Properties

Label 4017.2.a.e.1.2
Level $4017$
Weight $2$
Character 4017.1
Self dual yes
Analytic conductor $32.076$
Analytic rank $1$
Dimension $16$
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] = [4017,2,Mod(1,4017)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4017, 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("4017.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4017 = 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4017.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: \(32.0759064919\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 21 x^{14} - 3 x^{13} + 177 x^{12} + 45 x^{11} - 763 x^{10} - 251 x^{9} + 1771 x^{8} + 639 x^{7} - 2118 x^{6} - 710 x^{5} + 1113 x^{4} + 243 x^{3} - 183 x^{2} - 10 x + 7 \) 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 1.2
Root \(2.37644\) of defining polynomial
Character \(\chi\) \(=\) 4017.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.37644 q^{2} +1.00000 q^{3} +3.64749 q^{4} +1.54914 q^{5} -2.37644 q^{6} +1.30434 q^{7} -3.91517 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.37644 q^{2} +1.00000 q^{3} +3.64749 q^{4} +1.54914 q^{5} -2.37644 q^{6} +1.30434 q^{7} -3.91517 q^{8} +1.00000 q^{9} -3.68145 q^{10} +0.998212 q^{11} +3.64749 q^{12} -1.00000 q^{13} -3.09970 q^{14} +1.54914 q^{15} +2.00920 q^{16} -6.18975 q^{17} -2.37644 q^{18} -3.83531 q^{19} +5.65049 q^{20} +1.30434 q^{21} -2.37219 q^{22} +3.79897 q^{23} -3.91517 q^{24} -2.60015 q^{25} +2.37644 q^{26} +1.00000 q^{27} +4.75758 q^{28} +2.72405 q^{29} -3.68145 q^{30} -3.88694 q^{31} +3.05559 q^{32} +0.998212 q^{33} +14.7096 q^{34} +2.02062 q^{35} +3.64749 q^{36} -4.36970 q^{37} +9.11440 q^{38} -1.00000 q^{39} -6.06516 q^{40} +1.99305 q^{41} -3.09970 q^{42} -4.74274 q^{43} +3.64097 q^{44} +1.54914 q^{45} -9.02803 q^{46} -10.7060 q^{47} +2.00920 q^{48} -5.29868 q^{49} +6.17912 q^{50} -6.18975 q^{51} -3.64749 q^{52} +1.58834 q^{53} -2.37644 q^{54} +1.54637 q^{55} -5.10673 q^{56} -3.83531 q^{57} -6.47355 q^{58} -9.55468 q^{59} +5.65049 q^{60} +0.815146 q^{61} +9.23709 q^{62} +1.30434 q^{63} -11.2798 q^{64} -1.54914 q^{65} -2.37219 q^{66} -8.73068 q^{67} -22.5770 q^{68} +3.79897 q^{69} -4.80189 q^{70} +11.8921 q^{71} -3.91517 q^{72} -3.23415 q^{73} +10.3843 q^{74} -2.60015 q^{75} -13.9892 q^{76} +1.30201 q^{77} +2.37644 q^{78} -8.61766 q^{79} +3.11253 q^{80} +1.00000 q^{81} -4.73638 q^{82} +7.32306 q^{83} +4.75758 q^{84} -9.58881 q^{85} +11.2709 q^{86} +2.72405 q^{87} -3.90816 q^{88} -12.5308 q^{89} -3.68145 q^{90} -1.30434 q^{91} +13.8567 q^{92} -3.88694 q^{93} +25.4422 q^{94} -5.94145 q^{95} +3.05559 q^{96} +7.89294 q^{97} +12.5920 q^{98} +0.998212 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{3} + 10 q^{4} - 6 q^{5} - 13 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{3} + 10 q^{4} - 6 q^{5} - 13 q^{7} - 9 q^{8} + 16 q^{9} - 8 q^{10} - 5 q^{11} + 10 q^{12} - 16 q^{13} - 8 q^{14} - 6 q^{15} - 14 q^{16} - q^{17} + 6 q^{19} - 4 q^{20} - 13 q^{21} - 11 q^{22} - 21 q^{23} - 9 q^{24} - 10 q^{25} + 16 q^{27} - 10 q^{28} - 17 q^{29} - 8 q^{30} - 33 q^{31} - 18 q^{32} - 5 q^{33} - 5 q^{34} - 4 q^{35} + 10 q^{36} - 23 q^{37} - 28 q^{38} - 16 q^{39} - 12 q^{40} + 7 q^{41} - 8 q^{42} - 33 q^{43} + 11 q^{44} - 6 q^{45} - 15 q^{46} - 13 q^{47} - 14 q^{48} - 17 q^{49} + 35 q^{50} - q^{51} - 10 q^{52} - 20 q^{53} - 54 q^{55} + 12 q^{56} + 6 q^{57} - 33 q^{58} + 6 q^{59} - 4 q^{60} - 49 q^{61} - 13 q^{62} - 13 q^{63} - 35 q^{64} + 6 q^{65} - 11 q^{66} - 4 q^{67} - 14 q^{68} - 21 q^{69} - 33 q^{70} - 29 q^{71} - 9 q^{72} - 21 q^{73} + 22 q^{74} - 10 q^{75} + 10 q^{76} - 21 q^{77} - 70 q^{79} - 8 q^{80} + 16 q^{81} - 10 q^{82} + 5 q^{83} - 10 q^{84} + 14 q^{85} + 29 q^{86} - 17 q^{87} - 45 q^{88} - 8 q^{89} - 8 q^{90} + 13 q^{91} - 29 q^{92} - 33 q^{93} + 12 q^{94} - 45 q^{95} - 18 q^{96} - 30 q^{97} + 15 q^{98} - 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.37644 −1.68040 −0.840200 0.542277i \(-0.817562\pi\)
−0.840200 + 0.542277i \(0.817562\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.64749 1.82374
\(5\) 1.54914 0.692798 0.346399 0.938087i \(-0.387404\pi\)
0.346399 + 0.938087i \(0.387404\pi\)
\(6\) −2.37644 −0.970179
\(7\) 1.30434 0.492996 0.246498 0.969143i \(-0.420720\pi\)
0.246498 + 0.969143i \(0.420720\pi\)
\(8\) −3.91517 −1.38422
\(9\) 1.00000 0.333333
\(10\) −3.68145 −1.16418
\(11\) 0.998212 0.300972 0.150486 0.988612i \(-0.451916\pi\)
0.150486 + 0.988612i \(0.451916\pi\)
\(12\) 3.64749 1.05294
\(13\) −1.00000 −0.277350
\(14\) −3.09970 −0.828430
\(15\) 1.54914 0.399987
\(16\) 2.00920 0.502299
\(17\) −6.18975 −1.50123 −0.750617 0.660737i \(-0.770244\pi\)
−0.750617 + 0.660737i \(0.770244\pi\)
\(18\) −2.37644 −0.560133
\(19\) −3.83531 −0.879880 −0.439940 0.898027i \(-0.645000\pi\)
−0.439940 + 0.898027i \(0.645000\pi\)
\(20\) 5.65049 1.26349
\(21\) 1.30434 0.284631
\(22\) −2.37219 −0.505754
\(23\) 3.79897 0.792139 0.396070 0.918220i \(-0.370374\pi\)
0.396070 + 0.918220i \(0.370374\pi\)
\(24\) −3.91517 −0.799180
\(25\) −2.60015 −0.520031
\(26\) 2.37644 0.466059
\(27\) 1.00000 0.192450
\(28\) 4.75758 0.899099
\(29\) 2.72405 0.505843 0.252921 0.967487i \(-0.418609\pi\)
0.252921 + 0.967487i \(0.418609\pi\)
\(30\) −3.68145 −0.672139
\(31\) −3.88694 −0.698114 −0.349057 0.937101i \(-0.613498\pi\)
−0.349057 + 0.937101i \(0.613498\pi\)
\(32\) 3.05559 0.540157
\(33\) 0.998212 0.173766
\(34\) 14.7096 2.52268
\(35\) 2.02062 0.341547
\(36\) 3.64749 0.607915
\(37\) −4.36970 −0.718374 −0.359187 0.933266i \(-0.616946\pi\)
−0.359187 + 0.933266i \(0.616946\pi\)
\(38\) 9.11440 1.47855
\(39\) −1.00000 −0.160128
\(40\) −6.06516 −0.958985
\(41\) 1.99305 0.311262 0.155631 0.987815i \(-0.450259\pi\)
0.155631 + 0.987815i \(0.450259\pi\)
\(42\) −3.09970 −0.478295
\(43\) −4.74274 −0.723262 −0.361631 0.932321i \(-0.617780\pi\)
−0.361631 + 0.932321i \(0.617780\pi\)
\(44\) 3.64097 0.548896
\(45\) 1.54914 0.230933
\(46\) −9.02803 −1.33111
\(47\) −10.7060 −1.56163 −0.780814 0.624764i \(-0.785195\pi\)
−0.780814 + 0.624764i \(0.785195\pi\)
\(48\) 2.00920 0.290003
\(49\) −5.29868 −0.756955
\(50\) 6.17912 0.873859
\(51\) −6.18975 −0.866738
\(52\) −3.64749 −0.505816
\(53\) 1.58834 0.218176 0.109088 0.994032i \(-0.465207\pi\)
0.109088 + 0.994032i \(0.465207\pi\)
\(54\) −2.37644 −0.323393
\(55\) 1.54637 0.208513
\(56\) −5.10673 −0.682415
\(57\) −3.83531 −0.507999
\(58\) −6.47355 −0.850018
\(59\) −9.55468 −1.24391 −0.621957 0.783051i \(-0.713662\pi\)
−0.621957 + 0.783051i \(0.713662\pi\)
\(60\) 5.65049 0.729474
\(61\) 0.815146 0.104369 0.0521844 0.998637i \(-0.483382\pi\)
0.0521844 + 0.998637i \(0.483382\pi\)
\(62\) 9.23709 1.17311
\(63\) 1.30434 0.164332
\(64\) −11.2798 −1.40998
\(65\) −1.54914 −0.192148
\(66\) −2.37219 −0.291997
\(67\) −8.73068 −1.06662 −0.533311 0.845919i \(-0.679053\pi\)
−0.533311 + 0.845919i \(0.679053\pi\)
\(68\) −22.5770 −2.73787
\(69\) 3.79897 0.457342
\(70\) −4.80189 −0.573935
\(71\) 11.8921 1.41133 0.705667 0.708544i \(-0.250647\pi\)
0.705667 + 0.708544i \(0.250647\pi\)
\(72\) −3.91517 −0.461407
\(73\) −3.23415 −0.378529 −0.189265 0.981926i \(-0.560610\pi\)
−0.189265 + 0.981926i \(0.560610\pi\)
\(74\) 10.3843 1.20716
\(75\) −2.60015 −0.300240
\(76\) −13.9892 −1.60468
\(77\) 1.30201 0.148378
\(78\) 2.37644 0.269079
\(79\) −8.61766 −0.969563 −0.484781 0.874635i \(-0.661101\pi\)
−0.484781 + 0.874635i \(0.661101\pi\)
\(80\) 3.11253 0.347992
\(81\) 1.00000 0.111111
\(82\) −4.73638 −0.523045
\(83\) 7.32306 0.803810 0.401905 0.915681i \(-0.368348\pi\)
0.401905 + 0.915681i \(0.368348\pi\)
\(84\) 4.75758 0.519095
\(85\) −9.58881 −1.04005
\(86\) 11.2709 1.21537
\(87\) 2.72405 0.292048
\(88\) −3.90816 −0.416612
\(89\) −12.5308 −1.32826 −0.664129 0.747618i \(-0.731197\pi\)
−0.664129 + 0.747618i \(0.731197\pi\)
\(90\) −3.68145 −0.388059
\(91\) −1.30434 −0.136732
\(92\) 13.8567 1.44466
\(93\) −3.88694 −0.403056
\(94\) 25.4422 2.62416
\(95\) −5.94145 −0.609580
\(96\) 3.05559 0.311860
\(97\) 7.89294 0.801407 0.400703 0.916208i \(-0.368766\pi\)
0.400703 + 0.916208i \(0.368766\pi\)
\(98\) 12.5920 1.27199
\(99\) 0.998212 0.100324
\(100\) −9.48403 −0.948403
\(101\) 10.7172 1.06640 0.533200 0.845989i \(-0.320989\pi\)
0.533200 + 0.845989i \(0.320989\pi\)
\(102\) 14.7096 1.45647
\(103\) 1.00000 0.0985329
\(104\) 3.91517 0.383914
\(105\) 2.02062 0.197192
\(106\) −3.77461 −0.366623
\(107\) −19.7317 −1.90753 −0.953766 0.300549i \(-0.902830\pi\)
−0.953766 + 0.300549i \(0.902830\pi\)
\(108\) 3.64749 0.350980
\(109\) 2.32319 0.222521 0.111260 0.993791i \(-0.464511\pi\)
0.111260 + 0.993791i \(0.464511\pi\)
\(110\) −3.67487 −0.350385
\(111\) −4.36970 −0.414754
\(112\) 2.62068 0.247631
\(113\) −14.1085 −1.32721 −0.663607 0.748081i \(-0.730975\pi\)
−0.663607 + 0.748081i \(0.730975\pi\)
\(114\) 9.11440 0.853642
\(115\) 5.88515 0.548793
\(116\) 9.93593 0.922528
\(117\) −1.00000 −0.0924500
\(118\) 22.7062 2.09027
\(119\) −8.07357 −0.740103
\(120\) −6.06516 −0.553670
\(121\) −10.0036 −0.909416
\(122\) −1.93715 −0.175381
\(123\) 1.99305 0.179707
\(124\) −14.1776 −1.27318
\(125\) −11.7737 −1.05307
\(126\) −3.09970 −0.276143
\(127\) −3.67047 −0.325702 −0.162851 0.986651i \(-0.552069\pi\)
−0.162851 + 0.986651i \(0.552069\pi\)
\(128\) 20.6947 1.82917
\(129\) −4.74274 −0.417575
\(130\) 3.68145 0.322885
\(131\) 1.83891 0.160666 0.0803330 0.996768i \(-0.474402\pi\)
0.0803330 + 0.996768i \(0.474402\pi\)
\(132\) 3.64097 0.316905
\(133\) −5.00257 −0.433777
\(134\) 20.7480 1.79235
\(135\) 1.54914 0.133329
\(136\) 24.2339 2.07804
\(137\) −4.15752 −0.355201 −0.177601 0.984103i \(-0.556834\pi\)
−0.177601 + 0.984103i \(0.556834\pi\)
\(138\) −9.02803 −0.768517
\(139\) −16.1928 −1.37346 −0.686728 0.726915i \(-0.740954\pi\)
−0.686728 + 0.726915i \(0.740954\pi\)
\(140\) 7.37018 0.622894
\(141\) −10.7060 −0.901606
\(142\) −28.2609 −2.37161
\(143\) −0.998212 −0.0834747
\(144\) 2.00920 0.167433
\(145\) 4.21994 0.350447
\(146\) 7.68579 0.636080
\(147\) −5.29868 −0.437028
\(148\) −15.9384 −1.31013
\(149\) 16.1197 1.32058 0.660288 0.751013i \(-0.270434\pi\)
0.660288 + 0.751013i \(0.270434\pi\)
\(150\) 6.17912 0.504523
\(151\) 7.13374 0.580536 0.290268 0.956945i \(-0.406256\pi\)
0.290268 + 0.956945i \(0.406256\pi\)
\(152\) 15.0159 1.21795
\(153\) −6.18975 −0.500412
\(154\) −3.09416 −0.249334
\(155\) −6.02142 −0.483652
\(156\) −3.64749 −0.292033
\(157\) −3.13762 −0.250409 −0.125205 0.992131i \(-0.539959\pi\)
−0.125205 + 0.992131i \(0.539959\pi\)
\(158\) 20.4794 1.62925
\(159\) 1.58834 0.125964
\(160\) 4.73354 0.374220
\(161\) 4.95516 0.390521
\(162\) −2.37644 −0.186711
\(163\) 5.91150 0.463024 0.231512 0.972832i \(-0.425633\pi\)
0.231512 + 0.972832i \(0.425633\pi\)
\(164\) 7.26963 0.567663
\(165\) 1.54637 0.120385
\(166\) −17.4028 −1.35072
\(167\) 24.5246 1.89777 0.948886 0.315619i \(-0.102212\pi\)
0.948886 + 0.315619i \(0.102212\pi\)
\(168\) −5.10673 −0.393992
\(169\) 1.00000 0.0769231
\(170\) 22.7873 1.74770
\(171\) −3.83531 −0.293293
\(172\) −17.2991 −1.31904
\(173\) −7.07942 −0.538239 −0.269119 0.963107i \(-0.586733\pi\)
−0.269119 + 0.963107i \(0.586733\pi\)
\(174\) −6.47355 −0.490758
\(175\) −3.39150 −0.256373
\(176\) 2.00560 0.151178
\(177\) −9.55468 −0.718174
\(178\) 29.7787 2.23200
\(179\) −6.28398 −0.469686 −0.234843 0.972033i \(-0.575458\pi\)
−0.234843 + 0.972033i \(0.575458\pi\)
\(180\) 5.65049 0.421162
\(181\) −15.9763 −1.18751 −0.593754 0.804647i \(-0.702355\pi\)
−0.593754 + 0.804647i \(0.702355\pi\)
\(182\) 3.09970 0.229765
\(183\) 0.815146 0.0602573
\(184\) −14.8736 −1.09650
\(185\) −6.76929 −0.497688
\(186\) 9.23709 0.677296
\(187\) −6.17868 −0.451830
\(188\) −39.0499 −2.84801
\(189\) 1.30434 0.0948771
\(190\) 14.1195 1.02434
\(191\) 17.1115 1.23814 0.619072 0.785334i \(-0.287509\pi\)
0.619072 + 0.785334i \(0.287509\pi\)
\(192\) −11.2798 −0.814051
\(193\) −8.19444 −0.589849 −0.294925 0.955521i \(-0.595295\pi\)
−0.294925 + 0.955521i \(0.595295\pi\)
\(194\) −18.7571 −1.34668
\(195\) −1.54914 −0.110937
\(196\) −19.3269 −1.38049
\(197\) 22.4587 1.60011 0.800057 0.599924i \(-0.204803\pi\)
0.800057 + 0.599924i \(0.204803\pi\)
\(198\) −2.37219 −0.168585
\(199\) 18.5204 1.31288 0.656439 0.754379i \(-0.272062\pi\)
0.656439 + 0.754379i \(0.272062\pi\)
\(200\) 10.1800 0.719837
\(201\) −8.73068 −0.615815
\(202\) −25.4688 −1.79198
\(203\) 3.55310 0.249378
\(204\) −22.5770 −1.58071
\(205\) 3.08752 0.215642
\(206\) −2.37644 −0.165575
\(207\) 3.79897 0.264046
\(208\) −2.00920 −0.139313
\(209\) −3.82845 −0.264819
\(210\) −4.80189 −0.331362
\(211\) 21.9114 1.50845 0.754223 0.656619i \(-0.228014\pi\)
0.754223 + 0.656619i \(0.228014\pi\)
\(212\) 5.79347 0.397897
\(213\) 11.8921 0.814834
\(214\) 46.8912 3.20542
\(215\) −7.34719 −0.501074
\(216\) −3.91517 −0.266393
\(217\) −5.06990 −0.344167
\(218\) −5.52092 −0.373924
\(219\) −3.23415 −0.218544
\(220\) 5.64038 0.380274
\(221\) 6.18975 0.416368
\(222\) 10.3843 0.696952
\(223\) 6.00131 0.401878 0.200939 0.979604i \(-0.435601\pi\)
0.200939 + 0.979604i \(0.435601\pi\)
\(224\) 3.98554 0.266295
\(225\) −2.60015 −0.173344
\(226\) 33.5280 2.23025
\(227\) 7.52318 0.499331 0.249666 0.968332i \(-0.419679\pi\)
0.249666 + 0.968332i \(0.419679\pi\)
\(228\) −13.9892 −0.926461
\(229\) −9.53118 −0.629838 −0.314919 0.949119i \(-0.601977\pi\)
−0.314919 + 0.949119i \(0.601977\pi\)
\(230\) −13.9857 −0.922191
\(231\) 1.30201 0.0856661
\(232\) −10.6651 −0.700198
\(233\) 13.0635 0.855820 0.427910 0.903821i \(-0.359250\pi\)
0.427910 + 0.903821i \(0.359250\pi\)
\(234\) 2.37644 0.155353
\(235\) −16.5851 −1.08189
\(236\) −34.8506 −2.26858
\(237\) −8.61766 −0.559777
\(238\) 19.1864 1.24367
\(239\) −2.04869 −0.132518 −0.0662592 0.997802i \(-0.521106\pi\)
−0.0662592 + 0.997802i \(0.521106\pi\)
\(240\) 3.11253 0.200913
\(241\) 0.454757 0.0292935 0.0146467 0.999893i \(-0.495338\pi\)
0.0146467 + 0.999893i \(0.495338\pi\)
\(242\) 23.7729 1.52818
\(243\) 1.00000 0.0641500
\(244\) 2.97323 0.190342
\(245\) −8.20843 −0.524417
\(246\) −4.73638 −0.301980
\(247\) 3.83531 0.244035
\(248\) 15.2180 0.966344
\(249\) 7.32306 0.464080
\(250\) 27.9796 1.76959
\(251\) 29.5491 1.86512 0.932561 0.361011i \(-0.117568\pi\)
0.932561 + 0.361011i \(0.117568\pi\)
\(252\) 4.75758 0.299700
\(253\) 3.79217 0.238412
\(254\) 8.72267 0.547309
\(255\) −9.58881 −0.600475
\(256\) −26.6202 −1.66376
\(257\) 4.57116 0.285141 0.142571 0.989785i \(-0.454463\pi\)
0.142571 + 0.989785i \(0.454463\pi\)
\(258\) 11.2709 0.701694
\(259\) −5.69959 −0.354156
\(260\) −5.65049 −0.350428
\(261\) 2.72405 0.168614
\(262\) −4.37006 −0.269983
\(263\) −8.65184 −0.533495 −0.266748 0.963766i \(-0.585949\pi\)
−0.266748 + 0.963766i \(0.585949\pi\)
\(264\) −3.90816 −0.240531
\(265\) 2.46057 0.151152
\(266\) 11.8883 0.728920
\(267\) −12.5308 −0.766870
\(268\) −31.8451 −1.94525
\(269\) −5.54940 −0.338353 −0.169176 0.985586i \(-0.554111\pi\)
−0.169176 + 0.985586i \(0.554111\pi\)
\(270\) −3.68145 −0.224046
\(271\) 21.6564 1.31553 0.657766 0.753222i \(-0.271502\pi\)
0.657766 + 0.753222i \(0.271502\pi\)
\(272\) −12.4364 −0.754069
\(273\) −1.30434 −0.0789425
\(274\) 9.88012 0.596880
\(275\) −2.59550 −0.156515
\(276\) 13.8567 0.834075
\(277\) 4.52966 0.272161 0.136080 0.990698i \(-0.456549\pi\)
0.136080 + 0.990698i \(0.456549\pi\)
\(278\) 38.4813 2.30795
\(279\) −3.88694 −0.232705
\(280\) −7.91105 −0.472776
\(281\) −11.8177 −0.704983 −0.352491 0.935815i \(-0.614665\pi\)
−0.352491 + 0.935815i \(0.614665\pi\)
\(282\) 25.4422 1.51506
\(283\) −14.2567 −0.847471 −0.423735 0.905786i \(-0.639281\pi\)
−0.423735 + 0.905786i \(0.639281\pi\)
\(284\) 43.3763 2.57391
\(285\) −5.94145 −0.351941
\(286\) 2.37219 0.140271
\(287\) 2.59963 0.153451
\(288\) 3.05559 0.180052
\(289\) 21.3130 1.25371
\(290\) −10.0285 −0.588891
\(291\) 7.89294 0.462692
\(292\) −11.7965 −0.690340
\(293\) 3.73104 0.217970 0.108985 0.994043i \(-0.465240\pi\)
0.108985 + 0.994043i \(0.465240\pi\)
\(294\) 12.5920 0.734382
\(295\) −14.8016 −0.861781
\(296\) 17.1081 0.994388
\(297\) 0.998212 0.0579221
\(298\) −38.3075 −2.21910
\(299\) −3.79897 −0.219700
\(300\) −9.48403 −0.547561
\(301\) −6.18617 −0.356565
\(302\) −16.9529 −0.975532
\(303\) 10.7172 0.615687
\(304\) −7.70589 −0.441963
\(305\) 1.26278 0.0723065
\(306\) 14.7096 0.840892
\(307\) −2.73282 −0.155970 −0.0779851 0.996955i \(-0.524849\pi\)
−0.0779851 + 0.996955i \(0.524849\pi\)
\(308\) 4.74907 0.270604
\(309\) 1.00000 0.0568880
\(310\) 14.3096 0.812729
\(311\) −13.4725 −0.763955 −0.381978 0.924172i \(-0.624757\pi\)
−0.381978 + 0.924172i \(0.624757\pi\)
\(312\) 3.91517 0.221653
\(313\) −17.5971 −0.994647 −0.497323 0.867565i \(-0.665684\pi\)
−0.497323 + 0.867565i \(0.665684\pi\)
\(314\) 7.45637 0.420787
\(315\) 2.02062 0.113849
\(316\) −31.4328 −1.76823
\(317\) 13.0410 0.732457 0.366229 0.930525i \(-0.380649\pi\)
0.366229 + 0.930525i \(0.380649\pi\)
\(318\) −3.77461 −0.211670
\(319\) 2.71917 0.152245
\(320\) −17.4741 −0.976831
\(321\) −19.7317 −1.10131
\(322\) −11.7757 −0.656232
\(323\) 23.7396 1.32091
\(324\) 3.64749 0.202638
\(325\) 2.60015 0.144231
\(326\) −14.0484 −0.778066
\(327\) 2.32319 0.128472
\(328\) −7.80313 −0.430856
\(329\) −13.9643 −0.769876
\(330\) −3.67487 −0.202295
\(331\) 29.9797 1.64783 0.823916 0.566711i \(-0.191784\pi\)
0.823916 + 0.566711i \(0.191784\pi\)
\(332\) 26.7108 1.46594
\(333\) −4.36970 −0.239458
\(334\) −58.2814 −3.18902
\(335\) −13.5251 −0.738954
\(336\) 2.62068 0.142970
\(337\) −6.99406 −0.380991 −0.190495 0.981688i \(-0.561009\pi\)
−0.190495 + 0.981688i \(0.561009\pi\)
\(338\) −2.37644 −0.129262
\(339\) −14.1085 −0.766268
\(340\) −34.9751 −1.89679
\(341\) −3.87998 −0.210113
\(342\) 9.11440 0.492850
\(343\) −16.0417 −0.866172
\(344\) 18.5686 1.00115
\(345\) 5.88515 0.316846
\(346\) 16.8239 0.904456
\(347\) −35.9879 −1.93193 −0.965965 0.258675i \(-0.916714\pi\)
−0.965965 + 0.258675i \(0.916714\pi\)
\(348\) 9.93593 0.532622
\(349\) −2.81375 −0.150617 −0.0753084 0.997160i \(-0.523994\pi\)
−0.0753084 + 0.997160i \(0.523994\pi\)
\(350\) 8.05970 0.430809
\(351\) −1.00000 −0.0533761
\(352\) 3.05012 0.162572
\(353\) 4.97986 0.265051 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(354\) 22.7062 1.20682
\(355\) 18.4226 0.977770
\(356\) −45.7058 −2.42240
\(357\) −8.07357 −0.427298
\(358\) 14.9335 0.789261
\(359\) −14.2904 −0.754221 −0.377110 0.926168i \(-0.623082\pi\)
−0.377110 + 0.926168i \(0.623082\pi\)
\(360\) −6.06516 −0.319662
\(361\) −4.29040 −0.225811
\(362\) 37.9667 1.99549
\(363\) −10.0036 −0.525051
\(364\) −4.75758 −0.249365
\(365\) −5.01017 −0.262244
\(366\) −1.93715 −0.101256
\(367\) −16.8112 −0.877539 −0.438770 0.898600i \(-0.644586\pi\)
−0.438770 + 0.898600i \(0.644586\pi\)
\(368\) 7.63287 0.397891
\(369\) 1.99305 0.103754
\(370\) 16.0869 0.836316
\(371\) 2.07175 0.107560
\(372\) −14.1776 −0.735072
\(373\) −8.90595 −0.461132 −0.230566 0.973057i \(-0.574058\pi\)
−0.230566 + 0.973057i \(0.574058\pi\)
\(374\) 14.6833 0.759255
\(375\) −11.7737 −0.607993
\(376\) 41.9157 2.16164
\(377\) −2.72405 −0.140296
\(378\) −3.09970 −0.159432
\(379\) −6.07179 −0.311887 −0.155943 0.987766i \(-0.549842\pi\)
−0.155943 + 0.987766i \(0.549842\pi\)
\(380\) −21.6714 −1.11172
\(381\) −3.67047 −0.188044
\(382\) −40.6645 −2.08058
\(383\) 9.80223 0.500871 0.250435 0.968133i \(-0.419426\pi\)
0.250435 + 0.968133i \(0.419426\pi\)
\(384\) 20.6947 1.05607
\(385\) 2.01700 0.102796
\(386\) 19.4736 0.991182
\(387\) −4.74274 −0.241087
\(388\) 28.7894 1.46156
\(389\) −26.9054 −1.36416 −0.682078 0.731279i \(-0.738924\pi\)
−0.682078 + 0.731279i \(0.738924\pi\)
\(390\) 3.68145 0.186418
\(391\) −23.5147 −1.18919
\(392\) 20.7452 1.04779
\(393\) 1.83891 0.0927606
\(394\) −53.3718 −2.68883
\(395\) −13.3500 −0.671711
\(396\) 3.64097 0.182965
\(397\) −10.1821 −0.511026 −0.255513 0.966806i \(-0.582244\pi\)
−0.255513 + 0.966806i \(0.582244\pi\)
\(398\) −44.0128 −2.20616
\(399\) −5.00257 −0.250442
\(400\) −5.22422 −0.261211
\(401\) 13.1041 0.654387 0.327193 0.944957i \(-0.393897\pi\)
0.327193 + 0.944957i \(0.393897\pi\)
\(402\) 20.7480 1.03482
\(403\) 3.88694 0.193622
\(404\) 39.0908 1.94484
\(405\) 1.54914 0.0769776
\(406\) −8.44373 −0.419055
\(407\) −4.36189 −0.216211
\(408\) 24.2339 1.19976
\(409\) 38.0991 1.88388 0.941939 0.335783i \(-0.109001\pi\)
0.941939 + 0.335783i \(0.109001\pi\)
\(410\) −7.33733 −0.362365
\(411\) −4.15752 −0.205076
\(412\) 3.64749 0.179699
\(413\) −12.4626 −0.613244
\(414\) −9.02803 −0.443704
\(415\) 11.3445 0.556878
\(416\) −3.05559 −0.149812
\(417\) −16.1928 −0.792965
\(418\) 9.09810 0.445003
\(419\) 10.3491 0.505585 0.252792 0.967521i \(-0.418651\pi\)
0.252792 + 0.967521i \(0.418651\pi\)
\(420\) 7.37018 0.359628
\(421\) −20.5052 −0.999364 −0.499682 0.866209i \(-0.666550\pi\)
−0.499682 + 0.866209i \(0.666550\pi\)
\(422\) −52.0713 −2.53479
\(423\) −10.7060 −0.520543
\(424\) −6.21863 −0.302003
\(425\) 16.0943 0.780688
\(426\) −28.2609 −1.36925
\(427\) 1.06323 0.0514534
\(428\) −71.9711 −3.47885
\(429\) −0.998212 −0.0481941
\(430\) 17.4602 0.842005
\(431\) 25.7240 1.23908 0.619540 0.784965i \(-0.287319\pi\)
0.619540 + 0.784965i \(0.287319\pi\)
\(432\) 2.00920 0.0966675
\(433\) 11.9162 0.572654 0.286327 0.958132i \(-0.407566\pi\)
0.286327 + 0.958132i \(0.407566\pi\)
\(434\) 12.0483 0.578339
\(435\) 4.21994 0.202331
\(436\) 8.47379 0.405821
\(437\) −14.5702 −0.696988
\(438\) 7.68579 0.367241
\(439\) −1.60422 −0.0765653 −0.0382826 0.999267i \(-0.512189\pi\)
−0.0382826 + 0.999267i \(0.512189\pi\)
\(440\) −6.05431 −0.288628
\(441\) −5.29868 −0.252318
\(442\) −14.7096 −0.699664
\(443\) −25.1699 −1.19586 −0.597929 0.801549i \(-0.704010\pi\)
−0.597929 + 0.801549i \(0.704010\pi\)
\(444\) −15.9384 −0.756404
\(445\) −19.4119 −0.920215
\(446\) −14.2618 −0.675315
\(447\) 16.1197 0.762435
\(448\) −14.7128 −0.695114
\(449\) −15.1160 −0.713368 −0.356684 0.934225i \(-0.616093\pi\)
−0.356684 + 0.934225i \(0.616093\pi\)
\(450\) 6.17912 0.291286
\(451\) 1.98949 0.0936813
\(452\) −51.4605 −2.42050
\(453\) 7.13374 0.335172
\(454\) −17.8784 −0.839076
\(455\) −2.02062 −0.0947280
\(456\) 15.0159 0.703183
\(457\) −17.5065 −0.818921 −0.409460 0.912328i \(-0.634283\pi\)
−0.409460 + 0.912328i \(0.634283\pi\)
\(458\) 22.6503 1.05838
\(459\) −6.18975 −0.288913
\(460\) 21.4660 1.00086
\(461\) 28.8415 1.34328 0.671642 0.740876i \(-0.265589\pi\)
0.671642 + 0.740876i \(0.265589\pi\)
\(462\) −3.09416 −0.143953
\(463\) −29.9946 −1.39397 −0.696985 0.717086i \(-0.745475\pi\)
−0.696985 + 0.717086i \(0.745475\pi\)
\(464\) 5.47314 0.254084
\(465\) −6.02142 −0.279237
\(466\) −31.0448 −1.43812
\(467\) 29.6606 1.37253 0.686265 0.727352i \(-0.259249\pi\)
0.686265 + 0.727352i \(0.259249\pi\)
\(468\) −3.64749 −0.168605
\(469\) −11.3878 −0.525841
\(470\) 39.4136 1.81801
\(471\) −3.13762 −0.144574
\(472\) 37.4082 1.72185
\(473\) −4.73426 −0.217682
\(474\) 20.4794 0.940650
\(475\) 9.97239 0.457565
\(476\) −29.4482 −1.34976
\(477\) 1.58834 0.0727253
\(478\) 4.86859 0.222684
\(479\) 19.8032 0.904831 0.452416 0.891807i \(-0.350562\pi\)
0.452416 + 0.891807i \(0.350562\pi\)
\(480\) 4.73354 0.216056
\(481\) 4.36970 0.199241
\(482\) −1.08071 −0.0492248
\(483\) 4.95516 0.225468
\(484\) −36.4879 −1.65854
\(485\) 12.2273 0.555213
\(486\) −2.37644 −0.107798
\(487\) 43.3406 1.96395 0.981976 0.189005i \(-0.0605263\pi\)
0.981976 + 0.189005i \(0.0605263\pi\)
\(488\) −3.19143 −0.144469
\(489\) 5.91150 0.267327
\(490\) 19.5069 0.881231
\(491\) −29.4802 −1.33042 −0.665212 0.746654i \(-0.731659\pi\)
−0.665212 + 0.746654i \(0.731659\pi\)
\(492\) 7.26963 0.327740
\(493\) −16.8612 −0.759389
\(494\) −9.11440 −0.410076
\(495\) 1.54637 0.0695043
\(496\) −7.80962 −0.350662
\(497\) 15.5114 0.695782
\(498\) −17.4028 −0.779840
\(499\) −30.9431 −1.38520 −0.692602 0.721320i \(-0.743536\pi\)
−0.692602 + 0.721320i \(0.743536\pi\)
\(500\) −42.9446 −1.92054
\(501\) 24.5246 1.09568
\(502\) −70.2218 −3.13415
\(503\) −31.9197 −1.42323 −0.711614 0.702571i \(-0.752035\pi\)
−0.711614 + 0.702571i \(0.752035\pi\)
\(504\) −5.10673 −0.227472
\(505\) 16.6025 0.738800
\(506\) −9.01189 −0.400627
\(507\) 1.00000 0.0444116
\(508\) −13.3880 −0.593996
\(509\) −19.1063 −0.846873 −0.423436 0.905926i \(-0.639176\pi\)
−0.423436 + 0.905926i \(0.639176\pi\)
\(510\) 22.7873 1.00904
\(511\) −4.21845 −0.186613
\(512\) 21.8719 0.966613
\(513\) −3.83531 −0.169333
\(514\) −10.8631 −0.479151
\(515\) 1.54914 0.0682634
\(516\) −17.2991 −0.761551
\(517\) −10.6868 −0.470006
\(518\) 13.5448 0.595123
\(519\) −7.07942 −0.310752
\(520\) 6.06516 0.265975
\(521\) −31.6998 −1.38879 −0.694396 0.719593i \(-0.744328\pi\)
−0.694396 + 0.719593i \(0.744328\pi\)
\(522\) −6.47355 −0.283339
\(523\) −33.2108 −1.45221 −0.726104 0.687585i \(-0.758671\pi\)
−0.726104 + 0.687585i \(0.758671\pi\)
\(524\) 6.70739 0.293014
\(525\) −3.39150 −0.148017
\(526\) 20.5606 0.896485
\(527\) 24.0592 1.04803
\(528\) 2.00560 0.0872827
\(529\) −8.56785 −0.372515
\(530\) −5.84742 −0.253996
\(531\) −9.55468 −0.414638
\(532\) −18.2468 −0.791099
\(533\) −1.99305 −0.0863286
\(534\) 29.7787 1.28865
\(535\) −30.5672 −1.32154
\(536\) 34.1821 1.47644
\(537\) −6.28398 −0.271174
\(538\) 13.1878 0.568568
\(539\) −5.28921 −0.227822
\(540\) 5.65049 0.243158
\(541\) 2.37553 0.102132 0.0510661 0.998695i \(-0.483738\pi\)
0.0510661 + 0.998695i \(0.483738\pi\)
\(542\) −51.4652 −2.21062
\(543\) −15.9763 −0.685607
\(544\) −18.9133 −0.810902
\(545\) 3.59895 0.154162
\(546\) 3.09970 0.132655
\(547\) −11.0030 −0.470454 −0.235227 0.971941i \(-0.575583\pi\)
−0.235227 + 0.971941i \(0.575583\pi\)
\(548\) −15.1645 −0.647796
\(549\) 0.815146 0.0347896
\(550\) 6.16807 0.263007
\(551\) −10.4476 −0.445081
\(552\) −14.8736 −0.633062
\(553\) −11.2404 −0.477990
\(554\) −10.7645 −0.457339
\(555\) −6.76929 −0.287341
\(556\) −59.0631 −2.50483
\(557\) 14.7101 0.623288 0.311644 0.950199i \(-0.399120\pi\)
0.311644 + 0.950199i \(0.399120\pi\)
\(558\) 9.23709 0.391037
\(559\) 4.74274 0.200597
\(560\) 4.05982 0.171559
\(561\) −6.17868 −0.260864
\(562\) 28.0840 1.18465
\(563\) 24.0877 1.01518 0.507588 0.861600i \(-0.330537\pi\)
0.507588 + 0.861600i \(0.330537\pi\)
\(564\) −39.0499 −1.64430
\(565\) −21.8561 −0.919492
\(566\) 33.8802 1.42409
\(567\) 1.30434 0.0547773
\(568\) −46.5596 −1.95360
\(569\) 13.7005 0.574354 0.287177 0.957878i \(-0.407283\pi\)
0.287177 + 0.957878i \(0.407283\pi\)
\(570\) 14.1195 0.591402
\(571\) 34.7830 1.45562 0.727811 0.685778i \(-0.240538\pi\)
0.727811 + 0.685778i \(0.240538\pi\)
\(572\) −3.64097 −0.152236
\(573\) 17.1115 0.714843
\(574\) −6.17787 −0.257859
\(575\) −9.87789 −0.411937
\(576\) −11.2798 −0.469993
\(577\) −18.4334 −0.767391 −0.383696 0.923460i \(-0.625349\pi\)
−0.383696 + 0.923460i \(0.625349\pi\)
\(578\) −50.6492 −2.10673
\(579\) −8.19444 −0.340549
\(580\) 15.3922 0.639126
\(581\) 9.55180 0.396275
\(582\) −18.7571 −0.777508
\(583\) 1.58550 0.0656649
\(584\) 12.6622 0.523968
\(585\) −1.54914 −0.0640492
\(586\) −8.86661 −0.366276
\(587\) −41.5828 −1.71631 −0.858153 0.513394i \(-0.828388\pi\)
−0.858153 + 0.513394i \(0.828388\pi\)
\(588\) −19.3269 −0.797028
\(589\) 14.9076 0.614257
\(590\) 35.1751 1.44814
\(591\) 22.4587 0.923826
\(592\) −8.77958 −0.360839
\(593\) −15.6620 −0.643161 −0.321581 0.946882i \(-0.604214\pi\)
−0.321581 + 0.946882i \(0.604214\pi\)
\(594\) −2.37219 −0.0973323
\(595\) −12.5071 −0.512742
\(596\) 58.7963 2.40839
\(597\) 18.5204 0.757991
\(598\) 9.02803 0.369184
\(599\) −16.4211 −0.670948 −0.335474 0.942050i \(-0.608896\pi\)
−0.335474 + 0.942050i \(0.608896\pi\)
\(600\) 10.1800 0.415598
\(601\) 13.0892 0.533918 0.266959 0.963708i \(-0.413981\pi\)
0.266959 + 0.963708i \(0.413981\pi\)
\(602\) 14.7011 0.599172
\(603\) −8.73068 −0.355541
\(604\) 26.0202 1.05875
\(605\) −15.4970 −0.630042
\(606\) −25.4688 −1.03460
\(607\) 17.5766 0.713413 0.356707 0.934216i \(-0.383900\pi\)
0.356707 + 0.934216i \(0.383900\pi\)
\(608\) −11.7191 −0.475273
\(609\) 3.55310 0.143979
\(610\) −3.00092 −0.121504
\(611\) 10.7060 0.433118
\(612\) −22.5770 −0.912623
\(613\) 10.5398 0.425699 0.212849 0.977085i \(-0.431726\pi\)
0.212849 + 0.977085i \(0.431726\pi\)
\(614\) 6.49439 0.262092
\(615\) 3.08752 0.124501
\(616\) −5.09759 −0.205388
\(617\) 14.8994 0.599825 0.299913 0.953967i \(-0.403042\pi\)
0.299913 + 0.953967i \(0.403042\pi\)
\(618\) −2.37644 −0.0955946
\(619\) −24.6412 −0.990413 −0.495207 0.868775i \(-0.664908\pi\)
−0.495207 + 0.868775i \(0.664908\pi\)
\(620\) −21.9631 −0.882058
\(621\) 3.79897 0.152447
\(622\) 32.0166 1.28375
\(623\) −16.3444 −0.654826
\(624\) −2.00920 −0.0804322
\(625\) −5.23844 −0.209538
\(626\) 41.8185 1.67140
\(627\) −3.82845 −0.152894
\(628\) −11.4444 −0.456682
\(629\) 27.0473 1.07845
\(630\) −4.80189 −0.191312
\(631\) 34.9645 1.39192 0.695958 0.718082i \(-0.254980\pi\)
0.695958 + 0.718082i \(0.254980\pi\)
\(632\) 33.7396 1.34209
\(633\) 21.9114 0.870901
\(634\) −30.9913 −1.23082
\(635\) −5.68609 −0.225645
\(636\) 5.79347 0.229726
\(637\) 5.29868 0.209942
\(638\) −6.46197 −0.255832
\(639\) 11.8921 0.470445
\(640\) 32.0591 1.26725
\(641\) −32.0423 −1.26560 −0.632798 0.774317i \(-0.718094\pi\)
−0.632798 + 0.774317i \(0.718094\pi\)
\(642\) 46.8912 1.85065
\(643\) −37.0823 −1.46238 −0.731192 0.682172i \(-0.761036\pi\)
−0.731192 + 0.682172i \(0.761036\pi\)
\(644\) 18.0739 0.712211
\(645\) −7.34719 −0.289295
\(646\) −56.4159 −2.21965
\(647\) 30.1877 1.18680 0.593401 0.804907i \(-0.297785\pi\)
0.593401 + 0.804907i \(0.297785\pi\)
\(648\) −3.91517 −0.153802
\(649\) −9.53760 −0.374383
\(650\) −6.17912 −0.242365
\(651\) −5.06990 −0.198705
\(652\) 21.5621 0.844438
\(653\) −37.4278 −1.46466 −0.732331 0.680949i \(-0.761568\pi\)
−0.732331 + 0.680949i \(0.761568\pi\)
\(654\) −5.52092 −0.215885
\(655\) 2.84873 0.111309
\(656\) 4.00443 0.156347
\(657\) −3.23415 −0.126176
\(658\) 33.1854 1.29370
\(659\) −15.2056 −0.592327 −0.296164 0.955137i \(-0.595707\pi\)
−0.296164 + 0.955137i \(0.595707\pi\)
\(660\) 5.64038 0.219552
\(661\) 30.1880 1.17418 0.587088 0.809523i \(-0.300274\pi\)
0.587088 + 0.809523i \(0.300274\pi\)
\(662\) −71.2451 −2.76902
\(663\) 6.18975 0.240390
\(664\) −28.6710 −1.11265
\(665\) −7.74969 −0.300520
\(666\) 10.3843 0.402385
\(667\) 10.3486 0.400698
\(668\) 89.4533 3.46105
\(669\) 6.00131 0.232024
\(670\) 32.1416 1.24174
\(671\) 0.813688 0.0314121
\(672\) 3.98554 0.153746
\(673\) 40.3310 1.55465 0.777323 0.629102i \(-0.216577\pi\)
0.777323 + 0.629102i \(0.216577\pi\)
\(674\) 16.6210 0.640217
\(675\) −2.60015 −0.100080
\(676\) 3.64749 0.140288
\(677\) 17.2813 0.664175 0.332087 0.943249i \(-0.392247\pi\)
0.332087 + 0.943249i \(0.392247\pi\)
\(678\) 33.5280 1.28764
\(679\) 10.2951 0.395090
\(680\) 37.5418 1.43966
\(681\) 7.52318 0.288289
\(682\) 9.22057 0.353074
\(683\) −38.8489 −1.48651 −0.743256 0.669007i \(-0.766719\pi\)
−0.743256 + 0.669007i \(0.766719\pi\)
\(684\) −13.9892 −0.534892
\(685\) −6.44060 −0.246083
\(686\) 38.1223 1.45551
\(687\) −9.53118 −0.363637
\(688\) −9.52910 −0.363294
\(689\) −1.58834 −0.0605111
\(690\) −13.9857 −0.532427
\(691\) 32.3528 1.23076 0.615379 0.788231i \(-0.289003\pi\)
0.615379 + 0.788231i \(0.289003\pi\)
\(692\) −25.8221 −0.981609
\(693\) 1.30201 0.0494594
\(694\) 85.5231 3.24641
\(695\) −25.0850 −0.951528
\(696\) −10.6651 −0.404259
\(697\) −12.3365 −0.467278
\(698\) 6.68673 0.253096
\(699\) 13.0635 0.494108
\(700\) −12.3704 −0.467559
\(701\) 50.4385 1.90503 0.952517 0.304485i \(-0.0984844\pi\)
0.952517 + 0.304485i \(0.0984844\pi\)
\(702\) 2.37644 0.0896931
\(703\) 16.7591 0.632083
\(704\) −11.2597 −0.424364
\(705\) −16.5851 −0.624631
\(706\) −11.8344 −0.445392
\(707\) 13.9789 0.525731
\(708\) −34.8506 −1.30977
\(709\) 19.7175 0.740506 0.370253 0.928931i \(-0.379271\pi\)
0.370253 + 0.928931i \(0.379271\pi\)
\(710\) −43.7803 −1.64304
\(711\) −8.61766 −0.323188
\(712\) 49.0600 1.83860
\(713\) −14.7663 −0.553004
\(714\) 19.1864 0.718032
\(715\) −1.54637 −0.0578311
\(716\) −22.9207 −0.856588
\(717\) −2.04869 −0.0765096
\(718\) 33.9604 1.26739
\(719\) −31.6734 −1.18122 −0.590609 0.806958i \(-0.701112\pi\)
−0.590609 + 0.806958i \(0.701112\pi\)
\(720\) 3.11253 0.115997
\(721\) 1.30434 0.0485763
\(722\) 10.1959 0.379452
\(723\) 0.454757 0.0169126
\(724\) −58.2733 −2.16571
\(725\) −7.08294 −0.263054
\(726\) 23.7729 0.882296
\(727\) −37.7524 −1.40016 −0.700079 0.714065i \(-0.746852\pi\)
−0.700079 + 0.714065i \(0.746852\pi\)
\(728\) 5.10673 0.189268
\(729\) 1.00000 0.0370370
\(730\) 11.9064 0.440675
\(731\) 29.3564 1.08579
\(732\) 2.97323 0.109894
\(733\) 21.7447 0.803160 0.401580 0.915824i \(-0.368461\pi\)
0.401580 + 0.915824i \(0.368461\pi\)
\(734\) 39.9510 1.47462
\(735\) −8.20843 −0.302772
\(736\) 11.6081 0.427879
\(737\) −8.71507 −0.321024
\(738\) −4.73638 −0.174348
\(739\) −35.9828 −1.32365 −0.661824 0.749659i \(-0.730218\pi\)
−0.661824 + 0.749659i \(0.730218\pi\)
\(740\) −24.6909 −0.907656
\(741\) 3.83531 0.140894
\(742\) −4.92339 −0.180744
\(743\) −44.9581 −1.64935 −0.824676 0.565606i \(-0.808642\pi\)
−0.824676 + 0.565606i \(0.808642\pi\)
\(744\) 15.2180 0.557919
\(745\) 24.9717 0.914892
\(746\) 21.1645 0.774887
\(747\) 7.32306 0.267937
\(748\) −22.5367 −0.824022
\(749\) −25.7369 −0.940406
\(750\) 27.9796 1.02167
\(751\) −13.6203 −0.497012 −0.248506 0.968630i \(-0.579940\pi\)
−0.248506 + 0.968630i \(0.579940\pi\)
\(752\) −21.5104 −0.784404
\(753\) 29.5491 1.07683
\(754\) 6.47355 0.235753
\(755\) 11.0512 0.402194
\(756\) 4.75758 0.173032
\(757\) 30.3825 1.10427 0.552135 0.833755i \(-0.313813\pi\)
0.552135 + 0.833755i \(0.313813\pi\)
\(758\) 14.4293 0.524095
\(759\) 3.79217 0.137647
\(760\) 23.2617 0.843792
\(761\) 11.1454 0.404020 0.202010 0.979383i \(-0.435253\pi\)
0.202010 + 0.979383i \(0.435253\pi\)
\(762\) 8.72267 0.315989
\(763\) 3.03023 0.109702
\(764\) 62.4140 2.25806
\(765\) −9.58881 −0.346684
\(766\) −23.2945 −0.841663
\(767\) 9.55468 0.345000
\(768\) −26.6202 −0.960573
\(769\) 26.3612 0.950609 0.475304 0.879821i \(-0.342338\pi\)
0.475304 + 0.879821i \(0.342338\pi\)
\(770\) −4.79330 −0.172738
\(771\) 4.57116 0.164626
\(772\) −29.8891 −1.07573
\(773\) 1.19513 0.0429858 0.0214929 0.999769i \(-0.493158\pi\)
0.0214929 + 0.999769i \(0.493158\pi\)
\(774\) 11.2709 0.405123
\(775\) 10.1066 0.363041
\(776\) −30.9022 −1.10932
\(777\) −5.69959 −0.204472
\(778\) 63.9391 2.29233
\(779\) −7.64397 −0.273874
\(780\) −5.65049 −0.202320
\(781\) 11.8708 0.424772
\(782\) 55.8813 1.99831
\(783\) 2.72405 0.0973495
\(784\) −10.6461 −0.380218
\(785\) −4.86062 −0.173483
\(786\) −4.37006 −0.155875
\(787\) 21.6949 0.773340 0.386670 0.922218i \(-0.373625\pi\)
0.386670 + 0.922218i \(0.373625\pi\)
\(788\) 81.9177 2.91820
\(789\) −8.65184 −0.308014
\(790\) 31.7255 1.12874
\(791\) −18.4023 −0.654311
\(792\) −3.90816 −0.138871
\(793\) −0.815146 −0.0289467
\(794\) 24.1972 0.858727
\(795\) 2.46057 0.0872676
\(796\) 67.5531 2.39435
\(797\) 13.8975 0.492275 0.246137 0.969235i \(-0.420839\pi\)
0.246137 + 0.969235i \(0.420839\pi\)
\(798\) 11.8883 0.420842
\(799\) 66.2673 2.34437
\(800\) −7.94499 −0.280898
\(801\) −12.5308 −0.442753
\(802\) −31.1411 −1.09963
\(803\) −3.22837 −0.113927
\(804\) −31.8451 −1.12309
\(805\) 7.67626 0.270553
\(806\) −9.23709 −0.325363
\(807\) −5.54940 −0.195348
\(808\) −41.9596 −1.47613
\(809\) −30.4630 −1.07102 −0.535511 0.844529i \(-0.679881\pi\)
−0.535511 + 0.844529i \(0.679881\pi\)
\(810\) −3.68145 −0.129353
\(811\) −7.87483 −0.276523 −0.138261 0.990396i \(-0.544151\pi\)
−0.138261 + 0.990396i \(0.544151\pi\)
\(812\) 12.9599 0.454802
\(813\) 21.6564 0.759523
\(814\) 10.3658 0.363320
\(815\) 9.15776 0.320783
\(816\) −12.4364 −0.435362
\(817\) 18.1899 0.636384
\(818\) −90.5404 −3.16567
\(819\) −1.30434 −0.0455775
\(820\) 11.2617 0.393276
\(821\) 21.7907 0.760501 0.380250 0.924884i \(-0.375838\pi\)
0.380250 + 0.924884i \(0.375838\pi\)
\(822\) 9.88012 0.344609
\(823\) −29.5397 −1.02969 −0.514844 0.857284i \(-0.672150\pi\)
−0.514844 + 0.857284i \(0.672150\pi\)
\(824\) −3.91517 −0.136391
\(825\) −2.59550 −0.0903638
\(826\) 29.6167 1.03050
\(827\) −20.4485 −0.711062 −0.355531 0.934664i \(-0.615700\pi\)
−0.355531 + 0.934664i \(0.615700\pi\)
\(828\) 13.8567 0.481553
\(829\) −15.2614 −0.530049 −0.265024 0.964242i \(-0.585380\pi\)
−0.265024 + 0.964242i \(0.585380\pi\)
\(830\) −26.9595 −0.935779
\(831\) 4.52966 0.157132
\(832\) 11.2798 0.391058
\(833\) 32.7975 1.13637
\(834\) 38.4813 1.33250
\(835\) 37.9922 1.31477
\(836\) −13.9642 −0.482963
\(837\) −3.88694 −0.134352
\(838\) −24.5940 −0.849585
\(839\) −50.3094 −1.73687 −0.868437 0.495799i \(-0.834875\pi\)
−0.868437 + 0.495799i \(0.834875\pi\)
\(840\) −7.91105 −0.272957
\(841\) −21.5796 −0.744123
\(842\) 48.7295 1.67933
\(843\) −11.8177 −0.407022
\(844\) 79.9217 2.75102
\(845\) 1.54914 0.0532922
\(846\) 25.4422 0.874720
\(847\) −13.0481 −0.448338
\(848\) 3.19130 0.109590
\(849\) −14.2567 −0.489287
\(850\) −38.2472 −1.31187
\(851\) −16.6003 −0.569052
\(852\) 43.3763 1.48605
\(853\) 11.0233 0.377430 0.188715 0.982032i \(-0.439568\pi\)
0.188715 + 0.982032i \(0.439568\pi\)
\(854\) −2.52671 −0.0864622
\(855\) −5.94145 −0.203193
\(856\) 77.2528 2.64045
\(857\) 4.33332 0.148023 0.0740117 0.997257i \(-0.476420\pi\)
0.0740117 + 0.997257i \(0.476420\pi\)
\(858\) 2.37219 0.0809854
\(859\) 10.9554 0.373792 0.186896 0.982380i \(-0.440157\pi\)
0.186896 + 0.982380i \(0.440157\pi\)
\(860\) −26.7988 −0.913832
\(861\) 2.59963 0.0885950
\(862\) −61.1316 −2.08215
\(863\) −42.7748 −1.45607 −0.728035 0.685540i \(-0.759566\pi\)
−0.728035 + 0.685540i \(0.759566\pi\)
\(864\) 3.05559 0.103953
\(865\) −10.9670 −0.372891
\(866\) −28.3181 −0.962288
\(867\) 21.3130 0.723827
\(868\) −18.4924 −0.627674
\(869\) −8.60225 −0.291811
\(870\) −10.0285 −0.339996
\(871\) 8.73068 0.295828
\(872\) −9.09566 −0.308018
\(873\) 7.89294 0.267136
\(874\) 34.6253 1.17122
\(875\) −15.3570 −0.519161
\(876\) −11.7965 −0.398568
\(877\) 4.60259 0.155418 0.0777092 0.996976i \(-0.475239\pi\)
0.0777092 + 0.996976i \(0.475239\pi\)
\(878\) 3.81234 0.128660
\(879\) 3.73104 0.125845
\(880\) 3.10697 0.104736
\(881\) 12.9976 0.437901 0.218951 0.975736i \(-0.429737\pi\)
0.218951 + 0.975736i \(0.429737\pi\)
\(882\) 12.5920 0.423996
\(883\) 5.87431 0.197686 0.0988432 0.995103i \(-0.468486\pi\)
0.0988432 + 0.995103i \(0.468486\pi\)
\(884\) 22.5770 0.759348
\(885\) −14.8016 −0.497550
\(886\) 59.8149 2.00952
\(887\) 0.861770 0.0289354 0.0144677 0.999895i \(-0.495395\pi\)
0.0144677 + 0.999895i \(0.495395\pi\)
\(888\) 17.1081 0.574110
\(889\) −4.78756 −0.160570
\(890\) 46.1314 1.54633
\(891\) 0.998212 0.0334414
\(892\) 21.8897 0.732922
\(893\) 41.0607 1.37405
\(894\) −38.3075 −1.28120
\(895\) −9.73479 −0.325398
\(896\) 26.9930 0.901774
\(897\) −3.79897 −0.126844
\(898\) 35.9223 1.19874
\(899\) −10.5882 −0.353136
\(900\) −9.48403 −0.316134
\(901\) −9.83145 −0.327533
\(902\) −4.72791 −0.157422
\(903\) −6.18617 −0.205863
\(904\) 55.2371 1.83716
\(905\) −24.7495 −0.822703
\(906\) −16.9529 −0.563224
\(907\) −23.3940 −0.776786 −0.388393 0.921494i \(-0.626970\pi\)
−0.388393 + 0.921494i \(0.626970\pi\)
\(908\) 27.4407 0.910652
\(909\) 10.7172 0.355467
\(910\) 4.80189 0.159181
\(911\) 55.6209 1.84280 0.921401 0.388613i \(-0.127046\pi\)
0.921401 + 0.388613i \(0.127046\pi\)
\(912\) −7.70589 −0.255167
\(913\) 7.30997 0.241925
\(914\) 41.6033 1.37611
\(915\) 1.26278 0.0417462
\(916\) −34.7649 −1.14866
\(917\) 2.39857 0.0792077
\(918\) 14.7096 0.485489
\(919\) 35.9293 1.18520 0.592599 0.805498i \(-0.298102\pi\)
0.592599 + 0.805498i \(0.298102\pi\)
\(920\) −23.0413 −0.759650
\(921\) −2.73282 −0.0900494
\(922\) −68.5403 −2.25726
\(923\) −11.8921 −0.391434
\(924\) 4.74907 0.156233
\(925\) 11.3619 0.373577
\(926\) 71.2806 2.34243
\(927\) 1.00000 0.0328443
\(928\) 8.32356 0.273234
\(929\) 30.0078 0.984524 0.492262 0.870447i \(-0.336170\pi\)
0.492262 + 0.870447i \(0.336170\pi\)
\(930\) 14.3096 0.469230
\(931\) 20.3221 0.666030
\(932\) 47.6491 1.56080
\(933\) −13.4725 −0.441070
\(934\) −70.4868 −2.30640
\(935\) −9.57167 −0.313027
\(936\) 3.91517 0.127971
\(937\) 30.9226 1.01020 0.505098 0.863062i \(-0.331456\pi\)
0.505098 + 0.863062i \(0.331456\pi\)
\(938\) 27.0625 0.883622
\(939\) −17.5971 −0.574260
\(940\) −60.4940 −1.97310
\(941\) 16.6465 0.542662 0.271331 0.962486i \(-0.412536\pi\)
0.271331 + 0.962486i \(0.412536\pi\)
\(942\) 7.45637 0.242942
\(943\) 7.57154 0.246563
\(944\) −19.1972 −0.624817
\(945\) 2.02062 0.0657307
\(946\) 11.2507 0.365792
\(947\) −39.7813 −1.29272 −0.646360 0.763033i \(-0.723709\pi\)
−0.646360 + 0.763033i \(0.723709\pi\)
\(948\) −31.4328 −1.02089
\(949\) 3.23415 0.104985
\(950\) −23.6988 −0.768892
\(951\) 13.0410 0.422884
\(952\) 31.6094 1.02446
\(953\) 1.86533 0.0604238 0.0302119 0.999544i \(-0.490382\pi\)
0.0302119 + 0.999544i \(0.490382\pi\)
\(954\) −3.77461 −0.122208
\(955\) 26.5082 0.857784
\(956\) −7.47256 −0.241680
\(957\) 2.71917 0.0878984
\(958\) −47.0612 −1.52048
\(959\) −5.42284 −0.175113
\(960\) −17.4741 −0.563973
\(961\) −15.8917 −0.512637
\(962\) −10.3843 −0.334805
\(963\) −19.7317 −0.635844
\(964\) 1.65872 0.0534238
\(965\) −12.6944 −0.408646
\(966\) −11.7757 −0.378876
\(967\) −32.3041 −1.03883 −0.519414 0.854522i \(-0.673850\pi\)
−0.519414 + 0.854522i \(0.673850\pi\)
\(968\) 39.1656 1.25883
\(969\) 23.7396 0.762626
\(970\) −29.0575 −0.932980
\(971\) 20.8391 0.668758 0.334379 0.942439i \(-0.391474\pi\)
0.334379 + 0.942439i \(0.391474\pi\)
\(972\) 3.64749 0.116993
\(973\) −21.1210 −0.677108
\(974\) −102.997 −3.30023
\(975\) 2.60015 0.0832715
\(976\) 1.63779 0.0524243
\(977\) −5.98592 −0.191506 −0.0957532 0.995405i \(-0.530526\pi\)
−0.0957532 + 0.995405i \(0.530526\pi\)
\(978\) −14.0484 −0.449217
\(979\) −12.5083 −0.399769
\(980\) −29.9401 −0.956403
\(981\) 2.32319 0.0741736
\(982\) 70.0582 2.23565
\(983\) 39.7389 1.26747 0.633736 0.773549i \(-0.281520\pi\)
0.633736 + 0.773549i \(0.281520\pi\)
\(984\) −7.80313 −0.248755
\(985\) 34.7917 1.10856
\(986\) 40.0696 1.27608
\(987\) −13.9643 −0.444488
\(988\) 13.9892 0.445057
\(989\) −18.0175 −0.572924
\(990\) −3.67487 −0.116795
\(991\) −5.50656 −0.174922 −0.0874609 0.996168i \(-0.527875\pi\)
−0.0874609 + 0.996168i \(0.527875\pi\)
\(992\) −11.8769 −0.377091
\(993\) 29.9797 0.951377
\(994\) −36.8620 −1.16919
\(995\) 28.6908 0.909560
\(996\) 26.7108 0.846364
\(997\) −28.2905 −0.895968 −0.447984 0.894041i \(-0.647858\pi\)
−0.447984 + 0.894041i \(0.647858\pi\)
\(998\) 73.5346 2.32770
\(999\) −4.36970 −0.138251
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 4017.2.a.e.1.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4017.2.a.e.1.2 16 1.1 even 1 trivial