Properties

Label 4010.2.a.m.1.8
Level $4010$
Weight $2$
Character 4010.1
Self dual yes
Analytic conductor $32.020$
Analytic rank $0$
Dimension $20$
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] = [4010,2,Mod(1,4010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4010.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: \(32.0200112105\)
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: \( x^{20} - 4 x^{19} - 35 x^{18} + 154 x^{17} + 460 x^{16} - 2392 x^{15} - 2591 x^{14} + 19157 x^{13} + 2776 x^{12} - 83577 x^{11} + 34362 x^{10} + 190617 x^{9} - 150697 x^{8} + \cdots + 104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.323343\) of defining polynomial
Character \(\chi\) \(=\) 4010.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.323343 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.323343 q^{6} +4.27073 q^{7} -1.00000 q^{8} -2.89545 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.323343 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.323343 q^{6} +4.27073 q^{7} -1.00000 q^{8} -2.89545 q^{9} -1.00000 q^{10} -6.20730 q^{11} -0.323343 q^{12} +1.40339 q^{13} -4.27073 q^{14} -0.323343 q^{15} +1.00000 q^{16} +4.33608 q^{17} +2.89545 q^{18} -2.64762 q^{19} +1.00000 q^{20} -1.38091 q^{21} +6.20730 q^{22} -4.01015 q^{23} +0.323343 q^{24} +1.00000 q^{25} -1.40339 q^{26} +1.90625 q^{27} +4.27073 q^{28} -5.35613 q^{29} +0.323343 q^{30} -3.65307 q^{31} -1.00000 q^{32} +2.00709 q^{33} -4.33608 q^{34} +4.27073 q^{35} -2.89545 q^{36} +4.46073 q^{37} +2.64762 q^{38} -0.453778 q^{39} -1.00000 q^{40} -5.68834 q^{41} +1.38091 q^{42} +11.1937 q^{43} -6.20730 q^{44} -2.89545 q^{45} +4.01015 q^{46} +11.1258 q^{47} -0.323343 q^{48} +11.2391 q^{49} -1.00000 q^{50} -1.40204 q^{51} +1.40339 q^{52} +2.41534 q^{53} -1.90625 q^{54} -6.20730 q^{55} -4.27073 q^{56} +0.856091 q^{57} +5.35613 q^{58} +2.54166 q^{59} -0.323343 q^{60} +4.37565 q^{61} +3.65307 q^{62} -12.3657 q^{63} +1.00000 q^{64} +1.40339 q^{65} -2.00709 q^{66} +14.1842 q^{67} +4.33608 q^{68} +1.29666 q^{69} -4.27073 q^{70} +12.9847 q^{71} +2.89545 q^{72} -10.8750 q^{73} -4.46073 q^{74} -0.323343 q^{75} -2.64762 q^{76} -26.5097 q^{77} +0.453778 q^{78} +6.97469 q^{79} +1.00000 q^{80} +8.06997 q^{81} +5.68834 q^{82} +6.32611 q^{83} -1.38091 q^{84} +4.33608 q^{85} -11.1937 q^{86} +1.73187 q^{87} +6.20730 q^{88} -0.107411 q^{89} +2.89545 q^{90} +5.99352 q^{91} -4.01015 q^{92} +1.18120 q^{93} -11.1258 q^{94} -2.64762 q^{95} +0.323343 q^{96} +12.2056 q^{97} -11.2391 q^{98} +17.9729 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} - 9 q^{13} - 11 q^{14} + 4 q^{15} + 20 q^{16} - 11 q^{17} - 26 q^{18} + 17 q^{19} + 20 q^{20} - 2 q^{21} - 10 q^{22} - 3 q^{23} - 4 q^{24} + 20 q^{25} + 9 q^{26} - 2 q^{27} + 11 q^{28} + 6 q^{29} - 4 q^{30} + 28 q^{31} - 20 q^{32} + 2 q^{33} + 11 q^{34} + 11 q^{35} + 26 q^{36} + 33 q^{37} - 17 q^{38} + 36 q^{39} - 20 q^{40} + 32 q^{41} + 2 q^{42} + 30 q^{43} + 10 q^{44} + 26 q^{45} + 3 q^{46} + 13 q^{47} + 4 q^{48} + 43 q^{49} - 20 q^{50} + 43 q^{51} - 9 q^{52} + 2 q^{54} + 10 q^{55} - 11 q^{56} + 19 q^{57} - 6 q^{58} + 52 q^{59} + 4 q^{60} + 25 q^{61} - 28 q^{62} + 16 q^{63} + 20 q^{64} - 9 q^{65} - 2 q^{66} + 40 q^{67} - 11 q^{68} + 39 q^{69} - 11 q^{70} + 25 q^{71} - 26 q^{72} - 5 q^{73} - 33 q^{74} + 4 q^{75} + 17 q^{76} - 9 q^{77} - 36 q^{78} + 40 q^{79} + 20 q^{80} + 48 q^{81} - 32 q^{82} + 10 q^{83} - 2 q^{84} - 11 q^{85} - 30 q^{86} + 10 q^{87} - 10 q^{88} + 17 q^{89} - 26 q^{90} + 88 q^{91} - 3 q^{92} - 4 q^{93} - 13 q^{94} + 17 q^{95} - 4 q^{96} - 2 q^{97} - 43 q^{98} + 50 q^{99}+O(q^{100}) \) 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
\(3\) −0.323343 −0.186682 −0.0933412 0.995634i \(-0.529755\pi\)
−0.0933412 + 0.995634i \(0.529755\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0.323343 0.132004
\(7\) 4.27073 1.61418 0.807092 0.590426i \(-0.201040\pi\)
0.807092 + 0.590426i \(0.201040\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.89545 −0.965150
\(10\) −1.00000 −0.316228
\(11\) −6.20730 −1.87157 −0.935786 0.352569i \(-0.885308\pi\)
−0.935786 + 0.352569i \(0.885308\pi\)
\(12\) −0.323343 −0.0933412
\(13\) 1.40339 0.389232 0.194616 0.980880i \(-0.437654\pi\)
0.194616 + 0.980880i \(0.437654\pi\)
\(14\) −4.27073 −1.14140
\(15\) −0.323343 −0.0834869
\(16\) 1.00000 0.250000
\(17\) 4.33608 1.05165 0.525827 0.850591i \(-0.323756\pi\)
0.525827 + 0.850591i \(0.323756\pi\)
\(18\) 2.89545 0.682464
\(19\) −2.64762 −0.607406 −0.303703 0.952767i \(-0.598223\pi\)
−0.303703 + 0.952767i \(0.598223\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.38091 −0.301340
\(22\) 6.20730 1.32340
\(23\) −4.01015 −0.836174 −0.418087 0.908407i \(-0.637299\pi\)
−0.418087 + 0.908407i \(0.637299\pi\)
\(24\) 0.323343 0.0660022
\(25\) 1.00000 0.200000
\(26\) −1.40339 −0.275228
\(27\) 1.90625 0.366859
\(28\) 4.27073 0.807092
\(29\) −5.35613 −0.994609 −0.497305 0.867576i \(-0.665677\pi\)
−0.497305 + 0.867576i \(0.665677\pi\)
\(30\) 0.323343 0.0590341
\(31\) −3.65307 −0.656111 −0.328056 0.944658i \(-0.606393\pi\)
−0.328056 + 0.944658i \(0.606393\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00709 0.349389
\(34\) −4.33608 −0.743632
\(35\) 4.27073 0.721885
\(36\) −2.89545 −0.482575
\(37\) 4.46073 0.733339 0.366669 0.930351i \(-0.380498\pi\)
0.366669 + 0.930351i \(0.380498\pi\)
\(38\) 2.64762 0.429501
\(39\) −0.453778 −0.0726627
\(40\) −1.00000 −0.158114
\(41\) −5.68834 −0.888370 −0.444185 0.895935i \(-0.646507\pi\)
−0.444185 + 0.895935i \(0.646507\pi\)
\(42\) 1.38091 0.213079
\(43\) 11.1937 1.70702 0.853512 0.521073i \(-0.174468\pi\)
0.853512 + 0.521073i \(0.174468\pi\)
\(44\) −6.20730 −0.935786
\(45\) −2.89545 −0.431628
\(46\) 4.01015 0.591265
\(47\) 11.1258 1.62286 0.811430 0.584449i \(-0.198689\pi\)
0.811430 + 0.584449i \(0.198689\pi\)
\(48\) −0.323343 −0.0466706
\(49\) 11.2391 1.60559
\(50\) −1.00000 −0.141421
\(51\) −1.40204 −0.196325
\(52\) 1.40339 0.194616
\(53\) 2.41534 0.331772 0.165886 0.986145i \(-0.446952\pi\)
0.165886 + 0.986145i \(0.446952\pi\)
\(54\) −1.90625 −0.259408
\(55\) −6.20730 −0.836992
\(56\) −4.27073 −0.570700
\(57\) 0.856091 0.113392
\(58\) 5.35613 0.703295
\(59\) 2.54166 0.330896 0.165448 0.986219i \(-0.447093\pi\)
0.165448 + 0.986219i \(0.447093\pi\)
\(60\) −0.323343 −0.0417434
\(61\) 4.37565 0.560244 0.280122 0.959964i \(-0.409625\pi\)
0.280122 + 0.959964i \(0.409625\pi\)
\(62\) 3.65307 0.463941
\(63\) −12.3657 −1.55793
\(64\) 1.00000 0.125000
\(65\) 1.40339 0.174070
\(66\) −2.00709 −0.247056
\(67\) 14.1842 1.73287 0.866435 0.499290i \(-0.166406\pi\)
0.866435 + 0.499290i \(0.166406\pi\)
\(68\) 4.33608 0.525827
\(69\) 1.29666 0.156099
\(70\) −4.27073 −0.510450
\(71\) 12.9847 1.54100 0.770500 0.637441i \(-0.220007\pi\)
0.770500 + 0.637441i \(0.220007\pi\)
\(72\) 2.89545 0.341232
\(73\) −10.8750 −1.27282 −0.636411 0.771350i \(-0.719582\pi\)
−0.636411 + 0.771350i \(0.719582\pi\)
\(74\) −4.46073 −0.518549
\(75\) −0.323343 −0.0373365
\(76\) −2.64762 −0.303703
\(77\) −26.5097 −3.02106
\(78\) 0.453778 0.0513803
\(79\) 6.97469 0.784714 0.392357 0.919813i \(-0.371660\pi\)
0.392357 + 0.919813i \(0.371660\pi\)
\(80\) 1.00000 0.111803
\(81\) 8.06997 0.896664
\(82\) 5.68834 0.628172
\(83\) 6.32611 0.694380 0.347190 0.937795i \(-0.387136\pi\)
0.347190 + 0.937795i \(0.387136\pi\)
\(84\) −1.38091 −0.150670
\(85\) 4.33608 0.470314
\(86\) −11.1937 −1.20705
\(87\) 1.73187 0.185676
\(88\) 6.20730 0.661701
\(89\) −0.107411 −0.0113855 −0.00569276 0.999984i \(-0.501812\pi\)
−0.00569276 + 0.999984i \(0.501812\pi\)
\(90\) 2.89545 0.305207
\(91\) 5.99352 0.628292
\(92\) −4.01015 −0.418087
\(93\) 1.18120 0.122484
\(94\) −11.1258 −1.14754
\(95\) −2.64762 −0.271640
\(96\) 0.323343 0.0330011
\(97\) 12.2056 1.23929 0.619646 0.784881i \(-0.287276\pi\)
0.619646 + 0.784881i \(0.287276\pi\)
\(98\) −11.2391 −1.13532
\(99\) 17.9729 1.80635
\(100\) 1.00000 0.100000
\(101\) 0.337459 0.0335785 0.0167892 0.999859i \(-0.494656\pi\)
0.0167892 + 0.999859i \(0.494656\pi\)
\(102\) 1.40204 0.138823
\(103\) −9.67687 −0.953490 −0.476745 0.879042i \(-0.658183\pi\)
−0.476745 + 0.879042i \(0.658183\pi\)
\(104\) −1.40339 −0.137614
\(105\) −1.38091 −0.134763
\(106\) −2.41534 −0.234598
\(107\) 13.0574 1.26231 0.631153 0.775658i \(-0.282582\pi\)
0.631153 + 0.775658i \(0.282582\pi\)
\(108\) 1.90625 0.183429
\(109\) 4.28493 0.410422 0.205211 0.978718i \(-0.434212\pi\)
0.205211 + 0.978718i \(0.434212\pi\)
\(110\) 6.20730 0.591843
\(111\) −1.44235 −0.136901
\(112\) 4.27073 0.403546
\(113\) 5.51677 0.518974 0.259487 0.965747i \(-0.416447\pi\)
0.259487 + 0.965747i \(0.416447\pi\)
\(114\) −0.856091 −0.0801802
\(115\) −4.01015 −0.373949
\(116\) −5.35613 −0.497305
\(117\) −4.06346 −0.375667
\(118\) −2.54166 −0.233979
\(119\) 18.5182 1.69756
\(120\) 0.323343 0.0295171
\(121\) 27.5306 2.50278
\(122\) −4.37565 −0.396152
\(123\) 1.83929 0.165843
\(124\) −3.65307 −0.328056
\(125\) 1.00000 0.0894427
\(126\) 12.3657 1.10162
\(127\) −3.91161 −0.347099 −0.173550 0.984825i \(-0.555524\pi\)
−0.173550 + 0.984825i \(0.555524\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −3.61941 −0.318671
\(130\) −1.40339 −0.123086
\(131\) −5.58121 −0.487633 −0.243816 0.969821i \(-0.578399\pi\)
−0.243816 + 0.969821i \(0.578399\pi\)
\(132\) 2.00709 0.174695
\(133\) −11.3073 −0.980465
\(134\) −14.1842 −1.22532
\(135\) 1.90625 0.164064
\(136\) −4.33608 −0.371816
\(137\) −19.2302 −1.64295 −0.821473 0.570247i \(-0.806847\pi\)
−0.821473 + 0.570247i \(0.806847\pi\)
\(138\) −1.29666 −0.110379
\(139\) 2.58223 0.219022 0.109511 0.993986i \(-0.465072\pi\)
0.109511 + 0.993986i \(0.465072\pi\)
\(140\) 4.27073 0.360942
\(141\) −3.59744 −0.302959
\(142\) −12.9847 −1.08965
\(143\) −8.71129 −0.728475
\(144\) −2.89545 −0.241287
\(145\) −5.35613 −0.444803
\(146\) 10.8750 0.900021
\(147\) −3.63410 −0.299735
\(148\) 4.46073 0.366669
\(149\) 4.92135 0.403172 0.201586 0.979471i \(-0.435390\pi\)
0.201586 + 0.979471i \(0.435390\pi\)
\(150\) 0.323343 0.0264009
\(151\) −8.56366 −0.696900 −0.348450 0.937327i \(-0.613292\pi\)
−0.348450 + 0.937327i \(0.613292\pi\)
\(152\) 2.64762 0.214751
\(153\) −12.5549 −1.01500
\(154\) 26.5097 2.13621
\(155\) −3.65307 −0.293422
\(156\) −0.453778 −0.0363313
\(157\) 14.3778 1.14747 0.573735 0.819041i \(-0.305494\pi\)
0.573735 + 0.819041i \(0.305494\pi\)
\(158\) −6.97469 −0.554877
\(159\) −0.780983 −0.0619360
\(160\) −1.00000 −0.0790569
\(161\) −17.1263 −1.34974
\(162\) −8.06997 −0.634037
\(163\) 5.48063 0.429276 0.214638 0.976694i \(-0.431143\pi\)
0.214638 + 0.976694i \(0.431143\pi\)
\(164\) −5.68834 −0.444185
\(165\) 2.00709 0.156252
\(166\) −6.32611 −0.491001
\(167\) 3.20672 0.248144 0.124072 0.992273i \(-0.460405\pi\)
0.124072 + 0.992273i \(0.460405\pi\)
\(168\) 1.38091 0.106540
\(169\) −11.0305 −0.848499
\(170\) −4.33608 −0.332562
\(171\) 7.66606 0.586238
\(172\) 11.1937 0.853512
\(173\) −14.3715 −1.09265 −0.546323 0.837574i \(-0.683973\pi\)
−0.546323 + 0.837574i \(0.683973\pi\)
\(174\) −1.73187 −0.131293
\(175\) 4.27073 0.322837
\(176\) −6.20730 −0.467893
\(177\) −0.821829 −0.0617724
\(178\) 0.107411 0.00805078
\(179\) −9.78741 −0.731546 −0.365773 0.930704i \(-0.619195\pi\)
−0.365773 + 0.930704i \(0.619195\pi\)
\(180\) −2.89545 −0.215814
\(181\) 10.2982 0.765462 0.382731 0.923860i \(-0.374984\pi\)
0.382731 + 0.923860i \(0.374984\pi\)
\(182\) −5.99352 −0.444269
\(183\) −1.41484 −0.104588
\(184\) 4.01015 0.295632
\(185\) 4.46073 0.327959
\(186\) −1.18120 −0.0866096
\(187\) −26.9154 −1.96825
\(188\) 11.1258 0.811430
\(189\) 8.14109 0.592177
\(190\) 2.64762 0.192079
\(191\) 0.221079 0.0159967 0.00799835 0.999968i \(-0.497454\pi\)
0.00799835 + 0.999968i \(0.497454\pi\)
\(192\) −0.323343 −0.0233353
\(193\) 20.3644 1.46586 0.732929 0.680305i \(-0.238153\pi\)
0.732929 + 0.680305i \(0.238153\pi\)
\(194\) −12.2056 −0.876312
\(195\) −0.453778 −0.0324957
\(196\) 11.2391 0.802795
\(197\) −25.5964 −1.82367 −0.911835 0.410558i \(-0.865334\pi\)
−0.911835 + 0.410558i \(0.865334\pi\)
\(198\) −17.9729 −1.27728
\(199\) −11.9209 −0.845052 −0.422526 0.906351i \(-0.638857\pi\)
−0.422526 + 0.906351i \(0.638857\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.58635 −0.323496
\(202\) −0.337459 −0.0237436
\(203\) −22.8746 −1.60548
\(204\) −1.40204 −0.0981626
\(205\) −5.68834 −0.397291
\(206\) 9.67687 0.674219
\(207\) 11.6112 0.807033
\(208\) 1.40339 0.0973079
\(209\) 16.4346 1.13680
\(210\) 1.38091 0.0952919
\(211\) −8.03884 −0.553417 −0.276708 0.960954i \(-0.589244\pi\)
−0.276708 + 0.960954i \(0.589244\pi\)
\(212\) 2.41534 0.165886
\(213\) −4.19851 −0.287677
\(214\) −13.0574 −0.892585
\(215\) 11.1937 0.763404
\(216\) −1.90625 −0.129704
\(217\) −15.6013 −1.05908
\(218\) −4.28493 −0.290212
\(219\) 3.51636 0.237613
\(220\) −6.20730 −0.418496
\(221\) 6.08524 0.409337
\(222\) 1.44235 0.0968039
\(223\) −5.81581 −0.389456 −0.194728 0.980857i \(-0.562382\pi\)
−0.194728 + 0.980857i \(0.562382\pi\)
\(224\) −4.27073 −0.285350
\(225\) −2.89545 −0.193030
\(226\) −5.51677 −0.366970
\(227\) 13.2037 0.876358 0.438179 0.898888i \(-0.355624\pi\)
0.438179 + 0.898888i \(0.355624\pi\)
\(228\) 0.856091 0.0566960
\(229\) 25.2511 1.66864 0.834319 0.551281i \(-0.185861\pi\)
0.834319 + 0.551281i \(0.185861\pi\)
\(230\) 4.01015 0.264422
\(231\) 8.57173 0.563979
\(232\) 5.35613 0.351647
\(233\) 9.40454 0.616112 0.308056 0.951368i \(-0.400322\pi\)
0.308056 + 0.951368i \(0.400322\pi\)
\(234\) 4.06346 0.265637
\(235\) 11.1258 0.725765
\(236\) 2.54166 0.165448
\(237\) −2.25522 −0.146492
\(238\) −18.5182 −1.20036
\(239\) −7.68893 −0.497355 −0.248678 0.968586i \(-0.579996\pi\)
−0.248678 + 0.968586i \(0.579996\pi\)
\(240\) −0.323343 −0.0208717
\(241\) −22.3622 −1.44048 −0.720239 0.693726i \(-0.755968\pi\)
−0.720239 + 0.693726i \(0.755968\pi\)
\(242\) −27.5306 −1.76973
\(243\) −8.32813 −0.534250
\(244\) 4.37565 0.280122
\(245\) 11.2391 0.718041
\(246\) −1.83929 −0.117269
\(247\) −3.71566 −0.236422
\(248\) 3.65307 0.231970
\(249\) −2.04550 −0.129629
\(250\) −1.00000 −0.0632456
\(251\) 16.0991 1.01617 0.508083 0.861308i \(-0.330354\pi\)
0.508083 + 0.861308i \(0.330354\pi\)
\(252\) −12.3657 −0.778965
\(253\) 24.8922 1.56496
\(254\) 3.91161 0.245436
\(255\) −1.40204 −0.0877993
\(256\) 1.00000 0.0625000
\(257\) 21.0520 1.31319 0.656593 0.754245i \(-0.271997\pi\)
0.656593 + 0.754245i \(0.271997\pi\)
\(258\) 3.61941 0.225335
\(259\) 19.0506 1.18374
\(260\) 1.40339 0.0870349
\(261\) 15.5084 0.959947
\(262\) 5.58121 0.344808
\(263\) 30.9243 1.90687 0.953437 0.301594i \(-0.0975186\pi\)
0.953437 + 0.301594i \(0.0975186\pi\)
\(264\) −2.00709 −0.123528
\(265\) 2.41534 0.148373
\(266\) 11.3073 0.693294
\(267\) 0.0347306 0.00212548
\(268\) 14.1842 0.866435
\(269\) −12.5733 −0.766606 −0.383303 0.923623i \(-0.625214\pi\)
−0.383303 + 0.923623i \(0.625214\pi\)
\(270\) −1.90625 −0.116011
\(271\) −1.50082 −0.0911681 −0.0455841 0.998961i \(-0.514515\pi\)
−0.0455841 + 0.998961i \(0.514515\pi\)
\(272\) 4.33608 0.262914
\(273\) −1.93796 −0.117291
\(274\) 19.2302 1.16174
\(275\) −6.20730 −0.374314
\(276\) 1.29666 0.0780495
\(277\) 11.0854 0.666059 0.333030 0.942916i \(-0.391929\pi\)
0.333030 + 0.942916i \(0.391929\pi\)
\(278\) −2.58223 −0.154872
\(279\) 10.5773 0.633246
\(280\) −4.27073 −0.255225
\(281\) −26.8713 −1.60301 −0.801503 0.597990i \(-0.795966\pi\)
−0.801503 + 0.597990i \(0.795966\pi\)
\(282\) 3.59744 0.214225
\(283\) 14.6942 0.873479 0.436740 0.899588i \(-0.356133\pi\)
0.436740 + 0.899588i \(0.356133\pi\)
\(284\) 12.9847 0.770500
\(285\) 0.856091 0.0507104
\(286\) 8.71129 0.515110
\(287\) −24.2934 −1.43399
\(288\) 2.89545 0.170616
\(289\) 1.80161 0.105977
\(290\) 5.35613 0.314523
\(291\) −3.94660 −0.231354
\(292\) −10.8750 −0.636411
\(293\) 0.346545 0.0202454 0.0101227 0.999949i \(-0.496778\pi\)
0.0101227 + 0.999949i \(0.496778\pi\)
\(294\) 3.63410 0.211945
\(295\) 2.54166 0.147981
\(296\) −4.46073 −0.259274
\(297\) −11.8327 −0.686602
\(298\) −4.92135 −0.285086
\(299\) −5.62783 −0.325466
\(300\) −0.323343 −0.0186682
\(301\) 47.8053 2.75545
\(302\) 8.56366 0.492783
\(303\) −0.109115 −0.00626850
\(304\) −2.64762 −0.151852
\(305\) 4.37565 0.250549
\(306\) 12.5549 0.717716
\(307\) 17.5826 1.00349 0.501746 0.865015i \(-0.332691\pi\)
0.501746 + 0.865015i \(0.332691\pi\)
\(308\) −26.5097 −1.51053
\(309\) 3.12895 0.178000
\(310\) 3.65307 0.207481
\(311\) −12.9754 −0.735765 −0.367883 0.929872i \(-0.619917\pi\)
−0.367883 + 0.929872i \(0.619917\pi\)
\(312\) 0.453778 0.0256901
\(313\) −4.33743 −0.245166 −0.122583 0.992458i \(-0.539118\pi\)
−0.122583 + 0.992458i \(0.539118\pi\)
\(314\) −14.3778 −0.811384
\(315\) −12.3657 −0.696727
\(316\) 6.97469 0.392357
\(317\) 27.6801 1.55467 0.777335 0.629086i \(-0.216571\pi\)
0.777335 + 0.629086i \(0.216571\pi\)
\(318\) 0.780983 0.0437954
\(319\) 33.2471 1.86148
\(320\) 1.00000 0.0559017
\(321\) −4.22202 −0.235650
\(322\) 17.1263 0.954410
\(323\) −11.4803 −0.638781
\(324\) 8.06997 0.448332
\(325\) 1.40339 0.0778463
\(326\) −5.48063 −0.303544
\(327\) −1.38550 −0.0766185
\(328\) 5.68834 0.314086
\(329\) 47.5152 2.61960
\(330\) −2.00709 −0.110487
\(331\) −1.73730 −0.0954905 −0.0477453 0.998860i \(-0.515204\pi\)
−0.0477453 + 0.998860i \(0.515204\pi\)
\(332\) 6.32611 0.347190
\(333\) −12.9158 −0.707782
\(334\) −3.20672 −0.175464
\(335\) 14.1842 0.774963
\(336\) −1.38091 −0.0753349
\(337\) −13.3698 −0.728298 −0.364149 0.931341i \(-0.618640\pi\)
−0.364149 + 0.931341i \(0.618640\pi\)
\(338\) 11.0305 0.599979
\(339\) −1.78381 −0.0968833
\(340\) 4.33608 0.235157
\(341\) 22.6757 1.22796
\(342\) −7.66606 −0.414533
\(343\) 18.1042 0.977533
\(344\) −11.1937 −0.603524
\(345\) 1.29666 0.0698096
\(346\) 14.3715 0.772618
\(347\) −5.92373 −0.318002 −0.159001 0.987278i \(-0.550827\pi\)
−0.159001 + 0.987278i \(0.550827\pi\)
\(348\) 1.73187 0.0928380
\(349\) −8.56349 −0.458393 −0.229197 0.973380i \(-0.573610\pi\)
−0.229197 + 0.973380i \(0.573610\pi\)
\(350\) −4.27073 −0.228280
\(351\) 2.67523 0.142793
\(352\) 6.20730 0.330850
\(353\) 13.3185 0.708870 0.354435 0.935081i \(-0.384673\pi\)
0.354435 + 0.935081i \(0.384673\pi\)
\(354\) 0.821829 0.0436797
\(355\) 12.9847 0.689156
\(356\) −0.107411 −0.00569276
\(357\) −5.98774 −0.316905
\(358\) 9.78741 0.517281
\(359\) 19.2025 1.01347 0.506734 0.862102i \(-0.330853\pi\)
0.506734 + 0.862102i \(0.330853\pi\)
\(360\) 2.89545 0.152604
\(361\) −11.9901 −0.631058
\(362\) −10.2982 −0.541263
\(363\) −8.90183 −0.467225
\(364\) 5.99352 0.314146
\(365\) −10.8750 −0.569223
\(366\) 1.41484 0.0739546
\(367\) 20.1447 1.05154 0.525771 0.850626i \(-0.323777\pi\)
0.525771 + 0.850626i \(0.323777\pi\)
\(368\) −4.01015 −0.209044
\(369\) 16.4703 0.857410
\(370\) −4.46073 −0.231902
\(371\) 10.3153 0.535541
\(372\) 1.18120 0.0612422
\(373\) 19.7747 1.02390 0.511948 0.859017i \(-0.328924\pi\)
0.511948 + 0.859017i \(0.328924\pi\)
\(374\) 26.9154 1.39176
\(375\) −0.323343 −0.0166974
\(376\) −11.1258 −0.573768
\(377\) −7.51677 −0.387133
\(378\) −8.14109 −0.418733
\(379\) 9.72670 0.499627 0.249814 0.968294i \(-0.419631\pi\)
0.249814 + 0.968294i \(0.419631\pi\)
\(380\) −2.64762 −0.135820
\(381\) 1.26479 0.0647973
\(382\) −0.221079 −0.0113114
\(383\) 8.07832 0.412783 0.206391 0.978470i \(-0.433828\pi\)
0.206391 + 0.978470i \(0.433828\pi\)
\(384\) 0.323343 0.0165005
\(385\) −26.5097 −1.35106
\(386\) −20.3644 −1.03652
\(387\) −32.4108 −1.64753
\(388\) 12.2056 0.619646
\(389\) −6.99401 −0.354610 −0.177305 0.984156i \(-0.556738\pi\)
−0.177305 + 0.984156i \(0.556738\pi\)
\(390\) 0.453778 0.0229780
\(391\) −17.3883 −0.879366
\(392\) −11.2391 −0.567662
\(393\) 1.80465 0.0910324
\(394\) 25.5964 1.28953
\(395\) 6.97469 0.350935
\(396\) 17.9729 0.903173
\(397\) 7.39389 0.371089 0.185544 0.982636i \(-0.440595\pi\)
0.185544 + 0.982636i \(0.440595\pi\)
\(398\) 11.9209 0.597542
\(399\) 3.65613 0.183036
\(400\) 1.00000 0.0500000
\(401\) −1.00000 −0.0499376
\(402\) 4.58635 0.228746
\(403\) −5.12671 −0.255379
\(404\) 0.337459 0.0167892
\(405\) 8.06997 0.401000
\(406\) 22.8746 1.13525
\(407\) −27.6891 −1.37250
\(408\) 1.40204 0.0694115
\(409\) −12.4951 −0.617841 −0.308920 0.951088i \(-0.599968\pi\)
−0.308920 + 0.951088i \(0.599968\pi\)
\(410\) 5.68834 0.280927
\(411\) 6.21795 0.306709
\(412\) −9.67687 −0.476745
\(413\) 10.8547 0.534127
\(414\) −11.6112 −0.570659
\(415\) 6.32611 0.310536
\(416\) −1.40339 −0.0688071
\(417\) −0.834945 −0.0408874
\(418\) −16.4346 −0.803842
\(419\) 12.4835 0.609858 0.304929 0.952375i \(-0.401367\pi\)
0.304929 + 0.952375i \(0.401367\pi\)
\(420\) −1.38091 −0.0673816
\(421\) 14.8237 0.722463 0.361232 0.932476i \(-0.382356\pi\)
0.361232 + 0.932476i \(0.382356\pi\)
\(422\) 8.03884 0.391325
\(423\) −32.2141 −1.56630
\(424\) −2.41534 −0.117299
\(425\) 4.33608 0.210331
\(426\) 4.19851 0.203419
\(427\) 18.6872 0.904337
\(428\) 13.0574 0.631153
\(429\) 2.81674 0.135993
\(430\) −11.1937 −0.539808
\(431\) 8.10706 0.390503 0.195252 0.980753i \(-0.437448\pi\)
0.195252 + 0.980753i \(0.437448\pi\)
\(432\) 1.90625 0.0917147
\(433\) −7.33157 −0.352333 −0.176166 0.984360i \(-0.556370\pi\)
−0.176166 + 0.984360i \(0.556370\pi\)
\(434\) 15.6013 0.748886
\(435\) 1.73187 0.0830368
\(436\) 4.28493 0.205211
\(437\) 10.6174 0.507898
\(438\) −3.51636 −0.168018
\(439\) 18.4591 0.881005 0.440502 0.897751i \(-0.354800\pi\)
0.440502 + 0.897751i \(0.354800\pi\)
\(440\) 6.20730 0.295921
\(441\) −32.5423 −1.54963
\(442\) −6.08524 −0.289445
\(443\) −33.5527 −1.59414 −0.797069 0.603889i \(-0.793617\pi\)
−0.797069 + 0.603889i \(0.793617\pi\)
\(444\) −1.44235 −0.0684507
\(445\) −0.107411 −0.00509176
\(446\) 5.81581 0.275387
\(447\) −1.59128 −0.0752651
\(448\) 4.27073 0.201773
\(449\) 27.2104 1.28414 0.642070 0.766646i \(-0.278076\pi\)
0.642070 + 0.766646i \(0.278076\pi\)
\(450\) 2.89545 0.136493
\(451\) 35.3093 1.66265
\(452\) 5.51677 0.259487
\(453\) 2.76900 0.130099
\(454\) −13.2037 −0.619679
\(455\) 5.99352 0.280981
\(456\) −0.856091 −0.0400901
\(457\) −40.7803 −1.90762 −0.953812 0.300405i \(-0.902878\pi\)
−0.953812 + 0.300405i \(0.902878\pi\)
\(458\) −25.2511 −1.17991
\(459\) 8.26567 0.385809
\(460\) −4.01015 −0.186974
\(461\) 31.6278 1.47305 0.736527 0.676408i \(-0.236464\pi\)
0.736527 + 0.676408i \(0.236464\pi\)
\(462\) −8.57173 −0.398793
\(463\) −26.4982 −1.23148 −0.615738 0.787951i \(-0.711142\pi\)
−0.615738 + 0.787951i \(0.711142\pi\)
\(464\) −5.35613 −0.248652
\(465\) 1.18120 0.0547767
\(466\) −9.40454 −0.435657
\(467\) 1.68402 0.0779271 0.0389635 0.999241i \(-0.487594\pi\)
0.0389635 + 0.999241i \(0.487594\pi\)
\(468\) −4.06346 −0.187833
\(469\) 60.5767 2.79717
\(470\) −11.1258 −0.513194
\(471\) −4.64895 −0.214212
\(472\) −2.54166 −0.116989
\(473\) −69.4827 −3.19482
\(474\) 2.25522 0.103586
\(475\) −2.64762 −0.121481
\(476\) 18.5182 0.848782
\(477\) −6.99349 −0.320210
\(478\) 7.68893 0.351683
\(479\) 18.4258 0.841897 0.420948 0.907085i \(-0.361697\pi\)
0.420948 + 0.907085i \(0.361697\pi\)
\(480\) 0.323343 0.0147585
\(481\) 6.26016 0.285439
\(482\) 22.3622 1.01857
\(483\) 5.53766 0.251972
\(484\) 27.5306 1.25139
\(485\) 12.2056 0.554228
\(486\) 8.32813 0.377772
\(487\) 1.65439 0.0749674 0.0374837 0.999297i \(-0.488066\pi\)
0.0374837 + 0.999297i \(0.488066\pi\)
\(488\) −4.37565 −0.198076
\(489\) −1.77212 −0.0801382
\(490\) −11.2391 −0.507732
\(491\) 10.2927 0.464503 0.232252 0.972656i \(-0.425391\pi\)
0.232252 + 0.972656i \(0.425391\pi\)
\(492\) 1.83929 0.0829215
\(493\) −23.2246 −1.04598
\(494\) 3.71566 0.167175
\(495\) 17.9729 0.807823
\(496\) −3.65307 −0.164028
\(497\) 55.4541 2.48746
\(498\) 2.04550 0.0916612
\(499\) −22.2383 −0.995523 −0.497762 0.867314i \(-0.665845\pi\)
−0.497762 + 0.867314i \(0.665845\pi\)
\(500\) 1.00000 0.0447214
\(501\) −1.03687 −0.0463240
\(502\) −16.0991 −0.718538
\(503\) 8.91876 0.397668 0.198834 0.980033i \(-0.436285\pi\)
0.198834 + 0.980033i \(0.436285\pi\)
\(504\) 12.3657 0.550811
\(505\) 0.337459 0.0150167
\(506\) −24.8922 −1.10659
\(507\) 3.56663 0.158400
\(508\) −3.91161 −0.173550
\(509\) 14.1245 0.626058 0.313029 0.949743i \(-0.398656\pi\)
0.313029 + 0.949743i \(0.398656\pi\)
\(510\) 1.40204 0.0620835
\(511\) −46.4442 −2.05457
\(512\) −1.00000 −0.0441942
\(513\) −5.04704 −0.222832
\(514\) −21.0520 −0.928563
\(515\) −9.67687 −0.426414
\(516\) −3.61941 −0.159336
\(517\) −69.0610 −3.03730
\(518\) −19.0506 −0.837033
\(519\) 4.64693 0.203978
\(520\) −1.40339 −0.0615429
\(521\) −40.2938 −1.76531 −0.882653 0.470026i \(-0.844245\pi\)
−0.882653 + 0.470026i \(0.844245\pi\)
\(522\) −15.5084 −0.678785
\(523\) 36.5230 1.59704 0.798520 0.601968i \(-0.205617\pi\)
0.798520 + 0.601968i \(0.205617\pi\)
\(524\) −5.58121 −0.243816
\(525\) −1.38091 −0.0602679
\(526\) −30.9243 −1.34836
\(527\) −15.8400 −0.690002
\(528\) 2.00709 0.0873473
\(529\) −6.91868 −0.300812
\(530\) −2.41534 −0.104916
\(531\) −7.35925 −0.319364
\(532\) −11.3073 −0.490233
\(533\) −7.98299 −0.345782
\(534\) −0.0347306 −0.00150294
\(535\) 13.0574 0.564520
\(536\) −14.1842 −0.612662
\(537\) 3.16469 0.136567
\(538\) 12.5733 0.542072
\(539\) −69.7646 −3.00498
\(540\) 1.90625 0.0820321
\(541\) 4.55782 0.195956 0.0979780 0.995189i \(-0.468763\pi\)
0.0979780 + 0.995189i \(0.468763\pi\)
\(542\) 1.50082 0.0644656
\(543\) −3.32987 −0.142898
\(544\) −4.33608 −0.185908
\(545\) 4.28493 0.183546
\(546\) 1.93796 0.0829372
\(547\) −30.9905 −1.32506 −0.662528 0.749037i \(-0.730517\pi\)
−0.662528 + 0.749037i \(0.730517\pi\)
\(548\) −19.2302 −0.821473
\(549\) −12.6695 −0.540719
\(550\) 6.20730 0.264680
\(551\) 14.1810 0.604132
\(552\) −1.29666 −0.0551893
\(553\) 29.7870 1.26667
\(554\) −11.0854 −0.470975
\(555\) −1.44235 −0.0612242
\(556\) 2.58223 0.109511
\(557\) −42.7716 −1.81229 −0.906145 0.422968i \(-0.860988\pi\)
−0.906145 + 0.422968i \(0.860988\pi\)
\(558\) −10.5773 −0.447772
\(559\) 15.7092 0.664428
\(560\) 4.27073 0.180471
\(561\) 8.70290 0.367437
\(562\) 26.8713 1.13350
\(563\) −7.66268 −0.322943 −0.161472 0.986877i \(-0.551624\pi\)
−0.161472 + 0.986877i \(0.551624\pi\)
\(564\) −3.59744 −0.151480
\(565\) 5.51677 0.232092
\(566\) −14.6942 −0.617643
\(567\) 34.4647 1.44738
\(568\) −12.9847 −0.544825
\(569\) −17.2084 −0.721414 −0.360707 0.932679i \(-0.617465\pi\)
−0.360707 + 0.932679i \(0.617465\pi\)
\(570\) −0.856091 −0.0358577
\(571\) −14.5373 −0.608368 −0.304184 0.952613i \(-0.598384\pi\)
−0.304184 + 0.952613i \(0.598384\pi\)
\(572\) −8.71129 −0.364238
\(573\) −0.0714843 −0.00298630
\(574\) 24.2934 1.01399
\(575\) −4.01015 −0.167235
\(576\) −2.89545 −0.120644
\(577\) −11.5453 −0.480635 −0.240318 0.970694i \(-0.577252\pi\)
−0.240318 + 0.970694i \(0.577252\pi\)
\(578\) −1.80161 −0.0749369
\(579\) −6.58468 −0.273650
\(580\) −5.35613 −0.222401
\(581\) 27.0171 1.12086
\(582\) 3.94660 0.163592
\(583\) −14.9927 −0.620935
\(584\) 10.8750 0.450011
\(585\) −4.06346 −0.168003
\(586\) −0.346545 −0.0143156
\(587\) 22.7503 0.939007 0.469504 0.882931i \(-0.344433\pi\)
0.469504 + 0.882931i \(0.344433\pi\)
\(588\) −3.63410 −0.149868
\(589\) 9.67196 0.398526
\(590\) −2.54166 −0.104639
\(591\) 8.27643 0.340447
\(592\) 4.46073 0.183335
\(593\) 31.0911 1.27676 0.638380 0.769722i \(-0.279605\pi\)
0.638380 + 0.769722i \(0.279605\pi\)
\(594\) 11.8327 0.485501
\(595\) 18.5182 0.759173
\(596\) 4.92135 0.201586
\(597\) 3.85455 0.157756
\(598\) 5.62783 0.230139
\(599\) −35.5828 −1.45387 −0.726936 0.686705i \(-0.759056\pi\)
−0.726936 + 0.686705i \(0.759056\pi\)
\(600\) 0.323343 0.0132004
\(601\) −20.6968 −0.844240 −0.422120 0.906540i \(-0.638714\pi\)
−0.422120 + 0.906540i \(0.638714\pi\)
\(602\) −47.8053 −1.94840
\(603\) −41.0695 −1.67248
\(604\) −8.56366 −0.348450
\(605\) 27.5306 1.11928
\(606\) 0.109115 0.00443250
\(607\) 30.3161 1.23049 0.615247 0.788334i \(-0.289056\pi\)
0.615247 + 0.788334i \(0.289056\pi\)
\(608\) 2.64762 0.107375
\(609\) 7.39635 0.299715
\(610\) −4.37565 −0.177165
\(611\) 15.6139 0.631669
\(612\) −12.5549 −0.507502
\(613\) 28.6830 1.15850 0.579248 0.815151i \(-0.303346\pi\)
0.579248 + 0.815151i \(0.303346\pi\)
\(614\) −17.5826 −0.709575
\(615\) 1.83929 0.0741672
\(616\) 26.5097 1.06811
\(617\) −1.00329 −0.0403910 −0.0201955 0.999796i \(-0.506429\pi\)
−0.0201955 + 0.999796i \(0.506429\pi\)
\(618\) −3.12895 −0.125865
\(619\) 34.0162 1.36723 0.683613 0.729845i \(-0.260408\pi\)
0.683613 + 0.729845i \(0.260408\pi\)
\(620\) −3.65307 −0.146711
\(621\) −7.64437 −0.306758
\(622\) 12.9754 0.520265
\(623\) −0.458722 −0.0183783
\(624\) −0.453778 −0.0181657
\(625\) 1.00000 0.0400000
\(626\) 4.33743 0.173359
\(627\) −5.31401 −0.212221
\(628\) 14.3778 0.573735
\(629\) 19.3421 0.771219
\(630\) 12.3657 0.492660
\(631\) −5.20577 −0.207238 −0.103619 0.994617i \(-0.533042\pi\)
−0.103619 + 0.994617i \(0.533042\pi\)
\(632\) −6.97469 −0.277438
\(633\) 2.59930 0.103313
\(634\) −27.6801 −1.09932
\(635\) −3.91161 −0.155228
\(636\) −0.780983 −0.0309680
\(637\) 15.7729 0.624946
\(638\) −33.2471 −1.31627
\(639\) −37.5965 −1.48729
\(640\) −1.00000 −0.0395285
\(641\) −44.8305 −1.77070 −0.885348 0.464928i \(-0.846080\pi\)
−0.885348 + 0.464928i \(0.846080\pi\)
\(642\) 4.22202 0.166630
\(643\) −29.1809 −1.15078 −0.575391 0.817878i \(-0.695150\pi\)
−0.575391 + 0.817878i \(0.695150\pi\)
\(644\) −17.1263 −0.674870
\(645\) −3.61941 −0.142514
\(646\) 11.4803 0.451687
\(647\) −39.5448 −1.55467 −0.777334 0.629088i \(-0.783428\pi\)
−0.777334 + 0.629088i \(0.783428\pi\)
\(648\) −8.06997 −0.317018
\(649\) −15.7769 −0.619296
\(650\) −1.40339 −0.0550457
\(651\) 5.04457 0.197712
\(652\) 5.48063 0.214638
\(653\) 27.1640 1.06301 0.531506 0.847055i \(-0.321626\pi\)
0.531506 + 0.847055i \(0.321626\pi\)
\(654\) 1.38550 0.0541774
\(655\) −5.58121 −0.218076
\(656\) −5.68834 −0.222093
\(657\) 31.4880 1.22846
\(658\) −47.5152 −1.85233
\(659\) 12.3231 0.480038 0.240019 0.970768i \(-0.422846\pi\)
0.240019 + 0.970768i \(0.422846\pi\)
\(660\) 2.00709 0.0781258
\(661\) −3.78566 −0.147245 −0.0736225 0.997286i \(-0.523456\pi\)
−0.0736225 + 0.997286i \(0.523456\pi\)
\(662\) 1.73730 0.0675220
\(663\) −1.96762 −0.0764160
\(664\) −6.32611 −0.245501
\(665\) −11.3073 −0.438477
\(666\) 12.9158 0.500477
\(667\) 21.4789 0.831667
\(668\) 3.20672 0.124072
\(669\) 1.88050 0.0727045
\(670\) −14.1842 −0.547982
\(671\) −27.1609 −1.04854
\(672\) 1.38091 0.0532698
\(673\) 29.6559 1.14315 0.571575 0.820549i \(-0.306332\pi\)
0.571575 + 0.820549i \(0.306332\pi\)
\(674\) 13.3698 0.514984
\(675\) 1.90625 0.0733717
\(676\) −11.0305 −0.424249
\(677\) −40.2461 −1.54678 −0.773392 0.633928i \(-0.781441\pi\)
−0.773392 + 0.633928i \(0.781441\pi\)
\(678\) 1.78381 0.0685068
\(679\) 52.1269 2.00045
\(680\) −4.33608 −0.166281
\(681\) −4.26932 −0.163601
\(682\) −22.6757 −0.868299
\(683\) 32.1002 1.22828 0.614141 0.789197i \(-0.289503\pi\)
0.614141 + 0.789197i \(0.289503\pi\)
\(684\) 7.66606 0.293119
\(685\) −19.2302 −0.734748
\(686\) −18.1042 −0.691220
\(687\) −8.16477 −0.311505
\(688\) 11.1937 0.426756
\(689\) 3.38967 0.129136
\(690\) −1.29666 −0.0493628
\(691\) 20.3668 0.774791 0.387396 0.921914i \(-0.373375\pi\)
0.387396 + 0.921914i \(0.373375\pi\)
\(692\) −14.3715 −0.546323
\(693\) 76.7575 2.91578
\(694\) 5.92373 0.224862
\(695\) 2.58223 0.0979494
\(696\) −1.73187 −0.0656463
\(697\) −24.6651 −0.934258
\(698\) 8.56349 0.324133
\(699\) −3.04090 −0.115017
\(700\) 4.27073 0.161418
\(701\) −5.30031 −0.200190 −0.100095 0.994978i \(-0.531915\pi\)
−0.100095 + 0.994978i \(0.531915\pi\)
\(702\) −2.67523 −0.100970
\(703\) −11.8103 −0.445435
\(704\) −6.20730 −0.233946
\(705\) −3.59744 −0.135488
\(706\) −13.3185 −0.501247
\(707\) 1.44120 0.0542018
\(708\) −0.821829 −0.0308862
\(709\) −1.38308 −0.0519427 −0.0259713 0.999663i \(-0.508268\pi\)
−0.0259713 + 0.999663i \(0.508268\pi\)
\(710\) −12.9847 −0.487307
\(711\) −20.1949 −0.757367
\(712\) 0.107411 0.00402539
\(713\) 14.6494 0.548624
\(714\) 5.98774 0.224086
\(715\) −8.71129 −0.325784
\(716\) −9.78741 −0.365773
\(717\) 2.48616 0.0928474
\(718\) −19.2025 −0.716630
\(719\) 30.0613 1.12110 0.560549 0.828122i \(-0.310590\pi\)
0.560549 + 0.828122i \(0.310590\pi\)
\(720\) −2.89545 −0.107907
\(721\) −41.3273 −1.53911
\(722\) 11.9901 0.446225
\(723\) 7.23068 0.268912
\(724\) 10.2982 0.382731
\(725\) −5.35613 −0.198922
\(726\) 8.90183 0.330378
\(727\) 31.1256 1.15439 0.577193 0.816608i \(-0.304148\pi\)
0.577193 + 0.816608i \(0.304148\pi\)
\(728\) −5.99352 −0.222135
\(729\) −21.5171 −0.796929
\(730\) 10.8750 0.402502
\(731\) 48.5368 1.79520
\(732\) −1.41484 −0.0522938
\(733\) −36.1823 −1.33642 −0.668211 0.743972i \(-0.732940\pi\)
−0.668211 + 0.743972i \(0.732940\pi\)
\(734\) −20.1447 −0.743553
\(735\) −3.63410 −0.134046
\(736\) 4.01015 0.147816
\(737\) −88.0453 −3.24319
\(738\) −16.4703 −0.606281
\(739\) 22.1965 0.816512 0.408256 0.912868i \(-0.366137\pi\)
0.408256 + 0.912868i \(0.366137\pi\)
\(740\) 4.46073 0.163980
\(741\) 1.20143 0.0441358
\(742\) −10.3153 −0.378685
\(743\) 11.6611 0.427804 0.213902 0.976855i \(-0.431383\pi\)
0.213902 + 0.976855i \(0.431383\pi\)
\(744\) −1.18120 −0.0433048
\(745\) 4.92135 0.180304
\(746\) −19.7747 −0.724004
\(747\) −18.3169 −0.670181
\(748\) −26.9154 −0.984123
\(749\) 55.7646 2.03759
\(750\) 0.323343 0.0118068
\(751\) −4.32571 −0.157847 −0.0789237 0.996881i \(-0.525148\pi\)
−0.0789237 + 0.996881i \(0.525148\pi\)
\(752\) 11.1258 0.405715
\(753\) −5.20553 −0.189700
\(754\) 7.51677 0.273745
\(755\) −8.56366 −0.311663
\(756\) 8.14109 0.296089
\(757\) −39.5617 −1.43789 −0.718947 0.695065i \(-0.755375\pi\)
−0.718947 + 0.695065i \(0.755375\pi\)
\(758\) −9.72670 −0.353290
\(759\) −8.04873 −0.292150
\(760\) 2.64762 0.0960394
\(761\) −44.6905 −1.62003 −0.810015 0.586410i \(-0.800541\pi\)
−0.810015 + 0.586410i \(0.800541\pi\)
\(762\) −1.26479 −0.0458186
\(763\) 18.2998 0.662496
\(764\) 0.221079 0.00799835
\(765\) −12.5549 −0.453924
\(766\) −8.07832 −0.291882
\(767\) 3.56695 0.128795
\(768\) −0.323343 −0.0116676
\(769\) 0.326678 0.0117803 0.00589016 0.999983i \(-0.498125\pi\)
0.00589016 + 0.999983i \(0.498125\pi\)
\(770\) 26.5097 0.955343
\(771\) −6.80701 −0.245149
\(772\) 20.3644 0.732929
\(773\) 50.0407 1.79984 0.899920 0.436055i \(-0.143625\pi\)
0.899920 + 0.436055i \(0.143625\pi\)
\(774\) 32.4108 1.16498
\(775\) −3.65307 −0.131222
\(776\) −12.2056 −0.438156
\(777\) −6.15987 −0.220984
\(778\) 6.99401 0.250747
\(779\) 15.0606 0.539601
\(780\) −0.453778 −0.0162479
\(781\) −80.5999 −2.88409
\(782\) 17.3883 0.621806
\(783\) −10.2101 −0.364881
\(784\) 11.2391 0.401397
\(785\) 14.3778 0.513164
\(786\) −1.80465 −0.0643696
\(787\) −29.7447 −1.06028 −0.530141 0.847909i \(-0.677861\pi\)
−0.530141 + 0.847909i \(0.677861\pi\)
\(788\) −25.5964 −0.911835
\(789\) −9.99916 −0.355979
\(790\) −6.97469 −0.248148
\(791\) 23.5606 0.837719
\(792\) −17.9729 −0.638640
\(793\) 6.14076 0.218065
\(794\) −7.39389 −0.262399
\(795\) −0.780983 −0.0276986
\(796\) −11.9209 −0.422526
\(797\) −22.0554 −0.781242 −0.390621 0.920552i \(-0.627740\pi\)
−0.390621 + 0.920552i \(0.627740\pi\)
\(798\) −3.65613 −0.129426
\(799\) 48.2423 1.70669
\(800\) −1.00000 −0.0353553
\(801\) 0.311003 0.0109887
\(802\) 1.00000 0.0353112
\(803\) 67.5044 2.38218
\(804\) −4.58635 −0.161748
\(805\) −17.1263 −0.603622
\(806\) 5.12671 0.180581
\(807\) 4.06548 0.143112
\(808\) −0.337459 −0.0118718
\(809\) 51.7239 1.81852 0.909258 0.416232i \(-0.136650\pi\)
0.909258 + 0.416232i \(0.136650\pi\)
\(810\) −8.06997 −0.283550
\(811\) −43.3346 −1.52168 −0.760841 0.648938i \(-0.775213\pi\)
−0.760841 + 0.648938i \(0.775213\pi\)
\(812\) −22.8746 −0.802741
\(813\) 0.485279 0.0170195
\(814\) 27.6891 0.970502
\(815\) 5.48063 0.191978
\(816\) −1.40204 −0.0490813
\(817\) −29.6367 −1.03686
\(818\) 12.4951 0.436879
\(819\) −17.3539 −0.606395
\(820\) −5.68834 −0.198646
\(821\) −14.2942 −0.498872 −0.249436 0.968391i \(-0.580245\pi\)
−0.249436 + 0.968391i \(0.580245\pi\)
\(822\) −6.21795 −0.216876
\(823\) −19.2583 −0.671304 −0.335652 0.941986i \(-0.608957\pi\)
−0.335652 + 0.941986i \(0.608957\pi\)
\(824\) 9.67687 0.337110
\(825\) 2.00709 0.0698779
\(826\) −10.8547 −0.377685
\(827\) 46.3949 1.61331 0.806655 0.591023i \(-0.201276\pi\)
0.806655 + 0.591023i \(0.201276\pi\)
\(828\) 11.6112 0.403517
\(829\) 17.4837 0.607235 0.303618 0.952794i \(-0.401805\pi\)
0.303618 + 0.952794i \(0.401805\pi\)
\(830\) −6.32611 −0.219582
\(831\) −3.58440 −0.124341
\(832\) 1.40339 0.0486540
\(833\) 48.7338 1.68853
\(834\) 0.834945 0.0289118
\(835\) 3.20672 0.110973
\(836\) 16.4346 0.568402
\(837\) −6.96369 −0.240700
\(838\) −12.4835 −0.431235
\(839\) 4.25339 0.146843 0.0734216 0.997301i \(-0.476608\pi\)
0.0734216 + 0.997301i \(0.476608\pi\)
\(840\) 1.38091 0.0476460
\(841\) −0.311831 −0.0107528
\(842\) −14.8237 −0.510859
\(843\) 8.68865 0.299253
\(844\) −8.03884 −0.276708
\(845\) −11.0305 −0.379460
\(846\) 32.2141 1.10754
\(847\) 117.576 4.03995
\(848\) 2.41534 0.0829430
\(849\) −4.75127 −0.163063
\(850\) −4.33608 −0.148726
\(851\) −17.8882 −0.613199
\(852\) −4.19851 −0.143839
\(853\) −8.55908 −0.293057 −0.146529 0.989206i \(-0.546810\pi\)
−0.146529 + 0.989206i \(0.546810\pi\)
\(854\) −18.6872 −0.639463
\(855\) 7.66606 0.262174
\(856\) −13.0574 −0.446293
\(857\) −38.3832 −1.31114 −0.655572 0.755133i \(-0.727572\pi\)
−0.655572 + 0.755133i \(0.727572\pi\)
\(858\) −2.81674 −0.0961619
\(859\) 7.21381 0.246132 0.123066 0.992398i \(-0.460727\pi\)
0.123066 + 0.992398i \(0.460727\pi\)
\(860\) 11.1937 0.381702
\(861\) 7.85510 0.267701
\(862\) −8.10706 −0.276128
\(863\) 19.9722 0.679861 0.339931 0.940451i \(-0.389596\pi\)
0.339931 + 0.940451i \(0.389596\pi\)
\(864\) −1.90625 −0.0648521
\(865\) −14.3715 −0.488646
\(866\) 7.33157 0.249137
\(867\) −0.582537 −0.0197840
\(868\) −15.6013 −0.529542
\(869\) −43.2940 −1.46865
\(870\) −1.73187 −0.0587159
\(871\) 19.9060 0.674488
\(872\) −4.28493 −0.145106
\(873\) −35.3407 −1.19610
\(874\) −10.6174 −0.359138
\(875\) 4.27073 0.144377
\(876\) 3.51636 0.118807
\(877\) 46.7667 1.57920 0.789599 0.613623i \(-0.210289\pi\)
0.789599 + 0.613623i \(0.210289\pi\)
\(878\) −18.4591 −0.622965
\(879\) −0.112053 −0.00377945
\(880\) −6.20730 −0.209248
\(881\) 0.182159 0.00613710 0.00306855 0.999995i \(-0.499023\pi\)
0.00306855 + 0.999995i \(0.499023\pi\)
\(882\) 32.5423 1.09576
\(883\) 42.2776 1.42276 0.711378 0.702810i \(-0.248072\pi\)
0.711378 + 0.702810i \(0.248072\pi\)
\(884\) 6.08524 0.204669
\(885\) −0.821829 −0.0276255
\(886\) 33.5527 1.12723
\(887\) 44.0745 1.47988 0.739939 0.672674i \(-0.234854\pi\)
0.739939 + 0.672674i \(0.234854\pi\)
\(888\) 1.44235 0.0484020
\(889\) −16.7054 −0.560282
\(890\) 0.107411 0.00360042
\(891\) −50.0928 −1.67817
\(892\) −5.81581 −0.194728
\(893\) −29.4568 −0.985736
\(894\) 1.59128 0.0532205
\(895\) −9.78741 −0.327157
\(896\) −4.27073 −0.142675
\(897\) 1.81972 0.0607587
\(898\) −27.2104 −0.908024
\(899\) 19.5664 0.652574
\(900\) −2.89545 −0.0965150
\(901\) 10.4731 0.348910
\(902\) −35.3093 −1.17567
\(903\) −15.4575 −0.514394
\(904\) −5.51677 −0.183485
\(905\) 10.2982 0.342325
\(906\) −2.76900 −0.0919939
\(907\) 38.8630 1.29042 0.645212 0.764004i \(-0.276769\pi\)
0.645212 + 0.764004i \(0.276769\pi\)
\(908\) 13.2037 0.438179
\(909\) −0.977096 −0.0324082
\(910\) −5.99352 −0.198683
\(911\) 3.50179 0.116019 0.0580097 0.998316i \(-0.481525\pi\)
0.0580097 + 0.998316i \(0.481525\pi\)
\(912\) 0.856091 0.0283480
\(913\) −39.2681 −1.29958
\(914\) 40.7803 1.34889
\(915\) −1.41484 −0.0467730
\(916\) 25.2511 0.834319
\(917\) −23.8358 −0.787129
\(918\) −8.26567 −0.272808
\(919\) −38.0048 −1.25366 −0.626831 0.779155i \(-0.715648\pi\)
−0.626831 + 0.779155i \(0.715648\pi\)
\(920\) 4.01015 0.132211
\(921\) −5.68521 −0.187334
\(922\) −31.6278 −1.04161
\(923\) 18.2226 0.599806
\(924\) 8.57173 0.281989
\(925\) 4.46073 0.146668
\(926\) 26.4982 0.870784
\(927\) 28.0189 0.920261
\(928\) 5.35613 0.175824
\(929\) −50.3771 −1.65282 −0.826409 0.563070i \(-0.809620\pi\)
−0.826409 + 0.563070i \(0.809620\pi\)
\(930\) −1.18120 −0.0387330
\(931\) −29.7570 −0.975245
\(932\) 9.40454 0.308056
\(933\) 4.19549 0.137354
\(934\) −1.68402 −0.0551028
\(935\) −26.9154 −0.880227
\(936\) 4.06346 0.132818
\(937\) 1.47657 0.0482376 0.0241188 0.999709i \(-0.492322\pi\)
0.0241188 + 0.999709i \(0.492322\pi\)
\(938\) −60.5767 −1.97790
\(939\) 1.40248 0.0457682
\(940\) 11.1258 0.362883
\(941\) −29.4025 −0.958494 −0.479247 0.877680i \(-0.659090\pi\)
−0.479247 + 0.877680i \(0.659090\pi\)
\(942\) 4.64895 0.151471
\(943\) 22.8111 0.742832
\(944\) 2.54166 0.0827240
\(945\) 8.14109 0.264830
\(946\) 69.4827 2.25908
\(947\) −30.9155 −1.00462 −0.502309 0.864688i \(-0.667516\pi\)
−0.502309 + 0.864688i \(0.667516\pi\)
\(948\) −2.25522 −0.0732461
\(949\) −15.2619 −0.495423
\(950\) 2.64762 0.0859002
\(951\) −8.95018 −0.290230
\(952\) −18.5182 −0.600179
\(953\) 0.417113 0.0135116 0.00675581 0.999977i \(-0.497850\pi\)
0.00675581 + 0.999977i \(0.497850\pi\)
\(954\) 6.99349 0.226423
\(955\) 0.221079 0.00715394
\(956\) −7.68893 −0.248678
\(957\) −10.7502 −0.347506
\(958\) −18.4258 −0.595311
\(959\) −82.1270 −2.65202
\(960\) −0.323343 −0.0104359
\(961\) −17.6550 −0.569518
\(962\) −6.26016 −0.201836
\(963\) −37.8070 −1.21831
\(964\) −22.3622 −0.720239
\(965\) 20.3644 0.655552
\(966\) −5.53766 −0.178171
\(967\) 12.8291 0.412555 0.206277 0.978494i \(-0.433865\pi\)
0.206277 + 0.978494i \(0.433865\pi\)
\(968\) −27.5306 −0.884867
\(969\) 3.71208 0.119249
\(970\) −12.2056 −0.391899
\(971\) 15.7514 0.505487 0.252743 0.967533i \(-0.418667\pi\)
0.252743 + 0.967533i \(0.418667\pi\)
\(972\) −8.32813 −0.267125
\(973\) 11.0280 0.353541
\(974\) −1.65439 −0.0530100
\(975\) −0.453778 −0.0145325
\(976\) 4.37565 0.140061
\(977\) −14.7346 −0.471403 −0.235702 0.971825i \(-0.575739\pi\)
−0.235702 + 0.971825i \(0.575739\pi\)
\(978\) 1.77212 0.0566663
\(979\) 0.666731 0.0213088
\(980\) 11.2391 0.359021
\(981\) −12.4068 −0.396118
\(982\) −10.2927 −0.328454
\(983\) 6.04795 0.192900 0.0964498 0.995338i \(-0.469251\pi\)
0.0964498 + 0.995338i \(0.469251\pi\)
\(984\) −1.83929 −0.0586343
\(985\) −25.5964 −0.815570
\(986\) 23.2246 0.739623
\(987\) −15.3637 −0.489032
\(988\) −3.71566 −0.118211
\(989\) −44.8885 −1.42737
\(990\) −17.9729 −0.571217
\(991\) −31.7203 −1.00763 −0.503814 0.863812i \(-0.668070\pi\)
−0.503814 + 0.863812i \(0.668070\pi\)
\(992\) 3.65307 0.115985
\(993\) 0.561744 0.0178264
\(994\) −55.4541 −1.75890
\(995\) −11.9209 −0.377919
\(996\) −2.04550 −0.0648143
\(997\) −24.3015 −0.769635 −0.384817 0.922993i \(-0.625736\pi\)
−0.384817 + 0.922993i \(0.625736\pi\)
\(998\) 22.2383 0.703941
\(999\) 8.50328 0.269032
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 4010.2.a.m.1.8 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4010.2.a.m.1.8 20 1.1 even 1 trivial