Properties

Label 1932.2.k.a.1609.23
Level $1932$
Weight $2$
Character 1932.1609
Analytic conductor $15.427$
Analytic rank $0$
Dimension $32$
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] = [1932,2,Mod(1609,1932)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1932, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1932.1609");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1932.k (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: \(15.4270976705\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1609.23
Character \(\chi\) \(=\) 1932.1609
Dual form 1932.2.k.a.1609.21

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -0.248143 q^{5} +(-0.652227 - 2.56410i) q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -0.248143 q^{5} +(-0.652227 - 2.56410i) q^{7} -1.00000 q^{9} +4.05549i q^{11} -5.46658i q^{13} -0.248143i q^{15} +0.349644 q^{17} +3.87946 q^{19} +(2.56410 - 0.652227i) q^{21} +(0.300350 + 4.78642i) q^{23} -4.93843 q^{25} -1.00000i q^{27} -3.50660 q^{29} -7.87222i q^{31} -4.05549 q^{33} +(0.161846 + 0.636263i) q^{35} +3.76156i q^{37} +5.46658 q^{39} -5.40062i q^{41} -11.4582i q^{43} +0.248143 q^{45} -1.86784i q^{47} +(-6.14920 + 3.34475i) q^{49} +0.349644i q^{51} -9.71322i q^{53} -1.00634i q^{55} +3.87946i q^{57} -12.5854i q^{59} +4.03300 q^{61} +(0.652227 + 2.56410i) q^{63} +1.35649i q^{65} -4.70293i q^{67} +(-4.78642 + 0.300350i) q^{69} +13.9003 q^{71} -9.64759i q^{73} -4.93843i q^{75} +(10.3987 - 2.64510i) q^{77} +6.98013i q^{79} +1.00000 q^{81} -12.2060 q^{83} -0.0867616 q^{85} -3.50660i q^{87} +6.84062 q^{89} +(-14.0169 + 3.56545i) q^{91} +7.87222 q^{93} -0.962660 q^{95} +1.02306 q^{97} -4.05549i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 32 q^{9} + 12 q^{23} + 40 q^{25} - 16 q^{29} + 8 q^{35} + 32 q^{71} + 24 q^{77} + 32 q^{81} + 8 q^{85} + 8 q^{93} - 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1932\mathbb{Z}\right)^\times\).

\(n\) \(829\) \(925\) \(967\) \(1289\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −0.248143 −0.110973 −0.0554865 0.998459i \(-0.517671\pi\)
−0.0554865 + 0.998459i \(0.517671\pi\)
\(6\) 0 0
\(7\) −0.652227 2.56410i −0.246519 0.969138i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.05549i 1.22278i 0.791331 + 0.611388i \(0.209388\pi\)
−0.791331 + 0.611388i \(0.790612\pi\)
\(12\) 0 0
\(13\) 5.46658i 1.51616i −0.652163 0.758079i \(-0.726138\pi\)
0.652163 0.758079i \(-0.273862\pi\)
\(14\) 0 0
\(15\) 0.248143i 0.0640702i
\(16\) 0 0
\(17\) 0.349644 0.0848011 0.0424005 0.999101i \(-0.486499\pi\)
0.0424005 + 0.999101i \(0.486499\pi\)
\(18\) 0 0
\(19\) 3.87946 0.890008 0.445004 0.895528i \(-0.353202\pi\)
0.445004 + 0.895528i \(0.353202\pi\)
\(20\) 0 0
\(21\) 2.56410 0.652227i 0.559532 0.142328i
\(22\) 0 0
\(23\) 0.300350 + 4.78642i 0.0626272 + 0.998037i
\(24\) 0 0
\(25\) −4.93843 −0.987685
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −3.50660 −0.651158 −0.325579 0.945515i \(-0.605559\pi\)
−0.325579 + 0.945515i \(0.605559\pi\)
\(30\) 0 0
\(31\) 7.87222i 1.41389i −0.707267 0.706947i \(-0.750072\pi\)
0.707267 0.706947i \(-0.249928\pi\)
\(32\) 0 0
\(33\) −4.05549 −0.705970
\(34\) 0 0
\(35\) 0.161846 + 0.636263i 0.0273569 + 0.107548i
\(36\) 0 0
\(37\) 3.76156i 0.618396i 0.950998 + 0.309198i \(0.100061\pi\)
−0.950998 + 0.309198i \(0.899939\pi\)
\(38\) 0 0
\(39\) 5.46658 0.875354
\(40\) 0 0
\(41\) 5.40062i 0.843435i −0.906727 0.421718i \(-0.861427\pi\)
0.906727 0.421718i \(-0.138573\pi\)
\(42\) 0 0
\(43\) 11.4582i 1.74737i −0.486495 0.873683i \(-0.661725\pi\)
0.486495 0.873683i \(-0.338275\pi\)
\(44\) 0 0
\(45\) 0.248143 0.0369910
\(46\) 0 0
\(47\) 1.86784i 0.272452i −0.990678 0.136226i \(-0.956503\pi\)
0.990678 0.136226i \(-0.0434973\pi\)
\(48\) 0 0
\(49\) −6.14920 + 3.34475i −0.878457 + 0.477821i
\(50\) 0 0
\(51\) 0.349644i 0.0489599i
\(52\) 0 0
\(53\) 9.71322i 1.33421i −0.744962 0.667107i \(-0.767532\pi\)
0.744962 0.667107i \(-0.232468\pi\)
\(54\) 0 0
\(55\) 1.00634i 0.135695i
\(56\) 0 0
\(57\) 3.87946i 0.513847i
\(58\) 0 0
\(59\) 12.5854i 1.63848i −0.573453 0.819239i \(-0.694396\pi\)
0.573453 0.819239i \(-0.305604\pi\)
\(60\) 0 0
\(61\) 4.03300 0.516372 0.258186 0.966095i \(-0.416875\pi\)
0.258186 + 0.966095i \(0.416875\pi\)
\(62\) 0 0
\(63\) 0.652227 + 2.56410i 0.0821729 + 0.323046i
\(64\) 0 0
\(65\) 1.35649i 0.168252i
\(66\) 0 0
\(67\) 4.70293i 0.574554i −0.957848 0.287277i \(-0.907250\pi\)
0.957848 0.287277i \(-0.0927500\pi\)
\(68\) 0 0
\(69\) −4.78642 + 0.300350i −0.576217 + 0.0361578i
\(70\) 0 0
\(71\) 13.9003 1.64966 0.824829 0.565382i \(-0.191271\pi\)
0.824829 + 0.565382i \(0.191271\pi\)
\(72\) 0 0
\(73\) 9.64759i 1.12917i −0.825377 0.564583i \(-0.809037\pi\)
0.825377 0.564583i \(-0.190963\pi\)
\(74\) 0 0
\(75\) 4.93843i 0.570240i
\(76\) 0 0
\(77\) 10.3987 2.64510i 1.18504 0.301437i
\(78\) 0 0
\(79\) 6.98013i 0.785326i 0.919682 + 0.392663i \(0.128446\pi\)
−0.919682 + 0.392663i \(0.871554\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.2060 −1.33978 −0.669890 0.742461i \(-0.733659\pi\)
−0.669890 + 0.742461i \(0.733659\pi\)
\(84\) 0 0
\(85\) −0.0867616 −0.00941062
\(86\) 0 0
\(87\) 3.50660i 0.375947i
\(88\) 0 0
\(89\) 6.84062 0.725104 0.362552 0.931964i \(-0.381906\pi\)
0.362552 + 0.931964i \(0.381906\pi\)
\(90\) 0 0
\(91\) −14.0169 + 3.56545i −1.46937 + 0.373761i
\(92\) 0 0
\(93\) 7.87222 0.816312
\(94\) 0 0
\(95\) −0.962660 −0.0987668
\(96\) 0 0
\(97\) 1.02306 0.103876 0.0519379 0.998650i \(-0.483460\pi\)
0.0519379 + 0.998650i \(0.483460\pi\)
\(98\) 0 0
\(99\) 4.05549i 0.407592i
\(100\) 0 0
\(101\) 9.90088i 0.985174i 0.870263 + 0.492587i \(0.163949\pi\)
−0.870263 + 0.492587i \(0.836051\pi\)
\(102\) 0 0
\(103\) 6.54945 0.645337 0.322668 0.946512i \(-0.395420\pi\)
0.322668 + 0.946512i \(0.395420\pi\)
\(104\) 0 0
\(105\) −0.636263 + 0.161846i −0.0620929 + 0.0157945i
\(106\) 0 0
\(107\) 5.20163i 0.502860i −0.967875 0.251430i \(-0.919099\pi\)
0.967875 0.251430i \(-0.0809009\pi\)
\(108\) 0 0
\(109\) 6.98642i 0.669177i 0.942364 + 0.334589i \(0.108597\pi\)
−0.942364 + 0.334589i \(0.891403\pi\)
\(110\) 0 0
\(111\) −3.76156 −0.357031
\(112\) 0 0
\(113\) 5.89960i 0.554987i 0.960727 + 0.277494i \(0.0895038\pi\)
−0.960727 + 0.277494i \(0.910496\pi\)
\(114\) 0 0
\(115\) −0.0745296 1.18772i −0.00694992 0.110755i
\(116\) 0 0
\(117\) 5.46658i 0.505386i
\(118\) 0 0
\(119\) −0.228047 0.896521i −0.0209051 0.0821839i
\(120\) 0 0
\(121\) −5.44697 −0.495179
\(122\) 0 0
\(123\) 5.40062 0.486958
\(124\) 0 0
\(125\) 2.46615 0.220579
\(126\) 0 0
\(127\) −9.08378 −0.806055 −0.403028 0.915188i \(-0.632042\pi\)
−0.403028 + 0.915188i \(0.632042\pi\)
\(128\) 0 0
\(129\) 11.4582 1.00884
\(130\) 0 0
\(131\) 16.8087i 1.46858i −0.678835 0.734291i \(-0.737515\pi\)
0.678835 0.734291i \(-0.262485\pi\)
\(132\) 0 0
\(133\) −2.53029 9.94731i −0.219404 0.862541i
\(134\) 0 0
\(135\) 0.248143i 0.0213567i
\(136\) 0 0
\(137\) 14.9328i 1.27579i −0.770122 0.637897i \(-0.779805\pi\)
0.770122 0.637897i \(-0.220195\pi\)
\(138\) 0 0
\(139\) 0.0692691i 0.00587533i 0.999996 + 0.00293766i \(0.000935089\pi\)
−0.999996 + 0.00293766i \(0.999065\pi\)
\(140\) 0 0
\(141\) 1.86784 0.157300
\(142\) 0 0
\(143\) 22.1697 1.85392
\(144\) 0 0
\(145\) 0.870137 0.0722609
\(146\) 0 0
\(147\) −3.34475 6.14920i −0.275870 0.507177i
\(148\) 0 0
\(149\) 5.35388i 0.438607i 0.975657 + 0.219303i \(0.0703785\pi\)
−0.975657 + 0.219303i \(0.929622\pi\)
\(150\) 0 0
\(151\) 7.79280 0.634169 0.317085 0.948397i \(-0.397296\pi\)
0.317085 + 0.948397i \(0.397296\pi\)
\(152\) 0 0
\(153\) −0.349644 −0.0282670
\(154\) 0 0
\(155\) 1.95344i 0.156904i
\(156\) 0 0
\(157\) −8.35019 −0.666418 −0.333209 0.942853i \(-0.608131\pi\)
−0.333209 + 0.942853i \(0.608131\pi\)
\(158\) 0 0
\(159\) 9.71322 0.770309
\(160\) 0 0
\(161\) 12.0769 3.89196i 0.951797 0.306729i
\(162\) 0 0
\(163\) −3.50912 −0.274856 −0.137428 0.990512i \(-0.543884\pi\)
−0.137428 + 0.990512i \(0.543884\pi\)
\(164\) 0 0
\(165\) 1.00634 0.0783435
\(166\) 0 0
\(167\) 7.07693i 0.547629i −0.961782 0.273815i \(-0.911715\pi\)
0.961782 0.273815i \(-0.0882855\pi\)
\(168\) 0 0
\(169\) −16.8835 −1.29873
\(170\) 0 0
\(171\) −3.87946 −0.296669
\(172\) 0 0
\(173\) 10.8180i 0.822477i 0.911528 + 0.411238i \(0.134904\pi\)
−0.911528 + 0.411238i \(0.865096\pi\)
\(174\) 0 0
\(175\) 3.22098 + 12.6626i 0.243483 + 0.957203i
\(176\) 0 0
\(177\) 12.5854 0.945975
\(178\) 0 0
\(179\) 9.95147 0.743808 0.371904 0.928271i \(-0.378705\pi\)
0.371904 + 0.928271i \(0.378705\pi\)
\(180\) 0 0
\(181\) 6.14027 0.456402 0.228201 0.973614i \(-0.426716\pi\)
0.228201 + 0.973614i \(0.426716\pi\)
\(182\) 0 0
\(183\) 4.03300i 0.298128i
\(184\) 0 0
\(185\) 0.933404i 0.0686252i
\(186\) 0 0
\(187\) 1.41798i 0.103693i
\(188\) 0 0
\(189\) −2.56410 + 0.652227i −0.186511 + 0.0474426i
\(190\) 0 0
\(191\) 0.686980i 0.0497081i −0.999691 0.0248541i \(-0.992088\pi\)
0.999691 0.0248541i \(-0.00791211\pi\)
\(192\) 0 0
\(193\) −2.09247 −0.150619 −0.0753096 0.997160i \(-0.523995\pi\)
−0.0753096 + 0.997160i \(0.523995\pi\)
\(194\) 0 0
\(195\) −1.35649 −0.0971405
\(196\) 0 0
\(197\) 9.87120 0.703294 0.351647 0.936133i \(-0.385622\pi\)
0.351647 + 0.936133i \(0.385622\pi\)
\(198\) 0 0
\(199\) 14.7198 1.04346 0.521731 0.853110i \(-0.325286\pi\)
0.521731 + 0.853110i \(0.325286\pi\)
\(200\) 0 0
\(201\) 4.70293 0.331719
\(202\) 0 0
\(203\) 2.28710 + 8.99126i 0.160523 + 0.631062i
\(204\) 0 0
\(205\) 1.34013i 0.0935985i
\(206\) 0 0
\(207\) −0.300350 4.78642i −0.0208757 0.332679i
\(208\) 0 0
\(209\) 15.7331i 1.08828i
\(210\) 0 0
\(211\) −5.39018 −0.371075 −0.185538 0.982637i \(-0.559403\pi\)
−0.185538 + 0.982637i \(0.559403\pi\)
\(212\) 0 0
\(213\) 13.9003i 0.952431i
\(214\) 0 0
\(215\) 2.84328i 0.193910i
\(216\) 0 0
\(217\) −20.1852 + 5.13448i −1.37026 + 0.348551i
\(218\) 0 0
\(219\) 9.64759 0.651924
\(220\) 0 0
\(221\) 1.91136i 0.128572i
\(222\) 0 0
\(223\) 20.0807i 1.34470i −0.740233 0.672351i \(-0.765285\pi\)
0.740233 0.672351i \(-0.234715\pi\)
\(224\) 0 0
\(225\) 4.93843 0.329228
\(226\) 0 0
\(227\) −15.5820 −1.03421 −0.517107 0.855921i \(-0.672991\pi\)
−0.517107 + 0.855921i \(0.672991\pi\)
\(228\) 0 0
\(229\) 7.89135 0.521475 0.260738 0.965410i \(-0.416034\pi\)
0.260738 + 0.965410i \(0.416034\pi\)
\(230\) 0 0
\(231\) 2.64510 + 10.3987i 0.174035 + 0.684182i
\(232\) 0 0
\(233\) −15.5621 −1.01951 −0.509753 0.860321i \(-0.670263\pi\)
−0.509753 + 0.860321i \(0.670263\pi\)
\(234\) 0 0
\(235\) 0.463491i 0.0302348i
\(236\) 0 0
\(237\) −6.98013 −0.453408
\(238\) 0 0
\(239\) 13.6315 0.881748 0.440874 0.897569i \(-0.354669\pi\)
0.440874 + 0.897569i \(0.354669\pi\)
\(240\) 0 0
\(241\) −27.6041 −1.77814 −0.889069 0.457773i \(-0.848647\pi\)
−0.889069 + 0.457773i \(0.848647\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 1.52588 0.829976i 0.0974849 0.0530252i
\(246\) 0 0
\(247\) 21.2074i 1.34939i
\(248\) 0 0
\(249\) 12.2060i 0.773522i
\(250\) 0 0
\(251\) −19.8899 −1.25544 −0.627721 0.778439i \(-0.716012\pi\)
−0.627721 + 0.778439i \(0.716012\pi\)
\(252\) 0 0
\(253\) −19.4113 + 1.21806i −1.22037 + 0.0765790i
\(254\) 0 0
\(255\) 0.0867616i 0.00543322i
\(256\) 0 0
\(257\) 5.75374i 0.358909i 0.983766 + 0.179454i \(0.0574332\pi\)
−0.983766 + 0.179454i \(0.942567\pi\)
\(258\) 0 0
\(259\) 9.64500 2.45339i 0.599311 0.152446i
\(260\) 0 0
\(261\) 3.50660 0.217053
\(262\) 0 0
\(263\) 11.0037i 0.678516i 0.940693 + 0.339258i \(0.110176\pi\)
−0.940693 + 0.339258i \(0.889824\pi\)
\(264\) 0 0
\(265\) 2.41027i 0.148062i
\(266\) 0 0
\(267\) 6.84062i 0.418639i
\(268\) 0 0
\(269\) 18.3564i 1.11921i 0.828759 + 0.559606i \(0.189048\pi\)
−0.828759 + 0.559606i \(0.810952\pi\)
\(270\) 0 0
\(271\) 29.9547i 1.81962i −0.415027 0.909809i \(-0.636228\pi\)
0.415027 0.909809i \(-0.363772\pi\)
\(272\) 0 0
\(273\) −3.56545 14.0169i −0.215791 0.848339i
\(274\) 0 0
\(275\) 20.0277i 1.20772i
\(276\) 0 0
\(277\) −18.6594 −1.12113 −0.560567 0.828109i \(-0.689417\pi\)
−0.560567 + 0.828109i \(0.689417\pi\)
\(278\) 0 0
\(279\) 7.87222i 0.471298i
\(280\) 0 0
\(281\) 0.0173097i 0.00103261i 1.00000 0.000516305i \(0.000164345\pi\)
−1.00000 0.000516305i \(0.999836\pi\)
\(282\) 0 0
\(283\) 0.120744 0.00717752 0.00358876 0.999994i \(-0.498858\pi\)
0.00358876 + 0.999994i \(0.498858\pi\)
\(284\) 0 0
\(285\) 0.962660i 0.0570230i
\(286\) 0 0
\(287\) −13.8477 + 3.52243i −0.817405 + 0.207923i
\(288\) 0 0
\(289\) −16.8777 −0.992809
\(290\) 0 0
\(291\) 1.02306i 0.0599727i
\(292\) 0 0
\(293\) −28.5499 −1.66790 −0.833950 0.551839i \(-0.813926\pi\)
−0.833950 + 0.551839i \(0.813926\pi\)
\(294\) 0 0
\(295\) 3.12297i 0.181827i
\(296\) 0 0
\(297\) 4.05549 0.235323
\(298\) 0 0
\(299\) 26.1653 1.64189i 1.51318 0.0949527i
\(300\) 0 0
\(301\) −29.3801 + 7.47338i −1.69344 + 0.430759i
\(302\) 0 0
\(303\) −9.90088 −0.568790
\(304\) 0 0
\(305\) −1.00076 −0.0573033
\(306\) 0 0
\(307\) 9.73639i 0.555685i 0.960627 + 0.277843i \(0.0896194\pi\)
−0.960627 + 0.277843i \(0.910381\pi\)
\(308\) 0 0
\(309\) 6.54945i 0.372585i
\(310\) 0 0
\(311\) 24.8161i 1.40719i 0.710601 + 0.703596i \(0.248423\pi\)
−0.710601 + 0.703596i \(0.751577\pi\)
\(312\) 0 0
\(313\) 24.1571 1.36544 0.682719 0.730681i \(-0.260797\pi\)
0.682719 + 0.730681i \(0.260797\pi\)
\(314\) 0 0
\(315\) −0.161846 0.636263i −0.00911897 0.0358494i
\(316\) 0 0
\(317\) 0.882476 0.0495648 0.0247824 0.999693i \(-0.492111\pi\)
0.0247824 + 0.999693i \(0.492111\pi\)
\(318\) 0 0
\(319\) 14.2210i 0.796220i
\(320\) 0 0
\(321\) 5.20163 0.290327
\(322\) 0 0
\(323\) 1.35643 0.0754736
\(324\) 0 0
\(325\) 26.9963i 1.49749i
\(326\) 0 0
\(327\) −6.98642 −0.386350
\(328\) 0 0
\(329\) −4.78932 + 1.21825i −0.264044 + 0.0671645i
\(330\) 0 0
\(331\) 4.48030 0.246259 0.123130 0.992391i \(-0.460707\pi\)
0.123130 + 0.992391i \(0.460707\pi\)
\(332\) 0 0
\(333\) 3.76156i 0.206132i
\(334\) 0 0
\(335\) 1.16700i 0.0637599i
\(336\) 0 0
\(337\) 7.99001i 0.435243i −0.976033 0.217622i \(-0.930170\pi\)
0.976033 0.217622i \(-0.0698299\pi\)
\(338\) 0 0
\(339\) −5.89960 −0.320422
\(340\) 0 0
\(341\) 31.9257 1.72887
\(342\) 0 0
\(343\) 12.5869 + 13.5856i 0.679631 + 0.733554i
\(344\) 0 0
\(345\) 1.18772 0.0745296i 0.0639445 0.00401254i
\(346\) 0 0
\(347\) −8.18595 −0.439445 −0.219723 0.975562i \(-0.570515\pi\)
−0.219723 + 0.975562i \(0.570515\pi\)
\(348\) 0 0
\(349\) 23.3084i 1.24767i 0.781557 + 0.623834i \(0.214426\pi\)
−0.781557 + 0.623834i \(0.785574\pi\)
\(350\) 0 0
\(351\) −5.46658 −0.291785
\(352\) 0 0
\(353\) 20.6822i 1.10080i −0.834901 0.550400i \(-0.814475\pi\)
0.834901 0.550400i \(-0.185525\pi\)
\(354\) 0 0
\(355\) −3.44925 −0.183067
\(356\) 0 0
\(357\) 0.896521 0.228047i 0.0474489 0.0120695i
\(358\) 0 0
\(359\) 17.3509i 0.915748i 0.889017 + 0.457874i \(0.151389\pi\)
−0.889017 + 0.457874i \(0.848611\pi\)
\(360\) 0 0
\(361\) −3.94982 −0.207885
\(362\) 0 0
\(363\) 5.44697i 0.285892i
\(364\) 0 0
\(365\) 2.39398i 0.125307i
\(366\) 0 0
\(367\) 15.2971 0.798501 0.399250 0.916842i \(-0.369270\pi\)
0.399250 + 0.916842i \(0.369270\pi\)
\(368\) 0 0
\(369\) 5.40062i 0.281145i
\(370\) 0 0
\(371\) −24.9057 + 6.33523i −1.29304 + 0.328909i
\(372\) 0 0
\(373\) 19.8672i 1.02869i −0.857584 0.514343i \(-0.828036\pi\)
0.857584 0.514343i \(-0.171964\pi\)
\(374\) 0 0
\(375\) 2.46615i 0.127351i
\(376\) 0 0
\(377\) 19.1691i 0.987258i
\(378\) 0 0
\(379\) 2.96163i 0.152129i 0.997103 + 0.0760644i \(0.0242355\pi\)
−0.997103 + 0.0760644i \(0.975765\pi\)
\(380\) 0 0
\(381\) 9.08378i 0.465376i
\(382\) 0 0
\(383\) 6.82827 0.348908 0.174454 0.984665i \(-0.444184\pi\)
0.174454 + 0.984665i \(0.444184\pi\)
\(384\) 0 0
\(385\) −2.58036 + 0.656363i −0.131507 + 0.0334513i
\(386\) 0 0
\(387\) 11.4582i 0.582456i
\(388\) 0 0
\(389\) 13.7609i 0.697706i 0.937177 + 0.348853i \(0.113429\pi\)
−0.937177 + 0.348853i \(0.886571\pi\)
\(390\) 0 0
\(391\) 0.105015 + 1.67354i 0.00531085 + 0.0846346i
\(392\) 0 0
\(393\) 16.8087 0.847887
\(394\) 0 0
\(395\) 1.73207i 0.0871499i
\(396\) 0 0
\(397\) 26.0290i 1.30636i 0.757203 + 0.653180i \(0.226565\pi\)
−0.757203 + 0.653180i \(0.773435\pi\)
\(398\) 0 0
\(399\) 9.94731 2.53029i 0.497988 0.126673i
\(400\) 0 0
\(401\) 33.8292i 1.68935i 0.535279 + 0.844675i \(0.320207\pi\)
−0.535279 + 0.844675i \(0.679793\pi\)
\(402\) 0 0
\(403\) −43.0342 −2.14368
\(404\) 0 0
\(405\) −0.248143 −0.0123303
\(406\) 0 0
\(407\) −15.2549 −0.756160
\(408\) 0 0
\(409\) 24.2940i 1.20126i 0.799527 + 0.600630i \(0.205083\pi\)
−0.799527 + 0.600630i \(0.794917\pi\)
\(410\) 0 0
\(411\) 14.9328 0.736580
\(412\) 0 0
\(413\) −32.2702 + 8.20853i −1.58791 + 0.403915i
\(414\) 0 0
\(415\) 3.02883 0.148679
\(416\) 0 0
\(417\) −0.0692691 −0.00339212
\(418\) 0 0
\(419\) 21.5498 1.05277 0.526387 0.850245i \(-0.323546\pi\)
0.526387 + 0.850245i \(0.323546\pi\)
\(420\) 0 0
\(421\) 37.1329i 1.80975i −0.425682 0.904873i \(-0.639966\pi\)
0.425682 0.904873i \(-0.360034\pi\)
\(422\) 0 0
\(423\) 1.86784i 0.0908173i
\(424\) 0 0
\(425\) −1.72669 −0.0837567
\(426\) 0 0
\(427\) −2.63043 10.3410i −0.127295 0.500436i
\(428\) 0 0
\(429\) 22.1697i 1.07036i
\(430\) 0 0
\(431\) 35.6918i 1.71921i −0.510958 0.859606i \(-0.670709\pi\)
0.510958 0.859606i \(-0.329291\pi\)
\(432\) 0 0
\(433\) −17.2866 −0.830743 −0.415372 0.909652i \(-0.636348\pi\)
−0.415372 + 0.909652i \(0.636348\pi\)
\(434\) 0 0
\(435\) 0.870137i 0.0417199i
\(436\) 0 0
\(437\) 1.16519 + 18.5687i 0.0557387 + 0.888261i
\(438\) 0 0
\(439\) 28.6110i 1.36553i 0.730638 + 0.682765i \(0.239223\pi\)
−0.730638 + 0.682765i \(0.760777\pi\)
\(440\) 0 0
\(441\) 6.14920 3.34475i 0.292819 0.159274i
\(442\) 0 0
\(443\) 21.0921 1.00212 0.501058 0.865414i \(-0.332944\pi\)
0.501058 + 0.865414i \(0.332944\pi\)
\(444\) 0 0
\(445\) −1.69745 −0.0804669
\(446\) 0 0
\(447\) −5.35388 −0.253230
\(448\) 0 0
\(449\) 23.5513 1.11146 0.555728 0.831364i \(-0.312440\pi\)
0.555728 + 0.831364i \(0.312440\pi\)
\(450\) 0 0
\(451\) 21.9021 1.03133
\(452\) 0 0
\(453\) 7.79280i 0.366138i
\(454\) 0 0
\(455\) 3.47818 0.884742i 0.163060 0.0414774i
\(456\) 0 0
\(457\) 26.9541i 1.26086i 0.776247 + 0.630429i \(0.217121\pi\)
−0.776247 + 0.630429i \(0.782879\pi\)
\(458\) 0 0
\(459\) 0.349644i 0.0163200i
\(460\) 0 0
\(461\) 18.4634i 0.859924i −0.902847 0.429962i \(-0.858527\pi\)
0.902847 0.429962i \(-0.141473\pi\)
\(462\) 0 0
\(463\) 35.3889 1.64466 0.822331 0.569009i \(-0.192673\pi\)
0.822331 + 0.569009i \(0.192673\pi\)
\(464\) 0 0
\(465\) −1.95344 −0.0905885
\(466\) 0 0
\(467\) 14.9569 0.692123 0.346062 0.938212i \(-0.387519\pi\)
0.346062 + 0.938212i \(0.387519\pi\)
\(468\) 0 0
\(469\) −12.0588 + 3.06738i −0.556822 + 0.141638i
\(470\) 0 0
\(471\) 8.35019i 0.384757i
\(472\) 0 0
\(473\) 46.4688 2.13664
\(474\) 0 0
\(475\) −19.1584 −0.879048
\(476\) 0 0
\(477\) 9.71322i 0.444738i
\(478\) 0 0
\(479\) 12.1485 0.555078 0.277539 0.960714i \(-0.410481\pi\)
0.277539 + 0.960714i \(0.410481\pi\)
\(480\) 0 0
\(481\) 20.5629 0.937586
\(482\) 0 0
\(483\) 3.89196 + 12.0769i 0.177090 + 0.549520i
\(484\) 0 0
\(485\) −0.253865 −0.0115274
\(486\) 0 0
\(487\) 19.1745 0.868879 0.434440 0.900701i \(-0.356946\pi\)
0.434440 + 0.900701i \(0.356946\pi\)
\(488\) 0 0
\(489\) 3.50912i 0.158688i
\(490\) 0 0
\(491\) −13.0026 −0.586800 −0.293400 0.955990i \(-0.594787\pi\)
−0.293400 + 0.955990i \(0.594787\pi\)
\(492\) 0 0
\(493\) −1.22606 −0.0552189
\(494\) 0 0
\(495\) 1.00634i 0.0452316i
\(496\) 0 0
\(497\) −9.06614 35.6417i −0.406672 1.59875i
\(498\) 0 0
\(499\) −37.0374 −1.65802 −0.829011 0.559232i \(-0.811096\pi\)
−0.829011 + 0.559232i \(0.811096\pi\)
\(500\) 0 0
\(501\) 7.07693 0.316174
\(502\) 0 0
\(503\) −13.8200 −0.616206 −0.308103 0.951353i \(-0.599694\pi\)
−0.308103 + 0.951353i \(0.599694\pi\)
\(504\) 0 0
\(505\) 2.45683i 0.109328i
\(506\) 0 0
\(507\) 16.8835i 0.749823i
\(508\) 0 0
\(509\) 33.4182i 1.48123i 0.671927 + 0.740617i \(0.265467\pi\)
−0.671927 + 0.740617i \(0.734533\pi\)
\(510\) 0 0
\(511\) −24.7374 + 6.29242i −1.09432 + 0.278360i
\(512\) 0 0
\(513\) 3.87946i 0.171282i
\(514\) 0 0
\(515\) −1.62520 −0.0716149
\(516\) 0 0
\(517\) 7.57499 0.333148
\(518\) 0 0
\(519\) −10.8180 −0.474857
\(520\) 0 0
\(521\) 33.6964 1.47627 0.738133 0.674655i \(-0.235708\pi\)
0.738133 + 0.674655i \(0.235708\pi\)
\(522\) 0 0
\(523\) −3.03787 −0.132837 −0.0664185 0.997792i \(-0.521157\pi\)
−0.0664185 + 0.997792i \(0.521157\pi\)
\(524\) 0 0
\(525\) −12.6626 + 3.22098i −0.552641 + 0.140575i
\(526\) 0 0
\(527\) 2.75247i 0.119900i
\(528\) 0 0
\(529\) −22.8196 + 2.87520i −0.992156 + 0.125009i
\(530\) 0 0
\(531\) 12.5854i 0.546159i
\(532\) 0 0
\(533\) −29.5229 −1.27878
\(534\) 0 0
\(535\) 1.29075i 0.0558039i
\(536\) 0 0
\(537\) 9.95147i 0.429438i
\(538\) 0 0
\(539\) −13.5646 24.9380i −0.584268 1.07416i
\(540\) 0 0
\(541\) −23.9327 −1.02895 −0.514474 0.857506i \(-0.672013\pi\)
−0.514474 + 0.857506i \(0.672013\pi\)
\(542\) 0 0
\(543\) 6.14027i 0.263504i
\(544\) 0 0
\(545\) 1.73363i 0.0742605i
\(546\) 0 0
\(547\) −9.70243 −0.414846 −0.207423 0.978251i \(-0.566508\pi\)
−0.207423 + 0.978251i \(0.566508\pi\)
\(548\) 0 0
\(549\) −4.03300 −0.172124
\(550\) 0 0
\(551\) −13.6037 −0.579536
\(552\) 0 0
\(553\) 17.8977 4.55263i 0.761090 0.193598i
\(554\) 0 0
\(555\) 0.933404 0.0396208
\(556\) 0 0
\(557\) 12.9311i 0.547907i −0.961743 0.273953i \(-0.911669\pi\)
0.961743 0.273953i \(-0.0883314\pi\)
\(558\) 0 0
\(559\) −62.6375 −2.64928
\(560\) 0 0
\(561\) −1.41798 −0.0598670
\(562\) 0 0
\(563\) −28.3780 −1.19599 −0.597996 0.801499i \(-0.704036\pi\)
−0.597996 + 0.801499i \(0.704036\pi\)
\(564\) 0 0
\(565\) 1.46394i 0.0615886i
\(566\) 0 0
\(567\) −0.652227 2.56410i −0.0273910 0.107682i
\(568\) 0 0
\(569\) 3.41585i 0.143200i −0.997433 0.0716000i \(-0.977190\pi\)
0.997433 0.0716000i \(-0.0228105\pi\)
\(570\) 0 0
\(571\) 43.5325i 1.82178i 0.412651 + 0.910889i \(0.364603\pi\)
−0.412651 + 0.910889i \(0.635397\pi\)
\(572\) 0 0
\(573\) 0.686980 0.0286990
\(574\) 0 0
\(575\) −1.48325 23.6374i −0.0618560 0.985746i
\(576\) 0 0
\(577\) 39.0578i 1.62600i −0.582267 0.812998i \(-0.697834\pi\)
0.582267 0.812998i \(-0.302166\pi\)
\(578\) 0 0
\(579\) 2.09247i 0.0869600i
\(580\) 0 0
\(581\) 7.96107 + 31.2973i 0.330281 + 1.29843i
\(582\) 0 0
\(583\) 39.3919 1.63144
\(584\) 0 0
\(585\) 1.35649i 0.0560841i
\(586\) 0 0
\(587\) 10.8977i 0.449798i 0.974382 + 0.224899i \(0.0722052\pi\)
−0.974382 + 0.224899i \(0.927795\pi\)
\(588\) 0 0
\(589\) 30.5399i 1.25838i
\(590\) 0 0
\(591\) 9.87120i 0.406047i
\(592\) 0 0
\(593\) 34.7874i 1.42855i 0.699866 + 0.714275i \(0.253243\pi\)
−0.699866 + 0.714275i \(0.746757\pi\)
\(594\) 0 0
\(595\) 0.0565883 + 0.222465i 0.00231989 + 0.00912019i
\(596\) 0 0
\(597\) 14.7198i 0.602443i
\(598\) 0 0
\(599\) −18.1144 −0.740134 −0.370067 0.929005i \(-0.620665\pi\)
−0.370067 + 0.929005i \(0.620665\pi\)
\(600\) 0 0
\(601\) 0.729850i 0.0297712i 0.999889 + 0.0148856i \(0.00473841\pi\)
−0.999889 + 0.0148856i \(0.995262\pi\)
\(602\) 0 0
\(603\) 4.70293i 0.191518i
\(604\) 0 0
\(605\) 1.35163 0.0549515
\(606\) 0 0
\(607\) 20.2777i 0.823047i −0.911399 0.411523i \(-0.864997\pi\)
0.911399 0.411523i \(-0.135003\pi\)
\(608\) 0 0
\(609\) −8.99126 + 2.28710i −0.364344 + 0.0926779i
\(610\) 0 0
\(611\) −10.2107 −0.413080
\(612\) 0 0
\(613\) 29.7029i 1.19969i 0.800117 + 0.599844i \(0.204771\pi\)
−0.800117 + 0.599844i \(0.795229\pi\)
\(614\) 0 0
\(615\) −1.34013 −0.0540391
\(616\) 0 0
\(617\) 32.0102i 1.28868i 0.764739 + 0.644340i \(0.222868\pi\)
−0.764739 + 0.644340i \(0.777132\pi\)
\(618\) 0 0
\(619\) 47.0226 1.89000 0.944999 0.327073i \(-0.106062\pi\)
0.944999 + 0.327073i \(0.106062\pi\)
\(620\) 0 0
\(621\) 4.78642 0.300350i 0.192072 0.0120526i
\(622\) 0 0
\(623\) −4.46164 17.5400i −0.178752 0.702726i
\(624\) 0 0
\(625\) 24.0802 0.963207
\(626\) 0 0
\(627\) −15.7331 −0.628319
\(628\) 0 0
\(629\) 1.31521i 0.0524407i
\(630\) 0 0
\(631\) 34.8593i 1.38772i −0.720108 0.693862i \(-0.755908\pi\)
0.720108 0.693862i \(-0.244092\pi\)
\(632\) 0 0
\(633\) 5.39018i 0.214240i
\(634\) 0 0
\(635\) 2.25408 0.0894503
\(636\) 0 0
\(637\) 18.2844 + 33.6151i 0.724452 + 1.33188i
\(638\) 0 0
\(639\) −13.9003 −0.549886
\(640\) 0 0
\(641\) 23.7753i 0.939066i 0.882915 + 0.469533i \(0.155578\pi\)
−0.882915 + 0.469533i \(0.844422\pi\)
\(642\) 0 0
\(643\) −25.3414 −0.999367 −0.499684 0.866208i \(-0.666550\pi\)
−0.499684 + 0.866208i \(0.666550\pi\)
\(644\) 0 0
\(645\) −2.84328 −0.111954
\(646\) 0 0
\(647\) 47.5590i 1.86974i −0.354994 0.934868i \(-0.615517\pi\)
0.354994 0.934868i \(-0.384483\pi\)
\(648\) 0 0
\(649\) 51.0399 2.00349
\(650\) 0 0
\(651\) −5.13448 20.1852i −0.201236 0.791119i
\(652\) 0 0
\(653\) 8.23220 0.322151 0.161075 0.986942i \(-0.448504\pi\)
0.161075 + 0.986942i \(0.448504\pi\)
\(654\) 0 0
\(655\) 4.17096i 0.162973i
\(656\) 0 0
\(657\) 9.64759i 0.376388i
\(658\) 0 0
\(659\) 41.6169i 1.62117i 0.585624 + 0.810583i \(0.300849\pi\)
−0.585624 + 0.810583i \(0.699151\pi\)
\(660\) 0 0
\(661\) 4.58613 0.178380 0.0891898 0.996015i \(-0.471572\pi\)
0.0891898 + 0.996015i \(0.471572\pi\)
\(662\) 0 0
\(663\) 1.91136 0.0742309
\(664\) 0 0
\(665\) 0.627873 + 2.46835i 0.0243479 + 0.0957187i
\(666\) 0 0
\(667\) −1.05320 16.7840i −0.0407802 0.649880i
\(668\) 0 0
\(669\) 20.0807 0.776364
\(670\) 0 0
\(671\) 16.3558i 0.631407i
\(672\) 0 0
\(673\) 19.5644 0.754151 0.377076 0.926182i \(-0.376930\pi\)
0.377076 + 0.926182i \(0.376930\pi\)
\(674\) 0 0
\(675\) 4.93843i 0.190080i
\(676\) 0 0
\(677\) 8.22886 0.316261 0.158130 0.987418i \(-0.449453\pi\)
0.158130 + 0.987418i \(0.449453\pi\)
\(678\) 0 0
\(679\) −0.667266 2.62322i −0.0256073 0.100670i
\(680\) 0 0
\(681\) 15.5820i 0.597104i
\(682\) 0 0
\(683\) 45.1557 1.72783 0.863917 0.503634i \(-0.168004\pi\)
0.863917 + 0.503634i \(0.168004\pi\)
\(684\) 0 0
\(685\) 3.70546i 0.141578i
\(686\) 0 0
\(687\) 7.89135i 0.301074i
\(688\) 0 0
\(689\) −53.0981 −2.02288
\(690\) 0 0
\(691\) 18.1471i 0.690349i −0.938539 0.345174i \(-0.887820\pi\)
0.938539 0.345174i \(-0.112180\pi\)
\(692\) 0 0
\(693\) −10.3987 + 2.64510i −0.395013 + 0.100479i
\(694\) 0 0
\(695\) 0.0171886i 0.000652002i
\(696\) 0 0
\(697\) 1.88829i 0.0715242i
\(698\) 0 0
\(699\) 15.5621i 0.588612i
\(700\) 0 0
\(701\) 23.7085i 0.895458i −0.894169 0.447729i \(-0.852233\pi\)
0.894169 0.447729i \(-0.147767\pi\)
\(702\) 0 0
\(703\) 14.5928i 0.550378i
\(704\) 0 0
\(705\) −0.463491 −0.0174561
\(706\) 0 0
\(707\) 25.3868 6.45762i 0.954769 0.242864i
\(708\) 0 0
\(709\) 25.0512i 0.940819i −0.882448 0.470409i \(-0.844106\pi\)
0.882448 0.470409i \(-0.155894\pi\)
\(710\) 0 0
\(711\) 6.98013i 0.261775i
\(712\) 0 0
\(713\) 37.6797 2.36442i 1.41112 0.0885482i
\(714\) 0 0
\(715\) −5.50124 −0.205735
\(716\) 0 0
\(717\) 13.6315i 0.509078i
\(718\) 0 0
\(719\) 11.1554i 0.416028i −0.978126 0.208014i \(-0.933300\pi\)
0.978126 0.208014i \(-0.0667000\pi\)
\(720\) 0 0
\(721\) −4.27173 16.7934i −0.159088 0.625420i
\(722\) 0 0
\(723\) 27.6041i 1.02661i
\(724\) 0 0
\(725\) 17.3171 0.643139
\(726\) 0 0
\(727\) −23.8412 −0.884221 −0.442110 0.896961i \(-0.645770\pi\)
−0.442110 + 0.896961i \(0.645770\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 4.00630i 0.148179i
\(732\) 0 0
\(733\) 21.7749 0.804273 0.402137 0.915580i \(-0.368268\pi\)
0.402137 + 0.915580i \(0.368268\pi\)
\(734\) 0 0
\(735\) 0.829976 + 1.52588i 0.0306141 + 0.0562829i
\(736\) 0 0
\(737\) 19.0727 0.702550
\(738\) 0 0
\(739\) 37.8156 1.39107 0.695534 0.718493i \(-0.255168\pi\)
0.695534 + 0.718493i \(0.255168\pi\)
\(740\) 0 0
\(741\) 21.2074 0.779072
\(742\) 0 0
\(743\) 32.1362i 1.17896i 0.807782 + 0.589482i \(0.200668\pi\)
−0.807782 + 0.589482i \(0.799332\pi\)
\(744\) 0 0
\(745\) 1.32853i 0.0486735i
\(746\) 0 0
\(747\) 12.2060 0.446593
\(748\) 0 0
\(749\) −13.3375 + 3.39264i −0.487341 + 0.123965i
\(750\) 0 0
\(751\) 53.2188i 1.94198i 0.239116 + 0.970991i \(0.423143\pi\)
−0.239116 + 0.970991i \(0.576857\pi\)
\(752\) 0 0
\(753\) 19.8899i 0.724830i
\(754\) 0 0
\(755\) −1.93373 −0.0703756
\(756\) 0 0
\(757\) 0.976339i 0.0354857i 0.999843 + 0.0177428i \(0.00564801\pi\)
−0.999843 + 0.0177428i \(0.994352\pi\)
\(758\) 0 0
\(759\) −1.21806 19.4113i −0.0442129 0.704584i
\(760\) 0 0
\(761\) 42.1794i 1.52900i −0.644623 0.764500i \(-0.722986\pi\)
0.644623 0.764500i \(-0.277014\pi\)
\(762\) 0 0
\(763\) 17.9139 4.55673i 0.648525 0.164965i
\(764\) 0 0
\(765\) 0.0867616 0.00313687
\(766\) 0 0
\(767\) −68.7990 −2.48419
\(768\) 0 0
\(769\) −29.3735 −1.05924 −0.529618 0.848236i \(-0.677665\pi\)
−0.529618 + 0.848236i \(0.677665\pi\)
\(770\) 0 0
\(771\) −5.75374 −0.207216
\(772\) 0 0
\(773\) 21.7233 0.781332 0.390666 0.920532i \(-0.372245\pi\)
0.390666 + 0.920532i \(0.372245\pi\)
\(774\) 0 0
\(775\) 38.8764i 1.39648i
\(776\) 0 0
\(777\) 2.45339 + 9.64500i 0.0880149 + 0.346013i
\(778\) 0 0
\(779\) 20.9515i 0.750664i
\(780\) 0 0
\(781\) 56.3724i 2.01716i
\(782\) 0 0
\(783\) 3.50660i 0.125316i
\(784\) 0 0
\(785\) 2.07204 0.0739543
\(786\) 0 0
\(787\) 13.8818 0.494834 0.247417 0.968909i \(-0.420418\pi\)
0.247417 + 0.968909i \(0.420418\pi\)
\(788\) 0 0
\(789\) −11.0037 −0.391741
\(790\) 0 0
\(791\) 15.1271 3.84788i 0.537859 0.136815i
\(792\) 0 0
\(793\) 22.0467i 0.782902i
\(794\) 0 0
\(795\) −2.41027 −0.0854834
\(796\) 0 0
\(797\) 15.2781 0.541177 0.270588 0.962695i \(-0.412782\pi\)
0.270588 + 0.962695i \(0.412782\pi\)
\(798\) 0 0
\(799\) 0.653077i 0.0231042i
\(800\) 0 0
\(801\) −6.84062 −0.241701
\(802\) 0 0
\(803\) 39.1257 1.38072
\(804\) 0 0
\(805\) −2.99681 + 0.965762i −0.105624 + 0.0340386i
\(806\) 0 0
\(807\) −18.3564 −0.646177
\(808\) 0 0
\(809\) −22.7917 −0.801313 −0.400656 0.916228i \(-0.631218\pi\)
−0.400656 + 0.916228i \(0.631218\pi\)
\(810\) 0 0
\(811\) 38.2968i 1.34478i −0.740196 0.672391i \(-0.765268\pi\)
0.740196 0.672391i \(-0.234732\pi\)
\(812\) 0 0
\(813\) 29.9547 1.05056
\(814\) 0 0
\(815\) 0.870764 0.0305015
\(816\) 0 0
\(817\) 44.4518i 1.55517i
\(818\) 0 0
\(819\) 14.0169 3.56545i 0.489788 0.124587i
\(820\) 0 0
\(821\) −10.3811 −0.362304 −0.181152 0.983455i \(-0.557983\pi\)
−0.181152 + 0.983455i \(0.557983\pi\)
\(822\) 0 0
\(823\) 40.1306 1.39886 0.699432 0.714699i \(-0.253436\pi\)
0.699432 + 0.714699i \(0.253436\pi\)
\(824\) 0 0
\(825\) 20.0277 0.697276
\(826\) 0 0
\(827\) 17.4365i 0.606326i 0.952939 + 0.303163i \(0.0980426\pi\)
−0.952939 + 0.303163i \(0.901957\pi\)
\(828\) 0 0
\(829\) 2.92638i 0.101637i 0.998708 + 0.0508187i \(0.0161831\pi\)
−0.998708 + 0.0508187i \(0.983817\pi\)
\(830\) 0 0
\(831\) 18.6594i 0.647287i
\(832\) 0 0
\(833\) −2.15003 + 1.16947i −0.0744941 + 0.0405198i
\(834\) 0 0
\(835\) 1.75609i 0.0607720i
\(836\) 0 0
\(837\) −7.87222 −0.272104
\(838\) 0 0
\(839\) −53.2071 −1.83691 −0.918456 0.395524i \(-0.870563\pi\)
−0.918456 + 0.395524i \(0.870563\pi\)
\(840\) 0 0
\(841\) −16.7038 −0.575993
\(842\) 0 0
\(843\) −0.0173097 −0.000596178
\(844\) 0 0
\(845\) 4.18953 0.144124
\(846\) 0 0
\(847\) 3.55267 + 13.9666i 0.122071 + 0.479897i
\(848\) 0 0
\(849\) 0.120744i 0.00414394i
\(850\) 0 0
\(851\) −18.0044 + 1.12978i −0.617182 + 0.0387284i
\(852\) 0 0
\(853\) 33.3053i 1.14035i −0.821522 0.570176i \(-0.806875\pi\)
0.821522 0.570176i \(-0.193125\pi\)
\(854\) 0 0
\(855\) 0.962660 0.0329223
\(856\) 0 0
\(857\) 28.3676i 0.969018i −0.874786 0.484509i \(-0.838998\pi\)
0.874786 0.484509i \(-0.161002\pi\)
\(858\) 0 0
\(859\) 40.7811i 1.39143i 0.718316 + 0.695717i \(0.244913\pi\)
−0.718316 + 0.695717i \(0.755087\pi\)
\(860\) 0 0
\(861\) −3.52243 13.8477i −0.120044 0.471929i
\(862\) 0 0
\(863\) 37.0714 1.26193 0.630963 0.775813i \(-0.282660\pi\)
0.630963 + 0.775813i \(0.282660\pi\)
\(864\) 0 0
\(865\) 2.68441i 0.0912726i
\(866\) 0 0
\(867\) 16.8777i 0.573198i
\(868\) 0 0
\(869\) −28.3078 −0.960278
\(870\) 0 0
\(871\) −25.7089 −0.871114
\(872\) 0 0
\(873\) −1.02306 −0.0346253
\(874\) 0 0
\(875\) −1.60849 6.32345i −0.0543769 0.213772i
\(876\) 0 0
\(877\) −0.932860 −0.0315004 −0.0157502 0.999876i \(-0.505014\pi\)
−0.0157502 + 0.999876i \(0.505014\pi\)
\(878\) 0 0
\(879\) 28.5499i 0.962963i
\(880\) 0 0
\(881\) −38.0199 −1.28092 −0.640461 0.767991i \(-0.721257\pi\)
−0.640461 + 0.767991i \(0.721257\pi\)
\(882\) 0 0
\(883\) 19.9372 0.670941 0.335470 0.942051i \(-0.391105\pi\)
0.335470 + 0.942051i \(0.391105\pi\)
\(884\) 0 0
\(885\) −3.12297 −0.104978
\(886\) 0 0
\(887\) 15.8042i 0.530654i 0.964159 + 0.265327i \(0.0854799\pi\)
−0.964159 + 0.265327i \(0.914520\pi\)
\(888\) 0 0
\(889\) 5.92469 + 23.2917i 0.198708 + 0.781179i
\(890\) 0 0
\(891\) 4.05549i 0.135864i
\(892\) 0 0
\(893\) 7.24619i 0.242485i
\(894\) 0 0
\(895\) −2.46939 −0.0825425
\(896\) 0 0
\(897\) 1.64189 + 26.1653i 0.0548210 + 0.873635i
\(898\) 0 0
\(899\) 27.6047i 0.920668i
\(900\) 0 0
\(901\) 3.39617i 0.113143i
\(902\) 0 0
\(903\) −7.47338 29.3801i −0.248699 0.977708i
\(904\) 0 0
\(905\) −1.52366 −0.0506483
\(906\) 0 0
\(907\) 20.0631i 0.666185i −0.942894 0.333093i \(-0.891908\pi\)
0.942894 0.333093i \(-0.108092\pi\)
\(908\) 0 0
\(909\) 9.90088i 0.328391i
\(910\) 0 0
\(911\) 24.2258i 0.802637i −0.915939 0.401318i \(-0.868552\pi\)
0.915939 0.401318i \(-0.131448\pi\)
\(912\) 0 0
\(913\) 49.5011i 1.63825i
\(914\) 0 0
\(915\) 1.00076i 0.0330841i
\(916\) 0 0
\(917\) −43.0991 + 10.9631i −1.42326 + 0.362033i
\(918\) 0 0
\(919\) 28.8911i 0.953030i −0.879166 0.476515i \(-0.841900\pi\)
0.879166 0.476515i \(-0.158100\pi\)
\(920\) 0 0
\(921\) −9.73639 −0.320825
\(922\) 0 0
\(923\) 75.9870i 2.50114i
\(924\) 0 0
\(925\) 18.5762i 0.610781i
\(926\) 0 0
\(927\) −6.54945 −0.215112
\(928\) 0 0
\(929\) 40.6003i 1.33205i −0.745929 0.666026i \(-0.767994\pi\)
0.745929 0.666026i \(-0.232006\pi\)
\(930\) 0 0
\(931\) −23.8556 + 12.9758i −0.781834 + 0.425265i
\(932\) 0 0
\(933\) −24.8161 −0.812442
\(934\) 0 0
\(935\) 0.351861i 0.0115071i
\(936\) 0 0
\(937\) −33.1559 −1.08316 −0.541579 0.840650i \(-0.682173\pi\)
−0.541579 + 0.840650i \(0.682173\pi\)
\(938\) 0 0
\(939\) 24.1571i 0.788336i
\(940\) 0 0
\(941\) 20.1569 0.657095 0.328548 0.944487i \(-0.393441\pi\)
0.328548 + 0.944487i \(0.393441\pi\)
\(942\) 0 0
\(943\) 25.8496 1.62207i 0.841780 0.0528220i
\(944\) 0 0
\(945\) 0.636263 0.161846i 0.0206976 0.00526484i
\(946\) 0 0
\(947\) −13.5374 −0.439907 −0.219954 0.975510i \(-0.570591\pi\)
−0.219954 + 0.975510i \(0.570591\pi\)
\(948\) 0 0
\(949\) −52.7394 −1.71199
\(950\) 0 0
\(951\) 0.882476i 0.0286163i
\(952\) 0 0
\(953\) 29.0790i 0.941962i 0.882143 + 0.470981i \(0.156100\pi\)
−0.882143 + 0.470981i \(0.843900\pi\)
\(954\) 0 0
\(955\) 0.170469i 0.00551626i
\(956\) 0 0
\(957\) 14.2210 0.459698
\(958\) 0 0
\(959\) −38.2891 + 9.73956i −1.23642 + 0.314507i
\(960\) 0 0
\(961\) −30.9719 −0.999094
\(962\) 0 0
\(963\) 5.20163i 0.167620i
\(964\) 0 0
\(965\) 0.519231 0.0167146
\(966\) 0 0
\(967\) 8.24367 0.265099 0.132549 0.991176i \(-0.457684\pi\)
0.132549 + 0.991176i \(0.457684\pi\)
\(968\) 0 0
\(969\) 1.35643i 0.0435747i
\(970\) 0 0
\(971\) 44.4456 1.42633 0.713163 0.700998i \(-0.247262\pi\)
0.713163 + 0.700998i \(0.247262\pi\)
\(972\) 0 0
\(973\) 0.177613 0.0451792i 0.00569400 0.00144838i
\(974\) 0 0
\(975\) −26.9963 −0.864574
\(976\) 0 0
\(977\) 27.6030i 0.883098i −0.897237 0.441549i \(-0.854429\pi\)
0.897237 0.441549i \(-0.145571\pi\)
\(978\) 0 0
\(979\) 27.7420i 0.886639i
\(980\) 0 0
\(981\) 6.98642i 0.223059i
\(982\) 0 0
\(983\) 57.9436 1.84811 0.924057 0.382255i \(-0.124852\pi\)
0.924057 + 0.382255i \(0.124852\pi\)
\(984\) 0 0
\(985\) −2.44947 −0.0780465
\(986\) 0 0
\(987\) −1.21825 4.78932i −0.0387775 0.152446i
\(988\) 0 0
\(989\) 54.8440 3.44148i 1.74394 0.109433i
\(990\) 0 0
\(991\) 59.2749 1.88293 0.941464 0.337113i \(-0.109450\pi\)
0.941464 + 0.337113i \(0.109450\pi\)
\(992\) 0 0
\(993\) 4.48030i 0.142178i
\(994\) 0 0
\(995\) −3.65263 −0.115796
\(996\) 0 0
\(997\) 0.648259i 0.0205306i 0.999947 + 0.0102653i \(0.00326760\pi\)
−0.999947 + 0.0102653i \(0.996732\pi\)
\(998\) 0 0
\(999\) 3.76156 0.119010
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 1932.2.k.a.1609.23 yes 32
3.2 odd 2 5796.2.k.d.5473.17 32
7.6 odd 2 inner 1932.2.k.a.1609.22 yes 32
21.20 even 2 5796.2.k.d.5473.15 32
23.22 odd 2 inner 1932.2.k.a.1609.24 yes 32
69.68 even 2 5796.2.k.d.5473.16 32
161.160 even 2 inner 1932.2.k.a.1609.21 32
483.482 odd 2 5796.2.k.d.5473.18 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.k.a.1609.21 32 161.160 even 2 inner
1932.2.k.a.1609.22 yes 32 7.6 odd 2 inner
1932.2.k.a.1609.23 yes 32 1.1 even 1 trivial
1932.2.k.a.1609.24 yes 32 23.22 odd 2 inner
5796.2.k.d.5473.15 32 21.20 even 2
5796.2.k.d.5473.16 32 69.68 even 2
5796.2.k.d.5473.17 32 3.2 odd 2
5796.2.k.d.5473.18 32 483.482 odd 2