Properties

Label 5292.2.x.a.4409.5
Level $5292$
Weight $2$
Character 5292.4409
Analytic conductor $42.257$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5292,2,Mod(881,5292)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5292, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5292.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5292.x (of order \(6\), 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: \(42.2568327497\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 4409.5
Root \(1.08696 + 1.34852i\) of defining polynomial
Character \(\chi\) \(=\) 5292.4409
Dual form 5292.2.x.a.881.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.0382122 - 0.0661855i) q^{5} +O(q^{10})\) \(q+(-0.0382122 - 0.0661855i) q^{5} +(4.66300 + 2.69219i) q^{11} +(4.60313 - 2.65762i) q^{13} +3.78183 q^{17} -5.01070i q^{19} +(-2.02463 + 1.16892i) q^{23} +(2.49708 - 4.32507i) q^{25} +(-8.84430 - 5.10626i) q^{29} +(4.97636 - 2.87310i) q^{31} -0.708972 q^{37} +(3.29910 + 5.71422i) q^{41} +(0.716520 - 1.24105i) q^{43} +(-1.46192 + 2.53213i) q^{47} -12.1053i q^{53} -0.411498i q^{55} +(-0.289951 - 0.502210i) q^{59} +(-2.40641 - 1.38934i) q^{61} +(-0.351792 - 0.203107i) q^{65} +(-2.63593 - 4.56556i) q^{67} +3.32103i q^{71} +7.12826i q^{73} +(-0.469123 + 0.812544i) q^{79} +(-6.49790 + 11.2547i) q^{83} +(-0.144512 - 0.250303i) q^{85} +3.03588 q^{89} +(-0.331636 + 0.191470i) q^{95} +(-6.18183 - 3.56908i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{11} - 3 q^{13} + 18 q^{17} + 21 q^{23} - 8 q^{25} - 6 q^{29} + 6 q^{31} - 2 q^{37} + 6 q^{41} - 2 q^{43} - 18 q^{47} - 15 q^{59} + 3 q^{61} - 39 q^{65} - 7 q^{67} - q^{79} + 6 q^{85} + 42 q^{89} + 6 q^{95} - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\)

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
\(5\) −0.0382122 0.0661855i −0.0170890 0.0295991i 0.857354 0.514727i \(-0.172107\pi\)
−0.874443 + 0.485127i \(0.838773\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.66300 + 2.69219i 1.40595 + 0.811725i 0.994994 0.0999316i \(-0.0318624\pi\)
0.410954 + 0.911656i \(0.365196\pi\)
\(12\) 0 0
\(13\) 4.60313 2.65762i 1.27668 0.737091i 0.300442 0.953800i \(-0.402866\pi\)
0.976236 + 0.216709i \(0.0695324\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.78183 0.917229 0.458615 0.888635i \(-0.348346\pi\)
0.458615 + 0.888635i \(0.348346\pi\)
\(18\) 0 0
\(19\) 5.01070i 1.14953i −0.818317 0.574767i \(-0.805093\pi\)
0.818317 0.574767i \(-0.194907\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.02463 + 1.16892i −0.422164 + 0.243737i −0.696003 0.718039i \(-0.745040\pi\)
0.273839 + 0.961776i \(0.411707\pi\)
\(24\) 0 0
\(25\) 2.49708 4.32507i 0.499416 0.865014i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.84430 5.10626i −1.64235 0.948209i −0.979997 0.199013i \(-0.936226\pi\)
−0.662349 0.749196i \(-0.730440\pi\)
\(30\) 0 0
\(31\) 4.97636 2.87310i 0.893780 0.516024i 0.0186031 0.999827i \(-0.494078\pi\)
0.875177 + 0.483803i \(0.160745\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.708972 −0.116554 −0.0582771 0.998300i \(-0.518561\pi\)
−0.0582771 + 0.998300i \(0.518561\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.29910 + 5.71422i 0.515234 + 0.892411i 0.999844 + 0.0176805i \(0.00562816\pi\)
−0.484610 + 0.874730i \(0.661039\pi\)
\(42\) 0 0
\(43\) 0.716520 1.24105i 0.109268 0.189258i −0.806206 0.591635i \(-0.798483\pi\)
0.915474 + 0.402377i \(0.131816\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.46192 + 2.53213i −0.213244 + 0.369349i −0.952728 0.303825i \(-0.901736\pi\)
0.739484 + 0.673174i \(0.235069\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.1053i 1.66279i −0.555683 0.831394i \(-0.687543\pi\)
0.555683 0.831394i \(-0.312457\pi\)
\(54\) 0 0
\(55\) 0.411498i 0.0554863i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.289951 0.502210i −0.0377484 0.0653822i 0.846534 0.532335i \(-0.178685\pi\)
−0.884282 + 0.466953i \(0.845352\pi\)
\(60\) 0 0
\(61\) −2.40641 1.38934i −0.308109 0.177887i 0.337971 0.941156i \(-0.390259\pi\)
−0.646080 + 0.763270i \(0.723593\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.351792 0.203107i −0.0436344 0.0251923i
\(66\) 0 0
\(67\) −2.63593 4.56556i −0.322030 0.557771i 0.658877 0.752251i \(-0.271032\pi\)
−0.980907 + 0.194479i \(0.937698\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.32103i 0.394134i 0.980390 + 0.197067i \(0.0631416\pi\)
−0.980390 + 0.197067i \(0.936858\pi\)
\(72\) 0 0
\(73\) 7.12826i 0.834300i 0.908838 + 0.417150i \(0.136971\pi\)
−0.908838 + 0.417150i \(0.863029\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.469123 + 0.812544i −0.0527804 + 0.0914184i −0.891208 0.453594i \(-0.850142\pi\)
0.838428 + 0.545012i \(0.183475\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.49790 + 11.2547i −0.713238 + 1.23536i 0.250398 + 0.968143i \(0.419439\pi\)
−0.963635 + 0.267221i \(0.913895\pi\)
\(84\) 0 0
\(85\) −0.144512 0.250303i −0.0156746 0.0271491i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.03588 0.321802 0.160901 0.986971i \(-0.448560\pi\)
0.160901 + 0.986971i \(0.448560\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.331636 + 0.191470i −0.0340251 + 0.0196444i
\(96\) 0 0
\(97\) −6.18183 3.56908i −0.627670 0.362385i 0.152179 0.988353i \(-0.451371\pi\)
−0.779849 + 0.625967i \(0.784704\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.08628 + 7.07765i −0.406600 + 0.704252i −0.994506 0.104677i \(-0.966619\pi\)
0.587906 + 0.808929i \(0.299952\pi\)
\(102\) 0 0
\(103\) 6.46599 3.73314i 0.637113 0.367837i −0.146389 0.989227i \(-0.546765\pi\)
0.783502 + 0.621390i \(0.213432\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.61870i 0.446507i 0.974760 + 0.223253i \(0.0716677\pi\)
−0.974760 + 0.223253i \(0.928332\pi\)
\(108\) 0 0
\(109\) −10.4558 −1.00149 −0.500744 0.865595i \(-0.666940\pi\)
−0.500744 + 0.865595i \(0.666940\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.6379 9.60591i 1.56516 0.903648i 0.568445 0.822721i \(-0.307545\pi\)
0.996720 0.0809270i \(-0.0257881\pi\)
\(114\) 0 0
\(115\) 0.154731 + 0.0893340i 0.0144287 + 0.00833044i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 8.99573 + 15.5811i 0.817793 + 1.41646i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −0.763798 −0.0683162
\(126\) 0 0
\(127\) 1.26488 0.112240 0.0561198 0.998424i \(-0.482127\pi\)
0.0561198 + 0.998424i \(0.482127\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.24394 + 12.5469i 0.632906 + 1.09623i 0.986955 + 0.160998i \(0.0514714\pi\)
−0.354049 + 0.935227i \(0.615195\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.3414 7.70264i −1.13983 0.658081i −0.193442 0.981112i \(-0.561965\pi\)
−0.946389 + 0.323030i \(0.895298\pi\)
\(138\) 0 0
\(139\) 0.374701 0.216333i 0.0317817 0.0183492i −0.484025 0.875054i \(-0.660826\pi\)
0.515807 + 0.856705i \(0.327492\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 28.6192 2.39326
\(144\) 0 0
\(145\) 0.780486i 0.0648159i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.04535 + 2.33558i −0.331408 + 0.191338i −0.656466 0.754356i \(-0.727949\pi\)
0.325058 + 0.945694i \(0.394616\pi\)
\(150\) 0 0
\(151\) 4.12276 7.14083i 0.335506 0.581113i −0.648076 0.761575i \(-0.724426\pi\)
0.983582 + 0.180463i \(0.0577595\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.380316 0.219575i −0.0305477 0.0176367i
\(156\) 0 0
\(157\) 15.2334 8.79500i 1.21576 0.701917i 0.251749 0.967793i \(-0.418994\pi\)
0.964007 + 0.265875i \(0.0856609\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.5419 0.825709 0.412854 0.910797i \(-0.364532\pi\)
0.412854 + 0.910797i \(0.364532\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.59146 7.95265i −0.355298 0.615395i 0.631871 0.775074i \(-0.282287\pi\)
−0.987169 + 0.159679i \(0.948954\pi\)
\(168\) 0 0
\(169\) 7.62587 13.2084i 0.586605 1.01603i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.22358 + 2.11931i −0.0930274 + 0.161128i −0.908784 0.417268i \(-0.862988\pi\)
0.815756 + 0.578396i \(0.196321\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.83712i 0.436287i −0.975917 0.218143i \(-0.930000\pi\)
0.975917 0.218143i \(-0.0700001\pi\)
\(180\) 0 0
\(181\) 16.0704i 1.19451i −0.802053 0.597253i \(-0.796259\pi\)
0.802053 0.597253i \(-0.203741\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.0270914 + 0.0469237i 0.00199180 + 0.00344990i
\(186\) 0 0
\(187\) 17.6347 + 10.1814i 1.28958 + 0.744537i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.90415 3.98611i −0.499567 0.288425i 0.228968 0.973434i \(-0.426465\pi\)
−0.728535 + 0.685009i \(0.759798\pi\)
\(192\) 0 0
\(193\) −0.359027 0.621853i −0.0258433 0.0447620i 0.852814 0.522214i \(-0.174894\pi\)
−0.878658 + 0.477452i \(0.841560\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.5035i 0.962083i −0.876698 0.481042i \(-0.840259\pi\)
0.876698 0.481042i \(-0.159741\pi\)
\(198\) 0 0
\(199\) 24.5452i 1.73997i −0.493082 0.869983i \(-0.664130\pi\)
0.493082 0.869983i \(-0.335870\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0.252132 0.436706i 0.0176097 0.0305009i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 13.4897 23.3649i 0.933105 1.61618i
\(210\) 0 0
\(211\) −11.7838 20.4101i −0.811227 1.40509i −0.912005 0.410178i \(-0.865467\pi\)
0.100778 0.994909i \(-0.467867\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.109519 −0.00746916
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 17.4083 10.0507i 1.17101 0.676081i
\(222\) 0 0
\(223\) 6.47489 + 3.73828i 0.433590 + 0.250334i 0.700875 0.713284i \(-0.252793\pi\)
−0.267285 + 0.963618i \(0.586126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.318701 + 0.552006i −0.0211529 + 0.0366379i −0.876408 0.481569i \(-0.840067\pi\)
0.855255 + 0.518207i \(0.173400\pi\)
\(228\) 0 0
\(229\) 1.58351 0.914239i 0.104641 0.0604146i −0.446766 0.894651i \(-0.647424\pi\)
0.551407 + 0.834236i \(0.314091\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.1186i 1.31801i 0.752136 + 0.659007i \(0.229023\pi\)
−0.752136 + 0.659007i \(0.770977\pi\)
\(234\) 0 0
\(235\) 0.223454 0.0145765
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.41455 1.39404i 0.156184 0.0901730i −0.419871 0.907584i \(-0.637925\pi\)
0.576055 + 0.817411i \(0.304591\pi\)
\(240\) 0 0
\(241\) 20.0304 + 11.5645i 1.29027 + 0.744938i 0.978702 0.205286i \(-0.0658126\pi\)
0.311568 + 0.950224i \(0.399146\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −13.3165 23.0649i −0.847310 1.46758i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.6541 1.17743 0.588717 0.808339i \(-0.299633\pi\)
0.588717 + 0.808339i \(0.299633\pi\)
\(252\) 0 0
\(253\) −12.5878 −0.791388
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.43687 9.41694i −0.339143 0.587413i 0.645129 0.764074i \(-0.276804\pi\)
−0.984272 + 0.176661i \(0.943470\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.4519 + 9.49852i 1.01447 + 0.585704i 0.912497 0.409083i \(-0.134151\pi\)
0.101972 + 0.994787i \(0.467485\pi\)
\(264\) 0 0
\(265\) −0.801194 + 0.462570i −0.0492170 + 0.0284154i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.59576 0.524093 0.262046 0.965055i \(-0.415603\pi\)
0.262046 + 0.965055i \(0.415603\pi\)
\(270\) 0 0
\(271\) 1.83258i 0.111322i 0.998450 + 0.0556608i \(0.0177265\pi\)
−0.998450 + 0.0556608i \(0.982273\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 23.2878 13.4452i 1.40431 0.810776i
\(276\) 0 0
\(277\) −7.90931 + 13.6993i −0.475224 + 0.823113i −0.999597 0.0283760i \(-0.990966\pi\)
0.524373 + 0.851489i \(0.324300\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.95916 + 5.74992i 0.594114 + 0.343012i 0.766722 0.641979i \(-0.221886\pi\)
−0.172609 + 0.984990i \(0.555220\pi\)
\(282\) 0 0
\(283\) −8.59806 + 4.96409i −0.511101 + 0.295085i −0.733286 0.679920i \(-0.762014\pi\)
0.222185 + 0.975005i \(0.428681\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.69774 −0.158691
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.63598 + 14.9580i 0.504520 + 0.873854i 0.999986 + 0.00522664i \(0.00166370\pi\)
−0.495467 + 0.868627i \(0.665003\pi\)
\(294\) 0 0
\(295\) −0.0221594 + 0.0383812i −0.00129017 + 0.00223464i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.21308 + 10.7614i −0.359312 + 0.622346i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.212359i 0.0121596i
\(306\) 0 0
\(307\) 21.6425i 1.23520i 0.786490 + 0.617602i \(0.211896\pi\)
−0.786490 + 0.617602i \(0.788104\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.1016 + 17.4964i 0.572808 + 0.992133i 0.996276 + 0.0862215i \(0.0274793\pi\)
−0.423468 + 0.905911i \(0.639187\pi\)
\(312\) 0 0
\(313\) −18.9146 10.9203i −1.06911 0.617254i −0.141175 0.989985i \(-0.545088\pi\)
−0.927939 + 0.372731i \(0.878421\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −21.5288 12.4297i −1.20918 0.698120i −0.246599 0.969117i \(-0.579313\pi\)
−0.962580 + 0.270997i \(0.912647\pi\)
\(318\) 0 0
\(319\) −27.4940 47.6210i −1.53937 2.66626i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 18.9496i 1.05439i
\(324\) 0 0
\(325\) 26.5451i 1.47246i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8.07219 + 13.9814i −0.443688 + 0.768490i −0.997960 0.0638459i \(-0.979663\pi\)
0.554272 + 0.832336i \(0.312997\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.201449 + 0.348920i −0.0110063 + 0.0190635i
\(336\) 0 0
\(337\) −7.81522 13.5364i −0.425722 0.737372i 0.570765 0.821113i \(-0.306647\pi\)
−0.996488 + 0.0837408i \(0.973313\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 30.9397 1.67548
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.0445 16.1915i 1.50551 0.869206i 0.505529 0.862810i \(-0.331297\pi\)
0.999980 0.00639573i \(-0.00203584\pi\)
\(348\) 0 0
\(349\) 26.0421 + 15.0354i 1.39400 + 0.804827i 0.993755 0.111581i \(-0.0355915\pi\)
0.400246 + 0.916408i \(0.368925\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8.50607 + 14.7329i −0.452733 + 0.784156i −0.998555 0.0537453i \(-0.982884\pi\)
0.545822 + 0.837901i \(0.316217\pi\)
\(354\) 0 0
\(355\) 0.219804 0.126904i 0.0116660 0.00673537i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 29.1783i 1.53997i 0.638060 + 0.769987i \(0.279737\pi\)
−0.638060 + 0.769987i \(0.720263\pi\)
\(360\) 0 0
\(361\) −6.10712 −0.321427
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.471788 0.272387i 0.0246945 0.0142574i
\(366\) 0 0
\(367\) 15.6981 + 9.06329i 0.819433 + 0.473100i 0.850221 0.526426i \(-0.176468\pi\)
−0.0307880 + 0.999526i \(0.509802\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.1823 + 17.6362i 0.527219 + 0.913170i 0.999497 + 0.0317200i \(0.0100985\pi\)
−0.472278 + 0.881450i \(0.656568\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −54.2820 −2.79566
\(378\) 0 0
\(379\) −21.9961 −1.12986 −0.564931 0.825138i \(-0.691097\pi\)
−0.564931 + 0.825138i \(0.691097\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −16.3127 28.2544i −0.833538 1.44373i −0.895215 0.445634i \(-0.852978\pi\)
0.0616774 0.998096i \(-0.480355\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −13.6400 7.87504i −0.691574 0.399280i 0.112628 0.993637i \(-0.464073\pi\)
−0.804201 + 0.594357i \(0.797407\pi\)
\(390\) 0 0
\(391\) −7.65680 + 4.42066i −0.387221 + 0.223562i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.0717049 0.00360786
\(396\) 0 0
\(397\) 3.41635i 0.171462i −0.996318 0.0857308i \(-0.972678\pi\)
0.996318 0.0857308i \(-0.0273225\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.851348 + 0.491526i −0.0425143 + 0.0245456i −0.521106 0.853492i \(-0.674481\pi\)
0.478592 + 0.878037i \(0.341147\pi\)
\(402\) 0 0
\(403\) 15.2712 26.4505i 0.760713 1.31759i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.30594 1.90868i −0.163869 0.0946099i
\(408\) 0 0
\(409\) 25.0195 14.4450i 1.23714 0.714260i 0.268627 0.963244i \(-0.413430\pi\)
0.968508 + 0.248984i \(0.0800966\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0.993198 0.0487542
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.28926 + 10.8933i 0.307251 + 0.532174i 0.977760 0.209727i \(-0.0672577\pi\)
−0.670509 + 0.741901i \(0.733924\pi\)
\(420\) 0 0
\(421\) −13.0232 + 22.5568i −0.634710 + 1.09935i 0.351866 + 0.936050i \(0.385547\pi\)
−0.986576 + 0.163300i \(0.947786\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 9.44354 16.3567i 0.458079 0.793416i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.25676i 0.349546i −0.984609 0.174773i \(-0.944081\pi\)
0.984609 0.174773i \(-0.0559192\pi\)
\(432\) 0 0
\(433\) 8.29113i 0.398446i 0.979954 + 0.199223i \(0.0638419\pi\)
−0.979954 + 0.199223i \(0.936158\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.85710 + 10.1448i 0.280183 + 0.485292i
\(438\) 0 0
\(439\) −2.83357 1.63596i −0.135239 0.0780802i 0.430854 0.902422i \(-0.358212\pi\)
−0.566093 + 0.824341i \(0.691546\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.46737 + 1.42454i 0.117228 + 0.0676817i 0.557468 0.830199i \(-0.311773\pi\)
−0.440239 + 0.897880i \(0.645106\pi\)
\(444\) 0 0
\(445\) −0.116008 0.200931i −0.00549929 0.00952505i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 19.9802i 0.942925i 0.881886 + 0.471463i \(0.156274\pi\)
−0.881886 + 0.471463i \(0.843726\pi\)
\(450\) 0 0
\(451\) 35.5272i 1.67291i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9.15008 + 15.8484i −0.428023 + 0.741357i −0.996697 0.0812053i \(-0.974123\pi\)
0.568675 + 0.822563i \(0.307456\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.52954 7.84539i 0.210962 0.365396i −0.741054 0.671445i \(-0.765674\pi\)
0.952016 + 0.306049i \(0.0990071\pi\)
\(462\) 0 0
\(463\) 10.8227 + 18.7455i 0.502974 + 0.871176i 0.999994 + 0.00343694i \(0.00109401\pi\)
−0.497021 + 0.867739i \(0.665573\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.5523 1.27497 0.637484 0.770464i \(-0.279975\pi\)
0.637484 + 0.770464i \(0.279975\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.68227 3.85801i 0.307251 0.177392i
\(474\) 0 0
\(475\) −21.6716 12.5121i −0.994362 0.574095i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.47325 + 4.28380i −0.113006 + 0.195732i −0.916981 0.398931i \(-0.869381\pi\)
0.803975 + 0.594663i \(0.202715\pi\)
\(480\) 0 0
\(481\) −3.26349 + 1.88418i −0.148802 + 0.0859110i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.545530i 0.0247713i
\(486\) 0 0
\(487\) 9.57146 0.433724 0.216862 0.976202i \(-0.430418\pi\)
0.216862 + 0.976202i \(0.430418\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 33.0010 19.0531i 1.48931 0.859855i 0.489387 0.872067i \(-0.337221\pi\)
0.999925 + 0.0122119i \(0.00388725\pi\)
\(492\) 0 0
\(493\) −33.4477 19.3110i −1.50641 0.869725i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.4192 + 21.5107i 0.555960 + 0.962951i 0.997828 + 0.0658709i \(0.0209825\pi\)
−0.441868 + 0.897080i \(0.645684\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 27.2820 1.21645 0.608223 0.793766i \(-0.291883\pi\)
0.608223 + 0.793766i \(0.291883\pi\)
\(504\) 0 0
\(505\) 0.624584 0.0277936
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −20.8860 36.1757i −0.925758 1.60346i −0.790338 0.612671i \(-0.790095\pi\)
−0.135420 0.990788i \(-0.543238\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −0.494160 0.285303i −0.0217753 0.0125720i
\(516\) 0 0
\(517\) −13.6339 + 7.87154i −0.599619 + 0.346190i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.05257 0.177546 0.0887732 0.996052i \(-0.471705\pi\)
0.0887732 + 0.996052i \(0.471705\pi\)
\(522\) 0 0
\(523\) 30.3027i 1.32505i 0.749042 + 0.662523i \(0.230514\pi\)
−0.749042 + 0.662523i \(0.769486\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.8198 10.8656i 0.819801 0.473312i
\(528\) 0 0
\(529\) −8.76726 + 15.1853i −0.381185 + 0.660232i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 30.3724 + 17.5355i 1.31558 + 0.759548i
\(534\) 0 0
\(535\) 0.305691 0.176491i 0.0132162 0.00763037i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −17.6536 −0.758988 −0.379494 0.925194i \(-0.623902\pi\)
−0.379494 + 0.925194i \(0.623902\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.399541 + 0.692026i 0.0171145 + 0.0296431i
\(546\) 0 0
\(547\) −2.18319 + 3.78140i −0.0933466 + 0.161681i −0.908917 0.416976i \(-0.863090\pi\)
0.815571 + 0.578657i \(0.196423\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −25.5859 + 44.3161i −1.09000 + 1.88793i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17.0350i 0.721796i 0.932605 + 0.360898i \(0.117530\pi\)
−0.932605 + 0.360898i \(0.882470\pi\)
\(558\) 0 0
\(559\) 7.61695i 0.322163i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.45992 11.1889i −0.272253 0.471556i 0.697185 0.716891i \(-0.254436\pi\)
−0.969438 + 0.245335i \(0.921102\pi\)
\(564\) 0 0
\(565\) −1.27155 0.734127i −0.0534943 0.0308850i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.8280 10.8704i −0.789313 0.455710i 0.0504079 0.998729i \(-0.483948\pi\)
−0.839720 + 0.543019i \(0.817281\pi\)
\(570\) 0 0
\(571\) 16.8254 + 29.1425i 0.704122 + 1.21958i 0.967007 + 0.254748i \(0.0819925\pi\)
−0.262885 + 0.964827i \(0.584674\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.6755i 0.486904i
\(576\) 0 0
\(577\) 14.5028i 0.603760i −0.953346 0.301880i \(-0.902386\pi\)
0.953346 0.301880i \(-0.0976142\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 32.5897 56.4469i 1.34973 2.33779i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.8417 27.4386i 0.653857 1.13251i −0.328322 0.944566i \(-0.606483\pi\)
0.982179 0.187948i \(-0.0601837\pi\)
\(588\) 0 0
\(589\) −14.3963 24.9350i −0.593187 1.02743i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.08201 0.290823 0.145412 0.989371i \(-0.453549\pi\)
0.145412 + 0.989371i \(0.453549\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.20178 + 3.00325i −0.212539 + 0.122709i −0.602491 0.798126i \(-0.705825\pi\)
0.389952 + 0.920835i \(0.372492\pi\)
\(600\) 0 0
\(601\) 0.530083 + 0.306043i 0.0216225 + 0.0124838i 0.510772 0.859716i \(-0.329360\pi\)
−0.489150 + 0.872200i \(0.662693\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.687494 1.19077i 0.0279506 0.0484119i
\(606\) 0 0
\(607\) −1.77500 + 1.02480i −0.0720450 + 0.0415952i −0.535590 0.844478i \(-0.679911\pi\)
0.463545 + 0.886073i \(0.346577\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 15.5409i 0.628719i
\(612\) 0 0
\(613\) 9.86332 0.398376 0.199188 0.979961i \(-0.436170\pi\)
0.199188 + 0.979961i \(0.436170\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −23.2143 + 13.4028i −0.934571 + 0.539575i −0.888254 0.459352i \(-0.848082\pi\)
−0.0463170 + 0.998927i \(0.514748\pi\)
\(618\) 0 0
\(619\) −0.0603011 0.0348148i −0.00242370 0.00139933i 0.498788 0.866724i \(-0.333779\pi\)
−0.501211 + 0.865325i \(0.667112\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.4562 21.5748i −0.498248 0.862992i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.68121 −0.106907
\(630\) 0 0
\(631\) 11.8214 0.470603 0.235301 0.971922i \(-0.424392\pi\)
0.235301 + 0.971922i \(0.424392\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.0483338 0.0837165i −0.00191807 0.00332219i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −17.7673 10.2580i −0.701766 0.405165i 0.106239 0.994341i \(-0.466119\pi\)
−0.808005 + 0.589176i \(0.799453\pi\)
\(642\) 0 0
\(643\) 15.6081 9.01132i 0.615522 0.355372i −0.159602 0.987182i \(-0.551021\pi\)
0.775123 + 0.631810i \(0.217688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.2365 0.716952 0.358476 0.933539i \(-0.383296\pi\)
0.358476 + 0.933539i \(0.383296\pi\)
\(648\) 0 0
\(649\) 3.12241i 0.122565i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.79559 + 4.50079i −0.305065 + 0.176129i −0.644716 0.764422i \(-0.723024\pi\)
0.339651 + 0.940552i \(0.389691\pi\)
\(654\) 0 0
\(655\) 0.553614 0.958888i 0.0216315 0.0374669i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −30.4806 17.5980i −1.18735 0.685519i −0.229650 0.973273i \(-0.573758\pi\)
−0.957704 + 0.287754i \(0.907091\pi\)
\(660\) 0 0
\(661\) −10.8797 + 6.28141i −0.423172 + 0.244318i −0.696433 0.717621i \(-0.745231\pi\)
0.273262 + 0.961940i \(0.411898\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 23.8752 0.924452
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −7.48072 12.9570i −0.288790 0.500199i
\(672\) 0 0
\(673\) 23.8913 41.3810i 0.920942 1.59512i 0.122982 0.992409i \(-0.460754\pi\)
0.797960 0.602710i \(-0.205913\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.5235 + 32.0837i −0.711918 + 1.23308i 0.252219 + 0.967670i \(0.418840\pi\)
−0.964136 + 0.265407i \(0.914494\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25.0390i 0.958092i 0.877790 + 0.479046i \(0.159017\pi\)
−0.877790 + 0.479046i \(0.840983\pi\)
\(684\) 0 0
\(685\) 1.17734i 0.0449839i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −32.1712 55.7222i −1.22563 2.12285i
\(690\) 0 0
\(691\) 40.2655 + 23.2473i 1.53177 + 0.884370i 0.999280 + 0.0379352i \(0.0120781\pi\)
0.532493 + 0.846434i \(0.321255\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −0.0286363 0.0165332i −0.00108624 0.000627139i
\(696\) 0 0
\(697\) 12.4767 + 21.6102i 0.472587 + 0.818545i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 36.0041i 1.35986i −0.733279 0.679928i \(-0.762011\pi\)
0.733279 0.679928i \(-0.237989\pi\)
\(702\) 0 0
\(703\) 3.55244i 0.133983i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −15.9158 + 27.5670i −0.597731 + 1.03530i 0.395424 + 0.918499i \(0.370598\pi\)
−0.993155 + 0.116802i \(0.962736\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.71685 + 11.6339i −0.251548 + 0.435694i
\(714\) 0 0
\(715\) −1.09360 1.89418i −0.0408985 0.0708382i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 40.0541 1.49377 0.746883 0.664955i \(-0.231550\pi\)
0.746883 + 0.664955i \(0.231550\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −44.1699 + 25.5015i −1.64043 + 0.947101i
\(726\) 0 0
\(727\) 3.39242 + 1.95862i 0.125818 + 0.0726411i 0.561588 0.827417i \(-0.310191\pi\)
−0.435770 + 0.900058i \(0.643524\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.70976 4.69344i 0.100224 0.173593i
\(732\) 0 0
\(733\) −20.4239 + 11.7918i −0.754376 + 0.435539i −0.827273 0.561800i \(-0.810109\pi\)
0.0728971 + 0.997339i \(0.476776\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 28.3856i 1.04560i
\(738\) 0 0
\(739\) −33.7282 −1.24071 −0.620355 0.784321i \(-0.713011\pi\)
−0.620355 + 0.784321i \(0.713011\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −29.4003 + 16.9743i −1.07859 + 0.622725i −0.930516 0.366251i \(-0.880641\pi\)
−0.148076 + 0.988976i \(0.547308\pi\)
\(744\) 0 0
\(745\) 0.309164 + 0.178496i 0.0113269 + 0.00653958i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.69831 2.94157i −0.0619724 0.107339i 0.833375 0.552709i \(-0.186406\pi\)
−0.895347 + 0.445369i \(0.853072\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.630160 −0.0229339
\(756\) 0 0
\(757\) 29.1344 1.05891 0.529454 0.848339i \(-0.322397\pi\)
0.529454 + 0.848339i \(0.322397\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −8.36288 14.4849i −0.303154 0.525079i 0.673694 0.739010i \(-0.264706\pi\)
−0.976849 + 0.213931i \(0.931373\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.66937 1.54116i −0.0963852 0.0556480i
\(768\) 0 0
\(769\) 24.0816 13.9035i 0.868404 0.501373i 0.00158643 0.999999i \(-0.499495\pi\)
0.866818 + 0.498625i \(0.166162\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12.8448 0.461994 0.230997 0.972954i \(-0.425801\pi\)
0.230997 + 0.972954i \(0.425801\pi\)
\(774\) 0 0
\(775\) 28.6975i 1.03084i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 28.6322 16.5308i 1.02586 0.592278i
\(780\) 0 0
\(781\) −8.94083 + 15.4860i −0.319928 + 0.554132i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.16420 0.672153i −0.0415522 0.0239902i
\(786\) 0 0
\(787\) 6.55243 3.78305i 0.233569 0.134851i −0.378648 0.925541i \(-0.623611\pi\)
0.612217 + 0.790689i \(0.290278\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −14.7693 −0.524474
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.03362 6.98643i −0.142878 0.247472i 0.785701 0.618606i \(-0.212302\pi\)
−0.928579 + 0.371134i \(0.878969\pi\)
\(798\) 0 0
\(799\) −5.52875 + 9.57608i −0.195593 + 0.338777i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −19.1906 + 33.2391i −0.677222 + 1.17298i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.0980908i 0.00344869i 0.999999 + 0.00172435i \(0.000548876\pi\)
−0.999999 + 0.00172435i \(0.999451\pi\)
\(810\) 0 0
\(811\) 30.3085i 1.06428i −0.846658 0.532138i \(-0.821389\pi\)
0.846658 0.532138i \(-0.178611\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.402831 0.697724i −0.0141106 0.0244402i
\(816\) 0 0
\(817\) −6.21853 3.59027i −0.217559 0.125608i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −19.5499 11.2871i −0.682295 0.393923i 0.118424 0.992963i \(-0.462216\pi\)
−0.800719 + 0.599040i \(0.795549\pi\)
\(822\) 0 0
\(823\) 12.2655 + 21.2445i 0.427549 + 0.740536i 0.996655 0.0817282i \(-0.0260439\pi\)
−0.569106 + 0.822264i \(0.692711\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 40.3057i 1.40157i 0.713375 + 0.700783i \(0.247166\pi\)
−0.713375 + 0.700783i \(0.752834\pi\)
\(828\) 0 0
\(829\) 54.0493i 1.87721i 0.344993 + 0.938605i \(0.387881\pi\)
−0.344993 + 0.938605i \(0.612119\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −0.350900 + 0.607777i −0.0121434 + 0.0210330i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −11.8650 + 20.5507i −0.409624 + 0.709489i −0.994847 0.101383i \(-0.967673\pi\)
0.585224 + 0.810872i \(0.301007\pi\)
\(840\) 0 0
\(841\) 37.6478 + 65.2079i 1.29820 + 2.24855i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.16561 −0.0400980
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.43540 0.828731i 0.0492050 0.0284085i
\(852\) 0 0
\(853\) −48.0748 27.7560i −1.64605 0.950347i −0.978621 0.205674i \(-0.934061\pi\)
−0.667429 0.744673i \(-0.732605\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −15.3048 + 26.5088i −0.522803 + 0.905522i 0.476845 + 0.878988i \(0.341780\pi\)
−0.999648 + 0.0265343i \(0.991553\pi\)
\(858\) 0 0
\(859\) 36.4944 21.0700i 1.24517 0.718900i 0.275030 0.961436i \(-0.411312\pi\)
0.970143 + 0.242535i \(0.0779790\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 26.3020i 0.895330i 0.894201 + 0.447665i \(0.147744\pi\)
−0.894201 + 0.447665i \(0.852256\pi\)
\(864\) 0 0
\(865\) 0.187024 0.00635899
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4.37504 + 2.52593i −0.148413 + 0.0856863i
\(870\) 0 0
\(871\) −24.2670 14.0106i −0.822256 0.474730i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −17.8533 30.9228i −0.602863 1.04419i −0.992385 0.123172i \(-0.960693\pi\)
0.389522 0.921017i \(-0.372640\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −12.4482 −0.419392 −0.209696 0.977767i \(-0.567247\pi\)
−0.209696 + 0.977767i \(0.567247\pi\)
\(882\) 0 0
\(883\) 2.35637 0.0792982 0.0396491 0.999214i \(-0.487376\pi\)
0.0396491 + 0.999214i \(0.487376\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.7299 + 28.9770i 0.561734 + 0.972952i 0.997345 + 0.0728170i \(0.0231989\pi\)
−0.435611 + 0.900135i \(0.643468\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 12.6877 + 7.32526i 0.424579 + 0.245131i
\(894\) 0 0
\(895\) −0.386333 + 0.223049i −0.0129137 + 0.00745572i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −58.6832 −1.95719
\(900\) 0 0
\(901\) 45.7801i 1.52516i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.06363 + 0.614087i −0.0353563 + 0.0204130i
\(906\) 0 0
\(907\) 0.467962 0.810535i 0.0155384 0.0269134i −0.858152 0.513396i \(-0.828387\pi\)
0.873690 + 0.486483i \(0.161720\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −28.8739 16.6703i −0.956634 0.552313i −0.0614988 0.998107i \(-0.519588\pi\)
−0.895136 + 0.445794i \(0.852921\pi\)
\(912\) 0 0
\(913\) −60.5995 + 34.9871i −2.00555 + 1.15791i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 3.46967 0.114454 0.0572270 0.998361i \(-0.481774\pi\)
0.0572270 + 0.998361i \(0.481774\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.82603 + 15.2871i 0.290512 + 0.503182i
\(924\) 0 0
\(925\) −1.77036 + 3.06635i −0.0582090 + 0.100821i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.57680 13.1234i 0.248587 0.430565i −0.714547 0.699587i \(-0.753367\pi\)
0.963134 + 0.269022i \(0.0867005\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.55622i 0.0508937i
\(936\) 0 0
\(937\) 33.6651i 1.09979i 0.835233 + 0.549896i \(0.185333\pi\)
−0.835233 + 0.549896i \(0.814667\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −18.8980 32.7323i −0.616058 1.06704i −0.990198 0.139671i \(-0.955395\pi\)
0.374140 0.927372i \(-0.377938\pi\)
\(942\) 0 0
\(943\) −13.3589 7.71277i −0.435026 0.251162i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.47426 + 5.46997i 0.307872 + 0.177750i 0.645974 0.763360i \(-0.276452\pi\)
−0.338102 + 0.941110i \(0.609785\pi\)
\(948\) 0 0
\(949\) 18.9442 + 32.8123i 0.614955 + 1.06513i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 11.0914i 0.359284i 0.983732 + 0.179642i \(0.0574939\pi\)
−0.983732 + 0.179642i \(0.942506\pi\)
\(954\) 0 0
\(955\) 0.609273i 0.0197156i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00942 1.74838i 0.0325621 0.0563992i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.0274384 + 0.0475248i −0.000883275 + 0.00152988i
\(966\) 0 0
\(967\) 20.1446 + 34.8915i 0.647807 + 1.12203i 0.983646 + 0.180115i \(0.0576470\pi\)
−0.335839 + 0.941920i \(0.609020\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −47.6916 −1.53050 −0.765248 0.643736i \(-0.777384\pi\)
−0.765248 + 0.643736i \(0.777384\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −14.4540 + 8.34504i −0.462426 + 0.266982i −0.713064 0.701099i \(-0.752693\pi\)
0.250638 + 0.968081i \(0.419360\pi\)
\(978\) 0 0
\(979\) 14.1563 + 8.17314i 0.452437 + 0.261215i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −16.9255 + 29.3157i −0.539838 + 0.935027i 0.459074 + 0.888398i \(0.348181\pi\)
−0.998912 + 0.0466291i \(0.985152\pi\)
\(984\) 0 0
\(985\) −0.893735 + 0.515998i −0.0284768 + 0.0164411i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.35022i 0.106531i
\(990\) 0 0
\(991\) 8.19550 0.260339 0.130169 0.991492i \(-0.458448\pi\)
0.130169 + 0.991492i \(0.458448\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.62454 + 0.937928i −0.0515014 + 0.0297343i
\(996\) 0 0
\(997\) 18.7391 + 10.8190i 0.593472 + 0.342641i 0.766469 0.642281i \(-0.222012\pi\)
−0.172997 + 0.984922i \(0.555345\pi\)
\(998\) 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 5292.2.x.a.4409.5 16
3.2 odd 2 1764.2.x.a.1469.5 16
7.2 even 3 756.2.w.a.521.5 16
7.3 odd 6 756.2.bm.a.89.5 16
7.4 even 3 5292.2.bm.a.4625.4 16
7.5 odd 6 5292.2.w.b.521.4 16
7.6 odd 2 5292.2.x.b.4409.4 16
9.4 even 3 1764.2.x.b.293.4 16
9.5 odd 6 5292.2.x.b.881.4 16
21.2 odd 6 252.2.w.a.101.1 yes 16
21.5 even 6 1764.2.w.b.1109.8 16
21.11 odd 6 1764.2.bm.a.1685.6 16
21.17 even 6 252.2.bm.a.173.3 yes 16
21.20 even 2 1764.2.x.b.1469.4 16
28.3 even 6 3024.2.df.d.1601.5 16
28.23 odd 6 3024.2.ca.d.2033.5 16
63.2 odd 6 2268.2.t.b.1781.4 16
63.4 even 3 1764.2.w.b.509.8 16
63.5 even 6 5292.2.bm.a.2285.4 16
63.13 odd 6 1764.2.x.a.293.5 16
63.16 even 3 2268.2.t.a.1781.5 16
63.23 odd 6 756.2.bm.a.17.5 16
63.31 odd 6 252.2.w.a.5.1 16
63.32 odd 6 5292.2.w.b.1097.4 16
63.38 even 6 2268.2.t.a.2105.5 16
63.40 odd 6 1764.2.bm.a.1697.6 16
63.41 even 6 inner 5292.2.x.a.881.5 16
63.52 odd 6 2268.2.t.b.2105.4 16
63.58 even 3 252.2.bm.a.185.3 yes 16
63.59 even 6 756.2.w.a.341.5 16
84.23 even 6 1008.2.ca.d.353.8 16
84.59 odd 6 1008.2.df.d.929.6 16
252.23 even 6 3024.2.df.d.17.5 16
252.31 even 6 1008.2.ca.d.257.8 16
252.59 odd 6 3024.2.ca.d.2609.5 16
252.247 odd 6 1008.2.df.d.689.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.w.a.5.1 16 63.31 odd 6
252.2.w.a.101.1 yes 16 21.2 odd 6
252.2.bm.a.173.3 yes 16 21.17 even 6
252.2.bm.a.185.3 yes 16 63.58 even 3
756.2.w.a.341.5 16 63.59 even 6
756.2.w.a.521.5 16 7.2 even 3
756.2.bm.a.17.5 16 63.23 odd 6
756.2.bm.a.89.5 16 7.3 odd 6
1008.2.ca.d.257.8 16 252.31 even 6
1008.2.ca.d.353.8 16 84.23 even 6
1008.2.df.d.689.6 16 252.247 odd 6
1008.2.df.d.929.6 16 84.59 odd 6
1764.2.w.b.509.8 16 63.4 even 3
1764.2.w.b.1109.8 16 21.5 even 6
1764.2.x.a.293.5 16 63.13 odd 6
1764.2.x.a.1469.5 16 3.2 odd 2
1764.2.x.b.293.4 16 9.4 even 3
1764.2.x.b.1469.4 16 21.20 even 2
1764.2.bm.a.1685.6 16 21.11 odd 6
1764.2.bm.a.1697.6 16 63.40 odd 6
2268.2.t.a.1781.5 16 63.16 even 3
2268.2.t.a.2105.5 16 63.38 even 6
2268.2.t.b.1781.4 16 63.2 odd 6
2268.2.t.b.2105.4 16 63.52 odd 6
3024.2.ca.d.2033.5 16 28.23 odd 6
3024.2.ca.d.2609.5 16 252.59 odd 6
3024.2.df.d.17.5 16 252.23 even 6
3024.2.df.d.1601.5 16 28.3 even 6
5292.2.w.b.521.4 16 7.5 odd 6
5292.2.w.b.1097.4 16 63.32 odd 6
5292.2.x.a.881.5 16 63.41 even 6 inner
5292.2.x.a.4409.5 16 1.1 even 1 trivial
5292.2.x.b.881.4 16 9.5 odd 6
5292.2.x.b.4409.4 16 7.6 odd 2
5292.2.bm.a.2285.4 16 63.5 even 6
5292.2.bm.a.4625.4 16 7.4 even 3