Properties

Label 1792.2.m.d.1345.3
Level $1792$
Weight $2$
Character 1792.1345
Analytic conductor $14.309$
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] = [1792,2,Mod(449,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.m (of order \(4\), degree \(2\), 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: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1345.3
Root \(0.500000 + 1.44392i\) of defining polynomial
Character \(\chi\) \(=\) 1792.1345
Dual form 1792.2.m.d.449.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.23681 + 1.23681i) q^{3} +(-0.571717 + 0.571717i) q^{5} -1.00000i q^{7} +0.0594122i q^{9} +O(q^{10})\) \(q+(1.23681 + 1.23681i) q^{3} +(-0.571717 + 0.571717i) q^{5} -1.00000i q^{7} +0.0594122i q^{9} +(-4.55765 + 4.55765i) q^{11} +(-3.65103 - 3.65103i) q^{13} -1.41421 q^{15} +2.47363 q^{17} +(-5.54358 - 5.54358i) q^{19} +(1.23681 - 1.23681i) q^{21} -4.08402i q^{23} +4.34628i q^{25} +(3.63696 - 3.63696i) q^{27} +(-4.30205 - 4.30205i) q^{29} -1.02461 q^{31} -11.2739 q^{33} +(0.571717 + 0.571717i) q^{35} +(5.91245 - 5.91245i) q^{37} -9.03127i q^{39} -5.47010i q^{41} +(-4.35480 + 4.35480i) q^{43} +(-0.0339670 - 0.0339670i) q^{45} +6.47363 q^{47} -1.00000 q^{49} +(3.05941 + 3.05941i) q^{51} +(-1.94059 + 1.94059i) q^{53} -5.21137i q^{55} -13.7127i q^{57} +(-6.90191 + 6.90191i) q^{59} +(-7.84563 - 7.84563i) q^{61} +0.0594122 q^{63} +4.17471 q^{65} +(3.88118 + 3.88118i) q^{67} +(5.05117 - 5.05117i) q^{69} +9.44196i q^{71} -1.06608i q^{73} +(-5.37553 + 5.37553i) q^{75} +(4.55765 + 4.55765i) q^{77} -12.3400 q^{79} +9.17471 q^{81} +(5.68230 + 5.68230i) q^{83} +(-1.41421 + 1.41421i) q^{85} -10.6417i q^{87} -2.81900i q^{89} +(-3.65103 + 3.65103i) q^{91} +(-1.26725 - 1.26725i) q^{93} +6.33871 q^{95} -1.74754 q^{97} +(-0.270780 - 0.270780i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 4 q^{5} - 8 q^{11} - 12 q^{13} + 8 q^{17} + 4 q^{19} + 4 q^{21} - 8 q^{27} + 8 q^{31} - 16 q^{33} - 4 q^{35} + 8 q^{37} - 24 q^{43} - 12 q^{45} + 40 q^{47} - 8 q^{49} + 24 q^{51} - 16 q^{53} - 52 q^{59} + 20 q^{61} - 24 q^{65} + 32 q^{67} - 8 q^{69} - 28 q^{75} + 8 q^{77} + 16 q^{81} - 12 q^{83} - 12 q^{91} + 40 q^{93} + 80 q^{95} + 72 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 1.23681 + 1.23681i 0.714074 + 0.714074i 0.967385 0.253311i \(-0.0815195\pi\)
−0.253311 + 0.967385i \(0.581520\pi\)
\(4\) 0 0
\(5\) −0.571717 + 0.571717i −0.255680 + 0.255680i −0.823294 0.567615i \(-0.807866\pi\)
0.567615 + 0.823294i \(0.307866\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0.0594122i 0.0198041i
\(10\) 0 0
\(11\) −4.55765 + 4.55765i −1.37418 + 1.37418i −0.520041 + 0.854141i \(0.674083\pi\)
−0.854141 + 0.520041i \(0.825917\pi\)
\(12\) 0 0
\(13\) −3.65103 3.65103i −1.01261 1.01261i −0.999919 0.0126931i \(-0.995960\pi\)
−0.0126931 0.999919i \(-0.504040\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 0 0
\(17\) 2.47363 0.599942 0.299971 0.953948i \(-0.403023\pi\)
0.299971 + 0.953948i \(0.403023\pi\)
\(18\) 0 0
\(19\) −5.54358 5.54358i −1.27178 1.27178i −0.945151 0.326633i \(-0.894086\pi\)
−0.326633 0.945151i \(-0.605914\pi\)
\(20\) 0 0
\(21\) 1.23681 1.23681i 0.269895 0.269895i
\(22\) 0 0
\(23\) 4.08402i 0.851577i −0.904823 0.425789i \(-0.859997\pi\)
0.904823 0.425789i \(-0.140003\pi\)
\(24\) 0 0
\(25\) 4.34628i 0.869256i
\(26\) 0 0
\(27\) 3.63696 3.63696i 0.699933 0.699933i
\(28\) 0 0
\(29\) −4.30205 4.30205i −0.798871 0.798871i 0.184046 0.982918i \(-0.441080\pi\)
−0.982918 + 0.184046i \(0.941080\pi\)
\(30\) 0 0
\(31\) −1.02461 −0.184025 −0.0920126 0.995758i \(-0.529330\pi\)
−0.0920126 + 0.995758i \(0.529330\pi\)
\(32\) 0 0
\(33\) −11.2739 −1.96254
\(34\) 0 0
\(35\) 0.571717 + 0.571717i 0.0966378 + 0.0966378i
\(36\) 0 0
\(37\) 5.91245 5.91245i 0.972001 0.972001i −0.0276180 0.999619i \(-0.508792\pi\)
0.999619 + 0.0276180i \(0.00879219\pi\)
\(38\) 0 0
\(39\) 9.03127i 1.44616i
\(40\) 0 0
\(41\) 5.47010i 0.854285i −0.904184 0.427143i \(-0.859520\pi\)
0.904184 0.427143i \(-0.140480\pi\)
\(42\) 0 0
\(43\) −4.35480 + 4.35480i −0.664101 + 0.664101i −0.956344 0.292243i \(-0.905598\pi\)
0.292243 + 0.956344i \(0.405598\pi\)
\(44\) 0 0
\(45\) −0.0339670 0.0339670i −0.00506349 0.00506349i
\(46\) 0 0
\(47\) 6.47363 0.944275 0.472138 0.881525i \(-0.343483\pi\)
0.472138 + 0.881525i \(0.343483\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 3.05941 + 3.05941i 0.428403 + 0.428403i
\(52\) 0 0
\(53\) −1.94059 + 1.94059i −0.266560 + 0.266560i −0.827713 0.561152i \(-0.810358\pi\)
0.561152 + 0.827713i \(0.310358\pi\)
\(54\) 0 0
\(55\) 5.21137i 0.702701i
\(56\) 0 0
\(57\) 13.7127i 1.81630i
\(58\) 0 0
\(59\) −6.90191 + 6.90191i −0.898552 + 0.898552i −0.995308 0.0967561i \(-0.969153\pi\)
0.0967561 + 0.995308i \(0.469153\pi\)
\(60\) 0 0
\(61\) −7.84563 7.84563i −1.00453 1.00453i −0.999990 0.00454037i \(-0.998555\pi\)
−0.00454037 0.999990i \(-0.501445\pi\)
\(62\) 0 0
\(63\) 0.0594122 0.00748523
\(64\) 0 0
\(65\) 4.17471 0.517809
\(66\) 0 0
\(67\) 3.88118 + 3.88118i 0.474161 + 0.474161i 0.903258 0.429097i \(-0.141168\pi\)
−0.429097 + 0.903258i \(0.641168\pi\)
\(68\) 0 0
\(69\) 5.05117 5.05117i 0.608089 0.608089i
\(70\) 0 0
\(71\) 9.44196i 1.12055i 0.828305 + 0.560277i \(0.189305\pi\)
−0.828305 + 0.560277i \(0.810695\pi\)
\(72\) 0 0
\(73\) 1.06608i 0.124775i −0.998052 0.0623874i \(-0.980129\pi\)
0.998052 0.0623874i \(-0.0198714\pi\)
\(74\) 0 0
\(75\) −5.37553 + 5.37553i −0.620713 + 0.620713i
\(76\) 0 0
\(77\) 4.55765 + 4.55765i 0.519392 + 0.519392i
\(78\) 0 0
\(79\) −12.3400 −1.38836 −0.694179 0.719803i \(-0.744232\pi\)
−0.694179 + 0.719803i \(0.744232\pi\)
\(80\) 0 0
\(81\) 9.17471 1.01941
\(82\) 0 0
\(83\) 5.68230 + 5.68230i 0.623713 + 0.623713i 0.946479 0.322766i \(-0.104613\pi\)
−0.322766 + 0.946479i \(0.604613\pi\)
\(84\) 0 0
\(85\) −1.41421 + 1.41421i −0.153393 + 0.153393i
\(86\) 0 0
\(87\) 10.6417i 1.14091i
\(88\) 0 0
\(89\) 2.81900i 0.298814i −0.988776 0.149407i \(-0.952264\pi\)
0.988776 0.149407i \(-0.0477364\pi\)
\(90\) 0 0
\(91\) −3.65103 + 3.65103i −0.382732 + 0.382732i
\(92\) 0 0
\(93\) −1.26725 1.26725i −0.131408 0.131408i
\(94\) 0 0
\(95\) 6.33871 0.650338
\(96\) 0 0
\(97\) −1.74754 −0.177436 −0.0887179 0.996057i \(-0.528277\pi\)
−0.0887179 + 0.996057i \(0.528277\pi\)
\(98\) 0 0
\(99\) −0.270780 0.270780i −0.0272144 0.0272144i
\(100\) 0 0
\(101\) −5.10947 + 5.10947i −0.508411 + 0.508411i −0.914038 0.405628i \(-0.867053\pi\)
0.405628 + 0.914038i \(0.367053\pi\)
\(102\) 0 0
\(103\) 1.35833i 0.133840i −0.997758 0.0669202i \(-0.978683\pi\)
0.997758 0.0669202i \(-0.0213173\pi\)
\(104\) 0 0
\(105\) 1.41421i 0.138013i
\(106\) 0 0
\(107\) −0.348138 + 0.348138i −0.0336557 + 0.0336557i −0.723734 0.690079i \(-0.757576\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(108\) 0 0
\(109\) −6.70108 6.70108i −0.641847 0.641847i 0.309162 0.951009i \(-0.399951\pi\)
−0.951009 + 0.309162i \(0.899951\pi\)
\(110\) 0 0
\(111\) 14.6252 1.38816
\(112\) 0 0
\(113\) −14.1747 −1.33344 −0.666722 0.745306i \(-0.732303\pi\)
−0.666722 + 0.745306i \(0.732303\pi\)
\(114\) 0 0
\(115\) 2.33490 + 2.33490i 0.217731 + 0.217731i
\(116\) 0 0
\(117\) 0.216915 0.216915i 0.0200538 0.0200538i
\(118\) 0 0
\(119\) 2.47363i 0.226757i
\(120\) 0 0
\(121\) 30.5443i 2.77675i
\(122\) 0 0
\(123\) 6.76549 6.76549i 0.610023 0.610023i
\(124\) 0 0
\(125\) −5.34343 5.34343i −0.477931 0.477931i
\(126\) 0 0
\(127\) −0.457538 −0.0405999 −0.0203000 0.999794i \(-0.506462\pi\)
−0.0203000 + 0.999794i \(0.506462\pi\)
\(128\) 0 0
\(129\) −10.7721 −0.948435
\(130\) 0 0
\(131\) 5.75966 + 5.75966i 0.503224 + 0.503224i 0.912438 0.409215i \(-0.134197\pi\)
−0.409215 + 0.912438i \(0.634197\pi\)
\(132\) 0 0
\(133\) −5.54358 + 5.54358i −0.480689 + 0.480689i
\(134\) 0 0
\(135\) 4.15862i 0.357917i
\(136\) 0 0
\(137\) 5.60764i 0.479093i −0.970885 0.239546i \(-0.923001\pi\)
0.970885 0.239546i \(-0.0769987\pi\)
\(138\) 0 0
\(139\) 7.03239 7.03239i 0.596479 0.596479i −0.342894 0.939374i \(-0.611407\pi\)
0.939374 + 0.342894i \(0.111407\pi\)
\(140\) 0 0
\(141\) 8.00666 + 8.00666i 0.674283 + 0.674283i
\(142\) 0 0
\(143\) 33.2802 2.78303
\(144\) 0 0
\(145\) 4.91911 0.408510
\(146\) 0 0
\(147\) −1.23681 1.23681i −0.102011 0.102011i
\(148\) 0 0
\(149\) 11.4947 11.4947i 0.941683 0.941683i −0.0567079 0.998391i \(-0.518060\pi\)
0.998391 + 0.0567079i \(0.0180604\pi\)
\(150\) 0 0
\(151\) 0.863230i 0.0702487i −0.999383 0.0351243i \(-0.988817\pi\)
0.999383 0.0351243i \(-0.0111827\pi\)
\(152\) 0 0
\(153\) 0.146964i 0.0118813i
\(154\) 0 0
\(155\) 0.585786 0.585786i 0.0470515 0.0470515i
\(156\) 0 0
\(157\) −1.83778 1.83778i −0.146671 0.146671i 0.629958 0.776629i \(-0.283072\pi\)
−0.776629 + 0.629958i \(0.783072\pi\)
\(158\) 0 0
\(159\) −4.80029 −0.380688
\(160\) 0 0
\(161\) −4.08402 −0.321866
\(162\) 0 0
\(163\) 4.27078 + 4.27078i 0.334513 + 0.334513i 0.854298 0.519784i \(-0.173988\pi\)
−0.519784 + 0.854298i \(0.673988\pi\)
\(164\) 0 0
\(165\) 6.44549 6.44549i 0.501780 0.501780i
\(166\) 0 0
\(167\) 1.52637i 0.118114i 0.998255 + 0.0590572i \(0.0188094\pi\)
−0.998255 + 0.0590572i \(0.981191\pi\)
\(168\) 0 0
\(169\) 13.6600i 1.05077i
\(170\) 0 0
\(171\) 0.329356 0.329356i 0.0251865 0.0251865i
\(172\) 0 0
\(173\) 18.4232 + 18.4232i 1.40069 + 1.40069i 0.797905 + 0.602783i \(0.205941\pi\)
0.602783 + 0.797905i \(0.294059\pi\)
\(174\) 0 0
\(175\) 4.34628 0.328548
\(176\) 0 0
\(177\) −17.0727 −1.28327
\(178\) 0 0
\(179\) −9.23412 9.23412i −0.690190 0.690190i 0.272083 0.962274i \(-0.412287\pi\)
−0.962274 + 0.272083i \(0.912287\pi\)
\(180\) 0 0
\(181\) −15.7683 + 15.7683i −1.17205 + 1.17205i −0.190325 + 0.981721i \(0.560954\pi\)
−0.981721 + 0.190325i \(0.939046\pi\)
\(182\) 0 0
\(183\) 19.4072i 1.43462i
\(184\) 0 0
\(185\) 6.76049i 0.497041i
\(186\) 0 0
\(187\) −11.2739 + 11.2739i −0.824430 + 0.824430i
\(188\) 0 0
\(189\) −3.63696 3.63696i −0.264550 0.264550i
\(190\) 0 0
\(191\) −25.9178 −1.87535 −0.937674 0.347517i \(-0.887025\pi\)
−0.937674 + 0.347517i \(0.887025\pi\)
\(192\) 0 0
\(193\) −11.4884 −0.826954 −0.413477 0.910515i \(-0.635686\pi\)
−0.413477 + 0.910515i \(0.635686\pi\)
\(194\) 0 0
\(195\) 5.16333 + 5.16333i 0.369754 + 0.369754i
\(196\) 0 0
\(197\) 17.9241 17.9241i 1.27704 1.27704i 0.334723 0.942317i \(-0.391357\pi\)
0.942317 0.334723i \(-0.108643\pi\)
\(198\) 0 0
\(199\) 2.34538i 0.166259i −0.996539 0.0831297i \(-0.973508\pi\)
0.996539 0.0831297i \(-0.0264916\pi\)
\(200\) 0 0
\(201\) 9.60058i 0.677172i
\(202\) 0 0
\(203\) −4.30205 + 4.30205i −0.301945 + 0.301945i
\(204\) 0 0
\(205\) 3.12735 + 3.12735i 0.218423 + 0.218423i
\(206\) 0 0
\(207\) 0.242641 0.0168647
\(208\) 0 0
\(209\) 50.5313 3.49533
\(210\) 0 0
\(211\) 4.94725 + 4.94725i 0.340583 + 0.340583i 0.856586 0.516004i \(-0.172581\pi\)
−0.516004 + 0.856586i \(0.672581\pi\)
\(212\) 0 0
\(213\) −11.6779 + 11.6779i −0.800159 + 0.800159i
\(214\) 0 0
\(215\) 4.97943i 0.339594i
\(216\) 0 0
\(217\) 1.02461i 0.0695550i
\(218\) 0 0
\(219\) 1.31854 1.31854i 0.0890984 0.0890984i
\(220\) 0 0
\(221\) −9.03127 9.03127i −0.607509 0.607509i
\(222\) 0 0
\(223\) 5.47233 0.366454 0.183227 0.983071i \(-0.441346\pi\)
0.183227 + 0.983071i \(0.441346\pi\)
\(224\) 0 0
\(225\) −0.258222 −0.0172148
\(226\) 0 0
\(227\) −5.59986 5.59986i −0.371675 0.371675i 0.496412 0.868087i \(-0.334651\pi\)
−0.868087 + 0.496412i \(0.834651\pi\)
\(228\) 0 0
\(229\) −8.48888 + 8.48888i −0.560961 + 0.560961i −0.929580 0.368620i \(-0.879831\pi\)
0.368620 + 0.929580i \(0.379831\pi\)
\(230\) 0 0
\(231\) 11.2739i 0.741769i
\(232\) 0 0
\(233\) 0.673711i 0.0441363i 0.999756 + 0.0220682i \(0.00702508\pi\)
−0.999756 + 0.0220682i \(0.992975\pi\)
\(234\) 0 0
\(235\) −3.70108 + 3.70108i −0.241432 + 0.241432i
\(236\) 0 0
\(237\) −15.2623 15.2623i −0.991390 0.991390i
\(238\) 0 0
\(239\) −17.4603 −1.12941 −0.564706 0.825292i \(-0.691010\pi\)
−0.564706 + 0.825292i \(0.691010\pi\)
\(240\) 0 0
\(241\) 7.01519 0.451888 0.225944 0.974140i \(-0.427453\pi\)
0.225944 + 0.974140i \(0.427453\pi\)
\(242\) 0 0
\(243\) 0.436525 + 0.436525i 0.0280031 + 0.0280031i
\(244\) 0 0
\(245\) 0.571717 0.571717i 0.0365257 0.0365257i
\(246\) 0 0
\(247\) 40.4795i 2.57565i
\(248\) 0 0
\(249\) 14.0559i 0.890755i
\(250\) 0 0
\(251\) 8.38248 8.38248i 0.529097 0.529097i −0.391206 0.920303i \(-0.627942\pi\)
0.920303 + 0.391206i \(0.127942\pi\)
\(252\) 0 0
\(253\) 18.6135 + 18.6135i 1.17022 + 1.17022i
\(254\) 0 0
\(255\) −3.49824 −0.219068
\(256\) 0 0
\(257\) 9.28724 0.579322 0.289661 0.957129i \(-0.406457\pi\)
0.289661 + 0.957129i \(0.406457\pi\)
\(258\) 0 0
\(259\) −5.91245 5.91245i −0.367382 0.367382i
\(260\) 0 0
\(261\) 0.255594 0.255594i 0.0158209 0.0158209i
\(262\) 0 0
\(263\) 11.8547i 0.730993i 0.930813 + 0.365496i \(0.119101\pi\)
−0.930813 + 0.365496i \(0.880899\pi\)
\(264\) 0 0
\(265\) 2.21893i 0.136308i
\(266\) 0 0
\(267\) 3.48658 3.48658i 0.213375 0.213375i
\(268\) 0 0
\(269\) 0.902189 + 0.902189i 0.0550074 + 0.0550074i 0.734075 0.679068i \(-0.237616\pi\)
−0.679068 + 0.734075i \(0.737616\pi\)
\(270\) 0 0
\(271\) −19.2043 −1.16658 −0.583289 0.812264i \(-0.698235\pi\)
−0.583289 + 0.812264i \(0.698235\pi\)
\(272\) 0 0
\(273\) −9.03127 −0.546598
\(274\) 0 0
\(275\) −19.8088 19.8088i −1.19452 1.19452i
\(276\) 0 0
\(277\) −23.0461 + 23.0461i −1.38470 + 1.38470i −0.548655 + 0.836049i \(0.684860\pi\)
−0.836049 + 0.548655i \(0.815140\pi\)
\(278\) 0 0
\(279\) 0.0608743i 0.00364445i
\(280\) 0 0
\(281\) 19.3459i 1.15408i −0.816716 0.577039i \(-0.804208\pi\)
0.816716 0.577039i \(-0.195792\pi\)
\(282\) 0 0
\(283\) 12.1935 12.1935i 0.724828 0.724828i −0.244757 0.969584i \(-0.578708\pi\)
0.969584 + 0.244757i \(0.0787081\pi\)
\(284\) 0 0
\(285\) 7.83980 + 7.83980i 0.464390 + 0.464390i
\(286\) 0 0
\(287\) −5.47010 −0.322890
\(288\) 0 0
\(289\) −10.8812 −0.640069
\(290\) 0 0
\(291\) −2.16138 2.16138i −0.126702 0.126702i
\(292\) 0 0
\(293\) 14.5139 14.5139i 0.847909 0.847909i −0.141963 0.989872i \(-0.545341\pi\)
0.989872 + 0.141963i \(0.0453415\pi\)
\(294\) 0 0
\(295\) 7.89188i 0.459483i
\(296\) 0 0
\(297\) 33.1519i 1.92367i
\(298\) 0 0
\(299\) −14.9109 + 14.9109i −0.862318 + 0.862318i
\(300\) 0 0
\(301\) 4.35480 + 4.35480i 0.251007 + 0.251007i
\(302\) 0 0
\(303\) −12.6389 −0.726086
\(304\) 0 0
\(305\) 8.97096 0.513676
\(306\) 0 0
\(307\) 12.6975 + 12.6975i 0.724684 + 0.724684i 0.969555 0.244872i \(-0.0787459\pi\)
−0.244872 + 0.969555i \(0.578746\pi\)
\(308\) 0 0
\(309\) 1.68000 1.68000i 0.0955719 0.0955719i
\(310\) 0 0
\(311\) 17.7155i 1.00455i 0.864707 + 0.502277i \(0.167504\pi\)
−0.864707 + 0.502277i \(0.832496\pi\)
\(312\) 0 0
\(313\) 13.7873i 0.779306i 0.920962 + 0.389653i \(0.127405\pi\)
−0.920962 + 0.389653i \(0.872595\pi\)
\(314\) 0 0
\(315\) −0.0339670 + 0.0339670i −0.00191382 + 0.00191382i
\(316\) 0 0
\(317\) −8.95039 8.95039i −0.502704 0.502704i 0.409573 0.912277i \(-0.365678\pi\)
−0.912277 + 0.409573i \(0.865678\pi\)
\(318\) 0 0
\(319\) 39.2145 2.19559
\(320\) 0 0
\(321\) −0.861162 −0.0480654
\(322\) 0 0
\(323\) −13.7127 13.7127i −0.762997 0.762997i
\(324\) 0 0
\(325\) 15.8684 15.8684i 0.880219 0.880219i
\(326\) 0 0
\(327\) 16.5760i 0.916653i
\(328\) 0 0
\(329\) 6.47363i 0.356903i
\(330\) 0 0
\(331\) 12.0509 12.0509i 0.662377 0.662377i −0.293563 0.955940i \(-0.594841\pi\)
0.955940 + 0.293563i \(0.0948410\pi\)
\(332\) 0 0
\(333\) 0.351272 + 0.351272i 0.0192496 + 0.0192496i
\(334\) 0 0
\(335\) −4.43787 −0.242467
\(336\) 0 0
\(337\) −7.50100 −0.408605 −0.204303 0.978908i \(-0.565493\pi\)
−0.204303 + 0.978908i \(0.565493\pi\)
\(338\) 0 0
\(339\) −17.5315 17.5315i −0.952178 0.952178i
\(340\) 0 0
\(341\) 4.66981 4.66981i 0.252884 0.252884i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 5.77568i 0.310952i
\(346\) 0 0
\(347\) 16.9589 16.9589i 0.910402 0.910402i −0.0859018 0.996304i \(-0.527377\pi\)
0.996304 + 0.0859018i \(0.0273771\pi\)
\(348\) 0 0
\(349\) −12.4131 12.4131i −0.664458 0.664458i 0.291970 0.956428i \(-0.405689\pi\)
−0.956428 + 0.291970i \(0.905689\pi\)
\(350\) 0 0
\(351\) −26.5573 −1.41752
\(352\) 0 0
\(353\) 4.71903 0.251168 0.125584 0.992083i \(-0.459919\pi\)
0.125584 + 0.992083i \(0.459919\pi\)
\(354\) 0 0
\(355\) −5.39813 5.39813i −0.286503 0.286503i
\(356\) 0 0
\(357\) 3.05941 3.05941i 0.161921 0.161921i
\(358\) 0 0
\(359\) 6.09029i 0.321433i −0.987001 0.160717i \(-0.948619\pi\)
0.987001 0.160717i \(-0.0513805\pi\)
\(360\) 0 0
\(361\) 42.4625i 2.23487i
\(362\) 0 0
\(363\) 37.7776 37.7776i 1.98281 1.98281i
\(364\) 0 0
\(365\) 0.609494 + 0.609494i 0.0319024 + 0.0319024i
\(366\) 0 0
\(367\) −31.0286 −1.61968 −0.809841 0.586649i \(-0.800447\pi\)
−0.809841 + 0.586649i \(0.800447\pi\)
\(368\) 0 0
\(369\) 0.324990 0.0169183
\(370\) 0 0
\(371\) 1.94059 + 1.94059i 0.100750 + 0.100750i
\(372\) 0 0
\(373\) 4.96021 4.96021i 0.256830 0.256830i −0.566934 0.823763i \(-0.691870\pi\)
0.823763 + 0.566934i \(0.191870\pi\)
\(374\) 0 0
\(375\) 13.2176i 0.682556i
\(376\) 0 0
\(377\) 31.4138i 1.61789i
\(378\) 0 0
\(379\) −10.6815 + 10.6815i −0.548670 + 0.548670i −0.926056 0.377386i \(-0.876823\pi\)
0.377386 + 0.926056i \(0.376823\pi\)
\(380\) 0 0
\(381\) −0.565889 0.565889i −0.0289914 0.0289914i
\(382\) 0 0
\(383\) 9.43030 0.481866 0.240933 0.970542i \(-0.422547\pi\)
0.240933 + 0.970542i \(0.422547\pi\)
\(384\) 0 0
\(385\) −5.21137 −0.265596
\(386\) 0 0
\(387\) −0.258728 0.258728i −0.0131519 0.0131519i
\(388\) 0 0
\(389\) −6.33462 + 6.33462i −0.321178 + 0.321178i −0.849219 0.528041i \(-0.822927\pi\)
0.528041 + 0.849219i \(0.322927\pi\)
\(390\) 0 0
\(391\) 10.1023i 0.510897i
\(392\) 0 0
\(393\) 14.2472i 0.718678i
\(394\) 0 0
\(395\) 7.05498 7.05498i 0.354975 0.354975i
\(396\) 0 0
\(397\) 17.6080 + 17.6080i 0.883719 + 0.883719i 0.993910 0.110191i \(-0.0351463\pi\)
−0.110191 + 0.993910i \(0.535146\pi\)
\(398\) 0 0
\(399\) −13.7127 −0.686495
\(400\) 0 0
\(401\) 24.7529 1.23610 0.618051 0.786138i \(-0.287923\pi\)
0.618051 + 0.786138i \(0.287923\pi\)
\(402\) 0 0
\(403\) 3.74088 + 3.74088i 0.186346 + 0.186346i
\(404\) 0 0
\(405\) −5.24533 + 5.24533i −0.260643 + 0.260643i
\(406\) 0 0
\(407\) 53.8937i 2.67141i
\(408\) 0 0
\(409\) 21.6913i 1.07256i −0.844039 0.536282i \(-0.819828\pi\)
0.844039 0.536282i \(-0.180172\pi\)
\(410\) 0 0
\(411\) 6.93560 6.93560i 0.342108 0.342108i
\(412\) 0 0
\(413\) 6.90191 + 6.90191i 0.339621 + 0.339621i
\(414\) 0 0
\(415\) −6.49733 −0.318942
\(416\) 0 0
\(417\) 17.3955 0.851861
\(418\) 0 0
\(419\) −7.39320 7.39320i −0.361181 0.361181i 0.503066 0.864248i \(-0.332205\pi\)
−0.864248 + 0.503066i \(0.832205\pi\)
\(420\) 0 0
\(421\) −4.59821 + 4.59821i −0.224103 + 0.224103i −0.810224 0.586121i \(-0.800654\pi\)
0.586121 + 0.810224i \(0.300654\pi\)
\(422\) 0 0
\(423\) 0.384612i 0.0187005i
\(424\) 0 0
\(425\) 10.7511i 0.521503i
\(426\) 0 0
\(427\) −7.84563 + 7.84563i −0.379677 + 0.379677i
\(428\) 0 0
\(429\) 41.1614 + 41.1614i 1.98729 + 1.98729i
\(430\) 0 0
\(431\) 10.3842 0.500190 0.250095 0.968221i \(-0.419538\pi\)
0.250095 + 0.968221i \(0.419538\pi\)
\(432\) 0 0
\(433\) 27.6476 1.32866 0.664329 0.747441i \(-0.268718\pi\)
0.664329 + 0.747441i \(0.268718\pi\)
\(434\) 0 0
\(435\) 6.08402 + 6.08402i 0.291707 + 0.291707i
\(436\) 0 0
\(437\) −22.6401 + 22.6401i −1.08302 + 1.08302i
\(438\) 0 0
\(439\) 33.2404i 1.58648i 0.608911 + 0.793239i \(0.291607\pi\)
−0.608911 + 0.793239i \(0.708393\pi\)
\(440\) 0 0
\(441\) 0.0594122i 0.00282915i
\(442\) 0 0
\(443\) −21.8780 + 21.8780i −1.03946 + 1.03946i −0.0402671 + 0.999189i \(0.512821\pi\)
−0.999189 + 0.0402671i \(0.987179\pi\)
\(444\) 0 0
\(445\) 1.61167 + 1.61167i 0.0764006 + 0.0764006i
\(446\) 0 0
\(447\) 28.4336 1.34486
\(448\) 0 0
\(449\) 15.9670 0.753531 0.376765 0.926309i \(-0.377036\pi\)
0.376765 + 0.926309i \(0.377036\pi\)
\(450\) 0 0
\(451\) 24.9308 + 24.9308i 1.17394 + 1.17394i
\(452\) 0 0
\(453\) 1.06765 1.06765i 0.0501628 0.0501628i
\(454\) 0 0
\(455\) 4.17471i 0.195713i
\(456\) 0 0
\(457\) 8.09512i 0.378674i −0.981912 0.189337i \(-0.939366\pi\)
0.981912 0.189337i \(-0.0606338\pi\)
\(458\) 0 0
\(459\) 8.99647 8.99647i 0.419919 0.419919i
\(460\) 0 0
\(461\) 0.240227 + 0.240227i 0.0111885 + 0.0111885i 0.712679 0.701490i \(-0.247482\pi\)
−0.701490 + 0.712679i \(0.747482\pi\)
\(462\) 0 0
\(463\) −10.7400 −0.499129 −0.249564 0.968358i \(-0.580287\pi\)
−0.249564 + 0.968358i \(0.580287\pi\)
\(464\) 0 0
\(465\) 1.44902 0.0671965
\(466\) 0 0
\(467\) −6.77143 6.77143i −0.313344 0.313344i 0.532859 0.846204i \(-0.321117\pi\)
−0.846204 + 0.532859i \(0.821117\pi\)
\(468\) 0 0
\(469\) 3.88118 3.88118i 0.179216 0.179216i
\(470\) 0 0
\(471\) 4.54599i 0.209468i
\(472\) 0 0
\(473\) 39.6953i 1.82519i
\(474\) 0 0
\(475\) 24.0939 24.0939i 1.10551 1.10551i
\(476\) 0 0
\(477\) −0.115295 0.115295i −0.00527898 0.00527898i
\(478\) 0 0
\(479\) 20.7472 0.947962 0.473981 0.880535i \(-0.342816\pi\)
0.473981 + 0.880535i \(0.342816\pi\)
\(480\) 0 0
\(481\) −43.1730 −1.96852
\(482\) 0 0
\(483\) −5.05117 5.05117i −0.229836 0.229836i
\(484\) 0 0
\(485\) 0.999098 0.999098i 0.0453667 0.0453667i
\(486\) 0 0
\(487\) 5.57283i 0.252529i 0.991997 + 0.126265i \(0.0402988\pi\)
−0.991997 + 0.126265i \(0.959701\pi\)
\(488\) 0 0
\(489\) 10.5643i 0.477735i
\(490\) 0 0
\(491\) −16.4196 + 16.4196i −0.741005 + 0.741005i −0.972772 0.231766i \(-0.925550\pi\)
0.231766 + 0.972772i \(0.425550\pi\)
\(492\) 0 0
\(493\) −10.6417 10.6417i −0.479277 0.479277i
\(494\) 0 0
\(495\) 0.309619 0.0139163
\(496\) 0 0
\(497\) 9.44196 0.423530
\(498\) 0 0
\(499\) 17.7731 + 17.7731i 0.795631 + 0.795631i 0.982403 0.186772i \(-0.0598026\pi\)
−0.186772 + 0.982403i \(0.559803\pi\)
\(500\) 0 0
\(501\) −1.88784 + 1.88784i −0.0843424 + 0.0843424i
\(502\) 0 0
\(503\) 20.9839i 0.935625i −0.883828 0.467813i \(-0.845042\pi\)
0.883828 0.467813i \(-0.154958\pi\)
\(504\) 0 0
\(505\) 5.84234i 0.259981i
\(506\) 0 0
\(507\) −16.8948 + 16.8948i −0.750327 + 0.750327i
\(508\) 0 0
\(509\) 2.90544 + 2.90544i 0.128781 + 0.128781i 0.768560 0.639778i \(-0.220974\pi\)
−0.639778 + 0.768560i \(0.720974\pi\)
\(510\) 0 0
\(511\) −1.06608 −0.0471604
\(512\) 0 0
\(513\) −40.3235 −1.78033
\(514\) 0 0
\(515\) 0.776581 + 0.776581i 0.0342202 + 0.0342202i
\(516\) 0 0
\(517\) −29.5045 + 29.5045i −1.29761 + 1.29761i
\(518\) 0 0
\(519\) 45.5720i 2.00039i
\(520\) 0 0
\(521\) 1.02225i 0.0447854i −0.999749 0.0223927i \(-0.992872\pi\)
0.999749 0.0223927i \(-0.00712841\pi\)
\(522\) 0 0
\(523\) 22.2532 22.2532i 0.973064 0.973064i −0.0265830 0.999647i \(-0.508463\pi\)
0.999647 + 0.0265830i \(0.00846262\pi\)
\(524\) 0 0
\(525\) 5.37553 + 5.37553i 0.234608 + 0.234608i
\(526\) 0 0
\(527\) −2.53450 −0.110405
\(528\) 0 0
\(529\) 6.32077 0.274816
\(530\) 0 0
\(531\) −0.410058 0.410058i −0.0177950 0.0177950i
\(532\) 0 0
\(533\) −19.9715 + 19.9715i −0.865060 + 0.865060i
\(534\) 0 0
\(535\) 0.398072i 0.0172102i
\(536\) 0 0
\(537\) 22.8418i 0.985694i
\(538\) 0 0
\(539\) 4.55765 4.55765i 0.196312 0.196312i
\(540\) 0 0
\(541\) 11.1188 + 11.1188i 0.478036 + 0.478036i 0.904503 0.426467i \(-0.140242\pi\)
−0.426467 + 0.904503i \(0.640242\pi\)
\(542\) 0 0
\(543\) −39.0048 −1.67386
\(544\) 0 0
\(545\) 7.66224 0.328214
\(546\) 0 0
\(547\) −17.7261 17.7261i −0.757912 0.757912i 0.218030 0.975942i \(-0.430037\pi\)
−0.975942 + 0.218030i \(0.930037\pi\)
\(548\) 0 0
\(549\) 0.466126 0.466126i 0.0198938 0.0198938i
\(550\) 0 0
\(551\) 47.6975i 2.03198i
\(552\) 0 0
\(553\) 12.3400i 0.524750i
\(554\) 0 0
\(555\) −8.36147 + 8.36147i −0.354924 + 0.354924i
\(556\) 0 0
\(557\) 14.1751 + 14.1751i 0.600617 + 0.600617i 0.940476 0.339859i \(-0.110379\pi\)
−0.339859 + 0.940476i \(0.610379\pi\)
\(558\) 0 0
\(559\) 31.7990 1.34495
\(560\) 0 0
\(561\) −27.8874 −1.17741
\(562\) 0 0
\(563\) 0.484561 + 0.484561i 0.0204218 + 0.0204218i 0.717244 0.696822i \(-0.245403\pi\)
−0.696822 + 0.717244i \(0.745403\pi\)
\(564\) 0 0
\(565\) 8.10392 8.10392i 0.340934 0.340934i
\(566\) 0 0
\(567\) 9.17471i 0.385301i
\(568\) 0 0
\(569\) 9.22040i 0.386539i 0.981146 + 0.193270i \(0.0619092\pi\)
−0.981146 + 0.193270i \(0.938091\pi\)
\(570\) 0 0
\(571\) −1.47325 + 1.47325i −0.0616537 + 0.0616537i −0.737261 0.675608i \(-0.763881\pi\)
0.675608 + 0.737261i \(0.263881\pi\)
\(572\) 0 0
\(573\) −32.0555 32.0555i −1.33914 1.33914i
\(574\) 0 0
\(575\) 17.7503 0.740239
\(576\) 0 0
\(577\) 40.4518 1.68403 0.842014 0.539455i \(-0.181370\pi\)
0.842014 + 0.539455i \(0.181370\pi\)
\(578\) 0 0
\(579\) −14.2090 14.2090i −0.590507 0.590507i
\(580\) 0 0
\(581\) 5.68230 5.68230i 0.235742 0.235742i
\(582\) 0 0
\(583\) 17.6890i 0.732605i
\(584\) 0 0
\(585\) 0.248028i 0.0102547i
\(586\) 0 0
\(587\) −2.58467 + 2.58467i −0.106681 + 0.106681i −0.758432 0.651752i \(-0.774034\pi\)
0.651752 + 0.758432i \(0.274034\pi\)
\(588\) 0 0
\(589\) 5.68000 + 5.68000i 0.234040 + 0.234040i
\(590\) 0 0
\(591\) 44.3375 1.82380
\(592\) 0 0
\(593\) −37.8610 −1.55476 −0.777382 0.629029i \(-0.783453\pi\)
−0.777382 + 0.629029i \(0.783453\pi\)
\(594\) 0 0
\(595\) 1.41421 + 1.41421i 0.0579771 + 0.0579771i
\(596\) 0 0
\(597\) 2.90079 2.90079i 0.118722 0.118722i
\(598\) 0 0
\(599\) 1.42807i 0.0583493i 0.999574 + 0.0291747i \(0.00928790\pi\)
−0.999574 + 0.0291747i \(0.990712\pi\)
\(600\) 0 0
\(601\) 35.2990i 1.43988i 0.694038 + 0.719939i \(0.255830\pi\)
−0.694038 + 0.719939i \(0.744170\pi\)
\(602\) 0 0
\(603\) −0.230589 + 0.230589i −0.00939032 + 0.00939032i
\(604\) 0 0
\(605\) 17.4627 + 17.4627i 0.709959 + 0.709959i
\(606\) 0 0
\(607\) −23.4364 −0.951256 −0.475628 0.879647i \(-0.657779\pi\)
−0.475628 + 0.879647i \(0.657779\pi\)
\(608\) 0 0
\(609\) −10.6417 −0.431222
\(610\) 0 0
\(611\) −23.6354 23.6354i −0.956185 0.956185i
\(612\) 0 0
\(613\) 13.4964 13.4964i 0.545114 0.545114i −0.379910 0.925023i \(-0.624045\pi\)
0.925023 + 0.379910i \(0.124045\pi\)
\(614\) 0 0
\(615\) 7.73588i 0.311941i
\(616\) 0 0
\(617\) 0.504525i 0.0203114i 0.999948 + 0.0101557i \(0.00323272\pi\)
−0.999948 + 0.0101557i \(0.996767\pi\)
\(618\) 0 0
\(619\) 23.6377 23.6377i 0.950078 0.950078i −0.0487334 0.998812i \(-0.515518\pi\)
0.998812 + 0.0487334i \(0.0155185\pi\)
\(620\) 0 0
\(621\) −14.8534 14.8534i −0.596047 0.596047i
\(622\) 0 0
\(623\) −2.81900 −0.112941
\(624\) 0 0
\(625\) −15.6215 −0.624862
\(626\) 0 0
\(627\) 62.4978 + 62.4978i 2.49592 + 2.49592i
\(628\) 0 0
\(629\) 14.6252 14.6252i 0.583144 0.583144i
\(630\) 0 0
\(631\) 38.0329i 1.51407i −0.653376 0.757033i \(-0.726648\pi\)
0.653376 0.757033i \(-0.273352\pi\)
\(632\) 0 0
\(633\) 12.2376i 0.486403i
\(634\) 0 0
\(635\) 0.261582 0.261582i 0.0103806 0.0103806i
\(636\) 0 0
\(637\) 3.65103 + 3.65103i 0.144659 + 0.144659i
\(638\) 0 0
\(639\) −0.560967 −0.0221915
\(640\) 0 0
\(641\) 44.2113 1.74624 0.873122 0.487503i \(-0.162092\pi\)
0.873122 + 0.487503i \(0.162092\pi\)
\(642\) 0 0
\(643\) 7.98280 + 7.98280i 0.314811 + 0.314811i 0.846770 0.531959i \(-0.178544\pi\)
−0.531959 + 0.846770i \(0.678544\pi\)
\(644\) 0 0
\(645\) 6.15862 6.15862i 0.242495 0.242495i
\(646\) 0 0
\(647\) 46.5922i 1.83173i −0.401487 0.915865i \(-0.631507\pi\)
0.401487 0.915865i \(-0.368493\pi\)
\(648\) 0 0
\(649\) 62.9129i 2.46955i
\(650\) 0 0
\(651\) −1.26725 + 1.26725i −0.0496674 + 0.0496674i
\(652\) 0 0
\(653\) 28.1497 + 28.1497i 1.10158 + 1.10158i 0.994220 + 0.107363i \(0.0342407\pi\)
0.107363 + 0.994220i \(0.465759\pi\)
\(654\) 0 0
\(655\) −6.58579 −0.257328
\(656\) 0 0
\(657\) 0.0633379 0.00247105
\(658\) 0 0
\(659\) −12.5741 12.5741i −0.489819 0.489819i 0.418430 0.908249i \(-0.362580\pi\)
−0.908249 + 0.418430i \(0.862580\pi\)
\(660\) 0 0
\(661\) 7.13055 7.13055i 0.277346 0.277346i −0.554703 0.832049i \(-0.687168\pi\)
0.832049 + 0.554703i \(0.187168\pi\)
\(662\) 0 0
\(663\) 22.3400i 0.867613i
\(664\) 0 0
\(665\) 6.33871i 0.245805i
\(666\) 0 0
\(667\) −17.5697 + 17.5697i −0.680301 + 0.680301i
\(668\) 0 0
\(669\) 6.76825 + 6.76825i 0.261675 + 0.261675i
\(670\) 0 0
\(671\) 71.5152 2.76081
\(672\) 0 0
\(673\) −34.0500 −1.31253 −0.656265 0.754531i \(-0.727865\pi\)
−0.656265 + 0.754531i \(0.727865\pi\)
\(674\) 0 0
\(675\) 15.8072 + 15.8072i 0.608421 + 0.608421i
\(676\) 0 0
\(677\) 15.7666 15.7666i 0.605960 0.605960i −0.335928 0.941888i \(-0.609050\pi\)
0.941888 + 0.335928i \(0.109050\pi\)
\(678\) 0 0
\(679\) 1.74754i 0.0670644i
\(680\) 0 0
\(681\) 13.8519i 0.530808i
\(682\) 0 0
\(683\) 30.5756 30.5756i 1.16994 1.16994i 0.187719 0.982223i \(-0.439891\pi\)
0.982223 0.187719i \(-0.0601094\pi\)
\(684\) 0 0
\(685\) 3.20598 + 3.20598i 0.122494 + 0.122494i
\(686\) 0 0
\(687\) −20.9983 −0.801135
\(688\) 0 0
\(689\) 14.1703 0.539844
\(690\) 0 0
\(691\) −3.97140 3.97140i −0.151079 0.151079i 0.627521 0.778600i \(-0.284070\pi\)
−0.778600 + 0.627521i \(0.784070\pi\)
\(692\) 0 0
\(693\) −0.270780 + 0.270780i −0.0102861 + 0.0102861i
\(694\) 0 0
\(695\) 8.04107i 0.305015i
\(696\) 0 0
\(697\) 13.5310i 0.512522i
\(698\) 0 0
\(699\) −0.833255 + 0.833255i −0.0315166 + 0.0315166i
\(700\) 0 0
\(701\) 21.4532 + 21.4532i 0.810278 + 0.810278i 0.984675 0.174398i \(-0.0557978\pi\)
−0.174398 + 0.984675i \(0.555798\pi\)
\(702\) 0 0
\(703\) −65.5522 −2.47235
\(704\) 0 0
\(705\) −9.15509 −0.344801
\(706\) 0 0
\(707\) 5.10947 + 5.10947i 0.192161 + 0.192161i
\(708\) 0 0
\(709\) −2.30481 + 2.30481i −0.0865591 + 0.0865591i −0.749061 0.662502i \(-0.769495\pi\)
0.662502 + 0.749061i \(0.269495\pi\)
\(710\) 0 0
\(711\) 0.733146i 0.0274951i
\(712\) 0 0
\(713\) 4.18453i 0.156712i
\(714\) 0 0
\(715\) −19.0268 + 19.0268i −0.711564 + 0.711564i
\(716\) 0 0
\(717\) −21.5951 21.5951i −0.806484 0.806484i
\(718\) 0 0
\(719\) 27.8530 1.03874 0.519371 0.854549i \(-0.326166\pi\)
0.519371 + 0.854549i \(0.326166\pi\)
\(720\) 0 0
\(721\) −1.35833 −0.0505869
\(722\) 0 0
\(723\) 8.67647 + 8.67647i 0.322681 + 0.322681i
\(724\) 0 0
\(725\) 18.6979 18.6979i 0.694423 0.694423i
\(726\) 0 0
\(727\) 4.56027i 0.169131i −0.996418 0.0845656i \(-0.973050\pi\)
0.996418 0.0845656i \(-0.0269503\pi\)
\(728\) 0 0
\(729\) 26.4443i 0.979419i
\(730\) 0 0
\(731\) −10.7721 + 10.7721i −0.398422 + 0.398422i
\(732\) 0 0
\(733\) −29.8784 29.8784i −1.10358 1.10358i −0.993975 0.109608i \(-0.965041\pi\)
−0.109608 0.993975i \(-0.534959\pi\)
\(734\) 0 0
\(735\) 1.41421 0.0521641
\(736\) 0 0
\(737\) −35.3781 −1.30317
\(738\) 0 0
\(739\) −10.9312 10.9312i −0.402109 0.402109i 0.476866 0.878976i \(-0.341773\pi\)
−0.878976 + 0.476866i \(0.841773\pi\)
\(740\) 0 0
\(741\) −50.0656 + 50.0656i −1.83920 + 1.83920i
\(742\) 0 0
\(743\) 33.5354i 1.23029i −0.788412 0.615147i \(-0.789097\pi\)
0.788412 0.615147i \(-0.210903\pi\)
\(744\) 0 0
\(745\) 13.1434i 0.481538i
\(746\) 0 0
\(747\) −0.337598 + 0.337598i −0.0123521 + 0.0123521i
\(748\) 0 0
\(749\) 0.348138 + 0.348138i 0.0127207 + 0.0127207i
\(750\) 0 0
\(751\) −16.7562 −0.611442 −0.305721 0.952121i \(-0.598898\pi\)
−0.305721 + 0.952121i \(0.598898\pi\)
\(752\) 0 0
\(753\) 20.7351 0.755630
\(754\) 0 0
\(755\) 0.493523 + 0.493523i 0.0179611 + 0.0179611i
\(756\) 0 0
\(757\) −10.2722 + 10.2722i −0.373351 + 0.373351i −0.868696 0.495345i \(-0.835042\pi\)
0.495345 + 0.868696i \(0.335042\pi\)
\(758\) 0 0
\(759\) 46.0429i 1.67125i
\(760\) 0 0
\(761\) 1.54207i 0.0558998i −0.999609 0.0279499i \(-0.991102\pi\)
0.999609 0.0279499i \(-0.00889790\pi\)
\(762\) 0 0
\(763\) −6.70108 + 6.70108i −0.242595 + 0.242595i
\(764\) 0 0
\(765\) −0.0840215 0.0840215i −0.00303780 0.00303780i
\(766\) 0 0
\(767\) 50.3981 1.81977
\(768\) 0 0
\(769\) 32.1627 1.15982 0.579908 0.814682i \(-0.303089\pi\)
0.579908 + 0.814682i \(0.303089\pi\)
\(770\) 0 0
\(771\) 11.4866 + 11.4866i 0.413679 + 0.413679i
\(772\) 0 0
\(773\) 20.9593 20.9593i 0.753852 0.753852i −0.221344 0.975196i \(-0.571044\pi\)
0.975196 + 0.221344i \(0.0710443\pi\)
\(774\) 0 0
\(775\) 4.45324i 0.159965i
\(776\) 0 0
\(777\) 14.6252i 0.524676i
\(778\) 0 0
\(779\) −30.3239 + 30.3239i −1.08647 + 1.08647i
\(780\) 0 0
\(781\) −43.0331 43.0331i −1.53985 1.53985i
\(782\) 0 0
\(783\) −31.2928 −1.11831
\(784\) 0 0
\(785\) 2.10139 0.0750016
\(786\) 0 0
\(787\) −14.8158 14.8158i −0.528127 0.528127i 0.391887 0.920014i \(-0.371823\pi\)
−0.920014 + 0.391887i \(0.871823\pi\)
\(788\) 0 0
\(789\) −14.6621 + 14.6621i −0.521983 + 0.521983i
\(790\) 0 0
\(791\) 14.1747i 0.503995i
\(792\) 0 0
\(793\) 57.2892i 2.03440i
\(794\) 0 0
\(795\) 2.74441 2.74441i 0.0973340 0.0973340i
\(796\) 0 0
\(797\) −35.4119 35.4119i −1.25435 1.25435i −0.953749 0.300605i \(-0.902811\pi\)
−0.300605 0.953749i \(-0.597189\pi\)
\(798\) 0 0
\(799\) 16.0133 0.566511
\(800\) 0 0
\(801\) 0.167483 0.00591773
\(802\) 0 0
\(803\) 4.85880 + 4.85880i 0.171463 + 0.171463i
\(804\) 0 0
\(805\) 2.33490 2.33490i 0.0822946 0.0822946i
\(806\) 0 0
\(807\) 2.23168i 0.0785588i
\(808\) 0 0
\(809\) 17.4359i 0.613012i −0.951869 0.306506i \(-0.900840\pi\)
0.951869 0.306506i \(-0.0991599\pi\)
\(810\) 0 0
\(811\) −38.5546 + 38.5546i −1.35383 + 1.35383i −0.472506 + 0.881327i \(0.656651\pi\)
−0.881327 + 0.472506i \(0.843349\pi\)
\(812\) 0 0
\(813\) −23.7521 23.7521i −0.833024 0.833024i
\(814\) 0 0
\(815\) −4.88335 −0.171056
\(816\) 0 0
\(817\) 48.2824 1.68919
\(818\) 0 0
\(819\) −0.216915 0.216915i −0.00757964 0.00757964i
\(820\) 0 0
\(821\) −12.1020 + 12.1020i −0.422362 + 0.422362i −0.886016 0.463655i \(-0.846538\pi\)
0.463655 + 0.886016i \(0.346538\pi\)
\(822\) 0 0
\(823\) 32.3555i 1.12784i −0.825829 0.563921i \(-0.809292\pi\)
0.825829 0.563921i \(-0.190708\pi\)
\(824\) 0 0
\(825\) 48.9996i 1.70595i
\(826\) 0 0
\(827\) −11.2064 + 11.2064i −0.389684 + 0.389684i −0.874575 0.484891i \(-0.838859\pi\)
0.484891 + 0.874575i \(0.338859\pi\)
\(828\) 0 0
\(829\) −23.6494 23.6494i −0.821379 0.821379i 0.164927 0.986306i \(-0.447261\pi\)
−0.986306 + 0.164927i \(0.947261\pi\)
\(830\) 0 0
\(831\) −57.0073 −1.97756
\(832\) 0 0
\(833\) −2.47363 −0.0857061
\(834\) 0 0
\(835\) −0.872654 0.872654i −0.0301994 0.0301994i
\(836\) 0 0
\(837\) −3.72646 + 3.72646i −0.128805 + 0.128805i
\(838\) 0 0
\(839\) 28.6777i 0.990066i −0.868874 0.495033i \(-0.835156\pi\)
0.868874 0.495033i \(-0.164844\pi\)
\(840\) 0 0
\(841\) 8.01532i 0.276390i
\(842\) 0 0
\(843\) 23.9272 23.9272i 0.824098 0.824098i
\(844\) 0 0
\(845\) −7.80965 7.80965i −0.268660 0.268660i
\(846\) 0 0
\(847\) −30.5443 −1.04951
\(848\) 0 0
\(849\) 30.1621 1.03516
\(850\) 0 0
\(851\) −24.1466 24.1466i −0.827734 0.827734i
\(852\) 0 0
\(853\) 24.0950 24.0950i 0.824998 0.824998i −0.161822 0.986820i \(-0.551737\pi\)
0.986820 + 0.161822i \(0.0517370\pi\)
\(854\) 0 0
\(855\) 0.376597i 0.0128793i
\(856\) 0 0
\(857\) 38.8652i 1.32761i −0.747906 0.663805i \(-0.768941\pi\)
0.747906 0.663805i \(-0.231059\pi\)
\(858\) 0 0
\(859\) −10.2051 + 10.2051i −0.348195 + 0.348195i −0.859437 0.511242i \(-0.829186\pi\)
0.511242 + 0.859437i \(0.329186\pi\)
\(860\) 0 0
\(861\) −6.76549 6.76549i −0.230567 0.230567i
\(862\) 0 0
\(863\) 21.8471 0.743684 0.371842 0.928296i \(-0.378726\pi\)
0.371842 + 0.928296i \(0.378726\pi\)
\(864\) 0 0
\(865\) −21.0657 −0.716255
\(866\) 0 0
\(867\) −13.4580 13.4580i −0.457057 0.457057i
\(868\) 0 0
\(869\) 56.2413 56.2413i 1.90786 1.90786i
\(870\) 0 0
\(871\) 28.3405i 0.960283i
\(872\) 0 0
\(873\) 0.103825i 0.00351395i
\(874\) 0 0
\(875\) −5.34343 + 5.34343i −0.180641 + 0.180641i
\(876\) 0 0
\(877\) −26.6322 26.6322i −0.899306 0.899306i 0.0960685 0.995375i \(-0.469373\pi\)
−0.995375 + 0.0960685i \(0.969373\pi\)
\(878\) 0 0
\(879\) 35.9019 1.21094
\(880\) 0 0
\(881\) −4.11567 −0.138660 −0.0693302 0.997594i \(-0.522086\pi\)
−0.0693302 + 0.997594i \(0.522086\pi\)
\(882\) 0 0
\(883\) 0.315403 + 0.315403i 0.0106141 + 0.0106141i 0.712394 0.701780i \(-0.247611\pi\)
−0.701780 + 0.712394i \(0.747611\pi\)
\(884\) 0 0
\(885\) 9.76077 9.76077i 0.328105 0.328105i
\(886\) 0 0
\(887\) 20.5273i 0.689240i 0.938742 + 0.344620i \(0.111992\pi\)
−0.938742 + 0.344620i \(0.888008\pi\)
\(888\) 0 0
\(889\) 0.457538i 0.0153453i
\(890\) 0 0
\(891\) −41.8151 + 41.8151i −1.40086 + 1.40086i
\(892\) 0 0
\(893\) −35.8870 35.8870i −1.20091 1.20091i
\(894\) 0 0
\(895\) 10.5586 0.352935
\(896\) 0 0
\(897\) −36.8839 −1.23152
\(898\) 0 0
\(899\) 4.40792 + 4.40792i 0.147012 + 0.147012i
\(900\) 0 0
\(901\) −4.80029 + 4.80029i −0.159921 + 0.159921i
\(902\) 0 0
\(903\) 10.7721i 0.358475i
\(904\) 0 0
\(905\) 18.0300i 0.599337i
\(906\) 0 0
\(907\) −6.78157 + 6.78157i −0.225178 + 0.225178i −0.810675 0.585497i \(-0.800900\pi\)
0.585497 + 0.810675i \(0.300900\pi\)
\(908\) 0 0
\(909\) −0.303565 0.303565i −0.0100686 0.0100686i
\(910\) 0 0
\(911\) −30.5756 −1.01301 −0.506507 0.862236i \(-0.669064\pi\)
−0.506507 + 0.862236i \(0.669064\pi\)
\(912\) 0 0
\(913\) −51.7958 −1.71419
\(914\) 0 0
\(915\) 11.0954 + 11.0954i 0.366803 + 0.366803i
\(916\) 0 0
\(917\) 5.75966 5.75966i 0.190201 0.190201i
\(918\) 0 0
\(919\) 22.4777i 0.741470i −0.928739 0.370735i \(-0.879106\pi\)
0.928739 0.370735i \(-0.120894\pi\)
\(920\) 0 0
\(921\) 31.4088i 1.03496i
\(922\) 0 0
\(923\) 34.4728 34.4728i 1.13469 1.13469i
\(924\) 0 0
\(925\) 25.6972 + 25.6972i 0.844917 + 0.844917i
\(926\) 0 0
\(927\) 0.0807014 0.00265058
\(928\) 0 0
\(929\) 15.3955 0.505111 0.252555 0.967582i \(-0.418729\pi\)
0.252555 + 0.967582i \(0.418729\pi\)
\(930\) 0 0
\(931\) 5.54358 + 5.54358i 0.181683 + 0.181683i
\(932\) 0 0
\(933\) −21.9108 + 21.9108i −0.717326 + 0.717326i
\(934\) 0 0
\(935\) 12.8910i 0.421580i
\(936\) 0 0
\(937\) 29.4553i 0.962262i 0.876649 + 0.481131i \(0.159774\pi\)
−0.876649 + 0.481131i \(0.840226\pi\)
\(938\) 0 0
\(939\) −17.0524 + 17.0524i −0.556482 + 0.556482i
\(940\) 0 0
\(941\) 10.2529 + 10.2529i 0.334235 + 0.334235i 0.854192 0.519957i \(-0.174052\pi\)
−0.519957 + 0.854192i \(0.674052\pi\)
\(942\) 0 0
\(943\) −22.3400 −0.727490
\(944\) 0 0
\(945\) 4.15862 0.135280
\(946\) 0 0
\(947\) −31.6627 31.6627i −1.02890 1.02890i −0.999570 0.0293306i \(-0.990662\pi\)
−0.0293306 0.999570i \(-0.509338\pi\)
\(948\) 0 0
\(949\) −3.89227 + 3.89227i −0.126348 + 0.126348i
\(950\) 0 0
\(951\) 22.1399i 0.717936i
\(952\) 0 0
\(953\) 21.7024i 0.703008i 0.936186 + 0.351504i \(0.114330\pi\)
−0.936186 + 0.351504i \(0.885670\pi\)
\(954\) 0 0
\(955\) 14.8177 14.8177i 0.479488 0.479488i
\(956\) 0 0
\(957\) 48.5010 + 48.5010i 1.56781 + 1.56781i
\(958\) 0 0
\(959\) −5.60764 −0.181080
\(960\) 0 0
\(961\) −29.9502 −0.966135
\(962\) 0 0
\(963\) −0.0206836 0.0206836i −0.000666520 0.000666520i
\(964\) 0 0
\(965\) 6.56812 6.56812i 0.211435 0.211435i
\(966\) 0 0
\(967\) 16.1725i 0.520072i −0.965599 0.260036i \(-0.916266\pi\)
0.965599 0.260036i \(-0.0837344\pi\)
\(968\) 0 0
\(969\) 33.9202i 1.08967i
\(970\) 0 0
\(971\) −19.6139 + 19.6139i −0.629439 + 0.629439i −0.947927 0.318488i \(-0.896825\pi\)
0.318488 + 0.947927i \(0.396825\pi\)
\(972\) 0 0
\(973\) −7.03239 7.03239i −0.225448 0.225448i
\(974\) 0 0
\(975\) 39.2524 1.25708
\(976\) 0 0
\(977\) −49.3566 −1.57906 −0.789529 0.613713i \(-0.789675\pi\)
−0.789529 + 0.613713i \(0.789675\pi\)
\(978\) 0 0
\(979\) 12.8480 + 12.8480i 0.410625 + 0.410625i
\(980\) 0 0
\(981\) 0.398126 0.398126i 0.0127112 0.0127112i
\(982\) 0 0
\(983\) 0.208762i 0.00665848i −0.999994 0.00332924i \(-0.998940\pi\)
0.999994 0.00332924i \(-0.00105973\pi\)
\(984\) 0 0
\(985\) 20.4950i 0.653026i
\(986\) 0 0
\(987\) 8.00666 8.00666i 0.254855 0.254855i
\(988\) 0 0
\(989\) 17.7851 + 17.7851i 0.565533 + 0.565533i
\(990\) 0 0
\(991\) −23.4623 −0.745306 −0.372653 0.927971i \(-0.621552\pi\)
−0.372653 + 0.927971i \(0.621552\pi\)
\(992\) 0 0
\(993\) 29.8094 0.945972
\(994\) 0 0
\(995\) 1.34089 + 1.34089i 0.0425091 + 0.0425091i
\(996\) 0 0
\(997\) 19.7777 19.7777i 0.626366 0.626366i −0.320786 0.947152i \(-0.603947\pi\)
0.947152 + 0.320786i \(0.103947\pi\)
\(998\) 0 0
\(999\) 43.0066i 1.36067i
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 1792.2.m.d.1345.3 yes 8
4.3 odd 2 1792.2.m.b.1345.2 yes 8
8.3 odd 2 1792.2.m.c.1345.3 yes 8
8.5 even 2 1792.2.m.a.1345.2 yes 8
16.3 odd 4 1792.2.m.c.449.3 yes 8
16.5 even 4 inner 1792.2.m.d.449.3 yes 8
16.11 odd 4 1792.2.m.b.449.2 yes 8
16.13 even 4 1792.2.m.a.449.2 8
32.5 even 8 7168.2.a.x.1.1 4
32.11 odd 8 7168.2.a.s.1.1 4
32.21 even 8 7168.2.a.t.1.4 4
32.27 odd 8 7168.2.a.w.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.a.449.2 8 16.13 even 4
1792.2.m.a.1345.2 yes 8 8.5 even 2
1792.2.m.b.449.2 yes 8 16.11 odd 4
1792.2.m.b.1345.2 yes 8 4.3 odd 2
1792.2.m.c.449.3 yes 8 16.3 odd 4
1792.2.m.c.1345.3 yes 8 8.3 odd 2
1792.2.m.d.449.3 yes 8 16.5 even 4 inner
1792.2.m.d.1345.3 yes 8 1.1 even 1 trivial
7168.2.a.s.1.1 4 32.11 odd 8
7168.2.a.t.1.4 4 32.21 even 8
7168.2.a.w.1.4 4 32.27 odd 8
7168.2.a.x.1.1 4 32.5 even 8