Properties

Label 4140.2.i.a.1241.5
Level $4140$
Weight $2$
Character 4140.1241
Analytic conductor $33.058$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4140,2,Mod(1241,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.1241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.i (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: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 62x^{14} + 1303x^{12} + 12842x^{10} + 65359x^{8} + 170834x^{6} + 207293x^{4} + 91366x^{2} + 9604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1241.5
Root \(-2.31631i\) of defining polynomial
Character \(\chi\) \(=\) 4140.1241
Dual form 4140.2.i.a.1241.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} -1.73994i q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -1.73994i q^{7} -1.48546 q^{11} -2.14610 q^{13} +3.22657 q^{17} +5.06099i q^{19} +(4.42186 + 1.85665i) q^{23} +1.00000 q^{25} -5.21742i q^{29} -5.45116 q^{31} +1.73994i q^{35} -2.87861i q^{37} -10.9887i q^{41} +4.50476i q^{43} -4.76294i q^{47} +3.97260 q^{49} +0.239954 q^{53} +1.48546 q^{55} -5.20247i q^{59} +6.30502i q^{61} +2.14610 q^{65} +2.61074i q^{67} -3.17216i q^{71} -15.7765 q^{73} +2.58462i q^{77} -16.1921i q^{79} +0.583836 q^{83} -3.22657 q^{85} -14.7009 q^{89} +3.73410i q^{91} -5.06099i q^{95} -1.66138i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{5} - 8 q^{23} + 16 q^{25} - 8 q^{31} - 40 q^{49} + 4 q^{53} + 8 q^{73} + 20 q^{83} + 32 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.73994i 0.657637i −0.944393 0.328818i \(-0.893350\pi\)
0.944393 0.328818i \(-0.106650\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.48546 −0.447884 −0.223942 0.974603i \(-0.571893\pi\)
−0.223942 + 0.974603i \(0.571893\pi\)
\(12\) 0 0
\(13\) −2.14610 −0.595222 −0.297611 0.954687i \(-0.596190\pi\)
−0.297611 + 0.954687i \(0.596190\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.22657 0.782558 0.391279 0.920272i \(-0.372033\pi\)
0.391279 + 0.920272i \(0.372033\pi\)
\(18\) 0 0
\(19\) 5.06099i 1.16107i 0.814235 + 0.580535i \(0.197157\pi\)
−0.814235 + 0.580535i \(0.802843\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.42186 + 1.85665i 0.922022 + 0.387137i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.21742i 0.968851i −0.874832 0.484426i \(-0.839029\pi\)
0.874832 0.484426i \(-0.160971\pi\)
\(30\) 0 0
\(31\) −5.45116 −0.979057 −0.489528 0.871987i \(-0.662831\pi\)
−0.489528 + 0.871987i \(0.662831\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.73994i 0.294104i
\(36\) 0 0
\(37\) 2.87861i 0.473241i −0.971602 0.236620i \(-0.923960\pi\)
0.971602 0.236620i \(-0.0760398\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.9887i 1.71615i −0.513525 0.858074i \(-0.671661\pi\)
0.513525 0.858074i \(-0.328339\pi\)
\(42\) 0 0
\(43\) 4.50476i 0.686969i 0.939158 + 0.343485i \(0.111607\pi\)
−0.939158 + 0.343485i \(0.888393\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.76294i 0.694746i −0.937727 0.347373i \(-0.887074\pi\)
0.937727 0.347373i \(-0.112926\pi\)
\(48\) 0 0
\(49\) 3.97260 0.567514
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.239954 0.0329601 0.0164801 0.999864i \(-0.494754\pi\)
0.0164801 + 0.999864i \(0.494754\pi\)
\(54\) 0 0
\(55\) 1.48546 0.200300
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.20247i 0.677304i −0.940912 0.338652i \(-0.890029\pi\)
0.940912 0.338652i \(-0.109971\pi\)
\(60\) 0 0
\(61\) 6.30502i 0.807275i 0.914919 + 0.403638i \(0.132254\pi\)
−0.914919 + 0.403638i \(0.867746\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.14610 0.266191
\(66\) 0 0
\(67\) 2.61074i 0.318952i 0.987202 + 0.159476i \(0.0509805\pi\)
−0.987202 + 0.159476i \(0.949019\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.17216i 0.376466i −0.982124 0.188233i \(-0.939724\pi\)
0.982124 0.188233i \(-0.0602760\pi\)
\(72\) 0 0
\(73\) −15.7765 −1.84649 −0.923247 0.384207i \(-0.874475\pi\)
−0.923247 + 0.384207i \(0.874475\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.58462i 0.294545i
\(78\) 0 0
\(79\) 16.1921i 1.82175i −0.412682 0.910875i \(-0.635408\pi\)
0.412682 0.910875i \(-0.364592\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.583836 0.0640843 0.0320421 0.999487i \(-0.489799\pi\)
0.0320421 + 0.999487i \(0.489799\pi\)
\(84\) 0 0
\(85\) −3.22657 −0.349971
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.7009 −1.55829 −0.779147 0.626842i \(-0.784347\pi\)
−0.779147 + 0.626842i \(0.784347\pi\)
\(90\) 0 0
\(91\) 3.73410i 0.391440i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.06099i 0.519246i
\(96\) 0 0
\(97\) 1.66138i 0.168687i −0.996437 0.0843436i \(-0.973121\pi\)
0.996437 0.0843436i \(-0.0268793\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.21263i 0.618180i 0.951033 + 0.309090i \(0.100024\pi\)
−0.951033 + 0.309090i \(0.899976\pi\)
\(102\) 0 0
\(103\) 3.57047i 0.351809i 0.984407 + 0.175904i \(0.0562850\pi\)
−0.984407 + 0.175904i \(0.943715\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −18.4060 −1.77938 −0.889689 0.456566i \(-0.849079\pi\)
−0.889689 + 0.456566i \(0.849079\pi\)
\(108\) 0 0
\(109\) 17.6728i 1.69275i 0.532590 + 0.846374i \(0.321219\pi\)
−0.532590 + 0.846374i \(0.678781\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.9644 −1.12551 −0.562757 0.826622i \(-0.690259\pi\)
−0.562757 + 0.826622i \(0.690259\pi\)
\(114\) 0 0
\(115\) −4.42186 1.85665i −0.412341 0.173133i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.61405i 0.514639i
\(120\) 0 0
\(121\) −8.79340 −0.799400
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −15.2155 −1.35016 −0.675081 0.737744i \(-0.735891\pi\)
−0.675081 + 0.737744i \(0.735891\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 21.1936i 1.85170i −0.377897 0.925848i \(-0.623353\pi\)
0.377897 0.925848i \(-0.376647\pi\)
\(132\) 0 0
\(133\) 8.80583 0.763562
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −19.3615 −1.65417 −0.827084 0.562078i \(-0.810002\pi\)
−0.827084 + 0.562078i \(0.810002\pi\)
\(138\) 0 0
\(139\) −12.3341 −1.04617 −0.523083 0.852282i \(-0.675218\pi\)
−0.523083 + 0.852282i \(0.675218\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.18795 0.266590
\(144\) 0 0
\(145\) 5.21742i 0.433283i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.5487 1.02803 0.514015 0.857781i \(-0.328158\pi\)
0.514015 + 0.857781i \(0.328158\pi\)
\(150\) 0 0
\(151\) 17.3650 1.41314 0.706570 0.707643i \(-0.250242\pi\)
0.706570 + 0.707643i \(0.250242\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.45116 0.437847
\(156\) 0 0
\(157\) 17.5785i 1.40291i −0.712711 0.701457i \(-0.752533\pi\)
0.712711 0.701457i \(-0.247467\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.23046 7.69379i 0.254596 0.606356i
\(162\) 0 0
\(163\) 15.0296 1.17721 0.588606 0.808420i \(-0.299677\pi\)
0.588606 + 0.808420i \(0.299677\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.4559i 0.886485i 0.896402 + 0.443242i \(0.146172\pi\)
−0.896402 + 0.443242i \(0.853828\pi\)
\(168\) 0 0
\(169\) −8.39424 −0.645711
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.36507i 0.559956i −0.960006 0.279978i \(-0.909673\pi\)
0.960006 0.279978i \(-0.0903272\pi\)
\(174\) 0 0
\(175\) 1.73994i 0.131527i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.0710i 0.976975i 0.872571 + 0.488487i \(0.162451\pi\)
−0.872571 + 0.488487i \(0.837549\pi\)
\(180\) 0 0
\(181\) 19.0486i 1.41587i 0.706276 + 0.707937i \(0.250374\pi\)
−0.706276 + 0.707937i \(0.749626\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.87861i 0.211640i
\(186\) 0 0
\(187\) −4.79295 −0.350495
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.27526 0.309347 0.154673 0.987966i \(-0.450567\pi\)
0.154673 + 0.987966i \(0.450567\pi\)
\(192\) 0 0
\(193\) −18.8399 −1.35613 −0.678063 0.735004i \(-0.737180\pi\)
−0.678063 + 0.735004i \(0.737180\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.8545i 1.91330i −0.291233 0.956652i \(-0.594065\pi\)
0.291233 0.956652i \(-0.405935\pi\)
\(198\) 0 0
\(199\) 10.8944i 0.772282i −0.922440 0.386141i \(-0.873808\pi\)
0.922440 0.386141i \(-0.126192\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −9.07802 −0.637152
\(204\) 0 0
\(205\) 10.9887i 0.767485i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7.51790i 0.520024i
\(210\) 0 0
\(211\) 13.8858 0.955941 0.477970 0.878376i \(-0.341373\pi\)
0.477970 + 0.878376i \(0.341373\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.50476i 0.307222i
\(216\) 0 0
\(217\) 9.48470i 0.643864i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.92455 −0.465796
\(222\) 0 0
\(223\) −20.1599 −1.35001 −0.675003 0.737815i \(-0.735858\pi\)
−0.675003 + 0.737815i \(0.735858\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.4260 1.35572 0.677861 0.735190i \(-0.262907\pi\)
0.677861 + 0.735190i \(0.262907\pi\)
\(228\) 0 0
\(229\) 12.6822i 0.838064i −0.907971 0.419032i \(-0.862369\pi\)
0.907971 0.419032i \(-0.137631\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.4814i 1.21076i −0.795938 0.605378i \(-0.793022\pi\)
0.795938 0.605378i \(-0.206978\pi\)
\(234\) 0 0
\(235\) 4.76294i 0.310700i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.96320i 0.644466i 0.946660 + 0.322233i \(0.104433\pi\)
−0.946660 + 0.322233i \(0.895567\pi\)
\(240\) 0 0
\(241\) 4.57276i 0.294557i 0.989095 + 0.147279i \(0.0470514\pi\)
−0.989095 + 0.147279i \(0.952949\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.97260 −0.253800
\(246\) 0 0
\(247\) 10.8614i 0.691094i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.15628 0.136103 0.0680515 0.997682i \(-0.478322\pi\)
0.0680515 + 0.997682i \(0.478322\pi\)
\(252\) 0 0
\(253\) −6.56851 2.75798i −0.412959 0.173392i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.9962i 1.18495i −0.805589 0.592474i \(-0.798151\pi\)
0.805589 0.592474i \(-0.201849\pi\)
\(258\) 0 0
\(259\) −5.00862 −0.311221
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.44986 0.0894022 0.0447011 0.999000i \(-0.485766\pi\)
0.0447011 + 0.999000i \(0.485766\pi\)
\(264\) 0 0
\(265\) −0.239954 −0.0147402
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.9723i 0.668992i −0.942397 0.334496i \(-0.891434\pi\)
0.942397 0.334496i \(-0.108566\pi\)
\(270\) 0 0
\(271\) −18.9514 −1.15122 −0.575608 0.817726i \(-0.695235\pi\)
−0.575608 + 0.817726i \(0.695235\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.48546 −0.0895767
\(276\) 0 0
\(277\) 4.76990 0.286595 0.143298 0.989680i \(-0.454229\pi\)
0.143298 + 0.989680i \(0.454229\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.47200 0.505397 0.252699 0.967545i \(-0.418682\pi\)
0.252699 + 0.967545i \(0.418682\pi\)
\(282\) 0 0
\(283\) 3.48520i 0.207174i −0.994620 0.103587i \(-0.966968\pi\)
0.994620 0.103587i \(-0.0330320\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −19.1197 −1.12860
\(288\) 0 0
\(289\) −6.58925 −0.387603
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −16.5320 −0.965811 −0.482905 0.875672i \(-0.660419\pi\)
−0.482905 + 0.875672i \(0.660419\pi\)
\(294\) 0 0
\(295\) 5.20247i 0.302899i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.48977 3.98455i −0.548808 0.230433i
\(300\) 0 0
\(301\) 7.83802 0.451776
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.30502i 0.361024i
\(306\) 0 0
\(307\) −12.9569 −0.739491 −0.369746 0.929133i \(-0.620555\pi\)
−0.369746 + 0.929133i \(0.620555\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.07291i 0.117544i 0.998271 + 0.0587720i \(0.0187185\pi\)
−0.998271 + 0.0587720i \(0.981282\pi\)
\(312\) 0 0
\(313\) 9.21330i 0.520767i 0.965505 + 0.260383i \(0.0838489\pi\)
−0.965505 + 0.260383i \(0.916151\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.62477i 0.0912560i −0.998958 0.0456280i \(-0.985471\pi\)
0.998958 0.0456280i \(-0.0145289\pi\)
\(318\) 0 0
\(319\) 7.75028i 0.433933i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 16.3296i 0.908605i
\(324\) 0 0
\(325\) −2.14610 −0.119044
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.28725 −0.456891
\(330\) 0 0
\(331\) 15.7992 0.868402 0.434201 0.900816i \(-0.357031\pi\)
0.434201 + 0.900816i \(0.357031\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.61074i 0.142640i
\(336\) 0 0
\(337\) 2.58536i 0.140834i 0.997518 + 0.0704169i \(0.0224330\pi\)
−0.997518 + 0.0704169i \(0.977567\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.09748 0.438503
\(342\) 0 0
\(343\) 19.0917i 1.03085i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0.459390i 0.0246614i 0.999924 + 0.0123307i \(0.00392508\pi\)
−0.999924 + 0.0123307i \(0.996075\pi\)
\(348\) 0 0
\(349\) −20.8616 −1.11670 −0.558348 0.829607i \(-0.688565\pi\)
−0.558348 + 0.829607i \(0.688565\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 27.4459i 1.46080i −0.683020 0.730400i \(-0.739334\pi\)
0.683020 0.730400i \(-0.260666\pi\)
\(354\) 0 0
\(355\) 3.17216i 0.168361i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −14.9440 −0.788716 −0.394358 0.918957i \(-0.629033\pi\)
−0.394358 + 0.918957i \(0.629033\pi\)
\(360\) 0 0
\(361\) −6.61358 −0.348083
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 15.7765 0.825777
\(366\) 0 0
\(367\) 10.6253i 0.554636i −0.960778 0.277318i \(-0.910554\pi\)
0.960778 0.277318i \(-0.0894456\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.417505i 0.0216758i
\(372\) 0 0
\(373\) 14.2372i 0.737173i 0.929593 + 0.368587i \(0.120158\pi\)
−0.929593 + 0.368587i \(0.879842\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11.1971i 0.576681i
\(378\) 0 0
\(379\) 7.81778i 0.401572i −0.979635 0.200786i \(-0.935650\pi\)
0.979635 0.200786i \(-0.0643496\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 33.0636 1.68947 0.844737 0.535182i \(-0.179757\pi\)
0.844737 + 0.535182i \(0.179757\pi\)
\(384\) 0 0
\(385\) 2.58462i 0.131724i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.1348 0.615259 0.307630 0.951506i \(-0.400464\pi\)
0.307630 + 0.951506i \(0.400464\pi\)
\(390\) 0 0
\(391\) 14.2674 + 5.99060i 0.721536 + 0.302957i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 16.1921i 0.814711i
\(396\) 0 0
\(397\) −1.46588 −0.0735704 −0.0367852 0.999323i \(-0.511712\pi\)
−0.0367852 + 0.999323i \(0.511712\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.10416 −0.155014 −0.0775071 0.996992i \(-0.524696\pi\)
−0.0775071 + 0.996992i \(0.524696\pi\)
\(402\) 0 0
\(403\) 11.6987 0.582756
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.27607i 0.211957i
\(408\) 0 0
\(409\) −24.1259 −1.19295 −0.596475 0.802632i \(-0.703432\pi\)
−0.596475 + 0.802632i \(0.703432\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −9.05200 −0.445420
\(414\) 0 0
\(415\) −0.583836 −0.0286594
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −17.4432 −0.852158 −0.426079 0.904686i \(-0.640105\pi\)
−0.426079 + 0.904686i \(0.640105\pi\)
\(420\) 0 0
\(421\) 20.3486i 0.991729i 0.868400 + 0.495864i \(0.165149\pi\)
−0.868400 + 0.495864i \(0.834851\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.22657 0.156512
\(426\) 0 0
\(427\) 10.9704 0.530894
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.4974 1.46901 0.734505 0.678603i \(-0.237414\pi\)
0.734505 + 0.678603i \(0.237414\pi\)
\(432\) 0 0
\(433\) 19.0844i 0.917137i −0.888659 0.458568i \(-0.848362\pi\)
0.888659 0.458568i \(-0.151638\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −9.39646 + 22.3790i −0.449494 + 1.07053i
\(438\) 0 0
\(439\) 29.7420 1.41951 0.709753 0.704451i \(-0.248807\pi\)
0.709753 + 0.704451i \(0.248807\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 11.9964i 0.569965i −0.958533 0.284983i \(-0.908012\pi\)
0.958533 0.284983i \(-0.0919878\pi\)
\(444\) 0 0
\(445\) 14.7009 0.696890
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.01821i 0.0480525i 0.999711 + 0.0240263i \(0.00764853\pi\)
−0.999711 + 0.0240263i \(0.992351\pi\)
\(450\) 0 0
\(451\) 16.3233i 0.768635i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.73410i 0.175057i
\(456\) 0 0
\(457\) 26.4418i 1.23689i −0.785826 0.618447i \(-0.787762\pi\)
0.785826 0.618447i \(-0.212238\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 35.5680i 1.65657i −0.560309 0.828284i \(-0.689317\pi\)
0.560309 0.828284i \(-0.310683\pi\)
\(462\) 0 0
\(463\) 23.1095 1.07399 0.536995 0.843586i \(-0.319560\pi\)
0.536995 + 0.843586i \(0.319560\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.60164 0.259213 0.129607 0.991566i \(-0.458629\pi\)
0.129607 + 0.991566i \(0.458629\pi\)
\(468\) 0 0
\(469\) 4.54254 0.209755
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.69165i 0.307682i
\(474\) 0 0
\(475\) 5.06099i 0.232214i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.0470 −0.641825 −0.320912 0.947109i \(-0.603990\pi\)
−0.320912 + 0.947109i \(0.603990\pi\)
\(480\) 0 0
\(481\) 6.17780i 0.281683i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.66138i 0.0754392i
\(486\) 0 0
\(487\) −34.3972 −1.55868 −0.779342 0.626598i \(-0.784447\pi\)
−0.779342 + 0.626598i \(0.784447\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 39.4728i 1.78138i 0.454610 + 0.890691i \(0.349779\pi\)
−0.454610 + 0.890691i \(0.650221\pi\)
\(492\) 0 0
\(493\) 16.8344i 0.758182i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −5.51938 −0.247578
\(498\) 0 0
\(499\) 19.1017 0.855108 0.427554 0.903990i \(-0.359375\pi\)
0.427554 + 0.903990i \(0.359375\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10.1989 −0.454749 −0.227374 0.973807i \(-0.573014\pi\)
−0.227374 + 0.973807i \(0.573014\pi\)
\(504\) 0 0
\(505\) 6.21263i 0.276458i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.706053i 0.0312953i −0.999878 0.0156476i \(-0.995019\pi\)
0.999878 0.0156476i \(-0.00498100\pi\)
\(510\) 0 0
\(511\) 27.4501i 1.21432i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.57047i 0.157334i
\(516\) 0 0
\(517\) 7.07517i 0.311166i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 38.3159 1.67865 0.839325 0.543630i \(-0.182951\pi\)
0.839325 + 0.543630i \(0.182951\pi\)
\(522\) 0 0
\(523\) 28.1352i 1.23027i 0.788423 + 0.615133i \(0.210898\pi\)
−0.788423 + 0.615133i \(0.789102\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −17.5885 −0.766169
\(528\) 0 0
\(529\) 16.1057 + 16.4197i 0.700249 + 0.713898i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 23.5829i 1.02149i
\(534\) 0 0
\(535\) 18.4060 0.795762
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.90114 −0.254180
\(540\) 0 0
\(541\) −21.9400 −0.943272 −0.471636 0.881793i \(-0.656336\pi\)
−0.471636 + 0.881793i \(0.656336\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 17.6728i 0.757020i
\(546\) 0 0
\(547\) −0.611160 −0.0261313 −0.0130657 0.999915i \(-0.504159\pi\)
−0.0130657 + 0.999915i \(0.504159\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 26.4053 1.12490
\(552\) 0 0
\(553\) −28.1733 −1.19805
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.5939 1.04208 0.521039 0.853533i \(-0.325545\pi\)
0.521039 + 0.853533i \(0.325545\pi\)
\(558\) 0 0
\(559\) 9.66767i 0.408899i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.35384 −0.225637 −0.112819 0.993616i \(-0.535988\pi\)
−0.112819 + 0.993616i \(0.535988\pi\)
\(564\) 0 0
\(565\) 11.9644 0.503345
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.18277 −0.133429 −0.0667144 0.997772i \(-0.521252\pi\)
−0.0667144 + 0.997772i \(0.521252\pi\)
\(570\) 0 0
\(571\) 20.9849i 0.878191i −0.898440 0.439096i \(-0.855299\pi\)
0.898440 0.439096i \(-0.144701\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.42186 + 1.85665i 0.184404 + 0.0774275i
\(576\) 0 0
\(577\) −20.1387 −0.838384 −0.419192 0.907897i \(-0.637687\pi\)
−0.419192 + 0.907897i \(0.637687\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.01584i 0.0421442i
\(582\) 0 0
\(583\) −0.356442 −0.0147623
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.4769i 1.87703i 0.345237 + 0.938516i \(0.387799\pi\)
−0.345237 + 0.938516i \(0.612201\pi\)
\(588\) 0 0
\(589\) 27.5882i 1.13675i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 24.0282i 0.986722i 0.869825 + 0.493361i \(0.164232\pi\)
−0.869825 + 0.493361i \(0.835768\pi\)
\(594\) 0 0
\(595\) 5.61405i 0.230154i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.54733i 0.104081i 0.998645 + 0.0520406i \(0.0165725\pi\)
−0.998645 + 0.0520406i \(0.983427\pi\)
\(600\) 0 0
\(601\) 12.7927 0.521825 0.260913 0.965362i \(-0.415977\pi\)
0.260913 + 0.965362i \(0.415977\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8.79340 0.357503
\(606\) 0 0
\(607\) 22.4815 0.912497 0.456248 0.889852i \(-0.349193\pi\)
0.456248 + 0.889852i \(0.349193\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 10.2218i 0.413528i
\(612\) 0 0
\(613\) 1.99075i 0.0804055i −0.999192 0.0402027i \(-0.987200\pi\)
0.999192 0.0402027i \(-0.0128004\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −46.5579 −1.87435 −0.937175 0.348860i \(-0.886569\pi\)
−0.937175 + 0.348860i \(0.886569\pi\)
\(618\) 0 0
\(619\) 25.2164i 1.01353i −0.862084 0.506766i \(-0.830841\pi\)
0.862084 0.506766i \(-0.169159\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 25.5787i 1.02479i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9.28804i 0.370338i
\(630\) 0 0
\(631\) 49.3547i 1.96478i 0.186840 + 0.982390i \(0.440175\pi\)
−0.186840 + 0.982390i \(0.559825\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 15.2155 0.603810
\(636\) 0 0
\(637\) −8.52560 −0.337797
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7.85956 −0.310434 −0.155217 0.987880i \(-0.549608\pi\)
−0.155217 + 0.987880i \(0.549608\pi\)
\(642\) 0 0
\(643\) 8.93166i 0.352230i 0.984370 + 0.176115i \(0.0563531\pi\)
−0.984370 + 0.176115i \(0.943647\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.29757i 0.129641i −0.997897 0.0648204i \(-0.979353\pi\)
0.997897 0.0648204i \(-0.0206474\pi\)
\(648\) 0 0
\(649\) 7.72807i 0.303353i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 33.1565i 1.29752i 0.760995 + 0.648758i \(0.224711\pi\)
−0.760995 + 0.648758i \(0.775289\pi\)
\(654\) 0 0
\(655\) 21.1936i 0.828103i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −3.27163 −0.127444 −0.0637222 0.997968i \(-0.520297\pi\)
−0.0637222 + 0.997968i \(0.520297\pi\)
\(660\) 0 0
\(661\) 46.2040i 1.79713i 0.438844 + 0.898563i \(0.355388\pi\)
−0.438844 + 0.898563i \(0.644612\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.80583 −0.341475
\(666\) 0 0
\(667\) 9.68690 23.0707i 0.375078 0.893302i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9.36587i 0.361565i
\(672\) 0 0
\(673\) 42.4260 1.63540 0.817701 0.575643i \(-0.195248\pi\)
0.817701 + 0.575643i \(0.195248\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −30.8444 −1.18545 −0.592723 0.805407i \(-0.701947\pi\)
−0.592723 + 0.805407i \(0.701947\pi\)
\(678\) 0 0
\(679\) −2.89070 −0.110935
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.4892i 0.477887i −0.971033 0.238944i \(-0.923199\pi\)
0.971033 0.238944i \(-0.0768011\pi\)
\(684\) 0 0
\(685\) 19.3615 0.739767
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.514965 −0.0196186
\(690\) 0 0
\(691\) 28.2420 1.07438 0.537189 0.843462i \(-0.319486\pi\)
0.537189 + 0.843462i \(0.319486\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.3341 0.467859
\(696\) 0 0
\(697\) 35.4558i 1.34299i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 14.8720 0.561708 0.280854 0.959751i \(-0.409382\pi\)
0.280854 + 0.959751i \(0.409382\pi\)
\(702\) 0 0
\(703\) 14.5686 0.549466
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.8096 0.406538
\(708\) 0 0
\(709\) 24.4780i 0.919290i −0.888103 0.459645i \(-0.847977\pi\)
0.888103 0.459645i \(-0.152023\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −24.1043 10.1209i −0.902712 0.379029i
\(714\) 0 0
\(715\) −3.18795 −0.119223
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 14.4680i 0.539566i 0.962921 + 0.269783i \(0.0869520\pi\)
−0.962921 + 0.269783i \(0.913048\pi\)
\(720\) 0 0
\(721\) 6.21242 0.231362
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.21742i 0.193770i
\(726\) 0 0
\(727\) 36.4058i 1.35022i −0.737718 0.675109i \(-0.764097\pi\)
0.737718 0.675109i \(-0.235903\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 14.5349i 0.537593i
\(732\) 0 0
\(733\) 39.6907i 1.46601i −0.680223 0.733005i \(-0.738117\pi\)
0.680223 0.733005i \(-0.261883\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.87815i 0.142854i
\(738\) 0 0
\(739\) −26.6534 −0.980462 −0.490231 0.871592i \(-0.663088\pi\)
−0.490231 + 0.871592i \(0.663088\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −4.33994 −0.159217 −0.0796085 0.996826i \(-0.525367\pi\)
−0.0796085 + 0.996826i \(0.525367\pi\)
\(744\) 0 0
\(745\) −12.5487 −0.459749
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 32.0255i 1.17018i
\(750\) 0 0
\(751\) 22.0717i 0.805408i 0.915330 + 0.402704i \(0.131930\pi\)
−0.915330 + 0.402704i \(0.868070\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −17.3650 −0.631975
\(756\) 0 0
\(757\) 28.4839i 1.03527i −0.855603 0.517633i \(-0.826813\pi\)
0.855603 0.517633i \(-0.173187\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 29.8896i 1.08350i −0.840540 0.541749i \(-0.817762\pi\)
0.840540 0.541749i \(-0.182238\pi\)
\(762\) 0 0
\(763\) 30.7497 1.11321
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.1650i 0.403146i
\(768\) 0 0
\(769\) 1.87428i 0.0675881i −0.999429 0.0337941i \(-0.989241\pi\)
0.999429 0.0337941i \(-0.0107590\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6.80634 −0.244807 −0.122404 0.992480i \(-0.539060\pi\)
−0.122404 + 0.992480i \(0.539060\pi\)
\(774\) 0 0
\(775\) −5.45116 −0.195811
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 55.6137 1.99257
\(780\) 0 0
\(781\) 4.71212i 0.168613i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 17.5785i 0.627403i
\(786\) 0 0
\(787\) 33.9079i 1.20869i −0.796724 0.604344i \(-0.793435\pi\)
0.796724 0.604344i \(-0.206565\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 20.8173i 0.740179i
\(792\) 0 0
\(793\) 13.5312i 0.480508i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0.345368 0.0122336 0.00611678 0.999981i \(-0.498053\pi\)
0.00611678 + 0.999981i \(0.498053\pi\)
\(798\) 0 0
\(799\) 15.3680i 0.543679i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 23.4353 0.827014
\(804\) 0 0
\(805\) −3.23046 + 7.69379i −0.113859 + 0.271170i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 8.81115i 0.309784i −0.987931 0.154892i \(-0.950497\pi\)
0.987931 0.154892i \(-0.0495029\pi\)
\(810\) 0 0
\(811\) 9.96662 0.349975 0.174988 0.984571i \(-0.444011\pi\)
0.174988 + 0.984571i \(0.444011\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −15.0296 −0.526466
\(816\) 0 0
\(817\) −22.7985 −0.797619
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 28.9308i 1.00969i 0.863210 + 0.504845i \(0.168450\pi\)
−0.863210 + 0.504845i \(0.831550\pi\)
\(822\) 0 0
\(823\) 33.4278 1.16522 0.582610 0.812752i \(-0.302032\pi\)
0.582610 + 0.812752i \(0.302032\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.3409 0.394363 0.197182 0.980367i \(-0.436821\pi\)
0.197182 + 0.980367i \(0.436821\pi\)
\(828\) 0 0
\(829\) 0.900479 0.0312749 0.0156375 0.999878i \(-0.495022\pi\)
0.0156375 + 0.999878i \(0.495022\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 12.8179 0.444113
\(834\) 0 0
\(835\) 11.4559i 0.396448i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 36.8397 1.27185 0.635924 0.771751i \(-0.280619\pi\)
0.635924 + 0.771751i \(0.280619\pi\)
\(840\) 0 0
\(841\) 1.77850 0.0613275
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8.39424 0.288771
\(846\) 0 0
\(847\) 15.3000i 0.525715i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 5.34456 12.7288i 0.183209 0.436338i
\(852\) 0 0
\(853\) −16.7815 −0.574588 −0.287294 0.957843i \(-0.592756\pi\)
−0.287294 + 0.957843i \(0.592756\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 22.6280i 0.772957i −0.922298 0.386479i \(-0.873691\pi\)
0.922298 0.386479i \(-0.126309\pi\)
\(858\) 0 0
\(859\) −0.122682 −0.00418587 −0.00209293 0.999998i \(-0.500666\pi\)
−0.00209293 + 0.999998i \(0.500666\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12.3766i 0.421306i 0.977561 + 0.210653i \(0.0675590\pi\)
−0.977561 + 0.210653i \(0.932441\pi\)
\(864\) 0 0
\(865\) 7.36507i 0.250420i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 24.0527i 0.815932i
\(870\) 0 0
\(871\) 5.60291i 0.189847i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.73994i 0.0588208i
\(876\) 0 0
\(877\) −49.1426 −1.65943 −0.829714 0.558188i \(-0.811497\pi\)
−0.829714 + 0.558188i \(0.811497\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −10.8206 −0.364554 −0.182277 0.983247i \(-0.558347\pi\)
−0.182277 + 0.983247i \(0.558347\pi\)
\(882\) 0 0
\(883\) −40.8514 −1.37476 −0.687380 0.726298i \(-0.741239\pi\)
−0.687380 + 0.726298i \(0.741239\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 45.5013i 1.52779i 0.645343 + 0.763893i \(0.276714\pi\)
−0.645343 + 0.763893i \(0.723286\pi\)
\(888\) 0 0
\(889\) 26.4742i 0.887916i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 24.1052 0.806649
\(894\) 0 0
\(895\) 13.0710i 0.436916i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 28.4410i 0.948560i
\(900\) 0 0
\(901\) 0.774227 0.0257932
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.0486i 0.633198i
\(906\) 0 0
\(907\) 40.4240i 1.34226i −0.741341 0.671129i \(-0.765810\pi\)
0.741341 0.671129i \(-0.234190\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 14.5112 0.480776 0.240388 0.970677i \(-0.422725\pi\)
0.240388 + 0.970677i \(0.422725\pi\)
\(912\) 0 0
\(913\) −0.867265 −0.0287023
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −36.8757 −1.21774
\(918\) 0 0
\(919\) 57.0808i 1.88292i −0.337124 0.941460i \(-0.609454\pi\)
0.337124 0.941460i \(-0.390546\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.80778i 0.224081i
\(924\) 0 0
\(925\) 2.87861i 0.0946482i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.48787i 0.245669i 0.992427 + 0.122835i \(0.0391984\pi\)
−0.992427 + 0.122835i \(0.960802\pi\)
\(930\) 0 0
\(931\) 20.1053i 0.658923i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.79295 0.156746
\(936\) 0 0
\(937\) 5.76692i 0.188397i 0.995553 + 0.0941986i \(0.0300289\pi\)
−0.995553 + 0.0941986i \(0.969971\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 22.0222 0.717902 0.358951 0.933356i \(-0.383134\pi\)
0.358951 + 0.933356i \(0.383134\pi\)
\(942\) 0 0
\(943\) 20.4021 48.5906i 0.664385 1.58233i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40.0318i 1.30086i 0.759567 + 0.650429i \(0.225411\pi\)
−0.759567 + 0.650429i \(0.774589\pi\)
\(948\) 0 0
\(949\) 33.8579 1.09907
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 28.9291 0.937105 0.468553 0.883436i \(-0.344776\pi\)
0.468553 + 0.883436i \(0.344776\pi\)
\(954\) 0 0
\(955\) −4.27526 −0.138344
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 33.6880i 1.08784i
\(960\) 0 0
\(961\) −1.28490 −0.0414484
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 18.8399 0.606478
\(966\) 0 0
\(967\) 47.3299 1.52203 0.761014 0.648736i \(-0.224702\pi\)
0.761014 + 0.648736i \(0.224702\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 38.0244 1.22026 0.610131 0.792301i \(-0.291117\pi\)
0.610131 + 0.792301i \(0.291117\pi\)
\(972\) 0 0
\(973\) 21.4606i 0.687997i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −20.4601 −0.654575 −0.327288 0.944925i \(-0.606135\pi\)
−0.327288 + 0.944925i \(0.606135\pi\)
\(978\) 0 0
\(979\) 21.8376 0.697934
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 20.7804 0.662792 0.331396 0.943492i \(-0.392480\pi\)
0.331396 + 0.943492i \(0.392480\pi\)
\(984\) 0 0
\(985\) 26.8545i 0.855656i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.36374 + 19.9194i −0.265951 + 0.633401i
\(990\) 0 0
\(991\) 9.91895 0.315086 0.157543 0.987512i \(-0.449643\pi\)
0.157543 + 0.987512i \(0.449643\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 10.8944i 0.345375i
\(996\) 0 0
\(997\) 34.8855 1.10484 0.552418 0.833568i \(-0.313705\pi\)
0.552418 + 0.833568i \(0.313705\pi\)
\(998\) 0 0
\(999\) 0 0
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 4140.2.i.a.1241.5 16
3.2 odd 2 4140.2.i.b.1241.5 yes 16
23.22 odd 2 4140.2.i.b.1241.12 yes 16
69.68 even 2 inner 4140.2.i.a.1241.12 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.i.a.1241.5 16 1.1 even 1 trivial
4140.2.i.a.1241.12 yes 16 69.68 even 2 inner
4140.2.i.b.1241.5 yes 16 3.2 odd 2
4140.2.i.b.1241.12 yes 16 23.22 odd 2