Properties

Label 1045.2.a.g.1.6
Level $1045$
Weight $2$
Character 1045.1
Self dual yes
Analytic conductor $8.344$
Analytic rank $0$
Dimension $6$
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] = [1045,2,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, 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("1045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.131947641.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.56745\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.56745 q^{2} +0.903918 q^{3} +4.59179 q^{4} +1.00000 q^{5} +2.32076 q^{6} +2.16848 q^{7} +6.65430 q^{8} -2.18293 q^{9} +O(q^{10})\) \(q+2.56745 q^{2} +0.903918 q^{3} +4.59179 q^{4} +1.00000 q^{5} +2.32076 q^{6} +2.16848 q^{7} +6.65430 q^{8} -2.18293 q^{9} +2.56745 q^{10} -1.00000 q^{11} +4.15060 q^{12} -4.47137 q^{13} +5.56745 q^{14} +0.903918 q^{15} +7.90099 q^{16} +2.71222 q^{17} -5.60457 q^{18} -1.00000 q^{19} +4.59179 q^{20} +1.96012 q^{21} -2.56745 q^{22} -2.85635 q^{23} +6.01494 q^{24} +1.00000 q^{25} -11.4800 q^{26} -4.68494 q^{27} +9.95719 q^{28} -2.17663 q^{29} +2.32076 q^{30} -6.53387 q^{31} +6.97678 q^{32} -0.903918 q^{33} +6.96349 q^{34} +2.16848 q^{35} -10.0236 q^{36} +0.861702 q^{37} -2.56745 q^{38} -4.04175 q^{39} +6.65430 q^{40} +10.7487 q^{41} +5.03252 q^{42} +8.75038 q^{43} -4.59179 q^{44} -2.18293 q^{45} -7.33354 q^{46} -0.665870 q^{47} +7.14184 q^{48} -2.29772 q^{49} +2.56745 q^{50} +2.45163 q^{51} -20.5316 q^{52} -13.9364 q^{53} -12.0284 q^{54} -1.00000 q^{55} +14.4297 q^{56} -0.903918 q^{57} -5.58839 q^{58} +8.66353 q^{59} +4.15060 q^{60} +5.59008 q^{61} -16.7754 q^{62} -4.73364 q^{63} +2.11055 q^{64} -4.47137 q^{65} -2.32076 q^{66} +15.2043 q^{67} +12.4540 q^{68} -2.58191 q^{69} +5.56745 q^{70} -5.99426 q^{71} -14.5259 q^{72} -10.8568 q^{73} +2.21237 q^{74} +0.903918 q^{75} -4.59179 q^{76} -2.16848 q^{77} -10.3770 q^{78} +5.74408 q^{79} +7.90099 q^{80} +2.31399 q^{81} +27.5967 q^{82} -12.3220 q^{83} +9.00048 q^{84} +2.71222 q^{85} +22.4662 q^{86} -1.96750 q^{87} -6.65430 q^{88} +4.83970 q^{89} -5.60457 q^{90} -9.69605 q^{91} -13.1158 q^{92} -5.90608 q^{93} -1.70959 q^{94} -1.00000 q^{95} +6.30643 q^{96} +14.8355 q^{97} -5.89927 q^{98} +2.18293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 8 q^{4} + 6 q^{5} + 2 q^{6} + 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} + 8 q^{4} + 6 q^{5} + 2 q^{6} + 5 q^{7} + 9 q^{9} - 6 q^{11} + 19 q^{12} - 9 q^{13} + 18 q^{14} + 3 q^{15} + 4 q^{16} - 5 q^{17} + 2 q^{18} - 6 q^{19} + 8 q^{20} + 3 q^{21} + 8 q^{23} - 7 q^{24} + 6 q^{25} - 22 q^{26} + 30 q^{27} + 10 q^{28} - 5 q^{29} + 2 q^{30} - q^{31} + 15 q^{32} - 3 q^{33} - 22 q^{34} + 5 q^{35} + 12 q^{36} + 9 q^{37} - 32 q^{39} + 25 q^{41} + 11 q^{42} + 15 q^{43} - 8 q^{44} + 9 q^{45} - 16 q^{46} + 24 q^{47} - 4 q^{48} + 13 q^{49} - 27 q^{52} + 5 q^{53} - 11 q^{54} - 6 q^{55} - 12 q^{56} - 3 q^{57} + 13 q^{58} + 39 q^{59} + 19 q^{60} - 11 q^{61} - 42 q^{62} + 38 q^{63} - 14 q^{64} - 9 q^{65} - 2 q^{66} + 24 q^{67} + 45 q^{68} + 14 q^{69} + 18 q^{70} - 24 q^{71} - 61 q^{72} - 26 q^{73} + q^{74} + 3 q^{75} - 8 q^{76} - 5 q^{77} - 29 q^{78} + 11 q^{79} + 4 q^{80} + 30 q^{81} + 8 q^{82} + 39 q^{83} + 25 q^{84} - 5 q^{85} + 18 q^{86} - 16 q^{87} + 22 q^{89} + 2 q^{90} - 26 q^{91} - 11 q^{92} - 6 q^{93} - 30 q^{94} - 6 q^{95} - 15 q^{96} + 22 q^{97} + 33 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56745 1.81546 0.907730 0.419554i \(-0.137814\pi\)
0.907730 + 0.419554i \(0.137814\pi\)
\(3\) 0.903918 0.521877 0.260939 0.965355i \(-0.415968\pi\)
0.260939 + 0.965355i \(0.415968\pi\)
\(4\) 4.59179 2.29590
\(5\) 1.00000 0.447214
\(6\) 2.32076 0.947447
\(7\) 2.16848 0.819607 0.409803 0.912174i \(-0.365597\pi\)
0.409803 + 0.912174i \(0.365597\pi\)
\(8\) 6.65430 2.35265
\(9\) −2.18293 −0.727644
\(10\) 2.56745 0.811899
\(11\) −1.00000 −0.301511
\(12\) 4.15060 1.19818
\(13\) −4.47137 −1.24013 −0.620067 0.784549i \(-0.712895\pi\)
−0.620067 + 0.784549i \(0.712895\pi\)
\(14\) 5.56745 1.48796
\(15\) 0.903918 0.233391
\(16\) 7.90099 1.97525
\(17\) 2.71222 0.657810 0.328905 0.944363i \(-0.393320\pi\)
0.328905 + 0.944363i \(0.393320\pi\)
\(18\) −5.60457 −1.32101
\(19\) −1.00000 −0.229416
\(20\) 4.59179 1.02676
\(21\) 1.96012 0.427734
\(22\) −2.56745 −0.547382
\(23\) −2.85635 −0.595590 −0.297795 0.954630i \(-0.596251\pi\)
−0.297795 + 0.954630i \(0.596251\pi\)
\(24\) 6.01494 1.22779
\(25\) 1.00000 0.200000
\(26\) −11.4800 −2.25141
\(27\) −4.68494 −0.901618
\(28\) 9.95719 1.88173
\(29\) −2.17663 −0.404191 −0.202095 0.979366i \(-0.564775\pi\)
−0.202095 + 0.979366i \(0.564775\pi\)
\(30\) 2.32076 0.423711
\(31\) −6.53387 −1.17352 −0.586759 0.809762i \(-0.699596\pi\)
−0.586759 + 0.809762i \(0.699596\pi\)
\(32\) 6.97678 1.23333
\(33\) −0.903918 −0.157352
\(34\) 6.96349 1.19423
\(35\) 2.16848 0.366539
\(36\) −10.0236 −1.67060
\(37\) 0.861702 0.141663 0.0708314 0.997488i \(-0.477435\pi\)
0.0708314 + 0.997488i \(0.477435\pi\)
\(38\) −2.56745 −0.416495
\(39\) −4.04175 −0.647198
\(40\) 6.65430 1.05214
\(41\) 10.7487 1.67866 0.839332 0.543619i \(-0.182946\pi\)
0.839332 + 0.543619i \(0.182946\pi\)
\(42\) 5.03252 0.776534
\(43\) 8.75038 1.33442 0.667210 0.744869i \(-0.267488\pi\)
0.667210 + 0.744869i \(0.267488\pi\)
\(44\) −4.59179 −0.692239
\(45\) −2.18293 −0.325412
\(46\) −7.33354 −1.08127
\(47\) −0.665870 −0.0971272 −0.0485636 0.998820i \(-0.515464\pi\)
−0.0485636 + 0.998820i \(0.515464\pi\)
\(48\) 7.14184 1.03084
\(49\) −2.29772 −0.328245
\(50\) 2.56745 0.363092
\(51\) 2.45163 0.343296
\(52\) −20.5316 −2.84722
\(53\) −13.9364 −1.91431 −0.957155 0.289576i \(-0.906486\pi\)
−0.957155 + 0.289576i \(0.906486\pi\)
\(54\) −12.0284 −1.63685
\(55\) −1.00000 −0.134840
\(56\) 14.4297 1.92825
\(57\) −0.903918 −0.119727
\(58\) −5.58839 −0.733792
\(59\) 8.66353 1.12790 0.563948 0.825810i \(-0.309282\pi\)
0.563948 + 0.825810i \(0.309282\pi\)
\(60\) 4.15060 0.535841
\(61\) 5.59008 0.715736 0.357868 0.933772i \(-0.383504\pi\)
0.357868 + 0.933772i \(0.383504\pi\)
\(62\) −16.7754 −2.13048
\(63\) −4.73364 −0.596382
\(64\) 2.11055 0.263819
\(65\) −4.47137 −0.554605
\(66\) −2.32076 −0.285666
\(67\) 15.2043 1.85751 0.928753 0.370698i \(-0.120882\pi\)
0.928753 + 0.370698i \(0.120882\pi\)
\(68\) 12.4540 1.51027
\(69\) −2.58191 −0.310825
\(70\) 5.56745 0.665437
\(71\) −5.99426 −0.711388 −0.355694 0.934603i \(-0.615755\pi\)
−0.355694 + 0.934603i \(0.615755\pi\)
\(72\) −14.5259 −1.71189
\(73\) −10.8568 −1.27069 −0.635347 0.772226i \(-0.719143\pi\)
−0.635347 + 0.772226i \(0.719143\pi\)
\(74\) 2.21237 0.257183
\(75\) 0.903918 0.104375
\(76\) −4.59179 −0.526715
\(77\) −2.16848 −0.247121
\(78\) −10.3770 −1.17496
\(79\) 5.74408 0.646260 0.323130 0.946355i \(-0.395265\pi\)
0.323130 + 0.946355i \(0.395265\pi\)
\(80\) 7.90099 0.883357
\(81\) 2.31399 0.257110
\(82\) 27.5967 3.04755
\(83\) −12.3220 −1.35252 −0.676259 0.736664i \(-0.736400\pi\)
−0.676259 + 0.736664i \(0.736400\pi\)
\(84\) 9.00048 0.982033
\(85\) 2.71222 0.294182
\(86\) 22.4662 2.42259
\(87\) −1.96750 −0.210938
\(88\) −6.65430 −0.709351
\(89\) 4.83970 0.513007 0.256503 0.966543i \(-0.417430\pi\)
0.256503 + 0.966543i \(0.417430\pi\)
\(90\) −5.60457 −0.590773
\(91\) −9.69605 −1.01642
\(92\) −13.1158 −1.36741
\(93\) −5.90608 −0.612432
\(94\) −1.70959 −0.176331
\(95\) −1.00000 −0.102598
\(96\) 6.30643 0.643648
\(97\) 14.8355 1.50631 0.753157 0.657841i \(-0.228530\pi\)
0.753157 + 0.657841i \(0.228530\pi\)
\(98\) −5.89927 −0.595916
\(99\) 2.18293 0.219393
\(100\) 4.59179 0.459179
\(101\) 9.35188 0.930546 0.465273 0.885167i \(-0.345956\pi\)
0.465273 + 0.885167i \(0.345956\pi\)
\(102\) 6.29442 0.623241
\(103\) 8.08853 0.796987 0.398493 0.917171i \(-0.369533\pi\)
0.398493 + 0.917171i \(0.369533\pi\)
\(104\) −29.7538 −2.91760
\(105\) 1.96012 0.191288
\(106\) −35.7810 −3.47535
\(107\) −2.71050 −0.262034 −0.131017 0.991380i \(-0.541824\pi\)
−0.131017 + 0.991380i \(0.541824\pi\)
\(108\) −21.5123 −2.07002
\(109\) 0.495056 0.0474178 0.0237089 0.999719i \(-0.492453\pi\)
0.0237089 + 0.999719i \(0.492453\pi\)
\(110\) −2.56745 −0.244797
\(111\) 0.778907 0.0739306
\(112\) 17.1331 1.61892
\(113\) 20.6374 1.94140 0.970701 0.240292i \(-0.0772432\pi\)
0.970701 + 0.240292i \(0.0772432\pi\)
\(114\) −2.32076 −0.217359
\(115\) −2.85635 −0.266356
\(116\) −9.99465 −0.927980
\(117\) 9.76069 0.902376
\(118\) 22.2432 2.04765
\(119\) 5.88139 0.539146
\(120\) 6.01494 0.549086
\(121\) 1.00000 0.0909091
\(122\) 14.3522 1.29939
\(123\) 9.71594 0.876057
\(124\) −30.0022 −2.69428
\(125\) 1.00000 0.0894427
\(126\) −12.1534 −1.08271
\(127\) −15.2674 −1.35477 −0.677383 0.735631i \(-0.736886\pi\)
−0.677383 + 0.735631i \(0.736886\pi\)
\(128\) −8.53482 −0.754379
\(129\) 7.90962 0.696404
\(130\) −11.4800 −1.00686
\(131\) −14.5918 −1.27489 −0.637444 0.770497i \(-0.720008\pi\)
−0.637444 + 0.770497i \(0.720008\pi\)
\(132\) −4.15060 −0.361264
\(133\) −2.16848 −0.188031
\(134\) 39.0364 3.37223
\(135\) −4.68494 −0.403216
\(136\) 18.0479 1.54760
\(137\) −11.9896 −1.02434 −0.512169 0.858885i \(-0.671158\pi\)
−0.512169 + 0.858885i \(0.671158\pi\)
\(138\) −6.62891 −0.564291
\(139\) −7.16027 −0.607326 −0.303663 0.952779i \(-0.598210\pi\)
−0.303663 + 0.952779i \(0.598210\pi\)
\(140\) 9.95719 0.841536
\(141\) −0.601892 −0.0506885
\(142\) −15.3900 −1.29150
\(143\) 4.47137 0.373914
\(144\) −17.2473 −1.43728
\(145\) −2.17663 −0.180760
\(146\) −27.8743 −2.30690
\(147\) −2.07695 −0.171304
\(148\) 3.95676 0.325243
\(149\) 3.73998 0.306391 0.153196 0.988196i \(-0.451044\pi\)
0.153196 + 0.988196i \(0.451044\pi\)
\(150\) 2.32076 0.189489
\(151\) −7.70787 −0.627258 −0.313629 0.949546i \(-0.601545\pi\)
−0.313629 + 0.949546i \(0.601545\pi\)
\(152\) −6.65430 −0.539735
\(153\) −5.92060 −0.478652
\(154\) −5.56745 −0.448638
\(155\) −6.53387 −0.524813
\(156\) −18.5589 −1.48590
\(157\) −7.78415 −0.621242 −0.310621 0.950534i \(-0.600537\pi\)
−0.310621 + 0.950534i \(0.600537\pi\)
\(158\) 14.7476 1.17326
\(159\) −12.5973 −0.999035
\(160\) 6.97678 0.551563
\(161\) −6.19393 −0.488150
\(162\) 5.94106 0.466774
\(163\) −3.60748 −0.282560 −0.141280 0.989970i \(-0.545122\pi\)
−0.141280 + 0.989970i \(0.545122\pi\)
\(164\) 49.3558 3.85404
\(165\) −0.903918 −0.0703699
\(166\) −31.6362 −2.45544
\(167\) −0.588302 −0.0455242 −0.0227621 0.999741i \(-0.507246\pi\)
−0.0227621 + 0.999741i \(0.507246\pi\)
\(168\) 13.0432 1.00631
\(169\) 6.99312 0.537932
\(170\) 6.96349 0.534075
\(171\) 2.18293 0.166933
\(172\) 40.1800 3.06369
\(173\) 23.1505 1.76010 0.880050 0.474882i \(-0.157509\pi\)
0.880050 + 0.474882i \(0.157509\pi\)
\(174\) −5.05145 −0.382949
\(175\) 2.16848 0.163921
\(176\) −7.90099 −0.595559
\(177\) 7.83112 0.588623
\(178\) 12.4257 0.931343
\(179\) 11.2340 0.839666 0.419833 0.907601i \(-0.362089\pi\)
0.419833 + 0.907601i \(0.362089\pi\)
\(180\) −10.0236 −0.747113
\(181\) −7.18465 −0.534031 −0.267015 0.963692i \(-0.586037\pi\)
−0.267015 + 0.963692i \(0.586037\pi\)
\(182\) −24.8941 −1.84527
\(183\) 5.05297 0.373526
\(184\) −19.0070 −1.40122
\(185\) 0.861702 0.0633536
\(186\) −15.1636 −1.11185
\(187\) −2.71222 −0.198337
\(188\) −3.05754 −0.222994
\(189\) −10.1592 −0.738972
\(190\) −2.56745 −0.186262
\(191\) −13.9736 −1.01109 −0.505546 0.862800i \(-0.668709\pi\)
−0.505546 + 0.862800i \(0.668709\pi\)
\(192\) 1.90777 0.137681
\(193\) −12.8048 −0.921710 −0.460855 0.887475i \(-0.652457\pi\)
−0.460855 + 0.887475i \(0.652457\pi\)
\(194\) 38.0893 2.73465
\(195\) −4.04175 −0.289436
\(196\) −10.5506 −0.753617
\(197\) 3.78816 0.269895 0.134948 0.990853i \(-0.456913\pi\)
0.134948 + 0.990853i \(0.456913\pi\)
\(198\) 5.60457 0.398299
\(199\) 15.9061 1.12755 0.563777 0.825927i \(-0.309348\pi\)
0.563777 + 0.825927i \(0.309348\pi\)
\(200\) 6.65430 0.470530
\(201\) 13.7435 0.969390
\(202\) 24.0105 1.68937
\(203\) −4.71997 −0.331277
\(204\) 11.2574 0.788173
\(205\) 10.7487 0.750722
\(206\) 20.7669 1.44690
\(207\) 6.23522 0.433378
\(208\) −35.3282 −2.44957
\(209\) 1.00000 0.0691714
\(210\) 5.03252 0.347277
\(211\) 2.20422 0.151745 0.0758723 0.997118i \(-0.475826\pi\)
0.0758723 + 0.997118i \(0.475826\pi\)
\(212\) −63.9930 −4.39506
\(213\) −5.41832 −0.371257
\(214\) −6.95908 −0.475713
\(215\) 8.75038 0.596771
\(216\) −31.1750 −2.12119
\(217\) −14.1685 −0.961823
\(218\) 1.27103 0.0860851
\(219\) −9.81367 −0.663147
\(220\) −4.59179 −0.309579
\(221\) −12.1273 −0.815773
\(222\) 1.99980 0.134218
\(223\) −14.3670 −0.962086 −0.481043 0.876697i \(-0.659742\pi\)
−0.481043 + 0.876697i \(0.659742\pi\)
\(224\) 15.1290 1.01085
\(225\) −2.18293 −0.145529
\(226\) 52.9854 3.52454
\(227\) 21.6909 1.43968 0.719838 0.694142i \(-0.244216\pi\)
0.719838 + 0.694142i \(0.244216\pi\)
\(228\) −4.15060 −0.274880
\(229\) 9.39735 0.620995 0.310497 0.950574i \(-0.399504\pi\)
0.310497 + 0.950574i \(0.399504\pi\)
\(230\) −7.33354 −0.483559
\(231\) −1.96012 −0.128967
\(232\) −14.4840 −0.950919
\(233\) −22.6514 −1.48394 −0.741971 0.670432i \(-0.766109\pi\)
−0.741971 + 0.670432i \(0.766109\pi\)
\(234\) 25.0601 1.63823
\(235\) −0.665870 −0.0434366
\(236\) 39.7812 2.58953
\(237\) 5.19218 0.337268
\(238\) 15.1002 0.978798
\(239\) 12.3190 0.796853 0.398426 0.917200i \(-0.369556\pi\)
0.398426 + 0.917200i \(0.369556\pi\)
\(240\) 7.14184 0.461004
\(241\) −27.5506 −1.77469 −0.887345 0.461106i \(-0.847453\pi\)
−0.887345 + 0.461106i \(0.847453\pi\)
\(242\) 2.56745 0.165042
\(243\) 16.1465 1.03580
\(244\) 25.6685 1.64326
\(245\) −2.29772 −0.146796
\(246\) 24.9452 1.59045
\(247\) 4.47137 0.284506
\(248\) −43.4783 −2.76088
\(249\) −11.1381 −0.705848
\(250\) 2.56745 0.162380
\(251\) 11.9051 0.751443 0.375722 0.926733i \(-0.377395\pi\)
0.375722 + 0.926733i \(0.377395\pi\)
\(252\) −21.7359 −1.36923
\(253\) 2.85635 0.179577
\(254\) −39.1983 −2.45952
\(255\) 2.45163 0.153527
\(256\) −26.1338 −1.63336
\(257\) −8.19766 −0.511356 −0.255678 0.966762i \(-0.582299\pi\)
−0.255678 + 0.966762i \(0.582299\pi\)
\(258\) 20.3076 1.26429
\(259\) 1.86858 0.116108
\(260\) −20.5316 −1.27332
\(261\) 4.75144 0.294107
\(262\) −37.4636 −2.31451
\(263\) −13.1875 −0.813175 −0.406588 0.913612i \(-0.633281\pi\)
−0.406588 + 0.913612i \(0.633281\pi\)
\(264\) −6.01494 −0.370194
\(265\) −13.9364 −0.856106
\(266\) −5.56745 −0.341362
\(267\) 4.37469 0.267726
\(268\) 69.8152 4.26464
\(269\) 19.3988 1.18277 0.591384 0.806390i \(-0.298582\pi\)
0.591384 + 0.806390i \(0.298582\pi\)
\(270\) −12.0284 −0.732022
\(271\) 3.57069 0.216904 0.108452 0.994102i \(-0.465411\pi\)
0.108452 + 0.994102i \(0.465411\pi\)
\(272\) 21.4292 1.29934
\(273\) −8.76443 −0.530447
\(274\) −30.7826 −1.85964
\(275\) −1.00000 −0.0603023
\(276\) −11.8556 −0.713622
\(277\) −20.6671 −1.24176 −0.620882 0.783904i \(-0.713225\pi\)
−0.620882 + 0.783904i \(0.713225\pi\)
\(278\) −18.3836 −1.10258
\(279\) 14.2630 0.853904
\(280\) 14.4297 0.862338
\(281\) −6.89894 −0.411556 −0.205778 0.978599i \(-0.565973\pi\)
−0.205778 + 0.978599i \(0.565973\pi\)
\(282\) −1.54533 −0.0920229
\(283\) 23.5344 1.39898 0.699488 0.714644i \(-0.253411\pi\)
0.699488 + 0.714644i \(0.253411\pi\)
\(284\) −27.5244 −1.63327
\(285\) −0.903918 −0.0535435
\(286\) 11.4800 0.678827
\(287\) 23.3083 1.37584
\(288\) −15.2298 −0.897427
\(289\) −9.64385 −0.567285
\(290\) −5.58839 −0.328162
\(291\) 13.4100 0.786111
\(292\) −49.8523 −2.91738
\(293\) 8.70291 0.508429 0.254215 0.967148i \(-0.418183\pi\)
0.254215 + 0.967148i \(0.418183\pi\)
\(294\) −5.33245 −0.310995
\(295\) 8.66353 0.504410
\(296\) 5.73402 0.333283
\(297\) 4.68494 0.271848
\(298\) 9.60221 0.556241
\(299\) 12.7718 0.738612
\(300\) 4.15060 0.239635
\(301\) 18.9750 1.09370
\(302\) −19.7896 −1.13876
\(303\) 8.45333 0.485631
\(304\) −7.90099 −0.453153
\(305\) 5.59008 0.320087
\(306\) −15.2008 −0.868974
\(307\) −0.0345724 −0.00197315 −0.000986576 1.00000i \(-0.500314\pi\)
−0.000986576 1.00000i \(0.500314\pi\)
\(308\) −9.95719 −0.567364
\(309\) 7.31137 0.415929
\(310\) −16.7754 −0.952778
\(311\) −16.8869 −0.957570 −0.478785 0.877932i \(-0.658923\pi\)
−0.478785 + 0.877932i \(0.658923\pi\)
\(312\) −26.8950 −1.52263
\(313\) −1.47670 −0.0834682 −0.0417341 0.999129i \(-0.513288\pi\)
−0.0417341 + 0.999129i \(0.513288\pi\)
\(314\) −19.9854 −1.12784
\(315\) −4.73364 −0.266710
\(316\) 26.3756 1.48375
\(317\) 24.5279 1.37763 0.688813 0.724939i \(-0.258132\pi\)
0.688813 + 0.724939i \(0.258132\pi\)
\(318\) −32.3430 −1.81371
\(319\) 2.17663 0.121868
\(320\) 2.11055 0.117983
\(321\) −2.45007 −0.136750
\(322\) −15.9026 −0.886217
\(323\) −2.71222 −0.150912
\(324\) 10.6254 0.590299
\(325\) −4.47137 −0.248027
\(326\) −9.26202 −0.512976
\(327\) 0.447490 0.0247463
\(328\) 71.5251 3.94931
\(329\) −1.44392 −0.0796061
\(330\) −2.32076 −0.127754
\(331\) 19.8183 1.08931 0.544657 0.838659i \(-0.316660\pi\)
0.544657 + 0.838659i \(0.316660\pi\)
\(332\) −56.5802 −3.10524
\(333\) −1.88104 −0.103080
\(334\) −1.51044 −0.0826474
\(335\) 15.2043 0.830702
\(336\) 15.4869 0.844880
\(337\) 12.6702 0.690188 0.345094 0.938568i \(-0.387847\pi\)
0.345094 + 0.938568i \(0.387847\pi\)
\(338\) 17.9545 0.976595
\(339\) 18.6545 1.01317
\(340\) 12.4540 0.675411
\(341\) 6.53387 0.353829
\(342\) 5.60457 0.303060
\(343\) −20.1619 −1.08864
\(344\) 58.2277 3.13943
\(345\) −2.58191 −0.139005
\(346\) 59.4377 3.19539
\(347\) 6.29698 0.338040 0.169020 0.985613i \(-0.445940\pi\)
0.169020 + 0.985613i \(0.445940\pi\)
\(348\) −9.03434 −0.484292
\(349\) −4.90512 −0.262565 −0.131283 0.991345i \(-0.541910\pi\)
−0.131283 + 0.991345i \(0.541910\pi\)
\(350\) 5.56745 0.297593
\(351\) 20.9481 1.11813
\(352\) −6.97678 −0.371864
\(353\) 22.7868 1.21282 0.606409 0.795153i \(-0.292609\pi\)
0.606409 + 0.795153i \(0.292609\pi\)
\(354\) 20.1060 1.06862
\(355\) −5.99426 −0.318142
\(356\) 22.2229 1.17781
\(357\) 5.31629 0.281368
\(358\) 28.8426 1.52438
\(359\) 26.4835 1.39774 0.698872 0.715246i \(-0.253686\pi\)
0.698872 + 0.715246i \(0.253686\pi\)
\(360\) −14.5259 −0.765582
\(361\) 1.00000 0.0526316
\(362\) −18.4462 −0.969512
\(363\) 0.903918 0.0474434
\(364\) −44.5223 −2.33360
\(365\) −10.8568 −0.568272
\(366\) 12.9732 0.678122
\(367\) 14.4411 0.753818 0.376909 0.926250i \(-0.376987\pi\)
0.376909 + 0.926250i \(0.376987\pi\)
\(368\) −22.5680 −1.17644
\(369\) −23.4637 −1.22147
\(370\) 2.21237 0.115016
\(371\) −30.2207 −1.56898
\(372\) −27.1195 −1.40608
\(373\) −3.81445 −0.197505 −0.0987525 0.995112i \(-0.531485\pi\)
−0.0987525 + 0.995112i \(0.531485\pi\)
\(374\) −6.96349 −0.360074
\(375\) 0.903918 0.0466781
\(376\) −4.43090 −0.228506
\(377\) 9.73252 0.501250
\(378\) −26.0832 −1.34157
\(379\) −8.00109 −0.410988 −0.205494 0.978658i \(-0.565880\pi\)
−0.205494 + 0.978658i \(0.565880\pi\)
\(380\) −4.59179 −0.235554
\(381\) −13.8005 −0.707021
\(382\) −35.8764 −1.83560
\(383\) 13.5609 0.692932 0.346466 0.938063i \(-0.387382\pi\)
0.346466 + 0.938063i \(0.387382\pi\)
\(384\) −7.71478 −0.393693
\(385\) −2.16848 −0.110516
\(386\) −32.8757 −1.67333
\(387\) −19.1015 −0.970984
\(388\) 68.1214 3.45834
\(389\) −1.63547 −0.0829216 −0.0414608 0.999140i \(-0.513201\pi\)
−0.0414608 + 0.999140i \(0.513201\pi\)
\(390\) −10.3770 −0.525459
\(391\) −7.74706 −0.391786
\(392\) −15.2897 −0.772246
\(393\) −13.1898 −0.665335
\(394\) 9.72590 0.489984
\(395\) 5.74408 0.289016
\(396\) 10.0236 0.503704
\(397\) 9.24683 0.464085 0.232043 0.972706i \(-0.425459\pi\)
0.232043 + 0.972706i \(0.425459\pi\)
\(398\) 40.8381 2.04703
\(399\) −1.96012 −0.0981289
\(400\) 7.90099 0.395049
\(401\) −0.299914 −0.0149770 −0.00748851 0.999972i \(-0.502384\pi\)
−0.00748851 + 0.999972i \(0.502384\pi\)
\(402\) 35.2857 1.75989
\(403\) 29.2153 1.45532
\(404\) 42.9419 2.13644
\(405\) 2.31399 0.114983
\(406\) −12.1183 −0.601421
\(407\) −0.861702 −0.0427130
\(408\) 16.3138 0.807656
\(409\) 9.28447 0.459088 0.229544 0.973298i \(-0.426277\pi\)
0.229544 + 0.973298i \(0.426277\pi\)
\(410\) 27.5967 1.36291
\(411\) −10.8376 −0.534578
\(412\) 37.1409 1.82980
\(413\) 18.7867 0.924431
\(414\) 16.0086 0.786781
\(415\) −12.3220 −0.604865
\(416\) −31.1957 −1.52950
\(417\) −6.47230 −0.316950
\(418\) 2.56745 0.125578
\(419\) −9.33196 −0.455896 −0.227948 0.973673i \(-0.573202\pi\)
−0.227948 + 0.973673i \(0.573202\pi\)
\(420\) 9.00048 0.439179
\(421\) −25.1456 −1.22552 −0.612760 0.790269i \(-0.709941\pi\)
−0.612760 + 0.790269i \(0.709941\pi\)
\(422\) 5.65921 0.275486
\(423\) 1.45355 0.0706740
\(424\) −92.7369 −4.50370
\(425\) 2.71222 0.131562
\(426\) −13.9113 −0.674003
\(427\) 12.1219 0.586622
\(428\) −12.4461 −0.601604
\(429\) 4.04175 0.195137
\(430\) 22.4662 1.08341
\(431\) 37.5660 1.80949 0.904744 0.425955i \(-0.140062\pi\)
0.904744 + 0.425955i \(0.140062\pi\)
\(432\) −37.0157 −1.78092
\(433\) 9.24282 0.444181 0.222091 0.975026i \(-0.428712\pi\)
0.222091 + 0.975026i \(0.428712\pi\)
\(434\) −36.3770 −1.74615
\(435\) −1.96750 −0.0943343
\(436\) 2.27320 0.108866
\(437\) 2.85635 0.136638
\(438\) −25.1961 −1.20392
\(439\) 0.941826 0.0449509 0.0224754 0.999747i \(-0.492845\pi\)
0.0224754 + 0.999747i \(0.492845\pi\)
\(440\) −6.65430 −0.317231
\(441\) 5.01576 0.238846
\(442\) −31.1363 −1.48100
\(443\) 17.2727 0.820649 0.410324 0.911940i \(-0.365415\pi\)
0.410324 + 0.911940i \(0.365415\pi\)
\(444\) 3.57658 0.169737
\(445\) 4.83970 0.229424
\(446\) −36.8866 −1.74663
\(447\) 3.38063 0.159899
\(448\) 4.57668 0.216228
\(449\) −20.9917 −0.990661 −0.495330 0.868705i \(-0.664953\pi\)
−0.495330 + 0.868705i \(0.664953\pi\)
\(450\) −5.60457 −0.264202
\(451\) −10.7487 −0.506136
\(452\) 94.7626 4.45726
\(453\) −6.96728 −0.327351
\(454\) 55.6903 2.61368
\(455\) −9.69605 −0.454558
\(456\) −6.01494 −0.281675
\(457\) −28.9893 −1.35606 −0.678032 0.735032i \(-0.737167\pi\)
−0.678032 + 0.735032i \(0.737167\pi\)
\(458\) 24.1272 1.12739
\(459\) −12.7066 −0.593094
\(460\) −13.1158 −0.611526
\(461\) −24.7915 −1.15465 −0.577327 0.816513i \(-0.695904\pi\)
−0.577327 + 0.816513i \(0.695904\pi\)
\(462\) −5.03252 −0.234134
\(463\) −0.757122 −0.0351864 −0.0175932 0.999845i \(-0.505600\pi\)
−0.0175932 + 0.999845i \(0.505600\pi\)
\(464\) −17.1975 −0.798376
\(465\) −5.90608 −0.273888
\(466\) −58.1563 −2.69404
\(467\) −5.54461 −0.256574 −0.128287 0.991737i \(-0.540948\pi\)
−0.128287 + 0.991737i \(0.540948\pi\)
\(468\) 44.8191 2.07176
\(469\) 32.9702 1.52242
\(470\) −1.70959 −0.0788574
\(471\) −7.03623 −0.324212
\(472\) 57.6497 2.65354
\(473\) −8.75038 −0.402343
\(474\) 13.3307 0.612297
\(475\) −1.00000 −0.0458831
\(476\) 27.0061 1.23782
\(477\) 30.4222 1.39294
\(478\) 31.6285 1.44665
\(479\) −8.78759 −0.401515 −0.200758 0.979641i \(-0.564340\pi\)
−0.200758 + 0.979641i \(0.564340\pi\)
\(480\) 6.30643 0.287848
\(481\) −3.85298 −0.175681
\(482\) −70.7348 −3.22188
\(483\) −5.59880 −0.254754
\(484\) 4.59179 0.208718
\(485\) 14.8355 0.673644
\(486\) 41.4553 1.88045
\(487\) 36.8929 1.67178 0.835889 0.548898i \(-0.184953\pi\)
0.835889 + 0.548898i \(0.184953\pi\)
\(488\) 37.1980 1.68388
\(489\) −3.26086 −0.147461
\(490\) −5.89927 −0.266502
\(491\) −6.12358 −0.276353 −0.138177 0.990408i \(-0.544124\pi\)
−0.138177 + 0.990408i \(0.544124\pi\)
\(492\) 44.6136 2.01134
\(493\) −5.90351 −0.265881
\(494\) 11.4800 0.516510
\(495\) 2.18293 0.0981155
\(496\) −51.6240 −2.31799
\(497\) −12.9984 −0.583058
\(498\) −28.5965 −1.28144
\(499\) −16.9154 −0.757239 −0.378619 0.925552i \(-0.623601\pi\)
−0.378619 + 0.925552i \(0.623601\pi\)
\(500\) 4.59179 0.205351
\(501\) −0.531777 −0.0237580
\(502\) 30.5657 1.36422
\(503\) 4.84680 0.216108 0.108054 0.994145i \(-0.465538\pi\)
0.108054 + 0.994145i \(0.465538\pi\)
\(504\) −31.4990 −1.40308
\(505\) 9.35188 0.416153
\(506\) 7.33354 0.326015
\(507\) 6.32120 0.280735
\(508\) −70.1049 −3.11040
\(509\) −18.1418 −0.804122 −0.402061 0.915613i \(-0.631706\pi\)
−0.402061 + 0.915613i \(0.631706\pi\)
\(510\) 6.29442 0.278722
\(511\) −23.5427 −1.04147
\(512\) −50.0276 −2.21093
\(513\) 4.68494 0.206845
\(514\) −21.0471 −0.928346
\(515\) 8.08853 0.356423
\(516\) 36.3194 1.59887
\(517\) 0.665870 0.0292849
\(518\) 4.79748 0.210789
\(519\) 20.9261 0.918556
\(520\) −29.7538 −1.30479
\(521\) 36.0419 1.57902 0.789511 0.613736i \(-0.210334\pi\)
0.789511 + 0.613736i \(0.210334\pi\)
\(522\) 12.1991 0.533940
\(523\) 38.6617 1.69056 0.845279 0.534325i \(-0.179434\pi\)
0.845279 + 0.534325i \(0.179434\pi\)
\(524\) −67.0024 −2.92701
\(525\) 1.96012 0.0855468
\(526\) −33.8582 −1.47629
\(527\) −17.7213 −0.771952
\(528\) −7.14184 −0.310809
\(529\) −14.8413 −0.645272
\(530\) −35.7810 −1.55423
\(531\) −18.9119 −0.820707
\(532\) −9.95719 −0.431699
\(533\) −48.0614 −2.08177
\(534\) 11.2318 0.486047
\(535\) −2.71050 −0.117185
\(536\) 101.174 4.37006
\(537\) 10.1546 0.438202
\(538\) 49.8056 2.14727
\(539\) 2.29772 0.0989696
\(540\) −21.5123 −0.925742
\(541\) −39.8208 −1.71203 −0.856015 0.516951i \(-0.827067\pi\)
−0.856015 + 0.516951i \(0.827067\pi\)
\(542\) 9.16756 0.393780
\(543\) −6.49433 −0.278698
\(544\) 18.9226 0.811299
\(545\) 0.495056 0.0212059
\(546\) −22.5022 −0.963006
\(547\) 34.9397 1.49392 0.746958 0.664871i \(-0.231514\pi\)
0.746958 + 0.664871i \(0.231514\pi\)
\(548\) −55.0536 −2.35177
\(549\) −12.2028 −0.520801
\(550\) −2.56745 −0.109476
\(551\) 2.17663 0.0927277
\(552\) −17.1808 −0.731263
\(553\) 12.4559 0.529679
\(554\) −53.0616 −2.25437
\(555\) 0.778907 0.0330628
\(556\) −32.8785 −1.39436
\(557\) −1.09011 −0.0461896 −0.0230948 0.999733i \(-0.507352\pi\)
−0.0230948 + 0.999733i \(0.507352\pi\)
\(558\) 36.6195 1.55023
\(559\) −39.1262 −1.65486
\(560\) 17.1331 0.724005
\(561\) −2.45163 −0.103508
\(562\) −17.7127 −0.747164
\(563\) −2.39110 −0.100773 −0.0503864 0.998730i \(-0.516045\pi\)
−0.0503864 + 0.998730i \(0.516045\pi\)
\(564\) −2.76376 −0.116375
\(565\) 20.6374 0.868221
\(566\) 60.4234 2.53979
\(567\) 5.01784 0.210729
\(568\) −39.8876 −1.67365
\(569\) −27.9054 −1.16985 −0.584927 0.811086i \(-0.698877\pi\)
−0.584927 + 0.811086i \(0.698877\pi\)
\(570\) −2.32076 −0.0972060
\(571\) 9.33966 0.390853 0.195426 0.980718i \(-0.437391\pi\)
0.195426 + 0.980718i \(0.437391\pi\)
\(572\) 20.5316 0.858469
\(573\) −12.6310 −0.527666
\(574\) 59.8428 2.49779
\(575\) −2.85635 −0.119118
\(576\) −4.60719 −0.191966
\(577\) 24.2224 1.00839 0.504195 0.863590i \(-0.331789\pi\)
0.504195 + 0.863590i \(0.331789\pi\)
\(578\) −24.7601 −1.02988
\(579\) −11.5745 −0.481020
\(580\) −9.99465 −0.415005
\(581\) −26.7200 −1.10853
\(582\) 34.4296 1.42715
\(583\) 13.9364 0.577186
\(584\) −72.2445 −2.98950
\(585\) 9.76069 0.403555
\(586\) 22.3443 0.923033
\(587\) 3.72615 0.153795 0.0768973 0.997039i \(-0.475499\pi\)
0.0768973 + 0.997039i \(0.475499\pi\)
\(588\) −9.53691 −0.393296
\(589\) 6.53387 0.269224
\(590\) 22.2432 0.915737
\(591\) 3.42418 0.140852
\(592\) 6.80829 0.279819
\(593\) 14.1469 0.580944 0.290472 0.956883i \(-0.406188\pi\)
0.290472 + 0.956883i \(0.406188\pi\)
\(594\) 12.0284 0.493529
\(595\) 5.88139 0.241113
\(596\) 17.1732 0.703442
\(597\) 14.3778 0.588444
\(598\) 32.7909 1.34092
\(599\) −37.7856 −1.54388 −0.771940 0.635696i \(-0.780713\pi\)
−0.771940 + 0.635696i \(0.780713\pi\)
\(600\) 6.01494 0.245559
\(601\) −29.5251 −1.20436 −0.602178 0.798362i \(-0.705700\pi\)
−0.602178 + 0.798362i \(0.705700\pi\)
\(602\) 48.7173 1.98557
\(603\) −33.1901 −1.35160
\(604\) −35.3930 −1.44012
\(605\) 1.00000 0.0406558
\(606\) 21.7035 0.881644
\(607\) −5.19284 −0.210771 −0.105385 0.994431i \(-0.533608\pi\)
−0.105385 + 0.994431i \(0.533608\pi\)
\(608\) −6.97678 −0.282946
\(609\) −4.26647 −0.172886
\(610\) 14.3522 0.581105
\(611\) 2.97735 0.120451
\(612\) −27.1862 −1.09894
\(613\) −44.9254 −1.81452 −0.907260 0.420569i \(-0.861830\pi\)
−0.907260 + 0.420569i \(0.861830\pi\)
\(614\) −0.0887629 −0.00358218
\(615\) 9.71594 0.391784
\(616\) −14.4297 −0.581388
\(617\) −8.39687 −0.338045 −0.169023 0.985612i \(-0.554061\pi\)
−0.169023 + 0.985612i \(0.554061\pi\)
\(618\) 18.7716 0.755103
\(619\) 4.27492 0.171823 0.0859117 0.996303i \(-0.472620\pi\)
0.0859117 + 0.996303i \(0.472620\pi\)
\(620\) −30.0022 −1.20492
\(621\) 13.3818 0.536995
\(622\) −43.3563 −1.73843
\(623\) 10.4948 0.420464
\(624\) −31.9338 −1.27837
\(625\) 1.00000 0.0400000
\(626\) −3.79136 −0.151533
\(627\) 0.903918 0.0360990
\(628\) −35.7432 −1.42631
\(629\) 2.33713 0.0931873
\(630\) −12.1534 −0.484202
\(631\) 7.33731 0.292094 0.146047 0.989278i \(-0.453345\pi\)
0.146047 + 0.989278i \(0.453345\pi\)
\(632\) 38.2228 1.52042
\(633\) 1.99243 0.0791920
\(634\) 62.9742 2.50103
\(635\) −15.2674 −0.605869
\(636\) −57.8444 −2.29368
\(637\) 10.2739 0.407068
\(638\) 5.58839 0.221247
\(639\) 13.0851 0.517637
\(640\) −8.53482 −0.337368
\(641\) 13.0203 0.514272 0.257136 0.966375i \(-0.417221\pi\)
0.257136 + 0.966375i \(0.417221\pi\)
\(642\) −6.29044 −0.248264
\(643\) 33.7766 1.33202 0.666010 0.745943i \(-0.268001\pi\)
0.666010 + 0.745943i \(0.268001\pi\)
\(644\) −28.4412 −1.12074
\(645\) 7.90962 0.311441
\(646\) −6.96349 −0.273975
\(647\) 36.6351 1.44028 0.720138 0.693831i \(-0.244079\pi\)
0.720138 + 0.693831i \(0.244079\pi\)
\(648\) 15.3980 0.604891
\(649\) −8.66353 −0.340073
\(650\) −11.4800 −0.450283
\(651\) −12.8072 −0.501953
\(652\) −16.5648 −0.648728
\(653\) 0.637706 0.0249553 0.0124777 0.999922i \(-0.496028\pi\)
0.0124777 + 0.999922i \(0.496028\pi\)
\(654\) 1.14891 0.0449259
\(655\) −14.5918 −0.570147
\(656\) 84.9253 3.31578
\(657\) 23.6997 0.924614
\(658\) −3.70720 −0.144522
\(659\) −5.35534 −0.208614 −0.104307 0.994545i \(-0.533263\pi\)
−0.104307 + 0.994545i \(0.533263\pi\)
\(660\) −4.15060 −0.161562
\(661\) −38.4599 −1.49592 −0.747958 0.663746i \(-0.768966\pi\)
−0.747958 + 0.663746i \(0.768966\pi\)
\(662\) 50.8825 1.97761
\(663\) −10.9621 −0.425733
\(664\) −81.9945 −3.18200
\(665\) −2.16848 −0.0840899
\(666\) −4.82947 −0.187138
\(667\) 6.21723 0.240732
\(668\) −2.70136 −0.104519
\(669\) −12.9866 −0.502091
\(670\) 39.0364 1.50811
\(671\) −5.59008 −0.215803
\(672\) 13.6753 0.527538
\(673\) −34.8302 −1.34261 −0.671303 0.741183i \(-0.734265\pi\)
−0.671303 + 0.741183i \(0.734265\pi\)
\(674\) 32.5300 1.25301
\(675\) −4.68494 −0.180324
\(676\) 32.1110 1.23504
\(677\) 24.7488 0.951174 0.475587 0.879669i \(-0.342236\pi\)
0.475587 + 0.879669i \(0.342236\pi\)
\(678\) 47.8945 1.83938
\(679\) 32.1703 1.23458
\(680\) 18.0479 0.692107
\(681\) 19.6068 0.751334
\(682\) 16.7754 0.642363
\(683\) 7.74417 0.296322 0.148161 0.988963i \(-0.452665\pi\)
0.148161 + 0.988963i \(0.452665\pi\)
\(684\) 10.0236 0.383261
\(685\) −11.9896 −0.458098
\(686\) −51.7646 −1.97638
\(687\) 8.49443 0.324083
\(688\) 69.1366 2.63581
\(689\) 62.3147 2.37400
\(690\) −6.62891 −0.252358
\(691\) −29.3966 −1.11830 −0.559150 0.829066i \(-0.688873\pi\)
−0.559150 + 0.829066i \(0.688873\pi\)
\(692\) 106.302 4.04101
\(693\) 4.73364 0.179816
\(694\) 16.1672 0.613698
\(695\) −7.16027 −0.271605
\(696\) −13.0923 −0.496263
\(697\) 29.1529 1.10424
\(698\) −12.5936 −0.476677
\(699\) −20.4750 −0.774435
\(700\) 9.95719 0.376346
\(701\) −16.3181 −0.616324 −0.308162 0.951334i \(-0.599714\pi\)
−0.308162 + 0.951334i \(0.599714\pi\)
\(702\) 53.7832 2.02992
\(703\) −0.861702 −0.0324997
\(704\) −2.11055 −0.0795444
\(705\) −0.601892 −0.0226686
\(706\) 58.5039 2.20182
\(707\) 20.2793 0.762682
\(708\) 35.9589 1.35142
\(709\) −17.0214 −0.639251 −0.319625 0.947544i \(-0.603557\pi\)
−0.319625 + 0.947544i \(0.603557\pi\)
\(710\) −15.3900 −0.577575
\(711\) −12.5389 −0.470247
\(712\) 32.2048 1.20693
\(713\) 18.6630 0.698936
\(714\) 13.6493 0.510812
\(715\) 4.47137 0.167220
\(716\) 51.5840 1.92779
\(717\) 11.1354 0.415859
\(718\) 67.9950 2.53755
\(719\) −13.3760 −0.498842 −0.249421 0.968395i \(-0.580240\pi\)
−0.249421 + 0.968395i \(0.580240\pi\)
\(720\) −17.2473 −0.642770
\(721\) 17.5398 0.653216
\(722\) 2.56745 0.0955506
\(723\) −24.9035 −0.926170
\(724\) −32.9904 −1.22608
\(725\) −2.17663 −0.0808381
\(726\) 2.32076 0.0861316
\(727\) 34.6984 1.28689 0.643446 0.765492i \(-0.277504\pi\)
0.643446 + 0.765492i \(0.277504\pi\)
\(728\) −64.5204 −2.39129
\(729\) 7.65312 0.283449
\(730\) −27.8743 −1.03168
\(731\) 23.7330 0.877796
\(732\) 23.2022 0.857578
\(733\) −8.67688 −0.320488 −0.160244 0.987077i \(-0.551228\pi\)
−0.160244 + 0.987077i \(0.551228\pi\)
\(734\) 37.0767 1.36853
\(735\) −2.07695 −0.0766093
\(736\) −19.9281 −0.734561
\(737\) −15.2043 −0.560059
\(738\) −60.2418 −2.21753
\(739\) 50.7472 1.86677 0.933383 0.358882i \(-0.116842\pi\)
0.933383 + 0.358882i \(0.116842\pi\)
\(740\) 3.95676 0.145453
\(741\) 4.04175 0.148477
\(742\) −77.5901 −2.84842
\(743\) 24.0074 0.880745 0.440372 0.897815i \(-0.354846\pi\)
0.440372 + 0.897815i \(0.354846\pi\)
\(744\) −39.3008 −1.44084
\(745\) 3.73998 0.137022
\(746\) −9.79342 −0.358563
\(747\) 26.8982 0.984152
\(748\) −12.4540 −0.455362
\(749\) −5.87766 −0.214765
\(750\) 2.32076 0.0847423
\(751\) −6.31971 −0.230610 −0.115305 0.993330i \(-0.536784\pi\)
−0.115305 + 0.993330i \(0.536784\pi\)
\(752\) −5.26103 −0.191850
\(753\) 10.7612 0.392161
\(754\) 24.9878 0.910000
\(755\) −7.70787 −0.280518
\(756\) −46.6489 −1.69660
\(757\) 1.33647 0.0485748 0.0242874 0.999705i \(-0.492268\pi\)
0.0242874 + 0.999705i \(0.492268\pi\)
\(758\) −20.5424 −0.746133
\(759\) 2.58191 0.0937173
\(760\) −6.65430 −0.241377
\(761\) 26.0134 0.942986 0.471493 0.881870i \(-0.343715\pi\)
0.471493 + 0.881870i \(0.343715\pi\)
\(762\) −35.4321 −1.28357
\(763\) 1.07352 0.0388639
\(764\) −64.1637 −2.32136
\(765\) −5.92060 −0.214060
\(766\) 34.8170 1.25799
\(767\) −38.7378 −1.39874
\(768\) −23.6228 −0.852415
\(769\) 14.6805 0.529394 0.264697 0.964332i \(-0.414728\pi\)
0.264697 + 0.964332i \(0.414728\pi\)
\(770\) −5.56745 −0.200637
\(771\) −7.41001 −0.266865
\(772\) −58.7970 −2.11615
\(773\) −47.7853 −1.71872 −0.859358 0.511374i \(-0.829137\pi\)
−0.859358 + 0.511374i \(0.829137\pi\)
\(774\) −49.0421 −1.76278
\(775\) −6.53387 −0.234704
\(776\) 98.7197 3.54383
\(777\) 1.68904 0.0605940
\(778\) −4.19898 −0.150541
\(779\) −10.7487 −0.385112
\(780\) −18.5589 −0.664514
\(781\) 5.99426 0.214491
\(782\) −19.8902 −0.711271
\(783\) 10.1974 0.364426
\(784\) −18.1542 −0.648365
\(785\) −7.78415 −0.277828
\(786\) −33.8640 −1.20789
\(787\) −11.9838 −0.427177 −0.213589 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(788\) 17.3944 0.619651
\(789\) −11.9204 −0.424377
\(790\) 14.7476 0.524697
\(791\) 44.7516 1.59119
\(792\) 14.5259 0.516155
\(793\) −24.9953 −0.887609
\(794\) 23.7408 0.842528
\(795\) −12.5973 −0.446782
\(796\) 73.0375 2.58875
\(797\) 33.0078 1.16920 0.584598 0.811323i \(-0.301252\pi\)
0.584598 + 0.811323i \(0.301252\pi\)
\(798\) −5.03252 −0.178149
\(799\) −1.80599 −0.0638913
\(800\) 6.97678 0.246666
\(801\) −10.5647 −0.373286
\(802\) −0.770015 −0.0271902
\(803\) 10.8568 0.383129
\(804\) 63.1072 2.22562
\(805\) −6.19393 −0.218307
\(806\) 75.0089 2.64208
\(807\) 17.5350 0.617260
\(808\) 62.2302 2.18925
\(809\) −29.9288 −1.05224 −0.526120 0.850411i \(-0.676354\pi\)
−0.526120 + 0.850411i \(0.676354\pi\)
\(810\) 5.94106 0.208748
\(811\) −27.3858 −0.961647 −0.480824 0.876817i \(-0.659662\pi\)
−0.480824 + 0.876817i \(0.659662\pi\)
\(812\) −21.6731 −0.760578
\(813\) 3.22761 0.113197
\(814\) −2.21237 −0.0775437
\(815\) −3.60748 −0.126364
\(816\) 19.3703 0.678095
\(817\) −8.75038 −0.306137
\(818\) 23.8374 0.833456
\(819\) 21.1658 0.739594
\(820\) 49.3558 1.72358
\(821\) 17.1910 0.599971 0.299986 0.953944i \(-0.403018\pi\)
0.299986 + 0.953944i \(0.403018\pi\)
\(822\) −27.8249 −0.970506
\(823\) 5.25893 0.183315 0.0916575 0.995791i \(-0.470784\pi\)
0.0916575 + 0.995791i \(0.470784\pi\)
\(824\) 53.8235 1.87503
\(825\) −0.903918 −0.0314704
\(826\) 48.2338 1.67827
\(827\) −18.3379 −0.637672 −0.318836 0.947810i \(-0.603292\pi\)
−0.318836 + 0.947810i \(0.603292\pi\)
\(828\) 28.6309 0.994991
\(829\) 37.5846 1.30537 0.652684 0.757630i \(-0.273643\pi\)
0.652684 + 0.757630i \(0.273643\pi\)
\(830\) −31.6362 −1.09811
\(831\) −18.6813 −0.648048
\(832\) −9.43705 −0.327171
\(833\) −6.23192 −0.215923
\(834\) −16.6173 −0.575410
\(835\) −0.588302 −0.0203590
\(836\) 4.59179 0.158811
\(837\) 30.6108 1.05807
\(838\) −23.9593 −0.827662
\(839\) −52.4722 −1.81154 −0.905770 0.423769i \(-0.860707\pi\)
−0.905770 + 0.423769i \(0.860707\pi\)
\(840\) 13.0432 0.450035
\(841\) −24.2623 −0.836630
\(842\) −64.5599 −2.22488
\(843\) −6.23607 −0.214782
\(844\) 10.1213 0.348390
\(845\) 6.99312 0.240571
\(846\) 3.73192 0.128306
\(847\) 2.16848 0.0745097
\(848\) −110.111 −3.78123
\(849\) 21.2732 0.730093
\(850\) 6.96349 0.238846
\(851\) −2.46132 −0.0843730
\(852\) −24.8798 −0.852368
\(853\) −29.5852 −1.01298 −0.506489 0.862246i \(-0.669057\pi\)
−0.506489 + 0.862246i \(0.669057\pi\)
\(854\) 31.1225 1.06499
\(855\) 2.18293 0.0746547
\(856\) −18.0365 −0.616475
\(857\) −23.6715 −0.808601 −0.404301 0.914626i \(-0.632485\pi\)
−0.404301 + 0.914626i \(0.632485\pi\)
\(858\) 10.3770 0.354264
\(859\) −29.0072 −0.989714 −0.494857 0.868974i \(-0.664780\pi\)
−0.494857 + 0.868974i \(0.664780\pi\)
\(860\) 40.1800 1.37013
\(861\) 21.0688 0.718022
\(862\) 96.4487 3.28505
\(863\) −36.3796 −1.23837 −0.619187 0.785243i \(-0.712538\pi\)
−0.619187 + 0.785243i \(0.712538\pi\)
\(864\) −32.6858 −1.11199
\(865\) 23.1505 0.787140
\(866\) 23.7305 0.806394
\(867\) −8.71725 −0.296053
\(868\) −65.0590 −2.20825
\(869\) −5.74408 −0.194855
\(870\) −5.05145 −0.171260
\(871\) −67.9842 −2.30356
\(872\) 3.29425 0.111557
\(873\) −32.3848 −1.09606
\(874\) 7.33354 0.248061
\(875\) 2.16848 0.0733078
\(876\) −45.0624 −1.52252
\(877\) −56.9075 −1.92163 −0.960815 0.277192i \(-0.910596\pi\)
−0.960815 + 0.277192i \(0.910596\pi\)
\(878\) 2.41809 0.0816066
\(879\) 7.86671 0.265338
\(880\) −7.90099 −0.266342
\(881\) 53.1897 1.79201 0.896004 0.444047i \(-0.146458\pi\)
0.896004 + 0.444047i \(0.146458\pi\)
\(882\) 12.8777 0.433615
\(883\) 22.9447 0.772151 0.386075 0.922467i \(-0.373831\pi\)
0.386075 + 0.922467i \(0.373831\pi\)
\(884\) −55.6862 −1.87293
\(885\) 7.83112 0.263240
\(886\) 44.3467 1.48986
\(887\) 35.0993 1.17852 0.589260 0.807943i \(-0.299419\pi\)
0.589260 + 0.807943i \(0.299419\pi\)
\(888\) 5.18308 0.173933
\(889\) −33.1070 −1.11037
\(890\) 12.4257 0.416509
\(891\) −2.31399 −0.0775217
\(892\) −65.9704 −2.20885
\(893\) 0.665870 0.0222825
\(894\) 8.67961 0.290289
\(895\) 11.2340 0.375510
\(896\) −18.5075 −0.618294
\(897\) 11.5447 0.385465
\(898\) −53.8952 −1.79851
\(899\) 14.2218 0.474325
\(900\) −10.0236 −0.334119
\(901\) −37.7986 −1.25925
\(902\) −27.5967 −0.918871
\(903\) 17.1518 0.570777
\(904\) 137.327 4.56744
\(905\) −7.18465 −0.238826
\(906\) −17.8881 −0.594294
\(907\) 49.6640 1.64906 0.824532 0.565815i \(-0.191438\pi\)
0.824532 + 0.565815i \(0.191438\pi\)
\(908\) 99.6002 3.30535
\(909\) −20.4145 −0.677107
\(910\) −24.8941 −0.825232
\(911\) 44.2360 1.46560 0.732802 0.680442i \(-0.238212\pi\)
0.732802 + 0.680442i \(0.238212\pi\)
\(912\) −7.14184 −0.236490
\(913\) 12.3220 0.407800
\(914\) −74.4286 −2.46188
\(915\) 5.05297 0.167046
\(916\) 43.1507 1.42574
\(917\) −31.6419 −1.04491
\(918\) −32.6236 −1.07674
\(919\) −36.2925 −1.19718 −0.598590 0.801055i \(-0.704272\pi\)
−0.598590 + 0.801055i \(0.704272\pi\)
\(920\) −19.0070 −0.626643
\(921\) −0.0312506 −0.00102974
\(922\) −63.6508 −2.09623
\(923\) 26.8025 0.882216
\(924\) −9.00048 −0.296094
\(925\) 0.861702 0.0283326
\(926\) −1.94387 −0.0638796
\(927\) −17.6567 −0.579923
\(928\) −15.1859 −0.498501
\(929\) 20.6900 0.678818 0.339409 0.940639i \(-0.389773\pi\)
0.339409 + 0.940639i \(0.389773\pi\)
\(930\) −15.1636 −0.497233
\(931\) 2.29772 0.0753046
\(932\) −104.010 −3.40698
\(933\) −15.2644 −0.499734
\(934\) −14.2355 −0.465800
\(935\) −2.71222 −0.0886991
\(936\) 64.9506 2.12298
\(937\) −58.8084 −1.92119 −0.960594 0.277957i \(-0.910343\pi\)
−0.960594 + 0.277957i \(0.910343\pi\)
\(938\) 84.6494 2.76390
\(939\) −1.33482 −0.0435602
\(940\) −3.05754 −0.0997260
\(941\) −18.1745 −0.592471 −0.296235 0.955115i \(-0.595731\pi\)
−0.296235 + 0.955115i \(0.595731\pi\)
\(942\) −18.0652 −0.588595
\(943\) −30.7021 −0.999797
\(944\) 68.4504 2.22787
\(945\) −10.1592 −0.330478
\(946\) −22.4662 −0.730438
\(947\) 9.22596 0.299803 0.149902 0.988701i \(-0.452104\pi\)
0.149902 + 0.988701i \(0.452104\pi\)
\(948\) 23.8414 0.774333
\(949\) 48.5448 1.57583
\(950\) −2.56745 −0.0832990
\(951\) 22.1712 0.718951
\(952\) 39.1365 1.26842
\(953\) 34.8382 1.12852 0.564260 0.825597i \(-0.309162\pi\)
0.564260 + 0.825597i \(0.309162\pi\)
\(954\) 78.1074 2.52882
\(955\) −13.9736 −0.452174
\(956\) 56.5665 1.82949
\(957\) 1.96750 0.0636001
\(958\) −22.5617 −0.728935
\(959\) −25.9991 −0.839554
\(960\) 1.90777 0.0615729
\(961\) 11.6915 0.377145
\(962\) −9.89234 −0.318942
\(963\) 5.91685 0.190668
\(964\) −126.507 −4.07451
\(965\) −12.8048 −0.412201
\(966\) −14.3746 −0.462496
\(967\) −21.5696 −0.693631 −0.346816 0.937933i \(-0.612737\pi\)
−0.346816 + 0.937933i \(0.612737\pi\)
\(968\) 6.65430 0.213877
\(969\) −2.45163 −0.0787576
\(970\) 38.0893 1.22297
\(971\) 52.0792 1.67130 0.835650 0.549262i \(-0.185091\pi\)
0.835650 + 0.549262i \(0.185091\pi\)
\(972\) 74.1414 2.37809
\(973\) −15.5269 −0.497769
\(974\) 94.7207 3.03505
\(975\) −4.04175 −0.129440
\(976\) 44.1671 1.41376
\(977\) −14.7908 −0.473200 −0.236600 0.971607i \(-0.576033\pi\)
−0.236600 + 0.971607i \(0.576033\pi\)
\(978\) −8.37210 −0.267710
\(979\) −4.83970 −0.154677
\(980\) −10.5506 −0.337028
\(981\) −1.08067 −0.0345033
\(982\) −15.7220 −0.501708
\(983\) −35.1347 −1.12062 −0.560311 0.828282i \(-0.689318\pi\)
−0.560311 + 0.828282i \(0.689318\pi\)
\(984\) 64.6528 2.06105
\(985\) 3.78816 0.120701
\(986\) −15.1570 −0.482696
\(987\) −1.30519 −0.0415446
\(988\) 20.5316 0.653197
\(989\) −24.9942 −0.794768
\(990\) 5.60457 0.178125
\(991\) −50.5723 −1.60648 −0.803241 0.595654i \(-0.796893\pi\)
−0.803241 + 0.595654i \(0.796893\pi\)
\(992\) −45.5854 −1.44734
\(993\) 17.9141 0.568488
\(994\) −33.3727 −1.05852
\(995\) 15.9061 0.504257
\(996\) −51.1439 −1.62056
\(997\) −16.9360 −0.536370 −0.268185 0.963367i \(-0.586424\pi\)
−0.268185 + 0.963367i \(0.586424\pi\)
\(998\) −43.4295 −1.37474
\(999\) −4.03702 −0.127726
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 1045.2.a.g.1.6 6
3.2 odd 2 9405.2.a.w.1.1 6
5.4 even 2 5225.2.a.k.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.g.1.6 6 1.1 even 1 trivial
5225.2.a.k.1.1 6 5.4 even 2
9405.2.a.w.1.1 6 3.2 odd 2