Properties

Label 338.4.b.f.337.5
Level $338$
Weight $4$
Character 338.337
Analytic conductor $19.943$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,4,Mod(337,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, 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("338.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(19.9426455819\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.5
Root \(-0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 338.337
Dual form 338.4.b.f.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -3.66487 q^{3} -4.00000 q^{4} +8.53079i q^{5} -7.32975i q^{6} +4.20105i q^{7} -8.00000i q^{8} -13.5687 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} -3.66487 q^{3} -4.00000 q^{4} +8.53079i q^{5} -7.32975i q^{6} +4.20105i q^{7} -8.00000i q^{8} -13.5687 q^{9} -17.0616 q^{10} +65.3726i q^{11} +14.6595 q^{12} -8.40209 q^{14} -31.2643i q^{15} +16.0000 q^{16} +26.9019 q^{17} -27.1374i q^{18} -13.3730i q^{19} -34.1232i q^{20} -15.3963i q^{21} -130.745 q^{22} -159.929 q^{23} +29.3190i q^{24} +52.2255 q^{25} +148.679 q^{27} -16.8042i q^{28} -301.288 q^{29} +62.5286 q^{30} -73.0232i q^{31} +32.0000i q^{32} -239.582i q^{33} +53.8038i q^{34} -35.8383 q^{35} +54.2748 q^{36} +118.781i q^{37} +26.7459 q^{38} +68.2464 q^{40} -432.901i q^{41} +30.7926 q^{42} +356.508 q^{43} -261.490i q^{44} -115.752i q^{45} -319.858i q^{46} -588.614i q^{47} -58.6380 q^{48} +325.351 q^{49} +104.451i q^{50} -98.5921 q^{51} -269.462 q^{53} +297.358i q^{54} -557.680 q^{55} +33.6084 q^{56} +49.0102i q^{57} -602.576i q^{58} -230.340i q^{59} +125.057i q^{60} -380.816 q^{61} +146.046 q^{62} -57.0027i q^{63} -64.0000 q^{64} +479.164 q^{66} +435.848i q^{67} -107.608 q^{68} +586.120 q^{69} -71.6765i q^{70} +65.9622i q^{71} +108.550i q^{72} -885.517i q^{73} -237.562 q^{74} -191.400 q^{75} +53.4918i q^{76} -274.633 q^{77} -385.463 q^{79} +136.493i q^{80} -178.536 q^{81} +865.803 q^{82} -254.207i q^{83} +61.5852i q^{84} +229.495i q^{85} +713.016i q^{86} +1104.18 q^{87} +522.980 q^{88} +372.612i q^{89} +231.504 q^{90} +639.716 q^{92} +267.621i q^{93} +1177.23 q^{94} +114.082 q^{95} -117.276i q^{96} +1313.88i q^{97} +650.702i q^{98} -887.020i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24 q^{3} - 24 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 24 q^{3} - 24 q^{4} + 18 q^{9} + 48 q^{10} + 96 q^{12} + 108 q^{14} + 96 q^{16} + 180 q^{17} - 328 q^{22} - 38 q^{23} + 122 q^{25} - 138 q^{27} - 202 q^{29} - 360 q^{30} - 916 q^{35} - 72 q^{36} + 520 q^{38} - 192 q^{40} - 936 q^{42} + 2410 q^{43} - 384 q^{48} + 368 q^{49} + 414 q^{51} - 2190 q^{53} - 1052 q^{55} - 432 q^{56} - 2216 q^{61} + 2076 q^{62} - 384 q^{64} + 584 q^{66} - 720 q^{68} + 628 q^{69} - 336 q^{74} + 1010 q^{75} - 960 q^{77} - 3922 q^{79} + 1830 q^{81} + 748 q^{82} + 3272 q^{87} + 1312 q^{88} + 2580 q^{90} + 152 q^{92} + 2144 q^{94} - 3658 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) −3.66487 −0.705305 −0.352653 0.935754i \(-0.614720\pi\)
−0.352653 + 0.935754i \(0.614720\pi\)
\(4\) −4.00000 −0.500000
\(5\) 8.53079i 0.763017i 0.924365 + 0.381509i \(0.124595\pi\)
−0.924365 + 0.381509i \(0.875405\pi\)
\(6\) − 7.32975i − 0.498726i
\(7\) 4.20105i 0.226835i 0.993547 + 0.113418i \(0.0361798\pi\)
−0.993547 + 0.113418i \(0.963820\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) −13.5687 −0.502544
\(10\) −17.0616 −0.539535
\(11\) 65.3726i 1.79187i 0.444186 + 0.895935i \(0.353493\pi\)
−0.444186 + 0.895935i \(0.646507\pi\)
\(12\) 14.6595 0.352653
\(13\) 0 0
\(14\) −8.40209 −0.160397
\(15\) − 31.2643i − 0.538160i
\(16\) 16.0000 0.250000
\(17\) 26.9019 0.383804 0.191902 0.981414i \(-0.438534\pi\)
0.191902 + 0.981414i \(0.438534\pi\)
\(18\) − 27.1374i − 0.355352i
\(19\) − 13.3730i − 0.161472i −0.996736 0.0807360i \(-0.974273\pi\)
0.996736 0.0807360i \(-0.0257271\pi\)
\(20\) − 34.1232i − 0.381509i
\(21\) − 15.3963i − 0.159988i
\(22\) −130.745 −1.26704
\(23\) −159.929 −1.44989 −0.724946 0.688806i \(-0.758135\pi\)
−0.724946 + 0.688806i \(0.758135\pi\)
\(24\) 29.3190i 0.249363i
\(25\) 52.2255 0.417804
\(26\) 0 0
\(27\) 148.679 1.05975
\(28\) − 16.8042i − 0.113418i
\(29\) −301.288 −1.92923 −0.964616 0.263658i \(-0.915071\pi\)
−0.964616 + 0.263658i \(0.915071\pi\)
\(30\) 62.5286 0.380537
\(31\) − 73.0232i − 0.423076i −0.977370 0.211538i \(-0.932153\pi\)
0.977370 0.211538i \(-0.0678472\pi\)
\(32\) 32.0000i 0.176777i
\(33\) − 239.582i − 1.26382i
\(34\) 53.8038i 0.271390i
\(35\) −35.8383 −0.173079
\(36\) 54.2748 0.251272
\(37\) 118.781i 0.527769i 0.964554 + 0.263885i \(0.0850038\pi\)
−0.964554 + 0.263885i \(0.914996\pi\)
\(38\) 26.7459 0.114178
\(39\) 0 0
\(40\) 68.2464 0.269767
\(41\) − 432.901i − 1.64897i −0.565883 0.824486i \(-0.691465\pi\)
0.565883 0.824486i \(-0.308535\pi\)
\(42\) 30.7926 0.113129
\(43\) 356.508 1.26435 0.632174 0.774827i \(-0.282163\pi\)
0.632174 + 0.774827i \(0.282163\pi\)
\(44\) − 261.490i − 0.895935i
\(45\) − 115.752i − 0.383450i
\(46\) − 319.858i − 1.02523i
\(47\) − 588.614i − 1.82677i −0.407096 0.913385i \(-0.633459\pi\)
0.407096 0.913385i \(-0.366541\pi\)
\(48\) −58.6380 −0.176326
\(49\) 325.351 0.948546
\(50\) 104.451i 0.295432i
\(51\) −98.5921 −0.270699
\(52\) 0 0
\(53\) −269.462 −0.698366 −0.349183 0.937054i \(-0.613541\pi\)
−0.349183 + 0.937054i \(0.613541\pi\)
\(54\) 297.358i 0.749358i
\(55\) −557.680 −1.36723
\(56\) 33.6084 0.0801983
\(57\) 49.0102i 0.113887i
\(58\) − 602.576i − 1.36417i
\(59\) − 230.340i − 0.508265i −0.967169 0.254133i \(-0.918210\pi\)
0.967169 0.254133i \(-0.0817900\pi\)
\(60\) 125.057i 0.269080i
\(61\) −380.816 −0.799319 −0.399659 0.916664i \(-0.630872\pi\)
−0.399659 + 0.916664i \(0.630872\pi\)
\(62\) 146.046 0.299160
\(63\) − 57.0027i − 0.113995i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 479.164 0.893652
\(67\) 435.848i 0.794736i 0.917659 + 0.397368i \(0.130076\pi\)
−0.917659 + 0.397368i \(0.869924\pi\)
\(68\) −107.608 −0.191902
\(69\) 586.120 1.02262
\(70\) − 71.6765i − 0.122385i
\(71\) 65.9622i 0.110257i 0.998479 + 0.0551287i \(0.0175569\pi\)
−0.998479 + 0.0551287i \(0.982443\pi\)
\(72\) 108.550i 0.177676i
\(73\) − 885.517i − 1.41975i −0.704326 0.709876i \(-0.748751\pi\)
0.704326 0.709876i \(-0.251249\pi\)
\(74\) −237.562 −0.373189
\(75\) −191.400 −0.294680
\(76\) 53.4918i 0.0807360i
\(77\) −274.633 −0.406459
\(78\) 0 0
\(79\) −385.463 −0.548962 −0.274481 0.961592i \(-0.588506\pi\)
−0.274481 + 0.961592i \(0.588506\pi\)
\(80\) 136.493i 0.190754i
\(81\) −178.536 −0.244905
\(82\) 865.803 1.16600
\(83\) − 254.207i − 0.336179i −0.985772 0.168089i \(-0.946240\pi\)
0.985772 0.168089i \(-0.0537597\pi\)
\(84\) 61.5852i 0.0799940i
\(85\) 229.495i 0.292849i
\(86\) 713.016i 0.894029i
\(87\) 1104.18 1.36070
\(88\) 522.980 0.633522
\(89\) 372.612i 0.443785i 0.975071 + 0.221892i \(0.0712234\pi\)
−0.975071 + 0.221892i \(0.928777\pi\)
\(90\) 231.504 0.271140
\(91\) 0 0
\(92\) 639.716 0.724946
\(93\) 267.621i 0.298398i
\(94\) 1177.23 1.29172
\(95\) 114.082 0.123206
\(96\) − 117.276i − 0.124682i
\(97\) 1313.88i 1.37531i 0.726039 + 0.687653i \(0.241359\pi\)
−0.726039 + 0.687653i \(0.758641\pi\)
\(98\) 650.702i 0.670723i
\(99\) − 887.020i − 0.900494i
\(100\) −208.902 −0.208902
\(101\) −1463.72 −1.44204 −0.721019 0.692915i \(-0.756326\pi\)
−0.721019 + 0.692915i \(0.756326\pi\)
\(102\) − 197.184i − 0.191413i
\(103\) −210.886 −0.201740 −0.100870 0.994900i \(-0.532163\pi\)
−0.100870 + 0.994900i \(0.532163\pi\)
\(104\) 0 0
\(105\) 131.343 0.122074
\(106\) − 538.924i − 0.493820i
\(107\) 391.061 0.353321 0.176660 0.984272i \(-0.443471\pi\)
0.176660 + 0.984272i \(0.443471\pi\)
\(108\) −594.717 −0.529876
\(109\) 1331.40i 1.16996i 0.811049 + 0.584978i \(0.198897\pi\)
−0.811049 + 0.584978i \(0.801103\pi\)
\(110\) − 1115.36i − 0.966776i
\(111\) − 435.317i − 0.372238i
\(112\) 67.2167i 0.0567088i
\(113\) −711.912 −0.592665 −0.296332 0.955085i \(-0.595764\pi\)
−0.296332 + 0.955085i \(0.595764\pi\)
\(114\) −98.0204 −0.0805303
\(115\) − 1364.32i − 1.10629i
\(116\) 1205.15 0.964616
\(117\) 0 0
\(118\) 460.679 0.359398
\(119\) 113.016i 0.0870603i
\(120\) −250.114 −0.190268
\(121\) −2942.57 −2.21080
\(122\) − 761.631i − 0.565204i
\(123\) 1586.53i 1.16303i
\(124\) 292.093i 0.211538i
\(125\) 1511.87i 1.08181i
\(126\) 114.005 0.0806064
\(127\) 1171.70 0.818677 0.409338 0.912383i \(-0.365760\pi\)
0.409338 + 0.912383i \(0.365760\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) −1306.56 −0.891751
\(130\) 0 0
\(131\) −1128.61 −0.752726 −0.376363 0.926472i \(-0.622825\pi\)
−0.376363 + 0.926472i \(0.622825\pi\)
\(132\) 958.329i 0.631908i
\(133\) 56.1804 0.0366275
\(134\) −871.696 −0.561963
\(135\) 1268.35i 0.808610i
\(136\) − 215.215i − 0.135695i
\(137\) 1747.93i 1.09004i 0.838422 + 0.545022i \(0.183479\pi\)
−0.838422 + 0.545022i \(0.816521\pi\)
\(138\) 1172.24i 0.723099i
\(139\) 1676.54 1.02304 0.511518 0.859273i \(-0.329083\pi\)
0.511518 + 0.859273i \(0.329083\pi\)
\(140\) 143.353 0.0865396
\(141\) 2157.20i 1.28843i
\(142\) −131.924 −0.0779637
\(143\) 0 0
\(144\) −217.099 −0.125636
\(145\) − 2570.22i − 1.47204i
\(146\) 1771.03 1.00392
\(147\) −1192.37 −0.669014
\(148\) − 475.124i − 0.263885i
\(149\) − 896.961i − 0.493167i −0.969122 0.246584i \(-0.920692\pi\)
0.969122 0.246584i \(-0.0793080\pi\)
\(150\) − 382.800i − 0.208370i
\(151\) 2078.28i 1.12005i 0.828475 + 0.560027i \(0.189209\pi\)
−0.828475 + 0.560027i \(0.810791\pi\)
\(152\) −106.984 −0.0570890
\(153\) −365.024 −0.192879
\(154\) − 549.266i − 0.287410i
\(155\) 622.946 0.322814
\(156\) 0 0
\(157\) −3494.75 −1.77650 −0.888252 0.459357i \(-0.848080\pi\)
−0.888252 + 0.459357i \(0.848080\pi\)
\(158\) − 770.927i − 0.388175i
\(159\) 987.544 0.492562
\(160\) −272.985 −0.134884
\(161\) − 671.869i − 0.328887i
\(162\) − 357.071i − 0.173174i
\(163\) 451.254i 0.216840i 0.994105 + 0.108420i \(0.0345792\pi\)
−0.994105 + 0.108420i \(0.965421\pi\)
\(164\) 1731.61i 0.824486i
\(165\) 2043.83 0.964313
\(166\) 508.414 0.237714
\(167\) − 3203.35i − 1.48433i −0.670218 0.742164i \(-0.733800\pi\)
0.670218 0.742164i \(-0.266200\pi\)
\(168\) −123.170 −0.0565643
\(169\) 0 0
\(170\) −458.989 −0.207076
\(171\) 181.454i 0.0811468i
\(172\) −1426.03 −0.632174
\(173\) −1899.38 −0.834725 −0.417362 0.908740i \(-0.637045\pi\)
−0.417362 + 0.908740i \(0.637045\pi\)
\(174\) 2208.36i 0.962159i
\(175\) 219.402i 0.0947727i
\(176\) 1045.96i 0.447967i
\(177\) 844.165i 0.358482i
\(178\) −745.225 −0.313803
\(179\) −2650.89 −1.10691 −0.553455 0.832879i \(-0.686691\pi\)
−0.553455 + 0.832879i \(0.686691\pi\)
\(180\) 463.007i 0.191725i
\(181\) 2289.94 0.940387 0.470194 0.882563i \(-0.344184\pi\)
0.470194 + 0.882563i \(0.344184\pi\)
\(182\) 0 0
\(183\) 1395.64 0.563764
\(184\) 1279.43i 0.512614i
\(185\) −1013.30 −0.402697
\(186\) −535.242 −0.210999
\(187\) 1758.65i 0.687727i
\(188\) 2354.46i 0.913385i
\(189\) 624.608i 0.240389i
\(190\) 228.164i 0.0871198i
\(191\) −338.313 −0.128165 −0.0640825 0.997945i \(-0.520412\pi\)
−0.0640825 + 0.997945i \(0.520412\pi\)
\(192\) 234.552 0.0881632
\(193\) − 2684.05i − 1.00105i −0.865723 0.500524i \(-0.833141\pi\)
0.865723 0.500524i \(-0.166859\pi\)
\(194\) −2627.77 −0.972489
\(195\) 0 0
\(196\) −1301.40 −0.474273
\(197\) 1897.24i 0.686156i 0.939307 + 0.343078i \(0.111470\pi\)
−0.939307 + 0.343078i \(0.888530\pi\)
\(198\) 1774.04 0.636745
\(199\) 2415.60 0.860491 0.430245 0.902712i \(-0.358427\pi\)
0.430245 + 0.902712i \(0.358427\pi\)
\(200\) − 417.804i − 0.147716i
\(201\) − 1597.33i − 0.560532i
\(202\) − 2927.44i − 1.01967i
\(203\) − 1265.72i − 0.437618i
\(204\) 394.368 0.135350
\(205\) 3692.99 1.25819
\(206\) − 421.772i − 0.142652i
\(207\) 2170.03 0.728635
\(208\) 0 0
\(209\) 874.225 0.289337
\(210\) 262.685i 0.0863191i
\(211\) 2668.42 0.870623 0.435312 0.900280i \(-0.356638\pi\)
0.435312 + 0.900280i \(0.356638\pi\)
\(212\) 1077.85 0.349183
\(213\) − 241.743i − 0.0777651i
\(214\) 782.122i 0.249835i
\(215\) 3041.30i 0.964719i
\(216\) − 1189.43i − 0.374679i
\(217\) 306.774 0.0959685
\(218\) −2662.81 −0.827284
\(219\) 3245.31i 1.00136i
\(220\) 2230.72 0.683614
\(221\) 0 0
\(222\) 870.634 0.263212
\(223\) − 286.358i − 0.0859907i −0.999075 0.0429953i \(-0.986310\pi\)
0.999075 0.0429953i \(-0.0136901\pi\)
\(224\) −134.433 −0.0400992
\(225\) −708.632 −0.209965
\(226\) − 1423.82i − 0.419077i
\(227\) 5201.24i 1.52079i 0.649463 + 0.760393i \(0.274994\pi\)
−0.649463 + 0.760393i \(0.725006\pi\)
\(228\) − 196.041i − 0.0569435i
\(229\) − 890.458i − 0.256957i −0.991712 0.128478i \(-0.958991\pi\)
0.991712 0.128478i \(-0.0410093\pi\)
\(230\) 2728.64 0.782267
\(231\) 1006.50 0.286678
\(232\) 2410.30i 0.682087i
\(233\) −4753.11 −1.33642 −0.668212 0.743971i \(-0.732940\pi\)
−0.668212 + 0.743971i \(0.732940\pi\)
\(234\) 0 0
\(235\) 5021.35 1.39386
\(236\) 921.358i 0.254133i
\(237\) 1412.67 0.387186
\(238\) −226.032 −0.0615609
\(239\) − 2292.62i − 0.620491i −0.950656 0.310245i \(-0.899589\pi\)
0.950656 0.310245i \(-0.100411\pi\)
\(240\) − 500.229i − 0.134540i
\(241\) − 1975.21i − 0.527943i −0.964530 0.263972i \(-0.914967\pi\)
0.964530 0.263972i \(-0.0850325\pi\)
\(242\) − 5885.14i − 1.56327i
\(243\) −3360.03 −0.887020
\(244\) 1523.26 0.399659
\(245\) 2775.50i 0.723757i
\(246\) −3173.06 −0.822385
\(247\) 0 0
\(248\) −584.186 −0.149580
\(249\) 931.637i 0.237109i
\(250\) −3023.75 −0.764955
\(251\) 7465.74 1.87742 0.938711 0.344705i \(-0.112021\pi\)
0.938711 + 0.344705i \(0.112021\pi\)
\(252\) 228.011i 0.0569974i
\(253\) − 10455.0i − 2.59802i
\(254\) 2343.41i 0.578892i
\(255\) − 841.069i − 0.206548i
\(256\) 256.000 0.0625000
\(257\) 554.966 0.134700 0.0673499 0.997729i \(-0.478546\pi\)
0.0673499 + 0.997729i \(0.478546\pi\)
\(258\) − 2613.11i − 0.630563i
\(259\) −499.004 −0.119717
\(260\) 0 0
\(261\) 4088.08 0.969525
\(262\) − 2257.22i − 0.532258i
\(263\) −1993.07 −0.467292 −0.233646 0.972322i \(-0.575066\pi\)
−0.233646 + 0.972322i \(0.575066\pi\)
\(264\) −1916.66 −0.446826
\(265\) − 2298.72i − 0.532866i
\(266\) 112.361i 0.0258996i
\(267\) − 1365.58i − 0.313004i
\(268\) − 1743.39i − 0.397368i
\(269\) −5207.76 −1.18038 −0.590191 0.807263i \(-0.700948\pi\)
−0.590191 + 0.807263i \(0.700948\pi\)
\(270\) −2536.70 −0.571773
\(271\) − 4084.90i − 0.915646i −0.889043 0.457823i \(-0.848629\pi\)
0.889043 0.457823i \(-0.151371\pi\)
\(272\) 430.430 0.0959510
\(273\) 0 0
\(274\) −3495.87 −0.770778
\(275\) 3414.12i 0.748651i
\(276\) −2344.48 −0.511308
\(277\) −437.993 −0.0950051 −0.0475026 0.998871i \(-0.515126\pi\)
−0.0475026 + 0.998871i \(0.515126\pi\)
\(278\) 3353.07i 0.723395i
\(279\) 990.829i 0.212614i
\(280\) 286.706i 0.0611927i
\(281\) − 5462.34i − 1.15963i −0.814748 0.579815i \(-0.803125\pi\)
0.814748 0.579815i \(-0.196875\pi\)
\(282\) −4314.40 −0.911058
\(283\) 796.676 0.167341 0.0836704 0.996493i \(-0.473336\pi\)
0.0836704 + 0.996493i \(0.473336\pi\)
\(284\) − 263.849i − 0.0551287i
\(285\) −418.096 −0.0868978
\(286\) 0 0
\(287\) 1818.64 0.374045
\(288\) − 434.198i − 0.0888381i
\(289\) −4189.29 −0.852694
\(290\) 5140.45 1.04089
\(291\) − 4815.22i − 0.970011i
\(292\) 3542.07i 0.709876i
\(293\) − 1697.32i − 0.338426i −0.985580 0.169213i \(-0.945877\pi\)
0.985580 0.169213i \(-0.0541225\pi\)
\(294\) − 2384.74i − 0.473065i
\(295\) 1964.98 0.387815
\(296\) 950.247 0.186595
\(297\) 9719.54i 1.89894i
\(298\) 1793.92 0.348722
\(299\) 0 0
\(300\) 765.600 0.147340
\(301\) 1497.71i 0.286798i
\(302\) −4156.56 −0.791997
\(303\) 5364.36 1.01708
\(304\) − 213.967i − 0.0403680i
\(305\) − 3248.66i − 0.609894i
\(306\) − 730.047i − 0.136386i
\(307\) 9385.86i 1.74488i 0.488718 + 0.872442i \(0.337465\pi\)
−0.488718 + 0.872442i \(0.662535\pi\)
\(308\) 1098.53 0.203230
\(309\) 772.870 0.142288
\(310\) 1245.89i 0.228264i
\(311\) 3282.65 0.598527 0.299264 0.954170i \(-0.403259\pi\)
0.299264 + 0.954170i \(0.403259\pi\)
\(312\) 0 0
\(313\) −5924.67 −1.06991 −0.534955 0.844880i \(-0.679672\pi\)
−0.534955 + 0.844880i \(0.679672\pi\)
\(314\) − 6989.49i − 1.25618i
\(315\) 486.278 0.0869800
\(316\) 1541.85 0.274481
\(317\) − 6467.88i − 1.14597i −0.819566 0.572985i \(-0.805785\pi\)
0.819566 0.572985i \(-0.194215\pi\)
\(318\) 1975.09i 0.348294i
\(319\) − 19696.0i − 3.45693i
\(320\) − 545.971i − 0.0953772i
\(321\) −1433.19 −0.249199
\(322\) 1343.74 0.232558
\(323\) − 359.758i − 0.0619736i
\(324\) 714.143 0.122452
\(325\) 0 0
\(326\) −902.508 −0.153329
\(327\) − 4879.42i − 0.825177i
\(328\) −3463.21 −0.582999
\(329\) 2472.80 0.414376
\(330\) 4087.65i 0.681872i
\(331\) − 3509.24i − 0.582735i −0.956611 0.291368i \(-0.905890\pi\)
0.956611 0.291368i \(-0.0941103\pi\)
\(332\) 1016.83i 0.168089i
\(333\) − 1611.70i − 0.265227i
\(334\) 6406.70 1.04958
\(335\) −3718.13 −0.606398
\(336\) − 246.341i − 0.0399970i
\(337\) 1834.82 0.296584 0.148292 0.988944i \(-0.452622\pi\)
0.148292 + 0.988944i \(0.452622\pi\)
\(338\) 0 0
\(339\) 2609.07 0.418010
\(340\) − 917.978i − 0.146425i
\(341\) 4773.71 0.758097
\(342\) −362.907 −0.0573795
\(343\) 2807.77i 0.441999i
\(344\) − 2852.06i − 0.447014i
\(345\) 5000.07i 0.780274i
\(346\) − 3798.76i − 0.590239i
\(347\) −7259.30 −1.12305 −0.561527 0.827459i \(-0.689786\pi\)
−0.561527 + 0.827459i \(0.689786\pi\)
\(348\) −4416.73 −0.680349
\(349\) − 1795.24i − 0.275349i −0.990478 0.137674i \(-0.956037\pi\)
0.990478 0.137674i \(-0.0439628\pi\)
\(350\) −438.804 −0.0670144
\(351\) 0 0
\(352\) −2091.92 −0.316761
\(353\) 4552.38i 0.686398i 0.939263 + 0.343199i \(0.111511\pi\)
−0.939263 + 0.343199i \(0.888489\pi\)
\(354\) −1688.33 −0.253485
\(355\) −562.710 −0.0841283
\(356\) − 1490.45i − 0.221892i
\(357\) − 414.190i − 0.0614041i
\(358\) − 5301.78i − 0.782703i
\(359\) 10165.4i 1.49445i 0.664569 + 0.747227i \(0.268615\pi\)
−0.664569 + 0.747227i \(0.731385\pi\)
\(360\) −926.014 −0.135570
\(361\) 6680.16 0.973927
\(362\) 4579.88i 0.664954i
\(363\) 10784.2 1.55929
\(364\) 0 0
\(365\) 7554.16 1.08330
\(366\) 2791.28i 0.398641i
\(367\) −10401.6 −1.47945 −0.739723 0.672911i \(-0.765044\pi\)
−0.739723 + 0.672911i \(0.765044\pi\)
\(368\) −2558.87 −0.362473
\(369\) 5873.91i 0.828681i
\(370\) − 2026.59i − 0.284750i
\(371\) − 1132.02i − 0.158414i
\(372\) − 1070.48i − 0.149199i
\(373\) −7953.41 −1.10405 −0.552027 0.833826i \(-0.686145\pi\)
−0.552027 + 0.833826i \(0.686145\pi\)
\(374\) −3517.29 −0.486296
\(375\) − 5540.83i − 0.763006i
\(376\) −4708.91 −0.645861
\(377\) 0 0
\(378\) −1249.22 −0.169981
\(379\) − 9014.81i − 1.22179i −0.791710 0.610897i \(-0.790809\pi\)
0.791710 0.610897i \(-0.209191\pi\)
\(380\) −456.328 −0.0616030
\(381\) −4294.15 −0.577417
\(382\) − 676.627i − 0.0906263i
\(383\) 2496.24i 0.333034i 0.986039 + 0.166517i \(0.0532520\pi\)
−0.986039 + 0.166517i \(0.946748\pi\)
\(384\) 469.104i 0.0623408i
\(385\) − 2342.84i − 0.310135i
\(386\) 5368.10 0.707848
\(387\) −4837.35 −0.635391
\(388\) − 5255.54i − 0.687653i
\(389\) −10896.9 −1.42029 −0.710145 0.704056i \(-0.751371\pi\)
−0.710145 + 0.704056i \(0.751371\pi\)
\(390\) 0 0
\(391\) −4302.40 −0.556475
\(392\) − 2602.81i − 0.335362i
\(393\) 4136.21 0.530902
\(394\) −3794.48 −0.485186
\(395\) − 3288.31i − 0.418868i
\(396\) 3548.08i 0.450247i
\(397\) 10887.1i 1.37635i 0.725546 + 0.688174i \(0.241587\pi\)
−0.725546 + 0.688174i \(0.758413\pi\)
\(398\) 4831.21i 0.608459i
\(399\) −205.894 −0.0258336
\(400\) 835.609 0.104451
\(401\) 9968.76i 1.24144i 0.784034 + 0.620718i \(0.213159\pi\)
−0.784034 + 0.620718i \(0.786841\pi\)
\(402\) 3194.66 0.396356
\(403\) 0 0
\(404\) 5854.89 0.721019
\(405\) − 1523.05i − 0.186867i
\(406\) 2531.45 0.309442
\(407\) −7765.01 −0.945693
\(408\) 788.737i 0.0957066i
\(409\) − 4036.32i − 0.487979i −0.969778 0.243989i \(-0.921544\pi\)
0.969778 0.243989i \(-0.0784562\pi\)
\(410\) 7385.98i 0.889677i
\(411\) − 6405.96i − 0.768814i
\(412\) 843.543 0.100870
\(413\) 967.667 0.115292
\(414\) 4340.06i 0.515223i
\(415\) 2168.59 0.256510
\(416\) 0 0
\(417\) −6144.29 −0.721552
\(418\) 1748.45i 0.204592i
\(419\) 15194.9 1.77164 0.885821 0.464028i \(-0.153596\pi\)
0.885821 + 0.464028i \(0.153596\pi\)
\(420\) −525.371 −0.0610368
\(421\) − 10154.8i − 1.17556i −0.809019 0.587782i \(-0.800001\pi\)
0.809019 0.587782i \(-0.199999\pi\)
\(422\) 5336.84i 0.615624i
\(423\) 7986.73i 0.918033i
\(424\) 2155.69i 0.246910i
\(425\) 1404.97 0.160355
\(426\) 483.486 0.0549882
\(427\) − 1599.82i − 0.181314i
\(428\) −1564.24 −0.176660
\(429\) 0 0
\(430\) −6082.59 −0.682159
\(431\) 6616.84i 0.739494i 0.929133 + 0.369747i \(0.120556\pi\)
−0.929133 + 0.369747i \(0.879444\pi\)
\(432\) 2378.87 0.264938
\(433\) −5697.60 −0.632354 −0.316177 0.948700i \(-0.602399\pi\)
−0.316177 + 0.948700i \(0.602399\pi\)
\(434\) 613.548i 0.0678600i
\(435\) 9419.55i 1.03824i
\(436\) − 5325.61i − 0.584978i
\(437\) 2138.73i 0.234117i
\(438\) −6490.62 −0.708068
\(439\) −436.007 −0.0474020 −0.0237010 0.999719i \(-0.507545\pi\)
−0.0237010 + 0.999719i \(0.507545\pi\)
\(440\) 4461.44i 0.483388i
\(441\) −4414.59 −0.476686
\(442\) 0 0
\(443\) 6609.58 0.708873 0.354436 0.935080i \(-0.384673\pi\)
0.354436 + 0.935080i \(0.384673\pi\)
\(444\) 1741.27i 0.186119i
\(445\) −3178.68 −0.338616
\(446\) 572.715 0.0608046
\(447\) 3287.25i 0.347833i
\(448\) − 268.867i − 0.0283544i
\(449\) 8015.31i 0.842463i 0.906953 + 0.421231i \(0.138402\pi\)
−0.906953 + 0.421231i \(0.861598\pi\)
\(450\) − 1417.26i − 0.148468i
\(451\) 28299.9 2.95474
\(452\) 2847.65 0.296332
\(453\) − 7616.64i − 0.789980i
\(454\) −10402.5 −1.07536
\(455\) 0 0
\(456\) 392.082 0.0402652
\(457\) 4507.03i 0.461335i 0.973033 + 0.230668i \(0.0740910\pi\)
−0.973033 + 0.230668i \(0.925909\pi\)
\(458\) 1780.92 0.181696
\(459\) 3999.75 0.406737
\(460\) 5457.29i 0.553146i
\(461\) − 2103.55i − 0.212521i −0.994338 0.106260i \(-0.966112\pi\)
0.994338 0.106260i \(-0.0338877\pi\)
\(462\) 2012.99i 0.202712i
\(463\) 5468.28i 0.548883i 0.961604 + 0.274441i \(0.0884929\pi\)
−0.961604 + 0.274441i \(0.911507\pi\)
\(464\) −4820.60 −0.482308
\(465\) −2283.02 −0.227683
\(466\) − 9506.22i − 0.944994i
\(467\) 6043.79 0.598872 0.299436 0.954116i \(-0.403201\pi\)
0.299436 + 0.954116i \(0.403201\pi\)
\(468\) 0 0
\(469\) −1831.02 −0.180274
\(470\) 10042.7i 0.985606i
\(471\) 12807.8 1.25298
\(472\) −1842.72 −0.179699
\(473\) 23305.8i 2.26555i
\(474\) 2825.35i 0.273782i
\(475\) − 698.410i − 0.0674637i
\(476\) − 452.065i − 0.0435301i
\(477\) 3656.25 0.350960
\(478\) 4585.24 0.438753
\(479\) − 8481.30i − 0.809020i −0.914534 0.404510i \(-0.867442\pi\)
0.914534 0.404510i \(-0.132558\pi\)
\(480\) 1000.46 0.0951342
\(481\) 0 0
\(482\) 3950.42 0.373312
\(483\) 2462.32i 0.231965i
\(484\) 11770.3 1.10540
\(485\) −11208.5 −1.04938
\(486\) − 6720.05i − 0.627218i
\(487\) 6623.36i 0.616289i 0.951340 + 0.308145i \(0.0997081\pi\)
−0.951340 + 0.308145i \(0.900292\pi\)
\(488\) 3046.53i 0.282602i
\(489\) − 1653.79i − 0.152939i
\(490\) −5551.01 −0.511774
\(491\) −16379.4 −1.50548 −0.752739 0.658319i \(-0.771268\pi\)
−0.752739 + 0.658319i \(0.771268\pi\)
\(492\) − 6346.12i − 0.581514i
\(493\) −8105.21 −0.740447
\(494\) 0 0
\(495\) 7566.99 0.687093
\(496\) − 1168.37i − 0.105769i
\(497\) −277.110 −0.0250102
\(498\) −1863.27 −0.167661
\(499\) 7915.70i 0.710131i 0.934841 + 0.355065i \(0.115541\pi\)
−0.934841 + 0.355065i \(0.884459\pi\)
\(500\) − 6047.50i − 0.540905i
\(501\) 11739.9i 1.04690i
\(502\) 14931.5i 1.32754i
\(503\) −479.511 −0.0425056 −0.0212528 0.999774i \(-0.506765\pi\)
−0.0212528 + 0.999774i \(0.506765\pi\)
\(504\) −456.022 −0.0403032
\(505\) − 12486.7i − 1.10030i
\(506\) 20909.9 1.83708
\(507\) 0 0
\(508\) −4686.82 −0.409338
\(509\) − 1192.33i − 0.103829i −0.998652 0.0519145i \(-0.983468\pi\)
0.998652 0.0519145i \(-0.0165323\pi\)
\(510\) 1682.14 0.146052
\(511\) 3720.10 0.322050
\(512\) 512.000i 0.0441942i
\(513\) − 1988.28i − 0.171120i
\(514\) 1109.93i 0.0952471i
\(515\) − 1799.02i − 0.153931i
\(516\) 5226.23 0.445876
\(517\) 38479.2 3.27333
\(518\) − 998.008i − 0.0846524i
\(519\) 6961.00 0.588736
\(520\) 0 0
\(521\) −23238.5 −1.95412 −0.977062 0.212955i \(-0.931691\pi\)
−0.977062 + 0.212955i \(0.931691\pi\)
\(522\) 8176.16i 0.685557i
\(523\) −11026.3 −0.921884 −0.460942 0.887430i \(-0.652488\pi\)
−0.460942 + 0.887430i \(0.652488\pi\)
\(524\) 4514.44 0.376363
\(525\) − 804.080i − 0.0668437i
\(526\) − 3986.14i − 0.330425i
\(527\) − 1964.46i − 0.162378i
\(528\) − 3833.32i − 0.315954i
\(529\) 13410.3 1.10219
\(530\) 4597.45 0.376793
\(531\) 3125.41i 0.255426i
\(532\) −224.722 −0.0183138
\(533\) 0 0
\(534\) 2731.16 0.221327
\(535\) 3336.06i 0.269590i
\(536\) 3486.79 0.280982
\(537\) 9715.18 0.780709
\(538\) − 10415.5i − 0.834657i
\(539\) 21269.0i 1.69967i
\(540\) − 5073.41i − 0.404305i
\(541\) 8987.70i 0.714254i 0.934056 + 0.357127i \(0.116244\pi\)
−0.934056 + 0.357127i \(0.883756\pi\)
\(542\) 8169.80 0.647459
\(543\) −8392.35 −0.663260
\(544\) 860.861i 0.0678476i
\(545\) −11357.9 −0.892697
\(546\) 0 0
\(547\) −10734.8 −0.839101 −0.419550 0.907732i \(-0.637812\pi\)
−0.419550 + 0.907732i \(0.637812\pi\)
\(548\) − 6991.74i − 0.545022i
\(549\) 5167.17 0.401693
\(550\) −6828.23 −0.529376
\(551\) 4029.11i 0.311517i
\(552\) − 4688.96i − 0.361550i
\(553\) − 1619.35i − 0.124524i
\(554\) − 875.985i − 0.0671788i
\(555\) 3713.60 0.284024
\(556\) −6706.14 −0.511518
\(557\) 13228.3i 1.00628i 0.864204 + 0.503141i \(0.167822\pi\)
−0.864204 + 0.503141i \(0.832178\pi\)
\(558\) −1981.66 −0.150341
\(559\) 0 0
\(560\) −573.412 −0.0432698
\(561\) − 6445.22i − 0.485058i
\(562\) 10924.7 0.819983
\(563\) −10734.0 −0.803522 −0.401761 0.915745i \(-0.631602\pi\)
−0.401761 + 0.915745i \(0.631602\pi\)
\(564\) − 8628.79i − 0.644216i
\(565\) − 6073.18i − 0.452213i
\(566\) 1593.35i 0.118328i
\(567\) − 750.037i − 0.0555531i
\(568\) 527.698 0.0389819
\(569\) −5313.45 −0.391479 −0.195739 0.980656i \(-0.562711\pi\)
−0.195739 + 0.980656i \(0.562711\pi\)
\(570\) − 836.192i − 0.0614460i
\(571\) −4629.37 −0.339287 −0.169644 0.985505i \(-0.554262\pi\)
−0.169644 + 0.985505i \(0.554262\pi\)
\(572\) 0 0
\(573\) 1239.88 0.0903954
\(574\) 3637.28i 0.264490i
\(575\) −8352.38 −0.605771
\(576\) 868.397 0.0628180
\(577\) 504.750i 0.0364177i 0.999834 + 0.0182088i \(0.00579638\pi\)
−0.999834 + 0.0182088i \(0.994204\pi\)
\(578\) − 8378.58i − 0.602946i
\(579\) 9836.71i 0.706045i
\(580\) 10280.9i 0.736019i
\(581\) 1067.94 0.0762572
\(582\) 9630.44 0.685901
\(583\) − 17615.4i − 1.25138i
\(584\) −7084.14 −0.501958
\(585\) 0 0
\(586\) 3394.65 0.239303
\(587\) − 5532.93i − 0.389044i −0.980898 0.194522i \(-0.937684\pi\)
0.980898 0.194522i \(-0.0623155\pi\)
\(588\) 4769.49 0.334507
\(589\) −976.536 −0.0683149
\(590\) 3929.96i 0.274227i
\(591\) − 6953.15i − 0.483950i
\(592\) 1900.49i 0.131942i
\(593\) − 18079.8i − 1.25202i −0.779816 0.626009i \(-0.784687\pi\)
0.779816 0.626009i \(-0.215313\pi\)
\(594\) −19439.1 −1.34275
\(595\) −964.117 −0.0664285
\(596\) 3587.84i 0.246584i
\(597\) −8852.88 −0.606909
\(598\) 0 0
\(599\) −1837.55 −0.125342 −0.0626712 0.998034i \(-0.519962\pi\)
−0.0626712 + 0.998034i \(0.519962\pi\)
\(600\) 1531.20i 0.104185i
\(601\) −23487.3 −1.59412 −0.797060 0.603901i \(-0.793612\pi\)
−0.797060 + 0.603901i \(0.793612\pi\)
\(602\) −2995.41 −0.202797
\(603\) − 5913.89i − 0.399390i
\(604\) − 8313.12i − 0.560027i
\(605\) − 25102.5i − 1.68688i
\(606\) 10728.7i 0.719182i
\(607\) −16310.8 −1.09067 −0.545335 0.838218i \(-0.683598\pi\)
−0.545335 + 0.838218i \(0.683598\pi\)
\(608\) 427.935 0.0285445
\(609\) 4638.72i 0.308654i
\(610\) 6497.32 0.431260
\(611\) 0 0
\(612\) 1460.09 0.0964393
\(613\) − 8453.11i − 0.556962i −0.960442 0.278481i \(-0.910169\pi\)
0.960442 0.278481i \(-0.0898310\pi\)
\(614\) −18771.7 −1.23382
\(615\) −13534.4 −0.887411
\(616\) 2197.06i 0.143705i
\(617\) 13410.3i 0.875008i 0.899216 + 0.437504i \(0.144137\pi\)
−0.899216 + 0.437504i \(0.855863\pi\)
\(618\) 1545.74i 0.100613i
\(619\) − 890.135i − 0.0577990i −0.999582 0.0288995i \(-0.990800\pi\)
0.999582 0.0288995i \(-0.00920027\pi\)
\(620\) −2491.78 −0.161407
\(621\) −23778.1 −1.53653
\(622\) 6565.30i 0.423223i
\(623\) −1565.36 −0.100666
\(624\) 0 0
\(625\) −6369.30 −0.407635
\(626\) − 11849.3i − 0.756541i
\(627\) −3203.92 −0.204071
\(628\) 13979.0 0.888252
\(629\) 3195.43i 0.202560i
\(630\) 972.557i 0.0615041i
\(631\) − 10582.4i − 0.667639i −0.942637 0.333819i \(-0.891662\pi\)
0.942637 0.333819i \(-0.108338\pi\)
\(632\) 3083.71i 0.194087i
\(633\) −9779.42 −0.614055
\(634\) 12935.8 0.810323
\(635\) 9995.57i 0.624665i
\(636\) −3950.17 −0.246281
\(637\) 0 0
\(638\) 39391.9 2.44442
\(639\) − 895.021i − 0.0554092i
\(640\) 1091.94 0.0674419
\(641\) −26884.2 −1.65657 −0.828286 0.560305i \(-0.810684\pi\)
−0.828286 + 0.560305i \(0.810684\pi\)
\(642\) − 2866.38i − 0.176210i
\(643\) − 5691.12i − 0.349045i −0.984653 0.174523i \(-0.944162\pi\)
0.984653 0.174523i \(-0.0558382\pi\)
\(644\) 2687.48i 0.164443i
\(645\) − 11146.0i − 0.680422i
\(646\) 719.516 0.0438220
\(647\) −1809.79 −0.109969 −0.0549847 0.998487i \(-0.517511\pi\)
−0.0549847 + 0.998487i \(0.517511\pi\)
\(648\) 1428.29i 0.0865870i
\(649\) 15057.9 0.910745
\(650\) 0 0
\(651\) −1124.29 −0.0676871
\(652\) − 1805.02i − 0.108420i
\(653\) 9459.16 0.566869 0.283435 0.958992i \(-0.408526\pi\)
0.283435 + 0.958992i \(0.408526\pi\)
\(654\) 9758.85 0.583488
\(655\) − 9627.94i − 0.574343i
\(656\) − 6926.42i − 0.412243i
\(657\) 12015.3i 0.713488i
\(658\) 4945.59i 0.293008i
\(659\) 1015.32 0.0600171 0.0300085 0.999550i \(-0.490447\pi\)
0.0300085 + 0.999550i \(0.490447\pi\)
\(660\) −8175.31 −0.482157
\(661\) 23648.4i 1.39156i 0.718257 + 0.695778i \(0.244940\pi\)
−0.718257 + 0.695778i \(0.755060\pi\)
\(662\) 7018.49 0.412056
\(663\) 0 0
\(664\) −2033.66 −0.118857
\(665\) 479.264i 0.0279474i
\(666\) 3223.40 0.187544
\(667\) 48184.7 2.79718
\(668\) 12813.4i 0.742164i
\(669\) 1049.46i 0.0606497i
\(670\) − 7436.26i − 0.428788i
\(671\) − 24894.9i − 1.43228i
\(672\) 492.682 0.0282822
\(673\) −17747.0 −1.01649 −0.508243 0.861214i \(-0.669705\pi\)
−0.508243 + 0.861214i \(0.669705\pi\)
\(674\) 3669.63i 0.209717i
\(675\) 7764.85 0.442769
\(676\) 0 0
\(677\) 10754.2 0.610511 0.305256 0.952270i \(-0.401258\pi\)
0.305256 + 0.952270i \(0.401258\pi\)
\(678\) 5218.14i 0.295577i
\(679\) −5519.69 −0.311968
\(680\) 1835.96 0.103538
\(681\) − 19061.9i − 1.07262i
\(682\) 9547.42i 0.536055i
\(683\) − 23981.7i − 1.34354i −0.740762 0.671768i \(-0.765535\pi\)
0.740762 0.671768i \(-0.234465\pi\)
\(684\) − 725.815i − 0.0405734i
\(685\) −14911.3 −0.831723
\(686\) −5615.55 −0.312540
\(687\) 3263.42i 0.181233i
\(688\) 5704.13 0.316087
\(689\) 0 0
\(690\) −10000.1 −0.551737
\(691\) − 29001.4i − 1.59662i −0.602244 0.798312i \(-0.705727\pi\)
0.602244 0.798312i \(-0.294273\pi\)
\(692\) 7597.53 0.417362
\(693\) 3726.41 0.204264
\(694\) − 14518.6i − 0.794119i
\(695\) 14302.2i 0.780594i
\(696\) − 8833.46i − 0.481079i
\(697\) − 11645.9i − 0.632882i
\(698\) 3590.47 0.194701
\(699\) 17419.6 0.942587
\(700\) − 877.608i − 0.0473864i
\(701\) 31031.5 1.67196 0.835979 0.548761i \(-0.184900\pi\)
0.835979 + 0.548761i \(0.184900\pi\)
\(702\) 0 0
\(703\) 1588.45 0.0852199
\(704\) − 4183.84i − 0.223984i
\(705\) −18402.6 −0.983096
\(706\) −9104.75 −0.485357
\(707\) − 6149.16i − 0.327105i
\(708\) − 3376.66i − 0.179241i
\(709\) − 4382.59i − 0.232146i −0.993241 0.116073i \(-0.962969\pi\)
0.993241 0.116073i \(-0.0370307\pi\)
\(710\) − 1125.42i − 0.0594877i
\(711\) 5230.23 0.275878
\(712\) 2980.90 0.156902
\(713\) 11678.5i 0.613414i
\(714\) 828.380 0.0434192
\(715\) 0 0
\(716\) 10603.6 0.553455
\(717\) 8402.17i 0.437636i
\(718\) −20330.8 −1.05674
\(719\) 28764.6 1.49199 0.745994 0.665953i \(-0.231975\pi\)
0.745994 + 0.665953i \(0.231975\pi\)
\(720\) − 1852.03i − 0.0958625i
\(721\) − 885.941i − 0.0457617i
\(722\) 13360.3i 0.688670i
\(723\) 7238.89i 0.372361i
\(724\) −9159.77 −0.470194
\(725\) −15734.9 −0.806042
\(726\) 21568.3i 1.10258i
\(727\) −25408.5 −1.29621 −0.648107 0.761549i \(-0.724439\pi\)
−0.648107 + 0.761549i \(0.724439\pi\)
\(728\) 0 0
\(729\) 17134.5 0.870525
\(730\) 15108.3i 0.766006i
\(731\) 9590.74 0.485262
\(732\) −5582.57 −0.281882
\(733\) 22341.8i 1.12580i 0.826524 + 0.562902i \(0.190315\pi\)
−0.826524 + 0.562902i \(0.809685\pi\)
\(734\) − 20803.1i − 1.04613i
\(735\) − 10171.9i − 0.510470i
\(736\) − 5117.73i − 0.256307i
\(737\) −28492.5 −1.42406
\(738\) −11747.8 −0.585966
\(739\) − 8210.90i − 0.408718i −0.978896 0.204359i \(-0.934489\pi\)
0.978896 0.204359i \(-0.0655110\pi\)
\(740\) 4053.18 0.201349
\(741\) 0 0
\(742\) 2264.04 0.112016
\(743\) 32752.3i 1.61718i 0.588373 + 0.808590i \(0.299769\pi\)
−0.588373 + 0.808590i \(0.700231\pi\)
\(744\) 2140.97 0.105500
\(745\) 7651.79 0.376295
\(746\) − 15906.8i − 0.780684i
\(747\) 3449.26i 0.168945i
\(748\) − 7034.58i − 0.343863i
\(749\) 1642.87i 0.0801455i
\(750\) 11081.7 0.539527
\(751\) −8907.41 −0.432804 −0.216402 0.976304i \(-0.569432\pi\)
−0.216402 + 0.976304i \(0.569432\pi\)
\(752\) − 9417.83i − 0.456693i
\(753\) −27361.0 −1.32416
\(754\) 0 0
\(755\) −17729.4 −0.854620
\(756\) − 2498.43i − 0.120195i
\(757\) −10744.6 −0.515879 −0.257940 0.966161i \(-0.583044\pi\)
−0.257940 + 0.966161i \(0.583044\pi\)
\(758\) 18029.6 0.863938
\(759\) 38316.2i 1.83240i
\(760\) − 912.656i − 0.0435599i
\(761\) − 16997.8i − 0.809683i −0.914387 0.404841i \(-0.867327\pi\)
0.914387 0.404841i \(-0.132673\pi\)
\(762\) − 8588.30i − 0.408296i
\(763\) −5593.28 −0.265387
\(764\) 1353.25 0.0640825
\(765\) − 3113.94i − 0.147170i
\(766\) −4992.48 −0.235490
\(767\) 0 0
\(768\) −938.208 −0.0440816
\(769\) − 19032.4i − 0.892492i −0.894910 0.446246i \(-0.852761\pi\)
0.894910 0.446246i \(-0.147239\pi\)
\(770\) 4685.68 0.219299
\(771\) −2033.88 −0.0950045
\(772\) 10736.2i 0.500524i
\(773\) − 23750.6i − 1.10511i −0.833477 0.552555i \(-0.813653\pi\)
0.833477 0.552555i \(-0.186347\pi\)
\(774\) − 9674.69i − 0.449289i
\(775\) − 3813.68i − 0.176763i
\(776\) 10511.1 0.486244
\(777\) 1828.79 0.0844368
\(778\) − 21793.7i − 1.00430i
\(779\) −5789.17 −0.266263
\(780\) 0 0
\(781\) −4312.12 −0.197567
\(782\) − 8604.79i − 0.393487i
\(783\) −44795.2 −2.04451
\(784\) 5205.62 0.237136
\(785\) − 29813.0i − 1.35550i
\(786\) 8272.42i 0.375404i
\(787\) 17825.9i 0.807403i 0.914891 + 0.403702i \(0.132277\pi\)
−0.914891 + 0.403702i \(0.867723\pi\)
\(788\) − 7588.96i − 0.343078i
\(789\) 7304.34 0.329584
\(790\) 6576.62 0.296184
\(791\) − 2990.78i − 0.134437i
\(792\) −7096.16 −0.318373
\(793\) 0 0
\(794\) −21774.3 −0.973225
\(795\) 8424.53i 0.375833i
\(796\) −9662.42 −0.430245
\(797\) 26461.2 1.17604 0.588020 0.808846i \(-0.299908\pi\)
0.588020 + 0.808846i \(0.299908\pi\)
\(798\) − 411.788i − 0.0182671i
\(799\) − 15834.8i − 0.701122i
\(800\) 1671.22i 0.0738581i
\(801\) − 5055.87i − 0.223021i
\(802\) −19937.5 −0.877828
\(803\) 57888.5 2.54401
\(804\) 6389.32i 0.280266i
\(805\) 5731.58 0.250946
\(806\) 0 0
\(807\) 19085.8 0.832530
\(808\) 11709.8i 0.509837i
\(809\) 21873.3 0.950588 0.475294 0.879827i \(-0.342342\pi\)
0.475294 + 0.879827i \(0.342342\pi\)
\(810\) 3046.10 0.132135
\(811\) − 39113.8i − 1.69355i −0.531949 0.846776i \(-0.678540\pi\)
0.531949 0.846776i \(-0.321460\pi\)
\(812\) 5062.90i 0.218809i
\(813\) 14970.6i 0.645810i
\(814\) − 15530.0i − 0.668706i
\(815\) −3849.56 −0.165453
\(816\) −1577.47 −0.0676748
\(817\) − 4767.57i − 0.204157i
\(818\) 8072.65 0.345053
\(819\) 0 0
\(820\) −14772.0 −0.629097
\(821\) 26296.8i 1.11786i 0.829214 + 0.558931i \(0.188788\pi\)
−0.829214 + 0.558931i \(0.811212\pi\)
\(822\) 12811.9 0.543634
\(823\) −24681.1 −1.04536 −0.522678 0.852530i \(-0.675067\pi\)
−0.522678 + 0.852530i \(0.675067\pi\)
\(824\) 1687.09i 0.0713258i
\(825\) − 12512.3i − 0.528027i
\(826\) 1935.33i 0.0815241i
\(827\) − 3081.68i − 0.129578i −0.997899 0.0647888i \(-0.979363\pi\)
0.997899 0.0647888i \(-0.0206374\pi\)
\(828\) −8680.12 −0.364317
\(829\) 8498.53 0.356051 0.178026 0.984026i \(-0.443029\pi\)
0.178026 + 0.984026i \(0.443029\pi\)
\(830\) 4337.18i 0.181380i
\(831\) 1605.19 0.0670076
\(832\) 0 0
\(833\) 8752.57 0.364056
\(834\) − 12288.6i − 0.510215i
\(835\) 27327.1 1.13257
\(836\) −3496.90 −0.144668
\(837\) − 10857.0i − 0.448356i
\(838\) 30389.7i 1.25274i
\(839\) 18680.1i 0.768664i 0.923195 + 0.384332i \(0.125568\pi\)
−0.923195 + 0.384332i \(0.874432\pi\)
\(840\) − 1050.74i − 0.0431596i
\(841\) 66385.3 2.72194
\(842\) 20309.5 0.831249
\(843\) 20018.8i 0.817894i
\(844\) −10673.7 −0.435312
\(845\) 0 0
\(846\) −15973.5 −0.649147
\(847\) − 12361.9i − 0.501486i
\(848\) −4311.39 −0.174592
\(849\) −2919.72 −0.118026
\(850\) 2809.93i 0.113388i
\(851\) − 18996.5i − 0.765208i
\(852\) 966.973i 0.0388826i
\(853\) 24129.9i 0.968574i 0.874909 + 0.484287i \(0.160921\pi\)
−0.874909 + 0.484287i \(0.839079\pi\)
\(854\) 3199.65 0.128208
\(855\) −1547.94 −0.0619164
\(856\) − 3128.49i − 0.124918i
\(857\) 7739.30 0.308482 0.154241 0.988033i \(-0.450707\pi\)
0.154241 + 0.988033i \(0.450707\pi\)
\(858\) 0 0
\(859\) 4302.59 0.170899 0.0854496 0.996342i \(-0.472767\pi\)
0.0854496 + 0.996342i \(0.472767\pi\)
\(860\) − 12165.2i − 0.482360i
\(861\) −6665.08 −0.263816
\(862\) −13233.7 −0.522901
\(863\) 16253.2i 0.641096i 0.947232 + 0.320548i \(0.103867\pi\)
−0.947232 + 0.320548i \(0.896133\pi\)
\(864\) 4757.73i 0.187340i
\(865\) − 16203.2i − 0.636910i
\(866\) − 11395.2i − 0.447142i
\(867\) 15353.2 0.601410
\(868\) −1227.10 −0.0479842
\(869\) − 25198.7i − 0.983669i
\(870\) −18839.1 −0.734144
\(871\) 0 0
\(872\) 10651.2 0.413642
\(873\) − 17827.7i − 0.691152i
\(874\) −4277.45 −0.165546
\(875\) −6351.46 −0.245392
\(876\) − 12981.2i − 0.500680i
\(877\) 9441.33i 0.363525i 0.983342 + 0.181762i \(0.0581802\pi\)
−0.983342 + 0.181762i \(0.941820\pi\)
\(878\) − 872.014i − 0.0335183i
\(879\) 6220.48i 0.238693i
\(880\) −8922.88 −0.341807
\(881\) 11872.4 0.454021 0.227011 0.973892i \(-0.427105\pi\)
0.227011 + 0.973892i \(0.427105\pi\)
\(882\) − 8829.18i − 0.337068i
\(883\) 493.891 0.0188231 0.00941153 0.999956i \(-0.497004\pi\)
0.00941153 + 0.999956i \(0.497004\pi\)
\(884\) 0 0
\(885\) −7201.40 −0.273528
\(886\) 13219.2i 0.501249i
\(887\) 20543.6 0.777664 0.388832 0.921309i \(-0.372879\pi\)
0.388832 + 0.921309i \(0.372879\pi\)
\(888\) −3482.54 −0.131606
\(889\) 4922.38i 0.185705i
\(890\) − 6357.36i − 0.239437i
\(891\) − 11671.3i − 0.438838i
\(892\) 1145.43i 0.0429953i
\(893\) −7871.52 −0.294972
\(894\) −6574.50 −0.245955
\(895\) − 22614.2i − 0.844592i
\(896\) 537.734 0.0200496
\(897\) 0 0
\(898\) −16030.6 −0.595711
\(899\) 22001.0i 0.816212i
\(900\) 2834.53 0.104983
\(901\) −7249.03 −0.268036
\(902\) 56599.7i 2.08932i
\(903\) − 5488.90i − 0.202280i
\(904\) 5695.30i 0.209539i
\(905\) 19535.0i 0.717532i
\(906\) 15233.3 0.558600
\(907\) 5386.61 0.197199 0.0985994 0.995127i \(-0.468564\pi\)
0.0985994 + 0.995127i \(0.468564\pi\)
\(908\) − 20805.0i − 0.760393i
\(909\) 19860.8 0.724688
\(910\) 0 0
\(911\) 31793.5 1.15627 0.578137 0.815940i \(-0.303780\pi\)
0.578137 + 0.815940i \(0.303780\pi\)
\(912\) 784.164i 0.0284718i
\(913\) 16618.2 0.602389
\(914\) −9014.07 −0.326213
\(915\) 11905.9i 0.430162i
\(916\) 3561.83i 0.128478i
\(917\) − 4741.34i − 0.170745i
\(918\) 7999.51i 0.287607i
\(919\) −42991.4 −1.54315 −0.771575 0.636139i \(-0.780531\pi\)
−0.771575 + 0.636139i \(0.780531\pi\)
\(920\) −10914.6 −0.391134
\(921\) − 34398.0i − 1.23068i
\(922\) 4207.10 0.150275
\(923\) 0 0
\(924\) −4025.98 −0.143339
\(925\) 6203.40i 0.220504i
\(926\) −10936.6 −0.388119
\(927\) 2861.45 0.101383
\(928\) − 9641.21i − 0.341043i
\(929\) 40676.7i 1.43655i 0.695757 + 0.718277i \(0.255069\pi\)
−0.695757 + 0.718277i \(0.744931\pi\)
\(930\) − 4566.04i − 0.160996i
\(931\) − 4350.91i − 0.153164i
\(932\) 19012.4 0.668212
\(933\) −12030.5 −0.422145
\(934\) 12087.6i 0.423467i
\(935\) −15002.6 −0.524748
\(936\) 0 0
\(937\) 21008.3 0.732456 0.366228 0.930525i \(-0.380649\pi\)
0.366228 + 0.930525i \(0.380649\pi\)
\(938\) − 3662.04i − 0.127473i
\(939\) 21713.2 0.754614
\(940\) −20085.4 −0.696929
\(941\) − 4525.50i − 0.156777i −0.996923 0.0783885i \(-0.975023\pi\)
0.996923 0.0783885i \(-0.0249775\pi\)
\(942\) 25615.6i 0.885989i
\(943\) 69233.5i 2.39083i
\(944\) − 3685.43i − 0.127066i
\(945\) −5328.40 −0.183421
\(946\) −46611.7 −1.60198
\(947\) 27401.6i 0.940267i 0.882595 + 0.470134i \(0.155794\pi\)
−0.882595 + 0.470134i \(0.844206\pi\)
\(948\) −5650.70 −0.193593
\(949\) 0 0
\(950\) 1396.82 0.0477040
\(951\) 23704.0i 0.808259i
\(952\) 904.129 0.0307805
\(953\) −26226.2 −0.891449 −0.445725 0.895170i \(-0.647054\pi\)
−0.445725 + 0.895170i \(0.647054\pi\)
\(954\) 7312.49i 0.248166i
\(955\) − 2886.08i − 0.0977921i
\(956\) 9170.48i 0.310245i
\(957\) 72183.2i 2.43819i
\(958\) 16962.6 0.572063
\(959\) −7343.15 −0.247260
\(960\) 2000.91i 0.0672700i
\(961\) 24458.6 0.821007
\(962\) 0 0
\(963\) −5306.19 −0.177559
\(964\) 7900.83i 0.263972i
\(965\) 22897.1 0.763817
\(966\) −4924.63 −0.164024
\(967\) 20843.6i 0.693158i 0.938021 + 0.346579i \(0.112657\pi\)
−0.938021 + 0.346579i \(0.887343\pi\)
\(968\) 23540.6i 0.781635i
\(969\) 1318.47i 0.0437103i
\(970\) − 22417.0i − 0.742026i
\(971\) 32577.3 1.07668 0.538339 0.842729i \(-0.319052\pi\)
0.538339 + 0.842729i \(0.319052\pi\)
\(972\) 13440.1 0.443510
\(973\) 7043.20i 0.232060i
\(974\) −13246.7 −0.435782
\(975\) 0 0
\(976\) −6093.05 −0.199830
\(977\) − 43864.1i − 1.43637i −0.695851 0.718186i \(-0.744973\pi\)
0.695851 0.718186i \(-0.255027\pi\)
\(978\) 3307.58 0.108144
\(979\) −24358.6 −0.795204
\(980\) − 11102.0i − 0.361879i
\(981\) − 18065.4i − 0.587955i
\(982\) − 32758.7i − 1.06453i
\(983\) 14758.8i 0.478875i 0.970912 + 0.239437i \(0.0769630\pi\)
−0.970912 + 0.239437i \(0.923037\pi\)
\(984\) 12692.2 0.411193
\(985\) −16185.0 −0.523549
\(986\) − 16210.4i − 0.523575i
\(987\) −9062.49 −0.292262
\(988\) 0 0
\(989\) −57016.0 −1.83317
\(990\) 15134.0i 0.485848i
\(991\) 48622.4 1.55857 0.779284 0.626671i \(-0.215583\pi\)
0.779284 + 0.626671i \(0.215583\pi\)
\(992\) 2336.74 0.0747900
\(993\) 12860.9i 0.411006i
\(994\) − 554.220i − 0.0176849i
\(995\) 20607.0i 0.656569i
\(996\) − 3726.55i − 0.118554i
\(997\) −18812.3 −0.597586 −0.298793 0.954318i \(-0.596584\pi\)
−0.298793 + 0.954318i \(0.596584\pi\)
\(998\) −15831.4 −0.502138
\(999\) 17660.2i 0.559305i
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 338.4.b.f.337.5 6
13.2 odd 12 338.4.c.k.191.2 6
13.3 even 3 338.4.e.h.147.2 12
13.4 even 6 338.4.e.h.23.2 12
13.5 odd 4 338.4.a.k.1.2 yes 3
13.6 odd 12 338.4.c.k.315.2 6
13.7 odd 12 338.4.c.l.315.2 6
13.8 odd 4 338.4.a.j.1.2 3
13.9 even 3 338.4.e.h.23.5 12
13.10 even 6 338.4.e.h.147.5 12
13.11 odd 12 338.4.c.l.191.2 6
13.12 even 2 inner 338.4.b.f.337.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
338.4.a.j.1.2 3 13.8 odd 4
338.4.a.k.1.2 yes 3 13.5 odd 4
338.4.b.f.337.2 6 13.12 even 2 inner
338.4.b.f.337.5 6 1.1 even 1 trivial
338.4.c.k.191.2 6 13.2 odd 12
338.4.c.k.315.2 6 13.6 odd 12
338.4.c.l.191.2 6 13.11 odd 12
338.4.c.l.315.2 6 13.7 odd 12
338.4.e.h.23.2 12 13.4 even 6
338.4.e.h.23.5 12 13.9 even 3
338.4.e.h.147.2 12 13.3 even 3
338.4.e.h.147.5 12 13.10 even 6