Properties

Label 4032.2.b.r.3583.3
Level $4032$
Weight $2$
Character 4032.3583
Analytic conductor $32.196$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(3583,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.3583");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.29960650073923649536.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{30} \)
Twist minimal: no (minimal twist has level 2016)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3583.3
Root \(-1.32968 - 0.481610i\) of defining polynomial
Character \(\chi\) \(=\) 4032.3583
Dual form 4032.2.b.r.3583.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.33513i q^{5} +(0.662153 - 2.56155i) q^{7} +O(q^{10})\) \(q-3.33513i q^{5} +(0.662153 - 2.56155i) q^{7} -5.03680i q^{11} +6.04090i q^{13} -5.20798i q^{17} -4.71659 q^{19} +2.20837i q^{23} -6.12311 q^{25} -7.24517 q^{29} +6.04090 q^{31} +(-8.54312 - 2.20837i) q^{35} -5.12311 q^{37} -5.20798i q^{41} -9.12311i q^{43} +3.74571 q^{47} +(-6.12311 - 3.39228i) q^{49} -1.58831 q^{53} -16.7984 q^{55} +13.3405 q^{59} -12.8255i q^{61} +20.1472 q^{65} +9.12311i q^{67} +12.2820i q^{71} +12.0818i q^{73} +(-12.9020 - 3.33513i) q^{77} +5.12311i q^{79} -3.74571 q^{83} -17.3693 q^{85} +1.46228i q^{89} +(15.4741 + 4.00000i) q^{91} +15.7304i q^{95} -6.78456i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{25} - 16 q^{37} - 32 q^{49} - 80 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(5\) 3.33513i 1.49152i −0.666217 0.745758i \(-0.732087\pi\)
0.666217 0.745758i \(-0.267913\pi\)
\(6\) 0 0
\(7\) 0.662153 2.56155i 0.250270 0.968176i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.03680i 1.51865i −0.650711 0.759326i \(-0.725529\pi\)
0.650711 0.759326i \(-0.274471\pi\)
\(12\) 0 0
\(13\) 6.04090i 1.67544i 0.546098 + 0.837722i \(0.316113\pi\)
−0.546098 + 0.837722i \(0.683887\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.20798i 1.26312i −0.775326 0.631561i \(-0.782415\pi\)
0.775326 0.631561i \(-0.217585\pi\)
\(18\) 0 0
\(19\) −4.71659 −1.08206 −0.541030 0.841003i \(-0.681965\pi\)
−0.541030 + 0.841003i \(0.681965\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.20837i 0.460477i 0.973134 + 0.230238i \(0.0739506\pi\)
−0.973134 + 0.230238i \(0.926049\pi\)
\(24\) 0 0
\(25\) −6.12311 −1.22462
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.24517 −1.34539 −0.672697 0.739918i \(-0.734864\pi\)
−0.672697 + 0.739918i \(0.734864\pi\)
\(30\) 0 0
\(31\) 6.04090 1.08498 0.542488 0.840063i \(-0.317482\pi\)
0.542488 + 0.840063i \(0.317482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.54312 2.20837i −1.44405 0.373283i
\(36\) 0 0
\(37\) −5.12311 −0.842233 −0.421117 0.907006i \(-0.638362\pi\)
−0.421117 + 0.907006i \(0.638362\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.20798i 0.813351i −0.913573 0.406675i \(-0.866688\pi\)
0.913573 0.406675i \(-0.133312\pi\)
\(42\) 0 0
\(43\) 9.12311i 1.39126i −0.718400 0.695630i \(-0.755125\pi\)
0.718400 0.695630i \(-0.244875\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.74571 0.546367 0.273184 0.961962i \(-0.411923\pi\)
0.273184 + 0.961962i \(0.411923\pi\)
\(48\) 0 0
\(49\) −6.12311 3.39228i −0.874729 0.484612i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.58831 −0.218171 −0.109086 0.994032i \(-0.534792\pi\)
−0.109086 + 0.994032i \(0.534792\pi\)
\(54\) 0 0
\(55\) −16.7984 −2.26509
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 13.3405 1.73679 0.868394 0.495874i \(-0.165152\pi\)
0.868394 + 0.495874i \(0.165152\pi\)
\(60\) 0 0
\(61\) 12.8255i 1.64213i −0.570833 0.821066i \(-0.693380\pi\)
0.570833 0.821066i \(-0.306620\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 20.1472 2.49895
\(66\) 0 0
\(67\) 9.12311i 1.11456i 0.830323 + 0.557282i \(0.188156\pi\)
−0.830323 + 0.557282i \(0.811844\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.2820i 1.45760i 0.684726 + 0.728800i \(0.259922\pi\)
−0.684726 + 0.728800i \(0.740078\pi\)
\(72\) 0 0
\(73\) 12.0818i 1.41407i 0.707180 + 0.707033i \(0.249967\pi\)
−0.707180 + 0.707033i \(0.750033\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.9020 3.33513i −1.47032 0.380074i
\(78\) 0 0
\(79\) 5.12311i 0.576394i 0.957571 + 0.288197i \(0.0930559\pi\)
−0.957571 + 0.288197i \(0.906944\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.74571 −0.411145 −0.205572 0.978642i \(-0.565906\pi\)
−0.205572 + 0.978642i \(0.565906\pi\)
\(84\) 0 0
\(85\) −17.3693 −1.88397
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.46228i 0.155001i 0.996992 + 0.0775006i \(0.0246940\pi\)
−0.996992 + 0.0775006i \(0.975306\pi\)
\(90\) 0 0
\(91\) 15.4741 + 4.00000i 1.62212 + 0.419314i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.7304i 1.61391i
\(96\) 0 0
\(97\) 6.78456i 0.688868i −0.938811 0.344434i \(-0.888071\pi\)
0.938811 0.344434i \(-0.111929\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.08084i 0.704570i 0.935893 + 0.352285i \(0.114595\pi\)
−0.935893 + 0.352285i \(0.885405\pi\)
\(102\) 0 0
\(103\) 12.8255 1.26373 0.631865 0.775078i \(-0.282290\pi\)
0.631865 + 0.775078i \(0.282290\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.79668i 0.367039i −0.983016 0.183519i \(-0.941251\pi\)
0.983016 0.183519i \(-0.0587490\pi\)
\(108\) 0 0
\(109\) −8.24621 −0.789844 −0.394922 0.918715i \(-0.629228\pi\)
−0.394922 + 0.918715i \(0.629228\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −5.65685 −0.532152 −0.266076 0.963952i \(-0.585727\pi\)
−0.266076 + 0.963952i \(0.585727\pi\)
\(114\) 0 0
\(115\) 7.36520 0.686809
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −13.3405 3.44849i −1.22292 0.316122i
\(120\) 0 0
\(121\) −14.3693 −1.30630
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.74571i 0.335026i
\(126\) 0 0
\(127\) 2.87689i 0.255283i −0.991820 0.127642i \(-0.959259\pi\)
0.991820 0.127642i \(-0.0407407\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.3405 −1.16557 −0.582784 0.812627i \(-0.698037\pi\)
−0.582784 + 0.812627i \(0.698037\pi\)
\(132\) 0 0
\(133\) −3.12311 + 12.0818i −0.270808 + 1.04762i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.17662 0.271397 0.135699 0.990750i \(-0.456672\pi\)
0.135699 + 0.990750i \(0.456672\pi\)
\(138\) 0 0
\(139\) −5.29723 −0.449305 −0.224652 0.974439i \(-0.572125\pi\)
−0.224652 + 0.974439i \(0.572125\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 30.4268 2.54441
\(144\) 0 0
\(145\) 24.1636i 2.00668i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.58831 0.130120 0.0650598 0.997881i \(-0.479276\pi\)
0.0650598 + 0.997881i \(0.479276\pi\)
\(150\) 0 0
\(151\) 2.87689i 0.234118i 0.993125 + 0.117059i \(0.0373467\pi\)
−0.993125 + 0.117059i \(0.962653\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 20.1472i 1.61826i
\(156\) 0 0
\(157\) 11.3381i 0.904881i 0.891795 + 0.452440i \(0.149446\pi\)
−0.891795 + 0.452440i \(0.850554\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.65685 + 1.46228i 0.445823 + 0.115244i
\(162\) 0 0
\(163\) 1.12311i 0.0879684i 0.999032 + 0.0439842i \(0.0140051\pi\)
−0.999032 + 0.0439842i \(0.985995\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −23.4924 −1.80711
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.4214i 1.55261i 0.630359 + 0.776304i \(0.282908\pi\)
−0.630359 + 0.776304i \(0.717092\pi\)
\(174\) 0 0
\(175\) −4.05444 + 15.6847i −0.306487 + 1.18565i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 22.0074i 1.64491i −0.568832 0.822454i \(-0.692605\pi\)
0.568832 0.822454i \(-0.307395\pi\)
\(180\) 0 0
\(181\) 6.04090i 0.449016i 0.974472 + 0.224508i \(0.0720775\pi\)
−0.974472 + 0.224508i \(0.927922\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 17.0862i 1.25620i
\(186\) 0 0
\(187\) −26.2316 −1.91824
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.10534i 0.658839i −0.944184 0.329420i \(-0.893147\pi\)
0.944184 0.329420i \(-0.106853\pi\)
\(192\) 0 0
\(193\) −11.3693 −0.818381 −0.409191 0.912449i \(-0.634189\pi\)
−0.409191 + 0.912449i \(0.634189\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −7.24517 −0.516197 −0.258098 0.966119i \(-0.583096\pi\)
−0.258098 + 0.966119i \(0.583096\pi\)
\(198\) 0 0
\(199\) 8.10887 0.574823 0.287411 0.957807i \(-0.407205\pi\)
0.287411 + 0.957807i \(0.407205\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.79741 + 18.5589i −0.336712 + 1.30258i
\(204\) 0 0
\(205\) −17.3693 −1.21313
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 23.7565i 1.64327i
\(210\) 0 0
\(211\) 3.36932i 0.231953i 0.993252 + 0.115977i \(0.0369998\pi\)
−0.993252 + 0.115977i \(0.963000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −30.4268 −2.07509
\(216\) 0 0
\(217\) 4.00000 15.4741i 0.271538 1.05045i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 31.4609 2.11629
\(222\) 0 0
\(223\) 3.97292 0.266046 0.133023 0.991113i \(-0.457532\pi\)
0.133023 + 0.991113i \(0.457532\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.59482 0.636831 0.318415 0.947951i \(-0.396849\pi\)
0.318415 + 0.947951i \(0.396849\pi\)
\(228\) 0 0
\(229\) 11.3381i 0.749244i 0.927178 + 0.374622i \(0.122228\pi\)
−0.927178 + 0.374622i \(0.877772\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.9706 1.11178 0.555889 0.831256i \(-0.312378\pi\)
0.555889 + 0.831256i \(0.312378\pi\)
\(234\) 0 0
\(235\) 12.4924i 0.814916i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.1154i 1.36584i −0.730491 0.682922i \(-0.760709\pi\)
0.730491 0.682922i \(-0.239291\pi\)
\(240\) 0 0
\(241\) 18.8664i 1.21529i −0.794209 0.607644i \(-0.792115\pi\)
0.794209 0.607644i \(-0.207885\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −11.3137 + 20.4214i −0.722806 + 1.30467i
\(246\) 0 0
\(247\) 28.4924i 1.81293i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.0862 1.07847 0.539237 0.842154i \(-0.318713\pi\)
0.539237 + 0.842154i \(0.318713\pi\)
\(252\) 0 0
\(253\) 11.1231 0.699304
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.8028i 0.923374i −0.887043 0.461687i \(-0.847244\pi\)
0.887043 0.461687i \(-0.152756\pi\)
\(258\) 0 0
\(259\) −3.39228 + 13.1231i −0.210786 + 0.815430i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.7622i 0.910276i 0.890421 + 0.455138i \(0.150410\pi\)
−0.890421 + 0.455138i \(0.849590\pi\)
\(264\) 0 0
\(265\) 5.29723i 0.325406i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 17.4968i 1.06680i −0.845863 0.533400i \(-0.820914\pi\)
0.845863 0.533400i \(-0.179086\pi\)
\(270\) 0 0
\(271\) −0.743668 −0.0451746 −0.0225873 0.999745i \(-0.507190\pi\)
−0.0225873 + 0.999745i \(0.507190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 30.8408i 1.85977i
\(276\) 0 0
\(277\) −23.3693 −1.40413 −0.702063 0.712115i \(-0.747738\pi\)
−0.702063 + 0.712115i \(0.747738\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.65685 −0.337460 −0.168730 0.985662i \(-0.553967\pi\)
−0.168730 + 0.985662i \(0.553967\pi\)
\(282\) 0 0
\(283\) −7.36520 −0.437816 −0.218908 0.975746i \(-0.570249\pi\)
−0.218908 + 0.975746i \(0.570249\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −13.3405 3.44849i −0.787466 0.203558i
\(288\) 0 0
\(289\) −10.1231 −0.595477
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.15628i 0.242812i −0.992603 0.121406i \(-0.961260\pi\)
0.992603 0.121406i \(-0.0387404\pi\)
\(294\) 0 0
\(295\) 44.4924i 2.59045i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −13.3405 −0.771503
\(300\) 0 0
\(301\) −23.3693 6.04090i −1.34699 0.348191i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −42.7746 −2.44927
\(306\) 0 0
\(307\) 12.6624 0.722683 0.361342 0.932434i \(-0.382319\pi\)
0.361342 + 0.932434i \(0.382319\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 22.9354 1.30054 0.650272 0.759701i \(-0.274655\pi\)
0.650272 + 0.759701i \(0.274655\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −24.2157 −1.36009 −0.680045 0.733170i \(-0.738040\pi\)
−0.680045 + 0.733170i \(0.738040\pi\)
\(318\) 0 0
\(319\) 36.4924i 2.04318i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.5639i 1.36677i
\(324\) 0 0
\(325\) 36.9890i 2.05178i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.48023 9.59482i 0.136740 0.528980i
\(330\) 0 0
\(331\) 29.6155i 1.62782i −0.580993 0.813908i \(-0.697336\pi\)
0.580993 0.813908i \(-0.302664\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 30.4268 1.66239
\(336\) 0 0
\(337\) −8.24621 −0.449200 −0.224600 0.974451i \(-0.572108\pi\)
−0.224600 + 0.974451i \(0.572108\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 30.4268i 1.64770i
\(342\) 0 0
\(343\) −12.7439 + 13.4384i −0.688108 + 0.725608i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.97330i 0.374347i 0.982327 + 0.187173i \(0.0599326\pi\)
−0.982327 + 0.187173i \(0.940067\pi\)
\(348\) 0 0
\(349\) 0.743668i 0.0398076i −0.999802 0.0199038i \(-0.993664\pi\)
0.999802 0.0199038i \(-0.00633600\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.8782i 0.632215i −0.948723 0.316108i \(-0.897624\pi\)
0.948723 0.316108i \(-0.102376\pi\)
\(354\) 0 0
\(355\) 40.9620 2.17404
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 33.6693i 1.77700i −0.458881 0.888498i \(-0.651749\pi\)
0.458881 0.888498i \(-0.348251\pi\)
\(360\) 0 0
\(361\) 3.24621 0.170853
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 40.2944 2.10910
\(366\) 0 0
\(367\) −26.9752 −1.40810 −0.704048 0.710153i \(-0.748626\pi\)
−0.704048 + 0.710153i \(0.748626\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.05171 + 4.06854i −0.0546018 + 0.211228i
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 43.7673i 2.25413i
\(378\) 0 0
\(379\) 25.1231i 1.29049i −0.763977 0.645244i \(-0.776756\pi\)
0.763977 0.645244i \(-0.223244\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −37.9182 −1.93753 −0.968764 0.247984i \(-0.920232\pi\)
−0.968764 + 0.247984i \(0.920232\pi\)
\(384\) 0 0
\(385\) −11.1231 + 43.0299i −0.566886 + 2.19301i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 29.8726 1.51460 0.757300 0.653067i \(-0.226518\pi\)
0.757300 + 0.653067i \(0.226518\pi\)
\(390\) 0 0
\(391\) 11.5012 0.581638
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 17.0862 0.859702
\(396\) 0 0
\(397\) 7.52823i 0.377831i 0.981993 + 0.188916i \(0.0604973\pi\)
−0.981993 + 0.188916i \(0.939503\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −32.1573 −1.60586 −0.802929 0.596075i \(-0.796726\pi\)
−0.802929 + 0.596075i \(0.796726\pi\)
\(402\) 0 0
\(403\) 36.4924i 1.81782i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 25.8040i 1.27906i
\(408\) 0 0
\(409\) 17.3790i 0.859337i 0.902987 + 0.429669i \(0.141370\pi\)
−0.902987 + 0.429669i \(0.858630\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.83348 34.1725i 0.434667 1.68152i
\(414\) 0 0
\(415\) 12.4924i 0.613229i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 11.2371 0.548969 0.274485 0.961592i \(-0.411493\pi\)
0.274485 + 0.961592i \(0.411493\pi\)
\(420\) 0 0
\(421\) 5.12311 0.249685 0.124842 0.992177i \(-0.460157\pi\)
0.124842 + 0.992177i \(0.460157\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 31.8890i 1.54685i
\(426\) 0 0
\(427\) −32.8531 8.49242i −1.58987 0.410977i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 21.6592i 1.04329i 0.853164 + 0.521643i \(0.174681\pi\)
−0.853164 + 0.521643i \(0.825319\pi\)
\(432\) 0 0
\(433\) 1.48734i 0.0714768i 0.999361 + 0.0357384i \(0.0113783\pi\)
−0.999361 + 0.0357384i \(0.988622\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 10.4160i 0.498263i
\(438\) 0 0
\(439\) −16.0547 −0.766250 −0.383125 0.923697i \(-0.625152\pi\)
−0.383125 + 0.923697i \(0.625152\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 22.0074i 1.04560i 0.852455 + 0.522801i \(0.175113\pi\)
−0.852455 + 0.522801i \(0.824887\pi\)
\(444\) 0 0
\(445\) 4.87689 0.231187
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.83348 −0.416878 −0.208439 0.978035i \(-0.566838\pi\)
−0.208439 + 0.978035i \(0.566838\pi\)
\(450\) 0 0
\(451\) −26.2316 −1.23520
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 13.3405 51.6081i 0.625414 2.41942i
\(456\) 0 0
\(457\) 20.2462 0.947078 0.473539 0.880773i \(-0.342976\pi\)
0.473539 + 0.880773i \(0.342976\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 16.6757i 0.776663i −0.921520 0.388331i \(-0.873052\pi\)
0.921520 0.388331i \(-0.126948\pi\)
\(462\) 0 0
\(463\) 1.61553i 0.0750800i 0.999295 + 0.0375400i \(0.0119522\pi\)
−0.999295 + 0.0375400i \(0.988048\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17.0862 −0.790657 −0.395328 0.918540i \(-0.629369\pi\)
−0.395328 + 0.918540i \(0.629369\pi\)
\(468\) 0 0
\(469\) 23.3693 + 6.04090i 1.07909 + 0.278943i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −45.9512 −2.11284
\(474\) 0 0
\(475\) 28.8802 1.32511
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −37.9182 −1.73253 −0.866263 0.499589i \(-0.833485\pi\)
−0.866263 + 0.499589i \(0.833485\pi\)
\(480\) 0 0
\(481\) 30.9481i 1.41111i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −22.6274 −1.02746
\(486\) 0 0
\(487\) 9.61553i 0.435721i −0.975980 0.217861i \(-0.930092\pi\)
0.975980 0.217861i \(-0.0699078\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.10029i 0.139914i −0.997550 0.0699571i \(-0.977714\pi\)
0.997550 0.0699571i \(-0.0222862\pi\)
\(492\) 0 0
\(493\) 37.7327i 1.69940i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 31.4609 + 8.13254i 1.41121 + 0.364794i
\(498\) 0 0
\(499\) 31.8617i 1.42633i 0.700998 + 0.713164i \(0.252738\pi\)
−0.700998 + 0.713164i \(0.747262\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3.74571 −0.167013 −0.0835064 0.996507i \(-0.526612\pi\)
−0.0835064 + 0.996507i \(0.526612\pi\)
\(504\) 0 0
\(505\) 23.6155 1.05088
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.43854i 0.241059i 0.992710 + 0.120530i \(0.0384593\pi\)
−0.992710 + 0.120530i \(0.961541\pi\)
\(510\) 0 0
\(511\) 30.9481 + 8.00000i 1.36907 + 0.353899i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 42.7746i 1.88487i
\(516\) 0 0
\(517\) 18.8664i 0.829741i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8.13254i 0.356293i −0.984004 0.178147i \(-0.942990\pi\)
0.984004 0.178147i \(-0.0570101\pi\)
\(522\) 0 0
\(523\) 20.3537 0.890005 0.445002 0.895529i \(-0.353203\pi\)
0.445002 + 0.895529i \(0.353203\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 31.4609i 1.37046i
\(528\) 0 0
\(529\) 18.1231 0.787961
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 31.4609 1.36272
\(534\) 0 0
\(535\) −12.6624 −0.547445
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −17.0862 + 30.8408i −0.735956 + 1.32841i
\(540\) 0 0
\(541\) 9.61553 0.413404 0.206702 0.978404i \(-0.433727\pi\)
0.206702 + 0.978404i \(0.433727\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 27.5022i 1.17806i
\(546\) 0 0
\(547\) 30.8769i 1.32020i −0.751177 0.660100i \(-0.770514\pi\)
0.751177 0.660100i \(-0.229486\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 34.1725 1.45580
\(552\) 0 0
\(553\) 13.1231 + 3.39228i 0.558051 + 0.144255i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.54877 0.277480 0.138740 0.990329i \(-0.455695\pi\)
0.138740 + 0.990329i \(0.455695\pi\)
\(558\) 0 0
\(559\) 55.1117 2.33098
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 37.9182 1.59806 0.799030 0.601291i \(-0.205347\pi\)
0.799030 + 0.601291i \(0.205347\pi\)
\(564\) 0 0
\(565\) 18.8664i 0.793714i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.83348 −0.370319 −0.185159 0.982709i \(-0.559280\pi\)
−0.185159 + 0.982709i \(0.559280\pi\)
\(570\) 0 0
\(571\) 33.1231i 1.38616i 0.720861 + 0.693079i \(0.243746\pi\)
−0.720861 + 0.693079i \(0.756254\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.5221i 0.563910i
\(576\) 0 0
\(577\) 22.6762i 0.944025i −0.881592 0.472012i \(-0.843528\pi\)
0.881592 0.472012i \(-0.156472\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.48023 + 9.59482i −0.102897 + 0.398060i
\(582\) 0 0
\(583\) 8.00000i 0.331326i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.49141 −0.309204 −0.154602 0.987977i \(-0.549409\pi\)
−0.154602 + 0.987977i \(0.549409\pi\)
\(588\) 0 0
\(589\) −28.4924 −1.17401
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 28.1433i 1.15571i −0.816140 0.577854i \(-0.803890\pi\)
0.816140 0.577854i \(-0.196110\pi\)
\(594\) 0 0
\(595\) −11.5012 + 44.4924i −0.471501 + 1.82401i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 34.3657i 1.40414i −0.712107 0.702071i \(-0.752259\pi\)
0.712107 0.702071i \(-0.247741\pi\)
\(600\) 0 0
\(601\) 6.78456i 0.276748i 0.990380 + 0.138374i \(0.0441876\pi\)
−0.990380 + 0.138374i \(0.955812\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 47.9236i 1.94837i
\(606\) 0 0
\(607\) −9.59621 −0.389498 −0.194749 0.980853i \(-0.562389\pi\)
−0.194749 + 0.980853i \(0.562389\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 22.6274i 0.915407i
\(612\) 0 0
\(613\) 41.2311 1.66531 0.832653 0.553795i \(-0.186821\pi\)
0.832653 + 0.553795i \(0.186821\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −11.3137 −0.455473 −0.227736 0.973723i \(-0.573132\pi\)
−0.227736 + 0.973723i \(0.573132\pi\)
\(618\) 0 0
\(619\) 8.27190 0.332476 0.166238 0.986086i \(-0.446838\pi\)
0.166238 + 0.986086i \(0.446838\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.74571 + 0.968253i 0.150069 + 0.0387922i
\(624\) 0 0
\(625\) −18.1231 −0.724924
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 26.6811i 1.06384i
\(630\) 0 0
\(631\) 39.3693i 1.56727i −0.621223 0.783634i \(-0.713364\pi\)
0.621223 0.783634i \(-0.286636\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −9.59482 −0.380759
\(636\) 0 0
\(637\) 20.4924 36.9890i 0.811939 1.46556i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8.13709 −0.321396 −0.160698 0.987004i \(-0.551374\pi\)
−0.160698 + 0.987004i \(0.551374\pi\)
\(642\) 0 0
\(643\) −33.0161 −1.30203 −0.651014 0.759065i \(-0.725656\pi\)
−0.651014 + 0.759065i \(0.725656\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.49141 0.294518 0.147259 0.989098i \(-0.452955\pi\)
0.147259 + 0.989098i \(0.452955\pi\)
\(648\) 0 0
\(649\) 67.1935i 2.63758i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −21.0391 −0.823324 −0.411662 0.911337i \(-0.635052\pi\)
−0.411662 + 0.911337i \(0.635052\pi\)
\(654\) 0 0
\(655\) 44.4924i 1.73846i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.03680i 0.196206i 0.995176 + 0.0981029i \(0.0312774\pi\)
−0.995176 + 0.0981029i \(0.968723\pi\)
\(660\) 0 0
\(661\) 0.743668i 0.0289253i 0.999895 + 0.0144627i \(0.00460377\pi\)
−0.999895 + 0.0144627i \(0.995396\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 40.2944 + 10.4160i 1.56255 + 0.403914i
\(666\) 0 0
\(667\) 16.0000i 0.619522i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −64.5992 −2.49383
\(672\) 0 0
\(673\) 29.6155 1.14159 0.570797 0.821091i \(-0.306634\pi\)
0.570797 + 0.821091i \(0.306634\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 26.2705i 1.00966i −0.863220 0.504828i \(-0.831556\pi\)
0.863220 0.504828i \(-0.168444\pi\)
\(678\) 0 0
\(679\) −17.3790 4.49242i −0.666946 0.172403i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 23.2475i 0.889540i −0.895645 0.444770i \(-0.853285\pi\)
0.895645 0.444770i \(-0.146715\pi\)
\(684\) 0 0
\(685\) 10.5945i 0.404793i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.59482i 0.365534i
\(690\) 0 0
\(691\) −6.78456 −0.258097 −0.129048 0.991638i \(-0.541192\pi\)
−0.129048 + 0.991638i \(0.541192\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.6670i 0.670146i
\(696\) 0 0
\(697\) −27.1231 −1.02736
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −12.9020 −0.487303 −0.243651 0.969863i \(-0.578345\pi\)
−0.243651 + 0.969863i \(0.578345\pi\)
\(702\) 0 0
\(703\) 24.1636 0.911347
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.1379 + 4.68860i 0.682147 + 0.176333i
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.3405i 0.499607i
\(714\) 0 0
\(715\) 101.477i 3.79503i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 30.4268 1.13473 0.567363 0.823468i \(-0.307964\pi\)
0.567363 + 0.823468i \(0.307964\pi\)
\(720\) 0 0
\(721\) 8.49242 32.8531i 0.316274 1.22351i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 44.3629 1.64760
\(726\) 0 0
\(727\) 24.9073 0.923759 0.461879 0.886943i \(-0.347175\pi\)
0.461879 + 0.886943i \(0.347175\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −47.5130 −1.75733
\(732\) 0 0
\(733\) 0.743668i 0.0274680i 0.999906 + 0.0137340i \(0.00437181\pi\)
−0.999906 + 0.0137340i \(0.995628\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 45.9512 1.69264
\(738\) 0 0
\(739\) 25.1231i 0.924168i 0.886836 + 0.462084i \(0.152898\pi\)
−0.886836 + 0.462084i \(0.847102\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0023i 0.587068i 0.955949 + 0.293534i \(0.0948314\pi\)
−0.955949 + 0.293534i \(0.905169\pi\)
\(744\) 0 0
\(745\) 5.29723i 0.194075i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.72540 2.51398i −0.355358 0.0918590i
\(750\) 0 0
\(751\) 0.630683i 0.0230140i −0.999934 0.0115070i \(-0.996337\pi\)
0.999934 0.0115070i \(-0.00366287\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 9.59482 0.349191
\(756\) 0 0
\(757\) 30.9848 1.12616 0.563082 0.826401i \(-0.309616\pi\)
0.563082 + 0.826401i \(0.309616\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 42.3050i 1.53356i 0.641913 + 0.766778i \(0.278141\pi\)
−0.641913 + 0.766778i \(0.721859\pi\)
\(762\) 0 0
\(763\) −5.46026 + 21.1231i −0.197675 + 0.764708i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 80.5887i 2.90989i
\(768\) 0 0
\(769\) 43.0299i 1.55170i 0.630918 + 0.775850i \(0.282678\pi\)
−0.630918 + 0.775850i \(0.717322\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0.410574i 0.0147673i 0.999973 + 0.00738365i \(0.00235031\pi\)
−0.999973 + 0.00738365i \(0.997650\pi\)
\(774\) 0 0
\(775\) −36.9890 −1.32869
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.5639i 0.880094i
\(780\) 0 0
\(781\) 61.8617 2.21359
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 37.8141 1.34964
\(786\) 0 0
\(787\) 46.8398 1.66966 0.834830 0.550508i \(-0.185566\pi\)
0.834830 + 0.550508i \(0.185566\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.74571 + 14.4903i −0.133182 + 0.515217i
\(792\) 0 0
\(793\) 77.4773 2.75130
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.9300i 0.458002i −0.973426 0.229001i \(-0.926454\pi\)
0.973426 0.229001i \(-0.0735460\pi\)
\(798\) 0 0
\(799\) 19.5076i 0.690128i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 60.8535 2.14747
\(804\) 0 0
\(805\) 4.87689 18.8664i 0.171888 0.664952i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −54.0883 −1.90164 −0.950822 0.309738i \(-0.899759\pi\)
−0.950822 + 0.309738i \(0.899759\pi\)
\(810\) 0 0
\(811\) −41.5426 −1.45876 −0.729379 0.684110i \(-0.760191\pi\)
−0.729379 + 0.684110i \(0.760191\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.74571 0.131206
\(816\) 0 0
\(817\) 43.0299i 1.50543i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −53.1964 −1.85657 −0.928283 0.371874i \(-0.878716\pi\)
−0.928283 + 0.371874i \(0.878716\pi\)
\(822\) 0 0
\(823\) 19.8617i 0.692337i −0.938172 0.346168i \(-0.887483\pi\)
0.938172 0.346168i \(-0.112517\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.1941i 1.29336i −0.762759 0.646682i \(-0.776156\pi\)
0.762759 0.646682i \(-0.223844\pi\)
\(828\) 0 0
\(829\) 11.3381i 0.393789i −0.980425 0.196895i \(-0.936914\pi\)
0.980425 0.196895i \(-0.0630857\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −17.6670 + 31.8890i −0.612124 + 1.10489i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 34.1725 1.17976 0.589882 0.807489i \(-0.299174\pi\)
0.589882 + 0.807489i \(0.299174\pi\)
\(840\) 0 0
\(841\) 23.4924 0.810084
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 78.3503i 2.69533i
\(846\) 0 0
\(847\) −9.51469 + 36.8078i −0.326929 + 1.26473i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11.3137i 0.387829i
\(852\) 0 0
\(853\) 35.5017i 1.21556i −0.794107 0.607778i \(-0.792061\pi\)
0.794107 0.607778i \(-0.207939\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 27.3222i 0.933308i 0.884440 + 0.466654i \(0.154541\pi\)
−0.884440 + 0.466654i \(0.845459\pi\)
\(858\) 0 0
\(859\) 38.3134 1.30723 0.653617 0.756825i \(-0.273251\pi\)
0.653617 + 0.756825i \(0.273251\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 20.4190i 0.695072i −0.937667 0.347536i \(-0.887018\pi\)
0.937667 0.347536i \(-0.112982\pi\)
\(864\) 0 0
\(865\) 68.1080 2.31574
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 25.8040 0.875342
\(870\) 0 0
\(871\) −55.1117 −1.86739
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.59482 + 2.48023i 0.324364 + 0.0838471i
\(876\) 0 0
\(877\) 8.24621 0.278455 0.139227 0.990260i \(-0.455538\pi\)
0.139227 + 0.990260i \(0.455538\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 20.6519i 0.695781i 0.937535 + 0.347891i \(0.113102\pi\)
−0.937535 + 0.347891i \(0.886898\pi\)
\(882\) 0 0
\(883\) 7.86174i 0.264569i 0.991212 + 0.132284i \(0.0422312\pi\)
−0.991212 + 0.132284i \(0.957769\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 34.1725 1.14740 0.573700 0.819066i \(-0.305508\pi\)
0.573700 + 0.819066i \(0.305508\pi\)
\(888\) 0 0
\(889\) −7.36932 1.90495i −0.247159 0.0638898i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −17.6670 −0.591202
\(894\) 0 0
\(895\) −73.3974 −2.45341
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −43.7673 −1.45972
\(900\) 0 0
\(901\) 8.27190i 0.275577i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.1472 0.669715
\(906\) 0 0
\(907\) 18.1080i 0.601265i −0.953740 0.300632i \(-0.902802\pi\)
0.953740 0.300632i \(-0.0971977\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 12.2820i 0.406920i 0.979083 + 0.203460i \(0.0652186\pi\)
−0.979083 + 0.203460i \(0.934781\pi\)
\(912\) 0 0
\(913\) 18.8664i 0.624385i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −8.83348 + 34.1725i −0.291707 + 1.12847i
\(918\) 0 0
\(919\) 49.6155i 1.63667i −0.574745 0.818333i \(-0.694899\pi\)
0.574745 0.818333i \(-0.305101\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −74.1941 −2.44213
\(924\) 0 0
\(925\) 31.3693 1.03142
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 52.7210i 1.72972i −0.502014 0.864860i \(-0.667407\pi\)
0.502014 0.864860i \(-0.332593\pi\)
\(930\) 0 0
\(931\) 28.8802 + 16.0000i 0.946509 + 0.524379i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 87.4857i 2.86109i
\(936\) 0 0
\(937\) 8.27190i 0.270231i 0.990830 + 0.135116i \(0.0431406\pi\)
−0.990830 + 0.135116i \(0.956859\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 27.9128i 0.909931i −0.890509 0.454965i \(-0.849652\pi\)
0.890509 0.454965i \(-0.150348\pi\)
\(942\) 0 0
\(943\) 11.5012 0.374529
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 46.0276i 1.49569i −0.663871 0.747847i \(-0.731088\pi\)
0.663871 0.747847i \(-0.268912\pi\)
\(948\) 0 0
\(949\) −72.9848 −2.36919
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 25.8040 0.835875 0.417937 0.908476i \(-0.362753\pi\)
0.417937 + 0.908476i \(0.362753\pi\)
\(954\) 0 0
\(955\) −30.3675 −0.982670
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.10341 8.13709i 0.0679227 0.262760i
\(960\) 0 0
\(961\) 5.49242 0.177175
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 37.9182i 1.22063i
\(966\) 0 0
\(967\) 47.3693i 1.52329i −0.647992 0.761647i \(-0.724391\pi\)
0.647992 0.761647i \(-0.275609\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 36.2759 1.16415 0.582074 0.813136i \(-0.302241\pi\)
0.582074 + 0.813136i \(0.302241\pi\)
\(972\) 0 0
\(973\) −3.50758 + 13.5691i −0.112448 + 0.435006i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 49.1278 1.57174 0.785870 0.618392i \(-0.212216\pi\)
0.785870 + 0.618392i \(0.212216\pi\)
\(978\) 0 0
\(979\) 7.36520 0.235393
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −45.8707 −1.46305 −0.731524 0.681816i \(-0.761191\pi\)
−0.731524 + 0.681816i \(0.761191\pi\)
\(984\) 0 0
\(985\) 24.1636i 0.769916i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 20.1472 0.640643
\(990\) 0 0
\(991\) 39.3693i 1.25061i 0.780381 + 0.625304i \(0.215025\pi\)
−0.780381 + 0.625304i \(0.784975\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 27.0442i 0.857357i
\(996\) 0 0
\(997\) 50.5582i 1.60119i 0.599204 + 0.800597i \(0.295484\pi\)
−0.599204 + 0.800597i \(0.704516\pi\)
\(998\) 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 4032.2.b.r.3583.3 16
3.2 odd 2 inner 4032.2.b.r.3583.15 16
4.3 odd 2 inner 4032.2.b.r.3583.2 16
7.6 odd 2 inner 4032.2.b.r.3583.13 16
8.3 odd 2 2016.2.b.d.1567.14 yes 16
8.5 even 2 2016.2.b.d.1567.15 yes 16
12.11 even 2 inner 4032.2.b.r.3583.14 16
21.20 even 2 inner 4032.2.b.r.3583.1 16
24.5 odd 2 2016.2.b.d.1567.3 yes 16
24.11 even 2 2016.2.b.d.1567.2 yes 16
28.27 even 2 inner 4032.2.b.r.3583.16 16
56.13 odd 2 2016.2.b.d.1567.1 16
56.27 even 2 2016.2.b.d.1567.4 yes 16
84.83 odd 2 inner 4032.2.b.r.3583.4 16
168.83 odd 2 2016.2.b.d.1567.16 yes 16
168.125 even 2 2016.2.b.d.1567.13 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2016.2.b.d.1567.1 16 56.13 odd 2
2016.2.b.d.1567.2 yes 16 24.11 even 2
2016.2.b.d.1567.3 yes 16 24.5 odd 2
2016.2.b.d.1567.4 yes 16 56.27 even 2
2016.2.b.d.1567.13 yes 16 168.125 even 2
2016.2.b.d.1567.14 yes 16 8.3 odd 2
2016.2.b.d.1567.15 yes 16 8.5 even 2
2016.2.b.d.1567.16 yes 16 168.83 odd 2
4032.2.b.r.3583.1 16 21.20 even 2 inner
4032.2.b.r.3583.2 16 4.3 odd 2 inner
4032.2.b.r.3583.3 16 1.1 even 1 trivial
4032.2.b.r.3583.4 16 84.83 odd 2 inner
4032.2.b.r.3583.13 16 7.6 odd 2 inner
4032.2.b.r.3583.14 16 12.11 even 2 inner
4032.2.b.r.3583.15 16 3.2 odd 2 inner
4032.2.b.r.3583.16 16 28.27 even 2 inner