Properties

Label 128.4.g.a.81.9
Level $128$
Weight $4$
Character 128.81
Analytic conductor $7.552$
Analytic rank $0$
Dimension $44$
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] = [128,4,Mod(17,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.g (of order \(8\), degree \(4\), not 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: \(7.55224448073\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 81.9
Character \(\chi\) \(=\) 128.81
Dual form 128.4.g.a.49.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.56908 + 2.30679i) q^{3} +(6.28381 + 15.1704i) q^{5} +(16.6573 - 16.6573i) q^{7} +(6.60151 + 6.60151i) q^{9} +O(q^{10})\) \(q+(5.56908 + 2.30679i) q^{3} +(6.28381 + 15.1704i) q^{5} +(16.6573 - 16.6573i) q^{7} +(6.60151 + 6.60151i) q^{9} +(-3.11257 + 1.28927i) q^{11} +(-28.8862 + 69.7374i) q^{13} +98.9809i q^{15} -66.2302i q^{17} +(12.5299 - 30.2498i) q^{19} +(131.191 - 54.3410i) q^{21} +(63.7309 + 63.7309i) q^{23} +(-102.268 + 102.268i) q^{25} +(-40.7473 - 98.3726i) q^{27} +(190.908 + 79.0767i) q^{29} -123.811 q^{31} -20.3082 q^{33} +(357.370 + 148.028i) q^{35} +(-46.0901 - 111.271i) q^{37} +(-321.739 + 321.739i) q^{39} +(-100.187 - 100.187i) q^{41} +(-27.5836 + 11.4255i) q^{43} +(-58.6652 + 141.630i) q^{45} -394.293i q^{47} -211.932i q^{49} +(152.779 - 368.841i) q^{51} +(135.196 - 56.0002i) q^{53} +(-39.1175 - 39.1175i) q^{55} +(139.560 - 139.560i) q^{57} +(-297.070 - 717.190i) q^{59} +(-548.826 - 227.331i) q^{61} +219.927 q^{63} -1239.46 q^{65} +(-163.352 - 67.6626i) q^{67} +(207.909 + 501.937i) q^{69} +(-194.022 + 194.022i) q^{71} +(547.142 + 547.142i) q^{73} +(-805.449 + 333.628i) q^{75} +(-30.3713 + 73.3227i) q^{77} -715.813i q^{79} -893.911i q^{81} +(-54.6375 + 131.907i) q^{83} +(1004.74 - 416.178i) q^{85} +(880.769 + 880.769i) q^{87} +(220.576 - 220.576i) q^{89} +(680.471 + 1642.80i) q^{91} +(-689.512 - 285.605i) q^{93} +537.638 q^{95} +1364.39 q^{97} +(-29.0587 - 12.0365i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{19} - 4 q^{21} - 324 q^{23} - 4 q^{25} + 268 q^{27} - 4 q^{29} + 752 q^{31} - 8 q^{33} + 460 q^{35} - 4 q^{37} - 596 q^{39} - 4 q^{41} - 804 q^{43} + 104 q^{45} + 1384 q^{51} + 748 q^{53} + 292 q^{55} - 4 q^{57} - 1372 q^{59} - 1828 q^{61} - 2512 q^{63} - 8 q^{65} - 2036 q^{67} - 1060 q^{69} - 220 q^{71} - 4 q^{73} + 1712 q^{75} + 1900 q^{77} - 2436 q^{83} + 496 q^{85} + 1292 q^{87} - 4 q^{89} + 3604 q^{91} - 112 q^{93} + 6088 q^{95} - 8 q^{97} + 5424 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{3}{8}\right)\) \(1\)

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
\(3\) 5.56908 + 2.30679i 1.07177 + 0.443942i 0.847615 0.530612i \(-0.178038\pi\)
0.224155 + 0.974553i \(0.428038\pi\)
\(4\) 0 0
\(5\) 6.28381 + 15.1704i 0.562041 + 1.35689i 0.908131 + 0.418685i \(0.137509\pi\)
−0.346091 + 0.938201i \(0.612491\pi\)
\(6\) 0 0
\(7\) 16.6573 16.6573i 0.899411 0.899411i −0.0959733 0.995384i \(-0.530596\pi\)
0.995384 + 0.0959733i \(0.0305963\pi\)
\(8\) 0 0
\(9\) 6.60151 + 6.60151i 0.244500 + 0.244500i
\(10\) 0 0
\(11\) −3.11257 + 1.28927i −0.0853158 + 0.0353390i −0.424933 0.905225i \(-0.639702\pi\)
0.339617 + 0.940564i \(0.389702\pi\)
\(12\) 0 0
\(13\) −28.8862 + 69.7374i −0.616275 + 1.48782i 0.239723 + 0.970841i \(0.422943\pi\)
−0.855998 + 0.516979i \(0.827057\pi\)
\(14\) 0 0
\(15\) 98.9809i 1.70378i
\(16\) 0 0
\(17\) 66.2302i 0.944893i −0.881359 0.472447i \(-0.843371\pi\)
0.881359 0.472447i \(-0.156629\pi\)
\(18\) 0 0
\(19\) 12.5299 30.2498i 0.151292 0.365251i −0.830004 0.557758i \(-0.811662\pi\)
0.981296 + 0.192507i \(0.0616617\pi\)
\(20\) 0 0
\(21\) 131.191 54.3410i 1.36325 0.564676i
\(22\) 0 0
\(23\) 63.7309 + 63.7309i 0.577775 + 0.577775i 0.934290 0.356515i \(-0.116035\pi\)
−0.356515 + 0.934290i \(0.616035\pi\)
\(24\) 0 0
\(25\) −102.268 + 102.268i −0.818143 + 0.818143i
\(26\) 0 0
\(27\) −40.7473 98.3726i −0.290438 0.701178i
\(28\) 0 0
\(29\) 190.908 + 79.0767i 1.22244 + 0.506351i 0.898184 0.439619i \(-0.144887\pi\)
0.324255 + 0.945970i \(0.394887\pi\)
\(30\) 0 0
\(31\) −123.811 −0.717325 −0.358662 0.933467i \(-0.616767\pi\)
−0.358662 + 0.933467i \(0.616767\pi\)
\(32\) 0 0
\(33\) −20.3082 −0.107127
\(34\) 0 0
\(35\) 357.370 + 148.028i 1.72590 + 0.714892i
\(36\) 0 0
\(37\) −46.0901 111.271i −0.204788 0.494403i 0.787799 0.615932i \(-0.211220\pi\)
−0.992588 + 0.121529i \(0.961220\pi\)
\(38\) 0 0
\(39\) −321.739 + 321.739i −1.32101 + 1.32101i
\(40\) 0 0
\(41\) −100.187 100.187i −0.381624 0.381624i 0.490063 0.871687i \(-0.336974\pi\)
−0.871687 + 0.490063i \(0.836974\pi\)
\(42\) 0 0
\(43\) −27.5836 + 11.4255i −0.0978246 + 0.0405203i −0.431059 0.902324i \(-0.641860\pi\)
0.333234 + 0.942844i \(0.391860\pi\)
\(44\) 0 0
\(45\) −58.6652 + 141.630i −0.194340 + 0.469178i
\(46\) 0 0
\(47\) 394.293i 1.22369i −0.790976 0.611847i \(-0.790427\pi\)
0.790976 0.611847i \(-0.209573\pi\)
\(48\) 0 0
\(49\) 211.932i 0.617879i
\(50\) 0 0
\(51\) 152.779 368.841i 0.419478 1.01271i
\(52\) 0 0
\(53\) 135.196 56.0002i 0.350390 0.145136i −0.200545 0.979684i \(-0.564271\pi\)
0.550935 + 0.834548i \(0.314271\pi\)
\(54\) 0 0
\(55\) −39.1175 39.1175i −0.0959019 0.0959019i
\(56\) 0 0
\(57\) 139.560 139.560i 0.324301 0.324301i
\(58\) 0 0
\(59\) −297.070 717.190i −0.655512 1.58254i −0.804664 0.593731i \(-0.797654\pi\)
0.149152 0.988814i \(-0.452346\pi\)
\(60\) 0 0
\(61\) −548.826 227.331i −1.15197 0.477160i −0.276774 0.960935i \(-0.589265\pi\)
−0.875193 + 0.483775i \(0.839265\pi\)
\(62\) 0 0
\(63\) 219.927 0.439812
\(64\) 0 0
\(65\) −1239.46 −2.36517
\(66\) 0 0
\(67\) −163.352 67.6626i −0.297860 0.123378i 0.228749 0.973486i \(-0.426537\pi\)
−0.526609 + 0.850108i \(0.676537\pi\)
\(68\) 0 0
\(69\) 207.909 + 501.937i 0.362743 + 0.875740i
\(70\) 0 0
\(71\) −194.022 + 194.022i −0.324312 + 0.324312i −0.850419 0.526107i \(-0.823651\pi\)
0.526107 + 0.850419i \(0.323651\pi\)
\(72\) 0 0
\(73\) 547.142 + 547.142i 0.877235 + 0.877235i 0.993248 0.116013i \(-0.0370113\pi\)
−0.116013 + 0.993248i \(0.537011\pi\)
\(74\) 0 0
\(75\) −805.449 + 333.628i −1.24007 + 0.513654i
\(76\) 0 0
\(77\) −30.3713 + 73.3227i −0.0449497 + 0.108518i
\(78\) 0 0
\(79\) 715.813i 1.01943i −0.860342 0.509717i \(-0.829750\pi\)
0.860342 0.509717i \(-0.170250\pi\)
\(80\) 0 0
\(81\) 893.911i 1.22621i
\(82\) 0 0
\(83\) −54.6375 + 131.907i −0.0722560 + 0.174441i −0.955881 0.293754i \(-0.905096\pi\)
0.883625 + 0.468195i \(0.155096\pi\)
\(84\) 0 0
\(85\) 1004.74 416.178i 1.28211 0.531068i
\(86\) 0 0
\(87\) 880.769 + 880.769i 1.08538 + 1.08538i
\(88\) 0 0
\(89\) 220.576 220.576i 0.262708 0.262708i −0.563445 0.826153i \(-0.690524\pi\)
0.826153 + 0.563445i \(0.190524\pi\)
\(90\) 0 0
\(91\) 680.471 + 1642.80i 0.783877 + 1.89245i
\(92\) 0 0
\(93\) −689.512 285.605i −0.768807 0.318450i
\(94\) 0 0
\(95\) 537.638 0.580637
\(96\) 0 0
\(97\) 1364.39 1.42817 0.714087 0.700057i \(-0.246842\pi\)
0.714087 + 0.700057i \(0.246842\pi\)
\(98\) 0 0
\(99\) −29.0587 12.0365i −0.0295001 0.0122194i
\(100\) 0 0
\(101\) −349.311 843.310i −0.344136 0.830817i −0.997289 0.0735909i \(-0.976554\pi\)
0.653153 0.757226i \(-0.273446\pi\)
\(102\) 0 0
\(103\) −273.149 + 273.149i −0.261302 + 0.261302i −0.825583 0.564281i \(-0.809154\pi\)
0.564281 + 0.825583i \(0.309154\pi\)
\(104\) 0 0
\(105\) 1648.76 + 1648.76i 1.53240 + 1.53240i
\(106\) 0 0
\(107\) 675.362 279.744i 0.610185 0.252747i −0.0561229 0.998424i \(-0.517874\pi\)
0.666307 + 0.745677i \(0.267874\pi\)
\(108\) 0 0
\(109\) −602.506 + 1454.58i −0.529446 + 1.27820i 0.402441 + 0.915446i \(0.368162\pi\)
−0.931887 + 0.362749i \(0.881838\pi\)
\(110\) 0 0
\(111\) 726.000i 0.620801i
\(112\) 0 0
\(113\) 315.691i 0.262812i 0.991329 + 0.131406i \(0.0419491\pi\)
−0.991329 + 0.131406i \(0.958051\pi\)
\(114\) 0 0
\(115\) −566.354 + 1367.30i −0.459242 + 1.10871i
\(116\) 0 0
\(117\) −651.064 + 269.679i −0.514452 + 0.213093i
\(118\) 0 0
\(119\) −1103.22 1103.22i −0.849847 0.849847i
\(120\) 0 0
\(121\) −933.133 + 933.133i −0.701077 + 0.701077i
\(122\) 0 0
\(123\) −326.839 789.060i −0.239594 0.578432i
\(124\) 0 0
\(125\) −297.776 123.343i −0.213071 0.0882569i
\(126\) 0 0
\(127\) −356.698 −0.249227 −0.124613 0.992205i \(-0.539769\pi\)
−0.124613 + 0.992205i \(0.539769\pi\)
\(128\) 0 0
\(129\) −179.971 −0.122834
\(130\) 0 0
\(131\) 2481.91 + 1028.04i 1.65531 + 0.685652i 0.997705 0.0677095i \(-0.0215691\pi\)
0.657606 + 0.753362i \(0.271569\pi\)
\(132\) 0 0
\(133\) −295.166 712.594i −0.192437 0.464585i
\(134\) 0 0
\(135\) 1236.31 1236.31i 0.788181 0.788181i
\(136\) 0 0
\(137\) −811.069 811.069i −0.505798 0.505798i 0.407436 0.913234i \(-0.366423\pi\)
−0.913234 + 0.407436i \(0.866423\pi\)
\(138\) 0 0
\(139\) −2505.86 + 1037.96i −1.52910 + 0.633373i −0.979389 0.201984i \(-0.935261\pi\)
−0.549709 + 0.835356i \(0.685261\pi\)
\(140\) 0 0
\(141\) 909.552 2195.85i 0.543249 1.31152i
\(142\) 0 0
\(143\) 254.304i 0.148713i
\(144\) 0 0
\(145\) 3393.06i 1.94330i
\(146\) 0 0
\(147\) 488.883 1180.27i 0.274302 0.662224i
\(148\) 0 0
\(149\) −610.519 + 252.885i −0.335676 + 0.139041i −0.544154 0.838986i \(-0.683149\pi\)
0.208478 + 0.978027i \(0.433149\pi\)
\(150\) 0 0
\(151\) 1314.56 + 1314.56i 0.708457 + 0.708457i 0.966211 0.257753i \(-0.0829822\pi\)
−0.257753 + 0.966211i \(0.582982\pi\)
\(152\) 0 0
\(153\) 437.219 437.219i 0.231027 0.231027i
\(154\) 0 0
\(155\) −778.003 1878.26i −0.403166 0.973328i
\(156\) 0 0
\(157\) 2290.74 + 948.857i 1.16447 + 0.482338i 0.879360 0.476158i \(-0.157971\pi\)
0.285107 + 0.958496i \(0.407971\pi\)
\(158\) 0 0
\(159\) 882.101 0.439969
\(160\) 0 0
\(161\) 2123.17 1.03931
\(162\) 0 0
\(163\) 26.3934 + 10.9325i 0.0126828 + 0.00525338i 0.389016 0.921231i \(-0.372815\pi\)
−0.376333 + 0.926484i \(0.622815\pi\)
\(164\) 0 0
\(165\) −127.613 308.085i −0.0602100 0.145360i
\(166\) 0 0
\(167\) 111.233 111.233i 0.0515415 0.0515415i −0.680866 0.732408i \(-0.738397\pi\)
0.732408 + 0.680866i \(0.238397\pi\)
\(168\) 0 0
\(169\) −2475.38 2475.38i −1.12671 1.12671i
\(170\) 0 0
\(171\) 282.410 116.978i 0.126295 0.0523131i
\(172\) 0 0
\(173\) −835.349 + 2016.71i −0.367112 + 0.886287i 0.627109 + 0.778932i \(0.284238\pi\)
−0.994221 + 0.107355i \(0.965762\pi\)
\(174\) 0 0
\(175\) 3407.02i 1.47169i
\(176\) 0 0
\(177\) 4679.37i 1.98713i
\(178\) 0 0
\(179\) −539.124 + 1301.56i −0.225117 + 0.543481i −0.995571 0.0940151i \(-0.970030\pi\)
0.770454 + 0.637496i \(0.220030\pi\)
\(180\) 0 0
\(181\) −702.054 + 290.800i −0.288305 + 0.119420i −0.522149 0.852854i \(-0.674870\pi\)
0.233844 + 0.972274i \(0.424870\pi\)
\(182\) 0 0
\(183\) −2532.05 2532.05i −1.02281 1.02281i
\(184\) 0 0
\(185\) 1398.42 1398.42i 0.555749 0.555749i
\(186\) 0 0
\(187\) 85.3885 + 206.146i 0.0333916 + 0.0806144i
\(188\) 0 0
\(189\) −2317.36 959.883i −0.891870 0.369425i
\(190\) 0 0
\(191\) −2407.52 −0.912052 −0.456026 0.889966i \(-0.650728\pi\)
−0.456026 + 0.889966i \(0.650728\pi\)
\(192\) 0 0
\(193\) −1721.96 −0.642224 −0.321112 0.947041i \(-0.604057\pi\)
−0.321112 + 0.947041i \(0.604057\pi\)
\(194\) 0 0
\(195\) −6902.66 2859.18i −2.53492 1.05000i
\(196\) 0 0
\(197\) 160.639 + 387.816i 0.0580966 + 0.140258i 0.950262 0.311451i \(-0.100815\pi\)
−0.892166 + 0.451708i \(0.850815\pi\)
\(198\) 0 0
\(199\) 854.977 854.977i 0.304562 0.304562i −0.538234 0.842795i \(-0.680908\pi\)
0.842795 + 0.538234i \(0.180908\pi\)
\(200\) 0 0
\(201\) −753.637 753.637i −0.264465 0.264465i
\(202\) 0 0
\(203\) 4497.22 1862.81i 1.55489 0.644057i
\(204\) 0 0
\(205\) 890.326 2149.44i 0.303332 0.732308i
\(206\) 0 0
\(207\) 841.441i 0.282532i
\(208\) 0 0
\(209\) 110.309i 0.0365082i
\(210\) 0 0
\(211\) 819.136 1977.57i 0.267259 0.645221i −0.732093 0.681204i \(-0.761456\pi\)
0.999352 + 0.0359838i \(0.0114565\pi\)
\(212\) 0 0
\(213\) −1528.09 + 632.956i −0.491563 + 0.203612i
\(214\) 0 0
\(215\) −346.660 346.660i −0.109963 0.109963i
\(216\) 0 0
\(217\) −2062.36 + 2062.36i −0.645170 + 0.645170i
\(218\) 0 0
\(219\) 1784.94 + 4309.22i 0.550753 + 1.32964i
\(220\) 0 0
\(221\) 4618.72 + 1913.14i 1.40583 + 0.582314i
\(222\) 0 0
\(223\) −3155.56 −0.947587 −0.473793 0.880636i \(-0.657116\pi\)
−0.473793 + 0.880636i \(0.657116\pi\)
\(224\) 0 0
\(225\) −1350.24 −0.400073
\(226\) 0 0
\(227\) −329.052 136.298i −0.0962112 0.0398520i 0.334059 0.942552i \(-0.391582\pi\)
−0.430270 + 0.902700i \(0.641582\pi\)
\(228\) 0 0
\(229\) 1834.29 + 4428.38i 0.529317 + 1.27788i 0.931971 + 0.362532i \(0.118088\pi\)
−0.402654 + 0.915352i \(0.631912\pi\)
\(230\) 0 0
\(231\) −338.280 + 338.280i −0.0963515 + 0.0963515i
\(232\) 0 0
\(233\) −2068.42 2068.42i −0.581573 0.581573i 0.353762 0.935335i \(-0.384902\pi\)
−0.935335 + 0.353762i \(0.884902\pi\)
\(234\) 0 0
\(235\) 5981.61 2477.66i 1.66041 0.687766i
\(236\) 0 0
\(237\) 1651.23 3986.42i 0.452569 1.09260i
\(238\) 0 0
\(239\) 1635.41i 0.442620i 0.975204 + 0.221310i \(0.0710332\pi\)
−0.975204 + 0.221310i \(0.928967\pi\)
\(240\) 0 0
\(241\) 3798.84i 1.01537i 0.861542 + 0.507686i \(0.169499\pi\)
−0.861542 + 0.507686i \(0.830501\pi\)
\(242\) 0 0
\(243\) 961.887 2322.20i 0.253930 0.613042i
\(244\) 0 0
\(245\) 3215.11 1331.74i 0.838391 0.347273i
\(246\) 0 0
\(247\) 1747.60 + 1747.60i 0.450191 + 0.450191i
\(248\) 0 0
\(249\) −608.562 + 608.562i −0.154884 + 0.154884i
\(250\) 0 0
\(251\) 641.114 + 1547.79i 0.161222 + 0.389224i 0.983761 0.179485i \(-0.0574430\pi\)
−0.822539 + 0.568709i \(0.807443\pi\)
\(252\) 0 0
\(253\) −280.533 116.201i −0.0697113 0.0288754i
\(254\) 0 0
\(255\) 6555.52 1.60989
\(256\) 0 0
\(257\) −576.329 −0.139885 −0.0699424 0.997551i \(-0.522282\pi\)
−0.0699424 + 0.997551i \(0.522282\pi\)
\(258\) 0 0
\(259\) −2621.22 1085.75i −0.628860 0.260482i
\(260\) 0 0
\(261\) 738.255 + 1782.31i 0.175084 + 0.422690i
\(262\) 0 0
\(263\) −5045.11 + 5045.11i −1.18287 + 1.18287i −0.203873 + 0.978997i \(0.565353\pi\)
−0.978997 + 0.203873i \(0.934647\pi\)
\(264\) 0 0
\(265\) 1699.10 + 1699.10i 0.393867 + 0.393867i
\(266\) 0 0
\(267\) 1737.23 719.583i 0.398190 0.164936i
\(268\) 0 0
\(269\) 555.865 1341.98i 0.125991 0.304170i −0.848280 0.529548i \(-0.822362\pi\)
0.974271 + 0.225378i \(0.0723616\pi\)
\(270\) 0 0
\(271\) 6517.54i 1.46093i 0.682950 + 0.730465i \(0.260697\pi\)
−0.682950 + 0.730465i \(0.739303\pi\)
\(272\) 0 0
\(273\) 10718.6i 2.37626i
\(274\) 0 0
\(275\) 186.465 450.166i 0.0408882 0.0987129i
\(276\) 0 0
\(277\) −5983.48 + 2478.44i −1.29788 + 0.537599i −0.921324 0.388795i \(-0.872891\pi\)
−0.376555 + 0.926394i \(0.622891\pi\)
\(278\) 0 0
\(279\) −817.338 817.338i −0.175386 0.175386i
\(280\) 0 0
\(281\) 3663.87 3663.87i 0.777822 0.777822i −0.201638 0.979460i \(-0.564626\pi\)
0.979460 + 0.201638i \(0.0646264\pi\)
\(282\) 0 0
\(283\) −1065.93 2573.38i −0.223897 0.540535i 0.771516 0.636210i \(-0.219499\pi\)
−0.995413 + 0.0956751i \(0.969499\pi\)
\(284\) 0 0
\(285\) 2994.15 + 1240.22i 0.622309 + 0.257769i
\(286\) 0 0
\(287\) −3337.69 −0.686473
\(288\) 0 0
\(289\) 526.558 0.107177
\(290\) 0 0
\(291\) 7598.40 + 3147.36i 1.53067 + 0.634026i
\(292\) 0 0
\(293\) 1873.89 + 4523.96i 0.373630 + 0.902023i 0.993129 + 0.117025i \(0.0373358\pi\)
−0.619499 + 0.784997i \(0.712664\pi\)
\(294\) 0 0
\(295\) 9013.36 9013.36i 1.77891 1.77891i
\(296\) 0 0
\(297\) 253.657 + 253.657i 0.0495578 + 0.0495578i
\(298\) 0 0
\(299\) −6285.37 + 2603.49i −1.21569 + 0.503557i
\(300\) 0 0
\(301\) −269.151 + 649.787i −0.0515401 + 0.124429i
\(302\) 0 0
\(303\) 5502.25i 1.04322i
\(304\) 0 0
\(305\) 9754.44i 1.83127i
\(306\) 0 0
\(307\) −3251.76 + 7850.45i −0.604521 + 1.45944i 0.264361 + 0.964424i \(0.414839\pi\)
−0.868882 + 0.495019i \(0.835161\pi\)
\(308\) 0 0
\(309\) −2151.28 + 891.091i −0.396059 + 0.164053i
\(310\) 0 0
\(311\) −1620.08 1620.08i −0.295389 0.295389i 0.543815 0.839205i \(-0.316979\pi\)
−0.839205 + 0.543815i \(0.816979\pi\)
\(312\) 0 0
\(313\) 4904.15 4904.15i 0.885620 0.885620i −0.108479 0.994099i \(-0.534598\pi\)
0.994099 + 0.108479i \(0.0345979\pi\)
\(314\) 0 0
\(315\) 1381.98 + 3336.39i 0.247192 + 0.596775i
\(316\) 0 0
\(317\) −1326.36 549.398i −0.235003 0.0973415i 0.262074 0.965048i \(-0.415593\pi\)
−0.497077 + 0.867706i \(0.665593\pi\)
\(318\) 0 0
\(319\) −696.165 −0.122187
\(320\) 0 0
\(321\) 4406.46 0.766182
\(322\) 0 0
\(323\) −2003.45 829.856i −0.345124 0.142955i
\(324\) 0 0
\(325\) −4177.77 10086.0i −0.713049 1.72145i
\(326\) 0 0
\(327\) −6710.81 + 6710.81i −1.13489 + 1.13489i
\(328\) 0 0
\(329\) −6567.87 6567.87i −1.10060 1.10060i
\(330\) 0 0
\(331\) −3808.89 + 1577.69i −0.632494 + 0.261988i −0.675812 0.737074i \(-0.736207\pi\)
0.0433183 + 0.999061i \(0.486207\pi\)
\(332\) 0 0
\(333\) 430.295 1038.82i 0.0708109 0.170953i
\(334\) 0 0
\(335\) 2903.30i 0.473505i
\(336\) 0 0
\(337\) 528.946i 0.0855000i 0.999086 + 0.0427500i \(0.0136119\pi\)
−0.999086 + 0.0427500i \(0.986388\pi\)
\(338\) 0 0
\(339\) −728.232 + 1758.11i −0.116673 + 0.281674i
\(340\) 0 0
\(341\) 385.369 159.625i 0.0611992 0.0253495i
\(342\) 0 0
\(343\) 2183.23 + 2183.23i 0.343684 + 0.343684i
\(344\) 0 0
\(345\) −6308.14 + 6308.14i −0.984403 + 0.984403i
\(346\) 0 0
\(347\) −2777.92 6706.49i −0.429759 1.03753i −0.979364 0.202105i \(-0.935222\pi\)
0.549605 0.835425i \(-0.314778\pi\)
\(348\) 0 0
\(349\) −2949.74 1221.82i −0.452424 0.187400i 0.144823 0.989458i \(-0.453739\pi\)
−0.597247 + 0.802058i \(0.703739\pi\)
\(350\) 0 0
\(351\) 8037.28 1.22222
\(352\) 0 0
\(353\) 9005.55 1.35784 0.678919 0.734213i \(-0.262449\pi\)
0.678919 + 0.734213i \(0.262449\pi\)
\(354\) 0 0
\(355\) −4162.59 1724.20i −0.622331 0.257778i
\(356\) 0 0
\(357\) −3599.02 8688.80i −0.533558 1.28812i
\(358\) 0 0
\(359\) 7280.93 7280.93i 1.07040 1.07040i 0.0730705 0.997327i \(-0.476720\pi\)
0.997327 0.0730705i \(-0.0232798\pi\)
\(360\) 0 0
\(361\) 4091.99 + 4091.99i 0.596588 + 0.596588i
\(362\) 0 0
\(363\) −7349.24 + 3044.15i −1.06263 + 0.440156i
\(364\) 0 0
\(365\) −4862.26 + 11738.5i −0.697266 + 1.68335i
\(366\) 0 0
\(367\) 2730.08i 0.388308i −0.980971 0.194154i \(-0.937804\pi\)
0.980971 0.194154i \(-0.0621961\pi\)
\(368\) 0 0
\(369\) 1322.77i 0.186614i
\(370\) 0 0
\(371\) 1319.20 3184.82i 0.184607 0.445681i
\(372\) 0 0
\(373\) 807.251 334.374i 0.112059 0.0464162i −0.325950 0.945387i \(-0.605684\pi\)
0.438008 + 0.898971i \(0.355684\pi\)
\(374\) 0 0
\(375\) −1373.81 1373.81i −0.189182 0.189182i
\(376\) 0 0
\(377\) −11029.2 + 11029.2i −1.50672 + 1.50672i
\(378\) 0 0
\(379\) 511.994 + 1236.06i 0.0693914 + 0.167526i 0.954770 0.297344i \(-0.0961010\pi\)
−0.885379 + 0.464870i \(0.846101\pi\)
\(380\) 0 0
\(381\) −1986.48 822.826i −0.267114 0.110642i
\(382\) 0 0
\(383\) −7830.32 −1.04468 −0.522338 0.852739i \(-0.674940\pi\)
−0.522338 + 0.852739i \(0.674940\pi\)
\(384\) 0 0
\(385\) −1303.19 −0.172510
\(386\) 0 0
\(387\) −257.519 106.668i −0.0338254 0.0140109i
\(388\) 0 0
\(389\) 1347.70 + 3253.64i 0.175659 + 0.424077i 0.987047 0.160430i \(-0.0512880\pi\)
−0.811389 + 0.584507i \(0.801288\pi\)
\(390\) 0 0
\(391\) 4220.91 4220.91i 0.545936 0.545936i
\(392\) 0 0
\(393\) 11450.5 + 11450.5i 1.46972 + 1.46972i
\(394\) 0 0
\(395\) 10859.2 4498.03i 1.38326 0.572963i
\(396\) 0 0
\(397\) −23.4283 + 56.5609i −0.00296180 + 0.00715041i −0.925354 0.379105i \(-0.876232\pi\)
0.922392 + 0.386256i \(0.126232\pi\)
\(398\) 0 0
\(399\) 4649.38i 0.583359i
\(400\) 0 0
\(401\) 12034.3i 1.49866i −0.662195 0.749332i \(-0.730375\pi\)
0.662195 0.749332i \(-0.269625\pi\)
\(402\) 0 0
\(403\) 3576.42 8634.24i 0.442070 1.06725i
\(404\) 0 0
\(405\) 13561.0 5617.16i 1.66383 0.689183i
\(406\) 0 0
\(407\) 286.917 + 286.917i 0.0349434 + 0.0349434i
\(408\) 0 0
\(409\) 8440.37 8440.37i 1.02041 1.02041i 0.0206265 0.999787i \(-0.493434\pi\)
0.999787 0.0206265i \(-0.00656608\pi\)
\(410\) 0 0
\(411\) −2645.94 6387.87i −0.317554 0.766644i
\(412\) 0 0
\(413\) −16894.8 6998.07i −2.01293 0.833784i
\(414\) 0 0
\(415\) −2344.41 −0.277308
\(416\) 0 0
\(417\) −16349.7 −1.92002
\(418\) 0 0
\(419\) 14617.4 + 6054.71i 1.70431 + 0.705947i 0.999992 0.00393979i \(-0.00125408\pi\)
0.704315 + 0.709887i \(0.251254\pi\)
\(420\) 0 0
\(421\) −5214.86 12589.8i −0.603697 1.45745i −0.869749 0.493495i \(-0.835719\pi\)
0.266051 0.963959i \(-0.414281\pi\)
\(422\) 0 0
\(423\) 2602.93 2602.93i 0.299194 0.299194i
\(424\) 0 0
\(425\) 6773.23 + 6773.23i 0.773058 + 0.773058i
\(426\) 0 0
\(427\) −12928.7 + 5355.24i −1.46525 + 0.606928i
\(428\) 0 0
\(429\) 586.626 1416.24i 0.0660200 0.159386i
\(430\) 0 0
\(431\) 3490.31i 0.390075i 0.980796 + 0.195037i \(0.0624828\pi\)
−0.980796 + 0.195037i \(0.937517\pi\)
\(432\) 0 0
\(433\) 11691.1i 1.29755i 0.760980 + 0.648775i \(0.224718\pi\)
−0.760980 + 0.648775i \(0.775282\pi\)
\(434\) 0 0
\(435\) −7827.08 + 18896.2i −0.862712 + 2.08277i
\(436\) 0 0
\(437\) 2726.39 1129.31i 0.298446 0.123620i
\(438\) 0 0
\(439\) −6721.98 6721.98i −0.730804 0.730804i 0.239975 0.970779i \(-0.422861\pi\)
−0.970779 + 0.239975i \(0.922861\pi\)
\(440\) 0 0
\(441\) 1399.07 1399.07i 0.151072 0.151072i
\(442\) 0 0
\(443\) 3951.12 + 9538.86i 0.423755 + 1.02304i 0.981230 + 0.192842i \(0.0617704\pi\)
−0.557475 + 0.830194i \(0.688230\pi\)
\(444\) 0 0
\(445\) 4732.29 + 1960.18i 0.504117 + 0.208812i
\(446\) 0 0
\(447\) −3983.39 −0.421494
\(448\) 0 0
\(449\) −3683.45 −0.387155 −0.193578 0.981085i \(-0.562009\pi\)
−0.193578 + 0.981085i \(0.562009\pi\)
\(450\) 0 0
\(451\) 441.006 + 182.671i 0.0460448 + 0.0190724i
\(452\) 0 0
\(453\) 4288.47 + 10353.3i 0.444790 + 1.07382i
\(454\) 0 0
\(455\) −20646.1 + 20646.1i −2.12726 + 2.12726i
\(456\) 0 0
\(457\) −6693.56 6693.56i −0.685146 0.685146i 0.276009 0.961155i \(-0.410988\pi\)
−0.961155 + 0.276009i \(0.910988\pi\)
\(458\) 0 0
\(459\) −6515.24 + 2698.70i −0.662539 + 0.274432i
\(460\) 0 0
\(461\) −2097.99 + 5065.00i −0.211959 + 0.511715i −0.993724 0.111858i \(-0.964320\pi\)
0.781765 + 0.623573i \(0.214320\pi\)
\(462\) 0 0
\(463\) 17006.8i 1.70707i −0.521038 0.853534i \(-0.674455\pi\)
0.521038 0.853534i \(-0.325545\pi\)
\(464\) 0 0
\(465\) 12254.9i 1.22217i
\(466\) 0 0
\(467\) 7003.63 16908.3i 0.693981 1.67542i −0.0426191 0.999091i \(-0.513570\pi\)
0.736601 0.676328i \(-0.236430\pi\)
\(468\) 0 0
\(469\) −3848.08 + 1593.93i −0.378866 + 0.156931i
\(470\) 0 0
\(471\) 10568.5 + 10568.5i 1.03391 + 1.03391i
\(472\) 0 0
\(473\) 71.1253 71.1253i 0.00691404 0.00691404i
\(474\) 0 0
\(475\) 1812.18 + 4374.99i 0.175049 + 0.422607i
\(476\) 0 0
\(477\) 1262.19 + 522.815i 0.121156 + 0.0501846i
\(478\) 0 0
\(479\) 12250.2 1.16853 0.584263 0.811564i \(-0.301384\pi\)
0.584263 + 0.811564i \(0.301384\pi\)
\(480\) 0 0
\(481\) 9091.14 0.861789
\(482\) 0 0
\(483\) 11824.1 + 4897.71i 1.11391 + 0.461395i
\(484\) 0 0
\(485\) 8573.56 + 20698.4i 0.802691 + 1.93787i
\(486\) 0 0
\(487\) −6781.49 + 6781.49i −0.631003 + 0.631003i −0.948320 0.317317i \(-0.897218\pi\)
0.317317 + 0.948320i \(0.397218\pi\)
\(488\) 0 0
\(489\) 121.768 + 121.768i 0.0112608 + 0.0112608i
\(490\) 0 0
\(491\) 2990.34 1238.64i 0.274851 0.113847i −0.241000 0.970525i \(-0.577476\pi\)
0.515852 + 0.856678i \(0.327476\pi\)
\(492\) 0 0
\(493\) 5237.27 12643.9i 0.478447 1.15507i
\(494\) 0 0
\(495\) 516.469i 0.0468961i
\(496\) 0 0
\(497\) 6463.76i 0.583379i
\(498\) 0 0
\(499\) −672.528 + 1623.63i −0.0603336 + 0.145658i −0.951171 0.308663i \(-0.900118\pi\)
0.890838 + 0.454322i \(0.150118\pi\)
\(500\) 0 0
\(501\) 876.053 362.873i 0.0781221 0.0323592i
\(502\) 0 0
\(503\) 14586.7 + 14586.7i 1.29302 + 1.29302i 0.932911 + 0.360107i \(0.117260\pi\)
0.360107 + 0.932911i \(0.382740\pi\)
\(504\) 0 0
\(505\) 10598.4 10598.4i 0.933906 0.933906i
\(506\) 0 0
\(507\) −8075.40 19495.7i −0.707379 1.70776i
\(508\) 0 0
\(509\) 10300.8 + 4266.74i 0.897005 + 0.371552i 0.783068 0.621936i \(-0.213654\pi\)
0.113937 + 0.993488i \(0.463654\pi\)
\(510\) 0 0
\(511\) 18227.8 1.57799
\(512\) 0 0
\(513\) −3486.31 −0.300047
\(514\) 0 0
\(515\) −5860.20 2427.38i −0.501420 0.207695i
\(516\) 0 0
\(517\) 508.350 + 1227.26i 0.0432441 + 0.104400i
\(518\) 0 0
\(519\) −9304.25 + 9304.25i −0.786919 + 0.786919i
\(520\) 0 0
\(521\) 10841.1 + 10841.1i 0.911623 + 0.911623i 0.996400 0.0847773i \(-0.0270179\pi\)
−0.0847773 + 0.996400i \(0.527018\pi\)
\(522\) 0 0
\(523\) 7489.02 3102.05i 0.626142 0.259356i −0.0469711 0.998896i \(-0.514957\pi\)
0.673113 + 0.739540i \(0.264957\pi\)
\(524\) 0 0
\(525\) −7859.27 + 18974.0i −0.653346 + 1.57732i
\(526\) 0 0
\(527\) 8200.01i 0.677795i
\(528\) 0 0
\(529\) 4043.73i 0.332352i
\(530\) 0 0
\(531\) 2773.43 6695.64i 0.226660 0.547205i
\(532\) 0 0
\(533\) 9880.79 4092.76i 0.802973 0.332602i
\(534\) 0 0
\(535\) 8487.69 + 8487.69i 0.685897 + 0.685897i
\(536\) 0 0
\(537\) −6004.85 + 6004.85i −0.482548 + 0.482548i
\(538\) 0 0
\(539\) 273.238 + 659.654i 0.0218352 + 0.0527149i
\(540\) 0 0
\(541\) 20246.0 + 8386.17i 1.60895 + 0.666451i 0.992646 0.121054i \(-0.0386276\pi\)
0.616308 + 0.787505i \(0.288628\pi\)
\(542\) 0 0
\(543\) −4580.61 −0.362013
\(544\) 0 0
\(545\) −25852.6 −2.03194
\(546\) 0 0
\(547\) −10328.3 4278.13i −0.807325 0.334405i −0.0594388 0.998232i \(-0.518931\pi\)
−0.747886 + 0.663827i \(0.768931\pi\)
\(548\) 0 0
\(549\) −2122.35 5123.81i −0.164990 0.398322i
\(550\) 0 0
\(551\) 4784.11 4784.11i 0.369891 0.369891i
\(552\) 0 0
\(553\) −11923.5 11923.5i −0.916890 0.916890i
\(554\) 0 0
\(555\) 11013.7 4562.04i 0.842356 0.348915i
\(556\) 0 0
\(557\) 1262.85 3048.78i 0.0960657 0.231923i −0.868540 0.495619i \(-0.834941\pi\)
0.964606 + 0.263696i \(0.0849414\pi\)
\(558\) 0 0
\(559\) 2253.65i 0.170517i
\(560\) 0 0
\(561\) 1345.02i 0.101224i
\(562\) 0 0
\(563\) −3354.10 + 8097.52i −0.251081 + 0.606163i −0.998292 0.0584237i \(-0.981393\pi\)
0.747211 + 0.664587i \(0.231393\pi\)
\(564\) 0 0
\(565\) −4789.17 + 1983.74i −0.356605 + 0.147711i
\(566\) 0 0
\(567\) −14890.2 14890.2i −1.10287 1.10287i
\(568\) 0 0
\(569\) −8264.07 + 8264.07i −0.608872 + 0.608872i −0.942651 0.333780i \(-0.891676\pi\)
0.333780 + 0.942651i \(0.391676\pi\)
\(570\) 0 0
\(571\) 2298.13 + 5548.17i 0.168430 + 0.406626i 0.985446 0.169989i \(-0.0543732\pi\)
−0.817016 + 0.576615i \(0.804373\pi\)
\(572\) 0 0
\(573\) −13407.7 5553.64i −0.977510 0.404898i
\(574\) 0 0
\(575\) −13035.3 −0.945405
\(576\) 0 0
\(577\) −7818.78 −0.564125 −0.282062 0.959396i \(-0.591019\pi\)
−0.282062 + 0.959396i \(0.591019\pi\)
\(578\) 0 0
\(579\) −9589.72 3972.19i −0.688317 0.285110i
\(580\) 0 0
\(581\) 1287.10 + 3107.33i 0.0919067 + 0.221882i
\(582\) 0 0
\(583\) −348.609 + 348.609i −0.0247648 + 0.0247648i
\(584\) 0 0
\(585\) −8182.32 8182.32i −0.578286 0.578286i
\(586\) 0 0
\(587\) −12603.6 + 5220.58i −0.886211 + 0.367081i −0.778902 0.627145i \(-0.784223\pi\)
−0.107309 + 0.994226i \(0.534223\pi\)
\(588\) 0 0
\(589\) −1551.33 + 3745.25i −0.108526 + 0.262004i
\(590\) 0 0
\(591\) 2530.34i 0.176116i
\(592\) 0 0
\(593\) 22972.2i 1.59081i −0.606075 0.795407i \(-0.707257\pi\)
0.606075 0.795407i \(-0.292743\pi\)
\(594\) 0 0
\(595\) 9803.90 23668.7i 0.675497 1.63079i
\(596\) 0 0
\(597\) 6733.69 2789.19i 0.461628 0.191212i
\(598\) 0 0
\(599\) −7947.99 7947.99i −0.542147 0.542147i 0.382011 0.924158i \(-0.375232\pi\)
−0.924158 + 0.382011i \(0.875232\pi\)
\(600\) 0 0
\(601\) −10055.8 + 10055.8i −0.682506 + 0.682506i −0.960564 0.278058i \(-0.910309\pi\)
0.278058 + 0.960564i \(0.410309\pi\)
\(602\) 0 0
\(603\) −631.694 1525.04i −0.0426610 0.102993i
\(604\) 0 0
\(605\) −20019.7 8292.42i −1.34532 0.557248i
\(606\) 0 0
\(607\) −24716.9 −1.65277 −0.826383 0.563109i \(-0.809605\pi\)
−0.826383 + 0.563109i \(0.809605\pi\)
\(608\) 0 0
\(609\) 29342.5 1.95241
\(610\) 0 0
\(611\) 27497.0 + 11389.6i 1.82064 + 0.754132i
\(612\) 0 0
\(613\) −5388.03 13007.9i −0.355009 0.857067i −0.995986 0.0895074i \(-0.971471\pi\)
0.640977 0.767560i \(-0.278529\pi\)
\(614\) 0 0
\(615\) 9916.59 9916.59i 0.650204 0.650204i
\(616\) 0 0
\(617\) −1763.96 1763.96i −0.115096 0.115096i 0.647213 0.762309i \(-0.275934\pi\)
−0.762309 + 0.647213i \(0.775934\pi\)
\(618\) 0 0
\(619\) 21023.8 8708.35i 1.36513 0.565457i 0.424670 0.905348i \(-0.360390\pi\)
0.940465 + 0.339891i \(0.110390\pi\)
\(620\) 0 0
\(621\) 3672.52 8866.24i 0.237316 0.572931i
\(622\) 0 0
\(623\) 7348.41i 0.472565i
\(624\) 0 0
\(625\) 12786.1i 0.818312i
\(626\) 0 0
\(627\) −254.459 + 614.319i −0.0162075 + 0.0391284i
\(628\) 0 0
\(629\) −7369.53 + 3052.56i −0.467158 + 0.193503i
\(630\) 0 0
\(631\) −20760.0 20760.0i −1.30973 1.30973i −0.921605 0.388129i \(-0.873122\pi\)
−0.388129 0.921605i \(-0.626878\pi\)
\(632\) 0 0
\(633\) 9123.67 9123.67i 0.572881 0.572881i
\(634\) 0 0
\(635\) −2241.42 5411.26i −0.140076 0.338172i
\(636\) 0 0
\(637\) 14779.6 + 6121.92i 0.919293 + 0.380784i
\(638\) 0 0
\(639\) −2561.67 −0.158589
\(640\) 0 0
\(641\) 16062.4 0.989745 0.494872 0.868966i \(-0.335215\pi\)
0.494872 + 0.868966i \(0.335215\pi\)
\(642\) 0 0
\(643\) −24877.7 10304.7i −1.52579 0.632001i −0.547046 0.837103i \(-0.684248\pi\)
−0.978741 + 0.205101i \(0.934248\pi\)
\(644\) 0 0
\(645\) −1130.91 2730.25i −0.0690378 0.166672i
\(646\) 0 0
\(647\) 11101.6 11101.6i 0.674572 0.674572i −0.284195 0.958767i \(-0.591726\pi\)
0.958767 + 0.284195i \(0.0917262\pi\)
\(648\) 0 0
\(649\) 1849.30 + 1849.30i 0.111851 + 0.111851i
\(650\) 0 0
\(651\) −16242.8 + 6728.01i −0.977891 + 0.405056i
\(652\) 0 0
\(653\) 1485.55 3586.44i 0.0890262 0.214928i −0.873095 0.487550i \(-0.837891\pi\)
0.962121 + 0.272622i \(0.0878907\pi\)
\(654\) 0 0
\(655\) 44111.8i 2.63143i
\(656\) 0 0
\(657\) 7223.93i 0.428968i
\(658\) 0 0
\(659\) −10740.4 + 25929.6i −0.634881 + 1.53274i 0.198537 + 0.980093i \(0.436381\pi\)
−0.833417 + 0.552644i \(0.813619\pi\)
\(660\) 0 0
\(661\) −1477.94 + 612.182i −0.0869670 + 0.0360229i −0.425743 0.904844i \(-0.639987\pi\)
0.338776 + 0.940867i \(0.389987\pi\)
\(662\) 0 0
\(663\) 21308.8 + 21308.8i 1.24821 + 1.24821i
\(664\) 0 0
\(665\) 8955.61 8955.61i 0.522231 0.522231i
\(666\) 0 0
\(667\) 7127.12 + 17206.4i 0.413738 + 0.998851i
\(668\) 0 0
\(669\) −17573.6 7279.21i −1.01560 0.420673i
\(670\) 0 0
\(671\) 2001.35 0.115143
\(672\) 0 0
\(673\) −25713.9 −1.47281 −0.736404 0.676543i \(-0.763478\pi\)
−0.736404 + 0.676543i \(0.763478\pi\)
\(674\) 0 0
\(675\) 14227.5 + 5893.22i 0.811284 + 0.336045i
\(676\) 0 0
\(677\) −4314.50 10416.1i −0.244933 0.591321i 0.752827 0.658219i \(-0.228690\pi\)
−0.997760 + 0.0668981i \(0.978690\pi\)
\(678\) 0 0
\(679\) 22727.1 22727.1i 1.28451 1.28451i
\(680\) 0 0
\(681\) −1518.11 1518.11i −0.0854244 0.0854244i
\(682\) 0 0
\(683\) 1854.70 768.242i 0.103907 0.0430395i −0.330125 0.943937i \(-0.607091\pi\)
0.434031 + 0.900898i \(0.357091\pi\)
\(684\) 0 0
\(685\) 7207.68 17400.9i 0.402031 0.970589i
\(686\) 0 0
\(687\) 28893.3i 1.60458i
\(688\) 0 0
\(689\) 11045.9i 0.610761i
\(690\) 0 0
\(691\) 2881.88 6957.49i 0.158657 0.383032i −0.824483 0.565887i \(-0.808534\pi\)
0.983140 + 0.182855i \(0.0585338\pi\)
\(692\) 0 0
\(693\) −684.537 + 283.544i −0.0375230 + 0.0155425i
\(694\) 0 0
\(695\) −31492.7 31492.7i −1.71883 1.71883i
\(696\) 0 0
\(697\) −6635.41 + 6635.41i −0.360594 + 0.360594i
\(698\) 0 0
\(699\) −6747.78 16290.6i −0.365128 0.881497i
\(700\) 0 0
\(701\) 26418.3 + 10942.8i 1.42340 + 0.589592i 0.955713 0.294302i \(-0.0950870\pi\)
0.467688 + 0.883893i \(0.345087\pi\)
\(702\) 0 0
\(703\) −3943.44 −0.211564
\(704\) 0 0
\(705\) 39027.5 2.08491
\(706\) 0 0
\(707\) −19865.9 8228.71i −1.05676 0.437726i
\(708\) 0 0
\(709\) −1886.73 4554.97i −0.0999404 0.241277i 0.865999 0.500046i \(-0.166683\pi\)
−0.965939 + 0.258768i \(0.916683\pi\)
\(710\) 0 0
\(711\) 4725.45 4725.45i 0.249252 0.249252i
\(712\) 0 0
\(713\) −7890.58 7890.58i −0.414452 0.414452i
\(714\) 0 0
\(715\) 3857.91 1598.00i 0.201787 0.0835828i
\(716\) 0 0
\(717\) −3772.55 + 9107.75i −0.196497 + 0.474387i
\(718\) 0 0
\(719\) 29695.2i 1.54026i −0.637889 0.770129i \(-0.720192\pi\)
0.637889 0.770129i \(-0.279808\pi\)
\(720\) 0 0
\(721\) 9099.85i 0.470036i
\(722\) 0 0
\(723\) −8763.12 + 21156.0i −0.450766 + 1.08825i
\(724\) 0 0
\(725\) −27610.8 + 11436.8i −1.41440 + 0.585863i
\(726\) 0 0
\(727\) 26485.0 + 26485.0i 1.35113 + 1.35113i 0.884395 + 0.466739i \(0.154571\pi\)
0.466739 + 0.884395i \(0.345429\pi\)
\(728\) 0 0
\(729\) −6352.78 + 6352.78i −0.322755 + 0.322755i
\(730\) 0 0
\(731\) 756.713 + 1826.87i 0.0382873 + 0.0924338i
\(732\) 0 0
\(733\) 3763.54 + 1558.91i 0.189645 + 0.0785534i 0.475485 0.879724i \(-0.342273\pi\)
−0.285840 + 0.958277i \(0.592273\pi\)
\(734\) 0 0
\(735\) 20977.3 1.05273
\(736\) 0 0
\(737\) 595.679 0.0297722
\(738\) 0 0
\(739\) −14266.3 5909.30i −0.710142 0.294150i −0.00177857 0.999998i \(-0.500566\pi\)
−0.708363 + 0.705848i \(0.750566\pi\)
\(740\) 0 0
\(741\) 5701.18 + 13763.9i 0.282643 + 0.682360i
\(742\) 0 0
\(743\) −17888.3 + 17888.3i −0.883253 + 0.883253i −0.993864 0.110611i \(-0.964719\pi\)
0.110611 + 0.993864i \(0.464719\pi\)
\(744\) 0 0
\(745\) −7672.77 7672.77i −0.377327 0.377327i
\(746\) 0 0
\(747\) −1231.47 + 510.093i −0.0603176 + 0.0249844i
\(748\) 0 0
\(749\) 6589.94 15909.5i 0.321483 0.776130i
\(750\) 0 0
\(751\) 20239.7i 0.983432i −0.870756 0.491716i \(-0.836370\pi\)
0.870756 0.491716i \(-0.163630\pi\)
\(752\) 0 0
\(753\) 10098.7i 0.488732i
\(754\) 0 0
\(755\) −11682.0 + 28202.8i −0.563114 + 1.35948i
\(756\) 0 0
\(757\) −15064.3 + 6239.82i −0.723275 + 0.299591i −0.713785 0.700364i \(-0.753021\pi\)
−0.00949004 + 0.999955i \(0.503021\pi\)
\(758\) 0 0
\(759\) −1294.26 1294.26i −0.0618955 0.0618955i
\(760\) 0 0
\(761\) −3008.89 + 3008.89i −0.143328 + 0.143328i −0.775130 0.631802i \(-0.782316\pi\)
0.631802 + 0.775130i \(0.282316\pi\)
\(762\) 0 0
\(763\) 14193.2 + 34265.5i 0.673433 + 1.62581i
\(764\) 0 0
\(765\) 9380.21 + 3885.41i 0.443323 + 0.183631i
\(766\) 0 0
\(767\) 58596.1 2.75852
\(768\) 0 0
\(769\) 33250.5 1.55923 0.779613 0.626261i \(-0.215416\pi\)
0.779613 + 0.626261i \(0.215416\pi\)
\(770\) 0 0
\(771\) −3209.62 1329.47i −0.149924 0.0621007i
\(772\) 0 0
\(773\) −6032.84 14564.6i −0.280707 0.677686i 0.719146 0.694859i \(-0.244533\pi\)
−0.999853 + 0.0171732i \(0.994533\pi\)
\(774\) 0 0
\(775\) 12661.9 12661.9i 0.586875 0.586875i
\(776\) 0 0
\(777\) −12093.2 12093.2i −0.558355 0.558355i
\(778\) 0 0
\(779\) −4285.96 + 1775.30i −0.197125 + 0.0816520i
\(780\) 0 0
\(781\) 353.760 854.052i 0.0162081 0.0391298i
\(782\) 0 0
\(783\) 22002.3i 1.00421i
\(784\) 0 0
\(785\) 40714.0i 1.85114i
\(786\) 0 0
\(787\) 15236.1 36783.2i 0.690100 1.66605i −0.0544793 0.998515i \(-0.517350\pi\)
0.744579 0.667534i \(-0.232650\pi\)
\(788\) 0 0
\(789\) −39734.6 + 16458.6i −1.79289 + 0.742640i
\(790\) 0 0
\(791\) 5258.57 + 5258.57i 0.236376 + 0.236376i
\(792\) 0 0
\(793\) 31706.9 31706.9i 1.41986 1.41986i
\(794\) 0 0
\(795\) 5542.95 + 13381.9i 0.247281 + 0.596988i
\(796\) 0 0
\(797\) 4162.41 + 1724.12i 0.184994 + 0.0766269i 0.473258 0.880924i \(-0.343078\pi\)
−0.288264 + 0.957551i \(0.593078\pi\)
\(798\) 0 0
\(799\) −26114.1 −1.15626
\(800\) 0 0
\(801\) 2912.27 0.128464
\(802\) 0 0
\(803\) −2408.43 997.604i −0.105843 0.0438414i
\(804\) 0 0
\(805\) 13341.6 + 32209.5i 0.584136 + 1.41023i
\(806\) 0 0
\(807\) 6191.31 6191.31i 0.270068 0.270068i
\(808\) 0 0
\(809\) 24296.9 + 24296.9i 1.05591 + 1.05591i 0.998341 + 0.0575706i \(0.0183354\pi\)
0.0575706 + 0.998341i \(0.481665\pi\)
\(810\) 0 0
\(811\) 11241.7 4656.46i 0.486744 0.201616i −0.125795 0.992056i \(-0.540148\pi\)
0.612539 + 0.790440i \(0.290148\pi\)
\(812\) 0 0
\(813\) −15034.6 + 36296.7i −0.648568 + 1.56578i
\(814\) 0 0
\(815\) 469.098i 0.0201617i
\(816\) 0 0
\(817\) 977.558i 0.0418610i
\(818\) 0 0
\(819\) −6352.84 + 15337.1i −0.271045 + 0.654362i
\(820\) 0 0
\(821\) 3924.75 1625.68i 0.166839 0.0691069i −0.297701 0.954659i \(-0.596220\pi\)
0.464540 + 0.885552i \(0.346220\pi\)
\(822\) 0 0
\(823\) −14112.0 14112.0i −0.597708 0.597708i 0.341994 0.939702i \(-0.388898\pi\)
−0.939702 + 0.341994i \(0.888898\pi\)
\(824\) 0 0
\(825\) 2076.88 2076.88i 0.0876456 0.0876456i
\(826\) 0 0
\(827\) −8156.23 19690.9i −0.342950 0.827956i −0.997415 0.0718625i \(-0.977106\pi\)
0.654464 0.756093i \(-0.272894\pi\)
\(828\) 0 0
\(829\) −19708.8 8163.65i −0.825711 0.342021i −0.0705076 0.997511i \(-0.522462\pi\)
−0.755203 + 0.655491i \(0.772462\pi\)
\(830\) 0 0
\(831\) −39039.7 −1.62969
\(832\) 0 0
\(833\) −14036.3 −0.583830
\(834\) 0 0
\(835\) 2386.41 + 988.484i 0.0989044 + 0.0409675i
\(836\) 0 0
\(837\) 5044.95 + 12179.6i 0.208338 + 0.502973i
\(838\) 0 0
\(839\) −18262.0 + 18262.0i −0.751458 + 0.751458i −0.974751 0.223293i \(-0.928319\pi\)
0.223293 + 0.974751i \(0.428319\pi\)
\(840\) 0 0
\(841\) 12947.1 + 12947.1i 0.530859 + 0.530859i
\(842\) 0 0
\(843\) 28856.2 11952.6i 1.17895 0.488339i
\(844\) 0 0
\(845\) 21997.8 53107.3i 0.895558 2.16207i
\(846\) 0 0
\(847\) 31087.0i 1.26111i
\(848\) 0 0
\(849\) 16790.2i 0.678727i
\(850\) 0 0
\(851\) 4154.07 10028.8i 0.167332 0.403975i
\(852\) 0 0
\(853\) −2913.29 + 1206.73i −0.116939 + 0.0484379i −0.440386 0.897809i \(-0.645158\pi\)
0.323447 + 0.946246i \(0.395158\pi\)
\(854\) 0 0
\(855\) 3549.22 + 3549.22i 0.141966 + 0.141966i
\(856\) 0 0
\(857\) 7266.15 7266.15i 0.289623 0.289623i −0.547308 0.836931i \(-0.684347\pi\)
0.836931 + 0.547308i \(0.184347\pi\)
\(858\) 0 0
\(859\) 16409.5 + 39616.1i 0.651787 + 1.57355i 0.810183 + 0.586177i \(0.199368\pi\)
−0.158396 + 0.987376i \(0.550632\pi\)
\(860\) 0 0
\(861\) −18587.9 7699.35i −0.735741 0.304754i
\(862\) 0 0
\(863\) 29533.5 1.16493 0.582464 0.812856i \(-0.302089\pi\)
0.582464 + 0.812856i \(0.302089\pi\)
\(864\) 0 0
\(865\) −35843.6 −1.40892
\(866\) 0 0
\(867\) 2932.45 + 1214.66i 0.114869 + 0.0475802i
\(868\) 0 0
\(869\) 922.875 + 2228.02i 0.0360258 + 0.0869739i
\(870\) 0 0
\(871\) 9437.22 9437.22i 0.367128 0.367128i
\(872\) 0 0
\(873\) 9007.03 + 9007.03i 0.349189 + 0.349189i
\(874\) 0 0
\(875\) −7014.71 + 2905.59i −0.271018 + 0.112259i
\(876\) 0 0
\(877\) 12488.6 30150.2i 0.480856 1.16089i −0.478347 0.878171i \(-0.658764\pi\)
0.959203 0.282719i \(-0.0912363\pi\)
\(878\) 0 0
\(879\) 29517.0i 1.13263i
\(880\) 0 0
\(881\) 14625.2i 0.559292i −0.960103 0.279646i \(-0.909783\pi\)
0.960103 0.279646i \(-0.0902171\pi\)
\(882\) 0 0
\(883\) −18156.7 + 43834.1i −0.691983 + 1.67059i 0.0487652 + 0.998810i \(0.484471\pi\)
−0.740748 + 0.671783i \(0.765529\pi\)
\(884\) 0 0
\(885\) 70988.1 29404.2i 2.69631 1.11685i
\(886\) 0 0
\(887\) 3906.55 + 3906.55i 0.147879 + 0.147879i 0.777170 0.629291i \(-0.216654\pi\)
−0.629291 + 0.777170i \(0.716654\pi\)
\(888\) 0 0
\(889\) −5941.62 + 5941.62i −0.224157 + 0.224157i
\(890\) 0 0
\(891\) 1152.49 + 2782.36i 0.0433332 + 0.104616i
\(892\) 0 0
\(893\) −11927.3 4940.45i −0.446956 0.185135i
\(894\) 0 0
\(895\) −23133.0 −0.863967
\(896\) 0 0
\(897\) −41009.4 −1.52649
\(898\) 0 0
\(899\) −23636.5 9790.55i −0.876886 0.363218i
\(900\) 0 0
\(901\) −3708.91 8954.09i −0.137138 0.331081i
\(902\) 0 0
\(903\) −2997.84 + 2997.84i −0.110478 + 0.110478i
\(904\) 0 0
\(905\) −8823.14 8823.14i −0.324079 0.324079i
\(906\) 0 0
\(907\) −15093.4 + 6251.90i −0.552556 + 0.228876i −0.641450 0.767165i \(-0.721667\pi\)
0.0888937 + 0.996041i \(0.471667\pi\)
\(908\) 0 0
\(909\) 3261.14 7873.10i 0.118994 0.287276i
\(910\) 0 0
\(911\) 12262.7i 0.445971i 0.974822 + 0.222986i \(0.0715803\pi\)
−0.974822 + 0.222986i \(0.928420\pi\)
\(912\) 0 0
\(913\) 481.011i 0.0174361i
\(914\) 0 0
\(915\) 22501.4 54323.3i 0.812978 1.96270i
\(916\) 0 0
\(917\) 58466.4 24217.6i 2.10549 0.872121i
\(918\) 0 0
\(919\) −26825.0 26825.0i −0.962866 0.962866i 0.0364685 0.999335i \(-0.488389\pi\)
−0.999335 + 0.0364685i \(0.988389\pi\)
\(920\) 0 0
\(921\) −36218.7 + 36218.7i −1.29582 + 1.29582i
\(922\) 0 0
\(923\) −7926.02 19135.1i −0.282652 0.682383i
\(924\) 0 0
\(925\) 16093.0 + 6665.96i 0.572039 + 0.236946i
\(926\) 0 0
\(927\) −3606.39 −0.127777
\(928\) 0 0
\(929\) 27282.3 0.963512 0.481756 0.876305i \(-0.339999\pi\)
0.481756 + 0.876305i \(0.339999\pi\)
\(930\) 0 0
\(931\) −6410.91 2655.49i −0.225681 0.0934802i
\(932\) 0 0
\(933\) −5285.16 12759.5i −0.185454 0.447725i
\(934\) 0 0
\(935\) −2590.76 + 2590.76i −0.0906171 + 0.0906171i
\(936\) 0 0
\(937\) −24857.5 24857.5i −0.866658 0.866658i 0.125443 0.992101i \(-0.459965\pi\)
−0.992101 + 0.125443i \(0.959965\pi\)
\(938\) 0 0
\(939\) 38624.5 15998.8i 1.34235 0.556018i
\(940\) 0 0
\(941\) −16878.2 + 40747.6i −0.584712 + 1.41162i 0.303786 + 0.952740i \(0.401749\pi\)
−0.888498 + 0.458880i \(0.848251\pi\)
\(942\) 0 0
\(943\) 12770.0i 0.440985i
\(944\) 0 0
\(945\) 41187.2i 1.41780i
\(946\) 0 0
\(947\) 17615.8 42528.2i 0.604472 1.45933i −0.264461 0.964396i \(-0.585194\pi\)
0.868934 0.494929i \(-0.164806\pi\)
\(948\) 0 0
\(949\) −53961.1 + 22351.4i −1.84579 + 0.764550i
\(950\) 0 0
\(951\) −6119.28 6119.28i −0.208655 0.208655i
\(952\) 0 0
\(953\) 9439.64 9439.64i 0.320860 0.320860i −0.528237 0.849097i \(-0.677147\pi\)
0.849097 + 0.528237i \(0.177147\pi\)
\(954\) 0 0
\(955\) −15128.4 36523.1i −0.512610 1.23755i
\(956\) 0 0
\(957\) −3877.00 1605.91i −0.130957 0.0542440i
\(958\) 0 0
\(959\) −27020.5 −0.909840
\(960\) 0 0
\(961\) −14461.9 −0.485445
\(962\) 0 0
\(963\) 6305.14 + 2611.68i 0.210987 + 0.0873936i
\(964\) 0 0
\(965\) −10820.4 26122.9i −0.360956 0.871425i
\(966\) 0 0
\(967\) −25745.5 + 25745.5i −0.856173 + 0.856173i −0.990885 0.134711i \(-0.956989\pi\)
0.134711 + 0.990885i \(0.456989\pi\)
\(968\) 0 0
\(969\) −9243.07 9243.07i −0.306430 0.306430i
\(970\) 0 0
\(971\) −18385.2 + 7615.42i −0.607632 + 0.251689i −0.665216 0.746651i \(-0.731660\pi\)
0.0575837 + 0.998341i \(0.481660\pi\)
\(972\) 0 0
\(973\) −24451.3 + 59030.6i −0.805624 + 1.94495i
\(974\) 0 0
\(975\) 65807.1i 2.16155i
\(976\) 0 0
\(977\) 1781.76i 0.0583455i −0.999574 0.0291727i \(-0.990713\pi\)
0.999574 0.0291727i \(-0.00928729\pi\)
\(978\) 0 0
\(979\) −402.176 + 970.939i −0.0131293 + 0.0316970i
\(980\) 0 0
\(981\) −13579.9 + 5624.96i −0.441969 + 0.183069i
\(982\) 0 0
\(983\) 19228.0 + 19228.0i 0.623883 + 0.623883i 0.946522 0.322639i \(-0.104570\pi\)
−0.322639 + 0.946522i \(0.604570\pi\)
\(984\) 0 0
\(985\) −4873.92 + 4873.92i −0.157661 + 0.157661i
\(986\) 0 0
\(987\) −21426.3 51727.7i −0.690990 1.66820i
\(988\) 0 0
\(989\) −2486.09 1029.77i −0.0799322 0.0331090i
\(990\) 0 0
\(991\) −2811.67 −0.0901269 −0.0450634 0.998984i \(-0.514349\pi\)
−0.0450634 + 0.998984i \(0.514349\pi\)
\(992\) 0 0
\(993\) −24851.4 −0.794196
\(994\) 0 0
\(995\) 18342.9 + 7597.88i 0.584431 + 0.242079i
\(996\) 0 0
\(997\) 11942.0 + 28830.7i 0.379347 + 0.915824i 0.992088 + 0.125541i \(0.0400665\pi\)
−0.612742 + 0.790283i \(0.709933\pi\)
\(998\) 0 0
\(999\) −9068.01 + 9068.01i −0.287186 + 0.287186i
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 128.4.g.a.81.9 44
4.3 odd 2 32.4.g.a.29.2 yes 44
8.3 odd 2 256.4.g.b.161.9 44
8.5 even 2 256.4.g.a.161.3 44
32.5 even 8 256.4.g.a.97.3 44
32.11 odd 8 32.4.g.a.21.2 44
32.21 even 8 inner 128.4.g.a.49.9 44
32.27 odd 8 256.4.g.b.97.9 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.g.a.21.2 44 32.11 odd 8
32.4.g.a.29.2 yes 44 4.3 odd 2
128.4.g.a.49.9 44 32.21 even 8 inner
128.4.g.a.81.9 44 1.1 even 1 trivial
256.4.g.a.97.3 44 32.5 even 8
256.4.g.a.161.3 44 8.5 even 2
256.4.g.b.97.9 44 32.27 odd 8
256.4.g.b.161.9 44 8.3 odd 2