Properties

Label 169.4.b.f.168.2
Level $169$
Weight $4$
Character 169.168
Analytic conductor $9.971$
Analytic rank $0$
Dimension $4$
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] = [169,4,Mod(168,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.168");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\), degree \(1\), 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: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 168.2
Root \(-1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 169.168
Dual form 169.4.b.f.168.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.56155i q^{2} +8.68466 q^{3} +5.56155 q^{4} -3.56155i q^{5} -13.5616i q^{6} +27.1771i q^{7} -21.1771i q^{8} +48.4233 q^{9} +O(q^{10})\) \(q-1.56155i q^{2} +8.68466 q^{3} +5.56155 q^{4} -3.56155i q^{5} -13.5616i q^{6} +27.1771i q^{7} -21.1771i q^{8} +48.4233 q^{9} -5.56155 q^{10} -15.2614i q^{11} +48.3002 q^{12} +42.4384 q^{14} -30.9309i q^{15} +11.4233 q^{16} -44.5464 q^{17} -75.6155i q^{18} +23.9697i q^{19} -19.8078i q^{20} +236.024i q^{21} -23.8314 q^{22} -122.739 q^{23} -183.916i q^{24} +112.315 q^{25} +186.054 q^{27} +151.147i q^{28} -219.909 q^{29} -48.3002 q^{30} +27.0928i q^{31} -187.255i q^{32} -132.540i q^{33} +69.5616i q^{34} +96.7926 q^{35} +269.309 q^{36} -94.1922i q^{37} +37.4299 q^{38} -75.4233 q^{40} -160.354i q^{41} +368.563 q^{42} +151.302 q^{43} -84.8769i q^{44} -172.462i q^{45} +191.663i q^{46} -466.948i q^{47} +99.2074 q^{48} -395.594 q^{49} -175.386i q^{50} -386.870 q^{51} -120.847 q^{53} -290.533i q^{54} -54.3542 q^{55} +575.531 q^{56} +208.169i q^{57} +343.400i q^{58} +439.633i q^{59} -172.024i q^{60} -137.305 q^{61} +42.3068 q^{62} +1316.00i q^{63} -201.022 q^{64} -206.968 q^{66} +512.280i q^{67} -247.747 q^{68} -1065.94 q^{69} -151.147i q^{70} +410.719i q^{71} -1025.46i q^{72} +308.004i q^{73} -147.086 q^{74} +975.420 q^{75} +133.309i q^{76} +414.759 q^{77} -586.462 q^{79} -40.6847i q^{80} +308.386 q^{81} -250.401 q^{82} +1354.20i q^{83} +1312.66i q^{84} +158.654i q^{85} -236.266i q^{86} -1909.84 q^{87} -323.191 q^{88} -439.882i q^{89} -269.309 q^{90} -682.617 q^{92} +235.292i q^{93} -729.164 q^{94} +85.3693 q^{95} -1626.24i q^{96} -1511.27i q^{97} +617.740i q^{98} -739.006i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9} - 14 q^{10} + 86 q^{12} + 178 q^{14} - 78 q^{16} - 38 q^{17} + 284 q^{22} - 392 q^{23} + 474 q^{25} + 670 q^{27} - 88 q^{29} - 86 q^{30} + 214 q^{35} + 500 q^{36} + 628 q^{38} - 178 q^{40} + 394 q^{42} - 574 q^{43} + 570 q^{48} - 766 q^{49} - 962 q^{51} - 236 q^{53} - 36 q^{55} + 2030 q^{56} - 2116 q^{61} + 664 q^{62} - 1538 q^{64} - 1636 q^{66} - 422 q^{68} - 1592 q^{69} + 294 q^{74} + 1032 q^{75} - 1524 q^{77} - 2016 q^{79} + 244 q^{81} - 144 q^{82} - 5116 q^{87} + 2484 q^{88} - 500 q^{90} - 1576 q^{92} - 1622 q^{94} + 292 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.56155i − 0.552092i −0.961144 0.276046i \(-0.910976\pi\)
0.961144 0.276046i \(-0.0890243\pi\)
\(3\) 8.68466 1.67136 0.835682 0.549214i \(-0.185073\pi\)
0.835682 + 0.549214i \(0.185073\pi\)
\(4\) 5.56155 0.695194
\(5\) − 3.56155i − 0.318555i −0.987234 0.159277i \(-0.949084\pi\)
0.987234 0.159277i \(-0.0509165\pi\)
\(6\) − 13.5616i − 0.922747i
\(7\) 27.1771i 1.46742i 0.679460 + 0.733712i \(0.262214\pi\)
−0.679460 + 0.733712i \(0.737786\pi\)
\(8\) − 21.1771i − 0.935904i
\(9\) 48.4233 1.79346
\(10\) −5.56155 −0.175872
\(11\) − 15.2614i − 0.418316i −0.977882 0.209158i \(-0.932928\pi\)
0.977882 0.209158i \(-0.0670723\pi\)
\(12\) 48.3002 1.16192
\(13\) 0 0
\(14\) 42.4384 0.810154
\(15\) − 30.9309i − 0.532421i
\(16\) 11.4233 0.178489
\(17\) −44.5464 −0.635535 −0.317767 0.948169i \(-0.602933\pi\)
−0.317767 + 0.948169i \(0.602933\pi\)
\(18\) − 75.6155i − 0.990153i
\(19\) 23.9697i 0.289422i 0.989474 + 0.144711i \(0.0462253\pi\)
−0.989474 + 0.144711i \(0.953775\pi\)
\(20\) − 19.8078i − 0.221458i
\(21\) 236.024i 2.45260i
\(22\) −23.8314 −0.230949
\(23\) −122.739 −1.11273 −0.556365 0.830938i \(-0.687804\pi\)
−0.556365 + 0.830938i \(0.687804\pi\)
\(24\) − 183.916i − 1.56423i
\(25\) 112.315 0.898523
\(26\) 0 0
\(27\) 186.054 1.32615
\(28\) 151.147i 1.02014i
\(29\) −219.909 −1.40814 −0.704071 0.710130i \(-0.748636\pi\)
−0.704071 + 0.710130i \(0.748636\pi\)
\(30\) −48.3002 −0.293946
\(31\) 27.0928i 0.156968i 0.996915 + 0.0784840i \(0.0250080\pi\)
−0.996915 + 0.0784840i \(0.974992\pi\)
\(32\) − 187.255i − 1.03445i
\(33\) − 132.540i − 0.699158i
\(34\) 69.5616i 0.350874i
\(35\) 96.7926 0.467455
\(36\) 269.309 1.24680
\(37\) − 94.1922i − 0.418516i −0.977860 0.209258i \(-0.932895\pi\)
0.977860 0.209258i \(-0.0671049\pi\)
\(38\) 37.4299 0.159788
\(39\) 0 0
\(40\) −75.4233 −0.298137
\(41\) − 160.354i − 0.610808i −0.952223 0.305404i \(-0.901209\pi\)
0.952223 0.305404i \(-0.0987914\pi\)
\(42\) 368.563 1.35406
\(43\) 151.302 0.536589 0.268295 0.963337i \(-0.413540\pi\)
0.268295 + 0.963337i \(0.413540\pi\)
\(44\) − 84.8769i − 0.290811i
\(45\) − 172.462i − 0.571314i
\(46\) 191.663i 0.614329i
\(47\) − 466.948i − 1.44918i −0.689181 0.724589i \(-0.742030\pi\)
0.689181 0.724589i \(-0.257970\pi\)
\(48\) 99.2074 0.298320
\(49\) −395.594 −1.15333
\(50\) − 175.386i − 0.496067i
\(51\) −386.870 −1.06221
\(52\) 0 0
\(53\) −120.847 −0.313199 −0.156600 0.987662i \(-0.550053\pi\)
−0.156600 + 0.987662i \(0.550053\pi\)
\(54\) − 290.533i − 0.732158i
\(55\) −54.3542 −0.133257
\(56\) 575.531 1.37337
\(57\) 208.169i 0.483730i
\(58\) 343.400i 0.777424i
\(59\) 439.633i 0.970090i 0.874489 + 0.485045i \(0.161197\pi\)
−0.874489 + 0.485045i \(0.838803\pi\)
\(60\) − 172.024i − 0.370136i
\(61\) −137.305 −0.288198 −0.144099 0.989563i \(-0.546028\pi\)
−0.144099 + 0.989563i \(0.546028\pi\)
\(62\) 42.3068 0.0866609
\(63\) 1316.00i 2.63176i
\(64\) −201.022 −0.392621
\(65\) 0 0
\(66\) −206.968 −0.386000
\(67\) 512.280i 0.934104i 0.884230 + 0.467052i \(0.154684\pi\)
−0.884230 + 0.467052i \(0.845316\pi\)
\(68\) −247.747 −0.441820
\(69\) −1065.94 −1.85977
\(70\) − 151.147i − 0.258078i
\(71\) 410.719i 0.686526i 0.939239 + 0.343263i \(0.111532\pi\)
−0.939239 + 0.343263i \(0.888468\pi\)
\(72\) − 1025.46i − 1.67850i
\(73\) 308.004i 0.493823i 0.969038 + 0.246912i \(0.0794158\pi\)
−0.969038 + 0.246912i \(0.920584\pi\)
\(74\) −147.086 −0.231060
\(75\) 975.420 1.50176
\(76\) 133.309i 0.201205i
\(77\) 414.759 0.613847
\(78\) 0 0
\(79\) −586.462 −0.835217 −0.417608 0.908627i \(-0.637132\pi\)
−0.417608 + 0.908627i \(0.637132\pi\)
\(80\) − 40.6847i − 0.0568585i
\(81\) 308.386 0.423027
\(82\) −250.401 −0.337222
\(83\) 1354.20i 1.79088i 0.445182 + 0.895440i \(0.353139\pi\)
−0.445182 + 0.895440i \(0.646861\pi\)
\(84\) 1312.66i 1.70503i
\(85\) 158.654i 0.202453i
\(86\) − 236.266i − 0.296247i
\(87\) −1909.84 −2.35352
\(88\) −323.191 −0.391503
\(89\) − 439.882i − 0.523904i −0.965081 0.261952i \(-0.915634\pi\)
0.965081 0.261952i \(-0.0843662\pi\)
\(90\) −269.309 −0.315418
\(91\) 0 0
\(92\) −682.617 −0.773563
\(93\) 235.292i 0.262351i
\(94\) −729.164 −0.800080
\(95\) 85.3693 0.0921969
\(96\) − 1626.24i − 1.72894i
\(97\) − 1511.27i − 1.58192i −0.611869 0.790959i \(-0.709582\pi\)
0.611869 0.790959i \(-0.290418\pi\)
\(98\) 617.740i 0.636747i
\(99\) − 739.006i − 0.750231i
\(100\) 624.648 0.624648
\(101\) −336.260 −0.331278 −0.165639 0.986186i \(-0.552969\pi\)
−0.165639 + 0.986186i \(0.552969\pi\)
\(102\) 604.118i 0.586438i
\(103\) −322.712 −0.308716 −0.154358 0.988015i \(-0.549331\pi\)
−0.154358 + 0.988015i \(0.549331\pi\)
\(104\) 0 0
\(105\) 840.611 0.781288
\(106\) 188.708i 0.172915i
\(107\) 1434.62 1.29617 0.648083 0.761570i \(-0.275571\pi\)
0.648083 + 0.761570i \(0.275571\pi\)
\(108\) 1034.75 0.921933
\(109\) 849.147i 0.746179i 0.927795 + 0.373089i \(0.121702\pi\)
−0.927795 + 0.373089i \(0.878298\pi\)
\(110\) 84.8769i 0.0735699i
\(111\) − 818.027i − 0.699493i
\(112\) 310.452i 0.261919i
\(113\) 1614.53 1.34409 0.672044 0.740511i \(-0.265417\pi\)
0.672044 + 0.740511i \(0.265417\pi\)
\(114\) 325.066 0.267064
\(115\) 437.140i 0.354465i
\(116\) −1223.04 −0.978931
\(117\) 0 0
\(118\) 686.509 0.535579
\(119\) − 1210.64i − 0.932599i
\(120\) −655.026 −0.498295
\(121\) 1098.09 0.825012
\(122\) 214.409i 0.159112i
\(123\) − 1392.62i − 1.02088i
\(124\) 150.678i 0.109123i
\(125\) − 845.211i − 0.604784i
\(126\) 2055.01 1.45297
\(127\) −865.174 −0.604502 −0.302251 0.953228i \(-0.597738\pi\)
−0.302251 + 0.953228i \(0.597738\pi\)
\(128\) − 1184.13i − 0.817683i
\(129\) 1314.01 0.896836
\(130\) 0 0
\(131\) −281.400 −0.187680 −0.0938400 0.995587i \(-0.529914\pi\)
−0.0938400 + 0.995587i \(0.529914\pi\)
\(132\) − 737.127i − 0.486050i
\(133\) −651.426 −0.424705
\(134\) 799.953 0.515712
\(135\) − 662.641i − 0.422452i
\(136\) 943.363i 0.594799i
\(137\) 2641.43i 1.64725i 0.567137 + 0.823624i \(0.308051\pi\)
−0.567137 + 0.823624i \(0.691949\pi\)
\(138\) 1664.53i 1.02677i
\(139\) −1998.64 −1.21958 −0.609791 0.792562i \(-0.708747\pi\)
−0.609791 + 0.792562i \(0.708747\pi\)
\(140\) 538.317 0.324972
\(141\) − 4055.28i − 2.42210i
\(142\) 641.359 0.379026
\(143\) 0 0
\(144\) 553.153 0.320112
\(145\) 783.218i 0.448570i
\(146\) 480.964 0.272636
\(147\) −3435.60 −1.92764
\(148\) − 523.855i − 0.290950i
\(149\) − 1752.98i − 0.963824i −0.876220 0.481912i \(-0.839942\pi\)
0.876220 0.481912i \(-0.160058\pi\)
\(150\) − 1523.17i − 0.829109i
\(151\) 2794.64i 1.50613i 0.657949 + 0.753063i \(0.271424\pi\)
−0.657949 + 0.753063i \(0.728576\pi\)
\(152\) 507.608 0.270871
\(153\) −2157.08 −1.13980
\(154\) − 647.669i − 0.338900i
\(155\) 96.4924 0.0500030
\(156\) 0 0
\(157\) 3244.87 1.64949 0.824743 0.565508i \(-0.191320\pi\)
0.824743 + 0.565508i \(0.191320\pi\)
\(158\) 915.792i 0.461117i
\(159\) −1049.51 −0.523470
\(160\) −666.918 −0.329528
\(161\) − 3335.68i − 1.63285i
\(162\) − 481.562i − 0.233550i
\(163\) − 3281.47i − 1.57684i −0.615139 0.788418i \(-0.710900\pi\)
0.615139 0.788418i \(-0.289100\pi\)
\(164\) − 891.818i − 0.424630i
\(165\) −472.047 −0.222720
\(166\) 2114.66 0.988731
\(167\) 3126.52i 1.44873i 0.689418 + 0.724364i \(0.257866\pi\)
−0.689418 + 0.724364i \(0.742134\pi\)
\(168\) 4998.29 2.29540
\(169\) 0 0
\(170\) 247.747 0.111773
\(171\) 1160.69i 0.519066i
\(172\) 841.474 0.373034
\(173\) −97.5698 −0.0428792 −0.0214396 0.999770i \(-0.506825\pi\)
−0.0214396 + 0.999770i \(0.506825\pi\)
\(174\) 2982.31i 1.29936i
\(175\) 3052.40i 1.31851i
\(176\) − 174.335i − 0.0746648i
\(177\) 3818.06i 1.62137i
\(178\) −686.900 −0.289243
\(179\) 34.7150 0.0144956 0.00724782 0.999974i \(-0.497693\pi\)
0.00724782 + 0.999974i \(0.497693\pi\)
\(180\) − 959.157i − 0.397174i
\(181\) 1229.35 0.504843 0.252422 0.967617i \(-0.418773\pi\)
0.252422 + 0.967617i \(0.418773\pi\)
\(182\) 0 0
\(183\) −1192.45 −0.481684
\(184\) 2599.25i 1.04141i
\(185\) −335.471 −0.133320
\(186\) 367.420 0.144842
\(187\) 679.839i 0.265854i
\(188\) − 2596.96i − 1.00746i
\(189\) 5056.40i 1.94603i
\(190\) − 133.309i − 0.0509012i
\(191\) 4280.80 1.62172 0.810858 0.585243i \(-0.199001\pi\)
0.810858 + 0.585243i \(0.199001\pi\)
\(192\) −1745.81 −0.656212
\(193\) − 472.320i − 0.176157i −0.996114 0.0880786i \(-0.971927\pi\)
0.996114 0.0880786i \(-0.0280727\pi\)
\(194\) −2359.93 −0.873365
\(195\) 0 0
\(196\) −2200.12 −0.801791
\(197\) − 4484.37i − 1.62182i −0.585173 0.810908i \(-0.698974\pi\)
0.585173 0.810908i \(-0.301026\pi\)
\(198\) −1154.00 −0.414197
\(199\) 366.240 0.130463 0.0652314 0.997870i \(-0.479221\pi\)
0.0652314 + 0.997870i \(0.479221\pi\)
\(200\) − 2378.51i − 0.840931i
\(201\) 4448.98i 1.56123i
\(202\) 525.087i 0.182896i
\(203\) − 5976.49i − 2.06634i
\(204\) −2151.60 −0.738442
\(205\) −571.110 −0.194576
\(206\) 503.932i 0.170440i
\(207\) −5943.41 −1.99563
\(208\) 0 0
\(209\) 365.810 0.121070
\(210\) − 1312.66i − 0.431343i
\(211\) 2122.55 0.692524 0.346262 0.938138i \(-0.387451\pi\)
0.346262 + 0.938138i \(0.387451\pi\)
\(212\) −672.095 −0.217734
\(213\) 3566.95i 1.14743i
\(214\) − 2240.23i − 0.715603i
\(215\) − 538.870i − 0.170933i
\(216\) − 3940.08i − 1.24115i
\(217\) −736.303 −0.230339
\(218\) 1325.99 0.411960
\(219\) 2674.91i 0.825358i
\(220\) −302.294 −0.0926392
\(221\) 0 0
\(222\) −1277.39 −0.386185
\(223\) − 5926.42i − 1.77965i −0.456301 0.889826i \(-0.650826\pi\)
0.456301 0.889826i \(-0.349174\pi\)
\(224\) 5089.04 1.51797
\(225\) 5438.68 1.61146
\(226\) − 2521.17i − 0.742060i
\(227\) − 895.661i − 0.261881i −0.991390 0.130941i \(-0.958200\pi\)
0.991390 0.130941i \(-0.0417998\pi\)
\(228\) 1157.74i 0.336286i
\(229\) − 627.717i − 0.181138i −0.995890 0.0905692i \(-0.971131\pi\)
0.995890 0.0905692i \(-0.0288686\pi\)
\(230\) 682.617 0.195698
\(231\) 3602.04 1.02596
\(232\) 4657.03i 1.31788i
\(233\) −2303.72 −0.647734 −0.323867 0.946103i \(-0.604983\pi\)
−0.323867 + 0.946103i \(0.604983\pi\)
\(234\) 0 0
\(235\) −1663.06 −0.461643
\(236\) 2445.04i 0.674401i
\(237\) −5093.22 −1.39595
\(238\) −1890.48 −0.514881
\(239\) 544.622i 0.147400i 0.997280 + 0.0737001i \(0.0234808\pi\)
−0.997280 + 0.0737001i \(0.976519\pi\)
\(240\) − 353.332i − 0.0950313i
\(241\) − 5426.10i − 1.45031i −0.688584 0.725157i \(-0.741767\pi\)
0.688584 0.725157i \(-0.258233\pi\)
\(242\) − 1714.73i − 0.455483i
\(243\) −2345.23 −0.619121
\(244\) −763.629 −0.200354
\(245\) 1408.93i 0.367400i
\(246\) −2174.65 −0.563621
\(247\) 0 0
\(248\) 573.746 0.146907
\(249\) 11760.8i 2.99321i
\(250\) −1319.84 −0.333897
\(251\) 5221.22 1.31299 0.656494 0.754331i \(-0.272039\pi\)
0.656494 + 0.754331i \(0.272039\pi\)
\(252\) 7319.02i 1.82958i
\(253\) 1873.16i 0.465472i
\(254\) 1351.02i 0.333741i
\(255\) 1377.86i 0.338372i
\(256\) −3457.26 −0.844057
\(257\) −658.206 −0.159758 −0.0798789 0.996805i \(-0.525453\pi\)
−0.0798789 + 0.996805i \(0.525453\pi\)
\(258\) − 2051.89i − 0.495136i
\(259\) 2559.87 0.614141
\(260\) 0 0
\(261\) −10648.7 −2.52544
\(262\) 439.422i 0.103617i
\(263\) 3246.45 0.761160 0.380580 0.924748i \(-0.375724\pi\)
0.380580 + 0.924748i \(0.375724\pi\)
\(264\) −2806.81 −0.654344
\(265\) 430.401i 0.0997711i
\(266\) 1017.24i 0.234477i
\(267\) − 3820.23i − 0.875634i
\(268\) 2849.07i 0.649384i
\(269\) −2585.80 −0.586093 −0.293047 0.956098i \(-0.594669\pi\)
−0.293047 + 0.956098i \(0.594669\pi\)
\(270\) −1034.75 −0.233233
\(271\) − 988.933i − 0.221673i −0.993839 0.110836i \(-0.964647\pi\)
0.993839 0.110836i \(-0.0353530\pi\)
\(272\) −508.867 −0.113436
\(273\) 0 0
\(274\) 4124.74 0.909433
\(275\) − 1714.09i − 0.375866i
\(276\) −5928.30 −1.29290
\(277\) −8142.40 −1.76617 −0.883086 0.469211i \(-0.844538\pi\)
−0.883086 + 0.469211i \(0.844538\pi\)
\(278\) 3120.97i 0.673322i
\(279\) 1311.92i 0.281515i
\(280\) − 2049.78i − 0.437493i
\(281\) − 1534.21i − 0.325705i −0.986650 0.162853i \(-0.947930\pi\)
0.986650 0.162853i \(-0.0520695\pi\)
\(282\) −6332.54 −1.33722
\(283\) 6965.00 1.46299 0.731495 0.681847i \(-0.238823\pi\)
0.731495 + 0.681847i \(0.238823\pi\)
\(284\) 2284.23i 0.477269i
\(285\) 741.403 0.154095
\(286\) 0 0
\(287\) 4357.96 0.896314
\(288\) − 9067.49i − 1.85523i
\(289\) −2928.62 −0.596096
\(290\) 1223.04 0.247652
\(291\) − 13124.9i − 2.64396i
\(292\) 1712.98i 0.343303i
\(293\) − 640.029i − 0.127614i −0.997962 0.0638070i \(-0.979676\pi\)
0.997962 0.0638070i \(-0.0203242\pi\)
\(294\) 5364.87i 1.06424i
\(295\) 1565.77 0.309027
\(296\) −1994.72 −0.391691
\(297\) − 2839.44i − 0.554750i
\(298\) −2737.37 −0.532120
\(299\) 0 0
\(300\) 5424.85 1.04401
\(301\) 4111.95i 0.787404i
\(302\) 4363.99 0.831520
\(303\) −2920.30 −0.553686
\(304\) 273.813i 0.0516587i
\(305\) 489.019i 0.0918070i
\(306\) 3368.40i 0.629276i
\(307\) 100.406i 0.0186660i 0.999956 + 0.00933299i \(0.00297083\pi\)
−0.999956 + 0.00933299i \(0.997029\pi\)
\(308\) 2306.71 0.426743
\(309\) −2802.64 −0.515977
\(310\) − 150.678i − 0.0276062i
\(311\) 3878.92 0.707245 0.353623 0.935388i \(-0.384950\pi\)
0.353623 + 0.935388i \(0.384950\pi\)
\(312\) 0 0
\(313\) −3789.39 −0.684311 −0.342155 0.939643i \(-0.611157\pi\)
−0.342155 + 0.939643i \(0.611157\pi\)
\(314\) − 5067.04i − 0.910668i
\(315\) 4687.02 0.838360
\(316\) −3261.64 −0.580638
\(317\) 4406.81i 0.780791i 0.920647 + 0.390396i \(0.127662\pi\)
−0.920647 + 0.390396i \(0.872338\pi\)
\(318\) 1638.87i 0.289004i
\(319\) 3356.11i 0.589048i
\(320\) 715.950i 0.125071i
\(321\) 12459.2 2.16636
\(322\) −5208.84 −0.901482
\(323\) − 1067.76i − 0.183938i
\(324\) 1715.11 0.294086
\(325\) 0 0
\(326\) −5124.19 −0.870559
\(327\) 7374.55i 1.24714i
\(328\) −3395.83 −0.571657
\(329\) 12690.3 2.12656
\(330\) 737.127i 0.122962i
\(331\) − 4131.49i − 0.686064i −0.939324 0.343032i \(-0.888546\pi\)
0.939324 0.343032i \(-0.111454\pi\)
\(332\) 7531.47i 1.24501i
\(333\) − 4561.10i − 0.750591i
\(334\) 4882.23 0.799831
\(335\) 1824.51 0.297564
\(336\) 2696.17i 0.437762i
\(337\) 4560.82 0.737221 0.368611 0.929584i \(-0.379834\pi\)
0.368611 + 0.929584i \(0.379834\pi\)
\(338\) 0 0
\(339\) 14021.6 2.24646
\(340\) 882.365i 0.140744i
\(341\) 413.473 0.0656622
\(342\) 1812.48 0.286572
\(343\) − 1429.34i − 0.225007i
\(344\) − 3204.14i − 0.502196i
\(345\) 3796.41i 0.592441i
\(346\) 152.360i 0.0236733i
\(347\) 10069.4 1.55779 0.778896 0.627153i \(-0.215780\pi\)
0.778896 + 0.627153i \(0.215780\pi\)
\(348\) −10621.6 −1.63615
\(349\) − 5879.32i − 0.901757i −0.892585 0.450878i \(-0.851111\pi\)
0.892585 0.450878i \(-0.148889\pi\)
\(350\) 4766.49 0.727942
\(351\) 0 0
\(352\) −2857.76 −0.432725
\(353\) − 9142.56i − 1.37850i −0.724525 0.689249i \(-0.757941\pi\)
0.724525 0.689249i \(-0.242059\pi\)
\(354\) 5962.10 0.895147
\(355\) 1462.80 0.218696
\(356\) − 2446.43i − 0.364215i
\(357\) − 10514.0i − 1.55871i
\(358\) − 54.2093i − 0.00800293i
\(359\) 2754.32i 0.404924i 0.979290 + 0.202462i \(0.0648942\pi\)
−0.979290 + 0.202462i \(0.935106\pi\)
\(360\) −3652.24 −0.534695
\(361\) 6284.45 0.916235
\(362\) − 1919.69i − 0.278720i
\(363\) 9536.54 1.37889
\(364\) 0 0
\(365\) 1096.97 0.157310
\(366\) 1862.07i 0.265934i
\(367\) 3040.19 0.432416 0.216208 0.976347i \(-0.430631\pi\)
0.216208 + 0.976347i \(0.430631\pi\)
\(368\) −1402.08 −0.198610
\(369\) − 7764.88i − 1.09546i
\(370\) 523.855i 0.0736052i
\(371\) − 3284.26i − 0.459596i
\(372\) 1308.59i 0.182385i
\(373\) −5384.72 −0.747481 −0.373740 0.927533i \(-0.621925\pi\)
−0.373740 + 0.927533i \(0.621925\pi\)
\(374\) 1061.60 0.146776
\(375\) − 7340.37i − 1.01081i
\(376\) −9888.59 −1.35629
\(377\) 0 0
\(378\) 7895.84 1.07439
\(379\) − 3424.27i − 0.464097i −0.972704 0.232049i \(-0.925457\pi\)
0.972704 0.232049i \(-0.0745428\pi\)
\(380\) 474.786 0.0640948
\(381\) −7513.74 −1.01034
\(382\) − 6684.69i − 0.895336i
\(383\) − 382.985i − 0.0510956i −0.999674 0.0255478i \(-0.991867\pi\)
0.999674 0.0255478i \(-0.00813301\pi\)
\(384\) − 10283.8i − 1.36665i
\(385\) − 1477.19i − 0.195544i
\(386\) −737.553 −0.0972551
\(387\) 7326.54 0.962349
\(388\) − 8405.00i − 1.09974i
\(389\) −8588.34 −1.11940 −0.559699 0.828696i \(-0.689083\pi\)
−0.559699 + 0.828696i \(0.689083\pi\)
\(390\) 0 0
\(391\) 5467.56 0.707178
\(392\) 8377.52i 1.07941i
\(393\) −2443.87 −0.313681
\(394\) −7002.57 −0.895392
\(395\) 2088.72i 0.266063i
\(396\) − 4110.02i − 0.521556i
\(397\) 7239.16i 0.915171i 0.889166 + 0.457586i \(0.151286\pi\)
−0.889166 + 0.457586i \(0.848714\pi\)
\(398\) − 571.904i − 0.0720275i
\(399\) −5657.41 −0.709837
\(400\) 1283.01 0.160376
\(401\) − 4269.62i − 0.531708i −0.964013 0.265854i \(-0.914346\pi\)
0.964013 0.265854i \(-0.0856539\pi\)
\(402\) 6947.32 0.861942
\(403\) 0 0
\(404\) −1870.12 −0.230302
\(405\) − 1098.33i − 0.134757i
\(406\) −9332.60 −1.14081
\(407\) −1437.50 −0.175072
\(408\) 8192.78i 0.994125i
\(409\) 13562.5i 1.63967i 0.572602 + 0.819834i \(0.305934\pi\)
−0.572602 + 0.819834i \(0.694066\pi\)
\(410\) 891.818i 0.107424i
\(411\) 22939.9i 2.75315i
\(412\) −1794.78 −0.214618
\(413\) −11947.9 −1.42353
\(414\) 9280.95i 1.10177i
\(415\) 4823.06 0.570494
\(416\) 0 0
\(417\) −17357.5 −2.03837
\(418\) − 571.232i − 0.0668418i
\(419\) −14576.9 −1.69959 −0.849794 0.527114i \(-0.823274\pi\)
−0.849794 + 0.527114i \(0.823274\pi\)
\(420\) 4675.10 0.543147
\(421\) 15848.4i 1.83469i 0.398099 + 0.917343i \(0.369670\pi\)
−0.398099 + 0.917343i \(0.630330\pi\)
\(422\) − 3314.48i − 0.382337i
\(423\) − 22611.2i − 2.59904i
\(424\) 2559.18i 0.293124i
\(425\) −5003.24 −0.571042
\(426\) 5569.98 0.633490
\(427\) − 3731.55i − 0.422909i
\(428\) 7978.70 0.901087
\(429\) 0 0
\(430\) −841.474 −0.0943709
\(431\) 10694.7i 1.19524i 0.801781 + 0.597618i \(0.203886\pi\)
−0.801781 + 0.597618i \(0.796114\pi\)
\(432\) 2125.35 0.236703
\(433\) 16079.0 1.78454 0.892272 0.451498i \(-0.149110\pi\)
0.892272 + 0.451498i \(0.149110\pi\)
\(434\) 1149.78i 0.127168i
\(435\) 6801.98i 0.749724i
\(436\) 4722.57i 0.518739i
\(437\) − 2942.01i − 0.322049i
\(438\) 4177.01 0.455674
\(439\) −6035.80 −0.656203 −0.328101 0.944643i \(-0.606409\pi\)
−0.328101 + 0.944643i \(0.606409\pi\)
\(440\) 1151.06i 0.124715i
\(441\) −19156.0 −2.06845
\(442\) 0 0
\(443\) 10201.3 1.09409 0.547043 0.837105i \(-0.315753\pi\)
0.547043 + 0.837105i \(0.315753\pi\)
\(444\) − 4549.50i − 0.486283i
\(445\) −1566.66 −0.166892
\(446\) −9254.41 −0.982532
\(447\) − 15224.0i − 1.61090i
\(448\) − 5463.19i − 0.576141i
\(449\) 5822.54i 0.611988i 0.952033 + 0.305994i \(0.0989888\pi\)
−0.952033 + 0.305994i \(0.901011\pi\)
\(450\) − 8492.78i − 0.889675i
\(451\) −2447.22 −0.255511
\(452\) 8979.27 0.934402
\(453\) 24270.5i 2.51728i
\(454\) −1398.62 −0.144583
\(455\) 0 0
\(456\) 4408.40 0.452724
\(457\) 4621.60i 0.473062i 0.971624 + 0.236531i \(0.0760105\pi\)
−0.971624 + 0.236531i \(0.923990\pi\)
\(458\) −980.213 −0.100005
\(459\) −8288.03 −0.842816
\(460\) 2431.18i 0.246422i
\(461\) 5127.77i 0.518056i 0.965870 + 0.259028i \(0.0834023\pi\)
−0.965870 + 0.259028i \(0.916598\pi\)
\(462\) − 5624.78i − 0.566425i
\(463\) − 6486.27i − 0.651064i −0.945531 0.325532i \(-0.894457\pi\)
0.945531 0.325532i \(-0.105543\pi\)
\(464\) −2512.09 −0.251338
\(465\) 838.004 0.0835731
\(466\) 3597.39i 0.357609i
\(467\) −12978.0 −1.28598 −0.642990 0.765875i \(-0.722306\pi\)
−0.642990 + 0.765875i \(0.722306\pi\)
\(468\) 0 0
\(469\) −13922.3 −1.37073
\(470\) 2596.96i 0.254869i
\(471\) 28180.6 2.75689
\(472\) 9310.13 0.907910
\(473\) − 2309.08i − 0.224464i
\(474\) 7953.34i 0.770694i
\(475\) 2692.16i 0.260053i
\(476\) − 6733.04i − 0.648337i
\(477\) −5851.79 −0.561709
\(478\) 850.456 0.0813786
\(479\) 5808.96i 0.554109i 0.960854 + 0.277055i \(0.0893583\pi\)
−0.960854 + 0.277055i \(0.910642\pi\)
\(480\) −5791.95 −0.550761
\(481\) 0 0
\(482\) −8473.14 −0.800707
\(483\) − 28969.2i − 2.72908i
\(484\) 6107.09 0.573543
\(485\) −5382.46 −0.503928
\(486\) 3662.20i 0.341812i
\(487\) − 5387.14i − 0.501262i −0.968083 0.250631i \(-0.919362\pi\)
0.968083 0.250631i \(-0.0806381\pi\)
\(488\) 2907.72i 0.269726i
\(489\) − 28498.4i − 2.63547i
\(490\) 2200.12 0.202839
\(491\) −15259.1 −1.40251 −0.701255 0.712911i \(-0.747376\pi\)
−0.701255 + 0.712911i \(0.747376\pi\)
\(492\) − 7745.14i − 0.709711i
\(493\) 9796.16 0.894922
\(494\) 0 0
\(495\) −2632.01 −0.238990
\(496\) 309.489i 0.0280171i
\(497\) −11162.1 −1.00742
\(498\) 18365.1 1.65253
\(499\) 1856.04i 0.166509i 0.996528 + 0.0832544i \(0.0265314\pi\)
−0.996528 + 0.0832544i \(0.973469\pi\)
\(500\) − 4700.69i − 0.420442i
\(501\) 27152.8i 2.42135i
\(502\) − 8153.20i − 0.724891i
\(503\) 1049.46 0.0930283 0.0465142 0.998918i \(-0.485189\pi\)
0.0465142 + 0.998918i \(0.485189\pi\)
\(504\) 27869.1 2.46307
\(505\) 1197.61i 0.105530i
\(506\) 2925.04 0.256984
\(507\) 0 0
\(508\) −4811.71 −0.420246
\(509\) − 551.106i − 0.0479909i −0.999712 0.0239954i \(-0.992361\pi\)
0.999712 0.0239954i \(-0.00763872\pi\)
\(510\) 2151.60 0.186813
\(511\) −8370.64 −0.724649
\(512\) − 4074.36i − 0.351686i
\(513\) 4459.66i 0.383818i
\(514\) 1027.82i 0.0882010i
\(515\) 1149.36i 0.0983431i
\(516\) 7307.92 0.623475
\(517\) −7126.26 −0.606214
\(518\) − 3997.37i − 0.339063i
\(519\) −847.361 −0.0716667
\(520\) 0 0
\(521\) −8995.30 −0.756413 −0.378206 0.925721i \(-0.623459\pi\)
−0.378206 + 0.925721i \(0.623459\pi\)
\(522\) 16628.5i 1.39427i
\(523\) 2663.91 0.222724 0.111362 0.993780i \(-0.464479\pi\)
0.111362 + 0.993780i \(0.464479\pi\)
\(524\) −1565.02 −0.130474
\(525\) 26509.1i 2.20372i
\(526\) − 5069.51i − 0.420230i
\(527\) − 1206.89i − 0.0997586i
\(528\) − 1514.04i − 0.124792i
\(529\) 2897.77 0.238167
\(530\) 672.095 0.0550829
\(531\) 21288.5i 1.73981i
\(532\) −3622.94 −0.295253
\(533\) 0 0
\(534\) −5965.49 −0.483431
\(535\) − 5109.47i − 0.412900i
\(536\) 10848.6 0.874232
\(537\) 301.488 0.0242275
\(538\) 4037.86i 0.323577i
\(539\) 6037.30i 0.482458i
\(540\) − 3685.31i − 0.293686i
\(541\) 6169.23i 0.490270i 0.969489 + 0.245135i \(0.0788322\pi\)
−0.969489 + 0.245135i \(0.921168\pi\)
\(542\) −1544.27 −0.122384
\(543\) 10676.5 0.843776
\(544\) 8341.52i 0.657426i
\(545\) 3024.28 0.237699
\(546\) 0 0
\(547\) 5140.42 0.401807 0.200904 0.979611i \(-0.435612\pi\)
0.200904 + 0.979611i \(0.435612\pi\)
\(548\) 14690.5i 1.14516i
\(549\) −6648.76 −0.516871
\(550\) −2676.64 −0.207513
\(551\) − 5271.15i − 0.407547i
\(552\) 22573.6i 1.74057i
\(553\) − 15938.3i − 1.22562i
\(554\) 12714.8i 0.975090i
\(555\) −2913.45 −0.222827
\(556\) −11115.5 −0.847847
\(557\) − 2778.56i − 0.211367i −0.994400 0.105683i \(-0.966297\pi\)
0.994400 0.105683i \(-0.0337030\pi\)
\(558\) 2048.64 0.155422
\(559\) 0 0
\(560\) 1105.69 0.0834356
\(561\) 5904.17i 0.444339i
\(562\) −2395.75 −0.179819
\(563\) −4906.14 −0.367263 −0.183632 0.982995i \(-0.558785\pi\)
−0.183632 + 0.982995i \(0.558785\pi\)
\(564\) − 22553.7i − 1.68383i
\(565\) − 5750.22i − 0.428166i
\(566\) − 10876.2i − 0.807706i
\(567\) 8381.04i 0.620759i
\(568\) 8697.82 0.642522
\(569\) 9363.15 0.689849 0.344924 0.938631i \(-0.387905\pi\)
0.344924 + 0.938631i \(0.387905\pi\)
\(570\) − 1157.74i − 0.0850744i
\(571\) −7199.32 −0.527640 −0.263820 0.964572i \(-0.584982\pi\)
−0.263820 + 0.964572i \(0.584982\pi\)
\(572\) 0 0
\(573\) 37177.3 2.71048
\(574\) − 6805.18i − 0.494848i
\(575\) −13785.4 −0.999813
\(576\) −9734.14 −0.704148
\(577\) − 11449.6i − 0.826086i −0.910711 0.413043i \(-0.864466\pi\)
0.910711 0.413043i \(-0.135534\pi\)
\(578\) 4573.19i 0.329100i
\(579\) − 4101.94i − 0.294423i
\(580\) 4355.91i 0.311843i
\(581\) −36803.3 −2.62798
\(582\) −20495.2 −1.45971
\(583\) 1844.28i 0.131016i
\(584\) 6522.62 0.462171
\(585\) 0 0
\(586\) −999.439 −0.0704547
\(587\) − 5439.39i − 0.382466i −0.981545 0.191233i \(-0.938751\pi\)
0.981545 0.191233i \(-0.0612487\pi\)
\(588\) −19107.3 −1.34008
\(589\) −649.406 −0.0454301
\(590\) − 2445.04i − 0.170611i
\(591\) − 38945.2i − 2.71064i
\(592\) − 1075.99i − 0.0747006i
\(593\) 28405.8i 1.96709i 0.180651 + 0.983547i \(0.442180\pi\)
−0.180651 + 0.983547i \(0.557820\pi\)
\(594\) −4433.93 −0.306273
\(595\) −4311.76 −0.297084
\(596\) − 9749.30i − 0.670045i
\(597\) 3180.67 0.218051
\(598\) 0 0
\(599\) −10482.3 −0.715020 −0.357510 0.933909i \(-0.616374\pi\)
−0.357510 + 0.933909i \(0.616374\pi\)
\(600\) − 20656.6i − 1.40550i
\(601\) 3199.54 0.217158 0.108579 0.994088i \(-0.465370\pi\)
0.108579 + 0.994088i \(0.465370\pi\)
\(602\) 6421.02 0.434720
\(603\) 24806.3i 1.67527i
\(604\) 15542.6i 1.04705i
\(605\) − 3910.91i − 0.262812i
\(606\) 4560.20i 0.305686i
\(607\) 11342.8 0.758468 0.379234 0.925301i \(-0.376188\pi\)
0.379234 + 0.925301i \(0.376188\pi\)
\(608\) 4488.44 0.299392
\(609\) − 51903.7i − 3.45361i
\(610\) 763.629 0.0506859
\(611\) 0 0
\(612\) −11996.7 −0.792384
\(613\) 14385.4i 0.947831i 0.880570 + 0.473916i \(0.157160\pi\)
−0.880570 + 0.473916i \(0.842840\pi\)
\(614\) 156.789 0.0103053
\(615\) −4959.89 −0.325207
\(616\) − 8783.39i − 0.574502i
\(617\) 22056.8i 1.43918i 0.694401 + 0.719588i \(0.255669\pi\)
−0.694401 + 0.719588i \(0.744331\pi\)
\(618\) 4376.48i 0.284867i
\(619\) − 13621.4i − 0.884477i −0.896898 0.442238i \(-0.854185\pi\)
0.896898 0.442238i \(-0.145815\pi\)
\(620\) 536.648 0.0347618
\(621\) −22836.0 −1.47565
\(622\) − 6057.14i − 0.390465i
\(623\) 11954.7 0.768789
\(624\) 0 0
\(625\) 11029.2 0.705866
\(626\) 5917.34i 0.377803i
\(627\) 3176.94 0.202352
\(628\) 18046.5 1.14671
\(629\) 4195.92i 0.265982i
\(630\) − 7319.02i − 0.462852i
\(631\) 18737.5i 1.18214i 0.806622 + 0.591068i \(0.201293\pi\)
−0.806622 + 0.591068i \(0.798707\pi\)
\(632\) 12419.6i 0.781683i
\(633\) 18433.7 1.15746
\(634\) 6881.46 0.431069
\(635\) 3081.36i 0.192567i
\(636\) −5836.91 −0.363913
\(637\) 0 0
\(638\) 5240.75 0.325209
\(639\) 19888.4i 1.23125i
\(640\) −4217.35 −0.260477
\(641\) −29798.7 −1.83616 −0.918081 0.396394i \(-0.870261\pi\)
−0.918081 + 0.396394i \(0.870261\pi\)
\(642\) − 19455.6i − 1.19603i
\(643\) − 22983.5i − 1.40961i −0.709399 0.704807i \(-0.751034\pi\)
0.709399 0.704807i \(-0.248966\pi\)
\(644\) − 18551.5i − 1.13515i
\(645\) − 4679.90i − 0.285692i
\(646\) −1667.37 −0.101551
\(647\) 24905.4 1.51334 0.756672 0.653794i \(-0.226824\pi\)
0.756672 + 0.653794i \(0.226824\pi\)
\(648\) − 6530.72i − 0.395912i
\(649\) 6709.39 0.405804
\(650\) 0 0
\(651\) −6394.54 −0.384980
\(652\) − 18250.1i − 1.09621i
\(653\) 10077.8 0.603946 0.301973 0.953316i \(-0.402355\pi\)
0.301973 + 0.953316i \(0.402355\pi\)
\(654\) 11515.7 0.688534
\(655\) 1002.22i 0.0597864i
\(656\) − 1831.77i − 0.109022i
\(657\) 14914.6i 0.885650i
\(658\) − 19816.5i − 1.17406i
\(659\) 12334.6 0.729116 0.364558 0.931181i \(-0.381220\pi\)
0.364558 + 0.931181i \(0.381220\pi\)
\(660\) −2625.32 −0.154834
\(661\) 12749.1i 0.750202i 0.926984 + 0.375101i \(0.122392\pi\)
−0.926984 + 0.375101i \(0.877608\pi\)
\(662\) −6451.54 −0.378771
\(663\) 0 0
\(664\) 28678.1 1.67609
\(665\) 2320.09i 0.135292i
\(666\) −7122.40 −0.414395
\(667\) 26991.3 1.56688
\(668\) 17388.3i 1.00715i
\(669\) − 51468.9i − 2.97444i
\(670\) − 2849.07i − 0.164283i
\(671\) 2095.46i 0.120558i
\(672\) 44196.5 2.53708
\(673\) 13618.2 0.780007 0.390004 0.920813i \(-0.372474\pi\)
0.390004 + 0.920813i \(0.372474\pi\)
\(674\) − 7121.96i − 0.407014i
\(675\) 20896.7 1.19158
\(676\) 0 0
\(677\) 9655.67 0.548150 0.274075 0.961708i \(-0.411628\pi\)
0.274075 + 0.961708i \(0.411628\pi\)
\(678\) − 21895.5i − 1.24025i
\(679\) 41071.9 2.32135
\(680\) 3359.84 0.189476
\(681\) − 7778.51i − 0.437699i
\(682\) − 645.660i − 0.0362516i
\(683\) − 16316.8i − 0.914119i −0.889436 0.457060i \(-0.848903\pi\)
0.889436 0.457060i \(-0.151097\pi\)
\(684\) 6455.25i 0.360852i
\(685\) 9407.61 0.524739
\(686\) −2232.00 −0.124225
\(687\) − 5451.51i − 0.302748i
\(688\) 1728.37 0.0957753
\(689\) 0 0
\(690\) 5928.30 0.327082
\(691\) 2350.84i 0.129421i 0.997904 + 0.0647106i \(0.0206124\pi\)
−0.997904 + 0.0647106i \(0.979388\pi\)
\(692\) −542.640 −0.0298093
\(693\) 20084.0 1.10091
\(694\) − 15723.9i − 0.860045i
\(695\) 7118.24i 0.388504i
\(696\) 40444.7i 2.20266i
\(697\) 7143.20i 0.388189i
\(698\) −9180.88 −0.497853
\(699\) −20007.1 −1.08260
\(700\) 16976.1i 0.916623i
\(701\) 8076.90 0.435179 0.217589 0.976040i \(-0.430181\pi\)
0.217589 + 0.976040i \(0.430181\pi\)
\(702\) 0 0
\(703\) 2257.76 0.121128
\(704\) 3067.87i 0.164239i
\(705\) −14443.1 −0.771573
\(706\) −14276.6 −0.761058
\(707\) − 9138.55i − 0.486125i
\(708\) 21234.3i 1.12717i
\(709\) 13624.9i 0.721712i 0.932622 + 0.360856i \(0.117515\pi\)
−0.932622 + 0.360856i \(0.882485\pi\)
\(710\) − 2284.23i − 0.120741i
\(711\) −28398.4 −1.49792
\(712\) −9315.43 −0.490324
\(713\) − 3325.33i − 0.174663i
\(714\) −16418.2 −0.860553
\(715\) 0 0
\(716\) 193.069 0.0100773
\(717\) 4729.86i 0.246359i
\(718\) 4301.02 0.223555
\(719\) −16235.8 −0.842131 −0.421066 0.907030i \(-0.638344\pi\)
−0.421066 + 0.907030i \(0.638344\pi\)
\(720\) − 1970.09i − 0.101973i
\(721\) − 8770.37i − 0.453018i
\(722\) − 9813.51i − 0.505846i
\(723\) − 47123.8i − 2.42400i
\(724\) 6837.08 0.350964
\(725\) −24699.2 −1.26525
\(726\) − 14891.8i − 0.761277i
\(727\) −24181.2 −1.23361 −0.616803 0.787118i \(-0.711572\pi\)
−0.616803 + 0.787118i \(0.711572\pi\)
\(728\) 0 0
\(729\) −28693.9 −1.45780
\(730\) − 1712.98i − 0.0868496i
\(731\) −6739.96 −0.341021
\(732\) −6631.86 −0.334864
\(733\) 3053.70i 0.153876i 0.997036 + 0.0769379i \(0.0245143\pi\)
−0.997036 + 0.0769379i \(0.975486\pi\)
\(734\) − 4747.41i − 0.238733i
\(735\) 12236.1i 0.614060i
\(736\) 22983.4i 1.15106i
\(737\) 7818.10 0.390751
\(738\) −12125.3 −0.604793
\(739\) 8033.62i 0.399894i 0.979807 + 0.199947i \(0.0640770\pi\)
−0.979807 + 0.199947i \(0.935923\pi\)
\(740\) −1865.74 −0.0926836
\(741\) 0 0
\(742\) −5128.54 −0.253739
\(743\) − 16139.6i − 0.796912i −0.917187 0.398456i \(-0.869546\pi\)
0.917187 0.398456i \(-0.130454\pi\)
\(744\) 4982.79 0.245535
\(745\) −6243.33 −0.307031
\(746\) 8408.53i 0.412678i
\(747\) 65574.9i 3.21186i
\(748\) 3780.96i 0.184820i
\(749\) 38988.7i 1.90202i
\(750\) −11462.4 −0.558062
\(751\) 18491.1 0.898469 0.449235 0.893414i \(-0.351697\pi\)
0.449235 + 0.893414i \(0.351697\pi\)
\(752\) − 5334.08i − 0.258662i
\(753\) 45344.5 2.19448
\(754\) 0 0
\(755\) 9953.28 0.479784
\(756\) 28121.5i 1.35287i
\(757\) 160.630 0.00771227 0.00385613 0.999993i \(-0.498773\pi\)
0.00385613 + 0.999993i \(0.498773\pi\)
\(758\) −5347.17 −0.256224
\(759\) 16267.7i 0.777973i
\(760\) − 1807.87i − 0.0862874i
\(761\) − 26799.1i − 1.27656i −0.769803 0.638282i \(-0.779645\pi\)
0.769803 0.638282i \(-0.220355\pi\)
\(762\) 11733.1i 0.557803i
\(763\) −23077.3 −1.09496
\(764\) 23807.9 1.12741
\(765\) 7682.57i 0.363090i
\(766\) −598.052 −0.0282095
\(767\) 0 0
\(768\) −30025.1 −1.41073
\(769\) − 5145.82i − 0.241304i −0.992695 0.120652i \(-0.961501\pi\)
0.992695 0.120652i \(-0.0384986\pi\)
\(770\) −2306.71 −0.107958
\(771\) −5716.29 −0.267013
\(772\) − 2626.83i − 0.122463i
\(773\) 12810.6i 0.596072i 0.954555 + 0.298036i \(0.0963316\pi\)
−0.954555 + 0.298036i \(0.903668\pi\)
\(774\) − 11440.8i − 0.531306i
\(775\) 3042.94i 0.141039i
\(776\) −32004.3 −1.48052
\(777\) 22231.6 1.02645
\(778\) 13411.1i 0.618011i
\(779\) 3843.64 0.176781
\(780\) 0 0
\(781\) 6268.13 0.287185
\(782\) − 8537.89i − 0.390428i
\(783\) −40915.0 −1.86741
\(784\) −4518.98 −0.205857
\(785\) − 11556.8i − 0.525452i
\(786\) 3816.23i 0.173181i
\(787\) − 28073.0i − 1.27153i −0.771883 0.635764i \(-0.780685\pi\)
0.771883 0.635764i \(-0.219315\pi\)
\(788\) − 24940.0i − 1.12748i
\(789\) 28194.3 1.27217
\(790\) 3261.64 0.146891
\(791\) 43878.1i 1.97235i
\(792\) −15650.0 −0.702144
\(793\) 0 0
\(794\) 11304.3 0.505259
\(795\) 3737.89i 0.166754i
\(796\) 2036.87 0.0906970
\(797\) −30093.1 −1.33746 −0.668729 0.743507i \(-0.733161\pi\)
−0.668729 + 0.743507i \(0.733161\pi\)
\(798\) 8834.35i 0.391896i
\(799\) 20800.8i 0.921003i
\(800\) − 21031.6i − 0.929473i
\(801\) − 21300.6i − 0.939598i
\(802\) −6667.24 −0.293552
\(803\) 4700.56 0.206574
\(804\) 24743.2i 1.08536i
\(805\) −11880.2 −0.520151
\(806\) 0 0
\(807\) −22456.8 −0.979575
\(808\) 7120.99i 0.310044i
\(809\) −24337.1 −1.05766 −0.528831 0.848727i \(-0.677369\pi\)
−0.528831 + 0.848727i \(0.677369\pi\)
\(810\) −1715.11 −0.0743984
\(811\) 19078.7i 0.826071i 0.910715 + 0.413035i \(0.135531\pi\)
−0.910715 + 0.413035i \(0.864469\pi\)
\(812\) − 33238.5i − 1.43651i
\(813\) − 8588.54i − 0.370496i
\(814\) 2244.74i 0.0966559i
\(815\) −11687.1 −0.502309
\(816\) −4419.33 −0.189593
\(817\) 3626.66i 0.155301i
\(818\) 21178.6 0.905248
\(819\) 0 0
\(820\) −3176.26 −0.135268
\(821\) 2013.92i 0.0856104i 0.999083 + 0.0428052i \(0.0136295\pi\)
−0.999083 + 0.0428052i \(0.986371\pi\)
\(822\) 35821.9 1.51999
\(823\) 7692.10 0.325795 0.162898 0.986643i \(-0.447916\pi\)
0.162898 + 0.986643i \(0.447916\pi\)
\(824\) 6834.10i 0.288929i
\(825\) − 14886.2i − 0.628209i
\(826\) 18657.3i 0.785922i
\(827\) 4762.76i 0.200263i 0.994974 + 0.100131i \(0.0319263\pi\)
−0.994974 + 0.100131i \(0.968074\pi\)
\(828\) −33054.6 −1.38735
\(829\) −19977.7 −0.836976 −0.418488 0.908222i \(-0.637440\pi\)
−0.418488 + 0.908222i \(0.637440\pi\)
\(830\) − 7531.47i − 0.314965i
\(831\) −70714.0 −2.95191
\(832\) 0 0
\(833\) 17622.3 0.732984
\(834\) 27104.6i 1.12537i
\(835\) 11135.3 0.461499
\(836\) 2034.47 0.0841671
\(837\) 5040.72i 0.208164i
\(838\) 22762.6i 0.938330i
\(839\) 30615.8i 1.25980i 0.776676 + 0.629901i \(0.216905\pi\)
−0.776676 + 0.629901i \(0.783095\pi\)
\(840\) − 17801.7i − 0.731210i
\(841\) 23971.0 0.982861
\(842\) 24748.1 1.01292
\(843\) − 13324.1i − 0.544372i
\(844\) 11804.7 0.481439
\(845\) 0 0
\(846\) −35308.5 −1.43491
\(847\) 29842.9i 1.21064i
\(848\) −1380.47 −0.0559026
\(849\) 60488.6 2.44519
\(850\) 7812.83i 0.315268i
\(851\) 11561.0i 0.465696i
\(852\) 19837.8i 0.797690i
\(853\) − 5660.88i − 0.227227i −0.993525 0.113614i \(-0.963757\pi\)
0.993525 0.113614i \(-0.0362426\pi\)
\(854\) −5827.01 −0.233485
\(855\) 4133.86 0.165351
\(856\) − 30381.0i − 1.21309i
\(857\) −41346.1 −1.64802 −0.824012 0.566572i \(-0.808269\pi\)
−0.824012 + 0.566572i \(0.808269\pi\)
\(858\) 0 0
\(859\) −34810.5 −1.38268 −0.691339 0.722530i \(-0.742979\pi\)
−0.691339 + 0.722530i \(0.742979\pi\)
\(860\) − 2996.96i − 0.118832i
\(861\) 37847.4 1.49807
\(862\) 16700.4 0.659880
\(863\) − 8360.51i − 0.329774i −0.986312 0.164887i \(-0.947274\pi\)
0.986312 0.164887i \(-0.0527260\pi\)
\(864\) − 34839.5i − 1.37183i
\(865\) 347.500i 0.0136594i
\(866\) − 25108.2i − 0.985233i
\(867\) −25434.1 −0.996293
\(868\) −4094.99 −0.160130
\(869\) 8950.21i 0.349385i
\(870\) 10621.6 0.413917
\(871\) 0 0
\(872\) 17982.4 0.698352
\(873\) − 73180.6i − 2.83710i
\(874\) −4594.10 −0.177801
\(875\) 22970.4 0.887475
\(876\) 14876.6i 0.573784i
\(877\) − 40579.3i − 1.56245i −0.624251 0.781223i \(-0.714596\pi\)
0.624251 0.781223i \(-0.285404\pi\)
\(878\) 9425.22i 0.362285i
\(879\) − 5558.43i − 0.213289i
\(880\) −620.903 −0.0237848
\(881\) −10445.2 −0.399442 −0.199721 0.979853i \(-0.564004\pi\)
−0.199721 + 0.979853i \(0.564004\pi\)
\(882\) 29913.0i 1.14198i
\(883\) −18227.6 −0.694685 −0.347343 0.937738i \(-0.612916\pi\)
−0.347343 + 0.937738i \(0.612916\pi\)
\(884\) 0 0
\(885\) 13598.2 0.516496
\(886\) − 15929.9i − 0.604036i
\(887\) −23517.7 −0.890245 −0.445122 0.895470i \(-0.646840\pi\)
−0.445122 + 0.895470i \(0.646840\pi\)
\(888\) −17323.4 −0.654658
\(889\) − 23512.9i − 0.887061i
\(890\) 2446.43i 0.0921399i
\(891\) − 4706.40i − 0.176959i
\(892\) − 32960.1i − 1.23720i
\(893\) 11192.6 0.419424
\(894\) −23773.1 −0.889366
\(895\) − 123.639i − 0.00461766i
\(896\) 32181.2 1.19989
\(897\) 0 0
\(898\) 9092.21 0.337874
\(899\) − 5957.95i − 0.221033i
\(900\) 30247.5 1.12028
\(901\) 5383.28 0.199049
\(902\) 3821.47i 0.141065i
\(903\) 35710.9i 1.31604i
\(904\) − 34191.0i − 1.25794i
\(905\) − 4378.38i − 0.160820i
\(906\) 37899.7 1.38977
\(907\) 30564.6 1.11894 0.559471 0.828850i \(-0.311004\pi\)
0.559471 + 0.828850i \(0.311004\pi\)
\(908\) − 4981.26i − 0.182058i
\(909\) −16282.8 −0.594132
\(910\) 0 0
\(911\) −32766.5 −1.19166 −0.595831 0.803110i \(-0.703177\pi\)
−0.595831 + 0.803110i \(0.703177\pi\)
\(912\) 2377.97i 0.0863404i
\(913\) 20667.0 0.749154
\(914\) 7216.87 0.261174
\(915\) 4246.96i 0.153443i
\(916\) − 3491.08i − 0.125926i
\(917\) − 7647.64i − 0.275406i
\(918\) 12942.2i 0.465312i
\(919\) 20686.7 0.742538 0.371269 0.928525i \(-0.378923\pi\)
0.371269 + 0.928525i \(0.378923\pi\)
\(920\) 9257.35 0.331745
\(921\) 871.989i 0.0311976i
\(922\) 8007.28 0.286015
\(923\) 0 0
\(924\) 20033.0 0.713242
\(925\) − 10579.2i − 0.376047i
\(926\) −10128.6 −0.359447
\(927\) −15626.8 −0.553669
\(928\) 41179.0i 1.45665i
\(929\) 45632.2i 1.61156i 0.592212 + 0.805782i \(0.298255\pi\)
−0.592212 + 0.805782i \(0.701745\pi\)
\(930\) − 1308.59i − 0.0461401i
\(931\) − 9482.26i − 0.333801i
\(932\) −12812.3 −0.450301
\(933\) 33687.1 1.18206
\(934\) 20265.9i 0.709979i
\(935\) 2421.28 0.0846892
\(936\) 0 0
\(937\) −17761.4 −0.619253 −0.309626 0.950858i \(-0.600204\pi\)
−0.309626 + 0.950858i \(0.600204\pi\)
\(938\) 21740.4i 0.756768i
\(939\) −32909.6 −1.14373
\(940\) −9249.19 −0.320931
\(941\) − 44888.3i − 1.55507i −0.628841 0.777534i \(-0.716470\pi\)
0.628841 0.777534i \(-0.283530\pi\)
\(942\) − 44005.5i − 1.52206i
\(943\) 19681.7i 0.679664i
\(944\) 5022.05i 0.173150i
\(945\) 18008.6 0.619917
\(946\) −3605.74 −0.123925
\(947\) − 16069.6i − 0.551415i −0.961242 0.275708i \(-0.911088\pi\)
0.961242 0.275708i \(-0.0889122\pi\)
\(948\) −28326.2 −0.970457
\(949\) 0 0
\(950\) 4203.96 0.143573
\(951\) 38271.6i 1.30499i
\(952\) −25637.8 −0.872823
\(953\) −3512.03 −0.119377 −0.0596883 0.998217i \(-0.519011\pi\)
−0.0596883 + 0.998217i \(0.519011\pi\)
\(954\) 9137.88i 0.310115i
\(955\) − 15246.3i − 0.516605i
\(956\) 3028.94i 0.102472i
\(957\) 29146.7i 0.984513i
\(958\) 9071.00 0.305919
\(959\) −71786.5 −2.41721
\(960\) 6217.78i 0.209040i
\(961\) 29057.0 0.975361
\(962\) 0 0
\(963\) 69468.9 2.32461
\(964\) − 30177.5i − 1.00825i
\(965\) −1682.19 −0.0561158
\(966\) −45237.0 −1.50670
\(967\) − 37011.9i − 1.23084i −0.788199 0.615421i \(-0.788986\pi\)
0.788199 0.615421i \(-0.211014\pi\)
\(968\) − 23254.4i − 0.772132i
\(969\) − 9273.16i − 0.307427i
\(970\) 8405.00i 0.278215i
\(971\) 19532.3 0.645542 0.322771 0.946477i \(-0.395386\pi\)
0.322771 + 0.946477i \(0.395386\pi\)
\(972\) −13043.1 −0.430409
\(973\) − 54317.1i − 1.78965i
\(974\) −8412.30 −0.276743
\(975\) 0 0
\(976\) −1568.47 −0.0514402
\(977\) 30201.2i 0.988970i 0.869186 + 0.494485i \(0.164643\pi\)
−0.869186 + 0.494485i \(0.835357\pi\)
\(978\) −44501.8 −1.45502
\(979\) −6713.21 −0.219157
\(980\) 7835.83i 0.255415i
\(981\) 41118.5i 1.33824i
\(982\) 23827.8i 0.774315i
\(983\) 38774.9i 1.25812i 0.777359 + 0.629058i \(0.216559\pi\)
−0.777359 + 0.629058i \(0.783441\pi\)
\(984\) −29491.7 −0.955447
\(985\) −15971.3 −0.516638
\(986\) − 15297.2i − 0.494080i
\(987\) 110211. 3.55425
\(988\) 0 0
\(989\) −18570.6 −0.597079
\(990\) 4110.02i 0.131944i
\(991\) −27728.9 −0.888838 −0.444419 0.895819i \(-0.646590\pi\)
−0.444419 + 0.895819i \(0.646590\pi\)
\(992\) 5073.25 0.162375
\(993\) − 35880.6i − 1.14666i
\(994\) 17430.3i 0.556192i
\(995\) − 1304.38i − 0.0415596i
\(996\) 65408.2i 2.08086i
\(997\) −48918.2 −1.55392 −0.776958 0.629552i \(-0.783239\pi\)
−0.776958 + 0.629552i \(0.783239\pi\)
\(998\) 2898.31 0.0919283
\(999\) − 17524.8i − 0.555016i
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 169.4.b.f.168.2 4
13.2 odd 12 169.4.c.g.22.2 4
13.3 even 3 169.4.e.f.147.3 8
13.4 even 6 169.4.e.f.23.3 8
13.5 odd 4 13.4.a.b.1.1 2
13.6 odd 12 169.4.c.g.146.2 4
13.7 odd 12 169.4.c.j.146.1 4
13.8 odd 4 169.4.a.g.1.2 2
13.9 even 3 169.4.e.f.23.2 8
13.10 even 6 169.4.e.f.147.2 8
13.11 odd 12 169.4.c.j.22.1 4
13.12 even 2 inner 169.4.b.f.168.3 4
39.5 even 4 117.4.a.d.1.2 2
39.8 even 4 1521.4.a.r.1.1 2
52.31 even 4 208.4.a.h.1.1 2
65.18 even 4 325.4.b.e.274.3 4
65.44 odd 4 325.4.a.f.1.2 2
65.57 even 4 325.4.b.e.274.2 4
91.83 even 4 637.4.a.b.1.1 2
104.5 odd 4 832.4.a.s.1.1 2
104.83 even 4 832.4.a.z.1.2 2
143.109 even 4 1573.4.a.b.1.2 2
156.83 odd 4 1872.4.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.a.b.1.1 2 13.5 odd 4
117.4.a.d.1.2 2 39.5 even 4
169.4.a.g.1.2 2 13.8 odd 4
169.4.b.f.168.2 4 1.1 even 1 trivial
169.4.b.f.168.3 4 13.12 even 2 inner
169.4.c.g.22.2 4 13.2 odd 12
169.4.c.g.146.2 4 13.6 odd 12
169.4.c.j.22.1 4 13.11 odd 12
169.4.c.j.146.1 4 13.7 odd 12
169.4.e.f.23.2 8 13.9 even 3
169.4.e.f.23.3 8 13.4 even 6
169.4.e.f.147.2 8 13.10 even 6
169.4.e.f.147.3 8 13.3 even 3
208.4.a.h.1.1 2 52.31 even 4
325.4.a.f.1.2 2 65.44 odd 4
325.4.b.e.274.2 4 65.57 even 4
325.4.b.e.274.3 4 65.18 even 4
637.4.a.b.1.1 2 91.83 even 4
832.4.a.s.1.1 2 104.5 odd 4
832.4.a.z.1.2 2 104.83 even 4
1521.4.a.r.1.1 2 39.8 even 4
1573.4.a.b.1.2 2 143.109 even 4
1872.4.a.bb.1.2 2 156.83 odd 4