Properties

Label 1840.2.i.c.1471.2
Level $1840$
Weight $2$
Character 1840.1471
Analytic conductor $14.692$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(1471,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.1471");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.i (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 8x^{14} + 44x^{12} + 162x^{10} + 404x^{8} - 84x^{6} - 79x^{4} - 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.2
Root \(0.605201 + 2.04824i\) of defining polynomial
Character \(\chi\) \(=\) 1840.1471
Dual form 1840.2.i.c.1471.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.94245i q^{3} +1.00000i q^{5} +1.89011 q^{7} -5.65803 q^{9} +O(q^{10})\) \(q-2.94245i q^{3} +1.00000i q^{5} +1.89011 q^{7} -5.65803 q^{9} +2.17399 q^{11} -0.0572532 q^{13} +2.94245 q^{15} -1.76546i q^{17} -1.57399 q^{19} -5.56155i q^{21} +(1.82081 - 4.43674i) q^{23} -1.00000 q^{25} +7.82113i q^{27} -4.26116 q^{29} -3.54245i q^{31} -6.39687i q^{33} +1.89011i q^{35} -11.3161i q^{37} +0.168465i q^{39} +2.02663 q^{41} +5.72549 q^{43} -5.65803i q^{45} +6.49395i q^{47} -3.42749 q^{49} -5.19479 q^{51} -11.7239i q^{53} +2.17399i q^{55} +4.63140i q^{57} -3.46410i q^{59} -7.00308i q^{61} -10.6943 q^{63} -0.0572532i q^{65} -7.96878 q^{67} +(-13.0549 - 5.35764i) q^{69} -9.53373i q^{71} +7.21651 q^{73} +2.94245i q^{75} +4.10908 q^{77} +2.24329 q^{79} +6.03923 q^{81} +0.177514 q^{83} +1.76546 q^{85} +12.5383i q^{87} +10.1930i q^{89} -0.108215 q^{91} -10.4235 q^{93} -1.57399i q^{95} +7.95842i q^{97} -12.3005 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{13} - 16 q^{25} - 84 q^{29} + 24 q^{41} + 36 q^{49} - 12 q^{69} + 68 q^{73} - 48 q^{77} + 32 q^{81} + 4 q^{85} - 52 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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 0
\(3\) 2.94245i 1.69883i −0.527729 0.849413i \(-0.676956\pi\)
0.527729 0.849413i \(-0.323044\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 1.89011 0.714393 0.357197 0.934029i \(-0.383733\pi\)
0.357197 + 0.934029i \(0.383733\pi\)
\(8\) 0 0
\(9\) −5.65803 −1.88601
\(10\) 0 0
\(11\) 2.17399 0.655483 0.327742 0.944767i \(-0.393712\pi\)
0.327742 + 0.944767i \(0.393712\pi\)
\(12\) 0 0
\(13\) −0.0572532 −0.0158792 −0.00793960 0.999968i \(-0.502527\pi\)
−0.00793960 + 0.999968i \(0.502527\pi\)
\(14\) 0 0
\(15\) 2.94245 0.759738
\(16\) 0 0
\(17\) 1.76546i 0.428188i −0.976813 0.214094i \(-0.931320\pi\)
0.976813 0.214094i \(-0.0686798\pi\)
\(18\) 0 0
\(19\) −1.57399 −0.361099 −0.180550 0.983566i \(-0.557788\pi\)
−0.180550 + 0.983566i \(0.557788\pi\)
\(20\) 0 0
\(21\) 5.56155i 1.21363i
\(22\) 0 0
\(23\) 1.82081 4.43674i 0.379665 0.925124i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 7.82113i 1.50518i
\(28\) 0 0
\(29\) −4.26116 −0.791278 −0.395639 0.918406i \(-0.629477\pi\)
−0.395639 + 0.918406i \(0.629477\pi\)
\(30\) 0 0
\(31\) 3.54245i 0.636243i −0.948050 0.318121i \(-0.896948\pi\)
0.948050 0.318121i \(-0.103052\pi\)
\(32\) 0 0
\(33\) 6.39687i 1.11355i
\(34\) 0 0
\(35\) 1.89011i 0.319486i
\(36\) 0 0
\(37\) 11.3161i 1.86035i −0.367117 0.930175i \(-0.619655\pi\)
0.367117 0.930175i \(-0.380345\pi\)
\(38\) 0 0
\(39\) 0.168465i 0.0269760i
\(40\) 0 0
\(41\) 2.02663 0.316506 0.158253 0.987399i \(-0.449414\pi\)
0.158253 + 0.987399i \(0.449414\pi\)
\(42\) 0 0
\(43\) 5.72549 0.873129 0.436565 0.899673i \(-0.356195\pi\)
0.436565 + 0.899673i \(0.356195\pi\)
\(44\) 0 0
\(45\) 5.65803i 0.843450i
\(46\) 0 0
\(47\) 6.49395i 0.947240i 0.880729 + 0.473620i \(0.157053\pi\)
−0.880729 + 0.473620i \(0.842947\pi\)
\(48\) 0 0
\(49\) −3.42749 −0.489642
\(50\) 0 0
\(51\) −5.19479 −0.727417
\(52\) 0 0
\(53\) 11.7239i 1.61040i −0.593004 0.805200i \(-0.702058\pi\)
0.593004 0.805200i \(-0.297942\pi\)
\(54\) 0 0
\(55\) 2.17399i 0.293141i
\(56\) 0 0
\(57\) 4.63140i 0.613445i
\(58\) 0 0
\(59\) 3.46410i 0.450988i −0.974245 0.225494i \(-0.927600\pi\)
0.974245 0.225494i \(-0.0723995\pi\)
\(60\) 0 0
\(61\) 7.00308i 0.896652i −0.893870 0.448326i \(-0.852020\pi\)
0.893870 0.448326i \(-0.147980\pi\)
\(62\) 0 0
\(63\) −10.6943 −1.34735
\(64\) 0 0
\(65\) 0.0572532i 0.00710139i
\(66\) 0 0
\(67\) −7.96878 −0.973542 −0.486771 0.873530i \(-0.661825\pi\)
−0.486771 + 0.873530i \(0.661825\pi\)
\(68\) 0 0
\(69\) −13.0549 5.35764i −1.57163 0.644984i
\(70\) 0 0
\(71\) 9.53373i 1.13145i −0.824596 0.565723i \(-0.808597\pi\)
0.824596 0.565723i \(-0.191403\pi\)
\(72\) 0 0
\(73\) 7.21651 0.844628 0.422314 0.906449i \(-0.361218\pi\)
0.422314 + 0.906449i \(0.361218\pi\)
\(74\) 0 0
\(75\) 2.94245i 0.339765i
\(76\) 0 0
\(77\) 4.10908 0.468273
\(78\) 0 0
\(79\) 2.24329 0.252390 0.126195 0.992005i \(-0.459724\pi\)
0.126195 + 0.992005i \(0.459724\pi\)
\(80\) 0 0
\(81\) 6.03923 0.671025
\(82\) 0 0
\(83\) 0.177514 0.0194847 0.00974236 0.999953i \(-0.496899\pi\)
0.00974236 + 0.999953i \(0.496899\pi\)
\(84\) 0 0
\(85\) 1.76546 0.191491
\(86\) 0 0
\(87\) 12.5383i 1.34424i
\(88\) 0 0
\(89\) 10.1930i 1.08045i 0.841520 + 0.540226i \(0.181661\pi\)
−0.841520 + 0.540226i \(0.818339\pi\)
\(90\) 0 0
\(91\) −0.108215 −0.0113440
\(92\) 0 0
\(93\) −10.4235 −1.08087
\(94\) 0 0
\(95\) 1.57399i 0.161488i
\(96\) 0 0
\(97\) 7.95842i 0.808055i 0.914747 + 0.404028i \(0.132390\pi\)
−0.914747 + 0.404028i \(0.867610\pi\)
\(98\) 0 0
\(99\) −12.3005 −1.23625
\(100\) 0 0
\(101\) 1.53093 0.152333 0.0761664 0.997095i \(-0.475732\pi\)
0.0761664 + 0.997095i \(0.475732\pi\)
\(102\) 0 0
\(103\) −5.49949 −0.541881 −0.270941 0.962596i \(-0.587335\pi\)
−0.270941 + 0.962596i \(0.587335\pi\)
\(104\) 0 0
\(105\) 5.56155 0.542752
\(106\) 0 0
\(107\) 15.0953 1.45932 0.729660 0.683811i \(-0.239679\pi\)
0.729660 + 0.683811i \(0.239679\pi\)
\(108\) 0 0
\(109\) 8.86594i 0.849203i 0.905380 + 0.424602i \(0.139586\pi\)
−0.905380 + 0.424602i \(0.860414\pi\)
\(110\) 0 0
\(111\) −33.2970 −3.16041
\(112\) 0 0
\(113\) 10.0785i 0.948101i 0.880498 + 0.474050i \(0.157209\pi\)
−0.880498 + 0.474050i \(0.842791\pi\)
\(114\) 0 0
\(115\) 4.43674 + 1.82081i 0.413728 + 0.169791i
\(116\) 0 0
\(117\) 0.323941 0.0299483
\(118\) 0 0
\(119\) 3.33692i 0.305895i
\(120\) 0 0
\(121\) −6.27376 −0.570342
\(122\) 0 0
\(123\) 5.96326i 0.537689i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 12.0836i 1.07224i 0.844141 + 0.536121i \(0.180111\pi\)
−0.844141 + 0.536121i \(0.819889\pi\)
\(128\) 0 0
\(129\) 16.8470i 1.48329i
\(130\) 0 0
\(131\) 10.2561i 0.896077i 0.894014 + 0.448038i \(0.147877\pi\)
−0.894014 + 0.448038i \(0.852123\pi\)
\(132\) 0 0
\(133\) −2.97502 −0.257967
\(134\) 0 0
\(135\) −7.82113 −0.673136
\(136\) 0 0
\(137\) 1.11215i 0.0950176i 0.998871 + 0.0475088i \(0.0151282\pi\)
−0.998871 + 0.0475088i \(0.984872\pi\)
\(138\) 0 0
\(139\) 18.2638i 1.54911i −0.632505 0.774556i \(-0.717973\pi\)
0.632505 0.774556i \(-0.282027\pi\)
\(140\) 0 0
\(141\) 19.1082 1.60920
\(142\) 0 0
\(143\) −0.124468 −0.0104085
\(144\) 0 0
\(145\) 4.26116i 0.353870i
\(146\) 0 0
\(147\) 10.0852i 0.831817i
\(148\) 0 0
\(149\) 2.84074i 0.232723i 0.993207 + 0.116361i \(0.0371231\pi\)
−0.993207 + 0.116361i \(0.962877\pi\)
\(150\) 0 0
\(151\) 2.60552i 0.212034i −0.994364 0.106017i \(-0.966190\pi\)
0.994364 0.106017i \(-0.0338099\pi\)
\(152\) 0 0
\(153\) 9.98905i 0.807567i
\(154\) 0 0
\(155\) 3.54245 0.284536
\(156\) 0 0
\(157\) 8.07845i 0.644731i −0.946615 0.322365i \(-0.895522\pi\)
0.946615 0.322365i \(-0.104478\pi\)
\(158\) 0 0
\(159\) −34.4970 −2.73579
\(160\) 0 0
\(161\) 3.44152 8.38591i 0.271230 0.660903i
\(162\) 0 0
\(163\) 0.805532i 0.0630941i −0.999502 0.0315471i \(-0.989957\pi\)
0.999502 0.0315471i \(-0.0100434\pi\)
\(164\) 0 0
\(165\) 6.39687 0.497995
\(166\) 0 0
\(167\) 18.7694i 1.45242i 0.687471 + 0.726211i \(0.258721\pi\)
−0.687471 + 0.726211i \(0.741279\pi\)
\(168\) 0 0
\(169\) −12.9967 −0.999748
\(170\) 0 0
\(171\) 8.90571 0.681037
\(172\) 0 0
\(173\) −16.6125 −1.26302 −0.631511 0.775367i \(-0.717565\pi\)
−0.631511 + 0.775367i \(0.717565\pi\)
\(174\) 0 0
\(175\) −1.89011 −0.142879
\(176\) 0 0
\(177\) −10.1930 −0.766150
\(178\) 0 0
\(179\) 2.69564i 0.201482i 0.994913 + 0.100741i \(0.0321213\pi\)
−0.994913 + 0.100741i \(0.967879\pi\)
\(180\) 0 0
\(181\) 16.4282i 1.22110i −0.791978 0.610550i \(-0.790949\pi\)
0.791978 0.610550i \(-0.209051\pi\)
\(182\) 0 0
\(183\) −20.6062 −1.52326
\(184\) 0 0
\(185\) 11.3161 0.831973
\(186\) 0 0
\(187\) 3.83810i 0.280670i
\(188\) 0 0
\(189\) 14.7828i 1.07529i
\(190\) 0 0
\(191\) 0.863064 0.0624491 0.0312246 0.999512i \(-0.490059\pi\)
0.0312246 + 0.999512i \(0.490059\pi\)
\(192\) 0 0
\(193\) 1.05490 0.0759333 0.0379667 0.999279i \(-0.487912\pi\)
0.0379667 + 0.999279i \(0.487912\pi\)
\(194\) 0 0
\(195\) −0.168465 −0.0120640
\(196\) 0 0
\(197\) −3.86502 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(198\) 0 0
\(199\) 23.8947 1.69385 0.846924 0.531715i \(-0.178452\pi\)
0.846924 + 0.531715i \(0.178452\pi\)
\(200\) 0 0
\(201\) 23.4478i 1.65388i
\(202\) 0 0
\(203\) −8.05406 −0.565284
\(204\) 0 0
\(205\) 2.02663i 0.141546i
\(206\) 0 0
\(207\) −10.3022 + 25.1032i −0.716052 + 1.74479i
\(208\) 0 0
\(209\) −3.42185 −0.236694
\(210\) 0 0
\(211\) 2.75774i 0.189850i −0.995484 0.0949252i \(-0.969739\pi\)
0.995484 0.0949252i \(-0.0302612\pi\)
\(212\) 0 0
\(213\) −28.0525 −1.92213
\(214\) 0 0
\(215\) 5.72549i 0.390475i
\(216\) 0 0
\(217\) 6.69561i 0.454528i
\(218\) 0 0
\(219\) 21.2342i 1.43488i
\(220\) 0 0
\(221\) 0.101079i 0.00679928i
\(222\) 0 0
\(223\) 7.10572i 0.475834i 0.971285 + 0.237917i \(0.0764646\pi\)
−0.971285 + 0.237917i \(0.923535\pi\)
\(224\) 0 0
\(225\) 5.65803 0.377202
\(226\) 0 0
\(227\) 25.0094 1.65993 0.829965 0.557815i \(-0.188360\pi\)
0.829965 + 0.557815i \(0.188360\pi\)
\(228\) 0 0
\(229\) 21.3413i 1.41027i −0.709073 0.705135i \(-0.750886\pi\)
0.709073 0.705135i \(-0.249114\pi\)
\(230\) 0 0
\(231\) 12.0908i 0.795514i
\(232\) 0 0
\(233\) −0.615731 −0.0403379 −0.0201689 0.999797i \(-0.506420\pi\)
−0.0201689 + 0.999797i \(0.506420\pi\)
\(234\) 0 0
\(235\) −6.49395 −0.423619
\(236\) 0 0
\(237\) 6.60078i 0.428767i
\(238\) 0 0
\(239\) 2.67754i 0.173196i −0.996243 0.0865979i \(-0.972400\pi\)
0.996243 0.0865979i \(-0.0275995\pi\)
\(240\) 0 0
\(241\) 7.94675i 0.511895i 0.966691 + 0.255947i \(0.0823874\pi\)
−0.966691 + 0.255947i \(0.917613\pi\)
\(242\) 0 0
\(243\) 5.69326i 0.365223i
\(244\) 0 0
\(245\) 3.42749i 0.218975i
\(246\) 0 0
\(247\) 0.0901163 0.00573396
\(248\) 0 0
\(249\) 0.522327i 0.0331012i
\(250\) 0 0
\(251\) 11.9636 0.755135 0.377567 0.925982i \(-0.376761\pi\)
0.377567 + 0.925982i \(0.376761\pi\)
\(252\) 0 0
\(253\) 3.95842 9.64543i 0.248864 0.606403i
\(254\) 0 0
\(255\) 5.19479i 0.325311i
\(256\) 0 0
\(257\) 2.86994 0.179022 0.0895110 0.995986i \(-0.471470\pi\)
0.0895110 + 0.995986i \(0.471470\pi\)
\(258\) 0 0
\(259\) 21.3886i 1.32902i
\(260\) 0 0
\(261\) 24.1098 1.49236
\(262\) 0 0
\(263\) 28.5455 1.76019 0.880095 0.474797i \(-0.157479\pi\)
0.880095 + 0.474797i \(0.157479\pi\)
\(264\) 0 0
\(265\) 11.7239 0.720192
\(266\) 0 0
\(267\) 29.9923 1.83550
\(268\) 0 0
\(269\) 21.7450 1.32582 0.662908 0.748701i \(-0.269322\pi\)
0.662908 + 0.748701i \(0.269322\pi\)
\(270\) 0 0
\(271\) 21.7626i 1.32198i −0.750394 0.660991i \(-0.770136\pi\)
0.750394 0.660991i \(-0.229864\pi\)
\(272\) 0 0
\(273\) 0.318417i 0.0192715i
\(274\) 0 0
\(275\) −2.17399 −0.131097
\(276\) 0 0
\(277\) −12.3144 −0.739902 −0.369951 0.929051i \(-0.620625\pi\)
−0.369951 + 0.929051i \(0.620625\pi\)
\(278\) 0 0
\(279\) 20.0433i 1.19996i
\(280\) 0 0
\(281\) 17.6094i 1.05049i 0.850952 + 0.525244i \(0.176026\pi\)
−0.850952 + 0.525244i \(0.823974\pi\)
\(282\) 0 0
\(283\) 24.9913 1.48558 0.742789 0.669526i \(-0.233503\pi\)
0.742789 + 0.669526i \(0.233503\pi\)
\(284\) 0 0
\(285\) −4.63140 −0.274341
\(286\) 0 0
\(287\) 3.83054 0.226110
\(288\) 0 0
\(289\) 13.8831 0.816655
\(290\) 0 0
\(291\) 23.4173 1.37275
\(292\) 0 0
\(293\) 31.8697i 1.86185i 0.365210 + 0.930925i \(0.380997\pi\)
−0.365210 + 0.930925i \(0.619003\pi\)
\(294\) 0 0
\(295\) 3.46410 0.201688
\(296\) 0 0
\(297\) 17.0031i 0.986619i
\(298\) 0 0
\(299\) −0.104247 + 0.254018i −0.00602877 + 0.0146902i
\(300\) 0 0
\(301\) 10.8218 0.623758
\(302\) 0 0
\(303\) 4.50468i 0.258787i
\(304\) 0 0
\(305\) 7.00308 0.400995
\(306\) 0 0
\(307\) 15.7297i 0.897739i 0.893597 + 0.448870i \(0.148173\pi\)
−0.893597 + 0.448870i \(0.851827\pi\)
\(308\) 0 0
\(309\) 16.1820i 0.920562i
\(310\) 0 0
\(311\) 15.1574i 0.859498i −0.902948 0.429749i \(-0.858602\pi\)
0.902948 0.429749i \(-0.141398\pi\)
\(312\) 0 0
\(313\) 23.2519i 1.31427i −0.753772 0.657136i \(-0.771768\pi\)
0.753772 0.657136i \(-0.228232\pi\)
\(314\) 0 0
\(315\) 10.6943i 0.602555i
\(316\) 0 0
\(317\) −25.5984 −1.43775 −0.718875 0.695139i \(-0.755343\pi\)
−0.718875 + 0.695139i \(0.755343\pi\)
\(318\) 0 0
\(319\) −9.26373 −0.518669
\(320\) 0 0
\(321\) 44.4173i 2.47913i
\(322\) 0 0
\(323\) 2.77883i 0.154618i
\(324\) 0 0
\(325\) 0.0572532 0.00317584
\(326\) 0 0
\(327\) 26.0876 1.44265
\(328\) 0 0
\(329\) 12.2743i 0.676702i
\(330\) 0 0
\(331\) 3.32314i 0.182656i −0.995821 0.0913281i \(-0.970889\pi\)
0.995821 0.0913281i \(-0.0291112\pi\)
\(332\) 0 0
\(333\) 64.0266i 3.50864i
\(334\) 0 0
\(335\) 7.96878i 0.435381i
\(336\) 0 0
\(337\) 25.2267i 1.37418i 0.726571 + 0.687092i \(0.241113\pi\)
−0.726571 + 0.687092i \(0.758887\pi\)
\(338\) 0 0
\(339\) 29.6554 1.61066
\(340\) 0 0
\(341\) 7.70126i 0.417046i
\(342\) 0 0
\(343\) −19.7091 −1.06419
\(344\) 0 0
\(345\) 5.35764 13.0549i 0.288446 0.702852i
\(346\) 0 0
\(347\) 35.1015i 1.88435i −0.335122 0.942175i \(-0.608778\pi\)
0.335122 0.942175i \(-0.391222\pi\)
\(348\) 0 0
\(349\) −3.95985 −0.211966 −0.105983 0.994368i \(-0.533799\pi\)
−0.105983 + 0.994368i \(0.533799\pi\)
\(350\) 0 0
\(351\) 0.447785i 0.0239010i
\(352\) 0 0
\(353\) 19.7702 1.05226 0.526130 0.850404i \(-0.323642\pi\)
0.526130 + 0.850404i \(0.323642\pi\)
\(354\) 0 0
\(355\) 9.53373 0.505998
\(356\) 0 0
\(357\) −9.81872 −0.519662
\(358\) 0 0
\(359\) 21.1902 1.11838 0.559189 0.829040i \(-0.311113\pi\)
0.559189 + 0.829040i \(0.311113\pi\)
\(360\) 0 0
\(361\) −16.5225 −0.869607
\(362\) 0 0
\(363\) 18.4603i 0.968912i
\(364\) 0 0
\(365\) 7.21651i 0.377729i
\(366\) 0 0
\(367\) −6.94630 −0.362594 −0.181297 0.983428i \(-0.558030\pi\)
−0.181297 + 0.983428i \(0.558030\pi\)
\(368\) 0 0
\(369\) −11.4667 −0.596934
\(370\) 0 0
\(371\) 22.1594i 1.15046i
\(372\) 0 0
\(373\) 14.3466i 0.742837i −0.928466 0.371419i \(-0.878871\pi\)
0.928466 0.371419i \(-0.121129\pi\)
\(374\) 0 0
\(375\) −2.94245 −0.151948
\(376\) 0 0
\(377\) 0.243965 0.0125649
\(378\) 0 0
\(379\) 10.4125 0.534856 0.267428 0.963578i \(-0.413826\pi\)
0.267428 + 0.963578i \(0.413826\pi\)
\(380\) 0 0
\(381\) 35.5553 1.82155
\(382\) 0 0
\(383\) 33.9853 1.73657 0.868283 0.496070i \(-0.165224\pi\)
0.868283 + 0.496070i \(0.165224\pi\)
\(384\) 0 0
\(385\) 4.10908i 0.209418i
\(386\) 0 0
\(387\) −32.3950 −1.64673
\(388\) 0 0
\(389\) 11.4422i 0.580145i 0.957005 + 0.290072i \(0.0936794\pi\)
−0.957005 + 0.290072i \(0.906321\pi\)
\(390\) 0 0
\(391\) −7.83290 3.21457i −0.396127 0.162568i
\(392\) 0 0
\(393\) 30.1780 1.52228
\(394\) 0 0
\(395\) 2.24329i 0.112872i
\(396\) 0 0
\(397\) 25.4824 1.27892 0.639462 0.768822i \(-0.279157\pi\)
0.639462 + 0.768822i \(0.279157\pi\)
\(398\) 0 0
\(399\) 8.75385i 0.438241i
\(400\) 0 0
\(401\) 26.5789i 1.32729i −0.748050 0.663643i \(-0.769010\pi\)
0.748050 0.663643i \(-0.230990\pi\)
\(402\) 0 0
\(403\) 0.202817i 0.0101030i
\(404\) 0 0
\(405\) 6.03923i 0.300092i
\(406\) 0 0
\(407\) 24.6010i 1.21943i
\(408\) 0 0
\(409\) −31.8596 −1.57536 −0.787678 0.616088i \(-0.788717\pi\)
−0.787678 + 0.616088i \(0.788717\pi\)
\(410\) 0 0
\(411\) 3.27246 0.161418
\(412\) 0 0
\(413\) 6.54752i 0.322183i
\(414\) 0 0
\(415\) 0.177514i 0.00871383i
\(416\) 0 0
\(417\) −53.7403 −2.63167
\(418\) 0 0
\(419\) 1.34131 0.0655273 0.0327637 0.999463i \(-0.489569\pi\)
0.0327637 + 0.999463i \(0.489569\pi\)
\(420\) 0 0
\(421\) 4.16481i 0.202980i −0.994837 0.101490i \(-0.967639\pi\)
0.994837 0.101490i \(-0.0323610\pi\)
\(422\) 0 0
\(423\) 36.7430i 1.78651i
\(424\) 0 0
\(425\) 1.76546i 0.0856375i
\(426\) 0 0
\(427\) 13.2366i 0.640562i
\(428\) 0 0
\(429\) 0.366241i 0.0176823i
\(430\) 0 0
\(431\) 11.4329 0.550703 0.275351 0.961344i \(-0.411206\pi\)
0.275351 + 0.961344i \(0.411206\pi\)
\(432\) 0 0
\(433\) 24.0755i 1.15699i −0.815684 0.578497i \(-0.803639\pi\)
0.815684 0.578497i \(-0.196361\pi\)
\(434\) 0 0
\(435\) −12.5383 −0.601164
\(436\) 0 0
\(437\) −2.86594 + 6.98340i −0.137097 + 0.334061i
\(438\) 0 0
\(439\) 39.1931i 1.87058i 0.353878 + 0.935291i \(0.384863\pi\)
−0.353878 + 0.935291i \(0.615137\pi\)
\(440\) 0 0
\(441\) 19.3929 0.923470
\(442\) 0 0
\(443\) 7.46126i 0.354495i 0.984166 + 0.177248i \(0.0567194\pi\)
−0.984166 + 0.177248i \(0.943281\pi\)
\(444\) 0 0
\(445\) −10.1930 −0.483193
\(446\) 0 0
\(447\) 8.35876 0.395356
\(448\) 0 0
\(449\) 11.1122 0.524415 0.262207 0.965012i \(-0.415550\pi\)
0.262207 + 0.965012i \(0.415550\pi\)
\(450\) 0 0
\(451\) 4.40587 0.207464
\(452\) 0 0
\(453\) −7.66663 −0.360210
\(454\) 0 0
\(455\) 0.108215i 0.00507319i
\(456\) 0 0
\(457\) 23.2157i 1.08598i 0.839738 + 0.542992i \(0.182709\pi\)
−0.839738 + 0.542992i \(0.817291\pi\)
\(458\) 0 0
\(459\) 13.8079 0.644499
\(460\) 0 0
\(461\) 6.68313 0.311265 0.155632 0.987815i \(-0.450259\pi\)
0.155632 + 0.987815i \(0.450259\pi\)
\(462\) 0 0
\(463\) 19.5078i 0.906602i 0.891357 + 0.453301i \(0.149754\pi\)
−0.891357 + 0.453301i \(0.850246\pi\)
\(464\) 0 0
\(465\) 10.4235i 0.483378i
\(466\) 0 0
\(467\) −24.4054 −1.12935 −0.564674 0.825314i \(-0.690998\pi\)
−0.564674 + 0.825314i \(0.690998\pi\)
\(468\) 0 0
\(469\) −15.0619 −0.695492
\(470\) 0 0
\(471\) −23.7705 −1.09529
\(472\) 0 0
\(473\) 12.4472 0.572321
\(474\) 0 0
\(475\) 1.57399 0.0722198
\(476\) 0 0
\(477\) 66.3341i 3.03723i
\(478\) 0 0
\(479\) −5.95905 −0.272276 −0.136138 0.990690i \(-0.543469\pi\)
−0.136138 + 0.990690i \(0.543469\pi\)
\(480\) 0 0
\(481\) 0.647881i 0.0295408i
\(482\) 0 0
\(483\) −24.6752 10.1265i −1.12276 0.460773i
\(484\) 0 0
\(485\) −7.95842 −0.361373
\(486\) 0 0
\(487\) 6.21675i 0.281708i 0.990030 + 0.140854i \(0.0449848\pi\)
−0.990030 + 0.140854i \(0.955015\pi\)
\(488\) 0 0
\(489\) −2.37024 −0.107186
\(490\) 0 0
\(491\) 13.5325i 0.610713i −0.952238 0.305356i \(-0.901224\pi\)
0.952238 0.305356i \(-0.0987756\pi\)
\(492\) 0 0
\(493\) 7.52293i 0.338816i
\(494\) 0 0
\(495\) 12.3005i 0.552867i
\(496\) 0 0
\(497\) 18.0198i 0.808297i
\(498\) 0 0
\(499\) 31.2155i 1.39740i 0.715416 + 0.698698i \(0.246237\pi\)
−0.715416 + 0.698698i \(0.753763\pi\)
\(500\) 0 0
\(501\) 55.2282 2.46741
\(502\) 0 0
\(503\) −16.6976 −0.744508 −0.372254 0.928131i \(-0.621415\pi\)
−0.372254 + 0.928131i \(0.621415\pi\)
\(504\) 0 0
\(505\) 1.53093i 0.0681253i
\(506\) 0 0
\(507\) 38.2422i 1.69840i
\(508\) 0 0
\(509\) 9.16941 0.406427 0.203213 0.979134i \(-0.434861\pi\)
0.203213 + 0.979134i \(0.434861\pi\)
\(510\) 0 0
\(511\) 13.6400 0.603397
\(512\) 0 0
\(513\) 12.3104i 0.543518i
\(514\) 0 0
\(515\) 5.49949i 0.242337i
\(516\) 0 0
\(517\) 14.1178i 0.620900i
\(518\) 0 0
\(519\) 48.8814i 2.14565i
\(520\) 0 0
\(521\) 44.9177i 1.96788i 0.178501 + 0.983940i \(0.442875\pi\)
−0.178501 + 0.983940i \(0.557125\pi\)
\(522\) 0 0
\(523\) −0.824149 −0.0360375 −0.0180188 0.999838i \(-0.505736\pi\)
−0.0180188 + 0.999838i \(0.505736\pi\)
\(524\) 0 0
\(525\) 5.56155i 0.242726i
\(526\) 0 0
\(527\) −6.25407 −0.272431
\(528\) 0 0
\(529\) −16.3693 16.1569i −0.711709 0.702474i
\(530\) 0 0
\(531\) 19.6000i 0.850567i
\(532\) 0 0
\(533\) −0.116031 −0.00502586
\(534\) 0 0
\(535\) 15.0953i 0.652627i
\(536\) 0 0
\(537\) 7.93179 0.342282
\(538\) 0 0
\(539\) −7.45134 −0.320952
\(540\) 0 0
\(541\) 24.8572 1.06870 0.534348 0.845265i \(-0.320557\pi\)
0.534348 + 0.845265i \(0.320557\pi\)
\(542\) 0 0
\(543\) −48.3393 −2.07444
\(544\) 0 0
\(545\) −8.86594 −0.379775
\(546\) 0 0
\(547\) 29.8515i 1.27636i −0.769888 0.638179i \(-0.779688\pi\)
0.769888 0.638179i \(-0.220312\pi\)
\(548\) 0 0
\(549\) 39.6236i 1.69109i
\(550\) 0 0
\(551\) 6.70705 0.285730
\(552\) 0 0
\(553\) 4.24006 0.180306
\(554\) 0 0
\(555\) 33.2970i 1.41338i
\(556\) 0 0
\(557\) 33.4367i 1.41676i 0.705832 + 0.708380i \(0.250574\pi\)
−0.705832 + 0.708380i \(0.749426\pi\)
\(558\) 0 0
\(559\) −0.327803 −0.0138646
\(560\) 0 0
\(561\) −11.2934 −0.476809
\(562\) 0 0
\(563\) 20.7675 0.875245 0.437623 0.899159i \(-0.355821\pi\)
0.437623 + 0.899159i \(0.355821\pi\)
\(564\) 0 0
\(565\) −10.0785 −0.424004
\(566\) 0 0
\(567\) 11.4148 0.479376
\(568\) 0 0
\(569\) 31.2880i 1.31166i 0.754908 + 0.655831i \(0.227681\pi\)
−0.754908 + 0.655831i \(0.772319\pi\)
\(570\) 0 0
\(571\) −38.8456 −1.62564 −0.812818 0.582518i \(-0.802068\pi\)
−0.812818 + 0.582518i \(0.802068\pi\)
\(572\) 0 0
\(573\) 2.53953i 0.106090i
\(574\) 0 0
\(575\) −1.82081 + 4.43674i −0.0759329 + 0.185025i
\(576\) 0 0
\(577\) −40.6172 −1.69092 −0.845458 0.534043i \(-0.820672\pi\)
−0.845458 + 0.534043i \(0.820672\pi\)
\(578\) 0 0
\(579\) 3.10399i 0.128997i
\(580\) 0 0
\(581\) 0.335521 0.0139198
\(582\) 0 0
\(583\) 25.4876i 1.05559i
\(584\) 0 0
\(585\) 0.323941i 0.0133933i
\(586\) 0 0
\(587\) 10.6368i 0.439026i 0.975610 + 0.219513i \(0.0704468\pi\)
−0.975610 + 0.219513i \(0.929553\pi\)
\(588\) 0 0
\(589\) 5.57580i 0.229747i
\(590\) 0 0
\(591\) 11.3726i 0.467808i
\(592\) 0 0
\(593\) 22.3640 0.918380 0.459190 0.888338i \(-0.348140\pi\)
0.459190 + 0.888338i \(0.348140\pi\)
\(594\) 0 0
\(595\) 3.33692 0.136800
\(596\) 0 0
\(597\) 70.3089i 2.87755i
\(598\) 0 0
\(599\) 7.06196i 0.288544i 0.989538 + 0.144272i \(0.0460840\pi\)
−0.989538 + 0.144272i \(0.953916\pi\)
\(600\) 0 0
\(601\) 18.7506 0.764854 0.382427 0.923986i \(-0.375088\pi\)
0.382427 + 0.923986i \(0.375088\pi\)
\(602\) 0 0
\(603\) 45.0876 1.83611
\(604\) 0 0
\(605\) 6.27376i 0.255065i
\(606\) 0 0
\(607\) 35.7967i 1.45294i −0.687197 0.726471i \(-0.741159\pi\)
0.687197 0.726471i \(-0.258841\pi\)
\(608\) 0 0
\(609\) 23.6987i 0.960319i
\(610\) 0 0
\(611\) 0.371800i 0.0150414i
\(612\) 0 0
\(613\) 21.1735i 0.855190i 0.903971 + 0.427595i \(0.140639\pi\)
−0.903971 + 0.427595i \(0.859361\pi\)
\(614\) 0 0
\(615\) 5.96326 0.240462
\(616\) 0 0
\(617\) 5.76546i 0.232109i 0.993243 + 0.116054i \(0.0370247\pi\)
−0.993243 + 0.116054i \(0.962975\pi\)
\(618\) 0 0
\(619\) 22.9390 0.921998 0.460999 0.887401i \(-0.347491\pi\)
0.460999 + 0.887401i \(0.347491\pi\)
\(620\) 0 0
\(621\) 34.7003 + 14.2408i 1.39248 + 0.571463i
\(622\) 0 0
\(623\) 19.2658i 0.771867i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 10.0686i 0.402102i
\(628\) 0 0
\(629\) −19.9781 −0.796579
\(630\) 0 0
\(631\) −6.13384 −0.244184 −0.122092 0.992519i \(-0.538960\pi\)
−0.122092 + 0.992519i \(0.538960\pi\)
\(632\) 0 0
\(633\) −8.11451 −0.322523
\(634\) 0 0
\(635\) −12.0836 −0.479522
\(636\) 0 0
\(637\) 0.196235 0.00777512
\(638\) 0 0
\(639\) 53.9421i 2.13392i
\(640\) 0 0
\(641\) 40.9316i 1.61670i 0.588702 + 0.808350i \(0.299639\pi\)
−0.588702 + 0.808350i \(0.700361\pi\)
\(642\) 0 0
\(643\) 7.24432 0.285688 0.142844 0.989745i \(-0.454375\pi\)
0.142844 + 0.989745i \(0.454375\pi\)
\(644\) 0 0
\(645\) 16.8470 0.663350
\(646\) 0 0
\(647\) 36.5225i 1.43585i −0.696123 0.717923i \(-0.745093\pi\)
0.696123 0.717923i \(-0.254907\pi\)
\(648\) 0 0
\(649\) 7.53093i 0.295615i
\(650\) 0 0
\(651\) −19.7015 −0.772164
\(652\) 0 0
\(653\) 22.1890 0.868321 0.434161 0.900836i \(-0.357045\pi\)
0.434161 + 0.900836i \(0.357045\pi\)
\(654\) 0 0
\(655\) −10.2561 −0.400738
\(656\) 0 0
\(657\) −40.8312 −1.59298
\(658\) 0 0
\(659\) −9.47744 −0.369189 −0.184594 0.982815i \(-0.559097\pi\)
−0.184594 + 0.982815i \(0.559097\pi\)
\(660\) 0 0
\(661\) 0.151378i 0.00588791i −0.999996 0.00294396i \(-0.999063\pi\)
0.999996 0.00294396i \(-0.000937092\pi\)
\(662\) 0 0
\(663\) 0.297419 0.0115508
\(664\) 0 0
\(665\) 2.97502i 0.115366i
\(666\) 0 0
\(667\) −7.75876 + 18.9057i −0.300420 + 0.732031i
\(668\) 0 0
\(669\) 20.9082 0.808359
\(670\) 0 0
\(671\) 15.2246i 0.587740i
\(672\) 0 0
\(673\) 42.6643 1.64459 0.822293 0.569064i \(-0.192694\pi\)
0.822293 + 0.569064i \(0.192694\pi\)
\(674\) 0 0
\(675\) 7.82113i 0.301036i
\(676\) 0 0
\(677\) 17.9482i 0.689805i 0.938639 + 0.344902i \(0.112088\pi\)
−0.938639 + 0.344902i \(0.887912\pi\)
\(678\) 0 0
\(679\) 15.0423i 0.577269i
\(680\) 0 0
\(681\) 73.5889i 2.81993i
\(682\) 0 0
\(683\) 47.4418i 1.81531i 0.419719 + 0.907654i \(0.362129\pi\)
−0.419719 + 0.907654i \(0.637871\pi\)
\(684\) 0 0
\(685\) −1.11215 −0.0424932
\(686\) 0 0
\(687\) −62.7957 −2.39580
\(688\) 0 0
\(689\) 0.671230i 0.0255718i
\(690\) 0 0
\(691\) 20.9849i 0.798305i −0.916885 0.399153i \(-0.869304\pi\)
0.916885 0.399153i \(-0.130696\pi\)
\(692\) 0 0
\(693\) −23.2493 −0.883167
\(694\) 0 0
\(695\) 18.2638 0.692784
\(696\) 0 0
\(697\) 3.57794i 0.135524i
\(698\) 0 0
\(699\) 1.81176i 0.0685270i
\(700\) 0 0
\(701\) 31.1545i 1.17669i 0.808610 + 0.588346i \(0.200220\pi\)
−0.808610 + 0.588346i \(0.799780\pi\)
\(702\) 0 0
\(703\) 17.8114i 0.671770i
\(704\) 0 0
\(705\) 19.1082i 0.719655i
\(706\) 0 0
\(707\) 2.89362 0.108826
\(708\) 0 0
\(709\) 17.2194i 0.646687i −0.946282 0.323344i \(-0.895193\pi\)
0.946282 0.323344i \(-0.104807\pi\)
\(710\) 0 0
\(711\) −12.6926 −0.476010
\(712\) 0 0
\(713\) −15.7169 6.45012i −0.588604 0.241559i
\(714\) 0 0
\(715\) 0.124468i 0.00465484i
\(716\) 0 0
\(717\) −7.87854 −0.294229
\(718\) 0 0
\(719\) 36.8093i 1.37276i −0.727245 0.686378i \(-0.759200\pi\)
0.727245 0.686378i \(-0.240800\pi\)
\(720\) 0 0
\(721\) −10.3946 −0.387116
\(722\) 0 0
\(723\) 23.3829 0.869620
\(724\) 0 0
\(725\) 4.26116 0.158256
\(726\) 0 0
\(727\) −48.1891 −1.78724 −0.893618 0.448828i \(-0.851841\pi\)
−0.893618 + 0.448828i \(0.851841\pi\)
\(728\) 0 0
\(729\) 34.8698 1.29148
\(730\) 0 0
\(731\) 10.1081i 0.373863i
\(732\) 0 0
\(733\) 6.95064i 0.256728i −0.991727 0.128364i \(-0.959027\pi\)
0.991727 0.128364i \(-0.0409725\pi\)
\(734\) 0 0
\(735\) −10.0852 −0.372000
\(736\) 0 0
\(737\) −17.3241 −0.638140
\(738\) 0 0
\(739\) 49.3175i 1.81417i 0.420945 + 0.907086i \(0.361698\pi\)
−0.420945 + 0.907086i \(0.638302\pi\)
\(740\) 0 0
\(741\) 0.265163i 0.00974100i
\(742\) 0 0
\(743\) −2.47597 −0.0908347 −0.0454173 0.998968i \(-0.514462\pi\)
−0.0454173 + 0.998968i \(0.514462\pi\)
\(744\) 0 0
\(745\) −2.84074 −0.104077
\(746\) 0 0
\(747\) −1.00438 −0.0367484
\(748\) 0 0
\(749\) 28.5318 1.04253
\(750\) 0 0
\(751\) −6.42441 −0.234430 −0.117215 0.993107i \(-0.537397\pi\)
−0.117215 + 0.993107i \(0.537397\pi\)
\(752\) 0 0
\(753\) 35.2023i 1.28284i
\(754\) 0 0
\(755\) 2.60552 0.0948247
\(756\) 0 0
\(757\) 28.0485i 1.01944i −0.860340 0.509721i \(-0.829749\pi\)
0.860340 0.509721i \(-0.170251\pi\)
\(758\) 0 0
\(759\) −28.3812 11.6475i −1.03017 0.422776i
\(760\) 0 0
\(761\) −27.8815 −1.01070 −0.505352 0.862913i \(-0.668637\pi\)
−0.505352 + 0.862913i \(0.668637\pi\)
\(762\) 0 0
\(763\) 16.7576i 0.606665i
\(764\) 0 0
\(765\) −9.98905 −0.361155
\(766\) 0 0
\(767\) 0.198331i 0.00716132i
\(768\) 0 0
\(769\) 34.8642i 1.25723i −0.777715 0.628617i \(-0.783621\pi\)
0.777715 0.628617i \(-0.216379\pi\)
\(770\) 0 0
\(771\) 8.44467i 0.304127i
\(772\) 0 0
\(773\) 1.25666i 0.0451988i −0.999745 0.0225994i \(-0.992806\pi\)
0.999745 0.0225994i \(-0.00719423\pi\)
\(774\) 0 0
\(775\) 3.54245i 0.127249i
\(776\) 0 0
\(777\) −62.9349 −2.25778
\(778\) 0 0
\(779\) −3.18990 −0.114290
\(780\) 0 0
\(781\) 20.7262i 0.741643i
\(782\) 0 0
\(783\) 33.3271i 1.19101i
\(784\) 0 0
\(785\) 8.07845 0.288332
\(786\) 0 0
\(787\) 33.2400 1.18488 0.592439 0.805616i \(-0.298165\pi\)
0.592439 + 0.805616i \(0.298165\pi\)
\(788\) 0 0
\(789\) 83.9938i 2.99026i
\(790\) 0 0
\(791\) 19.0494i 0.677317i
\(792\) 0 0
\(793\) 0.400949i 0.0142381i
\(794\) 0 0
\(795\) 34.4970i 1.22348i
\(796\) 0 0
\(797\) 0.675939i 0.0239430i −0.999928 0.0119715i \(-0.996189\pi\)
0.999928 0.0119715i \(-0.00381073\pi\)
\(798\) 0 0
\(799\) 11.4648 0.405597
\(800\) 0 0
\(801\) 57.6721i 2.03774i
\(802\) 0 0
\(803\) 15.6886 0.553640
\(804\) 0 0
\(805\) 8.38591 + 3.44152i 0.295565 + 0.121298i
\(806\) 0 0
\(807\) 63.9836i 2.25233i
\(808\) 0 0
\(809\) −3.86911 −0.136031 −0.0680153 0.997684i \(-0.521667\pi\)
−0.0680153 + 0.997684i \(0.521667\pi\)
\(810\) 0 0
\(811\) 1.51005i 0.0530252i −0.999648 0.0265126i \(-0.991560\pi\)
0.999648 0.0265126i \(-0.00844020\pi\)
\(812\) 0 0
\(813\) −64.0354 −2.24582
\(814\) 0 0
\(815\) 0.805532 0.0282166
\(816\) 0 0
\(817\) −9.01189 −0.315286
\(818\) 0 0
\(819\) 0.612283 0.0213949
\(820\) 0 0
\(821\) −25.7380 −0.898264 −0.449132 0.893466i \(-0.648267\pi\)
−0.449132 + 0.893466i \(0.648267\pi\)
\(822\) 0 0
\(823\) 9.67020i 0.337082i 0.985695 + 0.168541i \(0.0539056\pi\)
−0.985695 + 0.168541i \(0.946094\pi\)
\(824\) 0 0
\(825\) 6.39687i 0.222710i
\(826\) 0 0
\(827\) 34.2044 1.18940 0.594702 0.803946i \(-0.297270\pi\)
0.594702 + 0.803946i \(0.297270\pi\)
\(828\) 0 0
\(829\) −21.9058 −0.760820 −0.380410 0.924818i \(-0.624217\pi\)
−0.380410 + 0.924818i \(0.624217\pi\)
\(830\) 0 0
\(831\) 36.2346i 1.25696i
\(832\) 0 0
\(833\) 6.05111i 0.209659i
\(834\) 0 0
\(835\) −18.7694 −0.649543
\(836\) 0 0
\(837\) 27.7060 0.957659
\(838\) 0 0
\(839\) −8.46985 −0.292412 −0.146206 0.989254i \(-0.546706\pi\)
−0.146206 + 0.989254i \(0.546706\pi\)
\(840\) 0 0
\(841\) −10.8425 −0.373879
\(842\) 0 0
\(843\) 51.8148 1.78460
\(844\) 0 0
\(845\) 12.9967i 0.447101i
\(846\) 0 0
\(847\) −11.8581 −0.407449
\(848\) 0 0
\(849\) 73.5357i 2.52374i
\(850\) 0 0
\(851\) −50.2064 20.6044i −1.72105 0.706309i
\(852\) 0 0
\(853\) 12.2525 0.419516 0.209758 0.977753i \(-0.432732\pi\)
0.209758 + 0.977753i \(0.432732\pi\)
\(854\) 0 0
\(855\) 8.90571i 0.304569i
\(856\) 0 0
\(857\) 30.4746 1.04099 0.520496 0.853864i \(-0.325747\pi\)
0.520496 + 0.853864i \(0.325747\pi\)
\(858\) 0 0
\(859\) 40.3587i 1.37702i −0.725226 0.688511i \(-0.758265\pi\)
0.725226 0.688511i \(-0.241735\pi\)
\(860\) 0 0
\(861\) 11.2712i 0.384121i
\(862\) 0 0
\(863\) 21.1628i 0.720391i 0.932877 + 0.360195i \(0.117290\pi\)
−0.932877 + 0.360195i \(0.882710\pi\)
\(864\) 0 0
\(865\) 16.6125i 0.564840i
\(866\) 0 0
\(867\) 40.8505i 1.38736i
\(868\) 0 0
\(869\) 4.87689 0.165437
\(870\) 0 0
\(871\) 0.456239 0.0154591
\(872\) 0 0
\(873\) 45.0290i 1.52400i
\(874\) 0 0
\(875\) 1.89011i 0.0638973i
\(876\) 0 0
\(877\) −57.4236 −1.93906 −0.969529 0.244976i \(-0.921220\pi\)
−0.969529 + 0.244976i \(0.921220\pi\)
\(878\) 0 0
\(879\) 93.7752 3.16296
\(880\) 0 0
\(881\) 40.2714i 1.35678i 0.734703 + 0.678389i \(0.237322\pi\)
−0.734703 + 0.678389i \(0.762678\pi\)
\(882\) 0 0
\(883\) 24.6562i 0.829747i 0.909879 + 0.414873i \(0.136174\pi\)
−0.909879 + 0.414873i \(0.863826\pi\)
\(884\) 0 0
\(885\) 10.1930i 0.342633i
\(886\) 0 0
\(887\) 51.5560i 1.73108i −0.500838 0.865541i \(-0.666975\pi\)
0.500838 0.865541i \(-0.333025\pi\)
\(888\) 0 0
\(889\) 22.8392i 0.766003i
\(890\) 0 0
\(891\) 13.1292 0.439846
\(892\) 0 0
\(893\) 10.2214i 0.342048i
\(894\) 0 0
\(895\) −2.69564 −0.0901053
\(896\) 0 0
\(897\) 0.747435 + 0.306742i 0.0249561 + 0.0102418i
\(898\) 0 0
\(899\) 15.0950i 0.503445i
\(900\) 0 0
\(901\) −20.6981 −0.689553
\(902\) 0 0
\(903\) 31.8426i 1.05966i
\(904\) 0 0
\(905\) 16.4282 0.546092
\(906\) 0 0
\(907\) −20.9567 −0.695855 −0.347928 0.937521i \(-0.613115\pi\)
−0.347928 + 0.937521i \(0.613115\pi\)
\(908\) 0 0
\(909\) −8.66203 −0.287301
\(910\) 0 0
\(911\) −36.8181 −1.21984 −0.609919 0.792464i \(-0.708798\pi\)
−0.609919 + 0.792464i \(0.708798\pi\)
\(912\) 0 0
\(913\) 0.385914 0.0127719
\(914\) 0 0
\(915\) 20.6062i 0.681221i
\(916\) 0 0
\(917\) 19.3851i 0.640151i
\(918\) 0 0
\(919\) −15.2312 −0.502431 −0.251215 0.967931i \(-0.580830\pi\)
−0.251215 + 0.967931i \(0.580830\pi\)
\(920\) 0 0
\(921\) 46.2838 1.52510
\(922\) 0 0
\(923\) 0.545837i 0.0179664i
\(924\) 0 0
\(925\) 11.3161i 0.372070i
\(926\) 0 0
\(927\) 31.1163 1.02199
\(928\) 0 0
\(929\) 46.2878 1.51865 0.759327 0.650709i \(-0.225528\pi\)
0.759327 + 0.650709i \(0.225528\pi\)
\(930\) 0 0
\(931\) 5.39486 0.176809
\(932\) 0 0
\(933\) −44.6000 −1.46014
\(934\) 0 0
\(935\) 3.83810 0.125519
\(936\) 0 0
\(937\) 30.9505i 1.01111i −0.862794 0.505555i \(-0.831288\pi\)
0.862794 0.505555i \(-0.168712\pi\)
\(938\) 0 0
\(939\) −68.4175 −2.23272
\(940\) 0 0
\(941\) 40.4680i 1.31922i −0.751608 0.659610i \(-0.770721\pi\)
0.751608 0.659610i \(-0.229279\pi\)
\(942\) 0 0
\(943\) 3.69010 8.99162i 0.120166 0.292807i
\(944\) 0 0
\(945\) −14.7828 −0.480884
\(946\) 0 0
\(947\) 7.85793i 0.255348i 0.991816 + 0.127674i \(0.0407512\pi\)
−0.991816 + 0.127674i \(0.959249\pi\)
\(948\) 0 0
\(949\) −0.413169 −0.0134120
\(950\) 0 0
\(951\) 75.3222i 2.44249i
\(952\) 0 0
\(953\) 4.97973i 0.161309i −0.996742 0.0806546i \(-0.974299\pi\)
0.996742 0.0806546i \(-0.0257011\pi\)
\(954\) 0 0
\(955\) 0.863064i 0.0279281i
\(956\) 0 0
\(957\) 27.2581i 0.881129i
\(958\) 0 0
\(959\) 2.10209i 0.0678799i
\(960\) 0 0
\(961\) 18.4510 0.595195
\(962\) 0 0
\(963\) −85.4098 −2.75229
\(964\) 0 0
\(965\) 1.05490i 0.0339584i
\(966\) 0 0
\(967\) 36.4682i 1.17274i 0.810045 + 0.586368i \(0.199443\pi\)
−0.810045 + 0.586368i \(0.800557\pi\)
\(968\) 0 0
\(969\) 8.17657 0.262669
\(970\) 0 0
\(971\) 48.3022 1.55009 0.775045 0.631905i \(-0.217727\pi\)
0.775045 + 0.631905i \(0.217727\pi\)
\(972\) 0 0
\(973\) 34.5205i 1.10668i
\(974\) 0 0
\(975\) 0.168465i 0.00539520i
\(976\) 0 0
\(977\) 39.2738i 1.25648i 0.778020 + 0.628239i \(0.216224\pi\)
−0.778020 + 0.628239i \(0.783776\pi\)
\(978\) 0 0
\(979\) 22.1594i 0.708217i
\(980\) 0 0
\(981\) 50.1638i 1.60161i
\(982\) 0 0
\(983\) 24.9582 0.796042 0.398021 0.917376i \(-0.369697\pi\)
0.398021 + 0.917376i \(0.369697\pi\)
\(984\) 0 0
\(985\) 3.86502i 0.123150i
\(986\) 0 0
\(987\) 36.1165 1.14960
\(988\) 0 0
\(989\) 10.4250 25.4025i 0.331496 0.807753i
\(990\) 0 0
\(991\) 24.7770i 0.787066i −0.919310 0.393533i \(-0.871253\pi\)
0.919310 0.393533i \(-0.128747\pi\)
\(992\) 0 0
\(993\) −9.77818 −0.310301
\(994\) 0 0
\(995\) 23.8947i 0.757511i
\(996\) 0 0
\(997\) 46.0627 1.45882 0.729410 0.684077i \(-0.239795\pi\)
0.729410 + 0.684077i \(0.239795\pi\)
\(998\) 0 0
\(999\) 88.5044 2.80016
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 1840.2.i.c.1471.2 yes 16
4.3 odd 2 inner 1840.2.i.c.1471.16 yes 16
23.22 odd 2 inner 1840.2.i.c.1471.1 16
92.91 even 2 inner 1840.2.i.c.1471.15 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1840.2.i.c.1471.1 16 23.22 odd 2 inner
1840.2.i.c.1471.2 yes 16 1.1 even 1 trivial
1840.2.i.c.1471.15 yes 16 92.91 even 2 inner
1840.2.i.c.1471.16 yes 16 4.3 odd 2 inner