Properties

Label 1008.3.cg.p.145.3
Level $1008$
Weight $3$
Character 1008.145
Analytic conductor $27.466$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,3,Mod(145,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.cg (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: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.35911766016.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.3
Root \(-1.90015 + 1.67440i\) of defining polynomial
Character \(\chi\) \(=\) 1008.145
Dual form 1008.3.cg.p.577.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.68140 + 2.70281i) q^{5} +(6.12873 + 3.38210i) q^{7} +O(q^{10})\) \(q+(4.68140 + 2.70281i) q^{5} +(6.12873 + 3.38210i) q^{7} +(-5.26719 - 9.12303i) q^{11} -12.0366i q^{13} +(20.8464 - 12.0357i) q^{17} +(-30.9077 - 17.8446i) q^{19} +(20.6552 - 35.7759i) q^{23} +(2.11033 + 3.65520i) q^{25} +28.6732 q^{29} +(4.50233 - 2.59942i) q^{31} +(19.5499 + 32.3978i) q^{35} +(1.90932 - 3.30704i) q^{37} +9.20741i q^{41} -54.2960 q^{43} +(14.3350 + 8.27632i) q^{47} +(26.1228 + 41.4560i) q^{49} +(9.58322 + 16.5986i) q^{53} -56.9447i q^{55} +(10.8261 - 6.25043i) q^{59} +(84.1179 + 48.5655i) q^{61} +(32.5327 - 56.3483i) q^{65} +(0.499078 + 0.864429i) q^{67} -14.8401 q^{71} +(-51.0987 + 29.5018i) q^{73} +(-1.42615 - 73.7268i) q^{77} +(-50.7558 + 87.9116i) q^{79} +22.7269i q^{83} +130.121 q^{85} +(98.7518 + 57.0144i) q^{89} +(40.7092 - 73.7694i) q^{91} +(-96.4608 - 167.075i) q^{95} -153.333i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{5} - 8 q^{7} - 22 q^{11} - 36 q^{17} - 42 q^{19} + 48 q^{23} + 42 q^{25} - 68 q^{29} + 60 q^{31} - 12 q^{35} - 118 q^{37} + 92 q^{43} - 12 q^{47} - 20 q^{49} - 10 q^{53} - 54 q^{59} + 24 q^{61} + 148 q^{65} - 22 q^{67} - 392 q^{71} - 138 q^{73} + 126 q^{77} - 164 q^{79} + 200 q^{85} + 60 q^{89} - 90 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 4.68140 + 2.70281i 0.936280 + 0.540561i 0.888792 0.458310i \(-0.151545\pi\)
0.0474876 + 0.998872i \(0.484879\pi\)
\(6\) 0 0
\(7\) 6.12873 + 3.38210i 0.875533 + 0.483158i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.26719 9.12303i −0.478835 0.829367i 0.520870 0.853636i \(-0.325607\pi\)
−0.999705 + 0.0242691i \(0.992274\pi\)
\(12\) 0 0
\(13\) 12.0366i 0.925896i −0.886386 0.462948i \(-0.846792\pi\)
0.886386 0.462948i \(-0.153208\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 20.8464 12.0357i 1.22626 0.707982i 0.260015 0.965605i \(-0.416272\pi\)
0.966246 + 0.257622i \(0.0829391\pi\)
\(18\) 0 0
\(19\) −30.9077 17.8446i −1.62672 0.939187i −0.985063 0.172195i \(-0.944914\pi\)
−0.641657 0.766992i \(-0.721753\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 20.6552 35.7759i 0.898053 1.55547i 0.0680720 0.997680i \(-0.478315\pi\)
0.829981 0.557792i \(-0.188351\pi\)
\(24\) 0 0
\(25\) 2.11033 + 3.65520i 0.0844132 + 0.146208i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 28.6732 0.988732 0.494366 0.869254i \(-0.335400\pi\)
0.494366 + 0.869254i \(0.335400\pi\)
\(30\) 0 0
\(31\) 4.50233 2.59942i 0.145236 0.0838522i −0.425621 0.904902i \(-0.639944\pi\)
0.570857 + 0.821049i \(0.306611\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 19.5499 + 32.3978i 0.558568 + 0.925650i
\(36\) 0 0
\(37\) 1.90932 3.30704i 0.0516033 0.0893795i −0.839070 0.544024i \(-0.816900\pi\)
0.890673 + 0.454644i \(0.150234\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.20741i 0.224571i 0.993676 + 0.112286i \(0.0358171\pi\)
−0.993676 + 0.112286i \(0.964183\pi\)
\(42\) 0 0
\(43\) −54.2960 −1.26270 −0.631349 0.775499i \(-0.717498\pi\)
−0.631349 + 0.775499i \(0.717498\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 14.3350 + 8.27632i 0.305000 + 0.176092i 0.644687 0.764447i \(-0.276988\pi\)
−0.339687 + 0.940539i \(0.610321\pi\)
\(48\) 0 0
\(49\) 26.1228 + 41.4560i 0.533118 + 0.846041i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.58322 + 16.5986i 0.180815 + 0.313182i 0.942158 0.335168i \(-0.108793\pi\)
−0.761343 + 0.648349i \(0.775460\pi\)
\(54\) 0 0
\(55\) 56.9447i 1.03536i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.8261 6.25043i 0.183493 0.105939i −0.405440 0.914122i \(-0.632882\pi\)
0.588933 + 0.808182i \(0.299548\pi\)
\(60\) 0 0
\(61\) 84.1179 + 48.5655i 1.37898 + 0.796156i 0.992037 0.125947i \(-0.0401969\pi\)
0.386945 + 0.922103i \(0.373530\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 32.5327 56.3483i 0.500503 0.866897i
\(66\) 0 0
\(67\) 0.499078 + 0.864429i 0.00744893 + 0.0129019i 0.869726 0.493535i \(-0.164296\pi\)
−0.862277 + 0.506437i \(0.830962\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.8401 −0.209015 −0.104507 0.994524i \(-0.533327\pi\)
−0.104507 + 0.994524i \(0.533327\pi\)
\(72\) 0 0
\(73\) −51.0987 + 29.5018i −0.699982 + 0.404135i −0.807341 0.590085i \(-0.799094\pi\)
0.107359 + 0.994220i \(0.465761\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.42615 73.7268i −0.0185214 0.957491i
\(78\) 0 0
\(79\) −50.7558 + 87.9116i −0.642478 + 1.11281i 0.342399 + 0.939555i \(0.388760\pi\)
−0.984878 + 0.173251i \(0.944573\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 22.7269i 0.273818i 0.990584 + 0.136909i \(0.0437168\pi\)
−0.990584 + 0.136909i \(0.956283\pi\)
\(84\) 0 0
\(85\) 130.121 1.53083
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 98.7518 + 57.0144i 1.10957 + 0.640611i 0.938718 0.344685i \(-0.112014\pi\)
0.170853 + 0.985297i \(0.445348\pi\)
\(90\) 0 0
\(91\) 40.7092 73.7694i 0.447353 0.810653i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −96.4608 167.075i −1.01538 1.75868i
\(96\) 0 0
\(97\) 153.333i 1.58075i −0.612621 0.790377i \(-0.709885\pi\)
0.612621 0.790377i \(-0.290115\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.2778 8.82063i 0.151265 0.0873329i −0.422457 0.906383i \(-0.638832\pi\)
0.573722 + 0.819050i \(0.305499\pi\)
\(102\) 0 0
\(103\) 119.710 + 69.1149i 1.16224 + 0.671018i 0.951839 0.306598i \(-0.0991907\pi\)
0.210398 + 0.977616i \(0.432524\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 41.0743 71.1428i 0.383872 0.664886i −0.607740 0.794136i \(-0.707924\pi\)
0.991612 + 0.129250i \(0.0412570\pi\)
\(108\) 0 0
\(109\) −45.1889 78.2695i −0.414577 0.718069i 0.580807 0.814042i \(-0.302737\pi\)
−0.995384 + 0.0959724i \(0.969404\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −206.994 −1.83180 −0.915901 0.401405i \(-0.868522\pi\)
−0.915901 + 0.401405i \(0.868522\pi\)
\(114\) 0 0
\(115\) 193.391 111.654i 1.68166 0.970905i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 168.468 3.25879i 1.41570 0.0273848i
\(120\) 0 0
\(121\) 5.01351 8.68366i 0.0414340 0.0717658i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 112.325i 0.898601i
\(126\) 0 0
\(127\) 155.159 1.22172 0.610862 0.791737i \(-0.290823\pi\)
0.610862 + 0.791737i \(0.290823\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 62.0353 + 35.8161i 0.473552 + 0.273405i 0.717725 0.696326i \(-0.245183\pi\)
−0.244174 + 0.969732i \(0.578517\pi\)
\(132\) 0 0
\(133\) −129.073 213.897i −0.970472 1.60825i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 76.1226 + 131.848i 0.555639 + 0.962395i 0.997853 + 0.0654859i \(0.0208597\pi\)
−0.442214 + 0.896909i \(0.645807\pi\)
\(138\) 0 0
\(139\) 119.318i 0.858401i −0.903209 0.429200i \(-0.858795\pi\)
0.903209 0.429200i \(-0.141205\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −109.811 + 63.3992i −0.767907 + 0.443351i
\(144\) 0 0
\(145\) 134.231 + 77.4982i 0.925729 + 0.534470i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −27.0510 + 46.8537i −0.181550 + 0.314455i −0.942409 0.334464i \(-0.891445\pi\)
0.760858 + 0.648918i \(0.224778\pi\)
\(150\) 0 0
\(151\) 65.8756 + 114.100i 0.436262 + 0.755629i 0.997398 0.0720954i \(-0.0229686\pi\)
−0.561135 + 0.827724i \(0.689635\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 28.1029 0.181309
\(156\) 0 0
\(157\) −208.180 + 120.193i −1.32599 + 0.765558i −0.984676 0.174393i \(-0.944204\pi\)
−0.341309 + 0.939951i \(0.610870\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 247.588 149.403i 1.53781 0.927968i
\(162\) 0 0
\(163\) −134.585 + 233.108i −0.825674 + 1.43011i 0.0757286 + 0.997128i \(0.475872\pi\)
−0.901403 + 0.432981i \(0.857462\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 123.116i 0.737219i −0.929584 0.368609i \(-0.879834\pi\)
0.929584 0.368609i \(-0.120166\pi\)
\(168\) 0 0
\(169\) 24.1192 0.142717
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −195.401 112.815i −1.12949 0.652109i −0.185680 0.982610i \(-0.559449\pi\)
−0.943805 + 0.330502i \(0.892782\pi\)
\(174\) 0 0
\(175\) 0.571395 + 29.5391i 0.00326511 + 0.168795i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 133.624 + 231.444i 0.746505 + 1.29298i 0.949488 + 0.313803i \(0.101603\pi\)
−0.202983 + 0.979182i \(0.565064\pi\)
\(180\) 0 0
\(181\) 64.5441i 0.356597i 0.983976 + 0.178299i \(0.0570593\pi\)
−0.983976 + 0.178299i \(0.942941\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 17.8766 10.3211i 0.0966302 0.0557895i
\(186\) 0 0
\(187\) −219.604 126.788i −1.17435 0.678013i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −74.3710 + 128.814i −0.389377 + 0.674421i −0.992366 0.123329i \(-0.960643\pi\)
0.602989 + 0.797750i \(0.293976\pi\)
\(192\) 0 0
\(193\) −14.9315 25.8622i −0.0773655 0.134001i 0.824747 0.565502i \(-0.191318\pi\)
−0.902112 + 0.431501i \(0.857984\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −191.905 −0.974138 −0.487069 0.873364i \(-0.661934\pi\)
−0.487069 + 0.873364i \(0.661934\pi\)
\(198\) 0 0
\(199\) −18.5965 + 10.7367i −0.0934495 + 0.0539531i −0.545996 0.837787i \(-0.683849\pi\)
0.452547 + 0.891741i \(0.350515\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 175.731 + 96.9758i 0.865668 + 0.477713i
\(204\) 0 0
\(205\) −24.8859 + 43.1036i −0.121394 + 0.210261i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 375.962i 1.79886i
\(210\) 0 0
\(211\) 179.565 0.851020 0.425510 0.904954i \(-0.360095\pi\)
0.425510 + 0.904954i \(0.360095\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −254.181 146.752i −1.18224 0.682565i
\(216\) 0 0
\(217\) 36.3851 0.703821i 0.167673 0.00324341i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −144.869 250.921i −0.655518 1.13539i
\(222\) 0 0
\(223\) 296.009i 1.32739i −0.748001 0.663697i \(-0.768986\pi\)
0.748001 0.663697i \(-0.231014\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 152.738 88.1834i 0.672855 0.388473i −0.124302 0.992244i \(-0.539669\pi\)
0.797158 + 0.603771i \(0.206336\pi\)
\(228\) 0 0
\(229\) −41.8862 24.1830i −0.182909 0.105603i 0.405750 0.913984i \(-0.367011\pi\)
−0.588659 + 0.808382i \(0.700344\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 229.214 397.011i 0.983753 1.70391i 0.336401 0.941719i \(-0.390790\pi\)
0.647352 0.762191i \(-0.275877\pi\)
\(234\) 0 0
\(235\) 44.7386 + 77.4895i 0.190377 + 0.329743i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 124.419 0.520581 0.260291 0.965530i \(-0.416182\pi\)
0.260291 + 0.965530i \(0.416182\pi\)
\(240\) 0 0
\(241\) −190.318 + 109.880i −0.789702 + 0.455935i −0.839858 0.542807i \(-0.817362\pi\)
0.0501556 + 0.998741i \(0.484028\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 10.2435 + 264.677i 0.0418101 + 1.08031i
\(246\) 0 0
\(247\) −214.789 + 372.025i −0.869589 + 1.50617i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 69.5582i 0.277124i 0.990354 + 0.138562i \(0.0442481\pi\)
−0.990354 + 0.138562i \(0.955752\pi\)
\(252\) 0 0
\(253\) −435.179 −1.72008
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 88.1589 + 50.8985i 0.343031 + 0.198049i 0.661611 0.749847i \(-0.269873\pi\)
−0.318581 + 0.947896i \(0.603206\pi\)
\(258\) 0 0
\(259\) 22.8865 13.8105i 0.0883648 0.0533222i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 105.787 + 183.229i 0.402233 + 0.696688i 0.993995 0.109425i \(-0.0349008\pi\)
−0.591762 + 0.806113i \(0.701568\pi\)
\(264\) 0 0
\(265\) 103.606i 0.390967i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −270.057 + 155.918i −1.00393 + 0.579620i −0.909409 0.415903i \(-0.863466\pi\)
−0.0945219 + 0.995523i \(0.530132\pi\)
\(270\) 0 0
\(271\) −222.199 128.287i −0.819924 0.473383i 0.0304661 0.999536i \(-0.490301\pi\)
−0.850390 + 0.526152i \(0.823634\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22.2310 38.5052i 0.0808400 0.140019i
\(276\) 0 0
\(277\) 139.547 + 241.702i 0.503778 + 0.872570i 0.999990 + 0.00436839i \(0.00139050\pi\)
−0.496212 + 0.868201i \(0.665276\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 84.1671 0.299527 0.149763 0.988722i \(-0.452149\pi\)
0.149763 + 0.988722i \(0.452149\pi\)
\(282\) 0 0
\(283\) −403.071 + 232.713i −1.42428 + 0.822309i −0.996661 0.0816501i \(-0.973981\pi\)
−0.427619 + 0.903959i \(0.640648\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −31.1404 + 56.4298i −0.108503 + 0.196619i
\(288\) 0 0
\(289\) 145.216 251.521i 0.502477 0.870316i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 289.553i 0.988237i 0.869395 + 0.494119i \(0.164509\pi\)
−0.869395 + 0.494119i \(0.835491\pi\)
\(294\) 0 0
\(295\) 67.5748 0.229067
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −430.621 248.619i −1.44021 0.831503i
\(300\) 0 0
\(301\) −332.766 183.635i −1.10553 0.610082i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 262.526 + 454.709i 0.860742 + 1.49085i
\(306\) 0 0
\(307\) 339.471i 1.10577i 0.833258 + 0.552884i \(0.186473\pi\)
−0.833258 + 0.552884i \(0.813527\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 85.5013 49.3642i 0.274924 0.158727i −0.356199 0.934410i \(-0.615928\pi\)
0.631123 + 0.775683i \(0.282594\pi\)
\(312\) 0 0
\(313\) 185.867 + 107.310i 0.593824 + 0.342845i 0.766608 0.642115i \(-0.221943\pi\)
−0.172784 + 0.984960i \(0.555276\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 96.8033 167.668i 0.305373 0.528922i −0.671971 0.740577i \(-0.734552\pi\)
0.977344 + 0.211655i \(0.0678855\pi\)
\(318\) 0 0
\(319\) −151.027 261.587i −0.473439 0.820021i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −859.086 −2.65971
\(324\) 0 0
\(325\) 43.9964 25.4013i 0.135373 0.0781579i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 59.8641 + 99.2058i 0.181958 + 0.301537i
\(330\) 0 0
\(331\) 244.085 422.768i 0.737417 1.27724i −0.216238 0.976341i \(-0.569379\pi\)
0.953655 0.300903i \(-0.0972879\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.39565i 0.0161064i
\(336\) 0 0
\(337\) 535.060 1.58771 0.793857 0.608104i \(-0.208070\pi\)
0.793857 + 0.608104i \(0.208070\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −47.4292 27.3832i −0.139088 0.0803028i
\(342\) 0 0
\(343\) 19.8910 + 342.423i 0.0579912 + 0.998317i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −167.692 290.451i −0.483262 0.837034i 0.516553 0.856255i \(-0.327215\pi\)
−0.999815 + 0.0192209i \(0.993881\pi\)
\(348\) 0 0
\(349\) 105.238i 0.301542i −0.988569 0.150771i \(-0.951824\pi\)
0.988569 0.150771i \(-0.0481756\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −284.108 + 164.030i −0.804840 + 0.464674i −0.845161 0.534512i \(-0.820495\pi\)
0.0403210 + 0.999187i \(0.487162\pi\)
\(354\) 0 0
\(355\) −69.4722 40.1098i −0.195696 0.112985i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −267.417 + 463.180i −0.744895 + 1.29020i 0.205349 + 0.978689i \(0.434167\pi\)
−0.950244 + 0.311507i \(0.899166\pi\)
\(360\) 0 0
\(361\) 456.356 + 790.432i 1.26414 + 2.18956i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −318.951 −0.873839
\(366\) 0 0
\(367\) −72.0037 + 41.5714i −0.196195 + 0.113274i −0.594880 0.803815i \(-0.702800\pi\)
0.398684 + 0.917088i \(0.369467\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.59476 + 134.140i 0.00699396 + 0.361563i
\(372\) 0 0
\(373\) 270.334 468.233i 0.724757 1.25532i −0.234317 0.972160i \(-0.575285\pi\)
0.959074 0.283156i \(-0.0913814\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 345.129i 0.915462i
\(378\) 0 0
\(379\) −74.8470 −0.197486 −0.0987428 0.995113i \(-0.531482\pi\)
−0.0987428 + 0.995113i \(0.531482\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 101.467 + 58.5818i 0.264926 + 0.152955i 0.626580 0.779357i \(-0.284454\pi\)
−0.361654 + 0.932313i \(0.617788\pi\)
\(384\) 0 0
\(385\) 192.593 348.999i 0.500241 0.906491i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −264.804 458.653i −0.680729 1.17906i −0.974759 0.223261i \(-0.928330\pi\)
0.294029 0.955796i \(-0.405004\pi\)
\(390\) 0 0
\(391\) 994.399i 2.54322i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −475.216 + 274.366i −1.20308 + 0.694598i
\(396\) 0 0
\(397\) 301.419 + 174.024i 0.759241 + 0.438348i 0.829023 0.559214i \(-0.188897\pi\)
−0.0697819 + 0.997562i \(0.522230\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −226.644 + 392.559i −0.565197 + 0.978950i 0.431834 + 0.901953i \(0.357866\pi\)
−0.997031 + 0.0769969i \(0.975467\pi\)
\(402\) 0 0
\(403\) −31.2883 54.1929i −0.0776384 0.134474i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −40.2270 −0.0988378
\(408\) 0 0
\(409\) −481.022 + 277.718i −1.17609 + 0.679018i −0.955108 0.296259i \(-0.904261\pi\)
−0.220986 + 0.975277i \(0.570927\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 87.4896 1.69237i 0.211839 0.00409775i
\(414\) 0 0
\(415\) −61.4264 + 106.394i −0.148016 + 0.256370i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 54.3155i 0.129631i −0.997897 0.0648156i \(-0.979354\pi\)
0.997897 0.0648156i \(-0.0206459\pi\)
\(420\) 0 0
\(421\) −578.890 −1.37504 −0.687518 0.726168i \(-0.741300\pi\)
−0.687518 + 0.726168i \(0.741300\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 87.9858 + 50.7986i 0.207025 + 0.119526i
\(426\) 0 0
\(427\) 351.283 + 582.140i 0.822676 + 1.36333i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 266.699 + 461.936i 0.618791 + 1.07178i 0.989707 + 0.143111i \(0.0457106\pi\)
−0.370916 + 0.928667i \(0.620956\pi\)
\(432\) 0 0
\(433\) 110.744i 0.255760i 0.991790 + 0.127880i \(0.0408173\pi\)
−0.991790 + 0.127880i \(0.959183\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1276.81 + 737.166i −2.92176 + 1.68688i
\(438\) 0 0
\(439\) −539.160 311.284i −1.22816 0.709076i −0.261512 0.965200i \(-0.584221\pi\)
−0.966644 + 0.256124i \(0.917554\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 211.173 365.762i 0.476688 0.825647i −0.522956 0.852360i \(-0.675171\pi\)
0.999643 + 0.0267129i \(0.00850399\pi\)
\(444\) 0 0
\(445\) 308.198 + 533.814i 0.692579 + 1.19958i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 365.850 0.814810 0.407405 0.913248i \(-0.366434\pi\)
0.407405 + 0.913248i \(0.366434\pi\)
\(450\) 0 0
\(451\) 83.9995 48.4971i 0.186252 0.107532i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 389.960 235.315i 0.857056 0.517176i
\(456\) 0 0
\(457\) 16.3276 28.2803i 0.0357279 0.0618825i −0.847609 0.530622i \(-0.821958\pi\)
0.883336 + 0.468740i \(0.155292\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 770.259i 1.67084i −0.549609 0.835422i \(-0.685223\pi\)
0.549609 0.835422i \(-0.314777\pi\)
\(462\) 0 0
\(463\) 308.030 0.665292 0.332646 0.943052i \(-0.392059\pi\)
0.332646 + 0.943052i \(0.392059\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −81.0507 46.7947i −0.173556 0.100203i 0.410705 0.911768i \(-0.365282\pi\)
−0.584262 + 0.811565i \(0.698616\pi\)
\(468\) 0 0
\(469\) 0.135131 + 6.98579i 0.000288125 + 0.0148951i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 285.987 + 495.344i 0.604624 + 1.04724i
\(474\) 0 0
\(475\) 150.632i 0.317119i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −66.5254 + 38.4084i −0.138884 + 0.0801846i −0.567832 0.823145i \(-0.692218\pi\)
0.428948 + 0.903329i \(0.358884\pi\)
\(480\) 0 0
\(481\) −39.8057 22.9818i −0.0827561 0.0477792i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 414.430 717.813i 0.854494 1.48003i
\(486\) 0 0
\(487\) 304.987 + 528.253i 0.626256 + 1.08471i 0.988296 + 0.152545i \(0.0487470\pi\)
−0.362040 + 0.932162i \(0.617920\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 132.287 0.269424 0.134712 0.990885i \(-0.456989\pi\)
0.134712 + 0.990885i \(0.456989\pi\)
\(492\) 0 0
\(493\) 597.734 345.102i 1.21244 0.700004i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −90.9508 50.1906i −0.183000 0.100987i
\(498\) 0 0
\(499\) −196.807 + 340.880i −0.394403 + 0.683126i −0.993025 0.117906i \(-0.962382\pi\)
0.598622 + 0.801032i \(0.295715\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 79.0334i 0.157124i 0.996909 + 0.0785620i \(0.0250329\pi\)
−0.996909 + 0.0785620i \(0.974967\pi\)
\(504\) 0 0
\(505\) 95.3618 0.188835
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 278.660 + 160.884i 0.547465 + 0.316079i 0.748099 0.663587i \(-0.230967\pi\)
−0.200634 + 0.979666i \(0.564300\pi\)
\(510\) 0 0
\(511\) −412.949 + 7.98794i −0.808119 + 0.0156320i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 373.608 + 647.108i 0.725453 + 1.25652i
\(516\) 0 0
\(517\) 174.372i 0.337276i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 433.474 250.266i 0.832004 0.480358i −0.0225341 0.999746i \(-0.507173\pi\)
0.854538 + 0.519388i \(0.173840\pi\)
\(522\) 0 0
\(523\) 115.191 + 66.5054i 0.220250 + 0.127161i 0.606066 0.795414i \(-0.292747\pi\)
−0.385816 + 0.922576i \(0.626080\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 62.5716 108.377i 0.118732 0.205649i
\(528\) 0 0
\(529\) −588.775 1019.79i −1.11300 1.92777i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 110.826 0.207929
\(534\) 0 0
\(535\) 384.571 222.032i 0.718823 0.415013i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 240.611 456.675i 0.446403 0.847264i
\(540\) 0 0
\(541\) −449.836 + 779.140i −0.831491 + 1.44018i 0.0653652 + 0.997861i \(0.479179\pi\)
−0.896856 + 0.442323i \(0.854155\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 488.548i 0.896418i
\(546\) 0 0
\(547\) 59.3373 0.108478 0.0542388 0.998528i \(-0.482727\pi\)
0.0542388 + 0.998528i \(0.482727\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −886.223 511.661i −1.60839 0.928604i
\(552\) 0 0
\(553\) −608.395 + 367.126i −1.10017 + 0.663880i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 348.483 + 603.590i 0.625643 + 1.08364i 0.988416 + 0.151767i \(0.0484965\pi\)
−0.362774 + 0.931877i \(0.618170\pi\)
\(558\) 0 0
\(559\) 653.541i 1.16913i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 74.3002 42.8972i 0.131972 0.0761940i −0.432561 0.901605i \(-0.642390\pi\)
0.564532 + 0.825411i \(0.309056\pi\)
\(564\) 0 0
\(565\) −969.020 559.464i −1.71508 0.990201i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −260.352 + 450.943i −0.457561 + 0.792518i −0.998831 0.0483301i \(-0.984610\pi\)
0.541271 + 0.840848i \(0.317943\pi\)
\(570\) 0 0
\(571\) −90.3719 156.529i −0.158270 0.274131i 0.775975 0.630763i \(-0.217258\pi\)
−0.934245 + 0.356632i \(0.883925\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 174.357 0.303230
\(576\) 0 0
\(577\) −448.898 + 259.171i −0.777986 + 0.449170i −0.835716 0.549162i \(-0.814947\pi\)
0.0577301 + 0.998332i \(0.481614\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −76.8647 + 139.287i −0.132297 + 0.239737i
\(582\) 0 0
\(583\) 100.953 174.856i 0.173162 0.299925i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1072.05i 1.82632i −0.407602 0.913160i \(-0.633635\pi\)
0.407602 0.913160i \(-0.366365\pi\)
\(588\) 0 0
\(589\) −185.542 −0.315012
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −230.489 133.073i −0.388682 0.224406i 0.292907 0.956141i \(-0.405377\pi\)
−0.681589 + 0.731735i \(0.738711\pi\)
\(594\) 0 0
\(595\) 797.475 + 440.081i 1.34029 + 0.739633i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 453.423 + 785.351i 0.756966 + 1.31110i 0.944391 + 0.328825i \(0.106653\pi\)
−0.187425 + 0.982279i \(0.560014\pi\)
\(600\) 0 0
\(601\) 472.642i 0.786426i −0.919448 0.393213i \(-0.871364\pi\)
0.919448 0.393213i \(-0.128636\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 46.9405 27.1011i 0.0775876 0.0447952i
\(606\) 0 0
\(607\) 19.5830 + 11.3063i 0.0322620 + 0.0186264i 0.516044 0.856562i \(-0.327404\pi\)
−0.483782 + 0.875188i \(0.660737\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 99.6191 172.545i 0.163043 0.282398i
\(612\) 0 0
\(613\) 2.39151 + 4.14221i 0.00390132 + 0.00675728i 0.867969 0.496618i \(-0.165425\pi\)
−0.864068 + 0.503375i \(0.832092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 537.983 0.871934 0.435967 0.899963i \(-0.356406\pi\)
0.435967 + 0.899963i \(0.356406\pi\)
\(618\) 0 0
\(619\) 363.363 209.788i 0.587016 0.338914i −0.176901 0.984229i \(-0.556607\pi\)
0.763917 + 0.645315i \(0.223274\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 412.395 + 683.415i 0.661951 + 1.09697i
\(624\) 0 0
\(625\) 356.351 617.219i 0.570162 0.987550i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 91.9200i 0.146137i
\(630\) 0 0
\(631\) −346.433 −0.549022 −0.274511 0.961584i \(-0.588516\pi\)
−0.274511 + 0.961584i \(0.588516\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 726.361 + 419.365i 1.14388 + 0.660417i
\(636\) 0 0
\(637\) 498.991 314.430i 0.783346 0.493611i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 413.473 + 716.157i 0.645044 + 1.11725i 0.984291 + 0.176552i \(0.0564944\pi\)
−0.339247 + 0.940697i \(0.610172\pi\)
\(642\) 0 0
\(643\) 1041.58i 1.61988i 0.586514 + 0.809939i \(0.300500\pi\)
−0.586514 + 0.809939i \(0.699500\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 464.511 268.186i 0.717946 0.414507i −0.0960500 0.995377i \(-0.530621\pi\)
0.813996 + 0.580870i \(0.197288\pi\)
\(648\) 0 0
\(649\) −114.046 65.8443i −0.175725 0.101455i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 398.410 690.067i 0.610123 1.05676i −0.381096 0.924535i \(-0.624453\pi\)
0.991219 0.132229i \(-0.0422134\pi\)
\(654\) 0 0
\(655\) 193.608 + 335.339i 0.295585 + 0.511968i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 668.820 1.01490 0.507451 0.861681i \(-0.330588\pi\)
0.507451 + 0.861681i \(0.330588\pi\)
\(660\) 0 0
\(661\) 227.421 131.302i 0.344056 0.198641i −0.318008 0.948088i \(-0.603014\pi\)
0.662064 + 0.749447i \(0.269681\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −26.1178 1350.20i −0.0392749 2.03037i
\(666\) 0 0
\(667\) 592.251 1025.81i 0.887933 1.53795i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1023.21i 1.52491i
\(672\) 0 0
\(673\) −234.458 −0.348378 −0.174189 0.984712i \(-0.555730\pi\)
−0.174189 + 0.984712i \(0.555730\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −453.701 261.944i −0.670164 0.386919i 0.125975 0.992033i \(-0.459794\pi\)
−0.796139 + 0.605114i \(0.793127\pi\)
\(678\) 0 0
\(679\) 518.588 939.738i 0.763753 1.38400i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −112.693 195.190i −0.164997 0.285783i 0.771657 0.636038i \(-0.219428\pi\)
−0.936654 + 0.350256i \(0.886095\pi\)
\(684\) 0 0
\(685\) 822.978i 1.20143i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 199.792 115.350i 0.289973 0.167416i
\(690\) 0 0
\(691\) −484.078 279.483i −0.700547 0.404461i 0.107004 0.994259i \(-0.465874\pi\)
−0.807551 + 0.589798i \(0.799208\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 322.493 558.574i 0.464018 0.803703i
\(696\) 0 0
\(697\) 110.818 + 191.942i 0.158992 + 0.275383i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 270.005 0.385171 0.192585 0.981280i \(-0.438313\pi\)
0.192585 + 0.981280i \(0.438313\pi\)
\(702\) 0 0
\(703\) −118.025 + 68.1420i −0.167888 + 0.0969303i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 123.466 2.38828i 0.174633 0.00337805i
\(708\) 0 0
\(709\) 626.647 1085.38i 0.883846 1.53087i 0.0368152 0.999322i \(-0.488279\pi\)
0.847031 0.531544i \(-0.178388\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 214.766i 0.301215i
\(714\) 0 0
\(715\) −685.424 −0.958634
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 513.681 + 296.574i 0.714438 + 0.412481i 0.812702 0.582680i \(-0.197996\pi\)
−0.0982644 + 0.995160i \(0.531329\pi\)
\(720\) 0 0
\(721\) 499.920 + 828.460i 0.693370 + 1.14904i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 60.5100 + 104.806i 0.0834620 + 0.144561i
\(726\) 0 0
\(727\) 752.400i 1.03494i −0.855702 0.517469i \(-0.826874\pi\)
0.855702 0.517469i \(-0.173126\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1131.88 + 653.490i −1.54840 + 0.893967i
\(732\) 0 0
\(733\) −636.602 367.542i −0.868488 0.501422i −0.00164274 0.999999i \(-0.500523\pi\)
−0.866846 + 0.498577i \(0.833856\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.25747 9.10621i 0.00713362 0.0123558i
\(738\) 0 0
\(739\) −34.3763 59.5415i −0.0465173 0.0805703i 0.841829 0.539744i \(-0.181479\pi\)
−0.888347 + 0.459174i \(0.848146\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1023.37 1.37735 0.688674 0.725071i \(-0.258193\pi\)
0.688674 + 0.725071i \(0.258193\pi\)
\(744\) 0 0
\(745\) −253.273 + 146.227i −0.339964 + 0.196278i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 492.346 297.098i 0.657338 0.396659i
\(750\) 0 0
\(751\) −36.8175 + 63.7697i −0.0490246 + 0.0849131i −0.889496 0.456942i \(-0.848945\pi\)
0.840472 + 0.541855i \(0.182278\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 712.196i 0.943307i
\(756\) 0 0
\(757\) 14.6651 0.0193727 0.00968634 0.999953i \(-0.496917\pi\)
0.00968634 + 0.999953i \(0.496917\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −692.428 399.774i −0.909893 0.525327i −0.0294962 0.999565i \(-0.509390\pi\)
−0.880397 + 0.474238i \(0.842724\pi\)
\(762\) 0 0
\(763\) −12.2354 632.527i −0.0160359 0.829000i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −75.2342 130.309i −0.0980889 0.169895i
\(768\) 0 0
\(769\) 665.988i 0.866044i 0.901383 + 0.433022i \(0.142553\pi\)
−0.901383 + 0.433022i \(0.857447\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 47.3482 27.3365i 0.0612525 0.0353642i −0.469061 0.883166i \(-0.655408\pi\)
0.530314 + 0.847802i \(0.322074\pi\)
\(774\) 0 0
\(775\) 19.0028 + 10.9713i 0.0245197 + 0.0141565i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 164.302 284.580i 0.210914 0.365314i
\(780\) 0 0
\(781\) 78.1653 + 135.386i 0.100084 + 0.173350i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1299.43 −1.65532
\(786\) 0 0
\(787\) −380.225 + 219.523i −0.483132 + 0.278936i −0.721721 0.692184i \(-0.756648\pi\)
0.238589 + 0.971121i \(0.423315\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1268.61 700.074i −1.60380 0.885049i
\(792\) 0 0
\(793\) 584.566 1012.50i 0.737157 1.27679i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 479.187i 0.601239i −0.953744 0.300619i \(-0.902807\pi\)
0.953744 0.300619i \(-0.0971934\pi\)
\(798\) 0 0
\(799\) 398.445 0.498680
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 538.293 + 310.783i 0.670352 + 0.387028i
\(804\) 0 0
\(805\) 1562.86 30.2315i 1.94145 0.0375547i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 370.116 + 641.060i 0.457498 + 0.792410i 0.998828 0.0484003i \(-0.0154123\pi\)
−0.541330 + 0.840810i \(0.682079\pi\)
\(810\) 0 0
\(811\) 157.469i 0.194166i −0.995276 0.0970831i \(-0.969049\pi\)
0.995276 0.0970831i \(-0.0309513\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1260.09 + 727.514i −1.54612 + 0.892655i
\(816\) 0 0
\(817\) 1678.16 + 968.888i 2.05405 + 1.18591i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −416.699 + 721.744i −0.507551 + 0.879104i 0.492411 + 0.870363i \(0.336116\pi\)
−0.999962 + 0.00874120i \(0.997218\pi\)
\(822\) 0 0
\(823\) 290.180 + 502.607i 0.352588 + 0.610701i 0.986702 0.162539i \(-0.0519684\pi\)
−0.634114 + 0.773240i \(0.718635\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 310.093 0.374962 0.187481 0.982268i \(-0.439968\pi\)
0.187481 + 0.982268i \(0.439968\pi\)
\(828\) 0 0
\(829\) −510.938 + 294.990i −0.616330 + 0.355838i −0.775439 0.631423i \(-0.782471\pi\)
0.159109 + 0.987261i \(0.449138\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1043.52 + 549.804i 1.25272 + 0.660029i
\(834\) 0 0
\(835\) 332.758 576.353i 0.398512 0.690243i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1658.50i 1.97676i −0.152018 0.988378i \(-0.548577\pi\)
0.152018 0.988378i \(-0.451423\pi\)
\(840\) 0 0
\(841\) −18.8466 −0.0224098
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 112.912 + 65.1895i 0.133623 + 0.0771474i
\(846\) 0 0
\(847\) 60.0955 36.2636i 0.0709510 0.0428142i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −78.8749 136.615i −0.0926849 0.160535i
\(852\) 0 0
\(853\) 921.244i 1.08001i −0.841663 0.540003i \(-0.818423\pi\)
0.841663 0.540003i \(-0.181577\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1107.38 639.346i 1.29216 0.746028i 0.313122 0.949713i \(-0.398625\pi\)
0.979037 + 0.203685i \(0.0652918\pi\)
\(858\) 0 0
\(859\) 800.548 + 462.197i 0.931953 + 0.538064i 0.887429 0.460945i \(-0.152489\pi\)
0.0445245 + 0.999008i \(0.485823\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −611.344 + 1058.88i −0.708394 + 1.22697i 0.257059 + 0.966396i \(0.417247\pi\)
−0.965453 + 0.260579i \(0.916087\pi\)
\(864\) 0 0
\(865\) −609.833 1056.26i −0.705009 1.22111i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1069.36 1.23056
\(870\) 0 0
\(871\) 10.4048 6.00723i 0.0119458 0.00689693i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 379.895 688.411i 0.434166 0.786755i
\(876\) 0 0
\(877\) −565.185 + 978.929i −0.644453 + 1.11622i 0.339975 + 0.940434i \(0.389581\pi\)
−0.984428 + 0.175790i \(0.943752\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 37.1187i 0.0421325i 0.999778 + 0.0210663i \(0.00670609\pi\)
−0.999778 + 0.0210663i \(0.993294\pi\)
\(882\) 0 0
\(883\) −1517.07 −1.71809 −0.859045 0.511901i \(-0.828942\pi\)
−0.859045 + 0.511901i \(0.828942\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1153.86 666.179i −1.30085 0.751047i −0.320302 0.947316i \(-0.603784\pi\)
−0.980550 + 0.196268i \(0.937118\pi\)
\(888\) 0 0
\(889\) 950.928 + 524.764i 1.06966 + 0.590285i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −295.374 511.604i −0.330766 0.572904i
\(894\) 0 0
\(895\) 1444.64i 1.61413i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 129.096 74.5337i 0.143600 0.0829074i
\(900\) 0 0
\(901\) 399.552 + 230.681i 0.443454 + 0.256028i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −174.450 + 302.157i −0.192763 + 0.333875i
\(906\) 0 0
\(907\) 32.7414 + 56.7098i 0.0360986 + 0.0625246i 0.883510 0.468412i \(-0.155174\pi\)
−0.847412 + 0.530936i \(0.821840\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −242.277 −0.265946 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(912\) 0 0
\(913\) 207.338 119.707i 0.227096 0.131114i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 259.064 + 429.317i 0.282513 + 0.468176i
\(918\) 0 0
\(919\) −401.585 + 695.566i −0.436981 + 0.756873i −0.997455 0.0712988i \(-0.977286\pi\)
0.560474 + 0.828172i \(0.310619\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 178.624i 0.193526i
\(924\) 0 0
\(925\) 16.1172 0.0174240
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 544.851 + 314.570i 0.586492 + 0.338611i 0.763709 0.645560i \(-0.223376\pi\)
−0.177217 + 0.984172i \(0.556709\pi\)
\(930\) 0 0
\(931\) −67.6298 1747.46i −0.0726421 1.87697i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −685.370 1187.09i −0.733016 1.26962i
\(936\) 0 0
\(937\) 217.501i 0.232124i 0.993242 + 0.116062i \(0.0370272\pi\)
−0.993242 + 0.116062i \(0.962973\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1549.14 + 894.399i −1.64628 + 0.950477i −0.667741 + 0.744394i \(0.732738\pi\)
−0.978534 + 0.206083i \(0.933928\pi\)
\(942\) 0 0
\(943\) 329.403 + 190.181i 0.349314 + 0.201677i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −542.609 + 939.826i −0.572976 + 0.992424i 0.423282 + 0.905998i \(0.360878\pi\)
−0.996258 + 0.0864261i \(0.972455\pi\)
\(948\) 0 0
\(949\) 355.103 + 615.057i 0.374187 + 0.648111i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1638.91 −1.71973 −0.859867 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(954\) 0 0
\(955\) −696.321 + 402.021i −0.729132 + 0.420964i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 20.6110 + 1065.52i 0.0214922 + 1.11107i
\(960\) 0 0
\(961\) −466.986 + 808.844i −0.485938 + 0.841669i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 161.428i 0.167283i
\(966\) 0 0
\(967\) −486.560 −0.503164 −0.251582 0.967836i \(-0.580951\pi\)
−0.251582 + 0.967836i \(0.580951\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 302.690 + 174.758i 0.311730 + 0.179978i 0.647700 0.761895i \(-0.275731\pi\)
−0.335970 + 0.941873i \(0.609064\pi\)
\(972\) 0 0
\(973\) 403.545 731.266i 0.414743 0.751559i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −120.146 208.099i −0.122974 0.212998i 0.797965 0.602704i \(-0.205910\pi\)
−0.920939 + 0.389706i \(0.872577\pi\)
\(978\) 0 0
\(979\) 1201.22i 1.22699i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −520.831 + 300.702i −0.529838 + 0.305902i −0.740950 0.671560i \(-0.765625\pi\)
0.211113 + 0.977462i \(0.432291\pi\)
\(984\) 0 0
\(985\) −898.384 518.682i −0.912065 0.526581i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1121.49 + 1942.49i −1.13397 + 1.96409i
\(990\) 0 0
\(991\) 225.882 + 391.238i 0.227933 + 0.394791i 0.957195 0.289443i \(-0.0934700\pi\)
−0.729262 + 0.684234i \(0.760137\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −116.077 −0.116660
\(996\) 0 0
\(997\) −442.742 + 255.617i −0.444074 + 0.256386i −0.705324 0.708885i \(-0.749199\pi\)
0.261250 + 0.965271i \(0.415865\pi\)
\(998\) 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 1008.3.cg.p.145.3 8
3.2 odd 2 336.3.bh.g.145.2 8
4.3 odd 2 504.3.by.c.145.3 8
7.3 odd 6 inner 1008.3.cg.p.577.3 8
12.11 even 2 168.3.z.b.145.2 yes 8
21.2 odd 6 2352.3.f.g.97.3 8
21.5 even 6 2352.3.f.g.97.6 8
21.17 even 6 336.3.bh.g.241.2 8
28.3 even 6 504.3.by.c.73.3 8
28.19 even 6 3528.3.f.b.2449.6 8
28.23 odd 6 3528.3.f.b.2449.3 8
84.11 even 6 1176.3.z.c.913.3 8
84.23 even 6 1176.3.f.c.97.7 8
84.47 odd 6 1176.3.f.c.97.2 8
84.59 odd 6 168.3.z.b.73.2 8
84.83 odd 2 1176.3.z.c.313.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.3.z.b.73.2 8 84.59 odd 6
168.3.z.b.145.2 yes 8 12.11 even 2
336.3.bh.g.145.2 8 3.2 odd 2
336.3.bh.g.241.2 8 21.17 even 6
504.3.by.c.73.3 8 28.3 even 6
504.3.by.c.145.3 8 4.3 odd 2
1008.3.cg.p.145.3 8 1.1 even 1 trivial
1008.3.cg.p.577.3 8 7.3 odd 6 inner
1176.3.f.c.97.2 8 84.47 odd 6
1176.3.f.c.97.7 8 84.23 even 6
1176.3.z.c.313.3 8 84.83 odd 2
1176.3.z.c.913.3 8 84.11 even 6
2352.3.f.g.97.3 8 21.2 odd 6
2352.3.f.g.97.6 8 21.5 even 6
3528.3.f.b.2449.3 8 28.23 odd 6
3528.3.f.b.2449.6 8 28.19 even 6