Properties

Label 5586.2.a.be.1.1
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $2$
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] = [5586,2,Mod(1,5586)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5586, 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("5586.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.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: \(44.6044345691\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 5586.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.23607 q^{10} -2.00000 q^{11} +1.00000 q^{12} -5.23607 q^{13} -1.23607 q^{15} +1.00000 q^{16} -2.47214 q^{17} -1.00000 q^{18} +1.00000 q^{19} -1.23607 q^{20} +2.00000 q^{22} -5.70820 q^{23} -1.00000 q^{24} -3.47214 q^{25} +5.23607 q^{26} +1.00000 q^{27} +8.47214 q^{29} +1.23607 q^{30} -7.70820 q^{31} -1.00000 q^{32} -2.00000 q^{33} +2.47214 q^{34} +1.00000 q^{36} +0.763932 q^{37} -1.00000 q^{38} -5.23607 q^{39} +1.23607 q^{40} +6.94427 q^{41} +1.52786 q^{43} -2.00000 q^{44} -1.23607 q^{45} +5.70820 q^{46} -2.76393 q^{47} +1.00000 q^{48} +3.47214 q^{50} -2.47214 q^{51} -5.23607 q^{52} +8.47214 q^{53} -1.00000 q^{54} +2.47214 q^{55} +1.00000 q^{57} -8.47214 q^{58} +12.9443 q^{59} -1.23607 q^{60} -12.4721 q^{61} +7.70820 q^{62} +1.00000 q^{64} +6.47214 q^{65} +2.00000 q^{66} +10.9443 q^{67} -2.47214 q^{68} -5.70820 q^{69} +14.4721 q^{71} -1.00000 q^{72} -6.00000 q^{73} -0.763932 q^{74} -3.47214 q^{75} +1.00000 q^{76} +5.23607 q^{78} +12.1803 q^{79} -1.23607 q^{80} +1.00000 q^{81} -6.94427 q^{82} -8.94427 q^{83} +3.05573 q^{85} -1.52786 q^{86} +8.47214 q^{87} +2.00000 q^{88} +2.94427 q^{89} +1.23607 q^{90} -5.70820 q^{92} -7.70820 q^{93} +2.76393 q^{94} -1.23607 q^{95} -1.00000 q^{96} -10.4721 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} + 2 q^{12} - 6 q^{13} + 2 q^{15} + 2 q^{16} + 4 q^{17} - 2 q^{18} + 2 q^{19} + 2 q^{20} + 4 q^{22} + 2 q^{23} - 2 q^{24} + 2 q^{25} + 6 q^{26} + 2 q^{27} + 8 q^{29} - 2 q^{30} - 2 q^{31} - 2 q^{32} - 4 q^{33} - 4 q^{34} + 2 q^{36} + 6 q^{37} - 2 q^{38} - 6 q^{39} - 2 q^{40} - 4 q^{41} + 12 q^{43} - 4 q^{44} + 2 q^{45} - 2 q^{46} - 10 q^{47} + 2 q^{48} - 2 q^{50} + 4 q^{51} - 6 q^{52} + 8 q^{53} - 2 q^{54} - 4 q^{55} + 2 q^{57} - 8 q^{58} + 8 q^{59} + 2 q^{60} - 16 q^{61} + 2 q^{62} + 2 q^{64} + 4 q^{65} + 4 q^{66} + 4 q^{67} + 4 q^{68} + 2 q^{69} + 20 q^{71} - 2 q^{72} - 12 q^{73} - 6 q^{74} + 2 q^{75} + 2 q^{76} + 6 q^{78} + 2 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} + 24 q^{85} - 12 q^{86} + 8 q^{87} + 4 q^{88} - 12 q^{89} - 2 q^{90} + 2 q^{92} - 2 q^{93} + 10 q^{94} + 2 q^{95} - 2 q^{96} - 12 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.23607 0.390879
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) −5.23607 −1.45222 −0.726112 0.687576i \(-0.758675\pi\)
−0.726112 + 0.687576i \(0.758675\pi\)
\(14\) 0 0
\(15\) −1.23607 −0.319151
\(16\) 1.00000 0.250000
\(17\) −2.47214 −0.599581 −0.299791 0.954005i \(-0.596917\pi\)
−0.299791 + 0.954005i \(0.596917\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) −1.23607 −0.276393
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −5.70820 −1.19024 −0.595121 0.803636i \(-0.702896\pi\)
−0.595121 + 0.803636i \(0.702896\pi\)
\(24\) −1.00000 −0.204124
\(25\) −3.47214 −0.694427
\(26\) 5.23607 1.02688
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.47214 1.57324 0.786618 0.617440i \(-0.211830\pi\)
0.786618 + 0.617440i \(0.211830\pi\)
\(30\) 1.23607 0.225674
\(31\) −7.70820 −1.38443 −0.692217 0.721689i \(-0.743366\pi\)
−0.692217 + 0.721689i \(0.743366\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) 2.47214 0.423968
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0.763932 0.125590 0.0627948 0.998026i \(-0.479999\pi\)
0.0627948 + 0.998026i \(0.479999\pi\)
\(38\) −1.00000 −0.162221
\(39\) −5.23607 −0.838442
\(40\) 1.23607 0.195440
\(41\) 6.94427 1.08451 0.542257 0.840213i \(-0.317570\pi\)
0.542257 + 0.840213i \(0.317570\pi\)
\(42\) 0 0
\(43\) 1.52786 0.232997 0.116499 0.993191i \(-0.462833\pi\)
0.116499 + 0.993191i \(0.462833\pi\)
\(44\) −2.00000 −0.301511
\(45\) −1.23607 −0.184262
\(46\) 5.70820 0.841629
\(47\) −2.76393 −0.403161 −0.201580 0.979472i \(-0.564608\pi\)
−0.201580 + 0.979472i \(0.564608\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 3.47214 0.491034
\(51\) −2.47214 −0.346168
\(52\) −5.23607 −0.726112
\(53\) 8.47214 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.47214 0.333343
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) −8.47214 −1.11245
\(59\) 12.9443 1.68520 0.842600 0.538539i \(-0.181024\pi\)
0.842600 + 0.538539i \(0.181024\pi\)
\(60\) −1.23607 −0.159576
\(61\) −12.4721 −1.59689 −0.798447 0.602066i \(-0.794345\pi\)
−0.798447 + 0.602066i \(0.794345\pi\)
\(62\) 7.70820 0.978943
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.47214 0.802770
\(66\) 2.00000 0.246183
\(67\) 10.9443 1.33706 0.668528 0.743687i \(-0.266925\pi\)
0.668528 + 0.743687i \(0.266925\pi\)
\(68\) −2.47214 −0.299791
\(69\) −5.70820 −0.687187
\(70\) 0 0
\(71\) 14.4721 1.71753 0.858763 0.512373i \(-0.171233\pi\)
0.858763 + 0.512373i \(0.171233\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −0.763932 −0.0888053
\(75\) −3.47214 −0.400928
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 5.23607 0.592868
\(79\) 12.1803 1.37040 0.685198 0.728357i \(-0.259716\pi\)
0.685198 + 0.728357i \(0.259716\pi\)
\(80\) −1.23607 −0.138197
\(81\) 1.00000 0.111111
\(82\) −6.94427 −0.766867
\(83\) −8.94427 −0.981761 −0.490881 0.871227i \(-0.663325\pi\)
−0.490881 + 0.871227i \(0.663325\pi\)
\(84\) 0 0
\(85\) 3.05573 0.331440
\(86\) −1.52786 −0.164754
\(87\) 8.47214 0.908308
\(88\) 2.00000 0.213201
\(89\) 2.94427 0.312092 0.156046 0.987750i \(-0.450125\pi\)
0.156046 + 0.987750i \(0.450125\pi\)
\(90\) 1.23607 0.130293
\(91\) 0 0
\(92\) −5.70820 −0.595121
\(93\) −7.70820 −0.799304
\(94\) 2.76393 0.285078
\(95\) −1.23607 −0.126818
\(96\) −1.00000 −0.102062
\(97\) −10.4721 −1.06328 −0.531642 0.846969i \(-0.678425\pi\)
−0.531642 + 0.846969i \(0.678425\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −3.47214 −0.347214
\(101\) 19.7082 1.96104 0.980520 0.196420i \(-0.0629317\pi\)
0.980520 + 0.196420i \(0.0629317\pi\)
\(102\) 2.47214 0.244778
\(103\) 12.6525 1.24669 0.623343 0.781949i \(-0.285774\pi\)
0.623343 + 0.781949i \(0.285774\pi\)
\(104\) 5.23607 0.513439
\(105\) 0 0
\(106\) −8.47214 −0.822887
\(107\) −8.94427 −0.864675 −0.432338 0.901712i \(-0.642311\pi\)
−0.432338 + 0.901712i \(0.642311\pi\)
\(108\) 1.00000 0.0962250
\(109\) −1.70820 −0.163616 −0.0818081 0.996648i \(-0.526069\pi\)
−0.0818081 + 0.996648i \(0.526069\pi\)
\(110\) −2.47214 −0.235709
\(111\) 0.763932 0.0725092
\(112\) 0 0
\(113\) 12.4721 1.17328 0.586640 0.809848i \(-0.300450\pi\)
0.586640 + 0.809848i \(0.300450\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 7.05573 0.657950
\(116\) 8.47214 0.786618
\(117\) −5.23607 −0.484075
\(118\) −12.9443 −1.19162
\(119\) 0 0
\(120\) 1.23607 0.112837
\(121\) −7.00000 −0.636364
\(122\) 12.4721 1.12917
\(123\) 6.94427 0.626144
\(124\) −7.70820 −0.692217
\(125\) 10.4721 0.936656
\(126\) 0 0
\(127\) 21.7082 1.92629 0.963146 0.268980i \(-0.0866865\pi\)
0.963146 + 0.268980i \(0.0866865\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.52786 0.134521
\(130\) −6.47214 −0.567644
\(131\) 3.05573 0.266980 0.133490 0.991050i \(-0.457382\pi\)
0.133490 + 0.991050i \(0.457382\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) −10.9443 −0.945441
\(135\) −1.23607 −0.106384
\(136\) 2.47214 0.211984
\(137\) −9.41641 −0.804498 −0.402249 0.915530i \(-0.631771\pi\)
−0.402249 + 0.915530i \(0.631771\pi\)
\(138\) 5.70820 0.485915
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) −2.76393 −0.232765
\(142\) −14.4721 −1.21447
\(143\) 10.4721 0.875724
\(144\) 1.00000 0.0833333
\(145\) −10.4721 −0.869664
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) 0.763932 0.0627948
\(149\) 12.1803 0.997852 0.498926 0.866644i \(-0.333728\pi\)
0.498926 + 0.866644i \(0.333728\pi\)
\(150\) 3.47214 0.283499
\(151\) −13.7082 −1.11556 −0.557779 0.829990i \(-0.688346\pi\)
−0.557779 + 0.829990i \(0.688346\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −2.47214 −0.199860
\(154\) 0 0
\(155\) 9.52786 0.765296
\(156\) −5.23607 −0.419221
\(157\) 4.47214 0.356915 0.178458 0.983948i \(-0.442889\pi\)
0.178458 + 0.983948i \(0.442889\pi\)
\(158\) −12.1803 −0.969016
\(159\) 8.47214 0.671884
\(160\) 1.23607 0.0977198
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −20.9443 −1.64048 −0.820241 0.572018i \(-0.806161\pi\)
−0.820241 + 0.572018i \(0.806161\pi\)
\(164\) 6.94427 0.542257
\(165\) 2.47214 0.192456
\(166\) 8.94427 0.694210
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 14.4164 1.10895
\(170\) −3.05573 −0.234364
\(171\) 1.00000 0.0764719
\(172\) 1.52786 0.116499
\(173\) −11.8885 −0.903869 −0.451935 0.892051i \(-0.649266\pi\)
−0.451935 + 0.892051i \(0.649266\pi\)
\(174\) −8.47214 −0.642271
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 12.9443 0.972951
\(178\) −2.94427 −0.220683
\(179\) −15.4164 −1.15228 −0.576138 0.817352i \(-0.695441\pi\)
−0.576138 + 0.817352i \(0.695441\pi\)
\(180\) −1.23607 −0.0921311
\(181\) 9.23607 0.686512 0.343256 0.939242i \(-0.388470\pi\)
0.343256 + 0.939242i \(0.388470\pi\)
\(182\) 0 0
\(183\) −12.4721 −0.921967
\(184\) 5.70820 0.420814
\(185\) −0.944272 −0.0694243
\(186\) 7.70820 0.565193
\(187\) 4.94427 0.361561
\(188\) −2.76393 −0.201580
\(189\) 0 0
\(190\) 1.23607 0.0896738
\(191\) −3.81966 −0.276381 −0.138190 0.990406i \(-0.544129\pi\)
−0.138190 + 0.990406i \(0.544129\pi\)
\(192\) 1.00000 0.0721688
\(193\) −0.472136 −0.0339851 −0.0169925 0.999856i \(-0.505409\pi\)
−0.0169925 + 0.999856i \(0.505409\pi\)
\(194\) 10.4721 0.751856
\(195\) 6.47214 0.463479
\(196\) 0 0
\(197\) 2.29180 0.163284 0.0816419 0.996662i \(-0.473984\pi\)
0.0816419 + 0.996662i \(0.473984\pi\)
\(198\) 2.00000 0.142134
\(199\) 5.52786 0.391860 0.195930 0.980618i \(-0.437227\pi\)
0.195930 + 0.980618i \(0.437227\pi\)
\(200\) 3.47214 0.245517
\(201\) 10.9443 0.771949
\(202\) −19.7082 −1.38666
\(203\) 0 0
\(204\) −2.47214 −0.173084
\(205\) −8.58359 −0.599504
\(206\) −12.6525 −0.881540
\(207\) −5.70820 −0.396748
\(208\) −5.23607 −0.363056
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −25.4164 −1.74974 −0.874869 0.484360i \(-0.839053\pi\)
−0.874869 + 0.484360i \(0.839053\pi\)
\(212\) 8.47214 0.581869
\(213\) 14.4721 0.991614
\(214\) 8.94427 0.611418
\(215\) −1.88854 −0.128798
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 1.70820 0.115694
\(219\) −6.00000 −0.405442
\(220\) 2.47214 0.166671
\(221\) 12.9443 0.870726
\(222\) −0.763932 −0.0512718
\(223\) −3.70820 −0.248320 −0.124160 0.992262i \(-0.539624\pi\)
−0.124160 + 0.992262i \(0.539624\pi\)
\(224\) 0 0
\(225\) −3.47214 −0.231476
\(226\) −12.4721 −0.829634
\(227\) 7.05573 0.468305 0.234153 0.972200i \(-0.424768\pi\)
0.234153 + 0.972200i \(0.424768\pi\)
\(228\) 1.00000 0.0662266
\(229\) 23.8885 1.57860 0.789300 0.614008i \(-0.210444\pi\)
0.789300 + 0.614008i \(0.210444\pi\)
\(230\) −7.05573 −0.465241
\(231\) 0 0
\(232\) −8.47214 −0.556223
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 5.23607 0.342292
\(235\) 3.41641 0.222862
\(236\) 12.9443 0.842600
\(237\) 12.1803 0.791198
\(238\) 0 0
\(239\) 22.6525 1.46527 0.732633 0.680623i \(-0.238291\pi\)
0.732633 + 0.680623i \(0.238291\pi\)
\(240\) −1.23607 −0.0797878
\(241\) 16.3607 1.05388 0.526942 0.849901i \(-0.323338\pi\)
0.526942 + 0.849901i \(0.323338\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) −12.4721 −0.798447
\(245\) 0 0
\(246\) −6.94427 −0.442751
\(247\) −5.23607 −0.333163
\(248\) 7.70820 0.489471
\(249\) −8.94427 −0.566820
\(250\) −10.4721 −0.662316
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 11.4164 0.717743
\(254\) −21.7082 −1.36209
\(255\) 3.05573 0.191357
\(256\) 1.00000 0.0625000
\(257\) 19.8885 1.24061 0.620307 0.784359i \(-0.287008\pi\)
0.620307 + 0.784359i \(0.287008\pi\)
\(258\) −1.52786 −0.0951207
\(259\) 0 0
\(260\) 6.47214 0.401385
\(261\) 8.47214 0.524412
\(262\) −3.05573 −0.188784
\(263\) −4.18034 −0.257771 −0.128885 0.991659i \(-0.541140\pi\)
−0.128885 + 0.991659i \(0.541140\pi\)
\(264\) 2.00000 0.123091
\(265\) −10.4721 −0.643298
\(266\) 0 0
\(267\) 2.94427 0.180187
\(268\) 10.9443 0.668528
\(269\) −28.4721 −1.73598 −0.867988 0.496584i \(-0.834587\pi\)
−0.867988 + 0.496584i \(0.834587\pi\)
\(270\) 1.23607 0.0752247
\(271\) −12.9443 −0.786309 −0.393154 0.919473i \(-0.628616\pi\)
−0.393154 + 0.919473i \(0.628616\pi\)
\(272\) −2.47214 −0.149895
\(273\) 0 0
\(274\) 9.41641 0.568866
\(275\) 6.94427 0.418755
\(276\) −5.70820 −0.343594
\(277\) 23.8885 1.43532 0.717662 0.696392i \(-0.245212\pi\)
0.717662 + 0.696392i \(0.245212\pi\)
\(278\) −16.0000 −0.959616
\(279\) −7.70820 −0.461478
\(280\) 0 0
\(281\) −5.41641 −0.323116 −0.161558 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(282\) 2.76393 0.164590
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 14.4721 0.858763
\(285\) −1.23607 −0.0732183
\(286\) −10.4721 −0.619230
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −10.8885 −0.640503
\(290\) 10.4721 0.614945
\(291\) −10.4721 −0.613887
\(292\) −6.00000 −0.351123
\(293\) 8.47214 0.494947 0.247474 0.968895i \(-0.420400\pi\)
0.247474 + 0.968895i \(0.420400\pi\)
\(294\) 0 0
\(295\) −16.0000 −0.931556
\(296\) −0.763932 −0.0444026
\(297\) −2.00000 −0.116052
\(298\) −12.1803 −0.705588
\(299\) 29.8885 1.72850
\(300\) −3.47214 −0.200464
\(301\) 0 0
\(302\) 13.7082 0.788818
\(303\) 19.7082 1.13221
\(304\) 1.00000 0.0573539
\(305\) 15.4164 0.882741
\(306\) 2.47214 0.141323
\(307\) −5.52786 −0.315492 −0.157746 0.987480i \(-0.550423\pi\)
−0.157746 + 0.987480i \(0.550423\pi\)
\(308\) 0 0
\(309\) 12.6525 0.719774
\(310\) −9.52786 −0.541146
\(311\) 17.2361 0.977368 0.488684 0.872461i \(-0.337477\pi\)
0.488684 + 0.872461i \(0.337477\pi\)
\(312\) 5.23607 0.296434
\(313\) −30.0000 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) −4.47214 −0.252377
\(315\) 0 0
\(316\) 12.1803 0.685198
\(317\) 12.4721 0.700505 0.350252 0.936655i \(-0.386096\pi\)
0.350252 + 0.936655i \(0.386096\pi\)
\(318\) −8.47214 −0.475094
\(319\) −16.9443 −0.948697
\(320\) −1.23607 −0.0690983
\(321\) −8.94427 −0.499221
\(322\) 0 0
\(323\) −2.47214 −0.137553
\(324\) 1.00000 0.0555556
\(325\) 18.1803 1.00846
\(326\) 20.9443 1.16000
\(327\) −1.70820 −0.0944639
\(328\) −6.94427 −0.383433
\(329\) 0 0
\(330\) −2.47214 −0.136087
\(331\) −18.9443 −1.04127 −0.520636 0.853779i \(-0.674305\pi\)
−0.520636 + 0.853779i \(0.674305\pi\)
\(332\) −8.94427 −0.490881
\(333\) 0.763932 0.0418632
\(334\) 8.00000 0.437741
\(335\) −13.5279 −0.739106
\(336\) 0 0
\(337\) 1.05573 0.0575092 0.0287546 0.999587i \(-0.490846\pi\)
0.0287546 + 0.999587i \(0.490846\pi\)
\(338\) −14.4164 −0.784149
\(339\) 12.4721 0.677393
\(340\) 3.05573 0.165720
\(341\) 15.4164 0.834845
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) −1.52786 −0.0823769
\(345\) 7.05573 0.379868
\(346\) 11.8885 0.639132
\(347\) 10.0000 0.536828 0.268414 0.963304i \(-0.413500\pi\)
0.268414 + 0.963304i \(0.413500\pi\)
\(348\) 8.47214 0.454154
\(349\) 22.3607 1.19694 0.598470 0.801145i \(-0.295776\pi\)
0.598470 + 0.801145i \(0.295776\pi\)
\(350\) 0 0
\(351\) −5.23607 −0.279481
\(352\) 2.00000 0.106600
\(353\) −8.00000 −0.425797 −0.212899 0.977074i \(-0.568290\pi\)
−0.212899 + 0.977074i \(0.568290\pi\)
\(354\) −12.9443 −0.687980
\(355\) −17.8885 −0.949425
\(356\) 2.94427 0.156046
\(357\) 0 0
\(358\) 15.4164 0.814782
\(359\) 14.2918 0.754292 0.377146 0.926154i \(-0.376905\pi\)
0.377146 + 0.926154i \(0.376905\pi\)
\(360\) 1.23607 0.0651465
\(361\) 1.00000 0.0526316
\(362\) −9.23607 −0.485437
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 7.41641 0.388193
\(366\) 12.4721 0.651929
\(367\) 19.0557 0.994701 0.497350 0.867550i \(-0.334306\pi\)
0.497350 + 0.867550i \(0.334306\pi\)
\(368\) −5.70820 −0.297561
\(369\) 6.94427 0.361504
\(370\) 0.944272 0.0490904
\(371\) 0 0
\(372\) −7.70820 −0.399652
\(373\) 1.70820 0.0884474 0.0442237 0.999022i \(-0.485919\pi\)
0.0442237 + 0.999022i \(0.485919\pi\)
\(374\) −4.94427 −0.255662
\(375\) 10.4721 0.540779
\(376\) 2.76393 0.142539
\(377\) −44.3607 −2.28469
\(378\) 0 0
\(379\) −10.9443 −0.562169 −0.281085 0.959683i \(-0.590694\pi\)
−0.281085 + 0.959683i \(0.590694\pi\)
\(380\) −1.23607 −0.0634089
\(381\) 21.7082 1.11214
\(382\) 3.81966 0.195431
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 0.472136 0.0240311
\(387\) 1.52786 0.0776657
\(388\) −10.4721 −0.531642
\(389\) −7.23607 −0.366883 −0.183442 0.983031i \(-0.558724\pi\)
−0.183442 + 0.983031i \(0.558724\pi\)
\(390\) −6.47214 −0.327729
\(391\) 14.1115 0.713647
\(392\) 0 0
\(393\) 3.05573 0.154141
\(394\) −2.29180 −0.115459
\(395\) −15.0557 −0.757536
\(396\) −2.00000 −0.100504
\(397\) 6.94427 0.348523 0.174262 0.984699i \(-0.444246\pi\)
0.174262 + 0.984699i \(0.444246\pi\)
\(398\) −5.52786 −0.277087
\(399\) 0 0
\(400\) −3.47214 −0.173607
\(401\) −20.4721 −1.02233 −0.511165 0.859483i \(-0.670786\pi\)
−0.511165 + 0.859483i \(0.670786\pi\)
\(402\) −10.9443 −0.545851
\(403\) 40.3607 2.01051
\(404\) 19.7082 0.980520
\(405\) −1.23607 −0.0614207
\(406\) 0 0
\(407\) −1.52786 −0.0757334
\(408\) 2.47214 0.122389
\(409\) −5.52786 −0.273335 −0.136668 0.990617i \(-0.543639\pi\)
−0.136668 + 0.990617i \(0.543639\pi\)
\(410\) 8.58359 0.423913
\(411\) −9.41641 −0.464477
\(412\) 12.6525 0.623343
\(413\) 0 0
\(414\) 5.70820 0.280543
\(415\) 11.0557 0.542704
\(416\) 5.23607 0.256719
\(417\) 16.0000 0.783523
\(418\) 2.00000 0.0978232
\(419\) 20.3607 0.994684 0.497342 0.867554i \(-0.334309\pi\)
0.497342 + 0.867554i \(0.334309\pi\)
\(420\) 0 0
\(421\) −11.2361 −0.547612 −0.273806 0.961785i \(-0.588283\pi\)
−0.273806 + 0.961785i \(0.588283\pi\)
\(422\) 25.4164 1.23725
\(423\) −2.76393 −0.134387
\(424\) −8.47214 −0.411443
\(425\) 8.58359 0.416365
\(426\) −14.4721 −0.701177
\(427\) 0 0
\(428\) −8.94427 −0.432338
\(429\) 10.4721 0.505599
\(430\) 1.88854 0.0910737
\(431\) −9.52786 −0.458941 −0.229471 0.973316i \(-0.573699\pi\)
−0.229471 + 0.973316i \(0.573699\pi\)
\(432\) 1.00000 0.0481125
\(433\) 13.5279 0.650108 0.325054 0.945696i \(-0.394618\pi\)
0.325054 + 0.945696i \(0.394618\pi\)
\(434\) 0 0
\(435\) −10.4721 −0.502100
\(436\) −1.70820 −0.0818081
\(437\) −5.70820 −0.273060
\(438\) 6.00000 0.286691
\(439\) −12.6525 −0.603870 −0.301935 0.953329i \(-0.597633\pi\)
−0.301935 + 0.953329i \(0.597633\pi\)
\(440\) −2.47214 −0.117854
\(441\) 0 0
\(442\) −12.9443 −0.615696
\(443\) −11.8885 −0.564842 −0.282421 0.959291i \(-0.591137\pi\)
−0.282421 + 0.959291i \(0.591137\pi\)
\(444\) 0.763932 0.0362546
\(445\) −3.63932 −0.172520
\(446\) 3.70820 0.175589
\(447\) 12.1803 0.576110
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 3.47214 0.163678
\(451\) −13.8885 −0.653986
\(452\) 12.4721 0.586640
\(453\) −13.7082 −0.644068
\(454\) −7.05573 −0.331142
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −23.8885 −1.11624
\(459\) −2.47214 −0.115389
\(460\) 7.05573 0.328975
\(461\) 25.2361 1.17536 0.587680 0.809093i \(-0.300041\pi\)
0.587680 + 0.809093i \(0.300041\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 8.47214 0.393309
\(465\) 9.52786 0.441844
\(466\) −6.00000 −0.277945
\(467\) −29.8885 −1.38308 −0.691538 0.722340i \(-0.743067\pi\)
−0.691538 + 0.722340i \(0.743067\pi\)
\(468\) −5.23607 −0.242037
\(469\) 0 0
\(470\) −3.41641 −0.157587
\(471\) 4.47214 0.206065
\(472\) −12.9443 −0.595808
\(473\) −3.05573 −0.140503
\(474\) −12.1803 −0.559462
\(475\) −3.47214 −0.159313
\(476\) 0 0
\(477\) 8.47214 0.387912
\(478\) −22.6525 −1.03610
\(479\) −1.23607 −0.0564774 −0.0282387 0.999601i \(-0.508990\pi\)
−0.0282387 + 0.999601i \(0.508990\pi\)
\(480\) 1.23607 0.0564185
\(481\) −4.00000 −0.182384
\(482\) −16.3607 −0.745209
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 12.9443 0.587769
\(486\) −1.00000 −0.0453609
\(487\) 19.2361 0.871669 0.435835 0.900027i \(-0.356453\pi\)
0.435835 + 0.900027i \(0.356453\pi\)
\(488\) 12.4721 0.564587
\(489\) −20.9443 −0.947133
\(490\) 0 0
\(491\) 23.5279 1.06180 0.530899 0.847435i \(-0.321854\pi\)
0.530899 + 0.847435i \(0.321854\pi\)
\(492\) 6.94427 0.313072
\(493\) −20.9443 −0.943283
\(494\) 5.23607 0.235582
\(495\) 2.47214 0.111114
\(496\) −7.70820 −0.346109
\(497\) 0 0
\(498\) 8.94427 0.400802
\(499\) 8.36068 0.374275 0.187138 0.982334i \(-0.440079\pi\)
0.187138 + 0.982334i \(0.440079\pi\)
\(500\) 10.4721 0.468328
\(501\) −8.00000 −0.357414
\(502\) 0 0
\(503\) 11.1246 0.496022 0.248011 0.968757i \(-0.420223\pi\)
0.248011 + 0.968757i \(0.420223\pi\)
\(504\) 0 0
\(505\) −24.3607 −1.08404
\(506\) −11.4164 −0.507521
\(507\) 14.4164 0.640255
\(508\) 21.7082 0.963146
\(509\) 24.4721 1.08471 0.542354 0.840150i \(-0.317533\pi\)
0.542354 + 0.840150i \(0.317533\pi\)
\(510\) −3.05573 −0.135310
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) −19.8885 −0.877246
\(515\) −15.6393 −0.689151
\(516\) 1.52786 0.0672605
\(517\) 5.52786 0.243115
\(518\) 0 0
\(519\) −11.8885 −0.521849
\(520\) −6.47214 −0.283822
\(521\) 28.8328 1.26319 0.631594 0.775299i \(-0.282401\pi\)
0.631594 + 0.775299i \(0.282401\pi\)
\(522\) −8.47214 −0.370815
\(523\) 2.47214 0.108099 0.0540495 0.998538i \(-0.482787\pi\)
0.0540495 + 0.998538i \(0.482787\pi\)
\(524\) 3.05573 0.133490
\(525\) 0 0
\(526\) 4.18034 0.182271
\(527\) 19.0557 0.830081
\(528\) −2.00000 −0.0870388
\(529\) 9.58359 0.416678
\(530\) 10.4721 0.454881
\(531\) 12.9443 0.561734
\(532\) 0 0
\(533\) −36.3607 −1.57496
\(534\) −2.94427 −0.127411
\(535\) 11.0557 0.477981
\(536\) −10.9443 −0.472721
\(537\) −15.4164 −0.665267
\(538\) 28.4721 1.22752
\(539\) 0 0
\(540\) −1.23607 −0.0531919
\(541\) 24.8328 1.06765 0.533823 0.845596i \(-0.320755\pi\)
0.533823 + 0.845596i \(0.320755\pi\)
\(542\) 12.9443 0.556004
\(543\) 9.23607 0.396358
\(544\) 2.47214 0.105992
\(545\) 2.11146 0.0904448
\(546\) 0 0
\(547\) −18.0000 −0.769624 −0.384812 0.922995i \(-0.625734\pi\)
−0.384812 + 0.922995i \(0.625734\pi\)
\(548\) −9.41641 −0.402249
\(549\) −12.4721 −0.532298
\(550\) −6.94427 −0.296105
\(551\) 8.47214 0.360925
\(552\) 5.70820 0.242957
\(553\) 0 0
\(554\) −23.8885 −1.01493
\(555\) −0.944272 −0.0400821
\(556\) 16.0000 0.678551
\(557\) −36.5410 −1.54829 −0.774146 0.633007i \(-0.781821\pi\)
−0.774146 + 0.633007i \(0.781821\pi\)
\(558\) 7.70820 0.326314
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 4.94427 0.208747
\(562\) 5.41641 0.228477
\(563\) 9.88854 0.416752 0.208376 0.978049i \(-0.433182\pi\)
0.208376 + 0.978049i \(0.433182\pi\)
\(564\) −2.76393 −0.116383
\(565\) −15.4164 −0.648573
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) −14.4721 −0.607237
\(569\) 13.0557 0.547325 0.273662 0.961826i \(-0.411765\pi\)
0.273662 + 0.961826i \(0.411765\pi\)
\(570\) 1.23607 0.0517732
\(571\) −26.4721 −1.10782 −0.553912 0.832575i \(-0.686866\pi\)
−0.553912 + 0.832575i \(0.686866\pi\)
\(572\) 10.4721 0.437862
\(573\) −3.81966 −0.159569
\(574\) 0 0
\(575\) 19.8197 0.826537
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 10.8885 0.452904
\(579\) −0.472136 −0.0196213
\(580\) −10.4721 −0.434832
\(581\) 0 0
\(582\) 10.4721 0.434084
\(583\) −16.9443 −0.701760
\(584\) 6.00000 0.248282
\(585\) 6.47214 0.267590
\(586\) −8.47214 −0.349981
\(587\) −19.4164 −0.801401 −0.400700 0.916209i \(-0.631233\pi\)
−0.400700 + 0.916209i \(0.631233\pi\)
\(588\) 0 0
\(589\) −7.70820 −0.317611
\(590\) 16.0000 0.658710
\(591\) 2.29180 0.0942719
\(592\) 0.763932 0.0313974
\(593\) 4.00000 0.164260 0.0821302 0.996622i \(-0.473828\pi\)
0.0821302 + 0.996622i \(0.473828\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 12.1803 0.498926
\(597\) 5.52786 0.226240
\(598\) −29.8885 −1.22223
\(599\) 32.3607 1.32222 0.661111 0.750288i \(-0.270085\pi\)
0.661111 + 0.750288i \(0.270085\pi\)
\(600\) 3.47214 0.141749
\(601\) 12.9443 0.528008 0.264004 0.964522i \(-0.414957\pi\)
0.264004 + 0.964522i \(0.414957\pi\)
\(602\) 0 0
\(603\) 10.9443 0.445685
\(604\) −13.7082 −0.557779
\(605\) 8.65248 0.351773
\(606\) −19.7082 −0.800591
\(607\) 42.5410 1.72669 0.863343 0.504617i \(-0.168366\pi\)
0.863343 + 0.504617i \(0.168366\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −15.4164 −0.624192
\(611\) 14.4721 0.585480
\(612\) −2.47214 −0.0999302
\(613\) −14.3607 −0.580022 −0.290011 0.957023i \(-0.593659\pi\)
−0.290011 + 0.957023i \(0.593659\pi\)
\(614\) 5.52786 0.223086
\(615\) −8.58359 −0.346124
\(616\) 0 0
\(617\) 26.9443 1.08474 0.542368 0.840141i \(-0.317528\pi\)
0.542368 + 0.840141i \(0.317528\pi\)
\(618\) −12.6525 −0.508957
\(619\) 26.8328 1.07850 0.539251 0.842145i \(-0.318707\pi\)
0.539251 + 0.842145i \(0.318707\pi\)
\(620\) 9.52786 0.382648
\(621\) −5.70820 −0.229062
\(622\) −17.2361 −0.691103
\(623\) 0 0
\(624\) −5.23607 −0.209610
\(625\) 4.41641 0.176656
\(626\) 30.0000 1.19904
\(627\) −2.00000 −0.0798723
\(628\) 4.47214 0.178458
\(629\) −1.88854 −0.0753012
\(630\) 0 0
\(631\) 32.3607 1.28826 0.644129 0.764917i \(-0.277220\pi\)
0.644129 + 0.764917i \(0.277220\pi\)
\(632\) −12.1803 −0.484508
\(633\) −25.4164 −1.01021
\(634\) −12.4721 −0.495332
\(635\) −26.8328 −1.06483
\(636\) 8.47214 0.335942
\(637\) 0 0
\(638\) 16.9443 0.670830
\(639\) 14.4721 0.572509
\(640\) 1.23607 0.0488599
\(641\) −29.0557 −1.14763 −0.573816 0.818984i \(-0.694538\pi\)
−0.573816 + 0.818984i \(0.694538\pi\)
\(642\) 8.94427 0.353002
\(643\) 48.7214 1.92138 0.960691 0.277618i \(-0.0895451\pi\)
0.960691 + 0.277618i \(0.0895451\pi\)
\(644\) 0 0
\(645\) −1.88854 −0.0743613
\(646\) 2.47214 0.0972649
\(647\) 4.87539 0.191671 0.0958356 0.995397i \(-0.469448\pi\)
0.0958356 + 0.995397i \(0.469448\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −25.8885 −1.01621
\(650\) −18.1803 −0.713092
\(651\) 0 0
\(652\) −20.9443 −0.820241
\(653\) −2.87539 −0.112523 −0.0562613 0.998416i \(-0.517918\pi\)
−0.0562613 + 0.998416i \(0.517918\pi\)
\(654\) 1.70820 0.0667961
\(655\) −3.77709 −0.147583
\(656\) 6.94427 0.271128
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 42.4721 1.65448 0.827240 0.561849i \(-0.189910\pi\)
0.827240 + 0.561849i \(0.189910\pi\)
\(660\) 2.47214 0.0962278
\(661\) −37.0132 −1.43964 −0.719822 0.694158i \(-0.755777\pi\)
−0.719822 + 0.694158i \(0.755777\pi\)
\(662\) 18.9443 0.736290
\(663\) 12.9443 0.502714
\(664\) 8.94427 0.347105
\(665\) 0 0
\(666\) −0.763932 −0.0296018
\(667\) −48.3607 −1.87253
\(668\) −8.00000 −0.309529
\(669\) −3.70820 −0.143367
\(670\) 13.5279 0.522627
\(671\) 24.9443 0.962963
\(672\) 0 0
\(673\) −36.4721 −1.40590 −0.702949 0.711240i \(-0.748134\pi\)
−0.702949 + 0.711240i \(0.748134\pi\)
\(674\) −1.05573 −0.0406651
\(675\) −3.47214 −0.133643
\(676\) 14.4164 0.554477
\(677\) 46.3607 1.78179 0.890893 0.454214i \(-0.150080\pi\)
0.890893 + 0.454214i \(0.150080\pi\)
\(678\) −12.4721 −0.478989
\(679\) 0 0
\(680\) −3.05573 −0.117182
\(681\) 7.05573 0.270376
\(682\) −15.4164 −0.590325
\(683\) 40.3607 1.54436 0.772179 0.635405i \(-0.219167\pi\)
0.772179 + 0.635405i \(0.219167\pi\)
\(684\) 1.00000 0.0382360
\(685\) 11.6393 0.444716
\(686\) 0 0
\(687\) 23.8885 0.911405
\(688\) 1.52786 0.0582493
\(689\) −44.3607 −1.69001
\(690\) −7.05573 −0.268607
\(691\) 13.8885 0.528345 0.264173 0.964475i \(-0.414901\pi\)
0.264173 + 0.964475i \(0.414901\pi\)
\(692\) −11.8885 −0.451935
\(693\) 0 0
\(694\) −10.0000 −0.379595
\(695\) −19.7771 −0.750188
\(696\) −8.47214 −0.321135
\(697\) −17.1672 −0.650253
\(698\) −22.3607 −0.846364
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −39.2361 −1.48193 −0.740963 0.671546i \(-0.765631\pi\)
−0.740963 + 0.671546i \(0.765631\pi\)
\(702\) 5.23607 0.197623
\(703\) 0.763932 0.0288122
\(704\) −2.00000 −0.0753778
\(705\) 3.41641 0.128669
\(706\) 8.00000 0.301084
\(707\) 0 0
\(708\) 12.9443 0.486476
\(709\) 10.5836 0.397475 0.198738 0.980053i \(-0.436316\pi\)
0.198738 + 0.980053i \(0.436316\pi\)
\(710\) 17.8885 0.671345
\(711\) 12.1803 0.456798
\(712\) −2.94427 −0.110341
\(713\) 44.0000 1.64781
\(714\) 0 0
\(715\) −12.9443 −0.484088
\(716\) −15.4164 −0.576138
\(717\) 22.6525 0.845972
\(718\) −14.2918 −0.533365
\(719\) −18.5410 −0.691463 −0.345732 0.938333i \(-0.612369\pi\)
−0.345732 + 0.938333i \(0.612369\pi\)
\(720\) −1.23607 −0.0460655
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) 16.3607 0.608460
\(724\) 9.23607 0.343256
\(725\) −29.4164 −1.09250
\(726\) 7.00000 0.259794
\(727\) 10.4721 0.388390 0.194195 0.980963i \(-0.437791\pi\)
0.194195 + 0.980963i \(0.437791\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −7.41641 −0.274494
\(731\) −3.77709 −0.139701
\(732\) −12.4721 −0.460983
\(733\) 19.3050 0.713045 0.356522 0.934287i \(-0.383962\pi\)
0.356522 + 0.934287i \(0.383962\pi\)
\(734\) −19.0557 −0.703360
\(735\) 0 0
\(736\) 5.70820 0.210407
\(737\) −21.8885 −0.806275
\(738\) −6.94427 −0.255622
\(739\) 32.9443 1.21187 0.605937 0.795512i \(-0.292798\pi\)
0.605937 + 0.795512i \(0.292798\pi\)
\(740\) −0.944272 −0.0347121
\(741\) −5.23607 −0.192352
\(742\) 0 0
\(743\) −35.0557 −1.28607 −0.643035 0.765837i \(-0.722325\pi\)
−0.643035 + 0.765837i \(0.722325\pi\)
\(744\) 7.70820 0.282596
\(745\) −15.0557 −0.551599
\(746\) −1.70820 −0.0625418
\(747\) −8.94427 −0.327254
\(748\) 4.94427 0.180780
\(749\) 0 0
\(750\) −10.4721 −0.382388
\(751\) −53.4853 −1.95171 −0.975853 0.218428i \(-0.929907\pi\)
−0.975853 + 0.218428i \(0.929907\pi\)
\(752\) −2.76393 −0.100790
\(753\) 0 0
\(754\) 44.3607 1.61552
\(755\) 16.9443 0.616665
\(756\) 0 0
\(757\) 39.3050 1.42856 0.714281 0.699859i \(-0.246754\pi\)
0.714281 + 0.699859i \(0.246754\pi\)
\(758\) 10.9443 0.397514
\(759\) 11.4164 0.414389
\(760\) 1.23607 0.0448369
\(761\) −16.3607 −0.593074 −0.296537 0.955021i \(-0.595832\pi\)
−0.296537 + 0.955021i \(0.595832\pi\)
\(762\) −21.7082 −0.786405
\(763\) 0 0
\(764\) −3.81966 −0.138190
\(765\) 3.05573 0.110480
\(766\) 24.0000 0.867155
\(767\) −67.7771 −2.44729
\(768\) 1.00000 0.0360844
\(769\) −23.8885 −0.861443 −0.430721 0.902485i \(-0.641741\pi\)
−0.430721 + 0.902485i \(0.641741\pi\)
\(770\) 0 0
\(771\) 19.8885 0.716268
\(772\) −0.472136 −0.0169925
\(773\) −31.5279 −1.13398 −0.566989 0.823725i \(-0.691892\pi\)
−0.566989 + 0.823725i \(0.691892\pi\)
\(774\) −1.52786 −0.0549179
\(775\) 26.7639 0.961389
\(776\) 10.4721 0.375928
\(777\) 0 0
\(778\) 7.23607 0.259426
\(779\) 6.94427 0.248804
\(780\) 6.47214 0.231740
\(781\) −28.9443 −1.03571
\(782\) −14.1115 −0.504625
\(783\) 8.47214 0.302769
\(784\) 0 0
\(785\) −5.52786 −0.197298
\(786\) −3.05573 −0.108994
\(787\) 38.8328 1.38424 0.692120 0.721782i \(-0.256677\pi\)
0.692120 + 0.721782i \(0.256677\pi\)
\(788\) 2.29180 0.0816419
\(789\) −4.18034 −0.148824
\(790\) 15.0557 0.535659
\(791\) 0 0
\(792\) 2.00000 0.0710669
\(793\) 65.3050 2.31905
\(794\) −6.94427 −0.246443
\(795\) −10.4721 −0.371408
\(796\) 5.52786 0.195930
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) 6.83282 0.241728
\(800\) 3.47214 0.122759
\(801\) 2.94427 0.104031
\(802\) 20.4721 0.722896
\(803\) 12.0000 0.423471
\(804\) 10.9443 0.385975
\(805\) 0 0
\(806\) −40.3607 −1.42164
\(807\) −28.4721 −1.00227
\(808\) −19.7082 −0.693332
\(809\) 33.4164 1.17486 0.587429 0.809276i \(-0.300140\pi\)
0.587429 + 0.809276i \(0.300140\pi\)
\(810\) 1.23607 0.0434310
\(811\) 31.4164 1.10318 0.551590 0.834116i \(-0.314021\pi\)
0.551590 + 0.834116i \(0.314021\pi\)
\(812\) 0 0
\(813\) −12.9443 −0.453975
\(814\) 1.52786 0.0535516
\(815\) 25.8885 0.906836
\(816\) −2.47214 −0.0865421
\(817\) 1.52786 0.0534532
\(818\) 5.52786 0.193277
\(819\) 0 0
\(820\) −8.58359 −0.299752
\(821\) −22.2918 −0.777989 −0.388995 0.921240i \(-0.627178\pi\)
−0.388995 + 0.921240i \(0.627178\pi\)
\(822\) 9.41641 0.328435
\(823\) 51.7771 1.80484 0.902418 0.430862i \(-0.141790\pi\)
0.902418 + 0.430862i \(0.141790\pi\)
\(824\) −12.6525 −0.440770
\(825\) 6.94427 0.241769
\(826\) 0 0
\(827\) −2.47214 −0.0859646 −0.0429823 0.999076i \(-0.513686\pi\)
−0.0429823 + 0.999076i \(0.513686\pi\)
\(828\) −5.70820 −0.198374
\(829\) −13.8197 −0.479977 −0.239988 0.970776i \(-0.577144\pi\)
−0.239988 + 0.970776i \(0.577144\pi\)
\(830\) −11.0557 −0.383750
\(831\) 23.8885 0.828684
\(832\) −5.23607 −0.181528
\(833\) 0 0
\(834\) −16.0000 −0.554035
\(835\) 9.88854 0.342207
\(836\) −2.00000 −0.0691714
\(837\) −7.70820 −0.266435
\(838\) −20.3607 −0.703348
\(839\) −41.8885 −1.44615 −0.723077 0.690768i \(-0.757273\pi\)
−0.723077 + 0.690768i \(0.757273\pi\)
\(840\) 0 0
\(841\) 42.7771 1.47507
\(842\) 11.2361 0.387220
\(843\) −5.41641 −0.186551
\(844\) −25.4164 −0.874869
\(845\) −17.8197 −0.613015
\(846\) 2.76393 0.0950259
\(847\) 0 0
\(848\) 8.47214 0.290934
\(849\) 8.00000 0.274559
\(850\) −8.58359 −0.294415
\(851\) −4.36068 −0.149482
\(852\) 14.4721 0.495807
\(853\) 37.4164 1.28111 0.640557 0.767911i \(-0.278704\pi\)
0.640557 + 0.767911i \(0.278704\pi\)
\(854\) 0 0
\(855\) −1.23607 −0.0422726
\(856\) 8.94427 0.305709
\(857\) −11.8885 −0.406105 −0.203052 0.979168i \(-0.565086\pi\)
−0.203052 + 0.979168i \(0.565086\pi\)
\(858\) −10.4721 −0.357513
\(859\) −53.8885 −1.83865 −0.919327 0.393495i \(-0.871266\pi\)
−0.919327 + 0.393495i \(0.871266\pi\)
\(860\) −1.88854 −0.0643988
\(861\) 0 0
\(862\) 9.52786 0.324520
\(863\) 44.9443 1.52992 0.764960 0.644077i \(-0.222759\pi\)
0.764960 + 0.644077i \(0.222759\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 14.6950 0.499647
\(866\) −13.5279 −0.459696
\(867\) −10.8885 −0.369794
\(868\) 0 0
\(869\) −24.3607 −0.826379
\(870\) 10.4721 0.355039
\(871\) −57.3050 −1.94170
\(872\) 1.70820 0.0578471
\(873\) −10.4721 −0.354428
\(874\) 5.70820 0.193083
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) −2.29180 −0.0773885 −0.0386942 0.999251i \(-0.512320\pi\)
−0.0386942 + 0.999251i \(0.512320\pi\)
\(878\) 12.6525 0.427000
\(879\) 8.47214 0.285758
\(880\) 2.47214 0.0833357
\(881\) 36.9443 1.24468 0.622342 0.782745i \(-0.286181\pi\)
0.622342 + 0.782745i \(0.286181\pi\)
\(882\) 0 0
\(883\) −0.360680 −0.0121378 −0.00606892 0.999982i \(-0.501932\pi\)
−0.00606892 + 0.999982i \(0.501932\pi\)
\(884\) 12.9443 0.435363
\(885\) −16.0000 −0.537834
\(886\) 11.8885 0.399403
\(887\) 43.1935 1.45030 0.725148 0.688593i \(-0.241771\pi\)
0.725148 + 0.688593i \(0.241771\pi\)
\(888\) −0.763932 −0.0256359
\(889\) 0 0
\(890\) 3.63932 0.121990
\(891\) −2.00000 −0.0670025
\(892\) −3.70820 −0.124160
\(893\) −2.76393 −0.0924915
\(894\) −12.1803 −0.407372
\(895\) 19.0557 0.636963
\(896\) 0 0
\(897\) 29.8885 0.997949
\(898\) −30.0000 −1.00111
\(899\) −65.3050 −2.17804
\(900\) −3.47214 −0.115738
\(901\) −20.9443 −0.697755
\(902\) 13.8885 0.462438
\(903\) 0 0
\(904\) −12.4721 −0.414817
\(905\) −11.4164 −0.379494
\(906\) 13.7082 0.455425
\(907\) 34.0000 1.12895 0.564476 0.825450i \(-0.309078\pi\)
0.564476 + 0.825450i \(0.309078\pi\)
\(908\) 7.05573 0.234153
\(909\) 19.7082 0.653680
\(910\) 0 0
\(911\) −22.4721 −0.744535 −0.372268 0.928125i \(-0.621420\pi\)
−0.372268 + 0.928125i \(0.621420\pi\)
\(912\) 1.00000 0.0331133
\(913\) 17.8885 0.592024
\(914\) 26.0000 0.860004
\(915\) 15.4164 0.509651
\(916\) 23.8885 0.789300
\(917\) 0 0
\(918\) 2.47214 0.0815926
\(919\) 22.8328 0.753185 0.376593 0.926379i \(-0.377096\pi\)
0.376593 + 0.926379i \(0.377096\pi\)
\(920\) −7.05573 −0.232620
\(921\) −5.52786 −0.182149
\(922\) −25.2361 −0.831106
\(923\) −75.7771 −2.49423
\(924\) 0 0
\(925\) −2.65248 −0.0872129
\(926\) 16.0000 0.525793
\(927\) 12.6525 0.415562
\(928\) −8.47214 −0.278111
\(929\) 39.4164 1.29321 0.646605 0.762825i \(-0.276188\pi\)
0.646605 + 0.762825i \(0.276188\pi\)
\(930\) −9.52786 −0.312431
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 17.2361 0.564284
\(934\) 29.8885 0.977983
\(935\) −6.11146 −0.199866
\(936\) 5.23607 0.171146
\(937\) 33.7771 1.10345 0.551725 0.834026i \(-0.313970\pi\)
0.551725 + 0.834026i \(0.313970\pi\)
\(938\) 0 0
\(939\) −30.0000 −0.979013
\(940\) 3.41641 0.111431
\(941\) 16.4721 0.536976 0.268488 0.963283i \(-0.413476\pi\)
0.268488 + 0.963283i \(0.413476\pi\)
\(942\) −4.47214 −0.145710
\(943\) −39.6393 −1.29083
\(944\) 12.9443 0.421300
\(945\) 0 0
\(946\) 3.05573 0.0993503
\(947\) 32.2492 1.04796 0.523979 0.851731i \(-0.324447\pi\)
0.523979 + 0.851731i \(0.324447\pi\)
\(948\) 12.1803 0.395599
\(949\) 31.4164 1.01982
\(950\) 3.47214 0.112651
\(951\) 12.4721 0.404437
\(952\) 0 0
\(953\) 18.5836 0.601982 0.300991 0.953627i \(-0.402683\pi\)
0.300991 + 0.953627i \(0.402683\pi\)
\(954\) −8.47214 −0.274296
\(955\) 4.72136 0.152780
\(956\) 22.6525 0.732633
\(957\) −16.9443 −0.547731
\(958\) 1.23607 0.0399355
\(959\) 0 0
\(960\) −1.23607 −0.0398939
\(961\) 28.4164 0.916658
\(962\) 4.00000 0.128965
\(963\) −8.94427 −0.288225
\(964\) 16.3607 0.526942
\(965\) 0.583592 0.0187865
\(966\) 0 0
\(967\) −19.4164 −0.624390 −0.312195 0.950018i \(-0.601064\pi\)
−0.312195 + 0.950018i \(0.601064\pi\)
\(968\) 7.00000 0.224989
\(969\) −2.47214 −0.0794164
\(970\) −12.9443 −0.415616
\(971\) 44.7214 1.43518 0.717588 0.696467i \(-0.245246\pi\)
0.717588 + 0.696467i \(0.245246\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −19.2361 −0.616363
\(975\) 18.1803 0.582237
\(976\) −12.4721 −0.399223
\(977\) 5.41641 0.173286 0.0866431 0.996239i \(-0.472386\pi\)
0.0866431 + 0.996239i \(0.472386\pi\)
\(978\) 20.9443 0.669724
\(979\) −5.88854 −0.188199
\(980\) 0 0
\(981\) −1.70820 −0.0545388
\(982\) −23.5279 −0.750804
\(983\) −17.3050 −0.551942 −0.275971 0.961166i \(-0.588999\pi\)
−0.275971 + 0.961166i \(0.588999\pi\)
\(984\) −6.94427 −0.221375
\(985\) −2.83282 −0.0902610
\(986\) 20.9443 0.667001
\(987\) 0 0
\(988\) −5.23607 −0.166582
\(989\) −8.72136 −0.277323
\(990\) −2.47214 −0.0785696
\(991\) 49.7082 1.57903 0.789517 0.613729i \(-0.210331\pi\)
0.789517 + 0.613729i \(0.210331\pi\)
\(992\) 7.70820 0.244736
\(993\) −18.9443 −0.601178
\(994\) 0 0
\(995\) −6.83282 −0.216615
\(996\) −8.94427 −0.283410
\(997\) −48.4721 −1.53513 −0.767564 0.640972i \(-0.778531\pi\)
−0.767564 + 0.640972i \(0.778531\pi\)
\(998\) −8.36068 −0.264653
\(999\) 0.763932 0.0241697
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 5586.2.a.be.1.1 2
7.6 odd 2 798.2.a.j.1.2 2
21.20 even 2 2394.2.a.z.1.1 2
28.27 even 2 6384.2.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.a.j.1.2 2 7.6 odd 2
2394.2.a.z.1.1 2 21.20 even 2
5586.2.a.be.1.1 2 1.1 even 1 trivial
6384.2.a.bn.1.2 2 28.27 even 2