Properties

Label 128.4.g.a.113.4
Level $128$
Weight $4$
Character 128.113
Analytic conductor $7.552$
Analytic rank $0$
Dimension $44$
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] = [128,4,Mod(17,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, 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("128.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.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: \(7.55224448073\)
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 113.4
Character \(\chi\) \(=\) 128.113
Dual form 128.4.g.a.17.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.64064 - 3.96085i) q^{3} +(-11.8087 - 4.89132i) q^{5} +(5.11236 + 5.11236i) q^{7} +(6.09524 - 6.09524i) q^{9} +O(q^{10})\) \(q+(-1.64064 - 3.96085i) q^{3} +(-11.8087 - 4.89132i) q^{5} +(5.11236 + 5.11236i) q^{7} +(6.09524 - 6.09524i) q^{9} +(-15.2446 + 36.8038i) q^{11} +(-73.4903 + 30.4407i) q^{13} +54.7973i q^{15} +66.8708i q^{17} +(37.0353 - 15.3405i) q^{19} +(11.8618 - 28.6368i) q^{21} +(-30.1143 + 30.1143i) q^{23} +(27.1316 + 27.1316i) q^{25} +(-141.085 - 58.4395i) q^{27} +(64.4434 + 155.580i) q^{29} -219.132 q^{31} +170.785 q^{33} +(-35.3641 - 85.3765i) q^{35} +(-286.081 - 118.499i) q^{37} +(241.142 + 241.142i) q^{39} +(64.2737 - 64.2737i) q^{41} +(200.870 - 484.942i) q^{43} +(-101.791 + 42.1630i) q^{45} +392.444i q^{47} -290.727i q^{49} +(264.865 - 109.711i) q^{51} +(107.214 - 258.838i) q^{53} +(360.038 - 360.038i) q^{55} +(-121.523 - 121.523i) q^{57} +(-237.764 - 98.4852i) q^{59} +(-43.9101 - 106.008i) q^{61} +62.3222 q^{63} +1016.72 q^{65} +(333.028 + 804.000i) q^{67} +(168.685 + 69.8717i) q^{69} +(-387.445 - 387.445i) q^{71} +(-518.132 + 518.132i) q^{73} +(62.9512 - 151.978i) q^{75} +(-266.091 + 110.218i) q^{77} -214.985i q^{79} +421.957i q^{81} +(-436.657 + 180.869i) q^{83} +(327.086 - 789.656i) q^{85} +(510.501 - 510.501i) q^{87} +(-877.926 - 877.926i) q^{89} +(-531.333 - 220.085i) q^{91} +(359.517 + 867.951i) q^{93} -512.374 q^{95} +43.7563 q^{97} +(131.408 + 317.248i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{8}\right)\) \(1\)

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
\(3\) −1.64064 3.96085i −0.315741 0.762266i −0.999471 0.0325307i \(-0.989643\pi\)
0.683730 0.729735i \(-0.260357\pi\)
\(4\) 0 0
\(5\) −11.8087 4.89132i −1.05620 0.437493i −0.214099 0.976812i \(-0.568682\pi\)
−0.842101 + 0.539319i \(0.818682\pi\)
\(6\) 0 0
\(7\) 5.11236 + 5.11236i 0.276042 + 0.276042i 0.831527 0.555485i \(-0.187467\pi\)
−0.555485 + 0.831527i \(0.687467\pi\)
\(8\) 0 0
\(9\) 6.09524 6.09524i 0.225750 0.225750i
\(10\) 0 0
\(11\) −15.2446 + 36.8038i −0.417858 + 1.00880i 0.565110 + 0.825016i \(0.308834\pi\)
−0.982967 + 0.183781i \(0.941166\pi\)
\(12\) 0 0
\(13\) −73.4903 + 30.4407i −1.56789 + 0.649440i −0.986437 0.164142i \(-0.947515\pi\)
−0.581450 + 0.813582i \(0.697515\pi\)
\(14\) 0 0
\(15\) 54.7973i 0.943240i
\(16\) 0 0
\(17\) 66.8708i 0.954033i 0.878894 + 0.477016i \(0.158282\pi\)
−0.878894 + 0.477016i \(0.841718\pi\)
\(18\) 0 0
\(19\) 37.0353 15.3405i 0.447184 0.185230i −0.147715 0.989030i \(-0.547192\pi\)
0.594899 + 0.803800i \(0.297192\pi\)
\(20\) 0 0
\(21\) 11.8618 28.6368i 0.123260 0.297575i
\(22\) 0 0
\(23\) −30.1143 + 30.1143i −0.273012 + 0.273012i −0.830311 0.557300i \(-0.811837\pi\)
0.557300 + 0.830311i \(0.311837\pi\)
\(24\) 0 0
\(25\) 27.1316 + 27.1316i 0.217053 + 0.217053i
\(26\) 0 0
\(27\) −141.085 58.4395i −1.00563 0.416544i
\(28\) 0 0
\(29\) 64.4434 + 155.580i 0.412649 + 0.996224i 0.984424 + 0.175813i \(0.0562553\pi\)
−0.571774 + 0.820411i \(0.693745\pi\)
\(30\) 0 0
\(31\) −219.132 −1.26959 −0.634796 0.772680i \(-0.718916\pi\)
−0.634796 + 0.772680i \(0.718916\pi\)
\(32\) 0 0
\(33\) 170.785 0.900907
\(34\) 0 0
\(35\) −35.3641 85.3765i −0.170789 0.412322i
\(36\) 0 0
\(37\) −286.081 118.499i −1.27112 0.526515i −0.357816 0.933792i \(-0.616478\pi\)
−0.913305 + 0.407277i \(0.866478\pi\)
\(38\) 0 0
\(39\) 241.142 + 241.142i 0.990092 + 0.990092i
\(40\) 0 0
\(41\) 64.2737 64.2737i 0.244826 0.244826i −0.574017 0.818843i \(-0.694616\pi\)
0.818843 + 0.574017i \(0.194616\pi\)
\(42\) 0 0
\(43\) 200.870 484.942i 0.712380 1.71984i 0.0184123 0.999830i \(-0.494139\pi\)
0.693967 0.720006i \(-0.255861\pi\)
\(44\) 0 0
\(45\) −101.791 + 42.1630i −0.337201 + 0.139673i
\(46\) 0 0
\(47\) 392.444i 1.21795i 0.793188 + 0.608977i \(0.208420\pi\)
−0.793188 + 0.608977i \(0.791580\pi\)
\(48\) 0 0
\(49\) 290.727i 0.847602i
\(50\) 0 0
\(51\) 264.865 109.711i 0.727227 0.301227i
\(52\) 0 0
\(53\) 107.214 258.838i 0.277868 0.670833i −0.721908 0.691989i \(-0.756735\pi\)
0.999776 + 0.0211562i \(0.00673472\pi\)
\(54\) 0 0
\(55\) 360.038 360.038i 0.882683 0.882683i
\(56\) 0 0
\(57\) −121.523 121.523i −0.282388 0.282388i
\(58\) 0 0
\(59\) −237.764 98.4852i −0.524649 0.217317i 0.104609 0.994513i \(-0.466641\pi\)
−0.629258 + 0.777197i \(0.716641\pi\)
\(60\) 0 0
\(61\) −43.9101 106.008i −0.0921658 0.222508i 0.871074 0.491153i \(-0.163424\pi\)
−0.963239 + 0.268645i \(0.913424\pi\)
\(62\) 0 0
\(63\) 62.3222 0.124633
\(64\) 0 0
\(65\) 1016.72 1.94013
\(66\) 0 0
\(67\) 333.028 + 804.000i 0.607251 + 1.46603i 0.865978 + 0.500083i \(0.166697\pi\)
−0.258727 + 0.965951i \(0.583303\pi\)
\(68\) 0 0
\(69\) 168.685 + 69.8717i 0.294309 + 0.121907i
\(70\) 0 0
\(71\) −387.445 387.445i −0.647624 0.647624i 0.304794 0.952418i \(-0.401412\pi\)
−0.952418 + 0.304794i \(0.901412\pi\)
\(72\) 0 0
\(73\) −518.132 + 518.132i −0.830723 + 0.830723i −0.987616 0.156892i \(-0.949852\pi\)
0.156892 + 0.987616i \(0.449852\pi\)
\(74\) 0 0
\(75\) 62.9512 151.978i 0.0969197 0.233985i
\(76\) 0 0
\(77\) −266.091 + 110.218i −0.393816 + 0.163124i
\(78\) 0 0
\(79\) 214.985i 0.306173i −0.988213 0.153086i \(-0.951079\pi\)
0.988213 0.153086i \(-0.0489213\pi\)
\(80\) 0 0
\(81\) 421.957i 0.578816i
\(82\) 0 0
\(83\) −436.657 + 180.869i −0.577462 + 0.239193i −0.652246 0.758007i \(-0.726173\pi\)
0.0747842 + 0.997200i \(0.476173\pi\)
\(84\) 0 0
\(85\) 327.086 789.656i 0.417382 1.00765i
\(86\) 0 0
\(87\) 510.501 510.501i 0.629097 0.629097i
\(88\) 0 0
\(89\) −877.926 877.926i −1.04562 1.04562i −0.998909 0.0467087i \(-0.985127\pi\)
−0.0467087 0.998909i \(-0.514873\pi\)
\(90\) 0 0
\(91\) −531.333 220.085i −0.612075 0.253530i
\(92\) 0 0
\(93\) 359.517 + 867.951i 0.400862 + 0.967767i
\(94\) 0 0
\(95\) −512.374 −0.553352
\(96\) 0 0
\(97\) 43.7563 0.0458019 0.0229009 0.999738i \(-0.492710\pi\)
0.0229009 + 0.999738i \(0.492710\pi\)
\(98\) 0 0
\(99\) 131.408 + 317.248i 0.133404 + 0.322067i
\(100\) 0 0
\(101\) 427.453 + 177.057i 0.421120 + 0.174434i 0.583172 0.812349i \(-0.301811\pi\)
−0.162052 + 0.986782i \(0.551811\pi\)
\(102\) 0 0
\(103\) −570.431 570.431i −0.545691 0.545691i 0.379500 0.925192i \(-0.376096\pi\)
−0.925192 + 0.379500i \(0.876096\pi\)
\(104\) 0 0
\(105\) −280.144 + 280.144i −0.260374 + 0.260374i
\(106\) 0 0
\(107\) −398.490 + 962.041i −0.360033 + 0.869196i 0.635261 + 0.772297i \(0.280892\pi\)
−0.995294 + 0.0968989i \(0.969108\pi\)
\(108\) 0 0
\(109\) −129.224 + 53.5263i −0.113554 + 0.0470357i −0.438737 0.898615i \(-0.644574\pi\)
0.325183 + 0.945651i \(0.394574\pi\)
\(110\) 0 0
\(111\) 1327.54i 1.13517i
\(112\) 0 0
\(113\) 772.521i 0.643121i 0.946889 + 0.321561i \(0.104207\pi\)
−0.946889 + 0.321561i \(0.895793\pi\)
\(114\) 0 0
\(115\) 502.909 208.312i 0.407796 0.168915i
\(116\) 0 0
\(117\) −262.398 + 633.484i −0.207339 + 0.500561i
\(118\) 0 0
\(119\) −341.868 + 341.868i −0.263353 + 0.263353i
\(120\) 0 0
\(121\) −180.963 180.963i −0.135960 0.135960i
\(122\) 0 0
\(123\) −360.028 149.129i −0.263924 0.109321i
\(124\) 0 0
\(125\) 423.735 + 1022.99i 0.303200 + 0.731990i
\(126\) 0 0
\(127\) −485.865 −0.339477 −0.169738 0.985489i \(-0.554292\pi\)
−0.169738 + 0.985489i \(0.554292\pi\)
\(128\) 0 0
\(129\) −2250.34 −1.53590
\(130\) 0 0
\(131\) 764.762 + 1846.30i 0.510058 + 1.23139i 0.943850 + 0.330375i \(0.107175\pi\)
−0.433792 + 0.901013i \(0.642825\pi\)
\(132\) 0 0
\(133\) 267.765 + 110.912i 0.174572 + 0.0723103i
\(134\) 0 0
\(135\) 1380.19 + 1380.19i 0.879908 + 0.879908i
\(136\) 0 0
\(137\) −1360.64 + 1360.64i −0.848519 + 0.848519i −0.989948 0.141430i \(-0.954830\pi\)
0.141430 + 0.989948i \(0.454830\pi\)
\(138\) 0 0
\(139\) −76.8204 + 185.461i −0.0468764 + 0.113170i −0.945583 0.325380i \(-0.894508\pi\)
0.898707 + 0.438550i \(0.144508\pi\)
\(140\) 0 0
\(141\) 1554.41 643.859i 0.928405 0.384558i
\(142\) 0 0
\(143\) 3168.78i 1.85305i
\(144\) 0 0
\(145\) 2152.41i 1.23274i
\(146\) 0 0
\(147\) −1151.53 + 476.978i −0.646098 + 0.267623i
\(148\) 0 0
\(149\) 687.916 1660.78i 0.378230 0.913128i −0.614068 0.789253i \(-0.710468\pi\)
0.992298 0.123875i \(-0.0395321\pi\)
\(150\) 0 0
\(151\) 1133.92 1133.92i 0.611107 0.611107i −0.332128 0.943235i \(-0.607766\pi\)
0.943235 + 0.332128i \(0.107766\pi\)
\(152\) 0 0
\(153\) 407.594 + 407.594i 0.215373 + 0.215373i
\(154\) 0 0
\(155\) 2587.67 + 1071.85i 1.34094 + 0.555437i
\(156\) 0 0
\(157\) −368.347 889.267i −0.187244 0.452046i 0.802183 0.597078i \(-0.203672\pi\)
−0.989427 + 0.145032i \(0.953672\pi\)
\(158\) 0 0
\(159\) −1201.12 −0.599087
\(160\) 0 0
\(161\) −307.911 −0.150725
\(162\) 0 0
\(163\) −988.667 2386.85i −0.475082 1.14695i −0.961889 0.273440i \(-0.911839\pi\)
0.486807 0.873509i \(-0.338161\pi\)
\(164\) 0 0
\(165\) −2016.75 835.365i −0.951538 0.394140i
\(166\) 0 0
\(167\) −373.791 373.791i −0.173202 0.173202i 0.615182 0.788385i \(-0.289082\pi\)
−0.788385 + 0.615182i \(0.789082\pi\)
\(168\) 0 0
\(169\) 2920.67 2920.67i 1.32939 1.32939i
\(170\) 0 0
\(171\) 132.235 319.244i 0.0591361 0.142767i
\(172\) 0 0
\(173\) −71.8748 + 29.7715i −0.0315870 + 0.0130837i −0.398421 0.917203i \(-0.630442\pi\)
0.366834 + 0.930286i \(0.380442\pi\)
\(174\) 0 0
\(175\) 277.414i 0.119831i
\(176\) 0 0
\(177\) 1103.33i 0.468538i
\(178\) 0 0
\(179\) 1660.73 687.899i 0.693459 0.287240i −0.00798155 0.999968i \(-0.502541\pi\)
0.701440 + 0.712728i \(0.252541\pi\)
\(180\) 0 0
\(181\) −757.224 + 1828.10i −0.310962 + 0.750728i 0.688708 + 0.725038i \(0.258178\pi\)
−0.999670 + 0.0256892i \(0.991822\pi\)
\(182\) 0 0
\(183\) −347.843 + 347.843i −0.140510 + 0.140510i
\(184\) 0 0
\(185\) 2798.63 + 2798.63i 1.11221 + 1.11221i
\(186\) 0 0
\(187\) −2461.10 1019.42i −0.962426 0.398650i
\(188\) 0 0
\(189\) −422.516 1020.04i −0.162611 0.392578i
\(190\) 0 0
\(191\) 3023.78 1.14551 0.572757 0.819725i \(-0.305874\pi\)
0.572757 + 0.819725i \(0.305874\pi\)
\(192\) 0 0
\(193\) 2155.34 0.803860 0.401930 0.915670i \(-0.368339\pi\)
0.401930 + 0.915670i \(0.368339\pi\)
\(194\) 0 0
\(195\) −1668.07 4027.07i −0.612578 1.47889i
\(196\) 0 0
\(197\) 2832.03 + 1173.07i 1.02423 + 0.424251i 0.830628 0.556828i \(-0.187982\pi\)
0.193606 + 0.981079i \(0.437982\pi\)
\(198\) 0 0
\(199\) −751.482 751.482i −0.267694 0.267694i 0.560476 0.828171i \(-0.310618\pi\)
−0.828171 + 0.560476i \(0.810618\pi\)
\(200\) 0 0
\(201\) 2638.15 2638.15i 0.925774 0.925774i
\(202\) 0 0
\(203\) −465.924 + 1124.84i −0.161091 + 0.388908i
\(204\) 0 0
\(205\) −1073.37 + 444.605i −0.365695 + 0.151476i
\(206\) 0 0
\(207\) 367.108i 0.123265i
\(208\) 0 0
\(209\) 1596.90i 0.528517i
\(210\) 0 0
\(211\) 469.343 194.408i 0.153132 0.0634295i −0.304801 0.952416i \(-0.598590\pi\)
0.457933 + 0.888987i \(0.348590\pi\)
\(212\) 0 0
\(213\) −898.956 + 2170.27i −0.289180 + 0.698143i
\(214\) 0 0
\(215\) −4744.01 + 4744.01i −1.50483 + 1.50483i
\(216\) 0 0
\(217\) −1120.29 1120.29i −0.350460 0.350460i
\(218\) 0 0
\(219\) 2902.31 + 1202.18i 0.895525 + 0.370939i
\(220\) 0 0
\(221\) −2035.59 4914.35i −0.619587 1.49582i
\(222\) 0 0
\(223\) 6411.24 1.92524 0.962619 0.270857i \(-0.0873072\pi\)
0.962619 + 0.270857i \(0.0873072\pi\)
\(224\) 0 0
\(225\) 330.748 0.0979993
\(226\) 0 0
\(227\) −139.729 337.336i −0.0408553 0.0986334i 0.902135 0.431455i \(-0.141999\pi\)
−0.942990 + 0.332821i \(0.891999\pi\)
\(228\) 0 0
\(229\) −2169.11 898.477i −0.625935 0.259271i 0.0470898 0.998891i \(-0.485005\pi\)
−0.673025 + 0.739620i \(0.735005\pi\)
\(230\) 0 0
\(231\) 873.117 + 873.117i 0.248688 + 0.248688i
\(232\) 0 0
\(233\) −1442.89 + 1442.89i −0.405696 + 0.405696i −0.880235 0.474539i \(-0.842615\pi\)
0.474539 + 0.880235i \(0.342615\pi\)
\(234\) 0 0
\(235\) 1919.57 4634.25i 0.532846 1.28640i
\(236\) 0 0
\(237\) −851.522 + 352.712i −0.233385 + 0.0966713i
\(238\) 0 0
\(239\) 2049.94i 0.554811i 0.960753 + 0.277406i \(0.0894746\pi\)
−0.960753 + 0.277406i \(0.910525\pi\)
\(240\) 0 0
\(241\) 3964.18i 1.05956i −0.848134 0.529782i \(-0.822274\pi\)
0.848134 0.529782i \(-0.177726\pi\)
\(242\) 0 0
\(243\) −2138.00 + 885.588i −0.564414 + 0.233788i
\(244\) 0 0
\(245\) −1422.04 + 3433.11i −0.370820 + 0.895238i
\(246\) 0 0
\(247\) −2254.76 + 2254.76i −0.580838 + 0.580838i
\(248\) 0 0
\(249\) 1432.79 + 1432.79i 0.364657 + 0.364657i
\(250\) 0 0
\(251\) 3653.50 + 1513.33i 0.918754 + 0.380560i 0.791401 0.611297i \(-0.209352\pi\)
0.127353 + 0.991858i \(0.459352\pi\)
\(252\) 0 0
\(253\) −649.240 1567.41i −0.161334 0.389494i
\(254\) 0 0
\(255\) −3664.34 −0.899882
\(256\) 0 0
\(257\) −6137.79 −1.48975 −0.744873 0.667206i \(-0.767490\pi\)
−0.744873 + 0.667206i \(0.767490\pi\)
\(258\) 0 0
\(259\) −856.743 2068.36i −0.205542 0.496223i
\(260\) 0 0
\(261\) 1341.10 + 555.500i 0.318053 + 0.131742i
\(262\) 0 0
\(263\) −1379.24 1379.24i −0.323374 0.323374i 0.526686 0.850060i \(-0.323434\pi\)
−0.850060 + 0.526686i \(0.823434\pi\)
\(264\) 0 0
\(265\) −2532.12 + 2532.12i −0.586969 + 0.586969i
\(266\) 0 0
\(267\) −2036.97 + 4917.69i −0.466894 + 1.12718i
\(268\) 0 0
\(269\) 136.201 56.4164i 0.0308711 0.0127872i −0.367194 0.930144i \(-0.619682\pi\)
0.398066 + 0.917357i \(0.369682\pi\)
\(270\) 0 0
\(271\) 1576.16i 0.353302i 0.984274 + 0.176651i \(0.0565264\pi\)
−0.984274 + 0.176651i \(0.943474\pi\)
\(272\) 0 0
\(273\) 2465.61i 0.546614i
\(274\) 0 0
\(275\) −1412.16 + 584.936i −0.309660 + 0.128265i
\(276\) 0 0
\(277\) −1137.39 + 2745.89i −0.246711 + 0.595613i −0.997921 0.0644505i \(-0.979471\pi\)
0.751210 + 0.660063i \(0.229471\pi\)
\(278\) 0 0
\(279\) −1335.67 + 1335.67i −0.286610 + 0.286610i
\(280\) 0 0
\(281\) 346.910 + 346.910i 0.0736474 + 0.0736474i 0.742971 0.669324i \(-0.233416\pi\)
−0.669324 + 0.742971i \(0.733416\pi\)
\(282\) 0 0
\(283\) −2926.90 1212.36i −0.614791 0.254655i 0.0534844 0.998569i \(-0.482967\pi\)
−0.668275 + 0.743914i \(0.732967\pi\)
\(284\) 0 0
\(285\) 840.620 + 2029.44i 0.174716 + 0.421802i
\(286\) 0 0
\(287\) 657.181 0.135164
\(288\) 0 0
\(289\) 441.294 0.0898217
\(290\) 0 0
\(291\) −71.7883 173.312i −0.0144615 0.0349132i
\(292\) 0 0
\(293\) 551.934 + 228.618i 0.110049 + 0.0455837i 0.437029 0.899448i \(-0.356031\pi\)
−0.326980 + 0.945031i \(0.606031\pi\)
\(294\) 0 0
\(295\) 2325.96 + 2325.96i 0.459060 + 0.459060i
\(296\) 0 0
\(297\) 4301.59 4301.59i 0.840416 0.840416i
\(298\) 0 0
\(299\) 1296.41 3129.81i 0.250747 0.605357i
\(300\) 0 0
\(301\) 3506.12 1452.28i 0.671393 0.278100i
\(302\) 0 0
\(303\) 1983.56i 0.376081i
\(304\) 0 0
\(305\) 1466.60i 0.275335i
\(306\) 0 0
\(307\) −3795.75 + 1572.25i −0.705651 + 0.292290i −0.706503 0.707710i \(-0.749729\pi\)
0.000852873 1.00000i \(0.499729\pi\)
\(308\) 0 0
\(309\) −1323.52 + 3195.26i −0.243665 + 0.588259i
\(310\) 0 0
\(311\) 2922.48 2922.48i 0.532857 0.532857i −0.388565 0.921421i \(-0.627029\pi\)
0.921421 + 0.388565i \(0.127029\pi\)
\(312\) 0 0
\(313\) 4410.61 + 4410.61i 0.796493 + 0.796493i 0.982541 0.186048i \(-0.0595678\pi\)
−0.186048 + 0.982541i \(0.559568\pi\)
\(314\) 0 0
\(315\) −735.943 304.838i −0.131637 0.0545259i
\(316\) 0 0
\(317\) −823.383 1987.82i −0.145886 0.352199i 0.833998 0.551767i \(-0.186046\pi\)
−0.979884 + 0.199568i \(0.936046\pi\)
\(318\) 0 0
\(319\) −6708.36 −1.17742
\(320\) 0 0
\(321\) 4464.28 0.776236
\(322\) 0 0
\(323\) 1025.83 + 2476.58i 0.176715 + 0.426628i
\(324\) 0 0
\(325\) −2819.82 1168.01i −0.481278 0.199352i
\(326\) 0 0
\(327\) 424.019 + 424.019i 0.0717074 + 0.0717074i
\(328\) 0 0
\(329\) −2006.32 + 2006.32i −0.336206 + 0.336206i
\(330\) 0 0
\(331\) −2294.33 + 5539.00i −0.380990 + 0.919791i 0.610785 + 0.791797i \(0.290854\pi\)
−0.991775 + 0.127995i \(0.959146\pi\)
\(332\) 0 0
\(333\) −2466.01 + 1021.46i −0.405816 + 0.168094i
\(334\) 0 0
\(335\) 11123.1i 1.81409i
\(336\) 0 0
\(337\) 4177.18i 0.675209i −0.941288 0.337604i \(-0.890383\pi\)
0.941288 0.337604i \(-0.109617\pi\)
\(338\) 0 0
\(339\) 3059.84 1267.43i 0.490229 0.203060i
\(340\) 0 0
\(341\) 3340.60 8064.91i 0.530509 1.28076i
\(342\) 0 0
\(343\) 3239.85 3239.85i 0.510015 0.510015i
\(344\) 0 0
\(345\) −1650.18 1650.18i −0.257516 0.257516i
\(346\) 0 0
\(347\) 6068.09 + 2513.48i 0.938767 + 0.388850i 0.798998 0.601334i \(-0.205364\pi\)
0.139769 + 0.990184i \(0.455364\pi\)
\(348\) 0 0
\(349\) 3219.82 + 7773.32i 0.493848 + 1.19225i 0.952747 + 0.303765i \(0.0982438\pi\)
−0.458899 + 0.888488i \(0.651756\pi\)
\(350\) 0 0
\(351\) 12147.3 1.84723
\(352\) 0 0
\(353\) −10007.7 −1.50895 −0.754473 0.656331i \(-0.772107\pi\)
−0.754473 + 0.656331i \(0.772107\pi\)
\(354\) 0 0
\(355\) 2680.10 + 6470.34i 0.400690 + 0.967352i
\(356\) 0 0
\(357\) 1914.97 + 793.206i 0.283896 + 0.117594i
\(358\) 0 0
\(359\) −3853.86 3853.86i −0.566572 0.566572i 0.364595 0.931166i \(-0.381208\pi\)
−0.931166 + 0.364595i \(0.881208\pi\)
\(360\) 0 0
\(361\) −3713.76 + 3713.76i −0.541443 + 0.541443i
\(362\) 0 0
\(363\) −419.873 + 1013.66i −0.0607097 + 0.146566i
\(364\) 0 0
\(365\) 8652.81 3584.11i 1.24085 0.513975i
\(366\) 0 0
\(367\) 2274.12i 0.323456i 0.986835 + 0.161728i \(0.0517067\pi\)
−0.986835 + 0.161728i \(0.948293\pi\)
\(368\) 0 0
\(369\) 783.528i 0.110539i
\(370\) 0 0
\(371\) 1871.39 775.156i 0.261881 0.108475i
\(372\) 0 0
\(373\) 4974.80 12010.2i 0.690577 1.66720i −0.0530384 0.998592i \(-0.516891\pi\)
0.743615 0.668608i \(-0.233109\pi\)
\(374\) 0 0
\(375\) 3356.70 3356.70i 0.462238 0.462238i
\(376\) 0 0
\(377\) −9471.92 9471.92i −1.29398 1.29398i
\(378\) 0 0
\(379\) −10078.6 4174.70i −1.36597 0.565805i −0.425280 0.905062i \(-0.639824\pi\)
−0.940694 + 0.339257i \(0.889824\pi\)
\(380\) 0 0
\(381\) 797.128 + 1924.44i 0.107187 + 0.258771i
\(382\) 0 0
\(383\) −14593.3 −1.94695 −0.973477 0.228784i \(-0.926525\pi\)
−0.973477 + 0.228784i \(0.926525\pi\)
\(384\) 0 0
\(385\) 3681.29 0.487314
\(386\) 0 0
\(387\) −1731.49 4180.19i −0.227433 0.549072i
\(388\) 0 0
\(389\) 5607.97 + 2322.90i 0.730939 + 0.302765i 0.716938 0.697137i \(-0.245543\pi\)
0.0140014 + 0.999902i \(0.495543\pi\)
\(390\) 0 0
\(391\) −2013.77 2013.77i −0.260462 0.260462i
\(392\) 0 0
\(393\) 6058.21 6058.21i 0.777599 0.777599i
\(394\) 0 0
\(395\) −1051.56 + 2538.69i −0.133948 + 0.323380i
\(396\) 0 0
\(397\) −1428.80 + 591.828i −0.180628 + 0.0748187i −0.471165 0.882045i \(-0.656166\pi\)
0.290536 + 0.956864i \(0.406166\pi\)
\(398\) 0 0
\(399\) 1242.54i 0.155902i
\(400\) 0 0
\(401\) 7265.44i 0.904786i 0.891819 + 0.452393i \(0.149429\pi\)
−0.891819 + 0.452393i \(0.850571\pi\)
\(402\) 0 0
\(403\) 16104.1 6670.54i 1.99058 0.824524i
\(404\) 0 0
\(405\) 2063.92 4982.75i 0.253228 0.611346i
\(406\) 0 0
\(407\) 8722.41 8722.41i 1.06229 1.06229i
\(408\) 0 0
\(409\) −7223.12 7223.12i −0.873252 0.873252i 0.119573 0.992825i \(-0.461847\pi\)
−0.992825 + 0.119573i \(0.961847\pi\)
\(410\) 0 0
\(411\) 7621.59 + 3156.97i 0.914709 + 0.378885i
\(412\) 0 0
\(413\) −712.046 1719.03i −0.0848365 0.204813i
\(414\) 0 0
\(415\) 6041.03 0.714561
\(416\) 0 0
\(417\) 860.618 0.101066
\(418\) 0 0
\(419\) 3225.87 + 7787.95i 0.376120 + 0.908034i 0.992685 + 0.120730i \(0.0385235\pi\)
−0.616566 + 0.787304i \(0.711476\pi\)
\(420\) 0 0
\(421\) 1956.30 + 810.327i 0.226471 + 0.0938074i 0.493034 0.870010i \(-0.335888\pi\)
−0.266563 + 0.963818i \(0.585888\pi\)
\(422\) 0 0
\(423\) 2392.04 + 2392.04i 0.274953 + 0.274953i
\(424\) 0 0
\(425\) −1814.31 + 1814.31i −0.207076 + 0.207076i
\(426\) 0 0
\(427\) 317.469 766.438i 0.0359799 0.0868631i
\(428\) 0 0
\(429\) −12551.1 + 5198.82i −1.41252 + 0.585085i
\(430\) 0 0
\(431\) 10185.7i 1.13835i 0.822216 + 0.569175i \(0.192737\pi\)
−0.822216 + 0.569175i \(0.807263\pi\)
\(432\) 0 0
\(433\) 4456.20i 0.494576i 0.968942 + 0.247288i \(0.0795394\pi\)
−0.968942 + 0.247288i \(0.920461\pi\)
\(434\) 0 0
\(435\) −8525.37 + 3531.32i −0.939678 + 0.389227i
\(436\) 0 0
\(437\) −653.325 + 1577.27i −0.0715166 + 0.172656i
\(438\) 0 0
\(439\) −2494.47 + 2494.47i −0.271195 + 0.271195i −0.829581 0.558386i \(-0.811421\pi\)
0.558386 + 0.829581i \(0.311421\pi\)
\(440\) 0 0
\(441\) −1772.05 1772.05i −0.191346 0.191346i
\(442\) 0 0
\(443\) −13843.2 5734.06i −1.48468 0.614974i −0.514526 0.857475i \(-0.672032\pi\)
−0.970151 + 0.242501i \(0.922032\pi\)
\(444\) 0 0
\(445\) 6072.93 + 14661.4i 0.646932 + 1.56183i
\(446\) 0 0
\(447\) −7706.71 −0.815469
\(448\) 0 0
\(449\) 6724.60 0.706801 0.353400 0.935472i \(-0.385025\pi\)
0.353400 + 0.935472i \(0.385025\pi\)
\(450\) 0 0
\(451\) 1385.69 + 3345.35i 0.144677 + 0.349282i
\(452\) 0 0
\(453\) −6351.64 2630.94i −0.658778 0.272875i
\(454\) 0 0
\(455\) 5197.83 + 5197.83i 0.535556 + 0.535556i
\(456\) 0 0
\(457\) −12569.7 + 12569.7i −1.28662 + 1.28662i −0.349795 + 0.936826i \(0.613749\pi\)
−0.936826 + 0.349795i \(0.886251\pi\)
\(458\) 0 0
\(459\) 3907.90 9434.49i 0.397396 0.959400i
\(460\) 0 0
\(461\) 5822.39 2411.71i 0.588234 0.243654i −0.0686568 0.997640i \(-0.521871\pi\)
0.656891 + 0.753986i \(0.271871\pi\)
\(462\) 0 0
\(463\) 15045.1i 1.51016i 0.655634 + 0.755079i \(0.272402\pi\)
−0.655634 + 0.755079i \(0.727598\pi\)
\(464\) 0 0
\(465\) 12007.9i 1.19753i
\(466\) 0 0
\(467\) −12797.5 + 5300.90i −1.26809 + 0.525260i −0.912383 0.409337i \(-0.865760\pi\)
−0.355708 + 0.934597i \(0.615760\pi\)
\(468\) 0 0
\(469\) −2407.78 + 5812.90i −0.237060 + 0.572313i
\(470\) 0 0
\(471\) −2917.93 + 2917.93i −0.285459 + 0.285459i
\(472\) 0 0
\(473\) 14785.5 + 14785.5i 1.43729 + 1.43729i
\(474\) 0 0
\(475\) 1421.04 + 588.616i 0.137267 + 0.0568580i
\(476\) 0 0
\(477\) −924.184 2231.18i −0.0887117 0.214169i
\(478\) 0 0
\(479\) −5991.54 −0.571525 −0.285762 0.958301i \(-0.592247\pi\)
−0.285762 + 0.958301i \(0.592247\pi\)
\(480\) 0 0
\(481\) 24631.4 2.33491
\(482\) 0 0
\(483\) 505.170 + 1219.59i 0.0475902 + 0.114893i
\(484\) 0 0
\(485\) −516.705 214.026i −0.0483760 0.0200380i
\(486\) 0 0
\(487\) −6120.86 6120.86i −0.569534 0.569534i 0.362464 0.931998i \(-0.381936\pi\)
−0.931998 + 0.362464i \(0.881936\pi\)
\(488\) 0 0
\(489\) −7831.92 + 7831.92i −0.724278 + 0.724278i
\(490\) 0 0
\(491\) 7107.17 17158.2i 0.653242 1.57707i −0.154803 0.987945i \(-0.549474\pi\)
0.808045 0.589121i \(-0.200526\pi\)
\(492\) 0 0
\(493\) −10403.8 + 4309.38i −0.950430 + 0.393681i
\(494\) 0 0
\(495\) 4389.04i 0.398531i
\(496\) 0 0
\(497\) 3961.52i 0.357543i
\(498\) 0 0
\(499\) −276.250 + 114.426i −0.0247828 + 0.0102654i −0.395040 0.918664i \(-0.629269\pi\)
0.370258 + 0.928929i \(0.379269\pi\)
\(500\) 0 0
\(501\) −867.274 + 2093.78i −0.0773392 + 0.186713i
\(502\) 0 0
\(503\) −3966.95 + 3966.95i −0.351645 + 0.351645i −0.860721 0.509076i \(-0.829987\pi\)
0.509076 + 0.860721i \(0.329987\pi\)
\(504\) 0 0
\(505\) −4181.61 4181.61i −0.368474 0.368474i
\(506\) 0 0
\(507\) −16360.1 6776.58i −1.43309 0.593606i
\(508\) 0 0
\(509\) −4067.11 9818.87i −0.354168 0.855037i −0.996096 0.0882726i \(-0.971865\pi\)
0.641928 0.766765i \(-0.278135\pi\)
\(510\) 0 0
\(511\) −5297.76 −0.458629
\(512\) 0 0
\(513\) −6121.64 −0.526856
\(514\) 0 0
\(515\) 3945.88 + 9526.19i 0.337624 + 0.815095i
\(516\) 0 0
\(517\) −14443.4 5982.67i −1.22867 0.508931i
\(518\) 0 0
\(519\) 235.841 + 235.841i 0.0199466 + 0.0199466i
\(520\) 0 0
\(521\) 9171.85 9171.85i 0.771259 0.771259i −0.207067 0.978327i \(-0.566392\pi\)
0.978327 + 0.207067i \(0.0663919\pi\)
\(522\) 0 0
\(523\) −6048.17 + 14601.6i −0.505675 + 1.22081i 0.440676 + 0.897666i \(0.354739\pi\)
−0.946351 + 0.323140i \(0.895261\pi\)
\(524\) 0 0
\(525\) 1098.79 455.135i 0.0913434 0.0378357i
\(526\) 0 0
\(527\) 14653.6i 1.21123i
\(528\) 0 0
\(529\) 10353.3i 0.850929i
\(530\) 0 0
\(531\) −2049.52 + 848.940i −0.167499 + 0.0693802i
\(532\) 0 0
\(533\) −2766.96 + 6680.03i −0.224860 + 0.542860i
\(534\) 0 0
\(535\) 9411.29 9411.29i 0.760534 0.760534i
\(536\) 0 0
\(537\) −5449.33 5449.33i −0.437907 0.437907i
\(538\) 0 0
\(539\) 10699.9 + 4432.04i 0.855059 + 0.354177i
\(540\) 0 0
\(541\) 1872.33 + 4520.21i 0.148795 + 0.359222i 0.980650 0.195771i \(-0.0627210\pi\)
−0.831855 + 0.554993i \(0.812721\pi\)
\(542\) 0 0
\(543\) 8483.17 0.670437
\(544\) 0 0
\(545\) 1787.78 0.140514
\(546\) 0 0
\(547\) 1131.56 + 2731.83i 0.0884499 + 0.213537i 0.961914 0.273351i \(-0.0881320\pi\)
−0.873464 + 0.486888i \(0.838132\pi\)
\(548\) 0 0
\(549\) −913.789 378.504i −0.0710375 0.0294247i
\(550\) 0 0
\(551\) 4773.36 + 4773.36i 0.369060 + 0.369060i
\(552\) 0 0
\(553\) 1099.08 1099.08i 0.0845165 0.0845165i
\(554\) 0 0
\(555\) 6493.41 15676.5i 0.496630 1.19897i
\(556\) 0 0
\(557\) −4125.91 + 1709.01i −0.313861 + 0.130005i −0.534053 0.845451i \(-0.679332\pi\)
0.220192 + 0.975457i \(0.429332\pi\)
\(558\) 0 0
\(559\) 41753.1i 3.15916i
\(560\) 0 0
\(561\) 11420.6i 0.859494i
\(562\) 0 0
\(563\) 13629.3 5645.44i 1.02026 0.422605i 0.191072 0.981576i \(-0.438804\pi\)
0.829187 + 0.558971i \(0.188804\pi\)
\(564\) 0 0
\(565\) 3778.65 9122.46i 0.281361 0.679265i
\(566\) 0 0
\(567\) −2157.20 + 2157.20i −0.159777 + 0.159777i
\(568\) 0 0
\(569\) −1198.17 1198.17i −0.0882777 0.0882777i 0.661589 0.749867i \(-0.269882\pi\)
−0.749867 + 0.661589i \(0.769882\pi\)
\(570\) 0 0
\(571\) −1981.86 820.913i −0.145251 0.0601649i 0.308874 0.951103i \(-0.400048\pi\)
−0.454125 + 0.890938i \(0.650048\pi\)
\(572\) 0 0
\(573\) −4960.93 11976.7i −0.361685 0.873186i
\(574\) 0 0
\(575\) −1634.10 −0.118516
\(576\) 0 0
\(577\) 1653.54 0.119303 0.0596515 0.998219i \(-0.481001\pi\)
0.0596515 + 0.998219i \(0.481001\pi\)
\(578\) 0 0
\(579\) −3536.14 8536.99i −0.253812 0.612755i
\(580\) 0 0
\(581\) −3157.02 1307.68i −0.225431 0.0933765i
\(582\) 0 0
\(583\) 7891.78 + 7891.78i 0.560625 + 0.560625i
\(584\) 0 0
\(585\) 6197.14 6197.14i 0.437983 0.437983i
\(586\) 0 0
\(587\) −5748.28 + 13877.6i −0.404186 + 0.975790i 0.582453 + 0.812865i \(0.302093\pi\)
−0.986638 + 0.162926i \(0.947907\pi\)
\(588\) 0 0
\(589\) −8115.65 + 3361.61i −0.567741 + 0.235166i
\(590\) 0 0
\(591\) 13141.8i 0.914692i
\(592\) 0 0
\(593\) 10098.5i 0.699322i −0.936876 0.349661i \(-0.886297\pi\)
0.936876 0.349661i \(-0.113703\pi\)
\(594\) 0 0
\(595\) 5709.19 2364.83i 0.393368 0.162938i
\(596\) 0 0
\(597\) −1743.60 + 4209.42i −0.119532 + 0.288576i
\(598\) 0 0
\(599\) −15005.1 + 15005.1i −1.02353 + 1.02353i −0.0238107 + 0.999716i \(0.507580\pi\)
−0.999716 + 0.0238107i \(0.992420\pi\)
\(600\) 0 0
\(601\) 11177.5 + 11177.5i 0.758632 + 0.758632i 0.976073 0.217442i \(-0.0697712\pi\)
−0.217442 + 0.976073i \(0.569771\pi\)
\(602\) 0 0
\(603\) 6930.46 + 2870.69i 0.468043 + 0.193870i
\(604\) 0 0
\(605\) 1251.79 + 3022.08i 0.0841197 + 0.203083i
\(606\) 0 0
\(607\) 16418.5 1.09787 0.548935 0.835865i \(-0.315033\pi\)
0.548935 + 0.835865i \(0.315033\pi\)
\(608\) 0 0
\(609\) 5219.73 0.347314
\(610\) 0 0
\(611\) −11946.3 28840.8i −0.790989 1.90962i
\(612\) 0 0
\(613\) −954.545 395.385i −0.0628935 0.0260513i 0.351015 0.936370i \(-0.385837\pi\)
−0.413908 + 0.910319i \(0.635837\pi\)
\(614\) 0 0
\(615\) 3522.03 + 3522.03i 0.230930 + 0.230930i
\(616\) 0 0
\(617\) 7933.23 7933.23i 0.517633 0.517633i −0.399221 0.916855i \(-0.630719\pi\)
0.916855 + 0.399221i \(0.130719\pi\)
\(618\) 0 0
\(619\) 5475.77 13219.7i 0.355557 0.858391i −0.640356 0.768078i \(-0.721213\pi\)
0.995913 0.0903128i \(-0.0287867\pi\)
\(620\) 0 0
\(621\) 6008.56 2488.83i 0.388269 0.160826i
\(622\) 0 0
\(623\) 8976.55i 0.577268i
\(624\) 0 0
\(625\) 18949.0i 1.21274i
\(626\) 0 0
\(627\) 6325.10 2619.94i 0.402871 0.166875i
\(628\) 0 0
\(629\) 7924.11 19130.5i 0.502313 1.21269i
\(630\) 0 0
\(631\) 14339.1 14339.1i 0.904647 0.904647i −0.0911870 0.995834i \(-0.529066\pi\)
0.995834 + 0.0911870i \(0.0290661\pi\)
\(632\) 0 0
\(633\) −1540.04 1540.04i −0.0967002 0.0967002i
\(634\) 0 0
\(635\) 5737.42 + 2376.52i 0.358555 + 0.148519i
\(636\) 0 0
\(637\) 8849.94 + 21365.6i 0.550467 + 1.32894i
\(638\) 0 0
\(639\) −4723.15 −0.292402
\(640\) 0 0
\(641\) −8662.28 −0.533759 −0.266879 0.963730i \(-0.585993\pi\)
−0.266879 + 0.963730i \(0.585993\pi\)
\(642\) 0 0
\(643\) −3070.12 7411.92i −0.188295 0.454584i 0.801337 0.598214i \(-0.204123\pi\)
−0.989632 + 0.143630i \(0.954123\pi\)
\(644\) 0 0
\(645\) 26573.5 + 11007.1i 1.62222 + 0.671945i
\(646\) 0 0
\(647\) −16839.6 16839.6i −1.02323 1.02323i −0.999724 0.0235089i \(-0.992516\pi\)
−0.0235089 0.999724i \(-0.507484\pi\)
\(648\) 0 0
\(649\) 7249.27 7249.27i 0.438457 0.438457i
\(650\) 0 0
\(651\) −2599.30 + 6275.26i −0.156489 + 0.377799i
\(652\) 0 0
\(653\) 5431.53 2249.81i 0.325501 0.134827i −0.213948 0.976845i \(-0.568632\pi\)
0.539449 + 0.842018i \(0.318632\pi\)
\(654\) 0 0
\(655\) 25543.0i 1.52374i
\(656\) 0 0
\(657\) 6316.28i 0.375071i
\(658\) 0 0
\(659\) −6283.92 + 2602.88i −0.371452 + 0.153860i −0.560597 0.828089i \(-0.689428\pi\)
0.189145 + 0.981949i \(0.439428\pi\)
\(660\) 0 0
\(661\) −6288.29 + 15181.3i −0.370024 + 0.893318i 0.623721 + 0.781647i \(0.285620\pi\)
−0.993745 + 0.111671i \(0.964380\pi\)
\(662\) 0 0
\(663\) −16125.4 + 16125.4i −0.944580 + 0.944580i
\(664\) 0 0
\(665\) −2619.44 2619.44i −0.152748 0.152748i
\(666\) 0 0
\(667\) −6625.86 2744.52i −0.384639 0.159323i
\(668\) 0 0
\(669\) −10518.5 25394.0i −0.607877 1.46754i
\(670\) 0 0
\(671\) 4570.91 0.262978
\(672\) 0 0
\(673\) −22750.9 −1.30309 −0.651546 0.758609i \(-0.725879\pi\)
−0.651546 + 0.758609i \(0.725879\pi\)
\(674\) 0 0
\(675\) −2242.32 5413.44i −0.127862 0.308686i
\(676\) 0 0
\(677\) 18090.7 + 7493.42i 1.02701 + 0.425400i 0.831631 0.555329i \(-0.187408\pi\)
0.195375 + 0.980729i \(0.437408\pi\)
\(678\) 0 0
\(679\) 223.698 + 223.698i 0.0126432 + 0.0126432i
\(680\) 0 0
\(681\) −1106.89 + 1106.89i −0.0622852 + 0.0622852i
\(682\) 0 0
\(683\) −4949.33 + 11948.7i −0.277278 + 0.669408i −0.999758 0.0219835i \(-0.993002\pi\)
0.722481 + 0.691391i \(0.243002\pi\)
\(684\) 0 0
\(685\) 22722.6 9412.02i 1.26743 0.524985i
\(686\) 0 0
\(687\) 10065.6i 0.558991i
\(688\) 0 0
\(689\) 22285.7i 1.23225i
\(690\) 0 0
\(691\) 8985.78 3722.03i 0.494696 0.204910i −0.121365 0.992608i \(-0.538727\pi\)
0.616062 + 0.787698i \(0.288727\pi\)
\(692\) 0 0
\(693\) −950.080 + 2293.69i −0.0520787 + 0.125729i
\(694\) 0 0
\(695\) 1814.30 1814.30i 0.0990218 0.0990218i
\(696\) 0 0
\(697\) 4298.04 + 4298.04i 0.233572 + 0.233572i
\(698\) 0 0
\(699\) 8082.35 + 3347.82i 0.437343 + 0.181153i
\(700\) 0 0
\(701\) −5041.79 12172.0i −0.271649 0.655818i 0.727905 0.685678i \(-0.240494\pi\)
−0.999554 + 0.0298593i \(0.990494\pi\)
\(702\) 0 0
\(703\) −12413.0 −0.665951
\(704\) 0 0
\(705\) −21504.9 −1.14882
\(706\) 0 0
\(707\) 1280.12 + 3090.47i 0.0680958 + 0.164398i
\(708\) 0 0
\(709\) −29815.3 12349.9i −1.57932 0.654176i −0.591014 0.806661i \(-0.701272\pi\)
−0.988306 + 0.152486i \(0.951272\pi\)
\(710\) 0 0
\(711\) −1310.38 1310.38i −0.0691185 0.0691185i
\(712\) 0 0
\(713\) 6599.03 6599.03i 0.346614 0.346614i
\(714\) 0 0
\(715\) −15499.5 + 37419.1i −0.810697 + 1.95720i
\(716\) 0 0
\(717\) 8119.52 3363.22i 0.422914 0.175177i
\(718\) 0 0
\(719\) 35519.5i 1.84236i −0.389141 0.921178i \(-0.627228\pi\)
0.389141 0.921178i \(-0.372772\pi\)
\(720\) 0 0
\(721\) 5832.50i 0.301267i
\(722\) 0 0
\(723\) −15701.5 + 6503.78i −0.807670 + 0.334548i
\(724\) 0 0
\(725\) −2472.69 + 5969.60i −0.126667 + 0.305800i
\(726\) 0 0
\(727\) 6873.52 6873.52i 0.350653 0.350653i −0.509699 0.860353i \(-0.670243\pi\)
0.860353 + 0.509699i \(0.170243\pi\)
\(728\) 0 0
\(729\) 15071.3 + 15071.3i 0.765702 + 0.765702i
\(730\) 0 0
\(731\) 32428.5 + 13432.3i 1.64078 + 0.679634i
\(732\) 0 0
\(733\) 6189.58 + 14943.0i 0.311893 + 0.752975i 0.999635 + 0.0270204i \(0.00860190\pi\)
−0.687742 + 0.725955i \(0.741398\pi\)
\(734\) 0 0
\(735\) 15931.1 0.799492
\(736\) 0 0
\(737\) −34667.2 −1.73268
\(738\) 0 0
\(739\) 8349.16 + 20156.7i 0.415601 + 1.00335i 0.983607 + 0.180325i \(0.0577148\pi\)
−0.568007 + 0.823024i \(0.692285\pi\)
\(740\) 0 0
\(741\) 12630.0 + 5231.53i 0.626148 + 0.259359i
\(742\) 0 0
\(743\) 6318.96 + 6318.96i 0.312006 + 0.312006i 0.845686 0.533680i \(-0.179191\pi\)
−0.533680 + 0.845686i \(0.679191\pi\)
\(744\) 0 0
\(745\) −16246.8 + 16246.8i −0.798974 + 0.798974i
\(746\) 0 0
\(747\) −1559.09 + 3763.97i −0.0763642 + 0.184360i
\(748\) 0 0
\(749\) −6955.53 + 2881.07i −0.339318 + 0.140550i
\(750\) 0 0
\(751\) 4475.99i 0.217485i 0.994070 + 0.108742i \(0.0346824\pi\)
−0.994070 + 0.108742i \(0.965318\pi\)
\(752\) 0 0
\(753\) 16953.8i 0.820493i
\(754\) 0 0
\(755\) −18936.5 + 7843.74i −0.912806 + 0.378097i
\(756\) 0 0
\(757\) −8674.08 + 20941.1i −0.416466 + 1.00544i 0.566897 + 0.823788i \(0.308144\pi\)
−0.983363 + 0.181650i \(0.941856\pi\)
\(758\) 0 0
\(759\) −5143.09 + 5143.09i −0.245958 + 0.245958i
\(760\) 0 0
\(761\) −6136.35 6136.35i −0.292303 0.292303i 0.545686 0.837989i \(-0.316269\pi\)
−0.837989 + 0.545686i \(0.816269\pi\)
\(762\) 0 0
\(763\) −934.286 386.994i −0.0443295 0.0183619i
\(764\) 0 0
\(765\) −2819.48 6806.82i −0.133253 0.321701i
\(766\) 0 0
\(767\) 20471.3 0.963725
\(768\) 0 0
\(769\) 13154.1 0.616838 0.308419 0.951251i \(-0.400200\pi\)
0.308419 + 0.951251i \(0.400200\pi\)
\(770\) 0 0
\(771\) 10069.9 + 24310.9i 0.470374 + 1.13558i
\(772\) 0 0
\(773\) −14475.7 5996.05i −0.673552 0.278995i 0.0195763 0.999808i \(-0.493768\pi\)
−0.693129 + 0.720814i \(0.743768\pi\)
\(774\) 0 0
\(775\) −5945.42 5945.42i −0.275569 0.275569i
\(776\) 0 0
\(777\) −6786.86 + 6786.86i −0.313356 + 0.313356i
\(778\) 0 0
\(779\) 1394.41 3366.39i 0.0641332 0.154831i
\(780\) 0 0
\(781\) 20165.9 8353.00i 0.923936 0.382707i
\(782\) 0 0
\(783\) 25716.1i 1.17371i
\(784\) 0 0
\(785\) 12302.8i 0.559369i
\(786\) 0 0
\(787\) −29783.0 + 12336.5i −1.34898 + 0.558768i −0.936010 0.351975i \(-0.885510\pi\)
−0.412975 + 0.910742i \(0.635510\pi\)
\(788\) 0 0
\(789\) −3200.12 + 7725.78i −0.144395 + 0.348599i
\(790\) 0 0
\(791\) −3949.41 + 3949.41i −0.177528 + 0.177528i
\(792\) 0 0
\(793\) 6453.93 + 6453.93i 0.289011 + 0.289011i
\(794\) 0 0
\(795\) 14183.6 + 5875.05i 0.632756 + 0.262096i
\(796\) 0 0
\(797\) −179.018 432.187i −0.00795626 0.0192081i 0.919851 0.392267i \(-0.128309\pi\)
−0.927808 + 0.373059i \(0.878309\pi\)
\(798\) 0 0
\(799\) −26243.1 −1.16197
\(800\) 0 0
\(801\) −10702.3 −0.472096
\(802\) 0 0
\(803\) −11170.5 26968.0i −0.490907 1.18516i
\(804\) 0 0
\(805\) 3636.02 + 1506.09i 0.159196 + 0.0659412i
\(806\) 0 0
\(807\) −446.914 446.914i −0.0194946 0.0194946i
\(808\) 0 0
\(809\) −1696.33 + 1696.33i −0.0737205 + 0.0737205i −0.743006 0.669285i \(-0.766600\pi\)
0.669285 + 0.743006i \(0.266600\pi\)
\(810\) 0 0
\(811\) 9036.86 21816.9i 0.391279 0.944631i −0.598383 0.801210i \(-0.704190\pi\)
0.989662 0.143421i \(-0.0458102\pi\)
\(812\) 0 0
\(813\) 6242.93 2585.91i 0.269310 0.111552i
\(814\) 0 0
\(815\) 33021.5i 1.41925i
\(816\) 0 0
\(817\) 21041.5i 0.901037i
\(818\) 0 0
\(819\) −4580.07 + 1897.13i −0.195410 + 0.0809415i
\(820\) 0 0
\(821\) −14759.0 + 35631.4i −0.627397 + 1.51467i 0.215450 + 0.976515i \(0.430878\pi\)
−0.842846 + 0.538155i \(0.819122\pi\)
\(822\) 0 0
\(823\) −26350.3 + 26350.3i −1.11606 + 1.11606i −0.123742 + 0.992314i \(0.539489\pi\)
−0.992314 + 0.123742i \(0.960511\pi\)
\(824\) 0 0
\(825\) 4633.69 + 4633.69i 0.195545 + 0.195545i
\(826\) 0 0
\(827\) −15258.6 6320.31i −0.641587 0.265754i 0.0380800 0.999275i \(-0.487876\pi\)
−0.679667 + 0.733521i \(0.737876\pi\)
\(828\) 0 0
\(829\) −12319.9 29742.8i −0.516148 1.24609i −0.940252 0.340479i \(-0.889411\pi\)
0.424104 0.905614i \(-0.360589\pi\)
\(830\) 0 0
\(831\) 12742.1 0.531912
\(832\) 0 0
\(833\) 19441.2 0.808640
\(834\) 0 0
\(835\) 2585.65 + 6242.31i 0.107162 + 0.258711i
\(836\) 0 0
\(837\) 30916.4 + 12806.0i 1.27673 + 0.528841i
\(838\) 0 0
\(839\) 17943.3 + 17943.3i 0.738347 + 0.738347i 0.972258 0.233911i \(-0.0751524\pi\)
−0.233911 + 0.972258i \(0.575152\pi\)
\(840\) 0 0
\(841\) −2806.57 + 2806.57i −0.115075 + 0.115075i
\(842\) 0 0
\(843\) 804.905 1943.21i 0.0328854 0.0793924i
\(844\) 0 0
\(845\) −48775.2 + 20203.4i −1.98570 + 0.822505i
\(846\) 0 0
\(847\) 1850.30i 0.0750614i
\(848\) 0 0
\(849\) 13582.0i 0.549039i
\(850\) 0 0
\(851\) 12183.7 5046.64i 0.490776 0.203286i
\(852\) 0 0
\(853\) 12194.4 29439.9i 0.489482 1.18171i −0.465500 0.885048i \(-0.654125\pi\)
0.954981 0.296666i \(-0.0958747\pi\)
\(854\) 0 0
\(855\) −3123.04 + 3123.04i −0.124919 + 0.124919i
\(856\) 0 0
\(857\) −26988.2 26988.2i −1.07573 1.07573i −0.996887 0.0788412i \(-0.974878\pi\)
−0.0788412 0.996887i \(-0.525122\pi\)
\(858\) 0 0
\(859\) −9332.38 3865.60i −0.370683 0.153542i 0.189562 0.981869i \(-0.439293\pi\)
−0.560245 + 0.828327i \(0.689293\pi\)
\(860\) 0 0
\(861\) −1078.20 2603.00i −0.0426769 0.103031i
\(862\) 0 0
\(863\) 42549.4 1.67833 0.839164 0.543878i \(-0.183044\pi\)
0.839164 + 0.543878i \(0.183044\pi\)
\(864\) 0 0
\(865\) 994.369 0.0390862
\(866\) 0 0
\(867\) −724.003 1747.90i −0.0283604 0.0684680i
\(868\) 0 0
\(869\) 7912.26 + 3277.36i 0.308866 + 0.127937i
\(870\) 0 0
\(871\) −48948.6 48948.6i −1.90420 1.90420i
\(872\) 0 0
\(873\) 266.705 266.705i 0.0103398 0.0103398i
\(874\) 0 0
\(875\) −3063.59 + 7396.17i −0.118364 + 0.285756i
\(876\) 0 0
\(877\) 29966.3 12412.4i 1.15381 0.477923i 0.277999 0.960581i \(-0.410329\pi\)
0.875809 + 0.482659i \(0.160329\pi\)
\(878\) 0 0
\(879\) 2561.21i 0.0982792i
\(880\) 0 0
\(881\) 14567.6i 0.557088i 0.960423 + 0.278544i \(0.0898519\pi\)
−0.960423 + 0.278544i \(0.910148\pi\)
\(882\) 0 0
\(883\) 43111.8 17857.5i 1.64306 0.680580i 0.646462 0.762946i \(-0.276248\pi\)
0.996602 + 0.0823661i \(0.0262477\pi\)
\(884\) 0 0
\(885\) 5396.72 13028.8i 0.204982 0.494870i
\(886\) 0 0
\(887\) 3731.29 3731.29i 0.141245 0.141245i −0.632949 0.774194i \(-0.718156\pi\)
0.774194 + 0.632949i \(0.218156\pi\)
\(888\) 0 0
\(889\) −2483.92 2483.92i −0.0937097 0.0937097i
\(890\) 0 0
\(891\) −15529.6 6432.58i −0.583908 0.241863i
\(892\) 0 0
\(893\) 6020.31 + 14534.3i 0.225601 + 0.544649i
\(894\) 0 0
\(895\) −22975.8 −0.858097
\(896\) 0 0
\(897\) −14523.7 −0.540614
\(898\) 0 0
\(899\) −14121.6 34092.6i −0.523896 1.26480i
\(900\) 0 0
\(901\) 17308.7 + 7169.50i 0.639996 + 0.265095i
\(902\) 0 0
\(903\) −11504.5 11504.5i −0.423973 0.423973i
\(904\) 0 0
\(905\) 17883.6 17883.6i 0.656876 0.656876i
\(906\) 0 0
\(907\) −2716.64 + 6558.56i −0.0994538 + 0.240103i −0.965773 0.259387i \(-0.916479\pi\)
0.866320 + 0.499490i \(0.166479\pi\)
\(908\) 0 0
\(909\) 3684.63 1526.22i 0.134446 0.0556894i
\(910\) 0 0
\(911\) 26795.2i 0.974496i 0.873264 + 0.487248i \(0.161999\pi\)
−0.873264 + 0.487248i \(0.838001\pi\)
\(912\) 0 0
\(913\) 18827.9i 0.682491i
\(914\) 0 0
\(915\) 5808.97 2406.15i 0.209878 0.0869345i
\(916\) 0 0
\(917\) −5529.21 + 13348.7i −0.199117 + 0.480712i
\(918\) 0 0
\(919\) 39104.1 39104.1i 1.40362 1.40362i 0.615423 0.788197i \(-0.288985\pi\)
0.788197 0.615423i \(-0.211015\pi\)
\(920\) 0 0
\(921\) 12454.9 + 12454.9i 0.445605 + 0.445605i
\(922\) 0 0
\(923\) 40267.6 + 16679.4i 1.43599 + 0.594808i
\(924\) 0 0
\(925\) −4546.79 10976.9i −0.161619 0.390183i
\(926\) 0 0
\(927\) −6953.83 −0.246379
\(928\) 0 0
\(929\) 48932.6 1.72812 0.864061 0.503386i \(-0.167913\pi\)
0.864061 + 0.503386i \(0.167913\pi\)
\(930\) 0 0
\(931\) −4459.92 10767.2i −0.157001 0.379034i
\(932\) 0 0
\(933\) −16370.2 6780.77i −0.574423 0.237934i
\(934\) 0 0
\(935\) 24076.1 + 24076.1i 0.842108 + 0.842108i
\(936\) 0 0
\(937\) 3286.41 3286.41i 0.114581 0.114581i −0.647492 0.762072i \(-0.724182\pi\)
0.762072 + 0.647492i \(0.224182\pi\)
\(938\) 0 0
\(939\) 10233.5 24706.0i 0.355654 0.858625i
\(940\) 0 0
\(941\) −12857.9 + 5325.93i −0.445438 + 0.184506i −0.594116 0.804379i \(-0.702498\pi\)
0.148678 + 0.988886i \(0.452498\pi\)
\(942\) 0 0
\(943\) 3871.12i 0.133681i
\(944\) 0 0
\(945\) 14112.0i 0.485782i
\(946\) 0 0
\(947\) −24989.2 + 10350.8i −0.857485 + 0.355182i −0.767723 0.640781i \(-0.778611\pi\)
−0.0897616 + 0.995963i \(0.528611\pi\)
\(948\) 0 0
\(949\) 22305.4 53850.0i 0.762975 1.84199i
\(950\) 0 0
\(951\) −6522.59 + 6522.59i −0.222407 + 0.222407i
\(952\) 0 0
\(953\) 16871.9 + 16871.9i 0.573488 + 0.573488i 0.933101 0.359614i \(-0.117092\pi\)
−0.359614 + 0.933101i \(0.617092\pi\)
\(954\) 0 0
\(955\) −35706.9 14790.3i −1.20989 0.501154i
\(956\) 0 0
\(957\) 11006.0 + 26570.8i 0.371759 + 0.897504i
\(958\) 0 0
\(959\) −13912.1 −0.468453
\(960\) 0 0
\(961\) 18228.0 0.611864
\(962\) 0 0
\(963\) 3434.98 + 8292.77i 0.114943 + 0.277498i
\(964\) 0 0
\(965\) −25451.8 10542.5i −0.849038 0.351683i
\(966\) 0 0
\(967\) 20339.3 + 20339.3i 0.676388 + 0.676388i 0.959181 0.282793i \(-0.0912610\pi\)
−0.282793 + 0.959181i \(0.591261\pi\)
\(968\) 0 0
\(969\) 8126.35 8126.35i 0.269408 0.269408i
\(970\) 0 0
\(971\) 9157.64 22108.5i 0.302660 0.730685i −0.697244 0.716834i \(-0.745591\pi\)
0.999904 0.0138517i \(-0.00440928\pi\)
\(972\) 0 0
\(973\) −1340.88 + 555.410i −0.0441794 + 0.0182997i
\(974\) 0 0
\(975\) 13085.1i 0.429805i
\(976\) 0 0
\(977\) 13599.6i 0.445334i −0.974895 0.222667i \(-0.928524\pi\)
0.974895 0.222667i \(-0.0714762\pi\)
\(978\) 0 0
\(979\) 45694.7 18927.4i 1.49173 0.617897i
\(980\) 0 0
\(981\) −461.395 + 1113.91i −0.0150165 + 0.0362531i
\(982\) 0 0
\(983\) −19227.9 + 19227.9i −0.623882 + 0.623882i −0.946522 0.322640i \(-0.895430\pi\)
0.322640 + 0.946522i \(0.395430\pi\)
\(984\) 0 0
\(985\) −27704.7 27704.7i −0.896189 0.896189i
\(986\) 0 0
\(987\) 11238.4 + 4655.08i 0.362433 + 0.150125i
\(988\) 0 0
\(989\) 8554.66 + 20652.8i 0.275048 + 0.664024i
\(990\) 0 0
\(991\) 10267.1 0.329106 0.164553 0.986368i \(-0.447382\pi\)
0.164553 + 0.986368i \(0.447382\pi\)
\(992\) 0 0
\(993\) 25703.3 0.821420
\(994\) 0 0
\(995\) 5198.28 + 12549.8i 0.165625 + 0.399853i
\(996\) 0 0
\(997\) −47357.4 19616.1i −1.50434 0.623117i −0.529957 0.848025i \(-0.677792\pi\)
−0.974380 + 0.224908i \(0.927792\pi\)
\(998\) 0 0
\(999\) 33436.9 + 33436.9i 1.05895 + 1.05895i
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 128.4.g.a.113.4 44
4.3 odd 2 32.4.g.a.5.11 44
8.3 odd 2 256.4.g.b.225.4 44
8.5 even 2 256.4.g.a.225.8 44
32.3 odd 8 256.4.g.b.33.4 44
32.13 even 8 inner 128.4.g.a.17.4 44
32.19 odd 8 32.4.g.a.13.11 yes 44
32.29 even 8 256.4.g.a.33.8 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.g.a.5.11 44 4.3 odd 2
32.4.g.a.13.11 yes 44 32.19 odd 8
128.4.g.a.17.4 44 32.13 even 8 inner
128.4.g.a.113.4 44 1.1 even 1 trivial
256.4.g.a.33.8 44 32.29 even 8
256.4.g.a.225.8 44 8.5 even 2
256.4.g.b.33.4 44 32.3 odd 8
256.4.g.b.225.4 44 8.3 odd 2