Properties

Label 384.3.h.g.65.7
Level $384$
Weight $3$
Character 384.65
Analytic conductor $10.463$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,3,Mod(65,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, 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("384.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.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: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 28x^{14} + 274x^{12} + 1236x^{10} + 2703x^{8} + 2676x^{6} + 946x^{4} + 64x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{46}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 65.7
Root \(0.736993i\) of defining polynomial
Character \(\chi\) \(=\) 384.65
Dual form 384.3.h.g.65.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.628052 + 2.93352i) q^{3} -6.51323 q^{5} -7.64344 q^{7} +(-8.21110 - 3.68481i) q^{9} +O(q^{10})\) \(q+(-0.628052 + 2.93352i) q^{3} -6.51323 q^{5} -7.64344 q^{7} +(-8.21110 - 3.68481i) q^{9} +17.8506 q^{11} +14.4222i q^{13} +(4.09065 - 19.1067i) q^{15} -18.6833i q^{17} -12.9726i q^{19} +(4.80048 - 22.4222i) q^{21} -28.1646i q^{23} +17.4222 q^{25} +(15.9665 - 21.7732i) q^{27} -3.08772 q^{29} +32.1873 q^{31} +(-11.2111 + 52.3652i) q^{33} +49.7835 q^{35} +1.57779i q^{37} +(-42.3079 - 9.05789i) q^{39} +18.1647i q^{41} +22.2296i q^{43} +(53.4808 + 24.0000i) q^{45} -86.4757i q^{47} +9.42221 q^{49} +(54.8079 + 11.7341i) q^{51} -25.7151 q^{53} -116.265 q^{55} +(38.0555 + 8.14748i) q^{57} -11.3049 q^{59} -91.2666i q^{61} +(62.7611 + 28.1646i) q^{63} -93.9352i q^{65} -17.6011i q^{67} +(82.6215 + 17.6888i) q^{69} +58.3111i q^{71} -104.533 q^{73} +(-10.9420 + 51.1084i) q^{75} -136.440 q^{77} +123.909 q^{79} +(53.8444 + 60.5126i) q^{81} -81.7164 q^{83} +121.689i q^{85} +(1.93925 - 9.05789i) q^{87} -142.097i q^{89} -110.235i q^{91} +(-20.2153 + 94.4222i) q^{93} +84.4938i q^{95} -58.1110 q^{97} +(-146.573 - 65.7760i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} + 48 q^{25} - 64 q^{33} - 80 q^{49} + 32 q^{57} - 288 q^{73} + 400 q^{81} + 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 0
\(3\) −0.628052 + 2.93352i −0.209351 + 0.977841i
\(4\) 0 0
\(5\) −6.51323 −1.30265 −0.651323 0.758800i \(-0.725786\pi\)
−0.651323 + 0.758800i \(0.725786\pi\)
\(6\) 0 0
\(7\) −7.64344 −1.09192 −0.545960 0.837811i \(-0.683835\pi\)
−0.545960 + 0.837811i \(0.683835\pi\)
\(8\) 0 0
\(9\) −8.21110 3.68481i −0.912345 0.409423i
\(10\) 0 0
\(11\) 17.8506 1.62278 0.811391 0.584503i \(-0.198711\pi\)
0.811391 + 0.584503i \(0.198711\pi\)
\(12\) 0 0
\(13\) 14.4222i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 4.09065 19.1067i 0.272710 1.27378i
\(16\) 0 0
\(17\) 18.6833i 1.09902i −0.835488 0.549509i \(-0.814815\pi\)
0.835488 0.549509i \(-0.185185\pi\)
\(18\) 0 0
\(19\) 12.9726i 0.682770i −0.939923 0.341385i \(-0.889104\pi\)
0.939923 0.341385i \(-0.110896\pi\)
\(20\) 0 0
\(21\) 4.80048 22.4222i 0.228594 1.06772i
\(22\) 0 0
\(23\) 28.1646i 1.22455i −0.790646 0.612274i \(-0.790255\pi\)
0.790646 0.612274i \(-0.209745\pi\)
\(24\) 0 0
\(25\) 17.4222 0.696888
\(26\) 0 0
\(27\) 15.9665 21.7732i 0.591350 0.806415i
\(28\) 0 0
\(29\) −3.08772 −0.106473 −0.0532365 0.998582i \(-0.516954\pi\)
−0.0532365 + 0.998582i \(0.516954\pi\)
\(30\) 0 0
\(31\) 32.1873 1.03830 0.519150 0.854683i \(-0.326248\pi\)
0.519150 + 0.854683i \(0.326248\pi\)
\(32\) 0 0
\(33\) −11.2111 + 52.3652i −0.339730 + 1.58682i
\(34\) 0 0
\(35\) 49.7835 1.42239
\(36\) 0 0
\(37\) 1.57779i 0.0426431i 0.999773 + 0.0213216i \(0.00678738\pi\)
−0.999773 + 0.0213216i \(0.993213\pi\)
\(38\) 0 0
\(39\) −42.3079 9.05789i −1.08482 0.232254i
\(40\) 0 0
\(41\) 18.1647i 0.443042i 0.975156 + 0.221521i \(0.0711022\pi\)
−0.975156 + 0.221521i \(0.928898\pi\)
\(42\) 0 0
\(43\) 22.2296i 0.516968i 0.966016 + 0.258484i \(0.0832229\pi\)
−0.966016 + 0.258484i \(0.916777\pi\)
\(44\) 0 0
\(45\) 53.4808 + 24.0000i 1.18846 + 0.533333i
\(46\) 0 0
\(47\) 86.4757i 1.83991i −0.392026 0.919954i \(-0.628226\pi\)
0.392026 0.919954i \(-0.371774\pi\)
\(48\) 0 0
\(49\) 9.42221 0.192290
\(50\) 0 0
\(51\) 54.8079 + 11.7341i 1.07467 + 0.230080i
\(52\) 0 0
\(53\) −25.7151 −0.485191 −0.242596 0.970128i \(-0.577999\pi\)
−0.242596 + 0.970128i \(0.577999\pi\)
\(54\) 0 0
\(55\) −116.265 −2.11391
\(56\) 0 0
\(57\) 38.0555 + 8.14748i 0.667641 + 0.142938i
\(58\) 0 0
\(59\) −11.3049 −0.191609 −0.0958045 0.995400i \(-0.530542\pi\)
−0.0958045 + 0.995400i \(0.530542\pi\)
\(60\) 0 0
\(61\) 91.2666i 1.49617i −0.663601 0.748087i \(-0.730973\pi\)
0.663601 0.748087i \(-0.269027\pi\)
\(62\) 0 0
\(63\) 62.7611 + 28.1646i 0.996208 + 0.447057i
\(64\) 0 0
\(65\) 93.9352i 1.44516i
\(66\) 0 0
\(67\) 17.6011i 0.262703i −0.991336 0.131352i \(-0.958068\pi\)
0.991336 0.131352i \(-0.0419317\pi\)
\(68\) 0 0
\(69\) 82.6215 + 17.6888i 1.19741 + 0.256360i
\(70\) 0 0
\(71\) 58.3111i 0.821283i 0.911797 + 0.410641i \(0.134695\pi\)
−0.911797 + 0.410641i \(0.865305\pi\)
\(72\) 0 0
\(73\) −104.533 −1.43196 −0.715981 0.698120i \(-0.754020\pi\)
−0.715981 + 0.698120i \(0.754020\pi\)
\(74\) 0 0
\(75\) −10.9420 + 51.1084i −0.145894 + 0.681446i
\(76\) 0 0
\(77\) −136.440 −1.77195
\(78\) 0 0
\(79\) 123.909 1.56846 0.784232 0.620468i \(-0.213057\pi\)
0.784232 + 0.620468i \(0.213057\pi\)
\(80\) 0 0
\(81\) 53.8444 + 60.5126i 0.664746 + 0.747070i
\(82\) 0 0
\(83\) −81.7164 −0.984535 −0.492268 0.870444i \(-0.663832\pi\)
−0.492268 + 0.870444i \(0.663832\pi\)
\(84\) 0 0
\(85\) 121.689i 1.43163i
\(86\) 0 0
\(87\) 1.93925 9.05789i 0.0222902 0.104114i
\(88\) 0 0
\(89\) 142.097i 1.59659i −0.602263 0.798297i \(-0.705734\pi\)
0.602263 0.798297i \(-0.294266\pi\)
\(90\) 0 0
\(91\) 110.235i 1.21138i
\(92\) 0 0
\(93\) −20.2153 + 94.4222i −0.217369 + 1.01529i
\(94\) 0 0
\(95\) 84.4938i 0.889408i
\(96\) 0 0
\(97\) −58.1110 −0.599083 −0.299541 0.954083i \(-0.596834\pi\)
−0.299541 + 0.954083i \(0.596834\pi\)
\(98\) 0 0
\(99\) −146.573 65.7760i −1.48054 0.664404i
\(100\) 0 0
\(101\) −119.650 −1.18466 −0.592328 0.805697i \(-0.701791\pi\)
−0.592328 + 0.805697i \(0.701791\pi\)
\(102\) 0 0
\(103\) 4.41634 0.0428771 0.0214386 0.999770i \(-0.493175\pi\)
0.0214386 + 0.999770i \(0.493175\pi\)
\(104\) 0 0
\(105\) −31.2666 + 146.041i −0.297777 + 1.39087i
\(106\) 0 0
\(107\) 5.75018 0.0537400 0.0268700 0.999639i \(-0.491446\pi\)
0.0268700 + 0.999639i \(0.491446\pi\)
\(108\) 0 0
\(109\) 155.267i 1.42446i −0.701944 0.712232i \(-0.747684\pi\)
0.701944 0.712232i \(-0.252316\pi\)
\(110\) 0 0
\(111\) −4.62850 0.990937i −0.0416982 0.00892736i
\(112\) 0 0
\(113\) 30.5156i 0.270050i −0.990842 0.135025i \(-0.956889\pi\)
0.990842 0.135025i \(-0.0431114\pi\)
\(114\) 0 0
\(115\) 183.443i 1.59515i
\(116\) 0 0
\(117\) 53.1430 118.422i 0.454214 1.01216i
\(118\) 0 0
\(119\) 142.805i 1.20004i
\(120\) 0 0
\(121\) 197.644 1.63342
\(122\) 0 0
\(123\) −53.2867 11.4084i −0.433225 0.0927512i
\(124\) 0 0
\(125\) 49.3559 0.394848
\(126\) 0 0
\(127\) 13.6733 0.107664 0.0538320 0.998550i \(-0.482856\pi\)
0.0538320 + 0.998550i \(0.482856\pi\)
\(128\) 0 0
\(129\) −65.2111 13.9614i −0.505512 0.108228i
\(130\) 0 0
\(131\) −110.872 −0.846351 −0.423175 0.906048i \(-0.639085\pi\)
−0.423175 + 0.906048i \(0.639085\pi\)
\(132\) 0 0
\(133\) 99.1556i 0.745531i
\(134\) 0 0
\(135\) −103.993 + 141.814i −0.770320 + 1.05047i
\(136\) 0 0
\(137\) 109.712i 0.800814i −0.916337 0.400407i \(-0.868869\pi\)
0.916337 0.400407i \(-0.131131\pi\)
\(138\) 0 0
\(139\) 210.301i 1.51295i 0.654020 + 0.756477i \(0.273081\pi\)
−0.654020 + 0.756477i \(0.726919\pi\)
\(140\) 0 0
\(141\) 253.678 + 54.3112i 1.79914 + 0.385186i
\(142\) 0 0
\(143\) 257.445i 1.80032i
\(144\) 0 0
\(145\) 20.1110 0.138697
\(146\) 0 0
\(147\) −5.91763 + 27.6402i −0.0402560 + 0.188029i
\(148\) 0 0
\(149\) 45.5926 0.305991 0.152995 0.988227i \(-0.451108\pi\)
0.152995 + 0.988227i \(0.451108\pi\)
\(150\) 0 0
\(151\) 22.9303 0.151856 0.0759282 0.997113i \(-0.475808\pi\)
0.0759282 + 0.997113i \(0.475808\pi\)
\(152\) 0 0
\(153\) −68.8444 + 153.411i −0.449963 + 1.00268i
\(154\) 0 0
\(155\) −209.644 −1.35254
\(156\) 0 0
\(157\) 123.489i 0.786552i 0.919420 + 0.393276i \(0.128658\pi\)
−0.919420 + 0.393276i \(0.871342\pi\)
\(158\) 0 0
\(159\) 16.1504 75.4359i 0.101575 0.474440i
\(160\) 0 0
\(161\) 215.274i 1.33711i
\(162\) 0 0
\(163\) 163.039i 1.00024i −0.865957 0.500119i \(-0.833290\pi\)
0.865957 0.500119i \(-0.166710\pi\)
\(164\) 0 0
\(165\) 73.0205 341.066i 0.442549 2.06707i
\(166\) 0 0
\(167\) 144.787i 0.866987i −0.901157 0.433493i \(-0.857281\pi\)
0.901157 0.433493i \(-0.142719\pi\)
\(168\) 0 0
\(169\) −39.0000 −0.230769
\(170\) 0 0
\(171\) −47.8016 + 106.520i −0.279542 + 0.622922i
\(172\) 0 0
\(173\) −168.331 −0.973010 −0.486505 0.873678i \(-0.661728\pi\)
−0.486505 + 0.873678i \(0.661728\pi\)
\(174\) 0 0
\(175\) −133.166 −0.760946
\(176\) 0 0
\(177\) 7.10008 33.1633i 0.0401134 0.187363i
\(178\) 0 0
\(179\) 105.317 0.588364 0.294182 0.955749i \(-0.404953\pi\)
0.294182 + 0.955749i \(0.404953\pi\)
\(180\) 0 0
\(181\) 30.2002i 0.166852i 0.996514 + 0.0834258i \(0.0265862\pi\)
−0.996514 + 0.0834258i \(0.973414\pi\)
\(182\) 0 0
\(183\) 267.733 + 57.3201i 1.46302 + 0.313225i
\(184\) 0 0
\(185\) 10.2765i 0.0555489i
\(186\) 0 0
\(187\) 333.509i 1.78347i
\(188\) 0 0
\(189\) −122.039 + 166.422i −0.645707 + 0.880541i
\(190\) 0 0
\(191\) 255.463i 1.33750i −0.743486 0.668752i \(-0.766829\pi\)
0.743486 0.668752i \(-0.233171\pi\)
\(192\) 0 0
\(193\) 286.222 1.48302 0.741508 0.670944i \(-0.234111\pi\)
0.741508 + 0.670944i \(0.234111\pi\)
\(194\) 0 0
\(195\) 275.561 + 58.9961i 1.41313 + 0.302544i
\(196\) 0 0
\(197\) −121.002 −0.614221 −0.307110 0.951674i \(-0.599362\pi\)
−0.307110 + 0.951674i \(0.599362\pi\)
\(198\) 0 0
\(199\) 243.401 1.22312 0.611560 0.791198i \(-0.290542\pi\)
0.611560 + 0.791198i \(0.290542\pi\)
\(200\) 0 0
\(201\) 51.6333 + 11.0544i 0.256882 + 0.0549971i
\(202\) 0 0
\(203\) 23.6008 0.116260
\(204\) 0 0
\(205\) 118.311i 0.577128i
\(206\) 0 0
\(207\) −103.781 + 231.262i −0.501358 + 1.11721i
\(208\) 0 0
\(209\) 231.569i 1.10799i
\(210\) 0 0
\(211\) 112.125i 0.531399i −0.964056 0.265700i \(-0.914397\pi\)
0.964056 0.265700i \(-0.0856029\pi\)
\(212\) 0 0
\(213\) −171.057 36.6224i −0.803084 0.171936i
\(214\) 0 0
\(215\) 144.787i 0.673427i
\(216\) 0 0
\(217\) −246.022 −1.13374
\(218\) 0 0
\(219\) 65.6523 306.651i 0.299782 1.40023i
\(220\) 0 0
\(221\) 269.455 1.21925
\(222\) 0 0
\(223\) −341.152 −1.52983 −0.764915 0.644131i \(-0.777219\pi\)
−0.764915 + 0.644131i \(0.777219\pi\)
\(224\) 0 0
\(225\) −143.056 64.1974i −0.635802 0.285322i
\(226\) 0 0
\(227\) 57.1247 0.251651 0.125825 0.992052i \(-0.459842\pi\)
0.125825 + 0.992052i \(0.459842\pi\)
\(228\) 0 0
\(229\) 254.644i 1.11198i 0.831188 + 0.555992i \(0.187661\pi\)
−0.831188 + 0.555992i \(0.812339\pi\)
\(230\) 0 0
\(231\) 85.6914 400.250i 0.370958 1.73268i
\(232\) 0 0
\(233\) 117.395i 0.503842i −0.967748 0.251921i \(-0.918938\pi\)
0.967748 0.251921i \(-0.0810623\pi\)
\(234\) 0 0
\(235\) 563.236i 2.39675i
\(236\) 0 0
\(237\) −77.8210 + 363.489i −0.328359 + 1.53371i
\(238\) 0 0
\(239\) 138.841i 0.580925i −0.956886 0.290463i \(-0.906191\pi\)
0.956886 0.290463i \(-0.0938092\pi\)
\(240\) 0 0
\(241\) −379.911 −1.57639 −0.788197 0.615423i \(-0.788985\pi\)
−0.788197 + 0.615423i \(0.788985\pi\)
\(242\) 0 0
\(243\) −211.332 + 119.949i −0.869680 + 0.493616i
\(244\) 0 0
\(245\) −61.3690 −0.250486
\(246\) 0 0
\(247\) 187.094 0.757466
\(248\) 0 0
\(249\) 51.3221 239.717i 0.206113 0.962719i
\(250\) 0 0
\(251\) 399.064 1.58990 0.794948 0.606678i \(-0.207498\pi\)
0.794948 + 0.606678i \(0.207498\pi\)
\(252\) 0 0
\(253\) 502.755i 1.98718i
\(254\) 0 0
\(255\) −356.977 76.4268i −1.39991 0.299713i
\(256\) 0 0
\(257\) 78.4728i 0.305342i −0.988277 0.152671i \(-0.951213\pi\)
0.988277 0.152671i \(-0.0487874\pi\)
\(258\) 0 0
\(259\) 12.0598i 0.0465629i
\(260\) 0 0
\(261\) 25.3536 + 11.3776i 0.0971401 + 0.0435925i
\(262\) 0 0
\(263\) 1.98187i 0.00753564i −0.999993 0.00376782i \(-0.998801\pi\)
0.999993 0.00376782i \(-0.00119934\pi\)
\(264\) 0 0
\(265\) 167.489 0.632033
\(266\) 0 0
\(267\) 416.845 + 89.2442i 1.56122 + 0.334248i
\(268\) 0 0
\(269\) −47.6194 −0.177024 −0.0885119 0.996075i \(-0.528211\pi\)
−0.0885119 + 0.996075i \(0.528211\pi\)
\(270\) 0 0
\(271\) −181.829 −0.670956 −0.335478 0.942048i \(-0.608898\pi\)
−0.335478 + 0.942048i \(0.608898\pi\)
\(272\) 0 0
\(273\) 323.378 + 69.2334i 1.18453 + 0.253602i
\(274\) 0 0
\(275\) 310.997 1.13090
\(276\) 0 0
\(277\) 71.8890i 0.259527i 0.991545 + 0.129763i \(0.0414218\pi\)
−0.991545 + 0.129763i \(0.958578\pi\)
\(278\) 0 0
\(279\) −264.293 118.604i −0.947288 0.425104i
\(280\) 0 0
\(281\) 459.509i 1.63526i 0.575742 + 0.817631i \(0.304713\pi\)
−0.575742 + 0.817631i \(0.695287\pi\)
\(282\) 0 0
\(283\) 343.678i 1.21441i −0.794545 0.607206i \(-0.792290\pi\)
0.794545 0.607206i \(-0.207710\pi\)
\(284\) 0 0
\(285\) −247.864 53.0665i −0.869700 0.186198i
\(286\) 0 0
\(287\) 138.841i 0.483767i
\(288\) 0 0
\(289\) −60.0665 −0.207842
\(290\) 0 0
\(291\) 36.4967 170.470i 0.125418 0.585807i
\(292\) 0 0
\(293\) 431.610 1.47307 0.736536 0.676399i \(-0.236460\pi\)
0.736536 + 0.676399i \(0.236460\pi\)
\(294\) 0 0
\(295\) 73.6316 0.249599
\(296\) 0 0
\(297\) 285.011 388.665i 0.959633 1.30864i
\(298\) 0 0
\(299\) 406.196 1.35851
\(300\) 0 0
\(301\) 169.911i 0.564488i
\(302\) 0 0
\(303\) 75.1466 350.997i 0.248008 1.15841i
\(304\) 0 0
\(305\) 594.441i 1.94899i
\(306\) 0 0
\(307\) 221.383i 0.721119i 0.932736 + 0.360559i \(0.117414\pi\)
−0.932736 + 0.360559i \(0.882586\pi\)
\(308\) 0 0
\(309\) −2.77369 + 12.9554i −0.00897634 + 0.0419270i
\(310\) 0 0
\(311\) 170.969i 0.549741i −0.961481 0.274871i \(-0.911365\pi\)
0.961481 0.274871i \(-0.0886350\pi\)
\(312\) 0 0
\(313\) −17.2666 −0.0551649 −0.0275825 0.999620i \(-0.508781\pi\)
−0.0275825 + 0.999620i \(0.508781\pi\)
\(314\) 0 0
\(315\) −408.778 183.443i −1.29771 0.582357i
\(316\) 0 0
\(317\) −103.874 −0.327678 −0.163839 0.986487i \(-0.552388\pi\)
−0.163839 + 0.986487i \(0.552388\pi\)
\(318\) 0 0
\(319\) −55.1176 −0.172783
\(320\) 0 0
\(321\) −3.61141 + 16.8683i −0.0112505 + 0.0525492i
\(322\) 0 0
\(323\) −242.372 −0.750377
\(324\) 0 0
\(325\) 251.267i 0.773128i
\(326\) 0 0
\(327\) 455.478 + 97.5154i 1.39290 + 0.298212i
\(328\) 0 0
\(329\) 660.972i 2.00903i
\(330\) 0 0
\(331\) 189.961i 0.573900i −0.957945 0.286950i \(-0.907359\pi\)
0.957945 0.286950i \(-0.0926414\pi\)
\(332\) 0 0
\(333\) 5.81387 12.9554i 0.0174591 0.0389052i
\(334\) 0 0
\(335\) 114.640i 0.342210i
\(336\) 0 0
\(337\) −157.889 −0.468513 −0.234257 0.972175i \(-0.575266\pi\)
−0.234257 + 0.972175i \(0.575266\pi\)
\(338\) 0 0
\(339\) 89.5182 + 19.1654i 0.264066 + 0.0565350i
\(340\) 0 0
\(341\) 574.563 1.68494
\(342\) 0 0
\(343\) 302.511 0.881955
\(344\) 0 0
\(345\) −538.133 115.211i −1.55981 0.333946i
\(346\) 0 0
\(347\) −258.632 −0.745336 −0.372668 0.927965i \(-0.621557\pi\)
−0.372668 + 0.927965i \(0.621557\pi\)
\(348\) 0 0
\(349\) 228.955i 0.656033i −0.944672 0.328016i \(-0.893620\pi\)
0.944672 0.328016i \(-0.106380\pi\)
\(350\) 0 0
\(351\) 314.018 + 230.271i 0.894637 + 0.656044i
\(352\) 0 0
\(353\) 523.133i 1.48196i −0.671525 0.740982i \(-0.734361\pi\)
0.671525 0.740982i \(-0.265639\pi\)
\(354\) 0 0
\(355\) 379.794i 1.06984i
\(356\) 0 0
\(357\) −418.921 89.6888i −1.17345 0.251229i
\(358\) 0 0
\(359\) 114.640i 0.319332i −0.987171 0.159666i \(-0.948958\pi\)
0.987171 0.159666i \(-0.0510418\pi\)
\(360\) 0 0
\(361\) 192.711 0.533825
\(362\) 0 0
\(363\) −124.131 + 579.794i −0.341958 + 1.59723i
\(364\) 0 0
\(365\) 680.849 1.86534
\(366\) 0 0
\(367\) −188.283 −0.513033 −0.256517 0.966540i \(-0.582575\pi\)
−0.256517 + 0.966540i \(0.582575\pi\)
\(368\) 0 0
\(369\) 66.9335 149.153i 0.181392 0.404207i
\(370\) 0 0
\(371\) 196.552 0.529790
\(372\) 0 0
\(373\) 30.6443i 0.0821562i 0.999156 + 0.0410781i \(0.0130792\pi\)
−0.999156 + 0.0410781i \(0.986921\pi\)
\(374\) 0 0
\(375\) −30.9981 + 144.787i −0.0826615 + 0.386098i
\(376\) 0 0
\(377\) 44.5317i 0.118121i
\(378\) 0 0
\(379\) 216.755i 0.571913i −0.958243 0.285956i \(-0.907689\pi\)
0.958243 0.285956i \(-0.0923112\pi\)
\(380\) 0 0
\(381\) −8.58756 + 40.1110i −0.0225395 + 0.105278i
\(382\) 0 0
\(383\) 653.731i 1.70687i 0.521199 + 0.853435i \(0.325485\pi\)
−0.521199 + 0.853435i \(0.674515\pi\)
\(384\) 0 0
\(385\) 888.666 2.30822
\(386\) 0 0
\(387\) 81.9119 182.530i 0.211659 0.471653i
\(388\) 0 0
\(389\) −227.288 −0.584287 −0.292143 0.956375i \(-0.594368\pi\)
−0.292143 + 0.956375i \(0.594368\pi\)
\(390\) 0 0
\(391\) −526.208 −1.34580
\(392\) 0 0
\(393\) 69.6333 325.245i 0.177184 0.827596i
\(394\) 0 0
\(395\) −807.046 −2.04315
\(396\) 0 0
\(397\) 338.244i 0.852000i 0.904723 + 0.426000i \(0.140078\pi\)
−0.904723 + 0.426000i \(0.859922\pi\)
\(398\) 0 0
\(399\) −290.875 62.2748i −0.729010 0.156077i
\(400\) 0 0
\(401\) 217.553i 0.542527i 0.962505 + 0.271264i \(0.0874415\pi\)
−0.962505 + 0.271264i \(0.912559\pi\)
\(402\) 0 0
\(403\) 464.212i 1.15189i
\(404\) 0 0
\(405\) −350.701 394.133i −0.865929 0.973168i
\(406\) 0 0
\(407\) 28.1646i 0.0692005i
\(408\) 0 0
\(409\) −225.822 −0.552131 −0.276066 0.961139i \(-0.589031\pi\)
−0.276066 + 0.961139i \(0.589031\pi\)
\(410\) 0 0
\(411\) 321.841 + 68.9045i 0.783069 + 0.167651i
\(412\) 0 0
\(413\) 86.4086 0.209222
\(414\) 0 0
\(415\) 532.238 1.28250
\(416\) 0 0
\(417\) −616.922 132.080i −1.47943 0.316738i
\(418\) 0 0
\(419\) −185.247 −0.442117 −0.221059 0.975260i \(-0.570951\pi\)
−0.221059 + 0.975260i \(0.570951\pi\)
\(420\) 0 0
\(421\) 72.1110i 0.171285i −0.996326 0.0856425i \(-0.972706\pi\)
0.996326 0.0856425i \(-0.0272943\pi\)
\(422\) 0 0
\(423\) −318.646 + 710.061i −0.753300 + 1.67863i
\(424\) 0 0
\(425\) 325.505i 0.765893i
\(426\) 0 0
\(427\) 697.591i 1.63370i
\(428\) 0 0
\(429\) −755.221 161.689i −1.76042 0.376897i
\(430\) 0 0
\(431\) 380.013i 0.881701i −0.897581 0.440850i \(-0.854677\pi\)
0.897581 0.440850i \(-0.145323\pi\)
\(432\) 0 0
\(433\) −451.800 −1.04342 −0.521709 0.853124i \(-0.674705\pi\)
−0.521709 + 0.853124i \(0.674705\pi\)
\(434\) 0 0
\(435\) −12.6308 + 58.9961i −0.0290362 + 0.135623i
\(436\) 0 0
\(437\) −365.369 −0.836085
\(438\) 0 0
\(439\) −173.421 −0.395036 −0.197518 0.980299i \(-0.563288\pi\)
−0.197518 + 0.980299i \(0.563288\pi\)
\(440\) 0 0
\(441\) −77.3667 34.7190i −0.175435 0.0787279i
\(442\) 0 0
\(443\) −466.893 −1.05394 −0.526968 0.849885i \(-0.676671\pi\)
−0.526968 + 0.849885i \(0.676671\pi\)
\(444\) 0 0
\(445\) 925.511i 2.07980i
\(446\) 0 0
\(447\) −28.6345 + 133.747i −0.0640593 + 0.299210i
\(448\) 0 0
\(449\) 694.394i 1.54654i 0.634080 + 0.773268i \(0.281379\pi\)
−0.634080 + 0.773268i \(0.718621\pi\)
\(450\) 0 0
\(451\) 324.252i 0.718962i
\(452\) 0 0
\(453\) −14.4014 + 67.2666i −0.0317912 + 0.148491i
\(454\) 0 0
\(455\) 717.988i 1.57800i
\(456\) 0 0
\(457\) −113.378 −0.248091 −0.124046 0.992277i \(-0.539587\pi\)
−0.124046 + 0.992277i \(0.539587\pi\)
\(458\) 0 0
\(459\) −406.796 298.306i −0.886265 0.649905i
\(460\) 0 0
\(461\) −247.213 −0.536253 −0.268126 0.963384i \(-0.586405\pi\)
−0.268126 + 0.963384i \(0.586405\pi\)
\(462\) 0 0
\(463\) 625.149 1.35021 0.675107 0.737720i \(-0.264098\pi\)
0.675107 + 0.737720i \(0.264098\pi\)
\(464\) 0 0
\(465\) 131.667 614.994i 0.283155 1.32257i
\(466\) 0 0
\(467\) −756.467 −1.61984 −0.809922 0.586538i \(-0.800490\pi\)
−0.809922 + 0.586538i \(0.800490\pi\)
\(468\) 0 0
\(469\) 134.533i 0.286851i
\(470\) 0 0
\(471\) −362.257 77.5572i −0.769123 0.164665i
\(472\) 0 0
\(473\) 396.812i 0.838927i
\(474\) 0 0
\(475\) 226.012i 0.475815i
\(476\) 0 0
\(477\) 211.150 + 94.7553i 0.442662 + 0.198648i
\(478\) 0 0
\(479\) 510.927i 1.06665i 0.845910 + 0.533326i \(0.179058\pi\)
−0.845910 + 0.533326i \(0.820942\pi\)
\(480\) 0 0
\(481\) −22.7553 −0.0473083
\(482\) 0 0
\(483\) −631.512 135.203i −1.30748 0.279924i
\(484\) 0 0
\(485\) 378.491 0.780393
\(486\) 0 0
\(487\) −368.923 −0.757542 −0.378771 0.925490i \(-0.623653\pi\)
−0.378771 + 0.925490i \(0.623653\pi\)
\(488\) 0 0
\(489\) 478.278 + 102.397i 0.978073 + 0.209400i
\(490\) 0 0
\(491\) 567.842 1.15650 0.578251 0.815859i \(-0.303736\pi\)
0.578251 + 0.815859i \(0.303736\pi\)
\(492\) 0 0
\(493\) 57.6888i 0.117016i
\(494\) 0 0
\(495\) 954.665 + 428.415i 1.92862 + 0.865484i
\(496\) 0 0
\(497\) 445.697i 0.896775i
\(498\) 0 0
\(499\) 28.6838i 0.0574826i 0.999587 + 0.0287413i \(0.00914990\pi\)
−0.999587 + 0.0287413i \(0.990850\pi\)
\(500\) 0 0
\(501\) 424.735 + 90.9335i 0.847775 + 0.181504i
\(502\) 0 0
\(503\) 573.201i 1.13957i 0.821795 + 0.569783i \(0.192973\pi\)
−0.821795 + 0.569783i \(0.807027\pi\)
\(504\) 0 0
\(505\) 779.310 1.54319
\(506\) 0 0
\(507\) 24.4940 114.407i 0.0483117 0.225656i
\(508\) 0 0
\(509\) 197.857 0.388716 0.194358 0.980931i \(-0.437738\pi\)
0.194358 + 0.980931i \(0.437738\pi\)
\(510\) 0 0
\(511\) 798.994 1.56359
\(512\) 0 0
\(513\) −282.456 207.127i −0.550596 0.403756i
\(514\) 0 0
\(515\) −28.7647 −0.0558537
\(516\) 0 0
\(517\) 1543.64i 2.98577i
\(518\) 0 0
\(519\) 105.720 493.802i 0.203700 0.951448i
\(520\) 0 0
\(521\) 375.018i 0.719803i −0.932990 0.359902i \(-0.882810\pi\)
0.932990 0.359902i \(-0.117190\pi\)
\(522\) 0 0
\(523\) 422.363i 0.807577i −0.914852 0.403789i \(-0.867693\pi\)
0.914852 0.403789i \(-0.132307\pi\)
\(524\) 0 0
\(525\) 83.6349 390.644i 0.159305 0.744084i
\(526\) 0 0
\(527\) 601.366i 1.14111i
\(528\) 0 0
\(529\) −264.245 −0.499517
\(530\) 0 0
\(531\) 92.8259 + 41.6565i 0.174813 + 0.0784491i
\(532\) 0 0
\(533\) −261.976 −0.491511
\(534\) 0 0
\(535\) −37.4523 −0.0700043
\(536\) 0 0
\(537\) −66.1446 + 308.950i −0.123174 + 0.575327i
\(538\) 0 0
\(539\) 168.192 0.312045
\(540\) 0 0
\(541\) 226.022i 0.417785i −0.977939 0.208893i \(-0.933014\pi\)
0.977939 0.208893i \(-0.0669859\pi\)
\(542\) 0 0
\(543\) −88.5928 18.9673i −0.163154 0.0349305i
\(544\) 0 0
\(545\) 1011.29i 1.85557i
\(546\) 0 0
\(547\) 581.814i 1.06365i −0.846855 0.531823i \(-0.821507\pi\)
0.846855 0.531823i \(-0.178493\pi\)
\(548\) 0 0
\(549\) −336.300 + 749.400i −0.612568 + 1.36503i
\(550\) 0 0
\(551\) 40.0558i 0.0726966i
\(552\) 0 0
\(553\) −947.088 −1.71264
\(554\) 0 0
\(555\) 30.1465 + 6.45420i 0.0543180 + 0.0116292i
\(556\) 0 0
\(557\) 960.871 1.72508 0.862541 0.505987i \(-0.168872\pi\)
0.862541 + 0.505987i \(0.168872\pi\)
\(558\) 0 0
\(559\) −320.600 −0.573525
\(560\) 0 0
\(561\) 978.355 + 209.461i 1.74395 + 0.373370i
\(562\) 0 0
\(563\) −376.454 −0.668657 −0.334328 0.942457i \(-0.608509\pi\)
−0.334328 + 0.942457i \(0.608509\pi\)
\(564\) 0 0
\(565\) 198.755i 0.351779i
\(566\) 0 0
\(567\) −411.557 462.525i −0.725849 0.815740i
\(568\) 0 0
\(569\) 596.515i 1.04836i −0.851608 0.524178i \(-0.824372\pi\)
0.851608 0.524178i \(-0.175628\pi\)
\(570\) 0 0
\(571\) 275.100i 0.481786i 0.970552 + 0.240893i \(0.0774403\pi\)
−0.970552 + 0.240893i \(0.922560\pi\)
\(572\) 0 0
\(573\) 749.407 + 160.444i 1.30787 + 0.280007i
\(574\) 0 0
\(575\) 490.689i 0.853373i
\(576\) 0 0
\(577\) 475.356 0.823840 0.411920 0.911220i \(-0.364858\pi\)
0.411920 + 0.911220i \(0.364858\pi\)
\(578\) 0 0
\(579\) −179.762 + 839.639i −0.310470 + 1.45015i
\(580\) 0 0
\(581\) 624.595 1.07503
\(582\) 0 0
\(583\) −459.031 −0.787360
\(584\) 0 0
\(585\) −346.133 + 771.311i −0.591680 + 1.31848i
\(586\) 0 0
\(587\) 621.408 1.05862 0.529308 0.848430i \(-0.322452\pi\)
0.529308 + 0.848430i \(0.322452\pi\)
\(588\) 0 0
\(589\) 417.554i 0.708921i
\(590\) 0 0
\(591\) 75.9952 354.961i 0.128587 0.600610i
\(592\) 0 0
\(593\) 260.529i 0.439341i 0.975574 + 0.219671i \(0.0704983\pi\)
−0.975574 + 0.219671i \(0.929502\pi\)
\(594\) 0 0
\(595\) 930.121i 1.56323i
\(596\) 0 0
\(597\) −152.868 + 714.022i −0.256061 + 1.19602i
\(598\) 0 0
\(599\) 205.080i 0.342370i 0.985239 + 0.171185i \(0.0547596\pi\)
−0.985239 + 0.171185i \(0.945240\pi\)
\(600\) 0 0
\(601\) 202.844 0.337511 0.168756 0.985658i \(-0.446025\pi\)
0.168756 + 0.985658i \(0.446025\pi\)
\(602\) 0 0
\(603\) −64.8568 + 144.525i −0.107557 + 0.239676i
\(604\) 0 0
\(605\) −1287.30 −2.12777
\(606\) 0 0
\(607\) −377.331 −0.621633 −0.310817 0.950470i \(-0.600603\pi\)
−0.310817 + 0.950470i \(0.600603\pi\)
\(608\) 0 0
\(609\) −14.8225 + 69.2334i −0.0243391 + 0.113684i
\(610\) 0 0
\(611\) 1247.17 2.04119
\(612\) 0 0
\(613\) 533.400i 0.870146i 0.900395 + 0.435073i \(0.143277\pi\)
−0.900395 + 0.435073i \(0.856723\pi\)
\(614\) 0 0
\(615\) 347.068 + 74.3055i 0.564339 + 0.120822i
\(616\) 0 0
\(617\) 545.242i 0.883698i 0.897089 + 0.441849i \(0.145677\pi\)
−0.897089 + 0.441849i \(0.854323\pi\)
\(618\) 0 0
\(619\) 843.941i 1.36339i 0.731634 + 0.681697i \(0.238758\pi\)
−0.731634 + 0.681697i \(0.761242\pi\)
\(620\) 0 0
\(621\) −613.234 449.689i −0.987494 0.724137i
\(622\) 0 0
\(623\) 1086.11i 1.74335i
\(624\) 0 0
\(625\) −757.022 −1.21124
\(626\) 0 0
\(627\) 679.314 + 145.438i 1.08344 + 0.231958i
\(628\) 0 0
\(629\) 29.4784 0.0468656
\(630\) 0 0
\(631\) −1150.93 −1.82398 −0.911991 0.410210i \(-0.865455\pi\)
−0.911991 + 0.410210i \(0.865455\pi\)
\(632\) 0 0
\(633\) 328.922 + 70.4204i 0.519624 + 0.111249i
\(634\) 0 0
\(635\) −89.0576 −0.140248
\(636\) 0 0
\(637\) 135.889i 0.213326i
\(638\) 0 0
\(639\) 214.865 478.798i 0.336252 0.749293i
\(640\) 0 0
\(641\) 1266.93i 1.97649i −0.152875 0.988246i \(-0.548853\pi\)
0.152875 0.988246i \(-0.451147\pi\)
\(642\) 0 0
\(643\) 996.810i 1.55025i 0.631808 + 0.775125i \(0.282313\pi\)
−0.631808 + 0.775125i \(0.717687\pi\)
\(644\) 0 0
\(645\) 424.735 + 90.9335i 0.658504 + 0.140982i
\(646\) 0 0
\(647\) 1061.91i 1.64128i −0.571445 0.820641i \(-0.693617\pi\)
0.571445 0.820641i \(-0.306383\pi\)
\(648\) 0 0
\(649\) −201.800 −0.310940
\(650\) 0 0
\(651\) 154.514 721.711i 0.237349 1.10862i
\(652\) 0 0
\(653\) −513.147 −0.785830 −0.392915 0.919575i \(-0.628533\pi\)
−0.392915 + 0.919575i \(0.628533\pi\)
\(654\) 0 0
\(655\) 722.135 1.10250
\(656\) 0 0
\(657\) 858.333 + 385.185i 1.30644 + 0.586278i
\(658\) 0 0
\(659\) 495.658 0.752136 0.376068 0.926592i \(-0.377276\pi\)
0.376068 + 0.926592i \(0.377276\pi\)
\(660\) 0 0
\(661\) 1150.42i 1.74043i −0.492675 0.870213i \(-0.663981\pi\)
0.492675 0.870213i \(-0.336019\pi\)
\(662\) 0 0
\(663\) −169.231 + 790.451i −0.255251 + 1.19223i
\(664\) 0 0
\(665\) 645.823i 0.971163i
\(666\) 0 0
\(667\) 86.9643i 0.130381i
\(668\) 0 0
\(669\) 214.261 1000.78i 0.320271 1.49593i
\(670\) 0 0
\(671\) 1629.16i 2.42797i
\(672\) 0 0
\(673\) 72.8225 0.108206 0.0541029 0.998535i \(-0.482770\pi\)
0.0541029 + 0.998535i \(0.482770\pi\)
\(674\) 0 0
\(675\) 278.171 379.337i 0.412105 0.561981i
\(676\) 0 0
\(677\) −927.386 −1.36985 −0.684923 0.728615i \(-0.740164\pi\)
−0.684923 + 0.728615i \(0.740164\pi\)
\(678\) 0 0
\(679\) 444.168 0.654151
\(680\) 0 0
\(681\) −35.8773 + 167.577i −0.0526832 + 0.246074i
\(682\) 0 0
\(683\) −709.265 −1.03846 −0.519228 0.854636i \(-0.673780\pi\)
−0.519228 + 0.854636i \(0.673780\pi\)
\(684\) 0 0
\(685\) 714.577i 1.04318i
\(686\) 0 0
\(687\) −747.005 159.930i −1.08734 0.232794i
\(688\) 0 0
\(689\) 370.869i 0.538271i
\(690\) 0 0
\(691\) 27.7067i 0.0400966i 0.999799 + 0.0200483i \(0.00638200\pi\)
−0.999799 + 0.0200483i \(0.993618\pi\)
\(692\) 0 0
\(693\) 1120.32 + 502.755i 1.61663 + 0.725477i
\(694\) 0 0
\(695\) 1369.74i 1.97085i
\(696\) 0 0
\(697\) 339.378 0.486912
\(698\) 0 0
\(699\) 344.381 + 73.7302i 0.492677 + 0.105480i
\(700\) 0 0
\(701\) −765.474 −1.09197 −0.545987 0.837794i \(-0.683845\pi\)
−0.545987 + 0.837794i \(0.683845\pi\)
\(702\) 0 0
\(703\) 20.4682 0.0291154
\(704\) 0 0
\(705\) −1652.27 353.741i −2.34364 0.501761i
\(706\) 0 0
\(707\) 914.540 1.29355
\(708\) 0 0
\(709\) 796.599i 1.12355i 0.827289 + 0.561776i \(0.189882\pi\)
−0.827289 + 0.561776i \(0.810118\pi\)
\(710\) 0 0
\(711\) −1017.43 456.579i −1.43098 0.642165i
\(712\) 0 0
\(713\) 906.543i 1.27145i
\(714\) 0 0
\(715\) 1676.80i 2.34518i
\(716\) 0 0
\(717\) 407.294 + 87.1994i 0.568052 + 0.121617i
\(718\) 0 0
\(719\) 108.695i 0.151175i −0.997139 0.0755874i \(-0.975917\pi\)
0.997139 0.0755874i \(-0.0240832\pi\)
\(720\) 0 0
\(721\) −33.7561 −0.0468184
\(722\) 0 0
\(723\) 238.604 1114.48i 0.330019 1.54146i
\(724\) 0 0
\(725\) −53.7949 −0.0741998
\(726\) 0 0
\(727\) 960.355 1.32098 0.660491 0.750834i \(-0.270348\pi\)
0.660491 + 0.750834i \(0.270348\pi\)
\(728\) 0 0
\(729\) −219.145 695.282i −0.300610 0.953747i
\(730\) 0 0
\(731\) 415.323 0.568158
\(732\) 0 0
\(733\) 519.667i 0.708959i 0.935064 + 0.354479i \(0.115342\pi\)
−0.935064 + 0.354479i \(0.884658\pi\)
\(734\) 0 0
\(735\) 38.5429 180.027i 0.0524393 0.244935i
\(736\) 0 0
\(737\) 314.191i 0.426311i
\(738\) 0 0
\(739\) 614.214i 0.831142i −0.909561 0.415571i \(-0.863582\pi\)
0.909561 0.415571i \(-0.136418\pi\)
\(740\) 0 0
\(741\) −117.505 + 548.844i −0.158576 + 0.740681i
\(742\) 0 0
\(743\) 20.2371i 0.0272370i 0.999907 + 0.0136185i \(0.00433504\pi\)
−0.999907 + 0.0136185i \(0.995665\pi\)
\(744\) 0 0
\(745\) −296.955 −0.398598
\(746\) 0 0
\(747\) 670.982 + 301.109i 0.898236 + 0.403091i
\(748\) 0 0
\(749\) −43.9512 −0.0586798
\(750\) 0 0
\(751\) 107.092 0.142599 0.0712995 0.997455i \(-0.477285\pi\)
0.0712995 + 0.997455i \(0.477285\pi\)
\(752\) 0 0
\(753\) −250.633 + 1170.66i −0.332845 + 1.55466i
\(754\) 0 0
\(755\) −149.351 −0.197815
\(756\) 0 0
\(757\) 181.177i 0.239336i −0.992814 0.119668i \(-0.961817\pi\)
0.992814 0.119668i \(-0.0381830\pi\)
\(758\) 0 0
\(759\) 1474.84 + 315.756i 1.94314 + 0.416016i
\(760\) 0 0
\(761\) 74.4193i 0.0977914i 0.998804 + 0.0488957i \(0.0155702\pi\)
−0.998804 + 0.0488957i \(0.984430\pi\)
\(762\) 0 0
\(763\) 1186.77i 1.55540i
\(764\) 0 0
\(765\) 448.400 999.199i 0.586143 1.30614i
\(766\) 0 0
\(767\) 163.042i 0.212571i
\(768\) 0 0
\(769\) 631.467 0.821153 0.410577 0.911826i \(-0.365327\pi\)
0.410577 + 0.911826i \(0.365327\pi\)
\(770\) 0 0
\(771\) 230.202 + 49.2850i 0.298576 + 0.0639234i
\(772\) 0 0
\(773\) −1030.15 −1.33267 −0.666334 0.745654i \(-0.732137\pi\)
−0.666334 + 0.745654i \(0.732137\pi\)
\(774\) 0 0
\(775\) 560.774 0.723579
\(776\) 0 0
\(777\) 35.3776 + 7.57417i 0.0455311 + 0.00974796i
\(778\) 0 0
\(779\) 235.645 0.302496
\(780\) 0 0
\(781\) 1040.89i 1.33276i
\(782\) 0 0
\(783\) −49.2999 + 67.2295i −0.0629628 + 0.0858614i
\(784\) 0 0
\(785\) 804.310i 1.02460i
\(786\) 0 0
\(787\) 157.433i 0.200042i −0.994985 0.100021i \(-0.968109\pi\)
0.994985 0.100021i \(-0.0318910\pi\)
\(788\) 0 0
\(789\) 5.81387 + 1.24472i 0.00736865 + 0.00157759i
\(790\) 0 0
\(791\) 233.244i 0.294873i
\(792\) 0 0
\(793\) 1316.27 1.65986
\(794\) 0 0
\(795\) −105.192 + 491.332i −0.132316 + 0.618027i
\(796\) 0 0
\(797\) 703.044 0.882113 0.441056 0.897479i \(-0.354604\pi\)
0.441056 + 0.897479i \(0.354604\pi\)
\(798\) 0 0
\(799\) −1615.65 −2.02209
\(800\) 0 0
\(801\) −523.600 + 1166.77i −0.653683 + 1.45664i
\(802\) 0 0
\(803\) −1865.98 −2.32376
\(804\) 0 0
\(805\) 1402.13i 1.74178i
\(806\) 0 0
\(807\) 29.9074 139.693i 0.0370600 0.173101i
\(808\) 0 0
\(809\) 1374.99i 1.69962i 0.527090 + 0.849809i \(0.323283\pi\)
−0.527090 + 0.849809i \(0.676717\pi\)
\(810\) 0 0
\(811\) 867.212i 1.06931i −0.845070 0.534656i \(-0.820441\pi\)
0.845070 0.534656i \(-0.179559\pi\)
\(812\) 0 0
\(813\) 114.198 533.400i 0.140465 0.656088i
\(814\) 0 0
\(815\) 1061.91i 1.30296i
\(816\) 0 0
\(817\) 288.377 0.352970
\(818\) 0 0
\(819\) −406.196 + 905.153i −0.495965 + 1.10519i
\(820\) 0 0
\(821\) −1577.36 −1.92126 −0.960632 0.277823i \(-0.910387\pi\)
−0.960632 + 0.277823i \(0.910387\pi\)
\(822\) 0 0
\(823\) 303.700 0.369016 0.184508 0.982831i \(-0.440931\pi\)
0.184508 + 0.982831i \(0.440931\pi\)
\(824\) 0 0
\(825\) −195.322 + 912.316i −0.236754 + 1.10584i
\(826\) 0 0
\(827\) 1302.89 1.57544 0.787718 0.616036i \(-0.211263\pi\)
0.787718 + 0.616036i \(0.211263\pi\)
\(828\) 0 0
\(829\) 238.866i 0.288138i 0.989568 + 0.144069i \(0.0460187\pi\)
−0.989568 + 0.144069i \(0.953981\pi\)
\(830\) 0 0
\(831\) −210.888 45.1500i −0.253776 0.0543321i
\(832\) 0 0
\(833\) 176.038i 0.211330i
\(834\) 0 0
\(835\) 943.030i 1.12938i
\(836\) 0 0
\(837\) 513.917 700.821i 0.613999 0.837301i
\(838\) 0 0
\(839\) 355.812i 0.424091i −0.977260 0.212045i \(-0.931988\pi\)
0.977260 0.212045i \(-0.0680124\pi\)
\(840\) 0 0
\(841\) −831.466 −0.988663
\(842\) 0 0
\(843\) −1347.98 288.595i −1.59903 0.342343i
\(844\) 0 0
\(845\) 254.016 0.300611
\(846\) 0 0
\(847\) −1510.68 −1.78357
\(848\) 0 0
\(849\) 1008.19 + 215.848i 1.18750 + 0.254238i
\(850\) 0 0
\(851\) 44.4380 0.0522185
\(852\) 0 0
\(853\) 719.755i 0.843792i −0.906644 0.421896i \(-0.861365\pi\)
0.906644 0.421896i \(-0.138635\pi\)
\(854\) 0 0
\(855\) 311.343 693.787i 0.364144 0.811447i
\(856\) 0 0
\(857\) 604.812i 0.705732i −0.935674 0.352866i \(-0.885207\pi\)
0.935674 0.352866i \(-0.114793\pi\)
\(858\) 0 0
\(859\) 309.453i 0.360248i 0.983644 + 0.180124i \(0.0576500\pi\)
−0.983644 + 0.180124i \(0.942350\pi\)
\(860\) 0 0
\(861\) 407.294 + 87.1994i 0.473047 + 0.101277i
\(862\) 0 0
\(863\) 1120.22i 1.29805i 0.760766 + 0.649027i \(0.224824\pi\)
−0.760766 + 0.649027i \(0.775176\pi\)
\(864\) 0 0
\(865\) 1096.38 1.26749
\(866\) 0 0
\(867\) 37.7248 176.206i 0.0435119 0.203237i
\(868\) 0 0
\(869\) 2211.84 2.54528
\(870\) 0 0
\(871\) 253.847 0.291443
\(872\) 0 0
\(873\) 477.156 + 214.128i 0.546570 + 0.245278i
\(874\) 0 0
\(875\) −377.249 −0.431142
\(876\) 0 0
\(877\) 1384.11i 1.57823i 0.614243 + 0.789117i \(0.289461\pi\)
−0.614243 + 0.789117i \(0.710539\pi\)
\(878\) 0 0
\(879\) −271.073 + 1266.14i −0.308388 + 1.44043i
\(880\) 0 0
\(881\) 807.327i 0.916376i −0.888855 0.458188i \(-0.848499\pi\)
0.888855 0.458188i \(-0.151501\pi\)
\(882\) 0 0
\(883\) 1276.73i 1.44590i 0.690899 + 0.722951i \(0.257215\pi\)
−0.690899 + 0.722951i \(0.742785\pi\)
\(884\) 0 0
\(885\) −46.2445 + 216.000i −0.0522536 + 0.244068i
\(886\) 0 0
\(887\) 670.005i 0.755361i 0.925936 + 0.377680i \(0.123278\pi\)
−0.925936 + 0.377680i \(0.876722\pi\)
\(888\) 0 0
\(889\) −104.511 −0.117561
\(890\) 0 0
\(891\) 961.156 + 1080.19i 1.07874 + 1.21233i
\(892\) 0 0
\(893\) −1121.82 −1.25623
\(894\) 0 0
\(895\) −685.956 −0.766431
\(896\) 0 0
\(897\) −255.112 + 1191.58i −0.284406 + 1.32841i
\(898\) 0 0
\(899\) −99.3854 −0.110551
\(900\) 0 0
\(901\) 480.444i 0.533234i
\(902\) 0 0
\(903\) 498.437 + 106.713i 0.551979 + 0.118176i
\(904\) 0 0
\(905\) 196.701i 0.217349i
\(906\) 0 0
\(907\) 1637.75i 1.80568i −0.429974 0.902841i \(-0.641477\pi\)
0.429974 0.902841i \(-0.358523\pi\)
\(908\) 0 0
\(909\) 982.461 + 440.888i 1.08082 + 0.485026i
\(910\) 0 0
\(911\) 1116.26i 1.22531i −0.790351 0.612654i \(-0.790102\pi\)
0.790351 0.612654i \(-0.209898\pi\)
\(912\) 0 0
\(913\) −1458.69 −1.59769
\(914\) 0 0
\(915\) −1743.80 373.339i −1.90580 0.408021i
\(916\) 0 0
\(917\) 847.443 0.924148
\(918\) 0 0
\(919\) 366.545 0.398852 0.199426 0.979913i \(-0.436092\pi\)
0.199426 + 0.979913i \(0.436092\pi\)
\(920\) 0 0
\(921\) −649.433 139.040i −0.705139 0.150967i
\(922\) 0 0
\(923\) −840.974 −0.911131
\(924\) 0 0
\(925\) 27.4887i 0.0297175i
\(926\) 0 0
\(927\) −36.2630 16.2734i −0.0391187 0.0175549i
\(928\) 0 0
\(929\) 488.045i 0.525345i −0.964885 0.262672i \(-0.915396\pi\)
0.964885 0.262672i \(-0.0846038\pi\)
\(930\) 0 0
\(931\) 122.231i 0.131290i
\(932\) 0 0
\(933\) 501.543 + 107.378i 0.537559 + 0.115089i
\(934\) 0 0
\(935\) 2172.22i 2.32323i
\(936\) 0 0
\(937\) −973.489 −1.03894 −0.519471 0.854488i \(-0.673871\pi\)
−0.519471 + 0.854488i \(0.673871\pi\)
\(938\) 0 0
\(939\) 10.8443 50.6520i 0.0115488 0.0539425i
\(940\) 0 0
\(941\) −232.064 −0.246615 −0.123307 0.992369i \(-0.539350\pi\)
−0.123307 + 0.992369i \(0.539350\pi\)
\(942\) 0 0
\(943\) 511.603 0.542527
\(944\) 0 0
\(945\) 794.866 1083.95i 0.841128 1.14703i
\(946\) 0 0
\(947\) 1005.38 1.06165 0.530826 0.847481i \(-0.321882\pi\)
0.530826 + 0.847481i \(0.321882\pi\)
\(948\) 0 0
\(949\) 1507.60i 1.58862i
\(950\) 0 0
\(951\) 65.2382 304.716i 0.0685996 0.320417i
\(952\) 0 0
\(953\) 937.182i 0.983402i 0.870764 + 0.491701i \(0.163625\pi\)
−0.870764 + 0.491701i \(0.836375\pi\)
\(954\) 0 0
\(955\) 1663.89i 1.74230i
\(956\) 0 0
\(957\) 34.6167 161.689i 0.0361721 0.168954i
\(958\) 0 0
\(959\) 838.574i 0.874425i
\(960\) 0 0
\(961\) 75.0234 0.0780681
\(962\) 0 0
\(963\) −47.2153 21.1883i −0.0490294 0.0220024i
\(964\) 0 0
\(965\) −1864.23 −1.93185
\(966\) 0 0
\(967\) 1748.82 1.80850 0.904249 0.427005i \(-0.140431\pi\)
0.904249 + 0.427005i \(0.140431\pi\)
\(968\) 0 0
\(969\) 152.222 711.003i 0.157092 0.733750i
\(970\) 0 0
\(971\) −845.706 −0.870964 −0.435482 0.900197i \(-0.643422\pi\)
−0.435482 + 0.900197i \(0.643422\pi\)
\(972\) 0 0
\(973\) 1607.42i 1.65203i
\(974\) 0 0
\(975\) −737.096 157.808i −0.755996 0.161855i
\(976\) 0 0
\(977\) 880.190i 0.900911i 0.892799 + 0.450456i \(0.148738\pi\)
−0.892799 + 0.450456i \(0.851262\pi\)
\(978\) 0 0
\(979\) 2536.52i 2.59093i
\(980\) 0 0
\(981\) −572.127 + 1274.91i −0.583208 + 1.29960i
\(982\) 0 0
\(983\) 1353.88i 1.37730i 0.725095 + 0.688648i \(0.241796\pi\)
−0.725095 + 0.688648i \(0.758204\pi\)
\(984\) 0 0
\(985\) 788.111 0.800113
\(986\) 0 0
\(987\) −1938.98 415.124i −1.96451 0.420592i
\(988\) 0 0
\(989\) 626.089 0.633052
\(990\) 0 0
\(991\) 1880.37 1.89745 0.948724 0.316106i \(-0.102376\pi\)
0.948724 + 0.316106i \(0.102376\pi\)
\(992\) 0 0
\(993\) 557.255 + 119.305i 0.561183 + 0.120146i
\(994\) 0 0
\(995\) −1585.33 −1.59329
\(996\) 0 0
\(997\) 786.688i 0.789055i −0.918884 0.394528i \(-0.870908\pi\)
0.918884 0.394528i \(-0.129092\pi\)
\(998\) 0 0
\(999\) 34.3536 + 25.1918i 0.0343880 + 0.0252170i
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 384.3.h.g.65.7 yes 16
3.2 odd 2 inner 384.3.h.g.65.12 yes 16
4.3 odd 2 inner 384.3.h.g.65.9 yes 16
8.3 odd 2 inner 384.3.h.g.65.8 yes 16
8.5 even 2 inner 384.3.h.g.65.10 yes 16
12.11 even 2 inner 384.3.h.g.65.6 yes 16
16.3 odd 4 768.3.e.o.257.8 8
16.5 even 4 768.3.e.p.257.8 8
16.11 odd 4 768.3.e.p.257.1 8
16.13 even 4 768.3.e.o.257.1 8
24.5 odd 2 inner 384.3.h.g.65.5 16
24.11 even 2 inner 384.3.h.g.65.11 yes 16
48.5 odd 4 768.3.e.p.257.7 8
48.11 even 4 768.3.e.p.257.2 8
48.29 odd 4 768.3.e.o.257.2 8
48.35 even 4 768.3.e.o.257.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.h.g.65.5 16 24.5 odd 2 inner
384.3.h.g.65.6 yes 16 12.11 even 2 inner
384.3.h.g.65.7 yes 16 1.1 even 1 trivial
384.3.h.g.65.8 yes 16 8.3 odd 2 inner
384.3.h.g.65.9 yes 16 4.3 odd 2 inner
384.3.h.g.65.10 yes 16 8.5 even 2 inner
384.3.h.g.65.11 yes 16 24.11 even 2 inner
384.3.h.g.65.12 yes 16 3.2 odd 2 inner
768.3.e.o.257.1 8 16.13 even 4
768.3.e.o.257.2 8 48.29 odd 4
768.3.e.o.257.7 8 48.35 even 4
768.3.e.o.257.8 8 16.3 odd 4
768.3.e.p.257.1 8 16.11 odd 4
768.3.e.p.257.2 8 48.11 even 4
768.3.e.p.257.7 8 48.5 odd 4
768.3.e.p.257.8 8 16.5 even 4