Properties

Label 768.3.l.e.703.7
Level $768$
Weight $3$
Character 768.703
Analytic conductor $20.926$
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] = [768,3,Mod(319,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.l (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 52 x^{14} + 1743 x^{12} - 34996 x^{10} + 513409 x^{8} - 5039424 x^{6} + 36142848 x^{4} - 155271168 x^{2} + 429981696 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 703.7
Root \(3.95096 - 2.28109i\) of defining polynomial
Character \(\chi\) \(=\) 768.703
Dual form 768.3.l.e.319.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.22474 - 1.22474i) q^{3} +(3.71984 - 3.71984i) q^{5} +5.63959 q^{7} -3.00000i q^{9} +O(q^{10})\) \(q+(1.22474 - 1.22474i) q^{3} +(3.71984 - 3.71984i) q^{5} +5.63959 q^{7} -3.00000i q^{9} +(5.26065 + 5.26065i) q^{11} +(1.74099 + 1.74099i) q^{13} -9.11171i q^{15} -1.43968 q^{17} +(22.1170 - 22.1170i) q^{19} +(6.90705 - 6.90705i) q^{21} +2.27362 q^{23} -2.67442i q^{25} +(-3.67423 - 3.67423i) q^{27} +(35.9507 + 35.9507i) q^{29} +13.3072i q^{31} +12.8859 q^{33} +(20.9784 - 20.9784i) q^{35} +(-36.1679 + 36.1679i) q^{37} +4.26454 q^{39} -63.2115i q^{41} +(-46.5429 - 46.5429i) q^{43} +(-11.1595 - 11.1595i) q^{45} -61.0614i q^{47} -17.1951 q^{49} +(-1.76324 + 1.76324i) q^{51} +(-49.0582 + 49.0582i) q^{53} +39.1375 q^{55} -54.1754i q^{57} +(41.6432 + 41.6432i) q^{59} +(40.4833 + 40.4833i) q^{61} -16.9188i q^{63} +12.9524 q^{65} +(36.0209 - 36.0209i) q^{67} +(2.78461 - 2.78461i) q^{69} -8.24767 q^{71} -122.728i q^{73} +(-3.27548 - 3.27548i) q^{75} +(29.6679 + 29.6679i) q^{77} -93.0832i q^{79} -9.00000 q^{81} +(21.6857 - 21.6857i) q^{83} +(-5.35538 + 5.35538i) q^{85} +88.0608 q^{87} +119.561i q^{89} +(9.81846 + 9.81846i) q^{91} +(16.2979 + 16.2979i) q^{93} -164.544i q^{95} -115.028 q^{97} +(15.7819 - 15.7819i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{5} + 32 q^{13} + 112 q^{17} - 72 q^{21} + 56 q^{29} - 48 q^{33} - 272 q^{37} + 24 q^{45} + 304 q^{49} - 504 q^{53} + 176 q^{61} + 624 q^{65} - 288 q^{69} + 848 q^{77} - 144 q^{81} - 880 q^{85} - 216 q^{93} - 160 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.22474 1.22474i 0.408248 0.408248i
\(4\) 0 0
\(5\) 3.71984 3.71984i 0.743968 0.743968i −0.229371 0.973339i \(-0.573667\pi\)
0.973339 + 0.229371i \(0.0736669\pi\)
\(6\) 0 0
\(7\) 5.63959 0.805655 0.402828 0.915276i \(-0.368027\pi\)
0.402828 + 0.915276i \(0.368027\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) 5.26065 + 5.26065i 0.478241 + 0.478241i 0.904569 0.426328i \(-0.140193\pi\)
−0.426328 + 0.904569i \(0.640193\pi\)
\(12\) 0 0
\(13\) 1.74099 + 1.74099i 0.133922 + 0.133922i 0.770890 0.636968i \(-0.219812\pi\)
−0.636968 + 0.770890i \(0.719812\pi\)
\(14\) 0 0
\(15\) 9.11171i 0.607447i
\(16\) 0 0
\(17\) −1.43968 −0.0846870 −0.0423435 0.999103i \(-0.513482\pi\)
−0.0423435 + 0.999103i \(0.513482\pi\)
\(18\) 0 0
\(19\) 22.1170 22.1170i 1.16405 1.16405i 0.180475 0.983580i \(-0.442237\pi\)
0.983580 0.180475i \(-0.0577634\pi\)
\(20\) 0 0
\(21\) 6.90705 6.90705i 0.328907 0.328907i
\(22\) 0 0
\(23\) 2.27362 0.0988532 0.0494266 0.998778i \(-0.484261\pi\)
0.0494266 + 0.998778i \(0.484261\pi\)
\(24\) 0 0
\(25\) 2.67442i 0.106977i
\(26\) 0 0
\(27\) −3.67423 3.67423i −0.136083 0.136083i
\(28\) 0 0
\(29\) 35.9507 + 35.9507i 1.23968 + 1.23968i 0.960132 + 0.279546i \(0.0901840\pi\)
0.279546 + 0.960132i \(0.409816\pi\)
\(30\) 0 0
\(31\) 13.3072i 0.429264i 0.976695 + 0.214632i \(0.0688551\pi\)
−0.976695 + 0.214632i \(0.931145\pi\)
\(32\) 0 0
\(33\) 12.8859 0.390482
\(34\) 0 0
\(35\) 20.9784 20.9784i 0.599382 0.599382i
\(36\) 0 0
\(37\) −36.1679 + 36.1679i −0.977510 + 0.977510i −0.999753 0.0222427i \(-0.992919\pi\)
0.0222427 + 0.999753i \(0.492919\pi\)
\(38\) 0 0
\(39\) 4.26454 0.109347
\(40\) 0 0
\(41\) 63.2115i 1.54174i −0.636990 0.770872i \(-0.719821\pi\)
0.636990 0.770872i \(-0.280179\pi\)
\(42\) 0 0
\(43\) −46.5429 46.5429i −1.08239 1.08239i −0.996286 0.0861060i \(-0.972558\pi\)
−0.0861060 0.996286i \(-0.527442\pi\)
\(44\) 0 0
\(45\) −11.1595 11.1595i −0.247989 0.247989i
\(46\) 0 0
\(47\) 61.0614i 1.29918i −0.760285 0.649589i \(-0.774941\pi\)
0.760285 0.649589i \(-0.225059\pi\)
\(48\) 0 0
\(49\) −17.1951 −0.350920
\(50\) 0 0
\(51\) −1.76324 + 1.76324i −0.0345733 + 0.0345733i
\(52\) 0 0
\(53\) −49.0582 + 49.0582i −0.925627 + 0.925627i −0.997420 0.0717927i \(-0.977128\pi\)
0.0717927 + 0.997420i \(0.477128\pi\)
\(54\) 0 0
\(55\) 39.1375 0.711592
\(56\) 0 0
\(57\) 54.1754i 0.950446i
\(58\) 0 0
\(59\) 41.6432 + 41.6432i 0.705818 + 0.705818i 0.965653 0.259835i \(-0.0836682\pi\)
−0.259835 + 0.965653i \(0.583668\pi\)
\(60\) 0 0
\(61\) 40.4833 + 40.4833i 0.663661 + 0.663661i 0.956241 0.292580i \(-0.0945139\pi\)
−0.292580 + 0.956241i \(0.594514\pi\)
\(62\) 0 0
\(63\) 16.9188i 0.268552i
\(64\) 0 0
\(65\) 12.9524 0.199268
\(66\) 0 0
\(67\) 36.0209 36.0209i 0.537626 0.537626i −0.385205 0.922831i \(-0.625869\pi\)
0.922831 + 0.385205i \(0.125869\pi\)
\(68\) 0 0
\(69\) 2.78461 2.78461i 0.0403567 0.0403567i
\(70\) 0 0
\(71\) −8.24767 −0.116164 −0.0580822 0.998312i \(-0.518499\pi\)
−0.0580822 + 0.998312i \(0.518499\pi\)
\(72\) 0 0
\(73\) 122.728i 1.68121i −0.541649 0.840604i \(-0.682200\pi\)
0.541649 0.840604i \(-0.317800\pi\)
\(74\) 0 0
\(75\) −3.27548 3.27548i −0.0436730 0.0436730i
\(76\) 0 0
\(77\) 29.6679 + 29.6679i 0.385297 + 0.385297i
\(78\) 0 0
\(79\) 93.0832i 1.17827i −0.808035 0.589134i \(-0.799469\pi\)
0.808035 0.589134i \(-0.200531\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) 21.6857 21.6857i 0.261273 0.261273i −0.564298 0.825571i \(-0.690853\pi\)
0.825571 + 0.564298i \(0.190853\pi\)
\(84\) 0 0
\(85\) −5.35538 + 5.35538i −0.0630044 + 0.0630044i
\(86\) 0 0
\(87\) 88.0608 1.01219
\(88\) 0 0
\(89\) 119.561i 1.34339i 0.740829 + 0.671693i \(0.234433\pi\)
−0.740829 + 0.671693i \(0.765567\pi\)
\(90\) 0 0
\(91\) 9.81846 + 9.81846i 0.107895 + 0.107895i
\(92\) 0 0
\(93\) 16.2979 + 16.2979i 0.175246 + 0.175246i
\(94\) 0 0
\(95\) 164.544i 1.73204i
\(96\) 0 0
\(97\) −115.028 −1.18585 −0.592926 0.805257i \(-0.702027\pi\)
−0.592926 + 0.805257i \(0.702027\pi\)
\(98\) 0 0
\(99\) 15.7819 15.7819i 0.159414 0.159414i
\(100\) 0 0
\(101\) −37.6316 + 37.6316i −0.372590 + 0.372590i −0.868420 0.495829i \(-0.834864\pi\)
0.495829 + 0.868420i \(0.334864\pi\)
\(102\) 0 0
\(103\) 130.007 1.26220 0.631100 0.775702i \(-0.282604\pi\)
0.631100 + 0.775702i \(0.282604\pi\)
\(104\) 0 0
\(105\) 51.3863i 0.489393i
\(106\) 0 0
\(107\) 28.6866 + 28.6866i 0.268099 + 0.268099i 0.828334 0.560235i \(-0.189289\pi\)
−0.560235 + 0.828334i \(0.689289\pi\)
\(108\) 0 0
\(109\) −41.4538 41.4538i −0.380310 0.380310i 0.490904 0.871214i \(-0.336667\pi\)
−0.871214 + 0.490904i \(0.836667\pi\)
\(110\) 0 0
\(111\) 88.5928i 0.798133i
\(112\) 0 0
\(113\) −16.4099 −0.145220 −0.0726101 0.997360i \(-0.523133\pi\)
−0.0726101 + 0.997360i \(0.523133\pi\)
\(114\) 0 0
\(115\) 8.45752 8.45752i 0.0735436 0.0735436i
\(116\) 0 0
\(117\) 5.22297 5.22297i 0.0446408 0.0446408i
\(118\) 0 0
\(119\) −8.11920 −0.0682285
\(120\) 0 0
\(121\) 65.6512i 0.542572i
\(122\) 0 0
\(123\) −77.4179 77.4179i −0.629414 0.629414i
\(124\) 0 0
\(125\) 83.0476 + 83.0476i 0.664381 + 0.664381i
\(126\) 0 0
\(127\) 96.2701i 0.758033i 0.925390 + 0.379016i \(0.123738\pi\)
−0.925390 + 0.379016i \(0.876262\pi\)
\(128\) 0 0
\(129\) −114.006 −0.883769
\(130\) 0 0
\(131\) 157.643 157.643i 1.20338 1.20338i 0.230249 0.973132i \(-0.426046\pi\)
0.973132 0.230249i \(-0.0739541\pi\)
\(132\) 0 0
\(133\) 124.731 124.731i 0.937826 0.937826i
\(134\) 0 0
\(135\) −27.3351 −0.202482
\(136\) 0 0
\(137\) 129.255i 0.943471i 0.881740 + 0.471735i \(0.156372\pi\)
−0.881740 + 0.471735i \(0.843628\pi\)
\(138\) 0 0
\(139\) 3.93636 + 3.93636i 0.0283191 + 0.0283191i 0.721125 0.692805i \(-0.243626\pi\)
−0.692805 + 0.721125i \(0.743626\pi\)
\(140\) 0 0
\(141\) −74.7846 74.7846i −0.530387 0.530387i
\(142\) 0 0
\(143\) 18.3175i 0.128094i
\(144\) 0 0
\(145\) 267.461 1.84456
\(146\) 0 0
\(147\) −21.0596 + 21.0596i −0.143262 + 0.143262i
\(148\) 0 0
\(149\) −170.277 + 170.277i −1.14280 + 1.14280i −0.154863 + 0.987936i \(0.549494\pi\)
−0.987936 + 0.154863i \(0.950506\pi\)
\(150\) 0 0
\(151\) −22.9645 −0.152083 −0.0760414 0.997105i \(-0.524228\pi\)
−0.0760414 + 0.997105i \(0.524228\pi\)
\(152\) 0 0
\(153\) 4.31904i 0.0282290i
\(154\) 0 0
\(155\) 49.5005 + 49.5005i 0.319358 + 0.319358i
\(156\) 0 0
\(157\) −28.6344 28.6344i −0.182385 0.182385i 0.610009 0.792394i \(-0.291166\pi\)
−0.792394 + 0.610009i \(0.791166\pi\)
\(158\) 0 0
\(159\) 120.168i 0.755771i
\(160\) 0 0
\(161\) 12.8223 0.0796416
\(162\) 0 0
\(163\) −156.712 + 156.712i −0.961421 + 0.961421i −0.999283 0.0378616i \(-0.987945\pi\)
0.0378616 + 0.999283i \(0.487945\pi\)
\(164\) 0 0
\(165\) 47.9335 47.9335i 0.290506 0.290506i
\(166\) 0 0
\(167\) −315.505 −1.88925 −0.944626 0.328148i \(-0.893575\pi\)
−0.944626 + 0.328148i \(0.893575\pi\)
\(168\) 0 0
\(169\) 162.938i 0.964130i
\(170\) 0 0
\(171\) −66.3511 66.3511i −0.388018 0.388018i
\(172\) 0 0
\(173\) 120.104 + 120.104i 0.694243 + 0.694243i 0.963163 0.268919i \(-0.0866665\pi\)
−0.268919 + 0.963163i \(0.586666\pi\)
\(174\) 0 0
\(175\) 15.0826i 0.0861863i
\(176\) 0 0
\(177\) 102.005 0.576298
\(178\) 0 0
\(179\) −181.133 + 181.133i −1.01192 + 1.01192i −0.0119883 + 0.999928i \(0.503816\pi\)
−0.999928 + 0.0119883i \(0.996184\pi\)
\(180\) 0 0
\(181\) 20.9668 20.9668i 0.115839 0.115839i −0.646811 0.762650i \(-0.723898\pi\)
0.762650 + 0.646811i \(0.223898\pi\)
\(182\) 0 0
\(183\) 99.1634 0.541877
\(184\) 0 0
\(185\) 269.077i 1.45447i
\(186\) 0 0
\(187\) −7.57365 7.57365i −0.0405008 0.0405008i
\(188\) 0 0
\(189\) −20.7212 20.7212i −0.109636 0.109636i
\(190\) 0 0
\(191\) 40.2387i 0.210674i 0.994437 + 0.105337i \(0.0335921\pi\)
−0.994437 + 0.105337i \(0.966408\pi\)
\(192\) 0 0
\(193\) −29.7137 −0.153957 −0.0769784 0.997033i \(-0.524527\pi\)
−0.0769784 + 0.997033i \(0.524527\pi\)
\(194\) 0 0
\(195\) 15.8634 15.8634i 0.0813507 0.0813507i
\(196\) 0 0
\(197\) 135.692 135.692i 0.688790 0.688790i −0.273175 0.961964i \(-0.588074\pi\)
0.961964 + 0.273175i \(0.0880737\pi\)
\(198\) 0 0
\(199\) −158.309 −0.795524 −0.397762 0.917489i \(-0.630213\pi\)
−0.397762 + 0.917489i \(0.630213\pi\)
\(200\) 0 0
\(201\) 88.2329i 0.438970i
\(202\) 0 0
\(203\) 202.747 + 202.747i 0.998753 + 0.998753i
\(204\) 0 0
\(205\) −235.137 235.137i −1.14701 1.14701i
\(206\) 0 0
\(207\) 6.82087i 0.0329511i
\(208\) 0 0
\(209\) 232.700 1.11340
\(210\) 0 0
\(211\) 36.9270 36.9270i 0.175010 0.175010i −0.614167 0.789176i \(-0.710508\pi\)
0.789176 + 0.614167i \(0.210508\pi\)
\(212\) 0 0
\(213\) −10.1013 + 10.1013i −0.0474239 + 0.0474239i
\(214\) 0 0
\(215\) −346.264 −1.61053
\(216\) 0 0
\(217\) 75.0469i 0.345838i
\(218\) 0 0
\(219\) −150.311 150.311i −0.686351 0.686351i
\(220\) 0 0
\(221\) −2.50647 2.50647i −0.0113415 0.0113415i
\(222\) 0 0
\(223\) 185.284i 0.830869i 0.909623 + 0.415435i \(0.136370\pi\)
−0.909623 + 0.415435i \(0.863630\pi\)
\(224\) 0 0
\(225\) −8.02325 −0.0356589
\(226\) 0 0
\(227\) −165.315 + 165.315i −0.728261 + 0.728261i −0.970273 0.242012i \(-0.922193\pi\)
0.242012 + 0.970273i \(0.422193\pi\)
\(228\) 0 0
\(229\) 291.789 291.789i 1.27419 1.27419i 0.330321 0.943869i \(-0.392843\pi\)
0.943869 0.330321i \(-0.107157\pi\)
\(230\) 0 0
\(231\) 72.6712 0.314594
\(232\) 0 0
\(233\) 265.531i 1.13962i 0.821778 + 0.569808i \(0.192982\pi\)
−0.821778 + 0.569808i \(0.807018\pi\)
\(234\) 0 0
\(235\) −227.139 227.139i −0.966547 0.966547i
\(236\) 0 0
\(237\) −114.003 114.003i −0.481026 0.481026i
\(238\) 0 0
\(239\) 284.025i 1.18839i 0.804321 + 0.594195i \(0.202529\pi\)
−0.804321 + 0.594195i \(0.797471\pi\)
\(240\) 0 0
\(241\) −39.1098 −0.162281 −0.0811406 0.996703i \(-0.525856\pi\)
−0.0811406 + 0.996703i \(0.525856\pi\)
\(242\) 0 0
\(243\) −11.0227 + 11.0227i −0.0453609 + 0.0453609i
\(244\) 0 0
\(245\) −63.9629 + 63.9629i −0.261073 + 0.261073i
\(246\) 0 0
\(247\) 77.0111 0.311786
\(248\) 0 0
\(249\) 53.1188i 0.213329i
\(250\) 0 0
\(251\) 304.283 + 304.283i 1.21228 + 1.21228i 0.970275 + 0.242007i \(0.0778057\pi\)
0.242007 + 0.970275i \(0.422194\pi\)
\(252\) 0 0
\(253\) 11.9607 + 11.9607i 0.0472756 + 0.0472756i
\(254\) 0 0
\(255\) 13.1179i 0.0514429i
\(256\) 0 0
\(257\) −425.226 −1.65458 −0.827288 0.561778i \(-0.810117\pi\)
−0.827288 + 0.561778i \(0.810117\pi\)
\(258\) 0 0
\(259\) −203.972 + 203.972i −0.787536 + 0.787536i
\(260\) 0 0
\(261\) 107.852 107.852i 0.413226 0.413226i
\(262\) 0 0
\(263\) 192.884 0.733399 0.366700 0.930339i \(-0.380488\pi\)
0.366700 + 0.930339i \(0.380488\pi\)
\(264\) 0 0
\(265\) 364.977i 1.37727i
\(266\) 0 0
\(267\) 146.432 + 146.432i 0.548435 + 0.548435i
\(268\) 0 0
\(269\) −111.994 111.994i −0.416335 0.416335i 0.467603 0.883938i \(-0.345118\pi\)
−0.883938 + 0.467603i \(0.845118\pi\)
\(270\) 0 0
\(271\) 201.397i 0.743161i 0.928401 + 0.371581i \(0.121184\pi\)
−0.928401 + 0.371581i \(0.878816\pi\)
\(272\) 0 0
\(273\) 24.0502 0.0880960
\(274\) 0 0
\(275\) 14.0692 14.0692i 0.0511606 0.0511606i
\(276\) 0 0
\(277\) −199.341 + 199.341i −0.719642 + 0.719642i −0.968532 0.248890i \(-0.919934\pi\)
0.248890 + 0.968532i \(0.419934\pi\)
\(278\) 0 0
\(279\) 39.9215 0.143088
\(280\) 0 0
\(281\) 526.542i 1.87381i 0.349578 + 0.936907i \(0.386325\pi\)
−0.349578 + 0.936907i \(0.613675\pi\)
\(282\) 0 0
\(283\) −113.365 113.365i −0.400585 0.400585i 0.477854 0.878439i \(-0.341415\pi\)
−0.878439 + 0.477854i \(0.841415\pi\)
\(284\) 0 0
\(285\) −201.524 201.524i −0.707102 0.707102i
\(286\) 0 0
\(287\) 356.487i 1.24211i
\(288\) 0 0
\(289\) −286.927 −0.992828
\(290\) 0 0
\(291\) −140.880 + 140.880i −0.484122 + 0.484122i
\(292\) 0 0
\(293\) 29.7487 29.7487i 0.101531 0.101531i −0.654516 0.756048i \(-0.727128\pi\)
0.756048 + 0.654516i \(0.227128\pi\)
\(294\) 0 0
\(295\) 309.812 1.05021
\(296\) 0 0
\(297\) 38.6577i 0.130161i
\(298\) 0 0
\(299\) 3.95836 + 3.95836i 0.0132387 + 0.0132387i
\(300\) 0 0
\(301\) −262.482 262.482i −0.872035 0.872035i
\(302\) 0 0
\(303\) 92.1783i 0.304219i
\(304\) 0 0
\(305\) 301.183 0.987485
\(306\) 0 0
\(307\) 52.8719 52.8719i 0.172221 0.172221i −0.615733 0.787954i \(-0.711140\pi\)
0.787954 + 0.615733i \(0.211140\pi\)
\(308\) 0 0
\(309\) 159.225 159.225i 0.515291 0.515291i
\(310\) 0 0
\(311\) −350.713 −1.12770 −0.563848 0.825879i \(-0.690680\pi\)
−0.563848 + 0.825879i \(0.690680\pi\)
\(312\) 0 0
\(313\) 393.542i 1.25732i 0.777679 + 0.628662i \(0.216397\pi\)
−0.777679 + 0.628662i \(0.783603\pi\)
\(314\) 0 0
\(315\) −62.9351 62.9351i −0.199794 0.199794i
\(316\) 0 0
\(317\) 330.939 + 330.939i 1.04397 + 1.04397i 0.998988 + 0.0449847i \(0.0143239\pi\)
0.0449847 + 0.998988i \(0.485676\pi\)
\(318\) 0 0
\(319\) 378.248i 1.18573i
\(320\) 0 0
\(321\) 70.2675 0.218902
\(322\) 0 0
\(323\) −31.8414 + 31.8414i −0.0985803 + 0.0985803i
\(324\) 0 0
\(325\) 4.65613 4.65613i 0.0143266 0.0143266i
\(326\) 0 0
\(327\) −101.541 −0.310522
\(328\) 0 0
\(329\) 344.361i 1.04669i
\(330\) 0 0
\(331\) 181.290 + 181.290i 0.547703 + 0.547703i 0.925776 0.378073i \(-0.123413\pi\)
−0.378073 + 0.925776i \(0.623413\pi\)
\(332\) 0 0
\(333\) 108.504 + 108.504i 0.325837 + 0.325837i
\(334\) 0 0
\(335\) 267.984i 0.799953i
\(336\) 0 0
\(337\) −136.580 −0.405282 −0.202641 0.979253i \(-0.564952\pi\)
−0.202641 + 0.979253i \(0.564952\pi\)
\(338\) 0 0
\(339\) −20.0979 + 20.0979i −0.0592859 + 0.0592859i
\(340\) 0 0
\(341\) −70.0043 + 70.0043i −0.205291 + 0.205291i
\(342\) 0 0
\(343\) −373.313 −1.08838
\(344\) 0 0
\(345\) 20.7166i 0.0600481i
\(346\) 0 0
\(347\) 135.037 + 135.037i 0.389157 + 0.389157i 0.874387 0.485230i \(-0.161264\pi\)
−0.485230 + 0.874387i \(0.661264\pi\)
\(348\) 0 0
\(349\) −53.9888 53.9888i −0.154696 0.154696i 0.625516 0.780212i \(-0.284889\pi\)
−0.780212 + 0.625516i \(0.784889\pi\)
\(350\) 0 0
\(351\) 12.7936i 0.0364490i
\(352\) 0 0
\(353\) 137.350 0.389092 0.194546 0.980893i \(-0.437677\pi\)
0.194546 + 0.980893i \(0.437677\pi\)
\(354\) 0 0
\(355\) −30.6800 + 30.6800i −0.0864226 + 0.0864226i
\(356\) 0 0
\(357\) −9.94394 + 9.94394i −0.0278542 + 0.0278542i
\(358\) 0 0
\(359\) 122.293 0.340649 0.170324 0.985388i \(-0.445518\pi\)
0.170324 + 0.985388i \(0.445518\pi\)
\(360\) 0 0
\(361\) 617.326i 1.71005i
\(362\) 0 0
\(363\) −80.4059 80.4059i −0.221504 0.221504i
\(364\) 0 0
\(365\) −456.529 456.529i −1.25077 1.25077i
\(366\) 0 0
\(367\) 59.0682i 0.160949i 0.996757 + 0.0804744i \(0.0256435\pi\)
−0.996757 + 0.0804744i \(0.974356\pi\)
\(368\) 0 0
\(369\) −189.634 −0.513915
\(370\) 0 0
\(371\) −276.668 + 276.668i −0.745736 + 0.745736i
\(372\) 0 0
\(373\) −101.421 + 101.421i −0.271906 + 0.271906i −0.829867 0.557961i \(-0.811584\pi\)
0.557961 + 0.829867i \(0.311584\pi\)
\(374\) 0 0
\(375\) 203.424 0.542465
\(376\) 0 0
\(377\) 125.180i 0.332041i
\(378\) 0 0
\(379\) −272.095 272.095i −0.717929 0.717929i 0.250252 0.968181i \(-0.419486\pi\)
−0.968181 + 0.250252i \(0.919486\pi\)
\(380\) 0 0
\(381\) 117.906 + 117.906i 0.309466 + 0.309466i
\(382\) 0 0
\(383\) 315.688i 0.824251i −0.911127 0.412125i \(-0.864787\pi\)
0.911127 0.412125i \(-0.135213\pi\)
\(384\) 0 0
\(385\) 220.719 0.573297
\(386\) 0 0
\(387\) −139.629 + 139.629i −0.360797 + 0.360797i
\(388\) 0 0
\(389\) 71.8537 71.8537i 0.184714 0.184714i −0.608692 0.793406i \(-0.708306\pi\)
0.793406 + 0.608692i \(0.208306\pi\)
\(390\) 0 0
\(391\) −3.27329 −0.00837159
\(392\) 0 0
\(393\) 386.145i 0.982556i
\(394\) 0 0
\(395\) −346.255 346.255i −0.876594 0.876594i
\(396\) 0 0
\(397\) 269.316 + 269.316i 0.678378 + 0.678378i 0.959633 0.281255i \(-0.0907508\pi\)
−0.281255 + 0.959633i \(0.590751\pi\)
\(398\) 0 0
\(399\) 305.527i 0.765732i
\(400\) 0 0
\(401\) 159.680 0.398204 0.199102 0.979979i \(-0.436197\pi\)
0.199102 + 0.979979i \(0.436197\pi\)
\(402\) 0 0
\(403\) −23.1676 + 23.1676i −0.0574880 + 0.0574880i
\(404\) 0 0
\(405\) −33.4786 + 33.4786i −0.0826631 + 0.0826631i
\(406\) 0 0
\(407\) −380.533 −0.934970
\(408\) 0 0
\(409\) 190.873i 0.466683i −0.972395 0.233342i \(-0.925034\pi\)
0.972395 0.233342i \(-0.0749661\pi\)
\(410\) 0 0
\(411\) 158.305 + 158.305i 0.385170 + 0.385170i
\(412\) 0 0
\(413\) 234.851 + 234.851i 0.568646 + 0.568646i
\(414\) 0 0
\(415\) 161.334i 0.388758i
\(416\) 0 0
\(417\) 9.64207 0.0231225
\(418\) 0 0
\(419\) 370.260 370.260i 0.883676 0.883676i −0.110230 0.993906i \(-0.535159\pi\)
0.993906 + 0.110230i \(0.0351588\pi\)
\(420\) 0 0
\(421\) −273.190 + 273.190i −0.648907 + 0.648907i −0.952729 0.303822i \(-0.901737\pi\)
0.303822 + 0.952729i \(0.401737\pi\)
\(422\) 0 0
\(423\) −183.184 −0.433059
\(424\) 0 0
\(425\) 3.85030i 0.00905954i
\(426\) 0 0
\(427\) 228.309 + 228.309i 0.534682 + 0.534682i
\(428\) 0 0
\(429\) 22.4342 + 22.4342i 0.0522942 + 0.0522942i
\(430\) 0 0
\(431\) 507.863i 1.17834i 0.808011 + 0.589168i \(0.200544\pi\)
−0.808011 + 0.589168i \(0.799456\pi\)
\(432\) 0 0
\(433\) 352.417 0.813895 0.406948 0.913452i \(-0.366593\pi\)
0.406948 + 0.913452i \(0.366593\pi\)
\(434\) 0 0
\(435\) 327.572 327.572i 0.753039 0.753039i
\(436\) 0 0
\(437\) 50.2858 50.2858i 0.115071 0.115071i
\(438\) 0 0
\(439\) 776.156 1.76801 0.884005 0.467477i \(-0.154837\pi\)
0.884005 + 0.467477i \(0.154837\pi\)
\(440\) 0 0
\(441\) 51.5852i 0.116973i
\(442\) 0 0
\(443\) −77.8171 77.8171i −0.175659 0.175659i 0.613801 0.789461i \(-0.289640\pi\)
−0.789461 + 0.613801i \(0.789640\pi\)
\(444\) 0 0
\(445\) 444.749 + 444.749i 0.999437 + 0.999437i
\(446\) 0 0
\(447\) 417.092i 0.933092i
\(448\) 0 0
\(449\) −135.576 −0.301952 −0.150976 0.988537i \(-0.548242\pi\)
−0.150976 + 0.988537i \(0.548242\pi\)
\(450\) 0 0
\(451\) 332.533 332.533i 0.737325 0.737325i
\(452\) 0 0
\(453\) −28.1257 + 28.1257i −0.0620875 + 0.0620875i
\(454\) 0 0
\(455\) 73.0462 0.160541
\(456\) 0 0
\(457\) 265.075i 0.580032i −0.957022 0.290016i \(-0.906339\pi\)
0.957022 0.290016i \(-0.0936607\pi\)
\(458\) 0 0
\(459\) 5.28972 + 5.28972i 0.0115244 + 0.0115244i
\(460\) 0 0
\(461\) 57.6181 + 57.6181i 0.124985 + 0.124985i 0.766832 0.641847i \(-0.221832\pi\)
−0.641847 + 0.766832i \(0.721832\pi\)
\(462\) 0 0
\(463\) 343.191i 0.741234i −0.928786 0.370617i \(-0.879146\pi\)
0.928786 0.370617i \(-0.120854\pi\)
\(464\) 0 0
\(465\) 121.251 0.260755
\(466\) 0 0
\(467\) 402.281 402.281i 0.861416 0.861416i −0.130086 0.991503i \(-0.541525\pi\)
0.991503 + 0.130086i \(0.0415255\pi\)
\(468\) 0 0
\(469\) 203.143 203.143i 0.433141 0.433141i
\(470\) 0 0
\(471\) −70.1397 −0.148917
\(472\) 0 0
\(473\) 489.691i 1.03529i
\(474\) 0 0
\(475\) −59.1502 59.1502i −0.124527 0.124527i
\(476\) 0 0
\(477\) 147.175 + 147.175i 0.308542 + 0.308542i
\(478\) 0 0
\(479\) 148.717i 0.310474i −0.987877 0.155237i \(-0.950386\pi\)
0.987877 0.155237i \(-0.0496141\pi\)
\(480\) 0 0
\(481\) −125.936 −0.261821
\(482\) 0 0
\(483\) 15.7040 15.7040i 0.0325135 0.0325135i
\(484\) 0 0
\(485\) −427.884 + 427.884i −0.882236 + 0.882236i
\(486\) 0 0
\(487\) −631.177 −1.29605 −0.648025 0.761619i \(-0.724405\pi\)
−0.648025 + 0.761619i \(0.724405\pi\)
\(488\) 0 0
\(489\) 383.864i 0.784997i
\(490\) 0 0
\(491\) −2.03842 2.03842i −0.00415157 0.00415157i 0.705028 0.709180i \(-0.250934\pi\)
−0.709180 + 0.705028i \(0.750934\pi\)
\(492\) 0 0
\(493\) −51.7575 51.7575i −0.104985 0.104985i
\(494\) 0 0
\(495\) 117.413i 0.237197i
\(496\) 0 0
\(497\) −46.5134 −0.0935884
\(498\) 0 0
\(499\) 366.034 366.034i 0.733535 0.733535i −0.237783 0.971318i \(-0.576421\pi\)
0.971318 + 0.237783i \(0.0764207\pi\)
\(500\) 0 0
\(501\) −386.413 + 386.413i −0.771284 + 0.771284i
\(502\) 0 0
\(503\) 835.830 1.66169 0.830845 0.556504i \(-0.187858\pi\)
0.830845 + 0.556504i \(0.187858\pi\)
\(504\) 0 0
\(505\) 279.967i 0.554391i
\(506\) 0 0
\(507\) −199.557 199.557i −0.393604 0.393604i
\(508\) 0 0
\(509\) 47.1170 + 47.1170i 0.0925679 + 0.0925679i 0.751874 0.659306i \(-0.229150\pi\)
−0.659306 + 0.751874i \(0.729150\pi\)
\(510\) 0 0
\(511\) 692.136i 1.35447i
\(512\) 0 0
\(513\) −162.526 −0.316815
\(514\) 0 0
\(515\) 483.603 483.603i 0.939036 0.939036i
\(516\) 0 0
\(517\) 321.222 321.222i 0.621320 0.621320i
\(518\) 0 0
\(519\) 294.194 0.566847
\(520\) 0 0
\(521\) 546.253i 1.04847i −0.851574 0.524235i \(-0.824351\pi\)
0.851574 0.524235i \(-0.175649\pi\)
\(522\) 0 0
\(523\) 331.446 + 331.446i 0.633740 + 0.633740i 0.949004 0.315264i \(-0.102093\pi\)
−0.315264 + 0.949004i \(0.602093\pi\)
\(524\) 0 0
\(525\) −18.4723 18.4723i −0.0351854 0.0351854i
\(526\) 0 0
\(527\) 19.1581i 0.0363531i
\(528\) 0 0
\(529\) −523.831 −0.990228
\(530\) 0 0
\(531\) 124.930 124.930i 0.235273 0.235273i
\(532\) 0 0
\(533\) 110.051 110.051i 0.206474 0.206474i
\(534\) 0 0
\(535\) 213.419 0.398914
\(536\) 0 0
\(537\) 443.684i 0.826226i
\(538\) 0 0
\(539\) −90.4572 90.4572i −0.167824 0.167824i
\(540\) 0 0
\(541\) 34.7238 + 34.7238i 0.0641844 + 0.0641844i 0.738470 0.674286i \(-0.235548\pi\)
−0.674286 + 0.738470i \(0.735548\pi\)
\(542\) 0 0
\(543\) 51.3580i 0.0945820i
\(544\) 0 0
\(545\) −308.403 −0.565877
\(546\) 0 0
\(547\) −11.2062 + 11.2062i −0.0204867 + 0.0204867i −0.717276 0.696789i \(-0.754611\pi\)
0.696789 + 0.717276i \(0.254611\pi\)
\(548\) 0 0
\(549\) 121.450 121.450i 0.221220 0.221220i
\(550\) 0 0
\(551\) 1590.24 2.88611
\(552\) 0 0
\(553\) 524.951i 0.949278i
\(554\) 0 0
\(555\) 329.551 + 329.551i 0.593786 + 0.593786i
\(556\) 0 0
\(557\) 485.377 + 485.377i 0.871414 + 0.871414i 0.992627 0.121213i \(-0.0386784\pi\)
−0.121213 + 0.992627i \(0.538678\pi\)
\(558\) 0 0
\(559\) 162.061i 0.289913i
\(560\) 0 0
\(561\) −18.5516 −0.0330688
\(562\) 0 0
\(563\) −140.768 + 140.768i −0.250032 + 0.250032i −0.820984 0.570951i \(-0.806574\pi\)
0.570951 + 0.820984i \(0.306574\pi\)
\(564\) 0 0
\(565\) −61.0421 + 61.0421i −0.108039 + 0.108039i
\(566\) 0 0
\(567\) −50.7563 −0.0895172
\(568\) 0 0
\(569\) 953.637i 1.67599i −0.545680 0.837993i \(-0.683729\pi\)
0.545680 0.837993i \(-0.316271\pi\)
\(570\) 0 0
\(571\) 51.7112 + 51.7112i 0.0905624 + 0.0905624i 0.750937 0.660374i \(-0.229602\pi\)
−0.660374 + 0.750937i \(0.729602\pi\)
\(572\) 0 0
\(573\) 49.2821 + 49.2821i 0.0860072 + 0.0860072i
\(574\) 0 0
\(575\) 6.08062i 0.0105750i
\(576\) 0 0
\(577\) 511.951 0.887263 0.443632 0.896209i \(-0.353690\pi\)
0.443632 + 0.896209i \(0.353690\pi\)
\(578\) 0 0
\(579\) −36.3917 + 36.3917i −0.0628526 + 0.0628526i
\(580\) 0 0
\(581\) 122.298 122.298i 0.210496 0.210496i
\(582\) 0 0
\(583\) −516.156 −0.885345
\(584\) 0 0
\(585\) 38.8572i 0.0664226i
\(586\) 0 0
\(587\) −738.225 738.225i −1.25762 1.25762i −0.952224 0.305399i \(-0.901210\pi\)
−0.305399 0.952224i \(-0.598790\pi\)
\(588\) 0 0
\(589\) 294.315 + 294.315i 0.499686 + 0.499686i
\(590\) 0 0
\(591\) 332.375i 0.562395i
\(592\) 0 0
\(593\) −712.871 −1.20214 −0.601071 0.799195i \(-0.705259\pi\)
−0.601071 + 0.799195i \(0.705259\pi\)
\(594\) 0 0
\(595\) −30.2021 + 30.2021i −0.0507599 + 0.0507599i
\(596\) 0 0
\(597\) −193.889 + 193.889i −0.324771 + 0.324771i
\(598\) 0 0
\(599\) −706.968 −1.18025 −0.590123 0.807313i \(-0.700921\pi\)
−0.590123 + 0.807313i \(0.700921\pi\)
\(600\) 0 0
\(601\) 405.158i 0.674139i −0.941480 0.337069i \(-0.890564\pi\)
0.941480 0.337069i \(-0.109436\pi\)
\(602\) 0 0
\(603\) −108.063 108.063i −0.179209 0.179209i
\(604\) 0 0
\(605\) −244.212 244.212i −0.403656 0.403656i
\(606\) 0 0
\(607\) 1065.53i 1.75541i −0.479202 0.877705i \(-0.659074\pi\)
0.479202 0.877705i \(-0.340926\pi\)
\(608\) 0 0
\(609\) 496.626 0.815479
\(610\) 0 0
\(611\) 106.307 106.307i 0.173989 0.173989i
\(612\) 0 0
\(613\) −71.1894 + 71.1894i −0.116133 + 0.116133i −0.762785 0.646652i \(-0.776169\pi\)
0.646652 + 0.762785i \(0.276169\pi\)
\(614\) 0 0
\(615\) −575.965 −0.936528
\(616\) 0 0
\(617\) 36.7627i 0.0595830i 0.999556 + 0.0297915i \(0.00948433\pi\)
−0.999556 + 0.0297915i \(0.990516\pi\)
\(618\) 0 0
\(619\) 227.779 + 227.779i 0.367980 + 0.367980i 0.866740 0.498760i \(-0.166211\pi\)
−0.498760 + 0.866740i \(0.666211\pi\)
\(620\) 0 0
\(621\) −8.35383 8.35383i −0.0134522 0.0134522i
\(622\) 0 0
\(623\) 674.277i 1.08231i
\(624\) 0 0
\(625\) 684.708 1.09553
\(626\) 0 0
\(627\) 284.998 284.998i 0.454542 0.454542i
\(628\) 0 0
\(629\) 52.0701 52.0701i 0.0827824 0.0827824i
\(630\) 0 0
\(631\) 994.938 1.57676 0.788382 0.615186i \(-0.210919\pi\)
0.788382 + 0.615186i \(0.210919\pi\)
\(632\) 0 0
\(633\) 90.4524i 0.142895i
\(634\) 0 0
\(635\) 358.110 + 358.110i 0.563952 + 0.563952i
\(636\) 0 0
\(637\) −29.9365 29.9365i −0.0469960 0.0469960i
\(638\) 0 0
\(639\) 24.7430i 0.0387215i
\(640\) 0 0
\(641\) 312.367 0.487311 0.243656 0.969862i \(-0.421653\pi\)
0.243656 + 0.969862i \(0.421653\pi\)
\(642\) 0 0
\(643\) −98.2755 + 98.2755i −0.152839 + 0.152839i −0.779385 0.626546i \(-0.784468\pi\)
0.626546 + 0.779385i \(0.284468\pi\)
\(644\) 0 0
\(645\) −424.085 + 424.085i −0.657496 + 0.657496i
\(646\) 0 0
\(647\) 123.488 0.190863 0.0954313 0.995436i \(-0.469577\pi\)
0.0954313 + 0.995436i \(0.469577\pi\)
\(648\) 0 0
\(649\) 438.141i 0.675102i
\(650\) 0 0
\(651\) 91.9133 + 91.9133i 0.141188 + 0.141188i
\(652\) 0 0
\(653\) −474.759 474.759i −0.727043 0.727043i 0.242987 0.970030i \(-0.421873\pi\)
−0.970030 + 0.242987i \(0.921873\pi\)
\(654\) 0 0
\(655\) 1172.81i 1.79055i
\(656\) 0 0
\(657\) −368.185 −0.560403
\(658\) 0 0
\(659\) −848.321 + 848.321i −1.28729 + 1.28729i −0.350856 + 0.936429i \(0.614109\pi\)
−0.936429 + 0.350856i \(0.885891\pi\)
\(660\) 0 0
\(661\) −734.819 + 734.819i −1.11168 + 1.11168i −0.118754 + 0.992924i \(0.537890\pi\)
−0.992924 + 0.118754i \(0.962110\pi\)
\(662\) 0 0
\(663\) −6.13957 −0.00926028
\(664\) 0 0
\(665\) 927.958i 1.39543i
\(666\) 0 0
\(667\) 81.7383 + 81.7383i 0.122546 + 0.122546i
\(668\) 0 0
\(669\) 226.925 + 226.925i 0.339201 + 0.339201i
\(670\) 0 0
\(671\) 425.937i 0.634779i
\(672\) 0 0
\(673\) −1060.24 −1.57540 −0.787700 0.616059i \(-0.788728\pi\)
−0.787700 + 0.616059i \(0.788728\pi\)
\(674\) 0 0
\(675\) −9.82644 + 9.82644i −0.0145577 + 0.0145577i
\(676\) 0 0
\(677\) 5.38518 5.38518i 0.00795448 0.00795448i −0.703118 0.711073i \(-0.748210\pi\)
0.711073 + 0.703118i \(0.248210\pi\)
\(678\) 0 0
\(679\) −648.708 −0.955388
\(680\) 0 0
\(681\) 404.938i 0.594622i
\(682\) 0 0
\(683\) −669.250 669.250i −0.979868 0.979868i 0.0199331 0.999801i \(-0.493655\pi\)
−0.999801 + 0.0199331i \(0.993655\pi\)
\(684\) 0 0
\(685\) 480.810 + 480.810i 0.701912 + 0.701912i
\(686\) 0 0
\(687\) 714.735i 1.04037i
\(688\) 0 0
\(689\) −170.820 −0.247924
\(690\) 0 0
\(691\) −394.907 + 394.907i −0.571501 + 0.571501i −0.932548 0.361047i \(-0.882419\pi\)
0.361047 + 0.932548i \(0.382419\pi\)
\(692\) 0 0
\(693\) 89.0036 89.0036i 0.128432 0.128432i
\(694\) 0 0
\(695\) 29.2852 0.0421370
\(696\) 0 0
\(697\) 91.0043i 0.130566i
\(698\) 0 0
\(699\) 325.207 + 325.207i 0.465246 + 0.465246i
\(700\) 0 0
\(701\) −769.452 769.452i −1.09765 1.09765i −0.994685 0.102963i \(-0.967168\pi\)
−0.102963 0.994685i \(-0.532832\pi\)
\(702\) 0 0
\(703\) 1599.85i 2.27575i
\(704\) 0 0
\(705\) −556.374 −0.789182
\(706\) 0 0
\(707\) −212.227 + 212.227i −0.300179 + 0.300179i
\(708\) 0 0
\(709\) −818.749 + 818.749i −1.15479 + 1.15479i −0.169215 + 0.985579i \(0.554123\pi\)
−0.985579 + 0.169215i \(0.945877\pi\)
\(710\) 0 0
\(711\) −279.250 −0.392756
\(712\) 0 0
\(713\) 30.2555i 0.0424341i
\(714\) 0 0
\(715\) 68.1381 + 68.1381i 0.0952980 + 0.0952980i
\(716\) 0 0
\(717\) 347.858 + 347.858i 0.485158 + 0.485158i
\(718\) 0 0
\(719\) 294.229i 0.409219i −0.978844 0.204610i \(-0.934408\pi\)
0.978844 0.204610i \(-0.0655925\pi\)
\(720\) 0 0
\(721\) 733.183 1.01690
\(722\) 0 0
\(723\) −47.8995 + 47.8995i −0.0662510 + 0.0662510i
\(724\) 0 0
\(725\) 96.1471 96.1471i 0.132617 0.132617i
\(726\) 0 0
\(727\) −720.844 −0.991532 −0.495766 0.868456i \(-0.665113\pi\)
−0.495766 + 0.868456i \(0.665113\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) 67.0068 + 67.0068i 0.0916646 + 0.0916646i
\(732\) 0 0
\(733\) −459.987 459.987i −0.627541 0.627541i 0.319908 0.947449i \(-0.396348\pi\)
−0.947449 + 0.319908i \(0.896348\pi\)
\(734\) 0 0
\(735\) 156.677i 0.213165i
\(736\) 0 0
\(737\) 378.987 0.514229
\(738\) 0 0
\(739\) 817.319 817.319i 1.10598 1.10598i 0.112306 0.993674i \(-0.464176\pi\)
0.993674 0.112306i \(-0.0358238\pi\)
\(740\) 0 0
\(741\) 94.3189 94.3189i 0.127286 0.127286i
\(742\) 0 0
\(743\) 874.150 1.17651 0.588257 0.808674i \(-0.299814\pi\)
0.588257 + 0.808674i \(0.299814\pi\)
\(744\) 0 0
\(745\) 1266.81i 1.70041i
\(746\) 0 0
\(747\) −65.0570 65.0570i −0.0870910 0.0870910i
\(748\) 0 0
\(749\) 161.780 + 161.780i 0.215995 + 0.215995i
\(750\) 0 0
\(751\) 1278.46i 1.70234i 0.524892 + 0.851169i \(0.324106\pi\)
−0.524892 + 0.851169i \(0.675894\pi\)
\(752\) 0 0
\(753\) 745.337 0.989824
\(754\) 0 0
\(755\) −85.4243 + 85.4243i −0.113145 + 0.113145i
\(756\) 0 0
\(757\) 543.081 543.081i 0.717412 0.717412i −0.250662 0.968075i \(-0.580648\pi\)
0.968075 + 0.250662i \(0.0806483\pi\)
\(758\) 0 0
\(759\) 29.2977 0.0386004
\(760\) 0 0
\(761\) 121.613i 0.159806i −0.996803 0.0799031i \(-0.974539\pi\)
0.996803 0.0799031i \(-0.0254611\pi\)
\(762\) 0 0
\(763\) −233.782 233.782i −0.306399 0.306399i
\(764\) 0 0
\(765\) 16.0661 + 16.0661i 0.0210015 + 0.0210015i
\(766\) 0 0
\(767\) 145.001i 0.189049i
\(768\) 0 0
\(769\) 368.174 0.478769 0.239385 0.970925i \(-0.423054\pi\)
0.239385 + 0.970925i \(0.423054\pi\)
\(770\) 0 0
\(771\) −520.793 + 520.793i −0.675478 + 0.675478i
\(772\) 0 0
\(773\) 871.868 871.868i 1.12790 1.12790i 0.137384 0.990518i \(-0.456131\pi\)
0.990518 0.137384i \(-0.0438695\pi\)
\(774\) 0 0
\(775\) 35.5889 0.0459212
\(776\) 0 0
\(777\) 499.627i 0.643020i
\(778\) 0 0
\(779\) −1398.05 1398.05i −1.79467 1.79467i
\(780\) 0 0
\(781\) −43.3881 43.3881i −0.0555545 0.0555545i
\(782\) 0 0
\(783\) 264.182i 0.337398i
\(784\) 0 0
\(785\) −213.031 −0.271377
\(786\) 0 0
\(787\) −138.396 + 138.396i −0.175853 + 0.175853i −0.789545 0.613692i \(-0.789684\pi\)
0.613692 + 0.789545i \(0.289684\pi\)
\(788\) 0 0
\(789\) 236.234 236.234i 0.299409 0.299409i
\(790\) 0 0
\(791\) −92.5449 −0.116997
\(792\) 0 0
\(793\) 140.962i 0.177758i
\(794\) 0 0
\(795\) 447.004 + 447.004i 0.562270 + 0.562270i
\(796\) 0 0
\(797\) −419.719 419.719i −0.526623 0.526623i 0.392940 0.919564i \(-0.371458\pi\)
−0.919564 + 0.392940i \(0.871458\pi\)
\(798\) 0 0
\(799\) 87.9088i 0.110024i
\(800\) 0 0
\(801\) 358.684 0.447796
\(802\) 0 0
\(803\) 645.630 645.630i 0.804023 0.804023i
\(804\) 0 0
\(805\) 47.6969 47.6969i 0.0592508 0.0592508i
\(806\) 0 0
\(807\) −274.328 −0.339936
\(808\) 0 0
\(809\) 340.167i 0.420478i −0.977650 0.210239i \(-0.932576\pi\)
0.977650 0.210239i \(-0.0674242\pi\)
\(810\) 0 0
\(811\) 808.987 + 808.987i 0.997518 + 0.997518i 0.999997 0.00247919i \(-0.000789151\pi\)
−0.00247919 + 0.999997i \(0.500789\pi\)
\(812\) 0 0
\(813\) 246.660 + 246.660i 0.303394 + 0.303394i
\(814\) 0 0
\(815\) 1165.88i 1.43053i
\(816\) 0 0
\(817\) −2058.78 −2.51993
\(818\) 0 0
\(819\) 29.4554 29.4554i 0.0359651 0.0359651i
\(820\) 0 0
\(821\) 504.072 504.072i 0.613973 0.613973i −0.330006 0.943979i \(-0.607051\pi\)
0.943979 + 0.330006i \(0.107051\pi\)
\(822\) 0 0
\(823\) −312.505 −0.379715 −0.189858 0.981812i \(-0.560803\pi\)
−0.189858 + 0.981812i \(0.560803\pi\)
\(824\) 0 0
\(825\) 34.4623i 0.0417725i
\(826\) 0 0
\(827\) 336.658 + 336.658i 0.407083 + 0.407083i 0.880720 0.473637i \(-0.157059\pi\)
−0.473637 + 0.880720i \(0.657059\pi\)
\(828\) 0 0
\(829\) −65.0235 65.0235i −0.0784360 0.0784360i 0.666800 0.745236i \(-0.267663\pi\)
−0.745236 + 0.666800i \(0.767663\pi\)
\(830\) 0 0
\(831\) 488.283i 0.587585i
\(832\) 0 0
\(833\) 24.7554 0.0297184
\(834\) 0 0
\(835\) −1173.63 + 1173.63i −1.40554 + 1.40554i
\(836\) 0 0
\(837\) 48.8937 48.8937i 0.0584154 0.0584154i
\(838\) 0 0
\(839\) −153.170 −0.182563 −0.0912815 0.995825i \(-0.529096\pi\)
−0.0912815 + 0.995825i \(0.529096\pi\)
\(840\) 0 0
\(841\) 1743.90i 2.07360i
\(842\) 0 0
\(843\) 644.879 + 644.879i 0.764981 + 0.764981i
\(844\) 0 0
\(845\) −606.103 606.103i −0.717282 0.717282i
\(846\) 0 0
\(847\) 370.245i 0.437126i
\(848\) 0 0
\(849\) −277.687 −0.327076
\(850\) 0 0
\(851\) −82.2321 + 82.2321i −0.0966300 + 0.0966300i
\(852\) 0 0
\(853\) 505.256 505.256i 0.592328 0.592328i −0.345932 0.938260i \(-0.612437\pi\)
0.938260 + 0.345932i \(0.112437\pi\)
\(854\) 0 0
\(855\) −493.631 −0.577346
\(856\) 0 0
\(857\) 390.376i 0.455514i 0.973718 + 0.227757i \(0.0731392\pi\)
−0.973718 + 0.227757i \(0.926861\pi\)
\(858\) 0 0
\(859\) −155.878 155.878i −0.181464 0.181464i 0.610529 0.791994i \(-0.290957\pi\)
−0.791994 + 0.610529i \(0.790957\pi\)
\(860\) 0 0
\(861\) −436.605 436.605i −0.507091 0.507091i
\(862\) 0 0
\(863\) 1035.85i 1.20029i −0.799893 0.600143i \(-0.795110\pi\)
0.799893 0.600143i \(-0.204890\pi\)
\(864\) 0 0
\(865\) 893.536 1.03299
\(866\) 0 0
\(867\) −351.413 + 351.413i −0.405320 + 0.405320i
\(868\) 0 0
\(869\) 489.678 489.678i 0.563496 0.563496i
\(870\) 0 0
\(871\) 125.424 0.144000
\(872\) 0 0
\(873\) 345.083i 0.395284i
\(874\) 0 0
\(875\) 468.354 + 468.354i 0.535262 + 0.535262i
\(876\) 0 0
\(877\) 501.050 + 501.050i 0.571323 + 0.571323i 0.932498 0.361175i \(-0.117624\pi\)
−0.361175 + 0.932498i \(0.617624\pi\)
\(878\) 0 0
\(879\) 72.8690i 0.0828999i
\(880\) 0 0
\(881\) 990.013 1.12374 0.561869 0.827226i \(-0.310083\pi\)
0.561869 + 0.827226i \(0.310083\pi\)
\(882\) 0 0
\(883\) 933.600 933.600i 1.05731 1.05731i 0.0590501 0.998255i \(-0.481193\pi\)
0.998255 0.0590501i \(-0.0188072\pi\)
\(884\) 0 0
\(885\) 379.441 379.441i 0.428747 0.428747i
\(886\) 0 0
\(887\) 572.252 0.645155 0.322578 0.946543i \(-0.395451\pi\)
0.322578 + 0.946543i \(0.395451\pi\)
\(888\) 0 0
\(889\) 542.924i 0.610713i
\(890\) 0 0
\(891\) −47.3458 47.3458i −0.0531379 0.0531379i
\(892\) 0 0
\(893\) −1350.50 1350.50i −1.51231 1.51231i
\(894\) 0 0
\(895\) 1347.57i 1.50567i
\(896\) 0 0
\(897\) 9.69595 0.0108093
\(898\) 0 0
\(899\) −478.402 + 478.402i −0.532149 + 0.532149i
\(900\) 0 0
\(901\) 70.6281 70.6281i 0.0783886 0.0783886i
\(902\) 0 0
\(903\) −642.948 −0.712013
\(904\) 0 0
\(905\) 155.986i 0.172361i
\(906\) 0 0
\(907\) −356.673 356.673i −0.393245 0.393245i 0.482598 0.875842i \(-0.339693\pi\)
−0.875842 + 0.482598i \(0.839693\pi\)
\(908\) 0 0
\(909\) 112.895 + 112.895i 0.124197 + 0.124197i
\(910\) 0 0
\(911\) 656.235i 0.720345i 0.932886 + 0.360173i \(0.117282\pi\)
−0.932886 + 0.360173i \(0.882718\pi\)
\(912\) 0 0
\(913\) 228.161 0.249903
\(914\) 0 0
\(915\) 368.872 368.872i 0.403139 0.403139i
\(916\) 0 0
\(917\) 889.040 889.040i 0.969510 0.969510i
\(918\) 0 0
\(919\) −947.482 −1.03099 −0.515496 0.856892i \(-0.672392\pi\)
−0.515496 + 0.856892i \(0.672392\pi\)
\(920\) 0 0
\(921\) 129.509i 0.140618i
\(922\) 0 0
\(923\) −14.3591 14.3591i −0.0155570 0.0155570i
\(924\) 0 0
\(925\) 96.7280 + 96.7280i 0.104571 + 0.104571i
\(926\) 0 0
\(927\) 390.020i 0.420733i
\(928\) 0 0
\(929\) 628.551 0.676589 0.338295 0.941040i \(-0.390150\pi\)
0.338295 + 0.941040i \(0.390150\pi\)
\(930\) 0 0
\(931\) −380.304 + 380.304i −0.408490 + 0.408490i
\(932\) 0 0
\(933\) −429.534 + 429.534i −0.460380 + 0.460380i
\(934\) 0 0
\(935\) −56.3455 −0.0602626
\(936\) 0 0
\(937\) 910.799i 0.972037i 0.873949 + 0.486018i \(0.161551\pi\)
−0.873949 + 0.486018i \(0.838449\pi\)
\(938\) 0 0
\(939\) 481.989 + 481.989i 0.513300 + 0.513300i
\(940\) 0 0
\(941\) 304.500 + 304.500i 0.323592 + 0.323592i 0.850143 0.526551i \(-0.176515\pi\)
−0.526551 + 0.850143i \(0.676515\pi\)
\(942\) 0 0
\(943\) 143.719i 0.152406i
\(944\) 0 0
\(945\) −154.159 −0.163131
\(946\) 0 0
\(947\) −720.759 + 720.759i −0.761097 + 0.761097i −0.976521 0.215423i \(-0.930887\pi\)
0.215423 + 0.976521i \(0.430887\pi\)
\(948\) 0 0
\(949\) 213.669 213.669i 0.225151 0.225151i
\(950\) 0 0
\(951\) 810.632 0.852400
\(952\) 0 0
\(953\) 372.120i 0.390473i 0.980756 + 0.195236i \(0.0625474\pi\)
−0.980756 + 0.195236i \(0.937453\pi\)
\(954\) 0 0
\(955\) 149.681 + 149.681i 0.156735 + 0.156735i
\(956\) 0 0
\(957\) 463.257 + 463.257i 0.484072 + 0.484072i
\(958\) 0 0
\(959\) 728.947i 0.760112i
\(960\) 0 0
\(961\) 783.919 0.815733
\(962\) 0 0
\(963\) 86.0597 86.0597i 0.0893663 0.0893663i
\(964\) 0 0
\(965\) −110.530 + 110.530i −0.114539 + 0.114539i
\(966\) 0 0
\(967\) −1575.78 −1.62956 −0.814780 0.579771i \(-0.803142\pi\)
−0.814780 + 0.579771i \(0.803142\pi\)
\(968\) 0 0
\(969\) 77.9953i 0.0804905i
\(970\) 0 0
\(971\) −48.8117 48.8117i −0.0502695 0.0502695i 0.681525 0.731795i \(-0.261317\pi\)
−0.731795 + 0.681525i \(0.761317\pi\)
\(972\) 0 0
\(973\) 22.1994 + 22.1994i 0.0228154 + 0.0228154i
\(974\) 0 0
\(975\) 11.4051i 0.0116976i
\(976\) 0 0
\(977\) −83.5759 −0.0855434 −0.0427717 0.999085i \(-0.513619\pi\)
−0.0427717 + 0.999085i \(0.513619\pi\)
\(978\) 0 0
\(979\) −628.970 + 628.970i −0.642462 + 0.642462i
\(980\) 0 0
\(981\) −124.361 + 124.361i −0.126770 + 0.126770i
\(982\) 0 0
\(983\) −1864.55 −1.89679 −0.948397 0.317086i \(-0.897296\pi\)
−0.948397 + 0.317086i \(0.897296\pi\)
\(984\) 0 0
\(985\) 1009.50i 1.02488i
\(986\) 0 0
\(987\) −421.754 421.754i −0.427309 0.427309i
\(988\) 0 0
\(989\) −105.821 105.821i −0.106998 0.106998i
\(990\) 0 0
\(991\) 491.954i 0.496422i 0.968706 + 0.248211i \(0.0798427\pi\)
−0.968706 + 0.248211i \(0.920157\pi\)
\(992\) 0 0
\(993\) 444.068 0.447198
\(994\) 0 0
\(995\) −588.885 + 588.885i −0.591844 + 0.591844i
\(996\) 0 0
\(997\) −708.553 + 708.553i −0.710685 + 0.710685i −0.966679 0.255993i \(-0.917597\pi\)
0.255993 + 0.966679i \(0.417597\pi\)
\(998\) 0 0
\(999\) 265.778 0.266044
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 768.3.l.e.703.7 yes 16
4.3 odd 2 inner 768.3.l.e.703.3 yes 16
8.3 odd 2 768.3.l.f.703.6 yes 16
8.5 even 2 768.3.l.f.703.2 yes 16
16.3 odd 4 768.3.l.f.319.2 yes 16
16.5 even 4 inner 768.3.l.e.319.3 16
16.11 odd 4 inner 768.3.l.e.319.7 yes 16
16.13 even 4 768.3.l.f.319.6 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
768.3.l.e.319.3 16 16.5 even 4 inner
768.3.l.e.319.7 yes 16 16.11 odd 4 inner
768.3.l.e.703.3 yes 16 4.3 odd 2 inner
768.3.l.e.703.7 yes 16 1.1 even 1 trivial
768.3.l.f.319.2 yes 16 16.3 odd 4
768.3.l.f.319.6 yes 16 16.13 even 4
768.3.l.f.703.2 yes 16 8.5 even 2
768.3.l.f.703.6 yes 16 8.3 odd 2