Properties

Label 4012.2.b.a.237.1
Level $4012$
Weight $2$
Character 4012.237
Analytic conductor $32.036$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4012,2,Mod(237,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4012.237");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.b (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: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 237.1
Character \(\chi\) \(=\) 4012.237
Dual form 4012.2.b.a.237.40

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.27953i q^{3} +4.21610i q^{5} +0.547734i q^{7} -7.75529 q^{9} +O(q^{10})\) \(q-3.27953i q^{3} +4.21610i q^{5} +0.547734i q^{7} -7.75529 q^{9} +0.860268i q^{11} +0.171895 q^{13} +13.8268 q^{15} +(-3.68648 + 1.84658i) q^{17} +0.806575 q^{19} +1.79631 q^{21} -1.57303i q^{23} -12.7755 q^{25} +15.5951i q^{27} +1.06772i q^{29} -4.51231i q^{31} +2.82127 q^{33} -2.30930 q^{35} -8.01843i q^{37} -0.563734i q^{39} -4.56732i q^{41} +10.0082 q^{43} -32.6971i q^{45} +2.00791 q^{47} +6.69999 q^{49} +(6.05592 + 12.0899i) q^{51} -4.26805 q^{53} -3.62697 q^{55} -2.64518i q^{57} -1.00000 q^{59} -4.06025i q^{61} -4.24784i q^{63} +0.724726i q^{65} -12.9086 q^{67} -5.15878 q^{69} -7.44608i q^{71} +5.64906i q^{73} +41.8975i q^{75} -0.471198 q^{77} -11.7532i q^{79} +27.8787 q^{81} -4.59528 q^{83} +(-7.78538 - 15.5426i) q^{85} +3.50160 q^{87} -10.9491 q^{89} +0.0941527i q^{91} -14.7982 q^{93} +3.40060i q^{95} -4.06247i q^{97} -6.67163i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times\).

\(n\) \(2007\) \(3129\) \(3777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 3.27953i 1.89344i −0.322065 0.946718i \(-0.604377\pi\)
0.322065 0.946718i \(-0.395623\pi\)
\(4\) 0 0
\(5\) 4.21610i 1.88550i 0.333505 + 0.942748i \(0.391769\pi\)
−0.333505 + 0.942748i \(0.608231\pi\)
\(6\) 0 0
\(7\) 0.547734i 0.207024i 0.994628 + 0.103512i \(0.0330080\pi\)
−0.994628 + 0.103512i \(0.966992\pi\)
\(8\) 0 0
\(9\) −7.75529 −2.58510
\(10\) 0 0
\(11\) 0.860268i 0.259380i 0.991555 + 0.129690i \(0.0413983\pi\)
−0.991555 + 0.129690i \(0.958602\pi\)
\(12\) 0 0
\(13\) 0.171895 0.0476751 0.0238375 0.999716i \(-0.492412\pi\)
0.0238375 + 0.999716i \(0.492412\pi\)
\(14\) 0 0
\(15\) 13.8268 3.57007
\(16\) 0 0
\(17\) −3.68648 + 1.84658i −0.894102 + 0.447863i
\(18\) 0 0
\(19\) 0.806575 0.185041 0.0925205 0.995711i \(-0.470508\pi\)
0.0925205 + 0.995711i \(0.470508\pi\)
\(20\) 0 0
\(21\) 1.79631 0.391987
\(22\) 0 0
\(23\) 1.57303i 0.327999i −0.986460 0.163999i \(-0.947561\pi\)
0.986460 0.163999i \(-0.0524395\pi\)
\(24\) 0 0
\(25\) −12.7755 −2.55510
\(26\) 0 0
\(27\) 15.5951i 3.00128i
\(28\) 0 0
\(29\) 1.06772i 0.198270i 0.995074 + 0.0991350i \(0.0316076\pi\)
−0.995074 + 0.0991350i \(0.968392\pi\)
\(30\) 0 0
\(31\) 4.51231i 0.810434i −0.914221 0.405217i \(-0.867196\pi\)
0.914221 0.405217i \(-0.132804\pi\)
\(32\) 0 0
\(33\) 2.82127 0.491120
\(34\) 0 0
\(35\) −2.30930 −0.390343
\(36\) 0 0
\(37\) 8.01843i 1.31822i −0.752046 0.659111i \(-0.770933\pi\)
0.752046 0.659111i \(-0.229067\pi\)
\(38\) 0 0
\(39\) 0.563734i 0.0902696i
\(40\) 0 0
\(41\) 4.56732i 0.713295i −0.934239 0.356647i \(-0.883920\pi\)
0.934239 0.356647i \(-0.116080\pi\)
\(42\) 0 0
\(43\) 10.0082 1.52624 0.763118 0.646259i \(-0.223667\pi\)
0.763118 + 0.646259i \(0.223667\pi\)
\(44\) 0 0
\(45\) 32.6971i 4.87419i
\(46\) 0 0
\(47\) 2.00791 0.292884 0.146442 0.989219i \(-0.453218\pi\)
0.146442 + 0.989219i \(0.453218\pi\)
\(48\) 0 0
\(49\) 6.69999 0.957141
\(50\) 0 0
\(51\) 6.05592 + 12.0899i 0.847999 + 1.69293i
\(52\) 0 0
\(53\) −4.26805 −0.586262 −0.293131 0.956072i \(-0.594697\pi\)
−0.293131 + 0.956072i \(0.594697\pi\)
\(54\) 0 0
\(55\) −3.62697 −0.489061
\(56\) 0 0
\(57\) 2.64518i 0.350363i
\(58\) 0 0
\(59\) −1.00000 −0.130189
\(60\) 0 0
\(61\) 4.06025i 0.519862i −0.965627 0.259931i \(-0.916300\pi\)
0.965627 0.259931i \(-0.0836998\pi\)
\(62\) 0 0
\(63\) 4.24784i 0.535177i
\(64\) 0 0
\(65\) 0.724726i 0.0898912i
\(66\) 0 0
\(67\) −12.9086 −1.57704 −0.788519 0.615010i \(-0.789152\pi\)
−0.788519 + 0.615010i \(0.789152\pi\)
\(68\) 0 0
\(69\) −5.15878 −0.621044
\(70\) 0 0
\(71\) 7.44608i 0.883687i −0.897092 0.441844i \(-0.854325\pi\)
0.897092 0.441844i \(-0.145675\pi\)
\(72\) 0 0
\(73\) 5.64906i 0.661172i 0.943776 + 0.330586i \(0.107246\pi\)
−0.943776 + 0.330586i \(0.892754\pi\)
\(74\) 0 0
\(75\) 41.8975i 4.83791i
\(76\) 0 0
\(77\) −0.471198 −0.0536980
\(78\) 0 0
\(79\) 11.7532i 1.32233i −0.750239 0.661167i \(-0.770061\pi\)
0.750239 0.661167i \(-0.229939\pi\)
\(80\) 0 0
\(81\) 27.8787 3.09763
\(82\) 0 0
\(83\) −4.59528 −0.504398 −0.252199 0.967675i \(-0.581154\pi\)
−0.252199 + 0.967675i \(0.581154\pi\)
\(84\) 0 0
\(85\) −7.78538 15.5426i −0.844443 1.68583i
\(86\) 0 0
\(87\) 3.50160 0.375411
\(88\) 0 0
\(89\) −10.9491 −1.16060 −0.580300 0.814403i \(-0.697065\pi\)
−0.580300 + 0.814403i \(0.697065\pi\)
\(90\) 0 0
\(91\) 0.0941527i 0.00986989i
\(92\) 0 0
\(93\) −14.7982 −1.53450
\(94\) 0 0
\(95\) 3.40060i 0.348894i
\(96\) 0 0
\(97\) 4.06247i 0.412482i −0.978501 0.206241i \(-0.933877\pi\)
0.978501 0.206241i \(-0.0661230\pi\)
\(98\) 0 0
\(99\) 6.67163i 0.670524i
\(100\) 0 0
\(101\) −7.93186 −0.789249 −0.394625 0.918842i \(-0.629125\pi\)
−0.394625 + 0.918842i \(0.629125\pi\)
\(102\) 0 0
\(103\) 7.67163 0.755909 0.377954 0.925824i \(-0.376628\pi\)
0.377954 + 0.925824i \(0.376628\pi\)
\(104\) 0 0
\(105\) 7.57341i 0.739090i
\(106\) 0 0
\(107\) 6.85449i 0.662648i 0.943517 + 0.331324i \(0.107495\pi\)
−0.943517 + 0.331324i \(0.892505\pi\)
\(108\) 0 0
\(109\) 16.7832i 1.60754i −0.594941 0.803769i \(-0.702825\pi\)
0.594941 0.803769i \(-0.297175\pi\)
\(110\) 0 0
\(111\) −26.2967 −2.49597
\(112\) 0 0
\(113\) 16.9165i 1.59137i −0.605709 0.795686i \(-0.707111\pi\)
0.605709 0.795686i \(-0.292889\pi\)
\(114\) 0 0
\(115\) 6.63203 0.618440
\(116\) 0 0
\(117\) −1.33309 −0.123245
\(118\) 0 0
\(119\) −1.01144 2.01921i −0.0927183 0.185101i
\(120\) 0 0
\(121\) 10.2599 0.932722
\(122\) 0 0
\(123\) −14.9786 −1.35058
\(124\) 0 0
\(125\) 32.7822i 2.93213i
\(126\) 0 0
\(127\) 4.73624 0.420274 0.210137 0.977672i \(-0.432609\pi\)
0.210137 + 0.977672i \(0.432609\pi\)
\(128\) 0 0
\(129\) 32.8222i 2.88983i
\(130\) 0 0
\(131\) 4.24147i 0.370579i −0.982684 0.185289i \(-0.940678\pi\)
0.982684 0.185289i \(-0.0593223\pi\)
\(132\) 0 0
\(133\) 0.441789i 0.0383080i
\(134\) 0 0
\(135\) −65.7505 −5.65890
\(136\) 0 0
\(137\) −2.44650 −0.209019 −0.104509 0.994524i \(-0.533327\pi\)
−0.104509 + 0.994524i \(0.533327\pi\)
\(138\) 0 0
\(139\) 16.8890i 1.43251i −0.697840 0.716254i \(-0.745855\pi\)
0.697840 0.716254i \(-0.254145\pi\)
\(140\) 0 0
\(141\) 6.58500i 0.554557i
\(142\) 0 0
\(143\) 0.147876i 0.0123660i
\(144\) 0 0
\(145\) −4.50160 −0.373837
\(146\) 0 0
\(147\) 21.9728i 1.81228i
\(148\) 0 0
\(149\) 1.35943 0.111369 0.0556844 0.998448i \(-0.482266\pi\)
0.0556844 + 0.998448i \(0.482266\pi\)
\(150\) 0 0
\(151\) −10.3037 −0.838504 −0.419252 0.907870i \(-0.637708\pi\)
−0.419252 + 0.907870i \(0.637708\pi\)
\(152\) 0 0
\(153\) 28.5897 14.3208i 2.31134 1.15777i
\(154\) 0 0
\(155\) 19.0243 1.52807
\(156\) 0 0
\(157\) −2.39530 −0.191166 −0.0955828 0.995421i \(-0.530471\pi\)
−0.0955828 + 0.995421i \(0.530471\pi\)
\(158\) 0 0
\(159\) 13.9972i 1.11005i
\(160\) 0 0
\(161\) 0.861600 0.0679036
\(162\) 0 0
\(163\) 19.1053i 1.49644i −0.663448 0.748222i \(-0.730908\pi\)
0.663448 0.748222i \(-0.269092\pi\)
\(164\) 0 0
\(165\) 11.8948i 0.926005i
\(166\) 0 0
\(167\) 2.55660i 0.197836i −0.995096 0.0989179i \(-0.968462\pi\)
0.995096 0.0989179i \(-0.0315381\pi\)
\(168\) 0 0
\(169\) −12.9705 −0.997727
\(170\) 0 0
\(171\) −6.25523 −0.478349
\(172\) 0 0
\(173\) 20.9882i 1.59570i 0.602855 + 0.797850i \(0.294030\pi\)
−0.602855 + 0.797850i \(0.705970\pi\)
\(174\) 0 0
\(175\) 6.99757i 0.528967i
\(176\) 0 0
\(177\) 3.27953i 0.246504i
\(178\) 0 0
\(179\) −16.7844 −1.25452 −0.627262 0.778808i \(-0.715824\pi\)
−0.627262 + 0.778808i \(0.715824\pi\)
\(180\) 0 0
\(181\) 15.2329i 1.13226i 0.824317 + 0.566128i \(0.191559\pi\)
−0.824317 + 0.566128i \(0.808441\pi\)
\(182\) 0 0
\(183\) −13.3157 −0.984325
\(184\) 0 0
\(185\) 33.8065 2.48550
\(186\) 0 0
\(187\) −1.58856 3.17136i −0.116167 0.231913i
\(188\) 0 0
\(189\) −8.54197 −0.621337
\(190\) 0 0
\(191\) 4.25965 0.308217 0.154109 0.988054i \(-0.450749\pi\)
0.154109 + 0.988054i \(0.450749\pi\)
\(192\) 0 0
\(193\) 1.48174i 0.106658i 0.998577 + 0.0533289i \(0.0169832\pi\)
−0.998577 + 0.0533289i \(0.983017\pi\)
\(194\) 0 0
\(195\) 2.37676 0.170203
\(196\) 0 0
\(197\) 6.10907i 0.435253i 0.976032 + 0.217627i \(0.0698315\pi\)
−0.976032 + 0.217627i \(0.930169\pi\)
\(198\) 0 0
\(199\) 19.2884i 1.36732i −0.729800 0.683661i \(-0.760387\pi\)
0.729800 0.683661i \(-0.239613\pi\)
\(200\) 0 0
\(201\) 42.3341i 2.98602i
\(202\) 0 0
\(203\) −0.584825 −0.0410467
\(204\) 0 0
\(205\) 19.2563 1.34492
\(206\) 0 0
\(207\) 12.1993i 0.847908i
\(208\) 0 0
\(209\) 0.693871i 0.0479960i
\(210\) 0 0
\(211\) 18.5418i 1.27647i 0.769841 + 0.638236i \(0.220336\pi\)
−0.769841 + 0.638236i \(0.779664\pi\)
\(212\) 0 0
\(213\) −24.4196 −1.67320
\(214\) 0 0
\(215\) 42.1956i 2.87771i
\(216\) 0 0
\(217\) 2.47154 0.167779
\(218\) 0 0
\(219\) 18.5262 1.25189
\(220\) 0 0
\(221\) −0.633687 + 0.317418i −0.0426264 + 0.0213519i
\(222\) 0 0
\(223\) 13.9857 0.936555 0.468277 0.883582i \(-0.344875\pi\)
0.468277 + 0.883582i \(0.344875\pi\)
\(224\) 0 0
\(225\) 99.0776 6.60517
\(226\) 0 0
\(227\) 7.46318i 0.495349i 0.968843 + 0.247674i \(0.0796663\pi\)
−0.968843 + 0.247674i \(0.920334\pi\)
\(228\) 0 0
\(229\) −16.2348 −1.07282 −0.536412 0.843956i \(-0.680221\pi\)
−0.536412 + 0.843956i \(0.680221\pi\)
\(230\) 0 0
\(231\) 1.54531i 0.101674i
\(232\) 0 0
\(233\) 28.2485i 1.85062i −0.379214 0.925309i \(-0.623805\pi\)
0.379214 0.925309i \(-0.376195\pi\)
\(234\) 0 0
\(235\) 8.46556i 0.552232i
\(236\) 0 0
\(237\) −38.5448 −2.50375
\(238\) 0 0
\(239\) −0.848385 −0.0548775 −0.0274387 0.999623i \(-0.508735\pi\)
−0.0274387 + 0.999623i \(0.508735\pi\)
\(240\) 0 0
\(241\) 13.0656i 0.841632i 0.907146 + 0.420816i \(0.138256\pi\)
−0.907146 + 0.420816i \(0.861744\pi\)
\(242\) 0 0
\(243\) 44.6435i 2.86388i
\(244\) 0 0
\(245\) 28.2478i 1.80469i
\(246\) 0 0
\(247\) 0.138646 0.00882184
\(248\) 0 0
\(249\) 15.0703i 0.955044i
\(250\) 0 0
\(251\) −6.08938 −0.384358 −0.192179 0.981360i \(-0.561555\pi\)
−0.192179 + 0.981360i \(0.561555\pi\)
\(252\) 0 0
\(253\) 1.35322 0.0850764
\(254\) 0 0
\(255\) −50.9722 + 25.5324i −3.19200 + 1.59890i
\(256\) 0 0
\(257\) 5.57147 0.347539 0.173769 0.984786i \(-0.444405\pi\)
0.173769 + 0.984786i \(0.444405\pi\)
\(258\) 0 0
\(259\) 4.39197 0.272904
\(260\) 0 0
\(261\) 8.28045i 0.512547i
\(262\) 0 0
\(263\) −24.0863 −1.48523 −0.742613 0.669721i \(-0.766414\pi\)
−0.742613 + 0.669721i \(0.766414\pi\)
\(264\) 0 0
\(265\) 17.9945i 1.10540i
\(266\) 0 0
\(267\) 35.9078i 2.19752i
\(268\) 0 0
\(269\) 14.8007i 0.902416i 0.892419 + 0.451208i \(0.149007\pi\)
−0.892419 + 0.451208i \(0.850993\pi\)
\(270\) 0 0
\(271\) 15.3764 0.934047 0.467023 0.884245i \(-0.345326\pi\)
0.467023 + 0.884245i \(0.345326\pi\)
\(272\) 0 0
\(273\) 0.308776 0.0186880
\(274\) 0 0
\(275\) 10.9903i 0.662742i
\(276\) 0 0
\(277\) 1.24867i 0.0750254i −0.999296 0.0375127i \(-0.988057\pi\)
0.999296 0.0375127i \(-0.0119435\pi\)
\(278\) 0 0
\(279\) 34.9942i 2.09505i
\(280\) 0 0
\(281\) 16.8069 1.00262 0.501308 0.865269i \(-0.332852\pi\)
0.501308 + 0.865269i \(0.332852\pi\)
\(282\) 0 0
\(283\) 19.8648i 1.18084i −0.807097 0.590419i \(-0.798963\pi\)
0.807097 0.590419i \(-0.201037\pi\)
\(284\) 0 0
\(285\) 11.1524 0.660609
\(286\) 0 0
\(287\) 2.50168 0.147669
\(288\) 0 0
\(289\) 10.1803 13.6148i 0.598838 0.800870i
\(290\) 0 0
\(291\) −13.3230 −0.781007
\(292\) 0 0
\(293\) 32.6794 1.90915 0.954576 0.297966i \(-0.0963082\pi\)
0.954576 + 0.297966i \(0.0963082\pi\)
\(294\) 0 0
\(295\) 4.21610i 0.245471i
\(296\) 0 0
\(297\) −13.4160 −0.778473
\(298\) 0 0
\(299\) 0.270395i 0.0156373i
\(300\) 0 0
\(301\) 5.48183i 0.315968i
\(302\) 0 0
\(303\) 26.0127i 1.49439i
\(304\) 0 0
\(305\) 17.1184 0.980198
\(306\) 0 0
\(307\) 2.26935 0.129519 0.0647594 0.997901i \(-0.479372\pi\)
0.0647594 + 0.997901i \(0.479372\pi\)
\(308\) 0 0
\(309\) 25.1593i 1.43126i
\(310\) 0 0
\(311\) 18.8502i 1.06890i 0.845200 + 0.534450i \(0.179481\pi\)
−0.845200 + 0.534450i \(0.820519\pi\)
\(312\) 0 0
\(313\) 20.9452i 1.18389i −0.805977 0.591946i \(-0.798360\pi\)
0.805977 0.591946i \(-0.201640\pi\)
\(314\) 0 0
\(315\) 17.9093 1.00908
\(316\) 0 0
\(317\) 17.6609i 0.991936i −0.868341 0.495968i \(-0.834813\pi\)
0.868341 0.495968i \(-0.165187\pi\)
\(318\) 0 0
\(319\) −0.918522 −0.0514274
\(320\) 0 0
\(321\) 22.4795 1.25468
\(322\) 0 0
\(323\) −2.97342 + 1.48941i −0.165446 + 0.0828730i
\(324\) 0 0
\(325\) −2.19604 −0.121814
\(326\) 0 0
\(327\) −55.0409 −3.04377
\(328\) 0 0
\(329\) 1.09980i 0.0606341i
\(330\) 0 0
\(331\) −21.1135 −1.16050 −0.580252 0.814437i \(-0.697046\pi\)
−0.580252 + 0.814437i \(0.697046\pi\)
\(332\) 0 0
\(333\) 62.1853i 3.40773i
\(334\) 0 0
\(335\) 54.4240i 2.97350i
\(336\) 0 0
\(337\) 11.1462i 0.607172i 0.952804 + 0.303586i \(0.0981840\pi\)
−0.952804 + 0.303586i \(0.901816\pi\)
\(338\) 0 0
\(339\) −55.4782 −3.01316
\(340\) 0 0
\(341\) 3.88179 0.210211
\(342\) 0 0
\(343\) 7.50395i 0.405175i
\(344\) 0 0
\(345\) 21.7499i 1.17098i
\(346\) 0 0
\(347\) 29.7640i 1.59782i 0.601452 + 0.798909i \(0.294589\pi\)
−0.601452 + 0.798909i \(0.705411\pi\)
\(348\) 0 0
\(349\) 21.1196 1.13051 0.565253 0.824917i \(-0.308778\pi\)
0.565253 + 0.824917i \(0.308778\pi\)
\(350\) 0 0
\(351\) 2.68072i 0.143086i
\(352\) 0 0
\(353\) −26.7091 −1.42158 −0.710790 0.703404i \(-0.751663\pi\)
−0.710790 + 0.703404i \(0.751663\pi\)
\(354\) 0 0
\(355\) 31.3934 1.66619
\(356\) 0 0
\(357\) −6.62205 + 3.31704i −0.350476 + 0.175556i
\(358\) 0 0
\(359\) −14.3817 −0.759037 −0.379518 0.925184i \(-0.623910\pi\)
−0.379518 + 0.925184i \(0.623910\pi\)
\(360\) 0 0
\(361\) −18.3494 −0.965760
\(362\) 0 0
\(363\) 33.6477i 1.76605i
\(364\) 0 0
\(365\) −23.8170 −1.24664
\(366\) 0 0
\(367\) 3.35410i 0.175083i 0.996161 + 0.0875414i \(0.0279010\pi\)
−0.996161 + 0.0875414i \(0.972099\pi\)
\(368\) 0 0
\(369\) 35.4209i 1.84394i
\(370\) 0 0
\(371\) 2.33776i 0.121370i
\(372\) 0 0
\(373\) −25.2802 −1.30896 −0.654479 0.756080i \(-0.727112\pi\)
−0.654479 + 0.756080i \(0.727112\pi\)
\(374\) 0 0
\(375\) −107.510 −5.55180
\(376\) 0 0
\(377\) 0.183535i 0.00945253i
\(378\) 0 0
\(379\) 20.8208i 1.06949i 0.845013 + 0.534746i \(0.179593\pi\)
−0.845013 + 0.534746i \(0.820407\pi\)
\(380\) 0 0
\(381\) 15.5326i 0.795761i
\(382\) 0 0
\(383\) −12.0017 −0.613260 −0.306630 0.951829i \(-0.599201\pi\)
−0.306630 + 0.951829i \(0.599201\pi\)
\(384\) 0 0
\(385\) 1.98662i 0.101247i
\(386\) 0 0
\(387\) −77.6165 −3.94547
\(388\) 0 0
\(389\) −6.06398 −0.307456 −0.153728 0.988113i \(-0.549128\pi\)
−0.153728 + 0.988113i \(0.549128\pi\)
\(390\) 0 0
\(391\) 2.90473 + 5.79893i 0.146898 + 0.293264i
\(392\) 0 0
\(393\) −13.9100 −0.701667
\(394\) 0 0
\(395\) 49.5525 2.49326
\(396\) 0 0
\(397\) 26.1300i 1.31143i −0.755009 0.655715i \(-0.772367\pi\)
0.755009 0.655715i \(-0.227633\pi\)
\(398\) 0 0
\(399\) 1.44886 0.0725336
\(400\) 0 0
\(401\) 2.13988i 0.106861i 0.998572 + 0.0534303i \(0.0170155\pi\)
−0.998572 + 0.0534303i \(0.982984\pi\)
\(402\) 0 0
\(403\) 0.775642i 0.0386375i
\(404\) 0 0
\(405\) 117.539i 5.84057i
\(406\) 0 0
\(407\) 6.89800 0.341921
\(408\) 0 0
\(409\) −1.99171 −0.0984836 −0.0492418 0.998787i \(-0.515681\pi\)
−0.0492418 + 0.998787i \(0.515681\pi\)
\(410\) 0 0
\(411\) 8.02338i 0.395764i
\(412\) 0 0
\(413\) 0.547734i 0.0269522i
\(414\) 0 0
\(415\) 19.3742i 0.951040i
\(416\) 0 0
\(417\) −55.3880 −2.71236
\(418\) 0 0
\(419\) 33.8010i 1.65129i 0.564192 + 0.825643i \(0.309188\pi\)
−0.564192 + 0.825643i \(0.690812\pi\)
\(420\) 0 0
\(421\) 1.94463 0.0947755 0.0473877 0.998877i \(-0.484910\pi\)
0.0473877 + 0.998877i \(0.484910\pi\)
\(422\) 0 0
\(423\) −15.5720 −0.757134
\(424\) 0 0
\(425\) 47.0966 23.5910i 2.28452 1.14433i
\(426\) 0 0
\(427\) 2.22394 0.107624
\(428\) 0 0
\(429\) 0.484962 0.0234142
\(430\) 0 0
\(431\) 34.7047i 1.67167i 0.548982 + 0.835834i \(0.315015\pi\)
−0.548982 + 0.835834i \(0.684985\pi\)
\(432\) 0 0
\(433\) −0.950499 −0.0456781 −0.0228390 0.999739i \(-0.507271\pi\)
−0.0228390 + 0.999739i \(0.507271\pi\)
\(434\) 0 0
\(435\) 14.7631i 0.707837i
\(436\) 0 0
\(437\) 1.26876i 0.0606932i
\(438\) 0 0
\(439\) 3.21395i 0.153394i 0.997054 + 0.0766968i \(0.0244373\pi\)
−0.997054 + 0.0766968i \(0.975563\pi\)
\(440\) 0 0
\(441\) −51.9604 −2.47430
\(442\) 0 0
\(443\) 25.6989 1.22099 0.610495 0.792020i \(-0.290970\pi\)
0.610495 + 0.792020i \(0.290970\pi\)
\(444\) 0 0
\(445\) 46.1624i 2.18831i
\(446\) 0 0
\(447\) 4.45829i 0.210870i
\(448\) 0 0
\(449\) 19.9069i 0.939464i −0.882809 0.469732i \(-0.844351\pi\)
0.882809 0.469732i \(-0.155649\pi\)
\(450\) 0 0
\(451\) 3.92911 0.185015
\(452\) 0 0
\(453\) 33.7913i 1.58765i
\(454\) 0 0
\(455\) −0.396957 −0.0186096
\(456\) 0 0
\(457\) 3.50137 0.163787 0.0818935 0.996641i \(-0.473903\pi\)
0.0818935 + 0.996641i \(0.473903\pi\)
\(458\) 0 0
\(459\) −28.7977 57.4910i −1.34416 2.68345i
\(460\) 0 0
\(461\) 32.9099 1.53277 0.766383 0.642384i \(-0.222055\pi\)
0.766383 + 0.642384i \(0.222055\pi\)
\(462\) 0 0
\(463\) −5.78529 −0.268865 −0.134433 0.990923i \(-0.542921\pi\)
−0.134433 + 0.990923i \(0.542921\pi\)
\(464\) 0 0
\(465\) 62.3908i 2.89330i
\(466\) 0 0
\(467\) 30.9601 1.43266 0.716332 0.697759i \(-0.245820\pi\)
0.716332 + 0.697759i \(0.245820\pi\)
\(468\) 0 0
\(469\) 7.07049i 0.326485i
\(470\) 0 0
\(471\) 7.85544i 0.361960i
\(472\) 0 0
\(473\) 8.60973i 0.395876i
\(474\) 0 0
\(475\) −10.3044 −0.472798
\(476\) 0 0
\(477\) 33.1000 1.51554
\(478\) 0 0
\(479\) 21.0902i 0.963635i −0.876272 0.481817i \(-0.839977\pi\)
0.876272 0.481817i \(-0.160023\pi\)
\(480\) 0 0
\(481\) 1.37833i 0.0628463i
\(482\) 0 0
\(483\) 2.82564i 0.128571i
\(484\) 0 0
\(485\) 17.1278 0.777732
\(486\) 0 0
\(487\) 13.5548i 0.614227i 0.951673 + 0.307113i \(0.0993631\pi\)
−0.951673 + 0.307113i \(0.900637\pi\)
\(488\) 0 0
\(489\) −62.6564 −2.83342
\(490\) 0 0
\(491\) 39.1903 1.76863 0.884317 0.466887i \(-0.154625\pi\)
0.884317 + 0.466887i \(0.154625\pi\)
\(492\) 0 0
\(493\) −1.97163 3.93611i −0.0887977 0.177274i
\(494\) 0 0
\(495\) 28.1282 1.26427
\(496\) 0 0
\(497\) 4.07847 0.182945
\(498\) 0 0
\(499\) 0.119101i 0.00533171i 0.999996 + 0.00266585i \(0.000848568\pi\)
−0.999996 + 0.00266585i \(0.999151\pi\)
\(500\) 0 0
\(501\) −8.38444 −0.374589
\(502\) 0 0
\(503\) 31.5384i 1.40623i −0.711076 0.703115i \(-0.751792\pi\)
0.711076 0.703115i \(-0.248208\pi\)
\(504\) 0 0
\(505\) 33.4415i 1.48813i
\(506\) 0 0
\(507\) 42.5369i 1.88913i
\(508\) 0 0
\(509\) −18.3532 −0.813493 −0.406746 0.913541i \(-0.633337\pi\)
−0.406746 + 0.913541i \(0.633337\pi\)
\(510\) 0 0
\(511\) −3.09418 −0.136879
\(512\) 0 0
\(513\) 12.5786i 0.555360i
\(514\) 0 0
\(515\) 32.3444i 1.42526i
\(516\) 0 0
\(517\) 1.72734i 0.0759684i
\(518\) 0 0
\(519\) 68.8312 3.02136
\(520\) 0 0
\(521\) 17.8386i 0.781523i 0.920492 + 0.390762i \(0.127788\pi\)
−0.920492 + 0.390762i \(0.872212\pi\)
\(522\) 0 0
\(523\) 21.6052 0.944729 0.472365 0.881403i \(-0.343401\pi\)
0.472365 + 0.881403i \(0.343401\pi\)
\(524\) 0 0
\(525\) −22.9487 −1.00156
\(526\) 0 0
\(527\) 8.33235 + 16.6345i 0.362963 + 0.724611i
\(528\) 0 0
\(529\) 20.5256 0.892417
\(530\) 0 0
\(531\) 7.75529 0.336551
\(532\) 0 0
\(533\) 0.785098i 0.0340064i
\(534\) 0 0
\(535\) −28.8992 −1.24942
\(536\) 0 0
\(537\) 55.0448i 2.37536i
\(538\) 0 0
\(539\) 5.76378i 0.248264i
\(540\) 0 0
\(541\) 30.8707i 1.32724i 0.748072 + 0.663618i \(0.230980\pi\)
−0.748072 + 0.663618i \(0.769020\pi\)
\(542\) 0 0
\(543\) 49.9568 2.14385
\(544\) 0 0
\(545\) 70.7596 3.03101
\(546\) 0 0
\(547\) 13.0862i 0.559526i −0.960069 0.279763i \(-0.909744\pi\)
0.960069 0.279763i \(-0.0902559\pi\)
\(548\) 0 0
\(549\) 31.4884i 1.34389i
\(550\) 0 0
\(551\) 0.861194i 0.0366881i
\(552\) 0 0
\(553\) 6.43761 0.273755
\(554\) 0 0
\(555\) 110.869i 4.70614i
\(556\) 0 0
\(557\) −19.7565 −0.837111 −0.418555 0.908191i \(-0.637463\pi\)
−0.418555 + 0.908191i \(0.637463\pi\)
\(558\) 0 0
\(559\) 1.72036 0.0727634
\(560\) 0 0
\(561\) −10.4006 + 5.20971i −0.439112 + 0.219954i
\(562\) 0 0
\(563\) −41.8113 −1.76213 −0.881067 0.472991i \(-0.843174\pi\)
−0.881067 + 0.472991i \(0.843174\pi\)
\(564\) 0 0
\(565\) 71.3217 3.00053
\(566\) 0 0
\(567\) 15.2701i 0.641284i
\(568\) 0 0
\(569\) −20.2849 −0.850386 −0.425193 0.905103i \(-0.639794\pi\)
−0.425193 + 0.905103i \(0.639794\pi\)
\(570\) 0 0
\(571\) 22.4670i 0.940213i −0.882610 0.470106i \(-0.844216\pi\)
0.882610 0.470106i \(-0.155784\pi\)
\(572\) 0 0
\(573\) 13.9696i 0.583589i
\(574\) 0 0
\(575\) 20.0962i 0.838068i
\(576\) 0 0
\(577\) −25.7152 −1.07054 −0.535269 0.844682i \(-0.679790\pi\)
−0.535269 + 0.844682i \(0.679790\pi\)
\(578\) 0 0
\(579\) 4.85940 0.201950
\(580\) 0 0
\(581\) 2.51699i 0.104422i
\(582\) 0 0
\(583\) 3.67167i 0.152065i
\(584\) 0 0
\(585\) 5.62046i 0.232377i
\(586\) 0 0
\(587\) 6.04268 0.249408 0.124704 0.992194i \(-0.460202\pi\)
0.124704 + 0.992194i \(0.460202\pi\)
\(588\) 0 0
\(589\) 3.63951i 0.149964i
\(590\) 0 0
\(591\) 20.0349 0.824124
\(592\) 0 0
\(593\) −0.888605 −0.0364906 −0.0182453 0.999834i \(-0.505808\pi\)
−0.0182453 + 0.999834i \(0.505808\pi\)
\(594\) 0 0
\(595\) 8.51319 4.26432i 0.349007 0.174820i
\(596\) 0 0
\(597\) −63.2570 −2.58894
\(598\) 0 0
\(599\) −29.5835 −1.20875 −0.604375 0.796700i \(-0.706577\pi\)
−0.604375 + 0.796700i \(0.706577\pi\)
\(600\) 0 0
\(601\) 24.6368i 1.00496i 0.864590 + 0.502478i \(0.167578\pi\)
−0.864590 + 0.502478i \(0.832422\pi\)
\(602\) 0 0
\(603\) 100.110 4.07680
\(604\) 0 0
\(605\) 43.2569i 1.75864i
\(606\) 0 0
\(607\) 20.5939i 0.835880i −0.908475 0.417940i \(-0.862752\pi\)
0.908475 0.417940i \(-0.137248\pi\)
\(608\) 0 0
\(609\) 1.91795i 0.0777192i
\(610\) 0 0
\(611\) 0.345150 0.0139633
\(612\) 0 0
\(613\) 24.9526 1.00783 0.503914 0.863754i \(-0.331893\pi\)
0.503914 + 0.863754i \(0.331893\pi\)
\(614\) 0 0
\(615\) 63.1514i 2.54651i
\(616\) 0 0
\(617\) 47.3422i 1.90593i −0.303086 0.952963i \(-0.598017\pi\)
0.303086 0.952963i \(-0.401983\pi\)
\(618\) 0 0
\(619\) 27.0719i 1.08811i −0.839049 0.544056i \(-0.816888\pi\)
0.839049 0.544056i \(-0.183112\pi\)
\(620\) 0 0
\(621\) 24.5315 0.984415
\(622\) 0 0
\(623\) 5.99719i 0.240272i
\(624\) 0 0
\(625\) 74.3356 2.97342
\(626\) 0 0
\(627\) 2.27557 0.0908774
\(628\) 0 0
\(629\) 14.8067 + 29.5598i 0.590382 + 1.17863i
\(630\) 0 0
\(631\) 44.9128 1.78795 0.893976 0.448115i \(-0.147905\pi\)
0.893976 + 0.448115i \(0.147905\pi\)
\(632\) 0 0
\(633\) 60.8084 2.41692
\(634\) 0 0
\(635\) 19.9685i 0.792424i
\(636\) 0 0
\(637\) 1.15169 0.0456318
\(638\) 0 0
\(639\) 57.7465i 2.28442i
\(640\) 0 0
\(641\) 0.605421i 0.0239127i −0.999929 0.0119563i \(-0.996194\pi\)
0.999929 0.0119563i \(-0.00380591\pi\)
\(642\) 0 0
\(643\) 22.5895i 0.890842i −0.895321 0.445421i \(-0.853054\pi\)
0.895321 0.445421i \(-0.146946\pi\)
\(644\) 0 0
\(645\) 138.381 5.44876
\(646\) 0 0
\(647\) −40.0415 −1.57419 −0.787096 0.616830i \(-0.788416\pi\)
−0.787096 + 0.616830i \(0.788416\pi\)
\(648\) 0 0
\(649\) 0.860268i 0.0337685i
\(650\) 0 0
\(651\) 8.10549i 0.317679i
\(652\) 0 0
\(653\) 22.1525i 0.866893i 0.901179 + 0.433447i \(0.142703\pi\)
−0.901179 + 0.433447i \(0.857297\pi\)
\(654\) 0 0
\(655\) 17.8825 0.698725
\(656\) 0 0
\(657\) 43.8101i 1.70919i
\(658\) 0 0
\(659\) −9.41116 −0.366607 −0.183303 0.983056i \(-0.558679\pi\)
−0.183303 + 0.983056i \(0.558679\pi\)
\(660\) 0 0
\(661\) −42.7745 −1.66374 −0.831869 0.554973i \(-0.812729\pi\)
−0.831869 + 0.554973i \(0.812729\pi\)
\(662\) 0 0
\(663\) 1.04098 + 2.07819i 0.0404284 + 0.0807103i
\(664\) 0 0
\(665\) −1.86263 −0.0722295
\(666\) 0 0
\(667\) 1.67955 0.0650323
\(668\) 0 0
\(669\) 45.8666i 1.77331i
\(670\) 0 0
\(671\) 3.49290 0.134842
\(672\) 0 0
\(673\) 14.6152i 0.563374i 0.959506 + 0.281687i \(0.0908940\pi\)
−0.959506 + 0.281687i \(0.909106\pi\)
\(674\) 0 0
\(675\) 199.235i 7.66856i
\(676\) 0 0
\(677\) 38.5940i 1.48329i −0.670794 0.741644i \(-0.734047\pi\)
0.670794 0.741644i \(-0.265953\pi\)
\(678\) 0 0
\(679\) 2.22516 0.0853936
\(680\) 0 0
\(681\) 24.4757 0.937911
\(682\) 0 0
\(683\) 34.2970i 1.31234i −0.754614 0.656169i \(-0.772176\pi\)
0.754614 0.656169i \(-0.227824\pi\)
\(684\) 0 0
\(685\) 10.3147i 0.394105i
\(686\) 0 0
\(687\) 53.2424i 2.03132i
\(688\) 0 0
\(689\) −0.733656 −0.0279501
\(690\) 0 0
\(691\) 11.1839i 0.425457i −0.977111 0.212729i \(-0.931765\pi\)
0.977111 0.212729i \(-0.0682350\pi\)
\(692\) 0 0
\(693\) 3.65428 0.138815
\(694\) 0 0
\(695\) 71.2057 2.70099
\(696\) 0 0
\(697\) 8.43394 + 16.8373i 0.319458 + 0.637759i
\(698\) 0 0
\(699\) −92.6416 −3.50403
\(700\) 0 0
\(701\) −38.1839 −1.44219 −0.721093 0.692839i \(-0.756360\pi\)
−0.721093 + 0.692839i \(0.756360\pi\)
\(702\) 0 0
\(703\) 6.46747i 0.243925i
\(704\) 0 0
\(705\) 27.7630 1.04562
\(706\) 0 0
\(707\) 4.34455i 0.163394i
\(708\) 0 0
\(709\) 12.4630i 0.468057i −0.972230 0.234028i \(-0.924809\pi\)
0.972230 0.234028i \(-0.0751908\pi\)
\(710\) 0 0
\(711\) 91.1492i 3.41836i
\(712\) 0 0
\(713\) −7.09797 −0.265821
\(714\) 0 0
\(715\) −0.623458 −0.0233160
\(716\) 0 0
\(717\) 2.78230i 0.103907i
\(718\) 0 0
\(719\) 13.2747i 0.495062i 0.968880 + 0.247531i \(0.0796193\pi\)
−0.968880 + 0.247531i \(0.920381\pi\)
\(720\) 0 0
\(721\) 4.20202i 0.156491i
\(722\) 0 0
\(723\) 42.8491 1.59358
\(724\) 0 0
\(725\) 13.6406i 0.506599i
\(726\) 0 0
\(727\) −41.1132 −1.52480 −0.762402 0.647104i \(-0.775980\pi\)
−0.762402 + 0.647104i \(0.775980\pi\)
\(728\) 0 0
\(729\) −62.7736 −2.32495
\(730\) 0 0
\(731\) −36.8950 + 18.4810i −1.36461 + 0.683544i
\(732\) 0 0
\(733\) −25.2280 −0.931818 −0.465909 0.884833i \(-0.654273\pi\)
−0.465909 + 0.884833i \(0.654273\pi\)
\(734\) 0 0
\(735\) 92.6394 3.41706
\(736\) 0 0
\(737\) 11.1049i 0.409053i
\(738\) 0 0
\(739\) 38.1797 1.40446 0.702231 0.711950i \(-0.252188\pi\)
0.702231 + 0.711950i \(0.252188\pi\)
\(740\) 0 0
\(741\) 0.454694i 0.0167036i
\(742\) 0 0
\(743\) 0.766960i 0.0281370i 0.999901 + 0.0140685i \(0.00447829\pi\)
−0.999901 + 0.0140685i \(0.995522\pi\)
\(744\) 0 0
\(745\) 5.73149i 0.209986i
\(746\) 0 0
\(747\) 35.6378 1.30392
\(748\) 0 0
\(749\) −3.75444 −0.137184
\(750\) 0 0
\(751\) 10.0416i 0.366424i 0.983073 + 0.183212i \(0.0586494\pi\)
−0.983073 + 0.183212i \(0.941351\pi\)
\(752\) 0 0
\(753\) 19.9703i 0.727757i
\(754\) 0 0
\(755\) 43.4415i 1.58100i
\(756\) 0 0
\(757\) 25.1639 0.914599 0.457299 0.889313i \(-0.348817\pi\)
0.457299 + 0.889313i \(0.348817\pi\)
\(758\) 0 0
\(759\) 4.43793i 0.161087i
\(760\) 0 0
\(761\) −12.4933 −0.452883 −0.226441 0.974025i \(-0.572709\pi\)
−0.226441 + 0.974025i \(0.572709\pi\)
\(762\) 0 0
\(763\) 9.19273 0.332799
\(764\) 0 0
\(765\) 60.3779 + 120.537i 2.18297 + 4.35803i
\(766\) 0 0
\(767\) −0.171895 −0.00620676
\(768\) 0 0
\(769\) −29.7374 −1.07236 −0.536179 0.844104i \(-0.680133\pi\)
−0.536179 + 0.844104i \(0.680133\pi\)
\(770\) 0 0
\(771\) 18.2718i 0.658042i
\(772\) 0 0
\(773\) −4.05880 −0.145985 −0.0729925 0.997332i \(-0.523255\pi\)
−0.0729925 + 0.997332i \(0.523255\pi\)
\(774\) 0 0
\(775\) 57.6469i 2.07074i
\(776\) 0 0
\(777\) 14.4036i 0.516726i
\(778\) 0 0
\(779\) 3.68388i 0.131989i
\(780\) 0 0
\(781\) 6.40562 0.229211
\(782\) 0 0
\(783\) −16.6512 −0.595064
\(784\) 0 0
\(785\) 10.0988i 0.360442i
\(786\) 0 0
\(787\) 3.55552i 0.126740i −0.997990 0.0633702i \(-0.979815\pi\)
0.997990 0.0633702i \(-0.0201849\pi\)
\(788\) 0 0
\(789\) 78.9917i 2.81218i
\(790\) 0 0
\(791\) 9.26576 0.329452
\(792\) 0 0
\(793\) 0.697936i 0.0247845i
\(794\) 0 0
\(795\) −59.0135 −2.09299
\(796\) 0 0
\(797\) 37.8515 1.34077 0.670384 0.742014i \(-0.266129\pi\)
0.670384 + 0.742014i \(0.266129\pi\)
\(798\) 0 0
\(799\) −7.40213 + 3.70778i −0.261868 + 0.131172i
\(800\) 0 0
\(801\) 84.9133 3.00026
\(802\) 0 0
\(803\) −4.85970 −0.171495
\(804\) 0 0
\(805\) 3.63259i 0.128032i
\(806\) 0 0
\(807\) 48.5394 1.70867
\(808\) 0 0
\(809\) 39.7579i 1.39781i −0.715212 0.698907i \(-0.753670\pi\)
0.715212 0.698907i \(-0.246330\pi\)
\(810\) 0 0
\(811\) 11.8644i 0.416617i 0.978063 + 0.208308i \(0.0667958\pi\)
−0.978063 + 0.208308i \(0.933204\pi\)
\(812\) 0 0
\(813\) 50.4271i 1.76856i
\(814\) 0 0
\(815\) 80.5499 2.82154
\(816\) 0 0
\(817\) 8.07237 0.282416
\(818\) 0 0
\(819\) 0.730182i 0.0255146i
\(820\) 0 0
\(821\) 37.6923i 1.31547i −0.753250 0.657734i \(-0.771515\pi\)
0.753250 0.657734i \(-0.228485\pi\)
\(822\) 0 0
\(823\) 9.43692i 0.328950i 0.986381 + 0.164475i \(0.0525931\pi\)
−0.986381 + 0.164475i \(0.947407\pi\)
\(824\) 0 0
\(825\) −36.0431 −1.25486
\(826\) 0 0
\(827\) 23.6807i 0.823460i −0.911306 0.411730i \(-0.864925\pi\)
0.911306 0.411730i \(-0.135075\pi\)
\(828\) 0 0
\(829\) −36.5718 −1.27019 −0.635095 0.772434i \(-0.719039\pi\)
−0.635095 + 0.772434i \(0.719039\pi\)
\(830\) 0 0
\(831\) −4.09505 −0.142056
\(832\) 0 0
\(833\) −24.6994 + 12.3721i −0.855782 + 0.428668i
\(834\) 0 0
\(835\) 10.7789 0.373019
\(836\) 0 0
\(837\) 70.3699 2.43234
\(838\) 0 0
\(839\) 49.1775i 1.69780i 0.528557 + 0.848898i \(0.322733\pi\)
−0.528557 + 0.848898i \(0.677267\pi\)
\(840\) 0 0
\(841\) 27.8600 0.960689
\(842\) 0 0
\(843\) 55.1187i 1.89839i
\(844\) 0 0
\(845\) 54.6847i 1.88121i
\(846\) 0 0
\(847\) 5.61972i 0.193096i
\(848\) 0 0
\(849\) −65.1470 −2.23584
\(850\) 0 0
\(851\) −12.6132 −0.432375
\(852\) 0 0
\(853\) 2.31165i 0.0791494i 0.999217 + 0.0395747i \(0.0126003\pi\)
−0.999217 + 0.0395747i \(0.987400\pi\)
\(854\) 0 0
\(855\) 26.3726i 0.901926i
\(856\) 0 0
\(857\) 46.4366i 1.58624i −0.609063 0.793122i \(-0.708454\pi\)
0.609063 0.793122i \(-0.291546\pi\)
\(858\) 0 0
\(859\) −8.87291 −0.302740 −0.151370 0.988477i \(-0.548368\pi\)
−0.151370 + 0.988477i \(0.548368\pi\)
\(860\) 0 0
\(861\) 8.20431i 0.279602i
\(862\) 0 0
\(863\) −16.6477 −0.566695 −0.283348 0.959017i \(-0.591445\pi\)
−0.283348 + 0.959017i \(0.591445\pi\)
\(864\) 0 0
\(865\) −88.4882 −3.00869
\(866\) 0 0
\(867\) −44.6501 33.3864i −1.51640 1.13386i
\(868\) 0 0
\(869\) 10.1109 0.342988
\(870\) 0 0
\(871\) −2.21892 −0.0751854
\(872\) 0 0
\(873\) 31.5057i 1.06630i
\(874\) 0 0
\(875\) 17.9559 0.607022
\(876\) 0 0
\(877\) 51.7439i 1.74727i −0.486584 0.873634i \(-0.661757\pi\)
0.486584 0.873634i \(-0.338243\pi\)
\(878\) 0 0
\(879\) 107.173i 3.61486i
\(880\) 0 0
\(881\) 9.30770i 0.313585i 0.987632 + 0.156792i \(0.0501153\pi\)
−0.987632 + 0.156792i \(0.949885\pi\)
\(882\) 0 0
\(883\) 27.3278 0.919652 0.459826 0.888009i \(-0.347912\pi\)
0.459826 + 0.888009i \(0.347912\pi\)
\(884\) 0 0
\(885\) −13.8268 −0.464783
\(886\) 0 0
\(887\) 35.3323i 1.18634i −0.805076 0.593172i \(-0.797876\pi\)
0.805076 0.593172i \(-0.202124\pi\)
\(888\) 0 0
\(889\) 2.59420i 0.0870068i
\(890\) 0 0
\(891\) 23.9831i 0.803465i
\(892\) 0 0
\(893\) 1.61953 0.0541956
\(894\) 0 0
\(895\) 70.7646i 2.36540i
\(896\) 0 0
\(897\) −0.886768 −0.0296083
\(898\) 0 0
\(899\) 4.81786 0.160685
\(900\) 0 0
\(901\) 15.7341 7.88132i 0.524178 0.262565i
\(902\) 0 0
\(903\) 17.9778 0.598264
\(904\) 0 0
\(905\) −64.2236 −2.13486
\(906\) 0 0
\(907\) 47.5465i 1.57875i −0.613908 0.789377i \(-0.710404\pi\)
0.613908 0.789377i \(-0.289596\pi\)
\(908\) 0 0
\(909\) 61.5139 2.04029
\(910\) 0 0
\(911\) 43.6406i 1.44588i 0.690912 + 0.722938i \(0.257209\pi\)
−0.690912 + 0.722938i \(0.742791\pi\)
\(912\) 0 0
\(913\) 3.95317i 0.130831i
\(914\) 0 0
\(915\) 56.1403i 1.85594i
\(916\) 0 0
\(917\) 2.32320 0.0767188
\(918\) 0 0
\(919\) −3.68843 −0.121670 −0.0608351 0.998148i \(-0.519376\pi\)
−0.0608351 + 0.998148i \(0.519376\pi\)
\(920\) 0 0
\(921\) 7.44240i 0.245236i
\(922\) 0 0
\(923\) 1.27994i 0.0421298i
\(924\) 0 0
\(925\) 102.439i 3.36819i
\(926\) 0 0
\(927\) −59.4958 −1.95410
\(928\) 0 0
\(929\) 1.94636i 0.0638579i −0.999490 0.0319289i \(-0.989835\pi\)
0.999490 0.0319289i \(-0.0101650\pi\)
\(930\) 0 0
\(931\) 5.40404 0.177110
\(932\) 0 0
\(933\) 61.8199 2.02389
\(934\) 0 0
\(935\) 13.3708 6.69751i 0.437271 0.219032i
\(936\) 0 0
\(937\) −34.4794 −1.12639 −0.563196 0.826323i \(-0.690428\pi\)
−0.563196 + 0.826323i \(0.690428\pi\)
\(938\) 0 0
\(939\) −68.6903 −2.24162
\(940\) 0 0
\(941\) 26.1931i 0.853870i −0.904282 0.426935i \(-0.859593\pi\)
0.904282 0.426935i \(-0.140407\pi\)
\(942\) 0 0
\(943\) −7.18451 −0.233960
\(944\) 0 0
\(945\) 36.0138i 1.17153i
\(946\) 0 0
\(947\) 14.1518i 0.459873i 0.973206 + 0.229937i \(0.0738519\pi\)
−0.973206 + 0.229937i \(0.926148\pi\)
\(948\) 0 0
\(949\) 0.971044i 0.0315214i
\(950\) 0 0
\(951\) −57.9194 −1.87817
\(952\) 0 0
\(953\) −32.4551 −1.05133 −0.525663 0.850693i \(-0.676183\pi\)
−0.525663 + 0.850693i \(0.676183\pi\)
\(954\) 0 0
\(955\) 17.9591i 0.581142i
\(956\) 0 0
\(957\) 3.01232i 0.0973744i
\(958\) 0 0
\(959\) 1.34003i 0.0432720i
\(960\) 0 0
\(961\) 10.6391 0.343197
\(962\) 0 0
\(963\) 53.1586i 1.71301i
\(964\) 0 0
\(965\) −6.24715 −0.201103
\(966\) 0 0
\(967\) −4.77681 −0.153612 −0.0768059 0.997046i \(-0.524472\pi\)
−0.0768059 + 0.997046i \(0.524472\pi\)
\(968\) 0 0
\(969\) 4.88456 + 9.75142i 0.156915 + 0.313261i
\(970\) 0 0
\(971\) 48.0344 1.54150 0.770748 0.637140i \(-0.219883\pi\)
0.770748 + 0.637140i \(0.219883\pi\)
\(972\) 0 0
\(973\) 9.25069 0.296564
\(974\) 0 0
\(975\) 7.20197i 0.230648i
\(976\) 0 0
\(977\) 37.4960 1.19960 0.599801 0.800149i \(-0.295246\pi\)
0.599801 + 0.800149i \(0.295246\pi\)
\(978\) 0 0
\(979\) 9.41914i 0.301037i
\(980\) 0 0
\(981\) 130.159i 4.15564i
\(982\) 0 0
\(983\) 6.83044i 0.217857i 0.994050 + 0.108929i \(0.0347420\pi\)
−0.994050 + 0.108929i \(0.965258\pi\)
\(984\) 0 0
\(985\) −25.7564 −0.820668
\(986\) 0 0
\(987\) 3.60683 0.114807
\(988\) 0 0
\(989\) 15.7432i 0.500603i
\(990\) 0 0
\(991\) 57.2231i 1.81775i −0.417068 0.908875i \(-0.636942\pi\)
0.417068 0.908875i \(-0.363058\pi\)
\(992\) 0 0
\(993\) 69.2424i 2.19734i
\(994\) 0 0
\(995\) 81.3220 2.57808
\(996\) 0 0
\(997\) 41.2750i 1.30719i 0.756843 + 0.653596i \(0.226741\pi\)
−0.756843 + 0.653596i \(0.773259\pi\)
\(998\) 0 0
\(999\) 125.048 3.95635
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 4012.2.b.a.237.1 40
17.16 even 2 inner 4012.2.b.a.237.40 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4012.2.b.a.237.1 40 1.1 even 1 trivial
4012.2.b.a.237.40 yes 40 17.16 even 2 inner