Properties

Label 1344.4.p.d.223.22
Level $1344$
Weight $4$
Character 1344.223
Analytic conductor $79.299$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,4,Mod(223,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.223");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.p (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: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 223.22
Character \(\chi\) \(=\) 1344.223
Dual form 1344.4.p.d.223.21

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +3.15485 q^{5} +(-6.11026 - 17.4833i) q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} +3.15485 q^{5} +(-6.11026 - 17.4833i) q^{7} -9.00000 q^{9} +60.9479 q^{11} +59.6460 q^{13} +9.46455i q^{15} -21.4250i q^{17} -95.5205i q^{19} +(52.4498 - 18.3308i) q^{21} +46.3236i q^{23} -115.047 q^{25} -27.0000i q^{27} -107.590i q^{29} +94.2549 q^{31} +182.844i q^{33} +(-19.2770 - 55.1571i) q^{35} +131.650i q^{37} +178.938i q^{39} -283.782i q^{41} -373.255 q^{43} -28.3937 q^{45} +136.224 q^{47} +(-268.330 + 213.655i) q^{49} +64.2751 q^{51} -298.133i q^{53} +192.281 q^{55} +286.561 q^{57} +468.435i q^{59} -563.070 q^{61} +(54.9923 + 157.349i) q^{63} +188.174 q^{65} +160.990 q^{67} -138.971 q^{69} +409.730i q^{71} -930.545i q^{73} -345.141i q^{75} +(-372.407 - 1065.57i) q^{77} -442.826i q^{79} +81.0000 q^{81} -190.709i q^{83} -67.5928i q^{85} +322.771 q^{87} -829.404i q^{89} +(-364.453 - 1042.81i) q^{91} +282.765i q^{93} -301.353i q^{95} -1030.40i q^{97} -548.531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 288 q^{9} + 224 q^{13} + 72 q^{21} + 1120 q^{25} - 752 q^{49} - 672 q^{57} + 544 q^{61} + 1536 q^{65} + 144 q^{69} + 1632 q^{77} + 2592 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 3.15485 0.282178 0.141089 0.989997i \(-0.454940\pi\)
0.141089 + 0.989997i \(0.454940\pi\)
\(6\) 0 0
\(7\) −6.11026 17.4833i −0.329923 0.944008i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 60.9479 1.67059 0.835294 0.549803i \(-0.185297\pi\)
0.835294 + 0.549803i \(0.185297\pi\)
\(12\) 0 0
\(13\) 59.6460 1.27253 0.636263 0.771472i \(-0.280479\pi\)
0.636263 + 0.771472i \(0.280479\pi\)
\(14\) 0 0
\(15\) 9.46455i 0.162916i
\(16\) 0 0
\(17\) 21.4250i 0.305667i −0.988252 0.152833i \(-0.951160\pi\)
0.988252 0.152833i \(-0.0488398\pi\)
\(18\) 0 0
\(19\) 95.5205i 1.15336i −0.816969 0.576682i \(-0.804347\pi\)
0.816969 0.576682i \(-0.195653\pi\)
\(20\) 0 0
\(21\) 52.4498 18.3308i 0.545023 0.190481i
\(22\) 0 0
\(23\) 46.3236i 0.419963i 0.977705 + 0.209981i \(0.0673403\pi\)
−0.977705 + 0.209981i \(0.932660\pi\)
\(24\) 0 0
\(25\) −115.047 −0.920375
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 107.590i 0.688932i −0.938799 0.344466i \(-0.888060\pi\)
0.938799 0.344466i \(-0.111940\pi\)
\(30\) 0 0
\(31\) 94.2549 0.546086 0.273043 0.962002i \(-0.411970\pi\)
0.273043 + 0.962002i \(0.411970\pi\)
\(32\) 0 0
\(33\) 182.844i 0.964515i
\(34\) 0 0
\(35\) −19.2770 55.1571i −0.0930971 0.266379i
\(36\) 0 0
\(37\) 131.650i 0.584949i 0.956273 + 0.292475i \(0.0944787\pi\)
−0.956273 + 0.292475i \(0.905521\pi\)
\(38\) 0 0
\(39\) 178.938i 0.734693i
\(40\) 0 0
\(41\) 283.782i 1.08096i −0.841358 0.540479i \(-0.818243\pi\)
0.841358 0.540479i \(-0.181757\pi\)
\(42\) 0 0
\(43\) −373.255 −1.32374 −0.661871 0.749618i \(-0.730237\pi\)
−0.661871 + 0.749618i \(0.730237\pi\)
\(44\) 0 0
\(45\) −28.3937 −0.0940595
\(46\) 0 0
\(47\) 136.224 0.422773 0.211387 0.977403i \(-0.432202\pi\)
0.211387 + 0.977403i \(0.432202\pi\)
\(48\) 0 0
\(49\) −268.330 + 213.655i −0.782302 + 0.622900i
\(50\) 0 0
\(51\) 64.2751 0.176477
\(52\) 0 0
\(53\) 298.133i 0.772674i −0.922358 0.386337i \(-0.873740\pi\)
0.922358 0.386337i \(-0.126260\pi\)
\(54\) 0 0
\(55\) 192.281 0.471404
\(56\) 0 0
\(57\) 286.561 0.665895
\(58\) 0 0
\(59\) 468.435i 1.03365i 0.856092 + 0.516823i \(0.172885\pi\)
−0.856092 + 0.516823i \(0.827115\pi\)
\(60\) 0 0
\(61\) −563.070 −1.18186 −0.590932 0.806721i \(-0.701240\pi\)
−0.590932 + 0.806721i \(0.701240\pi\)
\(62\) 0 0
\(63\) 54.9923 + 157.349i 0.109974 + 0.314669i
\(64\) 0 0
\(65\) 188.174 0.359079
\(66\) 0 0
\(67\) 160.990 0.293553 0.146777 0.989170i \(-0.453110\pi\)
0.146777 + 0.989170i \(0.453110\pi\)
\(68\) 0 0
\(69\) −138.971 −0.242466
\(70\) 0 0
\(71\) 409.730i 0.684873i 0.939541 + 0.342437i \(0.111252\pi\)
−0.939541 + 0.342437i \(0.888748\pi\)
\(72\) 0 0
\(73\) 930.545i 1.49195i −0.665976 0.745973i \(-0.731985\pi\)
0.665976 0.745973i \(-0.268015\pi\)
\(74\) 0 0
\(75\) 345.141i 0.531379i
\(76\) 0 0
\(77\) −372.407 1065.57i −0.551165 1.57705i
\(78\) 0 0
\(79\) 442.826i 0.630656i −0.948983 0.315328i \(-0.897885\pi\)
0.948983 0.315328i \(-0.102115\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 190.709i 0.252205i −0.992017 0.126102i \(-0.959753\pi\)
0.992017 0.126102i \(-0.0402468\pi\)
\(84\) 0 0
\(85\) 67.5928i 0.0862526i
\(86\) 0 0
\(87\) 322.771 0.397755
\(88\) 0 0
\(89\) 829.404i 0.987828i −0.869511 0.493914i \(-0.835566\pi\)
0.869511 0.493914i \(-0.164434\pi\)
\(90\) 0 0
\(91\) −364.453 1042.81i −0.419835 1.20127i
\(92\) 0 0
\(93\) 282.765i 0.315283i
\(94\) 0 0
\(95\) 301.353i 0.325454i
\(96\) 0 0
\(97\) 1030.40i 1.07857i −0.842123 0.539285i \(-0.818694\pi\)
0.842123 0.539285i \(-0.181306\pi\)
\(98\) 0 0
\(99\) −548.531 −0.556863
\(100\) 0 0
\(101\) 1784.29 1.75785 0.878926 0.476958i \(-0.158261\pi\)
0.878926 + 0.476958i \(0.158261\pi\)
\(102\) 0 0
\(103\) 288.271 0.275769 0.137885 0.990448i \(-0.455970\pi\)
0.137885 + 0.990448i \(0.455970\pi\)
\(104\) 0 0
\(105\) 165.471 57.8309i 0.153794 0.0537497i
\(106\) 0 0
\(107\) 292.479 0.264252 0.132126 0.991233i \(-0.457820\pi\)
0.132126 + 0.991233i \(0.457820\pi\)
\(108\) 0 0
\(109\) 518.765i 0.455860i −0.973678 0.227930i \(-0.926804\pi\)
0.973678 0.227930i \(-0.0731957\pi\)
\(110\) 0 0
\(111\) −394.950 −0.337721
\(112\) 0 0
\(113\) −1101.69 −0.917156 −0.458578 0.888654i \(-0.651641\pi\)
−0.458578 + 0.888654i \(0.651641\pi\)
\(114\) 0 0
\(115\) 146.144i 0.118504i
\(116\) 0 0
\(117\) −536.814 −0.424175
\(118\) 0 0
\(119\) −374.580 + 130.913i −0.288552 + 0.100847i
\(120\) 0 0
\(121\) 2383.64 1.79087
\(122\) 0 0
\(123\) 851.345 0.624091
\(124\) 0 0
\(125\) −757.312 −0.541889
\(126\) 0 0
\(127\) 1739.50i 1.21540i 0.794168 + 0.607698i \(0.207907\pi\)
−0.794168 + 0.607698i \(0.792093\pi\)
\(128\) 0 0
\(129\) 1119.77i 0.764263i
\(130\) 0 0
\(131\) 2546.59i 1.69845i −0.528033 0.849224i \(-0.677070\pi\)
0.528033 0.849224i \(-0.322930\pi\)
\(132\) 0 0
\(133\) −1670.01 + 583.655i −1.08878 + 0.380521i
\(134\) 0 0
\(135\) 85.1810i 0.0543053i
\(136\) 0 0
\(137\) −197.078 −0.122902 −0.0614509 0.998110i \(-0.519573\pi\)
−0.0614509 + 0.998110i \(0.519573\pi\)
\(138\) 0 0
\(139\) 810.877i 0.494803i 0.968913 + 0.247402i \(0.0795767\pi\)
−0.968913 + 0.247402i \(0.920423\pi\)
\(140\) 0 0
\(141\) 408.673i 0.244088i
\(142\) 0 0
\(143\) 3635.30 2.12587
\(144\) 0 0
\(145\) 339.432i 0.194402i
\(146\) 0 0
\(147\) −640.964 804.989i −0.359631 0.451662i
\(148\) 0 0
\(149\) 1838.58i 1.01089i 0.862860 + 0.505443i \(0.168671\pi\)
−0.862860 + 0.505443i \(0.831329\pi\)
\(150\) 0 0
\(151\) 43.6496i 0.0235242i 0.999931 + 0.0117621i \(0.00374408\pi\)
−0.999931 + 0.0117621i \(0.996256\pi\)
\(152\) 0 0
\(153\) 192.825i 0.101889i
\(154\) 0 0
\(155\) 297.360 0.154094
\(156\) 0 0
\(157\) −193.518 −0.0983721 −0.0491860 0.998790i \(-0.515663\pi\)
−0.0491860 + 0.998790i \(0.515663\pi\)
\(158\) 0 0
\(159\) 894.399 0.446103
\(160\) 0 0
\(161\) 809.888 283.049i 0.396448 0.138555i
\(162\) 0 0
\(163\) 1656.29 0.795893 0.397946 0.917409i \(-0.369723\pi\)
0.397946 + 0.917409i \(0.369723\pi\)
\(164\) 0 0
\(165\) 576.844i 0.272165i
\(166\) 0 0
\(167\) 4172.57 1.93343 0.966716 0.255851i \(-0.0823555\pi\)
0.966716 + 0.255851i \(0.0823555\pi\)
\(168\) 0 0
\(169\) 1360.65 0.619322
\(170\) 0 0
\(171\) 859.684i 0.384454i
\(172\) 0 0
\(173\) −3172.52 −1.39423 −0.697116 0.716958i \(-0.745534\pi\)
−0.697116 + 0.716958i \(0.745534\pi\)
\(174\) 0 0
\(175\) 702.966 + 2011.40i 0.303653 + 0.868842i
\(176\) 0 0
\(177\) −1405.31 −0.596776
\(178\) 0 0
\(179\) −137.757 −0.0575221 −0.0287611 0.999586i \(-0.509156\pi\)
−0.0287611 + 0.999586i \(0.509156\pi\)
\(180\) 0 0
\(181\) −978.118 −0.401674 −0.200837 0.979625i \(-0.564366\pi\)
−0.200837 + 0.979625i \(0.564366\pi\)
\(182\) 0 0
\(183\) 1689.21i 0.682350i
\(184\) 0 0
\(185\) 415.336i 0.165060i
\(186\) 0 0
\(187\) 1305.81i 0.510644i
\(188\) 0 0
\(189\) −472.048 + 164.977i −0.181674 + 0.0634937i
\(190\) 0 0
\(191\) 4051.60i 1.53489i 0.641116 + 0.767444i \(0.278472\pi\)
−0.641116 + 0.767444i \(0.721528\pi\)
\(192\) 0 0
\(193\) 1175.58 0.438444 0.219222 0.975675i \(-0.429648\pi\)
0.219222 + 0.975675i \(0.429648\pi\)
\(194\) 0 0
\(195\) 564.523i 0.207315i
\(196\) 0 0
\(197\) 3303.51i 1.19475i −0.801963 0.597374i \(-0.796211\pi\)
0.801963 0.597374i \(-0.203789\pi\)
\(198\) 0 0
\(199\) 4452.76 1.58617 0.793086 0.609110i \(-0.208473\pi\)
0.793086 + 0.609110i \(0.208473\pi\)
\(200\) 0 0
\(201\) 482.970i 0.169483i
\(202\) 0 0
\(203\) −1881.03 + 657.405i −0.650358 + 0.227295i
\(204\) 0 0
\(205\) 895.289i 0.305023i
\(206\) 0 0
\(207\) 416.912i 0.139988i
\(208\) 0 0
\(209\) 5821.77i 1.92680i
\(210\) 0 0
\(211\) 5875.61 1.91703 0.958516 0.285038i \(-0.0920061\pi\)
0.958516 + 0.285038i \(0.0920061\pi\)
\(212\) 0 0
\(213\) −1229.19 −0.395412
\(214\) 0 0
\(215\) −1177.57 −0.373532
\(216\) 0 0
\(217\) −575.921 1647.88i −0.180166 0.515510i
\(218\) 0 0
\(219\) 2791.63 0.861375
\(220\) 0 0
\(221\) 1277.92i 0.388969i
\(222\) 0 0
\(223\) −1909.89 −0.573524 −0.286762 0.958002i \(-0.592579\pi\)
−0.286762 + 0.958002i \(0.592579\pi\)
\(224\) 0 0
\(225\) 1035.42 0.306792
\(226\) 0 0
\(227\) 4211.95i 1.23153i −0.787931 0.615764i \(-0.788848\pi\)
0.787931 0.615764i \(-0.211152\pi\)
\(228\) 0 0
\(229\) 135.681 0.0391532 0.0195766 0.999808i \(-0.493768\pi\)
0.0195766 + 0.999808i \(0.493768\pi\)
\(230\) 0 0
\(231\) 3196.70 1117.22i 0.910509 0.318215i
\(232\) 0 0
\(233\) −613.936 −0.172619 −0.0863096 0.996268i \(-0.527507\pi\)
−0.0863096 + 0.996268i \(0.527507\pi\)
\(234\) 0 0
\(235\) 429.767 0.119298
\(236\) 0 0
\(237\) 1328.48 0.364109
\(238\) 0 0
\(239\) 4611.45i 1.24807i −0.781395 0.624037i \(-0.785492\pi\)
0.781395 0.624037i \(-0.214508\pi\)
\(240\) 0 0
\(241\) 4934.31i 1.31887i −0.751763 0.659433i \(-0.770796\pi\)
0.751763 0.659433i \(-0.229204\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) −846.540 + 674.048i −0.220749 + 0.175769i
\(246\) 0 0
\(247\) 5697.42i 1.46768i
\(248\) 0 0
\(249\) 572.126 0.145611
\(250\) 0 0
\(251\) 3655.89i 0.919353i −0.888086 0.459676i \(-0.847965\pi\)
0.888086 0.459676i \(-0.152035\pi\)
\(252\) 0 0
\(253\) 2823.32i 0.701585i
\(254\) 0 0
\(255\) 202.779 0.0497980
\(256\) 0 0
\(257\) 4419.08i 1.07259i −0.844032 0.536294i \(-0.819824\pi\)
0.844032 0.536294i \(-0.180176\pi\)
\(258\) 0 0
\(259\) 2301.67 804.415i 0.552197 0.192988i
\(260\) 0 0
\(261\) 968.314i 0.229644i
\(262\) 0 0
\(263\) 5949.45i 1.39490i 0.716633 + 0.697450i \(0.245682\pi\)
−0.716633 + 0.697450i \(0.754318\pi\)
\(264\) 0 0
\(265\) 940.565i 0.218032i
\(266\) 0 0
\(267\) 2488.21 0.570323
\(268\) 0 0
\(269\) −735.471 −0.166701 −0.0833503 0.996520i \(-0.526562\pi\)
−0.0833503 + 0.996520i \(0.526562\pi\)
\(270\) 0 0
\(271\) 8337.62 1.86891 0.934454 0.356084i \(-0.115888\pi\)
0.934454 + 0.356084i \(0.115888\pi\)
\(272\) 0 0
\(273\) 3128.42 1093.36i 0.693556 0.242392i
\(274\) 0 0
\(275\) −7011.86 −1.53757
\(276\) 0 0
\(277\) 3922.04i 0.850731i −0.905022 0.425366i \(-0.860145\pi\)
0.905022 0.425366i \(-0.139855\pi\)
\(278\) 0 0
\(279\) −848.294 −0.182029
\(280\) 0 0
\(281\) 6465.12 1.37251 0.686257 0.727359i \(-0.259253\pi\)
0.686257 + 0.727359i \(0.259253\pi\)
\(282\) 0 0
\(283\) 868.108i 0.182345i 0.995835 + 0.0911726i \(0.0290615\pi\)
−0.995835 + 0.0911726i \(0.970939\pi\)
\(284\) 0 0
\(285\) 904.059 0.187901
\(286\) 0 0
\(287\) −4961.43 + 1733.98i −1.02043 + 0.356633i
\(288\) 0 0
\(289\) 4453.97 0.906568
\(290\) 0 0
\(291\) 3091.20 0.622713
\(292\) 0 0
\(293\) −4595.00 −0.916186 −0.458093 0.888904i \(-0.651467\pi\)
−0.458093 + 0.888904i \(0.651467\pi\)
\(294\) 0 0
\(295\) 1477.84i 0.291673i
\(296\) 0 0
\(297\) 1645.59i 0.321505i
\(298\) 0 0
\(299\) 2763.02i 0.534413i
\(300\) 0 0
\(301\) 2280.69 + 6525.73i 0.436733 + 1.24962i
\(302\) 0 0
\(303\) 5352.86i 1.01490i
\(304\) 0 0
\(305\) −1776.40 −0.333497
\(306\) 0 0
\(307\) 3217.92i 0.598230i −0.954217 0.299115i \(-0.903309\pi\)
0.954217 0.299115i \(-0.0966914\pi\)
\(308\) 0 0
\(309\) 864.814i 0.159215i
\(310\) 0 0
\(311\) −7883.65 −1.43743 −0.718715 0.695305i \(-0.755269\pi\)
−0.718715 + 0.695305i \(0.755269\pi\)
\(312\) 0 0
\(313\) 2670.55i 0.482263i 0.970492 + 0.241131i \(0.0775185\pi\)
−0.970492 + 0.241131i \(0.922482\pi\)
\(314\) 0 0
\(315\) 173.493 + 496.414i 0.0310324 + 0.0887929i
\(316\) 0 0
\(317\) 5005.09i 0.886794i −0.896325 0.443397i \(-0.853773\pi\)
0.896325 0.443397i \(-0.146227\pi\)
\(318\) 0 0
\(319\) 6557.41i 1.15092i
\(320\) 0 0
\(321\) 877.436i 0.152566i
\(322\) 0 0
\(323\) −2046.53 −0.352545
\(324\) 0 0
\(325\) −6862.09 −1.17120
\(326\) 0 0
\(327\) 1556.30 0.263191
\(328\) 0 0
\(329\) −832.365 2381.65i −0.139483 0.399101i
\(330\) 0 0
\(331\) −10962.6 −1.82042 −0.910210 0.414148i \(-0.864080\pi\)
−0.910210 + 0.414148i \(0.864080\pi\)
\(332\) 0 0
\(333\) 1184.85i 0.194983i
\(334\) 0 0
\(335\) 507.900 0.0828344
\(336\) 0 0
\(337\) −10584.5 −1.71090 −0.855448 0.517888i \(-0.826718\pi\)
−0.855448 + 0.517888i \(0.826718\pi\)
\(338\) 0 0
\(339\) 3305.08i 0.529520i
\(340\) 0 0
\(341\) 5744.63 0.912285
\(342\) 0 0
\(343\) 5374.94 + 3385.79i 0.846121 + 0.532990i
\(344\) 0 0
\(345\) −438.432 −0.0684185
\(346\) 0 0
\(347\) −3267.07 −0.505434 −0.252717 0.967540i \(-0.581324\pi\)
−0.252717 + 0.967540i \(0.581324\pi\)
\(348\) 0 0
\(349\) −5781.37 −0.886732 −0.443366 0.896341i \(-0.646216\pi\)
−0.443366 + 0.896341i \(0.646216\pi\)
\(350\) 0 0
\(351\) 1610.44i 0.244898i
\(352\) 0 0
\(353\) 10650.3i 1.60583i 0.596093 + 0.802915i \(0.296719\pi\)
−0.596093 + 0.802915i \(0.703281\pi\)
\(354\) 0 0
\(355\) 1292.64i 0.193256i
\(356\) 0 0
\(357\) −392.738 1123.74i −0.0582238 0.166596i
\(358\) 0 0
\(359\) 9179.33i 1.34949i 0.738051 + 0.674745i \(0.235746\pi\)
−0.738051 + 0.674745i \(0.764254\pi\)
\(360\) 0 0
\(361\) −2265.16 −0.330247
\(362\) 0 0
\(363\) 7150.93i 1.03396i
\(364\) 0 0
\(365\) 2935.73i 0.420995i
\(366\) 0 0
\(367\) −2199.66 −0.312865 −0.156433 0.987689i \(-0.549999\pi\)
−0.156433 + 0.987689i \(0.549999\pi\)
\(368\) 0 0
\(369\) 2554.03i 0.360319i
\(370\) 0 0
\(371\) −5212.34 + 1821.67i −0.729410 + 0.254923i
\(372\) 0 0
\(373\) 1961.66i 0.272309i −0.990688 0.136154i \(-0.956526\pi\)
0.990688 0.136154i \(-0.0434743\pi\)
\(374\) 0 0
\(375\) 2271.94i 0.312859i
\(376\) 0 0
\(377\) 6417.34i 0.876684i
\(378\) 0 0
\(379\) 4839.03 0.655842 0.327921 0.944705i \(-0.393652\pi\)
0.327921 + 0.944705i \(0.393652\pi\)
\(380\) 0 0
\(381\) −5218.49 −0.701709
\(382\) 0 0
\(383\) 14701.3 1.96136 0.980678 0.195628i \(-0.0626744\pi\)
0.980678 + 0.195628i \(0.0626744\pi\)
\(384\) 0 0
\(385\) −1174.89 3361.71i −0.155527 0.445009i
\(386\) 0 0
\(387\) 3359.30 0.441248
\(388\) 0 0
\(389\) 15110.1i 1.96945i −0.174126 0.984723i \(-0.555710\pi\)
0.174126 0.984723i \(-0.444290\pi\)
\(390\) 0 0
\(391\) 992.486 0.128369
\(392\) 0 0
\(393\) 7639.77 0.980599
\(394\) 0 0
\(395\) 1397.05i 0.177958i
\(396\) 0 0
\(397\) 11309.2 1.42971 0.714854 0.699274i \(-0.246493\pi\)
0.714854 + 0.699274i \(0.246493\pi\)
\(398\) 0 0
\(399\) −1750.96 5010.03i −0.219694 0.628610i
\(400\) 0 0
\(401\) −12823.1 −1.59690 −0.798450 0.602061i \(-0.794346\pi\)
−0.798450 + 0.602061i \(0.794346\pi\)
\(402\) 0 0
\(403\) 5621.93 0.694909
\(404\) 0 0
\(405\) 255.543 0.0313532
\(406\) 0 0
\(407\) 8023.79i 0.977210i
\(408\) 0 0
\(409\) 3805.30i 0.460049i 0.973185 + 0.230025i \(0.0738807\pi\)
−0.973185 + 0.230025i \(0.926119\pi\)
\(410\) 0 0
\(411\) 591.235i 0.0709574i
\(412\) 0 0
\(413\) 8189.78 2862.26i 0.975770 0.341023i
\(414\) 0 0
\(415\) 601.658i 0.0711668i
\(416\) 0 0
\(417\) −2432.63 −0.285675
\(418\) 0 0
\(419\) 8027.30i 0.935941i −0.883744 0.467970i \(-0.844985\pi\)
0.883744 0.467970i \(-0.155015\pi\)
\(420\) 0 0
\(421\) 16646.2i 1.92705i 0.267622 + 0.963524i \(0.413762\pi\)
−0.267622 + 0.963524i \(0.586238\pi\)
\(422\) 0 0
\(423\) −1226.02 −0.140924
\(424\) 0 0
\(425\) 2464.89i 0.281328i
\(426\) 0 0
\(427\) 3440.50 + 9844.30i 0.389924 + 1.11569i
\(428\) 0 0
\(429\) 10905.9i 1.22737i
\(430\) 0 0
\(431\) 602.051i 0.0672849i −0.999434 0.0336425i \(-0.989289\pi\)
0.999434 0.0336425i \(-0.0107107\pi\)
\(432\) 0 0
\(433\) 9231.81i 1.02460i 0.858806 + 0.512301i \(0.171207\pi\)
−0.858806 + 0.512301i \(0.828793\pi\)
\(434\) 0 0
\(435\) 1018.30 0.112238
\(436\) 0 0
\(437\) 4424.85 0.484369
\(438\) 0 0
\(439\) −11079.3 −1.20452 −0.602262 0.798298i \(-0.705734\pi\)
−0.602262 + 0.798298i \(0.705734\pi\)
\(440\) 0 0
\(441\) 2414.97 1922.89i 0.260767 0.207633i
\(442\) 0 0
\(443\) −3757.07 −0.402943 −0.201471 0.979494i \(-0.564572\pi\)
−0.201471 + 0.979494i \(0.564572\pi\)
\(444\) 0 0
\(445\) 2616.65i 0.278744i
\(446\) 0 0
\(447\) −5515.73 −0.583635
\(448\) 0 0
\(449\) −5309.74 −0.558090 −0.279045 0.960278i \(-0.590018\pi\)
−0.279045 + 0.960278i \(0.590018\pi\)
\(450\) 0 0
\(451\) 17295.9i 1.80583i
\(452\) 0 0
\(453\) −130.949 −0.0135817
\(454\) 0 0
\(455\) −1149.79 3289.90i −0.118469 0.338974i
\(456\) 0 0
\(457\) 17826.5 1.82470 0.912350 0.409412i \(-0.134266\pi\)
0.912350 + 0.409412i \(0.134266\pi\)
\(458\) 0 0
\(459\) −578.476 −0.0588256
\(460\) 0 0
\(461\) 7613.64 0.769203 0.384601 0.923083i \(-0.374339\pi\)
0.384601 + 0.923083i \(0.374339\pi\)
\(462\) 0 0
\(463\) 831.086i 0.0834208i 0.999130 + 0.0417104i \(0.0132807\pi\)
−0.999130 + 0.0417104i \(0.986719\pi\)
\(464\) 0 0
\(465\) 892.080i 0.0889661i
\(466\) 0 0
\(467\) 11189.4i 1.10875i −0.832267 0.554374i \(-0.812958\pi\)
0.832267 0.554374i \(-0.187042\pi\)
\(468\) 0 0
\(469\) −983.691 2814.63i −0.0968500 0.277117i
\(470\) 0 0
\(471\) 580.554i 0.0567951i
\(472\) 0 0
\(473\) −22749.1 −2.21143
\(474\) 0 0
\(475\) 10989.3i 1.06153i
\(476\) 0 0
\(477\) 2683.20i 0.257558i
\(478\) 0 0
\(479\) −8503.67 −0.811154 −0.405577 0.914061i \(-0.632929\pi\)
−0.405577 + 0.914061i \(0.632929\pi\)
\(480\) 0 0
\(481\) 7852.40i 0.744363i
\(482\) 0 0
\(483\) 849.147 + 2429.66i 0.0799949 + 0.228889i
\(484\) 0 0
\(485\) 3250.76i 0.304349i
\(486\) 0 0
\(487\) 1829.08i 0.170192i −0.996373 0.0850962i \(-0.972880\pi\)
0.996373 0.0850962i \(-0.0271198\pi\)
\(488\) 0 0
\(489\) 4968.87i 0.459509i
\(490\) 0 0
\(491\) −15970.1 −1.46786 −0.733931 0.679224i \(-0.762316\pi\)
−0.733931 + 0.679224i \(0.762316\pi\)
\(492\) 0 0
\(493\) −2305.13 −0.210584
\(494\) 0 0
\(495\) −1730.53 −0.157135
\(496\) 0 0
\(497\) 7163.42 2503.55i 0.646526 0.225955i
\(498\) 0 0
\(499\) −1903.84 −0.170796 −0.0853982 0.996347i \(-0.527216\pi\)
−0.0853982 + 0.996347i \(0.527216\pi\)
\(500\) 0 0
\(501\) 12517.7i 1.11627i
\(502\) 0 0
\(503\) −5674.20 −0.502982 −0.251491 0.967860i \(-0.580921\pi\)
−0.251491 + 0.967860i \(0.580921\pi\)
\(504\) 0 0
\(505\) 5629.16 0.496028
\(506\) 0 0
\(507\) 4081.95i 0.357566i
\(508\) 0 0
\(509\) 10624.8 0.925219 0.462609 0.886562i \(-0.346913\pi\)
0.462609 + 0.886562i \(0.346913\pi\)
\(510\) 0 0
\(511\) −16269.0 + 5685.87i −1.40841 + 0.492227i
\(512\) 0 0
\(513\) −2579.05 −0.221965
\(514\) 0 0
\(515\) 909.453 0.0778161
\(516\) 0 0
\(517\) 8302.58 0.706280
\(518\) 0 0
\(519\) 9517.56i 0.804961i
\(520\) 0 0
\(521\) 695.563i 0.0584898i 0.999572 + 0.0292449i \(0.00931026\pi\)
−0.999572 + 0.0292449i \(0.990690\pi\)
\(522\) 0 0
\(523\) 11831.5i 0.989211i 0.869118 + 0.494605i \(0.164687\pi\)
−0.869118 + 0.494605i \(0.835313\pi\)
\(524\) 0 0
\(525\) −6034.19 + 2108.90i −0.501626 + 0.175314i
\(526\) 0 0
\(527\) 2019.42i 0.166921i
\(528\) 0 0
\(529\) 10021.1 0.823631
\(530\) 0 0
\(531\) 4215.92i 0.344549i
\(532\) 0 0
\(533\) 16926.5i 1.37555i
\(534\) 0 0
\(535\) 922.726 0.0745662
\(536\) 0 0
\(537\) 413.272i 0.0332104i
\(538\) 0 0
\(539\) −16354.1 + 13021.8i −1.30690 + 1.04061i
\(540\) 0 0
\(541\) 10553.5i 0.838687i 0.907828 + 0.419343i \(0.137740\pi\)
−0.907828 + 0.419343i \(0.862260\pi\)
\(542\) 0 0
\(543\) 2934.36i 0.231907i
\(544\) 0 0
\(545\) 1636.63i 0.128634i
\(546\) 0 0
\(547\) 1001.89 0.0783136 0.0391568 0.999233i \(-0.487533\pi\)
0.0391568 + 0.999233i \(0.487533\pi\)
\(548\) 0 0
\(549\) 5067.63 0.393955
\(550\) 0 0
\(551\) −10277.1 −0.794589
\(552\) 0 0
\(553\) −7742.05 + 2705.78i −0.595344 + 0.208068i
\(554\) 0 0
\(555\) −1246.01 −0.0952975
\(556\) 0 0
\(557\) 12853.2i 0.977752i −0.872353 0.488876i \(-0.837407\pi\)
0.872353 0.488876i \(-0.162593\pi\)
\(558\) 0 0
\(559\) −22263.2 −1.68450
\(560\) 0 0
\(561\) 3917.43 0.294820
\(562\) 0 0
\(563\) 10159.5i 0.760518i 0.924880 + 0.380259i \(0.124165\pi\)
−0.924880 + 0.380259i \(0.875835\pi\)
\(564\) 0 0
\(565\) −3475.68 −0.258802
\(566\) 0 0
\(567\) −494.931 1416.14i −0.0366581 0.104890i
\(568\) 0 0
\(569\) −9104.92 −0.670823 −0.335411 0.942072i \(-0.608875\pi\)
−0.335411 + 0.942072i \(0.608875\pi\)
\(570\) 0 0
\(571\) 21714.0 1.59143 0.795713 0.605674i \(-0.207097\pi\)
0.795713 + 0.605674i \(0.207097\pi\)
\(572\) 0 0
\(573\) −12154.8 −0.886168
\(574\) 0 0
\(575\) 5329.39i 0.386523i
\(576\) 0 0
\(577\) 12878.6i 0.929191i 0.885523 + 0.464595i \(0.153800\pi\)
−0.885523 + 0.464595i \(0.846200\pi\)
\(578\) 0 0
\(579\) 3526.73i 0.253136i
\(580\) 0 0
\(581\) −3334.21 + 1165.28i −0.238083 + 0.0832081i
\(582\) 0 0
\(583\) 18170.6i 1.29082i
\(584\) 0 0
\(585\) −1693.57 −0.119693
\(586\) 0 0
\(587\) 19024.7i 1.33771i 0.743394 + 0.668854i \(0.233215\pi\)
−0.743394 + 0.668854i \(0.766785\pi\)
\(588\) 0 0
\(589\) 9003.27i 0.629836i
\(590\) 0 0
\(591\) 9910.53 0.689788
\(592\) 0 0
\(593\) 25477.9i 1.76434i 0.470934 + 0.882169i \(0.343917\pi\)
−0.470934 + 0.882169i \(0.656083\pi\)
\(594\) 0 0
\(595\) −1181.74 + 413.010i −0.0814232 + 0.0284567i
\(596\) 0 0
\(597\) 13358.3i 0.915776i
\(598\) 0 0
\(599\) 12475.5i 0.850976i −0.904964 0.425488i \(-0.860102\pi\)
0.904964 0.425488i \(-0.139898\pi\)
\(600\) 0 0
\(601\) 947.642i 0.0643180i −0.999483 0.0321590i \(-0.989762\pi\)
0.999483 0.0321590i \(-0.0102383\pi\)
\(602\) 0 0
\(603\) −1448.91 −0.0978511
\(604\) 0 0
\(605\) 7520.04 0.505344
\(606\) 0 0
\(607\) 17586.1 1.17594 0.587971 0.808882i \(-0.299927\pi\)
0.587971 + 0.808882i \(0.299927\pi\)
\(608\) 0 0
\(609\) −1972.22 5643.10i −0.131229 0.375484i
\(610\) 0 0
\(611\) 8125.24 0.537990
\(612\) 0 0
\(613\) 9936.44i 0.654697i 0.944904 + 0.327348i \(0.106155\pi\)
−0.944904 + 0.327348i \(0.893845\pi\)
\(614\) 0 0
\(615\) 2685.87 0.176105
\(616\) 0 0
\(617\) −23950.9 −1.56277 −0.781383 0.624051i \(-0.785486\pi\)
−0.781383 + 0.624051i \(0.785486\pi\)
\(618\) 0 0
\(619\) 26145.1i 1.69767i 0.528654 + 0.848837i \(0.322697\pi\)
−0.528654 + 0.848837i \(0.677303\pi\)
\(620\) 0 0
\(621\) 1250.74 0.0808218
\(622\) 0 0
\(623\) −14500.7 + 5067.87i −0.932517 + 0.325907i
\(624\) 0 0
\(625\) 11991.7 0.767466
\(626\) 0 0
\(627\) 17465.3 1.11244
\(628\) 0 0
\(629\) 2820.61 0.178800
\(630\) 0 0
\(631\) 6978.35i 0.440260i −0.975471 0.220130i \(-0.929352\pi\)
0.975471 0.220130i \(-0.0706481\pi\)
\(632\) 0 0
\(633\) 17626.8i 1.10680i
\(634\) 0 0
\(635\) 5487.85i 0.342958i
\(636\) 0 0
\(637\) −16004.8 + 12743.6i −0.995499 + 0.792656i
\(638\) 0 0
\(639\) 3687.57i 0.228291i
\(640\) 0 0
\(641\) 9353.09 0.576326 0.288163 0.957581i \(-0.406956\pi\)
0.288163 + 0.957581i \(0.406956\pi\)
\(642\) 0 0
\(643\) 11216.2i 0.687906i 0.938987 + 0.343953i \(0.111766\pi\)
−0.938987 + 0.343953i \(0.888234\pi\)
\(644\) 0 0
\(645\) 3532.70i 0.215659i
\(646\) 0 0
\(647\) −8875.46 −0.539305 −0.269653 0.962958i \(-0.586909\pi\)
−0.269653 + 0.962958i \(0.586909\pi\)
\(648\) 0 0
\(649\) 28550.1i 1.72680i
\(650\) 0 0
\(651\) 4943.65 1727.76i 0.297630 0.104019i
\(652\) 0 0
\(653\) 12.5387i 0.000751419i 1.00000 0.000375710i \(0.000119592\pi\)
−1.00000 0.000375710i \(0.999880\pi\)
\(654\) 0 0
\(655\) 8034.11i 0.479265i
\(656\) 0 0
\(657\) 8374.90i 0.497315i
\(658\) 0 0
\(659\) −127.456 −0.00753414 −0.00376707 0.999993i \(-0.501199\pi\)
−0.00376707 + 0.999993i \(0.501199\pi\)
\(660\) 0 0
\(661\) 18113.0 1.06583 0.532916 0.846168i \(-0.321096\pi\)
0.532916 + 0.846168i \(0.321096\pi\)
\(662\) 0 0
\(663\) 3833.76 0.224571
\(664\) 0 0
\(665\) −5268.63 + 1841.34i −0.307231 + 0.107375i
\(666\) 0 0
\(667\) 4983.98 0.289326
\(668\) 0 0
\(669\) 5729.67i 0.331124i
\(670\) 0 0
\(671\) −34317.9 −1.97441
\(672\) 0 0
\(673\) −11689.6 −0.669540 −0.334770 0.942300i \(-0.608659\pi\)
−0.334770 + 0.942300i \(0.608659\pi\)
\(674\) 0 0
\(675\) 3106.27i 0.177126i
\(676\) 0 0
\(677\) 16101.4 0.914070 0.457035 0.889449i \(-0.348911\pi\)
0.457035 + 0.889449i \(0.348911\pi\)
\(678\) 0 0
\(679\) −18014.8 + 6296.02i −1.01818 + 0.355845i
\(680\) 0 0
\(681\) 12635.8 0.711023
\(682\) 0 0
\(683\) −23914.0 −1.33974 −0.669870 0.742478i \(-0.733650\pi\)
−0.669870 + 0.742478i \(0.733650\pi\)
\(684\) 0 0
\(685\) −621.753 −0.0346802
\(686\) 0 0
\(687\) 407.044i 0.0226051i
\(688\) 0 0
\(689\) 17782.4i 0.983247i
\(690\) 0 0
\(691\) 27330.4i 1.50463i −0.658805 0.752314i \(-0.728938\pi\)
0.658805 0.752314i \(-0.271062\pi\)
\(692\) 0 0
\(693\) 3351.66 + 9590.11i 0.183722 + 0.525683i
\(694\) 0 0
\(695\) 2558.20i 0.139623i
\(696\) 0 0
\(697\) −6080.04 −0.330413
\(698\) 0 0
\(699\) 1841.81i 0.0996617i
\(700\) 0 0
\(701\) 13641.6i 0.735002i −0.930023 0.367501i \(-0.880213\pi\)
0.930023 0.367501i \(-0.119787\pi\)
\(702\) 0 0
\(703\) 12575.3 0.674659
\(704\) 0 0
\(705\) 1289.30i 0.0688765i
\(706\) 0 0
\(707\) −10902.4 31195.1i −0.579956 1.65943i
\(708\) 0 0
\(709\) 25345.6i 1.34256i 0.741205 + 0.671278i \(0.234254\pi\)
−0.741205 + 0.671278i \(0.765746\pi\)
\(710\) 0 0
\(711\) 3985.43i 0.210219i
\(712\) 0 0
\(713\) 4366.23i 0.229336i
\(714\) 0 0
\(715\) 11468.8 0.599874
\(716\) 0 0
\(717\) 13834.3 0.720576
\(718\) 0 0
\(719\) 13143.0 0.681715 0.340857 0.940115i \(-0.389283\pi\)
0.340857 + 0.940115i \(0.389283\pi\)
\(720\) 0 0
\(721\) −1761.41 5039.93i −0.0909826 0.260328i
\(722\) 0 0
\(723\) 14802.9 0.761447
\(724\) 0 0
\(725\) 12377.9i 0.634076i
\(726\) 0 0
\(727\) 7313.19 0.373083 0.186541 0.982447i \(-0.440272\pi\)
0.186541 + 0.982447i \(0.440272\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 7997.02i 0.404624i
\(732\) 0 0
\(733\) 2577.28 0.129869 0.0649344 0.997890i \(-0.479316\pi\)
0.0649344 + 0.997890i \(0.479316\pi\)
\(734\) 0 0
\(735\) −2022.15 2539.62i −0.101480 0.127449i
\(736\) 0 0
\(737\) 9812.00 0.490407
\(738\) 0 0
\(739\) −5495.97 −0.273576 −0.136788 0.990600i \(-0.543678\pi\)
−0.136788 + 0.990600i \(0.543678\pi\)
\(740\) 0 0
\(741\) 17092.3 0.847368
\(742\) 0 0
\(743\) 16818.0i 0.830409i 0.909728 + 0.415204i \(0.136290\pi\)
−0.909728 + 0.415204i \(0.863710\pi\)
\(744\) 0 0
\(745\) 5800.43i 0.285250i
\(746\) 0 0
\(747\) 1716.38i 0.0840683i
\(748\) 0 0
\(749\) −1787.12 5113.48i −0.0871828 0.249456i
\(750\) 0 0
\(751\) 10217.3i 0.496450i −0.968702 0.248225i \(-0.920153\pi\)
0.968702 0.248225i \(-0.0798472\pi\)
\(752\) 0 0
\(753\) 10967.7 0.530789
\(754\) 0 0
\(755\) 137.708i 0.00663802i
\(756\) 0 0
\(757\) 2257.96i 0.108411i 0.998530 + 0.0542055i \(0.0172626\pi\)
−0.998530 + 0.0542055i \(0.982737\pi\)
\(758\) 0 0
\(759\) −8469.97 −0.405060
\(760\) 0 0
\(761\) 27933.8i 1.33061i 0.746570 + 0.665307i \(0.231699\pi\)
−0.746570 + 0.665307i \(0.768301\pi\)
\(762\) 0 0
\(763\) −9069.71 + 3169.79i −0.430335 + 0.150399i
\(764\) 0 0
\(765\) 608.336i 0.0287509i
\(766\) 0 0
\(767\) 27940.3i 1.31534i
\(768\) 0 0
\(769\) 33790.5i 1.58455i −0.610166 0.792274i \(-0.708897\pi\)
0.610166 0.792274i \(-0.291103\pi\)
\(770\) 0 0
\(771\) 13257.3 0.619259
\(772\) 0 0
\(773\) 18955.5 0.881996 0.440998 0.897508i \(-0.354625\pi\)
0.440998 + 0.897508i \(0.354625\pi\)
\(774\) 0 0
\(775\) −10843.7 −0.502604
\(776\) 0 0
\(777\) 2413.25 + 6905.02i 0.111422 + 0.318811i
\(778\) 0 0
\(779\) −27107.0 −1.24674
\(780\) 0 0
\(781\) 24972.2i 1.14414i
\(782\) 0 0
\(783\) −2904.94 −0.132585
\(784\) 0 0
\(785\) −610.520 −0.0277585
\(786\) 0 0
\(787\) 3001.54i 0.135951i −0.997687 0.0679754i \(-0.978346\pi\)
0.997687 0.0679754i \(-0.0216539\pi\)
\(788\) 0 0
\(789\) −17848.3 −0.805346
\(790\) 0 0
\(791\) 6731.63 + 19261.2i 0.302591 + 0.865803i
\(792\) 0 0
\(793\) −33584.9 −1.50395
\(794\) 0 0
\(795\) 2821.69 0.125881
\(796\) 0 0
\(797\) −14068.6 −0.625265 −0.312633 0.949874i \(-0.601211\pi\)
−0.312633 + 0.949874i \(0.601211\pi\)
\(798\) 0 0
\(799\) 2918.61i 0.129228i
\(800\) 0 0
\(801\) 7464.64i 0.329276i
\(802\) 0 0
\(803\) 56714.7i 2.49243i
\(804\) 0 0
\(805\) 2555.08 892.978i 0.111869 0.0390973i
\(806\) 0 0
\(807\) 2206.41i 0.0962446i
\(808\) 0 0
\(809\) −4579.59 −0.199023 −0.0995117 0.995036i \(-0.531728\pi\)
−0.0995117 + 0.995036i \(0.531728\pi\)
\(810\) 0 0
\(811\) 28588.1i 1.23781i 0.785465 + 0.618906i \(0.212424\pi\)
−0.785465 + 0.618906i \(0.787576\pi\)
\(812\) 0 0
\(813\) 25012.8i 1.07901i
\(814\) 0 0
\(815\) 5225.34 0.224584
\(816\) 0 0
\(817\) 35653.5i 1.52676i
\(818\) 0 0
\(819\) 3280.07 + 9385.27i 0.139945 + 0.400425i
\(820\) 0 0
\(821\) 13834.8i 0.588107i −0.955789 0.294054i \(-0.904996\pi\)
0.955789 0.294054i \(-0.0950045\pi\)
\(822\) 0 0
\(823\) 33259.1i 1.40867i −0.709866 0.704337i \(-0.751244\pi\)
0.709866 0.704337i \(-0.248756\pi\)
\(824\) 0 0
\(825\) 21035.6i 0.887716i
\(826\) 0 0
\(827\) −17593.0 −0.739744 −0.369872 0.929083i \(-0.620598\pi\)
−0.369872 + 0.929083i \(0.620598\pi\)
\(828\) 0 0
\(829\) 10912.0 0.457164 0.228582 0.973525i \(-0.426591\pi\)
0.228582 + 0.973525i \(0.426591\pi\)
\(830\) 0 0
\(831\) 11766.1 0.491170
\(832\) 0 0
\(833\) 4577.56 + 5748.97i 0.190400 + 0.239124i
\(834\) 0 0
\(835\) 13163.8 0.545573
\(836\) 0 0
\(837\) 2544.88i 0.105094i
\(838\) 0 0
\(839\) −24129.6 −0.992905 −0.496452 0.868064i \(-0.665364\pi\)
−0.496452 + 0.868064i \(0.665364\pi\)
\(840\) 0 0
\(841\) 12813.3 0.525372
\(842\) 0 0
\(843\) 19395.3i 0.792421i
\(844\) 0 0
\(845\) 4292.65 0.174759
\(846\) 0 0
\(847\) −14564.7 41673.9i −0.590848 1.69059i
\(848\) 0 0
\(849\) −2604.32 −0.105277
\(850\) 0 0
\(851\) −6098.50 −0.245657
\(852\) 0 0
\(853\) −15204.6 −0.610310 −0.305155 0.952303i \(-0.598708\pi\)
−0.305155 + 0.952303i \(0.598708\pi\)
\(854\) 0 0
\(855\) 2712.18i 0.108485i
\(856\) 0 0
\(857\) 13467.4i 0.536799i −0.963308 0.268400i \(-0.913505\pi\)
0.963308 0.268400i \(-0.0864948\pi\)
\(858\) 0 0
\(859\) 796.377i 0.0316322i 0.999875 + 0.0158161i \(0.00503463\pi\)
−0.999875 + 0.0158161i \(0.994965\pi\)
\(860\) 0 0
\(861\) −5201.94 14884.3i −0.205902 0.589147i
\(862\) 0 0
\(863\) 33680.2i 1.32849i 0.747514 + 0.664246i \(0.231247\pi\)
−0.747514 + 0.664246i \(0.768753\pi\)
\(864\) 0 0
\(865\) −10008.8 −0.393422
\(866\) 0 0
\(867\) 13361.9i 0.523407i
\(868\) 0 0
\(869\) 26989.3i 1.05357i
\(870\) 0 0
\(871\) 9602.42 0.373554
\(872\) 0 0
\(873\) 9273.61i 0.359524i
\(874\) 0 0
\(875\) 4627.37 + 13240.3i 0.178781 + 0.511547i
\(876\) 0 0
\(877\) 11140.6i 0.428954i −0.976729 0.214477i \(-0.931195\pi\)
0.976729 0.214477i \(-0.0688047\pi\)
\(878\) 0 0
\(879\) 13785.0i 0.528960i
\(880\) 0 0
\(881\) 2372.73i 0.0907370i −0.998970 0.0453685i \(-0.985554\pi\)
0.998970 0.0453685i \(-0.0144462\pi\)
\(882\) 0 0
\(883\) −17403.2 −0.663267 −0.331634 0.943408i \(-0.607600\pi\)
−0.331634 + 0.943408i \(0.607600\pi\)
\(884\) 0 0
\(885\) −4433.53 −0.168397
\(886\) 0 0
\(887\) 31804.3 1.20393 0.601965 0.798523i \(-0.294385\pi\)
0.601965 + 0.798523i \(0.294385\pi\)
\(888\) 0 0
\(889\) 30412.1 10628.8i 1.14734 0.400987i
\(890\) 0 0
\(891\) 4936.78 0.185621
\(892\) 0 0
\(893\) 13012.2i 0.487611i
\(894\) 0 0
\(895\) −434.603 −0.0162315
\(896\) 0 0
\(897\) −8289.06 −0.308544
\(898\) 0 0
\(899\) 10140.9i 0.376217i
\(900\) 0 0
\(901\) −6387.51 −0.236181
\(902\) 0 0
\(903\) −19577.2 + 6842.06i −0.721470 + 0.252148i
\(904\) 0 0
\(905\) −3085.82 −0.113344
\(906\) 0 0
\(907\) −1321.37 −0.0483741 −0.0241871 0.999707i \(-0.507700\pi\)
−0.0241871 + 0.999707i \(0.507700\pi\)
\(908\) 0 0
\(909\) −16058.6 −0.585951
\(910\) 0 0
\(911\) 15070.8i 0.548098i −0.961716 0.274049i \(-0.911637\pi\)
0.961716 0.274049i \(-0.0883631\pi\)
\(912\) 0 0
\(913\) 11623.3i 0.421330i
\(914\) 0 0
\(915\) 5329.20i 0.192544i
\(916\) 0 0
\(917\) −44522.7 + 15560.3i −1.60335 + 0.560357i
\(918\) 0 0
\(919\) 42833.7i 1.53749i 0.639556 + 0.768744i \(0.279118\pi\)
−0.639556 + 0.768744i \(0.720882\pi\)
\(920\) 0 0
\(921\) 9653.77 0.345388
\(922\) 0 0
\(923\) 24438.8i 0.871519i
\(924\) 0 0
\(925\) 15145.9i 0.538373i
\(926\) 0 0
\(927\) −2594.44 −0.0919231
\(928\) 0 0
\(929\) 19434.7i 0.686362i 0.939269 + 0.343181i \(0.111504\pi\)
−0.939269 + 0.343181i \(0.888496\pi\)
\(930\) 0 0
\(931\) 20408.4 + 25631.0i 0.718429 + 0.902278i
\(932\) 0 0
\(933\) 23651.0i 0.829901i
\(934\) 0 0
\(935\) 4119.64i 0.144093i
\(936\) 0 0
\(937\) 42125.9i 1.46872i −0.678759 0.734361i \(-0.737482\pi\)
0.678759 0.734361i \(-0.262518\pi\)
\(938\) 0 0
\(939\) −8011.64 −0.278435
\(940\) 0 0
\(941\) 1325.03 0.0459030 0.0229515 0.999737i \(-0.492694\pi\)
0.0229515 + 0.999737i \(0.492694\pi\)
\(942\) 0 0
\(943\) 13145.8 0.453962
\(944\) 0 0
\(945\) −1489.24 + 520.478i −0.0512646 + 0.0179166i
\(946\) 0 0
\(947\) −42522.6 −1.45913 −0.729566 0.683910i \(-0.760278\pi\)
−0.729566 + 0.683910i \(0.760278\pi\)
\(948\) 0 0
\(949\) 55503.3i 1.89854i
\(950\) 0 0
\(951\) 15015.3 0.511991
\(952\) 0 0
\(953\) −30507.0 −1.03696 −0.518478 0.855091i \(-0.673501\pi\)
−0.518478 + 0.855091i \(0.673501\pi\)
\(954\) 0 0
\(955\) 12782.2i 0.433112i
\(956\) 0 0
\(957\) 19672.2 0.664486
\(958\) 0 0
\(959\) 1204.20 + 3445.57i 0.0405481 + 0.116020i
\(960\) 0 0
\(961\) −20907.0 −0.701790
\(962\) 0 0
\(963\) −2632.31 −0.0880840
\(964\) 0 0
\(965\) 3708.77 0.123720
\(966\) 0 0
\(967\) 15544.9i 0.516950i 0.966018 + 0.258475i \(0.0832199\pi\)
−0.966018 + 0.258475i \(0.916780\pi\)
\(968\) 0 0
\(969\) 6139.59i 0.203542i
\(970\) 0 0
\(971\) 939.518i 0.0310510i 0.999879 + 0.0155255i \(0.00494213\pi\)
−0.999879 + 0.0155255i \(0.995058\pi\)
\(972\) 0 0
\(973\) 14176.8 4954.67i 0.467098 0.163247i
\(974\) 0 0
\(975\) 20586.3i 0.676193i
\(976\) 0 0
\(977\) 7969.39 0.260966 0.130483 0.991451i \(-0.458347\pi\)
0.130483 + 0.991451i \(0.458347\pi\)
\(978\) 0 0
\(979\) 50550.4i 1.65025i
\(980\) 0 0
\(981\) 4668.89i 0.151953i
\(982\) 0 0
\(983\) 35911.0 1.16519 0.582595 0.812762i \(-0.302037\pi\)
0.582595 + 0.812762i \(0.302037\pi\)
\(984\) 0 0
\(985\) 10422.1i 0.337132i
\(986\) 0 0
\(987\) 7144.94 2497.10i 0.230421 0.0805303i
\(988\) 0 0
\(989\) 17290.5i 0.555922i
\(990\) 0 0
\(991\) 29077.2i 0.932055i 0.884770 + 0.466027i \(0.154315\pi\)
−0.884770 + 0.466027i \(0.845685\pi\)
\(992\) 0 0
\(993\) 32887.8i 1.05102i
\(994\) 0 0
\(995\) 14047.8 0.447583
\(996\) 0 0
\(997\) −18798.3 −0.597141 −0.298570 0.954388i \(-0.596510\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(998\) 0 0
\(999\) 3554.55 0.112574
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 1344.4.p.d.223.22 yes 32
4.3 odd 2 inner 1344.4.p.d.223.3 yes 32
7.6 odd 2 1344.4.p.c.223.19 32
8.3 odd 2 1344.4.p.c.223.20 yes 32
8.5 even 2 1344.4.p.c.223.31 yes 32
28.27 even 2 1344.4.p.c.223.32 yes 32
56.13 odd 2 inner 1344.4.p.d.223.4 yes 32
56.27 even 2 inner 1344.4.p.d.223.21 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1344.4.p.c.223.19 32 7.6 odd 2
1344.4.p.c.223.20 yes 32 8.3 odd 2
1344.4.p.c.223.31 yes 32 8.5 even 2
1344.4.p.c.223.32 yes 32 28.27 even 2
1344.4.p.d.223.3 yes 32 4.3 odd 2 inner
1344.4.p.d.223.4 yes 32 56.13 odd 2 inner
1344.4.p.d.223.21 yes 32 56.27 even 2 inner
1344.4.p.d.223.22 yes 32 1.1 even 1 trivial