Properties

Label 2240.2.k.f.1791.3
Level $2240$
Weight $2$
Character 2240.1791
Analytic conductor $17.886$
Analytic rank $0$
Dimension $16$
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] = [2240,2,Mod(1791,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1791");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.k (of order \(2\), degree \(1\), 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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 6 x^{14} + 68 x^{13} - 126 x^{12} - 148 x^{11} + 1006 x^{10} - 1516 x^{9} - 179 x^{8} + 3992 x^{7} - 5596 x^{6} - 488 x^{5} + 16080 x^{4} - 33776 x^{3} + \cdots + 5956 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 1120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1791.3
Root \(2.23037 + 0.887196i\) of defining polynomial
Character \(\chi\) \(=\) 2240.1791
Dual form 2240.2.k.f.1791.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.77439 q^{3} -1.00000i q^{5} +(0.829909 - 2.51222i) q^{7} +0.148464 q^{9} +O(q^{10})\) \(q-1.77439 q^{3} -1.00000i q^{5} +(0.829909 - 2.51222i) q^{7} +0.148464 q^{9} +6.36631i q^{11} -0.563705i q^{13} +1.77439i q^{15} -6.12055i q^{17} +0.694886 q^{19} +(-1.47258 + 4.45766i) q^{21} +0.964933i q^{23} -1.00000 q^{25} +5.05974 q^{27} +2.02585 q^{29} +8.09745 q^{31} -11.2963i q^{33} +(-2.51222 - 0.829909i) q^{35} -1.89633 q^{37} +1.00023i q^{39} +1.97453i q^{41} -5.26331i q^{43} -0.148464i q^{45} +3.03286 q^{47} +(-5.62250 - 4.16983i) q^{49} +10.8603i q^{51} -1.73956 q^{53} +6.36631 q^{55} -1.23300 q^{57} -12.9168 q^{59} -9.96224i q^{61} +(0.123212 - 0.372974i) q^{63} -0.563705 q^{65} +0.714651i q^{67} -1.71217i q^{69} +7.90688i q^{71} +0.743450i q^{73} +1.77439 q^{75} +(15.9936 + 5.28346i) q^{77} +3.94493i q^{79} -9.42335 q^{81} -11.5667 q^{83} -6.12055 q^{85} -3.59465 q^{87} -16.6650i q^{89} +(-1.41615 - 0.467824i) q^{91} -14.3680 q^{93} -0.694886i q^{95} -9.54489i q^{97} +0.945168i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} + 16 q^{9} - 8 q^{19} - 4 q^{21} - 16 q^{25} + 48 q^{27} - 8 q^{29} - 16 q^{37} - 8 q^{47} - 4 q^{49} - 16 q^{53} + 8 q^{55} + 16 q^{57} - 8 q^{59} - 60 q^{63} + 8 q^{65} + 8 q^{77} + 40 q^{81} + 64 q^{83} - 16 q^{87} - 64 q^{91} + 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.77439 −1.02445 −0.512223 0.858853i \(-0.671178\pi\)
−0.512223 + 0.858853i \(0.671178\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 0.829909 2.51222i 0.313676 0.949530i
\(8\) 0 0
\(9\) 0.148464 0.0494880
\(10\) 0 0
\(11\) 6.36631i 1.91951i 0.280832 + 0.959757i \(0.409390\pi\)
−0.280832 + 0.959757i \(0.590610\pi\)
\(12\) 0 0
\(13\) 0.563705i 0.156344i −0.996940 0.0781718i \(-0.975092\pi\)
0.996940 0.0781718i \(-0.0249083\pi\)
\(14\) 0 0
\(15\) 1.77439i 0.458146i
\(16\) 0 0
\(17\) 6.12055i 1.48445i −0.670149 0.742226i \(-0.733770\pi\)
0.670149 0.742226i \(-0.266230\pi\)
\(18\) 0 0
\(19\) 0.694886 0.159418 0.0797088 0.996818i \(-0.474601\pi\)
0.0797088 + 0.996818i \(0.474601\pi\)
\(20\) 0 0
\(21\) −1.47258 + 4.45766i −0.321344 + 0.972741i
\(22\) 0 0
\(23\) 0.964933i 0.201202i 0.994927 + 0.100601i \(0.0320766\pi\)
−0.994927 + 0.100601i \(0.967923\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.05974 0.973747
\(28\) 0 0
\(29\) 2.02585 0.376191 0.188095 0.982151i \(-0.439769\pi\)
0.188095 + 0.982151i \(0.439769\pi\)
\(30\) 0 0
\(31\) 8.09745 1.45434 0.727172 0.686455i \(-0.240834\pi\)
0.727172 + 0.686455i \(0.240834\pi\)
\(32\) 0 0
\(33\) 11.2963i 1.96644i
\(34\) 0 0
\(35\) −2.51222 0.829909i −0.424643 0.140280i
\(36\) 0 0
\(37\) −1.89633 −0.311755 −0.155877 0.987776i \(-0.549820\pi\)
−0.155877 + 0.987776i \(0.549820\pi\)
\(38\) 0 0
\(39\) 1.00023i 0.160165i
\(40\) 0 0
\(41\) 1.97453i 0.308370i 0.988042 + 0.154185i \(0.0492752\pi\)
−0.988042 + 0.154185i \(0.950725\pi\)
\(42\) 0 0
\(43\) 5.26331i 0.802648i −0.915936 0.401324i \(-0.868550\pi\)
0.915936 0.401324i \(-0.131450\pi\)
\(44\) 0 0
\(45\) 0.148464i 0.0221317i
\(46\) 0 0
\(47\) 3.03286 0.442388 0.221194 0.975230i \(-0.429005\pi\)
0.221194 + 0.975230i \(0.429005\pi\)
\(48\) 0 0
\(49\) −5.62250 4.16983i −0.803214 0.595690i
\(50\) 0 0
\(51\) 10.8603i 1.52074i
\(52\) 0 0
\(53\) −1.73956 −0.238947 −0.119473 0.992837i \(-0.538121\pi\)
−0.119473 + 0.992837i \(0.538121\pi\)
\(54\) 0 0
\(55\) 6.36631 0.858433
\(56\) 0 0
\(57\) −1.23300 −0.163315
\(58\) 0 0
\(59\) −12.9168 −1.68162 −0.840810 0.541330i \(-0.817921\pi\)
−0.840810 + 0.541330i \(0.817921\pi\)
\(60\) 0 0
\(61\) 9.96224i 1.27553i −0.770229 0.637767i \(-0.779858\pi\)
0.770229 0.637767i \(-0.220142\pi\)
\(62\) 0 0
\(63\) 0.123212 0.372974i 0.0155232 0.0469904i
\(64\) 0 0
\(65\) −0.563705 −0.0699190
\(66\) 0 0
\(67\) 0.714651i 0.0873085i 0.999047 + 0.0436543i \(0.0139000\pi\)
−0.999047 + 0.0436543i \(0.986100\pi\)
\(68\) 0 0
\(69\) 1.71217i 0.206121i
\(70\) 0 0
\(71\) 7.90688i 0.938374i 0.883099 + 0.469187i \(0.155453\pi\)
−0.883099 + 0.469187i \(0.844547\pi\)
\(72\) 0 0
\(73\) 0.743450i 0.0870143i 0.999053 + 0.0435071i \(0.0138531\pi\)
−0.999053 + 0.0435071i \(0.986147\pi\)
\(74\) 0 0
\(75\) 1.77439 0.204889
\(76\) 0 0
\(77\) 15.9936 + 5.28346i 1.82264 + 0.602106i
\(78\) 0 0
\(79\) 3.94493i 0.443840i 0.975065 + 0.221920i \(0.0712324\pi\)
−0.975065 + 0.221920i \(0.928768\pi\)
\(80\) 0 0
\(81\) −9.42335 −1.04704
\(82\) 0 0
\(83\) −11.5667 −1.26962 −0.634808 0.772670i \(-0.718921\pi\)
−0.634808 + 0.772670i \(0.718921\pi\)
\(84\) 0 0
\(85\) −6.12055 −0.663867
\(86\) 0 0
\(87\) −3.59465 −0.385387
\(88\) 0 0
\(89\) 16.6650i 1.76648i −0.468919 0.883241i \(-0.655356\pi\)
0.468919 0.883241i \(-0.344644\pi\)
\(90\) 0 0
\(91\) −1.41615 0.467824i −0.148453 0.0490413i
\(92\) 0 0
\(93\) −14.3680 −1.48990
\(94\) 0 0
\(95\) 0.694886i 0.0712938i
\(96\) 0 0
\(97\) 9.54489i 0.969137i −0.874753 0.484568i \(-0.838977\pi\)
0.874753 0.484568i \(-0.161023\pi\)
\(98\) 0 0
\(99\) 0.945168i 0.0949929i
\(100\) 0 0
\(101\) 14.0637i 1.39939i −0.714441 0.699695i \(-0.753319\pi\)
0.714441 0.699695i \(-0.246681\pi\)
\(102\) 0 0
\(103\) −8.11405 −0.799501 −0.399751 0.916624i \(-0.630903\pi\)
−0.399751 + 0.916624i \(0.630903\pi\)
\(104\) 0 0
\(105\) 4.45766 + 1.47258i 0.435023 + 0.143709i
\(106\) 0 0
\(107\) 10.6050i 1.02522i −0.858621 0.512611i \(-0.828678\pi\)
0.858621 0.512611i \(-0.171322\pi\)
\(108\) 0 0
\(109\) 12.4913 1.19645 0.598225 0.801328i \(-0.295873\pi\)
0.598225 + 0.801328i \(0.295873\pi\)
\(110\) 0 0
\(111\) 3.36483 0.319376
\(112\) 0 0
\(113\) 14.9783 1.40904 0.704518 0.709686i \(-0.251163\pi\)
0.704518 + 0.709686i \(0.251163\pi\)
\(114\) 0 0
\(115\) 0.964933 0.0899805
\(116\) 0 0
\(117\) 0.0836899i 0.00773714i
\(118\) 0 0
\(119\) −15.3762 5.07951i −1.40953 0.465638i
\(120\) 0 0
\(121\) −29.5299 −2.68453
\(122\) 0 0
\(123\) 3.50359i 0.315908i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 7.51224i 0.666604i −0.942820 0.333302i \(-0.891837\pi\)
0.942820 0.333302i \(-0.108163\pi\)
\(128\) 0 0
\(129\) 9.33918i 0.822269i
\(130\) 0 0
\(131\) −9.12485 −0.797242 −0.398621 0.917116i \(-0.630511\pi\)
−0.398621 + 0.917116i \(0.630511\pi\)
\(132\) 0 0
\(133\) 0.576692 1.74571i 0.0500055 0.151372i
\(134\) 0 0
\(135\) 5.05974i 0.435473i
\(136\) 0 0
\(137\) −15.5030 −1.32451 −0.662255 0.749279i \(-0.730400\pi\)
−0.662255 + 0.749279i \(0.730400\pi\)
\(138\) 0 0
\(139\) 10.4290 0.884574 0.442287 0.896874i \(-0.354167\pi\)
0.442287 + 0.896874i \(0.354167\pi\)
\(140\) 0 0
\(141\) −5.38148 −0.453202
\(142\) 0 0
\(143\) 3.58872 0.300104
\(144\) 0 0
\(145\) 2.02585i 0.168238i
\(146\) 0 0
\(147\) 9.97652 + 7.39891i 0.822849 + 0.610252i
\(148\) 0 0
\(149\) −18.0129 −1.47567 −0.737837 0.674979i \(-0.764153\pi\)
−0.737837 + 0.674979i \(0.764153\pi\)
\(150\) 0 0
\(151\) 15.7987i 1.28568i −0.765999 0.642842i \(-0.777755\pi\)
0.765999 0.642842i \(-0.222245\pi\)
\(152\) 0 0
\(153\) 0.908682i 0.0734626i
\(154\) 0 0
\(155\) 8.09745i 0.650402i
\(156\) 0 0
\(157\) 22.6822i 1.81024i −0.425157 0.905119i \(-0.639781\pi\)
0.425157 0.905119i \(-0.360219\pi\)
\(158\) 0 0
\(159\) 3.08666 0.244788
\(160\) 0 0
\(161\) 2.42412 + 0.800807i 0.191048 + 0.0631124i
\(162\) 0 0
\(163\) 17.7258i 1.38839i −0.719788 0.694194i \(-0.755761\pi\)
0.719788 0.694194i \(-0.244239\pi\)
\(164\) 0 0
\(165\) −11.2963 −0.879417
\(166\) 0 0
\(167\) 16.6783 1.29061 0.645303 0.763927i \(-0.276731\pi\)
0.645303 + 0.763927i \(0.276731\pi\)
\(168\) 0 0
\(169\) 12.6822 0.975557
\(170\) 0 0
\(171\) 0.103166 0.00788927
\(172\) 0 0
\(173\) 16.8854i 1.28377i 0.766800 + 0.641886i \(0.221848\pi\)
−0.766800 + 0.641886i \(0.778152\pi\)
\(174\) 0 0
\(175\) −0.829909 + 2.51222i −0.0627353 + 0.189906i
\(176\) 0 0
\(177\) 22.9194 1.72273
\(178\) 0 0
\(179\) 10.2643i 0.767193i 0.923501 + 0.383596i \(0.125315\pi\)
−0.923501 + 0.383596i \(0.874685\pi\)
\(180\) 0 0
\(181\) 0.656666i 0.0488096i −0.999702 0.0244048i \(-0.992231\pi\)
0.999702 0.0244048i \(-0.00776906\pi\)
\(182\) 0 0
\(183\) 17.6769i 1.30672i
\(184\) 0 0
\(185\) 1.89633i 0.139421i
\(186\) 0 0
\(187\) 38.9653 2.84943
\(188\) 0 0
\(189\) 4.19913 12.7112i 0.305441 0.924602i
\(190\) 0 0
\(191\) 25.4758i 1.84337i −0.387943 0.921683i \(-0.626814\pi\)
0.387943 0.921683i \(-0.373186\pi\)
\(192\) 0 0
\(193\) 10.8050 0.777760 0.388880 0.921288i \(-0.372862\pi\)
0.388880 + 0.921288i \(0.372862\pi\)
\(194\) 0 0
\(195\) 1.00023 0.0716282
\(196\) 0 0
\(197\) 12.2699 0.874196 0.437098 0.899414i \(-0.356006\pi\)
0.437098 + 0.899414i \(0.356006\pi\)
\(198\) 0 0
\(199\) −17.0170 −1.20631 −0.603153 0.797626i \(-0.706089\pi\)
−0.603153 + 0.797626i \(0.706089\pi\)
\(200\) 0 0
\(201\) 1.26807i 0.0894428i
\(202\) 0 0
\(203\) 1.68127 5.08938i 0.118002 0.357204i
\(204\) 0 0
\(205\) 1.97453 0.137907
\(206\) 0 0
\(207\) 0.143258i 0.00995711i
\(208\) 0 0
\(209\) 4.42386i 0.306004i
\(210\) 0 0
\(211\) 16.4726i 1.13402i −0.823712 0.567009i \(-0.808101\pi\)
0.823712 0.567009i \(-0.191899\pi\)
\(212\) 0 0
\(213\) 14.0299i 0.961313i
\(214\) 0 0
\(215\) −5.26331 −0.358955
\(216\) 0 0
\(217\) 6.72015 20.3426i 0.456193 1.38094i
\(218\) 0 0
\(219\) 1.31917i 0.0891414i
\(220\) 0 0
\(221\) −3.45019 −0.232085
\(222\) 0 0
\(223\) 4.48939 0.300632 0.150316 0.988638i \(-0.451971\pi\)
0.150316 + 0.988638i \(0.451971\pi\)
\(224\) 0 0
\(225\) −0.148464 −0.00989760
\(226\) 0 0
\(227\) −14.6264 −0.970791 −0.485395 0.874295i \(-0.661324\pi\)
−0.485395 + 0.874295i \(0.661324\pi\)
\(228\) 0 0
\(229\) 1.05959i 0.0700195i −0.999387 0.0350098i \(-0.988854\pi\)
0.999387 0.0350098i \(-0.0111462\pi\)
\(230\) 0 0
\(231\) −28.3788 9.37492i −1.86719 0.616825i
\(232\) 0 0
\(233\) −0.856456 −0.0561083 −0.0280542 0.999606i \(-0.508931\pi\)
−0.0280542 + 0.999606i \(0.508931\pi\)
\(234\) 0 0
\(235\) 3.03286i 0.197842i
\(236\) 0 0
\(237\) 6.99986i 0.454690i
\(238\) 0 0
\(239\) 4.54240i 0.293824i −0.989150 0.146912i \(-0.953067\pi\)
0.989150 0.146912i \(-0.0469333\pi\)
\(240\) 0 0
\(241\) 18.1050i 1.16625i 0.812384 + 0.583123i \(0.198169\pi\)
−0.812384 + 0.583123i \(0.801831\pi\)
\(242\) 0 0
\(243\) 1.54149 0.0988866
\(244\) 0 0
\(245\) −4.16983 + 5.62250i −0.266401 + 0.359208i
\(246\) 0 0
\(247\) 0.391710i 0.0249239i
\(248\) 0 0
\(249\) 20.5239 1.30065
\(250\) 0 0
\(251\) 21.2709 1.34261 0.671303 0.741183i \(-0.265735\pi\)
0.671303 + 0.741183i \(0.265735\pi\)
\(252\) 0 0
\(253\) −6.14306 −0.386211
\(254\) 0 0
\(255\) 10.8603 0.680096
\(256\) 0 0
\(257\) 19.1582i 1.19506i 0.801848 + 0.597528i \(0.203850\pi\)
−0.801848 + 0.597528i \(0.796150\pi\)
\(258\) 0 0
\(259\) −1.57378 + 4.76400i −0.0977900 + 0.296020i
\(260\) 0 0
\(261\) 0.300766 0.0186169
\(262\) 0 0
\(263\) 20.9902i 1.29431i 0.762358 + 0.647155i \(0.224041\pi\)
−0.762358 + 0.647155i \(0.775959\pi\)
\(264\) 0 0
\(265\) 1.73956i 0.106860i
\(266\) 0 0
\(267\) 29.5702i 1.80966i
\(268\) 0 0
\(269\) 7.17835i 0.437672i 0.975762 + 0.218836i \(0.0702259\pi\)
−0.975762 + 0.218836i \(0.929774\pi\)
\(270\) 0 0
\(271\) −21.3174 −1.29494 −0.647470 0.762091i \(-0.724173\pi\)
−0.647470 + 0.762091i \(0.724173\pi\)
\(272\) 0 0
\(273\) 2.51281 + 0.830103i 0.152082 + 0.0502401i
\(274\) 0 0
\(275\) 6.36631i 0.383903i
\(276\) 0 0
\(277\) −11.2723 −0.677284 −0.338642 0.940915i \(-0.609968\pi\)
−0.338642 + 0.940915i \(0.609968\pi\)
\(278\) 0 0
\(279\) 1.20218 0.0719726
\(280\) 0 0
\(281\) −18.0380 −1.07606 −0.538029 0.842926i \(-0.680831\pi\)
−0.538029 + 0.842926i \(0.680831\pi\)
\(282\) 0 0
\(283\) −28.3421 −1.68476 −0.842380 0.538883i \(-0.818846\pi\)
−0.842380 + 0.538883i \(0.818846\pi\)
\(284\) 0 0
\(285\) 1.23300i 0.0730366i
\(286\) 0 0
\(287\) 4.96046 + 1.63868i 0.292806 + 0.0967283i
\(288\) 0 0
\(289\) −20.4612 −1.20360
\(290\) 0 0
\(291\) 16.9364i 0.992828i
\(292\) 0 0
\(293\) 23.6862i 1.38376i −0.722011 0.691881i \(-0.756782\pi\)
0.722011 0.691881i \(-0.243218\pi\)
\(294\) 0 0
\(295\) 12.9168i 0.752044i
\(296\) 0 0
\(297\) 32.2119i 1.86912i
\(298\) 0 0
\(299\) 0.543938 0.0314567
\(300\) 0 0
\(301\) −13.2226 4.36807i −0.762138 0.251772i
\(302\) 0 0
\(303\) 24.9545i 1.43360i
\(304\) 0 0
\(305\) −9.96224 −0.570436
\(306\) 0 0
\(307\) −8.87125 −0.506309 −0.253155 0.967426i \(-0.581468\pi\)
−0.253155 + 0.967426i \(0.581468\pi\)
\(308\) 0 0
\(309\) 14.3975 0.819045
\(310\) 0 0
\(311\) 0.465767 0.0264112 0.0132056 0.999913i \(-0.495796\pi\)
0.0132056 + 0.999913i \(0.495796\pi\)
\(312\) 0 0
\(313\) 15.7351i 0.889398i 0.895680 + 0.444699i \(0.146689\pi\)
−0.895680 + 0.444699i \(0.853311\pi\)
\(314\) 0 0
\(315\) −0.372974 0.123212i −0.0210147 0.00694219i
\(316\) 0 0
\(317\) 23.9122 1.34304 0.671521 0.740986i \(-0.265641\pi\)
0.671521 + 0.740986i \(0.265641\pi\)
\(318\) 0 0
\(319\) 12.8972i 0.722103i
\(320\) 0 0
\(321\) 18.8174i 1.05028i
\(322\) 0 0
\(323\) 4.25309i 0.236648i
\(324\) 0 0
\(325\) 0.563705i 0.0312687i
\(326\) 0 0
\(327\) −22.1645 −1.22570
\(328\) 0 0
\(329\) 2.51700 7.61921i 0.138767 0.420061i
\(330\) 0 0
\(331\) 1.69904i 0.0933874i −0.998909 0.0466937i \(-0.985132\pi\)
0.998909 0.0466937i \(-0.0148685\pi\)
\(332\) 0 0
\(333\) −0.281537 −0.0154281
\(334\) 0 0
\(335\) 0.714651 0.0390456
\(336\) 0 0
\(337\) 26.7339 1.45629 0.728145 0.685423i \(-0.240383\pi\)
0.728145 + 0.685423i \(0.240383\pi\)
\(338\) 0 0
\(339\) −26.5773 −1.44348
\(340\) 0 0
\(341\) 51.5508i 2.79163i
\(342\) 0 0
\(343\) −15.1417 + 10.6644i −0.817575 + 0.575822i
\(344\) 0 0
\(345\) −1.71217 −0.0921801
\(346\) 0 0
\(347\) 13.2342i 0.710448i −0.934781 0.355224i \(-0.884405\pi\)
0.934781 0.355224i \(-0.115595\pi\)
\(348\) 0 0
\(349\) 18.8288i 1.00788i −0.863738 0.503941i \(-0.831883\pi\)
0.863738 0.503941i \(-0.168117\pi\)
\(350\) 0 0
\(351\) 2.85220i 0.152239i
\(352\) 0 0
\(353\) 19.4969i 1.03771i −0.854861 0.518856i \(-0.826358\pi\)
0.854861 0.518856i \(-0.173642\pi\)
\(354\) 0 0
\(355\) 7.90688 0.419654
\(356\) 0 0
\(357\) 27.2834 + 9.01303i 1.44399 + 0.477020i
\(358\) 0 0
\(359\) 9.76471i 0.515362i 0.966230 + 0.257681i \(0.0829583\pi\)
−0.966230 + 0.257681i \(0.917042\pi\)
\(360\) 0 0
\(361\) −18.5171 −0.974586
\(362\) 0 0
\(363\) 52.3975 2.75016
\(364\) 0 0
\(365\) 0.743450 0.0389140
\(366\) 0 0
\(367\) 5.80647 0.303095 0.151548 0.988450i \(-0.451574\pi\)
0.151548 + 0.988450i \(0.451574\pi\)
\(368\) 0 0
\(369\) 0.293147i 0.0152606i
\(370\) 0 0
\(371\) −1.44368 + 4.37015i −0.0749519 + 0.226887i
\(372\) 0 0
\(373\) −18.7648 −0.971604 −0.485802 0.874069i \(-0.661472\pi\)
−0.485802 + 0.874069i \(0.661472\pi\)
\(374\) 0 0
\(375\) 1.77439i 0.0916292i
\(376\) 0 0
\(377\) 1.14198i 0.0588150i
\(378\) 0 0
\(379\) 5.91732i 0.303952i 0.988384 + 0.151976i \(0.0485637\pi\)
−0.988384 + 0.151976i \(0.951436\pi\)
\(380\) 0 0
\(381\) 13.3297i 0.682899i
\(382\) 0 0
\(383\) 26.2320 1.34039 0.670196 0.742184i \(-0.266210\pi\)
0.670196 + 0.742184i \(0.266210\pi\)
\(384\) 0 0
\(385\) 5.28346 15.9936i 0.269270 0.815108i
\(386\) 0 0
\(387\) 0.781413i 0.0397215i
\(388\) 0 0
\(389\) 31.9320 1.61902 0.809508 0.587109i \(-0.199734\pi\)
0.809508 + 0.587109i \(0.199734\pi\)
\(390\) 0 0
\(391\) 5.90593 0.298676
\(392\) 0 0
\(393\) 16.1911 0.816731
\(394\) 0 0
\(395\) 3.94493 0.198491
\(396\) 0 0
\(397\) 19.0500i 0.956094i −0.878334 0.478047i \(-0.841345\pi\)
0.878334 0.478047i \(-0.158655\pi\)
\(398\) 0 0
\(399\) −1.02328 + 3.09756i −0.0512279 + 0.155072i
\(400\) 0 0
\(401\) −30.6227 −1.52923 −0.764613 0.644489i \(-0.777070\pi\)
−0.764613 + 0.644489i \(0.777070\pi\)
\(402\) 0 0
\(403\) 4.56457i 0.227377i
\(404\) 0 0
\(405\) 9.42335i 0.468250i
\(406\) 0 0
\(407\) 12.0726i 0.598417i
\(408\) 0 0
\(409\) 29.2209i 1.44488i 0.691434 + 0.722440i \(0.256979\pi\)
−0.691434 + 0.722440i \(0.743021\pi\)
\(410\) 0 0
\(411\) 27.5084 1.35689
\(412\) 0 0
\(413\) −10.7198 + 32.4498i −0.527485 + 1.59675i
\(414\) 0 0
\(415\) 11.5667i 0.567789i
\(416\) 0 0
\(417\) −18.5051 −0.906198
\(418\) 0 0
\(419\) 4.35398 0.212706 0.106353 0.994328i \(-0.466083\pi\)
0.106353 + 0.994328i \(0.466083\pi\)
\(420\) 0 0
\(421\) 25.8255 1.25866 0.629328 0.777140i \(-0.283330\pi\)
0.629328 + 0.777140i \(0.283330\pi\)
\(422\) 0 0
\(423\) 0.450271 0.0218929
\(424\) 0 0
\(425\) 6.12055i 0.296890i
\(426\) 0 0
\(427\) −25.0273 8.26776i −1.21116 0.400105i
\(428\) 0 0
\(429\) −6.36779 −0.307440
\(430\) 0 0
\(431\) 4.84144i 0.233204i −0.993179 0.116602i \(-0.962800\pi\)
0.993179 0.116602i \(-0.0372002\pi\)
\(432\) 0 0
\(433\) 2.54916i 0.122505i −0.998122 0.0612524i \(-0.980491\pi\)
0.998122 0.0612524i \(-0.0195094\pi\)
\(434\) 0 0
\(435\) 3.59465i 0.172350i
\(436\) 0 0
\(437\) 0.670518i 0.0320752i
\(438\) 0 0
\(439\) 10.9908 0.524564 0.262282 0.964991i \(-0.415525\pi\)
0.262282 + 0.964991i \(0.415525\pi\)
\(440\) 0 0
\(441\) −0.834739 0.619070i −0.0397495 0.0294795i
\(442\) 0 0
\(443\) 36.3209i 1.72566i 0.505495 + 0.862830i \(0.331310\pi\)
−0.505495 + 0.862830i \(0.668690\pi\)
\(444\) 0 0
\(445\) −16.6650 −0.789995
\(446\) 0 0
\(447\) 31.9619 1.51175
\(448\) 0 0
\(449\) 8.05482 0.380131 0.190065 0.981771i \(-0.439130\pi\)
0.190065 + 0.981771i \(0.439130\pi\)
\(450\) 0 0
\(451\) −12.5705 −0.591920
\(452\) 0 0
\(453\) 28.0331i 1.31711i
\(454\) 0 0
\(455\) −0.467824 + 1.41615i −0.0219319 + 0.0663902i
\(456\) 0 0
\(457\) 14.7790 0.691333 0.345667 0.938357i \(-0.387653\pi\)
0.345667 + 0.938357i \(0.387653\pi\)
\(458\) 0 0
\(459\) 30.9684i 1.44548i
\(460\) 0 0
\(461\) 2.08795i 0.0972457i 0.998817 + 0.0486229i \(0.0154832\pi\)
−0.998817 + 0.0486229i \(0.984517\pi\)
\(462\) 0 0
\(463\) 37.0568i 1.72217i −0.508458 0.861087i \(-0.669784\pi\)
0.508458 0.861087i \(-0.330216\pi\)
\(464\) 0 0
\(465\) 14.3680i 0.666302i
\(466\) 0 0
\(467\) 4.16280 0.192631 0.0963156 0.995351i \(-0.469294\pi\)
0.0963156 + 0.995351i \(0.469294\pi\)
\(468\) 0 0
\(469\) 1.79536 + 0.593096i 0.0829020 + 0.0273866i
\(470\) 0 0
\(471\) 40.2471i 1.85449i
\(472\) 0 0
\(473\) 33.5079 1.54069
\(474\) 0 0
\(475\) −0.694886 −0.0318835
\(476\) 0 0
\(477\) −0.258262 −0.0118250
\(478\) 0 0
\(479\) 37.5746 1.71683 0.858414 0.512957i \(-0.171450\pi\)
0.858414 + 0.512957i \(0.171450\pi\)
\(480\) 0 0
\(481\) 1.06897i 0.0487409i
\(482\) 0 0
\(483\) −4.30135 1.42095i −0.195718 0.0646552i
\(484\) 0 0
\(485\) −9.54489 −0.433411
\(486\) 0 0
\(487\) 0.821825i 0.0372405i 0.999827 + 0.0186202i \(0.00592735\pi\)
−0.999827 + 0.0186202i \(0.994073\pi\)
\(488\) 0 0
\(489\) 31.4524i 1.42233i
\(490\) 0 0
\(491\) 15.6087i 0.704409i 0.935923 + 0.352204i \(0.114568\pi\)
−0.935923 + 0.352204i \(0.885432\pi\)
\(492\) 0 0
\(493\) 12.3993i 0.558437i
\(494\) 0 0
\(495\) 0.945168 0.0424821
\(496\) 0 0
\(497\) 19.8638 + 6.56199i 0.891014 + 0.294346i
\(498\) 0 0
\(499\) 35.6642i 1.59655i 0.602293 + 0.798275i \(0.294254\pi\)
−0.602293 + 0.798275i \(0.705746\pi\)
\(500\) 0 0
\(501\) −29.5938 −1.32216
\(502\) 0 0
\(503\) 34.1203 1.52135 0.760675 0.649133i \(-0.224868\pi\)
0.760675 + 0.649133i \(0.224868\pi\)
\(504\) 0 0
\(505\) −14.0637 −0.625826
\(506\) 0 0
\(507\) −22.5032 −0.999404
\(508\) 0 0
\(509\) 40.7563i 1.80649i 0.429124 + 0.903246i \(0.358822\pi\)
−0.429124 + 0.903246i \(0.641178\pi\)
\(510\) 0 0
\(511\) 1.86771 + 0.616996i 0.0826227 + 0.0272943i
\(512\) 0 0
\(513\) 3.51594 0.155233
\(514\) 0 0
\(515\) 8.11405i 0.357548i
\(516\) 0 0
\(517\) 19.3081i 0.849170i
\(518\) 0 0
\(519\) 29.9613i 1.31515i
\(520\) 0 0
\(521\) 11.0375i 0.483562i −0.970331 0.241781i \(-0.922268\pi\)
0.970331 0.241781i \(-0.0777315\pi\)
\(522\) 0 0
\(523\) −6.55759 −0.286744 −0.143372 0.989669i \(-0.545794\pi\)
−0.143372 + 0.989669i \(0.545794\pi\)
\(524\) 0 0
\(525\) 1.47258 4.45766i 0.0642688 0.194548i
\(526\) 0 0
\(527\) 49.5609i 2.15890i
\(528\) 0 0
\(529\) 22.0689 0.959518
\(530\) 0 0
\(531\) −1.91768 −0.0832201
\(532\) 0 0
\(533\) 1.11305 0.0482117
\(534\) 0 0
\(535\) −10.6050 −0.458494
\(536\) 0 0
\(537\) 18.2130i 0.785947i
\(538\) 0 0
\(539\) 26.5464 35.7946i 1.14344 1.54178i
\(540\) 0 0
\(541\) −20.7600 −0.892541 −0.446271 0.894898i \(-0.647248\pi\)
−0.446271 + 0.894898i \(0.647248\pi\)
\(542\) 0 0
\(543\) 1.16518i 0.0500027i
\(544\) 0 0
\(545\) 12.4913i 0.535069i
\(546\) 0 0
\(547\) 4.20675i 0.179868i 0.995948 + 0.0899338i \(0.0286655\pi\)
−0.995948 + 0.0899338i \(0.971334\pi\)
\(548\) 0 0
\(549\) 1.47903i 0.0631237i
\(550\) 0 0
\(551\) 1.40773 0.0599714
\(552\) 0 0
\(553\) 9.91055 + 3.27394i 0.421439 + 0.139222i
\(554\) 0 0
\(555\) 3.36483i 0.142829i
\(556\) 0 0
\(557\) 19.1465 0.811265 0.405633 0.914036i \(-0.367051\pi\)
0.405633 + 0.914036i \(0.367051\pi\)
\(558\) 0 0
\(559\) −2.96696 −0.125489
\(560\) 0 0
\(561\) −69.1397 −2.91908
\(562\) 0 0
\(563\) −34.3646 −1.44829 −0.724147 0.689646i \(-0.757766\pi\)
−0.724147 + 0.689646i \(0.757766\pi\)
\(564\) 0 0
\(565\) 14.9783i 0.630140i
\(566\) 0 0
\(567\) −7.82053 + 23.6735i −0.328431 + 0.994195i
\(568\) 0 0
\(569\) −0.965245 −0.0404652 −0.0202326 0.999795i \(-0.506441\pi\)
−0.0202326 + 0.999795i \(0.506441\pi\)
\(570\) 0 0
\(571\) 1.72034i 0.0719939i −0.999352 0.0359969i \(-0.988539\pi\)
0.999352 0.0359969i \(-0.0114607\pi\)
\(572\) 0 0
\(573\) 45.2041i 1.88843i
\(574\) 0 0
\(575\) 0.964933i 0.0402405i
\(576\) 0 0
\(577\) 1.83237i 0.0762826i −0.999272 0.0381413i \(-0.987856\pi\)
0.999272 0.0381413i \(-0.0121437\pi\)
\(578\) 0 0
\(579\) −19.1723 −0.796773
\(580\) 0 0
\(581\) −9.59935 + 29.0582i −0.398248 + 1.20554i
\(582\) 0 0
\(583\) 11.0746i 0.458661i
\(584\) 0 0
\(585\) −0.0836899 −0.00346015
\(586\) 0 0
\(587\) −7.03942 −0.290548 −0.145274 0.989391i \(-0.546406\pi\)
−0.145274 + 0.989391i \(0.546406\pi\)
\(588\) 0 0
\(589\) 5.62680 0.231848
\(590\) 0 0
\(591\) −21.7716 −0.895565
\(592\) 0 0
\(593\) 5.37760i 0.220832i −0.993885 0.110416i \(-0.964782\pi\)
0.993885 0.110416i \(-0.0352182\pi\)
\(594\) 0 0
\(595\) −5.07951 + 15.3762i −0.208239 + 0.630362i
\(596\) 0 0
\(597\) 30.1949 1.23579
\(598\) 0 0
\(599\) 19.2945i 0.788350i −0.919035 0.394175i \(-0.871030\pi\)
0.919035 0.394175i \(-0.128970\pi\)
\(600\) 0 0
\(601\) 19.2218i 0.784075i −0.919949 0.392038i \(-0.871770\pi\)
0.919949 0.392038i \(-0.128230\pi\)
\(602\) 0 0
\(603\) 0.106100i 0.00432073i
\(604\) 0 0
\(605\) 29.5299i 1.20056i
\(606\) 0 0
\(607\) −9.04592 −0.367163 −0.183581 0.983005i \(-0.558769\pi\)
−0.183581 + 0.983005i \(0.558769\pi\)
\(608\) 0 0
\(609\) −2.98323 + 9.03055i −0.120887 + 0.365936i
\(610\) 0 0
\(611\) 1.70964i 0.0691645i
\(612\) 0 0
\(613\) 16.7177 0.675221 0.337611 0.941286i \(-0.390381\pi\)
0.337611 + 0.941286i \(0.390381\pi\)
\(614\) 0 0
\(615\) −3.50359 −0.141278
\(616\) 0 0
\(617\) 25.6742 1.03361 0.516803 0.856104i \(-0.327122\pi\)
0.516803 + 0.856104i \(0.327122\pi\)
\(618\) 0 0
\(619\) 8.03441 0.322930 0.161465 0.986878i \(-0.448378\pi\)
0.161465 + 0.986878i \(0.448378\pi\)
\(620\) 0 0
\(621\) 4.88231i 0.195920i
\(622\) 0 0
\(623\) −41.8661 13.8304i −1.67733 0.554104i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.84965i 0.313485i
\(628\) 0 0
\(629\) 11.6066i 0.462785i
\(630\) 0 0
\(631\) 8.22789i 0.327547i 0.986498 + 0.163774i \(0.0523667\pi\)
−0.986498 + 0.163774i \(0.947633\pi\)
\(632\) 0 0
\(633\) 29.2288i 1.16174i
\(634\) 0 0
\(635\) −7.51224 −0.298114
\(636\) 0 0
\(637\) −2.35055 + 3.16943i −0.0931323 + 0.125577i
\(638\) 0 0
\(639\) 1.17389i 0.0464383i
\(640\) 0 0
\(641\) 37.4651 1.47978 0.739891 0.672727i \(-0.234877\pi\)
0.739891 + 0.672727i \(0.234877\pi\)
\(642\) 0 0
\(643\) −5.50809 −0.217218 −0.108609 0.994085i \(-0.534640\pi\)
−0.108609 + 0.994085i \(0.534640\pi\)
\(644\) 0 0
\(645\) 9.33918 0.367730
\(646\) 0 0
\(647\) −39.9344 −1.56998 −0.784992 0.619506i \(-0.787333\pi\)
−0.784992 + 0.619506i \(0.787333\pi\)
\(648\) 0 0
\(649\) 82.2322i 3.22789i
\(650\) 0 0
\(651\) −11.9242 + 36.0957i −0.467345 + 1.41470i
\(652\) 0 0
\(653\) 37.5904 1.47103 0.735513 0.677510i \(-0.236941\pi\)
0.735513 + 0.677510i \(0.236941\pi\)
\(654\) 0 0
\(655\) 9.12485i 0.356537i
\(656\) 0 0
\(657\) 0.110376i 0.00430616i
\(658\) 0 0
\(659\) 2.81823i 0.109783i −0.998492 0.0548913i \(-0.982519\pi\)
0.998492 0.0548913i \(-0.0174812\pi\)
\(660\) 0 0
\(661\) 19.0129i 0.739518i −0.929128 0.369759i \(-0.879440\pi\)
0.929128 0.369759i \(-0.120560\pi\)
\(662\) 0 0
\(663\) 6.12198 0.237758
\(664\) 0 0
\(665\) −1.74571 0.576692i −0.0676956 0.0223632i
\(666\) 0 0
\(667\) 1.95481i 0.0756905i
\(668\) 0 0
\(669\) −7.96594 −0.307981
\(670\) 0 0
\(671\) 63.4227 2.44841
\(672\) 0 0
\(673\) −35.0799 −1.35223 −0.676115 0.736796i \(-0.736338\pi\)
−0.676115 + 0.736796i \(0.736338\pi\)
\(674\) 0 0
\(675\) −5.05974 −0.194749
\(676\) 0 0
\(677\) 20.1811i 0.775623i −0.921739 0.387812i \(-0.873231\pi\)
0.921739 0.387812i \(-0.126769\pi\)
\(678\) 0 0
\(679\) −23.9789 7.92140i −0.920225 0.303995i
\(680\) 0 0
\(681\) 25.9530 0.994522
\(682\) 0 0
\(683\) 11.6160i 0.444473i −0.974993 0.222236i \(-0.928664\pi\)
0.974993 0.222236i \(-0.0713356\pi\)
\(684\) 0 0
\(685\) 15.5030i 0.592339i
\(686\) 0 0
\(687\) 1.88012i 0.0717312i
\(688\) 0 0
\(689\) 0.980597i 0.0373578i
\(690\) 0 0
\(691\) 21.5766 0.820812 0.410406 0.911903i \(-0.365387\pi\)
0.410406 + 0.911903i \(0.365387\pi\)
\(692\) 0 0
\(693\) 2.37447 + 0.784404i 0.0901986 + 0.0297970i
\(694\) 0 0
\(695\) 10.4290i 0.395594i
\(696\) 0 0
\(697\) 12.0852 0.457760
\(698\) 0 0
\(699\) 1.51969 0.0574799
\(700\) 0 0
\(701\) 14.7329 0.556453 0.278226 0.960516i \(-0.410253\pi\)
0.278226 + 0.960516i \(0.410253\pi\)
\(702\) 0 0
\(703\) −1.31773 −0.0496992
\(704\) 0 0
\(705\) 5.38148i 0.202678i
\(706\) 0 0
\(707\) −35.3311 11.6716i −1.32876 0.438956i
\(708\) 0 0
\(709\) 38.4666 1.44464 0.722321 0.691558i \(-0.243075\pi\)
0.722321 + 0.691558i \(0.243075\pi\)
\(710\) 0 0
\(711\) 0.585681i 0.0219648i
\(712\) 0 0
\(713\) 7.81349i 0.292618i
\(714\) 0 0
\(715\) 3.58872i 0.134210i
\(716\) 0 0
\(717\) 8.06000i 0.301006i
\(718\) 0 0
\(719\) 24.0175 0.895700 0.447850 0.894109i \(-0.352190\pi\)
0.447850 + 0.894109i \(0.352190\pi\)
\(720\) 0 0
\(721\) −6.73393 + 20.3843i −0.250785 + 0.759150i
\(722\) 0 0
\(723\) 32.1254i 1.19476i
\(724\) 0 0
\(725\) −2.02585 −0.0752381
\(726\) 0 0
\(727\) 9.65598 0.358121 0.179060 0.983838i \(-0.442694\pi\)
0.179060 + 0.983838i \(0.442694\pi\)
\(728\) 0 0
\(729\) 25.5348 0.945735
\(730\) 0 0
\(731\) −32.2144 −1.19149
\(732\) 0 0
\(733\) 15.2522i 0.563355i −0.959509 0.281677i \(-0.909109\pi\)
0.959509 0.281677i \(-0.0908907\pi\)
\(734\) 0 0
\(735\) 7.39891 9.97652i 0.272913 0.367989i
\(736\) 0 0
\(737\) −4.54969 −0.167590
\(738\) 0 0
\(739\) 19.4632i 0.715964i 0.933729 + 0.357982i \(0.116535\pi\)
−0.933729 + 0.357982i \(0.883465\pi\)
\(740\) 0 0
\(741\) 0.695048i 0.0255332i
\(742\) 0 0
\(743\) 29.0393i 1.06535i 0.846320 + 0.532674i \(0.178813\pi\)
−0.846320 + 0.532674i \(0.821187\pi\)
\(744\) 0 0
\(745\) 18.0129i 0.659941i
\(746\) 0 0
\(747\) −1.71725 −0.0628307
\(748\) 0 0
\(749\) −26.6421 8.80118i −0.973480 0.321588i
\(750\) 0 0
\(751\) 0.883907i 0.0322542i 0.999870 + 0.0161271i \(0.00513364\pi\)
−0.999870 + 0.0161271i \(0.994866\pi\)
\(752\) 0 0
\(753\) −37.7428 −1.37543
\(754\) 0 0
\(755\) −15.7987 −0.574975
\(756\) 0 0
\(757\) −28.1875 −1.02449 −0.512246 0.858839i \(-0.671186\pi\)
−0.512246 + 0.858839i \(0.671186\pi\)
\(758\) 0 0
\(759\) 10.9002 0.395652
\(760\) 0 0
\(761\) 22.5553i 0.817629i 0.912618 + 0.408814i \(0.134058\pi\)
−0.912618 + 0.408814i \(0.865942\pi\)
\(762\) 0 0
\(763\) 10.3667 31.3809i 0.375298 1.13607i
\(764\) 0 0
\(765\) −0.908682 −0.0328535
\(766\) 0 0
\(767\) 7.28125i 0.262911i
\(768\) 0 0
\(769\) 4.68731i 0.169029i −0.996422 0.0845144i \(-0.973066\pi\)
0.996422 0.0845144i \(-0.0269339\pi\)
\(770\) 0 0
\(771\) 33.9942i 1.22427i
\(772\) 0 0
\(773\) 38.9796i 1.40200i −0.713162 0.700999i \(-0.752738\pi\)
0.713162 0.700999i \(-0.247262\pi\)
\(774\) 0 0
\(775\) −8.09745 −0.290869
\(776\) 0 0
\(777\) 2.79250 8.45320i 0.100181 0.303257i
\(778\) 0 0
\(779\) 1.37207i 0.0491596i
\(780\) 0 0
\(781\) −50.3376 −1.80122
\(782\) 0 0
\(783\) 10.2503 0.366315
\(784\) 0 0
\(785\) −22.6822 −0.809563
\(786\) 0 0
\(787\) −11.6892 −0.416674 −0.208337 0.978057i \(-0.566805\pi\)
−0.208337 + 0.978057i \(0.566805\pi\)
\(788\) 0 0
\(789\) 37.2448i 1.32595i
\(790\) 0 0
\(791\) 12.4306 37.6287i 0.441981 1.33792i
\(792\) 0 0
\(793\) −5.61576 −0.199422
\(794\) 0 0
\(795\) 3.08666i 0.109472i
\(796\) 0 0
\(797\) 25.6177i 0.907427i 0.891148 + 0.453714i \(0.149901\pi\)
−0.891148 + 0.453714i \(0.850099\pi\)
\(798\) 0 0
\(799\) 18.5628i 0.656704i
\(800\) 0 0
\(801\) 2.47415i 0.0874197i
\(802\) 0 0
\(803\) −4.73303 −0.167025
\(804\) 0 0
\(805\) 0.800807 2.42412i 0.0282247 0.0854392i
\(806\) 0 0
\(807\) 12.7372i 0.448371i
\(808\) 0 0
\(809\) −41.7251 −1.46698 −0.733488 0.679702i \(-0.762109\pi\)
−0.733488 + 0.679702i \(0.762109\pi\)
\(810\) 0 0
\(811\) 25.9122 0.909899 0.454950 0.890517i \(-0.349657\pi\)
0.454950 + 0.890517i \(0.349657\pi\)
\(812\) 0 0
\(813\) 37.8254 1.32659
\(814\) 0 0
\(815\) −17.7258 −0.620906
\(816\) 0 0
\(817\) 3.65740i 0.127956i
\(818\) 0 0
\(819\) −0.210248 0.0694550i −0.00734664 0.00242696i
\(820\) 0 0
\(821\) −29.4953 −1.02939 −0.514697 0.857372i \(-0.672096\pi\)
−0.514697 + 0.857372i \(0.672096\pi\)
\(822\) 0 0
\(823\) 34.2637i 1.19436i −0.802109 0.597178i \(-0.796289\pi\)
0.802109 0.597178i \(-0.203711\pi\)
\(824\) 0 0
\(825\) 11.2963i 0.393287i
\(826\) 0 0
\(827\) 53.5268i 1.86131i 0.365899 + 0.930655i \(0.380762\pi\)
−0.365899 + 0.930655i \(0.619238\pi\)
\(828\) 0 0
\(829\) 33.4570i 1.16201i 0.813900 + 0.581005i \(0.197340\pi\)
−0.813900 + 0.581005i \(0.802660\pi\)
\(830\) 0 0
\(831\) 20.0014 0.693841
\(832\) 0 0
\(833\) −25.5217 + 34.4128i −0.884274 + 1.19233i
\(834\) 0 0
\(835\) 16.6783i 0.577177i
\(836\) 0 0
\(837\) 40.9710 1.41616
\(838\) 0 0
\(839\) 22.7494 0.785397 0.392699 0.919667i \(-0.371542\pi\)
0.392699 + 0.919667i \(0.371542\pi\)
\(840\) 0 0
\(841\) −24.8959 −0.858481
\(842\) 0 0
\(843\) 32.0065 1.10236
\(844\) 0 0
\(845\) 12.6822i 0.436282i
\(846\) 0 0
\(847\) −24.5071 + 74.1855i −0.842074 + 2.54905i
\(848\) 0 0
\(849\) 50.2899 1.72595
\(850\) 0 0
\(851\) 1.82983i 0.0627258i
\(852\) 0 0
\(853\) 39.0172i 1.33592i −0.744196 0.667961i \(-0.767167\pi\)
0.744196 0.667961i \(-0.232833\pi\)
\(854\) 0 0
\(855\) 0.103166i 0.00352819i
\(856\) 0 0
\(857\) 28.8131i 0.984238i −0.870528 0.492119i \(-0.836222\pi\)
0.870528 0.492119i \(-0.163778\pi\)
\(858\) 0 0
\(859\) 18.9381 0.646159 0.323079 0.946372i \(-0.395282\pi\)
0.323079 + 0.946372i \(0.395282\pi\)
\(860\) 0 0
\(861\) −8.80179 2.90766i −0.299964 0.0990928i
\(862\) 0 0
\(863\) 11.0291i 0.375435i −0.982223 0.187717i \(-0.939891\pi\)
0.982223 0.187717i \(-0.0601089\pi\)
\(864\) 0 0
\(865\) 16.8854 0.574121
\(866\) 0 0
\(867\) 36.3061 1.23302
\(868\) 0 0
\(869\) −25.1147 −0.851957
\(870\) 0 0
\(871\) 0.402852 0.0136501
\(872\) 0 0
\(873\) 1.41707i 0.0479607i
\(874\) 0 0
\(875\) 2.51222 + 0.829909i 0.0849285 + 0.0280561i
\(876\) 0 0
\(877\) 0.186515 0.00629815 0.00314908 0.999995i \(-0.498998\pi\)
0.00314908 + 0.999995i \(0.498998\pi\)
\(878\) 0 0
\(879\) 42.0286i 1.41759i
\(880\) 0 0
\(881\) 28.4366i 0.958055i 0.877800 + 0.479027i \(0.159011\pi\)
−0.877800 + 0.479027i \(0.840989\pi\)
\(882\) 0 0
\(883\) 6.76938i 0.227808i −0.993492 0.113904i \(-0.963664\pi\)
0.993492 0.113904i \(-0.0363356\pi\)
\(884\) 0 0
\(885\) 22.9194i 0.770428i
\(886\) 0 0
\(887\) −31.0752 −1.04340 −0.521701 0.853128i \(-0.674702\pi\)
−0.521701 + 0.853128i \(0.674702\pi\)
\(888\) 0 0
\(889\) −18.8724 6.23448i −0.632960 0.209098i
\(890\) 0 0
\(891\) 59.9919i 2.00981i
\(892\) 0 0
\(893\) 2.10749 0.0705245
\(894\) 0 0
\(895\) 10.2643 0.343099
\(896\) 0 0
\(897\) −0.965158 −0.0322257
\(898\) 0 0
\(899\) 16.4042 0.547111
\(900\) 0 0
\(901\) 10.6471i 0.354705i
\(902\) 0 0
\(903\) 23.4621 + 7.75067i 0.780769 + 0.257926i
\(904\) 0 0
\(905\) −0.656666 −0.0218283
\(906\) 0 0
\(907\) 39.5879i 1.31450i −0.753674 0.657248i \(-0.771720\pi\)
0.753674 0.657248i \(-0.228280\pi\)
\(908\) 0 0
\(909\) 2.08795i 0.0692531i
\(910\) 0 0
\(911\) 29.9338i 0.991750i 0.868394 + 0.495875i \(0.165153\pi\)
−0.868394 + 0.495875i \(0.834847\pi\)
\(912\) 0 0
\(913\) 73.6375i 2.43704i
\(914\) 0 0
\(915\) 17.6769 0.584381
\(916\) 0 0
\(917\) −7.57280 + 22.9236i −0.250076 + 0.757005i
\(918\) 0 0
\(919\) 18.1442i 0.598524i −0.954171 0.299262i \(-0.903260\pi\)
0.954171 0.299262i \(-0.0967404\pi\)
\(920\) 0 0
\(921\) 15.7411 0.518686
\(922\) 0 0
\(923\) 4.45715 0.146709
\(924\) 0 0
\(925\) 1.89633 0.0623509
\(926\) 0 0
\(927\) −1.20465 −0.0395657
\(928\) 0 0
\(929\) 4.33330i 0.142171i 0.997470 + 0.0710854i \(0.0226463\pi\)
−0.997470 + 0.0710854i \(0.977354\pi\)
\(930\) 0 0
\(931\) −3.90699 2.89756i −0.128047 0.0949635i
\(932\) 0 0
\(933\) −0.826453 −0.0270569
\(934\) 0 0
\(935\) 38.9653i 1.27430i
\(936\) 0 0
\(937\) 20.1249i 0.657451i −0.944426 0.328725i \(-0.893381\pi\)
0.944426 0.328725i \(-0.106619\pi\)
\(938\) 0 0
\(939\) 27.9201i 0.911139i
\(940\) 0 0
\(941\) 16.2874i 0.530955i −0.964117 0.265478i \(-0.914470\pi\)
0.964117 0.265478i \(-0.0855296\pi\)
\(942\) 0 0
\(943\) −1.90529 −0.0620448
\(944\) 0 0
\(945\) −12.7112 4.19913i −0.413495 0.136598i
\(946\) 0 0
\(947\) 18.2240i 0.592201i 0.955157 + 0.296101i \(0.0956864\pi\)
−0.955157 + 0.296101i \(0.904314\pi\)
\(948\) 0 0
\(949\) 0.419087 0.0136041
\(950\) 0 0
\(951\) −42.4296 −1.37587
\(952\) 0 0
\(953\) 14.0976 0.456667 0.228334 0.973583i \(-0.426672\pi\)
0.228334 + 0.973583i \(0.426672\pi\)
\(954\) 0 0
\(955\) −25.4758 −0.824379
\(956\) 0 0
\(957\) 22.8846i 0.739755i
\(958\) 0 0
\(959\) −12.8661 + 38.9469i −0.415467 + 1.25766i
\(960\) 0 0
\(961\) 34.5686 1.11512
\(962\) 0 0
\(963\) 1.57446i 0.0507362i
\(964\) 0 0
\(965\) 10.8050i 0.347825i
\(966\) 0 0
\(967\) 57.2233i 1.84018i 0.391709 + 0.920089i \(0.371884\pi\)
−0.391709 + 0.920089i \(0.628116\pi\)
\(968\) 0 0
\(969\) 7.54664i 0.242433i
\(970\) 0 0
\(971\) −30.0148 −0.963220 −0.481610 0.876386i \(-0.659948\pi\)
−0.481610 + 0.876386i \(0.659948\pi\)
\(972\) 0 0
\(973\) 8.65510 26.1999i 0.277470 0.839930i
\(974\) 0 0
\(975\) 1.00023i 0.0320331i
\(976\) 0 0
\(977\) 23.1481 0.740574 0.370287 0.928917i \(-0.379259\pi\)
0.370287 + 0.928917i \(0.379259\pi\)
\(978\) 0 0
\(979\) 106.094 3.39079
\(980\) 0 0
\(981\) 1.85451 0.0592100
\(982\) 0 0
\(983\) −17.3460 −0.553250 −0.276625 0.960978i \(-0.589216\pi\)
−0.276625 + 0.960978i \(0.589216\pi\)
\(984\) 0 0
\(985\) 12.2699i 0.390952i
\(986\) 0 0
\(987\) −4.46614 + 13.5195i −0.142159 + 0.430329i
\(988\) 0 0
\(989\) 5.07875 0.161495
\(990\) 0 0
\(991\) 29.7547i 0.945188i −0.881280 0.472594i \(-0.843318\pi\)
0.881280 0.472594i \(-0.156682\pi\)
\(992\) 0 0
\(993\) 3.01475i 0.0956703i
\(994\) 0 0
\(995\) 17.0170i 0.539476i
\(996\) 0 0
\(997\) 36.4069i 1.15302i −0.817091 0.576509i \(-0.804415\pi\)
0.817091 0.576509i \(-0.195585\pi\)
\(998\) 0 0
\(999\) −9.59494 −0.303570
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 2240.2.k.f.1791.3 16
4.3 odd 2 2240.2.k.g.1791.13 16
7.6 odd 2 2240.2.k.g.1791.14 16
8.3 odd 2 1120.2.k.b.671.4 yes 16
8.5 even 2 1120.2.k.a.671.14 yes 16
28.27 even 2 inner 2240.2.k.f.1791.4 16
56.13 odd 2 1120.2.k.b.671.3 yes 16
56.27 even 2 1120.2.k.a.671.13 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1120.2.k.a.671.13 16 56.27 even 2
1120.2.k.a.671.14 yes 16 8.5 even 2
1120.2.k.b.671.3 yes 16 56.13 odd 2
1120.2.k.b.671.4 yes 16 8.3 odd 2
2240.2.k.f.1791.3 16 1.1 even 1 trivial
2240.2.k.f.1791.4 16 28.27 even 2 inner
2240.2.k.g.1791.13 16 4.3 odd 2
2240.2.k.g.1791.14 16 7.6 odd 2