Properties

Label 896.2.q.d.703.6
Level $896$
Weight $2$
Character 896.703
Analytic conductor $7.155$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [896,2,Mod(703,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.q (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 24x^{14} + 226x^{12} - 972x^{10} + 1575x^{8} + 252x^{6} + 550x^{4} + 156x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 703.6
Root \(-0.526379 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 896.703
Dual form 896.2.q.d.831.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.20586 + 0.696202i) q^{3} +(2.07821 + 3.59957i) q^{5} +(0.632797 - 2.56896i) q^{7} +(-0.530605 - 0.919035i) q^{9} +O(q^{10})\) \(q+(1.20586 + 0.696202i) q^{3} +(2.07821 + 3.59957i) q^{5} +(0.632797 - 2.56896i) q^{7} +(-0.530605 - 0.919035i) q^{9} +(-2.36310 + 4.09302i) q^{11} +3.68997 q^{13} +5.78742i q^{15} +(2.28389 + 1.31860i) q^{17} +(-0.698106 + 0.403051i) q^{19} +(2.55158 - 2.65725i) q^{21} +(4.73969 - 2.73646i) q^{23} +(-6.13792 + 10.6312i) q^{25} -5.65485i q^{27} +3.88464i q^{29} +(0.515984 - 0.893710i) q^{31} +(-5.69914 + 3.29040i) q^{33} +(10.5622 - 3.06105i) q^{35} +(1.03237 - 0.596037i) q^{37} +(4.44957 + 2.56896i) q^{39} +6.47477i q^{41} -10.2758 q^{43} +(2.20542 - 3.81990i) q^{45} +(-3.98009 - 6.89371i) q^{47} +(-6.19914 - 3.25126i) q^{49} +(1.83603 + 3.18010i) q^{51} +(-2.32572 - 1.34275i) q^{53} -19.6441 q^{55} -1.12242 q^{57} +(2.38243 + 1.37550i) q^{59} +(0.819529 + 1.41947i) q^{61} +(-2.69673 + 0.781542i) q^{63} +(7.66853 + 13.2823i) q^{65} +(2.45914 - 4.25936i) q^{67} +7.62052 q^{69} +11.7693i q^{71} +(-10.5226 - 6.07521i) q^{73} +(-14.8029 + 8.54647i) q^{75} +(9.01944 + 8.66078i) q^{77} +(12.2126 - 7.05096i) q^{79} +(2.34510 - 4.06183i) q^{81} -7.37993i q^{83} +10.9614i q^{85} +(-2.70449 + 4.68432i) q^{87} +(15.8983 - 9.17887i) q^{89} +(2.33500 - 9.47938i) q^{91} +(1.24441 - 0.718458i) q^{93} +(-2.90162 - 1.67525i) q^{95} -1.83807i q^{97} +5.01550 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{3} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{3} + 8 q^{9} - 4 q^{11} - 12 q^{19} - 16 q^{25} + 24 q^{33} + 20 q^{35} + 16 q^{49} - 52 q^{51} + 48 q^{57} + 60 q^{59} + 24 q^{65} + 12 q^{67} - 24 q^{73} - 120 q^{75} - 32 q^{81} + 24 q^{89} + 72 q^{91} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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
\(3\) 1.20586 + 0.696202i 0.696202 + 0.401952i 0.805931 0.592009i \(-0.201665\pi\)
−0.109729 + 0.993962i \(0.534998\pi\)
\(4\) 0 0
\(5\) 2.07821 + 3.59957i 0.929404 + 1.60978i 0.784320 + 0.620356i \(0.213012\pi\)
0.145084 + 0.989419i \(0.453655\pi\)
\(6\) 0 0
\(7\) 0.632797 2.56896i 0.239175 0.970977i
\(8\) 0 0
\(9\) −0.530605 0.919035i −0.176868 0.306345i
\(10\) 0 0
\(11\) −2.36310 + 4.09302i −0.712503 + 1.23409i 0.251412 + 0.967880i \(0.419105\pi\)
−0.963915 + 0.266211i \(0.914228\pi\)
\(12\) 0 0
\(13\) 3.68997 1.02341 0.511706 0.859160i \(-0.329014\pi\)
0.511706 + 0.859160i \(0.329014\pi\)
\(14\) 0 0
\(15\) 5.78742i 1.49431i
\(16\) 0 0
\(17\) 2.28389 + 1.31860i 0.553925 + 0.319809i 0.750703 0.660639i \(-0.229715\pi\)
−0.196779 + 0.980448i \(0.563048\pi\)
\(18\) 0 0
\(19\) −0.698106 + 0.403051i −0.160156 + 0.0924663i −0.577936 0.816082i \(-0.696142\pi\)
0.417780 + 0.908548i \(0.362808\pi\)
\(20\) 0 0
\(21\) 2.55158 2.65725i 0.556800 0.579859i
\(22\) 0 0
\(23\) 4.73969 2.73646i 0.988294 0.570592i 0.0835302 0.996505i \(-0.473380\pi\)
0.904764 + 0.425913i \(0.140047\pi\)
\(24\) 0 0
\(25\) −6.13792 + 10.6312i −1.22758 + 2.12624i
\(26\) 0 0
\(27\) 5.65485i 1.08828i
\(28\) 0 0
\(29\) 3.88464i 0.721360i 0.932690 + 0.360680i \(0.117455\pi\)
−0.932690 + 0.360680i \(0.882545\pi\)
\(30\) 0 0
\(31\) 0.515984 0.893710i 0.0926734 0.160515i −0.815962 0.578106i \(-0.803792\pi\)
0.908635 + 0.417591i \(0.137125\pi\)
\(32\) 0 0
\(33\) −5.69914 + 3.29040i −0.992092 + 0.572785i
\(34\) 0 0
\(35\) 10.5622 3.06105i 1.78534 0.517412i
\(36\) 0 0
\(37\) 1.03237 0.596037i 0.169720 0.0979879i −0.412734 0.910852i \(-0.635426\pi\)
0.582454 + 0.812864i \(0.302093\pi\)
\(38\) 0 0
\(39\) 4.44957 + 2.56896i 0.712502 + 0.411363i
\(40\) 0 0
\(41\) 6.47477i 1.01119i 0.862771 + 0.505595i \(0.168727\pi\)
−0.862771 + 0.505595i \(0.831273\pi\)
\(42\) 0 0
\(43\) −10.2758 −1.56705 −0.783526 0.621359i \(-0.786581\pi\)
−0.783526 + 0.621359i \(0.786581\pi\)
\(44\) 0 0
\(45\) 2.20542 3.81990i 0.328765 0.569437i
\(46\) 0 0
\(47\) −3.98009 6.89371i −0.580555 1.00555i −0.995414 0.0956650i \(-0.969502\pi\)
0.414858 0.909886i \(-0.363831\pi\)
\(48\) 0 0
\(49\) −6.19914 3.25126i −0.885591 0.464466i
\(50\) 0 0
\(51\) 1.83603 + 3.18010i 0.257096 + 0.445303i
\(52\) 0 0
\(53\) −2.32572 1.34275i −0.319462 0.184441i 0.331691 0.943388i \(-0.392381\pi\)
−0.651153 + 0.758947i \(0.725714\pi\)
\(54\) 0 0
\(55\) −19.6441 −2.64881
\(56\) 0 0
\(57\) −1.12242 −0.148668
\(58\) 0 0
\(59\) 2.38243 + 1.37550i 0.310166 + 0.179074i 0.647001 0.762489i \(-0.276023\pi\)
−0.336835 + 0.941564i \(0.609356\pi\)
\(60\) 0 0
\(61\) 0.819529 + 1.41947i 0.104930 + 0.181744i 0.913710 0.406368i \(-0.133205\pi\)
−0.808780 + 0.588112i \(0.799871\pi\)
\(62\) 0 0
\(63\) −2.69673 + 0.781542i −0.339756 + 0.0984650i
\(64\) 0 0
\(65\) 7.66853 + 13.2823i 0.951164 + 1.64746i
\(66\) 0 0
\(67\) 2.45914 4.25936i 0.300432 0.520363i −0.675802 0.737083i \(-0.736202\pi\)
0.976234 + 0.216720i \(0.0695358\pi\)
\(68\) 0 0
\(69\) 7.62052 0.917403
\(70\) 0 0
\(71\) 11.7693i 1.39676i 0.715729 + 0.698378i \(0.246095\pi\)
−0.715729 + 0.698378i \(0.753905\pi\)
\(72\) 0 0
\(73\) −10.5226 6.07521i −1.23157 0.711049i −0.264215 0.964464i \(-0.585113\pi\)
−0.967358 + 0.253415i \(0.918446\pi\)
\(74\) 0 0
\(75\) −14.8029 + 8.54647i −1.70929 + 0.986862i
\(76\) 0 0
\(77\) 9.01944 + 8.66078i 1.02786 + 0.986987i
\(78\) 0 0
\(79\) 12.2126 7.05096i 1.37403 0.793295i 0.382594 0.923916i \(-0.375031\pi\)
0.991432 + 0.130622i \(0.0416973\pi\)
\(80\) 0 0
\(81\) 2.34510 4.06183i 0.260567 0.451315i
\(82\) 0 0
\(83\) 7.37993i 0.810053i −0.914305 0.405026i \(-0.867262\pi\)
0.914305 0.405026i \(-0.132738\pi\)
\(84\) 0 0
\(85\) 10.9614i 1.18893i
\(86\) 0 0
\(87\) −2.70449 + 4.68432i −0.289952 + 0.502212i
\(88\) 0 0
\(89\) 15.8983 9.17887i 1.68521 0.972958i 0.727116 0.686514i \(-0.240860\pi\)
0.958097 0.286444i \(-0.0924733\pi\)
\(90\) 0 0
\(91\) 2.33500 9.47938i 0.244775 0.993710i
\(92\) 0 0
\(93\) 1.24441 0.718458i 0.129039 0.0745006i
\(94\) 0 0
\(95\) −2.90162 1.67525i −0.297700 0.171877i
\(96\) 0 0
\(97\) 1.83807i 0.186628i −0.995637 0.0933139i \(-0.970254\pi\)
0.995637 0.0933139i \(-0.0297460\pi\)
\(98\) 0 0
\(99\) 5.01550 0.504077
\(100\) 0 0
\(101\) 1.39892 2.42300i 0.139198 0.241097i −0.787996 0.615681i \(-0.788881\pi\)
0.927193 + 0.374584i \(0.122214\pi\)
\(102\) 0 0
\(103\) 1.50146 + 2.60060i 0.147943 + 0.256244i 0.930467 0.366376i \(-0.119402\pi\)
−0.782524 + 0.622620i \(0.786068\pi\)
\(104\) 0 0
\(105\) 14.8677 + 3.66226i 1.45094 + 0.357400i
\(106\) 0 0
\(107\) −5.10982 8.85047i −0.493985 0.855607i 0.505991 0.862539i \(-0.331127\pi\)
−0.999976 + 0.00693167i \(0.997794\pi\)
\(108\) 0 0
\(109\) 4.70239 + 2.71493i 0.450407 + 0.260043i 0.708002 0.706210i \(-0.249597\pi\)
−0.257595 + 0.966253i \(0.582930\pi\)
\(110\) 0 0
\(111\) 1.65985 0.157546
\(112\) 0 0
\(113\) 8.70807 0.819186 0.409593 0.912268i \(-0.365671\pi\)
0.409593 + 0.912268i \(0.365671\pi\)
\(114\) 0 0
\(115\) 19.7002 + 11.3739i 1.83705 + 1.06062i
\(116\) 0 0
\(117\) −1.95792 3.39121i −0.181009 0.313517i
\(118\) 0 0
\(119\) 4.83268 5.03282i 0.443011 0.461358i
\(120\) 0 0
\(121\) −5.66853 9.81818i −0.515321 0.892562i
\(122\) 0 0
\(123\) −4.50775 + 7.80766i −0.406450 + 0.703993i
\(124\) 0 0
\(125\) −30.2415 −2.70488
\(126\) 0 0
\(127\) 7.76928i 0.689412i −0.938711 0.344706i \(-0.887979\pi\)
0.938711 0.344706i \(-0.112021\pi\)
\(128\) 0 0
\(129\) −12.3912 7.15407i −1.09099 0.629881i
\(130\) 0 0
\(131\) −0.595007 + 0.343528i −0.0519860 + 0.0300141i −0.525768 0.850628i \(-0.676222\pi\)
0.473782 + 0.880642i \(0.342889\pi\)
\(132\) 0 0
\(133\) 0.593665 + 2.04846i 0.0514773 + 0.177624i
\(134\) 0 0
\(135\) 20.3550 11.7520i 1.75188 1.01145i
\(136\) 0 0
\(137\) 0.222679 0.385692i 0.0190248 0.0329519i −0.856356 0.516385i \(-0.827277\pi\)
0.875381 + 0.483434i \(0.160611\pi\)
\(138\) 0 0
\(139\) 13.1355i 1.11414i 0.830465 + 0.557071i \(0.188075\pi\)
−0.830465 + 0.557071i \(0.811925\pi\)
\(140\) 0 0
\(141\) 11.0838i 0.933422i
\(142\) 0 0
\(143\) −8.71978 + 15.1031i −0.729184 + 1.26298i
\(144\) 0 0
\(145\) −13.9830 + 8.07310i −1.16123 + 0.670435i
\(146\) 0 0
\(147\) −5.21174 8.23641i −0.429857 0.679328i
\(148\) 0 0
\(149\) −4.83527 + 2.79164i −0.396120 + 0.228700i −0.684809 0.728723i \(-0.740114\pi\)
0.288688 + 0.957423i \(0.406781\pi\)
\(150\) 0 0
\(151\) −7.89592 4.55871i −0.642561 0.370983i 0.143040 0.989717i \(-0.454312\pi\)
−0.785600 + 0.618734i \(0.787646\pi\)
\(152\) 0 0
\(153\) 2.79863i 0.226256i
\(154\) 0 0
\(155\) 4.28929 0.344524
\(156\) 0 0
\(157\) 9.58580 16.6031i 0.765030 1.32507i −0.175200 0.984533i \(-0.556057\pi\)
0.940231 0.340539i \(-0.110609\pi\)
\(158\) 0 0
\(159\) −1.86965 3.23834i −0.148273 0.256817i
\(160\) 0 0
\(161\) −4.03061 13.9077i −0.317656 1.09608i
\(162\) 0 0
\(163\) −4.79414 8.30370i −0.375506 0.650396i 0.614896 0.788608i \(-0.289198\pi\)
−0.990403 + 0.138212i \(0.955864\pi\)
\(164\) 0 0
\(165\) −23.6880 13.6763i −1.84411 1.06470i
\(166\) 0 0
\(167\) 7.92465 0.613228 0.306614 0.951834i \(-0.400804\pi\)
0.306614 + 0.951834i \(0.400804\pi\)
\(168\) 0 0
\(169\) 0.615852 0.0473732
\(170\) 0 0
\(171\) 0.740837 + 0.427722i 0.0566532 + 0.0327088i
\(172\) 0 0
\(173\) −6.70800 11.6186i −0.510000 0.883346i −0.999933 0.0115858i \(-0.996312\pi\)
0.489933 0.871760i \(-0.337021\pi\)
\(174\) 0 0
\(175\) 23.4271 + 22.4955i 1.77092 + 1.70050i
\(176\) 0 0
\(177\) 1.91525 + 3.31730i 0.143959 + 0.249344i
\(178\) 0 0
\(179\) −6.67760 + 11.5659i −0.499107 + 0.864479i −0.999999 0.00103054i \(-0.999672\pi\)
0.500892 + 0.865510i \(0.333005\pi\)
\(180\) 0 0
\(181\) −3.49016 −0.259421 −0.129711 0.991552i \(-0.541405\pi\)
−0.129711 + 0.991552i \(0.541405\pi\)
\(182\) 0 0
\(183\) 2.28223i 0.168707i
\(184\) 0 0
\(185\) 4.29095 + 2.47738i 0.315477 + 0.182141i
\(186\) 0 0
\(187\) −10.7941 + 6.23200i −0.789346 + 0.455729i
\(188\) 0 0
\(189\) −14.5271 3.57837i −1.05669 0.260288i
\(190\) 0 0
\(191\) −15.8821 + 9.16956i −1.14919 + 0.663486i −0.948690 0.316207i \(-0.897590\pi\)
−0.200501 + 0.979693i \(0.564257\pi\)
\(192\) 0 0
\(193\) 4.91525 8.51345i 0.353807 0.612812i −0.633106 0.774065i \(-0.718220\pi\)
0.986913 + 0.161253i \(0.0515536\pi\)
\(194\) 0 0
\(195\) 21.3554i 1.52929i
\(196\) 0 0
\(197\) 14.7895i 1.05371i −0.849956 0.526853i \(-0.823372\pi\)
0.849956 0.526853i \(-0.176628\pi\)
\(198\) 0 0
\(199\) 0.975475 1.68957i 0.0691496 0.119771i −0.829378 0.558688i \(-0.811305\pi\)
0.898527 + 0.438918i \(0.144638\pi\)
\(200\) 0 0
\(201\) 5.93075 3.42412i 0.418323 0.241519i
\(202\) 0 0
\(203\) 9.97949 + 2.45819i 0.700423 + 0.172531i
\(204\) 0 0
\(205\) −23.3064 + 13.4560i −1.62779 + 0.939804i
\(206\) 0 0
\(207\) −5.02981 2.90396i −0.349596 0.201839i
\(208\) 0 0
\(209\) 3.80981i 0.263530i
\(210\) 0 0
\(211\) 15.5748 1.07222 0.536108 0.844149i \(-0.319894\pi\)
0.536108 + 0.844149i \(0.319894\pi\)
\(212\) 0 0
\(213\) −8.19380 + 14.1921i −0.561430 + 0.972425i
\(214\) 0 0
\(215\) −21.3554 36.9886i −1.45643 2.52260i
\(216\) 0 0
\(217\) −1.96939 1.89108i −0.133691 0.128375i
\(218\) 0 0
\(219\) −8.45914 14.6517i −0.571616 0.990068i
\(220\) 0 0
\(221\) 8.42748 + 4.86561i 0.566893 + 0.327296i
\(222\) 0 0
\(223\) −1.10497 −0.0739941 −0.0369970 0.999315i \(-0.511779\pi\)
−0.0369970 + 0.999315i \(0.511779\pi\)
\(224\) 0 0
\(225\) 13.0273 0.868484
\(226\) 0 0
\(227\) −8.71567 5.03199i −0.578479 0.333985i 0.182050 0.983289i \(-0.441727\pi\)
−0.760529 + 0.649304i \(0.775060\pi\)
\(228\) 0 0
\(229\) 6.34145 + 10.9837i 0.419055 + 0.725824i 0.995845 0.0910683i \(-0.0290281\pi\)
−0.576790 + 0.816893i \(0.695695\pi\)
\(230\) 0 0
\(231\) 4.84651 + 16.7230i 0.318877 + 1.10029i
\(232\) 0 0
\(233\) −10.4218 18.0511i −0.682756 1.18257i −0.974136 0.225961i \(-0.927448\pi\)
0.291381 0.956607i \(-0.405885\pi\)
\(234\) 0 0
\(235\) 16.5429 28.6532i 1.07914 1.86913i
\(236\) 0 0
\(237\) 19.6356 1.27547
\(238\) 0 0
\(239\) 19.4138i 1.25577i −0.778305 0.627886i \(-0.783920\pi\)
0.778305 0.627886i \(-0.216080\pi\)
\(240\) 0 0
\(241\) 10.6836 + 6.16820i 0.688193 + 0.397329i 0.802935 0.596067i \(-0.203271\pi\)
−0.114742 + 0.993395i \(0.536604\pi\)
\(242\) 0 0
\(243\) −9.03601 + 5.21694i −0.579660 + 0.334667i
\(244\) 0 0
\(245\) −1.17997 29.0710i −0.0753853 1.85728i
\(246\) 0 0
\(247\) −2.57599 + 1.48725i −0.163906 + 0.0946312i
\(248\) 0 0
\(249\) 5.13792 8.89915i 0.325603 0.563961i
\(250\) 0 0
\(251\) 26.4577i 1.67000i 0.550251 + 0.834999i \(0.314532\pi\)
−0.550251 + 0.834999i \(0.685468\pi\)
\(252\) 0 0
\(253\) 25.8662i 1.62619i
\(254\) 0 0
\(255\) −7.63132 + 13.2178i −0.477892 + 0.827733i
\(256\) 0 0
\(257\) 19.4985 11.2575i 1.21628 0.702222i 0.252164 0.967685i \(-0.418858\pi\)
0.964121 + 0.265462i \(0.0855246\pi\)
\(258\) 0 0
\(259\) −0.877918 3.02928i −0.0545512 0.188230i
\(260\) 0 0
\(261\) 3.57012 2.06121i 0.220985 0.127586i
\(262\) 0 0
\(263\) 13.7717 + 7.95110i 0.849200 + 0.490286i 0.860381 0.509652i \(-0.170226\pi\)
−0.0111808 + 0.999937i \(0.503559\pi\)
\(264\) 0 0
\(265\) 11.1621i 0.685682i
\(266\) 0 0
\(267\) 25.5614 1.56433
\(268\) 0 0
\(269\) 3.11103 5.38846i 0.189683 0.328540i −0.755462 0.655193i \(-0.772587\pi\)
0.945144 + 0.326653i \(0.105921\pi\)
\(270\) 0 0
\(271\) −7.49068 12.9742i −0.455027 0.788129i 0.543663 0.839304i \(-0.317037\pi\)
−0.998690 + 0.0511743i \(0.983704\pi\)
\(272\) 0 0
\(273\) 9.41525 9.80515i 0.569837 0.593435i
\(274\) 0 0
\(275\) −29.0091 50.2453i −1.74932 3.02990i
\(276\) 0 0
\(277\) −6.20167 3.58053i −0.372622 0.215133i 0.301981 0.953314i \(-0.402352\pi\)
−0.674603 + 0.738180i \(0.735685\pi\)
\(278\) 0 0
\(279\) −1.09513 −0.0655640
\(280\) 0 0
\(281\) −9.18363 −0.547850 −0.273925 0.961751i \(-0.588322\pi\)
−0.273925 + 0.961751i \(0.588322\pi\)
\(282\) 0 0
\(283\) −0.145830 0.0841948i −0.00866867 0.00500486i 0.495659 0.868517i \(-0.334926\pi\)
−0.504328 + 0.863512i \(0.668260\pi\)
\(284\) 0 0
\(285\) −2.33263 4.04023i −0.138173 0.239323i
\(286\) 0 0
\(287\) 16.6335 + 4.09722i 0.981842 + 0.241851i
\(288\) 0 0
\(289\) −5.02256 8.69934i −0.295445 0.511726i
\(290\) 0 0
\(291\) 1.27967 2.21645i 0.0750155 0.129931i
\(292\) 0 0
\(293\) 14.7893 0.864001 0.432001 0.901873i \(-0.357808\pi\)
0.432001 + 0.901873i \(0.357808\pi\)
\(294\) 0 0
\(295\) 11.4343i 0.665730i
\(296\) 0 0
\(297\) 23.1454 + 13.3630i 1.34303 + 0.775400i
\(298\) 0 0
\(299\) 17.4893 10.0975i 1.01143 0.583951i
\(300\) 0 0
\(301\) −6.50253 + 26.3983i −0.374800 + 1.52157i
\(302\) 0 0
\(303\) 3.37379 1.94786i 0.193819 0.111902i
\(304\) 0 0
\(305\) −3.40631 + 5.89990i −0.195045 + 0.337827i
\(306\) 0 0
\(307\) 12.2442i 0.698813i 0.936971 + 0.349407i \(0.113617\pi\)
−0.936971 + 0.349407i \(0.886383\pi\)
\(308\) 0 0
\(309\) 4.18127i 0.237864i
\(310\) 0 0
\(311\) −13.0325 + 22.5730i −0.739008 + 1.28000i 0.213935 + 0.976848i \(0.431372\pi\)
−0.952942 + 0.303151i \(0.901961\pi\)
\(312\) 0 0
\(313\) −13.7301 + 7.92710i −0.776074 + 0.448066i −0.835037 0.550194i \(-0.814554\pi\)
0.0589634 + 0.998260i \(0.481220\pi\)
\(314\) 0 0
\(315\) −8.41759 8.08286i −0.474278 0.455418i
\(316\) 0 0
\(317\) 23.8270 13.7565i 1.33826 0.772644i 0.351709 0.936109i \(-0.385601\pi\)
0.986549 + 0.163466i \(0.0522674\pi\)
\(318\) 0 0
\(319\) −15.8999 9.17981i −0.890224 0.513971i
\(320\) 0 0
\(321\) 14.2299i 0.794234i
\(322\) 0 0
\(323\) −2.12586 −0.118286
\(324\) 0 0
\(325\) −22.6487 + 39.2288i −1.25633 + 2.17602i
\(326\) 0 0
\(327\) 3.78028 + 6.54763i 0.209050 + 0.362085i
\(328\) 0 0
\(329\) −20.2283 + 5.86237i −1.11522 + 0.323203i
\(330\) 0 0
\(331\) −8.45914 14.6517i −0.464956 0.805328i 0.534243 0.845331i \(-0.320597\pi\)
−0.999200 + 0.0400026i \(0.987263\pi\)
\(332\) 0 0
\(333\) −1.09556 0.632521i −0.0600362 0.0346619i
\(334\) 0 0
\(335\) 20.4425 1.11689
\(336\) 0 0
\(337\) −17.2288 −0.938510 −0.469255 0.883063i \(-0.655478\pi\)
−0.469255 + 0.883063i \(0.655478\pi\)
\(338\) 0 0
\(339\) 10.5007 + 6.06258i 0.570319 + 0.329274i
\(340\) 0 0
\(341\) 2.43865 + 4.22386i 0.132060 + 0.228735i
\(342\) 0 0
\(343\) −12.2752 + 13.8680i −0.662797 + 0.748799i
\(344\) 0 0
\(345\) 15.8371 + 27.4306i 0.852639 + 1.47681i
\(346\) 0 0
\(347\) −15.0976 + 26.1497i −0.810479 + 1.40379i 0.102049 + 0.994779i \(0.467460\pi\)
−0.912529 + 0.409012i \(0.865873\pi\)
\(348\) 0 0
\(349\) −23.8353 −1.27587 −0.637936 0.770089i \(-0.720212\pi\)
−0.637936 + 0.770089i \(0.720212\pi\)
\(350\) 0 0
\(351\) 20.8662i 1.11376i
\(352\) 0 0
\(353\) 5.12282 + 2.95766i 0.272660 + 0.157421i 0.630096 0.776517i \(-0.283016\pi\)
−0.357436 + 0.933938i \(0.616349\pi\)
\(354\) 0 0
\(355\) −42.3643 + 24.4591i −2.24846 + 1.29815i
\(356\) 0 0
\(357\) 9.33139 2.70434i 0.493869 0.143129i
\(358\) 0 0
\(359\) 4.99982 2.88665i 0.263881 0.152352i −0.362223 0.932091i \(-0.617982\pi\)
0.626103 + 0.779740i \(0.284649\pi\)
\(360\) 0 0
\(361\) −9.17510 + 15.8917i −0.482900 + 0.836407i
\(362\) 0 0
\(363\) 15.7858i 0.828538i
\(364\) 0 0
\(365\) 50.5022i 2.64341i
\(366\) 0 0
\(367\) −4.37926 + 7.58509i −0.228595 + 0.395939i −0.957392 0.288791i \(-0.906747\pi\)
0.728797 + 0.684730i \(0.240080\pi\)
\(368\) 0 0
\(369\) 5.95055 3.43555i 0.309773 0.178848i
\(370\) 0 0
\(371\) −4.92119 + 5.12499i −0.255495 + 0.266076i
\(372\) 0 0
\(373\) 27.9028 16.1097i 1.44475 0.834127i 0.446589 0.894739i \(-0.352639\pi\)
0.998161 + 0.0606123i \(0.0193053\pi\)
\(374\) 0 0
\(375\) −36.4669 21.0542i −1.88314 1.08723i
\(376\) 0 0
\(377\) 14.3342i 0.738248i
\(378\) 0 0
\(379\) 32.1289 1.65035 0.825176 0.564876i \(-0.191076\pi\)
0.825176 + 0.564876i \(0.191076\pi\)
\(380\) 0 0
\(381\) 5.40899 9.36864i 0.277111 0.479970i
\(382\) 0 0
\(383\) −11.8334 20.4961i −0.604660 1.04730i −0.992105 0.125410i \(-0.959975\pi\)
0.387445 0.921893i \(-0.373358\pi\)
\(384\) 0 0
\(385\) −12.4307 + 50.4650i −0.633530 + 2.57194i
\(386\) 0 0
\(387\) 5.45242 + 9.44387i 0.277162 + 0.480059i
\(388\) 0 0
\(389\) −4.56952 2.63821i −0.231684 0.133763i 0.379665 0.925124i \(-0.376039\pi\)
−0.611349 + 0.791361i \(0.709373\pi\)
\(390\) 0 0
\(391\) 14.4332 0.729921
\(392\) 0 0
\(393\) −0.956659 −0.0482570
\(394\) 0 0
\(395\) 50.7608 + 29.3068i 2.55405 + 1.47458i
\(396\) 0 0
\(397\) −1.01243 1.75357i −0.0508122 0.0880093i 0.839501 0.543359i \(-0.182848\pi\)
−0.890313 + 0.455349i \(0.849514\pi\)
\(398\) 0 0
\(399\) −0.710265 + 2.88346i −0.0355577 + 0.144353i
\(400\) 0 0
\(401\) −10.6008 18.3610i −0.529376 0.916906i −0.999413 0.0342597i \(-0.989093\pi\)
0.470037 0.882647i \(-0.344241\pi\)
\(402\) 0 0
\(403\) 1.90396 3.29776i 0.0948432 0.164273i
\(404\) 0 0
\(405\) 19.4945 0.968687
\(406\) 0 0
\(407\) 5.63399i 0.279267i
\(408\) 0 0
\(409\) −28.0804 16.2122i −1.38849 0.801644i −0.395343 0.918534i \(-0.629374\pi\)
−0.993145 + 0.116890i \(0.962708\pi\)
\(410\) 0 0
\(411\) 0.537039 0.310060i 0.0264902 0.0152941i
\(412\) 0 0
\(413\) 5.04119 5.24996i 0.248061 0.258334i
\(414\) 0 0
\(415\) 26.5646 15.3371i 1.30400 0.752867i
\(416\) 0 0
\(417\) −9.14499 + 15.8396i −0.447832 + 0.775668i
\(418\) 0 0
\(419\) 9.20577i 0.449731i 0.974390 + 0.224866i \(0.0721943\pi\)
−0.974390 + 0.224866i \(0.927806\pi\)
\(420\) 0 0
\(421\) 16.7895i 0.818269i 0.912474 + 0.409135i \(0.134169\pi\)
−0.912474 + 0.409135i \(0.865831\pi\)
\(422\) 0 0
\(423\) −4.22371 + 7.31568i −0.205364 + 0.355700i
\(424\) 0 0
\(425\) −28.0367 + 16.1870i −1.35998 + 0.785184i
\(426\) 0 0
\(427\) 4.16515 1.20711i 0.201566 0.0584159i
\(428\) 0 0
\(429\) −21.0296 + 12.1415i −1.01532 + 0.586195i
\(430\) 0 0
\(431\) 18.3005 + 10.5658i 0.881502 + 0.508935i 0.871153 0.491011i \(-0.163373\pi\)
0.0103485 + 0.999946i \(0.496706\pi\)
\(432\) 0 0
\(433\) 26.7782i 1.28688i 0.765497 + 0.643439i \(0.222493\pi\)
−0.765497 + 0.643439i \(0.777507\pi\)
\(434\) 0 0
\(435\) −22.4820 −1.07793
\(436\) 0 0
\(437\) −2.20587 + 3.82068i −0.105521 + 0.182768i
\(438\) 0 0
\(439\) −0.0980721 0.169866i −0.00468073 0.00810725i 0.863676 0.504048i \(-0.168157\pi\)
−0.868356 + 0.495941i \(0.834823\pi\)
\(440\) 0 0
\(441\) 0.301267 + 7.42236i 0.0143460 + 0.353446i
\(442\) 0 0
\(443\) 2.20204 + 3.81404i 0.104622 + 0.181211i 0.913584 0.406651i \(-0.133303\pi\)
−0.808962 + 0.587861i \(0.799970\pi\)
\(444\) 0 0
\(445\) 66.0799 + 38.1513i 3.13249 + 1.80854i
\(446\) 0 0
\(447\) −7.77419 −0.367706
\(448\) 0 0
\(449\) −28.1195 −1.32704 −0.663520 0.748158i \(-0.730938\pi\)
−0.663520 + 0.748158i \(0.730938\pi\)
\(450\) 0 0
\(451\) −26.5014 15.3006i −1.24790 0.720476i
\(452\) 0 0
\(453\) −6.34757 10.9943i −0.298235 0.516558i
\(454\) 0 0
\(455\) 38.9743 11.2952i 1.82714 0.529526i
\(456\) 0 0
\(457\) 15.6581 + 27.1207i 0.732456 + 1.26865i 0.955830 + 0.293919i \(0.0949594\pi\)
−0.223374 + 0.974733i \(0.571707\pi\)
\(458\) 0 0
\(459\) 7.45651 12.9150i 0.348040 0.602823i
\(460\) 0 0
\(461\) −3.93305 −0.183181 −0.0915903 0.995797i \(-0.529195\pi\)
−0.0915903 + 0.995797i \(0.529195\pi\)
\(462\) 0 0
\(463\) 25.8507i 1.20138i 0.799481 + 0.600692i \(0.205108\pi\)
−0.799481 + 0.600692i \(0.794892\pi\)
\(464\) 0 0
\(465\) 5.17228 + 2.98622i 0.239859 + 0.138482i
\(466\) 0 0
\(467\) −2.95865 + 1.70818i −0.136910 + 0.0790450i −0.566891 0.823793i \(-0.691854\pi\)
0.429981 + 0.902838i \(0.358520\pi\)
\(468\) 0 0
\(469\) −9.38599 9.01275i −0.433405 0.416170i
\(470\) 0 0
\(471\) 23.1182 13.3473i 1.06523 0.615012i
\(472\) 0 0
\(473\) 24.2829 42.0592i 1.11653 1.93389i
\(474\) 0 0
\(475\) 9.89560i 0.454041i
\(476\) 0 0
\(477\) 2.84989i 0.130487i
\(478\) 0 0
\(479\) −0.336615 + 0.583034i −0.0153803 + 0.0266395i −0.873613 0.486621i \(-0.838229\pi\)
0.858233 + 0.513261i \(0.171563\pi\)
\(480\) 0 0
\(481\) 3.80940 2.19936i 0.173694 0.100282i
\(482\) 0 0
\(483\) 4.82225 19.5768i 0.219420 0.890777i
\(484\) 0 0
\(485\) 6.61626 3.81990i 0.300429 0.173453i
\(486\) 0 0
\(487\) 22.1931 + 12.8132i 1.00566 + 0.580620i 0.909919 0.414785i \(-0.136143\pi\)
0.0957449 + 0.995406i \(0.469477\pi\)
\(488\) 0 0
\(489\) 13.3508i 0.603743i
\(490\) 0 0
\(491\) −11.1931 −0.505136 −0.252568 0.967579i \(-0.581275\pi\)
−0.252568 + 0.967579i \(0.581275\pi\)
\(492\) 0 0
\(493\) −5.12230 + 8.87209i −0.230697 + 0.399579i
\(494\) 0 0
\(495\) 10.4233 + 18.0536i 0.468491 + 0.811451i
\(496\) 0 0
\(497\) 30.2348 + 7.44757i 1.35622 + 0.334069i
\(498\) 0 0
\(499\) 17.7962 + 30.8239i 0.796667 + 1.37987i 0.921775 + 0.387725i \(0.126739\pi\)
−0.125108 + 0.992143i \(0.539928\pi\)
\(500\) 0 0
\(501\) 9.55600 + 5.51716i 0.426931 + 0.246489i
\(502\) 0 0
\(503\) −35.2598 −1.57216 −0.786079 0.618127i \(-0.787892\pi\)
−0.786079 + 0.618127i \(0.787892\pi\)
\(504\) 0 0
\(505\) 11.6290 0.517483
\(506\) 0 0
\(507\) 0.742630 + 0.428757i 0.0329813 + 0.0190418i
\(508\) 0 0
\(509\) 5.76818 + 9.99078i 0.255670 + 0.442833i 0.965077 0.261965i \(-0.0843706\pi\)
−0.709407 + 0.704799i \(0.751037\pi\)
\(510\) 0 0
\(511\) −22.2656 + 23.1877i −0.984973 + 1.02576i
\(512\) 0 0
\(513\) 2.27919 + 3.94768i 0.100629 + 0.174294i
\(514\) 0 0
\(515\) −6.24068 + 10.8092i −0.274997 + 0.476309i
\(516\) 0 0
\(517\) 37.6214 1.65459
\(518\) 0 0
\(519\) 18.6805i 0.819983i
\(520\) 0 0
\(521\) 20.3305 + 11.7378i 0.890695 + 0.514243i 0.874170 0.485620i \(-0.161406\pi\)
0.0165252 + 0.999863i \(0.494740\pi\)
\(522\) 0 0
\(523\) −32.9353 + 19.0152i −1.44016 + 0.831477i −0.997860 0.0653897i \(-0.979171\pi\)
−0.442301 + 0.896867i \(0.645838\pi\)
\(524\) 0 0
\(525\) 12.5883 + 43.4363i 0.549399 + 1.89572i
\(526\) 0 0
\(527\) 2.35690 1.36076i 0.102668 0.0592755i
\(528\) 0 0
\(529\) 3.47646 6.02140i 0.151150 0.261800i
\(530\) 0 0
\(531\) 2.91938i 0.126690i
\(532\) 0 0
\(533\) 23.8917i 1.03486i
\(534\) 0 0
\(535\) 21.2386 36.7863i 0.918224 1.59041i
\(536\) 0 0
\(537\) −16.1045 + 9.29792i −0.694959 + 0.401235i
\(538\) 0 0
\(539\) 27.9567 17.6901i 1.20418 0.761966i
\(540\) 0 0
\(541\) 18.0532 10.4230i 0.776166 0.448120i −0.0589038 0.998264i \(-0.518761\pi\)
0.835070 + 0.550144i \(0.185427\pi\)
\(542\) 0 0
\(543\) −4.20863 2.42985i −0.180610 0.104275i
\(544\) 0 0
\(545\) 22.5688i 0.966740i
\(546\) 0 0
\(547\) −1.93035 −0.0825357 −0.0412678 0.999148i \(-0.513140\pi\)
−0.0412678 + 0.999148i \(0.513140\pi\)
\(548\) 0 0
\(549\) 0.869693 1.50635i 0.0371176 0.0642896i
\(550\) 0 0
\(551\) −1.56571 2.71189i −0.0667015 0.115530i
\(552\) 0 0
\(553\) −10.3855 35.8356i −0.441638 1.52388i
\(554\) 0 0
\(555\) 3.44952 + 5.97474i 0.146424 + 0.253614i
\(556\) 0 0
\(557\) 28.5515 + 16.4842i 1.20977 + 0.698458i 0.962708 0.270543i \(-0.0872031\pi\)
0.247057 + 0.969001i \(0.420536\pi\)
\(558\) 0 0
\(559\) −37.9175 −1.60374
\(560\) 0 0
\(561\) −17.3549 −0.732726
\(562\) 0 0
\(563\) −15.5561 8.98129i −0.655610 0.378516i 0.134993 0.990847i \(-0.456899\pi\)
−0.790602 + 0.612330i \(0.790232\pi\)
\(564\) 0 0
\(565\) 18.0972 + 31.3453i 0.761355 + 1.31871i
\(566\) 0 0
\(567\) −8.95072 8.59479i −0.375895 0.360947i
\(568\) 0 0
\(569\) 6.87571 + 11.9091i 0.288245 + 0.499254i 0.973391 0.229151i \(-0.0735951\pi\)
−0.685146 + 0.728406i \(0.740262\pi\)
\(570\) 0 0
\(571\) 0.935886 1.62100i 0.0391656 0.0678369i −0.845778 0.533535i \(-0.820863\pi\)
0.884944 + 0.465698i \(0.154197\pi\)
\(572\) 0 0
\(573\) −25.5355 −1.06676
\(574\) 0 0
\(575\) 67.1848i 2.80180i
\(576\) 0 0
\(577\) 2.28242 + 1.31775i 0.0950183 + 0.0548589i 0.546756 0.837292i \(-0.315862\pi\)
−0.451738 + 0.892151i \(0.649196\pi\)
\(578\) 0 0
\(579\) 11.8542 6.84401i 0.492643 0.284427i
\(580\) 0 0
\(581\) −18.9588 4.67000i −0.786542 0.193744i
\(582\) 0 0
\(583\) 10.9918 6.34613i 0.455235 0.262830i
\(584\) 0 0
\(585\) 8.13792 14.0953i 0.336462 0.582769i
\(586\) 0 0
\(587\) 2.51566i 0.103833i 0.998651 + 0.0519163i \(0.0165329\pi\)
−0.998651 + 0.0519163i \(0.983467\pi\)
\(588\) 0 0
\(589\) 0.831872i 0.0342767i
\(590\) 0 0
\(591\) 10.2965 17.8340i 0.423540 0.733593i
\(592\) 0 0
\(593\) −16.0848 + 9.28654i −0.660522 + 0.381352i −0.792476 0.609903i \(-0.791208\pi\)
0.131954 + 0.991256i \(0.457875\pi\)
\(594\) 0 0
\(595\) 28.1593 + 6.93632i 1.15442 + 0.284361i
\(596\) 0 0
\(597\) 2.35257 1.35826i 0.0962842 0.0555897i
\(598\) 0 0
\(599\) 16.4021 + 9.46973i 0.670170 + 0.386923i 0.796141 0.605111i \(-0.206871\pi\)
−0.125971 + 0.992034i \(0.540205\pi\)
\(600\) 0 0
\(601\) 16.1462i 0.658618i 0.944222 + 0.329309i \(0.106816\pi\)
−0.944222 + 0.329309i \(0.893184\pi\)
\(602\) 0 0
\(603\) −5.21933 −0.212548
\(604\) 0 0
\(605\) 23.5608 40.8085i 0.957883 1.65910i
\(606\) 0 0
\(607\) −9.71399 16.8251i −0.394279 0.682911i 0.598730 0.800951i \(-0.295672\pi\)
−0.993009 + 0.118040i \(0.962339\pi\)
\(608\) 0 0
\(609\) 10.3225 + 9.91197i 0.418287 + 0.401653i
\(610\) 0 0
\(611\) −14.6864 25.4376i −0.594147 1.02909i
\(612\) 0 0
\(613\) −13.2420 7.64527i −0.534839 0.308790i 0.208145 0.978098i \(-0.433257\pi\)
−0.742985 + 0.669308i \(0.766591\pi\)
\(614\) 0 0
\(615\) −37.4722 −1.51103
\(616\) 0 0
\(617\) 29.9971 1.20764 0.603818 0.797122i \(-0.293645\pi\)
0.603818 + 0.797122i \(0.293645\pi\)
\(618\) 0 0
\(619\) −8.33436 4.81184i −0.334986 0.193404i 0.323067 0.946376i \(-0.395286\pi\)
−0.658053 + 0.752972i \(0.728620\pi\)
\(620\) 0 0
\(621\) −15.4743 26.8022i −0.620961 1.07554i
\(622\) 0 0
\(623\) −13.5198 46.6504i −0.541659 1.86901i
\(624\) 0 0
\(625\) −32.1586 55.7004i −1.28634 2.22801i
\(626\) 0 0
\(627\) 2.65240 4.59409i 0.105927 0.183470i
\(628\) 0 0
\(629\) 3.14375 0.125349
\(630\) 0 0
\(631\) 16.6431i 0.662552i −0.943534 0.331276i \(-0.892521\pi\)
0.943534 0.331276i \(-0.107479\pi\)
\(632\) 0 0
\(633\) 18.7810 + 10.8432i 0.746479 + 0.430980i
\(634\) 0 0
\(635\) 27.9661 16.1462i 1.10980 0.640743i
\(636\) 0 0
\(637\) −22.8746 11.9971i −0.906325 0.475341i
\(638\) 0 0
\(639\) 10.8164 6.24484i 0.427890 0.247042i
\(640\) 0 0
\(641\) −4.44585 + 7.70044i −0.175601 + 0.304149i −0.940369 0.340156i \(-0.889520\pi\)
0.764768 + 0.644305i \(0.222853\pi\)
\(642\) 0 0
\(643\) 31.1319i 1.22772i −0.789413 0.613862i \(-0.789615\pi\)
0.789413 0.613862i \(-0.210385\pi\)
\(644\) 0 0
\(645\) 59.4707i 2.34165i
\(646\) 0 0
\(647\) −16.2830 + 28.2030i −0.640151 + 1.10877i 0.345247 + 0.938512i \(0.387795\pi\)
−0.985399 + 0.170263i \(0.945538\pi\)
\(648\) 0 0
\(649\) −11.2599 + 6.50088i −0.441988 + 0.255182i
\(650\) 0 0
\(651\) −1.05824 3.65147i −0.0414755 0.143112i
\(652\) 0 0
\(653\) −30.5163 + 17.6186i −1.19420 + 0.689469i −0.959255 0.282541i \(-0.908823\pi\)
−0.234940 + 0.972010i \(0.575489\pi\)
\(654\) 0 0
\(655\) −2.47310 1.42785i −0.0966321 0.0557906i
\(656\) 0 0
\(657\) 12.8941i 0.503048i
\(658\) 0 0
\(659\) −39.4975 −1.53861 −0.769303 0.638884i \(-0.779396\pi\)
−0.769303 + 0.638884i \(0.779396\pi\)
\(660\) 0 0
\(661\) −7.96057 + 13.7881i −0.309630 + 0.536295i −0.978281 0.207281i \(-0.933539\pi\)
0.668651 + 0.743576i \(0.266872\pi\)
\(662\) 0 0
\(663\) 6.77489 + 11.7345i 0.263115 + 0.455728i
\(664\) 0 0
\(665\) −6.13980 + 6.39406i −0.238091 + 0.247951i
\(666\) 0 0
\(667\) 10.6302 + 18.4120i 0.411602 + 0.712915i
\(668\) 0 0
\(669\) −1.33243 0.769280i −0.0515148 0.0297421i
\(670\) 0 0
\(671\) −7.74654 −0.299052
\(672\) 0 0
\(673\) −38.7175 −1.49245 −0.746224 0.665695i \(-0.768135\pi\)
−0.746224 + 0.665695i \(0.768135\pi\)
\(674\) 0 0
\(675\) 60.1178 + 34.7090i 2.31393 + 1.33595i
\(676\) 0 0
\(677\) −0.572823 0.992158i −0.0220154 0.0381317i 0.854808 0.518945i \(-0.173675\pi\)
−0.876823 + 0.480813i \(0.840342\pi\)
\(678\) 0 0
\(679\) −4.72193 1.16313i −0.181211 0.0446367i
\(680\) 0 0
\(681\) −7.00657 12.1357i −0.268492 0.465042i
\(682\) 0 0
\(683\) −18.4264 + 31.9154i −0.705066 + 1.22121i 0.261602 + 0.965176i \(0.415749\pi\)
−0.966668 + 0.256034i \(0.917584\pi\)
\(684\) 0 0
\(685\) 1.85110 0.0707268
\(686\) 0 0
\(687\) 17.6597i 0.673761i
\(688\) 0 0
\(689\) −8.58181 4.95471i −0.326941 0.188759i
\(690\) 0 0
\(691\) −3.66564 + 2.11636i −0.139448 + 0.0805102i −0.568101 0.822959i \(-0.692322\pi\)
0.428653 + 0.903469i \(0.358988\pi\)
\(692\) 0 0
\(693\) 3.17380 12.8846i 0.120563 0.489447i
\(694\) 0 0
\(695\) −47.2822 + 27.2984i −1.79352 + 1.03549i
\(696\) 0 0
\(697\) −8.53767 + 14.7877i −0.323387 + 0.560123i
\(698\) 0 0
\(699\) 29.0228i 1.09774i
\(700\) 0 0
\(701\) 23.6873i 0.894655i 0.894370 + 0.447328i \(0.147624\pi\)
−0.894370 + 0.447328i \(0.852376\pi\)
\(702\) 0 0
\(703\) −0.480467 + 0.832194i −0.0181212 + 0.0313868i
\(704\) 0 0
\(705\) 39.8968 23.0344i 1.50260 0.867527i
\(706\) 0 0
\(707\) −5.33936 5.12703i −0.200807 0.192822i
\(708\) 0 0
\(709\) −35.8535 + 20.7000i −1.34651 + 0.777405i −0.987753 0.156027i \(-0.950131\pi\)
−0.358753 + 0.933433i \(0.616798\pi\)
\(710\) 0 0
\(711\) −12.9602 7.48255i −0.486044 0.280618i
\(712\) 0 0
\(713\) 5.64788i 0.211515i
\(714\) 0 0
\(715\) −72.4862 −2.71083
\(716\) 0 0
\(717\) 13.5159 23.4102i 0.504761 0.874272i
\(718\) 0 0
\(719\) 7.97793 + 13.8182i 0.297527 + 0.515331i 0.975569 0.219691i \(-0.0705050\pi\)
−0.678043 + 0.735022i \(0.737172\pi\)
\(720\) 0 0
\(721\) 7.63095 2.21153i 0.284192 0.0823618i
\(722\) 0 0
\(723\) 8.58862 + 14.8759i 0.319414 + 0.553242i
\(724\) 0 0
\(725\) −41.2984 23.8436i −1.53378 0.885530i
\(726\) 0 0
\(727\) 19.2245 0.712998 0.356499 0.934296i \(-0.383970\pi\)
0.356499 + 0.934296i \(0.383970\pi\)
\(728\) 0 0
\(729\) −28.5988 −1.05921
\(730\) 0 0
\(731\) −23.4689 13.5498i −0.868029 0.501157i
\(732\) 0 0
\(733\) −11.3309 19.6257i −0.418516 0.724890i 0.577275 0.816550i \(-0.304116\pi\)
−0.995790 + 0.0916596i \(0.970783\pi\)
\(734\) 0 0
\(735\) 18.8164 35.8770i 0.694055 1.32334i
\(736\) 0 0
\(737\) 11.6224 + 20.1306i 0.428117 + 0.741521i
\(738\) 0 0
\(739\) −13.3722 + 23.1614i −0.491905 + 0.852005i −0.999957 0.00932191i \(-0.997033\pi\)
0.508051 + 0.861327i \(0.330366\pi\)
\(740\) 0 0
\(741\) −4.14170 −0.152149
\(742\) 0 0
\(743\) 21.4114i 0.785508i 0.919643 + 0.392754i \(0.128478\pi\)
−0.919643 + 0.392754i \(0.871522\pi\)
\(744\) 0 0
\(745\) −20.0974 11.6032i −0.736312 0.425110i
\(746\) 0 0
\(747\) −6.78242 + 3.91583i −0.248156 + 0.143273i
\(748\) 0 0
\(749\) −25.9700 + 7.52638i −0.948923 + 0.275008i
\(750\) 0 0
\(751\) 13.6677 7.89107i 0.498743 0.287949i −0.229451 0.973320i \(-0.573693\pi\)
0.728194 + 0.685371i \(0.240360\pi\)
\(752\) 0 0
\(753\) −18.4199 + 31.9043i −0.671260 + 1.16266i
\(754\) 0 0
\(755\) 37.8958i 1.37917i
\(756\) 0 0
\(757\) 17.1426i 0.623059i −0.950236 0.311530i \(-0.899159\pi\)
0.950236 0.311530i \(-0.100841\pi\)
\(758\) 0 0
\(759\) −18.0081 + 31.1909i −0.653653 + 1.13216i
\(760\) 0 0
\(761\) 10.7542 6.20893i 0.389839 0.225073i −0.292252 0.956341i \(-0.594404\pi\)
0.682090 + 0.731268i \(0.261071\pi\)
\(762\) 0 0
\(763\) 9.95021 10.3623i 0.360222 0.375139i
\(764\) 0 0
\(765\) 10.0739 5.81615i 0.364222 0.210283i
\(766\) 0 0
\(767\) 8.79108 + 5.07553i 0.317427 + 0.183267i
\(768\) 0 0
\(769\) 44.3368i 1.59883i −0.600782 0.799413i \(-0.705144\pi\)
0.600782 0.799413i \(-0.294856\pi\)
\(770\) 0 0
\(771\) 31.3499 1.12904
\(772\) 0 0
\(773\) −25.8172 + 44.7167i −0.928580 + 1.60835i −0.142879 + 0.989740i \(0.545636\pi\)
−0.785701 + 0.618607i \(0.787697\pi\)
\(774\) 0 0
\(775\) 6.33414 + 10.9711i 0.227529 + 0.394092i
\(776\) 0 0
\(777\) 1.05035 4.26409i 0.0376810 0.152973i
\(778\) 0 0
\(779\) −2.60967 4.52008i −0.0935010 0.161949i
\(780\) 0 0
\(781\) −48.1719 27.8120i −1.72373 0.995193i
\(782\) 0 0
\(783\) 21.9670 0.785038
\(784\) 0 0
\(785\) 79.6853 2.84409
\(786\) 0 0
\(787\) 10.8258 + 6.25029i 0.385899 + 0.222799i 0.680382 0.732858i \(-0.261814\pi\)
−0.294483 + 0.955657i \(0.595147\pi\)
\(788\) 0 0
\(789\) 11.0711 + 19.1758i 0.394143 + 0.682676i
\(790\) 0 0
\(791\) 5.51044 22.3707i 0.195929 0.795411i
\(792\) 0 0
\(793\) 3.02404 + 5.23778i 0.107387 + 0.185999i
\(794\) 0 0
\(795\) 7.77107 13.4599i 0.275612 0.477373i
\(796\) 0 0
\(797\) −26.1284 −0.925514 −0.462757 0.886485i \(-0.653140\pi\)
−0.462757 + 0.886485i \(0.653140\pi\)
\(798\) 0 0
\(799\) 20.9926i 0.742666i
\(800\) 0 0
\(801\) −16.8714 9.74071i −0.596122 0.344171i
\(802\) 0 0
\(803\) 49.7319 28.7127i 1.75500 1.01325i
\(804\) 0 0
\(805\) 41.6853 43.4116i 1.46921 1.53006i
\(806\) 0 0
\(807\) 7.50291 4.33181i 0.264115 0.152487i
\(808\) 0 0
\(809\) 14.0974 24.4174i 0.495638 0.858471i −0.504349 0.863500i \(-0.668267\pi\)
0.999987 + 0.00502905i \(0.00160080\pi\)
\(810\) 0 0
\(811\) 3.63616i 0.127683i 0.997960 + 0.0638414i \(0.0203352\pi\)
−0.997960 + 0.0638414i \(0.979665\pi\)
\(812\) 0 0
\(813\) 20.8601i 0.731596i
\(814\) 0 0
\(815\) 19.9265 34.5137i 0.697994 1.20896i
\(816\) 0 0
\(817\) 7.17363 4.14170i 0.250973 0.144900i
\(818\) 0 0
\(819\) −9.95085 + 2.88386i −0.347711 + 0.100770i
\(820\) 0 0
\(821\) 29.2291 16.8754i 1.02010 0.588956i 0.105968 0.994369i \(-0.466206\pi\)
0.914133 + 0.405413i \(0.132872\pi\)
\(822\) 0 0
\(823\) 32.7497 + 18.9080i 1.14158 + 0.659093i 0.946822 0.321758i \(-0.104274\pi\)
0.194760 + 0.980851i \(0.437607\pi\)
\(824\) 0 0
\(825\) 80.7848i 2.81257i
\(826\) 0 0
\(827\) −0.873832 −0.0303861 −0.0151931 0.999885i \(-0.504836\pi\)
−0.0151931 + 0.999885i \(0.504836\pi\)
\(828\) 0 0
\(829\) −7.21806 + 12.5020i −0.250693 + 0.434214i −0.963717 0.266926i \(-0.913992\pi\)
0.713023 + 0.701140i \(0.247325\pi\)
\(830\) 0 0
\(831\) −4.98555 8.63523i −0.172947 0.299553i
\(832\) 0 0
\(833\) −9.87101 15.5997i −0.342010 0.540499i
\(834\) 0 0
\(835\) 16.4691 + 28.5253i 0.569937 + 0.987160i
\(836\) 0 0
\(837\) −5.05379 2.91781i −0.174685 0.100854i
\(838\) 0 0
\(839\) 37.7316 1.30264 0.651319 0.758804i \(-0.274216\pi\)
0.651319 + 0.758804i \(0.274216\pi\)
\(840\) 0 0
\(841\) 13.9096 0.479640
\(842\) 0 0
\(843\) −11.0742 6.39366i −0.381414 0.220209i
\(844\) 0 0
\(845\) 1.27987 + 2.21680i 0.0440289 + 0.0762603i
\(846\) 0 0
\(847\) −28.8096 + 8.34932i −0.989909 + 0.286886i
\(848\) 0 0
\(849\) −0.117233 0.203054i −0.00402343 0.00696879i
\(850\) 0 0
\(851\) 3.26207 5.65006i 0.111822 0.193682i
\(852\) 0 0
\(853\) −27.7787 −0.951126 −0.475563 0.879682i \(-0.657756\pi\)
−0.475563 + 0.879682i \(0.657756\pi\)
\(854\) 0 0
\(855\) 3.55559i 0.121599i
\(856\) 0 0
\(857\) 12.6891 + 7.32607i 0.433452 + 0.250254i 0.700816 0.713342i \(-0.252819\pi\)
−0.267364 + 0.963596i \(0.586153\pi\)
\(858\) 0 0
\(859\) −4.00523 + 2.31242i −0.136657 + 0.0788988i −0.566770 0.823876i \(-0.691807\pi\)
0.430113 + 0.902775i \(0.358474\pi\)
\(860\) 0 0
\(861\) 17.2051 + 16.5209i 0.586348 + 0.563031i
\(862\) 0 0
\(863\) 24.2969 14.0278i 0.827076 0.477512i −0.0257746 0.999668i \(-0.508205\pi\)
0.852850 + 0.522155i \(0.174872\pi\)
\(864\) 0 0
\(865\) 27.8813 48.2918i 0.947992 1.64197i
\(866\) 0 0
\(867\) 13.9869i 0.475019i
\(868\) 0 0
\(869\) 66.6486i 2.26090i
\(870\) 0 0
\(871\) 9.07415 15.7169i 0.307466 0.532547i
\(872\) 0 0
\(873\) −1.68925 + 0.975290i −0.0571725 + 0.0330086i
\(874\) 0 0
\(875\) −19.1367 + 77.6893i −0.646940 + 2.62638i
\(876\) 0 0
\(877\) 31.1732 17.9978i 1.05264 0.607744i 0.129255 0.991611i \(-0.458741\pi\)
0.923388 + 0.383868i \(0.125408\pi\)
\(878\) 0 0
\(879\) 17.8338 + 10.2964i 0.601519 + 0.347287i
\(880\) 0 0
\(881\) 20.2724i 0.682994i −0.939883 0.341497i \(-0.889066\pi\)
0.939883 0.341497i \(-0.110934\pi\)
\(882\) 0 0
\(883\) 33.7734 1.13657 0.568283 0.822833i \(-0.307608\pi\)
0.568283 + 0.822833i \(0.307608\pi\)
\(884\) 0 0
\(885\) −7.96057 + 13.7881i −0.267592 + 0.463482i
\(886\) 0 0
\(887\) 1.52190 + 2.63600i 0.0511003 + 0.0885082i 0.890444 0.455093i \(-0.150394\pi\)
−0.839344 + 0.543601i \(0.817061\pi\)
\(888\) 0 0
\(889\) −19.9590 4.91638i −0.669403 0.164890i
\(890\) 0 0
\(891\) 11.0834 + 19.1971i 0.371309 + 0.643126i
\(892\) 0 0
\(893\) 5.55704 + 3.20836i 0.185959 + 0.107364i
\(894\) 0 0
\(895\) −55.5099 −1.85549
\(896\) 0 0
\(897\) 28.1195 0.938882
\(898\) 0 0
\(899\) 3.47174 + 2.00441i 0.115789 + 0.0668509i
\(900\) 0 0
\(901\) −3.54112 6.13340i −0.117972 0.204333i
\(902\) 0 0
\(903\) −26.2197 + 27.3055i −0.872535 + 0.908669i
\(904\) 0 0
\(905\) −7.25328 12.5631i −0.241107 0.417610i
\(906\) 0 0
\(907\) 11.9344 20.6710i 0.396276 0.686371i −0.596987 0.802251i \(-0.703636\pi\)
0.993263 + 0.115880i \(0.0369689\pi\)
\(908\) 0 0
\(909\) −2.96909 −0.0984786
\(910\) 0 0
\(911\) 13.6058i 0.450782i −0.974268 0.225391i \(-0.927634\pi\)
0.974268 0.225391i \(-0.0723659\pi\)
\(912\) 0 0
\(913\) 30.2062 + 17.4396i 0.999679 + 0.577165i
\(914\) 0 0
\(915\) −8.21505 + 4.74296i −0.271581 + 0.156797i
\(916\) 0 0
\(917\) 0.505991 + 1.74593i 0.0167093 + 0.0576558i
\(918\) 0 0
\(919\) −13.2691 + 7.66093i −0.437708 + 0.252711i −0.702625 0.711560i \(-0.747989\pi\)
0.264917 + 0.964271i \(0.414655\pi\)
\(920\) 0 0
\(921\) −8.52444 + 14.7648i −0.280890 + 0.486515i
\(922\) 0 0
\(923\) 43.4282i 1.42946i
\(924\) 0 0
\(925\) 14.6337i 0.481154i
\(926\) 0 0
\(927\) 1.59336 2.75978i 0.0523328 0.0906431i
\(928\) 0 0
\(929\) 45.5296 26.2865i 1.49378 0.862433i 0.493804 0.869573i \(-0.335606\pi\)
0.999975 + 0.00713973i \(0.00227267\pi\)
\(930\) 0 0
\(931\) 5.63808 0.228844i 0.184781 0.00750007i
\(932\) 0 0
\(933\) −31.4308 + 18.1466i −1.02900 + 0.594092i
\(934\) 0 0
\(935\) −44.8650 25.9028i −1.46724 0.847113i
\(936\) 0 0
\(937\) 55.5022i 1.81318i 0.422015 + 0.906589i \(0.361323\pi\)
−0.422015 + 0.906589i \(0.638677\pi\)
\(938\) 0 0
\(939\) −22.0755 −0.720405
\(940\) 0 0
\(941\) −11.0357 + 19.1144i −0.359753 + 0.623110i −0.987919 0.154969i \(-0.950472\pi\)
0.628166 + 0.778079i \(0.283806\pi\)
\(942\) 0 0
\(943\) 17.7180 + 30.6884i 0.576977 + 0.999353i
\(944\) 0 0
\(945\) −17.3098 59.7278i −0.563087 1.94295i
\(946\) 0 0
\(947\) 1.54086 + 2.66884i 0.0500712 + 0.0867258i 0.889975 0.456010i \(-0.150722\pi\)
−0.839904 + 0.542736i \(0.817389\pi\)
\(948\) 0 0
\(949\) −38.8279 22.4173i −1.26041 0.727697i
\(950\) 0 0
\(951\) 38.3093 1.24226
\(952\) 0 0
\(953\) 13.8174 0.447588 0.223794 0.974636i \(-0.428156\pi\)
0.223794 + 0.974636i \(0.428156\pi\)
\(954\) 0 0
\(955\) −66.0129 38.1126i −2.13613 1.23329i
\(956\) 0 0
\(957\) −12.7820 22.1391i −0.413184 0.715655i
\(958\) 0 0
\(959\) −0.849917 0.816119i −0.0274452 0.0263539i
\(960\) 0 0
\(961\) 14.9675 + 25.9245i 0.482823 + 0.836274i
\(962\) 0 0
\(963\) −5.42260 + 9.39221i −0.174741 + 0.302660i
\(964\) 0 0
\(965\) 40.8597 1.31532
\(966\) 0 0
\(967\) 42.3520i 1.36195i −0.732308 0.680974i \(-0.761557\pi\)
0.732308 0.680974i \(-0.238443\pi\)
\(968\) 0 0
\(969\) −2.56349 1.48003i −0.0823510 0.0475454i
\(970\) 0 0
\(971\) −16.0539 + 9.26873i −0.515195 + 0.297448i −0.734966 0.678104i \(-0.762802\pi\)
0.219772 + 0.975551i \(0.429469\pi\)
\(972\) 0 0
\(973\) 33.7447 + 8.31213i 1.08181 + 0.266475i
\(974\) 0 0
\(975\) −54.6223 + 31.5362i −1.74931 + 1.00997i
\(976\) 0 0
\(977\) −15.0075 + 25.9937i −0.480131 + 0.831612i −0.999740 0.0227924i \(-0.992744\pi\)
0.519609 + 0.854404i \(0.326078\pi\)
\(978\) 0 0
\(979\) 86.7625i 2.77294i
\(980\) 0 0
\(981\) 5.76222i 0.183973i
\(982\) 0 0
\(983\) 26.2129 45.4020i 0.836061 1.44810i −0.0571028 0.998368i \(-0.518186\pi\)
0.893164 0.449732i \(-0.148480\pi\)
\(984\) 0 0
\(985\) 53.2357 30.7357i 1.69623 0.979320i
\(986\) 0 0
\(987\) −28.4738 7.01378i −0.906331 0.223251i
\(988\) 0 0
\(989\) −48.7044 + 28.1195i −1.54871 + 0.894147i
\(990\) 0 0
\(991\) 45.7531 + 26.4156i 1.45340 + 0.839119i 0.998672 0.0515137i \(-0.0164046\pi\)
0.454724 + 0.890632i \(0.349738\pi\)
\(992\) 0 0
\(993\) 23.5571i 0.747562i
\(994\) 0 0
\(995\) 8.10898 0.257072
\(996\) 0 0
\(997\) 26.9722 46.7173i 0.854220 1.47955i −0.0231474 0.999732i \(-0.507369\pi\)
0.877367 0.479820i \(-0.159298\pi\)
\(998\) 0 0
\(999\) −3.37050 5.83787i −0.106638 0.184702i
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 896.2.q.d.703.6 yes 16
4.3 odd 2 896.2.q.a.703.4 yes 16
7.5 odd 6 inner 896.2.q.d.831.5 yes 16
8.3 odd 2 inner 896.2.q.d.703.5 yes 16
8.5 even 2 896.2.q.a.703.3 16
28.19 even 6 896.2.q.a.831.3 yes 16
56.5 odd 6 896.2.q.a.831.4 yes 16
56.19 even 6 inner 896.2.q.d.831.6 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
896.2.q.a.703.3 16 8.5 even 2
896.2.q.a.703.4 yes 16 4.3 odd 2
896.2.q.a.831.3 yes 16 28.19 even 6
896.2.q.a.831.4 yes 16 56.5 odd 6
896.2.q.d.703.5 yes 16 8.3 odd 2 inner
896.2.q.d.703.6 yes 16 1.1 even 1 trivial
896.2.q.d.831.5 yes 16 7.5 odd 6 inner
896.2.q.d.831.6 yes 16 56.19 even 6 inner