Properties

Label 4032.2.v.d.1583.8
Level $4032$
Weight $2$
Character 4032.1583
Analytic conductor $32.196$
Analytic rank $0$
Dimension $36$
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] = [4032,2,Mod(1583,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.1583");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.v (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: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1583.8
Character \(\chi\) \(=\) 4032.1583
Dual form 4032.2.v.d.3599.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.495166 - 0.495166i) q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+(-0.495166 - 0.495166i) q^{5} +1.00000 q^{7} +(-0.675994 + 0.675994i) q^{11} +(-2.39551 - 2.39551i) q^{13} +1.03686i q^{17} +(-0.913823 + 0.913823i) q^{19} +2.92895i q^{23} -4.50962i q^{25} +(-3.39091 + 3.39091i) q^{29} +9.43104i q^{31} +(-0.495166 - 0.495166i) q^{35} +(5.47539 - 5.47539i) q^{37} +6.91226 q^{41} +(2.30822 + 2.30822i) q^{43} +9.68077 q^{47} +1.00000 q^{49} +(8.02498 + 8.02498i) q^{53} +0.669458 q^{55} +(-1.14297 + 1.14297i) q^{59} +(-7.05839 - 7.05839i) q^{61} +2.37235i q^{65} +(5.66980 - 5.66980i) q^{67} +7.26670i q^{71} -6.66570i q^{73} +(-0.675994 + 0.675994i) q^{77} +2.26385i q^{79} +(-0.619284 - 0.619284i) q^{83} +(0.513417 - 0.513417i) q^{85} -3.99233 q^{89} +(-2.39551 - 2.39551i) q^{91} +0.904988 q^{95} +6.82547 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 36 q + 36 q^{7} - 16 q^{13} + 16 q^{19} + 20 q^{37} - 36 q^{43} + 36 q^{49} - 32 q^{55} + 112 q^{61} + 36 q^{67} - 96 q^{85} - 16 q^{91} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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
\(5\) −0.495166 0.495166i −0.221445 0.221445i 0.587662 0.809107i \(-0.300049\pi\)
−0.809107 + 0.587662i \(0.800049\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.675994 + 0.675994i −0.203820 + 0.203820i −0.801634 0.597815i \(-0.796036\pi\)
0.597815 + 0.801634i \(0.296036\pi\)
\(12\) 0 0
\(13\) −2.39551 2.39551i −0.664394 0.664394i 0.292019 0.956413i \(-0.405673\pi\)
−0.956413 + 0.292019i \(0.905673\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.03686i 0.251475i 0.992064 + 0.125738i \(0.0401297\pi\)
−0.992064 + 0.125738i \(0.959870\pi\)
\(18\) 0 0
\(19\) −0.913823 + 0.913823i −0.209645 + 0.209645i −0.804117 0.594471i \(-0.797361\pi\)
0.594471 + 0.804117i \(0.297361\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.92895i 0.610727i 0.952236 + 0.305364i \(0.0987780\pi\)
−0.952236 + 0.305364i \(0.901222\pi\)
\(24\) 0 0
\(25\) 4.50962i 0.901924i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.39091 + 3.39091i −0.629676 + 0.629676i −0.947987 0.318310i \(-0.896885\pi\)
0.318310 + 0.947987i \(0.396885\pi\)
\(30\) 0 0
\(31\) 9.43104i 1.69386i 0.531701 + 0.846932i \(0.321553\pi\)
−0.531701 + 0.846932i \(0.678447\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.495166 0.495166i −0.0836983 0.0836983i
\(36\) 0 0
\(37\) 5.47539 5.47539i 0.900148 0.900148i −0.0953004 0.995449i \(-0.530381\pi\)
0.995449 + 0.0953004i \(0.0303812\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.91226 1.07951 0.539757 0.841821i \(-0.318516\pi\)
0.539757 + 0.841821i \(0.318516\pi\)
\(42\) 0 0
\(43\) 2.30822 + 2.30822i 0.352000 + 0.352000i 0.860853 0.508853i \(-0.169930\pi\)
−0.508853 + 0.860853i \(0.669930\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.68077 1.41209 0.706043 0.708169i \(-0.250478\pi\)
0.706043 + 0.708169i \(0.250478\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.02498 + 8.02498i 1.10232 + 1.10232i 0.994131 + 0.108185i \(0.0345039\pi\)
0.108185 + 0.994131i \(0.465496\pi\)
\(54\) 0 0
\(55\) 0.669458 0.0902697
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.14297 + 1.14297i −0.148802 + 0.148802i −0.777583 0.628781i \(-0.783554\pi\)
0.628781 + 0.777583i \(0.283554\pi\)
\(60\) 0 0
\(61\) −7.05839 7.05839i −0.903734 0.903734i 0.0920229 0.995757i \(-0.470667\pi\)
−0.995757 + 0.0920229i \(0.970667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.37235i 0.294253i
\(66\) 0 0
\(67\) 5.66980 5.66980i 0.692677 0.692677i −0.270143 0.962820i \(-0.587071\pi\)
0.962820 + 0.270143i \(0.0870711\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.26670i 0.862399i 0.902257 + 0.431199i \(0.141909\pi\)
−0.902257 + 0.431199i \(0.858091\pi\)
\(72\) 0 0
\(73\) 6.66570i 0.780161i −0.920781 0.390081i \(-0.872447\pi\)
0.920781 0.390081i \(-0.127553\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.675994 + 0.675994i −0.0770366 + 0.0770366i
\(78\) 0 0
\(79\) 2.26385i 0.254704i 0.991858 + 0.127352i \(0.0406477\pi\)
−0.991858 + 0.127352i \(0.959352\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.619284 0.619284i −0.0679753 0.0679753i 0.672302 0.740277i \(-0.265306\pi\)
−0.740277 + 0.672302i \(0.765306\pi\)
\(84\) 0 0
\(85\) 0.513417 0.513417i 0.0556879 0.0556879i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.99233 −0.423186 −0.211593 0.977358i \(-0.567865\pi\)
−0.211593 + 0.977358i \(0.567865\pi\)
\(90\) 0 0
\(91\) −2.39551 2.39551i −0.251117 0.251117i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.904988 0.0928498
\(96\) 0 0
\(97\) 6.82547 0.693022 0.346511 0.938046i \(-0.387366\pi\)
0.346511 + 0.938046i \(0.387366\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.00990 2.00990i −0.199992 0.199992i 0.600004 0.799997i \(-0.295165\pi\)
−0.799997 + 0.600004i \(0.795165\pi\)
\(102\) 0 0
\(103\) 9.62061 0.947947 0.473973 0.880539i \(-0.342819\pi\)
0.473973 + 0.880539i \(0.342819\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.18658 4.18658i 0.404732 0.404732i −0.475165 0.879897i \(-0.657612\pi\)
0.879897 + 0.475165i \(0.157612\pi\)
\(108\) 0 0
\(109\) 1.12051 + 1.12051i 0.107326 + 0.107326i 0.758730 0.651405i \(-0.225820\pi\)
−0.651405 + 0.758730i \(0.725820\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.04200i 0.286167i 0.989711 + 0.143084i \(0.0457018\pi\)
−0.989711 + 0.143084i \(0.954298\pi\)
\(114\) 0 0
\(115\) 1.45031 1.45031i 0.135242 0.135242i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.03686i 0.0950486i
\(120\) 0 0
\(121\) 10.0861i 0.916915i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.70884 + 4.70884i −0.421171 + 0.421171i
\(126\) 0 0
\(127\) 2.15994i 0.191664i −0.995398 0.0958318i \(-0.969449\pi\)
0.995398 0.0958318i \(-0.0305511\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.29314 + 9.29314i 0.811945 + 0.811945i 0.984925 0.172980i \(-0.0553396\pi\)
−0.172980 + 0.984925i \(0.555340\pi\)
\(132\) 0 0
\(133\) −0.913823 + 0.913823i −0.0792385 + 0.0792385i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.8459 1.43924 0.719620 0.694368i \(-0.244316\pi\)
0.719620 + 0.694368i \(0.244316\pi\)
\(138\) 0 0
\(139\) 10.8283 + 10.8283i 0.918444 + 0.918444i 0.996916 0.0784726i \(-0.0250043\pi\)
−0.0784726 + 0.996916i \(0.525004\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.23870 0.270833
\(144\) 0 0
\(145\) 3.35812 0.278877
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.41586 5.41586i −0.443685 0.443685i 0.449564 0.893248i \(-0.351579\pi\)
−0.893248 + 0.449564i \(0.851579\pi\)
\(150\) 0 0
\(151\) −1.86764 −0.151987 −0.0759933 0.997108i \(-0.524213\pi\)
−0.0759933 + 0.997108i \(0.524213\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.66993 4.66993i 0.375098 0.375098i
\(156\) 0 0
\(157\) −6.27144 6.27144i −0.500515 0.500515i 0.411083 0.911598i \(-0.365151\pi\)
−0.911598 + 0.411083i \(0.865151\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.92895i 0.230833i
\(162\) 0 0
\(163\) 9.10671 9.10671i 0.713292 0.713292i −0.253930 0.967223i \(-0.581723\pi\)
0.967223 + 0.253930i \(0.0817233\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.44503i 0.730878i 0.930835 + 0.365439i \(0.119081\pi\)
−0.930835 + 0.365439i \(0.880919\pi\)
\(168\) 0 0
\(169\) 1.52309i 0.117161i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −16.9743 + 16.9743i −1.29053 + 1.29053i −0.356078 + 0.934456i \(0.615886\pi\)
−0.934456 + 0.356078i \(0.884114\pi\)
\(174\) 0 0
\(175\) 4.50962i 0.340895i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.830318 + 0.830318i 0.0620609 + 0.0620609i 0.737456 0.675395i \(-0.236027\pi\)
−0.675395 + 0.737456i \(0.736027\pi\)
\(180\) 0 0
\(181\) 3.74015 3.74015i 0.278003 0.278003i −0.554308 0.832311i \(-0.687017\pi\)
0.832311 + 0.554308i \(0.187017\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.42245 −0.398666
\(186\) 0 0
\(187\) −0.700910 0.700910i −0.0512556 0.0512556i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.3035 0.745533 0.372766 0.927925i \(-0.378409\pi\)
0.372766 + 0.927925i \(0.378409\pi\)
\(192\) 0 0
\(193\) 12.1382 0.873726 0.436863 0.899528i \(-0.356089\pi\)
0.436863 + 0.899528i \(0.356089\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.77271 + 8.77271i 0.625029 + 0.625029i 0.946813 0.321784i \(-0.104282\pi\)
−0.321784 + 0.946813i \(0.604282\pi\)
\(198\) 0 0
\(199\) −22.8194 −1.61762 −0.808812 0.588067i \(-0.799889\pi\)
−0.808812 + 0.588067i \(0.799889\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.39091 + 3.39091i −0.237995 + 0.237995i
\(204\) 0 0
\(205\) −3.42272 3.42272i −0.239053 0.239053i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.23548i 0.0854598i
\(210\) 0 0
\(211\) 18.3550 18.3550i 1.26361 1.26361i 0.314283 0.949329i \(-0.398236\pi\)
0.949329 0.314283i \(-0.101764\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.28590i 0.155897i
\(216\) 0 0
\(217\) 9.43104i 0.640221i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.48380 2.48380i 0.167079 0.167079i
\(222\) 0 0
\(223\) 7.85008i 0.525680i 0.964839 + 0.262840i \(0.0846591\pi\)
−0.964839 + 0.262840i \(0.915341\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.49085 8.49085i −0.563557 0.563557i 0.366759 0.930316i \(-0.380467\pi\)
−0.930316 + 0.366759i \(0.880467\pi\)
\(228\) 0 0
\(229\) 10.9830 10.9830i 0.725775 0.725775i −0.244000 0.969775i \(-0.578460\pi\)
0.969775 + 0.244000i \(0.0784598\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.0449847 0.00294704 0.00147352 0.999999i \(-0.499531\pi\)
0.00147352 + 0.999999i \(0.499531\pi\)
\(234\) 0 0
\(235\) −4.79359 4.79359i −0.312699 0.312699i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 30.1340 1.94921 0.974605 0.223933i \(-0.0718895\pi\)
0.974605 + 0.223933i \(0.0718895\pi\)
\(240\) 0 0
\(241\) 14.4042 0.927853 0.463927 0.885874i \(-0.346440\pi\)
0.463927 + 0.885874i \(0.346440\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.495166 0.495166i −0.0316350 0.0316350i
\(246\) 0 0
\(247\) 4.37814 0.278574
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.14495 + 7.14495i −0.450986 + 0.450986i −0.895682 0.444696i \(-0.853312\pi\)
0.444696 + 0.895682i \(0.353312\pi\)
\(252\) 0 0
\(253\) −1.97995 1.97995i −0.124478 0.124478i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.01111i 0.437341i −0.975799 0.218670i \(-0.929828\pi\)
0.975799 0.218670i \(-0.0701720\pi\)
\(258\) 0 0
\(259\) 5.47539 5.47539i 0.340224 0.340224i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 28.2013i 1.73897i 0.493964 + 0.869483i \(0.335548\pi\)
−0.493964 + 0.869483i \(0.664452\pi\)
\(264\) 0 0
\(265\) 7.94739i 0.488204i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.65309 1.65309i 0.100790 0.100790i −0.654913 0.755704i \(-0.727295\pi\)
0.755704 + 0.654913i \(0.227295\pi\)
\(270\) 0 0
\(271\) 19.8214i 1.20406i 0.798472 + 0.602032i \(0.205642\pi\)
−0.798472 + 0.602032i \(0.794358\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.04848 + 3.04848i 0.183830 + 0.183830i
\(276\) 0 0
\(277\) −3.17209 + 3.17209i −0.190592 + 0.190592i −0.795952 0.605360i \(-0.793029\pi\)
0.605360 + 0.795952i \(0.293029\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.10515 −0.0659280 −0.0329640 0.999457i \(-0.510495\pi\)
−0.0329640 + 0.999457i \(0.510495\pi\)
\(282\) 0 0
\(283\) 7.08295 + 7.08295i 0.421038 + 0.421038i 0.885561 0.464523i \(-0.153774\pi\)
−0.464523 + 0.885561i \(0.653774\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.91226 0.408018
\(288\) 0 0
\(289\) 15.9249 0.936760
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 23.5296 + 23.5296i 1.37461 + 1.37461i 0.853470 + 0.521142i \(0.174494\pi\)
0.521142 + 0.853470i \(0.325506\pi\)
\(294\) 0 0
\(295\) 1.13192 0.0659030
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.01631 7.01631i 0.405764 0.405764i
\(300\) 0 0
\(301\) 2.30822 + 2.30822i 0.133043 + 0.133043i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.99014i 0.400254i
\(306\) 0 0
\(307\) 8.29187 8.29187i 0.473242 0.473242i −0.429720 0.902962i \(-0.641388\pi\)
0.902962 + 0.429720i \(0.141388\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.1040i 0.572945i 0.958088 + 0.286472i \(0.0924827\pi\)
−0.958088 + 0.286472i \(0.907517\pi\)
\(312\) 0 0
\(313\) 0.519053i 0.0293386i −0.999892 0.0146693i \(-0.995330\pi\)
0.999892 0.0146693i \(-0.00466955\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.3517 + 10.3517i −0.581408 + 0.581408i −0.935290 0.353882i \(-0.884861\pi\)
0.353882 + 0.935290i \(0.384861\pi\)
\(318\) 0 0
\(319\) 4.58447i 0.256681i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −0.947505 0.947505i −0.0527206 0.0527206i
\(324\) 0 0
\(325\) −10.8028 + 10.8028i −0.599233 + 0.599233i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.68077 0.533718
\(330\) 0 0
\(331\) −23.4180 23.4180i −1.28717 1.28717i −0.936498 0.350672i \(-0.885953\pi\)
−0.350672 0.936498i \(-0.614047\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5.61498 −0.306779
\(336\) 0 0
\(337\) 12.8612 0.700597 0.350298 0.936638i \(-0.386080\pi\)
0.350298 + 0.936638i \(0.386080\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −6.37532 6.37532i −0.345243 0.345243i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.07714 + 6.07714i −0.326238 + 0.326238i −0.851154 0.524916i \(-0.824097\pi\)
0.524916 + 0.851154i \(0.324097\pi\)
\(348\) 0 0
\(349\) −15.0938 15.0938i −0.807952 0.807952i 0.176372 0.984324i \(-0.443564\pi\)
−0.984324 + 0.176372i \(0.943564\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0292i 0.959599i −0.877378 0.479799i \(-0.840709\pi\)
0.877378 0.479799i \(-0.159291\pi\)
\(354\) 0 0
\(355\) 3.59822 3.59822i 0.190974 0.190974i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.7990i 0.781059i −0.920590 0.390529i \(-0.872292\pi\)
0.920590 0.390529i \(-0.127708\pi\)
\(360\) 0 0
\(361\) 17.3299i 0.912098i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.30063 + 3.30063i −0.172763 + 0.172763i
\(366\) 0 0
\(367\) 15.4513i 0.806554i −0.915078 0.403277i \(-0.867871\pi\)
0.915078 0.403277i \(-0.132129\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.02498 + 8.02498i 0.416636 + 0.416636i
\(372\) 0 0
\(373\) −18.9447 + 18.9447i −0.980918 + 0.980918i −0.999821 0.0189036i \(-0.993982\pi\)
0.0189036 + 0.999821i \(0.493982\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.2459 0.836706
\(378\) 0 0
\(379\) −3.79368 3.79368i −0.194868 0.194868i 0.602928 0.797796i \(-0.294001\pi\)
−0.797796 + 0.602928i \(0.794001\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.0654 0.667608 0.333804 0.942642i \(-0.391668\pi\)
0.333804 + 0.942642i \(0.391668\pi\)
\(384\) 0 0
\(385\) 0.669458 0.0341187
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13.7690 + 13.7690i 0.698117 + 0.698117i 0.964004 0.265887i \(-0.0856647\pi\)
−0.265887 + 0.964004i \(0.585665\pi\)
\(390\) 0 0
\(391\) −3.03690 −0.153583
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.12098 1.12098i 0.0564028 0.0564028i
\(396\) 0 0
\(397\) −14.5406 14.5406i −0.729772 0.729772i 0.240802 0.970574i \(-0.422589\pi\)
−0.970574 + 0.240802i \(0.922589\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 31.9581i 1.59591i −0.602715 0.797956i \(-0.705915\pi\)
0.602715 0.797956i \(-0.294085\pi\)
\(402\) 0 0
\(403\) 22.5921 22.5921i 1.12539 1.12539i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.40266i 0.366936i
\(408\) 0 0
\(409\) 32.5777i 1.61087i 0.592687 + 0.805433i \(0.298067\pi\)
−0.592687 + 0.805433i \(0.701933\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.14297 + 1.14297i −0.0562420 + 0.0562420i
\(414\) 0 0
\(415\) 0.613296i 0.0301055i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −11.3571 11.3571i −0.554829 0.554829i 0.373002 0.927831i \(-0.378329\pi\)
−0.927831 + 0.373002i \(0.878329\pi\)
\(420\) 0 0
\(421\) 12.6460 12.6460i 0.616330 0.616330i −0.328258 0.944588i \(-0.606462\pi\)
0.944588 + 0.328258i \(0.106462\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.67584 0.226811
\(426\) 0 0
\(427\) −7.05839 7.05839i −0.341579 0.341579i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −31.2767 −1.50655 −0.753273 0.657708i \(-0.771526\pi\)
−0.753273 + 0.657708i \(0.771526\pi\)
\(432\) 0 0
\(433\) 28.6869 1.37861 0.689303 0.724473i \(-0.257917\pi\)
0.689303 + 0.724473i \(0.257917\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.67654 2.67654i −0.128036 0.128036i
\(438\) 0 0
\(439\) −30.0241 −1.43297 −0.716487 0.697601i \(-0.754251\pi\)
−0.716487 + 0.697601i \(0.754251\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.5402 + 16.5402i −0.785850 + 0.785850i −0.980811 0.194961i \(-0.937542\pi\)
0.194961 + 0.980811i \(0.437542\pi\)
\(444\) 0 0
\(445\) 1.97687 + 1.97687i 0.0937124 + 0.0937124i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 20.3748i 0.961546i 0.876845 + 0.480773i \(0.159644\pi\)
−0.876845 + 0.480773i \(0.840356\pi\)
\(450\) 0 0
\(451\) −4.67265 + 4.67265i −0.220026 + 0.220026i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.37235i 0.111217i
\(456\) 0 0
\(457\) 37.0265i 1.73203i 0.500020 + 0.866014i \(0.333326\pi\)
−0.500020 + 0.866014i \(0.666674\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 22.3029 22.3029i 1.03875 1.03875i 0.0395325 0.999218i \(-0.487413\pi\)
0.999218 0.0395325i \(-0.0125869\pi\)
\(462\) 0 0
\(463\) 21.5480i 1.00142i 0.865616 + 0.500709i \(0.166927\pi\)
−0.865616 + 0.500709i \(0.833073\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −21.7980 21.7980i −1.00869 1.00869i −0.999962 0.00873046i \(-0.997221\pi\)
−0.00873046 0.999962i \(-0.502779\pi\)
\(468\) 0 0
\(469\) 5.66980 5.66980i 0.261807 0.261807i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.12068 −0.143489
\(474\) 0 0
\(475\) 4.12100 + 4.12100i 0.189084 + 0.189084i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −29.5744 −1.35129 −0.675644 0.737228i \(-0.736134\pi\)
−0.675644 + 0.737228i \(0.736134\pi\)
\(480\) 0 0
\(481\) −26.2327 −1.19611
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.37974 3.37974i −0.153466 0.153466i
\(486\) 0 0
\(487\) −17.4819 −0.792179 −0.396089 0.918212i \(-0.629633\pi\)
−0.396089 + 0.918212i \(0.629633\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −19.1434 + 19.1434i −0.863929 + 0.863929i −0.991792 0.127863i \(-0.959188\pi\)
0.127863 + 0.991792i \(0.459188\pi\)
\(492\) 0 0
\(493\) −3.51589 3.51589i −0.158348 0.158348i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.26670i 0.325956i
\(498\) 0 0
\(499\) −1.34854 + 1.34854i −0.0603691 + 0.0603691i −0.736647 0.676278i \(-0.763592\pi\)
0.676278 + 0.736647i \(0.263592\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.93767i 0.443099i 0.975149 + 0.221549i \(0.0711114\pi\)
−0.975149 + 0.221549i \(0.928889\pi\)
\(504\) 0 0
\(505\) 1.99047i 0.0885745i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4.46530 + 4.46530i −0.197921 + 0.197921i −0.799108 0.601187i \(-0.794695\pi\)
0.601187 + 0.799108i \(0.294695\pi\)
\(510\) 0 0
\(511\) 6.66570i 0.294873i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.76380 4.76380i −0.209918 0.209918i
\(516\) 0 0
\(517\) −6.54414 + 6.54414i −0.287811 + 0.287811i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 20.2545 0.887366 0.443683 0.896184i \(-0.353672\pi\)
0.443683 + 0.896184i \(0.353672\pi\)
\(522\) 0 0
\(523\) 16.2772 + 16.2772i 0.711754 + 0.711754i 0.966902 0.255148i \(-0.0821241\pi\)
−0.255148 + 0.966902i \(0.582124\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.77865 −0.425965
\(528\) 0 0
\(529\) 14.4213 0.627012
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −16.5584 16.5584i −0.717223 0.717223i
\(534\) 0 0
\(535\) −4.14610 −0.179251
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.675994 + 0.675994i −0.0291171 + 0.0291171i
\(540\) 0 0
\(541\) 19.1382 + 19.1382i 0.822815 + 0.822815i 0.986511 0.163696i \(-0.0523417\pi\)
−0.163696 + 0.986511i \(0.552342\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.10968i 0.0475334i
\(546\) 0 0
\(547\) 1.15918 1.15918i 0.0495629 0.0495629i −0.681891 0.731454i \(-0.738842\pi\)
0.731454 + 0.681891i \(0.238842\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.19739i 0.264018i
\(552\) 0 0
\(553\) 2.26385i 0.0962689i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3.95406 + 3.95406i −0.167539 + 0.167539i −0.785897 0.618358i \(-0.787798\pi\)
0.618358 + 0.785897i \(0.287798\pi\)
\(558\) 0 0
\(559\) 11.0587i 0.467733i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −32.5658 32.5658i −1.37248 1.37248i −0.856746 0.515739i \(-0.827517\pi\)
−0.515739 0.856746i \(-0.672483\pi\)
\(564\) 0 0
\(565\) 1.50629 1.50629i 0.0633702 0.0633702i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 19.8687 0.832940 0.416470 0.909150i \(-0.363267\pi\)
0.416470 + 0.909150i \(0.363267\pi\)
\(570\) 0 0
\(571\) 4.59303 + 4.59303i 0.192212 + 0.192212i 0.796651 0.604439i \(-0.206603\pi\)
−0.604439 + 0.796651i \(0.706603\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.2084 0.550830
\(576\) 0 0
\(577\) −28.9231 −1.20408 −0.602041 0.798465i \(-0.705646\pi\)
−0.602041 + 0.798465i \(0.705646\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −0.619284 0.619284i −0.0256922 0.0256922i
\(582\) 0 0
\(583\) −10.8497 −0.449348
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.9197 14.9197i 0.615803 0.615803i −0.328649 0.944452i \(-0.606593\pi\)
0.944452 + 0.328649i \(0.106593\pi\)
\(588\) 0 0
\(589\) −8.61830 8.61830i −0.355111 0.355111i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17.3114i 0.710894i 0.934696 + 0.355447i \(0.115671\pi\)
−0.934696 + 0.355447i \(0.884329\pi\)
\(594\) 0 0
\(595\) 0.513417 0.513417i 0.0210480 0.0210480i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0.631879i 0.0258179i 0.999917 + 0.0129089i \(0.00410916\pi\)
−0.999917 + 0.0129089i \(0.995891\pi\)
\(600\) 0 0
\(601\) 45.0956i 1.83949i −0.392520 0.919744i \(-0.628397\pi\)
0.392520 0.919744i \(-0.371603\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.99427 4.99427i 0.203046 0.203046i
\(606\) 0 0
\(607\) 40.1367i 1.62910i 0.580094 + 0.814550i \(0.303016\pi\)
−0.580094 + 0.814550i \(0.696984\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −23.1904 23.1904i −0.938182 0.938182i
\(612\) 0 0
\(613\) 7.38922 7.38922i 0.298448 0.298448i −0.541958 0.840406i \(-0.682317\pi\)
0.840406 + 0.541958i \(0.182317\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.8450 0.597637 0.298818 0.954310i \(-0.403408\pi\)
0.298818 + 0.954310i \(0.403408\pi\)
\(618\) 0 0
\(619\) −24.1413 24.1413i −0.970322 0.970322i 0.0292499 0.999572i \(-0.490688\pi\)
−0.999572 + 0.0292499i \(0.990688\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.99233 −0.159949
\(624\) 0 0
\(625\) −17.8848 −0.715392
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.67720 + 5.67720i 0.226365 + 0.226365i
\(630\) 0 0
\(631\) 19.3942 0.772070 0.386035 0.922484i \(-0.373844\pi\)
0.386035 + 0.922484i \(0.373844\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.06953 + 1.06953i −0.0424429 + 0.0424429i
\(636\) 0 0
\(637\) −2.39551 2.39551i −0.0949134 0.0949134i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 48.4253i 1.91268i −0.292250 0.956342i \(-0.594404\pi\)
0.292250 0.956342i \(-0.405596\pi\)
\(642\) 0 0
\(643\) −2.42503 + 2.42503i −0.0956340 + 0.0956340i −0.753305 0.657671i \(-0.771542\pi\)
0.657671 + 0.753305i \(0.271542\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.9245i 1.29439i −0.762323 0.647197i \(-0.775941\pi\)
0.762323 0.647197i \(-0.224059\pi\)
\(648\) 0 0
\(649\) 1.54528i 0.0606577i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 16.0855 16.0855i 0.629476 0.629476i −0.318460 0.947936i \(-0.603166\pi\)
0.947936 + 0.318460i \(0.103166\pi\)
\(654\) 0 0
\(655\) 9.20329i 0.359602i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 32.1819 + 32.1819i 1.25363 + 1.25363i 0.954082 + 0.299546i \(0.0968353\pi\)
0.299546 + 0.954082i \(0.403165\pi\)
\(660\) 0 0
\(661\) −28.2861 + 28.2861i −1.10020 + 1.10020i −0.105818 + 0.994386i \(0.533746\pi\)
−0.994386 + 0.105818i \(0.966254\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.904988 0.0350939
\(666\) 0 0
\(667\) −9.93179 9.93179i −0.384561 0.384561i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9.54285 0.368398
\(672\) 0 0
\(673\) −2.63038 −0.101394 −0.0506968 0.998714i \(-0.516144\pi\)
−0.0506968 + 0.998714i \(0.516144\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −15.1362 15.1362i −0.581730 0.581730i 0.353649 0.935378i \(-0.384941\pi\)
−0.935378 + 0.353649i \(0.884941\pi\)
\(678\) 0 0
\(679\) 6.82547 0.261938
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.97609 5.97609i 0.228669 0.228669i −0.583468 0.812136i \(-0.698305\pi\)
0.812136 + 0.583468i \(0.198305\pi\)
\(684\) 0 0
\(685\) −8.34150 8.34150i −0.318712 0.318712i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 38.4478i 1.46474i
\(690\) 0 0
\(691\) −13.6369 + 13.6369i −0.518774 + 0.518774i −0.917200 0.398426i \(-0.869556\pi\)
0.398426 + 0.917200i \(0.369556\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.7236i 0.406769i
\(696\) 0 0
\(697\) 7.16704i 0.271471i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −23.6139 + 23.6139i −0.891886 + 0.891886i −0.994701 0.102815i \(-0.967215\pi\)
0.102815 + 0.994701i \(0.467215\pi\)
\(702\) 0 0
\(703\) 10.0071i 0.377424i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.00990 2.00990i −0.0755900 0.0755900i
\(708\) 0 0
\(709\) 13.4057 13.4057i 0.503460 0.503460i −0.409051 0.912511i \(-0.634140\pi\)
0.912511 + 0.409051i \(0.134140\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −27.6230 −1.03449
\(714\) 0 0
\(715\) −1.60369 1.60369i −0.0599746 0.0599746i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.0030 0.596810 0.298405 0.954439i \(-0.403545\pi\)
0.298405 + 0.954439i \(0.403545\pi\)
\(720\) 0 0
\(721\) 9.62061 0.358290
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 15.2917 + 15.2917i 0.567920 + 0.567920i
\(726\) 0 0
\(727\) −36.4754 −1.35280 −0.676399 0.736536i \(-0.736460\pi\)
−0.676399 + 0.736536i \(0.736460\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.39329 + 2.39329i −0.0885192 + 0.0885192i
\(732\) 0 0
\(733\) −2.37303 2.37303i −0.0876497 0.0876497i 0.661923 0.749572i \(-0.269741\pi\)
−0.749572 + 0.661923i \(0.769741\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.66550i 0.282362i
\(738\) 0 0
\(739\) −10.6601 + 10.6601i −0.392137 + 0.392137i −0.875449 0.483311i \(-0.839434\pi\)
0.483311 + 0.875449i \(0.339434\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.4432i 0.603244i −0.953428 0.301622i \(-0.902472\pi\)
0.953428 0.301622i \(-0.0975281\pi\)
\(744\) 0 0
\(745\) 5.36350i 0.196503i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.18658 4.18658i 0.152974 0.152974i
\(750\) 0 0
\(751\) 8.62234i 0.314634i 0.987548 + 0.157317i \(0.0502844\pi\)
−0.987548 + 0.157317i \(0.949716\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0.924793 + 0.924793i 0.0336567 + 0.0336567i
\(756\) 0 0
\(757\) 14.7724 14.7724i 0.536910 0.536910i −0.385710 0.922620i \(-0.626043\pi\)
0.922620 + 0.385710i \(0.126043\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −49.3931 −1.79050 −0.895249 0.445566i \(-0.853002\pi\)
−0.895249 + 0.445566i \(0.853002\pi\)
\(762\) 0 0
\(763\) 1.12051 + 1.12051i 0.0405653 + 0.0405653i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.47599 0.197727
\(768\) 0 0
\(769\) 20.4566 0.737685 0.368843 0.929492i \(-0.379754\pi\)
0.368843 + 0.929492i \(0.379754\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −30.7904 30.7904i −1.10745 1.10745i −0.993484 0.113967i \(-0.963644\pi\)
−0.113967 0.993484i \(-0.536356\pi\)
\(774\) 0 0
\(775\) 42.5304 1.52774
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.31659 + 6.31659i −0.226315 + 0.226315i
\(780\) 0 0
\(781\) −4.91224 4.91224i −0.175774 0.175774i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.21080i 0.221673i
\(786\) 0 0
\(787\) −4.61526 + 4.61526i −0.164516 + 0.164516i −0.784564 0.620048i \(-0.787113\pi\)
0.620048 + 0.784564i \(0.287113\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 3.04200i 0.108161i
\(792\) 0 0
\(793\) 33.8168i 1.20087i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −27.8979 + 27.8979i −0.988195 + 0.988195i −0.999931 0.0117360i \(-0.996264\pi\)
0.0117360 + 0.999931i \(0.496264\pi\)
\(798\) 0 0
\(799\) 10.0376i 0.355104i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.50597 + 4.50597i 0.159012 + 0.159012i
\(804\) 0 0
\(805\) 1.45031 1.45031i 0.0511168 0.0511168i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.6104 0.654308 0.327154 0.944971i \(-0.393910\pi\)
0.327154 + 0.944971i \(0.393910\pi\)
\(810\) 0 0
\(811\) 38.8480 + 38.8480i 1.36414 + 1.36414i 0.868568 + 0.495570i \(0.165041\pi\)
0.495570 + 0.868568i \(0.334959\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −9.01866 −0.315910
\(816\) 0 0
\(817\) −4.21861 −0.147590
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −37.4454 37.4454i −1.30685 1.30685i −0.923673 0.383181i \(-0.874829\pi\)
−0.383181 0.923673i \(-0.625171\pi\)
\(822\) 0 0
\(823\) −1.86401 −0.0649752 −0.0324876 0.999472i \(-0.510343\pi\)
−0.0324876 + 0.999472i \(0.510343\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.7774 31.7774i 1.10501 1.10501i 0.111211 0.993797i \(-0.464527\pi\)
0.993797 0.111211i \(-0.0354729\pi\)
\(828\) 0 0
\(829\) 14.5497 + 14.5497i 0.505333 + 0.505333i 0.913090 0.407757i \(-0.133689\pi\)
−0.407757 + 0.913090i \(0.633689\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.03686i 0.0359250i
\(834\) 0 0
\(835\) 4.67686 4.67686i 0.161849 0.161849i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 48.2843i 1.66696i 0.552549 + 0.833480i \(0.313655\pi\)
−0.552549 + 0.833480i \(0.686345\pi\)
\(840\) 0 0
\(841\) 6.00345i 0.207016i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −0.754183 + 0.754183i −0.0259447 + 0.0259447i
\(846\) 0 0
\(847\) 10.0861i 0.346561i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 16.0371 + 16.0371i 0.549745 + 0.549745i
\(852\) 0 0
\(853\) −29.1174 + 29.1174i −0.996962 + 0.996962i −0.999995 0.00303388i \(-0.999034\pi\)
0.00303388 + 0.999995i \(0.499034\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −30.4208 −1.03915 −0.519577 0.854424i \(-0.673910\pi\)
−0.519577 + 0.854424i \(0.673910\pi\)
\(858\) 0 0
\(859\) −12.4603 12.4603i −0.425141 0.425141i 0.461828 0.886969i \(-0.347194\pi\)
−0.886969 + 0.461828i \(0.847194\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −20.6468 −0.702826 −0.351413 0.936221i \(-0.614299\pi\)
−0.351413 + 0.936221i \(0.614299\pi\)
\(864\) 0 0
\(865\) 16.8102 0.571564
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.53035 1.53035i −0.0519136 0.0519136i
\(870\) 0 0
\(871\) −27.1641 −0.920421
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.70884 + 4.70884i −0.159188 + 0.159188i
\(876\) 0 0
\(877\) −16.2158 16.2158i −0.547570 0.547570i 0.378167 0.925737i \(-0.376554\pi\)
−0.925737 + 0.378167i \(0.876554\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 7.21499i 0.243079i −0.992587 0.121540i \(-0.961217\pi\)
0.992587 0.121540i \(-0.0387831\pi\)
\(882\) 0 0
\(883\) −19.5756 + 19.5756i −0.658771 + 0.658771i −0.955089 0.296318i \(-0.904241\pi\)
0.296318 + 0.955089i \(0.404241\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 22.3985i 0.752067i −0.926606 0.376034i \(-0.877288\pi\)
0.926606 0.376034i \(-0.122712\pi\)
\(888\) 0 0
\(889\) 2.15994i 0.0724421i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8.84652 + 8.84652i −0.296037 + 0.296037i
\(894\) 0 0
\(895\) 0.822290i 0.0274861i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −31.9798 31.9798i −1.06659 1.06659i
\(900\) 0 0
\(901\) −8.32077 + 8.32077i −0.277205 + 0.277205i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.70398 −0.123125
\(906\) 0 0
\(907\) 2.55150 + 2.55150i 0.0847212 + 0.0847212i 0.748197 0.663476i \(-0.230920\pi\)
−0.663476 + 0.748197i \(0.730920\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 28.3400 0.938946 0.469473 0.882947i \(-0.344444\pi\)
0.469473 + 0.882947i \(0.344444\pi\)
\(912\) 0 0
\(913\) 0.837264 0.0277094
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 9.29314 + 9.29314i 0.306886 + 0.306886i
\(918\) 0 0
\(919\) −14.6615 −0.483639 −0.241819 0.970321i \(-0.577744\pi\)
−0.241819 + 0.970321i \(0.577744\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 17.4074 17.4074i 0.572973 0.572973i
\(924\) 0 0
\(925\) −24.6919 24.6919i −0.811866 0.811866i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 38.0843i 1.24950i −0.780823 0.624752i \(-0.785200\pi\)
0.780823 0.624752i \(-0.214800\pi\)
\(930\) 0 0
\(931\) −0.913823 + 0.913823i −0.0299493 + 0.0299493i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.694133i 0.0227006i
\(936\) 0 0
\(937\) 31.6411i 1.03367i −0.856085 0.516835i \(-0.827110\pi\)
0.856085 0.516835i \(-0.172890\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 12.4691 12.4691i 0.406480 0.406480i −0.474029 0.880509i \(-0.657201\pi\)
0.880509 + 0.474029i \(0.157201\pi\)
\(942\) 0 0
\(943\) 20.2456i 0.659289i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −35.5835 35.5835i −1.15631 1.15631i −0.985263 0.171044i \(-0.945286\pi\)
−0.171044 0.985263i \(-0.554714\pi\)
\(948\) 0 0
\(949\) −15.9677 + 15.9677i −0.518335 + 0.518335i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.461925 −0.0149632 −0.00748160 0.999972i \(-0.502381\pi\)
−0.00748160 + 0.999972i \(0.502381\pi\)
\(954\) 0 0
\(955\) −5.10192 5.10192i −0.165094 0.165094i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 16.8459 0.543982
\(960\) 0 0
\(961\) −57.9445 −1.86918
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −6.01041 6.01041i −0.193482 0.193482i
\(966\) 0 0
\(967\) 31.2365 1.00450 0.502248 0.864723i \(-0.332506\pi\)
0.502248 + 0.864723i \(0.332506\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 21.5210 21.5210i 0.690642 0.690642i −0.271731 0.962373i \(-0.587596\pi\)
0.962373 + 0.271731i \(0.0875960\pi\)
\(972\) 0 0
\(973\) 10.8283 + 10.8283i 0.347139 + 0.347139i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 39.8191i 1.27393i 0.770895 + 0.636963i \(0.219809\pi\)
−0.770895 + 0.636963i \(0.780191\pi\)
\(978\) 0 0
\(979\) 2.69879 2.69879i 0.0862537 0.0862537i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 44.1551i 1.40833i −0.710037 0.704164i \(-0.751322\pi\)
0.710037 0.704164i \(-0.248678\pi\)
\(984\) 0 0
\(985\) 8.68789i 0.276819i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.76065 + 6.76065i −0.214976 + 0.214976i
\(990\) 0 0
\(991\) 19.2245i 0.610687i −0.952242 0.305343i \(-0.901229\pi\)
0.952242 0.305343i \(-0.0987713\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 11.2994 + 11.2994i 0.358214 + 0.358214i
\(996\) 0 0
\(997\) −25.3176 + 25.3176i −0.801818 + 0.801818i −0.983380 0.181562i \(-0.941885\pi\)
0.181562 + 0.983380i \(0.441885\pi\)
\(998\) 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.v.d.1583.8 36
3.2 odd 2 inner 4032.2.v.d.1583.11 36
4.3 odd 2 1008.2.v.d.323.11 yes 36
12.11 even 2 1008.2.v.d.323.8 36
16.5 even 4 1008.2.v.d.827.8 yes 36
16.11 odd 4 inner 4032.2.v.d.3599.11 36
48.5 odd 4 1008.2.v.d.827.11 yes 36
48.11 even 4 inner 4032.2.v.d.3599.8 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.v.d.323.8 36 12.11 even 2
1008.2.v.d.323.11 yes 36 4.3 odd 2
1008.2.v.d.827.8 yes 36 16.5 even 4
1008.2.v.d.827.11 yes 36 48.5 odd 4
4032.2.v.d.1583.8 36 1.1 even 1 trivial
4032.2.v.d.1583.11 36 3.2 odd 2 inner
4032.2.v.d.3599.8 36 48.11 even 4 inner
4032.2.v.d.3599.11 36 16.11 odd 4 inner