Properties

Label 2940.2.f.a.1469.31
Level $2940$
Weight $2$
Character 2940.1469
Analytic conductor $23.476$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2940,2,Mod(1469,2940)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2940.1469"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.4760181943\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1469.31
Character \(\chi\) \(=\) 2940.1469
Dual form 2940.2.f.a.1469.29

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.69264 + 0.367385i) q^{3} +(-2.03939 + 0.916997i) q^{5} +(2.73006 + 1.24370i) q^{9} -5.87781i q^{11} -3.54004 q^{13} +(-3.78885 + 0.802902i) q^{15} +3.10812i q^{17} +4.13866i q^{19} -7.15173 q^{23} +(3.31823 - 3.74023i) q^{25} +(4.16408 + 3.10812i) q^{27} +6.84761i q^{29} +4.50776i q^{31} +(2.15942 - 9.94902i) q^{33} -1.97369i q^{37} +(-5.99202 - 1.30056i) q^{39} -6.33142 q^{41} +3.88979i q^{43} +(-6.70812 - 0.0329421i) q^{45} +1.83399i q^{47} +(-1.14188 + 5.26093i) q^{51} -13.8078 q^{53} +(5.38994 + 11.9872i) q^{55} +(-1.52048 + 7.00526i) q^{57} -4.65168 q^{59} +0.810766i q^{61} +(7.21953 - 3.24621i) q^{65} +10.2094i q^{67} +(-12.1053 - 2.62744i) q^{69} +1.18684i q^{71} -2.17045 q^{73} +(6.99068 - 5.11179i) q^{75} +10.1046 q^{79} +(5.90641 + 6.79075i) q^{81} +5.31243i q^{83} +(-2.85014 - 6.33868i) q^{85} +(-2.51571 + 11.5905i) q^{87} +12.5797 q^{89} +(-1.65609 + 7.63002i) q^{93} +(-3.79514 - 8.44035i) q^{95} -4.50686 q^{97} +(7.31025 - 16.0468i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 2 q^{15} - 12 q^{25} - 48 q^{39} + 20 q^{51} + 56 q^{79} + 40 q^{81} - 4 q^{85} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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
\(3\) 1.69264 + 0.367385i 0.977246 + 0.212110i
\(4\) 0 0
\(5\) −2.03939 + 0.916997i −0.912044 + 0.410093i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.73006 + 1.24370i 0.910019 + 0.414567i
\(10\) 0 0
\(11\) 5.87781i 1.77223i −0.463467 0.886114i \(-0.653395\pi\)
0.463467 0.886114i \(-0.346605\pi\)
\(12\) 0 0
\(13\) −3.54004 −0.981831 −0.490916 0.871207i \(-0.663338\pi\)
−0.490916 + 0.871207i \(0.663338\pi\)
\(14\) 0 0
\(15\) −3.78885 + 0.802902i −0.978276 + 0.207309i
\(16\) 0 0
\(17\) 3.10812i 0.753830i 0.926248 + 0.376915i \(0.123015\pi\)
−0.926248 + 0.376915i \(0.876985\pi\)
\(18\) 0 0
\(19\) 4.13866i 0.949474i 0.880128 + 0.474737i \(0.157457\pi\)
−0.880128 + 0.474737i \(0.842543\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.15173 −1.49124 −0.745620 0.666372i \(-0.767846\pi\)
−0.745620 + 0.666372i \(0.767846\pi\)
\(24\) 0 0
\(25\) 3.31823 3.74023i 0.663647 0.748046i
\(26\) 0 0
\(27\) 4.16408 + 3.10812i 0.801378 + 0.598158i
\(28\) 0 0
\(29\) 6.84761i 1.27157i 0.771867 + 0.635785i \(0.219323\pi\)
−0.771867 + 0.635785i \(0.780677\pi\)
\(30\) 0 0
\(31\) 4.50776i 0.809618i 0.914401 + 0.404809i \(0.132662\pi\)
−0.914401 + 0.404809i \(0.867338\pi\)
\(32\) 0 0
\(33\) 2.15942 9.94902i 0.375907 1.73190i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.97369i 0.324472i −0.986752 0.162236i \(-0.948129\pi\)
0.986752 0.162236i \(-0.0518706\pi\)
\(38\) 0 0
\(39\) −5.99202 1.30056i −0.959490 0.208256i
\(40\) 0 0
\(41\) −6.33142 −0.988801 −0.494401 0.869234i \(-0.664612\pi\)
−0.494401 + 0.869234i \(0.664612\pi\)
\(42\) 0 0
\(43\) 3.88979i 0.593187i 0.955004 + 0.296594i \(0.0958507\pi\)
−0.955004 + 0.296594i \(0.904149\pi\)
\(44\) 0 0
\(45\) −6.70812 0.0329421i −0.999988 0.00491072i
\(46\) 0 0
\(47\) 1.83399i 0.267515i 0.991014 + 0.133758i \(0.0427044\pi\)
−0.991014 + 0.133758i \(0.957296\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1.14188 + 5.26093i −0.159895 + 0.736677i
\(52\) 0 0
\(53\) −13.8078 −1.89665 −0.948325 0.317302i \(-0.897223\pi\)
−0.948325 + 0.317302i \(0.897223\pi\)
\(54\) 0 0
\(55\) 5.38994 + 11.9872i 0.726779 + 1.61635i
\(56\) 0 0
\(57\) −1.52048 + 7.00526i −0.201393 + 0.927869i
\(58\) 0 0
\(59\) −4.65168 −0.605598 −0.302799 0.953054i \(-0.597921\pi\)
−0.302799 + 0.953054i \(0.597921\pi\)
\(60\) 0 0
\(61\) 0.810766i 0.103808i 0.998652 + 0.0519040i \(0.0165290\pi\)
−0.998652 + 0.0519040i \(0.983471\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.21953 3.24621i 0.895473 0.402643i
\(66\) 0 0
\(67\) 10.2094i 1.24727i 0.781714 + 0.623637i \(0.214346\pi\)
−0.781714 + 0.623637i \(0.785654\pi\)
\(68\) 0 0
\(69\) −12.1053 2.62744i −1.45731 0.316307i
\(70\) 0 0
\(71\) 1.18684i 0.140852i 0.997517 + 0.0704262i \(0.0224359\pi\)
−0.997517 + 0.0704262i \(0.977564\pi\)
\(72\) 0 0
\(73\) −2.17045 −0.254032 −0.127016 0.991901i \(-0.540540\pi\)
−0.127016 + 0.991901i \(0.540540\pi\)
\(74\) 0 0
\(75\) 6.99068 5.11179i 0.807214 0.590259i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.1046 1.13685 0.568426 0.822734i \(-0.307553\pi\)
0.568426 + 0.822734i \(0.307553\pi\)
\(80\) 0 0
\(81\) 5.90641 + 6.79075i 0.656268 + 0.754528i
\(82\) 0 0
\(83\) 5.31243i 0.583115i 0.956553 + 0.291558i \(0.0941735\pi\)
−0.956553 + 0.291558i \(0.905826\pi\)
\(84\) 0 0
\(85\) −2.85014 6.33868i −0.309141 0.687526i
\(86\) 0 0
\(87\) −2.51571 + 11.5905i −0.269713 + 1.24264i
\(88\) 0 0
\(89\) 12.5797 1.33345 0.666724 0.745305i \(-0.267696\pi\)
0.666724 + 0.745305i \(0.267696\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.65609 + 7.63002i −0.171728 + 0.791196i
\(94\) 0 0
\(95\) −3.79514 8.44035i −0.389373 0.865962i
\(96\) 0 0
\(97\) −4.50686 −0.457602 −0.228801 0.973473i \(-0.573481\pi\)
−0.228801 + 0.973473i \(0.573481\pi\)
\(98\) 0 0
\(99\) 7.31025 16.0468i 0.734708 1.61276i
\(100\) 0 0
\(101\) −1.79358 −0.178468 −0.0892340 0.996011i \(-0.528442\pi\)
−0.0892340 + 0.996011i \(0.528442\pi\)
\(102\) 0 0
\(103\) −12.6138 −1.24287 −0.621436 0.783465i \(-0.713450\pi\)
−0.621436 + 0.783465i \(0.713450\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.47951 −0.336377 −0.168189 0.985755i \(-0.553792\pi\)
−0.168189 + 0.985755i \(0.553792\pi\)
\(108\) 0 0
\(109\) −9.97605 −0.955532 −0.477766 0.878487i \(-0.658553\pi\)
−0.477766 + 0.878487i \(0.658553\pi\)
\(110\) 0 0
\(111\) 0.725103 3.34074i 0.0688238 0.317089i
\(112\) 0 0
\(113\) 2.54532 0.239444 0.119722 0.992807i \(-0.461800\pi\)
0.119722 + 0.992807i \(0.461800\pi\)
\(114\) 0 0
\(115\) 14.5852 6.55811i 1.36007 0.611547i
\(116\) 0 0
\(117\) −9.66452 4.40276i −0.893485 0.407035i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −23.5487 −2.14079
\(122\) 0 0
\(123\) −10.7168 2.32607i −0.966302 0.209735i
\(124\) 0 0
\(125\) −3.33740 + 10.6706i −0.298506 + 0.954408i
\(126\) 0 0
\(127\) 12.6616i 1.12353i −0.827295 0.561767i \(-0.810122\pi\)
0.827295 0.561767i \(-0.189878\pi\)
\(128\) 0 0
\(129\) −1.42905 + 6.58401i −0.125821 + 0.579690i
\(130\) 0 0
\(131\) 12.6954 1.10920 0.554601 0.832116i \(-0.312871\pi\)
0.554601 + 0.832116i \(0.312871\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −11.3423 2.52023i −0.976192 0.216906i
\(136\) 0 0
\(137\) −5.24782 −0.448352 −0.224176 0.974549i \(-0.571969\pi\)
−0.224176 + 0.974549i \(0.571969\pi\)
\(138\) 0 0
\(139\) 14.2842i 1.21157i −0.795627 0.605787i \(-0.792858\pi\)
0.795627 0.605787i \(-0.207142\pi\)
\(140\) 0 0
\(141\) −0.673782 + 3.10429i −0.0567427 + 0.261428i
\(142\) 0 0
\(143\) 20.8077i 1.74003i
\(144\) 0 0
\(145\) −6.27924 13.9650i −0.521462 1.15973i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.0991i 1.23697i −0.785798 0.618484i \(-0.787747\pi\)
0.785798 0.618484i \(-0.212253\pi\)
\(150\) 0 0
\(151\) −8.33958 −0.678665 −0.339333 0.940666i \(-0.610201\pi\)
−0.339333 + 0.940666i \(0.610201\pi\)
\(152\) 0 0
\(153\) −3.86558 + 8.48535i −0.312513 + 0.686000i
\(154\) 0 0
\(155\) −4.13361 9.19310i −0.332019 0.738407i
\(156\) 0 0
\(157\) 0.960596 0.0766639 0.0383319 0.999265i \(-0.487796\pi\)
0.0383319 + 0.999265i \(0.487796\pi\)
\(158\) 0 0
\(159\) −23.3717 5.07279i −1.85349 0.402298i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 18.6713i 1.46245i 0.682137 + 0.731224i \(0.261051\pi\)
−0.682137 + 0.731224i \(0.738949\pi\)
\(164\) 0 0
\(165\) 4.71931 + 22.2701i 0.367398 + 1.73373i
\(166\) 0 0
\(167\) 14.2212i 1.10047i 0.835011 + 0.550233i \(0.185461\pi\)
−0.835011 + 0.550233i \(0.814539\pi\)
\(168\) 0 0
\(169\) −0.468096 −0.0360074
\(170\) 0 0
\(171\) −5.14726 + 11.2988i −0.393621 + 0.864039i
\(172\) 0 0
\(173\) 0.714262i 0.0543043i −0.999631 0.0271522i \(-0.991356\pi\)
0.999631 0.0271522i \(-0.00864386\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −7.87362 1.70896i −0.591818 0.128453i
\(178\) 0 0
\(179\) 11.1834i 0.835886i −0.908473 0.417943i \(-0.862751\pi\)
0.908473 0.417943i \(-0.137249\pi\)
\(180\) 0 0
\(181\) 3.69700i 0.274796i 0.990516 + 0.137398i \(0.0438739\pi\)
−0.990516 + 0.137398i \(0.956126\pi\)
\(182\) 0 0
\(183\) −0.297864 + 1.37233i −0.0220187 + 0.101446i
\(184\) 0 0
\(185\) 1.80986 + 4.02512i 0.133064 + 0.295933i
\(186\) 0 0
\(187\) 18.2690 1.33596
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.3883i 0.751670i −0.926687 0.375835i \(-0.877356\pi\)
0.926687 0.375835i \(-0.122644\pi\)
\(192\) 0 0
\(193\) 5.92106i 0.426207i −0.977030 0.213104i \(-0.931643\pi\)
0.977030 0.213104i \(-0.0683572\pi\)
\(194\) 0 0
\(195\) 13.4127 2.84231i 0.960502 0.203542i
\(196\) 0 0
\(197\) −11.1053 −0.791221 −0.395610 0.918418i \(-0.629467\pi\)
−0.395610 + 0.918418i \(0.629467\pi\)
\(198\) 0 0
\(199\) 13.2406i 0.938600i −0.883039 0.469300i \(-0.844506\pi\)
0.883039 0.469300i \(-0.155494\pi\)
\(200\) 0 0
\(201\) −3.75077 + 17.2808i −0.264559 + 1.21889i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 12.9122 5.80589i 0.901830 0.405501i
\(206\) 0 0
\(207\) −19.5246 8.89462i −1.35706 0.618219i
\(208\) 0 0
\(209\) 24.3263 1.68268
\(210\) 0 0
\(211\) −19.9122 −1.37082 −0.685408 0.728160i \(-0.740376\pi\)
−0.685408 + 0.728160i \(0.740376\pi\)
\(212\) 0 0
\(213\) −0.436029 + 2.00890i −0.0298762 + 0.137647i
\(214\) 0 0
\(215\) −3.56692 7.93280i −0.243262 0.541013i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3.67378 0.797390i −0.248251 0.0538826i
\(220\) 0 0
\(221\) 11.0029i 0.740134i
\(222\) 0 0
\(223\) −10.4924 −0.702623 −0.351311 0.936259i \(-0.614264\pi\)
−0.351311 + 0.936259i \(0.614264\pi\)
\(224\) 0 0
\(225\) 13.7107 6.08415i 0.914046 0.405610i
\(226\) 0 0
\(227\) 16.4255i 1.09020i 0.838372 + 0.545098i \(0.183508\pi\)
−0.838372 + 0.545098i \(0.816492\pi\)
\(228\) 0 0
\(229\) 1.04366i 0.0689672i −0.999405 0.0344836i \(-0.989021\pi\)
0.999405 0.0344836i \(-0.0109786\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 28.9238 1.89486 0.947431 0.319960i \(-0.103669\pi\)
0.947431 + 0.319960i \(0.103669\pi\)
\(234\) 0 0
\(235\) −1.68177 3.74023i −0.109706 0.243986i
\(236\) 0 0
\(237\) 17.1034 + 3.71227i 1.11098 + 0.241138i
\(238\) 0 0
\(239\) 5.69731i 0.368528i 0.982877 + 0.184264i \(0.0589902\pi\)
−0.982877 + 0.184264i \(0.941010\pi\)
\(240\) 0 0
\(241\) 7.05058i 0.454168i 0.973875 + 0.227084i \(0.0729192\pi\)
−0.973875 + 0.227084i \(0.927081\pi\)
\(242\) 0 0
\(243\) 7.50260 + 13.6642i 0.481292 + 0.876560i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 14.6510i 0.932223i
\(248\) 0 0
\(249\) −1.95171 + 8.99203i −0.123685 + 0.569847i
\(250\) 0 0
\(251\) 3.93237 0.248209 0.124105 0.992269i \(-0.460394\pi\)
0.124105 + 0.992269i \(0.460394\pi\)
\(252\) 0 0
\(253\) 42.0366i 2.64282i
\(254\) 0 0
\(255\) −2.49552 11.7762i −0.156275 0.737454i
\(256\) 0 0
\(257\) 15.3057i 0.954746i −0.878701 0.477373i \(-0.841589\pi\)
0.878701 0.477373i \(-0.158411\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −8.51639 + 18.6944i −0.527151 + 1.15715i
\(262\) 0 0
\(263\) 19.5195 1.20363 0.601813 0.798637i \(-0.294445\pi\)
0.601813 + 0.798637i \(0.294445\pi\)
\(264\) 0 0
\(265\) 28.1595 12.6617i 1.72983 0.777803i
\(266\) 0 0
\(267\) 21.2929 + 4.62161i 1.30311 + 0.282838i
\(268\) 0 0
\(269\) −9.83730 −0.599791 −0.299895 0.953972i \(-0.596952\pi\)
−0.299895 + 0.953972i \(0.596952\pi\)
\(270\) 0 0
\(271\) 20.8793i 1.26833i 0.773198 + 0.634164i \(0.218656\pi\)
−0.773198 + 0.634164i \(0.781344\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −21.9844 19.5040i −1.32571 1.17613i
\(276\) 0 0
\(277\) 8.50375i 0.510941i 0.966817 + 0.255470i \(0.0822303\pi\)
−0.966817 + 0.255470i \(0.917770\pi\)
\(278\) 0 0
\(279\) −5.60632 + 12.3064i −0.335641 + 0.736768i
\(280\) 0 0
\(281\) 7.63690i 0.455579i −0.973710 0.227790i \(-0.926850\pi\)
0.973710 0.227790i \(-0.0731499\pi\)
\(282\) 0 0
\(283\) −9.00750 −0.535440 −0.267720 0.963497i \(-0.586270\pi\)
−0.267720 + 0.963497i \(0.586270\pi\)
\(284\) 0 0
\(285\) −3.32294 15.6807i −0.196834 0.928847i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 7.33958 0.431740
\(290\) 0 0
\(291\) −7.62848 1.65575i −0.447190 0.0970620i
\(292\) 0 0
\(293\) 27.6102i 1.61301i 0.591231 + 0.806503i \(0.298642\pi\)
−0.591231 + 0.806503i \(0.701358\pi\)
\(294\) 0 0
\(295\) 9.48660 4.26558i 0.552331 0.248352i
\(296\) 0 0
\(297\) 18.2690 24.4757i 1.06007 1.42022i
\(298\) 0 0
\(299\) 25.3174 1.46415
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −3.03589 0.658936i −0.174407 0.0378549i
\(304\) 0 0
\(305\) −0.743470 1.65347i −0.0425710 0.0946774i
\(306\) 0 0
\(307\) 9.99609 0.570507 0.285253 0.958452i \(-0.407922\pi\)
0.285253 + 0.958452i \(0.407922\pi\)
\(308\) 0 0
\(309\) −21.3506 4.63411i −1.21459 0.263625i
\(310\) 0 0
\(311\) 23.8672 1.35339 0.676693 0.736265i \(-0.263412\pi\)
0.676693 + 0.736265i \(0.263412\pi\)
\(312\) 0 0
\(313\) 5.59519 0.316259 0.158129 0.987418i \(-0.449454\pi\)
0.158129 + 0.987418i \(0.449454\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.7366 0.659191 0.329596 0.944122i \(-0.393088\pi\)
0.329596 + 0.944122i \(0.393088\pi\)
\(318\) 0 0
\(319\) 40.2490 2.25351
\(320\) 0 0
\(321\) −5.88956 1.27832i −0.328723 0.0713490i
\(322\) 0 0
\(323\) −12.8635 −0.715742
\(324\) 0 0
\(325\) −11.7467 + 13.2406i −0.651589 + 0.734455i
\(326\) 0 0
\(327\) −16.8859 3.66505i −0.933790 0.202678i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.52907 0.139010 0.0695050 0.997582i \(-0.477858\pi\)
0.0695050 + 0.997582i \(0.477858\pi\)
\(332\) 0 0
\(333\) 2.45468 5.38827i 0.134515 0.295276i
\(334\) 0 0
\(335\) −9.36196 20.8209i −0.511499 1.13757i
\(336\) 0 0
\(337\) 34.3299i 1.87007i −0.354560 0.935033i \(-0.615369\pi\)
0.354560 0.935033i \(-0.384631\pi\)
\(338\) 0 0
\(339\) 4.30831 + 0.935114i 0.233995 + 0.0507884i
\(340\) 0 0
\(341\) 26.4958 1.43483
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 27.0968 5.74214i 1.45884 0.309147i
\(346\) 0 0
\(347\) −0.595727 −0.0319803 −0.0159902 0.999872i \(-0.505090\pi\)
−0.0159902 + 0.999872i \(0.505090\pi\)
\(348\) 0 0
\(349\) 23.0392i 1.23326i 0.787253 + 0.616630i \(0.211503\pi\)
−0.787253 + 0.616630i \(0.788497\pi\)
\(350\) 0 0
\(351\) −14.7410 11.0029i −0.786818 0.587290i
\(352\) 0 0
\(353\) 10.8971i 0.579995i −0.957027 0.289997i \(-0.906346\pi\)
0.957027 0.289997i \(-0.0936544\pi\)
\(354\) 0 0
\(355\) −1.08833 2.42044i −0.0577627 0.128464i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.4306i 0.550508i −0.961372 0.275254i \(-0.911238\pi\)
0.961372 0.275254i \(-0.0887620\pi\)
\(360\) 0 0
\(361\) 1.87149 0.0984992
\(362\) 0 0
\(363\) −39.8595 8.65145i −2.09208 0.454083i
\(364\) 0 0
\(365\) 4.42639 1.99029i 0.231688 0.104177i
\(366\) 0 0
\(367\) 8.54942 0.446276 0.223138 0.974787i \(-0.428370\pi\)
0.223138 + 0.974787i \(0.428370\pi\)
\(368\) 0 0
\(369\) −17.2851 7.87440i −0.899828 0.409925i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6.62947i 0.343261i −0.985161 0.171630i \(-0.945096\pi\)
0.985161 0.171630i \(-0.0549035\pi\)
\(374\) 0 0
\(375\) −9.56923 + 16.8354i −0.494153 + 0.869375i
\(376\) 0 0
\(377\) 24.2408i 1.24847i
\(378\) 0 0
\(379\) 22.3165 1.14632 0.573162 0.819442i \(-0.305717\pi\)
0.573162 + 0.819442i \(0.305717\pi\)
\(380\) 0 0
\(381\) 4.65168 21.4315i 0.238313 1.09797i
\(382\) 0 0
\(383\) 28.1890i 1.44039i 0.693771 + 0.720196i \(0.255948\pi\)
−0.693771 + 0.720196i \(0.744052\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.83774 + 10.6193i −0.245916 + 0.539811i
\(388\) 0 0
\(389\) 4.13277i 0.209540i −0.994496 0.104770i \(-0.966589\pi\)
0.994496 0.104770i \(-0.0334106\pi\)
\(390\) 0 0
\(391\) 22.2284i 1.12414i
\(392\) 0 0
\(393\) 21.4887 + 4.66411i 1.08396 + 0.235273i
\(394\) 0 0
\(395\) −20.6072 + 9.26585i −1.03686 + 0.466216i
\(396\) 0 0
\(397\) 13.2378 0.664386 0.332193 0.943211i \(-0.392211\pi\)
0.332193 + 0.943211i \(0.392211\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.775246i 0.0387140i 0.999813 + 0.0193570i \(0.00616190\pi\)
−0.999813 + 0.0193570i \(0.993838\pi\)
\(402\) 0 0
\(403\) 15.9577i 0.794909i
\(404\) 0 0
\(405\) −18.2726 8.43284i −0.907972 0.419031i
\(406\) 0 0
\(407\) −11.6010 −0.575038
\(408\) 0 0
\(409\) 4.58524i 0.226726i −0.993554 0.113363i \(-0.963838\pi\)
0.993554 0.113363i \(-0.0361622\pi\)
\(410\) 0 0
\(411\) −8.88267 1.92797i −0.438150 0.0950999i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.87149 10.8341i −0.239132 0.531827i
\(416\) 0 0
\(417\) 5.24782 24.1781i 0.256987 1.18401i
\(418\) 0 0
\(419\) −11.9761 −0.585070 −0.292535 0.956255i \(-0.594499\pi\)
−0.292535 + 0.956255i \(0.594499\pi\)
\(420\) 0 0
\(421\) 25.5487 1.24517 0.622584 0.782553i \(-0.286083\pi\)
0.622584 + 0.782553i \(0.286083\pi\)
\(422\) 0 0
\(423\) −2.28094 + 5.00691i −0.110903 + 0.243444i
\(424\) 0 0
\(425\) 11.6251 + 10.3135i 0.563900 + 0.500277i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −7.64445 + 35.2200i −0.369078 + 1.70044i
\(430\) 0 0
\(431\) 14.5070i 0.698778i 0.936978 + 0.349389i \(0.113611\pi\)
−0.936978 + 0.349389i \(0.886389\pi\)
\(432\) 0 0
\(433\) −40.0655 −1.92543 −0.962713 0.270524i \(-0.912803\pi\)
−0.962713 + 0.270524i \(0.912803\pi\)
\(434\) 0 0
\(435\) −5.49796 25.9445i −0.263607 1.24395i
\(436\) 0 0
\(437\) 29.5986i 1.41589i
\(438\) 0 0
\(439\) 31.3891i 1.49812i −0.662502 0.749060i \(-0.730506\pi\)
0.662502 0.749060i \(-0.269494\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.0355 0.476802 0.238401 0.971167i \(-0.423377\pi\)
0.238401 + 0.971167i \(0.423377\pi\)
\(444\) 0 0
\(445\) −25.6550 + 11.5356i −1.21616 + 0.546838i
\(446\) 0 0
\(447\) 5.54719 25.5574i 0.262373 1.20882i
\(448\) 0 0
\(449\) 16.7060i 0.788405i 0.919024 + 0.394203i \(0.128979\pi\)
−0.919024 + 0.394203i \(0.871021\pi\)
\(450\) 0 0
\(451\) 37.2149i 1.75238i
\(452\) 0 0
\(453\) −14.1159 3.06384i −0.663223 0.143952i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 14.1481i 0.661821i 0.943662 + 0.330910i \(0.107356\pi\)
−0.943662 + 0.330910i \(0.892644\pi\)
\(458\) 0 0
\(459\) −9.66042 + 12.9425i −0.450910 + 0.604103i
\(460\) 0 0
\(461\) 4.22519 0.196787 0.0983935 0.995148i \(-0.468630\pi\)
0.0983935 + 0.995148i \(0.468630\pi\)
\(462\) 0 0
\(463\) 33.2490i 1.54521i 0.634886 + 0.772606i \(0.281047\pi\)
−0.634886 + 0.772606i \(0.718953\pi\)
\(464\) 0 0
\(465\) −3.61929 17.0792i −0.167841 0.792030i
\(466\) 0 0
\(467\) 36.8629i 1.70581i 0.522066 + 0.852905i \(0.325161\pi\)
−0.522066 + 0.852905i \(0.674839\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1.62594 + 0.352909i 0.0749195 + 0.0162612i
\(472\) 0 0
\(473\) 22.8635 1.05126
\(474\) 0 0
\(475\) 15.4795 + 13.7330i 0.710250 + 0.630115i
\(476\) 0 0
\(477\) −37.6961 17.1728i −1.72599 0.786289i
\(478\) 0 0
\(479\) 9.33593 0.426570 0.213285 0.976990i \(-0.431584\pi\)
0.213285 + 0.976990i \(0.431584\pi\)
\(480\) 0 0
\(481\) 6.98693i 0.318577i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.19124 4.13277i 0.417353 0.187660i
\(486\) 0 0
\(487\) 25.3389i 1.14822i 0.818779 + 0.574109i \(0.194651\pi\)
−0.818779 + 0.574109i \(0.805349\pi\)
\(488\) 0 0
\(489\) −6.85956 + 31.6038i −0.310200 + 1.42917i
\(490\) 0 0
\(491\) 35.4614i 1.60035i 0.599765 + 0.800176i \(0.295261\pi\)
−0.599765 + 0.800176i \(0.704739\pi\)
\(492\) 0 0
\(493\) −21.2832 −0.958547
\(494\) 0 0
\(495\) −0.193628 + 39.4291i −0.00870291 + 1.77221i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −29.4174 −1.31690 −0.658451 0.752624i \(-0.728788\pi\)
−0.658451 + 0.752624i \(0.728788\pi\)
\(500\) 0 0
\(501\) −5.22464 + 24.0713i −0.233420 + 1.07543i
\(502\) 0 0
\(503\) 20.9709i 0.935046i −0.883981 0.467523i \(-0.845147\pi\)
0.883981 0.467523i \(-0.154853\pi\)
\(504\) 0 0
\(505\) 3.65781 1.64471i 0.162771 0.0731886i
\(506\) 0 0
\(507\) −0.792318 0.171972i −0.0351881 0.00763753i
\(508\) 0 0
\(509\) −7.43642 −0.329613 −0.164807 0.986326i \(-0.552700\pi\)
−0.164807 + 0.986326i \(0.552700\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −12.8635 + 17.2337i −0.567936 + 0.760888i
\(514\) 0 0
\(515\) 25.7244 11.5668i 1.13355 0.509693i
\(516\) 0 0
\(517\) 10.7799 0.474098
\(518\) 0 0
\(519\) 0.262409 1.20899i 0.0115185 0.0530687i
\(520\) 0 0
\(521\) −5.10250 −0.223545 −0.111772 0.993734i \(-0.535653\pi\)
−0.111772 + 0.993734i \(0.535653\pi\)
\(522\) 0 0
\(523\) −16.4966 −0.721345 −0.360673 0.932692i \(-0.617453\pi\)
−0.360673 + 0.932692i \(0.617453\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −14.0107 −0.610315
\(528\) 0 0
\(529\) 28.1473 1.22379
\(530\) 0 0
\(531\) −12.6994 5.78531i −0.551105 0.251061i
\(532\) 0 0
\(533\) 22.4135 0.970836
\(534\) 0 0
\(535\) 7.09609 3.19070i 0.306791 0.137946i
\(536\) 0 0
\(537\) 4.10861 18.9294i 0.177300 0.816866i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 20.6154 0.886323 0.443162 0.896442i \(-0.353857\pi\)
0.443162 + 0.896442i \(0.353857\pi\)
\(542\) 0 0
\(543\) −1.35822 + 6.25769i −0.0582869 + 0.268543i
\(544\) 0 0
\(545\) 20.3451 9.14800i 0.871487 0.391857i
\(546\) 0 0
\(547\) 25.6070i 1.09488i 0.836846 + 0.547438i \(0.184397\pi\)
−0.836846 + 0.547438i \(0.815603\pi\)
\(548\) 0 0
\(549\) −1.00835 + 2.21344i −0.0430354 + 0.0944672i
\(550\) 0 0
\(551\) −28.3399 −1.20732
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1.58468 + 7.47799i 0.0672658 + 0.317423i
\(556\) 0 0
\(557\) 36.1972 1.53372 0.766862 0.641812i \(-0.221817\pi\)
0.766862 + 0.641812i \(0.221817\pi\)
\(558\) 0 0
\(559\) 13.7700i 0.582410i
\(560\) 0 0
\(561\) 30.9228 + 6.71175i 1.30556 + 0.283370i
\(562\) 0 0
\(563\) 30.6466i 1.29160i −0.763506 0.645801i \(-0.776524\pi\)
0.763506 0.645801i \(-0.223476\pi\)
\(564\) 0 0
\(565\) −5.19090 + 2.33405i −0.218383 + 0.0981943i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.2598i 0.639722i 0.947464 + 0.319861i \(0.103636\pi\)
−0.947464 + 0.319861i \(0.896364\pi\)
\(570\) 0 0
\(571\) −18.1046 −0.757652 −0.378826 0.925468i \(-0.623672\pi\)
−0.378826 + 0.925468i \(0.623672\pi\)
\(572\) 0 0
\(573\) 3.81650 17.5836i 0.159437 0.734566i
\(574\) 0 0
\(575\) −23.7311 + 26.7491i −0.989656 + 1.11552i
\(576\) 0 0
\(577\) −6.56672 −0.273376 −0.136688 0.990614i \(-0.543646\pi\)
−0.136688 + 0.990614i \(0.543646\pi\)
\(578\) 0 0
\(579\) 2.17531 10.0222i 0.0904028 0.416509i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 81.1598i 3.36129i
\(584\) 0 0
\(585\) 23.7470 + 0.116616i 0.981819 + 0.00482150i
\(586\) 0 0
\(587\) 9.35074i 0.385946i −0.981204 0.192973i \(-0.938187\pi\)
0.981204 0.192973i \(-0.0618130\pi\)
\(588\) 0 0
\(589\) −18.6561 −0.768712
\(590\) 0 0
\(591\) −18.7973 4.07993i −0.773217 0.167826i
\(592\) 0 0
\(593\) 26.9976i 1.10866i −0.832298 0.554329i \(-0.812975\pi\)
0.832298 0.554329i \(-0.187025\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.86439 22.4115i 0.199086 0.917243i
\(598\) 0 0
\(599\) 22.7726i 0.930461i −0.885189 0.465231i \(-0.845971\pi\)
0.885189 0.465231i \(-0.154029\pi\)
\(600\) 0 0
\(601\) 5.26871i 0.214915i 0.994210 + 0.107458i \(0.0342710\pi\)
−0.994210 + 0.107458i \(0.965729\pi\)
\(602\) 0 0
\(603\) −12.6974 + 27.8722i −0.517079 + 1.13504i
\(604\) 0 0
\(605\) 48.0250 21.5941i 1.95250 0.877925i
\(606\) 0 0
\(607\) 9.35499 0.379707 0.189854 0.981812i \(-0.439199\pi\)
0.189854 + 0.981812i \(0.439199\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.49242i 0.262655i
\(612\) 0 0
\(613\) 31.5591i 1.27466i −0.770591 0.637330i \(-0.780039\pi\)
0.770591 0.637330i \(-0.219961\pi\)
\(614\) 0 0
\(615\) 23.9888 5.08351i 0.967320 0.204987i
\(616\) 0 0
\(617\) 39.6450 1.59605 0.798023 0.602626i \(-0.205879\pi\)
0.798023 + 0.602626i \(0.205879\pi\)
\(618\) 0 0
\(619\) 38.3829i 1.54274i −0.636387 0.771370i \(-0.719572\pi\)
0.636387 0.771370i \(-0.280428\pi\)
\(620\) 0 0
\(621\) −29.7804 22.2284i −1.19505 0.891997i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −2.97865 24.8219i −0.119146 0.992877i
\(626\) 0 0
\(627\) 41.1756 + 8.93712i 1.64440 + 0.356914i
\(628\) 0 0
\(629\) 6.13446 0.244597
\(630\) 0 0
\(631\) 1.06381 0.0423495 0.0211748 0.999776i \(-0.493259\pi\)
0.0211748 + 0.999776i \(0.493259\pi\)
\(632\) 0 0
\(633\) −33.7042 7.31547i −1.33962 0.290764i
\(634\) 0 0
\(635\) 11.6106 + 25.8219i 0.460754 + 1.02471i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −1.47608 + 3.24015i −0.0583928 + 0.128178i
\(640\) 0 0
\(641\) 29.9812i 1.18419i −0.805870 0.592093i \(-0.798302\pi\)
0.805870 0.592093i \(-0.201698\pi\)
\(642\) 0 0
\(643\) −16.0861 −0.634373 −0.317187 0.948363i \(-0.602738\pi\)
−0.317187 + 0.948363i \(0.602738\pi\)
\(644\) 0 0
\(645\) −3.12312 14.7378i −0.122973 0.580301i
\(646\) 0 0
\(647\) 24.9639i 0.981431i −0.871320 0.490716i \(-0.836735\pi\)
0.871320 0.490716i \(-0.163265\pi\)
\(648\) 0 0
\(649\) 27.3417i 1.07326i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.39213 0.171877 0.0859387 0.996300i \(-0.472611\pi\)
0.0859387 + 0.996300i \(0.472611\pi\)
\(654\) 0 0
\(655\) −25.8909 + 11.6416i −1.01164 + 0.454877i
\(656\) 0 0
\(657\) −5.92544 2.69939i −0.231173 0.105313i
\(658\) 0 0
\(659\) 1.93959i 0.0755557i 0.999286 + 0.0377778i \(0.0120279\pi\)
−0.999286 + 0.0377778i \(0.987972\pi\)
\(660\) 0 0
\(661\) 24.9454i 0.970265i −0.874441 0.485132i \(-0.838771\pi\)
0.874441 0.485132i \(-0.161229\pi\)
\(662\) 0 0
\(663\) 4.04230 18.6239i 0.156990 0.723293i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 48.9723i 1.89621i
\(668\) 0 0
\(669\) −17.7598 3.85475i −0.686635 0.149033i
\(670\) 0 0
\(671\) 4.76553 0.183971
\(672\) 0 0
\(673\) 6.21519i 0.239578i −0.992799 0.119789i \(-0.961778\pi\)
0.992799 0.119789i \(-0.0382218\pi\)
\(674\) 0 0
\(675\) 25.4425 5.26116i 0.979282 0.202502i
\(676\) 0 0
\(677\) 16.6603i 0.640309i 0.947365 + 0.320154i \(0.103735\pi\)
−0.947365 + 0.320154i \(0.896265\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −6.03448 + 27.8024i −0.231242 + 1.06539i
\(682\) 0 0
\(683\) −22.2677 −0.852051 −0.426025 0.904711i \(-0.640087\pi\)
−0.426025 + 0.904711i \(0.640087\pi\)
\(684\) 0 0
\(685\) 10.7024 4.81224i 0.408916 0.183866i
\(686\) 0 0
\(687\) 0.383426 1.76654i 0.0146286 0.0673979i
\(688\) 0 0
\(689\) 48.8803 1.86219
\(690\) 0 0
\(691\) 27.7696i 1.05640i −0.849119 0.528202i \(-0.822866\pi\)
0.849119 0.528202i \(-0.177134\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13.0986 + 29.1312i 0.496858 + 1.10501i
\(696\) 0 0
\(697\) 19.6788i 0.745388i
\(698\) 0 0
\(699\) 48.9576 + 10.6262i 1.85175 + 0.401919i
\(700\) 0 0
\(701\) 0.789291i 0.0298111i 0.999889 + 0.0149056i \(0.00474476\pi\)
−0.999889 + 0.0149056i \(0.995255\pi\)
\(702\) 0 0
\(703\) 8.16842 0.308078
\(704\) 0 0
\(705\) −1.47252 6.94872i −0.0554582 0.261704i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 15.8686 0.595960 0.297980 0.954572i \(-0.403687\pi\)
0.297980 + 0.954572i \(0.403687\pi\)
\(710\) 0 0
\(711\) 27.5860 + 12.5671i 1.03456 + 0.471302i
\(712\) 0 0
\(713\) 32.2383i 1.20733i
\(714\) 0 0
\(715\) −19.0806 42.4351i −0.713574 1.58698i
\(716\) 0 0
\(717\) −2.09311 + 9.64349i −0.0781686 + 0.360143i
\(718\) 0 0
\(719\) −45.2605 −1.68793 −0.843967 0.536396i \(-0.819785\pi\)
−0.843967 + 0.536396i \(0.819785\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −2.59028 + 11.9341i −0.0963336 + 0.443834i
\(724\) 0 0
\(725\) 25.6116 + 22.7220i 0.951192 + 0.843873i
\(726\) 0 0
\(727\) −21.4724 −0.796369 −0.398185 0.917305i \(-0.630360\pi\)
−0.398185 + 0.917305i \(0.630360\pi\)
\(728\) 0 0
\(729\) 7.67916 + 25.8849i 0.284413 + 0.958702i
\(730\) 0 0
\(731\) −12.0899 −0.447162
\(732\) 0 0
\(733\) −14.7624 −0.545263 −0.272631 0.962119i \(-0.587894\pi\)
−0.272631 + 0.962119i \(0.587894\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 60.0088 2.21045
\(738\) 0 0
\(739\) −2.44131 −0.0898049 −0.0449024 0.998991i \(-0.514298\pi\)
−0.0449024 + 0.998991i \(0.514298\pi\)
\(740\) 0 0
\(741\) 5.38258 24.7989i 0.197734 0.911011i
\(742\) 0 0
\(743\) 32.2677 1.18379 0.591894 0.806016i \(-0.298380\pi\)
0.591894 + 0.806016i \(0.298380\pi\)
\(744\) 0 0
\(745\) 13.8458 + 30.7930i 0.507272 + 1.12817i
\(746\) 0 0
\(747\) −6.60708 + 14.5032i −0.241741 + 0.530646i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 14.0168 0.511480 0.255740 0.966746i \(-0.417681\pi\)
0.255740 + 0.966746i \(0.417681\pi\)
\(752\) 0 0
\(753\) 6.65608 + 1.44470i 0.242561 + 0.0526476i
\(754\) 0 0
\(755\) 17.0077 7.64737i 0.618972 0.278316i
\(756\) 0 0
\(757\) 41.4500i 1.50653i −0.657720 0.753263i \(-0.728479\pi\)
0.657720 0.753263i \(-0.271521\pi\)
\(758\) 0 0
\(759\) −15.4436 + 71.1527i −0.560568 + 2.58268i
\(760\) 0 0
\(761\) 40.1172 1.45425 0.727124 0.686506i \(-0.240856\pi\)
0.727124 + 0.686506i \(0.240856\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.102388 20.8497i 0.00370185 0.753821i
\(766\) 0 0
\(767\) 16.4672 0.594595
\(768\) 0 0
\(769\) 22.4441i 0.809355i −0.914459 0.404677i \(-0.867384\pi\)
0.914459 0.404677i \(-0.132616\pi\)
\(770\) 0 0
\(771\) 5.62310 25.9071i 0.202511 0.933021i
\(772\) 0 0
\(773\) 29.0928i 1.04640i −0.852211 0.523198i \(-0.824739\pi\)
0.852211 0.523198i \(-0.175261\pi\)
\(774\) 0 0
\(775\) 16.8601 + 14.9578i 0.605632 + 0.537301i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 26.2036i 0.938841i
\(780\) 0 0
\(781\) 6.97605 0.249623
\(782\) 0 0
\(783\) −21.2832 + 28.5140i −0.760600 + 1.01901i
\(784\) 0 0
\(785\) −1.95903 + 0.880863i −0.0699208 + 0.0314394i
\(786\) 0 0
\(787\) −33.8923 −1.20813 −0.604064 0.796936i \(-0.706453\pi\)
−0.604064 + 0.796936i \(0.706453\pi\)
\(788\) 0 0
\(789\) 33.0396 + 7.17120i 1.17624 + 0.255301i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.87015i 0.101922i
\(794\) 0 0
\(795\) 52.3157 11.0863i 1.85545 0.393192i
\(796\) 0 0
\(797\) 30.8362i 1.09227i 0.837696 + 0.546137i \(0.183902\pi\)
−0.837696 + 0.546137i \(0.816098\pi\)
\(798\) 0 0
\(799\) −5.70028 −0.201661
\(800\) 0 0
\(801\) 34.3433 + 15.6454i 1.21346 + 0.552804i
\(802\) 0 0
\(803\) 12.7575i 0.450202i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −16.6510 3.61408i −0.586143 0.127222i
\(808\) 0 0
\(809\) 37.1049i 1.30454i 0.757988 + 0.652269i \(0.226183\pi\)
−0.757988 + 0.652269i \(0.773817\pi\)
\(810\) 0 0
\(811\) 33.9956i 1.19375i 0.802336 + 0.596873i \(0.203590\pi\)
−0.802336 + 0.596873i \(0.796410\pi\)
\(812\) 0 0
\(813\) −7.67076 + 35.3412i −0.269025 + 1.23947i
\(814\) 0 0
\(815\) −17.1215 38.0781i −0.599741 1.33382i
\(816\) 0 0
\(817\) −16.0985 −0.563216
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 46.2531i 1.61424i −0.590385 0.807122i \(-0.701024\pi\)
0.590385 0.807122i \(-0.298976\pi\)
\(822\) 0 0
\(823\) 47.0648i 1.64058i −0.571951 0.820288i \(-0.693813\pi\)
0.571951 0.820288i \(-0.306187\pi\)
\(824\) 0 0
\(825\) −30.0462 41.0899i −1.04607 1.43057i
\(826\) 0 0
\(827\) −14.6102 −0.508046 −0.254023 0.967198i \(-0.581754\pi\)
−0.254023 + 0.967198i \(0.581754\pi\)
\(828\) 0 0
\(829\) 32.3173i 1.12243i −0.827672 0.561213i \(-0.810335\pi\)
0.827672 0.561213i \(-0.189665\pi\)
\(830\) 0 0
\(831\) −3.12415 + 14.3938i −0.108376 + 0.499315i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −13.0408 29.0025i −0.451294 1.00367i
\(836\) 0 0
\(837\) −14.0107 + 18.7707i −0.484280 + 0.648810i
\(838\) 0 0
\(839\) −15.0113 −0.518249 −0.259124 0.965844i \(-0.583434\pi\)
−0.259124 + 0.965844i \(0.583434\pi\)
\(840\) 0 0
\(841\) −17.8898 −0.616888
\(842\) 0 0
\(843\) 2.80569 12.9265i 0.0966329 0.445213i
\(844\) 0 0
\(845\) 0.954631 0.429242i 0.0328403 0.0147664i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −15.2464 3.30922i −0.523257 0.113572i
\(850\) 0 0
\(851\) 14.1153i 0.483865i
\(852\) 0 0
\(853\) −43.2807 −1.48190 −0.740952 0.671558i \(-0.765626\pi\)
−0.740952 + 0.671558i \(0.765626\pi\)
\(854\) 0 0
\(855\) 0.136336 27.7626i 0.00466260 0.949463i
\(856\) 0 0
\(857\) 21.1604i 0.722827i 0.932406 + 0.361413i \(0.117706\pi\)
−0.932406 + 0.361413i \(0.882294\pi\)
\(858\) 0 0
\(859\) 42.2472i 1.44146i 0.693218 + 0.720728i \(0.256192\pi\)
−0.693218 + 0.720728i \(0.743808\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −35.6914 −1.21495 −0.607475 0.794339i \(-0.707817\pi\)
−0.607475 + 0.794339i \(0.707817\pi\)
\(864\) 0 0
\(865\) 0.654976 + 1.45666i 0.0222698 + 0.0495279i
\(866\) 0 0
\(867\) 12.4233 + 2.69645i 0.421916 + 0.0915764i
\(868\) 0 0
\(869\) 59.3927i 2.01476i
\(870\) 0 0
\(871\) 36.1416i 1.22461i
\(872\) 0 0
\(873\) −12.3040 5.60519i −0.416426 0.189707i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.19992i 0.0742860i 0.999310 + 0.0371430i \(0.0118257\pi\)
−0.999310 + 0.0371430i \(0.988174\pi\)
\(878\) 0 0
\(879\) −10.1436 + 46.7341i −0.342135 + 1.57630i
\(880\) 0 0
\(881\) 29.2075 0.984025 0.492013 0.870588i \(-0.336261\pi\)
0.492013 + 0.870588i \(0.336261\pi\)
\(882\) 0 0
\(883\) 2.21069i 0.0743956i −0.999308 0.0371978i \(-0.988157\pi\)
0.999308 0.0371978i \(-0.0118432\pi\)
\(884\) 0 0
\(885\) 17.6245 3.73485i 0.592441 0.125546i
\(886\) 0 0
\(887\) 13.3627i 0.448675i 0.974512 + 0.224337i \(0.0720217\pi\)
−0.974512 + 0.224337i \(0.927978\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 39.9148 34.7168i 1.33720 1.16306i
\(892\) 0 0
\(893\) −7.59028 −0.253999
\(894\) 0 0
\(895\) 10.2551 + 22.8073i 0.342791 + 0.762364i
\(896\) 0 0
\(897\) 42.8533 + 9.30126i 1.43083 + 0.310560i
\(898\) 0 0
\(899\) −30.8674 −1.02949
\(900\) 0 0
\(901\) 42.9164i 1.42975i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.39014 7.53963i −0.112692 0.250626i
\(906\) 0 0
\(907\) 4.18890i 0.139090i 0.997579 + 0.0695451i \(0.0221548\pi\)
−0.997579 + 0.0695451i \(0.977845\pi\)
\(908\) 0 0
\(909\) −4.89658 2.23068i −0.162409 0.0739870i
\(910\) 0 0
\(911\) 22.5190i 0.746087i −0.927814 0.373043i \(-0.878314\pi\)
0.927814 0.373043i \(-0.121686\pi\)
\(912\) 0 0
\(913\) 31.2255 1.03341
\(914\) 0 0
\(915\) −0.650966 3.07187i −0.0215203 0.101553i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 14.5291 0.479270 0.239635 0.970863i \(-0.422972\pi\)
0.239635 + 0.970863i \(0.422972\pi\)
\(920\) 0 0
\(921\) 16.9198 + 3.67242i 0.557526 + 0.121010i
\(922\) 0 0
\(923\) 4.20148i 0.138293i
\(924\) 0 0
\(925\) −7.38204 6.54915i −0.242720 0.215335i
\(926\) 0 0
\(927\) −34.4363 15.6878i −1.13104 0.515254i
\(928\) 0 0
\(929\) 41.5577 1.36346 0.681732 0.731602i \(-0.261227\pi\)
0.681732 + 0.731602i \(0.261227\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 40.3986 + 8.76847i 1.32259 + 0.287067i
\(934\) 0 0
\(935\) −37.2576 + 16.7526i −1.21845 + 0.547868i
\(936\) 0 0
\(937\) 45.5010 1.48645 0.743226 0.669040i \(-0.233295\pi\)
0.743226 + 0.669040i \(0.233295\pi\)
\(938\) 0 0
\(939\) 9.47064 + 2.05559i 0.309063 + 0.0670817i
\(940\) 0 0
\(941\) 20.0487 0.653569 0.326785 0.945099i \(-0.394035\pi\)
0.326785 + 0.945099i \(0.394035\pi\)
\(942\) 0 0
\(943\) 45.2806 1.47454
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −29.4980 −0.958555 −0.479278 0.877663i \(-0.659101\pi\)
−0.479278 + 0.877663i \(0.659101\pi\)
\(948\) 0 0
\(949\) 7.68347 0.249416
\(950\) 0 0
\(951\) 19.8658 + 4.31184i 0.644192 + 0.139821i
\(952\) 0 0
\(953\) −1.26250 −0.0408964 −0.0204482 0.999791i \(-0.506509\pi\)
−0.0204482 + 0.999791i \(0.506509\pi\)
\(954\) 0 0
\(955\) 9.52602 + 21.1858i 0.308255 + 0.685555i
\(956\) 0 0
\(957\) 68.1270 + 14.7869i 2.20223 + 0.477992i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 10.6801 0.344518
\(962\) 0 0
\(963\) −9.49927 4.32748i −0.306110 0.139451i
\(964\) 0 0
\(965\) 5.42959 + 12.0754i 0.174785 + 0.388719i
\(966\) 0 0
\(967\) 26.0942i 0.839132i 0.907725 + 0.419566i \(0.137818\pi\)
−0.907725 + 0.419566i \(0.862182\pi\)
\(968\) 0 0
\(969\) −21.7732 4.72585i −0.699456 0.151816i
\(970\) 0 0
\(971\) −13.9063 −0.446276 −0.223138 0.974787i \(-0.571630\pi\)
−0.223138 + 0.974787i \(0.571630\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −24.7473 + 18.0960i −0.792548 + 0.579535i
\(976\) 0 0
\(977\) 2.83810 0.0907990 0.0453995 0.998969i \(-0.485544\pi\)
0.0453995 + 0.998969i \(0.485544\pi\)
\(978\) 0 0
\(979\) 73.9413i 2.36317i
\(980\) 0 0
\(981\) −27.2352 12.4072i −0.869552 0.396132i
\(982\) 0 0
\(983\) 49.1410i 1.56735i 0.621169 + 0.783677i \(0.286658\pi\)
−0.621169 + 0.783677i \(0.713342\pi\)
\(984\) 0 0
\(985\) 22.6481 10.1835i 0.721628 0.324475i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 27.8187i 0.884584i
\(990\) 0 0
\(991\) 20.6561 0.656163 0.328081 0.944649i \(-0.393598\pi\)
0.328081 + 0.944649i \(0.393598\pi\)
\(992\) 0 0
\(993\) 4.28080 + 0.929142i 0.135847 + 0.0294854i
\(994\) 0 0
\(995\) 12.1416 + 27.0027i 0.384914 + 0.856044i
\(996\) 0 0
\(997\) 41.6906 1.32035 0.660177 0.751110i \(-0.270481\pi\)
0.660177 + 0.751110i \(0.270481\pi\)
\(998\) 0 0
\(999\) 6.13446 8.21859i 0.194086 0.260025i
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 2940.2.f.a.1469.31 32
3.2 odd 2 inner 2940.2.f.a.1469.30 32
5.4 even 2 inner 2940.2.f.a.1469.1 32
7.2 even 3 420.2.bn.a.269.5 yes 32
7.3 odd 6 420.2.bn.a.89.10 yes 32
7.6 odd 2 inner 2940.2.f.a.1469.2 32
15.14 odd 2 inner 2940.2.f.a.1469.4 32
21.2 odd 6 420.2.bn.a.269.7 yes 32
21.17 even 6 420.2.bn.a.89.12 yes 32
21.20 even 2 inner 2940.2.f.a.1469.3 32
35.2 odd 12 2100.2.bi.n.101.13 32
35.3 even 12 2100.2.bi.n.1601.2 32
35.9 even 6 420.2.bn.a.269.12 yes 32
35.17 even 12 2100.2.bi.n.1601.15 32
35.23 odd 12 2100.2.bi.n.101.4 32
35.24 odd 6 420.2.bn.a.89.7 yes 32
35.34 odd 2 inner 2940.2.f.a.1469.32 32
105.2 even 12 2100.2.bi.n.101.15 32
105.17 odd 12 2100.2.bi.n.1601.13 32
105.23 even 12 2100.2.bi.n.101.2 32
105.38 odd 12 2100.2.bi.n.1601.4 32
105.44 odd 6 420.2.bn.a.269.10 yes 32
105.59 even 6 420.2.bn.a.89.5 32
105.104 even 2 inner 2940.2.f.a.1469.29 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bn.a.89.5 32 105.59 even 6
420.2.bn.a.89.7 yes 32 35.24 odd 6
420.2.bn.a.89.10 yes 32 7.3 odd 6
420.2.bn.a.89.12 yes 32 21.17 even 6
420.2.bn.a.269.5 yes 32 7.2 even 3
420.2.bn.a.269.7 yes 32 21.2 odd 6
420.2.bn.a.269.10 yes 32 105.44 odd 6
420.2.bn.a.269.12 yes 32 35.9 even 6
2100.2.bi.n.101.2 32 105.23 even 12
2100.2.bi.n.101.4 32 35.23 odd 12
2100.2.bi.n.101.13 32 35.2 odd 12
2100.2.bi.n.101.15 32 105.2 even 12
2100.2.bi.n.1601.2 32 35.3 even 12
2100.2.bi.n.1601.4 32 105.38 odd 12
2100.2.bi.n.1601.13 32 105.17 odd 12
2100.2.bi.n.1601.15 32 35.17 even 12
2940.2.f.a.1469.1 32 5.4 even 2 inner
2940.2.f.a.1469.2 32 7.6 odd 2 inner
2940.2.f.a.1469.3 32 21.20 even 2 inner
2940.2.f.a.1469.4 32 15.14 odd 2 inner
2940.2.f.a.1469.29 32 105.104 even 2 inner
2940.2.f.a.1469.30 32 3.2 odd 2 inner
2940.2.f.a.1469.31 32 1.1 even 1 trivial
2940.2.f.a.1469.32 32 35.34 odd 2 inner