Properties

Label 8022.2.a.z.1.9
Level $8022$
Weight $2$
Character 8022.1
Self dual yes
Analytic conductor $64.056$
Analytic rank $0$
Dimension $15$
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] = [8022,2,Mod(1,8022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8022, 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("8022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8022.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: \(64.0559925015\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 53 x^{13} - x^{12} + 1068 x^{11} + 45 x^{10} - 10139 x^{9} - 615 x^{8} + 45390 x^{7} + 2130 x^{6} - 84842 x^{5} + 7822 x^{4} + 62828 x^{3} - 16144 x^{2} + \cdots + 4704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-0.600780\) of defining polynomial
Character \(\chi\) \(=\) 8022.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.600780 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.600780 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.600780 q^{10} +3.98407 q^{11} +1.00000 q^{12} -2.21056 q^{13} -1.00000 q^{14} +0.600780 q^{15} +1.00000 q^{16} +0.291399 q^{17} -1.00000 q^{18} +0.811418 q^{19} +0.600780 q^{20} +1.00000 q^{21} -3.98407 q^{22} +1.18963 q^{23} -1.00000 q^{24} -4.63906 q^{25} +2.21056 q^{26} +1.00000 q^{27} +1.00000 q^{28} -9.44157 q^{29} -0.600780 q^{30} +7.61531 q^{31} -1.00000 q^{32} +3.98407 q^{33} -0.291399 q^{34} +0.600780 q^{35} +1.00000 q^{36} -5.07939 q^{37} -0.811418 q^{38} -2.21056 q^{39} -0.600780 q^{40} +2.80141 q^{41} -1.00000 q^{42} -4.76406 q^{43} +3.98407 q^{44} +0.600780 q^{45} -1.18963 q^{46} +5.87638 q^{47} +1.00000 q^{48} +1.00000 q^{49} +4.63906 q^{50} +0.291399 q^{51} -2.21056 q^{52} +11.5189 q^{53} -1.00000 q^{54} +2.39355 q^{55} -1.00000 q^{56} +0.811418 q^{57} +9.44157 q^{58} -5.71021 q^{59} +0.600780 q^{60} +2.62535 q^{61} -7.61531 q^{62} +1.00000 q^{63} +1.00000 q^{64} -1.32806 q^{65} -3.98407 q^{66} +14.9062 q^{67} +0.291399 q^{68} +1.18963 q^{69} -0.600780 q^{70} +11.6398 q^{71} -1.00000 q^{72} +16.8053 q^{73} +5.07939 q^{74} -4.63906 q^{75} +0.811418 q^{76} +3.98407 q^{77} +2.21056 q^{78} +1.60018 q^{79} +0.600780 q^{80} +1.00000 q^{81} -2.80141 q^{82} -5.30607 q^{83} +1.00000 q^{84} +0.175067 q^{85} +4.76406 q^{86} -9.44157 q^{87} -3.98407 q^{88} -6.96178 q^{89} -0.600780 q^{90} -2.21056 q^{91} +1.18963 q^{92} +7.61531 q^{93} -5.87638 q^{94} +0.487484 q^{95} -1.00000 q^{96} +4.49961 q^{97} -1.00000 q^{98} +3.98407 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9} - 7 q^{11} + 15 q^{12} + 10 q^{13} - 15 q^{14} + 15 q^{16} - 3 q^{17} - 15 q^{18} + 12 q^{19} + 15 q^{21} + 7 q^{22} - q^{23} - 15 q^{24} + 31 q^{25} - 10 q^{26} + 15 q^{27} + 15 q^{28} - 3 q^{29} + 19 q^{31} - 15 q^{32} - 7 q^{33} + 3 q^{34} + 15 q^{36} + 25 q^{37} - 12 q^{38} + 10 q^{39} + 8 q^{41} - 15 q^{42} + 25 q^{43} - 7 q^{44} + q^{46} + 11 q^{47} + 15 q^{48} + 15 q^{49} - 31 q^{50} - 3 q^{51} + 10 q^{52} - 4 q^{53} - 15 q^{54} + 9 q^{55} - 15 q^{56} + 12 q^{57} + 3 q^{58} + 27 q^{61} - 19 q^{62} + 15 q^{63} + 15 q^{64} - 2 q^{65} + 7 q^{66} + 31 q^{67} - 3 q^{68} - q^{69} - 16 q^{71} - 15 q^{72} + 26 q^{73} - 25 q^{74} + 31 q^{75} + 12 q^{76} - 7 q^{77} - 10 q^{78} + 32 q^{79} + 15 q^{81} - 8 q^{82} + 7 q^{83} + 15 q^{84} + 26 q^{85} - 25 q^{86} - 3 q^{87} + 7 q^{88} - 11 q^{89} + 10 q^{91} - q^{92} + 19 q^{93} - 11 q^{94} - 8 q^{95} - 15 q^{96} + 30 q^{97} - 15 q^{98} - 7 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0.600780 0.268677 0.134339 0.990935i \(-0.457109\pi\)
0.134339 + 0.990935i \(0.457109\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −0.600780 −0.189983
\(11\) 3.98407 1.20124 0.600620 0.799534i \(-0.294920\pi\)
0.600620 + 0.799534i \(0.294920\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.21056 −0.613100 −0.306550 0.951855i \(-0.599175\pi\)
−0.306550 + 0.951855i \(0.599175\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0.600780 0.155121
\(16\) 1.00000 0.250000
\(17\) 0.291399 0.0706746 0.0353373 0.999375i \(-0.488749\pi\)
0.0353373 + 0.999375i \(0.488749\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0.811418 0.186152 0.0930761 0.995659i \(-0.470330\pi\)
0.0930761 + 0.995659i \(0.470330\pi\)
\(20\) 0.600780 0.134339
\(21\) 1.00000 0.218218
\(22\) −3.98407 −0.849406
\(23\) 1.18963 0.248055 0.124028 0.992279i \(-0.460419\pi\)
0.124028 + 0.992279i \(0.460419\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.63906 −0.927813
\(26\) 2.21056 0.433527
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −9.44157 −1.75326 −0.876628 0.481169i \(-0.840212\pi\)
−0.876628 + 0.481169i \(0.840212\pi\)
\(30\) −0.600780 −0.109687
\(31\) 7.61531 1.36775 0.683875 0.729599i \(-0.260293\pi\)
0.683875 + 0.729599i \(0.260293\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.98407 0.693537
\(34\) −0.291399 −0.0499745
\(35\) 0.600780 0.101550
\(36\) 1.00000 0.166667
\(37\) −5.07939 −0.835046 −0.417523 0.908666i \(-0.637102\pi\)
−0.417523 + 0.908666i \(0.637102\pi\)
\(38\) −0.811418 −0.131629
\(39\) −2.21056 −0.353974
\(40\) −0.600780 −0.0949917
\(41\) 2.80141 0.437507 0.218754 0.975780i \(-0.429801\pi\)
0.218754 + 0.975780i \(0.429801\pi\)
\(42\) −1.00000 −0.154303
\(43\) −4.76406 −0.726513 −0.363256 0.931689i \(-0.618335\pi\)
−0.363256 + 0.931689i \(0.618335\pi\)
\(44\) 3.98407 0.600620
\(45\) 0.600780 0.0895591
\(46\) −1.18963 −0.175402
\(47\) 5.87638 0.857159 0.428579 0.903504i \(-0.359014\pi\)
0.428579 + 0.903504i \(0.359014\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 4.63906 0.656063
\(51\) 0.291399 0.0408040
\(52\) −2.21056 −0.306550
\(53\) 11.5189 1.58224 0.791120 0.611661i \(-0.209498\pi\)
0.791120 + 0.611661i \(0.209498\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.39355 0.322746
\(56\) −1.00000 −0.133631
\(57\) 0.811418 0.107475
\(58\) 9.44157 1.23974
\(59\) −5.71021 −0.743406 −0.371703 0.928352i \(-0.621226\pi\)
−0.371703 + 0.928352i \(0.621226\pi\)
\(60\) 0.600780 0.0775604
\(61\) 2.62535 0.336141 0.168071 0.985775i \(-0.446246\pi\)
0.168071 + 0.985775i \(0.446246\pi\)
\(62\) −7.61531 −0.967145
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −1.32806 −0.164726
\(66\) −3.98407 −0.490405
\(67\) 14.9062 1.82108 0.910540 0.413421i \(-0.135666\pi\)
0.910540 + 0.413421i \(0.135666\pi\)
\(68\) 0.291399 0.0353373
\(69\) 1.18963 0.143215
\(70\) −0.600780 −0.0718070
\(71\) 11.6398 1.38139 0.690693 0.723148i \(-0.257306\pi\)
0.690693 + 0.723148i \(0.257306\pi\)
\(72\) −1.00000 −0.117851
\(73\) 16.8053 1.96691 0.983455 0.181154i \(-0.0579832\pi\)
0.983455 + 0.181154i \(0.0579832\pi\)
\(74\) 5.07939 0.590467
\(75\) −4.63906 −0.535673
\(76\) 0.811418 0.0930761
\(77\) 3.98407 0.454026
\(78\) 2.21056 0.250297
\(79\) 1.60018 0.180035 0.0900173 0.995940i \(-0.471308\pi\)
0.0900173 + 0.995940i \(0.471308\pi\)
\(80\) 0.600780 0.0671693
\(81\) 1.00000 0.111111
\(82\) −2.80141 −0.309364
\(83\) −5.30607 −0.582417 −0.291209 0.956660i \(-0.594057\pi\)
−0.291209 + 0.956660i \(0.594057\pi\)
\(84\) 1.00000 0.109109
\(85\) 0.175067 0.0189887
\(86\) 4.76406 0.513722
\(87\) −9.44157 −1.01224
\(88\) −3.98407 −0.424703
\(89\) −6.96178 −0.737947 −0.368974 0.929440i \(-0.620291\pi\)
−0.368974 + 0.929440i \(0.620291\pi\)
\(90\) −0.600780 −0.0633278
\(91\) −2.21056 −0.231730
\(92\) 1.18963 0.124028
\(93\) 7.61531 0.789671
\(94\) −5.87638 −0.606103
\(95\) 0.487484 0.0500148
\(96\) −1.00000 −0.102062
\(97\) 4.49961 0.456866 0.228433 0.973560i \(-0.426640\pi\)
0.228433 + 0.973560i \(0.426640\pi\)
\(98\) −1.00000 −0.101015
\(99\) 3.98407 0.400414
\(100\) −4.63906 −0.463906
\(101\) 5.35438 0.532780 0.266390 0.963865i \(-0.414169\pi\)
0.266390 + 0.963865i \(0.414169\pi\)
\(102\) −0.291399 −0.0288528
\(103\) −9.04862 −0.891587 −0.445794 0.895136i \(-0.647079\pi\)
−0.445794 + 0.895136i \(0.647079\pi\)
\(104\) 2.21056 0.216764
\(105\) 0.600780 0.0586302
\(106\) −11.5189 −1.11881
\(107\) 15.9118 1.53825 0.769126 0.639097i \(-0.220692\pi\)
0.769126 + 0.639097i \(0.220692\pi\)
\(108\) 1.00000 0.0962250
\(109\) 1.54381 0.147870 0.0739349 0.997263i \(-0.476444\pi\)
0.0739349 + 0.997263i \(0.476444\pi\)
\(110\) −2.39355 −0.228216
\(111\) −5.07939 −0.482114
\(112\) 1.00000 0.0944911
\(113\) −17.9097 −1.68480 −0.842401 0.538851i \(-0.818859\pi\)
−0.842401 + 0.538851i \(0.818859\pi\)
\(114\) −0.811418 −0.0759963
\(115\) 0.714708 0.0666468
\(116\) −9.44157 −0.876628
\(117\) −2.21056 −0.204367
\(118\) 5.71021 0.525667
\(119\) 0.291399 0.0267125
\(120\) −0.600780 −0.0548435
\(121\) 4.87278 0.442980
\(122\) −2.62535 −0.237688
\(123\) 2.80141 0.252595
\(124\) 7.61531 0.683875
\(125\) −5.79096 −0.517959
\(126\) −1.00000 −0.0890871
\(127\) 19.1969 1.70345 0.851723 0.523992i \(-0.175558\pi\)
0.851723 + 0.523992i \(0.175558\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.76406 −0.419452
\(130\) 1.32806 0.116479
\(131\) 16.1818 1.41381 0.706906 0.707307i \(-0.250090\pi\)
0.706906 + 0.707307i \(0.250090\pi\)
\(132\) 3.98407 0.346768
\(133\) 0.811418 0.0703589
\(134\) −14.9062 −1.28770
\(135\) 0.600780 0.0517069
\(136\) −0.291399 −0.0249873
\(137\) −6.36691 −0.543962 −0.271981 0.962303i \(-0.587679\pi\)
−0.271981 + 0.962303i \(0.587679\pi\)
\(138\) −1.18963 −0.101268
\(139\) −12.0899 −1.02545 −0.512725 0.858553i \(-0.671364\pi\)
−0.512725 + 0.858553i \(0.671364\pi\)
\(140\) 0.600780 0.0507752
\(141\) 5.87638 0.494881
\(142\) −11.6398 −0.976788
\(143\) −8.80703 −0.736481
\(144\) 1.00000 0.0833333
\(145\) −5.67231 −0.471060
\(146\) −16.8053 −1.39082
\(147\) 1.00000 0.0824786
\(148\) −5.07939 −0.417523
\(149\) 4.06155 0.332735 0.166368 0.986064i \(-0.446796\pi\)
0.166368 + 0.986064i \(0.446796\pi\)
\(150\) 4.63906 0.378778
\(151\) 18.5807 1.51208 0.756040 0.654526i \(-0.227132\pi\)
0.756040 + 0.654526i \(0.227132\pi\)
\(152\) −0.811418 −0.0658147
\(153\) 0.291399 0.0235582
\(154\) −3.98407 −0.321045
\(155\) 4.57513 0.367483
\(156\) −2.21056 −0.176987
\(157\) 4.70929 0.375842 0.187921 0.982184i \(-0.439825\pi\)
0.187921 + 0.982184i \(0.439825\pi\)
\(158\) −1.60018 −0.127304
\(159\) 11.5189 0.913506
\(160\) −0.600780 −0.0474959
\(161\) 1.18963 0.0937562
\(162\) −1.00000 −0.0785674
\(163\) 22.9695 1.79911 0.899553 0.436811i \(-0.143892\pi\)
0.899553 + 0.436811i \(0.143892\pi\)
\(164\) 2.80141 0.218754
\(165\) 2.39355 0.186337
\(166\) 5.30607 0.411831
\(167\) 2.85628 0.221025 0.110513 0.993875i \(-0.464751\pi\)
0.110513 + 0.993875i \(0.464751\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −8.11341 −0.624108
\(170\) −0.175067 −0.0134270
\(171\) 0.811418 0.0620507
\(172\) −4.76406 −0.363256
\(173\) −23.1834 −1.76260 −0.881302 0.472553i \(-0.843332\pi\)
−0.881302 + 0.472553i \(0.843332\pi\)
\(174\) 9.44157 0.715763
\(175\) −4.63906 −0.350680
\(176\) 3.98407 0.300310
\(177\) −5.71021 −0.429205
\(178\) 6.96178 0.521807
\(179\) 19.7984 1.47980 0.739900 0.672717i \(-0.234873\pi\)
0.739900 + 0.672717i \(0.234873\pi\)
\(180\) 0.600780 0.0447795
\(181\) −7.54151 −0.560556 −0.280278 0.959919i \(-0.590427\pi\)
−0.280278 + 0.959919i \(0.590427\pi\)
\(182\) 2.21056 0.163858
\(183\) 2.62535 0.194071
\(184\) −1.18963 −0.0877009
\(185\) −3.05160 −0.224358
\(186\) −7.61531 −0.558382
\(187\) 1.16095 0.0848972
\(188\) 5.87638 0.428579
\(189\) 1.00000 0.0727393
\(190\) −0.487484 −0.0353658
\(191\) 1.00000 0.0723575
\(192\) 1.00000 0.0721688
\(193\) −9.25956 −0.666518 −0.333259 0.942835i \(-0.608148\pi\)
−0.333259 + 0.942835i \(0.608148\pi\)
\(194\) −4.49961 −0.323053
\(195\) −1.32806 −0.0951046
\(196\) 1.00000 0.0714286
\(197\) −10.0613 −0.716841 −0.358420 0.933560i \(-0.616685\pi\)
−0.358420 + 0.933560i \(0.616685\pi\)
\(198\) −3.98407 −0.283135
\(199\) 5.35710 0.379755 0.189877 0.981808i \(-0.439191\pi\)
0.189877 + 0.981808i \(0.439191\pi\)
\(200\) 4.63906 0.328031
\(201\) 14.9062 1.05140
\(202\) −5.35438 −0.376733
\(203\) −9.44157 −0.662668
\(204\) 0.291399 0.0204020
\(205\) 1.68303 0.117548
\(206\) 9.04862 0.630448
\(207\) 1.18963 0.0826852
\(208\) −2.21056 −0.153275
\(209\) 3.23274 0.223614
\(210\) −0.600780 −0.0414578
\(211\) 10.4729 0.720983 0.360491 0.932763i \(-0.382609\pi\)
0.360491 + 0.932763i \(0.382609\pi\)
\(212\) 11.5189 0.791120
\(213\) 11.6398 0.797544
\(214\) −15.9118 −1.08771
\(215\) −2.86215 −0.195197
\(216\) −1.00000 −0.0680414
\(217\) 7.61531 0.516961
\(218\) −1.54381 −0.104560
\(219\) 16.8053 1.13560
\(220\) 2.39355 0.161373
\(221\) −0.644156 −0.0433306
\(222\) 5.07939 0.340906
\(223\) −14.5234 −0.972562 −0.486281 0.873802i \(-0.661647\pi\)
−0.486281 + 0.873802i \(0.661647\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −4.63906 −0.309271
\(226\) 17.9097 1.19134
\(227\) −6.06557 −0.402586 −0.201293 0.979531i \(-0.564514\pi\)
−0.201293 + 0.979531i \(0.564514\pi\)
\(228\) 0.811418 0.0537375
\(229\) −9.54355 −0.630655 −0.315328 0.948983i \(-0.602114\pi\)
−0.315328 + 0.948983i \(0.602114\pi\)
\(230\) −0.714708 −0.0471264
\(231\) 3.98407 0.262132
\(232\) 9.44157 0.619869
\(233\) −6.93594 −0.454388 −0.227194 0.973849i \(-0.572955\pi\)
−0.227194 + 0.973849i \(0.572955\pi\)
\(234\) 2.21056 0.144509
\(235\) 3.53042 0.230299
\(236\) −5.71021 −0.371703
\(237\) 1.60018 0.103943
\(238\) −0.291399 −0.0188886
\(239\) −0.650366 −0.0420687 −0.0210343 0.999779i \(-0.506696\pi\)
−0.0210343 + 0.999779i \(0.506696\pi\)
\(240\) 0.600780 0.0387802
\(241\) 6.26146 0.403336 0.201668 0.979454i \(-0.435364\pi\)
0.201668 + 0.979454i \(0.435364\pi\)
\(242\) −4.87278 −0.313234
\(243\) 1.00000 0.0641500
\(244\) 2.62535 0.168071
\(245\) 0.600780 0.0383825
\(246\) −2.80141 −0.178611
\(247\) −1.79369 −0.114130
\(248\) −7.61531 −0.483573
\(249\) −5.30607 −0.336259
\(250\) 5.79096 0.366252
\(251\) −21.3290 −1.34627 −0.673137 0.739518i \(-0.735054\pi\)
−0.673137 + 0.739518i \(0.735054\pi\)
\(252\) 1.00000 0.0629941
\(253\) 4.73957 0.297974
\(254\) −19.1969 −1.20452
\(255\) 0.175067 0.0109631
\(256\) 1.00000 0.0625000
\(257\) −1.04463 −0.0651622 −0.0325811 0.999469i \(-0.510373\pi\)
−0.0325811 + 0.999469i \(0.510373\pi\)
\(258\) 4.76406 0.296598
\(259\) −5.07939 −0.315618
\(260\) −1.32806 −0.0823630
\(261\) −9.44157 −0.584418
\(262\) −16.1818 −0.999717
\(263\) 6.29202 0.387982 0.193991 0.981003i \(-0.437857\pi\)
0.193991 + 0.981003i \(0.437857\pi\)
\(264\) −3.98407 −0.245202
\(265\) 6.92032 0.425112
\(266\) −0.811418 −0.0497512
\(267\) −6.96178 −0.426054
\(268\) 14.9062 0.910540
\(269\) −24.7427 −1.50859 −0.754296 0.656535i \(-0.772021\pi\)
−0.754296 + 0.656535i \(0.772021\pi\)
\(270\) −0.600780 −0.0365623
\(271\) 21.0873 1.28096 0.640480 0.767975i \(-0.278735\pi\)
0.640480 + 0.767975i \(0.278735\pi\)
\(272\) 0.291399 0.0176687
\(273\) −2.21056 −0.133789
\(274\) 6.36691 0.384639
\(275\) −18.4823 −1.11453
\(276\) 1.18963 0.0716075
\(277\) 12.6960 0.762831 0.381415 0.924404i \(-0.375437\pi\)
0.381415 + 0.924404i \(0.375437\pi\)
\(278\) 12.0899 0.725103
\(279\) 7.61531 0.455917
\(280\) −0.600780 −0.0359035
\(281\) 6.29496 0.375526 0.187763 0.982214i \(-0.439876\pi\)
0.187763 + 0.982214i \(0.439876\pi\)
\(282\) −5.87638 −0.349934
\(283\) 28.4254 1.68971 0.844856 0.534994i \(-0.179686\pi\)
0.844856 + 0.534994i \(0.179686\pi\)
\(284\) 11.6398 0.690693
\(285\) 0.487484 0.0288761
\(286\) 8.80703 0.520771
\(287\) 2.80141 0.165362
\(288\) −1.00000 −0.0589256
\(289\) −16.9151 −0.995005
\(290\) 5.67231 0.333089
\(291\) 4.49961 0.263772
\(292\) 16.8053 0.983455
\(293\) 28.6147 1.67169 0.835844 0.548967i \(-0.184979\pi\)
0.835844 + 0.548967i \(0.184979\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −3.43058 −0.199736
\(296\) 5.07939 0.295233
\(297\) 3.98407 0.231179
\(298\) −4.06155 −0.235279
\(299\) −2.62976 −0.152083
\(300\) −4.63906 −0.267836
\(301\) −4.76406 −0.274596
\(302\) −18.5807 −1.06920
\(303\) 5.35438 0.307601
\(304\) 0.811418 0.0465380
\(305\) 1.57726 0.0903134
\(306\) −0.291399 −0.0166582
\(307\) 33.6273 1.91921 0.959606 0.281348i \(-0.0907814\pi\)
0.959606 + 0.281348i \(0.0907814\pi\)
\(308\) 3.98407 0.227013
\(309\) −9.04862 −0.514758
\(310\) −4.57513 −0.259850
\(311\) −7.23048 −0.410003 −0.205001 0.978762i \(-0.565720\pi\)
−0.205001 + 0.978762i \(0.565720\pi\)
\(312\) 2.21056 0.125149
\(313\) −23.1287 −1.30731 −0.653655 0.756793i \(-0.726765\pi\)
−0.653655 + 0.756793i \(0.726765\pi\)
\(314\) −4.70929 −0.265761
\(315\) 0.600780 0.0338501
\(316\) 1.60018 0.0900173
\(317\) 7.17192 0.402815 0.201408 0.979508i \(-0.435448\pi\)
0.201408 + 0.979508i \(0.435448\pi\)
\(318\) −11.5189 −0.645947
\(319\) −37.6158 −2.10608
\(320\) 0.600780 0.0335846
\(321\) 15.9118 0.888111
\(322\) −1.18963 −0.0662956
\(323\) 0.236446 0.0131562
\(324\) 1.00000 0.0555556
\(325\) 10.2549 0.568842
\(326\) −22.9695 −1.27216
\(327\) 1.54381 0.0853726
\(328\) −2.80141 −0.154682
\(329\) 5.87638 0.323976
\(330\) −2.39355 −0.131761
\(331\) 26.0184 1.43010 0.715051 0.699073i \(-0.246404\pi\)
0.715051 + 0.699073i \(0.246404\pi\)
\(332\) −5.30607 −0.291209
\(333\) −5.07939 −0.278349
\(334\) −2.85628 −0.156289
\(335\) 8.95534 0.489283
\(336\) 1.00000 0.0545545
\(337\) −16.0449 −0.874020 −0.437010 0.899457i \(-0.643963\pi\)
−0.437010 + 0.899457i \(0.643963\pi\)
\(338\) 8.11341 0.441311
\(339\) −17.9097 −0.972721
\(340\) 0.175067 0.00949433
\(341\) 30.3399 1.64300
\(342\) −0.811418 −0.0438765
\(343\) 1.00000 0.0539949
\(344\) 4.76406 0.256861
\(345\) 0.714708 0.0384786
\(346\) 23.1834 1.24635
\(347\) −2.80431 −0.150543 −0.0752717 0.997163i \(-0.523982\pi\)
−0.0752717 + 0.997163i \(0.523982\pi\)
\(348\) −9.44157 −0.506121
\(349\) −1.70382 −0.0912032 −0.0456016 0.998960i \(-0.514520\pi\)
−0.0456016 + 0.998960i \(0.514520\pi\)
\(350\) 4.63906 0.247968
\(351\) −2.21056 −0.117991
\(352\) −3.98407 −0.212351
\(353\) 16.7576 0.891918 0.445959 0.895053i \(-0.352863\pi\)
0.445959 + 0.895053i \(0.352863\pi\)
\(354\) 5.71021 0.303494
\(355\) 6.99294 0.371147
\(356\) −6.96178 −0.368974
\(357\) 0.291399 0.0154225
\(358\) −19.7984 −1.04638
\(359\) −13.1677 −0.694965 −0.347482 0.937686i \(-0.612963\pi\)
−0.347482 + 0.937686i \(0.612963\pi\)
\(360\) −0.600780 −0.0316639
\(361\) −18.3416 −0.965347
\(362\) 7.54151 0.396373
\(363\) 4.87278 0.255754
\(364\) −2.21056 −0.115865
\(365\) 10.0963 0.528464
\(366\) −2.62535 −0.137229
\(367\) 9.51113 0.496477 0.248239 0.968699i \(-0.420148\pi\)
0.248239 + 0.968699i \(0.420148\pi\)
\(368\) 1.18963 0.0620139
\(369\) 2.80141 0.145836
\(370\) 3.05160 0.158645
\(371\) 11.5189 0.598030
\(372\) 7.61531 0.394835
\(373\) 27.8130 1.44010 0.720052 0.693920i \(-0.244118\pi\)
0.720052 + 0.693920i \(0.244118\pi\)
\(374\) −1.16095 −0.0600314
\(375\) −5.79096 −0.299044
\(376\) −5.87638 −0.303051
\(377\) 20.8712 1.07492
\(378\) −1.00000 −0.0514344
\(379\) −0.528092 −0.0271263 −0.0135631 0.999908i \(-0.504317\pi\)
−0.0135631 + 0.999908i \(0.504317\pi\)
\(380\) 0.487484 0.0250074
\(381\) 19.1969 0.983485
\(382\) −1.00000 −0.0511645
\(383\) 1.79471 0.0917055 0.0458528 0.998948i \(-0.485399\pi\)
0.0458528 + 0.998948i \(0.485399\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 2.39355 0.121987
\(386\) 9.25956 0.471299
\(387\) −4.76406 −0.242171
\(388\) 4.49961 0.228433
\(389\) −23.0637 −1.16938 −0.584688 0.811258i \(-0.698783\pi\)
−0.584688 + 0.811258i \(0.698783\pi\)
\(390\) 1.32806 0.0672491
\(391\) 0.346658 0.0175312
\(392\) −1.00000 −0.0505076
\(393\) 16.1818 0.816265
\(394\) 10.0613 0.506883
\(395\) 0.961358 0.0483712
\(396\) 3.98407 0.200207
\(397\) 24.2450 1.21682 0.608412 0.793622i \(-0.291807\pi\)
0.608412 + 0.793622i \(0.291807\pi\)
\(398\) −5.35710 −0.268527
\(399\) 0.811418 0.0406217
\(400\) −4.63906 −0.231953
\(401\) −10.6619 −0.532428 −0.266214 0.963914i \(-0.585773\pi\)
−0.266214 + 0.963914i \(0.585773\pi\)
\(402\) −14.9062 −0.743453
\(403\) −16.8341 −0.838568
\(404\) 5.35438 0.266390
\(405\) 0.600780 0.0298530
\(406\) 9.44157 0.468577
\(407\) −20.2366 −1.00309
\(408\) −0.291399 −0.0144264
\(409\) −10.7685 −0.532469 −0.266234 0.963908i \(-0.585780\pi\)
−0.266234 + 0.963908i \(0.585780\pi\)
\(410\) −1.68303 −0.0831191
\(411\) −6.36691 −0.314056
\(412\) −9.04862 −0.445794
\(413\) −5.71021 −0.280981
\(414\) −1.18963 −0.0584672
\(415\) −3.18779 −0.156482
\(416\) 2.21056 0.108382
\(417\) −12.0899 −0.592044
\(418\) −3.23274 −0.158119
\(419\) 6.01704 0.293952 0.146976 0.989140i \(-0.453046\pi\)
0.146976 + 0.989140i \(0.453046\pi\)
\(420\) 0.600780 0.0293151
\(421\) 27.8137 1.35556 0.677778 0.735267i \(-0.262943\pi\)
0.677778 + 0.735267i \(0.262943\pi\)
\(422\) −10.4729 −0.509812
\(423\) 5.87638 0.285720
\(424\) −11.5189 −0.559406
\(425\) −1.35182 −0.0655728
\(426\) −11.6398 −0.563949
\(427\) 2.62535 0.127049
\(428\) 15.9118 0.769126
\(429\) −8.80703 −0.425207
\(430\) 2.86215 0.138025
\(431\) −28.1741 −1.35710 −0.678550 0.734554i \(-0.737391\pi\)
−0.678550 + 0.734554i \(0.737391\pi\)
\(432\) 1.00000 0.0481125
\(433\) 19.5526 0.939640 0.469820 0.882762i \(-0.344319\pi\)
0.469820 + 0.882762i \(0.344319\pi\)
\(434\) −7.61531 −0.365547
\(435\) −5.67231 −0.271966
\(436\) 1.54381 0.0739349
\(437\) 0.965289 0.0461761
\(438\) −16.8053 −0.802987
\(439\) −7.01706 −0.334906 −0.167453 0.985880i \(-0.553554\pi\)
−0.167453 + 0.985880i \(0.553554\pi\)
\(440\) −2.39355 −0.114108
\(441\) 1.00000 0.0476190
\(442\) 0.644156 0.0306394
\(443\) −33.9891 −1.61487 −0.807436 0.589956i \(-0.799145\pi\)
−0.807436 + 0.589956i \(0.799145\pi\)
\(444\) −5.07939 −0.241057
\(445\) −4.18250 −0.198270
\(446\) 14.5234 0.687705
\(447\) 4.06155 0.192105
\(448\) 1.00000 0.0472456
\(449\) −19.7565 −0.932368 −0.466184 0.884688i \(-0.654372\pi\)
−0.466184 + 0.884688i \(0.654372\pi\)
\(450\) 4.63906 0.218688
\(451\) 11.1610 0.525551
\(452\) −17.9097 −0.842401
\(453\) 18.5807 0.872999
\(454\) 6.06557 0.284672
\(455\) −1.32806 −0.0622606
\(456\) −0.811418 −0.0379981
\(457\) 39.2825 1.83756 0.918779 0.394772i \(-0.129177\pi\)
0.918779 + 0.394772i \(0.129177\pi\)
\(458\) 9.54355 0.445941
\(459\) 0.291399 0.0136013
\(460\) 0.714708 0.0333234
\(461\) −5.75616 −0.268091 −0.134045 0.990975i \(-0.542797\pi\)
−0.134045 + 0.990975i \(0.542797\pi\)
\(462\) −3.98407 −0.185355
\(463\) −36.4705 −1.69493 −0.847464 0.530853i \(-0.821872\pi\)
−0.847464 + 0.530853i \(0.821872\pi\)
\(464\) −9.44157 −0.438314
\(465\) 4.57513 0.212167
\(466\) 6.93594 0.321301
\(467\) −36.6938 −1.69799 −0.848993 0.528405i \(-0.822790\pi\)
−0.848993 + 0.528405i \(0.822790\pi\)
\(468\) −2.21056 −0.102183
\(469\) 14.9062 0.688304
\(470\) −3.53042 −0.162846
\(471\) 4.70929 0.216993
\(472\) 5.71021 0.262834
\(473\) −18.9803 −0.872717
\(474\) −1.60018 −0.0734988
\(475\) −3.76422 −0.172714
\(476\) 0.291399 0.0133562
\(477\) 11.5189 0.527413
\(478\) 0.650366 0.0297470
\(479\) 30.8394 1.40909 0.704545 0.709659i \(-0.251151\pi\)
0.704545 + 0.709659i \(0.251151\pi\)
\(480\) −0.600780 −0.0274217
\(481\) 11.2283 0.511967
\(482\) −6.26146 −0.285202
\(483\) 1.18963 0.0541301
\(484\) 4.87278 0.221490
\(485\) 2.70328 0.122750
\(486\) −1.00000 −0.0453609
\(487\) 10.3033 0.466886 0.233443 0.972371i \(-0.425001\pi\)
0.233443 + 0.972371i \(0.425001\pi\)
\(488\) −2.62535 −0.118844
\(489\) 22.9695 1.03871
\(490\) −0.600780 −0.0271405
\(491\) −20.2337 −0.913132 −0.456566 0.889690i \(-0.650921\pi\)
−0.456566 + 0.889690i \(0.650921\pi\)
\(492\) 2.80141 0.126297
\(493\) −2.75126 −0.123911
\(494\) 1.79369 0.0807020
\(495\) 2.39355 0.107582
\(496\) 7.61531 0.341938
\(497\) 11.6398 0.522115
\(498\) 5.30607 0.237771
\(499\) 41.8077 1.87157 0.935785 0.352572i \(-0.114693\pi\)
0.935785 + 0.352572i \(0.114693\pi\)
\(500\) −5.79096 −0.258980
\(501\) 2.85628 0.127609
\(502\) 21.3290 0.951959
\(503\) 28.3288 1.26312 0.631560 0.775327i \(-0.282415\pi\)
0.631560 + 0.775327i \(0.282415\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 3.21680 0.143146
\(506\) −4.73957 −0.210700
\(507\) −8.11341 −0.360329
\(508\) 19.1969 0.851723
\(509\) 12.4573 0.552161 0.276080 0.961135i \(-0.410964\pi\)
0.276080 + 0.961135i \(0.410964\pi\)
\(510\) −0.175067 −0.00775209
\(511\) 16.8053 0.743422
\(512\) −1.00000 −0.0441942
\(513\) 0.811418 0.0358250
\(514\) 1.04463 0.0460767
\(515\) −5.43624 −0.239549
\(516\) −4.76406 −0.209726
\(517\) 23.4119 1.02965
\(518\) 5.07939 0.223175
\(519\) −23.1834 −1.01764
\(520\) 1.32806 0.0582394
\(521\) −37.1101 −1.62582 −0.812912 0.582386i \(-0.802119\pi\)
−0.812912 + 0.582386i \(0.802119\pi\)
\(522\) 9.44157 0.413246
\(523\) −15.6601 −0.684770 −0.342385 0.939560i \(-0.611235\pi\)
−0.342385 + 0.939560i \(0.611235\pi\)
\(524\) 16.1818 0.706906
\(525\) −4.63906 −0.202465
\(526\) −6.29202 −0.274345
\(527\) 2.21909 0.0966652
\(528\) 3.98407 0.173384
\(529\) −21.5848 −0.938468
\(530\) −6.92032 −0.300599
\(531\) −5.71021 −0.247802
\(532\) 0.811418 0.0351794
\(533\) −6.19270 −0.268236
\(534\) 6.96178 0.301266
\(535\) 9.55950 0.413293
\(536\) −14.9062 −0.643849
\(537\) 19.7984 0.854363
\(538\) 24.7427 1.06674
\(539\) 3.98407 0.171606
\(540\) 0.600780 0.0258535
\(541\) −8.78722 −0.377792 −0.188896 0.981997i \(-0.560491\pi\)
−0.188896 + 0.981997i \(0.560491\pi\)
\(542\) −21.0873 −0.905775
\(543\) −7.54151 −0.323637
\(544\) −0.291399 −0.0124936
\(545\) 0.927488 0.0397292
\(546\) 2.21056 0.0946034
\(547\) 13.6520 0.583718 0.291859 0.956461i \(-0.405726\pi\)
0.291859 + 0.956461i \(0.405726\pi\)
\(548\) −6.36691 −0.271981
\(549\) 2.62535 0.112047
\(550\) 18.4823 0.788089
\(551\) −7.66106 −0.326372
\(552\) −1.18963 −0.0506341
\(553\) 1.60018 0.0680467
\(554\) −12.6960 −0.539403
\(555\) −3.05160 −0.129533
\(556\) −12.0899 −0.512725
\(557\) 4.32923 0.183435 0.0917177 0.995785i \(-0.470764\pi\)
0.0917177 + 0.995785i \(0.470764\pi\)
\(558\) −7.61531 −0.322382
\(559\) 10.5313 0.445425
\(560\) 0.600780 0.0253876
\(561\) 1.16095 0.0490154
\(562\) −6.29496 −0.265537
\(563\) −7.33859 −0.309285 −0.154642 0.987971i \(-0.549422\pi\)
−0.154642 + 0.987971i \(0.549422\pi\)
\(564\) 5.87638 0.247440
\(565\) −10.7598 −0.452668
\(566\) −28.4254 −1.19481
\(567\) 1.00000 0.0419961
\(568\) −11.6398 −0.488394
\(569\) 0.843687 0.0353692 0.0176846 0.999844i \(-0.494371\pi\)
0.0176846 + 0.999844i \(0.494371\pi\)
\(570\) −0.487484 −0.0204185
\(571\) 9.09512 0.380619 0.190309 0.981724i \(-0.439051\pi\)
0.190309 + 0.981724i \(0.439051\pi\)
\(572\) −8.80703 −0.368240
\(573\) 1.00000 0.0417756
\(574\) −2.80141 −0.116929
\(575\) −5.51878 −0.230149
\(576\) 1.00000 0.0416667
\(577\) −28.7246 −1.19582 −0.597910 0.801563i \(-0.704002\pi\)
−0.597910 + 0.801563i \(0.704002\pi\)
\(578\) 16.9151 0.703575
\(579\) −9.25956 −0.384814
\(580\) −5.67231 −0.235530
\(581\) −5.30607 −0.220133
\(582\) −4.49961 −0.186515
\(583\) 45.8920 1.90065
\(584\) −16.8053 −0.695408
\(585\) −1.32806 −0.0549087
\(586\) −28.6147 −1.18206
\(587\) 29.2293 1.20642 0.603212 0.797581i \(-0.293887\pi\)
0.603212 + 0.797581i \(0.293887\pi\)
\(588\) 1.00000 0.0412393
\(589\) 6.17920 0.254610
\(590\) 3.43058 0.141235
\(591\) −10.0613 −0.413868
\(592\) −5.07939 −0.208762
\(593\) 15.6551 0.642878 0.321439 0.946930i \(-0.395834\pi\)
0.321439 + 0.946930i \(0.395834\pi\)
\(594\) −3.98407 −0.163468
\(595\) 0.175067 0.00717704
\(596\) 4.06155 0.166368
\(597\) 5.35710 0.219251
\(598\) 2.62976 0.107539
\(599\) −46.2954 −1.89158 −0.945789 0.324781i \(-0.894709\pi\)
−0.945789 + 0.324781i \(0.894709\pi\)
\(600\) 4.63906 0.189389
\(601\) −20.1477 −0.821840 −0.410920 0.911671i \(-0.634792\pi\)
−0.410920 + 0.911671i \(0.634792\pi\)
\(602\) 4.76406 0.194169
\(603\) 14.9062 0.607027
\(604\) 18.5807 0.756040
\(605\) 2.92747 0.119019
\(606\) −5.35438 −0.217507
\(607\) −7.05084 −0.286185 −0.143092 0.989709i \(-0.545705\pi\)
−0.143092 + 0.989709i \(0.545705\pi\)
\(608\) −0.811418 −0.0329074
\(609\) −9.44157 −0.382592
\(610\) −1.57726 −0.0638612
\(611\) −12.9901 −0.525524
\(612\) 0.291399 0.0117791
\(613\) 23.4082 0.945447 0.472724 0.881211i \(-0.343271\pi\)
0.472724 + 0.881211i \(0.343271\pi\)
\(614\) −33.6273 −1.35709
\(615\) 1.68303 0.0678665
\(616\) −3.98407 −0.160523
\(617\) −27.3958 −1.10291 −0.551457 0.834203i \(-0.685928\pi\)
−0.551457 + 0.834203i \(0.685928\pi\)
\(618\) 9.04862 0.363989
\(619\) −2.18096 −0.0876600 −0.0438300 0.999039i \(-0.513956\pi\)
−0.0438300 + 0.999039i \(0.513956\pi\)
\(620\) 4.57513 0.183742
\(621\) 1.18963 0.0477383
\(622\) 7.23048 0.289916
\(623\) −6.96178 −0.278918
\(624\) −2.21056 −0.0884934
\(625\) 19.7162 0.788649
\(626\) 23.1287 0.924408
\(627\) 3.23274 0.129103
\(628\) 4.70929 0.187921
\(629\) −1.48013 −0.0590166
\(630\) −0.600780 −0.0239357
\(631\) −32.8276 −1.30685 −0.653423 0.756993i \(-0.726668\pi\)
−0.653423 + 0.756993i \(0.726668\pi\)
\(632\) −1.60018 −0.0636518
\(633\) 10.4729 0.416260
\(634\) −7.17192 −0.284833
\(635\) 11.5331 0.457677
\(636\) 11.5189 0.456753
\(637\) −2.21056 −0.0875857
\(638\) 37.6158 1.48922
\(639\) 11.6398 0.460462
\(640\) −0.600780 −0.0237479
\(641\) −15.0772 −0.595513 −0.297756 0.954642i \(-0.596238\pi\)
−0.297756 + 0.954642i \(0.596238\pi\)
\(642\) −15.9118 −0.627989
\(643\) −16.3606 −0.645199 −0.322599 0.946536i \(-0.604557\pi\)
−0.322599 + 0.946536i \(0.604557\pi\)
\(644\) 1.18963 0.0468781
\(645\) −2.86215 −0.112697
\(646\) −0.236446 −0.00930286
\(647\) 49.1364 1.93175 0.965875 0.259007i \(-0.0833953\pi\)
0.965875 + 0.259007i \(0.0833953\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −22.7498 −0.893009
\(650\) −10.2549 −0.402232
\(651\) 7.61531 0.298468
\(652\) 22.9695 0.899553
\(653\) −2.88589 −0.112934 −0.0564669 0.998404i \(-0.517984\pi\)
−0.0564669 + 0.998404i \(0.517984\pi\)
\(654\) −1.54381 −0.0603676
\(655\) 9.72173 0.379859
\(656\) 2.80141 0.109377
\(657\) 16.8053 0.655636
\(658\) −5.87638 −0.229085
\(659\) 13.5442 0.527607 0.263804 0.964576i \(-0.415023\pi\)
0.263804 + 0.964576i \(0.415023\pi\)
\(660\) 2.39355 0.0931687
\(661\) 45.4190 1.76660 0.883298 0.468813i \(-0.155318\pi\)
0.883298 + 0.468813i \(0.155318\pi\)
\(662\) −26.0184 −1.01123
\(663\) −0.644156 −0.0250169
\(664\) 5.30607 0.205916
\(665\) 0.487484 0.0189038
\(666\) 5.07939 0.196822
\(667\) −11.2320 −0.434905
\(668\) 2.85628 0.110513
\(669\) −14.5234 −0.561509
\(670\) −8.95534 −0.345975
\(671\) 10.4595 0.403786
\(672\) −1.00000 −0.0385758
\(673\) 11.9755 0.461620 0.230810 0.972999i \(-0.425862\pi\)
0.230810 + 0.972999i \(0.425862\pi\)
\(674\) 16.0449 0.618025
\(675\) −4.63906 −0.178558
\(676\) −8.11341 −0.312054
\(677\) −4.86100 −0.186823 −0.0934117 0.995628i \(-0.529777\pi\)
−0.0934117 + 0.995628i \(0.529777\pi\)
\(678\) 17.9097 0.687818
\(679\) 4.49961 0.172679
\(680\) −0.175067 −0.00671350
\(681\) −6.06557 −0.232433
\(682\) −30.3399 −1.16177
\(683\) −1.30802 −0.0500501 −0.0250251 0.999687i \(-0.507967\pi\)
−0.0250251 + 0.999687i \(0.507967\pi\)
\(684\) 0.811418 0.0310254
\(685\) −3.82511 −0.146150
\(686\) −1.00000 −0.0381802
\(687\) −9.54355 −0.364109
\(688\) −4.76406 −0.181628
\(689\) −25.4632 −0.970071
\(690\) −0.714708 −0.0272085
\(691\) −31.3551 −1.19280 −0.596402 0.802686i \(-0.703404\pi\)
−0.596402 + 0.802686i \(0.703404\pi\)
\(692\) −23.1834 −0.881302
\(693\) 3.98407 0.151342
\(694\) 2.80431 0.106450
\(695\) −7.26336 −0.275515
\(696\) 9.44157 0.357882
\(697\) 0.816328 0.0309206
\(698\) 1.70382 0.0644904
\(699\) −6.93594 −0.262341
\(700\) −4.63906 −0.175340
\(701\) −29.6420 −1.11956 −0.559781 0.828641i \(-0.689115\pi\)
−0.559781 + 0.828641i \(0.689115\pi\)
\(702\) 2.21056 0.0834324
\(703\) −4.12151 −0.155446
\(704\) 3.98407 0.150155
\(705\) 3.53042 0.132963
\(706\) −16.7576 −0.630681
\(707\) 5.35438 0.201372
\(708\) −5.71021 −0.214603
\(709\) 5.86898 0.220414 0.110207 0.993909i \(-0.464849\pi\)
0.110207 + 0.993909i \(0.464849\pi\)
\(710\) −6.99294 −0.262441
\(711\) 1.60018 0.0600115
\(712\) 6.96178 0.260904
\(713\) 9.05942 0.339278
\(714\) −0.291399 −0.0109053
\(715\) −5.29109 −0.197876
\(716\) 19.7984 0.739900
\(717\) −0.650366 −0.0242883
\(718\) 13.1677 0.491414
\(719\) 2.00973 0.0749501 0.0374751 0.999298i \(-0.488069\pi\)
0.0374751 + 0.999298i \(0.488069\pi\)
\(720\) 0.600780 0.0223898
\(721\) −9.04862 −0.336988
\(722\) 18.3416 0.682604
\(723\) 6.26146 0.232866
\(724\) −7.54151 −0.280278
\(725\) 43.8000 1.62669
\(726\) −4.87278 −0.180846
\(727\) 11.5401 0.428000 0.214000 0.976834i \(-0.431351\pi\)
0.214000 + 0.976834i \(0.431351\pi\)
\(728\) 2.21056 0.0819290
\(729\) 1.00000 0.0370370
\(730\) −10.0963 −0.373680
\(731\) −1.38824 −0.0513460
\(732\) 2.62535 0.0970356
\(733\) 26.4861 0.978285 0.489142 0.872204i \(-0.337310\pi\)
0.489142 + 0.872204i \(0.337310\pi\)
\(734\) −9.51113 −0.351062
\(735\) 0.600780 0.0221601
\(736\) −1.18963 −0.0438504
\(737\) 59.3872 2.18756
\(738\) −2.80141 −0.103121
\(739\) 6.26547 0.230479 0.115240 0.993338i \(-0.463236\pi\)
0.115240 + 0.993338i \(0.463236\pi\)
\(740\) −3.05160 −0.112179
\(741\) −1.79369 −0.0658929
\(742\) −11.5189 −0.422871
\(743\) 43.8083 1.60717 0.803585 0.595190i \(-0.202923\pi\)
0.803585 + 0.595190i \(0.202923\pi\)
\(744\) −7.61531 −0.279191
\(745\) 2.44010 0.0893984
\(746\) −27.8130 −1.01831
\(747\) −5.30607 −0.194139
\(748\) 1.16095 0.0424486
\(749\) 15.9118 0.581405
\(750\) 5.79096 0.211456
\(751\) −35.7985 −1.30631 −0.653153 0.757226i \(-0.726554\pi\)
−0.653153 + 0.757226i \(0.726554\pi\)
\(752\) 5.87638 0.214290
\(753\) −21.3290 −0.777272
\(754\) −20.8712 −0.760084
\(755\) 11.1629 0.406261
\(756\) 1.00000 0.0363696
\(757\) 14.8736 0.540590 0.270295 0.962778i \(-0.412879\pi\)
0.270295 + 0.962778i \(0.412879\pi\)
\(758\) 0.528092 0.0191812
\(759\) 4.73957 0.172036
\(760\) −0.487484 −0.0176829
\(761\) −14.0187 −0.508178 −0.254089 0.967181i \(-0.581776\pi\)
−0.254089 + 0.967181i \(0.581776\pi\)
\(762\) −19.1969 −0.695429
\(763\) 1.54381 0.0558895
\(764\) 1.00000 0.0361787
\(765\) 0.175067 0.00632955
\(766\) −1.79471 −0.0648456
\(767\) 12.6228 0.455782
\(768\) 1.00000 0.0360844
\(769\) −13.4016 −0.483273 −0.241636 0.970367i \(-0.577684\pi\)
−0.241636 + 0.970367i \(0.577684\pi\)
\(770\) −2.39355 −0.0862575
\(771\) −1.04463 −0.0376214
\(772\) −9.25956 −0.333259
\(773\) 25.5943 0.920564 0.460282 0.887773i \(-0.347748\pi\)
0.460282 + 0.887773i \(0.347748\pi\)
\(774\) 4.76406 0.171241
\(775\) −35.3279 −1.26902
\(776\) −4.49961 −0.161527
\(777\) −5.07939 −0.182222
\(778\) 23.0637 0.826874
\(779\) 2.27312 0.0814429
\(780\) −1.32806 −0.0475523
\(781\) 46.3736 1.65938
\(782\) −0.346658 −0.0123965
\(783\) −9.44157 −0.337414
\(784\) 1.00000 0.0357143
\(785\) 2.82925 0.100980
\(786\) −16.1818 −0.577187
\(787\) 33.1324 1.18104 0.590522 0.807022i \(-0.298922\pi\)
0.590522 + 0.807022i \(0.298922\pi\)
\(788\) −10.0613 −0.358420
\(789\) 6.29202 0.224002
\(790\) −0.961358 −0.0342036
\(791\) −17.9097 −0.636796
\(792\) −3.98407 −0.141568
\(793\) −5.80349 −0.206088
\(794\) −24.2450 −0.860424
\(795\) 6.92032 0.245438
\(796\) 5.35710 0.189877
\(797\) −22.9225 −0.811955 −0.405978 0.913883i \(-0.633069\pi\)
−0.405978 + 0.913883i \(0.633069\pi\)
\(798\) −0.811418 −0.0287239
\(799\) 1.71237 0.0605794
\(800\) 4.63906 0.164016
\(801\) −6.96178 −0.245982
\(802\) 10.6619 0.376484
\(803\) 66.9533 2.36273
\(804\) 14.9062 0.525701
\(805\) 0.714708 0.0251901
\(806\) 16.8341 0.592957
\(807\) −24.7427 −0.870986
\(808\) −5.35438 −0.188366
\(809\) 32.7460 1.15129 0.575644 0.817701i \(-0.304752\pi\)
0.575644 + 0.817701i \(0.304752\pi\)
\(810\) −0.600780 −0.0211093
\(811\) −25.3283 −0.889399 −0.444699 0.895680i \(-0.646689\pi\)
−0.444699 + 0.895680i \(0.646689\pi\)
\(812\) −9.44157 −0.331334
\(813\) 21.0873 0.739562
\(814\) 20.2366 0.709293
\(815\) 13.7996 0.483379
\(816\) 0.291399 0.0102010
\(817\) −3.86565 −0.135242
\(818\) 10.7685 0.376512
\(819\) −2.21056 −0.0772434
\(820\) 1.68303 0.0587741
\(821\) 6.88758 0.240378 0.120189 0.992751i \(-0.461650\pi\)
0.120189 + 0.992751i \(0.461650\pi\)
\(822\) 6.36691 0.222071
\(823\) −36.0166 −1.25546 −0.627730 0.778431i \(-0.716016\pi\)
−0.627730 + 0.778431i \(0.716016\pi\)
\(824\) 9.04862 0.315224
\(825\) −18.4823 −0.643472
\(826\) 5.71021 0.198684
\(827\) 28.3474 0.985735 0.492868 0.870104i \(-0.335949\pi\)
0.492868 + 0.870104i \(0.335949\pi\)
\(828\) 1.18963 0.0413426
\(829\) −27.8916 −0.968716 −0.484358 0.874870i \(-0.660947\pi\)
−0.484358 + 0.874870i \(0.660947\pi\)
\(830\) 3.18779 0.110650
\(831\) 12.6960 0.440421
\(832\) −2.21056 −0.0766375
\(833\) 0.291399 0.0100964
\(834\) 12.0899 0.418638
\(835\) 1.71600 0.0593845
\(836\) 3.23274 0.111807
\(837\) 7.61531 0.263224
\(838\) −6.01704 −0.207855
\(839\) 29.4805 1.01778 0.508890 0.860831i \(-0.330056\pi\)
0.508890 + 0.860831i \(0.330056\pi\)
\(840\) −0.600780 −0.0207289
\(841\) 60.1432 2.07390
\(842\) −27.8137 −0.958523
\(843\) 6.29496 0.216810
\(844\) 10.4729 0.360491
\(845\) −4.87438 −0.167684
\(846\) −5.87638 −0.202034
\(847\) 4.87278 0.167431
\(848\) 11.5189 0.395560
\(849\) 28.4254 0.975556
\(850\) 1.35182 0.0463670
\(851\) −6.04260 −0.207138
\(852\) 11.6398 0.398772
\(853\) −36.0783 −1.23530 −0.617649 0.786454i \(-0.711915\pi\)
−0.617649 + 0.786454i \(0.711915\pi\)
\(854\) −2.62535 −0.0898375
\(855\) 0.487484 0.0166716
\(856\) −15.9118 −0.543854
\(857\) −2.69038 −0.0919017 −0.0459509 0.998944i \(-0.514632\pi\)
−0.0459509 + 0.998944i \(0.514632\pi\)
\(858\) 8.80703 0.300667
\(859\) 25.2024 0.859894 0.429947 0.902854i \(-0.358532\pi\)
0.429947 + 0.902854i \(0.358532\pi\)
\(860\) −2.86215 −0.0975987
\(861\) 2.80141 0.0954719
\(862\) 28.1741 0.959615
\(863\) 6.36768 0.216758 0.108379 0.994110i \(-0.465434\pi\)
0.108379 + 0.994110i \(0.465434\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −13.9282 −0.473571
\(866\) −19.5526 −0.664426
\(867\) −16.9151 −0.574466
\(868\) 7.61531 0.258480
\(869\) 6.37523 0.216265
\(870\) 5.67231 0.192309
\(871\) −32.9511 −1.11650
\(872\) −1.54381 −0.0522798
\(873\) 4.49961 0.152289
\(874\) −0.965289 −0.0326514
\(875\) −5.79096 −0.195770
\(876\) 16.8053 0.567798
\(877\) −40.1524 −1.35585 −0.677924 0.735132i \(-0.737120\pi\)
−0.677924 + 0.735132i \(0.737120\pi\)
\(878\) 7.01706 0.236814
\(879\) 28.6147 0.965150
\(880\) 2.39355 0.0806865
\(881\) 18.2610 0.615229 0.307614 0.951511i \(-0.400469\pi\)
0.307614 + 0.951511i \(0.400469\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −50.7678 −1.70847 −0.854236 0.519886i \(-0.825975\pi\)
−0.854236 + 0.519886i \(0.825975\pi\)
\(884\) −0.644156 −0.0216653
\(885\) −3.43058 −0.115318
\(886\) 33.9891 1.14189
\(887\) 1.55143 0.0520919 0.0260459 0.999661i \(-0.491708\pi\)
0.0260459 + 0.999661i \(0.491708\pi\)
\(888\) 5.07939 0.170453
\(889\) 19.1969 0.643842
\(890\) 4.18250 0.140198
\(891\) 3.98407 0.133471
\(892\) −14.5234 −0.486281
\(893\) 4.76821 0.159562
\(894\) −4.06155 −0.135839
\(895\) 11.8945 0.397588
\(896\) −1.00000 −0.0334077
\(897\) −2.62976 −0.0878051
\(898\) 19.7565 0.659284
\(899\) −71.9005 −2.39802
\(900\) −4.63906 −0.154635
\(901\) 3.35659 0.111824
\(902\) −11.1610 −0.371621
\(903\) −4.76406 −0.158538
\(904\) 17.9097 0.595668
\(905\) −4.53079 −0.150609
\(906\) −18.5807 −0.617304
\(907\) −5.33208 −0.177049 −0.0885243 0.996074i \(-0.528215\pi\)
−0.0885243 + 0.996074i \(0.528215\pi\)
\(908\) −6.06557 −0.201293
\(909\) 5.35438 0.177593
\(910\) 1.32806 0.0440249
\(911\) 44.8143 1.48477 0.742383 0.669976i \(-0.233696\pi\)
0.742383 + 0.669976i \(0.233696\pi\)
\(912\) 0.811418 0.0268687
\(913\) −21.1397 −0.699623
\(914\) −39.2825 −1.29935
\(915\) 1.57726 0.0521425
\(916\) −9.54355 −0.315328
\(917\) 16.1818 0.534371
\(918\) −0.291399 −0.00961760
\(919\) 45.7974 1.51072 0.755358 0.655313i \(-0.227463\pi\)
0.755358 + 0.655313i \(0.227463\pi\)
\(920\) −0.714708 −0.0235632
\(921\) 33.6273 1.10806
\(922\) 5.75616 0.189569
\(923\) −25.7305 −0.846928
\(924\) 3.98407 0.131066
\(925\) 23.5636 0.774766
\(926\) 36.4705 1.19850
\(927\) −9.04862 −0.297196
\(928\) 9.44157 0.309935
\(929\) −1.60488 −0.0526543 −0.0263272 0.999653i \(-0.508381\pi\)
−0.0263272 + 0.999653i \(0.508381\pi\)
\(930\) −4.57513 −0.150024
\(931\) 0.811418 0.0265932
\(932\) −6.93594 −0.227194
\(933\) −7.23048 −0.236715
\(934\) 36.6938 1.20066
\(935\) 0.697477 0.0228100
\(936\) 2.21056 0.0722545
\(937\) −31.0306 −1.01373 −0.506863 0.862027i \(-0.669195\pi\)
−0.506863 + 0.862027i \(0.669195\pi\)
\(938\) −14.9062 −0.486704
\(939\) −23.1287 −0.754776
\(940\) 3.53042 0.115150
\(941\) 33.8741 1.10427 0.552133 0.833756i \(-0.313814\pi\)
0.552133 + 0.833756i \(0.313814\pi\)
\(942\) −4.70929 −0.153437
\(943\) 3.33265 0.108526
\(944\) −5.71021 −0.185851
\(945\) 0.600780 0.0195434
\(946\) 18.9803 0.617104
\(947\) −20.5142 −0.666621 −0.333310 0.942817i \(-0.608166\pi\)
−0.333310 + 0.942817i \(0.608166\pi\)
\(948\) 1.60018 0.0519715
\(949\) −37.1492 −1.20591
\(950\) 3.76422 0.122127
\(951\) 7.17192 0.232566
\(952\) −0.291399 −0.00944429
\(953\) 4.80130 0.155529 0.0777647 0.996972i \(-0.475222\pi\)
0.0777647 + 0.996972i \(0.475222\pi\)
\(954\) −11.5189 −0.372937
\(955\) 0.600780 0.0194408
\(956\) −0.650366 −0.0210343
\(957\) −37.6158 −1.21595
\(958\) −30.8394 −0.996377
\(959\) −6.36691 −0.205598
\(960\) 0.600780 0.0193901
\(961\) 26.9930 0.870741
\(962\) −11.2283 −0.362015
\(963\) 15.9118 0.512751
\(964\) 6.26146 0.201668
\(965\) −5.56296 −0.179078
\(966\) −1.18963 −0.0382758
\(967\) 31.4244 1.01054 0.505271 0.862961i \(-0.331393\pi\)
0.505271 + 0.862961i \(0.331393\pi\)
\(968\) −4.87278 −0.156617
\(969\) 0.236446 0.00759575
\(970\) −2.70328 −0.0867970
\(971\) −37.4560 −1.20202 −0.601010 0.799242i \(-0.705235\pi\)
−0.601010 + 0.799242i \(0.705235\pi\)
\(972\) 1.00000 0.0320750
\(973\) −12.0899 −0.387584
\(974\) −10.3033 −0.330138
\(975\) 10.2549 0.328421
\(976\) 2.62535 0.0840353
\(977\) −59.1854 −1.89351 −0.946755 0.321956i \(-0.895660\pi\)
−0.946755 + 0.321956i \(0.895660\pi\)
\(978\) −22.9695 −0.734482
\(979\) −27.7362 −0.886452
\(980\) 0.600780 0.0191912
\(981\) 1.54381 0.0492899
\(982\) 20.2337 0.645682
\(983\) −40.8301 −1.30228 −0.651138 0.758959i \(-0.725708\pi\)
−0.651138 + 0.758959i \(0.725708\pi\)
\(984\) −2.80141 −0.0893057
\(985\) −6.04466 −0.192599
\(986\) 2.75126 0.0876181
\(987\) 5.87638 0.187047
\(988\) −1.79369 −0.0570649
\(989\) −5.66748 −0.180215
\(990\) −2.39355 −0.0760720
\(991\) −40.0915 −1.27355 −0.636774 0.771051i \(-0.719731\pi\)
−0.636774 + 0.771051i \(0.719731\pi\)
\(992\) −7.61531 −0.241786
\(993\) 26.0184 0.825669
\(994\) −11.6398 −0.369191
\(995\) 3.21844 0.102031
\(996\) −5.30607 −0.168129
\(997\) −50.9674 −1.61415 −0.807077 0.590446i \(-0.798952\pi\)
−0.807077 + 0.590446i \(0.798952\pi\)
\(998\) −41.8077 −1.32340
\(999\) −5.07939 −0.160705
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 8022.2.a.z.1.9 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8022.2.a.z.1.9 15 1.1 even 1 trivial