Properties

Label 216.3.h.f.53.4
Level $216$
Weight $3$
Character 216.53
Analytic conductor $5.886$
Analytic rank $0$
Dimension $8$
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] = [216,3,Mod(53,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.h (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.242095489024.11
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 2x^{4} - 32x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 53.4
Root \(-0.926315 + 1.77255i\) of defining polynomial
Character \(\chi\) \(=\) 216.53
Dual form 216.3.h.f.53.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.926315 + 1.77255i) q^{2} +(-2.28388 - 3.28388i) q^{4} -5.39773 q^{5} +11.5678 q^{7} +(7.93645 - 1.00639i) q^{8} +O(q^{10})\) \(q+(-0.926315 + 1.77255i) q^{2} +(-2.28388 - 3.28388i) q^{4} -5.39773 q^{5} +11.5678 q^{7} +(7.93645 - 1.00639i) q^{8} +(5.00000 - 9.56776i) q^{10} +1.69247 q^{11} +9.56776i q^{13} +(-10.7154 + 20.5045i) q^{14} +(-5.56776 + 15.0000i) q^{16} +19.2578i q^{17} -1.86447i q^{19} +(12.3278 + 17.7255i) q^{20} +(-1.56776 + 3.00000i) q^{22} +30.3736i q^{23} +4.13553 q^{25} +(-16.9594 - 8.86276i) q^{26} +(-26.4194 - 37.9872i) q^{28} +49.3112 q^{29} +23.4066 q^{31} +(-21.4308 - 23.7639i) q^{32} +(-34.1355 - 17.8388i) q^{34} -62.4397 q^{35} +24.9744i q^{37} +(3.30487 + 1.72709i) q^{38} +(-42.8388 + 5.43224i) q^{40} -68.7984i q^{41} +83.6776i q^{43} +(-3.86541 - 5.55789i) q^{44} +(-53.8388 - 28.1355i) q^{46} +28.2699i q^{47} +84.8132 q^{49} +(-3.83080 + 7.33044i) q^{50} +(31.4194 - 21.8516i) q^{52} +61.3879 q^{53} -9.13553 q^{55} +(91.8069 - 11.6417i) q^{56} +(-45.6776 + 87.4066i) q^{58} -75.3389 q^{59} +4.70329i q^{61} +(-21.6819 + 41.4894i) q^{62} +(61.9744 - 15.9744i) q^{64} -51.6442i q^{65} -99.8132i q^{67} +(63.2405 - 43.9826i) q^{68} +(57.8388 - 110.678i) q^{70} -73.0058i q^{71} +8.72894 q^{73} +(-44.2683 - 23.1341i) q^{74} +(-6.12270 + 4.25823i) q^{76} +19.5782 q^{77} -3.83882 q^{79} +(30.0533 - 80.9660i) q^{80} +(121.949 + 63.7289i) q^{82} +37.0526 q^{83} -103.949i q^{85} +(-148.323 - 77.5118i) q^{86} +(13.4322 - 1.70329i) q^{88} -43.4112i q^{89} +110.678i q^{91} +(99.7434 - 69.3698i) q^{92} +(-50.1099 - 26.1868i) q^{94} +10.0639i q^{95} -86.9487 q^{97} +(-78.5637 + 150.336i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 48 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 48 q^{7} + 40 q^{10} + 32 q^{22} - 56 q^{25} - 100 q^{28} - 80 q^{31} - 184 q^{34} - 120 q^{40} - 208 q^{46} + 144 q^{49} + 140 q^{52} + 16 q^{55} + 80 q^{58} + 184 q^{64} + 240 q^{70} + 248 q^{73} + 196 q^{76} + 192 q^{79} + 352 q^{82} + 152 q^{88} - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.926315 + 1.77255i −0.463157 + 0.886276i
\(3\) 0 0
\(4\) −2.28388 3.28388i −0.570971 0.820971i
\(5\) −5.39773 −1.07955 −0.539773 0.841810i \(-0.681490\pi\)
−0.539773 + 0.841810i \(0.681490\pi\)
\(6\) 0 0
\(7\) 11.5678 1.65254 0.826269 0.563276i \(-0.190459\pi\)
0.826269 + 0.563276i \(0.190459\pi\)
\(8\) 7.93645 1.00639i 0.992056 0.125799i
\(9\) 0 0
\(10\) 5.00000 9.56776i 0.500000 0.956776i
\(11\) 1.69247 0.153861 0.0769307 0.997036i \(-0.475488\pi\)
0.0769307 + 0.997036i \(0.475488\pi\)
\(12\) 0 0
\(13\) 9.56776i 0.735982i 0.929829 + 0.367991i \(0.119954\pi\)
−0.929829 + 0.367991i \(0.880046\pi\)
\(14\) −10.7154 + 20.5045i −0.765385 + 1.46460i
\(15\) 0 0
\(16\) −5.56776 + 15.0000i −0.347985 + 0.937500i
\(17\) 19.2578i 1.13281i 0.824126 + 0.566407i \(0.191667\pi\)
−0.824126 + 0.566407i \(0.808333\pi\)
\(18\) 0 0
\(19\) 1.86447i 0.0981301i −0.998796 0.0490650i \(-0.984376\pi\)
0.998796 0.0490650i \(-0.0156242\pi\)
\(20\) 12.3278 + 17.7255i 0.616389 + 0.886276i
\(21\) 0 0
\(22\) −1.56776 + 3.00000i −0.0712620 + 0.136364i
\(23\) 30.3736i 1.32059i 0.751005 + 0.660296i \(0.229569\pi\)
−0.751005 + 0.660296i \(0.770431\pi\)
\(24\) 0 0
\(25\) 4.13553 0.165421
\(26\) −16.9594 8.86276i −0.652283 0.340875i
\(27\) 0 0
\(28\) −26.4194 37.9872i −0.943550 1.35668i
\(29\) 49.3112 1.70038 0.850192 0.526472i \(-0.176486\pi\)
0.850192 + 0.526472i \(0.176486\pi\)
\(30\) 0 0
\(31\) 23.4066 0.755051 0.377526 0.925999i \(-0.376775\pi\)
0.377526 + 0.925999i \(0.376775\pi\)
\(32\) −21.4308 23.7639i −0.669712 0.742621i
\(33\) 0 0
\(34\) −34.1355 17.8388i −1.00399 0.524671i
\(35\) −62.4397 −1.78399
\(36\) 0 0
\(37\) 24.9744i 0.674982i 0.941329 + 0.337491i \(0.109578\pi\)
−0.941329 + 0.337491i \(0.890422\pi\)
\(38\) 3.30487 + 1.72709i 0.0869703 + 0.0454497i
\(39\) 0 0
\(40\) −42.8388 + 5.43224i −1.07097 + 0.135806i
\(41\) 68.7984i 1.67801i −0.544124 0.839005i \(-0.683138\pi\)
0.544124 0.839005i \(-0.316862\pi\)
\(42\) 0 0
\(43\) 83.6776i 1.94599i 0.230823 + 0.972996i \(0.425858\pi\)
−0.230823 + 0.972996i \(0.574142\pi\)
\(44\) −3.86541 5.55789i −0.0878503 0.126316i
\(45\) 0 0
\(46\) −53.8388 28.1355i −1.17041 0.611642i
\(47\) 28.2699i 0.601487i 0.953705 + 0.300744i \(0.0972349\pi\)
−0.953705 + 0.300744i \(0.902765\pi\)
\(48\) 0 0
\(49\) 84.8132 1.73088
\(50\) −3.83080 + 7.33044i −0.0766160 + 0.146609i
\(51\) 0 0
\(52\) 31.4194 21.8516i 0.604219 0.420224i
\(53\) 61.3879 1.15826 0.579131 0.815235i \(-0.303392\pi\)
0.579131 + 0.815235i \(0.303392\pi\)
\(54\) 0 0
\(55\) −9.13553 −0.166101
\(56\) 91.8069 11.6417i 1.63941 0.207888i
\(57\) 0 0
\(58\) −45.6776 + 87.4066i −0.787546 + 1.50701i
\(59\) −75.3389 −1.27693 −0.638465 0.769651i \(-0.720430\pi\)
−0.638465 + 0.769651i \(0.720430\pi\)
\(60\) 0 0
\(61\) 4.70329i 0.0771032i 0.999257 + 0.0385516i \(0.0122744\pi\)
−0.999257 + 0.0385516i \(0.987726\pi\)
\(62\) −21.6819 + 41.4894i −0.349707 + 0.669184i
\(63\) 0 0
\(64\) 61.9744 15.9744i 0.968349 0.249599i
\(65\) 51.6442i 0.794527i
\(66\) 0 0
\(67\) 99.8132i 1.48975i −0.667205 0.744874i \(-0.732509\pi\)
0.667205 0.744874i \(-0.267491\pi\)
\(68\) 63.2405 43.9826i 0.930007 0.646804i
\(69\) 0 0
\(70\) 57.8388 110.678i 0.826269 1.58111i
\(71\) 73.0058i 1.02825i −0.857715 0.514125i \(-0.828117\pi\)
0.857715 0.514125i \(-0.171883\pi\)
\(72\) 0 0
\(73\) 8.72894 0.119575 0.0597873 0.998211i \(-0.480958\pi\)
0.0597873 + 0.998211i \(0.480958\pi\)
\(74\) −44.2683 23.1341i −0.598221 0.312623i
\(75\) 0 0
\(76\) −6.12270 + 4.25823i −0.0805619 + 0.0560294i
\(77\) 19.5782 0.254262
\(78\) 0 0
\(79\) −3.83882 −0.0485927 −0.0242963 0.999705i \(-0.507735\pi\)
−0.0242963 + 0.999705i \(0.507735\pi\)
\(80\) 30.0533 80.9660i 0.375666 1.01208i
\(81\) 0 0
\(82\) 121.949 + 63.7289i 1.48718 + 0.777182i
\(83\) 37.0526 0.446417 0.223208 0.974771i \(-0.428347\pi\)
0.223208 + 0.974771i \(0.428347\pi\)
\(84\) 0 0
\(85\) 103.949i 1.22293i
\(86\) −148.323 77.5118i −1.72469 0.901300i
\(87\) 0 0
\(88\) 13.4322 1.70329i 0.152639 0.0193556i
\(89\) 43.4112i 0.487767i −0.969805 0.243883i \(-0.921579\pi\)
0.969805 0.243883i \(-0.0784214\pi\)
\(90\) 0 0
\(91\) 110.678i 1.21624i
\(92\) 99.7434 69.3698i 1.08417 0.754019i
\(93\) 0 0
\(94\) −50.1099 26.1868i −0.533084 0.278583i
\(95\) 10.0639i 0.105936i
\(96\) 0 0
\(97\) −86.9487 −0.896378 −0.448189 0.893939i \(-0.647931\pi\)
−0.448189 + 0.893939i \(0.647931\pi\)
\(98\) −78.5637 + 150.336i −0.801670 + 1.53404i
\(99\) 0 0
\(100\) −9.44506 13.5806i −0.0944506 0.135806i
\(101\) 51.4149 0.509058 0.254529 0.967065i \(-0.418080\pi\)
0.254529 + 0.967065i \(0.418080\pi\)
\(102\) 0 0
\(103\) 47.2454 0.458693 0.229347 0.973345i \(-0.426341\pi\)
0.229347 + 0.973345i \(0.426341\pi\)
\(104\) 9.62892 + 75.9340i 0.0925858 + 0.730135i
\(105\) 0 0
\(106\) −56.8645 + 108.813i −0.536457 + 1.02654i
\(107\) −75.1570 −0.702402 −0.351201 0.936300i \(-0.614227\pi\)
−0.351201 + 0.936300i \(0.614227\pi\)
\(108\) 0 0
\(109\) 133.136i 1.22143i 0.791852 + 0.610713i \(0.209117\pi\)
−0.791852 + 0.610713i \(0.790883\pi\)
\(110\) 8.46237 16.1932i 0.0769307 0.147211i
\(111\) 0 0
\(112\) −64.4066 + 173.516i −0.575059 + 1.54925i
\(113\) 90.1599i 0.797875i 0.916978 + 0.398938i \(0.130621\pi\)
−0.916978 + 0.398938i \(0.869379\pi\)
\(114\) 0 0
\(115\) 163.949i 1.42564i
\(116\) −112.621 161.932i −0.970869 1.39597i
\(117\) 0 0
\(118\) 69.7875 133.542i 0.591420 1.13171i
\(119\) 222.770i 1.87202i
\(120\) 0 0
\(121\) −118.136 −0.976327
\(122\) −8.33683 4.35673i −0.0683347 0.0357109i
\(123\) 0 0
\(124\) −53.4579 76.8645i −0.431112 0.619875i
\(125\) 112.621 0.900967
\(126\) 0 0
\(127\) −111.355 −0.876813 −0.438407 0.898777i \(-0.644457\pi\)
−0.438407 + 0.898777i \(0.644457\pi\)
\(128\) −29.0924 + 124.650i −0.227284 + 0.973828i
\(129\) 0 0
\(130\) 91.5421 + 47.8388i 0.704170 + 0.367991i
\(131\) −92.4930 −0.706054 −0.353027 0.935613i \(-0.614848\pi\)
−0.353027 + 0.935613i \(0.614848\pi\)
\(132\) 0 0
\(133\) 21.5678i 0.162164i
\(134\) 176.924 + 92.4584i 1.32033 + 0.689988i
\(135\) 0 0
\(136\) 19.3809 + 152.839i 0.142507 + 1.12381i
\(137\) 199.396i 1.45544i −0.685872 0.727722i \(-0.740579\pi\)
0.685872 0.727722i \(-0.259421\pi\)
\(138\) 0 0
\(139\) 99.4908i 0.715761i −0.933767 0.357881i \(-0.883499\pi\)
0.933767 0.357881i \(-0.116501\pi\)
\(140\) 142.605 + 205.045i 1.01861 + 1.46460i
\(141\) 0 0
\(142\) 129.407 + 67.6263i 0.911314 + 0.476242i
\(143\) 16.1932i 0.113239i
\(144\) 0 0
\(145\) −266.168 −1.83564
\(146\) −8.08575 + 15.4725i −0.0553818 + 0.105976i
\(147\) 0 0
\(148\) 82.0128 57.0385i 0.554141 0.385395i
\(149\) 100.085 0.671714 0.335857 0.941913i \(-0.390974\pi\)
0.335857 + 0.941913i \(0.390974\pi\)
\(150\) 0 0
\(151\) −185.788 −1.23038 −0.615190 0.788379i \(-0.710921\pi\)
−0.615190 + 0.788379i \(0.710921\pi\)
\(152\) −1.87639 14.7973i −0.0123447 0.0973505i
\(153\) 0 0
\(154\) −18.1355 + 34.7033i −0.117763 + 0.225346i
\(155\) −126.343 −0.815113
\(156\) 0 0
\(157\) 201.575i 1.28392i −0.766739 0.641959i \(-0.778122\pi\)
0.766739 0.641959i \(-0.221878\pi\)
\(158\) 3.55596 6.80451i 0.0225061 0.0430665i
\(159\) 0 0
\(160\) 115.678 + 128.271i 0.722985 + 0.801694i
\(161\) 351.355i 2.18233i
\(162\) 0 0
\(163\) 19.8645i 0.121868i 0.998142 + 0.0609340i \(0.0194079\pi\)
−0.998142 + 0.0609340i \(0.980592\pi\)
\(164\) −225.926 + 157.127i −1.37760 + 0.958094i
\(165\) 0 0
\(166\) −34.3224 + 65.6776i −0.206761 + 0.395648i
\(167\) 22.3225i 0.133668i 0.997764 + 0.0668338i \(0.0212897\pi\)
−0.997764 + 0.0668338i \(0.978710\pi\)
\(168\) 0 0
\(169\) 77.4579 0.458331
\(170\) 184.254 + 96.2892i 1.08385 + 0.566407i
\(171\) 0 0
\(172\) 274.788 191.110i 1.59760 1.11110i
\(173\) −262.476 −1.51720 −0.758602 0.651555i \(-0.774117\pi\)
−0.758602 + 0.651555i \(0.774117\pi\)
\(174\) 0 0
\(175\) 47.8388 0.273365
\(176\) −9.42330 + 25.3871i −0.0535415 + 0.144245i
\(177\) 0 0
\(178\) 76.9487 + 40.2125i 0.432296 + 0.225913i
\(179\) −88.1037 −0.492200 −0.246100 0.969244i \(-0.579149\pi\)
−0.246100 + 0.969244i \(0.579149\pi\)
\(180\) 0 0
\(181\) 201.568i 1.11363i −0.830635 0.556817i \(-0.812022\pi\)
0.830635 0.556817i \(-0.187978\pi\)
\(182\) −196.182 102.522i −1.07792 0.563309i
\(183\) 0 0
\(184\) 30.5678 + 241.059i 0.166129 + 1.31010i
\(185\) 134.805i 0.728675i
\(186\) 0 0
\(187\) 32.5934i 0.174296i
\(188\) 92.8350 64.5651i 0.493803 0.343432i
\(189\) 0 0
\(190\) −17.8388 9.32236i −0.0938885 0.0490650i
\(191\) 28.0880i 0.147058i −0.997293 0.0735289i \(-0.976574\pi\)
0.997293 0.0735289i \(-0.0234261\pi\)
\(192\) 0 0
\(193\) 265.762 1.37700 0.688502 0.725234i \(-0.258269\pi\)
0.688502 + 0.725234i \(0.258269\pi\)
\(194\) 80.5419 154.121i 0.415164 0.794439i
\(195\) 0 0
\(196\) −193.703 278.516i −0.988282 1.42100i
\(197\) −27.9930 −0.142096 −0.0710482 0.997473i \(-0.522634\pi\)
−0.0710482 + 0.997473i \(0.522634\pi\)
\(198\) 0 0
\(199\) 7.34801 0.0369247 0.0184623 0.999830i \(-0.494123\pi\)
0.0184623 + 0.999830i \(0.494123\pi\)
\(200\) 32.8214 4.16196i 0.164107 0.0208098i
\(201\) 0 0
\(202\) −47.6263 + 91.1355i −0.235774 + 0.451166i
\(203\) 570.420 2.80995
\(204\) 0 0
\(205\) 371.355i 1.81149i
\(206\) −43.7641 + 83.7449i −0.212447 + 0.406529i
\(207\) 0 0
\(208\) −143.516 53.2711i −0.689983 0.256111i
\(209\) 3.15557i 0.0150984i
\(210\) 0 0
\(211\) 140.678i 0.666719i 0.942800 + 0.333359i \(0.108182\pi\)
−0.942800 + 0.333359i \(0.891818\pi\)
\(212\) −140.203 201.590i −0.661333 0.950898i
\(213\) 0 0
\(214\) 69.6191 133.220i 0.325323 0.622522i
\(215\) 451.670i 2.10079i
\(216\) 0 0
\(217\) 270.762 1.24775
\(218\) −235.990 123.325i −1.08252 0.565713i
\(219\) 0 0
\(220\) 20.8645 + 30.0000i 0.0948385 + 0.136364i
\(221\) −184.254 −0.833731
\(222\) 0 0
\(223\) −0.593414 −0.00266105 −0.00133052 0.999999i \(-0.500424\pi\)
−0.00133052 + 0.999999i \(0.500424\pi\)
\(224\) −247.906 274.895i −1.10672 1.22721i
\(225\) 0 0
\(226\) −159.813 83.5165i −0.707138 0.369542i
\(227\) 381.044 1.67861 0.839305 0.543661i \(-0.182962\pi\)
0.839305 + 0.543661i \(0.182962\pi\)
\(228\) 0 0
\(229\) 127.458i 0.556585i −0.960496 0.278292i \(-0.910232\pi\)
0.960496 0.278292i \(-0.0897684\pi\)
\(230\) 290.608 + 151.868i 1.26351 + 0.660296i
\(231\) 0 0
\(232\) 391.355 49.6263i 1.68688 0.213907i
\(233\) 427.153i 1.83327i 0.399722 + 0.916636i \(0.369107\pi\)
−0.399722 + 0.916636i \(0.630893\pi\)
\(234\) 0 0
\(235\) 152.593i 0.649334i
\(236\) 172.065 + 247.404i 0.729090 + 1.04832i
\(237\) 0 0
\(238\) −394.872 206.355i −1.65913 0.867039i
\(239\) 190.475i 0.796965i 0.917176 + 0.398483i \(0.130463\pi\)
−0.917176 + 0.398483i \(0.869537\pi\)
\(240\) 0 0
\(241\) −2.78024 −0.0115363 −0.00576814 0.999983i \(-0.501836\pi\)
−0.00576814 + 0.999983i \(0.501836\pi\)
\(242\) 109.431 209.401i 0.452193 0.865295i
\(243\) 0 0
\(244\) 15.4451 10.7418i 0.0632994 0.0440236i
\(245\) −457.799 −1.86857
\(246\) 0 0
\(247\) 17.8388 0.0722220
\(248\) 185.765 23.5562i 0.749053 0.0949847i
\(249\) 0 0
\(250\) −104.322 + 199.626i −0.417289 + 0.798505i
\(251\) 100.085 0.398747 0.199373 0.979924i \(-0.436109\pi\)
0.199373 + 0.979924i \(0.436109\pi\)
\(252\) 0 0
\(253\) 51.4066i 0.203188i
\(254\) 103.150 197.383i 0.406103 0.777099i
\(255\) 0 0
\(256\) −194.000 167.033i −0.757812 0.652472i
\(257\) 344.719i 1.34132i −0.741765 0.670660i \(-0.766011\pi\)
0.741765 0.670660i \(-0.233989\pi\)
\(258\) 0 0
\(259\) 288.897i 1.11543i
\(260\) −169.594 + 117.949i −0.652283 + 0.453651i
\(261\) 0 0
\(262\) 85.6776 163.949i 0.327014 0.625758i
\(263\) 169.801i 0.645632i −0.946462 0.322816i \(-0.895370\pi\)
0.946462 0.322816i \(-0.104630\pi\)
\(264\) 0 0
\(265\) −331.355 −1.25040
\(266\) 38.2300 + 19.9785i 0.143722 + 0.0751073i
\(267\) 0 0
\(268\) −327.775 + 227.962i −1.22304 + 0.850603i
\(269\) 263.666 0.980173 0.490086 0.871674i \(-0.336965\pi\)
0.490086 + 0.871674i \(0.336965\pi\)
\(270\) 0 0
\(271\) 55.6849 0.205479 0.102740 0.994708i \(-0.467239\pi\)
0.102740 + 0.994708i \(0.467239\pi\)
\(272\) −288.868 107.223i −1.06201 0.394203i
\(273\) 0 0
\(274\) 353.440 + 184.703i 1.28993 + 0.674100i
\(275\) 6.99928 0.0254519
\(276\) 0 0
\(277\) 244.051i 0.881052i −0.897740 0.440526i \(-0.854792\pi\)
0.897740 0.440526i \(-0.145208\pi\)
\(278\) 176.353 + 92.1598i 0.634362 + 0.331510i
\(279\) 0 0
\(280\) −495.549 + 62.8388i −1.76982 + 0.224424i
\(281\) 123.780i 0.440498i −0.975444 0.220249i \(-0.929313\pi\)
0.975444 0.220249i \(-0.0706871\pi\)
\(282\) 0 0
\(283\) 41.5092i 0.146676i −0.997307 0.0733378i \(-0.976635\pi\)
0.997307 0.0733378i \(-0.0233651\pi\)
\(284\) −239.742 + 166.737i −0.844164 + 0.587101i
\(285\) 0 0
\(286\) −28.7033 15.0000i −0.100361 0.0524476i
\(287\) 795.843i 2.77297i
\(288\) 0 0
\(289\) −81.8645 −0.283268
\(290\) 246.556 471.797i 0.850192 1.62689i
\(291\) 0 0
\(292\) −19.9359 28.6648i −0.0682735 0.0981672i
\(293\) −371.985 −1.26957 −0.634786 0.772688i \(-0.718912\pi\)
−0.634786 + 0.772688i \(0.718912\pi\)
\(294\) 0 0
\(295\) 406.659 1.37851
\(296\) 25.1340 + 198.208i 0.0849121 + 0.669620i
\(297\) 0 0
\(298\) −92.7106 + 177.407i −0.311109 + 0.595324i
\(299\) −290.608 −0.971932
\(300\) 0 0
\(301\) 967.963i 3.21582i
\(302\) 172.098 329.318i 0.569860 1.09046i
\(303\) 0 0
\(304\) 27.9671 + 10.3809i 0.0919969 + 0.0341478i
\(305\) 25.3871i 0.0832365i
\(306\) 0 0
\(307\) 215.033i 0.700433i 0.936669 + 0.350216i \(0.113892\pi\)
−0.936669 + 0.350216i \(0.886108\pi\)
\(308\) −44.7142 64.2923i −0.145176 0.208741i
\(309\) 0 0
\(310\) 117.033 223.949i 0.377526 0.722415i
\(311\) 200.539i 0.644819i 0.946600 + 0.322409i \(0.104493\pi\)
−0.946600 + 0.322409i \(0.895507\pi\)
\(312\) 0 0
\(313\) 142.033 0.453779 0.226890 0.973920i \(-0.427144\pi\)
0.226890 + 0.973920i \(0.427144\pi\)
\(314\) 357.302 + 186.722i 1.13791 + 0.594656i
\(315\) 0 0
\(316\) 8.76742 + 12.6062i 0.0277450 + 0.0398932i
\(317\) −3.20309 −0.0101044 −0.00505219 0.999987i \(-0.501608\pi\)
−0.00505219 + 0.999987i \(0.501608\pi\)
\(318\) 0 0
\(319\) 83.4579 0.261623
\(320\) −334.521 + 86.2253i −1.04538 + 0.269454i
\(321\) 0 0
\(322\) −622.795 325.465i −1.93415 1.01076i
\(323\) 35.9057 0.111163
\(324\) 0 0
\(325\) 39.5678i 0.121747i
\(326\) −35.2108 18.4008i −0.108009 0.0564440i
\(327\) 0 0
\(328\) −69.2381 546.015i −0.211092 1.66468i
\(329\) 327.020i 0.993981i
\(330\) 0 0
\(331\) 328.136i 0.991346i −0.868509 0.495673i \(-0.834921\pi\)
0.868509 0.495673i \(-0.165079\pi\)
\(332\) −84.6237 121.676i −0.254891 0.366495i
\(333\) 0 0
\(334\) −39.5678 20.6776i −0.118466 0.0619091i
\(335\) 538.765i 1.60825i
\(336\) 0 0
\(337\) 430.678 1.27798 0.638988 0.769217i \(-0.279354\pi\)
0.638988 + 0.769217i \(0.279354\pi\)
\(338\) −71.7504 + 137.298i −0.212279 + 0.406208i
\(339\) 0 0
\(340\) −341.355 + 237.407i −1.00399 + 0.698255i
\(341\) 39.6151 0.116173
\(342\) 0 0
\(343\) 414.278 1.20781
\(344\) 84.2125 + 664.103i 0.244804 + 1.93053i
\(345\) 0 0
\(346\) 243.136 465.253i 0.702704 1.34466i
\(347\) 152.418 0.439244 0.219622 0.975585i \(-0.429518\pi\)
0.219622 + 0.975585i \(0.429518\pi\)
\(348\) 0 0
\(349\) 370.718i 1.06223i −0.847300 0.531114i \(-0.821773\pi\)
0.847300 0.531114i \(-0.178227\pi\)
\(350\) −44.3138 + 84.7968i −0.126611 + 0.242277i
\(351\) 0 0
\(352\) −36.2711 40.2198i −0.103043 0.114261i
\(353\) 64.7728i 0.183492i 0.995782 + 0.0917462i \(0.0292448\pi\)
−0.995782 + 0.0917462i \(0.970755\pi\)
\(354\) 0 0
\(355\) 394.066i 1.11004i
\(356\) −142.557 + 99.1462i −0.400442 + 0.278500i
\(357\) 0 0
\(358\) 81.6118 156.168i 0.227966 0.436225i
\(359\) 375.093i 1.04483i 0.852692 + 0.522414i \(0.174968\pi\)
−0.852692 + 0.522414i \(0.825032\pi\)
\(360\) 0 0
\(361\) 357.524 0.990370
\(362\) 357.289 + 186.715i 0.986987 + 0.515788i
\(363\) 0 0
\(364\) 363.452 252.775i 0.998495 0.694436i
\(365\) −47.1165 −0.129086
\(366\) 0 0
\(367\) 367.839 1.00229 0.501143 0.865365i \(-0.332913\pi\)
0.501143 + 0.865365i \(0.332913\pi\)
\(368\) −455.604 169.113i −1.23806 0.459547i
\(369\) 0 0
\(370\) 238.949 + 124.872i 0.645807 + 0.337491i
\(371\) 710.120 1.91407
\(372\) 0 0
\(373\) 247.128i 0.662542i −0.943536 0.331271i \(-0.892523\pi\)
0.943536 0.331271i \(-0.107477\pi\)
\(374\) −57.7735 30.1918i −0.154475 0.0807266i
\(375\) 0 0
\(376\) 28.4506 + 224.363i 0.0756665 + 0.596709i
\(377\) 471.797i 1.25145i
\(378\) 0 0
\(379\) 401.776i 1.06010i −0.847968 0.530048i \(-0.822174\pi\)
0.847968 0.530048i \(-0.177826\pi\)
\(380\) 33.0487 22.9848i 0.0869703 0.0604863i
\(381\) 0 0
\(382\) 49.7875 + 26.0184i 0.130334 + 0.0681109i
\(383\) 319.475i 0.834138i −0.908875 0.417069i \(-0.863057\pi\)
0.908875 0.417069i \(-0.136943\pi\)
\(384\) 0 0
\(385\) −105.678 −0.274487
\(386\) −246.179 + 471.077i −0.637770 + 1.22041i
\(387\) 0 0
\(388\) 198.581 + 285.529i 0.511806 + 0.735900i
\(389\) 213.802 0.549618 0.274809 0.961499i \(-0.411385\pi\)
0.274809 + 0.961499i \(0.411385\pi\)
\(390\) 0 0
\(391\) −584.930 −1.49599
\(392\) 673.115 85.3553i 1.71713 0.217743i
\(393\) 0 0
\(394\) 25.9303 49.6191i 0.0658130 0.125937i
\(395\) 20.7209 0.0524581
\(396\) 0 0
\(397\) 421.136i 1.06079i 0.847749 + 0.530397i \(0.177957\pi\)
−0.847749 + 0.530397i \(0.822043\pi\)
\(398\) −6.80657 + 13.0247i −0.0171019 + 0.0327254i
\(399\) 0 0
\(400\) −23.0256 + 62.0329i −0.0575641 + 0.155082i
\(401\) 203.097i 0.506476i 0.967404 + 0.253238i \(0.0814956\pi\)
−0.967404 + 0.253238i \(0.918504\pi\)
\(402\) 0 0
\(403\) 223.949i 0.555704i
\(404\) −117.425 168.840i −0.290657 0.417922i
\(405\) 0 0
\(406\) −528.388 + 1011.10i −1.30145 + 2.49039i
\(407\) 42.2685i 0.103854i
\(408\) 0 0
\(409\) −622.267 −1.52144 −0.760718 0.649083i \(-0.775153\pi\)
−0.760718 + 0.649083i \(0.775153\pi\)
\(410\) −658.247 343.992i −1.60548 0.839005i
\(411\) 0 0
\(412\) −107.903 155.148i −0.261900 0.376574i
\(413\) −871.503 −2.11018
\(414\) 0 0
\(415\) −200.000 −0.481928
\(416\) 227.367 205.045i 0.546556 0.492896i
\(417\) 0 0
\(418\) 5.59341 + 2.92305i 0.0133814 + 0.00699295i
\(419\) −271.397 −0.647726 −0.323863 0.946104i \(-0.604982\pi\)
−0.323863 + 0.946104i \(0.604982\pi\)
\(420\) 0 0
\(421\) 610.381i 1.44984i −0.688835 0.724918i \(-0.741878\pi\)
0.688835 0.724918i \(-0.258122\pi\)
\(422\) −249.358 130.312i −0.590897 0.308796i
\(423\) 0 0
\(424\) 487.201 61.7802i 1.14906 0.145708i
\(425\) 79.6414i 0.187391i
\(426\) 0 0
\(427\) 54.4066i 0.127416i
\(428\) 171.650 + 246.807i 0.401051 + 0.576651i
\(429\) 0 0
\(430\) 800.608 + 418.388i 1.86188 + 0.972996i
\(431\) 303.282i 0.703669i 0.936062 + 0.351835i \(0.114442\pi\)
−0.936062 + 0.351835i \(0.885558\pi\)
\(432\) 0 0
\(433\) 224.659 0.518844 0.259422 0.965764i \(-0.416468\pi\)
0.259422 + 0.965764i \(0.416468\pi\)
\(434\) −250.811 + 479.940i −0.577905 + 1.10585i
\(435\) 0 0
\(436\) 437.201 304.066i 1.00276 0.697399i
\(437\) 56.6307 0.129590
\(438\) 0 0
\(439\) −828.930 −1.88822 −0.944112 0.329625i \(-0.893078\pi\)
−0.944112 + 0.329625i \(0.893078\pi\)
\(440\) −72.5036 + 9.19392i −0.164781 + 0.0208953i
\(441\) 0 0
\(442\) 170.678 326.601i 0.386149 0.738916i
\(443\) 286.716 0.647215 0.323608 0.946191i \(-0.395104\pi\)
0.323608 + 0.946191i \(0.395104\pi\)
\(444\) 0 0
\(445\) 234.322i 0.526567i
\(446\) 0.549688 1.05186i 0.00123248 0.00235842i
\(447\) 0 0
\(448\) 716.905 184.788i 1.60023 0.412472i
\(449\) 196.928i 0.438593i −0.975658 0.219297i \(-0.929624\pi\)
0.975658 0.219297i \(-0.0703763\pi\)
\(450\) 0 0
\(451\) 116.440i 0.258181i
\(452\) 296.075 205.915i 0.655032 0.455563i
\(453\) 0 0
\(454\) −352.967 + 675.421i −0.777461 + 1.48771i
\(455\) 597.408i 1.31299i
\(456\) 0 0
\(457\) 7.03293 0.0153893 0.00769467 0.999970i \(-0.497551\pi\)
0.00769467 + 0.999970i \(0.497551\pi\)
\(458\) 225.926 + 118.066i 0.493288 + 0.257786i
\(459\) 0 0
\(460\) −538.388 + 374.440i −1.17041 + 0.813999i
\(461\) 217.282 0.471326 0.235663 0.971835i \(-0.424274\pi\)
0.235663 + 0.971835i \(0.424274\pi\)
\(462\) 0 0
\(463\) −538.344 −1.16273 −0.581365 0.813643i \(-0.697481\pi\)
−0.581365 + 0.813643i \(0.697481\pi\)
\(464\) −274.553 + 739.667i −0.591709 + 1.59411i
\(465\) 0 0
\(466\) −757.150 395.678i −1.62479 0.849094i
\(467\) −612.637 −1.31186 −0.655928 0.754824i \(-0.727722\pi\)
−0.655928 + 0.754824i \(0.727722\pi\)
\(468\) 0 0
\(469\) 1154.62i 2.46187i
\(470\) 270.480 + 141.350i 0.575489 + 0.300744i
\(471\) 0 0
\(472\) −597.923 + 75.8205i −1.26679 + 0.160637i
\(473\) 141.622i 0.299413i
\(474\) 0 0
\(475\) 7.71057i 0.0162328i
\(476\) 731.551 508.781i 1.53687 1.06887i
\(477\) 0 0
\(478\) −337.626 176.440i −0.706331 0.369120i
\(479\) 103.652i 0.216393i −0.994130 0.108196i \(-0.965492\pi\)
0.994130 0.108196i \(-0.0345076\pi\)
\(480\) 0 0
\(481\) −238.949 −0.496775
\(482\) 2.57538 4.92812i 0.00534311 0.0102243i
\(483\) 0 0
\(484\) 269.808 + 387.943i 0.557454 + 0.801535i
\(485\) 469.326 0.967682
\(486\) 0 0
\(487\) 163.516 0.335763 0.167881 0.985807i \(-0.446307\pi\)
0.167881 + 0.985807i \(0.446307\pi\)
\(488\) 4.73336 + 37.3274i 0.00969950 + 0.0764906i
\(489\) 0 0
\(490\) 424.066 811.472i 0.865441 1.65607i
\(491\) −770.543 −1.56933 −0.784667 0.619917i \(-0.787166\pi\)
−0.784667 + 0.619917i \(0.787166\pi\)
\(492\) 0 0
\(493\) 949.626i 1.92622i
\(494\) −16.5244 + 31.6202i −0.0334501 + 0.0640086i
\(495\) 0 0
\(496\) −130.322 + 351.099i −0.262747 + 0.707860i
\(497\) 844.514i 1.69922i
\(498\) 0 0
\(499\) 345.253i 0.691889i −0.938255 0.345945i \(-0.887559\pi\)
0.938255 0.345945i \(-0.112441\pi\)
\(500\) −257.213 369.834i −0.514426 0.739667i
\(501\) 0 0
\(502\) −92.7106 + 177.407i −0.184682 + 0.353400i
\(503\) 358.445i 0.712614i 0.934369 + 0.356307i \(0.115964\pi\)
−0.934369 + 0.356307i \(0.884036\pi\)
\(504\) 0 0
\(505\) −277.524 −0.549552
\(506\) −91.1209 47.6187i −0.180081 0.0941081i
\(507\) 0 0
\(508\) 254.322 + 365.678i 0.500635 + 0.719838i
\(509\) 403.367 0.792469 0.396235 0.918149i \(-0.370317\pi\)
0.396235 + 0.918149i \(0.370317\pi\)
\(510\) 0 0
\(511\) 100.974 0.197601
\(512\) 475.780 189.150i 0.929257 0.369434i
\(513\) 0 0
\(514\) 611.033 + 319.319i 1.18878 + 0.621242i
\(515\) −255.018 −0.495181
\(516\) 0 0
\(517\) 47.8461i 0.0925457i
\(518\) −512.086 267.610i −0.988582 0.516621i
\(519\) 0 0
\(520\) −51.9744 409.872i −0.0999507 0.788215i
\(521\) 835.047i 1.60278i −0.598144 0.801389i \(-0.704095\pi\)
0.598144 0.801389i \(-0.295905\pi\)
\(522\) 0 0
\(523\) 21.1539i 0.0404472i −0.999795 0.0202236i \(-0.993562\pi\)
0.999795 0.0202236i \(-0.00643781\pi\)
\(524\) 211.243 + 303.736i 0.403136 + 0.579649i
\(525\) 0 0
\(526\) 300.982 + 157.289i 0.572208 + 0.299029i
\(527\) 450.760i 0.855333i
\(528\) 0 0
\(529\) −393.557 −0.743963
\(530\) 306.939 587.345i 0.579131 1.10820i
\(531\) 0 0
\(532\) −70.8260 + 49.2582i −0.133132 + 0.0925907i
\(533\) 658.247 1.23498
\(534\) 0 0
\(535\) 405.678 0.758276
\(536\) −100.451 792.162i −0.187409 1.47791i
\(537\) 0 0
\(538\) −244.238 + 467.363i −0.453974 + 0.868704i
\(539\) 143.544 0.266316
\(540\) 0 0
\(541\) 130.718i 0.241623i −0.992675 0.120811i \(-0.961450\pi\)
0.992675 0.120811i \(-0.0385496\pi\)
\(542\) −51.5818 + 98.7044i −0.0951693 + 0.182112i
\(543\) 0 0
\(544\) 457.641 412.711i 0.841252 0.758659i
\(545\) 718.630i 1.31859i
\(546\) 0 0
\(547\) 544.963i 0.996277i −0.867098 0.498138i \(-0.834017\pi\)
0.867098 0.498138i \(-0.165983\pi\)
\(548\) −654.792 + 455.397i −1.19488 + 0.831016i
\(549\) 0 0
\(550\) −6.48353 + 12.4066i −0.0117882 + 0.0225574i
\(551\) 91.9392i 0.166859i
\(552\) 0 0
\(553\) −44.4066 −0.0803012
\(554\) 432.594 + 226.068i 0.780855 + 0.408066i
\(555\) 0 0
\(556\) −326.716 + 227.225i −0.587619 + 0.408679i
\(557\) 796.393 1.42979 0.714895 0.699232i \(-0.246474\pi\)
0.714895 + 0.699232i \(0.246474\pi\)
\(558\) 0 0
\(559\) −800.608 −1.43221
\(560\) 347.650 936.596i 0.620803 1.67249i
\(561\) 0 0
\(562\) 219.407 + 114.659i 0.390403 + 0.204020i
\(563\) −519.646 −0.922994 −0.461497 0.887142i \(-0.652687\pi\)
−0.461497 + 0.887142i \(0.652687\pi\)
\(564\) 0 0
\(565\) 486.659i 0.861344i
\(566\) 73.5772 + 38.4506i 0.129995 + 0.0679339i
\(567\) 0 0
\(568\) −73.4724 579.407i −0.129353 1.02008i
\(569\) 676.128i 1.18827i 0.804364 + 0.594137i \(0.202506\pi\)
−0.804364 + 0.594137i \(0.797494\pi\)
\(570\) 0 0
\(571\) 1048.64i 1.83650i 0.396002 + 0.918249i \(0.370397\pi\)
−0.396002 + 0.918249i \(0.629603\pi\)
\(572\) 53.1766 36.9834i 0.0929660 0.0646562i
\(573\) 0 0
\(574\) 1410.67 + 737.201i 2.45762 + 1.28432i
\(575\) 125.611i 0.218454i
\(576\) 0 0
\(577\) −494.150 −0.856413 −0.428206 0.903681i \(-0.640854\pi\)
−0.428206 + 0.903681i \(0.640854\pi\)
\(578\) 75.8323 145.109i 0.131198 0.251054i
\(579\) 0 0
\(580\) 607.897 + 874.066i 1.04810 + 1.50701i
\(581\) 428.616 0.737720
\(582\) 0 0
\(583\) 103.897 0.178212
\(584\) 69.2768 8.78474i 0.118625 0.0150424i
\(585\) 0 0
\(586\) 344.575 659.363i 0.588012 1.12519i
\(587\) −737.611 −1.25658 −0.628289 0.777980i \(-0.716244\pi\)
−0.628289 + 0.777980i \(0.716244\pi\)
\(588\) 0 0
\(589\) 43.6409i 0.0740932i
\(590\) −376.694 + 720.825i −0.638465 + 1.22174i
\(591\) 0 0
\(592\) −374.615 139.051i −0.632796 0.234884i
\(593\) 528.883i 0.891877i −0.895064 0.445938i \(-0.852870\pi\)
0.895064 0.445938i \(-0.147130\pi\)
\(594\) 0 0
\(595\) 1202.45i 2.02093i
\(596\) −228.583 328.669i −0.383529 0.551457i
\(597\) 0 0
\(598\) 269.194 515.117i 0.450157 0.861400i
\(599\) 576.826i 0.962982i 0.876451 + 0.481491i \(0.159905\pi\)
−0.876451 + 0.481491i \(0.840095\pi\)
\(600\) 0 0
\(601\) 353.407 0.588031 0.294015 0.955801i \(-0.405008\pi\)
0.294015 + 0.955801i \(0.405008\pi\)
\(602\) −1715.77 896.639i −2.85011 1.48943i
\(603\) 0 0
\(604\) 424.317 + 610.104i 0.702511 + 1.01011i
\(605\) 637.664 1.05399
\(606\) 0 0
\(607\) 585.634 0.964800 0.482400 0.875951i \(-0.339765\pi\)
0.482400 + 0.875951i \(0.339765\pi\)
\(608\) −44.3071 + 39.9571i −0.0728735 + 0.0657189i
\(609\) 0 0
\(610\) 45.0000 + 23.5165i 0.0737705 + 0.0385516i
\(611\) −270.480 −0.442684
\(612\) 0 0
\(613\) 301.282i 0.491488i −0.969335 0.245744i \(-0.920968\pi\)
0.969335 0.245744i \(-0.0790323\pi\)
\(614\) −381.157 199.188i −0.620777 0.324411i
\(615\) 0 0
\(616\) 155.381 19.7033i 0.252242 0.0319859i
\(617\) 203.058i 0.329105i −0.986368 0.164552i \(-0.947382\pi\)
0.986368 0.164552i \(-0.0526180\pi\)
\(618\) 0 0
\(619\) 667.879i 1.07896i −0.841997 0.539482i \(-0.818620\pi\)
0.841997 0.539482i \(-0.181380\pi\)
\(620\) 288.551 + 414.894i 0.465406 + 0.669184i
\(621\) 0 0
\(622\) −355.465 185.762i −0.571487 0.298653i
\(623\) 502.171i 0.806053i
\(624\) 0 0
\(625\) −711.286 −1.13806
\(626\) −131.567 + 251.761i −0.210171 + 0.402174i
\(627\) 0 0
\(628\) −661.949 + 460.374i −1.05406 + 0.733079i
\(629\) −480.952 −0.764630
\(630\) 0 0
\(631\) 874.755 1.38630 0.693149 0.720794i \(-0.256223\pi\)
0.693149 + 0.720794i \(0.256223\pi\)
\(632\) −30.4666 + 3.86336i −0.0482066 + 0.00611291i
\(633\) 0 0
\(634\) 2.96707 5.67764i 0.00467992 0.00895527i
\(635\) 601.066 0.946561
\(636\) 0 0
\(637\) 811.472i 1.27390i
\(638\) −77.3083 + 147.933i −0.121173 + 0.231871i
\(639\) 0 0
\(640\) 157.033 672.828i 0.245364 1.05129i
\(641\) 160.738i 0.250761i 0.992109 + 0.125380i \(0.0400151\pi\)
−0.992109 + 0.125380i \(0.959985\pi\)
\(642\) 0 0
\(643\) 133.949i 0.208318i −0.994561 0.104159i \(-0.966785\pi\)
0.994561 0.104159i \(-0.0332151\pi\)
\(644\) 1153.81 802.453i 1.79163 1.24605i
\(645\) 0 0
\(646\) −33.2600 + 63.6447i −0.0514860 + 0.0985212i
\(647\) 383.417i 0.592607i −0.955094 0.296304i \(-0.904246\pi\)
0.955094 0.296304i \(-0.0957540\pi\)
\(648\) 0 0
\(649\) −127.509 −0.196470
\(650\) −70.1359 36.6522i −0.107901 0.0563880i
\(651\) 0 0
\(652\) 65.2326 45.3681i 0.100050 0.0695830i
\(653\) −367.323 −0.562516 −0.281258 0.959632i \(-0.590752\pi\)
−0.281258 + 0.959632i \(0.590752\pi\)
\(654\) 0 0
\(655\) 499.253 0.762218
\(656\) 1031.98 + 383.053i 1.57313 + 0.583922i
\(657\) 0 0
\(658\) −579.659 302.923i −0.880941 0.460369i
\(659\) −584.418 −0.886826 −0.443413 0.896317i \(-0.646232\pi\)
−0.443413 + 0.896317i \(0.646232\pi\)
\(660\) 0 0
\(661\) 526.059i 0.795853i −0.917417 0.397926i \(-0.869730\pi\)
0.917417 0.397926i \(-0.130270\pi\)
\(662\) 581.637 + 303.957i 0.878606 + 0.459149i
\(663\) 0 0
\(664\) 294.066 37.2894i 0.442870 0.0561588i
\(665\) 116.417i 0.175063i
\(666\) 0 0
\(667\) 1497.76i 2.24551i
\(668\) 73.3044 50.9819i 0.109737 0.0763202i
\(669\) 0 0
\(670\) −954.989 499.066i −1.42536 0.744874i
\(671\) 7.96021i 0.0118632i
\(672\) 0 0
\(673\) 952.538 1.41536 0.707681 0.706532i \(-0.249742\pi\)
0.707681 + 0.706532i \(0.249742\pi\)
\(674\) −398.943 + 763.399i −0.591904 + 1.13264i
\(675\) 0 0
\(676\) −176.905 254.363i −0.261693 0.376276i
\(677\) −425.772 −0.628910 −0.314455 0.949272i \(-0.601822\pi\)
−0.314455 + 0.949272i \(0.601822\pi\)
\(678\) 0 0
\(679\) −1005.80 −1.48130
\(680\) −104.613 824.983i −0.153843 1.21321i
\(681\) 0 0
\(682\) −36.6960 + 70.2198i −0.0538065 + 0.102962i
\(683\) 319.752 0.468158 0.234079 0.972218i \(-0.424793\pi\)
0.234079 + 0.972218i \(0.424793\pi\)
\(684\) 0 0
\(685\) 1076.29i 1.57122i
\(686\) −383.752 + 734.330i −0.559405 + 1.07045i
\(687\) 0 0
\(688\) −1255.16 465.897i −1.82437 0.677176i
\(689\) 587.345i 0.852459i
\(690\) 0 0
\(691\) 152.454i 0.220628i 0.993897 + 0.110314i \(0.0351857\pi\)
−0.993897 + 0.110314i \(0.964814\pi\)
\(692\) 599.465 + 861.941i 0.866278 + 1.24558i
\(693\) 0 0
\(694\) −141.187 + 270.168i −0.203439 + 0.389292i
\(695\) 537.025i 0.772698i
\(696\) 0 0
\(697\) 1324.91 1.90087
\(698\) 657.117 + 343.401i 0.941428 + 0.491979i
\(699\) 0 0
\(700\) −109.258 157.097i −0.156083 0.224424i
\(701\) 376.643 0.537294 0.268647 0.963239i \(-0.413424\pi\)
0.268647 + 0.963239i \(0.413424\pi\)
\(702\) 0 0
\(703\) 46.5640 0.0662361
\(704\) 104.890 27.0362i 0.148992 0.0384037i
\(705\) 0 0
\(706\) −114.813 60.0000i −0.162625 0.0849858i
\(707\) 594.755 0.841238
\(708\) 0 0
\(709\) 3.09151i 0.00436038i 0.999998 + 0.00218019i \(0.000693977\pi\)
−0.999998 + 0.00218019i \(0.999306\pi\)
\(710\) −698.502 365.029i −0.983806 0.514125i
\(711\) 0 0
\(712\) −43.6887 344.531i −0.0613606 0.483892i
\(713\) 710.943i 0.997115i
\(714\) 0 0
\(715\) 87.4066i 0.122247i
\(716\) 201.219 + 289.322i 0.281031 + 0.404081i
\(717\) 0 0
\(718\) −664.872 347.454i −0.926005 0.483919i
\(719\) 1103.79i 1.53517i 0.640947 + 0.767585i \(0.278542\pi\)
−0.640947 + 0.767585i \(0.721458\pi\)
\(720\) 0 0
\(721\) 546.524 0.758008
\(722\) −331.179 + 633.729i −0.458697 + 0.877742i
\(723\) 0 0
\(724\) −661.925 + 460.357i −0.914261 + 0.635852i
\(725\) 203.928 0.281280
\(726\) 0 0
\(727\) −788.051 −1.08398 −0.541989 0.840386i \(-0.682328\pi\)
−0.541989 + 0.840386i \(0.682328\pi\)
\(728\) 111.385 + 878.387i 0.153001 + 1.20658i
\(729\) 0 0
\(730\) 43.6447 83.5165i 0.0597873 0.114406i
\(731\) −1611.45 −2.20445
\(732\) 0 0
\(733\) 678.996i 0.926325i 0.886273 + 0.463162i \(0.153285\pi\)
−0.886273 + 0.463162i \(0.846715\pi\)
\(734\) −340.735 + 652.013i −0.464216 + 0.888302i
\(735\) 0 0
\(736\) 721.795 650.930i 0.980699 0.884416i
\(737\) 168.931i 0.229215i
\(738\) 0 0
\(739\) 789.744i 1.06867i 0.845274 + 0.534333i \(0.179437\pi\)
−0.845274 + 0.534333i \(0.820563\pi\)
\(740\) −442.683 + 307.878i −0.598221 + 0.416052i
\(741\) 0 0
\(742\) −657.795 + 1258.73i −0.886516 + 1.69640i
\(743\) 216.641i 0.291576i 0.989316 + 0.145788i \(0.0465717\pi\)
−0.989316 + 0.145788i \(0.953428\pi\)
\(744\) 0 0
\(745\) −540.234 −0.725147
\(746\) 438.048 + 228.919i 0.587195 + 0.306861i
\(747\) 0 0
\(748\) 107.033 74.4395i 0.143092 0.0995181i
\(749\) −869.399 −1.16075
\(750\) 0 0
\(751\) 932.681 1.24192 0.620959 0.783843i \(-0.286743\pi\)
0.620959 + 0.783843i \(0.286743\pi\)
\(752\) −424.049 157.400i −0.563894 0.209309i
\(753\) 0 0
\(754\) −836.286 437.033i −1.10913 0.579619i
\(755\) 1002.83 1.32825
\(756\) 0 0
\(757\) 361.451i 0.477478i −0.971084 0.238739i \(-0.923266\pi\)
0.971084 0.238739i \(-0.0767340\pi\)
\(758\) 712.170 + 372.171i 0.939538 + 0.490991i
\(759\) 0 0
\(760\) 10.1282 + 79.8718i 0.0133266 + 0.105094i
\(761\) 883.718i 1.16126i −0.814168 0.580629i \(-0.802807\pi\)
0.814168 0.580629i \(-0.197193\pi\)
\(762\) 0 0
\(763\) 1540.08i 2.01845i
\(764\) −92.2378 + 64.1498i −0.120730 + 0.0839657i
\(765\) 0 0
\(766\) 566.286 + 295.934i 0.739276 + 0.386337i
\(767\) 720.825i 0.939798i
\(768\) 0 0
\(769\) −177.103 −0.230302 −0.115151 0.993348i \(-0.536735\pi\)
−0.115151 + 0.993348i \(0.536735\pi\)
\(770\) 97.8908 187.319i 0.127131 0.243272i
\(771\) 0 0
\(772\) −606.969 872.731i −0.786229 1.13048i
\(773\) −327.067 −0.423114 −0.211557 0.977366i \(-0.567853\pi\)
−0.211557 + 0.977366i \(0.567853\pi\)
\(774\) 0 0
\(775\) 96.7986 0.124901
\(776\) −690.064 + 87.5045i −0.889257 + 0.112763i
\(777\) 0 0
\(778\) −198.047 + 378.974i −0.254560 + 0.487114i
\(779\) −128.273 −0.164663
\(780\) 0 0
\(781\) 123.560i 0.158208i
\(782\) 541.830 1036.82i 0.692877 1.32586i
\(783\) 0 0
\(784\) −472.220 + 1272.20i −0.602321 + 1.62270i
\(785\) 1088.05i 1.38605i
\(786\) 0 0
\(787\) 884.033i 1.12329i −0.827377 0.561647i \(-0.810168\pi\)
0.827377 0.561647i \(-0.189832\pi\)
\(788\) 63.9327 + 91.9257i 0.0811329 + 0.116657i
\(789\) 0 0
\(790\) −19.1941 + 36.7289i −0.0242963 + 0.0464923i
\(791\) 1042.95i 1.31852i
\(792\) 0 0
\(793\) −45.0000 −0.0567465
\(794\) −746.485 390.104i −0.940157 0.491315i
\(795\) 0 0
\(796\) −16.7820 24.1300i −0.0210829 0.0303141i
\(797\) 1262.52 1.58409 0.792043 0.610466i \(-0.209018\pi\)
0.792043 + 0.610466i \(0.209018\pi\)
\(798\) 0 0
\(799\) −544.417 −0.681373
\(800\) −88.6276 98.2762i −0.110785 0.122845i
\(801\) 0 0
\(802\) −360.000 188.132i −0.448878 0.234578i
\(803\) 14.7735 0.0183979
\(804\) 0 0
\(805\) 1896.52i 2.35593i
\(806\) −396.961 207.447i −0.492507 0.257378i
\(807\) 0 0
\(808\) 408.051 51.7435i 0.505014 0.0640390i
\(809\) 1136.54i 1.40487i −0.711750 0.702433i \(-0.752097\pi\)
0.711750 0.702433i \(-0.247903\pi\)
\(810\) 0 0
\(811\) 245.509i 0.302724i −0.988478 0.151362i \(-0.951634\pi\)
0.988478 0.151362i \(-0.0483659\pi\)
\(812\) −1302.77 1873.19i −1.60440 2.30689i
\(813\) 0 0
\(814\) −74.9231 39.1539i −0.0920431 0.0481006i
\(815\) 107.223i 0.131562i
\(816\) 0 0
\(817\) 156.015 0.190960
\(818\) 576.415 1103.00i 0.704664 1.34841i
\(819\) 0 0
\(820\) 1219.49 848.132i 1.48718 1.03431i
\(821\) 733.724 0.893695 0.446848 0.894610i \(-0.352547\pi\)
0.446848 + 0.894610i \(0.352547\pi\)
\(822\) 0 0
\(823\) 1289.78 1.56717 0.783584 0.621285i \(-0.213389\pi\)
0.783584 + 0.621285i \(0.213389\pi\)
\(824\) 374.961 47.5474i 0.455049 0.0577032i
\(825\) 0 0
\(826\) 807.286 1544.78i 0.977343 1.87020i
\(827\) −468.729 −0.566782 −0.283391 0.959004i \(-0.591459\pi\)
−0.283391 + 0.959004i \(0.591459\pi\)
\(828\) 0 0
\(829\) 1356.65i 1.63649i 0.574868 + 0.818246i \(0.305053\pi\)
−0.574868 + 0.818246i \(0.694947\pi\)
\(830\) 185.263 354.510i 0.223208 0.427121i
\(831\) 0 0
\(832\) 152.839 + 592.956i 0.183701 + 0.712687i
\(833\) 1633.32i 1.96077i
\(834\) 0 0
\(835\) 120.491i 0.144300i
\(836\) −10.3625 + 7.20695i −0.0123954 + 0.00862076i
\(837\) 0 0
\(838\) 251.399 481.066i 0.299999 0.574064i
\(839\) 1196.01i 1.42552i −0.701408 0.712760i \(-0.747445\pi\)
0.701408 0.712760i \(-0.252555\pi\)
\(840\) 0 0
\(841\) 1590.59 1.89131
\(842\) 1081.93 + 565.405i 1.28495 + 0.671502i
\(843\) 0 0
\(844\) 461.969 321.291i 0.547356 0.380677i
\(845\) −418.097 −0.494789
\(846\) 0 0
\(847\) −1366.56 −1.61342
\(848\) −341.793 + 920.818i −0.403058 + 1.08587i
\(849\) 0 0
\(850\) −141.168 73.7730i −0.166081 0.0867917i
\(851\) −758.561 −0.891376
\(852\) 0 0
\(853\) 310.242i 0.363706i 0.983326 + 0.181853i \(0.0582096\pi\)
−0.983326 + 0.181853i \(0.941790\pi\)
\(854\) −96.4385 50.3976i −0.112926 0.0590136i
\(855\) 0 0
\(856\) −596.480 + 75.6374i −0.696822 + 0.0883615i
\(857\) 1253.46i 1.46261i −0.682048 0.731307i \(-0.738911\pi\)
0.682048 0.731307i \(-0.261089\pi\)
\(858\) 0 0
\(859\) 979.740i 1.14056i −0.821451 0.570279i \(-0.806835\pi\)
0.821451 0.570279i \(-0.193165\pi\)
\(860\) −1483.23 + 1031.56i −1.72469 + 1.19949i
\(861\) 0 0
\(862\) −537.582 280.934i −0.623645 0.325910i
\(863\) 1240.83i 1.43781i 0.695111 + 0.718903i \(0.255355\pi\)
−0.695111 + 0.718903i \(0.744645\pi\)
\(864\) 0 0
\(865\) 1416.78 1.63789
\(866\) −208.105 + 398.220i −0.240306 + 0.459839i
\(867\) 0 0
\(868\) −618.388 889.150i −0.712429 1.02437i
\(869\) −6.49711 −0.00747654
\(870\) 0 0
\(871\) 954.989 1.09643
\(872\) 133.987 + 1056.62i 0.153654 + 1.21172i
\(873\) 0 0
\(874\) −52.4579 + 100.381i −0.0600205 + 0.114852i
\(875\) 1302.77 1.48888
\(876\) 0 0
\(877\) 320.586i 0.365549i 0.983155 + 0.182774i \(0.0585078\pi\)
−0.983155 + 0.182774i \(0.941492\pi\)
\(878\) 767.850 1469.32i 0.874545 1.67349i
\(879\) 0 0
\(880\) 50.8645 137.033i 0.0578005 0.155719i
\(881\) 208.356i 0.236500i −0.992984 0.118250i \(-0.962272\pi\)
0.992984 0.118250i \(-0.0377284\pi\)
\(882\) 0 0
\(883\) 624.590i 0.707350i 0.935368 + 0.353675i \(0.115068\pi\)
−0.935368 + 0.353675i \(0.884932\pi\)
\(884\) 420.816 + 605.070i 0.476036 + 0.684468i
\(885\) 0 0
\(886\) −265.590 + 508.220i −0.299763 + 0.573611i
\(887\) 1468.63i 1.65573i −0.560926 0.827866i \(-0.689555\pi\)
0.560926 0.827866i \(-0.310445\pi\)
\(888\) 0 0
\(889\) −1288.13 −1.44897
\(890\) −415.349 217.056i −0.466684 0.243883i
\(891\) 0 0
\(892\) 1.35529 + 1.94870i 0.00151938 + 0.00218464i
\(893\) 52.7084 0.0590240
\(894\) 0 0
\(895\) 475.560 0.531352
\(896\) −336.534 + 1441.92i −0.375596 + 1.60929i
\(897\) 0 0
\(898\) 349.066 + 182.418i 0.388715 + 0.203138i
\(899\) 1154.21 1.28388
\(900\) 0 0
\(901\) 1182.20i 1.31209i
\(902\) 206.395 + 107.860i 0.228819 + 0.119578i
\(903\) 0 0
\(904\) 90.7362 + 715.549i 0.100372 + 0.791537i
\(905\) 1088.01i 1.20222i
\(906\) 0 0
\(907\) 418.333i 0.461227i 0.973045 + 0.230614i \(0.0740733\pi\)
−0.973045 + 0.230614i \(0.925927\pi\)
\(908\) −870.261 1251.31i −0.958437 1.37809i
\(909\) 0 0
\(910\) 1058.94 + 553.388i 1.16367 + 0.608119i
\(911\) 69.9928i 0.0768307i 0.999262 + 0.0384154i \(0.0122310\pi\)
−0.999262 + 0.0384154i \(0.987769\pi\)
\(912\) 0 0
\(913\) 62.7106 0.0686863
\(914\) −6.51471 + 12.4662i −0.00712769 + 0.0136392i
\(915\) 0 0
\(916\) −418.557 + 291.099i −0.456940 + 0.317793i
\(917\) −1069.94 −1.16678
\(918\) 0 0
\(919\) 1272.39 1.38454 0.692268 0.721641i \(-0.256612\pi\)
0.692268 + 0.721641i \(0.256612\pi\)
\(920\) −164.997 1301.17i −0.179344 1.41432i
\(921\) 0 0
\(922\) −201.271 + 385.143i −0.218298 + 0.417725i
\(923\) 698.502 0.756774
\(924\) 0 0
\(925\) 103.282i 0.111656i
\(926\) 498.676 954.243i 0.538527 1.03050i
\(927\) 0 0
\(928\) −1056.78 1171.82i −1.13877 1.26274i
\(929\) 675.361i 0.726977i 0.931599 + 0.363488i \(0.118414\pi\)
−0.931599 + 0.363488i \(0.881586\pi\)
\(930\) 0 0
\(931\) 158.132i 0.169851i
\(932\) 1402.72 975.566i 1.50506 1.04674i
\(933\) 0 0
\(934\) 567.494 1085.93i 0.607596 1.16267i
\(935\) 175.931i 0.188161i
\(936\) 0 0
\(937\) −1790.67 −1.91107 −0.955534 0.294882i \(-0.904720\pi\)
−0.955534 + 0.294882i \(0.904720\pi\)
\(938\) 2046.62 + 1069.54i 2.18189 + 1.14023i
\(939\) 0 0
\(940\) −501.099 + 348.505i −0.533084 + 0.370750i
\(941\) −399.697 −0.424758 −0.212379 0.977187i \(-0.568121\pi\)
−0.212379 + 0.977187i \(0.568121\pi\)
\(942\) 0 0
\(943\) 2089.66 2.21597
\(944\) 419.469 1130.08i 0.444353 1.19712i
\(945\) 0 0
\(946\) −251.033 131.187i −0.265363 0.138675i
\(947\) −88.0644 −0.0929931 −0.0464965 0.998918i \(-0.514806\pi\)
−0.0464965 + 0.998918i \(0.514806\pi\)
\(948\) 0 0
\(949\) 83.5165i 0.0880047i
\(950\) 13.6674 + 7.14242i 0.0143867 + 0.00751834i
\(951\) 0 0
\(952\) 224.194 + 1768.00i 0.235498 + 1.85715i
\(953\) 1455.87i 1.52767i −0.645412 0.763835i \(-0.723314\pi\)
0.645412 0.763835i \(-0.276686\pi\)
\(954\) 0 0
\(955\) 151.612i 0.158756i
\(956\) 625.496 435.022i 0.654285 0.455044i
\(957\) 0 0
\(958\) 183.729 + 96.0146i 0.191784 + 0.100224i
\(959\) 2306.56i 2.40518i
\(960\) 0 0
\(961\) −413.132 −0.429898
\(962\) 221.342 423.549i 0.230085 0.440280i
\(963\) 0 0
\(964\) 6.34974 + 9.12999i 0.00658687 + 0.00947094i
\(965\) −1434.51 −1.48654
\(966\) 0 0
\(967\) −1118.86 −1.15704 −0.578520 0.815668i \(-0.696369\pi\)
−0.578520 + 0.815668i \(0.696369\pi\)
\(968\) −937.576 + 118.891i −0.968571 + 0.122821i
\(969\) 0 0
\(970\) −434.744 + 831.905i −0.448189 + 0.857634i
\(971\) 1097.06 1.12983 0.564915 0.825149i \(-0.308909\pi\)
0.564915 + 0.825149i \(0.308909\pi\)
\(972\) 0 0
\(973\) 1150.89i 1.18282i
\(974\) −151.468 + 289.841i −0.155511 + 0.297579i
\(975\) 0 0
\(976\) −70.5494 26.1868i −0.0722842 0.0268308i
\(977\) 590.176i 0.604069i −0.953297 0.302035i \(-0.902334\pi\)
0.953297 0.302035i \(-0.0976658\pi\)
\(978\) 0 0
\(979\) 73.4724i 0.0750485i
\(980\) 1045.56 + 1503.36i 1.06690 + 1.53404i
\(981\) 0 0
\(982\) 713.765 1365.83i 0.726849 1.39086i
\(983\) 1408.99i 1.43336i −0.697403 0.716679i \(-0.745661\pi\)
0.697403 0.716679i \(-0.254339\pi\)
\(984\) 0 0
\(985\) 151.099 0.153400
\(986\) −1683.26 879.653i −1.70716 0.892143i
\(987\) 0 0
\(988\) −40.7418 58.5806i −0.0412366 0.0592921i
\(989\) −2541.59 −2.56986
\(990\) 0 0
\(991\) −939.516 −0.948049 −0.474024 0.880512i \(-0.657199\pi\)
−0.474024 + 0.880512i \(0.657199\pi\)
\(992\) −501.621 556.231i −0.505667 0.560717i
\(993\) 0 0
\(994\) 1496.94 + 782.286i 1.50598 + 0.787008i
\(995\) −39.6626 −0.0398619
\(996\) 0 0
\(997\) 1927.67i 1.93347i −0.255779 0.966735i \(-0.582332\pi\)
0.255779 0.966735i \(-0.417668\pi\)
\(998\) 611.978 + 319.813i 0.613205 + 0.320454i
\(999\) 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 216.3.h.f.53.4 yes 8
3.2 odd 2 inner 216.3.h.f.53.5 yes 8
4.3 odd 2 864.3.h.e.593.2 8
8.3 odd 2 864.3.h.e.593.7 8
8.5 even 2 inner 216.3.h.f.53.6 yes 8
12.11 even 2 864.3.h.e.593.8 8
24.5 odd 2 inner 216.3.h.f.53.3 8
24.11 even 2 864.3.h.e.593.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.3.h.f.53.3 8 24.5 odd 2 inner
216.3.h.f.53.4 yes 8 1.1 even 1 trivial
216.3.h.f.53.5 yes 8 3.2 odd 2 inner
216.3.h.f.53.6 yes 8 8.5 even 2 inner
864.3.h.e.593.1 8 24.11 even 2
864.3.h.e.593.2 8 4.3 odd 2
864.3.h.e.593.7 8 8.3 odd 2
864.3.h.e.593.8 8 12.11 even 2