Properties

Label 256.4.g.a.33.1
Level $256$
Weight $4$
Character 256.33
Analytic conductor $15.104$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,4,Mod(33,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.33");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 256.g (of order \(8\), degree \(4\), 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: \(15.1044889615\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 33.1
Character \(\chi\) \(=\) 256.33
Dual form 256.4.g.a.225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.28810 + 7.93817i) q^{3} +(-11.2895 + 4.67626i) q^{5} +(11.8490 - 11.8490i) q^{7} +(-33.1111 - 33.1111i) q^{9} +O(q^{10})\) \(q+(-3.28810 + 7.93817i) q^{3} +(-11.2895 + 4.67626i) q^{5} +(11.8490 - 11.8490i) q^{7} +(-33.1111 - 33.1111i) q^{9} +(-23.5791 - 56.9250i) q^{11} +(13.6580 + 5.65734i) q^{13} -104.994i q^{15} +44.1663i q^{17} +(66.5184 + 27.5528i) q^{19} +(55.0988 + 133.020i) q^{21} +(-60.0240 - 60.0240i) q^{23} +(17.1966 - 17.1966i) q^{25} +(157.383 - 65.1901i) q^{27} +(14.3335 - 34.6041i) q^{29} +174.518 q^{31} +529.411 q^{33} +(-78.3602 + 189.178i) q^{35} +(118.428 - 49.0545i) q^{37} +(-89.8179 + 89.8179i) q^{39} +(15.5284 + 15.5284i) q^{41} +(-87.3822 - 210.959i) q^{43} +(528.642 + 218.971i) q^{45} -228.677i q^{47} +62.2017i q^{49} +(-350.600 - 145.223i) q^{51} +(-258.652 - 624.440i) q^{53} +(532.392 + 532.392i) q^{55} +(-437.438 + 437.438i) q^{57} +(456.272 - 188.994i) q^{59} +(-242.128 + 584.548i) q^{61} -784.667 q^{63} -180.647 q^{65} +(332.601 - 802.971i) q^{67} +(673.845 - 279.116i) q^{69} +(550.460 - 550.460i) q^{71} +(69.2096 + 69.2096i) q^{73} +(79.9655 + 193.054i) q^{75} +(-953.895 - 395.116i) q^{77} -518.934i q^{79} +199.379i q^{81} +(-595.241 - 246.557i) q^{83} +(-206.533 - 498.615i) q^{85} +(227.563 + 227.563i) q^{87} +(656.135 - 656.135i) q^{89} +(228.868 - 94.8003i) q^{91} +(-573.832 + 1385.35i) q^{93} -879.802 q^{95} -388.503 q^{97} +(-1104.12 + 2665.58i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 4 q^{3} + 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 44 q - 4 q^{3} + 4 q^{5} + 4 q^{7} - 4 q^{9} - 4 q^{11} + 4 q^{13} - 4 q^{19} + 4 q^{21} - 324 q^{23} - 4 q^{25} - 268 q^{27} + 4 q^{29} + 752 q^{31} - 8 q^{33} - 460 q^{35} + 4 q^{37} - 596 q^{39} - 4 q^{41} + 804 q^{43} - 104 q^{45} - 1384 q^{51} - 748 q^{53} + 292 q^{55} - 4 q^{57} + 1372 q^{59} + 1828 q^{61} - 2512 q^{63} - 8 q^{65} + 2036 q^{67} + 1060 q^{69} - 220 q^{71} - 4 q^{73} - 1712 q^{75} - 1900 q^{77} + 2436 q^{83} - 496 q^{85} + 1292 q^{87} - 4 q^{89} - 3604 q^{91} + 112 q^{93} + 6088 q^{95} - 8 q^{97} - 5424 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\) \(1\)

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
\(3\) −3.28810 + 7.93817i −0.632795 + 1.52770i 0.203300 + 0.979116i \(0.434833\pi\)
−0.836095 + 0.548585i \(0.815167\pi\)
\(4\) 0 0
\(5\) −11.2895 + 4.67626i −1.00976 + 0.418257i −0.825368 0.564595i \(-0.809032\pi\)
−0.184394 + 0.982852i \(0.559032\pi\)
\(6\) 0 0
\(7\) 11.8490 11.8490i 0.639787 0.639787i −0.310716 0.950503i \(-0.600569\pi\)
0.950503 + 0.310716i \(0.100569\pi\)
\(8\) 0 0
\(9\) −33.1111 33.1111i −1.22634 1.22634i
\(10\) 0 0
\(11\) −23.5791 56.9250i −0.646306 1.56032i −0.818029 0.575176i \(-0.804933\pi\)
0.171723 0.985145i \(-0.445067\pi\)
\(12\) 0 0
\(13\) 13.6580 + 5.65734i 0.291389 + 0.120697i 0.523590 0.851971i \(-0.324593\pi\)
−0.232200 + 0.972668i \(0.574593\pi\)
\(14\) 0 0
\(15\) 104.994i 1.80728i
\(16\) 0 0
\(17\) 44.1663i 0.630112i 0.949073 + 0.315056i \(0.102023\pi\)
−0.949073 + 0.315056i \(0.897977\pi\)
\(18\) 0 0
\(19\) 66.5184 + 27.5528i 0.803177 + 0.332687i 0.746228 0.665690i \(-0.231863\pi\)
0.0569490 + 0.998377i \(0.481863\pi\)
\(20\) 0 0
\(21\) 55.0988 + 133.020i 0.572549 + 1.38226i
\(22\) 0 0
\(23\) −60.0240 60.0240i −0.544168 0.544168i 0.380580 0.924748i \(-0.375724\pi\)
−0.924748 + 0.380580i \(0.875724\pi\)
\(24\) 0 0
\(25\) 17.1966 17.1966i 0.137573 0.137573i
\(26\) 0 0
\(27\) 157.383 65.1901i 1.12179 0.464661i
\(28\) 0 0
\(29\) 14.3335 34.6041i 0.0917813 0.221580i −0.871322 0.490712i \(-0.836737\pi\)
0.963103 + 0.269132i \(0.0867368\pi\)
\(30\) 0 0
\(31\) 174.518 1.01111 0.505554 0.862795i \(-0.331288\pi\)
0.505554 + 0.862795i \(0.331288\pi\)
\(32\) 0 0
\(33\) 529.411 2.79268
\(34\) 0 0
\(35\) −78.3602 + 189.178i −0.378437 + 0.913627i
\(36\) 0 0
\(37\) 118.428 49.0545i 0.526201 0.217960i −0.103737 0.994605i \(-0.533080\pi\)
0.629938 + 0.776645i \(0.283080\pi\)
\(38\) 0 0
\(39\) −89.8179 + 89.8179i −0.368779 + 0.368779i
\(40\) 0 0
\(41\) 15.5284 + 15.5284i 0.0591494 + 0.0591494i 0.736063 0.676913i \(-0.236683\pi\)
−0.676913 + 0.736063i \(0.736683\pi\)
\(42\) 0 0
\(43\) −87.3822 210.959i −0.309899 0.748163i −0.999708 0.0241712i \(-0.992305\pi\)
0.689809 0.723992i \(-0.257695\pi\)
\(44\) 0 0
\(45\) 528.642 + 218.971i 1.75123 + 0.725383i
\(46\) 0 0
\(47\) 228.677i 0.709703i −0.934923 0.354851i \(-0.884531\pi\)
0.934923 0.354851i \(-0.115469\pi\)
\(48\) 0 0
\(49\) 62.2017i 0.181346i
\(50\) 0 0
\(51\) −350.600 145.223i −0.962623 0.398731i
\(52\) 0 0
\(53\) −258.652 624.440i −0.670349 1.61837i −0.781017 0.624509i \(-0.785299\pi\)
0.110668 0.993857i \(-0.464701\pi\)
\(54\) 0 0
\(55\) 532.392 + 532.392i 1.30523 + 1.30523i
\(56\) 0 0
\(57\) −437.438 + 437.438i −1.01649 + 1.01649i
\(58\) 0 0
\(59\) 456.272 188.994i 1.00681 0.417033i 0.182518 0.983203i \(-0.441575\pi\)
0.824289 + 0.566169i \(0.191575\pi\)
\(60\) 0 0
\(61\) −242.128 + 584.548i −0.508218 + 1.22695i 0.436690 + 0.899612i \(0.356151\pi\)
−0.944908 + 0.327335i \(0.893849\pi\)
\(62\) 0 0
\(63\) −784.667 −1.56919
\(64\) 0 0
\(65\) −180.647 −0.344716
\(66\) 0 0
\(67\) 332.601 802.971i 0.606473 1.46416i −0.260337 0.965518i \(-0.583834\pi\)
0.866810 0.498639i \(-0.166166\pi\)
\(68\) 0 0
\(69\) 673.845 279.116i 1.17567 0.486980i
\(70\) 0 0
\(71\) 550.460 550.460i 0.920107 0.920107i −0.0769293 0.997037i \(-0.524512\pi\)
0.997037 + 0.0769293i \(0.0245116\pi\)
\(72\) 0 0
\(73\) 69.2096 + 69.2096i 0.110964 + 0.110964i 0.760409 0.649445i \(-0.224999\pi\)
−0.649445 + 0.760409i \(0.724999\pi\)
\(74\) 0 0
\(75\) 79.9655 + 193.054i 0.123115 + 0.297226i
\(76\) 0 0
\(77\) −953.895 395.116i −1.41177 0.584775i
\(78\) 0 0
\(79\) 518.934i 0.739046i −0.929222 0.369523i \(-0.879521\pi\)
0.929222 0.369523i \(-0.120479\pi\)
\(80\) 0 0
\(81\) 199.379i 0.273496i
\(82\) 0 0
\(83\) −595.241 246.557i −0.787183 0.326062i −0.0473726 0.998877i \(-0.515085\pi\)
−0.739810 + 0.672815i \(0.765085\pi\)
\(84\) 0 0
\(85\) −206.533 498.615i −0.263549 0.636263i
\(86\) 0 0
\(87\) 227.563 + 227.563i 0.280429 + 0.280429i
\(88\) 0 0
\(89\) 656.135 656.135i 0.781462 0.781462i −0.198615 0.980078i \(-0.563644\pi\)
0.980078 + 0.198615i \(0.0636445\pi\)
\(90\) 0 0
\(91\) 228.868 94.8003i 0.263647 0.109206i
\(92\) 0 0
\(93\) −573.832 + 1385.35i −0.639823 + 1.54467i
\(94\) 0 0
\(95\) −879.802 −0.950166
\(96\) 0 0
\(97\) −388.503 −0.406665 −0.203333 0.979110i \(-0.565177\pi\)
−0.203333 + 0.979110i \(0.565177\pi\)
\(98\) 0 0
\(99\) −1104.12 + 2665.58i −1.12089 + 2.70607i
\(100\) 0 0
\(101\) −658.558 + 272.784i −0.648802 + 0.268742i −0.682718 0.730682i \(-0.739202\pi\)
0.0339161 + 0.999425i \(0.489202\pi\)
\(102\) 0 0
\(103\) −467.607 + 467.607i −0.447327 + 0.447327i −0.894465 0.447138i \(-0.852443\pi\)
0.447138 + 0.894465i \(0.352443\pi\)
\(104\) 0 0
\(105\) −1244.07 1244.07i −1.15628 1.15628i
\(106\) 0 0
\(107\) 237.456 + 573.269i 0.214539 + 0.517944i 0.994111 0.108369i \(-0.0345630\pi\)
−0.779571 + 0.626314i \(0.784563\pi\)
\(108\) 0 0
\(109\) 117.568 + 48.6982i 0.103311 + 0.0427930i 0.433741 0.901038i \(-0.357193\pi\)
−0.330429 + 0.943831i \(0.607193\pi\)
\(110\) 0 0
\(111\) 1101.40i 0.941802i
\(112\) 0 0
\(113\) 1440.10i 1.19887i −0.800422 0.599437i \(-0.795391\pi\)
0.800422 0.599437i \(-0.204609\pi\)
\(114\) 0 0
\(115\) 958.327 + 396.952i 0.777083 + 0.321878i
\(116\) 0 0
\(117\) −264.911 639.553i −0.209325 0.505356i
\(118\) 0 0
\(119\) 523.327 + 523.327i 0.403137 + 0.403137i
\(120\) 0 0
\(121\) −1743.32 + 1743.32i −1.30978 + 1.30978i
\(122\) 0 0
\(123\) −174.326 + 72.2081i −0.127792 + 0.0529332i
\(124\) 0 0
\(125\) 470.807 1136.63i 0.336882 0.813305i
\(126\) 0 0
\(127\) −340.683 −0.238037 −0.119019 0.992892i \(-0.537975\pi\)
−0.119019 + 0.992892i \(0.537975\pi\)
\(128\) 0 0
\(129\) 1961.95 1.33907
\(130\) 0 0
\(131\) −460.465 + 1111.66i −0.307107 + 0.741421i 0.692689 + 0.721236i \(0.256426\pi\)
−0.999796 + 0.0201855i \(0.993574\pi\)
\(132\) 0 0
\(133\) 1114.65 461.704i 0.726711 0.301013i
\(134\) 0 0
\(135\) −1471.93 + 1471.93i −0.938394 + 0.938394i
\(136\) 0 0
\(137\) −109.800 109.800i −0.0684732 0.0684732i 0.672041 0.740514i \(-0.265418\pi\)
−0.740514 + 0.672041i \(0.765418\pi\)
\(138\) 0 0
\(139\) 199.529 + 481.706i 0.121754 + 0.293941i 0.972992 0.230840i \(-0.0741474\pi\)
−0.851238 + 0.524781i \(0.824147\pi\)
\(140\) 0 0
\(141\) 1815.28 + 751.914i 1.08421 + 0.449096i
\(142\) 0 0
\(143\) 910.879i 0.532668i
\(144\) 0 0
\(145\) 457.689i 0.262131i
\(146\) 0 0
\(147\) −493.767 204.525i −0.277042 0.114755i
\(148\) 0 0
\(149\) 917.500 + 2215.04i 0.504460 + 1.21787i 0.947031 + 0.321141i \(0.104066\pi\)
−0.442572 + 0.896733i \(0.645934\pi\)
\(150\) 0 0
\(151\) −191.904 191.904i −0.103423 0.103423i 0.653502 0.756925i \(-0.273299\pi\)
−0.756925 + 0.653502i \(0.773299\pi\)
\(152\) 0 0
\(153\) 1462.39 1462.39i 0.772729 0.772729i
\(154\) 0 0
\(155\) −1970.22 + 816.090i −1.02098 + 0.422903i
\(156\) 0 0
\(157\) 572.497 1382.13i 0.291021 0.702586i −0.708976 0.705233i \(-0.750842\pi\)
0.999996 + 0.00264665i \(0.000842456\pi\)
\(158\) 0 0
\(159\) 5807.38 2.89657
\(160\) 0 0
\(161\) −1422.45 −0.696303
\(162\) 0 0
\(163\) 1331.47 3214.44i 0.639807 1.54463i −0.187130 0.982335i \(-0.559919\pi\)
0.826937 0.562295i \(-0.190081\pi\)
\(164\) 0 0
\(165\) −5976.77 + 2475.66i −2.81995 + 1.16806i
\(166\) 0 0
\(167\) −214.549 + 214.549i −0.0994149 + 0.0994149i −0.755065 0.655650i \(-0.772395\pi\)
0.655650 + 0.755065i \(0.272395\pi\)
\(168\) 0 0
\(169\) −1398.98 1398.98i −0.636767 0.636767i
\(170\) 0 0
\(171\) −1290.19 3114.80i −0.576979 1.39295i
\(172\) 0 0
\(173\) −3387.83 1403.29i −1.48886 0.616705i −0.517789 0.855509i \(-0.673245\pi\)
−0.971068 + 0.238804i \(0.923245\pi\)
\(174\) 0 0
\(175\) 407.526i 0.176035i
\(176\) 0 0
\(177\) 4243.40i 1.80200i
\(178\) 0 0
\(179\) 1642.06 + 680.162i 0.685659 + 0.284009i 0.698190 0.715912i \(-0.253989\pi\)
−0.0125313 + 0.999921i \(0.503989\pi\)
\(180\) 0 0
\(181\) −1072.59 2589.47i −0.440471 1.06339i −0.975784 0.218736i \(-0.929807\pi\)
0.535313 0.844653i \(-0.320193\pi\)
\(182\) 0 0
\(183\) −3844.10 3844.10i −1.55281 1.55281i
\(184\) 0 0
\(185\) −1107.60 + 1107.60i −0.440175 + 0.440175i
\(186\) 0 0
\(187\) 2514.17 1041.40i 0.983177 0.407245i
\(188\) 0 0
\(189\) 1092.39 2637.27i 0.420423 1.01499i
\(190\) 0 0
\(191\) −3774.95 −1.43008 −0.715042 0.699082i \(-0.753592\pi\)
−0.715042 + 0.699082i \(0.753592\pi\)
\(192\) 0 0
\(193\) −1118.54 −0.417172 −0.208586 0.978004i \(-0.566886\pi\)
−0.208586 + 0.978004i \(0.566886\pi\)
\(194\) 0 0
\(195\) 593.986 1434.01i 0.218134 0.526623i
\(196\) 0 0
\(197\) 4025.41 1667.38i 1.45583 0.603024i 0.492252 0.870453i \(-0.336174\pi\)
0.963578 + 0.267429i \(0.0861740\pi\)
\(198\) 0 0
\(199\) 776.416 776.416i 0.276576 0.276576i −0.555165 0.831741i \(-0.687345\pi\)
0.831741 + 0.555165i \(0.187345\pi\)
\(200\) 0 0
\(201\) 5280.49 + 5280.49i 1.85302 + 1.85302i
\(202\) 0 0
\(203\) −240.186 579.861i −0.0830433 0.200484i
\(204\) 0 0
\(205\) −247.922 102.693i −0.0844665 0.0349872i
\(206\) 0 0
\(207\) 3974.92i 1.33467i
\(208\) 0 0
\(209\) 4436.23i 1.46823i
\(210\) 0 0
\(211\) −1455.68 602.962i −0.474944 0.196728i 0.132354 0.991203i \(-0.457746\pi\)
−0.607298 + 0.794474i \(0.707746\pi\)
\(212\) 0 0
\(213\) 2559.68 + 6179.61i 0.823410 + 1.98789i
\(214\) 0 0
\(215\) 1973.00 + 1973.00i 0.625849 + 0.625849i
\(216\) 0 0
\(217\) 2067.86 2067.86i 0.646893 0.646893i
\(218\) 0 0
\(219\) −776.966 + 321.830i −0.239737 + 0.0993025i
\(220\) 0 0
\(221\) −249.864 + 603.225i −0.0760528 + 0.183608i
\(222\) 0 0
\(223\) −1454.48 −0.436768 −0.218384 0.975863i \(-0.570079\pi\)
−0.218384 + 0.975863i \(0.570079\pi\)
\(224\) 0 0
\(225\) −1138.80 −0.337421
\(226\) 0 0
\(227\) 1309.75 3162.03i 0.382958 0.924542i −0.608433 0.793605i \(-0.708202\pi\)
0.991391 0.130937i \(-0.0417985\pi\)
\(228\) 0 0
\(229\) 5746.77 2380.39i 1.65833 0.686903i 0.660381 0.750930i \(-0.270395\pi\)
0.997948 + 0.0640278i \(0.0203946\pi\)
\(230\) 0 0
\(231\) 6273.00 6273.00i 1.78672 1.78672i
\(232\) 0 0
\(233\) 1497.10 + 1497.10i 0.420938 + 0.420938i 0.885526 0.464589i \(-0.153798\pi\)
−0.464589 + 0.885526i \(0.653798\pi\)
\(234\) 0 0
\(235\) 1069.35 + 2581.65i 0.296838 + 0.716631i
\(236\) 0 0
\(237\) 4119.39 + 1706.31i 1.12904 + 0.467665i
\(238\) 0 0
\(239\) 1039.75i 0.281404i −0.990052 0.140702i \(-0.955064\pi\)
0.990052 0.140702i \(-0.0449360\pi\)
\(240\) 0 0
\(241\) 120.503i 0.0322087i −0.999870 0.0161043i \(-0.994874\pi\)
0.999870 0.0161043i \(-0.00512639\pi\)
\(242\) 0 0
\(243\) 2666.64 + 1104.56i 0.703970 + 0.291594i
\(244\) 0 0
\(245\) −290.871 702.224i −0.0758492 0.183116i
\(246\) 0 0
\(247\) 752.635 + 752.635i 0.193883 + 0.193883i
\(248\) 0 0
\(249\) 3914.42 3914.42i 0.996250 0.996250i
\(250\) 0 0
\(251\) −5865.74 + 2429.67i −1.47507 + 0.610994i −0.968009 0.250916i \(-0.919268\pi\)
−0.507061 + 0.861910i \(0.669268\pi\)
\(252\) 0 0
\(253\) −2001.55 + 4832.18i −0.497378 + 1.20078i
\(254\) 0 0
\(255\) 4637.19 1.13879
\(256\) 0 0
\(257\) −4520.96 −1.09731 −0.548657 0.836048i \(-0.684861\pi\)
−0.548657 + 0.836048i \(0.684861\pi\)
\(258\) 0 0
\(259\) 822.008 1984.50i 0.197209 0.476104i
\(260\) 0 0
\(261\) −1620.37 + 671.180i −0.384286 + 0.159176i
\(262\) 0 0
\(263\) −449.651 + 449.651i −0.105425 + 0.105425i −0.757852 0.652427i \(-0.773751\pi\)
0.652427 + 0.757852i \(0.273751\pi\)
\(264\) 0 0
\(265\) 5840.08 + 5840.08i 1.35379 + 1.35379i
\(266\) 0 0
\(267\) 3051.07 + 7365.94i 0.699336 + 1.68835i
\(268\) 0 0
\(269\) 2436.90 + 1009.40i 0.552343 + 0.228788i 0.641357 0.767243i \(-0.278372\pi\)
−0.0890145 + 0.996030i \(0.528372\pi\)
\(270\) 0 0
\(271\) 1662.12i 0.372570i 0.982496 + 0.186285i \(0.0596448\pi\)
−0.982496 + 0.186285i \(0.940355\pi\)
\(272\) 0 0
\(273\) 2128.51i 0.471880i
\(274\) 0 0
\(275\) −1384.40 573.437i −0.303572 0.125744i
\(276\) 0 0
\(277\) 2447.54 + 5908.88i 0.530897 + 1.28170i 0.930930 + 0.365198i \(0.118999\pi\)
−0.400033 + 0.916501i \(0.631001\pi\)
\(278\) 0 0
\(279\) −5778.47 5778.47i −1.23996 1.23996i
\(280\) 0 0
\(281\) 3546.76 3546.76i 0.752960 0.752960i −0.222070 0.975031i \(-0.571281\pi\)
0.975031 + 0.222070i \(0.0712814\pi\)
\(282\) 0 0
\(283\) −2136.81 + 885.094i −0.448834 + 0.185913i −0.595639 0.803252i \(-0.703101\pi\)
0.146805 + 0.989165i \(0.453101\pi\)
\(284\) 0 0
\(285\) 2892.87 6984.02i 0.601260 1.45157i
\(286\) 0 0
\(287\) 367.992 0.0756860
\(288\) 0 0
\(289\) 2962.34 0.602959
\(290\) 0 0
\(291\) 1277.44 3084.00i 0.257336 0.621263i
\(292\) 0 0
\(293\) −7533.04 + 3120.29i −1.50200 + 0.622147i −0.973888 0.227029i \(-0.927099\pi\)
−0.528109 + 0.849177i \(0.677099\pi\)
\(294\) 0 0
\(295\) −4267.29 + 4267.29i −0.842208 + 0.842208i
\(296\) 0 0
\(297\) −7421.90 7421.90i −1.45004 1.45004i
\(298\) 0 0
\(299\) −480.234 1159.39i −0.0928851 0.224244i
\(300\) 0 0
\(301\) −3535.05 1464.27i −0.676934 0.280395i
\(302\) 0 0
\(303\) 6124.68i 1.16123i
\(304\) 0 0
\(305\) 7731.50i 1.45149i
\(306\) 0 0
\(307\) −3320.22 1375.28i −0.617247 0.255672i 0.0520765 0.998643i \(-0.483416\pi\)
−0.669324 + 0.742971i \(0.733416\pi\)
\(308\) 0 0
\(309\) −2174.41 5249.49i −0.400316 0.966449i
\(310\) 0 0
\(311\) −2136.23 2136.23i −0.389499 0.389499i 0.485010 0.874509i \(-0.338816\pi\)
−0.874509 + 0.485010i \(0.838816\pi\)
\(312\) 0 0
\(313\) −7352.03 + 7352.03i −1.32767 + 1.32767i −0.420274 + 0.907397i \(0.638066\pi\)
−0.907397 + 0.420274i \(0.861934\pi\)
\(314\) 0 0
\(315\) 8858.48 3669.30i 1.58450 0.656323i
\(316\) 0 0
\(317\) −321.560 + 776.314i −0.0569735 + 0.137546i −0.949803 0.312849i \(-0.898717\pi\)
0.892829 + 0.450395i \(0.148717\pi\)
\(318\) 0 0
\(319\) −2307.81 −0.405054
\(320\) 0 0
\(321\) −5331.49 −0.927023
\(322\) 0 0
\(323\) −1216.91 + 2937.87i −0.209630 + 0.506092i
\(324\) 0 0
\(325\) 332.159 137.585i 0.0566920 0.0234826i
\(326\) 0 0
\(327\) −773.148 + 773.148i −0.130750 + 0.130750i
\(328\) 0 0
\(329\) −2709.60 2709.60i −0.454058 0.454058i
\(330\) 0 0
\(331\) −383.476 925.792i −0.0636789 0.153735i 0.888837 0.458224i \(-0.151514\pi\)
−0.952516 + 0.304489i \(0.901514\pi\)
\(332\) 0 0
\(333\) −5545.52 2297.03i −0.912590 0.378007i
\(334\) 0 0
\(335\) 10620.4i 1.73211i
\(336\) 0 0
\(337\) 6360.72i 1.02816i 0.857742 + 0.514081i \(0.171867\pi\)
−0.857742 + 0.514081i \(0.828133\pi\)
\(338\) 0 0
\(339\) 11431.7 + 4735.17i 1.83152 + 0.758641i
\(340\) 0 0
\(341\) −4114.98 9934.43i −0.653485 1.57765i
\(342\) 0 0
\(343\) 4801.24 + 4801.24i 0.755809 + 0.755809i
\(344\) 0 0
\(345\) −6302.15 + 6302.15i −0.983467 + 0.983467i
\(346\) 0 0
\(347\) −10094.1 + 4181.11i −1.56161 + 0.646840i −0.985368 0.170438i \(-0.945482\pi\)
−0.576242 + 0.817279i \(0.695482\pi\)
\(348\) 0 0
\(349\) −697.242 + 1683.29i −0.106941 + 0.258179i −0.968287 0.249840i \(-0.919622\pi\)
0.861346 + 0.508019i \(0.169622\pi\)
\(350\) 0 0
\(351\) 2518.34 0.382961
\(352\) 0 0
\(353\) 3236.30 0.487963 0.243982 0.969780i \(-0.421546\pi\)
0.243982 + 0.969780i \(0.421546\pi\)
\(354\) 0 0
\(355\) −3640.32 + 8788.50i −0.544248 + 1.31393i
\(356\) 0 0
\(357\) −5875.01 + 2433.51i −0.870976 + 0.360770i
\(358\) 0 0
\(359\) −5424.53 + 5424.53i −0.797482 + 0.797482i −0.982698 0.185216i \(-0.940701\pi\)
0.185216 + 0.982698i \(0.440701\pi\)
\(360\) 0 0
\(361\) −1184.51 1184.51i −0.172694 0.172694i
\(362\) 0 0
\(363\) −8106.58 19571.0i −1.17214 2.82978i
\(364\) 0 0
\(365\) −1104.98 457.699i −0.158459 0.0656358i
\(366\) 0 0
\(367\) 10914.3i 1.55237i −0.630503 0.776187i \(-0.717151\pi\)
0.630503 0.776187i \(-0.282849\pi\)
\(368\) 0 0
\(369\) 1028.32i 0.145074i
\(370\) 0 0
\(371\) −10463.8 4334.23i −1.46429 0.606529i
\(372\) 0 0
\(373\) 3921.61 + 9467.61i 0.544379 + 1.31425i 0.921606 + 0.388127i \(0.126878\pi\)
−0.377227 + 0.926121i \(0.623122\pi\)
\(374\) 0 0
\(375\) 7474.69 + 7474.69i 1.02931 + 1.02931i
\(376\) 0 0
\(377\) 391.534 391.534i 0.0534881 0.0534881i
\(378\) 0 0
\(379\) 10042.4 4159.68i 1.36106 0.563769i 0.421710 0.906731i \(-0.361430\pi\)
0.939349 + 0.342962i \(0.111430\pi\)
\(380\) 0 0
\(381\) 1120.20 2704.40i 0.150629 0.363650i
\(382\) 0 0
\(383\) −1609.29 −0.214702 −0.107351 0.994221i \(-0.534237\pi\)
−0.107351 + 0.994221i \(0.534237\pi\)
\(384\) 0 0
\(385\) 12616.6 1.67014
\(386\) 0 0
\(387\) −4091.77 + 9878.41i −0.537458 + 1.29754i
\(388\) 0 0
\(389\) 3941.97 1632.82i 0.513793 0.212820i −0.110695 0.993854i \(-0.535308\pi\)
0.624488 + 0.781034i \(0.285308\pi\)
\(390\) 0 0
\(391\) 2651.04 2651.04i 0.342887 0.342887i
\(392\) 0 0
\(393\) −7310.49 7310.49i −0.938335 0.938335i
\(394\) 0 0
\(395\) 2426.67 + 5858.50i 0.309111 + 0.746261i
\(396\) 0 0
\(397\) −2995.05 1240.59i −0.378632 0.156835i 0.185247 0.982692i \(-0.440692\pi\)
−0.563879 + 0.825857i \(0.690692\pi\)
\(398\) 0 0
\(399\) 10366.4i 1.30068i
\(400\) 0 0
\(401\) 8084.70i 1.00681i 0.864051 + 0.503405i \(0.167919\pi\)
−0.864051 + 0.503405i \(0.832081\pi\)
\(402\) 0 0
\(403\) 2383.57 + 987.308i 0.294626 + 0.122038i
\(404\) 0 0
\(405\) −932.346 2250.88i −0.114392 0.276166i
\(406\) 0 0
\(407\) −5584.85 5584.85i −0.680174 0.680174i
\(408\) 0 0
\(409\) 4621.85 4621.85i 0.558767 0.558767i −0.370189 0.928956i \(-0.620707\pi\)
0.928956 + 0.370189i \(0.120707\pi\)
\(410\) 0 0
\(411\) 1232.64 510.577i 0.147936 0.0612771i
\(412\) 0 0
\(413\) 3166.98 7645.77i 0.377329 0.910954i
\(414\) 0 0
\(415\) 7872.92 0.931245
\(416\) 0 0
\(417\) −4479.93 −0.526099
\(418\) 0 0
\(419\) −2131.19 + 5145.14i −0.248485 + 0.599897i −0.998076 0.0620052i \(-0.980250\pi\)
0.749590 + 0.661902i \(0.230250\pi\)
\(420\) 0 0
\(421\) 8558.20 3544.92i 0.990739 0.410378i 0.172346 0.985036i \(-0.444865\pi\)
0.818393 + 0.574659i \(0.194865\pi\)
\(422\) 0 0
\(423\) −7571.75 + 7571.75i −0.870334 + 0.870334i
\(424\) 0 0
\(425\) 759.512 + 759.512i 0.0866864 + 0.0866864i
\(426\) 0 0
\(427\) 4057.35 + 9795.30i 0.459833 + 1.11014i
\(428\) 0 0
\(429\) 7230.71 + 2995.06i 0.813758 + 0.337069i
\(430\) 0 0
\(431\) 6610.79i 0.738818i 0.929267 + 0.369409i \(0.120440\pi\)
−0.929267 + 0.369409i \(0.879560\pi\)
\(432\) 0 0
\(433\) 8705.35i 0.966172i 0.875573 + 0.483086i \(0.160484\pi\)
−0.875573 + 0.483086i \(0.839516\pi\)
\(434\) 0 0
\(435\) −3633.21 1504.92i −0.400458 0.165875i
\(436\) 0 0
\(437\) −2338.87 5646.53i −0.256026 0.618101i
\(438\) 0 0
\(439\) 11344.9 + 11344.9i 1.23341 + 1.23341i 0.962648 + 0.270757i \(0.0872741\pi\)
0.270757 + 0.962648i \(0.412726\pi\)
\(440\) 0 0
\(441\) 2059.56 2059.56i 0.222391 0.222391i
\(442\) 0 0
\(443\) −1931.24 + 799.946i −0.207124 + 0.0857936i −0.483834 0.875160i \(-0.660756\pi\)
0.276709 + 0.960954i \(0.410756\pi\)
\(444\) 0 0
\(445\) −4339.16 + 10475.7i −0.462239 + 1.11594i
\(446\) 0 0
\(447\) −20600.2 −2.17977
\(448\) 0 0
\(449\) 1770.44 0.186085 0.0930426 0.995662i \(-0.470341\pi\)
0.0930426 + 0.995662i \(0.470341\pi\)
\(450\) 0 0
\(451\) 517.808 1250.10i 0.0540635 0.130521i
\(452\) 0 0
\(453\) 2154.36 892.366i 0.223445 0.0925541i
\(454\) 0 0
\(455\) −2140.49 + 2140.49i −0.220545 + 0.220545i
\(456\) 0 0
\(457\) −7694.10 7694.10i −0.787560 0.787560i 0.193534 0.981094i \(-0.438005\pi\)
−0.981094 + 0.193534i \(0.938005\pi\)
\(458\) 0 0
\(459\) 2879.21 + 6951.02i 0.292788 + 0.706854i
\(460\) 0 0
\(461\) 5291.40 + 2191.77i 0.534588 + 0.221433i 0.633611 0.773652i \(-0.281572\pi\)
−0.0990235 + 0.995085i \(0.531572\pi\)
\(462\) 0 0
\(463\) 13153.5i 1.32029i −0.751138 0.660145i \(-0.770495\pi\)
0.751138 0.660145i \(-0.229505\pi\)
\(464\) 0 0
\(465\) 18323.3i 1.82736i
\(466\) 0 0
\(467\) 7862.87 + 3256.91i 0.779122 + 0.322723i 0.736561 0.676371i \(-0.236448\pi\)
0.0425611 + 0.999094i \(0.486448\pi\)
\(468\) 0 0
\(469\) −5573.41 13455.4i −0.548734 1.32476i
\(470\) 0 0
\(471\) 9089.16 + 9089.16i 0.889185 + 0.889185i
\(472\) 0 0
\(473\) −9948.47 + 9948.47i −0.967085 + 0.967085i
\(474\) 0 0
\(475\) 1617.71 670.076i 0.156264 0.0647268i
\(476\) 0 0
\(477\) −12111.6 + 29240.1i −1.16259 + 2.80673i
\(478\) 0 0
\(479\) −15687.9 −1.49645 −0.748224 0.663446i \(-0.769093\pi\)
−0.748224 + 0.663446i \(0.769093\pi\)
\(480\) 0 0
\(481\) 1895.01 0.179636
\(482\) 0 0
\(483\) 4677.16 11291.7i 0.440617 1.06374i
\(484\) 0 0
\(485\) 4386.00 1816.74i 0.410635 0.170091i
\(486\) 0 0
\(487\) 13515.4 13515.4i 1.25758 1.25758i 0.305330 0.952247i \(-0.401233\pi\)
0.952247 0.305330i \(-0.0987668\pi\)
\(488\) 0 0
\(489\) 21138.8 + 21138.8i 1.95487 + 1.95487i
\(490\) 0 0
\(491\) −1348.48 3255.53i −0.123943 0.299226i 0.849713 0.527245i \(-0.176775\pi\)
−0.973657 + 0.228019i \(0.926775\pi\)
\(492\) 0 0
\(493\) 1528.33 + 633.056i 0.139620 + 0.0578325i
\(494\) 0 0
\(495\) 35256.1i 3.20130i
\(496\) 0 0
\(497\) 13044.8i 1.17734i
\(498\) 0 0
\(499\) −13952.2 5779.20i −1.25168 0.518462i −0.344332 0.938848i \(-0.611895\pi\)
−0.907345 + 0.420386i \(0.861895\pi\)
\(500\) 0 0
\(501\) −997.667 2408.58i −0.0889671 0.214785i
\(502\) 0 0
\(503\) −12832.8 12832.8i −1.13755 1.13755i −0.988889 0.148658i \(-0.952505\pi\)
−0.148658 0.988889i \(-0.547495\pi\)
\(504\) 0 0
\(505\) 6159.17 6159.17i 0.542732 0.542732i
\(506\) 0 0
\(507\) 15705.3 6505.34i 1.37573 0.569847i
\(508\) 0 0
\(509\) 294.015 709.814i 0.0256031 0.0618113i −0.910561 0.413375i \(-0.864350\pi\)
0.936164 + 0.351564i \(0.114350\pi\)
\(510\) 0 0
\(511\) 1640.13 0.141987
\(512\) 0 0
\(513\) 12265.0 1.05558
\(514\) 0 0
\(515\) 3092.39 7465.70i 0.264596 0.638792i
\(516\) 0 0
\(517\) −13017.5 + 5392.01i −1.10736 + 0.458685i
\(518\) 0 0
\(519\) 22279.0 22279.0i 1.88428 1.88428i
\(520\) 0 0
\(521\) −4736.98 4736.98i −0.398332 0.398332i 0.479312 0.877644i \(-0.340886\pi\)
−0.877644 + 0.479312i \(0.840886\pi\)
\(522\) 0 0
\(523\) 881.013 + 2126.95i 0.0736597 + 0.177830i 0.956421 0.291992i \(-0.0943180\pi\)
−0.882761 + 0.469822i \(0.844318\pi\)
\(524\) 0 0
\(525\) 3235.01 + 1339.99i 0.268929 + 0.111394i
\(526\) 0 0
\(527\) 7707.81i 0.637111i
\(528\) 0 0
\(529\) 4961.24i 0.407762i
\(530\) 0 0
\(531\) −21365.5 8849.86i −1.74611 0.723260i
\(532\) 0 0
\(533\) 124.238 + 299.937i 0.0100963 + 0.0243747i
\(534\) 0 0
\(535\) −5361.51 5361.51i −0.433268 0.433268i
\(536\) 0 0
\(537\) −10798.5 + 10798.5i −0.867763 + 0.867763i
\(538\) 0 0
\(539\) 3540.83 1466.66i 0.282958 0.117205i
\(540\) 0 0
\(541\) −8550.73 + 20643.3i −0.679528 + 1.64053i 0.0853508 + 0.996351i \(0.472799\pi\)
−0.764879 + 0.644174i \(0.777201\pi\)
\(542\) 0 0
\(543\) 24082.4 1.90327
\(544\) 0 0
\(545\) −1555.00 −0.122218
\(546\) 0 0
\(547\) −2552.53 + 6162.34i −0.199521 + 0.481687i −0.991696 0.128608i \(-0.958949\pi\)
0.792174 + 0.610295i \(0.208949\pi\)
\(548\) 0 0
\(549\) 27372.1 11337.9i 2.12789 0.881403i
\(550\) 0 0
\(551\) 1906.88 1906.88i 0.147433 0.147433i
\(552\) 0 0
\(553\) −6148.86 6148.86i −0.472832 0.472832i
\(554\) 0 0
\(555\) −5150.41 12434.2i −0.393915 0.950995i
\(556\) 0 0
\(557\) 2494.05 + 1033.07i 0.189724 + 0.0785863i 0.475523 0.879703i \(-0.342259\pi\)
−0.285799 + 0.958290i \(0.592259\pi\)
\(558\) 0 0
\(559\) 3375.64i 0.255411i
\(560\) 0 0
\(561\) 23382.1i 1.75970i
\(562\) 0 0
\(563\) 19457.5 + 8059.54i 1.45654 + 0.603320i 0.963745 0.266823i \(-0.0859741\pi\)
0.492798 + 0.870144i \(0.335974\pi\)
\(564\) 0 0
\(565\) 6734.26 + 16257.9i 0.501438 + 1.21058i
\(566\) 0 0
\(567\) 2362.44 + 2362.44i 0.174979 + 0.174979i
\(568\) 0 0
\(569\) 1740.16 1740.16i 0.128209 0.128209i −0.640090 0.768300i \(-0.721103\pi\)
0.768300 + 0.640090i \(0.221103\pi\)
\(570\) 0 0
\(571\) 16395.2 6791.10i 1.20160 0.497721i 0.310088 0.950708i \(-0.399642\pi\)
0.891517 + 0.452987i \(0.149642\pi\)
\(572\) 0 0
\(573\) 12412.4 29966.2i 0.904949 2.18474i
\(574\) 0 0
\(575\) −2064.42 −0.149726
\(576\) 0 0
\(577\) −13473.2 −0.972091 −0.486046 0.873933i \(-0.661561\pi\)
−0.486046 + 0.873933i \(0.661561\pi\)
\(578\) 0 0
\(579\) 3677.87 8879.15i 0.263984 0.637314i
\(580\) 0 0
\(581\) −9974.48 + 4131.56i −0.712239 + 0.295019i
\(582\) 0 0
\(583\) −29447.5 + 29447.5i −2.09192 + 2.09192i
\(584\) 0 0
\(585\) 5981.42 + 5981.42i 0.422737 + 0.422737i
\(586\) 0 0
\(587\) 873.151 + 2107.97i 0.0613949 + 0.148220i 0.951600 0.307340i \(-0.0994389\pi\)
−0.890205 + 0.455560i \(0.849439\pi\)
\(588\) 0 0
\(589\) 11608.6 + 4808.46i 0.812098 + 0.336382i
\(590\) 0 0
\(591\) 37436.9i 2.60566i
\(592\) 0 0
\(593\) 4205.87i 0.291255i 0.989339 + 0.145628i \(0.0465201\pi\)
−0.989339 + 0.145628i \(0.953480\pi\)
\(594\) 0 0
\(595\) −8355.30 3460.88i −0.575688 0.238458i
\(596\) 0 0
\(597\) 3610.39 + 8716.25i 0.247510 + 0.597541i
\(598\) 0 0
\(599\) 9771.17 + 9771.17i 0.666510 + 0.666510i 0.956906 0.290397i \(-0.0937872\pi\)
−0.290397 + 0.956906i \(0.593787\pi\)
\(600\) 0 0
\(601\) 15649.6 15649.6i 1.06216 1.06216i 0.0642295 0.997935i \(-0.479541\pi\)
0.997935 0.0642295i \(-0.0204590\pi\)
\(602\) 0 0
\(603\) −37600.0 + 15574.4i −2.53929 + 1.05181i
\(604\) 0 0
\(605\) 11529.0 27833.4i 0.774744 1.87040i
\(606\) 0 0
\(607\) 12239.3 0.818417 0.409209 0.912441i \(-0.365805\pi\)
0.409209 + 0.912441i \(0.365805\pi\)
\(608\) 0 0
\(609\) 5392.79 0.358829
\(610\) 0 0
\(611\) 1293.71 3123.29i 0.0856592 0.206800i
\(612\) 0 0
\(613\) 17070.1 7070.67i 1.12472 0.465876i 0.258739 0.965947i \(-0.416693\pi\)
0.865984 + 0.500072i \(0.166693\pi\)
\(614\) 0 0
\(615\) 1630.38 1630.38i 0.106900 0.106900i
\(616\) 0 0
\(617\) 11602.2 + 11602.2i 0.757031 + 0.757031i 0.975781 0.218750i \(-0.0701978\pi\)
−0.218750 + 0.975781i \(0.570198\pi\)
\(618\) 0 0
\(619\) −8701.45 21007.2i −0.565009 1.36405i −0.905717 0.423884i \(-0.860667\pi\)
0.340707 0.940169i \(-0.389333\pi\)
\(620\) 0 0
\(621\) −13359.7 5533.78i −0.863297 0.357589i
\(622\) 0 0
\(623\) 15549.1i 0.999938i
\(624\) 0 0
\(625\) 18073.5i 1.15670i
\(626\) 0 0
\(627\) 35215.6 + 14586.8i 2.24302 + 0.929089i
\(628\) 0 0
\(629\) 2166.56 + 5230.53i 0.137339 + 0.331566i
\(630\) 0 0
\(631\) −6552.95 6552.95i −0.413421 0.413421i 0.469508 0.882928i \(-0.344431\pi\)
−0.882928 + 0.469508i \(0.844431\pi\)
\(632\) 0 0
\(633\) 9572.83 9572.83i 0.601084 0.601084i
\(634\) 0 0
\(635\) 3846.14 1593.12i 0.240361 0.0995608i
\(636\) 0 0
\(637\) −351.896 + 849.553i −0.0218880 + 0.0528422i
\(638\) 0 0
\(639\) −36452.6 −2.25672
\(640\) 0 0
\(641\) 7331.07 0.451731 0.225866 0.974158i \(-0.427479\pi\)
0.225866 + 0.974158i \(0.427479\pi\)
\(642\) 0 0
\(643\) −6386.47 + 15418.3i −0.391692 + 0.945628i 0.597879 + 0.801586i \(0.296010\pi\)
−0.989572 + 0.144042i \(0.953990\pi\)
\(644\) 0 0
\(645\) −22149.4 + 9174.59i −1.35214 + 0.560076i
\(646\) 0 0
\(647\) 2351.92 2351.92i 0.142911 0.142911i −0.632031 0.774943i \(-0.717779\pi\)
0.774943 + 0.632031i \(0.217779\pi\)
\(648\) 0 0
\(649\) −21517.0 21517.0i −1.30141 1.30141i
\(650\) 0 0
\(651\) 9615.72 + 23214.4i 0.578909 + 1.39761i
\(652\) 0 0
\(653\) −18790.1 7783.11i −1.12605 0.466427i −0.259616 0.965712i \(-0.583596\pi\)
−0.866438 + 0.499285i \(0.833596\pi\)
\(654\) 0 0
\(655\) 14703.3i 0.877109i
\(656\) 0 0
\(657\) 4583.21i 0.272158i
\(658\) 0 0
\(659\) 438.021 + 181.434i 0.0258921 + 0.0107248i 0.395592 0.918426i \(-0.370539\pi\)
−0.369700 + 0.929151i \(0.620539\pi\)
\(660\) 0 0
\(661\) −10971.1 26486.6i −0.645577 1.55856i −0.819050 0.573722i \(-0.805499\pi\)
0.173473 0.984839i \(-0.444501\pi\)
\(662\) 0 0
\(663\) −3966.93 3966.93i −0.232372 0.232372i
\(664\) 0 0
\(665\) −10424.8 + 10424.8i −0.607904 + 0.607904i
\(666\) 0 0
\(667\) −2937.43 + 1216.72i −0.170521 + 0.0706322i
\(668\) 0 0
\(669\) 4782.48 11545.9i 0.276385 0.667251i
\(670\) 0 0
\(671\) 38984.6 2.24290
\(672\) 0 0
\(673\) 27506.8 1.57550 0.787749 0.615996i \(-0.211246\pi\)
0.787749 + 0.615996i \(0.211246\pi\)
\(674\) 0 0
\(675\) 1585.40 3827.51i 0.0904034 0.218253i
\(676\) 0 0
\(677\) −21591.8 + 8943.62i −1.22576 + 0.507727i −0.899237 0.437462i \(-0.855878\pi\)
−0.326524 + 0.945189i \(0.605878\pi\)
\(678\) 0 0
\(679\) −4603.38 + 4603.38i −0.260179 + 0.260179i
\(680\) 0 0
\(681\) 20794.1 + 20794.1i 1.17009 + 1.17009i
\(682\) 0 0
\(683\) −11875.5 28670.1i −0.665308 1.60619i −0.789368 0.613920i \(-0.789592\pi\)
0.124060 0.992275i \(-0.460408\pi\)
\(684\) 0 0
\(685\) 1753.03 + 726.130i 0.0977810 + 0.0405022i
\(686\) 0 0
\(687\) 53445.8i 2.96810i
\(688\) 0 0
\(689\) 9991.91i 0.552484i
\(690\) 0 0
\(691\) −2996.03 1241.00i −0.164941 0.0683209i 0.298685 0.954352i \(-0.403452\pi\)
−0.463626 + 0.886031i \(0.653452\pi\)
\(692\) 0 0
\(693\) 18501.7 + 44667.2i 1.01417 + 2.44843i
\(694\) 0 0
\(695\) −4505.16 4505.16i −0.245886 0.245886i
\(696\) 0 0
\(697\) −685.832 + 685.832i −0.0372708 + 0.0372708i
\(698\) 0 0
\(699\) −16806.9 + 6961.63i −0.909434 + 0.376700i
\(700\) 0 0
\(701\) −1477.30 + 3566.51i −0.0795959 + 0.192162i −0.958668 0.284528i \(-0.908163\pi\)
0.879072 + 0.476689i \(0.158163\pi\)
\(702\) 0 0
\(703\) 9229.23 0.495145
\(704\) 0 0
\(705\) −24009.7 −1.28264
\(706\) 0 0
\(707\) −4571.05 + 11035.5i −0.243157 + 0.587032i
\(708\) 0 0
\(709\) −15754.7 + 6525.83i −0.834530 + 0.345674i −0.758694 0.651447i \(-0.774162\pi\)
−0.0758356 + 0.997120i \(0.524162\pi\)
\(710\) 0 0
\(711\) −17182.5 + 17182.5i −0.906319 + 0.906319i
\(712\) 0 0
\(713\) −10475.3 10475.3i −0.550213 0.550213i
\(714\) 0 0
\(715\) 4259.50 + 10283.4i 0.222792 + 0.537868i
\(716\) 0 0
\(717\) 8253.69 + 3418.79i 0.429902 + 0.178071i
\(718\) 0 0
\(719\) 1013.44i 0.0525657i −0.999655 0.0262829i \(-0.991633\pi\)
0.999655 0.0262829i \(-0.00836706\pi\)
\(720\) 0 0
\(721\) 11081.4i 0.572388i
\(722\) 0 0
\(723\) 956.575 + 396.226i 0.0492053 + 0.0203815i
\(724\) 0 0
\(725\) −348.586 841.560i −0.0178568 0.0431100i
\(726\) 0 0
\(727\) −3353.13 3353.13i −0.171060 0.171060i 0.616385 0.787445i \(-0.288597\pi\)
−0.787445 + 0.616385i \(0.788597\pi\)
\(728\) 0 0
\(729\) −21342.8 + 21342.8i −1.08433 + 1.08433i
\(730\) 0 0
\(731\) 9317.30 3859.35i 0.471426 0.195271i
\(732\) 0 0
\(733\) 12032.3 29048.5i 0.606307 1.46375i −0.260681 0.965425i \(-0.583947\pi\)
0.866988 0.498329i \(-0.166053\pi\)
\(734\) 0 0
\(735\) 6530.79 0.327744
\(736\) 0 0
\(737\) −53551.6 −2.67652
\(738\) 0 0
\(739\) 827.863 1998.64i 0.0412090 0.0994873i −0.901935 0.431872i \(-0.857853\pi\)
0.943144 + 0.332385i \(0.107853\pi\)
\(740\) 0 0
\(741\) −8449.28 + 3499.81i −0.418883 + 0.173507i
\(742\) 0 0
\(743\) 15379.7 15379.7i 0.759391 0.759391i −0.216821 0.976211i \(-0.569569\pi\)
0.976211 + 0.216821i \(0.0695688\pi\)
\(744\) 0 0
\(745\) −20716.2 20716.2i −1.01877 1.01877i
\(746\) 0 0
\(747\) 11545.3 + 27872.8i 0.565489 + 1.36521i
\(748\) 0 0
\(749\) 9606.29 + 3979.06i 0.468633 + 0.194114i
\(750\) 0 0
\(751\) 23971.2i 1.16474i −0.812923 0.582371i \(-0.802125\pi\)
0.812923 0.582371i \(-0.197875\pi\)
\(752\) 0 0
\(753\) 54552.3i 2.64010i
\(754\) 0 0
\(755\) 3063.88 + 1269.10i 0.147690 + 0.0611753i
\(756\) 0 0
\(757\) −9946.89 24013.9i −0.477577 1.15297i −0.960742 0.277443i \(-0.910513\pi\)
0.483165 0.875529i \(-0.339487\pi\)
\(758\) 0 0
\(759\) −31777.4 31777.4i −1.51969 1.51969i
\(760\) 0 0
\(761\) 26392.4 26392.4i 1.25719 1.25719i 0.304763 0.952428i \(-0.401423\pi\)
0.952428 0.304763i \(-0.0985771\pi\)
\(762\) 0 0
\(763\) 1970.09 816.037i 0.0934757 0.0387189i
\(764\) 0 0
\(765\) −9671.13 + 23348.2i −0.457073 + 1.10347i
\(766\) 0 0
\(767\) 7300.99 0.343707
\(768\) 0 0
\(769\) −13563.7 −0.636046 −0.318023 0.948083i \(-0.603019\pi\)
−0.318023 + 0.948083i \(0.603019\pi\)
\(770\) 0 0
\(771\) 14865.3 35888.1i 0.694374 1.67637i
\(772\) 0 0
\(773\) −3266.20 + 1352.90i −0.151975 + 0.0629502i −0.457374 0.889274i \(-0.651210\pi\)
0.305399 + 0.952224i \(0.401210\pi\)
\(774\) 0 0
\(775\) 3001.12 3001.12i 0.139101 0.139101i
\(776\) 0 0
\(777\) 13050.5 + 13050.5i 0.602552 + 0.602552i
\(778\) 0 0
\(779\) 605.073 + 1460.77i 0.0278292 + 0.0671857i
\(780\) 0 0
\(781\) −44314.3 18355.6i −2.03033 0.840992i
\(782\) 0 0
\(783\) 6380.49i 0.291213i
\(784\) 0 0
\(785\) 18280.7i 0.831166i
\(786\) 0 0
\(787\) −10905.7 4517.30i −0.493960 0.204605i 0.121775 0.992558i \(-0.461141\pi\)
−0.615736 + 0.787953i \(0.711141\pi\)
\(788\) 0 0
\(789\) −2090.91 5047.90i −0.0943451 0.227769i
\(790\) 0 0
\(791\) −17063.7 17063.7i −0.767024 0.767024i
\(792\) 0 0
\(793\) −6613.98 + 6613.98i −0.296178 + 0.296178i
\(794\) 0 0
\(795\) −65562.3 + 27156.8i −2.92485 + 1.21151i
\(796\) 0 0
\(797\) −1753.95 + 4234.40i −0.0779523 + 0.188193i −0.958051 0.286597i \(-0.907476\pi\)
0.880099 + 0.474790i \(0.157476\pi\)
\(798\) 0 0
\(799\) 10099.8 0.447192
\(800\) 0 0
\(801\) −43450.6 −1.91667
\(802\) 0 0
\(803\) 2307.86 5571.66i 0.101423 0.244856i
\(804\) 0 0
\(805\) 16058.7 6651.74i 0.703100 0.291234i
\(806\) 0 0
\(807\) −16025.5 + 16025.5i −0.699039 + 0.699039i
\(808\) 0 0
\(809\) 7035.78 + 7035.78i 0.305766 + 0.305766i 0.843265 0.537499i \(-0.180631\pi\)
−0.537499 + 0.843265i \(0.680631\pi\)
\(810\) 0 0
\(811\) 12951.5 + 31267.6i 0.560773 + 1.35383i 0.909149 + 0.416472i \(0.136734\pi\)
−0.348375 + 0.937355i \(0.613266\pi\)
\(812\) 0 0
\(813\) −13194.2 5465.20i −0.569176 0.235760i
\(814\) 0 0
\(815\) 42515.7i 1.82731i
\(816\) 0 0
\(817\) 16440.3i 0.704007i
\(818\) 0 0
\(819\) −10717.0 4439.13i −0.457244 0.189397i
\(820\) 0 0
\(821\) 14509.7 + 35029.4i 0.616798 + 1.48908i 0.855402 + 0.517964i \(0.173310\pi\)
−0.238605 + 0.971117i \(0.576690\pi\)
\(822\) 0 0
\(823\) 20334.4 + 20334.4i 0.861254 + 0.861254i 0.991484 0.130230i \(-0.0415716\pi\)
−0.130230 + 0.991484i \(0.541572\pi\)
\(824\) 0 0
\(825\) 9104.08 9104.08i 0.384198 0.384198i
\(826\) 0 0
\(827\) 8362.20 3463.74i 0.351611 0.145642i −0.199885 0.979819i \(-0.564057\pi\)
0.551496 + 0.834177i \(0.314057\pi\)
\(828\) 0 0
\(829\) −10022.4 + 24196.3i −0.419895 + 1.01372i 0.562482 + 0.826809i \(0.309847\pi\)
−0.982377 + 0.186908i \(0.940153\pi\)
\(830\) 0 0
\(831\) −54953.5 −2.29400
\(832\) 0 0
\(833\) −2747.22 −0.114268
\(834\) 0 0
\(835\) 1418.86 3425.43i 0.0588044 0.141966i
\(836\) 0 0
\(837\) 27466.1 11376.8i 1.13425 0.469822i
\(838\) 0 0
\(839\) 25343.9 25343.9i 1.04287 1.04287i 0.0438328 0.999039i \(-0.486043\pi\)
0.999039 0.0438328i \(-0.0139569\pi\)
\(840\) 0 0
\(841\) 16253.6 + 16253.6i 0.666433 + 0.666433i
\(842\) 0 0
\(843\) 16492.7 + 39816.9i 0.673829 + 1.62677i
\(844\) 0 0
\(845\) 22335.7 + 9251.75i 0.909315 + 0.376651i
\(846\) 0 0
\(847\) 41313.3i 1.67597i
\(848\) 0 0
\(849\) 19872.6i 0.803328i
\(850\) 0 0
\(851\) −10053.0 4164.08i −0.404949 0.167735i
\(852\) 0 0
\(853\) −380.743 919.196i −0.0152830 0.0368964i 0.916054 0.401055i \(-0.131356\pi\)
−0.931337 + 0.364159i \(0.881356\pi\)
\(854\) 0 0
\(855\) 29131.2 + 29131.2i 1.16522 + 1.16522i
\(856\) 0 0
\(857\) 13094.2 13094.2i 0.521926 0.521926i −0.396227 0.918153i \(-0.629681\pi\)
0.918153 + 0.396227i \(0.129681\pi\)
\(858\) 0 0
\(859\) 11812.2 4892.78i 0.469182 0.194342i −0.135550 0.990770i \(-0.543280\pi\)
0.604732 + 0.796429i \(0.293280\pi\)
\(860\) 0 0
\(861\) −1209.99 + 2921.18i −0.0478937 + 0.115626i
\(862\) 0 0
\(863\) −10032.7 −0.395732 −0.197866 0.980229i \(-0.563401\pi\)
−0.197866 + 0.980229i \(0.563401\pi\)
\(864\) 0 0
\(865\) 44809.0 1.76133
\(866\) 0 0
\(867\) −9740.45 + 23515.5i −0.381549 + 0.921141i
\(868\) 0 0
\(869\) −29540.3 + 12236.0i −1.15315 + 0.477650i
\(870\) 0 0
\(871\) 9085.36 9085.36i 0.353439 0.353439i
\(872\) 0 0
\(873\) 12863.8 + 12863.8i 0.498708 + 0.498708i
\(874\) 0 0
\(875\) −7889.33 19046.5i −0.304809 0.735874i
\(876\) 0 0
\(877\) −9750.86 4038.94i −0.375442 0.155513i 0.186979 0.982364i \(-0.440130\pi\)
−0.562422 + 0.826850i \(0.690130\pi\)
\(878\) 0 0
\(879\) 70058.4i 2.68829i
\(880\) 0 0
\(881\) 16390.5i 0.626800i 0.949621 + 0.313400i \(0.101468\pi\)
−0.949621 + 0.313400i \(0.898532\pi\)
\(882\) 0 0
\(883\) 4222.47 + 1749.01i 0.160926 + 0.0666577i 0.461692 0.887040i \(-0.347242\pi\)
−0.300766 + 0.953698i \(0.597242\pi\)
\(884\) 0 0
\(885\) −19843.2 47905.8i −0.753698 1.81959i
\(886\) 0 0
\(887\) 34678.9 + 34678.9i 1.31274 + 1.31274i 0.919386 + 0.393358i \(0.128687\pi\)
0.393358 + 0.919386i \(0.371313\pi\)
\(888\) 0 0
\(889\) −4036.76 + 4036.76i −0.152293 + 0.152293i
\(890\) 0 0
\(891\) 11349.6 4701.18i 0.426742 0.176762i
\(892\) 0 0
\(893\) 6300.71 15211.3i 0.236109 0.570017i
\(894\) 0 0
\(895\) −21718.6 −0.811141
\(896\) 0 0
\(897\) 10782.5 0.401356
\(898\) 0 0
\(899\) 2501.45 6039.03i 0.0928008 0.224041i
\(900\) 0 0
\(901\) 27579.2 11423.7i 1.01975 0.422395i
\(902\) 0 0
\(903\) 23247.2 23247.2i 0.856720 0.856720i
\(904\) 0 0
\(905\) 24218.0 + 24218.0i 0.889541 + 0.889541i
\(906\) 0 0
\(907\) −6567.47 15855.3i −0.240429 0.580447i 0.756897 0.653535i \(-0.226715\pi\)
−0.997326 + 0.0730878i \(0.976715\pi\)
\(908\) 0 0
\(909\) 30837.7 + 12773.4i 1.12522 + 0.466080i
\(910\) 0 0
\(911\) 21528.6i 0.782958i 0.920187 + 0.391479i \(0.128036\pi\)
−0.920187 + 0.391479i \(0.871964\pi\)
\(912\) 0 0
\(913\) 39697.7i 1.43899i
\(914\) 0 0
\(915\) 61373.9 + 25421.9i 2.21744 + 0.918495i
\(916\) 0 0
\(917\) 7716.03 + 18628.1i 0.277869 + 0.670834i
\(918\) 0 0
\(919\) −27666.0 27666.0i −0.993054 0.993054i 0.00692164 0.999976i \(-0.497797\pi\)
−0.999976 + 0.00692164i \(0.997797\pi\)
\(920\) 0 0
\(921\) 21834.4 21834.4i 0.781181 0.781181i
\(922\) 0 0
\(923\) 10632.4 4404.06i 0.379164 0.157055i
\(924\) 0 0
\(925\) 1192.99 2880.13i 0.0424057 0.102376i
\(926\) 0 0
\(927\) 30965.9 1.09715
\(928\) 0 0
\(929\) 42072.0 1.48583 0.742915 0.669386i \(-0.233443\pi\)
0.742915 + 0.669386i \(0.233443\pi\)
\(930\) 0 0
\(931\) −1713.83 + 4137.55i −0.0603314 + 0.145653i
\(932\) 0 0
\(933\) 23981.8 9933.60i 0.841511 0.348565i
\(934\) 0 0
\(935\) −23513.8 + 23513.8i −0.822442 + 0.822442i
\(936\) 0 0
\(937\) −17029.9 17029.9i −0.593749 0.593749i 0.344893 0.938642i \(-0.387915\pi\)
−0.938642 + 0.344893i \(0.887915\pi\)
\(938\) 0 0
\(939\) −34187.5 82535.8i −1.18814 2.86843i
\(940\) 0 0
\(941\) 50185.8 + 20787.6i 1.73859 + 0.720146i 0.998885 + 0.0472119i \(0.0150336\pi\)
0.739702 + 0.672934i \(0.234966\pi\)
\(942\) 0 0
\(943\) 1864.15i 0.0643745i
\(944\) 0 0
\(945\) 34881.7i 1.20074i
\(946\) 0 0
\(947\) −4647.70 1925.14i −0.159483 0.0660599i 0.301514 0.953462i \(-0.402508\pi\)
−0.460997 + 0.887402i \(0.652508\pi\)
\(948\) 0 0
\(949\) 553.725 + 1336.81i 0.0189407 + 0.0457268i
\(950\) 0 0
\(951\) −5105.19 5105.19i −0.174077 0.174077i
\(952\) 0 0
\(953\) −444.450 + 444.450i −0.0151072 + 0.0151072i −0.714620 0.699513i \(-0.753400\pi\)
0.699513 + 0.714620i \(0.253400\pi\)
\(954\) 0 0
\(955\) 42617.2 17652.6i 1.44404 0.598142i
\(956\) 0 0
\(957\) 7588.29 18319.8i 0.256316 0.618802i
\(958\) 0 0
\(959\) −2602.04 −0.0876165
\(960\) 0 0
\(961\) 665.482 0.0223383
\(962\) 0 0
\(963\) 11119.1 26844.0i 0.372076 0.898271i
\(964\) 0 0
\(965\) 12627.7 5230.58i 0.421244 0.174485i
\(966\) 0 0
\(967\) −30115.1 + 30115.1i −1.00148 + 1.00148i −0.00148526 + 0.999999i \(0.500473\pi\)
−0.999999 + 0.00148526i \(0.999527\pi\)
\(968\) 0 0
\(969\) −19320.0 19320.0i −0.640504 0.640504i
\(970\) 0 0
\(971\) −2287.20 5521.79i −0.0755919 0.182495i 0.881566 0.472060i \(-0.156489\pi\)
−0.957158 + 0.289565i \(0.906489\pi\)
\(972\) 0 0
\(973\) 8071.96 + 3343.52i 0.265956 + 0.110163i
\(974\) 0 0
\(975\) 3089.13i 0.101468i
\(976\) 0 0
\(977\) 10157.2i 0.332608i 0.986074 + 0.166304i \(0.0531834\pi\)
−0.986074 + 0.166304i \(0.946817\pi\)
\(978\) 0 0
\(979\) −52821.5 21879.4i −1.72440 0.714268i
\(980\) 0 0
\(981\) −2280.34 5505.24i −0.0742159 0.179173i
\(982\) 0 0
\(983\) −3880.55 3880.55i −0.125911 0.125911i 0.641343 0.767254i \(-0.278377\pi\)
−0.767254 + 0.641343i \(0.778377\pi\)
\(984\) 0 0
\(985\) −37647.7 + 37647.7i −1.21782 + 1.21782i
\(986\) 0 0
\(987\) 30418.7 12599.8i 0.980991 0.406340i
\(988\) 0 0
\(989\) −7417.60 + 17907.7i −0.238489 + 0.575764i
\(990\) 0 0
\(991\) −15197.2 −0.487139 −0.243569 0.969884i \(-0.578318\pi\)
−0.243569 + 0.969884i \(0.578318\pi\)
\(992\) 0 0
\(993\) 8610.00 0.275156
\(994\) 0 0
\(995\) −5134.61 + 12396.0i −0.163596 + 0.394956i
\(996\) 0 0
\(997\) 1293.68 535.861i 0.0410946 0.0170219i −0.362041 0.932162i \(-0.617920\pi\)
0.403136 + 0.915140i \(0.367920\pi\)
\(998\) 0 0
\(999\) 15440.7 15440.7i 0.489010 0.489010i
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 256.4.g.a.33.1 44
4.3 odd 2 256.4.g.b.33.11 44
8.3 odd 2 32.4.g.a.13.10 yes 44
8.5 even 2 128.4.g.a.17.11 44
32.5 even 8 inner 256.4.g.a.225.1 44
32.11 odd 8 32.4.g.a.5.10 44
32.21 even 8 128.4.g.a.113.11 44
32.27 odd 8 256.4.g.b.225.11 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.g.a.5.10 44 32.11 odd 8
32.4.g.a.13.10 yes 44 8.3 odd 2
128.4.g.a.17.11 44 8.5 even 2
128.4.g.a.113.11 44 32.21 even 8
256.4.g.a.33.1 44 1.1 even 1 trivial
256.4.g.a.225.1 44 32.5 even 8 inner
256.4.g.b.33.11 44 4.3 odd 2
256.4.g.b.225.11 44 32.27 odd 8