Properties

Label 768.2.n.b.97.7
Level $768$
Weight $2$
Character 768.97
Analytic conductor $6.133$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,2,Mod(97,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.n (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: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 97.7
Character \(\chi\) \(=\) 768.97
Dual form 768.2.n.b.673.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.923880 - 0.382683i) q^{3} +(0.155637 - 0.375742i) q^{5} +(-0.709092 - 0.709092i) q^{7} +(0.707107 - 0.707107i) q^{9} +O(q^{10})\) \(q+(0.923880 - 0.382683i) q^{3} +(0.155637 - 0.375742i) q^{5} +(-0.709092 - 0.709092i) q^{7} +(0.707107 - 0.707107i) q^{9} +(-2.79128 - 1.15619i) q^{11} +(-2.58738 - 6.24650i) q^{13} -0.406700i q^{15} -1.05927i q^{17} +(-1.48755 - 3.59125i) q^{19} +(-0.926474 - 0.383758i) q^{21} +(-0.922176 + 0.922176i) q^{23} +(3.41858 + 3.41858i) q^{25} +(0.382683 - 0.923880i) q^{27} +(7.64351 - 3.16605i) q^{29} +1.88437 q^{31} -3.02126 q^{33} +(-0.376797 + 0.156074i) q^{35} +(-1.24806 + 3.01308i) q^{37} +(-4.78086 - 4.78086i) q^{39} +(5.11980 - 5.11980i) q^{41} +(-10.9474 - 4.53458i) q^{43} +(-0.155637 - 0.375742i) q^{45} +7.47912i q^{47} -5.99438i q^{49} +(-0.405364 - 0.978635i) q^{51} +(7.58131 + 3.14028i) q^{53} +(-0.868855 + 0.868855i) q^{55} +(-2.74863 - 2.74863i) q^{57} +(-4.13463 + 9.98188i) q^{59} +(-1.35860 + 0.562751i) q^{61} -1.00281 q^{63} -2.74976 q^{65} +(10.8701 - 4.50256i) q^{67} +(-0.499078 + 1.20488i) q^{69} +(-9.35165 - 9.35165i) q^{71} +(0.367639 - 0.367639i) q^{73} +(4.46658 + 1.85012i) q^{75} +(1.15943 + 2.79912i) q^{77} -5.87774i q^{79} -1.00000i q^{81} +(1.62041 + 3.91201i) q^{83} +(-0.398011 - 0.164861i) q^{85} +(5.85009 - 5.85009i) q^{87} +(8.33036 + 8.33036i) q^{89} +(-2.59465 + 6.26404i) q^{91} +(1.74093 - 0.721116i) q^{93} -1.58090 q^{95} -7.97088 q^{97} +(-2.79128 + 1.15619i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 16 q^{23} + 48 q^{31} - 48 q^{35} - 16 q^{43} + 16 q^{51} + 32 q^{53} - 32 q^{55} + 64 q^{59} + 32 q^{61} - 16 q^{63} + 16 q^{67} + 32 q^{69} - 64 q^{71} + 32 q^{75} + 32 q^{77} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{8}\right)\)

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
\(3\) 0.923880 0.382683i 0.533402 0.220942i
\(4\) 0 0
\(5\) 0.155637 0.375742i 0.0696031 0.168037i −0.885250 0.465116i \(-0.846013\pi\)
0.954853 + 0.297079i \(0.0960126\pi\)
\(6\) 0 0
\(7\) −0.709092 0.709092i −0.268012 0.268012i 0.560287 0.828299i \(-0.310691\pi\)
−0.828299 + 0.560287i \(0.810691\pi\)
\(8\) 0 0
\(9\) 0.707107 0.707107i 0.235702 0.235702i
\(10\) 0 0
\(11\) −2.79128 1.15619i −0.841603 0.348603i −0.0801174 0.996785i \(-0.525530\pi\)
−0.761485 + 0.648182i \(0.775530\pi\)
\(12\) 0 0
\(13\) −2.58738 6.24650i −0.717611 1.73247i −0.680043 0.733173i \(-0.738039\pi\)
−0.0375686 0.999294i \(-0.511961\pi\)
\(14\) 0 0
\(15\) 0.406700i 0.105009i
\(16\) 0 0
\(17\) 1.05927i 0.256910i −0.991715 0.128455i \(-0.958998\pi\)
0.991715 0.128455i \(-0.0410018\pi\)
\(18\) 0 0
\(19\) −1.48755 3.59125i −0.341266 0.823890i −0.997588 0.0694090i \(-0.977889\pi\)
0.656322 0.754481i \(-0.272111\pi\)
\(20\) 0 0
\(21\) −0.926474 0.383758i −0.202173 0.0837429i
\(22\) 0 0
\(23\) −0.922176 + 0.922176i −0.192287 + 0.192287i −0.796684 0.604397i \(-0.793414\pi\)
0.604397 + 0.796684i \(0.293414\pi\)
\(24\) 0 0
\(25\) 3.41858 + 3.41858i 0.683715 + 0.683715i
\(26\) 0 0
\(27\) 0.382683 0.923880i 0.0736475 0.177801i
\(28\) 0 0
\(29\) 7.64351 3.16605i 1.41936 0.587920i 0.464664 0.885487i \(-0.346175\pi\)
0.954701 + 0.297567i \(0.0961753\pi\)
\(30\) 0 0
\(31\) 1.88437 0.338442 0.169221 0.985578i \(-0.445875\pi\)
0.169221 + 0.985578i \(0.445875\pi\)
\(32\) 0 0
\(33\) −3.02126 −0.525934
\(34\) 0 0
\(35\) −0.376797 + 0.156074i −0.0636903 + 0.0263814i
\(36\) 0 0
\(37\) −1.24806 + 3.01308i −0.205179 + 0.495347i −0.992652 0.121003i \(-0.961389\pi\)
0.787473 + 0.616349i \(0.211389\pi\)
\(38\) 0 0
\(39\) −4.78086 4.78086i −0.765551 0.765551i
\(40\) 0 0
\(41\) 5.11980 5.11980i 0.799579 0.799579i −0.183450 0.983029i \(-0.558727\pi\)
0.983029 + 0.183450i \(0.0587266\pi\)
\(42\) 0 0
\(43\) −10.9474 4.53458i −1.66947 0.691517i −0.670729 0.741702i \(-0.734019\pi\)
−0.998740 + 0.0501855i \(0.984019\pi\)
\(44\) 0 0
\(45\) −0.155637 0.375742i −0.0232010 0.0560123i
\(46\) 0 0
\(47\) 7.47912i 1.09094i 0.838130 + 0.545471i \(0.183649\pi\)
−0.838130 + 0.545471i \(0.816351\pi\)
\(48\) 0 0
\(49\) 5.99438i 0.856339i
\(50\) 0 0
\(51\) −0.405364 0.978635i −0.0567623 0.137036i
\(52\) 0 0
\(53\) 7.58131 + 3.14028i 1.04137 + 0.431351i 0.836805 0.547501i \(-0.184421\pi\)
0.204569 + 0.978852i \(0.434421\pi\)
\(54\) 0 0
\(55\) −0.868855 + 0.868855i −0.117156 + 0.117156i
\(56\) 0 0
\(57\) −2.74863 2.74863i −0.364064 0.364064i
\(58\) 0 0
\(59\) −4.13463 + 9.98188i −0.538283 + 1.29953i 0.387637 + 0.921812i \(0.373291\pi\)
−0.925921 + 0.377718i \(0.876709\pi\)
\(60\) 0 0
\(61\) −1.35860 + 0.562751i −0.173951 + 0.0720529i −0.467959 0.883750i \(-0.655011\pi\)
0.294008 + 0.955803i \(0.405011\pi\)
\(62\) 0 0
\(63\) −1.00281 −0.126342
\(64\) 0 0
\(65\) −2.74976 −0.341066
\(66\) 0 0
\(67\) 10.8701 4.50256i 1.32800 0.550075i 0.397915 0.917422i \(-0.369734\pi\)
0.930084 + 0.367347i \(0.119734\pi\)
\(68\) 0 0
\(69\) −0.499078 + 1.20488i −0.0600820 + 0.145051i
\(70\) 0 0
\(71\) −9.35165 9.35165i −1.10984 1.10984i −0.993171 0.116666i \(-0.962779\pi\)
−0.116666 0.993171i \(-0.537221\pi\)
\(72\) 0 0
\(73\) 0.367639 0.367639i 0.0430289 0.0430289i −0.685265 0.728294i \(-0.740314\pi\)
0.728294 + 0.685265i \(0.240314\pi\)
\(74\) 0 0
\(75\) 4.46658 + 1.85012i 0.515757 + 0.213633i
\(76\) 0 0
\(77\) 1.15943 + 2.79912i 0.132130 + 0.318989i
\(78\) 0 0
\(79\) 5.87774i 0.661297i −0.943754 0.330649i \(-0.892733\pi\)
0.943754 0.330649i \(-0.107267\pi\)
\(80\) 0 0
\(81\) 1.00000i 0.111111i
\(82\) 0 0
\(83\) 1.62041 + 3.91201i 0.177863 + 0.429398i 0.987518 0.157507i \(-0.0503458\pi\)
−0.809655 + 0.586906i \(0.800346\pi\)
\(84\) 0 0
\(85\) −0.398011 0.164861i −0.0431703 0.0178817i
\(86\) 0 0
\(87\) 5.85009 5.85009i 0.627195 0.627195i
\(88\) 0 0
\(89\) 8.33036 + 8.33036i 0.883016 + 0.883016i 0.993840 0.110824i \(-0.0353489\pi\)
−0.110824 + 0.993840i \(0.535349\pi\)
\(90\) 0 0
\(91\) −2.59465 + 6.26404i −0.271993 + 0.656650i
\(92\) 0 0
\(93\) 1.74093 0.721116i 0.180526 0.0747762i
\(94\) 0 0
\(95\) −1.58090 −0.162197
\(96\) 0 0
\(97\) −7.97088 −0.809320 −0.404660 0.914467i \(-0.632610\pi\)
−0.404660 + 0.914467i \(0.632610\pi\)
\(98\) 0 0
\(99\) −2.79128 + 1.15619i −0.280534 + 0.116201i
\(100\) 0 0
\(101\) −3.63502 + 8.77571i −0.361698 + 0.873216i 0.633354 + 0.773862i \(0.281678\pi\)
−0.995052 + 0.0993539i \(0.968322\pi\)
\(102\) 0 0
\(103\) 2.44527 + 2.44527i 0.240940 + 0.240940i 0.817239 0.576299i \(-0.195504\pi\)
−0.576299 + 0.817239i \(0.695504\pi\)
\(104\) 0 0
\(105\) −0.288388 + 0.288388i −0.0281438 + 0.0281438i
\(106\) 0 0
\(107\) 14.7198 + 6.09714i 1.42302 + 0.589433i 0.955617 0.294612i \(-0.0951905\pi\)
0.467401 + 0.884045i \(0.345190\pi\)
\(108\) 0 0
\(109\) −3.64422 8.79792i −0.349053 0.842688i −0.996732 0.0807753i \(-0.974260\pi\)
0.647679 0.761913i \(-0.275740\pi\)
\(110\) 0 0
\(111\) 3.26133i 0.309552i
\(112\) 0 0
\(113\) 6.79924i 0.639619i 0.947482 + 0.319810i \(0.103619\pi\)
−0.947482 + 0.319810i \(0.896381\pi\)
\(114\) 0 0
\(115\) 0.202975 + 0.490025i 0.0189275 + 0.0456951i
\(116\) 0 0
\(117\) −6.24650 2.58738i −0.577489 0.239204i
\(118\) 0 0
\(119\) −0.751118 + 0.751118i −0.0688549 + 0.0688549i
\(120\) 0 0
\(121\) −1.32369 1.32369i −0.120336 0.120336i
\(122\) 0 0
\(123\) 2.77082 6.68934i 0.249836 0.603158i
\(124\) 0 0
\(125\) 3.69527 1.53063i 0.330515 0.136904i
\(126\) 0 0
\(127\) 5.17506 0.459212 0.229606 0.973284i \(-0.426256\pi\)
0.229606 + 0.973284i \(0.426256\pi\)
\(128\) 0 0
\(129\) −11.8494 −1.04328
\(130\) 0 0
\(131\) 0.0139993 0.00579870i 0.00122313 0.000506635i −0.382072 0.924133i \(-0.624789\pi\)
0.383295 + 0.923626i \(0.374789\pi\)
\(132\) 0 0
\(133\) −1.49172 + 3.60134i −0.129349 + 0.312276i
\(134\) 0 0
\(135\) −0.287580 0.287580i −0.0247510 0.0247510i
\(136\) 0 0
\(137\) 8.75347 8.75347i 0.747859 0.747859i −0.226217 0.974077i \(-0.572636\pi\)
0.974077 + 0.226217i \(0.0726360\pi\)
\(138\) 0 0
\(139\) −0.340231 0.140928i −0.0288581 0.0119534i 0.368208 0.929744i \(-0.379972\pi\)
−0.397066 + 0.917790i \(0.629972\pi\)
\(140\) 0 0
\(141\) 2.86213 + 6.90980i 0.241035 + 0.581910i
\(142\) 0 0
\(143\) 20.4272i 1.70821i
\(144\) 0 0
\(145\) 3.36474i 0.279426i
\(146\) 0 0
\(147\) −2.29395 5.53808i −0.189202 0.456773i
\(148\) 0 0
\(149\) −11.5258 4.77413i −0.944228 0.391112i −0.143170 0.989698i \(-0.545729\pi\)
−0.801059 + 0.598586i \(0.795729\pi\)
\(150\) 0 0
\(151\) 5.63305 5.63305i 0.458411 0.458411i −0.439723 0.898134i \(-0.644923\pi\)
0.898134 + 0.439723i \(0.144923\pi\)
\(152\) 0 0
\(153\) −0.749015 0.749015i −0.0605542 0.0605542i
\(154\) 0 0
\(155\) 0.293278 0.708035i 0.0235566 0.0568707i
\(156\) 0 0
\(157\) −15.1312 + 6.26756i −1.20760 + 0.500206i −0.893448 0.449166i \(-0.851721\pi\)
−0.314155 + 0.949372i \(0.601721\pi\)
\(158\) 0 0
\(159\) 8.20595 0.650775
\(160\) 0 0
\(161\) 1.30782 0.103070
\(162\) 0 0
\(163\) 12.4717 5.16594i 0.976858 0.404628i 0.163597 0.986527i \(-0.447690\pi\)
0.813261 + 0.581899i \(0.197690\pi\)
\(164\) 0 0
\(165\) −0.470221 + 1.13521i −0.0366066 + 0.0883762i
\(166\) 0 0
\(167\) 13.0928 + 13.0928i 1.01315 + 1.01315i 0.999912 + 0.0132424i \(0.00421531\pi\)
0.0132424 + 0.999912i \(0.495785\pi\)
\(168\) 0 0
\(169\) −23.1318 + 23.1318i −1.77937 + 1.77937i
\(170\) 0 0
\(171\) −3.59125 1.48755i −0.274630 0.113755i
\(172\) 0 0
\(173\) 3.08646 + 7.45138i 0.234659 + 0.566518i 0.996715 0.0809941i \(-0.0258095\pi\)
−0.762055 + 0.647512i \(0.775809\pi\)
\(174\) 0 0
\(175\) 4.84817i 0.366487i
\(176\) 0 0
\(177\) 10.8043i 0.812102i
\(178\) 0 0
\(179\) 3.47037 + 8.37820i 0.259387 + 0.626216i 0.998898 0.0469281i \(-0.0149432\pi\)
−0.739511 + 0.673145i \(0.764943\pi\)
\(180\) 0 0
\(181\) 20.8285 + 8.62745i 1.54817 + 0.641274i 0.982984 0.183689i \(-0.0588040\pi\)
0.565187 + 0.824963i \(0.308804\pi\)
\(182\) 0 0
\(183\) −1.03983 + 1.03983i −0.0768663 + 0.0768663i
\(184\) 0 0
\(185\) 0.937894 + 0.937894i 0.0689553 + 0.0689553i
\(186\) 0 0
\(187\) −1.22471 + 2.95671i −0.0895596 + 0.216216i
\(188\) 0 0
\(189\) −0.926474 + 0.383758i −0.0673911 + 0.0279143i
\(190\) 0 0
\(191\) 10.0099 0.724293 0.362147 0.932121i \(-0.382044\pi\)
0.362147 + 0.932121i \(0.382044\pi\)
\(192\) 0 0
\(193\) 9.51081 0.684603 0.342301 0.939590i \(-0.388794\pi\)
0.342301 + 0.939590i \(0.388794\pi\)
\(194\) 0 0
\(195\) −2.54045 + 1.05229i −0.181925 + 0.0753559i
\(196\) 0 0
\(197\) 5.27324 12.7307i 0.375703 0.907028i −0.617058 0.786918i \(-0.711675\pi\)
0.992761 0.120110i \(-0.0383246\pi\)
\(198\) 0 0
\(199\) 9.74284 + 9.74284i 0.690652 + 0.690652i 0.962375 0.271724i \(-0.0875937\pi\)
−0.271724 + 0.962375i \(0.587594\pi\)
\(200\) 0 0
\(201\) 8.31964 8.31964i 0.586822 0.586822i
\(202\) 0 0
\(203\) −7.66497 3.17494i −0.537976 0.222837i
\(204\) 0 0
\(205\) −1.12689 2.72055i −0.0787055 0.190012i
\(206\) 0 0
\(207\) 1.30415i 0.0906450i
\(208\) 0 0
\(209\) 11.7441i 0.812355i
\(210\) 0 0
\(211\) 2.30086 + 5.55477i 0.158398 + 0.382406i 0.983077 0.183195i \(-0.0586441\pi\)
−0.824679 + 0.565601i \(0.808644\pi\)
\(212\) 0 0
\(213\) −12.2185 5.06108i −0.837199 0.346779i
\(214\) 0 0
\(215\) −3.40766 + 3.40766i −0.232400 + 0.232400i
\(216\) 0 0
\(217\) −1.33619 1.33619i −0.0907065 0.0907065i
\(218\) 0 0
\(219\) 0.198965 0.480344i 0.0134448 0.0324586i
\(220\) 0 0
\(221\) −6.61671 + 2.74073i −0.445088 + 0.184361i
\(222\) 0 0
\(223\) −8.30716 −0.556288 −0.278144 0.960539i \(-0.589719\pi\)
−0.278144 + 0.960539i \(0.589719\pi\)
\(224\) 0 0
\(225\) 4.83460 0.322306
\(226\) 0 0
\(227\) −21.6055 + 8.94931i −1.43401 + 0.593987i −0.958338 0.285636i \(-0.907795\pi\)
−0.475672 + 0.879623i \(0.657795\pi\)
\(228\) 0 0
\(229\) −4.89379 + 11.8147i −0.323391 + 0.780734i 0.675662 + 0.737212i \(0.263858\pi\)
−0.999052 + 0.0435225i \(0.986142\pi\)
\(230\) 0 0
\(231\) 2.14235 + 2.14235i 0.140956 + 0.140956i
\(232\) 0 0
\(233\) −14.3237 + 14.3237i −0.938376 + 0.938376i −0.998208 0.0598327i \(-0.980943\pi\)
0.0598327 + 0.998208i \(0.480943\pi\)
\(234\) 0 0
\(235\) 2.81021 + 1.16403i 0.183318 + 0.0759329i
\(236\) 0 0
\(237\) −2.24931 5.43032i −0.146109 0.352737i
\(238\) 0 0
\(239\) 11.8083i 0.763816i −0.924200 0.381908i \(-0.875267\pi\)
0.924200 0.381908i \(-0.124733\pi\)
\(240\) 0 0
\(241\) 8.34145i 0.537320i −0.963235 0.268660i \(-0.913419\pi\)
0.963235 0.268660i \(-0.0865808\pi\)
\(242\) 0 0
\(243\) −0.382683 0.923880i −0.0245492 0.0592669i
\(244\) 0 0
\(245\) −2.25234 0.932948i −0.143896 0.0596039i
\(246\) 0 0
\(247\) −18.5839 + 18.5839i −1.18247 + 1.18247i
\(248\) 0 0
\(249\) 2.99412 + 2.99412i 0.189745 + 0.189745i
\(250\) 0 0
\(251\) 8.31548 20.0753i 0.524868 1.26714i −0.409979 0.912095i \(-0.634464\pi\)
0.934848 0.355049i \(-0.115536\pi\)
\(252\) 0 0
\(253\) 3.64026 1.50785i 0.228861 0.0947974i
\(254\) 0 0
\(255\) −0.430804 −0.0269780
\(256\) 0 0
\(257\) 10.8777 0.678532 0.339266 0.940690i \(-0.389821\pi\)
0.339266 + 0.940690i \(0.389821\pi\)
\(258\) 0 0
\(259\) 3.02154 1.25156i 0.187749 0.0777683i
\(260\) 0 0
\(261\) 3.16605 7.64351i 0.195973 0.473121i
\(262\) 0 0
\(263\) 2.59878 + 2.59878i 0.160248 + 0.160248i 0.782676 0.622429i \(-0.213854\pi\)
−0.622429 + 0.782676i \(0.713854\pi\)
\(264\) 0 0
\(265\) 2.35987 2.35987i 0.144966 0.144966i
\(266\) 0 0
\(267\) 10.8841 + 4.50836i 0.666098 + 0.275907i
\(268\) 0 0
\(269\) 2.53422 + 6.11815i 0.154514 + 0.373030i 0.982114 0.188289i \(-0.0602941\pi\)
−0.827600 + 0.561319i \(0.810294\pi\)
\(270\) 0 0
\(271\) 25.4870i 1.54823i −0.633046 0.774114i \(-0.718196\pi\)
0.633046 0.774114i \(-0.281804\pi\)
\(272\) 0 0
\(273\) 6.78015i 0.410353i
\(274\) 0 0
\(275\) −5.58969 13.4947i −0.337071 0.813762i
\(276\) 0 0
\(277\) −1.45033 0.600745i −0.0871416 0.0360952i 0.338687 0.940899i \(-0.390017\pi\)
−0.425828 + 0.904804i \(0.640017\pi\)
\(278\) 0 0
\(279\) 1.33245 1.33245i 0.0797716 0.0797716i
\(280\) 0 0
\(281\) −3.62873 3.62873i −0.216472 0.216472i 0.590538 0.807010i \(-0.298916\pi\)
−0.807010 + 0.590538i \(0.798916\pi\)
\(282\) 0 0
\(283\) 2.49752 6.02955i 0.148462 0.358419i −0.832101 0.554625i \(-0.812862\pi\)
0.980563 + 0.196205i \(0.0628619\pi\)
\(284\) 0 0
\(285\) −1.46056 + 0.604985i −0.0865162 + 0.0358362i
\(286\) 0 0
\(287\) −7.26082 −0.428593
\(288\) 0 0
\(289\) 15.8780 0.933997
\(290\) 0 0
\(291\) −7.36413 + 3.05032i −0.431693 + 0.178813i
\(292\) 0 0
\(293\) 0.743879 1.79588i 0.0434579 0.104917i −0.900660 0.434524i \(-0.856916\pi\)
0.944118 + 0.329608i \(0.106916\pi\)
\(294\) 0 0
\(295\) 3.10711 + 3.10711i 0.180903 + 0.180903i
\(296\) 0 0
\(297\) −2.13635 + 2.13635i −0.123964 + 0.123964i
\(298\) 0 0
\(299\) 8.14640 + 3.37435i 0.471118 + 0.195144i
\(300\) 0 0
\(301\) 4.54731 + 10.9782i 0.262103 + 0.632772i
\(302\) 0 0
\(303\) 9.49876i 0.545690i
\(304\) 0 0
\(305\) 0.598068i 0.0342453i
\(306\) 0 0
\(307\) 8.42212 + 20.3328i 0.480676 + 1.16045i 0.959289 + 0.282428i \(0.0911397\pi\)
−0.478613 + 0.878026i \(0.658860\pi\)
\(308\) 0 0
\(309\) 3.19490 + 1.32337i 0.181752 + 0.0752840i
\(310\) 0 0
\(311\) 13.1751 13.1751i 0.747092 0.747092i −0.226840 0.973932i \(-0.572839\pi\)
0.973932 + 0.226840i \(0.0728394\pi\)
\(312\) 0 0
\(313\) −17.0669 17.0669i −0.964680 0.964680i 0.0347173 0.999397i \(-0.488947\pi\)
−0.999397 + 0.0347173i \(0.988947\pi\)
\(314\) 0 0
\(315\) −0.156074 + 0.376797i −0.00879379 + 0.0212301i
\(316\) 0 0
\(317\) −2.20756 + 0.914403i −0.123989 + 0.0513580i −0.443815 0.896118i \(-0.646375\pi\)
0.319826 + 0.947476i \(0.396375\pi\)
\(318\) 0 0
\(319\) −24.9957 −1.39949
\(320\) 0 0
\(321\) 15.9326 0.889271
\(322\) 0 0
\(323\) −3.80409 + 1.57571i −0.211666 + 0.0876747i
\(324\) 0 0
\(325\) 12.5090 30.1993i 0.693872 1.67516i
\(326\) 0 0
\(327\) −6.73364 6.73364i −0.372371 0.372371i
\(328\) 0 0
\(329\) 5.30338 5.30338i 0.292385 0.292385i
\(330\) 0 0
\(331\) −5.52800 2.28977i −0.303846 0.125857i 0.225551 0.974231i \(-0.427582\pi\)
−0.529397 + 0.848374i \(0.677582\pi\)
\(332\) 0 0
\(333\) 1.24806 + 3.01308i 0.0683931 + 0.165116i
\(334\) 0 0
\(335\) 4.78513i 0.261440i
\(336\) 0 0
\(337\) 12.4013i 0.675540i −0.941229 0.337770i \(-0.890327\pi\)
0.941229 0.337770i \(-0.109673\pi\)
\(338\) 0 0
\(339\) 2.60196 + 6.28168i 0.141319 + 0.341174i
\(340\) 0 0
\(341\) −5.25979 2.17868i −0.284834 0.117982i
\(342\) 0 0
\(343\) −9.21421 + 9.21421i −0.497521 + 0.497521i
\(344\) 0 0
\(345\) 0.375049 + 0.375049i 0.0201920 + 0.0201920i
\(346\) 0 0
\(347\) −1.93560 + 4.67295i −0.103908 + 0.250857i −0.967280 0.253712i \(-0.918349\pi\)
0.863371 + 0.504569i \(0.168349\pi\)
\(348\) 0 0
\(349\) −2.69921 + 1.11805i −0.144485 + 0.0598478i −0.453754 0.891127i \(-0.649916\pi\)
0.309269 + 0.950975i \(0.399916\pi\)
\(350\) 0 0
\(351\) −6.76116 −0.360884
\(352\) 0 0
\(353\) 3.35055 0.178332 0.0891658 0.996017i \(-0.471580\pi\)
0.0891658 + 0.996017i \(0.471580\pi\)
\(354\) 0 0
\(355\) −4.96927 + 2.05834i −0.263741 + 0.109245i
\(356\) 0 0
\(357\) −0.406502 + 0.981383i −0.0215144 + 0.0519403i
\(358\) 0 0
\(359\) −11.9270 11.9270i −0.629483 0.629483i 0.318455 0.947938i \(-0.396836\pi\)
−0.947938 + 0.318455i \(0.896836\pi\)
\(360\) 0 0
\(361\) 2.75072 2.75072i 0.144775 0.144775i
\(362\) 0 0
\(363\) −1.72949 0.716378i −0.0907747 0.0376001i
\(364\) 0 0
\(365\) −0.0809190 0.195356i −0.00423549 0.0102254i
\(366\) 0 0
\(367\) 28.5325i 1.48939i −0.667407 0.744693i \(-0.732596\pi\)
0.667407 0.744693i \(-0.267404\pi\)
\(368\) 0 0
\(369\) 7.24049i 0.376925i
\(370\) 0 0
\(371\) −3.14910 7.60260i −0.163493 0.394707i
\(372\) 0 0
\(373\) 16.9506 + 7.02116i 0.877668 + 0.363542i 0.775592 0.631235i \(-0.217452\pi\)
0.102076 + 0.994777i \(0.467452\pi\)
\(374\) 0 0
\(375\) 2.82823 2.82823i 0.146049 0.146049i
\(376\) 0 0
\(377\) −39.5534 39.5534i −2.03710 2.03710i
\(378\) 0 0
\(379\) −4.48842 + 10.8360i −0.230555 + 0.556608i −0.996243 0.0866037i \(-0.972399\pi\)
0.765688 + 0.643212i \(0.222399\pi\)
\(380\) 0 0
\(381\) 4.78113 1.98041i 0.244945 0.101459i
\(382\) 0 0
\(383\) 3.94835 0.201751 0.100876 0.994899i \(-0.467836\pi\)
0.100876 + 0.994899i \(0.467836\pi\)
\(384\) 0 0
\(385\) 1.23220 0.0627985
\(386\) 0 0
\(387\) −10.9474 + 4.53458i −0.556490 + 0.230506i
\(388\) 0 0
\(389\) 6.63971 16.0297i 0.336647 0.812737i −0.661386 0.750045i \(-0.730032\pi\)
0.998033 0.0626914i \(-0.0199684\pi\)
\(390\) 0 0
\(391\) 0.976831 + 0.976831i 0.0494005 + 0.0494005i
\(392\) 0 0
\(393\) 0.0107146 0.0107146i 0.000540480 0.000540480i
\(394\) 0 0
\(395\) −2.20851 0.914795i −0.111122 0.0460283i
\(396\) 0 0
\(397\) 1.83049 + 4.41918i 0.0918695 + 0.221792i 0.963134 0.269020i \(-0.0866999\pi\)
−0.871265 + 0.490813i \(0.836700\pi\)
\(398\) 0 0
\(399\) 3.89806i 0.195147i
\(400\) 0 0
\(401\) 23.8940i 1.19321i −0.802536 0.596604i \(-0.796516\pi\)
0.802536 0.596604i \(-0.203484\pi\)
\(402\) 0 0
\(403\) −4.87558 11.7707i −0.242870 0.586340i
\(404\) 0 0
\(405\) −0.375742 0.155637i −0.0186708 0.00773368i
\(406\) 0 0
\(407\) 6.96736 6.96736i 0.345359 0.345359i
\(408\) 0 0
\(409\) 13.8337 + 13.8337i 0.684032 + 0.684032i 0.960906 0.276874i \(-0.0892985\pi\)
−0.276874 + 0.960906i \(0.589299\pi\)
\(410\) 0 0
\(411\) 4.73734 11.4370i 0.233676 0.564144i
\(412\) 0 0
\(413\) 10.0099 4.14624i 0.492556 0.204023i
\(414\) 0 0
\(415\) 1.72210 0.0845345
\(416\) 0 0
\(417\) −0.368264 −0.0180340
\(418\) 0 0
\(419\) 0.473345 0.196066i 0.0231244 0.00957845i −0.371091 0.928596i \(-0.621016\pi\)
0.394216 + 0.919018i \(0.371016\pi\)
\(420\) 0 0
\(421\) −1.25389 + 3.02716i −0.0611109 + 0.147535i −0.951485 0.307694i \(-0.900443\pi\)
0.890374 + 0.455229i \(0.150443\pi\)
\(422\) 0 0
\(423\) 5.28853 + 5.28853i 0.257137 + 0.257137i
\(424\) 0 0
\(425\) 3.62118 3.62118i 0.175653 0.175653i
\(426\) 0 0
\(427\) 1.36242 + 0.564331i 0.0659319 + 0.0273099i
\(428\) 0 0
\(429\) 7.81716 + 18.8723i 0.377416 + 0.911163i
\(430\) 0 0
\(431\) 29.1399i 1.40362i −0.712365 0.701810i \(-0.752376\pi\)
0.712365 0.701810i \(-0.247624\pi\)
\(432\) 0 0
\(433\) 5.66352i 0.272172i −0.990697 0.136086i \(-0.956548\pi\)
0.990697 0.136086i \(-0.0434523\pi\)
\(434\) 0 0
\(435\) −1.28763 3.10861i −0.0617371 0.149047i
\(436\) 0 0
\(437\) 4.68355 + 1.93999i 0.224045 + 0.0928023i
\(438\) 0 0
\(439\) 10.7013 10.7013i 0.510743 0.510743i −0.404011 0.914754i \(-0.632384\pi\)
0.914754 + 0.404011i \(0.132384\pi\)
\(440\) 0 0
\(441\) −4.23866 4.23866i −0.201841 0.201841i
\(442\) 0 0
\(443\) 14.3853 34.7291i 0.683464 1.65003i −0.0740866 0.997252i \(-0.523604\pi\)
0.757551 0.652776i \(-0.226396\pi\)
\(444\) 0 0
\(445\) 4.42658 1.83355i 0.209840 0.0869185i
\(446\) 0 0
\(447\) −12.4754 −0.590067
\(448\) 0 0
\(449\) −26.7000 −1.26005 −0.630025 0.776575i \(-0.716956\pi\)
−0.630025 + 0.776575i \(0.716956\pi\)
\(450\) 0 0
\(451\) −20.2102 + 8.37136i −0.951663 + 0.394192i
\(452\) 0 0
\(453\) 3.04858 7.35993i 0.143235 0.345800i
\(454\) 0 0
\(455\) 1.94984 + 1.94984i 0.0914097 + 0.0914097i
\(456\) 0 0
\(457\) 1.66332 1.66332i 0.0778069 0.0778069i −0.667132 0.744939i \(-0.732478\pi\)
0.744939 + 0.667132i \(0.232478\pi\)
\(458\) 0 0
\(459\) −0.978635 0.405364i −0.0456788 0.0189208i
\(460\) 0 0
\(461\) −3.49519 8.43813i −0.162787 0.393003i 0.821347 0.570429i \(-0.193223\pi\)
−0.984134 + 0.177426i \(0.943223\pi\)
\(462\) 0 0
\(463\) 12.1833i 0.566204i 0.959090 + 0.283102i \(0.0913635\pi\)
−0.959090 + 0.283102i \(0.908636\pi\)
\(464\) 0 0
\(465\) 0.766371i 0.0355396i
\(466\) 0 0
\(467\) −4.89494 11.8174i −0.226511 0.546846i 0.769237 0.638963i \(-0.220636\pi\)
−0.995748 + 0.0921176i \(0.970636\pi\)
\(468\) 0 0
\(469\) −10.9007 4.51520i −0.503346 0.208493i
\(470\) 0 0
\(471\) −11.5809 + 11.5809i −0.533621 + 0.533621i
\(472\) 0 0
\(473\) 25.3146 + 25.3146i 1.16396 + 1.16396i
\(474\) 0 0
\(475\) 7.19168 17.3623i 0.329977 0.796635i
\(476\) 0 0
\(477\) 7.58131 3.14028i 0.347125 0.143784i
\(478\) 0 0
\(479\) −19.9031 −0.909396 −0.454698 0.890646i \(-0.650253\pi\)
−0.454698 + 0.890646i \(0.650253\pi\)
\(480\) 0 0
\(481\) 22.0504 1.00541
\(482\) 0 0
\(483\) 1.20826 0.500480i 0.0549780 0.0227726i
\(484\) 0 0
\(485\) −1.24057 + 2.99499i −0.0563312 + 0.135995i
\(486\) 0 0
\(487\) 16.3518 + 16.3518i 0.740969 + 0.740969i 0.972765 0.231795i \(-0.0744599\pi\)
−0.231795 + 0.972765i \(0.574460\pi\)
\(488\) 0 0
\(489\) 9.54542 9.54542i 0.431659 0.431659i
\(490\) 0 0
\(491\) −7.50463 3.10852i −0.338679 0.140286i 0.206861 0.978370i \(-0.433675\pi\)
−0.545540 + 0.838085i \(0.683675\pi\)
\(492\) 0 0
\(493\) −3.35369 8.09652i −0.151042 0.364649i
\(494\) 0 0
\(495\) 1.22875i 0.0552280i
\(496\) 0 0
\(497\) 13.2624i 0.594899i
\(498\) 0 0
\(499\) −4.17340 10.0755i −0.186827 0.451041i 0.802518 0.596627i \(-0.203493\pi\)
−0.989345 + 0.145587i \(0.953493\pi\)
\(500\) 0 0
\(501\) 17.1066 + 7.08580i 0.764268 + 0.316570i
\(502\) 0 0
\(503\) −18.7140 + 18.7140i −0.834416 + 0.834416i −0.988117 0.153701i \(-0.950881\pi\)
0.153701 + 0.988117i \(0.450881\pi\)
\(504\) 0 0
\(505\) 2.73166 + 2.73166i 0.121557 + 0.121557i
\(506\) 0 0
\(507\) −12.5188 + 30.2231i −0.555981 + 1.34226i
\(508\) 0 0
\(509\) −37.2151 + 15.4150i −1.64953 + 0.683258i −0.997207 0.0746822i \(-0.976206\pi\)
−0.652324 + 0.757940i \(0.726206\pi\)
\(510\) 0 0
\(511\) −0.521380 −0.0230645
\(512\) 0 0
\(513\) −3.88714 −0.171622
\(514\) 0 0
\(515\) 1.29937 0.538215i 0.0572569 0.0237166i
\(516\) 0 0
\(517\) 8.64725 20.8763i 0.380306 0.918139i
\(518\) 0 0
\(519\) 5.70304 + 5.70304i 0.250336 + 0.250336i
\(520\) 0 0
\(521\) 9.78297 9.78297i 0.428600 0.428600i −0.459551 0.888151i \(-0.651990\pi\)
0.888151 + 0.459551i \(0.151990\pi\)
\(522\) 0 0
\(523\) 9.08765 + 3.76423i 0.397375 + 0.164598i 0.572417 0.819963i \(-0.306006\pi\)
−0.175042 + 0.984561i \(0.556006\pi\)
\(524\) 0 0
\(525\) −1.85531 4.47913i −0.0809726 0.195485i
\(526\) 0 0
\(527\) 1.99605i 0.0869491i
\(528\) 0 0
\(529\) 21.2992i 0.926051i
\(530\) 0 0
\(531\) 4.13463 + 9.98188i 0.179428 + 0.433177i
\(532\) 0 0
\(533\) −45.2277 18.7339i −1.95903 0.811457i
\(534\) 0 0
\(535\) 4.58190 4.58190i 0.198093 0.198093i
\(536\) 0 0
\(537\) 6.41240 + 6.41240i 0.276715 + 0.276715i
\(538\) 0 0
\(539\) −6.93062 + 16.7320i −0.298523 + 0.720698i
\(540\) 0 0
\(541\) 20.5649 8.51826i 0.884154 0.366229i 0.106048 0.994361i \(-0.466180\pi\)
0.778106 + 0.628132i \(0.216180\pi\)
\(542\) 0 0
\(543\) 22.5446 0.967482
\(544\) 0 0
\(545\) −3.87292 −0.165898
\(546\) 0 0
\(547\) −18.2473 + 7.55828i −0.780198 + 0.323169i −0.736996 0.675898i \(-0.763756\pi\)
−0.0432027 + 0.999066i \(0.513756\pi\)
\(548\) 0 0
\(549\) −0.562751 + 1.35860i −0.0240176 + 0.0579837i
\(550\) 0 0
\(551\) −22.7401 22.7401i −0.968763 0.968763i
\(552\) 0 0
\(553\) −4.16786 + 4.16786i −0.177235 + 0.177235i
\(554\) 0 0
\(555\) 1.22542 + 0.507585i 0.0520161 + 0.0215458i
\(556\) 0 0
\(557\) 2.57380 + 6.21371i 0.109056 + 0.263283i 0.968981 0.247137i \(-0.0794898\pi\)
−0.859925 + 0.510420i \(0.829490\pi\)
\(558\) 0 0
\(559\) 80.1159i 3.38854i
\(560\) 0 0
\(561\) 3.20032i 0.135118i
\(562\) 0 0
\(563\) −3.34911 8.08547i −0.141148 0.340762i 0.837459 0.546501i \(-0.184040\pi\)
−0.978607 + 0.205738i \(0.934040\pi\)
\(564\) 0 0
\(565\) 2.55476 + 1.05822i 0.107480 + 0.0445195i
\(566\) 0 0
\(567\) −0.709092 + 0.709092i −0.0297791 + 0.0297791i
\(568\) 0 0
\(569\) −5.86928 5.86928i −0.246053 0.246053i 0.573296 0.819349i \(-0.305665\pi\)
−0.819349 + 0.573296i \(0.805665\pi\)
\(570\) 0 0
\(571\) −4.92409 + 11.8878i −0.206067 + 0.497489i −0.992797 0.119808i \(-0.961772\pi\)
0.786730 + 0.617297i \(0.211772\pi\)
\(572\) 0 0
\(573\) 9.24797 3.83064i 0.386340 0.160027i
\(574\) 0 0
\(575\) −6.30506 −0.262939
\(576\) 0 0
\(577\) 12.4165 0.516905 0.258453 0.966024i \(-0.416787\pi\)
0.258453 + 0.966024i \(0.416787\pi\)
\(578\) 0 0
\(579\) 8.78684 3.63963i 0.365169 0.151258i
\(580\) 0 0
\(581\) 1.62496 3.92299i 0.0674145 0.162753i
\(582\) 0 0
\(583\) −17.5308 17.5308i −0.726052 0.726052i
\(584\) 0 0
\(585\) −1.94438 + 1.94438i −0.0803900 + 0.0803900i
\(586\) 0 0
\(587\) 11.0333 + 4.57014i 0.455392 + 0.188630i 0.598575 0.801067i \(-0.295734\pi\)
−0.143183 + 0.989696i \(0.545734\pi\)
\(588\) 0 0
\(589\) −2.80308 6.76724i −0.115499 0.278839i
\(590\) 0 0
\(591\) 13.7797i 0.566819i
\(592\) 0 0
\(593\) 24.1273i 0.990791i 0.868668 + 0.495395i \(0.164977\pi\)
−0.868668 + 0.495395i \(0.835023\pi\)
\(594\) 0 0
\(595\) 0.165324 + 0.399128i 0.00677764 + 0.0163627i
\(596\) 0 0
\(597\) 12.7296 + 5.27279i 0.520989 + 0.215801i
\(598\) 0 0
\(599\) 22.1187 22.1187i 0.903747 0.903747i −0.0920113 0.995758i \(-0.529330\pi\)
0.995758 + 0.0920113i \(0.0293296\pi\)
\(600\) 0 0
\(601\) 20.4369 + 20.4369i 0.833637 + 0.833637i 0.988012 0.154375i \(-0.0493365\pi\)
−0.154375 + 0.988012i \(0.549336\pi\)
\(602\) 0 0
\(603\) 4.50256 10.8701i 0.183358 0.442666i
\(604\) 0 0
\(605\) −0.703383 + 0.291351i −0.0285966 + 0.0118451i
\(606\) 0 0
\(607\) 38.6180 1.56745 0.783727 0.621105i \(-0.213316\pi\)
0.783727 + 0.621105i \(0.213316\pi\)
\(608\) 0 0
\(609\) −8.29651 −0.336191
\(610\) 0 0
\(611\) 46.7183 19.3513i 1.89002 0.782872i
\(612\) 0 0
\(613\) −10.6069 + 25.6072i −0.428407 + 1.03427i 0.551385 + 0.834251i \(0.314099\pi\)
−0.979793 + 0.200016i \(0.935901\pi\)
\(614\) 0 0
\(615\) −2.08222 2.08222i −0.0839633 0.0839633i
\(616\) 0 0
\(617\) −31.9989 + 31.9989i −1.28823 + 1.28823i −0.352364 + 0.935863i \(0.614622\pi\)
−0.935863 + 0.352364i \(0.885378\pi\)
\(618\) 0 0
\(619\) −29.4178 12.1852i −1.18240 0.489766i −0.297127 0.954838i \(-0.596029\pi\)
−0.885273 + 0.465072i \(0.846029\pi\)
\(620\) 0 0
\(621\) 0.499078 + 1.20488i 0.0200273 + 0.0483502i
\(622\) 0 0
\(623\) 11.8140i 0.473317i
\(624\) 0 0
\(625\) 22.5463i 0.901852i
\(626\) 0 0
\(627\) 4.49426 + 10.8501i 0.179484 + 0.433312i
\(628\) 0 0
\(629\) 3.19165 + 1.32203i 0.127260 + 0.0527126i
\(630\) 0 0
\(631\) 8.23619 8.23619i 0.327878 0.327878i −0.523901 0.851779i \(-0.675524\pi\)
0.851779 + 0.523901i \(0.175524\pi\)
\(632\) 0 0
\(633\) 4.25143 + 4.25143i 0.168979 + 0.168979i
\(634\) 0 0
\(635\) 0.805431 1.94448i 0.0319626 0.0771645i
\(636\) 0 0
\(637\) −37.4439 + 15.5098i −1.48358 + 0.614519i
\(638\) 0 0
\(639\) −13.2252 −0.523182
\(640\) 0 0
\(641\) 8.56808 0.338419 0.169209 0.985580i \(-0.445879\pi\)
0.169209 + 0.985580i \(0.445879\pi\)
\(642\) 0 0
\(643\) 3.61947 1.49923i 0.142738 0.0591240i −0.310171 0.950681i \(-0.600386\pi\)
0.452909 + 0.891557i \(0.350386\pi\)
\(644\) 0 0
\(645\) −1.84421 + 4.45232i −0.0726158 + 0.175310i
\(646\) 0 0
\(647\) 23.7346 + 23.7346i 0.933105 + 0.933105i 0.997899 0.0647935i \(-0.0206389\pi\)
−0.0647935 + 0.997899i \(0.520639\pi\)
\(648\) 0 0
\(649\) 23.0818 23.0818i 0.906041 0.906041i
\(650\) 0 0
\(651\) −1.74582 0.723141i −0.0684239 0.0283421i
\(652\) 0 0
\(653\) −9.65936 23.3198i −0.378000 0.912573i −0.992341 0.123532i \(-0.960578\pi\)
0.614341 0.789041i \(-0.289422\pi\)
\(654\) 0 0
\(655\) 0.00616262i 0.000240793i
\(656\) 0 0
\(657\) 0.519920i 0.0202840i
\(658\) 0 0
\(659\) 14.0711 + 33.9708i 0.548134 + 1.32331i 0.918865 + 0.394572i \(0.129107\pi\)
−0.370731 + 0.928740i \(0.620893\pi\)
\(660\) 0 0
\(661\) −20.5485 8.51149i −0.799246 0.331058i −0.0545915 0.998509i \(-0.517386\pi\)
−0.744654 + 0.667450i \(0.767386\pi\)
\(662\) 0 0
\(663\) −5.06421 + 5.06421i −0.196678 + 0.196678i
\(664\) 0 0
\(665\) 1.12100 + 1.12100i 0.0434707 + 0.0434707i
\(666\) 0 0
\(667\) −4.12901 + 9.96832i −0.159876 + 0.385975i
\(668\) 0 0
\(669\) −7.67481 + 3.17901i −0.296725 + 0.122908i
\(670\) 0 0
\(671\) 4.44288 0.171516
\(672\) 0 0
\(673\) −32.6628 −1.25906 −0.629529 0.776977i \(-0.716752\pi\)
−0.629529 + 0.776977i \(0.716752\pi\)
\(674\) 0 0
\(675\) 4.46658 1.85012i 0.171919 0.0712111i
\(676\) 0 0
\(677\) −18.6093 + 44.9269i −0.715215 + 1.72668i −0.0286762 + 0.999589i \(0.509129\pi\)
−0.686539 + 0.727093i \(0.740871\pi\)
\(678\) 0 0
\(679\) 5.65209 + 5.65209i 0.216907 + 0.216907i
\(680\) 0 0
\(681\) −16.5362 + 16.5362i −0.633667 + 0.633667i
\(682\) 0 0
\(683\) 7.54722 + 3.12616i 0.288786 + 0.119619i 0.522374 0.852717i \(-0.325047\pi\)
−0.233588 + 0.972336i \(0.575047\pi\)
\(684\) 0 0
\(685\) −1.92668 4.65141i −0.0736145 0.177721i
\(686\) 0 0
\(687\) 12.7881i 0.487896i
\(688\) 0 0
\(689\) 55.4818i 2.11369i
\(690\) 0 0
\(691\) −11.8386 28.5808i −0.450360 1.08727i −0.972185 0.234213i \(-0.924749\pi\)
0.521825 0.853053i \(-0.325251\pi\)
\(692\) 0 0
\(693\) 2.79912 + 1.15943i 0.106330 + 0.0440432i
\(694\) 0 0
\(695\) −0.105905 + 0.105905i −0.00401722 + 0.00401722i
\(696\) 0 0
\(697\) −5.42324 5.42324i −0.205420 0.205420i
\(698\) 0 0
\(699\) −7.75192 + 18.7148i −0.293205 + 0.707859i
\(700\) 0 0
\(701\) 39.3636 16.3049i 1.48674 0.615829i 0.516138 0.856505i \(-0.327369\pi\)
0.970606 + 0.240676i \(0.0773691\pi\)
\(702\) 0 0
\(703\) 12.6773 0.478132
\(704\) 0 0
\(705\) 3.04175 0.114559
\(706\) 0 0
\(707\) 8.80035 3.64523i 0.330971 0.137093i
\(708\) 0 0
\(709\) −9.95013 + 24.0218i −0.373685 + 0.902156i 0.619434 + 0.785049i \(0.287362\pi\)
−0.993119 + 0.117107i \(0.962638\pi\)
\(710\) 0 0
\(711\) −4.15619 4.15619i −0.155869 0.155869i
\(712\) 0 0
\(713\) −1.73772 + 1.73772i −0.0650781 + 0.0650781i
\(714\) 0 0
\(715\) 7.67536 + 3.17924i 0.287042 + 0.118897i
\(716\) 0 0
\(717\) −4.51884 10.9095i −0.168759 0.407421i
\(718\) 0 0
\(719\) 21.4598i 0.800314i 0.916447 + 0.400157i \(0.131044\pi\)
−0.916447 + 0.400157i \(0.868956\pi\)
\(720\) 0 0
\(721\) 3.46785i 0.129149i
\(722\) 0 0
\(723\) −3.19213 7.70649i −0.118717 0.286608i
\(724\) 0 0
\(725\) 36.9533 + 15.3066i 1.37241 + 0.568471i
\(726\) 0 0
\(727\) −29.1500 + 29.1500i −1.08111 + 1.08111i −0.0847075 + 0.996406i \(0.526996\pi\)
−0.996406 + 0.0847075i \(0.973004\pi\)
\(728\) 0 0
\(729\) −0.707107 0.707107i −0.0261891 0.0261891i
\(730\) 0 0
\(731\) −4.80333 + 11.5963i −0.177658 + 0.428903i
\(732\) 0 0
\(733\) 24.5981 10.1889i 0.908551 0.376334i 0.121049 0.992646i \(-0.461374\pi\)
0.787502 + 0.616312i \(0.211374\pi\)
\(734\) 0 0
\(735\) −2.43791 −0.0899237
\(736\) 0 0
\(737\) −35.5474 −1.30941
\(738\) 0 0
\(739\) 32.6336 13.5173i 1.20044 0.497240i 0.309302 0.950964i \(-0.399905\pi\)
0.891143 + 0.453723i \(0.149905\pi\)
\(740\) 0 0
\(741\) −10.0575 + 24.2810i −0.369473 + 0.891986i
\(742\) 0 0
\(743\) 1.74184 + 1.74184i 0.0639017 + 0.0639017i 0.738335 0.674434i \(-0.235612\pi\)
−0.674434 + 0.738335i \(0.735612\pi\)
\(744\) 0 0
\(745\) −3.58768 + 3.58768i −0.131442 + 0.131442i
\(746\) 0 0
\(747\) 3.91201 + 1.62041i 0.143133 + 0.0592876i
\(748\) 0 0
\(749\) −6.11427 14.7611i −0.223410 0.539360i
\(750\) 0 0
\(751\) 2.87921i 0.105064i 0.998619 + 0.0525319i \(0.0167291\pi\)
−0.998619 + 0.0525319i \(0.983271\pi\)
\(752\) 0 0
\(753\) 21.7294i 0.791863i
\(754\) 0 0
\(755\) −1.23986 2.99328i −0.0451231 0.108937i
\(756\) 0 0
\(757\) −5.34241 2.21290i −0.194173 0.0804292i 0.283478 0.958979i \(-0.408512\pi\)
−0.477651 + 0.878550i \(0.658512\pi\)
\(758\) 0 0
\(759\) 2.78614 2.78614i 0.101130 0.101130i
\(760\) 0 0
\(761\) 10.3052 + 10.3052i 0.373565 + 0.373565i 0.868774 0.495209i \(-0.164908\pi\)
−0.495209 + 0.868774i \(0.664908\pi\)
\(762\) 0 0
\(763\) −3.65445 + 8.82263i −0.132300 + 0.319401i
\(764\) 0 0
\(765\) −0.398011 + 0.164861i −0.0143901 + 0.00596058i
\(766\) 0 0
\(767\) 73.0497 2.63767
\(768\) 0 0
\(769\) −45.8055 −1.65179 −0.825894 0.563826i \(-0.809329\pi\)
−0.825894 + 0.563826i \(0.809329\pi\)
\(770\) 0 0
\(771\) 10.0497 4.16272i 0.361931 0.149917i
\(772\) 0 0
\(773\) −5.47235 + 13.2114i −0.196827 + 0.475182i −0.991220 0.132222i \(-0.957789\pi\)
0.794393 + 0.607404i \(0.207789\pi\)
\(774\) 0 0
\(775\) 6.44185 + 6.44185i 0.231398 + 0.231398i
\(776\) 0 0
\(777\) 2.31258 2.31258i 0.0829635 0.0829635i
\(778\) 0 0
\(779\) −26.0024 10.7706i −0.931634 0.385896i
\(780\) 0 0
\(781\) 15.2908 + 36.9153i 0.547149 + 1.32093i
\(782\) 0 0
\(783\) 8.27328i 0.295663i
\(784\) 0 0
\(785\) 6.66090i 0.237738i
\(786\) 0 0
\(787\) 15.7659 + 38.0621i 0.561992 + 1.35677i 0.908171 + 0.418600i \(0.137479\pi\)
−0.346179 + 0.938169i \(0.612521\pi\)
\(788\) 0 0
\(789\) 3.39547 + 1.40645i 0.120882 + 0.0500709i
\(790\) 0 0
\(791\) 4.82129 4.82129i 0.171425 0.171425i
\(792\) 0 0
\(793\) 7.03044 + 7.03044i 0.249658 + 0.249658i
\(794\) 0 0
\(795\) 1.27715 3.08332i 0.0452959 0.109354i
\(796\) 0 0
\(797\) −48.3932 + 20.0451i −1.71418 + 0.710035i −0.714226 + 0.699915i \(0.753221\pi\)
−0.999949 + 0.0101198i \(0.996779\pi\)
\(798\) 0 0
\(799\) 7.92238 0.280274
\(800\) 0 0
\(801\) 11.7809 0.416258
\(802\) 0 0
\(803\) −1.45124 + 0.601125i −0.0512133 + 0.0212132i
\(804\) 0 0
\(805\) 0.203545 0.491401i 0.00717402 0.0173196i
\(806\) 0 0
\(807\) 4.68263 + 4.68263i 0.164836 + 0.164836i
\(808\) 0 0
\(809\) −8.44314 + 8.44314i −0.296845 + 0.296845i −0.839777 0.542932i \(-0.817314\pi\)
0.542932 + 0.839777i \(0.317314\pi\)
\(810\) 0 0
\(811\) 23.8583 + 9.88243i 0.837778 + 0.347019i 0.759977 0.649950i \(-0.225210\pi\)
0.0778011 + 0.996969i \(0.475210\pi\)
\(812\) 0 0
\(813\) −9.75347 23.5470i −0.342069 0.825828i
\(814\) 0 0
\(815\) 5.49015i 0.192311i
\(816\) 0 0
\(817\) 46.0604i 1.61145i
\(818\) 0 0
\(819\) 2.59465 + 6.26404i 0.0906644 + 0.218883i
\(820\) 0 0
\(821\) 32.6979 + 13.5439i 1.14117 + 0.472686i 0.871562 0.490285i \(-0.163107\pi\)
0.269604 + 0.962971i \(0.413107\pi\)
\(822\) 0 0
\(823\) 36.5949 36.5949i 1.27562 1.27562i 0.332524 0.943095i \(-0.392100\pi\)
0.943095 0.332524i \(-0.107900\pi\)
\(824\) 0 0
\(825\) −10.3284 10.3284i −0.359589 0.359589i
\(826\) 0 0
\(827\) 3.44587 8.31906i 0.119824 0.289282i −0.852575 0.522605i \(-0.824960\pi\)
0.972399 + 0.233323i \(0.0749601\pi\)
\(828\) 0 0
\(829\) 31.0049 12.8427i 1.07685 0.446044i 0.227444 0.973791i \(-0.426963\pi\)
0.849402 + 0.527747i \(0.176963\pi\)
\(830\) 0 0
\(831\) −1.56982 −0.0544565
\(832\) 0 0
\(833\) −6.34964 −0.220002
\(834\) 0 0
\(835\) 6.95726 2.88179i 0.240766 0.0997285i
\(836\) 0 0
\(837\) 0.721116 1.74093i 0.0249254 0.0601753i
\(838\) 0 0
\(839\) −24.3721 24.3721i −0.841418 0.841418i 0.147625 0.989043i \(-0.452837\pi\)
−0.989043 + 0.147625i \(0.952837\pi\)
\(840\) 0 0
\(841\) 27.8933 27.8933i 0.961838 0.961838i
\(842\) 0 0
\(843\) −4.74116 1.96385i −0.163294 0.0676387i
\(844\) 0 0
\(845\) 5.09141 + 12.2917i 0.175150 + 0.422849i
\(846\) 0 0
\(847\) 1.87724i 0.0645028i
\(848\) 0 0
\(849\) 6.52633i 0.223983i
\(850\) 0 0
\(851\) −1.62766 3.92952i −0.0557955 0.134702i
\(852\) 0 0
\(853\) −33.3194 13.8013i −1.14083 0.472548i −0.269383 0.963033i \(-0.586820\pi\)
−0.871450 + 0.490485i \(0.836820\pi\)
\(854\) 0 0
\(855\) −1.11787 + 1.11787i −0.0382302 + 0.0382302i
\(856\) 0 0
\(857\) −18.3586 18.3586i −0.627119 0.627119i 0.320223 0.947342i \(-0.396242\pi\)
−0.947342 + 0.320223i \(0.896242\pi\)
\(858\) 0 0
\(859\) 3.92601 9.47823i 0.133954 0.323393i −0.842643 0.538473i \(-0.819001\pi\)
0.976596 + 0.215080i \(0.0690013\pi\)
\(860\) 0 0
\(861\) −6.70813 + 2.77860i −0.228612 + 0.0946943i
\(862\) 0 0
\(863\) −31.1176 −1.05925 −0.529627 0.848231i \(-0.677668\pi\)
−0.529627 + 0.848231i \(0.677668\pi\)
\(864\) 0 0
\(865\) 3.28016 0.111529
\(866\) 0 0
\(867\) 14.6693 6.07623i 0.498196 0.206360i
\(868\) 0 0
\(869\) −6.79576 + 16.4064i −0.230530 + 0.556550i
\(870\) 0 0
\(871\) −56.2504 56.2504i −1.90597 1.90597i
\(872\) 0 0
\(873\) −5.63626 + 5.63626i −0.190759 + 0.190759i
\(874\) 0 0
\(875\) −3.70564 1.53493i −0.125274 0.0518900i
\(876\) 0 0
\(877\) −1.50097 3.62367i −0.0506843 0.122363i 0.896509 0.443025i \(-0.146095\pi\)
−0.947194 + 0.320662i \(0.896095\pi\)
\(878\) 0 0
\(879\) 1.94385i 0.0655644i
\(880\) 0 0
\(881\) 5.19539i 0.175037i 0.996163 + 0.0875186i \(0.0278937\pi\)
−0.996163 + 0.0875186i \(0.972106\pi\)
\(882\) 0 0
\(883\) 16.3773 + 39.5382i 0.551139 + 1.33057i 0.916624 + 0.399750i \(0.130903\pi\)
−0.365486 + 0.930817i \(0.619097\pi\)
\(884\) 0 0
\(885\) 4.05963 + 1.68155i 0.136463 + 0.0565248i
\(886\) 0 0
\(887\) −16.0073 + 16.0073i −0.537473 + 0.537473i −0.922786 0.385313i \(-0.874093\pi\)
0.385313 + 0.922786i \(0.374093\pi\)
\(888\) 0 0
\(889\) −3.66959 3.66959i −0.123074 0.123074i
\(890\) 0 0
\(891\) −1.15619 + 2.79128i −0.0387337 + 0.0935114i
\(892\) 0 0
\(893\) 26.8594 11.1255i 0.898815 0.372302i
\(894\) 0 0
\(895\) 3.68816 0.123282
\(896\) 0 0
\(897\) 8.81760 0.294411
\(898\) 0 0
\(899\) 14.4032 5.96599i 0.480373 0.198977i
\(900\) 0 0
\(901\) 3.32640 8.03063i 0.110818 0.267539i
\(902\) 0 0
\(903\) 8.40234 + 8.40234i 0.279612 + 0.279612i
\(904\) 0 0
\(905\) 6.48339 6.48339i 0.215515 0.215515i
\(906\) 0 0
\(907\) 27.8131 + 11.5206i 0.923519 + 0.382534i 0.793216 0.608940i \(-0.208405\pi\)
0.130303 + 0.991474i \(0.458405\pi\)
\(908\) 0 0
\(909\) 3.63502 + 8.77571i 0.120566 + 0.291072i
\(910\) 0 0
\(911\) 39.3207i 1.30275i 0.758755 + 0.651376i \(0.225808\pi\)
−0.758755 + 0.651376i \(0.774192\pi\)
\(912\) 0 0
\(913\) 12.7930i 0.423386i
\(914\) 0 0
\(915\) 0.228871 + 0.552543i 0.00756623 + 0.0182665i
\(916\) 0 0
\(917\) −0.0140386 0.00581499i −0.000463596 0.000192028i
\(918\) 0 0
\(919\) 11.6061 11.6061i 0.382849 0.382849i −0.489279 0.872128i \(-0.662740\pi\)
0.872128 + 0.489279i \(0.162740\pi\)
\(920\) 0 0
\(921\) 15.5620 + 15.5620i 0.512787 + 0.512787i
\(922\) 0 0
\(923\) −34.2188 + 82.6114i −1.12632 + 2.71919i
\(924\) 0 0
\(925\) −14.5670 + 6.03385i −0.478960 + 0.198392i
\(926\) 0 0
\(927\) 3.45814 0.113580
\(928\) 0 0
\(929\) 0.158082 0.00518650 0.00259325 0.999997i \(-0.499175\pi\)
0.00259325 + 0.999997i \(0.499175\pi\)
\(930\) 0 0
\(931\) −21.5273 + 8.91691i −0.705529 + 0.292240i
\(932\) 0 0
\(933\) 7.13032 17.2141i 0.233436 0.563565i
\(934\) 0 0
\(935\) 0.920349 + 0.920349i 0.0300986 + 0.0300986i
\(936\) 0 0
\(937\) 4.69489 4.69489i 0.153376 0.153376i −0.626248 0.779624i \(-0.715410\pi\)
0.779624 + 0.626248i \(0.215410\pi\)
\(938\) 0 0
\(939\) −22.2990 9.23656i −0.727701 0.301424i
\(940\) 0 0
\(941\) −10.6171 25.6321i −0.346109 0.835581i −0.997072 0.0764715i \(-0.975635\pi\)
0.650963 0.759110i \(-0.274365\pi\)
\(942\) 0 0
\(943\) 9.44272i 0.307497i
\(944\) 0 0
\(945\) 0.407842i 0.0132671i
\(946\) 0 0
\(947\) −3.51311 8.48139i −0.114161 0.275608i 0.856464 0.516207i \(-0.172656\pi\)
−0.970625 + 0.240598i \(0.922656\pi\)
\(948\) 0 0
\(949\) −3.24768 1.34523i −0.105424 0.0436681i
\(950\) 0 0
\(951\) −1.68960 + 1.68960i −0.0547889 + 0.0547889i
\(952\) 0 0
\(953\) 26.3477 + 26.3477i 0.853487 + 0.853487i 0.990561 0.137074i \(-0.0437697\pi\)
−0.137074 + 0.990561i \(0.543770\pi\)
\(954\) 0 0
\(955\) 1.55792 3.76115i 0.0504131 0.121708i
\(956\) 0 0
\(957\) −23.0930 + 9.56545i −0.746492 + 0.309207i
\(958\) 0 0
\(959\) −12.4140 −0.400870
\(960\) 0 0
\(961\) −27.4492 −0.885457
\(962\) 0 0
\(963\) 14.7198 6.09714i 0.474339 0.196478i
\(964\) 0 0
\(965\) 1.48024 3.57361i 0.0476505 0.115038i
\(966\) 0 0
\(967\) −18.1509 18.1509i −0.583694 0.583694i 0.352222 0.935916i \(-0.385426\pi\)
−0.935916 + 0.352222i \(0.885426\pi\)
\(968\) 0 0
\(969\) −2.91153 + 2.91153i −0.0935318 + 0.0935318i
\(970\) 0 0
\(971\) −38.1928 15.8200i −1.22566 0.507687i −0.326458 0.945212i \(-0.605855\pi\)
−0.899206 + 0.437525i \(0.855855\pi\)
\(972\) 0 0
\(973\) 0.141324 + 0.341187i 0.00453065 + 0.0109379i
\(974\) 0 0
\(975\) 32.6875i 1.04684i
\(976\) 0 0
\(977\) 18.3863i 0.588229i −0.955770 0.294114i \(-0.904975\pi\)
0.955770 0.294114i \(-0.0950246\pi\)
\(978\) 0 0
\(979\) −13.6209 32.8838i −0.435327 1.05097i
\(980\) 0 0
\(981\) −8.79792 3.64422i −0.280896 0.116351i
\(982\) 0 0
\(983\) −29.1468 + 29.1468i −0.929639 + 0.929639i −0.997682 0.0680436i \(-0.978324\pi\)
0.0680436 + 0.997682i \(0.478324\pi\)
\(984\) 0 0
\(985\) −3.96275 3.96275i −0.126264 0.126264i
\(986\) 0 0
\(987\) 2.87017 6.92920i 0.0913585 0.220559i
\(988\) 0 0
\(989\) 14.2772 5.91379i 0.453987 0.188048i
\(990\) 0 0
\(991\) −29.7473 −0.944954 −0.472477 0.881343i \(-0.656640\pi\)
−0.472477 + 0.881343i \(0.656640\pi\)
\(992\) 0 0
\(993\) −5.98346 −0.189879
\(994\) 0 0
\(995\) 5.17714 2.14444i 0.164126 0.0679834i
\(996\) 0 0
\(997\) −13.8686 + 33.4817i −0.439223 + 1.06038i 0.536995 + 0.843585i \(0.319559\pi\)
−0.976218 + 0.216792i \(0.930441\pi\)
\(998\) 0 0
\(999\) 2.30611 + 2.30611i 0.0729621 + 0.0729621i
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 768.2.n.b.97.7 32
4.3 odd 2 768.2.n.a.97.3 32
8.3 odd 2 96.2.n.a.85.6 yes 32
8.5 even 2 384.2.n.a.49.2 32
24.5 odd 2 1152.2.v.c.433.6 32
24.11 even 2 288.2.v.d.181.3 32
32.3 odd 8 768.2.n.a.673.3 32
32.13 even 8 384.2.n.a.337.2 32
32.19 odd 8 96.2.n.a.61.6 32
32.29 even 8 inner 768.2.n.b.673.7 32
96.77 odd 8 1152.2.v.c.721.6 32
96.83 even 8 288.2.v.d.253.3 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.n.a.61.6 32 32.19 odd 8
96.2.n.a.85.6 yes 32 8.3 odd 2
288.2.v.d.181.3 32 24.11 even 2
288.2.v.d.253.3 32 96.83 even 8
384.2.n.a.49.2 32 8.5 even 2
384.2.n.a.337.2 32 32.13 even 8
768.2.n.a.97.3 32 4.3 odd 2
768.2.n.a.673.3 32 32.3 odd 8
768.2.n.b.97.7 32 1.1 even 1 trivial
768.2.n.b.673.7 32 32.29 even 8 inner
1152.2.v.c.433.6 32 24.5 odd 2
1152.2.v.c.721.6 32 96.77 odd 8