Properties

Label 380.3.g.c.189.10
Level $380$
Weight $3$
Character 380.189
Analytic conductor $10.354$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [380,3,Mod(189,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.189");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.g (of order \(2\), degree \(1\), 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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 20x^{10} + 44x^{8} - 270x^{6} + 36676x^{4} - 71664x^{2} + 687241 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{19} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 189.10
Root \(-1.63363 - 1.34185i\) of defining polynomial
Character \(\chi\) \(=\) 380.189
Dual form 380.3.g.c.189.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.26725 q^{3} +(4.26535 + 2.60898i) q^{5} -6.30368i q^{7} +1.67495 q^{9} +O(q^{10})\) \(q+3.26725 q^{3} +(4.26535 + 2.60898i) q^{5} -6.30368i q^{7} +1.67495 q^{9} +4.53070 q^{11} +11.5357 q^{13} +(13.9360 + 8.52420i) q^{15} +1.08572i q^{17} +(16.8558 - 8.76832i) q^{19} -20.5957i q^{21} -31.5184i q^{23} +(11.3865 + 22.2564i) q^{25} -23.9328 q^{27} +38.1324i q^{29} +58.2398i q^{31} +14.8030 q^{33} +(16.4462 - 26.8874i) q^{35} -40.2691 q^{37} +37.6901 q^{39} -18.0249i q^{41} -2.74764i q^{43} +(7.14424 + 4.36990i) q^{45} -76.1195i q^{47} +9.26364 q^{49} +3.54733i q^{51} +8.06781 q^{53} +(19.3251 + 11.8205i) q^{55} +(55.0720 - 28.6483i) q^{57} +95.3957i q^{59} -4.28507 q^{61} -10.5583i q^{63} +(49.2039 + 30.0964i) q^{65} -70.7364 q^{67} -102.979i q^{69} -93.3131i q^{71} -14.2908i q^{73} +(37.2025 + 72.7174i) q^{75} -28.5601i q^{77} -6.11815i q^{79} -93.2691 q^{81} +89.0257i q^{83} +(-2.83262 + 4.63098i) q^{85} +124.588i q^{87} -92.3366i q^{89} -72.7174i q^{91} +190.284i q^{93} +(94.7721 + 6.57633i) q^{95} -64.6843 q^{97} +7.58869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{5} + 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{5} + 76 q^{9} - 24 q^{11} + 68 q^{19} - 76 q^{25} - 32 q^{35} - 264 q^{39} + 220 q^{45} + 212 q^{49} + 176 q^{55} - 600 q^{61} + 492 q^{81} + 408 q^{85} + 124 q^{95} + 136 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 3.26725 1.08908 0.544542 0.838733i \(-0.316703\pi\)
0.544542 + 0.838733i \(0.316703\pi\)
\(4\) 0 0
\(5\) 4.26535 + 2.60898i 0.853070 + 0.521796i
\(6\) 0 0
\(7\) 6.30368i 0.900525i −0.892896 0.450263i \(-0.851330\pi\)
0.892896 0.450263i \(-0.148670\pi\)
\(8\) 0 0
\(9\) 1.67495 0.186105
\(10\) 0 0
\(11\) 4.53070 0.411882 0.205941 0.978564i \(-0.433974\pi\)
0.205941 + 0.978564i \(0.433974\pi\)
\(12\) 0 0
\(13\) 11.5357 0.887362 0.443681 0.896185i \(-0.353672\pi\)
0.443681 + 0.896185i \(0.353672\pi\)
\(14\) 0 0
\(15\) 13.9360 + 8.52420i 0.929066 + 0.568280i
\(16\) 0 0
\(17\) 1.08572i 0.0638659i 0.999490 + 0.0319330i \(0.0101663\pi\)
−0.999490 + 0.0319330i \(0.989834\pi\)
\(18\) 0 0
\(19\) 16.8558 8.76832i 0.887145 0.461491i
\(20\) 0 0
\(21\) 20.5957i 0.980748i
\(22\) 0 0
\(23\) 31.5184i 1.37036i −0.728372 0.685182i \(-0.759722\pi\)
0.728372 0.685182i \(-0.240278\pi\)
\(24\) 0 0
\(25\) 11.3865 + 22.2564i 0.455458 + 0.890257i
\(26\) 0 0
\(27\) −23.9328 −0.886400
\(28\) 0 0
\(29\) 38.1324i 1.31491i 0.753494 + 0.657454i \(0.228367\pi\)
−0.753494 + 0.657454i \(0.771633\pi\)
\(30\) 0 0
\(31\) 58.2398i 1.87870i 0.342955 + 0.939352i \(0.388572\pi\)
−0.342955 + 0.939352i \(0.611428\pi\)
\(32\) 0 0
\(33\) 14.8030 0.448575
\(34\) 0 0
\(35\) 16.4462 26.8874i 0.469890 0.768212i
\(36\) 0 0
\(37\) −40.2691 −1.08835 −0.544177 0.838971i \(-0.683158\pi\)
−0.544177 + 0.838971i \(0.683158\pi\)
\(38\) 0 0
\(39\) 37.6901 0.966413
\(40\) 0 0
\(41\) 18.0249i 0.439632i −0.975541 0.219816i \(-0.929454\pi\)
0.975541 0.219816i \(-0.0705456\pi\)
\(42\) 0 0
\(43\) 2.74764i 0.0638986i −0.999489 0.0319493i \(-0.989828\pi\)
0.999489 0.0319493i \(-0.0101715\pi\)
\(44\) 0 0
\(45\) 7.14424 + 4.36990i 0.158761 + 0.0971090i
\(46\) 0 0
\(47\) 76.1195i 1.61956i −0.586731 0.809782i \(-0.699586\pi\)
0.586731 0.809782i \(-0.300414\pi\)
\(48\) 0 0
\(49\) 9.26364 0.189054
\(50\) 0 0
\(51\) 3.54733i 0.0695554i
\(52\) 0 0
\(53\) 8.06781 0.152223 0.0761115 0.997099i \(-0.475750\pi\)
0.0761115 + 0.997099i \(0.475750\pi\)
\(54\) 0 0
\(55\) 19.3251 + 11.8205i 0.351365 + 0.214918i
\(56\) 0 0
\(57\) 55.0720 28.6483i 0.966176 0.502602i
\(58\) 0 0
\(59\) 95.3957i 1.61688i 0.588581 + 0.808438i \(0.299687\pi\)
−0.588581 + 0.808438i \(0.700313\pi\)
\(60\) 0 0
\(61\) −4.28507 −0.0702470 −0.0351235 0.999383i \(-0.511182\pi\)
−0.0351235 + 0.999383i \(0.511182\pi\)
\(62\) 0 0
\(63\) 10.5583i 0.167593i
\(64\) 0 0
\(65\) 49.2039 + 30.0964i 0.756982 + 0.463022i
\(66\) 0 0
\(67\) −70.7364 −1.05577 −0.527883 0.849317i \(-0.677014\pi\)
−0.527883 + 0.849317i \(0.677014\pi\)
\(68\) 0 0
\(69\) 102.979i 1.49244i
\(70\) 0 0
\(71\) 93.3131i 1.31427i −0.753773 0.657135i \(-0.771768\pi\)
0.753773 0.657135i \(-0.228232\pi\)
\(72\) 0 0
\(73\) 14.2908i 0.195765i −0.995198 0.0978824i \(-0.968793\pi\)
0.995198 0.0978824i \(-0.0312069\pi\)
\(74\) 0 0
\(75\) 37.2025 + 72.7174i 0.496033 + 0.969565i
\(76\) 0 0
\(77\) 28.5601i 0.370910i
\(78\) 0 0
\(79\) 6.11815i 0.0774449i −0.999250 0.0387224i \(-0.987671\pi\)
0.999250 0.0387224i \(-0.0123288\pi\)
\(80\) 0 0
\(81\) −93.2691 −1.15147
\(82\) 0 0
\(83\) 89.0257i 1.07260i 0.844028 + 0.536299i \(0.180178\pi\)
−0.844028 + 0.536299i \(0.819822\pi\)
\(84\) 0 0
\(85\) −2.83262 + 4.63098i −0.0333250 + 0.0544821i
\(86\) 0 0
\(87\) 124.588i 1.43205i
\(88\) 0 0
\(89\) 92.3366i 1.03749i −0.854929 0.518745i \(-0.826399\pi\)
0.854929 0.518745i \(-0.173601\pi\)
\(90\) 0 0
\(91\) 72.7174i 0.799092i
\(92\) 0 0
\(93\) 190.284i 2.04607i
\(94\) 0 0
\(95\) 94.7721 + 6.57633i 0.997601 + 0.0692245i
\(96\) 0 0
\(97\) −64.6843 −0.666849 −0.333424 0.942777i \(-0.608204\pi\)
−0.333424 + 0.942777i \(0.608204\pi\)
\(98\) 0 0
\(99\) 7.58869 0.0766535
\(100\) 0 0
\(101\) −155.672 −1.54131 −0.770654 0.637254i \(-0.780070\pi\)
−0.770654 + 0.637254i \(0.780070\pi\)
\(102\) 0 0
\(103\) 130.821 1.27010 0.635052 0.772469i \(-0.280978\pi\)
0.635052 + 0.772469i \(0.280978\pi\)
\(104\) 0 0
\(105\) 53.7338 87.8480i 0.511750 0.836648i
\(106\) 0 0
\(107\) −95.1516 −0.889268 −0.444634 0.895712i \(-0.646666\pi\)
−0.444634 + 0.895712i \(0.646666\pi\)
\(108\) 0 0
\(109\) 155.589i 1.42742i 0.700443 + 0.713709i \(0.252986\pi\)
−0.700443 + 0.713709i \(0.747014\pi\)
\(110\) 0 0
\(111\) −131.569 −1.18531
\(112\) 0 0
\(113\) −174.840 −1.54725 −0.773626 0.633642i \(-0.781559\pi\)
−0.773626 + 0.633642i \(0.781559\pi\)
\(114\) 0 0
\(115\) 82.2308 134.437i 0.715050 1.16902i
\(116\) 0 0
\(117\) 19.3217 0.165143
\(118\) 0 0
\(119\) 6.84404 0.0575129
\(120\) 0 0
\(121\) −100.473 −0.830353
\(122\) 0 0
\(123\) 58.8919i 0.478796i
\(124\) 0 0
\(125\) −9.49927 + 124.639i −0.0759941 + 0.997108i
\(126\) 0 0
\(127\) 80.8200 0.636378 0.318189 0.948027i \(-0.396925\pi\)
0.318189 + 0.948027i \(0.396925\pi\)
\(128\) 0 0
\(129\) 8.97723i 0.0695910i
\(130\) 0 0
\(131\) −129.957 −0.992039 −0.496020 0.868311i \(-0.665206\pi\)
−0.496020 + 0.868311i \(0.665206\pi\)
\(132\) 0 0
\(133\) −55.2727 106.253i −0.415584 0.798897i
\(134\) 0 0
\(135\) −102.082 62.4402i −0.756162 0.462520i
\(136\) 0 0
\(137\) 58.8037i 0.429224i −0.976699 0.214612i \(-0.931151\pi\)
0.976699 0.214612i \(-0.0688487\pi\)
\(138\) 0 0
\(139\) 2.48444 0.0178736 0.00893682 0.999960i \(-0.497155\pi\)
0.00893682 + 0.999960i \(0.497155\pi\)
\(140\) 0 0
\(141\) 248.702i 1.76384i
\(142\) 0 0
\(143\) 52.2649 0.365489
\(144\) 0 0
\(145\) −99.4865 + 162.648i −0.686114 + 1.12171i
\(146\) 0 0
\(147\) 30.2667 0.205896
\(148\) 0 0
\(149\) −90.8294 −0.609594 −0.304797 0.952417i \(-0.598589\pi\)
−0.304797 + 0.952417i \(0.598589\pi\)
\(150\) 0 0
\(151\) 44.1209i 0.292192i −0.989270 0.146096i \(-0.953329\pi\)
0.989270 0.146096i \(-0.0466708\pi\)
\(152\) 0 0
\(153\) 1.81853i 0.0118858i
\(154\) 0 0
\(155\) −151.946 + 248.413i −0.980300 + 1.60267i
\(156\) 0 0
\(157\) 9.36283i 0.0596358i −0.999555 0.0298179i \(-0.990507\pi\)
0.999555 0.0298179i \(-0.00949275\pi\)
\(158\) 0 0
\(159\) 26.3596 0.165784
\(160\) 0 0
\(161\) −198.682 −1.23405
\(162\) 0 0
\(163\) 35.8739i 0.220085i −0.993927 0.110043i \(-0.964901\pi\)
0.993927 0.110043i \(-0.0350987\pi\)
\(164\) 0 0
\(165\) 63.1399 + 38.6206i 0.382666 + 0.234064i
\(166\) 0 0
\(167\) 195.223 1.16900 0.584501 0.811393i \(-0.301290\pi\)
0.584501 + 0.811393i \(0.301290\pi\)
\(168\) 0 0
\(169\) −35.9274 −0.212588
\(170\) 0 0
\(171\) 28.2325 14.6865i 0.165102 0.0858859i
\(172\) 0 0
\(173\) −236.304 −1.36592 −0.682960 0.730455i \(-0.739308\pi\)
−0.682960 + 0.730455i \(0.739308\pi\)
\(174\) 0 0
\(175\) 140.297 71.7766i 0.801699 0.410152i
\(176\) 0 0
\(177\) 311.682i 1.76091i
\(178\) 0 0
\(179\) 125.945i 0.703604i −0.936074 0.351802i \(-0.885569\pi\)
0.936074 0.351802i \(-0.114431\pi\)
\(180\) 0 0
\(181\) 11.4185i 0.0630856i −0.999502 0.0315428i \(-0.989958\pi\)
0.999502 0.0315428i \(-0.0100421\pi\)
\(182\) 0 0
\(183\) −14.0004 −0.0765049
\(184\) 0 0
\(185\) −171.762 105.061i −0.928442 0.567898i
\(186\) 0 0
\(187\) 4.91908i 0.0263052i
\(188\) 0 0
\(189\) 150.865i 0.798226i
\(190\) 0 0
\(191\) 36.3733 0.190436 0.0952180 0.995456i \(-0.469645\pi\)
0.0952180 + 0.995456i \(0.469645\pi\)
\(192\) 0 0
\(193\) −150.814 −0.781422 −0.390711 0.920513i \(-0.627771\pi\)
−0.390711 + 0.920513i \(0.627771\pi\)
\(194\) 0 0
\(195\) 160.762 + 98.3326i 0.824418 + 0.504270i
\(196\) 0 0
\(197\) 257.585i 1.30754i 0.756695 + 0.653768i \(0.226813\pi\)
−0.756695 + 0.653768i \(0.773187\pi\)
\(198\) 0 0
\(199\) 128.758 0.647027 0.323513 0.946224i \(-0.395136\pi\)
0.323513 + 0.946224i \(0.395136\pi\)
\(200\) 0 0
\(201\) −231.114 −1.14982
\(202\) 0 0
\(203\) 240.374 1.18411
\(204\) 0 0
\(205\) 47.0266 76.8825i 0.229398 0.375037i
\(206\) 0 0
\(207\) 52.7917i 0.255032i
\(208\) 0 0
\(209\) 76.3685 39.7267i 0.365399 0.190080i
\(210\) 0 0
\(211\) 161.636i 0.766048i 0.923738 + 0.383024i \(0.125117\pi\)
−0.923738 + 0.383024i \(0.874883\pi\)
\(212\) 0 0
\(213\) 304.878i 1.43135i
\(214\) 0 0
\(215\) 7.16853 11.7196i 0.0333420 0.0545100i
\(216\) 0 0
\(217\) 367.125 1.69182
\(218\) 0 0
\(219\) 46.6918i 0.213204i
\(220\) 0 0
\(221\) 12.5246i 0.0566722i
\(222\) 0 0
\(223\) 386.480 1.73310 0.866548 0.499094i \(-0.166334\pi\)
0.866548 + 0.499094i \(0.166334\pi\)
\(224\) 0 0
\(225\) 19.0717 + 37.2784i 0.0847632 + 0.165682i
\(226\) 0 0
\(227\) −52.4767 −0.231175 −0.115587 0.993297i \(-0.536875\pi\)
−0.115587 + 0.993297i \(0.536875\pi\)
\(228\) 0 0
\(229\) 394.025 1.72063 0.860316 0.509762i \(-0.170266\pi\)
0.860316 + 0.509762i \(0.170266\pi\)
\(230\) 0 0
\(231\) 93.3131i 0.403953i
\(232\) 0 0
\(233\) 22.2691i 0.0955754i −0.998858 0.0477877i \(-0.984783\pi\)
0.998858 0.0477877i \(-0.0152171\pi\)
\(234\) 0 0
\(235\) 198.594 324.676i 0.845081 1.38160i
\(236\) 0 0
\(237\) 19.9895i 0.0843440i
\(238\) 0 0
\(239\) 83.5976 0.349781 0.174890 0.984588i \(-0.444043\pi\)
0.174890 + 0.984588i \(0.444043\pi\)
\(240\) 0 0
\(241\) 214.605i 0.890477i −0.895412 0.445238i \(-0.853119\pi\)
0.895412 0.445238i \(-0.146881\pi\)
\(242\) 0 0
\(243\) −89.3385 −0.367648
\(244\) 0 0
\(245\) 39.5127 + 24.1686i 0.161276 + 0.0986475i
\(246\) 0 0
\(247\) 194.443 101.149i 0.787219 0.409509i
\(248\) 0 0
\(249\) 290.870i 1.16815i
\(250\) 0 0
\(251\) −24.6770 −0.0983146 −0.0491573 0.998791i \(-0.515654\pi\)
−0.0491573 + 0.998791i \(0.515654\pi\)
\(252\) 0 0
\(253\) 142.801i 0.564429i
\(254\) 0 0
\(255\) −9.25490 + 15.1306i −0.0362937 + 0.0593357i
\(256\) 0 0
\(257\) 152.548 0.593573 0.296787 0.954944i \(-0.404085\pi\)
0.296787 + 0.954944i \(0.404085\pi\)
\(258\) 0 0
\(259\) 253.843i 0.980090i
\(260\) 0 0
\(261\) 63.8697i 0.244712i
\(262\) 0 0
\(263\) 345.570i 1.31395i 0.753911 + 0.656977i \(0.228165\pi\)
−0.753911 + 0.656977i \(0.771835\pi\)
\(264\) 0 0
\(265\) 34.4121 + 21.0488i 0.129857 + 0.0794293i
\(266\) 0 0
\(267\) 301.687i 1.12991i
\(268\) 0 0
\(269\) 1.39433i 0.00518338i 0.999997 + 0.00259169i \(0.000824962\pi\)
−0.999997 + 0.00259169i \(0.999175\pi\)
\(270\) 0 0
\(271\) 407.864 1.50503 0.752516 0.658574i \(-0.228840\pi\)
0.752516 + 0.658574i \(0.228840\pi\)
\(272\) 0 0
\(273\) 237.586i 0.870279i
\(274\) 0 0
\(275\) 51.5887 + 100.837i 0.187595 + 0.366681i
\(276\) 0 0
\(277\) 330.614i 1.19355i −0.802407 0.596777i \(-0.796448\pi\)
0.802407 0.596777i \(-0.203552\pi\)
\(278\) 0 0
\(279\) 97.5487i 0.349637i
\(280\) 0 0
\(281\) 79.1238i 0.281579i −0.990040 0.140790i \(-0.955036\pi\)
0.990040 0.140790i \(-0.0449641\pi\)
\(282\) 0 0
\(283\) 15.0236i 0.0530870i −0.999648 0.0265435i \(-0.991550\pi\)
0.999648 0.0265435i \(-0.00845006\pi\)
\(284\) 0 0
\(285\) 309.645 + 21.4865i 1.08647 + 0.0753913i
\(286\) 0 0
\(287\) −113.623 −0.395899
\(288\) 0 0
\(289\) 287.821 0.995921
\(290\) 0 0
\(291\) −211.340 −0.726255
\(292\) 0 0
\(293\) 77.7057 0.265207 0.132604 0.991169i \(-0.457666\pi\)
0.132604 + 0.991169i \(0.457666\pi\)
\(294\) 0 0
\(295\) −248.885 + 406.896i −0.843679 + 1.37931i
\(296\) 0 0
\(297\) −108.432 −0.365092
\(298\) 0 0
\(299\) 363.587i 1.21601i
\(300\) 0 0
\(301\) −17.3202 −0.0575423
\(302\) 0 0
\(303\) −508.620 −1.67861
\(304\) 0 0
\(305\) −18.2773 11.1796i −0.0599256 0.0366546i
\(306\) 0 0
\(307\) −212.669 −0.692734 −0.346367 0.938099i \(-0.612585\pi\)
−0.346367 + 0.938099i \(0.612585\pi\)
\(308\) 0 0
\(309\) 427.425 1.38325
\(310\) 0 0
\(311\) 104.314 0.335415 0.167707 0.985837i \(-0.446364\pi\)
0.167707 + 0.985837i \(0.446364\pi\)
\(312\) 0 0
\(313\) 489.532i 1.56400i 0.623279 + 0.781999i \(0.285800\pi\)
−0.623279 + 0.781999i \(0.714200\pi\)
\(314\) 0 0
\(315\) 27.5465 45.0350i 0.0874491 0.142968i
\(316\) 0 0
\(317\) 146.496 0.462133 0.231067 0.972938i \(-0.425778\pi\)
0.231067 + 0.972938i \(0.425778\pi\)
\(318\) 0 0
\(319\) 172.766i 0.541588i
\(320\) 0 0
\(321\) −310.885 −0.968488
\(322\) 0 0
\(323\) 9.51995 + 18.3006i 0.0294735 + 0.0566584i
\(324\) 0 0
\(325\) 131.351 + 256.744i 0.404157 + 0.789980i
\(326\) 0 0
\(327\) 508.347i 1.55458i
\(328\) 0 0
\(329\) −479.833 −1.45846
\(330\) 0 0
\(331\) 319.048i 0.963892i 0.876201 + 0.481946i \(0.160070\pi\)
−0.876201 + 0.481946i \(0.839930\pi\)
\(332\) 0 0
\(333\) −67.4486 −0.202548
\(334\) 0 0
\(335\) −301.716 184.550i −0.900644 0.550895i
\(336\) 0 0
\(337\) −386.447 −1.14673 −0.573363 0.819301i \(-0.694362\pi\)
−0.573363 + 0.819301i \(0.694362\pi\)
\(338\) 0 0
\(339\) −571.245 −1.68509
\(340\) 0 0
\(341\) 263.867i 0.773805i
\(342\) 0 0
\(343\) 367.275i 1.07077i
\(344\) 0 0
\(345\) 268.669 439.240i 0.778750 1.27316i
\(346\) 0 0
\(347\) 624.138i 1.79867i −0.437262 0.899334i \(-0.644052\pi\)
0.437262 0.899334i \(-0.355948\pi\)
\(348\) 0 0
\(349\) 509.597 1.46016 0.730081 0.683360i \(-0.239482\pi\)
0.730081 + 0.683360i \(0.239482\pi\)
\(350\) 0 0
\(351\) −276.082 −0.786558
\(352\) 0 0
\(353\) 65.8979i 0.186680i 0.995634 + 0.0933398i \(0.0297543\pi\)
−0.995634 + 0.0933398i \(0.970246\pi\)
\(354\) 0 0
\(355\) 243.452 398.013i 0.685780 1.12116i
\(356\) 0 0
\(357\) 22.3612 0.0626364
\(358\) 0 0
\(359\) −548.958 −1.52913 −0.764566 0.644546i \(-0.777047\pi\)
−0.764566 + 0.644546i \(0.777047\pi\)
\(360\) 0 0
\(361\) 207.233 295.593i 0.574053 0.818818i
\(362\) 0 0
\(363\) −328.270 −0.904325
\(364\) 0 0
\(365\) 37.2845 60.9554i 0.102149 0.167001i
\(366\) 0 0
\(367\) 234.257i 0.638303i 0.947704 + 0.319151i \(0.103398\pi\)
−0.947704 + 0.319151i \(0.896602\pi\)
\(368\) 0 0
\(369\) 30.1908i 0.0818178i
\(370\) 0 0
\(371\) 50.8569i 0.137081i
\(372\) 0 0
\(373\) 167.444 0.448911 0.224455 0.974484i \(-0.427940\pi\)
0.224455 + 0.974484i \(0.427940\pi\)
\(374\) 0 0
\(375\) −31.0365 + 407.226i −0.0827641 + 1.08594i
\(376\) 0 0
\(377\) 439.884i 1.16680i
\(378\) 0 0
\(379\) 69.3701i 0.183034i 0.995803 + 0.0915172i \(0.0291717\pi\)
−0.995803 + 0.0915172i \(0.970828\pi\)
\(380\) 0 0
\(381\) 264.059 0.693069
\(382\) 0 0
\(383\) 547.099 1.42846 0.714229 0.699912i \(-0.246777\pi\)
0.714229 + 0.699912i \(0.246777\pi\)
\(384\) 0 0
\(385\) 74.5127 121.819i 0.193539 0.316413i
\(386\) 0 0
\(387\) 4.60215i 0.0118919i
\(388\) 0 0
\(389\) 49.2968 0.126727 0.0633635 0.997991i \(-0.479817\pi\)
0.0633635 + 0.997991i \(0.479817\pi\)
\(390\) 0 0
\(391\) 34.2202 0.0875196
\(392\) 0 0
\(393\) −424.603 −1.08041
\(394\) 0 0
\(395\) 15.9621 26.0961i 0.0404104 0.0660660i
\(396\) 0 0
\(397\) 333.006i 0.838805i −0.907800 0.419402i \(-0.862240\pi\)
0.907800 0.419402i \(-0.137760\pi\)
\(398\) 0 0
\(399\) −180.590 347.156i −0.452606 0.870066i
\(400\) 0 0
\(401\) 338.509i 0.844161i −0.906558 0.422081i \(-0.861300\pi\)
0.906558 0.422081i \(-0.138700\pi\)
\(402\) 0 0
\(403\) 671.838i 1.66709i
\(404\) 0 0
\(405\) −397.825 243.337i −0.982285 0.600832i
\(406\) 0 0
\(407\) −182.447 −0.448273
\(408\) 0 0
\(409\) 162.742i 0.397903i −0.980009 0.198951i \(-0.936246\pi\)
0.980009 0.198951i \(-0.0637536\pi\)
\(410\) 0 0
\(411\) 192.127i 0.467461i
\(412\) 0 0
\(413\) 601.344 1.45604
\(414\) 0 0
\(415\) −232.266 + 379.726i −0.559677 + 0.915002i
\(416\) 0 0
\(417\) 8.11728 0.0194659
\(418\) 0 0
\(419\) −240.614 −0.574257 −0.287128 0.957892i \(-0.592701\pi\)
−0.287128 + 0.957892i \(0.592701\pi\)
\(420\) 0 0
\(421\) 753.729i 1.79033i 0.445735 + 0.895165i \(0.352943\pi\)
−0.445735 + 0.895165i \(0.647057\pi\)
\(422\) 0 0
\(423\) 127.496i 0.301409i
\(424\) 0 0
\(425\) −24.1643 + 12.3625i −0.0568571 + 0.0290883i
\(426\) 0 0
\(427\) 27.0117i 0.0632592i
\(428\) 0 0
\(429\) 170.763 0.398048
\(430\) 0 0
\(431\) 198.792i 0.461235i −0.973045 0.230617i \(-0.925925\pi\)
0.973045 0.230617i \(-0.0740745\pi\)
\(432\) 0 0
\(433\) 592.322 1.36795 0.683975 0.729506i \(-0.260250\pi\)
0.683975 + 0.729506i \(0.260250\pi\)
\(434\) 0 0
\(435\) −325.048 + 531.412i −0.747236 + 1.22164i
\(436\) 0 0
\(437\) −276.363 531.266i −0.632410 1.21571i
\(438\) 0 0
\(439\) 837.288i 1.90726i −0.300977 0.953631i \(-0.597313\pi\)
0.300977 0.953631i \(-0.402687\pi\)
\(440\) 0 0
\(441\) 15.5161 0.0351839
\(442\) 0 0
\(443\) 736.917i 1.66347i 0.555174 + 0.831734i \(0.312652\pi\)
−0.555174 + 0.831734i \(0.687348\pi\)
\(444\) 0 0
\(445\) 240.904 393.848i 0.541358 0.885052i
\(446\) 0 0
\(447\) −296.763 −0.663899
\(448\) 0 0
\(449\) 369.746i 0.823489i 0.911299 + 0.411744i \(0.135080\pi\)
−0.911299 + 0.411744i \(0.864920\pi\)
\(450\) 0 0
\(451\) 81.6655i 0.181076i
\(452\) 0 0
\(453\) 144.154i 0.318221i
\(454\) 0 0
\(455\) 189.718 310.165i 0.416963 0.681682i
\(456\) 0 0
\(457\) 799.508i 1.74947i 0.484601 + 0.874735i \(0.338965\pi\)
−0.484601 + 0.874735i \(0.661035\pi\)
\(458\) 0 0
\(459\) 25.9843i 0.0566108i
\(460\) 0 0
\(461\) 721.581 1.56525 0.782625 0.622493i \(-0.213880\pi\)
0.782625 + 0.622493i \(0.213880\pi\)
\(462\) 0 0
\(463\) 850.242i 1.83638i −0.396145 0.918188i \(-0.629652\pi\)
0.396145 0.918188i \(-0.370348\pi\)
\(464\) 0 0
\(465\) −496.448 + 811.630i −1.06763 + 1.74544i
\(466\) 0 0
\(467\) 125.893i 0.269579i −0.990874 0.134789i \(-0.956964\pi\)
0.990874 0.134789i \(-0.0430358\pi\)
\(468\) 0 0
\(469\) 445.899i 0.950745i
\(470\) 0 0
\(471\) 30.5907i 0.0649485i
\(472\) 0 0
\(473\) 12.4487i 0.0263187i
\(474\) 0 0
\(475\) 387.079 + 275.309i 0.814903 + 0.579597i
\(476\) 0 0
\(477\) 13.5132 0.0283295
\(478\) 0 0
\(479\) −549.803 −1.14782 −0.573908 0.818920i \(-0.694573\pi\)
−0.573908 + 0.818920i \(0.694573\pi\)
\(480\) 0 0
\(481\) −464.532 −0.965764
\(482\) 0 0
\(483\) −649.144 −1.34398
\(484\) 0 0
\(485\) −275.901 168.760i −0.568869 0.347959i
\(486\) 0 0
\(487\) −165.606 −0.340054 −0.170027 0.985439i \(-0.554385\pi\)
−0.170027 + 0.985439i \(0.554385\pi\)
\(488\) 0 0
\(489\) 117.209i 0.239692i
\(490\) 0 0
\(491\) −344.284 −0.701190 −0.350595 0.936527i \(-0.614021\pi\)
−0.350595 + 0.936527i \(0.614021\pi\)
\(492\) 0 0
\(493\) −41.4011 −0.0839779
\(494\) 0 0
\(495\) 32.3685 + 19.7987i 0.0653908 + 0.0399975i
\(496\) 0 0
\(497\) −588.216 −1.18353
\(498\) 0 0
\(499\) 136.376 0.273298 0.136649 0.990620i \(-0.456367\pi\)
0.136649 + 0.990620i \(0.456367\pi\)
\(500\) 0 0
\(501\) 637.844 1.27314
\(502\) 0 0
\(503\) 175.677i 0.349258i 0.984634 + 0.174629i \(0.0558726\pi\)
−0.984634 + 0.174629i \(0.944127\pi\)
\(504\) 0 0
\(505\) −663.996 406.145i −1.31484 0.804248i
\(506\) 0 0
\(507\) −117.384 −0.231527
\(508\) 0 0
\(509\) 374.070i 0.734912i −0.930041 0.367456i \(-0.880229\pi\)
0.930041 0.367456i \(-0.119771\pi\)
\(510\) 0 0
\(511\) −90.0848 −0.176291
\(512\) 0 0
\(513\) −403.406 + 209.851i −0.786366 + 0.409065i
\(514\) 0 0
\(515\) 557.997 + 341.309i 1.08349 + 0.662735i
\(516\) 0 0
\(517\) 344.875i 0.667069i
\(518\) 0 0
\(519\) −772.066 −1.48760
\(520\) 0 0
\(521\) 160.601i 0.308255i 0.988051 + 0.154127i \(0.0492566\pi\)
−0.988051 + 0.154127i \(0.950743\pi\)
\(522\) 0 0
\(523\) 671.370 1.28369 0.641845 0.766834i \(-0.278169\pi\)
0.641845 + 0.766834i \(0.278169\pi\)
\(524\) 0 0
\(525\) 458.387 234.512i 0.873118 0.446690i
\(526\) 0 0
\(527\) −63.2322 −0.119985
\(528\) 0 0
\(529\) −464.409 −0.877900
\(530\) 0 0
\(531\) 159.783i 0.300909i
\(532\) 0 0
\(533\) 207.930i 0.390112i
\(534\) 0 0
\(535\) −405.855 248.249i −0.758608 0.464016i
\(536\) 0 0
\(537\) 411.495i 0.766284i
\(538\) 0 0
\(539\) 41.9708 0.0778680
\(540\) 0 0
\(541\) 59.9401 0.110795 0.0553975 0.998464i \(-0.482357\pi\)
0.0553975 + 0.998464i \(0.482357\pi\)
\(542\) 0 0
\(543\) 37.3071i 0.0687056i
\(544\) 0 0
\(545\) −405.927 + 663.640i −0.744820 + 1.21769i
\(546\) 0 0
\(547\) −754.492 −1.37933 −0.689664 0.724130i \(-0.742242\pi\)
−0.689664 + 0.724130i \(0.742242\pi\)
\(548\) 0 0
\(549\) −7.17726 −0.0130733
\(550\) 0 0
\(551\) 334.357 + 642.750i 0.606818 + 1.16652i
\(552\) 0 0
\(553\) −38.5668 −0.0697411
\(554\) 0 0
\(555\) −561.189 343.261i −1.01115 0.618489i
\(556\) 0 0
\(557\) 758.951i 1.36257i 0.732018 + 0.681285i \(0.238579\pi\)
−0.732018 + 0.681285i \(0.761421\pi\)
\(558\) 0 0
\(559\) 31.6960i 0.0567012i
\(560\) 0 0
\(561\) 16.0719i 0.0286486i
\(562\) 0 0
\(563\) 200.922 0.356877 0.178438 0.983951i \(-0.442895\pi\)
0.178438 + 0.983951i \(0.442895\pi\)
\(564\) 0 0
\(565\) −745.752 456.153i −1.31992 0.807350i
\(566\) 0 0
\(567\) 587.938i 1.03693i
\(568\) 0 0
\(569\) 353.921i 0.622006i −0.950409 0.311003i \(-0.899335\pi\)
0.950409 0.311003i \(-0.100665\pi\)
\(570\) 0 0
\(571\) 820.990 1.43781 0.718905 0.695108i \(-0.244643\pi\)
0.718905 + 0.695108i \(0.244643\pi\)
\(572\) 0 0
\(573\) 118.841 0.207401
\(574\) 0 0
\(575\) 701.487 358.883i 1.21998 0.624144i
\(576\) 0 0
\(577\) 549.666i 0.952627i −0.879276 0.476314i \(-0.841973\pi\)
0.879276 0.476314i \(-0.158027\pi\)
\(578\) 0 0
\(579\) −492.749 −0.851035
\(580\) 0 0
\(581\) 561.189 0.965903
\(582\) 0 0
\(583\) 36.5529 0.0626979
\(584\) 0 0
\(585\) 82.4139 + 50.4099i 0.140878 + 0.0861708i
\(586\) 0 0
\(587\) 206.107i 0.351120i 0.984469 + 0.175560i \(0.0561736\pi\)
−0.984469 + 0.175560i \(0.943826\pi\)
\(588\) 0 0
\(589\) 510.666 + 981.676i 0.867004 + 1.66668i
\(590\) 0 0
\(591\) 841.594i 1.42402i
\(592\) 0 0
\(593\) 332.865i 0.561324i −0.959807 0.280662i \(-0.909446\pi\)
0.959807 0.280662i \(-0.0905540\pi\)
\(594\) 0 0
\(595\) 29.1922 + 17.8559i 0.0490626 + 0.0300100i
\(596\) 0 0
\(597\) 420.686 0.704667
\(598\) 0 0
\(599\) 77.8882i 0.130030i −0.997884 0.0650152i \(-0.979290\pi\)
0.997884 0.0650152i \(-0.0207096\pi\)
\(600\) 0 0
\(601\) 454.271i 0.755859i −0.925834 0.377929i \(-0.876636\pi\)
0.925834 0.377929i \(-0.123364\pi\)
\(602\) 0 0
\(603\) −118.480 −0.196484
\(604\) 0 0
\(605\) −428.552 262.131i −0.708350 0.433275i
\(606\) 0 0
\(607\) 874.377 1.44049 0.720245 0.693720i \(-0.244029\pi\)
0.720245 + 0.693720i \(0.244029\pi\)
\(608\) 0 0
\(609\) 785.363 1.28959
\(610\) 0 0
\(611\) 878.092i 1.43714i
\(612\) 0 0
\(613\) 980.965i 1.60027i −0.599821 0.800134i \(-0.704761\pi\)
0.599821 0.800134i \(-0.295239\pi\)
\(614\) 0 0
\(615\) 153.648 251.195i 0.249834 0.408447i
\(616\) 0 0
\(617\) 806.424i 1.30701i −0.756923 0.653504i \(-0.773298\pi\)
0.756923 0.653504i \(-0.226702\pi\)
\(618\) 0 0
\(619\) 221.549 0.357915 0.178957 0.983857i \(-0.442728\pi\)
0.178957 + 0.983857i \(0.442728\pi\)
\(620\) 0 0
\(621\) 754.323i 1.21469i
\(622\) 0 0
\(623\) −582.060 −0.934286
\(624\) 0 0
\(625\) −365.697 + 506.844i −0.585115 + 0.810950i
\(626\) 0 0
\(627\) 249.515 129.797i 0.397951 0.207013i
\(628\) 0 0
\(629\) 43.7210i 0.0695087i
\(630\) 0 0
\(631\) 590.815 0.936316 0.468158 0.883645i \(-0.344918\pi\)
0.468158 + 0.883645i \(0.344918\pi\)
\(632\) 0 0
\(633\) 528.107i 0.834292i
\(634\) 0 0
\(635\) 344.726 + 210.858i 0.542875 + 0.332059i
\(636\) 0 0
\(637\) 106.863 0.167759
\(638\) 0 0
\(639\) 156.295i 0.244592i
\(640\) 0 0
\(641\) 1110.32i 1.73217i 0.499896 + 0.866086i \(0.333372\pi\)
−0.499896 + 0.866086i \(0.666628\pi\)
\(642\) 0 0
\(643\) 430.490i 0.669503i 0.942306 + 0.334752i \(0.108652\pi\)
−0.942306 + 0.334752i \(0.891348\pi\)
\(644\) 0 0
\(645\) 23.4214 38.2911i 0.0363123 0.0593660i
\(646\) 0 0
\(647\) 102.290i 0.158099i −0.996871 0.0790497i \(-0.974811\pi\)
0.996871 0.0790497i \(-0.0251886\pi\)
\(648\) 0 0
\(649\) 432.210i 0.665962i
\(650\) 0 0
\(651\) 1199.49 1.84254
\(652\) 0 0
\(653\) 71.2294i 0.109080i −0.998512 0.0545401i \(-0.982631\pi\)
0.998512 0.0545401i \(-0.0173693\pi\)
\(654\) 0 0
\(655\) −554.313 339.055i −0.846279 0.517642i
\(656\) 0 0
\(657\) 23.9364i 0.0364329i
\(658\) 0 0
\(659\) 1003.48i 1.52273i −0.648325 0.761364i \(-0.724530\pi\)
0.648325 0.761364i \(-0.275470\pi\)
\(660\) 0 0
\(661\) 620.531i 0.938775i 0.882992 + 0.469388i \(0.155525\pi\)
−0.882992 + 0.469388i \(0.844475\pi\)
\(662\) 0 0
\(663\) 40.9209i 0.0617208i
\(664\) 0 0
\(665\) 41.4550 597.413i 0.0623384 0.898365i
\(666\) 0 0
\(667\) 1201.87 1.80190
\(668\) 0 0
\(669\) 1262.73 1.88749
\(670\) 0 0
\(671\) −19.4144 −0.0289335
\(672\) 0 0
\(673\) −973.102 −1.44592 −0.722959 0.690891i \(-0.757218\pi\)
−0.722959 + 0.690891i \(0.757218\pi\)
\(674\) 0 0
\(675\) −272.510 532.659i −0.403718 0.789124i
\(676\) 0 0
\(677\) 518.327 0.765623 0.382811 0.923827i \(-0.374956\pi\)
0.382811 + 0.923827i \(0.374956\pi\)
\(678\) 0 0
\(679\) 407.749i 0.600514i
\(680\) 0 0
\(681\) −171.455 −0.251769
\(682\) 0 0
\(683\) 423.707 0.620362 0.310181 0.950678i \(-0.399610\pi\)
0.310181 + 0.950678i \(0.399610\pi\)
\(684\) 0 0
\(685\) 153.418 250.818i 0.223967 0.366158i
\(686\) 0 0
\(687\) 1287.38 1.87391
\(688\) 0 0
\(689\) 93.0680 0.135077
\(690\) 0 0
\(691\) 789.329 1.14230 0.571150 0.820846i \(-0.306498\pi\)
0.571150 + 0.820846i \(0.306498\pi\)
\(692\) 0 0
\(693\) 47.8367i 0.0690284i
\(694\) 0 0
\(695\) 10.5970 + 6.48184i 0.0152475 + 0.00932639i
\(696\) 0 0
\(697\) 19.5700 0.0280775
\(698\) 0 0
\(699\) 72.7587i 0.104090i
\(700\) 0 0
\(701\) 307.669 0.438900 0.219450 0.975624i \(-0.429574\pi\)
0.219450 + 0.975624i \(0.429574\pi\)
\(702\) 0 0
\(703\) −678.766 + 353.092i −0.965527 + 0.502265i
\(704\) 0 0
\(705\) 648.857 1060.80i 0.920365 1.50468i
\(706\) 0 0
\(707\) 981.307i 1.38799i
\(708\) 0 0
\(709\) −767.955 −1.08315 −0.541576 0.840651i \(-0.682172\pi\)
−0.541576 + 0.840651i \(0.682172\pi\)
\(710\) 0 0
\(711\) 10.2476i 0.0144129i
\(712\) 0 0
\(713\) 1835.63 2.57451
\(714\) 0 0
\(715\) 222.928 + 136.358i 0.311788 + 0.190710i
\(716\) 0 0
\(717\) 273.134 0.380941
\(718\) 0 0
\(719\) −1329.40 −1.84895 −0.924476 0.381239i \(-0.875497\pi\)
−0.924476 + 0.381239i \(0.875497\pi\)
\(720\) 0 0
\(721\) 824.652i 1.14376i
\(722\) 0 0
\(723\) 701.169i 0.969804i
\(724\) 0 0
\(725\) −848.690 + 434.193i −1.17061 + 0.598886i
\(726\) 0 0
\(727\) 1117.94i 1.53775i −0.639399 0.768875i \(-0.720817\pi\)
0.639399 0.768875i \(-0.279183\pi\)
\(728\) 0 0
\(729\) 547.530 0.751070
\(730\) 0 0
\(731\) 2.98317 0.00408094
\(732\) 0 0
\(733\) 1173.92i 1.60153i 0.598976 + 0.800767i \(0.295575\pi\)
−0.598976 + 0.800767i \(0.704425\pi\)
\(734\) 0 0
\(735\) 129.098 + 78.9651i 0.175644 + 0.107436i
\(736\) 0 0
\(737\) −320.486 −0.434852
\(738\) 0 0
\(739\) −415.270 −0.561935 −0.280968 0.959717i \(-0.590655\pi\)
−0.280968 + 0.959717i \(0.590655\pi\)
\(740\) 0 0
\(741\) 635.295 330.479i 0.857348 0.445990i
\(742\) 0 0
\(743\) −1113.69 −1.49892 −0.749458 0.662052i \(-0.769686\pi\)
−0.749458 + 0.662052i \(0.769686\pi\)
\(744\) 0 0
\(745\) −387.420 236.972i −0.520026 0.318083i
\(746\) 0 0
\(747\) 149.113i 0.199616i
\(748\) 0 0
\(749\) 599.805i 0.800808i
\(750\) 0 0
\(751\) 434.422i 0.578458i 0.957260 + 0.289229i \(0.0933989\pi\)
−0.957260 + 0.289229i \(0.906601\pi\)
\(752\) 0 0
\(753\) −80.6259 −0.107073
\(754\) 0 0
\(755\) 115.111 188.191i 0.152464 0.249260i
\(756\) 0 0
\(757\) 90.2103i 0.119168i −0.998223 0.0595841i \(-0.981023\pi\)
0.998223 0.0595841i \(-0.0189774\pi\)
\(758\) 0 0
\(759\) 466.566i 0.614711i
\(760\) 0 0
\(761\) 846.768 1.11270 0.556352 0.830947i \(-0.312201\pi\)
0.556352 + 0.830947i \(0.312201\pi\)
\(762\) 0 0
\(763\) 980.780 1.28543
\(764\) 0 0
\(765\) −4.74450 + 7.75665i −0.00620195 + 0.0101394i
\(766\) 0 0
\(767\) 1100.46i 1.43475i
\(768\) 0 0
\(769\) 154.558 0.200986 0.100493 0.994938i \(-0.467958\pi\)
0.100493 + 0.994938i \(0.467958\pi\)
\(770\) 0 0
\(771\) 498.414 0.646452
\(772\) 0 0
\(773\) −1094.68 −1.41615 −0.708076 0.706137i \(-0.750436\pi\)
−0.708076 + 0.706137i \(0.750436\pi\)
\(774\) 0 0
\(775\) −1296.21 + 663.146i −1.67253 + 0.855672i
\(776\) 0 0
\(777\) 829.370i 1.06740i
\(778\) 0 0
\(779\) −158.048 303.823i −0.202886 0.390017i
\(780\) 0 0
\(781\) 422.774i 0.541324i
\(782\) 0 0
\(783\) 912.614i 1.16554i
\(784\) 0 0
\(785\) 24.4274 39.9358i 0.0311177 0.0508736i
\(786\) 0 0
\(787\) −1085.31 −1.37905 −0.689524 0.724262i \(-0.742180\pi\)
−0.689524 + 0.724262i \(0.742180\pi\)
\(788\) 0 0
\(789\) 1129.06i 1.43101i
\(790\) 0 0
\(791\) 1102.13i 1.39334i
\(792\) 0 0
\(793\) −49.4313 −0.0623345
\(794\) 0 0
\(795\) 112.433 + 68.7716i 0.141425 + 0.0865052i
\(796\) 0 0
\(797\) 788.528 0.989370 0.494685 0.869072i \(-0.335283\pi\)
0.494685 + 0.869072i \(0.335283\pi\)
\(798\) 0 0
\(799\) 82.6445 0.103435
\(800\) 0 0
\(801\) 154.659i 0.193082i
\(802\) 0 0
\(803\) 64.7475i 0.0806320i
\(804\) 0 0
\(805\) −847.448 518.357i −1.05273 0.643921i
\(806\) 0 0
\(807\) 4.55563i 0.00564514i
\(808\) 0 0
\(809\) 159.009 0.196550 0.0982752 0.995159i \(-0.468667\pi\)
0.0982752 + 0.995159i \(0.468667\pi\)
\(810\) 0 0
\(811\) 948.179i 1.16915i 0.811340 + 0.584574i \(0.198738\pi\)
−0.811340 + 0.584574i \(0.801262\pi\)
\(812\) 0 0
\(813\) 1332.59 1.63911
\(814\) 0 0
\(815\) 93.5943 153.015i 0.114840 0.187748i
\(816\) 0 0
\(817\) −24.0922 46.3135i −0.0294886 0.0566873i
\(818\) 0 0
\(819\) 121.798i 0.148715i
\(820\) 0 0
\(821\) 352.733 0.429638 0.214819 0.976654i \(-0.431084\pi\)
0.214819 + 0.976654i \(0.431084\pi\)
\(822\) 0 0
\(823\) 993.008i 1.20657i −0.797525 0.603286i \(-0.793858\pi\)
0.797525 0.603286i \(-0.206142\pi\)
\(824\) 0 0
\(825\) 168.553 + 329.461i 0.204307 + 0.399347i
\(826\) 0 0
\(827\) −1388.92 −1.67947 −0.839737 0.542994i \(-0.817291\pi\)
−0.839737 + 0.542994i \(0.817291\pi\)
\(828\) 0 0
\(829\) 1346.99i 1.62483i −0.583077 0.812417i \(-0.698151\pi\)
0.583077 0.812417i \(-0.301849\pi\)
\(830\) 0 0
\(831\) 1080.20i 1.29988i
\(832\) 0 0
\(833\) 10.0577i 0.0120741i
\(834\) 0 0
\(835\) 832.696 + 509.333i 0.997241 + 0.609980i
\(836\) 0 0
\(837\) 1393.84i 1.66528i
\(838\) 0 0
\(839\) 23.1544i 0.0275976i −0.999905 0.0137988i \(-0.995608\pi\)
0.999905 0.0137988i \(-0.00439244\pi\)
\(840\) 0 0
\(841\) −613.077 −0.728986
\(842\) 0 0
\(843\) 258.518i 0.306664i
\(844\) 0 0
\(845\) −153.243 93.7339i −0.181353 0.110928i
\(846\) 0 0
\(847\) 633.348i 0.747754i
\(848\) 0 0
\(849\) 49.0860i 0.0578163i
\(850\) 0 0
\(851\) 1269.22i 1.49144i
\(852\) 0 0
\(853\) 1344.49i 1.57619i 0.615551 + 0.788097i \(0.288933\pi\)
−0.615551 + 0.788097i \(0.711067\pi\)
\(854\) 0 0
\(855\) 158.738 + 11.0150i 0.185659 + 0.0128830i
\(856\) 0 0
\(857\) 612.032 0.714156 0.357078 0.934074i \(-0.383773\pi\)
0.357078 + 0.934074i \(0.383773\pi\)
\(858\) 0 0
\(859\) 405.145 0.471647 0.235824 0.971796i \(-0.424221\pi\)
0.235824 + 0.971796i \(0.424221\pi\)
\(860\) 0 0
\(861\) −371.236 −0.431168
\(862\) 0 0
\(863\) −1715.67 −1.98802 −0.994012 0.109267i \(-0.965150\pi\)
−0.994012 + 0.109267i \(0.965150\pi\)
\(864\) 0 0
\(865\) −1007.92 616.513i −1.16523 0.712732i
\(866\) 0 0
\(867\) 940.385 1.08464
\(868\) 0 0
\(869\) 27.7195i 0.0318982i
\(870\) 0 0
\(871\) −815.994 −0.936848
\(872\) 0 0
\(873\) −108.343 −0.124104
\(874\) 0 0
\(875\) 785.681 + 59.8803i 0.897921 + 0.0684347i
\(876\) 0 0
\(877\) −806.571 −0.919693 −0.459847 0.887998i \(-0.652096\pi\)
−0.459847 + 0.887998i \(0.652096\pi\)
\(878\) 0 0
\(879\) 253.884 0.288833
\(880\) 0 0
\(881\) −762.909 −0.865958 −0.432979 0.901404i \(-0.642538\pi\)
−0.432979 + 0.901404i \(0.642538\pi\)
\(882\) 0 0
\(883\) 292.155i 0.330866i 0.986221 + 0.165433i \(0.0529022\pi\)
−0.986221 + 0.165433i \(0.947098\pi\)
\(884\) 0 0
\(885\) −813.171 + 1329.43i −0.918838 + 1.50218i
\(886\) 0 0
\(887\) −6.76046 −0.00762171 −0.00381086 0.999993i \(-0.501213\pi\)
−0.00381086 + 0.999993i \(0.501213\pi\)
\(888\) 0 0
\(889\) 509.463i 0.573074i
\(890\) 0 0
\(891\) −422.575 −0.474270
\(892\) 0 0
\(893\) −667.440 1283.05i −0.747413 1.43679i
\(894\) 0 0
\(895\) 328.588 537.200i 0.367138 0.600224i
\(896\) 0 0
\(897\) 1187.93i 1.32434i
\(898\) 0 0
\(899\) −2220.82 −2.47032
\(900\) 0 0
\(901\) 8.75940i 0.00972186i
\(902\) 0 0
\(903\) −56.5896 −0.0626684
\(904\) 0 0
\(905\) 29.7906 48.7039i 0.0329178 0.0538165i
\(906\) 0 0
\(907\) −123.248 −0.135885 −0.0679425 0.997689i \(-0.521643\pi\)
−0.0679425 + 0.997689i \(0.521643\pi\)
\(908\) 0 0
\(909\) −260.743 −0.286846
\(910\) 0 0
\(911\) 102.149i 0.112129i −0.998427 0.0560645i \(-0.982145\pi\)
0.998427 0.0560645i \(-0.0178552\pi\)
\(912\) 0 0
\(913\) 403.349i 0.441784i
\(914\) 0 0
\(915\) −59.7166 36.5267i −0.0652641 0.0399199i
\(916\) 0 0
\(917\) 819.208i 0.893357i
\(918\) 0 0
\(919\) 793.186 0.863096 0.431548 0.902090i \(-0.357967\pi\)
0.431548 + 0.902090i \(0.357967\pi\)
\(920\) 0 0
\(921\) −694.844 −0.754446
\(922\) 0 0
\(923\) 1076.43i 1.16623i
\(924\) 0 0
\(925\) −458.522 896.246i −0.495700 0.968914i
\(926\) 0 0
\(927\) 219.118 0.236373
\(928\) 0 0
\(929\) −527.324 −0.567625 −0.283813 0.958880i \(-0.591599\pi\)
−0.283813 + 0.958880i \(0.591599\pi\)
\(930\) 0 0
\(931\) 156.146 81.2266i 0.167718 0.0872466i
\(932\) 0 0
\(933\) 340.820 0.365295
\(934\) 0 0
\(935\) −12.8338 + 20.9816i −0.0137260 + 0.0224402i
\(936\) 0 0
\(937\) 1358.67i 1.45002i 0.688736 + 0.725012i \(0.258166\pi\)
−0.688736 + 0.725012i \(0.741834\pi\)
\(938\) 0 0
\(939\) 1599.42i 1.70333i
\(940\) 0 0
\(941\) 767.460i 0.815579i −0.913076 0.407790i \(-0.866300\pi\)
0.913076 0.407790i \(-0.133700\pi\)
\(942\) 0 0
\(943\) −568.116 −0.602456
\(944\) 0 0
\(945\) −393.603 + 643.491i −0.416511 + 0.680943i
\(946\) 0 0
\(947\) 588.458i 0.621392i −0.950509 0.310696i \(-0.899438\pi\)
0.950509 0.310696i \(-0.100562\pi\)
\(948\) 0 0
\(949\) 164.855i 0.173714i
\(950\) 0 0
\(951\) 478.641 0.503302
\(952\) 0 0
\(953\) 1523.39 1.59852 0.799259 0.600987i \(-0.205226\pi\)
0.799259 + 0.600987i \(0.205226\pi\)
\(954\) 0 0
\(955\) 155.145 + 94.8971i 0.162455 + 0.0993687i
\(956\) 0 0
\(957\) 564.472i 0.589835i
\(958\) 0 0
\(959\) −370.679 −0.386527
\(960\) 0 0
\(961\) −2430.88 −2.52953
\(962\) 0 0
\(963\) −159.374 −0.165497
\(964\) 0 0
\(965\) −643.277 393.472i −0.666608 0.407743i
\(966\) 0 0
\(967\) 18.3510i 0.0189772i 0.999955 + 0.00948861i \(0.00302036\pi\)
−0.999955 + 0.00948861i \(0.996980\pi\)
\(968\) 0 0
\(969\) 31.1041 + 59.7929i 0.0320992 + 0.0617057i
\(970\) 0 0
\(971\) 1560.15i 1.60675i −0.595476 0.803373i \(-0.703036\pi\)
0.595476 0.803373i \(-0.296964\pi\)
\(972\) 0 0
\(973\) 15.6611i 0.0160957i
\(974\) 0 0
\(975\) 429.157 + 838.847i 0.440161 + 0.860356i
\(976\) 0 0
\(977\) 577.097 0.590683 0.295341 0.955392i \(-0.404567\pi\)
0.295341 + 0.955392i \(0.404567\pi\)
\(978\) 0 0
\(979\) 418.350i 0.427324i
\(980\) 0 0
\(981\) 260.603i 0.265650i
\(982\) 0 0
\(983\) 1162.84 1.18295 0.591474 0.806324i \(-0.298546\pi\)
0.591474 + 0.806324i \(0.298546\pi\)
\(984\) 0 0
\(985\) −672.033 + 1098.69i −0.682267 + 1.11542i
\(986\) 0 0
\(987\) −1567.74 −1.58838
\(988\) 0 0
\(989\) −86.6012 −0.0875644
\(990\) 0 0
\(991\) 469.236i 0.473497i −0.971571 0.236749i \(-0.923918\pi\)
0.971571 0.236749i \(-0.0760818\pi\)
\(992\) 0 0
\(993\) 1042.41i 1.04976i
\(994\) 0 0
\(995\) 549.200 + 335.928i 0.551959 + 0.337616i
\(996\) 0 0
\(997\) 1618.73i 1.62360i 0.583933 + 0.811802i \(0.301513\pi\)
−0.583933 + 0.811802i \(0.698487\pi\)
\(998\) 0 0
\(999\) 963.752 0.964717
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 380.3.g.c.189.10 yes 12
3.2 odd 2 3420.3.h.e.2089.2 12
5.2 odd 4 1900.3.e.e.1101.4 12
5.3 odd 4 1900.3.e.e.1101.9 12
5.4 even 2 inner 380.3.g.c.189.3 12
15.14 odd 2 3420.3.h.e.2089.3 12
19.18 odd 2 inner 380.3.g.c.189.4 yes 12
57.56 even 2 3420.3.h.e.2089.1 12
95.18 even 4 1900.3.e.e.1101.3 12
95.37 even 4 1900.3.e.e.1101.10 12
95.94 odd 2 inner 380.3.g.c.189.9 yes 12
285.284 even 2 3420.3.h.e.2089.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.g.c.189.3 12 5.4 even 2 inner
380.3.g.c.189.4 yes 12 19.18 odd 2 inner
380.3.g.c.189.9 yes 12 95.94 odd 2 inner
380.3.g.c.189.10 yes 12 1.1 even 1 trivial
1900.3.e.e.1101.3 12 95.18 even 4
1900.3.e.e.1101.4 12 5.2 odd 4
1900.3.e.e.1101.9 12 5.3 odd 4
1900.3.e.e.1101.10 12 95.37 even 4
3420.3.h.e.2089.1 12 57.56 even 2
3420.3.h.e.2089.2 12 3.2 odd 2
3420.3.h.e.2089.3 12 15.14 odd 2
3420.3.h.e.2089.4 12 285.284 even 2