Properties

Label 864.2.r.b.145.4
Level $864$
Weight $2$
Character 864.145
Analytic conductor $6.899$
Analytic rank $0$
Dimension $16$
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] = [864,2,Mod(145,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.145");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.4
Root \(-1.34532 - 0.436011i\) of defining polynomial
Character \(\chi\) \(=\) 864.145
Dual form 864.2.r.b.721.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.602794 - 0.348023i) q^{5} +(-0.795065 - 1.37709i) q^{7} +O(q^{10})\) \(q+(-0.602794 - 0.348023i) q^{5} +(-0.795065 - 1.37709i) q^{7} +(-2.37222 + 1.36960i) q^{11} +(4.76780 + 2.75269i) q^{13} +5.65175 q^{17} -0.963328i q^{19} +(3.28857 - 5.69597i) q^{23} +(-2.25776 - 3.91055i) q^{25} +(2.85076 - 1.64589i) q^{29} +(3.69844 - 6.40589i) q^{31} +1.10680i q^{35} -6.25538i q^{37} +(0.931886 - 1.61407i) q^{41} +(-2.99838 + 1.73111i) q^{43} +(3.85668 + 6.67997i) q^{47} +(2.23574 - 3.87242i) q^{49} +2.54179i q^{53} +1.90662 q^{55} +(4.62019 + 2.66747i) q^{59} +(-7.93715 + 4.58252i) q^{61} +(-1.91600 - 3.31861i) q^{65} +(5.95780 + 3.43974i) q^{67} +3.68351 q^{71} +2.83201 q^{73} +(3.77214 + 2.17785i) q^{77} +(-2.87870 - 4.98605i) q^{79} +(5.74968 - 3.31958i) q^{83} +(-3.40684 - 1.96694i) q^{85} +2.98701 q^{89} -8.75427i q^{91} +(-0.335261 + 0.580689i) q^{95} +(-1.24837 - 2.16224i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{7} + 28 q^{17} - 10 q^{23} + 2 q^{25} + 10 q^{31} + 8 q^{41} + 6 q^{47} + 18 q^{49} + 4 q^{55} + 14 q^{65} + 72 q^{71} - 44 q^{73} + 30 q^{79} - 64 q^{89} + 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.602794 0.348023i −0.269578 0.155641i 0.359118 0.933292i \(-0.383078\pi\)
−0.628696 + 0.777651i \(0.716411\pi\)
\(6\) 0 0
\(7\) −0.795065 1.37709i −0.300506 0.520492i 0.675745 0.737136i \(-0.263822\pi\)
−0.976251 + 0.216644i \(0.930489\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.37222 + 1.36960i −0.715252 + 0.412951i −0.813003 0.582260i \(-0.802169\pi\)
0.0977506 + 0.995211i \(0.468835\pi\)
\(12\) 0 0
\(13\) 4.76780 + 2.75269i 1.32235 + 0.763460i 0.984103 0.177597i \(-0.0568325\pi\)
0.338248 + 0.941057i \(0.390166\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.65175 1.37075 0.685375 0.728190i \(-0.259638\pi\)
0.685375 + 0.728190i \(0.259638\pi\)
\(18\) 0 0
\(19\) 0.963328i 0.221003i −0.993876 0.110501i \(-0.964754\pi\)
0.993876 0.110501i \(-0.0352457\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.28857 5.69597i 0.685714 1.18769i −0.287498 0.957781i \(-0.592823\pi\)
0.973212 0.229910i \(-0.0738432\pi\)
\(24\) 0 0
\(25\) −2.25776 3.91055i −0.451552 0.782111i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.85076 1.64589i 0.529373 0.305634i −0.211388 0.977402i \(-0.567798\pi\)
0.740761 + 0.671768i \(0.234465\pi\)
\(30\) 0 0
\(31\) 3.69844 6.40589i 0.664259 1.15053i −0.315226 0.949017i \(-0.602080\pi\)
0.979486 0.201514i \(-0.0645863\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.10680i 0.187084i
\(36\) 0 0
\(37\) 6.25538i 1.02838i −0.857677 0.514189i \(-0.828093\pi\)
0.857677 0.514189i \(-0.171907\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.931886 1.61407i 0.145536 0.252076i −0.784037 0.620714i \(-0.786843\pi\)
0.929573 + 0.368639i \(0.120176\pi\)
\(42\) 0 0
\(43\) −2.99838 + 1.73111i −0.457248 + 0.263992i −0.710886 0.703307i \(-0.751706\pi\)
0.253638 + 0.967299i \(0.418373\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.85668 + 6.67997i 0.562555 + 0.974374i 0.997273 + 0.0738070i \(0.0235149\pi\)
−0.434717 + 0.900567i \(0.643152\pi\)
\(48\) 0 0
\(49\) 2.23574 3.87242i 0.319392 0.553203i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.54179i 0.349141i 0.984645 + 0.174571i \(0.0558537\pi\)
−0.984645 + 0.174571i \(0.944146\pi\)
\(54\) 0 0
\(55\) 1.90662 0.257088
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.62019 + 2.66747i 0.601498 + 0.347275i 0.769631 0.638489i \(-0.220440\pi\)
−0.168133 + 0.985764i \(0.553774\pi\)
\(60\) 0 0
\(61\) −7.93715 + 4.58252i −1.01625 + 0.586731i −0.913015 0.407926i \(-0.866252\pi\)
−0.103233 + 0.994657i \(0.532919\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.91600 3.31861i −0.237651 0.411623i
\(66\) 0 0
\(67\) 5.95780 + 3.43974i 0.727861 + 0.420231i 0.817639 0.575731i \(-0.195283\pi\)
−0.0897783 + 0.995962i \(0.528616\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.68351 0.437153 0.218576 0.975820i \(-0.429859\pi\)
0.218576 + 0.975820i \(0.429859\pi\)
\(72\) 0 0
\(73\) 2.83201 0.331461 0.165731 0.986171i \(-0.447002\pi\)
0.165731 + 0.986171i \(0.447002\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.77214 + 2.17785i 0.429875 + 0.248189i
\(78\) 0 0
\(79\) −2.87870 4.98605i −0.323879 0.560975i 0.657406 0.753537i \(-0.271654\pi\)
−0.981285 + 0.192562i \(0.938320\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.74968 3.31958i 0.631110 0.364371i −0.150072 0.988675i \(-0.547951\pi\)
0.781182 + 0.624304i \(0.214617\pi\)
\(84\) 0 0
\(85\) −3.40684 1.96694i −0.369524 0.213345i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.98701 0.316622 0.158311 0.987389i \(-0.449395\pi\)
0.158311 + 0.987389i \(0.449395\pi\)
\(90\) 0 0
\(91\) 8.75427i 0.917697i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.335261 + 0.580689i −0.0343970 + 0.0595774i
\(96\) 0 0
\(97\) −1.24837 2.16224i −0.126753 0.219543i 0.795664 0.605738i \(-0.207122\pi\)
−0.922417 + 0.386196i \(0.873789\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.22136 + 4.74661i −0.818056 + 0.472305i −0.849746 0.527193i \(-0.823245\pi\)
0.0316896 + 0.999498i \(0.489911\pi\)
\(102\) 0 0
\(103\) −7.37220 + 12.7690i −0.726405 + 1.25817i 0.231989 + 0.972719i \(0.425477\pi\)
−0.958393 + 0.285451i \(0.907857\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.83384i 0.757325i −0.925535 0.378663i \(-0.876384\pi\)
0.925535 0.378663i \(-0.123616\pi\)
\(108\) 0 0
\(109\) 0.242400i 0.0232177i 0.999933 + 0.0116089i \(0.00369529\pi\)
−0.999933 + 0.0116089i \(0.996305\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.34789 + 7.53076i −0.409015 + 0.708435i −0.994780 0.102046i \(-0.967461\pi\)
0.585765 + 0.810481i \(0.300794\pi\)
\(114\) 0 0
\(115\) −3.96466 + 2.28900i −0.369706 + 0.213450i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.49350 7.78298i −0.411919 0.713464i
\(120\) 0 0
\(121\) −1.74837 + 3.02827i −0.158943 + 0.275297i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.62325i 0.592401i
\(126\) 0 0
\(127\) 1.72754 0.153295 0.0766473 0.997058i \(-0.475578\pi\)
0.0766473 + 0.997058i \(0.475578\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.74968 3.31958i −0.502352 0.290033i 0.227332 0.973817i \(-0.427000\pi\)
−0.729684 + 0.683784i \(0.760333\pi\)
\(132\) 0 0
\(133\) −1.32659 + 0.765908i −0.115030 + 0.0664127i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.81325 3.14063i −0.154916 0.268322i 0.778112 0.628125i \(-0.216177\pi\)
−0.933028 + 0.359803i \(0.882844\pi\)
\(138\) 0 0
\(139\) 14.9919 + 8.65556i 1.27159 + 0.734155i 0.975288 0.220937i \(-0.0709117\pi\)
0.296307 + 0.955093i \(0.404245\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −15.0804 −1.26109
\(144\) 0 0
\(145\) −2.29123 −0.190276
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.7251 10.8109i −1.53402 0.885665i −0.999171 0.0407158i \(-0.987036\pi\)
−0.534846 0.844949i \(-0.679631\pi\)
\(150\) 0 0
\(151\) 6.35019 + 10.9988i 0.516771 + 0.895073i 0.999810 + 0.0194749i \(0.00619944\pi\)
−0.483039 + 0.875599i \(0.660467\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.45880 + 2.57429i −0.358139 + 0.206772i
\(156\) 0 0
\(157\) 15.1285 + 8.73443i 1.20738 + 0.697083i 0.962187 0.272390i \(-0.0878140\pi\)
0.245197 + 0.969473i \(0.421147\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −10.4585 −0.824245
\(162\) 0 0
\(163\) 8.56748i 0.671057i −0.942030 0.335528i \(-0.891085\pi\)
0.942030 0.335528i \(-0.108915\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.97532 + 10.3496i −0.462384 + 0.800873i −0.999079 0.0429032i \(-0.986339\pi\)
0.536695 + 0.843776i \(0.319673\pi\)
\(168\) 0 0
\(169\) 8.65464 + 14.9903i 0.665741 + 1.15310i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.2973 6.52248i 0.858916 0.495895i −0.00473326 0.999989i \(-0.501507\pi\)
0.863649 + 0.504094i \(0.168173\pi\)
\(174\) 0 0
\(175\) −3.59013 + 6.21829i −0.271388 + 0.470058i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.31875i 0.248055i 0.992279 + 0.124028i \(0.0395811\pi\)
−0.992279 + 0.124028i \(0.960419\pi\)
\(180\) 0 0
\(181\) 14.9128i 1.10846i −0.832363 0.554231i \(-0.813013\pi\)
0.832363 0.554231i \(-0.186987\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.17702 + 3.77070i −0.160058 + 0.277228i
\(186\) 0 0
\(187\) −13.4072 + 7.74065i −0.980432 + 0.566053i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.65884 + 6.33729i 0.264744 + 0.458550i 0.967497 0.252884i \(-0.0813792\pi\)
−0.702752 + 0.711434i \(0.748046\pi\)
\(192\) 0 0
\(193\) −10.2354 + 17.7282i −0.736759 + 1.27610i 0.217189 + 0.976130i \(0.430311\pi\)
−0.953947 + 0.299974i \(0.903022\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.5437i 1.46368i −0.681479 0.731838i \(-0.738663\pi\)
0.681479 0.731838i \(-0.261337\pi\)
\(198\) 0 0
\(199\) −1.95597 −0.138655 −0.0693275 0.997594i \(-0.522085\pi\)
−0.0693275 + 0.997594i \(0.522085\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.53308 2.61718i −0.318160 0.183690i
\(204\) 0 0
\(205\) −1.12347 + 0.648636i −0.0784666 + 0.0453027i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.31938 + 2.28523i 0.0912633 + 0.158073i
\(210\) 0 0
\(211\) −9.10981 5.25955i −0.627145 0.362082i 0.152501 0.988303i \(-0.451267\pi\)
−0.779646 + 0.626221i \(0.784601\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.40987 0.164352
\(216\) 0 0
\(217\) −11.7620 −0.798456
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 26.9464 + 15.5575i 1.81261 + 1.04651i
\(222\) 0 0
\(223\) −1.93129 3.34510i −0.129329 0.224004i 0.794088 0.607803i \(-0.207949\pi\)
−0.923417 + 0.383799i \(0.874616\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −13.9183 + 8.03574i −0.923790 + 0.533351i −0.884842 0.465891i \(-0.845734\pi\)
−0.0389481 + 0.999241i \(0.512401\pi\)
\(228\) 0 0
\(229\) −7.46319 4.30888i −0.493182 0.284739i 0.232712 0.972546i \(-0.425240\pi\)
−0.725893 + 0.687807i \(0.758573\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.1535 −1.58235 −0.791176 0.611589i \(-0.790531\pi\)
−0.791176 + 0.611589i \(0.790531\pi\)
\(234\) 0 0
\(235\) 5.36886i 0.350226i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.01493 + 3.48996i −0.130335 + 0.225746i −0.923806 0.382862i \(-0.874939\pi\)
0.793471 + 0.608608i \(0.208272\pi\)
\(240\) 0 0
\(241\) 2.81649 + 4.87830i 0.181426 + 0.314239i 0.942366 0.334583i \(-0.108595\pi\)
−0.760940 + 0.648822i \(0.775262\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.69539 + 1.55618i −0.172202 + 0.0994209i
\(246\) 0 0
\(247\) 2.65175 4.59296i 0.168727 0.292243i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.8828i 0.876276i 0.898908 + 0.438138i \(0.144362\pi\)
−0.898908 + 0.438138i \(0.855638\pi\)
\(252\) 0 0
\(253\) 18.0161i 1.13267i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.42539 9.39705i 0.338427 0.586172i −0.645710 0.763582i \(-0.723439\pi\)
0.984137 + 0.177410i \(0.0567720\pi\)
\(258\) 0 0
\(259\) −8.61423 + 4.97343i −0.535262 + 0.309034i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −11.6051 20.1005i −0.715598 1.23945i −0.962728 0.270470i \(-0.912821\pi\)
0.247130 0.968982i \(-0.420513\pi\)
\(264\) 0 0
\(265\) 0.884601 1.53217i 0.0543406 0.0941207i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.01966i 0.245083i −0.992463 0.122541i \(-0.960896\pi\)
0.992463 0.122541i \(-0.0391044\pi\)
\(270\) 0 0
\(271\) 6.75621 0.410411 0.205205 0.978719i \(-0.434214\pi\)
0.205205 + 0.978719i \(0.434214\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.7118 + 6.18447i 0.645947 + 0.372938i
\(276\) 0 0
\(277\) −1.83595 + 1.05999i −0.110312 + 0.0636885i −0.554141 0.832423i \(-0.686953\pi\)
0.443829 + 0.896111i \(0.353620\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 13.0580 + 22.6171i 0.778976 + 1.34923i 0.932533 + 0.361086i \(0.117594\pi\)
−0.153557 + 0.988140i \(0.549073\pi\)
\(282\) 0 0
\(283\) −16.5376 9.54799i −0.983058 0.567569i −0.0798661 0.996806i \(-0.525449\pi\)
−0.903192 + 0.429237i \(0.858783\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.96364 −0.174938
\(288\) 0 0
\(289\) 14.9423 0.878956
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5.07116 2.92784i −0.296261 0.171046i 0.344501 0.938786i \(-0.388048\pi\)
−0.640762 + 0.767740i \(0.721381\pi\)
\(294\) 0 0
\(295\) −1.85668 3.21587i −0.108100 0.187235i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 31.3585 18.1048i 1.81351 1.04703i
\(300\) 0 0
\(301\) 4.76780 + 2.75269i 0.274812 + 0.158663i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.37929 0.365277
\(306\) 0 0
\(307\) 13.7071i 0.782305i 0.920326 + 0.391152i \(0.127923\pi\)
−0.920326 + 0.391152i \(0.872077\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.57980 + 16.5927i −0.543221 + 0.940886i 0.455496 + 0.890238i \(0.349462\pi\)
−0.998717 + 0.0506479i \(0.983871\pi\)
\(312\) 0 0
\(313\) −12.6102 21.8416i −0.712773 1.23456i −0.963812 0.266582i \(-0.914106\pi\)
0.251039 0.967977i \(-0.419228\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.13931 1.23513i 0.120156 0.0693719i −0.438718 0.898625i \(-0.644567\pi\)
0.558873 + 0.829253i \(0.311234\pi\)
\(318\) 0 0
\(319\) −4.50843 + 7.80883i −0.252424 + 0.437211i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.44449i 0.302939i
\(324\) 0 0
\(325\) 24.8597i 1.37897i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.13262 10.6220i 0.338103 0.585611i
\(330\) 0 0
\(331\) 24.4404 14.1107i 1.34336 0.775592i 0.356065 0.934461i \(-0.384118\pi\)
0.987300 + 0.158869i \(0.0507848\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.39422 4.14691i −0.130810 0.226570i
\(336\) 0 0
\(337\) 5.60565 9.70927i 0.305359 0.528897i −0.671982 0.740567i \(-0.734557\pi\)
0.977341 + 0.211670i \(0.0678902\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 20.2616i 1.09723i
\(342\) 0 0
\(343\) −18.2411 −0.984929
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −17.8303 10.2943i −0.957180 0.552628i −0.0618763 0.998084i \(-0.519708\pi\)
−0.895304 + 0.445455i \(0.853042\pi\)
\(348\) 0 0
\(349\) −2.93968 + 1.69723i −0.157358 + 0.0908505i −0.576611 0.817019i \(-0.695625\pi\)
0.419253 + 0.907869i \(0.362292\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.503241 + 0.871639i 0.0267848 + 0.0463926i 0.879107 0.476624i \(-0.158140\pi\)
−0.852322 + 0.523017i \(0.824806\pi\)
\(354\) 0 0
\(355\) −2.22040 1.28195i −0.117847 0.0680388i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 31.4772 1.66131 0.830653 0.556791i \(-0.187968\pi\)
0.830653 + 0.556791i \(0.187968\pi\)
\(360\) 0 0
\(361\) 18.0720 0.951158
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.70712 0.985604i −0.0893546 0.0515889i
\(366\) 0 0
\(367\) 8.66667 + 15.0111i 0.452397 + 0.783574i 0.998534 0.0541214i \(-0.0172358\pi\)
−0.546138 + 0.837695i \(0.683902\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.50027 2.02088i 0.181725 0.104919i
\(372\) 0 0
\(373\) −11.2742 6.50917i −0.583757 0.337032i 0.178868 0.983873i \(-0.442756\pi\)
−0.762625 + 0.646841i \(0.776090\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.1225 0.933357
\(378\) 0 0
\(379\) 22.8643i 1.17446i 0.809421 + 0.587229i \(0.199781\pi\)
−0.809421 + 0.587229i \(0.800219\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.0117 26.0010i 0.767061 1.32859i −0.172089 0.985081i \(-0.555052\pi\)
0.939150 0.343508i \(-0.111615\pi\)
\(384\) 0 0
\(385\) −1.51588 2.62559i −0.0772565 0.133812i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −32.9474 + 19.0222i −1.67050 + 0.964463i −0.703140 + 0.711051i \(0.748220\pi\)
−0.967358 + 0.253412i \(0.918447\pi\)
\(390\) 0 0
\(391\) 18.5862 32.1922i 0.939943 1.62803i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.00742i 0.201635i
\(396\) 0 0
\(397\) 37.4510i 1.87961i 0.341709 + 0.939806i \(0.388994\pi\)
−0.341709 + 0.939806i \(0.611006\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.35402 4.07728i 0.117554 0.203610i −0.801244 0.598338i \(-0.795828\pi\)
0.918798 + 0.394728i \(0.129161\pi\)
\(402\) 0 0
\(403\) 35.2669 20.3613i 1.75677 1.01427i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8.56739 + 14.8391i 0.424670 + 0.735549i
\(408\) 0 0
\(409\) 5.36377 9.29032i 0.265221 0.459377i −0.702400 0.711782i \(-0.747888\pi\)
0.967622 + 0.252405i \(0.0812216\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.48324i 0.417433i
\(414\) 0 0
\(415\) −4.62117 −0.226844
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3.57600 + 2.06460i 0.174699 + 0.100863i 0.584800 0.811178i \(-0.301173\pi\)
−0.410101 + 0.912040i \(0.634506\pi\)
\(420\) 0 0
\(421\) 13.7321 7.92824i 0.669262 0.386399i −0.126535 0.991962i \(-0.540386\pi\)
0.795797 + 0.605563i \(0.207052\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −12.7603 22.1015i −0.618965 1.07208i
\(426\) 0 0
\(427\) 12.6211 + 7.28679i 0.610778 + 0.352633i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.1853 0.779619 0.389810 0.920895i \(-0.372541\pi\)
0.389810 + 0.920895i \(0.372541\pi\)
\(432\) 0 0
\(433\) −32.8306 −1.57774 −0.788868 0.614563i \(-0.789332\pi\)
−0.788868 + 0.614563i \(0.789332\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.48709 3.16797i −0.262483 0.151545i
\(438\) 0 0
\(439\) −10.9273 18.9267i −0.521533 0.903321i −0.999686 0.0250450i \(-0.992027\pi\)
0.478154 0.878276i \(-0.341306\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −30.4500 + 17.5803i −1.44672 + 0.835265i −0.998284 0.0585501i \(-0.981352\pi\)
−0.448436 + 0.893815i \(0.648019\pi\)
\(444\) 0 0
\(445\) −1.80055 1.03955i −0.0853543 0.0492793i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.21851 −0.151891 −0.0759453 0.997112i \(-0.524197\pi\)
−0.0759453 + 0.997112i \(0.524197\pi\)
\(450\) 0 0
\(451\) 5.10526i 0.240397i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.04669 + 5.27703i −0.142831 + 0.247391i
\(456\) 0 0
\(457\) 4.05512 + 7.02368i 0.189691 + 0.328554i 0.945147 0.326645i \(-0.105918\pi\)
−0.755456 + 0.655199i \(0.772585\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.1813 10.4970i 0.846789 0.488894i −0.0127771 0.999918i \(-0.504067\pi\)
0.859566 + 0.511024i \(0.170734\pi\)
\(462\) 0 0
\(463\) 4.45005 7.70772i 0.206812 0.358208i −0.743897 0.668294i \(-0.767025\pi\)
0.950708 + 0.310086i \(0.100358\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.0527i 1.20557i −0.797902 0.602787i \(-0.794057\pi\)
0.797902 0.602787i \(-0.205943\pi\)
\(468\) 0 0
\(469\) 10.9392i 0.505128i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.74188 8.21317i 0.218032 0.377642i
\(474\) 0 0
\(475\) −3.76715 + 2.17496i −0.172849 + 0.0997942i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8.71143 + 15.0886i 0.398035 + 0.689418i 0.993483 0.113976i \(-0.0363588\pi\)
−0.595448 + 0.803394i \(0.703025\pi\)
\(480\) 0 0
\(481\) 17.2191 29.8244i 0.785125 1.35988i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.73785i 0.0789118i
\(486\) 0 0
\(487\) 29.7367 1.34750 0.673750 0.738959i \(-0.264682\pi\)
0.673750 + 0.738959i \(0.264682\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 20.6346 + 11.9134i 0.931229 + 0.537645i 0.887200 0.461385i \(-0.152647\pi\)
0.0440286 + 0.999030i \(0.485981\pi\)
\(492\) 0 0
\(493\) 16.1118 9.30215i 0.725639 0.418948i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.92863 5.07254i −0.131367 0.227534i
\(498\) 0 0
\(499\) 16.8622 + 9.73540i 0.754856 + 0.435816i 0.827446 0.561546i \(-0.189793\pi\)
−0.0725899 + 0.997362i \(0.523126\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.23494 0.0550631 0.0275316 0.999621i \(-0.491235\pi\)
0.0275316 + 0.999621i \(0.491235\pi\)
\(504\) 0 0
\(505\) 6.60772 0.294040
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.392870 0.226823i −0.0174136 0.0100538i 0.491268 0.871009i \(-0.336534\pi\)
−0.508682 + 0.860955i \(0.669867\pi\)
\(510\) 0 0
\(511\) −2.25163 3.89993i −0.0996061 0.172523i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8.88784 5.13140i 0.391645 0.226116i
\(516\) 0 0
\(517\) −18.2978 10.5643i −0.804737 0.464615i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.0873 1.27434 0.637170 0.770724i \(-0.280105\pi\)
0.637170 + 0.770724i \(0.280105\pi\)
\(522\) 0 0
\(523\) 2.95874i 0.129377i 0.997906 + 0.0646883i \(0.0206053\pi\)
−0.997906 + 0.0646883i \(0.979395\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.9026 36.2044i 0.910534 1.57709i
\(528\) 0 0
\(529\) −10.1294 17.5446i −0.440407 0.762808i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.88610 5.13039i 0.384900 0.222222i
\(534\) 0 0
\(535\) −2.72636 + 4.72219i −0.117871 + 0.204158i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.2483i 0.527573i
\(540\) 0 0
\(541\) 14.9753i 0.643838i 0.946767 + 0.321919i \(0.104328\pi\)
−0.946767 + 0.321919i \(0.895672\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.0843608 0.146117i 0.00361362 0.00625897i
\(546\) 0 0
\(547\) −15.7731 + 9.10661i −0.674409 + 0.389370i −0.797745 0.602995i \(-0.793974\pi\)
0.123336 + 0.992365i \(0.460641\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.58553 2.74622i −0.0675459 0.116993i
\(552\) 0 0
\(553\) −4.57750 + 7.92846i −0.194655 + 0.337153i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 35.7359i 1.51418i 0.653310 + 0.757090i \(0.273380\pi\)
−0.653310 + 0.757090i \(0.726620\pi\)
\(558\) 0 0
\(559\) −19.0609 −0.806190
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.04256 4.64337i −0.338953 0.195695i 0.320856 0.947128i \(-0.396030\pi\)
−0.659809 + 0.751433i \(0.729363\pi\)
\(564\) 0 0
\(565\) 5.24176 3.02633i 0.220523 0.127319i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.66727 + 9.81599i 0.237584 + 0.411508i 0.960021 0.279930i \(-0.0903111\pi\)
−0.722436 + 0.691437i \(0.756978\pi\)
\(570\) 0 0
\(571\) −37.7843 21.8148i −1.58122 0.912920i −0.994681 0.103002i \(-0.967155\pi\)
−0.586543 0.809918i \(-0.699511\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −29.6992 −1.23854
\(576\) 0 0
\(577\) 6.98123 0.290632 0.145316 0.989385i \(-0.453580\pi\)
0.145316 + 0.989385i \(0.453580\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −9.14274 5.27856i −0.379305 0.218992i
\(582\) 0 0
\(583\) −3.48124 6.02968i −0.144178 0.249724i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.34574 4.24107i 0.303191 0.175048i −0.340684 0.940178i \(-0.610659\pi\)
0.643876 + 0.765130i \(0.277325\pi\)
\(588\) 0 0
\(589\) −6.17097 3.56281i −0.254270 0.146803i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.40869 0.386368 0.193184 0.981163i \(-0.438119\pi\)
0.193184 + 0.981163i \(0.438119\pi\)
\(594\) 0 0
\(595\) 6.25538i 0.256445i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −14.9623 + 25.9155i −0.611344 + 1.05888i 0.379670 + 0.925122i \(0.376038\pi\)
−0.991014 + 0.133757i \(0.957296\pi\)
\(600\) 0 0
\(601\) 1.81973 + 3.15186i 0.0742282 + 0.128567i 0.900750 0.434337i \(-0.143017\pi\)
−0.826522 + 0.562904i \(0.809684\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.10782 1.21695i 0.0856950 0.0494760i
\(606\) 0 0
\(607\) −3.63358 + 6.29355i −0.147482 + 0.255447i −0.930296 0.366809i \(-0.880450\pi\)
0.782814 + 0.622256i \(0.213784\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 42.4651i 1.71795i
\(612\) 0 0
\(613\) 32.6469i 1.31859i 0.751882 + 0.659297i \(0.229146\pi\)
−0.751882 + 0.659297i \(0.770854\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.6751 + 27.1501i −0.631056 + 1.09302i 0.356280 + 0.934379i \(0.384045\pi\)
−0.987336 + 0.158642i \(0.949288\pi\)
\(618\) 0 0
\(619\) −1.72589 + 0.996445i −0.0693695 + 0.0400505i −0.534284 0.845305i \(-0.679419\pi\)
0.464914 + 0.885356i \(0.346085\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.37486 4.11339i −0.0951469 0.164799i
\(624\) 0 0
\(625\) −8.98375 + 15.5603i −0.359350 + 0.622413i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 35.3538i 1.40965i
\(630\) 0 0
\(631\) 15.4643 0.615623 0.307812 0.951447i \(-0.400403\pi\)
0.307812 + 0.951447i \(0.400403\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.04135 0.601225i −0.0413248 0.0238589i
\(636\) 0 0
\(637\) 21.3192 12.3086i 0.844697 0.487686i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.3638 + 21.4147i 0.488340 + 0.845829i 0.999910 0.0134123i \(-0.00426940\pi\)
−0.511570 + 0.859241i \(0.670936\pi\)
\(642\) 0 0
\(643\) −40.0176 23.1042i −1.57814 0.911141i −0.995119 0.0986850i \(-0.968536\pi\)
−0.583023 0.812456i \(-0.698130\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.36971 −0.250419 −0.125210 0.992130i \(-0.539960\pi\)
−0.125210 + 0.992130i \(0.539960\pi\)
\(648\) 0 0
\(649\) −14.6135 −0.573630
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 31.8848 + 18.4087i 1.24775 + 0.720389i 0.970660 0.240455i \(-0.0772968\pi\)
0.277090 + 0.960844i \(0.410630\pi\)
\(654\) 0 0
\(655\) 2.31058 + 4.00205i 0.0902820 + 0.156373i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −6.46565 + 3.73294i −0.251866 + 0.145415i −0.620618 0.784113i \(-0.713118\pi\)
0.368752 + 0.929528i \(0.379785\pi\)
\(660\) 0 0
\(661\) −2.51984 1.45483i −0.0980102 0.0565862i 0.450194 0.892931i \(-0.351355\pi\)
−0.548204 + 0.836345i \(0.684688\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.06622 0.0413461
\(666\) 0 0
\(667\) 21.6505i 0.838310i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.5525 21.7415i 0.484582 0.839321i
\(672\) 0 0
\(673\) 21.0527 + 36.4643i 0.811522 + 1.40560i 0.911799 + 0.410637i \(0.134694\pi\)
−0.100277 + 0.994960i \(0.531973\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 32.9941 19.0492i 1.26807 0.732119i 0.293445 0.955976i \(-0.405198\pi\)
0.974622 + 0.223858i \(0.0718650\pi\)
\(678\) 0 0
\(679\) −1.98507 + 3.43825i −0.0761801 + 0.131948i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 47.0728i 1.80119i −0.434659 0.900595i \(-0.643131\pi\)
0.434659 0.900595i \(-0.356869\pi\)
\(684\) 0 0
\(685\) 2.52421i 0.0964450i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.99676 + 12.1187i −0.266555 + 0.461687i
\(690\) 0 0
\(691\) 3.38522 1.95446i 0.128780 0.0743512i −0.434226 0.900804i \(-0.642978\pi\)
0.563006 + 0.826453i \(0.309645\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.02468 10.4350i −0.228529 0.395824i
\(696\) 0 0
\(697\) 5.26678 9.12234i 0.199494 0.345533i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 16.4480i 0.621231i −0.950536 0.310615i \(-0.899465\pi\)
0.950536 0.310615i \(-0.100535\pi\)
\(702\) 0 0
\(703\) −6.02598 −0.227274
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.0730 + 7.54772i 0.491662 + 0.283861i
\(708\) 0 0
\(709\) 6.84805 3.95372i 0.257184 0.148485i −0.365865 0.930668i \(-0.619227\pi\)
0.623049 + 0.782183i \(0.285894\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −24.3251 42.1324i −0.910984 1.57787i
\(714\) 0 0
\(715\) 9.09037 + 5.24833i 0.339961 + 0.196276i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −37.0556 −1.38194 −0.690970 0.722884i \(-0.742816\pi\)
−0.690970 + 0.722884i \(0.742816\pi\)
\(720\) 0 0
\(721\) 23.4455 0.873156
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.8727 7.43204i −0.478079 0.276019i
\(726\) 0 0
\(727\) −2.83467 4.90979i −0.105132 0.182094i 0.808660 0.588276i \(-0.200193\pi\)
−0.913792 + 0.406182i \(0.866860\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −16.9461 + 9.78381i −0.626773 + 0.361867i
\(732\) 0 0
\(733\) −10.8544 6.26677i −0.400915 0.231469i 0.285964 0.958240i \(-0.407686\pi\)
−0.686879 + 0.726772i \(0.741020\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.8443 −0.694139
\(738\) 0 0
\(739\) 1.83358i 0.0674492i −0.999431 0.0337246i \(-0.989263\pi\)
0.999431 0.0337246i \(-0.0107369\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −15.6588 + 27.1219i −0.574467 + 0.995006i 0.421632 + 0.906767i \(0.361457\pi\)
−0.996099 + 0.0882391i \(0.971876\pi\)
\(744\) 0 0
\(745\) 7.52491 + 13.0335i 0.275691 + 0.477511i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −10.7879 + 6.22841i −0.394182 + 0.227581i
\(750\) 0 0
\(751\) −3.64466 + 6.31274i −0.132996 + 0.230355i −0.924830 0.380381i \(-0.875793\pi\)
0.791834 + 0.610736i \(0.209126\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.84005i 0.321722i
\(756\) 0 0
\(757\) 12.8156i 0.465792i 0.972502 + 0.232896i \(0.0748202\pi\)
−0.972502 + 0.232896i \(0.925180\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −12.5800 + 21.7892i −0.456025 + 0.789859i −0.998747 0.0500541i \(-0.984061\pi\)
0.542721 + 0.839913i \(0.317394\pi\)
\(762\) 0 0
\(763\) 0.333807 0.192724i 0.0120846 0.00697706i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.6854 + 25.4359i 0.530261 + 0.918439i
\(768\) 0 0
\(769\) −10.7318 + 18.5880i −0.386998 + 0.670300i −0.992044 0.125890i \(-0.959821\pi\)
0.605046 + 0.796190i \(0.293155\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 20.2122i 0.726981i 0.931598 + 0.363491i \(0.118415\pi\)
−0.931598 + 0.363491i \(0.881585\pi\)
\(774\) 0 0
\(775\) −33.4007 −1.19979
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.55488 0.897712i −0.0557095 0.0321639i
\(780\) 0 0
\(781\) −8.73811 + 5.04495i −0.312674 + 0.180523i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −6.07957 10.5301i −0.216989 0.375836i
\(786\) 0 0
\(787\) 17.5726 + 10.1455i 0.626395 + 0.361649i 0.779355 0.626583i \(-0.215547\pi\)
−0.152960 + 0.988232i \(0.548880\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 13.8274 0.491646
\(792\) 0 0
\(793\) −50.4570 −1.79178
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −28.8758 16.6715i −1.02283 0.590533i −0.107910 0.994161i \(-0.534416\pi\)
−0.914924 + 0.403627i \(0.867749\pi\)
\(798\) 0 0
\(799\) 21.7970 + 37.7535i 0.771122 + 1.33562i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.71815 + 3.87873i −0.237078 + 0.136877i
\(804\) 0 0
\(805\) 6.30432 + 3.63980i 0.222198 + 0.128286i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.6920 −1.07907 −0.539536 0.841962i \(-0.681400\pi\)
−0.539536 + 0.841962i \(0.681400\pi\)
\(810\) 0 0
\(811\) 49.5457i 1.73978i 0.493241 + 0.869892i \(0.335812\pi\)
−0.493241 + 0.869892i \(0.664188\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.98168 + 5.16443i −0.104444 + 0.180902i
\(816\) 0 0
\(817\) 1.66763 + 2.88842i 0.0583430 + 0.101053i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −32.9739 + 19.0375i −1.15080 + 0.664414i −0.949082 0.315030i \(-0.897985\pi\)
−0.201716 + 0.979444i \(0.564652\pi\)
\(822\) 0 0
\(823\) −11.2626 + 19.5074i −0.392589 + 0.679984i −0.992790 0.119865i \(-0.961754\pi\)
0.600201 + 0.799849i \(0.295087\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 33.5317i 1.16601i 0.812468 + 0.583006i \(0.198124\pi\)
−0.812468 + 0.583006i \(0.801876\pi\)
\(828\) 0 0
\(829\) 37.7559i 1.31132i −0.755058 0.655658i \(-0.772391\pi\)
0.755058 0.655658i \(-0.227609\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 12.6359 21.8860i 0.437807 0.758304i
\(834\) 0 0
\(835\) 7.20378 4.15910i 0.249297 0.143932i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −9.90604 17.1578i −0.341994 0.592352i 0.642809 0.766027i \(-0.277769\pi\)
−0.984803 + 0.173675i \(0.944436\pi\)
\(840\) 0 0
\(841\) −9.08210 + 15.7307i −0.313176 + 0.542436i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.0481i 0.414466i
\(846\) 0 0
\(847\) 5.56028 0.191053
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −35.6304 20.5712i −1.22140 0.705173i
\(852\) 0 0
\(853\) 5.95424 3.43768i 0.203869 0.117704i −0.394590 0.918857i \(-0.629113\pi\)
0.598459 + 0.801153i \(0.295780\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.87316 + 6.70851i 0.132305 + 0.229158i 0.924565 0.381025i \(-0.124429\pi\)
−0.792260 + 0.610184i \(0.791096\pi\)
\(858\) 0 0
\(859\) 0.594592 + 0.343288i 0.0202872 + 0.0117128i 0.510109 0.860110i \(-0.329605\pi\)
−0.489822 + 0.871822i \(0.662938\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −42.9194 −1.46099 −0.730496 0.682917i \(-0.760711\pi\)
−0.730496 + 0.682917i \(0.760711\pi\)
\(864\) 0 0
\(865\) −9.07991 −0.308726
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 13.6578 + 7.88535i 0.463310 + 0.267492i
\(870\) 0 0
\(871\) 18.9371 + 32.8000i 0.641658 + 1.11138i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.12082 5.26591i 0.308340 0.178020i
\(876\) 0 0
\(877\) −14.7508 8.51640i −0.498100 0.287578i 0.229828 0.973231i \(-0.426183\pi\)
−0.727929 + 0.685653i \(0.759517\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 7.90546 0.266342 0.133171 0.991093i \(-0.457484\pi\)
0.133171 + 0.991093i \(0.457484\pi\)
\(882\) 0 0
\(883\) 7.53298i 0.253505i 0.991934 + 0.126752i \(0.0404554\pi\)
−0.991934 + 0.126752i \(0.959545\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7.02719 12.1715i 0.235950 0.408677i −0.723598 0.690221i \(-0.757513\pi\)
0.959548 + 0.281544i \(0.0908465\pi\)
\(888\) 0 0
\(889\) −1.37351 2.37899i −0.0460660 0.0797886i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.43501 3.71525i 0.215339 0.124326i
\(894\) 0 0
\(895\) 1.15500 2.00052i 0.0386075 0.0668701i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.3489i 0.812081i
\(900\) 0 0
\(901\) 14.3655i 0.478585i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.19001 + 8.98936i −0.172522 + 0.298816i
\(906\) 0 0
\(907\) 39.7958 22.9761i 1.32140 0.762910i 0.337447 0.941345i \(-0.390437\pi\)
0.983952 + 0.178435i \(0.0571034\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 25.7911 + 44.6715i 0.854497 + 1.48003i 0.877111 + 0.480288i \(0.159468\pi\)
−0.0226136 + 0.999744i \(0.507199\pi\)
\(912\) 0 0
\(913\) −9.09302 + 15.7496i −0.300935 + 0.521235i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.5571i 0.348627i
\(918\) 0 0
\(919\) 21.2048 0.699481 0.349741 0.936847i \(-0.386270\pi\)
0.349741 + 0.936847i \(0.386270\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 17.5623 + 10.1396i 0.578069 + 0.333748i
\(924\) 0 0
\(925\) −24.4620 + 14.1231i −0.804305 + 0.464366i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.70516 2.95343i −0.0559446 0.0968989i 0.836697 0.547666i \(-0.184484\pi\)
−0.892641 + 0.450767i \(0.851150\pi\)
\(930\) 0 0
\(931\) −3.73042 2.15376i −0.122259 0.0705865i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 10.7757 0.352403
\(936\) 0 0
\(937\) 29.4448 0.961919 0.480959 0.876743i \(-0.340288\pi\)
0.480959 + 0.876743i \(0.340288\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −40.5880 23.4335i −1.32313 0.763910i −0.338904 0.940821i \(-0.610056\pi\)
−0.984227 + 0.176911i \(0.943389\pi\)
\(942\) 0 0
\(943\) −6.12914 10.6160i −0.199592 0.345704i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 31.5821 18.2340i 1.02628 0.592524i 0.110365 0.993891i \(-0.464798\pi\)
0.915917 + 0.401367i \(0.131465\pi\)
\(948\) 0 0
\(949\) 13.5024 + 7.79564i 0.438308 + 0.253057i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −38.0590 −1.23285 −0.616426 0.787413i \(-0.711420\pi\)
−0.616426 + 0.787413i \(0.711420\pi\)
\(954\) 0 0
\(955\) 5.09344i 0.164820i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.88330 + 4.99401i −0.0931065 + 0.161265i
\(960\) 0 0
\(961\) −11.8569 20.5368i −0.382481 0.662477i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 12.3397 7.12430i 0.397228 0.229339i
\(966\) 0 0
\(967\) −11.4864 + 19.8951i −0.369378 + 0.639782i −0.989468 0.144749i \(-0.953763\pi\)
0.620090 + 0.784531i \(0.287096\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 53.8829i 1.72919i −0.502474 0.864593i \(-0.667577\pi\)
0.502474 0.864593i \(-0.332423\pi\)
\(972\) 0 0
\(973\) 27.5269i 0.882473i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −19.1024 + 33.0863i −0.611140 + 1.05853i 0.379909 + 0.925024i \(0.375955\pi\)
−0.991049 + 0.133501i \(0.957378\pi\)
\(978\) 0 0
\(979\) −7.08585 + 4.09102i −0.226465 + 0.130749i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −24.3307 42.1420i −0.776028 1.34412i −0.934215 0.356711i \(-0.883898\pi\)
0.158186 0.987409i \(-0.449435\pi\)
\(984\) 0 0
\(985\) −7.14968 + 12.3836i −0.227808 + 0.394574i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 22.7715i 0.724093i
\(990\) 0 0
\(991\) 12.7822 0.406040 0.203020 0.979175i \(-0.434924\pi\)
0.203020 + 0.979175i \(0.434924\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.17905 + 0.680723i 0.0373783 + 0.0215804i
\(996\) 0 0
\(997\) −21.1161 + 12.1914i −0.668752 + 0.386104i −0.795604 0.605817i \(-0.792846\pi\)
0.126851 + 0.991922i \(0.459513\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 864.2.r.b.145.4 16
3.2 odd 2 288.2.r.b.49.7 16
4.3 odd 2 216.2.n.b.37.2 16
8.3 odd 2 216.2.n.b.37.8 16
8.5 even 2 inner 864.2.r.b.145.5 16
9.2 odd 6 288.2.r.b.241.2 16
9.4 even 3 2592.2.d.k.1297.5 8
9.5 odd 6 2592.2.d.j.1297.4 8
9.7 even 3 inner 864.2.r.b.721.5 16
12.11 even 2 72.2.n.b.13.7 yes 16
24.5 odd 2 288.2.r.b.49.2 16
24.11 even 2 72.2.n.b.13.1 16
36.7 odd 6 216.2.n.b.181.8 16
36.11 even 6 72.2.n.b.61.1 yes 16
36.23 even 6 648.2.d.j.325.6 8
36.31 odd 6 648.2.d.k.325.3 8
72.5 odd 6 2592.2.d.j.1297.5 8
72.11 even 6 72.2.n.b.61.7 yes 16
72.13 even 6 2592.2.d.k.1297.4 8
72.29 odd 6 288.2.r.b.241.7 16
72.43 odd 6 216.2.n.b.181.2 16
72.59 even 6 648.2.d.j.325.5 8
72.61 even 6 inner 864.2.r.b.721.4 16
72.67 odd 6 648.2.d.k.325.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.n.b.13.1 16 24.11 even 2
72.2.n.b.13.7 yes 16 12.11 even 2
72.2.n.b.61.1 yes 16 36.11 even 6
72.2.n.b.61.7 yes 16 72.11 even 6
216.2.n.b.37.2 16 4.3 odd 2
216.2.n.b.37.8 16 8.3 odd 2
216.2.n.b.181.2 16 72.43 odd 6
216.2.n.b.181.8 16 36.7 odd 6
288.2.r.b.49.2 16 24.5 odd 2
288.2.r.b.49.7 16 3.2 odd 2
288.2.r.b.241.2 16 9.2 odd 6
288.2.r.b.241.7 16 72.29 odd 6
648.2.d.j.325.5 8 72.59 even 6
648.2.d.j.325.6 8 36.23 even 6
648.2.d.k.325.3 8 36.31 odd 6
648.2.d.k.325.4 8 72.67 odd 6
864.2.r.b.145.4 16 1.1 even 1 trivial
864.2.r.b.145.5 16 8.5 even 2 inner
864.2.r.b.721.4 16 72.61 even 6 inner
864.2.r.b.721.5 16 9.7 even 3 inner
2592.2.d.j.1297.4 8 9.5 odd 6
2592.2.d.j.1297.5 8 72.5 odd 6
2592.2.d.k.1297.4 8 72.13 even 6
2592.2.d.k.1297.5 8 9.4 even 3