Properties

Label 5290.2.a.u.1.1
Level $5290$
Weight $2$
Character 5290.1
Self dual yes
Analytic conductor $42.241$
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] = [5290,2,Mod(1,5290)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5290, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5290.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5290 = 2 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5290.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: \(42.2408626693\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.517638\) of defining polynomial
Character \(\chi\) \(=\) 5290.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.414214 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.414214 q^{6} +0.282561 q^{7} -1.00000 q^{8} -2.82843 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.414214 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.414214 q^{6} +0.282561 q^{7} -1.00000 q^{8} -2.82843 q^{9} +1.00000 q^{10} +5.27792 q^{11} -0.414214 q^{12} +1.86370 q^{13} -0.282561 q^{14} +0.414214 q^{15} +1.00000 q^{16} -5.24969 q^{17} +2.82843 q^{18} -0.946464 q^{19} -1.00000 q^{20} -0.117041 q^{21} -5.27792 q^{22} +0.414214 q^{24} +1.00000 q^{25} -1.86370 q^{26} +2.41421 q^{27} +0.282561 q^{28} -1.64689 q^{29} -0.414214 q^{30} +3.85374 q^{31} -1.00000 q^{32} -2.18618 q^{33} +5.24969 q^{34} -0.282561 q^{35} -2.82843 q^{36} +0.828427 q^{37} +0.946464 q^{38} -0.771971 q^{39} +1.00000 q^{40} -7.53465 q^{41} +0.117041 q^{42} -0.663902 q^{43} +5.27792 q^{44} +2.82843 q^{45} +0.353113 q^{47} -0.414214 q^{48} -6.92016 q^{49} -1.00000 q^{50} +2.17449 q^{51} +1.86370 q^{52} -10.2168 q^{53} -2.41421 q^{54} -5.27792 q^{55} -0.282561 q^{56} +0.392038 q^{57} +1.64689 q^{58} +3.00000 q^{59} +0.414214 q^{60} +9.75663 q^{61} -3.85374 q^{62} -0.799203 q^{63} +1.00000 q^{64} -1.86370 q^{65} +2.18618 q^{66} +15.0628 q^{67} -5.24969 q^{68} +0.282561 q^{70} +13.4448 q^{71} +2.82843 q^{72} -15.9089 q^{73} -0.828427 q^{74} -0.414214 q^{75} -0.946464 q^{76} +1.49133 q^{77} +0.771971 q^{78} -4.81116 q^{79} -1.00000 q^{80} +7.48528 q^{81} +7.53465 q^{82} +0.643238 q^{83} -0.117041 q^{84} +5.24969 q^{85} +0.663902 q^{86} +0.682163 q^{87} -5.27792 q^{88} -2.29326 q^{89} -2.82843 q^{90} +0.526610 q^{91} -1.59627 q^{93} -0.353113 q^{94} +0.946464 q^{95} +0.414214 q^{96} +11.6702 q^{97} +6.92016 q^{98} -14.9282 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} - 4 q^{6} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} - 4 q^{6} + 4 q^{7} - 4 q^{8} + 4 q^{10} + 4 q^{12} - 8 q^{13} - 4 q^{14} - 4 q^{15} + 4 q^{16} - 12 q^{17} + 8 q^{19} - 4 q^{20} + 4 q^{21} - 4 q^{24} + 4 q^{25} + 8 q^{26} + 4 q^{27} + 4 q^{28} - 12 q^{29} + 4 q^{30} + 28 q^{31} - 4 q^{32} - 16 q^{33} + 12 q^{34} - 4 q^{35} - 8 q^{37} - 8 q^{38} - 16 q^{39} + 4 q^{40} - 8 q^{41} - 4 q^{42} + 12 q^{43} - 4 q^{47} + 4 q^{48} + 12 q^{49} - 4 q^{50} - 16 q^{51} - 8 q^{52} - 20 q^{53} - 4 q^{54} - 4 q^{56} + 12 q^{57} + 12 q^{58} + 12 q^{59} - 4 q^{60} - 28 q^{62} + 4 q^{64} + 8 q^{65} + 16 q^{66} - 12 q^{67} - 12 q^{68} + 4 q^{70} + 20 q^{71} - 16 q^{73} + 8 q^{74} + 4 q^{75} + 8 q^{76} - 24 q^{77} + 16 q^{78} + 4 q^{79} - 4 q^{80} - 4 q^{81} + 8 q^{82} - 12 q^{83} + 4 q^{84} + 12 q^{85} - 12 q^{86} + 4 q^{87} - 40 q^{89} - 32 q^{91} + 36 q^{93} + 4 q^{94} - 8 q^{95} - 4 q^{96} - 16 q^{97} - 12 q^{98} - 32 q^{99}+O(q^{100}) \) 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
\(3\) −0.414214 −0.239146 −0.119573 0.992825i \(-0.538153\pi\)
−0.119573 + 0.992825i \(0.538153\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0.414214 0.169102
\(7\) 0.282561 0.106798 0.0533990 0.998573i \(-0.482994\pi\)
0.0533990 + 0.998573i \(0.482994\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.82843 −0.942809
\(10\) 1.00000 0.316228
\(11\) 5.27792 1.59135 0.795676 0.605723i \(-0.207116\pi\)
0.795676 + 0.605723i \(0.207116\pi\)
\(12\) −0.414214 −0.119573
\(13\) 1.86370 0.516898 0.258449 0.966025i \(-0.416789\pi\)
0.258449 + 0.966025i \(0.416789\pi\)
\(14\) −0.282561 −0.0755176
\(15\) 0.414214 0.106949
\(16\) 1.00000 0.250000
\(17\) −5.24969 −1.27324 −0.636618 0.771179i \(-0.719667\pi\)
−0.636618 + 0.771179i \(0.719667\pi\)
\(18\) 2.82843 0.666667
\(19\) −0.946464 −0.217134 −0.108567 0.994089i \(-0.534626\pi\)
−0.108567 + 0.994089i \(0.534626\pi\)
\(20\) −1.00000 −0.223607
\(21\) −0.117041 −0.0255404
\(22\) −5.27792 −1.12526
\(23\) 0 0
\(24\) 0.414214 0.0845510
\(25\) 1.00000 0.200000
\(26\) −1.86370 −0.365502
\(27\) 2.41421 0.464616
\(28\) 0.282561 0.0533990
\(29\) −1.64689 −0.305819 −0.152910 0.988240i \(-0.548864\pi\)
−0.152910 + 0.988240i \(0.548864\pi\)
\(30\) −0.414214 −0.0756247
\(31\) 3.85374 0.692151 0.346076 0.938207i \(-0.387514\pi\)
0.346076 + 0.938207i \(0.387514\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.18618 −0.380566
\(34\) 5.24969 0.900314
\(35\) −0.282561 −0.0477615
\(36\) −2.82843 −0.471405
\(37\) 0.828427 0.136193 0.0680963 0.997679i \(-0.478307\pi\)
0.0680963 + 0.997679i \(0.478307\pi\)
\(38\) 0.946464 0.153537
\(39\) −0.771971 −0.123614
\(40\) 1.00000 0.158114
\(41\) −7.53465 −1.17672 −0.588358 0.808601i \(-0.700225\pi\)
−0.588358 + 0.808601i \(0.700225\pi\)
\(42\) 0.117041 0.0180598
\(43\) −0.663902 −0.101244 −0.0506221 0.998718i \(-0.516120\pi\)
−0.0506221 + 0.998718i \(0.516120\pi\)
\(44\) 5.27792 0.795676
\(45\) 2.82843 0.421637
\(46\) 0 0
\(47\) 0.353113 0.0515069 0.0257534 0.999668i \(-0.491802\pi\)
0.0257534 + 0.999668i \(0.491802\pi\)
\(48\) −0.414214 −0.0597866
\(49\) −6.92016 −0.988594
\(50\) −1.00000 −0.141421
\(51\) 2.17449 0.304490
\(52\) 1.86370 0.258449
\(53\) −10.2168 −1.40339 −0.701694 0.712479i \(-0.747573\pi\)
−0.701694 + 0.712479i \(0.747573\pi\)
\(54\) −2.41421 −0.328533
\(55\) −5.27792 −0.711674
\(56\) −0.282561 −0.0377588
\(57\) 0.392038 0.0519267
\(58\) 1.64689 0.216247
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0.414214 0.0534747
\(61\) 9.75663 1.24921 0.624604 0.780941i \(-0.285260\pi\)
0.624604 + 0.780941i \(0.285260\pi\)
\(62\) −3.85374 −0.489425
\(63\) −0.799203 −0.100690
\(64\) 1.00000 0.125000
\(65\) −1.86370 −0.231164
\(66\) 2.18618 0.269101
\(67\) 15.0628 1.84021 0.920105 0.391671i \(-0.128103\pi\)
0.920105 + 0.391671i \(0.128103\pi\)
\(68\) −5.24969 −0.636618
\(69\) 0 0
\(70\) 0.282561 0.0337725
\(71\) 13.4448 1.59561 0.797805 0.602916i \(-0.205995\pi\)
0.797805 + 0.602916i \(0.205995\pi\)
\(72\) 2.82843 0.333333
\(73\) −15.9089 −1.86200 −0.931001 0.365017i \(-0.881063\pi\)
−0.931001 + 0.365017i \(0.881063\pi\)
\(74\) −0.828427 −0.0963027
\(75\) −0.414214 −0.0478293
\(76\) −0.946464 −0.108567
\(77\) 1.49133 0.169953
\(78\) 0.771971 0.0874085
\(79\) −4.81116 −0.541298 −0.270649 0.962678i \(-0.587238\pi\)
−0.270649 + 0.962678i \(0.587238\pi\)
\(80\) −1.00000 −0.111803
\(81\) 7.48528 0.831698
\(82\) 7.53465 0.832063
\(83\) 0.643238 0.0706046 0.0353023 0.999377i \(-0.488761\pi\)
0.0353023 + 0.999377i \(0.488761\pi\)
\(84\) −0.117041 −0.0127702
\(85\) 5.24969 0.569409
\(86\) 0.663902 0.0715904
\(87\) 0.682163 0.0731355
\(88\) −5.27792 −0.562628
\(89\) −2.29326 −0.243085 −0.121542 0.992586i \(-0.538784\pi\)
−0.121542 + 0.992586i \(0.538784\pi\)
\(90\) −2.82843 −0.298142
\(91\) 0.526610 0.0552037
\(92\) 0 0
\(93\) −1.59627 −0.165525
\(94\) −0.353113 −0.0364209
\(95\) 0.946464 0.0971051
\(96\) 0.414214 0.0422755
\(97\) 11.6702 1.18493 0.592466 0.805596i \(-0.298155\pi\)
0.592466 + 0.805596i \(0.298155\pi\)
\(98\) 6.92016 0.699042
\(99\) −14.9282 −1.50034
\(100\) 1.00000 0.100000
\(101\) −10.6357 −1.05829 −0.529145 0.848532i \(-0.677487\pi\)
−0.529145 + 0.848532i \(0.677487\pi\)
\(102\) −2.17449 −0.215307
\(103\) −12.2973 −1.21169 −0.605846 0.795582i \(-0.707165\pi\)
−0.605846 + 0.795582i \(0.707165\pi\)
\(104\) −1.86370 −0.182751
\(105\) 0.117041 0.0114220
\(106\) 10.2168 0.992345
\(107\) −9.07107 −0.876933 −0.438467 0.898747i \(-0.644478\pi\)
−0.438467 + 0.898747i \(0.644478\pi\)
\(108\) 2.41421 0.232308
\(109\) −6.21201 −0.595003 −0.297501 0.954721i \(-0.596153\pi\)
−0.297501 + 0.954721i \(0.596153\pi\)
\(110\) 5.27792 0.503230
\(111\) −0.343146 −0.0325700
\(112\) 0.282561 0.0266995
\(113\) −5.12925 −0.482519 −0.241260 0.970461i \(-0.577561\pi\)
−0.241260 + 0.970461i \(0.577561\pi\)
\(114\) −0.392038 −0.0367177
\(115\) 0 0
\(116\) −1.64689 −0.152910
\(117\) −5.27135 −0.487336
\(118\) −3.00000 −0.276172
\(119\) −1.48336 −0.135979
\(120\) −0.414214 −0.0378124
\(121\) 16.8564 1.53240
\(122\) −9.75663 −0.883324
\(123\) 3.12096 0.281407
\(124\) 3.85374 0.346076
\(125\) −1.00000 −0.0894427
\(126\) 0.799203 0.0711987
\(127\) 7.58114 0.672718 0.336359 0.941734i \(-0.390805\pi\)
0.336359 + 0.941734i \(0.390805\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.274997 0.0242122
\(130\) 1.86370 0.163458
\(131\) 19.8776 1.73671 0.868356 0.495941i \(-0.165177\pi\)
0.868356 + 0.495941i \(0.165177\pi\)
\(132\) −2.18618 −0.190283
\(133\) −0.267434 −0.0231894
\(134\) −15.0628 −1.30123
\(135\) −2.41421 −0.207782
\(136\) 5.24969 0.450157
\(137\) −16.6067 −1.41881 −0.709404 0.704802i \(-0.751036\pi\)
−0.709404 + 0.704802i \(0.751036\pi\)
\(138\) 0 0
\(139\) 16.1623 1.37087 0.685434 0.728135i \(-0.259613\pi\)
0.685434 + 0.728135i \(0.259613\pi\)
\(140\) −0.282561 −0.0238808
\(141\) −0.146264 −0.0123177
\(142\) −13.4448 −1.12827
\(143\) 9.83647 0.822567
\(144\) −2.82843 −0.235702
\(145\) 1.64689 0.136766
\(146\) 15.9089 1.31663
\(147\) 2.86642 0.236419
\(148\) 0.828427 0.0680963
\(149\) 4.83839 0.396377 0.198188 0.980164i \(-0.436494\pi\)
0.198188 + 0.980164i \(0.436494\pi\)
\(150\) 0.414214 0.0338204
\(151\) −3.19151 −0.259721 −0.129861 0.991532i \(-0.541453\pi\)
−0.129861 + 0.991532i \(0.541453\pi\)
\(152\) 0.946464 0.0767683
\(153\) 14.8484 1.20042
\(154\) −1.49133 −0.120175
\(155\) −3.85374 −0.309539
\(156\) −0.771971 −0.0618072
\(157\) −10.4288 −0.832311 −0.416155 0.909293i \(-0.636623\pi\)
−0.416155 + 0.909293i \(0.636623\pi\)
\(158\) 4.81116 0.382756
\(159\) 4.23194 0.335615
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −7.48528 −0.588099
\(163\) −3.87780 −0.303733 −0.151866 0.988401i \(-0.548528\pi\)
−0.151866 + 0.988401i \(0.548528\pi\)
\(164\) −7.53465 −0.588358
\(165\) 2.18618 0.170194
\(166\) −0.643238 −0.0499250
\(167\) 2.76681 0.214102 0.107051 0.994254i \(-0.465859\pi\)
0.107051 + 0.994254i \(0.465859\pi\)
\(168\) 0.117041 0.00902988
\(169\) −9.52661 −0.732816
\(170\) −5.24969 −0.402633
\(171\) 2.67700 0.204716
\(172\) −0.663902 −0.0506221
\(173\) 19.2015 1.45986 0.729931 0.683521i \(-0.239552\pi\)
0.729931 + 0.683521i \(0.239552\pi\)
\(174\) −0.682163 −0.0517146
\(175\) 0.282561 0.0213596
\(176\) 5.27792 0.397838
\(177\) −1.24264 −0.0934026
\(178\) 2.29326 0.171887
\(179\) 21.6401 1.61746 0.808729 0.588182i \(-0.200156\pi\)
0.808729 + 0.588182i \(0.200156\pi\)
\(180\) 2.82843 0.210819
\(181\) −16.4288 −1.22114 −0.610572 0.791960i \(-0.709061\pi\)
−0.610572 + 0.791960i \(0.709061\pi\)
\(182\) −0.526610 −0.0390349
\(183\) −4.04133 −0.298744
\(184\) 0 0
\(185\) −0.828427 −0.0609072
\(186\) 1.59627 0.117044
\(187\) −27.7074 −2.02617
\(188\) 0.353113 0.0257534
\(189\) 0.682163 0.0496200
\(190\) −0.946464 −0.0686637
\(191\) −24.6797 −1.78576 −0.892879 0.450296i \(-0.851318\pi\)
−0.892879 + 0.450296i \(0.851318\pi\)
\(192\) −0.414214 −0.0298933
\(193\) −11.1429 −0.802081 −0.401040 0.916060i \(-0.631351\pi\)
−0.401040 + 0.916060i \(0.631351\pi\)
\(194\) −11.6702 −0.837873
\(195\) 0.771971 0.0552820
\(196\) −6.92016 −0.494297
\(197\) −10.9262 −0.778460 −0.389230 0.921141i \(-0.627259\pi\)
−0.389230 + 0.921141i \(0.627259\pi\)
\(198\) 14.9282 1.06090
\(199\) −21.4600 −1.52126 −0.760629 0.649187i \(-0.775109\pi\)
−0.760629 + 0.649187i \(0.775109\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −6.23921 −0.440080
\(202\) 10.6357 0.748323
\(203\) −0.465346 −0.0326609
\(204\) 2.17449 0.152245
\(205\) 7.53465 0.526243
\(206\) 12.2973 0.856796
\(207\) 0 0
\(208\) 1.86370 0.129225
\(209\) −4.99536 −0.345536
\(210\) −0.117041 −0.00807657
\(211\) 2.88812 0.198826 0.0994132 0.995046i \(-0.468303\pi\)
0.0994132 + 0.995046i \(0.468303\pi\)
\(212\) −10.2168 −0.701694
\(213\) −5.56904 −0.381584
\(214\) 9.07107 0.620085
\(215\) 0.663902 0.0452778
\(216\) −2.41421 −0.164266
\(217\) 1.08892 0.0739204
\(218\) 6.21201 0.420730
\(219\) 6.58970 0.445291
\(220\) −5.27792 −0.355837
\(221\) −9.78386 −0.658134
\(222\) 0.343146 0.0230304
\(223\) −13.5347 −0.906347 −0.453174 0.891422i \(-0.649708\pi\)
−0.453174 + 0.891422i \(0.649708\pi\)
\(224\) −0.282561 −0.0188794
\(225\) −2.82843 −0.188562
\(226\) 5.12925 0.341193
\(227\) −7.15648 −0.474992 −0.237496 0.971388i \(-0.576327\pi\)
−0.237496 + 0.971388i \(0.576327\pi\)
\(228\) 0.392038 0.0259634
\(229\) 0.818459 0.0540853 0.0270427 0.999634i \(-0.491391\pi\)
0.0270427 + 0.999634i \(0.491391\pi\)
\(230\) 0 0
\(231\) −0.617731 −0.0406437
\(232\) 1.64689 0.108123
\(233\) −22.0311 −1.44331 −0.721654 0.692254i \(-0.756618\pi\)
−0.721654 + 0.692254i \(0.756618\pi\)
\(234\) 5.27135 0.344599
\(235\) −0.353113 −0.0230346
\(236\) 3.00000 0.195283
\(237\) 1.99285 0.129449
\(238\) 1.48336 0.0961518
\(239\) −27.0892 −1.75226 −0.876129 0.482077i \(-0.839882\pi\)
−0.876129 + 0.482077i \(0.839882\pi\)
\(240\) 0.414214 0.0267374
\(241\) 6.86078 0.441942 0.220971 0.975280i \(-0.429077\pi\)
0.220971 + 0.975280i \(0.429077\pi\)
\(242\) −16.8564 −1.08357
\(243\) −10.3431 −0.663513
\(244\) 9.75663 0.624604
\(245\) 6.92016 0.442113
\(246\) −3.12096 −0.198985
\(247\) −1.76393 −0.112236
\(248\) −3.85374 −0.244712
\(249\) −0.266438 −0.0168848
\(250\) 1.00000 0.0632456
\(251\) −6.15456 −0.388472 −0.194236 0.980955i \(-0.562223\pi\)
−0.194236 + 0.980955i \(0.562223\pi\)
\(252\) −0.799203 −0.0503451
\(253\) 0 0
\(254\) −7.58114 −0.475683
\(255\) −2.17449 −0.136172
\(256\) 1.00000 0.0625000
\(257\) −7.41297 −0.462408 −0.231204 0.972905i \(-0.574267\pi\)
−0.231204 + 0.972905i \(0.574267\pi\)
\(258\) −0.274997 −0.0171206
\(259\) 0.234081 0.0145451
\(260\) −1.86370 −0.115582
\(261\) 4.65810 0.288329
\(262\) −19.8776 −1.22804
\(263\) −22.4263 −1.38287 −0.691434 0.722440i \(-0.743021\pi\)
−0.691434 + 0.722440i \(0.743021\pi\)
\(264\) 2.18618 0.134550
\(265\) 10.2168 0.627614
\(266\) 0.267434 0.0163974
\(267\) 0.949898 0.0581328
\(268\) 15.0628 0.920105
\(269\) −4.25120 −0.259200 −0.129600 0.991566i \(-0.541369\pi\)
−0.129600 + 0.991566i \(0.541369\pi\)
\(270\) 2.41421 0.146924
\(271\) 13.9859 0.849583 0.424792 0.905291i \(-0.360347\pi\)
0.424792 + 0.905291i \(0.360347\pi\)
\(272\) −5.24969 −0.318309
\(273\) −0.218129 −0.0132018
\(274\) 16.6067 1.00325
\(275\) 5.27792 0.318270
\(276\) 0 0
\(277\) 18.4023 1.10569 0.552843 0.833286i \(-0.313543\pi\)
0.552843 + 0.833286i \(0.313543\pi\)
\(278\) −16.1623 −0.969349
\(279\) −10.9000 −0.652567
\(280\) 0.282561 0.0168863
\(281\) −5.00292 −0.298449 −0.149225 0.988803i \(-0.547678\pi\)
−0.149225 + 0.988803i \(0.547678\pi\)
\(282\) 0.146264 0.00870992
\(283\) 18.9114 1.12416 0.562082 0.827082i \(-0.310001\pi\)
0.562082 + 0.827082i \(0.310001\pi\)
\(284\) 13.4448 0.797805
\(285\) −0.392038 −0.0232223
\(286\) −9.83647 −0.581643
\(287\) −2.12900 −0.125671
\(288\) 2.82843 0.166667
\(289\) 10.5592 0.621131
\(290\) −1.64689 −0.0967085
\(291\) −4.83396 −0.283372
\(292\) −15.9089 −0.931001
\(293\) −32.9481 −1.92485 −0.962425 0.271546i \(-0.912465\pi\)
−0.962425 + 0.271546i \(0.912465\pi\)
\(294\) −2.86642 −0.167173
\(295\) −3.00000 −0.174667
\(296\) −0.828427 −0.0481513
\(297\) 12.7420 0.739367
\(298\) −4.83839 −0.280281
\(299\) 0 0
\(300\) −0.414214 −0.0239146
\(301\) −0.187593 −0.0108127
\(302\) 3.19151 0.183651
\(303\) 4.40544 0.253086
\(304\) −0.946464 −0.0542834
\(305\) −9.75663 −0.558663
\(306\) −14.8484 −0.848824
\(307\) 8.01942 0.457692 0.228846 0.973463i \(-0.426505\pi\)
0.228846 + 0.973463i \(0.426505\pi\)
\(308\) 1.49133 0.0849766
\(309\) 5.09372 0.289772
\(310\) 3.85374 0.218877
\(311\) 24.6291 1.39659 0.698293 0.715812i \(-0.253943\pi\)
0.698293 + 0.715812i \(0.253943\pi\)
\(312\) 0.771971 0.0437043
\(313\) 15.0842 0.852608 0.426304 0.904580i \(-0.359815\pi\)
0.426304 + 0.904580i \(0.359815\pi\)
\(314\) 10.4288 0.588533
\(315\) 0.799203 0.0450300
\(316\) −4.81116 −0.270649
\(317\) −23.2015 −1.30312 −0.651562 0.758595i \(-0.725886\pi\)
−0.651562 + 0.758595i \(0.725886\pi\)
\(318\) −4.23194 −0.237316
\(319\) −8.69213 −0.486666
\(320\) −1.00000 −0.0559017
\(321\) 3.75736 0.209715
\(322\) 0 0
\(323\) 4.96864 0.276462
\(324\) 7.48528 0.415849
\(325\) 1.86370 0.103380
\(326\) 3.87780 0.214771
\(327\) 2.57310 0.142293
\(328\) 7.53465 0.416032
\(329\) 0.0997761 0.00550083
\(330\) −2.18618 −0.120346
\(331\) 3.25120 0.178702 0.0893511 0.996000i \(-0.471521\pi\)
0.0893511 + 0.996000i \(0.471521\pi\)
\(332\) 0.643238 0.0353023
\(333\) −2.34315 −0.128404
\(334\) −2.76681 −0.151393
\(335\) −15.0628 −0.822967
\(336\) −0.117041 −0.00638509
\(337\) −23.5714 −1.28402 −0.642009 0.766697i \(-0.721899\pi\)
−0.642009 + 0.766697i \(0.721899\pi\)
\(338\) 9.52661 0.518179
\(339\) 2.12460 0.115393
\(340\) 5.24969 0.284704
\(341\) 20.3397 1.10146
\(342\) −2.67700 −0.144756
\(343\) −3.93330 −0.212378
\(344\) 0.663902 0.0357952
\(345\) 0 0
\(346\) −19.2015 −1.03228
\(347\) 13.0980 0.703138 0.351569 0.936162i \(-0.385648\pi\)
0.351569 + 0.936162i \(0.385648\pi\)
\(348\) 0.682163 0.0365678
\(349\) −2.69412 −0.144213 −0.0721065 0.997397i \(-0.522972\pi\)
−0.0721065 + 0.997397i \(0.522972\pi\)
\(350\) −0.282561 −0.0151035
\(351\) 4.49938 0.240159
\(352\) −5.27792 −0.281314
\(353\) 3.54727 0.188802 0.0944012 0.995534i \(-0.469906\pi\)
0.0944012 + 0.995534i \(0.469906\pi\)
\(354\) 1.24264 0.0660456
\(355\) −13.4448 −0.713578
\(356\) −2.29326 −0.121542
\(357\) 0.614427 0.0325189
\(358\) −21.6401 −1.14372
\(359\) 21.8220 1.15172 0.575861 0.817548i \(-0.304667\pi\)
0.575861 + 0.817548i \(0.304667\pi\)
\(360\) −2.82843 −0.149071
\(361\) −18.1042 −0.952853
\(362\) 16.4288 0.863480
\(363\) −6.98215 −0.366468
\(364\) 0.526610 0.0276019
\(365\) 15.9089 0.832712
\(366\) 4.04133 0.211244
\(367\) −9.98177 −0.521044 −0.260522 0.965468i \(-0.583895\pi\)
−0.260522 + 0.965468i \(0.583895\pi\)
\(368\) 0 0
\(369\) 21.3112 1.10942
\(370\) 0.828427 0.0430679
\(371\) −2.88687 −0.149879
\(372\) −1.59627 −0.0827627
\(373\) 13.8537 0.717319 0.358660 0.933468i \(-0.383234\pi\)
0.358660 + 0.933468i \(0.383234\pi\)
\(374\) 27.7074 1.43272
\(375\) 0.414214 0.0213899
\(376\) −0.353113 −0.0182104
\(377\) −3.06931 −0.158077
\(378\) −0.682163 −0.0350867
\(379\) −27.1798 −1.39613 −0.698067 0.716033i \(-0.745956\pi\)
−0.698067 + 0.716033i \(0.745956\pi\)
\(380\) 0.946464 0.0485526
\(381\) −3.14021 −0.160878
\(382\) 24.6797 1.26272
\(383\) −9.17260 −0.468698 −0.234349 0.972153i \(-0.575296\pi\)
−0.234349 + 0.972153i \(0.575296\pi\)
\(384\) 0.414214 0.0211377
\(385\) −1.49133 −0.0760054
\(386\) 11.1429 0.567157
\(387\) 1.87780 0.0954539
\(388\) 11.6702 0.592466
\(389\) −16.0048 −0.811476 −0.405738 0.913989i \(-0.632985\pi\)
−0.405738 + 0.913989i \(0.632985\pi\)
\(390\) −0.771971 −0.0390903
\(391\) 0 0
\(392\) 6.92016 0.349521
\(393\) −8.23357 −0.415328
\(394\) 10.9262 0.550455
\(395\) 4.81116 0.242076
\(396\) −14.9282 −0.750170
\(397\) 28.4776 1.42925 0.714626 0.699507i \(-0.246597\pi\)
0.714626 + 0.699507i \(0.246597\pi\)
\(398\) 21.4600 1.07569
\(399\) 0.110775 0.00554567
\(400\) 1.00000 0.0500000
\(401\) −13.9535 −0.696807 −0.348403 0.937345i \(-0.613276\pi\)
−0.348403 + 0.937345i \(0.613276\pi\)
\(402\) 6.23921 0.311183
\(403\) 7.18222 0.357772
\(404\) −10.6357 −0.529145
\(405\) −7.48528 −0.371947
\(406\) 0.465346 0.0230947
\(407\) 4.37237 0.216730
\(408\) −2.17449 −0.107653
\(409\) −19.8732 −0.982664 −0.491332 0.870972i \(-0.663490\pi\)
−0.491332 + 0.870972i \(0.663490\pi\)
\(410\) −7.53465 −0.372110
\(411\) 6.87873 0.339303
\(412\) −12.2973 −0.605846
\(413\) 0.847683 0.0417118
\(414\) 0 0
\(415\) −0.643238 −0.0315753
\(416\) −1.86370 −0.0913756
\(417\) −6.69464 −0.327838
\(418\) 4.99536 0.244331
\(419\) 22.7512 1.11147 0.555735 0.831360i \(-0.312437\pi\)
0.555735 + 0.831360i \(0.312437\pi\)
\(420\) 0.117041 0.00571100
\(421\) −40.7127 −1.98422 −0.992109 0.125379i \(-0.959985\pi\)
−0.992109 + 0.125379i \(0.959985\pi\)
\(422\) −2.88812 −0.140591
\(423\) −0.998756 −0.0485612
\(424\) 10.2168 0.496172
\(425\) −5.24969 −0.254647
\(426\) 5.56904 0.269821
\(427\) 2.75684 0.133413
\(428\) −9.07107 −0.438467
\(429\) −4.07440 −0.196714
\(430\) −0.663902 −0.0320162
\(431\) −24.2843 −1.16973 −0.584866 0.811130i \(-0.698853\pi\)
−0.584866 + 0.811130i \(0.698853\pi\)
\(432\) 2.41421 0.116154
\(433\) −17.2074 −0.826933 −0.413467 0.910519i \(-0.635682\pi\)
−0.413467 + 0.910519i \(0.635682\pi\)
\(434\) −1.08892 −0.0522696
\(435\) −0.682163 −0.0327072
\(436\) −6.21201 −0.297501
\(437\) 0 0
\(438\) −6.58970 −0.314868
\(439\) 36.9825 1.76508 0.882540 0.470238i \(-0.155832\pi\)
0.882540 + 0.470238i \(0.155832\pi\)
\(440\) 5.27792 0.251615
\(441\) 19.5732 0.932056
\(442\) 9.78386 0.465371
\(443\) −15.0698 −0.715989 −0.357994 0.933724i \(-0.616539\pi\)
−0.357994 + 0.933724i \(0.616539\pi\)
\(444\) −0.343146 −0.0162850
\(445\) 2.29326 0.108711
\(446\) 13.5347 0.640884
\(447\) −2.00413 −0.0947920
\(448\) 0.282561 0.0133498
\(449\) −18.4675 −0.871535 −0.435768 0.900059i \(-0.643523\pi\)
−0.435768 + 0.900059i \(0.643523\pi\)
\(450\) 2.82843 0.133333
\(451\) −39.7673 −1.87257
\(452\) −5.12925 −0.241260
\(453\) 1.32197 0.0621114
\(454\) 7.15648 0.335870
\(455\) −0.526610 −0.0246879
\(456\) −0.392038 −0.0183589
\(457\) −28.5891 −1.33734 −0.668672 0.743558i \(-0.733137\pi\)
−0.668672 + 0.743558i \(0.733137\pi\)
\(458\) −0.818459 −0.0382441
\(459\) −12.6739 −0.591566
\(460\) 0 0
\(461\) −7.13986 −0.332536 −0.166268 0.986081i \(-0.553172\pi\)
−0.166268 + 0.986081i \(0.553172\pi\)
\(462\) 0.617731 0.0287394
\(463\) 3.77073 0.175241 0.0876203 0.996154i \(-0.472074\pi\)
0.0876203 + 0.996154i \(0.472074\pi\)
\(464\) −1.64689 −0.0764548
\(465\) 1.59627 0.0740252
\(466\) 22.0311 1.02057
\(467\) 26.5448 1.22835 0.614173 0.789171i \(-0.289490\pi\)
0.614173 + 0.789171i \(0.289490\pi\)
\(468\) −5.27135 −0.243668
\(469\) 4.25615 0.196531
\(470\) 0.353113 0.0162879
\(471\) 4.31976 0.199044
\(472\) −3.00000 −0.138086
\(473\) −3.50402 −0.161115
\(474\) −1.99285 −0.0915346
\(475\) −0.946464 −0.0434267
\(476\) −1.48336 −0.0679896
\(477\) 28.8975 1.32313
\(478\) 27.0892 1.23903
\(479\) −21.1723 −0.967385 −0.483692 0.875238i \(-0.660705\pi\)
−0.483692 + 0.875238i \(0.660705\pi\)
\(480\) −0.414214 −0.0189062
\(481\) 1.54394 0.0703977
\(482\) −6.86078 −0.312500
\(483\) 0 0
\(484\) 16.8564 0.766200
\(485\) −11.6702 −0.529917
\(486\) 10.3431 0.469175
\(487\) −30.3727 −1.37632 −0.688159 0.725560i \(-0.741581\pi\)
−0.688159 + 0.725560i \(0.741581\pi\)
\(488\) −9.75663 −0.441662
\(489\) 1.60624 0.0726366
\(490\) −6.92016 −0.312621
\(491\) 28.0706 1.26681 0.633403 0.773822i \(-0.281658\pi\)
0.633403 + 0.773822i \(0.281658\pi\)
\(492\) 3.12096 0.140704
\(493\) 8.64564 0.389380
\(494\) 1.76393 0.0793628
\(495\) 14.9282 0.670973
\(496\) 3.85374 0.173038
\(497\) 3.79899 0.170408
\(498\) 0.266438 0.0119394
\(499\) 1.52719 0.0683666 0.0341833 0.999416i \(-0.489117\pi\)
0.0341833 + 0.999416i \(0.489117\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −1.14605 −0.0512018
\(502\) 6.15456 0.274691
\(503\) −30.1177 −1.34288 −0.671441 0.741058i \(-0.734324\pi\)
−0.671441 + 0.741058i \(0.734324\pi\)
\(504\) 0.799203 0.0355993
\(505\) 10.6357 0.473281
\(506\) 0 0
\(507\) 3.94605 0.175250
\(508\) 7.58114 0.336359
\(509\) 27.4988 1.21886 0.609432 0.792839i \(-0.291398\pi\)
0.609432 + 0.792839i \(0.291398\pi\)
\(510\) 2.17449 0.0962881
\(511\) −4.49525 −0.198858
\(512\) −1.00000 −0.0441942
\(513\) −2.28497 −0.100884
\(514\) 7.41297 0.326972
\(515\) 12.2973 0.541885
\(516\) 0.274997 0.0121061
\(517\) 1.86370 0.0819656
\(518\) −0.234081 −0.0102849
\(519\) −7.95351 −0.349120
\(520\) 1.86370 0.0817288
\(521\) −41.2097 −1.80543 −0.902715 0.430240i \(-0.858429\pi\)
−0.902715 + 0.430240i \(0.858429\pi\)
\(522\) −4.65810 −0.203879
\(523\) 2.30390 0.100743 0.0503714 0.998731i \(-0.483960\pi\)
0.0503714 + 0.998731i \(0.483960\pi\)
\(524\) 19.8776 0.868356
\(525\) −0.117041 −0.00510807
\(526\) 22.4263 0.977835
\(527\) −20.2309 −0.881272
\(528\) −2.18618 −0.0951415
\(529\) 0 0
\(530\) −10.2168 −0.443790
\(531\) −8.48528 −0.368230
\(532\) −0.267434 −0.0115947
\(533\) −14.0424 −0.608242
\(534\) −0.949898 −0.0411061
\(535\) 9.07107 0.392176
\(536\) −15.0628 −0.650613
\(537\) −8.96363 −0.386809
\(538\) 4.25120 0.183282
\(539\) −36.5240 −1.57320
\(540\) −2.41421 −0.103891
\(541\) 25.1430 1.08098 0.540492 0.841349i \(-0.318238\pi\)
0.540492 + 0.841349i \(0.318238\pi\)
\(542\) −13.9859 −0.600746
\(543\) 6.80504 0.292032
\(544\) 5.24969 0.225079
\(545\) 6.21201 0.266093
\(546\) 0.218129 0.00933506
\(547\) −43.1127 −1.84337 −0.921683 0.387944i \(-0.873185\pi\)
−0.921683 + 0.387944i \(0.873185\pi\)
\(548\) −16.6067 −0.709404
\(549\) −27.5959 −1.17777
\(550\) −5.27792 −0.225051
\(551\) 1.55872 0.0664036
\(552\) 0 0
\(553\) −1.35945 −0.0578096
\(554\) −18.4023 −0.781838
\(555\) 0.343146 0.0145657
\(556\) 16.1623 0.685434
\(557\) −24.3814 −1.03307 −0.516536 0.856265i \(-0.672779\pi\)
−0.516536 + 0.856265i \(0.672779\pi\)
\(558\) 10.9000 0.461434
\(559\) −1.23732 −0.0523329
\(560\) −0.282561 −0.0119404
\(561\) 11.4768 0.484550
\(562\) 5.00292 0.211035
\(563\) −42.5108 −1.79162 −0.895808 0.444441i \(-0.853402\pi\)
−0.895808 + 0.444441i \(0.853402\pi\)
\(564\) −0.146264 −0.00615884
\(565\) 5.12925 0.215789
\(566\) −18.9114 −0.794903
\(567\) 2.11505 0.0888237
\(568\) −13.4448 −0.564133
\(569\) 29.2954 1.22813 0.614065 0.789256i \(-0.289533\pi\)
0.614065 + 0.789256i \(0.289533\pi\)
\(570\) 0.392038 0.0164207
\(571\) 35.8550 1.50049 0.750243 0.661162i \(-0.229936\pi\)
0.750243 + 0.661162i \(0.229936\pi\)
\(572\) 9.83647 0.411284
\(573\) 10.2227 0.427058
\(574\) 2.12900 0.0888627
\(575\) 0 0
\(576\) −2.82843 −0.117851
\(577\) 14.7477 0.613955 0.306977 0.951717i \(-0.400682\pi\)
0.306977 + 0.951717i \(0.400682\pi\)
\(578\) −10.5592 −0.439206
\(579\) 4.61553 0.191815
\(580\) 1.64689 0.0683832
\(581\) 0.181754 0.00754043
\(582\) 4.83396 0.200374
\(583\) −53.9235 −2.23328
\(584\) 15.9089 0.658317
\(585\) 5.27135 0.217943
\(586\) 32.9481 1.36108
\(587\) −44.8649 −1.85177 −0.925887 0.377801i \(-0.876680\pi\)
−0.925887 + 0.377801i \(0.876680\pi\)
\(588\) 2.86642 0.118209
\(589\) −3.64742 −0.150289
\(590\) 3.00000 0.123508
\(591\) 4.52579 0.186166
\(592\) 0.828427 0.0340481
\(593\) −8.71603 −0.357924 −0.178962 0.983856i \(-0.557274\pi\)
−0.178962 + 0.983856i \(0.557274\pi\)
\(594\) −12.7420 −0.522811
\(595\) 1.48336 0.0608117
\(596\) 4.83839 0.198188
\(597\) 8.88901 0.363803
\(598\) 0 0
\(599\) −22.1390 −0.904574 −0.452287 0.891873i \(-0.649392\pi\)
−0.452287 + 0.891873i \(0.649392\pi\)
\(600\) 0.414214 0.0169102
\(601\) 40.3898 1.64753 0.823767 0.566928i \(-0.191868\pi\)
0.823767 + 0.566928i \(0.191868\pi\)
\(602\) 0.187593 0.00764572
\(603\) −42.6040 −1.73497
\(604\) −3.19151 −0.129861
\(605\) −16.8564 −0.685310
\(606\) −4.40544 −0.178959
\(607\) −17.8158 −0.723119 −0.361560 0.932349i \(-0.617756\pi\)
−0.361560 + 0.932349i \(0.617756\pi\)
\(608\) 0.946464 0.0383842
\(609\) 0.192753 0.00781073
\(610\) 9.75663 0.395034
\(611\) 0.658099 0.0266238
\(612\) 14.8484 0.600209
\(613\) 15.3499 0.619976 0.309988 0.950741i \(-0.399675\pi\)
0.309988 + 0.950741i \(0.399675\pi\)
\(614\) −8.01942 −0.323637
\(615\) −3.12096 −0.125849
\(616\) −1.49133 −0.0600876
\(617\) −20.2454 −0.815049 −0.407524 0.913194i \(-0.633608\pi\)
−0.407524 + 0.913194i \(0.633608\pi\)
\(618\) −5.09372 −0.204900
\(619\) −10.0426 −0.403646 −0.201823 0.979422i \(-0.564687\pi\)
−0.201823 + 0.979422i \(0.564687\pi\)
\(620\) −3.85374 −0.154770
\(621\) 0 0
\(622\) −24.6291 −0.987535
\(623\) −0.647985 −0.0259610
\(624\) −0.771971 −0.0309036
\(625\) 1.00000 0.0400000
\(626\) −15.0842 −0.602885
\(627\) 2.06914 0.0826337
\(628\) −10.4288 −0.416155
\(629\) −4.34898 −0.173405
\(630\) −0.799203 −0.0318410
\(631\) 24.4305 0.972561 0.486281 0.873803i \(-0.338353\pi\)
0.486281 + 0.873803i \(0.338353\pi\)
\(632\) 4.81116 0.191378
\(633\) −1.19630 −0.0475486
\(634\) 23.2015 0.921448
\(635\) −7.58114 −0.300848
\(636\) 4.23194 0.167807
\(637\) −12.8971 −0.511003
\(638\) 8.69213 0.344125
\(639\) −38.0278 −1.50436
\(640\) 1.00000 0.0395285
\(641\) −19.3395 −0.763864 −0.381932 0.924190i \(-0.624741\pi\)
−0.381932 + 0.924190i \(0.624741\pi\)
\(642\) −3.75736 −0.148291
\(643\) 35.7466 1.40971 0.704854 0.709352i \(-0.251012\pi\)
0.704854 + 0.709352i \(0.251012\pi\)
\(644\) 0 0
\(645\) −0.274997 −0.0108280
\(646\) −4.96864 −0.195488
\(647\) −31.8753 −1.25315 −0.626574 0.779362i \(-0.715543\pi\)
−0.626574 + 0.779362i \(0.715543\pi\)
\(648\) −7.48528 −0.294050
\(649\) 15.8338 0.621529
\(650\) −1.86370 −0.0731005
\(651\) −0.451044 −0.0176778
\(652\) −3.87780 −0.151866
\(653\) −31.3226 −1.22575 −0.612875 0.790180i \(-0.709987\pi\)
−0.612875 + 0.790180i \(0.709987\pi\)
\(654\) −2.57310 −0.100616
\(655\) −19.8776 −0.776682
\(656\) −7.53465 −0.294179
\(657\) 44.9973 1.75551
\(658\) −0.0997761 −0.00388968
\(659\) −14.1631 −0.551717 −0.275858 0.961198i \(-0.588962\pi\)
−0.275858 + 0.961198i \(0.588962\pi\)
\(660\) 2.18618 0.0850971
\(661\) −34.3365 −1.33554 −0.667768 0.744369i \(-0.732750\pi\)
−0.667768 + 0.744369i \(0.732750\pi\)
\(662\) −3.25120 −0.126361
\(663\) 4.05261 0.157390
\(664\) −0.643238 −0.0249625
\(665\) 0.267434 0.0103706
\(666\) 2.34315 0.0907951
\(667\) 0 0
\(668\) 2.76681 0.107051
\(669\) 5.60624 0.216750
\(670\) 15.0628 0.581926
\(671\) 51.4947 1.98793
\(672\) 0.117041 0.00451494
\(673\) −22.5656 −0.869841 −0.434921 0.900469i \(-0.643224\pi\)
−0.434921 + 0.900469i \(0.643224\pi\)
\(674\) 23.5714 0.907938
\(675\) 2.41421 0.0929231
\(676\) −9.52661 −0.366408
\(677\) −2.61539 −0.100518 −0.0502588 0.998736i \(-0.516005\pi\)
−0.0502588 + 0.998736i \(0.516005\pi\)
\(678\) −2.12460 −0.0815949
\(679\) 3.29755 0.126548
\(680\) −5.24969 −0.201316
\(681\) 2.96431 0.113593
\(682\) −20.3397 −0.778847
\(683\) 3.99911 0.153022 0.0765108 0.997069i \(-0.475622\pi\)
0.0765108 + 0.997069i \(0.475622\pi\)
\(684\) 2.67700 0.102358
\(685\) 16.6067 0.634510
\(686\) 3.93330 0.150174
\(687\) −0.339017 −0.0129343
\(688\) −0.663902 −0.0253110
\(689\) −19.0411 −0.725409
\(690\) 0 0
\(691\) −30.3616 −1.15501 −0.577505 0.816387i \(-0.695974\pi\)
−0.577505 + 0.816387i \(0.695974\pi\)
\(692\) 19.2015 0.729931
\(693\) −4.21813 −0.160233
\(694\) −13.0980 −0.497194
\(695\) −16.1623 −0.613070
\(696\) −0.682163 −0.0258573
\(697\) 39.5546 1.49824
\(698\) 2.69412 0.101974
\(699\) 9.12560 0.345162
\(700\) 0.282561 0.0106798
\(701\) 31.0073 1.17113 0.585564 0.810626i \(-0.300873\pi\)
0.585564 + 0.810626i \(0.300873\pi\)
\(702\) −4.49938 −0.169818
\(703\) −0.784076 −0.0295720
\(704\) 5.27792 0.198919
\(705\) 0.146264 0.00550863
\(706\) −3.54727 −0.133503
\(707\) −3.00523 −0.113023
\(708\) −1.24264 −0.0467013
\(709\) 28.9157 1.08595 0.542977 0.839748i \(-0.317297\pi\)
0.542977 + 0.839748i \(0.317297\pi\)
\(710\) 13.4448 0.504576
\(711\) 13.6080 0.510341
\(712\) 2.29326 0.0859435
\(713\) 0 0
\(714\) −0.614427 −0.0229943
\(715\) −9.83647 −0.367863
\(716\) 21.6401 0.808729
\(717\) 11.2207 0.419046
\(718\) −21.8220 −0.814391
\(719\) −2.06340 −0.0769519 −0.0384759 0.999260i \(-0.512250\pi\)
−0.0384759 + 0.999260i \(0.512250\pi\)
\(720\) 2.82843 0.105409
\(721\) −3.47475 −0.129406
\(722\) 18.1042 0.673769
\(723\) −2.84183 −0.105689
\(724\) −16.4288 −0.610572
\(725\) −1.64689 −0.0611638
\(726\) 6.98215 0.259132
\(727\) −7.61304 −0.282352 −0.141176 0.989985i \(-0.545088\pi\)
−0.141176 + 0.989985i \(0.545088\pi\)
\(728\) −0.526610 −0.0195175
\(729\) −18.1716 −0.673021
\(730\) −15.9089 −0.588817
\(731\) 3.48528 0.128908
\(732\) −4.04133 −0.149372
\(733\) 0.545866 0.0201620 0.0100810 0.999949i \(-0.496791\pi\)
0.0100810 + 0.999949i \(0.496791\pi\)
\(734\) 9.98177 0.368434
\(735\) −2.86642 −0.105730
\(736\) 0 0
\(737\) 79.5001 2.92842
\(738\) −21.3112 −0.784477
\(739\) −1.49333 −0.0549329 −0.0274664 0.999623i \(-0.508744\pi\)
−0.0274664 + 0.999623i \(0.508744\pi\)
\(740\) −0.828427 −0.0304536
\(741\) 0.730643 0.0268408
\(742\) 2.88687 0.105981
\(743\) 42.4290 1.55657 0.778285 0.627912i \(-0.216090\pi\)
0.778285 + 0.627912i \(0.216090\pi\)
\(744\) 1.59627 0.0585221
\(745\) −4.83839 −0.177265
\(746\) −13.8537 −0.507221
\(747\) −1.81935 −0.0665666
\(748\) −27.7074 −1.01308
\(749\) −2.56313 −0.0936548
\(750\) −0.414214 −0.0151249
\(751\) 32.0575 1.16979 0.584897 0.811107i \(-0.301135\pi\)
0.584897 + 0.811107i \(0.301135\pi\)
\(752\) 0.353113 0.0128767
\(753\) 2.54930 0.0929017
\(754\) 3.06931 0.111778
\(755\) 3.19151 0.116151
\(756\) 0.682163 0.0248100
\(757\) 24.4763 0.889608 0.444804 0.895628i \(-0.353273\pi\)
0.444804 + 0.895628i \(0.353273\pi\)
\(758\) 27.1798 0.987215
\(759\) 0 0
\(760\) −0.946464 −0.0343318
\(761\) −19.3184 −0.700290 −0.350145 0.936696i \(-0.613868\pi\)
−0.350145 + 0.936696i \(0.613868\pi\)
\(762\) 3.14021 0.113758
\(763\) −1.75527 −0.0635451
\(764\) −24.6797 −0.892879
\(765\) −14.8484 −0.536844
\(766\) 9.17260 0.331420
\(767\) 5.59111 0.201883
\(768\) −0.414214 −0.0149466
\(769\) 30.5045 1.10002 0.550010 0.835158i \(-0.314624\pi\)
0.550010 + 0.835158i \(0.314624\pi\)
\(770\) 1.49133 0.0537439
\(771\) 3.07055 0.110583
\(772\) −11.1429 −0.401040
\(773\) −18.7589 −0.674711 −0.337355 0.941377i \(-0.609532\pi\)
−0.337355 + 0.941377i \(0.609532\pi\)
\(774\) −1.87780 −0.0674961
\(775\) 3.85374 0.138430
\(776\) −11.6702 −0.418937
\(777\) −0.0969596 −0.00347841
\(778\) 16.0048 0.573800
\(779\) 7.13128 0.255504
\(780\) 0.771971 0.0276410
\(781\) 70.9608 2.53918
\(782\) 0 0
\(783\) −3.97594 −0.142088
\(784\) −6.92016 −0.247149
\(785\) 10.4288 0.372221
\(786\) 8.23357 0.293682
\(787\) −25.0638 −0.893428 −0.446714 0.894677i \(-0.647406\pi\)
−0.446714 + 0.894677i \(0.647406\pi\)
\(788\) −10.9262 −0.389230
\(789\) 9.28929 0.330708
\(790\) −4.81116 −0.171173
\(791\) −1.44933 −0.0515321
\(792\) 14.9282 0.530451
\(793\) 18.1835 0.645714
\(794\) −28.4776 −1.01063
\(795\) −4.23194 −0.150092
\(796\) −21.4600 −0.760629
\(797\) −30.1700 −1.06868 −0.534338 0.845271i \(-0.679439\pi\)
−0.534338 + 0.845271i \(0.679439\pi\)
\(798\) −0.110775 −0.00392138
\(799\) −1.85374 −0.0655805
\(800\) −1.00000 −0.0353553
\(801\) 6.48631 0.229183
\(802\) 13.9535 0.492717
\(803\) −83.9661 −2.96310
\(804\) −6.23921 −0.220040
\(805\) 0 0
\(806\) −7.18222 −0.252983
\(807\) 1.76090 0.0619868
\(808\) 10.6357 0.374162
\(809\) −23.4794 −0.825491 −0.412746 0.910846i \(-0.635430\pi\)
−0.412746 + 0.910846i \(0.635430\pi\)
\(810\) 7.48528 0.263006
\(811\) −0.772718 −0.0271338 −0.0135669 0.999908i \(-0.504319\pi\)
−0.0135669 + 0.999908i \(0.504319\pi\)
\(812\) −0.465346 −0.0163304
\(813\) −5.79315 −0.203175
\(814\) −4.37237 −0.153251
\(815\) 3.87780 0.135833
\(816\) 2.17449 0.0761225
\(817\) 0.628359 0.0219835
\(818\) 19.8732 0.694849
\(819\) −1.48948 −0.0520466
\(820\) 7.53465 0.263122
\(821\) 22.8683 0.798109 0.399055 0.916927i \(-0.369338\pi\)
0.399055 + 0.916927i \(0.369338\pi\)
\(822\) −6.87873 −0.239923
\(823\) 29.3381 1.02266 0.511331 0.859384i \(-0.329153\pi\)
0.511331 + 0.859384i \(0.329153\pi\)
\(824\) 12.2973 0.428398
\(825\) −2.18618 −0.0761132
\(826\) −0.847683 −0.0294947
\(827\) −42.1809 −1.46678 −0.733388 0.679811i \(-0.762062\pi\)
−0.733388 + 0.679811i \(0.762062\pi\)
\(828\) 0 0
\(829\) −3.69089 −0.128190 −0.0640949 0.997944i \(-0.520416\pi\)
−0.0640949 + 0.997944i \(0.520416\pi\)
\(830\) 0.643238 0.0223271
\(831\) −7.62247 −0.264421
\(832\) 1.86370 0.0646123
\(833\) 36.3287 1.25871
\(834\) 6.69464 0.231816
\(835\) −2.76681 −0.0957495
\(836\) −4.99536 −0.172768
\(837\) 9.30374 0.321584
\(838\) −22.7512 −0.785928
\(839\) 38.1981 1.31874 0.659372 0.751817i \(-0.270822\pi\)
0.659372 + 0.751817i \(0.270822\pi\)
\(840\) −0.117041 −0.00403829
\(841\) −26.2878 −0.906475
\(842\) 40.7127 1.40305
\(843\) 2.07228 0.0713730
\(844\) 2.88812 0.0994132
\(845\) 9.52661 0.327725
\(846\) 0.998756 0.0343379
\(847\) 4.76296 0.163657
\(848\) −10.2168 −0.350847
\(849\) −7.83334 −0.268839
\(850\) 5.24969 0.180063
\(851\) 0 0
\(852\) −5.56904 −0.190792
\(853\) −15.5099 −0.531049 −0.265525 0.964104i \(-0.585545\pi\)
−0.265525 + 0.964104i \(0.585545\pi\)
\(854\) −2.75684 −0.0943373
\(855\) −2.67700 −0.0915516
\(856\) 9.07107 0.310043
\(857\) −31.5049 −1.07619 −0.538093 0.842886i \(-0.680855\pi\)
−0.538093 + 0.842886i \(0.680855\pi\)
\(858\) 4.07440 0.139098
\(859\) −41.9647 −1.43182 −0.715908 0.698195i \(-0.753987\pi\)
−0.715908 + 0.698195i \(0.753987\pi\)
\(860\) 0.663902 0.0226389
\(861\) 0.881861 0.0300537
\(862\) 24.2843 0.827126
\(863\) −12.3497 −0.420387 −0.210194 0.977660i \(-0.567409\pi\)
−0.210194 + 0.977660i \(0.567409\pi\)
\(864\) −2.41421 −0.0821332
\(865\) −19.2015 −0.652870
\(866\) 17.2074 0.584730
\(867\) −4.37378 −0.148541
\(868\) 1.08892 0.0369602
\(869\) −25.3929 −0.861396
\(870\) 0.682163 0.0231275
\(871\) 28.0725 0.951202
\(872\) 6.21201 0.210365
\(873\) −33.0084 −1.11716
\(874\) 0 0
\(875\) −0.282561 −0.00955231
\(876\) 6.58970 0.222645
\(877\) 29.9235 1.01045 0.505223 0.862989i \(-0.331410\pi\)
0.505223 + 0.862989i \(0.331410\pi\)
\(878\) −36.9825 −1.24810
\(879\) 13.6476 0.460321
\(880\) −5.27792 −0.177919
\(881\) −8.56618 −0.288602 −0.144301 0.989534i \(-0.546093\pi\)
−0.144301 + 0.989534i \(0.546093\pi\)
\(882\) −19.5732 −0.659063
\(883\) 19.9994 0.673035 0.336517 0.941677i \(-0.390751\pi\)
0.336517 + 0.941677i \(0.390751\pi\)
\(884\) −9.78386 −0.329067
\(885\) 1.24264 0.0417709
\(886\) 15.0698 0.506281
\(887\) 53.1363 1.78414 0.892071 0.451896i \(-0.149252\pi\)
0.892071 + 0.451896i \(0.149252\pi\)
\(888\) 0.343146 0.0115152
\(889\) 2.14214 0.0718449
\(890\) −2.29326 −0.0768702
\(891\) 39.5067 1.32352
\(892\) −13.5347 −0.453174
\(893\) −0.334209 −0.0111839
\(894\) 2.00413 0.0670281
\(895\) −21.6401 −0.723349
\(896\) −0.282561 −0.00943970
\(897\) 0 0
\(898\) 18.4675 0.616269
\(899\) −6.34667 −0.211673
\(900\) −2.82843 −0.0942809
\(901\) 53.6351 1.78684
\(902\) 39.7673 1.32411
\(903\) 0.0777036 0.00258581
\(904\) 5.12925 0.170596
\(905\) 16.4288 0.546113
\(906\) −1.32197 −0.0439194
\(907\) 21.8713 0.726223 0.363112 0.931746i \(-0.381714\pi\)
0.363112 + 0.931746i \(0.381714\pi\)
\(908\) −7.15648 −0.237496
\(909\) 30.0822 0.997765
\(910\) 0.526610 0.0174570
\(911\) −55.8763 −1.85127 −0.925633 0.378423i \(-0.876467\pi\)
−0.925633 + 0.378423i \(0.876467\pi\)
\(912\) 0.392038 0.0129817
\(913\) 3.39496 0.112357
\(914\) 28.5891 0.945645
\(915\) 4.04133 0.133602
\(916\) 0.818459 0.0270427
\(917\) 5.61663 0.185478
\(918\) 12.6739 0.418300
\(919\) −19.4789 −0.642549 −0.321274 0.946986i \(-0.604111\pi\)
−0.321274 + 0.946986i \(0.604111\pi\)
\(920\) 0 0
\(921\) −3.32175 −0.109455
\(922\) 7.13986 0.235139
\(923\) 25.0572 0.824768
\(924\) −0.617731 −0.0203218
\(925\) 0.828427 0.0272385
\(926\) −3.77073 −0.123914
\(927\) 34.7821 1.14239
\(928\) 1.64689 0.0540617
\(929\) 49.9789 1.63976 0.819878 0.572539i \(-0.194041\pi\)
0.819878 + 0.572539i \(0.194041\pi\)
\(930\) −1.59627 −0.0523437
\(931\) 6.54968 0.214657
\(932\) −22.0311 −0.721654
\(933\) −10.2017 −0.333988
\(934\) −26.5448 −0.868572
\(935\) 27.7074 0.906130
\(936\) 5.27135 0.172299
\(937\) 26.2127 0.856330 0.428165 0.903701i \(-0.359160\pi\)
0.428165 + 0.903701i \(0.359160\pi\)
\(938\) −4.25615 −0.138968
\(939\) −6.24807 −0.203898
\(940\) −0.353113 −0.0115173
\(941\) −23.1300 −0.754015 −0.377008 0.926210i \(-0.623047\pi\)
−0.377008 + 0.926210i \(0.623047\pi\)
\(942\) −4.31976 −0.140745
\(943\) 0 0
\(944\) 3.00000 0.0976417
\(945\) −0.682163 −0.0221908
\(946\) 3.50402 0.113926
\(947\) −31.2797 −1.01645 −0.508227 0.861223i \(-0.669699\pi\)
−0.508227 + 0.861223i \(0.669699\pi\)
\(948\) 1.99285 0.0647247
\(949\) −29.6496 −0.962465
\(950\) 0.946464 0.0307073
\(951\) 9.61037 0.311637
\(952\) 1.48336 0.0480759
\(953\) 46.1117 1.49370 0.746852 0.664990i \(-0.231564\pi\)
0.746852 + 0.664990i \(0.231564\pi\)
\(954\) −28.8975 −0.935592
\(955\) 24.6797 0.798615
\(956\) −27.0892 −0.876129
\(957\) 3.60040 0.116384
\(958\) 21.1723 0.684044
\(959\) −4.69241 −0.151526
\(960\) 0.414214 0.0133687
\(961\) −16.1487 −0.520927
\(962\) −1.54394 −0.0497787
\(963\) 25.6569 0.826781
\(964\) 6.86078 0.220971
\(965\) 11.1429 0.358702
\(966\) 0 0
\(967\) −17.5750 −0.565175 −0.282587 0.959242i \(-0.591193\pi\)
−0.282587 + 0.959242i \(0.591193\pi\)
\(968\) −16.8564 −0.541785
\(969\) −2.05808 −0.0661150
\(970\) 11.6702 0.374708
\(971\) 21.9107 0.703148 0.351574 0.936160i \(-0.385647\pi\)
0.351574 + 0.936160i \(0.385647\pi\)
\(972\) −10.3431 −0.331757
\(973\) 4.56683 0.146406
\(974\) 30.3727 0.973204
\(975\) −0.771971 −0.0247229
\(976\) 9.75663 0.312302
\(977\) −24.4929 −0.783597 −0.391798 0.920051i \(-0.628147\pi\)
−0.391798 + 0.920051i \(0.628147\pi\)
\(978\) −1.60624 −0.0513618
\(979\) −12.1036 −0.386833
\(980\) 6.92016 0.221056
\(981\) 17.5702 0.560974
\(982\) −28.0706 −0.895767
\(983\) −11.9177 −0.380115 −0.190057 0.981773i \(-0.560867\pi\)
−0.190057 + 0.981773i \(0.560867\pi\)
\(984\) −3.12096 −0.0994924
\(985\) 10.9262 0.348138
\(986\) −8.64564 −0.275333
\(987\) −0.0413286 −0.00131550
\(988\) −1.76393 −0.0561180
\(989\) 0 0
\(990\) −14.9282 −0.474449
\(991\) 53.9572 1.71401 0.857004 0.515310i \(-0.172323\pi\)
0.857004 + 0.515310i \(0.172323\pi\)
\(992\) −3.85374 −0.122356
\(993\) −1.34669 −0.0427360
\(994\) −3.79899 −0.120497
\(995\) 21.4600 0.680327
\(996\) −0.266438 −0.00844241
\(997\) −9.63795 −0.305237 −0.152618 0.988285i \(-0.548771\pi\)
−0.152618 + 0.988285i \(0.548771\pi\)
\(998\) −1.52719 −0.0483425
\(999\) 2.00000 0.0632772
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 5290.2.a.u.1.1 4
23.22 odd 2 5290.2.a.v.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5290.2.a.u.1.1 4 1.1 even 1 trivial
5290.2.a.v.1.2 yes 4 23.22 odd 2