215 q^{83} +3.36997 q^{84} +11.5289 q^{85} +8.22881 q^{86} +1.00000 q^{87} +12.4318 q^{88} -8.72169 q^{89} -3.12993 q^{90} -12.3322 q^{91} -1.41838 q^{92} +0.195443 q^{93} -7.92214 q^{94} +2.64658 q^{95} +5.86114 q^{96} -13.7648 q^{97} -1.03338 q^{98} -4.76863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9} + 7 q^{10} + 30 q^{12} + 21 q^{13} - q^{14} - q^{15} + 58 q^{16} - 4 q^{17} + 2 q^{18} + 7 q^{19} - 20 q^{20} + 9 q^{21} + 7 q^{22} + 20 q^{23} + 6 q^{24} + 47 q^{25} + 8 q^{26} + 20 q^{27} + 11 q^{28} + 20 q^{29} + 7 q^{30} + 28 q^{31} + 14 q^{32} + 16 q^{34} + 9 q^{35} + 30 q^{36} + 14 q^{37} - 20 q^{38} + 21 q^{39} + 34 q^{40} + 7 q^{41} - q^{42} + 3 q^{43} - q^{44} - q^{45} + 2 q^{46} + 3 q^{47} + 58 q^{48} + 35 q^{49} - 24 q^{50} - 4 q^{51} + 73 q^{52} - 19 q^{53} + 2 q^{54} + 29 q^{55} - 30 q^{56} + 7 q^{57} + 2 q^{58} + 20 q^{59} - 20 q^{60} + 15 q^{61} + 12 q^{62} + 9 q^{63} + 82 q^{64} - 28 q^{65} + 7 q^{66} + 20 q^{67} - 23 q^{68} + 20 q^{69} - 24 q^{70} + 63 q^{71} + 6 q^{72} + 19 q^{73} + 16 q^{74} + 47 q^{75} - 44 q^{76} - 7 q^{77} + 8 q^{78} + 32 q^{79} - 56 q^{80} + 20 q^{81} - 20 q^{82} - 21 q^{83} + 11 q^{84} + 4 q^{85} - 6 q^{86} + 20 q^{87} + 55 q^{88} - 13 q^{89} + 7 q^{90} + 70 q^{91} + 30 q^{92} + 28 q^{93} - 12 q^{94} + 9 q^{95} + 14 q^{96} - 9 q^{97} + 31 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.762638 0.539266 0.269633 0.962963i \(-0.413098\pi\)
0.269633 + 0.962963i \(0.413098\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.41838 −0.709192
\(5\) −4.10409 −1.83540 −0.917702 0.397270i \(-0.869958\pi\)
−0.917702 + 0.397270i \(0.869958\pi\)
\(6\) 0.762638 0.311346
\(7\) −2.37592 −0.898013 −0.449007 0.893528i \(-0.648222\pi\)
−0.449007 + 0.893528i \(0.648222\pi\)
\(8\) −2.60699 −0.921710
\(9\) 1.00000 0.333333
\(10\) −3.12993 −0.989771
\(11\) −4.76863 −1.43780 −0.718898 0.695116i \(-0.755353\pi\)
−0.718898 + 0.695116i \(0.755353\pi\)
\(12\) −1.41838 −0.409452
\(13\) 5.19051 1.43959 0.719794 0.694188i \(-0.244236\pi\)
0.719794 + 0.694188i \(0.244236\pi\)
\(14\) −1.81197 −0.484268
\(15\) −4.10409 −1.05967
\(16\) 0.848580 0.212145
\(17\) −2.80912 −0.681312 −0.340656 0.940188i \(-0.610649\pi\)
−0.340656 + 0.940188i \(0.610649\pi\)
\(18\) 0.762638 0.179755
\(19\) −0.644865 −0.147942 −0.0739710 0.997260i \(-0.523567\pi\)
−0.0739710 + 0.997260i \(0.523567\pi\)
\(20\) 5.82117 1.30165
\(21\) −2.37592 −0.518468
\(22\) −3.63674 −0.775355
\(23\) 1.00000 0.208514
\(24\) −2.60699 −0.532149
\(25\) 11.8435 2.36871
\(26\) 3.95848 0.776321
\(27\) 1.00000 0.192450
\(28\) 3.36997 0.636864
\(29\) 1.00000 0.185695
\(30\) −3.12993 −0.571445
\(31\) 0.195443 0.0351026 0.0175513 0.999846i \(-0.494413\pi\)
0.0175513 + 0.999846i \(0.494413\pi\)
\(32\) 5.86114 1.03611
\(33\) −4.76863 −0.830111
\(34\) −2.14234 −0.367409
\(35\) 9.75098 1.64822
\(36\) −1.41838 −0.236397
\(37\) 4.46849 0.734616 0.367308 0.930099i \(-0.380280\pi\)
0.367308 + 0.930099i \(0.380280\pi\)
\(38\) −0.491798 −0.0797802
\(39\) 5.19051 0.831146
\(40\) 10.6993 1.69171
\(41\) 2.20248 0.343969 0.171985 0.985100i \(-0.444982\pi\)
0.171985 + 0.985100i \(0.444982\pi\)
\(42\) −1.81197 −0.279592
\(43\) 10.7899 1.64545 0.822725 0.568440i \(-0.192453\pi\)
0.822725 + 0.568440i \(0.192453\pi\)
\(44\) 6.76374 1.01967
\(45\) −4.10409 −0.611801
\(46\) 0.762638 0.112445
\(47\) −10.3878 −1.51522 −0.757610 0.652708i \(-0.773633\pi\)
−0.757610 + 0.652708i \(0.773633\pi\)
\(48\) 0.848580 0.122482
\(49\) −1.35501 −0.193572
\(50\) 9.03232 1.27736
\(51\) −2.80912 −0.393356
\(52\) −7.36213 −1.02094
\(53\) −3.71349 −0.510088 −0.255044 0.966930i \(-0.582090\pi\)
−0.255044 + 0.966930i \(0.582090\pi\)
\(54\) 0.762638 0.103782
\(55\) 19.5709 2.63893
\(56\) 6.19400 0.827707
\(57\) −0.644865 −0.0854144
\(58\) 0.762638 0.100139
\(59\) −6.79452 −0.884572 −0.442286 0.896874i \(-0.645832\pi\)
−0.442286 + 0.896874i \(0.645832\pi\)
\(60\) 5.82117 0.751510
\(61\) 12.5542 1.60740 0.803698 0.595037i \(-0.202863\pi\)
0.803698 + 0.595037i \(0.202863\pi\)
\(62\) 0.149052 0.0189296
\(63\) −2.37592 −0.299338
\(64\) 2.77276 0.346596
\(65\) −21.3023 −2.64222
\(66\) −3.63674 −0.447651
\(67\) −2.80619 −0.342831 −0.171415 0.985199i \(-0.554834\pi\)
−0.171415 + 0.985199i \(0.554834\pi\)
\(68\) 3.98441 0.483181
\(69\) 1.00000 0.120386
\(70\) 7.43647 0.888828
\(71\) 14.2651 1.69295 0.846476 0.532428i \(-0.178720\pi\)
0.846476 + 0.532428i \(0.178720\pi\)
\(72\) −2.60699 −0.307237
\(73\) −7.86909 −0.921007 −0.460504 0.887658i \(-0.652331\pi\)
−0.460504 + 0.887658i \(0.652331\pi\)
\(74\) 3.40784 0.396153
\(75\) 11.8435 1.36757
\(76\) 0.914665 0.104919
\(77\) 11.3299 1.29116
\(78\) 3.95848 0.448209
\(79\) −9.62495 −1.08289 −0.541446 0.840736i \(-0.682123\pi\)
−0.541446 + 0.840736i \(0.682123\pi\)
\(80\) −3.48264 −0.389371
\(81\) 1.00000 0.111111
\(82\) 1.67969 0.185491
\(83\) 14.6215 1.60492 0.802458 0.596708i \(-0.203525\pi\)
0.802458 + 0.596708i \(0.203525\pi\)
\(84\) 3.36997 0.367693
\(85\) 11.5289 1.25048
\(86\) 8.22881 0.887336
\(87\) 1.00000 0.107211
\(88\) 12.4318 1.32523
\(89\) −8.72169 −0.924497 −0.462249 0.886750i \(-0.652957\pi\)
−0.462249 + 0.886750i \(0.652957\pi\)
\(90\) −3.12993 −0.329924
\(91\) −12.3322 −1.29277
\(92\) −1.41838 −0.147877
\(93\) 0.195443 0.0202665
\(94\) −7.92214 −0.817107
\(95\) 2.64658 0.271533
\(96\) 5.86114 0.598200
\(97\) −13.7648 −1.39761 −0.698804 0.715314i \(-0.746284\pi\)
−0.698804 + 0.715314i \(0.746284\pi\)
\(98\) −1.03338 −0.104387
\(99\) −4.76863 −0.479265
\(100\) −16.7987 −1.67987
\(101\) −0.857700 −0.0853444 −0.0426722 0.999089i \(-0.513587\pi\)
−0.0426722 + 0.999089i \(0.513587\pi\)
\(102\) −2.14234 −0.212124
\(103\) 3.84676 0.379033 0.189516 0.981878i \(-0.439308\pi\)
0.189516 + 0.981878i \(0.439308\pi\)
\(104\) −13.5316 −1.32688
\(105\) 9.75098 0.951598
\(106\) −2.83205 −0.275073
\(107\) 18.4797 1.78650 0.893248 0.449565i \(-0.148421\pi\)
0.893248 + 0.449565i \(0.148421\pi\)
\(108\) −1.41838 −0.136484
\(109\) −3.88301 −0.371925 −0.185963 0.982557i \(-0.559540\pi\)
−0.185963 + 0.982557i \(0.559540\pi\)
\(110\) 14.9255 1.42309
\(111\) 4.46849 0.424131
\(112\) −2.01616 −0.190509
\(113\) 0.238682 0.0224533 0.0112267 0.999937i \(-0.496426\pi\)
0.0112267 + 0.999937i \(0.496426\pi\)
\(114\) −0.491798 −0.0460611
\(115\) −4.10409 −0.382708
\(116\) −1.41838 −0.131694
\(117\) 5.19051 0.479862
\(118\) −5.18176 −0.477020
\(119\) 6.67425 0.611827
\(120\) 10.6993 0.976709
\(121\) 11.7398 1.06726
\(122\) 9.57428 0.866815
\(123\) 2.20248 0.198591
\(124\) −0.277213 −0.0248944
\(125\) −28.0864 −2.51213
\(126\) −1.81197 −0.161423
\(127\) 6.03868 0.535846 0.267923 0.963440i \(-0.413663\pi\)
0.267923 + 0.963440i \(0.413663\pi\)
\(128\) −9.60766 −0.849205
\(129\) 10.7899 0.950001
\(130\) −16.2459 −1.42486
\(131\) 16.4899 1.44073 0.720364 0.693596i \(-0.243975\pi\)
0.720364 + 0.693596i \(0.243975\pi\)
\(132\) 6.76374 0.588708
\(133\) 1.53215 0.132854
\(134\) −2.14011 −0.184877
\(135\) −4.10409 −0.353224
\(136\) 7.32335 0.627972
\(137\) 4.51947 0.386124 0.193062 0.981187i \(-0.438158\pi\)
0.193062 + 0.981187i \(0.438158\pi\)
\(138\) 0.762638 0.0649200
\(139\) 23.3271 1.97858 0.989288 0.145976i \(-0.0466323\pi\)
0.989288 + 0.145976i \(0.0466323\pi\)
\(140\) −13.8306 −1.16890
\(141\) −10.3878 −0.874812
\(142\) 10.8791 0.912952
\(143\) −24.7516 −2.06983
\(144\) 0.848580 0.0707150
\(145\) −4.10409 −0.340826
\(146\) −6.00127 −0.496668
\(147\) −1.35501 −0.111759
\(148\) −6.33804 −0.520983
\(149\) −12.3873 −1.01481 −0.507403 0.861709i \(-0.669395\pi\)
−0.507403 + 0.861709i \(0.669395\pi\)
\(150\) 9.03232 0.737486
\(151\) −8.79395 −0.715642 −0.357821 0.933790i \(-0.616480\pi\)
−0.357821 + 0.933790i \(0.616480\pi\)
\(152\) 1.68115 0.136360
\(153\) −2.80912 −0.227104
\(154\) 8.64059 0.696279
\(155\) −0.802114 −0.0644274
\(156\) −7.36213 −0.589442
\(157\) −12.9451 −1.03313 −0.516564 0.856249i \(-0.672789\pi\)
−0.516564 + 0.856249i \(0.672789\pi\)
\(158\) −7.34035 −0.583967
\(159\) −3.71349 −0.294499
\(160\) −24.0546 −1.90168
\(161\) −2.37592 −0.187249
\(162\) 0.762638 0.0599185
\(163\) 8.06894 0.632008 0.316004 0.948758i \(-0.397659\pi\)
0.316004 + 0.948758i \(0.397659\pi\)
\(164\) −3.12396 −0.243940
\(165\) 19.5709 1.52359
\(166\) 11.1509 0.865477
\(167\) 17.7452 1.37317 0.686584 0.727051i \(-0.259109\pi\)
0.686584 + 0.727051i \(0.259109\pi\)
\(168\) 6.19400 0.477877
\(169\) 13.9413 1.07241
\(170\) 8.79236 0.674343
\(171\) −0.644865 −0.0493140
\(172\) −15.3043 −1.16694
\(173\) 23.9022 1.81725 0.908624 0.417615i \(-0.137134\pi\)
0.908624 + 0.417615i \(0.137134\pi\)
\(174\) 0.762638 0.0578154
\(175\) −28.1393 −2.12713
\(176\) −4.04656 −0.305021
\(177\) −6.79452 −0.510708
\(178\) −6.65149 −0.498550
\(179\) −17.4110 −1.30136 −0.650678 0.759353i \(-0.725515\pi\)
−0.650678 + 0.759353i \(0.725515\pi\)
\(180\) 5.82117 0.433884
\(181\) −8.86252 −0.658746 −0.329373 0.944200i \(-0.606837\pi\)
−0.329373 + 0.944200i \(0.606837\pi\)
\(182\) −9.40502 −0.697146
\(183\) 12.5542 0.928031
\(184\) −2.60699 −0.192190
\(185\) −18.3391 −1.34832
\(186\) 0.149052 0.0109290
\(187\) 13.3957 0.979587
\(188\) 14.7339 1.07458
\(189\) −2.37592 −0.172823
\(190\) 2.01838 0.146429
\(191\) 12.9011 0.933493 0.466746 0.884391i \(-0.345426\pi\)
0.466746 + 0.884391i \(0.345426\pi\)
\(192\) 2.77276 0.200107
\(193\) 3.46564 0.249463 0.124731 0.992191i \(-0.460193\pi\)
0.124731 + 0.992191i \(0.460193\pi\)
\(194\) −10.4976 −0.753683
\(195\) −21.3023 −1.52549
\(196\) 1.92192 0.137280
\(197\) −10.0118 −0.713310 −0.356655 0.934236i \(-0.616083\pi\)
−0.356655 + 0.934236i \(0.616083\pi\)
\(198\) −3.63674 −0.258452
\(199\) 9.14492 0.648266 0.324133 0.946012i \(-0.394927\pi\)
0.324133 + 0.946012i \(0.394927\pi\)
\(200\) −30.8759 −2.18326
\(201\) −2.80619 −0.197933
\(202\) −0.654115 −0.0460234
\(203\) −2.37592 −0.166757
\(204\) 3.98441 0.278965
\(205\) −9.03916 −0.631322
\(206\) 2.93369 0.204400
\(207\) 1.00000 0.0695048
\(208\) 4.40456 0.305401
\(209\) 3.07512 0.212710
\(210\) 7.43647 0.513165
\(211\) 3.26744 0.224940 0.112470 0.993655i \(-0.464124\pi\)
0.112470 + 0.993655i \(0.464124\pi\)
\(212\) 5.26716 0.361750
\(213\) 14.2651 0.977426
\(214\) 14.0933 0.963397
\(215\) −44.2828 −3.02006
\(216\) −2.60699 −0.177383
\(217\) −0.464356 −0.0315226
\(218\) −2.96133 −0.200567
\(219\) −7.86909 −0.531744
\(220\) −27.7590 −1.87151
\(221\) −14.5808 −0.980808
\(222\) 3.40784 0.228719
\(223\) −16.3511 −1.09495 −0.547477 0.836821i \(-0.684412\pi\)
−0.547477 + 0.836821i \(0.684412\pi\)
\(224\) −13.9256 −0.930442
\(225\) 11.8435 0.789569
\(226\) 0.182028 0.0121083
\(227\) 6.26229 0.415643 0.207822 0.978167i \(-0.433363\pi\)
0.207822 + 0.978167i \(0.433363\pi\)
\(228\) 0.914665 0.0605752
\(229\) −2.82769 −0.186859 −0.0934293 0.995626i \(-0.529783\pi\)
−0.0934293 + 0.995626i \(0.529783\pi\)
\(230\) −3.12993 −0.206382
\(231\) 11.3299 0.745451
\(232\) −2.60699 −0.171157
\(233\) −6.61733 −0.433515 −0.216758 0.976225i \(-0.569548\pi\)
−0.216758 + 0.976225i \(0.569548\pi\)
\(234\) 3.95848 0.258774
\(235\) 42.6325 2.78104
\(236\) 9.63724 0.627331
\(237\) −9.62495 −0.625207
\(238\) 5.09003 0.329938
\(239\) 17.3956 1.12523 0.562615 0.826719i \(-0.309796\pi\)
0.562615 + 0.826719i \(0.309796\pi\)
\(240\) −3.48264 −0.224804
\(241\) 3.26312 0.210196 0.105098 0.994462i \(-0.466484\pi\)
0.105098 + 0.994462i \(0.466484\pi\)
\(242\) 8.95322 0.575535
\(243\) 1.00000 0.0641500
\(244\) −17.8066 −1.13995
\(245\) 5.56106 0.355283
\(246\) 1.67969 0.107093
\(247\) −3.34717 −0.212976
\(248\) −0.509517 −0.0323544
\(249\) 14.6215 0.926599
\(250\) −21.4198 −1.35471
\(251\) −23.6292 −1.49146 −0.745731 0.666247i \(-0.767900\pi\)
−0.745731 + 0.666247i \(0.767900\pi\)
\(252\) 3.36997 0.212288
\(253\) −4.76863 −0.299801
\(254\) 4.60533 0.288964
\(255\) 11.5289 0.721966
\(256\) −12.8727 −0.804543
\(257\) 4.73508 0.295366 0.147683 0.989035i \(-0.452818\pi\)
0.147683 + 0.989035i \(0.452818\pi\)
\(258\) 8.22881 0.512304
\(259\) −10.6168 −0.659695
\(260\) 30.2148 1.87384
\(261\) 1.00000 0.0618984
\(262\) 12.5758 0.776937
\(263\) 31.4462 1.93906 0.969529 0.244978i \(-0.0787807\pi\)
0.969529 + 0.244978i \(0.0787807\pi\)
\(264\) 12.4318 0.765122
\(265\) 15.2405 0.936216
\(266\) 1.16847 0.0716437
\(267\) −8.72169 −0.533759
\(268\) 3.98025 0.243133
\(269\) −13.8706 −0.845704 −0.422852 0.906199i \(-0.638971\pi\)
−0.422852 + 0.906199i \(0.638971\pi\)
\(270\) −3.12993 −0.190482
\(271\) 14.7222 0.894311 0.447156 0.894456i \(-0.352437\pi\)
0.447156 + 0.894456i \(0.352437\pi\)
\(272\) −2.38376 −0.144537
\(273\) −12.3322 −0.746380
\(274\) 3.44672 0.208224
\(275\) −56.4774 −3.40571
\(276\) −1.41838 −0.0853767
\(277\) 28.6108 1.71906 0.859528 0.511088i \(-0.170757\pi\)
0.859528 + 0.511088i \(0.170757\pi\)
\(278\) 17.7901 1.06698
\(279\) 0.195443 0.0117009
\(280\) −25.4207 −1.51918
\(281\) −1.91196 −0.114058 −0.0570289 0.998373i \(-0.518163\pi\)
−0.0570289 + 0.998373i \(0.518163\pi\)
\(282\) −7.92214 −0.471757
\(283\) −13.5965 −0.808230 −0.404115 0.914708i \(-0.632420\pi\)
−0.404115 + 0.914708i \(0.632420\pi\)
\(284\) −20.2333 −1.20063
\(285\) 2.64658 0.156770
\(286\) −18.8765 −1.11619
\(287\) −5.23291 −0.308889
\(288\) 5.86114 0.345371
\(289\) −9.10883 −0.535814
\(290\) −3.12993 −0.183796
\(291\) −13.7648 −0.806909
\(292\) 11.1614 0.653171
\(293\) −2.08364 −0.121727 −0.0608637 0.998146i \(-0.519386\pi\)
−0.0608637 + 0.998146i \(0.519386\pi\)
\(294\) −1.03338 −0.0602678
\(295\) 27.8853 1.62355
\(296\) −11.6493 −0.677102
\(297\) −4.76863 −0.276704
\(298\) −9.44702 −0.547251
\(299\) 5.19051 0.300175
\(300\) −16.7987 −0.969871
\(301\) −25.6360 −1.47764
\(302\) −6.70660 −0.385921
\(303\) −0.857700 −0.0492736
\(304\) −0.547219 −0.0313852
\(305\) −51.5234 −2.95022
\(306\) −2.14234 −0.122470
\(307\) −30.9157 −1.76445 −0.882227 0.470824i \(-0.843957\pi\)
−0.882227 + 0.470824i \(0.843957\pi\)
\(308\) −16.0701 −0.915680
\(309\) 3.84676 0.218835
\(310\) −0.611722 −0.0347435
\(311\) 12.6680 0.718334 0.359167 0.933273i \(-0.383061\pi\)
0.359167 + 0.933273i \(0.383061\pi\)
\(312\) −13.5316 −0.766075
\(313\) 23.5046 1.32856 0.664280 0.747484i \(-0.268738\pi\)
0.664280 + 0.747484i \(0.268738\pi\)
\(314\) −9.87239 −0.557131
\(315\) 9.75098 0.549406
\(316\) 13.6519 0.767977
\(317\) −9.06839 −0.509332 −0.254666 0.967029i \(-0.581965\pi\)
−0.254666 + 0.967029i \(0.581965\pi\)
\(318\) −2.83205 −0.158814
\(319\) −4.76863 −0.266992
\(320\) −11.3797 −0.636143
\(321\) 18.4797 1.03143
\(322\) −1.81197 −0.100977
\(323\) 1.81150 0.100795
\(324\) −1.41838 −0.0787991
\(325\) 61.4739 3.40996
\(326\) 6.15368 0.340821
\(327\) −3.88301 −0.214731
\(328\) −5.74183 −0.317040
\(329\) 24.6806 1.36069
\(330\) 14.9255 0.821620
\(331\) −17.1154 −0.940748 −0.470374 0.882467i \(-0.655881\pi\)
−0.470374 + 0.882467i \(0.655881\pi\)
\(332\) −20.7389 −1.13819
\(333\) 4.46849 0.244872
\(334\) 13.5332 0.740503
\(335\) 11.5168 0.629232
\(336\) −2.01616 −0.109990
\(337\) 9.73964 0.530552 0.265276 0.964172i \(-0.414537\pi\)
0.265276 + 0.964172i \(0.414537\pi\)
\(338\) 10.6322 0.578315
\(339\) 0.238682 0.0129634
\(340\) −16.3524 −0.886832
\(341\) −0.931994 −0.0504703
\(342\) −0.491798 −0.0265934
\(343\) 19.8508 1.07184
\(344\) −28.1292 −1.51663
\(345\) −4.10409 −0.220957
\(346\) 18.2287 0.979981
\(347\) −0.652189 −0.0350113 −0.0175057 0.999847i \(-0.505573\pi\)
−0.0175057 + 0.999847i \(0.505573\pi\)
\(348\) −1.41838 −0.0760333
\(349\) 0.388734 0.0208085 0.0104042 0.999946i \(-0.496688\pi\)
0.0104042 + 0.999946i \(0.496688\pi\)
\(350\) −21.4601 −1.14709
\(351\) 5.19051 0.277049
\(352\) −27.9496 −1.48972
\(353\) 33.9129 1.80500 0.902501 0.430688i \(-0.141729\pi\)
0.902501 + 0.430688i \(0.141729\pi\)
\(354\) −5.18176 −0.275407
\(355\) −58.5450 −3.10725
\(356\) 12.3707 0.655646
\(357\) 6.67425 0.353239
\(358\) −13.2783 −0.701778
\(359\) 19.5828 1.03354 0.516770 0.856124i \(-0.327134\pi\)
0.516770 + 0.856124i \(0.327134\pi\)
\(360\) 10.6993 0.563903
\(361\) −18.5841 −0.978113
\(362\) −6.75890 −0.355240
\(363\) 11.7398 0.616180
\(364\) 17.4918 0.916821
\(365\) 32.2954 1.69042
\(366\) 9.57428 0.500456
\(367\) 11.8792 0.620091 0.310045 0.950722i \(-0.399656\pi\)
0.310045 + 0.950722i \(0.399656\pi\)
\(368\) 0.848580 0.0442353
\(369\) 2.20248 0.114656
\(370\) −13.9861 −0.727101
\(371\) 8.82296 0.458065
\(372\) −0.277213 −0.0143728
\(373\) 35.5289 1.83962 0.919808 0.392370i \(-0.128345\pi\)
0.919808 + 0.392370i \(0.128345\pi\)
\(374\) 10.2160 0.528258
\(375\) −28.0864 −1.45038
\(376\) 27.0809 1.39659
\(377\) 5.19051 0.267325
\(378\) −1.81197 −0.0931975
\(379\) 20.2070 1.03796 0.518982 0.854785i \(-0.326311\pi\)
0.518982 + 0.854785i \(0.326311\pi\)
\(380\) −3.75387 −0.192569
\(381\) 6.03868 0.309371
\(382\) 9.83889 0.503401
\(383\) −5.97382 −0.305248 −0.152624 0.988284i \(-0.548772\pi\)
−0.152624 + 0.988284i \(0.548772\pi\)
\(384\) −9.60766 −0.490289
\(385\) −46.4988 −2.36980
\(386\) 2.64303 0.134527
\(387\) 10.7899 0.548483
\(388\) 19.5238 0.991172
\(389\) −10.8957 −0.552436 −0.276218 0.961095i \(-0.589081\pi\)
−0.276218 + 0.961095i \(0.589081\pi\)
\(390\) −16.2459 −0.822644
\(391\) −2.80912 −0.142063
\(392\) 3.53248 0.178417
\(393\) 16.4899 0.831805
\(394\) −7.63537 −0.384664
\(395\) 39.5016 1.98754
\(396\) 6.76374 0.339891
\(397\) 24.4605 1.22764 0.613819 0.789447i \(-0.289632\pi\)
0.613819 + 0.789447i \(0.289632\pi\)
\(398\) 6.97426 0.349588
\(399\) 1.53215 0.0767033
\(400\) 10.0502 0.502509
\(401\) 18.6658 0.932126 0.466063 0.884751i \(-0.345672\pi\)
0.466063 + 0.884751i \(0.345672\pi\)
\(402\) −2.14011 −0.106739
\(403\) 1.01445 0.0505332
\(404\) 1.21655 0.0605255
\(405\) −4.10409 −0.203934
\(406\) −1.81197 −0.0899264
\(407\) −21.3086 −1.05623
\(408\) 7.32335 0.362560
\(409\) 23.3006 1.15214 0.576071 0.817400i \(-0.304585\pi\)
0.576071 + 0.817400i \(0.304585\pi\)
\(410\) −6.89360 −0.340451
\(411\) 4.51947 0.222929
\(412\) −5.45619 −0.268807
\(413\) 16.1432 0.794357
\(414\) 0.762638 0.0374816
\(415\) −60.0078 −2.94567
\(416\) 30.4223 1.49157
\(417\) 23.3271 1.14233
\(418\) 2.34520 0.114708
\(419\) −11.7063 −0.571890 −0.285945 0.958246i \(-0.592307\pi\)
−0.285945 + 0.958246i \(0.592307\pi\)
\(420\) −13.8306 −0.674866
\(421\) −8.76302 −0.427083 −0.213542 0.976934i \(-0.568500\pi\)
−0.213542 + 0.976934i \(0.568500\pi\)
\(422\) 2.49187 0.121302
\(423\) −10.3878 −0.505073
\(424\) 9.68104 0.470153
\(425\) −33.2699 −1.61383
\(426\) 10.8791 0.527093
\(427\) −29.8277 −1.44346
\(428\) −26.2112 −1.26697
\(429\) −24.7516 −1.19502
\(430\) −33.7718 −1.62862
\(431\) 25.2322 1.21539 0.607697 0.794169i \(-0.292094\pi\)
0.607697 + 0.794169i \(0.292094\pi\)
\(432\) 0.848580 0.0408273
\(433\) 24.3713 1.17121 0.585606 0.810596i \(-0.300857\pi\)
0.585606 + 0.810596i \(0.300857\pi\)
\(434\) −0.354136 −0.0169991
\(435\) −4.10409 −0.196776
\(436\) 5.50760 0.263766
\(437\) −0.644865 −0.0308481
\(438\) −6.00127 −0.286752
\(439\) 40.8264 1.94854 0.974270 0.225386i \(-0.0723643\pi\)
0.974270 + 0.225386i \(0.0723643\pi\)
\(440\) −51.0210 −2.43233
\(441\) −1.35501 −0.0645241
\(442\) −11.1198 −0.528917
\(443\) −27.5392 −1.30842 −0.654212 0.756311i \(-0.727000\pi\)
−0.654212 + 0.756311i \(0.727000\pi\)
\(444\) −6.33804 −0.300790
\(445\) 35.7946 1.69683
\(446\) −12.4700 −0.590472
\(447\) −12.3873 −0.585899
\(448\) −6.58786 −0.311247
\(449\) −27.1392 −1.28078 −0.640389 0.768051i \(-0.721227\pi\)
−0.640389 + 0.768051i \(0.721227\pi\)
\(450\) 9.03232 0.425788
\(451\) −10.5028 −0.494557
\(452\) −0.338543 −0.0159237
\(453\) −8.79395 −0.413176
\(454\) 4.77586 0.224142
\(455\) 50.6125 2.37275
\(456\) 1.68115 0.0787273
\(457\) −11.4419 −0.535230 −0.267615 0.963526i \(-0.586236\pi\)
−0.267615 + 0.963526i \(0.586236\pi\)
\(458\) −2.15650 −0.100767
\(459\) −2.80912 −0.131119
\(460\) 5.82117 0.271413
\(461\) 18.6583 0.869004 0.434502 0.900671i \(-0.356924\pi\)
0.434502 + 0.900671i \(0.356924\pi\)
\(462\) 8.64059 0.401997
\(463\) 14.6373 0.680254 0.340127 0.940380i \(-0.389530\pi\)
0.340127 + 0.940380i \(0.389530\pi\)
\(464\) 0.848580 0.0393943
\(465\) −0.802114 −0.0371971
\(466\) −5.04662 −0.233780
\(467\) 10.6700 0.493749 0.246874 0.969047i \(-0.420596\pi\)
0.246874 + 0.969047i \(0.420596\pi\)
\(468\) −7.36213 −0.340314
\(469\) 6.66728 0.307866
\(470\) 32.5132 1.49972
\(471\) −12.9451 −0.596477
\(472\) 17.7132 0.815318
\(473\) −51.4532 −2.36582
\(474\) −7.34035 −0.337153
\(475\) −7.63747 −0.350431
\(476\) −9.46664 −0.433903
\(477\) −3.71349 −0.170029
\(478\) 13.2666 0.606798
\(479\) 32.0589 1.46481 0.732405 0.680870i \(-0.238398\pi\)
0.732405 + 0.680870i \(0.238398\pi\)
\(480\) −24.0546 −1.09794
\(481\) 23.1937 1.05754
\(482\) 2.48857 0.113351
\(483\) −2.37592 −0.108108
\(484\) −16.6515 −0.756889
\(485\) 56.4921 2.56517
\(486\) 0.762638 0.0345940
\(487\) −23.8137 −1.07910 −0.539551 0.841953i \(-0.681406\pi\)
−0.539551 + 0.841953i \(0.681406\pi\)
\(488\) −32.7286 −1.48155
\(489\) 8.06894 0.364890
\(490\) 4.24107 0.191592
\(491\) −18.2787 −0.824905 −0.412453 0.910979i \(-0.635328\pi\)
−0.412453 + 0.910979i \(0.635328\pi\)
\(492\) −3.12396 −0.140839
\(493\) −2.80912 −0.126516
\(494\) −2.55268 −0.114851
\(495\) 19.5709 0.879645
\(496\) 0.165849 0.00744683
\(497\) −33.8926 −1.52029
\(498\) 11.1509 0.499684
\(499\) 4.85298 0.217249 0.108625 0.994083i \(-0.465355\pi\)
0.108625 + 0.994083i \(0.465355\pi\)
\(500\) 39.8373 1.78158
\(501\) 17.7452 0.792799
\(502\) −18.0205 −0.804295
\(503\) −12.1670 −0.542499 −0.271249 0.962509i \(-0.587437\pi\)
−0.271249 + 0.962509i \(0.587437\pi\)
\(504\) 6.19400 0.275902
\(505\) 3.52008 0.156641
\(506\) −3.63674 −0.161673
\(507\) 13.9413 0.619157
\(508\) −8.56517 −0.380018
\(509\) 13.0048 0.576429 0.288214 0.957566i \(-0.406938\pi\)
0.288214 + 0.957566i \(0.406938\pi\)
\(510\) 8.79236 0.389332
\(511\) 18.6963 0.827077
\(512\) 9.39811 0.415342
\(513\) −0.644865 −0.0284715
\(514\) 3.61115 0.159281
\(515\) −15.7874 −0.695678
\(516\) −15.3043 −0.673733
\(517\) 49.5356 2.17857
\(518\) −8.09676 −0.355751
\(519\) 23.9022 1.04919
\(520\) 55.5348 2.43536
\(521\) −24.7847 −1.08584 −0.542919 0.839785i \(-0.682681\pi\)
−0.542919 + 0.839785i \(0.682681\pi\)
\(522\) 0.762638 0.0333797
\(523\) 1.69891 0.0742879 0.0371440 0.999310i \(-0.488174\pi\)
0.0371440 + 0.999310i \(0.488174\pi\)
\(524\) −23.3890 −1.02175
\(525\) −28.1393 −1.22810
\(526\) 23.9821 1.04567
\(527\) −0.549023 −0.0239158
\(528\) −4.04656 −0.176104
\(529\) 1.00000 0.0434783
\(530\) 11.6230 0.504870
\(531\) −6.79452 −0.294857
\(532\) −2.17317 −0.0942189
\(533\) 11.4320 0.495174
\(534\) −6.65149 −0.287838
\(535\) −75.8421 −3.27894
\(536\) 7.31570 0.315990
\(537\) −17.4110 −0.751339
\(538\) −10.5782 −0.456060
\(539\) 6.46152 0.278317
\(540\) 5.82117 0.250503
\(541\) −24.7795 −1.06535 −0.532677 0.846319i \(-0.678814\pi\)
−0.532677 + 0.846319i \(0.678814\pi\)
\(542\) 11.2277 0.482272
\(543\) −8.86252 −0.380327
\(544\) −16.4646 −0.705916
\(545\) 15.9362 0.682633
\(546\) −9.40502 −0.402498
\(547\) −24.2508 −1.03689 −0.518444 0.855112i \(-0.673489\pi\)
−0.518444 + 0.855112i \(0.673489\pi\)
\(548\) −6.41034 −0.273836
\(549\) 12.5542 0.535799
\(550\) −43.0718 −1.83659
\(551\) −0.644865 −0.0274722
\(552\) −2.60699 −0.110961
\(553\) 22.8681 0.972451
\(554\) 21.8197 0.927029
\(555\) −18.3391 −0.778451
\(556\) −33.0867 −1.40319
\(557\) −21.1596 −0.896559 −0.448280 0.893893i \(-0.647963\pi\)
−0.448280 + 0.893893i \(0.647963\pi\)
\(558\) 0.149052 0.00630988
\(559\) 56.0052 2.36877
\(560\) 8.27448 0.349661
\(561\) 13.3957 0.565565
\(562\) −1.45813 −0.0615075
\(563\) −47.1640 −1.98773 −0.993864 0.110613i \(-0.964719\pi\)
−0.993864 + 0.110613i \(0.964719\pi\)
\(564\) 14.7339 0.620410
\(565\) −0.979572 −0.0412109
\(566\) −10.3692 −0.435851
\(567\) −2.37592 −0.0997793
\(568\) −37.1888 −1.56041
\(569\) −11.1557 −0.467673 −0.233836 0.972276i \(-0.575128\pi\)
−0.233836 + 0.972276i \(0.575128\pi\)
\(570\) 2.01838 0.0845407
\(571\) 16.5122 0.691013 0.345506 0.938416i \(-0.387707\pi\)
0.345506 + 0.938416i \(0.387707\pi\)
\(572\) 35.1072 1.46791
\(573\) 12.9011 0.538952
\(574\) −3.99082 −0.166573
\(575\) 11.8435 0.493909
\(576\) 2.77276 0.115532
\(577\) −6.85661 −0.285444 −0.142722 0.989763i \(-0.545586\pi\)
−0.142722 + 0.989763i \(0.545586\pi\)
\(578\) −6.94674 −0.288946
\(579\) 3.46564 0.144027
\(580\) 5.82117 0.241711
\(581\) −34.7395 −1.44124
\(582\) −10.4976 −0.435139
\(583\) 17.7083 0.733402
\(584\) 20.5146 0.848901
\(585\) −21.3023 −0.880741
\(586\) −1.58906 −0.0656435
\(587\) −16.3914 −0.676545 −0.338273 0.941048i \(-0.609843\pi\)
−0.338273 + 0.941048i \(0.609843\pi\)
\(588\) 1.92192 0.0792585
\(589\) −0.126034 −0.00519315
\(590\) 21.2664 0.875524
\(591\) −10.0118 −0.411830
\(592\) 3.79187 0.155845
\(593\) −40.1902 −1.65041 −0.825207 0.564830i \(-0.808942\pi\)
−0.825207 + 0.564830i \(0.808942\pi\)
\(594\) −3.63674 −0.149217
\(595\) −27.3917 −1.12295
\(596\) 17.5699 0.719693
\(597\) 9.14492 0.374277
\(598\) 3.95848 0.161874
\(599\) 28.8613 1.17924 0.589621 0.807680i \(-0.299277\pi\)
0.589621 + 0.807680i \(0.299277\pi\)
\(600\) −30.8759 −1.26050
\(601\) −24.5658 −1.00206 −0.501031 0.865429i \(-0.667046\pi\)
−0.501031 + 0.865429i \(0.667046\pi\)
\(602\) −19.5510 −0.796839
\(603\) −2.80619 −0.114277
\(604\) 12.4732 0.507527
\(605\) −48.1812 −1.95884
\(606\) −0.654115 −0.0265716
\(607\) 29.3832 1.19263 0.596314 0.802751i \(-0.296631\pi\)
0.596314 + 0.802751i \(0.296631\pi\)
\(608\) −3.77964 −0.153285
\(609\) −2.37592 −0.0962771
\(610\) −39.2937 −1.59095
\(611\) −53.9180 −2.18129
\(612\) 3.98441 0.161060
\(613\) 22.8059 0.921122 0.460561 0.887628i \(-0.347648\pi\)
0.460561 + 0.887628i \(0.347648\pi\)
\(614\) −23.5775 −0.951511
\(615\) −9.03916 −0.364494
\(616\) −29.5369 −1.19007
\(617\) 20.2767 0.816309 0.408154 0.912913i \(-0.366172\pi\)
0.408154 + 0.912913i \(0.366172\pi\)
\(618\) 2.93369 0.118010
\(619\) −20.2300 −0.813113 −0.406556 0.913626i \(-0.633271\pi\)
−0.406556 + 0.913626i \(0.633271\pi\)
\(620\) 1.13771 0.0456914
\(621\) 1.00000 0.0401286
\(622\) 9.66107 0.387374
\(623\) 20.7220 0.830211
\(624\) 4.40456 0.176323
\(625\) 56.0515 2.24206
\(626\) 17.9255 0.716448
\(627\) 3.07512 0.122808
\(628\) 18.3611 0.732686
\(629\) −12.5525 −0.500503
\(630\) 7.43647 0.296276
\(631\) 22.9942 0.915385 0.457693 0.889110i \(-0.348676\pi\)
0.457693 + 0.889110i \(0.348676\pi\)
\(632\) 25.0921 0.998111
\(633\) 3.26744 0.129869
\(634\) −6.91590 −0.274665
\(635\) −24.7833 −0.983494
\(636\) 5.26716 0.208856
\(637\) −7.03316 −0.278664
\(638\) −3.63674 −0.143980
\(639\) 14.2651 0.564317
\(640\) 39.4307 1.55863
\(641\) 30.1570 1.19113 0.595565 0.803307i \(-0.296928\pi\)
0.595565 + 0.803307i \(0.296928\pi\)
\(642\) 14.0933 0.556217
\(643\) −0.607328 −0.0239507 −0.0119753 0.999928i \(-0.503812\pi\)
−0.0119753 + 0.999928i \(0.503812\pi\)
\(644\) 3.36997 0.132795
\(645\) −44.2828 −1.74364
\(646\) 1.38152 0.0543552
\(647\) 11.4460 0.449989 0.224994 0.974360i \(-0.427764\pi\)
0.224994 + 0.974360i \(0.427764\pi\)
\(648\) −2.60699 −0.102412
\(649\) 32.4006 1.27183
\(650\) 46.8823 1.83888
\(651\) −0.464356 −0.0181996
\(652\) −11.4449 −0.448215
\(653\) −24.7401 −0.968153 −0.484077 0.875026i \(-0.660844\pi\)
−0.484077 + 0.875026i \(0.660844\pi\)
\(654\) −2.96133 −0.115797
\(655\) −67.6760 −2.64432
\(656\) 1.86898 0.0729713
\(657\) −7.86909 −0.307002
\(658\) 18.8224 0.733773
\(659\) 18.8865 0.735715 0.367858 0.929882i \(-0.380091\pi\)
0.367858 + 0.929882i \(0.380091\pi\)
\(660\) −27.7590 −1.08052
\(661\) −0.433103 −0.0168457 −0.00842287 0.999965i \(-0.502681\pi\)
−0.00842287 + 0.999965i \(0.502681\pi\)
\(662\) −13.0529 −0.507314
\(663\) −14.5808 −0.566270
\(664\) −38.1180 −1.47927
\(665\) −6.28806 −0.243841
\(666\) 3.40784 0.132051
\(667\) 1.00000 0.0387202
\(668\) −25.1696 −0.973839
\(669\) −16.3511 −0.632172
\(670\) 8.78318 0.339324
\(671\) −59.8661 −2.31111
\(672\) −13.9256 −0.537191
\(673\) 7.37791 0.284398 0.142199 0.989838i \(-0.454583\pi\)
0.142199 + 0.989838i \(0.454583\pi\)
\(674\) 7.42782 0.286109
\(675\) 11.8435 0.455858
\(676\) −19.7742 −0.760545
\(677\) −38.0288 −1.46156 −0.730782 0.682611i \(-0.760844\pi\)
−0.730782 + 0.682611i \(0.760844\pi\)
\(678\) 0.182028 0.00699075
\(679\) 32.7041 1.25507
\(680\) −30.0557 −1.15258
\(681\) 6.26229 0.239972
\(682\) −0.710774 −0.0272169
\(683\) 2.89906 0.110929 0.0554647 0.998461i \(-0.482336\pi\)
0.0554647 + 0.998461i \(0.482336\pi\)
\(684\) 0.914665 0.0349731
\(685\) −18.5483 −0.708694
\(686\) 15.1390 0.578009
\(687\) −2.82769 −0.107883
\(688\) 9.15612 0.349074
\(689\) −19.2749 −0.734316
\(690\) −3.12993 −0.119154
\(691\) 2.97532 0.113186 0.0565932 0.998397i \(-0.481976\pi\)
0.0565932 + 0.998397i \(0.481976\pi\)
\(692\) −33.9024 −1.28878
\(693\) 11.3299 0.430386
\(694\) −0.497384 −0.0188804
\(695\) −95.7363 −3.63149
\(696\) −2.60699 −0.0988176
\(697\) −6.18703 −0.234350
\(698\) 0.296463 0.0112213
\(699\) −6.61733 −0.250290
\(700\) 39.9123 1.50854
\(701\) −39.2450 −1.48226 −0.741132 0.671359i \(-0.765711\pi\)
−0.741132 + 0.671359i \(0.765711\pi\)
\(702\) 3.95848 0.149403
\(703\) −2.88157 −0.108681
\(704\) −13.2223 −0.498333
\(705\) 42.6325 1.60563
\(706\) 25.8633 0.973377
\(707\) 2.03783 0.0766404
\(708\) 9.63724 0.362190
\(709\) 21.1537 0.794445 0.397223 0.917722i \(-0.369974\pi\)
0.397223 + 0.917722i \(0.369974\pi\)
\(710\) −44.6487 −1.67563
\(711\) −9.62495 −0.360964
\(712\) 22.7373 0.852118
\(713\) 0.195443 0.00731939
\(714\) 5.09003 0.190490
\(715\) 101.583 3.79898
\(716\) 24.6954 0.922912
\(717\) 17.3956 0.649651
\(718\) 14.9346 0.557353
\(719\) −28.2170 −1.05232 −0.526159 0.850386i \(-0.676368\pi\)
−0.526159 + 0.850386i \(0.676368\pi\)
\(720\) −3.48264 −0.129790
\(721\) −9.13960 −0.340377
\(722\) −14.1730 −0.527463
\(723\) 3.26312 0.121357
\(724\) 12.5705 0.467177
\(725\) 11.8435 0.439858
\(726\) 8.95322 0.332285
\(727\) −1.17513 −0.0435833 −0.0217917 0.999763i \(-0.506937\pi\)
−0.0217917 + 0.999763i \(0.506937\pi\)
\(728\) 32.1500 1.19156
\(729\) 1.00000 0.0370370
\(730\) 24.6297 0.911587
\(731\) −30.3103 −1.12107
\(732\) −17.8066 −0.658152
\(733\) −30.9783 −1.14421 −0.572105 0.820181i \(-0.693873\pi\)
−0.572105 + 0.820181i \(0.693873\pi\)
\(734\) 9.05955 0.334394
\(735\) 5.56106 0.205123
\(736\) 5.86114 0.216044
\(737\) 13.3817 0.492920
\(738\) 1.67969 0.0618303
\(739\) 4.13263 0.152021 0.0760106 0.997107i \(-0.475782\pi\)
0.0760106 + 0.997107i \(0.475782\pi\)
\(740\) 26.0119 0.956215
\(741\) −3.34717 −0.122961
\(742\) 6.72872 0.247019
\(743\) 29.3385 1.07633 0.538163 0.842841i \(-0.319119\pi\)
0.538163 + 0.842841i \(0.319119\pi\)
\(744\) −0.509517 −0.0186798
\(745\) 50.8385 1.86258
\(746\) 27.0957 0.992043
\(747\) 14.6215 0.534972
\(748\) −19.0002 −0.694715
\(749\) −43.9062 −1.60430
\(750\) −21.4198 −0.782140
\(751\) −31.0677 −1.13368 −0.566838 0.823830i \(-0.691833\pi\)
−0.566838 + 0.823830i \(0.691833\pi\)
\(752\) −8.81489 −0.321446
\(753\) −23.6292 −0.861096
\(754\) 3.95848 0.144159
\(755\) 36.0911 1.31349
\(756\) 3.36997 0.122564
\(757\) −34.1761 −1.24215 −0.621076 0.783750i \(-0.713304\pi\)
−0.621076 + 0.783750i \(0.713304\pi\)
\(758\) 15.4106 0.559739
\(759\) −4.76863 −0.173090
\(760\) −6.89960 −0.250275
\(761\) 22.4770 0.814791 0.407395 0.913252i \(-0.366437\pi\)
0.407395 + 0.913252i \(0.366437\pi\)
\(762\) 4.60533 0.166833
\(763\) 9.22573 0.333994
\(764\) −18.2987 −0.662025
\(765\) 11.5289 0.416828
\(766\) −4.55586 −0.164610
\(767\) −35.2670 −1.27342
\(768\) −12.8727 −0.464503
\(769\) 18.3350 0.661175 0.330588 0.943775i \(-0.392753\pi\)
0.330588 + 0.943775i \(0.392753\pi\)
\(770\) −35.4617 −1.27795
\(771\) 4.73508 0.170530
\(772\) −4.91561 −0.176917
\(773\) 20.4086 0.734046 0.367023 0.930212i \(-0.380377\pi\)
0.367023 + 0.930212i \(0.380377\pi\)
\(774\) 8.22881 0.295779
\(775\) 2.31473 0.0831476
\(776\) 35.8848 1.28819
\(777\) −10.6168 −0.380875
\(778\) −8.30950 −0.297910
\(779\) −1.42030 −0.0508875
\(780\) 30.2148 1.08186
\(781\) −68.0248 −2.43412
\(782\) −2.14234 −0.0766100
\(783\) 1.00000 0.0357371
\(784\) −1.14983 −0.0410654
\(785\) 53.1276 1.89621
\(786\) 12.5758 0.448565
\(787\) −2.03075 −0.0723884 −0.0361942 0.999345i \(-0.511523\pi\)
−0.0361942 + 0.999345i \(0.511523\pi\)
\(788\) 14.2006 0.505874
\(789\) 31.4462 1.11952
\(790\) 30.1254 1.07181
\(791\) −0.567090 −0.0201634
\(792\) 12.4318 0.441743
\(793\) 65.1625 2.31399
\(794\) 18.6545 0.662024
\(795\) 15.2405 0.540525
\(796\) −12.9710 −0.459745
\(797\) 41.4799 1.46929 0.734647 0.678450i \(-0.237348\pi\)
0.734647 + 0.678450i \(0.237348\pi\)
\(798\) 1.16847 0.0413635
\(799\) 29.1806 1.03234
\(800\) 69.4165 2.45424
\(801\) −8.72169 −0.308166
\(802\) 14.2353 0.502664
\(803\) 37.5248 1.32422
\(804\) 3.98025 0.140373
\(805\) 9.75098 0.343677
\(806\) 0.773655 0.0272508
\(807\) −13.8706 −0.488267
\(808\) 2.23602 0.0786627
\(809\) −53.1717 −1.86942 −0.934710 0.355412i \(-0.884340\pi\)
−0.934710 + 0.355412i \(0.884340\pi\)
\(810\) −3.12993 −0.109975
\(811\) −52.5852 −1.84652 −0.923258 0.384181i \(-0.874484\pi\)
−0.923258 + 0.384181i \(0.874484\pi\)
\(812\) 3.36997 0.118263
\(813\) 14.7222 0.516331
\(814\) −16.2507 −0.569588
\(815\) −33.1156 −1.15999
\(816\) −2.38376 −0.0834484
\(817\) −6.95805 −0.243431
\(818\) 17.7699 0.621311
\(819\) −12.3322 −0.430923
\(820\) 12.8210 0.447729
\(821\) 47.8814 1.67107 0.835536 0.549436i \(-0.185157\pi\)
0.835536 + 0.549436i \(0.185157\pi\)
\(822\) 3.44672 0.120218
\(823\) 38.1344 1.32928 0.664641 0.747162i \(-0.268584\pi\)
0.664641 + 0.747162i \(0.268584\pi\)
\(824\) −10.0285 −0.349358
\(825\) −56.4774 −1.96629
\(826\) 12.3114 0.428370
\(827\) 43.3718 1.50818 0.754092 0.656769i \(-0.228077\pi\)
0.754092 + 0.656769i \(0.228077\pi\)
\(828\) −1.41838 −0.0492922
\(829\) −17.1770 −0.596581 −0.298290 0.954475i \(-0.596416\pi\)
−0.298290 + 0.954475i \(0.596416\pi\)
\(830\) −45.7642 −1.58850
\(831\) 28.6108 0.992498
\(832\) 14.3920 0.498954
\(833\) 3.80638 0.131883
\(834\) 17.7901 0.616021
\(835\) −72.8280 −2.52032
\(836\) −4.36170 −0.150853
\(837\) 0.195443 0.00675549
\(838\) −8.92766 −0.308401
\(839\) 50.2700 1.73551 0.867756 0.496991i \(-0.165562\pi\)
0.867756 + 0.496991i \(0.165562\pi\)
\(840\) −25.4207 −0.877097
\(841\) 1.00000 0.0344828
\(842\) −6.68301 −0.230312
\(843\) −1.91196 −0.0658513
\(844\) −4.63448 −0.159525
\(845\) −57.2165 −1.96831
\(846\) −7.92214 −0.272369
\(847\) −27.8928 −0.958409
\(848\) −3.15120 −0.108212
\(849\) −13.5965 −0.466632
\(850\) −25.3729 −0.870283
\(851\) 4.46849 0.153178
\(852\) −20.2333 −0.693182
\(853\) 18.6995 0.640258 0.320129 0.947374i \(-0.396274\pi\)
0.320129 + 0.947374i \(0.396274\pi\)
\(854\) −22.7477 −0.778411
\(855\) 2.64658 0.0905111
\(856\) −48.1762 −1.64663
\(857\) 6.86614 0.234543 0.117271 0.993100i \(-0.462585\pi\)
0.117271 + 0.993100i \(0.462585\pi\)
\(858\) −18.8765 −0.644433
\(859\) −30.9629 −1.05644 −0.528220 0.849108i \(-0.677140\pi\)
−0.528220 + 0.849108i \(0.677140\pi\)
\(860\) 62.8101 2.14181
\(861\) −5.23291 −0.178337
\(862\) 19.2430 0.655421
\(863\) 34.0856 1.16029 0.580143 0.814515i \(-0.302997\pi\)
0.580143 + 0.814515i \(0.302997\pi\)
\(864\) 5.86114 0.199400
\(865\) −98.0966 −3.33538
\(866\) 18.5865 0.631595
\(867\) −9.10883 −0.309352
\(868\) 0.658635 0.0223555
\(869\) 45.8978 1.55698
\(870\) −3.12993 −0.106115
\(871\) −14.5655 −0.493535
\(872\) 10.1230 0.342807
\(873\) −13.7648 −0.465869
\(874\) −0.491798 −0.0166353
\(875\) 66.7311 2.25592
\(876\) 11.1614 0.377108
\(877\) 14.0788 0.475406 0.237703 0.971338i \(-0.423605\pi\)
0.237703 + 0.971338i \(0.423605\pi\)
\(878\) 31.1358 1.05078
\(879\) −2.08364 −0.0702794
\(880\) 16.6074 0.559836
\(881\) −24.2508 −0.817030 −0.408515 0.912751i \(-0.633953\pi\)
−0.408515 + 0.912751i \(0.633953\pi\)
\(882\) −1.03338 −0.0347957
\(883\) 52.0180 1.75054 0.875272 0.483631i \(-0.160682\pi\)
0.875272 + 0.483631i \(0.160682\pi\)
\(884\) 20.6811 0.695581
\(885\) 27.8853 0.937355
\(886\) −21.0024 −0.705589
\(887\) 10.3246 0.346665 0.173333 0.984863i \(-0.444546\pi\)
0.173333 + 0.984863i \(0.444546\pi\)
\(888\) −11.6493 −0.390925
\(889\) −14.3474 −0.481197
\(890\) 27.2983 0.915041
\(891\) −4.76863 −0.159755
\(892\) 23.1922 0.776532
\(893\) 6.69874 0.224165
\(894\) −9.44702 −0.315956
\(895\) 71.4561 2.38851
\(896\) 22.8270 0.762597
\(897\) 5.19051 0.173306
\(898\) −20.6974 −0.690680
\(899\) 0.195443 0.00651838
\(900\) −16.7987 −0.559956
\(901\) 10.4317 0.347529
\(902\) −8.00983 −0.266698
\(903\) −25.6360 −0.853114
\(904\) −0.622242 −0.0206955
\(905\) 36.3726 1.20907
\(906\) −6.70660 −0.222812
\(907\) −36.6028 −1.21538 −0.607689 0.794175i \(-0.707903\pi\)
−0.607689 + 0.794175i \(0.707903\pi\)
\(908\) −8.88234 −0.294771
\(909\) −0.857700 −0.0284481
\(910\) 38.5990 1.27954
\(911\) −25.3597 −0.840205 −0.420102 0.907477i \(-0.638006\pi\)
−0.420102 + 0.907477i \(0.638006\pi\)
\(912\) −0.547219 −0.0181202
\(913\) −69.7244 −2.30754
\(914\) −8.72604 −0.288632
\(915\) −51.5234 −1.70331
\(916\) 4.01074 0.132519
\(917\) −39.1787 −1.29379
\(918\) −2.14234 −0.0707078
\(919\) −9.02212 −0.297612 −0.148806 0.988866i \(-0.547543\pi\)
−0.148806 + 0.988866i \(0.547543\pi\)
\(920\) 10.6993 0.352746
\(921\) −30.9157 −1.01871
\(922\) 14.2295 0.468624
\(923\) 74.0429 2.43715
\(924\) −16.0701 −0.528668
\(925\) 52.9227 1.74009
\(926\) 11.1630 0.366838
\(927\) 3.84676 0.126344
\(928\) 5.86114 0.192401
\(929\) 34.1265 1.11965 0.559827 0.828610i \(-0.310868\pi\)
0.559827 + 0.828610i \(0.310868\pi\)
\(930\) −0.611722 −0.0200592
\(931\) 0.873795 0.0286375
\(932\) 9.38591 0.307446
\(933\) 12.6680 0.414731
\(934\) 8.13735 0.266262
\(935\) −54.9769 −1.79794
\(936\) −13.5316 −0.442294
\(937\) −34.4584 −1.12571 −0.562854 0.826557i \(-0.690297\pi\)
−0.562854 + 0.826557i \(0.690297\pi\)
\(938\) 5.08472 0.166022
\(939\) 23.5046 0.767045
\(940\) −60.4693 −1.97229
\(941\) 9.00188 0.293453 0.146727 0.989177i \(-0.453126\pi\)
0.146727 + 0.989177i \(0.453126\pi\)
\(942\) −9.87239 −0.321660
\(943\) 2.20248 0.0717225
\(944\) −5.76570 −0.187657
\(945\) 9.75098 0.317199
\(946\) −39.2401 −1.27581
\(947\) 51.1277 1.66143 0.830713 0.556701i \(-0.187933\pi\)
0.830713 + 0.556701i \(0.187933\pi\)
\(948\) 13.6519 0.443392
\(949\) −40.8446 −1.32587
\(950\) −5.82462 −0.188976
\(951\) −9.06839 −0.294063
\(952\) −17.3997 −0.563927
\(953\) 7.73247 0.250479 0.125240 0.992127i \(-0.460030\pi\)
0.125240 + 0.992127i \(0.460030\pi\)
\(954\) −2.83205 −0.0916910
\(955\) −52.9473 −1.71334
\(956\) −24.6737 −0.798003
\(957\) −4.76863 −0.154148
\(958\) 24.4493 0.789922
\(959\) −10.7379 −0.346745
\(960\) −11.3797 −0.367277
\(961\) −30.9618 −0.998768
\(962\) 17.6884 0.570297
\(963\) 18.4797 0.595499
\(964\) −4.62835 −0.149069
\(965\) −14.2233 −0.457864
\(966\) −1.81197 −0.0582991
\(967\) 15.6598 0.503586 0.251793 0.967781i \(-0.418980\pi\)
0.251793 + 0.967781i \(0.418980\pi\)
\(968\) −30.6055 −0.983699
\(969\) 1.81150 0.0581939
\(970\) 43.0830 1.38331
\(971\) −21.6197 −0.693810 −0.346905 0.937900i \(-0.612767\pi\)
−0.346905 + 0.937900i \(0.612767\pi\)
\(972\) −1.41838 −0.0454947
\(973\) −55.4232 −1.77679
\(974\) −18.1612 −0.581924
\(975\) 61.4739 1.96874
\(976\) 10.6532 0.341001
\(977\) 5.57404 0.178329 0.0891647 0.996017i \(-0.471580\pi\)
0.0891647 + 0.996017i \(0.471580\pi\)
\(978\) 6.15368 0.196773
\(979\) 41.5905 1.32924
\(980\) −7.88772 −0.251964
\(981\) −3.88301 −0.123975
\(982\) −13.9400 −0.444843
\(983\) −43.2735 −1.38021 −0.690105 0.723710i \(-0.742435\pi\)
−0.690105 + 0.723710i \(0.742435\pi\)
\(984\) −5.74183 −0.183043
\(985\) 41.0892 1.30921
\(986\) −2.14234 −0.0682261
\(987\) 24.6806 0.785593
\(988\) 4.74758 0.151041
\(989\) 10.7899 0.343100
\(990\) 14.9255 0.474363
\(991\) 9.95405 0.316201 0.158100 0.987423i \(-0.449463\pi\)
0.158100 + 0.987423i \(0.449463\pi\)
\(992\) 1.14552 0.0363702
\(993\) −17.1154 −0.543141
\(994\) −25.8478 −0.819843
\(995\) −37.5315 −1.18983
\(996\) −20.7389 −0.657136
\(997\) 9.24724 0.292863 0.146431 0.989221i \(-0.453221\pi\)
0.146431 + 0.989221i \(0.453221\pi\)
\(998\) 3.70106 0.117155
\(999\) 4.46849 0.141377
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.o.1.13 20
3.2 odd 2 6003.2.a.s.1.8 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.o.1.13 20 1.1 even 1 trivial
6003.2.a.s.1.8 20 3.2 odd 2