Properties

Label 232.2.q.a.209.3
Level $232$
Weight $2$
Character 232.209
Analytic conductor $1.853$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [232,2,Mod(9,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 232.q (of order \(14\), degree \(6\), 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: \(1.85252932689\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 209.3
Character \(\chi\) \(=\) 232.209
Dual form 232.2.q.a.121.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.18604 - 0.270705i) q^{3} +(1.84749 - 2.31668i) q^{5} +(-0.161250 + 0.706484i) q^{7} +(-1.36950 - 0.659517i) q^{9} +O(q^{10})\) \(q+(-1.18604 - 0.270705i) q^{3} +(1.84749 - 2.31668i) q^{5} +(-0.161250 + 0.706484i) q^{7} +(-1.36950 - 0.659517i) q^{9} +(-1.22355 - 2.54073i) q^{11} +(3.17169 - 1.52740i) q^{13} +(-2.81833 + 2.24754i) q^{15} -4.89842i q^{17} +(3.80988 - 0.869580i) q^{19} +(0.382498 - 0.794266i) q^{21} +(-1.27957 - 1.60454i) q^{23} +(-0.841172 - 3.68541i) q^{25} +(4.29913 + 3.42844i) q^{27} +(1.02298 + 5.28711i) q^{29} +(-1.13218 - 0.902887i) q^{31} +(0.763388 + 3.34462i) q^{33} +(1.33879 + 1.67878i) q^{35} +(-1.92038 + 3.98772i) q^{37} +(-4.17522 + 0.952967i) q^{39} +6.63260i q^{41} +(-9.57251 + 7.63382i) q^{43} +(-4.05802 + 1.95424i) q^{45} +(-1.65877 - 3.44448i) q^{47} +(5.83366 + 2.80934i) q^{49} +(-1.32603 + 5.80971i) q^{51} +(2.21277 - 2.77473i) q^{53} +(-8.14653 - 1.85939i) q^{55} -4.75406 q^{57} +11.0729 q^{59} +(2.36693 + 0.540237i) q^{61} +(0.686771 - 0.861184i) q^{63} +(2.32115 - 10.1696i) q^{65} +(9.82946 + 4.73362i) q^{67} +(1.08327 + 2.24943i) q^{69} +(5.16682 - 2.48821i) q^{71} +(-12.5229 + 9.98665i) q^{73} +4.59875i q^{75} +(1.99228 - 0.454725i) q^{77} +(2.83014 - 5.87684i) q^{79} +(-1.32766 - 1.66484i) q^{81} +(-2.25710 - 9.88900i) q^{83} +(-11.3480 - 9.04976i) q^{85} +(0.217955 - 6.54764i) q^{87} +(5.08526 + 4.05536i) q^{89} +(0.567651 + 2.48704i) q^{91} +(1.09840 + 1.37735i) q^{93} +(5.02417 - 10.4328i) q^{95} +(2.88891 - 0.659375i) q^{97} +4.28648i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 2 q^{5} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 2 q^{5} - 4 q^{7} + 6 q^{9} + 10 q^{13} + 14 q^{15} + 14 q^{21} + 4 q^{23} - 48 q^{25} - 4 q^{29} + 10 q^{33} + 8 q^{35} - 38 q^{45} - 14 q^{47} - 18 q^{49} - 56 q^{51} - 48 q^{53} - 28 q^{55} - 12 q^{57} - 128 q^{59} - 28 q^{61} + 42 q^{63} - 28 q^{65} - 4 q^{67} + 28 q^{69} - 14 q^{71} - 28 q^{73} + 14 q^{77} - 32 q^{81} + 80 q^{83} - 112 q^{87} + 42 q^{89} - 28 q^{91} + 6 q^{93} + 70 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/232\mathbb{Z}\right)^\times\).

\(n\) \(89\) \(117\) \(175\)
\(\chi(n)\) \(e\left(\frac{3}{14}\right)\) \(1\) \(1\)

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
\(3\) −1.18604 0.270705i −0.684759 0.156292i −0.134032 0.990977i \(-0.542792\pi\)
−0.550727 + 0.834685i \(0.685650\pi\)
\(4\) 0 0
\(5\) 1.84749 2.31668i 0.826221 1.03605i −0.172476 0.985014i \(-0.555177\pi\)
0.998697 0.0510348i \(-0.0162519\pi\)
\(6\) 0 0
\(7\) −0.161250 + 0.706484i −0.0609469 + 0.267026i −0.996216 0.0869115i \(-0.972300\pi\)
0.935269 + 0.353937i \(0.115157\pi\)
\(8\) 0 0
\(9\) −1.36950 0.659517i −0.456501 0.219839i
\(10\) 0 0
\(11\) −1.22355 2.54073i −0.368914 0.766058i 0.631039 0.775751i \(-0.282629\pi\)
−0.999953 + 0.00969294i \(0.996915\pi\)
\(12\) 0 0
\(13\) 3.17169 1.52740i 0.879668 0.423626i 0.0611648 0.998128i \(-0.480518\pi\)
0.818503 + 0.574502i \(0.194804\pi\)
\(14\) 0 0
\(15\) −2.81833 + 2.24754i −0.727689 + 0.580312i
\(16\) 0 0
\(17\) 4.89842i 1.18804i −0.804450 0.594020i \(-0.797540\pi\)
0.804450 0.594020i \(-0.202460\pi\)
\(18\) 0 0
\(19\) 3.80988 0.869580i 0.874046 0.199495i 0.238111 0.971238i \(-0.423472\pi\)
0.635935 + 0.771743i \(0.280615\pi\)
\(20\) 0 0
\(21\) 0.382498 0.794266i 0.0834679 0.173323i
\(22\) 0 0
\(23\) −1.27957 1.60454i −0.266810 0.334569i 0.630320 0.776335i \(-0.282923\pi\)
−0.897130 + 0.441766i \(0.854352\pi\)
\(24\) 0 0
\(25\) −0.841172 3.68541i −0.168234 0.737083i
\(26\) 0 0
\(27\) 4.29913 + 3.42844i 0.827368 + 0.659804i
\(28\) 0 0
\(29\) 1.02298 + 5.28711i 0.189963 + 0.981791i
\(30\) 0 0
\(31\) −1.13218 0.902887i −0.203346 0.162163i 0.516522 0.856274i \(-0.327226\pi\)
−0.719868 + 0.694111i \(0.755798\pi\)
\(32\) 0 0
\(33\) 0.763388 + 3.34462i 0.132889 + 0.582224i
\(34\) 0 0
\(35\) 1.33879 + 1.67878i 0.226296 + 0.283766i
\(36\) 0 0
\(37\) −1.92038 + 3.98772i −0.315709 + 0.655577i −0.997080 0.0763612i \(-0.975670\pi\)
0.681371 + 0.731938i \(0.261384\pi\)
\(38\) 0 0
\(39\) −4.17522 + 0.952967i −0.668570 + 0.152597i
\(40\) 0 0
\(41\) 6.63260i 1.03584i 0.855429 + 0.517919i \(0.173293\pi\)
−0.855429 + 0.517919i \(0.826707\pi\)
\(42\) 0 0
\(43\) −9.57251 + 7.63382i −1.45979 + 1.16415i −0.506446 + 0.862272i \(0.669041\pi\)
−0.953348 + 0.301875i \(0.902387\pi\)
\(44\) 0 0
\(45\) −4.05802 + 1.95424i −0.604934 + 0.291321i
\(46\) 0 0
\(47\) −1.65877 3.44448i −0.241957 0.502429i 0.744259 0.667891i \(-0.232803\pi\)
−0.986216 + 0.165462i \(0.947088\pi\)
\(48\) 0 0
\(49\) 5.83366 + 2.80934i 0.833381 + 0.401335i
\(50\) 0 0
\(51\) −1.32603 + 5.80971i −0.185681 + 0.813522i
\(52\) 0 0
\(53\) 2.21277 2.77473i 0.303948 0.381138i −0.606277 0.795254i \(-0.707338\pi\)
0.910224 + 0.414116i \(0.135909\pi\)
\(54\) 0 0
\(55\) −8.14653 1.85939i −1.09848 0.250721i
\(56\) 0 0
\(57\) −4.75406 −0.629691
\(58\) 0 0
\(59\) 11.0729 1.44157 0.720784 0.693160i \(-0.243782\pi\)
0.720784 + 0.693160i \(0.243782\pi\)
\(60\) 0 0
\(61\) 2.36693 + 0.540237i 0.303055 + 0.0691702i 0.371344 0.928495i \(-0.378897\pi\)
−0.0682895 + 0.997666i \(0.521754\pi\)
\(62\) 0 0
\(63\) 0.686771 0.861184i 0.0865250 0.108499i
\(64\) 0 0
\(65\) 2.32115 10.1696i 0.287903 1.26139i
\(66\) 0 0
\(67\) 9.82946 + 4.73362i 1.20086 + 0.578304i 0.923921 0.382584i \(-0.124966\pi\)
0.276939 + 0.960888i \(0.410680\pi\)
\(68\) 0 0
\(69\) 1.08327 + 2.24943i 0.130410 + 0.270799i
\(70\) 0 0
\(71\) 5.16682 2.48821i 0.613189 0.295296i −0.101395 0.994846i \(-0.532330\pi\)
0.714584 + 0.699550i \(0.246616\pi\)
\(72\) 0 0
\(73\) −12.5229 + 9.98665i −1.46569 + 1.16885i −0.515617 + 0.856819i \(0.672437\pi\)
−0.950072 + 0.312029i \(0.898991\pi\)
\(74\) 0 0
\(75\) 4.59875i 0.531018i
\(76\) 0 0
\(77\) 1.99228 0.454725i 0.227042 0.0518208i
\(78\) 0 0
\(79\) 2.83014 5.87684i 0.318415 0.661196i −0.678915 0.734217i \(-0.737550\pi\)
0.997330 + 0.0730206i \(0.0232639\pi\)
\(80\) 0 0
\(81\) −1.32766 1.66484i −0.147518 0.184982i
\(82\) 0 0
\(83\) −2.25710 9.88900i −0.247749 1.08546i −0.933769 0.357876i \(-0.883501\pi\)
0.686020 0.727582i \(-0.259356\pi\)
\(84\) 0 0
\(85\) −11.3480 9.04976i −1.23087 0.981584i
\(86\) 0 0
\(87\) 0.217955 6.54764i 0.0233672 0.701980i
\(88\) 0 0
\(89\) 5.08526 + 4.05536i 0.539036 + 0.429867i 0.854790 0.518974i \(-0.173686\pi\)
−0.315754 + 0.948841i \(0.602257\pi\)
\(90\) 0 0
\(91\) 0.567651 + 2.48704i 0.0595060 + 0.260713i
\(92\) 0 0
\(93\) 1.09840 + 1.37735i 0.113899 + 0.142824i
\(94\) 0 0
\(95\) 5.02417 10.4328i 0.515468 1.07038i
\(96\) 0 0
\(97\) 2.88891 0.659375i 0.293324 0.0669494i −0.0733265 0.997308i \(-0.523362\pi\)
0.366651 + 0.930359i \(0.380504\pi\)
\(98\) 0 0
\(99\) 4.28648i 0.430808i
\(100\) 0 0
\(101\) −3.45601 + 2.75608i −0.343886 + 0.274240i −0.780168 0.625570i \(-0.784866\pi\)
0.436282 + 0.899810i \(0.356295\pi\)
\(102\) 0 0
\(103\) −13.6113 + 6.55485i −1.34116 + 0.645869i −0.960353 0.278786i \(-0.910068\pi\)
−0.380807 + 0.924655i \(0.624354\pi\)
\(104\) 0 0
\(105\) −1.13340 2.35352i −0.110608 0.229680i
\(106\) 0 0
\(107\) 13.5245 + 6.51305i 1.30746 + 0.629641i 0.952300 0.305164i \(-0.0987112\pi\)
0.355163 + 0.934805i \(0.384426\pi\)
\(108\) 0 0
\(109\) −0.0382839 + 0.167733i −0.00366693 + 0.0160659i −0.976728 0.214480i \(-0.931194\pi\)
0.973061 + 0.230546i \(0.0740513\pi\)
\(110\) 0 0
\(111\) 3.35715 4.20973i 0.318646 0.399570i
\(112\) 0 0
\(113\) −10.7922 2.46324i −1.01524 0.231722i −0.317643 0.948211i \(-0.602891\pi\)
−0.697599 + 0.716488i \(0.745748\pi\)
\(114\) 0 0
\(115\) −6.08118 −0.567073
\(116\) 0 0
\(117\) −5.35098 −0.494699
\(118\) 0 0
\(119\) 3.46065 + 0.789872i 0.317238 + 0.0724074i
\(120\) 0 0
\(121\) 1.90016 2.38273i 0.172742 0.216612i
\(122\) 0 0
\(123\) 1.79548 7.86652i 0.161893 0.709300i
\(124\) 0 0
\(125\) 3.25652 + 1.56826i 0.291272 + 0.140269i
\(126\) 0 0
\(127\) −5.78650 12.0158i −0.513469 1.06623i −0.983050 0.183339i \(-0.941309\pi\)
0.469581 0.882889i \(-0.344405\pi\)
\(128\) 0 0
\(129\) 13.4199 6.46267i 1.18155 0.569006i
\(130\) 0 0
\(131\) 13.7547 10.9690i 1.20175 0.958366i 0.201974 0.979391i \(-0.435264\pi\)
0.999779 + 0.0210245i \(0.00669279\pi\)
\(132\) 0 0
\(133\) 2.83184i 0.245552i
\(134\) 0 0
\(135\) 15.8852 3.62569i 1.36718 0.312050i
\(136\) 0 0
\(137\) −2.72107 + 5.65036i −0.232477 + 0.482743i −0.984273 0.176652i \(-0.943473\pi\)
0.751797 + 0.659395i \(0.229187\pi\)
\(138\) 0 0
\(139\) 5.72668 + 7.18103i 0.485731 + 0.609087i 0.962944 0.269700i \(-0.0869246\pi\)
−0.477214 + 0.878787i \(0.658353\pi\)
\(140\) 0 0
\(141\) 1.03493 + 4.53432i 0.0871567 + 0.381859i
\(142\) 0 0
\(143\) −7.76144 6.18954i −0.649044 0.517596i
\(144\) 0 0
\(145\) 14.1385 + 7.39795i 1.17413 + 0.614366i
\(146\) 0 0
\(147\) −6.15844 4.91119i −0.507940 0.405068i
\(148\) 0 0
\(149\) −5.27767 23.1230i −0.432363 1.89431i −0.447232 0.894418i \(-0.647590\pi\)
0.0148685 0.999889i \(-0.495267\pi\)
\(150\) 0 0
\(151\) −5.05791 6.34241i −0.411607 0.516138i 0.532208 0.846614i \(-0.321362\pi\)
−0.943815 + 0.330475i \(0.892791\pi\)
\(152\) 0 0
\(153\) −3.23059 + 6.70839i −0.261178 + 0.542341i
\(154\) 0 0
\(155\) −4.18339 + 0.954832i −0.336018 + 0.0766939i
\(156\) 0 0
\(157\) 1.65786i 0.132312i −0.997809 0.0661559i \(-0.978927\pi\)
0.997809 0.0661559i \(-0.0210735\pi\)
\(158\) 0 0
\(159\) −3.37557 + 2.69192i −0.267700 + 0.213483i
\(160\) 0 0
\(161\) 1.33991 0.645267i 0.105600 0.0508542i
\(162\) 0 0
\(163\) 0.522746 + 1.08549i 0.0409446 + 0.0850224i 0.920438 0.390889i \(-0.127832\pi\)
−0.879493 + 0.475912i \(0.842118\pi\)
\(164\) 0 0
\(165\) 9.15875 + 4.41062i 0.713008 + 0.343366i
\(166\) 0 0
\(167\) −3.89624 + 17.0706i −0.301500 + 1.32096i 0.566363 + 0.824156i \(0.308350\pi\)
−0.867863 + 0.496803i \(0.834507\pi\)
\(168\) 0 0
\(169\) −0.378725 + 0.474906i −0.0291327 + 0.0365312i
\(170\) 0 0
\(171\) −5.79114 1.32179i −0.442859 0.101080i
\(172\) 0 0
\(173\) −23.7453 −1.80532 −0.902661 0.430351i \(-0.858390\pi\)
−0.902661 + 0.430351i \(0.858390\pi\)
\(174\) 0 0
\(175\) 2.73933 0.207074
\(176\) 0 0
\(177\) −13.1329 2.99749i −0.987127 0.225305i
\(178\) 0 0
\(179\) 14.5274 18.2168i 1.08583 1.36159i 0.158495 0.987360i \(-0.449336\pi\)
0.927336 0.374229i \(-0.122093\pi\)
\(180\) 0 0
\(181\) 0.923443 4.04587i 0.0686390 0.300727i −0.928943 0.370222i \(-0.879282\pi\)
0.997582 + 0.0694949i \(0.0221388\pi\)
\(182\) 0 0
\(183\) −2.66103 1.28148i −0.196709 0.0947299i
\(184\) 0 0
\(185\) 5.69037 + 11.8162i 0.418364 + 0.868742i
\(186\) 0 0
\(187\) −12.4455 + 5.99346i −0.910109 + 0.438285i
\(188\) 0 0
\(189\) −3.11538 + 2.48443i −0.226610 + 0.180716i
\(190\) 0 0
\(191\) 15.2057i 1.10025i 0.835084 + 0.550123i \(0.185419\pi\)
−0.835084 + 0.550123i \(0.814581\pi\)
\(192\) 0 0
\(193\) 18.1816 4.14983i 1.30874 0.298711i 0.489419 0.872049i \(-0.337209\pi\)
0.819319 + 0.573337i \(0.194352\pi\)
\(194\) 0 0
\(195\) −5.50595 + 11.4332i −0.394289 + 0.818750i
\(196\) 0 0
\(197\) 4.33808 + 5.43978i 0.309075 + 0.387568i 0.911973 0.410250i \(-0.134559\pi\)
−0.602898 + 0.797819i \(0.705987\pi\)
\(198\) 0 0
\(199\) 0.623047 + 2.72975i 0.0441666 + 0.193507i 0.992198 0.124669i \(-0.0397869\pi\)
−0.948032 + 0.318176i \(0.896930\pi\)
\(200\) 0 0
\(201\) −10.3767 8.27514i −0.731916 0.583683i
\(202\) 0 0
\(203\) −3.90021 0.129829i −0.273741 0.00911218i
\(204\) 0 0
\(205\) 15.3656 + 12.2536i 1.07318 + 0.855832i
\(206\) 0 0
\(207\) 0.694161 + 3.04132i 0.0482475 + 0.211386i
\(208\) 0 0
\(209\) −6.87094 8.61589i −0.475273 0.595974i
\(210\) 0 0
\(211\) −11.0820 + 23.0121i −0.762919 + 1.58422i 0.0478528 + 0.998854i \(0.484762\pi\)
−0.810772 + 0.585363i \(0.800952\pi\)
\(212\) 0 0
\(213\) −6.80162 + 1.55243i −0.466039 + 0.106370i
\(214\) 0 0
\(215\) 36.2798i 2.47426i
\(216\) 0 0
\(217\) 0.820441 0.654280i 0.0556951 0.0444154i
\(218\) 0 0
\(219\) 17.5560 8.45454i 1.18633 0.571305i
\(220\) 0 0
\(221\) −7.48187 15.5363i −0.503285 1.04508i
\(222\) 0 0
\(223\) −6.89915 3.32246i −0.462002 0.222488i 0.188380 0.982096i \(-0.439676\pi\)
−0.650381 + 0.759608i \(0.725391\pi\)
\(224\) 0 0
\(225\) −1.27861 + 5.60195i −0.0852405 + 0.373463i
\(226\) 0 0
\(227\) 2.08040 2.60874i 0.138081 0.173148i −0.707983 0.706230i \(-0.750395\pi\)
0.846064 + 0.533081i \(0.178966\pi\)
\(228\) 0 0
\(229\) 25.7547 + 5.87834i 1.70192 + 0.388451i 0.959549 0.281541i \(-0.0908457\pi\)
0.742368 + 0.669993i \(0.233703\pi\)
\(230\) 0 0
\(231\) −2.48602 −0.163568
\(232\) 0 0
\(233\) 11.1059 0.727574 0.363787 0.931482i \(-0.381484\pi\)
0.363787 + 0.931482i \(0.381484\pi\)
\(234\) 0 0
\(235\) −11.0443 2.52079i −0.720450 0.164438i
\(236\) 0 0
\(237\) −4.94754 + 6.20402i −0.321377 + 0.402995i
\(238\) 0 0
\(239\) −5.18885 + 22.7339i −0.335639 + 1.47053i 0.472391 + 0.881389i \(0.343391\pi\)
−0.808030 + 0.589142i \(0.799466\pi\)
\(240\) 0 0
\(241\) 1.65248 + 0.795790i 0.106445 + 0.0512614i 0.486349 0.873765i \(-0.338328\pi\)
−0.379903 + 0.925026i \(0.624043\pi\)
\(242\) 0 0
\(243\) −6.03354 12.5288i −0.387052 0.803721i
\(244\) 0 0
\(245\) 17.2860 8.32448i 1.10436 0.531831i
\(246\) 0 0
\(247\) 10.7555 8.57726i 0.684359 0.545758i
\(248\) 0 0
\(249\) 12.3397i 0.781999i
\(250\) 0 0
\(251\) −9.33943 + 2.13166i −0.589500 + 0.134549i −0.506857 0.862030i \(-0.669193\pi\)
−0.0826421 + 0.996579i \(0.526336\pi\)
\(252\) 0 0
\(253\) −2.51107 + 5.21428i −0.157869 + 0.327819i
\(254\) 0 0
\(255\) 11.0094 + 13.8053i 0.689435 + 0.864524i
\(256\) 0 0
\(257\) −0.806692 3.53435i −0.0503200 0.220467i 0.943516 0.331327i \(-0.107496\pi\)
−0.993836 + 0.110861i \(0.964639\pi\)
\(258\) 0 0
\(259\) −2.50760 1.99974i −0.155815 0.124258i
\(260\) 0 0
\(261\) 2.08597 7.91538i 0.129118 0.489950i
\(262\) 0 0
\(263\) −9.44179 7.52958i −0.582206 0.464294i 0.287557 0.957763i \(-0.407157\pi\)
−0.869763 + 0.493470i \(0.835728\pi\)
\(264\) 0 0
\(265\) −2.34008 10.2525i −0.143750 0.629809i
\(266\) 0 0
\(267\) −4.93350 6.18641i −0.301925 0.378602i
\(268\) 0 0
\(269\) −8.54750 + 17.7491i −0.521151 + 1.08218i 0.459818 + 0.888013i \(0.347915\pi\)
−0.980968 + 0.194167i \(0.937800\pi\)
\(270\) 0 0
\(271\) −30.9096 + 7.05491i −1.87762 + 0.428556i −0.998817 0.0486215i \(-0.984517\pi\)
−0.878807 + 0.477177i \(0.841660\pi\)
\(272\) 0 0
\(273\) 3.10339i 0.187826i
\(274\) 0 0
\(275\) −8.33442 + 6.64648i −0.502584 + 0.400798i
\(276\) 0 0
\(277\) 17.7140 8.53060i 1.06433 0.512554i 0.182055 0.983288i \(-0.441725\pi\)
0.882275 + 0.470734i \(0.156011\pi\)
\(278\) 0 0
\(279\) 0.955059 + 1.98320i 0.0571779 + 0.118731i
\(280\) 0 0
\(281\) −6.38898 3.07677i −0.381135 0.183545i 0.233500 0.972357i \(-0.424982\pi\)
−0.614635 + 0.788812i \(0.710697\pi\)
\(282\) 0 0
\(283\) −0.804064 + 3.52283i −0.0477966 + 0.209411i −0.993187 0.116529i \(-0.962823\pi\)
0.945391 + 0.325939i \(0.105680\pi\)
\(284\) 0 0
\(285\) −8.78306 + 11.0136i −0.520264 + 0.652390i
\(286\) 0 0
\(287\) −4.68583 1.06951i −0.276596 0.0631312i
\(288\) 0 0
\(289\) −6.99449 −0.411441
\(290\) 0 0
\(291\) −3.60485 −0.211320
\(292\) 0 0
\(293\) 21.0660 + 4.80817i 1.23069 + 0.280897i 0.787941 0.615750i \(-0.211147\pi\)
0.442747 + 0.896647i \(0.354004\pi\)
\(294\) 0 0
\(295\) 20.4570 25.6523i 1.19105 1.49353i
\(296\) 0 0
\(297\) 3.45054 15.1178i 0.200221 0.877224i
\(298\) 0 0
\(299\) −6.50919 3.13466i −0.376436 0.181282i
\(300\) 0 0
\(301\) −3.84960 7.99378i −0.221887 0.460754i
\(302\) 0 0
\(303\) 4.84505 2.33325i 0.278341 0.134042i
\(304\) 0 0
\(305\) 5.62443 4.48533i 0.322054 0.256829i
\(306\) 0 0
\(307\) 22.3962i 1.27822i 0.769116 + 0.639109i \(0.220697\pi\)
−0.769116 + 0.639109i \(0.779303\pi\)
\(308\) 0 0
\(309\) 17.9179 4.08965i 1.01932 0.232652i
\(310\) 0 0
\(311\) 10.0436 20.8558i 0.569521 1.18262i −0.395016 0.918674i \(-0.629261\pi\)
0.964537 0.263948i \(-0.0850246\pi\)
\(312\) 0 0
\(313\) 10.1241 + 12.6952i 0.572246 + 0.717574i 0.980769 0.195175i \(-0.0625273\pi\)
−0.408523 + 0.912748i \(0.633956\pi\)
\(314\) 0 0
\(315\) −0.726282 3.18205i −0.0409214 0.179288i
\(316\) 0 0
\(317\) −19.9730 15.9279i −1.12180 0.894602i −0.126546 0.991961i \(-0.540389\pi\)
−0.995249 + 0.0973583i \(0.968961\pi\)
\(318\) 0 0
\(319\) 12.1814 9.06816i 0.682029 0.507719i
\(320\) 0 0
\(321\) −14.2774 11.3859i −0.796889 0.635498i
\(322\) 0 0
\(323\) −4.25956 18.6624i −0.237009 1.03840i
\(324\) 0 0
\(325\) −8.29705 10.4042i −0.460238 0.577120i
\(326\) 0 0
\(327\) 0.0908124 0.188574i 0.00502193 0.0104282i
\(328\) 0 0
\(329\) 2.70095 0.616473i 0.148908 0.0339873i
\(330\) 0 0
\(331\) 10.1111i 0.555755i −0.960617 0.277877i \(-0.910369\pi\)
0.960617 0.277877i \(-0.0896309\pi\)
\(332\) 0 0
\(333\) 5.25994 4.19466i 0.288243 0.229866i
\(334\) 0 0
\(335\) 29.1261 14.0264i 1.59133 0.766342i
\(336\) 0 0
\(337\) −8.74139 18.1517i −0.476174 0.988785i −0.991294 0.131667i \(-0.957967\pi\)
0.515120 0.857118i \(-0.327747\pi\)
\(338\) 0 0
\(339\) 12.1331 + 5.84300i 0.658980 + 0.317348i
\(340\) 0 0
\(341\) −0.908706 + 3.98130i −0.0492092 + 0.215600i
\(342\) 0 0
\(343\) −6.08813 + 7.63428i −0.328728 + 0.412212i
\(344\) 0 0
\(345\) 7.21252 + 1.64621i 0.388309 + 0.0886290i
\(346\) 0 0
\(347\) −17.8444 −0.957938 −0.478969 0.877832i \(-0.658989\pi\)
−0.478969 + 0.877832i \(0.658989\pi\)
\(348\) 0 0
\(349\) −7.19398 −0.385085 −0.192543 0.981289i \(-0.561673\pi\)
−0.192543 + 0.981289i \(0.561673\pi\)
\(350\) 0 0
\(351\) 18.8721 + 4.30744i 1.00732 + 0.229914i
\(352\) 0 0
\(353\) −12.0858 + 15.1551i −0.643260 + 0.806623i −0.991407 0.130817i \(-0.958240\pi\)
0.348146 + 0.937440i \(0.386811\pi\)
\(354\) 0 0
\(355\) 3.78126 16.5668i 0.200688 0.879274i
\(356\) 0 0
\(357\) −3.89064 1.87364i −0.205915 0.0991633i
\(358\) 0 0
\(359\) 1.44636 + 3.00339i 0.0763358 + 0.158513i 0.935644 0.352945i \(-0.114820\pi\)
−0.859308 + 0.511458i \(0.829106\pi\)
\(360\) 0 0
\(361\) −3.35941 + 1.61780i −0.176811 + 0.0851476i
\(362\) 0 0
\(363\) −2.89868 + 2.31162i −0.152141 + 0.121329i
\(364\) 0 0
\(365\) 47.4616i 2.48425i
\(366\) 0 0
\(367\) 32.9680 7.52474i 1.72092 0.392788i 0.755840 0.654756i \(-0.227229\pi\)
0.965077 + 0.261968i \(0.0843714\pi\)
\(368\) 0 0
\(369\) 4.37432 9.08336i 0.227718 0.472861i
\(370\) 0 0
\(371\) 1.60349 + 2.01071i 0.0832491 + 0.104391i
\(372\) 0 0
\(373\) −8.33564 36.5208i −0.431603 1.89098i −0.453577 0.891217i \(-0.649852\pi\)
0.0219744 0.999759i \(-0.493005\pi\)
\(374\) 0 0
\(375\) −3.43782 2.74157i −0.177528 0.141574i
\(376\) 0 0
\(377\) 11.3201 + 15.2066i 0.583016 + 0.783177i
\(378\) 0 0
\(379\) 9.74439 + 7.77089i 0.500535 + 0.399164i 0.840951 0.541111i \(-0.181996\pi\)
−0.340415 + 0.940275i \(0.610568\pi\)
\(380\) 0 0
\(381\) 3.61027 + 15.8176i 0.184960 + 0.810361i
\(382\) 0 0
\(383\) −5.70818 7.15783i −0.291674 0.365748i 0.614306 0.789068i \(-0.289436\pi\)
−0.905980 + 0.423320i \(0.860865\pi\)
\(384\) 0 0
\(385\) 2.62726 5.45557i 0.133898 0.278042i
\(386\) 0 0
\(387\) 18.1442 4.14130i 0.922322 0.210514i
\(388\) 0 0
\(389\) 14.6428i 0.742417i −0.928549 0.371209i \(-0.878943\pi\)
0.928549 0.371209i \(-0.121057\pi\)
\(390\) 0 0
\(391\) −7.85968 + 6.26789i −0.397481 + 0.316981i
\(392\) 0 0
\(393\) −19.2830 + 9.28618i −0.972697 + 0.468426i
\(394\) 0 0
\(395\) −8.38609 17.4139i −0.421950 0.876188i
\(396\) 0 0
\(397\) −1.83610 0.884217i −0.0921510 0.0443776i 0.387240 0.921979i \(-0.373428\pi\)
−0.479392 + 0.877601i \(0.659143\pi\)
\(398\) 0 0
\(399\) 0.766594 3.35867i 0.0383777 0.168144i
\(400\) 0 0
\(401\) −11.6940 + 14.6638i −0.583972 + 0.732277i −0.982785 0.184755i \(-0.940851\pi\)
0.398813 + 0.917032i \(0.369422\pi\)
\(402\) 0 0
\(403\) −4.97001 1.13437i −0.247574 0.0565071i
\(404\) 0 0
\(405\) −6.30972 −0.313533
\(406\) 0 0
\(407\) 12.4814 0.618680
\(408\) 0 0
\(409\) 11.6171 + 2.65153i 0.574430 + 0.131110i 0.499861 0.866106i \(-0.333385\pi\)
0.0745695 + 0.997216i \(0.476242\pi\)
\(410\) 0 0
\(411\) 4.75687 5.96493i 0.234639 0.294228i
\(412\) 0 0
\(413\) −1.78551 + 7.82282i −0.0878591 + 0.384936i
\(414\) 0 0
\(415\) −27.0796 13.0408i −1.32928 0.640149i
\(416\) 0 0
\(417\) −4.84812 10.0672i −0.237413 0.492994i
\(418\) 0 0
\(419\) −11.8835 + 5.72278i −0.580546 + 0.279576i −0.701021 0.713140i \(-0.747272\pi\)
0.120476 + 0.992716i \(0.461558\pi\)
\(420\) 0 0
\(421\) −12.1448 + 9.68512i −0.591899 + 0.472024i −0.873044 0.487642i \(-0.837857\pi\)
0.281145 + 0.959665i \(0.409286\pi\)
\(422\) 0 0
\(423\) 5.81121i 0.282551i
\(424\) 0 0
\(425\) −18.0527 + 4.12041i −0.875684 + 0.199869i
\(426\) 0 0
\(427\) −0.763337 + 1.58509i −0.0369405 + 0.0767077i
\(428\) 0 0
\(429\) 7.52982 + 9.44210i 0.363543 + 0.455869i
\(430\) 0 0
\(431\) 8.31072 + 36.4116i 0.400313 + 1.75389i 0.626131 + 0.779718i \(0.284637\pi\)
−0.225818 + 0.974169i \(0.572506\pi\)
\(432\) 0 0
\(433\) −18.1813 14.4991i −0.873739 0.696784i 0.0802018 0.996779i \(-0.474444\pi\)
−0.953941 + 0.299995i \(0.903015\pi\)
\(434\) 0 0
\(435\) −14.7661 12.6016i −0.707979 0.604201i
\(436\) 0 0
\(437\) −6.27029 5.00039i −0.299949 0.239201i
\(438\) 0 0
\(439\) 5.08084 + 22.2606i 0.242495 + 1.06244i 0.938737 + 0.344634i \(0.111997\pi\)
−0.696242 + 0.717807i \(0.745146\pi\)
\(440\) 0 0
\(441\) −6.13640 7.69481i −0.292210 0.366419i
\(442\) 0 0
\(443\) 0.245308 0.509388i 0.0116550 0.0242018i −0.895060 0.445945i \(-0.852868\pi\)
0.906715 + 0.421744i \(0.138582\pi\)
\(444\) 0 0
\(445\) 18.7899 4.28867i 0.890726 0.203302i
\(446\) 0 0
\(447\) 28.8534i 1.36472i
\(448\) 0 0
\(449\) 23.4023 18.6627i 1.10442 0.880746i 0.110836 0.993839i \(-0.464647\pi\)
0.993584 + 0.113093i \(0.0360757\pi\)
\(450\) 0 0
\(451\) 16.8516 8.11532i 0.793513 0.382136i
\(452\) 0 0
\(453\) 4.28194 + 8.89155i 0.201183 + 0.417761i
\(454\) 0 0
\(455\) 6.81040 + 3.27971i 0.319276 + 0.153755i
\(456\) 0 0
\(457\) −5.85881 + 25.6691i −0.274063 + 1.20075i 0.631105 + 0.775697i \(0.282602\pi\)
−0.905168 + 0.425053i \(0.860255\pi\)
\(458\) 0 0
\(459\) 16.7939 21.0589i 0.783874 0.982947i
\(460\) 0 0
\(461\) −4.06162 0.927038i −0.189168 0.0431765i 0.126887 0.991917i \(-0.459501\pi\)
−0.316055 + 0.948741i \(0.602359\pi\)
\(462\) 0 0
\(463\) 3.39217 0.157648 0.0788239 0.996889i \(-0.474884\pi\)
0.0788239 + 0.996889i \(0.474884\pi\)
\(464\) 0 0
\(465\) 5.22014 0.242078
\(466\) 0 0
\(467\) −30.1904 6.89076i −1.39704 0.318866i −0.543291 0.839544i \(-0.682822\pi\)
−0.853753 + 0.520678i \(0.825679\pi\)
\(468\) 0 0
\(469\) −4.92923 + 6.18106i −0.227611 + 0.285415i
\(470\) 0 0
\(471\) −0.448792 + 1.96629i −0.0206793 + 0.0906017i
\(472\) 0 0
\(473\) 31.1079 + 14.9808i 1.43034 + 0.688817i
\(474\) 0 0
\(475\) −6.40952 13.3095i −0.294089 0.610682i
\(476\) 0 0
\(477\) −4.86038 + 2.34063i −0.222541 + 0.107170i
\(478\) 0 0
\(479\) 7.50897 5.98820i 0.343094 0.273608i −0.436749 0.899583i \(-0.643870\pi\)
0.779843 + 0.625975i \(0.215299\pi\)
\(480\) 0 0
\(481\) 15.5810i 0.710433i
\(482\) 0 0
\(483\) −1.76386 + 0.402590i −0.0802585 + 0.0183185i
\(484\) 0 0
\(485\) 3.80966 7.91085i 0.172988 0.359213i
\(486\) 0 0
\(487\) −16.3438 20.4944i −0.740607 0.928692i 0.258698 0.965958i \(-0.416706\pi\)
−0.999305 + 0.0372663i \(0.988135\pi\)
\(488\) 0 0
\(489\) −0.326148 1.42895i −0.0147489 0.0646192i
\(490\) 0 0
\(491\) 11.7516 + 9.37162i 0.530344 + 0.422935i 0.851705 0.524022i \(-0.175569\pi\)
−0.321361 + 0.946957i \(0.604140\pi\)
\(492\) 0 0
\(493\) 25.8985 5.01099i 1.16641 0.225683i
\(494\) 0 0
\(495\) 9.93039 + 7.91922i 0.446338 + 0.355943i
\(496\) 0 0
\(497\) 0.924729 + 4.05150i 0.0414798 + 0.181735i
\(498\) 0 0
\(499\) −5.34886 6.70725i −0.239448 0.300258i 0.647558 0.762016i \(-0.275790\pi\)
−0.887006 + 0.461758i \(0.847219\pi\)
\(500\) 0 0
\(501\) 9.24218 19.1916i 0.412910 0.857417i
\(502\) 0 0
\(503\) −20.3227 + 4.63853i −0.906145 + 0.206822i −0.650105 0.759844i \(-0.725275\pi\)
−0.256040 + 0.966666i \(0.582418\pi\)
\(504\) 0 0
\(505\) 13.0983i 0.582866i
\(506\) 0 0
\(507\) 0.577741 0.460733i 0.0256584 0.0204619i
\(508\) 0 0
\(509\) −33.7769 + 16.2661i −1.49713 + 0.720982i −0.990024 0.140900i \(-0.955001\pi\)
−0.507110 + 0.861881i \(0.669286\pi\)
\(510\) 0 0
\(511\) −5.03609 10.4575i −0.222784 0.462615i
\(512\) 0 0
\(513\) 19.3605 + 9.32351i 0.854786 + 0.411643i
\(514\) 0 0
\(515\) −9.96122 + 43.6429i −0.438944 + 1.92314i
\(516\) 0 0
\(517\) −6.72189 + 8.42898i −0.295628 + 0.370706i
\(518\) 0 0
\(519\) 28.1628 + 6.42799i 1.23621 + 0.282157i
\(520\) 0 0
\(521\) −38.5983 −1.69102 −0.845512 0.533957i \(-0.820704\pi\)
−0.845512 + 0.533957i \(0.820704\pi\)
\(522\) 0 0
\(523\) −6.47374 −0.283077 −0.141538 0.989933i \(-0.545205\pi\)
−0.141538 + 0.989933i \(0.545205\pi\)
\(524\) 0 0
\(525\) −3.24894 0.741550i −0.141796 0.0323639i
\(526\) 0 0
\(527\) −4.42272 + 5.54591i −0.192657 + 0.241584i
\(528\) 0 0
\(529\) 4.18076 18.3171i 0.181772 0.796395i
\(530\) 0 0
\(531\) −15.1643 7.30277i −0.658077 0.316913i
\(532\) 0 0
\(533\) 10.1307 + 21.0366i 0.438808 + 0.911194i
\(534\) 0 0
\(535\) 40.0750 19.2991i 1.73259 0.834372i
\(536\) 0 0
\(537\) −22.1615 + 17.6732i −0.956338 + 0.762654i
\(538\) 0 0
\(539\) 18.2591i 0.786476i
\(540\) 0 0
\(541\) −28.7696 + 6.56646i −1.23690 + 0.282314i −0.790468 0.612503i \(-0.790163\pi\)
−0.446432 + 0.894818i \(0.647305\pi\)
\(542\) 0 0
\(543\) −2.19048 + 4.54857i −0.0940024 + 0.195198i
\(544\) 0 0
\(545\) 0.317853 + 0.398575i 0.0136153 + 0.0170731i
\(546\) 0 0
\(547\) 7.09123 + 31.0687i 0.303199 + 1.32840i 0.865267 + 0.501311i \(0.167149\pi\)
−0.562068 + 0.827091i \(0.689994\pi\)
\(548\) 0 0
\(549\) −2.88522 2.30089i −0.123138 0.0981995i
\(550\) 0 0
\(551\) 8.49499 + 19.2537i 0.361899 + 0.820234i
\(552\) 0 0
\(553\) 3.69553 + 2.94709i 0.157150 + 0.125323i
\(554\) 0 0
\(555\) −3.55029 15.5548i −0.150701 0.660266i
\(556\) 0 0
\(557\) −9.31590 11.6818i −0.394728 0.494973i 0.544263 0.838914i \(-0.316809\pi\)
−0.938991 + 0.343942i \(0.888238\pi\)
\(558\) 0 0
\(559\) −18.7011 + 38.8332i −0.790971 + 1.64247i
\(560\) 0 0
\(561\) 16.3834 3.73939i 0.691706 0.157877i
\(562\) 0 0
\(563\) 4.93691i 0.208066i −0.994574 0.104033i \(-0.966825\pi\)
0.994574 0.104033i \(-0.0331748\pi\)
\(564\) 0 0
\(565\) −25.6449 + 20.4511i −1.07889 + 0.860386i
\(566\) 0 0
\(567\) 1.39027 0.669517i 0.0583857 0.0281171i
\(568\) 0 0
\(569\) 0.879472 + 1.82624i 0.0368694 + 0.0765600i 0.918590 0.395212i \(-0.129329\pi\)
−0.881721 + 0.471772i \(0.843615\pi\)
\(570\) 0 0
\(571\) 40.4906 + 19.4992i 1.69448 + 0.816017i 0.994833 + 0.101526i \(0.0323726\pi\)
0.699645 + 0.714491i \(0.253342\pi\)
\(572\) 0 0
\(573\) 4.11627 18.0345i 0.171959 0.753404i
\(574\) 0 0
\(575\) −4.83704 + 6.06545i −0.201718 + 0.252947i
\(576\) 0 0
\(577\) 1.98664 + 0.453438i 0.0827050 + 0.0188769i 0.263673 0.964612i \(-0.415066\pi\)
−0.180968 + 0.983489i \(0.557923\pi\)
\(578\) 0 0
\(579\) −22.6874 −0.942857
\(580\) 0 0
\(581\) 7.35038 0.304945
\(582\) 0 0
\(583\) −9.75727 2.22703i −0.404105 0.0922343i
\(584\) 0 0
\(585\) −9.88587 + 12.3965i −0.408730 + 0.512532i
\(586\) 0 0
\(587\) 1.42398 6.23888i 0.0587741 0.257506i −0.937002 0.349325i \(-0.886411\pi\)
0.995776 + 0.0918188i \(0.0292681\pi\)
\(588\) 0 0
\(589\) −5.09862 2.45536i −0.210085 0.101172i
\(590\) 0 0
\(591\) −3.67255 7.62613i −0.151069 0.313697i
\(592\) 0 0
\(593\) −19.5222 + 9.40139i −0.801680 + 0.386069i −0.789418 0.613856i \(-0.789618\pi\)
−0.0122621 + 0.999925i \(0.503903\pi\)
\(594\) 0 0
\(595\) 8.22339 6.55793i 0.337126 0.268849i
\(596\) 0 0
\(597\) 3.40625i 0.139408i
\(598\) 0 0
\(599\) −6.55712 + 1.49662i −0.267917 + 0.0611502i −0.354369 0.935106i \(-0.615304\pi\)
0.0864521 + 0.996256i \(0.472447\pi\)
\(600\) 0 0
\(601\) 6.84283 14.2093i 0.279125 0.579609i −0.713525 0.700630i \(-0.752903\pi\)
0.992650 + 0.121021i \(0.0386168\pi\)
\(602\) 0 0
\(603\) −10.3396 12.9654i −0.421059 0.527992i
\(604\) 0 0
\(605\) −2.00948 8.80412i −0.0816971 0.357938i
\(606\) 0 0
\(607\) 37.9488 + 30.2632i 1.54029 + 1.22834i 0.877535 + 0.479512i \(0.159186\pi\)
0.662759 + 0.748832i \(0.269385\pi\)
\(608\) 0 0
\(609\) 4.59066 + 1.20979i 0.186023 + 0.0490232i
\(610\) 0 0
\(611\) −10.5222 8.39119i −0.425683 0.339471i
\(612\) 0 0
\(613\) 4.77215 + 20.9082i 0.192745 + 0.844473i 0.975122 + 0.221668i \(0.0711500\pi\)
−0.782377 + 0.622805i \(0.785993\pi\)
\(614\) 0 0
\(615\) −14.9070 18.6928i −0.601110 0.753768i
\(616\) 0 0
\(617\) 4.23374 8.79145i 0.170444 0.353930i −0.798197 0.602397i \(-0.794212\pi\)
0.968640 + 0.248467i \(0.0799267\pi\)
\(618\) 0 0
\(619\) 16.4871 3.76306i 0.662671 0.151250i 0.122060 0.992523i \(-0.461050\pi\)
0.540611 + 0.841272i \(0.318193\pi\)
\(620\) 0 0
\(621\) 11.2851i 0.452854i
\(622\) 0 0
\(623\) −3.68504 + 2.93872i −0.147638 + 0.117737i
\(624\) 0 0
\(625\) 26.6787 12.8478i 1.06715 0.513911i
\(626\) 0 0
\(627\) 5.81683 + 12.0788i 0.232302 + 0.482380i
\(628\) 0 0
\(629\) 19.5335 + 9.40685i 0.778852 + 0.375076i
\(630\) 0 0
\(631\) −2.13194 + 9.34066i −0.0848713 + 0.371846i −0.999471 0.0325143i \(-0.989649\pi\)
0.914600 + 0.404360i \(0.132506\pi\)
\(632\) 0 0
\(633\) 19.3732 24.2932i 0.770016 0.965570i
\(634\) 0 0
\(635\) −38.5272 8.79357i −1.52890 0.348962i
\(636\) 0 0
\(637\) 22.7936 0.903114
\(638\) 0 0
\(639\) −8.71699 −0.344839
\(640\) 0 0
\(641\) 9.79740 + 2.23619i 0.386974 + 0.0883243i 0.411582 0.911373i \(-0.364977\pi\)
−0.0246077 + 0.999697i \(0.507834\pi\)
\(642\) 0 0
\(643\) 29.7819 37.3453i 1.17448 1.47275i 0.324544 0.945871i \(-0.394789\pi\)
0.849938 0.526883i \(-0.176639\pi\)
\(644\) 0 0
\(645\) 9.82113 43.0292i 0.386707 1.69427i
\(646\) 0 0
\(647\) −14.0796 6.78036i −0.553525 0.266563i 0.136143 0.990689i \(-0.456529\pi\)
−0.689668 + 0.724126i \(0.742243\pi\)
\(648\) 0 0
\(649\) −13.5482 28.1332i −0.531815 1.10433i
\(650\) 0 0
\(651\) −1.15019 + 0.553903i −0.0450795 + 0.0217092i
\(652\) 0 0
\(653\) 3.04580 2.42894i 0.119191 0.0950520i −0.562080 0.827083i \(-0.689999\pi\)
0.681272 + 0.732031i \(0.261427\pi\)
\(654\) 0 0
\(655\) 52.1303i 2.03690i
\(656\) 0 0
\(657\) 23.7364 5.41769i 0.926047 0.211364i
\(658\) 0 0
\(659\) −6.96607 + 14.4652i −0.271360 + 0.563484i −0.991464 0.130382i \(-0.958380\pi\)
0.720104 + 0.693866i \(0.244094\pi\)
\(660\) 0 0
\(661\) −28.5993 35.8624i −1.11239 1.39489i −0.909510 0.415683i \(-0.863543\pi\)
−0.202876 0.979204i \(-0.565029\pi\)
\(662\) 0 0
\(663\) 4.66803 + 20.4520i 0.181291 + 0.794289i
\(664\) 0 0
\(665\) 6.56045 + 5.23178i 0.254403 + 0.202880i
\(666\) 0 0
\(667\) 7.17437 8.40666i 0.277793 0.325507i
\(668\) 0 0
\(669\) 7.28325 + 5.80820i 0.281587 + 0.224558i
\(670\) 0 0
\(671\) −1.52347 6.67474i −0.0588127 0.257675i
\(672\) 0 0
\(673\) 19.8371 + 24.8750i 0.764665 + 0.958860i 0.999915 0.0130749i \(-0.00416200\pi\)
−0.235249 + 0.971935i \(0.575591\pi\)
\(674\) 0 0
\(675\) 9.01892 18.7280i 0.347139 0.720841i
\(676\) 0 0
\(677\) −15.6226 + 3.56575i −0.600424 + 0.137043i −0.511920 0.859033i \(-0.671066\pi\)
−0.0885039 + 0.996076i \(0.528209\pi\)
\(678\) 0 0
\(679\) 2.14729i 0.0824055i
\(680\) 0 0
\(681\) −3.17363 + 2.53089i −0.121614 + 0.0969838i
\(682\) 0 0
\(683\) −11.3892 + 5.48473i −0.435794 + 0.209867i −0.638901 0.769289i \(-0.720611\pi\)
0.203107 + 0.979157i \(0.434896\pi\)
\(684\) 0 0
\(685\) 8.06290 + 16.7428i 0.308068 + 0.639709i
\(686\) 0 0
\(687\) −28.9547 13.9439i −1.10469 0.531991i
\(688\) 0 0
\(689\) 2.78009 12.1804i 0.105913 0.464035i
\(690\) 0 0
\(691\) 1.55072 1.94455i 0.0589923 0.0739740i −0.751460 0.659779i \(-0.770650\pi\)
0.810452 + 0.585805i \(0.199221\pi\)
\(692\) 0 0
\(693\) −3.02833 0.691197i −0.115037 0.0262564i
\(694\) 0 0
\(695\) 27.2161 1.03236
\(696\) 0 0
\(697\) 32.4893 1.23062
\(698\) 0 0
\(699\) −13.1721 3.00644i −0.498213 0.113714i
\(700\) 0 0
\(701\) 23.1761 29.0619i 0.875350 1.09765i −0.119146 0.992877i \(-0.538016\pi\)
0.994496 0.104777i \(-0.0334130\pi\)
\(702\) 0 0
\(703\) −3.84879 + 16.8627i −0.145160 + 0.635987i
\(704\) 0 0
\(705\) 12.4166 + 5.97950i 0.467635 + 0.225201i
\(706\) 0 0
\(707\) −1.38984 2.88604i −0.0522704 0.108541i
\(708\) 0 0
\(709\) −1.97970 + 0.953374i −0.0743493 + 0.0358047i −0.470689 0.882299i \(-0.655995\pi\)
0.396340 + 0.918104i \(0.370280\pi\)
\(710\) 0 0
\(711\) −7.75176 + 6.18182i −0.290714 + 0.231836i
\(712\) 0 0
\(713\) 2.97194i 0.111300i
\(714\) 0 0
\(715\) −28.6783 + 6.54564i −1.07251 + 0.244793i
\(716\) 0 0
\(717\) 12.3084 25.5586i 0.459664 0.954502i
\(718\) 0 0
\(719\) −12.7919 16.0406i −0.477059 0.598213i 0.483825 0.875165i \(-0.339247\pi\)
−0.960884 + 0.276952i \(0.910676\pi\)
\(720\) 0 0
\(721\) −2.43607 10.6731i −0.0907241 0.397488i
\(722\) 0 0
\(723\) −1.74447 1.39117i −0.0648777 0.0517382i
\(724\) 0 0
\(725\) 18.6247 8.21747i 0.691703 0.305189i
\(726\) 0 0
\(727\) −34.6619 27.6419i −1.28554 1.02518i −0.997719 0.0674968i \(-0.978499\pi\)
−0.287818 0.957685i \(-0.592930\pi\)
\(728\) 0 0
\(729\) 5.18591 + 22.7210i 0.192071 + 0.841517i
\(730\) 0 0
\(731\) 37.3936 + 46.8901i 1.38305 + 1.73429i
\(732\) 0 0
\(733\) −0.753825 + 1.56533i −0.0278431 + 0.0578169i −0.914427 0.404752i \(-0.867358\pi\)
0.886583 + 0.462569i \(0.153072\pi\)
\(734\) 0 0
\(735\) −22.7553 + 5.19374i −0.839341 + 0.191574i
\(736\) 0 0
\(737\) 30.7658i 1.13327i
\(738\) 0 0
\(739\) −8.56036 + 6.82666i −0.314898 + 0.251123i −0.768165 0.640252i \(-0.778830\pi\)
0.453267 + 0.891375i \(0.350258\pi\)
\(740\) 0 0
\(741\) −15.0784 + 7.26137i −0.553919 + 0.266753i
\(742\) 0 0
\(743\) −10.2649 21.3152i −0.376581 0.781979i −1.00000 9.08456e-5i \(-0.999971\pi\)
0.623419 0.781888i \(-0.285743\pi\)
\(744\) 0 0
\(745\) −63.3188 30.4927i −2.31982 1.11717i
\(746\) 0 0
\(747\) −3.43086 + 15.0316i −0.125529 + 0.549977i
\(748\) 0 0
\(749\) −6.78220 + 8.50461i −0.247816 + 0.310752i
\(750\) 0 0
\(751\) 3.40435 + 0.777021i 0.124227 + 0.0283539i 0.284182 0.958770i \(-0.408278\pi\)
−0.159956 + 0.987124i \(0.551135\pi\)
\(752\) 0 0
\(753\) 11.6540 0.424694
\(754\) 0 0
\(755\) −24.0377 −0.874822
\(756\) 0 0
\(757\) −4.29672 0.980698i −0.156167 0.0356441i 0.143723 0.989618i \(-0.454093\pi\)
−0.299890 + 0.953974i \(0.596950\pi\)
\(758\) 0 0
\(759\) 4.38975 5.50458i 0.159338 0.199803i
\(760\) 0 0
\(761\) 1.40630 6.16139i 0.0509782 0.223350i −0.943022 0.332731i \(-0.892030\pi\)
0.994000 + 0.109381i \(0.0348869\pi\)
\(762\) 0 0
\(763\) −0.112327 0.0540940i −0.00406652 0.00195833i
\(764\) 0 0
\(765\) 9.57269 + 19.8779i 0.346101 + 0.718687i
\(766\) 0 0
\(767\) 35.1198 16.9128i 1.26810 0.610685i
\(768\) 0 0
\(769\) 7.60924 6.06816i 0.274396 0.218824i −0.476617 0.879111i \(-0.658137\pi\)
0.751013 + 0.660288i \(0.229566\pi\)
\(770\) 0 0
\(771\) 4.41025i 0.158831i
\(772\) 0 0
\(773\) −10.0031 + 2.28313i −0.359785 + 0.0821186i −0.398593 0.917128i \(-0.630501\pi\)
0.0388074 + 0.999247i \(0.487644\pi\)
\(774\) 0 0
\(775\) −2.37515 + 4.93205i −0.0853179 + 0.177165i
\(776\) 0 0
\(777\) 2.43277 + 3.05059i 0.0872750 + 0.109439i
\(778\) 0 0
\(779\) 5.76758 + 25.2694i 0.206645 + 0.905371i
\(780\) 0 0
\(781\) −12.6437 10.0830i −0.452428 0.360800i
\(782\) 0 0
\(783\) −13.7286 + 26.2372i −0.490621 + 0.937641i
\(784\) 0 0
\(785\) −3.84073 3.06288i −0.137081 0.109319i
\(786\) 0 0
\(787\) 5.21319 + 22.8405i 0.185830 + 0.814176i 0.978784 + 0.204894i \(0.0656850\pi\)
−0.792954 + 0.609282i \(0.791458\pi\)
\(788\) 0 0
\(789\) 9.16002 + 11.4863i 0.326105 + 0.408923i
\(790\) 0 0
\(791\) 3.48048 7.22730i 0.123752 0.256973i
\(792\) 0 0
\(793\) 8.33233 1.90180i 0.295890 0.0675349i
\(794\) 0 0
\(795\) 12.7934i 0.453734i
\(796\) 0 0
\(797\) 8.38598 6.68760i 0.297047 0.236887i −0.463613 0.886038i \(-0.653447\pi\)
0.760660 + 0.649151i \(0.224876\pi\)
\(798\) 0 0
\(799\) −16.8725 + 8.12536i −0.596906 + 0.287455i
\(800\) 0 0
\(801\) −4.28969 8.90763i −0.151569 0.314736i
\(802\) 0 0
\(803\) 40.6957 + 19.5980i 1.43612 + 0.691599i
\(804\) 0 0
\(805\) 0.980593 4.29626i 0.0345614 0.151423i
\(806\) 0 0
\(807\) 14.9424 18.7372i 0.525999 0.659581i
\(808\) 0 0
\(809\) −17.9505 4.09708i −0.631106 0.144046i −0.105010 0.994471i \(-0.533487\pi\)
−0.526096 + 0.850425i \(0.676345\pi\)
\(810\) 0 0
\(811\) 25.8805 0.908789 0.454394 0.890801i \(-0.349856\pi\)
0.454394 + 0.890801i \(0.349856\pi\)
\(812\) 0 0
\(813\) 38.5698 1.35270
\(814\) 0 0
\(815\) 3.48050 + 0.794402i 0.121917 + 0.0278267i
\(816\) 0 0
\(817\) −29.8319 + 37.4080i −1.04369 + 1.30874i
\(818\) 0 0
\(819\) 0.862848 3.78038i 0.0301504 0.132097i
\(820\) 0 0
\(821\) 40.4099 + 19.4604i 1.41032 + 0.679173i 0.975225 0.221214i \(-0.0710018\pi\)
0.435091 + 0.900386i \(0.356716\pi\)
\(822\) 0 0
\(823\) −12.3181 25.5788i −0.429382 0.891622i −0.997633 0.0687661i \(-0.978094\pi\)
0.568250 0.822856i \(-0.307620\pi\)
\(824\) 0 0
\(825\) 11.6842 5.62680i 0.406791 0.195900i
\(826\) 0 0
\(827\) −9.96266 + 7.94496i −0.346436 + 0.276273i −0.781213 0.624265i \(-0.785399\pi\)
0.434777 + 0.900538i \(0.356827\pi\)
\(828\) 0 0
\(829\) 3.79481i 0.131799i 0.997826 + 0.0658996i \(0.0209917\pi\)
−0.997826 + 0.0658996i \(0.979008\pi\)
\(830\) 0 0
\(831\) −23.3187 + 5.32235i −0.808918 + 0.184630i
\(832\) 0 0
\(833\) 13.7613 28.5757i 0.476802 0.990090i
\(834\) 0 0
\(835\) 32.3487 + 40.5640i 1.11947 + 1.40377i
\(836\) 0 0
\(837\) −1.77191 7.76326i −0.0612463 0.268338i
\(838\) 0 0
\(839\) 2.43121 + 1.93883i 0.0839348 + 0.0669358i 0.664558 0.747236i \(-0.268620\pi\)
−0.580623 + 0.814172i \(0.697191\pi\)
\(840\) 0 0
\(841\) −26.9070 + 10.8172i −0.927828 + 0.373008i
\(842\) 0 0
\(843\) 6.74468 + 5.37870i 0.232299 + 0.185252i
\(844\) 0 0
\(845\) 0.400513 + 1.75476i 0.0137781 + 0.0603657i
\(846\) 0 0
\(847\) 1.37696 + 1.72665i 0.0473128 + 0.0593284i
\(848\) 0 0
\(849\) 1.90730 3.96055i 0.0654584 0.135926i
\(850\) 0 0
\(851\) 8.85571 2.02126i 0.303570 0.0692879i
\(852\) 0 0
\(853\) 0.125350i 0.00429189i 0.999998 + 0.00214594i \(0.000683076\pi\)
−0.999998 + 0.00214594i \(0.999317\pi\)
\(854\) 0 0
\(855\) −13.7612 + 10.9742i −0.470623 + 0.375310i
\(856\) 0 0
\(857\) −15.1606 + 7.30095i −0.517875 + 0.249396i −0.674516 0.738260i \(-0.735648\pi\)
0.156641 + 0.987656i \(0.449933\pi\)
\(858\) 0 0
\(859\) 8.35394 + 17.3471i 0.285033 + 0.591877i 0.993495 0.113871i \(-0.0363252\pi\)
−0.708463 + 0.705748i \(0.750611\pi\)
\(860\) 0 0
\(861\) 5.26805 + 2.53696i 0.179535 + 0.0864593i
\(862\) 0 0
\(863\) 5.00175 21.9141i 0.170262 0.745964i −0.815629 0.578575i \(-0.803609\pi\)
0.985891 0.167390i \(-0.0535338\pi\)
\(864\) 0 0
\(865\) −43.8692 + 55.0102i −1.49160 + 1.87040i
\(866\) 0 0
\(867\) 8.29573 + 1.89345i 0.281738 + 0.0643048i
\(868\) 0 0
\(869\) −18.3943 −0.623983
\(870\) 0 0
\(871\) 38.4061 1.30134
\(872\) 0 0
\(873\) −4.39124 1.00227i −0.148621 0.0339217i
\(874\) 0 0
\(875\) −1.63306 + 2.04780i −0.0552076 + 0.0692282i
\(876\) 0 0
\(877\) −3.91217 + 17.1403i −0.132105 + 0.578788i 0.864934 + 0.501886i \(0.167360\pi\)
−0.997039 + 0.0769025i \(0.975497\pi\)
\(878\) 0 0
\(879\) −23.6835 11.4054i −0.798823 0.384693i
\(880\) 0 0
\(881\) 20.0183 + 41.5684i 0.674434 + 1.40048i 0.904148 + 0.427218i \(0.140506\pi\)
−0.229715 + 0.973258i \(0.573779\pi\)
\(882\) 0 0
\(883\) −30.1751 + 14.5316i −1.01547 + 0.489027i −0.866162 0.499763i \(-0.833420\pi\)
−0.149312 + 0.988790i \(0.547706\pi\)
\(884\) 0 0
\(885\) −31.2070 + 24.8868i −1.04901 + 0.836560i
\(886\) 0 0
\(887\) 12.7174i 0.427007i −0.976942 0.213504i \(-0.931512\pi\)
0.976942 0.213504i \(-0.0684875\pi\)
\(888\) 0 0
\(889\) 9.42204 2.15052i 0.316005 0.0721261i
\(890\) 0 0
\(891\) −2.60543 + 5.41024i −0.0872853 + 0.181250i
\(892\) 0 0
\(893\) −9.31497 11.6806i −0.311714 0.390876i
\(894\) 0 0
\(895\) −15.3632 67.3107i −0.513536 2.24995i
\(896\) 0 0
\(897\) 6.87157 + 5.47990i 0.229435 + 0.182968i
\(898\) 0 0
\(899\) 3.61546 6.90962i 0.120582 0.230449i
\(900\) 0 0
\(901\) −13.5918 10.8391i −0.452808 0.361102i
\(902\) 0 0
\(903\) 2.40181 + 10.5230i 0.0799274 + 0.350185i
\(904\) 0 0
\(905\) −7.66692 9.61401i −0.254857 0.319580i
\(906\) 0 0
\(907\) −14.8072 + 30.7474i −0.491664 + 1.02095i 0.496570 + 0.867997i \(0.334593\pi\)
−0.988233 + 0.152953i \(0.951122\pi\)
\(908\) 0 0
\(909\) 6.55070 1.49515i 0.217273 0.0495911i
\(910\) 0 0
\(911\) 22.9200i 0.759372i −0.925115 0.379686i \(-0.876032\pi\)
0.925115 0.379686i \(-0.123968\pi\)
\(912\) 0 0
\(913\) −22.3636 + 17.8344i −0.740126 + 0.590231i
\(914\) 0 0
\(915\) −7.88499 + 3.79721i −0.260670 + 0.125532i
\(916\) 0 0
\(917\) 5.53148 + 11.4862i 0.182665 + 0.379309i
\(918\) 0 0
\(919\) −39.0233 18.7926i −1.28726 0.619912i −0.340015 0.940420i \(-0.610432\pi\)
−0.947246 + 0.320508i \(0.896146\pi\)
\(920\) 0 0
\(921\) 6.06277 26.5627i 0.199775 0.875272i
\(922\) 0 0
\(923\) 12.5870 15.7837i 0.414308 0.519525i
\(924\) 0 0
\(925\) 16.3118 + 3.72305i 0.536328 + 0.122413i
\(926\) 0 0
\(927\) 22.9637 0.754228
\(928\) 0 0
\(929\) −42.9957 −1.41064 −0.705321 0.708888i \(-0.749197\pi\)
−0.705321 + 0.708888i \(0.749197\pi\)
\(930\) 0 0
\(931\) 24.6685 + 5.63042i 0.808477 + 0.184530i
\(932\) 0 0
\(933\) −17.5579 + 22.0169i −0.574819 + 0.720800i
\(934\) 0 0
\(935\) −9.10808 + 39.9051i −0.297866 + 1.30504i
\(936\) 0 0
\(937\) 16.4992 + 7.94560i 0.539006 + 0.259571i 0.683521 0.729931i \(-0.260448\pi\)
−0.144515 + 0.989503i \(0.546162\pi\)
\(938\) 0 0
\(939\) −8.57087 17.7976i −0.279700 0.580803i
\(940\) 0 0
\(941\) −4.16806 + 2.00723i −0.135875 + 0.0654339i −0.500586 0.865687i \(-0.666882\pi\)
0.364711 + 0.931121i \(0.381168\pi\)
\(942\) 0 0
\(943\) 10.6423 8.48691i 0.346559 0.276372i
\(944\) 0 0
\(945\) 11.8073i 0.384091i
\(946\) 0 0
\(947\) 23.7036 5.41018i 0.770262 0.175807i 0.180709 0.983537i \(-0.442161\pi\)
0.589554 + 0.807729i \(0.299304\pi\)
\(948\) 0 0
\(949\) −24.4650 + 50.8020i −0.794166 + 1.64910i
\(950\) 0 0
\(951\) 19.3770 + 24.2979i 0.628341 + 0.787915i
\(952\) 0 0
\(953\) 0.259391 + 1.13647i 0.00840250 + 0.0368138i 0.978955 0.204074i \(-0.0654184\pi\)
−0.970553 + 0.240888i \(0.922561\pi\)
\(954\) 0 0
\(955\) 35.2267 + 28.0923i 1.13991 + 0.909046i
\(956\) 0 0
\(957\) −16.9024 + 7.45760i −0.546378 + 0.241070i
\(958\) 0 0
\(959\) −3.55311 2.83351i −0.114736 0.0914989i
\(960\) 0 0
\(961\) −6.43151 28.1783i −0.207468 0.908977i
\(962\) 0 0
\(963\) −14.2264 17.8393i −0.458438 0.574863i
\(964\) 0 0
\(965\) 23.9764 49.7876i 0.771829 1.60272i
\(966\) 0 0
\(967\) −24.6836 + 5.63387i −0.793770 + 0.181173i −0.600133 0.799900i \(-0.704886\pi\)
−0.193637 + 0.981073i \(0.562029\pi\)
\(968\) 0 0
\(969\) 23.2874i 0.748098i
\(970\) 0 0
\(971\) 19.1810 15.2963i 0.615547 0.490882i −0.265374 0.964145i \(-0.585496\pi\)
0.880921 + 0.473263i \(0.156924\pi\)
\(972\) 0 0
\(973\) −5.99671 + 2.88786i −0.192246 + 0.0925807i
\(974\) 0 0
\(975\) 7.02415 + 14.5858i 0.224953 + 0.467120i
\(976\) 0 0
\(977\) −13.2888 6.39953i −0.425145 0.204739i 0.209062 0.977902i \(-0.432959\pi\)
−0.634207 + 0.773163i \(0.718673\pi\)
\(978\) 0 0
\(979\) 4.08149 17.8822i 0.130445 0.571517i
\(980\) 0 0
\(981\) 0.163053 0.204461i 0.00520587 0.00652795i
\(982\) 0 0
\(983\) −25.1635 5.74340i −0.802590 0.183186i −0.198499 0.980101i \(-0.563607\pi\)
−0.604091 + 0.796915i \(0.706464\pi\)
\(984\) 0 0
\(985\) 20.6168 0.656904
\(986\) 0 0
\(987\) −3.37031 −0.107278
\(988\) 0 0
\(989\) 24.4975 + 5.59139i 0.778974 + 0.177796i
\(990\) 0 0
\(991\) −15.9358 + 19.9829i −0.506217 + 0.634776i −0.967619 0.252415i \(-0.918775\pi\)
0.461402 + 0.887191i \(0.347347\pi\)
\(992\) 0 0
\(993\) −2.73712 + 11.9921i −0.0868599 + 0.380558i
\(994\) 0 0
\(995\) 7.47501 + 3.59978i 0.236974 + 0.114121i
\(996\) 0 0
\(997\) 17.3639 + 36.0564i 0.549919 + 1.14192i 0.971918 + 0.235322i \(0.0756144\pi\)
−0.421999 + 0.906596i \(0.638671\pi\)
\(998\) 0 0
\(999\) −21.9277 + 10.5598i −0.693760 + 0.334097i
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 232.2.q.a.209.3 yes 48
4.3 odd 2 464.2.y.e.209.6 48
29.5 even 14 inner 232.2.q.a.121.3 48
29.11 odd 28 6728.2.a.bf.1.16 24
29.18 odd 28 6728.2.a.be.1.9 24
116.63 odd 14 464.2.y.e.353.6 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.2.q.a.121.3 48 29.5 even 14 inner
232.2.q.a.209.3 yes 48 1.1 even 1 trivial
464.2.y.e.209.6 48 4.3 odd 2
464.2.y.e.353.6 48 116.63 odd 14
6728.2.a.be.1.9 24 29.18 odd 28
6728.2.a.bf.1.16 24 29.11 odd 28