Properties

Label 4002.2.a.bb.1.2
Level $4002$
Weight $2$
Character 4002.1
Self dual yes
Analytic conductor $31.956$
Analytic rank $1$
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] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.23252.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 2 \) 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 \(-1.88474\) of defining polynomial
Character \(\chi\) \(=\) 4002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.82358 q^{5} -1.00000 q^{6} +2.70832 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.82358 q^{5} -1.00000 q^{6} +2.70832 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.82358 q^{10} -4.88474 q^{11} +1.00000 q^{12} -0.552237 q^{13} -2.70832 q^{14} -1.82358 q^{15} +1.00000 q^{16} -2.70832 q^{17} -1.00000 q^{18} +5.81279 q^{19} -1.82358 q^{20} +2.70832 q^{21} +4.88474 q^{22} +1.00000 q^{23} -1.00000 q^{24} -1.67455 q^{25} +0.552237 q^{26} +1.00000 q^{27} +2.70832 q^{28} -1.00000 q^{29} +1.82358 q^{30} +0.321712 q^{31} -1.00000 q^{32} -4.88474 q^{33} +2.70832 q^{34} -4.93884 q^{35} +1.00000 q^{36} -7.52816 q^{37} -5.81279 q^{38} -0.552237 q^{39} +1.82358 q^{40} -1.57007 q^{41} -2.70832 q^{42} +11.6613 q^{43} -4.88474 q^{44} -1.82358 q^{45} -1.00000 q^{46} +9.58227 q^{47} +1.00000 q^{48} +0.334998 q^{49} +1.67455 q^{50} -2.70832 q^{51} -0.552237 q^{52} -9.70458 q^{53} -1.00000 q^{54} +8.90772 q^{55} -2.70832 q^{56} +5.81279 q^{57} +1.00000 q^{58} -7.21724 q^{59} -1.82358 q^{60} -1.11526 q^{61} -0.321712 q^{62} +2.70832 q^{63} +1.00000 q^{64} +1.00705 q^{65} +4.88474 q^{66} -9.17268 q^{67} -2.70832 q^{68} +1.00000 q^{69} +4.93884 q^{70} -4.07709 q^{71} -1.00000 q^{72} -6.58601 q^{73} +7.52816 q^{74} -1.67455 q^{75} +5.81279 q^{76} -13.2294 q^{77} +0.552237 q^{78} -1.16937 q^{79} -1.82358 q^{80} +1.00000 q^{81} +1.57007 q^{82} +10.0601 q^{83} +2.70832 q^{84} +4.93884 q^{85} -11.6613 q^{86} -1.00000 q^{87} +4.88474 q^{88} -16.8091 q^{89} +1.82358 q^{90} -1.49563 q^{91} +1.00000 q^{92} +0.321712 q^{93} -9.58227 q^{94} -10.6001 q^{95} -1.00000 q^{96} -12.8091 q^{97} -0.334998 q^{98} -4.88474 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} + 3 q^{10} - 11 q^{11} + 4 q^{12} - q^{13} + 2 q^{14} - 3 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 8 q^{19} - 3 q^{20} - 2 q^{21} + 11 q^{22} + 4 q^{23} - 4 q^{24} + 3 q^{25} + q^{26} + 4 q^{27} - 2 q^{28} - 4 q^{29} + 3 q^{30} - 17 q^{31} - 4 q^{32} - 11 q^{33} - 2 q^{34} - 24 q^{35} + 4 q^{36} + 15 q^{37} - 8 q^{38} - q^{39} + 3 q^{40} + q^{41} + 2 q^{42} + 4 q^{43} - 11 q^{44} - 3 q^{45} - 4 q^{46} + 6 q^{47} + 4 q^{48} + 16 q^{49} - 3 q^{50} + 2 q^{51} - q^{52} + 2 q^{53} - 4 q^{54} + 13 q^{55} + 2 q^{56} + 8 q^{57} + 4 q^{58} - 13 q^{59} - 3 q^{60} - 13 q^{61} + 17 q^{62} - 2 q^{63} + 4 q^{64} - 13 q^{65} + 11 q^{66} - 13 q^{67} + 2 q^{68} + 4 q^{69} + 24 q^{70} - 15 q^{71} - 4 q^{72} - 22 q^{73} - 15 q^{74} + 3 q^{75} + 8 q^{76} - 12 q^{77} + q^{78} - 26 q^{79} - 3 q^{80} + 4 q^{81} - q^{82} - 22 q^{83} - 2 q^{84} + 24 q^{85} - 4 q^{86} - 4 q^{87} + 11 q^{88} - 24 q^{89} + 3 q^{90} + 30 q^{91} + 4 q^{92} - 17 q^{93} - 6 q^{94} - 4 q^{95} - 4 q^{96} - 8 q^{97} - 16 q^{98} - 11 q^{99}+O(q^{100}) \) 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
\(5\) −1.82358 −0.815531 −0.407765 0.913087i \(-0.633692\pi\)
−0.407765 + 0.913087i \(0.633692\pi\)
\(6\) −1.00000 −0.408248
\(7\) 2.70832 1.02365 0.511824 0.859090i \(-0.328970\pi\)
0.511824 + 0.859090i \(0.328970\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.82358 0.576667
\(11\) −4.88474 −1.47280 −0.736402 0.676544i \(-0.763477\pi\)
−0.736402 + 0.676544i \(0.763477\pi\)
\(12\) 1.00000 0.288675
\(13\) −0.552237 −0.153163 −0.0765814 0.997063i \(-0.524401\pi\)
−0.0765814 + 0.997063i \(0.524401\pi\)
\(14\) −2.70832 −0.723829
\(15\) −1.82358 −0.470847
\(16\) 1.00000 0.250000
\(17\) −2.70832 −0.656864 −0.328432 0.944528i \(-0.606520\pi\)
−0.328432 + 0.944528i \(0.606520\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.81279 1.33355 0.666773 0.745261i \(-0.267675\pi\)
0.666773 + 0.745261i \(0.267675\pi\)
\(20\) −1.82358 −0.407765
\(21\) 2.70832 0.591004
\(22\) 4.88474 1.04143
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −1.67455 −0.334910
\(26\) 0.552237 0.108303
\(27\) 1.00000 0.192450
\(28\) 2.70832 0.511824
\(29\) −1.00000 −0.185695
\(30\) 1.82358 0.332939
\(31\) 0.321712 0.0577812 0.0288906 0.999583i \(-0.490803\pi\)
0.0288906 + 0.999583i \(0.490803\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.88474 −0.850324
\(34\) 2.70832 0.464473
\(35\) −4.93884 −0.834817
\(36\) 1.00000 0.166667
\(37\) −7.52816 −1.23762 −0.618811 0.785540i \(-0.712385\pi\)
−0.618811 + 0.785540i \(0.712385\pi\)
\(38\) −5.81279 −0.942960
\(39\) −0.552237 −0.0884286
\(40\) 1.82358 0.288334
\(41\) −1.57007 −0.245204 −0.122602 0.992456i \(-0.539124\pi\)
−0.122602 + 0.992456i \(0.539124\pi\)
\(42\) −2.70832 −0.417903
\(43\) 11.6613 1.77833 0.889163 0.457591i \(-0.151288\pi\)
0.889163 + 0.457591i \(0.151288\pi\)
\(44\) −4.88474 −0.736402
\(45\) −1.82358 −0.271844
\(46\) −1.00000 −0.147442
\(47\) 9.58227 1.39772 0.698859 0.715260i \(-0.253692\pi\)
0.698859 + 0.715260i \(0.253692\pi\)
\(48\) 1.00000 0.144338
\(49\) 0.334998 0.0478568
\(50\) 1.67455 0.236817
\(51\) −2.70832 −0.379241
\(52\) −0.552237 −0.0765814
\(53\) −9.70458 −1.33303 −0.666513 0.745493i \(-0.732214\pi\)
−0.666513 + 0.745493i \(0.732214\pi\)
\(54\) −1.00000 −0.136083
\(55\) 8.90772 1.20112
\(56\) −2.70832 −0.361914
\(57\) 5.81279 0.769923
\(58\) 1.00000 0.131306
\(59\) −7.21724 −0.939604 −0.469802 0.882772i \(-0.655675\pi\)
−0.469802 + 0.882772i \(0.655675\pi\)
\(60\) −1.82358 −0.235423
\(61\) −1.11526 −0.142795 −0.0713973 0.997448i \(-0.522746\pi\)
−0.0713973 + 0.997448i \(0.522746\pi\)
\(62\) −0.321712 −0.0408575
\(63\) 2.70832 0.341216
\(64\) 1.00000 0.125000
\(65\) 1.00705 0.124909
\(66\) 4.88474 0.601270
\(67\) −9.17268 −1.12062 −0.560310 0.828283i \(-0.689318\pi\)
−0.560310 + 0.828283i \(0.689318\pi\)
\(68\) −2.70832 −0.328432
\(69\) 1.00000 0.120386
\(70\) 4.93884 0.590305
\(71\) −4.07709 −0.483862 −0.241931 0.970294i \(-0.577781\pi\)
−0.241931 + 0.970294i \(0.577781\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.58601 −0.770834 −0.385417 0.922742i \(-0.625942\pi\)
−0.385417 + 0.922742i \(0.625942\pi\)
\(74\) 7.52816 0.875131
\(75\) −1.67455 −0.193360
\(76\) 5.81279 0.666773
\(77\) −13.2294 −1.50763
\(78\) 0.552237 0.0625285
\(79\) −1.16937 −0.131564 −0.0657821 0.997834i \(-0.520954\pi\)
−0.0657821 + 0.997834i \(0.520954\pi\)
\(80\) −1.82358 −0.203883
\(81\) 1.00000 0.111111
\(82\) 1.57007 0.173386
\(83\) 10.0601 1.10424 0.552118 0.833766i \(-0.313820\pi\)
0.552118 + 0.833766i \(0.313820\pi\)
\(84\) 2.70832 0.295502
\(85\) 4.93884 0.535693
\(86\) −11.6613 −1.25747
\(87\) −1.00000 −0.107211
\(88\) 4.88474 0.520715
\(89\) −16.8091 −1.78176 −0.890878 0.454243i \(-0.849910\pi\)
−0.890878 + 0.454243i \(0.849910\pi\)
\(90\) 1.82358 0.192222
\(91\) −1.49563 −0.156785
\(92\) 1.00000 0.104257
\(93\) 0.321712 0.0333600
\(94\) −9.58227 −0.988336
\(95\) −10.6001 −1.08755
\(96\) −1.00000 −0.102062
\(97\) −12.8091 −1.30056 −0.650281 0.759694i \(-0.725349\pi\)
−0.650281 + 0.759694i \(0.725349\pi\)
\(98\) −0.334998 −0.0338399
\(99\) −4.88474 −0.490935
\(100\) −1.67455 −0.167455
\(101\) −7.42619 −0.738933 −0.369467 0.929244i \(-0.620460\pi\)
−0.369467 + 0.929244i \(0.620460\pi\)
\(102\) 2.70832 0.268164
\(103\) −13.4059 −1.32092 −0.660459 0.750862i \(-0.729638\pi\)
−0.660459 + 0.750862i \(0.729638\pi\)
\(104\) 0.552237 0.0541513
\(105\) −4.93884 −0.481982
\(106\) 9.70458 0.942592
\(107\) 8.45622 0.817493 0.408747 0.912648i \(-0.365966\pi\)
0.408747 + 0.912648i \(0.365966\pi\)
\(108\) 1.00000 0.0962250
\(109\) 13.1645 1.26093 0.630467 0.776216i \(-0.282863\pi\)
0.630467 + 0.776216i \(0.282863\pi\)
\(110\) −8.90772 −0.849318
\(111\) −7.52816 −0.714542
\(112\) 2.70832 0.255912
\(113\) 9.06380 0.852651 0.426326 0.904570i \(-0.359808\pi\)
0.426326 + 0.904570i \(0.359808\pi\)
\(114\) −5.81279 −0.544418
\(115\) −1.82358 −0.170050
\(116\) −1.00000 −0.0928477
\(117\) −0.552237 −0.0510543
\(118\) 7.21724 0.664401
\(119\) −7.33500 −0.672398
\(120\) 1.82358 0.166470
\(121\) 12.8607 1.16915
\(122\) 1.11526 0.100971
\(123\) −1.57007 −0.141569
\(124\) 0.321712 0.0288906
\(125\) 12.1716 1.08866
\(126\) −2.70832 −0.241276
\(127\) −13.4262 −1.19138 −0.595691 0.803214i \(-0.703122\pi\)
−0.595691 + 0.803214i \(0.703122\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.6613 1.02672
\(130\) −1.00705 −0.0883240
\(131\) −1.64452 −0.143682 −0.0718410 0.997416i \(-0.522887\pi\)
−0.0718410 + 0.997416i \(0.522887\pi\)
\(132\) −4.88474 −0.425162
\(133\) 15.7429 1.36508
\(134\) 9.17268 0.792399
\(135\) −1.82358 −0.156949
\(136\) 2.70832 0.232237
\(137\) −6.24727 −0.533740 −0.266870 0.963732i \(-0.585990\pi\)
−0.266870 + 0.963732i \(0.585990\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −4.83063 −0.409729 −0.204864 0.978790i \(-0.565675\pi\)
−0.204864 + 0.978790i \(0.565675\pi\)
\(140\) −4.93884 −0.417409
\(141\) 9.58227 0.806973
\(142\) 4.07709 0.342142
\(143\) 2.69753 0.225579
\(144\) 1.00000 0.0833333
\(145\) 1.82358 0.151440
\(146\) 6.58601 0.545062
\(147\) 0.334998 0.0276301
\(148\) −7.52816 −0.618811
\(149\) −8.88474 −0.727866 −0.363933 0.931425i \(-0.618566\pi\)
−0.363933 + 0.931425i \(0.618566\pi\)
\(150\) 1.67455 0.136726
\(151\) −5.79105 −0.471269 −0.235635 0.971842i \(-0.575717\pi\)
−0.235635 + 0.971842i \(0.575717\pi\)
\(152\) −5.81279 −0.471480
\(153\) −2.70832 −0.218955
\(154\) 13.2294 1.06606
\(155\) −0.586669 −0.0471224
\(156\) −0.552237 −0.0442143
\(157\) 9.18347 0.732920 0.366460 0.930434i \(-0.380570\pi\)
0.366460 + 0.930434i \(0.380570\pi\)
\(158\) 1.16937 0.0930300
\(159\) −9.70458 −0.769623
\(160\) 1.82358 0.144167
\(161\) 2.70832 0.213446
\(162\) −1.00000 −0.0785674
\(163\) −18.1372 −1.42061 −0.710306 0.703893i \(-0.751443\pi\)
−0.710306 + 0.703893i \(0.751443\pi\)
\(164\) −1.57007 −0.122602
\(165\) 8.90772 0.693465
\(166\) −10.0601 −0.780812
\(167\) 19.6735 1.52238 0.761189 0.648530i \(-0.224616\pi\)
0.761189 + 0.648530i \(0.224616\pi\)
\(168\) −2.70832 −0.208951
\(169\) −12.6950 −0.976541
\(170\) −4.93884 −0.378792
\(171\) 5.81279 0.444515
\(172\) 11.6613 0.889163
\(173\) −17.5964 −1.33783 −0.668913 0.743340i \(-0.733240\pi\)
−0.668913 + 0.743340i \(0.733240\pi\)
\(174\) 1.00000 0.0758098
\(175\) −4.53521 −0.342830
\(176\) −4.88474 −0.368201
\(177\) −7.21724 −0.542481
\(178\) 16.8091 1.25989
\(179\) 4.29059 0.320694 0.160347 0.987061i \(-0.448739\pi\)
0.160347 + 0.987061i \(0.448739\pi\)
\(180\) −1.82358 −0.135922
\(181\) 15.7019 1.16712 0.583558 0.812072i \(-0.301660\pi\)
0.583558 + 0.812072i \(0.301660\pi\)
\(182\) 1.49563 0.110864
\(183\) −1.11526 −0.0824426
\(184\) −1.00000 −0.0737210
\(185\) 13.7282 1.00932
\(186\) −0.321712 −0.0235891
\(187\) 13.2294 0.967432
\(188\) 9.58227 0.698859
\(189\) 2.70832 0.197001
\(190\) 10.6001 0.769013
\(191\) 9.36253 0.677449 0.338725 0.940886i \(-0.390005\pi\)
0.338725 + 0.940886i \(0.390005\pi\)
\(192\) 1.00000 0.0721688
\(193\) −11.3376 −0.816102 −0.408051 0.912959i \(-0.633791\pi\)
−0.408051 + 0.912959i \(0.633791\pi\)
\(194\) 12.8091 0.919636
\(195\) 1.00705 0.0721163
\(196\) 0.334998 0.0239284
\(197\) −10.5070 −0.748594 −0.374297 0.927309i \(-0.622116\pi\)
−0.374297 + 0.927309i \(0.622116\pi\)
\(198\) 4.88474 0.347143
\(199\) −5.78026 −0.409752 −0.204876 0.978788i \(-0.565679\pi\)
−0.204876 + 0.978788i \(0.565679\pi\)
\(200\) 1.67455 0.118408
\(201\) −9.17268 −0.646991
\(202\) 7.42619 0.522505
\(203\) −2.70832 −0.190087
\(204\) −2.70832 −0.189620
\(205\) 2.86316 0.199972
\(206\) 13.4059 0.934030
\(207\) 1.00000 0.0695048
\(208\) −0.552237 −0.0382907
\(209\) −28.3940 −1.96405
\(210\) 4.93884 0.340813
\(211\) 17.6708 1.21651 0.608254 0.793742i \(-0.291870\pi\)
0.608254 + 0.793742i \(0.291870\pi\)
\(212\) −9.70458 −0.666513
\(213\) −4.07709 −0.279358
\(214\) −8.45622 −0.578055
\(215\) −21.2653 −1.45028
\(216\) −1.00000 −0.0680414
\(217\) 0.871300 0.0591477
\(218\) −13.1645 −0.891615
\(219\) −6.58601 −0.445041
\(220\) 8.90772 0.600558
\(221\) 1.49563 0.100607
\(222\) 7.52816 0.505257
\(223\) −25.0996 −1.68080 −0.840398 0.541969i \(-0.817679\pi\)
−0.840398 + 0.541969i \(0.817679\pi\)
\(224\) −2.70832 −0.180957
\(225\) −1.67455 −0.111637
\(226\) −9.06380 −0.602915
\(227\) 15.8080 1.04921 0.524606 0.851345i \(-0.324213\pi\)
0.524606 + 0.851345i \(0.324213\pi\)
\(228\) 5.81279 0.384962
\(229\) −16.4454 −1.08674 −0.543372 0.839492i \(-0.682853\pi\)
−0.543372 + 0.839492i \(0.682853\pi\)
\(230\) 1.82358 0.120243
\(231\) −13.2294 −0.870433
\(232\) 1.00000 0.0656532
\(233\) −0.276491 −0.0181135 −0.00905677 0.999959i \(-0.502883\pi\)
−0.00905677 + 0.999959i \(0.502883\pi\)
\(234\) 0.552237 0.0361008
\(235\) −17.4741 −1.13988
\(236\) −7.21724 −0.469802
\(237\) −1.16937 −0.0759587
\(238\) 7.33500 0.475457
\(239\) 14.9002 0.963816 0.481908 0.876222i \(-0.339944\pi\)
0.481908 + 0.876222i \(0.339944\pi\)
\(240\) −1.82358 −0.117712
\(241\) −21.3111 −1.37277 −0.686384 0.727240i \(-0.740803\pi\)
−0.686384 + 0.727240i \(0.740803\pi\)
\(242\) −12.8607 −0.826715
\(243\) 1.00000 0.0641500
\(244\) −1.11526 −0.0713973
\(245\) −0.610896 −0.0390287
\(246\) 1.57007 0.100104
\(247\) −3.21004 −0.204250
\(248\) −0.321712 −0.0204287
\(249\) 10.0601 0.637531
\(250\) −12.1716 −0.769799
\(251\) −4.05411 −0.255893 −0.127946 0.991781i \(-0.540839\pi\)
−0.127946 + 0.991781i \(0.540839\pi\)
\(252\) 2.70832 0.170608
\(253\) −4.88474 −0.307101
\(254\) 13.4262 0.842434
\(255\) 4.93884 0.309282
\(256\) 1.00000 0.0625000
\(257\) −14.8091 −0.923763 −0.461882 0.886942i \(-0.652826\pi\)
−0.461882 + 0.886942i \(0.652826\pi\)
\(258\) −11.6613 −0.725998
\(259\) −20.3887 −1.26689
\(260\) 1.00705 0.0624545
\(261\) −1.00000 −0.0618984
\(262\) 1.64452 0.101599
\(263\) −2.43183 −0.149953 −0.0749765 0.997185i \(-0.523888\pi\)
−0.0749765 + 0.997185i \(0.523888\pi\)
\(264\) 4.88474 0.300635
\(265\) 17.6971 1.08712
\(266\) −15.7429 −0.965259
\(267\) −16.8091 −1.02870
\(268\) −9.17268 −0.560310
\(269\) −16.9678 −1.03454 −0.517272 0.855821i \(-0.673052\pi\)
−0.517272 + 0.855821i \(0.673052\pi\)
\(270\) 1.82358 0.110980
\(271\) 19.5078 1.18502 0.592508 0.805565i \(-0.298138\pi\)
0.592508 + 0.805565i \(0.298138\pi\)
\(272\) −2.70832 −0.164216
\(273\) −1.49563 −0.0905199
\(274\) 6.24727 0.377411
\(275\) 8.17973 0.493256
\(276\) 1.00000 0.0601929
\(277\) −22.1728 −1.33224 −0.666118 0.745846i \(-0.732045\pi\)
−0.666118 + 0.745846i \(0.732045\pi\)
\(278\) 4.83063 0.289722
\(279\) 0.321712 0.0192604
\(280\) 4.93884 0.295152
\(281\) −8.04332 −0.479824 −0.239912 0.970795i \(-0.577119\pi\)
−0.239912 + 0.970795i \(0.577119\pi\)
\(282\) −9.58227 −0.570616
\(283\) 13.1612 0.782354 0.391177 0.920315i \(-0.372068\pi\)
0.391177 + 0.920315i \(0.372068\pi\)
\(284\) −4.07709 −0.241931
\(285\) −10.6001 −0.627896
\(286\) −2.69753 −0.159508
\(287\) −4.25226 −0.251003
\(288\) −1.00000 −0.0589256
\(289\) −9.66500 −0.568530
\(290\) −1.82358 −0.107084
\(291\) −12.8091 −0.750880
\(292\) −6.58601 −0.385417
\(293\) 14.4129 0.842011 0.421005 0.907058i \(-0.361677\pi\)
0.421005 + 0.907058i \(0.361677\pi\)
\(294\) −0.334998 −0.0195375
\(295\) 13.1612 0.766276
\(296\) 7.52816 0.437566
\(297\) −4.88474 −0.283441
\(298\) 8.88474 0.514679
\(299\) −0.552237 −0.0319367
\(300\) −1.67455 −0.0966801
\(301\) 31.5824 1.82038
\(302\) 5.79105 0.333238
\(303\) −7.42619 −0.426623
\(304\) 5.81279 0.333387
\(305\) 2.03377 0.116453
\(306\) 2.70832 0.154824
\(307\) 23.2632 1.32770 0.663851 0.747865i \(-0.268921\pi\)
0.663851 + 0.747865i \(0.268921\pi\)
\(308\) −13.2294 −0.753817
\(309\) −13.4059 −0.762632
\(310\) 0.586669 0.0333205
\(311\) −29.7024 −1.68427 −0.842134 0.539268i \(-0.818701\pi\)
−0.842134 + 0.539268i \(0.818701\pi\)
\(312\) 0.552237 0.0312642
\(313\) −24.2473 −1.37054 −0.685268 0.728291i \(-0.740315\pi\)
−0.685268 + 0.728291i \(0.740315\pi\)
\(314\) −9.18347 −0.518253
\(315\) −4.93884 −0.278272
\(316\) −1.16937 −0.0657821
\(317\) 18.4543 1.03650 0.518249 0.855230i \(-0.326584\pi\)
0.518249 + 0.855230i \(0.326584\pi\)
\(318\) 9.70458 0.544206
\(319\) 4.88474 0.273493
\(320\) −1.82358 −0.101941
\(321\) 8.45622 0.471980
\(322\) −2.70832 −0.150929
\(323\) −15.7429 −0.875959
\(324\) 1.00000 0.0555556
\(325\) 0.924747 0.0512957
\(326\) 18.1372 1.00452
\(327\) 13.1645 0.728001
\(328\) 1.57007 0.0866929
\(329\) 25.9519 1.43077
\(330\) −8.90772 −0.490354
\(331\) 25.7072 1.41300 0.706499 0.707714i \(-0.250274\pi\)
0.706499 + 0.707714i \(0.250274\pi\)
\(332\) 10.0601 0.552118
\(333\) −7.52816 −0.412541
\(334\) −19.6735 −1.07648
\(335\) 16.7271 0.913901
\(336\) 2.70832 0.147751
\(337\) 7.17003 0.390576 0.195288 0.980746i \(-0.437436\pi\)
0.195288 + 0.980746i \(0.437436\pi\)
\(338\) 12.6950 0.690519
\(339\) 9.06380 0.492278
\(340\) 4.93884 0.267846
\(341\) −1.57148 −0.0851004
\(342\) −5.81279 −0.314320
\(343\) −18.0510 −0.974660
\(344\) −11.6613 −0.628733
\(345\) −1.82358 −0.0981784
\(346\) 17.5964 0.945987
\(347\) 1.74790 0.0938321 0.0469160 0.998899i \(-0.485061\pi\)
0.0469160 + 0.998899i \(0.485061\pi\)
\(348\) −1.00000 −0.0536056
\(349\) −13.8899 −0.743508 −0.371754 0.928331i \(-0.621244\pi\)
−0.371754 + 0.928331i \(0.621244\pi\)
\(350\) 4.53521 0.242417
\(351\) −0.552237 −0.0294762
\(352\) 4.88474 0.260357
\(353\) 5.81279 0.309384 0.154692 0.987963i \(-0.450562\pi\)
0.154692 + 0.987963i \(0.450562\pi\)
\(354\) 7.21724 0.383592
\(355\) 7.43491 0.394604
\(356\) −16.8091 −0.890878
\(357\) −7.33500 −0.388209
\(358\) −4.29059 −0.226765
\(359\) −17.0894 −0.901941 −0.450971 0.892539i \(-0.648922\pi\)
−0.450971 + 0.892539i \(0.648922\pi\)
\(360\) 1.82358 0.0961112
\(361\) 14.7886 0.778346
\(362\) −15.7019 −0.825275
\(363\) 12.8607 0.675010
\(364\) −1.49563 −0.0783925
\(365\) 12.0101 0.628639
\(366\) 1.11526 0.0582957
\(367\) −1.62559 −0.0848549 −0.0424275 0.999100i \(-0.513509\pi\)
−0.0424275 + 0.999100i \(0.513509\pi\)
\(368\) 1.00000 0.0521286
\(369\) −1.57007 −0.0817348
\(370\) −13.7282 −0.713696
\(371\) −26.2831 −1.36455
\(372\) 0.321712 0.0166800
\(373\) 14.4562 0.748515 0.374257 0.927325i \(-0.377898\pi\)
0.374257 + 0.927325i \(0.377898\pi\)
\(374\) −13.2294 −0.684078
\(375\) 12.1716 0.628538
\(376\) −9.58227 −0.494168
\(377\) 0.552237 0.0284416
\(378\) −2.70832 −0.139301
\(379\) −11.2970 −0.580287 −0.290143 0.956983i \(-0.593703\pi\)
−0.290143 + 0.956983i \(0.593703\pi\)
\(380\) −10.6001 −0.543774
\(381\) −13.4262 −0.687844
\(382\) −9.36253 −0.479029
\(383\) −5.02548 −0.256790 −0.128395 0.991723i \(-0.540983\pi\)
−0.128395 + 0.991723i \(0.540983\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 24.1250 1.22952
\(386\) 11.3376 0.577071
\(387\) 11.6613 0.592775
\(388\) −12.8091 −0.650281
\(389\) 29.6742 1.50454 0.752271 0.658854i \(-0.228958\pi\)
0.752271 + 0.658854i \(0.228958\pi\)
\(390\) −1.00705 −0.0509939
\(391\) −2.70832 −0.136966
\(392\) −0.334998 −0.0169199
\(393\) −1.64452 −0.0829548
\(394\) 10.5070 0.529336
\(395\) 2.13244 0.107295
\(396\) −4.88474 −0.245467
\(397\) −8.87395 −0.445371 −0.222685 0.974890i \(-0.571482\pi\)
−0.222685 + 0.974890i \(0.571482\pi\)
\(398\) 5.78026 0.289738
\(399\) 15.7429 0.788131
\(400\) −1.67455 −0.0837274
\(401\) −16.2073 −0.809352 −0.404676 0.914460i \(-0.632616\pi\)
−0.404676 + 0.914460i \(0.632616\pi\)
\(402\) 9.17268 0.457492
\(403\) −0.177661 −0.00884994
\(404\) −7.42619 −0.369467
\(405\) −1.82358 −0.0906145
\(406\) 2.70832 0.134412
\(407\) 36.7731 1.82277
\(408\) 2.70832 0.134082
\(409\) 23.1428 1.14434 0.572169 0.820136i \(-0.306102\pi\)
0.572169 + 0.820136i \(0.306102\pi\)
\(410\) −2.86316 −0.141401
\(411\) −6.24727 −0.308155
\(412\) −13.4059 −0.660459
\(413\) −19.5466 −0.961825
\(414\) −1.00000 −0.0491473
\(415\) −18.3454 −0.900538
\(416\) 0.552237 0.0270756
\(417\) −4.83063 −0.236557
\(418\) 28.3940 1.38879
\(419\) −25.8843 −1.26453 −0.632266 0.774752i \(-0.717875\pi\)
−0.632266 + 0.774752i \(0.717875\pi\)
\(420\) −4.93884 −0.240991
\(421\) 4.11417 0.200512 0.100256 0.994962i \(-0.468034\pi\)
0.100256 + 0.994962i \(0.468034\pi\)
\(422\) −17.6708 −0.860201
\(423\) 9.58227 0.465906
\(424\) 9.70458 0.471296
\(425\) 4.53521 0.219990
\(426\) 4.07709 0.197536
\(427\) −3.02049 −0.146172
\(428\) 8.45622 0.408747
\(429\) 2.69753 0.130238
\(430\) 21.2653 1.02550
\(431\) 13.4281 0.646808 0.323404 0.946261i \(-0.395173\pi\)
0.323404 + 0.946261i \(0.395173\pi\)
\(432\) 1.00000 0.0481125
\(433\) 8.56162 0.411445 0.205723 0.978610i \(-0.434046\pi\)
0.205723 + 0.978610i \(0.434046\pi\)
\(434\) −0.871300 −0.0418237
\(435\) 1.82358 0.0874341
\(436\) 13.1645 0.630467
\(437\) 5.81279 0.278064
\(438\) 6.58601 0.314692
\(439\) 8.42700 0.402199 0.201099 0.979571i \(-0.435549\pi\)
0.201099 + 0.979571i \(0.435549\pi\)
\(440\) −8.90772 −0.424659
\(441\) 0.334998 0.0159523
\(442\) −1.49563 −0.0711400
\(443\) −9.49563 −0.451151 −0.225576 0.974226i \(-0.572426\pi\)
−0.225576 + 0.974226i \(0.572426\pi\)
\(444\) −7.52816 −0.357271
\(445\) 30.6527 1.45308
\(446\) 25.0996 1.18850
\(447\) −8.88474 −0.420234
\(448\) 2.70832 0.127956
\(449\) −11.1165 −0.524620 −0.262310 0.964984i \(-0.584484\pi\)
−0.262310 + 0.964984i \(0.584484\pi\)
\(450\) 1.67455 0.0789389
\(451\) 7.66940 0.361138
\(452\) 9.06380 0.426326
\(453\) −5.79105 −0.272087
\(454\) −15.8080 −0.741904
\(455\) 2.72741 0.127863
\(456\) −5.81279 −0.272209
\(457\) −23.7977 −1.11321 −0.556604 0.830778i \(-0.687896\pi\)
−0.556604 + 0.830778i \(0.687896\pi\)
\(458\) 16.4454 0.768444
\(459\) −2.70832 −0.126414
\(460\) −1.82358 −0.0850250
\(461\) −39.2849 −1.82968 −0.914841 0.403814i \(-0.867684\pi\)
−0.914841 + 0.403814i \(0.867684\pi\)
\(462\) 13.2294 0.615489
\(463\) 22.5163 1.04642 0.523210 0.852204i \(-0.324734\pi\)
0.523210 + 0.852204i \(0.324734\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −0.586669 −0.0272061
\(466\) 0.276491 0.0128082
\(467\) −22.2582 −1.02999 −0.514994 0.857194i \(-0.672206\pi\)
−0.514994 + 0.857194i \(0.672206\pi\)
\(468\) −0.552237 −0.0255271
\(469\) −24.8425 −1.14712
\(470\) 17.4741 0.806018
\(471\) 9.18347 0.423152
\(472\) 7.21724 0.332200
\(473\) −56.9622 −2.61913
\(474\) 1.16937 0.0537109
\(475\) −9.73380 −0.446617
\(476\) −7.33500 −0.336199
\(477\) −9.70458 −0.444342
\(478\) −14.9002 −0.681521
\(479\) −12.4859 −0.570497 −0.285248 0.958454i \(-0.592076\pi\)
−0.285248 + 0.958454i \(0.592076\pi\)
\(480\) 1.82358 0.0832348
\(481\) 4.15733 0.189558
\(482\) 21.3111 0.970693
\(483\) 2.70832 0.123233
\(484\) 12.8607 0.584576
\(485\) 23.3584 1.06065
\(486\) −1.00000 −0.0453609
\(487\) −16.6942 −0.756487 −0.378244 0.925706i \(-0.623472\pi\)
−0.378244 + 0.925706i \(0.623472\pi\)
\(488\) 1.11526 0.0504855
\(489\) −18.1372 −0.820190
\(490\) 0.610896 0.0275975
\(491\) 5.82689 0.262964 0.131482 0.991319i \(-0.458026\pi\)
0.131482 + 0.991319i \(0.458026\pi\)
\(492\) −1.57007 −0.0707844
\(493\) 2.70832 0.121977
\(494\) 3.21004 0.144426
\(495\) 8.90772 0.400372
\(496\) 0.321712 0.0144453
\(497\) −11.0421 −0.495304
\(498\) −10.0601 −0.450802
\(499\) 22.6359 1.01332 0.506662 0.862145i \(-0.330879\pi\)
0.506662 + 0.862145i \(0.330879\pi\)
\(500\) 12.1716 0.544330
\(501\) 19.6735 0.878945
\(502\) 4.05411 0.180944
\(503\) 30.6386 1.36611 0.683054 0.730368i \(-0.260651\pi\)
0.683054 + 0.730368i \(0.260651\pi\)
\(504\) −2.70832 −0.120638
\(505\) 13.5423 0.602623
\(506\) 4.88474 0.217153
\(507\) −12.6950 −0.563806
\(508\) −13.4262 −0.595691
\(509\) 0.686742 0.0304393 0.0152197 0.999884i \(-0.495155\pi\)
0.0152197 + 0.999884i \(0.495155\pi\)
\(510\) −4.93884 −0.218696
\(511\) −17.8370 −0.789063
\(512\) −1.00000 −0.0441942
\(513\) 5.81279 0.256641
\(514\) 14.8091 0.653199
\(515\) 24.4467 1.07725
\(516\) 11.6613 0.513358
\(517\) −46.8069 −2.05856
\(518\) 20.3887 0.895827
\(519\) −17.5964 −0.772395
\(520\) −1.00705 −0.0441620
\(521\) 4.04332 0.177141 0.0885705 0.996070i \(-0.471770\pi\)
0.0885705 + 0.996070i \(0.471770\pi\)
\(522\) 1.00000 0.0437688
\(523\) −1.34844 −0.0589630 −0.0294815 0.999565i \(-0.509386\pi\)
−0.0294815 + 0.999565i \(0.509386\pi\)
\(524\) −1.64452 −0.0718410
\(525\) −4.53521 −0.197933
\(526\) 2.43183 0.106033
\(527\) −0.871300 −0.0379544
\(528\) −4.88474 −0.212581
\(529\) 1.00000 0.0434783
\(530\) −17.6971 −0.768713
\(531\) −7.21724 −0.313201
\(532\) 15.7429 0.682541
\(533\) 0.867053 0.0375562
\(534\) 16.8091 0.727399
\(535\) −15.4206 −0.666691
\(536\) 9.17268 0.396199
\(537\) 4.29059 0.185153
\(538\) 16.9678 0.731533
\(539\) −1.63638 −0.0704837
\(540\) −1.82358 −0.0784745
\(541\) −28.7442 −1.23581 −0.617904 0.786254i \(-0.712018\pi\)
−0.617904 + 0.786254i \(0.712018\pi\)
\(542\) −19.5078 −0.837933
\(543\) 15.7019 0.673834
\(544\) 2.70832 0.116118
\(545\) −24.0066 −1.02833
\(546\) 1.49563 0.0640072
\(547\) 24.0053 1.02639 0.513196 0.858271i \(-0.328461\pi\)
0.513196 + 0.858271i \(0.328461\pi\)
\(548\) −6.24727 −0.266870
\(549\) −1.11526 −0.0475982
\(550\) −8.17973 −0.348785
\(551\) −5.81279 −0.247633
\(552\) −1.00000 −0.0425628
\(553\) −3.16703 −0.134676
\(554\) 22.1728 0.942033
\(555\) 13.7282 0.582731
\(556\) −4.83063 −0.204864
\(557\) −26.5996 −1.12706 −0.563531 0.826095i \(-0.690557\pi\)
−0.563531 + 0.826095i \(0.690557\pi\)
\(558\) −0.321712 −0.0136192
\(559\) −6.43978 −0.272374
\(560\) −4.93884 −0.208704
\(561\) 13.2294 0.558547
\(562\) 8.04332 0.339287
\(563\) 26.4263 1.11374 0.556869 0.830601i \(-0.312003\pi\)
0.556869 + 0.830601i \(0.312003\pi\)
\(564\) 9.58227 0.403486
\(565\) −16.5286 −0.695363
\(566\) −13.1612 −0.553208
\(567\) 2.70832 0.113739
\(568\) 4.07709 0.171071
\(569\) 28.8306 1.20864 0.604321 0.796741i \(-0.293444\pi\)
0.604321 + 0.796741i \(0.293444\pi\)
\(570\) 10.6001 0.443990
\(571\) −7.98438 −0.334136 −0.167068 0.985945i \(-0.553430\pi\)
−0.167068 + 0.985945i \(0.553430\pi\)
\(572\) 2.69753 0.112789
\(573\) 9.36253 0.391125
\(574\) 4.25226 0.177486
\(575\) −1.67455 −0.0698335
\(576\) 1.00000 0.0416667
\(577\) 13.3761 0.556856 0.278428 0.960457i \(-0.410187\pi\)
0.278428 + 0.960457i \(0.410187\pi\)
\(578\) 9.66500 0.402011
\(579\) −11.3376 −0.471177
\(580\) 1.82358 0.0757201
\(581\) 27.2459 1.13035
\(582\) 12.8091 0.530952
\(583\) 47.4043 1.96329
\(584\) 6.58601 0.272531
\(585\) 1.00705 0.0416363
\(586\) −14.4129 −0.595391
\(587\) −7.28903 −0.300850 −0.150425 0.988621i \(-0.548064\pi\)
−0.150425 + 0.988621i \(0.548064\pi\)
\(588\) 0.334998 0.0138151
\(589\) 1.87005 0.0770539
\(590\) −13.1612 −0.541839
\(591\) −10.5070 −0.432201
\(592\) −7.52816 −0.309406
\(593\) −7.96813 −0.327212 −0.163606 0.986526i \(-0.552313\pi\)
−0.163606 + 0.986526i \(0.552313\pi\)
\(594\) 4.88474 0.200423
\(595\) 13.3760 0.548361
\(596\) −8.88474 −0.363933
\(597\) −5.78026 −0.236570
\(598\) 0.552237 0.0225826
\(599\) −38.9367 −1.59091 −0.795454 0.606013i \(-0.792768\pi\)
−0.795454 + 0.606013i \(0.792768\pi\)
\(600\) 1.67455 0.0683631
\(601\) −37.5992 −1.53370 −0.766851 0.641825i \(-0.778178\pi\)
−0.766851 + 0.641825i \(0.778178\pi\)
\(602\) −31.5824 −1.28720
\(603\) −9.17268 −0.373540
\(604\) −5.79105 −0.235635
\(605\) −23.4525 −0.953479
\(606\) 7.42619 0.301668
\(607\) 34.6628 1.40692 0.703460 0.710735i \(-0.251637\pi\)
0.703460 + 0.710735i \(0.251637\pi\)
\(608\) −5.81279 −0.235740
\(609\) −2.70832 −0.109747
\(610\) −2.03377 −0.0823450
\(611\) −5.29168 −0.214078
\(612\) −2.70832 −0.109477
\(613\) −29.8755 −1.20666 −0.603330 0.797491i \(-0.706160\pi\)
−0.603330 + 0.797491i \(0.706160\pi\)
\(614\) −23.2632 −0.938827
\(615\) 2.86316 0.115454
\(616\) 13.2294 0.533029
\(617\) 18.2994 0.736706 0.368353 0.929686i \(-0.379922\pi\)
0.368353 + 0.929686i \(0.379922\pi\)
\(618\) 13.4059 0.539262
\(619\) −23.3658 −0.939153 −0.469576 0.882892i \(-0.655593\pi\)
−0.469576 + 0.882892i \(0.655593\pi\)
\(620\) −0.586669 −0.0235612
\(621\) 1.00000 0.0401286
\(622\) 29.7024 1.19096
\(623\) −45.5243 −1.82389
\(624\) −0.552237 −0.0221072
\(625\) −13.8232 −0.552926
\(626\) 24.2473 0.969116
\(627\) −28.3940 −1.13395
\(628\) 9.18347 0.366460
\(629\) 20.3887 0.812950
\(630\) 4.93884 0.196768
\(631\) −29.2186 −1.16318 −0.581588 0.813484i \(-0.697568\pi\)
−0.581588 + 0.813484i \(0.697568\pi\)
\(632\) 1.16937 0.0465150
\(633\) 17.6708 0.702352
\(634\) −18.4543 −0.732914
\(635\) 24.4838 0.971608
\(636\) −9.70458 −0.384812
\(637\) −0.184998 −0.00732989
\(638\) −4.88474 −0.193389
\(639\) −4.07709 −0.161287
\(640\) 1.82358 0.0720834
\(641\) 35.9162 1.41860 0.709302 0.704905i \(-0.249010\pi\)
0.709302 + 0.704905i \(0.249010\pi\)
\(642\) −8.45622 −0.333740
\(643\) 22.0026 0.867700 0.433850 0.900985i \(-0.357155\pi\)
0.433850 + 0.900985i \(0.357155\pi\)
\(644\) 2.70832 0.106723
\(645\) −21.2653 −0.837319
\(646\) 15.7429 0.619396
\(647\) 1.65936 0.0652361 0.0326181 0.999468i \(-0.489616\pi\)
0.0326181 + 0.999468i \(0.489616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 35.2543 1.38385
\(650\) −0.924747 −0.0362715
\(651\) 0.871300 0.0341489
\(652\) −18.1372 −0.710306
\(653\) 9.53705 0.373214 0.186607 0.982435i \(-0.440251\pi\)
0.186607 + 0.982435i \(0.440251\pi\)
\(654\) −13.1645 −0.514774
\(655\) 2.99891 0.117177
\(656\) −1.57007 −0.0613011
\(657\) −6.58601 −0.256945
\(658\) −25.9519 −1.01171
\(659\) −39.7699 −1.54922 −0.774609 0.632441i \(-0.782053\pi\)
−0.774609 + 0.632441i \(0.782053\pi\)
\(660\) 8.90772 0.346733
\(661\) −30.1852 −1.17407 −0.587034 0.809562i \(-0.699704\pi\)
−0.587034 + 0.809562i \(0.699704\pi\)
\(662\) −25.7072 −0.999140
\(663\) 1.49563 0.0580856
\(664\) −10.0601 −0.390406
\(665\) −28.7085 −1.11327
\(666\) 7.52816 0.291710
\(667\) −1.00000 −0.0387202
\(668\) 19.6735 0.761189
\(669\) −25.0996 −0.970408
\(670\) −16.7271 −0.646226
\(671\) 5.44776 0.210309
\(672\) −2.70832 −0.104476
\(673\) 12.5020 0.481918 0.240959 0.970535i \(-0.422538\pi\)
0.240959 + 0.970535i \(0.422538\pi\)
\(674\) −7.17003 −0.276179
\(675\) −1.67455 −0.0644534
\(676\) −12.6950 −0.488271
\(677\) 12.4610 0.478917 0.239459 0.970907i \(-0.423030\pi\)
0.239459 + 0.970907i \(0.423030\pi\)
\(678\) −9.06380 −0.348093
\(679\) −34.6910 −1.33132
\(680\) −4.93884 −0.189396
\(681\) 15.8080 0.605762
\(682\) 1.57148 0.0601751
\(683\) −24.5625 −0.939859 −0.469930 0.882704i \(-0.655721\pi\)
−0.469930 + 0.882704i \(0.655721\pi\)
\(684\) 5.81279 0.222258
\(685\) 11.3924 0.435282
\(686\) 18.0510 0.689189
\(687\) −16.4454 −0.627432
\(688\) 11.6613 0.444581
\(689\) 5.35922 0.204170
\(690\) 1.82358 0.0694226
\(691\) 42.5671 1.61933 0.809664 0.586894i \(-0.199649\pi\)
0.809664 + 0.586894i \(0.199649\pi\)
\(692\) −17.5964 −0.668913
\(693\) −13.2294 −0.502545
\(694\) −1.74790 −0.0663493
\(695\) 8.80905 0.334146
\(696\) 1.00000 0.0379049
\(697\) 4.25226 0.161066
\(698\) 13.8899 0.525740
\(699\) −0.276491 −0.0104579
\(700\) −4.53521 −0.171415
\(701\) 26.6542 1.00672 0.503358 0.864078i \(-0.332098\pi\)
0.503358 + 0.864078i \(0.332098\pi\)
\(702\) 0.552237 0.0208428
\(703\) −43.7597 −1.65043
\(704\) −4.88474 −0.184100
\(705\) −17.4741 −0.658111
\(706\) −5.81279 −0.218767
\(707\) −20.1125 −0.756408
\(708\) −7.21724 −0.271240
\(709\) 50.7966 1.90771 0.953853 0.300275i \(-0.0970783\pi\)
0.953853 + 0.300275i \(0.0970783\pi\)
\(710\) −7.43491 −0.279027
\(711\) −1.16937 −0.0438548
\(712\) 16.8091 0.629946
\(713\) 0.321712 0.0120482
\(714\) 7.33500 0.274505
\(715\) −4.91917 −0.183967
\(716\) 4.29059 0.160347
\(717\) 14.9002 0.556460
\(718\) 17.0894 0.637769
\(719\) 13.5804 0.506462 0.253231 0.967406i \(-0.418507\pi\)
0.253231 + 0.967406i \(0.418507\pi\)
\(720\) −1.82358 −0.0679609
\(721\) −36.3073 −1.35216
\(722\) −14.7886 −0.550373
\(723\) −21.3111 −0.792567
\(724\) 15.7019 0.583558
\(725\) 1.67455 0.0621911
\(726\) −12.8607 −0.477304
\(727\) −21.3685 −0.792513 −0.396257 0.918140i \(-0.629691\pi\)
−0.396257 + 0.918140i \(0.629691\pi\)
\(728\) 1.49563 0.0554319
\(729\) 1.00000 0.0370370
\(730\) −12.0101 −0.444515
\(731\) −31.5824 −1.16812
\(732\) −1.11526 −0.0412213
\(733\) 12.0898 0.446546 0.223273 0.974756i \(-0.428326\pi\)
0.223273 + 0.974756i \(0.428326\pi\)
\(734\) 1.62559 0.0600015
\(735\) −0.610896 −0.0225332
\(736\) −1.00000 −0.0368605
\(737\) 44.8061 1.65045
\(738\) 1.57007 0.0577952
\(739\) 17.7767 0.653926 0.326963 0.945037i \(-0.393975\pi\)
0.326963 + 0.945037i \(0.393975\pi\)
\(740\) 13.7282 0.504660
\(741\) −3.21004 −0.117924
\(742\) 26.2831 0.964883
\(743\) 20.9015 0.766801 0.383401 0.923582i \(-0.374753\pi\)
0.383401 + 0.923582i \(0.374753\pi\)
\(744\) −0.321712 −0.0117945
\(745\) 16.2021 0.593597
\(746\) −14.4562 −0.529280
\(747\) 10.0601 0.368079
\(748\) 13.2294 0.483716
\(749\) 22.9021 0.836826
\(750\) −12.1716 −0.444444
\(751\) 33.8003 1.23339 0.616696 0.787202i \(-0.288471\pi\)
0.616696 + 0.787202i \(0.288471\pi\)
\(752\) 9.58227 0.349429
\(753\) −4.05411 −0.147740
\(754\) −0.552237 −0.0201113
\(755\) 10.5605 0.384335
\(756\) 2.70832 0.0985006
\(757\) 43.7565 1.59036 0.795179 0.606375i \(-0.207377\pi\)
0.795179 + 0.606375i \(0.207377\pi\)
\(758\) 11.2970 0.410325
\(759\) −4.88474 −0.177305
\(760\) 10.6001 0.384506
\(761\) −1.33500 −0.0483936 −0.0241968 0.999707i \(-0.507703\pi\)
−0.0241968 + 0.999707i \(0.507703\pi\)
\(762\) 13.4262 0.486379
\(763\) 35.6538 1.29075
\(764\) 9.36253 0.338725
\(765\) 4.93884 0.178564
\(766\) 5.02548 0.181578
\(767\) 3.98562 0.143913
\(768\) 1.00000 0.0360844
\(769\) 18.8799 0.680827 0.340413 0.940276i \(-0.389433\pi\)
0.340413 + 0.940276i \(0.389433\pi\)
\(770\) −24.1250 −0.869403
\(771\) −14.8091 −0.533335
\(772\) −11.3376 −0.408051
\(773\) 17.3328 0.623418 0.311709 0.950178i \(-0.399099\pi\)
0.311709 + 0.950178i \(0.399099\pi\)
\(774\) −11.6613 −0.419155
\(775\) −0.538723 −0.0193515
\(776\) 12.8091 0.459818
\(777\) −20.3887 −0.731440
\(778\) −29.6742 −1.06387
\(779\) −9.12652 −0.326991
\(780\) 1.00705 0.0360581
\(781\) 19.9155 0.712633
\(782\) 2.70832 0.0968493
\(783\) −1.00000 −0.0357371
\(784\) 0.334998 0.0119642
\(785\) −16.7468 −0.597719
\(786\) 1.64452 0.0586579
\(787\) 23.0963 0.823295 0.411648 0.911343i \(-0.364953\pi\)
0.411648 + 0.911343i \(0.364953\pi\)
\(788\) −10.5070 −0.374297
\(789\) −2.43183 −0.0865754
\(790\) −2.13244 −0.0758688
\(791\) 24.5477 0.872815
\(792\) 4.88474 0.173572
\(793\) 0.615889 0.0218708
\(794\) 8.87395 0.314925
\(795\) 17.6971 0.627651
\(796\) −5.78026 −0.204876
\(797\) −36.2791 −1.28507 −0.642537 0.766255i \(-0.722118\pi\)
−0.642537 + 0.766255i \(0.722118\pi\)
\(798\) −15.7429 −0.557293
\(799\) −25.9519 −0.918111
\(800\) 1.67455 0.0592042
\(801\) −16.8091 −0.593919
\(802\) 16.2073 0.572298
\(803\) 32.1709 1.13529
\(804\) −9.17268 −0.323495
\(805\) −4.93884 −0.174071
\(806\) 0.177661 0.00625785
\(807\) −16.9678 −0.597294
\(808\) 7.42619 0.261252
\(809\) 23.4620 0.824881 0.412440 0.910985i \(-0.364677\pi\)
0.412440 + 0.910985i \(0.364677\pi\)
\(810\) 1.82358 0.0640741
\(811\) −11.6395 −0.408719 −0.204359 0.978896i \(-0.565511\pi\)
−0.204359 + 0.978896i \(0.565511\pi\)
\(812\) −2.70832 −0.0950434
\(813\) 19.5078 0.684169
\(814\) −36.7731 −1.28890
\(815\) 33.0746 1.15855
\(816\) −2.70832 −0.0948102
\(817\) 67.7845 2.37148
\(818\) −23.1428 −0.809169
\(819\) −1.49563 −0.0522617
\(820\) 2.86316 0.0999859
\(821\) −1.96416 −0.0685497 −0.0342749 0.999412i \(-0.510912\pi\)
−0.0342749 + 0.999412i \(0.510912\pi\)
\(822\) 6.24727 0.217899
\(823\) −34.3932 −1.19887 −0.599436 0.800423i \(-0.704609\pi\)
−0.599436 + 0.800423i \(0.704609\pi\)
\(824\) 13.4059 0.467015
\(825\) 8.17973 0.284782
\(826\) 19.5466 0.680113
\(827\) 26.3857 0.917520 0.458760 0.888560i \(-0.348294\pi\)
0.458760 + 0.888560i \(0.348294\pi\)
\(828\) 1.00000 0.0347524
\(829\) 16.1731 0.561715 0.280858 0.959749i \(-0.409381\pi\)
0.280858 + 0.959749i \(0.409381\pi\)
\(830\) 18.3454 0.636777
\(831\) −22.1728 −0.769167
\(832\) −0.552237 −0.0191454
\(833\) −0.907281 −0.0314354
\(834\) 4.83063 0.167271
\(835\) −35.8762 −1.24155
\(836\) −28.3940 −0.982026
\(837\) 0.321712 0.0111200
\(838\) 25.8843 0.894159
\(839\) −3.90522 −0.134823 −0.0674117 0.997725i \(-0.521474\pi\)
−0.0674117 + 0.997725i \(0.521474\pi\)
\(840\) 4.93884 0.170406
\(841\) 1.00000 0.0344828
\(842\) −4.11417 −0.141784
\(843\) −8.04332 −0.277027
\(844\) 17.6708 0.608254
\(845\) 23.1504 0.796399
\(846\) −9.58227 −0.329445
\(847\) 34.8308 1.19680
\(848\) −9.70458 −0.333257
\(849\) 13.1612 0.451692
\(850\) −4.53521 −0.155556
\(851\) −7.52816 −0.258062
\(852\) −4.07709 −0.139679
\(853\) −43.7431 −1.49773 −0.748867 0.662720i \(-0.769402\pi\)
−0.748867 + 0.662720i \(0.769402\pi\)
\(854\) 3.02049 0.103359
\(855\) −10.6001 −0.362516
\(856\) −8.45622 −0.289028
\(857\) 8.81551 0.301132 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(858\) −2.69753 −0.0920922
\(859\) −14.2831 −0.487333 −0.243667 0.969859i \(-0.578350\pi\)
−0.243667 + 0.969859i \(0.578350\pi\)
\(860\) −21.2653 −0.725140
\(861\) −4.25226 −0.144917
\(862\) −13.4281 −0.457363
\(863\) −31.2535 −1.06388 −0.531942 0.846781i \(-0.678537\pi\)
−0.531942 + 0.846781i \(0.678537\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 32.0884 1.09104
\(866\) −8.56162 −0.290936
\(867\) −9.66500 −0.328241
\(868\) 0.871300 0.0295738
\(869\) 5.71206 0.193768
\(870\) −1.82358 −0.0618252
\(871\) 5.06549 0.171638
\(872\) −13.1645 −0.445808
\(873\) −12.8091 −0.433521
\(874\) −5.81279 −0.196621
\(875\) 32.9646 1.11441
\(876\) −6.58601 −0.222521
\(877\) 25.3863 0.857233 0.428617 0.903486i \(-0.359001\pi\)
0.428617 + 0.903486i \(0.359001\pi\)
\(878\) −8.42700 −0.284397
\(879\) 14.4129 0.486135
\(880\) 8.90772 0.300279
\(881\) 54.3583 1.83138 0.915689 0.401888i \(-0.131646\pi\)
0.915689 + 0.401888i \(0.131646\pi\)
\(882\) −0.334998 −0.0112800
\(883\) 24.6076 0.828111 0.414056 0.910252i \(-0.364112\pi\)
0.414056 + 0.910252i \(0.364112\pi\)
\(884\) 1.49563 0.0503036
\(885\) 13.1612 0.442410
\(886\) 9.49563 0.319012
\(887\) 11.9875 0.402502 0.201251 0.979540i \(-0.435499\pi\)
0.201251 + 0.979540i \(0.435499\pi\)
\(888\) 7.52816 0.252629
\(889\) −36.3624 −1.21956
\(890\) −30.6527 −1.02748
\(891\) −4.88474 −0.163645
\(892\) −25.0996 −0.840398
\(893\) 55.6997 1.86392
\(894\) 8.88474 0.297150
\(895\) −7.82424 −0.261536
\(896\) −2.70832 −0.0904786
\(897\) −0.552237 −0.0184386
\(898\) 11.1165 0.370963
\(899\) −0.321712 −0.0107297
\(900\) −1.67455 −0.0558183
\(901\) 26.2831 0.875617
\(902\) −7.66940 −0.255363
\(903\) 31.5824 1.05100
\(904\) −9.06380 −0.301458
\(905\) −28.6338 −0.951818
\(906\) 5.79105 0.192395
\(907\) 46.0795 1.53004 0.765022 0.644005i \(-0.222728\pi\)
0.765022 + 0.644005i \(0.222728\pi\)
\(908\) 15.8080 0.524606
\(909\) −7.42619 −0.246311
\(910\) −2.72741 −0.0904128
\(911\) 27.6380 0.915688 0.457844 0.889033i \(-0.348622\pi\)
0.457844 + 0.889033i \(0.348622\pi\)
\(912\) 5.81279 0.192481
\(913\) −49.1408 −1.62632
\(914\) 23.7977 0.787157
\(915\) 2.03377 0.0672344
\(916\) −16.4454 −0.543372
\(917\) −4.45387 −0.147080
\(918\) 2.70832 0.0893879
\(919\) 0.836122 0.0275811 0.0137906 0.999905i \(-0.495610\pi\)
0.0137906 + 0.999905i \(0.495610\pi\)
\(920\) 1.82358 0.0601217
\(921\) 23.2632 0.766549
\(922\) 39.2849 1.29378
\(923\) 2.25152 0.0741096
\(924\) −13.2294 −0.435216
\(925\) 12.6063 0.414492
\(926\) −22.5163 −0.739931
\(927\) −13.4059 −0.440306
\(928\) 1.00000 0.0328266
\(929\) −41.4694 −1.36057 −0.680283 0.732949i \(-0.738143\pi\)
−0.680283 + 0.732949i \(0.738143\pi\)
\(930\) 0.586669 0.0192376
\(931\) 1.94727 0.0638193
\(932\) −0.276491 −0.00905677
\(933\) −29.7024 −0.972413
\(934\) 22.2582 0.728311
\(935\) −24.1250 −0.788971
\(936\) 0.552237 0.0180504
\(937\) −21.9085 −0.715721 −0.357860 0.933775i \(-0.616494\pi\)
−0.357860 + 0.933775i \(0.616494\pi\)
\(938\) 24.8425 0.811138
\(939\) −24.2473 −0.791280
\(940\) −17.4741 −0.569941
\(941\) 32.2202 1.05035 0.525174 0.850995i \(-0.324000\pi\)
0.525174 + 0.850995i \(0.324000\pi\)
\(942\) −9.18347 −0.299214
\(943\) −1.57007 −0.0511287
\(944\) −7.21724 −0.234901
\(945\) −4.93884 −0.160661
\(946\) 56.9622 1.85200
\(947\) −13.0996 −0.425681 −0.212841 0.977087i \(-0.568272\pi\)
−0.212841 + 0.977087i \(0.568272\pi\)
\(948\) −1.16937 −0.0379793
\(949\) 3.63704 0.118063
\(950\) 9.73380 0.315806
\(951\) 18.4543 0.598422
\(952\) 7.33500 0.237729
\(953\) −23.2563 −0.753346 −0.376673 0.926346i \(-0.622932\pi\)
−0.376673 + 0.926346i \(0.622932\pi\)
\(954\) 9.70458 0.314197
\(955\) −17.0734 −0.552481
\(956\) 14.9002 0.481908
\(957\) 4.88474 0.157901
\(958\) 12.4859 0.403402
\(959\) −16.9196 −0.546363
\(960\) −1.82358 −0.0588559
\(961\) −30.8965 −0.996661
\(962\) −4.15733 −0.134038
\(963\) 8.45622 0.272498
\(964\) −21.3111 −0.686384
\(965\) 20.6751 0.665556
\(966\) −2.70832 −0.0871388
\(967\) 37.7354 1.21349 0.606745 0.794897i \(-0.292475\pi\)
0.606745 + 0.794897i \(0.292475\pi\)
\(968\) −12.8607 −0.413357
\(969\) −15.7429 −0.505735
\(970\) −23.3584 −0.749992
\(971\) 12.3447 0.396160 0.198080 0.980186i \(-0.436529\pi\)
0.198080 + 0.980186i \(0.436529\pi\)
\(972\) 1.00000 0.0320750
\(973\) −13.0829 −0.419418
\(974\) 16.6942 0.534917
\(975\) 0.924747 0.0296156
\(976\) −1.11526 −0.0356987
\(977\) −18.9184 −0.605253 −0.302627 0.953109i \(-0.597863\pi\)
−0.302627 + 0.953109i \(0.597863\pi\)
\(978\) 18.1372 0.579962
\(979\) 82.1078 2.62418
\(980\) −0.610896 −0.0195143
\(981\) 13.1645 0.420311
\(982\) −5.82689 −0.185944
\(983\) 44.2215 1.41045 0.705224 0.708985i \(-0.250847\pi\)
0.705224 + 0.708985i \(0.250847\pi\)
\(984\) 1.57007 0.0500521
\(985\) 19.1604 0.610501
\(986\) −2.70832 −0.0862505
\(987\) 25.9519 0.826057
\(988\) −3.21004 −0.102125
\(989\) 11.6613 0.370807
\(990\) −8.90772 −0.283106
\(991\) −57.0882 −1.81347 −0.906733 0.421705i \(-0.861432\pi\)
−0.906733 + 0.421705i \(0.861432\pi\)
\(992\) −0.321712 −0.0102144
\(993\) 25.7072 0.815794
\(994\) 11.0421 0.350233
\(995\) 10.5408 0.334165
\(996\) 10.0601 0.318765
\(997\) −30.0627 −0.952096 −0.476048 0.879419i \(-0.657931\pi\)
−0.476048 + 0.879419i \(0.657931\pi\)
\(998\) −22.6359 −0.716528
\(999\) −7.52816 −0.238181
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 4002.2.a.bb.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.bb.1.2 4 1.1 even 1 trivial