Properties

Label 1512.2.r.d.505.4
Level $1512$
Weight $2$
Character 1512.505
Analytic conductor $12.073$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1512,2,Mod(505,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.505");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.r (of order \(3\), 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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.508277025.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 505.4
Root \(-0.734668 - 0.348716i\) of defining polynomial
Character \(\chi\) \(=\) 1512.505
Dual form 1512.2.r.d.1009.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.21814 + 2.10988i) q^{5} +(-0.500000 + 0.866025i) q^{7} +O(q^{10})\) \(q+(1.21814 + 2.10988i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(0.379477 - 0.657274i) q^{11} +(-1.11414 - 1.92976i) q^{13} +7.04904 q^{17} -1.37172 q^{19} +(3.51814 + 6.09360i) q^{23} +(-0.467722 + 0.810117i) q^{25} +(0.418134 - 0.724229i) q^{29} +(0.265332 + 0.459569i) q^{31} -2.43628 q^{35} +4.53066 q^{37} +(4.42852 + 7.67042i) q^{41} +(-3.70161 + 6.41137i) q^{43} +(-3.39762 + 5.88485i) q^{47} +(-0.500000 - 0.866025i) q^{49} +0.607978 q^{53} +1.84902 q^{55} +(-0.581866 - 1.00782i) q^{59} +(-5.85680 + 10.1443i) q^{61} +(2.71436 - 4.70142i) q^{65} +(0.152801 + 0.264659i) q^{67} -12.6192 q^{71} -10.1499 q^{73} +(0.379477 + 0.657274i) q^{77} +(-7.62453 + 13.2061i) q^{79} +(8.18909 - 14.1839i) q^{83} +(8.58671 + 14.8726i) q^{85} +9.63151 q^{89} +2.22829 q^{91} +(-1.67094 - 2.89416i) q^{95} +(5.46718 - 9.46943i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{5} - 4 q^{7} + 6 q^{11} - 3 q^{13} + 16 q^{17} - 4 q^{19} + 5 q^{23} - 14 q^{25} - q^{29} + 11 q^{31} + 8 q^{35} + 54 q^{37} - 2 q^{41} - 11 q^{43} - 7 q^{47} - 4 q^{49} + 8 q^{53} + 12 q^{55} - 9 q^{59} - 7 q^{61} + 9 q^{65} - 12 q^{67} + 24 q^{71} + 26 q^{73} + 6 q^{77} - 22 q^{79} + 6 q^{83} - 11 q^{85} + 28 q^{89} + 6 q^{91} + 23 q^{95} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(5\) 1.21814 + 2.10988i 0.544768 + 0.943566i 0.998621 + 0.0524895i \(0.0167156\pi\)
−0.453853 + 0.891076i \(0.649951\pi\)
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.379477 0.657274i 0.114417 0.198176i −0.803130 0.595804i \(-0.796833\pi\)
0.917546 + 0.397629i \(0.130167\pi\)
\(12\) 0 0
\(13\) −1.11414 1.92976i −0.309008 0.535218i 0.669138 0.743139i \(-0.266664\pi\)
−0.978146 + 0.207921i \(0.933330\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.04904 1.70964 0.854822 0.518922i \(-0.173666\pi\)
0.854822 + 0.518922i \(0.173666\pi\)
\(18\) 0 0
\(19\) −1.37172 −0.314694 −0.157347 0.987543i \(-0.550294\pi\)
−0.157347 + 0.987543i \(0.550294\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.51814 + 6.09360i 0.733583 + 1.27060i 0.955342 + 0.295502i \(0.0954870\pi\)
−0.221759 + 0.975102i \(0.571180\pi\)
\(24\) 0 0
\(25\) −0.467722 + 0.810117i −0.0935443 + 0.162023i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.418134 0.724229i 0.0776455 0.134486i −0.824588 0.565733i \(-0.808593\pi\)
0.902234 + 0.431248i \(0.141926\pi\)
\(30\) 0 0
\(31\) 0.265332 + 0.459569i 0.0476551 + 0.0825411i 0.888869 0.458161i \(-0.151492\pi\)
−0.841214 + 0.540702i \(0.818158\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.43628 −0.411806
\(36\) 0 0
\(37\) 4.53066 0.744837 0.372418 0.928065i \(-0.378529\pi\)
0.372418 + 0.928065i \(0.378529\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.42852 + 7.67042i 0.691618 + 1.19792i 0.971307 + 0.237828i \(0.0764353\pi\)
−0.279689 + 0.960091i \(0.590231\pi\)
\(42\) 0 0
\(43\) −3.70161 + 6.41137i −0.564490 + 0.977725i 0.432607 + 0.901583i \(0.357594\pi\)
−0.997097 + 0.0761428i \(0.975740\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.39762 + 5.88485i −0.495594 + 0.858394i −0.999987 0.00508036i \(-0.998383\pi\)
0.504393 + 0.863474i \(0.331716\pi\)
\(48\) 0 0
\(49\) −0.500000 0.866025i −0.0714286 0.123718i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.607978 0.0835122 0.0417561 0.999128i \(-0.486705\pi\)
0.0417561 + 0.999128i \(0.486705\pi\)
\(54\) 0 0
\(55\) 1.84902 0.249322
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.581866 1.00782i −0.0757525 0.131207i 0.825661 0.564167i \(-0.190803\pi\)
−0.901413 + 0.432960i \(0.857469\pi\)
\(60\) 0 0
\(61\) −5.85680 + 10.1443i −0.749887 + 1.29884i 0.197990 + 0.980204i \(0.436559\pi\)
−0.947876 + 0.318638i \(0.896775\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.71436 4.70142i 0.336676 0.583139i
\(66\) 0 0
\(67\) 0.152801 + 0.264659i 0.0186676 + 0.0323333i 0.875208 0.483746i \(-0.160724\pi\)
−0.856541 + 0.516080i \(0.827391\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.6192 −1.49763 −0.748813 0.662781i \(-0.769376\pi\)
−0.748813 + 0.662781i \(0.769376\pi\)
\(72\) 0 0
\(73\) −10.1499 −1.18795 −0.593977 0.804482i \(-0.702443\pi\)
−0.593977 + 0.804482i \(0.702443\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.379477 + 0.657274i 0.0432455 + 0.0749033i
\(78\) 0 0
\(79\) −7.62453 + 13.2061i −0.857827 + 1.48580i 0.0161711 + 0.999869i \(0.494852\pi\)
−0.873998 + 0.485930i \(0.838481\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.18909 14.1839i 0.898869 1.55689i 0.0699272 0.997552i \(-0.477723\pi\)
0.828942 0.559335i \(-0.188943\pi\)
\(84\) 0 0
\(85\) 8.58671 + 14.8726i 0.931359 + 1.61316i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.63151 1.02094 0.510469 0.859896i \(-0.329472\pi\)
0.510469 + 0.859896i \(0.329472\pi\)
\(90\) 0 0
\(91\) 2.22829 0.233588
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.67094 2.89416i −0.171435 0.296935i
\(96\) 0 0
\(97\) 5.46718 9.46943i 0.555108 0.961474i −0.442788 0.896627i \(-0.646010\pi\)
0.997895 0.0648479i \(-0.0206562\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.750417 1.29976i 0.0746693 0.129331i −0.826273 0.563270i \(-0.809543\pi\)
0.900942 + 0.433939i \(0.142877\pi\)
\(102\) 0 0
\(103\) −5.13229 8.88938i −0.505699 0.875897i −0.999978 0.00659356i \(-0.997901\pi\)
0.494279 0.869303i \(-0.335432\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.84428 0.758335 0.379168 0.925328i \(-0.376210\pi\)
0.379168 + 0.925328i \(0.376210\pi\)
\(108\) 0 0
\(109\) 12.1754 1.16619 0.583096 0.812403i \(-0.301841\pi\)
0.583096 + 0.812403i \(0.301841\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.98586 + 3.43962i 0.186814 + 0.323572i 0.944186 0.329412i \(-0.106850\pi\)
−0.757372 + 0.652984i \(0.773517\pi\)
\(114\) 0 0
\(115\) −8.57117 + 14.8457i −0.799266 + 1.38437i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.52452 + 6.10465i −0.323092 + 0.559612i
\(120\) 0 0
\(121\) 5.21199 + 9.02744i 0.473818 + 0.820676i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.90238 0.885696
\(126\) 0 0
\(127\) 15.7579 1.39828 0.699142 0.714983i \(-0.253565\pi\)
0.699142 + 0.714983i \(0.253565\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.93789 10.2847i −0.518796 0.898581i −0.999761 0.0218413i \(-0.993047\pi\)
0.480966 0.876739i \(-0.340286\pi\)
\(132\) 0 0
\(133\) 0.685860 1.18794i 0.0594716 0.103008i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.1869 + 17.6443i −0.870328 + 1.50745i −0.00867008 + 0.999962i \(0.502760\pi\)
−0.861658 + 0.507490i \(0.830574\pi\)
\(138\) 0 0
\(139\) 3.13489 + 5.42979i 0.265898 + 0.460549i 0.967799 0.251726i \(-0.0809982\pi\)
−0.701900 + 0.712275i \(0.747665\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.69117 −0.141423
\(144\) 0 0
\(145\) 2.03738 0.169195
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.45679 4.25529i −0.201268 0.348607i 0.747669 0.664071i \(-0.231173\pi\)
−0.948937 + 0.315465i \(0.897840\pi\)
\(150\) 0 0
\(151\) 2.80878 4.86494i 0.228575 0.395903i −0.728811 0.684715i \(-0.759927\pi\)
0.957386 + 0.288811i \(0.0932601\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.646423 + 1.11964i −0.0519220 + 0.0899315i
\(156\) 0 0
\(157\) −10.3174 17.8702i −0.823416 1.42620i −0.903124 0.429379i \(-0.858732\pi\)
0.0797087 0.996818i \(-0.474601\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −7.03629 −0.554537
\(162\) 0 0
\(163\) 4.95096 0.387789 0.193895 0.981022i \(-0.437888\pi\)
0.193895 + 0.981022i \(0.437888\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.485864 0.841542i −0.0375973 0.0651205i 0.846614 0.532207i \(-0.178637\pi\)
−0.884212 + 0.467086i \(0.845304\pi\)
\(168\) 0 0
\(169\) 4.01736 6.95828i 0.309028 0.535252i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.90960 + 15.4319i −0.677384 + 1.17326i 0.298382 + 0.954447i \(0.403553\pi\)
−0.975766 + 0.218817i \(0.929780\pi\)
\(174\) 0 0
\(175\) −0.467722 0.810117i −0.0353564 0.0612391i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.6065 0.792764 0.396382 0.918086i \(-0.370266\pi\)
0.396382 + 0.918086i \(0.370266\pi\)
\(180\) 0 0
\(181\) 24.7360 1.83862 0.919308 0.393539i \(-0.128749\pi\)
0.919308 + 0.393539i \(0.128749\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.51898 + 9.55915i 0.405763 + 0.702802i
\(186\) 0 0
\(187\) 2.67495 4.63315i 0.195612 0.338810i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.25120 16.0235i 0.669393 1.15942i −0.308681 0.951166i \(-0.599888\pi\)
0.978074 0.208257i \(-0.0667791\pi\)
\(192\) 0 0
\(193\) −8.24505 14.2808i −0.593492 1.02796i −0.993758 0.111559i \(-0.964416\pi\)
0.400266 0.916399i \(-0.368918\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.68164 0.618541 0.309271 0.950974i \(-0.399915\pi\)
0.309271 + 0.950974i \(0.399915\pi\)
\(198\) 0 0
\(199\) −19.2352 −1.36355 −0.681774 0.731563i \(-0.738791\pi\)
−0.681774 + 0.731563i \(0.738791\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.418134 + 0.724229i 0.0293472 + 0.0508309i
\(204\) 0 0
\(205\) −10.7891 + 18.6873i −0.753543 + 1.30518i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.520536 + 0.901595i −0.0360063 + 0.0623647i
\(210\) 0 0
\(211\) −11.3584 19.6734i −0.781946 1.35437i −0.930807 0.365512i \(-0.880894\pi\)
0.148861 0.988858i \(-0.452439\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −18.0363 −1.23006
\(216\) 0 0
\(217\) −0.530665 −0.0360239
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.85365 13.6029i −0.528294 0.915032i
\(222\) 0 0
\(223\) −7.76320 + 13.4462i −0.519862 + 0.900427i 0.479871 + 0.877339i \(0.340683\pi\)
−0.999733 + 0.0230886i \(0.992650\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.13468 1.96533i 0.0753114 0.130443i −0.825910 0.563802i \(-0.809338\pi\)
0.901222 + 0.433358i \(0.142672\pi\)
\(228\) 0 0
\(229\) −4.61037 7.98540i −0.304662 0.527690i 0.672524 0.740075i \(-0.265210\pi\)
−0.977186 + 0.212385i \(0.931877\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.0842 1.57781 0.788905 0.614515i \(-0.210648\pi\)
0.788905 + 0.614515i \(0.210648\pi\)
\(234\) 0 0
\(235\) −16.5551 −1.07993
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.50000 + 4.33013i 0.161712 + 0.280093i 0.935483 0.353373i \(-0.114965\pi\)
−0.773771 + 0.633465i \(0.781632\pi\)
\(240\) 0 0
\(241\) 8.61761 14.9261i 0.555109 0.961477i −0.442786 0.896627i \(-0.646010\pi\)
0.997895 0.0648494i \(-0.0206567\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.21814 2.10988i 0.0778240 0.134795i
\(246\) 0 0
\(247\) 1.52829 + 2.64708i 0.0972430 + 0.168430i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −15.2725 −0.963994 −0.481997 0.876173i \(-0.660088\pi\)
−0.481997 + 0.876173i \(0.660088\pi\)
\(252\) 0 0
\(253\) 5.34022 0.335737
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.79524 10.0377i −0.361497 0.626131i 0.626710 0.779252i \(-0.284401\pi\)
−0.988207 + 0.153121i \(0.951068\pi\)
\(258\) 0 0
\(259\) −2.26533 + 3.92367i −0.140761 + 0.243805i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4.05602 + 7.02523i −0.250105 + 0.433194i −0.963554 0.267512i \(-0.913798\pi\)
0.713450 + 0.700707i \(0.247132\pi\)
\(264\) 0 0
\(265\) 0.740601 + 1.28276i 0.0454948 + 0.0787992i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −11.4598 −0.698717 −0.349358 0.936989i \(-0.613600\pi\)
−0.349358 + 0.936989i \(0.613600\pi\)
\(270\) 0 0
\(271\) −3.30404 −0.200706 −0.100353 0.994952i \(-0.531997\pi\)
−0.100353 + 0.994952i \(0.531997\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.354979 + 0.614842i 0.0214061 + 0.0370764i
\(276\) 0 0
\(277\) 5.12453 8.87595i 0.307903 0.533304i −0.670000 0.742361i \(-0.733706\pi\)
0.977903 + 0.209057i \(0.0670394\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.861334 + 1.49188i −0.0513829 + 0.0889978i −0.890573 0.454841i \(-0.849696\pi\)
0.839190 + 0.543838i \(0.183030\pi\)
\(282\) 0 0
\(283\) −12.0752 20.9148i −0.717795 1.24326i −0.961872 0.273501i \(-0.911818\pi\)
0.244077 0.969756i \(-0.421515\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.85704 −0.522814
\(288\) 0 0
\(289\) 32.6890 1.92288
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7.46533 12.9303i −0.436129 0.755398i 0.561258 0.827641i \(-0.310318\pi\)
−0.997387 + 0.0722432i \(0.976984\pi\)
\(294\) 0 0
\(295\) 1.41759 2.45533i 0.0825351 0.142955i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.83944 13.5783i 0.453367 0.785254i
\(300\) 0 0
\(301\) −3.70161 6.41137i −0.213357 0.369545i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −28.5376 −1.63406
\(306\) 0 0
\(307\) −17.9563 −1.02482 −0.512411 0.858741i \(-0.671247\pi\)
−0.512411 + 0.858741i \(0.671247\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.8931 + 18.8673i 0.617689 + 1.06987i 0.989906 + 0.141723i \(0.0452643\pi\)
−0.372217 + 0.928146i \(0.621402\pi\)
\(312\) 0 0
\(313\) −11.0624 + 19.1606i −0.625284 + 1.08302i 0.363202 + 0.931710i \(0.381683\pi\)
−0.988486 + 0.151313i \(0.951650\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.86480 4.96197i 0.160903 0.278692i −0.774290 0.632831i \(-0.781893\pi\)
0.935193 + 0.354139i \(0.115226\pi\)
\(318\) 0 0
\(319\) −0.317344 0.549657i −0.0177679 0.0307749i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −9.66931 −0.538015
\(324\) 0 0
\(325\) 2.08444 0.115624
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.39762 5.88485i −0.187317 0.324442i
\(330\) 0 0
\(331\) 15.8178 27.3973i 0.869427 1.50589i 0.00684339 0.999977i \(-0.497822\pi\)
0.862583 0.505915i \(-0.168845\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.372266 + 0.644784i −0.0203391 + 0.0352283i
\(336\) 0 0
\(337\) −11.7771 20.3985i −0.641539 1.11118i −0.985089 0.172044i \(-0.944963\pi\)
0.343550 0.939134i \(-0.388371\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.402751 0.0218102
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.81359 16.9976i −0.526821 0.912481i −0.999512 0.0312526i \(-0.990050\pi\)
0.472690 0.881229i \(-0.343283\pi\)
\(348\) 0 0
\(349\) 13.0363 22.5795i 0.697816 1.20865i −0.271406 0.962465i \(-0.587488\pi\)
0.969222 0.246188i \(-0.0791782\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.30878 5.73097i 0.176108 0.305029i −0.764436 0.644700i \(-0.776982\pi\)
0.940544 + 0.339671i \(0.110316\pi\)
\(354\) 0 0
\(355\) −15.3720 26.6250i −0.815859 1.41311i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 23.0527 1.21668 0.608338 0.793678i \(-0.291836\pi\)
0.608338 + 0.793678i \(0.291836\pi\)
\(360\) 0 0
\(361\) −17.1184 −0.900968
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −12.3640 21.4150i −0.647159 1.12091i
\(366\) 0 0
\(367\) −4.08931 + 7.08289i −0.213460 + 0.369724i −0.952795 0.303614i \(-0.901807\pi\)
0.739335 + 0.673338i \(0.235140\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −0.303989 + 0.526524i −0.0157823 + 0.0273358i
\(372\) 0 0
\(373\) −7.75341 13.4293i −0.401456 0.695343i 0.592446 0.805610i \(-0.298163\pi\)
−0.993902 + 0.110268i \(0.964829\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.86345 −0.0959723
\(378\) 0 0
\(379\) 18.5814 0.954461 0.477231 0.878778i \(-0.341641\pi\)
0.477231 + 0.878778i \(0.341641\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.30959 + 14.3926i 0.424600 + 0.735429i 0.996383 0.0849761i \(-0.0270814\pi\)
−0.571783 + 0.820405i \(0.693748\pi\)
\(384\) 0 0
\(385\) −0.924512 + 1.60130i −0.0471175 + 0.0816099i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −12.9112 + 22.3629i −0.654624 + 1.13384i 0.327364 + 0.944898i \(0.393840\pi\)
−0.981988 + 0.188944i \(0.939494\pi\)
\(390\) 0 0
\(391\) 24.7995 + 42.9541i 1.25417 + 2.17228i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −37.1509 −1.86927
\(396\) 0 0
\(397\) −0.729016 −0.0365883 −0.0182941 0.999833i \(-0.505824\pi\)
−0.0182941 + 0.999833i \(0.505824\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.67094 15.0185i −0.433006 0.749989i 0.564124 0.825690i \(-0.309214\pi\)
−0.997131 + 0.0757010i \(0.975881\pi\)
\(402\) 0 0
\(403\) 0.591238 1.02405i 0.0294516 0.0510117i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.71928 2.97789i 0.0852218 0.147608i
\(408\) 0 0
\(409\) 1.82351 + 3.15841i 0.0901668 + 0.156174i 0.907581 0.419877i \(-0.137927\pi\)
−0.817414 + 0.576050i \(0.804593\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.16373 0.0572635
\(414\) 0 0
\(415\) 39.9018 1.95870
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.117685 + 0.203836i 0.00574927 + 0.00995803i 0.868886 0.495013i \(-0.164837\pi\)
−0.863136 + 0.504971i \(0.831503\pi\)
\(420\) 0 0
\(421\) −10.8985 + 18.8767i −0.531158 + 0.919993i 0.468181 + 0.883633i \(0.344910\pi\)
−0.999339 + 0.0363601i \(0.988424\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.29699 + 5.71055i −0.159927 + 0.277002i
\(426\) 0 0
\(427\) −5.85680 10.1443i −0.283431 0.490916i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.28806 0.206549 0.103274 0.994653i \(-0.467068\pi\)
0.103274 + 0.994653i \(0.467068\pi\)
\(432\) 0 0
\(433\) −13.9263 −0.669257 −0.334628 0.942350i \(-0.608611\pi\)
−0.334628 + 0.942350i \(0.608611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.82591 8.35871i −0.230854 0.399851i
\(438\) 0 0
\(439\) 15.9397 27.6084i 0.760762 1.31768i −0.181696 0.983355i \(-0.558159\pi\)
0.942458 0.334324i \(-0.108508\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.56216 4.43780i 0.121732 0.210846i −0.798719 0.601705i \(-0.794488\pi\)
0.920451 + 0.390858i \(0.127822\pi\)
\(444\) 0 0
\(445\) 11.7325 + 20.3213i 0.556174 + 0.963322i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 33.9914 1.60415 0.802077 0.597221i \(-0.203728\pi\)
0.802077 + 0.597221i \(0.203728\pi\)
\(450\) 0 0
\(451\) 6.72209 0.316531
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.71436 + 4.70142i 0.127251 + 0.220406i
\(456\) 0 0
\(457\) −2.78826 + 4.82941i −0.130429 + 0.225910i −0.923842 0.382774i \(-0.874969\pi\)
0.793413 + 0.608684i \(0.208302\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.24904 2.16340i 0.0581734 0.100759i −0.835472 0.549533i \(-0.814806\pi\)
0.893646 + 0.448774i \(0.148139\pi\)
\(462\) 0 0
\(463\) 4.77517 + 8.27084i 0.221921 + 0.384378i 0.955391 0.295343i \(-0.0954340\pi\)
−0.733470 + 0.679722i \(0.762101\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18.2128 −0.842787 −0.421393 0.906878i \(-0.638459\pi\)
−0.421393 + 0.906878i \(0.638459\pi\)
\(468\) 0 0
\(469\) −0.305602 −0.0141114
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.80935 + 4.86594i 0.129174 + 0.223736i
\(474\) 0 0
\(475\) 0.641583 1.11125i 0.0294378 0.0509878i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.87648 + 4.98222i −0.131430 + 0.227643i −0.924228 0.381841i \(-0.875290\pi\)
0.792798 + 0.609484i \(0.208623\pi\)
\(480\) 0 0
\(481\) −5.04782 8.74308i −0.230161 0.398650i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 26.6391 1.20962
\(486\) 0 0
\(487\) 23.8918 1.08264 0.541321 0.840816i \(-0.317925\pi\)
0.541321 + 0.840816i \(0.317925\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.31437 + 12.6689i 0.330093 + 0.571738i 0.982530 0.186105i \(-0.0595865\pi\)
−0.652437 + 0.757843i \(0.726253\pi\)
\(492\) 0 0
\(493\) 2.94744 5.10512i 0.132746 0.229923i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.30961 10.9286i 0.283025 0.490213i
\(498\) 0 0
\(499\) 11.9579 + 20.7117i 0.535308 + 0.927181i 0.999148 + 0.0412621i \(0.0131379\pi\)
−0.463840 + 0.885919i \(0.653529\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −7.57496 −0.337751 −0.168875 0.985637i \(-0.554014\pi\)
−0.168875 + 0.985637i \(0.554014\pi\)
\(504\) 0 0
\(505\) 3.65645 0.162710
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.50078 6.06353i −0.155169 0.268761i 0.777951 0.628325i \(-0.216259\pi\)
−0.933121 + 0.359563i \(0.882926\pi\)
\(510\) 0 0
\(511\) 5.07494 8.79006i 0.224502 0.388849i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.5037 21.6570i 0.550978 0.954321i
\(516\) 0 0
\(517\) 2.57864 + 4.46633i 0.113408 + 0.196429i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −42.4564 −1.86005 −0.930025 0.367497i \(-0.880215\pi\)
−0.930025 + 0.367497i \(0.880215\pi\)
\(522\) 0 0
\(523\) −18.6096 −0.813743 −0.406871 0.913485i \(-0.633380\pi\)
−0.406871 + 0.913485i \(0.633380\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.87034 + 3.23952i 0.0814733 + 0.141116i
\(528\) 0 0
\(529\) −13.2547 + 22.9577i −0.576289 + 0.998163i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9.86802 17.0919i 0.427431 0.740333i
\(534\) 0 0
\(535\) 9.55542 + 16.5505i 0.413117 + 0.715539i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.758955 −0.0326905
\(540\) 0 0
\(541\) −0.694443 −0.0298564 −0.0149282 0.999889i \(-0.504752\pi\)
−0.0149282 + 0.999889i \(0.504752\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 14.8313 + 25.6886i 0.635304 + 1.10038i
\(546\) 0 0
\(547\) −8.77655 + 15.2014i −0.375258 + 0.649966i −0.990366 0.138477i \(-0.955779\pi\)
0.615107 + 0.788443i \(0.289113\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.573562 + 0.993439i −0.0244346 + 0.0423219i
\(552\) 0 0
\(553\) −7.62453 13.2061i −0.324228 0.561579i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14.2666 −0.604495 −0.302248 0.953229i \(-0.597737\pi\)
−0.302248 + 0.953229i \(0.597737\pi\)
\(558\) 0 0
\(559\) 16.4965 0.697728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −16.8742 29.2270i −0.711162 1.23177i −0.964421 0.264371i \(-0.914836\pi\)
0.253259 0.967399i \(-0.418498\pi\)
\(564\) 0 0
\(565\) −4.83811 + 8.37986i −0.203541 + 0.352543i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 19.9867 34.6180i 0.837886 1.45126i −0.0537729 0.998553i \(-0.517125\pi\)
0.891659 0.452708i \(-0.149542\pi\)
\(570\) 0 0
\(571\) −20.2266 35.0335i −0.846457 1.46611i −0.884350 0.466824i \(-0.845398\pi\)
0.0378934 0.999282i \(-0.487935\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.58204 −0.274490
\(576\) 0 0
\(577\) −31.6758 −1.31868 −0.659340 0.751845i \(-0.729164\pi\)
−0.659340 + 0.751845i \(0.729164\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.18909 + 14.1839i 0.339741 + 0.588448i
\(582\) 0 0
\(583\) 0.230714 0.399608i 0.00955519 0.0165501i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.65220 13.2540i 0.315840 0.547051i −0.663776 0.747932i \(-0.731047\pi\)
0.979616 + 0.200881i \(0.0643803\pi\)
\(588\) 0 0
\(589\) −0.363962 0.630400i −0.0149968 0.0259752i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −39.2583 −1.61214 −0.806072 0.591817i \(-0.798411\pi\)
−0.806072 + 0.591817i \(0.798411\pi\)
\(594\) 0 0
\(595\) −17.1734 −0.704041
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.77972 + 15.2069i 0.358730 + 0.621338i 0.987749 0.156052i \(-0.0498766\pi\)
−0.629019 + 0.777390i \(0.716543\pi\)
\(600\) 0 0
\(601\) −9.12937 + 15.8125i −0.372395 + 0.645007i −0.989933 0.141534i \(-0.954797\pi\)
0.617539 + 0.786541i \(0.288130\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −12.6979 + 21.9933i −0.516241 + 0.894156i
\(606\) 0 0
\(607\) 3.71415 + 6.43310i 0.150753 + 0.261112i 0.931504 0.363730i \(-0.118497\pi\)
−0.780752 + 0.624842i \(0.785164\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 15.1418 0.612570
\(612\) 0 0
\(613\) −39.8772 −1.61062 −0.805312 0.592851i \(-0.798002\pi\)
−0.805312 + 0.592851i \(0.798002\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.187473 + 0.324713i 0.00754739 + 0.0130725i 0.869774 0.493450i \(-0.164264\pi\)
−0.862227 + 0.506522i \(0.830931\pi\)
\(618\) 0 0
\(619\) 6.95166 12.0406i 0.279411 0.483954i −0.691828 0.722063i \(-0.743194\pi\)
0.971238 + 0.238109i \(0.0765275\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.81575 + 8.34113i −0.192939 + 0.334180i
\(624\) 0 0
\(625\) 14.4011 + 24.9434i 0.576043 + 0.997736i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 31.9368 1.27341
\(630\) 0 0
\(631\) 28.6049 1.13874 0.569372 0.822080i \(-0.307187\pi\)
0.569372 + 0.822080i \(0.307187\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 19.1953 + 33.2472i 0.761740 + 1.31937i
\(636\) 0 0
\(637\) −1.11414 + 1.92976i −0.0441440 + 0.0764597i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.9011 + 18.8812i −0.430566 + 0.745762i −0.996922 0.0783986i \(-0.975019\pi\)
0.566356 + 0.824161i \(0.308353\pi\)
\(642\) 0 0
\(643\) 20.9205 + 36.2354i 0.825025 + 1.42899i 0.901900 + 0.431945i \(0.142172\pi\)
−0.0768751 + 0.997041i \(0.524494\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17.2614 −0.678615 −0.339308 0.940675i \(-0.610193\pi\)
−0.339308 + 0.940675i \(0.610193\pi\)
\(648\) 0 0
\(649\) −0.883220 −0.0346694
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.237661 + 0.411641i 0.00930039 + 0.0161087i 0.870638 0.491924i \(-0.163706\pi\)
−0.861338 + 0.508033i \(0.830373\pi\)
\(654\) 0 0
\(655\) 14.4663 25.0564i 0.565247 0.979036i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 18.2344 31.5828i 0.710310 1.23029i −0.254431 0.967091i \(-0.581888\pi\)
0.964741 0.263201i \(-0.0847784\pi\)
\(660\) 0 0
\(661\) −5.04982 8.74655i −0.196415 0.340201i 0.750948 0.660361i \(-0.229597\pi\)
−0.947364 + 0.320160i \(0.896263\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.34189 0.129593
\(666\) 0 0
\(667\) 5.88422 0.227838
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.44505 + 7.69905i 0.171599 + 0.297218i
\(672\) 0 0
\(673\) 9.51044 16.4726i 0.366600 0.634971i −0.622431 0.782675i \(-0.713855\pi\)
0.989032 + 0.147704i \(0.0471883\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −22.6374 + 39.2091i −0.870024 + 1.50693i −0.00805435 + 0.999968i \(0.502564\pi\)
−0.861970 + 0.506959i \(0.830770\pi\)
\(678\) 0 0
\(679\) 5.46718 + 9.46943i 0.209811 + 0.363403i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 45.5318 1.74222 0.871112 0.491084i \(-0.163399\pi\)
0.871112 + 0.491084i \(0.163399\pi\)
\(684\) 0 0
\(685\) −49.6363 −1.89651
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.677375 1.17325i −0.0258059 0.0446972i
\(690\) 0 0
\(691\) 8.09446 14.0200i 0.307928 0.533347i −0.669981 0.742378i \(-0.733698\pi\)
0.977909 + 0.209031i \(0.0670311\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.63746 + 13.2285i −0.289706 + 0.501785i
\(696\) 0 0
\(697\) 31.2168 + 54.0691i 1.18242 + 2.04801i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.61120 0.0986238 0.0493119 0.998783i \(-0.484297\pi\)
0.0493119 + 0.998783i \(0.484297\pi\)
\(702\) 0 0
\(703\) −6.21480 −0.234396
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.750417 + 1.29976i 0.0282223 + 0.0488825i
\(708\) 0 0
\(709\) −23.1702 + 40.1319i −0.870175 + 1.50719i −0.00835896 + 0.999965i \(0.502661\pi\)
−0.861816 + 0.507222i \(0.830673\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.86696 + 3.23366i −0.0699180 + 0.121102i
\(714\) 0 0
\(715\) −2.06008 3.56816i −0.0770426 0.133442i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 46.2069 1.72323 0.861614 0.507564i \(-0.169454\pi\)
0.861614 + 0.507564i \(0.169454\pi\)
\(720\) 0 0
\(721\) 10.2646 0.382273
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.391140 + 0.677475i 0.0145266 + 0.0251608i
\(726\) 0 0
\(727\) 9.72988 16.8526i 0.360861 0.625030i −0.627242 0.778825i \(-0.715816\pi\)
0.988103 + 0.153795i \(0.0491494\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −26.0928 + 45.1940i −0.965077 + 1.67156i
\(732\) 0 0
\(733\) −7.68617 13.3128i −0.283895 0.491721i 0.688445 0.725288i \(-0.258293\pi\)
−0.972341 + 0.233567i \(0.924960\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.231938 0.00854356
\(738\) 0 0
\(739\) −32.4962 −1.19539 −0.597696 0.801723i \(-0.703917\pi\)
−0.597696 + 0.801723i \(0.703917\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 20.8664 + 36.1417i 0.765514 + 1.32591i 0.939974 + 0.341245i \(0.110849\pi\)
−0.174460 + 0.984664i \(0.555818\pi\)
\(744\) 0 0
\(745\) 5.98542 10.3671i 0.219289 0.379819i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3.92214 + 6.79335i −0.143312 + 0.248224i
\(750\) 0 0
\(751\) −5.02452 8.70273i −0.183347 0.317567i 0.759671 0.650308i \(-0.225360\pi\)
−0.943018 + 0.332741i \(0.892027\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 13.6859 0.498081
\(756\) 0 0
\(757\) −31.7989 −1.15575 −0.577876 0.816125i \(-0.696118\pi\)
−0.577876 + 0.816125i \(0.696118\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −27.4352 47.5192i −0.994526 1.72257i −0.587751 0.809042i \(-0.699987\pi\)
−0.406775 0.913528i \(-0.633347\pi\)
\(762\) 0 0
\(763\) −6.08770 + 10.5442i −0.220389 + 0.381726i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.29657 + 2.24572i −0.0468163 + 0.0810882i
\(768\) 0 0
\(769\) 5.39041 + 9.33646i 0.194383 + 0.336681i 0.946698 0.322122i \(-0.104396\pi\)
−0.752315 + 0.658804i \(0.771063\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 47.4885 1.70804 0.854022 0.520237i \(-0.174156\pi\)
0.854022 + 0.520237i \(0.174156\pi\)
\(774\) 0 0
\(775\) −0.496407 −0.0178315
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.07469 10.5217i −0.217648 0.376978i
\(780\) 0 0
\(781\) −4.78871 + 8.29428i −0.171353 + 0.296793i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 25.1360 43.5368i 0.897141 1.55389i
\(786\) 0 0
\(787\) 12.0563 + 20.8820i 0.429759 + 0.744364i 0.996852 0.0792896i \(-0.0252652\pi\)
−0.567093 + 0.823654i \(0.691932\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.97173 −0.141218
\(792\) 0 0
\(793\) 26.1013 0.926885
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7.54620 13.0704i −0.267300 0.462978i 0.700863 0.713295i \(-0.252798\pi\)
−0.968164 + 0.250318i \(0.919465\pi\)
\(798\) 0 0
\(799\) −23.9500 + 41.4826i −0.847289 + 1.46755i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.85165 + 6.67125i −0.135922 + 0.235423i
\(804\) 0 0
\(805\) −8.57117 14.8457i −0.302094 0.523242i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −20.1104 −0.707043 −0.353522 0.935426i \(-0.615016\pi\)
−0.353522 + 0.935426i \(0.615016\pi\)
\(810\) 0 0
\(811\) −40.9770 −1.43890 −0.719448 0.694546i \(-0.755605\pi\)
−0.719448 + 0.694546i \(0.755605\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.03095 + 10.4459i 0.211255 + 0.365904i
\(816\) 0 0
\(817\) 5.07757 8.79461i 0.177642 0.307684i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −11.9427 + 20.6854i −0.416803 + 0.721925i −0.995616 0.0935360i \(-0.970183\pi\)
0.578813 + 0.815461i \(0.303516\pi\)
\(822\) 0 0
\(823\) −14.9451 25.8857i −0.520954 0.902319i −0.999703 0.0243673i \(-0.992243\pi\)
0.478749 0.877952i \(-0.341090\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 44.7265 1.55529 0.777647 0.628702i \(-0.216413\pi\)
0.777647 + 0.628702i \(0.216413\pi\)
\(828\) 0 0
\(829\) 36.8943 1.28139 0.640695 0.767795i \(-0.278646\pi\)
0.640695 + 0.767795i \(0.278646\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.52452 6.10465i −0.122117 0.211514i
\(834\) 0 0
\(835\) 1.18370 2.05023i 0.0409636 0.0709511i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0.918446 1.59080i 0.0317083 0.0549204i −0.849736 0.527209i \(-0.823239\pi\)
0.881444 + 0.472288i \(0.156572\pi\)
\(840\) 0 0
\(841\) 14.1503 + 24.5091i 0.487942 + 0.845141i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 19.5748 0.673394
\(846\) 0 0
\(847\) −10.4240 −0.358172
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 15.9395 + 27.6081i 0.546400 + 0.946392i
\(852\) 0 0
\(853\) 26.0385 45.1000i 0.891542 1.54420i 0.0535152 0.998567i \(-0.482957\pi\)
0.838027 0.545629i \(-0.183709\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9.46663 + 16.3967i −0.323374 + 0.560100i −0.981182 0.193086i \(-0.938150\pi\)
0.657808 + 0.753186i \(0.271484\pi\)
\(858\) 0 0
\(859\) 4.00700 + 6.94033i 0.136717 + 0.236801i 0.926252 0.376905i \(-0.123012\pi\)
−0.789535 + 0.613706i \(0.789678\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 23.8647 0.812364 0.406182 0.913792i \(-0.366860\pi\)
0.406182 + 0.913792i \(0.366860\pi\)
\(864\) 0 0
\(865\) −43.4125 −1.47607
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 5.78667 + 10.0228i 0.196299 + 0.340001i
\(870\) 0 0
\(871\) 0.340485 0.589738i 0.0115369 0.0199825i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.95119 + 8.57572i −0.167381 + 0.289912i
\(876\) 0 0
\(877\) 17.0238 + 29.4861i 0.574853 + 0.995674i 0.996058 + 0.0887086i \(0.0282740\pi\)
−0.421205 + 0.906965i \(0.638393\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 38.8018 1.30727 0.653633 0.756812i \(-0.273244\pi\)
0.653633 + 0.756812i \(0.273244\pi\)
\(882\) 0 0
\(883\) −42.0944 −1.41659 −0.708294 0.705917i \(-0.750535\pi\)
−0.708294 + 0.705917i \(0.750535\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.1168 + 20.9869i 0.406841 + 0.704670i 0.994534 0.104414i \(-0.0332968\pi\)
−0.587692 + 0.809084i \(0.699964\pi\)
\(888\) 0 0
\(889\) −7.87893 + 13.6467i −0.264251 + 0.457696i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4.66058 8.07236i 0.155960 0.270131i
\(894\) 0 0
\(895\) 12.9201 + 22.3783i 0.431873 + 0.748025i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.443778 0.0148008
\(900\) 0 0
\(901\) 4.28566 0.142776
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 30.1319 + 52.1900i 1.00162 + 1.73485i
\(906\) 0 0
\(907\) 12.9424 22.4170i 0.429747 0.744343i −0.567104 0.823646i \(-0.691936\pi\)
0.996851 + 0.0793031i \(0.0252695\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −21.9571 + 38.0308i −0.727471 + 1.26002i 0.230478 + 0.973078i \(0.425971\pi\)
−0.957949 + 0.286939i \(0.907362\pi\)
\(912\) 0 0
\(913\) −6.21514 10.7649i −0.205691 0.356268i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11.8758 0.392173
\(918\) 0 0
\(919\) 41.4335 1.36677 0.683383 0.730060i \(-0.260508\pi\)
0.683383 + 0.730060i \(0.260508\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 14.0596 + 24.3520i 0.462779 + 0.801556i
\(924\) 0 0
\(925\) −2.11909 + 3.67037i −0.0696752 + 0.120681i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.56911 + 2.71778i −0.0514808 + 0.0891674i −0.890617 0.454753i \(-0.849727\pi\)
0.839137 + 0.543921i \(0.183061\pi\)
\(930\) 0 0
\(931\) 0.685860 + 1.18794i 0.0224781 + 0.0389333i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 13.0338 0.426252
\(936\) 0 0
\(937\) 10.2810 0.335866 0.167933 0.985798i \(-0.446291\pi\)
0.167933 + 0.985798i \(0.446291\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −7.23844 12.5373i −0.235966 0.408706i 0.723587 0.690234i \(-0.242492\pi\)
−0.959553 + 0.281528i \(0.909159\pi\)
\(942\) 0 0
\(943\) −31.1603 + 53.9713i −1.01472 + 1.75755i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.8714 25.7580i 0.483256 0.837024i −0.516559 0.856251i \(-0.672787\pi\)
0.999815 + 0.0192278i \(0.00612078\pi\)
\(948\) 0 0
\(949\) 11.3084 + 19.5868i 0.367088 + 0.635814i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −19.4869 −0.631242 −0.315621 0.948885i \(-0.602213\pi\)
−0.315621 + 0.948885i \(0.602213\pi\)
\(954\) 0 0
\(955\) 45.0769 1.45866
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.1869 17.6443i −0.328953 0.569763i
\(960\) 0 0
\(961\) 15.3592 26.6029i 0.495458 0.858158i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 20.0872 34.7921i 0.646631 1.12000i
\(966\) 0 0
\(967\) 6.94942 + 12.0368i 0.223478 + 0.387076i 0.955862 0.293817i \(-0.0949255\pi\)
−0.732384 + 0.680892i \(0.761592\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −19.2406 −0.617460 −0.308730 0.951150i \(-0.599904\pi\)
−0.308730 + 0.951150i \(0.599904\pi\)
\(972\) 0 0
\(973\) −6.26978 −0.201000
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.44479 + 4.23450i 0.0782158 + 0.135474i 0.902480 0.430731i \(-0.141744\pi\)
−0.824264 + 0.566205i \(0.808411\pi\)
\(978\) 0 0
\(979\) 3.65494 6.33054i 0.116812 0.202325i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −14.9670 + 25.9236i −0.477373 + 0.826834i −0.999664 0.0259332i \(-0.991744\pi\)
0.522291 + 0.852768i \(0.325078\pi\)
\(984\) 0 0
\(985\) 10.5754 + 18.3172i 0.336962 + 0.583634i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −52.0912 −1.65640
\(990\) 0 0
\(991\) −9.36157 −0.297380 −0.148690 0.988884i \(-0.547506\pi\)
−0.148690 + 0.988884i \(0.547506\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −23.4311 40.5839i −0.742817 1.28660i
\(996\) 0 0
\(997\) 23.8210 41.2592i 0.754420 1.30669i −0.191242 0.981543i \(-0.561252\pi\)
0.945662 0.325151i \(-0.105415\pi\)
\(998\) 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 1512.2.r.d.505.4 8
3.2 odd 2 504.2.r.d.169.3 8
4.3 odd 2 3024.2.r.l.2017.4 8
9.2 odd 6 4536.2.a.x.1.4 4
9.4 even 3 inner 1512.2.r.d.1009.4 8
9.5 odd 6 504.2.r.d.337.3 yes 8
9.7 even 3 4536.2.a.ba.1.1 4
12.11 even 2 1008.2.r.m.673.2 8
36.7 odd 6 9072.2.a.cl.1.1 4
36.11 even 6 9072.2.a.ce.1.4 4
36.23 even 6 1008.2.r.m.337.2 8
36.31 odd 6 3024.2.r.l.1009.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.d.169.3 8 3.2 odd 2
504.2.r.d.337.3 yes 8 9.5 odd 6
1008.2.r.m.337.2 8 36.23 even 6
1008.2.r.m.673.2 8 12.11 even 2
1512.2.r.d.505.4 8 1.1 even 1 trivial
1512.2.r.d.1009.4 8 9.4 even 3 inner
3024.2.r.l.1009.4 8 36.31 odd 6
3024.2.r.l.2017.4 8 4.3 odd 2
4536.2.a.x.1.4 4 9.2 odd 6
4536.2.a.ba.1.1 4 9.7 even 3
9072.2.a.ce.1.4 4 36.11 even 6
9072.2.a.cl.1.1 4 36.7 odd 6