Properties

Label 4026.2.a.q.1.1
Level $4026$
Weight $2$
Character 4026.1
Self dual yes
Analytic conductor $32.148$
Analytic rank $0$
Dimension $4$
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] = [4026,2,Mod(1,4026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4026, 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("4026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4026.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: \(32.1477718538\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.6809.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.06963\) of defining polynomial
Character \(\chi\) \(=\) 4026.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.06963 q^{5} +1.00000 q^{6} -4.35299 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.06963 q^{5} +1.00000 q^{6} -4.35299 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.06963 q^{10} -1.00000 q^{11} -1.00000 q^{12} -0.682998 q^{13} +4.35299 q^{14} +2.06963 q^{15} +1.00000 q^{16} -1.26945 q^{17} -1.00000 q^{18} +0.110461 q^{19} -2.06963 q^{20} +4.35299 q^{21} +1.00000 q^{22} +4.31700 q^{23} +1.00000 q^{24} -0.716640 q^{25} +0.682998 q^{26} -1.00000 q^{27} -4.35299 q^{28} -4.08354 q^{29} -2.06963 q^{30} -6.92552 q^{31} -1.00000 q^{32} +1.00000 q^{33} +1.26945 q^{34} +9.00906 q^{35} +1.00000 q^{36} -7.85008 q^{37} -0.110461 q^{38} +0.682998 q^{39} +2.06963 q^{40} -7.84523 q^{41} -4.35299 q^{42} -2.86309 q^{43} -1.00000 q^{44} -2.06963 q^{45} -4.31700 q^{46} -10.5062 q^{47} -1.00000 q^{48} +11.9485 q^{49} +0.716640 q^{50} +1.26945 q^{51} -0.682998 q^{52} -7.98214 q^{53} +1.00000 q^{54} +2.06963 q^{55} +4.35299 q^{56} -0.110461 q^{57} +4.08354 q^{58} -7.49224 q^{59} +2.06963 q^{60} +1.00000 q^{61} +6.92552 q^{62} -4.35299 q^{63} +1.00000 q^{64} +1.41355 q^{65} -1.00000 q^{66} -12.9346 q^{67} -1.26945 q^{68} -4.31700 q^{69} -9.00906 q^{70} +9.82898 q^{71} -1.00000 q^{72} -14.1599 q^{73} +7.85008 q^{74} +0.716640 q^{75} +0.110461 q^{76} +4.35299 q^{77} -0.682998 q^{78} -1.79299 q^{79} -2.06963 q^{80} +1.00000 q^{81} +7.84523 q^{82} +10.9592 q^{83} +4.35299 q^{84} +2.62728 q^{85} +2.86309 q^{86} +4.08354 q^{87} +1.00000 q^{88} -1.93037 q^{89} +2.06963 q^{90} +2.97308 q^{91} +4.31700 q^{92} +6.92552 q^{93} +10.5062 q^{94} -0.228614 q^{95} +1.00000 q^{96} +11.4250 q^{97} -11.9485 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} - 4 q^{11} - 4 q^{12} - 4 q^{13} + 2 q^{14} + 4 q^{16} - q^{17} - 4 q^{18} + 4 q^{19} + 2 q^{21} + 4 q^{22} + 16 q^{23} + 4 q^{24} - 10 q^{25} + 4 q^{26} - 4 q^{27} - 2 q^{28} - 5 q^{29} - 10 q^{31} - 4 q^{32} + 4 q^{33} + q^{34} + 7 q^{35} + 4 q^{36} - 10 q^{37} - 4 q^{38} + 4 q^{39} + 16 q^{41} - 2 q^{42} - 8 q^{43} - 4 q^{44} - 16 q^{46} - 7 q^{47} - 4 q^{48} - 2 q^{49} + 10 q^{50} + q^{51} - 4 q^{52} + 12 q^{53} + 4 q^{54} + 2 q^{56} - 4 q^{57} + 5 q^{58} + 2 q^{59} + 4 q^{61} + 10 q^{62} - 2 q^{63} + 4 q^{64} + 11 q^{65} - 4 q^{66} - 5 q^{67} - q^{68} - 16 q^{69} - 7 q^{70} + 15 q^{71} - 4 q^{72} - 28 q^{73} + 10 q^{74} + 10 q^{75} + 4 q^{76} + 2 q^{77} - 4 q^{78} + 3 q^{79} + 4 q^{81} - 16 q^{82} + 32 q^{83} + 2 q^{84} + 17 q^{85} + 8 q^{86} + 5 q^{87} + 4 q^{88} - 16 q^{89} - 3 q^{91} + 16 q^{92} + 10 q^{93} + 7 q^{94} + 15 q^{95} + 4 q^{96} + 2 q^{97} + 2 q^{98} - 4 q^{99}+O(q^{100}) \) 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
\(5\) −2.06963 −0.925566 −0.462783 0.886472i \(-0.653149\pi\)
−0.462783 + 0.886472i \(0.653149\pi\)
\(6\) 1.00000 0.408248
\(7\) −4.35299 −1.64527 −0.822637 0.568566i \(-0.807498\pi\)
−0.822637 + 0.568566i \(0.807498\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.06963 0.654474
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) −0.682998 −0.189429 −0.0947147 0.995504i \(-0.530194\pi\)
−0.0947147 + 0.995504i \(0.530194\pi\)
\(14\) 4.35299 1.16338
\(15\) 2.06963 0.534376
\(16\) 1.00000 0.250000
\(17\) −1.26945 −0.307886 −0.153943 0.988080i \(-0.549197\pi\)
−0.153943 + 0.988080i \(0.549197\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0.110461 0.0253415 0.0126708 0.999920i \(-0.495967\pi\)
0.0126708 + 0.999920i \(0.495967\pi\)
\(20\) −2.06963 −0.462783
\(21\) 4.35299 0.949900
\(22\) 1.00000 0.213201
\(23\) 4.31700 0.900157 0.450079 0.892989i \(-0.351396\pi\)
0.450079 + 0.892989i \(0.351396\pi\)
\(24\) 1.00000 0.204124
\(25\) −0.716640 −0.143328
\(26\) 0.682998 0.133947
\(27\) −1.00000 −0.192450
\(28\) −4.35299 −0.822637
\(29\) −4.08354 −0.758295 −0.379147 0.925336i \(-0.623783\pi\)
−0.379147 + 0.925336i \(0.623783\pi\)
\(30\) −2.06963 −0.377861
\(31\) −6.92552 −1.24386 −0.621930 0.783073i \(-0.713651\pi\)
−0.621930 + 0.783073i \(0.713651\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 1.26945 0.217708
\(35\) 9.00906 1.52281
\(36\) 1.00000 0.166667
\(37\) −7.85008 −1.29055 −0.645273 0.763952i \(-0.723256\pi\)
−0.645273 + 0.763952i \(0.723256\pi\)
\(38\) −0.110461 −0.0179192
\(39\) 0.682998 0.109367
\(40\) 2.06963 0.327237
\(41\) −7.84523 −1.22522 −0.612610 0.790386i \(-0.709880\pi\)
−0.612610 + 0.790386i \(0.709880\pi\)
\(42\) −4.35299 −0.671681
\(43\) −2.86309 −0.436617 −0.218308 0.975880i \(-0.570054\pi\)
−0.218308 + 0.975880i \(0.570054\pi\)
\(44\) −1.00000 −0.150756
\(45\) −2.06963 −0.308522
\(46\) −4.31700 −0.636507
\(47\) −10.5062 −1.53248 −0.766240 0.642554i \(-0.777875\pi\)
−0.766240 + 0.642554i \(0.777875\pi\)
\(48\) −1.00000 −0.144338
\(49\) 11.9485 1.70693
\(50\) 0.716640 0.101348
\(51\) 1.26945 0.177758
\(52\) −0.682998 −0.0947147
\(53\) −7.98214 −1.09643 −0.548216 0.836337i \(-0.684693\pi\)
−0.548216 + 0.836337i \(0.684693\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.06963 0.279069
\(56\) 4.35299 0.581692
\(57\) −0.110461 −0.0146309
\(58\) 4.08354 0.536195
\(59\) −7.49224 −0.975407 −0.487704 0.873009i \(-0.662165\pi\)
−0.487704 + 0.873009i \(0.662165\pi\)
\(60\) 2.06963 0.267188
\(61\) 1.00000 0.128037
\(62\) 6.92552 0.879542
\(63\) −4.35299 −0.548425
\(64\) 1.00000 0.125000
\(65\) 1.41355 0.175329
\(66\) −1.00000 −0.123091
\(67\) −12.9346 −1.58021 −0.790106 0.612971i \(-0.789974\pi\)
−0.790106 + 0.612971i \(0.789974\pi\)
\(68\) −1.26945 −0.153943
\(69\) −4.31700 −0.519706
\(70\) −9.00906 −1.07679
\(71\) 9.82898 1.16648 0.583242 0.812298i \(-0.301784\pi\)
0.583242 + 0.812298i \(0.301784\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.1599 −1.65729 −0.828645 0.559775i \(-0.810888\pi\)
−0.828645 + 0.559775i \(0.810888\pi\)
\(74\) 7.85008 0.912553
\(75\) 0.716640 0.0827505
\(76\) 0.110461 0.0126708
\(77\) 4.35299 0.496069
\(78\) −0.682998 −0.0773342
\(79\) −1.79299 −0.201727 −0.100864 0.994900i \(-0.532161\pi\)
−0.100864 + 0.994900i \(0.532161\pi\)
\(80\) −2.06963 −0.231391
\(81\) 1.00000 0.111111
\(82\) 7.84523 0.866361
\(83\) 10.9592 1.20292 0.601462 0.798901i \(-0.294585\pi\)
0.601462 + 0.798901i \(0.294585\pi\)
\(84\) 4.35299 0.474950
\(85\) 2.62728 0.284969
\(86\) 2.86309 0.308735
\(87\) 4.08354 0.437802
\(88\) 1.00000 0.106600
\(89\) −1.93037 −0.204619 −0.102310 0.994753i \(-0.532623\pi\)
−0.102310 + 0.994753i \(0.532623\pi\)
\(90\) 2.06963 0.218158
\(91\) 2.97308 0.311663
\(92\) 4.31700 0.450079
\(93\) 6.92552 0.718143
\(94\) 10.5062 1.08363
\(95\) −0.228614 −0.0234553
\(96\) 1.00000 0.102062
\(97\) 11.4250 1.16003 0.580014 0.814606i \(-0.303047\pi\)
0.580014 + 0.814606i \(0.303047\pi\)
\(98\) −11.9485 −1.20698
\(99\) −1.00000 −0.100504
\(100\) −0.716640 −0.0716640
\(101\) 10.2233 1.01725 0.508626 0.860987i \(-0.330153\pi\)
0.508626 + 0.860987i \(0.330153\pi\)
\(102\) −1.26945 −0.125694
\(103\) −14.9778 −1.47580 −0.737902 0.674908i \(-0.764183\pi\)
−0.737902 + 0.674908i \(0.764183\pi\)
\(104\) 0.682998 0.0669734
\(105\) −9.00906 −0.879195
\(106\) 7.98214 0.775294
\(107\) 11.9413 1.15441 0.577205 0.816599i \(-0.304143\pi\)
0.577205 + 0.816599i \(0.304143\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −15.4609 −1.48089 −0.740445 0.672117i \(-0.765385\pi\)
−0.740445 + 0.672117i \(0.765385\pi\)
\(110\) −2.06963 −0.197331
\(111\) 7.85008 0.745097
\(112\) −4.35299 −0.411319
\(113\) −7.82944 −0.736532 −0.368266 0.929720i \(-0.620048\pi\)
−0.368266 + 0.929720i \(0.620048\pi\)
\(114\) 0.110461 0.0103456
\(115\) −8.93459 −0.833155
\(116\) −4.08354 −0.379147
\(117\) −0.682998 −0.0631431
\(118\) 7.49224 0.689717
\(119\) 5.52589 0.506557
\(120\) −2.06963 −0.188930
\(121\) 1.00000 0.0909091
\(122\) −1.00000 −0.0905357
\(123\) 7.84523 0.707381
\(124\) −6.92552 −0.621930
\(125\) 11.8313 1.05823
\(126\) 4.35299 0.387795
\(127\) −0.0427077 −0.00378969 −0.00189485 0.999998i \(-0.500603\pi\)
−0.00189485 + 0.999998i \(0.500603\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.86309 0.252081
\(130\) −1.41355 −0.123977
\(131\) 18.6791 1.63200 0.815998 0.578054i \(-0.196188\pi\)
0.815998 + 0.578054i \(0.196188\pi\)
\(132\) 1.00000 0.0870388
\(133\) −0.480836 −0.0416938
\(134\) 12.9346 1.11738
\(135\) 2.06963 0.178125
\(136\) 1.26945 0.108854
\(137\) 13.4365 1.14796 0.573980 0.818869i \(-0.305399\pi\)
0.573980 + 0.818869i \(0.305399\pi\)
\(138\) 4.31700 0.367488
\(139\) 17.9952 1.52633 0.763164 0.646204i \(-0.223645\pi\)
0.763164 + 0.646204i \(0.223645\pi\)
\(140\) 9.00906 0.761405
\(141\) 10.5062 0.884778
\(142\) −9.82898 −0.824829
\(143\) 0.682998 0.0571151
\(144\) 1.00000 0.0833333
\(145\) 8.45141 0.701851
\(146\) 14.1599 1.17188
\(147\) −11.9485 −0.985496
\(148\) −7.85008 −0.645273
\(149\) −10.2098 −0.836419 −0.418209 0.908351i \(-0.637342\pi\)
−0.418209 + 0.908351i \(0.637342\pi\)
\(150\) −0.716640 −0.0585134
\(151\) 20.2294 1.64624 0.823121 0.567866i \(-0.192231\pi\)
0.823121 + 0.567866i \(0.192231\pi\)
\(152\) −0.110461 −0.00895958
\(153\) −1.26945 −0.102629
\(154\) −4.35299 −0.350774
\(155\) 14.3333 1.15127
\(156\) 0.682998 0.0546836
\(157\) 0.568594 0.0453787 0.0226894 0.999743i \(-0.492777\pi\)
0.0226894 + 0.999743i \(0.492777\pi\)
\(158\) 1.79299 0.142643
\(159\) 7.98214 0.633025
\(160\) 2.06963 0.163618
\(161\) −18.7919 −1.48101
\(162\) −1.00000 −0.0785674
\(163\) −21.5071 −1.68457 −0.842284 0.539034i \(-0.818789\pi\)
−0.842284 + 0.539034i \(0.818789\pi\)
\(164\) −7.84523 −0.612610
\(165\) −2.06963 −0.161120
\(166\) −10.9592 −0.850596
\(167\) −3.49709 −0.270613 −0.135307 0.990804i \(-0.543202\pi\)
−0.135307 + 0.990804i \(0.543202\pi\)
\(168\) −4.35299 −0.335840
\(169\) −12.5335 −0.964116
\(170\) −2.62728 −0.201503
\(171\) 0.110461 0.00844718
\(172\) −2.86309 −0.218308
\(173\) 23.2026 1.76406 0.882030 0.471192i \(-0.156176\pi\)
0.882030 + 0.471192i \(0.156176\pi\)
\(174\) −4.08354 −0.309572
\(175\) 3.11953 0.235814
\(176\) −1.00000 −0.0753778
\(177\) 7.49224 0.563152
\(178\) 1.93037 0.144688
\(179\) −10.7522 −0.803654 −0.401827 0.915716i \(-0.631625\pi\)
−0.401827 + 0.915716i \(0.631625\pi\)
\(180\) −2.06963 −0.154261
\(181\) −9.37831 −0.697084 −0.348542 0.937293i \(-0.613323\pi\)
−0.348542 + 0.937293i \(0.613323\pi\)
\(182\) −2.97308 −0.220379
\(183\) −1.00000 −0.0739221
\(184\) −4.31700 −0.318254
\(185\) 16.2467 1.19448
\(186\) −6.92552 −0.507804
\(187\) 1.26945 0.0928311
\(188\) −10.5062 −0.766240
\(189\) 4.35299 0.316633
\(190\) 0.228614 0.0165854
\(191\) 13.3539 0.966254 0.483127 0.875550i \(-0.339501\pi\)
0.483127 + 0.875550i \(0.339501\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 6.59317 0.474587 0.237293 0.971438i \(-0.423740\pi\)
0.237293 + 0.971438i \(0.423740\pi\)
\(194\) −11.4250 −0.820264
\(195\) −1.41355 −0.101226
\(196\) 11.9485 0.853464
\(197\) −23.0051 −1.63905 −0.819524 0.573045i \(-0.805762\pi\)
−0.819524 + 0.573045i \(0.805762\pi\)
\(198\) 1.00000 0.0710669
\(199\) −13.5790 −0.962592 −0.481296 0.876558i \(-0.659834\pi\)
−0.481296 + 0.876558i \(0.659834\pi\)
\(200\) 0.716640 0.0506741
\(201\) 12.9346 0.912336
\(202\) −10.2233 −0.719306
\(203\) 17.7756 1.24760
\(204\) 1.26945 0.0888791
\(205\) 16.2367 1.13402
\(206\) 14.9778 1.04355
\(207\) 4.31700 0.300052
\(208\) −0.682998 −0.0473574
\(209\) −0.110461 −0.00764076
\(210\) 9.00906 0.621685
\(211\) 11.9974 0.825937 0.412969 0.910745i \(-0.364492\pi\)
0.412969 + 0.910745i \(0.364492\pi\)
\(212\) −7.98214 −0.548216
\(213\) −9.82898 −0.673470
\(214\) −11.9413 −0.816291
\(215\) 5.92552 0.404117
\(216\) 1.00000 0.0680414
\(217\) 30.1467 2.04649
\(218\) 15.4609 1.04715
\(219\) 14.1599 0.956837
\(220\) 2.06963 0.139534
\(221\) 0.867029 0.0583227
\(222\) −7.85008 −0.526863
\(223\) −8.83707 −0.591774 −0.295887 0.955223i \(-0.595615\pi\)
−0.295887 + 0.955223i \(0.595615\pi\)
\(224\) 4.35299 0.290846
\(225\) −0.716640 −0.0477760
\(226\) 7.82944 0.520807
\(227\) 26.6459 1.76855 0.884275 0.466967i \(-0.154653\pi\)
0.884275 + 0.466967i \(0.154653\pi\)
\(228\) −0.110461 −0.00731547
\(229\) 0.702259 0.0464066 0.0232033 0.999731i \(-0.492613\pi\)
0.0232033 + 0.999731i \(0.492613\pi\)
\(230\) 8.93459 0.589129
\(231\) −4.35299 −0.286406
\(232\) 4.08354 0.268098
\(233\) −21.3697 −1.39998 −0.699989 0.714153i \(-0.746812\pi\)
−0.699989 + 0.714153i \(0.746812\pi\)
\(234\) 0.682998 0.0446489
\(235\) 21.7438 1.41841
\(236\) −7.49224 −0.487704
\(237\) 1.79299 0.116467
\(238\) −5.52589 −0.358190
\(239\) −6.78467 −0.438864 −0.219432 0.975628i \(-0.570420\pi\)
−0.219432 + 0.975628i \(0.570420\pi\)
\(240\) 2.06963 0.133594
\(241\) 2.23534 0.143991 0.0719953 0.997405i \(-0.477063\pi\)
0.0719953 + 0.997405i \(0.477063\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 1.00000 0.0640184
\(245\) −24.7290 −1.57987
\(246\) −7.84523 −0.500194
\(247\) −0.0754447 −0.00480043
\(248\) 6.92552 0.439771
\(249\) −10.9592 −0.694509
\(250\) −11.8313 −0.748278
\(251\) 12.8290 0.809758 0.404879 0.914370i \(-0.367314\pi\)
0.404879 + 0.914370i \(0.367314\pi\)
\(252\) −4.35299 −0.274212
\(253\) −4.31700 −0.271408
\(254\) 0.0427077 0.00267972
\(255\) −2.62728 −0.164527
\(256\) 1.00000 0.0625000
\(257\) −16.4533 −1.02633 −0.513164 0.858291i \(-0.671527\pi\)
−0.513164 + 0.858291i \(0.671527\pi\)
\(258\) −2.86309 −0.178248
\(259\) 34.1713 2.12330
\(260\) 1.41355 0.0876647
\(261\) −4.08354 −0.252765
\(262\) −18.6791 −1.15400
\(263\) −3.91114 −0.241171 −0.120586 0.992703i \(-0.538477\pi\)
−0.120586 + 0.992703i \(0.538477\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 16.5201 1.01482
\(266\) 0.480836 0.0294820
\(267\) 1.93037 0.118137
\(268\) −12.9346 −0.790106
\(269\) −0.0336427 −0.00205123 −0.00102562 0.999999i \(-0.500326\pi\)
−0.00102562 + 0.999999i \(0.500326\pi\)
\(270\) −2.06963 −0.125954
\(271\) 10.6071 0.644338 0.322169 0.946682i \(-0.395588\pi\)
0.322169 + 0.946682i \(0.395588\pi\)
\(272\) −1.26945 −0.0769715
\(273\) −2.97308 −0.179939
\(274\) −13.4365 −0.811730
\(275\) 0.716640 0.0432150
\(276\) −4.31700 −0.259853
\(277\) −18.6526 −1.12073 −0.560363 0.828247i \(-0.689338\pi\)
−0.560363 + 0.828247i \(0.689338\pi\)
\(278\) −17.9952 −1.07928
\(279\) −6.92552 −0.414620
\(280\) −9.00906 −0.538395
\(281\) 1.56160 0.0931571 0.0465785 0.998915i \(-0.485168\pi\)
0.0465785 + 0.998915i \(0.485168\pi\)
\(282\) −10.5062 −0.625633
\(283\) 4.83058 0.287148 0.143574 0.989640i \(-0.454141\pi\)
0.143574 + 0.989640i \(0.454141\pi\)
\(284\) 9.82898 0.583242
\(285\) 0.228614 0.0135419
\(286\) −0.682998 −0.0403865
\(287\) 34.1502 2.01582
\(288\) −1.00000 −0.0589256
\(289\) −15.3885 −0.905206
\(290\) −8.45141 −0.496284
\(291\) −11.4250 −0.669743
\(292\) −14.1599 −0.828645
\(293\) 30.3558 1.77340 0.886701 0.462342i \(-0.152991\pi\)
0.886701 + 0.462342i \(0.152991\pi\)
\(294\) 11.9485 0.696851
\(295\) 15.5062 0.902803
\(296\) 7.85008 0.456277
\(297\) 1.00000 0.0580259
\(298\) 10.2098 0.591437
\(299\) −2.94850 −0.170516
\(300\) 0.716640 0.0413752
\(301\) 12.4630 0.718354
\(302\) −20.2294 −1.16407
\(303\) −10.2233 −0.587311
\(304\) 0.110461 0.00633538
\(305\) −2.06963 −0.118507
\(306\) 1.26945 0.0725694
\(307\) −25.7561 −1.46998 −0.734991 0.678077i \(-0.762813\pi\)
−0.734991 + 0.678077i \(0.762813\pi\)
\(308\) 4.35299 0.248034
\(309\) 14.9778 0.852055
\(310\) −14.3333 −0.814074
\(311\) 1.94338 0.110199 0.0550995 0.998481i \(-0.482452\pi\)
0.0550995 + 0.998481i \(0.482452\pi\)
\(312\) −0.682998 −0.0386671
\(313\) 15.6473 0.884437 0.442218 0.896907i \(-0.354192\pi\)
0.442218 + 0.896907i \(0.354192\pi\)
\(314\) −0.568594 −0.0320876
\(315\) 9.00906 0.507603
\(316\) −1.79299 −0.100864
\(317\) −23.9518 −1.34526 −0.672632 0.739977i \(-0.734836\pi\)
−0.672632 + 0.739977i \(0.734836\pi\)
\(318\) −7.98214 −0.447616
\(319\) 4.08354 0.228634
\(320\) −2.06963 −0.115696
\(321\) −11.9413 −0.666499
\(322\) 18.7919 1.04723
\(323\) −0.140225 −0.00780231
\(324\) 1.00000 0.0555556
\(325\) 0.489463 0.0271505
\(326\) 21.5071 1.19117
\(327\) 15.4609 0.854992
\(328\) 7.84523 0.433180
\(329\) 45.7332 2.52135
\(330\) 2.06963 0.113929
\(331\) 16.2564 0.893535 0.446767 0.894650i \(-0.352575\pi\)
0.446767 + 0.894650i \(0.352575\pi\)
\(332\) 10.9592 0.601462
\(333\) −7.85008 −0.430182
\(334\) 3.49709 0.191352
\(335\) 26.7698 1.46259
\(336\) 4.35299 0.237475
\(337\) −8.57110 −0.466898 −0.233449 0.972369i \(-0.575001\pi\)
−0.233449 + 0.972369i \(0.575001\pi\)
\(338\) 12.5335 0.681733
\(339\) 7.82944 0.425237
\(340\) 2.62728 0.142484
\(341\) 6.92552 0.375038
\(342\) −0.110461 −0.00597306
\(343\) −21.5408 −1.16309
\(344\) 2.86309 0.154367
\(345\) 8.93459 0.481022
\(346\) −23.2026 −1.24738
\(347\) 17.2704 0.927121 0.463561 0.886065i \(-0.346572\pi\)
0.463561 + 0.886065i \(0.346572\pi\)
\(348\) 4.08354 0.218901
\(349\) −29.1284 −1.55921 −0.779604 0.626272i \(-0.784580\pi\)
−0.779604 + 0.626272i \(0.784580\pi\)
\(350\) −3.11953 −0.166746
\(351\) 0.682998 0.0364557
\(352\) 1.00000 0.0533002
\(353\) 2.84476 0.151411 0.0757057 0.997130i \(-0.475879\pi\)
0.0757057 + 0.997130i \(0.475879\pi\)
\(354\) −7.49224 −0.398208
\(355\) −20.3423 −1.07966
\(356\) −1.93037 −0.102310
\(357\) −5.52589 −0.292461
\(358\) 10.7522 0.568269
\(359\) −28.7346 −1.51655 −0.758277 0.651932i \(-0.773959\pi\)
−0.758277 + 0.651932i \(0.773959\pi\)
\(360\) 2.06963 0.109079
\(361\) −18.9878 −0.999358
\(362\) 9.37831 0.492913
\(363\) −1.00000 −0.0524864
\(364\) 2.97308 0.155832
\(365\) 29.3057 1.53393
\(366\) 1.00000 0.0522708
\(367\) −19.0137 −0.992510 −0.496255 0.868177i \(-0.665292\pi\)
−0.496255 + 0.868177i \(0.665292\pi\)
\(368\) 4.31700 0.225039
\(369\) −7.84523 −0.408406
\(370\) −16.2467 −0.844628
\(371\) 34.7462 1.80393
\(372\) 6.92552 0.359072
\(373\) −15.9432 −0.825505 −0.412753 0.910843i \(-0.635433\pi\)
−0.412753 + 0.910843i \(0.635433\pi\)
\(374\) −1.26945 −0.0656415
\(375\) −11.8313 −0.610967
\(376\) 10.5062 0.541814
\(377\) 2.78905 0.143643
\(378\) −4.35299 −0.223894
\(379\) 0.698347 0.0358717 0.0179358 0.999839i \(-0.494291\pi\)
0.0179358 + 0.999839i \(0.494291\pi\)
\(380\) −0.228614 −0.0117276
\(381\) 0.0427077 0.00218798
\(382\) −13.3539 −0.683245
\(383\) 18.1588 0.927872 0.463936 0.885869i \(-0.346437\pi\)
0.463936 + 0.885869i \(0.346437\pi\)
\(384\) 1.00000 0.0510310
\(385\) −9.00906 −0.459144
\(386\) −6.59317 −0.335584
\(387\) −2.86309 −0.145539
\(388\) 11.4250 0.580014
\(389\) −4.93409 −0.250168 −0.125084 0.992146i \(-0.539920\pi\)
−0.125084 + 0.992146i \(0.539920\pi\)
\(390\) 1.41355 0.0715779
\(391\) −5.48020 −0.277146
\(392\) −11.9485 −0.603490
\(393\) −18.6791 −0.942234
\(394\) 23.0051 1.15898
\(395\) 3.71082 0.186712
\(396\) −1.00000 −0.0502519
\(397\) 32.0590 1.60899 0.804497 0.593957i \(-0.202435\pi\)
0.804497 + 0.593957i \(0.202435\pi\)
\(398\) 13.5790 0.680655
\(399\) 0.480836 0.0240719
\(400\) −0.716640 −0.0358320
\(401\) 30.7476 1.53546 0.767730 0.640774i \(-0.221386\pi\)
0.767730 + 0.640774i \(0.221386\pi\)
\(402\) −12.9346 −0.645119
\(403\) 4.73012 0.235624
\(404\) 10.2233 0.508626
\(405\) −2.06963 −0.102841
\(406\) −17.7756 −0.882188
\(407\) 7.85008 0.389114
\(408\) −1.26945 −0.0628470
\(409\) 8.61224 0.425848 0.212924 0.977069i \(-0.431701\pi\)
0.212924 + 0.977069i \(0.431701\pi\)
\(410\) −16.2367 −0.801874
\(411\) −13.4365 −0.662775
\(412\) −14.9778 −0.737902
\(413\) 32.6136 1.60481
\(414\) −4.31700 −0.212169
\(415\) −22.6814 −1.11339
\(416\) 0.682998 0.0334867
\(417\) −17.9952 −0.881226
\(418\) 0.110461 0.00540283
\(419\) −15.6180 −0.762990 −0.381495 0.924371i \(-0.624591\pi\)
−0.381495 + 0.924371i \(0.624591\pi\)
\(420\) −9.00906 −0.439597
\(421\) −23.3500 −1.13801 −0.569005 0.822334i \(-0.692672\pi\)
−0.569005 + 0.822334i \(0.692672\pi\)
\(422\) −11.9974 −0.584026
\(423\) −10.5062 −0.510827
\(424\) 7.98214 0.387647
\(425\) 0.909737 0.0441287
\(426\) 9.82898 0.476215
\(427\) −4.35299 −0.210656
\(428\) 11.9413 0.577205
\(429\) −0.682998 −0.0329754
\(430\) −5.92552 −0.285754
\(431\) 15.0898 0.726850 0.363425 0.931623i \(-0.381607\pi\)
0.363425 + 0.931623i \(0.381607\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 21.7332 1.04443 0.522215 0.852814i \(-0.325106\pi\)
0.522215 + 0.852814i \(0.325106\pi\)
\(434\) −30.1467 −1.44709
\(435\) −8.45141 −0.405214
\(436\) −15.4609 −0.740445
\(437\) 0.476861 0.0228114
\(438\) −14.1599 −0.676586
\(439\) 34.3340 1.63867 0.819336 0.573313i \(-0.194342\pi\)
0.819336 + 0.573313i \(0.194342\pi\)
\(440\) −2.06963 −0.0986656
\(441\) 11.9485 0.568976
\(442\) −0.867029 −0.0412404
\(443\) −5.05846 −0.240335 −0.120167 0.992754i \(-0.538343\pi\)
−0.120167 + 0.992754i \(0.538343\pi\)
\(444\) 7.85008 0.372548
\(445\) 3.99515 0.189388
\(446\) 8.83707 0.418447
\(447\) 10.2098 0.482906
\(448\) −4.35299 −0.205659
\(449\) −12.2145 −0.576437 −0.288218 0.957565i \(-0.593063\pi\)
−0.288218 + 0.957565i \(0.593063\pi\)
\(450\) 0.716640 0.0337827
\(451\) 7.84523 0.369418
\(452\) −7.82944 −0.368266
\(453\) −20.2294 −0.950458
\(454\) −26.6459 −1.25055
\(455\) −6.15317 −0.288465
\(456\) 0.110461 0.00517282
\(457\) −6.55628 −0.306690 −0.153345 0.988173i \(-0.549005\pi\)
−0.153345 + 0.988173i \(0.549005\pi\)
\(458\) −0.702259 −0.0328144
\(459\) 1.26945 0.0592527
\(460\) −8.93459 −0.416577
\(461\) −19.2054 −0.894485 −0.447243 0.894413i \(-0.647594\pi\)
−0.447243 + 0.894413i \(0.647594\pi\)
\(462\) 4.35299 0.202519
\(463\) −17.6468 −0.820116 −0.410058 0.912060i \(-0.634491\pi\)
−0.410058 + 0.912060i \(0.634491\pi\)
\(464\) −4.08354 −0.189574
\(465\) −14.3333 −0.664689
\(466\) 21.3697 0.989934
\(467\) 26.6882 1.23498 0.617491 0.786578i \(-0.288149\pi\)
0.617491 + 0.786578i \(0.288149\pi\)
\(468\) −0.682998 −0.0315716
\(469\) 56.3041 2.59988
\(470\) −21.7438 −1.00297
\(471\) −0.568594 −0.0261994
\(472\) 7.49224 0.344858
\(473\) 2.86309 0.131645
\(474\) −1.79299 −0.0823548
\(475\) −0.0791609 −0.00363215
\(476\) 5.52589 0.253279
\(477\) −7.98214 −0.365477
\(478\) 6.78467 0.310324
\(479\) 31.4971 1.43914 0.719569 0.694421i \(-0.244339\pi\)
0.719569 + 0.694421i \(0.244339\pi\)
\(480\) −2.06963 −0.0944652
\(481\) 5.36158 0.244467
\(482\) −2.23534 −0.101817
\(483\) 18.7919 0.855059
\(484\) 1.00000 0.0454545
\(485\) −23.6454 −1.07368
\(486\) 1.00000 0.0453609
\(487\) 24.8696 1.12695 0.563475 0.826133i \(-0.309464\pi\)
0.563475 + 0.826133i \(0.309464\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 21.5071 0.972586
\(490\) 24.7290 1.11714
\(491\) −32.0193 −1.44501 −0.722506 0.691364i \(-0.757010\pi\)
−0.722506 + 0.691364i \(0.757010\pi\)
\(492\) 7.84523 0.353690
\(493\) 5.18384 0.233468
\(494\) 0.0754447 0.00339442
\(495\) 2.06963 0.0930229
\(496\) −6.92552 −0.310965
\(497\) −42.7854 −1.91919
\(498\) 10.9592 0.491092
\(499\) 35.7304 1.59951 0.799756 0.600325i \(-0.204962\pi\)
0.799756 + 0.600325i \(0.204962\pi\)
\(500\) 11.8313 0.529113
\(501\) 3.49709 0.156239
\(502\) −12.8290 −0.572585
\(503\) −31.5329 −1.40598 −0.702991 0.711199i \(-0.748153\pi\)
−0.702991 + 0.711199i \(0.748153\pi\)
\(504\) 4.35299 0.193897
\(505\) −21.1584 −0.941534
\(506\) 4.31700 0.191914
\(507\) 12.5335 0.556633
\(508\) −0.0427077 −0.00189485
\(509\) 24.1026 1.06833 0.534164 0.845381i \(-0.320626\pi\)
0.534164 + 0.845381i \(0.320626\pi\)
\(510\) 2.62728 0.116338
\(511\) 61.6378 2.72670
\(512\) −1.00000 −0.0441942
\(513\) −0.110461 −0.00487698
\(514\) 16.4533 0.725723
\(515\) 30.9984 1.36595
\(516\) 2.86309 0.126040
\(517\) 10.5062 0.462060
\(518\) −34.1713 −1.50140
\(519\) −23.2026 −1.01848
\(520\) −1.41355 −0.0619883
\(521\) 27.4191 1.20125 0.600627 0.799529i \(-0.294918\pi\)
0.600627 + 0.799529i \(0.294918\pi\)
\(522\) 4.08354 0.178732
\(523\) −30.8643 −1.34960 −0.674800 0.738001i \(-0.735770\pi\)
−0.674800 + 0.738001i \(0.735770\pi\)
\(524\) 18.6791 0.815998
\(525\) −3.11953 −0.136147
\(526\) 3.91114 0.170534
\(527\) 8.79158 0.382967
\(528\) 1.00000 0.0435194
\(529\) −4.36349 −0.189717
\(530\) −16.5201 −0.717586
\(531\) −7.49224 −0.325136
\(532\) −0.480836 −0.0208469
\(533\) 5.35827 0.232093
\(534\) −1.93037 −0.0835354
\(535\) −24.7141 −1.06848
\(536\) 12.9346 0.558689
\(537\) 10.7522 0.463990
\(538\) 0.0336427 0.00145044
\(539\) −11.9485 −0.514658
\(540\) 2.06963 0.0890626
\(541\) 30.3344 1.30418 0.652090 0.758142i \(-0.273893\pi\)
0.652090 + 0.758142i \(0.273893\pi\)
\(542\) −10.6071 −0.455616
\(543\) 9.37831 0.402462
\(544\) 1.26945 0.0544271
\(545\) 31.9984 1.37066
\(546\) 2.97308 0.127236
\(547\) 31.8114 1.36016 0.680079 0.733139i \(-0.261945\pi\)
0.680079 + 0.733139i \(0.261945\pi\)
\(548\) 13.4365 0.573980
\(549\) 1.00000 0.0426790
\(550\) −0.716640 −0.0305576
\(551\) −0.451073 −0.0192163
\(552\) 4.31700 0.183744
\(553\) 7.80487 0.331897
\(554\) 18.6526 0.792473
\(555\) −16.2467 −0.689636
\(556\) 17.9952 0.763164
\(557\) −29.2482 −1.23929 −0.619643 0.784884i \(-0.712722\pi\)
−0.619643 + 0.784884i \(0.712722\pi\)
\(558\) 6.92552 0.293181
\(559\) 1.95548 0.0827080
\(560\) 9.00906 0.380702
\(561\) −1.26945 −0.0535961
\(562\) −1.56160 −0.0658720
\(563\) −2.34530 −0.0988425 −0.0494212 0.998778i \(-0.515738\pi\)
−0.0494212 + 0.998778i \(0.515738\pi\)
\(564\) 10.5062 0.442389
\(565\) 16.2040 0.681709
\(566\) −4.83058 −0.203044
\(567\) −4.35299 −0.182808
\(568\) −9.82898 −0.412415
\(569\) 14.9966 0.628688 0.314344 0.949309i \(-0.398215\pi\)
0.314344 + 0.949309i \(0.398215\pi\)
\(570\) −0.228614 −0.00957557
\(571\) 2.92018 0.122206 0.0611028 0.998131i \(-0.480538\pi\)
0.0611028 + 0.998131i \(0.480538\pi\)
\(572\) 0.682998 0.0285576
\(573\) −13.3539 −0.557867
\(574\) −34.1502 −1.42540
\(575\) −3.09374 −0.129018
\(576\) 1.00000 0.0416667
\(577\) −19.1315 −0.796456 −0.398228 0.917286i \(-0.630375\pi\)
−0.398228 + 0.917286i \(0.630375\pi\)
\(578\) 15.3885 0.640077
\(579\) −6.59317 −0.274003
\(580\) 8.45141 0.350926
\(581\) −47.7051 −1.97914
\(582\) 11.4250 0.473580
\(583\) 7.98214 0.330587
\(584\) 14.1599 0.585940
\(585\) 1.41355 0.0584431
\(586\) −30.3558 −1.25399
\(587\) −25.9773 −1.07220 −0.536099 0.844155i \(-0.680103\pi\)
−0.536099 + 0.844155i \(0.680103\pi\)
\(588\) −11.9485 −0.492748
\(589\) −0.765002 −0.0315213
\(590\) −15.5062 −0.638378
\(591\) 23.0051 0.946304
\(592\) −7.85008 −0.322636
\(593\) −10.2954 −0.422783 −0.211391 0.977402i \(-0.567799\pi\)
−0.211391 + 0.977402i \(0.567799\pi\)
\(594\) −1.00000 −0.0410305
\(595\) −11.4365 −0.468852
\(596\) −10.2098 −0.418209
\(597\) 13.5790 0.555753
\(598\) 2.94850 0.120573
\(599\) 25.8919 1.05791 0.528956 0.848649i \(-0.322584\pi\)
0.528956 + 0.848649i \(0.322584\pi\)
\(600\) −0.716640 −0.0292567
\(601\) 27.9685 1.14086 0.570429 0.821347i \(-0.306777\pi\)
0.570429 + 0.821347i \(0.306777\pi\)
\(602\) −12.4630 −0.507953
\(603\) −12.9346 −0.526737
\(604\) 20.2294 0.823121
\(605\) −2.06963 −0.0841423
\(606\) 10.2233 0.415292
\(607\) 1.54043 0.0625241 0.0312621 0.999511i \(-0.490047\pi\)
0.0312621 + 0.999511i \(0.490047\pi\)
\(608\) −0.110461 −0.00447979
\(609\) −17.7756 −0.720304
\(610\) 2.06963 0.0837968
\(611\) 7.17568 0.290297
\(612\) −1.26945 −0.0513143
\(613\) 28.0848 1.13433 0.567166 0.823603i \(-0.308040\pi\)
0.567166 + 0.823603i \(0.308040\pi\)
\(614\) 25.7561 1.03943
\(615\) −16.2367 −0.654727
\(616\) −4.35299 −0.175387
\(617\) 4.66464 0.187791 0.0938957 0.995582i \(-0.470068\pi\)
0.0938957 + 0.995582i \(0.470068\pi\)
\(618\) −14.9778 −0.602494
\(619\) 18.7803 0.754844 0.377422 0.926041i \(-0.376811\pi\)
0.377422 + 0.926041i \(0.376811\pi\)
\(620\) 14.3333 0.575637
\(621\) −4.31700 −0.173235
\(622\) −1.94338 −0.0779224
\(623\) 8.40289 0.336655
\(624\) 0.682998 0.0273418
\(625\) −20.9032 −0.836129
\(626\) −15.6473 −0.625391
\(627\) 0.110461 0.00441139
\(628\) 0.568594 0.0226894
\(629\) 9.96526 0.397341
\(630\) −9.00906 −0.358930
\(631\) −25.8058 −1.02731 −0.513657 0.857996i \(-0.671710\pi\)
−0.513657 + 0.857996i \(0.671710\pi\)
\(632\) 1.79299 0.0713213
\(633\) −11.9974 −0.476855
\(634\) 23.9518 0.951246
\(635\) 0.0883890 0.00350761
\(636\) 7.98214 0.316513
\(637\) −8.16080 −0.323343
\(638\) −4.08354 −0.161669
\(639\) 9.82898 0.388828
\(640\) 2.06963 0.0818092
\(641\) −38.4359 −1.51813 −0.759063 0.651017i \(-0.774343\pi\)
−0.759063 + 0.651017i \(0.774343\pi\)
\(642\) 11.9413 0.471286
\(643\) −37.0553 −1.46132 −0.730659 0.682742i \(-0.760787\pi\)
−0.730659 + 0.682742i \(0.760787\pi\)
\(644\) −18.7919 −0.740503
\(645\) −5.92552 −0.233317
\(646\) 0.140225 0.00551706
\(647\) 35.1722 1.38276 0.691382 0.722490i \(-0.257002\pi\)
0.691382 + 0.722490i \(0.257002\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 7.49224 0.294096
\(650\) −0.489463 −0.0191983
\(651\) −30.1467 −1.18154
\(652\) −21.5071 −0.842284
\(653\) 38.1337 1.49229 0.746144 0.665784i \(-0.231903\pi\)
0.746144 + 0.665784i \(0.231903\pi\)
\(654\) −15.4609 −0.604571
\(655\) −38.6587 −1.51052
\(656\) −7.84523 −0.306305
\(657\) −14.1599 −0.552430
\(658\) −45.7332 −1.78286
\(659\) 23.5960 0.919169 0.459584 0.888134i \(-0.347998\pi\)
0.459584 + 0.888134i \(0.347998\pi\)
\(660\) −2.06963 −0.0805602
\(661\) −27.3960 −1.06558 −0.532790 0.846248i \(-0.678856\pi\)
−0.532790 + 0.846248i \(0.678856\pi\)
\(662\) −16.2564 −0.631824
\(663\) −0.867029 −0.0336726
\(664\) −10.9592 −0.425298
\(665\) 0.995152 0.0385903
\(666\) 7.85008 0.304184
\(667\) −17.6287 −0.682584
\(668\) −3.49709 −0.135307
\(669\) 8.83707 0.341661
\(670\) −26.7698 −1.03421
\(671\) −1.00000 −0.0386046
\(672\) −4.35299 −0.167920
\(673\) 43.6737 1.68350 0.841749 0.539868i \(-0.181526\pi\)
0.841749 + 0.539868i \(0.181526\pi\)
\(674\) 8.57110 0.330146
\(675\) 0.716640 0.0275835
\(676\) −12.5335 −0.482058
\(677\) −9.67437 −0.371816 −0.185908 0.982567i \(-0.559523\pi\)
−0.185908 + 0.982567i \(0.559523\pi\)
\(678\) −7.82944 −0.300688
\(679\) −49.7327 −1.90857
\(680\) −2.62728 −0.100752
\(681\) −26.6459 −1.02107
\(682\) −6.92552 −0.265192
\(683\) −13.9693 −0.534520 −0.267260 0.963624i \(-0.586118\pi\)
−0.267260 + 0.963624i \(0.586118\pi\)
\(684\) 0.110461 0.00422359
\(685\) −27.8086 −1.06251
\(686\) 21.5408 0.822430
\(687\) −0.702259 −0.0267929
\(688\) −2.86309 −0.109154
\(689\) 5.45178 0.207696
\(690\) −8.93459 −0.340134
\(691\) 22.6619 0.862101 0.431050 0.902328i \(-0.358143\pi\)
0.431050 + 0.902328i \(0.358143\pi\)
\(692\) 23.2026 0.882030
\(693\) 4.35299 0.165356
\(694\) −17.2704 −0.655574
\(695\) −37.2433 −1.41272
\(696\) −4.08354 −0.154786
\(697\) 9.95910 0.377228
\(698\) 29.1284 1.10253
\(699\) 21.3697 0.808278
\(700\) 3.11953 0.117907
\(701\) 13.6874 0.516965 0.258483 0.966016i \(-0.416778\pi\)
0.258483 + 0.966016i \(0.416778\pi\)
\(702\) −0.682998 −0.0257781
\(703\) −0.867129 −0.0327044
\(704\) −1.00000 −0.0376889
\(705\) −21.7438 −0.818920
\(706\) −2.84476 −0.107064
\(707\) −44.5017 −1.67366
\(708\) 7.49224 0.281576
\(709\) 13.4655 0.505707 0.252853 0.967505i \(-0.418631\pi\)
0.252853 + 0.967505i \(0.418631\pi\)
\(710\) 20.3423 0.763434
\(711\) −1.79299 −0.0672424
\(712\) 1.93037 0.0723438
\(713\) −29.8975 −1.11967
\(714\) 5.52589 0.206801
\(715\) −1.41355 −0.0528638
\(716\) −10.7522 −0.401827
\(717\) 6.78467 0.253378
\(718\) 28.7346 1.07237
\(719\) −22.8197 −0.851031 −0.425515 0.904951i \(-0.639907\pi\)
−0.425515 + 0.904951i \(0.639907\pi\)
\(720\) −2.06963 −0.0771305
\(721\) 65.1980 2.42810
\(722\) 18.9878 0.706653
\(723\) −2.23534 −0.0831330
\(724\) −9.37831 −0.348542
\(725\) 2.92643 0.108685
\(726\) 1.00000 0.0371135
\(727\) −46.8673 −1.73821 −0.869106 0.494626i \(-0.835305\pi\)
−0.869106 + 0.494626i \(0.835305\pi\)
\(728\) −2.97308 −0.110190
\(729\) 1.00000 0.0370370
\(730\) −29.3057 −1.08465
\(731\) 3.63454 0.134428
\(732\) −1.00000 −0.0369611
\(733\) −48.7433 −1.80038 −0.900188 0.435503i \(-0.856571\pi\)
−0.900188 + 0.435503i \(0.856571\pi\)
\(734\) 19.0137 0.701810
\(735\) 24.7290 0.912141
\(736\) −4.31700 −0.159127
\(737\) 12.9346 0.476452
\(738\) 7.84523 0.288787
\(739\) −16.1228 −0.593086 −0.296543 0.955020i \(-0.595834\pi\)
−0.296543 + 0.955020i \(0.595834\pi\)
\(740\) 16.2467 0.597242
\(741\) 0.0754447 0.00277153
\(742\) −34.7462 −1.27557
\(743\) −31.1003 −1.14096 −0.570480 0.821312i \(-0.693243\pi\)
−0.570480 + 0.821312i \(0.693243\pi\)
\(744\) −6.92552 −0.253902
\(745\) 21.1305 0.774160
\(746\) 15.9432 0.583720
\(747\) 10.9592 0.400975
\(748\) 1.26945 0.0464156
\(749\) −51.9804 −1.89932
\(750\) 11.8313 0.432019
\(751\) 18.2110 0.664529 0.332264 0.943186i \(-0.392187\pi\)
0.332264 + 0.943186i \(0.392187\pi\)
\(752\) −10.5062 −0.383120
\(753\) −12.8290 −0.467514
\(754\) −2.78905 −0.101571
\(755\) −41.8672 −1.52370
\(756\) 4.35299 0.158317
\(757\) −13.7746 −0.500648 −0.250324 0.968162i \(-0.580537\pi\)
−0.250324 + 0.968162i \(0.580537\pi\)
\(758\) −0.698347 −0.0253651
\(759\) 4.31700 0.156697
\(760\) 0.228614 0.00829268
\(761\) −39.2769 −1.42379 −0.711893 0.702287i \(-0.752162\pi\)
−0.711893 + 0.702287i \(0.752162\pi\)
\(762\) −0.0427077 −0.00154714
\(763\) 67.3013 2.43647
\(764\) 13.3539 0.483127
\(765\) 2.62728 0.0949896
\(766\) −18.1588 −0.656105
\(767\) 5.11718 0.184771
\(768\) −1.00000 −0.0360844
\(769\) −6.56137 −0.236609 −0.118305 0.992977i \(-0.537746\pi\)
−0.118305 + 0.992977i \(0.537746\pi\)
\(770\) 9.00906 0.324664
\(771\) 16.4533 0.592551
\(772\) 6.59317 0.237293
\(773\) −49.9381 −1.79615 −0.898075 0.439843i \(-0.855034\pi\)
−0.898075 + 0.439843i \(0.855034\pi\)
\(774\) 2.86309 0.102912
\(775\) 4.96311 0.178280
\(776\) −11.4250 −0.410132
\(777\) −34.1713 −1.22589
\(778\) 4.93409 0.176896
\(779\) −0.866594 −0.0310489
\(780\) −1.41355 −0.0506132
\(781\) −9.82898 −0.351708
\(782\) 5.48020 0.195972
\(783\) 4.08354 0.145934
\(784\) 11.9485 0.426732
\(785\) −1.17678 −0.0420010
\(786\) 18.6791 0.666260
\(787\) −21.8859 −0.780148 −0.390074 0.920783i \(-0.627551\pi\)
−0.390074 + 0.920783i \(0.627551\pi\)
\(788\) −23.0051 −0.819524
\(789\) 3.91114 0.139240
\(790\) −3.71082 −0.132025
\(791\) 34.0815 1.21180
\(792\) 1.00000 0.0355335
\(793\) −0.682998 −0.0242540
\(794\) −32.0590 −1.13773
\(795\) −16.5201 −0.585906
\(796\) −13.5790 −0.481296
\(797\) 24.0665 0.852479 0.426239 0.904610i \(-0.359838\pi\)
0.426239 + 0.904610i \(0.359838\pi\)
\(798\) −0.480836 −0.0170214
\(799\) 13.3370 0.471829
\(800\) 0.716640 0.0253371
\(801\) −1.93037 −0.0682063
\(802\) −30.7476 −1.08573
\(803\) 14.1599 0.499692
\(804\) 12.9346 0.456168
\(805\) 38.8922 1.37077
\(806\) −4.73012 −0.166611
\(807\) 0.0336427 0.00118428
\(808\) −10.2233 −0.359653
\(809\) −40.7754 −1.43359 −0.716793 0.697286i \(-0.754391\pi\)
−0.716793 + 0.697286i \(0.754391\pi\)
\(810\) 2.06963 0.0727193
\(811\) 38.6449 1.35701 0.678504 0.734597i \(-0.262629\pi\)
0.678504 + 0.734597i \(0.262629\pi\)
\(812\) 17.7756 0.623801
\(813\) −10.6071 −0.372009
\(814\) −7.85008 −0.275145
\(815\) 44.5117 1.55918
\(816\) 1.26945 0.0444395
\(817\) −0.316260 −0.0110645
\(818\) −8.61224 −0.301120
\(819\) 2.97308 0.103888
\(820\) 16.2367 0.567010
\(821\) −13.3780 −0.466896 −0.233448 0.972369i \(-0.575001\pi\)
−0.233448 + 0.972369i \(0.575001\pi\)
\(822\) 13.4365 0.468653
\(823\) 0.496155 0.0172949 0.00864745 0.999963i \(-0.497247\pi\)
0.00864745 + 0.999963i \(0.497247\pi\)
\(824\) 14.9778 0.521775
\(825\) −0.716640 −0.0249502
\(826\) −32.6136 −1.13477
\(827\) 8.46044 0.294198 0.147099 0.989122i \(-0.453006\pi\)
0.147099 + 0.989122i \(0.453006\pi\)
\(828\) 4.31700 0.150026
\(829\) −37.2678 −1.29436 −0.647182 0.762335i \(-0.724053\pi\)
−0.647182 + 0.762335i \(0.724053\pi\)
\(830\) 22.6814 0.787283
\(831\) 18.6526 0.647052
\(832\) −0.682998 −0.0236787
\(833\) −15.1680 −0.525540
\(834\) 17.9952 0.623121
\(835\) 7.23768 0.250470
\(836\) −0.110461 −0.00382038
\(837\) 6.92552 0.239381
\(838\) 15.6180 0.539516
\(839\) −21.8698 −0.755030 −0.377515 0.926004i \(-0.623221\pi\)
−0.377515 + 0.926004i \(0.623221\pi\)
\(840\) 9.00906 0.310842
\(841\) −12.3247 −0.424989
\(842\) 23.3500 0.804695
\(843\) −1.56160 −0.0537843
\(844\) 11.9974 0.412969
\(845\) 25.9397 0.892353
\(846\) 10.5062 0.361209
\(847\) −4.35299 −0.149570
\(848\) −7.98214 −0.274108
\(849\) −4.83058 −0.165785
\(850\) −0.909737 −0.0312037
\(851\) −33.8888 −1.16169
\(852\) −9.82898 −0.336735
\(853\) 37.1453 1.27183 0.635915 0.771759i \(-0.280623\pi\)
0.635915 + 0.771759i \(0.280623\pi\)
\(854\) 4.35299 0.148956
\(855\) −0.228614 −0.00781842
\(856\) −11.9413 −0.408146
\(857\) −5.46439 −0.186660 −0.0933299 0.995635i \(-0.529751\pi\)
−0.0933299 + 0.995635i \(0.529751\pi\)
\(858\) 0.682998 0.0233172
\(859\) 13.3188 0.454432 0.227216 0.973844i \(-0.427038\pi\)
0.227216 + 0.973844i \(0.427038\pi\)
\(860\) 5.92552 0.202059
\(861\) −34.1502 −1.16384
\(862\) −15.0898 −0.513960
\(863\) 22.6823 0.772115 0.386057 0.922475i \(-0.373837\pi\)
0.386057 + 0.922475i \(0.373837\pi\)
\(864\) 1.00000 0.0340207
\(865\) −48.0207 −1.63275
\(866\) −21.7332 −0.738523
\(867\) 15.3885 0.522621
\(868\) 30.1467 1.02325
\(869\) 1.79299 0.0608230
\(870\) 8.45141 0.286530
\(871\) 8.83429 0.299339
\(872\) 15.4609 0.523574
\(873\) 11.4250 0.386676
\(874\) −0.476861 −0.0161301
\(875\) −51.5016 −1.74107
\(876\) 14.1599 0.478418
\(877\) −52.7402 −1.78091 −0.890455 0.455071i \(-0.849614\pi\)
−0.890455 + 0.455071i \(0.849614\pi\)
\(878\) −34.3340 −1.15872
\(879\) −30.3558 −1.02387
\(880\) 2.06963 0.0697671
\(881\) 22.8631 0.770279 0.385139 0.922858i \(-0.374153\pi\)
0.385139 + 0.922858i \(0.374153\pi\)
\(882\) −11.9485 −0.402327
\(883\) 18.7989 0.632632 0.316316 0.948654i \(-0.397554\pi\)
0.316316 + 0.948654i \(0.397554\pi\)
\(884\) 0.867029 0.0291613
\(885\) −15.5062 −0.521234
\(886\) 5.05846 0.169942
\(887\) 2.53661 0.0851712 0.0425856 0.999093i \(-0.486440\pi\)
0.0425856 + 0.999093i \(0.486440\pi\)
\(888\) −7.85008 −0.263431
\(889\) 0.185906 0.00623508
\(890\) −3.99515 −0.133918
\(891\) −1.00000 −0.0335013
\(892\) −8.83707 −0.295887
\(893\) −1.16052 −0.0388354
\(894\) −10.2098 −0.341466
\(895\) 22.2530 0.743835
\(896\) 4.35299 0.145423
\(897\) 2.94850 0.0984476
\(898\) 12.2145 0.407602
\(899\) 28.2807 0.943213
\(900\) −0.716640 −0.0238880
\(901\) 10.1329 0.337576
\(902\) −7.84523 −0.261218
\(903\) −12.4630 −0.414742
\(904\) 7.82944 0.260403
\(905\) 19.4096 0.645197
\(906\) 20.2294 0.672075
\(907\) 2.23484 0.0742065 0.0371032 0.999311i \(-0.488187\pi\)
0.0371032 + 0.999311i \(0.488187\pi\)
\(908\) 26.6459 0.884275
\(909\) 10.2233 0.339084
\(910\) 6.15317 0.203976
\(911\) −41.2937 −1.36812 −0.684060 0.729426i \(-0.739787\pi\)
−0.684060 + 0.729426i \(0.739787\pi\)
\(912\) −0.110461 −0.00365774
\(913\) −10.9592 −0.362695
\(914\) 6.55628 0.216862
\(915\) 2.06963 0.0684198
\(916\) 0.702259 0.0232033
\(917\) −81.3097 −2.68508
\(918\) −1.26945 −0.0418980
\(919\) −12.5914 −0.415352 −0.207676 0.978198i \(-0.566590\pi\)
−0.207676 + 0.978198i \(0.566590\pi\)
\(920\) 8.93459 0.294565
\(921\) 25.7561 0.848694
\(922\) 19.2054 0.632496
\(923\) −6.71317 −0.220967
\(924\) −4.35299 −0.143203
\(925\) 5.62568 0.184971
\(926\) 17.6468 0.579909
\(927\) −14.9778 −0.491934
\(928\) 4.08354 0.134049
\(929\) 8.27151 0.271380 0.135690 0.990751i \(-0.456675\pi\)
0.135690 + 0.990751i \(0.456675\pi\)
\(930\) 14.3333 0.470006
\(931\) 1.31985 0.0432562
\(932\) −21.3697 −0.699989
\(933\) −1.94338 −0.0636234
\(934\) −26.6882 −0.873264
\(935\) −2.62728 −0.0859213
\(936\) 0.682998 0.0223245
\(937\) 11.1799 0.365230 0.182615 0.983184i \(-0.441544\pi\)
0.182615 + 0.983184i \(0.441544\pi\)
\(938\) −56.3041 −1.83839
\(939\) −15.6473 −0.510630
\(940\) 21.7438 0.709206
\(941\) 54.9239 1.79047 0.895234 0.445597i \(-0.147008\pi\)
0.895234 + 0.445597i \(0.147008\pi\)
\(942\) 0.568594 0.0185258
\(943\) −33.8679 −1.10289
\(944\) −7.49224 −0.243852
\(945\) −9.00906 −0.293065
\(946\) −2.86309 −0.0930870
\(947\) −2.46271 −0.0800272 −0.0400136 0.999199i \(-0.512740\pi\)
−0.0400136 + 0.999199i \(0.512740\pi\)
\(948\) 1.79299 0.0582336
\(949\) 9.67117 0.313939
\(950\) 0.0791609 0.00256832
\(951\) 23.9518 0.776689
\(952\) −5.52589 −0.179095
\(953\) −15.5621 −0.504106 −0.252053 0.967713i \(-0.581106\pi\)
−0.252053 + 0.967713i \(0.581106\pi\)
\(954\) 7.98214 0.258431
\(955\) −27.6376 −0.894331
\(956\) −6.78467 −0.219432
\(957\) −4.08354 −0.132002
\(958\) −31.4971 −1.01762
\(959\) −58.4890 −1.88871
\(960\) 2.06963 0.0667970
\(961\) 16.9629 0.547190
\(962\) −5.36158 −0.172864
\(963\) 11.9413 0.384803
\(964\) 2.23534 0.0719953
\(965\) −13.6454 −0.439261
\(966\) −18.7919 −0.604618
\(967\) −9.25738 −0.297697 −0.148849 0.988860i \(-0.547557\pi\)
−0.148849 + 0.988860i \(0.547557\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0.140225 0.00450466
\(970\) 23.6454 0.759208
\(971\) 2.88863 0.0927006 0.0463503 0.998925i \(-0.485241\pi\)
0.0463503 + 0.998925i \(0.485241\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −78.3327 −2.51123
\(974\) −24.8696 −0.796874
\(975\) −0.489463 −0.0156754
\(976\) 1.00000 0.0320092
\(977\) −20.8733 −0.667795 −0.333898 0.942609i \(-0.608364\pi\)
−0.333898 + 0.942609i \(0.608364\pi\)
\(978\) −21.5071 −0.687722
\(979\) 1.93037 0.0616950
\(980\) −24.7290 −0.789937
\(981\) −15.4609 −0.493630
\(982\) 32.0193 1.02178
\(983\) −17.1706 −0.547658 −0.273829 0.961778i \(-0.588290\pi\)
−0.273829 + 0.961778i \(0.588290\pi\)
\(984\) −7.84523 −0.250097
\(985\) 47.6120 1.51705
\(986\) −5.18384 −0.165087
\(987\) −45.7332 −1.45570
\(988\) −0.0754447 −0.00240022
\(989\) −12.3600 −0.393024
\(990\) −2.06963 −0.0657771
\(991\) −45.3067 −1.43921 −0.719607 0.694382i \(-0.755678\pi\)
−0.719607 + 0.694382i \(0.755678\pi\)
\(992\) 6.92552 0.219886
\(993\) −16.2564 −0.515883
\(994\) 42.7854 1.35707
\(995\) 28.1035 0.890942
\(996\) −10.9592 −0.347254
\(997\) −37.3621 −1.18327 −0.591634 0.806207i \(-0.701517\pi\)
−0.591634 + 0.806207i \(0.701517\pi\)
\(998\) −35.7304 −1.13103
\(999\) 7.85008 0.248366
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 4026.2.a.q.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4026.2.a.q.1.1 4 1.1 even 1 trivial