Properties

Label 4022.2.a.e.1.18
Level $4022$
Weight $2$
Character 4022.1
Self dual yes
Analytic conductor $32.116$
Analytic rank $0$
Dimension $46$
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] = [4022,2,Mod(1,4022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4022, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4022 = 2 \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4022.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: \(32.1158316930\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) \(=\) 4022.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.848137 q^{3} +1.00000 q^{4} +2.52415 q^{5} +0.848137 q^{6} +4.28843 q^{7} -1.00000 q^{8} -2.28066 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.848137 q^{3} +1.00000 q^{4} +2.52415 q^{5} +0.848137 q^{6} +4.28843 q^{7} -1.00000 q^{8} -2.28066 q^{9} -2.52415 q^{10} +1.39012 q^{11} -0.848137 q^{12} +3.63558 q^{13} -4.28843 q^{14} -2.14082 q^{15} +1.00000 q^{16} -0.705537 q^{17} +2.28066 q^{18} +6.44216 q^{19} +2.52415 q^{20} -3.63717 q^{21} -1.39012 q^{22} +3.81495 q^{23} +0.848137 q^{24} +1.37132 q^{25} -3.63558 q^{26} +4.47873 q^{27} +4.28843 q^{28} -6.00320 q^{29} +2.14082 q^{30} +1.54915 q^{31} -1.00000 q^{32} -1.17901 q^{33} +0.705537 q^{34} +10.8246 q^{35} -2.28066 q^{36} +4.11262 q^{37} -6.44216 q^{38} -3.08347 q^{39} -2.52415 q^{40} -8.44136 q^{41} +3.63717 q^{42} +3.11793 q^{43} +1.39012 q^{44} -5.75673 q^{45} -3.81495 q^{46} -1.17750 q^{47} -0.848137 q^{48} +11.3906 q^{49} -1.37132 q^{50} +0.598392 q^{51} +3.63558 q^{52} -2.19908 q^{53} -4.47873 q^{54} +3.50886 q^{55} -4.28843 q^{56} -5.46383 q^{57} +6.00320 q^{58} -0.124586 q^{59} -2.14082 q^{60} +12.4020 q^{61} -1.54915 q^{62} -9.78046 q^{63} +1.00000 q^{64} +9.17674 q^{65} +1.17901 q^{66} -7.81056 q^{67} -0.705537 q^{68} -3.23560 q^{69} -10.8246 q^{70} +13.0693 q^{71} +2.28066 q^{72} +6.28186 q^{73} -4.11262 q^{74} -1.16307 q^{75} +6.44216 q^{76} +5.96141 q^{77} +3.08347 q^{78} +7.77058 q^{79} +2.52415 q^{80} +3.04341 q^{81} +8.44136 q^{82} +1.46305 q^{83} -3.63717 q^{84} -1.78088 q^{85} -3.11793 q^{86} +5.09154 q^{87} -1.39012 q^{88} +16.1746 q^{89} +5.75673 q^{90} +15.5909 q^{91} +3.81495 q^{92} -1.31389 q^{93} +1.17750 q^{94} +16.2609 q^{95} +0.848137 q^{96} -18.6424 q^{97} -11.3906 q^{98} -3.17039 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9} - 14 q^{10} - 6 q^{11} + 8 q^{12} + 37 q^{13} - 28 q^{14} + 9 q^{15} + 46 q^{16} + 6 q^{17} - 58 q^{18} + 18 q^{19} + 14 q^{20} + 19 q^{21} + 6 q^{22} - 4 q^{23} - 8 q^{24} + 86 q^{25} - 37 q^{26} + 32 q^{27} + 28 q^{28} + 15 q^{29} - 9 q^{30} + 18 q^{31} - 46 q^{32} + 37 q^{33} - 6 q^{34} - 2 q^{35} + 58 q^{36} + 74 q^{37} - 18 q^{38} - 3 q^{39} - 14 q^{40} - 18 q^{41} - 19 q^{42} + 25 q^{43} - 6 q^{44} + 94 q^{45} + 4 q^{46} + 18 q^{47} + 8 q^{48} + 92 q^{49} - 86 q^{50} - 10 q^{51} + 37 q^{52} + 17 q^{53} - 32 q^{54} + 37 q^{55} - 28 q^{56} + 43 q^{57} - 15 q^{58} - 24 q^{59} + 9 q^{60} + 46 q^{61} - 18 q^{62} + 80 q^{63} + 46 q^{64} + 24 q^{65} - 37 q^{66} + 61 q^{67} + 6 q^{68} + 59 q^{69} + 2 q^{70} - 8 q^{71} - 58 q^{72} + 101 q^{73} - 74 q^{74} + 34 q^{75} + 18 q^{76} + 40 q^{77} + 3 q^{78} + 9 q^{79} + 14 q^{80} + 58 q^{81} + 18 q^{82} + 18 q^{83} + 19 q^{84} + 60 q^{85} - 25 q^{86} + 20 q^{87} + 6 q^{88} - 25 q^{89} - 94 q^{90} + 51 q^{91} - 4 q^{92} + 63 q^{93} - 18 q^{94} - 31 q^{95} - 8 q^{96} + 76 q^{97} - 92 q^{98} + 21 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.848137 −0.489672 −0.244836 0.969564i \(-0.578734\pi\)
−0.244836 + 0.969564i \(0.578734\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.52415 1.12883 0.564416 0.825490i \(-0.309101\pi\)
0.564416 + 0.825490i \(0.309101\pi\)
\(6\) 0.848137 0.346251
\(7\) 4.28843 1.62087 0.810436 0.585827i \(-0.199230\pi\)
0.810436 + 0.585827i \(0.199230\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.28066 −0.760221
\(10\) −2.52415 −0.798205
\(11\) 1.39012 0.419136 0.209568 0.977794i \(-0.432794\pi\)
0.209568 + 0.977794i \(0.432794\pi\)
\(12\) −0.848137 −0.244836
\(13\) 3.63558 1.00833 0.504164 0.863608i \(-0.331801\pi\)
0.504164 + 0.863608i \(0.331801\pi\)
\(14\) −4.28843 −1.14613
\(15\) −2.14082 −0.552758
\(16\) 1.00000 0.250000
\(17\) −0.705537 −0.171118 −0.0855589 0.996333i \(-0.527268\pi\)
−0.0855589 + 0.996333i \(0.527268\pi\)
\(18\) 2.28066 0.537557
\(19\) 6.44216 1.47793 0.738966 0.673743i \(-0.235314\pi\)
0.738966 + 0.673743i \(0.235314\pi\)
\(20\) 2.52415 0.564416
\(21\) −3.63717 −0.793697
\(22\) −1.39012 −0.296374
\(23\) 3.81495 0.795473 0.397736 0.917500i \(-0.369796\pi\)
0.397736 + 0.917500i \(0.369796\pi\)
\(24\) 0.848137 0.173125
\(25\) 1.37132 0.274264
\(26\) −3.63558 −0.712996
\(27\) 4.47873 0.861932
\(28\) 4.28843 0.810436
\(29\) −6.00320 −1.11477 −0.557383 0.830255i \(-0.688195\pi\)
−0.557383 + 0.830255i \(0.688195\pi\)
\(30\) 2.14082 0.390859
\(31\) 1.54915 0.278235 0.139118 0.990276i \(-0.455573\pi\)
0.139118 + 0.990276i \(0.455573\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.17901 −0.205239
\(34\) 0.705537 0.120999
\(35\) 10.8246 1.82969
\(36\) −2.28066 −0.380111
\(37\) 4.11262 0.676111 0.338056 0.941126i \(-0.390231\pi\)
0.338056 + 0.941126i \(0.390231\pi\)
\(38\) −6.44216 −1.04506
\(39\) −3.08347 −0.493750
\(40\) −2.52415 −0.399103
\(41\) −8.44136 −1.31832 −0.659159 0.752003i \(-0.729088\pi\)
−0.659159 + 0.752003i \(0.729088\pi\)
\(42\) 3.63717 0.561228
\(43\) 3.11793 0.475481 0.237740 0.971329i \(-0.423593\pi\)
0.237740 + 0.971329i \(0.423593\pi\)
\(44\) 1.39012 0.209568
\(45\) −5.75673 −0.858162
\(46\) −3.81495 −0.562484
\(47\) −1.17750 −0.171756 −0.0858780 0.996306i \(-0.527370\pi\)
−0.0858780 + 0.996306i \(0.527370\pi\)
\(48\) −0.848137 −0.122418
\(49\) 11.3906 1.62723
\(50\) −1.37132 −0.193934
\(51\) 0.598392 0.0837917
\(52\) 3.63558 0.504164
\(53\) −2.19908 −0.302067 −0.151033 0.988529i \(-0.548260\pi\)
−0.151033 + 0.988529i \(0.548260\pi\)
\(54\) −4.47873 −0.609478
\(55\) 3.50886 0.473134
\(56\) −4.28843 −0.573065
\(57\) −5.46383 −0.723702
\(58\) 6.00320 0.788259
\(59\) −0.124586 −0.0162197 −0.00810985 0.999967i \(-0.502581\pi\)
−0.00810985 + 0.999967i \(0.502581\pi\)
\(60\) −2.14082 −0.276379
\(61\) 12.4020 1.58791 0.793955 0.607977i \(-0.208019\pi\)
0.793955 + 0.607977i \(0.208019\pi\)
\(62\) −1.54915 −0.196742
\(63\) −9.78046 −1.23222
\(64\) 1.00000 0.125000
\(65\) 9.17674 1.13823
\(66\) 1.17901 0.145126
\(67\) −7.81056 −0.954212 −0.477106 0.878846i \(-0.658314\pi\)
−0.477106 + 0.878846i \(0.658314\pi\)
\(68\) −0.705537 −0.0855589
\(69\) −3.23560 −0.389521
\(70\) −10.8246 −1.29379
\(71\) 13.0693 1.55104 0.775519 0.631324i \(-0.217488\pi\)
0.775519 + 0.631324i \(0.217488\pi\)
\(72\) 2.28066 0.268779
\(73\) 6.28186 0.735236 0.367618 0.929977i \(-0.380173\pi\)
0.367618 + 0.929977i \(0.380173\pi\)
\(74\) −4.11262 −0.478083
\(75\) −1.16307 −0.134299
\(76\) 6.44216 0.738966
\(77\) 5.96141 0.679366
\(78\) 3.08347 0.349134
\(79\) 7.77058 0.874258 0.437129 0.899399i \(-0.355995\pi\)
0.437129 + 0.899399i \(0.355995\pi\)
\(80\) 2.52415 0.282208
\(81\) 3.04341 0.338157
\(82\) 8.44136 0.932192
\(83\) 1.46305 0.160591 0.0802954 0.996771i \(-0.474414\pi\)
0.0802954 + 0.996771i \(0.474414\pi\)
\(84\) −3.63717 −0.396848
\(85\) −1.78088 −0.193163
\(86\) −3.11793 −0.336216
\(87\) 5.09154 0.545870
\(88\) −1.39012 −0.148187
\(89\) 16.1746 1.71451 0.857254 0.514893i \(-0.172168\pi\)
0.857254 + 0.514893i \(0.172168\pi\)
\(90\) 5.75673 0.606812
\(91\) 15.5909 1.63437
\(92\) 3.81495 0.397736
\(93\) −1.31389 −0.136244
\(94\) 1.17750 0.121450
\(95\) 16.2609 1.66834
\(96\) 0.848137 0.0865627
\(97\) −18.6424 −1.89285 −0.946426 0.322922i \(-0.895335\pi\)
−0.946426 + 0.322922i \(0.895335\pi\)
\(98\) −11.3906 −1.15062
\(99\) −3.17039 −0.318636
\(100\) 1.37132 0.137132
\(101\) −11.9626 −1.19033 −0.595163 0.803605i \(-0.702913\pi\)
−0.595163 + 0.803605i \(0.702913\pi\)
\(102\) −0.598392 −0.0592497
\(103\) −17.7134 −1.74535 −0.872675 0.488302i \(-0.837616\pi\)
−0.872675 + 0.488302i \(0.837616\pi\)
\(104\) −3.63558 −0.356498
\(105\) −9.18076 −0.895951
\(106\) 2.19908 0.213593
\(107\) −20.0166 −1.93508 −0.967539 0.252723i \(-0.918674\pi\)
−0.967539 + 0.252723i \(0.918674\pi\)
\(108\) 4.47873 0.430966
\(109\) 2.26646 0.217087 0.108544 0.994092i \(-0.465381\pi\)
0.108544 + 0.994092i \(0.465381\pi\)
\(110\) −3.50886 −0.334557
\(111\) −3.48807 −0.331073
\(112\) 4.28843 0.405218
\(113\) −14.5570 −1.36941 −0.684703 0.728822i \(-0.740068\pi\)
−0.684703 + 0.728822i \(0.740068\pi\)
\(114\) 5.46383 0.511735
\(115\) 9.62950 0.897956
\(116\) −6.00320 −0.557383
\(117\) −8.29153 −0.766552
\(118\) 0.124586 0.0114691
\(119\) −3.02564 −0.277360
\(120\) 2.14082 0.195430
\(121\) −9.06758 −0.824325
\(122\) −12.4020 −1.12282
\(123\) 7.15943 0.645544
\(124\) 1.54915 0.139118
\(125\) −9.15933 −0.819235
\(126\) 9.78046 0.871312
\(127\) 3.25025 0.288413 0.144207 0.989548i \(-0.453937\pi\)
0.144207 + 0.989548i \(0.453937\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.64444 −0.232830
\(130\) −9.17674 −0.804853
\(131\) 1.95777 0.171051 0.0855255 0.996336i \(-0.472743\pi\)
0.0855255 + 0.996336i \(0.472743\pi\)
\(132\) −1.17901 −0.102620
\(133\) 27.6267 2.39554
\(134\) 7.81056 0.674729
\(135\) 11.3050 0.972977
\(136\) 0.705537 0.0604993
\(137\) 21.4160 1.82969 0.914845 0.403806i \(-0.132313\pi\)
0.914845 + 0.403806i \(0.132313\pi\)
\(138\) 3.23560 0.275433
\(139\) −8.89425 −0.754401 −0.377200 0.926132i \(-0.623113\pi\)
−0.377200 + 0.926132i \(0.623113\pi\)
\(140\) 10.8246 0.914847
\(141\) 0.998681 0.0841041
\(142\) −13.0693 −1.09675
\(143\) 5.05388 0.422627
\(144\) −2.28066 −0.190055
\(145\) −15.1530 −1.25839
\(146\) −6.28186 −0.519890
\(147\) −9.66079 −0.796809
\(148\) 4.11262 0.338056
\(149\) −15.8776 −1.30075 −0.650373 0.759615i \(-0.725387\pi\)
−0.650373 + 0.759615i \(0.725387\pi\)
\(150\) 1.16307 0.0949639
\(151\) 14.3115 1.16465 0.582326 0.812956i \(-0.302143\pi\)
0.582326 + 0.812956i \(0.302143\pi\)
\(152\) −6.44216 −0.522528
\(153\) 1.60909 0.130087
\(154\) −5.96141 −0.480384
\(155\) 3.91028 0.314081
\(156\) −3.08347 −0.246875
\(157\) 16.3527 1.30509 0.652543 0.757752i \(-0.273702\pi\)
0.652543 + 0.757752i \(0.273702\pi\)
\(158\) −7.77058 −0.618194
\(159\) 1.86512 0.147914
\(160\) −2.52415 −0.199551
\(161\) 16.3601 1.28936
\(162\) −3.04341 −0.239113
\(163\) −6.71678 −0.526099 −0.263049 0.964782i \(-0.584728\pi\)
−0.263049 + 0.964782i \(0.584728\pi\)
\(164\) −8.44136 −0.659159
\(165\) −2.97599 −0.231681
\(166\) −1.46305 −0.113555
\(167\) −11.3429 −0.877740 −0.438870 0.898551i \(-0.644621\pi\)
−0.438870 + 0.898551i \(0.644621\pi\)
\(168\) 3.63717 0.280614
\(169\) 0.217437 0.0167259
\(170\) 1.78088 0.136587
\(171\) −14.6924 −1.12355
\(172\) 3.11793 0.237740
\(173\) 2.96943 0.225761 0.112881 0.993609i \(-0.463992\pi\)
0.112881 + 0.993609i \(0.463992\pi\)
\(174\) −5.09154 −0.385989
\(175\) 5.88080 0.444546
\(176\) 1.39012 0.104784
\(177\) 0.105666 0.00794234
\(178\) −16.1746 −1.21234
\(179\) −14.4536 −1.08031 −0.540156 0.841565i \(-0.681635\pi\)
−0.540156 + 0.841565i \(0.681635\pi\)
\(180\) −5.75673 −0.429081
\(181\) −16.4568 −1.22322 −0.611612 0.791158i \(-0.709479\pi\)
−0.611612 + 0.791158i \(0.709479\pi\)
\(182\) −15.5909 −1.15568
\(183\) −10.5186 −0.777555
\(184\) −3.81495 −0.281242
\(185\) 10.3809 0.763217
\(186\) 1.31389 0.0963391
\(187\) −0.980779 −0.0717216
\(188\) −1.17750 −0.0858780
\(189\) 19.2067 1.39708
\(190\) −16.2609 −1.17969
\(191\) −6.67151 −0.482734 −0.241367 0.970434i \(-0.577596\pi\)
−0.241367 + 0.970434i \(0.577596\pi\)
\(192\) −0.848137 −0.0612090
\(193\) −22.3603 −1.60953 −0.804763 0.593596i \(-0.797708\pi\)
−0.804763 + 0.593596i \(0.797708\pi\)
\(194\) 18.6424 1.33845
\(195\) −7.78313 −0.557362
\(196\) 11.3906 0.813614
\(197\) 5.84586 0.416500 0.208250 0.978076i \(-0.433223\pi\)
0.208250 + 0.978076i \(0.433223\pi\)
\(198\) 3.17039 0.225310
\(199\) 27.1617 1.92544 0.962721 0.270496i \(-0.0871877\pi\)
0.962721 + 0.270496i \(0.0871877\pi\)
\(200\) −1.37132 −0.0969668
\(201\) 6.62443 0.467251
\(202\) 11.9626 0.841688
\(203\) −25.7443 −1.80689
\(204\) 0.598392 0.0418958
\(205\) −21.3072 −1.48816
\(206\) 17.7134 1.23415
\(207\) −8.70062 −0.604735
\(208\) 3.63558 0.252082
\(209\) 8.95535 0.619454
\(210\) 9.18076 0.633533
\(211\) −23.6704 −1.62954 −0.814768 0.579787i \(-0.803136\pi\)
−0.814768 + 0.579787i \(0.803136\pi\)
\(212\) −2.19908 −0.151033
\(213\) −11.0845 −0.759501
\(214\) 20.0166 1.36831
\(215\) 7.87013 0.536738
\(216\) −4.47873 −0.304739
\(217\) 6.64341 0.450984
\(218\) −2.26646 −0.153504
\(219\) −5.32788 −0.360025
\(220\) 3.50886 0.236567
\(221\) −2.56504 −0.172543
\(222\) 3.48807 0.234104
\(223\) 17.8834 1.19756 0.598779 0.800914i \(-0.295653\pi\)
0.598779 + 0.800914i \(0.295653\pi\)
\(224\) −4.28843 −0.286533
\(225\) −3.12751 −0.208501
\(226\) 14.5570 0.968317
\(227\) 23.6615 1.57047 0.785235 0.619197i \(-0.212542\pi\)
0.785235 + 0.619197i \(0.212542\pi\)
\(228\) −5.46383 −0.361851
\(229\) 5.12897 0.338932 0.169466 0.985536i \(-0.445796\pi\)
0.169466 + 0.985536i \(0.445796\pi\)
\(230\) −9.62950 −0.634951
\(231\) −5.05610 −0.332667
\(232\) 6.00320 0.394129
\(233\) 22.6401 1.48320 0.741601 0.670841i \(-0.234067\pi\)
0.741601 + 0.670841i \(0.234067\pi\)
\(234\) 8.29153 0.542034
\(235\) −2.97218 −0.193884
\(236\) −0.124586 −0.00810985
\(237\) −6.59052 −0.428100
\(238\) 3.02564 0.196123
\(239\) −0.757583 −0.0490040 −0.0245020 0.999700i \(-0.507800\pi\)
−0.0245020 + 0.999700i \(0.507800\pi\)
\(240\) −2.14082 −0.138190
\(241\) 15.3907 0.991400 0.495700 0.868494i \(-0.334912\pi\)
0.495700 + 0.868494i \(0.334912\pi\)
\(242\) 9.06758 0.582886
\(243\) −16.0174 −1.02752
\(244\) 12.4020 0.793955
\(245\) 28.7516 1.83687
\(246\) −7.15943 −0.456469
\(247\) 23.4210 1.49024
\(248\) −1.54915 −0.0983710
\(249\) −1.24087 −0.0786369
\(250\) 9.15933 0.579287
\(251\) −5.72343 −0.361260 −0.180630 0.983551i \(-0.557814\pi\)
−0.180630 + 0.983551i \(0.557814\pi\)
\(252\) −9.78046 −0.616111
\(253\) 5.30323 0.333411
\(254\) −3.25025 −0.203939
\(255\) 1.51043 0.0945868
\(256\) 1.00000 0.0625000
\(257\) −0.205041 −0.0127901 −0.00639504 0.999980i \(-0.502036\pi\)
−0.00639504 + 0.999980i \(0.502036\pi\)
\(258\) 2.64444 0.164635
\(259\) 17.6367 1.09589
\(260\) 9.17674 0.569117
\(261\) 13.6913 0.847469
\(262\) −1.95777 −0.120951
\(263\) −25.7819 −1.58978 −0.794891 0.606753i \(-0.792472\pi\)
−0.794891 + 0.606753i \(0.792472\pi\)
\(264\) 1.17901 0.0725630
\(265\) −5.55079 −0.340983
\(266\) −27.6267 −1.69390
\(267\) −13.7183 −0.839547
\(268\) −7.81056 −0.477106
\(269\) 16.2087 0.988261 0.494130 0.869388i \(-0.335487\pi\)
0.494130 + 0.869388i \(0.335487\pi\)
\(270\) −11.3050 −0.687998
\(271\) −16.8037 −1.02075 −0.510376 0.859951i \(-0.670494\pi\)
−0.510376 + 0.859951i \(0.670494\pi\)
\(272\) −0.705537 −0.0427795
\(273\) −13.2232 −0.800307
\(274\) −21.4160 −1.29379
\(275\) 1.90629 0.114954
\(276\) −3.23560 −0.194760
\(277\) 3.73395 0.224351 0.112176 0.993688i \(-0.464218\pi\)
0.112176 + 0.993688i \(0.464218\pi\)
\(278\) 8.89425 0.533442
\(279\) −3.53308 −0.211520
\(280\) −10.8246 −0.646895
\(281\) 32.0279 1.91062 0.955312 0.295600i \(-0.0955196\pi\)
0.955312 + 0.295600i \(0.0955196\pi\)
\(282\) −0.998681 −0.0594706
\(283\) 11.3164 0.672691 0.336345 0.941739i \(-0.390809\pi\)
0.336345 + 0.941739i \(0.390809\pi\)
\(284\) 13.0693 0.775519
\(285\) −13.7915 −0.816939
\(286\) −5.05388 −0.298842
\(287\) −36.2002 −2.13683
\(288\) 2.28066 0.134389
\(289\) −16.5022 −0.970719
\(290\) 15.1530 0.889813
\(291\) 15.8113 0.926877
\(292\) 6.28186 0.367618
\(293\) 31.4834 1.83928 0.919639 0.392765i \(-0.128481\pi\)
0.919639 + 0.392765i \(0.128481\pi\)
\(294\) 9.66079 0.563429
\(295\) −0.314473 −0.0183093
\(296\) −4.11262 −0.239041
\(297\) 6.22595 0.361267
\(298\) 15.8776 0.919766
\(299\) 13.8696 0.802098
\(300\) −1.16307 −0.0671496
\(301\) 13.3710 0.770694
\(302\) −14.3115 −0.823533
\(303\) 10.1460 0.582870
\(304\) 6.44216 0.369483
\(305\) 31.3044 1.79248
\(306\) −1.60909 −0.0919857
\(307\) −28.2044 −1.60971 −0.804856 0.593470i \(-0.797758\pi\)
−0.804856 + 0.593470i \(0.797758\pi\)
\(308\) 5.96141 0.339683
\(309\) 15.0234 0.854649
\(310\) −3.91028 −0.222089
\(311\) 7.16541 0.406313 0.203156 0.979146i \(-0.434880\pi\)
0.203156 + 0.979146i \(0.434880\pi\)
\(312\) 3.08347 0.174567
\(313\) 31.5578 1.78375 0.891875 0.452281i \(-0.149390\pi\)
0.891875 + 0.452281i \(0.149390\pi\)
\(314\) −16.3527 −0.922835
\(315\) −24.6873 −1.39097
\(316\) 7.77058 0.437129
\(317\) 7.84681 0.440721 0.220360 0.975419i \(-0.429277\pi\)
0.220360 + 0.975419i \(0.429277\pi\)
\(318\) −1.86512 −0.104591
\(319\) −8.34515 −0.467239
\(320\) 2.52415 0.141104
\(321\) 16.9768 0.947554
\(322\) −16.3601 −0.911715
\(323\) −4.54518 −0.252901
\(324\) 3.04341 0.169079
\(325\) 4.98553 0.276548
\(326\) 6.71678 0.372008
\(327\) −1.92227 −0.106302
\(328\) 8.44136 0.466096
\(329\) −5.04962 −0.278395
\(330\) 2.97599 0.163823
\(331\) 18.8782 1.03764 0.518821 0.854883i \(-0.326371\pi\)
0.518821 + 0.854883i \(0.326371\pi\)
\(332\) 1.46305 0.0802954
\(333\) −9.37951 −0.513994
\(334\) 11.3429 0.620656
\(335\) −19.7150 −1.07715
\(336\) −3.63717 −0.198424
\(337\) 29.7801 1.62222 0.811112 0.584891i \(-0.198863\pi\)
0.811112 + 0.584891i \(0.198863\pi\)
\(338\) −0.217437 −0.0118270
\(339\) 12.3463 0.670561
\(340\) −1.78088 −0.0965817
\(341\) 2.15350 0.116618
\(342\) 14.6924 0.794473
\(343\) 18.8288 1.01666
\(344\) −3.11793 −0.168108
\(345\) −8.16714 −0.439704
\(346\) −2.96943 −0.159637
\(347\) 13.5235 0.725978 0.362989 0.931793i \(-0.381756\pi\)
0.362989 + 0.931793i \(0.381756\pi\)
\(348\) 5.09154 0.272935
\(349\) −17.9453 −0.960592 −0.480296 0.877106i \(-0.659471\pi\)
−0.480296 + 0.877106i \(0.659471\pi\)
\(350\) −5.88080 −0.314342
\(351\) 16.2828 0.869110
\(352\) −1.39012 −0.0740935
\(353\) −1.58860 −0.0845525 −0.0422762 0.999106i \(-0.513461\pi\)
−0.0422762 + 0.999106i \(0.513461\pi\)
\(354\) −0.105666 −0.00561608
\(355\) 32.9888 1.75086
\(356\) 16.1746 0.857254
\(357\) 2.56616 0.135816
\(358\) 14.4536 0.763895
\(359\) −7.53761 −0.397820 −0.198910 0.980018i \(-0.563740\pi\)
−0.198910 + 0.980018i \(0.563740\pi\)
\(360\) 5.75673 0.303406
\(361\) 22.5014 1.18428
\(362\) 16.4568 0.864950
\(363\) 7.69055 0.403649
\(364\) 15.5909 0.817186
\(365\) 15.8563 0.829958
\(366\) 10.5186 0.549815
\(367\) −22.2254 −1.16016 −0.580078 0.814561i \(-0.696978\pi\)
−0.580078 + 0.814561i \(0.696978\pi\)
\(368\) 3.81495 0.198868
\(369\) 19.2519 1.00221
\(370\) −10.3809 −0.539676
\(371\) −9.43058 −0.489611
\(372\) −1.31389 −0.0681220
\(373\) −3.50558 −0.181512 −0.0907560 0.995873i \(-0.528928\pi\)
−0.0907560 + 0.995873i \(0.528928\pi\)
\(374\) 0.980779 0.0507149
\(375\) 7.76837 0.401157
\(376\) 1.17750 0.0607249
\(377\) −21.8251 −1.12405
\(378\) −19.2067 −0.987886
\(379\) 31.1315 1.59912 0.799560 0.600587i \(-0.205066\pi\)
0.799560 + 0.600587i \(0.205066\pi\)
\(380\) 16.2609 0.834169
\(381\) −2.75666 −0.141228
\(382\) 6.67151 0.341344
\(383\) 1.47085 0.0751571 0.0375786 0.999294i \(-0.488036\pi\)
0.0375786 + 0.999294i \(0.488036\pi\)
\(384\) 0.848137 0.0432813
\(385\) 15.0475 0.766891
\(386\) 22.3603 1.13811
\(387\) −7.11096 −0.361470
\(388\) −18.6424 −0.946426
\(389\) −6.12789 −0.310697 −0.155348 0.987860i \(-0.549650\pi\)
−0.155348 + 0.987860i \(0.549650\pi\)
\(390\) 7.78313 0.394114
\(391\) −2.69159 −0.136120
\(392\) −11.3906 −0.575312
\(393\) −1.66046 −0.0837589
\(394\) −5.84586 −0.294510
\(395\) 19.6141 0.986891
\(396\) −3.17039 −0.159318
\(397\) −24.6791 −1.23861 −0.619304 0.785151i \(-0.712585\pi\)
−0.619304 + 0.785151i \(0.712585\pi\)
\(398\) −27.1617 −1.36149
\(399\) −23.4312 −1.17303
\(400\) 1.37132 0.0685659
\(401\) −12.5963 −0.629030 −0.314515 0.949252i \(-0.601842\pi\)
−0.314515 + 0.949252i \(0.601842\pi\)
\(402\) −6.62443 −0.330396
\(403\) 5.63205 0.280552
\(404\) −11.9626 −0.595163
\(405\) 7.68202 0.381723
\(406\) 25.7443 1.27767
\(407\) 5.71703 0.283383
\(408\) −0.598392 −0.0296248
\(409\) −26.8009 −1.32522 −0.662610 0.748964i \(-0.730551\pi\)
−0.662610 + 0.748964i \(0.730551\pi\)
\(410\) 21.3072 1.05229
\(411\) −18.1637 −0.895948
\(412\) −17.7134 −0.872675
\(413\) −0.534277 −0.0262901
\(414\) 8.70062 0.427612
\(415\) 3.69296 0.181280
\(416\) −3.63558 −0.178249
\(417\) 7.54355 0.369409
\(418\) −8.95535 −0.438020
\(419\) −17.4599 −0.852971 −0.426486 0.904494i \(-0.640249\pi\)
−0.426486 + 0.904494i \(0.640249\pi\)
\(420\) −9.18076 −0.447975
\(421\) 27.1884 1.32508 0.662541 0.749025i \(-0.269478\pi\)
0.662541 + 0.749025i \(0.269478\pi\)
\(422\) 23.6704 1.15226
\(423\) 2.68548 0.130572
\(424\) 2.19908 0.106797
\(425\) −0.967515 −0.0469314
\(426\) 11.0845 0.537048
\(427\) 53.1849 2.57380
\(428\) −20.0166 −0.967539
\(429\) −4.28638 −0.206949
\(430\) −7.87013 −0.379531
\(431\) 1.99033 0.0958706 0.0479353 0.998850i \(-0.484736\pi\)
0.0479353 + 0.998850i \(0.484736\pi\)
\(432\) 4.47873 0.215483
\(433\) −13.7589 −0.661210 −0.330605 0.943769i \(-0.607253\pi\)
−0.330605 + 0.943769i \(0.607253\pi\)
\(434\) −6.64341 −0.318894
\(435\) 12.8518 0.616196
\(436\) 2.26646 0.108544
\(437\) 24.5765 1.17565
\(438\) 5.32788 0.254576
\(439\) 9.47256 0.452101 0.226050 0.974116i \(-0.427419\pi\)
0.226050 + 0.974116i \(0.427419\pi\)
\(440\) −3.50886 −0.167278
\(441\) −25.9781 −1.23705
\(442\) 2.56504 0.122006
\(443\) 24.4260 1.16052 0.580258 0.814433i \(-0.302952\pi\)
0.580258 + 0.814433i \(0.302952\pi\)
\(444\) −3.48807 −0.165536
\(445\) 40.8272 1.93539
\(446\) −17.8834 −0.846801
\(447\) 13.4664 0.636939
\(448\) 4.28843 0.202609
\(449\) −32.5829 −1.53768 −0.768841 0.639441i \(-0.779166\pi\)
−0.768841 + 0.639441i \(0.779166\pi\)
\(450\) 3.12751 0.147432
\(451\) −11.7345 −0.552555
\(452\) −14.5570 −0.684703
\(453\) −12.1381 −0.570297
\(454\) −23.6615 −1.11049
\(455\) 39.3538 1.84493
\(456\) 5.46383 0.255867
\(457\) 41.3262 1.93316 0.966580 0.256367i \(-0.0825255\pi\)
0.966580 + 0.256367i \(0.0825255\pi\)
\(458\) −5.12897 −0.239661
\(459\) −3.15991 −0.147492
\(460\) 9.62950 0.448978
\(461\) −19.6357 −0.914528 −0.457264 0.889331i \(-0.651171\pi\)
−0.457264 + 0.889331i \(0.651171\pi\)
\(462\) 5.05610 0.235231
\(463\) 42.8568 1.99172 0.995861 0.0908841i \(-0.0289693\pi\)
0.995861 + 0.0908841i \(0.0289693\pi\)
\(464\) −6.00320 −0.278692
\(465\) −3.31645 −0.153797
\(466\) −22.6401 −1.04878
\(467\) −33.6907 −1.55902 −0.779509 0.626391i \(-0.784531\pi\)
−0.779509 + 0.626391i \(0.784531\pi\)
\(468\) −8.29153 −0.383276
\(469\) −33.4950 −1.54666
\(470\) 2.97218 0.137097
\(471\) −13.8693 −0.639065
\(472\) 0.124586 0.00573453
\(473\) 4.33429 0.199291
\(474\) 6.59052 0.302712
\(475\) 8.83424 0.405343
\(476\) −3.02564 −0.138680
\(477\) 5.01535 0.229637
\(478\) 0.757583 0.0346511
\(479\) 12.7300 0.581647 0.290824 0.956777i \(-0.406071\pi\)
0.290824 + 0.956777i \(0.406071\pi\)
\(480\) 2.14082 0.0977148
\(481\) 14.9518 0.681742
\(482\) −15.3907 −0.701026
\(483\) −13.8757 −0.631364
\(484\) −9.06758 −0.412163
\(485\) −47.0562 −2.13671
\(486\) 16.0174 0.726565
\(487\) −13.2547 −0.600627 −0.300313 0.953841i \(-0.597091\pi\)
−0.300313 + 0.953841i \(0.597091\pi\)
\(488\) −12.4020 −0.561411
\(489\) 5.69675 0.257616
\(490\) −28.7516 −1.29886
\(491\) 35.4462 1.59967 0.799833 0.600223i \(-0.204921\pi\)
0.799833 + 0.600223i \(0.204921\pi\)
\(492\) 7.15943 0.322772
\(493\) 4.23548 0.190756
\(494\) −23.4210 −1.05376
\(495\) −8.00253 −0.359687
\(496\) 1.54915 0.0695588
\(497\) 56.0467 2.51404
\(498\) 1.24087 0.0556047
\(499\) −19.7405 −0.883706 −0.441853 0.897087i \(-0.645679\pi\)
−0.441853 + 0.897087i \(0.645679\pi\)
\(500\) −9.15933 −0.409618
\(501\) 9.62034 0.429805
\(502\) 5.72343 0.255449
\(503\) −43.3502 −1.93289 −0.966445 0.256875i \(-0.917307\pi\)
−0.966445 + 0.256875i \(0.917307\pi\)
\(504\) 9.78046 0.435656
\(505\) −30.1954 −1.34368
\(506\) −5.30323 −0.235757
\(507\) −0.184417 −0.00819023
\(508\) 3.25025 0.144207
\(509\) −11.3968 −0.505154 −0.252577 0.967577i \(-0.581278\pi\)
−0.252577 + 0.967577i \(0.581278\pi\)
\(510\) −1.51043 −0.0668830
\(511\) 26.9393 1.19172
\(512\) −1.00000 −0.0441942
\(513\) 28.8527 1.27388
\(514\) 0.205041 0.00904395
\(515\) −44.7111 −1.97021
\(516\) −2.64444 −0.116415
\(517\) −1.63686 −0.0719891
\(518\) −17.6367 −0.774912
\(519\) −2.51848 −0.110549
\(520\) −9.17674 −0.402427
\(521\) 31.7470 1.39086 0.695432 0.718592i \(-0.255213\pi\)
0.695432 + 0.718592i \(0.255213\pi\)
\(522\) −13.6913 −0.599251
\(523\) 2.45982 0.107560 0.0537802 0.998553i \(-0.482873\pi\)
0.0537802 + 0.998553i \(0.482873\pi\)
\(524\) 1.95777 0.0855255
\(525\) −4.98772 −0.217682
\(526\) 25.7819 1.12415
\(527\) −1.09298 −0.0476110
\(528\) −1.17901 −0.0513098
\(529\) −8.44613 −0.367223
\(530\) 5.55079 0.241111
\(531\) 0.284138 0.0123306
\(532\) 27.6267 1.19777
\(533\) −30.6892 −1.32930
\(534\) 13.7183 0.593650
\(535\) −50.5248 −2.18438
\(536\) 7.81056 0.337365
\(537\) 12.2586 0.528999
\(538\) −16.2087 −0.698806
\(539\) 15.8343 0.682030
\(540\) 11.3050 0.486488
\(541\) 32.2425 1.38621 0.693106 0.720836i \(-0.256242\pi\)
0.693106 + 0.720836i \(0.256242\pi\)
\(542\) 16.8037 0.721781
\(543\) 13.9576 0.598979
\(544\) 0.705537 0.0302496
\(545\) 5.72087 0.245055
\(546\) 13.2232 0.565902
\(547\) 4.14648 0.177291 0.0886454 0.996063i \(-0.471746\pi\)
0.0886454 + 0.996063i \(0.471746\pi\)
\(548\) 21.4160 0.914845
\(549\) −28.2847 −1.20716
\(550\) −1.90629 −0.0812846
\(551\) −38.6736 −1.64755
\(552\) 3.23560 0.137716
\(553\) 33.3235 1.41706
\(554\) −3.73395 −0.158640
\(555\) −8.80440 −0.373726
\(556\) −8.89425 −0.377200
\(557\) −37.4512 −1.58686 −0.793429 0.608663i \(-0.791706\pi\)
−0.793429 + 0.608663i \(0.791706\pi\)
\(558\) 3.53308 0.149567
\(559\) 11.3355 0.479441
\(560\) 10.8246 0.457424
\(561\) 0.831835 0.0351201
\(562\) −32.0279 −1.35101
\(563\) 15.3689 0.647722 0.323861 0.946105i \(-0.395019\pi\)
0.323861 + 0.946105i \(0.395019\pi\)
\(564\) 0.998681 0.0420521
\(565\) −36.7440 −1.54583
\(566\) −11.3164 −0.475664
\(567\) 13.0515 0.548110
\(568\) −13.0693 −0.548375
\(569\) −35.2378 −1.47725 −0.738623 0.674119i \(-0.764524\pi\)
−0.738623 + 0.674119i \(0.764524\pi\)
\(570\) 13.7915 0.577663
\(571\) −4.23440 −0.177204 −0.0886021 0.996067i \(-0.528240\pi\)
−0.0886021 + 0.996067i \(0.528240\pi\)
\(572\) 5.05388 0.211313
\(573\) 5.65836 0.236381
\(574\) 36.2002 1.51097
\(575\) 5.23151 0.218169
\(576\) −2.28066 −0.0950276
\(577\) 18.6658 0.777069 0.388534 0.921434i \(-0.372981\pi\)
0.388534 + 0.921434i \(0.372981\pi\)
\(578\) 16.5022 0.686402
\(579\) 18.9646 0.788141
\(580\) −15.1530 −0.629193
\(581\) 6.27419 0.260297
\(582\) −15.8113 −0.655401
\(583\) −3.05697 −0.126607
\(584\) −6.28186 −0.259945
\(585\) −20.9290 −0.865309
\(586\) −31.4834 −1.30057
\(587\) 13.8080 0.569917 0.284958 0.958540i \(-0.408020\pi\)
0.284958 + 0.958540i \(0.408020\pi\)
\(588\) −9.66079 −0.398404
\(589\) 9.97985 0.411213
\(590\) 0.314473 0.0129466
\(591\) −4.95809 −0.203949
\(592\) 4.11262 0.169028
\(593\) 34.9226 1.43410 0.717049 0.697023i \(-0.245492\pi\)
0.717049 + 0.697023i \(0.245492\pi\)
\(594\) −6.22595 −0.255454
\(595\) −7.63717 −0.313093
\(596\) −15.8776 −0.650373
\(597\) −23.0369 −0.942836
\(598\) −13.8696 −0.567169
\(599\) −2.59642 −0.106087 −0.0530433 0.998592i \(-0.516892\pi\)
−0.0530433 + 0.998592i \(0.516892\pi\)
\(600\) 1.16307 0.0474820
\(601\) −27.0250 −1.10237 −0.551186 0.834382i \(-0.685825\pi\)
−0.551186 + 0.834382i \(0.685825\pi\)
\(602\) −13.3710 −0.544963
\(603\) 17.8133 0.725412
\(604\) 14.3115 0.582326
\(605\) −22.8879 −0.930525
\(606\) −10.1460 −0.412151
\(607\) −31.8214 −1.29159 −0.645795 0.763511i \(-0.723474\pi\)
−0.645795 + 0.763511i \(0.723474\pi\)
\(608\) −6.44216 −0.261264
\(609\) 21.8347 0.884786
\(610\) −31.3044 −1.26748
\(611\) −4.28089 −0.173186
\(612\) 1.60909 0.0650437
\(613\) 37.5120 1.51510 0.757549 0.652779i \(-0.226397\pi\)
0.757549 + 0.652779i \(0.226397\pi\)
\(614\) 28.2044 1.13824
\(615\) 18.0715 0.728712
\(616\) −5.96141 −0.240192
\(617\) 6.35395 0.255800 0.127900 0.991787i \(-0.459176\pi\)
0.127900 + 0.991787i \(0.459176\pi\)
\(618\) −15.0234 −0.604328
\(619\) −38.1690 −1.53414 −0.767070 0.641563i \(-0.778286\pi\)
−0.767070 + 0.641563i \(0.778286\pi\)
\(620\) 3.91028 0.157040
\(621\) 17.0861 0.685643
\(622\) −7.16541 −0.287307
\(623\) 69.3638 2.77900
\(624\) −3.08347 −0.123438
\(625\) −29.9761 −1.19904
\(626\) −31.5578 −1.26130
\(627\) −7.59537 −0.303330
\(628\) 16.3527 0.652543
\(629\) −2.90161 −0.115695
\(630\) 24.6873 0.983566
\(631\) −15.5718 −0.619904 −0.309952 0.950752i \(-0.600313\pi\)
−0.309952 + 0.950752i \(0.600313\pi\)
\(632\) −7.77058 −0.309097
\(633\) 20.0757 0.797939
\(634\) −7.84681 −0.311637
\(635\) 8.20411 0.325570
\(636\) 1.86512 0.0739568
\(637\) 41.4114 1.64078
\(638\) 8.34515 0.330388
\(639\) −29.8066 −1.17913
\(640\) −2.52415 −0.0997757
\(641\) −26.1849 −1.03424 −0.517121 0.855913i \(-0.672996\pi\)
−0.517121 + 0.855913i \(0.672996\pi\)
\(642\) −16.9768 −0.670022
\(643\) 6.17217 0.243407 0.121703 0.992567i \(-0.461164\pi\)
0.121703 + 0.992567i \(0.461164\pi\)
\(644\) 16.3601 0.644680
\(645\) −6.67495 −0.262826
\(646\) 4.54518 0.178828
\(647\) −10.5034 −0.412931 −0.206465 0.978454i \(-0.566196\pi\)
−0.206465 + 0.978454i \(0.566196\pi\)
\(648\) −3.04341 −0.119557
\(649\) −0.173189 −0.00679826
\(650\) −4.98553 −0.195549
\(651\) −5.63452 −0.220834
\(652\) −6.71678 −0.263049
\(653\) 12.3841 0.484628 0.242314 0.970198i \(-0.422094\pi\)
0.242314 + 0.970198i \(0.422094\pi\)
\(654\) 1.92227 0.0751666
\(655\) 4.94169 0.193088
\(656\) −8.44136 −0.329580
\(657\) −14.3268 −0.558942
\(658\) 5.04962 0.196855
\(659\) −8.97817 −0.349740 −0.174870 0.984592i \(-0.555951\pi\)
−0.174870 + 0.984592i \(0.555951\pi\)
\(660\) −2.97599 −0.115840
\(661\) −26.8418 −1.04403 −0.522013 0.852938i \(-0.674819\pi\)
−0.522013 + 0.852938i \(0.674819\pi\)
\(662\) −18.8782 −0.733723
\(663\) 2.17550 0.0844895
\(664\) −1.46305 −0.0567774
\(665\) 69.7339 2.70416
\(666\) 9.37951 0.363449
\(667\) −22.9019 −0.886766
\(668\) −11.3429 −0.438870
\(669\) −15.1675 −0.586411
\(670\) 19.7150 0.761657
\(671\) 17.2402 0.665550
\(672\) 3.63717 0.140307
\(673\) 14.8725 0.573294 0.286647 0.958036i \(-0.407459\pi\)
0.286647 + 0.958036i \(0.407459\pi\)
\(674\) −29.7801 −1.14709
\(675\) 6.14176 0.236396
\(676\) 0.217437 0.00836297
\(677\) 2.44642 0.0940237 0.0470118 0.998894i \(-0.485030\pi\)
0.0470118 + 0.998894i \(0.485030\pi\)
\(678\) −12.3463 −0.474158
\(679\) −79.9467 −3.06807
\(680\) 1.78088 0.0682936
\(681\) −20.0682 −0.769016
\(682\) −2.15350 −0.0824616
\(683\) 7.15665 0.273842 0.136921 0.990582i \(-0.456279\pi\)
0.136921 + 0.990582i \(0.456279\pi\)
\(684\) −14.6924 −0.561777
\(685\) 54.0570 2.06541
\(686\) −18.8288 −0.718886
\(687\) −4.35007 −0.165965
\(688\) 3.11793 0.118870
\(689\) −7.99492 −0.304582
\(690\) 8.16714 0.310918
\(691\) −27.0859 −1.03040 −0.515198 0.857071i \(-0.672282\pi\)
−0.515198 + 0.857071i \(0.672282\pi\)
\(692\) 2.96943 0.112881
\(693\) −13.5960 −0.516468
\(694\) −13.5235 −0.513344
\(695\) −22.4504 −0.851592
\(696\) −5.09154 −0.192994
\(697\) 5.95569 0.225588
\(698\) 17.9453 0.679241
\(699\) −19.2019 −0.726283
\(700\) 5.88080 0.222273
\(701\) 50.3868 1.90308 0.951542 0.307518i \(-0.0994984\pi\)
0.951542 + 0.307518i \(0.0994984\pi\)
\(702\) −16.2828 −0.614554
\(703\) 26.4942 0.999246
\(704\) 1.39012 0.0523920
\(705\) 2.52082 0.0949395
\(706\) 1.58860 0.0597876
\(707\) −51.3009 −1.92937
\(708\) 0.105666 0.00397117
\(709\) 11.8120 0.443610 0.221805 0.975091i \(-0.428805\pi\)
0.221805 + 0.975091i \(0.428805\pi\)
\(710\) −32.9888 −1.23805
\(711\) −17.7221 −0.664629
\(712\) −16.1746 −0.606170
\(713\) 5.90993 0.221328
\(714\) −2.56616 −0.0960362
\(715\) 12.7567 0.477075
\(716\) −14.4536 −0.540156
\(717\) 0.642535 0.0239959
\(718\) 7.53761 0.281301
\(719\) 30.3062 1.13023 0.565115 0.825012i \(-0.308832\pi\)
0.565115 + 0.825012i \(0.308832\pi\)
\(720\) −5.75673 −0.214541
\(721\) −75.9624 −2.82899
\(722\) −22.5014 −0.837414
\(723\) −13.0534 −0.485461
\(724\) −16.4568 −0.611612
\(725\) −8.23230 −0.305740
\(726\) −7.69055 −0.285423
\(727\) 6.54366 0.242691 0.121345 0.992610i \(-0.461279\pi\)
0.121345 + 0.992610i \(0.461279\pi\)
\(728\) −15.5909 −0.577838
\(729\) 4.45473 0.164990
\(730\) −15.8563 −0.586869
\(731\) −2.19982 −0.0813632
\(732\) −10.5186 −0.388778
\(733\) −1.62664 −0.0600813 −0.0300406 0.999549i \(-0.509564\pi\)
−0.0300406 + 0.999549i \(0.509564\pi\)
\(734\) 22.2254 0.820354
\(735\) −24.3853 −0.899464
\(736\) −3.81495 −0.140621
\(737\) −10.8576 −0.399944
\(738\) −19.2519 −0.708672
\(739\) 40.7557 1.49922 0.749612 0.661878i \(-0.230240\pi\)
0.749612 + 0.661878i \(0.230240\pi\)
\(740\) 10.3809 0.381608
\(741\) −19.8642 −0.729730
\(742\) 9.43058 0.346208
\(743\) 5.56392 0.204120 0.102060 0.994778i \(-0.467457\pi\)
0.102060 + 0.994778i \(0.467457\pi\)
\(744\) 1.31389 0.0481695
\(745\) −40.0775 −1.46832
\(746\) 3.50558 0.128348
\(747\) −3.33673 −0.122085
\(748\) −0.980779 −0.0358608
\(749\) −85.8397 −3.13651
\(750\) −7.76837 −0.283661
\(751\) −26.1738 −0.955096 −0.477548 0.878606i \(-0.658474\pi\)
−0.477548 + 0.878606i \(0.658474\pi\)
\(752\) −1.17750 −0.0429390
\(753\) 4.85426 0.176899
\(754\) 21.8251 0.794824
\(755\) 36.1243 1.31470
\(756\) 19.2067 0.698541
\(757\) −8.97940 −0.326362 −0.163181 0.986596i \(-0.552175\pi\)
−0.163181 + 0.986596i \(0.552175\pi\)
\(758\) −31.1315 −1.13075
\(759\) −4.49787 −0.163262
\(760\) −16.2609 −0.589847
\(761\) 22.7813 0.825821 0.412911 0.910772i \(-0.364512\pi\)
0.412911 + 0.910772i \(0.364512\pi\)
\(762\) 2.75666 0.0998632
\(763\) 9.71954 0.351871
\(764\) −6.67151 −0.241367
\(765\) 4.06158 0.146847
\(766\) −1.47085 −0.0531441
\(767\) −0.452942 −0.0163548
\(768\) −0.848137 −0.0306045
\(769\) −9.53046 −0.343677 −0.171839 0.985125i \(-0.554971\pi\)
−0.171839 + 0.985125i \(0.554971\pi\)
\(770\) −15.0475 −0.542274
\(771\) 0.173903 0.00626295
\(772\) −22.3603 −0.804763
\(773\) −46.0981 −1.65803 −0.829016 0.559225i \(-0.811099\pi\)
−0.829016 + 0.559225i \(0.811099\pi\)
\(774\) 7.11096 0.255598
\(775\) 2.12437 0.0763098
\(776\) 18.6424 0.669224
\(777\) −14.9583 −0.536627
\(778\) 6.12789 0.219696
\(779\) −54.3806 −1.94839
\(780\) −7.78313 −0.278681
\(781\) 18.1678 0.650096
\(782\) 2.69159 0.0962511
\(783\) −26.8867 −0.960852
\(784\) 11.3906 0.406807
\(785\) 41.2766 1.47322
\(786\) 1.66046 0.0592265
\(787\) 27.5820 0.983193 0.491596 0.870823i \(-0.336414\pi\)
0.491596 + 0.870823i \(0.336414\pi\)
\(788\) 5.84586 0.208250
\(789\) 21.8666 0.778472
\(790\) −19.6141 −0.697838
\(791\) −62.4266 −2.21963
\(792\) 3.17039 0.112655
\(793\) 45.0883 1.60113
\(794\) 24.6791 0.875829
\(795\) 4.70784 0.166970
\(796\) 27.1617 0.962721
\(797\) 5.33961 0.189139 0.0945693 0.995518i \(-0.469853\pi\)
0.0945693 + 0.995518i \(0.469853\pi\)
\(798\) 23.4312 0.829457
\(799\) 0.830769 0.0293905
\(800\) −1.37132 −0.0484834
\(801\) −36.8889 −1.30341
\(802\) 12.5963 0.444791
\(803\) 8.73252 0.308164
\(804\) 6.62443 0.233626
\(805\) 41.2954 1.45547
\(806\) −5.63205 −0.198380
\(807\) −13.7472 −0.483924
\(808\) 11.9626 0.420844
\(809\) 54.8261 1.92758 0.963792 0.266656i \(-0.0859186\pi\)
0.963792 + 0.266656i \(0.0859186\pi\)
\(810\) −7.68202 −0.269919
\(811\) 21.9104 0.769377 0.384689 0.923046i \(-0.374309\pi\)
0.384689 + 0.923046i \(0.374309\pi\)
\(812\) −25.7443 −0.903447
\(813\) 14.2519 0.499834
\(814\) −5.71703 −0.200382
\(815\) −16.9541 −0.593878
\(816\) 0.598392 0.0209479
\(817\) 20.0862 0.702728
\(818\) 26.8009 0.937073
\(819\) −35.5576 −1.24248
\(820\) −21.3072 −0.744081
\(821\) −5.04146 −0.175948 −0.0879742 0.996123i \(-0.528039\pi\)
−0.0879742 + 0.996123i \(0.528039\pi\)
\(822\) 18.1637 0.633531
\(823\) −1.29802 −0.0452462 −0.0226231 0.999744i \(-0.507202\pi\)
−0.0226231 + 0.999744i \(0.507202\pi\)
\(824\) 17.7134 0.617074
\(825\) −1.61680 −0.0562897
\(826\) 0.534277 0.0185899
\(827\) 46.6641 1.62267 0.811336 0.584581i \(-0.198741\pi\)
0.811336 + 0.584581i \(0.198741\pi\)
\(828\) −8.70062 −0.302368
\(829\) −26.0194 −0.903691 −0.451845 0.892096i \(-0.649234\pi\)
−0.451845 + 0.892096i \(0.649234\pi\)
\(830\) −3.69296 −0.128184
\(831\) −3.16690 −0.109858
\(832\) 3.63558 0.126041
\(833\) −8.03649 −0.278448
\(834\) −7.54355 −0.261212
\(835\) −28.6312 −0.990822
\(836\) 8.95535 0.309727
\(837\) 6.93821 0.239820
\(838\) 17.4599 0.603142
\(839\) −14.8946 −0.514219 −0.257109 0.966382i \(-0.582770\pi\)
−0.257109 + 0.966382i \(0.582770\pi\)
\(840\) 9.18076 0.316766
\(841\) 7.03843 0.242704
\(842\) −27.1884 −0.936975
\(843\) −27.1640 −0.935579
\(844\) −23.6704 −0.814768
\(845\) 0.548843 0.0188808
\(846\) −2.68548 −0.0923287
\(847\) −38.8856 −1.33613
\(848\) −2.19908 −0.0755166
\(849\) −9.59787 −0.329398
\(850\) 0.967515 0.0331855
\(851\) 15.6895 0.537828
\(852\) −11.0845 −0.379750
\(853\) −3.99597 −0.136819 −0.0684097 0.997657i \(-0.521793\pi\)
−0.0684097 + 0.997657i \(0.521793\pi\)
\(854\) −53.1849 −1.81995
\(855\) −37.0857 −1.26831
\(856\) 20.0166 0.684153
\(857\) 18.8330 0.643324 0.321662 0.946854i \(-0.395758\pi\)
0.321662 + 0.946854i \(0.395758\pi\)
\(858\) 4.28638 0.146335
\(859\) 9.39350 0.320502 0.160251 0.987076i \(-0.448770\pi\)
0.160251 + 0.987076i \(0.448770\pi\)
\(860\) 7.87013 0.268369
\(861\) 30.7027 1.04635
\(862\) −1.99033 −0.0677908
\(863\) −4.32213 −0.147127 −0.0735636 0.997291i \(-0.523437\pi\)
−0.0735636 + 0.997291i \(0.523437\pi\)
\(864\) −4.47873 −0.152369
\(865\) 7.49527 0.254847
\(866\) 13.7589 0.467546
\(867\) 13.9961 0.475334
\(868\) 6.64341 0.225492
\(869\) 10.8020 0.366433
\(870\) −12.8518 −0.435717
\(871\) −28.3959 −0.962159
\(872\) −2.26646 −0.0767520
\(873\) 42.5171 1.43899
\(874\) −24.5765 −0.831313
\(875\) −39.2791 −1.32788
\(876\) −5.32788 −0.180012
\(877\) −33.1618 −1.11979 −0.559897 0.828562i \(-0.689159\pi\)
−0.559897 + 0.828562i \(0.689159\pi\)
\(878\) −9.47256 −0.319683
\(879\) −26.7022 −0.900643
\(880\) 3.50886 0.118284
\(881\) −13.3574 −0.450023 −0.225012 0.974356i \(-0.572242\pi\)
−0.225012 + 0.974356i \(0.572242\pi\)
\(882\) 25.9781 0.874729
\(883\) 20.9771 0.705936 0.352968 0.935635i \(-0.385172\pi\)
0.352968 + 0.935635i \(0.385172\pi\)
\(884\) −2.56504 −0.0862715
\(885\) 0.266716 0.00896557
\(886\) −24.4260 −0.820609
\(887\) −28.5477 −0.958537 −0.479269 0.877668i \(-0.659098\pi\)
−0.479269 + 0.877668i \(0.659098\pi\)
\(888\) 3.48807 0.117052
\(889\) 13.9385 0.467481
\(890\) −40.8272 −1.36853
\(891\) 4.23070 0.141734
\(892\) 17.8834 0.598779
\(893\) −7.58564 −0.253844
\(894\) −13.4664 −0.450384
\(895\) −36.4830 −1.21949
\(896\) −4.28843 −0.143266
\(897\) −11.7633 −0.392765
\(898\) 32.5829 1.08730
\(899\) −9.29985 −0.310167
\(900\) −3.12751 −0.104250
\(901\) 1.55153 0.0516890
\(902\) 11.7345 0.390715
\(903\) −11.3405 −0.377387
\(904\) 14.5570 0.484158
\(905\) −41.5394 −1.38082
\(906\) 12.1381 0.403261
\(907\) 9.71651 0.322632 0.161316 0.986903i \(-0.448426\pi\)
0.161316 + 0.986903i \(0.448426\pi\)
\(908\) 23.6615 0.785235
\(909\) 27.2827 0.904911
\(910\) −39.3538 −1.30456
\(911\) 5.99946 0.198771 0.0993855 0.995049i \(-0.468312\pi\)
0.0993855 + 0.995049i \(0.468312\pi\)
\(912\) −5.46383 −0.180926
\(913\) 2.03381 0.0673094
\(914\) −41.3262 −1.36695
\(915\) −26.5504 −0.877730
\(916\) 5.12897 0.169466
\(917\) 8.39574 0.277252
\(918\) 3.15991 0.104292
\(919\) −27.5373 −0.908372 −0.454186 0.890907i \(-0.650070\pi\)
−0.454186 + 0.890907i \(0.650070\pi\)
\(920\) −9.62950 −0.317475
\(921\) 23.9212 0.788232
\(922\) 19.6357 0.646669
\(923\) 47.5144 1.56396
\(924\) −5.05610 −0.166333
\(925\) 5.63972 0.185433
\(926\) −42.8568 −1.40836
\(927\) 40.3982 1.32685
\(928\) 6.00320 0.197065
\(929\) −47.4312 −1.55617 −0.778083 0.628161i \(-0.783808\pi\)
−0.778083 + 0.628161i \(0.783808\pi\)
\(930\) 3.31645 0.108751
\(931\) 73.3800 2.40493
\(932\) 22.6401 0.741601
\(933\) −6.07725 −0.198960
\(934\) 33.6907 1.10239
\(935\) −2.47563 −0.0809617
\(936\) 8.29153 0.271017
\(937\) 37.8678 1.23709 0.618543 0.785751i \(-0.287723\pi\)
0.618543 + 0.785751i \(0.287723\pi\)
\(938\) 33.4950 1.09365
\(939\) −26.7653 −0.873453
\(940\) −2.97218 −0.0969419
\(941\) −7.09904 −0.231422 −0.115711 0.993283i \(-0.536915\pi\)
−0.115711 + 0.993283i \(0.536915\pi\)
\(942\) 13.8693 0.451887
\(943\) −32.2034 −1.04869
\(944\) −0.124586 −0.00405492
\(945\) 48.4805 1.57707
\(946\) −4.33429 −0.140920
\(947\) 22.4187 0.728511 0.364255 0.931299i \(-0.381323\pi\)
0.364255 + 0.931299i \(0.381323\pi\)
\(948\) −6.59052 −0.214050
\(949\) 22.8382 0.741359
\(950\) −8.83424 −0.286621
\(951\) −6.65517 −0.215809
\(952\) 3.02564 0.0980617
\(953\) 26.0659 0.844357 0.422178 0.906513i \(-0.361266\pi\)
0.422178 + 0.906513i \(0.361266\pi\)
\(954\) −5.01535 −0.162378
\(955\) −16.8399 −0.544926
\(956\) −0.757583 −0.0245020
\(957\) 7.07783 0.228794
\(958\) −12.7300 −0.411287
\(959\) 91.8408 2.96569
\(960\) −2.14082 −0.0690948
\(961\) −28.6001 −0.922585
\(962\) −14.9518 −0.482065
\(963\) 45.6511 1.47109
\(964\) 15.3907 0.495700
\(965\) −56.4406 −1.81689
\(966\) 13.8757 0.446442
\(967\) −16.8371 −0.541444 −0.270722 0.962658i \(-0.587262\pi\)
−0.270722 + 0.962658i \(0.587262\pi\)
\(968\) 9.06758 0.291443
\(969\) 3.85494 0.123838
\(970\) 47.0562 1.51088
\(971\) 7.81667 0.250849 0.125424 0.992103i \(-0.459971\pi\)
0.125424 + 0.992103i \(0.459971\pi\)
\(972\) −16.0174 −0.513759
\(973\) −38.1423 −1.22279
\(974\) 13.2547 0.424707
\(975\) −4.22842 −0.135418
\(976\) 12.4020 0.396977
\(977\) −35.0066 −1.11996 −0.559980 0.828506i \(-0.689191\pi\)
−0.559980 + 0.828506i \(0.689191\pi\)
\(978\) −5.69675 −0.182162
\(979\) 22.4846 0.718612
\(980\) 28.7516 0.918435
\(981\) −5.16903 −0.165034
\(982\) −35.4462 −1.13113
\(983\) −53.0322 −1.69146 −0.845732 0.533608i \(-0.820836\pi\)
−0.845732 + 0.533608i \(0.820836\pi\)
\(984\) −7.15943 −0.228234
\(985\) 14.7558 0.470159
\(986\) −4.23548 −0.134885
\(987\) 4.28277 0.136322
\(988\) 23.4210 0.745120
\(989\) 11.8948 0.378232
\(990\) 8.00253 0.254337
\(991\) 37.6006 1.19442 0.597211 0.802084i \(-0.296276\pi\)
0.597211 + 0.802084i \(0.296276\pi\)
\(992\) −1.54915 −0.0491855
\(993\) −16.0113 −0.508104
\(994\) −56.0467 −1.77769
\(995\) 68.5601 2.17350
\(996\) −1.24087 −0.0393184
\(997\) −33.2451 −1.05288 −0.526441 0.850212i \(-0.676474\pi\)
−0.526441 + 0.850212i \(0.676474\pi\)
\(998\) 19.7405 0.624874
\(999\) 18.4193 0.582762
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 4022.2.a.e.1.18 46
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4022.2.a.e.1.18 46 1.1 even 1 trivial