Properties

Label 1127.2.a.d.1.2
Level $1127$
Weight $2$
Character 1127.1
Self dual yes
Analytic conductor $8.999$
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] = [1127,2,Mod(1,1127)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1127, 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("1127.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1127 = 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1127.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: \(8.99914030780\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 161)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1127.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +3.23607 q^{5} +0.618034 q^{6} -2.23607 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +3.23607 q^{5} +0.618034 q^{6} -2.23607 q^{8} -2.00000 q^{9} +2.00000 q^{10} +4.47214 q^{11} -1.61803 q^{12} -0.236068 q^{13} +3.23607 q^{15} +1.85410 q^{16} -1.23607 q^{18} +7.23607 q^{19} -5.23607 q^{20} +2.76393 q^{22} -1.00000 q^{23} -2.23607 q^{24} +5.47214 q^{25} -0.145898 q^{26} -5.00000 q^{27} -1.47214 q^{29} +2.00000 q^{30} +9.00000 q^{31} +5.61803 q^{32} +4.47214 q^{33} +3.23607 q^{36} -5.70820 q^{37} +4.47214 q^{38} -0.236068 q^{39} -7.23607 q^{40} +2.23607 q^{41} +2.47214 q^{43} -7.23607 q^{44} -6.47214 q^{45} -0.618034 q^{46} +3.47214 q^{47} +1.85410 q^{48} +3.38197 q^{50} +0.381966 q^{52} +11.2361 q^{53} -3.09017 q^{54} +14.4721 q^{55} +7.23607 q^{57} -0.909830 q^{58} +1.52786 q^{59} -5.23607 q^{60} -13.4164 q^{61} +5.56231 q^{62} -0.236068 q^{64} -0.763932 q^{65} +2.76393 q^{66} -12.1803 q^{67} -1.00000 q^{69} -10.2361 q^{71} +4.47214 q^{72} +6.70820 q^{73} -3.52786 q^{74} +5.47214 q^{75} -11.7082 q^{76} -0.145898 q^{78} -7.23607 q^{79} +6.00000 q^{80} +1.00000 q^{81} +1.38197 q^{82} -6.47214 q^{83} +1.52786 q^{86} -1.47214 q^{87} -10.0000 q^{88} -8.94427 q^{89} -4.00000 q^{90} +1.61803 q^{92} +9.00000 q^{93} +2.14590 q^{94} +23.4164 q^{95} +5.61803 q^{96} -3.70820 q^{97} -8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - q^{6} - 4 q^{9} + 4 q^{10} - q^{12} + 4 q^{13} + 2 q^{15} - 3 q^{16} + 2 q^{18} + 10 q^{19} - 6 q^{20} + 10 q^{22} - 2 q^{23} + 2 q^{25} - 7 q^{26} - 10 q^{27} + 6 q^{29} + 4 q^{30} + 18 q^{31} + 9 q^{32} + 2 q^{36} + 2 q^{37} + 4 q^{39} - 10 q^{40} - 4 q^{43} - 10 q^{44} - 4 q^{45} + q^{46} - 2 q^{47} - 3 q^{48} + 9 q^{50} + 3 q^{52} + 18 q^{53} + 5 q^{54} + 20 q^{55} + 10 q^{57} - 13 q^{58} + 12 q^{59} - 6 q^{60} - 9 q^{62} + 4 q^{64} - 6 q^{65} + 10 q^{66} - 2 q^{67} - 2 q^{69} - 16 q^{71} - 16 q^{74} + 2 q^{75} - 10 q^{76} - 7 q^{78} - 10 q^{79} + 12 q^{80} + 2 q^{81} + 5 q^{82} - 4 q^{83} + 12 q^{86} + 6 q^{87} - 20 q^{88} - 8 q^{90} + q^{92} + 18 q^{93} + 11 q^{94} + 20 q^{95} + 9 q^{96} + 6 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) −1.61803 −0.809017
\(5\) 3.23607 1.44721 0.723607 0.690212i \(-0.242483\pi\)
0.723607 + 0.690212i \(0.242483\pi\)
\(6\) 0.618034 0.252311
\(7\) 0 0
\(8\) −2.23607 −0.790569
\(9\) −2.00000 −0.666667
\(10\) 2.00000 0.632456
\(11\) 4.47214 1.34840 0.674200 0.738549i \(-0.264489\pi\)
0.674200 + 0.738549i \(0.264489\pi\)
\(12\) −1.61803 −0.467086
\(13\) −0.236068 −0.0654735 −0.0327367 0.999464i \(-0.510422\pi\)
−0.0327367 + 0.999464i \(0.510422\pi\)
\(14\) 0 0
\(15\) 3.23607 0.835549
\(16\) 1.85410 0.463525
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.23607 −0.291344
\(19\) 7.23607 1.66007 0.830034 0.557713i \(-0.188321\pi\)
0.830034 + 0.557713i \(0.188321\pi\)
\(20\) −5.23607 −1.17082
\(21\) 0 0
\(22\) 2.76393 0.589272
\(23\) −1.00000 −0.208514
\(24\) −2.23607 −0.456435
\(25\) 5.47214 1.09443
\(26\) −0.145898 −0.0286130
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −1.47214 −0.273369 −0.136684 0.990615i \(-0.543645\pi\)
−0.136684 + 0.990615i \(0.543645\pi\)
\(30\) 2.00000 0.365148
\(31\) 9.00000 1.61645 0.808224 0.588875i \(-0.200429\pi\)
0.808224 + 0.588875i \(0.200429\pi\)
\(32\) 5.61803 0.993137
\(33\) 4.47214 0.778499
\(34\) 0 0
\(35\) 0 0
\(36\) 3.23607 0.539345
\(37\) −5.70820 −0.938423 −0.469211 0.883086i \(-0.655462\pi\)
−0.469211 + 0.883086i \(0.655462\pi\)
\(38\) 4.47214 0.725476
\(39\) −0.236068 −0.0378011
\(40\) −7.23607 −1.14412
\(41\) 2.23607 0.349215 0.174608 0.984638i \(-0.444134\pi\)
0.174608 + 0.984638i \(0.444134\pi\)
\(42\) 0 0
\(43\) 2.47214 0.376997 0.188499 0.982073i \(-0.439638\pi\)
0.188499 + 0.982073i \(0.439638\pi\)
\(44\) −7.23607 −1.09088
\(45\) −6.47214 −0.964809
\(46\) −0.618034 −0.0911241
\(47\) 3.47214 0.506463 0.253232 0.967406i \(-0.418507\pi\)
0.253232 + 0.967406i \(0.418507\pi\)
\(48\) 1.85410 0.267617
\(49\) 0 0
\(50\) 3.38197 0.478282
\(51\) 0 0
\(52\) 0.381966 0.0529692
\(53\) 11.2361 1.54339 0.771696 0.635991i \(-0.219409\pi\)
0.771696 + 0.635991i \(0.219409\pi\)
\(54\) −3.09017 −0.420519
\(55\) 14.4721 1.95142
\(56\) 0 0
\(57\) 7.23607 0.958441
\(58\) −0.909830 −0.119467
\(59\) 1.52786 0.198911 0.0994555 0.995042i \(-0.468290\pi\)
0.0994555 + 0.995042i \(0.468290\pi\)
\(60\) −5.23607 −0.675973
\(61\) −13.4164 −1.71780 −0.858898 0.512148i \(-0.828850\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 5.56231 0.706414
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) −0.763932 −0.0947541
\(66\) 2.76393 0.340217
\(67\) −12.1803 −1.48807 −0.744033 0.668143i \(-0.767089\pi\)
−0.744033 + 0.668143i \(0.767089\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −10.2361 −1.21480 −0.607399 0.794397i \(-0.707787\pi\)
−0.607399 + 0.794397i \(0.707787\pi\)
\(72\) 4.47214 0.527046
\(73\) 6.70820 0.785136 0.392568 0.919723i \(-0.371587\pi\)
0.392568 + 0.919723i \(0.371587\pi\)
\(74\) −3.52786 −0.410106
\(75\) 5.47214 0.631868
\(76\) −11.7082 −1.34302
\(77\) 0 0
\(78\) −0.145898 −0.0165197
\(79\) −7.23607 −0.814121 −0.407061 0.913401i \(-0.633446\pi\)
−0.407061 + 0.913401i \(0.633446\pi\)
\(80\) 6.00000 0.670820
\(81\) 1.00000 0.111111
\(82\) 1.38197 0.152613
\(83\) −6.47214 −0.710409 −0.355205 0.934789i \(-0.615589\pi\)
−0.355205 + 0.934789i \(0.615589\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.52786 0.164754
\(87\) −1.47214 −0.157830
\(88\) −10.0000 −1.06600
\(89\) −8.94427 −0.948091 −0.474045 0.880500i \(-0.657207\pi\)
−0.474045 + 0.880500i \(0.657207\pi\)
\(90\) −4.00000 −0.421637
\(91\) 0 0
\(92\) 1.61803 0.168692
\(93\) 9.00000 0.933257
\(94\) 2.14590 0.221332
\(95\) 23.4164 2.40247
\(96\) 5.61803 0.573388
\(97\) −3.70820 −0.376511 −0.188256 0.982120i \(-0.560283\pi\)
−0.188256 + 0.982120i \(0.560283\pi\)
\(98\) 0 0
\(99\) −8.94427 −0.898933
\(100\) −8.85410 −0.885410
\(101\) 13.4164 1.33498 0.667491 0.744618i \(-0.267368\pi\)
0.667491 + 0.744618i \(0.267368\pi\)
\(102\) 0 0
\(103\) 15.7082 1.54778 0.773888 0.633323i \(-0.218309\pi\)
0.773888 + 0.633323i \(0.218309\pi\)
\(104\) 0.527864 0.0517613
\(105\) 0 0
\(106\) 6.94427 0.674487
\(107\) 11.2361 1.08623 0.543116 0.839658i \(-0.317244\pi\)
0.543116 + 0.839658i \(0.317244\pi\)
\(108\) 8.09017 0.778477
\(109\) −15.4164 −1.47662 −0.738312 0.674459i \(-0.764377\pi\)
−0.738312 + 0.674459i \(0.764377\pi\)
\(110\) 8.94427 0.852803
\(111\) −5.70820 −0.541799
\(112\) 0 0
\(113\) 2.47214 0.232559 0.116279 0.993217i \(-0.462903\pi\)
0.116279 + 0.993217i \(0.462903\pi\)
\(114\) 4.47214 0.418854
\(115\) −3.23607 −0.301765
\(116\) 2.38197 0.221160
\(117\) 0.472136 0.0436490
\(118\) 0.944272 0.0869273
\(119\) 0 0
\(120\) −7.23607 −0.660560
\(121\) 9.00000 0.818182
\(122\) −8.29180 −0.750704
\(123\) 2.23607 0.201619
\(124\) −14.5623 −1.30773
\(125\) 1.52786 0.136656
\(126\) 0 0
\(127\) −2.70820 −0.240314 −0.120157 0.992755i \(-0.538340\pi\)
−0.120157 + 0.992755i \(0.538340\pi\)
\(128\) −11.3820 −1.00603
\(129\) 2.47214 0.217659
\(130\) −0.472136 −0.0414091
\(131\) −3.94427 −0.344613 −0.172306 0.985043i \(-0.555122\pi\)
−0.172306 + 0.985043i \(0.555122\pi\)
\(132\) −7.23607 −0.629819
\(133\) 0 0
\(134\) −7.52786 −0.650308
\(135\) −16.1803 −1.39258
\(136\) 0 0
\(137\) 15.7082 1.34204 0.671021 0.741438i \(-0.265856\pi\)
0.671021 + 0.741438i \(0.265856\pi\)
\(138\) −0.618034 −0.0526105
\(139\) −2.52786 −0.214411 −0.107205 0.994237i \(-0.534190\pi\)
−0.107205 + 0.994237i \(0.534190\pi\)
\(140\) 0 0
\(141\) 3.47214 0.292407
\(142\) −6.32624 −0.530886
\(143\) −1.05573 −0.0882844
\(144\) −3.70820 −0.309017
\(145\) −4.76393 −0.395623
\(146\) 4.14590 0.343117
\(147\) 0 0
\(148\) 9.23607 0.759200
\(149\) 15.2361 1.24819 0.624094 0.781350i \(-0.285468\pi\)
0.624094 + 0.781350i \(0.285468\pi\)
\(150\) 3.38197 0.276136
\(151\) −15.1803 −1.23536 −0.617679 0.786430i \(-0.711927\pi\)
−0.617679 + 0.786430i \(0.711927\pi\)
\(152\) −16.1803 −1.31240
\(153\) 0 0
\(154\) 0 0
\(155\) 29.1246 2.33935
\(156\) 0.381966 0.0305818
\(157\) −15.4164 −1.23036 −0.615182 0.788385i \(-0.710917\pi\)
−0.615182 + 0.788385i \(0.710917\pi\)
\(158\) −4.47214 −0.355784
\(159\) 11.2361 0.891078
\(160\) 18.1803 1.43728
\(161\) 0 0
\(162\) 0.618034 0.0485573
\(163\) 19.1803 1.50232 0.751160 0.660120i \(-0.229495\pi\)
0.751160 + 0.660120i \(0.229495\pi\)
\(164\) −3.61803 −0.282521
\(165\) 14.4721 1.12665
\(166\) −4.00000 −0.310460
\(167\) −21.8885 −1.69379 −0.846893 0.531763i \(-0.821530\pi\)
−0.846893 + 0.531763i \(0.821530\pi\)
\(168\) 0 0
\(169\) −12.9443 −0.995713
\(170\) 0 0
\(171\) −14.4721 −1.10671
\(172\) −4.00000 −0.304997
\(173\) 3.52786 0.268219 0.134109 0.990967i \(-0.457183\pi\)
0.134109 + 0.990967i \(0.457183\pi\)
\(174\) −0.909830 −0.0689740
\(175\) 0 0
\(176\) 8.29180 0.625018
\(177\) 1.52786 0.114841
\(178\) −5.52786 −0.414331
\(179\) −18.7082 −1.39832 −0.699158 0.714967i \(-0.746442\pi\)
−0.699158 + 0.714967i \(0.746442\pi\)
\(180\) 10.4721 0.780547
\(181\) 5.05573 0.375789 0.187895 0.982189i \(-0.439834\pi\)
0.187895 + 0.982189i \(0.439834\pi\)
\(182\) 0 0
\(183\) −13.4164 −0.991769
\(184\) 2.23607 0.164845
\(185\) −18.4721 −1.35810
\(186\) 5.56231 0.407848
\(187\) 0 0
\(188\) −5.61803 −0.409737
\(189\) 0 0
\(190\) 14.4721 1.04992
\(191\) −24.1803 −1.74963 −0.874814 0.484459i \(-0.839016\pi\)
−0.874814 + 0.484459i \(0.839016\pi\)
\(192\) −0.236068 −0.0170367
\(193\) 4.41641 0.317900 0.158950 0.987287i \(-0.449189\pi\)
0.158950 + 0.987287i \(0.449189\pi\)
\(194\) −2.29180 −0.164541
\(195\) −0.763932 −0.0547063
\(196\) 0 0
\(197\) −21.4721 −1.52983 −0.764913 0.644133i \(-0.777218\pi\)
−0.764913 + 0.644133i \(0.777218\pi\)
\(198\) −5.52786 −0.392848
\(199\) −15.8885 −1.12631 −0.563155 0.826352i \(-0.690412\pi\)
−0.563155 + 0.826352i \(0.690412\pi\)
\(200\) −12.2361 −0.865221
\(201\) −12.1803 −0.859135
\(202\) 8.29180 0.583409
\(203\) 0 0
\(204\) 0 0
\(205\) 7.23607 0.505389
\(206\) 9.70820 0.676403
\(207\) 2.00000 0.139010
\(208\) −0.437694 −0.0303486
\(209\) 32.3607 2.23844
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −18.1803 −1.24863
\(213\) −10.2361 −0.701364
\(214\) 6.94427 0.474701
\(215\) 8.00000 0.545595
\(216\) 11.1803 0.760726
\(217\) 0 0
\(218\) −9.52786 −0.645308
\(219\) 6.70820 0.453298
\(220\) −23.4164 −1.57873
\(221\) 0 0
\(222\) −3.52786 −0.236775
\(223\) −3.41641 −0.228780 −0.114390 0.993436i \(-0.536491\pi\)
−0.114390 + 0.993436i \(0.536491\pi\)
\(224\) 0 0
\(225\) −10.9443 −0.729618
\(226\) 1.52786 0.101632
\(227\) 5.88854 0.390836 0.195418 0.980720i \(-0.437394\pi\)
0.195418 + 0.980720i \(0.437394\pi\)
\(228\) −11.7082 −0.775395
\(229\) −14.7639 −0.975628 −0.487814 0.872948i \(-0.662206\pi\)
−0.487814 + 0.872948i \(0.662206\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) 3.29180 0.216117
\(233\) −11.4721 −0.751565 −0.375782 0.926708i \(-0.622626\pi\)
−0.375782 + 0.926708i \(0.622626\pi\)
\(234\) 0.291796 0.0190753
\(235\) 11.2361 0.732960
\(236\) −2.47214 −0.160922
\(237\) −7.23607 −0.470033
\(238\) 0 0
\(239\) −15.7639 −1.01968 −0.509842 0.860268i \(-0.670296\pi\)
−0.509842 + 0.860268i \(0.670296\pi\)
\(240\) 6.00000 0.387298
\(241\) 21.4164 1.37955 0.689776 0.724023i \(-0.257709\pi\)
0.689776 + 0.724023i \(0.257709\pi\)
\(242\) 5.56231 0.357559
\(243\) 16.0000 1.02640
\(244\) 21.7082 1.38973
\(245\) 0 0
\(246\) 1.38197 0.0881109
\(247\) −1.70820 −0.108690
\(248\) −20.1246 −1.27791
\(249\) −6.47214 −0.410155
\(250\) 0.944272 0.0597210
\(251\) −8.29180 −0.523374 −0.261687 0.965153i \(-0.584279\pi\)
−0.261687 + 0.965153i \(0.584279\pi\)
\(252\) 0 0
\(253\) −4.47214 −0.281161
\(254\) −1.67376 −0.105021
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) −4.23607 −0.264239 −0.132119 0.991234i \(-0.542178\pi\)
−0.132119 + 0.991234i \(0.542178\pi\)
\(258\) 1.52786 0.0951207
\(259\) 0 0
\(260\) 1.23607 0.0766577
\(261\) 2.94427 0.182246
\(262\) −2.43769 −0.150601
\(263\) −26.9443 −1.66145 −0.830727 0.556679i \(-0.812075\pi\)
−0.830727 + 0.556679i \(0.812075\pi\)
\(264\) −10.0000 −0.615457
\(265\) 36.3607 2.23362
\(266\) 0 0
\(267\) −8.94427 −0.547381
\(268\) 19.7082 1.20387
\(269\) 9.18034 0.559735 0.279868 0.960039i \(-0.409709\pi\)
0.279868 + 0.960039i \(0.409709\pi\)
\(270\) −10.0000 −0.608581
\(271\) 16.9443 1.02929 0.514646 0.857403i \(-0.327924\pi\)
0.514646 + 0.857403i \(0.327924\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 9.70820 0.586494
\(275\) 24.4721 1.47573
\(276\) 1.61803 0.0973942
\(277\) 20.4164 1.22670 0.613352 0.789810i \(-0.289821\pi\)
0.613352 + 0.789810i \(0.289821\pi\)
\(278\) −1.56231 −0.0937009
\(279\) −18.0000 −1.07763
\(280\) 0 0
\(281\) 3.70820 0.221213 0.110606 0.993864i \(-0.464721\pi\)
0.110606 + 0.993864i \(0.464721\pi\)
\(282\) 2.14590 0.127786
\(283\) −18.9443 −1.12612 −0.563060 0.826416i \(-0.690376\pi\)
−0.563060 + 0.826416i \(0.690376\pi\)
\(284\) 16.5623 0.982792
\(285\) 23.4164 1.38707
\(286\) −0.652476 −0.0385817
\(287\) 0 0
\(288\) −11.2361 −0.662092
\(289\) −17.0000 −1.00000
\(290\) −2.94427 −0.172894
\(291\) −3.70820 −0.217379
\(292\) −10.8541 −0.635188
\(293\) 15.7082 0.917683 0.458842 0.888518i \(-0.348265\pi\)
0.458842 + 0.888518i \(0.348265\pi\)
\(294\) 0 0
\(295\) 4.94427 0.287867
\(296\) 12.7639 0.741888
\(297\) −22.3607 −1.29750
\(298\) 9.41641 0.545478
\(299\) 0.236068 0.0136522
\(300\) −8.85410 −0.511192
\(301\) 0 0
\(302\) −9.38197 −0.539871
\(303\) 13.4164 0.770752
\(304\) 13.4164 0.769484
\(305\) −43.4164 −2.48602
\(306\) 0 0
\(307\) 11.4164 0.651569 0.325784 0.945444i \(-0.394372\pi\)
0.325784 + 0.945444i \(0.394372\pi\)
\(308\) 0 0
\(309\) 15.7082 0.893609
\(310\) 18.0000 1.02233
\(311\) 2.88854 0.163794 0.0818971 0.996641i \(-0.473902\pi\)
0.0818971 + 0.996641i \(0.473902\pi\)
\(312\) 0.527864 0.0298844
\(313\) −2.76393 −0.156227 −0.0781133 0.996944i \(-0.524890\pi\)
−0.0781133 + 0.996944i \(0.524890\pi\)
\(314\) −9.52786 −0.537688
\(315\) 0 0
\(316\) 11.7082 0.658638
\(317\) 31.3050 1.75826 0.879131 0.476581i \(-0.158124\pi\)
0.879131 + 0.476581i \(0.158124\pi\)
\(318\) 6.94427 0.389415
\(319\) −6.58359 −0.368610
\(320\) −0.763932 −0.0427051
\(321\) 11.2361 0.627136
\(322\) 0 0
\(323\) 0 0
\(324\) −1.61803 −0.0898908
\(325\) −1.29180 −0.0716560
\(326\) 11.8541 0.656538
\(327\) −15.4164 −0.852529
\(328\) −5.00000 −0.276079
\(329\) 0 0
\(330\) 8.94427 0.492366
\(331\) 0.708204 0.0389264 0.0194632 0.999811i \(-0.493804\pi\)
0.0194632 + 0.999811i \(0.493804\pi\)
\(332\) 10.4721 0.574733
\(333\) 11.4164 0.625615
\(334\) −13.5279 −0.740212
\(335\) −39.4164 −2.15355
\(336\) 0 0
\(337\) −17.5967 −0.958556 −0.479278 0.877663i \(-0.659101\pi\)
−0.479278 + 0.877663i \(0.659101\pi\)
\(338\) −8.00000 −0.435143
\(339\) 2.47214 0.134268
\(340\) 0 0
\(341\) 40.2492 2.17962
\(342\) −8.94427 −0.483651
\(343\) 0 0
\(344\) −5.52786 −0.298042
\(345\) −3.23607 −0.174224
\(346\) 2.18034 0.117216
\(347\) −5.88854 −0.316114 −0.158057 0.987430i \(-0.550523\pi\)
−0.158057 + 0.987430i \(0.550523\pi\)
\(348\) 2.38197 0.127687
\(349\) −1.29180 −0.0691483 −0.0345741 0.999402i \(-0.511007\pi\)
−0.0345741 + 0.999402i \(0.511007\pi\)
\(350\) 0 0
\(351\) 1.18034 0.0630019
\(352\) 25.1246 1.33915
\(353\) 1.76393 0.0938846 0.0469423 0.998898i \(-0.485052\pi\)
0.0469423 + 0.998898i \(0.485052\pi\)
\(354\) 0.944272 0.0501875
\(355\) −33.1246 −1.75807
\(356\) 14.4721 0.767022
\(357\) 0 0
\(358\) −11.5623 −0.611087
\(359\) 4.76393 0.251431 0.125715 0.992066i \(-0.459877\pi\)
0.125715 + 0.992066i \(0.459877\pi\)
\(360\) 14.4721 0.762749
\(361\) 33.3607 1.75583
\(362\) 3.12461 0.164226
\(363\) 9.00000 0.472377
\(364\) 0 0
\(365\) 21.7082 1.13626
\(366\) −8.29180 −0.433419
\(367\) 4.58359 0.239262 0.119631 0.992818i \(-0.461829\pi\)
0.119631 + 0.992818i \(0.461829\pi\)
\(368\) −1.85410 −0.0966517
\(369\) −4.47214 −0.232810
\(370\) −11.4164 −0.593511
\(371\) 0 0
\(372\) −14.5623 −0.755020
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 1.52786 0.0788986
\(376\) −7.76393 −0.400394
\(377\) 0.347524 0.0178984
\(378\) 0 0
\(379\) 8.29180 0.425921 0.212960 0.977061i \(-0.431689\pi\)
0.212960 + 0.977061i \(0.431689\pi\)
\(380\) −37.8885 −1.94364
\(381\) −2.70820 −0.138745
\(382\) −14.9443 −0.764615
\(383\) 15.7082 0.802652 0.401326 0.915935i \(-0.368549\pi\)
0.401326 + 0.915935i \(0.368549\pi\)
\(384\) −11.3820 −0.580834
\(385\) 0 0
\(386\) 2.72949 0.138927
\(387\) −4.94427 −0.251331
\(388\) 6.00000 0.304604
\(389\) −7.41641 −0.376027 −0.188013 0.982166i \(-0.560205\pi\)
−0.188013 + 0.982166i \(0.560205\pi\)
\(390\) −0.472136 −0.0239075
\(391\) 0 0
\(392\) 0 0
\(393\) −3.94427 −0.198962
\(394\) −13.2705 −0.668559
\(395\) −23.4164 −1.17821
\(396\) 14.4721 0.727252
\(397\) 25.6525 1.28746 0.643730 0.765252i \(-0.277386\pi\)
0.643730 + 0.765252i \(0.277386\pi\)
\(398\) −9.81966 −0.492215
\(399\) 0 0
\(400\) 10.1459 0.507295
\(401\) −19.8885 −0.993186 −0.496593 0.867983i \(-0.665416\pi\)
−0.496593 + 0.867983i \(0.665416\pi\)
\(402\) −7.52786 −0.375456
\(403\) −2.12461 −0.105834
\(404\) −21.7082 −1.08002
\(405\) 3.23607 0.160802
\(406\) 0 0
\(407\) −25.5279 −1.26537
\(408\) 0 0
\(409\) −4.12461 −0.203949 −0.101974 0.994787i \(-0.532516\pi\)
−0.101974 + 0.994787i \(0.532516\pi\)
\(410\) 4.47214 0.220863
\(411\) 15.7082 0.774829
\(412\) −25.4164 −1.25218
\(413\) 0 0
\(414\) 1.23607 0.0607494
\(415\) −20.9443 −1.02811
\(416\) −1.32624 −0.0650242
\(417\) −2.52786 −0.123790
\(418\) 20.0000 0.978232
\(419\) −26.4721 −1.29325 −0.646624 0.762809i \(-0.723820\pi\)
−0.646624 + 0.762809i \(0.723820\pi\)
\(420\) 0 0
\(421\) −8.58359 −0.418339 −0.209169 0.977879i \(-0.567076\pi\)
−0.209169 + 0.977879i \(0.567076\pi\)
\(422\) −7.41641 −0.361025
\(423\) −6.94427 −0.337642
\(424\) −25.1246 −1.22016
\(425\) 0 0
\(426\) −6.32624 −0.306507
\(427\) 0 0
\(428\) −18.1803 −0.878780
\(429\) −1.05573 −0.0509710
\(430\) 4.94427 0.238434
\(431\) 18.1803 0.875716 0.437858 0.899044i \(-0.355737\pi\)
0.437858 + 0.899044i \(0.355737\pi\)
\(432\) −9.27051 −0.446028
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −4.76393 −0.228413
\(436\) 24.9443 1.19461
\(437\) −7.23607 −0.346148
\(438\) 4.14590 0.198099
\(439\) −30.3050 −1.44638 −0.723188 0.690651i \(-0.757324\pi\)
−0.723188 + 0.690651i \(0.757324\pi\)
\(440\) −32.3607 −1.54273
\(441\) 0 0
\(442\) 0 0
\(443\) −7.18034 −0.341148 −0.170574 0.985345i \(-0.554562\pi\)
−0.170574 + 0.985345i \(0.554562\pi\)
\(444\) 9.23607 0.438324
\(445\) −28.9443 −1.37209
\(446\) −2.11146 −0.0999803
\(447\) 15.2361 0.720641
\(448\) 0 0
\(449\) 32.8328 1.54948 0.774738 0.632282i \(-0.217882\pi\)
0.774738 + 0.632282i \(0.217882\pi\)
\(450\) −6.76393 −0.318855
\(451\) 10.0000 0.470882
\(452\) −4.00000 −0.188144
\(453\) −15.1803 −0.713235
\(454\) 3.63932 0.170802
\(455\) 0 0
\(456\) −16.1803 −0.757714
\(457\) 14.4721 0.676978 0.338489 0.940970i \(-0.390084\pi\)
0.338489 + 0.940970i \(0.390084\pi\)
\(458\) −9.12461 −0.426365
\(459\) 0 0
\(460\) 5.23607 0.244133
\(461\) 25.7639 1.19995 0.599973 0.800020i \(-0.295178\pi\)
0.599973 + 0.800020i \(0.295178\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −2.72949 −0.126713
\(465\) 29.1246 1.35062
\(466\) −7.09017 −0.328446
\(467\) 11.5279 0.533446 0.266723 0.963773i \(-0.414059\pi\)
0.266723 + 0.963773i \(0.414059\pi\)
\(468\) −0.763932 −0.0353128
\(469\) 0 0
\(470\) 6.94427 0.320315
\(471\) −15.4164 −0.710351
\(472\) −3.41641 −0.157253
\(473\) 11.0557 0.508343
\(474\) −4.47214 −0.205412
\(475\) 39.5967 1.81682
\(476\) 0 0
\(477\) −22.4721 −1.02893
\(478\) −9.74265 −0.445618
\(479\) 23.1246 1.05659 0.528295 0.849061i \(-0.322831\pi\)
0.528295 + 0.849061i \(0.322831\pi\)
\(480\) 18.1803 0.829815
\(481\) 1.34752 0.0614418
\(482\) 13.2361 0.602886
\(483\) 0 0
\(484\) −14.5623 −0.661923
\(485\) −12.0000 −0.544892
\(486\) 9.88854 0.448553
\(487\) 20.1246 0.911933 0.455967 0.889997i \(-0.349294\pi\)
0.455967 + 0.889997i \(0.349294\pi\)
\(488\) 30.0000 1.35804
\(489\) 19.1803 0.867365
\(490\) 0 0
\(491\) 11.7639 0.530899 0.265449 0.964125i \(-0.414480\pi\)
0.265449 + 0.964125i \(0.414480\pi\)
\(492\) −3.61803 −0.163114
\(493\) 0 0
\(494\) −1.05573 −0.0474995
\(495\) −28.9443 −1.30095
\(496\) 16.6869 0.749265
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −36.7082 −1.64328 −0.821642 0.570003i \(-0.806942\pi\)
−0.821642 + 0.570003i \(0.806942\pi\)
\(500\) −2.47214 −0.110557
\(501\) −21.8885 −0.977908
\(502\) −5.12461 −0.228723
\(503\) 15.5967 0.695425 0.347712 0.937601i \(-0.386959\pi\)
0.347712 + 0.937601i \(0.386959\pi\)
\(504\) 0 0
\(505\) 43.4164 1.93200
\(506\) −2.76393 −0.122872
\(507\) −12.9443 −0.574875
\(508\) 4.38197 0.194418
\(509\) 12.8197 0.568221 0.284111 0.958791i \(-0.408302\pi\)
0.284111 + 0.958791i \(0.408302\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 18.7082 0.826794
\(513\) −36.1803 −1.59740
\(514\) −2.61803 −0.115477
\(515\) 50.8328 2.23996
\(516\) −4.00000 −0.176090
\(517\) 15.5279 0.682915
\(518\) 0 0
\(519\) 3.52786 0.154856
\(520\) 1.70820 0.0749097
\(521\) −21.3050 −0.933387 −0.466693 0.884419i \(-0.654555\pi\)
−0.466693 + 0.884419i \(0.654555\pi\)
\(522\) 1.81966 0.0796444
\(523\) 7.70820 0.337056 0.168528 0.985697i \(-0.446099\pi\)
0.168528 + 0.985697i \(0.446099\pi\)
\(524\) 6.38197 0.278797
\(525\) 0 0
\(526\) −16.6525 −0.726082
\(527\) 0 0
\(528\) 8.29180 0.360854
\(529\) 1.00000 0.0434783
\(530\) 22.4721 0.976127
\(531\) −3.05573 −0.132607
\(532\) 0 0
\(533\) −0.527864 −0.0228643
\(534\) −5.52786 −0.239214
\(535\) 36.3607 1.57201
\(536\) 27.2361 1.17642
\(537\) −18.7082 −0.807319
\(538\) 5.67376 0.244613
\(539\) 0 0
\(540\) 26.1803 1.12662
\(541\) 21.0000 0.902861 0.451430 0.892306i \(-0.350914\pi\)
0.451430 + 0.892306i \(0.350914\pi\)
\(542\) 10.4721 0.449817
\(543\) 5.05573 0.216962
\(544\) 0 0
\(545\) −49.8885 −2.13699
\(546\) 0 0
\(547\) 43.1803 1.84626 0.923129 0.384490i \(-0.125623\pi\)
0.923129 + 0.384490i \(0.125623\pi\)
\(548\) −25.4164 −1.08574
\(549\) 26.8328 1.14520
\(550\) 15.1246 0.644916
\(551\) −10.6525 −0.453811
\(552\) 2.23607 0.0951734
\(553\) 0 0
\(554\) 12.6180 0.536089
\(555\) −18.4721 −0.784099
\(556\) 4.09017 0.173462
\(557\) −26.7639 −1.13402 −0.567012 0.823709i \(-0.691901\pi\)
−0.567012 + 0.823709i \(0.691901\pi\)
\(558\) −11.1246 −0.470942
\(559\) −0.583592 −0.0246833
\(560\) 0 0
\(561\) 0 0
\(562\) 2.29180 0.0966736
\(563\) 9.59675 0.404455 0.202227 0.979339i \(-0.435182\pi\)
0.202227 + 0.979339i \(0.435182\pi\)
\(564\) −5.61803 −0.236562
\(565\) 8.00000 0.336563
\(566\) −11.7082 −0.492133
\(567\) 0 0
\(568\) 22.8885 0.960382
\(569\) −8.18034 −0.342938 −0.171469 0.985190i \(-0.554851\pi\)
−0.171469 + 0.985190i \(0.554851\pi\)
\(570\) 14.4721 0.606171
\(571\) 16.2918 0.681790 0.340895 0.940101i \(-0.389270\pi\)
0.340895 + 0.940101i \(0.389270\pi\)
\(572\) 1.70820 0.0714236
\(573\) −24.1803 −1.01015
\(574\) 0 0
\(575\) −5.47214 −0.228204
\(576\) 0.472136 0.0196723
\(577\) 15.2918 0.636606 0.318303 0.947989i \(-0.396887\pi\)
0.318303 + 0.947989i \(0.396887\pi\)
\(578\) −10.5066 −0.437016
\(579\) 4.41641 0.183540
\(580\) 7.70820 0.320066
\(581\) 0 0
\(582\) −2.29180 −0.0949980
\(583\) 50.2492 2.08111
\(584\) −15.0000 −0.620704
\(585\) 1.52786 0.0631694
\(586\) 9.70820 0.401042
\(587\) −20.8885 −0.862162 −0.431081 0.902313i \(-0.641868\pi\)
−0.431081 + 0.902313i \(0.641868\pi\)
\(588\) 0 0
\(589\) 65.1246 2.68341
\(590\) 3.05573 0.125802
\(591\) −21.4721 −0.883246
\(592\) −10.5836 −0.434983
\(593\) −34.3607 −1.41102 −0.705512 0.708698i \(-0.749283\pi\)
−0.705512 + 0.708698i \(0.749283\pi\)
\(594\) −13.8197 −0.567028
\(595\) 0 0
\(596\) −24.6525 −1.00980
\(597\) −15.8885 −0.650275
\(598\) 0.145898 0.00596621
\(599\) −2.47214 −0.101009 −0.0505044 0.998724i \(-0.516083\pi\)
−0.0505044 + 0.998724i \(0.516083\pi\)
\(600\) −12.2361 −0.499535
\(601\) −22.2361 −0.907028 −0.453514 0.891249i \(-0.649830\pi\)
−0.453514 + 0.891249i \(0.649830\pi\)
\(602\) 0 0
\(603\) 24.3607 0.992044
\(604\) 24.5623 0.999426
\(605\) 29.1246 1.18408
\(606\) 8.29180 0.336831
\(607\) 33.3050 1.35181 0.675903 0.736990i \(-0.263754\pi\)
0.675903 + 0.736990i \(0.263754\pi\)
\(608\) 40.6525 1.64868
\(609\) 0 0
\(610\) −26.8328 −1.08643
\(611\) −0.819660 −0.0331599
\(612\) 0 0
\(613\) −22.1803 −0.895855 −0.447928 0.894070i \(-0.647838\pi\)
−0.447928 + 0.894070i \(0.647838\pi\)
\(614\) 7.05573 0.284746
\(615\) 7.23607 0.291786
\(616\) 0 0
\(617\) 35.2361 1.41855 0.709275 0.704932i \(-0.249022\pi\)
0.709275 + 0.704932i \(0.249022\pi\)
\(618\) 9.70820 0.390521
\(619\) 47.4853 1.90860 0.954298 0.298858i \(-0.0966058\pi\)
0.954298 + 0.298858i \(0.0966058\pi\)
\(620\) −47.1246 −1.89257
\(621\) 5.00000 0.200643
\(622\) 1.78522 0.0715807
\(623\) 0 0
\(624\) −0.437694 −0.0175218
\(625\) −22.4164 −0.896656
\(626\) −1.70820 −0.0682736
\(627\) 32.3607 1.29236
\(628\) 24.9443 0.995385
\(629\) 0 0
\(630\) 0 0
\(631\) 3.41641 0.136005 0.0680025 0.997685i \(-0.478337\pi\)
0.0680025 + 0.997685i \(0.478337\pi\)
\(632\) 16.1803 0.643619
\(633\) −12.0000 −0.476957
\(634\) 19.3475 0.768388
\(635\) −8.76393 −0.347786
\(636\) −18.1803 −0.720897
\(637\) 0 0
\(638\) −4.06888 −0.161089
\(639\) 20.4721 0.809865
\(640\) −36.8328 −1.45594
\(641\) 7.05573 0.278685 0.139342 0.990244i \(-0.455501\pi\)
0.139342 + 0.990244i \(0.455501\pi\)
\(642\) 6.94427 0.274069
\(643\) 17.7082 0.698343 0.349172 0.937059i \(-0.386463\pi\)
0.349172 + 0.937059i \(0.386463\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) 18.5279 0.728405 0.364203 0.931320i \(-0.381342\pi\)
0.364203 + 0.931320i \(0.381342\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 6.83282 0.268211
\(650\) −0.798374 −0.0313148
\(651\) 0 0
\(652\) −31.0344 −1.21540
\(653\) 24.5279 0.959849 0.479925 0.877310i \(-0.340664\pi\)
0.479925 + 0.877310i \(0.340664\pi\)
\(654\) −9.52786 −0.372569
\(655\) −12.7639 −0.498728
\(656\) 4.14590 0.161870
\(657\) −13.4164 −0.523424
\(658\) 0 0
\(659\) 6.76393 0.263485 0.131743 0.991284i \(-0.457943\pi\)
0.131743 + 0.991284i \(0.457943\pi\)
\(660\) −23.4164 −0.911482
\(661\) 41.7082 1.62226 0.811131 0.584865i \(-0.198853\pi\)
0.811131 + 0.584865i \(0.198853\pi\)
\(662\) 0.437694 0.0170115
\(663\) 0 0
\(664\) 14.4721 0.561628
\(665\) 0 0
\(666\) 7.05573 0.273404
\(667\) 1.47214 0.0570013
\(668\) 35.4164 1.37030
\(669\) −3.41641 −0.132086
\(670\) −24.3607 −0.941135
\(671\) −60.0000 −2.31627
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −10.8754 −0.418904
\(675\) −27.3607 −1.05311
\(676\) 20.9443 0.805549
\(677\) −23.2361 −0.893035 −0.446517 0.894775i \(-0.647336\pi\)
−0.446517 + 0.894775i \(0.647336\pi\)
\(678\) 1.52786 0.0586773
\(679\) 0 0
\(680\) 0 0
\(681\) 5.88854 0.225649
\(682\) 24.8754 0.952528
\(683\) −5.18034 −0.198220 −0.0991101 0.995076i \(-0.531600\pi\)
−0.0991101 + 0.995076i \(0.531600\pi\)
\(684\) 23.4164 0.895349
\(685\) 50.8328 1.94222
\(686\) 0 0
\(687\) −14.7639 −0.563279
\(688\) 4.58359 0.174748
\(689\) −2.65248 −0.101051
\(690\) −2.00000 −0.0761387
\(691\) −42.8328 −1.62944 −0.814719 0.579857i \(-0.803109\pi\)
−0.814719 + 0.579857i \(0.803109\pi\)
\(692\) −5.70820 −0.216993
\(693\) 0 0
\(694\) −3.63932 −0.138147
\(695\) −8.18034 −0.310298
\(696\) 3.29180 0.124775
\(697\) 0 0
\(698\) −0.798374 −0.0302189
\(699\) −11.4721 −0.433916
\(700\) 0 0
\(701\) −22.7639 −0.859782 −0.429891 0.902881i \(-0.641448\pi\)
−0.429891 + 0.902881i \(0.641448\pi\)
\(702\) 0.729490 0.0275328
\(703\) −41.3050 −1.55785
\(704\) −1.05573 −0.0397892
\(705\) 11.2361 0.423175
\(706\) 1.09017 0.0410291
\(707\) 0 0
\(708\) −2.47214 −0.0929086
\(709\) 34.2492 1.28626 0.643128 0.765758i \(-0.277636\pi\)
0.643128 + 0.765758i \(0.277636\pi\)
\(710\) −20.4721 −0.768306
\(711\) 14.4721 0.542748
\(712\) 20.0000 0.749532
\(713\) −9.00000 −0.337053
\(714\) 0 0
\(715\) −3.41641 −0.127766
\(716\) 30.2705 1.13126
\(717\) −15.7639 −0.588715
\(718\) 2.94427 0.109879
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) −12.0000 −0.447214
\(721\) 0 0
\(722\) 20.6180 0.767324
\(723\) 21.4164 0.796485
\(724\) −8.18034 −0.304020
\(725\) −8.05573 −0.299182
\(726\) 5.56231 0.206437
\(727\) −3.23607 −0.120019 −0.0600096 0.998198i \(-0.519113\pi\)
−0.0600096 + 0.998198i \(0.519113\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 13.4164 0.496564
\(731\) 0 0
\(732\) 21.7082 0.802358
\(733\) −12.9443 −0.478108 −0.239054 0.971006i \(-0.576837\pi\)
−0.239054 + 0.971006i \(0.576837\pi\)
\(734\) 2.83282 0.104561
\(735\) 0 0
\(736\) −5.61803 −0.207083
\(737\) −54.4721 −2.00651
\(738\) −2.76393 −0.101742
\(739\) 8.70820 0.320336 0.160168 0.987090i \(-0.448796\pi\)
0.160168 + 0.987090i \(0.448796\pi\)
\(740\) 29.8885 1.09872
\(741\) −1.70820 −0.0627524
\(742\) 0 0
\(743\) −25.5279 −0.936527 −0.468263 0.883589i \(-0.655120\pi\)
−0.468263 + 0.883589i \(0.655120\pi\)
\(744\) −20.1246 −0.737804
\(745\) 49.3050 1.80639
\(746\) 16.0689 0.588324
\(747\) 12.9443 0.473606
\(748\) 0 0
\(749\) 0 0
\(750\) 0.944272 0.0344799
\(751\) −8.58359 −0.313220 −0.156610 0.987661i \(-0.550057\pi\)
−0.156610 + 0.987661i \(0.550057\pi\)
\(752\) 6.43769 0.234759
\(753\) −8.29180 −0.302170
\(754\) 0.214782 0.00782189
\(755\) −49.1246 −1.78783
\(756\) 0 0
\(757\) −28.8328 −1.04795 −0.523973 0.851735i \(-0.675551\pi\)
−0.523973 + 0.851735i \(0.675551\pi\)
\(758\) 5.12461 0.186134
\(759\) −4.47214 −0.162328
\(760\) −52.3607 −1.89932
\(761\) 31.6525 1.14740 0.573701 0.819065i \(-0.305507\pi\)
0.573701 + 0.819065i \(0.305507\pi\)
\(762\) −1.67376 −0.0606340
\(763\) 0 0
\(764\) 39.1246 1.41548
\(765\) 0 0
\(766\) 9.70820 0.350772
\(767\) −0.360680 −0.0130234
\(768\) −6.56231 −0.236797
\(769\) 43.2361 1.55913 0.779566 0.626320i \(-0.215440\pi\)
0.779566 + 0.626320i \(0.215440\pi\)
\(770\) 0 0
\(771\) −4.23607 −0.152558
\(772\) −7.14590 −0.257186
\(773\) −40.6525 −1.46217 −0.731084 0.682288i \(-0.760985\pi\)
−0.731084 + 0.682288i \(0.760985\pi\)
\(774\) −3.05573 −0.109836
\(775\) 49.2492 1.76908
\(776\) 8.29180 0.297658
\(777\) 0 0
\(778\) −4.58359 −0.164330
\(779\) 16.1803 0.579721
\(780\) 1.23607 0.0442583
\(781\) −45.7771 −1.63803
\(782\) 0 0
\(783\) 7.36068 0.263049
\(784\) 0 0
\(785\) −49.8885 −1.78060
\(786\) −2.43769 −0.0869497
\(787\) 0.360680 0.0128568 0.00642842 0.999979i \(-0.497954\pi\)
0.00642842 + 0.999979i \(0.497954\pi\)
\(788\) 34.7426 1.23766
\(789\) −26.9443 −0.959241
\(790\) −14.4721 −0.514895
\(791\) 0 0
\(792\) 20.0000 0.710669
\(793\) 3.16718 0.112470
\(794\) 15.8541 0.562641
\(795\) 36.3607 1.28958
\(796\) 25.7082 0.911203
\(797\) 36.7639 1.30225 0.651123 0.758973i \(-0.274298\pi\)
0.651123 + 0.758973i \(0.274298\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 30.7426 1.08692
\(801\) 17.8885 0.632061
\(802\) −12.2918 −0.434038
\(803\) 30.0000 1.05868
\(804\) 19.7082 0.695055
\(805\) 0 0
\(806\) −1.31308 −0.0462514
\(807\) 9.18034 0.323163
\(808\) −30.0000 −1.05540
\(809\) −32.8328 −1.15434 −0.577170 0.816624i \(-0.695843\pi\)
−0.577170 + 0.816624i \(0.695843\pi\)
\(810\) 2.00000 0.0702728
\(811\) 45.3607 1.59283 0.796414 0.604751i \(-0.206727\pi\)
0.796414 + 0.604751i \(0.206727\pi\)
\(812\) 0 0
\(813\) 16.9443 0.594262
\(814\) −15.7771 −0.552987
\(815\) 62.0689 2.17418
\(816\) 0 0
\(817\) 17.8885 0.625841
\(818\) −2.54915 −0.0891289
\(819\) 0 0
\(820\) −11.7082 −0.408868
\(821\) 22.5836 0.788173 0.394086 0.919073i \(-0.371061\pi\)
0.394086 + 0.919073i \(0.371061\pi\)
\(822\) 9.70820 0.338612
\(823\) 25.5410 0.890304 0.445152 0.895455i \(-0.353150\pi\)
0.445152 + 0.895455i \(0.353150\pi\)
\(824\) −35.1246 −1.22362
\(825\) 24.4721 0.852010
\(826\) 0 0
\(827\) −17.3050 −0.601752 −0.300876 0.953663i \(-0.597279\pi\)
−0.300876 + 0.953663i \(0.597279\pi\)
\(828\) −3.23607 −0.112461
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) −12.9443 −0.449302
\(831\) 20.4164 0.708237
\(832\) 0.0557281 0.00193202
\(833\) 0 0
\(834\) −1.56231 −0.0540982
\(835\) −70.8328 −2.45127
\(836\) −52.3607 −1.81093
\(837\) −45.0000 −1.55543
\(838\) −16.3607 −0.565170
\(839\) −20.0689 −0.692855 −0.346427 0.938077i \(-0.612605\pi\)
−0.346427 + 0.938077i \(0.612605\pi\)
\(840\) 0 0
\(841\) −26.8328 −0.925270
\(842\) −5.30495 −0.182821
\(843\) 3.70820 0.127717
\(844\) 19.4164 0.668340
\(845\) −41.8885 −1.44101
\(846\) −4.29180 −0.147555
\(847\) 0 0
\(848\) 20.8328 0.715402
\(849\) −18.9443 −0.650166
\(850\) 0 0
\(851\) 5.70820 0.195675
\(852\) 16.5623 0.567415
\(853\) −44.8328 −1.53505 −0.767523 0.641021i \(-0.778511\pi\)
−0.767523 + 0.641021i \(0.778511\pi\)
\(854\) 0 0
\(855\) −46.8328 −1.60165
\(856\) −25.1246 −0.858742
\(857\) −9.54102 −0.325915 −0.162958 0.986633i \(-0.552103\pi\)
−0.162958 + 0.986633i \(0.552103\pi\)
\(858\) −0.652476 −0.0222752
\(859\) −11.5836 −0.395227 −0.197614 0.980280i \(-0.563319\pi\)
−0.197614 + 0.980280i \(0.563319\pi\)
\(860\) −12.9443 −0.441396
\(861\) 0 0
\(862\) 11.2361 0.382702
\(863\) 2.23607 0.0761166 0.0380583 0.999276i \(-0.487883\pi\)
0.0380583 + 0.999276i \(0.487883\pi\)
\(864\) −28.0902 −0.955647
\(865\) 11.4164 0.388170
\(866\) −8.65248 −0.294023
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) −32.3607 −1.09776
\(870\) −2.94427 −0.0998202
\(871\) 2.87539 0.0974288
\(872\) 34.4721 1.16737
\(873\) 7.41641 0.251007
\(874\) −4.47214 −0.151272
\(875\) 0 0
\(876\) −10.8541 −0.366726
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) −18.7295 −0.632090
\(879\) 15.7082 0.529825
\(880\) 26.8328 0.904534
\(881\) −25.5967 −0.862376 −0.431188 0.902262i \(-0.641905\pi\)
−0.431188 + 0.902262i \(0.641905\pi\)
\(882\) 0 0
\(883\) 9.88854 0.332776 0.166388 0.986060i \(-0.446790\pi\)
0.166388 + 0.986060i \(0.446790\pi\)
\(884\) 0 0
\(885\) 4.94427 0.166200
\(886\) −4.43769 −0.149087
\(887\) −16.8885 −0.567062 −0.283531 0.958963i \(-0.591506\pi\)
−0.283531 + 0.958963i \(0.591506\pi\)
\(888\) 12.7639 0.428330
\(889\) 0 0
\(890\) −17.8885 −0.599625
\(891\) 4.47214 0.149822
\(892\) 5.52786 0.185087
\(893\) 25.1246 0.840763
\(894\) 9.41641 0.314932
\(895\) −60.5410 −2.02366
\(896\) 0 0
\(897\) 0.236068 0.00788208
\(898\) 20.2918 0.677146
\(899\) −13.2492 −0.441886
\(900\) 17.7082 0.590273
\(901\) 0 0
\(902\) 6.18034 0.205783
\(903\) 0 0
\(904\) −5.52786 −0.183854
\(905\) 16.3607 0.543847
\(906\) −9.38197 −0.311695
\(907\) 12.5410 0.416418 0.208209 0.978084i \(-0.433237\pi\)
0.208209 + 0.978084i \(0.433237\pi\)
\(908\) −9.52786 −0.316193
\(909\) −26.8328 −0.889988
\(910\) 0 0
\(911\) −21.7771 −0.721507 −0.360754 0.932661i \(-0.617480\pi\)
−0.360754 + 0.932661i \(0.617480\pi\)
\(912\) 13.4164 0.444262
\(913\) −28.9443 −0.957916
\(914\) 8.94427 0.295850
\(915\) −43.4164 −1.43530
\(916\) 23.8885 0.789300
\(917\) 0 0
\(918\) 0 0
\(919\) 3.70820 0.122322 0.0611612 0.998128i \(-0.480520\pi\)
0.0611612 + 0.998128i \(0.480520\pi\)
\(920\) 7.23607 0.238566
\(921\) 11.4164 0.376183
\(922\) 15.9230 0.524396
\(923\) 2.41641 0.0795370
\(924\) 0 0
\(925\) −31.2361 −1.02704
\(926\) 0 0
\(927\) −31.4164 −1.03185
\(928\) −8.27051 −0.271493
\(929\) −51.6525 −1.69466 −0.847331 0.531065i \(-0.821792\pi\)
−0.847331 + 0.531065i \(0.821792\pi\)
\(930\) 18.0000 0.590243
\(931\) 0 0
\(932\) 18.5623 0.608029
\(933\) 2.88854 0.0945667
\(934\) 7.12461 0.233124
\(935\) 0 0
\(936\) −1.05573 −0.0345076
\(937\) 57.1246 1.86618 0.933090 0.359643i \(-0.117102\pi\)
0.933090 + 0.359643i \(0.117102\pi\)
\(938\) 0 0
\(939\) −2.76393 −0.0901975
\(940\) −18.1803 −0.592977
\(941\) −53.0132 −1.72818 −0.864090 0.503338i \(-0.832105\pi\)
−0.864090 + 0.503338i \(0.832105\pi\)
\(942\) −9.52786 −0.310435
\(943\) −2.23607 −0.0728164
\(944\) 2.83282 0.0922003
\(945\) 0 0
\(946\) 6.83282 0.222154
\(947\) −22.5967 −0.734296 −0.367148 0.930163i \(-0.619666\pi\)
−0.367148 + 0.930163i \(0.619666\pi\)
\(948\) 11.7082 0.380265
\(949\) −1.58359 −0.0514056
\(950\) 24.4721 0.793981
\(951\) 31.3050 1.01513
\(952\) 0 0
\(953\) 12.1115 0.392329 0.196164 0.980571i \(-0.437151\pi\)
0.196164 + 0.980571i \(0.437151\pi\)
\(954\) −13.8885 −0.449658
\(955\) −78.2492 −2.53209
\(956\) 25.5066 0.824942
\(957\) −6.58359 −0.212817
\(958\) 14.2918 0.461747
\(959\) 0 0
\(960\) −0.763932 −0.0246558
\(961\) 50.0000 1.61290
\(962\) 0.832816 0.0268511
\(963\) −22.4721 −0.724154
\(964\) −34.6525 −1.11608
\(965\) 14.2918 0.460069
\(966\) 0 0
\(967\) −31.0689 −0.999108 −0.499554 0.866283i \(-0.666503\pi\)
−0.499554 + 0.866283i \(0.666503\pi\)
\(968\) −20.1246 −0.646830
\(969\) 0 0
\(970\) −7.41641 −0.238127
\(971\) −33.7082 −1.08175 −0.540874 0.841104i \(-0.681906\pi\)
−0.540874 + 0.841104i \(0.681906\pi\)
\(972\) −25.8885 −0.830375
\(973\) 0 0
\(974\) 12.4377 0.398529
\(975\) −1.29180 −0.0413706
\(976\) −24.8754 −0.796242
\(977\) −18.6525 −0.596746 −0.298373 0.954449i \(-0.596444\pi\)
−0.298373 + 0.954449i \(0.596444\pi\)
\(978\) 11.8541 0.379052
\(979\) −40.0000 −1.27841
\(980\) 0 0
\(981\) 30.8328 0.984416
\(982\) 7.27051 0.232011
\(983\) 4.18034 0.133332 0.0666661 0.997775i \(-0.478764\pi\)
0.0666661 + 0.997775i \(0.478764\pi\)
\(984\) −5.00000 −0.159394
\(985\) −69.4853 −2.21399
\(986\) 0 0
\(987\) 0 0
\(988\) 2.76393 0.0879324
\(989\) −2.47214 −0.0786094
\(990\) −17.8885 −0.568535
\(991\) −48.9443 −1.55477 −0.777383 0.629028i \(-0.783453\pi\)
−0.777383 + 0.629028i \(0.783453\pi\)
\(992\) 50.5623 1.60535
\(993\) 0.708204 0.0224742
\(994\) 0 0
\(995\) −51.4164 −1.63001
\(996\) 10.4721 0.331822
\(997\) 27.3050 0.864756 0.432378 0.901692i \(-0.357675\pi\)
0.432378 + 0.901692i \(0.357675\pi\)
\(998\) −22.6869 −0.718142
\(999\) 28.5410 0.902998
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 1127.2.a.d.1.2 2
7.6 odd 2 161.2.a.b.1.2 2
21.20 even 2 1449.2.a.i.1.1 2
28.27 even 2 2576.2.a.s.1.1 2
35.34 odd 2 4025.2.a.i.1.1 2
161.160 even 2 3703.2.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.a.b.1.2 2 7.6 odd 2
1127.2.a.d.1.2 2 1.1 even 1 trivial
1449.2.a.i.1.1 2 21.20 even 2
2576.2.a.s.1.1 2 28.27 even 2
3703.2.a.b.1.2 2 161.160 even 2
4025.2.a.i.1.1 2 35.34 odd 2