Properties

Label 4032.2.j.f.2591.8
Level $4032$
Weight $2$
Character 4032.2591
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(2591,4032)]
 
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, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.2591");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.j (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.11007531417600000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.8
Root \(-0.159959 + 0.596975i\) of defining polynomial
Character \(\chi\) \(=\) 4032.2591
Dual form 4032.2.j.f.2591.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.51387 q^{5} +1.00000i q^{7} +O(q^{10})\) \(q-1.51387 q^{5} +1.00000i q^{7} -0.935622i q^{11} +2.14093i q^{13} -2.62210i q^{17} +3.46410 q^{19} -6.86474 q^{23} -2.70820 q^{25} +2.44949 q^{29} -2.00000i q^{31} -1.51387i q^{35} -2.14093i q^{37} -2.62210i q^{41} +7.74597 q^{43} -1.00000 q^{49} +10.3762 q^{53} +1.41641i q^{55} -9.79796i q^{59} +11.2101i q^{61} -3.24109i q^{65} -2.14093 q^{67} -1.62054 q^{71} -5.70820 q^{73} +0.935622 q^{77} +1.70820i q^{79} -1.87124i q^{83} +3.96951i q^{85} +2.62210i q^{89} -2.14093 q^{91} -5.24419 q^{95} +17.7082 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 64 q^{25} - 16 q^{49} + 16 q^{73} + 176 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.51387 −0.677022 −0.338511 0.940962i \(-0.609923\pi\)
−0.338511 + 0.940962i \(0.609923\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 0.935622i − 0.282101i −0.990002 0.141050i \(-0.954952\pi\)
0.990002 0.141050i \(-0.0450479\pi\)
\(12\) 0 0
\(13\) 2.14093i 0.593788i 0.954910 + 0.296894i \(0.0959508\pi\)
−0.954910 + 0.296894i \(0.904049\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.62210i − 0.635952i −0.948099 0.317976i \(-0.896997\pi\)
0.948099 0.317976i \(-0.103003\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.86474 −1.43140 −0.715698 0.698410i \(-0.753891\pi\)
−0.715698 + 0.698410i \(0.753891\pi\)
\(24\) 0 0
\(25\) −2.70820 −0.541641
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.44949 0.454859 0.227429 0.973795i \(-0.426968\pi\)
0.227429 + 0.973795i \(0.426968\pi\)
\(30\) 0 0
\(31\) − 2.00000i − 0.359211i −0.983739 0.179605i \(-0.942518\pi\)
0.983739 0.179605i \(-0.0574821\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 1.51387i − 0.255890i
\(36\) 0 0
\(37\) − 2.14093i − 0.351967i −0.984393 0.175984i \(-0.943689\pi\)
0.984393 0.175984i \(-0.0563106\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 2.62210i − 0.409503i −0.978814 0.204751i \(-0.934361\pi\)
0.978814 0.204751i \(-0.0656386\pi\)
\(42\) 0 0
\(43\) 7.74597 1.18125 0.590624 0.806947i \(-0.298881\pi\)
0.590624 + 0.806947i \(0.298881\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.3762 1.42528 0.712641 0.701529i \(-0.247499\pi\)
0.712641 + 0.701529i \(0.247499\pi\)
\(54\) 0 0
\(55\) 1.41641i 0.190988i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 9.79796i − 1.27559i −0.770208 0.637793i \(-0.779848\pi\)
0.770208 0.637793i \(-0.220152\pi\)
\(60\) 0 0
\(61\) 11.2101i 1.43530i 0.696403 + 0.717651i \(0.254783\pi\)
−0.696403 + 0.717651i \(0.745217\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 3.24109i − 0.402008i
\(66\) 0 0
\(67\) −2.14093 −0.261557 −0.130778 0.991412i \(-0.541748\pi\)
−0.130778 + 0.991412i \(0.541748\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.62054 −0.192323 −0.0961616 0.995366i \(-0.530657\pi\)
−0.0961616 + 0.995366i \(0.530657\pi\)
\(72\) 0 0
\(73\) −5.70820 −0.668095 −0.334047 0.942556i \(-0.608415\pi\)
−0.334047 + 0.942556i \(0.608415\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.935622 0.106624
\(78\) 0 0
\(79\) 1.70820i 0.192188i 0.995372 + 0.0960940i \(0.0306349\pi\)
−0.995372 + 0.0960940i \(0.969365\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 1.87124i − 0.205396i −0.994713 0.102698i \(-0.967253\pi\)
0.994713 0.102698i \(-0.0327475\pi\)
\(84\) 0 0
\(85\) 3.96951i 0.430554i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.62210i 0.277942i 0.990296 + 0.138971i \(0.0443794\pi\)
−0.990296 + 0.138971i \(0.955621\pi\)
\(90\) 0 0
\(91\) −2.14093 −0.224431
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.24419 −0.538043
\(96\) 0 0
\(97\) 17.7082 1.79800 0.898998 0.437953i \(-0.144296\pi\)
0.898998 + 0.437953i \(0.144296\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.38511 0.336831 0.168416 0.985716i \(-0.446135\pi\)
0.168416 + 0.985716i \(0.446135\pi\)
\(102\) 0 0
\(103\) − 9.41641i − 0.927826i −0.885881 0.463913i \(-0.846445\pi\)
0.885881 0.463913i \(-0.153555\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 8.86234i − 0.856754i −0.903600 0.428377i \(-0.859085\pi\)
0.903600 0.428377i \(-0.140915\pi\)
\(108\) 0 0
\(109\) 7.74597i 0.741929i 0.928647 + 0.370965i \(0.120973\pi\)
−0.928647 + 0.370965i \(0.879027\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 12.7279i − 1.19734i −0.800995 0.598671i \(-0.795696\pi\)
0.800995 0.598671i \(-0.204304\pi\)
\(114\) 0 0
\(115\) 10.3923 0.969087
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.62210 0.240367
\(120\) 0 0
\(121\) 10.1246 0.920419
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.6692 1.04373
\(126\) 0 0
\(127\) − 1.70820i − 0.151579i −0.997124 0.0757893i \(-0.975852\pi\)
0.997124 0.0757893i \(-0.0241476\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 12.8257i − 1.12059i −0.828294 0.560293i \(-0.810689\pi\)
0.828294 0.560293i \(-0.189311\pi\)
\(132\) 0 0
\(133\) 3.46410i 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 4.24264i − 0.362473i −0.983440 0.181237i \(-0.941990\pi\)
0.983440 0.181237i \(-0.0580100\pi\)
\(138\) 0 0
\(139\) 4.28187 0.363183 0.181592 0.983374i \(-0.441875\pi\)
0.181592 + 0.983374i \(0.441875\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00310 0.167508
\(144\) 0 0
\(145\) −3.70820 −0.307950
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.50496 0.696754 0.348377 0.937355i \(-0.386733\pi\)
0.348377 + 0.937355i \(0.386733\pi\)
\(150\) 0 0
\(151\) − 11.4164i − 0.929054i −0.885559 0.464527i \(-0.846224\pi\)
0.885559 0.464527i \(-0.153776\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.02774i 0.243194i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 6.86474i − 0.541017i
\(162\) 0 0
\(163\) 15.9973 1.25301 0.626504 0.779418i \(-0.284485\pi\)
0.626504 + 0.779418i \(0.284485\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.24419 −0.405808 −0.202904 0.979199i \(-0.565038\pi\)
−0.202904 + 0.979199i \(0.565038\pi\)
\(168\) 0 0
\(169\) 8.41641 0.647416
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.3118 0.860022 0.430011 0.902824i \(-0.358510\pi\)
0.430011 + 0.902824i \(0.358510\pi\)
\(174\) 0 0
\(175\) − 2.70820i − 0.204721i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 3.96336i − 0.296235i −0.988970 0.148118i \(-0.952679\pi\)
0.988970 0.148118i \(-0.0473214\pi\)
\(180\) 0 0
\(181\) 0.505406i 0.0375665i 0.999824 + 0.0187833i \(0.00597925\pi\)
−0.999824 + 0.0187833i \(0.994021\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.24109i 0.238290i
\(186\) 0 0
\(187\) −2.45329 −0.179402
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −15.3500 −1.11069 −0.555344 0.831621i \(-0.687413\pi\)
−0.555344 + 0.831621i \(0.687413\pi\)
\(192\) 0 0
\(193\) 10.2918 0.740820 0.370410 0.928868i \(-0.379217\pi\)
0.370410 + 0.928868i \(0.379217\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.2019 1.65307 0.826533 0.562888i \(-0.190310\pi\)
0.826533 + 0.562888i \(0.190310\pi\)
\(198\) 0 0
\(199\) − 15.4164i − 1.09284i −0.837511 0.546420i \(-0.815990\pi\)
0.837511 0.546420i \(-0.184010\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.44949i 0.171920i
\(204\) 0 0
\(205\) 3.96951i 0.277242i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 3.24109i − 0.224191i
\(210\) 0 0
\(211\) −12.0278 −0.828030 −0.414015 0.910270i \(-0.635874\pi\)
−0.414015 + 0.910270i \(0.635874\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −11.7264 −0.799732
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.61373 0.377620
\(222\) 0 0
\(223\) − 4.00000i − 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 28.6791i − 1.90350i −0.306874 0.951750i \(-0.599283\pi\)
0.306874 0.951750i \(-0.400717\pi\)
\(228\) 0 0
\(229\) − 9.06914i − 0.599305i −0.954048 0.299653i \(-0.903129\pi\)
0.954048 0.299653i \(-0.0968708\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.2163i 1.52095i 0.649367 + 0.760475i \(0.275034\pi\)
−0.649367 + 0.760475i \(0.724966\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.62054 0.104824 0.0524122 0.998626i \(-0.483309\pi\)
0.0524122 + 0.998626i \(0.483309\pi\)
\(240\) 0 0
\(241\) 6.29180 0.405290 0.202645 0.979252i \(-0.435046\pi\)
0.202645 + 0.979252i \(0.435046\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.51387 0.0967175
\(246\) 0 0
\(247\) 7.41641i 0.471895i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 13.9822i − 0.882548i −0.897372 0.441274i \(-0.854527\pi\)
0.897372 0.441274i \(-0.145473\pi\)
\(252\) 0 0
\(253\) 6.42280i 0.403798i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 24.8369i − 1.54928i −0.632402 0.774640i \(-0.717931\pi\)
0.632402 0.774640i \(-0.282069\pi\)
\(258\) 0 0
\(259\) 2.14093 0.133031
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 17.3531 1.07004 0.535020 0.844840i \(-0.320304\pi\)
0.535020 + 0.844840i \(0.320304\pi\)
\(264\) 0 0
\(265\) −15.7082 −0.964947
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −29.0365 −1.77039 −0.885193 0.465223i \(-0.845974\pi\)
−0.885193 + 0.465223i \(0.845974\pi\)
\(270\) 0 0
\(271\) 2.00000i 0.121491i 0.998153 + 0.0607457i \(0.0193479\pi\)
−0.998153 + 0.0607457i \(0.980652\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.53385i 0.152797i
\(276\) 0 0
\(277\) 22.1078i 1.32833i 0.747587 + 0.664164i \(0.231212\pi\)
−0.747587 + 0.664164i \(0.768788\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.48683i 0.565937i 0.959129 + 0.282969i \(0.0913192\pi\)
−0.959129 + 0.282969i \(0.908681\pi\)
\(282\) 0 0
\(283\) 13.0386 0.775067 0.387533 0.921856i \(-0.373327\pi\)
0.387533 + 0.921856i \(0.373327\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.62210 0.154777
\(288\) 0 0
\(289\) 10.1246 0.595565
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.3396 0.837726 0.418863 0.908049i \(-0.362429\pi\)
0.418863 + 0.908049i \(0.362429\pi\)
\(294\) 0 0
\(295\) 14.8328i 0.863600i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 14.6969i − 0.849946i
\(300\) 0 0
\(301\) 7.74597i 0.446470i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 16.9706i − 0.971732i
\(306\) 0 0
\(307\) 5.09963 0.291051 0.145526 0.989354i \(-0.453513\pi\)
0.145526 + 0.989354i \(0.453513\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −27.4589 −1.55705 −0.778527 0.627611i \(-0.784033\pi\)
−0.778527 + 0.627611i \(0.784033\pi\)
\(312\) 0 0
\(313\) −9.41641 −0.532247 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.60598 0.202532 0.101266 0.994859i \(-0.467711\pi\)
0.101266 + 0.994859i \(0.467711\pi\)
\(318\) 0 0
\(319\) − 2.29180i − 0.128316i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 9.08321i − 0.505403i
\(324\) 0 0
\(325\) − 5.79808i − 0.321620i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 30.1661 1.65808 0.829039 0.559190i \(-0.188888\pi\)
0.829039 + 0.559190i \(0.188888\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.24109 0.177080
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.87124 −0.101334
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.7336i 0.576209i 0.957599 + 0.288104i \(0.0930250\pi\)
−0.957599 + 0.288104i \(0.906975\pi\)
\(348\) 0 0
\(349\) 9.57454i 0.512513i 0.966609 + 0.256257i \(0.0824892\pi\)
−0.966609 + 0.256257i \(0.917511\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 5.86319i − 0.312066i −0.987752 0.156033i \(-0.950129\pi\)
0.987752 0.156033i \(-0.0498706\pi\)
\(354\) 0 0
\(355\) 2.45329 0.130207
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.86163 0.256587 0.128294 0.991736i \(-0.459050\pi\)
0.128294 + 0.991736i \(0.459050\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8.64147 0.452315
\(366\) 0 0
\(367\) − 27.4164i − 1.43112i −0.698549 0.715562i \(-0.746170\pi\)
0.698549 0.715562i \(-0.253830\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.3762i 0.538706i
\(372\) 0 0
\(373\) − 13.8564i − 0.717458i −0.933442 0.358729i \(-0.883210\pi\)
0.933442 0.358729i \(-0.116790\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.24419i 0.270090i
\(378\) 0 0
\(379\) −6.11044 −0.313872 −0.156936 0.987609i \(-0.550162\pi\)
−0.156936 + 0.987609i \(0.550162\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.24419 0.267966 0.133983 0.990984i \(-0.457223\pi\)
0.133983 + 0.990984i \(0.457223\pi\)
\(384\) 0 0
\(385\) −1.41641 −0.0721868
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −0.578246 −0.0293182 −0.0146591 0.999893i \(-0.504666\pi\)
−0.0146591 + 0.999893i \(0.504666\pi\)
\(390\) 0 0
\(391\) 18.0000i 0.910299i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 2.58600i − 0.130116i
\(396\) 0 0
\(397\) − 15.4919i − 0.777518i −0.921340 0.388759i \(-0.872904\pi\)
0.921340 0.388759i \(-0.127096\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 6.24574i − 0.311898i −0.987765 0.155949i \(-0.950157\pi\)
0.987765 0.155949i \(-0.0498435\pi\)
\(402\) 0 0
\(403\) 4.28187 0.213295
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.00310 −0.0992901
\(408\) 0 0
\(409\) −29.7082 −1.46898 −0.734488 0.678622i \(-0.762578\pi\)
−0.734488 + 0.678622i \(0.762578\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.79796 0.482126
\(414\) 0 0
\(415\) 2.83282i 0.139057i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 32.8634i 1.60548i 0.596329 + 0.802740i \(0.296625\pi\)
−0.596329 + 0.802740i \(0.703375\pi\)
\(420\) 0 0
\(421\) 30.6715i 1.49484i 0.664353 + 0.747419i \(0.268707\pi\)
−0.664353 + 0.747419i \(0.731293\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.10117i 0.344457i
\(426\) 0 0
\(427\) −11.2101 −0.542493
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.86474 −0.330663 −0.165331 0.986238i \(-0.552869\pi\)
−0.165331 + 0.986238i \(0.552869\pi\)
\(432\) 0 0
\(433\) 17.4164 0.836979 0.418490 0.908222i \(-0.362560\pi\)
0.418490 + 0.908222i \(0.362560\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −23.7801 −1.13756
\(438\) 0 0
\(439\) − 20.0000i − 0.954548i −0.878755 0.477274i \(-0.841625\pi\)
0.878755 0.477274i \(-0.158375\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 25.8723i − 1.22923i −0.788828 0.614614i \(-0.789312\pi\)
0.788828 0.614614i \(-0.210688\pi\)
\(444\) 0 0
\(445\) − 3.96951i − 0.188173i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 21.2132i − 1.00111i −0.865704 0.500556i \(-0.833129\pi\)
0.865704 0.500556i \(-0.166871\pi\)
\(450\) 0 0
\(451\) −2.45329 −0.115521
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.24109 0.151945
\(456\) 0 0
\(457\) −15.4164 −0.721149 −0.360575 0.932730i \(-0.617419\pi\)
−0.360575 + 0.932730i \(0.617419\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.4683 0.580708 0.290354 0.956919i \(-0.406227\pi\)
0.290354 + 0.956919i \(0.406227\pi\)
\(462\) 0 0
\(463\) 17.7082i 0.822970i 0.911417 + 0.411485i \(0.134990\pi\)
−0.911417 + 0.411485i \(0.865010\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.92672i 0.366805i 0.983038 + 0.183402i \(0.0587111\pi\)
−0.983038 + 0.183402i \(0.941289\pi\)
\(468\) 0 0
\(469\) − 2.14093i − 0.0988591i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 7.24730i − 0.333231i
\(474\) 0 0
\(475\) −9.38149 −0.430452
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −25.4558 −1.16311 −0.581554 0.813508i \(-0.697555\pi\)
−0.581554 + 0.813508i \(0.697555\pi\)
\(480\) 0 0
\(481\) 4.58359 0.208994
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −26.8079 −1.21728
\(486\) 0 0
\(487\) 32.0000i 1.45006i 0.688718 + 0.725029i \(0.258174\pi\)
−0.688718 + 0.725029i \(0.741826\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 32.6425i 1.47313i 0.676364 + 0.736567i \(0.263555\pi\)
−0.676364 + 0.736567i \(0.736445\pi\)
\(492\) 0 0
\(493\) − 6.42280i − 0.289268i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 1.62054i − 0.0726914i
\(498\) 0 0
\(499\) −13.0386 −0.583690 −0.291845 0.956466i \(-0.594269\pi\)
−0.291845 + 0.956466i \(0.594269\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3.24109 −0.144513 −0.0722565 0.997386i \(-0.523020\pi\)
−0.0722565 + 0.997386i \(0.523020\pi\)
\(504\) 0 0
\(505\) −5.12461 −0.228042
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −30.1930 −1.33828 −0.669140 0.743136i \(-0.733338\pi\)
−0.669140 + 0.743136i \(0.733338\pi\)
\(510\) 0 0
\(511\) − 5.70820i − 0.252516i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 14.2552i 0.628159i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 36.5632i 1.60186i 0.598755 + 0.800932i \(0.295662\pi\)
−0.598755 + 0.800932i \(0.704338\pi\)
\(522\) 0 0
\(523\) 4.28187 0.187233 0.0936164 0.995608i \(-0.470157\pi\)
0.0936164 + 0.995608i \(0.470157\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.24419 −0.228441
\(528\) 0 0
\(529\) 24.1246 1.04890
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.61373 0.243158
\(534\) 0 0
\(535\) 13.4164i 0.580042i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.935622i 0.0403001i
\(540\) 0 0
\(541\) − 23.7433i − 1.02080i −0.859936 0.510402i \(-0.829497\pi\)
0.859936 0.510402i \(-0.170503\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 11.7264i − 0.502303i
\(546\) 0 0
\(547\) −17.6329 −0.753927 −0.376963 0.926228i \(-0.623032\pi\)
−0.376963 + 0.926228i \(0.623032\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.48528 0.361485
\(552\) 0 0
\(553\) −1.70820 −0.0726402
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 29.2574 1.23967 0.619837 0.784730i \(-0.287199\pi\)
0.619837 + 0.784730i \(0.287199\pi\)
\(558\) 0 0
\(559\) 16.5836i 0.701411i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 41.5048i − 1.74922i −0.484827 0.874610i \(-0.661118\pi\)
0.484827 0.874610i \(-0.338882\pi\)
\(564\) 0 0
\(565\) 19.2684i 0.810627i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 21.2132i 0.889304i 0.895703 + 0.444652i \(0.146673\pi\)
−0.895703 + 0.444652i \(0.853327\pi\)
\(570\) 0 0
\(571\) −34.1356 −1.42853 −0.714265 0.699875i \(-0.753239\pi\)
−0.714265 + 0.699875i \(0.753239\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 18.5911 0.775303
\(576\) 0 0
\(577\) −21.4164 −0.891577 −0.445788 0.895138i \(-0.647077\pi\)
−0.445788 + 0.895138i \(0.647077\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.87124 0.0776323
\(582\) 0 0
\(583\) − 9.70820i − 0.402073i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 32.4216i 1.33818i 0.743180 + 0.669092i \(0.233317\pi\)
−0.743180 + 0.669092i \(0.766683\pi\)
\(588\) 0 0
\(589\) − 6.92820i − 0.285472i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 13.1105i 0.538383i 0.963087 + 0.269191i \(0.0867564\pi\)
−0.963087 + 0.269191i \(0.913244\pi\)
\(594\) 0 0
\(595\) −3.96951 −0.162734
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 29.0795 1.18816 0.594078 0.804408i \(-0.297517\pi\)
0.594078 + 0.804408i \(0.297517\pi\)
\(600\) 0 0
\(601\) −2.58359 −0.105387 −0.0526935 0.998611i \(-0.516781\pi\)
−0.0526935 + 0.998611i \(0.516781\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −15.3273 −0.623144
\(606\) 0 0
\(607\) 8.00000i 0.324710i 0.986732 + 0.162355i \(0.0519090\pi\)
−0.986732 + 0.162355i \(0.948091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 1.63553i − 0.0660583i −0.999454 0.0330292i \(-0.989485\pi\)
0.999454 0.0330292i \(-0.0105154\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.7279i 0.512407i 0.966623 + 0.256203i \(0.0824717\pi\)
−0.966623 + 0.256203i \(0.917528\pi\)
\(618\) 0 0
\(619\) 29.3483 1.17961 0.589805 0.807546i \(-0.299205\pi\)
0.589805 + 0.807546i \(0.299205\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.62210 −0.105052
\(624\) 0 0
\(625\) −4.12461 −0.164984
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5.61373 −0.223834
\(630\) 0 0
\(631\) − 28.5410i − 1.13620i −0.822960 0.568100i \(-0.807679\pi\)
0.822960 0.568100i \(-0.192321\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.58600i 0.102622i
\(636\) 0 0
\(637\) − 2.14093i − 0.0848268i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 34.9427i − 1.38015i −0.723737 0.690076i \(-0.757577\pi\)
0.723737 0.690076i \(-0.242423\pi\)
\(642\) 0 0
\(643\) −7.74597 −0.305471 −0.152736 0.988267i \(-0.548808\pi\)
−0.152736 + 0.988267i \(0.548808\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.7031 1.28569 0.642847 0.765995i \(-0.277753\pi\)
0.642847 + 0.765995i \(0.277753\pi\)
\(648\) 0 0
\(649\) −9.16718 −0.359843
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.9899 −0.625735 −0.312867 0.949797i \(-0.601290\pi\)
−0.312867 + 0.949797i \(0.601290\pi\)
\(654\) 0 0
\(655\) 19.4164i 0.758662i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 40.1275i 1.56314i 0.623815 + 0.781572i \(0.285582\pi\)
−0.623815 + 0.781572i \(0.714418\pi\)
\(660\) 0 0
\(661\) − 48.4974i − 1.88633i −0.332323 0.943166i \(-0.607832\pi\)
0.332323 0.943166i \(-0.392168\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 5.24419i − 0.203361i
\(666\) 0 0
\(667\) −16.8151 −0.651083
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.4884 0.404900
\(672\) 0 0
\(673\) 37.7082 1.45354 0.726772 0.686879i \(-0.241020\pi\)
0.726772 + 0.686879i \(0.241020\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −23.4228 −0.900210 −0.450105 0.892976i \(-0.648613\pi\)
−0.450105 + 0.892976i \(0.648613\pi\)
\(678\) 0 0
\(679\) 17.7082i 0.679578i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25.8723i 0.989975i 0.868900 + 0.494987i \(0.164827\pi\)
−0.868900 + 0.494987i \(0.835173\pi\)
\(684\) 0 0
\(685\) 6.42280i 0.245402i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 22.2148i 0.846315i
\(690\) 0 0
\(691\) 38.9229 1.48070 0.740348 0.672224i \(-0.234661\pi\)
0.740348 + 0.672224i \(0.234661\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.48218 −0.245883
\(696\) 0 0
\(697\) −6.87539 −0.260424
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 10.3762 0.391904 0.195952 0.980614i \(-0.437220\pi\)
0.195952 + 0.980614i \(0.437220\pi\)
\(702\) 0 0
\(703\) − 7.41641i − 0.279715i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.38511i 0.127310i
\(708\) 0 0
\(709\) − 19.9668i − 0.749871i −0.927051 0.374935i \(-0.877665\pi\)
0.927051 0.374935i \(-0.122335\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.7295i 0.514173i
\(714\) 0 0
\(715\) −3.03243 −0.113407
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 28.6969 1.07022 0.535108 0.844784i \(-0.320271\pi\)
0.535108 + 0.844784i \(0.320271\pi\)
\(720\) 0 0
\(721\) 9.41641 0.350685
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6.63372 −0.246370
\(726\) 0 0
\(727\) − 24.8328i − 0.920998i −0.887660 0.460499i \(-0.847670\pi\)
0.887660 0.460499i \(-0.152330\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 20.3107i − 0.751217i
\(732\) 0 0
\(733\) − 9.06914i − 0.334976i −0.985874 0.167488i \(-0.946434\pi\)
0.985874 0.167488i \(-0.0535656\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.00310i 0.0737853i
\(738\) 0 0
\(739\) −42.6993 −1.57072 −0.785360 0.619039i \(-0.787522\pi\)
−0.785360 + 0.619039i \(0.787522\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 15.3500 0.563138 0.281569 0.959541i \(-0.409145\pi\)
0.281569 + 0.959541i \(0.409145\pi\)
\(744\) 0 0
\(745\) −12.8754 −0.471718
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.86234 0.323823
\(750\) 0 0
\(751\) − 15.4164i − 0.562553i −0.959627 0.281276i \(-0.909242\pi\)
0.959627 0.281276i \(-0.0907578\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 17.2829i 0.628990i
\(756\) 0 0
\(757\) − 12.0278i − 0.437159i −0.975819 0.218579i \(-0.929858\pi\)
0.975819 0.218579i \(-0.0701423\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 49.0547i 1.77823i 0.457682 + 0.889116i \(0.348680\pi\)
−0.457682 + 0.889116i \(0.651320\pi\)
\(762\) 0 0
\(763\) −7.74597 −0.280423
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.9768 0.757427
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −48.6324 −1.74919 −0.874593 0.484857i \(-0.838872\pi\)
−0.874593 + 0.484857i \(0.838872\pi\)
\(774\) 0 0
\(775\) 5.41641i 0.194563i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 9.08321i − 0.325440i
\(780\) 0 0
\(781\) 1.51622i 0.0542545i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 11.2101 0.399596 0.199798 0.979837i \(-0.435971\pi\)
0.199798 + 0.979837i \(0.435971\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.7279 0.452553
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −8.28409 −0.293438 −0.146719 0.989178i \(-0.546871\pi\)
−0.146719 + 0.989178i \(0.546871\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.34072i 0.188470i
\(804\) 0 0
\(805\) 10.3923i 0.366281i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 47.4342i 1.66770i 0.551994 + 0.833848i \(0.313867\pi\)
−0.551994 + 0.833848i \(0.686133\pi\)
\(810\) 0 0
\(811\) −54.4148 −1.91076 −0.955381 0.295375i \(-0.904555\pi\)
−0.955381 + 0.295375i \(0.904555\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −24.2179 −0.848315
\(816\) 0 0
\(817\) 26.8328 0.938761
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −36.0276 −1.25737 −0.628686 0.777659i \(-0.716407\pi\)
−0.628686 + 0.777659i \(0.716407\pi\)
\(822\) 0 0
\(823\) − 1.70820i − 0.0595442i −0.999557 0.0297721i \(-0.990522\pi\)
0.999557 0.0297721i \(-0.00947816\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.4860i 1.09488i 0.836847 + 0.547438i \(0.184397\pi\)
−0.836847 + 0.547438i \(0.815603\pi\)
\(828\) 0 0
\(829\) − 53.9094i − 1.87235i −0.351533 0.936176i \(-0.614339\pi\)
0.351533 0.936176i \(-0.385661\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.62210i 0.0908502i
\(834\) 0 0
\(835\) 7.93901 0.274741
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 20.2117 0.697784 0.348892 0.937163i \(-0.386558\pi\)
0.348892 + 0.937163i \(0.386558\pi\)
\(840\) 0 0
\(841\) −23.0000 −0.793103
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.7413 −0.438315
\(846\) 0 0
\(847\) 10.1246i 0.347886i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 14.6969i 0.503805i
\(852\) 0 0
\(853\) 40.5584i 1.38869i 0.719641 + 0.694347i \(0.244307\pi\)
−0.719641 + 0.694347i \(0.755693\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 45.0485i 1.53883i 0.638751 + 0.769414i \(0.279452\pi\)
−0.638751 + 0.769414i \(0.720548\pi\)
\(858\) 0 0
\(859\) −45.6580 −1.55783 −0.778916 0.627128i \(-0.784230\pi\)
−0.778916 + 0.627128i \(0.784230\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −32.3206 −1.10021 −0.550103 0.835097i \(-0.685411\pi\)
−0.550103 + 0.835097i \(0.685411\pi\)
\(864\) 0 0
\(865\) −17.1246 −0.582254
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.59823 0.0542163
\(870\) 0 0
\(871\) − 4.58359i − 0.155309i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 11.6692i 0.394491i
\(876\) 0 0
\(877\) − 19.1491i − 0.646619i −0.946293 0.323309i \(-0.895205\pi\)
0.946293 0.323309i \(-0.104795\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 52.2958i − 1.76189i −0.473218 0.880945i \(-0.656908\pi\)
0.473218 0.880945i \(-0.343092\pi\)
\(882\) 0 0
\(883\) −49.9399 −1.68061 −0.840306 0.542113i \(-0.817625\pi\)
−0.840306 + 0.542113i \(0.817625\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −56.1559 −1.88553 −0.942765 0.333458i \(-0.891784\pi\)
−0.942765 + 0.333458i \(0.891784\pi\)
\(888\) 0 0
\(889\) 1.70820 0.0572913
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 6.00000i 0.200558i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 4.89898i − 0.163390i
\(900\) 0 0
\(901\) − 27.2074i − 0.906410i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 0.765117i − 0.0254334i
\(906\) 0 0
\(907\) 27.5198 0.913779 0.456889 0.889524i \(-0.348964\pi\)
0.456889 + 0.889524i \(0.348964\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 51.2942 1.69945 0.849727 0.527223i \(-0.176767\pi\)
0.849727 + 0.527223i \(0.176767\pi\)
\(912\) 0 0
\(913\) −1.75078 −0.0579422
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.8257 0.423542
\(918\) 0 0
\(919\) 0.583592i 0.0192509i 0.999954 + 0.00962546i \(0.00306393\pi\)
−0.999954 + 0.00962546i \(0.996936\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 3.46948i − 0.114199i
\(924\) 0 0
\(925\) 5.79808i 0.190640i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.86008i 0.126645i 0.997993 + 0.0633226i \(0.0201697\pi\)
−0.997993 + 0.0633226i \(0.979830\pi\)
\(930\) 0 0
\(931\) −3.46410 −0.113531
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3.71396 0.121459
\(936\) 0 0
\(937\) 29.4164 0.960992 0.480496 0.876997i \(-0.340457\pi\)
0.480496 + 0.876997i \(0.340457\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −44.8899 −1.46337 −0.731685 0.681643i \(-0.761266\pi\)
−0.731685 + 0.681643i \(0.761266\pi\)
\(942\) 0 0
\(943\) 18.0000i 0.586161i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.96336i − 0.128792i −0.997924 0.0643959i \(-0.979488\pi\)
0.997924 0.0643959i \(-0.0205120\pi\)
\(948\) 0 0
\(949\) − 12.2209i − 0.396707i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 28.4605i − 0.921926i −0.887419 0.460963i \(-0.847504\pi\)
0.887419 0.460963i \(-0.152496\pi\)
\(954\) 0 0
\(955\) 23.2379 0.751961
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.24264 0.137002
\(960\) 0 0
\(961\) 27.0000 0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −15.5804 −0.501551
\(966\) 0 0
\(967\) 42.2918i 1.36001i 0.733206 + 0.680006i \(0.238023\pi\)
−0.733206 + 0.680006i \(0.761977\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 8.64147i − 0.277318i −0.990340 0.138659i \(-0.955721\pi\)
0.990340 0.138659i \(-0.0442792\pi\)
\(972\) 0 0
\(973\) 4.28187i 0.137270i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 7.48373i − 0.239426i −0.992809 0.119713i \(-0.961803\pi\)
0.992809 0.119713i \(-0.0381974\pi\)
\(978\) 0 0
\(979\) 2.45329 0.0784075
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −42.4264 −1.35319 −0.676596 0.736354i \(-0.736546\pi\)
−0.676596 + 0.736354i \(0.736546\pi\)
\(984\) 0 0
\(985\) −35.1246 −1.11916
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −53.1740 −1.69084
\(990\) 0 0
\(991\) − 10.8328i − 0.344116i −0.985087 0.172058i \(-0.944958\pi\)
0.985087 0.172058i \(-0.0550416\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 23.3384i 0.739877i
\(996\) 0 0
\(997\) − 18.1383i − 0.574445i −0.957864 0.287222i \(-0.907268\pi\)
0.957864 0.287222i \(-0.0927319\pi\)
\(998\) 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.j.f.2591.8 yes 16
3.2 odd 2 inner 4032.2.j.f.2591.12 yes 16
4.3 odd 2 inner 4032.2.j.f.2591.6 yes 16
8.3 odd 2 inner 4032.2.j.f.2591.9 yes 16
8.5 even 2 inner 4032.2.j.f.2591.11 yes 16
12.11 even 2 inner 4032.2.j.f.2591.10 yes 16
24.5 odd 2 inner 4032.2.j.f.2591.7 yes 16
24.11 even 2 inner 4032.2.j.f.2591.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4032.2.j.f.2591.5 16 24.11 even 2 inner
4032.2.j.f.2591.6 yes 16 4.3 odd 2 inner
4032.2.j.f.2591.7 yes 16 24.5 odd 2 inner
4032.2.j.f.2591.8 yes 16 1.1 even 1 trivial
4032.2.j.f.2591.9 yes 16 8.3 odd 2 inner
4032.2.j.f.2591.10 yes 16 12.11 even 2 inner
4032.2.j.f.2591.11 yes 16 8.5 even 2 inner
4032.2.j.f.2591.12 yes 16 3.2 odd 2 inner