Properties

Label 1152.3.j.a.161.13
Level $1152$
Weight $3$
Character 1152.161
Analytic conductor $31.390$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,3,Mod(161,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.j (of order \(4\), degree \(2\), 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: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 161.13
Character \(\chi\) \(=\) 1152.161
Dual form 1152.3.j.a.737.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.90343 + 3.90343i) q^{5} -0.778757i q^{7} +O(q^{10})\) \(q+(3.90343 + 3.90343i) q^{5} -0.778757i q^{7} +(4.29098 + 4.29098i) q^{11} +(-6.44007 - 6.44007i) q^{13} -31.3383i q^{17} +(-1.11906 - 1.11906i) q^{19} +34.2197 q^{23} +5.47346i q^{25} +(8.77471 - 8.77471i) q^{29} +50.8507 q^{31} +(3.03982 - 3.03982i) q^{35} +(29.3064 - 29.3064i) q^{37} -31.4271 q^{41} +(-55.9022 + 55.9022i) q^{43} +26.5249i q^{47} +48.3935 q^{49} +(9.76389 + 9.76389i) q^{53} +33.4991i q^{55} +(-54.3706 - 54.3706i) q^{59} +(47.1104 + 47.1104i) q^{61} -50.2766i q^{65} +(66.1339 + 66.1339i) q^{67} +75.9403 q^{71} +24.1992i q^{73} +(3.34163 - 3.34163i) q^{77} -80.5454 q^{79} +(82.6319 - 82.6319i) q^{83} +(122.327 - 122.327i) q^{85} -82.6445 q^{89} +(-5.01524 + 5.01524i) q^{91} -8.73634i q^{95} +48.9478 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 64 q^{19} - 128 q^{43} - 224 q^{49} - 64 q^{61} + 64 q^{67} - 512 q^{79} - 320 q^{85} + 192 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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
\(5\) 3.90343 + 3.90343i 0.780685 + 0.780685i 0.979946 0.199261i \(-0.0638542\pi\)
−0.199261 + 0.979946i \(0.563854\pi\)
\(6\) 0 0
\(7\) 0.778757i 0.111251i −0.998452 0.0556255i \(-0.982285\pi\)
0.998452 0.0556255i \(-0.0177153\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.29098 + 4.29098i 0.390089 + 0.390089i 0.874719 0.484630i \(-0.161046\pi\)
−0.484630 + 0.874719i \(0.661046\pi\)
\(12\) 0 0
\(13\) −6.44007 6.44007i −0.495390 0.495390i 0.414610 0.909999i \(-0.363918\pi\)
−0.909999 + 0.414610i \(0.863918\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 31.3383i 1.84343i −0.387872 0.921713i \(-0.626790\pi\)
0.387872 0.921713i \(-0.373210\pi\)
\(18\) 0 0
\(19\) −1.11906 1.11906i −0.0588979 0.0588979i 0.677044 0.735942i \(-0.263261\pi\)
−0.735942 + 0.677044i \(0.763261\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 34.2197 1.48781 0.743906 0.668285i \(-0.232971\pi\)
0.743906 + 0.668285i \(0.232971\pi\)
\(24\) 0 0
\(25\) 5.47346i 0.218939i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.77471 8.77471i 0.302576 0.302576i −0.539445 0.842021i \(-0.681366\pi\)
0.842021 + 0.539445i \(0.181366\pi\)
\(30\) 0 0
\(31\) 50.8507 1.64035 0.820173 0.572116i \(-0.193877\pi\)
0.820173 + 0.572116i \(0.193877\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.03982 3.03982i 0.0868520 0.0868520i
\(36\) 0 0
\(37\) 29.3064 29.3064i 0.792065 0.792065i −0.189765 0.981830i \(-0.560773\pi\)
0.981830 + 0.189765i \(0.0607725\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −31.4271 −0.766513 −0.383257 0.923642i \(-0.625198\pi\)
−0.383257 + 0.923642i \(0.625198\pi\)
\(42\) 0 0
\(43\) −55.9022 + 55.9022i −1.30005 + 1.30005i −0.371696 + 0.928354i \(0.621224\pi\)
−0.928354 + 0.371696i \(0.878776\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 26.5249i 0.564359i 0.959362 + 0.282180i \(0.0910574\pi\)
−0.959362 + 0.282180i \(0.908943\pi\)
\(48\) 0 0
\(49\) 48.3935 0.987623
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.76389 + 9.76389i 0.184224 + 0.184224i 0.793194 0.608969i \(-0.208417\pi\)
−0.608969 + 0.793194i \(0.708417\pi\)
\(54\) 0 0
\(55\) 33.4991i 0.609074i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −54.3706 54.3706i −0.921535 0.921535i 0.0756026 0.997138i \(-0.475912\pi\)
−0.997138 + 0.0756026i \(0.975912\pi\)
\(60\) 0 0
\(61\) 47.1104 + 47.1104i 0.772302 + 0.772302i 0.978508 0.206207i \(-0.0661119\pi\)
−0.206207 + 0.978508i \(0.566112\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 50.2766i 0.773487i
\(66\) 0 0
\(67\) 66.1339 + 66.1339i 0.987073 + 0.987073i 0.999918 0.0128446i \(-0.00408867\pi\)
−0.0128446 + 0.999918i \(0.504089\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 75.9403 1.06958 0.534791 0.844985i \(-0.320391\pi\)
0.534791 + 0.844985i \(0.320391\pi\)
\(72\) 0 0
\(73\) 24.1992i 0.331497i 0.986168 + 0.165748i \(0.0530039\pi\)
−0.986168 + 0.165748i \(0.946996\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.34163 3.34163i 0.0433978 0.0433978i
\(78\) 0 0
\(79\) −80.5454 −1.01956 −0.509781 0.860304i \(-0.670274\pi\)
−0.509781 + 0.860304i \(0.670274\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 82.6319 82.6319i 0.995565 0.995565i −0.00442530 0.999990i \(-0.501409\pi\)
0.999990 + 0.00442530i \(0.00140862\pi\)
\(84\) 0 0
\(85\) 122.327 122.327i 1.43914 1.43914i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −82.6445 −0.928590 −0.464295 0.885681i \(-0.653692\pi\)
−0.464295 + 0.885681i \(0.653692\pi\)
\(90\) 0 0
\(91\) −5.01524 + 5.01524i −0.0551126 + 0.0551126i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.73634i 0.0919614i
\(96\) 0 0
\(97\) 48.9478 0.504617 0.252308 0.967647i \(-0.418810\pi\)
0.252308 + 0.967647i \(0.418810\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −41.2778 41.2778i −0.408691 0.408691i 0.472591 0.881282i \(-0.343319\pi\)
−0.881282 + 0.472591i \(0.843319\pi\)
\(102\) 0 0
\(103\) 173.236i 1.68190i 0.541111 + 0.840951i \(0.318004\pi\)
−0.541111 + 0.840951i \(0.681996\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 53.2761 + 53.2761i 0.497907 + 0.497907i 0.910786 0.412879i \(-0.135477\pi\)
−0.412879 + 0.910786i \(0.635477\pi\)
\(108\) 0 0
\(109\) 93.1537 + 93.1537i 0.854621 + 0.854621i 0.990698 0.136077i \(-0.0434494\pi\)
−0.136077 + 0.990698i \(0.543449\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 40.8189i 0.361229i −0.983554 0.180615i \(-0.942191\pi\)
0.983554 0.180615i \(-0.0578087\pi\)
\(114\) 0 0
\(115\) 133.574 + 133.574i 1.16151 + 1.16151i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −24.4049 −0.205083
\(120\) 0 0
\(121\) 84.1749i 0.695661i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 76.2204 76.2204i 0.609763 0.609763i
\(126\) 0 0
\(127\) 88.9136 0.700107 0.350054 0.936730i \(-0.386163\pi\)
0.350054 + 0.936730i \(0.386163\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −22.0108 + 22.0108i −0.168021 + 0.168021i −0.786109 0.618088i \(-0.787908\pi\)
0.618088 + 0.786109i \(0.287908\pi\)
\(132\) 0 0
\(133\) −0.871475 + 0.871475i −0.00655245 + 0.00655245i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 110.794 0.808715 0.404358 0.914601i \(-0.367495\pi\)
0.404358 + 0.914601i \(0.367495\pi\)
\(138\) 0 0
\(139\) −30.3862 + 30.3862i −0.218605 + 0.218605i −0.807911 0.589305i \(-0.799402\pi\)
0.589305 + 0.807911i \(0.299402\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 55.2684i 0.386493i
\(144\) 0 0
\(145\) 68.5029 0.472433
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −173.924 173.924i −1.16727 1.16727i −0.982846 0.184428i \(-0.940957\pi\)
−0.184428 0.982846i \(-0.559043\pi\)
\(150\) 0 0
\(151\) 26.2521i 0.173855i −0.996215 0.0869274i \(-0.972295\pi\)
0.996215 0.0869274i \(-0.0277048\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 198.492 + 198.492i 1.28059 + 1.28059i
\(156\) 0 0
\(157\) −75.8493 75.8493i −0.483117 0.483117i 0.423009 0.906126i \(-0.360974\pi\)
−0.906126 + 0.423009i \(0.860974\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 26.6488i 0.165520i
\(162\) 0 0
\(163\) −155.285 155.285i −0.952672 0.952672i 0.0462578 0.998930i \(-0.485270\pi\)
−0.998930 + 0.0462578i \(0.985270\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 236.902 1.41857 0.709286 0.704920i \(-0.249017\pi\)
0.709286 + 0.704920i \(0.249017\pi\)
\(168\) 0 0
\(169\) 86.0511i 0.509178i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 68.3708 68.3708i 0.395207 0.395207i −0.481332 0.876539i \(-0.659847\pi\)
0.876539 + 0.481332i \(0.159847\pi\)
\(174\) 0 0
\(175\) 4.26250 0.0243571
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 129.626 129.626i 0.724170 0.724170i −0.245282 0.969452i \(-0.578880\pi\)
0.969452 + 0.245282i \(0.0788804\pi\)
\(180\) 0 0
\(181\) −157.641 + 157.641i −0.870947 + 0.870947i −0.992576 0.121629i \(-0.961188\pi\)
0.121629 + 0.992576i \(0.461188\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 228.791 1.23671
\(186\) 0 0
\(187\) 134.472 134.472i 0.719101 0.719101i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 58.1217i 0.304302i −0.988357 0.152151i \(-0.951380\pi\)
0.988357 0.152151i \(-0.0486200\pi\)
\(192\) 0 0
\(193\) 312.491 1.61912 0.809561 0.587035i \(-0.199705\pi\)
0.809561 + 0.587035i \(0.199705\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −44.8306 44.8306i −0.227567 0.227567i 0.584109 0.811675i \(-0.301444\pi\)
−0.811675 + 0.584109i \(0.801444\pi\)
\(198\) 0 0
\(199\) 157.131i 0.789601i 0.918767 + 0.394801i \(0.129186\pi\)
−0.918767 + 0.394801i \(0.870814\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.83336 6.83336i −0.0336619 0.0336619i
\(204\) 0 0
\(205\) −122.673 122.673i −0.598406 0.598406i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.60374i 0.0459509i
\(210\) 0 0
\(211\) 68.3433 + 68.3433i 0.323902 + 0.323902i 0.850262 0.526360i \(-0.176444\pi\)
−0.526360 + 0.850262i \(0.676444\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −436.420 −2.02986
\(216\) 0 0
\(217\) 39.6003i 0.182490i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −201.820 + 201.820i −0.913214 + 0.913214i
\(222\) 0 0
\(223\) −3.75135 −0.0168222 −0.00841111 0.999965i \(-0.502677\pi\)
−0.00841111 + 0.999965i \(0.502677\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −224.963 + 224.963i −0.991028 + 0.991028i −0.999960 0.00893184i \(-0.997157\pi\)
0.00893184 + 0.999960i \(0.497157\pi\)
\(228\) 0 0
\(229\) −26.3030 + 26.3030i −0.114860 + 0.114860i −0.762201 0.647341i \(-0.775881\pi\)
0.647341 + 0.762201i \(0.275881\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −166.492 −0.714557 −0.357279 0.933998i \(-0.616295\pi\)
−0.357279 + 0.933998i \(0.616295\pi\)
\(234\) 0 0
\(235\) −103.538 + 103.538i −0.440587 + 0.440587i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 247.174i 1.03420i −0.855925 0.517100i \(-0.827011\pi\)
0.855925 0.517100i \(-0.172989\pi\)
\(240\) 0 0
\(241\) 14.7552 0.0612247 0.0306124 0.999531i \(-0.490254\pi\)
0.0306124 + 0.999531i \(0.490254\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 188.901 + 188.901i 0.771023 + 0.771023i
\(246\) 0 0
\(247\) 14.4136i 0.0583548i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 41.5560 + 41.5560i 0.165562 + 0.165562i 0.785025 0.619464i \(-0.212650\pi\)
−0.619464 + 0.785025i \(0.712650\pi\)
\(252\) 0 0
\(253\) 146.836 + 146.836i 0.580379 + 0.580379i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 189.265i 0.736439i 0.929739 + 0.368220i \(0.120033\pi\)
−0.929739 + 0.368220i \(0.879967\pi\)
\(258\) 0 0
\(259\) −22.8226 22.8226i −0.0881180 0.0881180i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −57.5273 −0.218735 −0.109368 0.994001i \(-0.534883\pi\)
−0.109368 + 0.994001i \(0.534883\pi\)
\(264\) 0 0
\(265\) 76.2252i 0.287642i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −121.413 + 121.413i −0.451350 + 0.451350i −0.895803 0.444452i \(-0.853398\pi\)
0.444452 + 0.895803i \(0.353398\pi\)
\(270\) 0 0
\(271\) −217.759 −0.803539 −0.401769 0.915741i \(-0.631605\pi\)
−0.401769 + 0.915741i \(0.631605\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −23.4865 + 23.4865i −0.0854056 + 0.0854056i
\(276\) 0 0
\(277\) −56.8037 + 56.8037i −0.205068 + 0.205068i −0.802167 0.597100i \(-0.796320\pi\)
0.597100 + 0.802167i \(0.296320\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −58.9828 −0.209903 −0.104952 0.994477i \(-0.533469\pi\)
−0.104952 + 0.994477i \(0.533469\pi\)
\(282\) 0 0
\(283\) 170.002 170.002i 0.600713 0.600713i −0.339789 0.940502i \(-0.610356\pi\)
0.940502 + 0.339789i \(0.110356\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.4740i 0.0852753i
\(288\) 0 0
\(289\) −693.086 −2.39822
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −46.1968 46.1968i −0.157668 0.157668i 0.623864 0.781533i \(-0.285562\pi\)
−0.781533 + 0.623864i \(0.785562\pi\)
\(294\) 0 0
\(295\) 424.463i 1.43886i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −220.377 220.377i −0.737046 0.737046i
\(300\) 0 0
\(301\) 43.5342 + 43.5342i 0.144632 + 0.144632i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 367.784i 1.20585i
\(306\) 0 0
\(307\) −361.635 361.635i −1.17796 1.17796i −0.980262 0.197701i \(-0.936653\pi\)
−0.197701 0.980262i \(-0.563347\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −175.216 −0.563395 −0.281697 0.959503i \(-0.590897\pi\)
−0.281697 + 0.959503i \(0.590897\pi\)
\(312\) 0 0
\(313\) 432.899i 1.38307i 0.722345 + 0.691533i \(0.243064\pi\)
−0.722345 + 0.691533i \(0.756936\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 180.916 180.916i 0.570714 0.570714i −0.361614 0.932328i \(-0.617774\pi\)
0.932328 + 0.361614i \(0.117774\pi\)
\(318\) 0 0
\(319\) 75.3043 0.236064
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −35.0694 + 35.0694i −0.108574 + 0.108574i
\(324\) 0 0
\(325\) 35.2495 35.2495i 0.108460 0.108460i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 20.6564 0.0627855
\(330\) 0 0
\(331\) −98.3525 + 98.3525i −0.297137 + 0.297137i −0.839892 0.542754i \(-0.817382\pi\)
0.542754 + 0.839892i \(0.317382\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 516.297i 1.54119i
\(336\) 0 0
\(337\) −384.266 −1.14026 −0.570128 0.821556i \(-0.693106\pi\)
−0.570128 + 0.821556i \(0.693106\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 218.200 + 218.200i 0.639881 + 0.639881i
\(342\) 0 0
\(343\) 75.8459i 0.221125i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −380.232 380.232i −1.09577 1.09577i −0.994900 0.100869i \(-0.967838\pi\)
−0.100869 0.994900i \(-0.532162\pi\)
\(348\) 0 0
\(349\) −260.199 260.199i −0.745555 0.745555i 0.228086 0.973641i \(-0.426753\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 97.9162i 0.277383i 0.990336 + 0.138692i \(0.0442897\pi\)
−0.990336 + 0.138692i \(0.955710\pi\)
\(354\) 0 0
\(355\) 296.427 + 296.427i 0.835006 + 0.835006i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 126.839 0.353311 0.176656 0.984273i \(-0.443472\pi\)
0.176656 + 0.984273i \(0.443472\pi\)
\(360\) 0 0
\(361\) 358.495i 0.993062i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −94.4600 + 94.4600i −0.258794 + 0.258794i
\(366\) 0 0
\(367\) −354.084 −0.964807 −0.482404 0.875949i \(-0.660236\pi\)
−0.482404 + 0.875949i \(0.660236\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.60369 7.60369i 0.0204951 0.0204951i
\(372\) 0 0
\(373\) −77.4388 + 77.4388i −0.207611 + 0.207611i −0.803251 0.595640i \(-0.796898\pi\)
0.595640 + 0.803251i \(0.296898\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −113.019 −0.299786
\(378\) 0 0
\(379\) 209.955 209.955i 0.553971 0.553971i −0.373614 0.927584i \(-0.621881\pi\)
0.927584 + 0.373614i \(0.121881\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 579.620i 1.51337i 0.653781 + 0.756684i \(0.273182\pi\)
−0.653781 + 0.756684i \(0.726818\pi\)
\(384\) 0 0
\(385\) 26.0876 0.0677601
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 420.113 + 420.113i 1.07998 + 1.07998i 0.996510 + 0.0834708i \(0.0266005\pi\)
0.0834708 + 0.996510i \(0.473399\pi\)
\(390\) 0 0
\(391\) 1072.38i 2.74267i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −314.403 314.403i −0.795957 0.795957i
\(396\) 0 0
\(397\) −13.0474 13.0474i −0.0328650 0.0328650i 0.690483 0.723348i \(-0.257398\pi\)
−0.723348 + 0.690483i \(0.757398\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 152.424i 0.380110i −0.981773 0.190055i \(-0.939133\pi\)
0.981773 0.190055i \(-0.0608666\pi\)
\(402\) 0 0
\(403\) −327.482 327.482i −0.812610 0.812610i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 251.507 0.617952
\(408\) 0 0
\(409\) 43.2426i 0.105728i −0.998602 0.0528639i \(-0.983165\pi\)
0.998602 0.0528639i \(-0.0168349\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −42.3415 + 42.3415i −0.102522 + 0.102522i
\(414\) 0 0
\(415\) 645.095 1.55445
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −81.3896 + 81.3896i −0.194247 + 0.194247i −0.797528 0.603281i \(-0.793860\pi\)
0.603281 + 0.797528i \(0.293860\pi\)
\(420\) 0 0
\(421\) −198.570 + 198.570i −0.471663 + 0.471663i −0.902452 0.430789i \(-0.858235\pi\)
0.430789 + 0.902452i \(0.358235\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 171.529 0.403597
\(426\) 0 0
\(427\) 36.6875 36.6875i 0.0859193 0.0859193i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 395.497i 0.917627i −0.888533 0.458813i \(-0.848275\pi\)
0.888533 0.458813i \(-0.151725\pi\)
\(432\) 0 0
\(433\) −672.683 −1.55354 −0.776770 0.629784i \(-0.783143\pi\)
−0.776770 + 0.629784i \(0.783143\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −38.2939 38.2939i −0.0876290 0.0876290i
\(438\) 0 0
\(439\) 26.8920i 0.0612574i 0.999531 + 0.0306287i \(0.00975095\pi\)
−0.999531 + 0.0306287i \(0.990249\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 524.214 + 524.214i 1.18333 + 1.18333i 0.978877 + 0.204451i \(0.0655408\pi\)
0.204451 + 0.978877i \(0.434459\pi\)
\(444\) 0 0
\(445\) −322.597 322.597i −0.724937 0.724937i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 518.381i 1.15452i −0.816559 0.577262i \(-0.804121\pi\)
0.816559 0.577262i \(-0.195879\pi\)
\(450\) 0 0
\(451\) −134.853 134.853i −0.299009 0.299009i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −39.1533 −0.0860511
\(456\) 0 0
\(457\) 853.516i 1.86765i 0.357729 + 0.933825i \(0.383551\pi\)
−0.357729 + 0.933825i \(0.616449\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 331.810 331.810i 0.719761 0.719761i −0.248795 0.968556i \(-0.580034\pi\)
0.968556 + 0.248795i \(0.0800345\pi\)
\(462\) 0 0
\(463\) −164.893 −0.356141 −0.178070 0.984018i \(-0.556986\pi\)
−0.178070 + 0.984018i \(0.556986\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −105.253 + 105.253i −0.225380 + 0.225380i −0.810760 0.585379i \(-0.800946\pi\)
0.585379 + 0.810760i \(0.300946\pi\)
\(468\) 0 0
\(469\) 51.5022 51.5022i 0.109813 0.109813i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −479.751 −1.01427
\(474\) 0 0
\(475\) 6.12514 6.12514i 0.0128950 0.0128950i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 678.054i 1.41556i 0.706432 + 0.707781i \(0.250304\pi\)
−0.706432 + 0.707781i \(0.749696\pi\)
\(480\) 0 0
\(481\) −377.470 −0.784762
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 191.064 + 191.064i 0.393947 + 0.393947i
\(486\) 0 0
\(487\) 7.72273i 0.0158578i −0.999969 0.00792888i \(-0.997476\pi\)
0.999969 0.00792888i \(-0.00252387\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −296.311 296.311i −0.603484 0.603484i 0.337751 0.941235i \(-0.390334\pi\)
−0.941235 + 0.337751i \(0.890334\pi\)
\(492\) 0 0
\(493\) −274.984 274.984i −0.557777 0.557777i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 59.1390i 0.118992i
\(498\) 0 0
\(499\) −555.542 555.542i −1.11331 1.11331i −0.992700 0.120612i \(-0.961514\pi\)
−0.120612 0.992700i \(-0.538486\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −322.916 −0.641979 −0.320990 0.947083i \(-0.604015\pi\)
−0.320990 + 0.947083i \(0.604015\pi\)
\(504\) 0 0
\(505\) 322.250i 0.638118i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −522.344 + 522.344i −1.02622 + 1.02622i −0.0265686 + 0.999647i \(0.508458\pi\)
−0.999647 + 0.0265686i \(0.991542\pi\)
\(510\) 0 0
\(511\) 18.8453 0.0368793
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −676.213 + 676.213i −1.31304 + 1.31304i
\(516\) 0 0
\(517\) −113.818 + 113.818i −0.220150 + 0.220150i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 80.4109 0.154340 0.0771698 0.997018i \(-0.475412\pi\)
0.0771698 + 0.997018i \(0.475412\pi\)
\(522\) 0 0
\(523\) −133.868 + 133.868i −0.255962 + 0.255962i −0.823410 0.567447i \(-0.807931\pi\)
0.567447 + 0.823410i \(0.307931\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1593.57i 3.02386i
\(528\) 0 0
\(529\) 641.985 1.21358
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 202.392 + 202.392i 0.379723 + 0.379723i
\(534\) 0 0
\(535\) 415.918i 0.777417i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 207.656 + 207.656i 0.385261 + 0.385261i
\(540\) 0 0
\(541\) 643.791 + 643.791i 1.19000 + 1.19000i 0.977066 + 0.212936i \(0.0683025\pi\)
0.212936 + 0.977066i \(0.431698\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 727.237i 1.33438i
\(546\) 0 0
\(547\) 543.953 + 543.953i 0.994429 + 0.994429i 0.999985 0.00555539i \(-0.00176834\pi\)
−0.00555539 + 0.999985i \(0.501768\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −19.6389 −0.0356422
\(552\) 0 0
\(553\) 62.7252i 0.113427i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −436.430 + 436.430i −0.783537 + 0.783537i −0.980426 0.196889i \(-0.936916\pi\)
0.196889 + 0.980426i \(0.436916\pi\)
\(558\) 0 0
\(559\) 720.027 1.28806
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 525.339 525.339i 0.933107 0.933107i −0.0647918 0.997899i \(-0.520638\pi\)
0.997899 + 0.0647918i \(0.0206383\pi\)
\(564\) 0 0
\(565\) 159.334 159.334i 0.282006 0.282006i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 848.985 1.49207 0.746033 0.665909i \(-0.231956\pi\)
0.746033 + 0.665909i \(0.231956\pi\)
\(570\) 0 0
\(571\) −503.454 + 503.454i −0.881706 + 0.881706i −0.993708 0.112002i \(-0.964274\pi\)
0.112002 + 0.993708i \(0.464274\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 187.300i 0.325739i
\(576\) 0 0
\(577\) 891.580 1.54520 0.772600 0.634894i \(-0.218956\pi\)
0.772600 + 0.634894i \(0.218956\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −64.3501 64.3501i −0.110758 0.110758i
\(582\) 0 0
\(583\) 83.7934i 0.143728i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −121.737 121.737i −0.207388 0.207388i 0.595768 0.803156i \(-0.296848\pi\)
−0.803156 + 0.595768i \(0.796848\pi\)
\(588\) 0 0
\(589\) −56.9050 56.9050i −0.0966129 0.0966129i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 207.925i 0.350632i 0.984512 + 0.175316i \(0.0560947\pi\)
−0.984512 + 0.175316i \(0.943905\pi\)
\(594\) 0 0
\(595\) −95.2626 95.2626i −0.160105 0.160105i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −689.528 −1.15113 −0.575566 0.817755i \(-0.695218\pi\)
−0.575566 + 0.817755i \(0.695218\pi\)
\(600\) 0 0
\(601\) 542.421i 0.902531i −0.892390 0.451266i \(-0.850973\pi\)
0.892390 0.451266i \(-0.149027\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 328.571 328.571i 0.543092 0.543092i
\(606\) 0 0
\(607\) −1018.93 −1.67864 −0.839318 0.543641i \(-0.817045\pi\)
−0.839318 + 0.543641i \(0.817045\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 170.822 170.822i 0.279578 0.279578i
\(612\) 0 0
\(613\) −169.481 + 169.481i −0.276478 + 0.276478i −0.831701 0.555223i \(-0.812633\pi\)
0.555223 + 0.831701i \(0.312633\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −834.597 −1.35267 −0.676335 0.736594i \(-0.736433\pi\)
−0.676335 + 0.736594i \(0.736433\pi\)
\(618\) 0 0
\(619\) 322.772 322.772i 0.521441 0.521441i −0.396565 0.918007i \(-0.629798\pi\)
0.918007 + 0.396565i \(0.129798\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 64.3600i 0.103307i
\(624\) 0 0
\(625\) 731.878 1.17100
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −918.411 918.411i −1.46011 1.46011i
\(630\) 0 0
\(631\) 1006.06i 1.59439i 0.603723 + 0.797194i \(0.293683\pi\)
−0.603723 + 0.797194i \(0.706317\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 347.068 + 347.068i 0.546563 + 0.546563i
\(636\) 0 0
\(637\) −311.658 311.658i −0.489258 0.489258i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 257.206i 0.401257i −0.979667 0.200628i \(-0.935702\pi\)
0.979667 0.200628i \(-0.0642984\pi\)
\(642\) 0 0
\(643\) 311.891 + 311.891i 0.485056 + 0.485056i 0.906742 0.421686i \(-0.138562\pi\)
−0.421686 + 0.906742i \(0.638562\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 855.255 1.32188 0.660939 0.750440i \(-0.270158\pi\)
0.660939 + 0.750440i \(0.270158\pi\)
\(648\) 0 0
\(649\) 466.607i 0.718962i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 171.197 171.197i 0.262170 0.262170i −0.563765 0.825935i \(-0.690648\pi\)
0.825935 + 0.563765i \(0.190648\pi\)
\(654\) 0 0
\(655\) −171.835 −0.262343
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 510.283 510.283i 0.774330 0.774330i −0.204530 0.978860i \(-0.565567\pi\)
0.978860 + 0.204530i \(0.0655667\pi\)
\(660\) 0 0
\(661\) −491.063 + 491.063i −0.742910 + 0.742910i −0.973137 0.230227i \(-0.926053\pi\)
0.230227 + 0.973137i \(0.426053\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.80348 −0.0102308
\(666\) 0 0
\(667\) 300.268 300.268i 0.450176 0.450176i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 404.300i 0.602534i
\(672\) 0 0
\(673\) −200.282 −0.297596 −0.148798 0.988868i \(-0.547540\pi\)
−0.148798 + 0.988868i \(0.547540\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −818.346 818.346i −1.20878 1.20878i −0.971422 0.237361i \(-0.923718\pi\)
−0.237361 0.971422i \(-0.576282\pi\)
\(678\) 0 0
\(679\) 38.1184i 0.0561391i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −559.041 559.041i −0.818508 0.818508i 0.167383 0.985892i \(-0.446468\pi\)
−0.985892 + 0.167383i \(0.946468\pi\)
\(684\) 0 0
\(685\) 432.476 + 432.476i 0.631352 + 0.631352i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 125.760i 0.182526i
\(690\) 0 0
\(691\) 558.744 + 558.744i 0.808602 + 0.808602i 0.984422 0.175820i \(-0.0562577\pi\)
−0.175820 + 0.984422i \(0.556258\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −237.220 −0.341324
\(696\) 0 0
\(697\) 984.869i 1.41301i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 452.428 452.428i 0.645404 0.645404i −0.306475 0.951879i \(-0.599149\pi\)
0.951879 + 0.306475i \(0.0991495\pi\)
\(702\) 0 0
\(703\) −65.5913 −0.0933019
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −32.1454 + 32.1454i −0.0454673 + 0.0454673i
\(708\) 0 0
\(709\) 693.360 693.360i 0.977941 0.977941i −0.0218205 0.999762i \(-0.506946\pi\)
0.999762 + 0.0218205i \(0.00694622\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1740.09 2.44052
\(714\) 0 0
\(715\) 215.736 215.736i 0.301729 0.301729i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 641.528i 0.892251i −0.894970 0.446125i \(-0.852804\pi\)
0.894970 0.446125i \(-0.147196\pi\)
\(720\) 0 0
\(721\) 134.909 0.187113
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 48.0281 + 48.0281i 0.0662456 + 0.0662456i
\(726\) 0 0
\(727\) 644.665i 0.886747i 0.896337 + 0.443373i \(0.146218\pi\)
−0.896337 + 0.443373i \(0.853782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1751.88 + 1751.88i 2.39655 + 2.39655i
\(732\) 0 0
\(733\) 748.983 + 748.983i 1.02180 + 1.02180i 0.999757 + 0.0220473i \(0.00701846\pi\)
0.0220473 + 0.999757i \(0.492982\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 567.559i 0.770093i
\(738\) 0 0
\(739\) 387.245 + 387.245i 0.524013 + 0.524013i 0.918781 0.394768i \(-0.129175\pi\)
−0.394768 + 0.918781i \(0.629175\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 245.165 0.329966 0.164983 0.986296i \(-0.447243\pi\)
0.164983 + 0.986296i \(0.447243\pi\)
\(744\) 0 0
\(745\) 1357.80i 1.82255i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 41.4891 41.4891i 0.0553926 0.0553926i
\(750\) 0 0
\(751\) −547.753 −0.729365 −0.364683 0.931132i \(-0.618822\pi\)
−0.364683 + 0.931132i \(0.618822\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 102.473 102.473i 0.135726 0.135726i
\(756\) 0 0
\(757\) 876.755 876.755i 1.15820 1.15820i 0.173334 0.984863i \(-0.444546\pi\)
0.984863 0.173334i \(-0.0554540\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −876.682 −1.15201 −0.576007 0.817445i \(-0.695390\pi\)
−0.576007 + 0.817445i \(0.695390\pi\)
\(762\) 0 0
\(763\) 72.5441 72.5441i 0.0950774 0.0950774i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 700.300i 0.913038i
\(768\) 0 0
\(769\) −155.364 −0.202034 −0.101017 0.994885i \(-0.532210\pi\)
−0.101017 + 0.994885i \(0.532210\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −336.932 336.932i −0.435876 0.435876i 0.454746 0.890621i \(-0.349730\pi\)
−0.890621 + 0.454746i \(0.849730\pi\)
\(774\) 0 0
\(775\) 278.330i 0.359135i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 35.1688 + 35.1688i 0.0451460 + 0.0451460i
\(780\) 0 0
\(781\) 325.858 + 325.858i 0.417232 + 0.417232i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 592.144i 0.754324i
\(786\) 0 0
\(787\) −201.318 201.318i −0.255804 0.255804i 0.567541 0.823345i \(-0.307895\pi\)
−0.823345 + 0.567541i \(0.807895\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −31.7880 −0.0401871
\(792\) 0 0
\(793\) 606.788i 0.765181i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 571.213 571.213i 0.716704 0.716704i −0.251225 0.967929i \(-0.580833\pi\)
0.967929 + 0.251225i \(0.0808333\pi\)
\(798\) 0 0
\(799\) 831.243 1.04035
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −103.839 + 103.839i −0.129313 + 0.129313i
\(804\) 0 0
\(805\) 104.022 104.022i 0.129219 0.129219i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −554.891 −0.685898 −0.342949 0.939354i \(-0.611426\pi\)
−0.342949 + 0.939354i \(0.611426\pi\)
\(810\) 0 0
\(811\) −570.087 + 570.087i −0.702944 + 0.702944i −0.965041 0.262098i \(-0.915586\pi\)
0.262098 + 0.965041i \(0.415586\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1212.29i 1.48747i
\(816\) 0 0
\(817\) 125.116 0.153141
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 150.698 + 150.698i 0.183554 + 0.183554i 0.792903 0.609348i \(-0.208569\pi\)
−0.609348 + 0.792903i \(0.708569\pi\)
\(822\) 0 0
\(823\) 742.019i 0.901603i 0.892624 + 0.450801i \(0.148862\pi\)
−0.892624 + 0.450801i \(0.851138\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 248.150 + 248.150i 0.300061 + 0.300061i 0.841037 0.540977i \(-0.181945\pi\)
−0.540977 + 0.841037i \(0.681945\pi\)
\(828\) 0 0
\(829\) −373.174 373.174i −0.450150 0.450150i 0.445255 0.895404i \(-0.353113\pi\)
−0.895404 + 0.445255i \(0.853113\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1516.57i 1.82061i
\(834\) 0 0
\(835\) 924.728 + 924.728i 1.10746 + 1.10746i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1227.49 1.46304 0.731520 0.681820i \(-0.238811\pi\)
0.731520 + 0.681820i \(0.238811\pi\)
\(840\) 0 0
\(841\) 687.009i 0.816895i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 335.894 335.894i 0.397508 0.397508i
\(846\) 0 0
\(847\) −65.5518 −0.0773929
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1002.86 1002.86i 1.17844 1.17844i
\(852\) 0 0
\(853\) −24.4613 + 24.4613i −0.0286768 + 0.0286768i −0.721300 0.692623i \(-0.756455\pi\)
0.692623 + 0.721300i \(0.256455\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 839.732 0.979850 0.489925 0.871764i \(-0.337024\pi\)
0.489925 + 0.871764i \(0.337024\pi\)
\(858\) 0 0
\(859\) 483.229 483.229i 0.562548 0.562548i −0.367482 0.930031i \(-0.619780\pi\)
0.930031 + 0.367482i \(0.119780\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 21.1881i 0.0245517i −0.999925 0.0122758i \(-0.996092\pi\)
0.999925 0.0122758i \(-0.00390761\pi\)
\(864\) 0 0
\(865\) 533.761 0.617064
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −345.619 345.619i −0.397720 0.397720i
\(870\) 0 0
\(871\) 851.813i 0.977971i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −59.3571 59.3571i −0.0678367 0.0678367i
\(876\) 0 0
\(877\) −350.528 350.528i −0.399690 0.399690i 0.478434 0.878124i \(-0.341205\pi\)
−0.878124 + 0.478434i \(0.841205\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 472.491i 0.536312i −0.963375 0.268156i \(-0.913586\pi\)
0.963375 0.268156i \(-0.0864143\pi\)
\(882\) 0 0
\(883\) −294.850 294.850i −0.333919 0.333919i 0.520154 0.854073i \(-0.325875\pi\)
−0.854073 + 0.520154i \(0.825875\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 189.405 0.213535 0.106767 0.994284i \(-0.465950\pi\)
0.106767 + 0.994284i \(0.465950\pi\)
\(888\) 0 0
\(889\) 69.2421i 0.0778876i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 29.6829 29.6829i 0.0332396 0.0332396i
\(894\) 0 0
\(895\) 1011.97 1.13070
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 446.200 446.200i 0.496330 0.496330i
\(900\) 0 0
\(901\) 305.983 305.983i 0.339604 0.339604i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1230.68 −1.35987
\(906\) 0 0
\(907\) −1025.20 + 1025.20i −1.13032 + 1.13032i −0.140200 + 0.990123i \(0.544774\pi\)
−0.990123 + 0.140200i \(0.955226\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1115.84i 1.22485i 0.790528 + 0.612426i \(0.209806\pi\)
−0.790528 + 0.612426i \(0.790194\pi\)
\(912\) 0 0
\(913\) 709.144 0.776719
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 17.1410 + 17.1410i 0.0186925 + 0.0186925i
\(918\) 0 0
\(919\) 1169.50i 1.27258i 0.771450 + 0.636290i \(0.219532\pi\)
−0.771450 + 0.636290i \(0.780468\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −489.060 489.060i −0.529859 0.529859i
\(924\) 0 0
\(925\) 160.408 + 160.408i 0.173414 + 0.173414i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 115.506i 0.124334i −0.998066 0.0621671i \(-0.980199\pi\)
0.998066 0.0621671i \(-0.0198012\pi\)
\(930\) 0 0
\(931\) −54.1553 54.1553i −0.0581689 0.0581689i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1049.80 1.12278
\(936\) 0 0
\(937\) 666.120i 0.710907i 0.934694 + 0.355454i \(0.115674\pi\)
−0.934694 + 0.355454i \(0.884326\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −559.735 + 559.735i −0.594830 + 0.594830i −0.938932 0.344102i \(-0.888183\pi\)
0.344102 + 0.938932i \(0.388183\pi\)
\(942\) 0 0
\(943\) −1075.42 −1.14043
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −871.120 + 871.120i −0.919873 + 0.919873i −0.997020 0.0771466i \(-0.975419\pi\)
0.0771466 + 0.997020i \(0.475419\pi\)
\(948\) 0 0
\(949\) 155.845 155.845i 0.164220 0.164220i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 264.384 0.277423 0.138711 0.990333i \(-0.455704\pi\)
0.138711 + 0.990333i \(0.455704\pi\)
\(954\) 0 0
\(955\) 226.874 226.874i 0.237564 0.237564i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 86.2816i 0.0899703i
\(960\) 0 0
\(961\) 1624.80 1.69073
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1219.78 + 1219.78i 1.26402 + 1.26402i
\(966\) 0 0
\(967\) 92.8445i 0.0960129i −0.998847 0.0480065i \(-0.984713\pi\)
0.998847 0.0480065i \(-0.0152868\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1086.25 1086.25i −1.11870 1.11870i −0.991933 0.126764i \(-0.959541\pi\)
−0.126764 0.991933i \(-0.540459\pi\)
\(972\) 0 0
\(973\) 23.6634 + 23.6634i 0.0243201 + 0.0243201i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 626.473i 0.641222i 0.947211 + 0.320611i \(0.103888\pi\)
−0.947211 + 0.320611i \(0.896112\pi\)
\(978\) 0 0
\(979\) −354.626 354.626i −0.362233 0.362233i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −204.267 −0.207800 −0.103900 0.994588i \(-0.533132\pi\)
−0.103900 + 0.994588i \(0.533132\pi\)
\(984\) 0 0
\(985\) 349.986i 0.355316i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1912.95 + 1912.95i −1.93423 + 1.93423i
\(990\) 0 0
\(991\) 1147.30 1.15772 0.578859 0.815428i \(-0.303498\pi\)
0.578859 + 0.815428i \(0.303498\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −613.348 + 613.348i −0.616430 + 0.616430i
\(996\) 0 0
\(997\) −514.690 + 514.690i −0.516239 + 0.516239i −0.916431 0.400192i \(-0.868943\pi\)
0.400192 + 0.916431i \(0.368943\pi\)
\(998\) 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 1152.3.j.a.161.13 32
3.2 odd 2 inner 1152.3.j.a.161.4 32
4.3 odd 2 1152.3.j.b.161.13 32
8.3 odd 2 576.3.j.a.17.4 32
8.5 even 2 144.3.j.a.125.12 yes 32
12.11 even 2 1152.3.j.b.161.4 32
16.3 odd 4 576.3.j.a.305.13 32
16.5 even 4 inner 1152.3.j.a.737.4 32
16.11 odd 4 1152.3.j.b.737.4 32
16.13 even 4 144.3.j.a.53.5 32
24.5 odd 2 144.3.j.a.125.5 yes 32
24.11 even 2 576.3.j.a.17.13 32
48.5 odd 4 inner 1152.3.j.a.737.13 32
48.11 even 4 1152.3.j.b.737.13 32
48.29 odd 4 144.3.j.a.53.12 yes 32
48.35 even 4 576.3.j.a.305.4 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.3.j.a.53.5 32 16.13 even 4
144.3.j.a.53.12 yes 32 48.29 odd 4
144.3.j.a.125.5 yes 32 24.5 odd 2
144.3.j.a.125.12 yes 32 8.5 even 2
576.3.j.a.17.4 32 8.3 odd 2
576.3.j.a.17.13 32 24.11 even 2
576.3.j.a.305.4 32 48.35 even 4
576.3.j.a.305.13 32 16.3 odd 4
1152.3.j.a.161.4 32 3.2 odd 2 inner
1152.3.j.a.161.13 32 1.1 even 1 trivial
1152.3.j.a.737.4 32 16.5 even 4 inner
1152.3.j.a.737.13 32 48.5 odd 4 inner
1152.3.j.b.161.4 32 12.11 even 2
1152.3.j.b.161.13 32 4.3 odd 2
1152.3.j.b.737.4 32 16.11 odd 4
1152.3.j.b.737.13 32 48.11 even 4