Properties

Label 1148.2.r.a.737.3
Level $1148$
Weight $2$
Character 1148.737
Analytic conductor $9.167$
Analytic rank $0$
Dimension $56$
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] = [1148,2,Mod(81,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.r (of order \(6\), degree \(2\), 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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 737.3
Character \(\chi\) \(=\) 1148.737
Dual form 1148.2.r.a.81.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.67190 + 1.54262i) q^{3} +(-1.10096 + 1.90691i) q^{5} +(2.60761 - 0.447649i) q^{7} +(3.25937 - 5.64540i) q^{9} +O(q^{10})\) \(q+(-2.67190 + 1.54262i) q^{3} +(-1.10096 + 1.90691i) q^{5} +(2.60761 - 0.447649i) q^{7} +(3.25937 - 5.64540i) q^{9} +(-3.16097 + 1.82499i) q^{11} -0.353091i q^{13} -6.79344i q^{15} +(-3.88342 + 2.24210i) q^{17} +(-5.86226 - 3.38458i) q^{19} +(-6.27671 + 5.21863i) q^{21} +(3.07187 - 5.32063i) q^{23} +(0.0757913 + 0.131274i) q^{25} +10.8562i q^{27} -0.443384i q^{29} +(-2.14434 - 3.71411i) q^{31} +(5.63054 - 9.75239i) q^{33} +(-2.01723 + 5.46532i) q^{35} +(0.730118 - 1.26460i) q^{37} +(0.544686 + 0.943424i) q^{39} +(-0.647886 + 6.37026i) q^{41} -6.86200 q^{43} +(7.17685 + 12.4307i) q^{45} +(-5.36060 - 3.09494i) q^{47} +(6.59922 - 2.33458i) q^{49} +(6.91742 - 11.9813i) q^{51} +(9.66478 - 5.57996i) q^{53} -8.03693i q^{55} +20.8845 q^{57} +(5.14222 + 8.90659i) q^{59} +(5.88811 - 10.1985i) q^{61} +(5.97201 - 16.1800i) q^{63} +(0.673313 + 0.388737i) q^{65} +(9.36850 - 5.40891i) q^{67} +18.9550i q^{69} -12.5157i q^{71} +(-1.23009 - 2.13058i) q^{73} +(-0.405014 - 0.233835i) q^{75} +(-7.42562 + 6.17386i) q^{77} +(-1.47630 - 0.852345i) q^{79} +(-6.96891 - 12.0705i) q^{81} -1.57776 q^{83} -9.87380i q^{85} +(0.683975 + 1.18468i) q^{87} +(15.1590 + 8.75205i) q^{89} +(-0.158061 - 0.920722i) q^{91} +(11.4589 + 6.61582i) q^{93} +(12.9082 - 7.45254i) q^{95} -15.0452i q^{97} +23.7933i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 4 q^{5} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q + 4 q^{5} + 32 q^{9} - 6 q^{21} + 2 q^{23} - 24 q^{25} - 4 q^{31} + 10 q^{33} + 10 q^{37} + 10 q^{39} + 20 q^{41} + 8 q^{43} - 22 q^{45} - 4 q^{49} + 18 q^{51} + 28 q^{57} - 16 q^{59} + 16 q^{61} + 2 q^{73} + 34 q^{77} - 28 q^{81} - 48 q^{83} - 2 q^{87} - 42 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\)

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
\(3\) −2.67190 + 1.54262i −1.54262 + 0.890634i −0.543951 + 0.839117i \(0.683072\pi\)
−0.998672 + 0.0515172i \(0.983594\pi\)
\(4\) 0 0
\(5\) −1.10096 + 1.90691i −0.492363 + 0.852797i −0.999961 0.00879665i \(-0.997200\pi\)
0.507599 + 0.861594i \(0.330533\pi\)
\(6\) 0 0
\(7\) 2.60761 0.447649i 0.985583 0.169195i
\(8\) 0 0
\(9\) 3.25937 5.64540i 1.08646 1.88180i
\(10\) 0 0
\(11\) −3.16097 + 1.82499i −0.953070 + 0.550255i −0.894033 0.448001i \(-0.852136\pi\)
−0.0590366 + 0.998256i \(0.518803\pi\)
\(12\) 0 0
\(13\) 0.353091i 0.0979297i −0.998801 0.0489649i \(-0.984408\pi\)
0.998801 0.0489649i \(-0.0155922\pi\)
\(14\) 0 0
\(15\) 6.79344i 1.75406i
\(16\) 0 0
\(17\) −3.88342 + 2.24210i −0.941869 + 0.543788i −0.890546 0.454894i \(-0.849677\pi\)
−0.0513232 + 0.998682i \(0.516344\pi\)
\(18\) 0 0
\(19\) −5.86226 3.38458i −1.34490 0.776476i −0.357374 0.933961i \(-0.616328\pi\)
−0.987521 + 0.157486i \(0.949661\pi\)
\(20\) 0 0
\(21\) −6.27671 + 5.21863i −1.36969 + 1.13880i
\(22\) 0 0
\(23\) 3.07187 5.32063i 0.640529 1.10943i −0.344786 0.938681i \(-0.612048\pi\)
0.985315 0.170748i \(-0.0546182\pi\)
\(24\) 0 0
\(25\) 0.0757913 + 0.131274i 0.0151583 + 0.0262549i
\(26\) 0 0
\(27\) 10.8562i 2.08928i
\(28\) 0 0
\(29\) 0.443384i 0.0823344i −0.999152 0.0411672i \(-0.986892\pi\)
0.999152 0.0411672i \(-0.0131076\pi\)
\(30\) 0 0
\(31\) −2.14434 3.71411i −0.385135 0.667073i 0.606653 0.794967i \(-0.292512\pi\)
−0.991788 + 0.127893i \(0.959178\pi\)
\(32\) 0 0
\(33\) 5.63054 9.75239i 0.980152 1.69767i
\(34\) 0 0
\(35\) −2.01723 + 5.46532i −0.340975 + 0.923807i
\(36\) 0 0
\(37\) 0.730118 1.26460i 0.120031 0.207899i −0.799749 0.600335i \(-0.795034\pi\)
0.919779 + 0.392436i \(0.128367\pi\)
\(38\) 0 0
\(39\) 0.544686 + 0.943424i 0.0872196 + 0.151069i
\(40\) 0 0
\(41\) −0.647886 + 6.37026i −0.101183 + 0.994868i
\(42\) 0 0
\(43\) −6.86200 −1.04645 −0.523223 0.852196i \(-0.675270\pi\)
−0.523223 + 0.852196i \(0.675270\pi\)
\(44\) 0 0
\(45\) 7.17685 + 12.4307i 1.06986 + 1.85306i
\(46\) 0 0
\(47\) −5.36060 3.09494i −0.781924 0.451444i 0.0551878 0.998476i \(-0.482424\pi\)
−0.837112 + 0.547032i \(0.815758\pi\)
\(48\) 0 0
\(49\) 6.59922 2.33458i 0.942746 0.333512i
\(50\) 0 0
\(51\) 6.91742 11.9813i 0.968633 1.67772i
\(52\) 0 0
\(53\) 9.66478 5.57996i 1.32756 0.766467i 0.342638 0.939468i \(-0.388680\pi\)
0.984922 + 0.173001i \(0.0553463\pi\)
\(54\) 0 0
\(55\) 8.03693i 1.08370i
\(56\) 0 0
\(57\) 20.8845 2.76622
\(58\) 0 0
\(59\) 5.14222 + 8.90659i 0.669460 + 1.15954i 0.978055 + 0.208346i \(0.0668080\pi\)
−0.308595 + 0.951194i \(0.599859\pi\)
\(60\) 0 0
\(61\) 5.88811 10.1985i 0.753896 1.30579i −0.192026 0.981390i \(-0.561506\pi\)
0.945921 0.324396i \(-0.105161\pi\)
\(62\) 0 0
\(63\) 5.97201 16.1800i 0.752402 2.03849i
\(64\) 0 0
\(65\) 0.673313 + 0.388737i 0.0835142 + 0.0482169i
\(66\) 0 0
\(67\) 9.36850 5.40891i 1.14454 0.660803i 0.196993 0.980405i \(-0.436882\pi\)
0.947552 + 0.319602i \(0.103549\pi\)
\(68\) 0 0
\(69\) 18.9550i 2.28191i
\(70\) 0 0
\(71\) 12.5157i 1.48534i −0.669660 0.742668i \(-0.733560\pi\)
0.669660 0.742668i \(-0.266440\pi\)
\(72\) 0 0
\(73\) −1.23009 2.13058i −0.143971 0.249366i 0.785017 0.619474i \(-0.212654\pi\)
−0.928989 + 0.370108i \(0.879321\pi\)
\(74\) 0 0
\(75\) −0.405014 0.233835i −0.0467670 0.0270009i
\(76\) 0 0
\(77\) −7.42562 + 6.17386i −0.846228 + 0.703577i
\(78\) 0 0
\(79\) −1.47630 0.852345i −0.166097 0.0958963i 0.414647 0.909982i \(-0.363905\pi\)
−0.580744 + 0.814086i \(0.697238\pi\)
\(80\) 0 0
\(81\) −6.96891 12.0705i −0.774324 1.34117i
\(82\) 0 0
\(83\) −1.57776 −0.173182 −0.0865908 0.996244i \(-0.527597\pi\)
−0.0865908 + 0.996244i \(0.527597\pi\)
\(84\) 0 0
\(85\) 9.87380i 1.07096i
\(86\) 0 0
\(87\) 0.683975 + 1.18468i 0.0733298 + 0.127011i
\(88\) 0 0
\(89\) 15.1590 + 8.75205i 1.60685 + 0.927716i 0.990069 + 0.140581i \(0.0448970\pi\)
0.616781 + 0.787135i \(0.288436\pi\)
\(90\) 0 0
\(91\) −0.158061 0.920722i −0.0165693 0.0965178i
\(92\) 0 0
\(93\) 11.4589 + 6.61582i 1.18824 + 0.686029i
\(94\) 0 0
\(95\) 12.9082 7.45254i 1.32435 0.764615i
\(96\) 0 0
\(97\) 15.0452i 1.52761i −0.645450 0.763803i \(-0.723330\pi\)
0.645450 0.763803i \(-0.276670\pi\)
\(98\) 0 0
\(99\) 23.7933i 2.39132i
\(100\) 0 0
\(101\) 2.65074 1.53041i 0.263759 0.152281i −0.362289 0.932066i \(-0.618005\pi\)
0.626048 + 0.779785i \(0.284671\pi\)
\(102\) 0 0
\(103\) 3.80895 6.59730i 0.375307 0.650051i −0.615066 0.788476i \(-0.710871\pi\)
0.990373 + 0.138425i \(0.0442039\pi\)
\(104\) 0 0
\(105\) −3.04108 17.7146i −0.296779 1.72877i
\(106\) 0 0
\(107\) −6.76934 + 11.7248i −0.654416 + 1.13348i 0.327623 + 0.944808i \(0.393752\pi\)
−0.982040 + 0.188674i \(0.939581\pi\)
\(108\) 0 0
\(109\) −3.94788 + 2.27931i −0.378139 + 0.218318i −0.677008 0.735976i \(-0.736724\pi\)
0.298870 + 0.954294i \(0.403390\pi\)
\(110\) 0 0
\(111\) 4.50519i 0.427613i
\(112\) 0 0
\(113\) 0.643235 0.0605104 0.0302552 0.999542i \(-0.490368\pi\)
0.0302552 + 0.999542i \(0.490368\pi\)
\(114\) 0 0
\(115\) 6.76399 + 11.7156i 0.630745 + 1.09248i
\(116\) 0 0
\(117\) −1.99334 1.15085i −0.184284 0.106397i
\(118\) 0 0
\(119\) −9.12277 + 7.58492i −0.836283 + 0.695308i
\(120\) 0 0
\(121\) 1.16118 2.01121i 0.105561 0.182838i
\(122\) 0 0
\(123\) −8.09583 18.0202i −0.729976 1.62482i
\(124\) 0 0
\(125\) −11.3433 −1.01458
\(126\) 0 0
\(127\) −11.6422 −1.03308 −0.516539 0.856264i \(-0.672780\pi\)
−0.516539 + 0.856264i \(0.672780\pi\)
\(128\) 0 0
\(129\) 18.3346 10.5855i 1.61427 0.932000i
\(130\) 0 0
\(131\) −1.73254 + 3.00085i −0.151373 + 0.262185i −0.931732 0.363146i \(-0.881703\pi\)
0.780360 + 0.625331i \(0.215036\pi\)
\(132\) 0 0
\(133\) −16.8016 6.20141i −1.45688 0.537731i
\(134\) 0 0
\(135\) −20.7018 11.9522i −1.78173 1.02868i
\(136\) 0 0
\(137\) −9.27868 + 5.35705i −0.792731 + 0.457683i −0.840923 0.541155i \(-0.817987\pi\)
0.0481921 + 0.998838i \(0.484654\pi\)
\(138\) 0 0
\(139\) −11.7524 −0.996824 −0.498412 0.866940i \(-0.666083\pi\)
−0.498412 + 0.866940i \(0.666083\pi\)
\(140\) 0 0
\(141\) 19.0973 1.60829
\(142\) 0 0
\(143\) 0.644387 + 1.11611i 0.0538863 + 0.0933339i
\(144\) 0 0
\(145\) 0.845495 + 0.488147i 0.0702145 + 0.0405384i
\(146\) 0 0
\(147\) −14.0311 + 16.4179i −1.15726 + 1.35413i
\(148\) 0 0
\(149\) −16.0374 9.25919i −1.31383 0.758542i −0.331105 0.943594i \(-0.607421\pi\)
−0.982729 + 0.185052i \(0.940755\pi\)
\(150\) 0 0
\(151\) −6.70226 + 3.86955i −0.545422 + 0.314900i −0.747274 0.664517i \(-0.768638\pi\)
0.201851 + 0.979416i \(0.435304\pi\)
\(152\) 0 0
\(153\) 29.2313i 2.36321i
\(154\) 0 0
\(155\) 9.44330 0.758504
\(156\) 0 0
\(157\) 9.88381 5.70642i 0.788814 0.455422i −0.0507309 0.998712i \(-0.516155\pi\)
0.839545 + 0.543290i \(0.182822\pi\)
\(158\) 0 0
\(159\) −17.2156 + 29.8182i −1.36528 + 2.36474i
\(160\) 0 0
\(161\) 5.62845 15.2492i 0.443584 1.20181i
\(162\) 0 0
\(163\) 6.26768 10.8559i 0.490923 0.850303i −0.509023 0.860753i \(-0.669993\pi\)
0.999945 + 0.0104499i \(0.00332635\pi\)
\(164\) 0 0
\(165\) 12.3980 + 21.4739i 0.965180 + 1.67174i
\(166\) 0 0
\(167\) 9.92899i 0.768328i −0.923265 0.384164i \(-0.874490\pi\)
0.923265 0.384164i \(-0.125510\pi\)
\(168\) 0 0
\(169\) 12.8753 0.990410
\(170\) 0 0
\(171\) −38.2146 + 22.0632i −2.92234 + 1.68722i
\(172\) 0 0
\(173\) −0.419924 + 0.727330i −0.0319263 + 0.0552979i −0.881547 0.472096i \(-0.843498\pi\)
0.849621 + 0.527394i \(0.176831\pi\)
\(174\) 0 0
\(175\) 0.256399 + 0.308384i 0.0193819 + 0.0233117i
\(176\) 0 0
\(177\) −27.4790 15.8650i −2.06545 1.19249i
\(178\) 0 0
\(179\) 15.3809 8.88017i 1.14962 0.663735i 0.200827 0.979627i \(-0.435637\pi\)
0.948795 + 0.315892i \(0.102304\pi\)
\(180\) 0 0
\(181\) 11.6515i 0.866049i 0.901382 + 0.433024i \(0.142554\pi\)
−0.901382 + 0.433024i \(0.857446\pi\)
\(182\) 0 0
\(183\) 36.3326i 2.68578i
\(184\) 0 0
\(185\) 1.60765 + 2.78454i 0.118197 + 0.204723i
\(186\) 0 0
\(187\) 8.18361 14.1744i 0.598444 1.03654i
\(188\) 0 0
\(189\) 4.85977 + 28.3087i 0.353496 + 2.05916i
\(190\) 0 0
\(191\) −7.32761 4.23060i −0.530207 0.306115i 0.210894 0.977509i \(-0.432363\pi\)
−0.741101 + 0.671394i \(0.765696\pi\)
\(192\) 0 0
\(193\) −10.2408 + 5.91251i −0.737147 + 0.425592i −0.821031 0.570884i \(-0.806601\pi\)
0.0838843 + 0.996476i \(0.473267\pi\)
\(194\) 0 0
\(195\) −2.39870 −0.171775
\(196\) 0 0
\(197\) −1.49928 −0.106819 −0.0534097 0.998573i \(-0.517009\pi\)
−0.0534097 + 0.998573i \(0.517009\pi\)
\(198\) 0 0
\(199\) 2.85211 1.64667i 0.202181 0.116729i −0.395491 0.918470i \(-0.629426\pi\)
0.597672 + 0.801740i \(0.296092\pi\)
\(200\) 0 0
\(201\) −16.6878 + 28.9041i −1.17707 + 2.03874i
\(202\) 0 0
\(203\) −0.198481 1.15617i −0.0139306 0.0811473i
\(204\) 0 0
\(205\) −11.4342 8.24884i −0.798602 0.576124i
\(206\) 0 0
\(207\) −20.0247 34.6839i −1.39182 2.41070i
\(208\) 0 0
\(209\) 24.7073 1.70904
\(210\) 0 0
\(211\) 8.28479i 0.570348i −0.958476 0.285174i \(-0.907949\pi\)
0.958476 0.285174i \(-0.0920514\pi\)
\(212\) 0 0
\(213\) 19.3070 + 33.4406i 1.32289 + 2.29131i
\(214\) 0 0
\(215\) 7.55477 13.0852i 0.515231 0.892406i
\(216\) 0 0
\(217\) −7.25421 8.72502i −0.492448 0.592293i
\(218\) 0 0
\(219\) 6.57337 + 3.79514i 0.444187 + 0.256451i
\(220\) 0 0
\(221\) 0.791663 + 1.37120i 0.0532530 + 0.0922370i
\(222\) 0 0
\(223\) −18.8726 −1.26380 −0.631900 0.775050i \(-0.717725\pi\)
−0.631900 + 0.775050i \(0.717725\pi\)
\(224\) 0 0
\(225\) 0.988129 0.0658753
\(226\) 0 0
\(227\) −1.20151 + 0.693691i −0.0797469 + 0.0460419i −0.539343 0.842086i \(-0.681327\pi\)
0.459596 + 0.888128i \(0.347994\pi\)
\(228\) 0 0
\(229\) 16.0766 + 9.28180i 1.06237 + 0.613359i 0.926086 0.377311i \(-0.123151\pi\)
0.136282 + 0.990670i \(0.456485\pi\)
\(230\) 0 0
\(231\) 10.3166 27.9509i 0.678782 1.83903i
\(232\) 0 0
\(233\) 5.33949 + 3.08276i 0.349802 + 0.201958i 0.664598 0.747201i \(-0.268603\pi\)
−0.314796 + 0.949159i \(0.601936\pi\)
\(234\) 0 0
\(235\) 11.8036 6.81479i 0.769980 0.444548i
\(236\) 0 0
\(237\) 5.25939 0.341634
\(238\) 0 0
\(239\) 2.10642i 0.136253i −0.997677 0.0681265i \(-0.978298\pi\)
0.997677 0.0681265i \(-0.0217022\pi\)
\(240\) 0 0
\(241\) 7.58032 + 13.1295i 0.488291 + 0.845746i 0.999909 0.0134675i \(-0.00428696\pi\)
−0.511618 + 0.859213i \(0.670954\pi\)
\(242\) 0 0
\(243\) 9.03526 + 5.21651i 0.579612 + 0.334639i
\(244\) 0 0
\(245\) −2.81361 + 15.1544i −0.179755 + 0.968180i
\(246\) 0 0
\(247\) −1.19506 + 2.06991i −0.0760400 + 0.131705i
\(248\) 0 0
\(249\) 4.21562 2.43389i 0.267154 0.154242i
\(250\) 0 0
\(251\) −18.6633 −1.17802 −0.589008 0.808127i \(-0.700481\pi\)
−0.589008 + 0.808127i \(0.700481\pi\)
\(252\) 0 0
\(253\) 22.4245i 1.40982i
\(254\) 0 0
\(255\) 15.2316 + 26.3818i 0.953837 + 1.65209i
\(256\) 0 0
\(257\) 0.499938 + 0.288639i 0.0311853 + 0.0180048i 0.515512 0.856883i \(-0.327602\pi\)
−0.484326 + 0.874887i \(0.660935\pi\)
\(258\) 0 0
\(259\) 1.33776 3.62442i 0.0831245 0.225210i
\(260\) 0 0
\(261\) −2.50308 1.44516i −0.154937 0.0894529i
\(262\) 0 0
\(263\) 13.0450 7.53154i 0.804389 0.464414i −0.0406143 0.999175i \(-0.512931\pi\)
0.845004 + 0.534760i \(0.179598\pi\)
\(264\) 0 0
\(265\) 24.5732i 1.50952i
\(266\) 0 0
\(267\) −54.0045 −3.30502
\(268\) 0 0
\(269\) −13.6509 23.6441i −0.832311 1.44160i −0.896201 0.443648i \(-0.853684\pi\)
0.0638904 0.997957i \(-0.479649\pi\)
\(270\) 0 0
\(271\) 15.6601 27.1240i 0.951281 1.64767i 0.208622 0.977996i \(-0.433102\pi\)
0.742658 0.669670i \(-0.233565\pi\)
\(272\) 0 0
\(273\) 1.84265 + 2.21625i 0.111522 + 0.134134i
\(274\) 0 0
\(275\) −0.479149 0.276637i −0.0288938 0.0166818i
\(276\) 0 0
\(277\) −14.6452 25.3663i −0.879946 1.52411i −0.851399 0.524518i \(-0.824245\pi\)
−0.0285468 0.999592i \(-0.509088\pi\)
\(278\) 0 0
\(279\) −27.9568 −1.67373
\(280\) 0 0
\(281\) 10.1121i 0.603240i −0.953428 0.301620i \(-0.902473\pi\)
0.953428 0.301620i \(-0.0975274\pi\)
\(282\) 0 0
\(283\) −8.19895 14.2010i −0.487377 0.844161i 0.512518 0.858677i \(-0.328713\pi\)
−0.999895 + 0.0145153i \(0.995379\pi\)
\(284\) 0 0
\(285\) −22.9929 + 39.8249i −1.36198 + 2.35903i
\(286\) 0 0
\(287\) 1.16221 + 16.9012i 0.0686030 + 0.997644i
\(288\) 0 0
\(289\) 1.55399 2.69159i 0.0914112 0.158329i
\(290\) 0 0
\(291\) 23.2090 + 40.1992i 1.36054 + 2.35652i
\(292\) 0 0
\(293\) 26.6413i 1.55640i −0.628016 0.778200i \(-0.716133\pi\)
0.628016 0.778200i \(-0.283867\pi\)
\(294\) 0 0
\(295\) −22.6454 −1.31847
\(296\) 0 0
\(297\) −19.8125 34.3162i −1.14964 1.99123i
\(298\) 0 0
\(299\) −1.87867 1.08465i −0.108646 0.0627268i
\(300\) 0 0
\(301\) −17.8934 + 3.07177i −1.03136 + 0.177054i
\(302\) 0 0
\(303\) −4.72168 + 8.17819i −0.271254 + 0.469825i
\(304\) 0 0
\(305\) 12.9651 + 22.4562i 0.742380 + 1.28584i
\(306\) 0 0
\(307\) −19.2979 −1.10139 −0.550696 0.834706i \(-0.685638\pi\)
−0.550696 + 0.834706i \(0.685638\pi\)
\(308\) 0 0
\(309\) 23.5031i 1.33705i
\(310\) 0 0
\(311\) −2.36240 + 1.36393i −0.133960 + 0.0773416i −0.565482 0.824761i \(-0.691310\pi\)
0.431523 + 0.902102i \(0.357977\pi\)
\(312\) 0 0
\(313\) −10.7098 6.18329i −0.605353 0.349500i 0.165792 0.986161i \(-0.446982\pi\)
−0.771144 + 0.636660i \(0.780315\pi\)
\(314\) 0 0
\(315\) 24.2790 + 29.2016i 1.36797 + 1.64532i
\(316\) 0 0
\(317\) −27.9405 16.1315i −1.56930 0.906034i −0.996251 0.0865117i \(-0.972428\pi\)
−0.573047 0.819523i \(-0.694239\pi\)
\(318\) 0 0
\(319\) 0.809172 + 1.40153i 0.0453049 + 0.0784704i
\(320\) 0 0
\(321\) 41.7701i 2.33138i
\(322\) 0 0
\(323\) 30.3542 1.68895
\(324\) 0 0
\(325\) 0.0463518 0.0267612i 0.00257113 0.00148445i
\(326\) 0 0
\(327\) 7.03224 12.1802i 0.388884 0.673566i
\(328\) 0 0
\(329\) −15.3638 5.67073i −0.847033 0.312637i
\(330\) 0 0
\(331\) 20.9546 + 12.0982i 1.15177 + 0.664975i 0.949318 0.314317i \(-0.101775\pi\)
0.202453 + 0.979292i \(0.435109\pi\)
\(332\) 0 0
\(333\) −4.75945 8.24361i −0.260816 0.451747i
\(334\) 0 0
\(335\) 23.8199i 1.30142i
\(336\) 0 0
\(337\) −5.11428 −0.278593 −0.139296 0.990251i \(-0.544484\pi\)
−0.139296 + 0.990251i \(0.544484\pi\)
\(338\) 0 0
\(339\) −1.71866 + 0.992269i −0.0933448 + 0.0538927i
\(340\) 0 0
\(341\) 13.5564 + 7.82680i 0.734121 + 0.423845i
\(342\) 0 0
\(343\) 16.1631 9.04181i 0.872725 0.488212i
\(344\) 0 0
\(345\) −36.1454 20.8686i −1.94600 1.12353i
\(346\) 0 0
\(347\) −28.2404 + 16.3046i −1.51603 + 0.875278i −0.516203 + 0.856466i \(0.672655\pi\)
−0.999823 + 0.0188119i \(0.994012\pi\)
\(348\) 0 0
\(349\) 29.5204 1.58019 0.790094 0.612986i \(-0.210032\pi\)
0.790094 + 0.612986i \(0.210032\pi\)
\(350\) 0 0
\(351\) 3.83322 0.204602
\(352\) 0 0
\(353\) −7.04235 12.1977i −0.374827 0.649219i 0.615474 0.788157i \(-0.288964\pi\)
−0.990301 + 0.138938i \(0.955631\pi\)
\(354\) 0 0
\(355\) 23.8663 + 13.7792i 1.26669 + 0.731324i
\(356\) 0 0
\(357\) 12.6745 34.3391i 0.670805 1.81742i
\(358\) 0 0
\(359\) 10.9820 19.0214i 0.579610 1.00391i −0.415914 0.909404i \(-0.636538\pi\)
0.995524 0.0945096i \(-0.0301283\pi\)
\(360\) 0 0
\(361\) 13.4107 + 23.2281i 0.705829 + 1.22253i
\(362\) 0 0
\(363\) 7.16502i 0.376066i
\(364\) 0 0
\(365\) 5.41711 0.283544
\(366\) 0 0
\(367\) 6.32376 + 10.9531i 0.330097 + 0.571745i 0.982531 0.186101i \(-0.0595851\pi\)
−0.652433 + 0.757846i \(0.726252\pi\)
\(368\) 0 0
\(369\) 33.8510 + 24.4206i 1.76221 + 1.27129i
\(370\) 0 0
\(371\) 22.7041 18.8768i 1.17874 0.980033i
\(372\) 0 0
\(373\) 0.728596 1.26197i 0.0377253 0.0653421i −0.846546 0.532315i \(-0.821322\pi\)
0.884272 + 0.466973i \(0.154655\pi\)
\(374\) 0 0
\(375\) 30.3083 17.4985i 1.56511 0.903618i
\(376\) 0 0
\(377\) −0.156555 −0.00806299
\(378\) 0 0
\(379\) −15.4850 −0.795410 −0.397705 0.917513i \(-0.630193\pi\)
−0.397705 + 0.917513i \(0.630193\pi\)
\(380\) 0 0
\(381\) 31.1068 17.9595i 1.59365 0.920095i
\(382\) 0 0
\(383\) 14.5637 + 8.40835i 0.744170 + 0.429647i 0.823583 0.567195i \(-0.191971\pi\)
−0.0794136 + 0.996842i \(0.525305\pi\)
\(384\) 0 0
\(385\) −3.59773 20.9572i −0.183357 1.06808i
\(386\) 0 0
\(387\) −22.3658 + 38.7388i −1.13692 + 1.96920i
\(388\) 0 0
\(389\) 16.4452 + 28.4839i 0.833803 + 1.44419i 0.895001 + 0.446063i \(0.147174\pi\)
−0.0611984 + 0.998126i \(0.519492\pi\)
\(390\) 0 0
\(391\) 27.5497i 1.39325i
\(392\) 0 0
\(393\) 10.6906i 0.539271i
\(394\) 0 0
\(395\) 3.25069 1.87679i 0.163560 0.0944315i
\(396\) 0 0
\(397\) −18.1591 10.4842i −0.911380 0.526185i −0.0305048 0.999535i \(-0.509711\pi\)
−0.880875 + 0.473349i \(0.843045\pi\)
\(398\) 0 0
\(399\) 54.4586 9.34893i 2.72634 0.468032i
\(400\) 0 0
\(401\) −18.1720 + 31.4748i −0.907466 + 1.57178i −0.0898942 + 0.995951i \(0.528653\pi\)
−0.817572 + 0.575826i \(0.804680\pi\)
\(402\) 0 0
\(403\) −1.31142 + 0.757147i −0.0653263 + 0.0377162i
\(404\) 0 0
\(405\) 30.6899 1.52499
\(406\) 0 0
\(407\) 5.32983i 0.264190i
\(408\) 0 0
\(409\) −13.8339 23.9611i −0.684043 1.18480i −0.973736 0.227678i \(-0.926887\pi\)
0.289693 0.957120i \(-0.406447\pi\)
\(410\) 0 0
\(411\) 16.5278 28.6270i 0.815257 1.41207i
\(412\) 0 0
\(413\) 17.3959 + 20.9230i 0.855997 + 1.02955i
\(414\) 0 0
\(415\) 1.73704 3.00865i 0.0852682 0.147689i
\(416\) 0 0
\(417\) 31.4012 18.1295i 1.53772 0.887806i
\(418\) 0 0
\(419\) 5.83932 0.285270 0.142635 0.989775i \(-0.454443\pi\)
0.142635 + 0.989775i \(0.454443\pi\)
\(420\) 0 0
\(421\) 18.3691i 0.895254i −0.894220 0.447627i \(-0.852269\pi\)
0.894220 0.447627i \(-0.147731\pi\)
\(422\) 0 0
\(423\) −34.9444 + 20.1752i −1.69905 + 0.980950i
\(424\) 0 0
\(425\) −0.588660 0.339863i −0.0285542 0.0164858i
\(426\) 0 0
\(427\) 10.7885 29.2295i 0.522093 1.41452i
\(428\) 0 0
\(429\) −3.44348 1.98809i −0.166253 0.0959860i
\(430\) 0 0
\(431\) −4.62032 8.00263i −0.222553 0.385473i 0.733029 0.680197i \(-0.238106\pi\)
−0.955583 + 0.294724i \(0.904772\pi\)
\(432\) 0 0
\(433\) −33.1993 −1.59545 −0.797727 0.603019i \(-0.793964\pi\)
−0.797727 + 0.603019i \(0.793964\pi\)
\(434\) 0 0
\(435\) −3.01211 −0.144419
\(436\) 0 0
\(437\) −36.0162 + 20.7940i −1.72289 + 0.994710i
\(438\) 0 0
\(439\) −6.49173 3.74800i −0.309833 0.178882i 0.337019 0.941498i \(-0.390581\pi\)
−0.646852 + 0.762616i \(0.723915\pi\)
\(440\) 0 0
\(441\) 8.32966 44.8645i 0.396651 2.13641i
\(442\) 0 0
\(443\) −15.5961 + 27.0133i −0.740995 + 1.28344i 0.211047 + 0.977476i \(0.432313\pi\)
−0.952043 + 0.305966i \(0.901021\pi\)
\(444\) 0 0
\(445\) −33.3788 + 19.2712i −1.58231 + 0.913545i
\(446\) 0 0
\(447\) 57.1337 2.70233
\(448\) 0 0
\(449\) −15.7738 −0.744409 −0.372205 0.928151i \(-0.621398\pi\)
−0.372205 + 0.928151i \(0.621398\pi\)
\(450\) 0 0
\(451\) −9.57771 21.3186i −0.450997 1.00385i
\(452\) 0 0
\(453\) 11.9385 20.6781i 0.560921 0.971543i
\(454\) 0 0
\(455\) 1.92975 + 0.712266i 0.0904682 + 0.0333916i
\(456\) 0 0
\(457\) −25.1813 14.5384i −1.17793 0.680078i −0.222395 0.974957i \(-0.571388\pi\)
−0.955535 + 0.294878i \(0.904721\pi\)
\(458\) 0 0
\(459\) −24.3407 42.1593i −1.13612 1.96783i
\(460\) 0 0
\(461\) 5.50886 0.256573 0.128286 0.991737i \(-0.459052\pi\)
0.128286 + 0.991737i \(0.459052\pi\)
\(462\) 0 0
\(463\) 3.43337i 0.159562i −0.996812 0.0797810i \(-0.974578\pi\)
0.996812 0.0797810i \(-0.0254221\pi\)
\(464\) 0 0
\(465\) −25.2316 + 14.5675i −1.17009 + 0.675550i
\(466\) 0 0
\(467\) −2.28786 + 3.96269i −0.105870 + 0.183372i −0.914093 0.405504i \(-0.867096\pi\)
0.808224 + 0.588876i \(0.200429\pi\)
\(468\) 0 0
\(469\) 22.0081 18.2981i 1.01624 0.844928i
\(470\) 0 0
\(471\) −17.6057 + 30.4940i −0.811229 + 1.40509i
\(472\) 0 0
\(473\) 21.6906 12.5231i 0.997336 0.575812i
\(474\) 0 0
\(475\) 1.02609i 0.0470801i
\(476\) 0 0
\(477\) 72.7487i 3.33094i
\(478\) 0 0
\(479\) 5.67862 3.27856i 0.259463 0.149801i −0.364627 0.931154i \(-0.618803\pi\)
0.624090 + 0.781353i \(0.285470\pi\)
\(480\) 0 0
\(481\) −0.446519 0.257798i −0.0203595 0.0117546i
\(482\) 0 0
\(483\) 8.48516 + 49.4271i 0.386088 + 2.24901i
\(484\) 0 0
\(485\) 28.6898 + 16.5641i 1.30274 + 0.752136i
\(486\) 0 0
\(487\) 16.6579 + 28.8523i 0.754842 + 1.30742i 0.945453 + 0.325759i \(0.105620\pi\)
−0.190611 + 0.981666i \(0.561047\pi\)
\(488\) 0 0
\(489\) 38.6747i 1.74893i
\(490\) 0 0
\(491\) 13.7108 0.618760 0.309380 0.950939i \(-0.399879\pi\)
0.309380 + 0.950939i \(0.399879\pi\)
\(492\) 0 0
\(493\) 0.994110 + 1.72185i 0.0447725 + 0.0775482i
\(494\) 0 0
\(495\) −45.3717 26.1954i −2.03931 1.17739i
\(496\) 0 0
\(497\) −5.60262 32.6359i −0.251312 1.46392i
\(498\) 0 0
\(499\) 27.6743 + 15.9778i 1.23887 + 0.715264i 0.968864 0.247594i \(-0.0796399\pi\)
0.270009 + 0.962858i \(0.412973\pi\)
\(500\) 0 0
\(501\) 15.3167 + 26.5293i 0.684299 + 1.18524i
\(502\) 0 0
\(503\) 7.02693i 0.313315i 0.987653 + 0.156658i \(0.0500719\pi\)
−0.987653 + 0.156658i \(0.949928\pi\)
\(504\) 0 0
\(505\) 6.73964i 0.299910i
\(506\) 0 0
\(507\) −34.4016 + 19.8618i −1.52783 + 0.882093i
\(508\) 0 0
\(509\) −30.6836 17.7152i −1.36003 0.785213i −0.370402 0.928872i \(-0.620780\pi\)
−0.989627 + 0.143659i \(0.954113\pi\)
\(510\) 0 0
\(511\) −4.16135 5.00507i −0.184087 0.221411i
\(512\) 0 0
\(513\) 36.7437 63.6419i 1.62227 2.80986i
\(514\) 0 0
\(515\) 8.38698 + 14.5267i 0.369574 + 0.640122i
\(516\) 0 0
\(517\) 22.5930 0.993637
\(518\) 0 0
\(519\) 2.59114i 0.113738i
\(520\) 0 0
\(521\) −27.3229 + 15.7749i −1.19704 + 0.691109i −0.959893 0.280365i \(-0.909544\pi\)
−0.237143 + 0.971475i \(0.576211\pi\)
\(522\) 0 0
\(523\) −5.13477 + 8.89368i −0.224528 + 0.388893i −0.956178 0.292787i \(-0.905417\pi\)
0.731650 + 0.681681i \(0.238751\pi\)
\(524\) 0 0
\(525\) −1.16079 0.428445i −0.0506612 0.0186989i
\(526\) 0 0
\(527\) 16.6548 + 9.61564i 0.725493 + 0.418864i
\(528\) 0 0
\(529\) −7.37277 12.7700i −0.320555 0.555218i
\(530\) 0 0
\(531\) 67.0417 2.90936
\(532\) 0 0
\(533\) 2.24928 + 0.228763i 0.0974272 + 0.00990881i
\(534\) 0 0
\(535\) −14.9055 25.8171i −0.644420 1.11617i
\(536\) 0 0
\(537\) −27.3975 + 47.4539i −1.18229 + 2.04779i
\(538\) 0 0
\(539\) −16.5994 + 19.4231i −0.714986 + 0.836611i
\(540\) 0 0
\(541\) −2.45504 + 4.25226i −0.105551 + 0.182819i −0.913963 0.405798i \(-0.866994\pi\)
0.808412 + 0.588616i \(0.200327\pi\)
\(542\) 0 0
\(543\) −17.9739 31.1316i −0.771332 1.33599i
\(544\) 0 0
\(545\) 10.0377i 0.429967i
\(546\) 0 0
\(547\) 16.8764i 0.721584i −0.932646 0.360792i \(-0.882507\pi\)
0.932646 0.360792i \(-0.117493\pi\)
\(548\) 0 0
\(549\) −38.3831 66.4815i −1.63815 2.83736i
\(550\) 0 0
\(551\) −1.50067 + 2.59923i −0.0639306 + 0.110731i
\(552\) 0 0
\(553\) −4.23117 1.56171i −0.179928 0.0664108i
\(554\) 0 0
\(555\) −8.59099 4.96001i −0.364667 0.210541i
\(556\) 0 0
\(557\) 6.00870 3.46912i 0.254597 0.146991i −0.367271 0.930114i \(-0.619708\pi\)
0.621867 + 0.783123i \(0.286374\pi\)
\(558\) 0 0
\(559\) 2.42291i 0.102478i
\(560\) 0 0
\(561\) 50.4969i 2.13198i
\(562\) 0 0
\(563\) 3.54394 2.04610i 0.149359 0.0862326i −0.423458 0.905916i \(-0.639184\pi\)
0.572817 + 0.819683i \(0.305850\pi\)
\(564\) 0 0
\(565\) −0.708173 + 1.22659i −0.0297931 + 0.0516031i
\(566\) 0 0
\(567\) −23.5755 28.3555i −0.990079 1.19082i
\(568\) 0 0
\(569\) −10.3866 + 17.9902i −0.435431 + 0.754188i −0.997331 0.0730169i \(-0.976737\pi\)
0.561900 + 0.827205i \(0.310071\pi\)
\(570\) 0 0
\(571\) −24.8882 + 14.3692i −1.04154 + 0.601333i −0.920269 0.391287i \(-0.872030\pi\)
−0.121270 + 0.992620i \(0.538697\pi\)
\(572\) 0 0
\(573\) 26.1049 1.09055
\(574\) 0 0
\(575\) 0.931285 0.0388372
\(576\) 0 0
\(577\) −8.33918 + 4.81463i −0.347165 + 0.200436i −0.663436 0.748233i \(-0.730902\pi\)
0.316271 + 0.948669i \(0.397569\pi\)
\(578\) 0 0
\(579\) 18.2416 31.5953i 0.758093 1.31306i
\(580\) 0 0
\(581\) −4.11418 + 0.706283i −0.170685 + 0.0293015i
\(582\) 0 0
\(583\) −20.3667 + 35.2762i −0.843505 + 1.46099i
\(584\) 0 0
\(585\) 4.38916 2.53408i 0.181469 0.104771i
\(586\) 0 0
\(587\) 12.3233i 0.508637i −0.967120 0.254319i \(-0.918149\pi\)
0.967120 0.254319i \(-0.0818512\pi\)
\(588\) 0 0
\(589\) 29.0308i 1.19619i
\(590\) 0 0
\(591\) 4.00593 2.31283i 0.164782 0.0951370i
\(592\) 0 0
\(593\) −0.787934 0.454914i −0.0323566 0.0186811i 0.483734 0.875215i \(-0.339280\pi\)
−0.516091 + 0.856534i \(0.672613\pi\)
\(594\) 0 0
\(595\) −4.42000 25.7470i −0.181202 1.05552i
\(596\) 0 0
\(597\) −5.08037 + 8.79947i −0.207926 + 0.360138i
\(598\) 0 0
\(599\) −7.72441 13.3791i −0.315611 0.546654i 0.663956 0.747771i \(-0.268876\pi\)
−0.979567 + 0.201117i \(0.935543\pi\)
\(600\) 0 0
\(601\) 37.8156i 1.54253i 0.636514 + 0.771265i \(0.280376\pi\)
−0.636514 + 0.771265i \(0.719624\pi\)
\(602\) 0 0
\(603\) 70.5186i 2.87174i
\(604\) 0 0
\(605\) 2.55681 + 4.42852i 0.103949 + 0.180045i
\(606\) 0 0
\(607\) −0.581403 + 1.00702i −0.0235984 + 0.0408737i −0.877583 0.479424i \(-0.840846\pi\)
0.853985 + 0.520298i \(0.174179\pi\)
\(608\) 0 0
\(609\) 2.31386 + 2.78300i 0.0937623 + 0.112773i
\(610\) 0 0
\(611\) −1.09280 + 1.89278i −0.0442098 + 0.0765736i
\(612\) 0 0
\(613\) −6.63135 11.4858i −0.267838 0.463908i 0.700465 0.713686i \(-0.252976\pi\)
−0.968303 + 0.249778i \(0.919642\pi\)
\(614\) 0 0
\(615\) 43.2760 + 4.40138i 1.74506 + 0.177481i
\(616\) 0 0
\(617\) 19.9252 0.802158 0.401079 0.916044i \(-0.368635\pi\)
0.401079 + 0.916044i \(0.368635\pi\)
\(618\) 0 0
\(619\) 20.3345 + 35.2203i 0.817311 + 1.41562i 0.907657 + 0.419713i \(0.137869\pi\)
−0.0903459 + 0.995910i \(0.528797\pi\)
\(620\) 0 0
\(621\) 57.7619 + 33.3488i 2.31790 + 1.33824i
\(622\) 0 0
\(623\) 43.4465 + 16.0360i 1.74065 + 0.642469i
\(624\) 0 0
\(625\) 12.1096 20.9744i 0.484382 0.838975i
\(626\) 0 0
\(627\) −66.0154 + 38.1140i −2.63640 + 1.52213i
\(628\) 0 0
\(629\) 6.54797i 0.261085i
\(630\) 0 0
\(631\) 17.4244 0.693655 0.346828 0.937929i \(-0.387259\pi\)
0.346828 + 0.937929i \(0.387259\pi\)
\(632\) 0 0
\(633\) 12.7803 + 22.1361i 0.507972 + 0.879833i
\(634\) 0 0
\(635\) 12.8176 22.2007i 0.508649 0.881006i
\(636\) 0 0
\(637\) −0.824320 2.33012i −0.0326608 0.0923229i
\(638\) 0 0
\(639\) −70.6559 40.7932i −2.79511 1.61375i
\(640\) 0 0
\(641\) 1.01731 0.587342i 0.0401812 0.0231986i −0.479775 0.877392i \(-0.659282\pi\)
0.519956 + 0.854193i \(0.325948\pi\)
\(642\) 0 0
\(643\) 38.7130i 1.52669i −0.645989 0.763347i \(-0.723555\pi\)
0.645989 0.763347i \(-0.276445\pi\)
\(644\) 0 0
\(645\) 46.6166i 1.83553i
\(646\) 0 0
\(647\) 12.2559 + 21.2278i 0.481828 + 0.834551i 0.999782 0.0208572i \(-0.00663952\pi\)
−0.517954 + 0.855408i \(0.673306\pi\)
\(648\) 0 0
\(649\) −32.5089 18.7690i −1.27608 0.736748i
\(650\) 0 0
\(651\) 32.8420 + 12.1219i 1.28718 + 0.475094i
\(652\) 0 0
\(653\) 3.37942 + 1.95111i 0.132247 + 0.0763527i 0.564664 0.825321i \(-0.309006\pi\)
−0.432417 + 0.901674i \(0.642339\pi\)
\(654\) 0 0
\(655\) −3.81490 6.60761i −0.149061 0.258181i
\(656\) 0 0
\(657\) −16.0373 −0.625675
\(658\) 0 0
\(659\) 0.994257i 0.0387308i 0.999812 + 0.0193654i \(0.00616458\pi\)
−0.999812 + 0.0193654i \(0.993835\pi\)
\(660\) 0 0
\(661\) −22.9645 39.7757i −0.893217 1.54710i −0.835996 0.548735i \(-0.815109\pi\)
−0.0572208 0.998362i \(-0.518224\pi\)
\(662\) 0 0
\(663\) −4.23049 2.44248i −0.164299 0.0948579i
\(664\) 0 0
\(665\) 30.3233 25.2116i 1.17589 0.977665i
\(666\) 0 0
\(667\) −2.35909 1.36202i −0.0913442 0.0527376i
\(668\) 0 0
\(669\) 50.4256 29.1132i 1.94957 1.12558i
\(670\) 0 0
\(671\) 42.9830i 1.65934i
\(672\) 0 0
\(673\) 42.9532i 1.65573i 0.560930 + 0.827863i \(0.310444\pi\)
−0.560930 + 0.827863i \(0.689556\pi\)
\(674\) 0 0
\(675\) −1.42514 + 0.822806i −0.0548538 + 0.0316698i
\(676\) 0 0
\(677\) 18.4167 31.8987i 0.707813 1.22597i −0.257854 0.966184i \(-0.583015\pi\)
0.965667 0.259784i \(-0.0836515\pi\)
\(678\) 0 0
\(679\) −6.73496 39.2319i −0.258464 1.50558i
\(680\) 0 0
\(681\) 2.14021 3.70695i 0.0820130 0.142051i
\(682\) 0 0
\(683\) 30.4639 17.5883i 1.16567 0.672999i 0.213012 0.977049i \(-0.431673\pi\)
0.952656 + 0.304051i \(0.0983392\pi\)
\(684\) 0 0
\(685\) 23.5915i 0.901385i
\(686\) 0 0
\(687\) −57.2733 −2.18511
\(688\) 0 0
\(689\) −1.97023 3.41254i −0.0750599 0.130008i
\(690\) 0 0
\(691\) −3.36680 1.94382i −0.128079 0.0739466i 0.434591 0.900628i \(-0.356893\pi\)
−0.562671 + 0.826681i \(0.690226\pi\)
\(692\) 0 0
\(693\) 10.6510 + 62.0435i 0.404600 + 2.35684i
\(694\) 0 0
\(695\) 12.9389 22.4108i 0.490799 0.850089i
\(696\) 0 0
\(697\) −11.7667 26.1911i −0.445696 0.992057i
\(698\) 0 0
\(699\) −19.0221 −0.719483
\(700\) 0 0
\(701\) −3.09568 −0.116922 −0.0584611 0.998290i \(-0.518619\pi\)
−0.0584611 + 0.998290i \(0.518619\pi\)
\(702\) 0 0
\(703\) −8.56028 + 4.94228i −0.322857 + 0.186402i
\(704\) 0 0
\(705\) −21.0253 + 36.4169i −0.791860 + 1.37154i
\(706\) 0 0
\(707\) 6.22701 5.17730i 0.234191 0.194712i
\(708\) 0 0
\(709\) 22.1444 + 12.7851i 0.831650 + 0.480153i 0.854417 0.519588i \(-0.173914\pi\)
−0.0227676 + 0.999741i \(0.507248\pi\)
\(710\) 0 0
\(711\) −9.62366 + 5.55622i −0.360915 + 0.208375i
\(712\) 0 0
\(713\) −26.3485 −0.986761
\(714\) 0 0
\(715\) −2.83777 −0.106126
\(716\) 0 0
\(717\) 3.24942 + 5.62815i 0.121352 + 0.210187i
\(718\) 0 0
\(719\) 12.2466 + 7.07060i 0.456723 + 0.263689i 0.710665 0.703530i \(-0.248394\pi\)
−0.253943 + 0.967219i \(0.581727\pi\)
\(720\) 0 0
\(721\) 6.97897 18.9082i 0.259911 0.704179i
\(722\) 0 0
\(723\) −40.5078 23.3872i −1.50650 0.869778i
\(724\) 0 0
\(725\) 0.0582050 0.0336047i 0.00216168 0.00124805i
\(726\) 0 0
\(727\) 45.4858i 1.68697i 0.537149 + 0.843487i \(0.319501\pi\)
−0.537149 + 0.843487i \(0.680499\pi\)
\(728\) 0 0
\(729\) 9.62503 0.356483
\(730\) 0 0
\(731\) 26.6481 15.3853i 0.985615 0.569045i
\(732\) 0 0
\(733\) 6.14908 10.6505i 0.227121 0.393386i −0.729832 0.683626i \(-0.760402\pi\)
0.956954 + 0.290240i \(0.0937352\pi\)
\(734\) 0 0
\(735\) −15.8599 44.8314i −0.585000 1.65363i
\(736\) 0 0
\(737\) −19.7424 + 34.1948i −0.727221 + 1.25958i
\(738\) 0 0
\(739\) 4.97466 + 8.61637i 0.182996 + 0.316958i 0.942899 0.333078i \(-0.108087\pi\)
−0.759903 + 0.650036i \(0.774754\pi\)
\(740\) 0 0
\(741\) 7.37413i 0.270895i
\(742\) 0 0
\(743\) −34.5550 −1.26770 −0.633849 0.773456i \(-0.718526\pi\)
−0.633849 + 0.773456i \(0.718526\pi\)
\(744\) 0 0
\(745\) 35.3129 20.3879i 1.29376 0.746955i
\(746\) 0 0
\(747\) −5.14251 + 8.90709i −0.188155 + 0.325893i
\(748\) 0 0
\(749\) −12.4032 + 33.6040i −0.453201 + 1.22786i
\(750\) 0 0
\(751\) −26.0938 15.0653i −0.952176 0.549739i −0.0584200 0.998292i \(-0.518606\pi\)
−0.893756 + 0.448553i \(0.851940\pi\)
\(752\) 0 0
\(753\) 49.8664 28.7904i 1.81723 1.04918i
\(754\) 0 0
\(755\) 17.0408i 0.620179i
\(756\) 0 0
\(757\) 25.4331i 0.924380i −0.886781 0.462190i \(-0.847064\pi\)
0.886781 0.462190i \(-0.152936\pi\)
\(758\) 0 0
\(759\) −34.5926 59.9161i −1.25563 2.17482i
\(760\) 0 0
\(761\) 11.7092 20.2809i 0.424457 0.735181i −0.571913 0.820315i \(-0.693798\pi\)
0.996370 + 0.0851335i \(0.0271317\pi\)
\(762\) 0 0
\(763\) −9.27419 + 7.71081i −0.335748 + 0.279150i
\(764\) 0 0
\(765\) −55.7415 32.1824i −2.01534 1.16356i
\(766\) 0 0
\(767\) 3.14483 1.81567i 0.113553 0.0655601i
\(768\) 0 0
\(769\) −28.7089 −1.03527 −0.517634 0.855602i \(-0.673187\pi\)
−0.517634 + 0.855602i \(0.673187\pi\)
\(770\) 0 0
\(771\) −1.78105 −0.0641428
\(772\) 0 0
\(773\) −3.81317 + 2.20153i −0.137150 + 0.0791837i −0.567005 0.823714i \(-0.691898\pi\)
0.429855 + 0.902898i \(0.358565\pi\)
\(774\) 0 0
\(775\) 0.325045 0.562994i 0.0116760 0.0202234i
\(776\) 0 0
\(777\) 2.01674 + 11.7477i 0.0723502 + 0.421448i
\(778\) 0 0
\(779\) 25.3587 35.1513i 0.908571 1.25943i
\(780\) 0 0
\(781\) 22.8410 + 39.5617i 0.817314 + 1.41563i
\(782\) 0 0
\(783\) 4.81347 0.172019
\(784\) 0 0
\(785\) 25.1301i 0.896931i
\(786\) 0 0
\(787\) −17.1731 29.7448i −0.612157 1.06029i −0.990876 0.134775i \(-0.956969\pi\)
0.378720 0.925511i \(-0.376364\pi\)
\(788\) 0 0
\(789\) −23.2366 + 40.2471i −0.827247 + 1.43283i
\(790\) 0 0
\(791\) 1.67730 0.287943i 0.0596380 0.0102381i
\(792\) 0 0
\(793\) −3.60100 2.07904i −0.127875 0.0738288i
\(794\) 0 0
\(795\) −37.9072 65.6571i −1.34443 2.32862i
\(796\) 0 0
\(797\) −32.9362 −1.16666 −0.583330 0.812235i \(-0.698251\pi\)
−0.583330 + 0.812235i \(0.698251\pi\)
\(798\) 0 0
\(799\) 27.7566 0.981960
\(800\) 0 0
\(801\) 98.8177 57.0524i 3.49155 2.01585i
\(802\) 0 0
\(803\) 7.77658 + 4.48981i 0.274429 + 0.158442i
\(804\) 0 0
\(805\) 22.8823 + 27.5217i 0.806494 + 0.970012i
\(806\) 0 0
\(807\) 72.9478 + 42.1164i 2.56788 + 1.48257i
\(808\) 0 0
\(809\) −32.4905 + 18.7584i −1.14230 + 0.659510i −0.947000 0.321233i \(-0.895903\pi\)
−0.195304 + 0.980743i \(0.562569\pi\)
\(810\) 0 0
\(811\) −10.3349 −0.362908 −0.181454 0.983399i \(-0.558080\pi\)
−0.181454 + 0.983399i \(0.558080\pi\)
\(812\) 0 0
\(813\) 96.6303i 3.38897i
\(814\) 0 0
\(815\) 13.8009 + 23.9038i 0.483424 + 0.837315i
\(816\) 0 0
\(817\) 40.2269 + 23.2250i 1.40736 + 0.812540i
\(818\) 0 0
\(819\) −5.71302 2.10866i −0.199629 0.0736825i
\(820\) 0 0
\(821\) −13.9409 + 24.1464i −0.486541 + 0.842714i −0.999880 0.0154718i \(-0.995075\pi\)
0.513339 + 0.858186i \(0.328408\pi\)
\(822\) 0 0
\(823\) 29.9752 17.3062i 1.04487 0.603255i 0.123660 0.992325i \(-0.460537\pi\)
0.921208 + 0.389070i \(0.127203\pi\)
\(824\) 0 0
\(825\) 1.70699 0.0594296
\(826\) 0 0
\(827\) 14.1260i 0.491211i −0.969370 0.245605i \(-0.921013\pi\)
0.969370 0.245605i \(-0.0789867\pi\)
\(828\) 0 0
\(829\) −12.1560 21.0548i −0.422195 0.731263i 0.573959 0.818884i \(-0.305407\pi\)
−0.996154 + 0.0876213i \(0.972073\pi\)
\(830\) 0 0
\(831\) 78.2612 + 45.1841i 2.71485 + 1.56742i
\(832\) 0 0
\(833\) −20.3932 + 23.8623i −0.706583 + 0.826779i
\(834\) 0 0
\(835\) 18.9337 + 10.9314i 0.655228 + 0.378296i
\(836\) 0 0
\(837\) 40.3211 23.2794i 1.39370 0.804654i
\(838\) 0 0
\(839\) 30.5112i 1.05336i −0.850063 0.526682i \(-0.823436\pi\)
0.850063 0.526682i \(-0.176564\pi\)
\(840\) 0 0
\(841\) 28.8034 0.993221
\(842\) 0 0
\(843\) 15.5992 + 27.0187i 0.537266 + 0.930572i
\(844\) 0 0
\(845\) −14.1752 + 24.5521i −0.487641 + 0.844618i
\(846\) 0 0
\(847\) 2.12757 5.76425i 0.0731042 0.198062i
\(848\) 0 0
\(849\) 43.8136 + 25.2958i 1.50368 + 0.868149i
\(850\) 0 0
\(851\) −4.48565 7.76938i −0.153766 0.266331i
\(852\) 0 0
\(853\) 48.8989 1.67427 0.837133 0.547000i \(-0.184230\pi\)
0.837133 + 0.547000i \(0.184230\pi\)
\(854\) 0 0
\(855\) 97.1625i 3.32289i
\(856\) 0 0
\(857\) −0.408558 0.707642i −0.0139561 0.0241726i 0.858963 0.512038i \(-0.171109\pi\)
−0.872919 + 0.487865i \(0.837776\pi\)
\(858\) 0 0
\(859\) 1.01695 1.76140i 0.0346978 0.0600983i −0.848155 0.529748i \(-0.822286\pi\)
0.882853 + 0.469650i \(0.155620\pi\)
\(860\) 0 0
\(861\) −29.1774 43.3654i −0.994364 1.47789i
\(862\) 0 0
\(863\) 2.18418 3.78310i 0.0743502 0.128778i −0.826453 0.563005i \(-0.809645\pi\)
0.900804 + 0.434227i \(0.142978\pi\)
\(864\) 0 0
\(865\) −0.924636 1.60152i −0.0314386 0.0544532i
\(866\) 0 0
\(867\) 9.58889i 0.325656i
\(868\) 0 0
\(869\) 6.22208 0.211070
\(870\) 0 0
\(871\) −1.90984 3.30793i −0.0647123 0.112085i
\(872\) 0 0
\(873\) −84.9360 49.0378i −2.87465 1.65968i
\(874\) 0 0
\(875\) −29.5789 + 5.07783i −0.999951 + 0.171662i
\(876\) 0 0
\(877\) 24.9397 43.1969i 0.842155 1.45866i −0.0459133 0.998945i \(-0.514620\pi\)
0.888069 0.459711i \(-0.152047\pi\)
\(878\) 0 0
\(879\) 41.0975 + 71.1829i 1.38618 + 2.40094i
\(880\) 0 0
\(881\) −29.4572 −0.992437 −0.496219 0.868198i \(-0.665279\pi\)
−0.496219 + 0.868198i \(0.665279\pi\)
\(882\) 0 0
\(883\) 28.0341i 0.943423i 0.881753 + 0.471711i \(0.156364\pi\)
−0.881753 + 0.471711i \(0.843636\pi\)
\(884\) 0 0
\(885\) 60.5064 34.9334i 2.03390 1.17427i
\(886\) 0 0
\(887\) −11.0159 6.36005i −0.369878 0.213549i 0.303527 0.952823i \(-0.401836\pi\)
−0.673405 + 0.739273i \(0.735169\pi\)
\(888\) 0 0
\(889\) −30.3583 + 5.21162i −1.01818 + 0.174792i
\(890\) 0 0
\(891\) 44.0571 + 25.4364i 1.47597 + 0.852151i
\(892\) 0 0
\(893\) 20.9502 + 36.2867i 0.701070 + 1.21429i
\(894\) 0 0
\(895\) 39.1067i 1.30719i
\(896\) 0 0
\(897\) 6.69282 0.223467
\(898\) 0 0
\(899\) −1.64678 + 0.950767i −0.0549231 + 0.0317099i
\(900\) 0 0
\(901\) −25.0216 + 43.3387i −0.833591 + 1.44382i
\(902\) 0 0
\(903\) 43.0708 35.8102i 1.43331 1.19169i
\(904\) 0 0
\(905\) −22.2184 12.8278i −0.738564 0.426410i
\(906\) 0 0
\(907\) 3.01389 + 5.22021i 0.100075 + 0.173334i 0.911715 0.410823i \(-0.134759\pi\)
−0.811641 + 0.584157i \(0.801425\pi\)
\(908\) 0 0
\(909\) 19.9527i 0.661788i
\(910\) 0 0
\(911\) 22.7915 0.755117 0.377558 0.925986i \(-0.376764\pi\)
0.377558 + 0.925986i \(0.376764\pi\)
\(912\) 0 0
\(913\) 4.98726 2.87940i 0.165054 0.0952941i
\(914\) 0 0
\(915\) −69.2830 40.0006i −2.29043 1.32238i
\(916\) 0 0
\(917\) −3.17446 + 8.60061i −0.104830 + 0.284017i
\(918\) 0 0
\(919\) 4.60146 + 2.65666i 0.151788 + 0.0876350i 0.573970 0.818876i \(-0.305402\pi\)
−0.422182 + 0.906511i \(0.638736\pi\)
\(920\) 0 0
\(921\) 51.5622 29.7695i 1.69903 0.980937i
\(922\) 0 0
\(923\) −4.41916 −0.145459
\(924\) 0 0
\(925\) 0.221346 0.00727782
\(926\) 0 0
\(927\) −24.8296 43.0061i −0.815511 1.41251i
\(928\) 0 0
\(929\) 37.1428 + 21.4444i 1.21862 + 0.703568i 0.964622 0.263637i \(-0.0849222\pi\)
0.253994 + 0.967206i \(0.418256\pi\)
\(930\) 0 0
\(931\) −46.5879 8.64963i −1.52686 0.283480i
\(932\) 0 0
\(933\) 4.20807 7.28860i 0.137766 0.238618i
\(934\) 0 0
\(935\) 18.0196 + 31.2108i 0.589303 + 1.02070i
\(936\) 0 0
\(937\) 46.1036i 1.50614i −0.657941 0.753070i \(-0.728572\pi\)
0.657941 0.753070i \(-0.271428\pi\)
\(938\) 0 0
\(939\) 38.1540 1.24511
\(940\) 0 0
\(941\) 21.6317 + 37.4673i 0.705175 + 1.22140i 0.966628 + 0.256182i \(0.0824648\pi\)
−0.261454 + 0.965216i \(0.584202\pi\)
\(942\) 0 0
\(943\) 31.9036 + 23.0158i 1.03892 + 0.749497i
\(944\) 0 0
\(945\) −59.3326 21.8995i −1.93009 0.712391i
\(946\) 0 0
\(947\) −14.0965 + 24.4158i −0.458073 + 0.793406i −0.998859 0.0477542i \(-0.984794\pi\)
0.540786 + 0.841160i \(0.318127\pi\)
\(948\) 0 0
\(949\) −0.752288 + 0.434334i −0.0244203 + 0.0140991i
\(950\) 0 0
\(951\) 99.5392 3.22778
\(952\) 0 0
\(953\) −44.9250 −1.45526 −0.727631 0.685968i \(-0.759379\pi\)
−0.727631 + 0.685968i \(0.759379\pi\)
\(954\) 0 0
\(955\) 16.1348 9.31541i 0.522109 0.301439i
\(956\) 0 0
\(957\) −4.32406 2.49649i −0.139777 0.0807002i
\(958\) 0 0
\(959\) −21.7971 + 18.1227i −0.703864 + 0.585211i
\(960\) 0 0
\(961\) 6.30360 10.9182i 0.203342 0.352199i
\(962\) 0 0
\(963\) 44.1276 + 76.4312i 1.42199 + 2.46296i
\(964\) 0 0
\(965\) 26.0377i 0.838182i
\(966\) 0 0
\(967\) 17.4158i 0.560056i −0.959992 0.280028i \(-0.909656\pi\)
0.959992 0.280028i \(-0.0903437\pi\)
\(968\) 0 0
\(969\) −81.1035 + 46.8251i −2.60542 + 1.50424i
\(970\) 0 0
\(971\) −0.489865 0.282823i −0.0157205 0.00907624i 0.492119 0.870528i \(-0.336222\pi\)
−0.507840 + 0.861452i \(0.669556\pi\)
\(972\) 0 0
\(973\) −30.6456 + 5.26094i −0.982453 + 0.168658i
\(974\) 0 0
\(975\) −0.0825650 + 0.143007i −0.00264420 + 0.00457988i
\(976\) 0 0
\(977\) −10.9454 + 6.31935i −0.350176 + 0.202174i −0.664763 0.747055i \(-0.731467\pi\)
0.314587 + 0.949229i \(0.398134\pi\)
\(978\) 0 0
\(979\) −63.8896 −2.04192
\(980\) 0 0
\(981\) 29.7165i 0.948775i
\(982\) 0 0
\(983\) 10.0513 + 17.4094i 0.320588 + 0.555274i 0.980609 0.195972i \(-0.0627862\pi\)
−0.660022 + 0.751247i \(0.729453\pi\)
\(984\) 0 0
\(985\) 1.65064 2.85900i 0.0525939 0.0910953i
\(986\) 0 0
\(987\) 49.7983 8.54890i 1.58510 0.272115i
\(988\) 0 0
\(989\) −21.0792 + 36.5102i −0.670279 + 1.16096i
\(990\) 0 0
\(991\) −21.5973 + 12.4692i −0.686060 + 0.396097i −0.802134 0.597144i \(-0.796302\pi\)
0.116074 + 0.993241i \(0.462969\pi\)
\(992\) 0 0
\(993\) −74.6517 −2.36900
\(994\) 0 0
\(995\) 7.25163i 0.229892i
\(996\) 0 0
\(997\) −17.4557 + 10.0781i −0.552828 + 0.319175i −0.750262 0.661141i \(-0.770072\pi\)
0.197434 + 0.980316i \(0.436739\pi\)
\(998\) 0 0
\(999\) 13.7288 + 7.92631i 0.434359 + 0.250777i
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 1148.2.r.a.737.3 yes 56
7.4 even 3 inner 1148.2.r.a.81.26 yes 56
41.40 even 2 inner 1148.2.r.a.737.26 yes 56
287.81 even 6 inner 1148.2.r.a.81.3 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.r.a.81.3 56 287.81 even 6 inner
1148.2.r.a.81.26 yes 56 7.4 even 3 inner
1148.2.r.a.737.3 yes 56 1.1 even 1 trivial
1148.2.r.a.737.26 yes 56 41.40 even 2 inner