Properties

Label 4032.2.v.d.1583.9
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 1583.9
Character \(\chi\) \(=\) 4032.1583
Dual form 4032.2.v.d.3599.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.270063 - 0.270063i) q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+(-0.270063 - 0.270063i) q^{5} +1.00000 q^{7} +(3.03491 - 3.03491i) q^{11} +(1.28727 + 1.28727i) q^{13} -5.15741i q^{17} +(5.55077 - 5.55077i) q^{19} +5.69943i q^{23} -4.85413i q^{25} +(-1.94351 + 1.94351i) q^{29} -0.936219i q^{31} +(-0.270063 - 0.270063i) q^{35} +(-1.52927 + 1.52927i) q^{37} -12.4273 q^{41} +(0.346427 + 0.346427i) q^{43} +9.71825 q^{47} +1.00000 q^{49} +(-7.69522 - 7.69522i) q^{53} -1.63923 q^{55} +(-2.03405 + 2.03405i) q^{59} +(5.86902 + 5.86902i) q^{61} -0.695288i q^{65} +(-10.7316 + 10.7316i) q^{67} -12.5046i q^{71} +7.37743i q^{73} +(3.03491 - 3.03491i) q^{77} -13.0724i q^{79} +(-2.53806 - 2.53806i) q^{83} +(-1.39283 + 1.39283i) q^{85} +7.04518 q^{89} +(1.28727 + 1.28727i) q^{91} -2.99812 q^{95} +0.334289 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 36 q^{7}+O(q^{10}) \) 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
\(5\) −0.270063 0.270063i −0.120776 0.120776i 0.644136 0.764911i \(-0.277217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.03491 3.03491i 0.915060 0.915060i −0.0816051 0.996665i \(-0.526005\pi\)
0.996665 + 0.0816051i \(0.0260046\pi\)
\(12\) 0 0
\(13\) 1.28727 + 1.28727i 0.357025 + 0.357025i 0.862715 0.505690i \(-0.168762\pi\)
−0.505690 + 0.862715i \(0.668762\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.15741i 1.25086i −0.780282 0.625428i \(-0.784924\pi\)
0.780282 0.625428i \(-0.215076\pi\)
\(18\) 0 0
\(19\) 5.55077 5.55077i 1.27343 1.27343i 0.329161 0.944274i \(-0.393234\pi\)
0.944274 0.329161i \(-0.106766\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.69943i 1.18841i 0.804313 + 0.594206i \(0.202534\pi\)
−0.804313 + 0.594206i \(0.797466\pi\)
\(24\) 0 0
\(25\) 4.85413i 0.970826i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.94351 + 1.94351i −0.360900 + 0.360900i −0.864144 0.503244i \(-0.832140\pi\)
0.503244 + 0.864144i \(0.332140\pi\)
\(30\) 0 0
\(31\) 0.936219i 0.168150i −0.996459 0.0840750i \(-0.973206\pi\)
0.996459 0.0840750i \(-0.0267935\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.270063 0.270063i −0.0456490 0.0456490i
\(36\) 0 0
\(37\) −1.52927 + 1.52927i −0.251411 + 0.251411i −0.821549 0.570138i \(-0.806890\pi\)
0.570138 + 0.821549i \(0.306890\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −12.4273 −1.94082 −0.970412 0.241457i \(-0.922375\pi\)
−0.970412 + 0.241457i \(0.922375\pi\)
\(42\) 0 0
\(43\) 0.346427 + 0.346427i 0.0528296 + 0.0528296i 0.733028 0.680198i \(-0.238106\pi\)
−0.680198 + 0.733028i \(0.738106\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.71825 1.41755 0.708776 0.705433i \(-0.249248\pi\)
0.708776 + 0.705433i \(0.249248\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.69522 7.69522i −1.05702 1.05702i −0.998273 0.0587471i \(-0.981289\pi\)
−0.0587471 0.998273i \(-0.518711\pi\)
\(54\) 0 0
\(55\) −1.63923 −0.221034
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.03405 + 2.03405i −0.264810 + 0.264810i −0.827005 0.562195i \(-0.809957\pi\)
0.562195 + 0.827005i \(0.309957\pi\)
\(60\) 0 0
\(61\) 5.86902 + 5.86902i 0.751451 + 0.751451i 0.974750 0.223299i \(-0.0716826\pi\)
−0.223299 + 0.974750i \(0.571683\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.695288i 0.0862399i
\(66\) 0 0
\(67\) −10.7316 + 10.7316i −1.31107 + 1.31107i −0.390439 + 0.920629i \(0.627677\pi\)
−0.920629 + 0.390439i \(0.872323\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.5046i 1.48402i −0.670389 0.742010i \(-0.733873\pi\)
0.670389 0.742010i \(-0.266127\pi\)
\(72\) 0 0
\(73\) 7.37743i 0.863463i 0.902002 + 0.431732i \(0.142097\pi\)
−0.902002 + 0.431732i \(0.857903\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.03491 3.03491i 0.345860 0.345860i
\(78\) 0 0
\(79\) 13.0724i 1.47076i −0.677658 0.735378i \(-0.737005\pi\)
0.677658 0.735378i \(-0.262995\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.53806 2.53806i −0.278588 0.278588i 0.553957 0.832545i \(-0.313117\pi\)
−0.832545 + 0.553957i \(0.813117\pi\)
\(84\) 0 0
\(85\) −1.39283 + 1.39283i −0.151073 + 0.151073i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.04518 0.746787 0.373394 0.927673i \(-0.378194\pi\)
0.373394 + 0.927673i \(0.378194\pi\)
\(90\) 0 0
\(91\) 1.28727 + 1.28727i 0.134943 + 0.134943i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.99812 −0.307600
\(96\) 0 0
\(97\) 0.334289 0.0339419 0.0169710 0.999856i \(-0.494598\pi\)
0.0169710 + 0.999856i \(0.494598\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.38370 7.38370i −0.734706 0.734706i 0.236842 0.971548i \(-0.423888\pi\)
−0.971548 + 0.236842i \(0.923888\pi\)
\(102\) 0 0
\(103\) 14.6297 1.44151 0.720755 0.693190i \(-0.243795\pi\)
0.720755 + 0.693190i \(0.243795\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.13731 4.13731i 0.399969 0.399969i −0.478253 0.878222i \(-0.658730\pi\)
0.878222 + 0.478253i \(0.158730\pi\)
\(108\) 0 0
\(109\) 10.4525 + 10.4525i 1.00117 + 1.00117i 0.999999 + 0.00116943i \(0.000372243\pi\)
0.00116943 + 0.999999i \(0.499628\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.90575i 0.837782i −0.908036 0.418891i \(-0.862419\pi\)
0.908036 0.418891i \(-0.137581\pi\)
\(114\) 0 0
\(115\) 1.53920 1.53920i 0.143532 0.143532i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.15741i 0.472779i
\(120\) 0 0
\(121\) 7.42135i 0.674668i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.66124 + 2.66124i −0.238028 + 0.238028i
\(126\) 0 0
\(127\) 13.8060i 1.22509i −0.790437 0.612543i \(-0.790147\pi\)
0.790437 0.612543i \(-0.209853\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.53288 8.53288i −0.745521 0.745521i 0.228113 0.973635i \(-0.426744\pi\)
−0.973635 + 0.228113i \(0.926744\pi\)
\(132\) 0 0
\(133\) 5.55077 5.55077i 0.481313 0.481313i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.08397 0.348917 0.174459 0.984664i \(-0.444182\pi\)
0.174459 + 0.984664i \(0.444182\pi\)
\(138\) 0 0
\(139\) 2.52222 + 2.52222i 0.213932 + 0.213932i 0.805935 0.592003i \(-0.201663\pi\)
−0.592003 + 0.805935i \(0.701663\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 7.81350 0.653397
\(144\) 0 0
\(145\) 1.04974 0.0871760
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.47402 + 4.47402i 0.366526 + 0.366526i 0.866208 0.499683i \(-0.166550\pi\)
−0.499683 + 0.866208i \(0.666550\pi\)
\(150\) 0 0
\(151\) 18.6642 1.51887 0.759434 0.650584i \(-0.225476\pi\)
0.759434 + 0.650584i \(0.225476\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.252838 + 0.252838i −0.0203085 + 0.0203085i
\(156\) 0 0
\(157\) −0.517468 0.517468i −0.0412985 0.0412985i 0.686156 0.727454i \(-0.259297\pi\)
−0.727454 + 0.686156i \(0.759297\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.69943i 0.449178i
\(162\) 0 0
\(163\) −2.16031 + 2.16031i −0.169209 + 0.169209i −0.786632 0.617423i \(-0.788177\pi\)
0.617423 + 0.786632i \(0.288177\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.45631i 0.112693i −0.998411 0.0563465i \(-0.982055\pi\)
0.998411 0.0563465i \(-0.0179451\pi\)
\(168\) 0 0
\(169\) 9.68587i 0.745067i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −15.6519 + 15.6519i −1.18999 + 1.18999i −0.212922 + 0.977069i \(0.568298\pi\)
−0.977069 + 0.212922i \(0.931702\pi\)
\(174\) 0 0
\(175\) 4.85413i 0.366938i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.3258 10.3258i −0.771788 0.771788i 0.206631 0.978419i \(-0.433750\pi\)
−0.978419 + 0.206631i \(0.933750\pi\)
\(180\) 0 0
\(181\) 9.01856 9.01856i 0.670344 0.670344i −0.287451 0.957795i \(-0.592808\pi\)
0.957795 + 0.287451i \(0.0928078\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.825999 0.0607287
\(186\) 0 0
\(187\) −15.6523 15.6523i −1.14461 1.14461i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.73439 −0.704356 −0.352178 0.935933i \(-0.614559\pi\)
−0.352178 + 0.935933i \(0.614559\pi\)
\(192\) 0 0
\(193\) −3.44692 −0.248115 −0.124057 0.992275i \(-0.539591\pi\)
−0.124057 + 0.992275i \(0.539591\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −0.585435 0.585435i −0.0417105 0.0417105i 0.685944 0.727654i \(-0.259390\pi\)
−0.727654 + 0.685944i \(0.759390\pi\)
\(198\) 0 0
\(199\) 2.57428 0.182486 0.0912429 0.995829i \(-0.470916\pi\)
0.0912429 + 0.995829i \(0.470916\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.94351 + 1.94351i −0.136407 + 0.136407i
\(204\) 0 0
\(205\) 3.35616 + 3.35616i 0.234405 + 0.234405i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 33.6922i 2.33054i
\(210\) 0 0
\(211\) 11.3783 11.3783i 0.783317 0.783317i −0.197072 0.980389i \(-0.563143\pi\)
0.980389 + 0.197072i \(0.0631432\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.187114i 0.0127611i
\(216\) 0 0
\(217\) 0.936219i 0.0635547i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.63898 6.63898i 0.446586 0.446586i
\(222\) 0 0
\(223\) 6.26410i 0.419475i 0.977758 + 0.209738i \(0.0672610\pi\)
−0.977758 + 0.209738i \(0.932739\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.38594 + 4.38594i 0.291105 + 0.291105i 0.837517 0.546412i \(-0.184007\pi\)
−0.546412 + 0.837517i \(0.684007\pi\)
\(228\) 0 0
\(229\) −17.8540 + 17.8540i −1.17982 + 1.17982i −0.200034 + 0.979789i \(0.564105\pi\)
−0.979789 + 0.200034i \(0.935895\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.0142 0.983614 0.491807 0.870704i \(-0.336336\pi\)
0.491807 + 0.870704i \(0.336336\pi\)
\(234\) 0 0
\(235\) −2.62454 2.62454i −0.171206 0.171206i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.62807 0.105311 0.0526554 0.998613i \(-0.483232\pi\)
0.0526554 + 0.998613i \(0.483232\pi\)
\(240\) 0 0
\(241\) −22.7132 −1.46308 −0.731542 0.681797i \(-0.761199\pi\)
−0.731542 + 0.681797i \(0.761199\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.270063 0.270063i −0.0172537 0.0172537i
\(246\) 0 0
\(247\) 14.2907 0.909295
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.1734 16.1734i 1.02085 1.02085i 0.0210767 0.999778i \(-0.493291\pi\)
0.999778 0.0210767i \(-0.00670942\pi\)
\(252\) 0 0
\(253\) 17.2972 + 17.2972i 1.08747 + 1.08747i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.1675i 1.44515i −0.691293 0.722574i \(-0.742959\pi\)
0.691293 0.722574i \(-0.257041\pi\)
\(258\) 0 0
\(259\) −1.52927 + 1.52927i −0.0950243 + 0.0950243i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.643221i 0.0396627i 0.999803 + 0.0198314i \(0.00631293\pi\)
−0.999803 + 0.0198314i \(0.993687\pi\)
\(264\) 0 0
\(265\) 4.15639i 0.255325i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.72444 + 7.72444i −0.470967 + 0.470967i −0.902228 0.431260i \(-0.858069\pi\)
0.431260 + 0.902228i \(0.358069\pi\)
\(270\) 0 0
\(271\) 20.3731i 1.23758i −0.785558 0.618788i \(-0.787624\pi\)
0.785558 0.618788i \(-0.212376\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −14.7319 14.7319i −0.888364 0.888364i
\(276\) 0 0
\(277\) 21.6135 21.6135i 1.29863 1.29863i 0.369336 0.929296i \(-0.379585\pi\)
0.929296 0.369336i \(-0.120415\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.42826 0.443133 0.221566 0.975145i \(-0.428883\pi\)
0.221566 + 0.975145i \(0.428883\pi\)
\(282\) 0 0
\(283\) 8.92425 + 8.92425i 0.530492 + 0.530492i 0.920719 0.390227i \(-0.127603\pi\)
−0.390227 + 0.920719i \(0.627603\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.4273 −0.733562
\(288\) 0 0
\(289\) −9.59890 −0.564641
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −9.06945 9.06945i −0.529843 0.529843i 0.390682 0.920526i \(-0.372239\pi\)
−0.920526 + 0.390682i \(0.872239\pi\)
\(294\) 0 0
\(295\) 1.09864 0.0639654
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −7.33670 + 7.33670i −0.424293 + 0.424293i
\(300\) 0 0
\(301\) 0.346427 + 0.346427i 0.0199677 + 0.0199677i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.17001i 0.181514i
\(306\) 0 0
\(307\) −15.2349 + 15.2349i −0.869504 + 0.869504i −0.992417 0.122913i \(-0.960776\pi\)
0.122913 + 0.992417i \(0.460776\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.1899i 1.20157i 0.799410 + 0.600785i \(0.205145\pi\)
−0.799410 + 0.600785i \(0.794855\pi\)
\(312\) 0 0
\(313\) 15.4334i 0.872347i 0.899863 + 0.436173i \(0.143667\pi\)
−0.899863 + 0.436173i \(0.856333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.23380 8.23380i 0.462456 0.462456i −0.437003 0.899460i \(-0.643960\pi\)
0.899460 + 0.437003i \(0.143960\pi\)
\(318\) 0 0
\(319\) 11.7967i 0.660490i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −28.6276 28.6276i −1.59288 1.59288i
\(324\) 0 0
\(325\) 6.24858 6.24858i 0.346609 0.346609i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.71825 0.535784
\(330\) 0 0
\(331\) 18.1815 + 18.1815i 0.999344 + 0.999344i 1.00000 0.000656172i \(-0.000208866\pi\)
−0.000656172 1.00000i \(0.500209\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.79639 0.316691
\(336\) 0 0
\(337\) 20.4267 1.11271 0.556357 0.830943i \(-0.312199\pi\)
0.556357 + 0.830943i \(0.312199\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.84134 2.84134i −0.153867 0.153867i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.37130 1.37130i 0.0736154 0.0736154i −0.669340 0.742956i \(-0.733423\pi\)
0.742956 + 0.669340i \(0.233423\pi\)
\(348\) 0 0
\(349\) 5.10409 + 5.10409i 0.273216 + 0.273216i 0.830393 0.557178i \(-0.188116\pi\)
−0.557178 + 0.830393i \(0.688116\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 32.0190i 1.70420i 0.523378 + 0.852101i \(0.324672\pi\)
−0.523378 + 0.852101i \(0.675328\pi\)
\(354\) 0 0
\(355\) −3.37702 + 3.37702i −0.179234 + 0.179234i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.8950i 0.575018i −0.957778 0.287509i \(-0.907173\pi\)
0.957778 0.287509i \(-0.0928272\pi\)
\(360\) 0 0
\(361\) 42.6222i 2.24327i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.99237 1.99237i 0.104286 0.104286i
\(366\) 0 0
\(367\) 3.57323i 0.186521i 0.995642 + 0.0932605i \(0.0297290\pi\)
−0.995642 + 0.0932605i \(0.970271\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −7.69522 7.69522i −0.399516 0.399516i
\(372\) 0 0
\(373\) −8.40095 + 8.40095i −0.434985 + 0.434985i −0.890320 0.455335i \(-0.849519\pi\)
0.455335 + 0.890320i \(0.349519\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.00363 −0.257700
\(378\) 0 0
\(379\) 7.51287 + 7.51287i 0.385910 + 0.385910i 0.873226 0.487316i \(-0.162024\pi\)
−0.487316 + 0.873226i \(0.662024\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 28.9165 1.47757 0.738783 0.673944i \(-0.235401\pi\)
0.738783 + 0.673944i \(0.235401\pi\)
\(384\) 0 0
\(385\) −1.63923 −0.0835431
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.26758 1.26758i −0.0642689 0.0642689i 0.674242 0.738511i \(-0.264471\pi\)
−0.738511 + 0.674242i \(0.764471\pi\)
\(390\) 0 0
\(391\) 29.3943 1.48653
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.53036 + 3.53036i −0.177632 + 0.177632i
\(396\) 0 0
\(397\) −2.08140 2.08140i −0.104463 0.104463i 0.652944 0.757406i \(-0.273534\pi\)
−0.757406 + 0.652944i \(0.773534\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.64862i 0.381954i 0.981595 + 0.190977i \(0.0611656\pi\)
−0.981595 + 0.190977i \(0.938834\pi\)
\(402\) 0 0
\(403\) 1.20517 1.20517i 0.0600336 0.0600336i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.28240i 0.460111i
\(408\) 0 0
\(409\) 36.0331i 1.78172i −0.454275 0.890862i \(-0.650102\pi\)
0.454275 0.890862i \(-0.349898\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.03405 + 2.03405i −0.100089 + 0.100089i
\(414\) 0 0
\(415\) 1.37087i 0.0672934i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.27031 + 4.27031i 0.208618 + 0.208618i 0.803680 0.595062i \(-0.202872\pi\)
−0.595062 + 0.803680i \(0.702872\pi\)
\(420\) 0 0
\(421\) −6.30843 + 6.30843i −0.307454 + 0.307454i −0.843921 0.536467i \(-0.819759\pi\)
0.536467 + 0.843921i \(0.319759\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −25.0348 −1.21436
\(426\) 0 0
\(427\) 5.86902 + 5.86902i 0.284022 + 0.284022i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.7409 −0.613708 −0.306854 0.951757i \(-0.599276\pi\)
−0.306854 + 0.951757i \(0.599276\pi\)
\(432\) 0 0
\(433\) 2.89975 0.139353 0.0696764 0.997570i \(-0.477803\pi\)
0.0696764 + 0.997570i \(0.477803\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 31.6362 + 31.6362i 1.51337 + 1.51337i
\(438\) 0 0
\(439\) −18.1968 −0.868487 −0.434243 0.900796i \(-0.642984\pi\)
−0.434243 + 0.900796i \(0.642984\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.90248 + 5.90248i −0.280435 + 0.280435i −0.833283 0.552847i \(-0.813541\pi\)
0.552847 + 0.833283i \(0.313541\pi\)
\(444\) 0 0
\(445\) −1.90264 1.90264i −0.0901938 0.0901938i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13.5014i 0.637172i −0.947894 0.318586i \(-0.896792\pi\)
0.947894 0.318586i \(-0.103208\pi\)
\(450\) 0 0
\(451\) −37.7158 + 37.7158i −1.77597 + 1.77597i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.695288i 0.0325956i
\(456\) 0 0
\(457\) 24.4732i 1.14481i −0.819972 0.572403i \(-0.806011\pi\)
0.819972 0.572403i \(-0.193989\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5.52778 + 5.52778i −0.257455 + 0.257455i −0.824018 0.566563i \(-0.808273\pi\)
0.566563 + 0.824018i \(0.308273\pi\)
\(462\) 0 0
\(463\) 8.48501i 0.394332i −0.980370 0.197166i \(-0.936826\pi\)
0.980370 0.197166i \(-0.0631738\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.61601 + 1.61601i 0.0747800 + 0.0747800i 0.743508 0.668728i \(-0.233161\pi\)
−0.668728 + 0.743508i \(0.733161\pi\)
\(468\) 0 0
\(469\) −10.7316 + 10.7316i −0.495537 + 0.495537i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.10275 0.0966844
\(474\) 0 0
\(475\) −26.9442 26.9442i −1.23628 1.23628i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −16.9452 −0.774247 −0.387123 0.922028i \(-0.626531\pi\)
−0.387123 + 0.922028i \(0.626531\pi\)
\(480\) 0 0
\(481\) −3.93717 −0.179520
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.0902791 0.0902791i −0.00409936 0.00409936i
\(486\) 0 0
\(487\) −6.48781 −0.293991 −0.146995 0.989137i \(-0.546960\pi\)
−0.146995 + 0.989137i \(0.546960\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.03691 + 3.03691i −0.137054 + 0.137054i −0.772305 0.635251i \(-0.780896\pi\)
0.635251 + 0.772305i \(0.280896\pi\)
\(492\) 0 0
\(493\) 10.0235 + 10.0235i 0.451434 + 0.451434i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.5046i 0.560907i
\(498\) 0 0
\(499\) −29.0548 + 29.0548i −1.30067 + 1.30067i −0.372735 + 0.927938i \(0.621580\pi\)
−0.927938 + 0.372735i \(0.878420\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 24.3803i 1.08706i 0.839388 + 0.543532i \(0.182913\pi\)
−0.839388 + 0.543532i \(0.817087\pi\)
\(504\) 0 0
\(505\) 3.98813i 0.177469i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.39973 6.39973i 0.283663 0.283663i −0.550905 0.834568i \(-0.685717\pi\)
0.834568 + 0.550905i \(0.185717\pi\)
\(510\) 0 0
\(511\) 7.37743i 0.326359i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.95095 3.95095i −0.174100 0.174100i
\(516\) 0 0
\(517\) 29.4940 29.4940i 1.29715 1.29715i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 24.3522 1.06689 0.533444 0.845835i \(-0.320897\pi\)
0.533444 + 0.845835i \(0.320897\pi\)
\(522\) 0 0
\(523\) 13.4483 + 13.4483i 0.588054 + 0.588054i 0.937104 0.349050i \(-0.113496\pi\)
−0.349050 + 0.937104i \(0.613496\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.82847 −0.210331
\(528\) 0 0
\(529\) −9.48348 −0.412325
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −15.9973 15.9973i −0.692921 0.692921i
\(534\) 0 0
\(535\) −2.23467 −0.0966131
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.03491 3.03491i 0.130723 0.130723i
\(540\) 0 0
\(541\) 2.15117 + 2.15117i 0.0924861 + 0.0924861i 0.751836 0.659350i \(-0.229168\pi\)
−0.659350 + 0.751836i \(0.729168\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.64567i 0.241834i
\(546\) 0 0
\(547\) −5.10155 + 5.10155i −0.218126 + 0.218126i −0.807708 0.589582i \(-0.799292\pi\)
0.589582 + 0.807708i \(0.299292\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 21.5759i 0.919165i
\(552\) 0 0
\(553\) 13.0724i 0.555893i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.0391 + 16.0391i −0.679599 + 0.679599i −0.959909 0.280310i \(-0.909563\pi\)
0.280310 + 0.959909i \(0.409563\pi\)
\(558\) 0 0
\(559\) 0.891890i 0.0377229i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.3434 + 20.3434i 0.857370 + 0.857370i 0.991028 0.133657i \(-0.0426721\pi\)
−0.133657 + 0.991028i \(0.542672\pi\)
\(564\) 0 0
\(565\) −2.40511 + 2.40511i −0.101184 + 0.101184i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 34.6243 1.45152 0.725762 0.687945i \(-0.241487\pi\)
0.725762 + 0.687945i \(0.241487\pi\)
\(570\) 0 0
\(571\) 30.1046 + 30.1046i 1.25984 + 1.25984i 0.951170 + 0.308667i \(0.0998828\pi\)
0.308667 + 0.951170i \(0.400117\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 27.6658 1.15374
\(576\) 0 0
\(577\) −4.97393 −0.207067 −0.103534 0.994626i \(-0.533015\pi\)
−0.103534 + 0.994626i \(0.533015\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.53806 2.53806i −0.105296 0.105296i
\(582\) 0 0
\(583\) −46.7086 −1.93447
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.50154 4.50154i 0.185798 0.185798i −0.608079 0.793877i \(-0.708059\pi\)
0.793877 + 0.608079i \(0.208059\pi\)
\(588\) 0 0
\(589\) −5.19674 5.19674i −0.214128 0.214128i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 38.9459i 1.59931i 0.600457 + 0.799657i \(0.294985\pi\)
−0.600457 + 0.799657i \(0.705015\pi\)
\(594\) 0 0
\(595\) −1.39283 + 1.39283i −0.0571003 + 0.0571003i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.5162i 0.511398i −0.966756 0.255699i \(-0.917694\pi\)
0.966756 0.255699i \(-0.0823056\pi\)
\(600\) 0 0
\(601\) 25.5234i 1.04112i −0.853825 0.520561i \(-0.825723\pi\)
0.853825 0.520561i \(-0.174277\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.00423 + 2.00423i −0.0814836 + 0.0814836i
\(606\) 0 0
\(607\) 14.1652i 0.574949i −0.957788 0.287475i \(-0.907184\pi\)
0.957788 0.287475i \(-0.0928157\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.5100 + 12.5100i 0.506101 + 0.506101i
\(612\) 0 0
\(613\) 28.9735 28.9735i 1.17023 1.17023i 0.188071 0.982155i \(-0.439776\pi\)
0.982155 0.188071i \(-0.0602236\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5.73722 0.230972 0.115486 0.993309i \(-0.463157\pi\)
0.115486 + 0.993309i \(0.463157\pi\)
\(618\) 0 0
\(619\) −27.3596 27.3596i −1.09968 1.09968i −0.994448 0.105229i \(-0.966442\pi\)
−0.105229 0.994448i \(-0.533558\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.04518 0.282259
\(624\) 0 0
\(625\) −22.8333 −0.913330
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.88708 + 7.88708i 0.314479 + 0.314479i
\(630\) 0 0
\(631\) 39.5537 1.57461 0.787303 0.616566i \(-0.211477\pi\)
0.787303 + 0.616566i \(0.211477\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.72850 + 3.72850i −0.147961 + 0.147961i
\(636\) 0 0
\(637\) 1.28727 + 1.28727i 0.0510035 + 0.0510035i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 42.5474i 1.68052i −0.542182 0.840261i \(-0.682402\pi\)
0.542182 0.840261i \(-0.317598\pi\)
\(642\) 0 0
\(643\) 13.6174 13.6174i 0.537017 0.537017i −0.385635 0.922652i \(-0.626017\pi\)
0.922652 + 0.385635i \(0.126017\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 29.7806i 1.17080i 0.810746 + 0.585398i \(0.199062\pi\)
−0.810746 + 0.585398i \(0.800938\pi\)
\(648\) 0 0
\(649\) 12.3463i 0.484635i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −13.9493 + 13.9493i −0.545880 + 0.545880i −0.925246 0.379367i \(-0.876142\pi\)
0.379367 + 0.925246i \(0.376142\pi\)
\(654\) 0 0
\(655\) 4.60883i 0.180082i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.9096 + 12.9096i 0.502888 + 0.502888i 0.912334 0.409446i \(-0.134278\pi\)
−0.409446 + 0.912334i \(0.634278\pi\)
\(660\) 0 0
\(661\) −19.0427 + 19.0427i −0.740674 + 0.740674i −0.972708 0.232033i \(-0.925462\pi\)
0.232033 + 0.972708i \(0.425462\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.99812 −0.116262
\(666\) 0 0
\(667\) −11.0769 11.0769i −0.428898 0.428898i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 35.6239 1.37524
\(672\) 0 0
\(673\) −20.4420 −0.787983 −0.393991 0.919114i \(-0.628906\pi\)
−0.393991 + 0.919114i \(0.628906\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −4.78059 4.78059i −0.183733 0.183733i 0.609247 0.792980i \(-0.291472\pi\)
−0.792980 + 0.609247i \(0.791472\pi\)
\(678\) 0 0
\(679\) 0.334289 0.0128288
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.19706 1.19706i 0.0458041 0.0458041i −0.683834 0.729638i \(-0.739689\pi\)
0.729638 + 0.683834i \(0.239689\pi\)
\(684\) 0 0
\(685\) −1.10293 1.10293i −0.0421408 0.0421408i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 19.8117i 0.754764i
\(690\) 0 0
\(691\) 2.98198 2.98198i 0.113440 0.113440i −0.648108 0.761548i \(-0.724440\pi\)
0.761548 + 0.648108i \(0.224440\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.36232i 0.0516757i
\(696\) 0 0
\(697\) 64.0929i 2.42769i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −25.9008 + 25.9008i −0.978261 + 0.978261i −0.999769 0.0215079i \(-0.993153\pi\)
0.0215079 + 0.999769i \(0.493153\pi\)
\(702\) 0 0
\(703\) 16.9773i 0.640310i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.38370 7.38370i −0.277693 0.277693i
\(708\) 0 0
\(709\) −2.86472 + 2.86472i −0.107587 + 0.107587i −0.758851 0.651264i \(-0.774239\pi\)
0.651264 + 0.758851i \(0.274239\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.33591 0.199832
\(714\) 0 0
\(715\) −2.11014 2.11014i −0.0789146 0.0789146i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.64941 −0.0615126 −0.0307563 0.999527i \(-0.509792\pi\)
−0.0307563 + 0.999527i \(0.509792\pi\)
\(720\) 0 0
\(721\) 14.6297 0.544839
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9.43403 + 9.43403i 0.350371 + 0.350371i
\(726\) 0 0
\(727\) 30.5976 1.13480 0.567400 0.823442i \(-0.307949\pi\)
0.567400 + 0.823442i \(0.307949\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.78667 1.78667i 0.0660822 0.0660822i
\(732\) 0 0
\(733\) 36.9502 + 36.9502i 1.36479 + 1.36479i 0.867704 + 0.497082i \(0.165595\pi\)
0.497082 + 0.867704i \(0.334405\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 65.1386i 2.39941i
\(738\) 0 0
\(739\) 16.0208 16.0208i 0.589335 0.589335i −0.348116 0.937451i \(-0.613179\pi\)
0.937451 + 0.348116i \(0.113179\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.0911i 1.25068i −0.780352 0.625341i \(-0.784960\pi\)
0.780352 0.625341i \(-0.215040\pi\)
\(744\) 0 0
\(745\) 2.41653i 0.0885349i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.13731 4.13731i 0.151174 0.151174i
\(750\) 0 0
\(751\) 37.9956i 1.38648i 0.720708 + 0.693239i \(0.243817\pi\)
−0.720708 + 0.693239i \(0.756183\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5.04050 5.04050i −0.183443 0.183443i
\(756\) 0 0
\(757\) −36.2650 + 36.2650i −1.31807 + 1.31807i −0.402774 + 0.915300i \(0.631954\pi\)
−0.915300 + 0.402774i \(0.868046\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 16.8705 0.611556 0.305778 0.952103i \(-0.401084\pi\)
0.305778 + 0.952103i \(0.401084\pi\)
\(762\) 0 0
\(763\) 10.4525 + 10.4525i 0.378406 + 0.378406i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.23674 −0.189088
\(768\) 0 0
\(769\) −46.7971 −1.68755 −0.843774 0.536698i \(-0.819671\pi\)
−0.843774 + 0.536698i \(0.819671\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 28.7793 + 28.7793i 1.03512 + 1.03512i 0.999360 + 0.0357608i \(0.0113854\pi\)
0.0357608 + 0.999360i \(0.488615\pi\)
\(774\) 0 0
\(775\) −4.54453 −0.163244
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −68.9813 + 68.9813i −2.47151 + 2.47151i
\(780\) 0 0
\(781\) −37.9502 37.9502i −1.35797 1.35797i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.279498i 0.00997571i
\(786\) 0 0
\(787\) −29.7259 + 29.7259i −1.05961 + 1.05961i −0.0615057 + 0.998107i \(0.519590\pi\)
−0.998107 + 0.0615057i \(0.980410\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 8.90575i 0.316652i
\(792\) 0 0
\(793\) 15.1100i 0.536573i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −21.6739 + 21.6739i −0.767728 + 0.767728i −0.977706 0.209978i \(-0.932661\pi\)
0.209978 + 0.977706i \(0.432661\pi\)
\(798\) 0 0
\(799\) 50.1210i 1.77315i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 22.3898 + 22.3898i 0.790121 + 0.790121i
\(804\) 0 0
\(805\) 1.53920 1.53920i 0.0542498 0.0542498i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −27.6371 −0.971669 −0.485834 0.874051i \(-0.661484\pi\)
−0.485834 + 0.874051i \(0.661484\pi\)
\(810\) 0 0
\(811\) −20.6941 20.6941i −0.726668 0.726668i 0.243286 0.969955i \(-0.421775\pi\)
−0.969955 + 0.243286i \(0.921775\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.16684 0.0408727
\(816\) 0 0
\(817\) 3.84587 0.134550
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.30451 + 8.30451i 0.289829 + 0.289829i 0.837013 0.547183i \(-0.184300\pi\)
−0.547183 + 0.837013i \(0.684300\pi\)
\(822\) 0 0
\(823\) −5.77989 −0.201474 −0.100737 0.994913i \(-0.532120\pi\)
−0.100737 + 0.994913i \(0.532120\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8.49162 8.49162i 0.295282 0.295282i −0.543880 0.839163i \(-0.683045\pi\)
0.839163 + 0.543880i \(0.183045\pi\)
\(828\) 0 0
\(829\) −22.2746 22.2746i −0.773629 0.773629i 0.205110 0.978739i \(-0.434245\pi\)
−0.978739 + 0.205110i \(0.934245\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.15741i 0.178694i
\(834\) 0 0
\(835\) −0.393296 + 0.393296i −0.0136106 + 0.0136106i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 31.0889i 1.07331i 0.843803 + 0.536653i \(0.180312\pi\)
−0.843803 + 0.536653i \(0.819688\pi\)
\(840\) 0 0
\(841\) 21.4456i 0.739502i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.61580 + 2.61580i −0.0899861 + 0.0899861i
\(846\) 0 0
\(847\) 7.42135i 0.255001i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8.71597 8.71597i −0.298780 0.298780i
\(852\) 0 0
\(853\) −13.2899 + 13.2899i −0.455037 + 0.455037i −0.897022 0.441985i \(-0.854274\pi\)
0.441985 + 0.897022i \(0.354274\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.3573 0.661234 0.330617 0.943765i \(-0.392743\pi\)
0.330617 + 0.943765i \(0.392743\pi\)
\(858\) 0 0
\(859\) 1.21680 + 1.21680i 0.0415165 + 0.0415165i 0.727560 0.686044i \(-0.240654\pi\)
−0.686044 + 0.727560i \(0.740654\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 34.8956 1.18786 0.593929 0.804517i \(-0.297576\pi\)
0.593929 + 0.804517i \(0.297576\pi\)
\(864\) 0 0
\(865\) 8.45399 0.287444
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −39.6734 39.6734i −1.34583 1.34583i
\(870\) 0 0
\(871\) −27.6288 −0.936167
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.66124 + 2.66124i −0.0899662 + 0.0899662i
\(876\) 0 0
\(877\) −11.2180 11.2180i −0.378804 0.378804i 0.491866 0.870671i \(-0.336315\pi\)
−0.870671 + 0.491866i \(0.836315\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 3.79230i 0.127766i 0.997957 + 0.0638829i \(0.0203484\pi\)
−0.997957 + 0.0638829i \(0.979652\pi\)
\(882\) 0 0
\(883\) −7.66911 + 7.66911i −0.258086 + 0.258086i −0.824275 0.566189i \(-0.808417\pi\)
0.566189 + 0.824275i \(0.308417\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 15.3130i 0.514161i −0.966390 0.257081i \(-0.917239\pi\)
0.966390 0.257081i \(-0.0827606\pi\)
\(888\) 0 0
\(889\) 13.8060i 0.463039i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 53.9438 53.9438i 1.80516 1.80516i
\(894\) 0 0
\(895\) 5.57724i 0.186427i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.81955 + 1.81955i 0.0606853 + 0.0606853i
\(900\) 0 0
\(901\) −39.6874 + 39.6874i −1.32218 + 1.32218i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.87116 −0.161923
\(906\) 0 0
\(907\) −32.8167 32.8167i −1.08966 1.08966i −0.995563 0.0940982i \(-0.970003\pi\)
−0.0940982 0.995563i \(-0.529997\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −26.8385 −0.889200 −0.444600 0.895729i \(-0.646654\pi\)
−0.444600 + 0.895729i \(0.646654\pi\)
\(912\) 0 0
\(913\) −15.4055 −0.509849
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −8.53288 8.53288i −0.281781 0.281781i
\(918\) 0 0
\(919\) −19.8195 −0.653785 −0.326893 0.945061i \(-0.606002\pi\)
−0.326893 + 0.945061i \(0.606002\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 16.0968 16.0968i 0.529831 0.529831i
\(924\) 0 0
\(925\) 7.42329 + 7.42329i 0.244076 + 0.244076i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 38.2590i 1.25524i 0.778521 + 0.627619i \(0.215970\pi\)
−0.778521 + 0.627619i \(0.784030\pi\)
\(930\) 0 0
\(931\) 5.55077 5.55077i 0.181919 0.181919i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 8.45420i 0.276482i
\(936\) 0 0
\(937\) 41.1963i 1.34582i 0.739722 + 0.672912i \(0.234957\pi\)
−0.739722 + 0.672912i \(0.765043\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −14.9904 + 14.9904i −0.488673 + 0.488673i −0.907887 0.419214i \(-0.862306\pi\)
0.419214 + 0.907887i \(0.362306\pi\)
\(942\) 0 0
\(943\) 70.8287i 2.30650i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.55196 + 2.55196i 0.0829275 + 0.0829275i 0.747354 0.664426i \(-0.231324\pi\)
−0.664426 + 0.747354i \(0.731324\pi\)
\(948\) 0 0
\(949\) −9.49675 + 9.49675i −0.308278 + 0.308278i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −16.1058 −0.521719 −0.260859 0.965377i \(-0.584006\pi\)
−0.260859 + 0.965377i \(0.584006\pi\)
\(954\) 0 0
\(955\) 2.62890 + 2.62890i 0.0850692 + 0.0850692i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.08397 0.131878
\(960\) 0 0
\(961\) 30.1235 0.971726
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0.930886 + 0.930886i 0.0299663 + 0.0299663i
\(966\) 0 0
\(967\) −21.1491 −0.680110 −0.340055 0.940406i \(-0.610446\pi\)
−0.340055 + 0.940406i \(0.610446\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 43.0192 43.0192i 1.38055 1.38055i 0.536915 0.843636i \(-0.319590\pi\)
0.843636 0.536915i \(-0.180410\pi\)
\(972\) 0 0
\(973\) 2.52222 + 2.52222i 0.0808587 + 0.0808587i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.96367i 0.0948161i −0.998876 0.0474081i \(-0.984904\pi\)
0.998876 0.0474081i \(-0.0150961\pi\)
\(978\) 0 0
\(979\) 21.3815 21.3815i 0.683355 0.683355i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 29.5904i 0.943787i 0.881655 + 0.471894i \(0.156429\pi\)
−0.881655 + 0.471894i \(0.843571\pi\)
\(984\) 0 0
\(985\) 0.316209i 0.0100752i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.97443 + 1.97443i −0.0627834 + 0.0627834i
\(990\) 0 0
\(991\) 25.3741i 0.806035i 0.915192 + 0.403017i \(0.132039\pi\)
−0.915192 + 0.403017i \(0.867961\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −0.695217 0.695217i −0.0220399 0.0220399i
\(996\) 0 0
\(997\) −3.08066 + 3.08066i −0.0975654 + 0.0975654i −0.754205 0.656639i \(-0.771977\pi\)
0.656639 + 0.754205i \(0.271977\pi\)
\(998\) 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.1583.9 36
3.2 odd 2 inner 4032.2.v.d.1583.10 36
4.3 odd 2 1008.2.v.d.323.13 yes 36
12.11 even 2 1008.2.v.d.323.6 36
16.5 even 4 1008.2.v.d.827.6 yes 36
16.11 odd 4 inner 4032.2.v.d.3599.10 36
48.5 odd 4 1008.2.v.d.827.13 yes 36
48.11 even 4 inner 4032.2.v.d.3599.9 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.v.d.323.6 36 12.11 even 2
1008.2.v.d.323.13 yes 36 4.3 odd 2
1008.2.v.d.827.6 yes 36 16.5 even 4
1008.2.v.d.827.13 yes 36 48.5 odd 4
4032.2.v.d.1583.9 36 1.1 even 1 trivial
4032.2.v.d.1583.10 36 3.2 odd 2 inner
4032.2.v.d.3599.9 36 48.11 even 4 inner
4032.2.v.d.3599.10 36 16.11 odd 4 inner